diff --git "a/data_tmp/process_27/tokenized_finally.jsonl" "b/data_tmp/process_27/tokenized_finally.jsonl"
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-{"id": "2695.png", "formula": "\\begin{align*} \\Delta { C } ^ 1 _ { t } = ( \\alpha _ t - \\beta _ t ) \\Big ( \\Delta { C } ^ 2 _ { t + 1 } + \\log \\big ( 1 + 2 ^ { \\Delta { C } ^ 1 _ { t + 1 } } \\big ) \\Big ) , ~ \\Delta { C } ^ 1 _ { n + 1 } = 0 , ~ t \\in \\{ n , \\ldots , 0 \\} , \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} { y ' = \\frac { \\sqrt { x } - 2 \\log x } { x } } \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} \\mathbb { E } ( T _ { ( k ) } ) = \\sum _ { j = 1 } ^ { k } \\mathbb { E } ( W _ { j } ) = \\sum _ { j = 1 } ^ { k } \\frac { 1 } { ( n - k + j ) } \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} - u '' ( x ) + V ( x ) u ( x ) = \\lambda u ( x ) , \\ ; \\ ; x \\in ( a , b ) \\subseteq \\R , \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} \\Phi _ p = e ^ 0 \\wedge \\varphi _ p + \\ast _ 7 \\varphi _ p . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} \\left ( e ^ { C _ 1 / 2 } - 1 \\right ) W ( o ) = \\eta ( o ) W ( o ) \\le \\eta ( p ) W ( p ) \\le C _ 2 \\left ( e ^ { C _ 1 } - 1 \\right ) . \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ \\gamma ( z , \\bar z ) = ( - 1 ) ^ { m + n } ( 1 - | z | ^ 2 ) ^ { - \\gamma } \\frac { \\partial ^ { m + n } } { \\partial z ^ m \\partial \\bar z ^ n } \\left ( ( 1 - | z | ^ 2 ) ^ { \\gamma + m + n } \\right ) . \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} \\Phi ( S \\circ T ) & = \\Phi ( S ) \\circ \\Phi ( T ) ( ) \\\\ \\Phi ( S ^ * ) & = \\Phi ( S ) ^ * \\\\ \\Phi ( S \\otimes T ) & = \\Phi ( S ) \\otimes \\Phi ( T ) . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{gather*} \\Delta _ { \\hat { S } ( \\gg ^ * ) } \\partial ^ \\mu = \\partial ^ \\mu \\otimes 1 + [ \\partial ^ \\mu , \\hat { y } _ \\alpha ] \\otimes \\partial ^ \\alpha + \\frac { 1 } { 2 } [ [ \\partial ^ \\mu , \\hat { y } _ \\alpha ] , \\hat { y } _ \\beta ] \\otimes \\partial ^ \\alpha \\partial ^ \\beta + \\cdots . \\end{gather*}"}
-{"id": "7947.png", "formula": "\\begin{align*} g _ B \\left ( \\hat { s } \\right ) & = \\int \\limits _ { B } \\prod \\limits _ { t = 1 } ^ { k } \\prod \\limits _ { j = 1 } ^ { n } \\ell ^ j ( S _ { t } ^ j | \\theta ) d \\mu _ 0 ^ j \\left ( \\theta \\right ) . \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} \\frac { \\norm { y _ { n _ k } - x _ { n _ k } } } { \\sup _ { k \\in \\N } \\gamma _ k } \\leq \\frac { \\norm { y _ { n _ k } - x _ { n _ k } } } { \\gamma _ { n _ k } } \\to 0 \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} \\mathrm { R } _ { \\mathcal { X } } ( \\le t , x _ 0 ) = \\{ x ( T ) \\mid x : [ 0 , T ] \\to \\R ^ n \\mbox { i s a t r a j e c t o r y o f } \\mathcal { X } , \\ x ( 0 ) = x _ 0 , \\ T \\le t \\} . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\ ; ( \\mathbf { u } , \\mathbf { H } ) ^ { T } ( t , \\cdot ) = \\big ( \\mathbf { 0 } , \\mathbf { F } ( \\mathbf { 0 } ) \\big ) ^ { T } L ^ { 2 } ( G , \\mathbb { R } ^ { k } ) \\times L ^ { 2 } \\big ( G , \\mathbb { R } ^ { ( k \\times d ) \\times ( k \\times d ) } \\big ) . \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { b } _ { s , s + k } \\left [ Z _ s , t + \\tau ; \\left \\{ t _ j + \\tau , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] = \\\\ & = \\tilde { b } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\end{aligned} \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} \\begin{aligned} d _ 1 + d _ { 1 1 1 2 2 } > 2 d _ { 1 1 2 } , \\ \\ d _ { 1 1 1 2 } + d _ { 1 1 1 2 2 } > 3 d _ { 1 1 2 } , \\ \\ d _ { 1 1 1 2 } + d _ { 1 2 } > 2 d _ { 1 1 2 } , \\\\ d _ { 1 1 1 2 } + d _ { 2 } > d _ { 1 1 2 } + d _ { 1 2 } , \\ \\ d _ { 1 1 2 } + d _ { 2 } > 2 d _ { 1 2 } , \\ \\ d _ { 1 1 1 2 2 } + d _ { 2 } > 3 d _ { 1 2 } . \\end{aligned} \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\mu = ( 0 , 0 , 1 ) , \\theta = \\begin{pmatrix} p & q & 0 \\\\ r & l & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} U _ { \\Delta } ^ { * } B ( \\alpha ) U _ { \\Delta } = B ( \\beta ) . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} \\mathbf { R } \\boldsymbol { \\theta } = \\mathbf { r } _ { - p } ^ { - 1 } , r _ 0 = \\boldsymbol { \\theta } ^ T \\mathbf { r } _ { - p } ^ { - 1 } + \\sigma ^ 2 _ { \\sf w } , \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} \\int _ M | u | ^ { p - 2 } u \\Big ( ( p - 1 ) \\frac { \\partial u } { \\partial t } d \\mu + \\frac { \\partial } { \\partial t } ( u d \\mu ) \\Big ) = 0 . \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} [ \\varphi ( \\xi ) ] ^ 2 + | \\Psi ( \\xi ) | ^ 2 = | \\xi | ^ 2 \\xi \\in G r _ k ( \\R ^ n ) . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P - I A } } = \\frac { ( 1 - \\mu _ 1 M ) K } { M r } \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} B _ { M , m } ( x | a ) \\triangleq \\frac { d ^ m } { d t ^ m } \\Big | _ { t = 0 } \\bigl [ f _ M ( t | a ) e ^ { - x t } \\bigr ] . \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} u _ { 0 . 6 } ^ 1 ( f _ 1 ) = [ I - 0 . 6 P ( f _ 1 ) ] ^ { - 1 } \\bar { r } ^ 1 ( f _ 1 ) = ( 6 , 4 . 5 ) . \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} H ( X ) & = \\tfrac { 1 } { ( g ! ) ^ { 2 } } \\int _ { X ^ { g } } \\log \\| \\theta \\| ( P _ { 1 } + \\dots + P _ { g } - Q ) \\Phi ^ { * } \\nu ^ { g } \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} P _ { n } \\stackrel { ( d ) } { = } Q _ { U _ { n } } ' + Q _ { n - 2 - U _ { n } } '' , , Q _ { n } \\stackrel { ( d ) } { = } P _ { V _ { n } } ' \\cdot Q _ { n - V _ { n } } ''' , , \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} Y = X + N \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} ( \\forall \\ , k \\in \\N ) v _ k : = W _ k \\frac { x _ { k } - y _ { k } } { \\gamma _ { k } } + \\nabla f ( y _ { k } ) - \\nabla f ( x _ { k } ) \\in \\partial ( f + g ) ( y _ { k } ) . \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} \\eta ( x , t ) = \\int _ { t } ^ { T } h _ { m } ( \\sigma - t ) \\varphi ( \\sigma , x ) d \\sigma = \\int _ { 0 } ^ { T - t } h _ { m } ( \\sigma ) \\varphi ( \\sigma + t , x ) d \\sigma , \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + \\delta + t , x \\right \\rangle } ) = \\sum _ { \\gamma _ { 1 } , \\gamma _ { 2 } , . . . , \\gamma _ { m } } \\left ( \\frac { q _ { \\gamma _ { 1 } } q _ { \\gamma _ { 2 } } . . . q _ { \\gamma _ { m + 1 } } ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + \\delta - \\gamma ( m + 1 ) + t , x \\right \\rangle } ) } { d ( \\gamma , \\delta ) d ( \\gamma , \\delta - \\gamma _ { 1 } ) . . . d ( \\gamma , \\delta - \\gamma ( m ) ) } \\right ) . \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} \\big | A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\bigl ( 1 - e ^ { i \\lambda \\mu } \\bigr ) ^ { n } \\beta ^ 2 f ^ 0 ( \\lambda ) - \\lambda ^ { 2 n } C _ { \\mu , N } ^ { \\beta , 0 } \\bigl ( e ^ { i \\lambda } \\bigr ) \\big | = \\alpha _ 2 | \\lambda | ^ { n } \\big | 1 - e ^ { i \\lambda \\mu } \\big | ^ { n } , \\label { D 1 r i v n 1 _ c o i _ i _ s t . n _ d } \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} \\eta _ { 2 , 1 } ( q | a , b ) = \\frac { \\Gamma _ 2 ( q + b _ 0 \\ , | \\ , a ) } { \\Gamma _ 2 ( b _ 0 \\ , | \\ , a ) } \\frac { \\Gamma _ 2 ( b _ 0 + b _ 1 \\ , | \\ , a ) } { \\Gamma _ 2 ( q + b _ 0 + b _ 1 \\ , | \\ , a ) } . \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} y _ 1 & = \\sqrt { ( 1 - a ) \\xi } h _ 1 x _ 1 + \\sqrt { a \\xi } h _ 1 x _ 2 + z _ 1 \\\\ y _ 2 & = \\sqrt { a \\xi } h _ 2 x _ 2 + \\sqrt { ( 1 - a ) \\xi } h _ 2 x _ 1 + z _ 2 . \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} \\widehat { g } _ { k } = \\widehat { \\varphi } _ { k } - \\sum _ { j = 1 } ^ { k - 1 } b ^ { ( k ) } _ { j } \\widehat { g } _ { j } , 1 < k \\leq N \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} & { w ( T , { X } _ T ) } - w ( s , { X } _ s ) - \\int ^ T _ s \\nabla w ( r , { X } _ r ) \\d { W } _ r \\\\ & = - w ( s , { X } _ s ) - \\int ^ T _ s \\nabla w ( r , { X } _ r ) \\d { W } _ r \\\\ & = \\int ^ T _ s \\nabla u ( r , { X } _ r ) b ( r , { X } _ r ) \\d r \\\\ & = \\int ^ T _ s { Z _ r b ( r , { X } _ r ) } \\d r , \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} d + 1 - 2 s - C ( \\Delta ) \\geq 4 s - p - 2 s - \\left \\lfloor \\sum _ { i = 1 } ^ p \\frac { 2 ^ { u _ i + 1 } - 1 } { 3 } \\right \\rfloor \\geq 2 s - p - \\frac { 4 s - p } { 3 } = \\frac { 2 s - 2 p } { 3 } . \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\dot { V } ( x ) = \\dfrac { \\left \\langle \\nabla V _ N ( x ) ( \\gamma - R ( x ) ) + V _ N ( x ) \\nabla { R } , f ( x ) \\right \\rangle } { ( \\gamma - R ( x ) ) ^ 2 } \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} \\mathbb { E } e ^ { - ( n - m + j ) X _ { r } } & = \\left ( \\frac { s } { s + n - m + j } \\right ) ^ { r } . \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} \\mathcal { C } ^ { } = \\mathcal { C } ^ { } = \\mathcal { C } ^ { } , \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} d \\sigma ^ { - 1 } \\beta = 0 . \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} \\inf \\left \\{ \\max _ { g \\in K } | \\alpha ( g ) f - f | _ { \\infty } ^ - \\mid f \\in { \\mathcal F } ( G ) , \\ f ( e ) = 1 \\right \\} = 0 . \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} & \\mbox { I f $ M = 0 $ t h e n } ~ { \\bf P } _ { Y _ t | Y _ { t - M } ^ { t - 1 } , X _ t } | _ { M = 0 } = { \\bf P } _ { Y _ t | X _ t } , ~ \\mbox { i . e . , t h e c h a n n e l i s m e m o r y l e s s } , ~ t = 0 , \\ldots , n . \\\\ & \\mbox { I f $ N = 0 $ t h e n } ~ { \\gamma } _ { t } ( x _ t , y ^ { t - 1 } _ { t - N } ) | _ { N = 0 } = \\gamma _ t ( x _ t ) , ~ t = 0 , \\ldots , n . \\end{align*}"}
-{"id": "9927.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\delta _ 1 , \\delta _ 2 } V ^ { \\delta _ 1 , \\delta _ 2 } V ^ { \\delta _ 1 , \\delta _ 2 } : = \\left \\{ v \\in V : a _ 1 ( t ) v = e ^ { \\delta _ 1 t } v , a _ 2 ( t ) v = e ^ { \\delta _ 2 t } v t \\in \\R \\right \\} . \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\frac { \\tilde { \\mu } - \\mu } { 2 } \\left ( \\sum _ { j = 1 } ^ N \\frac { 1 } { a _ j ^ 2 } \\right ) \\min _ { j = 1 , \\hdots , N } \\norm { x _ j - y _ j } ^ 2 & \\le \\frac { \\tilde { \\mu } } { 2 } \\norm { x ^ * - v _ 0 } ^ 2 + \\rho M ^ 2 \\left ( \\sum _ { j = 1 } ^ N \\frac { 1 } { a _ j } \\right ) + \\frac { N r M ^ 2 } { 2 } . \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\bar { P } _ { V , t } ( Y ) = C _ { V , t } \\int _ { W \\in \\mathbb { R } ^ { p _ 1 \\times r } : \\sigma _ { \\min } ( W ) \\geq 1 / 2 } & \\frac { 1 } { ( 2 \\pi ) ^ { p _ 1 p _ 2 / 2 } } \\exp ( - \\| Y - 2 t W V ^ { \\intercal } \\| _ F ^ 2 / 2 ) \\\\ & \\cdot ( \\frac { p _ 1 } { 2 \\pi } ) ^ { p _ 1 r / 2 } \\exp ( - p _ 1 \\| W \\| _ F ^ 2 / 2 ) d W . \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} \\tilde { M } _ { r } '' ( z ) = \\frac { 1 } { 1 - z } \\tilde { M } _ { r } ' ( z ) + \\frac { 2 } { ( 2 - z ) ^ { 2 } } \\tilde { M } _ { r } ( z ) + R _ { r } ( z ) , \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} ( \\lambda + \\mu + ( n - 1 ) \\xi ) p _ { 1 , n } = \\lambda p _ { 1 , n - 1 } + \\gamma p _ { 0 , n } + ( \\mu + n \\xi ) p _ { 1 , n + 1 } . \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} [ \\bar { h } _ { 0 , k } , \\bar { f } _ { 0 , l } ] = - ( 2 - d ^ k - d ^ { - k } ) \\bar { f } _ { 0 , l + k } , \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} v _ { \\mu , N } ( k ) = \\sum _ { l = [ - \\frac { k } { \\mu } ] ' } ^ { \\min \\{ [ \\frac { N - k } { \\mu } ] , n \\} } ( - 1 ) ^ l { n \\choose l } b _ { \\mu , N } ( l \\mu + k ) , k = - \\mu n , - \\mu n + 1 , \\ldots , - 1 , \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} \\min \\sigma \\big ( \\mathbf { H } ( t , \\cdot ) \\big ) & \\geq \\alpha \\exp \\big ( - t / \\tau \\big ) + \\omega \\int _ { 0 } ^ { t } \\exp \\big ( - ( t - s ) / \\tau \\big ) \\mathrm { d } s \\\\ & \\geq \\alpha \\exp \\big ( - t / \\tau \\big ) + \\frac { \\omega } { \\tau } \\big ( 1 - \\exp \\big ( - t / \\tau \\big ) \\big ) \\\\ & \\geq \\min \\big \\{ \\alpha \\exp ( - 1 ) , \\omega / ( 2 \\tau ) \\big \\} = : \\kappa > 0 \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} \\begin{aligned} | \\alpha ( g _ p ) ( f \\circ \\rho ) - f \\circ \\rho | _ { \\infty } ^ - & \\le \\left | \\left ( \\sum _ { t \\in K ^ M } | \\alpha ( t ) f - f | \\right ) \\circ \\rho \\right | _ { \\infty } ^ - \\\\ & \\le \\sum _ { t \\in K ^ M } | \\alpha ( t ) f - f | _ { \\infty } ^ - \\\\ & \\le M | K | ^ M \\max _ { k \\in K } | \\alpha ( k ) f - f | _ { \\infty } ^ - . \\end{aligned} \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} a ( \\bigvee _ { z \\in A } h ( z ) \\otimes \\psi ( z ) , x ) = \\bigwedge _ { z \\in A } a ( h ( z ) \\otimes \\psi ( z ) , x ) = \\bigwedge _ { z \\in A } \\hom ( \\psi ( z ) , a ( h ( z ) , x ) ) . \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} ( [ i _ 1 , \\ldots , i _ { r - 1 } ] , S ) \\sim ( [ j _ 1 , \\ldots , j _ { r - 1 } ] , S ' ) \\Leftrightarrow [ i _ 1 , \\ldots , i _ { r - 1 } ] & = [ j _ 1 , \\ldots , j _ { r - 1 } ] \\\\ { } ^ { k _ S } H _ { [ i _ 1 ] } & < { } ^ { k _ { S ' } } H , \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} \\mathcal { Z } _ N = \\int _ { \\mathbb { R } ^ { 2 d N } } \\mathbf { 1 } _ { Z _ N \\in \\mathcal { D } _ N } f _ 0 ^ { \\otimes N } ( Z _ N ) d Z _ N \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} U D _ { \\rm r } U ^ * = - \\frac { d ^ 2 } { d r ^ 2 } , \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} B ^ { ( f ) } _ { k } ( x ) \\triangleq \\frac { d ^ m } { d t ^ k } | _ { t = 0 } \\bigl [ f ( t ) e ^ { - x t } \\bigr ] . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} f ' ( c ) = \\frac { f ( b ) - f ( a ) } { b - a } . \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\varphi _ n ( x ) : = \\sum \\limits _ { p = 0 } ^ \\infty S _ { n , p } x ^ p = \\sum \\limits _ { k = 0 } ^ n f _ n ( k ) \\sum \\limits _ { p = 0 } ^ \\infty \\frac { k ^ p } { n ^ p } x ^ p = \\sum \\limits _ { k = 0 } ^ n \\frac { f _ n ( k ) } { 1 - \\frac { k } { n } x } . \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\hom ( u , v ) = \\begin{cases} 1 & \\\\ v & \\end{cases} \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\varphi ( ( g r ) ( \\omega ) ) = \\varphi ( r ( \\omega ) ) \\omega . \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} \\tau '^ { q ^ { - m } } = ( \\xi ^ { q ^ { - m } } ) ^ { - ( 1 + q + \\cdots + q ^ { n - 1 } ) } \\tau ^ { q ^ { - m } } \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} \\sum _ { { t = 3 } \\atop { t \\neq 5 , 6 } } ^ { 8 } f _ t ( p ' ) = q ^ 3 ( q ^ 2 - q + 1 ) \\left ( \\frac { ( q ^ 2 - 1 ) r } { | G _ { p ' } | } - k _ 5 - k _ 6 \\right ) . \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} \\qquad \\quad \\textrm { s c w } _ m ( t ) = \\left \\{ \\begin{array} { l l } \\frac { 1 } { \\sqrt { 2 } } , & m = 0 , \\\\ \\textrm { c o s } ( 2 m \\pi t ) , & m = 1 , 2 , \\ldots , M , \\\\ \\textrm { s i n } ( 2 ( m - M ) \\pi t ) , & m = M + 1 , M + 2 , \\ldots , 2 M , \\end{array} \\right . \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} A _ i = \\begin{pmatrix} 1 & \\cdots & 1 \\\\ e _ n ^ 1 - z _ { i , 1 } & \\cdots & e _ n ^ 1 - z _ { i , n } \\\\ e _ n ^ 2 - z _ { i , 1 } e _ n ^ 1 + z _ { i , 1 } ^ 2 & \\cdots & e _ n ^ 2 - z _ { i , n } e _ n ^ 1 + z _ { i , n } ^ 2 \\\\ \\vdots & \\ddots & \\vdots \\\\ \\sum _ { d = 0 } ^ { n - h - 2 } ( - z _ { i , 1 } ) ^ d e _ n ^ { n - h - 2 - d } & \\cdots & \\sum _ { d = 0 } ^ { n - h - 2 } ( - z _ { i , n } ) ^ d e _ n ^ { n - h - 2 - d } \\end{pmatrix} . \\end{align*}"}
-{"id": "9896.png", "formula": "\\begin{align*} & \\Big \\{ \\frac { \\| V \\| ( B _ \\rho ( x ) ) + \\| V \\| ( \\tilde { B } _ \\rho ( x ) ) + 2 \\sigma \\mathcal { H } ^ n \\lfloor _ { B ^ + } ( B _ \\rho ( x ) ) } { \\omega _ n \\rho ^ n } \\Big \\} ^ { \\frac 1 p } \\Big ( 1 + C \\varkappa \\rho \\Big ( 1 + \\dfrac { 1 } { p - n } \\Big ) \\Big ) \\\\ & + \\dfrac { \\Gamma \\rho ^ { 1 - \\frac { n } { p } } } { p - n } \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} H ( p ) \\ ; = \\ ; \\sup _ { s \\in C } \\ , \\langle s , p \\rangle . \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} \\left | \\left ( Z _ { s , s + k } \\left [ \\cdot \\right ] - Z _ { s , s + k } ^ 0 \\left [ \\cdot \\right ] \\right ) \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\right | _ \\infty \\leq k \\varepsilon \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\xi _ s ( s , y ) + L ^ b \\xi ( s , y ) - ( \\lambda + 1 ) \\xi ( s , y ) = - b ( s , y ) , \\\\ \\xi ( T , y ) = 0 , \\\\ \\forall ( s , y ) \\in [ 0 , T ] \\times \\mathbb R ^ d . \\end{array} \\right . \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} \\alpha + \\beta & = a + b + A + B \\\\ \\alpha \\cdot \\beta & = a b + a B + b A + A B . \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} c ( \\alpha ) : = \\lim _ { n \\rightarrow \\infty } \\frac { c _ n ( \\alpha ) } { n } = \\frac { 1 } { j _ { ( \\alpha - 1 ) / 2 , 1 } } \\ , , \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} r _ { \\mu } ( t ) ^ { 2 } = \\mu ^ { 2 } v ( t ) ^ { 2 } + v ' ( t ) ^ { 2 } \\geq \\biggl { ( } \\dfrac { 8 k \\Vert b \\Vert _ { L ^ { 1 } ( \\mathopen { [ } 0 , T \\mathclose { ] } ) } } { \\pi } \\biggr { ) } ^ { 2 } , t \\in \\mathopen { [ } 0 , k T \\mathclose { ] } . \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} C _ { p , g } ^ { ( n ) } & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = 0 } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } 2 ^ { p - j - 2 t } \\binom { n } { p - j - 2 t } \\sum _ { i = 0 } ^ { j - 1 } \\binom { 0 } { p + i - j - t - g } \\\\ & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = p - g - j } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } 2 ^ { p - j - 2 t } \\binom { n } { p - j - 2 t } \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} B ( G _ { n , k , b } ) = k \\binom { b } { k } . \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} h ( z ) = - \\bar \\pi ^ { - q } \\left | \\begin{matrix} z & 1 \\\\ \\eta _ 1 ( z ) & \\eta _ 2 ( z ) \\end{matrix} \\right | ^ { - 1 } , \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} ( f ( 2 m _ 0 k ) - a ) ( f ( 2 m _ 0 k ) + 2 a f ( 2 m _ 0 k ) + a ) = 0 . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} f ( q ) \\cdot _ q g ( q ) : = ( f \\cdot g ) ( q ) . \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} \\partial _ t \\omega + y \\partial _ x \\omega & = \\nu \\Delta \\omega \\\\ \\Delta \\psi & = \\omega . \\end{align*}"}
-{"id": "9920.png", "formula": "\\begin{align*} \\lambda ^ { \\max } ( \\sigma ( r ) v ) = - \\lambda ^ { \\min } ( v ) ( \\sigma ( r ) v ) ^ { \\max } = \\sigma ( r ) v ^ { \\min } . \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} \\gamma _ n ( a ^ { ( 1 ) } \\otimes \\cdots \\otimes a ^ { ( n ) } ) = a ^ { ( 1 ) } _ 1 \\otimes a ^ { ( 1 ) } _ 2 \\rightharpoonup a ^ { ( 2 ) } _ 1 \\otimes \\cdots \\otimes ( a ^ { ( 1 ) } _ n \\circ \\cdots \\circ a ^ { ( n - 1 ) } _ 2 ) \\rightharpoonup a ^ { ( n ) } . \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} = E ^ z \\left [ P ^ { Z ( V _ k ^ n ) } \\left ( \\left ( Z ( \\cdot \\wedge \\tau _ 0 ) , Z ( \\cdot \\wedge \\tau _ 0 ) - F _ { \\gamma ^ z } \\left ( Z ( \\cdot ) \\right ) ( t \\wedge \\tau _ 0 ) \\right ) \\in { \\mathcal A } \\right ) \\right ] . \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} d = \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } \\binom { N _ R - r - 1 } { t - 1 } t } { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } \\binom { N _ R - r - 1 } { t - 1 } t + \\binom { N _ R - 1 } { r + 1 } \\binom { N _ R - r - 2 } { t - 1 } \\binom { N _ T } { t - 1 } } \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} \\phi \\big ( M ( D _ i , D _ j ) \\big ) & = \\prod _ { k = 1 } ^ K \\sum _ { \\pi \\in N _ { o } } \\kappa _ { \\pi } [ a _ { k ; i } ^ { n _ 1 } , a _ { k ; j } ^ { n _ 2 } , \\cdots , a _ { k ; i } ^ { n _ { 2 t - 1 } } , a _ { k ; j } ^ { n _ { 2 t } } ] . \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} \\phi ( S ) = \\frac { T } { \\varpi ' } = \\Bigl ( \\frac { T ^ { 1 / p ^ m } } { \\varpi '^ { 1 / p ^ m } } \\Bigr ) ^ { p ^ m } = \\frac { T ^ { 1 / p ^ m } } { \\varpi '^ { 1 / p ^ m } } . \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\hat { f } _ t & = P _ t f _ 0 + \\int _ 0 ^ t P _ { t - s } F ( s , u _ s ) d s , \\\\ \\hat { g } _ t & = P _ t g _ 0 + \\int _ 0 ^ t P _ { t - s } G ( s , u _ s , \\hat { f } _ s , g _ s ) d s , \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} R _ { a c h } = & \\log _ { 2 } \\bigg ( 1 + \\frac { \\mathbf { h } _ { s , k } ^ { H } \\mathbf { Q } ^ { * } _ { s } \\mathbf { h } _ { s , k } } { \\mathbf { h } _ { s , k } ^ { H } \\mathbf { V } ^ { * } \\mathbf { h } _ { s , k } + \\sigma _ { s a , k } ^ { 2 } + \\frac { \\sigma _ { s p , k } ^ { 2 } } { \\rho ^ { * } _ { s , k } } } \\bigg ) \\\\ & - \\log _ { 2 } \\bigg | \\mathbf { I } \\ ! + \\ ! ( \\mathbf { H } _ { e , l } ^ { H } \\mathbf { V } ^ { * } \\mathbf { H } _ { e , l } ) ^ { - 1 } \\mathbf { H } _ { e , l } ^ { H } \\mathbf { Q } ^ { * } _ { s } \\mathbf { H } _ { e , l } \\bigg | \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} h ' = e ^ { C ( A - d ) } \\ge 1 h '' = - C h ' , \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} A _ 1 = \\frac { k _ 1 } { 3 } , A _ 2 = \\frac { 2 - k } { 3 } , M = \\frac { 1 } { \\alpha ^ 3 } \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} \\partial _ t u = L u + N ( u ) , \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} w ( x , t ) \\leq u ( x , t ) \\leq \\liminf _ { ( y , s ) \\to ( x , t ) } u ( y , s ) = \\liminf _ { ( y , s ) \\to ( x , t ) } w ( y , s ) \\leq w ( x , t ) . \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} \\begin{cases} \\varphi '' _ a ( r ) - \\varphi ' _ a ( r ) / r = - a \\zeta _ a ( r ) , \\\\ \\varphi _ a ' ( 0 ) = 0 = \\lim _ { r \\to \\infty } \\varphi _ a '' ( r ) . \\end{cases} \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} \\Lambda ^ 2 _ - T ^ * L \\to \\Lambda ^ 2 _ 7 T ^ * M ^ 8 | _ { L } , \\ : \\alpha \\mapsto 2 \\pi _ 7 ( \\alpha ) = { 1 \\over 2 } ( \\alpha - * ( \\alpha \\wedge \\Phi ) ) , \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} \\lambda a _ 1 = a _ 2 , \\lambda a _ 2 = a _ 3 , \\ldots , \\lambda a _ { n - 1 } = a _ n . \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} | u ^ * | = \\lim _ { k \\to \\infty } | \\varphi _ k | \\leq \\sup _ { \\Omega _ T } | \\psi | \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} \\lambda = \\mid a _ { i } + n _ { 1 } v _ { k } + t \\mid ^ { 2 } = \\mid a _ { l , j } + n _ { l } v _ { k } + t \\mid ^ { 2 } \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} h ( z ) = | c | ^ 2 e ^ { \\eta ( 1 ) \\left ( 1 + R \\right ) } \\cdot ( - 1 ) = - 1 , \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} \\ln \\det ( D , \\Lambda ) = - \\zeta ' ( 0 , D ) - \\ln ( \\Lambda ^ 2 ) \\zeta ( 0 , D ) \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} & \\alpha = \\frac { 1 } { 1 - p } s _ f ( 1 - p ) , a ( t ) = { \\bf 1 } _ { [ 0 , 1 - p ] } ( t ) ( f ( t ) - \\alpha ) , \\\\ & \\beta = \\frac { 1 } { 2 p - 1 } ( s _ f ( p ) - s _ f ( 1 - p ) ) , b ( t ) = { \\bf 1 } _ { ( 1 - p , p ) } ( t ) ( f ( t ) - \\beta ) , \\\\ & \\gamma = \\frac { 1 } { 1 - p } ( s _ f ( 1 ) - s _ f ( p ) ) , c ( t ) = { \\bf 1 } _ { [ p , 1 ] } ( t ) ( f ( t ) - \\gamma ) . \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 2 n - m + 1 } ^ m { m \\choose i } \\frac { ( m + i ) ! } { ( m + i - 2 n - 1 ) ! } \\ : x ^ { i } = x ^ { 2 n - m + 1 } \\frac { d ^ { 2 n + 1 } } { d x ^ { 2 n + 1 } } \\left [ ( x ^ 2 + x ) ^ { m } \\right ] . \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} \\begin{array} { l l l } n = 0 ( \\lambda + c \\gamma ) p _ { 0 , 0 } = \\xi p _ { 0 , 1 } + \\mu p _ { 1 , 1 } \\\\ \\\\ n \\geq 1 ( \\lambda + c \\gamma + n \\xi ) p _ { 0 , n } = \\lambda p _ { 0 , n - 1 } + ( n + 1 ) \\xi p _ { 0 , n + 1 } . \\end{array} \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial a } = \\frac { \\partial } { \\partial z } \\ , + \\ , \\frac { \\partial } { \\partial \\overline { z } } , \\frac { \\partial } { \\partial b } = i \\ , \\Big ( \\frac { \\partial } { \\partial z } \\ , - \\ , \\frac { \\partial } { \\partial \\overline { z } } \\Big ) , \\displaystyle \\frac { \\partial \\overline { f } } { \\partial \\overline { z } } = \\overline { \\displaystyle \\frac { \\partial f } { \\partial z } } . \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} J ^ { ( n ) } y ( x , t ) & = \\int ^ t _ 0 d \\xi _ 1 \\int ^ { \\xi _ 1 } _ 0 d \\xi _ 2 \\cdots \\int ^ { \\xi _ { n - 1 } } _ 0 y ( x , \\xi _ n ) d \\xi _ n , \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} \\big [ \\delta , \\widehat { f } \\big ] ( L _ n , z ) & = 0 \\\\ & \\Rightarrow - ( - 1 ) ^ { | x | | z | } \\delta ( \\widehat { f } ( z ) , \\alpha ( L _ n ) ) = 0 , \\\\ & \\Rightarrow [ \\widehat { f } ( z ) , L _ { n } ] = 0 , \\\\ & \\Rightarrow \\widehat { f } ( z ) = 0 . \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} A _ { q ^ { 2 } } \\left ( - c ^ { 2 } \\right ) = \\left ( c ^ { 2 } q ; q ^ { 2 } \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { ( 3 n ^ { 2 } - n ) / 2 } c ^ { 2 n } } { \\left ( q , c q ^ { 1 / 2 } , - c q ^ { 1 / 2 } ; q \\right ) _ { n } } . \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} J ( x , f ) = H J ( x , f ) ^ { \\frac { n + 1 } { n } } . \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} \\mathbf { F } = \\mathbf { E } + i \\mathbf { H } , \\qquad \\mathbf { G } = \\mathbf { D } + i \\mathbf { B } \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} \\frac { d t } { d \\tau } = \\Theta \\frac { \\sinh \\Theta } { \\cosh \\Theta } . \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} \\theta _ k = \\frac { \\sqrt { L } - \\sqrt { \\ell } } { \\sqrt { L } + \\sqrt { \\ell } } . \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} U _ { \\sigma } ( x ) : = \\frac { 1 } { 4 | x | ^ 2 } \\left ( \\log \\frac { | x | } { \\rho } + \\frac { 1 } { \\rho \\sigma _ 0 } \\right ) ^ { - 2 } \\ , , \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} a _ x f ( x ) + b _ x \\mathbf n _ x f = 0 . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} & \\ ; \\ ; \\| x \\| _ { 1 } \\\\ & \\ ; \\ ; y = A x . \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( \\sup _ { y \\in \\mathbb { R } ^ N } \\int _ { y + B _ { R } ( 0 ) } | w _ n ( x ) | ^ { 2 } \\ d x \\right ) = 0 , \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} \\eta ( \\omega ) \\in i \\R \\ \\ \\omega \\in \\Gamma \\cap i \\R , \\\\ \\eta ( \\tilde \\omega ) \\in \\R \\ \\tilde \\omega \\in \\Gamma \\cap \\R . \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} & A _ { i k , j k } x ^ i x ^ j \\\\ = & \\sigma _ 1 ( A ) _ { , i j } x ^ i x ^ j + R _ { l i j k } A _ { l k } x ^ i x ^ j + R _ { l j } A _ { i l } x ^ i x ^ j \\\\ = & ( \\sigma _ 1 ( A ) _ { , i j } ( p ) + O ( r ) ) x ^ i x ^ j + ( W _ { l i j k } ( p ) + O ( r ) ) ( A _ { l k , m } ( p ) x ^ m + O ( r ^ 2 ) ) x ^ i x ^ j + O ( r ^ 4 ) \\\\ = & \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} J _ s ( x , y ) = J _ { s + 2 } ( x , y ) + \\frac { s + 1 } { \\sqrt { x y } } J _ { s + 1 } ( \\sqrt { x } ) J _ { s + 1 } ( \\sqrt { y } ) . \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} F _ + ( t ) = \\sum _ j S _ j ^ \\ell e ^ { - i t H _ 0 } P _ j ^ \\ell ( t ) . \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} & \\underline { f } ( t , x , \\cdot , \\cdot , \\cdot ) = f _ 1 ( t , x , \\cdot + u _ 2 , \\cdot + \\nabla u _ 2 , \\cdot + v _ 2 ) - f _ 2 ( t , x , u _ 2 , \\nabla u _ 2 , v _ 2 ) \\\\ & \\underline { g } ( t , x , \\cdot , \\cdot , \\cdot ) = g ( t , x , \\cdot + u _ 2 , \\cdot + \\nabla u _ 2 , \\cdot + v _ 2 ) - g ( t , x , u _ 2 , \\nabla u _ 2 , v _ 2 ) \\\\ & \\underline { G } = G _ 1 - G _ 2 . \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} \\int \\psi _ 1 ^ 2 ( x ) d ( \\mu x ) = \\left ( \\int \\psi _ 1 ( x ) d ( \\mu x ) \\right ) ^ 2 , \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} \\frac { ( a ( 1 - b ) q ; q ) _ n } { ( a q ; q ) _ n } = \\sum _ { \\pi \\in \\mathcal { U } _ n } a ^ { \\nu ( \\pi ) } b ^ { \\nu _ d ( \\pi ) } q ^ { | \\pi | } , \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} h _ { n } ( x ) : = e ^ { - x ^ 2 } \\left [ H _ { n + 1 } ^ { 2 } ( x ) - H _ { n } ( x ) H _ { n + 2 } ( x ) \\right ] \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} x _ { i _ 0 , 1 } + x _ { i _ 0 , 2 } + \\cdots + x _ { i _ 0 , t - 1 } = x _ { i _ 1 , 1 } + x _ { i _ 1 , 2 } + \\cdots + x _ { i _ 1 , t - 1 } . \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} \\rho ^ { \\tilde { A } } _ { 2 t } ( S _ { ( A , \\gamma _ 1 \\gamma _ 2 ) } ) = & e ^ { 2 \\pi \\sqrt { - 1 } 2 t } S _ { ( A , \\gamma _ 1 \\gamma _ 2 ) } \\\\ = & P _ A e ^ { 2 \\pi \\sqrt { - 1 } t } S _ { \\gamma _ 1 } e ^ { 2 \\pi \\sqrt { - 1 } t } S _ { \\gamma _ 2 } \\\\ = & P _ A \\rho ^ Z _ t ( S _ { \\gamma _ 1 } ) \\rho ^ Z _ t ( S _ { \\gamma _ 2 } ) \\\\ = & \\rho ^ Z _ t ( P _ A S _ { \\gamma _ 1 } S _ { \\gamma _ 2 } ) \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} A _ 0 X _ 0 = B _ 0 . \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} \\mathbb { P } ( T _ { ( k ) } < X _ { r } ) & = \\mathbb { E } \\left [ \\mathbb { P } ( T _ { ( k ) } < X _ { r } ) | X _ { r } \\right ] \\\\ & = \\mathbb { E } \\left ( \\sum _ { m = k } ^ { n } \\binom { n } { m } ( 1 - e ^ { - X _ { r } } ) ^ { m } e ^ { - ( n - m ) X _ { r } } \\right ) ~ ( ~ \\eqref { o r d e r 1 2 } ) \\\\ & = \\sum _ { m = k } ^ { n } \\binom { n } { m } \\left ( \\sum _ { j = 0 } ^ { m } ( - 1 ) ^ { j } \\binom { m } { j } \\mathbb { E } e ^ { - ( n - m + j ) X _ { r } } \\right ) \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} ( b _ 1 , b _ 2 , b _ 3 ) = \\begin{cases} ( 4 8 + \\epsilon , 6 0 + \\epsilon , 2 4 + \\epsilon ) & , \\\\ ( 2 4 + \\epsilon , 1 2 + \\epsilon , 1 2 + \\epsilon ) & \\end{cases} \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} \\tilde { u } = u \\Theta ^ { - 1 } \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\kappa _ \\lambda = \\det \\left ( \\begin{array} { c c c c } \\kappa _ { ( \\lambda _ 1 ) } & \\kappa _ { ( \\lambda _ 1 + 1 ) } & \\cdots & \\kappa _ { ( \\lambda _ 1 + k - 1 ) } \\\\ \\kappa _ { ( \\lambda _ 2 - 1 ) } & \\kappa _ { ( \\lambda _ 2 ) } & \\cdots & \\kappa _ { ( \\lambda _ 2 + k - 2 ) } \\\\ & \\multicolumn { 2 } { c } { \\dotfill } \\\\ \\kappa _ { ( \\lambda _ k - k + 1 ) } & \\kappa _ { ( \\lambda _ k - k + 2 ) } & \\cdots & \\kappa _ { ( \\lambda _ k ) } \\end{array} \\right ) . \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} \\sum _ { n \\leq x } \\rho ^ { \\omega ( n ) } = x ( \\log x ) ^ { \\rho - 1 } F ( \\rho ) \\left ( 1 + O _ B \\left ( \\frac { 1 } { \\log x } \\right ) \\right ) , \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} x _ j = \\tilde { h } _ j ^ k \\tilde { x } _ k , \\end{align*}"}
-{"id": "10009.png", "formula": "\\begin{align*} \\Sigma _ A & : = \\{ \\underline { x } \\in \\{ 1 , \\ldots , k \\} ^ { \\mathbb { Z } } : A _ { x _ n , x _ { n + 1 } } \\forall n \\in \\mathbb { Z } \\} \\\\ \\Sigma _ A ^ + & : = \\{ \\underline { x } \\in \\{ 1 , \\ldots , k \\} ^ { \\mathbb { N } } : A _ { x _ n , x _ { n + 1 } } \\forall n \\in \\mathbb { N } \\} . \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} Z = Z _ 1 \\cup _ Y Z _ 2 . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} C _ { \\frac { 1 } { \\sqrt { p } } ( \\nabla _ p B ) _ { [ p , 1 ] } } ( s , s + t ) = \\left ( 1 - \\frac { t } { p } \\right ) ^ + \\quad , p \\leq s \\leq s + t \\leq 1 . \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] \\\\ & \\in \\mathcal { G } _ { s + k } \\cap \\hat { \\mathcal { U } } ^ \\eta _ { s + k } \\end{aligned} \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} R ^ * = \\sum _ { \\beta \\neq 0 } r ^ * _ { \\alpha , \\beta } ( x , \\xi ) \\ , D ^ \\alpha _ x D ^ \\beta _ { \\xi } , \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} { \\rm H o m } _ { G ( \\R ) } ( \\mathbf { 1 } , L ^ 2 ) ^ { K ( N ) } = { \\rm H o m } _ { G ( \\R ) } ( \\mathbf { 1 } , L ^ 2 ( B ) ) ^ { K ( N ) } = \\bigoplus _ { \\chi \\in \\widehat { ( \\Z / N \\Z ) ^ \\times } \\atop \\chi ^ 2 = 1 } \\C . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} t r ( f ^ * g ) = \\langle g , \\ f \\rangle = \\int _ 0 ^ { 2 \\pi } \\overline { ( U f ) ( e ^ { i \\theta } ) } ( U g ) ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } . \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\delta ^ { '' } _ { \\mathsf { A c h } } ( \\mu , r ) = \\frac { M ( 1 - \\mu ) } { ( M - 1 ) } \\delta _ { \\mathsf { C a - I A } } + \\frac { ( \\mu M - 1 ) } { ( M - 1 ) } \\delta _ { \\mathsf { C a - Z F } } , \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} \\theta ( H ) [ H , \\i A ] _ { \\circ } \\theta ( H ) & = \\theta ( \\Delta ) [ H , \\i A ] _ { \\circ } \\theta ( \\Delta ) + \\Omega [ H , \\i A ] _ { \\circ } \\theta ( H ) + \\theta ( \\Delta ) [ H , \\i A ] _ { \\circ } \\Omega \\\\ & = \\theta ( \\Delta ) \\Delta ( 4 - \\Delta ) \\theta ( \\Delta ) + K . \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} { \\cal P } _ { 0 , n } \\triangleq \\Big \\{ { \\bf P } _ { X _ t | X ^ { t - 1 } , Y ^ { t - 1 } } = p _ t ( d x _ t | x ^ { t - 1 } , y ^ { t - 1 } ) : t = 0 , 1 , \\ldots , n \\Big \\} . \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 m : n ) } = \\overrightarrow { C } _ { ( m : n ) } \\otimes \\overrightarrow { C } _ { ( 4 : n ) } \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} y = 1 + h z , \\mbox { a n d } y = \\frac { 1 + h z } { 1 - h z } , \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{gather*} x _ { 0 } = k - k ^ { \\prime } + \\alpha - \\alpha ^ { \\prime } , x _ { 1 } = k ^ { \\prime } - k + \\ell - \\ell ^ { \\prime } + \\beta - \\beta ^ { \\prime } , x _ { 2 } = \\ell ^ { \\prime } - \\ell . \\end{gather*}"}
-{"id": "393.png", "formula": "\\begin{align*} Y _ { t } = g ( \\eta _ { T } ) + \\int _ t ^ T f ( s , \\eta _ { s } ) d s - \\int _ t ^ T Z _ { s } d B _ { s } ^ { H } , \\end{align*}"}
-{"id": "10131.png", "formula": "\\begin{align*} y ^ { 2 ( p + q ) } - s ^ { - p } = \\prod _ { k = 0 } ^ { m - 1 } ( y ^ { 2 ( p + q ) / m } - s ^ { - p / m } e ^ { 2 \\pi i k / m } ) , \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\mathbf { Y } ^ { T _ E } _ { [ \\ell + 1 : K ] } + \\tilde { \\mathbf { n } } ^ { T _ E } _ { [ \\ell + 1 : K ] } & = \\mathbf { H } _ 3 \\begin{bmatrix} \\mathbf { X } _ { [ 1 : ( M - \\ell ) ] } ^ { T _ E } \\\\ \\mathbf { H _ 1 } ^ { \\dagger } \\tilde { \\mathbf { Y } } ^ { T _ E } _ { [ 1 : \\ell ] } \\end{bmatrix} + \\mathbf { n } _ { [ \\ell + 1 : K ] } ^ { T _ E } , \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} M _ 1 = x _ 0 x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } = \\sum _ { i = 1 } ^ { r _ 1 } L ^ \\prime _ i , \\ \\ \\ \\ M _ 2 = y _ 0 ^ { b _ 0 } y _ 1 ^ { b _ 1 } \\cdots y _ m ^ { b _ m } = \\sum _ { i = 1 } ^ { r _ 2 } L ^ { \\prime \\prime } _ j ; \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} t h ' ( t ) = \\int _ 0 ^ t s \\Delta h ( s ) d s . \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} \\| g ^ { - 1 } f ^ * h \\| ^ { k / 2 } = \\lambda _ k ^ { k / 2 } \\leq H ^ { k - 1 } ( \\lambda _ 1 ^ { k - 1 } \\lambda _ k ) ^ { 1 / 2 } \\leq H ^ { k - 1 } \\det ( g ^ { - 1 } f ^ * h ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } R = ( n - 1 ) \\Delta R + R ^ 2 \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\Delta _ 1 ( L , B ) = \\sum _ { q = 0 } ^ p v _ L ^ q \\pi _ { L ^ \\perp } ^ * T ^ { ( p - q ) } ( L , B ) \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} g ^ { \\alpha \\alpha } e _ { \\alpha \\mu \\nu \\tau } \\partial ^ { \\nu } Q ^ { \\tau \\beta } - g ^ { \\alpha \\alpha } e _ { \\beta \\mu \\nu \\tau } \\partial ^ { \\nu } Q ^ { \\tau \\alpha } = - i \\partial _ { \\mu } Q ^ { \\alpha \\beta } \\end{align*}"}
-{"id": "10078.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 6 , 6 u + 3 , 6 v + 2 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 6 , 6 u + 3 , 6 v + 4 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "7584.png", "formula": "\\begin{align*} A _ { n , 1 } ( x ) = ( - 1 ) ^ n \\Gamma ( \\mu + \\nu + 1 + n ) \\sum _ { i = 0 } ^ { \\lfloor \\frac { n } { 2 } \\rfloor } a _ { i , n } x ^ i , \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} d \\bar { s } ^ 2 & = \\frac { r ^ 2 } { 1 - r ^ 2 } \\{ d \\tau ^ 2 + \\sigma _ { i j } d \\xi ^ i d \\xi ^ j \\} \\equiv e ^ { 2 \\psi } \\{ d \\tau ^ 2 + \\sigma _ { i j } d \\xi ^ i d \\xi ^ j \\} , \\\\ d \\tilde { s } ^ 2 & = r ^ 2 \\{ d \\tilde { \\tau } ^ 2 + \\sigma _ { i j } d \\xi ^ i d \\xi ^ j \\} \\equiv e ^ { 2 \\tilde { \\psi } } \\{ d \\tilde { \\tau } ^ 2 + \\sigma _ { i j } d \\xi ^ i d \\xi ^ j \\} . \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} \\eta = \\{ \\eta _ { g , h } : g _ * ( L _ h ) \\to L _ { g h } \\} _ { ( g , h ) \\in G \\times G } \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q \\ , | \\ , a , \\ , b _ 0 + x , b _ 1 \\cdots b _ N ) \\ , \\eta _ { M , N } ( x \\ , | \\ , a , \\ , b ) = \\eta _ { M , N } ( q + x \\ , | \\ , a , \\ , b ) , \\ ; x > 0 , \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} & \\tbinom { 2 g + 2 } { g } g \\log \\pi + \\tfrac { g + 1 } { 4 } \\log \\| \\varphi _ { g } \\| ( X ) \\\\ = & \\tbinom { 2 g + 2 } { g } B ( X ) + \\tbinom { 2 g } { g - 2 } g \\sum _ { 1 \\le k < l \\le 2 g + 2 } g ( W _ { k } , W _ { l } ) . \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} 9 t _ { 2 } + 7 t _ { 3 } + t _ { 4 } = 9 d + \\sum _ { r \\geq 5 } ( 6 r - 2 5 ) t _ { r } . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} r ^ * ( Y ) - r ^ * ( X ) & = r ( E - Y ) + | | Y | | _ r - r ( E ) - r ( E - X ) - | | X | | _ r + r ( E ) \\\\ & = r ( E - Y ) - r ( E - X ) + ( | | Y | | _ r - | | X | | _ r ) \\\\ & = | | Y - X | | _ r - ( r ( E - X ) - r ( E - Y ) ) . \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} k _ 1 + \\sum _ { i = i } ^ t k _ { s _ i } s _ i \\equiv r \\mod n - 1 . \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} T _ 2 [ X , Y ; u , v ] = u v ( v - 1 ) X Y , \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} L _ 1 : = \\langle \\nabla _ \\xi \\phi \\rangle ^ { - 2 } ( 1 - \\nabla _ \\xi \\phi \\cdot D _ \\xi ) . \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} f _ 0 ( x , v ) = \\int _ { \\mathcal { J } } d \\mu ( \\alpha ) \\rho _ \\alpha ( x ) m _ \\alpha ( v ) \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} \\partial _ { v v } \\Delta _ t ^ { - 1 } f _ 0 = ( \\Delta _ t - ( a ^ 2 - 1 ) \\partial _ { v v } - b \\partial _ v ) \\Delta _ t ^ { - 1 } f _ 0 , \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} g _ n \\circ u _ n ( x ) & = \\min \\{ g ( x , | u _ n ( x ) | ) , n \\} \\ , s i g n ( g ) \\\\ & = \\min \\{ g ( x , u _ n ( x ) ) , n \\} \\ , s i g n ( g ) \\\\ & = g ( x , u _ n ( x ) ) , \\ , \\ , \\ , \\mbox { f o r } \\ , \\ , n \\geq k ( x ) \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} \\frac { 1 } { \\prod _ i ( 1 - t ^ { a _ i } ) } = \\sum _ i c _ i t ^ i . \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} I = \\sum _ { \\alpha \\in \\mathcal { A } } I _ \\alpha = \\left \\{ \\sum _ { \\beta \\in \\mathcal { B } } f _ \\beta \\bigg \\rvert f _ \\beta \\in I _ \\beta \\mathcal { B } \\subset \\mathcal { A } \\right \\} \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} i _ { n - s - 1 } ( \\sigma _ { s + 1 } ) = d \\tau _ s = d \\omega = 0 . \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} e ^ { - \\frac { \\lambda } { \\xi } z } z ^ { \\frac { \\mu } { \\xi } - 1 } P _ { 1 } ( z ) = \\int _ { 0 } ^ { z } e ^ { - \\frac { \\lambda } { \\xi } s } s ^ { \\frac { \\mu } { \\xi } - 1 } \\left [ - \\frac { \\gamma } { \\xi ( 1 - s ) } P _ { 0 } ( s ) + \\frac { \\gamma } { \\xi s ( 1 - s ) } p _ { 0 , 0 } + \\frac { \\mu - \\xi } { \\xi s ( 1 - s ) } p _ { 1 , 1 } \\right ] d s . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i { 2 m - 1 \\choose m - i } { m + i \\choose i } S _ { n , m + i - 1 } = 0 . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} r ^ 1 ( s , a ^ 1 , a ^ 2 ) = r _ 1 ^ 1 ( s , a ^ 1 ) + r ^ 1 _ 2 ( s , a ^ 2 ) , \\ \\forall \\ s \\in S , a ^ 1 \\in A ^ 1 ( s ) , a ^ 2 \\in A ^ 2 ( s ) . \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q | a , b ) = & \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ \\frac { \\bigl ( \\mathcal { S } _ { M - 1 } e ^ { - x t } \\bigr ) ( q \\ , | \\ , b ) - \\bigl ( \\mathcal { S } _ { M - 1 } e ^ { - x t } \\bigr ) ( 0 \\ , | \\ , b ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } + \\\\ & + q e ^ { - t } \\frac { \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j } { \\prod \\limits _ { i = 1 } ^ M a _ i } \\Bigr ] \\Bigl ) , \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\phi _ k ( n ) = \\prod _ { p | n \\ ; } ( 1 - p ^ { - k } ) . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\d y ( t ) = \\kappa _ t ^ { - 1 } ( \\mathcal { T } _ t ) \\beta ( t , y ( t ) \\kappa _ t ( \\mathcal { T } _ t ) , v ( t ) ) \\d W ( t ) + \\kappa _ t ^ { - 1 } ( \\mathcal { T } _ t ) b ( t , y ( t ) \\kappa _ t ( \\mathcal { T } _ t ) , v ( t ) ) \\d t , \\ t \\in [ 0 , T ] , \\\\ y ( 0 ) = x _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\left ( z \\alpha q ^ { 1 - n + k } ; q \\right ) _ { \\infty } = ( - 1 ) ^ { n } \\alpha ^ { n } z ^ { n } q ^ { - \\frac { 1 } { 2 } n ( n - 1 ) + k n } \\left ( q ^ { - k } z ^ { - 1 } \\alpha ^ { - 1 } ; q \\right ) _ { n } \\left ( q ^ { k + 1 } z \\alpha ; q \\right ) _ { \\ ! \\infty } . \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} & h \\rightharpoonup k = S ( h _ 1 ) ( h _ 2 \\circ k ) , & & h \\leftharpoonup k = T ( h _ 1 \\rightharpoonup k _ 1 ) \\circ h _ 2 \\circ k _ 2 , & & h , k \\in H . \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} h _ { \\pi ( 1 ) } + \\dots + h _ { \\pi ( i ) } & = h _ { \\pi ' ( 1 ) } + \\dots + h _ { \\pi ' ( i ) } \\\\ & \\leq c _ 1 ( \\pi ' ) + \\dots + c _ i ( \\pi ' ) . \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\beta ( u ) = \\phi ( u ) + c ^ { 2 } \\phi ^ { \\prime \\prime } ( u ) . \\end{align*}"}
-{"id": "9887.png", "formula": "\\begin{align*} z ^ { ( i + 1 ) } = z ^ { ( i ) } - \\frac { \\mathcal { W } _ \\sigma ( z ^ { ( i ) } ) } { \\mathcal { W } _ \\sigma ^ { \\prime } ( z ^ { ( i ) } ) } \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} \\tilde { F } ( \\kappa _ i ) = \\frac { 1 } { F ( \\kappa _ i ^ { - 1 } ) } . \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} \\int _ { E \\to M } \\rho _ i \\omega = \\int _ { M \\times F \\to M } \\rho _ i h _ i ^ * \\omega . \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\varphi ( x , \\xi ) = ( \\phi _ + ( x , \\xi ) & - x \\cdot \\xi ) \\chi _ + ( x , \\xi ) \\\\ & + ( \\phi _ - ( x , \\xi ) - x \\cdot \\xi ) \\chi _ - ( x , \\xi ) + x \\cdot \\xi , \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} \\varphi _ { v } ( t ) = \\norm { \\nabla _ H v } ^ 4 + C \\norm { \\nabla _ H \\zeta } ^ 4 + \\norm { \\nabla v } _ { L _ z ^ 2 L _ { x y } ^ 3 } ^ { 6 } + 1 \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} i = 0 \\begin{array} { | c | c | c | c | c | } \\hline & & & & \\\\ \\hline j = 0 & 0 & 1 & 2 & 3 \\\\ \\hline j = 1 & 3 & 2 & 1 & 0 \\\\ \\hline j = 2 & 1 & 0 & 3 & 2 \\\\ \\hline j = 3 & 2 & 3 & 0 & 1 \\\\ \\hline \\end{array} \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} E ' ( z ) = \\frac { 2 p } { 1 - z } E ( z ) + ( 1 - p ) \\big ( E ( z ) \\big ) ^ { 2 } , E ( 0 ) = 1 . \\end{align*}"}
-{"id": "10128.png", "formula": "\\begin{align*} ( y ^ 2 + a x ^ 2 + b x ) ^ q - x ^ { - p } = 0 \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} g ^ { i j } = \\vartheta ^ { - 2 } \\tilde { g } ^ { i j } \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\hat W \\xi ( s , t ) = \\xi ( t s , t ) ( \\xi \\in L ^ 2 ( G \\times G ) ) . \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} h = \\inf \\frac { P ( E ) } { \\mu ( E ) } \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} r = \\tanh \\varrho = 1 - \\frac { 2 } { e ^ { 2 \\varrho } + 1 } . \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} { n \\brack i } = \\frac { 4 ^ { n - i + 1 } - 1 } { n - i + 1 } B _ { 2 ( n - i + 1 ) } { 2 n \\choose 2 i - 1 } , \\quad 0 \\leq i \\leq n . \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} C ( s , t ) = p ( 0 , 0 | s , t ) + p ( 1 , 1 | s , t ) - p ( 0 , 1 | s , t ) - p ( 1 , 0 | s , t ) s \\in [ m ] , t \\in [ n ] , \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} \\omega _ { 1 , \\infty } = \\omega _ { 1 , \\infty } ^ \\mathrm { p p } + \\omega _ { 1 , \\infty } ^ \\mathrm { a c } \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} p ^ { \\lambda \\nu } = \\left ( \\begin{array} [ c ] { c c c c } 0 & - p _ { 1 } & - p _ { 2 } & - p _ { 3 } \\\\ p _ { 1 } & 0 & m _ { 3 } & - m _ { 2 } \\\\ p _ { 2 } & - m _ { 3 } & 0 & m _ { 1 } \\\\ p _ { 3 } & m _ { 2 } & - m _ { 1 } & 0 \\end{array} \\right ) \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\frac { d ^ 4 } { d u ^ 4 } g ( u ) & = u ( 5 - u ^ 2 ) \\psi ( u ) \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} v _ 0 ( t ) & : = e ^ { t A _ p } a + \\int _ 0 ^ t e ^ { ( t - s ) A _ p } P _ p f ( s ) d s , \\\\ \\zeta _ 0 ( t ) & : = e ^ { t \\Delta _ { \\zeta } } b + \\int _ 0 ^ t e ^ { ( t - s ) \\Delta _ { \\zeta } } g ( s ) d s , \\\\ v _ { m + 1 } ( t ) & : = v _ 0 ( t ) + \\int _ 0 ^ t e ^ { ( t - s ) A _ p } F _ p ( v _ m ( s ) , \\zeta _ m ( s ) ) d s , \\\\ \\zeta _ { m + 1 } ( t ) & : = \\zeta _ 0 ( t ) + \\int _ 0 ^ t e ^ { ( t - s ) \\Delta _ { \\zeta } } G _ q ( v _ m ( s ) , \\zeta _ m ( s ) ) d s . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} g ^ { T Z } \\big | _ { ( - 1 , 1 ) \\times Y } = d u ^ 2 + g ^ { T Y } . \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} \\Pi \\tilde L _ a f ( \\xi ) = & ( 2 \\pi ) ^ { - d } \\sum _ { x \\in \\mathbb { Z } ^ d } \\int _ { \\mathbb { T } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - \\varphi _ a ( x , \\eta ) ) } f ( \\eta ) d \\eta = L _ a f ( \\xi ) . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} u _ { \\min } = - u ^ * _ { \\max } \\forall t \\in [ t _ \\delta , T ^ * ) . \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} \\delta + \\delta ^ { t r } - h \\sigma ^ { - 1 } \\Delta + \\sigma ^ { - 1 } \\Delta h ^ { t r } = 0 . \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} v _ i = \\mathcal { L } v _ i . \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { n } \\sum _ { j = 0 } ^ { m } ( - 1 ) ^ { j } \\binom { n } { m } \\binom { m } { j } \\left ( \\frac { s } { s + n - m + j } \\right ) = \\frac { n } { s + n } , \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} & \\left | D _ t ^ \\sigma ( - \\Delta ) ^ { \\gamma / 2 } q _ { \\alpha , \\beta } ( 1 , x ) \\right | + \\left | D _ t ^ \\sigma ( - \\Delta ) ^ { \\gamma / 2 } \\partial _ t q _ { \\alpha , \\beta } ( 1 , x ) \\right | \\\\ & \\leq N \\left ( \\{ | x | ^ { 1 - \\gamma } ( 1 + \\ln | x | 1 _ { \\gamma = 1 } ) \\} \\wedge | x | ^ { - 1 - \\gamma } \\right ) \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} \\min _ { i = 1 , \\ldots , N } & \\| \\mathcal { G } _ { 1 / \\tilde \\mu } ( y _ j ) \\| ^ 2 \\leq \\frac { 8 \\cdot 2 4 L \\tilde { \\mu } \\sum _ { j = 1 } ^ N \\frac { \\varepsilon _ j } { a _ j ^ 2 } } { N ( N + 1 ) ( 2 N + 1 ) } \\\\ & + \\frac { 4 8 \\tilde \\mu ^ 2 } { \\tilde { \\mu } - \\mu } \\left ( \\frac { \\| x ^ * - v _ 0 \\| ^ 2 } { N ( N + 1 ) ( 2 N + 1 ) } + \\frac { M ^ 2 ( r + \\frac { \\rho } { 2 } ( N + 3 ) ) } { ( N + 1 ) ( 2 N + 1 ) } + \\frac { 4 L \\sum _ { j = 1 } ^ N \\frac { \\varepsilon _ j + \\delta _ j } { a _ j ^ 2 } } { N ( N + 1 ) ( 2 N + 1 ) } \\right ) \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} h _ N ( z ) = \\sum _ { r = 1 } ^ N H _ N ^ { ( r ) } z ^ { r - 1 } . \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { \\mathcal { D } } _ s = \\left \\{ \\left . Z _ s = ( X _ s , V _ s ) \\in \\mathbb { R } ^ { d s } \\times \\mathbb { R } ^ { d s } \\right | \\ ; \\forall 1 \\leq i < j \\leq m - 1 , | x _ i - x _ j | > \\varepsilon \\right \\} \\\\ \\end{aligned} \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} \\widetilde { D } = \\left \\{ ( z , w ) \\in \\mathbb { C } ^ { n + m } \\mbox { s u c h t h a t } \\rho _ { 0 } ( z ) + \\sum \\left | w _ { i } \\right | ^ { 2 q _ { i } } < 0 \\right \\} , \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} x = \\varpi _ E ^ { v _ E ( x ) } \\zeta _ x ( 1 + u _ x \\varpi _ E ) , \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{gather*} D _ { 0 } = D _ { 1 } = \\ \\ldots \\ = D _ { n _ { 0 } - 1 } = 0 \\ , \\ D _ { n _ { 0 } } = ( - 1 ) ^ { \\tfrac { n _ { 0 } ( n _ { 0 } + 1 ) } { 2 } } s _ { n _ { 0 } } ^ { n _ { 0 } + 1 } = ( - 1 ) ^ { \\tfrac { n _ { 0 } ( n _ { 0 } + 1 ) } { 2 } } \\Delta _ { 0 } = t _ { n _ { 0 } } \\ . \\end{gather*}"}
-{"id": "9645.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } / 2 } \\left ( - e ^ { \\xi } t \\right ) ^ { n } h _ { n } \\left ( \\sinh \\left ( \\xi + \\log q ^ { n / 2 } \\right ) \\vert q \\right ) } { \\left ( q ; q \\right ) _ { n } } = \\frac { A _ { q } \\left ( t e ^ { 2 \\xi } \\right ) } { \\left ( t ; q \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} H _ f ( x ) \\leq \\frac { L _ f ( x ) } { l _ f ( x ) } = \\frac { \\| D f ( x ) \\| } { \\| D f ( x ) \\| _ s } \\leq K . \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} C _ m = \\hat { C } _ { h _ m } \\setminus \\bigcup _ { n = m + 1 } ^ { \\infty } ( \\hat { C } _ { h _ n } \\cup \\hat { D } _ { h _ n } ) \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\left ( - \\Delta \\right ) ^ \\alpha u _ 1 = \\lambda _ 1 u _ 1 + f _ 1 ( u _ 1 ) + \\partial _ 1 F ( u _ 1 , u _ 2 ) \\ \\ \\textrm { i n } \\ \\mathbb { R } ^ N , \\\\ & \\left ( - \\Delta \\right ) ^ \\alpha u _ 2 = \\lambda _ 1 u _ 2 + f _ 1 ( u _ 2 ) + \\partial _ 2 F ( u _ 1 , u _ 2 ) \\ \\ \\textrm { i n } \\ \\mathbb { R } ^ N . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} G ( \\pi ) & = - \\mathbb { E } _ { \\pi } \\log \\ell ^ i \\left ( s _ { k } ^ i | \\cdot \\right ) + \\sum _ { j = 1 } ^ { n } a _ { i j } D _ { K L } \\left ( \\pi | | \\bar { \\mu } _ { k - 1 } ^ j \\right ) \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{gather*} \\tilde \\phi \\big ( \\overline N _ { n - 1 } v _ { 0 } ^ { ( 1 ) } \\otimes \\overline N _ { n - 2 } v _ { 0 } ^ { ( 1 ) } \\otimes \\overline N _ { 0 } v _ { 0 } ^ { ( 1 ) } \\big ) = N _ { n - 1 } N _ { n - 2 } \\cdots N _ { 0 } v _ { 0 } ^ { ( n ) } . \\end{gather*}"}
-{"id": "9207.png", "formula": "\\begin{align*} M _ X ( a , b ) = \\frac { 1 } { N } \\sum _ { r , l \\leq k } c _ { r l } \\mu _ r ( a ) \\cdot \\mu _ l ( b ) . \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} a ^ * _ { 1 , 1 } = \\frac { \\mu _ R } { 3 } , a ^ * _ { 0 , 2 } = 1 - 2 \\mu _ R - \\mu _ T , a ^ * _ { 0 , 3 } = 3 \\mu _ R + 3 \\mu _ T - 2 , \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} | [ A ( n ) S _ j ] | _ { \\mathcal I } = | S _ j ( A ( n ) - S _ j ^ * A ( n ) S _ j ) | _ { \\mathcal I } \\le | A ( n ) - S _ j ^ * A ( n ) S _ j | _ { \\mathcal I } \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} \\Omega _ 1 \\mid _ { A _ 0 } = \\frac { d y _ 3 \\wedge d y _ 2 } { f _ { y _ 0 } } , ~ \\Omega _ 1 \\mid _ { A _ 2 } = \\frac { d y _ 0 \\wedge d y _ 3 } { f _ { y _ 2 } } , ~ \\Omega _ 1 \\mid _ { A _ 3 } = \\frac { d y _ 2 \\wedge d y _ 0 } { f _ { y _ 3 } } , \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} F _ { d , \\ell } ( z ; \\tau ) = \\sum _ { a \\geq 0 } \\frac { ( 2 \\pi i \\tau ) ^ a } { a ! } \\sum _ { n \\geq 0 } ( \\ell n + d ) ^ { 2 a } \\zeta ^ { \\ell n + d } = \\sum _ { a \\geq 0 } \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\zeta ^ d } { 1 - \\zeta ^ \\ell } \\right ) \\frac { ( 2 \\pi i \\tau ) ^ a } { a ! } . \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} \\omega = \\frac { z ^ 1 z ^ 2 \\cdots z ^ n } { z ^ i } \\partial h ^ i . \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\begin{aligned} A _ { n , i } ( x ) & = A _ { n , i } ^ { \\mu , \\nu , a , b } ( x ) , Q _ n ( x ) = Q _ n ^ { \\mu , \\nu , a , b } ( x ) , \\\\ B _ { n , i } ( x ) & = B _ { n , i } ^ { \\mu , \\nu , a , b } ( x ) , P _ n ( x ) = P _ n ^ { \\mu , \\nu , a , b } ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} & \\Delta ( P \\star ( H _ 1 H _ 2 ) ) = \\Delta ( ( H _ 1 H _ 2 ) \\star ( H _ 1 H _ 2 ) ( P ) ) = \\Delta ( H _ 1 H _ 2 ) \\star \\Delta ( ( H _ 1 H _ 2 ) ( P ) ) . \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} f ^ j \\rightharpoonup x ^ a & = \\sum _ { t = 0 } ^ { \\lfloor j / 2 \\rfloor } \\alpha ^ t ( 1 - \\alpha ^ { - 1 } ) ^ { j - t } \\binom { a } { t } \\binom { a - t } { j - 2 t } z ^ { j - 2 t } y ^ t x ^ { a - j + t } , & j , a & \\in \\N . \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} Q _ \\sigma [ w f , w g ] - \\lambda \\int _ M U \\overline { f } g \\ , w ^ 2 d x \\ , = \\ , \\widehat Q _ \\sigma [ f , g ] , \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} & T _ { j _ { 1 } } \\circ \\dots \\circ T _ { j _ { s } } ( I ) \\cap T _ { i _ { 1 } } \\circ \\dots \\circ T _ { i _ { \\ell } } ( I ) = \\emptyset , \\quad \\mbox { a n d } \\\\ & T _ { j _ { 1 } } \\circ \\dots \\circ T _ { j _ { s } } ( I ) \\cup T _ { i _ { 1 } } \\circ \\dots \\circ T _ { i _ { \\ell } } ( I ) \\subset J . \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\inf _ { \\psi \\in C X , \\psi ( x ) = 1 } ( \\varphi _ 1 ( y ) \\otimes \\psi ( y ) ) = \\varphi _ 1 ( y ) \\otimes \\left ( \\inf _ { \\psi \\in C X , \\psi ( x ) = 1 } \\psi ( y ) \\right ) = \\varphi _ 1 ( y ) \\otimes a _ 0 ( y , x ) \\le \\varphi _ 1 ( x ) < u . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} P _ { t } ( \\tau ( \\xi _ t ( y ) ) ) = \\varphi _ { | L _ t } \\cdot | \\xi _ t ( y ) | . \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} L _ v ^ { ( 3 ) } = v ( v - 1 ) ( 1 - v ( q + t + 4 ) ) \\ , m _ { ( 1 , 1 , 1 ) } [ X ] \\ , m _ { ( 1 , 1 , 1 ) } [ Y ] \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\R \\R } \\omega ( \\pi ) q ^ { | \\pi | } = \\sum _ { \\pi \\in \\mathcal { D } } \\omega ( \\pi ) q ^ { | \\pi | } . \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} \\liminf _ { s \\to \\infty } \\frac { s ^ 2 } { G ( s ) } = \\left ( \\limsup _ { s \\to \\infty } \\frac { s } { G ^ { - 1 } ( s ^ 2 ) } \\right ) ^ { - 2 } , \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} f _ i ( x ) = ( x - z _ { i , 1 } ) ( x - z _ { i , 2 } ) \\cdots ( x - z _ { i , n } ) \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} \\hat \\theta _ T ^ { ( 2 ) } = \\frac { X _ T ^ 2 } { 2 \\int _ 0 ^ T X _ t ^ 2 d t } \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} \\frac { H ( a ) } { f ( a ) } = \\frac { H ( b ) } { f ( b ) } \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} u _ 1 & = - \\frac { 1 } { 2 } a ^ 4 + 2 a ^ 3 + a ^ 2 + 2 a - \\frac { 1 } { 2 } + \\frac { 1 } { 2 } ( a + 1 ) ( a - 1 ) ^ 2 \\sqrt { a ^ 2 - 6 a + 1 } , \\\\ u _ 2 & = - \\frac { 1 } { 2 } a ^ 4 + 2 a ^ 3 + a ^ 2 + 2 a - \\frac { 1 } { 2 } - \\frac { 1 } { 2 } ( a + 1 ) ( a - 1 ) ^ 2 \\sqrt { a ^ 2 - 6 a + 1 } , \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} \\mathcal P \\colon J _ { x _ 0 } ( D ) \\to P J _ { x _ 0 } ( D ) , \\mathcal P ( \\mu ) \\overset { \\eqref { d f : V m u } } { : = } V _ { \\mu } , \\mu \\in J _ { z _ 0 } ( D ) , \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} O _ v ^ { \\times ^ 2 } = \\bigsqcup _ { ( \\overline { a } , \\overline { b } ) \\in ( \\mathbb { F } _ q ^ { \\times } ) ^ 2 } ( a , b ) + ( \\pi O _ v ) ^ 2 , \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} W _ { r e n } ( \\mu ) = - \\frac { 1 } { 2 } \\zeta _ \\Delta ' ( 0 ) - \\frac { 1 } { 2 } \\ln ( \\mu ^ 2 ) \\zeta _ \\Delta ( 0 ) \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} \\Delta { C } ^ \\infty = & \\big ( \\mu _ 1 ( \\beta - 1 ) - \\mu _ 0 ( \\alpha - 1 ) \\big ) + H ( \\alpha ) - H ( \\beta ) + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu _ 1 + \\Delta { C } ^ { \\infty } } } { 1 + 2 ^ { \\mu _ 0 + \\Delta { C } ^ { \\infty } } } \\Big ) . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} & P ^ { ' } _ { c } ( z ) - \\left [ ( \\frac { \\lambda } { \\xi } - \\frac { 1 } { z } \\left ( \\frac { \\mu j } { \\xi } - c \\right ) \\right ] P _ { c } ( z ) \\\\ & = - \\frac { \\gamma } { \\xi ( 1 - z ) } P _ { c - 1 } ( z ) + \\frac { \\mu } { \\xi ( 1 - z ) } \\sum _ { n = 1 } ^ { c } n z ^ { n } p _ { c , n } - \\frac { c ( \\xi - \\mu ) } { \\xi z } \\sum _ { n = 0 } ^ { c } z ^ { n } p _ { c , n } + \\frac { 1 } { z } \\sum _ { n = 1 } ^ { c } n z ^ { n } p _ { c , n } . \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{gather*} a ( z ) = \\sum _ { i \\in \\mathbb { Z } } a _ { i } z ^ { - i - 1 } , b ( z ) = \\sum _ { i \\in \\mathbb { Z } } b _ { i } z ^ { - i - 1 } \\end{gather*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\delta _ \\infty ( x ) = \\lim _ { n \\to \\infty } \\delta _ n ( x ) . \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} C _ f ^ * ( \\chi , s ) = C _ f ^ * ( \\underline T , s ) | _ { T _ j = \\chi ( c _ j ^ * ) - 1 \\textrm { f o r a l l } j } . \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} x - \\frac { \\xi ( u ) } { u } y - \\xi ( u ) = 0 , \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} a _ 1 = \\sum _ { k = 3 } ^ { g + 1 } A _ { k } \\cdot B _ { g , g + 1 - k } \\cdot \\tbinom { g } { k - 1 } \\cdot \\tbinom { g + 1 } { k } , \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} A = A _ { - } \\oplus 0 + 0 \\oplus A _ { + } , \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} \\mathcal { D } u : = \\nabla _ x \\cdot \\left ( \\frac { \\mathfrak { D } } { \\bar { \\sigma } ( x ) } \\nabla _ x u \\right ) \\mbox { a n d } \\rho ^ { i n } ( x ) : = \\int _ { \\mathcal { V } } f ^ { i n } ( x , v ) \\ , { \\rm d } \\mu ( v ) . \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} F _ \\alpha ( \\tilde Z ( \\cdot ) ) = \\tilde X ( \\cdot \\wedge \\tilde \\tau _ 0 ) - z , \\ Q ^ z \\hbox { - a . s . } \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} \\mathcal { W } ( \\theta _ 1 , \\theta _ 2 ) = \\{ r e ^ { i \\theta } : r > 0 , \\theta \\in [ \\theta _ 1 , \\theta _ 2 ] \\} \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} \\Delta \\tilde { h } ( t , x , y ) = ( \\partial _ z ^ 2 + a ^ 2 \\partial ^ L _ { v v } + b \\partial ^ L _ { v } ) h = \\Delta _ t h . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} \\mathcal { F } _ 1 ( t ) = \\rho \\int _ { \\Omega } \\dot { u } _ i u _ i \\ , \\mathrm { d } x \\mathcal { F } _ 2 ( t ) = a \\int _ { \\Omega } \\dot { \\tau } \\tau \\ , \\mathrm { d } x , \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { ( 2 w ) ^ { n } \\Gamma \\left ( \\frac { n + 1 } { 2 } \\right ) } { n ! } = \\sqrt { \\pi } e ^ { w ^ 2 } \\left ( 1 + \\operatorname { e r f } \\left ( w \\right ) \\right ) . \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} \\chi = \\frac { v } { \\sinh u } \\equiv v \\eta ( r ) , \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} & \\lambda _ { 0 } \\int _ { 0 } ^ { T } v ( t ) f ( u ( t ) ) ~ \\ ! d t = \\int _ { 0 } ^ { T } \\bigl { ( } u '' ( t ) f ' ( u ( t ) ) v ( t ) - v '' ( t ) f ( u ( t ) ) \\bigr { ) } ~ \\ ! d t \\\\ & = \\Bigl { [ } - v ' ( t ) f ( u ( t ) ) \\Bigr { ] } _ { t = 0 } ^ { t = T } + \\int _ { 0 } ^ { T } \\bigl { ( } u '' ( t ) f ' ( u ( t ) ) v ( t ) + v ' ( t ) f ' ( u ( t ) ) u ' ( t ) \\bigr { ) } ~ \\ ! d t \\\\ & = \\int _ { 0 } ^ { T } \\bigl { ( } u '' ( t ) f ' ( u ( t ) ) v ( t ) + v ' ( t ) f ' ( u ( t ) ) u ' ( t ) \\bigr { ) } ~ \\ ! d t . \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} Z _ { \\alpha , \\varepsilon } ( \\beta ) = \\sum \\limits _ { j = - N / 2 } ^ { N / 2 } | 1 + e ^ { i \\psi _ j } | ^ { 2 \\alpha \\ , \\beta } e ^ { \\beta V _ \\varepsilon ( \\psi _ j ) } . \\end{align*}"}
-{"id": "9703.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int g \\ , d ( \\mathfrak { S } ^ { n } ( { \\widehat { \\mu } } ) ) _ { j } = \\int _ { [ j ] } ( g \\circ \\pi ) \\ , d \\ , \\mathbb { P } ^ { - } . \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} \\beta _ 0 ^ 1 = \\max _ { s \\in S , a ^ 1 \\in A ^ 1 ( s ) , a ^ 1 \\neq a _ s ^ 1 } \\{ 0 , \\beta _ { s , a ^ 1 } ^ 1 \\} , \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} \\int ( P _ { t / 2 } \\mathbf 1 _ E ) ^ 2 d \\mu & \\le \\left ( \\int _ E \\left ( \\int p _ { t / 2 } ( x , y ) ^ 2 d \\mu ( y ) \\right ) ^ { \\frac { 1 } { 2 } } d \\mu ( x ) \\right ) ^ 2 \\\\ & = \\left ( \\int _ E p _ t ( x , x ) ^ { \\frac { 1 } { 2 } } d \\mu ( x ) \\right ) ^ 2 \\le \\frac { A } { t ^ { Q / 2 } } \\mu ( E ) ^ 2 . \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\frac 1 2 - \\varepsilon < 1 - \\prod _ { j = 1 } ^ J \\left ( 1 - \\frac 1 { \\ell _ j } \\right ) < \\frac 1 2 . \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} \\left ( a ; q \\right ) _ { n + m } = \\left ( a ; q \\right ) _ { n } \\left ( a q ^ { n } ; q \\right ) _ { m } , a , n , m \\in \\mathbb { C } . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} V ( x ) = \\dfrac { V _ N ( x ) } { \\gamma - R ( x ) } \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} - \\varepsilon u ^ { \\prime \\prime } + u ^ { \\prime } + \\xi - f ( v ) & = 0 ( 0 , T ) , \\\\ \\xi & \\in \\partial \\phi ( u ) ( 0 , T ) , \\\\ \\varepsilon u ^ { \\prime } ( T ) & = 0 , \\\\ u ( 0 ) & = u _ { 0 } . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} C ^ u ( O ^ + _ F ) = C ^ * \\big ( \\{ u _ { i j } \\} _ { 1 \\le i , j \\le N } \\ | \\ U = [ u _ { i j } ] \\& \\ U = F \\bar U F ^ { - 1 } \\big ) , \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} \\hat \\Delta \\circ \\hat \\sigma _ t = ( \\hat \\tau _ { t } \\otimes \\hat \\sigma _ { t } ) \\hat \\Delta , \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} | P _ t f ( x ) | & = \\left | \\int p _ t ( x , y ) f ( y ) d \\mu ( y ) \\right | \\\\ & \\le \\sqrt { \\int p _ t ( x , y ) ^ 2 d \\mu ( y ) } \\| f \\| _ 2 \\\\ & \\le \\sqrt { p _ { 2 t } ( x , x ) } \\| f \\| _ 2 . \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} \\big \\lVert \\ , \\cdot \\ , \\big \\rVert ^ \\mathrm { R S } _ { \\det H ^ \\bullet ( Z _ R , F ) } = \\big \\lVert \\ , \\cdot \\ , \\big \\rVert ^ { L ^ 2 } _ { \\det H ^ \\bullet ( Z _ R , F ) } \\exp \\Big ( \\frac { 1 } { 2 } { \\theta _ R } ' ( 0 ) \\Big ) . \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} \\varepsilon _ { n } = \\omega _ n - \\sqrt { 2 } \\ , \\frac { \\Gamma \\left ( n + \\frac { 3 } { 2 } \\right ) } { \\Gamma ( n + 1 ) } \\geq 0 \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} \\Theta _ { p + 2 } ( t + h , t ) = \\Theta _ p \\left ( t + h , t + ( 1 - \\gamma _ p ) h \\right ) \\circ \\Theta _ p \\left ( t + ( 1 - \\gamma _ p ) h , t + \\gamma _ p h \\right ) \\circ \\Theta _ p \\left ( t + \\gamma _ p h , t \\right ) , \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} C ( v , \\alpha ) = \\{ y \\in \\bar M \\setminus \\{ x \\} : \\sphericalangle ( v , \\dot \\gamma ^ { x , y } _ { 0 } ) < \\alpha \\} \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a ; q \\right ) _ { n } q ^ { n } } { \\left ( q ; q \\right ) _ { n } } = \\frac { \\left ( a q ; q \\right ) _ { \\infty } } { \\left ( q ; q \\right ) _ { \\infty } } = \\frac { 1 } { \\sqrt { 2 \\pi \\log q ^ { - 1 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } \\right ) d x } { \\left ( q / a , - q ^ { 1 / 2 } / ( a e ^ { i x } ) , - q ^ { 1 / 2 } ( a e ^ { i x } ) ; q \\right ) _ { \\infty } } , \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\abs { D \\varphi } ^ 2 = v ^ 2 - 1 \\leq c . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( \\| E _ 1 \\tilde v _ k \\| ^ 2 + \\| E _ 2 \\tilde w _ k \\| ^ 2 ) = \\frac { 1 } { 2 \\alpha ^ 2 } \\| E _ 2 w _ k \\| ^ 2 \\rightarrow 0 \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} ( \\partial ^ * \\star f ) ( \\phi ) = \\int \\phi \\ d \\nu _ { f , h } + \\sum _ { i = 0 } ^ 2 \\phi ( p _ i ) f ( p _ i ) \\mathbf { n } _ i h . \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} & ( 1 - \\beta ) \\tau _ 6 ( d v o l _ 6 - g _ n d v o l _ 6 ) \\\\ = & \\tau ( d v o l _ 6 - g _ n d v o l _ 6 ) - \\partial \\tau _ 7 ( d v o l _ 6 - g _ n d v o l _ 6 ) \\\\ = & 0 \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} Y _ 1 = t \\partial _ t - x \\partial _ x , & Y _ 2 = z \\partial _ t - x \\partial _ y , Y _ 3 = t \\partial _ z - y \\partial _ x , \\\\ Y _ 4 = z \\partial _ z - y \\partial _ y , & Y _ 5 = x \\partial _ x + y \\partial _ y , Y _ 6 = z \\partial _ x - t \\partial _ y . \\end{align*}"}
-{"id": "9975.png", "formula": "\\begin{align*} \\sum _ { b \\in A } Z ( a , b ) w _ b = 1 a \\in A \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} g = m + g _ 0 ( t , r ) + g _ 1 ( t , x ) \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} & \\mathbf { H } _ 1 = \\mathbf { H } _ { [ 1 : \\ell ] } ^ { [ ( M - \\ell ) + 1 : M ] } ; ~ ~ ~ \\mathbf { H } _ 2 = \\mathbf { H } _ { [ \\ell + 1 : K ] } ^ { [ ( M - \\ell ) + 1 : M ] } . \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} x _ { 1 } = \\frac { \\gamma _ { 3 } + ( 1 + \\gamma _ { 5 } ) d } { \\beta _ { 5 } d + \\beta _ { 3 } + 1 } . \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} | \\lambda ' | ^ 2 = 1 , \\delta \\Re { \\lambda ' } \\geq \\epsilon . \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} C ^ { N F B , A . 1 } = C ^ { F B , A . 1 } = \\frac { \\gamma } { \\alpha + \\gamma } . \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} g ( x ) = \\left \\{ \\begin{array} { l l } e ^ { - \\frac { C } { \\lambda } | x | ^ { \\lambda } } , \\lambda : = \\alpha - \\frac { \\beta } { p - 1 } + 1 \\neq 0 , \\\\ \\frac { 1 } { | x | ^ { C } } , \\alpha - \\frac { \\beta } { p - 1 } + 1 = 0 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} J _ \\sigma ( u ) = \\int _ \\Omega \\dfrac { ( \\Delta u ) ^ 2 } { 2 } - ( 1 - \\sigma ) \\int _ \\Omega d e t ( \\nabla ^ 2 u ) - \\int _ \\Omega F ( x , u ) , \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{gather*} \\frac { Q _ { r } ( x ) } { P _ { r } ( x ) } = \\sum _ { n = 1 } ^ { r } \\frac { Q _ { r } ( \\lambda _ { n } ) } { P _ { r } ^ { \\ , \\prime } ( \\lambda _ { n } ) ( x - \\lambda _ { n } ) } = \\sum _ { m = 0 } ^ { \\infty } \\frac { 1 } { x ^ { m + 1 } } \\sum _ { n = 1 } ^ { r } \\mu _ { n } \\lambda _ { n } ^ { m } \\ , \\ \\ \\ \\ | x | > \\max _ { 1 \\leq n \\leq r } \\ , | \\lambda _ { n } | \\ . \\end{gather*}"}
-{"id": "5031.png", "formula": "\\begin{align*} \\Psi _ { j } ( x , y ) : = \\big ( L ( u _ j ( x ) , u _ j ( y ) ) - L ( u ( x ) , u ( y ) ) \\big ) \\big ( \\phi ( x ) - \\phi ( y ) \\big ) K ( x , y ) , \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} Q ( \\rho , \\theta ) : = B _ \\rho ( x _ 0 ) \\times ( t _ 0 - \\theta , t _ 0 ) \\Subset \\Omega _ T \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} u = - 2 ( a ^ 3 ( s - 2 ) - a ^ 2 ( s ^ 2 - 3 s + 2 ) - a ( 2 s ^ 2 - 3 s + 2 + t ) - s ^ 2 + s - t ) \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} & \\frac { d } { d t } A _ 1 ( t ) + B _ 1 ( t ) \\leq K ( t ) \\left ( \\log \\sum _ { i = 1 } ^ n A _ i ( t ) \\right ) A _ 1 ( t ) , \\\\ & \\frac { d } { d t } A _ i ( t ) + B _ i ( t ) \\leq K ( t ) \\left ( \\log \\sum _ { i = 1 } ^ n A _ i ( t ) \\right ) A _ i ( t ) + \\zeta A _ { i - 1 } ^ \\alpha ( t ) B _ { i - 1 } ( t ) , \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} s _ b ( t ) = \\left \\{ \\begin{array} { l l } \\displaystyle A ( b ) + B ( b ) | t | & | t | \\leq t _ 0 \\\\ \\displaystyle C \\ , p _ N ( t ) & | t | \\geq t _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} \\begin{aligned} v _ 1 ^ * & = v _ 1 + \\omega \\omega \\cdot ( v _ 2 - v _ 1 ) \\\\ v _ 2 ^ * & = v _ 2 - \\omega \\omega \\cdot ( v _ 2 - v _ 1 ) \\end{aligned} \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} W ^ \\eta _ t : = W _ { k \\eta } + \\frac { W _ { ( k + 1 ) \\eta } - W _ { k \\eta } } { \\eta } ( t - k \\eta ) . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} \\mu ( P _ 1 P _ 2 P _ 4 P _ 3 ) = \\frac { 1 } { ( k - 2 ) ! } \\frac { 1 } { t - s } \\left ( \\frac { 1 } { k } ( v - u ) ( t ^ k - s ^ k ) + \\frac { t u - s v } { k - 1 } ( t ^ { k - 1 } - s ^ { k - 1 } ) \\right ) . \\end{align*}"}
-{"id": "9903.png", "formula": "\\begin{align*} \\nabla g ( x ) \\cdot S = & \\ , n ( \\gamma ( r ) + \\gamma ( \\tilde { r } ) ) + r \\gamma ^ { \\prime } ( r ) ( 1 - | S ^ { \\perp } ( \\nabla r ) | ^ 2 ) \\\\ & \\ , + \\tilde { r } \\gamma ^ { \\prime } ( \\tilde { r } ) \\left ( 1 - \\left | S ^ { \\perp } \\left ( i _ x \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\right ) \\right ) \\right | ^ 2 \\right ) + \\gamma ( \\tilde { r } ) \\varepsilon _ 1 ( x , S ) - \\tilde { r } \\gamma ^ { \\prime } ( \\tilde { r } ) \\varepsilon _ 2 ( x , S ) , \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } } { \\left ( q , e ; q \\right ) _ { n } } \\left ( \\frac { e } { q } \\right ) ^ { n } = \\frac { 1 } { \\left ( e ; q \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} \\partial _ t \\psi ( t , x ) = J * \\psi ( t , \\cdot ) ( x ) - \\psi ( t , x ) + e ^ { - t } J ( x ) . \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{align*} s _ n = L ( x ^ n ) = L ( \\lim \\limits _ { k \\to + \\infty } g _ { n , k } ( x ) ) & = \\lim \\limits _ { k \\to + \\infty } L ( g _ { n , k } ( x ) ) \\\\ & = \\lim \\limits _ { k \\to + \\infty } \\int _ \\mathbb { R } g _ { n , k } ( x ) d \\mu ( x ) = \\int _ \\mathbb { R } \\lim \\limits _ { k \\to + \\infty } g _ { n , k } ( x ) d \\mu ( x ) \\\\ & = \\int _ \\mathbb { R } x ^ n d \\mu ( x ) . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} z ^ i z ^ k \\Gamma ( h ^ i , h ^ k ) = \\Gamma ( h ^ i ) \\Gamma ( h ^ k ) , \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } L ( s , S _ f ^ \\nu ) V ( s ) X ^ s d s & = \\sum _ { n \\geq 1 } S _ f ^ \\nu ( n ) v ( \\tfrac { n } { X } ) \\\\ & = \\sum _ { n \\leq X } S _ f ^ \\nu ( n ) + \\sum _ { X < n \\leq X + X / Y } S _ f ^ \\nu ( n ) v ( \\tfrac { n } { X } ) \\\\ & = \\sum _ { n \\leq X } S _ f ^ \\nu ( n ) + O \\bigg ( \\frac { X } { Y } X ^ { \\kappa ( f ) + \\frac { 1 } { 3 } + \\frac { 1 } { 3 } \\alpha ( f ) + \\epsilon - \\nu } \\bigg ) , \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} \\mathcal { P } _ k = & \\{ p _ k ( y ) | \\ p _ k ( y ) = \\sum _ { | \\beta | \\leq k } a _ \\beta y ^ { \\beta } , \\\\ & a _ \\beta = 0 \\sum _ { i = 1 } ^ { n - 1 } 2 \\beta _ i + \\beta _ n + \\beta _ { n + 1 } > k \\} . \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} \\frac { 1 } { n \\log \\alpha } \\log \\frac { \\sigma ( x ) ( B \\cap B _ { \\alpha ^ n } ^ T ) } { \\sigma ( x ) ( B ) } \\geq \\frac { 1 } { n \\log \\alpha } \\sum _ { i = 0 } ^ { p - 1 } \\tau ( \\alpha _ X ^ { - i } x ) - \\frac { 1 } { n \\log \\alpha } \\sum _ { i = 0 } ^ { p - 1 } \\phi _ { \\alpha ^ { n - i } } ( \\alpha _ X ^ { - i } x ) \\text . \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} E _ { k , l } ( \\cdot ) : = \\langle e _ { k } , E _ { B } ( \\cdot ) e _ { l } \\rangle , k , l \\in \\Z , \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} b ( f ) = \\sum _ { s \\in S } \\big ( \\deg f - | f ^ { - 1 } ( s ) | \\big ) = \\sum _ { s \\in S } \\left ( \\sum _ { r \\in f ^ { - 1 } ( s ) } ( v _ f ( r ) - 1 ) \\right ) \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} \\varphi ^ i = \\sigma ^ { i k } \\varphi _ k . \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} I _ { - m } ^ { ( 2 ) } ( z ; q ) = I _ { m } ^ { ( 2 ) } ( z ; q ) = I _ { | m | } ^ { ( 2 ) } ( z ; q ) , z \\in \\mathbb { C } , \\ m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} \\left \\vert B _ { \\ell } \\right \\vert \\leq C \\sum _ { i = 1 } ^ { J } \\left \\vert a _ { i } \\right \\vert \\mu \\left ( z _ { i } \\right ) . \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} \\hat { \\mathcal { U } } _ s ^ \\eta = \\left \\{ Z _ s = ( X _ s , V _ s ) \\in \\mathcal { U } _ s ^ \\eta \\left | \\forall \\tau , \\tau ^ \\prime > 0 , ( \\psi _ s ^ { - \\tau } Z _ s , \\psi _ s ^ { - \\tau ^ \\prime } Z _ s ) \\in \\mathcal { V } _ s ^ \\eta \\right . \\right \\} \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} \\partial _ t \\tau - \\Delta \\tau = g _ { \\tau } , \\tau ( 0 ) = b _ { \\tau } , \\partial _ t \\sigma - \\Delta \\sigma = g _ { \\sigma } , \\sigma ( 0 ) = b _ { \\tau } , \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} \\sigma _ { [ i _ 1 , \\ldots , i _ l ] } = H _ { [ i _ 1 , \\ldots , i _ l ] } < H _ { [ i _ 1 , \\ldots , i _ { l - 1 } ] } < \\ldots < H _ { [ i _ 1 , i _ 2 ] } < H _ { [ i _ 1 ] } < H . \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} L ^ { - 1 } ( a ) ~ = ~ \\sum _ { i : L ( G _ i ) \\le a } ( D _ i - G _ i ) , ~ a \\geq 0 . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} \\begin{aligned} a y ^ 2 x + b y x y + a x y ^ 2 + c x ^ 3 & = 0 \\\\ a x ^ 2 y + b x y x + a y x ^ 2 + c y ^ 3 & = 0 \\end{aligned} . \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} ( h \\otimes u ) ( x ) = h ( x ) \\otimes u , \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} C ^ { F B , A . 1 } ( \\kappa ) = & \\nu _ 0 \\Big ( H ( \\nu _ { 0 | 0 } ) - H ( \\gamma ) \\Big ) + ( 1 - \\nu _ 0 ) \\Big ( H ( \\nu _ { 0 | 1 } ) - H ( \\delta ) \\Big ) + \\xi _ 0 \\Big ( H ( \\gamma ) - H ( \\alpha ) \\Big ) \\\\ & + \\xi _ 1 \\Big ( H ( \\delta ) - H ( \\beta ) \\Big ) \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} \\delta _ F ^ { ( 2 ) } = \\frac { K } { M r } ; ~ ~ ~ ~ ~ \\delta _ E ^ { ( 2 ) } = \\frac { K } { \\min \\{ M , K \\} } . \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} \\mathbf { K } _ { d , e } = \\left [ \\begin{matrix} \\Phi ( k _ { d , e , 1 } ) & \\mathbf { 0 } & \\cdots & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\Phi ( k _ { d , e , 2 } ) & \\cdots & \\cdots \\\\ \\cdots & \\cdots & \\ddots & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\mathbf { 0 } & \\mathbf { 0 } & \\Phi ( k _ { d , e , m } ) \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} \\Phi ^ * \\gamma _ { 1 k } \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 j } \\Phi = \\frac { 1 } { i } ( A _ 1 A _ k ^ * A _ j ^ * - A _ k A _ j A _ 1 ^ * ) + A _ 1 \\Phi ^ * \\sigma _ k \\sigma _ 1 ^ { - 1 } \\sigma _ j \\Phi A _ 1 ^ * \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} E _ { \\pm } ( k ) : = 2 \\pm 2 \\cos \\left ( k / 2 \\right ) \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} \\sigma ^ 2 _ r ( G ) = t ^ 2 > C _ { \\rm g a p } ( ( p d ) ^ { 1 \\over 2 } + d + p ^ { 3 / 2 } n ^ { - { 1 \\over 2 } } ) , n \\geq C _ 0 p , \\end{align*}"}
-{"id": "10076.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 6 , 6 u + 2 , 6 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 6 , 6 u + 4 , 6 v + 5 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "9161.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\begin{array} { l l l l l l l l l } x _ { i } & = & \\dfrac { h N p _ { i } } { h N q } & + & \\varepsilon \\phi & = & \\dfrac { H _ { i } } { K } & + & \\varepsilon \\phi \\end{array} & ; & i = 1 , 2 , . . . , n \\\\ \\begin{array} { l l l l l l l l l } x _ { n + 1 } & = & \\dfrac { h M } { h N q } & + & \\varepsilon a & = & \\dfrac { H _ { n + 1 } } { K } & + & \\varepsilon a \\end{array} & & \\end{array} \\right . \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} ( X ^ k , Y ^ k ) & \\sim \\prod _ { i = 1 } ^ k p ( x _ { i } ) p ( y _ { i } ) \\\\ & = \\prod _ { i = 1 } ^ k \\left ( \\prod _ { n = 1 } ^ N p ( x ^ { ( n ) } _ { i } ) \\prod _ { \\ell = 1 } ^ L p ( y _ { \\ell i } ) \\right ) . \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} S _ m : = \\sup _ { s \\geq s _ 0 + m } \\frac { s } { G ^ { - 1 } ( s ^ 2 ) } , \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} A ' = \\bigcup _ { A _ 0 \\in \\mathcal { D } } A _ 0 \\cong \\varinjlim _ { A _ 0 \\in \\mathcal { D } } A _ 0 , A / \\mathfrak { m } A \\cong \\varinjlim _ { A _ 0 \\in \\mathcal { D } } A _ 0 / \\mathfrak { m } A _ 0 . \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} \\dim S _ { \\iota , \\ell , \\mathbf { w } } = ( m - 2 ) ( h + 1 ) . \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { m } } \\sum _ { i = 1 } ^ m W _ i \\stackrel { } { \\longrightarrow } \\varGamma = ( \\varGamma _ 1 , \\dots , \\varGamma _ 5 ) m \\to \\infty , \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 3 } \\ [ \\bar { e } _ { i , k _ { \\pi ( 1 ) } } , [ \\bar { e } _ { i , k _ { \\pi ( 2 ) } } , [ \\bar { e } _ { i , k _ { \\pi ( 3 ) } } , \\bar { e } _ { i + 1 , l } ] ] ] = 0 , \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} ( T _ 1 f ) ( k ) = \\begin{cases} \\langle \\psi _ 1 - \\alpha _ 1 , f \\rangle & k = 0 \\\\ 0 & k = 1 \\\\ \\frac { k } { k - 1 } f ( k - 1 ) & k \\ge 2 \\end{cases} \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} \\begin{cases} z _ 1 = \\frac { g _ 3 ( a + \\pi x , b + \\pi y ) - g _ 3 ( a , b ) } { \\pi } & \\\\ z _ 2 = y . & \\end{cases} \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} a _ { 0 , m + 1 } = - \\Big ( 1 + \\frac { \\alpha } { m + 1 } \\Big ) \\ , a _ { 0 , m } \\ , , m \\in \\mathbb { N } _ 0 \\ , , \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 0 } ^ { m _ 1 } \\sum _ { \\beta = 0 } ^ { m _ 2 } N ( 2 m _ 1 + 2 m _ 2 - 2 \\alpha - 2 \\beta , m _ 1 + 2 m _ 2 - \\alpha - 2 \\beta ) . \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} a \\nabla _ { \\nu } b - r _ 0 ( \\sqrt { a } - \\sqrt { b } ) ^ 2 & = a \\nabla _ { \\nu } b - \\nu ( \\sqrt { a } - \\sqrt { b } ) ^ 2 \\\\ & = 2 \\nu \\sqrt { a b } + ( 1 - 2 \\nu ) a \\\\ & = a \\nabla _ { 2 \\nu } \\sqrt { a b } \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} q ( p ^ { - 1 } ( U _ 2 \\cdot [ \\phi _ 6 ] ) ) & = ( U _ 2 \\cdot [ x y ] ) \\cup ( U _ 2 \\cdot [ x ^ 2 - y ^ 2 ] ) \\cup ( U _ 2 \\cdot [ x ^ 2 + y ^ 2 ] ) = \\\\ & = \\{ x ( s x + y ) \\} \\cup \\{ x ^ 2 - ( s x + y ) ^ 2 \\} \\cup \\{ x ^ 2 + ( s x + y ) ^ 2 \\} , \\end{align*}"}
-{"id": "9929.png", "formula": "\\begin{align*} a _ 1 ( t ) w _ i = e ^ { ( r - 2 i ) t } w _ i , 0 \\leq i \\leq r . \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} d O \\mathcal { J } _ { ( p , j ) } \\left ( X + Y \\right ) & = d O \\left ( j X + j Y \\right ) \\\\ & = O j O ^ t \\ , O X + O j O ^ t \\ , O Y O ^ t = \\mathcal { J } _ { O \\cdot ( p , j ) } d O ( X + Y ) . \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = \\omega ( \\lambda ) + \\widetilde { \\xi } ( y , z , \\lambda ) + \\widetilde { f } ( x , y , z , \\lambda ) , \\\\ \\dot { y } & = \\sigma ( \\lambda ) + \\widetilde { \\eta } ( y , z , \\lambda ) + \\widetilde { g } ( x , y , z , \\lambda ) , \\\\ \\dot { z } & = \\widetilde { Q } ( \\lambda ) z + \\widetilde { \\zeta } ( y , z , \\lambda ) + \\widetilde { h } ( x , y , z , \\lambda ) \\end{aligned} \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} \\tau _ { m , n } = \\frac { 1 - \\mu _ R } { d _ { m , n } } , \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} \\epsilon ^ 1 _ { j 1 } = \\epsilon ^ 2 _ { j 2 } = \\epsilon ^ 3 _ { j 3 } = \\epsilon ^ 4 _ { j 4 } = 0 , 2 \\leq j \\leq 4 . \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ d } \\ , \\epsilon ^ d \\ , \\bigg | \\sum _ { i = 1 } ^ { N } \\ , \\hat { F _ i } ( \\epsilon \\lambda ) \\ , e ^ { - 2 \\pi i x _ i \\cdot \\lambda } \\bigg | ^ 2 \\ , d \\mu ( \\lambda ) \\leq \\left ( G ( \\delta , \\epsilon ) + I ( \\delta , \\epsilon ) \\right ) ^ 2 \\\\ & \\le \\left ( \\sqrt { \\mathcal { D } ^ { + } _ N ( \\check { \\mu } ) + \\rho ' } + \\sqrt { \\rho ' } \\right ) ^ 2 \\ , \\| f \\| _ 2 ^ 2 \\leq ( \\mathcal { D } ^ { + } _ N ( \\check { \\mu } ) + \\rho ) \\ , \\| f \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} q ^ { \\binom { n } { 2 } } = \\sum _ { k = 0 } ^ { \\left \\lfloor n / 2 \\right \\rfloor } \\frac { \\left ( q ^ { - n } , q ^ { 1 - n } ; q ^ { 2 } \\right ) _ { k } } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { k } } \\left ( - q ^ { 2 n - k } \\right ) ^ { k } . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} f _ S ( d _ 0 , d _ 1 , \\dots , d _ { n - 1 } ) = n ! { \\bf 1 } _ { \\{ d _ 0 + d _ 1 + \\cdots + d _ { n - 1 } < 1 \\} } { \\bf 1 } _ { \\{ \\min \\{ d _ 0 , d _ 1 , \\dots , d _ { n - 1 } \\} > 0 \\} } \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} \\log \\Gamma _ M ( w | a ) = \\int \\limits _ 0 ^ \\infty \\frac { d t } { t ^ { M + 1 } } \\Bigl ( e ^ { - w t } \\ , f _ M ( t | a ) - \\sum \\limits _ { k = 0 } ^ { M - 1 } \\frac { t ^ k } { k ! } \\ , B _ { M , k } ( w | a ) - \\frac { t ^ M \\ , e ^ { - t } } { M ! } \\ , B _ { M , M } ( w | a ) \\Bigr ) . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} C _ { p , 0 } ^ { ( \\ell ) } & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { \\beta = 0 } ^ { p - j } 2 ^ { p - j - \\beta } \\binom { n - \\ell } { \\beta } \\binom { \\ell } { p - j - \\beta } \\binom { \\beta } { p - j } \\\\ & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\binom { n - \\ell } { p - j } = ( - 1 ) ^ { p - 1 } \\sum _ { j = 0 } ^ { p - 1 } ( - 1 ) ^ { j } \\binom { n - \\ell } { j } = \\binom { n - \\ell - 1 } { p - 1 } = 0 \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} h \\left ( \\mathbf { Y } _ { [ 1 : \\ell ] } ^ { T _ E } \\right ) \\leq \\sum _ { k = 1 } ^ \\ell \\sum _ { t = 1 } ^ { T _ E } h \\Big ( { Y } _ k [ t ] \\Big ) . \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} y _ N ( x , t ) = \\sum _ { n = 0 } ^ { 2 ^ k - 1 } \\sum _ { m = 0 } ^ { 2 M } \\sum _ { l = - 1 } ^ { N _ h + 1 } c _ { n , m , l } J ^ { ( 1 ) } \\psi _ { n , m } ( t ) B _ l ( x ) + \\varphi ( x ) , \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} f ( a , b ) = [ d ( a ) , b ] + [ a , d ( b ) ] - d ( [ a , b ] ) \\ \\mbox { f o r s o m e l i n e a r m a p } \\ d . \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} A _ i = \\begin{pmatrix} \\alpha _ i & \\beta _ i \\\\ \\gamma _ i & \\delta _ i \\end{pmatrix} , B _ i = \\begin{pmatrix} 0 & 0 \\\\ b _ i & c _ i \\end{pmatrix} , C _ i = \\begin{pmatrix} 0 & 0 \\\\ e _ i & f _ i \\end{pmatrix} , \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} \\mathbf { g } ( \\mathbf { s } ) = \\left ( 1 + \\frac { | \\mathbf { s } | } { \\lambda } \\right ) ^ { - 1 } \\mathbf { I } \\lambda > 0 . \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} f ( x | s , r ) = \\frac { s ^ r } { \\Gamma ( r ) } e ^ { - s x } x ^ { r - 1 } , ~ ~ x > 0 . \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} F F P _ { \\mathbf { w } } ( \\{ V _ k ( t ) \\} _ { k = 1 } ^ K ) < F F P _ { \\mathbf { w } } ( \\mathbf { W } ) \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\| a \\| _ s = \\sup _ { n } \\left \\{ | a _ n | \\omega _ n ^ s \\right \\} , \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} A _ \\alpha \\ = \\frac 1 { ( 2 \\alpha - 1 ) ! ! } \\ = \\frac 1 { ( 2 \\alpha - 1 ) ( 2 \\alpha - 3 ) \\cdots 1 } \\ , . \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} P ^ z ( F _ \\gamma ( \\omega ) = W ( \\cdot , \\omega ) ) = 1 ; \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} \\delta _ F = \\lim _ { \\substack { P \\rightarrow \\infty } } \\frac { T _ F \\log ( P ) } { L } , \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} \\frac { k \\binom { b } { k } + O ( \\frac { b ^ { k + 1 } } { n } ) } { 1 + O ( \\frac { b } { n } ) } \\sim k \\binom { b } { k } , \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} \\| h \\| _ { \\infty , B } ^ { ( \\rho ) } : = \\sup _ { 0 \\le t \\le T } e ^ { - \\rho t } \\| h ( t ) \\| _ { B } . \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{gather*} U ^ L _ 0 ( x _ 0 , x _ 1 ) = \\varnothing \\ , . \\end{gather*}"}
-{"id": "5393.png", "formula": "\\begin{align*} X = \\begin{pmatrix} d \\\\ 0 \\end{pmatrix} , d = \\begin{pmatrix} x & y & z \\\\ 0 & w & u \\end{pmatrix} , \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} K ^ s _ { t } ( 0 ) & = \\mu _ 0 ( t ) ( \\alpha _ { t } - 1 ) + K ^ s _ { t + 1 } ( 0 ) + \\log ( 1 + 2 ^ { \\mu _ 0 ( t ) + \\Delta { K } ^ s _ { t + 1 } } ) - H ( \\alpha _ { t } ) , ~ K ^ s _ { n + 1 } ( 0 ) = 0 , \\\\ ~ K ^ s _ { t } ( 1 ) & = \\mu _ 1 ( t ) ( \\beta _ { t } - 1 ) + { K } ^ s _ { t + 1 } ( 0 ) + \\log ( 1 + 2 ^ { \\mu _ 1 ( t ) + \\Delta { K } ^ s _ { t + 1 } } ) - H ( \\beta _ { t } ) , ~ K ^ s _ { n + 1 } ( 1 ) = 0 , ~ t \\in \\{ n , \\ldots , 0 \\} . \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} \\begin{cases} { \\rm ( a ) } \\ i + j = k k \\le n , \\\\ { \\rm ( b ) } \\ ( 2 n + 1 - i ^ - ) + ( 2 n + 1 - j ^ - ) = 2 n + 1 - k ^ - , \\ k \\le n \\min \\{ i , j \\} \\le n , \\\\ { \\rm ( c ) } \\ i ^ - + j ^ - = k ^ - , \\ k \\ge n + 2 \\max \\{ i , j \\} \\ge n + 2 , \\\\ { \\rm ( d ) } \\ ( 2 n + 2 - i ) + ( 2 n + 2 - j ) = 2 n + 2 - k k \\ge n + 2 . \\end{cases} \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} - \\int _ { \\Omega } h ( v _ \\rho ) d x \\le & \\sum _ { i = 0 } ^ { n } \\overline \\sigma _ i \\| v _ \\rho \\| ^ { q _ i } _ { q _ n + 1 } , \\\\ k _ n \\| v _ \\rho \\| _ { q _ n + 1 } ^ { q _ n + 1 } \\le & \\sum _ { i = 0 } ^ { n - 1 } \\sigma _ i \\| v _ \\rho \\| ^ { q _ i + 1 } _ { q _ n + 1 } + \\sum _ { i = 0 } ^ { n } \\overline \\sigma _ i \\| v _ \\rho \\| ^ { q _ i } _ { q _ n + 1 } , \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} \\begin{aligned} & I _ s ( ( X _ s - ( T - t ) V _ s , V _ s ) ) \\geq I _ s ( ( X _ s - ( T - t ) V _ s ^ * , V _ s ^ * ) ) \\\\ & \\textnormal { w h e n e v e r } t \\in [ 0 , T ] \\textnormal { a n d } Z _ s = ( X _ s , V _ s ) \\in \\partial \\mathcal { D } _ s \\textnormal { i s \\emph { p r e } - c o l l i s i o n a l } \\end{aligned} \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} E _ { A D C } = P _ { A D C } \\times t _ { D e l } = c B R \\times t _ { D e l } \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} q ^ { - n + 1 } \\psi _ { n - 1 } + q ^ { - n } \\psi _ { n + 1 } = x \\psi _ { n } , \\forall n \\in \\Z . \\end{align*}"}
-{"id": "10040.png", "formula": "\\begin{align*} G ( z ) = f ( z ) + \\alpha z f ' ( z ) + \\beta z ^ 2 f '' ( z ) \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} \\psi _ T ^ z \\big ( \\lambda _ T \\ , h ( z ) ^ { - 1 } \\big ) = 0 . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} J _ \\alpha p _ \\alpha = \\frac { 2 n } { 2 n - 1 } J _ 1 p _ 1 , \\alpha = 2 , 3 . \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\widehat { \\varphi } _ { j } = \\widehat { \\phi } _ { j } - \\frac { [ \\widehat { \\phi } _ { j } , \\widehat { \\phi } _ { N + 1 } ] } { [ \\widehat { \\phi } _ { N + 1 } , \\widehat { \\phi } _ { N + 1 } ] } \\widehat { \\phi } _ { N + 1 } , 1 \\leq j \\leq N . \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } U & \\tilde U \\end{array} \\right ] = \\left [ \\begin{array} { c c } V & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } \\tilde \\Lambda _ { 1 1 } & \\tilde \\Lambda _ { 1 2 } \\\\ \\tilde \\Lambda _ { 2 1 } & \\tilde \\Lambda _ { 2 2 } \\end{array} \\right ] \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ { I I } } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq C _ d T R ^ d \\alpha \\end{aligned} \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} w _ { k , \\theta } ( z ) = ( \\mathcal { D } _ k - d _ 0 ) + \\varepsilon _ { k } z , k = 1 , 2 , \\cdots , r . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} F ( u ) = 0 . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{gather*} e \\big ( e _ { a } ^ { k } \\big ) \\alpha = e _ { a } ^ { k } \\wedge \\alpha , i \\big ( e _ { a } ^ { k } \\big ) \\alpha = \\beta , \\alpha = e _ { a } ^ { k } \\wedge \\beta , \\end{gather*}"}
-{"id": "3503.png", "formula": "\\begin{align*} \\mathbf { x } _ { { \\mathcal { R } } , { [ N _ T ] } } = ( x _ { { \\mathcal { R } } , { [ N _ T ] } } ^ 1 , x _ { { \\mathcal { R } } , { [ N _ T ] } } ^ 2 , \\ldots , x _ { { \\mathcal { R } } , { [ N _ T ] } } ^ \\rho ) ^ T \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} W = - \\frac { 1 } { 2 } \\frac { d } { d s } \\zeta _ \\Delta ( s ) | _ { s = 0 } = - \\frac { 1 } { 2 } \\zeta _ \\Delta ' ( 0 ) \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} R ( C , D ) = \\bigoplus _ { n \\geqslant 0 } \\mathcal { L } ( C , n D ) \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} \\kappa = ( 1 + I _ 1 + I _ 2 ) \\kappa _ { ( 2 ) } + ( I _ 1 + I _ 2 ) \\kappa _ { ( 1 , 1 ) } . \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} Q ( \\omega , \\mu ) & = \\widehat { Q } ( \\omega , 0 , \\mu ) , \\\\ \\zeta ( y , z , \\omega , \\sigma , \\mu ) & = \\widehat { \\zeta } ( y , z , \\omega , \\sigma , \\mu ) + \\left [ \\widehat { Q } ( \\omega , \\sigma , \\mu ) - \\widehat { Q } ( \\omega , 0 , \\mu ) \\right ] z . \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} M _ { 2 } ^ h = \\sum _ { f \\in H _ { 2 k } ^ { * } ( N ) } ^ { h } L ^ { 2 } _ { f } ( 1 / 2 ) X ^ 2 ( f ) = \\sum _ { \\substack { d \\leq L \\\\ l _ 1 d \\leq L \\\\ l _ 2 d \\leq L } } \\frac { x _ { d l _ 1 } x _ { d l _ 2 } } { d \\sqrt { l _ 1 l _ 2 } } M _ 2 ( l _ 1 l _ 2 , 0 , 0 ) . \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\Delta { C } ^ { 1 , \\infty } = & ( \\alpha - \\beta ) \\Big ( \\Delta { C } ^ { 2 , \\infty } + \\log \\big ( 1 + 2 ^ { \\Delta { C } ^ { 1 , \\infty } } \\big ) \\Big ) \\\\ \\Delta { C } ^ { 2 , \\infty } = & ( \\beta - \\gamma ) \\Big ( \\Delta { C } ^ { 2 , \\infty } + \\log \\big ( 1 + 2 ^ { \\Delta { C } ^ { 1 , \\infty } } \\big ) \\Big ) . \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\bar { X } _ A = \\{ ( x _ n ) _ { n \\in \\Z } \\in \\{ 1 , \\dots , N \\} ^ { \\Z } \\mid A ( x _ n , x _ { n + 1 } ) = 1 n \\in { \\Z } \\} \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} \\dim \\delta H _ 3 ^ g = & \\ ; 8 + 1 2 k + 6 k ^ 2 + { k ^ 3 } - { ( k + 2 ) ^ 3 } \\\\ = & \\ ; 0 , \\\\ \\dim \\delta E _ 3 ^ g = & \\ ; I _ M ( V _ 3 ^ g \\times W _ 3 ^ g ) = \\ ; 0 . \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\Omega ^ { p } ( Y , F [ d u ] ) = \\Omega ^ { p } ( Y , F ) \\oplus \\Omega ^ { p - 1 } ( Y , F ) d u . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} \\psi ' ( t ) = \\frac { C K } { ( 1 + K t ) \\log ( 1 + K ) } \\psi '' ( t ) = - \\frac { \\log ( 1 + K ) \\psi ' ( d ) ^ 2 } { C } , \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} S ( g ; \\phi ) = \\int _ \\Sigma d v \\ , g _ { i j } \\partial ^ \\mu \\phi ^ i \\partial _ \\mu \\phi ^ j = \\langle d \\phi , d \\phi \\rangle _ g \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} T = \\frac { 1 } { 2 } \\sum _ { \\gamma \\in S } \\lambda ( \\gamma ) + \\lambda ( \\gamma ) ^ * \\in C ^ * _ r ( \\Gamma ) | S | \\in \\sigma ( T ) . \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} U ( t ) + A \\int ^ t _ 0 k _ 1 ( t - s ) U ( s ) d s = \\int ^ t _ 0 k _ 2 ( t - s ) G ( s ) d W _ s . \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} z = \\Big ( \\frac { y ^ { ( 1 + \\delta ) / ( 1 + \\epsilon ) } } { h _ H M _ { \\delta } } \\Big ) ^ { 1 / ( 2 + 2 \\delta ) } \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\exp _ 2 \\{ \\lambda L _ { \\mbox { \\tiny L Z } } ( x _ { n _ { i - 1 } + 1 } ^ { n _ i } ) \\} \\le ( 2 \\alpha ) ^ { \\lambda } 2 ^ { ( \\lambda + 1 ) \\log c } . \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} \\mbox { \\bf T e r m 2 } = \\left ( F _ i ( x _ i ^ k , N \\hat v _ i ^ k ) - F _ i ( x _ i ^ k , N y ^ k ) \\right ) ^ T ( x _ i ^ k - x ^ * _ i ) + \\left ( F _ i ( x ^ k _ i , N y ^ k ) - F _ i ( x ^ * _ i , \\bar { x } ^ * ) \\right ) ^ T ( x _ i ^ k - x ^ * _ i ) . \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{gather*} I _ A = \\operatorname { K e r } ( H \\otimes H \\to H \\otimes _ A H ) \\end{gather*}"}
-{"id": "600.png", "formula": "\\begin{align*} \\pi ( s = 0 | q _ i ) p ( x = 0 | s = 0 , q _ i ) + \\pi ( s = 1 | q _ i ) p ( x = 1 | s = 0 , q _ i ) & = 0 . 5 , \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} x \\sim _ \\infty y \\Leftrightarrow \\exists n \\in \\N _ 0 : T ^ n x = T ^ n y . \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} \\gamma _ x ( u ^ * ( m ) a , u ^ * ( n ) b ) = \\gamma _ x ( u ^ * ( m ) , u ^ * ( n ) ) \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} S _ 2 ( l , u , v ; N ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { 1 / 2 + u + v } } \\Delta _ { 2 k , N } ( l , n p ) . \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { \\Omega } \\big ( E ^ 2 ( | \\dot { u } | ) | \\dot { u } | ^ 2 + | \\dot { u } | ^ 2 \\big ) \\mathrm { d } x \\mathrm { d } t & = \\int _ { Q _ 1 } \\big ( E ^ 2 ( | \\dot { u } | ) | \\dot { u } | ^ 2 + | \\dot { u } | ^ 2 \\big ) \\mathrm { d } x \\mathrm { d } t \\\\ & + \\int _ { Q _ 2 } \\big ( E ^ 2 ( | \\dot { u } | ) | \\dot { u } | ^ 2 + | \\dot { u } | ^ 2 \\big ) \\mathrm { d } x \\mathrm { d } t . \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} \\mathcal { M } ' _ a : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\begin{aligned} \\norm { x _ { k + 1 } - x } ^ 2 \\leq \\norm { x _ k - x } ^ 2 & + 2 \\gamma _ k \\lambda _ k \\big ( ( f + g ) ( x ) - ( f + g ) ( x _ { k } ) \\big ) \\\\ [ 0 . 5 e x ] & + \\frac { 2 \\gamma _ k } { 1 - \\delta } \\big ( ( f + g ) ( x _ { k } ) - ( f + g ) ( x _ { k + 1 } ) \\big ) - \\norm { x _ { k + 1 } - x _ k } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} \\frac { ( 1 + t z x ) } { ( 1 - z x ) } = 1 + ( 1 + t ) \\sum _ { \\ell \\geq 1 } ( z x ) ^ \\ell \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} \\partial e ^ { t \\Delta } f = e ^ { t \\vec \\Delta } \\partial f , t \\ge 0 . \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} \\varphi _ s ( \\lambda ) = \\frac { \\lambda - 1 + s } { \\lambda - 1 - s } = \\frac { 1 - s } { 1 + s } - \\frac { 2 s } { 1 + s } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\lambda ^ n } { ( 1 + s ) ^ n } . \\end{align*}"}
-{"id": "10057.png", "formula": "\\begin{align*} q + r = m + n + l + k = q + r + 2 \\gcd ( q , r ) \\geq q + r + 2 , \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} A _ { n , 2 } ( x ) = ( - 1 ) ^ { n + 1 } \\Gamma ( \\mu + \\nu + 1 + n ) a \\sum _ { i = 0 } ^ { \\lfloor \\frac { n - 1 } { 2 } \\rfloor } \\tilde a _ { i , n } x ^ i , \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} P _ \\omega ( x , y ) = \\sum _ { l \\geq 0 } \\binom { x } { l } \\binom { y } { l } \\omega ^ l . \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{gather*} \\frac { s _ { 0 } } { x } + \\frac { s _ { 1 } } { x ^ { 2 } } + . . . + \\frac { s _ { 2 r - 1 } } { x ^ { 2 r } } + \\sum _ { k \\geq 2 r } \\frac { s _ { k } ^ { ( r ) } } { x ^ { k + 1 } } = \\frac { Q _ { r } ( x ) } { P _ { r } ( x ) } \\ , \\end{gather*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\varphi _ \\sigma ( f ) = \\begin{cases} d _ \\sigma & , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} \\sup _ { \\theta \\le 0 } \\P ( Z ( t ) > c ) = 1 - g ( 0 , x _ 0 , t , c ) . \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} I _ 5 = & \\delta T _ 4 d ( r ^ { 6 - n } ) \\\\ = & - ( ( T _ 4 ) _ { i j } { r ^ { 6 - n } } _ { , j } ) _ { , i } \\\\ = & - ( T _ 4 ) _ { i j , i } ( r ^ { 6 - n } ) _ { , j } - ( T _ 4 ) _ { i j } ( r ^ { 6 - n } ) _ { , j i } \\\\ = & ( n - 6 ) \\big [ r ^ { 4 - n } ( T _ 4 ) _ { i j , i } x ^ j - ( n - 4 ) r ^ { 2 - n } ( T _ 4 ) _ { i j } x ^ i x ^ j + r ^ { 4 - n } { \\rm t r } ( T _ 4 ) \\big ] \\\\ : = & ( n - 6 ) [ I _ 1 ^ { ( 5 ) } + I _ 2 ^ { ( 5 ) } + I _ 3 ^ { ( 5 ) } ] . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} ( D d f ) ( B ) & = \\sum _ { i , k = 1 } ^ n \\left ( \\hat { \\kappa } ( A _ i ) A _ k ^ * ( B ) \\right ) \\hat { \\nabla } _ { A _ i } d f ( A _ k ) + \\sum _ { i = 1 } ^ n \\sum _ { s = 1 } ^ \\nu \\left ( \\hat { \\kappa } ( A _ i ) Z _ s ^ * ( B ) \\right ) \\hat { \\nabla } _ { A _ i } d f ( Z _ s ) . \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} q _ n [ u , u ] = \\int _ { M } \\ , | \\nabla u | ^ 2 \\ , d x - \\lambda \\int _ M \\ , U _ { \\frac 1 n } \\ , | u | ^ 2 \\ , d x , u \\in H ^ 1 _ 0 ( M ) \\ , , \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} A _ { \\alpha } = \\frac { d ^ { \\alpha / 2 } } { d x ^ { \\alpha / 2 } } + x \\mbox { a n d } B _ { \\alpha } = - \\frac { d ^ { \\alpha / 2 } } { d x ^ { \\alpha / 2 } } + x \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} f _ N ( t , Z _ N ) = f _ N \\left ( 0 , \\psi _ N ^ { - t } Z _ N \\right ) \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} \\left \\Vert \\mu \\right \\Vert = \\sum _ { j = 1 } ^ { \\infty } \\mu \\left ( z _ { j } \\right ) = \\sum _ { j = 1 } ^ { \\infty } \\left ( \\log \\frac { 1 } { 1 - \\left \\vert z _ { j } \\right \\vert ^ { 2 } } \\right ) ^ { - 1 } < \\varepsilon . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} b _ { i , n } = a _ { - i , n + i } , i \\in \\{ - 2 , - 1 , 0 , 1 , 2 \\} . \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} & \\textbf { A } = \\textrm { t r i } \\bigg [ \\frac { s - p h } { 2 ( p h c - s ) } , 1 , \\frac { s - p h } { 2 ( p h c - s ) } \\bigg ] , \\\\ & \\textbf { K } _ 1 = \\textrm { t r i } \\bigg [ \\frac { p ( 1 - c ) } { 2 ( p h c - s ) } , 0 , - \\frac { p ( 1 - c ) } { 2 ( p h c - s ) } \\bigg ] , \\\\ & \\textbf { K } _ 2 = \\textrm { t r i } \\bigg [ \\frac { p ^ 2 s } { 2 ( p h c - s ) } , - \\frac { p ^ 2 s } { p h c - s } , \\frac { p ^ 2 s } { 2 ( p h c - s ) } \\bigg ] , \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x _ { i } = \\dfrac { p _ { i } N } { q N } + \\varepsilon \\phi = \\dfrac { P _ { i } } { Q } + \\varepsilon \\phi & ; & i = 1 , 2 , . . . , n \\\\ x _ { n + 1 } = \\dfrac { M } { q N } + \\varepsilon a = \\dfrac { P _ { n + 1 } } { Q } + \\varepsilon \\phi & & \\\\ \\varepsilon Q = \\varepsilon q N \\cong 0 & & \\end{array} \\right . \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} K _ { N } = \\frac { g _ { 1 1 } ( L _ { 1 2 } ^ { 1 } L _ { 2 2 } ^ { 2 } - L _ { 1 2 } ^ { 2 } L _ { 2 2 } ^ { 1 } ) - g _ { 1 2 } ( L _ { 1 1 } ^ { 1 } L _ { 2 2 } ^ { 2 } - L _ { 1 1 } ^ { 2 } L _ { 2 2 } ^ { 1 } ) + g _ { 2 2 } ( L _ { 1 1 } ^ { 1 } L _ { 1 2 } ^ { 2 } - L _ { 1 1 } ^ { 2 } L _ { 1 2 } ^ { 1 } ) } { W ^ { 3 } } . \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} \\partial _ r X _ { i r } - \\frac 1 2 H ^ { j k } H _ { k i , r } X _ { j r } - ( \\partial _ r f ) X _ { i r } = \\frac { 1 } { 2 } \\partial _ i S . \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} \\left < \\Phi _ N ( t ) , F _ N ( 0 ) \\right > = \\left < \\Phi _ N ( 0 ) , F _ N ( t ) \\right > \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} \\hat w ( x , t ) : = w _ * ( 0 , t _ 0 ) + \\delta + S ( t , t _ 0 ) \\hat \\eta ( x ) + a ( t - t _ 0 ) - \\gamma ( | t - t _ 0 | ^ 2 + \\delta ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} A ^ { ( 1 ) } = P , x _ { 1 } = Y , A ^ { ( 2 ) } = - & \\frac { 1 } { 2 } ( P \\Omega _ { [ 1 ] } ) _ X , x _ { 2 } = X \\\\ & A ^ { ( 1 ) } = P , x _ { 1 } = T , A ^ { ( 2 ) } = \\Delta , x _ { 2 } = X . \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ { \\infty } z ^ { k } q ^ { k ( k - 1 ) / 2 } A _ { q } \\left ( z q ^ { k + \\ell } \\right ) A _ { q } \\left ( z q ^ { k - \\ell } \\right ) = ( q ; q ) _ { \\infty } ^ { 2 } \\ , \\theta _ { q } \\left ( - z \\right ) \\delta _ { 0 , \\ell } \\end{align*}"}
-{"id": "9904.png", "formula": "\\begin{align*} & \\varepsilon _ 1 ( x , S ) = 2 ( S \\cdot Q ( x ) ) + \\langle S \\circ \\{ \\tau ( \\xi ( x ) ) + Q ( x ) \\} , \\nabla _ { \\partial \\Omega } ( \\tau - \\nu ) ( \\xi ( x ) ) , \\tilde { x } - a \\rangle , \\\\ & \\varepsilon _ 2 ( x , S ) = - 2 \\left \\{ Q ( x ) \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\right ) \\right \\} \\cdot \\left \\{ S \\circ i _ x \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\right ) \\right \\} \\end{align*}"}
-{"id": "9853.png", "formula": "\\begin{align*} s _ q ^ + : = \\frac { 1 } { q } \\sum _ { i = 1 } ^ n x _ i ^ q s _ q ^ - : = \\frac { 1 } { q } \\sum _ { i = 1 } ^ n x _ i ^ { - q } . \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} K _ { i , j + n } { } ^ { k } & = \\left \\{ \\begin{array} [ c ] { l l } 1 , & k \\equiv i + j + n \\ \\left ( \\operatorname { m o d } 2 n \\right ) \\\\ 0 , & o t h e r w i s e \\end{array} \\right . \\\\ & = \\left \\{ \\begin{array} [ c ] { l l } 1 , & k \\equiv ( i + n ) + j \\ \\left ( \\operatorname { m o d } 2 n \\right ) \\\\ 0 , & o t h e r w i s e \\end{array} \\right . \\\\ & = K _ { i + n , j } { } ^ { k } . \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} A _ n ( r ) = A _ n \\binom { 2 n - r - 1 } { n - 1 } \\binom { n + r - 2 } { n - 1 } \\binom { 3 n - 2 } { n - 1 } ^ { - 1 } . \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} u ^ { \\lambda } = \\left ( \\gamma , \\gamma \\mathbf { v } / c \\right ) , u ^ { \\lambda } u _ { \\lambda } = 1 \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} \\frac { f ( a + b ) - f ( b ) } { f ( a ) } = G ( b ) \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} P _ { f } \\equiv \\left \\{ x \\in S | e \\left ( x \\right ) = 0 \\wedge \\left \\vert x \\right \\vert < f \\right \\} . \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} T _ 2 = \\left ( \\begin{array} { c | c | c } I _ { n - 1 } & & \\\\ \\hline & 1 & \\\\ \\hline \\mathbf a '' & - 1 & 1 \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { C } _ { i , s + 1 } ^ - g _ \\varepsilon ^ { ( s + 1 ) } ( t , Z _ s ) = \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\left [ \\omega \\cdot ( v _ { s + 1 } - v _ i ) \\right ] _ - \\times \\\\ & \\qquad \\times g _ \\varepsilon ^ { ( s + 1 ) } \\left ( t , x _ 1 , v _ 1 , \\dots , x _ i , v _ i , \\dots , x _ s , v _ s , x _ i + \\varepsilon \\omega , v _ { s + 1 } \\right ) d \\omega d v _ { s + 1 } \\end{aligned} \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} u _ { 0 . 6 } ^ 2 ( g _ 2 ) = [ I - 0 . 6 P ( g _ 2 ) ] ^ { - 1 } \\bar { r } ^ 2 ( g _ 2 ) = ( 6 . 6 , 6 ) . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } { \\rm I _ 1 } = 0 . \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} \\rho ( a ) = \\pi ( a ) = \\left ( \\begin{matrix} 0 & I \\\\ I & 0 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} u '' + a ( t ) g ( u ) = 0 . \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} b _ j ^ { t r } ( c _ j - f _ j ) = 0 , d _ j ^ { t r } d _ j + c _ j ^ { t r } c _ j = g _ j ^ { t r } g _ j + f _ j ^ { t r } f _ j . \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} V = V _ 0 \\oplus V _ 1 \\oplus \\cdots \\oplus V _ s , G = G _ 0 \\times S p ( V _ 1 ) \\times \\cdots \\times S p ( V _ s ) \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} & \\widetilde { f } _ n ( t _ 1 , x _ 1 , \\ldots , t _ n , t , x ) \\\\ & = \\eta \\frac { \\lambda ^ n } { n ! } \\sum _ { \\rho \\in S _ n } G ( t - t _ { \\rho ( n ) } , x - x _ { \\rho ( n ) } ) \\ldots G ( t _ { \\rho ( 2 ) } - t _ { \\rho ( 1 ) } , x _ { \\rho ( 2 ) } - x _ { \\rho ( 1 ) } ) 1 _ { \\{ 0 < t _ { \\rho ( 1 ) } < \\ldots < t _ { \\rho ( n ) } < t \\} } . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} u _ { t } & = \\mathrm { d i v } \\ , \\big ( g ( v ) \\nabla u \\big ) ( 0 , \\infty ) \\times G , \\\\ v _ { t } + v & = F \\big ( | \\nabla u | \\big ) ( 0 , \\infty ) \\times G , \\\\ \\frac { \\partial u } { \\partial \\mathbf { n } } & = 0 ( 0 , \\infty ) \\times \\Gamma , \\\\ u ( 0 , \\cdot ) = u ^ { 0 } , v ( 0 , \\cdot ) & = v ^ { 0 } \\Omega \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} { 2 m - 1 \\choose m - i } { m + i \\choose i } = \\frac { ( 2 m - 1 ) ! } { m ! ( m - 1 ) ! } Q _ i ^ { m - 1 } . \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} B _ j = \\begin{pmatrix} 0 & 0 \\\\ 0 & \\sigma _ j \\end{pmatrix} , A _ j = \\begin{pmatrix} I & 0 \\\\ 0 & \\Delta _ j \\end{pmatrix} , \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} E _ + ( s ) u [ x ] = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i \\Phi ( x , y , \\xi ; s ) } p _ + ( y , \\xi ) u [ y ] d \\xi , \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} \\tilde { p } _ { k , \\varphi } ( \\omega , \\lambda ) : = p _ k ( x , Y + \\lambda \\nu _ k + i \\tau d \\varphi _ k ( x ) ) , \\ \\varphi _ k = \\varphi _ { | \\Omega _ k } , \\ \\nu _ k = ( - 1 ) ^ k \\nu \\in N _ x ^ * ( S ) . \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} K ( h ^ { \\frac { 1 } { 2 ^ { n - 1 } } } , 2 ) ^ { r _ n } a ^ { 2 } \\sharp _ { \\nu } b ^ { 2 } & \\leqslant a ^ { 2 } \\nabla _ { \\nu } b ^ { 2 } - \\sum _ { k = 0 } ^ { n - 1 } r _ { k } \\big [ a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } - a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } \\\\ & \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ { n - 1 } } } , 2 ) ^ { R _ n } a ^ { 2 } \\sharp _ { \\nu } b ^ { 2 } \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} k = \\frac { \\partial X } { \\partial x } \\frac { \\partial x } { \\partial y } - \\frac { \\partial Y } { \\partial x } \\frac { \\partial Y } { \\partial y } , \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} L u = N u , u \\in \\ , L . \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} \\sum _ { U ( D ) = G } | P _ { \\mathbf { D } } ( D ) - P _ { \\mathbf { D } ' } ( D ) | = | P _ { \\mathbf { G } } ( G ) - P _ { \\mathbf { G } ' } ( G ) | \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} \\left ( \\mathbf { I } + K ( \\lambda , 0 ) \\right ) e ^ { i k \\frac { 2 \\pi } { P _ { 0 } } x } & = \\{ \\mathbf { I } + \\partial _ { x } ^ { - 2 } \\rho ( \\lambda , 0 ) \\} e ^ { i k \\frac { 2 \\pi } { P _ { 0 } } x } \\\\ & = \\left \\{ 1 - \\frac { 4 \\pi ^ { 2 } } { P _ { 0 } ^ { 2 } k ^ { 2 } } \\int \\frac { v [ \\mu _ { + } ^ { \\prime } + \\mu _ { - } ^ { \\prime } ] } { v + \\frac { \\lambda } { i k } } d v \\right \\} e ^ { i k \\frac { 2 \\pi } { P _ { 0 } } x } , k \\neq 0 . \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} A = \\left ( B \\rtimes _ { \\alpha ^ \\| , \\theta } \\Z ^ { d - 1 } \\right ) \\rtimes _ { \\alpha _ d } \\Z = C \\rtimes _ { \\alpha _ d } \\Z \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} v _ { { \\mathcal { R } } , { \\mathcal { T } } , p , n } ^ i ( u ) = \\alpha _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { \\bar { \\mathcal { R } } _ i } ( u ) c _ p ( u ) z _ { { \\mathcal { R } } , { \\mathcal { T } } , n } ^ { \\bar { \\mathcal { R } } _ i } ( u ) , \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} b _ { i , j } \\left ( z \\right ) & = b _ { i - 1 , j } \\left ( z \\right ) - b _ { i - 1 , j } \\varphi _ { z _ { i - 1 } } \\left ( z \\right ) , \\\\ b _ { i , j } & = b _ { i , j } \\left ( z _ { i } \\right ) , \\\\ b _ { i , j } ^ { \\ast } & = b _ { i , j } \\left ( z _ { i + 1 } \\right ) , \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\mathcal { I } = \\{ \\rho > 0 ~ : ~ p ( \\rho ) < 0 \\} . \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} ( \\lambda + \\mu + ( n - 1 ) \\xi ) p _ { 1 , n } = \\lambda p _ { 1 , n - 1 } + \\gamma p _ { 0 , n } + ( \\mu + n \\xi ) p _ { 1 , n + 1 } , n \\geq 2 . \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} Q _ \\sigma [ w f , w g ] - \\lambda \\int _ M U f g \\ , w ^ 2 d x \\ , \\geq \\ , \\int _ M \\nabla f \\cdot \\nabla g \\ w ^ 2 d x = : \\widehat Q _ \\sigma [ f , g ] \\ . \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} A _ n = \\prod _ { j = 0 } ^ { n - 1 } \\frac { ( 3 j + 1 ) ! } { ( n + j ) ! } . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} b _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] = b _ { s , s + k } ^ 0 \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} Q _ t = \\big ( h ^ { W ^ \\bullet } _ t \\big ) ^ { - 1 } \\frac { \\partial } { \\partial t } h ^ { W ^ \\bullet } _ t \\in \\mathrm { E n d } \\big ( W ^ \\bullet \\big ) . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} y ^ { j } . x _ 1 & = y ^ { j + 1 } \\\\ y ^ { j } . x _ 2 & = y ^ { j } ( x _ 1 + x _ 2 - y ) . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} ( \\Lambda _ { N } ( t ) - \\mid \\gamma + t \\mid ^ { 2 } ) ( \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) = \\sum _ { \\gamma _ { 1 } } \\dfrac { q _ { \\gamma _ { 1 } } ( q \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma - \\gamma _ { 1 } + t , x \\right \\rangle } ) } { \\Lambda _ { N } ( t ) - \\mid \\gamma - \\gamma _ { 1 } + t \\mid ^ { 2 } } . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( [ F ] , C , [ L ] ) = \\mathbb { P } \\mathrm { H o m } ( F , C , L ) _ e \\overset { \\textrm { o p e n } } { \\hookrightarrow } \\mathbb { P } ( \\mathrm { H o m } ( F , L ( 2 ) ) ) . \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} \\mathcal { M } _ { d , \\ell } ( z ; i t ) = \\left ( 2 \\ell ^ 2 t \\right ) ^ { - \\frac { 1 } { 2 } } e ^ { \\frac { 2 \\pi i j d } { \\ell } - \\frac { \\pi z _ 0 ^ 2 } { 2 \\ell ^ 2 t } } \\sum _ { n \\in \\Z } e ^ { - \\frac { \\pi n ^ 2 } { 2 \\ell ^ 2 t } - \\frac { \\pi z _ 0 n } { \\ell ^ 2 t } - \\frac { 2 \\pi i n d } { \\ell } } . \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} d _ { r , t } = \\left \\{ \\begin{array} { l l } 1 , & r + t \\ge N _ R \\\\ \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } t } { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } t + 1 } , & r + t = N _ R - 1 \\\\ \\max \\left \\{ d _ { 1 } , \\frac { r + t } { N _ R } \\right \\} , & r + t \\le N _ R - 2 \\end{array} \\right . \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} P _ q ^ { \\perp } ( X _ * ) & = ( I - U _ 1 U ^ { \\top } _ 1 ) X _ * - ( I - U _ 1 U ^ { \\top } _ 1 ) X _ * V ^ { ( q ) } _ 0 ( V ^ { ( q ) } _ 0 ) ^ { \\top } = \\\\ & = ( I - U _ 1 U ^ { \\top } _ 1 ) X _ * + ( I - U _ 1 U ^ { \\top } _ 1 ) H V ^ { ( q ) } _ 0 ( V ^ { ( q ) } _ 0 ) ^ { \\top } . \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} F _ 1 ( x ) = \\frac { - 1 } { \\sqrt { 2 } \\sin ( x ) } \\left ( \\frac { - \\sin ( x ) } { 1 + \\cos ( x ) } \\right ) ^ { - \\frac { 1 } { \\sqrt { 2 } } } , F _ 2 ( x ) = \\frac { - 1 } { \\sqrt { 2 } \\sin ( x ) } \\left ( \\frac { - \\sin ( x ) } { 1 + \\cos ( x ) } \\right ) ^ { \\frac { 1 } { \\sqrt { 2 } } } . \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\mathcal { R } ^ 1 _ { 2 2 } = \\left \\{ ( \\mu _ R , \\mu _ T ) : \\mu _ R + \\mu _ T \\ge 1 , \\mu _ R \\le 1 , \\mu _ T \\le 1 \\right \\} \\\\ \\mathcal { R } ^ 2 _ { 2 2 } = \\left \\{ ( \\mu _ R , \\mu _ T ) : \\mu _ R + \\mu _ T < 1 , \\mu _ R \\ge 0 , \\mu _ R + 2 \\mu _ T \\ge 1 \\right \\} \\end{array} \\right . \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} s ( \\gamma ) = \\gamma ^ { - 1 } \\gamma r ( \\gamma ) = \\gamma \\gamma ^ { - 1 } . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} \\operatorname { R e } \\int _ { \\R ^ { d - 1 } } g ( t ^ { \\prime } ) \\langle L _ { \\tilde { u } } ( t ^ { \\prime } ) , h \\rangle d t ^ { \\prime } = \\int _ { B _ { \\delta } } g ( t ^ { \\prime } ) \\operatorname { R e } \\langle L _ { \\tilde { u } } ( t ^ { \\prime } ) , h \\rangle d t ^ { \\prime } > 0 . \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} \\lambda = \\frac { 1 } { W ^ 3 } \\Lambda = \\frac { 1 } { W } \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} \\mathbb { Q } : = ( \\omega ) * \\dot { \\mathbb { C } } \\big ( [ \\omega _ 2 ] ^ \\omega - V \\big ) . \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} e ^ { i \\theta } \\overline p \\sqrt { 1 + | m | ^ 2 } = m \\overline n \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\sum _ { \\eta \\in E _ Z } S _ \\eta S _ \\eta ^ * = 1 , S _ \\gamma ^ * S _ \\gamma = \\sum _ { \\eta \\in E _ Z } Z ^ G ( \\gamma , \\eta ) S _ \\eta S _ \\eta ^ * \\gamma \\in E _ Z \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} c _ j = f _ j , 2 \\leq j \\leq 7 ; \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ r ( a _ k ^ H Z a _ k - \\| E a _ k \\| ^ 2 ) > 0 . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} F ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) & = ( x _ { 1 } - x _ { 3 } , \\ , x _ { 2 } , \\ , \\alpha x _ { 3 } + x _ { 2 } ( x _ { 1 } - x _ { 3 } ) ) , \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} f _ { n - 7 } = & A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\Big ( \\sum _ { k = 1 } ^ { n - 7 } \\psi _ k \\Big ) + K _ { 6 - n } \\Big ( \\sum _ { k = 1 } ^ { n - 7 } \\psi _ k \\Big ) + f = O ( r ^ { n - 6 } ) \\\\ : = & b _ { n - 6 } + O ( r ^ { n - 5 } ) . \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} \\delta ( X ) = \\tfrac { 4 ( g - 1 ) } { g } S _ { g } ( X ) - \\tfrac { 4 ( g - 1 ) ^ { 2 } } { g ^ { 2 } } H ( X ) - \\tfrac { 3 g - 1 } { 2 g n } \\log \\| \\Delta _ { g } \\| ( X ) - 8 g \\log 2 \\pi , \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { w _ { \\alpha } } : = \\Big ( \\int _ { 0 } ^ { \\infty } t ^ { \\alpha } e ^ { - t } | f ( t ) | ^ 2 \\ , d t \\Big ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} \\mathbb { H } ^ { n + 1 } = \\{ ( x ^ 0 , r , \\xi ^ i ) : r = \\sqrt { \\abs { x ^ 0 } ^ 2 - 1 } , x ^ 0 > 0 , \\xi \\in \\mathbb { S } ^ n \\} , \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} | \\tilde { f } ( y ) | \\leq C d _ G ( y , y _ 0 ) ^ { \\eta _ 0 } , \\eta _ 0 = \\min \\left \\{ 1 + 4 \\alpha , 3 - \\frac { 2 ( n + 1 ) } { p } \\right \\} . \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} R ^ { \\lambda \\nu } & = \\epsilon ^ { \\lambda \\nu \\sigma \\tau } F _ { \\sigma \\tau } = \\frac { 1 } { 2 } \\left ( \\epsilon ^ { \\lambda \\nu \\sigma \\tau } - \\epsilon ^ { \\lambda \\nu \\tau \\sigma } \\right ) F _ { \\sigma \\tau } \\\\ & = \\frac { 1 } { 4 } \\left ( \\epsilon ^ { \\lambda \\nu \\sigma \\tau } - \\epsilon ^ { \\lambda \\nu \\tau \\sigma } + \\epsilon ^ { \\nu \\lambda \\tau \\sigma } - \\epsilon ^ { \\nu \\lambda \\sigma \\tau } \\right ) F _ { \\sigma \\tau } \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} \\big \\langle \\alpha _ p ( \\omega ) , \\mu \\big \\rangle _ Z = \\big \\langle \\omega ' , \\mu \\big \\rangle _ Z = \\big \\langle \\omega ' , \\mu \\big \\rangle _ { Z _ 1 } = \\big \\langle \\omega , \\mu \\big \\rangle _ { Z _ 1 } . \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dfrac { \\partial g } { \\partial t } = \\left ( r - R \\right ) g \\mbox { i n } M \\times \\left ( 0 , T \\right ) \\\\ k _ g = \\dfrac { \\gamma } { \\sqrt { \\phi \\left ( t \\right ) } } \\mbox { o n } \\partial M \\times \\left ( 0 , \\tilde { T } \\right ) \\\\ g = g _ 0 \\mbox { i n } M , \\end{array} \\right . \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} h _ 0 ( \\xi ) = - \\frac { 1 } { 3 } ( \\cos \\xi _ 1 + \\cos \\xi _ 2 + \\cos ( \\xi _ 1 - \\xi _ 2 ) ) \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { k = 1 } ^ n \\left ( \\mathcal { R } [ q , \\varphi _ k ] - \\mathcal { R } [ q , \\psi _ k ] \\right ) = 0 . \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} H _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) ( j , \\beta , \\delta ) & = H _ { ( 2 x ) } ( i , \\alpha ) ( j , \\beta ) \\otimes H _ { y } ( \\gamma , \\gamma ) \\\\ & = H _ x ( i , j ) \\otimes \\Lambda ( \\alpha , \\beta ) \\otimes T _ { y } ( \\gamma ) \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} \\lim _ { a \\to + \\infty , \\ ; b - a = c } \\frac { 2 \\sqrt { \\pi a } } { e ^ { 2 a x } } Q _ n ( x ^ 2 ) = x ^ { \\mu - \\frac { 1 } { 2 } } L _ n ^ { ( \\mu + \\nu ) } ( 2 c x ) , \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} \\left ( \\oplus _ i G _ i \\right ) \\otimes \\left ( \\oplus _ j H _ j \\right ) = \\oplus _ i \\left ( G _ i \\otimes \\oplus _ j H _ j \\right ) = \\oplus _ i \\oplus _ j \\left ( G _ i \\otimes H _ j \\right ) \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} \\norm { h _ { n + 1 } } & = \\norm { h _ n } + \\norm { g _ n } \\norm { h _ { \\frac { g _ n } { \\norm { g _ n } } , \\max } } \\ge \\norm { h _ n } + ( 1 - \\norm { h _ n } ) \\beta = \\beta + ( 1 - \\beta ) \\norm { h _ n } \\\\ & \\ge \\beta + ( 1 - \\beta ) \\big ( 1 - \\delta ( 1 - \\beta ) ^ n \\big ) = 1 - \\delta ( 1 - \\beta ) ^ { n + 1 } . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} { \\rm g p h \\ , \\Xi } : = \\{ ( x , u ) : u \\in \\Xi ( x ) \\} { \\rm a n d } { \\rm d o m } \\ , \\Xi : = \\{ x : \\Xi ( x ) \\neq \\emptyset \\} . \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} \\frac { \\delta _ { \\mathsf { P , A c h } } ( \\mu , r ) } { \\delta _ { \\mathsf { P } } ^ * ( \\mu , r ) } & \\leq \\frac { M + K - 1 } { M } \\times \\frac { \\min \\{ M , K \\} } { K } \\\\ & \\leq \\frac { M + K } { \\max \\{ M , K \\} } ~ = 1 + \\frac { \\min \\{ M , K \\} } { \\max \\{ M , K \\} } \\leq 2 . \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\int _ 0 ^ 1 f ' _ i ( x ) d x = 0 , \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} L ^ p _ { \\overline { \\sigma } } ( \\Omega ) & = \\overline { \\{ v \\in C ^ { \\infty } _ { p e r } ( \\Omega ) ^ 2 \\mid \\div _ H \\overline { v } = 0 \\} } ^ { L ^ { p } ( \\Omega ) ^ 2 } \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} Y ( k ) = \\sum \\limits _ { t = 0 } ^ { l - 1 } { y _ t e ^ { - i 2 \\pi t k / l } = } \\sum \\limits _ { t = 0 } ^ { l - 1 } { y _ t \\cos ( 2 \\pi t k / l ) - i } \\sum \\limits _ { t = 0 } ^ { l - 1 } { y _ t \\sin ( 2 \\pi t k / l ) } \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} f _ { N , n , R } ^ { ( s ) } ( 0 , Z _ s ) = f _ N ^ { ( s ) } ( 0 , Z _ s ) \\mathbf { 1 } _ { 1 \\leq s \\leq n } \\chi \\left ( \\frac { 1 } { R ^ 2 } E _ s ( Z _ s ) \\right ) \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} \\rho = e ^ { \\beta ( F - H ) } \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} \\int P _ { \\mathbf { x } } ( D ) d ( \\mu \\mathbf { x } ) = p _ e ^ { n _ e } ( 1 - p _ e ) ^ { 3 - n _ e } ( 1 - p _ d ) ^ { n _ { a s } } ( 2 p _ d - 1 ) ^ { n _ s } , \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} \\Theta ( z ; \\tau ) : = \\sum _ { n \\in \\Z } ( - 1 ) ^ n \\zeta ^ n q ^ { n ^ 2 } , \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nabla _ x q ( t , s ) = I + \\int _ s ^ t A ( p ( \\tau , s ) ) \\nabla _ x p ( \\tau , s ) d \\tau , \\\\ \\nabla _ x p ( t , s ) = - \\int _ s ^ t \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , s ) ) \\nabla _ x q ( \\tau , s ) d \\tau . \\end{array} \\right . \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} \\frac { F ( x _ N ) - F ( x ^ * ) } { a _ N ^ 2 } & + \\frac { \\tilde { \\mu } } { 2 } \\norm { x ^ * - v _ N } ^ 2 \\leq \\frac { \\tilde { \\mu } } { 2 } \\norm { x ^ * - v _ 0 } ^ 2 - \\frac { \\tilde { \\mu } - \\mu } { 2 } \\sum _ { j = 1 } ^ N \\frac { \\norm { x _ j - y _ j } ^ 2 } { a _ j ^ 2 } . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} w ( x ) = \\int _ \\Omega \\Big [ \\frac { v ( y ) } { | x - y | ^ { n - 1 } } g \\Big ( x , \\frac { y - x } { | y - x | } \\Big ) + v ( y ) h ( x , y ) \\Big ] \\ , d y \\hbox { f o r $ x \\in \\Omega $ } , \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} \\delta _ F ^ { ( 1 ) } = 0 ; ~ ~ ~ ~ ~ \\delta _ E ^ { ( 1 ) } = \\delta _ { \\mathsf { C a - Z F } } = \\frac { K } { \\min \\{ M , K \\} } ; \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} a ^ * _ { 1 , 1 } = \\frac { \\mu _ R } { 3 } , a ^ * _ { 0 , 1 } = \\frac { 2 } { 3 } - \\mu _ R - \\mu _ T , a ^ * _ { 0 , 2 } = \\mu _ T - \\frac { 1 } { 3 } , \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} \\beta \\partial _ t e - e \\Delta e - \\Delta e = 0 \\quad \\partial _ n e = 0 \\Gamma _ T . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} p ( t ) = \\begin{pmatrix} p ^ { 0 } ( t ) \\\\ \\vdots \\\\ p ^ { n } ( t ) \\end{pmatrix} \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} a ^ 2 + b ^ 2 = c ^ 2 \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} F = - T \\ln Z _ { b u l k } = - T \\ln Z _ { C F T } = - \\tau \\ln Z _ { C F T } \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} | D f ( z ) | ^ { n } \\ = \\ K _ f ( z ) J _ { f } ( z ) . \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} \\delta ( Z ) = \\mu \\sigma ( A ) Z , \\ \\ \\sigma ( Z ) = \\sigma ( B ) Z \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} \\hom ( \\psi ( x ) , \\psi ( y ) ) = ( u \\pitchfork \\psi ) ( y ) \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} \\tilde L _ a f ( \\xi ) = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { T } ^ d } e ^ { i y \\cdot ( \\xi - \\eta ) } \\left | \\det \\left ( \\frac { d x } { d y } \\right ) \\right | f ( \\eta ) d \\eta d y . \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} \\langle u , v \\rangle = 0 \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} \\tau & = \\lim _ { P \\to \\infty } \\lim _ { F \\to \\infty } \\frac { T \\log P } { F } \\\\ & \\ge \\frac { 1 } { l } \\left \\{ ( s _ 1 + s _ 2 ) - ( N _ T - l ) s _ 2 \\mu _ T - \\left ( \\frac { 2 s _ 2 + s _ 1 + 1 } { 2 } \\cdot s _ 1 + s _ 2 ^ 2 \\right ) \\mu _ R \\right \\} . \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} \\theta : = 1 _ { \\alpha \\in ( 0 , 1 ] } \\tfrac { \\alpha + \\bar \\alpha } { 2 } + 1 _ { \\alpha \\in ( 1 , 2 ) } \\tfrac { \\alpha + \\bar \\alpha \\vee 1 } { 2 } , \\ \\ q > \\tfrac { d } { \\gamma _ 0 ( \\gamma _ \\sigma ( \\alpha \\wedge 1 ) \\wedge \\gamma _ \\nu ) } \\vee \\tfrac { d } { \\theta - \\bar \\alpha } \\vee \\tfrac { \\alpha } { \\alpha - \\theta } \\vee p . \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) : = \\int _ { D } \\alpha \\wedge \\star \\beta , \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} [ L ^ p ( \\tilde { \\Omega } ) , H _ { b . c . } ^ { 2 , p } ( \\tilde { \\Omega } ) ] _ { \\theta } = \\begin{cases} H ^ { 2 \\theta , p } ( \\tilde { \\Omega } ) \\hbox { a l l b . c . } , & 1 + 1 / p < \\theta \\leq 2 , \\\\ H ^ { 2 \\theta , p } ( \\tilde { \\Omega } ) \\hbox { o n l y D i r i c h l e t p a r t } , & 1 / p < \\theta \\leq 2 , \\\\ H ^ { 2 \\theta , p } ( \\tilde { \\Omega } ) \\hbox { w i t h o u t b . c . } , & 0 \\leq \\theta \\leq 1 / p . \\end{cases} \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } y _ 2 ^ H \\\\ \\vdots \\\\ y _ n ^ H \\end{array} \\right ] = \\left [ \\begin{array} { c } y _ 1 ^ H \\\\ \\vdots \\\\ y _ { n - 1 } ^ H \\end{array} \\right ] \\Lambda , \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} \\xi ^ 2 & = \\frac { \\rho _ k - \\theta _ k } { ( 1 + \\rho _ { k + 1 } ) \\nu _ k - \\theta _ k } \\\\ & = \\frac { \\sqrt { L / \\ell } - 1 - ( \\sqrt { L / \\ell } - 1 ) / ( \\sqrt { L / \\ell } + 1 ) } { ( \\sqrt { L / \\ell } - 1 ) \\cdot ( \\sqrt { L / \\ell } - 1 ) / ( \\sqrt { L / \\ell } + 1 ) } \\\\ & = 1 + \\frac { 1 } { \\sqrt { L / \\ell } - 1 } . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} \\textrm { C o v . r a t e } = \\textrm { l o g } _ 2 \\bigg ( \\frac { e _ \\nu ( t , N _ h ) } { e _ \\nu ( t , 2 N _ h ) } \\bigg ) , \\nu = 1 , \\infty . \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\varphi ^ { \\prime } \\colon \\Lambda ^ { q + 1 } \\cong ( \\Lambda \\oplus P ) ^ { \\ast } & \\longrightarrow \\ker ( d _ { n - 1 } ) = H _ { n - 1 } ( \\widetilde { K } ^ { [ n - 1 ] } ) \\xrightarrow [ ] { \\ h _ { n - 1 } ^ { - 1 } \\ } \\pi _ { n - 1 } ( \\widetilde { K } ^ { [ n - 1 ] } ) \\\\ x & \\longmapsto h _ { n - 1 } ^ { - 1 } \\big ( [ d _ { n } ( x ) ] \\big ) \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } \\frac { \\Psi ' ( t ) } { t } d t \\ = \\ \\infty , \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { P _ { \\beta } } \\rho ( \\lambda , \\varepsilon ) \\phi \\ d x & = \\int \\int \\left ( g _ { + } - g _ { - } \\right ) \\ d x d v \\\\ & = \\int \\int g _ { + } d I _ { + } d \\theta _ { + } - \\int \\int g _ { - } d I _ { - } d \\theta _ { - } = 0 , \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} v ( x , y , z ) = \\Phi \\left ( \\dfrac { k } { c } ( z - h ( x , y ) ) \\right ) . \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} & a _ { N _ R , 0 } = \\mu _ R , a _ { 0 , 1 } = \\frac { N _ R ( 1 - \\mu _ R ) - N _ T \\mu _ T } { N _ T ( N _ R - 1 ) } , a _ { 0 , N _ R } = \\frac { N _ T \\mu _ T + \\mu _ R - 1 } { \\binom { N _ T } { N _ R } ( N _ R - 1 ) } , \\textrm { a n d o t h e r s b e i n g 0 , } \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} \\left \\| ( \\frac { \\lambda _ u } { \\lambda } ) ^ { s } \\ss \\frac { 1 } { \\lambda } \\sum _ { m = s + 1 } ^ { t - 1 } \\frac { \\lambda _ u ^ { m - s } \\cdot { m } ^ { d _ u } } { \\lambda ^ { m - s } } \\frac { 1 } { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } M ^ { m } \\vec u _ { t - m } \\right \\| \\leq \\epsilon . \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} z _ { i j } : = \\| Z _ { i j } \\| \\quad \\mbox { f o r } i , \\ ; j = 1 , 2 . \\end{align*}"}
-{"id": "9987.png", "formula": "\\begin{align*} \\mu _ { B _ 1 \\cup \\cdots \\cup B _ m } = \\sum _ { k \\geq 0 } ( - 1 ) ^ k \\ ! \\ ! \\sum _ { 1 \\leq j _ 0 < \\cdots < j _ k \\leq m } \\mu _ { B _ { j _ 0 } \\cap \\cdots \\cap B _ { j _ k } } . \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} g _ { p , x _ { i } } ( x _ { p } ) v _ { \\delta ^ { p } } ( x _ { i } ) = f _ { p , 0 } ( x _ { p } ) \\left [ u _ { \\delta ^ { p } } ( 0 ) + u _ { \\delta ^ { i , p } } ( 0 ) f _ { i , 0 } ( x _ { i } ) \\right ] \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\tau = \\frac { 1 - \\mu _ R } { d } = 1 - \\mu _ R . \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} \\begin{pmatrix} u _ { t } \\\\ v _ { t } \\end{pmatrix} + \\partial \\phi \\begin{pmatrix} u \\\\ v \\end{pmatrix} \\ni \\tilde { f } \\begin{pmatrix} u \\\\ v \\end{pmatrix} \\qquad ( 0 , T ) , \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} h _ { \\mu , g } \\bigl ( f ^ 0 , g ^ 0 \\bigr ) = \\frac { | A _ N ( e ^ { i \\lambda } ) ( 1 - e ^ { i \\lambda \\mu } ) ^ { n } f ^ 0 ( \\lambda ) - \\lambda ^ { 2 n } C ^ { \\mu , 0 } _ { N } ( e ^ { i \\lambda } ) | } { | 1 - e ^ { i \\lambda \\mu } | ^ { n } p ^ 0 ( \\lambda ) } , \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{gather*} \\big ( \\tau _ { k , 0 } ^ { ( \\alpha , \\beta ) } \\big ) ^ { 2 } = \\tau _ { k , 0 } ^ { ( \\alpha , \\beta + 1 ) } \\tau _ { k , 0 } ^ { ( \\alpha , \\beta - 1 ) } - \\tau _ { k - 1 , 0 } ^ { ( \\alpha , \\beta - 1 ) } \\tau _ { k + 1 , 0 } ^ { ( \\alpha , \\beta + 1 ) } , \\end{gather*}"}
-{"id": "1092.png", "formula": "\\begin{align*} ( \\lambda - \\mid a + n v _ { k } + t \\mid ^ { 2 } ) c ( a , n ) = q _ { a - \\delta + \\left ( n - n _ { 2 } \\right ) v _ { k } } + { \\textstyle \\sum \\limits _ { m = 1 } ^ { \\infty } } \\left ( { \\textstyle \\sum \\limits _ { u \\in \\Gamma ( k ) } } c ( a - u , n - m ) q _ { u + m v _ { k } } \\right ) \\end{align*}"}
-{"id": "9919.png", "formula": "\\begin{align*} \\sigma ( r ) V ^ { \\lambda } ( A ) = V ^ { - \\lambda } ( A ) \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} r ^ * ( X ) = r ( E - X ) + | | X | | _ r - r ( E ) \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} \\varphi ' = - e ^ { - \\varphi } \\tilde { F } \\Theta ^ { - 1 } \\frac { \\sinh \\Theta } { \\cosh \\Theta } v + 1 . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\begin{cases} & | \\dot { u } | + | \\nabla u | < A , \\\\ & \\frac { 1 } { A } \\leq \\left . \\int _ M u d \\mu \\right | _ { t } \\leq A , \\forall \\ ; t \\in [ - 1 , 1 ] , \\\\ & \\frac { 1 } { A } \\leq \\frac { | B _ { g ( t ) } ( x , r ) | } { r ^ m } , \\forall \\ ; x \\in M , \\ ; t \\in [ - 1 , 1 ] , r \\in ( 0 , 1 ) . \\end{cases} \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} \\tau _ { y + z } = \\tau _ y \\tau _ z = \\tau _ z \\tau _ y y , z \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\| ( 1 - u _ { i i } ) \\xi _ k \\| ^ 2 = 2 \\langle ( 1 - \\Re u _ { i i } ) \\xi _ k | \\xi _ k \\rangle + \\| u _ { i i } \\xi \\| ^ 2 - 1 . \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} P _ { \\mathbf { G } } ( G ) = \\sum _ { U ( D ) = G } P _ { \\mathbf { D } } ( D ) G \\in \\mathcal { G } _ n . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\frac d { d x } y & = & A ( x ) y \\\\ y ( x + 1 ) & = & T _ A ( x ) y ( x ) \\end{array} \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} \\omega _ 1 ' = m \\ , \\varphi ' { \\rm a n d } \\psi ' = e ^ { i \\theta } \\sqrt { 1 + | m | ^ 2 } \\ , \\varphi ^ \\prime \\ , , \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} f _ { z _ 1 } d z _ 1 \\wedge d z _ 2 = f _ { z _ 3 } d z _ 2 \\wedge d z _ 3 ~ ~ f _ { z _ 1 } d z _ 1 \\wedge d z _ 3 = - f _ { z _ 2 } d z _ 2 \\wedge d z _ 3 ~ ~ \\widetilde { U } _ 0 . \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda \\ell } \\log _ 2 \\left [ \\frac { \\ell } { n } \\sum _ { t = 0 } ^ { n / \\ell - 1 } 2 ^ { \\lambda L ( y _ { t \\ell + 1 } ^ { t \\ell + \\ell } ) } \\right ] , \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} \\epsilon ^ { \\lambda \\nu \\sigma \\tau } = \\frac { 1 } { \\mu } \\left ( g ^ { \\lambda \\sigma } + \\kappa u ^ { \\lambda } u ^ { \\sigma } \\right ) \\left ( g ^ { \\nu \\tau } + \\kappa u ^ { \\nu } u ^ { \\tau } \\right ) = \\epsilon ^ { \\nu \\lambda \\tau \\sigma } \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} \\rho _ k = \\sqrt { L / \\ell } - 1 \\quad \\mbox { f o r $ k = 1 , 2 , \\ldots $ } . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\sqrt { \\pi t } } & \\int \\limits _ { - \\infty } ^ { + \\infty } \\Lambda _ n ( s _ 2 ) \\exp \\left \\{ - \\frac { ( s _ 2 - x _ 2 ) ^ 2 } { 4 t } \\right \\} \\ , d s _ 2 = \\\\ & = \\exp ( - \\eta _ 2 ^ 2 ) \\sum \\limits _ { m = 1 } ^ { N } t ^ { - m / 2 } Q _ { n , m - 1 } ( \\eta _ 2 ) + O ( ( x _ 2 ^ 2 + t ) ^ { - \\rho _ 3 N } ) , \\rho _ 3 > 0 , \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} v ^ 2 = 1 + \\sinh ^ { - 2 } u \\ , \\sigma ^ { i j } u _ i u _ j \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} h _ { i j } v ^ { - 1 } = - u _ { i j } - \\dot \\vartheta \\vartheta \\sigma _ { i j } , \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{gather*} s _ { 0 } = s _ { 1 } = \\ \\ldots \\ = s _ { n _ { 0 } - 1 } = 0 \\ , \\ s _ { n _ { 0 } } = ( - 1 ) ^ { \\frac { n _ { 0 } } { 2 } } t _ { n _ { 0 } } ^ { \\tfrac { 1 } { n _ { 0 } + 1 } } \\ . \\end{gather*}"}
-{"id": "1904.png", "formula": "\\begin{align*} | p _ t ( x , y ) - 1 | & \\le \\sum _ { j = 1 } ^ { + \\infty } e ^ { - \\lambda _ j t } | \\phi _ j ( x ) \\phi _ j ( y ) | \\\\ & \\le M ^ 2 \\sum _ { j = 1 } ^ { + \\infty } e ^ { - \\lambda _ j ( t - 2 t _ 0 ) } , \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} p _ n \\sum _ { \\ell = 0 } ^ { L - 1 } ( \\ell + 1 ) ( 1 - p _ n ) ^ \\ell & \\le \\lambda ^ \\star ; \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{gather*} p _ { n } ( x ) : = { P _ { n } ( x ) } / { D _ { n - 1 } } \\ , \\ n \\geq 0 \\ , \\end{gather*}"}
-{"id": "5083.png", "formula": "\\begin{align*} z = f ( x , y ) \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} i x ^ { \\prime } ( t ) + A _ 1 x ( t ) = \\Phi ^ * u ( t ) \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} Y _ { k + 1 } = I ( Y _ k , \\Phi ( Y _ k ) - Y _ k ) , k = 0 , \\ldots , \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} \\mathbb { E } ( N _ { 0 } ) = P ' _ { 0 } ( 1 ) = \\frac { \\lambda } { \\gamma + \\xi } P _ { 0 } ( 1 ) . \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} \\lambda _ { p , 1 } ( u ( t ) , t ) : = - \\int _ M u ( t ) \\Delta _ p u ( t ) d \\mu _ { g ( t ) } . \\end{align*}"}
-{"id": "9915.png", "formula": "\\begin{align*} A = \\left \\{ a ( t ) : = \\begin{bmatrix} e ^ t & \\\\ & e ^ { - t } \\end{bmatrix} : t \\in \\R \\right \\} , \\ U = \\left \\{ u ( s ) : = \\begin{bmatrix} 1 & s \\\\ 0 & 1 \\end{bmatrix} : s \\in \\R \\right \\} , \\ U ^ { - } = \\left \\{ u ^ { - } ( s ) : = \\begin{bmatrix} 1 & 0 \\\\ s & 1 \\end{bmatrix} : s \\in \\R \\right \\} . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} \\log x _ n , \\ , \\log y _ n = n \\log ( x + \\sqrt { x ^ 2 - 1 } ) + O ( 1 ) \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} \\mathsf { d } _ { g _ H } ( x , y ) : = \\sup \\left \\{ | f ( x ) - f ( y ) | \\ , \\colon \\ f \\in C _ c ^ \\infty ( M ) , \\ \\sigma ( \\Delta _ { H } ) ( d f , d f ) \\leq 1 \\right \\} , \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { m } d \\left ( z _ { j _ { k } } \\right ) ^ { - 1 } \\leq C \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} k _ { \\mathcal I } ( S _ 1 , S _ 2 ) = 0 . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} \\int _ { \\partial B _ R } \\bigg ( ( \\Delta u ) _ n + ( 1 - \\sigma ) \\bigg ( 1 - \\dfrac { 1 - \\sigma } { 2 } \\bigg ) u _ n \\bigg ) u _ n = - \\bigg ( 1 + \\dfrac { 2 } { p + 1 } \\bigg ) \\int _ { B _ R } u ^ { p + 1 } . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} \\int _ { L ^ 1 } f ( l ) \\ , \\dd \\nu ^ x ( l ) = \\int _ { H ^ 1 } \\int _ { L ^ 1 } f ( l ) \\ , \\dd \\dot \\lambda ^ { ( h , x ) } ( l ) \\ , \\dd \\mu ^ { F ^ 0 ( x ) } ( h ) \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{gather*} s _ { 2 n _ { k } + 2 } = s _ { 2 n _ { k } + 2 } ^ { ( n _ { k } + 1 ) } \\ , , \\ s _ { 2 n _ { k } + 3 } = s _ { 2 n _ { k } + 3 } ^ { ( n _ { k } + 1 ) } \\ , , \\ \\ldots \\ , \\ s _ { n _ { k + 1 } + n _ { k } } = s _ { n _ { k + 1 } + n _ { k } } ^ { ( n _ { k } + 1 ) } \\ , \\end{gather*}"}
-{"id": "3142.png", "formula": "\\begin{gather*} \\tau _ { k } ^ { ( \\alpha ) } = \\big \\langle T ^ { k } v _ { 0 } , g _ { C } ^ { ( \\alpha ) } v _ { 0 } \\big \\rangle . \\end{gather*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\eta ( v ) = v \\otimes 1 + a \\cdot 1 \\otimes x + p \\cdot v \\otimes x \\ \\ \\ \\ { \\rm a n d } \\ \\ \\ a ^ 2 + p a m - p ^ 2 b \\in R ^ * . \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} \\mathcal { B } _ { I I I } ^ - = \\left \\{ \\begin{aligned} & ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { A } ^ - \\textnormal { s u c h t h a t } \\\\ & \\exists \\ ; i \\in \\left \\{ 1 , 2 , \\dots , s + k \\right \\} , \\ ; t ^ \\prime \\geq 0 \\ ; : \\ ; \\left | v _ { s + k + 1 } - v _ i ^ \\prime ( \\tau ; t ^ \\prime ) \\right | \\leq \\eta \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} \\overleftarrow { P } _ { 0 , n } ( C | { y } ^ { n - 1 } ) & \\triangleq \\int _ { C _ 0 } p _ 0 ( d x _ 0 | x ^ { - 1 } , y ^ { - 1 } ) \\ldots \\int _ { C _ n } p _ n ( d x _ n | x ^ { n - 1 } , y ^ { n - 1 } ) , ~ C = \\times _ { t = 0 } ^ n { C } _ t \\in { \\cal B } ( { \\cal X } _ { 0 , n } ) \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} \\mathbf { x } _ { i } \\in X _ { i } , \\ i = 1 , \\ldots , n ; \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} \\widetilde { P } ( f \\circ \\pi ) ( x , \\xi ) = ( P f ) ( x ) , , \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} \\tilde { Z } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] = \\left ( X _ { s + k } ^ \\prime , V _ { s + k } ^ \\prime \\right ) \\in \\tilde { \\mathcal { D } } _ { s + k } \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} \\hat \\theta _ T = \\frac { X _ T ^ 2 } { 2 \\int _ 0 ^ T X _ t ^ 2 d t } \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} H _ N ^ { ( r ) } : = \\langle \\chi _ { e _ { ( r + 1 , 3 / 2 ) } } \\rangle . \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} \\begin{cases} v _ t + a v _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ v ( 0 , t ) = h _ 0 ( t ) , \\ , \\ , v ( L , t ) = h _ 1 ( t ) , \\ , \\ , v _ { x } ( L , t ) = h _ 2 ( t ) , & \\ , \\ , ( 0 , T ) , \\\\ v ( x , 0 ) = 0 , & \\ , \\ , ( 0 , L ) . \\end{cases} \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ n { n \\brack i } x ^ { 2 i - 1 } = x ^ { 2 n } - E _ { 2 n } ( x ) \\quad \\mbox { a n d } \\sum \\limits _ { i = 0 } ^ n { n \\brace i } x ^ { 2 i } = x ^ { 2 n + 1 } - E _ { 2 n + 1 } ( x ) . \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\inf _ { \\vert x \\vert \\leq R } u ( t , x ) = 1 , R \\geq 0 . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} S = \\cup _ { k = 1 } ^ { d } \\left ( S ( k + ) \\cup S ( k - ) \\right ) . \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} \\partial _ x p ( s + t , x , y ) = \\int p ( s , y , z ) \\partial _ x p ( t , x , z ) \\ d \\mu ( z ) . \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} W ( \\psi ^ { + } , \\psi ^ { - } ) = q ^ { - n } \\left ( \\psi _ { n + 1 } ^ { + } \\psi _ { n } ^ { - } - \\psi _ { n } ^ { + } \\psi _ { n + 1 } ^ { - } \\right ) \\ ! , \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} \\tilde { y } ( t ) = u ( t ) - i \\Phi x ( t ) \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| u _ i ^ { n } \\| _ { L ^ 2 ( \\mathbb { R } ^ N ) } = \\sqrt { \\tau _ i } \\ \\ \\textrm { a n d } \\ \\ \\lim _ { n \\to \\infty } E ( u _ 1 ^ { n } , u _ 2 ^ { n } ) = E _ \\tau . \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} \\frac { q } { f ( q ) } \\sum _ { n \\ge 1 \\atop ( n , P _ { \\Omega } ) = 1 } \\frac { f ( n ) } { n } ~ + ~ \\delta _ { \\Omega } \\log q ~ - ~ \\frac { r } { g ( r ) } \\sum _ { m \\ge 1 \\atop ( m , P _ { \\Omega } ) = 1 } \\frac { g ( m ) } { m } ~ - ~ \\delta _ { \\Omega } \\log r ~ = ~ 0 . \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} \\sigma _ E ( \\kappa _ E ( x ) ) = \\kappa _ E ( \\sigma _ { E _ \\curlyvee } ( x ) ) \\sigma _ { E _ \\curlyvee } ^ { m ( y ) } ( \\kappa _ E ^ { - 1 } ( y ) ) = \\kappa _ E ^ { - 1 } ( \\sigma _ { E } ( y ) ) . \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 2 } { k } ( \\phi ( x ) - \\phi ( x ^ * ) ) ^ T ( x ^ k - x ^ * ) < \\infty \\mbox { { \\it a . s . } } \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } G _ { n } ( \\xi ) = g \\circ \\pi ( \\xi ) \\mbox { f o r $ \\mathbb { P } ^ { - } $ - a l m o s t e v e r y $ \\xi $ } . \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\delta _ { \\min , i } : = \\tau \\delta _ { \\min , i } , \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P , A c h } } ( \\mu , r ) & \\leq \\max \\left \\{ \\frac { ( 1 - \\mu M ) K } { M r } ~ , ~ \\frac { ( 1 - \\mu M ) K } { \\min \\{ M , K \\} } + \\mu ( M + K - 1 ) \\right \\} \\\\ & = \\frac { ( 1 - \\mu M ) K } { \\min \\{ M , K \\} } + \\mu ( M + K - 1 ) , \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} d ^ { m + n - 1 } ( f \\bullet g ) = ( - 1 ) ^ { n } \\left ( g \\smile f - ( - 1 ) ^ { m n } f \\smile g \\right ) \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} F _ 1 ( x , y , \\eta ) ) & = y _ 1 - x _ 1 , \\\\ F _ i ( x , y , \\eta ) & = y _ i - x _ i + p _ i ( x , y , \\eta ) ( i = 2 , \\ldots , n ) , \\\\ F _ { n + k } ( x , y , \\eta ) & = \\eta _ k + q _ k ( x , y , \\eta ) , ( k = 1 , \\ldots , p ) , \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} W a ^ { i i } W _ { i ; i } = A + a ^ { i i } \\abs { \\nabla u } u _ { 1 ; i i } , \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} C _ { p , g + 1 } ^ { ( \\ell ) } & = ( - 1 ) ^ { p - 1 } \\sum _ { \\gamma = 0 } ^ { g + 1 } ( - 2 ) ^ { \\gamma } \\binom { \\ell } { \\gamma } \\binom { n - \\gamma } { g + 1 - \\gamma } \\sum _ { k = 0 } ^ { p - 2 g - 3 } ( - 1 ) ^ { k } \\binom { n - \\ell } { k } \\\\ & = ( - 1 ) ^ { p - 1 } \\sum _ { \\gamma = 0 } ^ { g + 1 } ( - 2 ) ^ { \\gamma } \\binom { \\ell } { \\gamma } \\binom { n - \\gamma } { g + 1 - \\gamma } ( - 1 ) ^ { p - 2 g - 3 } \\binom { n - \\ell - 1 } { p - 2 g - 3 } . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} v ( 0 , 0 ) & \\le \\mathbf { E } \\bigg [ \\int _ 0 ^ 1 e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ t \\| \\beta _ s \\| ^ 2 d s } \\langle \\alpha ^ * _ t ( \\beta ) + c \\beta _ t , \\beta _ t \\rangle \\ , d t + e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\| \\beta _ t \\| ^ 2 d t } f ( X _ 1 ^ \\beta ) \\bigg ] \\\\ & = J _ f [ \\alpha ^ * ( \\beta ) + c \\beta , \\beta ] \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} q ^ \\ell ( x , y , \\xi ; t ) : = e ^ { - i t h _ 0 ( \\xi ) } ( { } ^ t L _ 2 ) ^ \\ell ( s _ - ( x , \\xi ) p _ + ( y , \\xi ) ) \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} f _ \\infty ^ { ( s ) } ( t ) = \\int _ { \\mathcal { P } \\left ( \\mathbb { R } ^ { 2 d } \\right ) } h ^ { \\otimes s } ( Z _ s ) d \\pi _ t ( h ) \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} ( \\Lambda ( x , y ) f ) ( \\tau ) = \\left ( \\pi ( \\tau x + y ) f \\right ) ( 0 ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ { \\infty } e ^ { i \\tau ( s \\alpha ( x ) + \\beta ( x ) ) } e ^ { i ( s \\alpha ( y ) + \\beta ( y ) ) } \\widehat { f } ( s ) d s . \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} \\mathcal { A } ^ - = \\left \\{ \\begin{aligned} & \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\subset [ 0 , \\infty ) \\times \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } \\textnormal { s u c h t h a t } \\\\ & \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime \\right ) \\leq 0 \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} \\int _ { \\Gamma } f \\cdot \\omega = \\int _ { \\sigma ( \\Gamma ) } \\sigma ^ * ( f \\cdot \\omega ) . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} h ( x , \\nabla _ x \\phi _ \\pm ( x , \\xi ) ) = \\lim _ { t \\to \\pm \\infty } h _ \\rho ( t , y ( 0 , t ; x , \\xi ) , \\xi ) = h _ 0 ( \\xi ) . \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} \\mu \\left ( \\left [ a _ { 1 } , \\dots , a _ { n } \\right ] \\right ) \\coloneqq \\pi _ { a _ { 1 } } \\cdot \\prod _ { i = 1 } ^ { n - 1 } p _ { a _ { i } , a _ { i + 1 } } \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} [ \\varphi , \\varphi ' ] = \\inf _ { x \\in X } \\hom ( \\varphi ' ( x ) , \\varphi ( x ) ) , \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} \\frac { H ^ { n - 1 } ( \\{ u = 0 \\} \\cap B _ g ( x , \\rho ) ) } { \\rho ^ { n - 1 } } \\leq 2 F ( N ) . \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} \\{ r _ 1 , r _ 2 m \\} = \\{ r _ 1 , r _ 2 \\} m + r _ 2 \\{ r _ 1 , m \\} , \\{ r _ 1 r _ 2 , m \\} = r _ 1 \\{ r _ 2 , m \\} + r _ 2 \\{ r _ 1 , m \\} \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} \\mathcal { V } : = \\big ( H ^ { 1 } ( G , \\mathbb { R } ^ { k } ) \\cap \\mathcal { H } \\big ) \\times \\big ( H ^ { 1 } ( G , \\mathbb { R } ^ { k } ) \\cap \\mathcal { H } \\big ) . \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} \\tilde { Z } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\in \\overline { \\tilde { \\mathcal { D } } } _ { s + k } \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\big | \\phi \\big ( M ( D _ i ) \\big ) \\big | ^ 2 = \\prod _ { k = 1 } ^ K \\big | \\phi _ k \\big ( M ( a _ { k ; i } ) \\big ) \\big | ^ 2 = \\big | \\phi _ 1 \\big ( M ( a _ { 1 ; i } ) \\big ) \\big | ^ 2 . \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\frac { 4 } { q } + \\frac { 3 } { r } = \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{gather*} \\big ( \\tau _ { k + 1 , \\ell } ^ { ( \\alpha , \\beta ) } \\big ) ^ { 2 } = \\tau _ { k + 1 , \\ell } ^ { ( \\alpha + 1 , \\beta ) } \\tau _ { k + 1 , \\ell } ^ { ( \\alpha - 1 , \\beta ) } + \\tau _ { k + 2 , \\ell + 1 } ^ { ( \\alpha - 1 , \\beta ) } \\tau _ { k , \\ell - 1 } ^ { ( \\alpha + 1 , \\beta ) } - \\tau _ { k + 2 , \\ell } ^ { ( \\alpha - 1 , \\beta ) } \\tau _ { k , \\ell } ^ { ( \\alpha + 1 , \\beta ) } . \\end{gather*}"}
-{"id": "9859.png", "formula": "\\begin{align*} ( - 1 ) ^ n \\sum _ { J \\subseteq I } ( - 1 ) ^ { | J | } ( 1 + t ) ^ { s ( \\lambda ; \\mu ) + 1 - | J | } x ^ { | \\lambda | - | \\mu | } & = ( - 1 ) ^ n ( 1 + t ) ^ { s ( \\lambda ; \\mu ) + 1 - | \\mathcal { I } | } x ^ { | \\lambda | - | \\mu | } \\left ( \\sum _ { J \\subseteq I } ( - 1 ) ^ { | J | } ( 1 + t ) ^ { | \\mathcal { I } | - | J | } \\right ) \\\\ & = ( - 1 ) ^ n t ^ { | \\mathcal { I } | } ( 1 + t ) ^ { s ( \\lambda ; \\mu ) + 1 - | \\mathcal { I } | } x ^ { | \\lambda | - | \\mu | } . \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} n ^ { \\alpha / 2 } \\Big ( \\sum _ { i = 1 } ^ { n } | | \\mathbf { d _ i } | | ^ { 6 + 2 \\alpha } \\Big ) ^ { 1 / 2 } & \\leq n ^ { \\alpha / 2 } | | \\mathbf { D ^ { - 1 } } | | ^ { 3 + \\alpha } \\Big { ( } \\sum _ { i = 1 } ^ { n } | | \\mathbf { x _ { i } } | | ^ { 6 + 2 \\alpha } \\Big { ) } ^ { 1 / 2 } \\\\ & = O ( n ^ { - 1 } ) \\ ; \\ ; \\ ; w . p . 1 \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\tau } ( g ) : = \\{ \\phi _ { \\tau , u } ( g ) : u \\in \\mathcal { U } \\} \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\mathcal { A } ( g ) : = \\bigcup _ { \\tau > 0 } \\mathcal { A } _ { \\tau } ( g ) , \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} \\delta _ { \\mathbb { Z } / 2 } ( X ) = \\frac { 1 } { 2 } ( \\mathrm { m i n } \\{ m \\mid \\exists x \\in H ^ m _ { \\mathbb { Z } / 2 } ( X ) , W ^ \\ell x \\neq 0 \\ ; \\mathrm { f o r \\ ; a l l } \\ ; \\ell \\geq 0 \\} ) \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} \\phi _ { 1 } ^ { ( \\alpha ) } ( k ) = - 2 i | k | ^ { \\alpha / 2 } \\mbox { s g n } ( k ) \\exp \\left ( - \\frac { 2 | k | ^ { \\alpha / 2 + 1 } } { \\alpha + 2 } \\right ) , \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} \\| \\mathbf { H } \\| _ { L ( \\mathbb { R } ^ { k \\times d } ) } = \\sup _ { \\| \\mathbf { D } \\| _ { \\mathbb { R } ^ { k \\times d } } = 1 } \\| \\mathbf { H } \\mathbf { D } \\| _ { \\mathbb { R } ^ { k \\times d } } . \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} C ^ { F B , A . 2 } _ { X ^ n \\rightarrow { Y ^ n } } = \\sum _ { y _ { - 1 } \\in \\{ 0 , 1 \\} , y _ { - 2 } \\in \\{ 0 , 1 \\} } C _ t ( y ^ { - 1 } _ { - 2 } ) \\mu ( y _ { - 2 } ^ { - 1 } ) , ~ \\mu ( y _ { - 2 } ^ { - 1 } ) ~ \\mbox { i s f i x e d } . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} [ x _ j , \\frac { \\nabla _ k } { \\sqrt { - \\Delta + m ^ 2 } } ] & = [ x _ j , \\nabla _ k ] \\frac { 1 } { \\sqrt { - \\Delta + m ^ 2 } } + \\nabla _ k [ x _ j , \\frac { 1 } { \\sqrt { - \\Delta + m ^ 2 } } ] \\\\ & = - \\frac { \\delta _ { j k } } { \\sqrt { - \\Delta + m ^ 2 } } - \\frac { \\nabla _ k \\nabla _ j } { ( { - \\Delta + m ^ 2 } ) ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { N _ T } a _ { p } c _ { p } = \\det ( \\mathbf { H } _ { \\bar { \\mathcal { R } } _ i } ) . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} C _ t \\psi _ t ( x ) = C _ t \\langle \\pi _ t ^ 0 ( x ) u _ { 0 0 } ^ { ( 0 ) } | u _ { 0 0 } ^ { ( 0 ) } \\rangle = \\langle \\rho _ q ^ 0 ( E ( x ) ) \\eta _ t | \\eta _ { t } \\rangle = \\theta _ t ( x ) . \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} S _ f ( n ) : = \\sum _ { m \\leq n } A _ f ( m ) , \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} d \\left ( \\underbrace { \\left \\lceil \\frac { d _ 0 } { d } \\right \\rceil + \\cdots + \\left \\lceil \\frac { d _ 0 } { d } \\right \\rceil } _ { \\omega - 1 } + \\left \\lceil \\frac { ( a - 1 ) ( b - 1 ) d _ 0 } { d } \\right \\rceil - \\omega + 1 \\right ) + 2 = \\frac { p ^ L - 1 } { d ' } d ' + 2 > p ^ L , \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} \\nu _ i ^ { \\alpha } = h _ i ^ k x _ k ^ \\alpha , \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} - L u + g \\circ u & = \\mu \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = \\nu \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega , \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{gather*} g ^ { [ k ] ( \\alpha ) } _ { - } = \\sum _ { a , b = 0 } ^ { 1 } g ^ { [ k ] ( \\alpha ) } _ { a b } ( z ) E _ { a b } , g ^ { [ k ] ( \\alpha ) } _ { a b } ( z ) = \\big \\langle Q _ { b } ^ { - 1 } v _ { 0 } , \\psi _ { a } ^ { - } ( z ) T ^ { - k } g _ { C } ^ { ( \\alpha ) } v _ { 0 } \\big \\rangle / \\tau _ { k } ^ { ( \\alpha ) } , \\end{gather*}"}
-{"id": "8221.png", "formula": "\\begin{align*} F '' ( z , v ) = v e ^ { F ( z , v ) } e ^ { N ( z ) } = \\frac { v } { 1 - z } e ^ { F ( z , v ) } , F ( 0 , v ) = 0 , F ' ( 0 , v ) = v , \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} H ^ * _ G ( M , \\Z ) = H ^ * ( ( M \\times E G ) / G , \\Z ) \\simeq H ^ * ( M / G , \\Z ) . \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} R ( \\mathbf { r } ) = \\sum _ { n = 1 } ^ N \\kappa ( r _ n , p _ n ) . \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { N } \\mu \\left ( z _ { n } \\right ) & = \\sum _ { n = 1 } ^ { N } \\frac { 1 } { d \\left ( z _ { n } \\right ) } = \\sum _ { n = 1 } ^ { d \\left ( z _ { 0 } \\right ) ^ { 2 } } \\frac { 1 } { d \\left ( z _ { 0 } \\right ) + n + d \\left ( z _ { 0 } \\right ) ^ { 2 } } \\\\ & \\approx \\log \\frac { d \\left ( z _ { 0 } \\right ) + 2 d \\left ( z _ { 0 } \\right ) ^ { 2 } } { d \\left ( z _ { 0 } \\right ) + 1 + d \\left ( z _ { 0 } \\right ) ^ { 2 } } \\approx \\log 2 , \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} v _ { e a } ^ 1 ( f , g _ { a v g } ^ * ) & = \\lim _ { \\beta \\uparrow 1 } ( 1 - \\beta ) v _ \\beta ^ 1 ( f , g _ \\beta ^ * ) \\\\ & \\leq \\lim _ { \\beta \\uparrow 1 } ( 1 - \\beta ) v _ \\beta ^ 1 ( f _ \\beta ^ * , g _ \\beta ^ * ) \\\\ & = v _ { e a } ^ 1 ( f ^ * _ { a v g } , g ^ * _ { a v g } ) . \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\xi _ { q } ( z ) = q ^ { 1 / 2 } \\frac { \\left ( q ; q \\right ) _ { \\infty } ^ { 2 } } { \\left ( q ^ { 1 / 2 } ; q \\right ) _ { \\infty } ^ { \\ ! 2 } } \\frac { \\theta _ { q } \\left ( - z ^ { - 1 } \\right ) } { \\theta _ { q } \\left ( - q ^ { 1 / 2 } z \\right ) } = \\frac { \\left ( q ; q \\right ) _ { \\infty } ^ { 2 } } { \\left ( q ^ { 1 / 2 } ; q \\right ) _ { \\infty } ^ { \\ ! 2 } } \\frac { \\theta _ { q } \\left ( - z \\right ) } { \\theta _ { q } \\left ( - q ^ { - 1 / 2 } z \\right ) } \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} \\mathcal { F } _ k = \\mathcal { F } _ 0 \\cup \\{ I ^ l , J ^ l ; 1 \\leq l \\leq k - 1 \\} \\mbox { f o r a l l } k \\geq 2 , \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} J = \\Big | \\frac { \\partial ( x , y ) } { \\partial ( a , b ) } \\Big | = x _ a \\ , y _ b - y _ a \\ , x _ b \\ , \\neq 0 . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } ( u ( x , t ) - u _ { 0 } ( x ) ) + ( - \\Delta ) ^ { \\beta } u ( x , t ) & = f ( x , t ) \\Omega \\times [ 0 , \\infty ) , \\\\ u ( x , t ) & = 0 \\quad \\quad \\quad \\ , \\mathbb { R } ^ { N } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = u _ { 0 } ( x ) \\quad \\ , \\ , \\Omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } \\frac { t ^ { 1 + C _ 4 } b ( t ) } { f _ a ' ( t ) } = 0 . \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} \\Delta _ G \\tilde { v } _ \\lambda ( \\xi ) = f _ { \\lambda } ( \\xi ) , \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} \\phi = \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum ^ { N - 1 } _ { n = 0 } \\theta \\circ T ^ n \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ { \\infty } n ( 2 n - 1 ) ( | a _ n | + | b _ n | ) \\leq 1 - | b _ 1 | . \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{gather*} \\tilde { \\mathcal { F } _ l } \\big ( x _ \\mu \\otimes 1 - \\big ( \\tilde \\phi ^ { - 1 } \\big ) ^ \\tau _ \\mu \\otimes \\hat { x } _ \\tau \\big ) = ( x _ \\mu \\otimes 1 - 1 \\otimes x _ \\mu ) \\cdot \\tilde { \\mathcal { F } } _ L \\in I _ { U ( \\gg ) } , \\end{gather*}"}
-{"id": "6542.png", "formula": "\\begin{align*} f ( x ) = ( x - 2 ) ^ { - r } \\sum \\limits _ { i = 0 } ^ { \\lfloor - r / 2 \\rfloor } \\delta _ i \\left ( \\frac { x } { x - 2 } \\right ) ^ { 2 i } = x ^ { - r } \\sum \\limits _ { i = 0 } ^ { \\lfloor - r / 2 \\rfloor } \\delta _ i \\left ( \\frac { x - 2 } { x } \\right ) ^ { - r - 2 i } \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} \\norm { f } _ p = \\left ( \\sum _ { n = 0 } ^ \\infty | f ( n ) | ^ p \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\frac { n ^ { - 2 m } | K _ n ( z + \\frac { u } { \\sqrt { n } } , z + \\frac { v } { \\sqrt { n } } ) | ^ 2 e ^ { - 2 n \\varphi _ n ( z + \\frac { u } { \\sqrt { n } } ) } e ^ { - 2 n \\varphi _ n ( z + \\frac { v } { \\sqrt { n } } ) } } { \\det ( d d ^ c ( \\varphi _ n ( z ) ) ^ 2 \\exp ( - \\sum _ { j = 1 } ^ m \\lambda _ j ^ n | u _ j - v _ j | ^ 2 ) } \\to 1 \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} \\varrho _ { \\Delta } ^ A ( E ) = E ( 4 - E ) . \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} e _ i ( u ) = \\frac { 1 } { 2 } \\| D ^ \\alpha u \\| _ { L ^ 2 ( \\mathbb { R } ^ N ) } ^ 2 - \\frac { \\mu _ i } { p _ i } \\| u \\| _ { L ^ { p _ i } ( \\mathbb { R } ^ N ) } ^ { p _ i } \\ \\textrm { f o r } \\ i = 1 , 2 , \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} y _ t = \\sum \\limits _ { j = 0 } ^ { n - 1 } { x _ { j l + t } } , t = 0 , 1 , \\ldots l - 1 \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\| T ( x ) - T ( y ) \\| = \\| D T ( z ) ( x - y ) \\| \\le \\| D T ( z ) \\| \\| x - y \\| < \\kappa ( \\rho ) \\| x - y \\| . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} E = \\left \\{ f \\in S ( 0 , 1 ) \\ ; \\biggm | \\ ; \\sup _ { 0 < t < 1 } \\frac 1 { \\psi ( t ) } \\int _ 0 ^ t f ^ * ( s ) \\ , d s < \\infty \\right \\} , \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} D _ E ( x ; g , \\tau ) & : = \\sum _ { p \\leq x } \\frac { 1 - ( g ( p ) p ^ { - i \\tau } ) } { p } , \\\\ \\Delta _ E ( x ; g , T ) & : = \\min _ { | \\tau | \\leq T } D _ E ( x ; g , \\tau ) . \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} J _ { m , n , h } : = \\begin{pmatrix} A _ 1 & - A _ 2 & 0 & 0 & \\cdots & 0 \\\\ A _ 1 & 0 & - A _ 3 & 0 & \\cdots & 0 \\\\ A _ 1 & 0 & 0 & - A _ 4 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ 1 & 0 & 0 & 0 & \\cdots & - A _ m \\end{pmatrix} . \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} \\left ( \\partial _ t + v \\cdot \\nabla _ x \\right ) f ( t ) = \\ell ^ { - 1 } Q ( f ( t ) , f ( t ) ) \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{gather*} I ( s , f ^ { ( n ) } , \\chi ) : = \\int \\limits _ { O _ v ^ { \\times 2 } } \\chi ( a c ( ( y ^ 3 - x ^ 2 ) ^ 2 + \\pi ^ { 8 n } x ^ 4 y ^ 4 ) ) \\ | ( y ^ 3 - x ^ 2 ) ^ 2 + \\pi ^ { 8 n } x ^ 4 y ^ 4 | ^ s \\ | d x d y | \\\\ = \\int \\limits _ { O _ v ^ { \\times 2 } } \\chi ( a c ( x ^ { 1 2 } y ^ 4 \\widetilde { f ^ { ( n ) } } ( x , y ) ) ) \\ | \\widetilde { f ^ { ( n ) } } ( x , y ) | ^ s \\ | d x d y | . \\end{gather*}"}
-{"id": "4905.png", "formula": "\\begin{align*} | B | = k + k _ 1 , \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} D _ { a , b } ( { \\cal B } ) = w ( e ( a , j _ 1 ) ) + w ( e ( j _ 1 , j _ 2 ) ) + . . . + w ( e ( j _ t , b ) ) = D _ { a , j _ 1 } + D _ { j _ 1 , j _ 2 } + . . . + D _ { j _ t , b } \\geq D _ { a , b } , \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} & \\textrm { r a n k } ( \\mathbf { Q } _ { s } ) = \\textrm { r a n k } \\{ [ 1 + \\tilde { f } ^ { * } ( t ) ] [ \\mathbf { I } + \\mathbf { Y } + \\sum _ { l = 1 } ^ { L } t ^ { - 1 } \\mathbf { H } _ { e , l } \\mathbf { A } _ { l } \\mathbf { H } ^ { H } _ { e , l } ] ^ { - 1 } \\\\ & ( \\sum _ { k = 1 } ^ { K } \\mu _ { k } \\mathbf { h } _ { s , k } \\mathbf { h } ^ { H } _ { s , k } ) \\mathbf { Q } _ { s } \\} \\leq \\textrm { r a n k } ( \\sum _ { k = 1 } ^ { K } \\mu _ { k } \\mathbf { h } _ { s , k } \\mathbf { h } ^ { H } _ { s , k } ) \\leq K . \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} t ^ * ( u ) : = \\bigg ( \\dfrac { \\int _ \\Omega ( \\Delta u ) ^ 2 - 2 ( 1 - \\sigma ) \\int _ \\Omega d e t ( \\nabla ^ 2 u ) } { \\int _ \\Omega g ( x ) | u | ^ { p + 1 } } \\bigg ) ^ { \\frac { 1 } { p - 1 } } . \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} & \\phi ' > 0 , \\phi ( 0 ) = 0 , \\phi ( \\pm R ) = \\pm R ' , \\\\ & \\phi ' ( u ) = 1 u \\in [ - R , - 2 R / 3 ] \\ , \\cup \\ , [ - R / 3 , R / 3 ] \\ , \\cup \\ , [ 2 R / 3 , R ] . \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} P ^ 0 \\left ( F _ { \\gamma ^ z } ( \\omega ) = W ( \\cdot , \\omega ) \\right ) = 1 . \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\begin{cases} \\epsilon _ A ( x ) = \\frac { x } { 2 n } \\mbox { i f } A = 0 , \\\\ \\epsilon _ A ( x ) = \\frac { 1 } { A } ( \\log ( A x + 2 n ) - \\log ( 2 n ) ) \\mbox { i f } A > 0 . \\end{cases} \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} Y ( q ) > Y ( q + 1 ) Y ( n ) = X ( n ) . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} u '' + \\vartheta a ( t ) f ( u ) = 0 , \\end{align*}"}
-{"id": "10029.png", "formula": "\\begin{align*} \\sigma _ q ( d _ { s + 1 } , \\dim W ) = q ^ { \\dim W } + 1 > q ^ { d _ s } + 1 = \\sigma _ q ( d _ { s + 1 } , d _ s ) . \\end{align*}"}
-{"id": "3808.png", "formula": "\\begin{align*} [ \\phi , \\psi ] & = \\phi \\circ \\psi - ( - 1 ) ^ { | \\psi | | \\phi | } \\psi \\circ \\phi . \\\\ \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} \\| \\frac { 1 } { \\sqrt { p } } \\nabla _ p s _ { f } \\| _ { p - S l } = & \\frac { 1 } { \\sqrt { p } } \\inf _ { g \\in L ^ 2 [ 0 , 1 ] : \\nabla _ p s _ { f } = \\nabla _ p s _ { g } } \\| s _ g \\| _ { \\mathcal { H } _ { B _ { [ 0 , 1 ] } } } \\\\ = & \\frac { 1 } { \\sqrt { p } } \\inf _ { g \\in L ^ 2 [ 0 , 1 ] : \\nabla _ p s _ { f } = \\nabla _ p s _ { g } } \\| g \\| _ { L ^ 2 [ 0 , 1 ] } , \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} B \\dot { w } ( t ) + A { w } ( t ) = f ( t ) . \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\int _ { \\epsilon \\leq | \\eta | \\leq R } F ( \\eta ) G ( | \\eta | ) d \\eta & = - \\int _ { \\epsilon } ^ { R } G ' ( \\rho ) \\left [ \\int _ { | \\eta | \\leq \\rho } F ( \\eta ) d \\eta \\right ] d \\rho \\\\ & \\qquad + G ( R ) \\int _ { | \\eta | \\leq R } F ( \\eta ) d \\eta - G ( \\epsilon ) \\int _ { | \\eta | \\leq \\epsilon } F ( \\eta ) d \\eta . \\end{align*}"}
-{"id": "10014.png", "formula": "\\begin{align*} h ( \\underline { a } ) = h ^ { + } ( \\underline { a } ) \\cap h ^ { - } ( \\underline { a } ) . \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} G = G _ { \\ell } = \\{ \\textup { $ A \\in \\mathrm { G L } _ 2 ( \\mathbb { F } _ { \\ell } ) $ : $ \\det A $ i s a $ ( k - 1 ) $ - t h p o w e r i n $ \\mathbb { F } _ { \\ell } ^ \\times $ } \\} . \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} \\big \\lVert \\omega \\big \\rVert _ { Z _ R } ^ 2 = \\big \\lVert \\omega \\big \\rVert _ { Z _ { 1 , 0 } \\cup Z _ { 2 , 0 } } ^ 2 + \\big \\lVert \\omega ^ \\mathrm { z m } \\big \\rVert _ { Y _ { [ - R , R ] } } ^ 2 + \\big \\lVert \\omega ^ \\mathrm { n z } \\big \\rVert _ { Y _ { [ - R , R ] } } ^ 2 . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} g _ s ( x ) = ( x - 2 ) ^ { - r - 2 s } x ^ { 2 s } , 0 \\le s \\le - k . \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } ' : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} E _ a : \\ , \\ , v ^ 2 = u ^ 3 + ( a ^ 4 - 4 a ^ 3 - 2 a ^ 2 - 4 a + 1 ) u ^ 2 + 1 6 a ^ 4 u \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta _ p u & = \\lambda | u | ^ { p - 2 } u & & { \\rm i n } \\ B _ 1 , \\\\ u & = 0 & & { \\rm o n } \\ \\partial B _ 1 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} \\Theta _ 2 ( t + h , t ) : = \\psi _ 1 \\left ( t + h , t + h / 2 \\right ) \\circ \\bar { \\psi } _ 2 ( t + h , t ) \\circ \\psi _ 1 \\left ( t + h / 2 , t \\right ) . \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} \\theta ^ 3 _ 4 ( v ) = - \\theta ^ 3 _ 4 ( e _ l ) , \\theta ^ { 3 ' } _ { 4 ' } ( v ) = \\theta ^ { 3 ' } _ { 4 ' } ( e _ { l ' } ) \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} r a n k ( \\mathbf { B } _ { \\omega + 1 } ' - \\mathbf { B } _ { 1 } '^ j ) = r a n k ( \\mathbf { A } ( \\mathbf { B } _ { \\omega + 1 } - \\mathbf { B } _ { 1 } ^ j ) \\mathbf { A } ^ { - 1 } ) = 5 ~ ~ ~ \\forall 1 \\leq j \\leq 3 1 . \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} X _ { z _ i } = \\frac { \\Omega _ { [ i ] } } { 2 } , i = 1 , \\dots , n . \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} & . . . \\to \\mathrm { E x t } ^ 1 ( L ( 2 ) , L ( 2 ) ) \\overset { \\delta } { \\to } \\mathrm { E x t } ^ 2 ( L ( 2 ) , E ) \\\\ & \\to \\mathrm { E x t } ^ 2 ( L ( 2 ) , F ) \\to \\mathrm { E x t } ^ 2 ( L ( 2 ) , L ( 2 ) ) \\to \\mathrm { E x t } ^ 3 ( L ( 2 ) , E ) \\\\ & \\to \\mathrm { E x t } ^ 3 ( L ( 2 ) , F ) \\to . . . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} \\left [ Y , W \\right ] _ s : = \\lim _ { \\varepsilon \\to 0 } \\frac { 1 } { \\varepsilon } \\int _ 0 ^ s ( Y _ { r + \\varepsilon } - Y _ r ) ( W _ { r + \\varepsilon } - W _ r ) \\mathrm d r , \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { r } X _ { A _ { 0 } \\cdots A _ { r } } ^ { \\left ( p \\right ) } = 0 , \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} \\zeta ( z ; 1 - \\beta , \\beta , 0 ) = 1 + \\frac { \\beta } { 2 } - \\frac { 2 z - 1 } { 2 \\sqrt { z ^ 2 - z + 1 } } + \\frac { \\beta } { 2 } \\frac { 1 - \\beta z } { \\sqrt { ( 1 - \\beta z ) ^ 2 } } \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} C _ h = \\bigcup _ { i = 0 } ^ h T ^ i \\hat { A } . \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} \\kappa ( A , B ) = \\min \\{ \\mu ( X ) : A \\subseteq X \\subseteq E - B \\} . \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} & \\int _ { ( P _ { k + 1 } , \\dots , P _ { g } ) \\in X ^ { g - k } } \\Phi ^ { * } \\nu ^ { g } ( P _ { 1 } , \\dots , P _ { g } ) \\\\ = & ( g - k ) ! \\left ( \\tfrac { i } { 2 } \\right ) ^ { k } g ! \\sum _ { 1 \\le j _ { 1 } < \\dots < j _ { k } \\le g } \\sum _ { \\rho , \\tau \\in \\mathrm { S y m } ( k ) } \\bigwedge _ { m = 1 } ^ { k } \\psi _ { j _ { m } } ( P _ { \\rho ( m ) } ) \\wedge \\bar { \\psi } _ { j _ { m } } ( P _ { \\tau ( m ) } ) , \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} D _ { a , b } ( { \\cal B } ) \\leq D _ { a , z } + D _ { b , z } = D _ { a , b } , \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} \\omega \\big | _ { Y _ { [ 0 , 2 R ] } } = e ^ { - i \\lambda u _ 1 } e ^ { i \\lambda R } \\phi _ 1 + e ^ { i \\lambda u _ 1 } e ^ { - i \\lambda R } \\phi ' _ 1 + \\omega ^ \\mathrm { n z } . \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} \\left ( w q ^ { 4 } ; q ^ { 4 } \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 1 2 n ^ { 2 } } \\left ( - w ^ { 2 } \\right ) ^ { n } } { \\left ( q ^ { 8 } , w q ^ { 4 } , w q ^ { 8 } ; q ^ { 8 } \\right ) _ { n } } = \\left ( - w q ^ { 2 } ; q ^ { 4 } \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 3 n ^ { 2 } } \\left ( - q ^ { - 1 } w \\right ) ^ { n } } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } \\left ( - w q ^ { 2 } ; q ^ { 4 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} \\star \\partial e ^ { t \\Delta } = \\star e ^ { t \\vec { \\Delta } } \\partial . \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} \\sum _ { j : z _ { j } \\in S \\left ( t \\right ) } \\mu \\left ( j \\right ) & \\leq C \\beta \\left ( 0 , t \\right ) ^ { - 1 } \\approx C \\left ( \\log \\frac { 1 } { \\left ( 1 - \\left \\vert z _ { k } \\right \\vert ^ { 2 } \\right ) ^ { \\beta } } \\right ) ^ { - 1 } \\\\ & \\approx C \\left ( \\log \\frac { 1 } { 1 - \\left \\vert z _ { k } \\right \\vert ^ { 2 } } \\right ) ^ { - 1 } = C \\mu \\left ( z _ { k } \\right ) , \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} \\varphi _ { p , q } ( t ) = \\sum _ { i : G _ i < t } ( D _ i - G _ i ) ^ q , ~ t \\in \\Lambda . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} P _ p = \\frac { 1 } { 2 } ( 1 + S ^ p _ 2 ) , Q _ p = \\frac { 1 } { 2 } ( 1 - S ^ p _ 1 ) . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l l } $ u _ t + u _ { x x x } + a v _ { x x x } = 0 $ , & i n $ ( 0 , L ) \\times ( 0 , T ) $ , \\\\ $ v _ t + \\frac { r } { c } v _ x + \\frac { a b } { c } u _ { x x x } + \\frac { 1 } { c } v _ { x x x } = 0 $ , & i n $ ( 0 , L ) \\times ( 0 , T ) $ , \\\\ $ u ( 0 , t ) = u ( L , t ) = u _ { x } ( L , t ) = 0 $ , & i n $ ( 0 , T ) $ , \\\\ $ v ( 0 , t ) = v ( L , t ) = v _ { x } ( L , t ) = 0 $ , & i n $ ( 0 , T ) $ , \\\\ $ u ( x , 0 ) = u ^ 0 ( x ) , v ( x , 0 ) = v ^ 0 ( x ) $ , & i n $ ( 0 , L ) $ . \\end{tabular} \\right . \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} \\frac { d } { d s } \\zeta _ \\Delta ( s ) = \\sum _ { k = 1 } ^ \\infty \\frac { - \\ln \\lambda _ k } { \\lambda _ k ^ { s } } \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} R ^ + ( n , k ) = 2 k R ^ + ( n - 1 , k ) + ( 2 n - 4 k + 2 ) R ^ + ( n - 1 , k - 1 ) + R ( n - 1 , k ) , \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} M ( t ) = \\mathrm { g r a p h } \\ , u ( t , \\cdot ) , \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} \\nabla _ x p ( t , s ; x , \\eta ( t , s ) ) + \\nabla _ \\xi p ( t , s ; x , \\eta ( t , s ) ) \\nabla _ x \\eta ( t , s ) = 0 . \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} P N : = \\sum _ { \\substack { f \\in H _ { 2 k } ^ { * } ( N ) \\\\ L _ f ( 1 / 2 ) \\neq 0 } } ^ { h } 1 \\geq c - \\epsilon , \\epsilon > 0 . \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} \\phi _ 0 ( x , \\xi ) : = e ^ { i \\xi \\cdot x } , ( x , \\xi ) \\in \\R ^ { 2 n } . \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} \\beta _ 0 = \\max _ { s \\in S } \\max _ { a ^ 1 \\in A ^ 1 ( s ) ; a ^ 1 \\neq a ^ 1 _ s } \\max _ { a ^ 2 \\in A ^ 2 ( s ) ; a ^ 2 \\neq a ^ 2 _ s } \\{ 0 , \\beta _ { s , a ^ 1 } ^ 1 , \\beta _ { s , a ^ 2 } ^ 2 \\} \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { K } \\sum _ { j = 1 } ^ { N } d _ { i j } ^ { [ \\sf d ] } + \\sum _ { k = 1 } ^ { K } \\sum _ { j = 1 } ^ { N } d _ { k j } ^ { [ \\sf u ] } \\leq K \\min \\{ M , N \\} , \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to 0 ^ + } \\norm { \\nabla f ( J ( x , \\gamma , \\lambda ) ) - \\nabla f ( x ) } = 0 , \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { g \\in K } | \\alpha ( g ) f _ n - f _ n | _ { \\infty } ^ - = 0 . \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} \\max _ { \\mathbf { q } , \\mathbf { V } , \\rho _ { s , k } } & \\min _ { k , l } R _ { k } = R _ { s , k } - R _ { e , l } ^ { \\textrm { U P } } \\\\ s . t . & ~ \\min _ { k } E _ { s , k } \\geq \\bar { E } _ { s } , ~ \\min _ { l } E _ { e , l } \\geq \\bar { E } _ { e } , ~ \\forall k , l , \\\\ & ~ \\| \\mathbf { q } \\| ^ { 2 } + \\textrm { t r } ( \\mathbf { V } ) \\leq P _ { \\textrm { t o t a l } } , \\\\ & ~ 0 < \\rho _ { s , k } \\leq 1 , \\mathbf { V } \\succeq \\mathbf { 0 } . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} \\| u - \\sum _ { j = 0 } ^ { k } w _ j \\| _ { L ^ 2 ( ( \\frac 1 4 ) ^ { k + 1 } B _ 1 ) } \\le 4 ^ { - \\frac { ( k + 1 ) ( n + 2 ) } 2 } \\varphi ( 4 ^ { - ( k + 1 ) } ) . \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} \\frac { d } { d r } f ( r x ) & = \\frac { d } { d r } f ( { \\exp } _ { \\mathbb { G } } \\left ( r ^ { \\nu _ { 1 } } e _ { 1 } ( x ) X _ { 1 } + \\ldots + r ^ { \\nu _ { n } } e _ { n } ( x ) X _ { n } \\right ) ) \\\\ & = \\left [ ( \\nu _ { 1 } r ^ { \\nu _ { 1 } - 1 } e _ { 1 } ( x ) X _ { 1 } + \\ldots + \\nu _ { n } r ^ { \\nu _ { n } - 1 } e _ { n } ( x ) X _ { n } ) f \\right ] ( r x ) \\\\ & = \\frac { 1 } { r } \\left ( e ( r x ) \\cdot A \\nabla \\right ) f ( r x ) . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} u _ n = \\sum _ { k = 1 } ^ { N _ D } U _ k \\otimes s _ { n _ k , n } , & w _ n = P _ C \\otimes s _ { { n _ 0 } , n } + u _ n , \\\\ t _ n = \\sum _ { l = 1 } ^ { N _ C } T _ l \\otimes s _ { n _ l , n } , & z _ n = P _ D \\otimes s _ { { n _ 0 } , n } + t _ n . \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} \\widehat { f _ 1 } ( v ) = \\frac { \\sqrt { 2 } } { \\sqrt { \\pi } ( 1 + { v } ^ { 2 } ) } \\qquad ( v \\in \\R ) , \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} p _ { 1 , 1 } = \\frac { \\xi - \\gamma A } { \\mu A } p _ { 0 , 0 } \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\rho ( i , \\alpha , \\gamma ) = ( \\psi _ { \\gamma } ( i , \\alpha ) , \\sigma _ { i , \\alpha } ( \\gamma ) ) \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} H ^ 1 ( G , H ^ 2 ( M ' ) ) = H ^ 1 ( G , H ^ 2 ( M ) ) . \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\mathrm { I } _ { j } = \\int _ { \\R ^ n } \\Gamma ( x ; y ) \\ , P ^ * \\varphi ( y ) \\ , \\d y . \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} \\Gamma ( x ; y ) = \\Gamma ( y ; x ) , . \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} S _ { \\alpha p } ^ i \\pm \\sqrt { - 1 } T _ { \\alpha p } ^ i = c _ i ( S _ { \\alpha p } ^ 2 \\pm \\sqrt { - 1 } T _ { \\alpha p } ^ 2 ) , \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\Psi _ j ^ { \\ast } A _ j = \\Psi _ 1 ^ { \\ast } A _ 1 + i \\delta \\xi _ j + i \\eta _ j . \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} \\left \\vert \\sum _ { j \\neq k } a _ { j } \\overline { a _ { k } } \\int _ { \\mathbb { D } } \\varphi _ { z _ { j } } ^ { \\prime } \\left ( z \\right ) \\overline { \\varphi _ { z _ { k } } ^ { \\prime } \\left ( z \\right ) } d z \\right \\vert < \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ { \\infty } \\left \\vert a _ { j } \\right \\vert ^ { 2 } \\mu \\left ( z _ { j } \\right ) . \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} K ^ { ( \\rho ) } = \\biguplus _ { m \\in \\omega } K _ m ^ { ( \\rho ) } \\uplus \\{ b \\} . \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} q _ { \\alpha , \\beta } ( t , x ) = t ^ { - \\frac { \\alpha d } { 2 } + \\alpha - \\beta } q _ { \\alpha , \\beta } ( 1 , x t ^ { - \\frac { \\alpha } { 2 } } ) , \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} L _ 2 : = \\langle \\nabla _ \\xi \\Phi \\rangle ^ { - 2 } ( 1 + \\nabla _ \\xi \\Phi \\cdot D _ \\xi ) , \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} ( s _ { \\eta } \\zeta ) ( t ) : = \\zeta ( t + \\eta ) , ( s _ { \\eta } v ) ( t ) : = v ( t + \\eta ) \\hbox { f o r } t \\in ( 0 , T - \\eta ] , \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} \\sum _ { 0 \\le i \\le z } \\binom n i < 2 ^ { n H ( z / n ) } \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} \\left ( C _ { \\alpha _ d } \\Z \\hat \\otimes C \\ell _ { 0 , 1 } , \\ , \\ell ^ 2 ( \\Z , C ) _ { C } \\otimes \\bigwedge \\nolimits ^ { \\ ! \\ast } \\R , \\ , D = X _ 1 \\otimes \\gamma ^ 1 \\ , , \\gamma _ { \\bigwedge ^ * \\R } \\right ) . \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} M = \\Gamma ^ 0 M \\supset \\Gamma ^ 1 M \\supset \\cdots \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N f ( X _ i ) = \\int f ( x ) d \\mu ( x ) \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} H _ { \\nu + m } ^ { ( 1 ) } ( i y ) = - h _ { m - 2 , \\nu + 2 } ( - i / y ) H _ { \\nu } ^ { ( 1 ) } ( i y ) + h _ { m - 1 , \\nu + 1 } ( - i / y ) H _ { \\nu + 1 } ^ { ( 1 ) } ( i y ) . \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} A _ { q } \\left ( c \\right ) = \\left ( c q ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 3 n ^ { 2 } } \\left ( - c ^ { 2 } \\right ) ^ { n } } { \\left ( q ^ { 2 } , c q , c q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} \\begin{aligned} & \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\\\ & \\ ; \\ ; \\in \\left ( \\left ( [ 0 , T _ L ] \\times B _ { 2 R } ^ d \\times \\mathbb { S } ^ { d - 1 } \\times \\mathbb { N } \\right ) ^ k \\backslash \\mathcal { B } _ k \\right ) \\bigcap \\left \\{ 0 \\leq t _ k \\leq \\dots \\leq t _ 1 \\leq t \\right \\} \\end{aligned} \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} f ( y ) & = \\sum _ { j = 0 } ^ { i - 1 } \\omega ^ { \\beta _ j } \\cdot p _ j + 1 + f _ i ( y ) \\le \\sum _ { j = 0 } ^ { i - 1 } \\omega ^ { \\beta _ j } \\cdot p _ j + 1 + \\omega ^ { \\beta _ i } \\cdot p _ i \\\\ & = \\sum _ { j = 0 } ^ { i } \\omega ^ { \\beta _ j } \\cdot p _ j \\le \\tau < \\tau + 1 . \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} g ( \\omega , t , x ) = Y ( \\omega ) 1 _ { ( a , b ] } ( t ) 1 _ { ( v , w ] } ( x ) , \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} \\gamma _ { j } = \\frac { g _ { B ( j ) , j } \\ell ( r _ { B ( j ) , j } ) P _ B } { I ^ \\psi _ { j } + I ^ \\varphi _ { j } + \\sigma ^ 2 } . \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} \\omega _ { \\mu + j , a } ( x ) = x ^ { \\frac { \\mu + j } { 2 } } I _ { \\mu + j } ( 2 a \\sqrt { x } ) = ( a x ) ^ { \\mu + j } \\sum _ { k = 0 } ^ \\infty \\frac { ( a ^ 2 x ) ^ k } { k ! \\Gamma ( \\mu + j + k + 1 ) } , \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} K ( x , y ) = \\mathrm { V o l } _ N \\sum _ { i \\geqslant 0 } \\widehat { k _ S \\otimes k _ \\xi } ( \\psi _ i ) \\psi _ i ( x ) \\overline { \\psi _ i ( y ) } \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} \\Sigma ( N , y ) = \\check { w } _ 1 ( 1 + 2 \\pi i y ) N ^ { 1 + 2 \\pi i y } + O ( N ^ { \\varepsilon } ) . \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} F ( s e _ { 1 } + t e _ { 2 } ) - F \\left ( 0 \\right ) = f \\left ( s \\right ) u _ { 1 } + g \\left ( t \\right ) u _ { 2 } + f \\left ( s \\right ) g \\left ( t \\right ) u _ { 3 } \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\sum _ { a \\in A , b \\in B } M _ { ( a , s ) , ( b , t ) } = \\sum _ { a , a ' \\in A } M _ { ( a , s ) , ( a ' , s ' ) } = \\sum _ { b , b ' \\in B } M _ { ( b , t ) , ( b ' , t ' ) } = 1 \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} \\left ( F \\circ A \\right ) \\left ( x _ { 1 } , \\dots , x _ { n } \\right ) & = \\sum _ { \\delta \\in \\left \\{ 0 , 1 \\right \\} ^ { n } } u _ { \\delta } \\prod _ { i = 1 } ^ { n } x _ { i } ^ { \\delta _ { i } } \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} { \\sum \\limits _ { k = \\tau _ i - \\tau _ j } ^ { K - 1 } x _ { k , \\tau _ i } ^ { I M } = \\sum \\limits _ { k = \\tau _ i - \\tau _ j } ^ { K - 1 } x _ { k , \\tau _ i } ^ { B } , j = 1 , . . . , i - 1 . } \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} \\varprojlim _ { K ' \\lhd K _ 0 } X _ { K ' } = \\varprojlim _ { K ' \\lhd K _ 0 , K ' \\subset K } X _ { K ' } \\to X _ K . \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} \\| u \\| _ { \\mathcal { V } _ 2 ( \\mathcal { O } _ t ) } + \\| v \\| _ { 0 , 2 ; \\mathcal { O } _ t } \\leq & C \\left ( \\textrm { e s s s u p } _ { \\omega \\in \\Omega } \\| G ( \\omega , \\cdot ) \\| _ { L ^ 2 ( \\mathcal { O } ) } + \\textrm { e s s s u p } _ { \\omega \\in \\Omega } \\| \\hat { G } ( \\omega , \\cdot ) \\| _ { L ^ 2 ( \\mathcal { O } ) } \\right . \\\\ & \\left . + B _ 2 ( f _ 0 , g _ 0 ; \\mathcal { O } _ t ) + B _ 2 ( \\hat { f } , \\hat { g } ; \\mathcal { O } _ t ) \\right ) , \\\\ \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\begin{array} { l l } \\pi \\circ f _ B - f _ { B _ 1 } \\circ \\pi & = c ^ a _ n v ^ n _ b + c ^ k _ i B ^ i _ j v ^ j _ k \\delta ^ a _ b + c ^ d _ n B ^ n _ j v ^ j _ d \\delta ^ a _ b + c ^ d _ i B ^ i _ n v ^ n _ d \\delta ^ a _ b \\\\ & = c ^ k _ i B ^ i _ n v ^ n _ k \\delta ^ a _ b v ^ j _ k = \\delta ^ j _ n \\\\ & = 0 . \\end{array} \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} T ^ \\mu _ \\mu ( x ) = a _ 2 ( x , D ) = \\frac { 1 } { 2 4 \\pi } R \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} \\mathbf { P _ * } \\Big ( \\big { | } \\sum _ { i = 1 } ^ { n } | | \\mathbf { x _ i } | | ^ { 3 + \\alpha } ( G _ i ^ * - E G _ i ^ * ) \\big { | } > n \\epsilon \\Big ) = o ( n ^ { - 1 / 2 } ) \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} E _ k ( u ) F _ l ( v ) & = F _ l ( v ) E _ k ( u ) , \\ , \\ \\\\ E _ k ( u ) F _ k ( v ) & - F _ k ( v ) E _ k ( u ) = - \\hbar \\left ( \\frac { H _ k ( u ) - H _ k ( v ) } { u - v } \\right ) . \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} C = \\frac { 2 \\pi } { e ^ { \\kappa } \\Gamma ( 1 - \\beta ^ 2 ) } , \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} y _ q ( u ) & = \\sum _ { \\mathcal { R } : | \\mathcal { R } | = r + 1 } \\sum _ { \\mathcal { T } : | \\mathcal { T } | = t } \\sum _ { i = 1 } ^ \\varrho \\left [ \\sum _ { p \\in \\mathcal { T } } h _ { q p } ( u ) \\left ( \\mathbf { v } _ { { \\mathcal { R } } , { \\mathcal { T } } , p } ^ i ( u ) \\right ) ^ T \\right ] \\mathbf { x } _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ i , \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} K _ { n _ j + 1 } \\subseteq \\bigcup _ { i = 1 } ^ r U _ { x _ i } . \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} \\begin{cases} \\varphi ( 0 , t ) = \\varphi ( L , t ) = \\varphi _ x ( L , t ) = 0 , & \\ , \\ , ( 0 , T ) , \\\\ \\psi ( 0 , t ) = \\psi ( L , t ) = \\psi _ x ( L , t ) = 0 , & \\ , \\ , ( 0 , T ) . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( x : n ) } = \\bigoplus _ { i = 0 } ^ { x - 1 } T _ x ( i ) , \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} ( a , b ) _ 2 = \\begin{cases} 1 & p \\equiv 3 ~ { \\rm m o d } ~ 4 . \\\\ - 1 & p \\equiv 1 ~ { \\rm m o d } ~ 4 . \\end{cases} \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} \\norm { \\Psi ( a ) } = \\norm { a } = \\sup _ { x \\in X _ A } \\norm { a ( x ) } = \\sup _ { x \\in X _ A } \\norm { \\Psi ( a ) ( x ) } \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} X _ { M , p } : = \\left . \\frac { d } { d t } \\sigma _ { \\exp t X } ( p ) \\right | _ { t = 0 } \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} u _ { 0 . 6 7 } ^ 2 ( g _ 1 ) = [ I - 0 . 6 7 P ^ 2 ( g _ 1 ) ] ^ { - 1 } \\bar { r } ^ 2 ( g _ 1 ) = ( 6 , 8 ) . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} a \\cdot b = : a b : = a _ { ( - 1 ) } b , \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot { \\bar { V } } & = & \\dfrac { \\gamma ^ { - 1 } } { ( 1 - \\gamma ^ { - 1 } D ( x ) ) ^ { 2 } } \\left [ \\dot { V } ( x ) - \\dot { V } ( x ) D ( x ) + V ( x ) \\dot { D } ( x ) \\right ] \\end{array} \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} [ T _ n ^ { \\prime } + k T _ n ^ * , T _ n ^ { \\prime } + ( k + 1 ) T _ n ^ * ] \\cap [ T _ n ^ { \\prime } , T _ { n + 1 } ] , k = 0 , 1 , 2 , \\ldots k _ n , \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\lambda ( t ) = a _ n ( g , D ) \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} R ( t ) : = - \\int _ s ^ t \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , s ) ) d \\tau . \\end{align*}"}
-{"id": "10084.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q } { z ^ { p + 2 q } } , \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} \\alpha = 2 - \\sigma , \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} \\phi _ { R , R ' } ( u ) = \\left \\{ \\begin{array} { l l } u - R ' + R & u \\in [ - R , - \\frac { 1 } { 8 } R ] , \\\\ u - ( R ' - R ) \\chi _ { 1 , R / 8 } ( u ) & u \\in [ - \\frac { 1 } { 8 } R , 0 ] . \\end{array} \\right . \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } A _ { q } \\left ( q ^ { 2 \\alpha } u \\right ) A _ { q } \\left ( v q ^ { 2 \\alpha } \\right ) q ^ { 2 \\alpha ^ { 2 } } d \\alpha = \\sqrt { \\frac { \\pi } { \\log q ^ { - 2 } } } \\frac { \\left ( q ^ { 1 / 2 } u , q ^ { 1 / 2 } v ; q \\right ) _ { \\infty } } { \\left ( u v ; q \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} A ^ + _ { r , 2 r } ( x _ 0 ) & : = \\{ ( x '' , x _ n , x _ { n + 1 } ) \\in B _ 1 ^ + | \\ | x '' - x '' _ 0 | \\leq r , \\\\ & r \\leq \\sqrt { ( x _ n - ( x _ 0 ) _ n ) ^ 2 + ( x _ { n + 1 } - ( x _ 0 ) _ { n + 1 } ) ^ 2 } \\leq 2 r \\} . \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} \\mathcal { F } ( g _ { i j } ( t ) , u ( t ) ) = \\int _ { M } ( 4 | \\nabla u | ^ 2 + R u ^ 2 ) d \\mu \\ \\ \\ \\ \\ \\ w i t h \\ \\ \\ \\int _ M u ^ 2 d \\mu = 1 . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} \\tau _ \\delta = - \\log \\Theta ( t _ \\delta , T ^ * ) , Q ( \\tau _ \\delta , \\infty ) = [ \\tau _ \\delta , \\infty ) \\times \\mathbb { S } ^ n . \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} 1 6 ( \\sum _ { a = 5 } ^ 8 ( q ^ * _ a | _ V ) ^ 2 ) = | x y - y x | ^ 2 | z | ^ 2 . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} \\Phi ( Y ) & = X _ * + \\delta \\Vert Y - X _ * \\Vert _ F X _ { \\perp } , \\\\ X _ * & = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} , X _ { \\perp } = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} . \\\\ \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} M = \\langle [ g , g ' ] \\mid g ' \\in G \\rangle ^ G = [ G , G ] \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\inf _ { \\vert x \\vert \\leq R } u ( t , x ) = 1 , R \\geq 0 . \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} v _ { A } ^ * v _ { A } = v _ { B } ^ * v _ { B } = 1 \\otimes 1 , v _ { A } v _ { A } ^ * = P _ C \\otimes 1 , v _ { B } v _ { B } ^ * = P _ D \\otimes 1 . \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} g ( t ) = \\int ^ t _ 0 P _ p ( t - r ) h ( r ) \\d r . \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} \\int ^ S _ X C [ X , X ^ * , Z ] : = 1 + \\sum _ i ( F _ i , G _ i ) ^ S H _ i [ Z ] , \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} \\tau \\big ( A \\ , 1 _ { [ a , b ] } ( A ) \\big ) & = \\int _ c ^ d \\lambda ( t , A ) \\ , d t , \\\\ \\tau \\big ( 1 _ { [ a , b ] } ( A ) \\big ) & = d - c , \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} \\forall m \\in \\mathcal { M } , \\ \\exists \\lambda _ { m } \\ \\mbox { s u c h t h a t } \\ g _ { p } ( \\mathbf { u } ^ { * } ) \\left \\{ \\begin{array} { l l } \\geq \\lambda _ { m } & \\mbox { i f } u ^ { * } _ { p } = 0 , \\\\ = \\lambda _ { m } & \\mbox { i f } u ^ { * } _ { p } > 0 , \\end{array} \\right . \\forall p \\in \\mathcal { P } _ { m } ; \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} g _ m ( L _ 0 , W _ 0 ) = b _ m L _ 0 + d _ m , \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} \\begin{aligned} w & = a y _ 0 y _ 1 x _ 2 x _ 3 + b y _ 0 x _ 1 y _ 2 x _ 3 + a x _ 0 y _ 1 y _ 2 x _ 3 + c x _ 0 x _ 1 x _ 2 x _ 3 \\\\ & \\quad + a x _ 0 x _ 1 y _ 2 y _ 3 + b x _ 0 y _ 1 x _ 2 y _ 3 + a y _ 0 x _ 1 x _ 2 y _ 3 + c y _ 0 y _ 1 y _ 2 y _ 3 . \\end{aligned} \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} \\textbf { A } ( x , y , z ) = B _ 0 \\begin{bmatrix} - \\frac { ( \\sqrt { x ^ 2 + y ^ 2 } - R ) ^ 2 + z ^ 2 } { 2 Q ( x ^ 2 + y ^ 2 ) } y , & \\frac { ( \\sqrt { x ^ 2 + y ^ 2 } - R ) ^ 2 + z ^ 2 } { 2 Q ( x ^ 2 + y ^ 2 ) } x , & - R \\log \\left ( \\frac { \\sqrt { x ^ 2 + y ^ 2 } } { R } \\right ) \\end{bmatrix} . \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} R ^ \\star _ { \\mathsf { u } } ( R _ { \\mathsf { c } } ) & \\le \\sum _ { n = 1 } ^ N \\sum _ { j = 1 } ^ { L - r _ n } \\frac { j } { j + r _ n } \\binom { L - r _ n } { j } p _ n ^ { j } ( 1 - p _ n ) ^ { L - r _ n - j } , \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} F _ 1 ( z _ 0 , z _ 1 , z _ 2 ) = \\big ( z _ 0 ( z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 ) , z _ 1 ^ 2 z _ 2 , ( z _ 0 ^ 2 - z _ 2 ^ 2 ) z _ 2 \\big ) : = ( z _ 0 ' , z _ 1 ' , z _ 2 ' ) . \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} \\| u \\| _ { H _ { p } ^ { \\gamma } } : = \\| ( 1 - \\Delta ) ^ { \\gamma / 2 } u \\| _ { L _ { p } } < \\infty , \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} I ( X ^ n \\rightarrow Y ^ n ) \\triangleq & \\sum _ { t = 0 } ^ n { \\bf E } ^ { { p } } \\Big \\{ \\log \\Big ( \\frac { d q _ t ( \\cdot | Y ^ { t - 1 } , X ^ t ) } { d \\nu _ t ^ { { p } } ( \\cdot | Y ^ { t - 1 } ) } ( Y _ t ) \\Big ) \\Big \\} \\\\ = & \\sum _ { t = 0 } ^ n \\int _ { { \\cal X } ^ { t } \\times { \\cal Y } ^ { t } } \\log \\Big ( \\frac { d q _ t ( \\cdot | y ^ { t - 1 } , x ^ t ) } { d \\nu _ t ^ { { p } } ( \\cdot | y ^ { t - 1 } ) } ( y _ t ) \\Big ) { \\bf P } ^ { p } ( d x ^ t , d y ^ t ) . \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ { V } ^ + } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq C _ { d } T \\eta ^ { d } \\end{aligned} \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} \\begin{aligned} x \\frac { d z ( x ) } { d x } & = & { \\tilde { A } } z ( x ) , \\\\ z ( q x ) & = & \\tilde { B } z ( x ) \\end{aligned} \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} a \\ , d x ^ m + b \\ , d x ^ n = a \\ , d x ^ m \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} \\Delta { C } ^ { 1 , \\infty } = \\Big ( { \\frac { \\alpha - \\beta } { 1 - ( \\beta - \\gamma ) } } \\Big ) \\log ( 1 + 2 ^ { \\Delta { C } ^ { 1 , \\infty } } ) . \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\theta _ { s , a ^ 1 } ^ 1 = r ^ 1 ( s , a ^ 1 , a _ s ^ 2 ) + \\beta \\sum _ { s ' \\in S } p ( s ' | s , a ^ 1 , a _ s ^ 2 ) v _ \\beta ^ { 1 * } ( s ' ) - v _ \\beta ^ { 1 * } ( s ) . \\end{align*}"}
-{"id": "10069.png", "formula": "\\begin{align*} p = m + n + l , \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} E ^ z \\left [ P ^ { Z ( \\tau ) } \\left ( F _ { \\gamma ^ z } ( \\omega ) = W ( \\cdot , \\omega ) \\right ) \\right ] = 1 ; \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} u _ { \\rm N W } ( x ) = \\frac { \\sin | x | } { | x | ( 1 + g ( | x | ) ^ 2 ) } . \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { * } \\frac { 1 } { t } \\frac { 1 } { \\log _ { ( 1 ) } ( \\frac { 1 } { t } ) } \\cdots \\frac { 1 } { \\log _ { ( n + 2 ) } ( \\frac { 1 } { t } ) } \\ , \\mathrm { d } t \\\\ = \\ & \\int _ { * } ^ { + \\infty } \\frac { 1 } { s } \\frac { 1 } { \\log _ { ( 1 ) } ( s ) } \\cdots \\frac { 1 } { \\log _ { ( n + 2 ) } ( s ) } \\ , \\mathrm { d } s \\\\ = \\ & \\cdots \\ = \\ \\int _ { * } ^ { + \\infty } \\frac { 1 } { \\log ( x ) } \\ , \\mathrm { d } x \\ = \\ \\infty . \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} v _ { n - 1 } + \\left ( \\alpha q ^ { - n } - \\mu ( z ) \\right ) v _ { n } + v _ { n + 1 } = 0 , n \\in \\Z , \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { X } _ s ^ { t , x } = \\ x + \\int _ t ^ s \\mathrm d W _ r , \\\\ { Y } _ s ^ { t , x } = \\ \\Phi ( { X } _ T ^ { t , x } ) - \\int _ s ^ T { Z } _ r ^ { t , x } \\mathrm d { W } _ r + \\int _ s ^ T f ( r , { X } _ r ^ { t , x } , { Y } _ r ^ { t , x } , Z _ r ^ { t , x } ) \\mathrm d r \\\\ + \\int ^ T _ s { Z } _ r ^ { t , x } b ( r , { X } _ r ^ { t , x } ) \\mathrm { d } r , \\\\ { \\forall } s \\in [ t , T ] , \\end{array} \\right . \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\vec v _ t = \\lambda ' \\vec v \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} \\| g \\| ^ 2 = \\frac { 1 - p } { 2 p - 1 } ( \\alpha ^ 2 ( 3 p - 1 ) - 2 \\alpha ( s _ { g _ 0 } ( p ) - ( 2 p - 1 ) \\delta _ 0 ) ) + \\frac { s _ { g _ 0 } ( p ) ^ 2 } { 2 p - 1 } + \\delta _ 0 ^ 2 ( 1 - p ) . \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { 8 } f _ t ( p ' ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ { p ' } | } , \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 } { \\partial x ^ 2 } E _ { \\alpha , \\beta } ^ \\gamma \\left ( t { \\rm e } ^ { { \\rm i } x } \\right ) \\Big | _ { x = 0 } + t \\ , \\left ( E _ { \\alpha , \\beta } ^ \\gamma ( t ) \\right ) ' + t ^ 2 \\ , \\left ( E _ { \\alpha , \\beta } ^ \\gamma ( t ) \\right ) '' = 0 \\ , . \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} H ^ \\bullet _ \\mathrm { r e l } ( Z _ { 1 , R } , F ) = \\Big ( H ^ \\bullet ( Z _ { 1 , R } ^ \\mathrm { d b } , F ) \\Big ) ^ - , H ^ \\bullet _ \\mathrm { a b s } ( Z _ { 2 , R } , F ) = \\Big ( H ^ \\bullet ( Z _ { 2 , R } ^ \\mathrm { d b } , F ) \\Big ) ^ + . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\eta _ { Y } ^ { - 1 } \\circ _ 1 { \\Psi '' } ( \\alpha ) \\circ _ 1 \\beta & = & \\mathrm { i d } _ Y , \\\\ \\beta \\circ _ 1 \\eta _ { Y } ^ { - 1 } \\circ _ 1 { \\Psi '' } ( \\alpha ) & = & \\mathrm { i d } _ { \\Psi '' ( X ) } , \\\\ \\eta _ { X } ^ { - 1 } \\circ _ 1 { \\Psi '' } ( \\beta ) \\circ _ 1 \\alpha & = & \\mathrm { i d } _ X , \\\\ \\alpha \\circ _ 1 \\eta _ { X } ^ { - 1 } \\circ _ 1 { \\Psi '' } ( \\beta ) & = & \\mathrm { i d } _ { \\Psi '' ( Y ) } . \\end{array} \\right . \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} \\xi _ { \\underline { k } } = \\xi _ { 3 , k _ 1 - 3 , k _ 2 - 3 } , \\ \\underline { k } = ( k _ 1 , k _ 2 ) , \\ k _ 1 \\ge k _ 2 \\ge 3 \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} w = u ^ * \\Theta ^ { - 1 } \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} \\limsup _ { N \\rightarrow \\infty } \\left \\Vert \\left ( f _ N ^ { ( s ) } ( 0 , Z _ s ) - f _ 0 ^ { \\otimes s } ( Z _ s ) \\right ) \\mathbf { 1 } _ { Z _ s \\in \\mathcal { K } _ s \\cap \\mathcal { U } _ s ^ { \\eta ( \\varepsilon ) } } \\mathbf { 1 } _ { E _ s ( Z _ s ) \\leq R ^ 2 } \\right \\Vert _ { L ^ \\infty _ { Z _ s } } = 0 \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{gather*} \\frac { h ^ { ( \\alpha ) } _ { k } } { h _ { k - 1 } ^ { ( \\alpha + 1 ) } } + \\frac { h _ { k } ^ { ( \\alpha + 1 ) } } { h ^ { ( \\alpha ) } _ { k } } = \\frac { h _ { k + 1 } ^ { ( \\alpha - 1 ) } } { h _ { k } ^ { ( \\alpha ) } } + \\frac { h _ { k } ^ { ( \\alpha ) } } { h _ { k } ^ { ( \\alpha - 1 ) } } . \\end{gather*}"}
-{"id": "9684.png", "formula": "\\begin{align*} A _ { \\mathrm { t } } = \\{ x \\in X \\colon \\mbox { t h e r e i s $ \\xi \\in \\Sigma _ k ^ + $ w i t h $ \\{ x \\} = \\bigcap _ n T _ { \\xi _ 0 } \\circ \\dots \\circ T _ { \\xi _ n } ( X ) $ } \\} , \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} \\Gamma _ 2 ( f , g ) = \\frac { 1 } { 2 } ( \\Delta \\Gamma ( f , g ) - \\Gamma ( f , \\Delta g ) - \\Gamma ( g , \\Delta f ) ) . \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} D _ m = \\hat { D } _ { h _ m } \\setminus \\bigcup _ { n = m + 1 } ^ { \\infty } ( \\hat { C } _ { h _ n } \\cup \\hat { D } _ { h _ n } ) . \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } D ( S _ { \\phi } ) & = \\ \\left \\{ x \\in { \\cal H } \\ ; \\ \\lim _ { n \\rightarrow \\infty } \\sum _ { k = 0 } ^ { n } ( x | \\phi _ { k } ) \\phi _ { k } \\ { \\rm e x i s t s } \\ { \\rm i n } \\ { \\cal H } \\right \\} \\\\ \\\\ S _ { \\phi } x & = \\ \\sum _ { n = 0 } ^ { \\infty } ( x | \\phi _ { n } ) \\phi _ { n } , \\ ; \\ ; \\ ; x \\in D ( S _ { \\phi } ) \\\\ \\end{array} \\right . \\\\ \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - d u ( t , x ) & = [ \\partial _ j ( a ^ { i j } \\partial _ i u ( t , x ) + \\sigma ^ { j r } v ^ r ( t , x ) ) + f ( t , x , u ( t , x ) , \\nabla u ( t , x ) , v ( t , x ) ) \\\\ & + \\nabla \\cdot g ( t , x , u ( t , x ) , \\nabla u ( t , x ) , v ( t , x ) ) ] \\ , d t - v ^ r ( t , x ) \\ , d W ^ r _ t , ( t , x ) \\in Q , \\\\ u ( T , x ) & = G ( x ) , ~ ~ ~ x \\in \\mathcal { O } . \\end{array} \\right . \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} \\frac 1 n \\sum _ { k = 0 } ^ { n ^ m - 1 } X _ { k / n } ^ 2 = \\int _ 0 ^ { n ^ { m - 1 } } X ^ 2 _ t \\ , d t + \\vartheta _ n , \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} \\rho ( \\mathcal { T } ) = \\max \\{ x ^ { T } ( \\mathcal { T } x ) \\ , | \\ , x \\in \\mathbb { R } _ { + } ^ { n } , \\sum _ { i = 1 } ^ { n } x _ { i } ^ { k } = 1 \\} . \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} f _ { 1 } ( u , v ) & = \\alpha - u - h ( u , v ) , \\\\ f _ { 2 } ( u , v ) & = \\gamma ( \\beta - v ) - h ( u , v ) , \\\\ h ( u , v ) & = \\frac { \\rho u v } { 1 + u + \\delta u ^ { 2 } } \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} A _ { 2 1 } ( x ^ q ) B _ { 1 1 } + A _ { 2 2 } B _ { 2 1 } ( x ) = B _ { 2 1 } ( x ^ p ) A _ { 1 1 } + B _ { 2 2 } A _ { 2 1 } ( x ) . \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{gather*} d _ { r } \\ : = \\ \\inf \\left \\{ \\ m \\geq 0 \\ \\ \\big | \\ \\ s _ { 2 r + m } \\neq { s } _ { 2 r + m } ^ { ( r ) } \\ \\right \\} \\ \\in \\ \\{ 0 , 1 , 2 , . . . . \\} \\cup \\left \\{ \\infty \\right \\} \\ , \\end{gather*}"}
-{"id": "322.png", "formula": "\\begin{align*} F = - \\frac { 1 } { \\beta } \\ln Z , \\langle E \\rangle = - \\frac { \\partial } { \\partial \\beta } \\ln Z , S = \\ln Z - \\beta \\frac { \\partial } { \\partial \\beta } \\ln Z \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} H ( x ) = \\frac { 1 } { 3 } [ H ( x ^ { - 1 } ) + H ( x _ { + } ) + H ( x _ { - } ) ] \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} v = u + \\int _ 0 ^ t \\exp \\left ( - c \\left ( t \\right ) \\right ) \\left [ r ' - 2 \\left \\| \\nabla \\psi \\right \\| _ { \\infty } \\left ( r + \\frac { 1 } { 4 } \\right ) \\right ] \\ , d \\tau ; \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { \\Q } \\Sigma ( \\nabla ^ { s } u _ { h } ) : \\nabla \\varphi = \\int _ 0 ^ T \\int _ { \\Q } f \\cdot \\varphi \\qquad \\forall \\varphi \\in L ^ { 2 } ( 0 , T ; Y _ { h } ) \\ , . \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} \\| g \\| _ { N _ k } = \\| r ^ { - \\alpha q } \\psi _ R ^ q Z ^ { \\le k } g \\| _ { L ^ 1 L ^ 1 L ^ { 2 } } + \\| Z ^ { \\le k } g \\| _ { L ^ 1 L ^ 2 L ^ 2 } \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} \\alpha _ { k } ^ { L P } = \\underset { Y = [ y _ 1 , . . . , y _ n ] \\in \\mathbb { R } ^ { m \\times n } } { } \\ ; \\bigg \\{ \\underset { 1 \\leq j \\leq n } { } \\ ; | | ( I - Y ^ T A ) e _ j | | _ { k , 1 } \\ ; \\bigg \\} , \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} f _ { n + d + p } \\left ( x \\right ) = T _ { p } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) f _ { n + d } \\left ( x \\right ) + . . . + T _ { p } ^ { \\left ( d + 1 \\right ) } \\left ( x \\right ) f _ { n } \\left ( x \\right ) , \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} \\begin{cases} u _ { x x } ( 0 , t ) = h _ 0 ( t ) , \\ , \\ , u _ x ( L , t ) = h _ 1 ( t ) , \\ , \\ , u _ { x x } ( L , t ) = h _ 2 ( t ) , \\\\ v _ { x x } ( 0 , t ) = g _ 0 ( t ) , \\ , \\ , v _ x ( L , t ) = g _ 1 ( t ) , \\ , \\ , v _ { x x } ( L , t ) = g _ 2 ( t ) , \\end{cases} \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} P ^ S _ t f = ( P ^ S _ t f _ 1 , \\cdots , P ^ S _ t f _ n ) , \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} \\delta _ k = \\sup \\limits _ { z \\in \\partial \\Omega } ( u _ { 0 , k } ( z ) - u _ 0 ( z ) ) \\stackrel { k \\rightarrow \\infty } { \\longrightarrow } 0 . \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} l _ { i ( n _ 0 + 1 ) } - \\frac { 1 } { \\sigma } \\sum \\limits _ { k = 1 } ^ { n _ 0 } t _ k l _ { i k } + \\frac { 1 } { \\sigma } z _ i \\ge 0 , ~ i = 1 , \\dots , ( n _ 0 + 1 ) , \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} s ( x ; a ) : = \\phi ( x ; a ) - | x | , \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} \\int _ \\sigma t ^ k d \\mu = \\varphi ( \\chi ^ k ) = ( \\varphi \\otimes ) ( U ^ { \\otimes k } ) = ( ( \\varphi \\otimes \\iota ) U ^ { \\otimes k } ) = \\dim ( ( 1 , U ^ { \\otimes k } ) ) , \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} w & = ( S _ F ) ^ { - 1 } S _ K f , \\\\ f & = \\lambda ^ { - 1 } S w . \\end{align*}"}
-{"id": "10122.png", "formula": "\\begin{align*} C \\ : : \\ : ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q - z ^ { p + 2 q } x ^ { - p } = 0 . \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} \\widehat A _ \\sigma \\ , u \\ , = w ^ { - 2 } \\ \\nabla \\cdot ( w ^ 2 \\ , \\nabla u ) . \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ M u ( 4 \\pi \\tau ) ^ { - \\frac { m } { 2 } } e ^ { - l } d \\mu = - \\int _ M u \\square ^ * \\left \\{ ( 4 \\pi \\tau ) ^ { - \\frac { m } { 2 } } e ^ { - l } \\right \\} d \\mu \\geq 0 . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} s _ j = \\rho ( \\alpha ) + i \\frac { 2 \\pi j } { \\log ( p / q ) } . \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} { \\Delta { C } _ { n } = { C } _ n ( 1 ) - C _ n ( 0 ) = \\big ( \\mu _ 1 ( n ) ( \\beta _ n - 1 ) - \\mu _ 0 ( n ) ( \\alpha _ n - 1 ) \\big ) + H ( \\alpha _ n ) - H ( \\beta _ n ) + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu _ 1 ( n ) } } { 1 + 2 ^ { \\mu _ 0 ( n ) } } \\Big ) } . \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} \\dd X _ t = Q X _ t \\ , \\dd t + \\dd M ^ X _ t , \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} \\beta ^ { \\kappa } _ { M , N } ( \\kappa \\ , a , \\kappa \\ , b ) \\overset { { \\rm i n \\ , l a w } } { = } \\beta _ { M , N } ( a , \\ , b ) . \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} \\hat { v } = \\left [ \\begin{matrix} - \\operatorname { I m } ( v ) \\\\ \\operatorname { R e } ( v ) \\\\ \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( \\lambda ^ 2 H ) - ( \\lambda ^ 2 H ) [ | A | ^ 2 - { \\rm R i c } ^ N ( \\xi , \\xi ) ] = 0 , \\\\ A ( { \\rm g r a d } ( \\lambda ^ 2 H ) ) + ( \\lambda ^ 2 H ) [ { \\rm g r a d } H - \\ , ( { \\rm R i c } ^ N \\ , ( \\xi ) ) ^ { \\top } ] = 0 . \\end{cases} \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} G = \\begin{bmatrix} 1 & 0 & \\dots & 0 & 0 \\\\ 0 & 1 & \\dots & 0 & 0 \\\\ \\hdotsfor { 5 } \\\\ a _ { r - 1 , 0 } & a _ { r - 1 , 1 } & \\dots & 1 & 0 \\\\ a _ { r , 0 } & a _ { r , 1 } & \\dots & 0 & 1 \\end{bmatrix} \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} v = \\frac { x _ i y _ i k _ i } { z _ i n } = \\frac { x _ i ' z _ i n y _ i ' z _ i n k _ i } { z _ i n } = x _ i ' y _ i ' z _ i n k _ i \\end{align*}"}
-{"id": "9916.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\lambda \\in \\R } V ^ { \\lambda } ( A ) V ^ { \\lambda } ( A ) : = \\{ v \\in V : a ( t ) v = e ^ { \\lambda t } v : \\forall t \\in \\R \\} . \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} j ( f ) ( \\kappa ) = \\beta = ( \\dot { \\beta } _ f ) _ U \\in N \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} \\lambda _ 0 : = \\left ( \\frac { 1 } { \\lambda } \\lim _ { t \\to \\infty } \\frac { 1 } { t ^ { d _ u + 1 } } \\sum _ { k = s ( \\epsilon ) } ^ { t - 1 } { k } ^ { d _ u } \\right ) \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} & \\nabla _ x q ( s , t ; y ( s , t ) , \\xi ) \\partial _ x ^ \\alpha \\nabla _ x y ( s , t ) \\\\ & = - \\sum _ { 0 \\lneq \\alpha ^ \\prime \\leq \\alpha } \\binom { \\alpha } { \\alpha ^ \\prime } \\partial _ x ^ { \\alpha ^ \\prime } \\left [ \\nabla _ x q ( s , t ; y ( s , t ) , \\xi ) \\right ] \\partial _ x ^ { \\alpha - \\alpha ^ \\prime } \\nabla _ x y ( s , t ) . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} a _ 0 ( r ) = O \\left ( \\frac { 1 } { r ^ 2 ( \\log r ) ^ { \\alpha } } \\right ) \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} d _ { g ( t ) } ( x _ 0 , y _ 0 ) + C ' = d _ { g ( N \\theta _ 0 ) } ( x _ 0 , y _ 0 ) + C ' \\leq 4 0 ^ { N } \\left \\{ d _ { g ( 0 ) } ( x _ 0 , y _ 0 ) + C ' \\right \\} . \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} d _ k ( x ) = \\sum _ { i = 0 } ^ { k - 3 } ( i + 1 ) x ^ i + \\sum _ { i = k - 2 } ^ { 2 k - 3 } ( 2 k + 2 - i ) x ^ i . \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} U ^ + _ 0 ( x , t ) = \\sum \\limits _ { n = 1 } ^ { N - 1 } t ^ { - n / 2 } e ^ { - \\eta ^ 2 } [ P _ { n - 1 } ( \\eta ) + \\widetilde { P } _ { n - 1 } ( \\eta ) \\ln t ] + V _ { 0 } ( \\mu , \\eta , t ) + O ( \\sigma ^ { - \\rho N } ) . \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} - \\sum _ { a = 0 } ^ N \\frac { ( - 2 \\pi \\ell ^ 2 t ) ^ a } { a ! } \\sum _ { b \\geq 0 } \\frac { ( 2 \\pi i \\ell z ) ^ b } { b ! } \\frac { B _ { 2 a + b + 1 } \\left ( \\frac { d } { \\ell } \\right ) } { 2 a + b + 1 } . \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} W _ 1 = \\{ 2 m _ 0 k | k \\in \\mathbb Z , k \\neq 0 \\} , ~ ~ U _ 1 = \\{ 2 m _ 0 k + 1 | k \\in \\mathbb Z \\} . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} H _ { ( x y ) } ( i , \\alpha ) ( j , \\beta ) = T _ { ( x y ) } ( i , \\alpha ) \\oplus F _ n ( T _ { ( x y ) } ( i , \\alpha ) ) \\oplus F _ n ( T _ { ( x y ) } ( j , \\beta ) ) \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} & H ( W _ { d _ 1 \\sim d _ { s _ 1 } } , W _ { d _ { l + 1 } \\sim d _ { l + s _ 2 } } | Y _ { 1 \\sim l } , U _ { l + 1 \\sim N _ T } , V _ { 1 \\sim s _ 1 } , V _ { l + 1 \\sim l + s _ 2 } , W _ { d _ { s _ 1 + 1 } \\sim d _ { l } } , W _ { d _ { l + s _ 2 + 1 } \\sim d _ { N _ R } } , W _ { 1 \\sim L } \\setminus W _ \\mathbf { d } ) \\\\ = & F \\varepsilon _ F + T \\varepsilon _ P \\log P , \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} \\lim _ { n \\to + \\infty } \\lambda _ 1 ^ { 1 / { p _ n } } ( B _ { \\rho _ n } ^ { \\pi / 4 } ; { p _ n } ) = \\frac { 1 } { r _ 2 } \\lim _ { n \\to + \\infty } \\lambda _ 1 ^ { 1 / { p _ n } } ( B _ 1 ^ { \\pi / 4 } \\setminus \\overline { B _ { \\rho _ n } ^ { \\pi / 4 } } ; { p _ n } ) = \\frac { 1 } { r _ 2 } . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} \\mu _ A ( C _ 0 ( U _ i ) ) \\cdot A & = \\mu _ A ( C _ 0 ( U _ i ) ) \\cdot \\left ( \\mu _ B ( C _ 0 ( X ) ) \\cdot B \\right ) \\\\ & = \\mu _ B \\left ( C _ 0 ( X ) \\right ) \\cdot \\left ( \\mu _ { B } \\left ( C _ 0 ( V _ { y _ i } ) \\right ) \\cdot B \\right ) \\\\ & = \\mu _ { B } \\left ( C _ 0 ( X ) \\right ) \\cdot C _ 0 ( V _ { y _ i } , C ) \\\\ & = C _ 0 ( U _ i , C ) , \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} r ^ { * , \\pi } _ n ( x _ n | y ^ { n - 1 } _ { n - M } , y _ n ) = \\Big ( \\frac { q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) } { \\nu ^ { \\pi } _ { n } ( y _ n | y ^ { n - 1 } _ { n - J } ) } \\Big ) { \\pi } _ { n } ( x _ n | y ^ { n - 1 } _ { n - J } ) \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} T _ \\mu = \\frac { \\sum _ { n = 0 } ^ { \\mu - 1 } \\gamma ^ { n } \\delta _ { y _ { \\mu - n } , + 1 } } { \\sum _ { n = 0 } ^ { k - 1 } \\gamma ^ { n } } , \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} \\pi _ f ( x ; \\ell , a ) = \\# \\{ \\textup { $ p \\leq x $ : $ p \\nmid \\ell N $ , $ a _ f ( p ) \\equiv a \\pmod \\ell $ , $ \\ell $ s p l i t s i n $ \\mathbb { Q } ( ( a _ f ( p ) ^ 2 - 4 p ^ { k - 1 } ) ^ { 1 / 2 } ) $ } \\} . \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} \\int _ { \\Omega } v \\nabla _ H \\zeta \\cdot \\zeta = & \\frac { 1 } { 2 } \\int _ { \\Omega } v _ 1 \\partial _ x \\zeta ^ 2 + v _ 2 \\partial _ y \\zeta ^ 2 = - \\frac { 1 } { 2 } \\int _ { \\Omega } \\left ( \\partial _ x v _ 1 + \\partial _ y v _ 2 \\right ) \\zeta ^ 2 = - \\frac { 1 } { 2 } \\int _ { \\Omega } ( \\div _ H v ) \\zeta ^ 2 , \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} \\varphi = \\log \\tilde { u } , w = \\tfrac { 1 } { 2 } | D \\varphi | ^ 2 , \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{align*} \\alpha ( G [ ( \\cup _ { i = 1 } ^ { j - 1 } V _ i ) \\cup \\{ v \\} ] ) = \\max _ { \\substack { v ' \\in V _ { j - 1 } \\cap N ( v ) \\\\ v '' \\in V _ { j - 1 } - N ( v ) } } \\{ \\alpha ( G [ ( \\cup _ { i = 1 } ^ { j - 2 } V _ i ) \\cup \\{ v ' \\} ] ) , \\alpha ( G [ ( \\cup _ { i = 1 } ^ { j - 2 } V _ i ) \\cup \\{ v '' \\} ] ) + 1 \\} . \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} K = \\{ ( x , y , z ) : 0 < x < 1 , \\ ; 0 < y < 1 , \\ ; 0 < z < 1 , \\ ; x + y + z < 1 \\} \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{gather*} \\exp \\bigg ( \\sum _ \\alpha \\partial ^ \\alpha \\otimes x _ \\alpha \\bigg ) ( \\cdot \\otimes \\blacktriangleright _ \\gg ) ( 1 \\otimes 1 ) = \\exp \\bigg ( \\sum _ \\alpha \\partial ^ \\alpha \\otimes \\hat { x } _ \\alpha \\bigg ) , \\\\ \\exp \\bigg ( \\sum _ \\alpha \\partial ^ \\alpha \\otimes \\hat { x } _ \\alpha \\bigg ) ( \\cdot \\otimes \\triangleright ) ( 1 \\otimes 1 ) = \\exp \\bigg ( \\sum _ \\alpha \\partial ^ \\alpha \\otimes x _ \\alpha \\bigg ) . \\end{gather*}"}
-{"id": "3187.png", "formula": "\\begin{gather*} \\tau _ { k , \\ell - 1 } ^ { ( \\alpha , \\beta ) } \\tau _ { k , \\ell } ^ { ( \\alpha + 1 , \\beta ) } = \\tau _ { k , \\ell - 1 } ^ { ( \\alpha + 1 , \\beta ) } \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta ) } + \\tau _ { k + 1 , \\ell } ^ { ( \\alpha , \\beta ) } \\tau _ { k - 1 , \\ell - 1 } ^ { ( \\alpha + 1 , \\beta ) } , \\end{gather*}"}
-{"id": "1608.png", "formula": "\\begin{align*} \\frac { 1 - t ^ 6 } { ( 1 - t ) ( 1 - t ^ 2 ) ( 1 - t ^ 3 ) } = 1 + t + 2 t ^ 2 + \\ldots = P _ { C , p } ( t ) \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} h ( \\Omega ) : = \\inf \\frac { | \\partial A | } { | A | } . \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} Z _ { C F T } = Z _ { b u l k } \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} P _ { 0 } ( z ) = e ^ { \\frac { \\lambda } { \\xi } z } ( 1 - z ) ^ { - \\frac { \\gamma } { \\xi } } \\left [ P _ { 0 } ( 0 ) - \\frac { C _ { \\lambda , \\mu } } { \\xi } \\int _ { s = 0 } ^ { z } e ^ { - \\frac { \\lambda } { \\xi } z } ( 1 - s ) ^ { \\frac { \\gamma } { \\xi } - 1 } d s \\right ] , \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} \\Psi ( p , q , X ) = \\frac { \\Phi ( p , r _ 1 ) } { \\Phi ( q , r _ 1 ) } \\cdot \\frac { \\Phi ( p , r _ 2 ) } { \\Phi ( q , r _ 2 ) } \\cdot . . . \\cdot \\frac { \\Phi ( p , r _ l ) } { \\Phi ( q , r _ l ) } \\times \\Psi ( p , q , X \\backslash \\{ r _ 1 , . . . , r _ l \\} ) ; \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} P _ { \\mu \\lambda } \\left ( \\frac { \\partial \\overset { \\ast } { \\left . Q ^ { \\lambda \\nu } \\right . } } { \\partial x ^ { \\nu } } = - \\frac { 4 \\pi } { c } j ^ { \\lambda } \\right ) \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\Delta = \\frac { a ^ 2 + b ^ 2 - c ^ 2 } { 2 a b } , t = \\frac { b } { a } . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} & \\tilde { f } ( L _ n , L _ p ) = a _ { s , n , p } L _ { s + n + p } , \\ \\tilde { f } ( L _ n , G _ p ) = b _ { s , n , p } G _ { s + n + p } , \\ \\tilde { f } ( G _ n , G _ p ) = c _ { s , n , p } L _ { s + n + p } , \\\\ & \\hat { f } ( 1 , L _ p ) = a _ { s , p } ' L _ { s + p } \\textrm { a n d } \\hat { f } ( 1 , G _ p ) = b _ { s , p } ' G _ { s + p } . \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} \\begin{aligned} d _ Y ( G ( y , x ) , G ( v , u ) ) \\leq a \\ d _ Y ( y , G ( v , u ) ) + & \\ b \\ d _ Y ( v , G ( y , x ) ) + c \\ d _ Y ( y , v ) ; \\\\ & \\forall x \\leq _ { P _ 1 } u , \\ y \\geq _ { P 2 } v ; \\ 2 a + c < 1 \\end{aligned} \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} \\mathcal { Z } ^ { ( m , r ) } : = \\mathcal { Z } ^ { m } t ^ { r } \\mapsto \\mathcal { Z } ^ { T ^ { 0 } _ { \\gamma } ( m , r ) } : = \\mathcal { Z } ^ { ( m , r ) } t ^ { Z ( \\gamma , m ) } . \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} S _ { n } \\left ( - q ^ { - n + 1 / 2 } ; q \\right ) = \\frac { \\left ( - 1 \\right ) ^ { n } q ^ { - \\left ( n ^ { 2 } - n \\right ) / 4 } } { \\left ( q ^ { 1 / 2 } ; q ^ { 1 / 2 } \\right ) _ { n } } \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} F ^ { \\perp } : ( X _ 0 + Y _ 0 ) + ( X _ 0 + Y _ 0 ) & = \\big ( ( X _ 0 + Y _ 0 ) \\circ F \\big ) ^ { \\perp } + ( X _ 0 + Y _ 0 ) \\\\ & = \\left ( x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } + y _ 0 ^ { b _ 0 - 1 } y _ 1 ^ { b _ 1 } \\cdots y _ m ^ { b _ m } \\right ) ^ { \\perp } + ( X _ 0 + Y _ 0 ) \\\\ & = \\left ( x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } + y _ 0 ^ { b _ 0 - 1 } y _ 1 ^ { b _ 1 } \\cdots y _ m ^ { b _ m } \\right ) ^ { \\perp } + ( Y _ 0 ) , \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\left \\| \\lambda ^ { k + 1 } - \\lambda ^ k \\right \\| ^ 2 \\leq \\frac { 3 L ^ 2 } { \\sigma _ N } \\left \\| x _ N ^ k - x _ N ^ { k + 1 } \\right \\| ^ 2 + \\frac { 6 L ^ 2 } { \\sigma _ N } \\left \\| x _ N ^ { k - 1 } - x _ N ^ k \\right \\| ^ 2 + \\frac { 3 L ^ 2 } { \\sigma _ N } \\sum _ { i = 1 } ^ { N - 1 } \\left \\| x _ i ^ k - x _ i ^ { k + 1 } \\right \\| ^ 2 . \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} \\mu ( P _ 1 P _ 2 P _ 4 P _ 3 ) = \\mu ( Q _ 1 Q _ 2 Q _ 4 Q _ 3 ) . \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} u ( x ) = \\left ( \\log \\frac { | x | } { \\rho } + \\frac { 1 } { 2 \\rho \\sigma _ 0 } \\right ) ^ { \\frac 1 2 } \\ , f ( x ) \\ , . \\end{align*}"}
-{"id": "9994.png", "formula": "\\begin{align*} \\norm { f _ R } _ { H ^ { ( n + 1 ) / 2 } } ^ 2 = \\norm { f _ 0 } _ { H ^ { ( n + 1 ) / 2 } } ^ 2 + o ( 1 ) = n ! \\omega _ n + o ( 1 ) \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} Q ^ { \\mathcal { X } } _ \\omega : = \\bigcup _ { n \\in \\omega } Q ^ { \\mathcal { X } } _ n \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} & \\lim _ { a \\to 0 , \\ ; b / \\sqrt { a } \\to k } Q _ n \\left ( \\frac { x } { a } \\right ) \\\\ & = ( - 1 ) ^ n \\Gamma ( \\mu + \\nu + 1 + n ) \\sum _ { j = 0 } ^ n \\frac { ( - n ) _ j } { j ! } \\frac { x ^ { \\mu + j } } { \\Gamma ( \\mu + \\nu + 1 + j ) \\Gamma ( \\mu + 1 + j ) } k ^ { 2 j } \\\\ & = ( - 1 ) ^ n \\frac { ( \\mu + \\nu + 1 ) _ n } { \\Gamma ( \\mu + 1 ) } { \\ ; } _ 1 F _ 2 \\left ( { - n \\atop \\mu + \\nu + 1 , \\mu + 1 } \\Big { | } k ^ 2 x \\right ) x ^ \\mu , \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} a \\ast b : = \\sum _ { i \\geq 0 } { \\Delta _ a \\choose i } a _ { ( i - 1 ) } b \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} \\begin{array} { c } h \\gamma ^ { \\lambda } h ^ { - 1 } C = h \\cup _ { \\kappa < \\lambda } \\gamma ^ { \\kappa } h ^ { - 1 } C = \\cup _ { \\kappa < \\lambda } ( h \\gamma ^ { \\kappa } h ^ { - 1 } C ) \\subset \\cup _ { \\kappa < \\lambda } e C = e C \\ , . \\end{array} \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} ( \\tilde { B } \\Sigma _ \\theta , \\Phi ) & : = b ( { \\Sigma } _ \\theta , \\Phi ) , \\\\ ( \\tilde { A } \\Sigma _ \\theta , \\Phi ) & : = a ( \\Sigma _ \\theta , \\Phi ) , \\\\ ( \\tilde { f } , \\Phi ) & : = \\left ( f ( t ) , \\Phi \\right ) + c \\left ( \\theta , \\Phi \\right ) , \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} | \\nabla F | ^ 2 ( x ) = g ^ 2 | x | ^ { 2 g - 2 } , ( \\Delta F ) ( x ) = ( m _ { - } - m _ { + } ) g ^ 2 | x | ^ { g - 2 } / 2 , \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} a _ 0 \\Delta _ { h _ n } \\Delta _ { 2 h _ { n - 1 } } \\cdots \\Delta _ { ( n - 1 ) h _ 2 } \\Delta _ { n h _ 1 } ( f ) ( x ) = 0 h _ 1 , \\cdots , h _ n \\in B _ { d } ( \\delta / 2 ^ n ) . \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} | \\nabla _ x \\eta ( t , s ) | & = | ( \\nabla _ \\xi p ( t , s ; x , \\eta ( t , s ) ) ) ^ { - 1 } \\nabla _ x p ( t , s ; x , \\eta ( t , s ) ) | \\\\ & \\leq C | \\nabla _ x p ( t , s ; x , \\eta ( t , s ) ) | \\\\ & \\leq C \\rho ^ { \\varepsilon _ 0 } \\langle s \\rangle ^ { - 1 - \\varepsilon _ 1 } , \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} \\delta ^ { ''' } _ { \\mathsf { A c h } } ( \\mu , r ) = \\mu \\delta _ { \\mathsf { C a - Z F } } + ( 1 - \\mu ) \\delta _ { \\mathsf { C l - S f } } \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} y _ 1 = \\iota x _ 1 , y _ 2 = - x _ 2 , z _ 2 = \\sigma ^ { - 1 } ( \\Delta + \\iota I ) x _ 2 , \\iota = \\pm \\sqrt { - 1 } . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} \\mathbf { H } ( t , \\cdot ) \\in L ^ { \\infty } \\big ( G , \\mathcal { S } _ { \\geq \\kappa } ( \\mathbb { R } ^ { k \\times d } ) \\big ) t \\in [ 0 , T ] \\kappa : = \\alpha \\exp ( - T / \\tau ) . \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} G _ E ^ L ( x , y ) = & L ^ { - 2 } \\sum _ { \\xi ' \\in L ^ { - 1 } \\Z ^ 2 } \\frac { e _ { \\xi ' } ( x - y ) } { | \\xi ' | ^ 2 - E } \\\\ = & \\sum _ { \\xi \\in \\Z ^ 2 } \\frac { e _ { \\xi } ( ( x - y ) / L ) } { | \\xi | ^ 2 - E L ^ 2 } \\\\ = & G _ \\lambda \\left ( \\frac { x } { L } , \\frac { y } { L } \\right ) , \\lambda = E L ^ 2 . \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} C _ { t } ( 0 ) & = \\mu _ 0 ( t ) ( \\alpha _ { t } - 1 ) + C _ { t + 1 } ( 0 ) + \\log ( 1 + 2 ^ { \\mu _ 0 ( t ) + \\Delta { C } _ { t + 1 } } ) - H ( \\alpha _ { t } ) , ~ C _ { n + 1 } ( 0 ) = 0 , \\\\ ~ C _ { t } ( 1 ) & = \\mu _ 1 ( t ) ( \\beta _ { t } - 1 ) + { C } _ { t + 1 } ( 0 ) + \\log ( 1 + 2 ^ { \\mu _ 1 ( t ) + \\Delta { C } _ { t + 1 } } ) - H ( \\beta _ { t } ) , ~ C _ { n + 1 } ( 1 ) = 0 , ~ t \\in \\{ n , \\ldots , 0 \\} . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} K = \\{ ( x , y , z ) : 0 < x < 1 , \\ ; 0 < y < 1 , \\ ; 0 < z < 1 , \\ ; x + z < 1 , \\ ; y + z < 1 \\} \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} & \\psi _ { x x x } - \\psi _ x - \\lambda ( \\beta _ { x x } - \\beta ) \\psi = 0 , \\\\ & \\lambda \\psi _ t + \\psi _ { x x } + \\lambda \\beta \\psi _ x - ( 1 + \\lambda \\beta _ x ) \\psi = 0 \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} D = \\frac { { { w } ^ { H } } B w } { { { w } ^ { H } } A w } , \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} \\nu _ 0 = e _ n , b _ n ( 0 ) = b _ { n + 1 } ( 0 ) = 1 \\ ( a ( 0 ) = 2 / 3 ) . \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\tilde v ( x ) = \\frac { \\tilde u ( | x | ) } { \\sqrt { 4 \\pi } | x | } \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} \\sigma _ p ( a ) _ { ( n ) } \\sigma _ q ( b ) : = \\sigma _ { p + q - n + 1 } ( a _ { ( n ) } b ) n \\geq 0 . \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} \\sum \\limits _ { m = k } ^ { K - 1 } x _ { m , \\tau _ i } ^ { I M } \\leq \\sum \\limits _ { m = k } ^ { K - 1 } x _ { m , \\tau _ i } ^ { B } , k = 1 , . . . , K - 1 . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} T ^ { - 1 } \\{ y \\} = \\emptyset . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + { \\rm c o n s t } ) } \\ , { \\bf E } \\bigl [ M ^ q _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\bigr ] , \\ ; N \\rightarrow \\infty . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} F \\left ( x \\right ) = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } u _ { \\delta } \\prod _ { i = 1 } ^ { n } f _ { i } ^ { \\delta _ { i } } ( \\alpha _ { i } ) \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} H ^ 0 ( Y , R ^ i f _ * ( K _ X \\otimes F \\otimes \\mathcal J ( h ) \\otimes N _ 1 ) ) \\to H ^ 0 ( Y , R ^ i f _ * ( K _ X \\otimes F \\otimes \\mathcal J ( h ) \\otimes N _ 1 \\otimes N _ 2 ) ) \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} & \\sum _ { m = 1 } ^ M H \\left ( S _ { m , n } \\right ) \\leq M \\mu L , ~ ~ \\forall n \\in [ 1 : N ] , \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} & \\phi \\circ \\psi ( X , Y ) = \\phi ( \\psi ( X ) , \\gamma ^ r ( Y ) , ) - ( - 1 ) ^ { | X | | Y | } \\phi ( \\psi ( Y ) , \\gamma ^ r ( X ) , ) \\ \\forall X , Y \\in K _ { 0 } \\cup K _ { 1 } \\\\ \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} u _ { t } & = D _ { 1 } \\Delta _ { p } u + | u | ^ { m - 2 } u - | u | ^ { q - 2 } u + f _ { 1 } ( u , v ) \\qquad \\Omega \\times ( 0 , T ) \\\\ v _ { t } & = D _ { 2 } \\Delta _ { p } v + | v | ^ { m - 2 } v - | v | ^ { q - 2 } v + f _ { 2 } ( u , v ) \\qquad \\Omega \\times ( 0 , T ) \\\\ \\partial _ { n } u & = \\partial _ { n } v = 0 \\qquad \\partial \\Omega \\times ( 0 , T ) \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\frac { d } { d t } \\lambda _ { p , 1 } ( t ) = \\frac { d } { d t } \\lambda _ { p , 1 } ( u ( t ) , t ) = - \\frac { \\partial } { \\partial t } \\int _ M u ( t , x ) \\Delta _ p u ( t , x ) d \\mu _ { g ( t ) } , \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} \\dim M _ { 1 2 } ( \\Gamma _ 0 ( N ) ) + g ( \\Gamma _ 0 ( N ) ) - 1 = \\left [ S L _ 2 ( \\mathbb { Z } ) : \\Gamma _ 0 ( N ) \\right ] . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} ^ { C \\ ! } D _ { 0 + } ^ \\alpha x ( t ) = A x ( t ) , t \\in \\R _ + , \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} f = ( f _ i ) _ { i \\ge 1 } \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} G _ v = \\langle { a , b , c } \\mid { a ^ 2 = b ^ 2 = c ^ 2 = 1 , ~ c = a b } \\rangle \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} u ( a _ { m , j } , c ) \\geq v _ { \\frac { 1 + \\epsilon } { 2 ( c - \\epsilon ) } } + O ( 1 ) = : v _ l + O ( 1 ) , \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} \\frac { b _ { p , q } ( x _ n ) - b _ { p , q } ( x ) } { x _ n - x } & = \\frac { \\pi _ q ( 0 ^ m 1 b _ { n , m + 2 } b _ { n , m + 3 } \\cdots ) } { \\pi _ p ( 0 ^ m 1 b _ { n , m + 2 } b _ { n , m + 3 } \\cdots ) } \\ge \\left ( \\frac { p } { q } \\right ) ^ m \\frac { \\pi _ q ( 1 0 ^ { \\infty } ) } { \\pi _ p ( 1 ^ { \\infty } ) } \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} g ^ \\varepsilon : = \\det ( g ^ \\varepsilon _ { i j } ) . \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} \\| \\tilde { g } _ 1 \\| _ { C ^ { 0 , \\gamma } ( \\mathcal { N } _ { 0 } \\cap ( B _ { 1 } \\setminus B _ { 1 / 4 } ) ) } \\leq C \\lambda ^ { \\frac { 1 } { 2 } - \\alpha } \\lambda ^ { 1 - \\frac { n + 1 } { p } } \\lambda ^ { - \\frac { 1 } { 2 } } = C \\lambda ^ { 1 - \\alpha - \\frac { n + 1 } { p } } . \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} a = \\sum _ { i = 1 } ^ n \\lambda ^ { n - i } \\omega _ { [ i ] } , b = \\sum _ { i = 1 } ^ n \\lambda ^ { n - i } v _ { [ i ] } , I = \\sqrt { - 1 } i = 1 , \\dots , n , \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} x _ i ^ { k + 1 } & = \\Pi _ { K _ i } [ x _ i ^ k - \\alpha _ k F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) ] , \\\\ v _ i ^ { k + 1 } & = \\hat v _ i ^ k + h _ i ( x _ i ^ { k + 1 } ) - h _ i ( x _ i ^ k ) , \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} \\langle \\gamma , \\ t r \\omega _ R \\rangle - \\langle \\tilde { \\gamma } , \\ t r \\omega _ R \\rangle & = \\int _ { \\gamma } t r \\omega - \\int _ { \\tilde { \\gamma } } t r \\omega _ R \\\\ & = \\int _ { \\Delta } d [ t r \\omega _ R ] = 0 . \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{gather*} \\alpha _ { i } = \\delta _ { i } - \\delta _ { i - 1 } , i = 1 , 2 , \\dots , n - 1 , \\end{gather*}"}
-{"id": "9663.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } A _ { q } \\left ( a b q ^ { - \\alpha - 1 } \\right ) = \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( a b q ^ { - 1 / 2 } e ^ { - i x } ; q \\right ) _ { \\infty } \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i \\alpha x \\right ) } { \\sqrt { \\pi \\log q ^ { - 2 } } } d x , \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} n : = \\ell _ 1 \\cdot \\ell _ 2 \\cdot \\ldots \\cdot \\ell _ J \\cdot P , \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} H _ { ( 2 x ) } ( i , \\alpha ) ( j , \\beta ) = H _ x ( i , j ) \\otimes \\Lambda ( \\alpha , \\beta ) \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} H _ \\omega = - \\Delta + \\alpha \\sum _ { \\omega _ \\xi \\in \\omega } \\delta ( x - \\xi - \\omega _ \\xi ) , \\alpha \\in \\R \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} \\mathcal { B } ^ + _ I = \\left \\{ \\begin{aligned} & \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { A } ^ + \\textnormal { s u c h t h a t } \\\\ & \\inf _ { i \\in \\left \\{ 1 , \\dots , s , s + 1 , \\dots , s + k \\right \\} \\backslash \\left \\{ i _ { k + 1 } \\right \\} } \\left | \\left ( x _ { i _ { k + 1 } } ^ \\prime - x _ i ^ \\prime \\right ) - \\tau \\left ( v _ { i _ { k + 1 } } ^ \\prime - v _ i ^ \\prime \\right ) \\right | \\leq y \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} A ^ { \\pm } _ { 2 1 } ( x ^ q ) B _ { 1 1 } + A _ { 2 2 } B ^ { \\pm } _ { 2 1 } ( x ) = B ^ { \\pm } _ { 2 1 } ( x ^ p ) A _ { 1 1 } + B _ { 2 2 } A ^ { \\pm } _ { 2 1 } ( x ) . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} y ^ { 2 } = ( x - a _ { 1 } ) \\cdot ( x - a _ { 2 } ) \\cdot \\ldots \\cdot ( x - a _ { 2 g + 1 } ) ( = : f ( x ) ) \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} ( J _ a J _ a ^ * - I ) u [ x ] & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - \\varphi _ a ( y , \\xi ) ) } u \\left [ y \\right ] d \\xi - u [ x ] \\\\ & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( x - y ) \\cdot \\eta } \\left ( \\left | \\det \\left ( \\frac { d \\xi } { d \\eta } \\right ) \\right | - 1 \\right ) u \\left [ y \\right ] d \\eta . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} \\nabla _ v \\Phi = \\lambda ^ 4 ( T ( v ) ) = e ^ 0 \\wedge ( \\imath _ { T ( v ) } \\ast \\varphi ) - ( T ( v ) ) ^ \\flat \\wedge \\varphi \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} d _ { \\phi } ( a ) = \\sum _ { j \\leq k } \\phi _ a ( j ) . \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} F ( s ) + \\frac { \\delta ( \\chi ) } { s - 1 } = \\sum _ { | 1 + i \\tau - \\rho | < 1 / 2 } \\frac { 1 } { s - \\rho } + G ( s ) \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} r ^ * ( X ) = r ( E - X ) + | | X | | _ r - r ( E ) . \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} S = \\sum z _ { 1 , 2 } ^ { ( i _ 1 ) } z _ { 3 , 4 } ^ { ( i _ 3 ) } \\cdots z _ { K - 1 , K } ^ { ( i _ { K - 1 } ) } , \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} \\left . \\begin{aligned} \\bar { r } ^ 1 ( s , a ^ 1 ) & = \\frac { r ^ 1 ( s , a ^ 1 , g ) } { | | \\mu | | + \\alpha } , \\ \\forall \\ s \\in S , a ^ 1 \\in A ^ 1 ( s ) , \\\\ p ^ 1 ( s ' | s , a ^ 1 ) & = \\frac { \\mu ( s ' , s , a ^ 1 , g ) } { | | \\mu | | } + \\delta ( s , s ' ) , \\ \\forall \\ s , s ' \\in S , a ^ 1 \\in A ^ 1 ( s ) , \\\\ \\beta & = \\frac { | | \\mu | | } { \\alpha + | | \\mu | | } , \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\tilde { t } ^ j _ { k , \\varphi } ( \\omega , ( - 1 ) ^ k \\xi _ n ) = t ^ j _ { k , \\varphi } ( x , \\xi ) , \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} F ( s ) = \\frac { L ' } { L } ( s , \\chi ^ * ) = - \\sum _ { \\mathfrak { n } } \\chi ^ * ( \\mathfrak { n } ) \\Lambda _ K ( \\mathfrak { n } ) ( \\N \\mathfrak { n } ) ^ { - s } \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} q _ t ( d y _ t | x _ t , y _ { t - 1 } ) = \\bordermatrix { & 0 , 0 & 0 , 1 & 1 , 0 & 1 , 1 \\cr 0 & \\alpha _ t & \\beta _ t & \\gamma _ t & \\delta _ t \\cr 1 & 1 - \\alpha _ t & 1 - \\beta _ t & 1 - \\gamma _ t & 1 - \\delta _ t \\cr } , ~ \\alpha _ t , \\beta _ t , \\gamma _ t , \\delta _ t \\in [ 0 , 1 ] , \\alpha _ t \\neq { \\gamma _ t } , \\beta _ t \\neq { \\delta _ t } . \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} P ( \\{ ( 1 , 2 ) , ( 1 , 3 ) \\} \\subset A ( \\mathbf { D } ) ) \\geq P ( ( 1 , 2 ) \\in A ( \\mathbf { D } ) ) P ( ( 1 , 3 ) \\in A ( \\mathbf { D } ) ) = P ( ( 1 , 2 ) \\in A ( \\mathbf { D } ) ) ^ 2 . \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} & \\| u \\| _ { L ^ \\infty ( ( 0 , T ) , H ^ 1 ) } ^ 2 + \\| u \\| _ { L ^ 2 ( ( 0 , T ) , H ^ 3 ) } ^ 2 + \\| u \\| _ { L ^ 4 ( ( 0 , T ) , L ^ 4 ) } ^ 4 \\\\ & \\quad \\le C ( 1 + T e ^ { \\omega T } ) \\left ( \\| u _ 0 \\| _ { H ^ 1 } ^ 2 + \\| f \\| _ { L ^ 2 ( ( 0 , T ) , L ^ 2 ) } ^ 2 \\right ) . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} a \\circ m = \\sum _ { i \\geq 0 } \\begin{pmatrix} \\Delta _ a \\\\ i \\end{pmatrix} a _ { ( i - 2 ) } m \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} \\Theta ( t _ 2 , T ^ * ) = \\tfrac { 1 } { 2 } \\Theta ( t _ 1 , T ^ * ) . \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} \\left \\| \\begin{pmatrix} \\sqrt { \\sigma _ 1 } u \\\\ \\vdots \\\\ \\sqrt { \\sigma _ d } u \\end{pmatrix} \\right \\| ^ 2 \\geq \\epsilon \\| u \\| ^ 2 . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} \\langle { a , b , t , u } \\mid { a ^ 2 = b ^ 2 = ( a b ) ^ 2 = 1 } , ~ a t b t ^ { - 1 } = t b t ^ { - 1 } a = u a b u ^ { - 1 } \\rangle . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} P _ t f = P ^ S _ t f _ { \\mathfrak { L } } + P _ t ^ D f _ { \\mathfrak { L } ^ \\perp } . \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\begin{cases} P u = F _ p ( u ) , ( t , x ) \\in M \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} \\log \\| \\Lambda \\| ( D ) = \\log \\| \\theta \\| ( D + P - Q ) - g ( P , Q ) - g ( D , Q ) - g ( \\sigma ( D ) , P ) \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\begin{aligned} & f _ N ^ { ( s ) } ( t ) = \\sum _ { k = 0 } ^ { N - s } a _ { N , k , s } \\times \\\\ & \\ ; \\ ; \\times \\int _ 0 ^ t \\int _ 0 ^ { t _ 1 } \\dots \\int _ 0 ^ { t _ { k - 1 } } T _ s ( t - t _ 1 ) C _ { s + 1 } \\dots T _ { s + k } ( t _ k ) f _ N ^ { ( s + k ) } ( 0 ) d t _ k \\dots d t _ 1 \\end{aligned} \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} H _ { ( x y ) } ( i , \\alpha ) ( j , \\beta ) = H _ x ( i , j ) \\otimes H _ y ( \\alpha , \\alpha ) = H _ x ( i , j ) \\otimes T _ y ( \\alpha ) \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} | x | = \\begin{cases} \\infty & \\\\ n & . \\end{cases} \\end{align*}"}
-{"id": "10102.png", "formula": "\\begin{align*} H _ { \\alpha } = \\frac { y ^ 2 + a x ^ 2 + \\alpha x z + c z ^ 2 } { x z ^ 3 } \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\lambda _ { n } = \\begin{cases} 0 & n = 5 N \\mod 6 \\\\ 7 & n = 5 N + 1 \\mod 6 \\\\ 6 & n = 5 N + 2 \\mod 6 \\\\ 3 & n = 5 N + 3 \\mod 6 \\\\ 4 & n = 5 N + 4 \\mod 6 \\\\ 1 1 & n = 5 N + 5 \\mod 6 \\end{cases} \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} G _ i = \\{ | x _ 0 : x _ 1 : x _ 2 | \\in C : x _ i = 0 , ~ x _ { i + 1 } , x _ { i + 2 } \\neq 0 \\} . \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} 0 \\equiv \\frac { \\partial ^ { 2 } Q ^ { \\mu \\nu } } { \\partial x ^ { \\mu } \\partial x ^ { \\nu } } = - \\frac { 4 \\pi } { c } \\frac { \\partial j ^ { \\mu } } { \\partial x ^ { \\mu } } , \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} A ^ G ( i , j ) = \\begin{cases} 1 & t ( a _ i ) = s ( a _ j ) , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & 0 & 0 & \\cdots & 0 \\\\ 1 & 0 & 0 & \\cdots & 0 \\\\ 1 & 1 & 0 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & 1 & 1 & \\cdots & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} K _ { N } = \\left \\{ \\overset { n - 2 } { \\underset { 1 = \\alpha < \\beta } { \\sum } } \\left \\langle R ^ { \\bot } ( X _ { 1 } , X _ { 2 } ) N _ { \\alpha } , N _ { \\beta } \\right \\rangle ^ { 2 } \\right \\} ^ { 1 / 2 } . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} \\frac { 2 } { 9 } d ( d - 3 ) = t _ { 3 } . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} | \\{ X \\in V _ { n , k , b } : \\ , \\underline { X } = j \\} | = \\binom { b } { k - 1 } . \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} g _ 1 d g _ k = b . \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} \\tau ( 1 _ { [ 0 , \\mu ( r , T ) ] } ( | T | ) T ) = \\int _ { r ' } ^ 1 \\mu ( t , T ) \\ , d t , \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } + V _ s \\cdot \\nabla _ { X _ s } \\right ) f _ \\infty ^ { ( s ) } ( t , Z _ s ) = \\ell ^ { - 1 } C _ { s + 1 } ^ 0 f _ \\infty ^ { ( s + 1 ) } ( t , Z _ s ) \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} c _ { \\sigma } ^ { + } ( M ( \\chi ^ { ( r , \\varphi ) } ) ) _ { \\varphi , 1 } \\sim \\delta _ { \\sigma } ( M ) _ { \\varphi } P _ { \\sigma } ( \\chi ^ { ( r , \\varphi ) } ) _ { 1 } \\prod _ { i = 1 } ^ { j } Q _ { i , \\sigma , \\varphi } . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} P _ { \\mathbf { x } } ( D ) = \\prod _ { ( i , j ) \\in A } \\phi ( x _ i , x _ j ) \\times \\prod _ { ( i , j ) \\notin A } ( 1 - \\phi ( x _ i , x _ j ) ) . \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} C _ { p , g + 1 } ^ { ( \\ell ) } & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = 0 } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } \\sum _ { \\gamma = 0 } ^ { p - j - 2 t } 2 ^ { \\gamma } \\binom { \\ell } { \\gamma } \\binom { n - \\ell } { p - j - 2 t - \\gamma } \\binom { p - j - 2 t - \\gamma } { p - j - t - g - 1 } . \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} \\mathcal { S } _ { \\geq \\kappa } ( \\mathbb { R } ^ { k \\times d } ) : = \\big \\{ \\mathbf { H } \\in \\mathcal { S } ( \\mathbb { R } ^ { k \\times d } ) \\ , | \\ , \\lambda _ { \\min } ( \\mathbf { H } ) \\geq \\kappa \\big \\} , \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} F ( x , u ) = \\dfrac { g ( x ) | u | ^ { p + 1 } } { p + 1 } , \\quad \\mbox { w h e r e } \\ ; \\ , g \\in L ^ 1 ( \\Omega ) \\ ; \\ , \\mbox { a n d } \\ ; \\ , g > 0 \\ ; \\ , \\mbox { i n } \\ , \\ , \\Omega . \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} R ^ + ( x ; t ) & = \\frac { \\sqrt { 2 x - 1 } } { 2 \\sqrt { x } p ( x ) } \\left ( p ^ 2 ( x ) F _ 2 ( u ( x , t ) ) + F _ 1 ( u ( x , t ) ) \\right ) , \\\\ R ^ - ( x ; t ) & = \\frac { - \\sqrt { 2 x - 1 } } { 2 p ( x ) } ( p ^ 2 ( x ) F _ 2 ( u ( x , t ) ) - F _ 1 ( u ( x , t ) ) ) . \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} - L u + g \\circ u & = \\mu \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega \\\\ u & = \\nu \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\tau = \\frac { N _ T } { \\frac { N _ T } { N _ T + N _ R - 1 } } \\frac { 1 - \\mu _ R } { N _ T } = \\frac { N _ T + N _ R - 1 } { N _ T } ( 1 - \\mu _ R ) . \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x } \\ , E _ { \\alpha , \\beta } ^ \\gamma \\left ( t { \\rm e } ^ { { \\rm i } x } \\right ) \\Big | _ { x = 0 } = { \\rm i } \\ , t \\ , \\left ( E _ { \\alpha , \\beta } ^ \\gamma ( t ) \\right ) ' \\ , . \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\sum _ { f \\in M _ { n } } \\alpha _ { 2 } ( f ) = \\frac { q ^ { n / 2 } } { \\zeta _ { q } ( 3 ) } \\sum _ { \\substack { j = n \\mod 2 \\\\ 0 \\leq j \\leq \\lfloor \\dfrac { n } { 3 } \\rfloor - 2 } } q ^ { \\frac { - j } { 2 } } + q ^ { \\lfloor { \\frac { n - \\lfloor n / 3 \\rfloor } { 2 } \\rfloor } } \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} S _ 2 ( l , u , v ; N ) = S _ 2 ( l , u , v ; N / p ) = 0 . \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{gather*} \\Delta _ H ( h ) = \\mathcal { F } _ l ^ { - 1 } \\Delta _ 0 ( h ) \\mathcal { F } _ l = \\mathcal { F } _ r ^ { - 1 } \\Delta _ 0 ( h ) \\mathcal { F } _ r . \\end{gather*}"}
-{"id": "9970.png", "formula": "\\begin{align*} \\abs { P [ a , b ] } = d ( a , x _ 1 ) + d ( x _ 1 , x _ 2 ) + d ( x _ 2 , b ) \\leq d ( a , u _ 1 ) + d ( u _ 1 , u _ 2 ) + d ( u _ 2 , b ) + 1 2 N \\leq d ( a , b ) + 1 2 N . \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} H ^ r ( X _ { m , n , h } , \\Q _ \\ell ) = \\begin{cases} \\Q _ \\ell ( - r / 2 ) & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} \\pi ( T ^ { - 1 } ( x , \\xi ) ) = x ; \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} { T ^ 1 _ 1 u _ 1 } _ { | S } + { T ^ 1 _ 2 u _ 2 } _ { | S } = g _ 1 , { T ^ 2 _ 1 u _ 1 } _ { | S } + { T ^ 2 _ 2 u _ 2 } _ { | S } = g _ 2 , \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{gather*} \\int \\limits _ { [ 0 , \\ , 1 ] ^ l } \\prod _ { i = 1 } ^ l s _ i ^ { \\lambda _ 1 } ( 1 - s _ i ) ^ { \\lambda _ 2 } \\ , \\prod \\limits _ { i < j } ^ l | s _ i - s _ j | ^ { - 2 / \\tau } d s _ 1 \\cdots d s _ l = \\\\ \\\\ \\prod _ { k = 0 } ^ { l - 1 } \\frac { \\Gamma ( 1 - ( k + 1 ) / \\tau ) \\Gamma ( 1 + \\lambda _ 1 - k / \\tau ) \\Gamma ( 1 + \\lambda _ 2 - k / \\tau ) } { \\Gamma ( 1 - 1 / \\tau ) \\Gamma ( 2 + \\lambda _ 1 + \\lambda _ 2 - ( l + k - 1 ) / \\tau ) } , \\end{gather*}"}
-{"id": "5116.png", "formula": "\\begin{align*} \\int _ { \\vert x \\vert \\leq R } u ^ { 1 + p } ( t , \\cdot ) = B _ N R ^ { N } \\int _ { \\vert x \\vert \\leq R } \\frac { 1 } { B _ N R ^ { N } } u ^ { 1 + p } ( t , \\cdot ) \\geq \\frac { 1 } { B _ N ^ { p } R ^ { N p } } m ^ { 1 + p } ( t ) , \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} A _ N ^ { - 1 } \\begin{pmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ p _ 1 ^ { n _ 1 - 1 } \\cdots p _ s ^ { s _ 1 - 1 } ( p _ 1 + 1 ) \\cdots ( p _ s + 1 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} \\partial e ^ { t \\Delta } f & = \\sum _ { j = 1 } ^ { + \\infty } \\langle e ^ { t \\Delta } f , \\phi _ j \\rangle _ \\mathcal { H } \\partial \\phi _ j \\\\ & = \\sum _ { j = 1 } ^ { + \\infty } e ^ { - \\lambda _ j t } \\langle f , \\phi _ j \\rangle _ \\mathcal { H } \\partial \\phi _ j \\\\ & = - \\sum _ { j = 1 } ^ { + \\infty } \\frac { 1 } { \\lambda _ j } e ^ { - \\lambda _ j t } \\langle \\partial f , \\partial \\phi _ j \\rangle _ \\mathcal { H } \\partial \\phi _ j \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} u '' + a ( t ) \\biggl { ( } \\dfrac { u ^ { \\gamma } } { 1 - u ^ { \\sigma } } \\biggr { ) } = 0 , \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} c \\phi ' + L \\phi + N ( \\phi ) = 0 . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} H _ { i j } = \\langle \\lambda p _ { i - 1 } ( \\lambda ) , p _ { j - 1 } ( \\lambda ) \\rangle , G _ { i j } = \\langle p _ { i - 1 } ( \\lambda ) , p _ { j - 1 } ( \\lambda ) \\rangle , g _ j = \\langle p _ { n - 1 } ( \\lambda ) , p _ { j - 1 } ( \\lambda ) \\rangle , \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} \\mu ( \\bigcup _ { i = 1 } ^ N T ^ { k _ i } A ) > 1 - \\epsilon . \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} P _ { n } \\left ( x \\right ) = Q _ { n } \\left ( x \\right ) + \\sum \\nolimits _ { i = 1 } ^ { d l } a _ { n , i } Q _ { n - i } \\left ( x \\right ) , \\ \\ n \\geq d l , \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} \\begin{cases} [ ( i \\xi ) ^ 3 + \\lambda ] \\hat { \\varphi } ( \\xi ) + \\dfrac { a b } { c } ( i \\xi ) ^ 3 \\hat { \\psi } ( \\xi ) = \\varphi '' ( 0 ) + \\dfrac { a b } { c } \\psi '' ( 0 ) - \\left ( \\varphi '' ( L ) + \\dfrac { a b } { c } \\psi '' ( L ) \\right ) e ^ { - i L \\xi } , \\\\ \\dfrac { 1 } { c } [ ( i \\xi ) ^ 3 + r ( i \\xi ) + c \\lambda ] \\hat { \\psi } ( \\xi ) + a ( i \\xi ) ^ 3 \\hat { \\varphi } ( \\xi ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} U \\cap U ' = H \\cap U ' \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} s = \\frac { u ( a + 1 ) ^ 2 + v } { 2 u ( a + 1 ) } \\end{align*}"}
-{"id": "10166.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow + \\infty } ( \\mathbb E [ Z _ n ^ * ] ) ^ { - 1 } \\mathbb E \\left [ ( Z _ { n + \\lfloor \\varepsilon n \\rfloor } ^ * - Z _ { n + \\lfloor \\varepsilon n \\rfloor } ) \\mathbf 1 _ { \\{ \\sup _ { k = 1 , \\ldots , \\lfloor \\varepsilon n \\rfloor } ( Z _ { k } - Z _ { k + 1 } ) > b _ { d n } \\} } \\right ] < \\rho - 1 , \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\varphi \\otimes \\tilde { \\theta } _ n ) J _ n ( a ) = \\omega _ \\xi ( a ) \\ , \\varphi ( p _ \\xi ) . \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { m } \\sum _ { b = 1 } ^ { \\alpha _ j + 1 } \\frac { ( - i ) ^ { b - 1 } L _ { \\alpha _ { j + 1 } , b } } { b ! } \\ , x ^ { b - 1 } \\frac { \\sin b x } { \\sin ^ { b } x } \\ = \\ \\prod _ { j = 1 } ^ { m } \\Bigg ( \\frac { ( - 1 ) ^ { \\alpha _ j } } { ( 2 \\alpha _ j + 1 ) ! ! } \\ , x ^ { 2 \\alpha _ j } + O ( x ^ { 2 \\alpha _ j + 1 } ) \\Bigg ) . \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} w _ k ( \\underline \\pi ) \\equiv \\prod _ { i = 0 } ^ { a - 1 } \\bigg ( \\det \\left [ \\begin{array} { c c c c c c c c c c } 0 & 1 & \\cdots & k - 1 \\\\ 0 & 1 & \\cdots & k - 1 \\end{array} \\right ] _ { \\sigma ^ i ( N ) } \\bigg ) \\bmod \\ { I ^ { \\lambda ' _ k + 1 } } . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} m n - 1 - ( m - 2 ) ( n - h - 1 ) - n - ( h + 2 ) = n + ( m - 3 ) ( h + 1 ) - 2 . \\end{align*}"}
-{"id": "9882.png", "formula": "\\begin{align*} \\lambda _ { m , n } ( c ^ 2 = 0 ) = n ( n + 1 ) \\ , . \\end{align*}"}
-{"id": "10143.png", "formula": "\\begin{align*} X ( M e _ 1 r + M e _ 2 , \\ , M ^ * e _ 3 a ) = X ( M e _ 1 , \\ , M ^ * e _ 3 a r ) X ( M e _ 2 , \\ , M ^ * e _ 3 a ) \\ , \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = F ( \\sigma , \\mu ) + \\xi ( y , \\sigma , \\mu ) + f ( x , y , \\sigma , \\mu ) , \\\\ \\dot { y } & = \\sigma + \\eta ( y , \\sigma , \\mu ) + g ( x , y , \\sigma , \\mu ) \\end{aligned} \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} U ^ * ( \\lambda - V ( r ) ) u ) ( x ) = ( \\lambda - W ( x ) ) v ( x ) . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} 1 & \\ge 1 - g ( \\theta _ 1 , x _ 0 , t , c _ t ) \\ge 1 - g ( \\theta _ 1 , x _ 0 , t , c ) = 2 - \\Phi \\left ( \\frac { e ^ { t ^ c } - x _ 0 e ^ { \\theta _ 1 t } } { \\sqrt { v ( \\theta _ 1 , t ) } } \\right ) - \\Phi \\left ( \\frac { e ^ { t ^ c } + x _ 0 e ^ { \\theta _ 1 t } } { \\sqrt { v ( \\theta _ 1 , t ) } } \\right ) \\\\ & \\to 2 - \\Phi \\left ( - \\frac { x _ 0 \\theta _ 1 ^ H } { \\sqrt { H \\Gamma ( 2 H ) } } \\right ) - \\Phi \\left ( \\frac { x _ 0 \\theta _ 1 ^ H } { \\sqrt { H \\Gamma ( 2 H ) } } \\right ) = 1 . \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{gather*} S \\hat { y } _ \\mu = \\hat { x } _ \\mu , S \\partial ^ \\nu = S _ 0 \\partial ^ \\nu = - \\partial ^ \\nu . \\end{gather*}"}
-{"id": "1317.png", "formula": "\\begin{align*} X ( t ) = \\mu + \\eta \\Theta + \\int _ 0 ^ t b ( s ) X ( s ) \\ , d s + \\int _ 0 ^ t \\tau ( s ) \\ , d V ( s ) , \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} F \\left ( { t , s } \\right ) = t s f \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} e ^ { H [ \\boldsymbol { t } ] } \\psi ( z ) e ^ { - H [ \\boldsymbol { t } ] } & = e ^ { - \\sum _ { q \\geq 1 } t _ q z ^ q } \\psi ( z ) \\\\ e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast ( z ) e ^ { - H [ \\boldsymbol { t } ] } & = e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\psi ^ \\ast ( z ) . \\end{align*}"}
-{"id": "601.png", "formula": "\\begin{align*} \\delta & = 2 ( 1 - p ) [ \\alpha _ 1 - \\alpha _ 2 + \\alpha _ 1 \\alpha _ 3 - \\alpha _ 1 \\alpha _ 2 + \\alpha _ 2 \\alpha _ 3 ] + 4 \\alpha _ 1 p - 2 \\alpha _ 3 + 2 \\\\ \\kappa _ 1 & = \\frac { ( 1 - \\alpha _ 3 ) ( 1 - \\alpha _ 2 ( 1 - p ) ) } { \\delta } \\\\ \\kappa _ 2 & = \\frac { \\alpha _ 1 ( p + \\alpha _ 3 ( 1 - p ) ) } { \\delta } \\\\ \\kappa _ 3 & = \\frac { \\alpha _ 1 ( 1 - \\alpha _ 2 ( 1 - p ) ) } { \\delta } . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} \\phi _ k ( n ) = \\prod _ { p | n \\ ; } ( 1 - p ^ { - k } ) . \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} ( f ( 2 m _ 0 k ) - f ( 2 m _ 0 l ) ) ( f ( 2 m _ 0 k ) + 2 f ( 2 m _ 0 k ) f ( 2 m _ 0 l ) + f ( 2 m _ 0 l ) ) = 0 , \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} D ^ * \\Xi ( x ) ( y ) = \\partial ^ L \\langle y , \\Xi ( x ) \\rangle \\forall y \\in \\Re ^ m . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} \\prod \\nolimits _ { \\nu = 0 } ^ { n } \\gamma _ { \\nu d + r + 1 } ^ 0 = \\frac { \\left \\langle u _ { r } , x ^ { n + 1 } P _ { \\left ( n + 1 \\right ) d + r } \\right \\rangle } { \\left \\langle u _ { r } , P _ { r } \\right \\rangle } , \\ \\ 0 \\leq r \\leq d - 1 . \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} \\Im \\alpha _ 1 ^ 1 = \\tau \\gamma _ 1 - A _ 1 , \\Im \\alpha _ 2 ^ 1 = - \\tau \\gamma _ 2 + A _ 2 . \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} p ^ 4 \\tilde h _ { n } = ( j _ n * \\tilde h ^ { ( 3 ) } ) ' . \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\varphi ^ { ( k ) } ( A ) f = \\lim \\limits _ { R \\to \\infty } \\frac { \\i ( k ! ) } { 2 \\pi } \\int _ { \\C } \\frac { \\partial ( \\tilde { \\varphi \\theta _ R } ) _ N } { \\partial \\overline { z } } ( z ) ( z - A ) ^ { - 1 - k } f d z \\wedge d \\overline { z } , \\ f \\in \\mathcal { D } ( \\langle A \\rangle ^ { \\rho } ) . \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{gather*} \\Delta _ H \\colon \\ H _ \\gg \\to H _ \\gg \\hat \\otimes _ { U ( \\gg ) } H _ \\gg , u \\sharp P \\stackrel { \\Delta _ H } \\mapsto u \\sharp P _ { ( 1 ) } \\otimes _ { U ( \\gg ) } P _ { ( 2 ) } , \\\\ \\Delta _ { \\hat { S } ( \\gg ^ * ) } ( P ) = P _ { ( 1 ) } \\otimes P _ { ( 2 ) } , u \\in U ( \\gg ) , P \\in \\hat { S } ( \\gg ^ * ) . \\end{gather*}"}
-{"id": "7056.png", "formula": "\\begin{align*} \\bigoplus _ { ( i , \\alpha ) } T _ { ( x y ) } ( i , \\alpha ) = \\bigoplus _ { ( i , \\alpha ) } H _ { ( x y ) } ( i , \\alpha ) \\psi ( i , \\alpha ) \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} \\min | a ( t - p ) | ^ 2 + | c ( t ) | ^ 2 & = | \\frac { 1 } { 2 } ( - c ( t ) + a ( t - p ) ) | ^ 2 + | \\frac { 1 } { 2 } ( c ( t ) - a ( t - p ) ) | ^ 2 \\\\ & = \\frac { 1 } { 2 } | ( c ( t ) - a ( t - p ) ) | ^ 2 = \\frac { 1 } { 2 } | f ( t ) - f ( t - p ) - \\delta | ^ 2 , \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\alpha , \\beta , z _ { 2 1 } , z _ { 1 2 } , \\tilde { z } _ { 2 1 } , \\tilde { z } _ { 1 2 } } = \\big \\{ ( X , Z ) : & \\| Z _ { 2 1 } \\| _ F \\leq \\tilde { z } _ { 2 1 } , \\| Z _ { 1 2 } \\| _ F \\leq \\tilde { z } _ { 1 2 } , \\\\ & ( X , Z ) \\in \\mathcal { F } _ { r , \\alpha , \\beta , z _ { 2 1 } , z _ { 1 2 } } \\big \\} . \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} J ^ { \\psi _ n } _ n \\varphi _ a ( Q _ n ( z ) ) = \\varphi \\otimes \\psi _ n ( J _ n ( a ) \\cdot Q _ n ( z ) ) = \\varphi \\otimes \\psi _ n ( Q _ n J _ n ( a ) \\cdot z ) \\ , . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} d Z _ t & = - a Z _ t d t + \\sigma d W _ t , t \\in [ 0 , T ] \\\\ Z _ 0 & = x _ 0 - d ( 0 ) , \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} ( \\psi \\circ \\phi ) ( f ) = f \\circ \\sigma _ A , ( \\phi \\circ \\psi ) ( g ) = g \\circ \\sigma _ B \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} x _ 0 & : = \\{ s \\in S \\ | \\ s = s _ 0 , s _ 1 , \\dots s _ t \\in a _ 1 t s _ t \\in a _ 2 t \\} \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{gather*} V ^ { - 1 } _ { \\mathcal { F } } = \\big ( S _ { \\mathcal { F } } \\overline { \\mathcal { F } } ^ { ( 1 ) } \\big ) \\overline { \\mathcal { F } } ^ { ( 2 ) } , \\end{gather*}"}
-{"id": "6815.png", "formula": "\\begin{align*} & \\delta ^ * ( \\mu , r ) = \\max \\Big ( 1 + \\mu + \\frac { 1 - 2 \\mu } { r } , 2 - \\mu \\Big ) . \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} ( 2 L ( q ^ { 2 } ) - 7 L ( q ^ { 7 } ) ) ^ { 2 } & = \\sum _ { \\delta | 1 4 } x _ { \\delta } M ( q ^ { \\delta } ) + \\sum _ { j = 1 } ^ { 4 } y _ { j } A _ { j } ( q ) \\\\ & = \\sum _ { \\delta | 1 4 } x _ { \\delta } + \\sum _ { i = 1 } ^ { \\infty } \\biggl ( \\ , 2 4 0 \\ , \\sum _ { \\delta | 1 4 } x _ { \\delta } \\sigma _ { 3 } ( \\frac { n } { \\delta } ) + \\sum _ { j = 1 } ^ { 4 } y _ { j } a _ { j } ( n ) \\ , \\biggr ) \\ , q ^ { n } . \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} 1 - \\mu _ R \\le & \\sum _ { t = 1 } ^ { N _ T } \\binom { N _ T } { t } a _ { 0 , t } + \\sum _ { r = 1 } ^ { N _ R - 1 } \\sum _ { t = 1 } ^ { N _ T } \\left [ \\binom { N _ R } { r } - \\binom { N _ R - 1 } { r - 1 } \\right ] \\binom { N _ T } { t } a _ { r , t } \\\\ = & \\sum _ { r = 0 } ^ { N _ R - 1 } \\sum _ { t = 1 } ^ { N _ T } \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } a _ { r , t } . \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\Delta _ g u = \\alpha u \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} \\lambda G - F = \\left [ \\begin{array} { c c c c c c } \\lambda - \\rho _ 1 & \\cdots & 0 & 0 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\vdots & & \\vdots \\\\ 0 & \\cdots & \\lambda - \\rho _ t & 0 & \\cdots & 0 \\\\ 0 & \\cdots & 0 & L _ { \\epsilon _ 1 } ( \\lambda ) & \\cdots & 0 \\\\ \\vdots & & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 & 0 & \\cdots & L _ { \\epsilon _ r } ( \\lambda ) \\end{array} \\right ] \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} \\int _ { t _ 1 } ^ { t _ 2 } \\psi ( t ) d t & = \\int _ { t _ 1 } ^ { t _ 2 } \\Big ( \\frac { 1 } { \\psi ^ { - 1 } _ 0 - \\frac { 2 } { n } t } \\Big ) d t \\\\ \\displaystyle & = - \\frac { n } { 2 } \\log ( \\psi ^ { - 1 } _ 0 - \\frac { 2 } { n } t ) \\Big | _ { t _ 1 } ^ { t _ 2 } = \\log \\Bigg ( \\frac { \\psi ^ { - 1 } _ 0 - \\frac { 2 } { n } t _ 1 } { \\psi ^ { - 1 } _ 0 - \\frac { 2 } { n } t _ 2 } \\Bigg ) ^ { \\frac { n } { 2 } } . \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{gather*} \\xi _ 0 + \\xi _ 1 + \\xi _ 2 + \\xi _ 3 + \\xi _ 4 + \\xi _ 5 = 0 , \\end{gather*}"}
-{"id": "5753.png", "formula": "\\begin{align*} 2 k - 1 > \\lim \\limits _ { n \\to + \\infty } \\mu _ k ^ { 1 / { p _ n } } ( p _ n ) = \\lim \\limits _ { n \\to + \\infty } \\lambda _ 1 ^ { 1 / { p _ n } } ( B _ { r _ { i + 1 } ^ n } \\setminus \\overline { B _ { r _ { i } ^ n } } ; p _ n ) = \\frac { 2 } { r _ { i + 1 } - r _ i } \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} [ 4 ( m - 4 ) y ^ 3 - 3 ( m - 1 ) y ^ 2 - 1 ] \\cdot y ' = y ^ 3 - y ^ 4 . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} G _ t ( w ) : = \\frac { ( w - 1 ) ^ 2 ( w - \\frac 3 2 + i t ) \\zeta ( w - \\frac 1 2 + i t ) } { ( w + 1 ) ^ 2 ( w - \\frac 1 2 + i t - \\rho _ 0 ) ( w + i t + 1 ) ^ 4 } . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} e ^ { \\mu \\nu \\sigma \\tau } e _ { \\mu \\nu \\sigma \\rho } = - 6 \\delta _ { \\rho } ^ { \\tau } , e ^ { \\mu \\nu \\sigma \\tau } e _ { \\mu \\nu \\sigma \\rho } = - 2 4 . \\end{align*}"}
-{"id": "1018.png", "formula": "\\begin{align*} D ( x _ 1 , \\dots , x _ r ) = C \\prod _ { 1 \\leq \\alpha < \\beta \\leq s } ( x _ { i _ \\alpha } - x _ { i _ \\beta } ) \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} f _ \\infty ^ { \\left ( s \\right ) } \\left ( t \\right ) = T _ { s } ^ { 0 } \\left ( t \\right ) f _ \\infty ^ { \\left ( s \\right ) } \\left ( 0 \\right ) + \\ell ^ { - 1 } \\int _ { 0 } ^ { t } T _ { s } ^ { 0 } \\left ( t - \\tau \\right ) C _ { s + 1 } ^ 0 f _ \\infty ^ { \\left ( s + 1 \\right ) } \\left ( \\tau \\right ) d \\tau \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} ( \\sigma _ { 1 } - \\sigma _ { 2 } ) \\cdot ( \\varepsilon _ { 1 } - \\varepsilon _ { 2 } ) \\geq 0 , \\quad \\forall \\ , \\varepsilon _ { i } \\in d o m ( f ) , \\quad \\sigma _ { i } \\in f ( \\varepsilon _ { i } ) , \\ ( i = 1 , 2 ) . \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} H _ { m , n } ( z , \\bar z ) : = ( - 1 ) ^ { m + n } e ^ { | z | ^ 2 } \\frac { \\partial ^ { m + n } } { \\partial z ^ m \\partial \\bar z ^ n } \\left ( e ^ { - | z | ^ 2 } \\right ) . \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} C _ i ^ = \\frac { 1 } { 2 } \\log _ 2 \\left ( 1 + \\xi | h _ i | ^ 2 \\right ) , \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} \\rho _ { \\nu , b } ( x ) = x ^ { \\frac { \\nu } { 2 } } K _ { \\nu } ( 2 b \\sqrt { x } ) , x > 0 . \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} \\widehat { g _ 0 } ( \\eta ) = \\widehat { f _ 0 } ( \\eta \\omega ) \\eta ^ { d - 1 } . \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} \\left ( \\tilde { T } _ s ( t ) g ^ { ( s ) } \\right ) ( Z _ s ) = g ^ { ( s ) } \\left ( \\tilde { \\psi } _ s ^ { - t } Z _ s \\right ) \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} \\mathfrak { L } _ \\nu \\nabla _ { \\nu - 1 } = \\left ( \\nabla _ { \\nu - 1 } \\nabla _ { \\nu - 1 } ^ { * } + \\nu \\right ) \\nabla _ { \\nu - 1 } = \\nabla _ { \\nu - 1 } \\left ( \\mathfrak { L } _ { \\nu - 1 } + ( 2 \\nu - 1 ) \\right ) , \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} \\begin{aligned} h \\big ( c ( y _ k ) + a _ k \\nabla c ( y _ k ) ( v _ k - v _ { k - 1 } ) \\big ) & + a _ k g ( v _ k ) \\le h \\big ( c ( y _ k ) + a _ k \\nabla c ( y _ k ) ( x - v _ { k - 1 } ) \\big ) + a _ k g ( x ) \\\\ & + \\frac { \\tilde { \\mu } _ k a _ k ^ 2 } { 2 } \\left ( \\norm { x - v _ { k - 1 } } ^ 2 - \\norm { x - v _ k } ^ 2 - \\norm { v _ k - v _ { k - 1 } } ^ 2 \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} \\begin{aligned} \\begin{array} { l @ { \\quad } l @ { } r @ { \\quad } l } & \\underset { w } { m i n } \\ , { { w } ^ { H } } A w \\\\ s . t . & { { v } ^ { H } } ( { { \\theta } _ { 0 } } , { { \\phi } _ { 0 } } ) w = 1 \\\\ & \\operatorname { R e } [ { { v } _ { \\theta } } ^ { H } ( { { \\theta } _ { 0 } } , { { \\phi } _ { 0 } } ) w ] = 0 \\\\ & \\operatorname { R e } [ { { v } _ { \\phi } } ^ { H } ( { { \\theta } _ { 0 } } , { { \\phi } _ { 0 } } ) w ] = 0 \\end{array} \\end{aligned} \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} b _ 2 ^ { t r } \\sigma ^ { - 1 } ( d _ 2 ^ { t r } g _ 2 + g _ 2 ^ { t r } d _ 2 ) = 0 . \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} G ( x , y ) = \\rho ( x , y ) y ^ d + \\sum _ { i = 2 } ^ d c _ i ( x ) y ^ { d - i } \\ , , \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} \\big \\langle \\omega _ i , \\omega _ { 1 , \\infty } \\big \\rangle _ { Z _ { 1 , R _ i } } = \\big \\langle \\omega _ i , \\omega _ { 1 , \\infty } \\big \\rangle _ { Z _ { 1 , 0 } } + \\big \\langle \\omega _ i ^ \\mathrm { n z } , \\omega _ { 1 , \\infty } ^ \\mathrm { n z } \\big \\rangle _ { Y _ { [ 0 , R _ i ] } } + \\big \\langle \\omega _ i ^ \\mathrm { z m } , \\omega _ { 1 , \\infty } ^ \\mathrm { z m } \\big \\rangle _ { Y _ { [ 0 , R _ i ] } } . \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} f _ { T _ { 1 } } ( t ) = e ^ { - t } , ~ ~ t > 0 . \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} ( f \\bullet g ) \\bullet h - ( - 1 ) ^ { ( n - 1 ) ( p - 1 ) } \\left ( ( f \\bullet h ) \\bullet g \\right ) = f \\bullet ( g \\bullet h ) - ( - 1 ) ^ { ( n - 1 ) ( p - 1 ) } \\left ( f \\bullet ( h \\bullet g ) \\right ) \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} \\omega _ 1 \\star a & = ( \\iota \\otimes c \\cdot \\omega _ 1 ' ) \\Delta ( a ) = ( \\iota \\otimes \\omega _ 1 ' ) ( \\Delta ( a ) ( 1 \\otimes c ) ) , \\\\ a \\star \\omega _ 2 ^ \\sharp & = ( d \\cdot \\omega _ 2 ' \\otimes \\iota ) \\Delta ( a ) = ( d \\cdot \\omega _ 2 ' \\otimes \\iota ) ( \\Delta ( a ) ( d \\otimes 1 ) ) . \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} \\prod _ { j = 2 } ^ { n } | \\eta _ j - \\eta _ { j - 1 } | ^ { 1 - 2 H } \\leq \\sum _ { \\alpha \\in D _ n } \\prod _ { j = 1 } ^ { n } | \\eta _ j | ^ { \\alpha _ j } . \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} F ( y ) = ( \\alpha - \\gamma y ) ^ g P ( { \\delta y - \\beta } / { \\alpha - \\gamma y } ) \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} \\Delta { C } ^ { \\infty } = & \\big ( \\mu _ 1 ( \\beta - 1 ) - \\mu _ 0 ( \\alpha - 1 ) \\big ) + H ( \\alpha ) - H ( \\beta ) + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu _ 1 + \\Delta { C } ^ { \\infty } } } { 1 + 2 ^ { \\mu _ 0 + \\Delta { C } ^ { \\infty } } } \\Big ) , \\end{align*}"}
-{"id": "9985.png", "formula": "\\begin{align*} w _ { a _ 1 } = \\frac { 1 } { 2 } \\left ( 1 + \\tanh \\frac { a _ 2 - a _ 1 } { 2 } \\right ) , w _ { a _ N } = \\frac { 1 } { 2 } \\left ( 1 + \\tanh \\frac { a _ N - a _ { N - 1 } } { 2 } \\right ) . \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} \\mathfrak { B } _ { 5 } = \\left ( S _ { E } ^ { ( 3 ) } \\times A d S _ { 5 } \\right ) _ { R , 0 _ { S } } , \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} \\phi ( x , y ) \\phi ( y , x ) = p _ a ^ 2 ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( C _ { n } z _ { \\delta , n } ( 0 ) \\right ) : \\psi = \\int _ { \\Omega } \\xi ( 0 ) : \\psi \\qquad \\forall \\psi \\in L _ { s o l , n } ^ { 2 } ( \\Omega ) \\ , . \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} h ( n ) \\le ( 1 - \\epsilon ) \\binom { n - 1 } 2 , \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} P ( f ) : = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\sum _ { C \\in C _ { n } } \\exp \\left ( \\sup _ { x \\in C } S _ { n } f ( x ) \\right ) . \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} y _ q ( u ) = & \\sum _ { \\mathcal { R } : \\mathcal { R } \\ni q , | \\mathcal { R } | = r + 1 } \\sum _ { i = 1 } ^ \\rho \\left [ \\sum _ { p \\in [ N _ T ] } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { [ N _ T ] } , p } ^ i ( u ) \\right ] x _ { { \\mathcal { R } } , { [ N _ T ] } } ^ i \\\\ & + \\sum _ { \\mathcal { R } , \\bar { \\mathcal { R } } _ i : \\mathcal { R } \\cup \\bar { \\mathcal { R } } _ i \\not \\ni q } \\left [ \\sum _ { p \\in [ N _ T ] } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { [ N _ T ] } , p } ^ i ( u ) \\right ] x _ { { \\mathcal { R } } , { [ N _ T ] } } ^ i , \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} S ^ { \\prime } _ { t } = S ^ { 2 } _ { t } + R _ t , \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ n \\ell ^ h \\ , N _ { \\mathcal L _ { \\Gamma } } ( k , \\ell ) = \\sum _ { \\ell = 0 } ^ n \\ell ^ h \\ , N _ { \\mathcal L _ { \\Gamma ' } } ( k , \\ell ) \\quad k \\geq 0 , \\qquad 0 \\leq h \\leq p _ 0 . \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} a _ j ^ 0 ( k + i ) & : = \\begin{cases} g & \\neg x _ j \\in C _ i , \\\\ e & \\end{cases} \\\\ a _ j ^ 1 ( k + i ) & : = \\begin{cases} g & x _ j \\in C _ i , \\\\ e & \\end{cases} \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} n ^ 2 _ k ( x ) - ( 1 - 4 x ) = 4 x ^ 3 d _ k ( x ) . \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} I _ \\alpha = \\int _ { \\alpha / 2 } ^ \\infty r ^ { d - 1 } l ( r ) d r < \\infty . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} f ( t ) = ( 2 \\pi ) ^ N \\int _ { \\R ^ { N } } e ^ { - t } \\left ( \\delta _ 0 + \\sum _ { k = 1 } ^ { + \\infty } \\frac { t ^ { k } } { k ! } J ^ { * ( k ) } ( x ) \\right ) u _ 0 ( x ) d x = ( 2 \\pi ) ^ N \\int _ { \\R ^ { N } } K ( t , x ) u _ 0 ( x ) d x , \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} \\Delta _ k ( L , B ) ( y , \\dots , y , x _ 1 , \\dots , x _ { p - 2 q } ) = \\binom { p } { 2 q } ^ { - 1 } T ^ { ( p - 2 q ) } ( L , B ) ( x _ 1 , \\dots , x _ { p - 2 q } ) . \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} g ( X , X ) = \\delta _ { i j } \\left ( - p ^ i p ^ j - q ^ i q ^ j + r ^ i r ^ j + s ^ i s ^ j \\right ) , \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} \\inf \\ \\{ g _ { k ; \\ast } ( x _ 1 ) + g _ { k ; \\ast } ( x _ 2 ) : ( x _ 1 , x _ 2 ) \\in R \\} & = \\inf \\ \\{ g _ { k ; \\ast } ( x ) + g _ { k ; \\ast } ( \\lambda - x ) : \\lambda \\in \\mathcal { I } , x \\in [ 0 , \\lambda ] \\} \\\\ & \\geqslant \\inf \\ \\{ f _ k ( \\lambda ) + \\lambda : \\lambda \\in \\mathcal { I } \\} \\\\ & \\geqslant \\epsilon + E - \\epsilon ' . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} { \\cal C } _ { 0 , n } \\triangleq \\Big \\{ { \\bf P } _ { Y _ t | Y ^ { t - 1 } , X ^ t } = q _ t ( d y _ t | y ^ { t - 1 } , x ^ { t } ) : t = 0 , 1 , \\ldots , n \\Big \\} . \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} A _ 0 : = \\i \\sum _ { i = 1 } ^ d \\left ( 2 ^ { - 1 } ( S _ i ^ * + S _ i ) - ( S _ i ^ * - S _ i ) N _ i \\right ) = \\i \\sum _ { i = 1 } ^ d 2 ^ { - 1 } \\big ( ( S _ i - S _ i ^ * ) N _ i + N _ i ( S _ i - S _ i ^ * ) \\big ) \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} g ( \\theta _ 0 , x _ 0 , t , 0 ) = \\alpha \\quad g ( \\theta _ 0 , x _ 0 , t , 1 ) = \\alpha . \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} K _ { \\alpha \\beta } { } ^ { \\gamma } = K _ { \\beta \\alpha } { } ^ { \\gamma } . \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} L _ { \\mathrm { A d S } } ^ { \\left ( 5 \\right ) } = \\kappa \\left ( \\frac { 1 } { 5 l ^ { 5 } } \\epsilon _ { a _ { 1 } \\cdots a _ { 5 } } e ^ { a _ { 1 } } \\cdots e ^ { a _ { 5 } } + \\frac { 2 } { 3 l ^ { 3 } } \\epsilon _ { a _ { 1 } \\cdots a _ { 5 } } R ^ { a _ { 1 } a _ { 2 } } e ^ { a _ { 3 } } \\cdots e ^ { a _ { 5 } } + \\frac { 1 } { l } \\epsilon _ { a _ { 1 } \\cdots a _ { 5 } } R ^ { a _ { 1 } a _ { 2 } } R ^ { a _ { 3 } a _ { 4 } } e ^ { a _ { 5 } } \\right ) , \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} D = \\left \\{ [ 1 , z _ 2 , z _ 3 ] \\in \\mathbb { C } P ^ 2 : ~ g _ { z _ 2 } = 0 \\right \\} , \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\alpha _ { \\theta ^ k ( i j ) } ^ * & = y _ i - \\theta ^ { - k } y _ j , x _ i , & \\alpha _ { \\theta ^ k ( i j ) } & = x _ i - \\theta ^ k x _ j . \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} A _ m ^ \\prime { = } \\underbrace { \\frac { ( K { - } m ) ( m { - } 1 ) } { m ( K { - } m { + } 1 ) } } _ { B _ m } A _ { m { + } 1 } ^ \\prime { + } \\underbrace { \\frac { K { - } m } { ( K { - } m { + } 1 ) ( m { + } 1 ) } } _ { C _ m } , \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot { V } ( x ) & = & \\frac { d } { d t } \\left ( - \\ln ( W ( x ) + 1 ) \\right ) \\\\ & = & - \\dfrac { 1 } { ( W ( x ) + 1 ) } \\dot { W } ( x ) \\\\ & = & - \\dfrac { 1 } { ( W ( x ) + 1 ) } ( 1 + W ( x ) ) \\phi ( x ) \\\\ & = & - \\phi ( x ) \\\\ \\end{array} \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{gather*} \\pi \\big ( g ^ { [ k , \\ell ] ( \\alpha , \\beta ) } _ { - } \\big ) = \\big ( \\Sigma \\mathcal { T } _ { k , \\ell } ^ { ( \\alpha , \\beta ) } \\big ) / \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta ) } , \\end{gather*}"}
-{"id": "35.png", "formula": "\\begin{align*} - \\sum _ { m = 0 } ^ { N + \\mu n } T ^ { \\mu } _ { l , m } a _ { \\mu , N } ( m ) = \\sum _ { k = 0 } ^ { N + \\mu n } P _ { l , k } ^ { \\mu } c _ { \\mu } ( k ) , N + 1 \\leq l \\leq N + \\mu n , \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} E ( F _ h ( G ) ) = \\left \\{ \\left ( ( g _ 1 , h - 1 ) ( g _ 2 , h ) \\right ) | \\{ ( g _ 1 , h - 1 ) ( g _ 2 , h ) \\} \\in E ( G ) \\right \\} . \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} Z _ { s , s } ^ 0 \\left [ Z _ s , t \\right ] = \\left ( X _ s - V _ s t , V _ s \\right ) \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} e ( P ) = \\frac { 1 } { 2 } ( | P | + h - \\alpha ( G _ P ) - 2 ) . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { d } } \\int _ { - \\infty } ^ { T } | \\mathcal { T } g ( t , x ) | ^ { 2 } d t d x & = T ^ { 1 + \\frac { \\alpha d } { 2 } } \\int _ { \\mathbb { R } ^ { d } } \\int _ { - \\infty } ^ { 1 } | \\mathcal { T } \\tilde { g } ( t , x ) | ^ { 2 } d t d x \\\\ & \\leq N T ^ { 1 + \\frac { \\alpha d } { 2 } } \\int _ { \\mathbb { R } ^ { d } } \\int _ { - \\infty } ^ { 1 } | \\tilde { g } ( t , x ) | ^ { 2 } d t d x \\\\ & = N \\int _ { \\mathbb { R } ^ { d } } \\int _ { - \\infty } ^ { T } | g ( t , x ) | ^ { 2 } d t d x . \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } E _ \\infty ^ i ( C ^ { \\infty } ( S ^ 3 \\times S ^ { 6 } \\times S ^ { 8 } ) \\rtimes _ \\beta \\mathbb { Z } ) \\cong \\mathbb { C } , & i = 1 , 3 , 7 , 9 \\\\ E _ \\infty ^ i ( S ^ 3 \\times S ^ { 6 } \\times S ^ { 8 } ) \\rtimes _ \\beta \\mathbb { Z } ) \\cong \\{ 0 \\} , & i = \\ e l s e . \\end{array} \\right . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} \\mathbf { Q } _ { s } = \\frac { \\mathbf { \\tilde { Q } } _ { s } } { \\xi } , ~ \\mathbf { V } = \\frac { \\mathbf { \\tilde { V } } } { \\xi } , ~ \\rho _ { s , k } = \\frac { \\tilde { \\rho } _ { s , k } } { \\xi } , \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} v ^ 2 = 1 + \\sigma ^ { i j } u _ i u _ j , \\\\ \\tilde { v } ^ 2 = 1 + \\dot { \\varphi } ^ 2 \\sigma ^ { i j } u _ i u _ j . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} C ^ { F B , A . 1 } = \\sum _ { y \\in \\{ 0 , e , 1 \\} } \\bigg ( \\sum _ { x \\in \\{ 0 , 1 \\} , z \\in \\{ 0 , e , 1 \\} } \\log \\Big ( \\frac { q ( z | y , x ) } { \\nu ^ { * , \\infty } ( z | y ) } \\Big ) q ( z | y , x ) \\pi ^ { * , \\infty } ( x | y ) \\bigg ) \\nu ^ { { \\pi ^ { * , \\infty } } } ( y ) . \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} q t \\frac { D _ \\lambda \\bar D _ \\lambda - 1 } { S _ { q , t } } = \\sum _ { s \\in \\lambda } q ^ { - a ( s ) } t ^ { l ( s ) + 1 } + t ^ { - l ( s ) } q ^ { a ( s ) + 1 } , \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} e ^ { i t H } E _ \\pm ( t ) - P _ \\pm = E _ \\pm ( 0 ) - P _ \\pm + i \\int _ 0 ^ t e ^ { i \\tau H } G _ \\pm ( \\tau ) d \\tau . \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} L ( M ( \\chi ) , s ) _ { 1 } = \\prod _ { j } L ( M ( \\chi ) _ { K _ { j } } , s ) _ { 1 } ^ { n _ { j } } . \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} B _ { i j } x ^ i x ^ j = & [ C _ { i j k , k } - A _ { k l } W _ { k i j l } ] x ^ i x ^ j = [ ( A _ { i j , k } - A _ { i k , j } ) _ { , k } - A _ { k l } W _ { k i j l } ] x ^ i x ^ j \\\\ = & [ \\Delta A _ { i j } - A _ { i k , j k } + O ( r ) ] x ^ i x ^ j \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( \\tfrac { 1 } { 2 } + \\nu , - \\tfrac { 1 } { 2 } ; \\tfrac { 1 } { 2 } ; 1 ) = \\frac { \\Gamma ( \\tfrac { 1 } { 2 } ) \\Gamma ( \\tfrac { 1 } { 2 } - \\nu ) } { \\Gamma ( - \\nu ) } , \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P } } ^ * ( \\mu , r ) = \\begin{cases} ~ \\dfrac { 1 - 2 \\mu } { r } , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mu \\in [ 0 , \\mu _ 1 = ( 1 - r ) / ( 2 + r ) ] \\\\ ~ 1 , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mu \\in [ \\mu _ 2 = ( 1 - r ) , 1 ] . \\end{cases} \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} \\xi ^ l : = \\delta ^ l \\alpha ^ l \\gamma ^ l = \\left ( \\frac { \\delta ^ 1 } { l } \\right ) \\left ( \\frac { 2 } { l + 1 } \\right ) \\left ( \\frac { l ( l + 1 ) } { 2 \\delta ^ 1 } \\right ) = 1 . \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} \\tau = \\sum _ { k \\in \\omega } \\omega ^ { \\beta _ k } \\cdot p _ k \\le \\omega ^ \\alpha . \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 3 } \\ [ \\bar { x } ^ { \\pm } _ { i , r _ { \\pi ( 1 ) } } , [ \\bar { x } ^ { \\pm } _ { i , r _ { \\pi ( 2 ) } } , [ \\bar { x } ^ { \\pm } _ { i , r _ { \\pi ( 3 ) } } , \\bar { x } ^ { \\pm } _ { i + 1 , s } ] ] ] = 0 . \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} P ( \\varphi ) = \\sup \\left \\{ h _ { \\eta } + \\eta ( \\varphi ) : \\eta \\in \\mathcal { M } _ { \\theta } ( X ) \\right \\} = h _ { \\mu } + \\mu ( \\varphi ) , \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} C _ p = \\left ( \\int _ G \\varrho ( y ) \\cdot \\vartheta ^ q ( y ) \\ , d \\mu ( y ) , \\right ) ^ { 1 / q } , \\frac { 1 } { p } + \\frac { 1 } { q } = 1 . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} \\xi _ { q } ( z ) = \\chi _ { q } ( z ) + z ^ { - 1 } \\chi _ { q } \\left ( z ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} c _ n ( 0 ) = \\Big ( 2 \\sin \\frac { \\pi } { 4 n + 2 } \\Big ) ^ { - 1 } \\ , . \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} f _ m : = \\frac { 4 ^ { m } - 1 } { m } B _ { 2 m } \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t q ( t , s ) = \\nabla _ { \\xi } h _ \\rho ( t , p ( t , s ) , q ( t , s ) ) , \\\\ \\partial _ t p ( t , s ) = - \\nabla _ x h _ \\rho ( t , p ( t , s ) , q ( t , s ) ) , \\\\ ( q , p ) ( s , s ) = ( x , \\xi ) . \\end{array} \\right . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } + V _ s \\cdot \\nabla _ { X _ s } \\right ) f _ N ^ { ( s ) } ( t , Z _ s ) = ( N - s ) \\varepsilon ^ { d - 1 } C _ { s + 1 } f _ N ^ { ( s + 1 ) } ( t , Z _ s ) \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} ( f \\bullet _ j h ) \\bullet _ { i + p - 1 } g = ( f \\bullet _ { i + p - 1 } g ) \\bullet _ { j + n - 1 } h 1 \\leq i + p - 1 \\leq j - 1 \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} \\eta _ { M , N } ( q + a _ i \\ , | \\ , a , \\ , b ) = \\eta _ { M , N } ( q \\ , | \\ , a , \\ , b ) \\ , \\exp \\bigl ( - ( \\mathcal { S } _ N \\log \\Gamma _ { M - 1 } ) ( q \\ , | \\ , \\hat { a } _ i , b ) \\bigr ) . \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} w ( t ) = \\Big ( h _ t ^ { W ^ \\bullet } \\big ( P ( t ) w , P ( t ) w \\big ) \\Big ) ^ { - 1 / 2 } P ( t ) w \\in W ^ \\bullet . \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} s _ { \\mu - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } } ( z ) = \\frac { z ^ { \\mu + \\frac { 1 } { 2 } } } { \\mu ( \\mu + 1 ) } \\prod \\limits _ { n \\geq 1 } \\left ( 1 - \\frac { z ^ { 2 } } { \\xi _ { \\mu , n } ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} h ( z ) ^ { q - 1 } = - \\Delta ( z ) . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} \\lambda = \\lim _ { z \\to c } f ( z ) \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} c _ \\pi = \\pm 1 . \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { r _ 1 } \\left ( L _ i ^ \\prime \\right ) ^ d & = \\sum _ { i = 1 } ^ { r _ 1 } \\big ( \\alpha _ { i , 0 } ( x _ 0 - a ' _ 1 y _ 1 + \\ldots - a ' _ m y _ m ) + \\alpha _ { i , 1 } x _ 1 \\ldots + \\alpha _ { i , n } x _ n \\big ) ^ d \\\\ & = ( x _ 0 - a ' _ 1 y _ 1 + \\ldots - a ' _ m y _ m ) x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } . \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} \\mathcal { L } v _ i = v _ i - \\int _ { \\mathcal { V } } w _ i \\ , { \\rm d } \\mu ( w ) = v _ i \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k + 1 } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau , 0 ; v _ { s + 1 } , \\dots , v _ { s + k } , v _ { s + k + 1 } ; \\right . \\\\ & \\left . \\qquad \\omega _ 1 , \\dots , \\omega _ k , \\omega _ { k + 1 } ; i _ 1 , \\dots , i _ k , i _ { k + 1 } \\right ] \\\\ & \\in \\mathcal { G } _ { s + k + 1 } \\cap \\hat { \\mathcal { U } } _ { s + k + 1 } ^ \\eta \\end{aligned} \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} \\int _ { \\frac { i } { k } } ^ { \\frac { i + 1 } { k } } r | u _ { \\varepsilon , i } ' | ^ p \\ , d r = k ^ { p - 2 } \\int _ 0 ^ 1 ( s + i ) | v _ \\varepsilon ' | ^ p \\ , d s = \\frac { k ^ { p - 2 } } { \\varepsilon ^ { p - 1 } } ( 2 i + 1 ) \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} K _ H = \\prod _ { v \\nmid \\infty } K _ { H , v } K _ H ^ S = \\prod _ { v \\notin S \\cup \\infty } K _ { H , v } , \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} \\int _ X F [ X ] G [ X ^ * ] = ( F [ X ] , G [ X ] ) _ X . \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} \\mathcal { A } \\rho ( x ) = g ( x ) \\mbox { i n } \\Omega . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} \\begin{aligned} & d z _ { \\hat i } = A ^ { ( j ) } d x _ { i } + A ^ { ( j ' ) } d x _ { i ' } , \\\\ & d z _ i = d x _ i , \\end{aligned} \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} g _ \\varphi ( C r _ \\varphi ( x , y ) , z ) = \\varphi ( x , y , z ) , g _ \\varphi ( \\chi _ \\varphi ( x , y , z ) , w ) = * \\varphi ( x , y , z , w ) . \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} a + d x = a \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} M _ { ( a , s ) , ( b , t ) } = p ( a , b | s , t ) a \\in A , b \\in B , s \\in S , t \\in T , \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} \\left ( \\frac { t } { 2 } \\ , ( \\varphi ^ 2 ) ' + \\varphi ^ 2 + 1 \\right ) ^ 2 - 4 c t ^ 2 ( \\varphi ^ 2 + 1 ) = a ^ 2 ( \\varphi ^ 2 + 1 ) . \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} \\min \\ \\to \\ 0 . 5 \\tau \\left \\| \\sum \\limits _ { i = 1 } ^ { n } A _ { i } \\mathbf { x } _ { i } - \\mathbf { b } _ { 0 } \\right \\| ^ { 2 } - \\sum \\limits _ { i = 1 } ^ { n } \\langle \\mathbf { c } _ { i } , \\mathbf { x } _ { i } \\rangle \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\eta ^ x = \\sum _ { j = - \\infty } ^ { 1 } \\eta ^ x _ { j } \\lambda ^ { j } d x \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; \\eta ^ y = \\sum _ { j = - 1 } ^ { \\infty } \\eta ^ y _ { j } \\lambda ^ { j } d y , \\end{align*}"}
-{"id": "9946.png", "formula": "\\begin{align*} \\sigma ( \\vect { \\delta } - j \\vect { e } _ { i } ) = \\sigma ( \\vect { \\delta } ) + j ( k + 1 ) \\geq \\sigma ( \\vect { \\delta } ) - ( k + 1 ) \\delta _ { i } \\geq \\sigma ( \\vect { \\delta } ) ( 1 - ( k + 1 ) / k ) \\geq 0 . \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} s _ { m , \\mu } ( x ) & = ( - i ) ^ { m - 1 } x ^ { ( m - 1 ) / 2 } h _ { m - 1 , \\mu + 1 } ( ( 2 \\sqrt { x } i ) ^ { - 1 } ) \\\\ & = ( - 1 ) ^ { m - 1 } ( \\mu + 1 ) _ { m - 1 } { \\ ; } _ 2 F _ 3 \\left ( { - ( m - 1 ) / 2 , - ( m - 2 ) / 2 \\atop \\mu + 1 , - m + 1 , 1 - m - \\mu } \\Big { | } 4 x \\right ) \\\\ & = ( - 1 ) ^ { m + 1 } \\sum _ { j = 0 } ^ { \\lfloor ( m - 1 ) / 2 \\rfloor } \\binom { m - j - 1 } { j } ( \\mu + j + 1 ) _ { m - 2 j - 1 } x ^ { j } . \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} K _ W & : = 2 ^ { - 1 } W \\sum _ { i = 1 } ^ d ( S _ i ^ * + S _ i ) + 2 ^ { - 1 } \\sum _ { i = 1 } ^ d ( S _ i ^ * + S _ i ) W , \\\\ B _ W & : = \\sum _ { i = 1 } ^ d U _ i \\tilde { W } ( S _ i ^ * - S _ i ) - \\sum _ { i = 1 } ^ d ( S _ i ^ * - S _ i ) \\tilde { W } U _ i , \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} t \\log ( \\mu ( A ) ) + ( 1 - t ) \\log ( \\mu ( B ) ) = \\log ( \\ell ) + ( 1 - t ) \\ln ( 2 ) \\geq \\log ( \\ell ) = \\log ( \\mu ( [ A , B ] _ t ) ) . \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} a \\nabla _ { \\nu } b \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ n } } , 2 ) ^ { - r _ n } a \\sharp _ { \\nu } b + ( \\sqrt { a } - \\sqrt { b } ) ^ 2 - \\sum _ { k = 0 } ^ { n - 1 } r _ { k } \\big [ \\big ( a ^ { \\frac { m _ k } { 2 ^ k } } b ^ { 1 - \\frac { m _ k } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } - \\big ( a ^ { \\frac { m _ k + 1 } { 2 ^ k } } b ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ] ^ { 2 } , \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} \\lambda \\{ x \\in X \\ ; \\ \\theta ( x ) \\geq \\varepsilon \\} & \\leq a ^ * \\lambda \\{ ( g , x ' ) \\in U ^ 2 \\times \\Sigma \\ ; \\ \\theta ( g x ' ) \\geq \\varepsilon \\} \\\\ & \\leq a ^ * \\lambda \\{ ( g , x ' ) \\in U ^ 2 \\times \\Sigma \\ ; \\ \\theta _ r ' ( g x ' ) \\geq \\varepsilon \\} \\\\ & = \\int _ { \\theta _ r ' \\geq \\varepsilon } d \\lambda ( x ) \\kappa ( x ) \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} A _ { j , \\theta } : = \\Big \\{ x \\in D _ 1 \\ , : \\ , | u _ j ( x ) - u ( x ) | < \\theta \\Big \\} . \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} & a _ { 1 _ \\circ } \\otimes ( a _ { 2 _ \\circ } ) _ { 1 _ \\cdot } \\otimes ( a _ { 2 _ \\circ } ) _ { 2 _ \\cdot } = ( a _ { 1 \\cdot } ) _ { 1 _ \\circ } S ( a _ { 2 \\cdot } ) ( a _ { 3 \\cdot } ) _ { 1 _ \\circ } \\otimes ( a _ { 1 \\cdot } ) _ { 2 _ { \\circ } } \\otimes ( a _ { 3 \\cdot } ) _ { 2 _ \\circ } \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} \\psi _ i ( \\vect { x } _ 1 ) ) = \\psi _ i ( \\vect { x } _ 2 ) , i = 1 , \\dots , K \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\sum _ { i \\in [ 2 ] } w ^ { ( i , l ) } _ { 0 , 0 } = 0 , \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} Q _ i = N _ t \\int _ { i / N _ t } ^ { ( i + 1 ) / N _ t } b _ i ( \\xi ) \\psi _ { n , m } ( \\xi ) d \\xi \\cong \\psi _ { n , m } \\bigg ( \\frac { 2 i + 1 } { 2 N _ t } \\bigg ) , \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\pi ^ { - 1 } ( s _ i ) = \\hat s _ i e \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} \\lambda _ { p , 1 } = \\inf _ { 0 \\neq f \\in W ^ { 1 , p } ( M ) } \\Bigg \\{ \\frac { \\int _ M | \\nabla f | _ g ^ p \\ d \\mu _ g } { \\int _ M | f | ^ p _ g \\ d \\mu _ g } \\ \\ \\Big | \\ \\ f \\neq 0 , \\ \\ f \\in W ^ { 1 , p } ( M ) \\Bigg \\} , \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} \\partial _ { \\nu } S ( \\nu , \\varepsilon ) & = \\left ( \\partial _ { \\nu } \\left [ \\bar { K } ( \\nu , \\varepsilon ) - \\bar { K } ( \\nu , 0 ) \\right ] \\left [ Z ^ { \\perp } ( \\nu , \\varepsilon ) + \\mathbf { I } \\right ] r _ { 0 } , r _ { 0 } \\right ) \\\\ & + \\left ( \\left [ \\bar { K } ( \\nu , \\varepsilon ) - \\bar { K } ( \\nu , 0 ) \\right ] \\partial _ { \\nu } Z ^ { \\perp } ( \\nu , \\varepsilon ) r _ { 0 } , r _ { 0 } \\right ) . \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { c ( u ^ n ) } \\exp _ 2 \\{ ( \\lambda + 1 ) \\log c _ k ( x ^ n | u ^ n ) \\} . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} ( \\delta \\times \\delta ) _ ! \\Delta _ ! F ( U _ { I } \\subseteq M , V _ { J } \\subseteq M ) & \\overset { ( \\ref { l e m : d i a g o n a l p u s h f o r m u l a 1 } ) } { \\simeq } \\begin{cases} \\Delta _ ! F ( U _ I \\subseteq M , V _ J \\subseteq M ) & | I | = | J | = 1 \\\\ 0 & \\end{cases} \\\\ & \\overset { ( \\ref { l e m : d i a g o n a l p u s h f o r m u l a 2 } ) } { \\simeq } \\begin{cases} F ( U _ I \\subseteq M ) & U _ I = V _ J , \\ , | I | = | J | = 1 \\\\ 0 & \\quad \\end{cases} \\end{align*}"}
-{"id": "9881.png", "formula": "\\begin{align*} z = ( n + m + 1 ) ( n - m ) \\ , , \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} W _ { \\nu } ( z ) = \\prod \\limits _ { n \\geq 1 } \\left ( 1 + \\frac { z } { d _ { \\nu , n } } \\right ) , \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} F P S ( l / k ) = z _ { l , 0 } + 2 \\sum \\limits _ { q = 1 } ^ { \\frac { l } { 2 } - 1 } { z _ { l , q } \\cos ( ( q ) \\frac { { 2 k \\pi } } { l } ) + } 2 \\cos ( ( \\frac { l } { 2 } ) \\frac { { 2 k \\pi } } { l } ) \\sum \\limits _ { t = 0 } ^ { \\frac { l } { 2 } - 1 } { y _ t y _ { t + \\frac { l } { 2 } } } \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} \\int _ { \\mathcal { V } } \\phi ( v ) \\mathcal { L } \\phi ( v ) \\ , { \\rm d } \\mu ( v ) = \\int _ { \\mathcal { V } } | \\phi ( v ) | ^ 2 \\ , { \\rm d } \\mu ( v ) - \\iint _ { \\mathcal { V } \\times \\mathcal { V } } \\phi ( v ) \\phi ( w ) \\ , { \\rm d } \\mu ( w ) \\ , { \\rm d } \\mu ( v ) . \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} & [ L _ n + z , L _ m + z ' ] = ( \\{ m \\} - \\{ n \\} ) L _ { n + m } + c \\frac { 1 } { q ^ { n - 2 } } \\frac { 1 + q ^ { 2 } } { 1 + q ^ { n } } \\frac { \\{ n + 1 \\} \\{ n \\} \\{ n - 1 \\} } { \\{ 1 2 \\} } \\delta _ { n + m , 0 } , \\forall n > 0 \\\\ & [ L _ n + z , G _ m + z ' ] = ( \\{ m + 1 \\} - \\{ n \\} ) G _ { n + m } \\\\ & [ G _ n + z , G _ m + z ' ] = 0 . \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} P ^ j _ U : = i _ * i ^ * ( P ^ j ) , \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} B _ j ( x _ k ) = \\left \\{ \\begin{aligned} & 1 , & \\textrm { i f } \\ \\ k = j , \\\\ & \\frac { s - p h } { 2 ( p h c - s ) } , & \\textrm { i f } \\ \\ k = j \\pm 1 , \\\\ & 0 , & \\textrm { i f } \\ \\ k = j \\pm 2 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\bar { b } _ 0 = | a | - \\sum _ { j = 0 } ^ { M - 1 } b _ j > 0 . \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} H ^ j _ { G } ( \\P ^ 1 ) = \\left \\{ \\begin{array} { l l } \\Z \\cdot 1 , & j = 0 \\\\ \\Z / m \\cdot x \\oplus \\Z \\cdot h , & j = 2 \\\\ \\Z / m \\cdot x ^ k \\oplus \\Z / m \\cdot x ^ { k - 1 } h , & j = 2 k > 2 \\\\ 0 , & j \\ ; \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "10028.png", "formula": "\\begin{align*} | \\widehat { S T } | = | \\widehat { S T ' } | + | S T | \\geq \\sigma _ q ( d _ { s + 1 } , d _ s ) + \\sigma _ q ( d _ s , d _ { s - 1 } ) . \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{gather*} \\sum \\limits _ { k = 0 } ^ { r - 1 } \\ d _ { k } \\ s _ { k + m } = \\ s _ { r + m } \\ , \\ \\ \\ \\ \\ m \\geq 0 \\ . \\end{gather*}"}
-{"id": "6096.png", "formula": "\\begin{align*} \\delta = \\frac { 1 } { 2 } \\mathrm { m i n } \\big \\{ \\delta _ Y , \\delta _ { Z _ 1 } , \\delta _ { Z _ 2 } , \\delta _ { C _ 1 } , \\delta _ { C _ 2 } \\big \\} . \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} F S & = \\frac { 1 } { 4 } \\left ( F S \\right ) I = \\left ( \\mathbf { E } \\cdot \\mathbf { B } \\right ) I , \\\\ G R & = \\frac { 1 } { 4 } \\left ( G R \\right ) I = \\left ( \\mathbf { H } \\cdot \\mathbf { D } \\right ) I . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} R ^ \\star _ { \\mathsf { u } } ( N - 1 / L ) & \\le \\frac { 1 } { 1 + r _ N } p _ N \\\\ & = \\frac { p _ N } { L } . \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} F _ { d , \\ell } \\left ( z ; i t \\right ) & = \\sum _ { a = 0 } ^ N \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\zeta ^ d } { 1 - \\zeta ^ \\ell } \\right ) \\frac { ( - 2 \\pi t ) ^ a } { a ! } + O \\left ( t ^ { N + 1 } \\right ) . \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\Omega _ 0 \\mid _ { V _ 1 } = \\frac { d z _ 2 \\wedge d z _ 3 } { f _ { z _ 1 } } , ~ \\Omega _ 0 \\mid _ { V _ 2 } = \\frac { d z _ 3 \\wedge d z _ 1 } { f _ { z _ 2 } } , ~ \\Omega _ 0 \\mid _ { V _ 3 } = \\frac { d z _ 1 \\wedge d z _ 2 } { f _ { z _ 3 } } , \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} \\tilde { D } = - \\Delta _ { G } + m ^ 2 \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} \\gamma _ i \\leq \\frac { 1 } { \\binom { k - 1 } { l - 1 } } , \\ ; i = 1 , 2 , . . . , \\binom { n } { l } . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} \\Lambda ^ 4 _ 7 W _ 8 = \\{ ( u ^ \\flat \\wedge v ^ \\flat ) \\cdot \\Phi \\mid u , v \\in W _ 8 \\} = \\{ u ^ \\flat \\wedge ( \\imath _ v \\Phi ) - v ^ \\flat \\wedge ( \\imath _ u \\Phi ) \\mid u , v \\in W _ 8 \\} . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} \\langle a ^ * _ k ( s ) a _ { k ' } ( s ' ) \\rangle = \\delta _ { k , k ' } e ^ { ( s - s ' ) \\varepsilon ( k ) } f ( k ) \\quad \\langle a _ k ( s ) a ^ * _ { k ' } ( s ' ) \\rangle = \\delta _ { k , k ' } e ^ { - ( s - s ' ) \\varepsilon ( k ) } ( 1 + f ( k ) ) . \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} \\beta _ { s , a ^ 1 } ^ 1 = \\frac { r ^ 1 ( s , a ^ 1 , a _ s ^ 2 ) - r ^ 1 ( s , a _ s ^ 1 , a _ s ^ 2 ) } { r ^ 1 ( s , a ^ 1 , a _ s ^ 2 ) - \\sum _ { s ' \\in S } \\left ( \\frac { \\mu ( s ' , s , a ^ 1 , a _ s ^ 2 ) } { | | \\mu | | } + \\delta ( s , s ' ) \\right ) r ^ 1 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) } , \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} \\left ( \\alpha _ { 0 } , \\alpha _ { 1 } , \\alpha _ { 2 } , \\alpha _ { 3 } \\right ) = \\alpha _ { 0 } \\left ( 1 , - 1 , - 1 , - 1 \\right ) , \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} Q ^ H \\left [ \\begin{array} { c c } x & \\beta + \\j z \\\\ \\beta - \\j z & y \\end{array} \\right ] Q = Q ^ H \\left [ \\begin{array} { c c } x & \\j z \\\\ - \\j z & y \\end{array} \\right ] Q + \\left [ \\begin{array} { c c } 0 & \\beta \\\\ \\beta & 0 \\end{array} \\right ] = \\left [ \\begin{array} { c c } \\alpha & \\beta \\\\ \\beta & \\gamma \\end{array} \\right ] . \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} Z _ { j , R ' } \\subseteq Z _ { j , R } , j = 1 , 2 , \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} & \\limsup _ { j \\to \\infty } \\ , \\int _ { \\mathbb { R } ^ s } \\ , \\phi _ 2 ( \\xi ) - H ( \\xi ) \\ , \\chi _ F ( \\xi ) \\ , d \\tau _ j ( \\xi ) \\le \\lim _ { j \\to \\infty } \\ , \\int _ { \\mathbb { R } ^ s } \\ , \\phi _ 2 ( \\xi ) - \\phi _ 1 ( \\xi ) \\ , d \\tau _ j ( \\xi ) \\\\ & = \\int _ { \\mathbb { R } ^ s } \\ , G ( \\xi ) \\ , \\left ( \\phi _ 2 ( \\xi ) - \\phi _ 1 ( \\xi ) \\right ) \\ , d \\xi \\le B \\ , \\int _ { \\mathbb { R } ^ s } \\ , \\left ( \\phi _ 2 ( \\xi ) - \\phi _ 1 ( \\xi ) \\right ) \\ , d \\xi \\le B \\ , \\epsilon \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} \\delta ^ r = \\oplus _ { \\lambda \\in \\mathcal { P } ( k , r ) } \\delta _ \\lambda ^ r , \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} C _ { p , 0 } ^ { ( \\ell ) } & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = 0 } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } \\sum _ { \\beta = 0 } ^ { p - j - 2 t } 2 ^ { p - j - 2 t - \\beta } \\binom { n - \\ell } { \\beta } \\binom { \\ell } { p - j - 2 t - \\beta } \\\\ & \\sum _ { i = 0 } ^ { j - 1 } \\binom { \\beta } { p + i - j - t } . \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} S ( a _ 1 \\circ b ) a _ 2 = S ( a _ 1 ) ( a _ 2 \\circ S ( b ) ) \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} \\gamma ( X ) : = \\sup _ { x _ 0 \\in X ^ \\times } \\inf _ { x _ 0 \\ne x \\in X } { \\rm o r d } ( x - x _ 0 ) , \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} \\Delta _ { F } \\tilde H _ { \\lambda } = F [ B _ \\lambda ] , B _ \\lambda = \\sum _ { ( c , r ) \\in \\lambda } q ^ c t ^ r , \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} \\alpha ^ * _ t & = \\frac { F '' ( v ( t , X _ t ) ) } { F ' ( v ( t , X _ t ) ) } \\nabla v ( t , X _ t ) , \\\\ \\beta ^ * _ t & = \\frac { F ' ( v ( t , X _ t ) ) F ''' ( v ( t , X _ t ) ) } { F '' ( v ( t , X _ t ) ) ^ 2 } - 1 , \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{gather*} S _ { \\mathcal { F } } h = V ^ { - 1 } _ { \\mathcal { F } } ( S h ) V _ { \\mathcal { F } } = V ^ { - 1 } _ { \\mathcal { F } } ( S h ) \\big ( S \\mathcal { F } ^ { ( 1 ) } \\big ) \\mathcal { F } ^ { ( 2 ) } . \\end{gather*}"}
-{"id": "1025.png", "formula": "\\begin{align*} e _ n ^ j ( w _ 1 , \\dots , w _ n ) = c _ j \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} a ^ * _ { 2 , 0 } = \\mu _ R , a ^ * _ { 0 , 2 } = 1 - \\mu _ R \\ \\textrm { a n d o t h e r r a t i o s a r e 0 } . \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\lambda ^ { \\star } = \\Lambda _ 0 < \\Lambda _ 1 < \\Lambda _ 2 < \\dots < \\Lambda _ M = \\Lambda . \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} \\varphi _ { \\pm } ( t , \\xi ) = x _ 0 \\cdot \\xi \\pm \\lambda t \\xi _ \\varepsilon , \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} & \\lambda _ { P / A } ( \\{ e \\} ) \\\\ = & r _ { P / A } ( \\{ e \\} ) + r _ { P / A } ( ( E - A ) - \\{ e \\} ) - r _ { P / A } ( E - A ) \\\\ = & r _ P ( A \\cup \\{ e \\} ) - r _ P ( A ) + r _ P ( E - \\{ e \\} ) - r _ P ( A ) - ( r _ P ( E ) - r _ P ( A ) ) \\\\ = & r _ P ( A \\cup \\{ e \\} ) - r _ P ( A ) \\\\ = & r _ { P / A } ( \\{ e \\} ) , \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\mathbf { H _ 1 } ^ { \\dagger } \\tilde { \\mathbf { Y } } ^ { T _ E } _ { [ 1 : \\ell ] } = \\mathbf { X } _ { [ ( M - \\ell ) + 1 : M ] } ^ { T _ E } + \\mathbf { H _ 1 } ^ { \\dagger } \\mathbf { n } _ { [ 1 : \\ell ] } ^ { T _ E } . \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} \\gamma _ 2 \\gamma _ 2 ^ { t r } + \\delta _ 2 \\delta _ 2 ^ { t r } + 2 b _ 2 b _ 2 ^ { t r } = I ; \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} \\delta ^ * ( \\mu , r ) = ( M + K - 1 ) \\mu + \\frac { K \\left ( 1 - \\mu M \\right ) } { M } \\left ( 1 + \\frac { 1 } { r } \\right ) . \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} & C ^ { 0 , \\alpha } = \\{ h \\in { C ^ { 0 , 0 } ( \\mathbb R ^ d ; \\mathbb R ^ m ) } : \\| h \\| _ { C ^ { 0 , \\alpha } } < \\infty \\} \\\\ & C ^ { 1 , \\alpha } = \\{ h \\in { C ^ { 1 , 0 } ( \\mathbb R ^ d ; \\mathbb R ^ m ) } : \\| h \\| _ { C ^ { 1 , \\alpha } } < \\infty \\} , \\end{align*}"}
-{"id": "10035.png", "formula": "\\begin{align*} R \\subseteq \\bigcap _ { H \\in \\mathcal { H } _ 0 } H = W , \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} \\tau _ { P y } = ( O _ { P } ) ^ { - 1 } \\tau _ y O _ { P } = O _ { P ^ t } \\tau _ y O _ { P } y \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} P _ { m + d + 1 } \\left ( x \\right ) = \\left ( x - \\beta _ { m + d } \\right ) P _ { m + d } \\left ( x \\right ) - \\sum \\nolimits _ { \\nu = 0 } ^ { d - 1 } \\gamma _ { m + d - \\nu } ^ { d - 1 - \\nu } P _ { m + d - 1 - \\nu } \\left ( x \\right ) , \\ \\ m \\geq 0 , \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} \\mu _ { k } ^ i \\left ( B \\right ) & \\geq 1 \\textendash C _ 3 \\sum \\limits _ { \\theta \\in B ^ c } \\prod \\limits _ { t = 1 } ^ { k } \\prod \\limits _ { j = 1 } ^ { n } \\left ( \\frac { \\ell ^ j ( s _ { t } ^ j | \\theta ) } { \\ell ^ j ( s _ { t } ^ j | \\theta ^ * ) } \\right ) ^ { \\frac { 1 } { n } } \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} \\Lambda = Q _ v \\left [ \\begin{array} { c c } \\tilde \\Lambda _ { 1 1 } & \\tilde \\Lambda _ { 1 2 } \\\\ - \\tilde \\Lambda _ { 1 2 } ^ H & W ^ H \\Delta _ 1 ^ H W \\end{array} \\right ] Q _ v ^ H \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} P _ { \\tilde { i } , 1 } ( m ) \\alpha _ { \\tilde { i } } ^ m = Q ( n ) \\beta _ { j ^ * } ^ n \\end{align*}"}
-{"id": "10016.png", "formula": "\\begin{align*} \\begin{cases} & \\mbox { $ n \\geq d _ i + d _ j $ i f $ n _ { d _ i } $ + $ n _ { d _ j } \\geq 2 $ a n d $ i \\not = j $ } ; \\mbox { a n d } \\\\ & \\mbox { $ n \\geq 2 d _ i $ i f $ n _ { d _ i } \\geq 2 $ } . \\end{cases} \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} \\lim _ { D \\ni x ' \\to x } v ( x ' ) = 0 \\sup _ { x \\in D \\setminus K } v ( x ) < + \\infty . \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} \\xi _ \\varepsilon = \\left \\{ \\begin{array} { l l } \\xi ( 1 + \\varepsilon ^ { 2 } | \\xi | ^ { 2 } ) ^ { 1 / 2 } & d = 1 , \\\\ | \\xi | ( 1 + \\varepsilon ^ { 2 } | \\xi | ^ { 2 } ) ^ { 1 / 2 } & d = 2 , 3 . \\end{array} \\right . \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} ( g \\phi ) ( w ) = ( g , h _ g ) \\phi ( h _ g ^ { - 1 } w ) ( w \\in W ) . \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} i ( n - 1 ) - 2 i ( k - 1 ) - i ( n - 2 k + 1 ) = i ( n - 1 ) - i ( n - 1 ) = 0 . \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} \\Sigma _ 1 & : = \\sum _ { p ^ { \\nu } \\leq y \\atop p \\in E _ j } \\prod _ { 0 \\leq l \\leq n } \\left ( 1 - \\frac { E _ l ( x ) - E _ l ( x / p ^ { \\nu } ) } { E _ l ( x ) } \\right ) ^ { k _ l } \\left ( 1 - \\frac { \\log p ^ { \\nu } } { \\log x } \\right ) ^ { - 1 } , \\\\ \\Sigma _ 2 & : = \\sum _ { y < p ^ { \\nu } \\leq \\sqrt { x } } \\prod _ { 0 \\leq l \\leq n } \\left ( 1 - \\frac { E _ l ( x ) - E _ l ( x / p ^ { \\nu } ) } { E _ l ( x ) } \\right ) ^ { k _ l } \\left ( 1 - \\frac { \\log p ^ { \\nu } } { \\log x } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} A \\sharp _ { \\nu } B \\leqslant A \\sharp _ { \\nu } B + \\sum _ { k = 0 } ^ { n } r _ { k } ( A \\sharp _ { \\frac { m _ k } { 2 ^ k } } B - 2 A \\sharp _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } B + A \\sharp _ { \\frac { m _ k + 1 } { 2 ^ k } } B ) \\leqslant A \\nabla _ { \\nu } B . \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} \\partial _ t \\delta q _ { e \\varepsilon } & + \\delta u _ \\varepsilon \\cdot \\nabla \\delta q _ { e \\varepsilon } + \\delta u _ \\varepsilon \\cdot \\nabla q _ e + u \\cdot \\nabla \\delta q _ { e \\varepsilon } \\\\ & + ( \\bar Q + \\alpha ) \\nabla \\cdot \\delta v _ \\varepsilon = - \\frac { 1 + \\alpha } { \\varepsilon } q _ { e \\varepsilon } ^ + - ( \\bar Q + \\alpha ) \\nabla \\cdot v \\chi _ { \\mathcal O } ( x , y , t ) , \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} v _ { m , j } = ( - 1 ) ^ { j } \\alpha ^ { - j } q ^ { \\frac { 1 } { 2 } j ( j + 1 ) } \\ , _ { 1 } \\tilde { \\phi } _ { 1 } \\left ( 0 ; q ^ { - m + j + 1 } ; q , \\alpha ^ { - 2 } q ^ { m + j + 1 } \\right ) \\ ! . \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} { \\ ; } _ p F _ q \\left ( { a _ 1 , \\ldots , a _ p \\atop b _ 1 , \\ldots , b _ q } \\Big { | } z \\right ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( a _ 1 ) _ k \\cdots ( a _ p ) _ k } { ( b _ 1 ) _ k \\cdots ( b _ q ) _ k } \\frac { z ^ k } { k ! } \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} ( v , p ) = ( 0 , p _ 0 ) \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n } { 2 } } } { \\left ( q ; q \\right ) _ { n } } \\left ( c z \\right ) ^ { n } A _ { q } \\left ( c q ^ { n - 1 } \\right ) = \\frac { \\left ( c ; q \\right ) _ { \\infty } } { \\left ( c z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } \\prod _ { j = 0 } ^ { 2 n - 1 } \\left ( z - q ^ { j } \\right ) \\left ( - c ^ { 2 } \\right ) ^ { n } } { \\left ( q ^ { 2 } , c , c q ; q ^ { 2 } \\right ) _ { n } } \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\bar g ( s ) : = \\frac { k } { \\rho } G \\left ( \\frac { k } { \\rho } \\right ) ^ { - 1 } g \\left ( \\frac { k } { \\rho } s \\right ) , \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\mathbf { B } _ { i } & = \\mathbf { B } _ { i - 1 } ^ { \\ast } - \\mathbf { B } _ { i - 1 } + \\left ( 1 - \\beta _ { i , i } \\right ) \\mathbf { B } _ { i - 1 } + \\beta _ { i , i } \\mathbf { E } _ { i } \\\\ & = \\left \\{ \\left ( 1 - \\beta _ { i , i } \\right ) \\mathbf { B } _ { i - 1 } + \\beta _ { i , i } \\mathbf { E } _ { i } \\right \\} - \\left ( \\mathbf { B } _ { i - 1 } - \\mathbf { B } _ { i - 1 } ^ { \\ast } \\right ) \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} | v | _ { H ^ { k , 2 k } ( D _ T ) } : = \\Big ( \\sum _ { \\ell = 1 } ^ { 2 k } \\norm { D ^ \\ell v } { L ^ 2 ( D _ T ) } ^ 2 \\Big ) ^ { 1 / 2 } + \\sum _ { \\ell = 1 } ^ k \\norm { \\partial _ t ^ \\ell v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 \\ell } ( D ) ) } , \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} S _ { \\rho , n } ( e ) \\approx s _ { \\rho , n } ( e ) - \\frac { 1 } { 2 } \\frac { \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\rho ^ { ( 1 ) } \\left ( \\frac { r _ i ( e ) } { \\sigma _ n } \\right ) ^ 2 \\right ) \\chi ^ 2 _ { k - k ( e ) } } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\rho ^ { ( 2 ) } \\left ( \\frac { r _ i ( e ) } { \\sigma _ n } \\right ) } \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} B _ { W ' } : = B _ { W _ 1 ' } \\otimes W _ 2 ' \\otimes . . . \\otimes W _ d ' \\ + \\ W _ 1 ' \\otimes B _ { W _ 2 ' } \\otimes . . . \\otimes W _ d ' \\ + \\ . . . \\ + \\ W _ 1 ' \\otimes . . . \\otimes W _ { d - 1 } ' \\otimes B _ { W _ d ' } , \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { ( 2 n + 1 ) n } } { ( q ^ 2 ; q ^ 2 ) _ { n } } \\frac { ( - q ^ { 2 n + 2 } ; q ^ 2 ) _ \\infty } { ( q ^ { 2 n + 2 } ; q ^ 2 ) _ \\infty } ( - q ^ { 2 n + 3 } ; q ^ 2 ) _ \\infty = \\sum _ { j = 0 } ^ \\infty \\frac { q ^ { ( 3 j ^ 2 + j ) / 2 } } { ( q ; q ) _ { 2 j } ( q ^ { 2 j + 2 } ; q ) _ \\infty } \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} B F ( \\sp \\{ e _ { 1 } \\} ) = \\sp \\{ e _ { 1 } \\} , \\ ; B F ( \\sp \\{ e _ { 2 } \\} ) = \\sp \\{ e _ { 2 } \\} , \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} b _ { s , s } ^ 0 \\left [ Z _ s , t \\right ] = 1 \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} ( \\ ; + \\ ; ) = k , \\leq k ( k \\in \\Z _ { > 0 } ) . \\end{align*}"}
-{"id": "10044.png", "formula": "\\begin{align*} \\frac { z ^ k f ' ( z ) + z ^ { k + 1 } f '' ( z ) ( \\lambda - \\mu + 2 \\lambda \\mu ) + \\lambda \\mu z ^ { k + 2 } f ''' ( z ) } { g _ k ( z ) } = \\varphi ( w ( z ) ) . \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} d \\tau = \\frac { 1 } { r \\sqrt { 1 - r ^ 2 } } d r , d \\tilde { \\tau } = r ^ { - 2 } d r , \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} f ( x ) & = T ( p ) h ( x ) + 2 \\sqrt { p ^ 2 + m ^ 2 } h ( x ) \\\\ & = ( \\widehat { T } * h ) ( x ) + 2 \\sqrt { p ^ 2 + m ^ 2 } h ( x ) \\geq 2 \\sqrt { p ^ 2 + m ^ 2 } h ( x ) \\geq \\frac { 2 } { 1 + x ^ 2 } . \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{gather*} V ^ { ( k ) } _ { \\{ z _ { i } \\} } = \\begin{bmatrix} 1 & 1 & \\dots & 1 \\\\ z _ { 1 } & z _ { 2 } & \\dots & z _ { k } \\\\ z _ { 1 } ^ { 2 } & z _ { 2 } ^ { 2 } & \\dots & z _ { k } ^ { 2 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ z _ { 1 } ^ { k - 1 } & z _ { 2 } ^ { k - 1 } & \\dots & z _ { k } ^ { k - 1 } \\\\ \\end{bmatrix} . \\end{gather*}"}
-{"id": "7099.png", "formula": "\\begin{align*} \\left ( \\mathcal { A } ( G ) x \\right ) _ { v } = \\sum _ { v \\in e } x ^ { e \\backslash \\{ v \\} } = \\mu x _ { v } ^ { t - 1 } . \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} U ( 0 ) = \\phi ( 0 ) . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} \\norm { \\mu _ A ( f ) a } = \\sup _ { x \\in X } \\norm { ( \\mu _ A ( f ) a ) ( x ) } \\geq 1 . \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} \\delta ^ * _ \\lambda ( \\xi ) = ( \\lambda ^ { \\sigma ^ * _ 1 } \\xi _ 1 , \\ldots , \\lambda ^ { \\sigma ^ * _ p } \\xi _ p ) , \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} \\widehat { V } _ { t } ^ { k } ( R ^ x _ t , I ^ x _ t ) : = \\max _ { j \\leq k } \\big \\{ \\alpha ^ j _ { t } ( I ^ x _ t ) + \\langle \\beta ^ j _ { t } ( I ^ x _ t ) , R ^ x _ t - R ^ { x , j } _ t \\rangle \\} . \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\int \\frac { [ \\mu _ { + } ^ { \\prime } + \\mu _ { - } ^ { \\prime } ] } { v ^ { 2 } } d v & = 2 \\int _ { v > 0 } \\frac { [ \\mu _ { + } ^ { \\prime } + \\mu _ { - } ^ { \\prime } ] } { v ^ { 2 } } d v \\\\ & \\geq c _ { 0 } \\left ( \\frac { 1 } { 2 \\sigma _ { 0 } } - \\frac { 1 } { a } \\right ) + \\int _ { v > a } \\frac { [ \\mu _ { + } ^ { \\prime } + \\mu _ { - } ^ { \\prime } ] } { v ^ { 2 } } d v \\\\ & > 0 , \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} ( f , A _ { \\Omega , 2 m } ( a , b , q ) f ) _ { L ^ 2 ( { \\Omega } ) } = \\big \\| A _ { \\Omega , 2 } ( a , b , q ) ^ { \\ell } f \\big \\| ^ 2 _ { L ^ 2 ( { \\Omega } ) } \\geq \\varepsilon _ { \\ell } ^ 2 \\| f \\| ^ 2 _ { L ^ 2 ( { \\Omega } ) } = \\varepsilon _ { m } \\| f \\| ^ 2 _ { L ^ 2 ( { \\Omega } ) } , \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } S & T \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & I \\\\ I & 0 \\end{array} \\right ] \\left [ \\begin{array} { c } S ^ H \\\\ T ^ H \\end{array} \\right ] = \\left [ \\begin{array} { c c } U & V \\end{array} \\right ] \\left [ \\begin{array} { c c } \\Phi _ { 1 1 } I & \\Phi _ { 2 1 } I \\\\ \\Phi _ { 1 2 } I & \\Phi _ { 2 2 } I \\end{array} \\right ] \\left [ \\begin{array} { c } U ^ H \\\\ V ^ H \\end{array} \\right ] = 0 \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} R ^ { ( 2 ) } = \\bigoplus _ { i \\geqslant 0 } R _ { 2 i } = \\bigoplus _ { i \\geqslant 0 } \\{ f \\in k [ x , y ] \\mid \\deg f = 2 i \\} . \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } \\alpha _ { 0 } + \\alpha _ { 1 } = 0 , \\\\ \\alpha _ { 1 } - \\alpha _ { 2 } = 0 , \\\\ \\alpha _ { 2 } - \\alpha _ { 3 } = 0 , \\end{array} \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} d \\omega & = \\left ( \\frac { 1 } { 2 } \\omega ^ { a b } J _ { a b } \\right ) \\propto J _ { a b } \\\\ \\left [ \\omega , \\omega \\right ] = \\frac { 1 } { 4 } \\omega ^ { a b } \\omega ^ { c d } \\left [ J _ { a b } , J _ { c d } \\right ] & \\propto J _ { a b } . \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\langle { x , y } | x ^ { 2 ^ { i - 1 } } = 1 , ~ x ^ { 2 ^ { i - 2 } } = y ^ 2 , ~ y x y ^ { - 1 } = x ^ { - 1 } \\rangle , \\end{align*}"}
-{"id": "10133.png", "formula": "\\begin{align*} ( y ^ 2 + b z ) ^ q - z ^ { p + 2 q } = 0 \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} \\sigma ( B _ K ) = \\sigma ( A _ 0 [ 1 / \\varpi ] ) = \\sigma ( A _ 0 ) [ 1 / \\varpi ] \\subset A _ { K ' } [ 1 / \\varpi ] = B _ { K ' } , \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} { \\bf E } [ \\beta _ { 1 , 0 } ( a , b ) ^ q ] = & \\frac { \\Gamma _ 1 ( q + b _ 0 \\ , | \\ , a ) } { \\Gamma _ 1 ( b _ 0 \\ , | \\ , a ) } , \\\\ = & a ^ { \\frac { q } { a } } \\frac { \\Gamma ( \\frac { q + b _ 0 } { a } ) } { \\Gamma ( \\frac { b _ 0 } { a } ) } , \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} x ( t ) = x _ 0 + \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ 0 ^ t ( t - \\tau ) ^ { \\alpha - 1 } f ( \\tau , x ( \\tau ) ) d \\tau . \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} \\begin{aligned} Z _ { s , s + k + 1 } & \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau , 0 ; v _ { s + 1 } , \\dots , v _ { s + k } , v _ { s + k + 1 } ; \\right . \\\\ & \\left . \\omega _ 1 , \\dots , \\omega _ k , \\omega _ { k + 1 } ; i _ 1 , \\dots , i _ k , i _ { k + 1 } \\right ] \\\\ & \\in \\mathcal { K } _ { s + k + 1 } \\cap \\mathcal { U } _ { s + k + 1 } ^ \\eta \\end{aligned} \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ g ( - 1 ) ^ k f ( k ) \\tbinom { g } { k - 1 } = 0 . \\end{align*}"}
-{"id": "10112.png", "formula": "\\begin{align*} q + r = m + n + l + k , \\end{align*} % \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} m _ \\mu = \\begin{cases} \\frac { 1 } { 4 } , & \\mbox { i f } ~ \\mu \\ge \\frac { 1 + \\sqrt { 2 } } { 2 } , \\\\ \\frac { 4 \\mu + 1 } { 1 6 \\mu ^ 2 } , & \\mbox { i f } ~ \\frac { 1 } { 2 } < \\mu < \\frac { 1 + \\sqrt { 2 } } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} & B _ t ( P ( \\cdot ) ) ( s ) : = - \\int _ t ^ s \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , t ) ) \\left [ \\int _ t ^ \\tau A ( p ( \\sigma , t ) ) P ( \\sigma ) d \\sigma \\right ] d \\tau , \\\\ & R _ 0 ( s ) : = - \\int _ t ^ s \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , t ) ) d \\tau . \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} P _ A S _ \\mu = P _ A S _ { \\mu _ 1 } \\cdots S _ { \\mu _ m } = \\begin{cases} S _ { \\mu _ 1 } \\cdots S _ { \\mu _ m } P _ A & m \\\\ S _ { \\mu _ 1 } \\cdots S _ { \\mu _ m } P _ B & m \\end{cases} \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { k ^ { 2 } } \\left ( - q ^ { - 2 n - 1 } \\right ) ^ { k } } { \\left ( q , - q ; q \\right ) _ { k } } = 0 \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} F \\left ( q , q , \\dots , q \\right ) = u _ { 0 } + q \\left ( u _ { 1 } + \\dots + u _ { n } \\right ) . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} G \\otimes \\left ( H _ 1 \\oplus H _ 2 \\right ) = \\left ( G \\otimes H _ 1 \\right ) \\oplus \\left ( G \\otimes H _ 2 \\right ) , \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} | \\{ n \\in \\omega : \\beta _ n = \\widetilde { \\alpha } \\} | = \\aleph _ 0 . \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} & \\pi _ { n } ( x _ n | y ^ { n - 1 } _ { n - J } ) \\\\ & = \\exp { \\Big \\{ \\sum _ { y _ { n } } \\log \\big ( r _ { n } ( x _ n | y _ n , y ^ { n - 1 } _ { n - M } \\big ) q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) - 1 - s \\gamma _ n ( x _ n , y ^ { n - 1 } _ { n - N } ) + \\lambda _ n ( y ^ { n - 1 } _ { n - J } ) \\Big \\} } , ~ \\forall { x _ n } \\in { \\cal X } _ n . \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} \\bar { P } _ { l , t } ( Y ) = C _ { l , t } \\int _ { \\mu _ 0 \\in \\mathbb { R } ^ { p } : \\| \\mu _ 0 \\| \\geq 1 / 2 } & \\frac { 1 } { ( 2 \\pi ) ^ { p n / 2 } } \\exp ( - \\| Y - 2 t \\mu _ 0 l ^ { \\intercal } \\| _ F ^ 2 / 2 ) \\\\ & \\cdot \\left ( \\frac { p } { 2 \\pi } \\right ) ^ { p _ 1 r / 2 } \\exp ( - p \\| \\mu _ 0 \\| _ 2 ^ 2 / 2 ) d \\mu _ 0 . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} - L u + g _ n \\circ u & = f \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega , \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} \\Vert g _ n \\Vert _ { L _ { \\alpha + 1 } ^ 2 } ^ 2 & = b _ { \\alpha } \\int _ { a _ n } ^ { a _ { n - 1 } } \\big \\vert x ^ { \\alpha / 2 } \\mu ^ { - ( \\alpha + 2 ) / 2 } e ^ { - i t \\log _ { \\mu } x } \\big \\vert ^ 2 x ^ { - ( \\alpha + 1 ) } \\ , d x = b _ { \\alpha } \\mu ^ { - ( \\alpha + 2 ) } \\int _ { a _ n } ^ { a _ { n - 1 } } x ^ { - 1 } \\ , d x \\\\ & = b _ { \\alpha } \\mu ^ { - ( \\alpha + 2 ) } n . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} \\sigma _ E ^ { l ( x ) } ( \\kappa ( x ) ) & = \\begin{cases} \\kappa ( x ) & \\\\ \\sigma _ E ( \\kappa ( x ) ) & \\end{cases} \\\\ & = r ( x ) . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} P _ { N _ { t } } ^ { \\varepsilon } \\leq C \\zeta ^ { \\left ( \\frac { t } { \\sup \\varphi } \\right ) } = C \\zeta _ { 1 } ^ { t } \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} S _ \\gamma ^ * S _ \\gamma & = \\sum _ { \\eta \\in E _ Z } Z ^ G ( \\gamma , \\eta ) S _ { \\eta } S _ { \\eta } ^ * \\\\ & = \\sum _ { \\eta \\in E _ Z } Z ^ G ( \\gamma , \\eta ) ( P _ B S _ { \\eta } S _ { \\eta } ^ * P _ B + P _ A S _ { \\eta } S _ { \\eta } ^ * P _ A ) \\\\ & = \\sum _ { \\eta \\in E _ Z } Z ^ G ( \\gamma , \\eta ) S _ { ( B , \\eta ) } S _ { ( B , \\eta ) } ^ * + \\sum _ { \\eta \\in E _ Z } Z ^ G ( \\gamma , \\eta ) S _ { ( A , \\eta ) } S _ { ( A , \\eta ) } ^ * . \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} r \\iota _ i ( e _ j ) = \\iota _ i ( e _ j ) r . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} P _ { \\Delta , t } ( x , y ) = t ^ { x + y + 1 } \\sum _ { l \\geq 0 } \\binom { x } { l } \\binom { y } { l } \\left ( \\frac { t ^ 2 - 2 \\Delta t + 1 } { t ^ 2 } \\right ) ^ { l + 1 / 2 } . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i & = \\left ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } ) _ { , K } \\right ) _ { , J } - E ( | \\dot { u } | ) \\dot { u } _ i , \\\\ a \\ddot { \\tau } & = - \\beta _ { K i } \\dot { u } _ { i , K } + m _ { I J } q _ { I , J } + M _ { j L K I } u _ { j , L K I } + K _ { I J } \\tau _ { , I J } , \\\\ \\kappa \\dot { q } _ i & = \\dot { \\tau } _ { , i } - q _ { i } \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} P ( D ) = P _ { \\mathbf { G } } ( G ) ( 1 - p _ d ) ^ { n _ { a s } } > 0 , \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} \\tilde { \\jmath } = \\left ( { \\lambda } \\ , { \\lambda ^ { - 1 } _ T } \\right ) ^ { ( q - 1 ) / 2 } g ^ { ( q + 1 ) / 2 } \\quad \\in F . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} \\Gamma = \\{ 1 \\} \\cup \\Psi _ r ^ n \\cup \\bigcup _ { \\psi \\in \\Psi ^ n } J _ { \\psi } . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { m , n } ^ { \\gamma } ) \\right ] ( z ) & = \\dfrac { - ( \\gamma + 1 ) _ { m + n } ( - m - 1 ) _ { n } } { ( \\gamma + 1 ) _ { n } ( m + 1 ) } \\overline { z } ^ { m + 1 - n } \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + 1 } { _ 2 F _ 1 } \\left ( \\begin{array} { c } 1 - n , \\gamma + m + 2 \\\\ m - n + 2 \\end{array} \\bigg | | z | ^ 2 \\right ) . \\end{align*}"}
-{"id": "9936.png", "formula": "\\begin{align*} v = \\xi v ^ 0 ( A ) \\xi \\in P ^ { - } ( A ) . \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{gather*} s _ { 2 r } \\ ! = \\ ! s _ { 2 r } ^ { ( r ) } \\ , , \\ s _ { 2 r + 1 } \\ ! = \\ ! s _ { 2 r + 1 } ^ { ( r ) } \\ , , \\ \\ldots \\ , s _ { 2 r + d - 1 } \\ ! = \\ ! s _ { 2 r + d - 1 } ^ { ( r ) } \\ . \\end{gather*}"}
-{"id": "8856.png", "formula": "\\begin{align*} a \\nabla _ { \\nu } b - r _ 0 ( \\sqrt { a } - \\sqrt { b } ) ^ 2 & = a \\nabla _ { \\nu } b - \\nu ( \\sqrt { a } - \\sqrt { b } ) ^ 2 \\\\ & = 2 \\nu \\sqrt { a b } + ( 1 - 2 \\nu ) a \\\\ & = a \\nabla _ { 2 \\nu } \\sqrt { a b } \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ 1 _ { \\mathbf { A } } ( \\O ) } : = \\sqrt { \\| u \\| ^ 2 + \\| ( - i \\nabla + \\mathbf { A } ) u \\| ^ 2 } \\ , . \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} f ( x ) = ( x - 2 ) ^ { - r } \\sum \\limits _ { i = 0 } ^ { \\lfloor - r / 2 \\rfloor } \\delta _ i \\left ( \\frac { x } { x - 2 } \\right ) ^ { 2 i } = x ^ { - r } \\sum \\limits _ { i = 0 } ^ { \\lfloor - r / 2 \\rfloor } \\delta _ i \\left ( \\frac { x - 2 } { x } \\right ) ^ { - r - 2 i } . \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} \\{ \\lambda _ 1 , \\lambda _ 2 \\} = & \\frac { a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 \\pm \\sqrt { ( a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 ) ^ 2 - 4 ( a d - b c ) ^ 2 } } { 2 } \\\\ = & \\frac { a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 \\pm \\sqrt { ( a ^ 2 + c ^ 2 - b ^ 2 - d ^ 2 ) ^ 2 + 4 ( a b - c d ) ^ 2 } } { 2 } \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} ( Z _ 0 ) _ { r e d } = \\ell _ 0 \\cup \\ell _ 1 \\cup \\ell _ 2 , \\ \\ \\ q _ 1 : = \\ell _ 0 \\cap \\ell _ 1 = \\{ \\mathrm { p t } \\} , \\ q _ 2 : = \\ell _ 1 \\cap \\ell _ 2 = \\{ \\mathrm { p t } \\} , \\ q _ 1 \\ne q _ 2 . \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} A _ 1 = k \\lambda , \\ , A _ { - 1 } = k \\lambda ^ { - 1 } \\mbox { \\ f o r s o m e \\ } k \\in \\C ^ * . \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} \\lim _ { r \\longrightarrow \\infty } \\frac { \\log \\Pi ( p _ { r } ) } { r \\cdot \\log \\log \\Pi ( p _ { r } ) } = 1 ; \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} & \\int _ 0 ^ \\infty \\left \\| \\frac { d } { d t } \\left ( e ^ { i t H } J e ^ { - i t H _ 0 } E _ { H _ 0 } ( \\Gamma ) u \\right ) \\right \\| d t \\\\ & = \\int _ 0 ^ \\infty \\left \\| \\frac { d } { d t } \\left ( e ^ { i t H } J _ a e ^ { - i t H _ 0 } u \\right ) \\right \\| d t \\\\ & = \\int _ 0 ^ \\infty \\| e ^ { i t H } ( H J _ a - J _ a H _ 0 ) e ^ { - i t H _ 0 } u \\| d t \\\\ & = \\int _ 0 ^ \\infty \\| ( H J _ a - J _ a H _ 0 ) e ^ { - i t H _ 0 } u \\| d t \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\beta } v ( x ) = c _ { N , \\beta } \\int _ { \\mathbb { R } ^ { n } } \\frac { v ( x ) - v ( y ) } { | x - y | ^ { N + 2 \\beta } } d y , \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} \\begin{cases} \\varphi _ x ( L , t ) = \\psi _ x ( L , t ) = 0 , & \\ , \\ , ( 0 , T ) , \\\\ a \\varphi _ { x x } ( 0 , \\cdot ) + \\frac { 1 } { c } \\psi _ { x x } ( 0 , \\cdot ) = 0 , & \\ , \\ , ( 0 , T ) , \\\\ a \\varphi _ { x x } ( L , \\cdot ) + \\frac { 1 } { c } \\psi _ { x x } ( L , \\cdot ) = 0 , & \\ , \\ , ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "10003.png", "formula": "\\begin{align*} C _ { x _ { \\alpha _ 0 } ^ { i _ 0 } , \\ldots x _ { \\alpha _ r } ^ { i _ r } } = \\theta [ x _ { \\alpha _ 0 } ^ { i _ 0 } , \\ldots x _ { \\alpha _ r } ^ { i _ r } ] . \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} d _ c ( x , y ) : = \\abs { r ( e ) - x } + \\abs { y } + \\sum _ { k = 1 } ^ n r ( e _ k ) . \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} x \\ominus y = \\begin{cases} x - y , & x - y \\geq 0 \\\\ x - y + 1 , & x - y < 0 \\end{cases} \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { 2 } ( \\left ( \\mathbf { I } + K ( \\lambda , 0 ) \\right ) r _ { j } , r _ { i } ) a _ { j } + \\sum _ { j } ^ { 2 } B _ { i j } ( \\lambda , \\varepsilon ) a _ { j } = 0 , \\ i = 1 , 2 , \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} ( f \\bullet g ) \\bullet \\mu - ( - 1 ) ^ { n - 1 } ( f \\bullet \\mu ) \\bullet g = f \\bullet ( g \\bullet \\mu - ( - 1 ) ^ { n - 1 } \\mu \\bullet g ) \\end{align*}"}
-{"id": "9914.png", "formula": "\\begin{align*} \\mu _ { \\infty } ( \\pi ( N ( L , W ) ) ) > 0 \\mu _ { \\infty } ( \\pi ( S ( L , W ) ) ) = 0 . \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} J _ s = \\left [ \\begin{array} { c c c c c c } 0 & 1 & 0 & \\cdots & 0 & 0 \\\\ 0 & 0 & 1 & \\cdots & 0 & 0 \\\\ 0 & 0 & 0 & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & 0 & 1 \\\\ 0 & 0 & 0 & \\cdots & 0 & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} \\dd \\nu _ t = \\lambda ^ \\top X _ t A \\nu _ t \\ , \\dd t + \\dd M _ t , \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} D _ { \\rm s p } ( \\hat V , V ) = \\inf _ { O \\in \\mathbb { O } _ { r } } \\| \\hat V - V O \\| , D _ { F } ( \\hat V , V ) = \\inf _ { O \\in \\mathbb { O } _ r } \\| \\hat V - V O \\| _ F , \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} \\gamma ' & = S _ u u ' + S _ v v ' , \\\\ \\gamma '' & = S _ { u u } u '^ 2 + S _ { v v } v '^ 2 + S _ u u '' + S _ v v '' + 2 S _ { u v } u ' v ' \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\partial _ t q + u \\cdot \\nabla q + \\bar Q \\nabla \\cdot v = - P , \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} w ' = & 2 e ^ { - \\varphi } w \\tilde { F } \\Theta ^ { - 1 } \\frac { \\sinh \\Theta } { \\cosh \\Theta } v - e ^ { - \\varphi } \\tilde { F } \\Theta ^ { - 1 } \\frac { \\sinh \\Theta } { \\cosh \\Theta } v ^ { - 1 } \\sinh ^ { - 2 } u \\ , u ^ 2 w _ k \\varphi ^ k \\\\ & + R _ 1 + R _ 2 , \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} K ( x ) & = R ^ X ( E _ 1 , E _ 2 , E _ 2 , E _ 1 ) + \\langle ( D _ { E _ 1 } E _ 1 ) ^ \\perp , ( D _ { E _ 2 } E _ 2 ) ^ \\perp \\rangle - \\langle ( D _ { E _ 1 } E _ 2 ) ^ \\perp , ( D _ { E _ 1 } E _ 2 ) ^ \\perp \\rangle \\\\ & = R ^ X ( E _ 1 , E _ 2 , E _ 2 , E _ 1 ) + \\langle A _ { 1 1 } , A _ { 2 2 } \\rangle - \\langle A _ { 1 2 } , A _ { 1 2 } \\rangle . \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\psi \\big ( x ^ n \\big ) = \\sum _ { m = 0 } ^ \\infty e _ { m p - n } ( \\underline \\pi ) x ^ m , \\end{align*}"}
-{"id": "10048.png", "formula": "\\begin{align*} p + q + r = m + n + l , \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to 0 } \\gamma f _ \\gamma ( z ) = z , \\lim _ { \\gamma \\to + \\infty } \\gamma f _ \\gamma ( z ) = g _ \\infty ( z ) , \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} \\bar u ( x , t ) : = \\frac { u \\left ( \\rho x , \\theta t \\right ) } { k } \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} s _ 1 & = ( \\varphi ^ 1 , \\varphi ^ 2 ) , \\ s _ 2 = ( \\psi ^ 1 , \\psi ^ 2 ) \\ \\ \\\\ \\varphi ^ 2 & = \\frac { 1 } { g } \\varphi ^ 1 \\ \\ \\psi ^ 2 = \\frac { 1 } { g } \\psi ^ 1 \\ \\ M _ 1 \\cap M _ 2 . \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} \\| P _ - v _ s \\| ^ 2 = & ( P _ - ^ * P _ - v _ s , v _ s ) \\\\ = & ( ( P _ - ^ * - P _ - ) P _ - v _ s , v _ s ) \\\\ & + ( ( P _ - - e ^ { - i s H } E _ - ( - s ) ) P _ - v _ s , v _ s ) \\\\ & + ( P _ - v _ s , E _ - ( - s ) ^ * v ) . \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 + \\sqrt { 1 - \\lambda } } { 2 } \\ , \\beta = \\frac { 1 + \\sqrt { 1 - \\lambda } } { 2 \\rho \\sigma _ 0 } \\ , \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} b = a _ 1 ^ { z _ 1 } \\cdots a _ k ^ { z _ k } . \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial P _ { \\mu \\nu } } { \\partial x ^ { \\lambda } } + \\frac { \\partial P _ { \\nu \\lambda } } { \\partial x ^ { \\mu } } + \\frac { \\partial P _ { \\lambda \\mu } } { \\partial x ^ { \\nu } } = i e _ { \\mu \\nu \\lambda \\sigma } \\frac { \\partial Q ^ { \\sigma \\tau } } { \\partial x ^ { \\tau } } \\right ) \\overset { \\ast } { \\left . Q ^ { \\lambda \\nu } \\right . } , \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} d _ { k + 1 } ( x ) = d _ k ( x ) - x ^ { k - 2 } n _ k ( x ) + x ^ { 2 k - 1 } . \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} \\gamma ' \\times n = \\gamma ' \\times ( S _ u \\times S _ v ) = \\langle \\gamma ' , S _ v \\rangle S _ u - \\langle \\gamma ' , S _ u \\rangle S _ v \\ ; , \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} & \\| \\sin \\Theta ( \\hat V , V ) \\| _ F ^ 2 = \\sum _ { i = 1 } ^ r \\sin ^ 2 ( \\cos ^ { - 1 } ( \\sigma _ r ) ) \\\\ = & \\sum _ { i = 1 } ^ r ( 1 - \\sigma _ r ^ 2 ) = r - \\sum _ { i = 1 } ^ r \\sigma _ i ^ 2 = r - \\| \\hat V ^ { \\intercal } V \\| _ F ^ 2 . \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} R e \\ , \\big ( F ' _ t \\ , \\overline { F ' } \\ , - \\ , G ' \\ , \\overline { G ' _ t } \\big ) = 0 \\ , . \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} \\Delta _ k ( \\vartheta L , \\vartheta B ) = \\vartheta \\Delta _ k ( L , B ) \\quad \\mbox { f o r } \\vartheta \\in { \\rm S O } ( n ) . \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} h ( z ) = c \\sum _ { u , v \\in V , \\ ; \\langle u , v \\rangle = 1 } E _ u ^ q ( z ) E _ v ( z ) , \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} ( \\tau _ u F ) [ X ] = F [ X + u ] , \\tau : = \\tau _ 1 , \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} & I _ { k , j } '' \\left ( m _ { 1 } , \\dots , m _ { k } \\mid n _ { 1 } , \\dots , n _ { j } \\right ) = \\\\ & \\overline { \\left \\{ ( X , \\ell , p _ { 1 } , \\dots , p _ { k } , \\ell ' , q _ { 1 } , \\dots , q _ { j } ) \\mid X \\cap \\ell = \\sum _ { i } m _ { i } p _ { i } , \\ , \\ , \\ , X \\cap \\ell ' = \\sum _ { i } n _ { i } q _ { i } \\right \\} } \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} P _ { T o t } ^ { P S N } & = \\ N _ { M S } ( P _ { L N A } + P _ { S P } + N _ { C } P _ { P S } ) \\\\ & + N _ { C } P _ C + P _ { R F } + P _ { C o m p } + P _ { S w } + 2 P _ { A D C } \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\frac { \\phi _ { \\nu } ' ( z ) } { \\phi _ { \\nu } ( z ) } = \\frac { 1 } { z } + \\sum _ { n \\geq 1 } \\frac { 1 } { z - j _ { \\nu , n } ^ 2 } , \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} Z ( s , f , \\chi ) = \\frac { ( 1 - q ^ { - 1 } ) ^ 2 q ^ { - 6 - 1 4 s } } { ( 1 - q ^ { - 1 - 2 s } ) ( 1 - q ^ { - 5 - 1 2 s } ) } - \\frac { ( 1 - q ^ { - 1 } ) ^ 2 q ^ { - 9 - 2 0 s } } { ( 1 - q ^ { - 1 - 2 s } ) ( 1 - q ^ { - 9 - 2 0 s } ) } . \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} P _ U ^ C \\ ! = \\ ! \\Pr \\{ \\gamma _ U \\ ! \\ge \\ ! \\beta _ U \\} \\ ! = \\ ! \\ ! \\ ! \\int _ 0 ^ \\infty \\ ! \\ ! \\ ! \\ ! \\Pr \\{ \\gamma _ U \\ge \\beta _ U \\ ! \\mid \\ ! r \\} f _ { U , B ( U ) } ( r ) \\mathrm { d } r . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} ( \\forall \\ , \\lambda > 0 ) \\ \\lim _ { \\gamma \\to 0 ^ + } \\frac { \\gamma \\norm { \\nabla f ( J ( x , \\gamma , \\lambda ) ) - \\nabla f ( x ) } } { \\norm { J ( x , \\gamma ) - x } } = 0 . \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} & r a n k \\left ( \\mathbf { A } _ { j k _ 1 } - \\mathbf { A } _ { j k _ 2 } \\right ) = L ~ ~ ~ \\forall 1 \\leq j \\leq \\omega , 1 \\leq k _ 1 < k _ 2 \\leq d _ j , \\\\ & r a n k \\left ( \\mathbf { I } + ( - 1 ) ^ { \\omega - 1 } \\mathbf { M } _ \\omega \\mathbf { M } _ { \\omega - 1 } \\cdots \\mathbf { M } _ 1 \\right ) = L \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\forall ~ \\mathbf { M } _ j \\in \\left \\{ \\mathbf { A } _ { j 1 } , \\cdots , \\mathbf { A } _ { j d _ j } \\right \\} , 1 \\leq j \\leq \\omega . \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} \\theta _ { s , a ^ 1 } ^ 1 = \\bar { r } ^ 1 ( s , a ^ 1 ) + \\beta \\sum _ { s ' \\in S } p ^ 1 ( s ' | s , a ^ 1 ) u ^ { 1 * } ( s ' ) - u _ \\beta ^ { 1 * } ( s ) . \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} g ( 2 ) = ( q ^ 2 + 1 ) ( q - 1 ) ( r - 1 - \\alpha q ) . \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} \\sum _ { i \\in ( 0 . 5 \\pm \\epsilon ' ) | R | } \\binom { | R | } { i } 2 ^ { i } \\leq 2 ^ { ( 1 . 5 + \\epsilon ' ) | R | } \\leq 2 ^ { 1 . 5 | R | + n \\sqrt { \\ln ( 2 ) \\epsilon / 2 } } \\ , . \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} A d x + B d y + C d z = 0 \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} X = U _ \\alpha ( \\psi _ 1 ) \\cup \\dots \\cup U _ \\alpha ( \\psi _ n ) . \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} T _ { 1 , \\bullet } = 1 . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} \\frac { \\partial W ( g ( \\Lambda ) ) } { \\partial \\ln \\Lambda } + \\beta ( g ) \\frac { \\partial W ( g ) } { \\partial g } = 0 \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} \\sum _ { i } ( u _ { i l } ^ * u _ { i j } \\xi _ k - \\delta _ { i j } u _ { i l } ^ * \\xi _ k ) = \\delta _ { l j } \\xi _ k - u _ { j l } ^ * \\xi _ k \\to 0 ( 1 \\le l , j \\le n ) . \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} Q ^ { ( 5 ) } ( A , \\bar { A } ) = Q ^ { ( 5 ) } ( A , A _ { 2 } ) + Q ^ { ( 5 ) } ( A _ { 2 } , A _ { 1 } ) + Q ^ { ( 5 ) } ( A _ { 1 } , \\bar { A } ) + d B , \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} F _ { \\beta _ { s } } F _ { \\beta _ { l + 2 } } - q ^ { ( \\beta _ s , \\beta _ { l + 2 } ) } F _ { \\beta _ { l + 2 } } F _ { \\beta _ { s } } = \\sum _ { n _ { s + 1 } , \\ldots , n _ { l + 1 } \\geq 0 } c ( n _ { s + 1 } , \\ldots , n _ { l + 1 } ) F _ { \\beta _ { s + 1 } } ^ { n _ { s + 1 } } \\ldots F _ { \\beta _ { l + 1 } } ^ { n _ { l + 1 } } . \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} & \\int _ t ^ T \\int _ { \\mathcal { O } } ( u ( s , x ) - \\hat { \\xi } ( s , x ) ) \\ , \\mu ( d s d x ) \\\\ = & \\int _ t ^ T \\int _ { \\mathcal { O } } ( u ( s , x ) - \\xi ( s , x ) + \\xi ( s , x ) - \\hat { \\xi } ( s , x ) ) \\ , \\mu ( d s d x ) \\\\ \\leq & 0 . \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} \\psi ( s , \\cdot ) : = \\varphi ^ { - 1 } ( s , \\cdot ) \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} \\frac { \\partial Q ^ { \\mu \\nu } } { \\partial x ^ { \\nu } } = - \\frac { 4 \\pi } { c } j ^ { \\mu } . \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} U = \\{ \\emptyset \\} \\ \\cup \\ \\{ \\nu \\in S - \\{ \\emptyset \\} \\ : \\ \\nu \\neq \\psi ( \\nu ^ - ) \\} . \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} h _ + = \\inf \\frac { \\mu _ + ( E ) } { \\mu ( E ) } \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} K _ { \\alpha \\beta } ^ { \\ \\ \\ \\gamma } = \\left \\{ \\begin{array} [ c ] { c c } 1 & \\ \\ \\lambda _ { \\alpha } \\lambda _ { \\beta } = \\lambda _ { \\gamma } \\\\ 0 & o t h e r w i s e , \\end{array} \\right . \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} M = s I - P , P \\in \\mathbb { R } _ { \\geq 0 } ^ { n \\times n } , s \\geq 0 . \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a ; q \\right ) _ { n } S _ { n } \\left ( c x ; q \\right ) z ^ { n } } { \\left ( c q ; q \\right ) _ { n } } = \\frac { \\left ( a z ; q \\right ) _ { \\infty } } { \\left ( z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } \\left ( a ; q \\right ) _ { n } \\left ( c z \\right ) ^ { n } } { \\left ( c q , a z ; q \\right ) _ { n } } S _ { n } \\left ( x q ^ { - n } ; q \\right ) . \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} U _ x = \\{ z \\in X \\mid u < \\hom ( \\psi ( x ) , \\psi ( z ) ) \\} \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} & r _ { p _ 1 ^ i p _ 2 ^ j } = 0 , i < n _ 1 - 1 j < n _ 2 - 1 \\\\ & r _ { p _ 1 ^ { n _ 1 - 1 } p _ 2 ^ { n _ 2 - 1 } } = \\frac { 2 4 } { ( p _ 1 - 1 ) ( p _ 2 - 1 ) } = x \\\\ & r _ { p _ 1 ^ { n _ 1 - 1 } p _ 2 ^ { n _ 2 } } = - p _ 2 x \\\\ & r _ { p _ 1 ^ { n _ 1 } p _ 2 ^ { n _ 2 - 1 } } = - p _ 1 x \\\\ & r _ { p _ 1 ^ { n _ 1 } p _ 2 ^ { n _ 2 } } = p _ 1 p _ 2 x \\\\ \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} \\frac { { g \\left ( { x _ 1 } \\right ) f \\left ( { x _ 2 } \\right ) - g \\left ( { x _ 2 } \\right ) f \\left ( { x _ 1 } \\right ) } } { { g \\left ( { x _ 2 } \\right ) - g \\left ( { x _ 1 } \\right ) } } = f \\left ( \\xi \\right ) - \\frac { { g \\left ( \\xi \\right ) } } { { g ' \\left ( \\xi \\right ) } } f ' \\left ( \\xi \\right ) . \\end{align*}"}
-{"id": "10154.png", "formula": "\\begin{align*} \\alpha ( y _ k - x _ k ) + \\beta ( y _ k - x ^ * ) = \\beta ( z _ k - x ^ * ) . \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} f ( y ) = f _ 0 ( y '' ) y _ n + r ( y ) ^ { 1 + 2 \\alpha - \\epsilon } f _ 1 ( y ) , r ( y ) = \\sqrt { y _ n ^ 2 + y _ { n + 1 } ^ 2 } , \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} \\theta ^ i _ j = \\sum _ k F ^ i _ { j k } \\omega ^ k . \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} \\mu ( B ( x + 2 m k z , r ) ) = m \\Lambda _ B + \\mu ( B ( x , r ) ) , \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} - \\mathcal { T } _ 0 = \\frac { 1 } { 4 \\pi ^ 2 } \\sum _ { k \\neq 0 } \\int \\ ! \\ ! \\ ! \\int \\ ! \\ ! \\ ! \\int \\left ( A ( k , \\xi ) - A ( k , \\xi - \\eta ) \\right ) \\frac { \\eta } { \\eta ^ 2 } \\hat { f } ( 0 , \\eta ) k \\hat { f } ( k , \\xi - \\eta ) A ( k , \\xi ) \\bar { \\hat { f } } ( k , \\xi ) \\ , d \\eta d \\xi d t . \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\begin{aligned} F ( z ) & \\geq F _ t ( z ; x ) - \\tfrac { \\mu + t ^ { - 1 } } { 2 } \\| z - x \\| ^ 2 \\geq F _ { t } ( { \\bar x } ; x ) + \\tfrac { 1 } { 2 t } \\| { \\bar x } - z \\| ^ 2 - \\tfrac { \\mu + t ^ { - 1 } } { 2 } \\| z - x \\| ^ 2 \\\\ & \\geq F ( { \\bar x } ) + \\frac { 1 } { 2 t } \\| { \\bar x } - z \\| ^ 2 - \\tfrac { \\mu + t ^ { - 1 } } { 2 } \\| z - x \\| ^ 2 + \\tfrac { t ^ { - 1 } - \\mu } { 2 } \\| { \\bar x } - x \\| ^ 2 , \\end{aligned} \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} \\frac { ( 1 - \\sigma ) \\delta ^ l } { \\alpha ^ l } = \\frac { ( l + 1 ) \\zeta _ i } { l } > \\zeta _ i . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\Lambda ^ 2 _ 7 W _ 8 ^ \\ast & = \\{ \\alpha ^ 2 \\in \\Lambda ^ 2 W _ 8 ^ \\ast \\mid \\Phi \\wedge \\alpha ^ 2 = 3 \\ast \\alpha ^ 2 \\} , \\mbox { a n d } \\\\ \\Lambda ^ 2 _ { 2 1 } W _ 8 ^ \\ast & = \\{ \\alpha ^ 2 \\in \\Lambda ^ 2 W _ 8 ^ \\ast \\mid \\Phi \\wedge \\alpha ^ 2 = - \\ast \\alpha ^ 2 \\} . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} \\eta ( \\omega _ 2 ) \\ , \\omega _ 1 - \\eta ( \\omega _ 1 ) \\ , \\omega _ 2 = 2 \\pi i , \\ \\ \\Im \\left ( \\frac { \\omega _ 1 } { \\omega _ 2 } \\right ) > 0 . \\end{align*}"}
-{"id": "10152.png", "formula": "\\begin{align*} \\varphi ( y ) & = \\ell ( y ; y ) \\geq \\ell ( p ( y ) ; y ) \\\\ & = g ( p ( y ) ) + f ( e ( y ) + \\nabla e ( y ) ( p ( y ) - y ) ) + \\frac { L } { 2 } \\| p ( y ) - y \\| ^ 2 \\\\ & \\geq ( g ( y ) + \\langle z , p ( y ) - y \\rangle ) + ( f ( e ( y ) ) + \\langle w , \\nabla e ( y ) ( p ( y ) - y ) \\rangle ) + \\frac { L } { 2 } \\| p ( y ) - y \\| ^ 2 \\\\ & = \\varphi ( y ) + \\langle z + \\nabla e ( y ) ^ T w , p ( y ) - y \\rangle + \\frac { L } { 2 } \\| p ( y ) - y \\| ^ 2 , \\end{align*}"}
-{"id": "10047.png", "formula": "\\begin{align*} ( 1 + | B | ) \\displaystyle \\sum ^ \\infty _ { n = 2 } n [ 1 + ( n - 1 ) ( \\lambda - \\mu + n \\lambda \\mu ) ] | a _ n | + ( 1 + | A | ) \\displaystyle \\sum ^ \\infty _ { n = 2 } | B _ n | \\leq A - B \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} 8 \\sin ( k / 2 ) \\sin ( ( \\varphi - \\phi + k ) / 2 ) \\cos ( ( \\phi + \\varphi ) / 2 ) = 0 . \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} g _ { n } ^ { ( t , x ) } ( \\cdot , s , y ) = 1 _ { [ 0 , t ] } ( s ) \\lambda G ( t - s , x - y ) \\widetilde { f } _ n ( \\cdot , s , y ) , \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\omega ^ { \\mathrm { z m } , - } \\big | _ { \\partial Z _ { 1 , 0 } } = e ^ { i \\lambda R } \\phi _ 1 = : \\phi , \\omega ^ { \\mathrm { z m } , + } \\big | _ { \\partial Z _ { 1 , 0 } } = e ^ { - i \\lambda R } \\phi ' _ 1 = : \\phi ' . \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} \\xi _ { n } = \\sum _ { \\overline { A } \\in \\overline { \\mathfrak { B } } _ n , \\ ; \\varepsilon ( \\overline { A } ) \\ge 2 } \\left ( - 1 \\right ) ^ { \\varepsilon ( \\overline { A } ) } \\left | \\overline { A } \\right | \\prod _ { i = 0 } ^ { n - 2 } \\left [ \\left ( n - i \\right ) ! \\right ] ^ { \\psi _ i ( \\overline { A } ) } \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} \\Lambda _ f ( s ) = \\left ( \\frac { \\sqrt { N } } { 2 \\pi } \\right ) ^ s \\Gamma \\left ( s + \\frac { 2 k - 1 } { 2 } \\right ) L _ f ( s ) \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} u '' + a ( t ) f ( u ) = 0 , \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum \\limits _ { i = 1 } ^ n { 2 n \\choose 2 i } B _ { 2 n - 2 i } \\gamma _ { 2 i + 1 } & = \\frac { 1 } { n } \\sum \\limits _ { i = 1 } ^ n { 2 n \\choose 2 i } B _ { 2 n - 2 i } \\sum \\limits _ { j = 0 } ^ { i } { i \\brack j } \\gamma _ { 2 j } \\\\ & = \\frac { 1 } { n } \\sum \\limits _ { j = 0 } ^ { n } \\gamma _ { 2 j } \\sum \\limits _ { i = j } ^ n { 2 n \\choose 2 i } B _ { 2 n - 2 i } { i \\brack j } . \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} a _ 0 ( y , x ) = \\inf _ { \\psi \\in C X , \\psi ( x ) = 1 } \\psi ( y ) . \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} \\phi \\big ( M ( a _ { 1 ; i } \\otimes \\cdots \\otimes a _ { K ; i } : i \\in I ) \\big ) = \\phi _ 1 \\big ( M ( a _ { 1 ; i } : i \\in I ) \\big ) \\cdots \\phi _ K \\big ( M ( a _ { K ; i } : i \\in I ) \\big ) \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} c _ \\Gamma [ S ] : = S \\ , \\frac { S ^ \\tau } { \\prod _ { v \\in V } p _ { m ( v ) } [ S ] } = S \\ , \\frac { \\prod _ { e \\in E } p _ { m ( e ) } [ S ] } { \\prod _ { v \\in V } p _ { m ( v ) } [ S ] } . \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\delta _ 2 = 0 ; \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} [ \\xi _ { i } ( u ) , \\xi _ { i } ( v ) ] = 0 , \\ , \\ [ \\xi _ { i } ( u ) , h ] = 0 , \\ , \\ [ h , h ' ] = 0 \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} H ( t ) = P \\left ( t \\varphi + \\psi \\right ) . \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} L i n ( W _ 8 , W _ 7 ) : = W _ 8 ^ * \\otimes W _ 7 = W _ 8 \\oplus W _ { 4 8 } . \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ { I V } ^ + } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq C _ { d , s , k } C _ { d , \\alpha } T R \\eta ^ { d - 1 } \\end{aligned} \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { \\left ( b \\pm \\frac { t } { 2 } \\sqrt { a ^ 2 + 4 c t ^ 2 } \\pm \\frac { a ^ 2 } { 4 \\sqrt { c } } \\ln | 2 \\sqrt { c } t + \\sqrt { a ^ 2 + 4 c t ^ 2 } | \\right ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} \\beta ( u ) = ( \\gamma + c ^ { 2 } \\gamma ^ { \\prime \\prime } ) ( u ) = \\left ( ( f _ { 1 } + c ^ { 2 } f _ { 1 } ^ { \\prime \\prime } ) ( u ) , . . . , ( f _ { n + 1 } + c ^ { 2 } f _ { n + 1 } ^ { \\prime \\prime } ) ( u ) \\right ) , \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} \\mu ( \\overset { h _ { N } - 1 } { \\underset { i = 0 } { \\cup } } ( T ^ { q } ) ^ { i } ( A ^ { * } _ { N } ) ) & > ( 1 - \\frac { ( q - 1 ) ( q - 2 ) } { 2 } ) \\frac { 1 } { b ^ { N - 1 } } \\mu ( \\overset { h _ { N } - 1 } { \\underset { i = 0 } { \\cup } } T ^ { i } ( B ^ { * } _ { N } ) ) \\\\ & = ( 1 - \\frac { ( q - 1 ) ( q - 2 ) } { 2 } \\frac { 1 } { b ^ { N - 1 } } ) h _ { N } \\mu ( B ^ { * } _ { N } ) \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} - u '' ( x ) + \\left [ x ^ 2 + q ( x ) \\right ] u ( x ) = \\lambda u ( x ) , \\ ; \\ ; x \\in \\R , \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} \\mathcal { B } ^ - _ { I I } = \\left \\{ \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { A } ^ - \\left | \\inf _ { 1 \\leq i \\leq s + k } \\left | v _ { s + k + 1 } - v _ i ^ \\prime \\right | \\leq \\eta \\right . \\right \\} \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} F _ 2 ( V , W ) ( ( v \\otimes g ) \\otimes ( w \\otimes h ) ) = v \\otimes \\gamma _ g ( w ) \\otimes g h ( v \\in V _ g , w \\in W _ h ) . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} { C } _ { X ^ n \\rightarrow Y ^ n } ^ { F B , A . B } ( \\kappa ) = \\sup _ { { \\cal P } _ { 0 , n } ^ { B } ( \\kappa ) } \\sum _ { t = 0 } ^ n { \\bf E } ^ { \\pi } \\left \\{ \\log \\Big ( \\frac { q _ t ( \\cdot | Y _ { t - M } ^ { t - 1 } , X _ t ) } { \\nu _ t ^ { { \\pi } } ( \\cdot | Y ^ { t - 1 } ) } ( Y _ t ) \\Big ) \\right \\} , \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} { Y } _ s = & \\ \\Phi ( { X } _ T ) - \\int ^ T _ s { Z } _ r \\d { W } _ r + \\int ^ T _ s f ( r , { X } _ r , { Y } _ r , { Z } _ r ) \\d r \\\\ & - w ( s , { X } _ s ) - \\int ^ T _ s \\nabla w ( r , { X } _ r ) \\d { W } _ r \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} \\sigma _ j \\sigma _ 1 ^ { - 1 } \\sigma _ k \\sigma _ 2 ^ { - 1 } \\sigma _ 1 = \\sigma _ k \\sigma _ 1 ^ { - 1 } \\sigma _ j \\sigma _ 2 ^ { - 1 } \\sigma _ 1 . \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} u ^ i = g ^ { i j } u _ j = v ^ { - 2 } \\bar { g } ^ { i j } u _ j . \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} n ^ * { = } a r g \\max _ { n { = } 2 { , } \\cdots { , } K } \\frac { n ^ 2 } { 1 { + } \\frac { n ( n { - } 1 ) } { d _ 2 ( K { , } 1 { , } K ) } } . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} \\mathbf { m } _ \\ell = f _ { \\ell , i _ { \\sigma ( \\ell ) } - 1 } \\ ; , \\ ; \\mathbf { m } _ j = f _ { j , i _ { \\sigma ( j ) } - 1 } . \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} f _ 1 ( \\vect { x } ) & = \\begin{bmatrix} y ^ 2 + 1 0 y + 2 5 \\\\ 2 x y + 1 0 x - 4 0 y - 2 0 0 \\end{bmatrix} , \\\\ f _ 2 ( \\vect { x } ) & = \\begin{bmatrix} - y ^ 2 - 1 0 y - 2 5 \\\\ 8 x y + 4 0 x - 1 6 0 y - 8 0 0 \\end{bmatrix} \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} C ^ { F B , A . 1 } _ { X ^ n \\rightarrow { Y ^ n } } ( \\kappa ) = \\inf _ { s \\geq { 0 } } \\sum _ { y _ { - 1 } \\in \\{ 0 , 1 \\} } \\Big ( K ^ s _ 0 ( y _ { - 1 } ) \\mu ( y _ { - 1 } ) + ( n + 1 ) \\kappa \\Big ) , ~ \\mu ( y _ { - 1 } ) ~ \\mbox { i s f i x e d } . \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} { \\bf { E } } \\Bigl [ \\Bigl ( \\int _ 0 ^ 1 s ^ { \\lambda _ 1 } ( 1 - s ) ^ { \\lambda _ 2 } \\ , M _ \\beta ( d s ) \\Bigr ) ^ n \\Bigr ] = \\int \\limits _ { [ 0 , \\ , 1 ] ^ n } \\prod _ { i = 1 } ^ n s _ i ^ { \\lambda _ 1 } ( 1 - s _ i ) ^ { \\lambda _ 2 } \\prod _ { i < j } ^ n | s _ i - s _ j | ^ { - 2 \\beta ^ 2 } d s _ 1 \\cdots d s _ n . \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} h ( z ) = i \\frac { 1 + z } { 1 - z } , \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} \\phi _ { 2 } ^ { ( \\alpha ) } ( k ) = ( \\alpha | k | ^ { \\alpha / 2 - 1 } - 4 | k | ^ { \\alpha } ) \\exp \\left ( - \\frac { 2 | k | ^ { \\alpha / 2 + 1 } } { \\alpha + 2 } \\right ) , \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\kappa = \\sqrt { ( f _ { 1 } { } ^ { \\prime \\prime } ) ^ { 2 } + ( f _ { 2 } { } ^ { \\prime \\prime } ) ^ { 2 } + \\frac { \\lambda ^ { 2 } } { c ^ { 4 } } \\cos ^ { 2 } \\left ( \\frac { u } { c } \\right ) } , \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} \\tau = R / d . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\nabla ^ i X _ { i r } - ( \\nabla ^ i f ) X _ { i r } = \\frac { 1 } { 2 } \\partial _ { r } S \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\partial ^ L \\varphi ( x ) : = \\left \\{ v \\in \\Re ^ n : \\exists ( x ^ k , v ^ k ) \\to ( x , v ) \\ { \\rm w i t h } \\ \\lim _ { y \\to x ^ k } \\frac { f ( y ) - f ( x ^ k ) - \\langle v ^ k , y - x ^ k \\rangle } { \\| y - x ^ k \\| } \\geq 0 \\ \\forall k \\right \\} . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\left ( ( u - l ) ^ + - ( u - k ) ^ + \\right ) 1 _ { ( u > k ) } = ( k - l ) 1 _ { ( u > k ) } , \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\mathcal { A } ( x ) = \\frac { \\zeta ( 3 / 2 ) } { \\zeta ( 3 ) } x ^ { 1 / 2 } + O _ { \\epsilon } ( x ^ { 1 / 3 + \\epsilon } ) . \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{gather*} [ ( \\Delta \\otimes _ A \\mathrm { i d } ) ( \\mathcal { F } ) ] ( \\mathcal { F } \\otimes _ A 1 ) = [ ( \\mathrm { i d } \\otimes _ A \\Delta ) ( \\mathcal { F } ) ] ( 1 \\otimes _ A \\mathcal { F } ) \\end{gather*}"}
-{"id": "2755.png", "formula": "\\begin{align*} & \\underset { y _ { i _ t } \\in \\mathbb { R } ^ { m \\times 1 } } { } \\ ; | | e _ { i _ t } - A ^ T y _ { i _ t } | | _ { \\infty } \\ ; \\\\ & = \\underset { x } { } \\big \\{ e _ { i _ t } ^ T x \\ ; : \\ ; A x = 0 , \\ ; | | x | | _ 1 \\leq 1 \\big \\} \\\\ & = \\alpha _ { 1 , \\{ i _ t \\} } . \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} \\sigma h + h ^ { t r } \\sigma = 0 . \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} \\begin{cases} v _ i ^ * = v _ i + \\omega \\omega \\cdot \\left ( v _ j - v _ i \\right ) \\\\ v _ j ^ * = v _ j - \\omega \\omega \\cdot \\left ( v _ j - v _ i \\right ) \\end{cases} \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} | A | = \\int _ 0 ^ { 2 ^ { n + 1 } } N ( x ) \\ , d x . \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\gamma _ { 2 n } = \\sum _ { i = 0 } ^ n { n \\brack i } _ r \\gamma _ { 2 i + 1 } \\mbox { f o r a l l } n \\ge 0 \\mbox { w i t h } n \\ne - r / 2 \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{align*} \\beta _ \\alpha : = \\alpha 1 _ { \\alpha \\in ( 0 , 1 ) } + \\beta 1 _ { \\alpha = 1 } + 1 _ { \\alpha \\in ( 1 , 2 ) } . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} b _ 1 & = 2 ( a _ 1 + a _ 2 + a _ 3 + a _ 4 + a _ 5 ) , & b _ 2 & = 2 ( a _ 1 + a _ 2 + 2 a _ 3 + 2 a _ 4 + a _ 5 ) , \\\\ b _ 3 & = 2 ( a _ 3 + a _ 4 ) , & b _ 4 & = b _ 5 = 2 ( a _ 1 + a _ 2 + a _ 5 ) . \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\hat { R } \\left ( t \\right ) \\ , d t = \\infty , \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} F \\left ( x _ { 1 } v _ { 1 } + x _ { 2 } v _ { 2 } \\right ) = u _ { 0 } + f \\left ( x _ { 1 } \\right ) u _ { 1 } + g \\left ( x _ { 2 } \\right ) u _ { 2 } \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} \\mathcal { F } ^ { ( \\delta ) } = \\biguplus _ { n \\in I } K _ n ^ { ( \\delta ) } \\cup F . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} e ^ { - \\beta F } = \\ , e ^ { - \\beta H } \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} u '' + c u ' + a ( t ) u ^ { p } = 0 , \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} B ^ s _ { p , \\infty } ( \\mathbb { R } ^ d ) = \\left \\{ f \\in \\mathcal { S } ' ( \\mathbb { R } ^ d ) \\ , \\ , | \\ , \\ , f \\stackrel { \\mathcal { S } ' } { = } \\sum _ { j = 0 } ^ { \\infty } a _ j ( x ) ; \\ , \\ , \\mathrm { s u p p } F a _ j \\subset M _ j ; \\right . \\\\ \\left . \\| \\{ a _ j \\} \\| _ { l ^ s _ \\infty ( L _ p ) } = \\sup _ { j \\in \\mathbb { Z } _ + } 2 ^ { s j } \\| a _ j \\| _ { L _ p ( \\mathbb { R } ^ d ) } < \\infty \\right \\} , \\ , \\ , \\ , \\ , \\ , \\ , q = \\infty \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} \\min _ { j = 1 , \\ldots , N } \\norm { \\mathcal { G } _ { 1 / \\tilde { \\mu } } ( y _ j ) } ^ 2 \\le \\frac { 2 4 \\tilde { \\mu } ^ 2 } { \\tilde { \\mu } - \\mu } \\left ( \\frac { \\tilde { \\mu } \\norm { x ^ * - v _ 0 } ^ 2 } { N ( N + 1 ) ( 2 N + 1 ) } + \\frac { M ^ 2 ( r + \\frac { \\rho } { 2 } ( N + 3 ) ) } { ( N + 1 ) ( 2 N + 1 ) } \\right ) . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} h _ { a , b , c } ( z ) : = \\sum _ { r = 1 } ^ { a + 2 b + c } H _ { a , b , c } ^ { ( r ) } z ^ { r - 1 } \\end{align*}"}
-{"id": "10139.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { z ^ { - ( p + 2 q ) } ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q } { x ^ { - p } } , \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} u ( t ) = \\frac { h ( t ) } { \\Big ( \\int _ M | h ( t ) | ^ p d \\mu _ { g ( t ) } \\Big ) ^ { \\frac { 1 } { p } } } \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} r ( x , y , \\eta ) = p _ \\pm ( y , \\xi ( \\eta ) ) \\left | \\det \\left ( \\frac { d \\xi } { d \\eta } \\right ) \\right | - p _ \\pm ( y , \\eta ) . \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} - \\Re \\Big \\{ \\frac { L ' } { L } ( s , \\chi ) \\Big \\} = - \\sum _ { | s - \\rho | < R } \\Re \\Big \\{ \\frac { 1 } { s - \\rho } - \\frac { s - \\rho } { R ^ 2 } \\Big \\} - J \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} c _ { \\Gamma } [ S ] = S \\ , p _ { m ( e ) } [ S ] p _ g \\left [ \\frac { c _ { \\Gamma '' } [ S ] } { S } \\right ] = S \\ , \\frac { p _ { m ( e ) } [ S ] } { p _ g [ S ] } p _ g [ c _ { \\Gamma '' } [ S ] ] . \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} u _ { t } = \\mathrm { d i v } \\ , \\Big ( \\frac { \\nabla u } { | \\nabla u | } \\Big ) ( 0 , \\infty ) \\times G \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} \\Omega F = \\frac { 1 } { 1 2 } ( ( k _ 1 - 1 ) ^ 2 + ( k _ 2 - 2 ) ^ 2 - 5 ) F , \\Delta F = ( ( k _ 1 - 1 ) ( k _ 2 - 2 ) ) ^ 2 F , \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} f \\bullet g = \\sum _ { i = 1 } ^ { m } ( - 1 ) ^ { ( i - 1 ) ( n - 1 ) } f \\bullet _ i g . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } \\ , \\frac { 1 } { R } \\ , \\# \\left \\{ \\lambda \\in \\Lambda , \\ , \\ , t \\le \\lambda \\le t + R , \\ , \\ , \\lambda \\ , ( { \\rm m o d } \\ , a ^ { - 1 } ) \\in I \\right \\} = a \\ , | I | \\ , A , \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l l l l } $ u ( 0 , t ) = 0 $ , & $ u ( L , t ) = 0 $ , & $ u _ { x } ( L , t ) = h _ 2 ( t ) $ & i n $ ( 0 , T ) $ , \\\\ $ v ( 0 , t ) = g _ 0 ( t ) $ , & $ v ( L , t ) = g _ 1 ( t ) $ , & $ v _ { x } ( L , t ) = g _ 2 ( t ) $ & i n $ ( 0 , T ) $ , \\end{tabular} \\right . \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} D ' = \\min _ { M \\in \\Pi _ 0 } D ( M | p ' ) , \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} x _ { i j } \\cdot x _ { d - i w } = x _ { i j } x _ { d - i w } = m = x _ { d - i w } x _ { i j ' } = x _ { d - i w } \\cdot x _ { i j ' } . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} \\mathcal { S } : = \\left \\{ x \\in \\R ^ n \\left | \\begin{array} { c } \\phi ( t , x ) \\forall t \\ge 0 , \\\\ \\lim _ { t \\rightarrow \\infty } \\phi ( t , x ) = x ^ { * } \\end{array} \\right . \\right \\} , \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} \\abs { \\frac { \\log ^ + \\log \\abs { X _ t } } { \\log t } - 1 } = \\abs { \\frac { \\log \\log \\abs { X _ t } } { \\log t } - 1 } = \\abs { \\frac { \\log \\log \\abs { X _ t } - \\log t } { \\log t } } = \\abs { \\frac { \\log \\frac { \\log \\abs { X _ t } } { t } } { \\log t } } \\to 0 \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} \\sup _ { y \\in \\bar { B } ( x , r ) } d ( f ( x ) , f ( y ) ) & = \\sup _ { y \\in \\bar { B } ( x , r ) } \\big ( h _ { f ( x ) } ( f _ * V , f _ * V ) ^ { 1 / 2 } + o ( d _ g ( x , y ) ) \\big ) \\\\ & = \\sup _ { | V | _ { g ( x ) } \\leq r } | D f V | _ { h ( f ( x ) ) } + o ( r ) \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\beta } u ( x _ { 0 } , t _ { 0 } ) = c _ { n , \\beta } \\int _ { \\mathbb { R } ^ { n } } \\frac { u ( x _ { 0 } , t _ { 0 } ) - u ( x , t _ { 0 } ) } { | x _ { 0 } - x | ^ { N + 2 \\beta } } d x \\leq 0 . \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} \\overline \\nabla _ X Y = \\nabla _ X Y + h ( X , Y ) , \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { 2 m } u = \\lambda \\ , ( - \\Delta ) ^ m u , u \\in W _ 0 ^ { 2 m , 2 } ( \\Omega ) , \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} v ^ { k + 1 } ( \\ell ) = W ( k ) v ^ k ( \\ell ) + \\zeta ^ { k + 1 } ( \\ell ) \\hbox { w i t h } \\zeta ^ { k + 1 } ( \\ell ) = x ^ { k + 1 } ( \\ell ) - x ^ k ( \\ell ) \\qquad \\hbox { f o r a l l } \\ell = 1 , \\ldots , n . \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{gather*} \\left \\langle Q ^ { m - n - 1 } v _ 0 , \\psi ^ { - } ( z ) \\prod _ { i = 1 } ^ { m } \\psi ^ { + } ( w _ i ) \\prod _ { i = 1 } ^ { n } \\psi ^ { - } ( y _ i ) v _ 0 \\right \\rangle \\\\ { } = \\frac { \\prod \\limits _ { i = 1 } ^ { m } ( z - y _ { i } ) \\prod \\limits _ { 1 \\le i < j \\le m } ( w _ i - w _ j ) \\prod \\limits _ { 1 \\le i < j \\le n } ( y _ i - y _ j ) } { \\prod \\limits _ { i = 1 } ^ { n } ( z - w _ i ) \\prod \\limits _ { i = 1 } ^ { m } \\prod \\limits _ { j = 1 } ^ { n } ( w _ i - y _ j ) } . \\end{gather*}"}
-{"id": "3562.png", "formula": "\\begin{align*} A d = 0 \\ \\ \\ { \\rm a n d } \\ \\ \\ H _ 1 ( d ) \\le 1 . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} ( \\Delta f ) | \\tau ( \\phi ) | ^ 2 + ( { \\rm g r a d } \\ , f ) | \\tau ( \\phi ) | ^ 2 = 0 \\ ; \\ ; \\forall \\ ; f : ( U , g | _ U ) \\longrightarrow ( 0 , \\infty ) . \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} \\mathfrak { a } ( x , T ^ { \\psi } _ { \\varphi _ i ( h ) } \\nabla u ) - \\mathfrak { a } ( x , \\nabla u ) \\leq \\left [ T ^ { \\psi } _ { \\varphi _ i ( h ) } \\mathfrak { a } ( x , \\nabla u ) \\right ] - \\mathfrak { a } ( x , \\nabla u ) = \\psi \\left [ \\mathfrak { a } ( x , \\nabla u _ { \\varphi _ i ( h ) } ) - \\mathfrak { a } ( x , \\nabla u ) \\right ] ; \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a , b ; q \\right ) _ { n } z ^ { n } } { \\left ( q , c ; q \\right ) _ { n } } = \\frac { \\left ( c / b , b z ; q \\right ) _ { \\infty } } { \\left ( c , z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a b z / c , b ; q \\right ) _ { n } } { \\left ( q , b z ; q \\right ) _ { n } } \\left ( \\frac { c } { b } \\right ) ^ { n } \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ A ( \\Omega ) } = \\inf \\left \\{ \\lambda > 0 : \\ , \\ , \\int _ \\Omega A \\Big ( \\frac { | u ( x ) | } { \\lambda } \\Big ) \\ , d x \\leq 1 \\right \\} \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} \\rho _ { \\mathrm { e x t } } \\hat \\otimes 1 ( \\omega _ 1 \\hat \\otimes \\omega _ 2 ) & = ( e _ 1 \\wedge \\omega _ 1 - \\iota ( e _ 1 ) \\omega _ 1 ) \\hat \\otimes \\omega _ 2 \\\\ 1 \\hat \\otimes \\rho ^ j ( \\omega _ 1 \\hat \\otimes \\omega _ 2 ) & = ( - 1 ) ^ { | \\omega _ 1 | } \\omega _ 1 \\hat \\otimes ( e _ j \\wedge \\omega _ 2 - \\iota ( e _ j ) \\omega _ 2 ) , \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} M ( t ) = \\mathrm { g r a p h } \\ , u ( t , \\cdot ) . \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} y ^ { [ \\sf d ] } _ { i j } [ t ] & = \\sum _ { k = 1 } ^ { K } \\mathbf { { g } } _ { i j , k } [ t ] { \\mathbf { x } } ^ { [ \\sf b s ] } _ { k } [ t ] + \\sum _ { k = 1 } ^ { K } \\sum _ { l = 1 } ^ { N } h _ { i j , k l } [ t ] x ^ { [ \\sf u ] } _ { k l } [ t ] + z _ { i j } ^ { [ \\sf d ] } [ t ] \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\frac { \\Gamma _ 2 ( z + 1 - k \\ , | \\ , \\tau ) } { \\Gamma _ 2 ( z + 1 \\ , | \\ , \\tau ) } = \\bigl ( \\frac { 1 } { 2 \\pi \\tau } \\bigr ) ^ { k / 2 } \\tau ^ { \\sum \\limits _ { j = 0 } ^ { k - 1 } ( z - j ) / \\tau } \\ ; \\prod \\limits _ { j = 0 } ^ { k - 1 } \\Gamma \\bigl ( \\frac { z } { \\tau } - \\frac { j } { \\tau } \\bigr ) . \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} 0 = \\frac { \\partial \\lambda } { \\partial \\tau } = \\frac { \\partial } { \\partial \\tau } \\lambda ( \\tau ) + a _ n ( g , D ) \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { A ( k ) } v ^ 2 + \\int _ { A ( k ) } | D v | ^ 2 & \\leq \\sigma p \\int _ { A ( k ) } H ^ 2 f _ \\sigma ^ p + 5 C p \\int _ { A ( k ) } f _ \\sigma ^ p \\\\ & \\leq C ( p ) \\int _ { A ( k ) } H ^ 2 f _ \\sigma ^ p . \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} a = c \\ , ( p _ 0 ( \\lambda ) , \\ , p _ 1 ( \\lambda ) , \\ , p _ 2 ( \\lambda ) , \\ldots , \\ , p _ { n - 1 } ( \\lambda ) ) , a = c \\ , ( 0 , 0 , \\ldots , 0 , 1 ) \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} H ( \\textbf { q } , \\textbf { p } , t ) = \\frac { 1 } { 2 m } \\| \\textbf { p } - e \\textbf { A } ( \\textbf { q } , t ) \\| _ 2 ^ 2 + e \\phi ( \\textbf { q } , t ) , \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} x _ 0 = x _ 1 = \\cdots = x _ { 1 6 } = l ^ \\prime = q _ 0 = q ^ \\prime = 0 . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} \\tilde { \\Psi } ( y _ c ( B ) ) = \\sum _ { x \\in \\eta } \\tilde { l } _ \\alpha ( | x - y _ c ( B ) | ) \\geq \\sup _ { y \\in B } \\sum _ { x \\in \\eta } l ( | x - y | ) = \\sup _ { y \\in B } \\Psi ( y ) . \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} u ( t , x ) - w ( t , x ) = \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { d } } q ( t - s , x - y ) \\Delta w ( s , y ) d y d s \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} | d e t R ( z ) | = ( \\det R ( z ) ) ^ { 2 n } = \\prod _ { k = 0 } ^ { n - 1 } | z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\frac { 2 k \\pi } { n } | . \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } f ( \\pm 1 ) = f ( t _ 0 ) = 0 \\\\ f ( t ) < 0 , \\forall t \\in ( - 1 , t _ 0 ) \\\\ f ( t ) > 0 , \\forall t \\in ( t _ 0 , 1 ) \\\\ f ' ( \\pm 1 ) < 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\dd N _ t = \\lambda ( n - N _ t ) \\ , \\dd t + \\dd m _ t , \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} & { \\bf U } _ 1 = \\left [ { { \\bf U } _ { 2 , 1 } ^ p } ^ T ~ ~ { { \\bf U } _ { 3 , 1 } ^ p } ^ T ~ ~ { { \\bf U } _ { 3 , 1 } ^ c } ^ T ~ ~ { { \\bf U } _ { 3 , 1 } ^ r } ^ T \\right ] ^ T , \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} \\Delta { C } _ { t } \\triangleq { C } _ t ( 1 ) - C _ t ( 0 ) , ~ { t } \\in \\mathbb { N } _ 0 ^ { n + 1 } , ~ \\Delta { C } _ { n + 1 } ( 0 ) = \\Delta { C } _ { n + 1 } ( 1 ) = 0 . \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{gather*} p _ { n _ { k + 1 } } ( x ) = a _ { k } ( x ) p _ { n _ { k } } ( x ) - \\beta _ { k } p _ { n _ { k - 1 } } ( x ) \\ , \\ 1 \\leq k < m \\ , \\ \\end{gather*}"}
-{"id": "7832.png", "formula": "\\begin{align*} \\Phi _ { i } ( \\mathbf { x } , \\mathbf { y } _ { i } ) = \\langle \\mathbf { g } _ { i } ( \\mathbf { x } ) , \\mathbf { y } _ { i } \\rangle + h _ { i } ( \\mathbf { y } _ { i } ) \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} \\rho ( x ) - \\frac { { \\rm d } } { { \\rm d } x } \\left ( \\frac { \\mathfrak { D } } { \\sigma ^ * } \\frac { { \\rm d } \\rho } { { \\rm d } x } \\right ) = g ( x ) \\mbox { f o r } x \\in ( - \\ell , + \\ell ) , \\\\ \\rho ( x ) = 0 \\mbox { f o r } x = \\pm \\ell , \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} { \\mathrm { E } } [ \\widehat { s } ] = \\sum _ { i = 1 } ^ { n } { \\mathrm { E } } [ \\psi _ { i } X _ { i } ^ { \\prime } ( \\widehat { \\beta } - \\beta ) ] = \\mathcal { I } _ { 1 } + \\mathcal { I } _ { 2 } , \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} a \\rightharpoonup ( b c ) & = S ( a _ 1 ) ( a _ 2 \\circ ( b c ) ) = S ( a _ 1 ) ( a _ 2 \\circ b ) S ( a _ 3 ) ( a _ 4 \\circ c ) = ( a _ 1 \\rightharpoonup b ) ( a _ 2 \\rightharpoonup c ) \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} \\phi _ F ( g z _ f ) = \\lambda _ { \\underline { k } } ( J ( g , I ) ) ^ { - 1 } F _ a ( g _ \\infty I ) \\tilde \\chi ( z _ f ) . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} \\widetilde { F } ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) & = a _ { 1 } ( x _ { 1 } - x _ { 3 } ) + a _ { 2 } x _ { 2 } + a _ { 4 } ( \\gamma _ { 3 } x _ { 3 } + x _ { 2 } ( x _ { 1 } - x _ { 3 } ) ) \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\ln \\det ( g , D , t ) = a _ n ( g , D ) = \\zeta ( 0 , g , D ) \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} q _ { i j } & = \\xi ^ i , & \\forall i , j \\in I , \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} F _ \\gamma ( x ) = \\inf _ y \\left \\{ f ( y ) + P ( y ) + D _ \\phi ( y , x ) \\right \\} . \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} \\nabla ^ F = d u \\wedge \\frac { \\partial } { \\partial u } + \\nabla ^ F \\big | _ { Y } . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} B ( G _ { n , k , b } ) = \\left \\lceil \\frac { ( n + 1 ) \\binom { b } { k - 1 } - ( k - 1 ) \\binom { b + 1 } { k } + \\binom { 2 b - n + 1 } { k } - 2 } { 2 } \\right \\rceil . \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} \\exp _ { q ^ { 2 } } ( z ) = { } _ { 1 } \\phi _ { 1 } \\ ! \\left ( 0 ; - q ^ { 1 / 2 } ; q ^ { 1 / 2 } , - ( 1 - q ) z \\right ) \\ ! , \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} x _ 0 = \\bigvee _ { z \\in A } h ( z ) \\otimes \\psi ( z ) \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} \\inf _ { \\psi \\in C X } \\hom ( \\psi ( x ) , \\Phi ( \\psi ) ) = \\inf _ { \\psi \\in C X , \\psi ( x ) = 1 } \\Phi ( \\psi ) . \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} \\frac 1 { | x - y | ^ { n } } f \\Big ( x , \\frac { y - x } { | y - x | } \\Big ) = \\bigg [ \\frac { \\partial } { \\partial x _ h } \\bigg ( \\frac 1 { | x - y | ^ { n - 1 } } g \\Big ( z , \\frac { y - x } { | y - x | } \\Big ) \\bigg ) \\bigg ] _ { \\lfloor z = x } \\quad ( x , y ) \\in \\Omega \\times \\Omega , \\ , \\ , x \\neq y . \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} \\left ( \\bar { U } ^ { \\intercal } A W _ { \\perp } \\right ) ^ { \\intercal } = [ y ^ { ( 1 ) } ~ y ^ { ( 2 ) } \\cdots y ^ { ( r ) } ] , y ^ { ( 1 ) } , \\cdots , y ^ { ( r ) } \\in \\mathbb { R } ^ { p _ 2 - r } . \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { q ^ { n ^ 2 } ( - q ; q ^ 2 ) _ n } { ( q ^ 2 ; q ^ 2 ) ^ 2 _ n } & = \\frac { ( - q ; q ^ 2 ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } , \\\\ \\sum _ { n \\geq 0 } \\frac { q ^ { n ^ 2 + 2 n } ( - q ; q ^ 2 ) _ n } { ( q ^ 2 ; q ^ 2 ) ^ 2 _ n } & = \\frac { ( - q ; q ^ 2 ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\sum _ { n \\geq 0 } \\frac { ( - 1 ) ^ n q ^ { n ^ 2 + n } } { ( - q ; q ^ 2 ) _ { n + 1 } } = \\frac { ( - q ; q ^ 2 ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\sum _ { j \\geq 0 } q ^ { 3 j ^ 2 + 2 j } ( 1 - q ^ { 2 j + 1 } ) . \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} I ( z ) - z = - \\bar z - z + \\frac { \\tau } { 2 } = - 2 \\Re ( z ) + \\frac { \\tau } { 2 } = n + m \\tau \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} f _ { \\mathcal { Q } } ( x , y ) : = \\sum \\limits _ { w _ { j , \\theta } ( \\mathcal { Q } ) = 0 } f _ j ( x , y ) , \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} \\psi ^ z = h ( z ) ^ { - 1 } \\cdot \\rho \\cdot h ( z ) , \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{gather*} E ( u \\pi ^ { - m } , f ) = Z ( 0 , f , \\chi _ { t r i v } ) + _ { t ^ { m - 1 } } \\frac { ( t - q ) Z ( s , f , \\chi _ { t r i v } ) } { ( q - 1 ) ( 1 - t ) } \\\\ + \\sum _ { \\chi \\neq \\chi _ { t r i v } } g _ { \\chi ^ { - 1 } } \\chi ( u ) _ { t ^ { m - c ( \\chi ) } } Z ( s , f , \\chi ) , \\end{gather*}"}
-{"id": "7822.png", "formula": "\\begin{align*} n ^ + = 1 + ( q - 1 ) ( q ^ 2 + 1 ) = q ^ 3 + q - q ^ 2 = n + n ^ { \\frac { 1 } { 3 } } - n ^ { \\frac { 2 } { 3 } } \\approx n . \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { m } p _ { r , r - ( m - k ) } a _ { k } = q _ { r , r - 1 - m } \\ , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0 \\leq m \\leq r - 1 \\ , \\\\ & D _ { r - 1 } a _ { m + r } + \\sum \\nolimits _ { k = 0 } ^ { r - 1 } \\ p _ { r , k } a _ { m + k } = 0 \\ , \\ \\ \\ \\ \\ \\ m \\geq 0 \\ . \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} \\Gamma ( f , g ) = \\Gamma _ 1 ( f ( . , y ) , g ( . , y ) ) ( x ) + \\Gamma _ 2 ( f ( x , \\cdot ) , g ( x , \\cdot ) ) ( y ) . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} g ( 0 , x _ 0 , t , c ) = \\Phi \\left ( \\frac { e ^ { t ^ c } - x _ 0 } { t ^ H } \\right ) + \\Phi \\left ( \\frac { e ^ { t ^ c } + x _ 0 } { t ^ H } \\right ) - 1 . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\frac { 1 } { t } \\log \\left ( 1 - C \\zeta _ { 3 } ^ { t } \\right ) = 0 . \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} & \\Omega ^ { p } _ \\mathrm { a b s } ( X , F ) = \\big \\{ \\sigma \\in \\Omega ^ { p } ( X , F ) \\ ; : \\ ; \\left ( i _ { e _ \\mathfrak { n } } \\sigma \\right ) | _ { Y } = 0 \\big \\} , \\\\ & \\Omega ^ { p } _ \\mathrm { r e l } ( X , F ) = \\big \\{ \\sigma \\in \\Omega ^ { p } ( X , F ) \\ ; : \\ ; \\left ( e ^ \\mathfrak { n } \\wedge \\sigma \\right ) | _ { Y } = 0 \\big \\} . \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} \\begin{array} { c } \\displaystyle \\sum \\limits _ { i = 1 } ^ { n } \\left [ \\langle \\mathbf { g } _ { i } ( \\mathbf { x } ) , \\mathbf { y } _ { i } - \\mathbf { y } _ { i } ( \\mathbf { x } ) \\rangle + h _ { i } ( \\mathbf { y } _ { i } ) - h _ { i } ( \\mathbf { y } _ { i } ( \\mathbf { x } ) ) \\right ] \\geq 0 \\\\ \\displaystyle \\forall \\mathbf { y } _ { i } \\in X _ { i } , \\mbox { f o r } \\ i = 1 , \\dots , n . \\end{array} \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} W _ N ^ { ( i , j ) } ( Z _ N ) = \\left | \\left ( x _ j - x _ i \\right ) \\cdot \\left ( v _ j - v _ i \\right ) \\right | \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} \\Lambda ( x ) = \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { \\Lambda ^ { \\pm } _ n } { x ^ { n } } , x \\to \\pm \\infty . \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} A : = \\frac { 1 } { h ^ 2 } V , B : = 2 \\frac { 1 } { h ^ 2 } V + \\frac { 1 } { h } D , C : = \\frac { 1 } { h ^ 2 } V + \\frac { 1 } { h } D + Q . \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} k = \\inf \\frac { P ( E ) } { \\mu ( E ) \\sqrt { - \\ln \\mu ( E ) } } \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} Z _ { S , a _ S } : = { } ^ { k _ S } ( Z _ { H , a _ S } ) , \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} d _ G \\omega = - \\Phi ^ \\ast \\eta _ G , \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} \\phi ' ( t ) & = a ( ( 3 / 4 ) ^ 2 - | \\phi ( t ) | ^ 2 ) ^ 5 _ + \\eta ( y _ n , y _ { n + 1 } ) , \\\\ \\phi ( 0 ) & = y '' . \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} v _ s - e ^ { i ( t - s ) H } E _ + ( t - s ) v _ s = & \\gamma ( H ) v _ s - e ^ { i ( t - s ) H } E _ + ( t - s ) v _ s \\\\ = & ( \\gamma ( H ) - \\gamma ( H _ 0 ) ) v _ s \\\\ & + ( \\gamma ( H _ 0 ) - P _ + - P _ - ) v _ s \\\\ & + ( P _ + - e ^ { i ( t - s ) H } E _ + ( t - s ) ) v _ s + P _ - v _ s . \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} u _ { 0 . 6 7 } ^ 1 ( f _ 1 ) = [ I - 0 . 6 7 P ^ 1 ( f _ 1 ) ] ^ { - 1 } \\bar { r } ^ 1 ( f _ 1 ) = ( 8 , 1 0 ) . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} G = \\langle x , y \\mid x ^ { - 1 } y ^ 2 x = y ^ { - 2 } , y ^ { - 1 } x ^ 2 y = x ^ { - 2 } \\rangle . \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dfrac { \\partial R } { \\partial t } = \\Delta R + R \\left ( R - r \\right ) \\quad \\mbox { i n } M \\times \\left ( 0 , T \\right ) \\\\ \\dfrac { \\partial R } { \\partial \\eta } = k _ g R \\quad \\mbox { o n } \\partial M \\times \\left ( 0 , T \\right ) . \\end{array} \\right . \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} P _ { A D C } = c B 2 ^ b \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} g _ 0 = d r ^ 2 + f \\left ( \\theta \\right ) ^ 2 d \\theta ^ 2 , \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{gather*} s _ { 2 n _ { k + 1 } } = s _ { 2 n _ { k + 1 } } ^ { ( n _ { k } + 1 ) } + \\frac { t _ { n _ { k + 1 } } } { t _ { n _ { k } } } \\ , \\end{gather*}"}
-{"id": "8192.png", "formula": "\\begin{align*} \\begin{array} { c } p _ { x x } + 2 q _ { x y } + r _ { y y } = 0 , \\\\ \\ \\\\ m _ x + n _ y = 0 , \\\\ \\ \\\\ m _ z - q m _ x - r m _ y + ( q _ x + r _ y ) m = n _ t - p n _ x - q n _ y + ( p _ x + q _ y ) n , \\end{array} \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\left ( \\partial _ t + V _ { m - 1 } \\cdot \\nabla _ { X _ { m - 1 } } \\right ) u _ \\varepsilon ^ { ( m - 1 ) } ( t ) = \\ell ^ { - 1 } \\tilde { C } _ m \\left ( u _ \\varepsilon ^ { ( m - 1 ) } ( t ) \\otimes g _ { \\varepsilon } ( t ) \\right ) \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} \\zeta : = ( \\tau , \\sigma ) , g : = ( g _ { \\tau } , g _ { \\sigma } ) \\hbox { a n d } b : = ( b _ { \\tau } , b _ { \\sigma } ) , \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} B _ I : = \\lim _ { h \\to 0 ^ + } \\frac { 1 } { | I | } \\int _ { I + i h } b ( x + i h ) d x \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} \\left ( z ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n } { 2 } } z ^ { n } } { ( q , z ; q ) _ { n } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ { 2 } - n } z ^ { 2 n } } { ( q ^ { 2 } ; q ^ { 2 } ) _ { n } } . \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} f r ( J ' ) = f r ( J ) . \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { m , n } ^ { \\gamma } ) \\right ] ( z ) = \\overline { \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + 1 } \\mathcal { Z } _ { n , m - 1 } ^ { \\gamma + 1 } ( \\overline { z } , z ) } = \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + 1 } \\mathcal { Z } _ { n , m - 1 } ^ { \\gamma + 1 } ( z , \\bar { z } ) , \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} d _ 1 \\triangleq \\max \\limits _ { 1 \\le t ' \\le t } \\left \\{ \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t ' } \\binom { N _ R - r - 1 } { t ' - 1 } t ' } { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t ' } \\binom { N _ R - r - 1 } { t ' - 1 } t ' + \\binom { N _ R - 1 } { r + 1 } \\binom { N _ R - r - 2 } { t ' - 1 } \\binom { N _ T } { t ' - 1 } } \\right \\} \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} H _ k ( \\kappa _ 1 , \\dots , \\kappa _ n ) = \\sum _ { 1 \\leq i _ 1 < \\cdots < i _ k \\leq n } \\kappa _ { i _ 1 } \\dots \\kappa _ { i _ k } , 1 \\leq k \\leq n . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} S ( \\omega ) = \\frac { \\sigma _ { \\sf w } ^ 2 } { | 1 - \\sum _ { \\ell = 1 } ^ p \\theta _ \\ell e ^ { - j \\ell \\omega } | ^ 2 } . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} \\lim _ { l \\to e } \\eta ( t , l ) = e \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} P \\cap H = c _ 0 ( P ) \\ ; \\ ; \\ ; \\ ; Q \\cap H = c _ 0 ( Q ) , \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} \\langle \\phi , \\phi \\rangle _ { \\Delta g } : = \\int _ \\Sigma d v \\ , ( \\partial ^ \\mu \\partial _ \\mu g _ { i j } ) \\phi ^ i \\phi ^ j \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} \\bigcup _ { n = 1 } ^ \\infty K ( c _ n ) = X _ A . \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} \\begin{cases} \\sigma _ k ( u _ f ) = 0 , & , \\\\ u _ f = f , & . \\end{cases} \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} \\int _ t ^ T \\int _ { \\mathcal { O } } ( u - k _ m ) ^ + \\ , \\mu ( d x d s ) \\leq \\int _ t ^ T \\int _ { \\mathcal { O } } ( u - \\xi ) ^ + \\ , \\mu ( d x d s ) + \\int _ t ^ T \\int _ { \\mathcal { O } } ( \\xi - \\hat { \\xi } ^ + ) ^ + \\ , \\mu ( d x d s ) = 0 . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } ^ - } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , s , k } T R ^ d \\left [ \\alpha + \\frac { y } { \\eta T } + \\left ( \\frac { \\eta } { R } \\right ) ^ d + \\theta ^ { d - 1 } \\right ] \\end{aligned} \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} \\frac { \\partial P _ { \\mu \\nu } } { \\partial x ^ { \\lambda } } + \\frac { \\partial P _ { \\nu \\lambda } } { \\partial x ^ { \\mu } } + \\frac { \\partial P _ { \\lambda \\mu } } { \\partial x ^ { \\nu } } = i e _ { \\mu \\nu \\lambda \\sigma } \\frac { \\partial Q ^ { \\sigma \\tau } } { \\partial x ^ { \\tau } } \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} J ^ { ( \\mu ) } y ( x , t ) = \\frac { 1 } { \\Gamma ( \\mu ) } \\int ^ t _ 0 \\frac { y ( x , \\xi ) d \\xi } { ( t - \\xi ) ^ { 1 - \\mu } } , \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} Y = X ^ { \\prime } \\beta + \\varepsilon , { \\mathrm { E } } [ \\varepsilon \\mid X ] = 0 , \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} W _ { p } = S _ { p } \\times V _ { p } , p \\in I . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} I ( X ^ n \\rightarrow { Y } ^ n ) = & \\sup _ { { \\cal S } _ { 0 , n } \\otimes { \\cal R } _ { 0 , n } } \\sum ^ n _ { t = 0 } \\int _ { { { \\cal X } ^ { t } } \\times { \\cal Y } ^ { t } } \\log \\Bigg ( \\frac { d { r } _ t ( \\cdot | x ^ { t - 1 } , y ^ { t } ) } { d p _ t ( \\cdot | x ^ { t - 1 } , y ^ { t - 1 } ) } ( x _ t ) \\frac { d s _ t ( \\cdot | y ^ { t - 1 } , x ^ { t - 1 } ) } { d \\nu _ { t } ^ p ( \\cdot | y ^ { t - 1 } ) } ( y _ t ) \\Bigg ) { \\bf P } ^ p ( d x ^ t , d y ^ t ) \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} \\mathcal { B } ^ + _ { V I } = \\bigcup _ { i \\in \\left \\{ 1 , \\dots , s , s + 1 , \\dots , s + k \\right \\} \\backslash \\left \\{ i _ { k + 1 } \\right \\} } \\mathcal { B } ^ + _ { V I , i } \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} \\| D _ H f ( x ) \\| : = \\max _ { v } \\Big \\{ | D _ H f ( x ) v | _ g : v \\in H T _ x U \\ \\ | v | _ g = 1 \\Big \\} . \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\dfrac { 1 } { 4 } = \\dfrac { \\mu } { 2 \\pi } \\int _ { t _ { 1 } } ^ { t _ { 2 } } \\dfrac { v ' ( t ) ^ { 2 } - v ( t ) v '' ( t ) } { \\mu ^ { 2 } v ( t ) ^ { 2 } + v ' ( t ) ^ { 2 } } ~ \\ ! d t , \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( x : n ) } = \\bigoplus _ { i = 0 } ^ { x - 1 } H _ x ( i , \\phi _ s ( i ) ) \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} { \\mathrm { E } } [ f ^ { \\prime } ( \\varepsilon ) ] = - { \\mathrm { E } } [ f ( \\varepsilon ) \\chi ^ { \\prime } ( \\varepsilon ) / \\chi ( \\varepsilon ) ] . \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\Theta _ { \\mathbb { J } } : = \\{ ( C , [ L ] ) \\in \\mathbb { J } \\ | \\ L \\ \\textrm { i s a t h e t a - c h a r a c t e r i s t i c \\ o n } \\ C \\} . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { p \\in \\{ 1 , 2 \\} } | \\alpha ( g _ p ) ( f _ n \\circ \\rho ) - f _ n \\circ \\rho | _ { \\infty } ^ - = 0 . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} \\mathbb { R } _ + ^ 2 = \\{ ( 0 , 0 ) \\} \\cup \\bigcup \\limits _ { \\tau \\subset \\Gamma ^ { g e o m } ( f ) } \\Delta _ \\tau , \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} \\varphi ( x , t ) = - \\min _ v \\left \\{ J ^ * ( v ) + t H ( v ) - \\langle x , v \\rangle \\right \\} . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} F F P _ { \\mathbf { w } } ( \\mathbf { W } ) = ( S _ { \\mathbf { W } , \\mathbf { w } } ^ { 2 } ) . \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} K _ { n _ j + 1 } \\subseteq \\bigcup _ { i = 1 } ^ r U _ { x _ i } \\subseteq \\bigcup _ { i = 1 } ^ r \\overline { U } _ { x _ i } \\subseteq K _ { n _ { j + 1 } } . \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} P = \\sum _ { | \\alpha | \\leq d } a _ { \\alpha } ( x ) \\ , D _ x ^ { \\alpha } , \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} \\mathcal { M } = \\sum _ { i = 1 } ^ k ( d - y _ i ( \\pi _ f ^ * ) ) \\end{align*}"}
-{"id": "10088.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) z ^ 2 } { x ^ { 4 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z } { x ^ { 3 } } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 2 z } { x ^ { 5 } } . \\\\ \\end{gather*}"}
-{"id": "10030.png", "formula": "\\begin{align*} | B _ H ( U ) | = \\left \\lbrace \\begin{array} { l l l } 0 & \\hbox { i f $ U \\subseteq H $ } , \\\\ q ^ { \\dim U - 1 } & \\hbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} J _ { n } ( z ; q ) = ( - 1 ) ^ { n } q ^ { - n / 2 } J _ { - n } \\left ( z q ^ { - n / 2 } ; q \\right ) \\ ! , n \\in \\Z . \\end{align*}"}
-{"id": "10153.png", "formula": "\\begin{align*} \\ell _ { g _ \\lambda } ( x ; y ) = \\left \\{ \\begin{array} { l l } ( y + 2 ) x - \\frac { 1 } { 2 } y ^ 2 + 4 , ~ ~ & ~ ~ y \\leq - 3 , \\\\ - x - \\frac { 1 } { 2 } , & ~ ~ - 3 \\leq y \\leq - 1 , \\\\ y x - \\frac { 1 } { 2 } y ^ 2 , & ~ ~ - 1 \\leq y \\leq 1 , \\\\ x - \\frac { 1 } { 2 } , & ~ ~ 1 \\leq y \\leq 3 , \\\\ ( y - 2 ) x - \\frac { 1 } { 2 } y ^ 2 + 4 , & ~ ~ y \\geq 3 \\end{array} \\right . \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\nabla _ A B = \\nabla _ A ^ { g } B - \\frac { 1 } { 2 } \\sharp \\iota _ { A \\wedge B } \\zeta \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} X _ t = x _ 0 e ^ { \\theta t } + \\theta e ^ { \\theta t } \\int _ 0 ^ t e ^ { - \\theta s } B ^ H _ s \\ , d s + B ^ H _ t , t \\ge 0 . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} c ( \\epsilon ) : = C \\frac { \\epsilon } { \\eta ( \\epsilon ) } . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} [ \\bar { e } _ { i , k + 1 } , \\bar { e } _ { j , l } ] = d ^ { - m _ { i , j } } [ \\bar { e } _ { i , k } , \\bar { e } _ { j , l + 1 } ] , \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} & \\nabla ^ a \\left ( R _ { a b } + \\nabla _ a \\nabla _ b f + \\frac { 1 } { 2 } g _ { a b } \\right ) - ( \\nabla ^ a f ) \\left ( R _ { a b } + \\nabla _ a \\nabla _ b f + \\frac { 1 } { 2 } g _ { a b } \\right ) = \\\\ & \\frac { 1 } { 2 } \\nabla _ { b } \\left ( R + 2 \\Delta f - | \\nabla f | ^ 2 - f \\right ) . \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} & \\Delta ( ( H _ 1 \\star P _ 1 ) \\star ( H _ 2 \\star P _ 2 ) ) = \\Delta ( H _ 1 H _ 2 \\star H _ 2 ( P _ 1 ) \\star P _ 2 ) = \\Delta ( H _ 1 ) \\star \\Delta ( H _ 2 ) \\star \\Delta ( H _ 2 ( P _ 1 ) ) \\star \\Delta ( P _ 2 ) . \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { m , n } ^ { \\gamma } ) \\right ] ( z ) & = \\dfrac { ( - 1 ) ^ { n } ( \\gamma + 1 ) _ { m + n } ( \\gamma + m + 2 ) _ { n - 1 } ( - m - 1 ) _ { n } } { ( \\gamma + 1 ) _ { n } ( m - n + 2 ) _ { n - 1 } ( m + 1 ) } \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + 1 } \\\\ & \\frac { ( \\gamma + 2 ) _ { m } ( \\gamma + 2 ) _ { n - 1 } } { \\left ( ( \\gamma + 2 ) _ { m + n - 1 } \\right ) ^ 2 } \\mathcal { Z } _ { m , n - 1 } ^ { \\gamma + 1 } ( z , \\bar { z } ) , \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { 8 } f _ t ( p ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ p | } . \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } D _ t = \\frac { \\partial } { \\partial t } \\partial ^ * _ t = \\partial ^ * _ t Q _ t - Q _ t \\partial ^ * _ t . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} f ( x ) = \\left [ \\begin{array} { c } - x _ 1 ( 1 - x _ 1 x _ 2 ) \\\\ - x _ 2 \\end{array} \\right ] . \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { C } _ { i , s + 1 } ^ + g _ \\varepsilon ^ { ( s + 1 ) } ( t , Z _ s ) = \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\left [ \\omega \\cdot ( v _ { s + 1 } - v _ i ) \\right ] _ + \\times \\\\ & \\qquad \\times g _ \\varepsilon ^ { ( s + 1 ) } \\left ( t , x _ 1 , v _ 1 , \\dots , x _ i , v _ i ^ * , \\dots , x _ { s } , v _ { s } , x _ i + \\varepsilon \\omega , v _ { s + 1 } ^ * \\right ) d \\omega d v _ { s + 1 } \\end{aligned} \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} \\lambda _ { \\alpha } \\lambda _ { \\beta } = \\lambda _ { \\gamma \\equiv \\alpha + \\beta \\ \\left ( \\operatorname { m o d } 2 n \\right ) } \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} | S _ { m , N } | ^ { \\frac { 1 } { q } - \\frac { 1 } { p } } \\frac { \\gamma } { 2 } \\left | \\left \\{ k \\in S _ { m , N } : | x _ { k } - x ^ { ( n _ { 0 } ) } _ { k } | > \\frac { \\gamma } { 2 } \\right \\} \\right | ^ { \\frac { 1 } { p } } = 0 . \\end{align*}"}
-{"id": "9938.png", "formula": "\\begin{align*} [ \\mathcal { A } _ i , \\mathfrak { n } ( \\varphi ( s _ i ) ) ] = 2 \\mathfrak { n } ( \\varphi ( s _ i ) ) [ \\mathcal { A } _ j , \\mathfrak { n } ( \\varphi ( s _ i ) ) ] = \\mathfrak { n } ( \\varphi ( s _ i ) ) . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} N _ { m i n } = \\min _ { j \\leq k } w _ j \\cdot N , \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } \\frac { \\psi ( x ) } { \\log | x | } = \\infty . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} v \\partial _ { x } \\mp \\beta _ { x } \\partial _ { v } = \\left \\{ \\begin{array} [ c ] { c c } \\omega _ { \\pm } ( I _ { \\pm } ) \\partial _ { \\theta _ { \\pm } } & v > 0 \\\\ - \\omega _ { \\pm } ( I _ { \\pm } ) \\partial _ { \\theta _ { \\pm } } & v < 0 \\end{array} \\right . \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} \\alpha _ k \\triangleq \\underset { \\{ z : \\ ; A z = 0 , \\ ; z \\neq 0 \\} } { } \\underset { { \\{ K : \\ ; | K | \\leq k \\} } } { } \\ ; \\ ; \\frac { \\| z _ K \\| _ { 1 } } { \\| z \\| _ { 1 } } . \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} f _ { n } ( 1 ) = ( q ; q ) _ { \\infty } + o ( 1 ) . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} ( \\Psi , e ^ { i 2 \\pi ( n + p ) x } ) = 0 \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} \\beta _ { s , a ^ 2 } ^ 2 = \\frac { \\left [ r ^ 2 ( s , a _ s ^ 1 , a ^ 2 ) - r ^ 2 ( s , a _ s ^ 1 , a _ s ^ 2 ) \\right ] } { \\left [ r ^ 2 ( s , a _ { s } ^ 1 , a ^ 2 ) - \\sum _ { s ' \\in S } p ( s ' | s , a _ s ^ 1 , a ^ 2 ) r ^ 2 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) \\right ] } . \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} F _ { t } ( A _ { 1 } , \\bar { A } ) & = d A _ { t } ( A _ { 1 } , \\bar { A } ) + \\frac { 1 } { 2 } \\left [ A _ { t } ( A _ { 1 } , \\bar { A } ) , A _ { t } ( A _ { 1 } , \\bar { A } ) \\right ] , \\\\ & = t d \\omega + \\frac { 1 } { 2 } \\left [ t \\omega , t \\omega \\right ] , \\\\ & = t d \\omega + \\frac { t ^ { 2 } } { 2 } \\left [ \\omega , \\omega \\right ] , \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} \\bar { a } \\cdot \\bar { b } : = \\overline { a _ { ( - 1 ) } b } , \\quad \\{ \\bar { a } , \\bar { b } \\} = \\overline { a _ { ( 0 ) } b } , \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} x _ { \\hat i } = X ( z _ { 1 } , \\dots , z _ { n } ) \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} \\omega _ { 1 , \\infty } ^ + = 0 \\in \\Omega ^ \\bullet ( Y _ { [ 0 , + \\infty ) } , F ) . \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} N ( q , f , A ) = \\{ f ^ { - 1 } ( q ) \\cap A \\} \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} L _ { a b c } : = \\partial ^ { 2 } + \\frac { c - ( a + b + 1 ) x } { x ( 1 - x ) } \\partial - \\frac { a b } { x ( 1 - x ) } . \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} \\mathbf { P _ * } \\Big ( \\big { | } \\sum _ { i = 1 } ^ { n } x _ { i j } x _ { i k } \\psi ' ( \\bar { \\epsilon } _ i ) ( G _ i ^ * - E G _ i ^ * ) \\big { | } > n { \\epsilon } \\Big ) = o ( n ^ { - 1 / 2 } ) , \\ ; \\ ; \\ ; \\ ; j , k \\in \\{ 1 , \\ldots , p \\} \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} F ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } ) = \\sum _ { \\underset { \\delta _ { i } = 0 } { \\delta \\in \\{ 0 , 1 \\} ^ { n } } } v _ { \\delta } ( x _ { i } ) \\prod _ { k \\neq i } g _ { k , x _ { i } } ^ { \\delta _ { k } } ( x _ { k } ) . \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} \\mathcal { C } ( d _ 1 , d _ 2 , c ) : = f ( W ( d _ 1 , d _ 2 , c ) ) \\end{align*}"}
-{"id": "10063.png", "formula": "\\begin{align*} 1 \\leq q < p < 2 p - q = 1 + k + n \\leq 6 , \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} P ^ x ( X ( \\tau _ D ) \\in \\partial D ) = 0 , x \\in D . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} d = \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ R - r - 1 } { N _ T - 1 } } { \\binom { N _ R - 1 } { r } \\binom { N _ R - r - 1 } { N _ T - 1 } + \\binom { N _ R - 1 } { r + 1 } \\binom { N _ R - r - 2 } { N _ T - 1 } } \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\widetilde { F } ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) = a _ { 1 } x _ { 1 } + a _ { 2 } x _ { 2 } + a _ { 3 } x _ { 3 } + a _ { 4 } ( x _ { 1 } x _ { 2 } - x _ { 2 } x _ { 3 } ) + a _ { 5 } ( x _ { 1 } x _ { 3 } - x _ { 2 } x _ { 3 } ) . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} \\mu _ { k } ^ i \\left ( B \\right ) & = \\frac { 1 } { Z _ { k } ^ i } \\sum \\limits _ { \\theta \\in B } \\prod \\limits _ { j = 1 } ^ { n } \\mu _ 0 ^ j \\left ( \\theta \\right ) ^ { \\left [ A ^ { k } \\right ] _ { i j } } \\prod \\limits _ { t = 1 } ^ { k } \\prod \\limits _ { j = 1 } ^ { n } \\ell ^ j ( s _ { t } ^ j | \\theta ) ^ { \\left [ A ^ { k - t } \\right ] _ { i j } } \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} \\mathbf { f } _ { n } : = \\mathcal { A } \\big ( \\mathbf { H } ( \\tilde { \\mathbf { u } } ) \\big ) - \\mathcal { A } \\big ( \\mathbf { H } ( \\tilde { \\mathbf { u } } _ { n } ) \\big ) \\tilde { \\mathbf { u } } _ { n } = \\mathrm { d i v } \\Big ( \\big ( \\mathbf { H } ( \\tilde { \\mathbf { u } } ) - \\mathbf { H } ( \\tilde { \\mathbf { u } } _ { n } ) \\big ) \\nabla \\tilde { \\mathbf { u } } _ { n } \\Big ) n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} \\tau ^ * \\ge ( N _ T + N _ R - 1 ) \\mu _ T = \\frac { N _ T + N _ R - 1 } { N _ T } ( 1 - \\mu _ R ) . \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} C ^ u ( U ^ + _ F ) = C ^ * \\big ( \\{ v _ { i j } \\} _ { 1 \\le i , j \\le N } \\ | \\ V = [ v _ { i j } ] \\& \\ F \\bar V F ^ { - 1 } \\big ) , \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} \\bigg | \\underbrace { \\big \\{ [ \\sigma _ 0 x _ 0 ^ { 1 / a _ 0 } : \\sigma _ 1 x _ 1 ^ { 1 / a _ 1 } : \\sigma _ 2 x _ 2 ^ { 1 / a _ 2 } ] \\mid \\sigma _ i \\in \\mu ^ { a _ i } \\big \\} } _ { \\pi ^ { - 1 } ( | x _ 0 : x _ 1 : x _ 2 | ) } \\bigg | < \\underbrace { a _ 0 a _ 1 a _ 2 } _ { \\deg \\pi } \\iff x _ i = 0 i \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} \\Psi ( t , x , u ; \\lambda , \\upsilon , \\mu , \\nu ) : = g ( t , x , u ) ^ \\top \\lambda + h ( t , x , u ) ^ \\top \\upsilon - G ( t , x , u ) ^ \\top \\mu - H ( t , x , u ) ^ \\top \\nu . \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} r ( X ) + r ( Y ) & = \\lambda ( X ) + \\lambda ( Y ) + | | X | | _ \\lambda + | | Y | | _ \\lambda \\\\ & \\geq \\lambda ( X \\cup Y ) + \\lambda ( X \\cap Y ) + | | X \\cup Y | | _ \\lambda + | | X \\cap Y | | _ \\lambda \\\\ & = r ( X \\cup Y ) + r ( X \\cap Y ) . \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} \\psi ( t ) = \\frac { C \\log ( 1 + K t ) } { \\log ( 1 + K ) } , \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} m ( n , k ) = 2 k \\cdot m ( n - 1 , k ) + ( n - 2 ( k - 1 ) ) \\cdot m ( n - 1 , k - 1 ) , \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} \\theta _ > ( h ) \\leq \\theta _ { \\geq } ( h ) \\leq \\liminf _ { g \\uparrow h } \\theta _ { > } ( g ) = \\theta _ > ( h ) , \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} U ( x , y ; z ) = x - \\frac { z ( t ^ 2 - 2 \\Delta t + 1 ) } { ( z - 1 ) ( t ^ 2 z - 2 \\Delta t + 1 ) } y - r ( z ) \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} K _ 1 & = K \\cap ( - \\infty , z _ 1 ] , \\\\ K _ { k } & = K \\cap [ z _ { k - 1 } , z _ { k } ] , \\ ; \\ ; k \\in \\{ 2 , \\ldots , p - 1 \\} , \\\\ K _ p & = K \\cap [ z _ { p - 1 } , + \\infty ) . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} y ^ 2 + x _ { 1 } ^ 3 x _ { 2 } + x _ { 2 } ^ 3 x _ { 3 } + x _ { 3 } ^ 3 x _ { 4 } + x _ { 4 } ^ 3 x _ { 1 } = 0 , \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} \\begin{aligned} & C _ { i , s + 1 } ^ + f _ N ^ { ( s + 1 ) } ( t , Z _ s ) = \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { Z _ { s + 1 } \\in \\partial \\mathcal { D } _ { s + 1 } } \\left [ \\omega \\cdot ( v _ { s + 1 } - v _ i ) \\right ] _ { + } \\times \\\\ & \\ ; \\ ; \\times f _ N ^ { ( s + 1 ) } ( t , x _ 1 , v _ 1 , \\dots , x _ i , v _ i ^ * , \\dots , x _ s , v _ s , x _ i + \\varepsilon \\omega , v _ { s + 1 } ^ * ) d \\omega d v _ { s + 1 } \\end{aligned} \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} \\tilde { p } _ { k , \\varphi } ( \\omega , ( - 1 ) ^ k \\xi _ n ) & = p _ k ( x , Y + ( - 1 ) ^ k \\xi _ n \\nu _ k + i \\tau d \\varphi _ k ( x ) ) \\\\ & = p _ k ( x , \\xi + i \\tau d \\varphi _ k ( x ) ) \\\\ & = p _ { k , \\varphi } ( x , \\xi , \\tau ) , \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\left ( 1 + \\gamma \\mu _ { w } c _ { o } ^ { \\nu - 1 } \\right ) ^ { \\nu p \\cdot \\sup \\varphi } & = e ^ { \\log \\left ( 1 + \\gamma \\mu _ { w } c _ { o } ^ { \\nu - 1 } \\right ) \\cdot \\nu p \\cdot \\sup \\varphi } \\\\ & \\leq e ^ { \\gamma \\mu _ { w } c _ { o } ^ { \\nu - 1 } \\cdot \\nu p \\cdot \\sup \\varphi } \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} z ( x ^ p ) = A _ 0 z ( x ) , \\ \\ \\ z ( x ^ q ) = B _ 0 z ( x ) . \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} \\eta _ { s p } & = \\frac { 1 } { 2 \\theta } \\left [ - ( \\xi + u ) + \\sqrt { ( \\xi + u ) ^ 2 + 4 \\theta \\xi u } \\right ] , & \\theta & : = \\frac { 2 \\Delta t - 1 } { t ^ 2 - 2 \\Delta t + 1 } , \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\sup _ { \\overline { \\mathcal K } ^ p } | u | = \\sup _ { \\partial _ p \\mathcal K } | u | . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\delta _ t ^ { ( n ) } = 0 . \\end{align*}"}
-{"id": "10021.png", "formula": "\\begin{align*} \\alpha _ 0 = \\theta _ { n - t } - \\theta _ { n - t - a } . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} \\star \\partial e ^ { t \\Delta } f = e ^ { t \\Delta ^ \\perp } \\star \\partial f , t \\ge 0 . \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} f _ { n - 6 } = & f _ { n - 7 } + A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\psi _ { n - 6 } ^ { ( 0 ) } + K _ { 6 - n } \\psi _ { n - 6 } ^ { ( 0 ) } \\\\ = & O ( r ^ { n - 5 } ) + \\Big ( K _ { 6 - n } \\beta _ { n - 6 } ^ { ( 0 ) } \\Big ) \\log r \\\\ : = & b _ { n - 5 } + O ( r ^ { n - 4 } ) \\log r + O ( r ^ { n - 4 } ) . \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } q ^ { n ^ { 2 } / 2 } S _ { n } \\left ( x q ^ { n } ; q \\right ) t ^ { n } = \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { 5 k ^ { 2 } / 2 } \\left ( - x t \\right ) ^ { k } } { \\left ( q ; q \\right ) _ { k } } \\left ( - t q ^ { 2 k + 1 / 2 } ; q \\right ) _ { \\infty } . \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} { \\bf P } ^ { \\pi } ( y ^ t , x ^ t ) = & \\prod _ { i = 0 } ^ t q _ i ( y _ i | y _ { i - M } ^ { i - 1 } , x _ i ) \\pi _ i ( x _ i | y _ { i - J } ^ { i - 1 } ) , \\\\ \\nu _ t ^ { \\pi } ( y _ t | y _ { t - J } ^ { t - 1 } ) = & \\sum _ { x _ t \\in { \\cal X } _ t } q _ t ( y _ t | y _ { t - M } ^ { t - 1 } , x _ t ) \\pi _ t ( x _ t | y _ { t - J } ^ { t - 1 } ) , ~ t \\in \\mathbb { N } _ 0 ^ { n } . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} ( x _ 1 , x _ 2 , \\dots , x _ k ) < _ { l e x } ( y _ 1 , y _ 2 , \\dots , y _ k ) \\exists i \\in [ k ] ( \\forall j \\in [ i - 1 ] : \\ , x _ j = y _ j ) \\wedge x _ i < y _ i . \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} \\left \\| \\nabla \\psi \\left ( t \\right ) \\right \\| _ { \\infty } = \\max \\left \\{ \\beta _ 1 , \\beta _ 2 \\right \\} \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} E ( f ) = E ( v _ 1 \\cdots v _ { m + 1 } ) & = E ( v _ 1 ) \\cup \\bigcup _ { \\mathclap { r = 2 } } ^ m E ( x _ i v _ r ) \\cup E ( x _ i v _ { m + 1 } ) \\\\ & = E ( v _ 1 ) \\cup \\bigcup _ { \\mathclap { r \\in R } } E ( x _ i v _ r ) \\cup E ( x _ i v _ { m + 1 } ) = E ( g ) . \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\tilde \\rho _ j = \\frac { \\rho _ j + \\rho _ { j + 1 } } { 2 } , \\tilde \\theta _ j = \\frac { \\theta _ j + \\theta _ { j + 1 } } { 2 } , \\widetilde Q _ j = Q ( \\tilde \\rho _ j , \\tilde \\theta _ j ) . \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( \\varphi \\otimes \\theta _ n ) J _ n ( a ) = \\omega _ \\xi ( a ) , \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} \\beta - d \\sigma ^ { - 1 } \\Delta + g \\sigma ^ { - 1 } = 0 , \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| Q _ n J _ n ( a ) \\| _ { \\psi _ n } = \\| a \\| _ \\varphi \\ , . \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} \\min \\ , & \\tau _ 2 = 3 a _ { 0 , 1 } + a _ { 0 , 2 } + 2 a _ { 1 , 1 } + a _ { 1 , 2 } \\\\ \\textrm { s . t . } & 2 a _ { 0 , 1 } + a _ { 0 , 2 } + 4 a _ { 1 , 1 } + 2 a _ { 1 , 2 } + a _ { 2 , 0 } + 2 a _ { 2 , 1 } + a _ { 2 , 2 } = 1 , \\\\ & 2 a _ { 1 , 1 } + a _ { 1 , 2 } + a _ { 2 , 0 } + 2 a _ { 2 , 1 } + a _ { 2 , 2 } \\le \\mu _ R , \\\\ & a _ { 0 , 1 } + a _ { 0 , 2 } + 2 a _ { 1 , 1 } + 2 a _ { 1 , 2 } + a _ { 2 , 1 } + a _ { 2 , 2 } \\le \\mu _ T . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} & \\big | A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\bigl ( 1 - e ^ { i \\lambda \\mu } \\bigr ) ^ { n } \\lambda ^ { 2 n } g ( \\lambda ) + \\lambda ^ { 2 n } C ^ { \\mu , 0 } _ { N } \\bigl ( e ^ { i \\lambda } \\bigr ) \\big | ^ 2 \\\\ & = \\alpha _ 1 | \\lambda | ^ { 2 n } \\big | 1 - e ^ { i \\lambda \\mu } \\big | ^ { 2 n } \\bigl ( f ^ 0 ( \\lambda ) - f _ 1 ( \\lambda ) \\bigr ) \\bigl ( f ^ 0 ( \\lambda ) + \\lambda ^ { 2 n } g ( \\lambda ) \\bigr ) ^ 2 \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} \\dim H _ { \\mathrm { L T } } ^ { K _ m } = n \\dim H ^ { n - 1 } _ c ( M _ m ^ { ( 0 ) } \\otimes _ { \\breve { F } } C , \\overline { \\Q } _ \\ell ) = n [ \\breve { F } _ m : \\breve { F } ] \\dim H ^ { n - 1 } _ c ( M _ { m , C } ^ { ( 0 ) } , \\overline { \\Q } _ \\ell ) . \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ u } \\sum _ { j = 0 } ^ { t _ 0 - 1 } \\left ( \\frac { \\lambda } { \\lambda _ u } \\right ) ^ j \\left ( \\frac { { t - j - 1 } } { t } \\right ) ^ { d _ u } \\vec u _ { \\infty } ^ { j + 1 } \\ , , \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} f ( h ) = f \\circ ( I + g ) \\circ ( I + \\beta ) ( h ) \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\lambda _ k ( q ) \\leq \\sum _ { k = 1 } ^ { n } \\mathcal { R } [ q , \\phi _ k ] \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} a \\in \\{ u \\in H _ { p e r } ^ { 2 / p , p } ( \\Omega ) ^ 2 \\cap L ^ p _ { \\overline { \\sigma } } ( \\Omega ) \\mid v \\mid _ { \\Gamma _ b } = 0 \\} , b _ { \\tau } \\in H _ { p e r } ^ { 2 / { q _ { \\tau } } , q _ { \\tau } } ( \\Omega ) , b _ { \\sigma } \\in H _ { p e r } ^ { 2 / { q _ { \\sigma } } , q _ { \\sigma } } ( \\Omega ) . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} \\left [ \\vert k \\vert ^ { \\alpha } - \\frac { d ^ 2 } { d k ^ 2 } - \\frac { \\alpha } { 2 } \\vert k \\vert ^ { \\alpha / 2 - 1 } \\right ] \\phi _ 0 ^ { ( \\alpha ) } = 0 . \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v - \\Delta v + \\nabla _ H p + f _ 0 k \\times v = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\\\ \\partial _ z p = T , \\\\ \\partial _ t T + v \\cdot \\nabla _ H T + w \\left ( \\partial _ z T + \\frac 1 h \\right ) - \\Delta _ H T = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} U ( s , x ) = \\sum _ { n \\ge 0 } u _ n ( x ) s ^ n . \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} \\frac { t } { e ^ t - 1 } = \\sum \\limits _ { n = 0 } ^ \\infty B _ n \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} W ' = \\{ ( i , j ) : 1 \\leq i \\leq r ' , \\ ; 1 \\leq j \\leq l ( \\lambda '^ { ( i ) } ) \\} \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} J _ a u \\left [ x \\right ] : = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - y \\cdot \\xi ) } u [ y ] d \\xi . \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { m } ( - 1 ) ^ i { 2 m - 1 \\choose m - i } { m + i \\choose i } \\gamma _ { m + i - 1 } = 0 \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} S i n e _ { \\beta } [ \\lambda _ 1 , \\lambda _ 2 ] \\stackrel { d } { = } \\frac { \\alpha _ { \\infty } ( \\lambda _ 2 ) - \\alpha _ { \\infty } ( \\lambda _ 1 ) } { 2 \\pi } . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\begin{cases} & \\mathbf { s c r } ( M , g ( t ) ) \\geq 1 , \\forall t \\in [ - T + 1 , T - 1 ] ; \\\\ & | R | ( x , t ) + \\frac { 2 } { T } \\leq 1 , \\forall ( x , t ) \\in M \\times [ - T + 1 , T - 1 ] . \\end{cases} \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} \\left [ - { \\cal D } _ { \\alpha } \\hbar ^ { \\alpha } \\frac { d ^ { \\alpha } } { d x ^ { \\alpha } } + V ( x ) \\right ] \\psi ( x ) = E \\psi ( x ) , \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} F ( \\theta , \\phi ) = { { w } ^ { H } } v ( \\theta , \\phi ) , \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} \\begin{aligned} \\nu _ k \\norm { x _ { k + 1 } - x } ^ 2 \\leq \\mu _ k \\norm { x _ k - x } ^ 2 & + 2 \\gamma _ k \\lambda _ k \\big ( ( f + g ) ( x ) - ( f + g ) ( x _ { k + 1 } ) \\big ) \\\\ [ 0 . 5 e x ] & + \\frac { 2 \\gamma _ k } { 1 - \\delta } \\big ( ( f + g ) ( x _ { k } ) - ( f + g ) ( x _ { k + 1 } ) \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} k _ U = \\log _ { 1 / p } n + ( 1 + \\epsilon ) \\psi ( n ) , \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} R _ { H ^ { 2 } } f _ { i j } ( z _ { i } ) = 1 , R _ { H ^ { 2 } } f _ { i j } ( z _ { j } ) = 0 \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} \\otimes _ { t = 0 } ^ n \\Big ( s _ t ( d y _ t | y ^ { t - 1 } , x ^ { t - 1 } ) \\otimes { r } _ t ( d x _ t | x ^ { t - 1 } , y ^ t ) \\Big ) = { \\bf P } ^ p ( d x ^ n , d y ^ n ) . \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} x _ 1 ^ * \\geq L = \\frac { 1 } { 4 } \\left ( 4 - \\sqrt { 5 + \\sqrt { 3 } } \\right ) \\approx 0 . 3 5 1 3 . \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} a _ { N , k , s } = \\frac { ( N - s ) ! } { ( N - s - k ) ! } \\varepsilon ^ { k ( d - 1 ) } \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} & \\mathbf { A } \\cdot \\left ( \\nabla \\times \\left ( f \\nabla \\times \\mathbf { B } \\right ) \\right ) - \\mathbf { B } \\cdot \\left ( \\nabla \\times \\left ( f \\nabla \\times \\mathbf { A } \\right ) \\right ) \\\\ & \\quad = \\nabla \\cdot \\left ( f \\left ( \\mathbf { B } \\times \\left ( \\nabla \\times \\mathbf { A } \\right ) - \\mathbf { A } \\times \\left ( \\nabla \\times \\mathbf { B } \\right ) \\right ) \\right ) . \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} \\begin{aligned} & b _ { s , s + k } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & = b _ { s , s + k } \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] \\end{aligned} \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} ( \\alpha _ 1 - \\beta _ 1 ) v _ 1 + \\sum \\limits _ { i = 2 } ^ m { \\alpha _ i v _ i } - \\sum \\limits _ { i = 2 } ^ { 2 d - m } { \\beta _ i w _ i } + \\left ( { \\sum \\limits _ { i = 1 } ^ m { \\alpha _ i K _ i } } \\right ) u _ 0 = 0 . \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} \\Delta \\pi ( v x ) & = \\Delta \\big ( \\pi ( v ) \\pi ( x ) + v \\rightharpoonup \\pi ( x ) \\big ) = ( \\pi ( v ) \\otimes 1 + 1 \\otimes \\pi ( v ) ) \\Delta \\pi ( x ) + \\Delta ( v \\rightharpoonup \\pi ( x ) ) \\\\ & = \\pi ( v x _ 1 ) \\otimes \\pi ( x _ 2 ) + \\pi ( x _ 1 ) \\otimes \\pi ( v x _ 2 ) = ( \\pi \\otimes \\pi ) \\Delta ( v x ) , \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} \\beta ( \\xi ) = \\prod _ { \\alpha \\in \\Delta ^ + _ \\R } ( 1 + | \\langle \\alpha , \\xi \\rangle | ) ^ { m ( \\alpha ) } . \\end{align*}"}
-{"id": "10111.png", "formula": "\\begin{align*} q + r = \\gcd ( - p , q ) + \\gcd ( - p , r ) + \\gcd ( q , p + q + r ) + \\gcd ( r , p + q + r ) . \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} P _ { 0 } ( 1 ) = \\frac { \\xi } { A \\gamma } p _ { 0 , 0 } . \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} \\aligned d & = \\begin{pmatrix} x & y & z \\\\ 0 & 0 & u \\end{pmatrix} , & g & = \\begin{pmatrix} - p / \\sqrt { 2 } & - z t - s q & y t \\\\ 0 & 0 & 0 \\\\ 0 & 0 & - s \\end{pmatrix} , \\\\ \\beta & = \\begin{pmatrix} p & q & 0 \\\\ 0 & l & 0 \\\\ 0 & 0 & 1 \\end{pmatrix} , \\quad & \\gamma ^ { t r } & = \\begin{pmatrix} - \\sqrt { 2 } x & - 2 s y & 0 \\\\ 0 & 0 & \\pm 1 \\\\ 0 & - t & 0 \\end{pmatrix} . \\endaligned \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} & \\forall \\ , i = 1 , 2 , \\sum _ { k = 0 } ^ { 4 } w ^ { k } _ { s _ i } \\leq t _ { s _ i } , \\\\ & \\forall \\ , i = 1 , 2 , w ^ { 1 } _ { r , } + w ^ { 2 } _ { r , } + w ^ { i } _ { r , } \\leq t _ r , \\\\ & t _ { s _ 1 } + t _ { s _ 2 } + t _ r \\leq 1 . \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} P ^ \\perp _ t f = P ^ S _ t f _ { \\mathfrak { L } } + P _ t ^ N f _ { \\mathfrak { L } ^ \\perp } . \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} a = c \\ , ( 1 , \\lambda , \\lambda ^ 2 , \\ldots , \\lambda ^ { n - 1 } ) , a = c \\ , ( 0 , 0 , \\ldots , 0 , 1 ) \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} \\int _ s ^ t r ^ { - \\theta _ 1 } ( t - r ) ^ { - \\theta _ 2 } d r & = \\int _ 0 ^ { t - s } ( t - r ) ^ { - \\theta _ 1 } r ^ { - \\theta _ 2 } d r \\le \\int _ 0 ^ { t - s } ( t - s - r ) ^ { - \\theta _ 1 } r ^ { - \\theta _ 2 } d r \\\\ & \\lesssim ( t - s ) ^ { 1 - \\theta _ 1 - \\theta _ 2 } . \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} & \\P \\left ( Q ^ c \\right ) \\leq C \\exp \\left \\{ - c \\frac { \\sigma _ r ^ 4 ( X ) } { \\sigma _ r ^ 2 ( X ) + p _ 1 } \\right \\} . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} \\bar m ^ * = h - \\vartheta - m ^ * \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} D ^ r _ { s h } ( U , F ) = \\left ( H ^ r ( U , F ) \\rightarrow \\prod _ { v \\in X ^ { ( 1 ) } _ 0 } H ^ r ( K _ v , F ) \\right ) , \\\\ \\mathcal { D } ^ { d + 4 - r } ( U , F ' ) : = \\left ( H ^ { d + 4 - r } _ c ( U , F ' ) \\rightarrow H ^ { d + 4 - r } ( K _ 0 , F ' ) \\right ) . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} - \\mathcal { T } _ 0 = \\frac { 1 } { 4 \\pi ^ 2 } \\sum _ { k \\neq 0 } \\int _ t \\int _ \\xi \\int _ \\eta \\left ( A ( k , \\xi ) - A ( k , \\xi - \\eta ) \\right ) \\widehat { u _ 0 ^ z } ( \\eta ) i k \\hat { f } ( k , \\xi - \\eta ) A ( k , \\xi ) \\bar { \\hat { f } } ( k , \\xi ) \\ , d \\eta d \\xi d t . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} a _ 0 & = \\dfrac { c } { \\lambda _ 0 } , c \\in ( 0 , 1 ) \\\\ a _ 1 & = \\dfrac { 1 - a _ 0 \\lambda _ 0 } { \\lambda _ 1 } , \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\det \\left [ \\begin{array} { c c c c c c c c c c } 0 & 1 & \\cdots & k - 1 \\\\ 0 & 1 & \\cdots & k - 1 \\end{array} \\right ] _ N \\bmod p = \\det ( \\bar N _ k ^ \\mathrm { T } ) . \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( B ) & = & \\mu \\ , \\sigma ( A ) B - B A . \\end{aligned} \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} a _ { i j } = \\left \\{ \\begin{array} { l l } \\frac { 1 } { \\max \\left \\{ d ^ i + 1 , d ^ j + 1 \\right \\} } & \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} C ^ { \\pi } _ t ( y _ { t - J } ^ { t - 1 } ) = { \\bf E } ^ { \\pi } \\bigg \\{ \\sum _ { i = t } ^ n \\log \\Big ( \\frac { q _ i ( Y _ i | y _ { i - M } ^ { i - 1 } , X _ i ) } { { \\nu } ^ { \\pi } _ { i } ( Y _ i | y _ { i - J } ^ { i - 1 } ) } \\Big ) \\Big { | } Y _ { t - J } ^ { t - 1 } = y _ { t - J } ^ { t - 1 } \\bigg \\} , ~ t \\in \\mathbb { N } _ 0 ^ n , ~ \\forall { y ^ { t - 1 } _ { t - J } \\in { \\cal Y } ^ { t - 1 } _ { t - J } } . \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} ( a * b ) \\cdot X = ( a \\cdot X ) \\diamond ( b \\cdot X ) . \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} & e _ 2 ( t , N _ h ) = \\sqrt { \\frac { \\ell } { N _ h } \\sum ^ { N _ h - 1 } _ { j = 1 } \\Big | y ( x _ j , t ) - y _ N ( x _ j , t ) \\Big | ^ 2 } , \\\\ & e _ \\infty ( t , N _ h ) = \\max \\limits _ { 1 \\leq j \\leq N _ h - 1 } \\Big | y ( x _ j , t ) - y _ N ( x _ j , t ) \\Big | , \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} \\sup _ { t , x } \\left | \\int _ { \\mathbb { R } ^ d } Z _ { k + 1 } ( t , x - \\xi ) u _ { k } ( t , \\xi ) d \\xi \\right | \\leq C \\ , k = 0 , 1 . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} g _ 2 = \\begin{pmatrix} d \\Delta - \\theta \\sigma \\\\ - \\mu \\sigma \\end{pmatrix} , \\gamma _ 2 ^ { t r } = \\begin{pmatrix} - d \\sigma ^ { - 1 } ( I - \\Delta ^ { t r } \\Delta ) + \\theta \\Delta \\\\ \\mu \\Delta \\end{pmatrix} . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} p _ { i } ^ { M ' } ( \\sigma ' , \\varphi ) = p _ { i } ( \\sigma ' | _ { K } , \\varphi ) . \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} \\frac { \\min \\{ m \\equiv 2 \\mu ( X ) + \\mathrm { d e g } \\ , e \\ , \\bmod { ( \\deg P ) } \\mid \\exists x \\in \\tilde { H } ^ m _ H ( X ) , \\ ; \\iota ^ * x = P ^ k e , k \\} - \\mathrm { d e g } \\ ; e } { 2 } . \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\vartheta _ { \\mathcal L } ( z ) & = \\frac { 1 } { ( 1 - z ^ q ) ^ n } \\sum _ { \\ell = 0 } ^ n \\sum _ { t = 0 } ^ n z ^ { t q } \\sum _ { s = 0 } ^ { n } \\Phi _ { \\mathcal L } ^ { ( \\ell + s ) } ( z ) \\ , 2 ^ s \\binom { \\ell + s } { s } \\binom { \\ell } { t - s } ( - 1 ) ^ { t - s } \\\\ & = \\sum _ { t = 0 } ^ n \\frac { z ^ { t q } } { ( 1 - z ^ q ) ^ n } \\sum _ { s = 0 } ^ { n } 2 ^ s ( - 1 ) ^ { t - s } \\sum _ { \\ell = s } ^ { n } \\Phi _ { \\mathcal L } ^ { ( \\ell ) } ( z ) \\binom { \\ell } { s } \\binom { \\ell - s } { t - s } . \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} s ( x , y ) = i ( \\phi ( x , y ) - p _ e p _ d ) = - i ( \\phi ( y , x ) - p _ e p _ d ) = \\overline { s ( y , x ) } \\ \\ \\mu ^ 2 \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} \\mathcal { T } _ 0 & = \\iint A ( - \\partial _ v \\partial _ v ^ { - 2 } f _ 0 \\ , \\partial _ z f ) A f \\ , d V d t = \\iint A ( - \\partial _ v \\partial _ v ^ { - 2 } f _ 0 \\ , \\partial _ z \\ne { f } ) A \\ne { f } \\ , d V d t . \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} H ^ { n - 1 } ( \\{ u = 0 \\} \\cap B _ g ( x , \\rho ) ) \\geq 2 ^ { c _ 4 \\log \\tilde N / \\log \\log \\tilde N } F ( N ) \\rho ^ { n - 1 } \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} F _ { i j } \\xi ^ i \\xi ^ j = \\varphi _ \\alpha f _ { i j } ^ \\alpha \\xi ^ i \\xi ^ j + \\varphi _ { \\alpha \\beta } f _ i ^ \\alpha f _ j ^ \\beta \\xi ^ i \\xi ^ j < 0 , \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} | \\xi _ 0 ( x ) | = 1 \\frac { d } { d t } | _ { t = 0 } | ( \\xi _ t ( x ) ) | = 0 . \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} G _ \\infty \\left ( t \\right ) = G _ \\infty \\left ( 0 \\right ) + \\ell ^ { - 1 } \\int _ { 0 } ^ { t } V ^ { 0 } \\left ( \\tau \\right ) G _ \\infty \\left ( \\tau \\right ) d \\tau \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} M _ { H , \\pi } = \\sum _ { i , j \\leq k } p _ { i j } \\hat { \\phi } _ i \\otimes \\nu _ j . \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { m = - \\infty } ^ { \\infty } ( a ; q ) _ { m } q ^ { m + \\ell ( | m | + m ) / 2 } S _ { \\ell } \\left ( - q ^ { - | m | - \\ell } ; q \\right ) \\\\ & = \\frac { \\left ( a q , q ^ { \\ell + 1 } , 1 / ( a q ^ { \\ell } ) ; q \\right ) _ { \\infty } } { \\left ( 1 / a , q ; q \\right ) _ { \\infty } } . \\end{aligned} \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\left [ S L _ 2 ( \\mathbb { Z } ) : \\Gamma _ 0 ( N ) \\right ] = N \\prod \\limits _ { p | N } ( 1 + 1 / p ) = \\Psi ( N ) . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} J _ 1 = \\int _ { I _ 1 } + \\int _ { I _ 2 } \\ll n _ K \\int _ { I _ 1 } \\cos \\theta \\log ( 1 / \\cos \\theta ) d \\theta + n _ K \\log ( 1 / ( \\sigma - 1 ) ) \\int _ { I _ 2 } \\cos \\theta d \\theta \\ll _ { \\epsilon } n _ K . \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} q _ { 2 n } + { \\textstyle \\sum \\limits _ { p = 1 } ^ { 2 n - 1 } } \\frac { q _ { 2 n - p } } { 4 \\pi ^ { 2 } p ( 2 n - p ) } \\left ( q _ { p } + \\sum _ { k = 1 } ^ { p - 1 } \\sum _ { n _ { 1 } , n _ { 2 } . . . , n _ { k } } \\frac { q _ { n _ { 1 } } q _ { n _ { 1 } } . . . q _ { n _ { k } } q _ { p - n ( k ) } } { b ( n , p , k ) } \\right ) = 0 . \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\frac { \\lambda _ { u } ^ m \\cdot m ^ { d _ u } } { \\lambda ^ m } \\frac { 1 } { \\lambda _ { u } ^ m \\cdot m ^ { d _ u } } M ^ m \\vec { u } = \\sum _ { m = 0 } ^ { \\infty } \\left ( \\frac { \\lambda _ u } { \\lambda } \\right ) ^ m \\cdot m ^ { d _ u } \\frac { 1 } { \\lambda _ { u } ^ m \\cdot m ^ { d _ u } } M ^ m \\vec { u } \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} L ( u , w ^ 1 , \\dotsc , w ^ k ) : = \\int _ X u \\sigma _ k ( D ^ 2 w ^ 1 , \\dotsc , D ^ 2 w ^ k ) d x + \\oint _ M u A _ k ( w ^ 1 , \\dotsc , w ^ k ) d \\mu \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} \\begin{cases} \\underset { 0 \\leq x \\leq L } { \\sup } \\| \\partial ^ k _ x \\varphi ( x , \\cdot ) \\| _ { H ^ { \\frac { 1 - k } 3 } ( 0 , T ) } \\le C _ T \\| \\varphi ^ 0 \\| _ { L ^ 2 ( 0 , L ) } , \\\\ \\underset { 0 \\leq x \\leq L } { \\sup } \\| \\partial ^ k _ x \\psi ( x , \\cdot ) \\| _ { H ^ { \\frac { 1 - k } 3 } ( 0 , T ) } \\le C _ T \\| \\psi ^ 0 \\| _ { L ^ 2 ( 0 , L ) } , \\end{cases} \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} \\gamma ( x _ i ) = \\sum _ { n \\geq 0 } \\gamma _ { i , ( - n - 1 ) } t ^ { n } \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} a _ { \\tau } ^ { ( r , \\varphi ) } = w ( M ) - 2 p _ { r } ( \\sigma , \\varphi ) + 1 , \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} v ^ * ( X - Y ) v = - v ^ * Y v < 0 \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} g ^ { ( k ) } ( 0 ) & = ( \\mathcal { S } _ { N } B ^ { ( f ) } _ k ) ( q \\ , | \\ , b ) , \\\\ & = 0 , \\ ; { \\rm i f } \\ ; k < N , \\\\ & = f ( 0 ) \\ , N ! \\prod \\limits _ { j = 1 } ^ N b _ j , \\ ; { \\rm i f } \\ ; k = N . \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} & \\delta ^ * ( \\mu , 0 ) = \\begin{cases} \\delta _ { \\mathsf { C a - I A } } ~ ~ ~ ~ ~ \\mu = 1 / M , \\\\ \\delta _ { \\mathsf { C a - Z F } } ~ ~ ~ ~ ~ \\mu = 1 , \\end{cases} \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} X _ 1 : = \\frac { 1 } { \\sqrt { 2 } } ( A - i \\sqrt { \\tau } B ) , X _ 2 : = \\frac { 1 } { \\sqrt { 2 } } ( A ^ * - i \\sqrt { \\tau } B ^ * ) , 0 < \\tau < 1 , \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} \\varphi ( a _ 1 , \\ldots , a _ m ) = \\left ( \\prod _ { 0 < i \\le l _ 1 } a _ { i } , \\prod _ { l _ 1 < i \\le l _ 2 } a _ i , \\ldots , \\prod _ { l _ { n - 1 } < i \\le l _ n } a _ { i } \\right ) \\textup { w i t h $ 0 < l _ 1 < \\cdots < l _ n \\le m $ , } \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} B _ { n , 1 } ( x ) = ( - 1 ) ^ n \\frac { 2 ( b ^ 2 - a ^ 2 ) ^ { \\mu + \\nu + 1 } } { a ^ \\mu b ^ \\nu n ! } \\sum _ { i = 0 } ^ { \\lfloor \\frac { n } { 2 } \\rfloor } b _ { i , n } x ^ i , \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} { \\big \\| } \\sum a _ i \\vec e _ i { \\big \\| } = \\sum | a _ i | \\ , . \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a ; q \\right ) _ { n } S _ { n } \\left ( x ; q \\right ) z ^ { n } } { \\left ( c q ; q \\right ) _ { n } } = \\frac { \\left ( a z ; q \\right ) _ { \\infty } } { \\left ( z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } \\left ( a ; q \\right ) _ { n } \\left ( - x z \\right ) ^ { n } } { \\left ( c q , a z ; q \\right ) _ { n } } S _ { n } \\left ( \\frac { c } { x q ^ { n } } ; q \\right ) . \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} d u _ i = ( \\Delta u _ i + ( V u ) _ i ) \\ , d t + ( G u ) _ i \\ , d W ( t ) , \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} \\exp ^ x ( p + q ) - \\exp ^ x ( p ) \\exp ^ x ( q ) = \\sum _ { n \\geq 2 } \\frac { x ^ n } { n ! } \\sum _ { k = 0 } ^ n { n \\choose k } p ^ k q ^ { n - k } - \\sum _ { n \\geq 2 } \\sum _ { k = 0 } ^ n \\frac { x ^ k } { k ! } p ^ k \\frac { x ^ { n - k } } { ( n - k ) ! } q ^ { n - k } = 0 . \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} & \\P \\left ( \\left | u ^ { \\intercal } M ^ { \\intercal } Y _ 1 ^ { \\intercal } Y _ 1 M u - u ^ { \\intercal } I _ r u \\right | > x \\right ) \\leq C \\exp \\left ( - c \\left ( \\sigma _ r ^ 2 ( X ) + p _ 1 \\right ) x ^ 2 \\wedge x \\right ) , \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} \\vartheta _ { m } ( w ) & = \\sum _ { \\gamma \\in M _ { k } } \\mathcal { Z } ^ { m + \\gamma } t ^ { \\varphi ( m + \\gamma ) } \\\\ & = \\sum _ { \\gamma \\in M _ { k } } \\exp \\pi \\sqrt { - 1 } ( 2 \\langle w , m + \\gamma \\rangle + \\frac { \\log t } { 2 \\pi \\sqrt { - 1 } } Z ( m + \\gamma , m + \\gamma ) ) , \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} \\min _ { j = 0 , \\hdots , N - 1 } \\norm { \\mathcal { G } _ t ( x _ j ) } ^ 2 \\le \\frac { 1 } { N } \\sum _ { j = 0 } ^ { N - 1 } \\norm { \\mathcal { G } _ t ( x _ j ) } ^ 2 & \\le \\frac { 2 t ^ { - 1 } \\big ( F ( x _ 0 ) - F ^ * \\big ) } { N } , \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} \\underline { c } ( \\alpha ) : = \\frac { \\sqrt { 2 } } { \\sqrt { ( \\alpha + 1 ) ( \\alpha + 5 ) } } \\leq c ( \\alpha ) \\leq \\frac { 1 } { \\sqrt { \\alpha + 1 } \\sqrt [ 6 ] { ( \\alpha + 3 ) ( \\alpha + 5 ) } } = : \\overline { c } ( \\alpha ) \\ , . \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} w _ 0 ( x , y , z ) = - \\int _ 0 ^ z \\nabla _ H \\cdot v _ 0 ( x , y , z ' ) d z ' , \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{align*} \\sigma _ w ( \\zeta ^ k ) & = \\sum _ { 0 \\leq i _ 1 < \\cdots < i _ w \\leq n - 1 } \\zeta ^ { k \\left ( q ^ { i _ 1 } + \\cdots + q ^ { i _ w } \\right ) } = \\sum _ { j \\in \\mathbb { Z } _ { q ^ n - 1 } } \\delta _ w ( j ) \\zeta ^ { k j } = \\mathcal { F } _ \\zeta [ \\delta _ w ] ( k ) . \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} S ( x ^ i y ^ { n + 2 - i } , g ) & = - s x ^ { i - 1 } y ^ { n + 3 - i } - x ^ { i + 1 } y ^ { n + 1 - i } + x ^ { i - 1 } y ^ { n + 3 - i } \\\\ & = ( - s + 1 ) x ^ { i - 1 } y ^ { n + 3 - i } + ( - 1 ) x ^ { i + 1 } y ^ { n + 1 - i } . \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} L ( t ) = L ( s ) + L ( t - s , Z ( s + \\cdot ) ) , \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} \\widehat { \\mathcal { K } } ^ k _ { \\mathcal { X } } ( x _ 0 ) = \\bigcup _ { \\alpha \\ge 0 } \\alpha \\left ( \\mathcal { K } ^ k _ { \\mathcal { X } } ( x _ 0 ) \\right ) . \\end{align*}"}
-{"id": "9981.png", "formula": "\\begin{align*} \\abs { f ( x ) } \\le \\norm { f } _ { \\mathcal { H } } \\norm { k _ x } _ { \\mathcal { H } } = \\norm { f } _ { \\mathcal { H } } \\sqrt { K ( x , x ) } \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} & \\widetilde { \\mathbf { s c r } } ( x _ i , 0 ) = 2 ; \\\\ & \\widetilde { \\mathbf { s c r } } ( x , t ) \\geq 1 , \\forall \\ ; x \\in M _ i , \\ ; t \\in [ - 2 ^ { i } , 2 ^ { i } ] . \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} a ( x \\otimes \\bot , y ) = \\hom ( \\bot , a ( x , y ) ) = \\top \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} \\mathcal { B } _ I = \\left \\{ \\begin{aligned} & ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in [ 0 , \\infty ) \\times \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } \\textnormal { s u c h t h a t } \\tau = 0 \\textnormal { o r } \\\\ & \\exists t ^ \\prime , t ^ { \\prime \\prime } \\geq 0 \\ ; : \\ ; d _ X \\left ( Z _ { s + k } ^ \\prime ( \\tau ; t ^ \\prime ) , Z _ { s + k } ^ \\prime ( \\tau ; t ^ { \\prime \\prime } ) \\right ) \\leq y \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} 1 + | \\psi | ^ 2 = | \\omega _ 1 | ^ 2 + | \\varphi | ^ 2 \\ , . \\end{align*}"}
-{"id": "9921.png", "formula": "\\begin{align*} \\lambda ^ { \\min } ( u ( r ) v ) = \\lambda ^ { \\min } ( v ) . \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} { \\cal P } ^ { A . 1 , \\infty } ( \\kappa ) = & \\Big \\{ \\pi ^ { \\infty } ( x _ t | y _ { t - 1 } ) , ~ t \\in \\mathbb { N } _ 0 : ~ \\lim _ { n \\longrightarrow \\infty } \\frac { 1 } { n + 1 } { \\bf E } ^ { \\pi ^ \\infty } \\big \\{ \\sum _ { t = 0 } ^ n \\gamma ( X _ t , Y _ { t - 1 } ) \\big \\} \\leq \\kappa \\Big \\} \\\\ \\bar { \\cal P } ^ { A . 1 , \\infty } ( \\kappa ) = & \\Big \\{ \\pi ^ { \\infty } ( x _ 0 | y _ { - 1 } ) : ~ { \\bf E } ^ { \\pi ^ \\infty } \\big \\{ \\gamma ( X _ 0 , Y _ { - 1 } ) \\big \\} \\leq \\kappa \\Big \\} . \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} f ( x , y ) = \\sum _ { j = 0 } ^ { l _ f } f _ j ( x , y ) , \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{gather*} R + ( - 1 ) ^ { m + 1 } R ^ { \\prime } = \\sum _ { i = 1 } ^ { m } \\delta ( z , w _ { i } ) W _ { i } , \\end{gather*}"}
-{"id": "7984.png", "formula": "\\begin{align*} \\phi ( x , y ) = p _ e p _ d \\pm \\sqrt { ( 1 - \\sqrt { 1 - p _ e } - p _ e p _ d ) ( 1 + \\sqrt { 1 - p _ e } - p _ e p _ d ) } ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\frac { \\gamma _ { 3 } ( 1 + \\gamma _ { 5 } ) d } { \\beta _ { 5 } d + \\beta _ { 3 } + 1 } + x _ { 3 } ( \\beta _ { 5 } d + \\beta _ { 3 } ) = d . \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} [ \\bar { h } _ { i , k } , \\bar { f } _ { j , l } ] = - a _ { i , j } d ^ { - k m _ { i , j } } \\bar { f } _ { j , l + k } , \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} \\eta _ { s p } = \\frac { \\gamma \\ , \\xi } { \\alpha + \\beta + \\gamma - \\xi } , \\qquad \\xi \\in [ 0 , \\alpha ] . \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} \\Phi ( U _ 1 ) = U _ 2 , \\Phi ( \\iota _ { \\C ^ { N _ 1 } } ) = \\iota _ { \\C ^ { N _ 2 } } , \\Phi \\Big ( \\frac { 1 } { ( F _ 1 ^ * F _ 1 ) ^ { 1 / 2 } } t _ { F _ 1 } \\Big ) = \\frac { 1 } { ( F _ 2 ^ * F _ 2 ) ^ { 1 / 2 } } t _ { F _ 2 } . \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} K _ { W _ i ' } = 2 ^ { - 1 } W _ i ' ( S _ i ^ * + S _ i ) + 2 ^ { - 1 } ( S _ i ^ * + S _ i ) W _ i ' , B _ { W _ i ' } = \\tilde { W _ i ' } ( S _ i ^ * - S _ i ) - ( S _ i ^ * - S _ i ) \\tilde { W _ i ' } . \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle \\frac { \\partial h } { \\partial t } = \\Delta h - 2 \\left | M _ { i j } \\right | ^ 2 + r h - r ' - 2 \\left < \\nabla \\psi , \\nabla f \\right > \\quad \\mbox { i n } M \\times \\left ( 0 , T \\right ) \\\\ \\displaystyle \\frac { \\partial h } { \\partial \\eta } = k _ g R \\mbox { o n } \\partial M \\times \\left ( 0 , T \\right ) . \\end{array} \\right . \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\left ( \\tilde { u } _ { \\nu } \\right ) ^ { \\left ( 1 \\right ) } : = \\tilde { u } _ { \\nu } ^ { \\left ( 1 \\right ) } = \\omega _ { \\nu } u _ { \\nu } ^ { \\left ( 1 \\right ) } . \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} F _ i & = ( i / ( q + 1 ) , i / ( q + 1 ) ) , \\\\ G _ i & = ( i ( 1 - \\beta ) / ( q + 1 ) , i ( 1 - \\beta ) / ( q + 1 ) + \\beta ) . \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} \\| y - \\bar x \\| \\le \\| y - u \\| + \\| u - \\bar x \\| = { \\rm d i s t } ( u , { \\cal X } ) + \\| u - \\bar x \\| \\le 2 \\| u - \\bar x \\| < \\epsilon . \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} \\tilde { \\mathbf { u } } ^ { 0 } ( \\mathbf { x } ) = \\mathbf { u } ^ { 0 } ( \\mathbf { x } ) + \\boldsymbol { \\varepsilon } ( \\mathbf { x } ) \\mathbf { x } \\in G . \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} T _ E { M - 1 \\choose K - 1 } / { M \\choose K } = T _ E \\frac { K } { M } , \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} H ( z ) = \\frac { \\sigma ^ 2 _ { \\sf w } } { 1 - \\sum _ { \\ell = 1 } ^ p \\theta _ \\ell z ^ { - \\ell } } . \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} e ^ { - ( q + b _ 0 ) t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) = & e ^ { - ( q + b _ 0 ) t } \\sum \\limits _ { p = 0 } ^ N ( - 1 ) ^ p \\sum \\limits _ { k _ 1 < \\cdots < k _ p = 1 } ^ N \\exp \\bigl ( - ( b _ { k _ 1 } + \\cdots + b _ { k _ p } ) t \\bigr ) , \\\\ = & \\bigl ( \\mathcal { S } _ { N } e ^ { - x t } \\bigr ) ( q | b ) , \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} \\Sigma _ { 3 \\text a } ( r _ 1 , r _ 2 ) = M _ { 3 \\text a } ( r _ 1 , r _ 2 ) + R _ { 3 \\text a } ( r _ 1 , r _ 2 ) , \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} m ( t ) : = \\Vert u ( t , \\cdot ) \\Vert _ { L ^ { 1 } } \\leq C , t \\geq 0 . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} \\left \\| \\frac { f } { | x | ^ { k + \\alpha } } \\right \\| _ { L ^ { 2 } ( \\mathbb { G } ) } \\leq \\left [ \\prod _ { j = 0 } ^ { k - 1 } \\left | \\frac { Q - 2 } { 2 } - ( \\alpha + j ) \\right | \\right ] ^ { - 1 } \\left \\| \\frac { 1 } { | x | ^ { \\alpha } } \\mathcal { R } ^ { k } f \\right \\| _ { L ^ { 2 } ( \\mathbb { G } ) } , \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} \\beta ( g ) = - \\frac { \\partial W ( g ( \\Lambda ) ) } { \\partial \\ln \\Lambda } / \\frac { \\partial W ( g ) } { \\partial g } \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} \\eta _ i = \\left ( - r ^ { - 2 } ( d ^ 2 ) _ i + C u _ i \\right ) g ' \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} \\frac { \\partial \\lambda } { \\partial \\rho } = - \\frac { 1 } { 2 } \\zeta ' ( 0 , D ) + \\rho \\zeta ( 0 , D ) \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} A _ j - A _ j ^ * = i \\Phi ^ * \\sigma _ j \\Phi . \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} | 1 : 0 : 2 | = 3 \\cdot | 1 : 0 : 2 | = | 3 : 0 : 1 8 | . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} \\begin{aligned} F ( x _ k ) \\le a _ k F ( x ) + & ( 1 - a _ k ) F ( x _ { k - 1 } ) + \\frac { \\tilde { \\mu } _ k a _ k ^ 2 } { 2 } \\left ( \\norm { x - v _ { k - 1 } } ^ 2 - \\norm { x - v _ k } ^ 2 \\right ) \\\\ & - \\frac { ( \\alpha ^ { - 1 } - 1 ) } { 2 t _ k } \\| y _ k - x _ k \\| ^ 2 + \\rho a _ k ( 1 - a _ k ) \\| x - x _ { k - 1 } \\| ^ 2 + \\frac { r a _ k ^ 2 } { 2 } \\| x - v _ { k - 1 } \\| ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } \\frac { \\mathcal { A } ( t ) } { t ^ 2 } \\ , \\mathrm { d } t = \\infty . \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} C = 1 + \\frac { 1 } { \\sqrt { L / \\ell } - 1 } . \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} v _ i & = F ^ 1 _ i , \\\\ \\frac { 1 } { \\rho } \\left ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\theta - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } - B _ { i J K j } v _ { j } ) _ { , K } \\right ) _ { , J } & = F ^ 2 _ i , \\\\ \\theta & = F ^ 3 , \\\\ \\frac { 1 } { a } \\left ( - \\beta _ { K i } v _ { i , K } + ( m _ { I J } \\theta _ { , J } + M _ { j L K I } u _ { j , L K } + K _ { I J } \\tau _ { , J } ) _ { , I } \\right ) & = F ^ 4 . \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} a _ m ( W _ 0 ) = A _ m W _ 0 + B _ m , \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\tilde { \\varphi } _ N ( x + \\i 0 ) = \\varphi ( x ) \\ \\forall x \\in \\R ; \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} ( \\Phi , e ^ { i 2 \\pi ( n + p ) x } ) = 0 \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} p _ k ( y ) = y _ n \\left ( a ^ 0 _ k + \\sum \\limits _ { j = 1 } ^ { n - 1 } a _ k ^ j y _ j \\right ) + b _ k ( y _ n ^ 3 - 3 y _ n y _ { n + 1 } ^ 2 ) , \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} x y x z x = x y z x . \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} \\sigma ( h , t ) = ( \\phi ( h ) , t ) , ( h , t ) \\in G . \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { \\beta t } & p _ D ( t , x , x ) \\d t = \\int _ { 0 } ^ { t _ 0 } e ^ { \\beta t } p _ D ( t , x , x ) \\d t + \\int _ { t _ 0 } ^ { \\infty } e ^ { \\beta t } p _ D ( t , x , x ) \\d t , \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} 0 = \\sum _ { k = 0 } ^ { n - 1 } a _ k \\Delta _ { ( n - k ) h _ 1 } ( \\tau _ { k y } f ) ( x ) y , h _ 1 \\in B _ { d } ( \\delta / 2 ) . \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\dot { p } ( t ) = \\lambda _ t A p ( t ) , \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} h _ \\xi ^ 0 ( \\nu ) = \\sum _ { w \\in W _ \\R } h _ 0 ( w \\nu - \\xi ) , \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} \\tilde { \\kappa } _ { ( r ) , B } = \\sum _ { i = 0 } ^ { r } I _ i ( B ) \\ \\kappa _ { ( r - i ) } \\cdot \\kappa _ { ( 1 ) } ^ i \\ = \\sum _ { i = 0 } ^ { r } I _ i ( B ' ) \\ \\kappa _ { ( r - i ) } \\cdot \\kappa _ { ( 1 ) } ^ i \\ \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} J _ { \\nu } ( z ; q ) : = \\frac { z ^ { \\nu } } { ( q ; q ) _ { \\infty } } { } _ { 1 } \\tilde { \\phi } _ { 1 } \\left ( 0 ; q ^ { \\nu + 1 } ; q , q z ^ { 2 } \\right ) \\ ! , \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} f _ 0 = n _ { 0 } + { I _ { \\varepsilon } } | E _ { 0 } | ^ { 2 } , \\nabla \\cdot f _ 1 = n _ { 1 } + 2 I m \\{ E _ { 0 } \\Delta _ \\varepsilon \\overline { E } _ { 0 } \\} , \\quad \\omega _ \\varepsilon = \\sqrt { - \\Delta _ \\varepsilon } . \\end{align*}"}
-{"id": "10175.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } n ^ { - \\frac 3 4 } \\mathbb E \\left [ \\max _ { k = 1 , \\ldots , n } M _ k ^ { ( 1 ) } \\right ] = K _ p \\mathbb E \\left [ \\sup _ { t \\in [ 0 , 1 ] } \\Delta _ t ^ { ( 0 ) } \\right ] . \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} w ( H ' ) > D _ { k , t } ( { \\cal G } ) = w ( J ) ; \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} & \\int _ { B _ \\rho } | \\nabla \\Delta \\varphi | _ g ^ 2 d \\mu _ g = \\int _ { B _ \\rho } g ^ { i j } ( \\Delta \\varphi ) _ { , i } ( \\Delta \\varphi ) _ { , j } d \\mu _ g \\\\ & = \\int _ { B _ \\rho } ( \\delta ^ { i j } + O ( r ^ 2 ) ) ( \\Delta _ 0 \\varphi + O ( r ^ { N - 1 } ) \\varphi ' ) _ { , i } ( \\Delta _ 0 \\varphi + O ( r ^ { N - 1 } ) \\varphi ' ) _ { , j } ( 1 + O ( r ^ N ) ) d x \\\\ & = \\int _ { B _ \\rho } | ( \\nabla \\Delta ) _ 0 \\varphi | ^ 2 d x + \\int _ { B _ \\rho } ( \\Delta _ 0 \\varphi ) ' ( O ( r ^ { N - 2 } ) \\varphi ' + O ( r ^ { N - 1 } ) \\varphi '' ) d x \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty e ^ { \\beta t } & \\int _ { B _ R ( 0 ) \\times B _ R ( 0 ) } p _ D ( t , x _ 1 , y _ 1 ) p _ D ( t , x _ 2 , y _ 2 ) f ( y _ 1 , y _ 1 ) \\d t \\ , \\d y _ 1 \\ , \\d y _ 2 \\\\ & = \\int _ 0 ^ { t _ 0 } e ^ { \\beta t } \\int _ { B _ R ( 0 ) \\times B _ R ( 0 ) } p _ D ( t , x _ 1 , y _ 1 ) p _ D ( t , x _ 2 , y _ 2 ) f ( y _ 1 , y _ 1 ) \\d t \\ , \\d y _ 1 \\ , \\d y _ 2 \\\\ & + \\int _ { t _ 0 } ^ \\infty e ^ { \\beta t } \\int _ { B _ R ( 0 ) \\times B _ R ( 0 ) } p _ D ( t , x _ 1 , y _ 1 ) p _ D ( t , x _ 2 , y _ 2 ) f ( y _ 1 , y _ 1 ) \\d t \\ , \\d y _ 1 \\ , \\d y _ 2 \\\\ & : = I _ 1 + I _ 2 . \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} \\frac { \\sinh \\Theta } { \\cosh \\Theta } F v = \\varPhi ( x , \\tau , \\tilde { u } , \\tilde { u } e ^ { - \\tau } , D \\tilde { u } , D ^ 2 \\tilde { u } ) , \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\{ \\alpha , \\beta \\} _ { \\pi , N ^ { * } } = \\{ N ^ { * } ( \\alpha ) , \\beta \\} _ { \\pi } + \\{ \\alpha , N ^ { * } ( \\beta ) \\} _ { \\pi } - N ^ { * } ( \\{ \\alpha , \\beta \\} _ { \\pi } ) \\ , . \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} Y _ q ( t ) = \\sum _ { p = 1 } ^ { N _ T } h _ { q p } ( t ) X _ p ( t ) + Z _ q ( t ) , q = 1 , 2 , \\cdots , N _ R , \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { i , j \\leq k } c _ { i j } \\log p _ { i j } + \\frac { 1 } { N } \\sum _ { j = 1 } ^ k \\sum _ { i } \\log \\mu _ j ( x _ i ^ j ) . \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} \\Gamma _ u = G \\times \\{ 0 \\} , \\Gamma _ b = G \\times \\{ - h \\} \\hbox { a n d } \\Gamma _ l = \\partial G \\times ( - h , 0 ) , \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\lambda _ { \\ell + 1 + r ( k m + i ) } = \\alpha _ { i + 1 } \\lambda _ { k m + i + 1 } ^ { r } , 0 \\leqslant i < k , \\\\ \\lambda _ { \\ell + 1 + r m + j } = \\beta _ { j } , 1 \\leqslant j < r , \\end{array} \\right . \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{gather*} \\deg ( E _ { a a - 1 } ( z ) ) = \\alpha _ a . \\end{gather*}"}
-{"id": "1037.png", "formula": "\\begin{align*} \\varphi _ \\sigma ( f ) = \\sum _ { \\lambda \\vdash n } \\hat { \\varphi } _ \\lambda X ^ \\lambda ( f ) + b ( f ) = \\sum _ { \\lambda \\vdash n } X ^ \\lambda ( \\sigma ) X ^ \\lambda ( f ) + b ( f ) , \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { N _ T } \\frac { \\binom { N _ T } { t } } { d _ { 0 , t } } a _ { 0 , t } \\ge \\frac { 1 } { \\min \\{ 1 , \\frac { N _ T } { N _ R } \\} } \\sum _ { t = 1 } ^ { N _ T } \\binom { N _ T } { t } a _ { 0 , t } = \\frac { N _ R } { \\min \\{ N _ T , N _ R \\} } . \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} \\mathfrak { a } = \\mathfrak { a } _ 1 \\oplus \\cdots \\oplus \\mathfrak { a } _ { r } , \\begin{cases} \\mathfrak { a } _ 1 : = \\mathrm { s p a n } \\big \\{ X _ 1 , \\ldots , X _ m \\} , \\\\ \\mathfrak { a } _ k : = [ \\mathfrak { a } _ 1 , \\mathfrak { a } _ { k - 1 } ] ; \\\\ [ \\mathfrak { a } _ 1 , \\mathfrak { a } _ { r } ] = \\{ 0 \\} . \\end{cases} \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} Y _ \\mathrm m = \\left [ \\begin{array} { c c c c c } y _ \\mathrm { m } ( 0 ) & y _ \\mathrm { m } ( 1 ) & \\cdots & y _ \\mathrm { m } ( m - 1 ) \\\\ y _ \\mathrm { m } ( 1 ) & y _ \\mathrm { m } ( 2 ) & \\cdots & y _ \\mathrm { m } ( m ) \\\\ \\vdots & \\vdots & & \\vdots \\\\ y _ \\mathrm { m } ( n - 1 ) & y _ \\mathrm { m } ( n ) & \\cdots & y _ \\mathrm { m } ( m + n - 1 ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} H ^ { l ( \\alpha ) } ( K _ { \\beta } \\mathcal { S } ^ * \\otimes K _ { \\gamma } \\mathcal { Q } ^ * ) = K _ { \\tilde { \\alpha } } { V } ^ * . \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} A ^ { \\mu } \\left ( x \\right ) = \\left ( \\frac { \\kappa } { 1 + \\kappa } u ^ { \\mu } u _ { \\lambda } - \\delta _ { \\lambda } ^ { \\mu } \\right ) \\partial _ { \\sigma } Z ^ { \\lambda \\sigma } \\left ( x \\right ) \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P , A c h } } ( \\mu , r ) \\leq \\delta _ { \\mathsf { P - Z F } } = \\frac { K } { \\min \\{ M , K \\} } , \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} \\ln \\det ( D , \\Lambda ) - \\ln \\det ( D , \\Lambda ' ) = - 2 \\ln ( \\Lambda / \\Lambda ' ) \\zeta ( 0 , D ) \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q V _ N } ] = \\lim _ { \\beta \\rightarrow \\infty } \\Bigl [ { \\bf E } \\bigl [ X ^ { \\frac { q } { \\beta } } \\bigr ] \\Gamma ( 1 - \\frac { q } { \\beta } ) \\Bigr ] . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} \\Gamma _ 1 : = & A \\cap A _ r \\cap H ^ c , \\ \\ \\ \\Gamma _ 2 : = ( \\underline { A } \\setminus A _ H ) \\cap H ^ c \\\\ \\Gamma _ 3 : = & ( \\underline { A _ H } \\setminus A ) \\cap H ^ c , \\ \\Gamma _ 4 : = H ^ c \\setminus ( A \\cup A _ H ) . \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} \\norm { \\frac { 1 } { N } \\sum _ { r \\neq l } c _ { r l } \\mu _ r \\otimes \\mu _ l } _ { T V } = \\norm { M _ X - \\sum _ { i \\leq k } w _ i \\mu _ i \\otimes \\mu _ i } _ { T V } \\leq \\frac { 1 } { m } . \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} F = \\left ( \\begin{array} { c c c c } \\bar { y } _ 1 - \\bar { y } _ 2 + \\bar { u } _ 2 & 0 & \\bar { y } _ 2 - \\bar { u } _ 2 & 0 \\\\ 0 & \\bar { u } _ 2 & 0 & \\bar { y } _ 2 - \\bar { u } _ 2 \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ n \\setminus \\overline { B _ \\rho } } | ( \\nabla \\Delta ) _ 0 \\varphi | ^ 2 d x = & ( n - 6 ) ^ 2 ( n - 4 ) ^ 2 \\int _ { \\mathbb { R } ^ n \\setminus \\overline { B _ \\rho } } \\frac { u _ \\epsilon ^ 2 r ^ 2 } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 6 } \\Big [ ( n + 2 ) \\epsilon ^ 2 + 4 r ^ 2 \\Big ] ^ 2 d x \\\\ \\leq & C \\int _ { \\rho / \\epsilon } ^ \\infty \\sigma ^ { 5 - n } d \\sigma = O \\big ( \\epsilon ^ { n - 6 } \\big ) . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} w ( x , t ) : = \\inf _ { v \\in \\mathcal { S } _ \\psi } v ( x , t ) \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} \\mu ^ s _ 0 ( \\alpha _ t , \\gamma _ t ) \\longmapsto \\mu ^ s _ 0 ( \\alpha , \\gamma ) = \\frac { H ( \\gamma ) - H ( \\alpha ) } { \\gamma - \\alpha } \\equiv { \\mu } ^ s _ 0 , ~ ~ ~ \\mu ^ s _ 1 ( \\beta _ t , \\delta _ t ) \\longmapsto \\mu ^ s _ 1 ( \\beta , \\delta ) = \\frac { H ( \\beta ) - H ( \\delta ) } { \\beta - \\delta } \\equiv \\mu ^ s _ 1 , ~ \\forall { t } . \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} ( F [ S X ] , G [ X ] ) _ X = ( F [ X ] , G [ S X ] ) _ X \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} g _ C = \\frac { ( d - 1 ) ( d - 2 ) } { 2 } . \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\frac { d ^ { r + 1 } } { d x ^ { r + 1 } } \\left [ x ^ r \\left ( \\frac { x } { x - a } \\right ) ^ k \\right ] = ( - a ) ^ { r + 1 } k ^ { ( r + 1 ) } \\frac { x ^ { k - 1 } } { ( x - a ) ^ { k + r + 1 } } . \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} \\left \\langle J _ { ( a b , i ) } , J _ { ( c d , j ) } , P _ { ( e , k ) } \\right \\rangle & = \\alpha _ { \\gamma } K _ { i j k } ^ { \\gamma } \\left \\langle \\tilde { J } _ { a b } , \\tilde { J } _ { c d } , \\tilde { P } _ { e } \\right \\rangle , \\\\ & = \\frac { 1 } { 8 } \\left ( \\alpha _ { 0 } K _ { i j k } ^ { 0 } + \\alpha _ { 1 } K _ { i j k } ^ { 1 } + \\alpha _ { 2 } K _ { i j k } ^ { 2 } + \\alpha _ { 3 } K _ { i j k } ^ { 3 } \\right ) \\varepsilon _ { a b c d e } . \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} d _ \\ell = 2 \\sum _ { k = 0 } ^ { n _ 1 - 1 } ( \\omega ^ { n _ 2 } ) ^ { k \\ell ' } a ' _ k , \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{gather*} \\tau _ { k + 1 , \\ell + 1 } ^ { ( \\alpha , \\beta ) } \\tau _ { k + 1 , \\ell } ^ { ( \\alpha - 1 , \\beta ) } = \\tau _ { k + 1 , \\ell } ^ { ( \\alpha , \\beta ) } \\tau _ { k + 1 , \\ell + 1 } ^ { ( \\alpha - 1 , \\beta ) } + \\tau _ { k + 2 , \\ell + 1 } ^ { ( \\alpha - 1 , \\beta ) } \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta ) } , \\end{gather*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\frac { 2 e ^ { x t } } { e ^ t + 1 } = \\sum \\limits _ { n = 0 } ^ \\infty E _ n ( x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} z & = x - t v \\\\ v & = y . \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} \\pi ( x ^ m y ^ n ) & = \\sum _ { j = 1 } ^ n \\tau _ n ( j ) \\alpha ^ { n - j } \\ , x ^ m y ^ { j } , & & m , n \\in \\N _ 0 . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} K ( \\beta , q ) : = \\left \\{ \\kappa = ( \\delta , p ) : \\ ; \\beta < \\delta < 1 - \\beta , \\ , \\frac d \\delta < p < q \\right \\} . \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 3 } \\ [ \\bar { f } _ { i , k _ { \\pi ( 1 ) } } , [ \\bar { f } _ { i , k _ { \\pi ( 2 ) } } , [ \\bar { f } _ { i , k _ { \\pi ( 3 ) } } , \\bar { f } _ { i + 1 , l } ] ] ] = 0 . \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} | { \\rm c o m } _ 1 | & = \\left | M ( k , \\xi ) \\frac { N } { 2 } ( 1 + k ^ 2 + ( \\xi - \\theta \\eta ) ^ 2 ) ^ { N / 2 - 1 } ) 2 ( \\xi - \\theta \\eta ) \\eta \\right | \\\\ & \\lesssim \\left ( ( 1 + k ^ 2 + ( \\xi - \\eta ) ^ 2 ) ^ { ( N - 1 ) / 2 } + ( 1 + k ^ 2 + \\xi ^ 2 ) ^ { ( N - 1 ) / 2 } \\right ) | \\eta | . \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} F _ \\Gamma ^ { 0 } ( z ) = \\frac { 1 } { z } \\left ( \\frac { \\vartheta _ { \\mathcal L _ \\Gamma } ( z ) } { ( 1 - z ^ 2 ) ^ { n - 1 } } - 1 \\right ) , \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} & A ^ { i j k l , \\varepsilon } : = \\lambda g ^ { i j , \\varepsilon } g ^ { k l , \\varepsilon } + \\mu ( g ^ { i k , \\varepsilon } g ^ { j l , \\varepsilon } + g ^ { i l , \\varepsilon } g ^ { j k , \\varepsilon } ) , \\\\ & B ^ { i j k l , \\varepsilon } : = \\theta g ^ { i j , \\varepsilon } g ^ { k l , \\varepsilon } + \\frac { \\rho } { 2 } ( g ^ { i k , \\varepsilon } g ^ { j l , \\varepsilon } + g ^ { i l , \\varepsilon } g ^ { j k , \\varepsilon } ) , \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} S _ { i j } = h _ { i j } - g _ { i j } \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\overline { d } ( X ) & = \\mathrm { m i n } \\{ r \\equiv 2 \\mu ( X ) \\ ; \\mathrm { m o d } \\ ; 2 \\mid \\exists \\ , x \\in \\tilde { H } ^ r _ { \\mathbb { Z } / 2 } ( X ) , U ^ l x \\neq 0 \\ ; \\mathrm { f o r \\ ; a l l } \\ ; l \\geq 0 \\} , \\\\ \\underline { d } ( X ) & = \\mathrm { m i n } \\{ r \\equiv 2 \\mu ( X ) + 1 \\ ; \\mathrm { m o d } \\ ; 2 \\mid \\exists \\ , x \\in \\tilde { H } ^ r _ { \\mathbb { Z } / 2 } ( X ) , U ^ l x \\neq 0 \\ ; \\mathrm { f o r \\ ; a l l } \\ ; l \\geq 0 \\} - 1 , \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\widetilde { \\omega } _ 1 ( \\pi ) : = 2 \\lambda _ { \\nu ( \\pi ) } \\cdot \\prod _ { i = 1 } ^ { \\nu ( \\pi ) - 1 } ( 2 \\lambda _ i - 2 \\lambda _ { i + 1 } - 1 ) . \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} v ( t ) & = e ^ { t A _ p } a + \\int _ 0 ^ t e ^ { ( t - s ) A _ p } \\left ( P _ p f ( s ) + F _ p ( v ( s ) , \\zeta ( s ) ) \\right ) d s , \\\\ \\zeta ( t ) & = e ^ { t \\Delta _ { \\zeta } } b + \\int _ 0 ^ t e ^ { ( t - s ) \\Delta _ { \\zeta } } \\left ( g ( s ) + G _ q ( v ( s ) , \\zeta ( s ) ) \\right ) d s . \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} s = ( \\beta - 1 ) + ( | V _ { n , k , b } | - \\alpha ) = | V _ { n , k , b } | + ( \\beta - \\alpha ) - 1 \\geq | V _ { n , k , b } | + | C | - 2 . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} ( u - v \\mp a ) x _ i ^ { \\pm } ( u ) x _ j ^ { \\pm } ( v ) = ( u - v \\pm a ) x _ j ^ { \\pm } ( v ) x _ i ^ { \\pm } ( u ) + \\hbar \\Big ( [ x _ { i , 0 } ^ { \\pm } , x _ j ^ { \\pm } ( v ) ] - [ x _ i ^ { \\pm } ( u ) , x _ { j , 0 } ^ { \\pm } ] \\Big ) \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} X _ t = x _ 0 + \\theta \\int _ 0 ^ t X _ s \\ , d s + B ^ H _ t , t \\ge 0 , \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} u ( x , t ) = \\Lambda ^ { - } _ 0 \\mathrm { e r f c } ( \\eta ) + \\Lambda ^ { + } _ 0 \\mathrm { e r f c } ( - \\eta ) + \\sum \\limits _ { n = 1 } ^ { \\infty } t ^ { - n / 2 } \\left ( H _ { n , 0 } ( \\eta ) + H _ { n , 1 } ( \\eta ) \\ln t \\right ) , \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} & A ^ { - 1 } \\sharp _ { \\nu } B ^ { - 1 } \\\\ & \\leqslant A ^ { - 1 } \\sharp _ { \\nu } B ^ { - 1 } + \\sum _ { k = 0 } ^ { \\infty } r _ { k } ( A ^ { - 1 } \\sharp _ { \\frac { m _ k } { 2 ^ k } } B ^ { - 1 } - 2 A ^ { - 1 } \\sharp _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } B ^ { - 1 } + A ^ { - 1 } \\sharp _ { \\frac { m _ k + 1 } { 2 ^ k } } B ^ { - 1 } ) \\\\ & \\leqslant A ^ { - 1 } \\nabla _ { \\nu } B ^ { - 1 } . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} d ( e _ 2 ) = d ( [ e _ 1 , e _ 1 ] ) = [ d ( e _ 1 ) , e _ 1 ] + [ e _ 1 , d ( e _ 1 ) ] = 2 a _ 1 e _ 2 + \\sum \\limits _ { i = 3 } ^ { n - 2 k } a _ { i - 1 } e _ { i } + \\sum \\limits _ { i = 1 } ^ { k } b _ i f _ { k + i } . \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} \\tilde { A } = \\left [ \\begin{matrix} \\operatorname { R e } ( A ) & - \\operatorname { I m } ( A ) \\\\ \\operatorname { I m } ( A ) & \\operatorname { R e } ( A ) \\\\ \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} H _ { \\Lambda _ L } = S \\sum _ { \\langle x , y \\rangle \\subset \\Lambda _ L } \\left ( - a ^ * _ x \\sqrt { 1 - \\frac { n _ x } { 2 S } } \\sqrt { 1 - \\frac { n _ y } { 2 S } } a _ y - a ^ * _ y \\sqrt { 1 - \\frac { n _ y } { 2 S } } \\sqrt { 1 - \\frac { n _ x } { 2 S } } a _ x + n _ x + n _ y - \\frac { 1 } { S } n _ x n _ y \\right ) . \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ n e _ i ( \\boldsymbol { x } ) ( t _ n x _ n ) ^ { n - i } s _ { \\lambda - \\rho } ( \\boldsymbol { x } ) = \\sum _ { i = 0 } ^ n ( t _ n x _ n ) ^ { n - i } \\sum _ { \\mu \\in ( \\lambda - \\rho ) \\otimes 1 ^ i } s _ { \\mu } ( \\boldsymbol { x } ) , \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} \\widehat { g _ 2 } ( \\eta ) = \\eta ^ { d - 1 } ( 1 + \\varepsilon ^ 2 \\eta ^ 2 ) ^ { - 3 / 2 } \\omega \\cdot \\widehat { f _ 2 } ( s , \\eta \\omega ) . \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} a \\circ ( b c ) = \\pi ( \\pi ^ { - 1 } ( a ) \\pi ^ { - 1 } ( b c ) ) = \\pi ( \\pi ^ { - 1 } ( a ) _ 1 ) ( \\pi ^ { - 1 } ( a ) _ 2 \\rightharpoonup ( b c ) ) . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} \\frac { 1 } { m } \\sum _ { i = 1 } ^ m \\sup _ { ( \\| X \\| \\leq \\| X _ 0 \\| + 1 ) } \\big \\| s _ x ^ { \\prime \\prime } ( a _ i , b _ i ; X ) \\big \\| . \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} ( a , b ) _ 2 = \\begin{cases} 1 & p \\equiv 3 , 5 ~ { \\rm m o d } ~ 8 . \\\\ - 1 & p \\equiv 1 , 7 ~ { \\rm m o d } ~ 8 . \\end{cases} \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} & : ( \\alpha _ 0 \\partial _ z + \\frac { 1 } { \\sqrt { 2 } } b ( z ) ) ( \\alpha _ 0 \\partial _ z - \\frac { 1 } { \\sqrt { 2 } } b ( z ) ) : \\\\ & = \\alpha _ 0 ^ 2 \\partial _ z ^ 2 - L ( z ) , \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} v _ { n - 1 } + \\left ( \\alpha q ^ { - n } - x \\right ) v _ { n } + v _ { n + 1 } = 0 , n \\in \\Z , \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} \\mathcal { K } _ s = \\left \\{ Z _ s = \\left ( X _ s , V _ s \\right ) \\in \\overline { \\mathcal { D } _ s } \\left | \\psi _ s ^ { - \\tau } Z _ s = \\left ( X _ s - V _ s \\tau , V _ s \\right ) \\ ; \\forall \\ ; \\tau > 0 \\right . \\right \\} \\subset \\mathbb { R } ^ { 2 d s } \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} \\left \\Vert \\left \\{ \\xi _ { j } - \\varphi \\left ( z _ { j } \\right ) \\right \\} _ { j = 1 } ^ { J } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } < \\delta \\left \\Vert \\left \\{ \\xi _ { j } \\right \\} _ { j = 1 } ^ { J } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} J _ { k } g \\left ( \\alpha \\right ) & = \\sum _ { m _ { k - 1 } \\left ( \\alpha \\right ) \\leq \\beta \\leq m _ { k } \\left ( \\alpha \\right ) } g \\left ( A \\beta \\right ) , \\\\ J _ { \\infty } g \\left ( \\alpha \\right ) & = g \\left ( \\alpha \\right ) + \\sum _ { m _ { N \\left ( \\alpha \\right ) } \\left ( \\alpha \\right ) \\leq \\beta \\leq \\alpha } g \\left ( A \\beta \\right ) \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ { \\infty } n | a _ n | + \\sum _ { n = 1 } ^ { \\infty } n | b _ n | \\leq \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} [ L _ m , L _ n ] = ( m - n ) L _ { m + n } + \\frac { m ^ 3 - m } { 1 2 } \\delta _ { m , n } c , \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + \\ , { \\rm c o n s t } ) } \\ , F ( q \\ , | \\ , \\beta = 1 , \\lambda _ 1 , \\lambda _ 2 ) , \\ ; N \\rightarrow \\infty \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} { \\cal P } _ { 0 , n } ^ { B } ( \\kappa ) \\triangleq \\Big \\{ \\pi _ t ( x _ t | y ^ { t - 1 } ) , ~ t = 0 , \\ldots , n : \\frac { 1 } { n + 1 } { \\bf E } ^ { \\pi } \\Big ( c ^ { B } _ { 0 , n } ( X ^ n , Y ^ { n - 1 } ) \\Big ) \\leq \\kappa \\Big \\} , ~ \\kappa \\in [ 0 , \\infty ) \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} & \\textrm { r a n k } ( [ { \\bf V } _ { 2 , 3 } ^ p ~ { \\bf V } _ { 2 , 3 } ^ c ~ { \\bf V } _ { 2 , 3 } ^ r ] ) = d _ { 2 , 3 } , \\\\ & \\textrm { r a n k } ( [ { \\bf V } _ { 3 , 1 } ^ p ~ { \\bf V } _ { 3 , 1 } ^ c ~ { \\bf V } _ { 3 , 1 } ^ r ] ) = d _ { 3 , 1 } . \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} p _ { i } : = \\prod _ { j \\neq i } ( z - \\alpha _ { j } ) ^ { n _ { j } - 1 } . \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} \\phi _ { A } \\left ( x \\right ) = \\prod \\limits _ { i \\in \\left [ k \\right ] } \\left ( \\phi _ { A _ { i } } \\left ( x \\right ) \\right ) ^ { \\left ( r - 1 \\right ) ^ { n - n _ { i } } } . \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} \\Delta _ N ^ \\perp = \\Delta _ D . \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\jmath ^ \\natural ( F \\otimes ^ { \\mathrm { R D } } G ) ( U _ I ) & = ( F \\otimes ^ { \\mathrm { R D } } G ) ( \\jmath ( U _ I ) ) \\\\ & \\simeq \\bigoplus _ { I _ 1 \\sqcup I _ 2 = I , I _ j \\neq \\varnothing } F ( \\jmath ( U _ { I _ 1 } ) ) \\otimes G ( \\jmath ( U _ { I _ 2 } ) ) \\\\ & = \\bigoplus _ { I _ 1 \\sqcup I _ 2 = I , I _ j \\neq \\varnothing } \\jmath ^ \\natural F ( U _ { I _ 1 } ) \\otimes \\jmath ^ \\natural G ( U _ { I _ 2 } ) \\\\ & \\simeq ( \\jmath ^ \\natural F \\otimes ^ \\mathrm { R D } \\jmath ^ \\natural G ) ( U _ I ) , \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} \\omega _ m ^ { - 1 } r ^ { - m } \\left | B _ { g ( b ) } \\left ( x _ 0 , r \\right ) \\right | _ { g ( b ) } \\geq e ^ { - \\frac { \\delta _ 0 } { 1 0 0 } } \\cdot \\left ( 1 - \\frac { \\delta _ 0 } { 1 0 0 } \\right ) \\cdot e ^ { - \\frac { m \\delta _ 0 } { 1 0 0 m } } = e ^ { - \\frac { \\delta _ 0 } { 5 0 } } \\cdot \\left ( 1 - \\frac { \\delta _ 0 } { 1 0 0 } \\right ) > 1 - \\frac { \\delta _ 0 } { 1 0 } . \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} A _ N : = & \\sqrt { \\tfrac { 2 } { \\tilde \\mu } } \\sum _ { i = 1 } ^ N \\sqrt { \\tfrac { \\delta _ i } { a _ i } } & + \\left ( \\| x ^ * - v _ 0 \\| ^ 2 + \\tfrac { M ^ 2 N ( r + \\frac { \\rho } { 2 } ( N + 3 ) ) } { \\tilde \\mu } + \\tfrac { 2 } { \\tilde \\mu } \\sum ^ N _ { i = 1 } \\tfrac { \\delta _ i a _ i + 2 \\varepsilon _ i } { a _ i ^ 2 } + \\tfrac { 2 } { \\tilde \\mu } \\left ( \\sum _ { i = 1 } ^ N \\sqrt { \\tfrac { \\delta _ i } { a _ i } } \\right ) ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} \\begin{aligned} \\xi & = C z - v ^ s + p & & \\mbox { a . e . i n } [ 0 , T ] \\times \\Omega \\ , , \\\\ \\partial _ { t } p & \\in \\partial \\Psi ( z - B p ) & & \\mbox { a . e . i n } [ 0 , T ] \\times \\Omega \\ , . \\end{aligned} \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\psi _ S ( x ) : = \\inf _ { y \\in S } \\{ \\nabla f ( x ) ^ { \\top } ( y - x ) + r ( y ) - r ( x ) \\} \\geq - \\epsilon . \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} F ( x ) & : T _ { f ( x ) } X ^ 4 \\to T _ { f ( x ) } X ^ 4 , \\\\ F ( x ) E _ 1 & = E _ 2 , \\ \\ F ( x ) E _ 2 = - E _ 1 , \\\\ F ( x ) N _ 1 & = - N _ 2 , \\ \\ F ( x ) N _ 2 = N _ 1 , \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} \\left [ H _ 0 - E - \\sum _ { j = 1 } ^ { n } \\lambda _ { j } ( \\epsilon ) | a _ { j } ^ { \\epsilon } \\rangle \\langle a _ { j } ^ { \\epsilon } | \\right ] | \\psi \\rangle = | \\chi \\rangle \\ ; , \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ( a , b ) ^ q \\bigr ] = & \\frac { \\Gamma _ 2 ( q + b _ 0 \\ , | \\ , a ) } { \\Gamma _ 2 ( b _ 0 \\ , | \\ , a ) } \\frac { \\Gamma _ 2 ( b _ 0 + b _ 1 \\ , | \\ , a ) } { \\Gamma _ 2 ( q + b _ 0 + b _ 1 \\ , | \\ , a ) } \\frac { \\Gamma _ 2 ( b _ 0 + b _ 2 \\ , | \\ , a ) } { \\Gamma _ 2 ( q + b _ 0 + b _ 2 \\ , | \\ , a ) } \\times \\\\ & \\times \\frac { \\Gamma _ 2 ( q + b _ 0 + b _ 1 + b _ 2 \\ , | \\ , a ) } { \\Gamma _ 2 ( b _ 0 + b _ 1 + b _ 2 \\ , | \\ , a ) } . \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} w _ k : = ( \\eta - u _ k - \\varepsilon ) _ + \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} C ^ { F B , A . 1 } _ { X ^ n \\rightarrow { Y ^ n } } = \\sum _ { y _ { - 1 } \\in \\{ 0 , 1 \\} } C _ 0 ( y _ { - 1 } ) { \\bf P } _ { Y _ { - 1 } } ( d y _ { - 1 } ) , ~ { \\bf P } _ { Y _ { - 1 } } ( d y _ { - 1 } ) \\equiv \\mu ( d y _ { - 1 } ) ~ \\mbox { i s f i x e d } . \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} \\varphi _ { l , j } ( x ) = e ^ { i \\left \\langle a _ { l , j } + n _ { l } v _ { k } + t , x \\right \\rangle } + { \\textstyle \\sum \\limits _ { n = n _ { l } + 1 } ^ { \\infty } } \\left ( { \\textstyle \\sum \\limits _ { a \\in \\Gamma ( k ) } } c ( a , n ) e ^ { i \\left \\langle a + n v _ { k } + t , x \\right \\rangle } \\right ) \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} x ( t ) = x ( t , 0 , \\ldots , 0 ) = e ^ { i t A _ 1 } \\left ( h - i \\int _ 0 ^ t e ^ { - i s A _ 1 } \\Phi ^ * u ( s ) d s \\right ) . \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\left | \\Sigma _ { n ^ 2 } \\right | = \\left ( n ! \\right ) ^ { 2 n } \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} F _ 2 ' = - \\frac { n } { m } \\ , F _ 1 ' + \\frac { 1 } { m } \\ , G _ 1 ' = - \\frac { e ^ { - i \\theta } e ^ { i s } \\sqrt { 1 + | m | ^ 2 } } { \\overline m } \\ , F _ 1 ' + \\frac { 1 } { m } \\ , G _ 1 ' \\ , . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} a _ k ( f , D ) = R e s _ { s = ( n - k ) / 2 } ( \\Gamma ( s ) \\zeta ( s , f , D ) ) \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\gamma ( u ) = \\left ( f _ { 1 } ( u ) , . . . , f _ { n } ( u ) , \\lambda \\cos \\left ( \\frac { u } { c } \\right ) \\right ) . \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\tilde { G } = d \\rho ^ 2 + g _ \\rho \\end{align*}"}
-{"id": "10001.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - u '' + c u = f \\mbox { i n } ( 0 , 1 ) \\\\ u ( 0 ) = g _ 0 , u ( 1 ) = g _ 1 \\end{array} \\right . \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} x K _ { n } & = K _ { n + 1 } + \\left ( l _ { n + 1 , n + 1 } + m _ { n + 1 } \\right ) K _ { n } + \\sum \\nolimits _ { i = 0 } ^ { d - 2 } \\left ( l _ { n + 1 , n - i } + l _ { n , n - i } m _ { n + 1 } \\right ) K _ { n - i - 1 } \\\\ & + l _ { n , n - d + 1 } m _ { n + 1 } K _ { n - d } \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\norm { \\nabla \\zeta ( t ) } ^ 2 + \\alpha \\norm { \\tau ( t ) } ^ 2 _ { L ^ 2 ( \\Gamma _ u ) } & \\leq \\left ( \\norm { \\nabla b } ^ 2 + \\alpha \\norm { b _ { \\tau } } ^ 2 _ { L ^ 2 ( \\Gamma _ u ) } + 3 \\int _ 0 ^ t \\norm { g ( s ) } ^ 2 d s \\right ) e ^ { \\norm { \\int _ 0 ^ t \\Delta v ( s ) } ^ 2 d s } \\\\ & \\leq \\left ( C \\norm { b } _ { H ^ 1 } ^ 2 + 3 \\int _ 0 ^ t \\norm { g ( s ) } ^ 2 d s \\right ) e ^ { B _ { H ^ 1 } ^ v ( t ) } \\\\ & = : \\tilde { B } _ { H ^ 1 } ^ { \\tau } ( t ) , \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} F ^ { \\bf a } _ { k } = a _ 0 F ^ 0 _ { k } + \\sum _ { i = 1 } ^ { n - 1 } a _ { - i } \\sum ^ { n - i } _ { j = 1 } b _ { i + j } F ^ 0 _ { k - j } . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} f ^ * P _ h & = \\det ( T ) \\omega ^ 1 \\wedge \\cdots \\wedge \\omega ^ n = \\det ( T _ 1 ) \\cdots \\det ( T _ m ) \\omega ^ 1 \\wedge \\cdots \\wedge \\omega ^ n \\\\ & = ( \\det ( T _ 1 ) \\wedge \\omega ( 1 ) ) \\wedge ( \\det ( T _ 2 ) \\wedge \\omega ( 2 ) ) \\wedge \\cdots \\wedge ( \\det ( T _ m ) \\wedge \\omega ( m ) ) . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} W ( \\gamma ) & = \\frac { 1 } { 2 \\pi i } \\int _ { 0 } ^ 1 \\frac { d } { d s } \\log L _ x ( z ( s ) ) d s \\\\ & = \\frac { 1 } { 2 \\pi i } \\int _ { 0 } ^ 1 \\frac { - 2 ( 1 + e ^ { 2 \\pi i s } / 2 ) \\pi i e ^ { 2 \\pi i s } d s } { - e ^ { i 2 \\pi s } - e ^ { 4 \\pi i s } / 4 } \\\\ & = \\int _ { 0 } ^ 1 \\frac { 1 + e ^ { 2 \\pi i s } / 2 d s } { 1 + e ^ { 2 \\pi i s } / 4 } . \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} \\lambda _ n = \\sqrt { \\big ( 8 ( a _ 3 + a _ 4 ) n ^ 4 + 2 \\mu n ^ 2 - 1 \\big ) / \\rho } . \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} V _ x = \\{ z \\in X \\mid u < \\hom ( \\psi ( z ) , \\phi _ x ( z ) ) \\} \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} ( f \\bullet _ i g ) \\bullet _ j h = ( f \\bullet _ { j - n + 1 } h ) \\bullet _ { i } g i + n \\leq j \\leq m + n - 1 \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\mu _ 1 = \\left ( \\frac { K - \\max \\{ M , K \\} r } { K M + M r \\left [ \\min \\{ M , K \\} - 1 \\right ] } \\right ) ^ + , ~ ~ \\mu _ 2 = \\left ( 1 - \\frac { M r } { \\min \\{ M , K \\} } \\right ) ^ + , \\end{align*}"}
-{"id": "10002.png", "formula": "\\begin{align*} [ x _ { \\alpha _ 0 } ^ { i _ 0 } , \\ldots x _ { \\alpha _ r } ^ { i _ r } ] : = \\{ \\underline { a } \\in \\Sigma _ { A } : a _ 0 = x _ { \\alpha _ 0 } ^ { i _ 0 } , \\ldots a _ r = x _ { \\alpha _ r } ^ { i _ r } \\} \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} Z ( t ) = T r ( e ^ { - t \\Delta } ) = \\sum _ k e ^ { - t \\lambda _ k } = \\int _ M d V \\ , K ( t ; x , x ; \\Delta ) = K ( t , \\Delta ) \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\frac { d } { d t } \\exp ( t X ) ( x _ 0 ) & = X ( t , \\exp ( t X ) ( x _ 0 ) ) , \\mbox { f o r a l m o s t e v e r y } t \\in \\R \\\\ \\exp ( 0 X ) ( x _ 0 ) & = x _ 0 , \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } n ! \\| f _ n ( \\cdot , \\cdot , t , x ) \\| ^ 2 _ { \\mathcal { H } ^ { \\otimes n } } < \\infty \\ , . \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} \\max _ { \\vec { \\pmb { \\alpha } } } & \\sum _ { i = 1 } ^ M \\alpha _ i \\log _ 2 \\left ( 1 + \\frac { 1 - \\alpha _ i } { \\alpha _ i } \\frac { h _ { s s } ^ i \\eta P _ p } { \\sigma _ s ^ 2 + h _ { p s } ^ i P _ p } \\right ) \\\\ & h _ { s p } ^ i ( 1 - \\alpha _ i ) \\eta P _ p \\leq \\alpha _ i P _ { i n t } , i = 1 , \\ldots , M . \\\\ & \\vec { 0 } \\prec \\vec { \\pmb { \\alpha } } \\prec \\vec { 1 } \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} \\nu _ S \\ \\geq \\ \\nu _ { U ' } / 2 = \\frac { 1 } { 2 \\det ( A ) } \\ \\geq \\ \\nu _ U / 2 \\ . \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} \\partial \\Upsilon _ { n } ^ { \\delta } ( z _ { 0 } ) = P _ { n } \\partial \\Psi ^ { \\delta } ( z _ { 0 } ) \\ , . \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} s = \\left ( \\begin{array} { c c c } 0 & 0 & 1 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{array} \\right ) , \\ , t = \\left ( \\begin{array} { c c c } 0 & - 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) , \\ , u = \\left ( \\begin{array} { c c c } i & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{gather*} W _ { i } = \\frac { W } { \\prod \\limits _ { 1 \\le j \\le m \\atop j \\ne i } ( w _ { j } - w _ { i } ) } = ( - 1 ) ^ { i + m } \\prod \\limits _ { 1 \\le j < k \\le m \\atop j , k \\ne i } ( w _ { j } - w _ { k } ) . \\end{gather*}"}
-{"id": "4710.png", "formula": "\\begin{align*} \\forall s , t \\in \\mathbb { R } _ { \\geq 0 } : \\quad \\Phi _ { t } \\circ \\Phi _ { s } = \\Phi _ { s + t } . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} H _ { ( 2 x ) } ( i , \\alpha ) ( j , \\beta ) = H _ x ( i , j ) \\otimes \\Lambda ( \\alpha , \\beta ) \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} \\sum _ { \\chi \\pmod { H } } N ( \\sigma , T , \\chi ) = \\sum _ { \\chi \\pmod { H ' } } N ( \\sigma , T , \\chi ) \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} \\tilde { v } = \\left [ \\begin{matrix} \\operatorname { R e } ( v ) \\\\ \\operatorname { I m } ( v ) \\\\ \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} s : = a _ 1 q _ { 3 4 5 } t _ { 1 2 } + a _ 2 q _ { 1 4 5 } t _ { 2 3 } + a _ 3 q _ { 1 2 5 } t _ { 3 4 } \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\psi ( x ) & = \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - \\frac { x ^ 2 } { 2 } } \\\\ Q ( x ) & = \\int _ { x } ^ { + \\infty } \\psi ( t ) d t \\\\ g ( u ) & \\triangleq u ^ 2 Q ( u ) - u \\psi ( u ) . \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} M _ { \\phi , H } ( a , b ) = \\phi _ a \\cdot p \\cdot \\chi _ b , \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} D _ { x , i } ( { \\cal B } ) = D _ { x , j _ 1 } ( { \\cal B } ) + D _ { j _ 1 , j _ 2 } ( { \\cal B } ) + . . . + D _ { j _ t , i } ( { \\cal B } ) \\textrm { a n d t h e e l e m e n t s o f t h e s u m a r e i n d e c o m p o s a b l e } \\Big \\} \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} M \\vec v = \\lambda \\vec v + \\vec { u } \\ , . \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} \\gamma ( p , n ) = \\max \\left \\{ \\frac { n - 1 } { 2 } \\left ( \\frac { 1 } 2 - \\frac 1 p \\right ) \\ , , \\ , \\frac { n - 1 } { 2 } - \\frac { n } { p } \\right \\} \\ , . \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} F _ { t s } \\left ( { t , s } \\right ) & = \\frac { 1 } { { t s } } f _ { s t } \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) - \\frac { 1 } { t } f _ t \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) - \\frac { 1 } { s } f _ s \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) + f \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) \\\\ & = F _ { s t } \\left ( { t , s } \\right ) . \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} \\frac { 1 } { r ^ { \\star } ( \\phi _ { \\nu } ) } = \\sum _ { n \\geq 1 } \\frac { 1 } { j _ { \\nu , n } ^ 2 - r ^ { \\star } ( \\phi _ { \\nu } ) } > \\sum _ { n \\geq 1 } \\frac { 1 } { j _ { \\nu , n } ^ 2 } = \\frac { 1 } { 4 ( \\nu + 1 ) } . \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} C _ Y ( H ) = \\{ \\prod _ { h \\in H '' } { } ^ h x \\} . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} \\| P ^ \\prime ( \\cdot ) \\| _ 1 = \\| ( 1 - B _ s ) ^ { - 1 } R ^ \\prime ( \\cdot ) \\| _ 1 \\leq \\frac { 1 } { 1 - \\| B _ s \\| _ 1 } \\| R ^ \\prime ( \\cdot ) \\| _ 1 \\leq 2 C \\rho ^ { \\varepsilon _ 0 } \\langle s \\rangle ^ { - \\varepsilon _ 1 } . \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} G _ + ( t ) u [ x ] & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i \\Phi ( x , y , \\xi ; t ) } ( s _ + p _ + + s _ - p _ + ) ( x , y , \\xi ) u [ y ] d \\xi \\\\ & = : ( F _ + ( t ) + F _ - ( t ) ) u [ x ] . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} \\lambda ( \\Lambda ' ) = \\lambda ( \\Lambda ) + \\ln \\det ( g D ) | _ { \\Lambda ' } ^ \\Lambda \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { h _ u ( t ) } { h ( t ) } = \\lim _ { t \\to \\infty } \\frac { \\lambda _ u ^ t t ^ { d _ u } } { \\lambda ^ t } = 0 . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} E _ k : = \\overline Q ^ p \\cap \\{ \\varphi _ k > h - \\varepsilon \\} . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} g _ { k } ( \\alpha ^ { - 1 } q ^ { m } ) & = \\alpha ^ { k } q ^ { - m k } \\sum _ { j = k - m } ^ { \\infty } ( q ^ { m - k + j + 1 } ; q ) _ { \\infty } \\frac { ( - 1 ) ^ { j } q ^ { \\frac { 1 } { 2 } j ( j + 1 ) + 2 m j } } { ( q ; q ) _ { j } } \\alpha ^ { - 2 j } \\\\ & = ( - 1 ) ^ { m + k } \\alpha ^ { 2 m - k } q ^ { - \\frac { 1 } { 2 } m ( 3 m + 1 ) + \\frac { 1 } { 2 } k ( k + 1 ) } \\sum _ { j = 0 } ^ { \\infty } ( q ^ { j + 1 } ; q ) _ { \\infty } \\frac { ( - 1 ) ^ { j } q ^ { \\frac { 1 } { 2 } j ( j + 1 ) + ( m + k ) j } } { ( q ; q ) _ { k - m + j } } \\alpha ^ { - 2 j } , \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} G _ i ^ m = 2 \\cdot 1 \\cdot 3 \\cdot 5 \\cdot \\ldots \\cdot ( 2 m - 1 ) { 2 m + 1 \\choose 2 i + 1 } B _ { 2 m - 2 i } . \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} \\nu ^ \\pi _ \\beta ( \\xi ) & = \\sum _ { n = 1 } ^ \\infty \\beta ^ n R ( \\pi _ { n } ( \\xi ) , f _ \\beta ) \\\\ \\nu ^ \\ast _ \\beta & = \\sum _ { n = 1 } ^ \\infty \\beta ^ n R ( \\pi _ { f _ \\beta } , f _ \\beta ) . \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u = L _ \\lambda u + f + \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } ( \\Lambda ^ k _ \\lambda u + g ^ { k } ) d w _ { s } ^ { k } , \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} f r ( I _ { B , A } ) = 1 + \\varepsilon _ B . \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} \\varphi _ j ( \\tau _ 1 , t _ 2 , \\ldots , t _ d ) = e ^ { i t _ j \\left ( \\alpha _ j \\tau _ 1 + \\beta _ j \\right ) } C _ j ( \\tau _ 1 , t _ 2 , \\ldots , t _ { j - 1 } , t _ { j + 1 } , \\ldots , t _ d ) , \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} \\textbf { y } = \\sqrt { P } \\textbf { H s } \\ + \\ \\textbf { n } \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} ( a , a ^ { \\prime } ) ( b , b ^ { \\prime } ) = ( a b , a ^ { \\prime } b ^ { \\prime } ) \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} \\left [ J _ { a b } , \\bar { P } _ { c } \\right ] & = \\frac { 1 } { \\sqrt { 2 } } \\left [ J _ { a b } , P _ { c } \\right ] + \\frac { 1 } { \\sqrt { 2 } } \\left [ J _ { a b } , Z _ { c } \\right ] , \\\\ & = \\frac { 1 } { \\sqrt { 2 } } f _ { a b , c } ^ { d } P _ { d } + \\frac { 1 } { \\sqrt { 2 } } f _ { a b , c } ^ { d } Z _ { d } , \\\\ & = f _ { a b , c } ^ { d } \\bar { P } _ { d } , \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\mathcal { B } _ r ^ c ( \\theta ^ * ) & = \\bigcup _ { l = 1 } ^ { \\infty } \\{ \\mathcal { B } _ { r _ { l + 1 } } ( \\theta ^ * ) \\backslash \\mathcal { B } _ { r _ { l } } ( \\theta ^ * ) \\} \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( \\pm a t + c ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} { \\rm d i m } _ { k } { \\rm H o m } _ { { \\rm r e p } ( Q ) } ( X , Y ) = \\left \\{ \\begin{array} { l l } 1 & \\mbox { $ Y \\in \\mathcal { W } ( Z ) ; $ } \\\\ 0 & \\mbox { $ Y \\notin \\mathcal { W } ( Z ) . $ } \\end{array} \\right . \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} \\sqrt { \\nu } \\frac { \\norm { y _ k - x _ k } } { \\gamma _ k } \\leq \\frac { \\norm { y _ k - x _ k } _ k } { \\gamma _ k } = \\frac { \\norm { x _ { k + 1 } - x _ k } _ k } { \\gamma _ k \\lambda _ k } \\leq \\frac { \\norm { \\tilde { x } _ { k } - x _ k } _ k } { \\gamma _ k \\lambda _ k } . \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} L ( q ^ { \\alpha } ) L ( q ^ { \\beta } ) = 1 - 2 4 \\ , \\sum _ { n = 1 } ^ { \\infty } \\sigma ( \\frac { n } { \\alpha } ) q ^ { n } - 2 4 \\ , \\sum _ { n = 1 } ^ { \\infty } \\sigma ( \\frac { n } { \\beta } ) q ^ { n } + 5 7 6 \\ , \\sum _ { n = 1 } ^ { \\infty } W _ { ( \\alpha , \\beta ) } ( n ) q ^ { n } . \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} \\inf _ { \\beta \\in \\mathbb C ^ 2 , \\| \\beta \\| = 1 } \\Big \\{ 1 - u ^ 2 \\| A _ { 1 } ^ * \\beta \\| ^ 2 - v ^ 2 \\| A _ { 2 } ^ * \\beta \\| ^ 2 + u ^ 2 v ^ 2 \\Big ( \\| A _ { 1 } ^ * \\beta \\| ^ 2 \\| A _ { 2 } ^ * \\beta \\| ^ 2 - | \\left \\langle A _ 1 A _ { 2 } ^ * \\beta , \\beta \\right \\rangle | ^ 2 \\Big ) \\Big \\} \\geq 0 . \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\partial _ t v - \\Delta v + \\nabla _ H \\pi _ s & = f , \\\\ \\div _ H \\overline { v } & = 0 \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\rho ( C _ { \\tau } ) = \\Vert C _ { \\tau } \\Vert = \\Vert C _ { \\tau } \\Vert _ { e s s } = \\big ( \\tau ' ( \\infty ) \\big ) ^ { ( \\alpha + 2 ) / 2 } . \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\overline J _ \\alpha ( X , X ^ \\prime ) = ( J _ \\alpha X , J _ \\alpha ^ \\prime X ^ \\prime ) , \\alpha = 1 , 2 , 3 , \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\tilde { Q } [ \\psi ] + b & = \\frac { 1 } { W } \\left ( \\sigma ^ { i j } - \\frac { ( h ' ) ^ 2 d ^ i d ^ j } { W ^ 2 } \\right ) \\left ( h '' d _ i d _ j + h ' d _ { i ; j } \\right ) \\\\ & = \\frac { 1 } { W } \\left ( h '' + h ' \\Delta d - \\frac { ( h ' ) ^ 2 h '' } { W ^ 2 } \\right ) \\\\ & = \\frac { 1 } { W } \\left ( \\frac { h '' } { W ^ 2 } - h ' H ( x ) \\right ) \\\\ & = \\frac { h '' } { W ^ 3 } - \\frac { h ' } { W } H ( x ) , \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\tilde { H } ^ K _ * ( X ) & = \\tilde { H } _ * ( E K _ + \\wedge _ K X ) , \\\\ \\tilde { H } ^ * _ K ( X ) & = \\tilde { H } ^ * ( E K _ + \\wedge _ K X ) . \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} \\nabla \\psi = A R _ { 3 } ^ { \\delta } \\nabla ( \\rho ^ { - \\delta } ) + A \\nabla h , \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} \\int _ { S _ { \\infty } } \\lambda _ { \\xi \\eta \\xi ^ { - 1 } , z } ^ 2 \\ d \\nu _ { \\xi \\eta ^ { - 1 } \\xi ^ { - 1 } } = 1 - \\sum \\nolimits _ { \\psi \\in \\{ \\xi ^ 2 , \\xi \\eta ^ { - 1 } \\xi ^ { - 1 } , \\xi \\eta ^ { - 1 } \\xi , \\xi \\eta ^ { - 2 } , \\xi \\eta ^ 2 , \\xi \\eta \\xi ^ { - 1 } , \\xi \\eta \\xi \\} } \\nu _ { \\psi } ( S _ { \\infty } ) . \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} \\mathcal { H } _ { 0 } = \\big \\{ \\{ r _ n ( t ) _ { n \\in \\Z } \\mid r _ n ( t ) = \\sum _ { i = 0 } ^ { \\infty } \\xi _ i r _ { n , i } ( t ) \\in \\C [ t ] , \\xi _ i \\in \\C \\big \\} . \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} e ^ { - S _ { e f f } [ \\Lambda , g ] } = \\int _ { [ 0 , \\Lambda ' ] } D \\psi \\int _ { ( \\Lambda ' , \\Lambda ] } D \\chi \\ , \\ , e x p \\left \\{ - \\langle { \\psi + \\chi } , L ( \\Lambda , { \\psi + \\chi } , g ) ( { \\psi + \\chi } ) \\rangle \\right \\} \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} \\sum a ^ - _ 1 \\star b ^ + _ 2 \\cdot ( a _ 2 , b _ 1 ) = \\sum b ^ + _ 1 \\star a ^ - _ 2 \\cdot ( b _ 2 , a _ 1 ) , \\ , \\ \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} \\sum _ { n = m + 1 } ^ { N } \\mu \\left ( z _ { n } \\right ) & = \\sum _ { n = d \\left ( z _ { 0 } \\right ) ^ { 2 } - R C + 1 } ^ { d \\left ( z _ { 0 } \\right ) ^ { 2 } } \\frac { 1 } { d \\left ( z _ { 0 } \\right ) + n + d \\left ( z _ { 0 } \\right ) ^ { 2 } } \\\\ & \\approx \\log \\frac { d \\left ( z _ { 0 } \\right ) + 2 d \\left ( z _ { 0 } \\right ) ^ { 2 } } { d \\left ( z _ { 0 } \\right ) + 2 d \\left ( z _ { 0 } \\right ) ^ { 2 } - R C + 1 } \\approx \\frac { R C } { d \\left ( z _ { 0 } \\right ) ^ { 2 } } , \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} u ( t , \\theta ) = \\sum \\limits _ { k \\in \\Z } \\alpha _ k ( t ) u _ { k } ( \\theta ) , \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} [ a _ 0 \\otimes a _ 1 \\otimes \\cdots \\otimes a _ r ] [ a _ { r + 1 } \\otimes \\cdots \\otimes a _ { s } ] = \\sum _ { i = 0 } ^ { r } ( - 1 ) ^ { r - i } [ a _ 0 \\otimes \\cdots \\otimes a _ i a _ { i + 1 } \\otimes \\cdots \\otimes a _ s ] \\ , , \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} q = f _ 0 ( x , y ) + \\sum _ { \\alpha = 1 } ^ { k + 1 } f _ \\alpha ( x , y ) z ^ \\alpha . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\| u _ k \\| _ { L ^ \\infty ( B _ { \\frac { 1 } { \\lambda _ k } } ( 0 ) ) } = u _ k ( 0 ) = \\lambda _ k ^ { \\frac { 4 } { p - 1 } } v _ k ( 0 ) = 1 , \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\varPhi ^ m = g ^ { m j } \\varPhi _ j , \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int \\widehat { f } \\ , d \\mathfrak { S } ^ { n } ( { \\widehat { \\mu } } ) = \\int \\widehat { f } \\ , d \\varpi _ { * } \\mathbb { P } ^ { - } . \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} - L ( - \\tilde { u } _ n ) + g _ n \\circ ( - \\tilde { u } _ n ) & = - | f | \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ - ( \\tilde { u } _ n ) & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} ( P E s ) ~ \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v - \\Delta v + \\nabla _ H p = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\\\ \\partial _ z p = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} P \\cap H = c _ h ( P ) = c _ h ( Q ) = Q \\cap H , \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} | f _ 0 ( y _ 0 ) - f _ 0 ( y _ 1 ) | \\leq C r ^ { 2 \\alpha } = | y _ 0 - y _ 1 | ^ \\alpha . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} \\liminf _ { \\rho ( x ) \\to \\infty } \\frac { | h ( x ) | } { \\rho ^ { - 1 } ( x ) \\bigl ( \\log r ( x ) \\bigr ) ^ { - 1 } } = 0 , \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} ( F _ { + } D ) ^ { - 1 } G _ { - } = V _ { - } V _ { + } ^ { - 1 } , \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u = L _ { \\lambda } u + f + \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } ( \\Lambda ^ k _ { \\lambda } u + g ^ { k } ) d w _ { s } ^ { k } \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} \\Delta ^ { m } p ( y ) = \\frac { n ! } { ( n - 2 m ) ! } \\sum _ { i _ 1 = 1 } ^ d \\dots \\sum _ { i _ m = 1 } ^ d \\phi ( e _ { i _ 1 } , e _ { i _ 1 } , \\dots , e _ { i _ m } , e _ { i _ m } , y , \\dots , y ) , \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} Q = \\sum _ { j = 1 } ^ k j ( \\mathrm { r a n k } \\ , \\mathfrak { g } _ j ) . \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} \\delta _ Y = \\min \\big \\{ \\abs { \\lambda } \\ ; : \\ ; 0 \\neq \\lambda \\in \\mathrm { S p } \\big ( D ^ F _ Y \\big ) \\big \\} . \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} & \\int _ 0 ^ { \\infty } \\langle \\sigma ( t ) u _ t ( w ) , u _ t ( w ) \\rangle d w = \\int _ 0 ^ { \\infty } \\langle u ( w ) , u ( w ) \\rangle d w , \\\\ & \\int _ { - \\infty } ^ 0 \\langle \\sigma ( t ) y _ t ( w ) , y _ t ( w ) \\rangle d w = \\int _ { - \\infty } ^ 0 \\langle y ( w ) , y ( w ) \\rangle d w . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { c _ k ( x ^ n | u ^ n ) } \\exp _ 2 \\{ \\lambda L ( y _ { n _ { i - 1 } + 1 } ^ { n _ i } ) \\} \\le ( 2 \\alpha ) ^ { \\lambda } 2 ^ { ( \\lambda + 1 ) \\log c _ k ( x ^ n | u ^ n ) } . \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} \\mathfrak { L } = \\left \\{ f \\in L ^ 2 ( X , \\mu ) , t \\ge 0 , e ^ { t \\Delta } f = e ^ { t \\Delta ^ \\perp } f \\right \\} . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} X + Y = ( X + y _ 0 ) + ( { - y _ 0 } + Y ) = X _ 0 + Y _ 0 . \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { i = 0 \\\\ \\mbox { \\scriptsize $ m + i $ e v e n } } } ^ m ( - 1 ) ^ i { m \\choose i } ( m + i ) \\sum \\limits _ j { \\frac { m + i } { 2 } - 1 \\brace j - 1 } \\gamma _ { 2 j - 2 } \\\\ \\quad + \\sum \\limits _ { \\substack { i = 0 \\\\ \\mbox { \\scriptsize $ m + i $ o d d } } } ^ m ( - 1 ) ^ i { m \\choose i } ( m + i ) \\gamma _ { m + i - 1 } & = 0 . \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} & \\deg \\left ( _ \\infty \\left ( \\frac { \\Delta } { \\Delta _ { N , 1 2 } } \\right ) \\right ) = p ^ { n - 1 } q ^ { m - 1 } ( p + 1 ) ( q + 1 ) - 1 \\\\ & \\deg \\left ( _ \\infty \\left ( \\frac { \\Delta _ N } { \\Delta _ { N , 1 2 } } \\right ) \\right ) = p ^ { n - 1 } q ^ { m - 1 } ( ( p + 1 ) ( q + 1 ) - p q ) = p ^ { n - 1 } q ^ { m - 1 } ( p + q + 1 ) . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} \\frac { \\partial ^ 2 F } { \\partial \\kappa ^ i \\partial \\kappa ^ j } = n ^ { - 1 / r } ( 1 - r ) \\left ( \\sum _ { l } \\kappa _ l ^ r \\right ) ^ { \\frac { 1 } { r } - 2 } \\kappa _ i ^ { r - 2 } ( \\kappa _ i \\kappa _ j ^ { r - 1 } - \\sum _ { l } \\kappa _ l ^ r \\delta _ { i j } ) . \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{gather*} E _ { a a } v _ { 0 } = \\sum _ { l \\ge 0 } v _ { 0 } , \\end{gather*}"}
-{"id": "1038.png", "formula": "\\begin{align*} \\dim \\mathcal { H } = \\sum _ { \\substack { \\lambda \\vdash n \\\\ \\lambda _ 1 \\leq n - h - 2 } } \\dim ( V _ \\lambda ) ^ 2 . \\end{align*}"}
-{"id": "9933.png", "formula": "\\begin{align*} p _ W ( u ' ( \\psi ( s ) ) v ) = \\sum _ { k \\geq i } a _ k w _ k , a _ { k } \\in \\R . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} p _ { 1 , 1 } = \\frac { \\xi } { \\mu A _ { c } } p _ { 0 , 0 } , \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} R _ j ( \\theta ) \\leq S _ { j + 1 } ( \\theta ) \\leq R _ { j + 1 } ( \\theta ) \\forall \\ j = 0 , 1 , \\dots , N ( \\theta ) \\ , , \\forall \\ \\theta \\in [ 0 , 2 \\pi ] \\ , , \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} \\# \\{ \\mathfrak { p } : \\mathrm { d e g } ( \\mathfrak { p } ) = 1 , [ \\tfrac { L / F } { \\mathfrak { p } } ] = C , \\N _ { F / \\mathbb { Q } } ~ \\mathfrak { p } \\leq x \\} \\gg ( D _ K Q n _ K ^ { n _ K } ) ^ { - 5 } \\frac { x } { h _ H \\log x } . \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{gather*} D _ { r - 1 } ^ { 2 } \\sum _ { k = 0 } ^ { r - 1 } \\frac { P _ { k } ( x ) ^ { 2 } } { D _ { k } D _ { k - 1 } } = P _ { r } ^ { \\ , \\prime } ( x ) P _ { r - 1 } ( x ) - P _ { r } ( x ) P _ { r - 1 } ^ { \\ , \\prime } ( x ) \\ , \\ \\end{gather*}"}
-{"id": "8314.png", "formula": "\\begin{align*} & \\int _ M \\varphi P _ g \\varphi d \\mu _ g \\\\ = & \\int _ M \\Big ( | \\nabla \\Delta \\varphi | _ g ^ 2 - 2 T _ 2 ( \\nabla \\Delta \\varphi , \\nabla \\varphi ) - \\frac { n - 2 } { 2 } \\sigma _ 1 ( A ) ( \\Delta _ g \\varphi ) ^ 2 - T _ 4 ( \\nabla \\varphi , \\nabla \\varphi ) + \\frac { n - 6 } { 2 } Q _ g \\varphi ^ 2 \\Big ) d \\mu _ g . \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} & \\nabla _ \\xi p ( t , s ; x , \\eta ( t , s ) ) \\partial _ \\xi ^ \\beta \\nabla _ \\xi \\eta ( t , s ) \\\\ & = - \\sum _ { 0 \\lneq \\beta ^ \\prime \\leq \\beta } \\binom { \\beta } { \\beta ^ \\prime } \\partial _ \\xi ^ { \\beta ^ \\prime } [ \\nabla _ \\xi p ( t , s ; x , \\eta ( t , s ) ) ] \\partial _ \\xi ^ { \\beta - \\beta ^ \\prime } \\nabla _ \\xi \\eta ( t , s ) , \\beta \\neq 0 . \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\phi \\big ( M ( D _ i ) N ( D _ j ) M ( D _ i ) ^ * N ( D _ j ) ^ * \\big ) = \\prod _ { k = 1 } ^ K \\phi _ k \\big ( M ( a _ { k ; i } ) N ( a _ { k ; j } ) M ( a _ { k ; i } ) ^ * N ( a _ { k ; j } ) ^ * \\big ) . \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} q ^ { \\beta ^ { 2 } / 2 } \\left ( q ^ { \\alpha + \\beta + n + 1 } ; q \\right ) _ { \\infty } L _ { n } ^ { ( \\alpha + \\beta ) } \\left ( x ; q \\right ) = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( x q ^ { \\alpha + 1 / 2 } e ^ { i y } ; q \\right ) _ { n } } { \\left ( q ; q \\right ) _ { n } } \\frac { \\exp \\left ( \\frac { y ^ { 2 } } { \\log q ^ { 2 } } + i \\beta y \\right ) } { \\left ( - q ^ { \\alpha + 1 / 2 } e ^ { i y } ; q \\right ) _ { \\infty } } d y , \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} a ( n _ 1 , \\ldots , n _ d ) = ( \\alpha _ 2 d _ 1 + i \\beta _ 2 ) ^ { n _ 2 } \\cdots ( \\alpha _ d d _ 1 + i \\beta _ d ) ^ { n _ d } b ( n _ 1 ) . \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} F ' ( t , z ) = f ( t , z ) { \\rm a n d } G ' ( t , z ) = g ( t , z ) \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} - \\sum _ { i = 1 } ^ { [ n s ] } r ' ( i / n ) Y _ i \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\dot { \\tilde { x } } = \\varPhi x + \\varPhi ^ m x _ m = \\varPhi x + \\varPhi ^ m \\tilde { h } _ m ^ k \\tilde { x } _ k , \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} V _ { \\tilde { A } } & : = V _ { \\tilde { A } s } = V _ { \\tilde { A } t } = V _ { \\tilde { C } s } = V _ { \\tilde { D } t } , \\\\ V _ { \\tilde { B } } & : = V _ { \\tilde { B } s } = V _ { \\tilde { B } t } = V _ { \\tilde { D } s } = V _ { \\tilde { C } t } . \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} | \\{ X \\in V _ { n , k , b } : \\ , \\underline { X } = j \\} | \\le \\binom { b } { k - 1 } . \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} x P _ n ( x ) = b _ { 2 , n } P _ { n + 2 } ( x ) + b _ { 1 , n } P _ { n + 1 } ( x ) + b _ { 0 , n } P _ n ( x ) + b _ { - 1 , n } P _ { n - 1 } ( x ) + b _ { - 2 , n } P _ { n - 2 } ( x ) \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 g ( x , y ) ~ d y = \\int _ 0 ^ 1 g ( x , y ) ~ d x = \\frac { 1 } { 2 } 1 + \\frac { 1 } { 2 } ( 2 p _ d - 1 ) = p _ d , \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} n = \\frac { e ^ { - i \\theta } e ^ { i s } m \\sqrt { 1 + | m | ^ 2 } } { \\overline m } . \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} \\mathbf { P _ * } \\Big ( | | \\Delta _ n ^ * | | > C . n ^ { - 1 / 2 } ( l o g n ) ^ { 1 / 2 } \\Big ) = o ( n ^ { - 1 / 2 } ) \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} \\left ( \\Pi _ { \\mathcal { E } _ D } ( w ) \\right ) _ i \\ ; = \\ ; \\frac { d _ i ^ 2 w _ i } { d _ i ^ 2 + \\bar { \\mu } } , \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} Q ( x , d x ) = \\frac { 1 } { 6 } [ G _ { x x } + E _ { y y } - 2 F _ { x y } ] . \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} e ^ { - \\theta t } X _ t = x _ 0 + \\theta \\int _ 0 ^ t e ^ { - \\theta s } B ^ H _ s \\ , d s + e ^ { - \\theta t } B ^ H _ t \\to x _ 0 + \\theta \\int _ 0 ^ \\infty e ^ { - \\theta s } B ^ H _ s \\ , d s \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} \\mathcal L _ \\Gamma = \\left \\{ a _ 1 \\varepsilon _ 1 + \\dots + a _ n \\varepsilon _ n \\in P ( G ) \\simeq \\Z ^ n : a _ 1 s _ 1 + \\dots + a _ n s _ n \\equiv 0 \\pmod q \\right \\} . \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} R _ V ( T ) = X { \\rm i . e . } { \\rm i m } \\ , \\Lambda _ { T } ^ V = X \\ , . \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} C _ { \\gamma } \\left ( \\overline { z } ^ { m - j } z ^ { n - j } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { j } \\right ) = \\dfrac { 1 } { \\pi } \\int _ 0 ^ 1 r ^ { m + n - 2 j + 1 } ( 1 - r ^ 2 ) ^ { \\gamma + j } \\left ( \\int _ 0 ^ { 2 \\pi } \\dfrac { e ^ { i ( n - m ) \\theta } } { r e ^ { i \\theta } - z } d \\theta \\right ) d r . \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} \\xi ( 0 , \\alpha ; 1 - s ) = \\frac { \\Gamma ( s ) } { ( 2 \\pi ) ^ s } \\left ( e \\left ( \\frac { s } { 4 } \\right ) \\xi ( \\alpha , 0 ; s ) + e \\left ( - \\frac { s } { 4 } \\right ) \\xi ( - \\alpha , 0 ; s ) \\right ) . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } \\Sigma _ 1 & 0 \\\\ 0 & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } \\tilde \\Lambda _ { 1 1 } + \\tilde \\Lambda _ { 1 1 } ^ H & \\tilde \\Lambda _ { 1 2 } + \\tilde \\Lambda _ { 2 1 } ^ H \\\\ \\tilde \\Lambda _ { 2 1 } + \\tilde \\Lambda _ { 1 2 } ^ H & \\tilde \\Lambda _ { 2 2 } + \\tilde \\Lambda _ { 2 2 } ^ H \\end{array} \\right ] \\left [ \\begin{array} { c c } \\Sigma _ 1 & 0 \\\\ 0 & 0 \\end{array} \\right ] = 0 \\end{align*}"}
-{"id": "9961.png", "formula": "\\begin{align*} r a n k ( H H ^ t ) & = r a n k ( K { K } ^ t ) \\\\ & = n - k - m \\\\ & = n - k - \\dim ( H u l l ( C ) ) \\\\ & = n - k - \\dim ( H u l l ( C ^ { \\bot } ) ) , \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} \\left | \\frac { d p _ { \\eta } } { d \\eta } ( t ) \\right | \\leq \\frac { 1 } { \\alpha } \\frac { 1 } { \\eta ^ { 1 + \\frac { 1 } { \\alpha } } } p _ N \\left ( \\frac { t } { \\sqrt [ \\alpha ] { \\eta } } \\right ) + \\frac { 1 } { \\alpha } \\frac { | t | } { \\eta ^ { 1 + \\frac { 2 } { \\alpha } } } \\left | \\frac { d p _ N } { d u } \\right | _ { u = \\frac { t } { \\sqrt [ \\alpha ] { \\eta } } } . \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\beta = v / c , \\qquad \\gamma = \\left ( 1 - \\beta ^ { 2 } \\right ) ^ { - 1 / 2 } , v = \\left \\vert \\mathbf { v } \\right \\vert , \\qquad \\kappa = \\varepsilon \\mu - 1 , \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} \\left [ T _ { ( A , i + n ) } , T _ { ( B , j + n ) } \\right ] & = K _ { i + n , j + n } ^ { \\gamma } C _ { A B } { } ^ { C } T _ { ( C , \\gamma ) } , \\\\ & = K _ { i + n , j + n } ^ { k } C _ { A B } { } ^ { C } T _ { ( C , k ) } + K _ { i + n , j + n } ^ { k + n } C _ { A B } { } ^ { C } T _ { ( C , k + n ) } , \\\\ & = - K _ { i + n , j + n } ^ { k } C _ { A B } { } ^ { C } T _ { ( C , k + n ) } + K _ { i + n , j + n } ^ { k + n } C _ { A B } { } ^ { C } T _ { ( C , k + n ) } , \\\\ & = - \\left ( K _ { i j } ^ { k } - K _ { i j } ^ { k + n } \\right ) C _ { A B } { } ^ { C } T _ { ( C , k + n ) } . \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} f = F ^ { - 2 } \\{ \\abs { A } ^ 2 - n F ^ 2 \\} . \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { T _ 1 } ^ t \\left < \\varphi , \\mathcal { T } ( t - \\tau ) C _ 2 f _ N ^ { ( 2 ) } ( \\tau ) \\right > d \\tau & = \\int _ { T _ 1 } ^ t \\left < \\mathcal { T } ( - ( t - \\tau ) ) \\varphi , C _ 2 f _ N ^ { ( 2 ) } ( \\tau ) \\right > d \\tau \\\\ & = \\int _ { T _ 1 } ^ t \\left < \\Phi ( \\tau ) , C _ 2 f _ N ^ { ( 2 ) } ( \\tau ) \\right > d \\tau \\end{aligned} \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} O _ { \\gamma _ v } ( \\phi _ v ) = \\int _ { I _ { \\gamma _ v } ( F _ v ) \\backslash G ( F _ v ) } \\phi _ v ( x _ v ^ { - 1 } \\gamma _ v x _ v ) d \\mu _ { \\gamma , v } ( x _ v ) \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} d _ X ( Z _ s ^ 0 , Z _ s ^ 1 ) = \\min \\left ( \\inf _ { 1 \\leq i \\neq j \\leq s } | x _ i ^ 0 - x _ j ^ 1 | , \\inf _ { 1 \\leq i \\leq s \\ ; : \\ ; ( x _ i ^ 0 , v _ i ^ 0 ) \\neq ( x _ i ^ 1 , v _ i ^ 1 ) } | x _ i ^ 0 - x _ i ^ 1 | \\right ) \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\textbf { p } ( t ) = \\left ( \\textbf { p } ( t _ 0 ) - \\frac { \\partial \\bar { c } } { \\partial \\textbf { x } } ( t , t _ 0 , \\textbf { x } ) \\right ) \\left [ \\frac { \\partial \\bar { \\textbf { g } } } { \\partial \\textbf { x } } ( t , t _ 0 , \\textbf { x } ) \\right ] ^ { - 1 } . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} \\alpha _ t + ( \\alpha \\beta ) _ x = 0 , \\alpha _ T + ( \\alpha \\epsilon ) _ x = 0 , \\beta _ T - \\epsilon _ t + \\epsilon \\beta _ x - \\epsilon _ x \\beta = 0 \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} G _ j ( z ) : = & \\frac { F ( x _ j , \\xi _ j + \\lambda _ j z ) - F ( x _ j , \\xi _ j ) - \\lambda _ j F _ { z } ( x _ j , \\xi _ j ) [ z ] } { \\lambda _ j ^ 2 } \\\\ = & \\int _ 0 ^ 1 ( 1 - t ) F _ { z z } ( x _ j , \\xi _ j + t \\lambda _ j z ) [ z , z ] . \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{gather*} \\exp ( \\partial ^ \\rho \\otimes x _ \\rho ) ( x _ \\mu \\otimes 1 ) \\exp ( - \\partial ^ \\sigma \\otimes x _ \\sigma ) = x _ \\mu \\otimes 1 + 1 \\otimes x _ \\mu , \\\\ \\exp ( \\partial ^ \\rho \\otimes x _ \\rho ) ( 1 \\otimes x _ \\mu ) \\exp ( - \\partial ^ \\sigma \\otimes x _ \\sigma ) = 1 \\otimes x _ \\mu . \\end{gather*}"}
-{"id": "8016.png", "formula": "\\begin{align*} R _ { s , k } = \\log _ { 2 } \\bigg ( 1 + \\frac { \\mathbf { h } _ { s , k } ^ { H } \\mathbf { q } \\mathbf { q } ^ { H } \\mathbf { h } _ { s , k } } { \\mathbf { h } _ { s , k } ^ { H } \\mathbf { V } \\mathbf { h } _ { s , k } + \\sigma _ { s a , k } ^ { 2 } + \\frac { \\sigma _ { s p , k } ^ { 2 } } { \\rho _ { s , k } } } \\bigg ) \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} \\lambda G _ k - F _ k = L _ { \\eta _ k } ( \\lambda ) ^ T = \\lambda \\left [ \\begin{array} { c } I _ { \\eta _ k } \\\\ 0 _ { 1 \\times \\eta _ k } \\end{array} \\right ] - \\left [ \\begin{array} { c } 0 _ { 1 \\times \\eta _ k } \\\\ I _ { \\eta _ k } \\end{array} \\right ] . \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} \\gamma _ { 2 n + 1 } = \\sum \\limits _ { i = - k } ^ n { n + k \\brack i + k } \\gamma _ { 2 i } , \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} \\abs { \\prod _ { i = 1 } ^ n \\lambda _ { n } ( g _ i ) } \\le \\frac { \\delta ^ n } { n ! } \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} \\sup _ { ( 0 , T ] } \\sup _ { \\Gamma _ i } w _ t ( x ) = \\alpha , \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\left ( \\left ( \\mathbf { I } + K ( \\lambda , 0 ) \\right ) r _ { 1 } , r _ { 1 } \\right ) & = ( \\left ( \\mathbf { I } + \\partial _ { x } ^ { - 2 } \\rho ( \\lambda , 0 ) \\right ) r _ { 1 } , r _ { 1 } ) \\\\ & = 1 - F \\left ( i \\lambda \\right ) \\backsim a _ { 0 } \\lambda ^ { 2 } . \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} \\Phi ( y ) = h ( c ( y ) ) \\geq h ( c ( x ) ) + \\langle w , c ( y ) - c ( x ) \\rangle & \\geq \\Phi ( x ) + \\langle w , \\nabla c ( x ) ( y - x ) \\rangle - \\frac { \\beta \\| w \\| } { 2 } \\| y - x \\| ^ 2 \\\\ & \\geq \\Phi ( x ) + \\langle v , y - x \\rangle - \\frac { \\mu } { 2 } \\| y - x \\| ^ 2 . \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} ( u - k _ j ) _ + \\geq k _ { j + 1 } - k _ j = 2 ^ { - j - 1 } k , \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} \\Gamma ^ \\varepsilon _ + : = \\omega \\times \\{ \\varepsilon \\} , \\Gamma ^ \\varepsilon _ - : = \\omega \\times \\{ - \\varepsilon \\} , \\Gamma _ 0 ^ \\varepsilon : = \\gamma _ 0 \\times [ - \\varepsilon , \\varepsilon ] . \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} X ( u , v ) = \\phi ( u ) + \\lambda \\cos \\left ( \\frac { u } { c } \\right ) \\rho ( v ) , \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} \\dot { x } & = \\omega + \\xi ( y , z , \\omega , \\sigma , \\mu ) + f ( x , y , z , \\omega , \\sigma , \\mu ) , \\\\ \\dot { y } & = \\sigma + \\eta ( y , z , \\omega , \\sigma , \\mu ) + g ( x , y , z , \\omega , \\sigma , \\mu ) , \\\\ \\dot { z } & = \\widehat { Q } ( \\omega , \\sigma , \\mu ) z + \\widehat { \\zeta } ( y , z , \\omega , \\sigma , \\mu ) + h ( x , y , z , \\omega , \\sigma , \\mu ) , \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ { \\infty } n ( 2 n - 1 ) ( | A _ n | + | B _ n | ) \\leq 1 - | B _ 1 | . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} \\left [ \\partial ^ { \\tau } \\partial _ { \\tau } + \\kappa \\left ( u ^ { \\tau } \\partial _ { \\tau } \\right ) ^ { 2 } \\right ] \\mathbf { Z } ^ { \\left ( e \\right ) } = 4 \\pi \\mu \\mathbf { p } , \\qquad \\left [ \\partial ^ { \\tau } \\partial _ { \\tau } + \\kappa \\left ( u ^ { \\tau } \\partial _ { \\tau } \\right ) ^ { 2 } \\right ] \\mathbf { Z } ^ { \\left ( m \\right ) } = 4 \\pi \\mu \\mathbf { m } \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} L ( x ) \\ ; = \\ ; \\begin{cases} \\frac { 1 } { 2 m } \\left ( \\left ( \\frac { \\langle x , \\ , A x \\rangle } { \\| x \\| _ 2 } \\right ) ^ { 2 m } - 1 \\right ) & \\mbox { i f } x \\neq 0 , \\\\ 0 & \\mbox { i f } x = 0 . \\end{cases} \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} \\mathcal { T } g ( T t , x ) = \\mathcal { T } \\tilde { g } ( t , T ^ { - \\frac { \\alpha } { 2 } } x ) , \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} \\mathcal { D } : = \\{ \\bold { d } \\in \\mathbb { R } _ { \\geq 0 } ^ { \\Delta _ + } \\mid d _ \\alpha + d _ \\beta > d _ \\gamma \\} . \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\lambda = \\mid b _ { 1 } + t \\mid ^ { 2 } = \\mid b _ { 2 } + t \\mid ^ { 2 } = . . . = \\mid b _ { m } + t \\mid ^ { 2 } . \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} & D _ { \\eta } ( v \\cdot \\nabla _ H \\zeta + w \\partial _ z \\zeta ) = \\left ( s _ { \\eta } v \\cdot \\nabla _ H D _ { \\eta } \\zeta + ( s _ { \\eta } w ) \\partial _ z D _ { \\eta } \\zeta \\right ) + \\left ( D _ { \\eta } v \\cdot \\nabla _ H \\zeta + ( D _ { \\eta } w ) \\partial _ z \\zeta \\right ) . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} w = \\left [ \\begin{array} { c } ( \\lambda I - A ) ^ { - 1 } B u \\\\ u \\end{array} \\right ] \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} \\pi ( [ x , y ] ) & = \\rho ( x ) \\pi ( y ) - \\rho ( y ) \\pi ( x ) , & x , y & \\in \\mathfrak { g } . \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\lim _ { N ^ \\prime } \\int _ { T _ 1 } ^ t \\left < \\varphi , \\mathcal { T } ( t - \\tau ) C _ 2 f _ N ^ { ( 2 ) } ( \\tau ) \\right > d \\tau = 0 \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} \\int P _ { \\mathbf { y } } ( D ' ) \\mu ( d \\mathbf { y } ) = \\int P _ { \\mathbf { x } } ( D ' ) \\mu ( d \\mathbf { x } ) = P ( D ' ) , \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} B \\left ( Z \\right ) = \" \\exists \\left ( \\dfrac { P _ { i } } { Q } \\right ) _ { i = 1 , 2 , . . . , s } \\forall ^ { s t } m \\in \\mathbb { N } ^ { \\ast } G \\left ( Z \\left ( \\dfrac { P _ { i } } { Q } \\right ) _ { i = 1 , 2 , . . . , s } m \\right ) \" \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} \\phi ( x ; a ) : = \\dfrac { | x | } { 1 + a | x | / 2 } , a \\geqslant 0 . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} H _ f ( x ) = \\limsup _ { r \\to 0 } \\frac { L _ f ( x , r ) } { l _ f ( x , r ) } = \\limsup _ { r \\to 0 } \\frac { L _ f ( x , r ) / r } { l _ f ( x , r ) / r } \\leq \\frac { L _ f ( x ) } { l _ f ( x ) } , \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} C _ 1 ( \\Gamma g K , \\Gamma g ' K ) : = c ( g ^ { - 1 } g ' ) \\in \\Z / d \\Z \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} S ( z , \\zeta ) = \\chi ( z , \\zeta ) S _ { 0 } ( z , \\zeta ) - \\left ( 1 - \\chi ( z , \\zeta ) \\right ) \\left | z - \\zeta \\right | ^ { 2 } = \\sum _ { i = 1 } ^ { n } Q _ { i } ( z , \\zeta ) \\left ( z _ { i } - \\zeta _ { i } \\right ) , \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n } \\lambda _ k ( q _ 1 ) = \\sum _ { k = 1 } ^ n \\mathcal { R } [ q _ 1 , \\psi _ k ] . \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} \\mathcal { T } x ^ { k - 1 } = \\lambda x ^ { [ k - 1 ] } . \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\begin{cases} u ( 0 , t ) = h _ 0 ( t ) , \\ , \\ , u ( L , t ) = h _ 1 ( t ) , \\ , \\ , u _ { x } ( L , t ) = h _ 2 ( t ) , \\\\ v ( 0 , t ) = g _ 0 ( t ) , \\ , \\ , v ( L , t ) = g _ 1 ( t ) , \\ , \\ , v _ { x } ( L , t ) = g _ 2 ( t ) , \\end{cases} \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} \\varPhi ( r ) = r . \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} ( \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma - \\gamma _ { 1 } + t , x \\right \\rangle } ) = \\dfrac { ( q \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma - \\gamma _ { 1 } + t , x \\right \\rangle } ) } { \\Lambda _ { N } ( t ) - \\mid \\gamma - \\gamma _ { 1 } + t \\mid ^ { 2 } } . \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} [ \\bar { \\xi } _ { i , 0 } , \\bar { x } ^ { \\pm } _ { j , s } ] = \\pm a _ { i , j } \\bar { x } ^ { \\pm } _ { j , s } , \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} | g _ r ( t ) | = \\frac { 1 } { \\sqrt t } 0 \\leq t \\leq r ^ 2 \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} Z _ i = X _ i + R _ i ( i = 1 , \\ldots , N ) , \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} a = c \\ , ( 1 , \\ , \\cos \\omega , \\ , \\cos { 2 \\omega } , \\ , \\ldots , \\ , \\cos { ( n - 1 ) \\omega } ) , \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} v _ \\alpha ^ i ( s , f , g ) = E _ { f , g } ^ s \\int _ 0 ^ \\infty e ^ { - \\alpha t } r ^ i ( s _ t , x _ t ^ 1 , x _ t ^ 2 ) d t , \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} \\frac { d } { d t } \\varphi _ t ^ \\ast e = \\varphi _ t ^ \\ast ( \\square _ t e ) , \\ \\ e \\in \\Gamma ( E ) . \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} \\lim \\limits _ { k \\rightarrow \\infty } \\lambda _ { k } = 0 . \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{gather*} U _ { k } ^ { ( \\alpha ) } = \\Gamma _ { [ k ] ( \\alpha ) } ^ { [ k + 1 ] ( \\alpha ) } , V _ { k } ^ { ( \\alpha ) } = \\Gamma _ { [ k ] ( \\alpha ) } ^ { [ k ] ( \\alpha + 1 ) } , W _ { k } ^ { ( \\alpha ) } = \\Gamma _ { [ k ] ( \\alpha ) } ^ { [ k - 1 ] ( \\alpha + 1 ) } , \\end{gather*}"}
-{"id": "1033.png", "formula": "\\begin{align*} H ^ r ( X , \\Q _ \\ell ) = \\begin{cases} \\Q _ \\ell ( - r / 2 ) & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} I ( X ^ n \\rightarrow Y ^ n ) \\triangleq \\sum _ { t = 0 } ^ n I ( X ^ t ; Y _ t | Y ^ { t - 1 } ) = \\sum _ { t = 0 } ^ n { \\bf E } \\Big \\{ \\log \\Big ( \\frac { d { \\bf P } _ { Y _ t | Y ^ { t - 1 } , X ^ t } ( \\cdot | Y ^ { t - 1 } , X ^ t ) } { d { \\bf P } _ { Y _ t | Y ^ { t - 1 } } ( \\cdot | Y ^ { t - 1 } ) } ( Y _ t ) \\Big ) \\Big \\} . \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} \\beta _ 0 ^ i = \\max _ { s \\in S , a ^ i \\in A ^ i ( s ) , a ^ i \\neq a _ s ^ i } \\{ 0 , \\beta _ { s , a ^ i } ^ i \\} , \\ i = 1 , 2 , \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { n _ 0 + 1 } l _ { i j } = 0 , ~ j = 1 , \\dots , n _ 0 ~ \\mbox { a n d } \\sum \\limits _ { i = 1 } ^ { n _ 0 + 1 } l _ { i ( n _ 0 + 1 ) } = 1 . \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{gather*} t _ { n _ { 0 } } = ( - 1 ) ^ { \\frac { n _ { 0 } ( n _ { 0 } + 1 ) } { 2 } } s _ { n _ { 0 } } ^ { n _ { 0 } + 1 } \\ . \\end{gather*}"}
-{"id": "5615.png", "formula": "\\begin{align*} z ( x , y , t ) : = u ( x , t ) - v ( y , t ) \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\rho \\left ( \\frac { x } { x - 1 } \\right ) & = - \\frac { x - 2 } { x } \\int \\limits _ 0 ^ { \\frac { x } { x - 1 } } \\frac { 1 } { 1 - \\xi } \\chi ( \\xi ) \\ , d \\xi = - \\frac { x - 2 } { x } \\int \\limits _ 0 ^ { \\frac { x } { x - 1 } } \\frac { 1 } { 1 - \\xi } \\chi \\left ( \\frac { \\xi } { \\xi - 1 } \\right ) d \\xi \\\\ & = \\frac { x - 2 } { x } \\int \\limits _ 0 ^ { x } \\frac { 1 } { 1 - \\zeta } \\chi ( \\zeta ) \\ , d \\zeta = \\rho ( x ) . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} ( T g ) ( x ) = \\int h ( x , y ) g ( y ) d ( \\mu y ) . \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} | a _ { n } ^ { i j } ( t , x ) - a _ { n } ^ { i j } ( t , y ) | & = | \\eta _ n ( x ) ( a ^ { i j } ( x ) - a ^ { i j } ( x _ n ) ) - \\eta _ n ( y ) ( a ^ { i j } ( y ) - a ^ { i j } ( x _ n ) ) | \\\\ & \\leq | \\eta _ n ( x ) ( a ^ { i j } ( x ) - a ^ { i j } ( x _ n ) ) | + | \\eta _ n ( y ) ( a ^ { i j } ( y ) - a ^ { i j } ( x _ n ) ) | \\leq \\varepsilon _ { 1 } . \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} \\phi ( x , y ) \\phi ( y , x ) = p _ e ( 2 p _ d - 1 ) ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} \\begin{aligned} & C _ { i , s + 1 } ^ { 0 , - } f _ \\infty ^ { ( s + 1 ) } ( t , Z _ s ) = \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\left [ \\omega \\cdot ( v _ { s + 1 } - v _ i ) \\right ] _ { - } \\times \\\\ & \\ ; \\ ; \\times f _ \\infty ^ { ( s + 1 ) } ( t , x _ 1 , v _ 1 , \\dots , x _ i , v _ i , \\dots , x _ s , v _ s , x _ i , v _ { s + 1 } ) d \\omega d v _ { s + 1 } \\end{aligned} \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\mu _ { k } ^ i \\left ( \\mathcal { B } _ r ( \\theta ^ * ) \\right ) & \\geq 1 - C _ 3 \\sum \\limits _ { \\theta \\in \\mathcal { B } _ r ^ c ( \\theta ^ * ) } \\exp \\left ( - \\frac { k } { 2 } \\gamma ( \\theta ) \\right ) \\\\ & \\geq 1 - \\sum \\limits _ { \\theta \\in \\mathcal { B } _ r ^ c ( \\theta ^ * ) } \\sigma { \\prod _ { i = 1 } ^ n \\mu _ 0 ^ i \\left ( \\theta \\right ) ^ { \\frac { 1 } { n } } } \\\\ & \\geq 1 - \\sigma \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} \\left \\| A \\left ( A ^ { \\intercal } A \\right ) ^ { \\dagger } \\right \\| = \\left \\| U _ A \\Sigma _ A V _ A ^ { \\intercal } \\left ( V _ A \\Sigma _ A ^ 2 V _ A ^ { \\intercal } \\right ) ^ { \\dagger } \\right \\| = \\left \\| U _ A \\Sigma _ A ^ { \\dagger } V _ A ^ { \\intercal } \\right \\| \\leq \\sigma _ { \\min } ^ { - 1 } ( A ) . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} q ( 0 ) + q ( \\pi ) = q _ { 0 } + 2 \\sum _ { k = 1 } ^ { \\infty } q _ { 2 k } , \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} N _ \\lambda ( u ; q , t ) = ( - u ) ^ { - | \\lambda | } q ^ { n ( \\lambda ' ) } t ^ { n ( \\lambda ) } \\prod _ { s \\in \\lambda } ( 1 - u q ^ { - a ( s ) } t ^ { l ( s ) + 1 } ) ( 1 - u t ^ { - l ( s ) } q ^ { a ( s ) + 1 } ) , \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} \\rho ( t ) = \\left ( \\begin{matrix} \\rho ( a ) & 0 \\\\ 0 & \\rho ( t ) \\end{matrix} \\right ) \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\left | \\delta \\right | = n \\\\ \\delta _ { i _ { 1 } } = \\dots = \\delta _ { i _ { n - 2 } } = 1 } } \\alpha _ { \\delta } = 0 , \\ , \\ , \\ , i _ { 1 } , \\dots , i _ { n - 2 } \\in \\left \\{ 1 , \\dots , 2 n \\right \\} . \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\beta ( x ) = \\varepsilon \\cos \\frac { 2 \\pi } { P _ { \\beta } } x + O ( \\varepsilon ^ { 2 } ) . \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} \\psi ^ N _ k ( x ) = \\cos ( \\lambda ( \\tilde h ( x ) + k ) ) \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} c _ 2 = f _ 2 = 0 . \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } v ( t ) : = \\frac { d } { d t } ( g _ { 1 - \\alpha } * v ( \\cdot ) ) ( t ) , \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} A ^ { k } ( i _ { 1 } , \\dots , i _ { k - 2 } ) : = \\sum _ { \\substack { | \\delta | = k , \\\\ \\delta _ { i _ { 1 } } = \\dots = \\delta _ { i _ { k - 2 } } = 1 } } u _ { \\delta } = 0 . \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} ( d ^ 2 \\varphi ) ( a , b , c ) = [ a , \\varphi ( b , c ) ] - [ \\varphi ( a , b ) , c ] + [ \\varphi ( a , c ) , b ] + \\varphi ( a , [ b , c ] ) - \\varphi ( [ a , b ] , c ) + \\varphi ( [ a , c ] , b ) = 0 \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} Q ^ { \\mathcal { X } } _ { n + 1 } \\cap s _ n = Q ^ { \\mathcal { X } } _ n \\cap \\omega _ 2 . \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} \\| ( y , h , u ) \\| = \\| \\rho ( t ) ( y , h , u ) \\| . \\end{align*}"}
-{"id": "10065.png", "formula": "\\begin{align*} \\alpha ( p - q ) - p + 2 q = 1 + n + k \\leq 1 + n + q . \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} P \\c { k } { l } { m } { a } { b } { c } { x } = ( \\alpha _ { 1 } ) _ { i _ { 1 } } ( \\alpha _ { 2 } ) _ { i _ { 2 } } \\dotsm ( \\alpha _ { n } ) _ { i _ { n } } ( - 1 ) ^ { j _ { 1 } } x ^ { j _ { 2 } } ( 1 - x ) ^ { j _ { 3 } } \\left ( \\varphi P \\right ) \\c { k } { l } { m } { a } { b } { c } { x } , \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} ( q ^ 2 - q + 1 ) \\left ( \\frac { q ^ 3 ( q ^ 2 - 1 ) r } { 8 } - \\sum _ { t = 2 } ^ { 6 } k _ t \\right ) = f _ 7 ( p ' ) + f _ 8 ( p ' ) , \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } \\Big ( \\frac { \\partial F } { \\partial e _ i } ( e ) \\Big ) ^ { 2 } & = \\sum _ { i = 1 } ^ { n } \\Big ( \\frac { \\partial f } { \\partial \\varepsilon _ i } ( \\varepsilon ) \\Big ) ^ { 2 } ( Q ^ { \\prime } _ i ( e _ { i } ) ) ^ { 2 } \\\\ & \\leq \\max _ { 1 \\leq i \\leq n } ( Q ^ { \\prime } _ i ( e _ { i } ) ) ^ { 2 } \\sum _ { i = 1 } ^ { n } \\Big ( \\frac { \\partial f } { \\partial \\varepsilon _ i } ( \\varepsilon ) \\Big ) ^ { 2 } \\leq L ^ { 2 } \\max _ { 1 \\leq i \\leq n } ( Q ^ { \\prime } _ i ( e _ { i } ) ) ^ { 2 } . \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} | \\P _ S ( n ) | \\ = \\ p _ S ( n ) 2 ^ { n - | S | - 1 } \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} z _ 1 = , ~ ~ \\phi ( S ^ 2 ) \\cap V _ 1 . \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\mathfrak { L } = \\{ f \\in L ^ 2 ( X ) , f _ 1 = \\cdots = f _ n \\} , \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\forall ~ \\varphi \\in \\mathcal { W } _ T \\cap \\mathcal { C } ( Q ) , ~ \\int _ 0 ^ T - \\langle \\partial _ t \\varphi ( t ) , \\upsilon ( t ) \\rangle + \\int _ 0 ^ T \\langle \\partial _ i \\varphi ( t ) , \\partial _ i \\upsilon ( t ) \\rangle \\ , d t = \\int _ 0 ^ T \\int _ { \\mathcal { O } } \\varphi ( t , x ) \\mu ^ { \\upsilon } ( d t , d x ) \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} \\ ; \\ ; \\alpha _ { l , L _ 1 } \\geq \\alpha _ { l , L _ 2 } \\geq \\cdots \\geq \\alpha _ { l , L _ i } \\geq \\cdots \\geq \\alpha _ { l , L _ { \\binom { n } { l } } } . \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} \\delta ^ * ( \\mu , r ) = \\inf \\left \\{ \\delta ( \\mu , r ) : \\delta ( \\mu , r ) ~ \\right \\} . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} t r \\omega _ R ( z ( s ) ) = \\frac { d } { d s } \\big ( \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } \\log ( 1 - z _ 1 ^ 2 ( s ) - z _ 2 ^ 2 ( s ) - 2 z _ 1 ( s ) z _ 2 ( s ) \\cos \\theta ) d \\theta \\big ) d s . \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\pi _ { \\Omega } ( y ) \\ ; = \\ ; y - s \\frac { \\nabla _ x \\psi ( y , s ) } { \\| \\nabla _ x \\psi ( y , s ) \\| _ 2 } . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\varphi ( x ) = \\sum _ { i = 0 } ^ \\infty \\delta _ i \\ , x ^ { 2 i } \\ , ( x - 2 ) ^ { - r - 2 i } \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\gamma = \\beta ( 1 - 1 / q ) \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} \\frac { 1 } { \\phi ( 2 N + 1 ) } \\left ( \\frac { 1 } { | S _ { m , N } | } \\sum _ { k \\in S _ { m , N } } | \\xi ^ { m _ 0 , N _ 0 } _ k | ^ p \\right ) ^ \\frac 1 p & \\le \\frac { C _ 2 ( 2 N + 1 ) ^ \\frac 1 p } { ( 2 N _ 0 + 1 ) ^ { \\frac 1 p } \\phi ( 2 N _ 0 + 1 ) } \\left ( \\frac { | S _ { m _ 0 , N _ 0 } | } { | S _ { m _ 0 , N } | } \\right ) ^ \\frac 1 p \\\\ & = \\frac { C _ 2 } { \\phi ( 2 N _ 0 + 1 ) } . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} B _ p ^ \\# = \\begin{pmatrix} 0 & 0 \\\\ 0 & c _ p ^ \\# \\end{pmatrix} , \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} ( \\alpha + 1 ) [ ( r - 3 ) \\binom { m _ 2 } { 2 } - m _ 2 r + r - 2 ] + ( \\varepsilon _ 2 + 1 ) [ ( r - 3 ) m _ 2 - r ] - r + 6 + ( m _ 2 + \\mu _ 2 ) ( r - 3 ) \\ge 0 . \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} g _ B ^ i \\left ( \\hat { s } \\right ) & = \\frac { 1 } { \\mu _ 0 ^ i \\left ( B \\right ) } \\int \\limits _ { B } \\prod \\limits _ { t = 1 } ^ { k } \\prod \\limits _ { j = 1 } ^ { n } \\ell ^ j ( S _ { t } ^ j | \\theta ) ^ { \\left [ A ^ { t - k } \\right ] _ { i j } } d \\mu _ 0 ^ j \\left ( \\theta \\right ) \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} - L ^ { * } \\varphi & = 1 \\ , \\ , \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ \\varphi & = 0 \\ , \\ , \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} R ^ \\star _ { \\mathsf { u } } ( R _ { \\mathsf { c } } ) & = N ( 1 - ( 1 - 1 / N ) ^ L ) . \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c } M & E \\\\ E ^ T & B \\end{array} \\right ) , \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\mathbf { U } _ m ^ { T _ F } = \\left \\{ U _ m [ t ] \\right \\} _ { t = 1 } ^ { T _ F } = \\pi _ f ^ m \\left ( \\{ F _ { [ 1 : N ] } \\} , S _ m , \\mathbf { D } , \\mathbf { H } \\right ) , \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} \\zeta _ q ( s ) : = \\sum _ { k = 1 } ^ \\infty \\lambda _ k ^ { - s } ( q ) . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} \\norm { ( \\mu _ A ( f ) a ) ( x _ 0 ) } = \\norm { f ( x _ 0 ) a ( x _ 0 ) } = \\norm { a ( x _ 0 ) } = 1 , \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} \\kappa = ( 1 + I _ 1 + I _ 2 + I _ 3 ) \\kappa _ { ( 3 ) } + ( I _ 1 + 2 I _ 2 + 2 I _ 3 ) \\kappa _ { ( 2 , 1 ) } + ( I _ 2 + I _ 3 ) \\kappa _ { ( 1 , 1 , 1 ) } . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} i \\left [ k ^ { \\alpha / 2 } \\mbox { s g n } ( k ) + \\frac { d } { d k } \\right ] \\phi _ { 0 } ^ { ( \\alpha ) } ( k ) = 0 . \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} 2 \\lambda a _ 1 = a _ 2 , 2 \\lambda a _ 2 = a _ 1 + a _ 3 , \\ldots , 2 \\lambda a _ { n - 1 } = a _ { n - 2 } + a _ n . \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} S _ 2 = r W ( x ) E _ k ( r \\log x ) - r ^ 2 \\int _ y ^ x W ( u ) E _ k ' ( r \\log u ) \\frac { d u } { u } \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} C _ { X ^ \\infty \\rightarrow { Y } ^ { \\infty } } ^ { F B , A . 1 } & = \\lim _ { n \\longrightarrow \\infty } \\frac { 1 } { n + 1 } C _ { X ^ n \\rightarrow { Y } ^ { n } } ^ { F B , A . 1 } = \\nu _ 0 \\Big ( H ( \\nu _ { 0 | 0 } ) - H ( \\gamma ) \\Big ) + ( 1 - \\nu _ 0 ) \\Big ( H ( \\nu _ { 0 | 1 } ) - H ( \\delta ) \\Big ) \\\\ & \\qquad \\qquad + \\xi _ 0 \\Big ( H ( \\gamma ) - H ( \\alpha ) \\Big ) + \\xi _ 1 \\Big ( H ( \\delta ) - H ( \\beta ) \\Big ) \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} \\int _ { | u | < 1 } | u | ^ p \\nu ( d u ) = \\infty . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} D _ { x , i } ( { \\cal B } ) = D _ { x , j _ 1 } ( { \\cal B } ) + D _ { j _ 1 , j _ 2 } ( { \\cal B } ) + . . . + D _ { j _ t , i } ( { \\cal B } ) \\textrm { a n d t h e e l e m e n t s o f t h e s u m a r e i n d e c o m p o s a b l e } \\Big \\} . \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} \\mathbf { x } _ { { \\mathcal { R } } , { \\mathcal { T } } } = ( ( \\mathbf { x } _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { 1 } ) ^ T , ( \\mathbf { x } _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { 2 } ) ^ T , \\ldots , ( \\mathbf { x } _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { { \\varrho } } ) ^ T ) ^ T , \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} \\tilde { n } ( \\theta ) = c _ 1 ( \\theta ) { \\tilde n } _ 1 + \\cdots + c _ 7 ( \\theta ) { \\tilde n } _ 7 \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} F ( x ) = \\sup _ { t \\in \\R } \\abs { \\bar { \\nabla } f ( x , t ) } \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{gather*} \\theta _ g ( x , y ) \\theta _ g ( x y , z ) = \\theta _ g ( x , y z ) \\theta _ { g ^ x } ( y , z ) \\\\ \\gamma _ x ( g , h ) \\gamma _ x ( g h , k ) \\omega ( g ^ x , h ^ x , k ^ x ) = \\gamma _ x ( h , k ) \\gamma _ x ( g , h k ) \\omega ( g , h , k ) \\\\ \\theta _ g ( x , y ) \\theta _ h ( x , y ) \\gamma _ x ( g , h ) \\gamma _ y ( g ^ x , h ^ x ) = \\theta _ { g h } ( x , y ) \\gamma _ { x y } ( g , h ) . \\end{gather*}"}
-{"id": "3596.png", "formula": "\\begin{align*} J ^ \\psi \\varphi _ a = ( \\varphi \\otimes \\psi ) _ { J ( a ) } . \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} \\psi ^ b ( f ) ( y ) = f ( \\psi ^ * ( y ) ) , \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} v _ 0 = \\begin{cases} d , & d \\ge 0 , \\\\ 0 , & d < 0 , \\end{cases} \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} f _ { n - 4 } ^ { ( 0 ) } = & f _ { n - 4 } ^ { ( 1 ) } + A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\psi _ { n - 4 } ^ { ( 0 ) } + K _ { 6 - n } \\psi _ { n - 4 } ^ { ( 0 ) } \\\\ = & O ( r ^ { n - 3 } ) \\log r + O ( r ^ { n - 3 } ) + O ( r ^ { n - 2 } ) \\log ^ 2 r . \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} \\begin{cases} c _ 0 | p | ^ q \\le H ^ * ( p , y ) \\le C _ 0 | p | ^ q , & | D _ y H ^ * ( p , y ) | \\le C | p | ^ q , \\\\ [ 1 m m ] c _ 0 | p | ^ { q - 1 } \\le | D _ p H ^ * ( p , y ) | \\le C _ 0 | p | ^ { q - 1 } , & | D ^ 2 _ { p y } H ^ * ( p , y ) | \\le C | p | ^ { q - 1 } , \\\\ [ 1 m m ] c _ 0 | p | ^ { q - 2 } \\ ; I _ d \\le D ^ 2 _ p H ^ * ( p , y ) \\le C _ 0 | p | ^ { q - 2 } \\ ; I _ d , & | D ^ 2 _ { y } H ^ * ( p , y ) | \\le C | p | ^ q . \\end{cases} \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} \\gamma ^ l = \\sum _ { j = 1 } ^ l \\alpha ^ j \\gamma ^ j . \\end{align*}"}
-{"id": "10124.png", "formula": "\\begin{align*} ( y ^ 2 + a x ^ 2 + b x + c ) ^ q = x ^ { - p } \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} \\bigoplus _ { i = 0 } ^ { x - 1 } H _ x ( i , \\phi ( i ) ) & = \\bigoplus _ { i = 0 } ^ { x - 1 } \\left ( T _ x ( i ) \\oplus F _ n ( T _ x ( i ) ) \\oplus F _ n ( T _ x ( \\phi ( i ) ) ) \\right ) \\\\ & = \\bigoplus _ { i = 0 } ^ { x - 1 } T _ x ( i ) \\bigoplus _ { i = 0 } ^ { x - 1 } F _ n ( T _ x ( i ) ) \\bigoplus _ { i = 0 } ^ { x - 1 } F _ n ( T _ x ( \\phi ( i ) ) ) \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} \\begin{aligned} E \\left ( u _ 1 ^ { 1 , n } , u _ 1 ^ { 2 , n } \\right ) & + E \\left ( u _ 2 ^ { 1 , n } , u _ 2 ^ { 2 , n } \\right ) \\leq E \\left ( u _ 1 ^ { n } , u _ 2 ^ { n } \\right ) + \\gamma _ 1 \\int _ { \\mathbb { R } ^ N } U ^ { p _ 1 } | u _ 1 ^ { n } | ^ { p _ 1 } \\ d x \\\\ \\ & + \\gamma _ 2 \\int _ { \\mathbb { R } ^ N } U ^ { p _ 2 } | u _ 2 ^ { n } | ^ { p _ 2 } \\ d x + \\int _ { \\mathbb { R } ^ N } U ^ { r _ 1 + r _ 2 } F ( u _ 1 ^ { n } , u _ 2 ^ { n } ) \\ d x + C \\varepsilon , \\end{aligned} \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} \\big \\lVert \\omega _ i \\big \\rVert ^ 2 _ { Z _ { 1 , 0 } } + \\big \\lVert \\omega _ i \\big \\rVert ^ 2 _ { Z _ { 2 , 0 } } = 1 . \\end{align*}"}
-{"id": "10135.png", "formula": "\\begin{align*} p + q = \\gcd ( 2 q , - p ) + \\gcd ( q , - 2 p ) . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} h ( t ) = \\frac { a } { ( t - 1 ) ^ { 2 } } + \\frac { b } { t - 1 } = \\sum _ { n = 0 } ^ { \\infty } \\left ( a ( n + 1 ) - b \\right ) t ^ { n } \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} b _ { 2 , 1 } & = b _ { 1 , 1 } ^ { \\ast } - b _ { 1 , 1 } \\beta _ { 2 , 2 } = \\varphi _ { z _ { 0 } } \\left ( z _ { 2 } \\right ) - \\varphi _ { z _ { 0 } } \\left ( z _ { 1 } \\right ) \\varphi _ { z _ { 1 } } \\left ( z _ { 2 } \\right ) , \\\\ b _ { 2 , 2 } & = \\beta _ { 2 , 2 } = \\varphi _ { z _ { 1 } } \\left ( z _ { 2 } \\right ) . \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} h ( p _ \\gamma ) + \\frac { 1 } { \\gamma } \\norm { x - p _ \\gamma } ^ 2 = \\Big \\langle x - p _ \\gamma , \\frac { x - p _ \\gamma } { \\gamma } \\Big \\rangle + h ( p _ \\gamma ) \\leq h ( x ) . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} I _ { \\gamma ( n ) } ( \\omega ( T + \\cdot ) ) = I _ { \\gamma ( n ) } ( \\omega ) ( T + \\cdot ) - I _ { \\gamma ( n ) } ( \\omega ) ( T ) . \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } | u _ 1 | ^ 2 \\ d x = \\tau _ 1 \\ \\ \\textrm { a n d } \\ \\ \\int _ { \\mathbb { R } ^ N } | u _ 2 | ^ 2 \\ d x = \\tau _ 2 \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) ( 2 \\pi ) ^ { - \\frac { q } { \\tau } } \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) \\Gamma ( 1 - \\frac { q } { \\tau } ) = & \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) ( 2 \\pi ) ^ { - q } \\times \\\\ & \\times \\Gamma ^ { q } ( 1 - \\frac { 1 } { \\tau } ) \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} \\| E ( t ) \\| _ { L _ { x } ^ { 2 } } = \\| E _ { 0 } \\| _ { L _ { x } ^ { 2 } } \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} \\langle F \\rangle : = \\int _ M F ( x ) d x = 0 . \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} \\det R ( z ) & = | \\exp \\big ( \\int _ 0 ^ 1 t r \\omega _ R ( \\gamma ( s ) \\big ) | \\\\ & = \\exp \\big ( R e \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } \\log ( 1 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta ) d \\theta \\big ) \\\\ & = \\exp \\big ( \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } \\log | 1 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta | d \\theta \\big ) . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} M _ \\zeta \\cdot \\vec v ( w ) = \\vec v ( { \\zeta ( w ) } ) \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} - \\Delta u + | u | ^ { p - 1 } u & = f ( x ) \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega , \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} f = x _ { 0 } q _ 0 + l ^ \\prime q ^ \\prime + h ( x _ 1 , \\dots , x _ { 1 6 } ) . \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v - \\Delta v + \\nabla _ H p + f _ 0 k \\times v = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\\\ \\partial _ z p = T , \\\\ \\partial _ t T + v \\cdot \\nabla _ H T + w \\partial _ z T - \\Delta _ H T = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} \\eta _ j ( ( x _ { m n } ) ) = \\otimes \\omega _ j ( ( x _ { m n } ) ) , \\mu _ j = . \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ + _ { V I } } & d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , \\alpha } \\left ( s + k - 1 \\right ) T R ^ d \\theta ^ { ( d - 1 ) / 2 } \\end{aligned} \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\tau & = \\lim _ { P \\to \\infty } \\lim _ { F \\to \\infty } \\frac { T \\log P } { F } \\\\ & \\ge \\frac { 1 } { l } \\left \\{ ( s _ 1 + s _ 2 ) - ( N _ T - l ) s _ 2 \\mu _ T - \\left ( \\frac { 2 s _ 2 + s _ 1 + 1 } { 2 } \\cdot s _ 1 + s _ 2 ^ 2 \\right ) \\mu _ R \\right . \\\\ & \\qquad \\quad \\left . + \\left ( \\frac { 2 s _ 2 + s _ 1 } { 2 } ( s _ 1 - 1 ) + s _ 2 ^ 2 \\right ) ( 1 - N _ T \\mu _ T ) ^ + \\right \\} . \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} \\begin{aligned} & \\limsup _ { N \\rightarrow \\infty } \\left \\Vert \\left ( f _ N ^ { ( s ) } ( 0 , Z _ s ) - \\left ( f _ N ^ { ( 2 ) } ( 0 ) \\otimes f _ 0 ^ { \\otimes ( s - 2 ) } \\right ) ( Z _ s ) \\right ) \\mathbf { 1 } _ { Z _ s \\in \\mathcal { G } _ s \\cap \\hat { \\mathcal { U } } _ s ^ { \\eta ( \\varepsilon ) } } \\mathbf { 1 } _ { E _ s ( Z _ s ) \\leq R ^ 2 } \\right \\Vert _ { L ^ \\infty _ { Z _ s } } \\\\ & = 0 \\end{aligned} \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} \\varphi _ { j , R } = \\varphi _ R \\big | _ { Z _ { j , R } } : Z _ { j , R } \\rightarrow Z _ j , \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} W ( \\gamma ) & = \\frac { 1 } { 2 \\pi i } \\int _ { L _ x ( \\gamma ) } \\frac { 1 } { w } d w \\\\ & = \\frac { 1 } { 2 \\pi i } \\int _ { 0 } ^ 1 \\frac { 1 } { L _ x ( z ( s ) ) } d L _ x ( z ( s ) ) \\\\ & = \\frac { 1 } { 2 \\pi i } \\int _ { 0 } ^ 1 \\frac { d } { d s } \\log L _ x ( z ( s ) ) d s . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} \\Delta ^ \\perp f = \\Delta ^ S f _ { \\mathfrak { L } } + \\Delta ^ N f _ { \\mathfrak { L } ^ \\perp } . \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} B _ m = X \\times \\ldots \\times X \\times A _ m ^ { c } \\times X \\times \\ldots \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } ( t ) = \\frac { q ^ { 1 / 2 } } { \\vartheta _ { 2 } \\left ( 0 \\mid q ^ { 2 } \\right ) \\vartheta _ { 3 } \\left ( 0 \\mid q ^ { 2 } \\right ) } \\int _ { t } ^ { \\infty } \\left [ \\left ( \\frac { q \\vartheta _ { 3 } ^ { 2 } \\left ( 0 \\mid q ^ { 2 } \\right ) } { \\vartheta _ { 2 } ^ { 2 } \\left ( 0 \\mid q ^ { 2 } \\right ) } + x ^ { 2 } \\right ) \\left ( q + x ^ { 2 } \\right ) \\right ] ^ { - 1 / 2 } \\ ! \\ ! \\ ! \\dd x \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} B _ j = \\begin{pmatrix} 0 & d _ j \\\\ b _ j & c _ j \\end{pmatrix} , C _ j = \\begin{pmatrix} 0 & g _ j \\\\ e _ j & f _ j \\end{pmatrix} , \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} u '' + a ( t ) u ^ { p } = 0 , \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{gather*} \\det \\big ( T ^ { ( \\alpha ) } _ { k } \\big ) = \\frac { 1 } { k ! } \\prod _ { i = 1 } ^ { k } c ^ { ( \\alpha ) } _ { i } \\big ( \\det \\big ( V ^ { ( k ) } _ { \\{ z _ { i } \\} } \\big ) ^ { 2 } \\big ) . \\end{gather*}"}
-{"id": "7046.png", "formula": "\\begin{align*} i = 3 \\begin{array} { | c | c | c | c | c | } \\hline & & & & \\\\ \\hline j = 0 & 2 & 3 & 0 & 1 \\\\ \\hline j = 1 & 1 & 0 & 3 & 2 \\\\ \\hline j = 2 & 3 & 2 & 1 & 0 \\\\ \\hline j = 3 & 0 & 1 & 2 & 3 \\\\ \\hline \\end{array} \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\rho ( P _ { 1 } \\otimes A _ { i } ) = ( 0 , \\dots , k [ z ] / ( z - \\alpha _ { i } ) ^ { n _ { i } - 1 } , 0 , \\dots ) . \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} \\mu _ b ( z ) : = ( 1 / \\overline { z } ) \\cdot 2 b ' / \\rho . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { t } a _ { p } c _ { p } = \\det ( \\mathbf { H } _ { \\bar { \\mathcal { R } } _ i , \\mathcal { T } } ) . \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} \\Gamma _ k ( i ) \\geq k p _ i - \\frac { 1 } { N ^ 2 } k ^ { \\frac { 1 } { 2 } + q } = \\left ( p _ i k ^ { \\frac { 1 } { 2 } - q } - \\frac { 1 } { N ^ 2 } \\right ) k ^ { \\frac { 1 } { 2 } + q } . \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\norm { T _ { \\psi _ 1 } } ^ 2 = \\sup _ { x \\in L ^ 0 } \\int _ { H ^ { F ^ 0 ( x ) } } \\abs { \\psi _ 1 ( h , x ) } ^ 2 \\ , \\dd \\mu ^ { F ^ 0 ( x ) } ( h ) . \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { h _ u ( t ) } { h ( t ) } = \\lim _ { t \\to \\infty } \\frac { \\lambda _ u ^ t t ^ { d _ u } } { \\lambda ^ t t ^ d } = \\lim _ { t \\to \\infty } \\frac { 1 } { t } = 0 . \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} a * m = \\sum _ { i \\geq 0 } \\begin{pmatrix} \\Delta _ a \\\\ i \\end{pmatrix} a _ { ( i - 1 ) } b , m * a = \\sum _ { i \\geq 0 } \\begin{pmatrix} \\Delta _ a - 1 \\\\ i \\end{pmatrix} a _ { ( i - 1 ) } m \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} ( f , \\omega ) \\mapsto f \\star \\omega = ( \\omega \\otimes \\iota ) \\Delta ( f ) ( \\omega \\in L ^ 1 ( G ) , \\ f \\in L ^ \\infty ( G ) ) , \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} \\| J ^ \\psi \\sigma \\| = \\| \\sigma \\| \\ , . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} D = \\norm { M _ X - M _ H } _ { T V } = \\inf _ { \\phi \\in P } \\norm { M _ X - M _ { \\phi , H } } _ { T V } - 3 / m . \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} M = 1 \\cdot A _ 1 + 2 \\cdot A _ 2 + \\cdots + n ^ 2 \\cdot A _ { n ^ 2 } \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} \\pi \\colon p = | p _ 0 : p _ 1 : p _ 2 | \\mapsto | \\Pi _ 0 ( p ) : \\Pi _ 1 ( p ) : \\Pi _ 2 ( p ) | = | p _ 0 ^ { a _ 0 } : p _ 1 ^ { a _ 1 } : p _ 2 ^ { a _ 2 } | . \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{gather*} T _ { 1 0 } ^ { \\alpha } T _ { 2 0 } ^ { \\beta } T _ { 2 1 } ^ { \\gamma } = ( - 1 ) ^ { \\frac { \\beta ( \\beta - 1 ) } { 2 } + \\alpha \\beta + \\alpha \\gamma + \\beta \\gamma } T _ { 2 } ^ { \\beta + \\gamma } T _ { 1 } ^ { \\alpha + \\beta } . \\end{gather*}"}
-{"id": "1003.png", "formula": "\\begin{align*} P _ 1 & = ( ( k - 1 ) ( k + 3 ) ( k ^ 2 - 2 k + 5 ) ( k ^ 4 + 1 2 k ^ 3 - 1 8 k ^ 2 - 4 k - 7 ) , \\\\ & 4 ( k - 1 ) ( k - 2 ) ( k + 3 ) ( k ^ 2 - 2 k + 5 ) ( k ^ 2 - 4 k - 1 ) ( k ^ 4 + 1 2 k ^ 3 - 1 8 k ^ 2 - 4 k - 7 ) ) . \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} \\left [ \\vert k \\vert ^ { \\alpha } - \\frac { d ^ { 2 } } { d k ^ { 2 } } \\right ] \\phi ( k ) = E \\phi ( k ) . \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} \\lambda < c \\max _ { 1 \\leq j \\leq p } \\left \\vert \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } X _ { i j } \\varepsilon _ { i } \\right \\vert , \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { t ^ 2 - ( a - c t ) ^ 2 } , a = c o n s t , \\ ; c = c o n s t \\neq 0 , \\ ; c ^ 2 \\neq \\kappa ^ 2 , \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} \\begin{aligned} d _ X ( F ( x , y ) , F ( u , v ) ) \\leq a \\ d _ X ( x , F ( u , v ) ) + & \\ b \\ d _ X ( u , F ( x , y ) ) + c \\ d _ X ( x , u ) ; \\\\ & \\forall x \\geq _ { P _ 1 } u , \\ y \\leq _ { P _ 2 } v ; \\ 2 b + c < 1 \\end{aligned} \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} s = - \\log _ { p / q } \\log n + O ( \\log \\log \\log n ) . \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} f _ \\alpha ( x ) = \\sum _ { | \\beta | \\leq m - | \\alpha | } \\frac { f _ { \\alpha + \\beta } ( y ) } { \\beta ! } ( x - y ) ^ \\beta + R _ \\alpha ( x , y ) \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} q = \\sum _ { \\alpha = 0 } ^ { k + 1 } f _ \\alpha ( x , y ) z ^ \\alpha , \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\left ( E _ { 1 , j } + 2 E _ { 2 , j } \\right ) = 0 , \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} \\begin{cases} J ^ 1 V _ 1 = \\sqrt { - 1 } V _ 1 , ~ J ^ 1 V _ 2 = \\sqrt { - 1 } V _ 2 , \\\\ J ^ 2 V _ 1 = \\tau \\overline { V } _ 1 + \\lambda e ^ { - i \\theta _ 1 } \\overline { V } _ 2 , ~ J ^ 2 { V } _ 2 = - \\mu e ^ { - i \\theta _ 1 } \\overline { V } _ 1 - \\overline { \\tau } e ^ { - 2 i \\theta _ 1 } \\overline { V } _ 2 , \\\\ J ^ 3 = J ^ 1 J ^ 2 . \\end{cases} \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} d _ 2 = \\begin{pmatrix} d \\\\ 0 \\end{pmatrix} , d = \\begin{pmatrix} p & y & z \\\\ 0 & q & w \\\\ 0 & 0 & r \\end{pmatrix} . \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} v '' + \\bigl { ( } \\lambda _ { 0 } + a ( t ) f ' ( u ( t ) ) \\bigr { ) } v = 0 , \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\left [ \\bar { Z } _ { a } , \\bar { Z } _ { b } \\right ] & = \\frac { 1 } { 2 } \\left [ P _ { a } - Z _ { a } , P _ { b } - Z _ { b } \\right ] , \\\\ & = \\frac { 1 } { 2 } \\left ( \\left [ P _ { a } , P _ { b } \\right ] - \\left [ P _ { a } , Z _ { b } \\right ] - \\left [ Z _ { a } , P _ { b } \\right ] + \\left [ Z _ { a } , Z _ { b } \\right ] \\right ) , \\\\ & = \\frac { 1 } { 2 } \\left ( J _ { a b } - Z _ { a b } + Z _ { b a } - J _ { a b } \\right ) , \\\\ & = - Z _ { a b } . \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\langle p r _ 1 ^ * T , p r _ { 1 , 2 } ^ * \\O ( \\Delta ) , p r _ 2 ^ * T \\rangle = \\langle \\langle p r _ 1 ^ * T , p r _ { 1 , 2 } ^ * \\O ( \\Delta ) \\rangle , T \\rangle , \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} I _ \\Delta \\big ( D ^ F _ { X _ \\infty } \\big ) = \\int _ \\Delta \\Psi ( \\lambda ) d \\lambda . \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\mathbf { H } _ { [ a : b ] } ^ { [ c : d ] } = \\begin{bmatrix} h _ { a , c } & h _ { a , c + 1 } & \\cdots & h _ { a , d } \\\\ h _ { a + 1 , c } & h _ { a + 1 , c + 1 } & \\cdots & h _ { a + 1 , d } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ h _ { b , c } & h _ { a , c + 1 } & \\cdots & h _ { b , d } \\end{bmatrix} . \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} c '' _ i = \\frac { \\langle \\mathrm { T r } ^ N _ M ( ( g E _ { \\lambda , N , \\overline { \\chi } } ) ^ \\mu ) , f _ i \\rangle } { \\langle f _ i , f _ i \\rangle } . \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{gather*} D _ { n } = \\prod \\limits _ { k = 0 } ^ { n } b _ { k } ^ { n + 1 - k } \\ , \\ \\ D _ { n + 1 } ^ { \\ , \\prime } = \\left ( \\sum \\limits _ { k = 0 } ^ { n } a _ k \\right ) \\prod \\limits _ { k = 0 } ^ { n } b _ { k } ^ { n + 1 - k } \\ , \\ \\ \\ \\ n \\geq 0 \\ . \\end{gather*}"}
-{"id": "5594.png", "formula": "\\begin{align*} A _ { \\alpha } \\psi _ { 0 } ^ { ( \\alpha ) } ( x ) = \\left [ \\frac { d ^ { \\alpha / 2 } } { d x ^ { \\alpha / 2 } } + x \\right ] \\psi _ { 0 } ^ { ( \\alpha ) } ( x ) = 0 . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} B _ a = \\begin{pmatrix} 0 & \\cdot \\\\ \\cdot & \\cdot \\end{pmatrix} , \\forall a = 1 , \\cdots , 7 . \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} a ^ \\ast \\circ \\phi _ V ( z ) = - \\pi \\sum _ k | z _ k | ^ 2 \\alpha _ k ^ - , \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} \\begin{aligned} F ( x _ k ) & \\le h ( c ( y _ k ) + a _ k \\nabla c ( y _ k ) ( v _ k - v _ { k - 1 } ) ) + a _ k g ( v _ k ) + ( 1 - a _ k ) g ( x _ { k - 1 } ) \\\\ & + \\tfrac { \\tilde { \\mu } a _ k ^ 2 } { 2 } \\norm { v _ k - v _ { k - 1 } } ^ 2 - \\tfrac { \\tilde { \\mu } - \\mu } { 2 } \\norm { x _ k - y _ k } ^ 2 + \\varepsilon _ k + \\tfrac { \\tilde { \\mu } } { 2 } \\norm { \\gamma _ k } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ { V I I } ^ + } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq C _ { d , s , k } C _ { d , \\alpha } T R ^ d \\theta ^ { ( d - 1 ) / 2 } \\end{aligned} \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} I _ 5 = & \\Big [ - \\frac { n ^ 2 - 8 n + 8 } { 3 ( n - 1 ) ( n - 2 ) } | W ( p ) | ^ 2 r ^ { 4 - n } - \\frac { 3 2 } { 9 ( n - 2 ) } \\sum _ { k , l , s } \\big ( ( W _ { i k l s } ( p ) + W _ { i l k s } ( p ) ) x ^ i \\big ) ^ 2 r ^ { 2 - n } \\\\ & ~ ~ - \\frac { 1 6 ( 7 n - 8 ) } { n - 2 } \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j r ^ { 2 - n } \\Big ] ( n - 6 ) + O ( r ^ { 5 - n } ) . \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} p ( n + 1 , k ) = \\frac { 2 k } { n } p ( n , k ) + \\frac { n - 2 k + 2 } { n } p ( n , k - 1 ) , \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} r a n k ( \\mathbf { A } _ i - \\mathbf { A } _ j ) = L ' , ~ ~ \\forall 1 \\leq i < j \\leq n - 1 , \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} \\sup _ { \\{ p ( x _ t | s _ { t - 1 } , y ^ { t - 1 } ) \\} _ { t = 1 } ^ N } \\sum _ { i = 1 } ^ N I ( X _ i , S _ { i - 1 } ; Y _ i | Q _ { i - 1 } ) & = \\sup _ { \\{ p ( x _ t | s _ { t - 1 } , q _ { t - 1 } ) \\} _ { t = 1 } ^ N } \\sum _ { i = 1 } ^ N I ( X _ i , S _ { i - 1 } ; Y _ i | Q _ { i - 1 } ) . \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} P _ { n } ( x ) = Q _ { n } ( x ) + a _ { n , 1 } Q _ { n - 1 } ( x ) + . . . + a _ { n , s } Q _ { n - s } ( x ) , \\ \\ 1 \\leq s \\leq d - 1 , \\ n \\geq 0 . \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} y _ q ( u ) & = \\sum _ { \\mathcal { R } : | \\mathcal { R } | = r + 1 } \\sum _ { \\mathcal { T } : | \\mathcal { T } | = t } \\sum _ { i = 1 } ^ t \\left [ \\sum _ { p \\in \\mathcal { T } } h _ { q p } ( u ) \\left ( \\mathbf { v } _ { { \\mathcal { R } } , { \\mathcal { T } } , p } ^ i ( u ) \\right ) ^ T \\right ] \\mathbf { x } _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ i , \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} \\square ^ * \\{ ( 4 \\pi \\tau ) ^ { - \\frac { m } { 2 } } e ^ { - l } \\} = ( - \\partial _ t - \\Delta + R ) \\{ ( 4 \\pi \\tau ) ^ { - \\frac { m } { 2 } } e ^ { - l } \\} \\leq 0 \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} \\zeta _ { d } = \\begin{cases} - 1 + \\frac { \\alpha } { 2 } , & d = 1 ; \\\\ \\alpha - \\frac \\alpha 2 \\gamma + \\nu _ 0 \\alpha - 2 , & d \\geq 2 \\ , . \\end{cases} \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} F ' = v \\cdot e ^ { u F + N } , \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} ( T _ 4 ) _ { i j } x ^ i x ^ j = ( n - 6 ) \\Delta \\sigma _ 1 ( A ) r ^ 2 - \\frac { 1 6 } { n - 4 } B _ { i j } x ^ i x ^ j + O ( r ^ 4 ) . \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} \\left ( - e ^ { \\xi } \\right ) ^ { n } h _ { n } \\left ( \\sinh \\xi \\vert q \\right ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { \\binom { k + 1 } { 2 } } e ^ { 2 k \\xi } } { \\left ( q ; q \\right ) _ { k } } A _ { k } \\left ( q ^ { \\left ( k - n \\right ) / 2 } e ^ { 2 \\xi } \\right ) , \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\nu ^ { \\pi ^ * } _ n ( 0 | 0 ) \\equiv { c } _ 0 ( n ) = \\frac { 1 } { 1 + 2 ^ { \\mu _ 0 ( n ) } } , ~ \\mu _ 0 ( n ) \\triangleq \\frac { H ( \\gamma _ n ) - H ( \\alpha _ n ) } { \\gamma _ n - \\alpha _ n } . \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} \\int \\phi ( x , y ) \\phi ( y , z ) d ( \\mu y ) = ( p _ e p _ d ) ^ 2 \\ \\ \\mu ^ 2 \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} \\begin{gathered} a _ m ( q ) = \\sum _ { j \\ge 0 } q ^ { j ^ 2 + j } \\bmatrix m - j - 2 \\\\ j \\endbmatrix _ q , b _ m ( q ) = \\sum _ { j \\ge 0 } q ^ { j ^ 2 } \\bmatrix m - j - 1 \\\\ j \\endbmatrix _ q . \\end{gathered} \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} Z _ j ( t , x - \\xi ) = & Z _ j ^ 0 ( t , x - \\xi , \\xi ) + V _ { Z _ j } ( t , x ; \\xi ) , j = 1 , 2 . \\\\ Y ( t , x - \\xi ) = & Y _ 0 ( t , x - \\xi , \\xi ) + V _ Y ( t , x ; \\xi ) . \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} \\gamma _ x ( u ^ * ( m ) k , u ^ * ( n ) l ) = \\gamma ' _ { v ( x ) } ( m , n ) . \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} \\textbf { w } _ { M S } ^ \\star = \\ & \\operatorname * { a r g \\ , m a x } _ { \\textbf { w } _ { M S } ^ i } | { \\textbf { w } _ { M S } ^ i } ^ H \\textbf { a } _ { M S } ( \\phi _ { C I } ) | \\\\ & s . t . \\ { \\textbf { w } _ { M S } ^ i } \\in \\textbf { W } _ { M S } \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} \\tilde \\omega ( \\theta ^ * y \\mu ) & = \\tilde \\omega ( y _ { ( 2 ) } \\theta ^ * ( y _ { ( 1 ) } \\cdot S ( y _ { ( 3 ) } ) ) \\mu ) = \\omega ( y _ { ( 2 ) } ) \\theta ^ * ( y _ { ( 1 ) } S ( y _ { ( 3 ) } ) ) \\mu ( 1 ) \\\\ & = \\omega ( y _ { ( 1 ) } ) \\theta ^ * ( y _ { ( 2 ) } S ( y _ { ( 3 ) } ) ) \\mu ( 1 ) \\\\ & = \\omega ( y _ { ( 1 ) } ) \\theta ^ * ( \\epsilon ( y _ { ( 2 ) } ) 1 ) \\mu ( 1 ) \\\\ & = \\omega ( y ) \\overline { \\theta ( 1 ) } \\mu ( 1 ) \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} c _ 1 ( \\beta , k ) & = \\frac { \\beta ^ k } { k ! } \\left ( k - \\frac { k - 1 } { q } \\right ) , \\\\ c _ 2 ( \\beta , k ) & = \\frac { \\beta ^ { k - 1 } } { ( q + 1 ) k ! } \\left ( k - ( k - 1 ) \\beta \\right ) , \\\\ c _ 3 ( \\beta , k ) & = \\frac { ( \\beta - r ) ^ k } { ( q + 1 ) k ! } q ^ { k - 1 } . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} c ^ { ( n ) } ( \\mu ) = c \\mu ^ n , \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} E ( n , k ) - \\widetilde { E } ( n , k ) = ( - 1 ) ^ k \\binom { n } { k } . \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} \\frac { \\partial M _ { p q } } { \\partial x _ { q } } & = e _ { p r s } T _ { s r } + e _ { p r s } x _ { r } \\frac { \\partial T _ { s q } } { \\partial x _ { q } } \\\\ & = e _ { p q r } x _ { q } \\mathcal { F } _ { r } + \\frac { \\partial } { \\partial t } \\left ( e _ { p q r } x _ { q } \\mathcal { G } _ { r } \\right ) , \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { j = - I } ^ I r _ { j } ( x ) ( \\log ( x ) ) ^ j \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} \\frac { | B _ { \\sigma r } ( x _ 0 ) | } { | B _ { 2 \\rho } ( z ) | } = \\left ( \\frac { \\sigma r } { 2 \\rho } \\right ) ^ { n } = \\left ( \\frac { \\sigma } { \\sigma - \\sigma ' } \\right ) ^ { n } , \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} p _ { l ^ * } = \\sum _ { \\alpha = 1 , p = 1 } ^ { 6 , 5 } ( S ^ l _ { \\alpha p } + \\pm \\sqrt { - 1 } T ^ l _ { \\alpha p } ) x _ \\alpha z _ p + \\ ; { \\rm o t h e r \\ ; t e r m s } . \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} \\varphi ( x , T ) = \\varphi ^ 1 ( x ) , \\psi ( x , T ) = \\psi ^ 1 ( x ) , \\ , \\ , ( 0 , L ) . \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} \\begin{bmatrix} - \\frac { \\delta } { q _ 0 } & \\frac { \\delta } { q _ 1 } & 0 & 0 & \\cdots & 0 \\\\ - \\frac { \\delta } { q _ 0 } & 0 & \\frac { \\delta } { q _ 2 } & 0 & \\cdots & 0 \\\\ - \\frac { \\delta } { q _ 0 } & 0 & 0 & \\frac { \\delta } { q _ 3 } & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\ddots & 0 \\\\ - \\frac { \\delta } { q _ 0 } & 0 & 0 & 0 & \\cdots & \\frac { \\delta } { q _ n } \\end{bmatrix} . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} L _ f ^ * ( \\chi , s ) = L _ f ^ * ( \\underline { \\pi } , s ) | _ { E ( \\pi _ j ) = \\chi ( c _ j ^ * ) \\textrm { f o r a l l } j } , \\ \\ C _ f ^ * ( \\chi , s ) = C _ f ^ * ( \\underline { \\pi } , s ) | _ { E ( \\pi _ j ) = \\chi ( c _ j ^ * ) \\textrm { f o r a l l } j } . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} \\widetilde { F } ( x _ { 1 } , \\dots , x _ { n } ) = F ( f _ { 1 } ^ { - 1 } ( x _ { 1 } ) , \\dots , f _ { n } ^ { - 1 } ( x _ { n } ) ) . \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} H _ 0 = \\frac { 1 } { \\sqrt { \\varepsilon \\left ( ( f f '' + ( f ' ) ^ 2 + 1 ) ^ 2 - \\kappa ^ 2 ( f '^ 2 + 1 ) \\right ) } } \\left ( - \\kappa \\sqrt { f '^ 2 + 1 } \\ , n _ 1 - ( f f '' + ( f ' ) ^ 2 + 1 ) \\ , n _ 2 \\right ) . \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} \\lambda _ { 0 } ( q ) = \\inf _ { v \\in H ^ { 1 } _ { T } } \\dfrac { \\int _ { 0 } ^ { T } \\bigl { ( } v ' ( t ) ^ { 2 } - q ( t ) v ( t ) ^ { 2 } \\bigr { ) } ~ \\ ! d t } { \\int _ { 0 } ^ { T } v ( t ) ^ { 2 } ~ \\ ! d t } \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} f _ N ( 0 , Z _ N ) = \\frac { 1 } { N ! } \\sum _ { \\sigma \\in \\mathcal { S } _ N } \\delta _ { Z _ N = \\sigma Z _ N ^ 0 } \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} ( \\rho \\otimes \\omega _ { \\delta _ j } ) \\ , J ( a ) = \\omega _ { \\delta _ j } ( a ) \\quad ( j = 0 , 1 ) . \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} \\omega _ { k } = f _ { k - 1 } \\left ( z _ { k } \\right ) = \\sum _ { i = 1 } ^ { k } a _ { i - 1 } \\varphi _ { z _ { i - 1 } } \\left ( z _ { k } \\right ) = \\sum _ { i = 1 } ^ { k } \\beta _ { k , i } a _ { i - 1 } , \\ ; \\ ; \\ ; \\ ; \\ ; k \\geq 1 . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} y = 1 . 7 6 3 2 0 8 1 9 x ^ { 0 . 4 3 2 8 5 2 } - ( \\log x ) ^ { 2 } \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} \\nu ^ k = \\frac { u ^ k } { W } = \\frac { \\abs { \\nabla u } \\delta ^ { k 1 } } { W } . \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} G _ 2 ( \\vec { x } ) = \\dfrac { \\alpha _ 2 } { 2 } \\| \\vec { y - x } \\| _ 2 ^ 2 + \\lambda _ 1 \\sum _ { i = 1 } ^ { m n } s ( x _ i , a _ 1 ) . \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} H = - \\frac { \\kappa } { 2 f } \\ , n _ 1 + \\frac { f f '' + ( f ' ) ^ 2 - 1 } { 2 f \\sqrt { f '^ 2 - 1 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { m , n } ^ { \\gamma } ) \\right ] ( z ) & = \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + 1 } \\mathcal { Z } _ { m , n - 1 } ^ { \\gamma + 1 } ( z , \\bar { z } ) \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} W ( \\omega ) \\ = \\ C \\cup \\bigcup _ { n : \\omega _ n = 1 } G _ n . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} \\ , A ( c '' t _ m ) \\geq \\ , c \\ , t _ m \\ , \\sum _ { k = 1 } ^ { m - 1 } \\frac { B \\big ( \\tfrac { 1 } { 2 C } 2 ^ { - k } t _ m \\big ) } { \\tfrac { 1 } { 2 C } 2 ^ { - k } t _ m } \\geq c ' \\ , t _ m \\int _ { 2 ^ { - m } \\frac { t _ m } { 4 C } } ^ { \\frac { t _ m } { 4 C } } \\frac { B ( s ) } { s ^ 2 } \\ , d s \\geq c ' \\ , t _ m \\int _ { \\frac { t _ 0 } { 2 C } } ^ { \\frac { t _ m } { 4 C } } \\frac { B ( s ) } { s ^ 2 } \\ , d s . \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{align*} d _ { H } ( \\overline { A _ { \\mathrm { t } } } , \\mathcal { B } ^ { n } ( K ) ) = \\max \\{ h _ { s } ( \\overline { A _ { \\mathrm { t } } } , \\mathcal { B } ^ { n } ( K ) ) , h _ { s } ( \\mathcal { B } ^ { n } ( K ) , \\overline { A _ { \\mathrm { t } } } ) \\} \\leq \\epsilon . \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} U ( t ) = S ( t ) U _ { 0 } + \\int _ { 0 } ^ { t } S ( t - s ) \\mathcal { N } \\big ( U ( s ) \\big ) \\mathrm { d } s \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} \\begin{array} { c } e C \\subset \\cap _ { \\alpha \\in J } h ^ { - 1 } ( \\varphi _ { \\alpha } ^ { - 1 } d _ { \\alpha } \\varphi _ { \\alpha } ) h C = h ^ { - 1 } ( \\cap _ { \\alpha \\in J } \\varphi _ { \\alpha } ^ { - 1 } d _ { \\alpha } \\varphi _ { \\alpha } ) h C = h ^ { - 1 } c h C \\ , , \\end{array} \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\lim _ { z \\to + \\infty } ( \\xi _ { s p } + \\eta _ { s p } ) = \\frac { \\alpha + 2 \\beta + \\gamma } { \\alpha + \\beta + \\gamma } = 1 + \\beta . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} \\sum _ { l = 2 } ^ 7 x _ l ^ * + \\left ( 2 - \\sum _ { l = 2 } ^ 7 x _ l ^ * \\right ) x _ 1 ^ * - ( x _ 1 ^ * ) ^ 2 \\geq \\frac { 3 } { 4 } . \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} r ( X ) = \\lambda ( X ) + | | X | | _ \\lambda \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} B _ 1 = C _ 1 = \\begin{pmatrix} 0 & 0 \\\\ 0 & \\sigma \\end{pmatrix} , \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } & \\sum _ { l = 1 } ^ \\mathcal { L } \\left ( c _ { i l } ( g _ { i l } ) - p _ l ( \\bar s _ l ) s _ { i l } \\right ) \\cr \\mbox { s u b j e c t t o } & \\sum _ { l = 1 } ^ \\mathcal { L } g _ { i l } { \\geq } \\sum _ { l = 1 } ^ \\mathcal { L } s _ { i l } , \\\\ & g _ { i l } , s _ { i l } \\geq 0 , g _ { i l } \\leq \\mathrm { c a p } _ { i l } , l = 1 , \\hdots , \\mathcal { L } . \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} v ( t _ 1 ) = w ( t _ 1 ) , \\qquad \\hbox { a n d } v ( t ) < w ( t ) \\quad \\hbox { f o r a l l } 0 \\leq t < t _ 1 . \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} & \\P \\left ( \\left \\| \\hat { U } _ { [ ( r + 1 ) : p , : ] } P _ { \\hat { U } _ { [ 1 : r , : ] } } \\right \\| > C ( d / n ) ^ { 1 \\over 2 } \\right ) \\\\ = & \\P \\left ( \\left \\| R ^ { \\intercal } \\hat { U } _ { [ ( r + 1 ) : n , : ] } P _ { \\hat { U } _ { [ 1 : r , : ] } } \\right \\| > C ( d / n ) ^ { 1 \\over 2 } \\right ) \\leq C \\exp ( - c d ) . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} \\mathbb { E } ( e ^ { - s T _ { ( k ) } } ) & = \\prod _ { j = n - k + 1 } ^ { n } \\mathbb { E } ( e ^ { - s Y _ { j } } ) \\\\ & = \\prod _ { j = n - k + 1 } ^ { n } \\mathbb { E } ( e ^ { - s \\sum _ { j = n - k + 1 } ^ { n } Y _ { j } } ) , \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} f _ { n } ( - 1 ) = \\frac { \\theta _ { q } ( - \\alpha ) } { \\left ( q ; q \\right ) _ { \\infty } } n + O ( 1 ) . \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} W _ { ( \\alpha , \\beta ) } ( n ) = \\sum _ { \\substack { { ( l , m ) \\in \\mathbb { N } _ { 0 } ^ { 2 } } \\\\ { \\alpha \\ , l + \\beta \\ , m = n } } } \\sigma ( l ) \\sigma ( m ) . \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} y _ k ^ n e _ k = & \\sum _ { j = 1 } ^ { k - 1 } \\sum _ { r = 0 } ^ { n - k + 1 } ( - 1 ) ^ { j + r + 1 } E _ j ( x _ 1 , \\ldots , x _ { k - 1 } ) E _ r ( x _ k , \\ldots , x _ n ) . y _ k ^ { n - j - r } e _ k + \\\\ & \\sum _ { j = 1 } ^ { n - k + 1 } ( - 1 ) ^ { j + 1 } E _ j ( x _ k , \\ldots , x _ n ) . ( y _ k ^ { n - j } e _ k ) . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} [ \\bar { \\xi } _ { i , 0 } , \\bar { x } ^ { \\pm } _ { j , s } ] = \\pm a _ { i , j } \\bar { x } ^ { \\pm } _ { j , s } , \\ [ \\bar { \\xi } _ { i , 1 } , \\bar { x } ^ { \\pm } _ { i + 1 , s } ] = \\mp 2 \\bar { x } ^ { \\pm } _ { i + 1 , s + 1 } , \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} \\rho ^ { \\omega } _ { K , \\mathbf { a } , f } ( X _ { \\mathbf { u } } ) = \\rho ^ { \\omega } _ { K , \\mathbf { a } , f } ( X _ 0 + \\sum _ { i = 1 } ^ { m } u _ i X _ i ) \\le ( m + 1 ) \\max \\{ \\rho ^ { \\omega } _ { K , \\mathbf { a } , f } ( X _ i ) \\mid i \\in \\{ 0 , 1 , \\ldots , m \\} \\} . \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\left | \\frac { | \\zeta _ { n } ^ { t } ( w _ 1 ) | _ w } { | \\zeta _ { n } ^ { t } ( w _ 1 ) | } - \\frac { | \\zeta ^ { t } ( x _ 1 ) | _ w } { | \\zeta ^ { t } ( x _ 1 ) | } \\right | \\le \\lim _ { t \\to \\infty } \\left | \\frac { n } { | \\zeta ^ { t } ( x _ 1 ) | } \\right | = 0 \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} & \\frac { ( z _ 0 ' ) ^ 2 - ( z _ 1 ' ) ^ 2 - ( z _ 2 ' ) ^ 2 } { 2 z _ 1 ' z _ 2 ' } \\\\ & = \\frac { \\big ( z _ 0 ( z _ 0 - z _ 2 ) - z _ 1 ^ 2 \\big ) \\big ( z _ 0 ( z _ 0 + z _ 2 ) - z _ 1 ^ 2 \\big ) - ( z _ 0 ^ 2 - z _ 2 ^ 2 ) z _ 2 ^ 2 } { 2 z _ 1 ^ 2 z _ 2 ^ 2 } \\\\ & = \\frac { z _ 0 ^ 4 + z _ 1 ^ 4 + z _ 2 ^ 4 - 2 z _ 0 ^ 2 z _ 1 ^ 2 - 2 z _ 0 ^ 2 z _ 2 ^ 2 } { 2 z _ 1 ^ 2 z _ 2 ^ 2 } \\\\ & = \\frac { ( z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 ) ^ 2 - 2 z _ 1 ^ 2 z _ 2 ^ 2 } { 2 z _ 1 ^ 2 z _ 2 ^ 2 } \\\\ & = 2 \\big ( \\frac { z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 } { 2 z _ 1 z _ 2 } \\big ) ^ 2 - 1 . \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} & \\left \\{ \\left ( \\sum _ { p \\in [ N _ T ] } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { [ N _ T ] } , p } ^ i ( u ) \\right ) _ { u = 1 } ^ S : \\mathcal { R } \\ni q , | \\mathcal { R } | = r + 1 , i \\in [ \\rho ] \\right \\} \\\\ & \\cup \\left \\{ \\left ( \\sum _ { p \\in [ N _ T ] } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { [ N _ T ] } , p } ^ i ( u ) \\right ) _ { u = 1 } ^ S : \\mathcal { R } \\cup \\bar { \\mathcal { R } } _ i \\not \\ni q \\right \\} \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} & \\textrm { e s s s u p } _ { ( \\omega , t , x ) \\in \\Omega \\times \\mathcal { O } _ t } u ^ { \\pm } \\\\ \\leq \\ , & C \\left ( \\textrm { e s s s u p } _ { ( \\omega , t , x ) \\in \\Omega \\times \\partial _ p \\mathcal { O } _ t } u ^ { \\pm } + A _ p ( f _ 0 ^ { \\pm } , g _ 0 ; \\mathcal { O } _ t ) + B _ 2 ( f _ 0 ^ { \\pm } , g _ 0 ; \\mathcal { O } _ t ) \\right ) \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} & u ( t , x , y ) = u _ 0 \\big ( x - t U ( y ) , y \\big ) + t U ' ( y ) \\int _ 0 ^ y u _ { 0 x } \\big ( x - t U ( z ) , z \\big ) d z , \\\\ & v ( t , x , y ) = - \\int _ 0 ^ y \\Big \\{ u _ { 0 x } \\big ( x - t U ( z ) , z \\big ) + t \\big [ U ( y ) - U ( z ) \\big ] u _ { 0 x x } \\big ( x - t U ( z ) , z \\big ) \\Big \\} d z , \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} & \\ \\int _ { U _ { n - m } ( F ) \\backslash U _ { n - m } ( \\mathbb { A } ) } \\int U _ \\xi \\left [ \\left ( \\begin{array} { c c } I _ { m } & x \\\\ 0 & u \\end{array} \\right ) g \\right ] \\overline { \\psi } _ { U _ { n - m } } ( u ) d x d u \\\\ = & \\ \\int _ { U _ { n - m } ( F ) \\backslash U _ { n - m } ( \\mathbb { A } ) } \\int V _ \\xi ^ m \\left [ \\left ( \\begin{array} { c c } I _ { m } & x \\\\ 0 & u \\end{array} \\right ) g \\right ] \\overline { \\psi } _ { U _ { n - m } } ( u ) d x d u \\ , , \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} h _ 0 ( \\xi ) : = \\sum _ { x \\in \\mathbb { Z } ^ d } e ^ { - i x \\cdot \\xi } f [ x ] , \\xi \\in \\mathbb { T } ^ d = [ - \\pi , \\pi ) . \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} V _ { n } : = \\left \\{ f \\in C \\left ( X \\right ) : \\sup _ { x \\in K _ { n } } \\left | f \\left ( x \\right ) \\right | \\leq \\frac { 1 } { n } \\right \\} . \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} h ( J _ \\alpha X , J _ \\alpha Y ) = - h ( X , Y ) , \\alpha = 2 , 3 . \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{gather*} A = \\det \\big ( z _ { i } ^ { i + j - 2 } \\big ) = \\prod _ { s = 1 } ^ { k } z _ { s } ^ { s - 1 } \\det \\big ( z _ { i } ^ { j - 1 } \\big ) = \\prod _ { s = 1 } ^ { k } z _ { s } ^ { s - 1 } \\det \\big ( V ^ { ( k ) } _ { \\{ z _ { i } \\} } \\big ) . \\end{gather*}"}
-{"id": "8961.png", "formula": "\\begin{align*} p ( t , t ^ \\prime ) - \\xi = - \\int _ { t ^ \\prime } ^ t \\nabla _ x V _ \\rho ( \\tau , q ( \\tau , t ^ \\prime ) ) d \\tau , t , t ^ \\prime \\in \\mathbb { R } . \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} \\beta _ { s , a ^ 1 } ^ 1 & = \\frac { \\bar { r } ^ 1 ( s , a ^ 1 ) - \\bar { r } ^ 1 ( s , a _ s ^ 1 ) } { \\sum _ { s ' \\in S } p _ { s ' } \\bar { r } ^ 1 ( s ' , a _ { s ' } ^ 1 ) - \\sum _ { s ' \\in S } p ^ 1 ( s ' | s , a ^ 1 ) \\bar { r } ^ 1 ( s ' , a _ { s ' } ^ 1 ) } \\\\ & = \\frac { r ^ 1 ( s , a ^ 1 , a _ s ^ 2 ) - r ^ 1 ( s , a _ s ^ 1 , a _ s ^ 2 ) } { \\sum _ { s ' \\in S } p _ { s ' } r ^ 1 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) - \\sum _ { s ' \\in S } \\left ( \\frac { \\mu ( s ' , s , a ^ 1 , a _ s ^ 2 ) } { | | \\mu | | } + \\delta ( s , s ' ) \\right ) r ^ 1 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) } . \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} \\left \\langle u _ { t } , \\pi _ { \\sigma } P _ n \\right \\rangle = \\sum _ { i = 0 } ^ { d - 1 } \\left \\langle v _ i , \\chi _ { t } ^ i P _ n \\right \\rangle = 0 , \\ n \\geq d \\big ( d ( l - 1 ) + r \\big ) + t + 1 . \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} \\left . \\begin{array} { l l l } X _ j = \\partial _ { x _ j } + 2 y _ j \\partial _ t , j = 1 , . . . , n , \\\\ Y _ j = \\partial _ { y _ j } - 2 x _ j \\partial _ t , j = 1 , . . . , n . \\end{array} \\right . \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} d \\| \\psi u \\| ^ 2 = 2 ( \\psi u , d ( \\psi u ) ) + \\| G \\psi u \\| ^ 2 \\ , d t , \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} a _ { N , k , s } = \\frac { ( N - s ) ! } { ( N - s - k ) ! } \\varepsilon ^ { k ( d - 1 ) } \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} \\frac { \\partial \\varphi } { \\partial n _ { L ^ { * } } } = \\nabla \\varphi A \\cdot { \\bf { n } } = \\nabla \\varphi \\cdot { \\bf { n } } A ^ T \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} F ' ( z , v ) = \\frac { v } { v ( 1 - z ) + ( 1 - v ) ( 1 - z ) ^ { 1 - p } } . \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} s ^ { n } + a _ { n - 2 } s ^ { n - 2 } t ^ { 2 } + a _ { n - 3 } s ^ { n - 3 } t ^ { 3 } + \\dots + a _ { 0 } t ^ { n } = 0 , \\end{align*}"}
-{"id": "9895.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mathbf n _ V = \\nu _ { \\partial \\Omega } & \\mbox { o n } \\partial M \\setminus \\partial B ^ + , \\\\ \\mathbf n _ V - ( \\mathbf n _ V \\cdot \\nu _ { \\partial \\Omega } ) \\nu _ { \\partial \\Omega } = - \\sigma \\mathbf n _ { B ^ + } & \\mbox { o n } \\partial B ^ + . \\end{array} \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} \\frac { 2 ( 2 n ) ! } { ( 2 n + 2 ) ! } \\sum \\limits _ { i = 0 } ^ n { 2 ( n + 1 ) \\choose 2 ( i + 1 ) } B _ { 2 n - 2 i } { i + 1 \\brack 1 } , \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} \\frac { \\partial ^ \\alpha y ( x , t ) } { \\partial t ^ \\alpha } + a ( x ) \\frac { \\partial y ( x , t ) } { \\partial x } + b ( x ) \\frac { \\partial ^ 2 y ( x , t ) } { \\partial x ^ 2 } = f ( x , t ) , 0 \\leq x \\leq \\ell , \\ 0 < t \\leq 1 , \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} ( T g ) ( x ) = \\int s ( x , y ) g ( y ) d ( \\mu y ) . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} \\lambda _ i ( \\tau _ { y + z } ) = \\lambda _ i ( \\tau _ y ) \\lambda _ i ( \\tau _ z ) , \\ \\ i = 1 , \\cdots , N : = \\dim V . \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} \\| \\eta \\| ( P _ 1 + \\dots + P _ { g - 1 } ) = \\exp \\left ( - \\tfrac { 1 } { 4 } \\delta ( X ) \\right ) \\prod _ { j = 1 } ^ { g - 1 } \\frac { \\| \\theta \\| ( P _ 1 + \\dots + P _ { g - 1 } + P _ j - Q ) } { G ( P _ j , Q ) ^ g } . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} D _ x \\Phi _ \\lambda ( z _ \\delta , w _ \\delta , t _ \\delta ) = \\frac { x _ \\delta - z _ \\delta } { \\delta } - \\beta z _ \\delta , D _ y \\Phi _ \\lambda ( z _ \\delta , w _ \\delta , t _ \\delta ) = \\frac { y _ \\delta - w _ \\delta } { \\delta } - \\beta w _ \\delta , \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} \\delta _ { \\alpha + \\beta , n - 1 } + \\sum _ { \\substack { \\ell \\ge \\alpha , m \\ge \\beta , \\\\ \\ell + m = n } } { \\ell \\brack \\alpha } _ r { m \\brack \\beta } _ s = \\sum _ { k = 0 } ^ n { n \\brack k } _ { r + s } \\left ( { k - \\beta \\brack \\alpha } _ { r } + { k - \\alpha \\brack \\beta } _ { s } \\right ) . \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} V _ { N } ( u , v , \\lambda ; q ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { 1 / 2 + u + v } } K l ( l , n ; q ) J _ { 2 \\lambda - 1 } \\left ( 4 \\pi \\frac { \\sqrt { l n } } { q } \\right ) . \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} \\sum \\limits _ { l = 1 } ^ N | \\langle \\phi _ j , \\phi _ l \\rangle | ^ 2 = \\frac { N } { M } , \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} \\dd Y _ t = \\lambda ^ \\top X _ t ( 1 - Y _ t ) \\ , \\dd t + \\dd m _ t , \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} \\mu _ { k } ^ i \\left ( B \\right ) & \\geq 1 \\frac { 1 } { \\epsilon } \\sum \\limits _ { \\theta \\in B ^ c } \\prod \\limits _ { t = 1 } ^ { k } \\prod \\limits _ { j = 1 } ^ { n } \\left ( \\frac { \\ell ^ j ( s _ { t } ^ j | \\theta ) } { \\ell ^ j ( s _ { t } ^ j | \\theta ^ * ) } \\right ) ^ { \\left [ A ^ { k t } \\right ] _ { i j } } \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} d z _ 0 = u \\ , d x - u \\omega _ { [ 1 ] } \\ , d y + \\delta \\ , d t \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } E _ \\infty ^ i ( C ^ { \\infty } ( S ^ 3 \\times S ^ { 6 } \\times S ^ { 8 } ) \\rtimes _ \\beta \\mathbb { Z } ) \\cong \\mathbb { C } , & i = 1 , 3 , 7 , 9 \\\\ E _ \\infty ^ i ( S ^ 3 \\times S ^ { 6 } \\times S ^ { 8 } ) \\rtimes _ \\beta \\mathbb { Z } ) \\cong \\{ 0 \\} , & i = \\ e l s e . \\end{array} \\right . \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nabla _ x \\phi ( t ; x , \\xi ) = \\eta ( t , 0 ; x , \\xi ) , \\\\ \\nabla _ \\xi \\phi ( t ; x , \\xi ) = y ( 0 , t ; x , \\xi ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} \\sigma _ A : = a _ { i j } e ^ { i j } \\in \\Lambda ^ 2 V _ 7 ^ \\ast . \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( x : n ) } = \\bigoplus _ { i = 0 } ^ { x - 1 } T _ x ( i ) = \\bigoplus _ { i = 0 } ^ { x - 1 } H _ x ( i , \\phi ( i ) ) \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} & S _ \\omega : = \\{ ( k _ 1 , s _ 1 ) \\dots ( k _ n , s _ n ) \\in ( \\omega \\times S ) ^ * \\ | \\ s _ i { R } s _ { i + 1 } ~ ( 0 \\leq i < n ) \\} , \\\\ & R _ \\omega [ ( k _ 1 , s _ 1 ) \\cdots ( k _ n , s _ n ) ] : = \\{ ( k _ 1 , s _ 1 ) \\cdots ( k _ n , s _ n ) ( k _ { n + 1 } , s _ { n + 1 } ) \\ : \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ : \\ k _ { n + 1 } \\in \\omega , s _ n { R } s _ { n + 1 } \\} . \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} \\frac { ( m - 1 ) ! ( k - m ) ! } { ( k - 1 ) ! } \\sum \\limits _ { i = 0 } ^ { m - 1 } ( - 1 ) ^ i \\frac { ( k - 1 ) ! } { ( m - i - 1 ) ! i ! ( k - m ) ! } \\gamma _ { m - r + i } = 0 , \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\left ( a ; q \\right ) _ { \\infty } = \\prod _ { k = 0 } ^ { \\infty } \\left ( 1 - a q ^ { k } \\right ) , \\left ( a ; q \\right ) _ { n } = \\frac { \\left ( a ; q \\right ) _ { \\infty } } { \\left ( a q ^ { n } ; q \\right ) _ { \\infty } } , \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} \\pi ^ * _ { n - 1 } ( 0 | 0 ) \\equiv { { d _ 0 ( n - 1 ) } = \\frac { 1 - \\gamma _ { n - 1 } ( 1 + 2 ^ { \\mu _ 0 ( n - 1 ) + \\Delta { C } _ n } ) } { ( \\alpha _ { n - 1 } - \\gamma _ { n - 1 } ) ( 1 + 2 ^ { \\mu _ 0 ( n - 1 ) + \\Delta { C } _ n } ) } } . \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} \\lim _ { c \\to \\infty } V _ c ( x ) = \\frac { 1 } { 2 m } - \\frac { 1 } { 2 m } \\frac { - \\tfrac { d ^ 2 } { d x ^ 2 } u _ \\infty ( x ) } { u _ \\infty ( x ) } , \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} \\hat { Z } _ { t | s } & = \\exp ( Q ( t - s ) ) \\hat { X } _ s - \\exp ( Q _ \\lambda ( t - s ) ) \\hat { X } _ { s } ( 1 - Y _ s ) . \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} b ( x ) : = x _ 1 b _ 4 + \\cdots + x _ 4 b _ 7 = \\begin{pmatrix} s x _ 1 & \\pm s x _ 2 & 0 \\\\ s x _ 2 & \\mp s x _ 1 & 0 \\\\ s x _ 3 & \\pm s x _ 4 & 0 \\\\ s x _ 4 & \\mp s x _ 3 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} ( X , Y , Z ) = ( - t Z , - u Z , Z ) , \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a , b ; q \\right ) _ { n } } { \\left ( q , c ; \\right ) _ { n } } \\left ( \\frac { c } { a b } \\right ) ^ { n } = \\frac { \\left ( c / a , c / b ; q \\right ) _ { \\infty } } { \\left ( c , c / a b ; q \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} C _ { i , s + 1 } = C _ { i , s + 1 } ^ + - C _ { i , s + 1 } ^ - \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} d _ 2 ^ { t r } d _ 2 = g _ 2 ^ { t r } g _ 2 , \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} \\begin{dcases} \\frac { d \\tilde X } { d t } = - D _ p H ( \\tilde P , \\tilde X ) , & \\tilde X ( x , p , 0 ) = x , \\\\ \\frac { d \\tilde P } { d t } = D _ x H ( \\tilde P , \\tilde X ) , & \\tilde P ( x , p , 0 ) = p . \\end{dcases} \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\widetilde { \\mathbf { s c r } } ( x _ i , 0 ) = \\infty , \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} & [ d , f ] ( x _ n , x _ m , 1 ) = 0 \\Rightarrow \\\\ & - [ \\alpha ( x _ n ) , g ( x _ m ) ] + [ \\alpha ( x _ m ) , g ( x _ n ) ] + g ( [ x _ n , x _ m ] ) + \\bar { f } ( 1 , \\varphi ( x _ n , x _ m ) ) \\\\ & - \\varphi ( \\alpha ( x _ n ) , g ( x _ m ) ) + \\varphi ( \\alpha ( x _ m ) , g ( x _ n ) ) + \\hat { v } ( 1 , [ x _ n , x _ m ] ) + \\bar { v } ( 1 , \\varphi ( x _ n , x _ m ) ) . \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} U _ \\alpha ( \\psi ' ) = \\{ x \\in X \\mid \\psi ( x ) \\otimes \\Phi ( \\psi ' ) < u \\} \\cap \\{ x \\in X \\mid \\psi ' ( x ) > \\alpha \\} . \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\Sigma ( N , y ) : = \\sum _ n w _ 1 \\left ( \\frac { n } { N } \\right ) n ^ { 2 \\pi i y } = \\int _ { ( 2 ) } \\check { w } _ 1 ( s ) N ^ s \\zeta ( s - 2 \\pi i y ) \\frac { d s } { 2 \\pi i } . \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} F = \\left [ \\begin{array} { c c } 0 & I _ { n - 1 } \\end{array} \\right ] , G = \\left [ \\begin{array} { c c } I _ { n - 1 } & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & \\ ; \\ ; = \\left ( X _ { s + k } ^ \\prime , V _ { s + k } ^ \\prime \\right ) \\in \\mathcal { D } _ { s + k } \\end{aligned} \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} & ~ \\delta ^ * ( 1 , 0 ) \\leq \\delta _ { \\mathsf { C a - Z F } } = \\frac { K } { \\min \\{ M , K \\} } . \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} \\psi _ \\varphi ( g ; h ) : = \\sum _ { f \\sim g } \\varphi ( f ) \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} \\Phi ^ { * } _ { 2 } \\nu ^ { 2 } = \\tfrac { 1 } { 2 } \\sum _ { p \\neq q } ( & \\psi _ { p } ( P _ { 1 } ) \\wedge \\bar { \\psi } _ { q } ( P _ { 1 } ) \\wedge \\psi _ { q } ( P _ { 2 } ) \\wedge \\bar { \\psi } _ { p } ( P _ { 2 } ) \\\\ - & \\psi _ { p } ( P _ { 1 } ) \\wedge \\bar { \\psi } _ { p } ( P _ { 1 } ) \\wedge \\psi _ { q } ( P _ { 2 } ) \\wedge \\bar { \\psi } _ { q } ( P _ { 2 } ) ) . \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\left \\langle Z _ { a b } , Z _ { c d } , \\bar { P } _ { e } \\right \\rangle & = \\frac { 1 } { \\sqrt { 2 } } \\left \\langle Z _ { a b } , Z _ { c d } , P _ { e } \\right \\rangle + \\frac { 1 } { \\sqrt { 2 } } \\left \\langle Z _ { a b } , Z _ { c d } , Z _ { e } \\right \\rangle , \\\\ & = \\left ( \\alpha _ { 2 } + \\alpha _ { 3 } \\right ) \\ , \\varepsilon _ { a b c d e } , \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{gather*} g = g _ { - } g _ { 0 + } , g \\in \\widetilde { \\rm G L } _ n . \\end{gather*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u _ t ( t , x ) + L ^ b u ( t , x ) + f ( t , x , u ( t , x ) , \\nabla u ( t , x ) ) = 0 , \\\\ u ( T , x ) = \\Phi ( x ) , \\\\ \\forall ( t , x ) \\in [ 0 , T ] \\times \\mathbb R ^ d . \\end{array} \\right . \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{gather*} S ^ { \\pm } ( z ) = \\left ( 1 - \\frac { S ^ { + } } { z } \\right ) ^ { \\pm 1 } , \\end{gather*}"}
-{"id": "7922.png", "formula": "\\begin{align*} | S | \\ge | Y _ 1 | + | X | \\ge | Y _ 1 | + | Y _ 2 | + 1 = | Y | + 1 = b ( G ' ) + 1 . \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} \\| F _ { i l } ( x _ { i l } , u _ l ) - F _ { i l } ( x _ { i l } , z _ l ) \\| & = \\sqrt { | c _ { i l } ' ( g _ { i l } ) - c _ { i l } ' ( g _ { i l } ) | ^ 2 + | p _ l ( u _ l ) + p ' _ l ( u _ l ) s _ { i l } - p _ l ( z _ l ) - p ' _ l ( z _ l ) s _ { i l } | ^ 2 } \\cr & = \\sqrt { | ( p _ l ( u _ l ) - p _ l ( z _ l ) ) + ( p ' _ l ( u _ l ) - p ' _ l ( z _ l ) ) s _ { i l } | ^ 2 } \\cr & \\le \\sqrt { 2 } \\sqrt { | p _ l ( u _ l ) - p _ l ( z _ l ) | ^ 2 + | p ' _ l ( u _ l ) - p ' _ l ( z _ l ) | ^ 2 s _ { i l } ^ 2 } , \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\pi i } \\partial \\overline { \\partial } H ( X ) = \\tfrac { 1 } { 2 } \\omega _ { \\mathrm { H d g } } - \\tfrac { 1 } { 8 } e _ 1 ^ A + \\tfrac { ( - 1 ) ^ { g + 1 } } { ( g ! ) ^ 2 } \\left ( c _ 1 ( \\widetilde { \\mathcal { L } } ^ { \\langle g + 1 \\rangle } ) + g \\cdot c _ 1 ( \\widetilde { \\mathcal { L } } '^ { \\langle g \\rangle } ) \\right ) \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{gather*} S _ 0 \\hat { y } _ \\mu = S _ 0 ( x _ \\rho \\phi ( - \\partial ) ^ \\rho _ \\mu ) = \\phi ( \\partial ) ^ \\rho _ \\mu x _ \\rho = x _ \\rho \\phi ( \\partial ) ^ \\rho _ \\mu + \\partial _ \\rho \\phi ( \\partial ) ^ \\rho _ \\mu , \\end{gather*}"}
-{"id": "1521.png", "formula": "\\begin{align*} y _ { 1 } = h _ d x _ 1 + h _ c x _ { 2 } + z _ 1 ; y _ 2 = h _ d x _ 2 + z _ 2 , \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} \\begin{array} { l l } S ( g ; \\phi ) = & - 2 \\int _ \\Sigma d v \\ , \\phi ^ i \\Delta _ g \\phi ^ j - S ( g ; \\phi ) + \\int _ \\Sigma d v \\ , \\phi ^ i \\phi ^ j \\partial _ \\mu \\partial ^ \\mu g _ { i j } ( \\phi ) \\end{array} \\end{align*}"}
-{"id": "9943.png", "formula": "\\begin{align*} \\pi _ { p } \\circ q ( \\vect { \\delta } ) = q ( \\vect { \\delta } ) \\circ \\pi _ { p } . \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} \\begin{array} { l l } ( \\overline \\nabla _ X J _ 1 ) Y = \\frac { 1 } { 2 ( 2 n - 1 ) } \\left [ g ( X , Y ) p _ 1 - \\overline \\theta _ 1 ( Y ) X + g ( X , J _ 1 Y ) J _ 1 p _ 1 \\right . \\\\ \\qquad \\qquad \\quad \\left . - \\overline \\theta _ 1 ( J _ 1 Y ) J _ 1 X \\right ] , \\end{array} \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} A ( z ) = \\frac { \\theta _ { q } \\left ( \\alpha z ^ { - 1 } \\right ) } { \\theta _ { q } \\left ( z ^ { - 2 } \\right ) } . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( x y : n ) } = \\bigoplus _ { ( i , \\alpha ) } H _ { ( x y ) } ( i , \\alpha ) \\psi ( i , \\alpha ) . \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} \\phi = \\sum _ { k \\in \\mathbf { Z } } \\phi _ { k } ^ { \\pm } ( I _ { \\pm } ) e ^ { i k \\theta _ { \\pm } } , \\phi _ { k } ^ { \\pm } ( I _ { \\pm } ) = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\phi ( x ) e ^ { - i k \\theta _ { \\pm } } d \\theta _ { \\pm } . \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} \\int f _ { 0 , + } \\left ( v \\right ) d v = \\int f _ { 0 , - } \\left ( v \\right ) d v , \\ \\int \\frac { f _ { 0 , + } ^ { \\prime } \\left ( v \\right ) + f _ { 0 , - } ^ { \\prime } \\left ( v \\right ) } { v } = \\left ( \\frac { 2 \\pi } { P _ { 0 } } \\right ) ^ { 2 } , \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} P \\rho ( t ) ( 0 , h , 0 ) = ( 0 , e ^ { i t A _ 1 } h , 0 ) . \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} R ^ { ( d ) } = \\frac { k [ y _ 0 , \\ldots , y _ N ] } { J } \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} V _ y = \\{ z \\in X \\mid \\varphi _ 1 ( z ) \\otimes \\psi _ y ( z ) < u \\} \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 4 } \\left ( a , a q ; c q , c q ^ { 2 } , a z , a z q ; q ^ { 2 } , c ^ { 2 } z ^ { 2 } q ^ { 3 } \\right ) = \\frac { \\left ( z ; q \\right ) _ { \\infty } } { \\left ( a z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a ; q \\right ) _ { n } S _ { n } \\left ( c ; q \\right ) z ^ { n } } { \\left ( c q ; q \\right ) _ { n } } . \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} \\tau _ { m , n } = \\frac { N _ R ( 1 - \\mu _ R ) } { ( m + 1 ) d _ { \\textrm { s u m } } } . \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} H = - \\frac { \\kappa } { 2 f } \\ , n _ 1 - \\frac { f f '' + ( f ' ) ^ 2 + 1 } { 2 f \\sqrt { f '^ 2 + 1 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} ~ b _ j = e _ j , \\quad \\sigma ( c _ j - f _ j ) = - ( c _ j - f _ j ) ^ { t r } \\sigma , 2 \\leq j \\leq 7 , \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} f = ( f _ 1 , \\cdots , f _ n ) \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} r _ { \\mathsf { t h } } = \\frac { K ( M - 1 ) } { M \\left ( \\min \\{ M , K \\} - 1 \\right ) } , \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} H = H _ 0 - \\sum _ { i = 1 } ^ { n + 1 } { \\lambda _ i \\over L _ i } | \\Gamma _ i \\rangle \\langle \\Gamma _ i | \\ ; , \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} \\bar \\omega \\nabla ^ { - 1 } = \\nabla \\bar \\omega , \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} d = \\begin{pmatrix} p & 0 \\\\ 0 & 0 \\end{pmatrix} , g = \\begin{pmatrix} u & v \\\\ w & z \\end{pmatrix} . \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} V ^ { - 1 } _ { i j } = { n - j \\choose n - i } I . \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} B _ W = U \\left ( \\tilde { W } ( S ^ * - S ) - ( S ^ * - S ) \\tilde { W } \\right ) + ( \\delta _ { \\{ 0 \\} } S ^ * + \\delta _ { \\{ 1 \\} } S ^ * - \\delta _ { \\{ 0 \\} } S - \\delta _ { - 1 } S ) \\tilde { W } \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} \\hat \\theta _ n ^ { ( 4 ) } ( m ) = \\frac { n X _ { n ^ { m - 1 } } ^ 2 } { 2 \\sum _ { k = 0 } ^ { n ^ m - 1 } X _ { k / n } ^ 2 } \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} \\frac { q ^ { n ^ { 2 } } } { \\left ( - q ^ { 2 } \\right ) ^ { n } } = \\sum _ { k = 0 } ^ { n } \\frac { \\left ( q ^ { - n } , - q ^ { - n } ; q \\right ) _ { k } q ^ { 2 n k } } { \\left ( q ; q \\right ) _ { k } q ^ { \\binom { k + 1 } { 2 } } } . \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} P _ { c } ( z ) = e ^ { \\frac { \\lambda } { \\xi } z } z ^ { - ( \\frac { \\mu j } { \\xi } - c ) } \\left \\lbrace - \\frac { \\gamma } { \\xi } \\ell _ { 1 } ( z ) + \\frac { \\mu } { \\xi } \\sum _ { n = 1 } ^ { c } n p _ { c , n } \\ell _ { 2 } ( z ) - \\frac { c ( \\xi - \\mu ) } { \\xi } \\sum _ { n = 0 } ^ { c } p _ { c , n } \\ell _ { 3 } ( z ) + \\sum _ { n = 1 } ^ { c } n p _ { c , n } \\ell _ { 3 } ( z ) \\right \\rbrace . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} M ( D _ i ) ^ \\circ = M ( a _ { 1 ; i } ) ^ \\circ \\otimes M ( a _ { 2 ; i } ) \\otimes \\cdots \\otimes M ( a _ { K ; i } ) , \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} { \\rm l e n g t h } ~ T / F ^ \\perp = { \\rm l e n g t h } ~ T / ( M _ 1 ^ \\perp \\cap M _ 2 ^ \\perp ) - 1 . \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} \\psi _ s ( u _ { i j } ^ k ) = \\frac { \\delta _ { i j } \\mu _ k ( s ) } { \\mu _ k ( | q | + | q | ^ { - 1 } ) } ( U ^ k = [ u _ { i j } ^ k ] ) . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} & \\mu _ n ( a ^ { ( 1 ) } \\otimes \\cdots \\otimes a ^ { ( n ) } ) = a ^ { ( 1 ) } _ 1 \\otimes a ^ { ( 1 ) } _ 2 \\rightharpoonup a ^ { ( 2 ) } \\otimes \\cdots \\otimes a ^ { ( 1 ) } _ n \\rightharpoonup a ^ { ( n ) } , \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} \\dot { y } ( t ) = \\lambda ( 1 - y ( t ) ) , \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} C _ 0 \\ = \\ [ 0 , 1 ] \\setminus \\bigcup _ { k \\in \\mathbb { N } } I _ k \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\langle T _ { ( A _ { 1 } , i _ { 1 } ) } \\cdots T _ { ( A _ { n } , i _ { n } ) } \\rangle = \\alpha _ { j } K _ { i _ { a } \\cdots i _ { n } } ^ { \\ \\ \\ \\ \\ j } \\langle T _ { A _ { 1 } } \\cdots T _ { A _ { n } } \\rangle \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} & \\mathbf { F } \\operatorname { d i v } \\mathbf { G } ^ { \\ast } - \\mathbf { G } ^ { \\ast } \\times \\operatorname { c u r l } \\mathbf { F } \\\\ & \\quad = 4 \\pi \\rho \\mathbf { F } + \\frac { i } { c } \\left ( \\frac { \\partial \\mathbf { G } } { \\partial t } \\times \\mathbf { G } ^ { \\ast } + 4 \\pi \\mathbf { j \\times G } ^ { \\ast } \\right ) \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} \\langle u , [ W , \\i A ] v \\rangle = \\langle u , K _ { W } v \\rangle + \\langle u , B _ { W } v \\rangle . \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} s _ { 2 i + 1 } - s _ i & = \\frac { \\omega ^ { a _ { h ( 2 i + 1 ) } ( 2 i + 1 ) } } { 2 ^ { h ( 2 i + 1 ) } } \\Biggl [ z ^ { h ( 2 i + 1 ) } + \\frac { 1 } { 2 } \\left ( \\sum _ { j = 0 } ^ { k - h ( 2 i + 1 ) - 1 } z ^ j \\right ) ( z ^ { h ( 2 i + 1 ) + 1 } - 1 ) \\\\ & \\quad \\qquad \\qquad \\qquad \\qquad - \\omega ^ { - 1 } \\left ( \\sum _ { j = 0 } ^ { k - h ( 2 i + 1 ) } z ^ j \\right ) ( z ^ { h ( 2 i + 1 ) } - 1 ) \\biggr ] . \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\kappa _ { 1 } ( u ) = - \\frac { \\lambda } { c ^ { 2 } } \\phi ^ { \\prime } { } ( u ) \\cos \\left ( \\frac { u } { c } \\right ) + \\frac { \\lambda } { c } \\phi { } ^ { \\prime \\prime } ( u ) \\sin \\left ( \\frac { u } { c } \\right ) , \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\tau \\partial _ { t } \\big ( \\mathbf { H } ( t , \\cdot ) - \\mathbf { F } ( \\mathbf { 0 } ) \\big ) + \\big ( \\mathbf { H } ( t , \\cdot ) - \\mathbf { F } ( \\mathbf { 0 } ) \\big ) = \\mathbf { F } \\big ( \\nabla \\mathbf { u } ( t , \\cdot ) \\big ) - \\mathbf { F } ( \\mathbf { 0 } ) t > 0 . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} | | u | | _ { V ^ G ( \\Omega _ T ) } : = | | u | | _ { L ^ G ( \\Omega _ T ) } + | | D u | | _ { L ^ G ( \\Omega _ T ) } . \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} h ( \\mathbf { x } ) = \\sum \\limits ^ { n } _ { i = 1 } h _ { i } ( \\mathbf { x } _ { i } ) , \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} \\phi ( x ) = F ( x , \\bar x ) \\qquad \\hbox { f o r a l l $ x \\in K $ } . \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} f = ( f _ 1 , f _ 2 ) = \\left ( \\frac { \\varphi ^ 1 - \\varphi ^ 1 \\circ I } { g - g \\circ I } , \\frac { \\bar \\psi ^ 1 - \\bar \\psi ^ 1 \\circ I } { \\bar g - \\bar g \\circ I } \\right ) = \\left ( \\frac { \\varphi ^ 1 - \\bar \\psi ^ 2 } { g + \\frac { 1 } { \\bar g } } , \\frac { \\bar \\psi ^ 1 + \\varphi ^ 2 } { \\bar g + \\frac { 1 } { g } } \\right ) . \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} f ( y _ 0 , 1 , y _ 2 , y _ 3 ) = \\frac { 1 } { x _ 1 ^ 4 } f ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) . \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\psi \\circ I \\circ \\psi ^ { - 1 } ( z ) = \\psi ( - \\bar \\tau \\bar z + \\frac { \\tau } { 2 } ) = - \\frac { \\bar \\tau } { \\tau } \\bar z + \\frac { 1 } { 2 } = \\bar z + \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} v _ \\beta ^ 2 ( g _ 2 ) = P ^ * ( g _ 2 ) \\tilde { r } ( g _ 2 ) = \\left [ 6 - p , 6 - p \\right ] ^ T . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} \\phi ( u ) = \\int \\sqrt { 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) } d u , \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} z _ a = - z _ a ^ { t r } , z _ a z _ b + z _ b z _ a = - 2 \\delta _ { a b } I , 2 \\leq a , b \\leq 7 - r . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\kappa _ { 1 } ( u ) = f _ { 1 } { } ^ { \\prime } ( u ) f _ { 2 } { } ^ { \\prime \\prime } ( u ) - f _ { 1 } { } ^ { \\prime \\prime } ( u ) f _ { 2 } { } ^ { \\prime } ( u ) , \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{gather*} P _ { r } ^ { \\star } ( x ) \\ ! = \\ ! D _ { r - 1 } \\ ! + \\ ! \\sum \\limits _ { k = 0 } ^ { r - 1 } p _ { r , k } \\ , x ^ { r - k } \\ , \\ \\ P _ { r } ^ { \\star } ( 0 ) \\ ! = \\ ! D _ { r - 1 } \\neq 0 \\ , \\ \\ Q _ { r } ^ { \\star } ( x ) \\ ! = \\ ! \\sum _ { k = 0 } ^ { r - 1 } q _ { r , k } x ^ { r - 1 - k } \\ , \\end{gather*}"}
-{"id": "8291.png", "formula": "\\begin{align*} H ^ p ( Y , R ^ q f _ * ( K _ X \\otimes F \\otimes \\mathcal J ( h ) ) \\otimes H ^ { \\otimes m + 1 } ) = 0 \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} \\nabla \\mathfrak { L } ( \\boldsymbol { \\theta } ) = \\frac { 2 } { n } \\mathbf { X } ^ T ( \\mathbf { x } _ 1 ^ n - \\mathbf { X } \\boldsymbol { \\theta } ) , \\end{align*}"}
-{"id": "10049.png", "formula": "\\begin{align*} p + q + r = m + n + l \\iff \\left \\{ \\begin{aligned} m = \\gcd ( q , p + r ) & = q , \\\\ n = \\gcd ( p , q + r ) & = p , \\\\ l = \\gcd ( r , p + q ) & = r . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} q : = q _ { \\tau } = q _ { \\sigma } , \\hat { V } _ { q , \\theta } : = \\hat { V } ^ { \\tau } _ { q , \\theta } \\times \\hat { V } ^ { \\sigma } _ { q , \\theta } \\hbox { a n d } \\Delta _ { \\zeta } : = \\Delta _ { \\tau } \\oplus \\Delta _ { \\sigma } , \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} \\frac { \\left ( a q , q / a ; q \\right ) _ { \\infty } } { \\left ( q , q ; q \\right ) _ { \\infty } } = \\frac { 1 } { \\sqrt { 2 \\pi \\log q ^ { - 1 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } \\right ) d x } { \\left ( q , - q ^ { 1 / 2 } / ( a e ^ { i x } ) , - q ^ { 1 / 2 } ( a e ^ { i x } ) ; q \\right ) _ { \\infty } } , \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} r v _ { \\delta } ( F ( x , y ) ) = \\infty \\Leftrightarrow F ( x , y ) = 0 . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} ( x t ^ m | f t ^ n ) = \\delta _ { m + n , 0 } f ( x ) , \\ ( x t ^ m | y t ^ n ) = 0 = ( f t ^ m | g t ^ n ) \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} G ( \\vec { X } ) : & = \\dfrac { \\alpha } { 2 } \\| \\vec { Y } - \\vec { X } \\| _ F ^ 2 + \\lambda _ 0 \\sum _ { i = 1 } ^ { m } s \\bigl ( \\sigma _ i ( \\vec { X } ) ; a _ 0 \\bigr ) \\\\ & + \\dfrac { 1 - \\alpha } { 2 } \\| \\vec { Y } - \\vec { X } \\| _ F ^ 2 + \\lambda _ 1 \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } s ( \\vec { X } _ { i j } ; a _ 1 ) , \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{gather*} p _ { n _ c , n _ d , n _ e } = \\left \\langle { T _ { 1 } ^ { k } T _ { 2 } ^ { \\ell } v _ { 0 } , \\prod _ { i = 1 } ^ { n _ c } \\psi _ { 1 } ^ { + } ( x _ { i } ) \\psi _ { 0 } ^ { - } ( x _ { i } ) \\prod _ { i = 1 } ^ { n _ d } \\psi _ { 2 } ^ { + } ( y _ { i } ) \\psi _ { 0 } ^ { - } ( y _ { i } ) \\prod _ { i = 1 } ^ { n _ e } \\psi _ { 2 } ^ { + } ( z _ { i } ) \\psi _ { 1 } ^ { - } ( z _ { i } ) v _ { 0 } } \\right \\rangle , \\end{gather*}"}
-{"id": "327.png", "formula": "\\begin{align*} S = \\ln Z = - \\beta F \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} P _ + & = \\{ p \\in P \\mid \\phi ( p ) \\geq c \\} \\\\ P _ - & = \\{ p \\in P \\mid \\phi ( p ) \\leq c \\} . \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} m ( e ^ { r T } ) = m ( e ^ T ) ^ r \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} \\sum _ { d = 0 } ^ { s } d = \\frac { ( s - 1 ) s } { 2 } = \\binom { s } { 2 } . \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\frac { \\varphi _ k '' ( x ) } { \\varphi _ k ( x ) } = C _ { k , j } + \\frac { \\varphi _ j '' ( x ) } { \\varphi _ j ( x ) } , \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} \\dim ( ( 1 , U ^ { \\otimes 2 m } ) ) = C _ { m } = \\frac { 1 } { m + 1 } { 2 m \\choose m } ( m \\in \\N ) . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} N _ { t } = \\frac { S _ { N _ { t } } \\varphi } { \\mu \\left ( \\varphi \\right ) + R _ { N _ { t } } } \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} F _ - ( t ) u [ x ] & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i \\Phi ( x , y , \\xi ; t ) } ( { } ^ t L _ 2 ) ^ \\ell ( s _ - ( x , \\xi ) p _ + ( y , \\xi ) ) u [ y ] d \\xi \\\\ & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - \\varphi _ a ( y , \\xi ) ) } e ^ { - i t h _ 0 ( \\xi ) } ( { } ^ t L _ 2 ) ^ \\ell ( s _ - p _ + ) u [ y ] d \\xi \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} \\eta ( \\xi ; x , y ) : = \\int _ 0 ^ 1 \\nabla _ x \\varphi _ a ( y + \\theta ( x - y ) , \\xi ) d \\theta . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\big \\langle \\omega ^ \\mathrm { z m } _ u , \\omega ^ \\mu _ u \\big \\rangle _ Y = 0 , \\big \\langle \\omega ^ \\mu _ u , \\omega ^ \\nu _ u \\big \\rangle _ Y = 0 . \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} Z _ { 2 n } & = \\left \\{ \\lambda _ { 0 } , \\ldots , \\lambda _ { n - 1 } \\right \\} \\cup \\left \\{ \\lambda _ { n } , \\ldots , \\lambda _ { 2 n - 1 } \\right \\} , n \\in N \\\\ Z _ { 2 n } & = \\left \\{ \\lambda _ { i } \\right \\} \\cup \\left \\{ \\lambda _ { i + n } \\right \\} , \\quad 0 \\leq i \\leq n - 1 \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } u _ { \\delta } \\ , x _ { 1 } ^ { \\delta _ { 1 } } \\cdots \\ , x _ { k - 2 } ^ { \\delta _ { k - 2 } } \\cdot x _ { n } ^ { \\sum _ { k - 1 } ^ { n } \\delta _ { i } } = \\sum _ { \\substack { \\delta \\in \\{ 0 , 1 \\} ^ { n } , \\\\ \\delta _ { k - 1 } = \\cdots = \\delta _ { n - 1 } = 0 } } v _ { \\delta } \\ , x _ { n } ^ { \\delta _ { n } } \\prod _ { i = 1 } ^ { k - 2 } \\left ( x _ { i } - x _ { n } \\right ) ^ { \\delta _ { i } } \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\sigma = \\langle ( E - \\langle E \\rangle ) ^ 2 \\rangle = \\frac { \\partial ^ 2 } { \\partial \\beta ^ 2 } \\ln Z \\geq 0 \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} J _ { \\log F } ( z ) = | z | ^ { 4 ( p - 1 ) } J _ { \\log G } ( z ) + 2 ( p - 1 ) | \\log G ( z ) | ^ { 2 } | z | ^ { 2 ( 2 p - 3 ) } { \\rm R e } \\left \\{ \\mathcal { L } [ \\log ( \\log G ( z ) ) ] \\right \\} . \\end{align*}"}
-{"id": "10141.png", "formula": "\\begin{align*} x _ \\alpha ( r ) x _ \\alpha ( s ) & = x _ \\alpha ( r + s ) , & \\\\ [ x _ \\alpha ( r ) , x _ \\beta ( s ) ] & = x _ { \\alpha + \\beta } ( N _ { \\alpha , \\beta } \\cdot r s ) , & \\ \\alpha , \\beta \\in \\Phi , \\ \\alpha + \\beta \\in \\Phi , \\\\ [ x _ \\alpha ( r ) , x _ \\beta ( s ) ] & = 1 , & \\ \\alpha , \\beta \\in \\Phi , \\ \\alpha + \\beta \\not \\in \\Phi \\cup \\{ 0 \\} . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} D _ { 2 \\rightarrow 1 } ( t ) = \\int _ 0 ^ t R _ { 2 \\rightarrow 1 } ( s ) \\ , d s , \\quad R _ { 2 \\rightarrow 1 } ( t ) = \\lim _ { \\delta \\downarrow 0 } \\delta ^ { - 1 } T _ { 2 \\rightarrow 1 } ( t , t + \\delta ) . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( w ) & = & A w \\\\ \\sigma ( w ) & = & B w \\end{aligned} \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} | Y ( t , x - \\xi ) | \\leq C \\prod _ { i = 1 } ^ { d } t ^ { \\zeta _ d / d } | x _ i - \\xi _ i | ^ { \\kappa _ d / d } p ( t , x _ i - \\xi _ i ) \\ , , \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} x _ { i , k } ^ b & = x _ { i , ( k - 1 ) \\mathrm { m o d } \\ , p } x _ { i , k } ^ a = \\omega ^ { i k } x _ { i , k } x _ { i , k } ^ c = \\omega ^ { i } x _ { i , k } . \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} | \\{ u > \\frac { 1 - \\rho } { 2 } \\} \\cap C _ { \\rho } ( Z ) | \\leq \\mu _ { 1 } \\rho ^ { 2 n + 2 } | C _ { \\rho } ( Z ) | , \\ \\rho = 2 ^ { - j } . \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} \\begin{aligned} & C _ { i , s + 1 } ^ { 0 , + } f _ \\infty ^ { ( s + 1 ) } ( t , Z _ s ) = \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\left [ \\omega \\cdot ( v _ { s + 1 } - v _ i ) \\right ] _ { + } \\times \\\\ & \\ ; \\ ; \\times f _ \\infty ^ { ( s + 1 ) } ( t , x _ 1 , v _ 1 , \\dots , x _ i , v _ i ^ * , \\dots , x _ s , v _ s , x _ i , v _ { s + 1 } ^ * ) d \\omega d v _ { s + 1 } \\end{aligned} \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} \\zeta _ K ( s ) = \\frac { \\kappa _ K } { s - 1 } + \\kappa _ K \\gamma _ K + O _ K ( | s - 1 | ) . \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} { C } _ { X ^ n \\rightarrow Y ^ n } ^ { F B , A . J } ( \\kappa ) = \\sup _ { { \\cal P } _ { 0 , n } ^ { A . J } ( \\kappa ) } \\sum _ { t = 0 } ^ n { \\bf E } ^ { \\pi } \\left \\{ \\log \\Big ( \\frac { q _ t ( Y _ t | Y _ { t - M } ^ { t - 1 } , X _ t ) } { \\nu _ t ^ { { \\pi } } ( Y _ t | Y _ { t - J } ^ { t - 1 } ) } \\Big ) \\right \\} , ~ J = \\max \\{ M , N \\} \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{gather*} \\overline N _ { a } = \\psi ^ { + } ( z ^ { a } _ { 1 } ) \\psi ^ { + } ( z ^ { a } _ { 2 } ) \\psi ^ { - } ( z ^ { a } _ { 3 } ) . \\end{gather*}"}
-{"id": "3124.png", "formula": "\\begin{align*} T _ 1 = \\left ( \\begin{array} { c c c c c } 1 & & & & \\\\ t _ { 1 1 } & 1 & & & \\\\ \\vdots & \\ddots & \\multicolumn { 1 } { l } { \\ddots } & \\multicolumn { 1 } { l } { } & \\\\ t _ { d - 1 , 1 } & \\cdots & t _ { d - 1 , d - 1 } & 1 & \\\\ 0 & \\ddots & & & \\ddots \\end{array} \\right ) . \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} \\nu ( P , x ) \\ = \\ \\lim _ { n \\rightarrow \\infty } \\frac { \\sharp \\{ y \\in ( \\Lambda - x ) \\cap B _ n ( 0 ) \\mid P ( \\varrho , y ) = P \\} } { \\lambda ( B _ n ( 0 ) ) } \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} A e _ { n } = q ^ { - n + 1 } e _ { n - 1 } + q ^ { - n } e _ { n + 1 } , n \\in \\Z . \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} \\frac { \\beta } { \\alpha } \\ , \\kappa = \\frac { f f '' + ( f ' ) ^ 2 + 1 } { \\sqrt { f '^ 2 + 1 } } , \\alpha \\neq 0 , \\ ; \\beta \\neq 0 . \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} \\left ( \\Pi _ H \\tau ( \\nu , v ) \\right ) ( 1 ) = \\frac { { \\rm v o l } ( H _ v \\cap K _ v \\nu ( \\varpi _ v ) K _ v ) } { { \\rm v o l } ( K _ v \\nu ( \\varpi _ v ) K _ v ) ^ { 1 / 2 } } \\end{align*}"}
-{"id": "9986.png", "formula": "\\begin{align*} A = [ a , b ] \\setminus \\bigcup _ i ( a _ i , b _ i ) , \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} B = \\left \\{ X _ { \\mathbf { u } } \\mid \\mathbf { u } \\in [ - 1 , 1 ] ^ m \\right \\} \\bigcup \\left \\{ Y _ { \\mathbf { u } } \\mid \\mathbf { u } \\in [ - 1 , 1 ] ^ m \\right \\} . \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} A _ { n , 1 } ( x ) & = ( - 1 ) ^ n \\sum _ { j = 0 } ^ n \\binom { n } { j } \\frac { \\Gamma ( \\mu + \\nu + 1 + n ) } { \\Gamma ( \\mu + \\nu + 1 + j ) } \\left ( \\frac { a ^ 2 - b ^ 2 } { a ^ 2 } \\right ) ^ j r _ { j , \\mu } ( a ^ 2 x ) , \\\\ A _ { n , 2 } ( x ) & = ( - 1 ) ^ n a \\sum _ { j = 0 } ^ n \\binom { n } { j } \\frac { \\Gamma ( \\mu + \\nu + 1 + n ) } { \\Gamma ( \\mu + \\nu + 1 + j ) } \\left ( \\frac { a ^ 2 - b ^ 2 } { a ^ 2 } \\right ) ^ j s _ { j , \\mu } ( a ^ 2 x ) . \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} F _ n ( T _ x ( i ) \\otimes \\Gamma ( \\alpha ) ) = F _ n ( T _ x ( i ) ) \\otimes \\Gamma ( \\alpha ) = T _ x ( i ) \\otimes F _ n ( \\Gamma ( \\alpha ) ) \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} ( \\Delta f ) | \\tau ( \\phi ) | ^ 2 = 0 \\ ; \\ ; \\forall \\ ; f : ( V , V | g ) \\longrightarrow ( 0 , \\infty ) . \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} C _ 0 ( g K ) = c ( g ) \\qquad \\forall g \\in G \\ . \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} \\mathbb { P } ^ q = \\int \\limits _ { \\mathrm { C o n f } ( C ) } \\mathbb { P } _ { ( \\cdot | Y ; C ) } ^ q \\cdot \\frac { d \\xi \\mathbb { P } _ { ( \\cdot | Y ; C ) } } { d \\xi \\mathbb { P } } ( q ) \\cdot d \\overline { \\mathbb { P } _ C } ( Y ) . \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} u \\left ( x , t \\right ) = \\exp \\left ( \\int _ 0 ^ t r - R \\left ( x , t \\right ) \\ , d t \\right ) , \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} \\alpha ( f ) = \\sum _ { \\lambda \\vdash n } \\hat { \\alpha } _ \\lambda X ^ \\lambda ( f ) + b ( f ) , \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} \\widetilde { F } ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) & = a _ { 1 } x _ { 1 } + a _ { 2 } ( x _ { 2 } - x _ { 3 } ) + a _ { 4 } ( \\gamma _ { 3 } x _ { 3 } + x _ { 1 } ( x _ { 2 } - x _ { 3 } ) ) . \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\sigma ( L ) ( d f , d \\phi ) = \\frac { 1 } { 2 } \\left ( L ( f \\phi ) - f L \\phi - \\phi L f \\right ) , f , \\phi \\in C ^ \\infty ( M ) . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} x \\theta _ { q ^ { 4 } } \\left ( q ^ { 2 } x ^ { 2 } \\right ) + t \\theta _ { q ^ { 4 } } \\left ( x ^ { 2 } \\right ) = 0 . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\hat { f } ( s ) : = \\begin{cases} \\ , f ( s ) , & s \\in \\mathopen { [ } 0 , \\rho \\mathclose { ] } ; \\\\ \\ , f ( \\rho ) + f ' ( \\rho ) ( s - \\rho ) , & s \\in \\mathopen { ] } \\rho , + \\infty \\mathclose { [ } . \\end{cases} \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} K ^ S _ { U V } [ X , Y ] = \\left ( K ^ S _ U [ X , Z ] , K ^ S _ V [ Z , Y ] \\right ) ^ S _ Z . \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} - L w + g \\circ w & = \\alpha _ k \\mu _ n \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ w & = \\alpha _ k \\nu _ n \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\displaystyle \\frac { \\partial } { \\partial t } g _ { i j } ( x , t ) = - R ( x , t ) g _ { i j } ( x , t ) , \\ \\ ( x , t ) \\in M \\times [ 0 , T ] \\\\ \\ \\\\ g _ { i j } ( x , 0 ) = g _ 0 ( x ) \\end{array} \\right . \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} \\| u - u _ h \\| _ { N ^ { s t + 1 - t } _ 2 ( \\mathbb { R } ^ d ) } & = \\| u - u _ h \\| _ { ( H ^ 1 ( \\mathbb { R } ^ d ) , N ^ s _ 2 ( \\mathbb { R } ^ d ) ) _ { t , \\infty } } \\leq \\\\ C _ { t , s } | h | ^ { ( 1 - s ) t } | h | ^ { 1 - t } \\| u \\| _ { ( H ^ 2 ( \\mathbb { R } ^ d ) , H ^ 1 ( \\mathbb { R } ^ d ) ) _ { t , \\infty } } & = C _ { t , s } | h | ^ { 1 - s t } \\| u \\| _ { N ^ { 2 - t } _ 2 ( \\Omega ) } , \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} \\begin{array} { l l } & \\langle { \\psi + \\chi } , L ( \\Lambda , \\psi + \\chi , g ) ( { \\psi + \\chi } ) \\rangle \\\\ & = \\langle { \\psi } , L ( \\Lambda , \\psi , g ) { \\psi } \\rangle + \\langle { \\chi } , L ( \\Lambda , \\chi , g ) { \\chi } \\rangle + S ^ { i n t } _ \\Lambda [ \\psi , \\chi ] \\end{array} \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} \\langle \\kappa ( v ) \\alpha , \\alpha \\rangle _ { g _ H ^ * } = 0 , v \\in T M , \\alpha \\in T ^ * M . \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 4 } \\left ( \\sqrt { a } , - \\sqrt { a } ; i \\sqrt { b } q ^ { 3 / 4 } , - i \\sqrt { b } q ^ { 3 / 4 } , \\sqrt { a z } , - \\sqrt { a z } ; \\sqrt { q } , - b z q \\right ) = \\frac { \\left ( z ; q \\right ) _ { \\infty } } { \\left ( a z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a ; q \\right ) _ { n } S _ { n } \\left ( b ; q \\right ) z ^ { n } } { \\left ( - b q ^ { 3 / 2 } ; q \\right ) _ { n } } . \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} C _ { s + 1 } = \\sum _ { i = 1 } ^ s C _ { i , s + 1 } \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} S _ 1 ( \\textbf { q } ) = e \\int ^ { q _ 1 } A _ 1 ( \\hat { q } _ 1 , q _ 2 , q _ 3 ) d \\hat { q } _ 1 . \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} h ^ { \\beta } _ { \\mu , N } ( \\lambda ) = B _ N ^ { \\mu } \\big ( e ^ { i \\lambda } \\big ) \\frac { ( 1 - e ^ { - i \\lambda \\mu } ) ^ n } { ( i \\lambda ) ^ n } - A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\frac { p ( \\lambda ) - \\beta ^ 2 f ( \\lambda ) } { ( i \\lambda ) ^ n p ( \\lambda ) } - \\frac { ( - i \\lambda ) ^ { n } C _ { \\mu , N } ^ { \\beta } ( e ^ { i \\lambda } ) } { ( 1 - e ^ { i \\lambda \\mu } ) ^ n p ( \\lambda ) } , \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} v ( x , t ) = \\sum _ { m = 0 } ^ 2 v _ m ( x , t ) , \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} T = \\sum _ { j = 0 } ^ { \\lfloor r / 2 \\rfloor } Q _ L ^ j \\pi _ { L ^ \\perp } ^ * T ^ { ( r - 2 j ) } \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} \\begin{array} { l } h \\gamma h ^ { - 1 } e C \\subset e ( e C ) = e C \\ , . \\end{array} \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} a _ { 0 , 1 } = \\frac { 1 - \\mu _ R } { N _ T } , a _ { N _ R , 0 } = \\mu _ R , \\textrm { a n d o t h e r s b e i n g 0 . } \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} g ( t ) = \\alpha { \\bf 1 } _ { [ 0 , 1 - p ] } ( t ) + \\beta { \\bf 1 } _ { ( 1 - p , p ) } ( t ) + ( \\alpha + \\delta ) { \\bf 1 } _ { [ p , 1 ] } ( t ) , t \\in [ 0 , 1 ] , \\alpha , \\beta , \\delta \\in \\R . \\end{align*}"}
-{"id": "9885.png", "formula": "\\begin{align*} W \\left [ w _ \\sigma , \\ , w _ { \\rm r e g } \\right ] ( 0 ) = - \\ , \\delta _ { \\sigma , 0 } \\ , \\left ( \\sum _ { j = 0 } ^ \\infty j \\ , b _ j \\right ) - \\delta _ { \\sigma , 1 } \\ , \\left ( \\sum _ { j = 0 } ^ \\infty b _ j \\right ) \\ , . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} f ( y ) = \\sum _ { j = 0 } ^ { i - 1 } \\omega ^ { \\beta _ j } \\cdot p _ j + 1 + f _ i ( y ) \\ge \\sum _ { j = 0 } ^ { m } \\omega ^ { \\beta _ j } \\cdot p _ j > \\mu . \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} \\psi _ { a , b } ( t ) = | a | ^ { - \\frac { 1 } { 2 } } \\psi \\bigg ( \\frac { t - b } { a } \\bigg ) , a , b \\in \\mathbb { R } , a \\neq 0 , \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} V _ { \\rm N W } ( x ) = - 3 2 \\frac { \\sin | x | \\left ( g ( | x | ) ^ 3 \\cos | x | - 3 g ( | x | ) ^ 2 \\sin ^ 3 | x | + g ( | x | ) \\cos | x | + \\sin ^ 3 | x | \\right ) } { ( 1 + g ( | x | ) ^ 2 ) ^ 2 } , \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} C & = \\sup _ { F _ X ( x ) } I ( X ; Y ) \\\\ & = \\sup _ { F _ X ( x ) } \\int { D \\left ( f _ { Y | X } ( . | x ) | | f _ Y ( . ) \\right ) d F _ X ( x ) } \\\\ & \\leq \\sup _ { F _ X ( x ) } \\int { D \\left ( f _ { Y | X } ( . | x ) | | q _ Y ( . ) \\right ) d F _ X ( x ) } \\\\ & \\leq \\sup _ { x \\in \\mathbb { S } } D \\left ( f _ { Y | X } ( . | x ) | | q _ Y ( . ) \\right ) \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} \\mathcal { L } _ { _ { 5 } , \\mathfrak { C } _ { 5 } } = \\alpha _ { 0 } \\ , \\varepsilon _ { a b c d e } \\left ( R ^ { a b } e ^ { c } e ^ { d } e ^ { e } + \\frac { 3 } { 1 0 \\ell ^ { 2 } } e ^ { a } e ^ { b } e ^ { c } e ^ { d } e ^ { e } + \\frac { 3 } { 4 } \\ell ^ { 2 } R ^ { a b } R ^ { c d } h ^ { e } + \\frac { 1 } { \\ell ^ { 2 } } e ^ { a } h ^ { b } e ^ { c } h ^ { d } h ^ { e } + \\frac { 3 } { 2 } R ^ { a b } e ^ { c } h ^ { d } h ^ { e } \\right ) , \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} - \\int _ { \\Omega } u _ n L ^ * \\varphi + \\int _ { \\Omega } ( g _ n \\circ u _ n ) v = \\int _ { \\Omega } f \\varphi \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} P ^ { \\perp } ( X _ * ) = ( I - U _ 1 U ^ { \\top } _ 1 ) X _ * ( I - V _ 0 V _ 0 ^ { \\top } ) . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} h ( x ) = \\min \\bigl ( 1 , L \\sphericalangle ( v _ { 0 } , \\dot \\gamma ^ { o , x } _ { 0 } ) \\bigr ) , \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} \\rho ^ \\prime \\left ( \\frac { x } { x - 1 } \\right ) = - ( 1 - x ) ^ 2 \\rho ^ \\prime ( x ) . \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} { \\rm q d i m } [ M ^ \\varepsilon _ { r , s } ] = s , \\ \\ { \\rm q d i m } [ F ^ \\varepsilon _ \\lambda ] = e ^ { \\frac { 2 \\pi i k \\lambda } { \\sqrt { 2 p } } } p ; \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} w _ i ( \\cdot , 0 ) = \\dot { w } _ i ( \\cdot , 0 ) = 0 , \\chi ( \\cdot , 0 ) = \\dot { \\chi } ( \\cdot , 0 ) = 0 , p _ i ( \\cdot , 0 ) = 0 \\Omega . \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} W ( P ) & = \\frac { 1 } { 2 } \\left ( \\log \\sigma ^ { 2 } ( P ) + \\frac { 1 } { \\sigma ^ { 2 } ( P ) } - 1 \\right ) \\left ( \\frac { 1 } { 2 } + Q ( 2 \\sqrt { P } ) \\right ) \\\\ & \\ \\ \\ + \\frac { g ( 2 \\sqrt { P } ) } { 2 \\sigma ^ { 2 } ( P ) } \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} \\mathbf { F } \\cdot \\mathbf { G } & = \\left ( \\mathbf { E } + i \\mathbf { H } \\right ) \\cdot \\left ( \\mathbf { D } + i \\mathbf { B } \\right ) \\\\ & = \\left ( \\mathbf { E } \\cdot \\mathbf { D } - \\mathbf { H } \\cdot \\mathbf { B } \\right ) + i \\left ( \\mathbf { E } \\cdot \\mathbf { B } + \\mathbf { H } \\cdot \\mathbf { D } \\right ) . \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} r ^ \\flat ( X ) = r ( X ) + \\sum _ { x \\in X } ( \\lambda ( \\{ x \\} ) - r ( \\{ x \\} ) ) \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} H \\coloneqq \\limsup _ { r \\to \\infty } \\Big \\{ \\int _ { r } ^ \\infty & \\frac { d s } { f _ a ^ { n - 1 } ( s ) } \\int _ 0 ^ { r } a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t \\\\ & - \\int _ 0 ^ { r } \\int _ t ^ \\infty \\frac { d s } { f _ a ^ { n - 1 } ( s ) } a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t \\Big \\} \\le 0 ; \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{gather*} \\mathfrak { M } ( q \\ , | \\tau , 0 , 0 ) = \\frac { ( 2 \\pi ) ^ q } { \\Gamma ^ q \\bigl ( 1 - \\frac { 1 } { \\tau } \\bigr ) } \\Gamma \\bigl ( 1 - \\frac { q } { \\tau } \\bigr ) , \\\\ M _ { ( \\tau , 0 , 0 ) } = \\frac { 2 \\pi \\tau ^ { 1 / \\tau } } { \\Gamma ( 1 - 1 / \\tau ) } \\beta _ { 1 , 0 } ^ { - 1 } ( \\tau , b _ 0 = \\tau ) . \\end{gather*}"}
-{"id": "509.png", "formula": "\\begin{align*} V _ i = \\underbrace { V _ { \\omega ^ i } \\oplus \\ldots \\oplus V _ { \\omega ^ i } } _ { n _ i } . \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} M ( x , \\beta ) = \\lambda ( \\beta , x ) f ( x ) f ( x ) ^ T . \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} \\tilde A ( x ) = \\left ( \\begin{matrix} A _ { 1 1 } ( x ) & 0 \\\\ A _ { 2 1 } ( x ) & A _ { 2 2 } ( x ) \\end{matrix} \\right ) , \\ \\ \\tilde B ( x ) = \\left ( \\begin{matrix} B _ { 1 1 } ( x ) & 0 \\\\ B _ { 2 1 } ( x ) & B _ { 2 2 } ( x ) \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} A _ { q } \\left ( z \\right ) = \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { k ^ { 2 } } } { \\left ( q ; q \\right ) _ { k } } \\left ( - z \\right ) ^ { k } \\left ( w ; q \\right ) _ { k } A _ { q } \\left ( w z q ^ { 2 k } \\right ) . \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} a ' _ 1 & = \\sum _ { k = 3 } ^ g \\tbinom { k - 1 } { 2 } \\tfrac { k ! ( k - 1 ) ! } { 1 2 } ( - 1 ) ^ { g - k } ( g - k ) ! \\tfrac { g ! } { k ! } \\tbinom { g - 1 } { k - 1 } \\tbinom { g } { k } \\\\ & = \\tfrac { g ! ( g - 1 ) ! } { 1 2 } ( - 1 ) ^ { g - 1 } \\sum _ { k = 3 } ^ { g } ( - 1 ) ^ { k - 1 } \\tbinom { k - 1 } { 2 } \\tbinom { g } { k } = \\tfrac { g ! ( g - 1 ) ! } { 1 2 } ( - 1 ) ^ { g - 1 } . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{gather*} X ^ { ( \\alpha ) } ( z ) = ( - 1 ) ^ \\alpha z ^ { \\alpha } X ( z ) = \\sum _ { i \\in \\mathbb { Z } } ( - 1 ) ^ \\alpha x _ { i + \\alpha } z ^ { - i - 1 } , X = C , D , E , x = c , d , e . \\end{gather*}"}
-{"id": "9805.png", "formula": "\\begin{align*} | R | = | ^ { 2 } G _ { 2 } ( q ) | & = q ^ { 3 } ( q ^ { 3 } + 1 ) ( q - 1 ) \\\\ & = | R _ { 2 } | \\cdot | R _ { 3 } | \\cdot | H _ { 1 } | \\cdot | H _ { 2 } | \\cdot | H _ { 3 } | \\cdot | H _ { 4 } | , \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\tau : = t _ { i j } e ^ { i j } = \\dfrac 1 2 ( t _ { i j } - t _ { j i } ) e ^ { i j } = e ^ i \\wedge T ( e _ i ) ^ \\flat \\in \\Lambda ^ 2 V _ 7 ^ * . \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{gather*} a ( z ) b ( z ) = \\sum _ { k \\in \\mathbb { Z } } \\left ( \\sum _ { i \\in \\mathbb { Z } } a _ { i } b _ { k - i - 1 } \\right ) z ^ { - k - 1 } \\end{gather*}"}
-{"id": "3955.png", "formula": "\\begin{align*} W ( f , g ) = - z ^ { - 1 } \\theta _ { q } \\left ( \\alpha z \\right ) \\ ! . \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} \\pi ( v _ { i _ 1 } \\dots v _ { i _ n } ) & = w _ { i _ 1 } \\pi ( v _ { i _ 2 } \\dots v _ { i _ n } ) + v _ { i _ 1 } \\rightharpoonup \\pi ( v _ { i _ 2 } \\dots v _ { i _ n } ) . \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} C r ( x , y ) : = ( x \\cdot y ) _ { I m \\O } \\mbox { a n d } \\chi ( x , y , z ) : = ( ( x \\cdot y ) \\cdot z - x \\cdot ( y \\cdot z ) ) _ { I m \\O } . \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} Z _ 1 ( s ) & = N _ { c , a , b } ( s ) , & Z _ 2 ( r , s ) & = \\binom { a - s } { a - r + 1 } . \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} & K ( x , y , \\eta ) = | y _ 1 - x _ 1 | + | y _ 2 - x _ 2 + x _ 1 \\eta _ 1 ( x _ 1 - y _ 1 ) - 2 \\ , x _ 1 \\eta _ 2 | ^ { 1 / 3 } + | \\eta _ 1 | + | \\eta _ 2 - \\tfrac { 1 } { 2 } x _ 1 \\eta _ 1 | ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } & \\alpha _ t ( t , x ) + \\frac 1 2 \\Delta \\alpha ( t , x ) + { \\nabla u ( t , x ) b ( t , x ) } + f ( t , \\alpha ( t , x ) , \\nabla \\alpha ( t , x ) ) = 0 , \\\\ & \\alpha ( T , x ) = \\Phi ( x ) , \\\\ & \\forall ( t , x ) \\in [ 0 , T ] \\times \\mathbb R ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} \\mathbf { p } _ { \\mathcal { B } } ^ { - 1 } ( [ F ] , C , [ L ] ) = \\mathbb { P } ( \\mathrm { H o m } ( F , L ( 2 ) ) ) . \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} { } [ \\chi , \\chi ] ^ { F N } _ p = \\gamma _ j \\otimes e _ j , \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} \\langle H , H \\rangle = \\frac { \\left ( f f '' + ( f ' ) ^ 2 + 1 \\right ) ^ 2 - \\kappa ^ 2 ( f '^ 2 + 1 ) } { 4 f ^ 2 ( f '^ 2 + 1 ) } . \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} A ( \\gamma ) \\Psi _ { \\gamma + t } = \\Psi _ { \\gamma + t } - e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) . \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} a ^ p = b ^ p = c ^ p = 1 [ a , b ] = c [ a , c ] = [ b , c ] = 1 \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} \\int A ( - \\partial _ v \\partial _ v ^ { - 2 } f _ 0 \\partial _ z \\ne { f } ) A f _ 0 \\ , d V = 0 . \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} T ( a \\cdot b ) = ( T a ) \\cdot b + a \\cdot ( T b ) . \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} P ^ 1 ( f ^ * ) = \\frac { Q ( f ^ * , g ^ * ) } { | | \\mu | | } + I = \\begin{matrix} \\begin{pmatrix} p _ 1 & p _ 2 & \\cdots & p _ { | S | } \\\\ p _ 1 & p _ 2 & \\cdots & p _ { | S | } \\\\ \\vdots & \\vdots & & \\vdots \\\\ p _ 1 & p _ 2 & \\cdots & p _ { | S | } \\end{pmatrix} \\end{matrix} . \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{gather*} D _ { 0 } = D _ { 1 } = \\ \\ldots \\ = D _ { n - 1 } = 0 \\ , \\ D _ { n } \\neq 0 \\ . \\end{gather*}"}
-{"id": "8980.png", "formula": "\\begin{align*} q ( t , s ; x , \\xi ) & = x + \\int _ s ^ t v ( p ( \\tau , s ) ) d \\tau \\\\ & = x + \\int _ s ^ t v \\left ( p ( t , s ) + \\int _ \\tau ^ t \\nabla _ x V _ \\rho ( \\sigma , q ( \\sigma , s ) ) d \\sigma \\right ) d \\tau . \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} S ( \\xi , \\eta ; u ) : = & \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\ln \\left [ \\binom { k - 1 } { l } \\binom { L } { l } t ^ { L - 2 l } ( t ^ 2 - 2 \\Delta t + 1 ) ^ { l } H _ N ^ { ( k ) } \\right ] \\\\ = & \\ell ( \\xi ) - \\ell ( \\eta ) - \\ell ( \\xi - \\eta ) + \\ell ( u ) - \\ell ( \\eta ) - \\ell ( u - \\eta ) - 2 \\eta \\ln t \\\\ & \\qquad \\qquad + \\eta \\ln ( t ^ 2 - 2 \\Delta t + 1 ) + \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\ln \\left [ H _ N ^ { ( \\xi N ) } \\right ] . \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} ( u '' \\sqrt { h } ( - h ^ { - 1 } ( v ' - u ' ( v '' / u '' ) ) + ( R / u '' ) ( v ' f _ u - u ' f _ v ) ) ) | _ { t = t _ j } \\ ; . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} \\Sigma ( Q _ u ^ H Q _ v + Q _ v ^ H Q _ u ) \\Sigma ^ T = 0 . \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} 4 j ^ 2 + 2 j - 1 & = 2 + 4 + 6 + \\dots + ( 4 j - 2 ) + ( 4 j - 1 ) , \\\\ 4 j ^ 2 + 2 j & = 2 + 4 + \\dots + 4 j . \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} \\lambda _ { Z } = \\sum _ { j = 1 } ^ { \\infty } \\left \\Vert \\tilde { k } _ { z } \\right \\Vert _ { H ^ { 2 } } ^ { - 2 } \\delta _ { z _ { j } } . \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} & v _ { 2 n - 1 } v _ { 2 n - 1 } ^ * + v _ { 2 n } v _ { 2 n } ^ * \\\\ = & v _ { 2 n - 1 } v _ { 2 n - 1 } ^ * + ( P _ C \\otimes f _ n - v _ { 2 n - 1 } v _ { 2 n - 1 } ^ * ) ( P _ C \\otimes f _ { n } ) ( P _ C \\otimes f _ n - v _ { 2 n - 1 } v _ { 2 n - 1 } ^ * ) \\\\ = & v _ { 2 n - 1 } v _ { 2 n - 1 } ^ * + P _ C \\otimes f _ n - v _ { 2 n - 1 } v _ { 2 n - 1 } ^ * \\\\ = & P _ C \\otimes f _ n . \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} P _ { n + 1 } ^ { \\left ( r \\right ) } ( x ) = \\left ( x - \\beta _ { r } \\right ) P _ { n } ^ { \\left ( r + 1 \\right ) } ( x ) - \\sum \\nolimits _ { i = 1 } ^ { d } \\gamma _ { r + 1 } ^ { d - i } P _ { n - i } ^ { \\left ( r + 1 + i \\right ) } ( x ) , \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} ( \\overline \\nabla _ X J _ \\alpha ) N = - A _ { J _ \\alpha N } X + J _ \\alpha ( A _ N X ) + D _ X J _ \\alpha N - J _ \\alpha ( D _ X N ) , \\alpha = 1 , 2 , 3 . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} F ( z , u , v ) = \\frac { 1 } { u } \\log \\left ( \\frac { 1 } { 1 - v + v ( 1 - z ( 1 + u ) ) ^ { \\frac { u } { 1 + u } } } \\right ) . \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} D _ m : = \\sup _ { s \\geq s _ 0 + m } \\frac { s ^ 2 } { G ( s ) } \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} F ( s ) & = \\left ( \\sup _ { r > 0 } \\left ( s ^ q r - G ( r ^ { 1 / q } ) \\right ) \\right ) ^ { 1 / q } = \\sup _ { r > 0 } \\big ( ( s r ) ^ q - G ( r ) \\big ) ^ { 1 / q } \\\\ & = \\sup _ { \\substack { r > 0 \\\\ ( s r ) ^ q \\geq G ( r ) } } \\big ( ( s r ) ^ q - G ( r ) \\big ) ^ { 1 / q } \\geq \\sup _ { \\substack { r > 0 \\\\ ( s r ) ^ q \\geq G ( r ) } } \\big ( s r - G ( r ) ^ { 1 / q } \\big ) \\\\ & = \\sup _ { r > 0 } \\big ( s r - G ( r ) ^ { 1 / q } \\big ) , \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} U = \\{ x \\in X \\mid \\psi ( x ) < u \\} . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} \\mathbb { V } : = \\{ \\mathbf { h } \\in \\mathbb { R } ^ p | \\| \\mathbf { h } _ { S ^ c } \\| _ 1 \\leq 3 \\| \\mathbf { h } _ S \\| _ 1 + 4 \\| \\boldsymbol { \\theta } _ { S ^ c } \\| _ 1 \\} . \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} d \\omega = \\theta \\wedge \\omega , \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} j ( \\pi ) ( \\langle j ` ` \\kappa ^ { + 4 } , u ^ * _ i , H ^ * _ i \\mid i < \\bar { \\lambda } \\rangle ) = \\langle \\kappa , u _ i , \\mathcal { F } _ i \\mid i < \\bar { \\lambda } \\rangle . \\end{align*}"}
-{"id": "10051.png", "formula": "\\begin{align*} ( \\alpha + 1 ) q & = ( \\beta + 1 ) p ; \\\\ ( \\beta + 1 ) p & = ( \\gamma + 1 ) r ; \\\\ ( \\gamma \\beta - 1 ) p & = ( \\gamma + 1 ) q , \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} \\left [ Y , W \\right ] _ s & = \\left [ \\alpha ( \\cdot , X ^ { t , x } ) , W \\right ] _ s = \\int _ 0 ^ s \\nabla \\alpha ( r , X _ r ^ { t , x } ) \\mathrm d r . \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} p _ n [ S ] = p _ m [ p _ k [ S ] ] = p _ m [ Q S ] = p _ m [ Q ] p _ m [ S ] . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} p _ { k + 1 } ( y '' , y _ n , y _ { n + 1 } ) = p _ k ( y '' , y _ { n } , y _ { n + 1 } ) + r _ 0 ^ { ( 3 + { 2 \\alpha } ) k } q \\left ( \\frac { y '' } { r _ 0 ^ { 2 k } } , \\frac { y _ n } { r _ 0 ^ k } , \\frac { y _ { n + 1 } } { r _ 0 ^ k } \\right ) , \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} L & = \\frac { 1 } { 4 } F _ { \\sigma \\tau } R ^ { \\tau \\sigma } - \\frac { 4 \\pi } { c } j ^ { \\sigma } A _ { \\sigma } \\\\ & = \\frac { 1 } { 4 } F _ { \\sigma \\tau } \\epsilon ^ { \\tau \\sigma \\lambda \\rho } F _ { \\lambda \\rho } - \\frac { 4 \\pi } { c } j ^ { \\sigma } A _ { \\sigma } , \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} g \\left ( - 1 + a \\right ) = g \\left ( - \\left ( 1 - a \\right ) \\right ) = - g \\left ( 1 - a \\right ) = - g \\left ( 1 \\right ) - g \\left ( a \\right ) = g \\left ( - 1 \\right ) + g \\left ( a \\right ) , \\end{align*}"}
-{"id": "10060.png", "formula": "\\begin{align*} q + r = m + n + l + k \\leq r + \\dfrac { q } { 2 } , \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} \\int _ M | \\Delta \\nabla u | _ g ^ 2 d \\mu _ g = \\int _ M | \\nabla ^ 3 u | _ g ^ 2 d \\mu _ g + \\int _ M O \\big ( | { \\rm R m } | | \\nabla ^ 2 u | _ g + | \\nabla { \\rm R m } | | \\nabla u | _ g \\big ) | \\nabla ^ 2 u | _ g d \\mu _ g . \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} h = \\eta W \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} x \\otimes y = \\begin{cases} a _ i + ( b _ i - a _ i ) \\cdot \\left ( \\left ( \\frac { x - a _ i } { b _ i - a _ i } \\right ) \\otimes _ i \\left ( \\frac { y - a _ i } { b _ i - a _ i } \\right ) \\right ) & \\\\ x \\wedge y & \\end{cases} \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} & W ' ( f ( x ) , g ( x ) ) = f _ { n + 1 } ' ( x ) g _ { n } ( x ) + f _ { n + 1 } ( x ) g _ { n } ' ( x ) - f _ { n } ' ( x ) g _ { n + 1 } ( x ) - f _ { n } ( x ) g _ { n + 1 } ' ( x ) \\\\ & = A W _ { n } ( f ' ( x ) , f ( x ) ) - A ^ { - 1 } W _ { n } ( g ' ( x ) , g ( x ) ) = \\frac { 1 - x ^ { 2 } } { x ^ { 2 } } \\left ( A \\sum _ { j = n + 1 } ^ { \\infty } f _ { j } ^ { 2 } ( x ) + A ^ { - 1 } \\sum _ { j = - \\infty } ^ { n } g _ { j } ^ { 2 } ( x ) \\right ) \\ ! \\ ! . \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{gather*} \\big ( \\tau _ { k } ^ { ( \\alpha ) } \\big ) ^ 2 \\big ( \\tau _ { k + 2 } ^ { ( \\alpha - 1 ) } \\tau _ { k } ^ { ( \\alpha + 1 ) } - \\tau _ { k + 1 } ^ { ( \\alpha - 1 ) } \\tau _ { k + 1 } ^ { ( \\alpha + 1 ) } \\big ) = \\big ( \\tau _ { k + 1 } ^ { ( \\alpha ) } \\big ) ^ 2 \\big ( \\tau _ { k + 1 } ^ { ( \\alpha - 1 ) } \\tau _ { k - 1 } ^ { ( \\alpha + 1 ) } - \\tau _ { k } ^ { ( \\alpha - 1 ) } \\tau _ { k } ^ { ( \\alpha + 1 ) } \\big ) . \\end{gather*}"}
-{"id": "8266.png", "formula": "\\begin{align*} ( \\overline \\nabla _ X J _ \\alpha ) N = - \\frac { 1 } { 4 n } \\left [ \\overline \\theta _ \\alpha ( N ) X + \\overline \\theta _ \\alpha ( J _ \\alpha N ) J _ \\alpha X \\right ] , \\alpha = 2 , 3 . \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} 1 \\le \\delta _ n ( x ) \\le \\delta _ { n - 1 } ( x ) \\le \\dots \\le \\delta _ 1 ( x ) \\le \\delta _ 0 ( x ) = x . \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} \\tilde { P } _ t ( x , A ) = \\left \\{ \\begin{array} { l l } \\displaystyle \\tilde { P } _ t ( x , A \\cap \\R ^ d _ + ) + 1 _ A ( \\partial ) ( 1 - \\tilde { P } _ t ( x , \\R ^ d _ + ) ) , & x \\in \\R ^ d _ + , \\\\ \\displaystyle 1 _ A ( \\partial ) , & x = \\partial , \\end{array} \\right . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} H _ f ( x , r ) : = \\frac { L _ f ( x , r ) } { l _ f ( x , r ) } , \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} \\texttt { \\rm d e g } _ { S V } \\left ( \\widetilde { T _ \\lambda } - p _ 0 , B _ { \\R ^ { 2 n } } ( \\widetilde \\eta , R _ 0 ) , 0 \\right ) = 1 , \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} \\lambda B - A = \\left [ \\begin{array} { c c c c } ( \\lambda - \\rho _ 1 ) I - J _ { s _ 1 } & 0 & \\cdots & 0 \\\\ 0 & ( \\lambda - \\rho _ 2 ) I - J _ { s _ 2 } & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & ( \\lambda - \\rho _ t ) I - J _ { s _ t } \\end{array} \\right ] , \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{gather*} E _ { a b } ( z _ { 1 } ) = \\colon \\psi _ { a } ^ { + } ( z _ { 1 } ) \\psi _ { b } ^ { - } ( z _ { 1 } ) \\colon . \\end{gather*}"}
-{"id": "657.png", "formula": "\\begin{align*} P _ { \\mu \\lambda } ^ { \\ast } \\left ( \\frac { \\partial Q ^ { \\lambda \\nu } } { \\partial x ^ { \\nu } } = - \\frac { 4 \\pi } { c } j ^ { \\lambda } \\right ) \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} \\| \\tilde { G } \\| _ { C ^ { 0 , \\gamma } ( \\mathcal { N } _ { 0 } \\cap ( B _ { 1 } \\setminus B _ { 1 / 4 } ) ) } \\leq C \\lambda ^ { - \\frac { 3 } { 2 } - \\alpha } \\lambda ^ { 2 - \\frac { n + 1 } { p } } = C \\lambda ^ { \\frac { 1 } { 2 } - \\alpha - \\frac { n + 1 } { p } } . \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} \\frac { \\left ( q ^ { \\nu + 1 } ; q \\right ) _ { \\infty } } { \\left ( q ^ { \\nu + 1 } ; q \\right ) _ { n } } = \\sqrt { \\log q ^ { - 1 } / \\left ( 2 \\pi \\right ) } \\int _ { - \\infty } ^ { \\infty } q ^ { \\left ( n - \\alpha \\right ) ^ { 2 } / 2 } A _ { q } ( q ^ { \\alpha + \\nu + 1 / 2 } ) d \\alpha \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\{ j _ 1 , \\ldots , j _ p \\} : = \\{ 1 , \\ldots , r \\} \\setminus \\{ i _ 1 , \\ldots , i _ n \\} ( p = N - n ) , \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} \\pi _ { S | Q = g ( q , y ) } & = B ( \\pi _ { S | Q = q } , y ) , \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} \\Phi _ { \\nu } ( z ) = \\prod \\limits _ { n \\geq 1 } \\left ( 1 + \\frac { z } { b _ { \\nu , n } } \\right ) , \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} p ^ * ( v , v ) = - \\sqrt { 2 } ( x z + z y ) , \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} h = \\ell / N _ h , p = \\max \\limits _ { 0 \\leq j \\leq N _ h } p _ j , s = \\sinh ( p h ) , c = \\cosh ( p h ) , \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} \\left | K _ { N } ^ { 1 } ( z , \\zeta ) \\right | \\lesssim \\frac { \\rho ( \\zeta ) ^ { N - 1 } \\left ( \\left | \\rho ( \\zeta ) \\right | + \\varepsilon \\right ) \\varepsilon ^ { n - 1 } } { \\prod _ { i = 1 } ^ { n - 1 } \\tau _ { i } ( z , \\varepsilon ) \\left | \\frac { 1 } { K _ { 0 } } S ( z , \\zeta ) + \\rho ( \\zeta ) \\right | ^ { N + n - 1 } } \\frac { 1 } { \\left | z - \\zeta \\right | } . \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} F ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) & = ( x _ { 1 } + \\alpha x _ { 3 } ( x _ { 1 } - x _ { 2 } ) , x _ { 2 } + \\alpha x _ { 3 } ( x _ { 1 } - x _ { 2 } ) , x _ { 3 } ) , \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} \\psi ( b _ 1 b _ 2 b _ 1 ^ * b _ 2 ^ * ) = | \\psi ( b _ 1 ) | ^ 2 \\psi ( b _ 2 b _ 2 ^ * ) + | \\psi ( b _ 2 ) | ^ 2 \\psi ( b _ 1 b _ 1 ^ * ) - | \\psi ( b _ 1 ) | ^ 2 | \\psi ( b _ 2 ) | ^ 2 . \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\Delta _ { \\mathsf { P } } ( \\mu , C _ F , P ) & = \\limsup _ { L \\rightarrow \\infty } ~ \\frac { ( B + 1 ) \\max \\left ( T _ F ^ { ( B ) } , T _ E ^ { ( B ) } \\right ) } { L } = \\limsup _ { L \\rightarrow \\infty } ~ \\frac { ( B + 1 ) } { B } \\frac { \\max \\left ( T _ F , T _ E \\right ) } { L } . \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} \\begin{aligned} & v _ { s + k + 1 } ^ * = v _ { s + k + 1 } - \\omega _ { k + 1 } \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime ( \\tau ; 0 ) \\right ) \\\\ & v _ { i _ { k + 1 } } ^ { \\prime * } = v _ { i _ { k + 1 } } ^ \\prime ( \\tau ; 0 ) + \\omega _ { k + 1 } \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime ( \\tau ; 0 ) \\right ) \\end{aligned} \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} \\frac { \\left ( q ; q \\right ) _ { n } L _ { n } ^ { ( \\alpha ) } \\left ( a b q ^ { - n - \\alpha } ; q \\right ) } { \\left ( - a b \\right ) ^ { n } \\left ( \\frac { q ^ { \\alpha + 1 } } { a } ; q \\right ) _ { n } } = \\sum _ { k = 0 } ^ { n } \\frac { \\left ( q ^ { - n } ; q \\right ) _ { k } q ^ { \\binom { k + 1 } { 2 } } } { \\left ( \\frac { q ^ { \\alpha + 1 } } { a } ; q \\right ) _ { k } \\left ( a b \\right ) ^ { k } } L _ { k } ^ { ( \\alpha + n - k ) } \\left ( b ; q \\right ) . \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{gather*} \\tau _ { k } ^ { ( \\alpha ) } = \\frac 1 { k ! } \\prod _ { i = 1 } ^ { k } c ^ { ( \\alpha ) } _ { i } \\big ( \\det \\big ( V ^ { ( k ) } _ { \\{ z _ { i } \\} } \\big ) ^ { 2 } \\big ) . \\end{gather*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\displaystyle \\check { \\mu } ( E ) & = \\begin{bmatrix} \\int _ E \\ , 1 \\ , d \\mu ( \\xi ) & \\int _ E \\ , e ^ { - 2 \\pi i ( x _ 1 - x _ 2 ) \\cdot \\xi } \\ , d \\mu ( \\xi ) \\\\ \\ , & \\ , \\\\ \\int _ E \\ , e ^ { - 2 \\pi i ( x _ 2 - x _ 1 ) \\cdot \\xi } \\ , d \\mu ( \\xi ) & \\int _ E \\ , 1 \\ , d \\mu ( \\xi ) \\end{bmatrix} \\\\ \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} \\alpha ( f ^ * ) = \\alpha ( f ) , { \\rm i f } \\ ; f ( 0 ) \\neq 0 \\ ; . \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} \\begin{aligned} d _ Y ( G ( y , x ) , G ( v , u ) ) \\leq a \\ d _ Y ( y , G ( v , u ) ) + & b \\ d _ Y ( v , G ( y , x ) ) + c \\ d _ Y ( y , v ) ; \\\\ & \\forall x \\leq _ { P _ 1 } u , \\ y \\geq _ { P _ 2 } v ; \\ 2 a + c < 1 \\end{aligned} \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} & H ^ 1 ( X ^ { r e g } , \\C ) \\oplus \\bigoplus _ { i = 1 } ^ k H ^ 1 ( Y _ i , \\C ) \\rightarrow \\bigoplus _ { i = 1 } ^ k H ^ 1 ( Y _ i ^ \\times , \\C ) \\rightarrow H ^ 2 ( X ^ 1 , \\C ) \\\\ & \\rightarrow H ^ 2 ( X ^ { r e g } , \\C ) \\oplus \\bigoplus _ { i = 1 } ^ k H ^ 2 ( Y _ i , \\C ) \\rightarrow \\bigoplus _ { i = 1 } ^ k H ^ 2 ( Y _ i ^ \\times , \\C ) . \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} c _ { 2 i } = p _ { 2 i } + \\frac 1 2 \\omega ^ { a _ { h ( 2 i ) } ( 2 i ) } \\left ( \\frac { z } { 2 } \\right ) ^ { h ( 2 i ) } \\left ( z ^ 1 + z ^ 2 + \\dots + z ^ { k - h ( 2 i ) } \\right ) , \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} f f '' + ( f ' ) ^ 2 + 1 = \\pm a \\sqrt { f '^ 2 + 1 } . \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} C ( z ) = \\int _ { 0 } ^ { z } e ^ { - \\frac { \\lambda } { \\xi } s } s ^ { \\frac { \\mu } { \\xi } - 2 } d s , \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} \\delta & = 3 \\alpha _ 1 - \\alpha _ 2 + \\alpha _ 1 \\alpha _ 3 - \\alpha _ 1 \\alpha _ 2 + \\alpha _ 2 \\alpha _ 3 - 2 \\alpha _ 3 + 2 \\\\ \\kappa _ 1 & = \\frac { ( 1 - \\alpha _ 3 ) ( 1 - 0 . 5 \\alpha _ 2 ) } { \\delta } \\\\ \\kappa _ 2 & = \\frac { 0 . 5 \\alpha _ 1 ( 1 + \\alpha _ 3 ) } { \\delta } \\\\ \\kappa _ 3 & = \\frac { \\alpha _ 1 ( 1 - 0 . 5 \\alpha _ 2 ) } { \\delta } . \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} { } _ { 1 } \\phi _ { 0 } ( 0 ; - ; q , z ) = \\frac { 1 } { ( z ; q ) _ { \\infty } } \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} P ^ 0 ( Z ( t ) \\in \\partial S ) = 0 . \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\partial _ t \\norm { D _ { \\eta } \\zeta } ^ 2 + \\norm { \\nabla D _ { \\eta } \\zeta } ^ 2 + \\alpha \\norm { D _ { \\eta } \\tau } ^ 2 _ { L ^ 2 ( \\Gamma _ u ) } & = - \\int _ { \\Omega } \\left ( D _ { \\eta } v \\cdot \\nabla _ H \\right ) \\zeta \\cdot D _ { \\eta } \\zeta - \\int _ { \\Omega } D _ { \\eta } w \\partial _ z \\zeta \\cdot D _ { \\eta } \\zeta - \\int _ { \\Omega } D _ { \\eta } g \\cdot D _ { \\eta } \\zeta \\\\ & = : I _ 1 + I _ 2 + I _ 3 . \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} v _ { i , j , b } = \\begin{cases} x _ { i , j } ^ b & b \\leq t - 1 , \\\\ \\langle \\vec { u } _ { b - t + 1 } , \\vec { w } _ { x _ { i , j } } \\rangle & b \\geq t , \\end{cases} \\end{align*}"}
-{"id": "10018.png", "formula": "\\begin{align*} \\alpha _ i \\not = 0 \\ ; \\Rightarrow \\ ; q ^ { t - a } \\mbox { d i v i d e s $ i $ } . \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} f _ { n } ( z ) = \\frac { \\theta _ { q } \\left ( \\alpha z ^ { - 1 } \\right ) } { \\theta _ { q } \\left ( z ^ { - 2 } \\right ) } g _ { n } ( z ) + \\frac { \\theta _ { q } \\left ( \\alpha z \\right ) } { \\theta _ { q } \\left ( z ^ { 2 } \\right ) } g _ { n } \\left ( z ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} | \\Gamma | : = \\sum _ { e \\in E } m ( e ) , \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} \\Sigma _ X = I _ { p _ 1 } , \\quad \\Sigma _ Y = ( I _ { p _ 2 } - \\Sigma _ S ^ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} A _ m ^ \\prime { = } A _ K ^ \\prime \\prod _ { i { = } m } ^ { K { - } 1 } B _ i { + } \\sum _ { l { = } m } ^ { K { - } 1 } C _ l \\prod _ { i { = } m } ^ { l { - } 1 } B _ i . \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} e _ 3 \\circ f _ 1 = - \\cos ( \\beta ) u _ 3 \\pm \\sin ( \\beta ) u _ 7 , e _ 3 \\circ f _ 2 = \\cos ( \\mu ) u _ 4 \\pm \\sin ( \\mu ) u _ 8 . \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} W _ { n } \\left ( \\varphi ' , \\varphi \\right ) = a ' a W _ { n } \\left ( \\psi , \\psi \\right ) + a ' \\bar { a } W _ { n } \\left ( \\psi , \\bar { \\psi } \\right ) + a ^ { 2 } W _ { n } \\left ( \\psi ' , \\psi \\right ) + a \\bar { a } W _ { n } \\left ( \\psi ' , \\bar { \\psi } \\right ) + \\mbox { c . c . } \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} T ( \\bar { M ' } p ' ) ( a , b ) = \\phi _ a \\cdot p \\cdot \\chi _ b . \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} U ^ * ( 0 , t _ 0 ) & \\le U ^ * ( y , s ) - S ( s , t _ 0 ) \\phi _ 2 ( y ) - a ( s - t _ 0 ) - 2 \\delta | s - t _ 0 | \\\\ [ 1 . 2 m m ] & \\le U ^ * ( 0 , t _ 0 ) + S ( s , t _ 0 ) \\phi _ 1 ( y ) - S ( s , t _ 0 ) \\phi _ 2 ( y ) - \\delta | s - t _ 0 | \\le U ^ * ( 0 , t _ 0 ) - \\delta | s - t _ 0 | , \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{gather*} P _ 1 ( x ) \\ ! = \\ ! D _ { 0 } x \\ ! - \\ ! D _ 1 ^ { \\ , \\prime } \\ , , \\ , P _ n ( x ) \\ ! = \\ ! D _ { n - 1 } x ^ { n } \\ ! - \\ ! D _ n ^ { \\ , \\prime } x ^ { n - 1 } \\ ! + \\ ! \\sum \\limits _ { k = 1 } ^ { n - 1 } ( - 1 ) ^ { k - 1 } x ^ { n - 1 - k } \\ , D _ { n } ^ { n - k , n + 1 } \\ , , \\ , n \\geq 2 \\ , . \\end{gather*}"}
-{"id": "960.png", "formula": "\\begin{align*} g ( a \\circ b ) & = g ( \\pi ( \\pi ^ { - 1 } ( a ) \\pi ^ { - 1 } ( b ) ) = \\eta f ( \\pi ^ { - 1 } ( a ) \\pi ^ { - 1 } ( b ) ) \\\\ & = \\eta ( f \\pi ^ { - 1 } ( a ) f \\pi ^ { - 1 } ( b ) ) = \\eta ( \\eta ^ { - 1 } g ( a ) \\eta ^ { - 1 } g ( b ) ) = g ( a ) \\circledast g ( b ) . \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} e _ { \\mu \\nu \\sigma \\tau } Q ^ { \\sigma \\tau } = 2 i P _ { \\mu \\nu } , \\qquad 2 i Q ^ { \\mu \\nu } = e ^ { \\mu \\nu \\sigma \\tau } P _ { \\sigma \\tau } . \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} u _ { \\delta ^ { i _ { 1 } , \\dots , i _ { k } } } = A ^ { k } - \\sum _ { j = k + 1 } ^ { n } A ^ { k } ( i _ { j } ) + \\sum _ { \\substack { j , l = k + 1 \\\\ j < l } } ^ { n } A ^ { k } ( i _ { j } , i _ { l } ) + \\cdots + \\left ( - 1 \\right ) ^ { n - k } A ^ { k } ( i _ { k + 1 } , \\dots , i _ { n } ) = 0 \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} & \\phi _ { x _ 1 x _ 1 x _ 1 } - \\phi _ { x _ 3 } + 3 H _ { x _ 1 } \\phi _ { x _ 1 } - \\frac { k - 5 } { 2 } H _ { x _ 1 x _ 1 } \\phi = 0 , \\\\ & \\phi _ { x _ 1 x _ 2 } + H _ { x _ 2 } \\phi + \\frac { k - 5 } { 6 } \\frac { H _ { x _ 1 x _ 2 } } { H _ { x _ 2 } } \\phi _ { x _ 2 } = 0 . \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { m - 1 } ( - 1 ) ^ i { m - 1 \\choose i } \\overline { \\gamma } _ { m + i } = 0 , \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} - L v & = 0 \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ v & = \\eta \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\lim \\limits _ { p \\to + \\infty } \\lambda _ 1 ^ { 1 / p } ( \\Omega ; p ) = \\frac { 1 } { \\max \\limits _ { x \\in \\Omega } ( x , \\partial \\Omega ) } \\equiv \\frac { 1 } { r _ \\Omega } , \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ - _ { I I I } } & d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ d \\left ( s + k - 1 \\right ) T R ^ d \\theta ^ { d - 1 } \\end{aligned} \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} \\Re \\Big \\{ \\sum _ { n = 1 } ^ { \\infty } z _ n ^ m \\Big \\} \\ll \\alpha ^ { - 2 m - 1 } m ( 1 - \\beta _ 1 ) . \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} \\left ( H _ k / H _ l \\right ) ^ { 1 / ( k - l ) } = \\left ( H _ { l + 1 } / H _ l \\right ) ^ { 1 / ( k - l ) } \\cdots \\left ( H _ { k } / H _ { k - 1 } \\right ) ^ { 1 / ( k - l ) } 0 \\leq l < k \\leq n \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} \\prod _ { v : v \\in e _ { m } } B ( v , e _ { m } ) & = \\frac { 2 \\alpha } { 1 - \\alpha ^ { \\frac 1 3 } } \\cdot \\left ( 1 - 2 \\alpha ^ { \\frac 2 3 } \\right ) ^ 2 \\\\ & = \\frac { 2 \\alpha \\left ( 4 \\alpha ^ { \\frac 4 3 } - 4 \\alpha ^ { \\frac 2 3 } + 1 \\right ) } { 1 - \\alpha ^ { \\frac 1 3 } } \\\\ & = \\alpha + \\frac { \\alpha ( 2 \\alpha ^ { \\frac 1 3 } - 1 ) ( \\alpha ^ { \\frac 1 3 } + 1 ) ( 4 \\alpha ^ { \\frac 2 3 } - 2 \\alpha ^ { \\frac 1 3 } - 1 ) } { 1 - \\alpha ^ { \\frac 1 3 } } . \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a ; q \\right ) _ { n } z ^ { n } } { \\left ( q , c q ; q \\right ) _ { n } } = \\frac { \\left ( a z ; q \\right ) _ { \\infty } } { \\left ( z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } \\left ( a ; q \\right ) _ { n } \\left ( c z \\right ) ^ { n } } { \\left ( q , c q , a z ; q \\right ) _ { n } } \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} \\mathcal { H } = \\left \\{ U \\ , | \\ , U = ( u , v , \\tau , \\theta ) ^ { T } \\in \\left ( H ^ 2 _ { 0 } ( \\Omega ) \\right ) ^ { d } \\times \\left ( L ^ 2 ( \\Omega ) \\right ) ^ { d } \\times H _ 0 ^ 1 ( \\Omega ) \\times L ^ 2 ( \\Omega ) \\right \\} , \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} \\hat \\theta _ T = - \\left ( \\frac { 1 } { H \\Gamma ( 2 H ) T } \\int _ 0 ^ T X _ t ^ 2 d t \\right ) ^ { - \\frac { 1 } { 2 H } } . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} & 0 = f ( \\tilde u ) - \\check v , ~ 0 = h ( \\tilde v ) + c g ( \\tilde u ) , \\\\ & 0 = f ( \\check u ) - \\tilde v , ~ 0 = h ( \\check v ) + c g ( \\check u ) . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ { I I I } ^ + } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq C _ { d , s , k } C _ { d , \\alpha } T R \\eta ^ { d - 1 } \\end{aligned} \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} H ( x , r _ 2 ) / H ( x , r _ 1 ) = \\exp ( 2 \\int _ { r _ 1 } ^ { r _ 2 } \\beta ( x , r ) d \\log r ) \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} \\int _ { ( F \\backslash \\mathbb { A } ) ^ 3 } ( U _ \\xi - V _ \\xi ) \\left ( \\begin{array} { c c c } 1 & x e _ 0 & z \\\\ 0 & I _ { n - 2 } & y f _ 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) \\psi ( - \\alpha x - \\beta y ) d z d x d y = 0 \\ , , \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} K ( u ) = \\dfrac { 1 } { 2 } \\int _ { \\partial \\Omega } \\kappa u _ n ^ 2 . \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{gather*} g ^ { ( \\alpha ) } _ { C } = \\exp \\big ( \\Gamma ^ { ( \\alpha ) } _ { C } \\big ) , \\end{gather*}"}
-{"id": "2122.png", "formula": "\\begin{align*} \\begin{cases} u _ t + u u _ x + u _ { x x x } + a v _ { x x x } + a _ 1 v v _ x + a _ 2 ( u v ) _ x = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ c v _ t + r v _ x + v v _ x + a b u _ { x x x } + v _ { x x x } + a _ 2 b u u _ x + a _ 1 b ( u v ) _ x = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ u ( x , 0 ) = u ^ 0 ( x ) , v ( x , 0 ) = v ^ 0 ( x ) , & \\ , \\ , ( 0 , L ) , \\end{cases} \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} \\sum _ { y _ { t } } \\Big ( \\log \\Big ( \\frac { q _ t ( y _ t | y ^ { t - 1 } _ { t - M } , x _ t ) } { \\nu ^ { \\pi } _ { t } ( y _ t | y ^ { t - 1 } _ { t - J } ) } \\Big ) + C _ { t + 1 } ( y ^ t _ { t + 1 - J } ) \\Big ) q _ t ( y _ t | y ^ { t - 1 } _ { t - M } , x _ t ) - s \\gamma _ t ( x _ t , y ^ { t - 1 } _ { t - N } ) = 1 - \\lambda _ t ( y ^ { t - 1 } _ { t - J } ) , ~ \\forall { x _ t } \\in { \\cal X } _ t . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} w = \\tilde v + \\xi _ 1 ( \\tilde \\eta ) , \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} \\lambda _ 1 ^ n = ( \\lambda ' _ 1 ) ^ n \\det ( \\zeta ( x ) ) \\overline { \\det ( \\zeta ( x - \\theta ) ) } , & \\ \\ x \\in C , \\\\ \\lambda _ 2 ^ n = ( \\lambda _ 2 ' ) ^ n \\det ( \\zeta ( x ) ) \\overline { \\det ( \\zeta ( x - \\theta ) ) } , & \\ \\ x \\in D . \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} A _ 1 = \\begin{pmatrix} I & 0 \\\\ 0 & \\Delta \\end{pmatrix} , \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} \\xi _ i ( x , y ) : = \\bigwedge _ j P _ { ( i , j ) } ( x , y ) = 0 \\wedge Q _ { ( i , j ) } ( x , y ) \\neq 0 \\wedge \\theta _ { ( i , j ) } ( t _ { 1 } ^ { ( i , j ) } , \\ldots , t _ { n _ { ( i , j ) } } ^ { ( i , j ) } ) , \\end{align*}"}
-{"id": "9892.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } g \\cdot \\ , d \\ , ( \\delta V ) + \\sigma \\int _ { B ^ + } { \\rm d i v } _ { \\partial \\Omega } \\ , g \\ , d \\mathcal H ^ { n } = 0 \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} \\psi _ { n , m } ( t ) = \\left \\{ \\begin{array} { l c } 2 ^ { \\frac { k + 1 } { 2 } } \\textrm { s c w } _ m ( 2 ^ k t - n ) , & \\quad \\frac { n } { 2 ^ k } \\leq t < \\frac { n + 1 } { 2 ^ k } , \\\\ 0 , & \\textrm { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} \\eta _ { v } \\ = \\ \\min \\left \\{ d ( l ^ * _ u , 0 ) / 2 \\mid u \\in \\mathcal { N } _ { \\kappa ( \\| v \\| _ \\infty ) } \\cap \\Z ^ { N } \\right \\} \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\sqrt { \\nu } \\frac { \\norm { y _ k - x _ k } } { \\gamma _ k } \\leq \\frac { \\norm { y _ k - x _ k } _ k } { \\gamma _ k } = \\frac { \\norm { x _ { k + 1 } - x _ k } _ k } { \\gamma _ k \\lambda _ k } \\leq \\frac { \\norm { \\tilde { x } _ { k } - x _ k } _ k } { \\gamma _ k \\lambda _ k } . \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} \\Psi _ 2 ( b ; \\ , b , \\ , c ; \\ , x , \\ , y ) = \\exp ( x + y ) \\ , \\Phi _ 2 ( c - b ; \\ ; c ; \\ ; - y , x y ) \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} v _ n = \\sum _ { i = 0 } ^ { q ' } Q _ i ( n ) \\beta _ i ^ n \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} ( \\iota \\otimes \\Delta ) \\circ \\Delta = ( \\Delta \\otimes \\iota ) \\circ \\Delta , \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} \\rho ' : P _ { 1 } \\otimes V _ { r - 1 } \\to k [ z ] / ( z - \\alpha _ { 1 } ) ^ { n _ { 1 } } \\oplus _ { i = 2 } ^ { k } k [ z ] / ( z - \\alpha _ { i } ) ^ { n _ { i } - 1 } \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} \\lambda a _ 1 = a _ 2 , 2 \\lambda a _ 2 = a _ 1 + a _ 3 , \\ldots , 2 \\lambda a _ { n - 1 } = a _ { n - 2 } + a _ n . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} Z _ 2 : = \\sum _ { \\chi \\pmod { H } } \\ , \\ , \\ , \\sum _ { \\substack { \\Lambda < \\lambda \\leq R \\\\ | \\gamma | \\leq 1 } } e ^ { - B ' \\lambda } \\leq e ^ { 1 8 8 - ( B ' - 1 6 2 ) R } + B ' \\int _ { \\Lambda } ^ { \\infty } e ^ { 1 8 8 - ( B ' - 1 6 2 ) \\lambda } d \\lambda \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{gather*} a _ { k } ( x ) \\ , \\ \\ \\deg a _ { k } ( x ) = n _ { k + 1 } - n _ { k } \\geq 1 \\ , \\ \\ \\ 0 \\leq k < m \\ , \\end{gather*}"}
-{"id": "4413.png", "formula": "\\begin{align*} | x _ i | ^ 2 + | x _ i - ( v _ { s + 1 } - v _ i ) \\tau | ^ 2 - | x _ i - ( v _ i ^ * - v _ i ) \\tau | ^ 2 - | x _ i - ( v _ { s + 1 } ^ * - v _ i ) \\tau | ^ 2 = 0 \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} \\left \\{ \\gamma _ { 1 } , \\gamma _ { 2 } , . . . , \\gamma _ { j } , \\delta - \\gamma ( j ) \\right \\} \\in \\Gamma ( k + ) , \\forall j = 1 , 2 , . . . , p - 1 \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} F _ p ( v , \\zeta ) & : = - P _ p \\left ( v \\cdot \\nabla _ H v + w \\partial _ z v - \\Pi ( \\zeta ) \\right ) , \\\\ G _ { q } ( v , \\zeta ) & : = - \\left ( v \\cdot \\nabla _ H \\zeta + w \\partial _ z \\zeta \\right ) . \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} F ( 1 , \\dots , 1 ) = 1 . \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} f _ { n + 1 } ( u ) = \\lambda \\cos \\left ( \\frac { u } { c } \\right ) , \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} \\begin{array} { c } e C \\subset g _ { \\alpha } ^ { - 1 } d _ { \\alpha } g _ { \\alpha } C = ( U g _ { \\alpha } ) ^ { - 1 } d _ { \\alpha } ( U g _ { \\alpha } ) C = \\\\ ( \\varphi _ { \\alpha } h ) ^ { - 1 } d _ { \\alpha } ( \\varphi _ { \\alpha } h ) C = h ^ { - 1 } ( \\varphi _ { \\alpha } ^ { - 1 } d _ { \\alpha } \\varphi _ { \\alpha } ) h C \\ , . \\end{array} \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} + \\sum \\limits _ { n = 1 } ^ { N - 1 } \\frac { \\Lambda ^ + _ n } { \\sqrt { \\pi } 2 ^ { n } } t ^ { - n / 2 } \\left [ \\int \\limits _ { 1 } ^ { + \\infty } z ^ { - n } e ^ { - ( z - \\eta ) ^ 2 } d z + \\int \\limits _ { 0 } ^ { 1 } \\Psi _ n ( z , \\eta ) d z \\right ] + \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} T ( t , s ) u _ 0 ~ : = ~ u ( t , \\cdot ) , \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} V _ g ( \\xi ) = \\left ( \\frac { g ( | \\xi | ) } { | \\xi | } \\right ) ^ { \\frac { 1 } { 2 } } \\xi \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} R _ T = R [ X _ t ] _ { t \\in T } / ( X _ t ^ p - \\hat t ) T \\subseteq S R ' = \\lim _ T R _ T \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sum _ { k = 1 } ^ n \\left ( \\langle \\varphi _ k , H \\varphi _ k \\rangle - \\langle \\psi _ k , H \\psi _ k \\rangle \\right ) = 0 . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} d \\phi \\circ J _ { S ^ 2 } = - a _ { i j } x ^ { j } J ^ { i } \\circ d \\phi , \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} \\gamma _ j = 2 ( ( \\imath _ { e _ k } \\ast \\varphi ) \\wedge ( \\imath _ { e _ j } \\nabla _ { e _ k } \\ast \\varphi ) + ( \\imath _ { e _ j } \\imath _ { e _ k } \\ast \\varphi ) \\wedge e ^ l \\wedge ( \\imath _ { e _ k } \\nabla _ { e _ l } \\ast \\varphi ) ) . \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} d _ { } ( \\mathbf { D _ 1 } , \\mathbf { D _ 2 } ) = \\frac { 1 } { 2 } \\sum _ { D \\in \\mathcal { D } _ n } | P _ 1 ( D ) - P _ 2 ( D ) | . \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} \\overline F \\big ( a ( x ) \\big ) = O \\big ( \\overline H ( x ) \\big ) , \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} \\Lambda ( A _ t , \\Theta _ t ) = & \\int _ { \\Theta _ 1 \\times A _ 2 } \\left ( g _ 2 \\log | t | + O ( 1 ) \\right ) \\tfrac { g _ 1 } { g \\cdot g _ 1 ! g _ 2 ! } \\nu _ 1 ^ { g _ 1 - 1 } \\nu _ 2 ^ { g _ 2 } \\\\ & + \\int _ { A _ 1 \\times \\Theta _ 2 } \\left ( g _ 1 \\log | t | + O ( 1 ) \\right ) \\tfrac { g _ 2 } { g \\cdot g _ 1 ! g _ 2 ! } \\nu _ 1 ^ { g _ 1 } \\nu _ 2 ^ { g _ 2 - 1 } \\\\ = & \\tfrac { 2 g _ 1 g _ 2 } { g } \\log | t | + O ( 1 ) . \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } u _ { \\delta } \\prod _ { i = 1 } ^ { n } g _ { i } ^ { \\delta _ { i } } ( x _ { i } ) . \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} & \\varphi ( L _ 1 , x ) = 0 , & \\forall x \\in \\mathcal { W } ^ { q } , \\\\ & \\varphi ( L _ { n } , L _ k ) = 0 , & \\forall n + k \\neq 0 , \\\\ & \\varphi ( L _ { n } , G _ k ) = 0 , & \\forall n + k \\neq - 1 . \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} M ( u _ 1 , \\ldots , u _ p ) = \\begin{pmatrix} M ( u _ 1 + 1 ) & & & { \\bf O } \\\\ & M ( u _ 2 + 1 ) & & \\\\ & & \\ddots & \\\\ { \\bf O } & & & M ( u _ p + 1 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{gather*} \\Psi ^ { [ k ] ( \\alpha + 1 ) } = \\Psi ^ { [ k ] ( \\alpha ) } V ^ { ( \\alpha ) } _ { k } , \\Psi ^ { [ k - 1 ] ( \\alpha + 1 ) } = \\Psi ^ { [ k ] ( \\alpha ) } W ^ { ( \\alpha ) } _ { k } . \\end{gather*}"}
-{"id": "3433.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } E _ \\infty ^ i ( C ^ { \\infty } ( S ^ 3 \\times S ^ { 6 } \\times S ^ { 8 } ) \\rtimes _ \\alpha \\mathbb { Z } ) \\cong \\mathbb { C } , & i = 1 , 3 , 9 , 1 1 \\\\ E _ \\infty ^ i ( C ^ { \\infty } ( S ^ 3 \\times S ^ { 6 } \\times S ^ { 8 } ) \\rtimes _ \\alpha \\mathbb { Z } ) \\cong \\{ 0 \\} , & i = \\ e l s e . \\end{array} \\right . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} x _ { 1 } - 5 \\ , x _ { 2 } + 6 \\ , x _ { 3 } + 5 \\ , x _ { 4 } & = 0 \\\\ x _ { 2 } - 2 \\ , x _ { 3 } - 3 \\ , x _ { 4 } & = 0 \\\\ x _ { 3 } + x _ { 4 } & = 0 \\\\ x _ { 3 } - 6 \\ , x _ { 4 } & = 0 \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} F _ { p , i } ( x _ { p } , x _ { i } ) = v _ { 0 } ( x _ { i } ) + v _ { \\delta ^ { p } } ( x _ { i } ) g _ { p , x _ { i } } ( x _ { p } ) \\underset { ( \\ref { e q : d - d e l t a - 0 } ) } { = } u _ { 0 } ( 0 ) + u _ { \\delta ^ { i } } ( 0 ) f _ { i , 0 } ( x _ { i } ) + v _ { \\delta ^ { p } } ( x _ { i } ) g _ { p , x _ { i } } ( x _ { p } ) \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ + _ I } & d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ d \\left ( s + k - 1 \\right ) R ^ d \\eta ^ { - 1 } y \\end{aligned} \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} \\begin{cases} \\cos \\alpha _ 1 = - a _ { 1 j } x ^ j , \\\\ \\cos \\alpha _ 2 = - 2 \\frac { a _ { 2 j } x ^ j } { \\rho S } , \\\\ \\cos \\alpha _ 2 = - 2 \\frac { a _ { 2 j } x ^ j } { \\rho S } . \\end{cases} \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} \\mathcal { B } ^ + _ { I I } = \\left \\{ \\begin{aligned} & \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { A } ^ + \\textnormal { s u c h t h a t } \\\\ & \\left | \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime \\right ) \\right | \\leq \\left ( \\sin \\alpha \\right ) \\left | v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime \\right | \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "9969.png", "formula": "\\begin{align*} d = d ( y _ 1 , y _ 2 ) \\geq d ( y _ 1 , x _ 1 ) + d ( x _ 1 , x _ 2 ) + d ( x _ 2 , y _ 2 ) - 1 2 N ( 3 , 0 ) . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } x _ { n + 1 } = \\dfrac { h _ { 1 } } { k _ { 1 } } + \\varepsilon \\phi _ { 1 } & = & \\dfrac { h _ { 2 } } { k _ { 2 } } - \\varepsilon \\phi _ { 2 } \\\\ \\varepsilon k _ { 1 } \\cong \\varepsilon k _ { 2 } \\cong 0 & ; & k _ { 1 } \\cong k _ { 2 } \\cong + \\infty \\end{array} \\right . \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} \\chi _ f \\ge 1 + \\max \\left ( \\frac { n ^ + } { n ^ - } , \\frac { n ^ - } { n ^ + } \\right ) = 1 + \\max \\left ( \\frac { 1 + f } { g } , \\frac { g } { 1 + f } \\right ) = \\max \\left ( \\frac { n } { g } , \\frac { n } { 1 + f } \\right ) . \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\dim V _ { \\pi _ { k , p } } ^ \\Gamma = \\sum _ { \\mu \\in \\mathcal L _ \\Gamma } m _ { \\pi _ { k , p } } ( \\mu ) , \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} E _ { 0 } ( k , \\alpha ) = 1 \\left ( \\frac { \\alpha } { 2 | k | ^ { 1 - \\alpha / 2 } } \\right ) . \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} \\Delta - \\Delta ' & = a ' E \\otimes \\frac { 1 } { a a ' } - a p ^ { e - c } E \\otimes \\frac { 1 } { a a ' } \\\\ & = ( m ( p ^ { e } + 1 ) - m p ^ c p ^ { e - c } ) E \\otimes \\frac { 1 } { a a ' } = m E \\otimes \\frac { 1 } { a a ' } = \\frac { m } { a ' } \\Delta = \\frac { 1 } { p ^ e + 1 } \\Delta \\ge 0 , \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} C d \\left ( z _ { k } \\right ) & \\leq d \\left ( z _ { j } , z _ { k } \\right ) \\leq d \\left ( z _ { j } , \\xi _ { k } \\right ) + d \\left ( \\xi _ { k } , z _ { k } \\right ) \\\\ & = d \\left ( z _ { j } \\right ) - d \\left ( \\xi _ { k } \\right ) + d \\left ( z _ { k } \\right ) - d \\left ( \\xi _ { k } \\right ) \\\\ & \\leq 2 d \\left ( z _ { k } \\right ) - 2 d \\left ( \\xi _ { k } \\right ) = 2 d \\left ( z _ { k } , \\xi _ { k } \\right ) . \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{gather*} \\sum \\limits _ { k = 0 } ^ { r - 1 } \\ d _ { n , k } \\ s _ { k + m } = \\ s _ { r + n + m } \\ , \\ \\ \\ \\ \\ m \\geq 0 \\ , \\end{gather*}"}
-{"id": "468.png", "formula": "\\begin{align*} \\gamma _ { 3 } - \\alpha _ { 3 } \\beta _ { 3 } d - 2 \\alpha _ { 3 } \\beta _ { 3 } x _ { 3 } = 0 . \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} x _ s = \\frac { x ' _ s \\oplus x _ s '' } { \\sqrt { 2 } } y _ t = \\frac { y _ t ' \\oplus y _ t '' } { \\sqrt { 2 } } , \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} C _ { p , g } ^ { ( n ) } & = \\sum _ { k = 0 } ^ { 2 g } ( - 1 ) ^ { p - k - 1 } \\sum _ { t = k - g } ^ { \\lfloor k / 2 \\rfloor } \\binom { n - k + 2 t } { t } 2 ^ { k - 2 t } \\binom { n } { k - 2 t } . \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} \\beta _ { s , a ^ 2 } ^ 2 = \\frac { \\left [ r ^ 2 ( s , a _ s ^ 1 , a ^ 2 ) - r ^ 2 ( s , a _ s ^ 1 , a _ s ^ 2 ) \\right ] } { \\left [ \\sum _ { s ' \\in S } p _ { s ' } \\ r ^ 2 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) - \\sum _ { s ' \\in S } p ( s ' | s , a _ s ^ 1 , a ^ 2 ) r ^ 2 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) \\right ] } . \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} \\theta = \\frac { k ^ 2 } { G \\left ( \\frac { k } { r } \\right ) } = \\frac { s _ { m ^ * } ^ 2 } { G \\left ( s _ { m ^ * } \\right ) } r ^ 2 \\leq M r ^ 2 \\leq r \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} w ^ { \\rho _ { i _ k } } = & [ - d _ 1 \\Delta + I ] ^ { - 1 } \\left \\{ w ^ { \\rho _ { i _ k } } + w ^ { \\rho _ { i _ k } } q ( \\rho _ { i _ k } w ^ { \\rho _ { i _ k } } ) \\left [ f ( \\rho _ { i _ k } w ^ { \\rho _ { i _ k } } ) - v ^ { \\rho _ { i _ k } } \\right ] \\right \\} , \\\\ v ^ { \\rho _ { i _ k } } = & [ - d _ 2 \\Delta + I ] ^ { - 1 } \\left \\{ v ^ { \\rho _ { i _ k } } + v ^ { \\rho _ { i _ k } } \\left [ f ( v ^ { \\rho _ { i _ k } } ) + q ( \\rho _ { i _ k } w ^ { \\rho _ { i _ k } } ) w ^ { \\rho _ { i _ k } } \\right ] \\right \\} , \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\varphi _ 0 ( f ) = \\log m ( e ^ T ) . \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} x \\mapsto \\frac { \\partial ^ { j - 1 } } { \\partial y ^ { j - 1 } } I _ { \\kappa } ( 2 y \\sqrt { x } ) | _ { y = \\alpha } , j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v - \\Delta v + \\nabla _ H p + f _ 0 k \\times v = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\\\ \\partial _ z p = T , \\\\ \\partial _ t T + v \\cdot \\nabla _ H T + w \\partial _ z T - \\partial _ z ^ 2 T = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { l l } \\dot { u } = \\log \\det ( u _ { \\alpha \\bar { \\beta } } ) - A u + f ( z , t ) \\ ; \\ ; \\ ; & \\mbox { o n } \\ ; \\Omega \\times ( 0 , T ) , \\\\ u = \\varphi & \\mbox { o n } \\ ; \\partial \\Omega \\times [ 0 , T ) , \\\\ u = u _ 0 & \\mbox { o n } \\ ; \\bar { \\Omega } \\times \\{ 0 \\} , \\\\ \\end{array} \\end{cases} \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} 8 = m _ { - } \\geq 1 2 - j - c _ j , \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} \\alpha _ t f _ 0 = T _ t f _ 0 = T _ { \\frac { t } { 2 } } T _ { \\frac { t } { 2 } } f _ 0 = \\alpha _ { \\frac { t } { 2 } } ^ 2 f _ 0 \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\delta = \\delta _ g : \\Lambda ^ k V ^ * \\longrightarrow \\Lambda ^ { k - 1 } V ^ * \\otimes V , \\delta _ g ( \\alpha ^ k ) : = ( \\imath _ { e _ i } \\alpha ^ k ) \\otimes ( e ^ i ) ^ \\# , \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} \\sum _ { k > n } \\frac { 2 a _ k } { ( k - n ) } t ^ { ( k - n ) / 2 } | _ { \\Lambda ^ { - 2 } } ^ { \\Lambda '^ { - 2 } } = \\sum _ { k > n } \\frac { 2 a _ k } { ( k - n ) } \\frac { e ^ { \\tau ( k - n ) / 2 } - 1 } { \\Lambda ^ { k - n } } \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\mathcal { F } _ k & : = \\{ \\mathcal { A } \\subset \\mathcal { S } : ~ \\mathcal { A } = h ( \\mathcal { S } ^ { k - 1 } ) , h : \\mathcal { S } ^ { k - 1 } \\to \\mathcal { S } \\ , \\} , \\\\ \\mathcal { S } & : = \\{ u \\in W _ 0 ^ { 1 , p } ( \\Omega ) : ~ \\| u \\| _ { L ^ p ( \\Omega ) } = 1 \\} \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ { \\gamma } ( 0 , 0 ) & = ( - 1 ) ^ m ( \\gamma + m + 1 ) _ { m } m ! \\delta _ { m , n } . \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} g ( x ) = \\inf { \\{ t \\geq 0 \\mid x \\in t C \\} } . \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} [ \\bar { h } _ { 0 , k } , \\bar { e } _ { 0 , l } ] = ( 2 - d ^ k - d ^ { - k } ) \\bar { e } _ { 0 , l + k } , \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} \\langle \\mathbf { H } , \\hat { \\mathbf { H } } \\rangle _ { \\mathbb { R } ^ { ( k \\times d ) \\times ( k \\times d ) } } \\equiv \\mathbf { H } : \\hat { \\mathbf { H } } : = \\sum _ { i , I = 1 } ^ { k } \\sum _ { j , J = 1 } ^ { d } H _ { i j I J } \\hat { H } _ { i j I J } \\mathbf { H } , \\hat { \\mathbf { H } } \\in \\mathbb { R } ^ { ( k \\times d ) \\times ( k \\times d ) } . \\end{align*}"}
-{"id": "10167.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n r ( i - j ) = n ^ { 2 H } \\ell ( n ) , \\end{align*}"}
-{"id": "9976.png", "formula": "\\begin{align*} \\sum _ { a \\in A } v _ a Z ( a , b ) = 1 b \\in A . \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} & x \\rightharpoonup ( a b ) = ( x _ 1 \\rightharpoonup a _ 1 ) \\left ( ( x _ 2 \\leftharpoonup a _ 2 ) \\rightharpoonup b \\right ) , \\\\ & ( x y ) \\leftharpoonup a = ( x \\leftharpoonup ( y _ 1 \\rightharpoonup a _ 1 ) ) ( y _ 2 \\leftharpoonup a _ 2 ) \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\mu = \\frac { \\sigma } { 2 \\sqrt { t } } , \\eta _ 1 = \\frac { x _ 1 } { 2 \\sqrt { t } } \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} \\Phi _ { t } \\left ( x , s \\right ) \\coloneqq \\begin{cases} \\left ( x , s + t \\right ) & , 0 \\leq t < \\varphi \\left ( x \\right ) - s \\\\ \\left ( \\theta ^ { N _ { s + t } \\varphi \\left ( x \\right ) - 1 } \\left ( x \\right ) , s + t - S _ { N _ { t + s } - 1 } \\varphi \\left ( x \\right ) \\right ) & , \\varphi \\left ( x \\right ) - s \\leq t . \\end{cases} \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ { \\infty } \\varphi _ { k } ^ { 2 } ( x ) = \\lim _ { n \\to \\infty } W _ { n } \\left ( \\varphi ' ( x ) , \\varphi ( x ) \\right ) , x \\neq 0 . \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} \\lambda ^ { - 1 - 2 \\alpha + \\epsilon } \\sum \\limits _ { i , j = 1 } ^ { n + 1 } [ Y _ i Y _ j \\tilde { v } ] _ { C ^ { 0 , \\epsilon } ( \\mathcal { B } _ { \\lambda / 4 } ( y ) ) } \\leq C \\left ( \\| v \\| _ { L ^ 2 ( \\mathcal { B } _ 1 ^ + ) } + \\| f \\| _ { Y _ { \\alpha , \\epsilon } } \\right ) . \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} H ( x ) \\ge \\sup _ { s \\in \\R } \\abs { \\bar { \\nabla } f ( x , s ) } : = F ( x ) \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} & \\dot { \\Theta } = - \\coth \\Theta , \\\\ & \\Theta ( 0 ) = r _ 0 . \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} Z _ N ( \\boldsymbol { \\lambda } ) = \\frac { \\prod _ { j = 1 } ^ N [ a ( \\lambda _ j ) b ( \\lambda _ j ) ] ^ N } { \\prod _ { n = 0 } ^ { N - 1 } n ! \\prod _ { 1 \\leq j < k \\leq N } d ( \\lambda _ k , \\lambda _ j ) } \\det \\left [ \\partial _ { \\lambda _ k } ^ { j - 1 } \\varphi ( \\lambda _ k ) \\right ] _ { j , k = 1 , \\dots , N } . \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\log \\| \\varphi _ { g } \\| ( X ) = 4 \\tbinom { 2 g + 2 } { g + 1 } S _ { g } ( X ) + 4 \\tbinom { 2 g } { g - 1 } \\sum _ { 1 \\le k < l \\le 2 g + 2 } g ( W _ { k } , W _ { l } ) . \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\lambda = ( \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\lambda _ { l ( \\lambda ) } ) ( \\lambda _ 1 \\geq \\lambda _ 2 \\geq \\ldots \\geq \\lambda _ { l ( \\lambda ) } > 0 , \\ ; l ( \\lambda ) \\geq 0 ) . \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} \\vartheta _ { \\mathcal L _ \\Gamma } ^ { ( \\ell ) } ( z ) = \\sum _ { k \\geq 0 } N _ { \\mathcal L _ \\Gamma } ( k , \\ell ) z ^ k . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} D = D ( H ) = \\inf _ { \\phi \\in P } \\norm { M _ X - M _ { \\phi , H } } _ { T V } - \\frac { 3 } { m } , \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\left ( 1 , \\tfrac { 1 } { 2 } + \\nu ; \\tfrac { 1 } { 2 } ; - x ^ 2 \\right ) = ( 1 + x ^ 2 ) ^ { - \\nu - \\tfrac { 1 } { 2 } } { } _ 2 F _ 1 ( \\tfrac { 1 } { 2 } + \\nu , - \\tfrac { 1 } { 2 } ; \\tfrac { 1 } { 2 } ; \\tfrac { x ^ 2 } { 1 + x ^ 2 } ) . \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} \\frac { \\partial X _ i ' ( \\widehat \\beta - \\beta ) } { \\partial \\varepsilon _ j } = X _ { i \\widehat T } ' \\left ( \\sum _ { l = 1 } ^ n X _ { l \\widehat T } X _ { l \\widehat T } ' \\right ) ^ { - 1 } X _ { j \\widehat T } , i , j = 1 , \\dots , n ; \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} \\zeta ^ 0 ( x ) = \\frac { \\mathfrak { D } } { \\sigma ^ * } \\frac { { \\rm d } \\rho } { { \\rm d } x } \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} \\int \\phi ( x , y ) d ( \\mu y ) = p _ e p _ d \\ \\ \\mu \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} M = ( a \\partial _ \\theta + \\lambda s \\partial _ s - L ) ^ { - 1 } . \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} x _ s = \\begin{pmatrix} \\Re ( X _ s \\psi ) \\\\ \\Im ( X _ s \\psi ) \\end{pmatrix} y _ t = \\begin{pmatrix} \\Re ( Y _ t \\psi ) \\\\ \\Im ( Y _ t \\psi ) \\end{pmatrix} . \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} d _ 2 = \\begin{pmatrix} p & q & 0 \\\\ 0 & r & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , b _ 2 = \\begin{pmatrix} 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 \\\\ a & 0 & 0 & 0 \\end{pmatrix} , \\delta _ 2 = \\begin{pmatrix} 0 & x & y \\\\ - x & 0 & w \\\\ - y & - w & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\sigma ^ 2 ( b ) = \\limsup _ { r \\to 1 } \\frac { 1 } { 2 \\pi | \\log ( 1 - r ) | } \\int _ { | z | = r } | b ( z ) | ^ 2 \\ , | d z | . \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} \\operatorname { d i v } \\left ( \\mathbf { A } \\times \\mathbf { B } \\right ) = \\mathbf { B } \\cdot \\operatorname { c u r l } \\mathbf { A } - \\mathbf { A } \\cdot \\operatorname { c u r l } \\mathbf { B } . \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} \\inf \\left \\{ \\max _ { g \\in K } | \\alpha ( g ) f - f | _ { \\infty } ^ - \\mid f \\in { \\mathcal F } ( G ) , \\ f ( e ) = 1 \\right \\} > 0 \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} ( f * g ) ( \\gamma ) & = \\int f ( \\gamma _ 1 ) g ( \\gamma _ { 1 } ^ { - 1 } \\gamma ) d \\lambda ^ { r ( \\gamma ) } ( \\gamma _ 1 ) \\\\ f ^ * ( \\gamma ) & = \\overline { f ( \\gamma ^ { - 1 } ) } . \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} | V _ { n , k , b } | = ( n - b + 1 ) \\binom { b } { k - 1 } + \\binom { b } { k } = ( n + 1 ) \\binom { b } { k - 1 } - ( k - 1 ) \\binom { b + 1 } { k } . \\end{align*}"}
-{"id": "9897.png", "formula": "\\begin{align*} Q ( x ) \\circ \\nu ( \\xi ( x ) ) = 0 \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} \\mathcal { B } ^ - _ I = \\left \\{ \\begin{aligned} & \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { A } ^ - \\textnormal { s u c h t h a t } \\\\ & \\inf _ { i \\in \\left \\{ 1 , \\dots , s , s + 1 , \\dots , s + k \\right \\} \\backslash \\left \\{ i _ { k + 1 } \\right \\} } \\left | \\left ( x _ { i _ { k + 1 } } ^ \\prime - x _ i ^ \\prime \\right ) - \\tau \\left ( v _ { i _ { k + 1 } } ^ \\prime - v _ i ^ \\prime \\right ) \\right | \\leq y \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{gather*} \\colon \\psi _ { a } ^ { + } ( z ) \\psi _ { b } ^ { - } ( w ) \\colon = \\psi _ { a , } ^ { + } ( z ) \\psi _ { b } ^ { - } ( w ) - \\psi _ { b } ^ { - } ( w ) \\psi _ { a , } ^ { + } ( z ) , \\end{gather*}"}
-{"id": "9371.png", "formula": "\\begin{align*} u ( p t ) = \\bar A ( t ) u ( t ) , \\ \\ \\ u ( q t ) = \\bar B ( t ) u ( t ) \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} F \\left ( t , t , t \\right ) & = a _ { 1 } \\left [ f _ { 1 } \\left ( t \\right ) + \\alpha f _ { 3 } \\left ( t \\right ) g \\left ( t \\right ) \\right ] + a _ { 2 } \\left [ f _ { 2 } \\left ( t \\right ) + \\beta f _ { 3 } \\left ( t \\right ) g \\left ( t \\right ) \\right ] + a _ { 3 } \\left [ f _ { 3 } \\left ( t \\right ) + \\gamma f _ { 3 } \\left ( t \\right ) g \\left ( t \\right ) \\right ] . \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} \\mathrm { p _ a } ( i , j ) = P ( ( i , j ) \\in A ( \\mathbf { D } ) ) = \\sum _ { ( i , j ) \\in A ( D ) } P ( D ) . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} p _ n \\sum _ { \\ell = 0 } ^ { L - 1 } ( \\ell + 1 ) ( 1 - p _ n ) ^ \\ell ( 1 - r _ n ) ^ \\ell & = \\lambda ^ \\star . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} & a ^ * _ { 0 , 1 } = a ^ * _ { 0 , 2 } = a ^ * _ { 3 , 1 } = a ^ * _ { 3 , 2 } = a ^ * _ { 3 , 3 } = a ^ * _ { 2 , 2 } = a ^ * _ { 2 , 3 } = a ^ * _ { 1 , 3 } = 0 , \\\\ & a ^ * _ { 1 , 1 } = \\frac { 1 } { 3 } - \\frac { \\mu _ R } { 3 } - \\frac { \\mu _ T } { 3 } , \\\\ & a ^ * _ { 3 , 0 } + 6 a ^ * _ { 2 , 1 } + 3 a ^ * _ { 1 , 2 } = 2 \\mu _ R + \\mu _ T - 1 , \\\\ & 3 a ^ * _ { 2 , 1 } + 6 a ^ * _ { 1 , 2 } + a ^ * _ { 0 , 3 } = \\mu _ R + 2 \\mu _ T - 1 . \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} D _ v ( \\gamma ) \\leqslant \\begin{cases} q _ v ^ { A \\kappa + B } , & v \\in T , \\\\ C , & v \\mid \\infty , \\\\ 1 , & v \\notin T \\cup \\infty . \\end{cases} \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} \\rho \\ddot { u } = \\mu ( 0 ) \\triangle u - l _ { 2 } ( 0 ) \\triangle ^ { 2 } u + \\int _ { 0 } ^ { \\infty } \\big ( \\mu ' ( s ) \\triangle u ( t - s ) - l _ { 2 } ' ( s ) \\triangle ^ { 2 } u ( t - s ) \\big ) \\mathrm { d } s = 0 \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} \\arg \\min _ { \\vec { X } \\in \\mathbb { R } ^ { m \\times n } } \\Biggl \\lbrace F ( \\vec { X } ) : = & \\dfrac { 1 } { 2 } \\| \\vec { Y - X } \\| _ F ^ 2 + \\lambda _ 0 \\sum _ { i = 1 } ^ { k } \\phi ( \\sigma _ i ( \\vec { X } ) ; a _ 0 ) \\\\ & + \\lambda _ 1 \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } \\phi ( \\vec { X } _ { i , j } ; a _ 1 ) \\Biggr \\rbrace , \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} C _ { \\left ( A , i \\right ) \\left ( B , j \\right ) } { } ^ { \\left ( C , k \\right ) } = \\left ( K _ { i j } { } ^ { k } - K _ { i j } { } ^ { k + n } \\right ) C _ { A B } { } ^ { C } \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} c \\phi ' + L \\phi + N ( \\phi ) = 0 , \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} P _ t = P _ { t - \\frac { \\tau } { n } } P _ { \\frac { \\tau } { n } } = P _ { t - \\frac { \\tau } { n } } P _ { ( n + 1 ) \\frac { \\tau } { n } } = P _ { t + \\tau } . \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} - \\varepsilon u _ { \\varepsilon } ^ { \\prime \\prime } + u _ { \\varepsilon } ^ { \\prime } + \\xi _ { \\varepsilon } - f ( u _ { \\varepsilon } ) & = 0 ( 0 , T ) , \\\\ \\xi _ { \\varepsilon } & \\in \\partial \\phi ( u _ { \\varepsilon } ) ( 0 , T ) , \\\\ u _ { \\varepsilon } ^ { \\prime } ( T ) & = 0 , \\\\ u _ { \\varepsilon } ( 0 ) & = u _ { 0 \\varepsilon } . \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty \\gamma _ i x ^ { i + 1 } = \\frac { d ^ { 2 k - 1 } } { d x ^ { 2 k - 1 } } \\left [ \\left ( \\frac { x ^ { 2 k } } { 1 - x } \\right ) \\psi \\left ( \\frac { x ^ 2 } { 1 - x } \\right ) \\right ] . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} v _ k ^ 1 = \\varphi _ k ^ 1 \\leq \\varphi _ k ^ 2 = v _ k ^ 2 \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| x ^ { k + 1 } - x ^ * \\| ^ 2 ] \\leq q \\mathbb { E } [ \\| x ^ k - x ^ * \\| ^ 2 ] + 4 p _ { \\max } \\alpha _ { \\max } ^ 2 C ^ 2 N + 2 p _ { \\max } \\alpha _ { \\max } B \\sum _ { j = 1 } ^ N \\mathbb { E } [ \\| v ^ k _ j - y ^ k \\| ] , \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } S ( t ) = 0 . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} \\langle T _ \\mu ^ \\mu \\rangle = - \\frac { c } { 2 4 \\pi } R \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} K ^ { ( \\alpha ) } & = \\biguplus _ { m \\in \\omega } K _ m ^ { ( \\alpha ) } \\uplus \\{ b \\} \\\\ & = \\biguplus _ { m \\in \\omega } \\{ x _ m \\} \\uplus \\{ b \\} \\\\ & = \\{ x _ m : m \\in \\omega \\} \\uplus \\{ b \\} . \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} - \\tilde { A } ' = \\int \\tilde { R } \\ , d A _ { \\tilde { g } } = 2 \\int _ { \\partial M } k _ { \\tilde { g } } \\ , d s _ { \\tilde { g } } = - 2 k _ 1 \\tilde { l } _ 1 - 2 k _ 2 \\tilde { l } _ 2 . \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} \\chi _ f ( G ) \\ge \\frac { n } { \\alpha } \\ge \\frac { n } { \\min ( n ^ + , n ^ - ) } = \\frac { n ^ + + n ^ - } { \\min ( n ^ + , n ^ - ) } = 1 + \\max \\left ( \\frac { n ^ + } { n ^ - } , \\frac { n ^ - } { n ^ + } \\right ) . \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n } \\left ( z - \\alpha _ { i } - r _ { j } \\right ) = 0 . \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} W _ i = \\frac { u ^ k u _ { k ; i } } { W } = \\frac { \\abs { \\nabla u } u _ { 1 ; i } } { W } \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\langle \\mathcal { L } _ 0 , \\dots , \\mathcal { L } _ g \\rangle = \\langle \\mathcal { L } _ 0 , \\dots , \\mathcal { L } _ q , \\langle \\mathcal { L } _ { q + 1 } , \\dots , \\mathcal { L } _ { g } \\rangle ( \\mathcal { X } _ g ^ { g + 1 } / \\mathcal { X } _ g ^ { q + 2 } ) \\rangle ( \\mathcal { X } _ g ^ { q + 2 } / \\mathcal { X } _ g ) . \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} R ( t , x , \\xi ) : = \\phi ( t ; x , \\xi ) - \\phi ( t ; 0 , \\xi ) . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} c _ i = f _ i = \\begin{pmatrix} 0 & p _ i \\\\ - p _ i & 0 \\end{pmatrix} , 2 \\leq i \\leq 3 , \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{gather*} \\big \\langle Q ^ { m - n } v _ 0 , \\psi ^ { + } ( w _ 1 ) \\cdots \\psi ^ { + } ( w _ m ) \\psi ^ { - } ( y _ 1 ) \\cdots \\psi ^ { - } ( y _ n ) v _ 0 \\big \\rangle \\\\ \\qquad { } = \\big \\langle \\psi _ { ( m - n ) } ^ { - } \\psi _ { ( m - n + 1 ) } ^ { - } \\cdots \\psi _ { ( - 1 ) } ^ { - } v _ 0 , \\psi ^ { + } ( w _ 1 ) \\cdots \\psi ^ { + } ( w _ m ) \\psi ^ { - } ( y _ 1 ) \\cdots \\psi ^ { - } ( y _ n ) v _ 0 \\big \\rangle , \\end{gather*}"}
-{"id": "2414.png", "formula": "\\begin{align*} \\mathbb { P } ( X _ { r } > x ) = \\sum _ { j = 0 } ^ { r - 1 } \\frac { e ^ { - s x } ( s x ) ^ { j } } { j ! } . \\end{align*}"}
-{"id": "10024.png", "formula": "\\begin{align*} \\ell = q ^ { r _ d } \\sum \\limits _ { i = 0 } ^ { k - 2 } q ^ { i d } \\Rightarrow \\ell q ^ { d } = \\frac { q ^ { d } ( q ^ { n - { d } } - q ^ { r _ d } ) } { q ^ { d } - 1 } . \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\begin{aligned} f _ N ^ { ( s ) } ( 0 , Z _ s ) \\geq \\mathcal { Z } _ N ^ { - 1 } & \\mathbf { 1 } _ { Z _ s \\in \\mathcal { D } _ s } f _ 0 ^ { \\otimes s } ( Z _ s ) \\times \\\\ & \\times \\left [ \\mathcal { Z } _ { N - s } - s ( N - s ) \\mathcal { Z } _ { N - s - 1 } \\varepsilon ^ d | B _ 1 ^ d | \\left \\Vert f _ 0 \\right \\Vert _ { L ^ \\infty _ x L ^ 1 _ v } \\right ] \\end{aligned} \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\frac { 2 } { q } + \\frac { 3 } { r } = \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} T _ { 1 , \\bullet } \\ , T _ { 2 , \\bullet } ^ { - 1 } \\ , T _ { 3 , \\bullet } = \\prod _ { k = 1 } ^ { 3 n + 3 } T _ { \\bullet , k } ^ { ( - 1 ) ^ { k + 1 } } . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\pi i } \\partial \\overline { \\partial } H ( X ) = \\tfrac { 1 } { 2 } \\omega _ { \\mathrm { H d g } } + \\tfrac { 1 } { 1 2 } \\int _ { \\pi _ 2 } h ^ 3 - \\tfrac { 1 } { 8 } e _ 1 ^ A \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\left . Q ^ { \\prime } \\right . ^ { \\mu \\nu } \\left ( x ^ { \\prime } \\right ) = \\Lambda _ { \\ \\sigma } ^ { \\mu } \\Lambda _ { \\ \\tau } ^ { \\nu } Q ^ { \\sigma \\tau } \\left ( x \\right ) , x ^ { \\prime } = \\Lambda x . \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} \\partial _ { t } u & = \\mathrm { d i v } \\ , \\big ( \\mathbf { g } ( \\nabla u ) \\nabla u \\big ) ( 0 , \\infty ) \\times G , \\\\ \\mathbf { g } ( \\nabla u ) \\cdot \\mathbf { n } & = 0 ( 0 , \\infty ) \\times \\Gamma , \\\\ u ( 0 , \\cdot ) & = \\tilde { u } ^ { 0 } \\Omega , \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\langle \\gamma '' , \\gamma ' \\rangle = 0 \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} S ( a _ 1 \\circ b ) a _ 2 & = S ( a _ 1 \\circ b _ 1 \\epsilon ( b _ 2 ) ) a _ 2 = S ( a _ 1 \\circ b _ 1 ) \\epsilon ( b _ 2 ) a _ 2 \\\\ & = S ( a _ 1 \\circ b _ 1 ) ( a _ 2 \\circ b _ 2 ) S ( a _ 3 ) ( a _ 4 \\circ S ( b _ 3 ) ) \\\\ & = \\epsilon ( a _ 1 \\circ b _ 1 ) S ( a _ 2 ) ( a _ 3 \\circ S ( b _ 2 ) ) = S ( a _ 1 ) ( a _ 2 \\circ S ( b ) ) . \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} \\tau ^ * ( \\mu _ R , \\mu _ T ) \\ge \\tau _ { L 2 } \\triangleq \\max \\limits _ { \\substack { l = 1 , \\cdots , \\min \\{ N _ T , N _ R \\} \\\\ s _ 1 = 0 , 1 , \\ldots , l \\\\ s _ 2 = 0 , 1 , \\ldots , N _ R - l } } \\frac { 1 } { l } \\Big \\{ & ( s _ 1 + s _ 2 ) - ( N _ T - l ) s _ 2 \\mu _ T \\\\ & \\left . - \\left ( \\frac { 2 s _ 2 + s _ 1 + 1 } { 2 } \\cdot s _ 1 + s _ 2 ^ 2 \\right ) \\mu _ R \\right . \\\\ & \\left . + \\left ( \\frac { 2 s _ 2 + s _ 1 } { 2 } ( s _ 1 - 1 ) + s _ 2 ^ 2 \\right ) ( 1 - N _ T \\mu _ T ) ^ + \\right \\} \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} A _ p v : = P _ p \\Delta v , D ( A _ p ) : = \\{ v \\in H _ { p e r } ^ { 2 , p } ( \\Omega ) ^ 2 \\mid ( \\partial _ z v ) \\vert _ { \\Gamma _ u } = 0 , v \\vert _ { \\Gamma _ b } = 0 \\} \\cap L ^ p _ { \\overline { \\sigma } } ( \\Omega ) . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} \\int _ { \\gamma } \\rho ' d s = \\int _ \\gamma ( \\rho \\circ f ) g _ f d s \\geq \\int _ { f \\circ \\gamma } \\rho d s \\geq 1 \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} P _ { i } = R [ y ] / ( y - x _ 1 ) \\cdots ( y - x _ { i + 1 } ) \\langle - i \\rangle \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} k _ { 0 } \\coloneqq \\sum _ { i = 1 } ^ { m - n } k _ { \\left [ a _ { i } , \\dots , a _ { i + n - 1 } \\right ] } = S _ { m - n } \\varphi \\left ( x \\right ) , \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} C _ { \\alpha , 2 j } : = ( - 1 ) ^ j 2 ^ { 2 j - \\alpha } \\int _ 0 ^ \\infty \\frac { \\sin ^ { 2 j } u } { u ^ { 1 + \\alpha } } \\ , d u , \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} S : = \\left \\{ { \\frac { { u } _ { 2 } { u } _ { 6 } - { u } _ { 4 } { u } _ { 5 } } { { u } _ { 6 } } } , { \\frac { - { u } _ { 3 } { u } _ { 5 } + { u } _ { 5 } { u } _ { 6 } } { { u } _ { 6 } } } , { \\frac { - { u } _ { 2 } { u } _ { 5 } + { u } _ { 4 } { u } _ { 6 } } { { u } _ { 6 } } } , \\right . \\\\ \\left . { \\frac { 2 \\ , { u } _ { 3 } { u } _ { 6 } - { u } _ { 5 } ^ 2 + 2 \\ , { u } _ { 6 } ^ 2 } { { u } _ { 6 } } } , - { \\frac { { u } _ { 1 } { u } _ { 5 } } { { u } _ { 6 } } } \\right \\} \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} a _ n : = 2 \\sqrt { n } - \\sum _ { k = 1 } ^ n \\frac { 1 } { \\sqrt { k } } \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} | \\gamma | _ { g ( s _ 0 ) } \\leq e ^ { C r ^ { - 1 } \\sqrt { S } ( s _ 0 - t _ 1 ) } | \\gamma | _ { g ( t _ 1 ) } = e ^ { C r ^ { - 1 } \\sqrt { S } ( s _ 0 - t _ 1 ) } \\cdot \\frac { r } { 2 K } . \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} k ( j ( f ) ( \\kappa ) ) & = j ( k ( f ) ) ( \\kappa ) \\\\ & = j ( k ) ( j ( f ) ) ( \\kappa ) \\\\ & = l ( j ( f ) ) ( \\kappa ) \\\\ & = l ( j ( f ) ) ( l ( \\kappa ) ) \\\\ & = l ( j ( f ) ( \\kappa ) ) \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} \\Psi \\theta ( t ) = \\int _ 0 ^ t e ^ { - k ( t - s ) } \\Sigma _ \\theta ( s ) d s , \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } ( F _ n ) _ * ( A ) = \\Phi _ 1 ^ * ( A ' ) = : A _ 1 . \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} F _ t \\left ( { t , s } \\right ) = s f \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) - \\frac { s } { t } f _ t \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) , \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} & f ^ 0 ( \\lambda ) \\big | A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\bigl ( 1 - e ^ { i \\lambda \\mu } \\bigr ) ^ { n } \\bigl [ p ^ 0 ( \\lambda ) - \\beta ^ 2 f ^ 0 ( \\lambda ) \\bigr ] _ + + \\lambda ^ { 2 n } C _ { \\mu , N } ^ { \\beta , 0 } \\bigl ( e ^ { i \\lambda } \\bigr ) \\big | \\\\ & = | \\lambda | ^ { n } \\big | 1 - e ^ { i \\lambda \\mu } \\big | ^ { n } \\bigl ( \\alpha _ 1 p ^ 0 ( \\lambda ) + \\alpha _ 2 | \\beta | f ^ 0 ( \\lambda ) \\bigr ) , \\label { D 1 r i v n 2 _ c o i _ i _ s t . n _ d } \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} F u ( \\xi ) = ( 2 \\pi ) ^ { - \\frac { d } { 2 } } \\sum _ { x \\in \\mathbb { Z } ^ d } e ^ { - i x \\cdot \\xi } u [ x ] , \\xi \\in \\mathbb { T } ^ d = [ - \\pi , \\pi ) ^ d \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} = \\sqrt { \\pi } \\ , \\mathrm { e r f c } ( - \\eta ) + \\sum \\limits _ { r = 1 } ^ { N - 1 } \\mu ^ r e ^ { - \\eta ^ 2 } P _ { r - 1 } ( \\eta ) + O ( \\sigma ^ { - \\gamma N } ) , \\sigma \\to \\infty , \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} C _ { s , t } = \\left \\langle \\begin{pmatrix} a _ s \\\\ \\sqrt { 1 - \\| a _ s \\| ^ 2 } \\\\ 0 \\end{pmatrix} , \\begin{pmatrix} b _ t \\\\ 0 \\\\ \\sqrt { 1 - \\| b _ t \\| ^ 2 } \\end{pmatrix} \\right \\rangle \\ \\ ( s , t ) \\in [ m ] \\times [ n ] . \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q | a , b , \\bar { b } ) = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl ( ( e ^ { - t q } - 1 ) e ^ { - b _ 0 t } + ( e ^ { t q } - 1 ) e ^ { - \\bar { b } _ 0 t } \\Bigr ) \\frac { \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } \\Bigr ) . \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} - \\varepsilon u _ { \\varepsilon } ^ { \\prime \\prime } + u _ { \\varepsilon } ^ { \\prime } + \\xi _ { \\varepsilon } - f ( u _ { \\varepsilon } ) & = 0 ( 0 , T ) , \\\\ \\xi _ { \\varepsilon } & \\in \\partial \\phi ( u _ { \\varepsilon } ) ( 0 , T ) , \\\\ u _ { \\varepsilon } ^ { \\prime } ( T ) & = 0 , \\\\ u _ { \\varepsilon } ( 0 ) & = u _ { 0 } , \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} h ^ i ( L ) = h ^ i ( L ^ { - 1 } \\otimes \\omega _ C ) = 0 , \\ i = 0 , 1 . \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} S _ { n } \\left ( - q ^ { - n - 1 / 2 } ; q \\right ) = \\frac { \\left ( - 1 \\right ) ^ { n } q ^ { - \\left ( n ^ { 2 } + n \\right ) / 4 } } { \\left ( q ^ { 1 / 2 } ; q ^ { 1 / 2 } \\right ) _ { n } } \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} y _ t + p _ t = d _ t , ~ \\mbox { f o r } t = 0 , \\dots , T . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{gather*} R = \\frac { W } { \\prod \\limits _ { 1 \\le j \\le m } ( w _ { j } - z ) } . \\end{gather*}"}
-{"id": "4671.png", "formula": "\\begin{align*} H _ { \\mathrm { L T } } ^ { K _ m } = H ^ { n - 1 } _ c ( M _ { m , \\varpi ^ \\Z } \\otimes _ { \\breve { F } } C , \\overline { \\Q } _ \\ell ) = \\bigoplus _ { 0 \\le \\delta < n } H ^ { n - 1 } _ c ( M _ m ^ { ( \\delta ) } \\otimes _ { \\breve { F } } C , \\overline { \\Q } _ \\ell ) . \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ N H \\left ( S _ { m , n } \\right ) \\leq N \\mu L , ~ ~ \\forall m \\in [ 1 : M ] , \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma _ j ( Y ) & = & B _ j \\ , Y , \\ j = 1 , 2 \\end{aligned} \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} D F ( x ) E _ 2 & = D F ( x ) F ( x ) E _ 1 = - F ( x ) D F ( x ) E _ 1 \\\\ \\langle N _ i , D F ( x ) E _ j \\rangle & = - \\langle E _ j , D F ( x ) N _ i \\rangle i , j = 1 , 2 . \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} Z ^ 2 = \\frac { 1 } { 1 + t ^ 2 + u ^ 2 } . \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{gather*} s _ { m } = \\sum _ { n = 1 } ^ { r } \\mu _ { n } \\lambda _ { n } ^ { m } \\ , \\ \\ m \\geq 0 \\ , \\ \\mu _ { n } = \\left ( \\sum \\limits _ { k = 0 } ^ { r - 1 } \\dfrac { P _ { k } ( \\lambda _ { n } ) ^ { 2 } } { D _ { k } D _ { k - 1 } } \\right ) ^ { - 1 } > 0 \\ , \\ P _ { r } ( \\lambda _ { n } ) = 0 \\ , \\ 1 \\leq n \\leq r \\ . \\end{gather*}"}
-{"id": "8840.png", "formula": "\\begin{align*} \\# \\lbrace \\chi | \\chi ^ { d } = 1 \\rbrace \\leq d ^ { n } . \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} w ( a , b ) : = \\left ( \\begin{array} { c c c c } 1 & 0 & 0 & 0 \\\\ a & 1 & 0 & 0 \\\\ b & a \\pi & 1 & 0 \\\\ a ^ { 2 } ( a \\pi ) + a b + b \\pi & a ( a \\pi ) + b & a & 1 \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\Psi _ j ^ { \\ast } \\Omega _ j = \\Psi _ 1 ^ { \\ast } \\Omega _ 1 + \\mathrm { d } \\zeta _ j . \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{gather*} \\pi ( g _ { C , D , E } ) = \\begin{bmatrix} 1 & 0 & 0 \\\\ C ( z ) & 1 & 0 \\\\ D ( z ) & E ( z ) & 1 \\end{bmatrix} , \\end{gather*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\dim \\mathrm { E x t } ^ 2 ( E , E ) = \\dim \\mathrm { E x t } ^ 2 ( L , L ) - 4 d _ 1 d _ 2 . \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\sum _ { i , j = 1 } ^ N a _ { i j } \\frac { \\partial u } { \\partial x _ j } \\frac { \\partial u } { \\partial x _ i } \\leq \\lim _ { m \\rightarrow \\infty } \\inf \\int _ { \\Omega } \\sum _ { i , j = 1 } ^ N a _ { i j } \\frac { \\partial v _ m } { \\partial x _ j } \\frac { \\partial v _ m } { \\partial x _ i } . \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} \\mathcal { \\tilde { B } } _ { c r } ( y _ 0 ) \\subseteq \\mathcal { B } _ r ( y _ 0 ) \\subseteq \\mathcal { \\tilde { B } } _ { C r } ( y _ 0 ) , \\mathcal { \\tilde { B } } _ r ( y _ 0 ) = \\{ y | d _ G ( y , y _ 0 ) < r \\} . \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\xi _ { i } & = & \\dfrac { p _ { i } } { q } + \\theta l _ { i } \\\\ \\theta q & \\cong & 0 \\end{array} \\right . \\end{align*}"}
-{"id": "10168.png", "formula": "\\begin{align*} D _ n : = \\int _ 0 ^ { \\sigma _ n / 2 } \\sqrt { \\log N ( n , t ) } \\dd t , \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l l l l } $ u ( 0 , t ) = 0 $ & $ u ( L , t ) = 0 , $ & $ u _ { x } ( L , t ) = h _ 2 ( t ) $ , & i n $ ( 0 , T ) $ , \\\\ $ v ( 0 , t ) = 0 , $ & $ v ( L , t ) = 0 , $ & $ v _ { x } ( L , t ) = 0 $ , & i n $ ( 0 , T ) $ . \\end{tabular} \\right . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} ( H _ k ) _ { i , j } = e ^ { 2 \\pi \\mathbf { i } ( i - 1 ) ( j - 1 ) / k } i , j \\in [ k ] . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} \\left . \\begin{aligned} p _ 0 & = 0 , p _ 1 = 1 , \\\\ p _ { 2 i } & = p _ i + \\frac 1 2 ( p _ i - p _ { \\lfloor i / 2 \\rfloor } ) \\omega ^ { - 1 } z , i \\geq 1 \\\\ p _ { 2 i + 1 } & = p _ i + \\frac 1 2 ( p _ i - p _ { \\lfloor i / 2 \\rfloor } ) \\omega z , i \\geq 1 . \\\\ \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { m } { m \\choose i } { 1 - r \\choose m + i } ^ { - 1 } \\gamma _ { m + i - 1 } = 0 , \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} P _ { A c c E r } = \\Pr ( S N R < \\Theta ) \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} T _ 2 [ X , Y ; 1 , v ] = v ( v - 1 ) ( 1 - v ( q + t ) ) e _ 2 [ X ] e _ 2 [ Y ] - v ^ 2 ( v - 1 ) ( e _ 2 [ Y ] X ^ 2 + e _ 2 [ X ] Y ^ 2 ) \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} E _ n ( x + 1 ) + E _ n ( x ) = 2 x ^ n . \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} f f '' + ( f ' ) ^ 2 + 1 = c \\sqrt { f '^ 2 + 1 } . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} D _ t : = \\{ x \\in \\mathbb { Z } ^ d \\mid \\exists \\xi \\in \\operatorname { s u p p } F u \\ ( \\ref { s t a t i o n a r y p o i n t } ) \\} . \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( m : n ) } & = \\overrightarrow { C } _ { ( z : n ) } \\otimes \\overrightarrow { C } _ { ( w : n ) } \\\\ & = \\left ( \\bigoplus _ i H _ z ( i , \\phi ( i ) ) \\right ) \\otimes \\left ( \\bigoplus _ j H _ w ( j , j ) \\right ) \\\\ & = \\bigoplus _ i \\bigoplus _ j H _ z ( i , \\phi ( i ) ) \\otimes H _ w ( j , j ) \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} d _ 2 = \\begin{pmatrix} p & q & 0 \\\\ 0 & r & u \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , b _ 2 = \\begin{pmatrix} b _ { 1 1 } & b _ { 1 2 } & b _ { 1 3 } & b _ { 1 4 } \\\\ b _ { 2 1 } & b _ { 2 2 } & b _ { 2 3 } & b _ { 2 4 } \\\\ b _ { 3 1 } & b _ { 3 2 } & b _ { 3 3 } & b _ { 3 4 } , \\end{pmatrix} . \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} h _ { i j } \\geq \\bar { h } _ { i j } , g _ { i j } = \\bar { g } _ { i j } , \\kappa _ i \\geq \\bar { \\kappa } _ i = \\frac { \\cosh u } { \\sinh u } , \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} G _ { k } ^ { \\pm , + } \\left ( \\omega _ { \\pm } \\right ) & = H _ { k } ^ { \\pm , + ; \\varepsilon } \\left ( \\omega _ { \\pm } \\right ) - H _ { k } ^ { \\pm , + ; 0 } \\left ( \\omega _ { \\pm } \\right ) , \\\\ G _ { k } ^ { \\pm , - } \\left ( \\omega _ { \\pm } \\right ) & = H _ { k } ^ { \\pm , - ; \\varepsilon } \\left ( \\omega _ { \\pm } \\right ) - H _ { k } ^ { \\pm , - ; 0 } \\left ( \\omega _ { \\pm } \\right ) , \\end{align*}"}
-{"id": "9900.png", "formula": "\\begin{align*} \\nabla ( \\gamma ( \\tilde { r } ) ( i _ x ( \\tilde { x } - a ) ) ) = \\left ( \\dfrac { \\partial } { \\partial x _ i } ( \\gamma ( \\tilde { r } ) ) ( i _ x ( \\tilde { x } - a ) ) _ j \\right ) + \\left ( \\gamma ( \\tilde { r } ) \\dfrac { \\partial ( i _ x ( \\tilde { x } - a ) ) _ j } { \\partial x _ i } \\right ) = : ( M _ { i j } ) + ( N _ { i j } ) . \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\overline F _ \\alpha ( \\overline X , \\overline Y , \\overline Z ) = \\overline F _ \\alpha ( X ^ \\prime , Y ^ \\prime , Z ^ \\prime ) = F _ \\alpha ^ \\prime ( X ^ \\prime , Y ^ \\prime , Z ^ \\prime ) , \\alpha = 1 , 2 , 3 . \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} \\begin{aligned} h ( c ( y _ k ) + \\nabla c ( y _ k ) ( \\hat x - y _ { k } ) ) & \\leq ( h \\circ c ) ( \\hat x ) + \\frac { r } { 2 } \\| \\hat x - y _ k \\| ^ 2 \\\\ & \\leq a _ k h ( c ( x ) ) + ( 1 - a _ k ) h ( c ( x _ { k - 1 } ) ) \\\\ & + \\rho a _ k ( 1 - a _ k ) \\| x - x _ { k - 1 } \\| ^ 2 + \\frac { r a _ k ^ 2 } { 2 } \\| x - v _ { k - 1 } \\| ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} \\begin{cases} \\Big ( \\tau + U _ s '' ( a ) \\frac { z ^ 2 } { 2 } \\Big ) V ' - U _ s '' ( a ) z V + i V ^ { ( 3 ) } = 0 , z \\neq 0 , \\\\ [ V ] \\big | _ { z = 0 } = - \\tau , \\quad ~ [ V ' ] \\big | _ { z = 0 } = 0 , \\quad ~ [ V '' ] \\big | _ { z = 0 } = - U _ s '' ( a ) , \\\\ \\lim \\limits _ { z \\rightarrow \\pm \\infty } V ~ = ~ 0 , e x p o n e n t i a l l y , \\end{cases} \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} \\xi ( s , \\chi ) = \\big [ s ( s - 1 ) \\big ] ^ { \\delta ( \\chi ) } D _ { \\chi } ^ { s / 2 } \\gamma _ { \\chi } ( s ) L ( s , \\chi ) , \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} s _ i = q _ i - \\lambda ( q _ i - s _ { \\lfloor i / 2 \\rfloor } ) , 0 < \\lambda < 1 . \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} B ^ { ( f ) } _ { 1 } ( x ) - B ^ { ( f ) } _ { 1 } ( x + y ) = f ( 0 ) \\ , y , \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} S _ G = \\begin{pmatrix} S _ \\Gamma & S _ { 1 2 } \\\\ S _ { 2 1 } & S _ { 2 2 } \\end{pmatrix} . \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} \\tau _ U ( 0 , \\mu _ T ) = \\left \\{ \\begin{array} { l l } 1 3 / 6 - 3 \\mu _ T / 2 , & 1 / 3 \\le \\mu _ T \\le 2 / 3 \\\\ 3 / 2 - \\mu _ T / 2 , & 2 / 3 < \\mu _ T \\le 1 \\end{array} . \\right . \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} f ( x ) = g ( x ) + \\mathcal { O } ( x ^ k ) \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} d f _ 4 + ( n - 1 ) f _ 4 \\wedge f _ 8 = k F ; \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} \\phi _ { j , L } ( \\sum _ q z _ q \\otimes \\lambda _ q ) \\otimes 1 - 1 \\otimes \\phi _ { j , L } ( \\sum _ q z _ q \\otimes \\lambda _ q ) = \\sum _ q \\lambda _ q ( z _ q \\otimes 1 - 1 \\otimes z _ q ) ^ { p ^ j } \\in J ^ { p ^ j } \\end{align*}"}
-{"id": "10164.png", "formula": "\\begin{align*} \\mathbb P ( Z _ n ^ * < 0 ) = \\mathbb P ( Z _ { k + 1 , n + k } ^ * < Z _ k ) \\ , . \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} \\star e ^ { t \\vec { \\Delta } } \\star = e ^ { t \\Delta ^ \\perp } , \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} \\mathbb { E } ( S _ { 1 , 0 } ) = \\frac { 1 } { \\mu } . \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{gather*} s _ { n } = \\int \\nolimits _ { - \\infty } ^ { + \\infty } x ^ { n } d \\mu ( x ) \\ , \\ \\ \\ \\ n \\geq 0 \\ , \\ \\ \\ \\ \\mu : = \\sum _ { k = 1 } ^ { n _ { 0 } } \\mu _ { k } \\delta _ { x _ { k } } \\ , \\end{gather*}"}
-{"id": "8809.png", "formula": "\\begin{align*} { J _ { \\sigma } } _ { | _ { H ^ 2 _ 0 ( \\Omega ) } } = J _ { D I R } , \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} P \\mu ( z ) = \\frac { 1 } { \\pi } \\int _ { \\mathbb { D } } \\frac { \\mu ( w ) } { ( 1 - z \\overline { w } ) ^ 2 } \\ , | d w | ^ 2 . \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} P _ { \\mu \\nu } = \\left ( \\begin{array} [ c ] { c c } 0 & F _ { q } \\medskip \\\\ - F _ { p } & i e _ { p q r } G _ { r } \\end{array} \\right ) , Q ^ { \\mu \\nu } = \\left ( \\begin{array} [ c ] { c c } 0 & - G _ { q } \\medskip \\\\ G _ { p } & i e _ { p q r } F _ { r } \\end{array} \\right ) , \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\nu ^ { \\pi ^ * } _ n ( 0 | 0 ) = \\sum _ { x _ n \\in \\{ 0 , 1 \\} } q _ n ( 0 | x _ n , 0 ) \\pi ^ * _ n ( x _ n | 0 ) = q _ n ( 0 | 0 , 0 ) \\pi _ n ( 0 | 0 ) + q _ n ( 0 | 1 , 0 ) \\pi ^ * _ n ( 1 | 0 ) . \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} { \\cal P } _ { 0 , n } ^ { A . J } ( \\kappa ) \\triangleq \\Big \\{ \\pi _ t ( x _ t | y _ { t - J } ^ { t - 1 } ) , ~ t = 0 , 1 , \\ldots , n : \\frac { 1 } { n + 1 } { \\bf E } ^ { \\pi } \\Big ( c ^ { A . N } _ { 0 , n } ( X ^ n , Y ^ { n - 1 } ) \\Big ) \\leq \\kappa \\Big \\} , ~ \\kappa \\in [ 0 , \\infty ) \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} H _ n = \\log _ { 2 } n + \\sqrt { 2 \\log _ { 2 } n } + o ( \\sqrt { \\log n } ) . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} F ( x ) D _ v F ( x ) = D _ { I ( v ) } F ( x ) v \\in T _ x M . \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } | a _ i | + | b | \\leq C \\bar c , \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( ( w w ^ * ) * ( R ( w ) ^ * R ( w ) ) ) & = \\varphi ( w w ^ * ) \\varphi ( R ( w ) ^ * R ( w ) ) \\\\ & = \\| w \\| _ 2 ^ 2 \\| R ( w ) \\| _ 2 ^ 2 = \\| w \\| ^ 4 _ 2 \\end{aligned} \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} | Y ( t , x - \\xi ) | \\leq \\left \\{ \\begin{array} { l l } $ $ C t ^ { - 1 + \\frac { \\alpha } { 2 } } p ( t , x - \\xi ) $ $ , & \\hbox { $ d = 1 $ ; } \\\\ $ $ C t ^ { \\alpha - \\frac \\alpha 2 \\gamma + \\nu _ 0 \\alpha - 2 } | x - \\xi | ^ { - d + \\gamma - 2 \\nu _ 0 + \\frac { 2 } { \\alpha } } p ( t , x - \\xi ) $ $ , & \\hbox { $ d \\geq 2 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} h ( J _ 1 X , J _ 1 Y ) + h ( X , Y ) = \\frac { 1 } { 2 n - 1 } g ( X , Y ) J _ 1 ( p ^ \\bot _ 1 ) , \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} I _ { \\mu } ( z ) = \\left ( \\frac { z } { 2 } \\right ) ^ \\mu \\sum _ { k = 0 } ^ \\infty \\frac { ( z ^ 2 / 4 ) ^ k } { k ! \\Gamma ( \\mu + k + 1 ) } , \\mu \\in \\mathbb { R } , \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } J _ { \\nu - \\alpha } ^ { ( 2 ) } \\left ( - 2 i q ^ { n / 2 } ; q \\right ) \\left ( i q ^ { - n / 2 } \\right ) ^ { \\nu - \\alpha } = \\frac { 1 } { \\sqrt { 2 \\pi \\log q ^ { - 1 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + \\alpha x i \\right ) } { \\left ( q ; q \\right ) _ { \\infty } \\left ( - q ^ { \\nu + 1 / 2 } e ^ { - i x } ; q \\right ) _ { n } } d x , \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} \\Phi ( u , v ) & = \\Phi _ 0 ( u , v ) + \\Phi _ r ( u , v ) , \\\\ \\Phi _ 0 ( u , v ) = \\int _ M \\textbf { A } ( \\nabla u , \\nabla v ) d \\mu , \\ , \\ , \\ , \\ , & \\Phi _ r ( u , v ) = \\int _ \\Omega \\textbf { b } ( \\nabla u ) v d \\mu + \\int _ \\Omega c u v d \\mu . \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} H ( p ) \\ ; = \\ ; \\max _ { c \\in C } \\left \\{ \\langle - f ( c ) , p \\rangle - L ( c ) \\right \\} . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} K ( h ^ { \\frac { 1 } { 2 ^ n } } , 2 ) ^ { r _ n } H _ { \\nu } ( a , b ) & \\leqslant a \\nabla b - \\sum _ { k = 0 } ^ { n - 1 } r _ { k } \\big [ H _ { \\frac { m _ k } { 2 ^ k } } ( a , b ) - 2 H _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } ( a , b ) + H _ { \\frac { m _ k + 1 } { 2 ^ k } } ( a , b ) \\big ] \\\\ & \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ n } } , 2 ) ^ { R _ n } H _ { \\nu } ( a , b ) , \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} K ^ { ( \\delta ) } = \\biguplus _ { m \\in \\omega } K _ m ^ { ( \\delta ) } \\uplus \\{ b \\} , \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} \\int _ Q ( \\tilde { u } ( t , x ) - \\xi ( t , x ) ) \\mu ( d t , d x ) = 0 ~ ~ \\mathbb { P } \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} \\eta \\in H _ { e } ^ { 1 , \\beta } ( Q _ { T } ) : = W ^ { 1 , 2 } ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) \\cap L ^ { 2 } ( [ 0 , T ] ; H _ { e } ^ { \\beta } ( \\Omega ) ) \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} g _ n ( s , y , t , x ) : = \\frac { 1 } { n ! } Y ( t - s _ { \\sigma ( n ) } , x - y _ { \\sigma ( n ) } ) \\cdots Y ( s _ { \\sigma ( 2 ) } - s _ { \\sigma ( 1 ) } , y _ { \\sigma ( 2 ) } - y _ { \\sigma ( 1 ) } ) \\ , ; \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} \\sigma _ 1 ( U ) = \\sigma _ 2 ^ N ( B _ 1 ) \\ , U \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} \\frac { g ( x + y ) - g ( y ) - g \\left ( x \\right ) } { g ( x ) } = g ( \\frac { 1 } { 1 + y } ) + g ( \\frac { y } { 1 + y } ) - 1 = : H \\left ( y \\right ) \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\norm { b ( y ) } = \\inf _ W \\sup _ { x \\in W \\cap \\mathbb { Q } } \\norm { b ( x ) } , \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} L _ { y _ 0 } v = L _ { y _ 0 } v _ { y _ 0 } + \\tilde { f } \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} a \\nabla _ { \\nu } b \\geqslant a \\sharp _ { \\nu } b + \\sum _ { k = 0 } ^ { \\infty } r _ { k } \\big [ \\big ( a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } - \\big ( a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ] ^ { 2 } . \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} \\sigma _ E ^ { m ( x ) } ( \\kappa ( \\sigma _ E ( x ) ) ) = \\sigma _ E ^ { l ( x ) } ( \\kappa ( x ) ) \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} S = - \\langle \\ln P \\rangle = \\beta \\langle E \\rangle + \\ln Z \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} c _ i = \\frac 1 2 ( p _ i + q _ i ) = p _ i + \\frac 1 2 \\omega ^ { a _ { h ( i ) } ( i ) } \\left ( \\frac { z } { 2 } \\right ) ^ { h ( i ) } \\sum _ { j = 1 } ^ { k - h ( i ) } z ^ j \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\dim \\pi ( \\mathcal { Z } ) = 3 g - 3 + \\rho + h ^ 1 ( N _ C ) = 4 d - 1 5 + h ^ 1 ( N _ C ) \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} \\Psi _ { \\gamma + t } ( x ) & = e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } + \\sum \\limits _ { \\delta \\in \\Gamma ( k + ) } c ( \\gamma , \\delta ) e ^ { i \\left \\langle \\gamma + \\delta + t , x \\right \\rangle } = \\\\ & e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } + \\sum _ { p \\in \\mathbb { N } } \\sum \\limits _ { \\delta \\in \\Gamma ( k , p ) } c ( \\gamma , \\delta ) e ^ { i \\left \\langle \\gamma + \\delta + t , x \\right \\rangle } , \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} & \\left [ \\mathbf { A } \\operatorname { d i v } \\mathbf { B } - \\mathbf { B } \\operatorname { d i v } \\mathbf { A } + \\mathbf { A } \\times \\operatorname { c u r l } \\mathbf { B } - \\mathbf { B } \\times \\operatorname { c u r l } \\mathbf { A } - \\operatorname { c u r l } \\left ( \\mathbf { A } \\times \\mathbf { B } \\right ) \\right ] _ { p } \\\\ & \\quad = A _ { q } \\frac { \\partial B _ { q } } { \\partial x _ { p } } - B _ { q } \\frac { \\partial A _ { q } } { \\partial x _ { p } } , \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{gather*} \\deg P _ { n _ { k } } = n _ { k } \\ , \\ \\ P _ { n _ { k } + 1 } \\equiv 0 \\ , \\ \\ . . . \\ \\ , \\ P _ { n _ { k + 1 } - 2 } \\equiv 0 \\ , \\ \\ P _ { n _ { k + 1 } - 1 } = \\gamma _ { k } P _ { n _ { k } } \\ , \\ \\ \\deg P _ { n _ { k + 1 } } = n _ { k + 1 } \\ , \\end{gather*}"}
-{"id": "4850.png", "formula": "\\begin{align*} \\langle \\psi , \\psi ' \\rangle = \\tfrac { i } { 2 } \\int _ { X } \\psi \\wedge \\overline { \\psi ' } . \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{gather*} g ^ { [ k ] ( \\alpha ) } = g ^ { [ k ] ( \\alpha ) } _ { - } g ^ { [ k ] ( \\alpha ) } _ { 0 + } , \\end{gather*}"}
-{"id": "950.png", "formula": "\\begin{align*} a \\rightharpoonup b = S ( a _ 1 ) ( a _ 2 \\circ b ) , a , b \\in A . \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} ( K _ { X } + \\beta C . C ) = ( K _ { \\tilde { X } } + \\beta \\tilde { C } + \\sum a _ { i } E _ { i } . C ) \\le - 1 - \\beta + \\sum ( E _ { i } . \\tilde { C } ) \\le - \\beta < 0 \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} X = X _ { 1 } \\times \\dots \\times X _ { n } = \\prod \\limits _ { i = 1 } ^ { n } X _ { i } , \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} S \\in \\mathbb { R } ^ { p _ 1 \\times p _ 2 } S _ { i j } = \\left \\{ \\begin{array} { l l } \\sigma _ i ( S ) , & i = j = 1 , \\cdots , r \\\\ 0 , & \\end{array} \\right . \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{gather*} \\sum \\nolimits _ { k = 0 } ^ { r - 1 } \\ d _ { n , k } ^ { ( r ) } \\ s _ { k + m } ^ { ( r ) } = \\ s _ { r + n + m } ^ { ( r ) } \\ , \\ \\ \\ \\ \\ m \\geq 0 \\ , \\end{gather*}"}
-{"id": "1705.png", "formula": "\\begin{align*} H _ n ^ { \\alpha _ n } H _ { n - 1 } ^ { \\alpha _ { n - 1 } - \\alpha _ n } \\cdots H _ 2 ^ { \\alpha _ 2 - \\alpha _ 3 } H _ 1 ^ { \\alpha _ 1 - \\alpha _ 2 } = \\left ( \\frac { H _ 1 } { H _ 0 } \\right ) ^ { \\alpha _ 1 } \\left ( \\frac { H _ 2 } { H _ 1 } \\right ) ^ { \\alpha _ 2 } \\cdots \\left ( \\frac { H _ n } { H _ { n - 1 } } \\right ) ^ { \\alpha _ n } \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} F \\left ( { t , s } \\right ) = t s H \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} \\Phi ( x , y , t ) = S ^ + ( t , t _ 0 ) S ^ - ( t , t _ 0 ) \\Phi ( \\cdot , t _ 0 ) ( x , y ) = S ^ - ( t , t _ 0 ) S ^ + ( t , t _ 0 ) \\Phi ( \\cdot , t _ 0 ) ( x , y ) , \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{gather*} D _ { r - 1 } = \\begin{vmatrix} s _ { 0 } & s _ { 1 } & \\ldots & s _ { r - 2 } & s _ { r - 1 } \\\\ s _ { 1 } & s _ { 2 } & \\ldots & s _ { r - 1 } & s _ { r } \\\\ \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\ s _ { r - 2 } & s _ { r - 1 } & \\ldots & s _ { 2 r - 4 } & s _ { 2 r - 3 } \\\\ s _ { r - 1 } & s _ { r } & \\ldots & s _ { 2 r - 3 } & s _ { 2 r - 2 } \\\\ \\end{vmatrix} \\neq 0 \\ , \\end{gather*}"}
-{"id": "6045.png", "formula": "\\begin{align*} \\big \\langle \\alpha _ p ( \\omega ) , \\mu \\big \\rangle _ { Z } = \\big \\langle \\omega , \\mu \\big \\rangle _ { Z _ { 1 } } , \\big \\langle \\beta _ p ( \\mu ) , \\nu \\big \\rangle _ { Z _ { 2 } } = \\big \\langle \\mu , \\nu \\big \\rangle _ { Z _ { 2 } } , \\big \\langle \\delta _ p ( \\nu ) , \\omega ' \\big \\rangle _ { Z _ 1 } = \\big \\langle \\nu , i _ { e _ { \\mathfrak { n } } } \\omega ' \\big \\rangle _ Y . \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} Z ( \\mu ) : = \\{ \\mu x \\mid x \\in X , r ( \\mu ) = s ( x ) \\} \\subseteq X . \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} \\chi ' ( C _ { 1 2 } ) - \\chi ' ( C _ { 1 , \\mathrm { a b s } } ) - \\chi ' ( C _ { 2 , \\mathrm { r e l } } ) = 2 \\chi ' . \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} d _ 1 ( t ) & : = 1 0 0 + 2 0 t , \\\\ d _ 2 ( t ) & : = 5 0 0 \\left ( 1 - \\tfrac { 2 } { T } | t - T / 2 | \\right ) , \\\\ d _ 3 ( t ) & : = 2 5 0 ( 1 + \\sin ( 2 \\pi t / T ) ) , \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} v _ { e a } ^ i ( f ^ * _ { a v g } , g ^ * _ { a v g } ) = \\lim _ { \\beta \\uparrow 1 } ( 1 - \\beta ) v _ \\beta ^ i ( f _ \\beta ^ * , g _ \\beta ^ * ) , \\ \\forall \\ i = 1 , 2 . \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} [ \\bar { \\xi } _ { 0 , 0 } , \\bar { x } ^ { \\pm } _ { 0 , s } ] = 0 , \\ [ \\bar { \\xi } _ { 0 , 1 } , \\bar { x } ^ { \\pm } _ { 0 , s } ] = 0 , \\ [ \\bar { \\xi } _ { 0 , 2 } , \\bar { x } ^ { \\pm } _ { 0 , s } ] = \\mp 2 \\beta ^ 2 \\bar { x } ^ { \\pm } _ { 0 , s } , \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( q ; q ^ { 2 } \\right ) _ { n } q ^ { 2 n ^ { 2 } } c ^ { 2 n } } { \\left ( q ^ { 2 } , c q , c q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } } = \\frac { 1 } { \\left ( c q ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } c ^ { n } } { \\left ( q ; q \\right ) _ { n } } A _ { q } ^ { 2 } \\left ( c q ^ { n } \\right ) . \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} A _ { _ { L F } } u : = ( A _ { _ { S H } } + B ) u , u \\in D ( A _ { _ { L F } } ) : = W ^ { 4 , p } ( \\mathbb { R } ^ n ) \\cap L ^ p _ \\sigma ( \\mathbb { R } ^ n ) , \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{gather*} \\big ( \\Gamma ^ { ( \\alpha ) } _ { C } \\big ) ^ { \\ell } = \\prod _ { i = 1 } ^ { \\ell } c ^ { ( \\alpha ) } _ { i } \\prod _ { j = 1 } ^ { \\ell } E _ { 1 0 } ( z _ { j } ) , \\end{gather*}"}
-{"id": "911.png", "formula": "\\begin{align*} H ^ * ( B G , \\Z ) = \\Z [ x ] / m x . \\end{align*}"}
-{"id": "10082.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p - q - r } y ^ q ( a x + b y ) ^ r } { z ^ { p } } = \\dfrac { y ^ q ( a x + b y ) ^ r } { x ^ { - p ' } z ^ { p ' + q + r } } , p ' = p - q - r . \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} \\begin{cases} \\alpha ^ { \\frac 1 3 } > \\frac 1 2 & ~ m = 5 , \\\\ \\alpha ^ { \\frac 1 3 } = \\frac 1 2 & ~ m = 6 , \\\\ \\alpha ^ { \\frac 1 3 } < \\frac 1 2 & ~ m \\geqslant 7 . \\end{cases} \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} v _ { 2 n - 1 } ( P _ C \\otimes 1 ) & = ( P _ C \\otimes s _ { n _ 0 , n } ) ( 1 \\otimes f _ n - v _ { 2 n - 2 } ^ * v _ { 2 n - 2 } ) , \\\\ v _ { 2 n - 1 } ( P _ D \\otimes 1 ) & = \\sum _ { k = 1 } ^ { N _ D } ( S _ { c ( k ) } S _ { d _ k } S _ { d _ k } ^ * \\otimes s _ { n _ k , n } ) ( 1 \\otimes f _ n - v _ { 2 n - 2 } ^ * v _ { 2 n - 2 } ) , \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\lambda _ { \\ominus } ( 2 ) & = \\lambda _ 2 ( B _ 1 ; 2 ) = \\lambda _ 3 ( B _ 1 ; 2 ) , \\\\ \\lambda _ { \\oplus } ( 2 ) & = \\lambda _ 4 ( B _ 1 ; 2 ) = \\lambda _ 5 ( B _ 1 ; 2 ) , \\\\ \\lambda _ { \\circledcirc } ( 2 ) & = \\lambda _ 6 ( B _ 1 ; 2 ) , \\\\ \\lambda _ { \\circledast } ( 2 ) & = \\lambda _ 7 ( B _ 1 ; 2 ) = \\lambda _ 8 ( B _ 1 ; 2 ) , \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} K _ n = \\max _ { 1 \\leq i \\leq n } \\max \\left \\{ \\left | \\frac { i } { n } - F _ 0 ( e _ i ) \\right | , \\left | \\frac { i - 1 } { n } - F _ 0 ( e _ i ) \\right | \\right \\} , \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} \\widetilde \\psi : = \\left \\{ \\begin{array} { l l l } h & \\textrm { i n } \\ , \\ , Q _ r \\\\ \\psi & \\textrm { i n } \\ , \\ , \\Omega _ T \\setminus Q _ r \\end{array} \\right . , \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} u '' + h ( t , u ) = 0 , \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + \\delta + t , x \\right \\rangle } ) = c ( \\gamma , \\delta ) ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) . \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} \\mathfrak { L } \\Big ( \\widehat { \\boldsymbol { \\theta } } \\Big ) : = \\frac { 1 } { n } \\left \\| \\mathbf { x } _ 1 ^ n - \\mathbf { X } \\widehat { \\boldsymbol { \\theta } } \\right \\| _ 2 ^ 2 , \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} \\mathcal { F } ( t ) = \\mathcal { F } _ 1 ( t ) + \\mathcal { F } _ 2 ( t ) + N \\mathcal { E } ( t ) \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} T = \\prod _ { i = 1 } ^ k \\P _ i ^ { s _ i } , \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} u ^ { \\prime } + \\mathrm { D } \\phi ( u ) & \\ni f ( u ) \\qquad ( 0 , T ) , \\\\ u ( 0 ) & = u _ { 0 } \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ X ^ { \\frac { q } { \\beta } } \\bigr ] \\Gamma ( 1 - \\frac { q } { \\beta } ) = { \\bf E } \\bigl [ X ^ { \\frac { q } { \\beta } } \\bigr ] \\Gamma ( 1 - \\frac { q } { \\beta } ) \\Big | _ { \\beta = 1 } . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} N ( \\sigma , T , \\chi ) : = \\# \\{ \\rho = \\beta + i \\gamma : L ( \\rho , \\chi ) = 0 , \\sigma < \\beta < 1 , | \\gamma | \\leq T \\} \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\frac { d p _ t ( \\cdot | x ^ { t - 1 } , y ^ { t - 1 } ) } { d { r } _ t ( \\cdot | x ^ { t - 1 } , y ^ t ) } ( x _ t ) . \\frac { d { q } _ t ( \\cdot | y ^ { t - 1 } , x ^ { t } ) } { d s _ t ( \\cdot | y ^ { t - 1 } , x ^ { t - 1 } ) } ( y _ t ) = 1 - a . a . ~ ( x ^ t , y ^ t ) , ~ t \\in \\mathbb { N } _ 0 ^ n . \\end{align*}"}
-{"id": "10006.png", "formula": "\\begin{align*} \\phi _ { \\mu , i ' } ^ + ( x ) = p r o j _ { D _ { i ' } } ( f x ) . \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} - 8 \\sin ( k / 2 ) \\sin ( ( \\phi + \\varphi - k ) / 2 ) \\cos ( ( \\phi - \\varphi ) / 2 ) = 0 . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} \\frac { \\partial S } { \\partial t } \\propto a _ n ( g , D ) = \\zeta ( 0 , g , D ) \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\zeta ^ i _ { t | s } & = \\sum _ { j = 0 } ^ n F ^ \\infty _ { i j } ( t - s ) \\zeta ^ j _ s \\\\ & = \\sum _ { j = 0 } ^ n f ^ \\infty _ { i j } ( t - s ) \\nu ^ j _ s \\pi \\\\ & = \\sum _ { j = 0 } ^ i { n - j \\choose n - i } \\exp ( - ( n - i ) \\lambda _ \\infty \\ , ( t - s ) ) ( 1 - \\exp ( - \\lambda _ \\infty \\ , ( t - s ) ) ) ^ { i - j } \\nu ^ j _ s \\pi \\\\ & = { n - N _ s \\choose n - i } \\exp ( - ( n - i ) \\lambda _ \\infty \\ , ( t - s ) ) ( 1 - \\exp ( - \\lambda _ \\infty \\ , ( t - s ) ) ) ^ { i - N _ s } \\pi , \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} f ^ * = \\begin{cases} f _ 1 & \\ q < \\frac { 2 } { 3 } \\\\ f _ 2 & \\ q > \\frac { 2 } { 3 } \\\\ \\big \\{ ( p , 1 - p ) : 0 \\leq p \\leq 1 \\big \\} & \\ q = \\frac { 2 } { 3 } . \\end{cases} \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} & \\begin{cases} \\varphi _ t + \\varphi _ { x x x } + \\frac { a b } { c } \\psi _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ \\psi _ t + \\frac { r } { c } \\psi _ x + a \\varphi _ { x x x } + \\frac { 1 } { c } \\psi _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ \\varphi ( x , 0 ) = \\varphi ^ 0 ( x ) , \\ , \\ , \\psi ( x , 0 ) = \\psi ^ 0 ( x ) , & \\ , \\ , ( 0 , L ) , \\end{cases} \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} F ( x , y ; z ) = x - \\frac { z ( t ^ 2 - 2 \\Delta t + 1 ) } { ( z - 1 ) ( t ^ 2 z - 2 \\Delta t + 1 ) } y - r ( z ) . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} i _ { 0 , * } , i _ { \\infty , * } : H ^ * ( B G , \\Z ) \\to H ^ * _ G ( \\P ^ 1 ) \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} \\| \\mu \\| ^ * & = \\underset { w \\in W } { \\max } \\langle w \\mu , \\rho \\rangle \\\\ \\| \\mu \\| ^ * _ H & = \\underset { w \\in W _ H } { \\max } \\langle w \\mu , \\rho _ H \\rangle \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} \\mathbf { u } ( 0 , \\cdot ) = \\tilde { \\mathbf { u } } ^ { 0 } , \\mathbf { H } ( 0 , \\cdot ) = \\tilde { \\mathbf { H } } ^ { 0 } G . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} f _ { m } = c f _ { g _ 1 } f _ { g _ 2 } \\cdots f _ { g _ w } \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} f _ { 1 } ( u , v ) & = p - u g ( v ) , \\\\ f _ { 2 } ( u , v ) & = k ( u g ( v ) - v ) , \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} A \\cdot ( c _ t , \\ldots , c _ { h + g - 1 } ) ^ T = \\vec { 0 } . \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\tilde { C } : = \\frac { \\| \\mathbf { u } ^ { 0 } \\| _ { \\mathcal { H } } } { \\kappa } \\geq \\| \\nabla \\mathbf { u } _ { 2 } \\| _ { L ^ { 2 } ( ( 0 , T ) \\times G , \\mathbb { R } ^ { k \\times d } ) } \\| \\nabla \\bar { \\mathbf { u } } \\| _ { L ^ { 2 } ( ( 0 , T ) \\times G , \\mathbb { R } ^ { k \\times d } ) } \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} \\mathcal { J } = \\left ( \\begin{array} [ c ] { c c } \\mathcal { J } _ { + } & 0 \\\\ 0 & \\mathcal { J } _ { - } \\end{array} \\right ) , \\ \\mathcal { L = } \\left ( \\begin{array} [ c ] { c c } \\mathcal { L } _ { + } - \\mathcal { B } & \\mathcal { B } \\\\ \\mathcal { B } & \\mathcal { L } _ { - } - \\mathcal { B } \\end{array} \\right ) \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} \\mu ( \\overset { h _ { N } - 1 } { \\underset { i = 0 } { \\cup } } ( T ^ { q } ) ^ { i } ( A ^ { * } _ { N } ) ) & > ( 1 - \\frac { ( q - 1 ) ( q - 2 ) } { 2 } \\frac { 1 } { b ^ { N - 1 } } ) ( 1 - \\frac { 2 } { h _ { N + 1 } } ) \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} d m _ t + \\frac 1 2 [ m _ t , m _ t ] & = 0 & \\frac { \\partial m _ t } { \\partial t } + d h _ t + [ h _ t , m _ t ] & = 0 . \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\gamma _ 2 = \\frac { 1 } { | G | } \\left ( \\sum \\limits _ { g \\in G _ { n - 1 } } \\frac { - \\mu _ 1 ( g ) } { \\big ( \\mu _ 1 ( g ) - 1 \\big ) ^ 2 } + \\sum \\limits _ { g \\in G _ { n - 2 } } \\frac { 1 } { \\big ( \\mu _ 1 ( g ) - 1 \\big ) \\big ( \\mu _ 2 ( g ) - 1 \\big ) } \\right ) , \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} H ( D _ x L _ \\lambda , x ) = H ( - D _ y L _ \\lambda , y ) , \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} - H ^ { j k } H _ { j k , r r } + \\frac { 1 } { 2 } H ^ { i j } H _ { j k , r } H ^ { k l } H _ { l i , r } + 2 f _ { , r r } + 1 = 0 \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} K = \\{ ( x , y , z ) : 0 < x < 1 , \\ ; 0 < y < 1 , \\ ; 0 < z < 1 \\} \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} \\dot { u } ( t ) + \\mathcal { A } ( t ) u ( t ) = f ( t ) L ^ { 2 } ( 0 , T ; \\mathcal { V } ' ) , u ( 0 ) = u ^ { 0 } \\in \\mathcal { H } . \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} \\left \\langle v _ { t } , P _ n \\right \\rangle = 0 , \\ \\ n \\geq d ( l - 1 ) + r + t + 1 , \\ \\ 1 \\leq r \\leq d , \\ \\ 0 \\leq t \\leq d - 1 . \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ A ( \\Omega ) } = \\| u ^ { \\ast } \\| _ { L ^ A ( 0 , | \\Omega | ) } \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} \\mathfrak { C } _ { 5 } = \\left ( Z _ { 4 } \\times A d S _ { 5 } \\right ) _ { H } . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} \\zeta ( t ) ( f g ) = \\zeta ( t ) ( f ) \\zeta ( t ) ( g ) , \\qquad \\forall f , g \\in C ^ { \\omega } ( M ) . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} \\tau _ n ( 1 ) & = \\tau _ n ( n ) = 1 , & \\tau _ { n + 1 } ( j ) & = \\tau _ n ( j - 1 ) + j \\tau _ n ( j ) , 2 \\le j \\le n . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} \\begin{array} { c } c { } \\gamma ^ { \\infty } { } { } \\\\ { } P ( X ) { } \\gamma A : = \\left [ \\cup _ { \\alpha \\in J } \\psi _ { \\alpha } d _ { \\alpha } \\psi _ { \\alpha } ^ { - 1 } ( A ) \\right ] \\cup A . \\end{array} \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} u = i ^ { * } ( f _ { \\epsilon } ( u ) ) \\ \\ \\ u \\in { H } _ { \\epsilon } \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} \\begin{cases} - d _ 1 \\Delta u = g ( u ) \\left ( f ( u ) - v \\right ) , & x \\in \\Omega , \\\\ - d _ 2 \\Delta v = v h ( v ) , & x \\in \\Omega , \\\\ \\partial _ \\nu u = \\partial _ \\nu v = 0 , & x \\in \\partial \\Omega , \\\\ \\end{cases} \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} & ( T _ 2 ) _ { i j } ( r ^ { 6 - n } ) _ { , j i } \\\\ = & [ ( n - 2 ) \\sigma _ 1 ( A ) g _ { i j } - 8 A _ { i j } ] ( 6 - n ) [ ( 4 - n ) r ^ { 2 - n } x ^ i x ^ j + r ^ { 4 - n } \\delta _ { i j } + O ( r ^ { 6 - n } ) ] \\\\ = & ( 6 - n ) [ 4 ( n - 4 ) \\sigma _ 1 ( A ) r ^ { 4 - n } - 8 ( 4 - n ) A _ { i j } x ^ i x ^ j r ^ { 2 - n } ] + O ( r ^ { 7 - n } ) . \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k + 1 } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau , 0 ; v _ { s + 1 } , \\dots , v _ { s + k } , v _ { s + k + 1 } ; \\right . \\\\ & \\left . \\omega _ 1 , \\dots , \\omega _ k , \\omega _ { k + 1 } ; i _ 1 , \\dots , i _ k , i _ { k + 1 } \\right ] \\\\ & \\in \\mathcal { K } _ { s + k + 1 } \\cap \\mathcal { U } _ { s + k + 1 } ^ \\eta \\end{aligned} \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} I ^ { \\alpha } \\left ( \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { \\cdot } h ^ { k } ( s ) d w _ { s } ^ { k } \\right ) ( t ) = \\sum _ { k = 1 } ^ { \\infty } \\left ( I ^ { \\alpha } \\int _ { 0 } ^ { \\cdot } h ^ { k } ( s ) d w _ { s } ^ { k } \\right ) ( t ) \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} \\frac { \\tilde \\mu } { 2 } \\| x ^ * - v _ N \\| ^ 2 \\leq & \\frac { \\tilde \\mu } { 2 } \\| x ^ * - v _ 0 \\| ^ 2 + \\frac { \\rho M ^ 2 N ( N + 3 ) } { 4 } + \\frac { N r M ^ 2 } { 2 } + \\sum ^ N _ { i = 1 } \\frac { \\delta _ i } { a _ i } \\\\ & + 2 \\sum _ { i = 1 } ^ N \\frac { \\varepsilon _ i } { a _ i ^ 2 } + \\sqrt { 2 \\tilde \\mu } \\sum _ { i = 1 } ^ N \\| x ^ * - v _ { i } \\| \\cdot \\sqrt { \\frac { \\delta _ i } { a _ i } } . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nabla _ \\xi q ( t , s ) = \\int _ s ^ t A ( p ( \\tau , s ) ) \\nabla _ \\xi p ( \\tau , s ) d \\tau , \\\\ \\nabla _ \\xi p ( t , s ) = I - \\int _ s ^ t \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , s ) ) \\nabla _ \\xi q ( \\tau , s ) d \\tau , \\end{array} \\right . \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} C _ { } & = \\lim _ { N \\rightarrow \\infty } \\sup _ { \\{ p ( x _ t | s _ { t - 1 } , y ^ { t - 1 } ) \\} _ { t = 1 } ^ N } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N I ( X _ i , S _ { i - 1 } ; Y _ i | Y ^ { i - 1 } ) . \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} R ' & = \\{ X \\in R : \\ , \\underline { X } + \\overline { X } \\neq n \\} , \\\\ R '' & = \\{ X \\in R : \\ , \\underline { X } + \\overline { X } = n \\} . \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} x _ 0 & : = \\{ s \\in S \\ | \\ n \\geq 0 s = s _ 0 , \\dots , s _ n s _ t \\in a _ 2 t < n s _ n \\in a _ 1 \\} \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} \\chi _ { R a _ ! b ^ * E ^ { \\theta } , \\varphi } ( g ) & = \\sum _ { i \\in \\mathbb { Z } } ( - 1 ) ^ i \\cdot \\mathrm { T r } \\left ( \\varphi _ g , \\mathcal { H } ^ i ( R a _ ! ( b ^ * E ^ { \\theta } ) ) _ g \\right ) \\\\ & = \\sum _ { i \\in \\mathbb { Z } } ( - 1 ) ^ i \\cdot \\mathrm { T r } \\left ( \\varphi , { H } _ c ^ i ( a ^ { - 1 } ( g ) , b ^ * E ^ { \\theta } ) \\right ) . \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} ( \\sqrt { a } - \\sqrt { b } ) ^ 2 - a \\nabla _ { \\nu } b & = b \\nabla _ { \\nu } a - 2 \\sqrt { a b } \\\\ & \\geq - a \\sharp _ { \\nu } b + \\sum _ { k = 0 } ^ { \\infty } r _ { k } \\big [ \\big ( a ^ { \\frac { m _ k } { 2 ^ k } } b ^ { 1 - \\frac { m _ k } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } - \\big ( a ^ { \\frac { m _ k + 1 } { 2 ^ k } } b ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ] ^ { 2 } . \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{gather*} T _ { 1 } ^ { k } T _ { 2 } ^ { \\ell } = ( \\pm 1 ) T _ { 1 } ^ { n _ { c } } T _ { 2 } ^ { n _ { e } } T _ { 3 } ^ { n _ { d } } , \\end{gather*}"}
-{"id": "2287.png", "formula": "\\begin{align*} \\epsilon : = \\liminf _ { n \\to \\infty } \\mu ( T ^ n C _ h \\cap C _ h ^ c ) > 0 . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} \\partial _ z p = \\theta , \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} I _ \\delta ( \\omega ( T + \\cdot ) ) ( t ) = I _ \\delta ( \\omega ) ( T + t ) - I _ \\delta ( \\omega ) ( T ) , \\ t \\ge 0 . \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } g _ { i j } = - R _ { i j } + g _ { i j } , \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} ( \\Delta ( z ) ) & = p \\mathfrak { a } _ { \\mathfrak { 0 } } + \\mathfrak { a } _ \\infty , \\\\ ( \\Delta _ p ( z ) ) & = \\mathfrak { a } _ { \\mathfrak { 0 } } + p \\mathfrak { a } _ \\infty , \\\\ ( \\eta ( z ) ^ { 1 2 } \\eta ( p z ) ^ { 1 2 } ) & = \\frac { p + 1 } { 2 } \\mathfrak { a } _ { \\mathfrak { 0 } } + \\frac { p + 1 } { 2 } \\mathfrak { a } _ \\infty , \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} d X _ s ^ { t , x } & = a ( s , X ^ { t , x } _ s ) d s + \\sigma ( s , X ^ { t , x } _ s ) d W _ s , \\forall s \\in [ t , T ] , \\\\ X ^ { t , x } _ s & = x , \\forall s \\in [ 0 , t ] . \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} h \\left ( f ( \\tilde u ) \\right ) - c g ( \\tilde u ) = h \\left ( f ( \\check u ) \\right ) - c g ( \\check u ) . \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{gather*} \\delta _ { a } = \\begin{bmatrix} 0 & 0 & \\dots & 1 & \\dots & 0 \\end{bmatrix} , \\end{gather*}"}
-{"id": "834.png", "formula": "\\begin{align*} I _ { k } = \\frac { 1 } { k } a ^ { k } ( k \\geq 1 ) \\ , . \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\pi _ { \\alpha C } ( z ) \\ ; = \\ ; \\mbox { s h r i n k } _ 1 ( z , \\bar { \\mu } ) , \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\cfrac { v _ { p , k } ( n ) } { n } = \\cfrac { 1 } { p ^ { k - 1 } } , \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} \\tilde { a } ^ { i j } ( y ) = a ^ { i j } ( \\xi ( y ) ) & = a ^ { i j } ( \\xi _ 0 ) + ( a ^ { i j } ( \\xi ) - a ^ { i j } ( \\xi _ 0 ) ) \\\\ & = a ^ { i j } ( \\xi _ 0 ) + \\int \\limits _ { 0 } ^ { 1 } \\nabla _ x a ^ { i j } ( ( 1 - t ) \\xi _ 0 + t \\xi ( y ) ) d t \\cdot ( \\xi _ 0 - \\xi ( y ) ) . \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} \\frac { R ' _ { \\sf u p p e r } ( 0 ) } { R _ { \\sf l o w e r } ( 0 ) } & = \\frac { \\min \\{ L , N \\} } { ( 1 - ( 1 - 1 / N ) ^ { \\overline { L } } ) N } \\\\ & \\le \\frac { 4 \\kappa / N } { 1 - ( 1 - 1 / N ) ^ { \\kappa } } \\\\ & \\stackrel { ( a ) } { \\le } 4 \\cdot \\frac { \\kappa / N } { 1 - e ^ { - \\kappa / N } } \\\\ & \\stackrel { ( b ) } { \\le } \\frac { 1 } { 1 - e ^ { - 1 / 4 } } \\approx 4 . 5 2 1 , \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} \\begin{aligned} a _ 1 ( a _ 2 \\rightharpoonup b ) & = a _ 1 \\circ ( T ( a _ 2 ) \\rightharpoonup ( a _ 3 \\rightharpoonup b ) ) \\\\ & = a _ 1 \\circ ( ( T ( a _ 2 ) \\circ a _ 3 ) \\rightharpoonup b ) = a \\circ ( 1 \\circ b ) = a \\circ b . \\end{aligned} \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} \\partial _ t S = \\Delta S + 2 | S _ { i j } | ^ 2 + 2 \\alpha | \\tau _ g \\varphi | ^ 2 - 2 \\dot { \\alpha } | \\nabla \\varphi | ^ 2 . \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 4 } \\left ( \\sqrt { a } , - \\sqrt { a } ; i \\sqrt { b } q ^ { 1 / 4 } , - i \\sqrt { b } q ^ { 1 / 4 } , \\sqrt { a z } , - \\sqrt { a z } ; \\sqrt { q } , - b z \\sqrt { q } \\right ) = \\frac { \\left ( z ; q \\right ) _ { \\infty } } { \\left ( a z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a ; q \\right ) _ { n } S _ { n } \\left ( b ; q \\right ) z ^ { n } } { \\left ( - b q ^ { 1 / 2 } ; q \\right ) _ { n } } . \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} \\Delta ( \\Omega ) = \\exp \\left ( \\sum _ { j \\geq 2 } \\frac { ( - 1 ) ^ j } { j } \\sum _ { \\substack { ( { \\bf l } _ i = ( l _ i ^ 1 , l _ i ^ 2 ) \\in \\Z ^ { 2 } \\setminus 0 ) _ { i = 1 } ^ j \\\\ \\textrm { w i t h } \\sum _ { i = 1 } ^ j { \\bf l } _ i = 0 } } \\exp \\left ( \\sum _ { k = 1 } ^ j \\pi i { \\bf l } _ k \\cdot \\Omega \\cdot { \\bf l } ^ T _ k \\right ) \\right ) . \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} h \\left ( \\mathbf { Y } ^ { T _ E } _ { [ 1 : \\ell ] } \\right ) & ~ \\leq ~ \\sum _ { k = 1 } ^ \\ell T _ E \\log \\Big ( 2 \\pi e \\big ( { \\Lambda } P + 1 \\big ) \\Big ) ~ = ~ \\ell { T _ E } \\log \\Big ( 2 \\pi e \\big ( { \\Lambda } P + 1 \\big ) \\Big ) , \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} 1 + | b c | \\geq \\tfrac { 1 } { 2 } \\sinh ^ 2 x _ 0 + 1 = \\tfrac { 1 } { 2 } \\cosh ^ 2 x _ 0 + \\tfrac { 1 } { 2 } \\geq \\tfrac { 1 } { 2 } \\cosh ^ 2 x _ 0 . \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} \\tau = \\frac { N _ T } { \\frac { N _ T } { N _ T + N _ R - 1 } } a ^ * _ { 0 , 1 } = \\frac { N _ T + N _ R - 1 } { N _ T } , \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} \\prod \\limits _ { \\delta | N } \\delta '^ { r _ \\delta } & = ( p _ 1 p _ 2 ) ^ r p _ 2 ^ { - p _ 1 r } p _ 1 ^ { - p _ 2 r } = p _ 1 ^ { r ( 1 - p _ 2 ) } p _ 2 ^ { r ( 1 - p _ 1 ) } \\\\ & = p _ 1 ^ { \\frac { - 2 4 } { p _ 1 - 1 } } p _ 2 ^ { \\frac { - 2 4 } { p _ 2 - 1 } } . \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} \\bar { g } _ { i j } = \\sinh ^ 2 r \\sigma _ { i j } , \\ , \\bar { h } _ { i j } = \\tfrac { 1 } { 2 } \\dot { \\bar { g } } _ { i j } = \\coth r \\bar { g } _ { i j } , \\ , \\bar { \\kappa } _ i = \\coth r , \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} \\nabla \\times ( v _ 1 , v _ 2 ) = - \\partial _ { x _ 2 } v _ 1 + \\partial _ { x _ 1 } v _ 2 = \\nabla \\cdot ( v _ 2 , - v _ 1 ) = \\nabla \\cdot ( v _ 1 , v _ 2 ) ^ { \\mathrm { r o t } } , \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{gather*} \\sum _ { i = 1 } ^ { m } \\delta ( z , w _ { i } ) ( - 1 ) ^ { m + 1 } W _ { i } + ( - 1 ) ^ { m } \\frac { W } { \\prod \\limits _ { j = 1 } ^ { m } ( w _ { j } - z ) } = \\frac { W } { \\prod \\limits _ { i = 1 } ^ { m } ( z - w _ { i } ) } . \\end{gather*}"}
-{"id": "9289.png", "formula": "\\begin{align*} T ( T ^ { - 1 } B ) = B . \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} | | \\mu | | = \\sup _ { s \\in S , a ^ 1 \\in A ^ 1 ( s ) , a ^ 2 \\in A ^ 2 ( s ) } \\left ( \\sum _ { s ' \\in S ; s ' \\neq s } \\mu ( s ' , s , a ^ 1 , a ^ 2 ) \\right ) . \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} Z _ R = Z _ { 1 , R } \\cup _ Y Z _ { 2 , R } , \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} f _ n ( s , y , t , x ) : = g _ n ( s , y , t , x ) J _ 0 ( s _ { \\sigma ( 1 ) } , x _ { \\sigma ( 1 ) } ) , \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} & \\bullet [ \\sigma _ 1 ^ { - 1 } \\sigma _ j , \\sigma _ 1 ^ { - 1 } \\sigma _ k ] = 0 , \\\\ & \\bullet [ \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 j } , \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } ] = 0 , \\\\ & \\bullet [ \\sigma _ 1 ^ { - 1 } \\sigma _ k , \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 j } ] = [ \\sigma _ 1 ^ { - 1 } \\sigma _ j , \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } ] . \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } A ( t ) | _ { t = 0 } = \\widetilde { F } \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} 2 k - 1 > \\lim \\limits _ { n \\to + \\infty } \\mu _ k ^ { 1 / { p _ n } } ( p _ n ) = \\lim \\limits _ { n \\to + \\infty } \\lambda _ 1 ^ { 1 / { p _ n } } ( B _ { r _ { 1 } ^ { n } } ; p _ n ) = \\frac { 1 } { r _ 1 } \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} \\sigma ( z ) = z \\prod _ { 0 \\neq \\omega \\in \\Gamma } \\left ( 1 - \\frac { z } { \\omega } \\right ) e ^ { \\frac { z } { \\omega } + \\frac { 1 } { 2 } \\left ( \\frac { z } { \\omega } \\right ) ^ 2 } . \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\vert \\mathcal { M } _ { n } ^ { \\varepsilon } - \\mathcal { M } _ { n - 1 } ^ { \\varepsilon } \\right \\vert ^ { 2 } = \\mathbb { E } \\left [ V ^ { \\varepsilon } \\left ( \\eta _ { 0 } ^ { \\varepsilon } \\right ) \\left ( I + q ^ { \\varepsilon } \\right ) \\left ( I - q ^ { \\varepsilon } \\right ) ^ { - 1 } V ^ { \\varepsilon } \\left ( \\eta _ { 0 } ^ { \\varepsilon } \\right ) \\right ] . \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } S _ { n } \\left ( x q ^ { \\alpha - 1 / 2 } ; q \\right ) = \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( x e ^ { i y } ; q \\right ) _ { n } } { \\left ( q ; q \\right ) _ { n } } \\frac { \\exp \\left ( y ^ { 2 } / \\log q ^ { 2 } + i \\alpha y \\right ) d y } { \\sqrt { \\pi \\log q ^ { - 2 } } } . \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} \\partial _ r u \\ , \\partial _ r v = & \\left ( \\log \\frac { r } { \\rho } + \\beta \\right ) ^ { 2 \\alpha } \\ , \\partial _ r f \\ , \\partial _ r g + \\frac { \\alpha ^ 2 } { r ^ 2 } \\ , \\left ( \\log \\frac { r } { \\rho } + \\beta \\right ) ^ { 2 \\alpha - 2 } \\ , f g \\\\ & + \\frac \\alpha r \\ , ( f \\ , \\partial _ r g + g \\ , \\partial _ r f ) \\left ( \\log \\frac { r } { \\rho } + \\beta \\right ) ^ { 2 \\alpha - 1 } . \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} G _ { N , s } ^ { ( r _ 1 , \\dots , r _ s ) } = \\frac { 1 } { Z _ N } \\sum _ { \\mathcal { C } } W ( \\mathcal { C } ) \\prod _ { j = 1 } ^ s \\chi _ { e _ j } ( \\mathcal { C } ) , \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} \\| \\xi ^ { m _ 0 , N _ 0 } \\| _ { \\ell ^ p _ \\phi } & = \\sup _ { m \\in \\mathbb { Z } , N \\in \\omega } \\frac { 1 } { \\phi ( 2 N + 1 ) } \\left ( \\frac { 1 } { | S _ { m , N } | } \\sum _ { k \\in S _ { m , N } } | \\xi ^ { m _ 0 , N _ 0 } _ k | ^ p \\right ) ^ \\frac 1 p \\\\ & \\geq \\frac { 1 } { \\phi ( 2 N _ 0 + 1 ) } \\left ( \\frac { | S _ { m _ 0 , N _ 0 } | } { | S _ { m _ 0 , N _ 0 } | } \\right ) ^ \\frac 1 p = \\frac { 1 } { \\phi ( 2 N _ 0 + 1 ) } . \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\frac { 1 } { 2 } | \\gamma ' | ^ 2 = \\langle \\nabla _ { \\gamma ' } \\gamma ' , \\gamma ' \\rangle = \\kappa \\langle J ^ { 9 0 } _ \\gamma ( \\gamma ' ) , \\gamma ' \\rangle = 0 . \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} H ^ * _ G ( M , \\Z ) = H ^ * ( ( M \\times E G ) / G , \\Z ) . \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} k \\ge \\log \\left ( \\frac { c + s ^ 2 } { s ^ 2 } \\right ) - 2 = \\log \\left ( \\frac { c + s ^ 2 } { 2 s ^ 2 } \\right ) - 1 , \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} \\| \\phi \\| _ { A ( G ) } = \\inf \\{ \\| \\xi \\| _ 2 \\| \\eta \\| _ 2 : \\phi = \\phi _ { \\xi , \\eta } \\} . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{gather*} \\delta ( z , w ) = \\frac 1 { z - w } + \\frac 1 { w - z } . \\end{gather*}"}
-{"id": "622.png", "formula": "\\begin{align*} \\operatorname { d i v } \\mathbf { G } = 4 \\pi \\rho , \\qquad \\rho = \\rho ^ { \\ast } , \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ { s _ i } \\lambda _ j ^ i \\cdot P _ j ^ i = 0 . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} P _ s ^ { i * } = \\left \\{ \\begin{array} { c l } \\left [ \\frac { 1 } { \\ln 2 ( \\sum _ { j = i } ^ M \\lambda _ j ^ * + \\gamma _ i ^ * h _ { s p } ^ i ) } - \\frac { 1 } { \\theta _ i } \\right ] ^ + , & 0 < \\alpha _ i \\leq 1 \\\\ 0 , & \\alpha _ i = 0 \\end{array} \\right . \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} M ^ t \\vec v _ 0 = \\lambda ^ t \\vec v _ t + \\sum _ { k = 0 } ^ { t - 1 } \\lambda ^ { k } M ^ { t - k - 1 } \\vec u _ { k + 1 } = \\lambda ^ t \\vec v _ t + \\sum _ { m = 0 } ^ { t - 1 } \\lambda ^ { t - m - 1 } M ^ { m } \\vec u _ { t - m } . \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} \\frac { p _ t ( x _ j , y _ k ) } { \\abs { p ^ \\perp _ t ( x _ j , y _ k } } = \\begin{cases} 1 & j \\neq k \\\\ \\displaystyle \\frac { 1 - K e ^ { - x _ j y _ k / 2 t } } { 1 + K e ^ { - x _ j y _ k / 2 t } } & j = k . \\end{cases} \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( f \\ , H ) - ( f \\ , H ) [ | A | ^ { 2 } - \\lambda ] = 0 , \\\\ A \\ , ( { \\rm g r a d } ( f \\ , H ) ) + \\frac { m } { 2 } ( f \\ , H ) { \\rm g r a d } \\ , H = 0 . \\end{cases} \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} g ( x , y ) + g ( y , x ) = 2 p _ d \\ \\ g ( x , y ) g ( y , x ) = 2 p _ d - 1 , \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} e ^ { H [ s ( t ) ] } \\psi ( z ) e ^ { - H [ s ( t ) ] } & = e ^ { - \\sum _ { q \\geq 1 } s _ q ( t ) z ^ q } \\psi ( z ) \\\\ [ 5 p t ] e ^ { H [ s ( t ) ] } \\psi ^ \\ast ( z ) e ^ { - H [ s ( t ) ] } & = e ^ { \\sum _ { q \\geq 1 } s _ q ( t ) z ^ q } \\psi ^ \\ast ( z ) . \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} h ( z ) = - t ( z ) \\prod _ { a \\in A _ + } f _ a ( t ( z ) ) ^ { q ^ 2 - 1 } , \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } u ^ { - } \\partial _ { t } \\left [ g _ { 1 - \\alpha , m } * ( u - u _ { 0 } ) \\right ] d x + a ( h _ { m } * u , u ^ { - } ) \\ , d t < 0 \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } \\beta _ { 0 } = \\left ( \\alpha _ { 0 } + \\alpha _ { 1 } \\right ) / 2 , \\\\ \\beta _ { 1 } = \\left ( \\alpha _ { 1 } + \\alpha _ { 2 } \\right ) / ( 3 \\ell ^ { 2 } ) , \\\\ \\beta _ { 2 } = \\left ( \\alpha _ { 2 } + \\alpha _ { 3 } \\right ) / ( 1 0 \\ell ^ { 4 } ) . \\end{array} \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} u '' + a ( t ) g ( u ) = 0 , \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} x _ { i j } = g _ { i j } x - h _ { i j } \\tilde { x } , \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} ( x - y ) \\sum _ { k = - \\infty } ^ { n } \\varphi _ { k } ( x ) \\varphi _ { k } ( y ) = W _ { n } \\left ( \\varphi ( x ) , \\varphi ( y ) \\right ) \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} F _ { i j } ( t ) & = \\sum _ { k = j } ^ { i } V _ { i k } \\exp ( Q _ { ( n - k ) \\lambda } t ) V _ { k j } \\\\ & = \\sum _ { k = j } ^ { i } { n - k \\choose n - i } ( - 1 ) ^ { i - k } \\exp ( Q _ { ( n - k ) \\lambda } t ) { n - j \\choose n - k } \\\\ & = { n - j \\choose n - i } \\sum _ { k = j } ^ { i } ( - 1 ) ^ { i - k } { i - j \\choose i - k } \\exp ( Q _ { ( n - k ) \\lambda } t ) , \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} d u = a _ { i j } J ^ { j } \\circ d u \\circ I ^ i , \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} \\zeta ^ i _ { t | s } = { n - N _ s \\choose n - i } \\exp ( - ( n - i ) \\lambda _ \\infty \\ , ( t - s ) ) ( 1 - \\exp ( - \\lambda _ \\infty \\ , ( t - s ) ) ) ^ { i - N _ s } \\pi . \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} y _ q ( u ) & = \\sum _ { \\mathcal { R } : | \\mathcal { R } | = r + 1 } \\sum _ { i = 1 } ^ \\rho \\left [ \\sum _ { p \\in [ N _ T ] } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { [ N _ T ] } , p } ^ i ( u ) \\right ] x _ { { \\mathcal { R } } , { [ N _ T ] } } ^ i \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ \\infty \\gamma _ i x ^ i = \\frac { 1 } { x } \\cdot \\frac { d ^ { 2 k - 1 } } { d x ^ { 2 k - 1 } } \\left [ x ^ { 2 k - 2 } \\rho \\left ( \\frac { x ^ 2 } { 1 - x } \\right ) \\right ] \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} B ( [ h , t ] , [ h ' , t ' ] ) \\vcentcolon = B ( t , t ' ) \\qquad \\left ( [ h , t ] , [ h ' , t ' ] \\in \\Gamma _ \\ell \\right ) . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\# ( G / P _ \\lambda ) ( k ) = \\# G ( k ) / \\# P _ \\lambda ( k ) \\quad \\# P _ \\lambda ( k ) = \\# M _ \\lambda ( k ) q ^ { \\dim U _ \\lambda } , \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} ( \\Delta ^ 2 - \\iota _ \\lambda ^ 2 I ) x _ 2 = ( \\Delta - \\iota _ \\lambda I ) \\sigma z _ 2 , ( \\Delta ^ 2 - \\iota _ \\lambda ^ 2 I ) y _ 2 = - ( \\Delta - \\iota _ \\lambda I ) \\sigma z _ 2 , \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} { L } _ { \\rm N } = { U } _ { \\rm g } ^ { - 1 } ( H - E ) { U } _ { \\rm g } \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} M _ { X , m i x e d } = \\frac { 1 } { N } \\sum _ { r \\neq l } c _ { r l } \\mu _ r \\otimes \\mu _ l \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} \\varphi ( x ) = \\frac { 1 } { x } \\int \\limits _ 0 ^ x \\frac { 1 } { 1 - \\xi } \\chi ( \\xi ) \\ , d \\xi . \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\norm { v ( t ) } ^ 2 + \\int _ 0 ^ t \\norm { \\nabla v ( s ) } ^ 2 d s \\leq \\norm { a } ^ 2 + \\frac { 1 } { \\lambda _ 1 } \\int _ 0 ^ t \\norm { f ( s ) } ^ 2 d s + \\frac { C } { \\lambda _ 1 } B _ { L ^ 2 } ^ { \\zeta } ( t ) = : B _ { L ^ 2 } ^ { v } ( t ) . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\overline { \\gamma } _ { 2 n + 1 } = \\sum \\limits _ { i = 0 } ^ n { n \\brack i } \\overline { \\gamma } _ { 2 i } . \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{gather*} \\Psi ^ { [ \\ell ] ( \\beta ) } = \\Psi ^ { [ k ] ( \\alpha ) } \\Gamma _ { [ k ] ( \\alpha ) } ^ { [ \\ell ] ( \\beta ) } . \\end{gather*}"}
-{"id": "3013.png", "formula": "\\begin{align*} x \\vee x & = x , & x \\vee y & = y \\vee x , & x \\vee ( y \\vee z ) & = ( x \\vee y ) \\vee z , & x \\vee \\bot & = x , \\\\ x \\otimes k & = x , & ( x \\otimes u ) \\otimes v & = x \\otimes ( u \\otimes v ) , & \\bot \\otimes u & = \\bot , & ( x \\vee y ) \\otimes u & = ( x \\otimes u ) \\vee ( y \\otimes u ) ; \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} \\xi ( x , y ) : = \\bigvee _ { i \\in I } \\bigwedge _ { j \\in J } P _ { w } ( x , y ) = 0 \\wedge Q _ { w } ( x , y ) \\neq 0 \\wedge \\theta _ { w } ( t _ { 1 } ^ w , \\ldots , t _ { n _ w } ^ w ) \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} H ^ j ( B G , \\Z ) = \\left \\{ \\begin{array} { l l } \\Z , & j = 0 \\\\ 0 , & j \\ ; o d d \\\\ \\Z / m , & j \\ ; e v e n , \\ ; j > 0 \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} f ( \\check u ) = \\tilde v = h ^ { - 1 } \\left ( - c g ( \\tilde u ) \\right ) < h ^ { - 1 } \\left ( - c _ 1 g ( a ) \\right ) < f ( 0 ) \\ ; \\ ; \\ ; \\ ; c \\in ( 0 , c _ 1 ) . \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} P _ 1 u _ 1 = f _ 1 \\ \\ \\Omega _ 1 , P _ 2 u _ 2 = f _ 2 \\ \\ \\Omega _ 2 , \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} P _ g u = f u _ + ^ { \\frac { n + 6 } { n - 6 } } \\hbox { ~ ~ i n ~ ~ } M . \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} & \\cos ^ 2 \\phi _ { L i , k } = \\Vert U _ i U _ i ^ { \\top } U _ * e _ k \\Vert _ F ^ 2 , & \\cos ^ 2 \\phi _ { R i , k } = \\Vert e _ k ^ { \\top } V _ * ^ { \\top } V _ i V _ i ^ { \\top } \\Vert _ F ^ 2 , \\\\ & \\sin ^ 2 \\phi _ { L i , k } = \\Vert ( I - U _ i U _ i ^ { \\top } ) U _ * e _ k \\Vert _ F ^ 2 , & \\sin ^ 2 \\phi _ { R i , k } = \\Vert e _ k ^ { \\top } V _ * ^ { \\top } ( I - V _ i V _ i ^ { \\top } ) \\Vert _ F ^ 2 . \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} Q ^ { \\mu \\nu } = - Q ^ { \\nu \\mu } = \\left ( \\begin{array} [ c ] { c c c c } 0 & - G _ { 1 } & - G _ { 2 } & - G _ { 3 } \\\\ G _ { 1 } & 0 & i F _ { 3 } & - i F _ { 2 } \\\\ G _ { 2 } & - i F _ { 3 } & 0 & i F _ { 1 } \\\\ G _ { 3 } & i F _ { 2 } & - i F _ { 1 } & 0 \\end{array} \\right ) \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} n = 1 + k + \\frac { k ( k - 1 - \\lambda ) } { \\mu } . \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} \\sum _ { z _ { k } \\in V _ { z _ { j } } ^ { \\alpha } } \\mu \\left ( z _ { k } \\right ) \\leq \\left \\Vert R \\varphi _ { z _ { j } } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } ^ { 2 } \\leq C \\left \\Vert \\varphi _ { z _ { j } } \\right \\Vert _ { B _ { 2 , Z } } ^ { 2 } = C \\mu \\left ( z _ { j } \\right ) \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\mathcal { B } ^ + _ { I I I } = \\left \\{ \\begin{aligned} & \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { A } ^ + \\backslash \\mathcal { B } ^ + _ { I I } \\ ; \\textnormal { s u c h t h a t } \\\\ & \\inf _ { i \\in \\left \\{ 1 , \\dots , s , s + 1 , \\dots , s + k \\right \\} \\backslash \\left \\{ i _ { k + 1 } \\right \\} } \\left | v _ { s + k + 1 } ^ * - v _ i ^ \\prime \\right | \\leq \\eta \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} { \\bf P } \\bigl ( V _ N < s \\bigr ) = \\lim \\limits _ { \\beta \\rightarrow \\infty } { \\bf E } \\Bigl [ \\exp \\Bigl ( - e ^ { - \\beta s } \\ , Z _ { \\lambda _ 1 , \\lambda _ 2 , \\varepsilon } ( \\beta ) / C \\Bigr ) \\Bigr ] , \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} \\check \\mu _ { \\mathbf { v } } : = \\sum _ { i , j = 1 } ^ N \\ , \\mu _ { i , j } \\ , v _ i \\overline { v _ j } \\ge 0 , \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} U ^ \\epsilon _ t + | D U ^ \\epsilon | + f ( x / \\epsilon ) \\cdot \\xi = 0 , \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} L = L \\big ( u ( o ) , m _ u , r , R _ { \\Omega } , \\norm { f } _ { C ^ 2 ( \\Omega \\times ( - \\infty , m _ u ) ) } \\big ) \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} f '' _ A ( x ) & \\geq f '' _ A ( 0 ) \\\\ & = 2 g '' ( A ) \\\\ & = 2 [ 2 Q ( A ) - A \\psi ( A ) ] \\\\ & > \\frac { 2 A ( 1 - A ^ 2 ) } { 1 + A ^ 2 } \\psi ( A ) \\\\ & > 0 \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} ( \\Phi ^ { * } ) ' ( s ) & \\ = \\ t ( s ) \\ + \\ ( s \\ - \\ \\Phi ' ( t ( s ) ) ) \\frac { d ( \\Phi ' ) ^ { - 1 } ( s ) } { d s } \\\\ & \\ = \\ ( \\Phi ' ) ^ { - 1 } ( s ) . \\\\ \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} K ( h ^ { \\frac { 1 } { 2 ^ { n - 1 } } } , 2 ) ^ { r _ { n } } ( a \\sharp _ { \\nu } b ) ^ { 2 } & \\leqslant ( a \\nabla _ { \\nu } b ) ^ { 2 } - r _ { 0 } ^ { 2 } ( a - b ) ^ { 2 } - \\sum _ { k = 1 } ^ { n - 1 } r _ { k } \\big [ a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } - a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } \\\\ & \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ { n - 1 } } } , 2 ) ^ { R _ { n } } ( a \\sharp _ { \\nu } b ) ^ { 2 } , \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { u } ( t ) + \\tilde { \\mathcal { A } } ( t ) \\mathbf { u } ( t ) = \\mathbf { 0 } t \\in ( 0 , T ) , \\mathbf { u } ( 0 ) = \\tilde { \\mathbf { u } } ^ { 0 } , \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} x _ { i } = \\dfrac { R T _ { i , i _ { 0 } } + j _ { 0 } H _ { i } } { R l _ { i _ { 0 } } t _ { i _ { 0 } } + j _ { 0 } K } + \\varepsilon \\phi \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} f r ( J ' ) = 1 + f r ( J ) . \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} g ^ { ( n ) } ( 0 ) & = \\sum \\limits _ { m = 0 } ^ n \\binom { n } { m } B ^ { ( f ) } _ { n - m } ( q ) \\frac { d ^ { m + r } } { d t ^ { m + r } } | _ { t = 0 } \\bigl [ e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) \\bigr ] , \\\\ & = ( - 1 ) ^ r \\sum \\limits _ { p = 0 } ^ { M - 1 } ( - 1 ) ^ p \\sum \\limits _ { k _ 1 < \\cdots < k _ p = 1 } ^ { M - 1 } \\bigl ( b _ 0 + \\sum b _ { k _ j } \\bigr ) ^ r \\ , B ^ { ( f ) } _ n \\bigl ( q + b _ 0 + \\sum b _ { k _ j } \\bigr ) . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} a = \\sigma , \\ \\ \\ \\ t = ( a , \\ t ) , \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dfrac { x _ { n + 1 } + \\xi } { 2 } = \\dfrac { s } { l } - \\varepsilon \\phi _ { 4 } \\\\ \\varepsilon l \\cong 0 , l \\cong + \\infty , \\phi _ { 4 } \\geq 0 \\phi _ { 4 } \\cong 0 \\end{array} \\right . \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} P _ { 0 } ( z ) = p _ { 0 , 0 } e ^ { \\frac { \\lambda } { \\xi } z } ( 1 - z ) ^ { - \\frac { \\gamma } { \\xi } } \\left [ 1 - \\dfrac { A ( z ) } { A } \\right ] , \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} a _ { ( n ) } ( b \\ * m ) = ( a _ { ( n ) } b ) \\ * m + \\delta _ { n , 0 } b \\ * \\{ a , m \\} \\end{align*}"}
-{"id": "9939.png", "formula": "\\begin{align*} \\mathcal { A } _ j \\mathfrak { n } ( \\varphi ( s _ i ) ) w = \\mathfrak { n } ( \\varphi ( s _ i ) ) \\mathcal { A } _ j w + [ \\mathcal { A } _ j , \\mathfrak { n } ( \\varphi ( s _ i ) ) ] w = \\begin{cases} ( \\delta _ j + 1 ) w & j \\neq i \\\\ ( \\delta _ i + 2 ) w & j = i . \\end{cases} \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} \\mathcal { Z } _ E ( \\mu \\setminus F ) = \\mathcal { Z } _ E ( \\mu ) \\setminus \\left ( \\bigcup _ { e \\in F } \\mathcal { Z } _ E ( \\mu e ) \\right ) . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} \\xi _ { s p } & = \\frac { ( \\gamma - \\beta ) z + \\alpha + \\beta - \\sqrt { [ ( \\gamma - \\beta ) z + \\alpha + \\beta ] ^ 2 - 4 \\gamma \\beta z ( 1 - z ) } } { 2 ( 1 - z ) } \\\\ \\eta _ { s p } & = \\frac { 1 - \\sqrt { z ^ 2 - z + 1 } } { 1 - z } \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} x ^ * _ i = \\begin{cases} - \\frac { v _ i } { \\| v _ { { \\cal I } } \\| } & { \\rm i f } \\ i \\in { \\cal I } , \\\\ 0 & { \\rm o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} J : \\mathcal { A } _ { \\alpha } ^ 2 ( \\mathbb { D } ) \\longrightarrow \\mathcal { A } _ { \\alpha } ^ 2 ( \\Pi ^ + ) , ( J f ) ( w ) = f \\Big ( \\frac { w - i } { w + i } \\Big ) \\frac { c _ { \\alpha } } { ( w + i ) ^ { \\alpha + 2 } } , \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} \\tilde { P } ^ x ( \\tau _ D \\in A , \\tilde { X } ( \\tau _ D ) \\in B ) = \\int _ A \\int _ B \\tilde { h } _ D ( x , s , z ) \\ , d z \\ , d s , x \\in D _ + . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} { } _ { 0 } \\phi _ { 0 } ( - ; - ; q , z ) = ( z ; q ) _ { \\infty } \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} g = d t d x + d z d y + p \\ , d t ^ 2 + 2 q \\ , d t d z + r \\ , d z ^ 2 . \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} G ( x , y ) = y ^ d + a _ 1 ( x ) y ^ { d - 1 } + \\cdots + a _ d ( x ) , \\end{align*}"}
-{"id": "9925.png", "formula": "\\begin{align*} u ^ { - } ( r ^ { - 1 } ) v = ( u ^ { - } ( r ^ { - 1 } ) v ) ^ { \\max } = v ^ { \\max } v = u ^ { - } ( - r ^ { - 1 } ) v ^ { \\max } , \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\frac { 4 } { 3 } I = P _ 1 + P _ 2 . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} \\overline F \\big ( x / a ( x ) \\big ) = O \\big ( \\overline H ( x ) \\big ) , \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ { 6 } x _ \\alpha ^ 2 = 0 , \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} | f | _ { \\infty } ^ - = \\sum _ { k \\in { \\mathbb N } } f ^ * ( k ) k ^ { - 1 } \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ { r } p _ { r , k } s _ { k + m } = 0 \\ , \\ \\ \\ \\ 0 \\leq m \\leq r - 1 \\ , \\end{gather*}"}
-{"id": "2509.png", "formula": "\\begin{align*} I m \\{ z \\} = I m \\{ w \\} \\quad { \\rm a n d } R e \\{ z \\} = R e \\{ w \\} + \\frac { 2 m \\pi } { k } \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ n B _ { \\Gamma , p } ^ { ( \\ell ) } ( z ) & = \\sum _ { g = 0 } ^ { \\lfloor \\frac { p - 1 } { 2 } \\rfloor } z ^ { 2 g - p } \\ ; \\Upsilon _ { p - 1 - 2 g } ^ { ( n ) } ( z ) \\ ; C _ { p , g } ^ { ( n ) } . \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} e ^ { \\frac { - 1 } { x ^ 2 } } & x \\ne 0 , \\\\ 0 & x = 0 . \\end{cases} \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} & \\tau ^ * ( \\mu _ R , \\mu _ T ) \\\\ \\le & \\tau _ U ( \\mu _ R , \\mu _ T ) \\\\ \\le & \\tau _ U ( \\mu _ R , 1 / N _ T ) \\\\ \\le & \\tau _ U ( 1 , 1 / N _ T ) + \\frac { \\tau _ U ( 1 / 4 , 1 / N _ T ) - \\tau _ U ( 1 , 1 / N _ T ) } { 3 / 4 } ( 1 - \\mu _ R ) \\\\ = & \\frac { \\tau _ U ( 1 / 4 , 1 / N _ T ) } { 3 / 4 } ( 1 - \\mu _ R ) . \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\nabla ^ \\kappa _ v : = \\nabla _ v + \\kappa ( v ) , v \\in T M . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} f _ { \\infty , n , R } ^ { ( s ) } ( 0 , Z _ s ) = f _ \\infty ^ { ( s ) } ( 0 , Z _ s ) \\mathbf { 1 } _ { 1 \\leq s \\leq n } \\chi \\left ( \\frac { 1 } { R ^ 2 } E _ s ( Z _ s ) \\right ) \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} p _ t ( x , y ) = \\sum _ { c \\in C ( x , y ) } S ( c ) g _ t ( d _ c ( x , y ) ) . \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} - \\varepsilon ^ 2 \\Delta v + v = v ^ { q - 1 } \\ \\ \\mathcal { D } , v > 0 \\ \\mathcal { D } , \\partial _ \\nu v = 0 \\ \\partial \\mathcal { D } \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} 1 + | \\psi | ^ 2 = | \\omega _ 1 | ^ 2 + | k | ^ 2 \\ , . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} H _ { \\pi _ 7 ( [ P , P ] ^ { F N } ) } = \\tau _ 3 ( T ) , \\mbox { w h e r e } \\tau _ 3 ( T ) = - \\dfrac 2 3 ( 4 T + \\phi _ \\sigma ( T ) ) . \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} S = \\iint A ( b \\partial _ z \\Delta _ t ^ { - 1 } \\ne { f } ) A \\ne { f } \\ , d V d t . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} u _ t = r ^ { - n } u _ y , \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} \\mathbb { P } \\{ D _ { n } = m \\} = [ z ^ { n - 1 } v ^ { m } ] F ' ( z , v ) = \\sum _ { j = 0 } ^ { m - 1 } \\binom { m - 1 } { j } ( - 1 ) ^ { n + j - 1 } \\binom { p ( j + 1 ) - 1 } { n - 1 } . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} [ Q ( r _ 1 , \\dots , r _ n ) = 0 ] \\leq \\frac { d _ 0 } { | \\mathbb { S } | } . \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} E _ { e , l } \\ ! \\ ! = \\ ! \\ ! \\eta _ { e , l } ( 1 \\ ! \\ ! - \\ ! \\ ! \\rho _ { e , l } ) \\ ! \\bigg [ \\textrm { t r } ( \\mathbf { H } _ { e , l } ^ { H } \\mathbf { q } \\mathbf { q } ^ { H } \\mathbf { H } _ { e , l } ) \\ ! \\ ! + \\ ! \\ ! \\textrm { t r } ( \\mathbf { H } _ { e , l } ^ { H } \\mathbf { V } \\mathbf { H } _ { e , l } \\ ! \\ ! + \\ ! \\ ! N _ { E } \\sigma _ { e a , l } ^ { 2 } ) \\bigg ] \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} 1 + \\max \\left ( \\frac { n ^ + } { n ^ - } , \\frac { n ^ - } { n ^ + } \\right ) = 1 + \\frac { n ^ + } { n ^ - } = 1 + \\frac { p - k } { k } = \\frac { p } { k } = \\chi _ f . \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} \\varphi ( y + \\gamma ) = \\varphi ( y ) + \\alpha _ { \\gamma } ( y ) \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} c _ { i j k l } = c _ { j i k l } = c _ { j i l k } , \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} | \\gamma ' | ^ 2 & = c \\\\ \\frac { 1 } { | n | } \\langle \\gamma '' , \\gamma ' \\times n \\rangle & = \\kappa | \\gamma ' | ^ 2 , \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\frac { d ^ { 2 n + 1 } } { d x ^ { 2 n + 1 } } \\left [ ( x ^ 2 + x ) ^ { m } \\right ] & = \\sum \\limits _ { i = 0 } ^ m { m \\choose i } ( m + i ) _ { 2 n + 1 } x ^ { m - 2 n + i - 1 } \\\\ & = \\sum \\limits _ { i = 2 n - m + 1 } ^ m { m \\choose i } \\frac { ( m + i ) ! } { ( m + i - 2 n - 1 ) ! } \\ : x ^ { m - 2 n + i - 1 } \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} S = \\frac { | f _ { z _ 1 } | ^ 2 + | f _ { z _ 2 } | ^ 2 + | f _ { z _ 3 } | ^ 2 } { | f _ { z _ 1 } | ^ 2 } = 1 , ~ ~ \\phi ( S ^ 2 ) \\cap V _ 1 \\subset \\widetilde { U } _ 0 \\subset X , \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{gather*} S ^ { \\pm } ( z ) f = \\sum _ { n = 0 } ^ { \\infty } S ^ { \\pm } [ n ] f z ^ { - n } . \\end{gather*}"}
-{"id": "5460.png", "formula": "\\begin{align*} d ( x ) b ( x ) = 0 \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} u _ n ( t , x ) = \\sum _ { k = 0 } ^ n I _ k ( f _ k ( \\cdot , t , x ) ) , \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} g ( t , x , r , 0 , 0 ) = g ( t , x , 0 , 0 ) ~ \\textrm { a n d } ~ f ( t , x , r , 0 , 0 ) ~ \\textrm { i s } ~ \\textrm { n o n - i n c r e a s i n g } ~ w . r . t . ~ \\textrm { r } , \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} \\alpha _ { k , i } = \\left \\{ \\begin{array} { l l } \\frac { 9 } { \\Gamma _ k ( i ) } & \\mbox { f o r a d i m i n i s h i n g s t e p s i z e } \\\\ \\alpha _ i & \\mbox { f o r a c o n s t a n t s t e p s i z e , } \\end{array} \\right . \\end{align*}"}
-{"id": "10013.png", "formula": "\\begin{align*} h ^ { u } ( \\underline { a } ) = \\bigcap _ { n \\geq 0 } ( \\phi _ { a [ - n , 0 ] } ^ { - } ) ^ { - 1 } ( R _ { a _ { - n } } ) = \\bigcap _ { n \\geq 0 } W _ { a [ - n , 0 ] } \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} R ( C , 2 p ) = \\bigoplus _ { n \\geqslant 0 } \\mathcal { L } ( 2 n p ) = \\bigoplus _ { \\substack { m \\geqslant 0 \\\\ 2 \\mid m } } \\mathcal { L } ( m p ) \\cong \\bigg ( \\bigoplus _ { m \\geqslant 0 } \\mathcal { L } ( m p ) \\bigg ) ^ { ( 2 ) } = R ( C , p ) ^ { ( 2 ) } . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\frac { m ! ( k - m ) ! } { ( 2 k ) ! } \\sum \\limits _ { i = 0 } ^ { m } \\frac { ( m + i ) ! ( 2 k - m - i ) ! } { ( m - i ) ! i ! ( k - m ) ! } \\gamma _ { m + i - 1 } = 0 , \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} h \\leq \\varphi _ { k _ 0 } + \\varepsilon \\leq \\varphi _ { k _ 1 } + \\varepsilon = v _ { k _ 1 } + \\varepsilon \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z } J _ { j } ^ { 2 } \\left ( \\alpha ^ { - 1 } q ^ { m } ; q \\right ) = \\frac { 1 } { 1 - \\alpha ^ { - 2 } q ^ { 2 m } } . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{gather*} \\theta _ { 1 2 } ( 1 ) ( h _ 0 \\wedge h _ 1 ) = 0 \\ , , \\ \\ \\theta _ { 1 2 } ( 1 ) ( h _ 0 \\wedge h _ 2 ) = 2 h _ 0 h _ 2 \\ , , \\ \\ \\theta _ { 1 2 } ( 1 ) ( h _ 0 \\wedge h _ 3 ) = 2 h _ 0 h _ 3 \\\\ \\theta _ { 1 2 } ( 1 ) ( h _ 1 \\wedge h _ 2 ) = 2 h _ 1 h _ 2 \\ , , \\ \\ \\theta _ { 1 2 } ( 1 ) ( h _ 1 \\wedge h _ 3 ) = 2 h _ 1 h _ 3 \\ , , \\ \\ \\theta _ { 1 2 } ( 1 ) ( h _ 2 \\wedge h _ 3 ) = 0 \\ , . \\end{gather*}"}
-{"id": "5219.png", "formula": "\\begin{align*} \\varphi ( A ) f = \\lim \\limits _ { R \\to \\infty } \\frac { \\i } { 2 \\pi } \\int _ { \\C } \\frac { \\partial ( \\tilde { \\varphi \\theta _ R } ) _ N } { \\partial \\overline { z } } ( z ) ( z - A ) ^ { - 1 } f d z \\wedge d \\overline { z } , \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} ( p _ \\lambda , p _ \\mu ) ^ { S _ { q , t } } = ( p _ \\lambda , p _ \\mu ) \\prod _ i S _ { q ^ { \\lambda _ i } , t ^ { \\lambda _ i } } = ( p _ \\lambda , p _ \\mu ) \\prod _ i - ( q ^ { \\lambda _ i } - 1 ) ( t ^ { \\lambda _ i } - 1 ) , \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} & \\rho _ + ( \\sigma ) + \\rho _ - ( \\sigma ) = 1 , \\\\ & \\rho _ + ( \\sigma ) = 1 , \\sigma \\in \\left [ \\frac { 1 } { 4 } , 1 \\right ] , \\\\ & \\rho _ - ( \\sigma ) = 1 , \\sigma \\in \\left [ - 1 , - \\frac { 1 } { 4 } \\right ] . \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{gather*} g = \\pi \\big ( g ^ { [ k ] ( \\alpha ) } \\big ) = \\pi \\big ( T ^ { - k } \\big ) \\pi \\big ( g ^ { ( \\alpha ) } _ { C } \\big ) , \\end{gather*}"}
-{"id": "4287.png", "formula": "\\begin{align*} 9 t _ { 2 } + \\frac { 2 7 } { 4 } t _ { 3 } = - t _ { 4 } - \\frac { 1 } { 4 } t _ { 3 } + 9 d + \\sum _ { r \\geq 5 } ( 6 r - 2 5 ) t _ { r } . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} t _ \\delta = T ^ * - \\delta , \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} H ( w ) = H ( v ) ^ { - 1 } , A ( v ) = - H ( w ) ^ { - 1 } A ( w ) . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} \\delta ^ { t r } _ i \\Delta - \\Delta \\delta _ i = - 2 s ( f _ i + f _ i ^ { t r } ) = - 2 s ( c _ i + c _ i ^ { t r } ) = - \\delta _ i \\Delta + \\Delta \\delta _ i ^ { t r } . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } ( A u '' + B v '' ) = \\cos \\theta \\cdot R | _ { t = 0 } \\ ; , \\ ; \\ ; \\ ; \\ ; A = ( 1 - f _ u ^ 2 ) u ' - f _ u f _ v v ' \\ ; , B = ( 1 - f _ v ^ 2 ) v ' - f _ u f _ v u ' \\ ; . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} d _ X ( F ( x , y ) , x ) & \\leq d _ X ( F ( x , y ) , F ^ { j + 1 } ( x _ 0 , y _ 0 ) ) + d _ X ( F ^ { j + 1 } ( x _ 0 , y _ 0 ) , x ) \\\\ & = d _ X ( F ( x , y ) , F ( F ^ j ( x _ 0 , y _ 0 ) , G ^ j ( y _ 0 , x _ 0 ) ) ) + d _ X ( F ^ { j + 1 } ( x _ 0 , y _ 0 ) , x ) \\\\ & \\leq k \\ d _ X ( x , F ^ j ( x _ 0 , y _ 0 ) ) + l \\ d _ Y ( y , G ^ j ( y _ 0 , x _ 0 ) ) + d _ X ( F ^ { j + 1 } ( x _ 0 , y _ 0 ) , x ) \\\\ & \\rightarrow 0 \\ \\ j \\rightarrow \\infty \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} \\rho ( P _ { 1 } \\otimes B _ { 1 , 2 } ) = ( k [ z ] / ( z - \\alpha _ { 1 } ) ^ { n _ { 1 } - 1 } , k [ z ] / ( z - \\alpha _ { 2 } ) ^ { n _ { 2 } - 1 } , 0 , 0 , \\dots ) . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} q _ { 1 } = ( z - \\alpha _ { 2 } ) ^ { n _ { 2 } - 2 } \\prod _ { j \\neq 1 , 2 } ( z - \\alpha _ { j } ) ^ { n _ { j } - 1 } \\\\ q _ { 2 } = ( z - \\alpha _ { 1 } ) ^ { n _ { 1 } - 2 } \\prod _ { j \\neq 1 , 2 } ( z - \\alpha _ { j } ) ^ { n _ { j } - 1 } . \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} o ( 1 ) = & \\langle u _ k - u , D E _ f [ u _ k ] \\rangle = \\langle u _ k - u , D E _ f [ u _ k ] - D E _ f [ u ] \\rangle \\\\ = & \\int _ M | \\nabla \\Delta ( u - u _ k ) | _ g ^ 2 d \\mu _ g - \\int _ M f | u - u _ k | ^ { 2 ^ \\sharp } d \\mu _ g + o ( 1 ) . \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} N ( H _ { [ i ] } ) \\cdot a = 0 \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} \\ ; \\sigma ^ { - 1 } \\beta \\subset \\cap _ { j = i } ^ 7 \\ ; ( d _ j ) , \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} E : = | G ( x _ 0 , w ) - G ( x , z ) | < \\frac { \\varepsilon } { 4 } | V ( z ) | ^ 2 . \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} S ( \\chi _ b ) S ( \\chi _ c ) & = | A | ^ 2 \\sum _ a p ( a , b ) p ( a , c ) e _ a \\\\ & = S ( \\chi _ c \\chi _ b ) \\\\ & = S ( \\chi _ { c b } ) \\\\ & = | A | \\sum _ a p ( a , c b ) e _ a . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} \\zeta _ n : = \\frac { 1 } { n ^ { m - 1 } } \\int _ 0 ^ { n ^ { m - 1 } } X ^ 2 _ t \\ , d t - \\frac { 1 } { n ^ m } \\sum _ { k = 0 } ^ { n ^ m - 1 } X _ { k / n } ^ 2 \\to 0 \\quad n \\to \\infty . \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} ( \\lambda z - \\mu + \\xi ) ( 1 - z ) P _ { 1 } ( z ) - \\xi z ( 1 - z ) P ' _ { 1 } ( z ) = \\gamma z P _ { 0 } ( z ) + ( \\xi - \\mu ) ( 1 - z ) p _ { 1 , 0 } - \\mu z p _ { 1 , 1 } . \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} \\begin{cases} \\sigma _ k ( D ^ 2 u ) = F ( x , u ) , & , \\\\ u = f ( x ) , & \\end{cases} \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} d z _ { \\hat i } = \\frac { d x _ { \\hat i } } { X _ { z _ { \\hat i } } } - \\sum _ { i \\neq \\hat { i } = 1 } ^ n \\frac { X _ { z _ { i } } } { X _ { z _ { \\hat i } } } d x _ i , \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} B _ { j } ( z ) = z ^ { j } A _ { q ^ { 2 } } \\left ( z ^ { 2 } q ^ { 2 j + 1 } \\right ) \\ ! , j \\in \\Z . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ { \\infty } \\omega ^ { 2 n } q ^ { n ( n - 1 ) } \\left [ \\varphi _ { k } \\left ( \\omega q ^ { n } \\right ) \\varphi _ { \\ell } \\left ( \\omega q ^ { n } \\right ) + \\varphi _ { k } \\left ( - \\omega ^ { - 1 } q ^ { - n + 1 } \\right ) \\varphi _ { \\ell } \\left ( - \\omega ^ { - 1 } q ^ { - n + 1 } \\right ) \\right ] = 2 \\| \\varphi \\left ( \\omega \\right ) \\ ! \\| ^ { 2 } \\delta _ { k , \\ell } \\end{align*}"}
-{"id": "9906.png", "formula": "\\begin{align*} & | \\varepsilon _ i ( x ) | \\le C \\varkappa \\rho , \\mbox { f o r } \\ ; x \\in \\tilde { B } _ \\rho ( a ) , \\ ; i = 1 , 2 , \\\\ & | \\varepsilon _ 3 ( x ) | \\le C \\varkappa \\rho \\mbox { f o r } \\ ; x \\in B ^ + \\cap B _ \\rho ( a ) \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ { I V } ^ - } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , s , k } T R ^ d \\theta ^ { d - 1 } \\end{aligned} \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} \\| \\varphi _ { g } \\| ( X ) = \\prod _ { \\lbrace j _ { 1 } , \\dots , j _ { g + 1 } \\rbrace \\in \\mathcal { U } _ { g + 1 } } \\| \\theta \\| ( W _ { j _ { 1 } } + \\dots + W _ { j _ { g } } - W _ { j _ { g + 1 } } ) ^ { 4 } , \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ p | } - f _ 8 ( p ) = q ^ 3 ( q ^ 2 - 1 ) ( q + \\sqrt { 3 q } + 1 ) \\left ( \\frac { ( q - \\sqrt { 3 q } + 1 ) r } { | G _ { p } | } - k \\right ) , \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} \\frac { F ( x _ N ) - F ( x ^ * ) } { a _ N ^ 2 } + \\frac { \\tilde { \\mu } } { 2 } \\| x ^ * - v _ N & \\| ^ 2 \\le \\frac { 1 - a _ 1 } { a _ 1 ^ 2 } \\big ( F ( x _ 0 ) - F ( x ^ * ) \\big ) + \\frac { \\tilde { \\mu } } { 2 } \\norm { x ^ * - v _ 0 } ^ 2 \\\\ & + \\rho M ^ 2 \\left ( \\sum _ { j = 1 } ^ N \\frac { 1 } { a _ j } \\right ) + \\frac { N r M ^ 2 } { 2 } - \\frac { \\tilde { \\mu } - \\mu } { 2 } \\sum _ { j = 1 } ^ N \\frac { \\norm { x _ j - y _ j } ^ 2 } { a _ j ^ 2 } . \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & = ( X _ { s + k } ^ \\prime , V _ { s + k } ^ \\prime ) \\in \\mathcal { G } _ { s + k } \\cap \\hat { \\mathcal { U } } ^ \\eta _ { s + k } \\end{aligned} \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} U _ 1 , S ' & = \\mathrm { Q R } ( ( A _ 0 + D ) V _ 0 ) , \\\\ V _ 1 , S ^ { \\top } _ 1 & = \\mathrm { Q R } ( ( A _ 0 + D ^ { \\top } ) U _ 1 ) . \\\\ \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} 1 + \\left | \\frac { \\omega _ 1 ' } { \\varphi ' } \\right | ^ 2 = \\left | \\frac { \\psi ' } { \\varphi ' } \\right | ^ 2 \\ , . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} | x - y _ j | \\le \\frac { j + M } { j ^ 2 } = : r _ j < \\infty , \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} \\mathcal { T } = \\{ h _ 0 ( \\xi ) \\mid \\xi \\in \\mathbb { T } ^ d , v ( \\xi ) = 0 \\} . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} b ( x ) = \\left \\langle \\bar { \\nabla } f \\bigl ( x , u ( x ) \\bigr ) , \\nu ( x ) \\right \\rangle , \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} | I | \\ge | H ( u _ 5 ) | + | H ( v _ 5 ) | - | H ( u _ 5 ) \\cup H ( v _ 5 ) | \\ge 2 \\delta ( H ) - \\binom n 2 \\ge . 5 9 8 \\binom n 2 + O ( n ) . \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} C _ 3 : = \\dfrac { C _ 1 ^ 2 } { 1 2 + \\frac { 2 } { 3 } C _ 1 \\sqrt { c ( q ) } } , \\ \\ \\ \\mbox { a n d } \\ \\ \\ C _ 4 : = \\frac { C _ 2 ^ 2 } { 1 2 + \\frac { 2 } { 3 } C _ 2 \\sqrt { c ( q ) } } . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} Q ^ { ( 5 ) } \\left ( A _ { 2 } , A _ { 1 } \\right ) = 3 \\ , \\varepsilon _ { a b c d e } \\left ( - \\frac { 1 } { 3 \\ell ^ { 3 } } \\alpha _ { 0 } \\ , R ^ { a b } e ^ { c } e ^ { d } e ^ { e } - \\frac { 1 } { 1 0 \\ell ^ { 5 } } \\alpha _ { 0 } \\ , e ^ { a } e ^ { b } e ^ { c } e ^ { d } e ^ { e } \\right ) . \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} F = \\sum _ { i = 1 } ^ { r _ 1 } ( L ^ \\prime _ i ) ^ d + \\sum _ { j = 1 } ^ { r _ 2 } ( L ^ { \\prime \\prime } _ j ) ^ d , \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} T ( v , \\alpha , r ) = C ( v , \\alpha ) \\setminus \\bar B ( x , r ) , \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} G ( s ) L ( s , f ) = \\epsilon ( f ) G ( \\delta - s ) L ( \\delta - s , g ) \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { 1 } { h ( t ) } M ^ t ( \\vec v ) = \\lambda ( \\vec v ^ { \\ , \\rm P F } _ i + \\vec w _ i ) \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} \\rho ^ { Z , f \\oplus 0 } _ t ( S _ c ) = \\exp { ( 2 \\pi \\sqrt { - 1 } t f ) } S _ c c \\in E _ C , \\rho ^ { Z , f \\oplus 0 } _ t ( S _ d ) = S _ d d \\in E _ D . \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} & r ^ 1 _ 1 ( 1 , 1 ) + r _ 2 ^ 1 ( 1 , 1 ) = 4 \\\\ & r ^ 1 _ 1 ( 1 , 1 ) + r _ 2 ^ 1 ( 1 , 2 ) = 6 \\\\ & r ^ 1 _ 1 ( 1 , 2 ) + r _ 2 ^ 1 ( 1 , 1 ) = 5 \\\\ & r ^ 1 _ 1 ( 1 , 2 ) + r _ 2 ^ 1 ( 1 , 2 ) = 4 . \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} L ( x ^ k P _ n ( x ) ) = 0 0 \\le k \\le n - 1 , \\ ; n \\ge 1 \\ , \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} P ( \\{ ( 1 , 2 ) , ( 2 , 3 ) , ( 1 , 4 ) , ( 4 , 3 ) \\} \\subset A ( \\mathbf { D } ) ) = ( p _ e p _ d ) ^ 4 = \\left ( P ( \\{ ( 1 , 2 ) , ( 2 , 3 ) \\} \\subset A ( \\mathbf { D } ) ) \\right ) ^ 2 , \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} \\left ( z ^ { 2 } q ; q ^ { 2 } \\right ) _ { \\infty } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } ( - 1 ) ^ { n } z ^ { 2 n } } { ( q ^ { 2 } ; q ^ { 2 } ) _ { n } } = \\sum _ { k = 0 } ^ { \\infty } \\left ( - z \\right ) ^ { k } q ^ { \\binom { k } { 2 } } S _ { k } \\left ( q ^ { - k } ; q \\right ) . \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( x _ i ) & = \\varphi ' ( x _ i ) & & i < j , \\\\ \\varphi ( x _ j ) & = \\ 0 , \\ \\varphi ' ( x _ j ) = \\ 1 , & & \\\\ \\varphi ( x _ i ) & = \\ 1 , \\ \\varphi ' ( x _ i ) = \\ 0 & & j < i \\leq n . \\end{aligned} \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} \\int s ( x , y ) s ( y , z ) d ( \\mu y ) = 0 \\ \\ \\mu ^ 2 \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} \\langle \\mathcal { A } ( t ) u , v \\rangle _ { \\mathcal { V } ' ; \\mathcal { V } } : = a ( u , v ; t ) u , v \\in \\mathcal { V } . \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} v = \\left \\{ \\begin{array} { l l } v _ 1 & \\textrm { i n } \\ , \\ , Q _ 1 \\setminus Q _ 2 \\\\ v _ 2 & \\textrm { i n } \\ , \\ , Q _ 1 \\cap Q _ 2 \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} \\Delta ( z ) = T \\prod _ { v \\in V ' } E _ v ( z ) . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} d _ 1 = s U \\begin{pmatrix} - x _ 3 & - x _ 4 \\\\ x _ 4 & - x _ 3 \\end{pmatrix} , d _ 2 = s U \\begin{pmatrix} x _ 1 & x _ 2 \\\\ - x _ 2 & x _ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} \\frac { \\Phi _ { p r i m } ( Q ) } { \\Phi ( Q ) } = 1 + O ( \\frac { 1 } { q } ) \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 x : n ) } = \\bigoplus _ { ( i , \\alpha ) } H _ { ( 2 x ) } ( i , \\alpha ) \\varphi ( i , \\alpha ) \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} - d _ 2 \\Delta v _ \\rho = v _ \\rho \\left [ h ( v _ \\rho ) + q ( \\rho w _ \\rho ) w _ \\rho \\right ] \\le v _ \\rho \\left [ h ( v _ \\rho ) + \\max _ { u \\in [ 0 , a ] } q ( u ) C _ 3 \\right ] . \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\frac { q ^ { m ^ { 2 } + m \\nu } } { \\left ( q ; q \\right ) _ { m } z ^ { m } } I _ { \\nu + m } ^ { ( 3 ) } ( 2 z q ^ { m / 2 } ; q ) = \\frac { z ^ { \\nu } } { ( q ; q ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n } { 2 } } ( q z ^ { 2 } ) ^ { n } } { \\left ( q ; q \\right ) _ { n } } = \\frac { \\left ( - q z ^ { 2 } ; q \\right ) _ { \\infty } } { ( q ; q ) _ { \\infty } } z ^ { \\nu } , \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} \\alpha _ { 2 } ( n ) = \\begin{cases} 1 & \\mbox { i f } n \\mbox { i s s q u a r e - f u l l } \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} w ( \\Gamma _ 1 \\sqcup \\Gamma _ 2 ) = w ( \\Gamma _ 1 ) w ( \\Gamma _ 2 ) , \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} { \\rm l e n g t h } ~ T / F ^ \\perp = { \\rm l e n g t h } ~ T / M _ 1 ^ \\perp + { \\rm l e n g t h } ~ T / M _ 2 ^ \\perp - { \\rm l e n g t h } ~ T / ( M _ 1 ^ \\perp + M _ 2 ^ \\perp ) - 1 . \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } & \\sum _ { l = 1 } ^ { 1 0 } \\left ( c _ { i l } ( g _ { i l } ) - p _ l ( \\bar s _ l ) s _ { i l } \\right ) \\cr \\mbox { s u b j e c t t o } & \\sum _ { l = 1 } ^ { 1 0 } g _ { i l } = \\sum _ { l = 1 } ^ { 1 0 } s _ { i l } , \\\\ & g _ { i l } , s _ { i l } \\geq 0 , g _ { i l } \\leq \\mathrm { c a p } _ { i l } , l = 1 , \\hdots , { 1 0 } , \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} \\kappa _ E ( x ) = \\begin{cases} x e _ { r ( x ) } ^ \\infty , & \\\\ x , & \\end{cases} \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\frac { d } { d t } A ( t ) | _ { t = 0 } = H \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} \\mathbb { R } _ + ^ 2 = \\{ ( 0 , 0 ) \\} \\cup \\bigcup \\limits _ { \\gamma \\subset \\Gamma ^ { \\textit { g e o m } } ( f ) } \\Delta _ { \\gamma } , \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} 4 M = x _ { z _ 0 } - m . \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} \\big | \\phi \\big ( M ( D _ j ) \\big ) \\big | ^ 2 = \\big | \\phi _ 1 \\big ( M ( a _ { 1 ; j } ) \\big ) \\big | ^ 2 . \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} F ( q \\ , | \\ , \\beta , \\lambda _ 1 , \\lambda _ 2 ) = F ( q \\ , \\big | \\ , \\frac { 1 } { \\beta } , \\lambda _ 1 , \\lambda _ 2 ) , \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} \\nabla _ x R ( t , x , \\xi ) & = \\eta ( t , 0 ; x , \\xi ) = p ( 0 , t ; y ( 0 , t ; x , \\xi ) , \\xi ) \\\\ & = \\xi + \\int _ 0 ^ t ( \\nabla _ x V _ \\rho ) ( \\tau , q ( \\tau , t ; y ( 0 , t ; x , \\xi ) , \\xi ) ) d \\tau \\\\ & = \\xi + \\int _ 0 ^ t ( \\nabla _ x V _ \\rho ) ( \\tau , q ( \\tau , 0 ; x , \\eta ( t , 0 ; x , \\xi ) ) ) d \\tau . \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\mu ( A _ m ) = 1 - \\frac { 1 } { \\sqrt { r _ m } } . \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} Z _ { \\mathrm { m } } ( u ) = \\overline { T _ 2 \\cdot [ \\phi _ { 6 , u } ] } \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} K _ { i i } = \\begin{cases} d _ { i i } & d _ { i i } > 0 , \\\\ 2 \\abs { d _ { i i } } & d _ { i i } < 0 . \\end{cases} \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{gather*} \\psi _ { a } ^ { \\pm } ( z ) Q _ { a } = z ^ { \\pm } Q _ { a } \\psi _ { a } ^ { \\pm } ( z ) , \\\\ \\psi _ { a } ^ { \\pm } ( z ) Q _ { b } = - Q _ { b } \\psi _ { a } ^ { \\pm } ( z ) , a \\ne b , \\\\ Q _ { a } Q _ { b } = - Q _ { b } Q _ { a } , a \\ne b . \\end{gather*}"}
-{"id": "267.png", "formula": "\\begin{align*} S ( g ; \\phi ) = S ( \\delta ; \\phi ) + S ( h ; \\phi ) \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} \\left \\vert b _ { j _ { \\ell } } \\right \\vert \\leq C _ { 0 } \\mu \\left ( z _ { j } \\right ) ^ { \\frac { 1 } { 2 } } \\ell \\exp \\left ( C \\sum \\nolimits _ { k = 0 } ^ { \\ell - 1 } d \\left ( z _ { j _ { k } } \\right ) ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} v _ \\beta ^ i ( s , f , g ) = \\sum _ { t = 0 } ^ \\infty \\beta ^ t [ P ^ t ( f , g ) ] _ s r ^ i ( f , g ) , \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P , A c h } } ( \\mu , r ) \\leq \\delta _ { \\mathsf { P - Z F } } = \\frac { K } { \\min \\{ M , K \\} } . \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} B ( Q ) = \\alpha _ { j , J } + \\alpha _ { 1 , Q \\setminus J } + \\sum _ { i = 1 } ^ { t } \\alpha _ { 1 , \\{ i + m a x ( Q ) \\} } , \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ N p _ n \\sum _ { \\ell = 0 } ^ { L - 1 } \\left ( 1 - p _ n \\right ) ^ \\ell \\left ( 1 - r _ n \\right ) ^ { \\ell + 1 } , \\\\ & 0 \\le r _ n \\le 1 , \\forall n \\in [ 1 : N ] , \\\\ & \\sum _ { n = 1 } ^ N r _ n = R _ { \\sf c } . \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} \\Delta { C } _ { n + 1 } = & 0 , \\\\ \\Delta { C } _ { t } = & \\big ( \\mu _ 1 ( t ) ( \\gamma _ t - 1 ) - \\mu _ 0 ( t ) ( \\alpha _ t - 1 ) \\big ) + H ( \\alpha _ t ) - H ( \\gamma _ t ) \\\\ & \\qquad \\qquad + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu _ 1 ( t ) + \\Delta { C } _ { t + 1 } } } { 1 + 2 ^ { \\mu _ 0 ( t ) + \\Delta { C } _ { t + 1 } } } \\Big ) , ~ ~ t \\in \\{ n , \\ldots , 0 \\} . \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{gather*} g ^ { [ k , \\ell ] ( \\alpha , \\beta ) } = g ^ { [ k , \\ell ] ( \\alpha , \\beta ) } _ { - } g ^ { [ k , \\ell ] ( \\alpha , \\beta ) } _ { 0 + } , \\end{gather*}"}
-{"id": "1659.png", "formula": "\\begin{align*} J _ 1 : = & - 2 E \\left [ \\int _ t ^ T \\langle \\partial _ j ( u - k _ m ) ^ + ( s ) , a ^ { i j } \\partial _ i ( u - k _ m ) ^ + ( s ) + \\sigma ^ { j r } ( s ) v ^ { r , k _ m } ( s ) \\rangle \\ , d s | \\mathcal { F } _ t \\right ] \\\\ \\leq & - \\lambda E \\left [ \\int _ t ^ T \\| \\nabla ( u - k _ m ) ^ + ( s ) \\| ^ 2 \\ , d s | \\mathcal { F } _ t \\right ] + \\frac { 1 } { \\varrho } E \\left [ \\int _ t ^ T \\| v ^ { k _ m } ( s ) \\| ^ 2 \\ , d s | \\mathcal { F } _ t \\right ] . \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} H _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) ( j , \\beta , \\delta ) = T _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) \\oplus F _ n ( T _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) ) \\oplus F _ n ( T _ { ( 2 x y ) } ( j , \\beta , \\delta ) \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} \\{ k \\in S _ { m , N } : | x _ { k } - x _ { k } ^ { ( j ) } | > \\gamma \\} = \\bigcup _ { M = 1 } ^ { \\infty } \\bigcap _ { i = M } ^ { \\infty } \\{ k \\in S _ { m , N } : | x _ { k } ^ { ( i ) } - x _ { k } ^ { ( j ) } | > \\gamma \\} . \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} \\left ( u ( t ) - u _ 0 - t \\partial _ t v _ 0 1 _ { \\alpha > 1 } , \\phi \\right ) = I _ { t } ^ { \\alpha } \\left ( f ( t ) , \\phi \\right ) + \\sum _ { k = 1 } ^ { \\infty } I _ { t } ^ { \\alpha - \\beta } \\int _ { 0 } ^ { t } \\left ( g ^ { k } ( s ) , \\phi \\right ) d w _ { s } ^ { k } \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} b = a \\partial _ v a . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} { } _ { 1 } \\tilde { \\phi } _ { 1 } \\left ( 0 ; z ^ { - 1 } \\xi q ^ { - n } ; q , z \\xi q ^ { - n } \\right ) = ( - z \\xi ) ^ { n } q ^ { - \\frac { 1 } { 2 } n ( n + 1 ) } \\left [ A + z ^ { - 2 n } B + o ( 1 ) \\right ] \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} & g ^ 0 ( \\lambda ) \\big | A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\bigl ( 1 - e ^ { i \\lambda \\mu } \\bigr ) ^ { n } f ^ 0 ( \\lambda ) - \\lambda ^ { 2 n } C ^ { \\mu , 0 } _ { N } \\bigl ( e ^ { i \\lambda } \\bigr ) \\big | \\\\ & = \\alpha _ 2 \\big | 1 - e ^ { i \\lambda \\mu } \\big | ^ { n } \\bigl ( f ^ 0 ( \\lambda ) + \\lambda ^ { 2 n } g ^ 0 ( \\lambda ) \\bigr ) , \\label { D 1 r i v n 2 _ i _ s t . n _ d } \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{gather*} M _ { a } ^ { \\pm } = \\mathop { \\overleftarrow \\prod } \\limits _ { i = 1 } ^ { t } \\psi _ { a } ^ { \\pm } ( z _ { i } ) = \\psi _ { a } ^ { \\pm } ( z _ { t } ) \\psi _ { a } ^ { \\pm } ( z _ { t - 1 } ) \\cdots \\psi ^ { \\pm } _ { a } ( z _ { 2 } ) \\psi _ { a } ^ { \\pm } ( z _ { 1 } ) . \\end{gather*}"}
-{"id": "1321.png", "formula": "\\begin{align*} & \\pi ( b ) ( \\delta _ j \\otimes \\xi ) = \\delta _ j \\otimes \\rho ( \\alpha ^ { - j } ( b ) ) \\xi , & & \\pi ( S ) ( \\delta _ j \\otimes \\xi ) = \\delta _ { j + 1 } \\otimes \\xi \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{gather*} I ( s , f ^ { ( n ) } , \\chi ) = \\\\ \\sum _ { y _ 0 \\not \\equiv 1 \\bmod { \\pi } } \\sum _ { j = 0 } ^ { \\infty } q ^ { - 1 - j } \\int \\limits _ { O _ v ^ { \\times 2 } } \\chi ( a c ( x ^ { 1 2 } [ y _ 0 + \\pi ^ { j + 1 } y ] ^ 4 \\widetilde { f ^ { ( n ) } } ( x , y _ 0 + \\pi ^ { j + 1 } y ) ) ) \\ | d x d y | \\\\ + \\sum _ { j = 0 } ^ { \\infty } q ^ { - 1 - j } \\int \\limits _ { O _ v ^ { \\times 2 } } \\mathcal { X } ( x ^ { 1 2 } [ 1 + \\pi ^ { j + 1 } y ] ^ 4 \\widetilde { f ^ { ( n ) } } ( x , 1 + \\pi ^ { j + 1 } y ) ) \\ | d x d y | , \\end{gather*}"}
-{"id": "8397.png", "formula": "\\begin{align*} C _ f ^ * ( \\chi , s ) = L _ f ^ * ( \\chi , s ) L _ f ^ * ( \\chi , q s ) L _ f ^ * ( \\chi , q ^ 2 s ) \\cdots , \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} y ( t ) = 1 - \\exp ( - \\int _ 0 ^ t \\lambda _ s \\ , \\dd s ) . \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} F ( p t ) = F ( t ) , \\ \\ F ( q t ) = \\tilde B _ 1 F ( t ) \\mbox { f o r } t \\in S \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} u _ { i j } \\mu \\varphi = u _ { i j } \\mu \\varphi = u _ { i j } \\mu ( 1 ) \\varphi . \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { + \\infty } a _ i \\int _ 0 ^ 1 f ' _ i ( x ) d x = 0 , \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} h ( Y , J _ \\alpha X ) - J _ \\alpha h ( X , Y ) = \\frac { 1 } { 4 n } \\left [ g ( X , Y ) p _ \\alpha ^ \\bot + g ( X , J _ \\alpha Y ) J _ \\alpha ( p _ \\alpha ^ \\bot ) \\right ] , \\ \\alpha = 2 , 3 . \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} F P S ( l / k ) = z _ { l , 0 } + 2 \\sum \\limits _ { q = 1 } ^ { \\frac { { l - 1 } } { 2 } } { z _ { l , q } \\cos ( ( q ) } \\frac { { 2 k \\pi } } { l } ) \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} q _ { \\alpha , \\beta } ( t , x ) : = \\begin{cases} I _ { t } ^ { \\alpha - \\beta } p ( t , x ) & : \\alpha \\geq \\beta \\\\ D _ { t } ^ { \\beta - \\alpha } p ( t , x ) & : \\alpha < \\beta , \\end{cases} \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} b ^ { t r } ( d ^ { t r } g + g ^ { t r } d ) = 0 . \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} F _ i \\left ( x _ i , \\sum _ { i = 1 } ^ N h _ i ( x _ i ) \\right ) = \\nabla _ { x _ i } f _ i \\left ( x _ i , \\sum _ { i = 1 } ^ N h _ i ( x _ i ) \\right ) , \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} y ( t ) = 1 - \\exp ( - \\lambda t ) . \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - d u ( t , x ) & = [ \\partial _ j ( a ^ { i j } \\partial _ i u ( t , x ) + \\sigma ^ { j r } v ^ r ( t , x ) ) + \\bar { f } ( t , x ) + \\nabla \\cdot \\bar { g } ( t , x ) ] \\ , d t \\\\ & + \\mu ( d t , x ) - v ^ r ( t , x ) \\ , d W ^ r _ t , ~ ~ ~ ( t , x ) \\in Q , \\\\ u ( T , x ) & = G ( x ) , ~ ~ ~ x \\in \\mathcal { O } , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} d ^ { t r } d = \\begin{pmatrix} ( 1 - 2 \\epsilon ^ 2 ) / 2 & 0 & 0 \\\\ 0 & 1 / 2 & 0 \\\\ 0 & 0 & 1 / 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} \\lim _ { g \\downarrow h } \\theta _ { > } ( g ) = \\theta _ { > } ( h ) , \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} \\int _ 0 ^ s e ^ { 2 \\pi i H _ n ( u ) } d u = \\int _ 0 ^ s e ^ { 2 \\pi i n u } d u = ( 1 / 2 \\pi n ) [ \\sin ( 2 \\pi n s ) - i \\{ \\cos ( 2 \\pi n s ) - 1 \\} ] , 0 \\leq s \\leq 1 . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} D : = \\left \\{ z : 0 < r < R , \\vert \\theta _ i \\vert < \\theta ^ * : = \\arccos \\frac { 1 } { 2 ^ { \\frac 1 { N - 1 } } } \\right \\} \\subset \\left \\{ z : \\vert z - y \\vert < R \\right \\} . \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} \\Im \\beta _ 3 = \\Im \\beta _ 4 \\geq 0 \\Rightarrow \\Im \\alpha _ 2 < 0 . \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} X ( n ) = X ( m / l ) = Y ( 1 ) \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} A _ 1 = 2 ( 4 e ( 1 + 1 / \\alpha ) ) ^ { \\alpha } ( 1 + \\eta ) . \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } Q ( t ) = \\int _ s ^ t A ( p ( \\tau , s ) ) P ( \\tau ) d \\tau , \\\\ P ( t ) = - \\int _ s ^ t \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , s ) ) Q ( \\tau ) d \\tau - \\int _ s ^ t \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , s ) ) d \\tau . \\end{array} \\right . \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} u = 0 , w ^ 2 = p ^ 2 = s ^ 2 . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} y _ 0 = x _ 0 ^ 2 , ~ ~ y _ 1 = x _ 0 x _ 1 , ~ ~ y _ 2 = x _ 1 ^ 2 , ~ ~ y _ 3 = x _ 2 \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t u + ( u \\cdot \\nabla ) u - \\Delta u + \\nabla p + \\nabla \\cdot ( v \\otimes v ) = 0 , \\\\ \\nabla \\cdot u = 0 , \\\\ \\partial _ t v + ( u \\cdot \\nabla ) v - \\Delta v + ( v \\cdot \\nabla ) u = \\nabla \\theta , \\\\ \\partial _ t \\theta + u \\cdot \\nabla \\theta - \\nabla \\cdot v = S _ \\theta , \\end{array} \\right . \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} \\dot { \\tilde { x } } = - \\varPhi ( \\tilde { F } ^ { - 1 } ) \\tilde { \\nu } \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} D _ \\lambda ( a ) \\cdot X = \\Delta _ \\lambda ( a \\cdot X ) . \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} [ g , x ] = y ^ { - 4 \\beta } z ^ { - 4 \\gamma } , [ g , y ] = x ^ { - 4 \\alpha } z ^ { - 4 \\gamma } , g ^ 2 = x ^ { 4 \\alpha } y ^ { 4 \\beta } z ^ { 4 \\gamma } \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} a \\nabla _ { \\nu } b \\leqslant a \\sharp _ { \\nu } b + ( \\sqrt { a } - \\sqrt { b } ) ^ 2 - \\sum _ { k = 0 } ^ { \\infty } r _ { k } \\big [ \\big ( a ^ { \\frac { m _ k } { 2 ^ k } } b ^ { 1 - \\frac { m _ k } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } - \\big ( a ^ { \\frac { m _ k + 1 } { 2 ^ k } } b ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ] ^ { 2 } . \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} r = \\int R \\ , d A _ g , \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\omega : \\alpha _ { k , i } \\leq \\frac { 2 } { k p _ i } \\mbox { f o r } k \\geq \\tilde k ( \\omega ) \\right ] = 1 . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} N = \\Bigg \\{ \\begin{pmatrix} 1 & & & \\\\ & \\ddots & \\ast & \\\\ & 0 & \\ddots & \\\\ & & & 1 \\end{pmatrix} \\Bigg \\} . \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} \\psi ( f ) ( y ) = \\left \\{ \\begin{array} { r c l } f ( \\psi ^ { \\ast } ( y ) ) & \\mbox { i f } & y \\in U _ { \\psi } , \\\\ 0 & \\mbox { i f } & y \\not \\in U _ { \\psi } \\end{array} \\right . \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} F R + G S & = 2 F R - \\frac { 1 } { 2 } \\left ( F R \\right ) I \\\\ & = 2 F R - \\left ( \\mathbf { E } \\cdot \\mathbf { D } - \\mathbf { H } \\cdot \\mathbf { B } \\right ) I . \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} \\mathbf { m } _ e = [ \\mathbf { m _ d } ] _ { d \\in O u t ( s ) } \\mathbf { F } _ e . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} \\hom ( u , a ( y , x ) ) = a ( y , x \\pitchfork u ) , \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i { m \\choose i } \\frac { m + i } m \\ : \\gamma _ { m - r + i } = 0 . \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} ( W u ) ( n ) : = \\frac { q \\sin ( k ( n _ 1 + . . . + n _ d ) ) } { | n | } u ( n ) , \\ n \\in \\Z ^ d \\ \\ u \\in \\mathcal { H } , \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} \\left | \\frac { d } { d t } \\| \\Delta v ( t ) \\| _ { L ^ 2 } ^ 2 \\right | & = \\left | 2 \\langle \\Delta v ( t ) , \\Delta v _ t ( t ) \\rangle \\right | \\\\ & = \\left | - 2 \\langle \\Delta ^ 2 \\exp ( - t A _ o ) v _ 0 , A _ o v ( t ) \\rangle \\right | \\leq C \\| v _ 0 \\| _ { H ^ 4 } ^ 2 , \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} ( H _ 1 + ( X _ 0 + Y _ 0 ) G _ 1 ) \\circ F & = \\big ( X _ 1 ^ { a _ 1 } \\cdots X _ n ^ { a _ n } - u _ 1 ^ { a _ 1 } \\cdots u _ n ^ { a _ n } X _ 0 ^ { d - 1 } + ( X _ 0 + Y _ 0 ) G _ 1 \\big ) \\circ F = \\\\ & = \\prod _ { i = 1 } ^ n ( a _ i ! ) x _ 0 + G _ 1 \\circ \\big ( x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } + y _ 0 ^ { b _ 0 - 1 } y _ 1 ^ { b _ 1 } \\ldots y _ m ^ { b _ m } \\big ) = 0 , \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} p = e ^ { i s } m \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} \\rho _ E ^ \\lambda ( x ^ n | u ^ n ) = \\frac { c } { n \\lambda } \\log _ 2 \\left [ \\frac { 1 } { c } \\sum _ { i = 1 } ^ c \\exp _ 2 \\{ \\lambda L ( y _ { n _ { i - 1 } + 1 } ^ { n _ i } ) \\} \\right ] , ~ ~ ~ ~ n _ 0 \\equiv 0 , ~ ~ n _ c \\equiv n . \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} G = \\langle x , y \\mid x ^ { - 1 } y ^ 2 x = y ^ { - 2 } , y ^ { - 1 } x ^ 2 y = x ^ { - 2 } \\rangle \\cong ( C _ 2 \\times C _ 2 ) . \\mathbb { Z } ^ 3 \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\| \\mbox { s h r i n k } _ 1 ( z , \\bar { \\mu } ) \\| _ 1 \\ ; = \\ ; \\alpha . \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} K _ { i + n , j + n } { } ^ { \\gamma } & = \\left \\{ \\begin{array} [ c ] { l l } 1 , & \\gamma \\equiv i + n + j + n \\ \\left ( \\operatorname { m o d } 2 n \\right ) \\\\ 0 , & o t h e r w i s e \\end{array} \\right . \\\\ & = \\left \\{ \\begin{array} [ c ] { l l } 1 , & \\gamma \\equiv i + j + { 2 n } \\ \\left ( \\operatorname { m o d } 2 n \\right ) \\\\ 0 , & o t h e r w i s e \\end{array} \\right . \\\\ & = K _ { i j } { } ^ { \\gamma } . \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & 0 & \\cdots & 0 & - a _ 1 \\\\ 1 & 0 & \\cdots & 0 & - a _ 2 \\\\ 0 & 1 & \\cdot & 0 & - a _ 3 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & & 1 & - a _ n \\end{pmatrix} \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} \\Phi ( G / N ) \\cong \\Phi \\left ( L \\big / N \\cap \\Phi ( L ) \\right ) = \\Phi ( L ) / N \\cap \\Phi ( L ) \\cong \\Phi ( L ) N \\big / N = \\Phi ( G ) N \\big / N . \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} I _ { n + 1 } ( g ^ { \\otimes ( n + 1 ) } ) = I _ n ( g ^ { \\otimes n } ) I _ 1 ( g ) - n \\ , \\| g \\| _ { \\mathcal H } ^ 2 \\ , I _ { n - 1 } ( g ^ { \\otimes ( n - 1 ) } ) , \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} \\mathcal { L } _ { _ { 3 } , \\mathfrak { c _ { 3 } } } = \\kappa \\ , Q ^ { ( 3 ) } ( A , 0 ) = \\kappa \\ , Q ^ { ( 3 ) } ( A ) , \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } \\big \\| \\mathbf { u } ( t , \\cdot ) \\big \\| _ { \\mathcal { H } } ^ { 2 } - \\tfrac { 1 } { 2 } \\big \\| \\mathbf { u } ( 0 , \\cdot ) \\big \\| _ { \\mathcal { H } } ^ { 2 } + \\int _ { 0 } ^ { t } a \\big ( \\mathbf { u } ( s , \\cdot ) , \\mathbf { u } ( s , \\cdot ) ; \\big ( \\mathbf { H } ( \\mathbf { u } ) \\big ) ( s , \\cdot ) \\big ) \\mathrm { d } s = 0 . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} \\ln \\lambda = - \\int _ 0 ^ \\infty \\frac { d t } { t } e ^ { - t \\lambda } \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} f _ n ( t ) \\leq C _ n + \\sum _ { j = 1 } ^ { n - 1 } C _ j a _ { n - j } + C _ 0 a _ n M . \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} p _ d = \\frac { 1 } { 1 + \\sqrt { 1 - p _ e } } p _ a = 1 - \\sqrt { 1 - p _ e } . \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} \\mathcal { B } _ { r } ^ c ( \\theta ^ * ) & = \\bigcup _ { l = 1 } ^ { { { L _ r } } - 1 } \\{ \\mathcal { B } _ { r _ { l } } \\backslash \\mathcal { B } _ { r _ { l + 1 } } \\} \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} \\rho = R _ { \\rm i n } ( K ) : = \\sup _ { y \\in K } { \\rm d i s t } ( y , \\partial K ) . \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} & \\big | A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\bigl ( 1 - e ^ { i \\lambda \\mu } \\bigr ) ^ { n } f ^ 0 ( \\lambda ) - \\lambda ^ { 2 n } C ^ { \\mu , 0 } _ { N } \\bigl ( e ^ { i \\lambda } \\bigr ) \\big | ^ 2 \\\\ & = \\alpha _ 2 \\gamma ( \\lambda ) \\big | 1 - e ^ { i \\lambda \\mu } \\big | ^ { 2 n } \\bigl ( f ^ 0 ( \\lambda ) + \\lambda ^ { 2 n } g ^ 0 ( \\lambda ) \\bigr ) ^ 2 , \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} S _ { n , 2 m + 1 } = \\sum \\limits _ { i = 0 } ^ m { m \\brace i } S _ { n , 2 i } . \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} Y ( x ) = \\Phi ( x ) x ^ L \\ , e ^ { Q ( x ) } x ^ { D x } , \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} | | g ^ { - 1 } f ^ * h ( x ) | | ^ { Q / 2 } = \\| D f ( x ) \\| ^ { Q } \\leq K J _ f ( x ) = K \\det ( \\overline { g } ^ { - 1 } \\overline { f ^ * h } ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} \\langle { \\psi } , L ( \\Lambda ' , \\psi , g ) { \\psi } \\rangle = \\langle { \\psi } , L ( \\Lambda , \\psi , g ) { \\psi } \\rangle - \\ln \\int _ { \\Lambda ' } ^ \\Lambda D \\chi \\ , e x p \\left \\{ - \\langle { \\chi } , L ( \\Lambda , \\chi , g ) { \\chi } \\rangle \\right \\} \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} \\int _ E P _ t \\mathbf 1 _ E d \\mu = \\int \\left ( P _ { t / 2 } \\mathbf 1 _ E \\right ) ^ 2 d \\mu . \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} \\sigma _ 1 ( Y ) = B _ 1 \\ , Y \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\pi _ d ( s , q ) & = \\begin{cases} \\frac { \\pi ( s , q ) } { \\sum _ { ( s , q ) \\in A _ d } \\pi ( s , q ) } & \\mbox { i f } ( s , q ) \\in A _ d ; \\\\ 0 & \\mbox { o t h e r w i s e . } \\end{cases} . \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} & \\lambda _ { P ^ * } ( X ) \\\\ = & r ^ * ( X ) + r ^ * ( E - X ) - r ^ * ( E ) \\\\ = & r ( E - X ) + | | X | | _ r - r ( E ) + r ( X ) + | | E - X | | _ r - r ( E ) \\\\ & - r ( \\emptyset ) - | | E | | _ r + r ( E ) \\\\ = & \\lambda _ P ( X ) . \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\mathrm { d } A = F _ A = i \\Omega . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} \\tau ^ * ( \\mu _ R , \\mu _ T ) \\ge \\tau _ { L 1 } \\triangleq \\max \\limits _ { \\substack { l = 1 , \\cdots , \\min \\{ N _ T , N _ R \\} \\\\ s _ 1 = 0 , 1 , \\ldots , l \\\\ s _ 2 = 0 , 1 , \\ldots , N _ R - l } } \\frac { 1 } { l } \\Big \\{ & ( s _ 1 + s _ 2 ) - ( N _ T - l ) s _ 2 \\mu _ T \\\\ & \\left . - \\left ( \\frac { 2 s _ 2 + s _ 1 + 1 } { 2 } \\cdot s _ 1 + s _ 2 ^ 2 \\right ) \\mu _ R \\right \\} , \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} \\mathbb { E } ( N _ 1 ) = \\frac { \\lambda - \\mu + \\xi } { \\xi } + \\left [ \\frac { \\lambda } { ( \\lambda + \\xi ) A } - \\frac { \\gamma } { \\lambda } + \\frac { \\mu \\gamma } { \\lambda \\xi } \\right ] p _ { 0 , 0 } , \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} v _ i = 0 , \\ v _ { i , J } = 0 , \\omega = 0 \\Gamma \\times ( 0 , \\infty ) \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} \\int _ \\varepsilon ^ R G ( \\rho ) \\frac { d } { d \\rho } \\left ( \\int _ { B _ \\rho ( 0 ) } F ( z ) \\ , d z \\right ) d \\rho & = \\int _ \\varepsilon ^ R G ( \\rho ) \\left ( \\int _ { \\partial B _ \\rho ( 0 ) } F ( s ) \\ , d S _ { \\rho } \\right ) d \\rho \\\\ & = \\int _ { R \\geq | z | \\geq \\varepsilon } F ( z ) G ( | z | ) \\ , d z . \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } W _ { n } \\left ( \\psi ' , \\psi \\right ) = \\lim _ { n \\to \\infty } W _ { n } \\left ( \\psi ' , \\bar { \\psi } \\right ) = 0 . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\phi _ \\pm ( x , \\xi ) : = \\lim _ { t \\to \\pm \\infty } ( \\phi ( t ; x , \\xi ) - \\phi ( t ; 0 , \\xi ) ) \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} \\widetilde { \\omega } _ { 1 , j } ^ \\mathrm { z m } = e ^ { - i \\lambda _ j u _ 1 } \\varphi _ j + e ^ { i \\lambda _ j u _ 1 } C _ 1 ( \\lambda _ j ) \\varphi _ j . \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} \\sum _ { j = - \\infty } ^ { n } g _ { j } ^ { 2 } ( x ) = \\frac { x ^ { 2 } } { x ^ { 2 } - 1 } W _ { n } ( g ' ( x ) , g ( x ) ) , \\mbox { f o r } 0 < | x | < 1 . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} \\langle \\alpha _ 1 , f \\rangle = \\sum _ { k = 2 } ^ \\infty \\frac { k } { k - 1 } f ( k - 1 ) \\frac { 1 } { k ^ p } = \\sum _ { k = 1 } ^ \\infty \\frac { k + 1 } { k } f ( k ) \\frac { 1 } { ( k + 1 ) ^ p } \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} | \\widetilde { R m } | ( y _ { \\infty } , s _ { \\infty } ) = | \\widetilde { R m } | ( y _ { \\infty } , 0 ) < c _ a ^ { - 2 } . \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} g ( 0 , x _ 0 , t , 1 ) = \\Phi \\left ( \\frac { e ^ { t } - x _ 0 } { t ^ H } \\right ) + \\Phi \\left ( \\frac { e ^ { t } + x _ 0 } { t ^ H } \\right ) - 1 \\to 2 \\Phi ( \\infty ) - 1 = 1 , \\quad t \\to \\infty . \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} \\mathcal { C } & \\coloneqq \\left \\{ w \\in \\Sigma _ { P } ^ { * } \\middle | \\ ; \\left | w \\right | \\geq m \\ , \\ , \\mathrm { a n d } \\ , \\ , \\forall 0 \\leq k \\leq \\left | w \\right | - m : \\ , \\theta ^ { k } \\left ( \\left [ w \\right ] \\right ) \\cap A = \\emptyset \\right \\} , \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} 2 ^ { \\lfloor ( m + 3 ) / 2 \\rfloor } S _ { n , 2 m + 1 } = \\sum \\limits _ { i = 0 } ^ m A _ i ^ m S _ { n , 2 i } , \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\pi i } \\partial \\overline { \\partial } H ( X ) = \\tfrac { 1 } { 2 } \\omega _ { \\mathrm { H d g } } + b _ 1 \\cdot \\int _ { \\pi _ 2 } h ^ 3 + b _ 2 \\cdot e _ 1 ^ A + b _ 3 \\cdot e ^ A , \\end{align*}"}
-{"id": "10146.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - h \\cdot \\nabla f ( x _ k ) , ~ k \\geq 0 , \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} ( u - v ) [ x ^ + _ i ( u ) , x ^ - _ j ( v ) ] = - \\delta _ { i j } \\hbar ( \\xi _ i ( u ) - \\xi _ i ( v ) ) \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} P _ { \\sigma } ( \\chi ^ { ( r , \\varphi ) } ) = ( 2 \\pi i ) ^ { - \\lceil \\frac { n } { 2 } \\rceil w _ { 0 } } G _ { \\sigma } ( \\chi ^ { ( r , \\varphi ) } ) ^ { r } a _ { \\sigma } ^ { * } ( \\chi ^ { ( r , \\varphi ) } ) Q _ { \\sigma } ( \\chi ^ { ( r , \\varphi ) } ) ^ { r - \\lceil \\frac { n } { 2 } \\rceil } . \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} Q ( \\vec { z } ) = \\lambda _ { \\sigma } ( \\vec { z } ) \\ , ( \\sigma Q ) ( \\vec { z } ) . \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} C _ { B D } { } ^ { C } C _ { C A } { } ^ { E } + C _ { A B } { } ^ { C } C _ { C D } { } ^ { E } + C _ { D A } { } ^ { C } C _ { C B } { } ^ { E } = 0 , \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\overrightarrow { Q } _ { 0 , n } ( d y ^ n | x ^ n ) & \\triangleq \\otimes _ { t = 0 } ^ n { q } _ { t } ( d y _ t | y ^ { t - 1 } , x ^ { t } ) \\in { \\cal M } ( { \\cal Y } ^ n ) \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} \\tau = - \\log \\Theta , \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} \\delta ^ 2 \\Vert Y _ 0 - X _ * \\Vert _ F ^ 2 + \\sum _ { k = 1 } ^ r \\limits s _ k ^ 2 \\sin ^ 2 \\phi _ { R 0 , k } \\leq s _ r . \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 3 } \\ [ \\bar { f } _ { 0 , k _ { \\pi ( 1 ) } } , [ \\bar { f } _ { 0 , k _ { \\pi ( 2 ) } + 1 } , \\bar { f } _ { 0 , k _ { \\pi ( 3 ) } - 1 } ] ] = 0 . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} \\bar { P } _ { a } & = \\frac { 1 } { \\sqrt { 2 } } P _ { a } + \\frac { 1 } { \\sqrt { 2 } } Z _ { a } , \\\\ \\bar { Z } _ { a } & = \\frac { 1 } { \\sqrt { 2 } } P _ { a } - \\frac { 1 } { \\sqrt { 2 } } Z _ { a } . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u ( t , x ) = \\Delta u ( t , x ) + \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } g ^ { k } ( s , x ) d w _ { s } ^ { k } , t > 0 \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} g ( r ) = \\frac { 1 } { f _ a ^ { n - 1 } ( r ) } \\int _ 0 ^ r a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t . \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a \\ , t ^ 2 ) ^ 2 + t ^ 2 } , a = c o n s t \\neq 0 , c = c o n s t . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} \\mu _ { \\mathbf { v } } ( t ) = f ^ t _ { \\mathcal { X } , \\mathcal { Y } } ( \\gamma _ { \\mathbf { v } } ( t ) ) = \\gamma _ { \\mathbf { v } } ( t ) + \\mathcal { O } ( t ^ { N + 1 } ) = x _ 0 + \\tfrac { C } { 2 } t ^ N \\mathbf { v } + \\mathcal { O } ( t ^ { N + 1 } ) . \\end{align*}"}
-{"id": "10017.png", "formula": "\\begin{align*} b _ { H , a } = q ^ t + ( 1 - b _ { H , t } ) q ^ { t - a } - c _ H q ^ { d - a } , \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} c _ j ^ + ( n + j ) & : = t , c _ j ^ - ( n + j ) : = s . \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} \\alpha _ 2 = \\begin{pmatrix} \\alpha & 0 \\\\ 0 & 0 \\end{pmatrix} , \\quad \\beta _ 2 = \\begin{pmatrix} 0 \\\\ \\beta \\end{pmatrix} , \\quad \\gamma _ 2 = \\begin{pmatrix} 0 & \\gamma \\end{pmatrix} , \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} \\sigma ( t ) = \\begin{bmatrix} \\sigma _ { 1 1 } ( t ) & 0 \\\\ \\sigma _ { 2 1 } ( t ) & I _ { n _ 2 } \\end{bmatrix} \\quad a ^ W ( t ) = \\begin{bmatrix} I _ k & 0 \\\\ 0 & \\alpha ( t ) \\end{bmatrix} , \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} \\textbf { w } _ 1 = ( \\omega _ { \\mu , a } , \\omega _ { \\mu + 1 , a } ) , \\textbf { w } _ 2 = ( \\rho _ { \\nu , b } , \\rho _ { \\nu + 1 , b } ) . \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\begin{cases} \\Big ( \\tilde \\tau + s i g n ( U _ s '' ( a ) ) z ^ 2 \\Big ) ^ 2 \\frac { d } { d z } W + i \\frac { d ^ 3 } { d z ^ 3 } \\Big ( \\big ( \\tilde \\tau + s i g n ( U _ s '' ( a ) ) z ^ 2 \\big ) W \\Big ) = 0 , \\\\ \\lim \\limits _ { z \\rightarrow - \\infty } W ~ = ~ 0 , \\quad \\lim \\limits _ { z \\rightarrow + \\infty } W ~ = ~ 1 . \\end{cases} \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} \\begin{array} { l } x \\in A \\Longrightarrow \\exists M \\in \\mu { } x \\in M { } k M = M \\cup \\{ 0 \\} \\subset A \\ , . \\end{array} \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} \\delta ( X ) = \\frac { 1 } { 2 } ( \\mathrm { m i n } \\{ m \\equiv 2 \\mu ( X ) + 1 \\bmod { 2 } \\mid \\exists x \\in H ^ m _ { \\mathbb { Z } / 4 } ( X ) , \\ ; U ^ \\ell x \\neq 0 \\ ; \\mathrm { f o r \\ ; a l l } \\ ; \\ell \\geq 0 , \\ ; Q x = 0 \\} - 1 ) . \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} \\nu _ e = \\nu ^ { { \\pi ^ { * , \\infty } } } ( e ) , ~ ~ ~ \\nu _ 0 = \\nu ^ { { \\pi ^ { * , \\infty } } } ( 0 ) . \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} P ( D ) = \\int P _ { \\mathbf { x } } ( D ) d ( \\mu \\mathbf { x } ) , D \\in \\mathcal { D } _ n , \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} \\Psi ( A ) = \\mu _ B ( C _ 0 ( X _ A ) ) \\cdot \\Psi ( A ) \\subseteq \\mu _ B ( C _ 0 ( X _ A ) ) \\cdot B , \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} F ^ T _ { \\gamma } ( y , y _ i ^ T ) : = T y + W ( y ) + { \\gamma \\over 2 } ( y - y ^ T _ { i } ) ^ 2 \\ , , \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} u ( t ) \\geq \\dfrac { \\varepsilon \\rho ^ { * } } { \\displaystyle \\max _ { i = 1 , \\ldots , m } | I ^ { + } _ { i } | } = M _ { 1 } \\rho ^ { * } , \\forall \\ , t \\in I ^ { + } _ { i } . \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u _ t ( x ) = - { \\nu } ( - \\Delta ) ^ { \\frac { \\alpha } { 2 } } u _ t ( x ) + \\lambda u _ t ( x ) + \\xi \\sigma ( u _ t ( x ) ) \\dot W ( t , x ) , \\ \\ \\ \\ x \\in B _ R ( 0 ) , \\ \\ \\ t > 0 \\\\ u _ t ( x ) = 0 , \\ \\ \\ x \\in B _ R ( 0 ) ^ { c } , \\ \\ t > 0 \\end{cases} \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} N = q _ 0 + q _ 0 ^ { - 1 } \\& ( F ^ * F ) = q + q ^ { - 1 } , \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} c = & 2 u '^ 2 ( 1 - \\cos u ) + v '^ 2 ( 2 + \\cos u ) ^ 2 , \\\\ c \\kappa = & ( u '' v ' - v '' u ' ) \\sqrt { 2 } \\sqrt { 1 - \\cos u } ( 2 + \\cos u ) + v '^ 3 \\frac { \\sin u ( 2 + \\cos u ) ^ 2 } { \\sqrt { 2 } \\sqrt { 1 - \\cos u } } \\\\ & + u '^ 2 v ' \\frac { ( 6 - 3 \\cos u ) \\sin u } { \\sqrt { 2 } \\sqrt { 1 - \\cos u } } . \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} A _ i & = ( i r , i r + q ( \\beta - r ) ) , \\\\ B _ i & = ( r - \\beta + i \\beta , r - \\beta + i \\beta ) , \\\\ C _ i & = ( i \\beta , i \\beta ) . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} \\frac { \\bar { P } } { 1 + \\sigma ^ 2 G } = \\frac { P ( 1 - 2 ^ { - B } ) } { 1 + 2 ^ { - B } P G } . \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} & \\mathrm { s u p p } \\ , \\Lambda \\subset \\{ ( x _ 1 , x _ 2 ) : x _ 1 > 0 , \\ | x _ 2 | < | x _ 1 | ^ { \\nu } , \\ \\nu > 0 \\} , \\\\ & \\mathrm { s u p p } \\ , \\Lambda _ n \\subset [ - R _ n , R _ n ] , \\ R _ n > 0 . \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} \\| T _ t x - T _ t y \\| ^ 2 & = \\| t T x + ( 1 - t ) x - t T y - ( 1 - t ) y \\| ^ 2 \\\\ & = \\| ( x - y ) + ( - t ) ( ( x - T x ) - ( y - T y ) ) \\| ^ 2 \\\\ & \\leq \\| x - y \\| ^ 2 - 2 t j ( x - y ) ( ( x - T x ) - ( y - T y ) ) + d t ^ 2 \\| ( x - T x ) - ( y - T y ) \\| ^ 2 \\\\ & \\leq \\| x - y \\| ^ 2 - t ( 1 - k ) \\| ( x - T x ) - ( y - T y ) \\| ^ 2 + d t ^ 2 \\| ( x - T x ) - ( y - T y ) \\| ^ 2 \\\\ & = \\| x - y \\| ^ 2 + t ( d t - ( 1 - k ) ) \\| ( x - T x ) - ( y - T y ) \\| ^ 2 \\\\ & \\leq \\| x - y \\| ^ 2 . \\\\ \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} & \\left [ ( \\lambda z - \\mu j ) ( 1 - z ) + c \\xi ( 1 - z ) \\right ] P _ { c } ( z ) - \\xi z ( 1 - z ) P ^ { ' } _ { c } ( z ) \\\\ & = \\gamma z P _ { c - 1 } ( z ) - \\mu z \\sum _ { n = 1 } ^ { c } n z ^ { n } p _ { c , n } + c ( 1 - z ) ( \\xi - \\mu ) \\sum _ { n = 0 } ^ { c } z ^ { n } p _ { c , n } - \\xi ( 1 - z ) \\sum _ { n = 1 } ^ { c } n z ^ { n } p _ { c , n } . \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} \\Pr _ { a \\sim _ { \\rho } b } [ a \\in A , b \\in B ] & = 2 ^ { - n } \\sum _ { d = 0 } ^ { n } \\left ( \\frac { 1 + \\rho } { 2 } \\right ) ^ { n - d } \\left ( \\frac { 1 - \\rho } { 2 } \\right ) ^ d | W _ d | \\\\ & \\le 2 ^ { - 2 n } \\sum _ { d = 0 } ^ { n } ( 1 + \\rho ) ^ { n - d } ( 1 - \\rho ) ^ d \\binom { n } { d } 2 ^ { d } \\\\ & = 2 ^ { - 2 n } ( 3 - \\rho ) ^ n \\ , , \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} K _ { k + n , l } { } ^ { m } & = \\left \\{ \\begin{array} [ c ] { l l } 1 , & m + n \\equiv k + n + l \\ \\left ( \\operatorname { m o d } 2 n \\right ) \\\\ 0 , & o t h e r w i s e \\end{array} \\right . \\\\ & = \\left \\{ \\begin{array} [ c ] { l l } 1 , & m + { 2 n } \\equiv k + { 2 n } + l \\ \\left ( \\operatorname { m o d } 2 n \\right ) \\\\ 0 , & o t h e r w i s e \\end{array} \\right . \\\\ & = K _ { k l } { } ^ { m } . \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} \\frac { 1 } { \\kappa _ K } \\sum _ { \\mathrm { N } \\mathfrak { n } < x } \\frac { 1 } { \\mathrm { N } \\mathfrak { n } } \\geq \\log x - \\sum _ { j = 1 } ^ { n _ K } \\frac { 1 } { j } + \\gamma _ K + O _ { \\epsilon } \\left ( x ^ { - \\tfrac { \\epsilon } { 8 } } \\log x \\right ) , \\end{align*}"}
-{"id": "9923.png", "formula": "\\begin{align*} \\lambda ^ { \\max } ( u ( r ) v ) = \\lambda ^ { \\max } ( \\sigma ( r ) u ( - r ) u ^ { - } ( r ^ { - 1 } ) v ) & = - \\lambda ^ { \\min } ( u ( - r ) u ^ { - } ( r ^ { - 1 } ) v ) \\\\ & = - \\lambda ^ { \\min } ( u ^ { - } ( r ^ { - 1 } ) v ) \\geq - \\lambda ^ { \\max } ( u ^ { - } ( r ^ { - 1 } ) v ) = - \\lambda ^ { \\max } ( v ) . \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} \\mathbf { K } _ { d , e } = \\Phi ( k _ { d , e } ) , \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} \\frac { \\partial { f ( z ) } } { \\partial { \\bar { z } } } = \\mu ( z ) \\frac { \\partial { f ( z ) } } { \\partial z } \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} \\dim \\left ( Z _ { 2 n } \\times \\mathfrak { g } \\right ) _ { H } = n \\dim \\mathfrak { g } . \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} \\beta = s ^ { - 1 } \\begin{pmatrix} - u & p t \\\\ - w & 0 \\end{pmatrix} , \\gamma ^ { t r } = s ^ { - 1 } \\begin{pmatrix} - p & - t u \\\\ 0 & - t w \\end{pmatrix} . \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} h ( x , \\nabla _ x \\phi _ \\pm ( x , \\xi ) ) = \\lim _ { t \\to \\pm \\infty } h _ \\rho ( 0 , x , p ( 0 , t ; y ( 0 , t ; x , \\xi ) , \\xi ) ) . \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} \\Gamma _ M ( \\kappa w \\ , | \\ , \\kappa a ) = \\kappa ^ { - B _ { M , M } ( w \\ , | \\ , a ) / M ! } \\ , \\Gamma _ M ( w \\ , | \\ , a ) . \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( b ; q \\right ) _ { n } q ^ { n ^ { 2 } / 2 } S _ { n } \\left ( a q ^ { - n - 1 / 2 } ; q \\right ) } { \\left ( c ; q \\right ) _ { n } } \\left ( \\frac { c } { a b } \\right ) ^ { n } = \\frac { \\left ( c / b ; q \\right ) _ { \\infty } } { \\left ( c ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } / 2 } } { \\left ( q ; q \\right ) _ { n } } \\left ( \\frac { c } { a b } \\right ) ^ { n } A _ { q } \\left ( \\frac { c q ^ { n - 1 / 2 } } { a } \\right ) . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\rho ( r ) \\approx \\rho _ q ( r ) = \\frac { a _ 1 r ^ 2 + a _ 2 r ^ 4 + \\cdots + a _ q r ^ { 2 q } } { 1 + b _ 1 r ^ 2 + b _ 2 r ^ 4 + \\cdots + b _ q r ^ { 2 q } } . \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\partial _ t \\left ( \\norm { \\nabla \\zeta } ^ 2 + \\alpha \\norm { \\tau } ^ 2 _ { L ^ 2 ( \\Gamma _ u ) } \\right ) + \\norm { \\Delta \\zeta } ^ 2 = \\int _ { \\Omega } ( v \\nabla _ H \\zeta \\cdot \\Delta \\zeta + w \\partial _ z \\zeta \\cdot \\Delta \\zeta ) - \\int _ { \\Omega } g \\cdot \\Delta \\zeta . \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} \\tilde { b } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\end{align*}"}
-{"id": "9889.png", "formula": "\\begin{align*} S _ { m , n } ( c , \\eta ) = \\mathcal { N } \\ , e ^ { i \\theta } \\ , ( 1 - \\eta ^ 2 ) ^ { m / 2 } \\ , \\sum _ { j = 0 } ^ \\infty b _ j \\ , ( 1 - \\eta ) ^ j \\ , , \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} m ( t ) : = \\int _ { \\vert x \\vert \\leq R } u ( t , x ) d x , \\end{align*}"}
-{"id": "10080.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 3 } { x ^ 2 z ^ 2 } , f ( x , y , z ) = \\dfrac { y ^ 2 ( a x + b y + c z ) ^ 2 } { x z ^ { 3 } } , \\\\ f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 4 } { x ^ 2 z ^ { 3 } } , f ( x , y , z ) = \\dfrac { y ^ 2 ( a x + b y + c z ) ^ 3 } { x z ^ { 4 } } , \\\\ f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 6 } { x ^ 3 z ^ { 4 } } , f ( x , y , z ) = \\dfrac { y ^ 3 ( a x + b y + c z ) ^ 4 } { x z ^ { 6 } } , \\end{gather*}"}
-{"id": "108.png", "formula": "\\begin{align*} A _ 1 ^ { - ( k + 1 ) } = \\Big ( \\frac { \\alpha } { 4 e ( 1 + \\alpha ) } \\Big ) ^ { \\alpha k } \\frac { 1 } { 2 ^ { k } } \\cdot \\frac { 1 } { A _ 1 ( 1 + \\eta ) ^ { k } } \\leq \\Big ( \\frac { \\alpha } { 4 e ( 1 + \\alpha ) } \\Big ) ^ { 2 \\phi A \\lambda + 8 } \\frac { 1 } { 2 ^ { k + 1 } } \\cdot \\frac { 2 } { A _ 1 ( 1 + \\eta ) ^ { k } } \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} \\bar A ( q t ) \\bar B ( t ) = \\bar B ( p t ) \\bar A ( t ) . \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } f _ N ^ { ( 1 ) } ( t ) = f ( t ) \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} \\Delta ^ \\perp = \\star \\vec { \\Delta } \\star \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} i \\dot \\Psi ( t , x ) + \\Delta \\Psi ( t , x ) - u ( t ) B ( x ) \\Psi ( t , x ) = 0 , \\Psi ( x , 0 ) = \\Psi _ 0 ( x ) , \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} \\hat v _ i ^ k = \\sum _ { j = 1 } ^ { N } [ W ( k ) ] _ { i j } v ^ k _ j \\quad \\hbox { f o r a l l $ k \\ge 1 $ , } \\mbox { w i t h } v _ i ^ 0 = x _ i ^ 0 , \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { C l - H f } } = \\min \\left \\{ \\frac { K } { \\min \\{ M , K \\} } + \\frac { K } { r } , ~ \\frac { M + K - 1 } { M } + \\frac { K } { M r } \\right \\} , \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} \\eta _ n ^ 0 : = \\eta ( B _ n ^ c ) + \\sum _ { k = 1 } ^ { Y ^ 0 } \\delta _ { U _ k } , \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} \\Delta _ \\gamma u = \\frac { 1 } { \\sqrt { | \\gamma | } } \\partial _ \\mu ( \\sqrt { | \\gamma | } \\gamma ^ { \\mu \\nu } \\partial _ \\nu u ) \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} \\beta ^ { \\kappa } _ { M , M - 1 } ( \\kappa \\ , a , \\kappa \\ , b ) \\overset { { \\rm i n \\ , l a w } } { = } \\kappa ^ { \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j / \\prod \\limits _ { i = 1 } ^ M a _ i } \\ ; \\beta _ { M , M - 1 } ( a , \\ , b ) . \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} ( \\sqrt { - \\Delta + m ^ 2 } \\ , v ) ( x ) & = ( U ^ * U \\sqrt { - \\Delta + m ^ 2 } \\ , U ^ * U v ) ( x ) \\\\ & = ( U ^ * \\sqrt { - d ^ 2 / d r ^ 2 + m ^ 2 } u ) ( x ) \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} d \\varphi _ p = T _ p ^ \\top ( e _ i ) ^ \\flat \\wedge ( \\imath _ { e _ i } \\ast \\varphi _ p ) \\mbox { a n d } d \\ast \\varphi _ p = - \\tau _ p \\wedge \\varphi _ p , \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} C _ { n - 1 } ( y _ { n - 2 } ) = \\sum _ { y _ { n - 1 } \\in \\{ 0 , 1 \\} } \\Big ( \\log \\big ( \\frac { q _ { n - 1 } ( y _ { n - 1 } | x _ { n - 1 } , y _ { n - 2 } ) } { \\nu ^ { \\pi ^ * } _ { n - 1 } ( y _ { n - 1 } | y _ { n - 2 } ) } \\big ) + C _ n ( y _ { n - 1 } ) \\Big ) q _ { n - 1 } ( y _ { n - 1 } | x _ { n - 1 } , y _ { n - 2 } ) , ~ \\forall { x _ { n - 1 } } . \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} S ( u , v ) = ( ( \\sqrt { 2 } + \\cos u ) \\cos v , ( \\sqrt { 2 } + \\cos u ) \\sin v , \\sin u ) \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} T _ { n + m } ( x ) = 2 T _ n ( x ) T _ m ( x ) - T _ { n - m } ( x ) , n \\geq m \\geq 0 , \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{gather*} R \\big ( \\zeta ^ 1 , \\dots , \\zeta ^ n \\big ) = \\int _ \\Gamma \\sum F _ \\mu \\big ( \\xi ^ 1 , \\dots , \\xi ^ n \\big ) { \\rm d } \\xi ^ \\mu , \\end{gather*}"}
-{"id": "5738.png", "formula": "\\begin{gather*} \\Delta _ { k + 1 } = ( - 1 ) ^ { \\tfrac { n _ { k + 1 } - n _ { k } } { 2 } } t _ { n _ { k + 1 } } t _ { n _ { k } } \\ , \\ \\ \\Delta _ { k + 1 } > 0 \\ . \\end{gather*}"}
-{"id": "7799.png", "formula": "\\begin{align*} \\alpha ^ 2 \\gamma \\ , t ^ 2 \\ , \\left ( E _ { \\alpha , \\alpha + \\beta + 2 } ^ { \\gamma + 1 } ( t ) \\right ) ' & = \\alpha ^ 2 \\gamma ( \\gamma + 1 ) \\ , t ^ 2 \\ , E _ { \\alpha , 2 \\alpha + \\beta + 2 } ^ { \\gamma + 2 } ( t ) \\\\ & = E _ { \\alpha , \\beta } ^ \\gamma ( t ) - ( \\alpha + 2 \\beta + 1 ) \\ , E _ { \\alpha , \\beta + 1 } ^ \\gamma ( t ) \\\\ & + ( \\alpha + \\beta + 1 ) ( \\beta + 1 ) \\ , E _ { \\alpha , \\beta + 2 } ^ \\gamma ( t ) \\ , , \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} \\Gamma _ + : = \\omega \\times \\{ 1 \\} , \\Gamma _ - : = \\omega \\times \\{ - 1 \\} , \\Gamma _ 0 : = \\gamma _ 0 \\times [ - 1 , 1 ] . \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} d \\theta _ { t } \\left ( x \\right ) = - \\sum _ { b \\ni x } V ^ { ^ { \\prime } } \\left ( \\eta _ { t } \\left ( b \\right ) \\right ) d t + \\sqrt { 2 } d B _ { x } \\left ( t \\right ) x \\in \\mathbb { Z } ^ { d } , \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} \\pi _ { \\sigma } ( x ) Q _ n ( x ) & = \\sum _ { v = n - \\rho } ^ { n + \\sigma } a _ { n , v } P _ v ( x ) . \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| Q _ n J _ n ( p _ \\xi ) - J _ n ( p _ \\xi ) \\| _ { \\varphi \\otimes \\psi _ n } = 0 \\ , , \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} ( f _ n \\circ \\rho ) ( e ) = 1 , \\ n \\in { \\mathbb N } \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\R \\R } \\omega ( \\pi ) q ^ { | \\pi | } = \\sum _ { \\pi \\in \\mathcal { U } } q ^ { | \\pi | } , \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} \\begin{cases} & \\\\ [ 1 m m ] c | \\xi | \\eta ^ { - \\theta } t \\le \\int _ 0 ^ t \\abs { \\dot W ^ \\eta _ s \\cdot \\xi } \\ ; d s \\int _ 0 ^ t \\abs { \\dot W ^ \\eta _ s } \\ ; d s \\le C \\eta ^ { - \\theta } t . & \\end{cases} \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} m _ { n } = - P _ { n } ( 0 ) / P _ { n - 1 } ( 0 ) . \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} a ^ { i j } = \\frac { 1 } { W } \\left ( \\delta ^ { i j } - \\frac { \\abs { \\nabla u } ^ 2 \\delta ^ { 1 i } \\delta ^ { 1 j } } { W ^ 2 } \\right ) , \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} \\lambda _ 2 ( \\cdot ) & = \\frac { \\alpha _ 2 \\alpha _ 3 - \\alpha _ 1 \\alpha _ 2 - \\alpha _ 1 \\alpha _ 3 - 2 \\alpha _ 3 - \\alpha _ 1 \\alpha _ 3 ^ 2 - \\alpha _ 2 \\alpha _ 3 ^ 2 + 2 \\alpha _ 3 ^ 2 - \\alpha _ 1 \\alpha _ 2 \\alpha _ 3 } { \\delta } . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} h = \\frac { e ^ { A C } } { C } \\big ( 1 - e ^ { - C d } \\big ) , \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} \\nu _ j ( n , s ) = - \\log _ { 2 } n + \\psi ( n ) ^ 2 / 2 + o ( \\psi ( n ) ^ 2 ) . \\end{align*}"}
-{"id": "9973.png", "formula": "\\begin{align*} E ( x , y ) = \\frac { 1 } { 4 \\pi ^ 2 } \\int _ { [ - \\pi , \\pi ] ^ 2 } \\frac { 2 } { i \\sin { u } - \\sin { v } } e ^ { i ( u x + v y ) } d u d v . \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\frac { \\Phi _ { d - p r i m } ^ { d - o d d } ( Q ) } { \\Phi ( Q ) } = 1 + O ( \\frac { 1 } { q } ) \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} S ( l , u , v ; N ) = S _ 1 ( l , u , v ; N ) - \\frac { 1 } { p ^ { 1 / 2 + u + v } } S _ 2 ( l , u , v ; N ) , \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} \\hbox { s h r i n k } _ 2 ( x , \\alpha ) \\ ; = \\ ; \\begin{cases} \\frac { x } { \\| x \\| _ 2 } \\max ( \\| x \\| _ 2 - \\alpha , 0 ) & \\mbox { i f } x \\neq 0 , \\\\ 0 & \\mbox { i f } x = 0 . \\end{cases} \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} \\mathbb { E } _ T & : = W ^ { 1 , p } ( ( 0 , T ) , L ^ p _ \\sigma ( \\mathbb { R } ^ n ) ) \\cap L ^ p ( ( 0 , T ) , W ^ { 4 , p } ( \\mathbb { R } ^ n ) ) , \\\\ \\mathbb { F } _ T ^ 1 & : = L ^ p ( ( 0 , T ) , L ^ p _ \\sigma ( \\mathbb { R } ^ n ) ) , \\mathbb { F } ^ 2 : = W ^ { 4 - 4 / p } _ p ( \\mathbb { R } ^ n ) , \\\\ \\mathbb { F } _ T & : = \\mathbb { F } ^ 1 _ T \\times \\mathbb { F } ^ 2 , \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\alpha _ { 2 } ( f ) = \\begin{cases} 1 & \\mbox { i f } f \\mbox { i s s q u a r e - f u l l } \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "10031.png", "formula": "\\begin{align*} i \\geq n _ { d } - q ^ { ( k - 1 ) { d } + r _ d } = \\ell q ^ { d } - q ^ { ( k - 1 ) { d } + r _ d } = \\ell - q ^ { r _ d } . \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} - \\Delta u + \\sum _ { i = 1 } ^ { m } \\frac { M _ { i } - 1 } { x _ { i } } \\frac { \\partial u } { \\partial x _ { i } } + u = u ^ { q - 1 } \\ \\ \\Omega , u > 0 \\ \\ \\Omega , \\partial _ \\nu u = 0 \\ \\ \\partial \\Omega . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} w ( t ) [ x ] = ( 2 \\pi ) ^ { - \\frac { d } { 2 } } \\int _ { \\mathbb { T } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - t h _ 0 ( \\xi ) ) } s _ a ( x , \\xi ) F u ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} F _ n ( T _ x ( i ) \\otimes T _ y ( \\alpha ) ) = F _ n ( T _ x ( i ) ) \\otimes T _ y ( \\alpha ) = T _ x ( i ) \\otimes F _ n ( T _ y ( \\alpha ) ) \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} \\liminf _ { \\rho ( x ) \\to \\infty } \\frac { | h ( x ) | } { \\rho ^ { - 2 } ( x ) \\bigl ( \\log \\rho ( x ) \\bigr ) ^ { - 1 } } = 0 , \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} R _ 1 & = - e ^ { - \\varphi } \\frac { \\sinh \\Theta } { \\cosh \\Theta } v F _ k \\varphi ^ k , \\\\ R _ 2 & = e ^ { - \\varphi } \\tilde { F } \\frac { \\sinh \\Theta } { \\Theta \\cosh \\Theta } v ^ { - 1 } | D \\varphi | ^ 4 \\sinh ^ { - 3 } u \\{ u ^ 3 \\cosh u - u ^ 2 \\sinh u \\} \\geq 0 . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} P ' _ { 1 } ( z ) - \\left [ \\frac { \\lambda } { \\xi } - \\left ( \\frac { \\mu } { \\xi } - 1 \\right ) \\frac { 1 } { z } \\right ] P _ { 1 } ( z ) = - \\dfrac { \\gamma } { \\xi ( 1 - z ) } P _ { 0 } ( z ) + \\left [ \\dfrac { ( \\mu - \\xi ) \\gamma } { \\xi \\lambda z } + \\dfrac { 1 } { ( 1 - z ) A } \\right ] p _ { 0 , 0 } . \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} \\varphi = - \\int _ u ^ { \\Theta ( 0 , T ^ * ) } \\vartheta ^ { - 1 } , \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} P _ i = \\frac { 1 } { Z } e ^ { - \\beta E _ i } \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} z ^ { \\mu } I _ { \\nu - \\mu } ^ { ( 3 ) } ( 2 z ; q ) = \\sum _ { m = 0 } ^ { \\infty } \\frac { \\left ( q ^ { \\mu } ; q \\right ) _ { m } } { \\left ( q ; q \\right ) _ { m } } I _ { \\nu + m } ^ { ( 3 ) } ( 2 z q ^ { m / 2 } ; q ) q ^ { \\binom { m + 1 } { 2 } } \\left ( - \\frac { q ^ { \\nu - \\mu } } { z } \\right ) ^ { m } , \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} \\widehat { f _ 2 } ( v ) = \\frac { 2 \\sqrt { 2 } ( - 3 v ^ { 2 } + 1 ) } { \\sqrt { \\pi } ( 1 + v ^ { 2 } ) ^ { 3 } } ( v \\in \\R ) . \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} u ( t , x ) & = \\frac { \\partial } { \\partial t } \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { d } } p ( t - s , x - y ) w ( s , y ) d y d s , ~ ( a . e . ) \\ , \\ , t \\leq T . \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} \\delta _ { l } = \\nu ^ { l } \\delta _ { 0 } , l = 0 , 1 , \\ldots ; \\nu \\in ( 0 , 1 ) , \\delta _ { 0 } > 0 . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} \\mathcal { V } _ s ^ \\eta = \\left \\{ ( Z _ s , Z _ s ^ \\prime ) \\in \\overline { \\mathcal { D } _ s } \\times \\overline { \\mathcal { D } _ s } \\left | \\begin{aligned} & \\inf _ { 1 \\leq i \\neq j \\leq s } | v _ i - v _ j ^ \\prime | > \\eta \\\\ & \\textnormal { a n d } \\\\ & \\inf _ { 1 \\leq i \\leq s \\ ; : \\ ; | v _ i - v _ i ^ \\prime | \\neq 0 } | v _ i - v _ i ^ \\prime | > \\eta \\end{aligned} \\right . \\right \\} \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} H _ 1 ( M , \\Z ) = 0 \\implies H ^ 1 ( M , \\Z ) = H ^ 3 ( M , \\Z ) = 0 . \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u + u \\partial _ x u + v \\partial _ y u - \\partial _ y ^ 2 u = 0 , & \\\\ \\partial _ x u + \\partial _ y v = 0 , & { \\mbox i n } \\quad \\Omega , \\\\ ( u , v ) | _ { y = 0 } = 0 , \\lim \\limits _ { y \\to + \\infty } u = U _ 0 . \\end{cases} \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} a + d x = a \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} \\mathcal { T } = \\begin{pmatrix} - 1 & \\zeta _ 1 & - \\sigma _ 1 ^ + & 0 \\\\ 0 & a _ { n n } ^ { ( 1 ) } & 1 & 0 \\\\ 1 & \\zeta _ 2 & 0 & - \\sigma _ 2 ^ + \\\\ 0 & a _ { n n } ^ { ( 2 ) } & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} \\frac { \\partial g _ { i j } } { \\partial t } = - 2 R _ { i j } \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { k ^ { 2 } } \\left ( - q ^ { - 2 n } \\right ) ^ { k } } { \\left ( q , - q ; q \\right ) _ { k } } = \\frac { \\left ( - 1 \\right ) ^ { n } \\left ( q ; q ^ { 2 } \\right ) _ { n } } { \\left ( - q ; q \\right ) _ { \\infty } q ^ { n ^ { 2 } } } . \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\triangleq \\lim \\limits _ { \\tau \\downarrow 1 } \\frac { \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) } { 2 \\pi } \\ , M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } , \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} \\int _ { B _ \\rho } \\frac { r ^ 3 u _ \\epsilon ^ 2 } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 2 } d x \\\\ = \\left \\{ \\begin{array} { l l } O ( \\epsilon ^ 4 ) & \\hbox { ~ ~ i f ~ ~ } n = 1 0 , \\\\ O ( \\epsilon ^ 5 | \\log \\epsilon | ) & \\hbox { ~ ~ i f ~ ~ } n = 1 1 , \\\\ O ( \\epsilon ^ 5 ) & \\hbox { ~ ~ i f ~ ~ } n \\geq 1 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial x _ 4 } - \\frac { \\partial f } { \\partial x _ 5 } = 0 . \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} B = \\sum _ { i = 1 } ^ { n ^ + } \\mu _ i v _ i v _ i ^ * \\quad \\mathrm { a n d } C = \\sum _ { i = n - n ^ - + 1 } ^ n ( - \\mu _ i ) v _ i v _ i ^ * . \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} \\widehat { R } _ { i , i + k } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n x _ i x _ { i + k } = \\frac { 1 } { n } \\sum _ { i , j = 1 , j - i = k } ^ { n + k } x _ i x _ j = \\frac { n + k } { n } \\widehat { r } ^ b _ k , \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} C ( \\Delta ( r + 2 ) ) & \\geq c _ r + d _ r \\geq 4 ( c _ { r - 2 } + d _ { r - 2 } ) + 1 = 4 \\cdot C ( \\Delta ( r ) ) + 1 \\\\ & \\geq 4 \\cdot \\frac { 2 ^ r - 2 ^ { r - 3 } - 1 } { 3 } + 1 = \\frac { 2 ^ { r + 2 } - 2 ^ { r - 1 } - 1 } { 3 } \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} v ( \\theta , t ) = H \\int _ 0 ^ t s ^ { 2 H - 1 } \\left ( e ^ { \\theta s } + e ^ { \\theta ( 2 t - s ) } \\right ) d s , \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} g ^ { i k } \\frac { \\partial } { \\partial x ^ k } | \\tau ( \\phi ) | ^ 2 = 0 , \\ ; \\ ; \\forall \\ ; i = 1 , 2 , \\cdots , m . \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} \\mathcal { L } g ( v ) : = g ( v ) - \\int _ { \\mathcal { V } } g ( w ) \\ , { \\rm d } \\mu ( w ) \\mbox { f o r a n y } g \\in L ^ 1 ( \\mathcal { V } ; { \\rm d } \\mu ) . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} \\dd Y _ t = \\lambda ( 1 - Y _ t ) \\ , \\dd t + \\dd m _ t , \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{gather*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\end{bmatrix} = \\begin{bmatrix} y ^ 2 - 2 y \\\\ x ^ 2 + 2 x \\end{bmatrix} , \\end{gather*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\inf \\ \\Big \\{ \\sum _ { i = 1 } ^ d g _ { k } ( x _ i , y _ i ) : ( x _ i ) _ { i = 1 } ^ d \\in R , ( y _ i ) _ { i = 1 } ^ d \\in \\{ 0 , 1 \\} ^ d \\Big \\} & \\geqslant \\inf \\ \\Big \\{ \\sum _ { i = 1 } ^ d g _ { k ; \\ast } ( x _ i ) : ( x _ i ) _ { i = 1 } ^ d \\in R \\Big \\} \\\\ & \\geqslant \\inf \\ \\{ g _ { k ; \\ast } ( x ) + g _ { k ; \\ast } ( \\lambda - x ) : \\lambda \\in \\mathcal { I } , x \\in [ 0 , \\lambda ] \\} \\\\ & \\geqslant E + \\epsilon / 2 . \\end{align*}"}
-{"id": "9991.png", "formula": "\\begin{align*} \\widehat { G } ( x ) = \\prod _ { i = 1 } ^ n \\frac { 2 } { 1 + 4 \\pi ^ 2 x _ i ^ 2 } , \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} \\langle \\mu ; n - 1 \\mid e ^ { H _ + [ s ( t ) ] } \\mid \\lambda ; n \\rangle = \\det _ { 1 \\leq p , q \\leq n } h _ { \\lambda _ q - \\mu _ p - q + p } [ x \\mid y ] = s _ { \\lambda / \\mu } ( x | t x ) , \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\partial _ t \\zeta \\cdot \\zeta - \\int _ { \\Omega } \\Delta \\zeta \\cdot \\zeta = - \\int _ { \\Omega } ( v \\nabla _ H \\zeta \\cdot \\zeta + w \\partial _ z \\zeta \\cdot \\zeta ) + \\int _ { \\Omega } g \\cdot \\zeta . \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} I _ 1 ( t ) = \\{ | \\xi | \\leq ( \\lambda t ) ^ { - 1 } \\} , I _ 2 ( t ) = \\{ ( \\lambda t ) ^ { - 1 } < | \\xi | \\leq 1 \\} , I _ 3 ( t ) = \\{ | \\xi | > 1 \\} . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{gather*} E _ { a b } ( z _ { 1 } ) = \\sum _ { l \\in \\mathbb { Z } } E _ { a b } z ^ { l } z _ { 1 } ^ { - l - 1 } . \\end{gather*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\| P _ { \\mathbf A ^ { \\rm t } } ^ { ( 2 ) } ( Z _ 1 , Z _ 2 ) \\| _ { \\rm o p } ^ 2 & = \\left \\| \\left ( \\begin{smallmatrix} ( Z _ 1 + Z _ 2 ) ( Z _ 1 + Z _ 2 ) ^ * + | c | ^ 2 Z _ 2 Z _ 2 ^ { * } ~ ~ & ~ ~ b ( Z _ 1 + Z _ 2 ) Z _ 2 ^ { * } + c \\bar { d } Z _ 2 Z _ 1 ^ { * } \\\\ b Z _ 2 ( Z _ 1 + Z _ 2 ) ^ { * } + \\bar { c } d Z _ 1 Z _ 2 ^ { * } & b ^ 2 Z _ 2 Z _ 2 ^ { * } + | d | ^ 2 Z _ 1 Z _ 1 ^ { * } \\end{smallmatrix} \\right ) \\right \\| _ { \\rm o p } . \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} H [ t ] : = \\sum _ { n = 1 } ^ \\infty t _ n J _ n e ^ { H [ t ] } : = \\sum _ { k = 1 } ^ \\infty \\frac { H [ t ] ^ k } { k ! } \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\langle \\phi , \\Delta \\phi \\rangle _ g : = \\int _ \\Sigma d v \\ , g _ { i j } \\phi ^ i \\partial ^ \\mu \\partial _ \\mu \\phi ^ j \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{gather*} \\Delta _ { \\mathcal { F } } ( h ) = \\mathcal { F } ^ { \\sharp - 1 } \\Delta ( h ) \\mathcal { F } \\end{gather*}"}
-{"id": "5476.png", "formula": "\\begin{align*} \\begin{pmatrix} a & p \\\\ q & a \\end{pmatrix} , a = s / \\sqrt { 2 } . \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\mathbf { y } _ { 1 } = \\mathbf { D } ^ { q - m } \\mathbf { x } _ { 1 } \\oplus \\mathbf { D } ^ { q - n } \\mathbf { x } _ { 2 } ; \\ \\mathbf { y } _ { 2 } = \\mathbf { D } ^ { q - m } \\mathbf { x } _ { 2 } , \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} h _ { N , s } ( z _ 1 , \\dots , z _ s ) = \\frac { \\det \\left [ z _ j ^ { k - 1 } ( z _ j - 1 ) ^ { s - k } h _ { N - k + 1 } ( z _ j ) \\right ] _ { j , k = 1 , \\dots , s } } { \\prod _ { 1 \\leq j < k \\leq s } ^ { } ( z _ j - z _ k ) } . \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} P _ e = \\max _ { \\mathbf { D } } \\max _ { k \\in [ 1 : K ] } \\mathbb { P } \\left ( \\widehat { F } _ { d _ k } \\neq { F } _ { d _ k } \\right ) , \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} E ( \\tau ) & : = h _ \\rho ( \\tau , q ( \\tau , t ; y ( 0 , t ; x , \\xi ) , \\xi ) , p ( \\tau , t ; y ( 0 , t ; x , \\xi ) , \\xi ) ) \\\\ & = h _ \\rho ( \\tau , q ( \\tau , 0 ; x , \\eta ( t , 0 ; x , \\xi ) ) , p ( \\tau , 0 ; x , \\eta ( t , 0 ; x , \\xi ) ) ) \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{gather*} \\big ( \\mathcal { F } ^ { - 1 } \\otimes _ { A _ \\star } 1 \\big ) ( \\Delta \\otimes _ { A _ \\star } \\mathrm { i d } ) \\big ( \\mathcal { F } ^ { - 1 } \\big ) = \\big ( 1 \\otimes _ { A _ \\star } \\mathcal { F } ^ { - 1 } \\big ) ( \\mathrm { i d } \\otimes _ { A _ \\star } \\Delta ) \\big ( \\mathcal { F } ^ { - 1 } \\big ) . \\end{gather*}"}
-{"id": "5222.png", "formula": "\\begin{align*} [ T , ( z - A ) ^ { - 1 } ] _ { \\circ } = ( z - A ) ^ { - 1 } [ T , A ] _ { \\circ } ( z - A ) ^ { - 1 } , \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{gather*} \\hat { x } ^ \\phi _ \\rho : = \\sum _ \\tau x _ \\tau \\phi ^ \\tau _ \\rho , \\hat { y } ^ \\phi _ \\rho : = \\sum _ \\tau x _ \\tau \\tilde \\phi ^ \\tau _ \\rho . \\end{gather*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\Delta ( A _ { i j } x ^ i x ^ j ) = & ( A _ { i j , k } x ^ i x ^ j + A _ { i j } ( x ^ i \\delta _ { j k } + x ^ j \\delta _ { i k } ) ) _ { , k } \\\\ = & ( \\Delta A _ { i j } ) x ^ i x ^ j + 2 A _ { i j , k } ( x ^ i \\delta _ { j k } + x ^ j \\delta _ { i k } ) + 2 \\sigma _ 1 ( A ) + O ( r ^ 3 ) \\\\ = & ( \\Delta A _ { i j } ) x ^ i x ^ j + 4 \\sigma _ 1 ( A ) _ { , i } x ^ i + 2 \\sigma _ 1 ( A ) + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{gather*} \\det \\big ( V ^ { ( k ) } _ { \\{ z _ { i } \\} } \\big ) ^ { 2 } = \\sum _ { \\sigma \\in \\mathfrak { S } _ { k } } \\det \\big ( z _ { \\sigma ( i ) } ^ { i + j - 2 } \\big ) _ { i , j = 1 } ^ { k } . \\end{gather*}"}
-{"id": "3038.png", "formula": "\\begin{align*} a _ 0 ( y , x ) = \\inf _ { \\psi \\in C X } \\hom ( \\psi ( x ) , \\psi ( y ) ) \\le \\inf _ { \\psi \\in C X , \\psi ( x ) = 1 } \\psi ( y ) . \\end{align*}"}
-{"id": "10011.png", "formula": "\\begin{align*} W ^ s _ { l o c } ( \\underline { a } ) & = \\{ \\underline { b } \\in \\Sigma _ { A ^ 1 } : d ( \\sigma ^ n \\underline { a } , \\sigma ^ n \\underline { b } ) \\leq 1 / 3 \\ \\forall \\ n \\geq 0 \\} \\\\ W ^ u _ { l o c } ( \\underline { a } ) & = \\{ \\underline { b } \\in \\Sigma _ { A ^ 1 } : d ( \\sigma ^ n \\underline { a } , \\sigma ^ n \\underline { b } ) \\leq 1 / 3 \\ \\forall \\ n \\leq 0 \\} . \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} Y _ G = V _ 0 \\oplus \\bigoplus _ { i = 1 } ^ { m / p - 1 } V _ { i p } . \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} \\Delta ( x \\rightharpoonup h y ) & = \\Delta \\big ( ( x \\rightharpoonup h ) y + h ( x \\rightharpoonup y ) \\big ) = x \\rightharpoonup ( h y ) _ 1 \\otimes ( h y ) _ 2 + ( h y ) _ 1 \\otimes x \\rightharpoonup ( h y ) _ 2 , \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} \\varphi _ a ( r ) = \\begin{cases} ( 2 r ^ { 2 - a } - a ) / ( 2 - a ) , & \\mbox { i f } ~ r \\ge \\max \\{ \\gamma , 2 \\} , \\\\ ( 1 + O ( a ) ) r ^ 2 , & \\mbox { i f } ~ 0 \\le r \\le \\max \\{ \\gamma , 2 \\} . \\end{cases} \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} [ y ^ { k + 1 } ] _ \\ell = \\frac { 1 } { N } \\ , \\mathbf { 1 } v ^ { k + 1 } ( \\ell ) = \\frac { 1 } { N } \\left ( \\mathbf { 1 } ^ T W ( k ) v ^ k ( \\ell ) + \\mathbf { 1 } ^ T \\zeta ^ { k + 1 } ( \\ell ) \\right ) , \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} D E & = \\nu \\iint A ( ( a ^ 2 - 1 ) \\partial _ { v v } f _ 0 ) A f _ 0 \\ , d V d t + \\nu \\iint A ( ( a ^ 2 - 1 ) \\partial ^ L _ { v v } \\ne { f } ) A \\ne { f } \\ , d V d t , \\\\ & = D E _ 0 + \\ne { D E } . \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{gather*} \\Psi ^ { [ k + 1 ] ( \\alpha ) } = \\Psi ^ { [ k ] ( \\alpha ) } U _ { k } ^ { ( \\alpha ) } , \\Psi ^ { [ k ] ( \\alpha + 1 ) } = \\Psi ^ { [ k ] ( \\alpha ) } V _ { k } ^ { ( \\alpha ) } , \\Psi ^ { [ k - 1 ] ( \\alpha + 1 ) } = \\Psi ^ { [ k ] ( \\alpha ) } W _ { k } ^ { ( \\alpha ) } . \\end{gather*}"}
-{"id": "161.png", "formula": "\\begin{align*} \\widehat { ( - \\Delta ) ^ \\alpha u } ( \\xi ) = | \\xi | ^ { 2 \\alpha } \\widehat { u } ( \\xi ) , \\ u \\in \\mathcal { S } ( \\mathbb { R } ^ N ) , \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} \\norm { J _ k ( x _ k , \\tilde { \\gamma } _ k , \\lambda _ k ) - x _ k } _ k \\leq \\frac { \\tilde { \\gamma } _ k } { \\gamma _ k } \\norm { J _ k ( x _ k , \\gamma _ k , \\lambda _ k ) - x _ k } _ k = \\frac { 1 } { \\theta } \\norm { x _ { k + 1 } - x _ k } _ k . \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} \\| x ^ { k + 1 } _ i - x ^ * _ i \\| ^ 2 & \\leq \\| x _ i ^ k - \\alpha _ { i } F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) - x ^ k _ i - x ^ * _ i + \\alpha _ { i } F _ i ( x ^ * _ i , \\bar x ^ * ) \\| ^ 2 \\cr & = \\| x _ i ^ k - x ^ * _ i \\| ^ 2 + \\alpha _ { i } ^ 2 \\| F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) - F _ i ( x ^ * _ i , \\bar x ^ * ) \\| ^ 2 \\cr & - 2 \\alpha _ { i } ( F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) - F _ i ( x ^ * _ i , \\bar x ^ * ) ) ^ T ( x ^ k _ i - x ^ * _ i ) . \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} \\sum ^ n _ { i = 1 } V a r [ Z _ { i n } ] \\leq \\delta _ n , \\max _ { 1 \\leq i \\leq n } | Z _ { i n } | \\leq \\frac { \\log 2 } { 2 } \\frac { \\sqrt { \\delta _ n } } { \\sqrt { \\log n } } . \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} ( - 1 ; q ) _ { \\infty } \\ , _ { 0 } \\phi _ { 1 } \\left ( - ; 0 ; q ^ { 2 } , q ^ { 5 } x ^ { - 2 } \\right ) \\ ! = \\theta _ { q } \\ ! \\left ( - q ^ { - 1 } x \\right ) { } _ { 1 } \\phi _ { 1 } \\left ( 0 ; - q ; q , x \\right ) + \\theta _ { q } \\ ! \\left ( q ^ { - 1 } x \\right ) { } _ { 1 } \\phi _ { 1 } \\left ( 0 ; - q ; q , - x \\right ) \\ ! , \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} d _ { g ( t ) } ( x _ 0 , y _ 0 ) \\leq 4 0 ^ { N } \\left \\{ d _ { g ( 0 ) } ( x _ 0 , y _ 0 ) + C \\right \\} \\leq C 4 0 ^ { N } d _ { g ( 0 ) } ( x _ 0 , y _ 0 ) = C 4 0 ^ { \\frac { t - t _ 0 } { \\epsilon } } d _ { g ( 0 ) } ( x _ 0 , y _ 0 ) . \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} & a _ 2 = \\sum _ { k = 2 } ^ { g + 1 } ( A ' _ k + A '' _ k ) \\cdot B _ { g , g + 1 - k } \\cdot \\tbinom { g } { k - 1 } \\tbinom { g + 1 } { k } \\\\ & a ' _ 2 = \\sum _ { k = 2 } ^ { g } ( A ' _ k + A '' _ k ) \\cdot B _ { g , g - k } \\cdot \\tbinom { g - 1 } { k - 1 } \\tbinom { g } { k } , \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} C ^ { 0 , \\alpha } _ \\ast ( \\overline { \\Omega } ) : = \\left \\{ u : \\overline { \\Omega } \\rightarrow \\R | \\ \\sup _ { x , y \\in \\overline { \\Omega } } \\frac { | u ( x ) - u ( y ) | } { d _ G ( x , y ) ^ \\alpha } < \\infty \\right \\} . \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} 0 = - c ^ { \\beta _ k } _ { \\beta _ { k + 1 } } ( s _ 1 ) - \\ldots - c ^ { \\beta _ { j - 1 } } _ { \\beta _ { j } } ( s _ 1 ) \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\arg \\min _ { \\vec { X } } & \\Biggl \\lbrace \\dfrac { 1 } { 2 } \\| \\vec { Y } - \\vec { X } \\| _ F ^ 2 + \\lambda _ 0 \\sum _ { i = 1 } ^ { m } \\phi ( \\sigma _ i ( \\vec { \\vec { Z } } ) ; a _ 0 ) \\\\ & \\qquad + \\lambda _ 1 \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } \\phi ( \\vec { X } _ { i , j } ; a _ 1 ) \\Biggr \\rbrace , \\\\ \\mbox { s . t . } & \\vec { X } = \\vec { Z } . \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} f _ n & = f + f \\circ T + \\dotsb + f \\circ T ^ { n - 1 } \\\\ g _ n & = g + g \\circ S + \\dotsb + f \\circ S ^ { n - 1 } . \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} \\eta ^ { \\varepsilon } \\left ( b _ { 1 } \\right ) + \\eta ^ { \\varepsilon } \\left ( b _ { 2 } \\right ) + \\eta ^ { \\varepsilon } \\left ( b _ { 3 } \\right ) + \\eta ^ { \\varepsilon } \\left ( b _ { 4 } \\right ) = 2 k _ { P } \\pi , \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\mathcal { E } _ n = \\Big \\{ E \\in \\S ^ n : E _ { i i } = 1 i = 1 , \\ldots , n \\Big \\} , \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} b _ { i , j + 1 } b _ { i + 1 , j } - b _ { i j } b _ { i + 1 , j + 1 } = - 1 . \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} \\int _ { \\mathbb S ^ { n - 1 } } N ( x , z ) \\ , d \\mathcal H ^ { n - 1 } ( z ) = 0 \\hbox { f o r $ x \\in \\Omega $ ; } \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} & - \\frac { 1 } { 4 \\pi c } \\frac { \\partial } { \\partial t } \\left [ e _ { p q r } x _ { q } \\left ( \\mathbf { D } \\times \\mathbf { B } \\right ) _ { r } \\right ] + \\frac { \\partial } { \\partial x _ { s } } \\left ( e _ { p q r } x _ { q } \\widetilde { T } _ { r s } \\right ) \\\\ & \\qquad + e _ { p q r } \\widetilde { T } _ { q r } + e _ { p q r } x _ { q } \\left ( X _ { r } + Y _ { r } \\right ) = 0 , \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { d - 2 } c _ i ( x ) H ( x ) ^ i + \\rho ( x , H ( x ) ) H ( x ) ^ d = 0 , \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\gamma ( u ) = \\left ( \\int \\sqrt { 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) } d u , \\lambda \\cos \\left ( \\frac { u } { c } \\right ) \\right ) . \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} _ { 1 } \\phi _ { 1 } \\left ( \\begin{array} { c c c } \\begin{array} { c } - a \\\\ b \\end{array} & \\vert & q , - b q ^ { \\alpha } \\end{array} \\right ) q ^ { \\alpha ^ { 2 } / 2 } = \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( - a b q ^ { - 1 / 2 } e ^ { i x } ; q \\right ) _ { \\infty } \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i \\alpha x \\right ) d x } { \\sqrt { \\pi \\log q ^ { - 2 } } \\left ( b ; q \\right ) _ { \\infty } \\left ( b e ^ { 2 i x } q ^ { - 1 } ; q ^ { 2 } \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} P _ s = \\left ( \\begin{array} { c | c } I _ s & \\\\ \\hline \\mathbf a & \\\\ 0 & \\\\ \\vdots & E _ { n - s + 1 } \\\\ 0 & \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "10074.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 4 , 4 u + 1 , 4 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 4 , 4 u + 3 , 4 v + 3 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "1924.png", "formula": "\\begin{align*} \\mathcal { E } ( f , g ) = \\int _ X f ' ( x ) g ' ( x ) d x , f , g \\in C _ 0 ^ \\infty ( X ) . \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} \\mathsf { h } _ { B _ { n + 1 } } ^ \\vee = \\mathsf { h } _ { A _ { 2 n } } ^ \\vee = 2 n + 1 , \\ \\mathsf { h } _ { C _ n } ^ \\vee = \\dfrac { 1 } { 2 } \\mathsf { h } _ { D _ { n + 1 } } ^ \\vee = \\mathsf { h } ^ \\vee _ { A _ n } = n + 1 . \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} \\overline G \\big ( a ( x ) \\big ) = O \\big ( \\overline H ( x ) \\big ) \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} | x | + | \\tau v ( \\xi ) | \\geq & \\left ( \\frac { 1 } { \\varepsilon _ 1 } | x | \\right ) ^ { \\varepsilon _ 1 } \\left ( \\frac { 1 } { 1 - \\varepsilon _ 1 } | \\tau v ( \\xi ) | \\right ) ^ { 1 - \\varepsilon _ 1 } = \\frac { | x | ^ { \\varepsilon _ 1 } | v ( \\xi ) | ^ { 1 - \\varepsilon _ 1 } } { \\varepsilon _ 1 ^ { \\varepsilon _ 1 } ( 1 - \\varepsilon _ 1 ) ^ { 1 - \\varepsilon _ 1 } } | \\tau | ^ { 1 - \\varepsilon _ 1 } , \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} \\rho \\left ( B _ { m } ^ { L } ( 2 ) \\right ) - \\rho \\left ( B _ { m } ^ { P } \\right ) & \\geqslant x ^ { T } \\left ( \\mathcal { A } ( B _ { m } ^ { L } ( 2 ) ) x \\right ) - x ^ { T } \\left ( \\mathcal { A } ( B _ { m } ^ { P } ) x \\right ) \\\\ & = x _ { v } x _ { a } x _ { b } ( x _ { a _ { 1 } } ) ^ { k - 3 } + x _ { v } x _ { a _ { 1 } } ( x _ { b _ { 1 } } ) ^ { k - 2 } - x _ { v } ( x _ { a _ { 1 } } ) ^ { k - 2 } x _ { a } - x _ { v } ( x _ { b _ { 1 } } ) ^ { k - 2 } x _ { b } \\\\ & = x _ { v } ( x _ { a _ { 1 } } ) ^ { k - 3 } \\left ( x _ a - x _ { a _ 1 } \\right ) ^ { 2 } \\\\ & > 0 . \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} \\left ( \\mathbf { I } + K ( \\lambda , \\varepsilon ) \\right ) r = \\{ \\mathbf { I } + K ( \\lambda , 0 ) + [ K ( \\lambda , \\varepsilon ) - K ( \\lambda , 0 ) ] \\} r = 0 . \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} \\left [ ( u - k _ { m - 1 } ) ^ + - ( u - k _ m ) ^ + \\right ] 1 _ { \\{ u > k _ m \\} } = ( k _ m - k _ { m - 1 } ) 1 _ { \\{ u > k _ m \\} } = 2 ^ { - m } k 1 _ { \\{ u > k _ m \\} } \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} c = & ( u '^ 2 + v '^ 2 ) \\cosh ^ 2 u , \\\\ c \\kappa = & ( u '' v ' - v '' u ' ) \\cosh ^ 2 u - \\sinh u \\cosh u ( v '^ 3 + u '^ 2 v ' ) . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} \\widehat { V } _ { N , \\iota } = ( B _ \\iota ( 1 , d , r , c ) ) \\ln N \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} F _ 0 ' = A e ^ { i k z } \\ , , \\b ( t ) \\equiv \\lambda \\ , , { \\rm a n d } \\nu _ 0 = 0 \\ , , \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} 2 = - K _ Y \\cdot L = - \\frac 1 2 \\pi ^ * K _ { Y ' } \\cdot \\pi ^ * L ' = - K _ { Y ' } \\cdot L ' . \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{align*} D ( P _ V | | P _ { V ' } ) = \\frac { 1 } { 2 } \\left \\| t U _ 0 V ^ { \\intercal } - t U _ 0 ( V ' ) ^ { \\intercal } \\right \\| _ F ^ 2 = \\frac { t ^ 2 } { 2 } \\left \\| V - V ' \\right \\| _ F ^ 2 . \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} a _ { j k } = a ( \\lambda _ j , \\nu _ k ) , b _ { j k } = b ( \\lambda _ j , \\nu _ k ) , c _ { j k } = c , \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{gather*} \\left \\{ \\ P _ n \\ | \\ \\deg P _ { n } = n \\ , \\ n \\geq 0 \\ \\right \\} = \\left \\{ P _ { n _ { k } } \\right \\} _ { 0 \\leq k < m + 1 } = \\left \\{ P _ { n _ { 0 } } \\equiv 1 \\ , , \\ , P _ { n _ { 1 } } \\ , , \\ , \\ldots \\ , , \\ , P _ { n _ { k } } \\ , , \\ , P _ { n _ { k + 1 } } \\ , , \\ , \\ldots \\ \\right \\} \\ , \\\\ [ 0 . 2 c m ] \\deg P _ { n _ { k } } = n _ { k } \\ , \\ 0 \\leq k < m + 1 \\ . \\end{gather*}"}
-{"id": "689.png", "formula": "\\begin{align*} j ^ { \\lambda } = c \\partial _ { \\sigma } p ^ { \\lambda \\sigma } , p ^ { \\lambda \\sigma } = - p ^ { \\sigma \\lambda } \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} \\begin{cases} u _ t + u u _ x + u _ { x x x } + a v _ { x x x } + a _ 1 v v _ x + a _ 2 ( u v ) _ x = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ c v _ t + r v _ x + v v _ x + a b u _ { x x x } + v _ { x x x } + a _ 2 b u u _ x + a _ 1 b ( u v ) _ x = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ u ( x , 0 ) = u ^ 0 ( x ) , v ( x , 0 ) = v ^ 0 ( x ) , & \\ , \\ , ( 0 , L ) , \\end{cases} \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} v ^ { k + 1 } ( \\ell ) - [ y ^ { k + 1 } ] _ \\ell \\mathbf { 1 } = D ( k ) ( v ^ k ( \\ell ) - [ y ^ { k } ] _ \\ell \\mathbf { 1 } ) + \\left ( \\mathbb { I } - \\frac { 1 } { N } \\mathbf { 1 } \\mathbf { 1 } ^ T \\right ) \\zeta ^ { k + 1 } ( \\ell ) . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{gather*} S _ { x } ^ { \\pm } ( z ) = \\left ( 1 - \\frac { S _ { x } ^ { + } } { z } \\right ) ^ { \\pm 1 } . \\end{gather*}"}
-{"id": "1280.png", "formula": "\\begin{align*} \\mathcal { A } : [ 1 , \\infty ) \\rightarrow [ 0 , \\infty ) \\mbox { i s a s m o o t h i n c r e a s i n g f u n c t i o n w i t h } \\mathcal { A } ( 1 ) = 0 . \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} A _ N : = & \\sqrt { \\frac { 2 } { \\tilde \\mu } } \\sum _ { i = 1 } ^ N \\sqrt { \\frac { \\delta _ i } { a _ i } } + \\\\ & + \\left ( \\| x ^ * - v _ 0 \\| ^ 2 + \\frac { M ^ 2 N ( r + \\frac { \\rho } { 2 } ( N + 3 ) ) } { \\tilde \\mu } + \\frac { 2 } { \\tilde \\mu } \\sum ^ N _ { i = 1 } \\frac { \\delta _ i } { a _ i } + \\frac { 4 } { \\tilde \\mu } \\sum _ { i = 1 } ^ N \\frac { \\varepsilon _ i } { a _ i ^ 2 } + \\frac { 2 } { \\mu } \\left ( \\sum _ { i = 1 } ^ N \\sqrt { \\frac { \\delta _ i } { a _ i } } \\right ) ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} \\frac { d } { d r } f ( r x ) = \\frac { 1 } { r } \\left ( e ( r x ) \\cdot A \\nabla \\right ) f ( r x ) = \\mathcal { R } f ( r x ) . \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\ , _ { 2 } \\phi _ { 1 } \\left ( a , b ; 0 ; q , z \\right ) = \\frac { ( b z ; q ) _ { \\infty } } { ( z ; q ) _ { \\infty } } \\ , _ { 1 } \\phi _ { 1 } \\left ( b ; b z ; q , a z \\right ) \\ ! . \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} F _ { \\sigma \\tau } = \\frac { \\partial A _ { \\tau } } { \\partial x ^ { \\sigma } } - \\frac { \\partial A _ { \\sigma } } { \\partial x ^ { \\tau } } , \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\left ( 1 + \\tau \\right ) ^ { - d } e ^ { - \\left ( \\beta ( \\tau ) - \\beta ( t ) \\right ) E _ s ( Z _ s ) } e ^ { - \\left ( \\mu ( \\tau ) - \\mu ( t ) \\right ) s } d \\tau \\leq \\frac { \\sum _ { k = 0 } ^ \\infty r _ k ^ { - 1 } \\left ( 1 + k \\right ) ^ { - d } } { s + \\frac { \\beta _ 0 } { 2 } E _ s ( Z _ s ) } \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} { } ^ { c _ i } ( { } ^ { c _ 1 ^ { \\delta _ 1 } \\ldots c _ { i - 1 } ^ { \\delta _ { i - 1 } } c _ { i + 1 } ^ { \\delta _ { i + 1 } } \\ldots c _ r ^ { \\delta _ r } } H _ { [ i ] } ) = { } ^ { c _ 1 ^ { \\delta _ 1 } \\ldots c _ { i - 1 } ^ { \\delta _ { i - 1 } } c _ i c _ { i + 1 } ^ { \\delta _ { i + 1 } } \\ldots c _ r ^ { \\delta _ r } } H _ { [ i ] } = { } ^ { c _ 1 ^ { \\delta _ 1 } \\ldots c _ { i - 1 } ^ { \\delta _ { i - 1 } } c _ { i + 1 } ^ { \\delta _ { i + 1 } } \\ldots c _ r ^ { \\delta _ r } c _ i } H _ { [ i ] } . \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} u _ x + v _ y = 0 , \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} 2 \\frac { \\partial } { \\partial x _ { q } } \\left ( A _ { p } B _ { q } \\right ) = \\frac { \\partial } { \\partial x _ { q } } \\left ( A _ { p } B _ { q } + A _ { q } B _ { p } \\right ) + \\left [ \\operatorname { c u r l } \\left ( \\mathbf { A \\times B } \\right ) \\right ] _ { p } , \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} H ^ 1 ( G , H ^ 2 ( M \\times \\P ^ n ) ) = H ^ 1 ( G , H ^ 2 ( M ) \\oplus H ^ 2 ( \\P ^ n ) ) = H ^ 1 ( G , H ^ 2 ( M ) ) . \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} u = { \\rm p r o x } _ g ( x ) \\ \\ \\Longleftrightarrow \\ \\ x - u \\in \\partial g ( u ) . \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} \\left ( T _ s ^ 0 ( t ) f _ \\infty ^ { ( s ) } \\right ) ( X _ s , V _ s ) = f _ \\infty ^ { ( s ) } ( X _ s - V _ s t , V _ s ) \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} \\left ( \\log \\frac { 1 } { 1 - R ^ { 2 } } \\right ) ^ { - 1 } , \\ ; 1 - R ^ { 2 } , \\ ; \\sum _ { j = 1 } ^ { \\infty } \\left ( 1 - \\left \\vert z _ { j } \\right \\vert \\right ) ^ { \\sigma } < \\varepsilon . \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} f ( n ) = \\sum _ { t = 3 } ^ { 8 } f _ t ( n ) , . \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} g _ { i j } & = \\frac { r ^ 2 } { 1 - r ^ 2 } \\{ u _ i u _ j + \\sigma _ { i j } \\} , \\\\ \\tilde { g } _ { i j } & = r ^ 2 \\{ \\dot { \\varphi } ^ 2 u _ i u _ j + \\sigma _ { i j } \\} . \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} c _ 1 ( \\langle \\mathcal { K } _ 1 , \\phi ^ * \\mathcal { L } _ 0 , \\dots , \\phi ^ * \\mathcal { L } _ { n - 1 } \\rangle ( \\mathfrak { X } / S ) ) = \\deg ( \\mathcal { K } _ 1 ) \\cdot c _ 1 ( \\langle \\mathcal { L } _ 0 , \\dots , \\mathcal { L } _ { n - 1 } \\rangle ( \\mathcal { Y } / S ) ) . \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} \\left ( - q ^ { \\ell + 1 } ; q \\right ) _ { \\infty } = ( - 1 ) ^ { \\ell } q ^ { - \\binom { \\ell } { 2 } } \\sum _ { n = 0 } ^ { \\infty } \\left [ \\frac { a _ { 2 n + \\ell } ( q ) q ^ { 2 n } } { ( q , q ^ { 4 } ; q ^ { 5 } ) _ { \\infty } ( q ^ { 2 } ; q ^ { 2 } ) _ { n } } - \\frac { b _ { 2 n + \\ell } ( q ) q ^ { 2 n } } { ( q ^ { 2 } , q ^ { 3 } ; q ^ { 5 } ) _ { \\infty } ( q ^ { 2 } ; q ^ { 2 } ) _ { n } } \\right ] . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} d q _ { l - l _ 0 } + t ( - r _ { l - l _ 0 } - l _ 0 , 1 ) \\overset { \\textup { ( \\ref { a l i g n : d 2 } ) } } { \\le } & q _ { l - l _ 0 } t ( - d , 0 ) + t ( - r _ { l - l _ 0 } - l _ 0 , l ) \\\\ \\overset { \\textup { ( \\ref { a l i g n : d 3 } ) } } { \\le } & t ( - d q _ { l - l _ 0 } - r _ { l - l _ 0 } - l _ 0 , 1 ) = t ( - l , 1 ) . \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} \\tilde { P } _ { n } \\left ( x \\right ) = P _ { n } \\left ( x \\right ) + \\gamma _ { 1 } ^ { 0 } \\left ( 1 - \\lambda \\right ) P _ { n - ( d + 1 ) } ^ { \\left ( d + 1 \\right ) } \\left ( x \\right ) , d \\geq 1 , n \\geq 0 . \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} N _ i & \\ge \\left ( \\frac 5 9 + \\lambda \\right ) ( n / t ) \\binom { n - 1 } 2 - \\frac { \\lambda } { 1 2 } \\binom { t - 1 } 2 ( n / t ) ^ 3 - 3 \\binom { n / t } 3 - 3 \\binom { n / t } 2 ( t - 1 ) ( n / t ) , \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} \\dim \\delta H _ 2 ^ g = 4 ( k + 1 ) + k ^ 2 - ( k + 2 ) ^ 2 = 0 . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\rho ( T ) = \\lim _ { n \\longrightarrow \\infty } \\Vert T ^ n \\Vert ^ { 1 / n } . \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} \\mathcal { V } = \\left ( H _ 0 ^ 2 ( \\Omega ) \\right ) ^ d \\times H _ 0 ^ 1 ( \\Omega ) \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} V _ { 1 / 2 } ( x ) : = - \\frac { 1 } { \\pi } \\left ( \\frac { 1 } { \\sqrt { 1 + x ^ 2 } } - \\frac { 2 | x | \\ , \\mathrm { a r c s i n h } | x | } { 1 + x ^ 2 } \\right ) \\mbox { a n d } u _ { 1 / 2 } ( x ) : = \\frac { 1 } { \\sqrt { 1 + x ^ 2 } } , \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\beta ( x ) = s ^ { - 1 } ( d ( x ) \\Delta - g ( x ) ) , \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} \\kappa _ E ( \\mathcal { Z } _ { E _ \\curlyvee } ( \\mu \\setminus F ) ) = \\mathcal { Z } _ { E } ( \\mu ) \\setminus \\left ( \\bigcup _ { e \\in F } \\mathcal { Z } _ { E } ( \\mu e ) \\right ) = \\mathcal Z _ E ( \\mu \\setminus F ) , \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} \\tau _ 1 = | \\cos ( \\beta ) | = | \\cos ( \\mu ) | = \\tau _ 2 , \\sigma _ 1 = \\sigma _ 2 = | \\sin ( \\beta ) | = | \\sin ( \\mu ) | . \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} Q ( x , d x ) = \\frac { 1 } { 3 } [ c _ { 2 2 1 1 } + c _ { 1 1 2 2 } - c _ { 2 1 1 2 } - c _ { 1 2 2 1 } ] ( x ^ 1 d x ^ 2 - x ^ 2 d x ^ 1 ) ^ 2 . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} | C | = \\binom { 2 b - n + 1 } { k } . \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ n \\tilde W _ i ( z ) ( \\alpha _ 0 \\partial _ z ) ^ { n - i } = : ( \\alpha _ 0 \\partial _ z + b _ 1 ( z ) ) ( \\alpha _ 0 \\partial _ z + b _ 2 ( z ) ) \\dots ( \\alpha _ 0 \\partial _ z + b _ n ( z ) ) : , \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\partial _ \\xi ^ \\beta R ( t , x , \\xi ) = x \\cdot \\int _ 0 ^ 1 \\nabla _ x \\partial _ \\xi ^ \\beta R ( t , \\theta x , \\xi ) d \\theta \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} \\mathbf { x } _ k ^ { [ \\sf b s ] } [ t ] = \\phi _ t ( W _ { k 1 } ^ { [ { \\sf d } ] } , \\cdots , W _ { k N } ^ { [ { \\sf d } ] } , \\mathbf { y } ^ { [ { \\sf b s } ] } _ k [ 1 ] , \\cdots \\mathbf { y } _ k ^ { [ { \\sf b s } ] } [ t - 1 ] ) . \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} P Q & = \\left ( F + i G \\right ) \\left ( R + i S \\right ) = \\left ( F R - G S \\right ) + i \\left ( F S + G R \\right ) , \\\\ P ^ { \\ast } Q & = \\left ( F - i G \\right ) \\left ( R + i S \\right ) = \\left ( F R + G S \\right ) + i \\left ( F S - G R \\right ) . \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} \\varphi _ a ( x , \\xi ) = ( \\varphi ( x , \\xi ) - x \\cdot \\xi ) \\chi \\left ( \\frac { v ( \\xi ) } { a } \\right ) + x \\cdot \\xi . \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} f _ { a _ i } ( a ) : = \\lim _ { t \\to \\infty } \\frac { | \\zeta ^ { t } ( a ) | _ { a _ i } } { | \\zeta ^ { t } ( a ) | } \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} x \\otimes v = \\bigvee _ { u \\in S } ( x \\otimes u ) , \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} h _ { \\mu } ( \\alpha \\vee \\beta , G ) = h _ { ( \\sigma _ { i } ) _ { i } , \\mu } ( \\alpha \\vee \\beta : \\gamma ) & \\leq h _ { ( \\sigma _ { i } ) _ { i } , \\mu } ( \\beta : \\gamma ) + h _ { ( \\sigma _ { i } ) _ { i } , \\mu } ( \\alpha | \\beta : \\gamma ) \\\\ & = h _ { \\mu } ( \\beta , G ) + h _ { ( \\sigma _ { i } ) _ { i } , \\mu } ( \\alpha | \\beta : \\gamma ) . \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} \\partial _ t u = J * u - u + u ^ { 1 + p } ( 1 - u ) ( 0 , \\infty ) \\times \\R ^ { N } , \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} ( u ^ \\alpha _ { i j } \\otimes \\omega ) ( U _ { k l } ^ \\beta \\otimes \\mu ) = \\sum _ { a , b } u ^ \\alpha _ { i j } u ^ \\beta _ { a b } \\otimes \\omega ( u ^ \\beta _ { k a } \\cdot S ( u ^ \\beta _ { b l } ) ) \\mu . \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\frac { L ( u ^ { 2 } , \\chi ^ { 2 } ) L ( u ^ { 3 } , \\chi ^ { 3 } ) } { L ( u ^ { 6 } , \\chi ^ { 6 } ) } = \\sum _ { j = 0 } ^ { \\deg Q - 1 } \\sum _ { l = 0 } ^ { \\deg Q - 1 } \\sum _ { k = 0 } ^ { \\infty } u ^ { 2 j + 3 l + 6 k } q ^ { \\frac { j + k + l } { 2 } } \\Lambda _ { j } ( \\chi ^ { 2 } ) \\Lambda _ { l } ( \\chi ^ { 3 } ) S y m ^ { k } ( \\chi ^ { 6 } ) \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} \\left ( \\frac { | x | + \\delta s } { \\rho } \\right ) ^ { \\frac 1 \\delta } & = \\left ( \\frac { | x | } { \\rho } \\right ) ^ { \\frac 1 \\delta } \\ , \\left ( 1 + \\frac { \\delta s } { | x | } \\right ) ^ { \\frac 1 \\delta } \\ , \\geq \\ , \\frac { | x | } { \\rho } \\ , \\left ( 1 + \\frac { s } { | x | } \\right ) = \\frac { | x | + s } { \\rho } \\ , , \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} \\begin{cases} | \\partial _ \\xi ^ \\beta s _ j ^ \\ell ( x , \\xi ) | \\leq C _ \\beta \\langle x \\rangle ^ { \\ell - 1 - \\varepsilon } , \\\\ | \\partial _ \\xi ^ \\beta p _ j ^ \\ell ( y , \\xi ; t ) | \\leq C _ \\beta \\langle | y | + t | v ( \\xi ) | \\rangle ^ { - \\ell } . \\end{cases} \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} W _ { 2 n } ( p , \\psi ) - t W _ { 2 n } ( q , \\psi ) = ( - 1 ) ^ { n + 1 } q ^ { - n } \\left ( \\psi _ { 2 n + 1 } + t \\psi _ { 2 n } \\right ) \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} \\varepsilon D ^ \\prime \\cdot H _ F ( C ^ \\prime ) + \\lambda \\nabla { F } ( B ) + \\mu \\nabla { F } ( C ^ \\prime ) = 0 . \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} D \\Phi _ 0 ( x _ 0 , t _ 0 ) = D \\eta ( x _ 0 ) , D ^ 2 \\Phi _ 0 ( x _ 0 , t _ 0 ) \\le D ^ 2 \\eta ( x _ 0 ) , \\psi ' ( t _ 0 ) = - C _ 0 , \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\langle x _ r ' , x _ s ' \\rangle = \\sum _ { t \\in T } \\lambda ^ { r } _ t \\langle y _ t ' , x _ s ' \\rangle = \\sum _ { t \\in T } \\lambda ^ { r } _ t C _ { s , t } = \\sum _ { t \\in T } \\lambda ^ { r } _ t \\langle y _ t '' , x _ s '' \\rangle = \\langle x _ r '' , x _ s '' \\rangle \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} Y = \\tau ^ { 1 / \\tau } \\ , \\beta ^ { - 1 } _ { 1 , 0 } ( a = \\tau , b _ 0 = \\tau ) . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\theta _ { u ^ * ( h ) } ( x , y ) = \\theta ' _ h ( v ( x ) , v ( y ) ) \\end{align*}"}
-{"id": "6079.png", "formula": "\\begin{align*} \\big \\langle \\omega _ i , \\omega _ { 1 , \\infty } \\big \\rangle _ { Z _ { 1 , R _ i } } = \\big \\langle \\omega _ i , \\omega _ { 1 , \\infty } \\big \\rangle _ { Z _ { 1 , 0 } } + \\big \\langle \\omega _ i ^ \\mathrm { n z } , \\omega _ { 1 , \\infty } ^ \\mathrm { n z } \\big \\rangle _ { Y _ { [ 0 , R _ i ] } } . \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c c } \\vec { h } _ 1 \\\\ \\vec { g } _ 1 \\end{array} \\right ) = \\Lambda _ 1 \\left ( \\left ( \\begin{array} { c c } u ^ 0 \\\\ v ^ 0 \\end{array} \\right ) , \\left ( \\begin{array} { c c } u ^ 1 \\\\ v ^ 1 \\end{array} \\right ) + \\left ( \\begin{array} { c c } v \\\\ \\nu ( T , u , v ) \\end{array} \\right ) \\right ) , \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} R = & a ^ 6 \\left ( D ^ 6 + \\tilde { R } ^ { ( 5 ) } D ^ 5 + \\tilde { R } ^ { ( 4 ) } D ^ 4 + \\tilde { R } ^ { ( 3 ) } D ^ 3 + \\tilde { R } ^ { ( 2 ) } D ^ 2 + \\tilde { R } ^ { ( 1 ) } D + \\tilde { R } ^ { ( 0 ) } \\right ) \\\\ & + \\sigma ^ { ( 1 ) } D ^ { - 1 } \\gamma ^ { ( 1 ) } + \\sigma ^ { ( 2 ) } D ^ { - 1 } \\gamma ^ { ( 2 ) } . \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{gather*} \\frac { Q _ { r } ( x ) } { P _ { r } ( x ) } = \\sum _ { m \\geq 0 } \\frac { a _ { m } } { x ^ { m + 1 } } \\ , \\ \\ \\ \\ | x | > \\rho _ { r } \\ . \\end{gather*}"}
-{"id": "8638.png", "formula": "\\begin{gather*} H \\times _ A H = \\left \\lbrace \\sum _ i b _ i \\otimes b ' _ i \\in H \\otimes _ A H \\ , | \\ , \\sum _ i b _ i \\otimes b ' _ i \\alpha ( a ) = \\sum _ i b _ i \\beta ( a ) \\otimes b ' _ i , \\ \\forall \\ , a \\in A \\right \\rbrace \\end{gather*}"}
-{"id": "8265.png", "formula": "\\begin{align*} ( \\overline \\nabla _ X J _ 1 ) N = - \\frac { 1 } { 2 ( 2 n - 1 ) } \\left [ \\overline \\theta _ 1 ( N ) X + \\overline \\theta _ 1 ( J _ 1 N ) J _ 1 X \\right ] , \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} \\psi ( n ) = - \\frac { s } { 2 } = \\frac { 1 } { 2 } \\log _ { p / q } \\log n + O ( \\log \\log \\log n ) , \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} B _ K ^ \\circ = \\{ x \\in B _ K \\mid v ( x ) \\le 1 \\ \\} . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\partial _ t v \\cdot v - \\int _ { \\Omega } \\Delta v \\cdot v = - \\int _ { \\Omega } ( v \\nabla _ H v \\cdot v + w \\partial _ z v \\cdot v ) + \\int _ { \\Omega } P _ 2 \\left ( f + \\Pi ( \\zeta ) \\right ) \\cdot v . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } \\left \\langle J _ { a b } , P _ { c } \\right \\rangle = \\alpha _ { 0 } \\ , \\varepsilon _ { a b c } , \\\\ \\left \\langle J _ { a b } , Z _ { c } \\right \\rangle = \\alpha _ { 1 } \\ , \\varepsilon _ { a b c } , \\\\ \\left \\langle Z _ { a b } , P _ { c } \\right \\rangle = \\alpha _ { 1 } \\ , \\varepsilon _ { a b c } , \\\\ \\left \\langle Z _ { a b } , Z _ { c } \\right \\rangle = \\alpha _ { 2 } \\ , \\varepsilon _ { a b c } . \\end{array} \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} \\omega ( x ) : = \\left ( x + \\frac \\alpha \\sigma \\right ) ^ \\alpha , \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} C r _ \\varphi : = \\delta _ { g _ \\varphi } ( \\varphi ) \\mbox { a n d } \\chi _ \\varphi : = - \\delta _ g ( \\ast \\varphi ) , \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} B e _ { n } = e _ { n - 1 } + \\alpha q ^ { - n } e _ { n } + e _ { n + 1 } , n \\in \\Z , \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} \\| \\xi \\| _ \\infty : = \\sup _ { t \\in \\R _ + } \\| \\xi ( t ) \\| < \\infty . \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } I _ n ( f _ n ( \\cdot , t , x ) ) = \\eta + \\sum _ { n \\geq 0 } I _ { n + 1 } ( g _ n ^ { ( t , x ) } ) . \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} \\| b \\| _ { \\mathcal B / \\mathcal B _ 0 , m } = \\limsup _ { | z | \\to 1 } ( 1 - | z | ^ 2 ) ^ m | b ^ { ( m ) } ( z ) | , m \\ge 2 , \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} \\left \\langle J _ { a b } , J _ { c d } , \\bar { Z } _ { e } \\right \\rangle & = \\frac { 1 } { \\sqrt { 2 } } \\left \\langle J _ { a b } , J _ { c d } , P _ { e } \\right \\rangle - \\frac { 1 } { \\sqrt { 2 } } \\left \\langle J _ { a b } , J _ { c d } , Z _ { e } \\right \\rangle , \\\\ & = \\left ( \\alpha _ { 0 } - \\alpha _ { 1 } \\right ) \\varepsilon _ { a b c d e } , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} C _ n ( 0 ) & = \\sum _ { y _ n \\in \\{ 0 , 1 \\} } \\log \\Big ( \\frac { q _ n ( y _ n | 1 , 0 ) } { \\nu ^ { \\pi ^ * } _ n ( y _ n | 0 ) } \\Big ) q _ n ( y _ n | 1 , 0 ) = \\log \\Big ( \\frac { q _ n ( 0 | 1 , 0 ) } { \\nu ^ { \\pi ^ * } _ n ( 0 | 0 ) } \\Big ) q _ n ( 0 | 1 , 0 ) + \\log \\Big ( \\frac { q _ n ( 1 | 1 , 0 ) } { \\nu ^ { \\pi ^ * } _ n ( 1 | 0 ) } \\Big ) q _ n ( 1 | 1 , 0 ) \\\\ & = \\gamma _ n \\log \\big ( \\frac { 1 - c _ 0 ( n ) } { c _ 0 ( n ) } \\big ) + \\log \\big ( \\frac { 1 } { 1 - c _ 0 ( n ) } \\big ) - H ( \\gamma _ n ) . \\end{align*}"}
-{"id": "10045.png", "formula": "\\begin{align*} h ( z ) = \\tfrac { \\displaystyle 1 + w ( z ) } { \\displaystyle 1 - w ( z ) } = 1 + d _ 1 z + d _ 2 z ^ 2 + \\cdots , \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} C _ m : \\underline y _ m ^ { F ^ \\ell } - \\underline y _ m = \\sum _ { i = 0 } ^ d ( \\bar a _ i x ^ i , 0 , 0 , \\dots ) _ m , \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} \\tilde { t } _ { k , \\varphi } ^ 1 ( \\omega , \\lambda ) = ( - 1 ) ^ k , \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} A _ { q } \\left ( a c \\right ) = \\left ( c q ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } S _ { n } \\left ( a q ^ { - n } ; q \\right ) c ^ { n } } { \\left ( c q ; q \\right ) _ { n } } , \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } | | \\mathbf { \\tilde { z } _ i } | | ^ 4 & \\leq | | \\mathbf { L } | | ^ 4 \\sum _ { i = 1 } ^ { n } | | \\mathbf { z _ i } | | ^ 4 \\\\ & = O ( n ^ { - 1 } ) \\ ; \\ ; \\ ; w . p . 1 \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} - \\vec \\Delta \\eta = \\sum _ { j = 1 } ^ { + \\infty } \\langle \\eta , \\partial \\phi _ j \\rangle _ \\mathcal { H } \\partial \\phi _ j . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} \\| \\boldsymbol { \\theta } _ 1 - \\boldsymbol { \\theta } _ 2 \\| _ 2 \\geq \\frac { 1 } { 4 } \\sqrt { s } e ^ { - m } = : \\alpha , \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} F '' ( z , v ) = \\big ( A ( z , v ) \\big ) ^ { 2 } , A ' ( z , v ) = \\frac { 1 } { 1 - z } \\cdot F ' ( z , v ) . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} \\lim _ { z \\to 0 } \\left ( \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\zeta ^ d } { 1 - \\zeta ^ \\ell } \\right ) + \\frac { ( 2 a ) ! } { \\ell \\left ( 2 \\pi i z \\right ) ^ { 2 a + 1 } } \\right ) = - \\ell ^ { 2 a } \\frac { B _ { 2 a + 1 } \\left ( \\frac { d } { \\ell } \\right ) } { 2 a + 1 } . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\omega \\ , \\mathrm { t r i v i a l } \\\\ L ( \\omega , \\chi ^ * ) = 0 } } \\frac { 1 } { | \\sigma + i t - \\omega | ^ 2 } & \\leq a ( \\chi ) \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { ( \\sigma + 2 k ) ^ 2 + t ^ 2 } + b ( \\chi ) \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { ( \\sigma + 2 k + 1 ) ^ 2 + t ^ 2 } \\\\ & \\leq n _ K \\sum _ { k = 0 } ^ { \\infty } \\frac { 1 } { ( \\sigma + 2 k ) ^ 2 } \\leq \\Big ( \\frac { 1 } { 2 \\sigma } + \\frac { 1 } { \\sigma ^ 2 } \\Big ) n _ K \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} \\Psi _ { p , q } ^ { \\Pi } ( X ) = \\prod \\limits _ { x \\in X } \\left ( \\frac { x - p } { x - q } \\right ) ^ 2 \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} 2 \\Big ( \\sum _ { \\substack { \\rho \\\\ \\zeta _ K ( \\rho ) = 0 } } \\frac { 1 } { | \\sigma - \\rho | ^ 2 } + \\sum _ { \\substack { \\rho \\\\ L ( \\rho , \\chi ) = 0 } } \\frac { 1 } { | \\sigma + i t - \\rho | ^ 2 } \\Big ) . \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} u ( x ) & \\leq \\lim _ { j \\to \\infty } \\left ( \\inf _ { y \\in B _ { r _ j } ( x ) } \\min \\big \\{ j , u ( y ) \\big \\} - \\frac 1 { j } \\right ) \\\\ & \\le \\lim _ { j \\to \\infty } \\left ( \\inf _ { y \\in B _ { r _ j } ( x ) } \\Big \\{ \\min \\big \\{ j , u ( y ) \\big \\} + j ^ 2 | x - y | \\Big \\} - \\frac 1 { j } \\right ) = \\lim _ { j \\to \\infty } \\widetilde \\psi _ j ( x ) . \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } f ( t , z ) = e ^ { i \\nu _ 0 t } \\ , F _ 0 ' ( z ) \\ , , \\\\ g ( t , z ) = e ^ { i ( \\xi _ 0 - \\nu _ 0 ) t } \\ , G _ 0 ' ( z ) \\ , , \\end{array} \\right . \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq 1 } \\lambda _ f ( n ) n ^ { ( 2 k - 1 ) / 2 } e ( n z ) \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} M _ 1 ( l , 0 , 0 ) = \\frac { 1 } { \\sqrt { l } } \\frac { \\phi ( N ) } { N } + O _ { k , p } \\left ( \\frac { \\sqrt { l } \\log { ^ 2 } ( 2 l ) } { N ^ { 1 / 4 - \\epsilon } } \\right ) ; \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} z = 4 ^ k \\xi ( T x ) + ( 4 ^ k - 1 ) / 3 , \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} \\alpha _ 0 = \\min \\{ \\alpha _ 0 ^ 1 , \\alpha _ 0 ^ 2 \\} . \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} E _ { ( d _ 1 d _ 2 ) } = E ( \\mathbf { w } ) \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} V = F ^ 0 V \\supset F ^ 1 V \\supset \\dots . \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\textbf { H } = \\sqrt { \\dfrac { N _ { B S } N _ { M S } } { \\rho L } } \\sum _ { l = 1 } ^ { L } \\eta _ l \\textbf { a } _ { M S } ( \\phi _ { l } ) \\textbf { a } _ { B S } ^ H ( \\theta _ { l } ) \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{gather*} p _ { 0 } ( x ) = 1 \\ , \\ \\ p _ { n _ { k } } ( x ) : = \\frac { P _ { n _ { k } } ( x ) } { D _ { n _ { k } - 1 } } \\ , \\ \\ 0 \\leq k < m + 1 \\ , \\end{gather*}"}
-{"id": "4929.png", "formula": "\\begin{align*} ( K \\cap F ) ^ { ( \\alpha ) } = K ^ { ( \\alpha ) } \\cap F . \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} \\left | \\nabla _ { z } K _ { N } ^ { 1 } ( z , \\zeta ) \\right | \\lesssim \\frac { \\left | \\rho ( \\zeta ) \\right | ^ { k } } { \\varepsilon ^ { k + 1 } } \\frac { 1 } { \\prod _ { i = 1 } ^ { n - 1 } \\tau _ { i } } \\frac { 1 } { \\left | z - \\zeta \\right | } , \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} E _ { \\lambda } ( t ) - E _ { \\infty } ( t ) = \\int _ { 0 } ^ { t } i U _ { \\varepsilon } ( t - s ) \\Big ( \\big [ \\{ { I _ { \\varepsilon } } | E _ { \\lambda } | ^ { 2 } \\} E _ { \\lambda } - \\{ { I _ { \\varepsilon } } | E _ { \\infty } | ^ { 2 } \\} E _ { \\infty } \\big ] - ( Q _ { \\lambda } E _ { \\lambda } ) \\Big ) d s , \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} \\mu \\left ( \\sum _ { j = 1 } ^ r a _ j ( v _ { \\lambda _ j } + v _ { - \\lambda _ j } ) \\right ) = \\sum _ { j = 1 } ^ r a _ j ^ 2 \\lambda _ j , \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } u ^ { - } \\partial _ { t } \\left [ k _ { m } * ( u - u _ { 0 } ) \\right ] d x + & a ( h _ { m } * u , u ^ { - } ) \\ , d t \\\\ & \\geq \\int _ { 0 } ^ { T } \\int _ { \\Omega } ( h _ { m } * f ) u ^ { - } d x d t . \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} L ( u , \\chi _ { 0 } ) = Z ( u ) \\prod _ { P | Q } ( 1 - u ^ { \\deg P } ) \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 t r \\omega _ R ( \\gamma ( s ) ) & = \\int _ 0 ^ 1 \\partial \\big ( \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } \\log ( 1 - z _ 1 ( s ) ^ 2 - z _ 2 ( s ) ^ 2 - 2 z _ 1 ( s ) z _ 2 ( s ) \\cos \\theta ) d \\theta \\big ) \\\\ & = \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } \\big ( \\int _ 0 ^ 1 \\frac { d } { d s } \\log ( 1 - z _ 1 ( s ) ^ 2 - z _ 2 ( s ) ^ 2 - 2 z _ 1 ( s ) z _ 2 ( s ) \\cos \\theta ) d s \\big ) d \\theta \\\\ & = \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } \\log ( 1 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta ) d \\theta . \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{gather*} X ( z ) = \\sum _ { i \\in \\mathbb { Z } } x _ { i } z ^ { - i - 1 } , X = C , D , E , x = c , d , e , \\end{gather*}"}
-{"id": "682.png", "formula": "\\begin{align*} Q ^ { \\mu \\nu } = \\left ( \\epsilon ^ { \\mu \\nu \\sigma \\tau } - \\frac { i } { 2 } e ^ { \\mu \\nu \\sigma \\tau } \\right ) F _ { \\sigma \\tau } , P _ { \\mu \\nu } = \\left ( \\delta _ { \\mu } ^ { \\lambda } \\delta _ { \\nu } ^ { \\rho } - \\frac { i } { 2 } e _ { \\mu \\nu \\sigma \\tau } \\epsilon ^ { \\sigma \\tau \\lambda \\rho } \\right ) F _ { \\lambda \\rho } , \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{gather*} E _ { \\Delta _ \\tau } : = \\{ ( x , y ) \\in O _ v ^ 2 \\mid ( v ( x ) , v ( y ) ) \\in \\Delta _ \\tau \\} , \\\\ Z ( s , f , \\chi , \\Delta _ \\tau ) : = \\int _ { E _ { \\Delta _ \\tau } } \\chi ( a c \\ f ( x , y ) ) \\ | f ( x , y ) | ^ s \\ | d x d y | , \\end{gather*}"}
-{"id": "9332.png", "formula": "\\begin{align*} D ^ { + } ( n , k ) & = ( k + 1 ) D ^ + ( n - 1 , k ) + ( n - 2 k + 1 ) D ^ + ( n - 1 , k - 1 ) + D ^ - ( n - 1 , k ) + \\\\ & k T ^ + ( n - 1 , k ) + ( n - 2 k + 1 ) T ^ + ( n - 1 , k - 1 ) \\\\ & = k ( D ^ + ( n - 1 , k ) + T ^ + ( n - 1 , k ) ) + ( n - 2 k + 1 ) ( D ^ + ( n - 1 , k - 1 ) + T ^ + ( n - 1 , k - 1 ) ) \\\\ & + D ^ + ( n - 1 , k ) + D ^ - ( n - 1 , k ) \\\\ & = k R ^ + ( n - 1 , k ) + ( n - 2 k + 1 ) R ^ + ( n - 1 , k - 1 ) + D ( n - 1 , k ) , \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} [ \\mathbf { m } _ e ] _ { e \\in I n ( t ) } \\mathbf { D } _ t & = \\left ( [ \\mathbf { m _ d } ] _ { d \\in O u t ( s ) } [ \\mathbf { F } _ e ] _ { e \\in I n ( t ) } \\right ) \\mathbf { D } _ t \\\\ & = [ \\mathbf { m _ d } ] _ { d \\in O u t ( s ) } \\left ( [ \\mathbf { F } _ e ] _ { e \\in I n ( t ) } \\mathbf { D } _ t \\right ) \\\\ & = [ \\mathbf { m _ d } ] _ { d \\in O u t ( s ) } \\mathbf { I } = [ \\mathbf { m _ d } ] _ { d \\in O u t ( s ) } . \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} \\delta ( \\mu , r ) = \\frac { K } { \\min \\{ M , K \\} } \\left ( 1 + \\frac { 1 } { r } \\right ) , \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} a ( \\mathbf { u } , \\mathbf { v } ; t ) : = \\int _ { G } ( \\mathbf { H } \\nabla \\mathbf { u } ) : ( \\nabla \\mathbf { v } ) \\mathrm { d } \\mathbf { x } . \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} D ^ { \\alpha } _ t I _ { t } ^ { \\beta } \\varphi = \\begin{cases} D _ { t } ^ { \\alpha - \\beta } \\varphi & : \\alpha > \\beta \\\\ I _ { t } ^ { \\beta - \\alpha } \\varphi & : \\alpha \\leq \\beta , \\end{cases} \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} h ( x ) = h ( x _ { + } ) + h ( x _ { - } ) \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\kappa = \\sqrt { ( \\varphi \\prime ) ^ { 2 } + \\varphi ^ { 2 } \\left ( ( \\alpha ^ { \\prime } ) ^ { 2 } + \\frac { 1 } { c ^ { 2 } } \\right ) + \\frac { \\lambda ^ { 2 } } { c ^ { 4 } } \\left ( 1 - \\frac { c ^ { 2 } } { \\lambda ^ { 2 } } \\right ) } , \\kappa _ { 1 } = \\varphi ^ { 2 } \\alpha ^ { \\prime } , \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} g _ s ( x ) = ( x - 2 ) ^ { - r - 2 s } x ^ { 2 s } , 0 \\le s \\le \\lfloor - r / 2 \\rfloor . \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} d _ { k + 1 } ( x ) & = \\sum _ { i = 0 } ^ { ( k + 1 ) - 3 } ( i + 1 ) x ^ i + \\sum _ { i = k - 1 } ^ { 2 ( k + 1 ) - 3 } ( 2 ( k + 1 ) + 2 - i ) x ^ i \\\\ & = \\sum _ { i = 0 } ^ { k - 3 } ( i + 1 ) x ^ i + \\sum _ { i = k - 2 } ^ { 2 k - 3 } ( 2 k + 2 - i ) x ^ i - x ^ { k - 2 } ( 1 - 2 \\sum _ { i = 1 } ^ k x ^ i ) + x ^ { 2 k - 1 } \\\\ & = d _ k ( x ) - x ^ { k - 2 } n _ k ( x ) + x ^ { 2 k - 1 } . \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 + 1 6 \\mu ^ 2 m _ \\mu + \\sqrt { 1 + 3 2 \\mu ^ 2 m _ \\mu } } { 1 6 \\mu ^ 2 } , \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\frac { a \\pm n d x } { a } = 1 . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\int _ X f d p _ { \\omega _ 0 } = \\int _ { A _ { \\omega _ 0 } } f \\circ h _ { \\omega _ 0 } d \\omega _ 1 \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} z _ { 1 , j } & = z _ { 2 , j } , & 1 \\leq j \\leq n ; \\\\ z _ { 1 , n - h - 2 } & = z _ { 1 , j } , & n - h - 1 \\leq j \\leq n . \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} \\xi _ { q } ( z ) = \\frac { \\left ( q ; q \\right ) _ { \\infty } ^ { 2 } } { \\left ( q ^ { 1 / 2 } ; q \\right ) _ { \\infty } ^ { \\ ! 2 } } \\frac { \\theta _ { q } \\left ( - z \\right ) } { \\theta _ { q } \\left ( - q ^ { - 1 / 2 } z \\right ) } . \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} S ( \\delta ; \\phi ) = \\int _ \\Sigma d \\sigma \\ , \\sqrt { | \\gamma | } \\gamma ^ { \\mu \\nu } \\partial _ \\mu \\phi ^ i ( x ) \\partial _ \\nu \\phi _ i ( x ) \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ { r - 1 } \\left ( - p _ { r , k } \\right ) s _ { k + m } = D _ { r - 1 } s _ { r + m } \\ , \\ \\ \\ \\ 0 \\leq m \\leq r - 1 \\ , \\end{gather*}"}
-{"id": "2407.png", "formula": "\\begin{align*} T _ { ( k ) } - T _ { ( k - 1 ) } & \\stackrel { d } { = } Y _ { n - k + 1 } \\stackrel { d } { = } \\frac { T _ { 1 } } { ( n - k + 1 ) } \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} \\Big \\{ { \\bf P } _ { X _ t | X ^ { t - 1 } , Y ^ { t - 1 } } = { \\bf P } _ { X _ t | Y _ { t - M } ^ { t - 1 } } \\equiv \\pi _ t ( d x _ t | y _ { t - M } ^ { t - 1 } ) , ~ t = 0 , \\ldots , n \\Big \\} \\subset { \\cal P } _ { 0 , n } \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} k + n = \\frac { p } { q } , \\ p , q \\in \\N , \\ ( p , q ) = 1 , \\ p \\geq n , \\ q \\geq n . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} w ( x ) : = \\left ( \\log \\frac { | x | } { \\rho } + \\beta \\right ) ^ \\alpha , x \\in M , \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} \\mathcal { Z } _ s = \\int _ { \\mathbb { R } ^ { 2 d s } } \\mathbf { 1 } _ { Z _ s \\in \\mathcal { D } _ s } f _ 0 ^ { \\otimes s } ( Z _ s ) d Z _ s \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} z ' + \\frac { 1 } { t } \\ , z = \\pm 2 a . \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} \\ 0 : = s \\quad \\ 1 & : = s t . \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} 0 & < \\delta < 1 , 1 < \\delta ^ 2 + \\delta ^ 6 , d _ * = \\frac { 1 } { \\sqrt { 1 - \\delta ^ 2 } } , \\\\ 0 & < q _ { m a x } , 0 < s , \\frac { 1 } { \\delta ^ 4 d _ * ^ 2 } \\Big ( 1 + \\frac { q _ { m a x } } { \\delta ^ 2 d _ * ^ 2 } \\Big ) \\leq \\delta ^ 2 - \\frac { s } { \\delta ^ 2 d _ * ^ 2 } . \\\\ \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ { \\infty } \\varphi _ { n } ^ { 2 } ( z ) = 4 \\frac { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { \\infty } ^ { \\ ! 2 } } { \\left ( q ; q ^ { 2 } \\right ) _ { \\infty } ^ { 2 } } \\theta _ { q ^ { 2 } } \\left ( - z ^ { 2 } \\right ) \\ ! . \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\ \\ \\sum _ { \\alpha = 1 } ^ { \\infty } ( - v ) ^ { \\alpha - 1 } \\sum _ { n = 1 } ^ { \\alpha } c _ { n } ( \\alpha ) \\frac { x ^ n \\sin ( n x ) } { \\sin ^ n x } \\ = \\ \\sum _ { n = 1 } ^ { \\infty } \\frac { x ^ n \\sin ( n x ) } { \\sin ^ n x } \\ , \\frac { v ^ n } { v ( v + 1 ) \\cdots ( v + n ) } \\ , . \\end{align*}"}
-{"id": "10113.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } y ^ { q } ( b y + c z ) ^ r } { z ^ { p + q + r } } . \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} X _ i = \\left [ X _ i ^ { ( 1 ) } , \\ldots , X _ i ^ { ( N ) } \\right ] , \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} L _ E ( y ^ n ) = \\sum _ { i = 1 } ^ n l ( y _ i ) , \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\Gamma ( f , g ) = \\frac 1 2 ( \\Delta ( f g ) - f \\Delta g - g \\Delta f ) . \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} C ^ { \\infty } _ { p e r } ( G ) = \\{ \\varphi \\in C ^ { \\infty } ( \\overline { G } ) \\mid \\hbox { $ \\varphi $ p e r i o d i c o f o r d e r $ m $ o n $ \\Gamma _ l $ f o r a l l $ m \\in \\N $ } \\} , \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} \\Big ( x _ a \\ , x _ { b t } \\ , + \\ , y _ a \\ , y _ { b t } \\ , - \\ , x _ b \\ , x _ { a t } \\ , - \\ , y _ b \\ , y _ { a t } \\Big ) _ t = 0 . \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} U P _ 0 + X U P _ 1 + X ^ 2 U P _ 2 + \\dots + X ^ g U P _ g = 0 \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} \\overline H ( x ) = O \\big ( \\overline F ( x / t ) \\big ) \\ \\ t \\ge 1 ; \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m x _ j v _ j = \\sum _ { j = 1 } ^ m x _ j ( v _ j - w _ j ) . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} u ^ p = p \\pi ^ { - 1 } e _ \\infty , v ^ p = - p \\pi ^ { - 1 } e _ 0 [ u , v ] = \\textstyle \\sum _ { i \\in I } e _ i . \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} \\Pr _ { a \\sim ^ L _ { \\rho } b } [ a \\in A , b \\in B \\ , | \\ , a _ { L } \\in A _ { L } , b _ L \\in B _ { L } ] \\geq 2 ^ { - | R | - \\epsilon n } / 2 = 2 ^ { ( \\lambda - 1 - \\epsilon - o ( 1 ) ) n } . \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} \\dot { x } = H ( Y , \\lambda ) , \\dot { Y } = 0 \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} \\mathbf { D } = \\varepsilon \\mathbf { E } , \\qquad \\mathbf { B } = \\mu \\mathbf { H } , \\mathbf { j } = \\sigma \\mathbf { E } , \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} \\| f ( T _ L + \\cdot ) \\| _ { \\cal { E } _ { T _ * - T _ L } ^ { \\frac { \\beta - \\gamma } 2 } \\cal { C } ^ { \\beta + 1 } } & = \\sup _ { T _ L < t \\le T _ * } ( t - T _ L ) ^ { \\frac { \\beta - \\gamma } 2 } \\| f ( t ) \\| _ { \\cal { C } ^ { \\beta + 1 } } \\\\ & \\le \\sup _ { T _ L < t \\le T _ * } t ^ { \\frac { \\beta - \\gamma } 2 } \\| f ( t ) \\| _ { \\cal { C } ^ { \\beta + 1 } } \\le \\| f \\| _ { \\cal { E } ^ { \\frac { \\beta - \\gamma } 2 } _ { T _ * } \\cal { C } ^ { \\beta + 1 } } . \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\left \\| \\sum _ { i = 1 } ^ N A _ i x _ i ^ { \\hat { K } + 1 } + y ^ { \\hat { K } + 1 } - b \\right \\| ^ 2 = \\frac { 1 } { \\beta ( \\epsilon ) ^ 2 } \\| \\lambda ^ { \\hat { K } + 1 } - \\lambda ^ { \\hat { K } } \\| ^ 2 = \\frac { \\mu ( \\epsilon ) ^ 2 } { \\beta ( \\epsilon ) ^ 2 } \\| y ^ { \\hat { K } + 1 } - y ^ { \\hat { K } } \\| ^ 2 \\leq \\frac { 1 } { 9 } \\theta _ { \\hat { K } } = O ( \\epsilon ^ 4 ) . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} \\mathrm { E x t } ^ 3 ( F , E ) = 0 . \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} \\ast _ { g _ \\varphi } \\varphi = e ^ { 4 5 6 7 } + e ^ { 2 3 6 7 } + e ^ { 2 3 4 5 } + e ^ { 1 3 5 7 } - e ^ { 1 3 4 6 } - e ^ { 1 2 5 6 } - e ^ { 1 2 4 7 } . \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} W _ 8 \\otimes W _ 8 & = \\odot ^ 2 W _ 8 \\oplus \\Lambda ^ 2 W _ 8 \\cong ( W _ 1 \\oplus W _ { 3 5 } ) \\oplus ( W _ 7 \\oplus W _ { 2 1 } ) , \\\\ W _ 7 \\otimes W _ 7 & = \\odot ^ 2 W _ 7 \\oplus \\Lambda ^ 2 W _ 7 \\cong ( W _ 1 \\oplus W _ { 2 7 } ) \\oplus W _ { 2 1 } , \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x _ { i } = \\dfrac { p _ { i } } { q } + \\varepsilon \\phi & ; & i = 1 , 2 , . . . , n \\\\ \\varepsilon q \\cong 0 & & \\end{array} \\right . \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\begin{array} { l l l l l } x _ { i } & = & \\dfrac { T _ { i , 2 } } { l _ { 2 } t _ { 2 } } & + & \\varepsilon \\pounds \\end{array} & ; & i = 1 , 2 , . . . , n \\\\ \\begin{array} { l l l l l } x _ { n + 1 } & = & \\dfrac { T _ { n + 1 , 2 } } { l _ { 2 } t _ { 2 } } & - & \\lambda _ { 2 } \\end{array} & & \\end{array} \\right . \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} \\varphi ( x , y ) = \\rho \\ast { h } ( x , y ) . \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ j \\left ( \\gamma ^ l d ^ l - ( 1 - \\alpha ^ l ) \\gamma ^ l d ^ { l - 1 } \\right ) = \\sum _ { l = 1 } ^ j \\left ( \\gamma ^ l d ^ l - \\gamma ^ { l - 1 } d ^ { l - 1 } \\right ) = \\gamma ^ j d ^ j . \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} \\eta _ i W = - \\eta W _ i \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} \\mathbb { E } ( N _ { 0 } ) = \\sum _ { n = 0 } ^ { \\infty } n \\mathbb { P } ( J = 0 , N = n ) = \\sum _ { n = 1 } ^ { \\infty } n p _ { 0 , n } = P ^ { ' } _ { 0 } ( 1 ) \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} x _ { i , j } = s _ j \\circ \\alpha _ i f _ j , \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} h = \\Bigg \\lfloor { \\log \\Bigl ( ( { 2 + \\gamma \\over \\gamma } ) { T - T _ \\gamma \\over T + 1 } \\Bigr ) \\over \\log \\Bigl ( { \\gamma \\over 2 + \\gamma } \\Bigr ) } \\Bigg \\rfloor + 1 \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} \\Phi ( \\psi ) ( x ) = \\sup _ { y \\in X } ( \\varphi ( y ) \\otimes \\psi ( y ) ) = \\sup _ { y \\ge x } ( \\psi _ 0 ( x ) \\otimes \\psi ( y ) ) = \\psi _ 0 ( x ) \\otimes \\psi ( x ) , \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} S ( u , v ) = ( \\cos u \\cos v , \\cos u \\sin v , \\sin u ) \\ ; , \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} y _ k ^ n e _ k & = \\sum _ { p = 0 } ^ { n - 1 } ( - 1 ) ^ { n - 1 + p } \\sum _ { \\gamma } E _ { \\gamma } ( x _ 1 , \\ldots , x _ { k - 1 } ) E _ { n - p - \\gamma } ( x _ k , \\ldots , x _ n ) . y _ k ^ p e _ k \\\\ & = \\sum _ { p = 0 } ^ { n - 1 } ( - 1 ) ^ { n - 1 + p } E _ { n - p } ( x _ 1 , \\ldots , x _ n ) . y _ k ^ p e _ k \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} \\tilde { u } ( t ) = y ( t ) + i \\Phi x ( t ) . \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} M ( v | u ) = \\frac { M ( u , v ) } { \\bar { M } ( u ) } . \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} d _ h ( f ( p ) , f ( y ) ) = d _ h ( f ( p ) , D f ( 0 ) y ) + o \\left ( d _ g ( 0 , y ) \\right ) . \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} V _ { p } : = & \\Big \\{ u \\in L ^ { 2 p } ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) \\cap L ^ { 2 } ( [ 0 , T ] ; H _ { e } ^ { \\beta } ( \\Omega ) ) \\\\ & g _ { 1 - \\alpha } * ( u - u _ { 0 } ) \\in C ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) , ( g _ { 1 - \\alpha } * ( u - u _ { 0 } ) ) | _ { t = 0 } = 0 \\Big \\} , \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} & \\phi \\big ( M _ 1 ( D _ { i ( 1 ) } ) ^ \\circ \\cdots M _ t ( D _ { i ( t ) } ) ^ \\circ \\big ) \\\\ & \\qquad = \\phi \\left ( \\prod _ { s = 1 } ^ t \\Big ( M _ s ( a _ { 1 ; i ( s ) } ) ^ \\circ \\otimes M _ s ( a _ { 2 ; i ( s ) } ) \\otimes \\cdots \\otimes M _ s ( a _ { K ; i ( s ) } ) \\Big ) \\right ) \\\\ & \\qquad = \\phi _ 1 \\big ( M _ 1 ( a _ { 1 ; i ( 1 ) } ) ^ \\circ \\cdots M _ t ( a _ { 1 ; i ( t ) } ) ^ \\circ \\big ) \\prod _ { k = 2 } ^ K \\phi _ k \\big ( M _ 1 ( a _ { k ; i ( 1 ) } ) \\cdots M _ t ( a _ { k ; i ( t ) } ) \\big ) . \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} a _ m ( W _ 0 ) = A _ m W _ 0 + B _ m , \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} \\big ( \\sigma \\otimes ( \\theta \\otimes \\chi ^ n ) \\big ) J _ { n + m } = ( \\big [ ( \\sigma \\otimes \\theta ) J _ n \\big ] \\otimes \\chi ) J _ m \\ , . \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} r ( E ) - r ( E - \\{ e \\} ) & = | | E | | _ \\lambda - \\lambda ( E - \\{ e \\} ) - | | E - \\{ e \\} | | _ \\lambda \\\\ & = | | \\{ e \\} | | _ \\lambda - \\lambda ( \\{ e \\} ) \\\\ & = 0 . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} x - { \\rm p r o x } _ { \\gamma P } ( x - \\gamma \\nabla f ( x ) ) = 0 , \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ n \\frac { 4 ( - r - 2 j ) ! \\gamma _ { 2 j } } { ( - r - 2 n ) ! ( 2 n - 2 j + 2 ) ! } \\sum \\limits _ { s = 1 } ^ \\ell { 2 \\ell \\choose 2 s } ( 2 ^ { 2 s } - 1 ) B _ { 2 ( \\ell - s ) } B _ { 2 s } . \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} \\sum _ { \\{ i : \\ ; B \\subseteq L _ i , \\ ; 1 \\leq i \\leq \\binom { n } { l } , \\ ; | B | = 1 \\} } \\gamma _ i \\leq 1 . \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\begin{cases} z _ 1 = x & \\\\ z _ 2 = \\frac { g ( a + \\pi x , b + \\pi y ) - g ( a , b ) } { \\pi } . & \\end{cases} \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\mu ^ \\prime ( t ) = ( d - 1 ) [ 1 + ( T - t ) ] ^ { - d } \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} u _ 1 = A \\sin ( n x ) \\sin ( n y ) , u _ 2 = B \\cos ( n x ) \\cos ( n y ) , \\tau = C \\cos ( n x ) \\sin ( n y ) , \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} \\frac { d g _ { i j } } { d t } = - \\beta ( g _ { i j } ) = - 2 \\alpha ' R _ { i j } - { \\alpha '^ 2 } R _ i ^ { \\ , k l m } R _ { j k l m } + O ( \\alpha '^ 3 ) \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} \\begin{aligned} & \\limsup _ { N \\rightarrow \\infty } \\underset { \\substack { 0 \\leq t \\leq T _ L \\\\ Z _ s \\in \\mathbb { R } ^ { 2 d s } } } { \\sup } \\left | f _ N ^ { ( s ) } ( t , Z _ s ) - f _ t ^ { \\otimes s } ( Z _ s ) \\right | \\mathbf { 1 } _ { Z _ s \\in \\mathcal { K } _ s \\cap \\mathcal { U } _ s ^ { \\eta ( \\varepsilon ) } } \\mathbf { 1 } _ { E _ s ( Z _ s ) \\leq 2 R ^ 2 } = 0 \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} \\dd Y ^ i _ t = \\lambda _ t ( 1 - Y ^ i _ t ) \\ , \\dd t + \\dd m ^ i _ t , \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} \\Big ( | \\phi _ k ( a _ { k ; j } ) | ^ 2 - | \\phi ( D _ j ) | ^ 2 \\Big ) \\Big ( \\phi _ k \\big ( ( a _ { k ; i } a _ { k ; i } ^ * ) ^ 2 \\big ) - 1 \\Big ) = 0 \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } q ^ { 1 / 2 } \\Im \\eta _ { 2 n + 1 } ( x ) + t \\Re \\eta _ { 2 n } ( x ) = 0 \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} A _ N \\xi = A _ N \\zeta - A _ N \\eta = B _ N \\zeta - A _ N \\eta - V _ N \\zeta = H _ N \\xi - V _ N \\zeta , \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} u ^ * : = \\lim _ { k \\to \\infty } \\varphi _ k . \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} g ^ * = \\begin{cases} g _ 1 & \\ p > \\frac { 4 + \\alpha } { 1 2 + 7 \\alpha } \\\\ g _ 2 & \\ p < \\frac { 4 + \\alpha } { 1 2 + 7 \\alpha } \\\\ \\big \\{ ( q , 1 - q ) : 0 \\leq q \\leq 1 \\big \\} & \\ p = \\frac { 4 + \\alpha } { 1 2 + 7 \\alpha } . \\end{cases} \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\partial _ { \\sigma } \\left \\{ \\left [ \\partial ^ { \\tau } \\partial _ { \\tau } + \\kappa \\left ( u ^ { \\tau } \\partial _ { \\tau } \\right ) ^ { 2 } \\right ] Z ^ { \\lambda \\sigma } + 4 \\pi \\mu p ^ { \\lambda \\sigma } \\right \\} = 0 . \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} \\left ( \\sum ^ \\infty _ { n = 0 } a _ n \\right ) ^ p \\leq 2 ^ { p - 1 } \\sum ^ \\infty _ { n = 0 } \\big ( 2 ^ { p - 1 } \\big ) ^ n \\ , a _ n ^ p . \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} \\Vert \\omega _ { \\rm i n } \\Vert _ { H ^ N ( \\Omega ) } + \\norm { \\bar { v } _ { \\rm i n } } _ { L ^ 2 ( \\Omega ) } = \\varepsilon \\leq C ^ { - 1 } \\nu ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } F ' ( t , z ) \\\\ G ' ( t , z ) \\end{array} \\right ) = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & e ^ { i \\xi _ 0 t } \\end{array} \\right ) \\left ( \\begin{array} { c c } e ^ { i c _ 3 t } & 0 \\\\ 0 & e ^ { - i c _ 3 t } \\end{array} \\right ) \\left ( \\begin{array} { c } F _ 0 ' ( z ) \\\\ G _ 0 ' ( z ) \\end{array} \\right ) \\ , , \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\bar q : = \\min \\left \\{ \\frac { n ( p - 1 ) } { n - s } , p \\right \\} . \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Q } \\Sigma ( \\nabla ^ { s } u ) : \\nabla \\varphi = \\int _ { 0 } ^ { T } \\int _ { \\Q } f \\cdot \\varphi \\forall \\varphi \\in L ^ { 2 } ( 0 , T ; H _ { 0 } ^ { 1 } ( \\Q ) ) \\ , . \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} \\frac { 1 } { 4 } P _ { \\mu \\nu } Q ^ { \\mu \\nu } = \\mathbf { H } \\cdot \\mathbf { B } - \\mathbf { E } \\cdot \\mathbf { D } - i \\left ( \\mathbf { E } \\cdot \\mathbf { B } + \\mathbf { H } \\cdot \\mathbf { D } \\right ) . \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} \\dot { x } = \\tilde { F } ^ { - 1 } \\nu , \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} z ^ { \\mu } J _ { \\nu - \\mu } ^ { ( 3 ) } ( 2 z ; q ) = \\sum _ { m = 0 } ^ { \\infty } \\frac { \\left ( q ^ { \\mu } ; q \\right ) _ { m } } { \\left ( q ; q \\right ) _ { m } } J _ { \\nu + m } ^ { ( 3 ) } ( 2 z q ^ { m / 2 } ; q ) q ^ { \\binom { m + 1 } { 2 } } \\left ( - \\frac { q ^ { \\nu - \\mu } } { z } \\right ) ^ { m } \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} ( A + B ) \\big ( - \\lambda _ n ^ 2 \\rho + 2 \\mu n ^ 2 + 8 ( a _ 3 + a _ 4 ) n ^ 4 \\big ) = 2 \\rho , \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} x ^ { r } B _ { n } = B _ { n + r } + \\sum _ { \\substack { k = 1 \\\\ 1 \\leq i _ { 1 } \\leq \\dots \\leq i _ { k } \\leq r - k + 1 } } ^ { r } \\rho _ { n + i _ { 1 } - d } \\rho _ { n + i _ { 2 } - 2 d } \\dots \\rho _ { n + i _ { k } - k d } B _ { n + r - k ( d + 1 ) } . \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} d \\omega ^ { ( C , \\gamma ) } + \\frac { 1 } { 2 } K _ { \\alpha \\beta } ^ { \\gamma } C _ { A B } ^ { C } \\omega ^ { ( A , \\alpha ) } \\omega ^ { ( B , \\beta ) } = 0 \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} \\varphi ^ { \\prime } \\colon ( \\Lambda \\oplus P ) ^ { \\ast } \\oplus \\Lambda ^ { \\infty } & \\longrightarrow \\ker ( d _ { n - 1 } ) = H _ { n - 1 } ( \\widetilde { K ' } ^ { [ n - 1 ] } ) \\xrightarrow [ ] { \\ h _ { n - 1 } ^ { - 1 } \\ } \\pi _ { n - 1 } ( \\widetilde { K ' } ^ { [ n - 1 ] } ) \\\\ x & \\longmapsto h _ { n - 1 } ^ { - 1 } \\big ( [ d _ { n } ( x ) ] \\big ) \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda ^ t } M ^ t \\vec v _ 0 = v _ t + \\frac { \\lambda ' } { \\lambda } \\vec w + \\vec w ^ * _ t , \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} \\lambda ( [ ( n , x ) ] ) = [ ( \\Psi ( x ) , \\kappa ( x ) ) ] \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} ( \\sqrt { a } - \\sqrt { b } ) ^ 2 - a \\nabla _ { \\nu } b & = b \\nabla _ { \\nu } a - 2 \\sqrt { a b } - K ( h ^ { \\frac { 1 } { 2 ^ n } } , 2 ) ^ { - r _ n } a \\sharp _ { \\nu } b + \\sum _ { k = 0 } ^ { n - 1 } r _ { k } \\big [ \\big ( a ^ { \\frac { m _ k } { 2 ^ k } } b ^ { 1 - \\frac { m _ k } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } - \\big ( a ^ { \\frac { m _ k + 1 } { 2 ^ k } } b ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ] ^ { 2 } . \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} 2 G _ { \\mu \\nu } = - e _ { \\mu \\nu \\sigma \\tau } R ^ { \\sigma \\tau } , \\qquad 2 F _ { \\mu \\nu } = e _ { \\mu \\nu \\sigma \\tau } S ^ { \\sigma \\tau } . \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N _ A } S _ { a _ j } S _ { a _ j } ^ * = 1 , S _ { a _ i } ^ * S _ { a _ i } = \\sum _ { j = 1 } ^ { N _ A } A ^ G ( i , j ) S _ { a _ j } S _ { a _ j } ^ * i = 1 , \\dots , N _ A . \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} \\pi _ { * } \\mathfrak { b } ( A _ \\mathrm { t } ) = \\mathfrak { b } ( \\pi ^ { - 1 } ( A _ \\mathrm { t } ) ) = \\mathfrak { b } ( S _ \\mathrm { t } ) = 1 \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} O ^ G _ \\gamma ( f \\| X ( \\cdot ) \\| _ 0 ^ { - \\eta } ) \\leqslant O ^ G _ \\gamma ( h \\| X ( \\cdot ) \\| _ 0 ^ { - \\eta } ) = C \\cdot O ^ { \\overline { G } } _ { \\overline { \\gamma } } ( \\overline { h } \\| X ( \\cdot ) \\| ^ { - \\eta } ) . \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} \\varphi _ P ( f ) = \\sum _ { I \\in P } m ( I ) \\varphi _ I ( f ) \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} d ( \\zeta , z ) = \\inf \\left \\{ \\varepsilon \\mbox { s u c h t h a t } z \\in P _ { \\varepsilon } ( \\zeta ) \\right \\} . \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} F _ { t , \\varsigma } \\big ( \\sqrt { t } z \\big ) + G _ { t , \\varsigma } \\big ( z \\big ) = \\exp \\big ( - t z ^ 2 \\big ) . \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} \\tau ^ * \\ge \\max \\limits _ { l = 1 , 2 , \\cdots , \\min \\{ N _ T , N _ R \\} } \\frac { 1 } { l } & \\left ( N _ R - ( N _ T - l ) ( N _ R - l ) \\mu _ T \\right ) . \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\begin{array} { c } f _ { 1 } ( u ) = \\int \\sqrt { 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) } \\cos \\alpha ( u ) d u , \\\\ f _ { 2 } ( u ) = \\int \\sqrt { 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) } \\sin \\alpha ( u ) d u . \\end{array} \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} F ( s , t , r ) = \\left ( \\begin{array} { c } s \\\\ t \\\\ s t - r \\end{array} \\right ) \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} \\mathbb { M } ^ 0 : = \\overline { \\mathbb { M } } _ s \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} - \\Delta \\varphi ( t , x ) = \\lambda ( t ) \\varphi ( t , x ) . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} a ( y + z ) = a ( y ) \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} \\frac { ( - q ; q ) _ n } { ( q ; q ) _ n } = \\sum _ { \\pi \\in \\mathcal { U } _ n } 2 ^ { \\nu _ d ( \\pi ) } q ^ { | \\pi | } , \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} \\pi ^ i ( t ) : = \\{ \\hat { t } | \\hat { t } < t , \\hat { t } \\in T ^ i \\} . \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} [ x , y ] = x ^ { - 1 } y ^ { - 1 } x y = x ^ { - 1 } y ^ { - 1 } z ^ 2 y ^ { - 1 } x ^ { - 1 } = x ^ { - 2 } y ^ 2 z ^ 2 . \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} v _ { \\delta ^ { p } } ( x _ { i } ) g _ { p , x _ { i } } ( x _ { p } ) = u _ { \\delta ^ { p } } ( 0 ) f _ { p , 0 } ( x _ { p } ) + u _ { \\delta ^ { i , p } } ( 0 ) f _ { i , 0 } ( x _ { i } ) f _ { p , 0 } ( x _ { p } ) \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} f ^ * ( t ) = & \\left ( \\frac { 1 } { 3 p - 1 } ( s _ { f } ( p ) + \\frac { 1 - p } { 2 } \\delta ) + \\frac { 1 } { 2 } ( - f ( t + p ) + f ( t ) ) \\right ) { \\bf 1 } _ { [ 0 , 1 - p ] } ( t ) \\\\ & + \\frac { 1 } { 2 p - 1 } ( s _ f ( p ) - \\frac { 1 - p } { 3 p - 1 } ( s _ { f } ( p ) - ( 2 p - 1 ) \\delta ) ) { \\bf 1 } _ { ( 1 - p , p ) } ( t ) \\\\ & + \\left ( \\frac { 1 } { 3 p - 1 } ( s _ { f } ( p ) + \\frac { 1 - p } { 2 } \\delta ) + \\frac { 1 } { 2 } ( f ( t + p ) - f ( t ) ) \\right ) { \\bf 1 } _ { [ p , 1 ] } ( t ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} \\underset { r \\rightarrow 0 ^ + } { \\liminf } \\frac { \\nu ( { B [ x _ 0 , r ] } ) } { r ^ n } = 0 \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { Z } _ { s , s + k } \\left [ Z _ s , t + \\tau ; \\left \\{ t _ j + \\tau , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] = \\\\ & = \\tilde { \\psi } _ { s + k } ^ { - \\tau } \\tilde { Z } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + k } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\end{aligned} \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\dfrac { 1 } { 2 ( 1 + \\xi x ) ^ { 1 / 2 } } = 0 . \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { u } - \\mathrm { d i v } ( \\mathbf { H } \\nabla \\mathbf { u } ) & = \\mathbf { 0 } ( 0 , \\infty ) \\times G , \\\\ \\tau \\partial _ { t } \\mathbf { H } + \\mathbf { H } - \\mathbf { F } ( \\nabla \\mathbf { u } ) & = \\mathbf { 0 } ( 0 , \\infty ) \\times G . \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} \\begin{aligned} F ( x _ { k + 1 } ) & \\le h \\big ( \\zeta _ { k + 1 } + c ( x _ k ) \\big ) + g ( x _ k ) + L \\cdot \\varepsilon _ { k + 1 } - \\tfrac { 1 } { 2 t } \\norm { x _ { k + 1 } - x _ k } ^ 2 \\\\ & \\le F ( x _ k ) + 2 L \\cdot \\varepsilon _ { k + 1 } - \\tfrac { 1 } { 2 t } \\norm { x _ { k + 1 } - x _ k } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} - \\beta _ { x x } = 1 - \\int _ { v > 0 } \\mu _ { - , + } ( e _ { - } ) d v - \\int _ { v < 0 } \\mu _ { - , - } ( e _ { - } ) d v \\equiv h \\left ( \\beta \\right ) , \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} \\theta = \\frac { k ^ 2 } { G \\left ( \\frac { k } { r } \\right ) } = \\frac { s _ { m ^ * } ^ 2 } { G ( s _ { m ^ * } ) } r ^ 2 \\leq r ^ 2 . \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} | u _ k ( x ) | = | u _ k ( x ) - u _ k ( x _ 0 ) | \\geq \\min _ { y \\in [ x _ 0 , x ] } | ( u _ k ) _ n ( y ) | | x - x _ 0 | > \\dfrac { \\alpha } { 3 } | x - x _ 0 | > 0 . \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} h _ m ( z ) & = \\lambda _ m ^ { - 1 } \\left ( f _ m \\circ \\eta _ { v _ m } ( z ) - f _ m \\circ \\eta _ { v _ m } ( 0 ) \\right ) \\\\ k _ m ( z ) & = \\lambda _ m ^ { - 1 } \\left ( f _ m \\circ \\eta _ { \\frac { b _ m } { 2 } + v _ m } ( z ) - f _ m \\circ \\eta _ { \\frac { b _ m } { 2 } + v _ m } ( 0 ) \\right ) . \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} B ( G _ { n , k , b } ) = \\left \\lceil \\frac { ( n + 1 ) \\binom { b } { k - 1 } - ( k - 1 ) \\binom { b + 1 } { k } + \\binom { 2 b - n + 1 } { k } - 2 } { 2 } \\right \\rceil . \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} V _ s = \\begin{cases} X _ s ^ { n - 2 } \\ & \\ s \\leq n - 2 , \\\\ \\ \\P ^ { n - 1 } \\ & \\ s = n , \\\\ \\ \\P ^ { n - 2 } \\ & \\ s = n - 1 , \\end{cases} \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} \\chi _ \\pm ( x , \\xi ) = \\chi \\left ( \\mu | v ( \\xi ) | ^ \\ell x \\right ) \\psi _ \\pm ( \\cos ( x , v ( \\xi ) ) ) \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} ( f \\bullet g ) \\bullet h = \\sum ^ { m + n - 1 } _ { j = 1 } \\sum ^ { m } _ { i = 1 } ( - 1 ) ^ { ( j - 1 ) ( p - 1 ) + ( i - 1 ) ( n - 1 ) } ( f \\bullet _ i g ) \\bullet _ j h \\\\ f \\bullet ( g \\bullet h ) = \\sum ^ { m } _ { \\xi = 1 } \\sum ^ { n } _ { \\omega = 1 } ( - 1 ) ^ { ( \\xi - 1 ) ( n + p - 2 ) + ( \\omega - 1 ) ( p - 1 ) } f \\bullet _ { \\xi } ( g \\bullet _ { \\omega } h ) \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} \\| \\nabla _ p s _ f \\| _ { p - S l } ^ 2 = & \\frac { 1 } { 3 p - 1 } ( 2 s _ { f } ( p ) ^ 2 + 2 ( 1 - p ) s _ { f } ( p ) \\delta + \\delta ^ 2 ( 1 - p ) p ) \\\\ & + \\min ( \\| a ( t ) \\| ^ 2 + \\| b ( t ) \\| ^ 2 + \\| c ( t ) \\| ^ 2 ) \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} | X ^ n _ r - X ^ n _ s | ^ { a } & = | \\psi _ n ( r , V ^ n _ r ) - \\psi _ n ( s , V ^ n _ s ) | ^ { a } \\\\ & \\leq C ( | V _ r ^ n - V _ s ^ n | + | r - s | ^ \\gamma ) ^ { a } \\\\ & \\leq C ( | V _ r ^ n - V _ s ^ n | ^ { a } + | r - s | ^ { a \\gamma } ) , \\end{align*}"}
-{"id": "9999.png", "formula": "\\begin{align*} \\Phi ( t ) : = \\frac { t ^ r } { \\varphi ( t ) } , t > 0 . \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} \\varPhi ^ { i j } & \\equiv \\frac { \\partial \\varPhi } { \\partial ( - \\tilde { u } _ { i j } ) } = F ^ { i j } \\Theta \\frac { \\sinh \\Theta } { \\cosh \\Theta } , \\\\ \\varPhi ^ { i j , k l } & = F ^ { i j , k l } v ^ { - 1 } \\Theta ^ 2 \\frac { \\sinh \\Theta } { \\cosh \\Theta } . \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} u ^ { \\prime } + \\xi & = f ( u ) + \\eta ( 0 , T ) , \\\\ \\xi & \\in \\partial \\varphi _ { 1 } ( u ) ( 0 , T ) , \\\\ \\eta & \\in \\partial \\varphi _ { 2 } ( u ) ( 0 , T ) , \\\\ u ( 0 ) & = u _ { 0 } \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} \\det ( A ( m , n ) ) = \\left ( { \\begin{array} { c c c c | c c c c } 1 & 1 & 1 & \\cdots & & & \\\\ & 1 & 1 & \\cdots & & & * \\\\ & & \\ddots & & & & \\\\ 0 & & & 1 & & \\\\ \\hline & & & & 1 & & & \\\\ & & & & & 1 & & \\\\ & & 0 & & & & \\ddots & \\\\ & & & & & & & 1 \\\\ \\end{array} } \\right ) = 1 . \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\varphi = \\int _ { r _ 2 } ^ u \\vartheta ^ { - 1 } , \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle \\frac { \\partial \\tilde { g } } { \\partial \\tilde { t } } = - \\tilde { R } \\tilde { g } \\mbox { i n } M \\times \\left ( 0 , \\tilde { T } \\right ) \\\\ k _ { \\tilde { g } } = \\gamma \\mbox { o n } \\partial M \\times \\left ( 0 , \\tilde { T } \\right ) \\\\ \\tilde { g } = g _ 0 \\mbox { i n } M , \\end{array} \\right . \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} P ' _ { 0 } ( 1 ) & = \\lim _ { z \\rightarrow 1 } \\dfrac { \\left [ \\lambda ( 1 - z ) + \\gamma \\right ] P _ { 0 ( z ) } - \\mu p _ { 1 1 } } { \\xi ( 1 - z ) } \\\\ & = \\lim _ { z \\rightarrow 1 } \\dfrac { - \\lambda P _ { 0 } ( z ) + \\left [ \\lambda ( 1 - z ) + \\gamma \\right ] P ^ { ' } _ { 0 } ( z ) } { - \\xi } \\\\ & = \\dfrac { - \\lambda P _ { 0 } ( 1 ) + \\gamma P ^ { ' } _ { 0 } ( 1 ) } { - \\xi } . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} & a _ { ( n ) } ^ M m = 0 \\quad n \\gg 0 , \\\\ & ( T a ) _ { ( n ) } ^ M = - n a ^ M _ { ( n - 1 ) } , \\\\ & a _ { ( n ) } ^ M ( b v ) = ( a _ { ( n ) } b ) v + b ( a _ { ( n ) } ^ M v ) , \\\\ & [ a ^ M _ { ( m ) } , b ^ M _ { ( n ) } ] = \\sum _ { i \\geq 0 } \\begin{pmatrix} m \\\\ i \\end{pmatrix} ( a _ { ( i ) } b ) ^ M _ { ( m + n - i ) } , \\\\ & ( a b ) ^ M _ { ( n ) } = \\sum _ { i = 0 } ^ { \\infty } ( a _ { ( - i - 1 ) } ^ M b _ { ( n + i ) } ^ M + b _ { ( - i - 1 ) } ^ M a _ { ( n + i ) } ^ M ) \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} X _ i = \\bar { X } _ i + Z _ i , \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} D _ { r , p } ( \\varphi _ { 1 } ) & = \\{ u \\in \\overline { D ( \\partial \\varphi _ { 1 } ) } : \\varepsilon \\longmapsto \\varepsilon ^ { - r } | u - J _ { \\varepsilon } u | \\in L ^ { p } ( 0 , 1 , \\varepsilon ^ { - 1 } \\mathrm { d } \\varepsilon ) \\} \\\\ J _ { \\varepsilon } & = ( i d + \\varepsilon \\partial \\varphi _ { 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} g : A _ 0 \\cup A _ 1 \\to [ 0 , 1 ] , \\ , x \\mapsto \\begin{cases} 0 & \\\\ 1 & \\end{cases} \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} \\sigma = v \\bigl ( \\Phi ( \\mu _ 0 ) , \\mu _ 0 \\bigr ) , \\mu = \\mu _ 0 + w \\bigl ( \\Phi ( \\mu _ 0 ) , \\mu _ 0 \\bigr ) \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} \\omega _ { \\pm } ( I _ { \\pm } ) = \\frac { 2 \\pi } { P _ { \\beta } } \\sqrt { 2 e _ { \\pm } } + O ( \\frac { \\varepsilon } { \\sqrt { 2 e _ { \\pm } } } ) . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} \\delta _ F ^ { ( 2 ) } = \\frac { K } { M r } ; ~ ~ ~ ~ ~ \\delta _ E ^ { ( 2 ) } = \\frac { K } { \\min \\{ M , K \\} } . \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} N _ { \\chi } ( r ; s ) : = \\# \\{ \\rho = \\beta + i \\gamma : 0 < \\beta < 1 , L ( \\rho , \\chi ) = 0 , | s - \\rho | \\leq r \\} . \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} \\begin{cases} \\varphi ( 0 , t ) = \\varphi ( L , t ) = \\varphi _ x ( 0 , t ) = 0 , \\ , \\ , ( 0 , T ) , \\\\ \\psi ( 0 , t ) = \\psi ( L , t ) = \\psi _ x ( 0 , t ) = 0 , \\ , \\ , ( 0 , T ) \\end{cases} \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} P ( D ) & = p _ e ^ { n _ e } ( 1 - p _ e ) ^ { { n \\choose 2 } - n _ e } ( 1 - p _ d ) ^ { n _ { a s } } ( 2 p _ d - 1 ) ^ { n _ s } \\\\ & = p _ e ^ { n _ e } ( 1 - p _ e ) ^ { ( n ( n - 1 ) - 2 n _ e ) } \\frac { p _ a ^ { n _ { a s } } ( 1 - p _ a ) ^ { n _ { a s } } } { p _ e ^ { n _ { a s } } } \\frac { p _ a ^ { 2 n _ s } } { p _ e ^ { n _ s } } \\\\ & = p _ e ^ { n _ e - n _ { a s } - n _ s } p _ a ^ { n _ { a s } + 2 n _ s } ( 1 - p _ a ) ^ { n ( n - 1 ) - 2 n _ e + n _ { a s } } \\\\ & = p _ a ^ { n _ a } ( 1 - p _ a ) ^ { n ( n - 1 ) - n _ a } , \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} \\begin{cases} v _ i ^ * = v _ i + \\omega \\omega \\cdot \\left ( v _ j - v _ i \\right ) \\\\ v _ j ^ * = v _ j - \\omega \\omega \\cdot \\left ( v _ j - v _ i \\right ) \\end{cases} \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} \\eta _ a \\eta _ b + \\eta _ b \\eta _ a = 0 , \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} f ( \\omega _ A ( h ) ) = \\omega _ { \\widetilde { A } } ( f \\cdot h ) . \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} P ( G ) = p _ e ^ { | E ( G ) | } ( 1 - p _ e ) ^ { { n \\choose 2 } - | E ( G ) | } G \\in \\mathcal { G } _ n . \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\mathcal J ( \\pi ^ * \\varphi | _ { f '^ { - 1 } ( A ) } ) = \\mathcal J ( \\pi ^ * \\varphi ) | _ { f '^ { - 1 } ( A ) } . \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} \\alpha _ 2 \\gamma _ 2 ^ { t r } = - \\beta _ 2 \\delta _ 2 ^ { t r } , \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} B _ { n + 1 } ^ i = B _ { n + 1 } ^ { i + 1 } + \\rho _ { ( d + 1 ) n + i + 2 } B _ n ^ { i + 1 } , \\ \\ 0 \\leq i \\leq d - 1 . \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} \\rho _ c ( u ) = \\left \\{ \\begin{array} { l @ { \\quad : \\quad } l } \\frac { 1 } { 2 } u ^ 2 & \\vert u \\vert \\le c \\\\ c \\vert u \\vert - \\frac { 1 } { 2 } c ^ 2 & \\vert u \\vert > c \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} d u ( \\tau , a ( t ) x ) = a ' ( t ) \\nabla u ( \\tau , a ( t ) x ) \\cdot x \\ , d t , \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} E _ { n , k , b } = \\left \\{ \\{ X , Y \\} \\in V _ { n , k , b } : \\overline { X \\cup Y } - \\underline { X \\cup Y } \\leq b \\right \\} . \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} f ( 2 ) = & ( q ^ 2 + 1 ) ( q - 1 ) + \\alpha q ( q ^ 2 + 1 ) ( q - 1 ) + g ( 2 ) , \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} \\zeta = ( A \\alpha ( z ) + B \\beta ( z ) , \\bar { B } \\alpha ( z ) - \\bar { A } \\beta ( z ) ) \\ , . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} \\theta _ { q } ' ( z ) \\theta _ { q } ( - z ) + \\theta _ { q } ' ( - z ) \\theta _ { q } ( z ) = - 2 \\theta _ { q } ( z ) \\theta _ { q } ( - z ) \\sum _ { k \\in \\mathbb { Z } } \\frac { q ^ { k } } { 1 - z ^ { 2 } q ^ { 2 k } } , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} & ( h \\# k ) ( h ' \\# k ' ) = h ( k _ 1 \\cdot h ' ) \\# k _ 2 k ' , \\\\ & ( h \\# k ) \\circ ( h ' \\# k ' ) = h ( k _ 1 \\rightharpoonup h ' ) \\# k _ 2 k ' , \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} & \\Delta ( H _ 1 \\star P _ 1 ) \\star \\Delta ( H _ 2 \\star P _ 2 ) = \\Delta ( H _ 1 ) \\star \\Delta ( P _ 1 ) \\star \\Delta ( H _ 2 ) \\star \\Delta ( P _ 2 ) = \\Delta ( H _ 1 ) \\star \\Delta ( P _ 1 \\star H _ 2 ) \\star \\Delta ( P _ 2 ) \\\\ = & \\Delta ( H _ 1 ) \\star \\Delta ( H _ 2 \\star H _ 2 ( P _ 1 ) ) \\star \\Delta ( P _ 2 ) = \\Delta ( H _ 1 ) \\star \\Delta ( H _ 2 ) \\star \\Delta ( H _ 2 ( P _ 1 ) ) \\star \\Delta ( P _ 2 ) . \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} \\tilde L _ a f ( \\xi ) = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { T } ^ d } e ^ { i \\int _ 0 ^ 1 \\nabla _ \\xi \\varphi _ a ( x , \\eta + \\theta ( \\xi - \\eta ) ) d \\theta \\cdot ( \\xi - \\eta ) } f ( \\eta ) d \\eta d x . \\end{align*}"}
-{"id": "10037.png", "formula": "\\begin{align*} g _ k ( z ) = \\prod _ { v = 0 } ^ { k - 1 } \\varepsilon ^ { - v } g ( \\varepsilon ^ v z ) \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} \\displaystyle { \\lim _ { | x | \\to + \\infty } | V ( x ) | = + \\infty } \\ , , \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} \\hat { f } ( \\xi ) = ( 2 \\pi ) ^ { - \\frac { n } { 2 } } \\int _ { \\mathbb { R } ^ n } e ^ { - i \\xi \\cdot x } f ( x ) \\ , d x . \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\omega \\cdot v ^ 0 = \\int _ { G } \\omega ( g ) ( g \\cdot v ^ 0 ) d g . \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} f _ { n } ( 1 ) = \\frac { \\theta _ { q } ( \\alpha ) } { \\left ( q ; q \\right ) _ { \\infty } } n + O ( 1 ) , \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} e = \\sum _ { i = 1 } ^ { n - 1 } i ( n - i ) e _ { i , i + 1 } , h = \\sum _ { i = 1 } ^ n ( n + 1 - 2 i ) e _ { i , i } , \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } S _ f ^ \\nu ( n ) S _ g ^ \\nu ( n ) = c _ { f , \\overline { g } } X ^ { 2 \\kappa ( f ) + \\frac { 3 } { 2 } - 2 \\nu } + O ( X ^ { 2 \\kappa ( f ) + 1 - 2 \\nu } \\log ^ 2 X ) \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\mu _ { k } ^ i \\left ( B \\right ) & = \\frac { 1 } { Z _ { k } ^ i } \\int \\limits _ { \\theta \\in B } \\prod \\limits _ { t = 1 } ^ { k } \\prod \\limits _ { j = 1 } ^ { n } \\ell ^ j ( s _ { t } ^ j | \\theta ) ^ { \\left [ A ^ { k - t } \\right ] _ { i j } } d \\mu _ 0 ^ j \\left ( \\theta \\right ) \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} & \\ ; \\ ; \\| x \\| _ { 0 } \\\\ & \\ ; \\ ; y = A x , \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} \\omega _ 1 = m \\varphi + n { \\rm a n d } \\psi = e ^ { i \\theta } \\sqrt { 1 + | m | ^ 2 } \\varphi + p \\ , . \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} \\widetilde { g } ( s , v ; q ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { K l ( l , n p ; q ) } { n ^ { s + v } } . \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + \\delta + t , x \\right \\rangle } ) = 0 , \\forall \\delta \\in \\Gamma ( k , p ) , p \\leq 0 , \\delta \\neq 0 . \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\bar \\psi + K _ { 6 - n } \\bar \\psi + f = O ( r ^ { n + 1 } ) , \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} T _ N ( f ) : = \\sum _ { | k | > N } f ( h k ) . \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{gather*} \\psi _ { } ( z ) = \\sum _ { k \\ge 0 } \\psi _ { ( - k - 1 ) } z ^ { k } , \\psi _ { } ( z ) = \\sum _ { k \\ge 0 } \\psi _ { ( k ) } z ^ { - k - 1 } . \\end{gather*}"}
-{"id": "5003.png", "formula": "\\begin{align*} f _ { 1 } ( u , v ) & = A U \\left ( 1 - \\frac { U } { K } \\right ) - \\frac { B U V } { 1 + E U } , \\\\ f _ { 2 } ( u , v ) & = \\frac { C U V } { 1 + E U } - D V \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} & \\Delta ( H _ k ( w ) ) = H _ k ( w ) \\otimes H _ k ( w ) , k \\in I . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} A _ k ( s ) = \\sum _ { j = 0 } ^ k T ( s ) ^ { - j } \\sum _ { m = j } ^ \\infty T ( - m ) ( \\mu _ { m , j } - \\mu _ { m , j - 1 } ) \\frac { \\Gamma ( m + s ) } { \\Gamma ( m + 1 ) \\Gamma ( s + 1 ) } , \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} A ( t ) = \\int _ 0 ^ t a ( r ) \\ , d r { \\rm f o r } \\ , \\ , t \\geq 0 . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} D ^ { F } _ { X } = d ^ { F } + d ^ { F , * } . \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} R ^ { ( d ) } = \\bigoplus _ { i \\geqslant 0 } R _ { d i } . \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\gamma ^ l ( \\alpha ^ l ) ^ 2 \\ge \\Theta : = \\frac { 1 - \\sigma } { \\eta \\zeta _ i + ( 1 - \\sigma ) \\delta _ { \\max } } . \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} q ^ { n ^ { 2 } / 2 } S _ { n } \\left ( x q ^ { - n - 1 / 2 } ; q \\right ) = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( x e ^ { i y } ; q \\right ) _ { n } } { \\left ( q ; q \\right ) _ { n } } \\exp \\left ( \\frac { y ^ { 2 } } { \\log q ^ { 2 } } - i n y \\right ) d y , \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} ( 1 - \\phi ( x , y ) ) ( 1 - \\phi ( y , x ) ) = 1 - p _ e ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} \\star \\partial f = ( f ' _ i ) _ { i \\ge 1 } . \\end{align*}"}
-{"id": "10007.png", "formula": "\\begin{align*} \\phi _ { \\mu , k } ^ { - } ( x ) = p r o j _ { D _ { k } } ( f ^ { - 1 } x ) . \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\hat \\Delta ( u ^ \\pi _ { i j } ) = \\sum _ { k = 1 } ^ { n ( \\pi ) } u ^ \\pi _ { i k } \\otimes \\hat u ^ \\pi _ { k j } . \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} & h _ { 0 } = 1 ; \\\\ & h _ { k + 1 } = b ^ { k + 1 } h _ { k } + 1 \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} E _ \\pm ( 0 ) u [ x ] & = J _ a \\tilde P _ \\pm u [ x ] \\\\ & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - \\varphi _ a ( y , \\xi ) ) } p _ \\pm ( y , \\xi ) u \\left [ y \\right ] d \\xi \\\\ & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( x - y ) \\cdot \\eta } p _ \\pm ( y , \\xi ( \\eta ) ) \\left | \\det \\left ( \\frac { d \\xi } { d \\eta } \\right ) \\right | u \\left [ y \\right ] d \\eta . \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} & \\Phi ^ { - 1 } \\bigg ( \\int \\Phi ( f ) \\ , d \\gamma _ n \\bigg ) = \\\\ & \\sup _ \\alpha \\inf _ \\beta \\mathbf { E } \\bigg [ \\int _ 0 ^ 1 e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ t \\| \\beta _ s \\| ^ 2 d s } \\langle \\alpha _ t , \\beta _ t \\rangle \\ , d t + e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\| \\beta _ t \\| ^ 2 d t } f \\bigg ( W _ 1 + \\int _ 0 ^ 1 \\alpha _ t \\ , d t \\bigg ) \\bigg ] . \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} \\frac { R _ n } { { n \\choose 2 } } = \\frac { 1 } { { n \\choose 2 } } \\sum _ { 1 \\leq i < j \\leq n } { \\bf 1 } _ { A ( i , j ) } \\frac { Q _ n } { { n \\choose 2 } } = \\frac { 1 } { { n \\choose 2 } } \\sum _ { 1 \\leq i < j \\leq n } { \\bf 1 } _ { B ( i , j ) } \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} r ( x , y , \\eta ) = p _ \\pm ( x , \\xi ( \\eta ; x , y ) ) p _ \\pm ( y , \\xi ( \\eta ; x , y ) ) \\left | \\det \\left ( \\frac { d \\xi } { d \\eta } \\right ) \\right | . \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} a ^ * _ { 1 , 1 } = \\frac { \\mu _ R } { 3 } + \\mu _ T - \\frac { 1 } { 3 } , a ^ * _ { 0 , 1 } = 1 - \\mu _ R - 2 \\mu _ T , a ^ * _ { 3 , 0 } = 1 - 3 \\mu _ T , \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} B _ { n , i } ( x ) & = B _ { \\textbf { n } , \\textbf { n - 1 } , i } ( x ) , ~ ~ i = 1 , 2 , \\\\ P _ n ( x ) & = B _ { n , 1 } ( x ) \\rho _ { \\nu , b } ( x ) + B _ { n , 2 } ( x ) \\rho _ { \\nu + 1 , b } ( x ) . \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} \\norm { \\partial _ t v } _ { L ^ p } + \\norm { \\Delta v } _ { L ^ p } = O ( e ^ { - \\beta _ v t } ) , \\norm { \\partial _ t \\tau } _ { L ^ { q _ { \\tau } } } + \\norm { \\Delta \\tau } _ { L ^ { q _ { \\tau } } } = O ( e ^ { - \\beta _ { \\tau } t } ) , \\norm { \\nabla _ H \\pi _ s } _ { L ^ p } = O ( e ^ { - \\beta t } ) \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\sum _ { u = 0 } ^ m ( - 1 ) ^ u \\binom { a } { u } = ( - 1 ) ^ m \\binom { a - 1 } { m } , a > 0 . \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} K _ { W ' } : = 2 ^ { - 1 } W ' ( S ^ * + S ) + 2 ^ { - 1 } ( S ^ * + S ) W ' , B _ { W ' } : = \\tilde { W } ( S ^ * - S ) - ( S ^ * - S ) \\tilde { W } . \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} Q _ n ( t ) = \\sup _ { y \\in \\mathbb { R } } \\int _ { y + B _ t ( 0 ) } \\left ( | u _ 1 ^ n ( x ) | ^ 2 + | u _ 2 ^ n ( x ) | ^ 2 \\right ) \\ d x , \\ n \\geq 1 , \\ t > 0 . \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} & \\big | A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\bigl ( 1 - e ^ { i \\lambda \\mu } \\bigr ) ^ { n } \\beta ^ 2 f ^ 0 ( \\lambda ) - \\lambda ^ { 2 n } C _ { \\mu , N } ^ { \\beta , 0 } \\bigl ( e ^ { i \\lambda } \\bigr ) \\big | ^ 2 \\\\ & = \\alpha _ 2 \\lambda ^ { 2 n } \\gamma ( \\lambda ) \\big | 1 - e ^ { i \\lambda \\mu } \\big | ^ { 2 n } \\bigl ( p ^ 0 ( \\lambda ) \\bigr ) ^ 2 , \\label { D 2 r i v n 1 _ c o i _ i _ s t . n _ d } \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\xi ^ { m _ 0 , N _ 0 } _ k : = \\left \\{ \\begin{array} { l l } 1 , & \\ k \\in S _ { m _ 0 , N _ 0 } , \\\\ 0 , & \\end{array} \\right . \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} L = \\limsup _ { R \\to \\infty } \\ , \\sup _ { t \\in \\mathbb { R } } \\ , a _ 1 \\dots a _ s \\ , \\int _ { [ 0 , a _ 1 ^ { - 1 } ) \\times \\dots \\times [ 0 , a _ s ^ { - 1 } ) } \\ , \\big | Q ( \\xi ) \\big | ^ 2 \\ , d \\sigma _ { t , R } ( \\xi ) \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} Q _ i ^ m : = { m \\choose i } + 2 { m \\choose i - 1 } = \\frac { m ! ( m + i + 1 ) } { i ! ( m - i + 1 ) ! } \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} C _ { i , s + 1 } ^ 0 = C _ { i , s + 1 } ^ { 0 , + } - C _ { i , s + 1 } ^ { 0 , - } \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} F _ { i j } ( t ) = { n - j \\choose n - i } \\sum _ { k = j } ^ i ( - 1 ) ^ { i - k } { i - j \\choose i - k } \\exp ( Q _ { ( n - k ) \\lambda } t ) . \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} \\mu ( H ) = \\mu ( H ' ) & : = ( 0 , 4 ) \\setminus \\{ E _ { \\pm } ( k ) \\} \\ \\ \\ \\ d = 1 , \\\\ \\mu ( H ) & : = [ 0 , E ( k ) ) \\cup ( 4 d - E ( k ) , 4 d ] \\ \\ \\ \\ d \\geqslant 2 , \\\\ \\mu ( H ' ) & : = [ 0 , E ' ( k ) ) \\cup ( 4 d - E ' ( k ) , 4 d ] \\ \\ \\ \\ d \\geqslant 2 . \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} q _ 1 = \\frac { q } { \\delta ^ 4 d _ * ^ 2 ( 1 + p ) } \\leq \\frac { q } { \\delta ^ 4 d _ * ^ 2 } < q \\leq q _ { m a x } , \\end{align*}"}
-{"id": "10087.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z ^ 2 } { x ^ { 4 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z } { x ^ { 3 } } , \\\\ \\end{gather*}"}
-{"id": "1446.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\int _ { \\Q } \\Sigma ( \\nabla ^ { s } u ) : \\nabla \\varphi = \\int _ 0 ^ T \\int _ { \\Q } f \\cdot \\varphi \\qquad \\forall \\varphi \\in L ^ { 2 } ( 0 , T ; H _ { 0 } ^ { 1 } ( \\Q ) ) \\ , . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} d \\Big ( ( x , u ) ; ( y , v ) \\Big ) = \\Big ( [ x , y ] , [ x , v ] _ { V } - ( - 1 ) ^ { | u | | y | } [ y , u ] _ { V } + \\varphi ( x , y ) \\Big ) \\ \\ \\ \\forall x , y \\in \\mathcal { G } \\ v , w \\in V . \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} \\nu _ 0 = 0 \\ , , \\xi _ 0 = \\sqrt { k \\ , \\mathfrak { g } } \\ , , F _ 0 ' = 1 \\ , , { \\rm a n d } G _ 0 ' ( z ) = - e ^ { - i k z } \\ , , \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} e ^ { - S ( \\psi , g ) } = e x p \\left \\{ - \\langle { \\psi } , L ( \\Lambda , \\psi , g ) { \\psi } \\rangle \\right \\} \\int _ { [ \\Lambda ' , \\Lambda ] } D \\chi \\ , e x p \\left \\{ - \\langle { \\chi } , L ( \\Lambda , \\chi , g ) { \\chi } \\rangle \\right \\} \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\frac { \\mathbf { H } _ { \\nu - 1 } ( z ) } { z \\mathbf { H } _ { \\nu } ( z ) } - \\frac { 2 \\nu + 1 } { z ^ 2 } = \\sum _ { n \\geq 1 } \\frac { 2 } { z ^ 2 - h _ { \\nu , n } ^ 2 } , \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} g _ { \\pm } = \\sum g _ { k } ^ { \\pm } ( I _ { \\pm } ) e ^ { i k \\theta _ { \\pm } } , \\ k \\in \\mathbf { Z } . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} | h ^ { ( n + 1 ) } ( x ) | & = | ( h ' ) ^ { ( n ) } ( x ) | = 2 | ( h ^ 2 g g ' ) ^ { ( n ) } ( x ) | \\\\ & \\leq 2 h ^ 2 ( x ) ( g g ' ) ^ { ( n ) } ( x ) + 2 \\sum _ { k = 0 } ^ { n - 1 } \\binom { n } { k } \\Big | ( h ^ 2 ) ^ { ( n - k ) } ( x ) \\ , ( g g ' ) ^ { ( k ) } ( x ) \\Big | . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} F _ { i j ; i j } = 1 , F _ { i j ; k l } = F _ { k l ; i j } , F _ { i j ; k l } = - F _ { k j ; i l } , F _ { i j ; k l } = - F _ { i l ; k j } , \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} E ( \\omega \\otimes \\iota ) \\bar \\Delta ( x ) & = E ( \\omega \\otimes \\iota ) ( W ^ * ( 1 \\otimes x ) W ) \\\\ & = \\lim _ \\alpha \\sum _ { i = 1 } ^ { n _ \\alpha } ( \\omega \\otimes \\iota ) ( W ^ * ( 1 \\otimes a _ { i , \\alpha } x b _ { i , \\alpha } ) W ) \\\\ & = ( \\omega \\otimes \\iota ) W ^ * ( 1 \\otimes E ( x ) ) W \\\\ & = ( \\omega \\otimes \\iota ) ( 1 \\otimes E ( x ) ) , \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} \\| u \\| _ \\mu = \\sum | a _ n | W _ n ^ \\mu \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} E ' ( k ) : = \\min \\{ \\ell ( k _ i ) : 1 \\leqslant i \\leqslant d \\} , \\ell ( k _ i ) : = \\begin{cases} 2 - 2 \\cos ( k _ i / 2 ) , \\ \\ k _ i \\in ( 0 , 2 \\pi / 3 ] \\\\ 2 + 2 \\cos ( k _ i ) , \\ \\ k _ i \\in ( 2 \\pi / 3 , \\pi ) \\cup ( \\pi , 4 \\pi / 3 ] \\\\ 2 + 2 \\cos ( k _ i / 2 ) , \\ \\ k _ i \\in ( 4 \\pi / 3 , 2 \\pi ) . \\end{cases} \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\Big ( \\phi _ k \\big ( ( a _ { k ; j } a _ { k ; j } ^ * ) ^ 2 \\big ) - 1 \\Big ) \\Big ( \\phi _ 1 \\big ( ( a _ { 1 ; i } a _ { 1 ; i } ^ * ) ^ 2 \\big ) - 1 \\Big ) = 0 . \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} \\xi _ { j } & = \\varphi \\left ( z _ { j } \\right ) , \\ ; \\ ; \\ ; \\ ; \\ ; 1 \\leq j \\leq J , \\\\ \\left \\Vert \\left \\{ a _ { i } \\right \\} _ { i = 1 } ^ { J } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } & \\leq C , \\\\ \\left \\Vert \\varphi \\right \\Vert _ { B _ { 2 } } & \\leq C , \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} & e ^ { 4 i R \\lambda _ j } C _ { 1 2 } ( \\lambda _ j ) w _ j - w _ j = 0 , \\\\ & \\big | \\lambda _ j - \\lambda _ 0 \\big | < \\epsilon ^ { 1 / 2 } \\big \\lVert v \\big \\rVert _ Y ^ { - 1 / 2 } , \\big \\lVert v _ j - w _ j \\big \\rVert _ Y < \\epsilon ^ { 1 / 2 } \\big \\lVert v \\big \\rVert _ Y ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} \\langle D Y , \\mathbf { 1 } _ { ( a \\wedge t , b \\wedge t ] } \\mathbf { 1 } _ { ( v , w ] } \\rangle _ { \\mathcal H } & = \\langle D \\big ( E [ Y | \\mathcal F _ a ] \\big ) , \\mathbf { 1 } _ { ( a \\wedge t , b \\wedge t ] } \\mathbf { 1 } _ { ( v , w ] } \\rangle _ { \\mathcal H } \\\\ & = \\langle \\mathbf { 1 } _ { [ 0 , a ] } E [ D Y | \\mathcal F _ a ] , \\mathbf { 1 } _ { ( a \\wedge t , b \\wedge t ] } \\mathbf { 1 } _ { ( v , w ] } \\rangle _ { \\mathcal H } = 0 , \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} \\Pi _ { d , \\xi } ^ { \\perp } \\left \\{ P U _ { \\delta , \\xi } + \\phi - i ^ { \\ast } \\left [ f _ { \\epsilon } \\ ( P U _ { \\delta , \\xi } + \\phi \\ ) \\right ] \\right \\} = 0 \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} \\Phi ( x , y ) : = \\nu ( x + y ) - \\nu ( x ) - \\nu ( y ) = a \\sum _ { i \\neq j } x _ i y _ j . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} F ( Z ) | [ \\gamma ] _ { \\underline { k } } : = \\lambda _ { \\underline { k } } ( \\nu ( \\gamma ) J ( \\gamma , z ) ^ { - 1 } ) F ( \\gamma Z ) . \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| \\exp ( - t A _ o ) \\Delta v _ 0 \\| _ { L ^ 2 } = \\lim _ { t \\to \\infty } \\| \\Delta v ( t ) \\| _ { L ^ 2 } = 0 . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} u _ n ( x ) : = u ( x ) \\varphi _ n ( x ) \\ , , \\varphi _ n ( x ) : = \\varphi \\left ( \\frac x n \\right ) \\ , . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} G _ 1 ( \\vec { X } ) : = \\dfrac { \\alpha _ 1 } { 2 } \\| \\vec { Y } - \\vec { X } \\| _ F ^ 2 + \\lambda _ 0 \\sum _ { i = 1 } ^ { k } s \\bigl ( \\sigma _ i ( \\vec { X } ) ; a _ 0 \\bigr ) , \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { p } } ( \\nabla _ p B ) _ { [ p , 1 ] } : = \\frac { 1 } { \\sqrt { p } } ( \\nabla _ p B _ t ) _ { t \\in [ p , 1 ] } = \\frac { 1 } { \\sqrt { p } } \\left ( B _ { t } - B _ { t - p } \\right ) _ { t \\in [ p , 1 ] } \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} P = P ( n ) = d _ 1 d _ 2 \\cdot n \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} A ( x ^ q ) B ( x ) = B ( x ^ p ) A ( x ) . \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} K _ n ( x , y ) = \\sum _ { k = 0 } ^ { n - 1 } \\mathcal { Q } _ k ( x ) \\mathcal { P } _ k ( y ) , \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle \\frac { \\partial } { \\partial t } g ( x , t ) = - 2 R c ( x , t ) + 2 \\alpha \\nabla \\varphi ( x , t ) \\otimes \\nabla \\varphi ( x , t ) \\\\ \\ \\\\ \\displaystyle \\frac { \\partial } { \\partial t } \\varphi ( x , t ) = \\tau _ g \\varphi ( x , t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} & \\pi ( h k ) = \\pi ( h _ 1 ) ( h _ 2 \\rightharpoonup \\pi ( k ) ) & & h , k \\in H . \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} 0 & \\to K _ X \\otimes F \\otimes \\mathcal J ( h ) \\otimes N \\to K _ X \\otimes F \\otimes \\mathcal J ( h ) \\otimes N \\otimes f ^ * H ^ { \\otimes m } \\\\ & \\to K _ D \\otimes F | _ D \\otimes \\mathcal J ( h | _ D ) \\otimes N | _ D \\to 0 . \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} \\frac { \\xi _ { s p } } { u } & = \\frac { z } { [ ( 1 + \\theta ) z - \\theta ] ( z - 1 ) } \\\\ & = \\frac { ( t ^ 2 - 2 \\Delta t + 1 ) z } { ( t ^ 2 z - 2 \\Delta t + 1 ) ( z - 1 ) } . \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} \\phi ( g ) = ( A _ { y x } ) . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} \\tau ( x , y , z , w ) : = - ( \\varphi ( y , z , w ) x + \\varphi ( z , x , w ) y + \\varphi ( x , y , w ) z + \\varphi ( y , x , z ) w ) . \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} a ^ z = a ^ { i \\omega } = ( 1 + k \\omega ) ^ i . \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} T ( I - A _ r ) f _ 2 ( x ) = \\int k _ r ( x , y ) f ( y ) \\ d \\mu ( y ) \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} \\epsilon _ { [ i _ 1 , \\ldots , i _ l ] } = ( - 1 ) ^ { n + m } , \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} ( \\lambda + \\gamma + n \\xi ) p _ { 0 , n } = \\lambda p _ { 0 , n - 1 } + ( n + 1 ) \\xi p _ { 0 , n + 1 } . \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} \\xi ( 1 - z ) P ' _ { 0 } ( z ) - \\left [ \\lambda ( 1 - z ) + \\gamma \\right ] P _ { 0 } ( z ) = - \\mu p _ { 1 , 1 } . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} ( \\{ m \\} - \\{ n \\} ) a _ { s , n + m } ' & = ( \\{ m \\} - \\{ n + s \\} ) a _ { s , n } ' + ( \\{ m + s \\} - \\{ n \\} ) a _ { s , m } ' . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} \\beta ( t ) = \\beta _ 0 - \\frac { 1 } { 2 } \\beta _ 0 \\left ( 1 - \\left [ 1 + ( T - t ) \\right ] ^ { - ( d - 1 ) } \\right ) \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} L ^ p _ { \\overline { \\sigma } } ( \\Omega ) = \\{ v \\in L ^ p ( \\Omega ) ^ 2 \\mid \\langle \\overline { v } , \\nabla _ H \\pi _ s \\rangle _ { L ^ { p ^ { \\prime } } ( G ) } = 0 \\hbox { f o r a l l } \\pi _ s \\in H _ { p e r } ^ { 1 , p ^ { \\prime } } ( G ) \\} , \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} F _ { 2 } ^ { \\prime } & = \\gamma F _ { 2 } + i \\beta \\gamma G _ { 3 } = \\frac { F _ { 2 } - \\dfrac { i } { c } \\left ( \\mathbf { v \\times G } \\right ) _ { 2 } } { \\sqrt { 1 - v ^ { 2 } / c ^ { 2 } } } , \\\\ F _ { 3 } ^ { \\prime } & = \\gamma F _ { 3 } - i \\beta \\gamma G _ { 2 } = \\frac { F _ { 3 } - \\dfrac { i } { c } \\left ( \\mathbf { v \\times G } \\right ) _ { 3 } } { \\sqrt { 1 - v ^ { 2 } / c ^ { 2 } } } . \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} B u : = \\lambda _ 0 ( V \\cdot \\nabla ) u + P M u - \\Gamma _ 0 \\Delta u , \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} s ' = s ^ 2 + r , \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} \\kappa _ \\lambda = \\frac { n _ \\lambda } { r ! } \\sum _ { g , h , \\tau } ( h ) \\delta ^ { a _ { \\tau ( 1 ) } } _ { b _ { \\tau \\rho ( 1 ) } } \\cdots \\delta ^ { a _ { \\tau ( r ) } } _ { b _ { \\tau \\rho ( r ) } } v _ { a _ 1 } \\otimes \\cdots \\otimes v ^ { b _ r } , \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} & H ^ \\bullet _ \\mathrm { a b s } ( X , F ) = H ^ \\bullet ( X , F ) \\simeq H ^ \\bullet \\big ( \\Omega ^ \\bullet ( X , F ) , d ^ F \\big ) , \\\\ & H ^ \\bullet _ \\mathrm { r e l } ( X , F ) = H ^ \\bullet ( X , \\partial X , F ) \\simeq H ^ \\bullet \\big ( \\Omega ^ \\bullet _ \\mathrm { c } ( X , F ) , d ^ F \\big ) . \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} \\Lambda _ R ( C ) & = \\big \\{ \\rho > 0 : \\ ; \\det \\big ( e ^ { 4 i R \\rho } C ( \\rho ) - \\mathrm { I d } \\big ) = 0 \\big \\} , \\\\ \\Lambda ^ * _ R ( C ) & = \\big \\{ \\lambda > 0 : \\ ; \\det \\big ( e ^ { 4 i R \\lambda } C ( 0 ) - \\mathrm { I d } \\big ) = 0 \\big \\} . \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} \\begin{array} { c } \\mu ( \\gamma _ 0 ) = \\{ A _ 0 \\subset X _ 0 \\mid A _ 0 \\subset \\gamma _ 0 A _ 0 \\} = \\{ A _ 0 \\subset X _ 0 \\mid A _ 0 \\subset \\gamma ( A _ 0 ) \\cap X _ 0 \\} = \\\\ \\{ A _ 0 \\subset X _ 0 \\mid A _ 0 \\subset \\gamma ( A _ 0 ) \\} = \\{ \\emptyset \\} \\end{array} \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} \\mathcal { R G } ^ * _ t ( d \\rho ( \\phi ; g ) ) = \\mathcal { R G } _ { - t } ( D \\phi ) e ^ { - \\widetilde { \\mathcal { R G } _ { - t } } S ( \\phi ; g ) } \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} \\begin{gathered} A ^ { ( j ) } _ { x _ { i } } \\left ( x _ { i _ 1 } , u _ j , ( u _ j ) _ { x _ { i _ 1 } } , ( u _ j ) _ { x _ { i _ 1 } ^ { j _ 1 } , x _ { i _ 2 } ^ { j _ 2 } } , \\dots \\right ) = A ^ { ( j ' ) } _ { x _ { i ' } } \\left ( x _ i , u _ j , ( u _ j ) _ { x _ { i _ 1 } ^ { j _ 1 } } , ( u _ j ) _ { x _ { i _ 1 } ^ { j _ 1 } , x _ { i _ 2 } ^ { j _ 2 } } , \\dots \\right ) , \\\\ x _ { i } \\neq x _ { i ' } , A ^ { ( j ) } \\neq A ^ { ( j ' ) } , \\\\ 1 \\leq i \\leq n , j _ 1 + \\dots + j _ n \\leq p , 1 \\leq j , j ' \\leq 2 n . \\end{gathered} \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} & f = f _ { 1 } + f _ { 2 } , \\\\ & f _ { 1 } \\\\ & - f _ { 2 } \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} \\tilde { \\varphi } _ i | \\sqcup D ^ n = H _ i ( \\cdot , 1 ) \\circ \\varphi _ i | \\sqcup D ^ n . \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} \\begin{aligned} u ( s , { X } _ s ) = \\ & \\Phi ( { X } _ T ) - w ( s , { X } _ s ) \\\\ & + \\int ^ T _ s f ( r , { X } _ r , u ( r , { X } _ r ) , \\nabla u ( r , { X } _ r ) ) \\d r \\\\ & - \\int ^ T _ s \\nabla w ( r , { X } _ r ) \\d { W } _ r - \\int ^ T _ s \\nabla u ( r , { X } _ r ) \\d { W } _ r . \\end{aligned} \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\gamma ( u ) = \\left ( f _ { 1 } ( u ) , . . . , f _ { n + 1 } ( u ) \\right ) , \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} \\phi ( \\frac 1 z , \\frac 1 w ) & = \\phi _ 0 ( z , w ) - \\phi _ 0 ( z , 0 ) - \\phi _ 0 ( w , 0 ) - \\int _ u ( \\phi _ 0 ( z , u ) - \\phi _ 0 ( z , 0 ) - \\phi _ 0 ( u , 0 ) ) \\omega ( u ) \\\\ & - \\int _ u ( \\phi _ 0 ( w , u ) - \\phi _ 0 ( w , 0 ) - \\phi _ 0 ( w , 0 ) ) ) \\omega ( u ) \\\\ & = \\phi ( z , w ) - \\phi _ 0 ( z , 0 ) - \\phi _ 0 ( w , 0 ) + \\phi _ 0 ( z , 0 ) \\int _ u \\omega ( u ) + \\phi _ 0 ( w , 0 ) \\int _ u \\omega ( u ) \\\\ & = \\phi ( z , w ) . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} ( a \\nabla _ { \\nu } b ) ^ { 2 } \\geqslant ( a \\sharp _ { \\nu } b ) ^ { 2 } + r _ { 0 } ^ { 2 } ( a - b ) ^ { 2 } + \\sum _ { k = 1 } ^ { \\infty } r _ { k } \\big [ a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } - a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } . \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} ( d + 2 ) \\left ( \\beta _ { d } ( \\Sigma ) + 1 \\right ) = ( d + 2 ) \\left ( \\beta _ { d + 1 } ( M ) + \\beta _ d ( \\Sigma ) \\right ) > ( d + 2 ) f _ { d + 1 } ( M ) = 2 f _ { d } ( M ) , \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} \\big ( x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } \\big ) ^ { \\perp } & \\cap \\left ( ( y _ 0 ^ { b _ 0 - 1 } y _ 1 ^ { b _ 1 } \\cdots y _ m ^ { b _ m } ) ^ \\perp + ( Y _ 0 ) \\right ) + \\big ( X _ 1 ^ { a _ 1 } \\cdots X _ n ^ { a _ n } \\big ) \\\\ & = \\left ( ( x _ 1 ^ { a _ 1 } \\cdots x _ n ^ { a _ n } ) ^ { \\perp } + ( X _ 1 ^ { a _ 1 } \\cdots X _ n ^ { a _ n } ) \\right ) \\cap \\left ( ( y _ 0 ^ { b _ 0 - 1 } y _ 1 ^ { b _ 1 } \\cdots y _ m ^ { b _ m } ) ^ \\perp + ( Y _ 0 ) \\right ) . \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} \\dot { X } ( t ) = F ( t , X ( t ) ) , X ( 0 ) = X _ 0 , & X ( t ) \\in \\mathbb { R } ^ { n \\times m } , & t \\in [ 0 , T ] , \\\\ \\Vert F ( t , X _ 1 ) - F ( t , X _ 2 ) \\Vert \\leq L , & \\forall X _ 1 , X _ 2 \\in \\mathbb { R } ^ { n \\times m } , & \\forall t \\in [ 0 , T ] , \\\\ \\Vert F ( t , X ) \\Vert \\leq B , & \\forall X \\in \\mathbb { R } ^ { n \\times m } , & \\forall t \\in [ 0 , T ] . \\\\ \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{gather*} \\psi _ { a } ^ { - } ( z ) T ^ { - k } = ( - 1 ) ^ { k } z ^ { k ( 2 a - 1 ) } T ^ { - k } \\psi _ { a } ^ { - } ( z ) . \\end{gather*}"}
-{"id": "6475.png", "formula": "\\begin{align*} I _ { \\pm } ( e _ { \\pm } ) & = \\frac { 1 } { 2 \\pi } \\int _ { - \\min \\beta } ^ { e _ { \\pm } } T _ { \\pm } ( e _ { \\pm } ^ { \\prime } ) d e _ { \\pm } ^ { \\prime } , \\\\ \\theta _ { \\pm } & = \\frac { 2 \\pi } { T _ { \\pm } ( e _ { \\pm } ) } \\int _ { 0 } ^ { x } \\frac { d x ^ { \\prime } } { \\sqrt { 2 ( e _ { \\pm } \\mp \\beta ( x ^ { \\prime } ) ) } } , \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} a ( n + e _ k ) = \\sigma _ 1 ^ { - 1 } \\sigma _ k a ( n + e _ 1 ) + i \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } a ( n ) . \\end{align*}"}
-{"id": "10068.png", "formula": "\\begin{align*} 2 ( \\alpha - 1 ) + ( 2 - \\alpha ) = 1 + 1 + k \\leq 2 + 1 , \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} \\Psi _ { \\gamma + t } ( x ) = ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } + \\sum _ { \\delta \\in \\Gamma \\backslash \\left \\{ 0 \\right \\} } ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + \\delta + t , x \\right \\rangle } ) e ^ { i \\left \\langle \\gamma + \\delta + t , x \\right \\rangle } . \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} \\sigma _ j ( z ) = \\tilde B _ j z , \\ j = 1 , 2 \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} \\begin{array} { l l } ( \\nabla _ X J _ \\alpha ) Y = \\frac { 1 } { 4 n } \\left [ g ( X , Y ) p _ \\alpha ^ \\top + \\overline \\theta _ \\alpha ( Y ) X + g ( X , J _ \\alpha Y ) J _ \\alpha ( p _ \\alpha ^ \\top ) \\right . \\\\ \\\\ \\qquad \\quad \\left . + \\overline \\theta _ \\alpha ( J _ \\alpha Y ) J _ \\alpha X \\right ] , \\alpha = 2 , 3 , \\end{array} \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} \\tau \\partial _ { t } \\mathbf { H } + \\mathbf { H } = \\mathbf { F } ( \\nabla \\mathbf { u } ) ( 0 , \\infty ) \\times G . \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} \\phi ( \\frac 1 z , \\frac 1 w ) & = \\phi _ 0 ( \\frac 1 z , \\frac 1 w ) - \\int _ u \\phi _ 0 ( \\frac 1 z , u ) \\omega ( u ) - \\int _ u \\phi _ 0 ( \\frac 1 w , u ) \\omega ( u ) . \\end{align*}"}
-{"id": "9902.png", "formula": "\\begin{align*} & \\tilde { \\varepsilon } _ 1 ( x ) = 2 Q ( x ) + \\{ \\tau ( \\xi ( x ) ) + Q ( x ) \\} \\circ \\nabla _ { \\partial \\Omega } ( \\tau - \\nu ) ( \\xi ( x ) ) \\circ ( \\tilde { x } - a ) , \\\\ & \\tilde { \\varepsilon } _ 2 ( x ) = - 2 Q ( x ) \\circ \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\otimes i _ x \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\right ) \\right ) . \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} T = \\lim _ { s \\to 0 + } \\varphi _ s ( C ( s ) ) , \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} \\Big ( \\frac { \\partial } { \\partial t } D _ t \\Big ) w ( t ) + D _ t \\Big ( \\frac { \\partial } { \\partial t } w _ t \\Big ) = \\Big ( \\frac { \\partial } { \\partial t } \\lambda ( t ) \\Big ) w ( t ) + \\lambda ( t ) \\Big ( \\frac { \\partial } { \\partial t } w ( t ) \\Big ) . \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { ( 2 n + 1 ) n } } { ( - q ; q ) _ { 2 n + 1 } } = \\sum _ { j = 0 } ^ \\infty q ^ { ( 3 j ^ 2 + j ) / 2 } ( 1 - q ^ { 2 j + 1 } ) . \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} & \\Delta { C } _ { n + 1 } = 0 , \\\\ & \\Delta { C } _ { t } = \\Big ( \\mu _ 1 ( t ) ( \\beta _ t - 1 ) - \\mu _ 0 ( t ) ( \\alpha _ t - 1 ) \\Big ) + H ( \\alpha _ t ) - H ( \\beta _ t ) + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu _ 1 ( t ) + \\Delta { C } _ { t + 1 } } } { 1 + 2 ^ { \\mu _ 0 ( t ) + \\Delta { C } _ { t + 1 } } } \\Big ) , ~ t \\in \\{ n , \\ldots , 0 \\} . \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } x = 1 + \\beta - \\zeta ( z ; \\alpha , \\beta , \\gamma ) \\\\ \\\\ y = \\zeta ( \\frac { z } { z - 1 } ; \\beta , \\alpha , \\gamma ) \\end{array} \\right . z \\in [ 1 , \\infty ) \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} t r R ^ { - 1 } ( z ) = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\frac { d \\theta } { 1 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta } . \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} \\dim H _ 2 ( K ) - \\dim E _ 2 ( K ) + \\dim W _ 2 ( K ) = 1 , \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} [ \\bar { e } _ { 0 , k } , \\bar { f } _ { 0 , l } ] = \\bar { h } _ { 0 , k + l } + k \\delta _ { k , - l } \\bar { c } , \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} T _ { \\xi _ { 0 } } \\circ \\cdots \\circ T _ { \\xi _ { \\ell } } ( I ) \\cap T _ { \\omega _ { 0 } } \\circ \\cdots \\circ T _ { \\omega _ { s } } ( I ) = \\emptyset . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} P ( G ) = \\int P _ { \\mathbf { x } } ( G ) d ( \\mu \\mathbf { x } ) , G \\in \\mathcal { G } _ n , \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\nabla { f } ( C ^ \\prime ) = D ^ \\prime \\cdot H _ F ( C ^ \\prime ) , \\nabla { g } ( C ^ \\prime ) = \\nabla { F } ( B ) . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 x y : n ) } = \\bigoplus _ { ( i , \\alpha , \\gamma ) } T _ { ( 2 x y ) } ( i , \\alpha ) \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} T x = T ( S x ) . \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} \\epsilon _ k = \\left \\lfloor \\frac { a - 1 - k } { p } \\right \\rfloor = \\left \\{ \\begin{array} { l l } 0 & ( 0 \\le k \\le a - 1 ) \\\\ - 1 & ( a \\le k \\le p - 1 ) \\end{array} \\right . , \\ \\ \\delta _ l = \\left \\lfloor \\frac { b - 1 - l } { q } \\right \\rfloor = \\left \\{ \\begin{array} { l l } 0 & ( 0 \\le k \\le b - 1 ) \\\\ - 1 & ( b \\le k \\le q - 1 ) . \\end{array} \\right . \\\\ \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} \\frac { 1 } { Z _ n } \\det \\left [ x _ j ^ { \\frac { \\kappa + i - 1 } { 2 } } I _ { \\kappa + i - 1 } ( 2 \\alpha \\sqrt { x _ j } ) \\right ] _ { i , j = 1 } ^ n \\det \\left [ x _ j ^ { \\frac { \\nu - \\kappa + i - 1 } { 2 } } K _ { \\nu - \\kappa + i - 1 } ( 2 \\beta \\sqrt { x _ j } ) \\right ] _ { i , j = 1 } ^ n , \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} ( g ^ { i j } \\nabla _ i S _ { j k } - \\frac { 1 } { 2 } \\nabla _ k S ) X _ j = - \\alpha \\tau _ g \\varphi \\nabla _ j \\varphi X _ j . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} F ( z , u , v ) : = \\sum _ { n } \\sum _ { k } \\sum _ { m } T _ { n } \\binom { n - 1 } { k } \\mathbb { P } \\{ D _ { n , k } = m \\} \\frac { z ^ { n } } { n ! } u ^ { k } v ^ { m } , \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} K ( u _ k ) = \\dfrac { 1 } { 2 } \\int _ { \\partial \\Omega } [ \\kappa ( u _ k ) _ n ^ 2 - ( ( u _ k ) _ { n \\tau } + ( u _ k ) _ { \\tau n } ) ( u _ k ) _ \\tau ] . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { \\mathbb { R } ^ { 2 d } } \\left ( | x | ^ 2 + | v | ^ 2 \\right ) \\left | f _ N ^ { ( 1 ) } ( t ) - f ( t ) \\right | d x d v \\lesssim \\\\ & \\lesssim \\left ( C \\log \\frac { 1 } { \\varepsilon } \\right ) ^ { d + 1 } \\left \\| f _ N ^ { ( 1 ) } ( t , x , v ) - f ( t , x , v ) \\right \\| _ { L ^ \\infty _ { x , v } } + \\varepsilon ^ { k C } \\end{aligned} \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} \\frac { \\partial y _ { N } ( x , t ) } { \\partial t } = \\sum _ { n = 0 } ^ { 2 ^ k - 1 } \\sum _ { m = 0 } ^ { 2 M } \\sum _ { l = - 1 } ^ { N _ h + 1 } c _ { n , m , l } \\psi _ { n , m } ( t ) B _ l ( x ) . \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} Y \\overset { { \\rm i n \\ , l a w } } { = } Y ' = \\beta _ { 1 , 0 } ^ { - 1 } \\bigl ( \\tau = 1 , b _ 0 = 1 \\bigr ) . \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} \\varphi ( x ) = \\frac { 1 } { x } \\int \\limits _ 0 ^ x \\frac { 1 } { 1 - \\xi } \\chi ( \\xi ) \\ , d \\xi , \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} S v = \\{ \\chi _ i v \\} _ { i \\in I } , R u = \\sum _ { i \\in I } \\chi _ i u _ i , \\hbox { w h e r e } \\chi _ i : = \\sqrt { \\varphi _ i } , \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} \\psi ( y ) & : = \\Bigl ( T y _ { 1 , - 1 } + W ( y _ { 1 , - 1 } ; - 1 ) + { \\gamma \\over 2 } ( y _ { 1 , - 1 } - y ) ^ 2 \\Bigr ) \\\\ & { } \\qquad - \\Bigl ( T y _ { 1 , 0 } + W ( y _ { 1 , 0 } ; 0 ) + { \\gamma \\over 2 } ( y _ { 1 , 0 } - y ) ^ 2 \\Bigr ) \\ , . \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\left \\| I _ { t } ^ { \\beta } \\left ( \\varphi - \\sum _ { k = 0 } ^ m \\frac { \\varphi ^ { ( k ) } ( 0 ) } { k ! } t ^ k \\right ) \\right \\| _ { \\mathcal { C } ^ { \\alpha + \\beta } ( [ 0 , T ] ) } \\leq N ( \\beta ) \\left \\| \\varphi - \\sum _ { k = 0 } ^ m \\frac { \\varphi ^ { ( k ) } ( 0 ) } { k ! } t ^ k \\right \\| _ { \\mathcal { C } ^ { \\alpha } ( [ 0 , T ] ) } \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} - \\Delta v + v = v ^ { q - 1 } \\ \\ \\mathcal D , v > 0 \\ \\ \\mathcal D , \\partial _ \\nu v = 0 \\ \\ \\partial \\mathcal D \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} \\delta _ ! F ( U _ I \\subseteq M ) \\simeq \\begin{cases} F ( U _ I \\subseteq M ) & | I | = 1 \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} P _ { 0 } ( z ) = p _ { 0 , 0 } e ^ { \\frac { \\lambda } { \\xi } z } ( 1 - z ) ^ { - \\frac { \\gamma } { \\xi } } \\left [ 1 - \\dfrac { A ( z ) } { A } \\right ] , \\end{align*}"}
-{"id": "10093.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p - q - r } y ^ q ( a x + b y ) ^ r } { z ^ { p } } = \\dfrac { y ^ q ( a x + b y ) ^ r } { x ^ { - p ' } z ^ { p ' + q + r } } , p ' = p - q - r . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} M _ X = \\frac { 1 } { N } \\sum _ { r , l \\leq k } c _ { r l } \\mu _ r \\otimes \\mu _ l . \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} \\varphi : \\R \\to \\R , \\varphi ( t ) : = \\int _ { - \\infty } ^ t \\langle x \\rangle ^ { - 2 s } d x . \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} a : = \\det ( a _ { \\alpha \\beta } ) . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\begin{array} { l c } F _ \\alpha ( X , Y , Z ) = \\frac { 1 } { 4 n } \\left [ g ( X , Y ) \\theta _ \\alpha ( Z ) + g ( X , Z ) \\theta _ \\alpha ( Y ) \\right . \\\\ \\qquad \\qquad \\left . + g ( X , J _ \\alpha Y ) \\theta _ \\alpha ( J _ \\alpha Z ) + g ( X , J _ \\alpha Z ) \\theta _ \\alpha ( J _ \\alpha Y ) \\right ] , \\end{array} \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} s ( z , \\zeta ) = \\sum _ { i = 1 } ^ { n } \\left ( \\overline { \\zeta _ { i } } - \\overline { z _ { i } } \\right ) d \\left ( \\zeta _ { i } - z _ { i } \\right ) \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} K = D ( \\lambda / 1 2 ) \\cap R ( \\varepsilon ) . \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\pi _ 7 ( - [ C r , \\chi ] ^ { F N } _ p ) = \\dfrac 1 3 b _ { i j } ( \\ast \\varphi \\wedge e ^ i ) \\otimes e _ j . \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} F \\left ( x _ { 1 } , \\dots , x _ { n } \\right ) = \\sum _ { i = 1 } ^ { n } f \\left ( x _ { i } \\right ) v _ { i } , \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} \\varphi ( t ) ( x , y ) : = B ( x , y ) ( t ) \\ \\ \\left ( t \\in K \\ \\mbox { a n d } \\ ( x , y ) \\in X \\times Y \\right ) . \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} G _ \\lambda ( x , y ) = & ( - \\Delta - \\lambda ) ^ { - 1 } \\delta ( x - y ) = \\sum _ { \\xi \\in \\Z ^ 2 } c _ \\lambda ( \\xi ) e _ { \\xi } ( x - y ) \\\\ c _ \\lambda ( \\xi ) = & \\frac { 1 } { | \\xi | ^ 2 - \\lambda } \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} W ( g ( \\Lambda ) ) = - \\frac { 1 } { 2 } [ \\zeta ' ( 0 , g D ) + \\ln \\Lambda ^ 2 \\ , a _ n ( g , D ) ] \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} & M _ { X } ( a , b ) = \\\\ & \\left ( \\begin{array} { l l l } u _ 1 \\mu _ 1 ( a ) , \\ldots , u _ k \\mu _ k ( a ) \\end{array} \\right ) \\left ( \\begin{array} { l l l } \\hdots & \\hdots & \\hdots \\\\ \\hdots & u _ { i j } & \\hdots \\\\ \\hdots & \\hdots & \\hdots \\end{array} \\right ) \\left ( \\begin{array} { l } \\mu _ 1 ( b ) \\\\ \\vdots \\\\ \\mu _ k ( b ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} w _ { \\kappa } ( x , t ) : = \\begin{cases} \\max ( \\hat { w } ( x , t ) , w ( x , t ) ) & ( x , t ) \\in N _ { 7 r / 8 , s } ( 0 , t _ 0 ) , \\\\ [ 1 . 2 m m ] w ( x , t ) & ( x , t ) \\notin N _ { 7 r / 8 , s } ( 0 , t _ 0 ) . \\end{cases} \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} \\psi _ 1 ( h ) : [ \\textbf { q } , \\textbf { p } ] \\mapsto [ \\textbf { Q } , \\textbf { P } ] : = [ \\textbf { q } + h \\textbf { p } / m , \\textbf { p } ] . \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} \\dim S _ \\iota \\leq \\dim T _ { n - h - 2 } + \\dim S _ { \\iota , \\ell , \\mathbf { w } } = ( n - h - 2 ) + ( m - 2 ) ( h + 1 ) . \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} - L u _ { n + 1 } + g _ { n + 1 } \\circ u _ { n + 1 } & = f \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u _ { n + 1 } & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\mathcal { B } ^ + _ { V I I } = \\bigcup _ { i \\in \\left \\{ 1 , \\dots , s , s + 1 , \\dots , s + k \\right \\} \\backslash \\left \\{ i _ { k + 1 } \\right \\} } \\mathcal { B } ^ + _ { V I I , i } \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} F _ { d , \\ell } \\left ( z ; \\tau \\right ) : = \\sum _ { n \\geq 0 } \\zeta ^ { \\ell n + d } q ^ { ( \\ell n + d ) ^ 2 } , \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} \\left . \\begin{aligned} \\bar { r } ^ 2 ( s , a ^ 2 ) & = \\frac { r ^ 2 ( s , f , a ^ 2 ) } { | | \\mu | | + \\alpha } , \\ \\forall \\ s \\in S , a ^ 1 \\in A ^ 1 ( s ) , \\\\ p ^ 2 ( s ' | s , a ^ 2 ) & = \\frac { \\mu ( s ' , s , f , a ^ 2 ) } { | | \\mu | | } + \\delta ( s , s ' ) , \\ \\forall \\ s , s ' \\in S , a ^ 2 \\in A ^ 2 ( s ) , \\\\ \\beta & = \\frac { | | \\mu | | } { \\alpha + | | \\mu | | } . \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} p _ { n + 1 } - p _ { n } = O ( ( \\log p _ { n } ) ^ { 2 } ) \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} C a p ( z , U _ { + } \\cup U _ { - } ) & = h ( z _ { + } ) = h ( \\zeta ) = h ( \\zeta _ { + } ) + h ( \\zeta _ { - } ) \\\\ & = H ( \\zeta ) [ h _ { + } ( \\zeta _ { + } ) + h _ { - } ( \\zeta _ { - } ) ] \\\\ & = ( 1 - d ( z , \\zeta ) h ( \\zeta _ { + } ) ) [ C a p ( \\zeta , U _ { + } ) + C a p ( \\zeta , U _ { - } ) ] \\\\ & = ( 1 - d ( z , \\zeta ) \\cdot C a p ( z , U _ { + } \\cup U _ { - } ) ) [ C a p ( \\zeta , U _ { + } ) + C a p ( \\zeta , U _ { - } ) ] . \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{gather*} \\sum _ { \\sigma \\in \\mathfrak { S } _ { k } } { \\sigma } \\det \\begin{bmatrix} 1 & z _ { 1 } & z _ { 1 } ^ { 2 } & \\dots & z _ { 1 } ^ { k - 1 } \\\\ z _ { 2 } & z _ { 2 } ^ { 2 } & z _ { 2 } ^ { 3 } & \\dots & z _ { 2 } ^ { k } \\\\ z _ { 3 } ^ { 2 } & z _ { 3 } ^ { 3 } & z _ { 3 } ^ { 4 } & \\dots & z _ { 3 } ^ { k + 1 } \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ z _ { k } ^ { k - 1 } & z _ { k } ^ { k } & z _ { k } ^ { k + 1 } & \\dots & z _ { k } ^ { 2 k - 2 } \\end{bmatrix} . \\end{gather*}"}
-{"id": "288.png", "formula": "\\begin{align*} d \\rho ( \\phi ; g ) : = D \\phi \\ , e ^ { - S ( \\phi ; g ) } \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} \\lim _ { P \\rightarrow \\infty } \\lim _ { L \\rightarrow \\infty } \\frac { T _ E \\log \\left ( ( P - 1 ) ( 1 + G ) \\right ) } { L } = \\frac { K } { \\min \\{ M , K \\} } , \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} P _ { \\textbf { n } _ 2 + \\textbf { e } _ i , \\textbf { n } _ 1 - \\textbf { e } _ j } ( x ) : = \\sum _ { k = 1 } ^ q B _ { \\textbf { n } _ 2 + \\textbf { e } _ i , \\textbf { n } _ 1 - \\textbf { e } _ j , k } ( x ) w _ { 2 , k } ( x ) , \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} F ( u ) : = L u + ( H ( u ) , 0 ) , \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} \\Psi = \\Psi _ \\mathrm { p p } + \\Psi _ \\mathrm { s c } + \\Psi _ \\mathrm { a c } , \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\gamma ( s , \\ell ) = 2 \\log s + \\log \\left [ 1 + \\log \\left ( \\frac { s ^ 2 + \\alpha ^ \\ell } { s ^ 2 } \\right ) \\right ] . \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} & \\pi _ \\omega ( f ) \\delta _ n = f ( \\alpha ^ { n } ( \\omega ) ) \\delta _ n , & & \\pi _ \\omega ( S _ i ) \\delta _ n = \\theta ( n , e _ i ) ( \\omega ) \\delta _ { n + e _ i } \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} 8 = m _ { - } \\geq 2 k - j - c _ j , \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} F ( t ) = \\sum _ { \\ell = - \\infty } ^ \\infty F _ \\ell \\ , t ^ { \\frac { 2 \\pi i } { \\log ( p ) } \\ell } \\mbox { f o r } t \\in S . \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } B _ n = A _ 1 \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} \\lambda _ 1 = \\inf _ { 0 \\neq u \\in W _ 0 ^ { 1 , 2 } ( M ) } \\Bigg \\{ \\frac { \\int _ M | \\nabla u | _ g ^ 2 \\ d \\mu _ g } { \\int _ M | u | ^ 2 _ g \\ d \\mu _ g } \\ \\ \\Big | \\ \\ u \\neq 0 , \\ \\ u \\in W _ 0 ^ { 1 , 2 } ( M , g ) \\Bigg \\} , \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} c _ 1 ( \\langle \\mathcal { L } _ 0 , \\dots , \\mathcal { L } _ n \\rangle ) = \\int _ q \\bigwedge _ { i = 0 } ^ n c _ 1 ( \\mathcal { L } _ i ) . \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\left ( \\mathcal { R } [ q , \\psi _ k ] - k ^ 2 \\right ) \\longrightarrow - \\frac { q ( 0 ) + q ( 4 ) } { 2 } \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} K \\left ( k ' ( q ^ { 2 } ) \\right ) = - \\ln ( q ) \\ , \\vartheta _ { 3 } ^ { 2 } \\left ( 0 \\mid q ^ { 2 } \\right ) \\ ! , \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} v _ { m , j } = ( - 1 ) ^ { m } \\alpha ^ { - m } q ^ { m ( m + 1 ) / 2 } ( q ; q ) _ { \\infty } \\ , J _ { m - j } \\left ( \\alpha ^ { - 1 } q ^ { m } ; q \\right ) \\ ! , \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} T ( k ) : = \\sqrt { ( k + 1 ) ^ 2 + m ^ 2 } + \\sqrt { ( k - 1 ) ^ 2 + m ^ 2 } - 2 \\sqrt { k ^ 2 + m ^ 2 } , \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} i \\partial _ t \\mathcal { N } - \\lambda \\sqrt { I _ \\varepsilon ^ { - 1 } ( - \\Delta ) } \\mathcal { N } = \\lambda \\sqrt { I _ \\varepsilon ( - \\Delta ) } | E | ^ 2 . \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} \\hat { \\varphi } ( \\xi ) = \\frac { \\left ( \\alpha + \\beta e ^ { - i L \\xi } \\right ) } { ( i \\xi ) ^ 3 + \\lambda } - \\frac { a b ( i \\xi ) ^ 3 \\hat { \\psi } ( \\xi ) } { c \\left ( ( i \\xi ) ^ 3 + \\lambda \\right ) } , \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} \\partial _ { t } u - \\mathrm { d i v } \\ , \\big ( \\mathbf { L } \\nabla u ) & = 0 ( 0 , \\infty ) \\times G \\\\ \\partial _ { t } \\mathbf { L } + \\mathbf { L } & = \\mathbf { F } ( \\nabla _ { \\sigma } u ) ( 0 , \\infty ) \\times G , \\\\ u ( 0 , \\cdot ) = u ^ { 0 } , \\mathbf { L } ( 0 , \\cdot ) & = \\mathbf { L } ^ { 0 } G \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} h ( Y ) = \\frac { 1 } { 2 } \\inf _ { V , W } g ( \\left [ \\begin{array} { c c } V & Y \\\\ Y ^ H & W \\end{array} \\right ] ) \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} & \\mathbb { E } [ u ( V - \\mathbb { E } V ) ] = \\mathbb { E } \\Big [ u ( F ( e ) - \\mathbb { E } F ( e ) ) \\Big ] = \\mathbb { E } \\Big [ u ( F ( e ) - \\mathbb { E } F ( \\tilde e ) ) \\Big ] \\\\ & = \\mathbb { E } \\Big [ u ( \\mathbb { E } [ F ( e ) - F ( \\tilde e ) \\mid e ] ) \\Big ] \\leq \\mathbb { E } \\Big [ \\mathbb { E } \\Big [ u ( F ( e ) - F ( \\tilde e ) ) \\mid e \\Big ] \\Big ] \\\\ & = \\mathbb { E } \\Big [ u ( F ( e ) - F ( \\tilde e ) ) \\Big ] . \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} \\Gamma = \\{ 1 \\} \\cup \\Psi _ r \\cup \\bigcup _ { \\psi \\in \\Psi } J _ { \\psi } , \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} a F _ \\gamma ( \\omega ) ( \\lambda \\cdot ) = F _ { a \\gamma } ( a \\omega ( \\lambda \\cdot ) ) , \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} \\rho ( x ) - \\nabla \\cdot \\left ( \\frac { \\mathfrak { D } } { \\bar { \\sigma } ( x ) } \\nabla \\rho ( x ) \\right ) = g ( x ) \\mbox { f o r } x \\in \\Omega , \\\\ \\rho ( x ) = 0 \\mbox { f o r } x \\in \\partial \\Omega . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} \\tau ' ( t ) = 1 + o ( 1 ) , n \\to \\infty \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\overline { \\mathrm { c o } } ( V \\cup \\{ a v _ 1 , a \\geq 0 \\} \\cup \\{ a v _ 2 , a \\geq 0 \\} ) = { \\mathbb R } ^ 2 . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\pi i } \\partial \\overline { \\partial } \\delta = e _ 1 ^ A - 1 2 \\omega _ { \\mathrm { H d g } } , \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} f ( 1 + 2 m _ 0 k ) = f ( 1 - 2 m _ 0 k ) = - 1 - f ( 2 m _ 0 k ) . \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} \\nabla _ v \\varphi = \\imath _ { T ( v ) } \\ast \\varphi \\mbox { a n d } \\nabla _ v * \\varphi = - ( T ( v ) ) ^ \\flat \\wedge \\varphi . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} H ^ * _ G ( M , \\Z ) = H ^ * ( ( M \\times E G ) / G , \\Z ) = H ^ * ( M \\times B G , \\Z ) . \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 2 } ^ q ( 1 , b _ 0 , b _ 1 , b _ 2 ) \\bigr ] = & \\frac { G ( b _ 0 ) } { G ( q + b _ 0 ) } \\frac { G ( q + b _ 0 + b _ 1 ) } { G ( b _ 0 + b _ 1 ) } \\frac { G ( q + b _ 0 + b _ 2 ) } { G ( b _ 0 + b _ 2 ) } \\times \\\\ & \\times \\frac { G ( b _ 0 + b _ 1 + b _ 2 ) } { G ( q + b _ 0 + b _ 1 + b _ 2 ) } . \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} F = ( x _ 0 - a ' _ 1 y _ 1 + \\ldots - a ' _ m y _ m ) x _ 1 ^ { a _ 1 } \\cdots & x _ n ^ { a _ n } + ( y _ 0 - b '' _ 1 x _ 1 + \\ldots - b '' _ n x _ n ) y _ 1 ^ { b _ 1 } \\cdots y _ m ^ { b _ m } \\\\ & - \\sum _ { i = 1 } ^ { r _ 1 } ( \\beta _ { i , 1 } y _ 1 + \\ldots + \\beta _ { i , m } y _ m ) ^ d - \\sum _ { j = 1 } ^ { r _ 2 } ( \\gamma _ { j , 1 } x _ 1 + \\ldots + \\gamma _ { j , n } x _ n ) ^ d . \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} V _ 1 ( z ) ~ : = ~ W _ 1 ( z ) - 1 _ { \\R ^ + } , \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} ( - 1 ) ^ { ( p - 1 ) ( j - n ) + ( n - 1 ) ( i - 1 ) } & = ( - 1 ) ^ { ( n - 1 ) ( i - 1 ) + ( p - 1 ) ( j - 1 ) } ( - 1 ) ^ { - ( p - 1 ) ( n - 1 ) } \\\\ & = ( - 1 ) ^ { ( n - 1 ) ( i - 1 ) + ( p - 1 ) ( j - 1 ) } ( - 1 ) ^ { ( p - 1 ) ( n - 1 ) } . \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} \\varGamma _ { i j } ^ k = \\tfrac { 1 } { 2 } g ^ { k m } ( g _ { m i : j } + g _ { m j : i } - g _ { i j : m } ) . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} \\tilde { Z } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] = \\left ( X _ { s + k } ^ \\prime , V _ { s + k } ^ \\prime \\right ) \\in \\tilde { \\mathcal { D } } _ { s + k } \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} F ^ { k l } W _ { i j ; k l } - \\dot { W } _ { i j } = & - F ^ { k l } h _ { r k } h ^ r _ l W _ { i j } - F ^ { k l } g _ { k l } W _ { i j } + 2 F h _ i ^ k W _ { k j } \\\\ & \\ , - F ^ { k l , r s } W _ { k l ; i } W _ { r s ; j } + 2 F \\epsilon g _ { i j } \\\\ & \\ , - ( 1 - \\epsilon ) \\{ F ^ { k l } h _ { r k } h ^ r _ l - 2 F + F ^ { k l } g _ { k l } \\} g _ { i j } . \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} B _ \\iota ( k , d , r , c ) = [ 0 , c | k | ^ { - \\frac { d } { r } } ] ^ r \\subset \\R ^ r \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} d ( \\Omega _ 0 \\mid _ { V _ 1 } ) = d ( \\Omega _ 0 \\mid _ { V _ 2 } ) = d ( \\Omega _ 0 \\mid _ { V _ 3 } ) = 0 . \\end{align*}"}
-{"id": "9693.png", "formula": "\\begin{align*} & T _ { 1 } \\circ T _ { \\xi _ { 1 } } \\circ \\dots \\circ T _ { \\xi _ { \\ell } } ( I ) \\cap T _ { 1 } \\circ T _ { \\omega _ { 1 } } \\circ \\dots \\circ T _ { \\omega _ { s } } ( I ) = \\emptyset \\mbox { a n d } \\\\ & T _ { 1 } \\circ T _ { \\xi _ { 1 } } \\circ \\dots \\circ T _ { \\xi _ { \\ell } } ( I ) \\cup T _ { 1 } \\circ T _ { \\omega _ { 1 } } \\circ \\dots \\circ T _ { \\omega _ { s } } ( I ) \\subset J . \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} c _ i & = p _ i + ( q _ { 2 i } - p _ i ) + ( q _ { 2 i + 1 } - p _ i ) \\\\ \\intertext { a n d } q _ i & = p _ i + 2 ( c _ i - p _ i ) \\\\ & = p _ i + 2 ( q _ { 2 i } - p _ i ) + 2 ( q _ { 2 i + 1 } - p _ i ) . \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} & B _ { n + 1 } ^ i = B _ { n + 1 } ^ { i + k } + \\\\ & \\sum \\limits _ { \\substack { t = 1 \\\\ 2 \\leq j _ 1 < j _ 2 < \\dots < j _ t \\leq k + 1 } } ^ { k } \\rho _ { ( d + 1 ) n + i + j _ 1 } \\rho _ { ( d + 1 ) ( n - 1 ) + i + j _ 2 } \\dots \\rho _ { ( d + 1 ) ( n - t + 1 ) + i + j _ t } \\ B _ { n + 1 - t } ^ { i + k } , \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} \\left ( z ^ { - 1 } \\alpha ^ { - 1 } q ^ { 1 + n + k } ; q \\right ) _ { \\infty } = ( - 1 ) ^ { n } \\alpha ^ { n } z ^ { n } q ^ { - \\frac { 1 } { 2 } n ( n + 1 ) - k n } \\left ( q ^ { - k } z \\alpha ; q \\right ) _ { - n } \\left ( q ^ { k + 1 } z ^ { - 1 } \\alpha ^ { - 1 } ; q \\right ) _ { \\ ! \\infty } , \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} \\Lambda _ { n - 1 } : = \\left \\{ \\phi ^ { - 1 } ( S ) \\cap \\left ( \\overline { \\mathbb { Z } } _ q ^ * ( ( t ^ { - 1 } ) ) \\right ) ^ d \\ : \\ S \\in \\Sigma _ { n - 1 } \\right \\} , \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} \\mathrm { E x t } ^ 2 ( F , E ) \\simeq H ^ 2 ( F ^ { \\vee } \\otimes E ) = 0 . \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} \\det R ( z ) & = \\exp \\big ( \\frac { 1 } { 2 n } \\sum _ { k = 0 } ^ { n - 1 } \\log | z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta _ k | \\big ) \\\\ & = \\big ( \\prod _ { k = 0 } ^ { n - 1 } | z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta _ k | \\big ) ^ { 1 / 2 n } . \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} { } _ { 1 } \\tilde { \\phi } _ { 1 } ( a ; b ; q , z ) : = ( b ; q ) _ { \\infty } \\ , { } _ { 1 } \\phi _ { 1 } ( a ; b ; q , z ) . \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} \\left \\lVert \\Big ( h ^ { \\Omega ^ \\bullet ( Z _ R , F ) } _ { \\epsilon , t } \\Big ) ^ { - 1 } \\frac { \\partial } { \\partial t } h ^ { \\Omega ^ \\bullet ( Z _ R , F ) } _ { \\epsilon , t } \\Big | _ { t = 0 } \\right \\rVert \\leqslant R ^ { - 1 } . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\sigma _ E ^ { m ( x ) } ( \\kappa ( \\sigma _ E ( x ) ) ) = \\sigma _ E ^ { | x | - 1 } ( \\sigma _ E ( x ) ) = r ( x ) \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} h ( z ) = - \\sum _ { a \\in A _ + } a ^ q t ( a z ) . \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} K = \\Bigg \\{ ( r \\cos \\theta , r \\sin \\theta ) \\ , : \\ , \\theta \\in [ 0 , 2 \\pi ] , \\ r \\in \\big [ 0 , R _ 0 ( \\theta ) \\big ) \\ , \\cup \\bigcup _ { j = 1 } ^ { N ( \\theta ) } \\big ( S _ j ( \\theta ) , \\ , R _ j ( \\theta ) \\big ) \\Bigg \\} . \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} \\max \\limits _ { j \\neq l } | \\langle \\phi _ j , \\phi _ l \\rangle | ^ 2 & = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\max \\limits _ { j \\neq l } \\langle y _ j , y _ l \\rangle \\\\ & \\geq \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\left [ \\frac { 1 } { 2 } \\csc ^ 2 \\left ( \\frac { N \\pi } { 6 ( N - 2 ) } \\right ) - 1 \\right ] . \\\\ \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{gather*} \\pi \\big ( g _ { C } ^ { ( \\alpha ) } \\big ) = \\begin{bmatrix} 1 & 0 \\\\ C ^ { ( \\alpha ) } ( z ) & 1 \\end{bmatrix} = \\begin{bmatrix} 1 & 0 \\\\ ( - 1 ) ^ \\alpha z ^ { \\alpha } C ( z ) & 1 \\end{bmatrix} . \\end{gather*}"}
-{"id": "9022.png", "formula": "\\begin{align*} \\| \\tilde P _ \\pm u \\| ^ 2 = | ( \\tilde P _ \\pm ^ * \\tilde P _ \\pm u , u ) | \\leq \\| \\tilde P _ \\pm ^ * \\tilde P _ \\pm \\| \\| u \\| ^ 2 , \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} P _ { \\mu \\lambda } ^ { \\ast } \\frac { \\partial Q ^ { \\lambda \\nu } } { \\partial x ^ { \\nu } } - \\frac { \\partial P _ { \\mu \\lambda } } { \\partial x ^ { \\nu } } \\overset { \\ast } { \\left . Q ^ { \\lambda \\nu } \\right . } = - \\frac { 1 } { 2 } \\overset { \\ast } { \\left . Q ^ { \\sigma \\tau } \\right . } \\frac { \\partial P _ { \\tau \\sigma } } { \\partial x ^ { \\mu } } . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} \\min _ y ~ \\left \\{ h _ { \\nu } ( y ) + \\frac { 1 } { 2 t } \\| y - x \\| ^ 2 \\right \\} & = \\min _ y \\min _ z ~ \\left \\{ h ( z ) + \\frac { 1 } { 2 \\nu } \\| z - y \\| ^ 2 + \\frac { 1 } { 2 t } \\| y - x \\| ^ 2 \\right \\} \\\\ & = \\min _ z \\left \\{ h ( z ) + \\frac { 1 } { 2 ( t + \\nu ) } \\left \\| z - x \\right \\| ^ 2 \\right \\} , \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\left ( C _ 1 \\mathbb { E } \\left [ \\sum _ { i = 1 } ^ N \\alpha _ { k , i } ^ 2 \\mid \\mathcal { F } _ { k } \\right ] + 2 C _ 2 \\mathbb { E } \\left [ \\sum _ { i = 1 } ^ N \\left | \\alpha _ { k , i } - \\frac { 1 } { k p _ i } \\right | \\mid \\mathcal { F } _ { k } \\right ] \\right ) < \\infty . \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} q _ i = p _ i + \\omega ^ { a _ { h ( i ) } ( i ) } \\left ( \\frac { z } { 2 } \\right ) ^ { h ( i ) } \\sum _ { j = 1 } ^ { k - h ( i ) } z ^ j \\quad ( i \\geq 1 ) . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} v _ r = \\sum _ { i = 0 } ^ { r } ( a _ r ^ i x - b _ r ^ i ) u _ i - \\sum _ { j = r + 1 } ^ { d - 1 } b _ r ^ j u _ j . \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} \\nu ^ { \\pi ^ * } _ n ( y _ n | y _ { n - 1 } ) = \\sum _ { x _ n \\in \\{ 0 , 1 \\} } q _ n ( y _ n | x _ n , y _ { n - 1 } ) \\pi ^ * _ n ( x _ n | y _ { n - 1 } ) . \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} q _ v ^ { \\| \\nu \\| ^ * } \\int _ { H _ v } \\tau ( v , \\nu ) ( x ) d x = ( H _ v \\cap K _ v \\nu ( \\varpi _ v ) K _ v ) \\geqslant ( K _ { H , v } \\nu ( \\varpi _ v ) K _ { H , v } ) \\gg q _ v ^ { 2 \\| \\nu \\| _ H ^ * } , \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} \\left [ H _ 0 - E - \\sum _ { j = 1 } ^ { n } \\lambda _ j | a _ j \\rangle \\langle a _ j | \\right ] | \\psi \\rangle = \\lambda _ { n + 1 } | a _ { n + 1 } \\rangle \\langle a _ { n + 1 } | \\psi \\rangle + | \\chi \\rangle \\ ; . \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} 1 + \\max \\left ( \\frac { n ^ + } { n ^ - } , \\frac { n ^ - } { n ^ + } \\right ) = 1 + \\frac { n + 1 } { n } = 2 + \\frac { 1 } { n } = \\chi _ f . \\end{align*}"}
-{"id": "10072.png", "formula": "\\begin{align*} ( \\beta - 2 ) \\gamma = 2 \\beta . \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\left | \\int \\limits _ { \\mu } ^ { + \\infty } R _ N ( z \\sqrt { t } ) e ^ { - ( z - \\eta ) ^ 2 } d z \\right | \\leqslant k _ N t ^ { - N / 2 } \\int \\limits _ { \\mu } ^ { + \\infty } z ^ { - N } d z = \\tilde { k } _ N \\mu \\sigma ^ { - N } . \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} J ^ \\psi \\varphi _ a ( J ( x ) ) = ( \\varphi \\otimes \\psi ) _ { J ( a ) } ( J ( x ) ) = ( \\varphi \\otimes \\psi ) ( J ( a x ) ) = \\varphi ( a x ) = \\varphi _ a ( x ) . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} \\lambda ( \\Lambda ) = \\lambda ( \\Lambda / \\Lambda ' ) + 2 \\ln ( \\Lambda / \\Lambda ' ) a _ n ( g , D ) \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} F ( \\tau _ 1 + \\tau _ 2 , \\tau _ 5 + \\tau _ 6 ) & = - 2 a _ { i j k } b _ { h j k } e ^ i \\wedge e ^ { n + h } , \\\\ \\overline { F } ( \\tau _ 1 + \\tau _ 2 , \\tau _ 5 + \\tau _ 6 ) & = - 4 a _ { i j k } b _ { i j h } e ^ k \\wedge e ^ { n + h } ; \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\dot { S } ( t ) + q \\big ( S ( t ) \\big ) = 0 t > 0 , S ( 0 ) = s _ 0 . \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} v _ { \\delta ^ { p } } ( x _ { i } ) = u _ { \\delta ^ { p } } ( 0 ) + u _ { \\delta ^ { i , p } } ( 0 ) f _ { i , 0 } ( x _ { i } ) \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} \\mathcal { K } _ s = \\left \\{ Z _ s = \\left ( X _ s , V _ s \\right ) \\in \\overline { \\mathcal { D } _ s } \\left | \\psi _ s ^ { - \\tau } Z _ s = \\left ( X _ s - V _ s \\tau , V _ s \\right ) \\ ; \\forall \\ ; \\tau > 0 \\right . \\right \\} \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} C \\big ( D ^ F _ { X _ \\infty } , D ^ F _ { Y _ { \\R _ + } } \\big ) E _ 0 ( \\phi , \\lambda ) = E _ 0 ( C ( \\lambda ) \\phi , \\lambda ) . \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} ( \\lambda - & \\mid a + n _ { 1 } v _ { k } + t \\mid ^ { 2 } ) c ( a , n _ { 1 } ) = q _ { a - \\delta + \\left ( n _ { 1 } - n _ { 2 } \\right ) v _ { k } } + \\\\ & { \\textstyle \\sum \\limits _ { m = 1 } ^ { n _ { 1 } - n _ { 2 } - 1 } } { \\textstyle \\sum \\limits _ { u \\in \\Gamma ( k ) } } c ( a - u , n - m ) q _ { u + m v _ { k } } + ( \\Psi , e ^ { i ( a + n _ { 1 } v _ { k } } ) . \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} \\tau _ U = \\min \\sum _ { t = 1 } ^ { N _ T } \\frac { \\binom { N _ T } { t } } { d _ { 0 , t } } a _ { 0 , t } , \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} \\mathbf { J } ( t , \\mathbf { x } ) = - \\mathbf { H } ( t , \\mathbf { x } ) \\nabla \\mathbf { u } ( t , \\mathbf { x } ) = - \\Big ( \\sum _ { I = 1 } ^ { d } \\sum _ { J = 1 } ^ { k } H _ { i j I J } ( t , \\mathbf { x } ) \\partial _ { x _ { J } } u _ { I } ( t , \\mathbf { x } ) \\Big ) _ { i = 1 , \\dots , k } ^ { j = 1 , \\dots , d } \\mathbf { x } \\in G . \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} - \\frac { ( p / q ) ^ s \\log ( p / q ) } { \\log ( 1 / p ) } \\log _ { 1 / p } n - \\psi ( n ) ( 1 + O ( ( p / q ) ^ s ) ) - s ( 1 + O ( ( p / q ) ^ s ) ) + O ( ( p / q ) ^ s ) = 0 \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} & \\lim _ { R \\to \\infty } \\ , \\frac { 1 } { R } \\ , \\# \\left \\{ \\lambda \\in \\Lambda , \\ , \\ , t \\le \\lambda \\le t + R , \\ , \\ , \\lambda \\ , ( { \\rm m o d } \\ , a ^ { - 1 } ) \\in I \\right \\} \\\\ & = \\lim _ { R \\to \\infty } \\ , \\frac { 1 } { R } \\ , \\# \\left \\{ \\lambda _ n , \\ , \\ , M ( t , R ) \\le n \\le M ( t , R ) + N ( t , R ) - 1 , \\ , \\ , \\lambda _ n \\ , ( { \\rm m o d } \\ , a ^ { - 1 } ) \\in I \\right \\} & = | I | \\ , a \\ , A , \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a \\ , t ^ 2 ) ^ 2 - t ^ 2 } , a = c o n s t \\neq 0 , c = c o n s t , \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} \\left ( u p \\right ) \\left ( x \\right ) = \\sum \\nolimits _ { n = 0 } ^ { m } \\left ( \\sum \\nolimits _ { \\nu = n } ^ { m } a _ { \\nu } \\left ( u \\right ) _ { \\nu - n } \\right ) x ^ { n } = \\left \\langle u , \\frac { x P _ { n + 1 } \\left ( x \\right ) - \\xi P _ { n + 1 } \\left ( \\xi \\right ) } { x - \\xi } \\right \\rangle , \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} F = - \\frac { 1 } { \\beta } \\ln Z \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} c _ 1 = 1 , \\ ; c _ r = 4 c _ { r - 2 } + 1 r \\geq 3 , d _ 1 = 1 , \\ ; d _ r = 4 d _ { r - 2 } r \\geq 3 , \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} \\bar { x } _ 1 = \\lambda _ 1 \\hat { x } _ 1 + \\lambda _ 3 \\hat { x } _ 3 , \\ \\ \\bar { x } _ 2 = \\lambda _ 2 \\hat { x } _ 2 + \\lambda _ 3 \\hat { x } _ 4 , \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} ( L \\otimes \\iota ) \\hat W = ( \\iota \\otimes \\iota \\otimes \\omega ) ( U _ { 2 3 } ^ * \\hat W _ { 1 2 } . ) = ( 1 \\otimes a ) \\hat W . \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} b _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} P _ { n } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) = \\left ( u _ { 0 } \\theta _ { 0 } P _ { n + 1 } \\right ) \\left ( x \\right ) , n \\geq 0 . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{gather*} f _ j ( x , y ) : = c _ j x ^ { u _ j } y ^ { v _ j } \\prod \\limits _ { i = 1 } ^ { l _ j } ( y ^ a - \\alpha _ { i , j } x ^ b ) ^ { e _ { i , j } } , \\ \\ c _ j \\in L ^ { \\times } . \\end{gather*}"}
-{"id": "8160.png", "formula": "\\begin{align*} I _ { \\delta } ( \\omega ) ( \\cdot ) = W ^ \\delta ( \\cdot , \\omega ) . \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} \\langle a _ { n + 1 } | \\psi \\rangle = { \\langle a _ { n + 1 } | G ^ { ( n ) } | \\chi \\rangle \\over 1 - \\lambda _ { n + 1 } \\langle a _ { n + 1 } | G ^ { ( n ) } | a _ { n + 1 } \\rangle } \\ ; . \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\mathcal J ( \\varphi | _ { f ^ { - 1 } ( Q ) } ) = \\mathcal J ( \\varphi | _ { X _ 1 } ) | _ { f ^ { - 1 } ( Q ) } \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} & [ Q , \\theta _ 0 ( x _ a ) ] = \\sum _ { b \\in \\Delta _ - \\sqcup I , \\alpha \\in \\Delta _ + } c _ { \\alpha , a } ^ b \\theta _ 0 ( x _ b ) ( 1 \\ * x _ \\alpha ^ * ) - 1 \\ * \\sum _ { \\beta , \\gamma \\in \\Delta _ + } c _ { a , \\beta } ^ { \\gamma } \\chi ( x _ { \\gamma } ) x _ { \\beta } ^ * \\\\ & [ Q , 1 \\ * x _ { \\alpha } ^ * ] = - 1 \\ * \\frac { 1 } { 2 } \\sum _ { \\beta , \\gamma \\in \\Delta _ + } c _ { \\beta , \\gamma } ^ { \\alpha } x _ { \\beta } ^ * x _ { \\gamma } ^ * \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} \\psi _ { j + 1 } ( x ) = \\psi _ j ( x ) + a _ { j + 1 } [ f ( x _ { j + 1 } ) + \\langle \\nabla f ( x _ { j + 1 } ) , x - x _ { j + 1 } \\rangle + p ( x ) ] . \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} \\mathbf { F } \\cdot \\frac { \\partial \\mathbf { G } ^ { \\ast } } { \\partial t } + \\mathbf { F } ^ { \\ast } \\cdot \\frac { \\partial \\mathbf { G } } { \\partial t } + 4 \\pi \\mathbf { j } \\cdot \\left ( \\mathbf { F } + \\mathbf { F } ^ { \\ast } \\right ) = \\frac { c } { i } \\operatorname { d i v } \\left ( \\mathbf { F } \\times \\mathbf { F } ^ { \\ast } \\right ) \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} u ^ 1 ( f ^ * ) = \\bar { r } ^ 1 ( f ^ * ) + \\frac { \\beta } { 1 - \\beta } \\sum _ { s \\in S } p _ s \\bar { r } ^ 1 ( s , a _ s ^ 1 ) \\textbf { 1 } _ { | S | } . \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} \\left [ T _ { ( A , 0 ) } , T _ { ( B , 0 ) } \\right ] & = C _ { A B } { } ^ { C } T _ { ( C , 0 ) } , \\quad \\left [ T _ { ( A , 0 ) } , T _ { ( B , 1 ) } \\right ] = C _ { A B } { } ^ { C } T _ { ( C , 1 ) } , \\\\ \\quad \\left [ T _ { ( A , 1 ) } , T _ { ( B , 1 ) } \\right ] & = - C _ { A B } { } ^ { C } T _ { ( C , 0 ) } . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} \\left ( x , v \\right ) \\in \\Omega _ { + } = \\left \\{ e _ { \\pm } > - \\min \\beta , v > 0 \\right \\} , \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} Z ( s , f , \\chi ) = \\frac { ( 1 - q ^ { - 1 } ) ( q ^ { - 1 0 - 2 0 s } ) } { ( 1 - q ^ { - 9 - 2 0 s } ) } . \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} d _ 2 = \\begin{pmatrix} 0 \\\\ X \\end{pmatrix} , g _ 2 = \\begin{pmatrix} 0 \\\\ Y \\end{pmatrix} , \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} \\left ( \\Psi ^ { \\ast } \\left ( \\partial _ { t } p \\right ) + \\Psi \\left ( z - B p \\right ) \\right ) = \\partial _ { t } p : \\left ( z - B p \\right ) \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} \\phi ( x , y ) + \\phi ( y , x ) = 2 p _ e p _ d ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} | S | ^ { - 1 } S w = ( S ^ * S ) ^ { - 1 / 2 } S w = \\lambda ^ { - 1 } ( S ^ * S ) ^ { 1 / 2 } w = \\lambda ^ { - 1 } | S | w \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} \\begin{aligned} \\dfrac { d ^ { B ^ { ( 1 ) } _ { n + 1 } } _ { k , l } ( z ) } { ( z - q ^ { \\mathtt { h } ^ \\vee } ) ^ { \\delta _ { k l } } } & = D _ { k , l } ( z ) \\times D _ { k , l ^ * } ( z ) = D _ { k , l } ( z ) \\times D _ { k ^ * , l } ( z ) \\\\ & = D _ { k ^ * , l ^ * } ( z ) \\times D _ { k , l ^ * } ( z ) = D _ { k ^ * , l ^ * } ( z ) \\times D _ { k ^ * , l } ( z ) \\end{aligned} \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} \\phi _ k ( a _ { k ; j } ) \\neq 0 \\phi _ k \\big ( ( a _ { k ; i } a _ { k ; i } ^ * ) ^ 2 \\big ) = 1 . \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} R _ k ( x ) = \\frac { x ^ { k - 2 } \\left ( n _ k ( x ) - \\sqrt { 1 - 4 x } \\right ) } { 2 d _ k ( x ) } , \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} \\frac { H \\left ( y \\right ) } { g \\left ( y \\right ) } = \\frac { H \\left ( a \\right ) } { g \\left ( a \\right ) } . \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\sum _ { a \\in \\{ 0 , 1 \\} } \\tilde X _ s ^ a = \\sum _ { b \\in \\{ 0 , 1 \\} } \\tilde Y _ t ^ b = I s \\in S , t \\in T , \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} \\mathcal B _ \\Omega f ( x ) = \\int _ { \\Omega } f ( y ) \\bigg ( \\frac { x - y } { | x - y | ^ { n } } \\int _ { | x - y | } ^ \\infty \\omega \\Big ( y + r \\frac { x - y } { | x - y | } \\Big ) \\zeta ^ { n - 1 } \\ , d r \\bigg ) \\ , d y \\hbox { f o r $ x \\in \\Omega $ , } \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} a ( t ) = t ^ { - \\alpha } ( 1 + o ( 1 ) ) , a ' ( t ) = O ( t ^ { - \\alpha - 1 } ) , t \\to \\infty , \\alpha > 0 . \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} \\widehat { A } _ 1 \\xi & = - \\frac { 1 0 6 } { 8 5 } \\xi ^ { ( 1 ) } ( - 1 ; 1 ) - \\frac { 4 } { 8 5 } \\xi ^ { ( 1 ) } ( 3 ; 1 ) - 3 \\xi ( - 1 ) \\\\ & = \\frac { 1 0 6 } { 8 5 } \\xi ( - 2 ) + \\frac { 1 4 9 } { 8 5 } \\xi ( - 1 ) + \\frac { 4 } { 8 5 } \\xi ( 2 ) - \\frac { 4 } { 8 5 } \\xi ( 3 ) . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} f \\left ( q , \\theta \\right ) = f \\left ( s , \\theta \\right ) e ^ { \\int _ { s } ^ r \\varphi \\left ( \\sigma , \\theta \\right ) \\ , d \\sigma } \\leq f \\left ( s , \\theta \\right ) e ^ { 2 \\rho \\left ( \\alpha \\rho + C \\right ) } , \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } f _ N ^ { ( 1 ) } ( t , x , v ) = f ( t , x , v ) \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} f _ i ( 1 ) = f _ i ( 0 ) = f _ j ( 0 ) = f _ j ( 1 ) . \\end{align*}"}
-{"id": "9968.png", "formula": "\\begin{align*} ( x \\cdot y ) _ z = \\frac { 1 } { 2 } \\left ( d ( z , x ) + d ( z , y ) - d ( x , y ) \\right ) . \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} \\beta _ j = ( d _ j \\Delta - g _ j ) \\sigma ^ { - 1 } , \\gamma _ j ^ { t r } = - ( d _ j + g _ j \\Delta ) \\sigma ^ { - 1 } , j \\geq 2 ; \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } W _ i ^ { \\delta ( n ) } = W _ i , ~ i = 1 , 2 , \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} | T ^ x \\cap H ' | \\ge | T ^ x | + | H ' | - | T ^ x \\cup H ' | \\ge ( . 2 9 6 2 6 2 + . 7 0 9 - 1 ) \\binom { n } 3 > 0 . 0 0 5 \\binom { n } 3 . \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} \\sup _ { \\| b \\| _ { \\mathcal B } = 1 } \\sigma ^ 2 ( b ) \\ , \\le \\ , \\inf _ { R > 0 } \\alpha ( R ) . \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{gather*} \\Gamma = \\begin{bmatrix} z ^ { k } & 0 \\\\ ( z ^ { - k } C ( z ) ) _ { - } & z ^ { - k } \\end{bmatrix} = \\begin{bmatrix} z ^ { k } & 0 \\\\ \\gamma _ { 0 } z ^ { - 1 } + \\gamma _ { 1 } z ^ { - 2 } + \\cdots & z ^ { - k } \\end{bmatrix} , \\end{gather*}"}
-{"id": "10046.png", "formula": "\\begin{align*} \\varphi ( w ( z ) ) & = \\varphi \\Bigg ( \\tfrac { \\displaystyle h ( z ) - 1 } { \\displaystyle h ( z ) + 1 } \\Bigg ) \\\\ & = 1 + \\tfrac { \\displaystyle 1 } { \\displaystyle 2 } Q _ 1 d _ 1 z + \\tfrac { \\displaystyle 1 } { \\displaystyle 2 } Q _ 1 \\Big ( d _ 2 - \\tfrac { \\displaystyle d _ 1 ^ 2 } { \\displaystyle 2 } \\Big ) z ^ 2 + \\tfrac { \\displaystyle 1 } { \\displaystyle 4 } Q _ 2 d _ 1 ^ 2 z ^ 2 + \\cdots , \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} d s ^ 2 = d x ^ 2 + d y ^ 2 + d z ^ 2 \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} - \\beta ( g ) = \\frac { 1 } { W } \\frac { \\partial W ( g ( \\Lambda ) ) } { \\partial \\ln \\Lambda } \\sim - a _ 2 ( g , D ) = - \\frac { 1 } { 6 } \\int _ M d V \\ , T r ( g R ) - \\frac { 1 } { 2 } \\int _ M d V \\ , T r ( \\Delta g ) \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\sum _ { m = k } ^ { n } \\binom { n } { m } \\sum _ { j = 0 } ^ { m } ( - 1 ) ^ { j } \\binom { m } { j } \\left ( \\frac { s } { s + n - m + j } \\right ) ^ { 2 } & = f _ { n , k } ( s ) - s f ^ { ( 1 ) } _ { n , k } ( s ) \\\\ & = \\prod _ { j = n - k + 1 } ^ { n } \\left ( \\frac { j } { s + j } \\right ) \\left [ 1 + \\sum _ { j = n - k + 1 } ^ { n } \\frac { s } { s + j } \\right ] \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} p _ { 1 } = p _ { 2 } = . . . = p _ { s } > p _ { s + 1 } \\geq p _ { s + 2 } \\geq . . . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\left \\vert x _ { i } - \\dfrac { p _ { i } } { q } \\right \\vert \\leq \\varepsilon t _ { i } \\\\ \\varepsilon q \\leq t _ { i } \\end{array} \\right \\vert i = 1 , 2 , . . . , n \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} \\| x _ i ^ { k + 1 } - x _ i ^ * \\| ^ 2 & = \\| \\Pi _ { K _ i } [ x _ i ^ k - \\alpha _ k F _ i ( x _ i ^ k , N { \\hat v _ i ^ k } ) ] - x ^ * _ i \\| ^ 2 \\cr & = \\| \\Pi _ { K _ i } [ x _ i ^ k - \\alpha _ k F _ i ( x _ i ^ k , N { \\hat v _ i ^ k } ) ] - \\Pi _ { K _ i } [ x ^ * _ i - \\alpha _ k F _ i ( x ^ * _ i , \\bar { x } ^ * ) ] \\| ^ 2 \\cr & \\leq \\| x _ i ^ k - x ^ * _ i - \\alpha _ k ( F _ i ( x _ i ^ k , N { \\hat v _ i ^ k } ) - F _ i ( x ^ * _ i , \\bar { x } ^ * ) ) \\| ^ 2 . \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} \\Phi ^ { \\bf a } = \\lim _ { k _ 0 < k \\to \\infty } \\big ( F ^ { \\bf a } _ { k + 1 } / F ^ { \\bf a } _ { k } \\big ) , \\ , \\ , \\ , \\ , \\ , \\ , F ^ { \\bf a } _ k \\neq 0 \\ , \\ , \\ , \\ , \\ , \\ , k > k _ 0 . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} \\eta ( x ) = g \\bigl ( \\phi ( x ) \\bigr ) , \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} S _ * ^ { \\epsilon , R } ( t ) : = \\big \\{ ( x , u ) \\in S ( t ) : \\| x - x ^ * ( t ) \\| \\leq \\epsilon , \\| u - u ^ * ( t ) \\| \\leq R ( t ) \\big \\} . \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} t _ { 2 } + t _ { 3 } = d + \\sum _ { r \\geq 5 } ( r - 4 ) t _ { r } . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ { \\infty } \\ ! q ^ { n ( n - 1 ) / 2 } \\left [ z ^ { n } A _ { q } \\ ! \\left ( z q ^ { n + m } \\right ) \\ ! A _ { q } \\ ! \\left ( z q ^ { n - m - 1 } \\right ) - z ^ { 1 - n } A _ { q } \\ ! \\left ( z ^ { - 1 } q ^ { n + m } \\right ) \\ ! A _ { q } \\ ! \\left ( z ^ { - 1 } q ^ { n - m - 1 } \\right ) \\right ] = 0 , \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} \\left ( c q ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( - \\sqrt { q } ; \\sqrt { q } \\right ) _ { n } q ^ { \\binom { n } { 2 } } c ^ { n } } { \\left ( \\sqrt { q } , \\sqrt { c q } , - \\sqrt { c q } ; \\sqrt { q } \\right ) _ { n } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } - n } c ^ { n } A _ { q } ^ { 2 } \\left ( - c q ^ { n - 1 / 2 } \\right ) } { \\left ( q ; q \\right ) _ { n } } . \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{gather*} Z ( s , f , \\chi , \\Delta _ 3 ) = \\sum _ { n = 1 } ^ { \\infty } \\int _ { \\pi ^ { n } O _ v ^ \\times \\times \\pi ^ { n } O _ v ^ \\times } \\chi ( a c \\ f ( x , y ) ) | f ( x , y ) | ^ s | d x d y | = \\\\ \\sum _ { n = 1 } ^ { \\infty } q ^ { - 2 n - 4 n s } \\int _ { O _ v ^ { \\times 2 } } \\chi ( a c \\ ( \\pi ^ { n } y ^ 3 - x ^ 2 ) ^ 2 + \\pi ^ { 4 n } x ^ 4 y ^ 4 ) | ( \\pi ^ { n } y ^ 3 - x ^ 2 ) ^ 2 + \\pi ^ { 4 n } x ^ 4 y ^ 4 | ^ s | d x d y | . \\end{gather*}"}
-{"id": "9525.png", "formula": "\\begin{align*} C a p ( z , U _ { + } \\cup U _ { - } ) = \\frac { C a p ( \\zeta , U _ { + } ) + C a p ( \\zeta , U _ { - } ) } { 1 + d ( z , \\zeta ) [ C a p ( \\zeta , U _ { + } ) + C a p ( \\zeta , U _ { - } ) ] } . \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} X ^ l = G \\cap O . \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} T ^ * _ { v } = 3 2 C ^ 3 ( \\norm { a } _ { V _ { \\delta + \\varepsilon } } + C \\max _ { t \\in [ 0 , T ] } \\norm { f } _ { L ^ p ( \\Omega ) ^ 2 } ) ^ { - 1 / \\varepsilon } , T ^ * _ { \\tau } = 3 2 C ^ 3 ( \\norm { b } _ { \\hat { V } _ { \\delta + \\varepsilon } } + C \\max _ { t \\in [ 0 , T ] } \\norm { g } _ { L ^ p ( \\Omega ) ^ 2 } ) ) ^ { - 1 / \\varepsilon } . \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} \\mathcal { A } ^ + = \\left \\{ \\begin{aligned} & ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in [ 0 , \\infty ) \\times \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } \\textnormal { s u c h t h a t } \\\\ & \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime ( \\tau ; 0 ) \\right ) > 0 \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\begin{cases} \\Box _ g u : = \\nabla ^ \\alpha \\partial _ \\alpha u = F _ p ( u ) , ( t , x ) \\in M , \\\\ u ( 0 , x ) = u _ 0 ( x ) , \\partial _ t u ( 0 , x ) = u _ 1 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} L ( p , q , \\beta ) = A J _ 0 - J _ q \\leq A J _ 0 - J _ 0 ^ { - q + 1 } J _ 1 ^ q . \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} ( J ^ \\psi \\sigma ) ( J ( x ) ) = \\sigma ( x ) . \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} \\left \\{ \\begin{tabular} { l } $ U _ { 0 } = \\dfrac { h _ { 2 } } { k _ { 2 } } $ , $ U _ { 1 } = \\dfrac { h _ { 2 } + h _ { 1 } } { k _ { 2 } + k _ { 1 } } $ , . . . , $ U _ { i } = \\dfrac { h _ { 2 } + i h _ { 1 } } { k _ { 2 } + i k _ { 1 } } $ , . . . \\\\ $ V _ { 0 } = \\dfrac { h _ { 1 } } { k _ { 1 } } $ , $ V _ { 1 } = \\dfrac { h _ { 1 } + h _ { 2 } } { k _ { 1 } + k _ { 2 } } $ , . . . , $ V _ { j } = \\dfrac { h _ { 1 } + j h _ { 2 } } { k _ { 1 } + j k _ { 2 } } $ , . . . \\end{tabular} \\right . \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} \\rho ( \\alpha ) = - \\frac { 1 } { \\log ( p / q ) } \\log \\left ( \\frac { \\alpha \\log ( 1 / q ) - 1 } { 1 - \\alpha \\log ( 1 / p ) } \\right ) , & & \\beta ( \\alpha ) = \\alpha \\log ( T ( \\rho ( \\alpha ) ) ) - \\rho ( \\alpha ) , \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} u = \\left ( \\begin{array} { c c } a ^ + & s ^ * b ^ + \\\\ b s & a \\\\ \\end{array} \\right ) \\ , . \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} F ^ { \\prime \\prime } \\left ( 0 \\right ) = \\frac { 8 \\pi ^ { 2 } } { P _ { 0 } ^ { 2 } } \\int \\frac { \\mu _ { + } ^ { \\prime } + \\mu _ { - } ^ { \\prime } } { v ^ { 2 } } d v > 0 . \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} b _ { 2 , 1 } \\left ( z \\right ) & = \\varphi _ { z _ { 0 } } \\left ( z \\right ) - b _ { 1 , 1 } \\varphi _ { z _ { 1 } } \\left ( z \\right ) , \\\\ b _ { 2 , 2 } \\left ( z \\right ) & = \\varphi _ { z _ { 1 } } \\left ( z \\right ) , \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} \\cosh R ( s ) = \\frac { 1 } { \\sin \\Delta \\theta _ s } , \\sinh R ( s ) = - \\frac { \\cos \\Delta \\theta _ s } { \\sin \\Delta \\theta _ s } . \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} \\omega _ B ( ( T \\underline { \\smash { \\pi } } ) \\xi _ 1 , ( T \\underline { \\smash { \\pi } } ) \\xi _ 2 ) = \\pi ( \\omega _ S ( \\xi _ 1 , \\xi _ 2 ) ) , \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\kappa _ { \\gamma } ( u ) = \\frac { f _ { 1 } ^ { \\prime } ( u ) } { f _ { 2 } ( u ) } , \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} P _ g u = f | u | ^ { 2 ^ \\sharp - 2 } u \\hbox { ~ ~ i n ~ ~ } M . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} P _ 1 u _ 1 = f _ 1 \\ \\ \\Omega _ 1 , P _ 2 u _ 2 = f _ 2 \\ \\ \\Omega _ 2 , \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} V _ N = & 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\mathcal { N } ( 0 , \\ , 4 \\log 2 ) + \\log X _ 1 + \\log X _ 2 + \\log X _ 3 + \\\\ & + \\log Y + \\log Y ' + o ( 1 ) , \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} \\mathbf { G } ^ { \\prime } = \\gamma \\left ( \\mathbf { G } - \\dfrac { i } { c } \\mathbf { v \\times F } \\right ) - \\left ( \\gamma - 1 \\right ) \\frac { \\mathbf { v } \\cdot \\mathbf { G } } { v ^ { 2 } } \\mathbf { v } , \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} \\dot { U } ( t ) + \\mathcal { K } \\big ( U ( t ) \\big ) = 0 t > 0 , U ( 0 ) = U _ 0 . \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} F ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) & = a _ { 1 } f _ { 1 } ( x _ { 1 } ) + a _ { 2 } f _ { 2 } ( x _ { 2 } ) + a _ { 3 } f _ { 3 } ( x _ { 3 } ) \\\\ & + a _ { 4 } f _ { 1 } ( x _ { 1 } ) f _ { 2 } ( x _ { 2 } ) + a _ { 5 } f _ { 1 } ( x _ { 1 } ) f _ { 3 } ( x _ { 3 } ) + a _ { 6 } f _ { 2 } ( x _ { 2 } ) f _ { 3 } ( x _ { 3 } ) \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} F P S ( l / k ) = z _ { l , 0 } + 2 \\sum \\limits _ { q = 1 } ^ { \\frac { l } { 2 } - 1 } { z _ { l , q } \\cos ( ( q ) \\frac { { 2 k \\pi } } { l } ) + } 2 \\cos ( ( \\frac { l } { 2 } ) \\frac { { 2 k \\pi } } { l } ) \\sum \\limits _ { t = 0 } ^ { \\frac { l } { 2 } - 1 } { y _ t y _ { t + \\frac { l } { 2 } } } \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} \\rho ( x ) = \\d ( y ) , \\ : y \\in \\phi ^ { - 1 } ( x ) \\cap \\Q , \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} Y = - 3 9 . 6 9 + 0 . 8 3 2 \\cdot A i r . F l o w + 0 . 5 7 4 \\cdot W a t e r . T e m p - 0 . 0 6 1 \\cdot A c i d . C o n c + \\varepsilon \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} N _ 0 + N _ 1 = M . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} P ( z ) & = V \\frac { ( y - 1 ) ^ 2 } { h ^ 2 } - D \\frac { y - 1 } { h } + Q \\\\ & = \\frac { 1 } { h ^ 2 } V y ^ 2 - \\left ( 2 \\frac { 1 } { h ^ 2 } V + \\frac { 1 } { h } D \\right ) y + \\left ( \\frac { 1 } { h ^ 2 } V + \\frac { 1 } { h } D + Q \\right ) , \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\xi ( t ) ( f g ) = \\xi ( t ) ( f ) \\xi ( t ) ( g ) , \\quad \\ \\mbox { a . e . } t \\in [ t _ 0 , T ] . \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} \\partial _ x ^ k u , \\partial _ x ^ k v \\in L ^ { \\infty } _ x ( 0 , L ; H ^ { \\frac { 1 - k } { 3 } } ( 0 , T ) ) , k = 0 , 1 , 2 . \\end{align*}"}
-{"id": "10094.png", "formula": "\\begin{align*} F [ x : y : z ] = [ a _ { 1 1 } x + a _ { 1 2 } y + a _ { 1 3 } z : a _ { 2 1 } x + a _ { 2 2 } y + a _ { 2 3 } z : a _ { 3 1 } x + a _ { 3 2 } y + a _ { 3 3 } z ] , \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} G _ 1 = \\left [ \\begin{array} { c c } I _ { n _ 1 - 1 } & 0 \\end{array} \\right ] , F _ 1 = \\left [ \\begin{array} { c c } 0 & I _ { n _ 1 - 1 } \\end{array} \\right ] , G _ 2 = \\left [ \\begin{array} { c c } 0 & I _ { n _ 2 - 1 } \\end{array} \\right ] , F _ 2 = \\left [ \\begin{array} { c c } I _ { n _ 2 - 1 } & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } } { \\left ( q , q ; q \\right ) _ { n } } = \\frac { 1 } { \\left ( q ; q \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { 1 } { \\lambda ^ t } M ^ t \\vec v _ { 0 } = \\lambda ' ( \\vec v + \\frac { 1 } { \\lambda } \\vec w ) \\ , , \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} \\widehat { \\gamma } _ { W , c } & = \\sum _ { w \\in W } ( - 1 ) ^ w \\mathcal { F } _ \\zeta [ \\delta _ w ] - \\mathcal { F } _ \\zeta [ c \\delta _ 0 ] = \\mathcal { F } _ \\zeta \\left [ \\sum _ { w \\in W } ( - 1 ) ^ w \\delta _ w - c \\delta _ 0 \\right ] = \\mathcal { F } _ \\zeta \\left [ \\sum _ { w \\in W } ( - 1 ) ^ w \\delta _ w - c \\delta _ 0 \\right ] \\\\ & = \\mathcal { F } _ \\zeta \\left [ \\gamma _ { W , c } \\right ] . \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} \\mathcal { U } _ s ^ \\eta = \\left \\{ Z _ s = \\left ( X _ s , V _ s \\right ) \\in \\overline { \\mathcal { D } _ s } \\left | \\inf _ { 1 \\leq i < j \\leq s } \\left | v _ i - v _ j \\right | > \\eta \\right . \\right \\} \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} \\varphi \\left ( j _ { 1 } \\right ) + \\cdots + \\varphi \\left ( j _ { r } \\right ) = r / 2 ( \\operatorname { m o d } r ) . \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} \\sigma _ { k } ( n ) = \\sum _ { 0 < d | n } d ^ { k } . \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} \\big \\| \\mathcal { N } ( U ^ { 1 } ) - \\mathcal { N } ( U ^ { 2 } ) \\big \\| _ { \\mathcal { H } _ { \\kappa } } & = \\Big ( \\rho \\int _ { \\Omega } \\big | E ( | v ^ { 1 } | ) v ^ { 1 } - E ( | v ^ { 2 } | ) v ^ { 2 } \\big | ^ { 2 } \\ , \\mathrm { d } x \\Big ) ^ { 1 / 2 } \\\\ & \\leq L \\sqrt { \\rho } \\| v ^ { 1 } - v ^ { 2 } \\| _ { L ^ { 2 } } \\leq L \\big \\| U ^ { 1 } - U ^ { 2 } \\| _ { \\mathcal { H } _ { \\kappa } } \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} d ^ F \\omega = d ^ { F , * } \\omega = 0 . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} \\widehat { h } _ { j } = \\widehat { \\varphi } _ { j } - \\widehat { P _ { G ^ { ( j ) } } ( \\varphi _ { j } ) } , 1 \\leq j \\leq N . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\gamma ( u ) = \\phi ( u ) + \\lambda \\cos \\left ( \\frac { u } { c } \\right ) e _ { n + 1 } , \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} A _ { J _ \\alpha N } X = J _ \\alpha ( A _ N X ) - \\frac { 1 } { 4 n } \\left [ \\overline \\theta _ \\alpha ( N ) X + \\overline \\theta _ \\alpha ( J _ \\alpha N ) J _ \\alpha X \\right ] , \\alpha = 2 , 3 , \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial \\tau } = - \\frac { 1 } { 2 } \\zeta ' ( 0 , D ) - \\tau \\zeta ( 0 , D ) \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} s ^ { \\lambda } = \\sum _ { \\alpha \\in A _ r } s ^ { \\lambda } _ { \\alpha } I ^ { \\alpha } , \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} \\langle U , U ^ { \\ast } \\rangle _ { \\mathcal { H } } & = \\int _ { \\Omega } \\big ( \\rho v _ { i } v _ { i } ^ * + a \\theta \\theta ^ * + A _ { i K L j } u _ { i , K } u ^ * _ { j , L } + C _ { i I J K L j } u _ { i , J I } u ^ * _ { j , L K } + K _ { I J } \\tau _ { , J } \\tau ^ * _ { , I } \\big ) \\mathrm { d } x \\\\ & + \\int _ { \\Omega } \\big ( M _ { i J K L } u _ { i , K J } \\tau ^ * _ { , L } + M _ { j L K I } u ^ * _ { j , K L } \\tau _ { , I } \\big ) \\mathrm { d } x , \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} S _ { l ( k ) } \\varphi \\left ( x \\right ) = \\sum _ { j = 0 } ^ { l ( k ) - 1 } \\varphi \\circ \\theta ^ { j } \\left ( x \\right ) = k \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} \\overrightarrow { \\mathcal { L } } = \\frac { 1 } { 4 \\pi c } \\mathbf { r \\times } \\left ( \\mathbf { D } \\times \\mathbf { B } \\right ) \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} \\Phi _ \\mathrm u = \\left [ \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ] , \\Phi _ \\mathrm i = \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ] , \\Phi _ \\mathrm r = \\left [ \\begin{array} { c c } 0 & \\j \\\\ - \\j & 0 \\end{array} \\right ] , \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} \\| ( \\sum _ { n , j = 1 } ^ \\infty | M _ { n , j } x _ j | ^ 2 ) ^ \\frac 1 2 \\| _ { p } \\leq C { p ^ 2 c _ p ^ 2 } \\| ( \\sum _ j | x _ j | ^ 2 ) ^ \\frac 1 2 \\| _ { p } \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} \\vec w : = \\sum _ { m = 0 } ^ { \\infty } \\frac { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } { \\lambda ^ { m } } \\frac { 1 } { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } M ^ { m } \\vec u = \\sum _ { m = 0 } ^ { \\infty } \\frac { 1 } { \\lambda ^ m } M ^ m \\vec { u } . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{gather*} B = \\mathbb { C } [ [ c _ { k } , d _ { k } , e _ { k } ] ] _ { k \\in \\mathbb { Z } } \\end{gather*}"}
-{"id": "606.png", "formula": "\\begin{align*} \\rho ^ { \\ast } & = \\sup _ { \\pi } \\liminf _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { t = 1 } ^ { N } I ( X _ t , S _ { t - 1 } ; Y _ { t } | Q _ { t - 1 } ) , \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} a ^ { 2 } \\nabla _ { \\nu } b ^ { 2 } \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ { n - 1 } } } , 2 ) ^ { - r _ { n } } a ^ { 2 } \\sharp _ { \\nu } b ^ { 2 } + ( a - b ) ^ 2 - \\sum _ { k = 0 } ^ { n - 1 } r _ { k } \\big [ a ^ { \\frac { m _ k } { 2 ^ k } } b ^ { 1 - \\frac { m _ k } { 2 ^ k } } - a ^ { \\frac { m _ k + 1 } { 2 ^ k } } b ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } , \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} \\Gamma = \\{ m \\omega + n \\omega ' : m , n \\in \\mathbb Z \\} \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} u ( y ) = \\sum \\limits _ { k = 0 } ^ { \\infty } \\alpha _ k p _ k ( y ) \\mbox { w i t h } \\sum \\limits _ { k = 0 } ^ { \\infty } | \\alpha _ k | ^ 2 \\leq \\bar { c } ^ 2 . \\end{align*}"}
-{"id": "10042.png", "formula": "\\begin{align*} n | a _ n | \\big [ 1 + ( n - 1 ) ( \\lambda - \\mu + n \\lambda \\mu ) \\big ] = B _ n + p _ { n - 1 } + p _ 1 B _ { n - 1 } + \\cdots + p _ { n - 2 } B _ 2 . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} \\sum _ { j = - \\infty } ^ { \\infty } z ^ { j } q ^ { j ( j - 1 ) / 2 } A _ { q } \\left ( z q ^ { j + ( m - n ) } \\right ) A _ { q } \\left ( z q ^ { j - ( m - n ) } \\right ) = \\frac { 4 \\left ( q ; q \\right ) _ { \\infty } ^ { 2 } } { \\left ( - 1 ; q ^ { 1 / 2 } \\right ) _ { \\ ! \\infty } ^ { \\ ! 2 } \\left ( q ^ { 1 / 2 } ; q \\right ) _ { \\ ! \\infty } ^ { \\ ! 2 } } \\theta _ { q } \\left ( - z \\right ) \\delta _ { m , n } \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} R _ { i j k l } = \\sigma \\{ h _ { i k } h _ { j l } - h _ { i l } h _ { j k } \\} + \\bar { R } _ { \\alpha \\beta \\gamma \\delta } x _ i ^ \\alpha x _ j ^ \\beta x _ k ^ \\gamma x _ l ^ \\delta . \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} u _ { i } = 0 , u _ { i , A } = 0 , \\tau = 0 \\Gamma \\times ( 0 , \\infty ) , \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} F = x _ 0 ^ 2 x _ { n + 1 } + x _ 0 q ( x _ 1 , \\dotsc , x _ { n + 1 } ) + c ( x _ 1 , \\dotsc , x _ { n + 1 } ) \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} F ( z , v ) : = \\sum _ { n } \\sum _ { m } T _ { n } \\mathbb { P } \\{ D _ { n } = m \\} \\frac { z ^ { n } } { n ! } v ^ { m } = \\sum _ { n } \\sum _ { m } \\mathbb { P } \\{ D _ { n } = m \\} \\frac { z ^ { n } } { n } v ^ { m } . \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { A } & = \\delta ( G ) G ^ { - 1 } + G A G ^ { - 1 } \\\\ \\tilde { B } & = \\sigma ( G ) B G ^ { - 1 } \\end{aligned} \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} \\norm { \\zeta ( t ) } ^ 2 & \\leq \\left ( \\norm { b } + \\int _ 0 ^ t \\norm { g ( s ) } ^ 2 d s \\right ) e ^ { 2 t } = : B _ { L ^ 2 , 1 } ^ { \\zeta } ( t ) \\\\ \\int _ 0 ^ t \\norm { \\nabla \\zeta ( s ) } ^ 2 d s & \\leq \\frac { 1 } { 2 } \\left ( \\norm { b } + \\int _ 0 ^ t \\norm { g ( s ) } ^ 2 d s + \\int _ 0 ^ t \\norm { \\zeta ( s ) } ^ 2 d s \\right ) \\\\ & \\leq \\frac { 1 } { 2 } \\left ( \\norm { b } + \\int _ 0 ^ t \\norm { g ( s ) } ^ 2 d s + \\int _ 0 ^ t B _ { L ^ 2 } ^ { \\zeta } ( s ) d s \\right ) = : B _ { L ^ 2 , 2 } ^ { \\zeta } ( t ) . \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} \\{ f _ i \\in C ^ \\infty ( [ 0 , 1 ] , \\mathbb { R } ) , f _ i ( 1 ) = f _ i ( 0 ) = f _ j ( 0 ) = f _ j ( 1 ) \\} . \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} { \\cal P } ^ { A . M } _ { 0 , n } = \\Big \\{ \\pi _ t ( d x _ t | y _ { t - M } ^ { t - 1 } ) : ~ t = 0 , \\ldots , n \\Big \\} . \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} F \\left ( x \\right ) = \\left ( f _ { 1 } \\left ( x _ { 1 } \\right ) , \\dots , f _ { n } \\left ( x _ { n } \\right ) \\right ) ^ { T } \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} D _ { x } \\mathcal { T } g ( t , x ) & = D _ { x } \\left [ \\int _ { - \\infty } ^ { - 3 ^ { \\frac { 2 } { \\alpha } } } \\left | ( - \\Delta ) ^ { { c _ 1 } / 2 } T _ { t - s } ^ { \\alpha , \\beta } g ( s , \\cdot ) ( x ) \\right | _ { H } ^ { 2 } d s \\right ] ^ { 1 / 2 } \\\\ & \\leq \\left [ \\int _ { - \\infty } ^ { - 3 ^ { \\frac { 2 } { \\alpha } } } \\left | D _ { x } ( - \\Delta ) ^ { { c _ 1 } / 2 } T _ { t - s } ^ { \\alpha , \\beta } g ( s , \\cdot ) ( x ) \\right | _ { H } ^ { 2 } d s \\right ] ^ { 1 / 2 } , \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} \\mathcal { Q } ( L / K ) = \\max \\{ \\mathrm { N } _ { K / \\mathbb { Q } } \\mathfrak { f } _ { \\chi } : \\chi \\in \\widehat { A } \\} . \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} P _ { 1 } ( z ) = p _ { 0 , 0 } e ^ { \\frac { \\lambda } { \\xi } z } z ^ { - ( \\frac { \\mu } { \\xi } - 1 ) } \\left [ - \\frac { \\gamma } { \\xi } B ( z ) + \\frac { ( \\mu - \\xi ) \\gamma } { \\xi \\lambda } C ( z ) + \\frac { D ( z ) } { A } \\right ] , \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} \\frac { p ^ { m + 1 } ( p - 1 ) ( q - 2 ) } { q ^ { m + 1 } ( q - 1 ) ( p - 2 ) } & \\abs { I _ { m , k } } ^ { 1 - \\alpha } \\\\ & = \\left ( \\frac { p } { q } \\right ) ^ { m + 1 } \\cdot \\frac { p - 1 } { q - 1 } \\cdot \\frac { q - 2 } { p - 2 } \\left ( \\frac { p - 2 } { p ^ { m + 1 } ( p - 1 ) } \\right ) ^ { 1 - \\alpha } \\\\ & = \\frac { q - 2 } { q - 1 } \\left ( \\frac { p - 1 } { p - 2 } \\right ) ^ { \\alpha } = : c _ 2 . \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} R _ { j + 1 } & \\leq \\widetilde C R _ j \\Big ( ( m _ { j , 0 } + R _ j ) ( 1 + m _ j ) + R _ j ^ 2 + m _ 0 \\sum _ { i = 0 } ^ { j - 1 } R _ i + \\big | \\sum _ { i = 0 } ^ { j - 1 } R _ i \\big | ^ 2 \\Big ) \\\\ & = : \\widetilde C Q _ j R _ j , \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} - L u _ n + g _ { n + 1 } \\circ u _ n & = f + g _ { n + 1 } \\circ u _ n - g _ n \\circ u _ n \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u _ n & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k + 1 } ^ 0 \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k , 0 ; v _ { s + 1 } , \\dots , v _ { s + k } , v _ { s + k + 1 } ; \\right . \\\\ & \\left . \\omega _ 1 , \\dots , \\omega _ k , \\omega _ { k + 1 } ; i _ 1 , \\dots , i _ k , i _ { k + 1 } \\right ] = \\\\ & = \\left ( x _ 1 ^ \\prime , v _ 1 ^ \\prime , \\dots , x _ { i _ { k + 1 } } ^ \\prime , v _ { i _ { k + 1 } } ^ \\prime , \\dots , x _ s ^ \\prime , v _ s ^ \\prime , x _ { i _ { k + 1 } } ^ \\prime , v _ { s + k + 1 } \\right ) \\end{aligned} \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} p ^ * ( v , v ) = - \\sqrt { 2 } ( x z + y \\circ z ) , \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} \\langle 2 d i v \\ h - \\nabla \\mathcal { H } , \\nabla f \\rangle & = \\langle 2 g ^ { i j } \\nabla _ i ( \\frac { 1 } { 2 } R g _ { i j } ) - \\frac { n } { 2 } \\nabla _ k R , \\nabla f \\rangle \\\\ \\displaystyle & = \\frac { 2 n - n } { 2 } \\langle \\nabla R , \\nabla f \\rangle \\end{align*}"}
-{"id": "9696.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } F _ { n } ( \\xi ) = f \\circ \\pi ( \\xi ) \\mbox { f o r $ \\mathfrak { b } $ - a . e . $ \\xi $ } . \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} \\Delta _ u ^ { - 1 } = \\sum _ { n = 0 } ^ \\infty u ^ n \\Delta _ { h _ n } . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} \\frac { \\Phi _ { d - p r i m } ( Q ) } { \\Phi ( Q ) } = 1 + O ( \\frac { 1 } { q } ) \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} d U = \\sum _ { i = 1 } ^ m H ^ i ( D _ x U , x ) \\cdot d W ^ i d U = - \\sum _ { i = 1 } ^ m H ^ i ( - D _ y U , y ) \\cdot d W ^ i . \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\hat f ( r ) & = \\frac \\pi { \\sinh ( \\pi r ) } \\int _ 0 ^ \\infty \\ ! \\frac { J _ { 2 i r } ( \\xi ) - J _ { - 2 i r } ( \\xi ) } { 2 i } f ( \\xi ) \\ , \\frac { d \\xi } \\xi , \\\\ \\check f ( r ) & = \\frac 4 \\pi \\cosh ( \\pi r ) \\int _ 0 ^ \\infty \\ ! K _ { 2 i r } ( \\xi ) f ( \\xi ) \\ , \\frac { d \\xi } \\xi , \\\\ \\tilde f ( \\ell ) & = \\int _ 0 ^ \\infty \\ ! J _ \\ell ( \\xi ) f ( \\xi ) \\ , \\frac { d \\xi } \\xi . \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} D ( \\zeta , z ) = \\left | z - \\zeta \\right | ^ { 2 } \\left ( \\frac { 1 } { K _ { 0 } } S ( z , \\zeta ) + \\rho ( \\zeta ) \\right ) ^ { n + N - 1 } , \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} \\partial ^ * \\star f = - ( f ' _ 1 , \\cdots , f ' _ n ) , \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} S = - \\frac { \\partial F } { \\partial T } \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\langle ( g , h ) \\rangle ^ { G \\times H } = \\langle g \\rangle ^ G \\times \\langle h \\rangle ^ H = \\langle ( g , h ^ { - 1 } ) \\rangle ^ { G \\times H } . \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} S _ 0 = \\begin{pmatrix} I d & 0 & 0 \\\\ 0 & - I d & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , S _ a = \\begin{pmatrix} 0 & A _ a & B _ a \\\\ A _ a ^ { t r } & 0 & C _ a \\\\ B _ a ^ { t r } & C _ a ^ { t r } & 0 \\end{pmatrix} , 1 \\leq a \\leq m _ { + } , \\end{align*}"}
-{"id": "10115.png", "formula": "\\begin{align*} p = \\gcd ( p , r ) + \\gcd ( p , q ) + \\gcd ( p , q + r ) . \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} G _ { p } = & \\{ x \\in X \\ | \\ \\sum _ { i = 0 } ^ { N - 1 } \\chi _ { C _ 1 } ( x _ i ) = p \\ { \\rm m o d } \\ n \\} , \\\\ H _ p = & \\{ x \\in X \\ | \\ \\sum _ { i = - N } ^ { - 1 } \\chi _ { C _ 1 } ( x _ i ) = p \\ { \\rm m o d } \\ n \\} , \\\\ L _ p = & \\{ x \\in X \\ | \\ \\sum _ { i = N } ^ { N + n - 1 } \\chi _ { C _ 1 } ( x _ i ) = p \\ { \\rm m o d } \\ n \\} \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\left \\vert Y _ i - \\frac { b Z _ i } { \\sqrt { n } } \\right \\vert \\approx \\sum _ { i = 1 } ^ n \\vert Y _ i \\vert - b N ( 0 , 1 ) + f ( 0 ) b ^ 2 \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} E _ 2 ^ { p , q } = H ^ p ( G , H ^ q ( M , \\Z ) ) \\implies H ^ { p + q } _ G ( M , \\Z ) \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} f ( w _ 0 , w _ 1 , 1 , w _ 3 ) = \\frac { 1 } { x _ 2 ^ 4 } f ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) . \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} - \\sum _ { i = 0 } ^ { k - 1 } ( \\frac { k - i } { n } ) r ' ( i / n ) Y _ i , \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} | \\P _ S ( q + 1 ) | & = 2 | \\P _ S ( q ) | + 2 \\displaystyle \\sum _ { \\ell = 1 } ^ { s } | \\P _ { S _ { i _ \\ell } } ( q ) | + \\displaystyle \\sum _ { \\ell = 1 } ^ { s } | \\P _ { \\widehat { S } _ { i _ \\ell } } ( q ) | . \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} B _ j = \\begin{pmatrix} 0 & d \\\\ b & c \\end{pmatrix} , d _ { 4 \\times 4 } \\neq 0 . \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} \\tilde { \\varphi } _ N ( x + \\i y ) : = \\sum \\limits _ { n = 0 } ^ { N } \\varphi ^ { ( n ) } ( x ) \\frac { ( \\i y ) ^ n } { n ! } \\theta \\left ( \\frac { y } { \\langle x \\rangle } \\right ) . \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} \\rho \\left ( A , \\varphi \\right ) = \\lim _ { t \\rightarrow \\infty } - \\frac { 1 } { t } \\log \\left ( \\mu \\left ( \\left \\{ x \\in X \\middle | \\ ; N _ { A } \\geq N _ { t } \\right \\} \\right ) \\right ) . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} V _ s ( y ) : = \\begin{cases} - \\frac { x } { s } & y \\in [ 0 , s ] , \\\\ [ 1 m m ] \\frac { x - 1 } { 1 - s } & y \\in [ s , 1 ] . \\end{cases} \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\left ( \\hbox { s t r e t c h } _ 1 ( x , \\alpha ) \\right ) _ i \\ ; = \\ ; \\begin{cases} x _ i + \\alpha & \\hbox { i f } \\ x _ i > 0 , \\\\ 0 & \\hbox { i f } x _ i = 0 , \\\\ x _ i - \\alpha & \\hbox { i f } \\ x _ i < 0 . \\end{cases} \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in \\omega q ^ { 2 \\Z } } \\frac { 1 } { \\| \\varphi \\left ( \\lambda \\right ) \\| ^ { 2 } } \\left ( \\varphi _ { k } \\left ( \\lambda \\right ) \\varphi _ { \\ell } \\left ( \\lambda \\right ) + \\lambda ^ { 2 } \\varphi _ { k } \\left ( - \\lambda ^ { - 1 } \\right ) \\varphi _ { \\ell } \\left ( - \\lambda ^ { - 1 } \\right ) \\right ) = \\delta _ { k , \\ell } . \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} [ s \\alpha _ j + \\beta _ j , s \\alpha _ k + \\beta _ k ] = 0 . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} u _ { n } ^ { \\left ( r + 1 \\right ) } & = \\left ( x u _ { n + r + 1 } \\right ) u _ { r } ^ { - 1 } , n , r \\geq 0 . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} g ( 1 , z _ 2 , z _ 3 ) = f ( 1 , \\sigma , z _ 2 , z _ 3 ) . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } \\frac { \\partial R _ { \\epsilon , t } } { \\partial t } \\Big | _ { t = 0 } = 1 . \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\mathcal { U } g : = \\{ K g \\ , \\mid \\ , K \\in \\mathcal { U } \\} \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x _ { i } = \\dfrac { N p _ { i } } { N q } + \\varepsilon \\phi & ; & i = 1 , 2 , . . . , n \\\\ x _ { n + 1 } = \\dfrac { M } { N q } + \\varepsilon a & & \\\\ \\varepsilon N q \\cong 0 & & \\end{array} \\right . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} \\tilde { P } _ { V , t } ( Y ) = \\int _ { \\mathbb { R } ^ { p _ 2 \\times r } } & \\frac { 1 } { ( 2 \\pi ) ^ { p _ 1 p _ 2 / 2 } } \\exp ( - \\| Y - 2 t W V ^ { \\intercal } \\| _ F ^ 2 / 2 ) \\\\ & \\cdot ( \\frac { p _ 1 } { 2 \\pi } ) ^ { p _ 1 r / 2 } \\exp ( - p _ 1 \\| W \\| _ F ^ 2 / 2 ) d W . \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} \\mathbb { E } ^ { \\delta } \\prod _ { b \\in \\mathcal { C } } \\exp \\left [ \\frac { \\beta } { 4 } \\eta \\left ( b \\right ) ^ { 2 } \\right ] = \\frac { Z _ { \\beta / 4 } } { Z } \\leq e ^ { \\left \\vert \\mathcal { C } \\right \\vert } . \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\kappa _ { k , \\varphi } ( \\omega _ 0 , \\lambda ) = \\tilde { p } _ { k , \\varphi } ^ + ( \\omega _ 0 , \\lambda ) \\tilde { p } _ { k , \\varphi } ^ 0 ( \\omega _ 0 , \\lambda ) . \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\zeta _ { \\Delta } ( s ) : = \\sum _ { k = 1 } ^ \\infty \\lambda _ k ^ { - s } = T r ( \\Delta ^ { - s } ) \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ M u d \\mu = \\int _ M ( \\square u - R u ) d \\mu \\leq \\int _ M u d \\mu , \\Rightarrow \\left . \\int _ M u d \\mu \\right | _ { t } \\leq \\left . e ^ { t + K ^ { - 2 } r ^ 2 } \\int _ M u d \\mu \\right | _ { s _ 0 } < 2 e ^ { t + K ^ { - 2 } r ^ 2 } . \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} \\| x _ i ^ { s } - x _ i ^ { s - 1 } \\| & = \\| \\Pi _ { K _ i } [ x _ i ^ { s - 1 } - \\alpha _ { s - 1 } F _ i ( x _ i ^ { s - 1 } , N \\hat v _ i ^ { s - 1 } ) ] - x _ i ^ { s - 1 } \\| \\\\ & \\leq \\| x _ i ^ { s - 1 } - \\alpha _ { s - 1 } F _ i ( x _ i ^ { s - 1 } , N \\hat v _ i ^ { s - 1 } ) - x _ i ^ { s - 1 } \\| \\\\ & = \\alpha _ { s - 1 } \\| F _ i ( x _ i ^ { s - 1 } , N \\hat v _ i ^ { s - 1 } ) \\| \\cr & \\le C \\alpha _ { s - 1 } , \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} \\dot { z } = Q ( \\mu ) z + \\Psi ( \\mu ) \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} g = u . ( \\alpha , \\beta , \\gamma , \\delta ) h ^ { - 1 } g = ( 0 , \\beta - \\beta ' , \\gamma - \\gamma ' , \\delta - \\delta ' ) . \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} F ( \\lambda , v ) = \\left ( \\begin{array} { c c } L v + A v - \\lambda v \\\\ \\langle v , v \\rangle - 1 \\end{array} \\right ) = 0 . \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j \\leq 2 \\prod \\limits _ { i = 1 } ^ M a _ i . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} \\tilde { a } ^ { i j } & = \\tilde { a } ^ { i j } ( y _ 0 ) + E _ 2 ^ { y _ 0 , i j } ( y ) , \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} C ^ { \\lambda } _ { y } ( v ) = \\{ y + t w \\mid t > 0 , \\ w \\in \\mathbb { R } ^ d , \\ \\| w - v \\| < \\lambda \\} . \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{gather*} \\sum \\nolimits _ { k = 0 } ^ { r - 1 } \\ d _ { k } ^ { ( r ) } s _ { k + m } ^ { ( r ) } = s _ { r + m } ^ { ( r ) } \\ , \\ \\ m \\geq 0 \\ , \\end{gather*}"}
-{"id": "5997.png", "formula": "\\begin{align*} f _ 2 ( x ) : = x ^ 2 e ^ { - \\abs { x } } \\qquad ( x \\in \\R ) \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ 2 ( \\Omega ) } ^ 2 : = \\| u \\| _ 2 ^ 2 + \\| | \\nabla u | \\| _ 2 ^ 2 + \\| | \\nabla ^ 2 u | \\| _ 2 ^ 2 \\leq C _ 0 \\| | \\nabla ^ 2 u | \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} [ d - \\sum _ { k = 1 } ^ { j + 1 } L _ { k } ] c _ { j + 1 } & = - L _ { j + 1 } c _ { j + 1 } + ( d - \\sum _ { k = 1 } ^ { j } L _ { k } ) c _ { j + 1 } \\leq - L _ { j + 1 } c _ { j + 1 } + ( d - \\sum _ { k = 1 } ^ { j } L _ { k } ) c _ { j } \\\\ & \\leq - L _ { j + 1 } c _ { j + 1 } + \\sum _ { k = j + 1 } ^ K L _ { k } c _ k = \\sum _ { k = j + 2 } ^ K L _ { k } c _ k \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} h ( t ) : = \\frac { a } { t - 1 } + \\frac { b } { t - z ^ { 2 } } = \\sum _ { n = 0 } ^ { \\infty } \\left ( - a - \\frac { b } { z ^ { 2 n + 2 } } \\right ) t ^ { n } , \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{gather*} \\Psi ( z ; t _ { 1 } , t _ { 3 } , \\dots ) = ( \\Gamma ( z , t ) \\circ \\tau ( t ) ) / \\tau ( t ) , \\end{gather*}"}
-{"id": "3575.png", "formula": "\\begin{align*} \\theta _ k = \\theta ' _ \\ell \\otimes ( \\theta '' _ m ) ^ { n ' ( \\ell ) } \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} K : = \\biguplus _ { m \\in \\omega } K _ m \\uplus \\{ b \\} . \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{gather*} \\tau _ { k + 1 , \\ell + 1 } ^ { ( \\alpha , \\beta ) } \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta + 1 ) } = \\tau _ { k + 1 , \\ell + 1 } ^ { ( \\alpha , \\beta + 1 ) } \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta ) } - \\tau _ { k + 1 , \\ell } ^ { ( \\alpha , \\beta + 1 ) } \\tau _ { k , \\ell + 1 } ^ { ( \\alpha , \\beta ) } , \\end{gather*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ F \\in \\mathcal { B } : \\ \\mbox { t h e r e e x i s t s } \\ i \\ \\mbox { s u c h t h a t e i t h e r } \\ F \\sim F _ i \\ \\mbox { o r } \\ F \\sim F _ i ^ c \\} . \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} y _ k ^ n e _ k = & \\sum _ { j = 1 } ^ { k - 1 } ( - 1 ) ^ { j + 1 } E _ j ( x _ 1 , \\ldots , x _ { k - 1 } ) . y _ k ^ { k - j - 1 } ( \\psi _ k \\cdots \\psi _ n ) ( \\psi _ n \\cdots \\psi _ k ) e _ k + \\\\ & \\sum _ { j = 1 } ^ { n - k + 1 } ( - 1 ) ^ { j + 1 } E _ j ( x _ k , \\ldots , x _ n ) . ( y _ k ^ { n - j } e _ k ) . \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} H ( { \\rm d i v } ; \\Omega ) : = \\left \\{ u \\in [ L ^ 2 ( \\Omega ) ] ^ d \\mbox { s u c h t h a t } \\nabla _ x \\cdot u \\in L ^ 2 ( \\Omega ) \\right \\} . \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} H ^ { 2 n - 1 } ( X ^ \\circ / X , \\Z ) = \\bigoplus _ { i = 1 } ^ r \\Z / m _ i \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\omega _ { \\nu } = 1 0 \\leq \\nu \\leq d - 1 . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} \\cos \\left ( i \\theta \\right ) = \\gamma = \\frac { 1 } { \\sqrt { 1 - \\beta ^ { 2 } } } , \\sin \\left ( i \\theta \\right ) = i \\beta \\gamma = \\frac { i \\beta } { \\sqrt { 1 - \\beta ^ { 2 } } } , \\qquad \\beta = \\frac { v } { c } . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) ( 2 \\pi ) ^ { - \\frac { q } { \\tau } } \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) \\Gamma ( 1 - \\frac { q } { \\tau } ) = & \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) ( 2 \\pi ) ^ { - q } \\times \\\\ & \\times \\Gamma ^ { q } ( 1 - \\frac { 1 } { \\tau } ) \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "9958.png", "formula": "\\begin{align*} \\R = \\Q ^ \\N _ C / \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} \\eta _ \\theta = \\theta \\int _ { - \\infty } ^ 0 e ^ { - \\theta s } B ^ H _ s \\ , d s - x _ 0 \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} s _ { 2 i } = p _ { 2 i } + \\frac { \\omega ^ { a _ { h ( 2 i ) } ( 2 i ) } } { 2 ^ { h ( 2 i ) + 1 } } ( 1 + z + \\dots + z ^ { k - h ( 2 i ) - 1 } ) ( z ^ { h ( 2 i ) + 1 } - 1 ) \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} \\mathcal { M } ' : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} \\partial _ t v ( t ) - A _ 2 v + P _ 2 ( v \\nabla _ H v + w \\partial _ z v ) = P _ 2 ( f + \\Pi ( \\zeta ) ) , \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} [ H _ { 0 } , ( \\gamma \\cdot x ) ] = [ - \\frac { 1 } { 2 } \\Delta , ( \\gamma \\cdot x ) ] = - \\gamma \\cdot \\nabla . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} - P _ g ( r ^ { 6 - n } ) = & \\Big [ \\Delta ^ 3 + \\Delta \\delta T _ 2 d + \\delta T _ 2 d \\Delta + \\frac { n - 2 } { 2 } \\Delta ( \\sigma _ 1 ( A ) \\Delta ) + \\delta T _ 4 d - \\frac { n - 6 } { 2 } Q _ g \\Big ] ( r ^ { 6 - n } ) \\\\ : = & \\sum _ { k = 1 } ^ 6 I _ k . \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i Q _ i ^ { m - 1 } S _ { n , m + i - 1 } = 0 . \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} J _ \\lambda ( [ h , t ] , z ) = \\frac { \\Psi _ \\lambda ( [ 0 , t ] z ) } { \\Psi _ \\lambda ( z ) } \\left ( [ h , t ] \\in \\Gamma _ \\ell \\right ) . \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{gather*} Q _ { 0 } ^ { \\beta } Q _ { 1 } ^ { - \\alpha } E _ { 1 0 } ( z ) Q _ { 1 } ^ { \\alpha } Q _ { 0 } ^ { - \\beta } = ( - 1 ) ^ { \\alpha + \\beta } z ^ { \\alpha + \\beta } E _ { 1 0 } ( z ) . \\end{gather*}"}
-{"id": "4082.png", "formula": "\\begin{align*} & \\sigma _ { \\min } ^ 2 ( \\hat { V } ^ { \\intercal } V ) = \\min _ { x \\in \\mathbb { R } ^ r } \\frac { \\| \\hat { V } ^ { \\intercal } V x \\| _ 2 ^ 2 } { \\| x \\| _ 2 ^ 2 } \\\\ = & \\min _ { x \\in \\mathbb { R } ^ r } \\frac { \\| V x \\| _ 2 ^ 2 - \\| \\hat { V } ^ { \\intercal } _ { \\perp } V x \\| _ 2 ^ 2 } { \\| x \\| _ 2 ^ 2 } = 1 - \\max _ { x \\in \\mathbb { R } ^ r } \\frac { \\| \\hat { V } _ { \\perp } V x \\| _ 2 ^ 2 } { \\| x \\| _ 2 ^ 2 } = 1 - \\| \\hat { V } _ { \\perp } ^ { \\intercal } V \\| ^ 2 , \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} f ( z ) = \\frac { z - \\frac 1 2 z ^ 2 + \\frac 1 6 z ^ 3 } { ( 1 - z ) ^ 3 } \\quad { \\rm a n d } g ( z ) = \\frac { \\frac 1 2 z ^ 2 + \\frac 1 6 z ^ 3 } { ( 1 - z ) ^ 3 } \\ , , z \\in \\mathbb { D } \\ , , \\end{align*}"}
-{"id": "9908.png", "formula": "\\begin{align*} \\int f \\dd \\lambda ^ { J } _ t : = \\frac { 1 } { \\abs { J } } \\int _ { s \\in J } f ( z ( s ) a ( t ) u ( \\varphi ( s ) ) x ) \\dd s . \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} f = a c _ 1 ^ * \\cdots c _ n ^ * d _ { 1 \\dagger } \\cdots d _ { m \\dagger } \\cdot ( \\prod _ { \\mathclap { \\varphi \\neq \\ 0 } } b _ { j _ \\varphi } c _ { j _ \\varphi } ^ * \\cdots c _ n ^ * d _ { 1 \\dagger } \\cdots d _ { m \\dagger } ) \\cdot e _ { i _ 1 } \\cdots e _ { i _ p } , \\end{align*}"}
-{"id": "9874.png", "formula": "\\begin{align*} ( 1 - \\eta ^ 2 ) \\ , \\frac { d ^ 2 } { d \\eta ^ 2 } w ( \\eta ) - 2 ( m + 1 ) \\ , \\eta \\ , \\frac { d } { d \\eta } w ( \\eta ) + \\left ( z - c ^ 2 \\ , \\eta ^ 2 \\right ) w ( \\eta ) = 0 \\ , , \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} P _ { T o t } ^ { D B F } = N _ { M S } ( P _ { L N A } + P _ { R F } + 2 P _ { A D C } ) \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} x _ 1 + x _ 2 + \\ldots + x _ k = k x _ { k + 1 } . \\end{align*}"}
-{"id": "9872.png", "formula": "\\begin{align*} \\frac { d } { d \\eta } \\left [ ( 1 - \\eta ^ 2 ) \\frac { d } { d \\eta } S _ { m , n } ( c , \\eta ) \\right ] + \\left ( \\lambda _ { m , n } - c ^ 2 \\eta ^ 2 - \\frac { m ^ 2 } { 1 - \\eta ^ 2 } \\right ) S _ { m , n } ( c , \\eta ) = 0 \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} C _ k : = \\mathbb P ^ 1 _ { \\mathfrak g ^ 1 _ { k + 1 } } \\cup \\mathbb P ^ 1 _ \\xi . \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} \\eqref { e q q 3 5 5 } = \\frac { 2 } { \\pi } \\bigg ( \\frac { 1 } { x ^ 2 + 1 } - \\frac { | x | \\sinh ^ { - 1 } | x | } { \\left ( x ^ 2 + 1 \\right ) ^ { 3 / 2 } } \\bigg ) . \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} P = \\prod _ { i = 1 } ^ { k } \\left ( x _ { r - 1 } - \\alpha _ { i } x _ { r } \\right ) ^ { m _ { i } } . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} M _ { H , \\pi } ( a , b ) = \\sum _ { i \\leq k } \\sum _ { j \\leq k } p _ { i j } \\phi _ a ( i ) \\nu _ j ( b ) , \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\textit { S } ( n ) : = \\sum _ { \\substack { 2 j + 3 l + 6 k = n \\\\ 0 \\leq j \\leq N \\\\ 0 \\leq l \\leq N \\\\ 0 \\leq k } } q ^ { j + k + l } \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} \\delta ( X ) = 8 ( g - 1 ) S _ { g } ( X ) - 8 B ( X ) . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} - \\frac { \\dot { M _ 2 } } { M _ 2 } & = \\frac { \\nu ^ { 1 / 3 } } { ( \\nu ^ { 1 / 3 } \\left | t - \\xi / k \\right | ) ^ { 2 } + 1 } \\\\ M _ 2 ( 0 , k , \\eta ) & = 1 . \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} \\phi _ F ( \\beta ( ( \\mu , 0 , \\mu ) ) ) = \\lambda ( [ ( s _ \\mu s _ \\mu ^ * , \\mu ) ] ) = [ ( \\Psi ( s _ \\mu s _ \\mu ^ * ) , \\kappa ( \\mu ) ) ] . \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} C [ X , Y , Z ] = 1 + \\sum _ i F _ i [ X ] G _ i [ Y ] H _ i [ Z ] , \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} r v _ \\rho ( A ) = \\{ ( a , b ) \\in R V _ \\rho ^ 2 : \\exists x , y \\in K ( r v _ \\rho ( x ) = a \\wedge r v _ \\rho ( y ) = b \\wedge ( x , y ) \\in A \\} . \\end{align*}"}
-{"id": "9762.png", "formula": "\\begin{align*} \\frac { 2 n + 1 + a _ { i _ t } - b _ { i _ t } } { 2 } - \\frac { 2 n + 1 + a _ { i ^ \\vee _ t } - b _ { i ^ \\vee _ t } } { 2 } = 1 \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} g _ { k } ( \\alpha ^ { - 1 } q ^ { m } ) = ( - 1 ) ^ { m + k } \\alpha ^ { 2 m - k } q ^ { - \\frac { 1 } { 2 } m ( 3 m + 1 ) + \\frac { 1 } { 2 } k ( k + 1 ) } \\left ( 1 + o \\left ( 1 \\right ) \\right ) \\ ! , k \\to \\infty . \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} 1 . E _ j ( x _ 2 , \\ldots , x _ n ) = ( - 1 ) ^ j x _ { i + 1 } ^ j + ( - 1 ) ^ { j - 1 } x _ { i + 1 } ^ { j - 1 } E _ 1 ( x _ 1 , \\ldots , x _ n ) + \\cdots + ( - 1 ) ^ 0 x _ { i + 1 } ^ 0 E _ j ( x _ 1 , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} \\dot \\vartheta \\vartheta \\sigma _ { i j } u ^ i u ^ j \\Theta ^ { - 3 } & = ( \\dot \\vartheta \\Theta ^ { - 1 } ) \\vartheta ^ { - 1 } v ^ { - 2 } \\norm { D u } ^ 2 \\Theta ^ { - 2 } . \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} J _ { \\sigma _ k } ( u _ k ) = \\bigg ( \\dfrac { 1 } { 2 } - \\dfrac { 1 } { p + 1 } \\bigg ) \\int _ \\Omega g ( x ) | u _ k | ^ { p + 1 } \\leq \\bigg ( \\dfrac { 1 } { 2 } - \\dfrac { 1 } { p + 1 } \\bigg ) \\int _ \\Omega g ( x ) | u _ \\infty | ^ { p + 1 } = J _ { N A V } ( u _ \\infty ) , \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} e _ 1 ^ \\prime & = e _ 1 + \\sum \\limits _ { i = 1 } ^ { k } \\sum \\limits _ { j = 1 } ^ { i } ( - 1 ) ^ { i + j - 1 } b _ { j } f _ { i } + \\sum \\limits _ { i = 1 } ^ { k } \\sum \\limits _ { j = 1 } ^ { i } ( - 1 ) ^ { i + j - 1 } b _ { k + j } f _ { k + i } , \\\\ x ^ \\prime & = x + \\sum \\limits _ { i = 1 } ^ { k } \\sum \\limits _ { j = 1 } ^ { i } ( - 1 ) ^ { i + j - 1 } \\theta _ { k + j } f _ { k + i } , \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} a : = \\liminf _ { s \\to \\infty } \\frac { s ^ 2 } { G ( s ) } , A : = \\limsup _ { s \\to \\infty } \\frac { s ^ 2 } { G ( s ) } . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} N ( \\theta ) ( a , a ^ { * } ) = ( a \\theta ( a ) , [ a ^ { * } , \\theta ( a ) ] + \\theta ( a ^ { * } ) a ) \\ , . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} \\psi _ \\pm ( \\sigma ) = \\begin{cases} 1 , \\pm \\sigma \\geq \\frac { 1 } { 2 } , \\\\ 0 , \\pm \\sigma \\leq 0 . \\end{cases} \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} \\langle T _ { \\varphi _ j } ( \\mu ) , \\psi \\rangle _ { \\mu } = 0 \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} \\varphi ( \\pm \\eta + \\epsilon ) = \\frac { 1 } { \\epsilon } \\pm \\cot 2 \\eta + O ( \\epsilon ) , \\epsilon \\to 0 , \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} f _ { n - 5 } = & f _ { n - 6 } + A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\psi _ { n - 5 } ^ { ( 0 ) } + K _ { 6 - n } \\psi _ { n - 5 } ^ { ( 0 ) } \\\\ = & O ( r ^ { n - 4 } ) \\log r + O ( r ^ { n - 4 } ) : = b _ { n - 4 } ^ { ( 1 ) } \\log r + O ( r ^ { n - 4 } ) + O ( r ^ { n - 3 } ) \\log r . \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} \\partial = \\sum \\partial ^ { 2 n - 2 } _ { P _ i } : \\bigoplus _ { i = 1 } ^ r \\Z / m _ i \\to H ^ { 2 n - 1 } ( X , \\Z ) \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { A c h } } ( \\mu , r ) \\leq ( M + K - 1 ) \\mu + ( 1 - \\mu M ) \\left ( 1 + \\frac { K } { M r } \\right ) = 1 + \\frac { K \\left ( 1 - \\mu M \\right ) } { M r } + ( K - 1 ) \\mu . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} \\left ( \\partial _ t + V _ s \\cdot \\nabla _ { X _ s } \\right ) g _ \\varepsilon ^ { ( s ) } ( t ) = \\ell ^ { - 1 } \\tilde { C } _ { s + 1 } g _ \\varepsilon ^ { ( s + 1 ) } ( t ) \\textnormal { ( i f } s \\geq m - 1 \\textnormal { ) } \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} v _ j ( x , y , z ) = \\Phi \\left ( \\dfrac { k } { c } ( z - h _ j ( x , y ) ) \\right ) = \\Phi \\left ( \\dfrac { k } { c } ( z - a _ j { x } - b _ j { y } ) \\right ) . \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} \\frac { \\left ( f f '' + ( f ' ) ^ 2 + 1 \\right ) ^ 2 - 4 c f ^ 2 ( f '^ 2 + 1 ) } { f '^ 2 + 1 } = \\kappa ^ 2 . \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} H _ n = \\log _ { 1 / p } n + \\frac { 1 } { 2 } \\log _ { p / q } \\log n + o ( \\log \\log n ) , \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} f _ c ( x ) & : = \\frac { 1 } { 2 m c } \\left ( \\sqrt { ( p + 1 ) ^ 2 + m ^ 2 c ^ 2 } + \\sqrt { ( p - 1 ) ^ 2 + m ^ 2 c ^ 2 } \\right ) h ( x ) , \\\\ u _ c ( x ) & : = f _ c ( x ) \\sin x , \\\\ \\lambda _ c & : = c \\left ( \\sqrt { 1 + m ^ 2 c ^ 2 } - m c \\right ) , \\\\ V _ c ( x ) & : = \\lambda _ c - c \\frac { \\left ( \\sqrt { p ^ 2 + m ^ 2 c ^ 2 } - m c \\right ) u _ c ( x ) } { u _ c ( x ) } . \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} \\frac { d \\mu \\circ T } { d \\mu } ( { \\bf p } ) = \\frac { F ( T { \\bf p } ) } { F ( { \\bf p } ) } . \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { k } { ( \\alpha _ i - \\beta _ i ) v _ i } + \\sum \\limits _ { i = k + 1 } ^ m { \\alpha _ i v _ i } - \\sum \\limits _ { i = k + 1 } ^ { 2 d - m } { \\beta _ i w _ i } + \\left ( { \\sum \\limits _ { i = 1 } ^ m { \\alpha _ i K _ i } } \\right ) u _ 0 = 0 \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} F ( X , p , r , x , t ) = F _ 1 ( X , p , r , t ) + F _ 0 ( p , x , r , t ) \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} \\vec { X } & \\gets \\arg \\min _ { \\vec { X } } \\Biggl \\lbrace \\dfrac { 1 } { 2 } \\biggl \\| \\dfrac { 1 } { 1 + \\mu } \\left ( \\vec { Y } + \\mu ( \\vec { Z } + \\vec { D } ) \\right ) - \\vec { X } \\biggr \\| _ F ^ 2 \\\\ & + \\dfrac { \\lambda _ 1 } { 1 + \\mu } \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } \\phi ( \\vec { X } _ { i , j } ; a _ 1 ) \\Biggr \\rbrace . \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} d \\left ( w _ { n } , z _ { n } \\right ) & = b , \\\\ w _ { n + 1 } & \\notin \\left [ w _ { n } , z _ { n } \\right ] , \\ \\ \\ \\ \\ 1 \\leq n < N . \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} H _ \\sigma ( \\lambda , U ) = - \\Delta _ \\sigma - \\lambda \\ , U , \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} \\omega ^ { \\mathrm { z m } , - } \\big | _ { \\partial Z _ { 1 , 0 } } = \\sum _ { j = 1 } ^ m \\phi _ j , e ^ { 4 i R \\lambda _ j } C _ { 1 2 } ( \\lambda _ j ) \\varphi _ j = \\varphi _ j , \\big \\lVert \\varphi _ j - \\phi _ j \\big \\rVert _ Y < e ^ { - a R } \\big \\lVert \\omega ^ { \\mathrm { z m } , - } \\big \\rVert _ Y . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\dot { x } = - \\varPhi \\tilde { x } , \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} \\frac { t } { 4 } \\min _ { j = 0 , \\hdots , N - 1 } \\mathbb { E } [ \\norm { \\mathcal { G } _ t ( x _ { j } ) } ^ 2 ] \\le \\frac { t } { 4 N } \\sum _ { j = 0 } ^ { N - 1 } \\mathbb { E } [ \\norm { \\mathcal { G } _ t ( x _ { j } ) } ^ 2 ] \\le \\frac { 1 } { N } \\sum _ { j = 0 } ^ { N - 1 } \\mathbb { E } [ F ( x _ { j } ) - F ( x _ { j + 1 } ) ] \\le \\frac { F ( x _ 0 ) - \\inf F } { N } , \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} \\frac { M _ { a , b , c } ( r ) } { M _ { a , b , c } } \\equiv \\frac { M _ { a , b , c } ( r ) } { M _ { a , b , c } ( a + 1 ) } = \\binom { a } { r - 1 } \\binom { b + c - 1 } { c } \\binom { a + b + c - r } { c } ^ { - 1 } , \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} \\mathbb { E } [ C _ 2 ^ ( a _ ) ] = & \\int _ 0 ^ \\infty \\int _ { x _ 1 } ^ \\infty \\frac { 2 } { \\beta ^ 2 } e ^ { - \\frac { x _ 1 + x _ 2 } { \\beta } } \\\\ & \\cdot \\log _ 2 \\left ( 1 + ( \\sqrt { 1 + \\xi x _ 1 } - 1 ) \\frac { x _ 2 } { x _ 1 } \\right ) d x _ 2 d x _ 1 . \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} M = \\mathrm { g r a p h } \\ , u = \\{ \\tau = u ( x ) : x \\in \\mathbb { S } ^ n \\} . \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} | v _ { 1 2 } | = | v _ { 2 1 } | \\geq \\frac { 1 } { \\sqrt { 2 } } . \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} \\mu _ { k } ^ i \\left ( B \\right ) & \\geq 1 \\sum \\limits _ { \\theta \\in B ^ c } { \\prod \\limits _ { j = 1 } ^ { n } \\left ( \\frac { \\mu _ 0 ^ j ( \\theta ) } { \\mu _ 0 ^ j ( \\theta ^ * ) } \\right ) ^ { \\left [ A ^ { k } \\right ] _ { i j } } } \\prod \\limits _ { t = 1 } ^ { k } \\prod \\limits _ { j = 1 } ^ { n } \\left ( \\frac { \\ell ^ j ( s _ { t } ^ j | \\theta ) } { \\ell ^ j ( s _ { t } ^ j | \\theta ^ * ) } \\right ) ^ { \\left [ A ^ { k t } \\right ] _ { i j } } \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} & [ i _ { \\vartheta L ^ \\perp } ^ * \\vartheta \\pi _ { L ^ \\perp } ^ * T ^ { ( p - 2 j ) } ( L , B ) ] ( \\vartheta x _ 1 , \\ldots , \\vartheta x _ { p - 2 j } ) \\\\ & = [ \\vartheta \\pi _ { L ^ \\perp } ^ * T ^ { ( p - 2 j ) } ( L , B ) ] ( \\vartheta x _ 1 , \\ldots , \\vartheta x _ { p - 2 j } ) \\\\ & = [ \\pi _ { L ^ \\perp } ^ * T ^ { ( p - 2 j ) } ( L , B ) ] ( x _ 1 , \\ldots , x _ { p - 2 j } ) \\\\ & = T ^ { ( p - 2 j ) } ( L , B ) ( x _ 1 , \\ldots , x _ { p - 2 j } ) . \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} \\widetilde { V } _ p ( 0 , v , k ) = - \\frac { 1 } { p } \\frac { \\Gamma ( k - v ) } { \\Gamma ( k + v ) } \\left ( \\frac { \\sqrt { p } } { 2 \\pi \\sqrt { l } } \\right ) ^ { 1 - 2 v } + O \\left ( \\frac { 1 } { \\sqrt { p } } V _ p ( 0 , v , k ) \\right ) , \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{gather*} ( u \\sharp f ) ( v \\sharp g ) = \\sum u \\cdot v _ { ( 1 ) } \\sharp { \\boldsymbol \\phi } ( v _ { ( 2 ) } ) ( f ) \\cdot g u , v \\in U ( \\gg ) , f , g \\in \\hat { S } ( \\gg ^ * ) , \\end{gather*}"}
-{"id": "9791.png", "formula": "\\begin{align*} \\frac { t } { 2 } \\ , ( \\varphi ^ 2 ) ' + \\varphi ^ 2 + 1 = \\pm a \\sqrt { \\varphi ^ 2 + 1 } . \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} \\Psi ^ l = \\Psi ^ l \\left ( x _ { i } , u _ j , ( u _ j ) _ { x _ { i _ 1 } } , ( u _ j ) _ { x _ { i _ 1 } ^ { j _ 1 } , x _ { i _ 2 } ^ { j _ 2 } } , \\dots , ( u _ j ) _ { x _ { i _ 1 } ^ { j _ 1 } , x _ { i _ 2 } ^ { j _ 2 } , x _ { i _ 3 } ^ { j _ 3 } , \\dots , x _ { i _ n } ^ { j _ n } } \\right ) , \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} \\langle a _ { i } ^ { \\epsilon } | G ^ { ( n ) } _ { 0 } | a _ { j } ^ { \\epsilon } \\rangle = G ^ { ( n ) } _ { 0 } ( a _ { i } ^ { \\epsilon } , a _ { j } ^ { \\epsilon } ) = \\int _ { M \\times M } d _ g \\mu ( x ) \\ ; d _ g \\mu ( y ) \\ ; K _ \\epsilon ( x , a _ i ) G ^ { ( n ) } _ { 0 } ( x , y ) K _ \\epsilon ( a _ j , y ) \\ ; . \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} \\eta ( \\omega _ 0 ) = m \\eta ( \\tau ) - 2 \\eta ( 1 ) \\ \\ \\eta ( \\tau ) \\in i \\R \\ \\ \\eta ( 1 ) \\in \\R . \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} \\mu ( \\mathbf { x } ) = f ( \\mathbf { x } ) + h ( \\mathbf { x } ) , \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} \\nabla \\psi = A \\bigl ( - R _ 3 ^ { \\delta } \\delta \\rho ^ { - \\delta - 1 } \\nabla \\rho + \\nabla h \\bigr ) \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} \\R { k } { l } { m } { a } { b } { c } { x } = \\frac { c ( c + 1 ) } { ( a + 1 ) ( b + 1 ) x ( 1 - x ) } \\Q { k - 1 } { l - 1 } { m - 1 } { a + 1 } { b + 1 } { c + 1 } { x } . \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} \\Lambda ^ 2 _ 7 V _ 7 ^ \\ast & = \\{ \\alpha ^ 2 \\in \\Lambda ^ 2 V _ 7 ^ \\ast \\mid \\ast ( \\alpha ^ 2 \\wedge \\varphi ) = 2 \\alpha ^ 2 \\} , \\mbox { a n d } \\\\ \\Lambda ^ 2 _ { 1 4 } V _ 7 ^ \\ast & = \\{ \\alpha ^ 2 \\in \\Lambda ^ 2 V _ 7 ^ \\ast \\mid \\ast ( \\alpha ^ 2 \\wedge \\varphi ) = - \\alpha ^ 2 \\} . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ + _ { I I I } } & d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , \\alpha } ( s + k - 1 ) T R \\eta ^ { d - 1 } \\end{aligned} \\end{align*}"}
-{"id": "10163.png", "formula": "\\begin{align*} \\kappa : = \\limsup _ { n \\rightarrow + \\infty } ( \\mathbb E [ Z _ n ^ * ] ) ^ { - 1 } \\Vert Z ^ * _ { n + \\lfloor \\varepsilon n \\rfloor } - Z _ { n + \\lfloor \\varepsilon n \\rfloor } \\Vert _ p < + \\infty \\ , . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{gather*} \\sigma ( A ) = ( - 1 ) ^ { \\abs { \\sigma } } \\prod _ { s = 1 } ^ { k } z _ { \\sigma ( s ) } ^ { s - 1 } \\det \\big ( V ^ { ( k ) } _ { \\{ z _ { i } \\} } \\big ) . \\end{gather*}"}
-{"id": "3804.png", "formula": "\\begin{align*} & [ L _ n , L _ m ] = ( \\{ m \\} - \\{ n \\} ) L _ { n + m } + D \\frac { 1 } { q ^ { n - 2 } } \\frac { 1 + q ^ { 2 } } { 1 + q ^ { n } } \\frac { \\{ n + 1 \\} \\{ n \\} \\{ n - 1 \\} } { \\{ 1 2 \\} } \\delta _ { n + m , 0 } , & \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\forall ( n , m ) \\in ( \\Z _ + \\times \\Z ) \\cup ( \\Z _ - \\times \\Z _ - ) & \\\\ & [ L _ n , F _ m ] = ( \\{ m + \\frac { 1 } { 2 } \\} - \\{ n \\} ) F _ { n + m } & \\\\ & [ F _ n , F _ m ] = [ F _ n , D ] = [ L _ n , D ] = 0 & . \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} \\varphi ( \\kappa _ n , \\cdots , \\kappa _ n ) = 0 , \\varphi _ j ( \\kappa _ n , \\cdots , \\kappa _ n ) = 0 \\forall j = 1 , \\dots n . \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} g \\circ I = p \\circ F \\circ I = p \\circ J \\circ F = - \\frac { 1 } { \\bar g } , \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} & \\left | { \\nabla a ( x ) \\over a ( x ) } \\nabla P U _ { \\delta , \\xi } \\right | _ { 2 n \\over n + 2 } = O \\ ( \\delta \\ ) \\ \\hbox { ( b e c a u s e $ n \\ge 5 $ ) . } \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} ( \\mathcal { F } T _ k \\Delta \\mathcal { F } ^ { - 1 } f ) ( \\xi ) & = f ( \\xi + k ) \\sum _ { i = 1 } ^ d ( 2 - 2 \\cos ( \\xi _ i + k ) ) \\\\ & = f ( \\xi + k ) \\sum _ { i = 1 } ^ d \\big [ 2 - 2 \\cos ( k ) \\cos ( \\xi _ i ) - 2 \\sin ( k ) \\sqrt { 1 - \\cos ^ 2 ( \\xi _ i ) } ( 2 \\ 1 _ { [ 0 , \\pi ] } ( \\xi _ i ) - 1 ) \\big ] . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} P _ { 0 } ( 0 ) = P _ { 0 , 0 } = \\frac { C _ { \\lambda , \\mu } } { \\xi } A , \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} \\Phi _ \\alpha ( \\mu ) = \\Phi ( \\alpha ^ { - 1 } \\mu ) , \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} \\pi ^ { N } ( \\alpha , \\beta ) = \\pi ( N ^ { * } ( \\alpha ) , \\beta ) = \\pi ( \\alpha , N ^ { * } ( \\beta ) ) . \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\| P _ 0 ( \\cdot ) \\| _ 0 = \\| ( 1 - { B _ t } ) ^ { - 1 } ( R _ 0 ( \\cdot ) ) \\| _ 0 \\leq \\frac { 1 } { 1 - \\| B _ t \\| _ 0 } \\| R _ 0 ( \\cdot ) \\| _ 0 \\leq 2 C \\rho ^ { \\varepsilon _ 0 } , \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} y ' = 0 . 7 6 3 2 0 8 1 9 x ^ { - 0 . 5 6 7 1 4 8 } - \\frac { 2 } { x } \\log x \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{gather*} \\tau _ { k , \\ell + 1 } ^ { ( \\alpha , \\beta - 1 ) } \\tau _ { k - 1 , \\ell } ^ { ( \\alpha , \\beta ) } = \\tau _ { k , \\ell + 1 } ^ { ( \\alpha , \\beta ) } \\tau _ { k - 1 , \\ell } ^ { ( \\alpha , \\beta - 1 ) } - \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta ) } \\tau _ { k - 1 , \\ell + 1 } ^ { ( \\alpha , \\beta - 1 ) } , \\end{gather*}"}
-{"id": "7176.png", "formula": "\\begin{align*} D _ { n } = ( p _ { n + 1 } ) ^ { a } - ( p _ { n } ) ^ { a } < \\frac { 1 } { n } \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} \\omega _ 2 ( \\pi ) : = \\prod _ { i = 1 } ^ { \\nu ( \\pi ) } \\frac { \\lambda _ i - \\lambda _ { i + 1 } - \\delta _ { \\lambda _ i , e } - \\delta _ { \\lambda _ { i + 1 } , e } } { 2 } . \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} ( S _ { ( A , \\gamma _ 1 ) } S _ { ( B , \\gamma _ 2 ) } ) ^ * ( S _ { ( A , \\gamma _ 1 ) } S _ { ( B , \\gamma _ 2 ) } ) & = P _ A S _ { \\gamma _ 2 } ^ * S _ { \\gamma _ 1 } ^ * S _ { \\gamma _ 1 } S _ { \\gamma _ 2 } P _ A \\\\ & = \\sum _ { \\eta _ 1 \\in E _ Z } Z ^ G ( \\gamma _ 1 , \\eta _ 1 ) P _ A S _ { \\gamma _ 2 } ^ * S _ { \\eta _ 1 } S _ { \\eta _ 1 } ^ * S _ { \\gamma _ 2 } P _ A \\\\ & = Z ^ G ( \\gamma _ 1 , \\gamma _ 2 ) S _ { ( B , \\gamma _ 2 ) } ^ * S _ { ( B , \\gamma _ 2 ) } . \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u + u ^ { 1 + p } ( 0 , \\infty ) \\times \\R ^ { N } , \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} H ^ 1 ( G , H ^ 2 ( \\P ^ N ) ) = 0 \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{gather*} \\tau _ { k } ^ { ( \\alpha ) } = \\frac 1 { k ! } \\big \\langle T ^ { k } v _ { 0 } , \\big ( \\Gamma _ { C } ^ { ( \\alpha ) } \\big ) ^ { k } v _ { 0 } \\big \\rangle . \\end{gather*}"}
-{"id": "3087.png", "formula": "\\begin{align*} B _ { n } ^ { \\left ( r \\right ) } = \\left \\vert \\mathbf { P } _ { n , - 1 } ^ { r } \\ \\mathbf { P } _ { n + 1 , - 1 } ^ { r } . . . \\mathbf { P } _ { n + d - 1 , - 1 } ^ { r } \\right \\vert ^ { T } . \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ - _ I } & d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ d \\left ( s + k - 1 \\right ) R ^ d \\eta ^ { - 1 } y \\end{aligned} \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\nabla ^ a \\left ( R _ { a b } + \\nabla _ a \\nabla _ b f \\right ) - ( \\nabla ^ a f ) \\left ( R _ { a b } + \\nabla _ a \\nabla _ b f \\right ) = \\frac { 1 } { 2 } \\nabla _ { b } \\left ( R + 2 \\Delta f - | \\nabla f | ^ 2 \\right ) . \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} F \\left ( x \\right ) = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } u _ { \\delta } \\prod _ { i = 1 } ^ { n } f _ { i } ^ { \\delta _ { i } } ( \\alpha _ { i } ) \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\dot { \\varphi } ^ 2 = 1 - r ^ 2 . \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} s _ a ( x , \\xi ) & : = e ^ { - i \\varphi _ a ( x , \\xi ) } H ( e ^ { i \\varphi _ a ( \\cdot , \\xi ) } ) \\left [ x \\right ] - h _ 0 ( \\xi ) \\\\ & = \\sum _ { z \\in \\mathbb { Z } ^ d } f [ z ] e ^ { i ( \\varphi _ a ( x - z , \\xi ) - \\varphi _ a ( x , \\xi ) ) } + V \\left [ x \\right ] - h _ 0 ( \\xi ) \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} \\widetilde { \\omega } ^ \\mathrm { z m } = \\sum _ { j = 1 } ^ m \\widetilde { \\omega } _ { 1 , j } ^ \\mathrm { z m } = \\sum _ { j = 1 } ^ m \\Big ( e ^ { - i \\lambda _ j u _ 1 } \\varphi _ j + e ^ { i \\lambda _ j u _ 1 } C _ 1 ( \\lambda _ j ) \\varphi _ j \\Big ) . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} n \\lambda u _ n = ( L + D N ( u _ 0 ) ) u _ n + R _ n ( u _ 0 , u _ 1 , \\ldots , u _ { n - 1 } ) , \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} r _ { Z _ N } ( t ) = r _ N \\left ( t , \\psi _ N ^ t Z _ N \\right ) \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} & R ^ + ( x ; t ) = \\frac { \\sqrt { 2 x - 1 } } { 2 \\sqrt { x } p ( x ) } \\left ( p ^ 2 ( x ) F _ 2 ( u ( x , t ) ) + F _ 1 ( u ( x , t ) ) \\right ) , \\\\ & R ^ - ( x ; t ) = \\frac { - \\sqrt { 2 x - 1 } } { 2 p ( x ) } \\left ( p ^ 2 ( x ) F _ 2 ( u ( x , t ) ) - F _ 1 ( u ( x , t ) ) \\right ) , \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} \\mu _ { Z } = \\sum _ { j = 1 } ^ { \\infty } \\left \\Vert k _ { z } \\right \\Vert ^ { - 2 } \\delta _ { z _ { j } } . \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot { \\bar { V } } & = & \\dfrac { \\gamma ^ { - 1 } } { ( 1 - \\gamma ^ { - 1 } V ( x ) ) ^ { 2 } } \\dot { V } ( x ) \\end{array} \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} \\Phi _ 3 ( b ; \\ , c ; \\ , x , \\ , x ^ 2 ) = \\exp \\left ( 2 x \\right ) \\ , { } _ 2 F _ 2 \\left [ \\begin{array} { c } c - \\frac { b } { 2 } , \\ , c - \\frac { b } { 2 } - \\frac { 1 } { 2 } \\\\ \\ , c , \\ , 2 c - b - 1 \\end{array} \\ , ; \\ , - 4 x \\ , \\right ] \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\limsup _ { t \\to \\infty } \\| v _ s - e ^ { i ( t - s ) H } E _ + ( t - s ) v _ s \\| = 0 . \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} \\lambda _ t = \\exp ( - B t ) \\left ( \\lambda _ 0 + \\int _ 0 ^ t \\exp ( B s ) a \\ , \\dd s + \\int _ 0 ^ t \\exp ( B s ) c \\ , \\dd N _ s \\right ) . \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} d _ { \\Sigma } = \\max _ { ( d ^ { [ { \\sf d } ] } _ { 1 1 } , \\cdots , d _ { K N } ^ { [ { \\sf d } ] } , d ^ { [ { \\sf u } ] } _ { 1 1 } , \\cdots , d _ { K N } ^ { [ { \\sf u } ] } ) \\in \\mathcal { D } } \\left \\{ \\sum _ { k = 1 } ^ K \\sum _ { i = 1 } ^ { N } \\left ( d _ { k i } ^ { [ { \\sf d } ] } + d _ { k i } ^ { [ { \\sf u } ] } \\right ) \\right \\} , \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} E _ 2 ^ { 4 , 0 } = H ^ 4 ( G , H ^ 0 ( M , \\Z ) ) = H ^ 4 ( G , \\Z ) = \\Z / m \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} C ^ { F B , A . 1 } _ { X ^ n \\rightarrow { Y ^ n } } = \\sum _ { y _ { - 1 } \\in \\{ 0 , e , 1 \\} } C _ 0 ( y _ { - 1 } ) \\mu ( y _ { - 1 } ) , ~ \\mu ( y _ { - 1 } ) ~ \\mbox { i s f i x e d } . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 1 } \\left ( a , b ; c ; q , z \\right ) = \\frac { \\left ( c / b , b z ; q \\right ) _ { \\infty } } { \\left ( c , z ; q \\right ) _ { \\infty } } { } _ { 2 } \\phi _ { 1 } \\left ( a b z / c , b ; b z ; q , c / b \\right ) , \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} a = a ( \\lambda ) \\equiv a ( \\lambda , 0 ) , b = b ( \\lambda ) \\equiv b ( \\lambda , 0 ) , \\varphi = \\varphi ( \\lambda ) \\equiv \\varphi ( \\lambda , 0 ) = c / a b . \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} g _ { 1 1 } = \\left \\langle X _ { u } , X _ { u } \\right \\rangle , g _ { 1 2 } = \\left \\langle X _ { u } , X _ { v } \\right \\rangle , g _ { 2 2 } = \\left \\langle X _ { v } , X _ { v } \\right \\rangle , \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} \\delta ^ * ( 1 / M , 0 ) \\geq K - \\frac { ( M - 1 ) ( K - 1 ) } { M } = \\frac { M + K - 1 } { M } . \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} \\hom ( u , \\psi ( x ) ) = 1 > \\hom ( u , \\Phi ( \\psi ) ) = u \\pitchfork \\Phi ( \\psi ) \\ge \\Phi ( u \\pitchfork \\psi ) , \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} & \\Lambda _ { T } ^ V g = ( w _ g ( T ) , w _ g ' ( T ) ) \\ , , \\\\ & R ( T ) = \\left \\{ ( u _ g ( T ) , u _ g ' ( T ) ) \\ , , \\ g \\in L ^ 2 ( 0 , T ; L ^ 2 ( \\Gamma ) ) \\right \\} \\ , , \\\\ & R _ V ( T ) = \\left \\{ ( w _ g ( T ) , w _ g ' ( T ) ) \\ , , \\ g \\in L ^ 2 ( 0 , T ; L ^ 2 ( \\Gamma ) ) \\right \\} \\ , . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} I _ { \\varepsilon , v } ( u ) = \\int _ { 0 } ^ { T } \\exp ( - t / \\varepsilon ) \\left ( \\frac { \\varepsilon } { 2 } | u ^ { \\prime } | ^ { 2 } + \\phi ( u ) - ( f ( v ) , u ) \\right ) \\mathrm { { d } } t \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} \\sigma _ k a ( n + e _ j ) - \\sigma _ j a ( n + e _ k ) + i \\gamma _ { j k } a ( n ) = 0 . \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} v = \\alpha ( 1 , e ^ { \\j \\omega } , \\ldots , e ^ { \\j ( m - 1 ) \\omega } ) , w = \\beta ( 1 , e ^ { - \\j \\omega } , \\ldots , e ^ { - \\j ( n - 1 ) \\omega } ) . \\end{align*}"}
-{"id": "10156.png", "formula": "\\begin{align*} z _ { k + 1 } - x ^ * = z _ k - x ^ * - \\gamma \\cdot G ( y _ k ) . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} e ^ { 4 i R \\lambda } C _ { 1 2 } ( \\lambda ) \\phi _ 2 = \\phi _ 2 . \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} \\sigma ( a , u , b , v ) : = \\frac 1 2 \\ast ( a ^ \\flat \\wedge b ^ \\flat \\wedge \\lambda ^ 4 ( u ) \\wedge \\lambda ^ 2 ( v ) ) . \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\dd N _ t & = \\lambda _ t ^ \\top X _ t \\ , \\dd t + \\dd m _ t , \\\\ \\dd \\lambda _ t & = ( a - B \\lambda _ t ) \\ , \\dd t + c \\ , \\dd N _ t . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} & \\tilde { S } _ d b = \\alpha _ d ( b ) \\tilde { S } _ d , & & \\tilde { S } _ d ^ * b = \\alpha ^ { - 1 } _ d ( b ) \\tilde { S } _ d , & & \\tilde { S } _ d ^ * \\tilde { S } _ d = 1 , & & \\tilde { S } _ d \\tilde { S } _ d ^ * = 1 - p \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} | h ' ( x ) | \\leq 8 h ( x ) \\frac { | g ( x ) | } { 1 + g ( x ) ^ 2 } = 8 h ( x ) \\frac { | g ( x ) | } { \\sqrt { 1 + g ( x ) ^ 2 } } \\frac { 1 } { \\sqrt { 1 + g ( x ) ^ 2 } } \\leq 8 h ( x ) ^ { 3 / 2 } . \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} \\left [ T _ { ( A , 0 ) } , T _ { ( B , 0 ) } \\right ] & = \\left ( K _ { 0 0 } { } ^ { k } - K _ { 0 0 } { } ^ { k + 1 } \\right ) C _ { A B } { } ^ { C } T _ { ( C , k ) } , \\\\ & = \\left ( K _ { 0 0 } { } ^ { 0 } - K _ { 0 0 } { } ^ { 1 } \\right ) C _ { A B } { } ^ { C } T _ { ( C , 0 ) } , \\\\ & = C _ { A B } { } ^ { C } T _ { ( C , 0 ) } , \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} \\delta _ p \\left ( \\big [ F _ { Z _ { 2 , R } } ( \\omega , 0 ) \\big ] \\right ) = 0 \\in H ^ { p + 1 } _ \\mathrm { r e l } ( Z _ 1 , F ) . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} c _ n ^ G ( E \\otimes \\rho ) = \\sum _ { i = 0 } ^ n c _ { i } ( E ) c _ 1 ( \\rho ) ^ { n - i } \\in H ^ { 2 n } _ G ( Z , \\Z ) . \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} [ \\pi ( q _ 1 ) , \\pi ( q _ 2 ) , \\pi ( q _ 3 ) , \\pi ( q _ 4 ) ] & = \\left [ \\frac { 1 - \\alpha } { 4 - 2 \\alpha } , \\frac { 1 } { 4 - 2 \\alpha } , \\frac { 1 } { 4 - 2 \\alpha } , \\frac { 1 - \\alpha } { 4 - 2 \\alpha } \\right ] , \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} [ L ^ p ( \\Omega ) , H _ { p e r , b . c . } ^ { 2 , p } ( \\Omega ) ] _ { \\theta } & = [ R ( \\oplus _ { i \\in I } L ^ p ( \\tilde { \\Omega } _ i ) ) , R ( \\oplus _ { i \\in I } H _ { b . c . } ^ { 2 , p } ( \\tilde { \\Omega } _ i ) ) ] _ { \\theta } \\\\ & = R ( \\oplus _ { i \\in I } [ L ^ p ( \\tilde { \\Omega } _ i ) , H _ { b . c . } ^ { 2 , p } ( \\tilde { \\Omega } _ i ) ] _ { \\theta } ) . \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} g _ { k , j } = g _ { k + 1 , j } + ( - 1 ) ^ { k - j } E _ { k - 1 - j } ( x _ 1 , \\ldots , x _ { k - 1 } ) g _ { k + 1 , k - 1 } . \\end{align*}"}
-{"id": "10032.png", "formula": "\\begin{align*} x = \\sum _ { i = \\delta } ^ \\ell i \\alpha _ i , y = \\sum _ { i = \\delta } ^ { \\ell } { i \\choose 2 } \\alpha _ i , \\mbox { a n d } z = \\sum _ { i = \\delta } ^ { \\ell } \\alpha _ i . \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} P _ { j } ( z ) & = e ^ { \\frac { \\lambda } { \\xi } z } z ^ { - ( \\frac { \\mu j } { \\xi } - 1 ) } ( 1 - z ) ^ { - \\frac { ( c - j ) \\gamma } { \\xi } } \\{ - \\frac { ( c - j + 1 ) \\gamma } { \\xi } k _ { 1 } ( z ) - j \\left ( 1 + \\frac { \\mu } { \\xi } \\right ) \\sum _ { n = 0 } ^ { j } p _ { j , n } k _ { 2 } ( z ) + \\frac { \\mu } { \\xi } \\sum _ { n = 1 } ^ { j } n p _ { j , n } k _ { 3 } ( z ) \\\\ & - \\frac { \\mu } { \\xi } \\sum _ { n = 1 } ^ { j + 1 } n p _ { j + 1 , n } k _ { 4 } ( z ) \\} . \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} & ( B _ { 6 - n } A _ { 4 - n } A _ { 2 - n } + A _ { 6 - n } B _ { 4 - n } A _ { 2 - n } + A _ { 6 - n } A _ { 4 - n } B _ { 2 - n } ) | _ { r ^ { n - 6 } \\mathcal { H } _ 2 } = 8 ( n + 2 ) n ( n - 2 ) \\neq 0 ; \\\\ & ( B _ { 6 - n } A _ { 4 - n } A _ { 2 - n } + A _ { 6 - n } B _ { 4 - n } A _ { 2 - n } + A _ { 6 - n } A _ { 4 - n } B _ { 2 - n } ) | _ { r ^ { n - 4 } \\mathcal { H } _ 0 } = - 4 n ( n - 2 ) ( n - 4 ) \\neq 0 \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} & \\frac { 1 } { 5 } \\int _ 0 ^ { \\rho / \\epsilon } \\Big \\{ \\sigma ^ 2 [ ( 4 \\sigma ^ 2 + 1 0 ) ^ 2 - 1 8 \\sigma ^ 2 ( 4 \\sigma ^ 2 + 1 2 ) ] + 5 ( 4 \\sigma ^ 2 + 1 0 ) ( 1 + \\sigma ^ 2 ) ^ 2 \\Big \\} ( 1 + \\sigma ^ 2 ) ^ { - 8 } \\sigma ^ 9 d \\sigma \\\\ = & \\frac { 1 } { 5 } \\int _ 0 ^ { \\rho / \\epsilon } ( - 3 6 \\sigma ^ 6 - 4 6 \\sigma ^ 4 + 2 2 0 \\sigma ^ 2 + 5 0 ) ( 1 + \\sigma ^ 2 ) ^ { - 8 } \\sigma ^ 9 d \\sigma , \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } 0 _ { 1 \\times \\eta _ k } \\\\ I _ { \\eta _ k } \\end{array} \\right ] X _ { k k } \\left [ \\begin{array} { c c } 0 _ { \\eta _ k \\times 1 } & I _ { \\eta _ k } \\end{array} \\right ] = \\left [ \\begin{array} { c } I _ { \\eta _ k } \\\\ 0 _ { 1 \\times \\eta _ k } \\end{array} \\right ] X _ { k k } \\left [ \\begin{array} { c c } I _ { \\eta _ k } & 0 _ { \\eta _ k \\times 1 } \\end{array} \\right ] . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\forall \\ , a , \\ , b , \\ , c \\ , \\in A , \\ a \\ast ( b \\ast c ) = ( a \\ast b ) \\ast c . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} \\phi ( t ; x , \\xi ) : = u ( t ; x , \\eta ( t , 0 ; x , \\xi ) ) , \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} B ' _ j ( x _ k ) = \\left \\{ \\begin{aligned} & 0 & \\textrm { i f } \\ \\ k = j , \\\\ & \\frac { \\mp p ( 1 - c ) } { 2 ( p h c - s ) } , & \\textrm { i f } \\ \\ k = j \\pm 1 , \\\\ & 0 , & \\textrm { i f } \\ \\ k = j \\pm 2 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} \\varphi = \\sqrt { 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) } . \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { \\psi ( 2 t ) } { \\psi ( t ) } = 1 . \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} \\begin{array} { l } \\displaystyle s ^ { \\lambda , B } = \\sum _ { \\alpha \\in A _ r } s ^ { \\lambda , n } _ { \\alpha } I ^ { \\alpha } , \\\\ \\displaystyle s ^ { \\lambda , B _ 1 } = \\sum _ { \\alpha \\in A _ r } s ^ { \\lambda , n - 1 } _ { \\alpha } I ^ { \\alpha } . \\end{array} \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} \\sigma ( \\Im \\alpha ) = \\begin{cases} \\phantom { - } 1 & \\\\ - 1 & \\quad \\end{cases} \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} \\zeta _ a ( r ) = \\begin{cases} 0 , & r \\le \\max \\{ \\gamma , 2 \\} - 1 , \\\\ 2 r ^ { - a } , & r \\ge \\max \\{ \\gamma , 2 \\} . \\end{cases} \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} \\begin{array} { c c c c c c c c c c } \\left \\lceil \\cfrac { n - 2 } { p } \\right \\rceil + 1 & \\leq & v _ { p , 2 } ( n ) & \\leq & \\left \\lceil \\cfrac { n - 2 } { p } \\right \\rceil + 2 & \\textrm { f o r } & n \\not \\equiv 2 \\textrm { m o d } p \\\\ \\\\ & & v _ { p , 2 } ( n ) & = & \\left \\lceil \\cfrac { n - 2 } { p } \\right \\rceil + 2 & \\textrm { f o r } & n \\equiv 2 \\textrm { m o d } p . \\end{array} \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} S _ { n } \\left ( a q ^ { - n - 1 } ; q \\right ) q ^ { \\binom { n + 1 } { 2 } } = \\sum _ { k = 0 } ^ { n } \\left ( - 1 \\right ) ^ { k } S _ { k } \\left ( a q ^ { - k } ; q \\right ) q ^ { \\binom { k } { 2 } } . \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 : n ) } = \\Gamma ( 0 ) \\oplus \\Gamma ( 1 ) \\oplus \\Gamma ( 2 ) \\oplus \\Gamma ( 3 ) = \\bigoplus _ { j = 0 } ^ 3 \\Gamma ( j ) \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} \\Lambda ( n ) = \\begin{cases} \\log ( p ) & n = p ^ k , \\ p , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} T _ { i j } = h _ { i j } - g _ { i j } - \\epsilon ( H - n ) g _ { i j } \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\chi - 1 } U _ i A U _ i ^ * = - A . \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} d ( u , v ) : = \\sum _ { i = 1 } ^ { \\infty } \\frac { 1 } { 2 ^ { i } } \\left | \\left \\langle u - v , \\phi _ { i } \\right \\rangle \\right | \\ , . \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} Q _ { \\textbf { n } _ 1 , \\textbf { n } _ 2 } ( x ) : = \\sum _ { i = 1 } ^ p A _ { \\textbf { n } _ 1 , \\textbf { n } _ 2 , i } ( x ) w _ { 1 , i } ( x ) \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} - L u & = \\lambda \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = \\mu \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega , \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 3 } \\ [ \\bar { e } _ { 0 , k _ { \\pi ( 1 ) } } , [ \\bar { e } _ { 0 , k _ { \\pi ( 2 ) } + 1 } , \\bar { e } _ { 0 , k _ { \\pi ( 3 ) } - 1 } ] ] = 0 , \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} \\mathcal { N } ( U ) = \\Big ( 0 , \\frac { 1 } { \\rho } E ( | v | ) v _ i , 0 , 0 , \\frac { 1 } { \\kappa } q _ { i } \\Big ) ^ { T } u \\in D ( \\mathcal { N } ) \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} & \\deg \\left ( _ \\infty \\left ( \\frac { \\Delta } { \\Delta _ { p ^ n , 1 2 } } \\right ) \\right ) = p ^ { n - 1 } ( p + 1 ) - 1 \\\\ & \\deg \\left ( _ \\infty \\left ( \\frac { \\Delta _ p ^ n } { \\Delta _ { p ^ n , 1 2 } } \\right ) \\right ) = p ^ { n - 1 } ( p + 1 - p ) = p ^ { n - 1 } . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} \\begin{gathered} \\tau _ 1 + \\tau _ 2 = a _ { i j k } e ^ i \\otimes e ^ { j k } \\in V ^ * \\otimes \\Lambda ^ 2 V ^ * , \\tau _ 3 + \\tau _ 4 = c _ { i j k } e ^ i \\otimes e _ { j k } \\in V ^ * \\otimes \\Lambda ^ 2 V , \\\\ \\tau _ 5 + \\tau _ 6 = b _ { i j k } e ^ { n + i } \\otimes e _ { j k } \\in V \\otimes \\Lambda ^ 2 V , \\tau _ 7 + \\tau _ 8 = d _ { i j k } e ^ { n + i } \\otimes e ^ { j k } \\in V \\otimes \\Lambda ^ 2 V ^ * . \\end{gathered} \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} B '' = \\begin{pmatrix} ( 1 + ( k + 1 ) \\varepsilon _ 2 ) B + \\varepsilon _ 2 I & 0 \\\\ 0 & \\varepsilon _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} m ( y ) = \\begin{cases} 0 , & \\\\ 1 , & \\end{cases} \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} \\left | \\frac { p } { Q - \\gamma } \\right | ^ { p } \\frac { \\left | \\mathcal { R } g ( x ) \\right | ^ { p } } { | x | ^ { \\alpha p } } = \\frac { | g ( x ) | ^ { p } } { | x | ^ { \\frac { \\beta p } { p - 1 } } } , \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} a _ { 0 } \\frac { \\partial u } { \\partial t } - \\frac { \\partial } { \\partial x _ { j } } ( a ^ { i j } \\frac { \\partial u } { \\partial x _ { i } } ) = f , \\end{align*}"}
-{"id": "9917.png", "formula": "\\begin{align*} v = u ^ { - } ( - r ^ { - 1 } ) v ^ { \\max } ( u ( r ) v ) ^ { \\max } = \\sigma ( r ) v ^ { \\max } \\sigma ( r ) = \\begin{bmatrix} 0 & r \\\\ - r ^ { - 1 } & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} h ( Y , J _ 1 X ) - J _ 1 h ( X , Y ) = \\frac { 1 } { 2 ( 2 n - 1 ) } \\left [ g ( X , Y ) p ^ \\bot _ 1 - g ( X , J _ 1 Y ) J _ 1 ( p ^ \\bot _ 1 ) \\right ] , \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} V & = u ( V _ 1 , s _ 1 ) \\times \\dots \\times u ( V _ \\ell , s _ \\ell ) \\\\ & \\quad \\times ( \\nu ^ { \\alpha _ 1 } u ( U _ 1 , t _ 1 ) \\times \\nu ^ { - \\alpha _ 1 } u ( U _ 1 , t _ 1 ) ) \\times \\dots \\times ( \\nu ^ { \\alpha _ m } u ( U _ m , t _ m ) \\times \\nu ^ { - \\alpha _ m } u ( U _ m , t _ m ) ) \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) : = \\frac { 1 } { 2 } \\int _ { G } \\big \\| \\mathbf { u } ( t , \\cdot ) \\big \\| _ { \\mathbb { R } ^ { k } } ^ { 2 } \\mathrm { d } \\mathbf { x } + \\frac { \\tau } { 2 } \\int _ { G } \\big \\| \\mathbf { H } ( t , \\cdot ) - \\mathbf { F } ( \\mathbf { 0 } ) \\big \\| _ { \\mathbb { R } ^ { ( k \\times d ) \\times ( k \\times d ) } } ^ { 2 } \\mathrm { d } \\mathbf { x } . \\end{align*}"}
-{"id": "10149.png", "formula": "\\begin{align*} \\langle G _ { \\varphi } ( x ) , x - x _ p \\rangle & \\geq \\varphi ( x ^ + ) - \\min \\varphi + \\frac { t } { 2 } \\| G _ { \\varphi } ( x ) \\| ^ 2 \\\\ & \\geq \\varphi ( x ^ + ) - \\min \\varphi + \\frac { t \\epsilon } { 2 } ( \\varphi ( x ) - \\varphi ( x ^ + ) ) \\\\ & = \\frac { t \\epsilon } { 2 } ( \\varphi ( x ) - \\min \\varphi ) + ( 1 - \\frac { t \\epsilon } { 2 } ) ( \\varphi ( x ^ + ) - \\min \\varphi ) \\\\ & \\geq \\frac { t \\epsilon } { 2 } ( \\varphi ( x ) - \\min \\varphi ) , \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} \\phi ( x ) \\triangleq \\left ( \\begin{array} { c } \\nabla _ { x _ 1 } f _ 1 ( x _ 1 , \\bar x ) \\\\ \\vdots \\\\ \\nabla _ { x _ N } f _ N ( x _ N , \\bar x ) \\end{array} \\right ) , K = \\prod _ { i = 1 } ^ { N } K _ i , \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} f ( y ) = f ( y _ 0 ) + D f ( y _ 0 ) \\cdot ( y - y _ 0 ) + | y - y _ 0 | \\phi ( y - y _ 0 ) . \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} f \\in H ^ 1 ( M , w ^ 2 d x ) : = \\left \\{ f \\in H ^ 1 ( M ) \\ , : \\ , \\int _ M ( | \\nabla f | ^ 2 + | f | ^ 2 ) \\ , w ^ 2 \\ , d x < \\infty \\right \\} . \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} \\| \\eta \\| ( D ) = \\exp \\left ( - \\tfrac { 1 } { 4 } \\delta ( X ) \\right ) G ( D , \\sigma ( D ) ) \\left ( \\frac { \\| \\theta \\| ( D + R - Q ) } { G ( R , Q ) G ( D , Q ) G ( \\sigma ( D ) , R ) } \\right ) ^ { g - 1 } . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} \\int _ v \\phi ( z , v ) \\omega ( v ) & = \\int _ v \\phi _ 0 ( z , v ) \\omega ( v ) - \\int _ v \\int _ u \\phi _ 0 ( z , u ) \\omega ( u ) \\omega ( v ) - \\int _ v \\int _ u \\phi _ 0 ( v , u ) \\omega ( u ) \\omega ( v ) \\\\ & = \\int _ v \\phi _ 0 ( z , v ) \\omega ( v ) - \\int _ u \\phi _ 0 ( z , u ) \\omega ( u ) - 0 \\\\ & = 0 . \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} U ^ 0 = 1 , U ^ 1 = [ u _ { i j } ] _ { 1 \\le i , j \\le N } , \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} M = \\begin{bmatrix} I & \\Psi \\\\ 0 & I \\end{bmatrix} , \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} { \\tilde p } ( e ) = 1 - \\Big ( \\sum _ { j \\in S } \\chi ^ 2 _ 1 ( j ) , \\vert S \\vert \\Big ) \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} v _ { ( - 1 ) } \\gamma _ g ( w ) _ { ( - 1 ) } \\otimes g h \\otimes & v _ { ( 0 ) } \\otimes \\gamma _ g ( w ) _ { ( 0 ) } \\otimes g h = v _ { ( - 1 ) } \\alpha _ g ( w _ { ( - 1 ) } ) \\otimes g h \\otimes v _ { ( 0 ) } \\otimes \\gamma _ g ( w _ { ( 0 ) } ) \\otimes g h \\\\ & = \\left ( ( v _ { ( - 1 ) } \\otimes g ) ( w _ { ( - 1 ) } \\otimes h ) \\right ) \\otimes F _ 2 ( V , W ) ( v _ { ( 0 ) } \\otimes g \\otimes w _ { ( 0 ) } \\otimes h ) . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} \\left \\langle ( x - \\lambda ) ^ { - 1 } u , p \\right \\rangle : = \\left \\langle u , \\theta _ { \\lambda } p \\right \\rangle , \\left \\langle u v , p \\right \\rangle : = \\left \\langle u , v p \\right \\rangle . \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u = L _ 0 u + f + \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } ( \\Lambda ^ k _ 0 u + g ^ { k } ) d w _ { s } ^ { k } \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} \\sum _ { k \\in S _ { m , N } } | x _ k | ^ { p _ 1 } & \\leq \\left ( \\sum _ { k \\in S _ { m , N } } | x _ k | ^ { p _ 2 } \\right ) ^ \\frac { p _ 1 } { p _ 2 } \\left ( \\sum _ { k \\in S _ { m , N } } 1 \\right ) ^ { 1 - \\frac { p _ 1 } { p _ 2 } } \\\\ & = | S _ { m , N } | ^ { 1 - \\frac { p _ 1 } { p _ 2 } } \\left ( \\sum _ { k \\in S _ { m , N } } | x _ k | ^ { p _ 2 } \\right ) ^ \\frac { p _ 1 } { p _ 2 } \\\\ & = | S _ { m , N } | \\left ( \\frac { 1 } { | S _ { m , N } | } \\sum _ { k \\in S _ { m , N } } | x _ k | ^ { p _ 2 } \\right ) ^ \\frac { p _ 1 } { p _ 2 } . \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( \\lambda ^ 2 H ) = ( \\lambda ^ 2 H ) | A | ^ 2 , \\\\ A ( { \\rm g r a d } ( \\lambda ^ 2 H ) ) + ( \\lambda ^ 2 H ) { \\rm g r a d } H = 0 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} \\int _ \\Omega | u _ 1 - u _ 2 | d x \\leq \\int _ \\Omega ( f _ 1 - f _ 2 ) \\varphi _ 0 ( u _ 1 - u _ 2 ) d x & + \\int _ \\Omega ( - g \\circ u _ 1 + g \\circ u _ 2 ) s i g n ( u _ 1 - u _ 2 ) \\varphi _ 0 d x \\\\ & - \\int _ { \\partial \\Omega } \\frac { \\partial \\varphi } { \\partial { \\bf { n } } _ { L ^ { * } } } d | \\eta _ 1 - \\eta _ 2 | \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{gather*} T _ { 1 0 } ^ { \\alpha } T _ { 2 0 } ^ { \\beta } T _ { 2 1 } ^ { \\gamma } = ( - 1 ) ^ { \\frac { \\alpha ( \\alpha - 1 ) } { 2 } + \\frac { \\beta ( \\beta - 1 ) } { 2 } + \\frac { \\gamma ( \\gamma - 1 ) } { 2 } + \\alpha \\gamma } Q _ { 2 } ^ { \\alpha + \\beta } Q _ { 1 } ^ { \\alpha - \\gamma } Q _ { 0 } ^ { - \\alpha - \\beta } . \\end{gather*}"}
-{"id": "6843.png", "formula": "\\begin{align*} & \\delta ^ * ( 1 / M , 0 ) \\leq \\delta _ { \\mathsf { C a - I A } } = \\frac { M + K - 1 } { M } . \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} \\log \\| \\varphi _ { g } \\| ( X ) = 4 \\tbinom { 2 g } { g - 1 } \\left ( \\tfrac { g + 1 } { g } B ( X ) - ( g - 1 ) S _ { g } ( X ) - ( g + 1 ) \\log \\pi \\right ) . \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} A _ { q } ( c ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 3 n ^ { 2 } + n } \\left ( - c ^ { 2 } \\right ) ^ { n } } { \\left ( q ^ { 2 } , c q , c q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } a ( n ) ^ 2 v _ Y ( \\frac { n } { X } ) = V _ Y ( \\gamma _ { 1 } ) R _ 1 X ^ { \\gamma _ { 1 } } + O ( Y ^ { \\eta _ { 1 } + \\epsilon } X ^ { \\sigma _ 1 } ) . \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{gather*} \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta ) } = \\big \\langle T _ { 1 } ^ { k } T _ { 2 } ^ { \\ell } v _ { 0 } , g ^ { ( \\alpha , \\beta ) } v _ { 0 } \\big \\rangle . \\end{gather*}"}
-{"id": "5451.png", "formula": "\\begin{align*} p ^ * ( v , v ) = - \\sqrt { 2 } ( x z + y z ) , \\end{align*}"}
-{"id": "10125.png", "formula": "\\begin{align*} \\omega = ( 2 i p \\sqrt { c } y + b ( p + q ) x + a ( p + 2 q ) x ^ 2 ) d x + ( 2 q i \\sqrt { c } x + 2 q x y ) d y \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} \\Delta ( z ) = T \\ , M \\big ( E _ u ( z ) ^ { - 1 } , E _ v ( z ) ^ { - 1 } \\big ) ^ { 1 - q } \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\omega _ { \\mu , a } ( x ) = x ^ { \\frac { \\mu } { 2 } } I _ { \\mu } ( 2 a \\sqrt { x } ) , x > 0 . \\end{align*}"}
-{"id": "10151.png", "formula": "\\begin{align*} \\varphi ( x ^ { ( t ) } ) - \\min \\varphi = & \\left ( \\varphi ( x ^ { ( t ) } ) - \\varphi ( x ^ { ( t + 1 ) } ) \\right ) + \\left ( \\varphi ( x ^ { ( t + 1 ) } ) - \\min \\varphi \\right ) \\\\ \\geq & \\frac { L _ { \\min } } { 2 } \\| x ^ { ( t ) } - x ^ { ( t + 1 ) } \\| ^ 2 + \\left ( \\varphi ( x ^ { ( t + 1 ) } ) - \\min \\varphi \\right ) \\\\ \\geq & \\left ( \\frac { \\eta ^ 2 L _ { \\min } } { 4 p L ^ 2 + 4 L _ { \\max } ^ 2 } + 1 \\right ) \\left ( \\varphi ( x ^ { ( t + 1 ) } ) - \\min \\varphi \\right ) , \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} V ( z ) ~ & : = ~ \\big ( \\tau - z ^ 2 \\big ) W ( z ) - 1 _ { \\R ^ + } \\big ( \\tau - z ^ 2 \\big ) . \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} F ^ { \\nu } ( x ) : = g ( x ) + h _ { \\nu } ( c ( x ) ) , \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} G _ \\lambda ( x , y ) = \\sum _ { \\xi \\in \\N ^ 2 } \\frac { e _ \\xi ( x - y ) } { | \\xi | ^ 2 - \\lambda } . \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} \\{ f \\in C ^ \\infty ( [ 0 , 1 ] , \\mathbb { R } ^ n ) , f _ i ( 1 ) = f _ i ( 0 ) = f _ j ( 0 ) = f _ j ( 1 ) , ~ \\forall i \\neq j \\} . \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{gather*} s _ { 0 } = s _ { 1 } = s _ { 2 } = \\ \\ldots \\ = s _ { n - 1 } = 0 \\ , \\ \\ s _ { n } \\neq 0 \\ , \\end{gather*}"}
-{"id": "9223.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { j = 1 } ^ k \\sum _ { i } \\log \\mu _ j ( x _ i ^ j ) = \\sum _ { j = 1 } ^ k \\frac { c _ { j j } } { N } \\frac { 1 } { c _ { j j } } \\sum _ { i } \\log \\mu _ j ( x _ i ^ j ) . \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} \\operatorname { c u r l } \\mathbf { E } = - \\frac { 1 } { c } \\frac { \\partial \\mathbf { B } } { \\partial t } \\ \\left ( \\right ) , \\qquad \\qquad \\operatorname { d i v } \\mathbf { B } = 0 \\ \\left ( \\right ) \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} J _ s ( z ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ n } { \\Gamma ( n + 1 ) \\Gamma ( n + 1 + s ) } \\left ( \\frac { z } { 2 } \\right ) ^ { s + 2 n } \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} \\mathbb { E } [ C _ 2 ^ ] = \\frac { e ^ { \\frac { 1 } { \\beta \\xi } } } { \\ln ( 2 ) } E _ 1 \\left ( \\frac { 1 } { \\beta \\xi } \\right ) - \\frac { e ^ { \\frac { 2 } { \\beta \\xi } } } { \\ln ( 4 ) } E _ 1 \\left ( \\frac { 2 } { \\beta \\xi } \\right ) \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} \\frac { \\partial \\lambda } { \\partial t } = a _ n ( { g } , D ) = \\zeta ( 0 , g , D ) \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} a ^ { 2 } \\nabla _ { \\nu } b ^ { 2 } \\leqslant a ^ { 2 } \\sharp _ { \\nu } b ^ { 2 } + ( a - b ) ^ 2 - \\sum _ { k = 0 } ^ { \\infty } r _ { k } \\big [ a ^ { \\frac { m _ k } { 2 ^ k } } b ^ { 1 - \\frac { m _ k } { 2 ^ k } } - a ^ { \\frac { m _ k + 1 } { 2 ^ k } } b ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } . \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { t } a _ { p } c _ { p } = \\det ( \\mathbf { H } _ { \\bar { \\mathcal { R } } _ i , \\mathcal { T } } ) . \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} G _ 1 ' = m \\ , F _ 2 ' + n \\ , F _ 1 ' \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\tilde { \\kappa } _ { ( r ) } = \\sum _ { \\lambda , \\alpha } s ^ { \\lambda } _ { \\alpha } I ^ \\alpha \\kappa _ { \\lambda } = \\sum _ { i = 0 } ^ { r } I _ i \\ \\kappa _ { ( r - i ) } \\cdot \\kappa _ { ( 1 ) } ^ i \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} \\dim _ E ( \\Lambda _ \\Gamma ^ c ) = \\left \\{ \\begin{array} { c c c } \\delta _ \\Gamma & & \\delta _ \\Gamma \\leq 2 ( n - 1 ) \\\\ 2 ( n - 1 ) + \\frac { 1 } { 2 } ( \\delta _ \\Gamma - 2 ( n - 1 ) ) & & \\delta _ \\Gamma > 2 ( n - 1 ) \\end{array} \\right . \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} \\iota ( x , v ) = \\inf _ { \\tau \\in \\mathbb { R } } \\left | x - v \\tau \\right | \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} K ( t ; x , y ; \\Delta ) = \\sum _ k e ^ { - \\lambda _ k t } \\psi _ k ( x ) \\psi _ k ( y ) \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} \\| m _ { \\mathbf { B } , V } ^ \\frac 1 2 ( - i \\nabla + \\mathbf { A } ) u \\| ^ 2 = \\langle ( - i \\nabla m _ { \\mathbf { B } , V } ) ( - i \\nabla + \\mathbf { A } ) u , u \\rangle + \\langle m _ { \\mathbf { B } , V } ( - i \\nabla + \\mathbf { A } ) ^ 2 u , u \\rangle \\ , . \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\delta ( K ' ) \\ge \\left ( \\frac 5 9 + \\frac 1 2 \\lambda \\right ) \\binom { t ' - 1 } 2 . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} X _ { \\alpha } ^ { \\varepsilon } \\left ( t \\right ) = M _ { \\alpha } ^ { \\varepsilon } \\left ( t \\right ) + \\int _ { 0 } ^ { t } \\left ( D _ { \\alpha } ^ { \\ast } \\tilde { V } ^ { \\prime \\prime } \\right ) \\left ( \\hat { \\eta } _ { s } ^ { \\varepsilon } \\left ( 0 , \\varepsilon e _ { \\alpha } \\right ) \\right ) d s , \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} V _ n ( \\mathbf { x } ) = \\prod _ { 1 \\leq i < j \\leq n } \\frac { x _ j - x _ i } { j - i } . \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} \\phi _ T ( X ) & = T X + g X ^ q - h ^ { q - 1 } X ^ { q ^ 2 } , \\\\ \\psi _ T ( X ) & = T X + h ^ { q - 1 } X ^ { q - 1 } , \\\\ \\lambda & = \\lambda _ T \\ , h ^ { - 1 } \\in \\psi [ T ] , \\\\ \\tilde { \\jmath } ( \\phi , \\lambda ) & = g ^ { ( q + 1 ) / 2 } / h ^ { ( q - 1 ) / 2 } \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} g _ 2 ^ n ( t ) = \\int ^ t _ 0 P _ p ( { t - r } ) f ( r , u ^ n ( r ) , \\nabla u ^ n ( r ) ) \\d r . \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { n - 1 } | a _ i | + | b | \\leq C . \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} \\mathbf { B } \\triangleq \\left \\{ \\cup _ { j = 0 } ^ { N } \\mathbf { B } _ j \\right \\} \\cap B _ { g ( 0 ) } ( x _ 0 , r ) = \\cup _ { j = 0 } ^ { N } \\left \\{ \\mathbf { B } _ j \\cap B _ { g ( 0 ) } ( x _ 0 , r ) \\right \\} . \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} P _ { s e r } = \\mathbb { P } ( J = 1 ) = 1 - P _ { v a c } = 1 - p _ { 0 , 0 } \\frac { \\xi } { \\gamma A } , \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} L _ v [ X , Z + 1 ] \\ ; \\approx \\ ; X ( Z + 1 ) ( 1 - v ) = X ( 1 - v ) + X ( 1 - v ) Z , \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} J ( X _ i ) = Y _ i , J ( Y _ i ) = - X _ i J ( \\xi ) = \\partial _ t , \\ i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} \\dd \\nu _ t = ( J - I ) \\nu _ { t - } \\ , \\dd N _ t , \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} P _ { k } = \\sum _ { 1 \\leq i , j \\leq n } D _ i a _ { i j } ^ { ( k ) } ( x ) D _ { j } , k = 1 , 2 , \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} F _ p ( v , \\zeta ) = F ^ * _ p ( v ) + P _ p \\Pi ( \\zeta ) , \\end{align*}"}
-{"id": "9990.png", "formula": "\\begin{align*} \\widehat { F } ( x ) = \\frac { n ! \\omega _ n } { ( 1 + 4 \\pi ^ 2 \\norm { x } _ 2 ^ 2 ) ^ { ( n + 1 ) / 2 } } , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} V ( x ) = \\dfrac { V _ N ( x ) } { \\gamma - R _ M ( x ) } \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} X = \\bigcup _ { \\psi ' \\in C X } U _ \\alpha ( \\psi ' ) . \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} \\hat { W } _ { { \\mathcal { R } } , { \\mathcal { T } ' } } = \\left \\{ W _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { \\mathcal { T } ' } : \\mathcal { T } \\supset \\mathcal { T } ' \\right \\} . \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 x : n ) } = \\bigoplus _ { ( i , \\alpha ) } T _ { ( 2 x ) } ( i , \\alpha ) \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} p ( n , k ) = \\frac { 2 } { n - 1 } p ( n - 1 , k ) + \\sum _ { i = 2 } ^ { n - 2 } \\sum _ { j = 1 } ^ { k - 1 } \\frac { p ( i , j ) p ( n - i , k - j ) } { n - 1 } , \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} Z _ { t | s } = \\exp ( Q _ \\lambda ( t - s ) ) Z _ s + \\big ( \\exp ( Q ( t - s ) ) - \\exp ( Q _ \\lambda ( t - s ) ) \\big ) X _ s . \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} p _ 0 : = 2 , p _ { n + 1 } : = \\frac { 3 p _ n } { 2 p _ n + 1 } \\hbox { a n d } q _ 0 : = 2 , q _ { n + 1 } : = \\frac { 3 q _ n } { 2 q _ n + 1 } , n \\in \\N _ 0 , \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} f ( n ) = \\left \\{ \\begin{array} { l l } x _ 0 & \\mbox { i f } n = s _ k , \\\\ f _ 0 ( n ) & \\mbox { e l s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} \\left [ a _ { 1 } , \\dots , a _ { n } \\right ] \\coloneqq \\left \\{ \\left ( x _ { i } \\right ) _ { i \\in \\mathbb { N } } \\in S ^ { \\mathbb { N } } \\middle | \\ ; \\forall 1 \\leq i \\leq n : x _ { i } = a _ { i } \\right \\} \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} \\frak { a } ( V , V ^ * ) & = \\int _ { \\Omega } ( A _ { i J K j } u _ { s , R } u ^ * _ { i , J } + C _ { i I J K L j } u _ { j , L K } u ^ * _ { i , I J } ) \\ \\mathrm { d } x \\\\ & + \\int _ { \\Omega } ( M _ { i J K L } \\tau _ { , L } u ^ * _ { i , J K } + M _ { j L K I } u _ { j , K L } \\tau ^ * _ { , I } + K _ { I J } \\tau _ { , I } \\tau ^ * _ { , J } ) \\ \\mathrm { d } x , \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} \\eta _ n ^ 1 : = \\eta ( B _ n ^ c ) + \\sum _ { k = 1 } ^ { Y ^ 1 } \\delta _ { U _ k } . \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} & D _ { 1 } ( L _ { n } ) = n L _ n , & \\ & D _ { 1 } ( G _ n ) = n G _ n , & \\\\ & D _ { 2 } ( L _ { n } ) = 0 , & \\ & D _ { 2 } ( G _ n ) = G _ n , & \\\\ & D _ { 3 } ( L _ { n } ) = n G _ { n - 1 } , & \\ & D _ { 3 } ( G _ n ) = 0 , & \\\\ & D _ { 4 } ( L _ { n } ) = 0 , & \\ & D _ { 4 } ( G _ n ) = L _ { n + 1 } . & \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} J : = \\int _ 0 ^ { 2 \\pi } \\frac { \\cos \\theta } { \\pi R } \\cdot \\log | L ( s + R e ^ { i \\theta } , \\chi ) | d \\theta . \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} \\phi _ \\pm ( x , \\xi + 2 \\pi m ) = \\phi _ \\pm ( x , \\xi ) + 2 \\pi x \\cdot m , x , \\xi \\in \\mathbb { R } ^ d , \\ m \\in \\mathbb { Z } ^ d . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} \\rho _ - ( t ) = & \\sup \\{ r : B _ r ( y ) \\textrm { i s e n c l o s e d b y } \\hat { M } ( t ) \\textrm { f o r s o m e } y \\in \\mathbb { H } ^ { n + 1 } \\} , \\\\ \\rho _ + ( t ) = & \\inf \\{ r : B _ r ( y ) \\textrm { e n c l o s e s } \\hat { M } ( t ) \\textrm { f o r s o m e } y \\in \\mathbb { H } ^ { n + 1 } \\} . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} A _ { q } \\left ( x c \\right ) = \\left ( c q ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } c ^ { n } S _ { n } \\left ( x q ^ { - n } ; q \\right ) } { \\left ( c q ; q \\right ) _ { n } } . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\Omega = C ( v _ { 0 } , 1 / L ) \\cap M \\ \\ \\ell \\Omega = C ( v _ { 0 } , \\ell / L ) \\cap M \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} \\Delta _ { K _ h } ( \\phi ( \\delta _ m ( B _ { A } ) ) ) = \\Delta _ { K _ h } ( \\phi ( \\delta _ m ( B _ { A ' } ) ) ) = c . \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} u ^ * = - \\mathrm { a r c s i n h } ( \\tilde { v } \\sinh u ) = - \\mathrm { a r c s i n h } \\tilde { \\chi } . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} c _ 2 ^ \\# = a \\ , I d + b \\begin{pmatrix} I & 0 \\\\ 0 & \\pm I \\end{pmatrix} , I = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , \\ ; \\ ; b \\neq 0 , \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} p _ n = \\frac { n ^ { - \\alpha } } { \\sum _ { \\tilde { n } = 1 } ^ N \\tilde { n } ^ { - \\alpha } } , \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{gather*} h _ { \\underline { k } , \\ell } = \\frac { \\tau _ { k + 1 , \\ell } } { \\tau _ { k , \\ell } } , h _ { { k } , \\underline { \\ell } } = \\frac { \\tau _ { k , \\ell + 1 } } { \\tau _ { k , \\ell } } , h _ { \\underline { k } , \\underline { \\ell } } = \\frac { \\tau _ { k + 1 , \\ell + 1 } } { \\tau _ { k , \\ell } } . \\end{gather*}"}
-{"id": "7026.png", "formula": "\\begin{align*} G \\otimes H = \\left ( \\oplus _ i G _ i \\right ) \\otimes \\left ( \\oplus _ j H _ j \\right ) \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} \\phi ( Z ( t ) ) = \\phi ( Z ( 0 ) ) + M ( t ) + A ( t ) , \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} I m ( d _ 3 ) = K e r ( \\phi ) = 0 . \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} | \\nabla \\varphi | ^ 2 + \\partial _ t \\varphi = 0 , \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} h _ i ( T ^ { a _ 1 \\cdots a _ r } _ { b _ 1 \\cdots b _ r } ) = T ^ { a _ 1 \\cdots a _ r } _ { b _ 1 \\cdots b _ r } + \\varepsilon T ^ { a _ 1 \\cdots a _ { i - 1 } c a _ { i + 1 } \\cdots a _ r } _ { b _ 1 \\cdots b _ { i - 1 } c b _ { i + 1 } \\cdots b _ r } \\delta ^ { a _ i } _ { b _ i } . \\end{align*}"}
-{"id": "9890.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 | S _ { m , n } ( c , \\eta ) | ^ 2 \\ , d \\eta = 1 \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} \\bar { u } ( x ) : = \\frac { u ( x _ 0 + \\lambda x ) - \\mathcal { W } _ { x _ 0 } ( x _ 0 + \\lambda x ) } { \\lambda ^ { 3 / 2 + \\alpha } } , \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} p _ { k + 1 } & \\leq s p _ k + \\frac { ( s p _ k - q _ { k + 1 } ) q _ k } { 1 - q _ k - q _ { k + 1 } } = \\\\ & = s p _ k \\frac { 1 - q _ k } { 1 - q _ k - q _ { k + 1 } } - \\frac { q _ { k + 1 } q _ k } { 1 - q _ k - q _ { k + 1 } } \\leq s \\frac { p _ k } { 1 - \\frac { q _ { k + 1 } } { 1 - q _ k } } . \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} \\frac { \\left ( - q ; q \\right ) _ { \\infty } ^ { 2 } } { \\left ( q ; q \\right ) _ { \\infty } } = \\frac { 1 } { \\sqrt { 2 \\pi \\log q ^ { - 1 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } \\right ) d x } { \\left | \\left ( q ^ { 1 / 2 } e ^ { i x } ; q \\right ) _ { \\infty } \\right | ^ { 2 } } \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} \\omega _ \\ell & = \\begin{cases} N & \\ell = 0 , \\\\ \\frac { N \\left ( 1 - \\frac { 1 } { N } \\right ) ^ \\ell } { N + ( \\ell + 1 - N ) \\left ( 1 - \\frac { 1 } { N } \\right ) ^ \\ell } & \\ell \\in [ \\overline { L } - 1 ] , \\\\ 0 & \\ell = \\overline { L } . \\end{cases} \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\tan \\theta = \\frac { - f _ { u v } \\pm \\sqrt { f _ { u v } ^ 2 - f _ { u u } f _ { v v } } } { f _ { v v } } \\bigg | _ { u = v = 0 } \\ ; , \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\cot \\theta = \\frac { - f _ { u v } \\pm \\sqrt { f _ { u v } ^ 2 - f _ { u u } f _ { v v } } } { f _ { u u } } \\bigg | _ { u = v = 0 } \\ ; . \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} \\Delta { K } ^ s _ { n + 1 } & = 0 \\\\ \\Delta { K } ^ s _ { t } & = \\big ( \\mu ^ s _ 1 ( t ) ( \\beta _ t - 1 ) - \\mu ^ s _ 0 ( t ) ( \\alpha _ t - 1 ) \\big ) + H ( \\alpha _ t ) - H ( \\beta _ t ) \\\\ & \\qquad \\qquad + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu ^ s _ 1 ( t ) + \\Delta { K } ^ s _ { t + 1 } } } { 1 + 2 ^ { \\mu ^ s _ 0 ( t ) + \\Delta { K } ^ s _ { t + 1 } } } \\Big ) + s , ~ t \\in \\{ n , \\ldots , 0 \\} . \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} \\frac { \\partial \\rho } { \\partial t } + \\operatorname { d i v } \\mathbf { j } = 0 , \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\mathcal { C } _ { r , s } ( \\varepsilon ; \\tau ) = F _ { 2 p - s - p r , 2 p } \\left ( \\frac { - i \\varepsilon } { \\sqrt { 2 p } } ; \\frac { \\tau } { 4 p } \\right ) - F _ { 2 p + s - p r , 2 p } \\left ( \\frac { - i \\varepsilon } { \\sqrt { 2 p } } ; \\frac { \\tau } { 4 p } \\right ) . \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} { \\cal E } _ { [ 0 , n ] } ^ { F B } ( \\kappa ) \\triangleq & \\Big \\{ g _ t : { \\cal M } _ n \\times { \\cal Y } ^ { t - 1 } \\longmapsto { \\cal X } _ t , ~ ~ x _ 0 = g _ 0 ( w , y ^ { - 1 } ) , x _ t = e _ t ( w , y ^ { i - 1 } ) , ~ ~ w \\in { \\cal M } _ n , ~ t = 0 , \\ldots , n : \\\\ & \\frac { 1 } { n + 1 } { \\bf E } ^ g \\Big ( c _ { 0 , n } ( X ^ n , Y ^ { n - 1 } ) \\Big ) \\leq \\kappa \\Big \\} . \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} \\min _ { i = 1 , \\ldots , N } \\| \\mathcal { G } _ { 1 / \\tilde \\mu } ( y _ j ) \\| ^ 2 \\leq & \\frac { 4 8 \\tilde \\mu ^ 2 } { \\tilde { \\mu } - \\mu } \\left ( \\frac { \\| x ^ * - v _ 0 \\| ^ 2 } { N ( N + 1 ) ( 2 N + 1 ) } + \\frac { M ^ 2 ( r + \\frac { \\rho } { 2 } ( N + 3 ) ) } { ( N + 1 ) ( 2 N + 1 ) } + \\frac { 4 L \\sum _ { j = 1 } ^ N \\tfrac { 2 \\varepsilon _ j + \\delta _ j } { a _ j ^ 2 } } { N ( N + 1 ) ( 2 N + 1 ) } \\right ) . \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} \\oplus : \\mathcal { I } _ 0 \\times \\mathcal { I } _ 0 \\longrightarrow \\mathcal { I } _ 0 \\ \\ \\ \\ \\ \\ a ^ 0 \\oplus b ^ 0 = [ \\mu ( a ^ 0 \\cup b ^ 1 ) ] ^ 0 \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} \\varDelta _ { \\mathcal { D } } ( f , p ) = \\widetilde { \\varDelta } ( f , p ) + \\varDelta ( f , p | \\mathcal { D } _ f \\times \\mathcal { D } _ p ) \\to \\inf , \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{gather*} \\left | \\begin{array} { l l l l l l } s _ { 0 } & s _ { 1 } & \\ldots & s _ { n } \\\\ s _ { 1 } & s _ { 2 } & \\ldots & s _ { n + 1 } \\\\ \\ldots & \\ldots & \\ldots & \\ldots \\\\ s _ { n } & s _ { n + 1 } & \\ldots & s _ { 2 n } \\end{array} \\right | = t _ { n } \\ , \\ \\ \\ \\ \\ n \\geq 0 \\ . \\end{gather*}"}
-{"id": "6626.png", "formula": "\\begin{align*} \\Gamma _ { M } \\bigl ( w \\ , | \\ , a , a _ { M } \\bigr ) = & e ^ { \\phi _ { M } ( w , x \\ , | \\ , a , a _ { M } ) } \\ , \\Gamma _ { M - 1 } ( w \\ , | \\ , a ) \\prod \\limits _ { k = 1 } ^ \\infty \\frac { \\Gamma _ { M - 1 } ( w + k a _ { M } \\ , | \\ , a ) } { \\Gamma _ { M - 1 } ( x + k a _ { M } \\ , | \\ , a ) } \\times \\\\ & \\times \\exp \\Bigl ( \\Psi _ { M } ( x , k a _ { M } \\ , | \\ , a ) - \\Psi _ { M } ( w , k a _ { M } \\ , | \\ , a ) \\Bigr ) . \\end{align*}"}
-{"id": "9980.png", "formula": "\\begin{align*} d _ { G H } ( A , B ) = \\inf d _ H \\bigl ( \\varphi ( A ) , \\psi ( B ) \\bigr ) , \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} \\dot { x } = H ( Y , \\lambda ) + O ( z ) , \\dot { Y } = O ( z ) , \\dot { z } = Q ( x , Y , \\lambda ) z + O _ 2 ( z ) \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x ^ { \\nu } } T _ { \\mu } ^ { \\ \\nu } = \\frac { \\partial T _ { \\mu } ^ { \\ 0 } } { \\partial x _ { 0 } } + \\frac { \\partial T _ { \\mu } ^ { \\ q } } { \\partial x _ { q } } \\qquad \\left ( \\mu , \\nu = 0 , 1 , 2 , 3 ; p , q = 1 , 2 , 3 \\right ) , \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} & [ L _ n , G _ m ] _ t = [ L _ n , G _ m ] + \\sum _ { i \\geq p } a _ i \\varphi _ 1 ( L _ n , G _ m ) t ^ i ; \\\\ & [ L _ n , L _ m ] _ t = [ L _ n , L _ m ] + \\varphi _ 0 ( L _ n , L _ m ) ; \\\\ & [ L _ n , G _ m ] _ t ' = [ L _ n , G _ m ] + \\lambda \\varphi _ 1 ( L _ n , G _ m ) t ; \\\\ & [ L _ n , L _ m ] _ t ' = [ L _ n , L _ m ] + \\varphi _ 0 ( L _ n , L _ m ) . \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} r ^ { * k } ( X ) = r ( E - X ) + k | X | - r ( E ) . \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} g _ 1 = s W \\begin{pmatrix} - x _ 3 & - x _ 4 \\\\ x _ 4 & - x _ 3 \\end{pmatrix} , g _ 2 = s W \\begin{pmatrix} x _ 1 & x _ 2 \\\\ - x _ 2 & x _ 1 \\end{pmatrix} \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + { \\rm c o n s t } ) } \\ , { \\bf E } \\bigl [ M ^ q _ { ( \\tau = 1 , \\alpha , \\alpha ) } \\bigr ] , \\ ; N \\rightarrow \\infty . \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} 0 = W _ { i - 1 } H ^ i ( X , \\C ) \\subset W _ i H ^ i ( X , \\C ) \\subset \\cdots W _ { 2 i } H ^ i ( X , \\C ) = H ^ i ( X , \\C ) . \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} \\varphi _ p = e ^ { 1 2 3 } + e ^ { 1 4 5 } + e ^ { 1 6 7 } + e ^ { 2 4 6 } - e ^ { 2 5 7 } - e ^ { 3 4 7 } - e ^ { 3 5 6 } , \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{gather*} \\sum _ { k \\geq 0 } \\frac { s _ { k } } { z ^ { k + 1 } } = \\frac { q ( z ) } { p ( z ) } \\ , \\ \\ \\ \\ | z | > R \\ , \\end{gather*}"}
-{"id": "2406.png", "formula": "\\begin{align*} \\mathbb { V } a r ( T _ { ( k ) } ) = \\sum _ { j = 1 } ^ { k } \\mathbb { V } a r ( W _ { j } ) = \\sum _ { j = 1 } ^ { k } \\frac { 1 } { ( n - k + j ) ^ 2 } , \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} G _ 1 ( q t ) = \\bar B ( t ) G _ 1 ( t ) B _ 1 ^ { - 1 } \\mbox { f o r s m a l l } t . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} y [ n ] = \\sum _ { k = 0 } ^ { J - 1 } f _ \\mathit { k } \\ , x [ n - k ] , \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} & F _ { t , \\varsigma } \\big ( \\sqrt { t } D ^ F _ { Z _ { j , R } } \\big ) ( x , y ) , G _ { t , \\varsigma } \\big ( D ^ F _ { Z _ { j , R } } \\big ) ( x , y ) \\\\ & \\in \\left ( \\Lambda ^ \\bullet \\big ( T ^ * Z _ { j , R } \\big ) \\otimes F \\right ) _ x \\otimes \\left ( \\Lambda ^ \\bullet \\big ( T ^ * Z _ { j , R } \\big ) \\otimes F \\right ) ^ * _ y , x , y \\in Z _ R , \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{gather*} V _ A \\hat { \\Delta } = - \\frac { 1 } { \\sqrt { m } } y _ m + \\mathit { r e s t } _ 2 , \\\\ \\| \\mathit { r e s t } _ 2 \\| \\leq \\| \\hat { \\Delta } \\| \\cdot o _ p ( 1 ) . \\end{gather*}"}
-{"id": "3660.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { q ^ { ( n ^ 2 + n ) / { 2 } } ( - q ; q ) _ n } { ( q ; q ) ^ 2 _ n } = \\sum _ { \\pi \\in \\mathcal { D } } \\widetilde { \\omega } _ 2 ( \\pi ) q ^ { | \\pi | } . \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} \\Gamma \\ni I ( z + \\omega ) - I ( z ) = \\psi ( \\bar z + \\bar \\omega ) - \\psi ( \\bar z ) = a \\bar \\omega \\ \\ \\ \\ \\forall \\omega \\in \\Gamma , \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{gather*} \\mu _ { n } : = \\frac { Q _ { r } ( \\lambda _ { n } ) } { P _ { r } ^ { \\ , \\prime } ( \\lambda _ { n } ) } = \\left ( \\sum \\limits _ { k = 0 } ^ { r - 1 } \\dfrac { P _ { k } ( \\lambda _ { n } ) ^ { 2 } } { D _ { k } D _ { k - 1 } } \\right ) ^ { - 1 } \\in ( 0 , + \\infty ) \\ , \\ \\ \\ \\ 1 \\leq n \\leq r \\ , \\end{gather*}"}
-{"id": "749.png", "formula": "\\begin{align*} L ^ 2 ( Y _ N ) = \\bigoplus _ \\pi m ( \\pi ) \\pi ^ { K ( N ) K _ H K _ \\infty } , \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} \\sum ^ { n - 1 } _ { j = 1 } \\Big | \\sum ^ { n - 1 } _ { i = j } \\frac { b _ { 1 + i } } { \\lambda ^ { 1 + i } } \\Big | < 1 , \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { k } u _ i = \\prod _ { i = 1 } ^ { k } \\left ( \\sum _ { \\chi \\in \\widehat { G / N } } \\tau _ { \\chi } ( u _ i ) \\right ) = \\sum _ { \\chi _ 1 , \\ldots , \\chi _ { k } \\in \\widehat { G / N } } \\tau _ { \\chi _ 1 } ( u _ 1 ) \\cdots \\tau _ { \\chi _ { k } } ( u _ { k } ) . \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} 0 & = \\left ( \\left ( \\mathbf { I } + \\bar { K } ( \\nu , 0 ) \\right ) r _ { 0 } , r _ { 0 } \\right ) \\\\ & \\ + \\left ( \\left ( \\bar { K } ( \\nu , \\varepsilon ) - \\bar { K } ( \\nu , 0 ) \\right ) \\left [ Z ^ { \\perp } ( \\nu , \\varepsilon ) + \\mathbf { I } \\right ] r _ { 0 } , r _ { 0 } \\right ) , \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} \\langle \\mathcal { K } _ 1 , \\dots \\mathcal { K } _ { m _ 2 - 1 } , \\phi ^ * \\mathcal { L } _ 0 , \\dots , \\phi ^ * \\mathcal { L } _ { m _ 1 + 1 } \\rangle ( \\mathfrak { X } / S ) = \\O _ S . \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} ( x , y ) ( x ' , y ' ) : = ( x x ' - y y ' , x y ' + y x ' ) . \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} d ( p t ) = A _ 1 d ( t ) , \\ \\ d ( q t ) = B _ 1 d ( t ) \\mbox { f o r } t \\in S . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} K _ 0 & = K \\cap \\big ( ( - \\infty , x - r _ 0 ] \\cup [ x + r _ 0 , + \\infty ) \\big ) , \\\\ K _ { k } & = K \\cap \\big ( [ x - r _ { k - 1 } , x - r _ { k } ] \\cup [ x + r _ { k } , x + r _ { k - 1 } ] \\big ) , \\ ; \\ ; k \\in \\omega \\smallsetminus \\{ 0 \\} . \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} \\Lambda ( x _ 1 , x _ 2 ) = x _ 1 ^ p \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { \\Lambda _ n ( x _ 2 ) } { x _ 1 ^ { n } } , x _ 1 \\to + \\infty , \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { A c h } } ( \\mu , r ) = \\frac { K } { \\min \\{ M , K \\} } + \\frac { ( 1 - \\mu ) K } { M r } . \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{gather*} \\Psi ^ { [ k , \\ell ] ( \\alpha , \\beta ) } = T _ { 1 } ^ { k } T _ { 2 } ^ { \\ell } Q _ 0 ^ { - \\alpha } Q _ 1 ^ { - \\beta } g _ { - } ^ { [ k , \\ell ] ( \\alpha , \\beta ) } . \\end{gather*}"}
-{"id": "2821.png", "formula": "\\begin{align*} \\partial _ p \\mathcal K : = \\big ( \\overline { \\mathcal D } \\times \\{ t _ 1 \\} \\big ) \\cup \\big ( \\partial \\mathcal D \\times ( t _ 1 , t _ 2 ) \\big ) . \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} H ^ { i j } \\left [ \\nabla _ j X _ { i r } - ( \\partial _ j f ) X _ { i r } \\right ] = \\frac { 1 } { 2 } \\partial _ { r } S \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\nu ^ { \\pi ^ * } _ { n - 1 } ( 1 | 1 ) \\equiv { { c } _ 1 ( n - 1 ) = \\frac { 2 ^ { \\mu _ 1 ( n - 1 ) } } { 2 ^ { \\mu _ 1 ( n - 1 ) + \\Delta { C } _ n } } } , ~ \\pi ^ * _ { n - 1 } ( 1 | 1 ) \\equiv { { d _ 1 ( n - 1 ) } = \\frac { \\beta _ { n - 1 } ( 1 + 2 ^ { \\mu _ 1 ( n - 1 ) + \\Delta { C } _ n } ) - 1 } { ( \\beta _ { n - 1 } - \\delta _ { n - 1 } ) ( 1 + 2 ^ { \\mu _ 1 ( n - 1 ) + \\Delta { C } _ n } ) } } \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} C _ { A B } { } ^ { C } C _ { C D } { } ^ { E } + C _ { B D } { } ^ { C } C _ { C A } { } ^ { E } + C _ { D A } { } ^ { C } C _ { C B } { } ^ { E } = 0 , \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} K _ { W } : = 2 ^ { - 1 } W ( S ^ * + S ) + 2 ^ { - 1 } ( S ^ * + S ) W , B _ W : = U \\tilde { W } ( S ^ * - S ) - ( S ^ * - S ) \\tilde { W } U , \\end{align*}"}
-{"id": "10062.png", "formula": "\\begin{align*} ( \\alpha - 1 ) p + ( 2 - \\alpha ) q = 1 + n + k . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} f ( M ) = \\sum _ { n = 0 } ^ \\infty \\left \\lfloor \\frac { M } { 2 ^ n } \\right \\rfloor = \\sum _ { n = 0 } ^ { \\lfloor \\log _ 2 M \\rfloor } \\left \\lfloor \\frac { M } { 2 ^ n } \\right \\rfloor \\leq \\sum _ { n = 0 } ^ { \\lfloor \\log _ 2 M \\rfloor } \\frac { M } { 2 ^ n } \\leq M \\cdot \\frac { 1 - \\frac { 1 } { 2 M } } { 1 - \\frac { 1 } { 2 } } = 2 M - 1 . \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} \\Theta _ { e _ l , e _ m } \\ , e _ n b = \\Theta _ { e _ l , e _ m } ( \\delta _ n \\otimes b ) & = ( \\delta _ l \\otimes 1 _ B ) \\cdot \\left ( \\delta _ m \\otimes 1 _ B \\mid \\delta _ n \\otimes b \\right ) _ B \\\\ & = \\delta _ { m , n } \\ , \\delta _ l \\otimes b = \\delta _ { m , n } \\ , e _ l \\cdot b . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} H ( p ) \\ ; = \\ ; \\max _ { c \\ , \\in \\ , C } \\ , \\langle c , p \\rangle . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} F & = A _ { L } , & f & = \\frac { | L | - \\epsilon n - 1 } { | L | } = 1 - \\frac { \\epsilon } { \\lambda } - o ( 1 ) , \\\\ G & = B _ { L } , & g & = \\frac { \\pi } { \\lambda } , \\\\ \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} \\sqrt { - d ^ 2 / d x ^ 2 } \\ , { u } _ { 1 / 2 } ( x ) = \\widehat { ( | k | \\widehat { u } _ { 1 / 2 } ( k ) ) } ( x ) = \\frac { 2 } { \\pi } \\frac { d } { d x } \\int _ 0 ^ \\infty K _ 0 ( k ) \\sin k x d k . \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} F ( z , v ) : = \\sum _ { n \\ge 1 } \\sum _ { m \\ge 0 } T _ { n } \\mathbb { P } \\{ L _ { n } = m \\} \\frac { z ^ { n } } { n ! } v ^ { m } = \\sum _ { n \\ge 1 } \\sum _ { m \\ge 0 } \\mathbb { P } \\{ L _ { n } = m \\} \\frac { z ^ { n } } { n } v ^ { m } , \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} \\delta ( [ \\alpha , v ] ) \\ , & = \\ , \\delta ( \\alpha \\cdot v - ( - 1 ) ^ { | \\alpha | | v | } v \\cdot \\alpha ) \\\\ & \\ , = \\ , \\delta ( \\alpha ) \\cdot v + ( - 1 ) ^ { | \\alpha | | \\delta | } \\alpha \\cdot \\delta ( v ) - ( - 1 ) ^ { | \\alpha | | v | } \\delta ( v ) \\cdot \\alpha - ( - 1 ) ^ { | \\alpha | | v | + | \\delta | | v | } v \\cdot \\delta ( \\alpha ) \\\\ & \\ , = \\ , [ \\delta ( \\alpha ) , v ] + ( - 1 ) ^ { | \\alpha | | \\delta | } [ \\alpha , \\delta ( v ) ] \\ , \\\\ \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} \\bar K \\triangleq \\sum _ { i = 1 } ^ N K _ i . \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} \\phi ( x ) = \\lim _ { n \\to \\infty } \\phi \\Bigl ( \\sum _ { m = 0 } ^ n \\varpi ^ m x _ m \\Bigr ) = \\lim _ { n \\to \\infty } ( a _ 0 - \\varpi ^ { n + 1 } a _ { n + 1 } ) = a _ 0 . \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} n ( k ) = \\mu _ k ( q _ 0 + q _ o ^ { - 1 } ) \\& d ( k ) = \\mu _ k ( q + q ^ { - 1 } ) ( k \\in \\N _ 0 ) . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} M _ { z _ 0 , z _ { i + 1 } } + M _ { z _ 0 , z _ 0 , z _ 0 , z _ i } + 4 M _ { z _ i } M _ { z _ 0 , z _ 0 } + 8 M _ { z _ 0 } M _ { z _ 0 , z _ i } = 0 , \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} \\Omega _ 2 \\mid _ { B _ 0 } = \\frac { d w _ 1 \\wedge d w _ 3 } { f _ { w _ 0 } } , ~ \\Omega _ 2 \\mid _ { B _ 1 } = \\frac { d w _ 3 \\wedge d w _ 0 } { f _ { w _ 1 } } , ~ \\Omega _ 2 \\mid _ { B _ 3 } = \\frac { d w _ 0 \\wedge d w _ 1 } { f _ { w _ 3 } } , \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} K = - \\frac { \\kappa _ { \\gamma } \\phi ^ { \\prime } ( u ) } { \\lambda \\cos \\left ( \\frac { u } { c } \\right ) { } } , \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\ & \\left \\vert \\left ( \\left ( \\bar { K } ( \\nu , \\varepsilon ) - \\bar { K } ( \\nu , 0 ) \\right ) \\partial _ { \\nu } Z ^ { \\perp } ( \\nu , \\varepsilon ) r _ { 0 } , r _ { 0 } \\right ) \\right \\vert \\\\ & = \\left \\vert \\left ( \\left ( K ( \\lambda , \\varepsilon ) - K ( \\lambda , 0 ) \\right ) \\partial _ { \\nu } Z ^ { \\perp } ( \\nu , \\varepsilon ) r _ { 0 } , r _ { 0 } \\right ) \\right \\vert \\lesssim \\sqrt { \\varepsilon } . \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} \\dot { G } = \\mathbf { Q } ^ V \\ , G , \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} \\Psi ( a ) ( y ) = \\left \\{ \\begin{array} { r c l } \\Psi _ y ( a ( \\psi ^ * ( y ) ) ) & \\mbox { i f } & y \\in U _ { \\psi } \\\\ 0 & \\mbox { } & y \\not \\in U _ { \\psi } \\end{array} \\right . \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} \\alpha _ { \\gamma } ( \\cdot ) = Z ( \\gamma , \\cdot ) + \\frac { 1 } { 2 } Z ( \\gamma , \\gamma ) , \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{gather*} Q _ { 2 } ^ { \\beta + \\gamma } Q _ { 1 } ^ { \\alpha - \\gamma } Q _ { 0 } ^ { - \\alpha - \\beta } = ( - 1 ) ^ { \\frac { \\alpha ( \\alpha - 1 ) } { 2 } + \\frac { \\gamma ( \\gamma - 1 ) } { 2 } + \\alpha \\beta + \\beta \\gamma } T _ { 2 } ^ { \\beta + \\gamma } T _ { 1 } ^ { \\alpha + \\beta } , \\end{gather*}"}
-{"id": "2761.png", "formula": "\\begin{align*} ( \\psi \\circ \\phi ) ( f ) = f \\circ \\sigma _ A , ( \\phi \\circ \\psi ) ( g ) = g \\circ \\sigma _ B \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} \\tau ( w ) = \\mu w + w _ 0 , \\textnormal { w h e r e } \\mu > 0 \\textnormal { a n d I m } \\ , w _ 0 \\geq 0 . \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} C _ f ^ * ( \\underline { \\pi } , s ) = \\det \\big ( I - s \\sigma ^ { a - 1 } ( N ) \\cdots \\sigma ( N ) N \\big ) . \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} \\pi _ 7 ( [ C r , \\chi ] _ p ^ { F N } ) = 2 \\ast \\varphi \\wedge \\left ( \\left ( T _ p ^ \\top - 2 T _ p - t r ( T _ p ) \\right ) e _ i \\right ) ^ \\flat \\otimes e _ i , \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} \\delta H _ k ^ { 2 , I I } = \\oplus _ { i = 3 } ^ { n e } \\Psi _ { i , k } , \\end{align*}"}
-{"id": "10022.png", "formula": "\\begin{align*} q ^ t + 1 = | S T | \\geq \\frac { \\left | \\sum \\limits _ { X \\in S T } | X ^ * | \\right | } { | V ( t , q ) ^ * | } = \\frac { | V ( n , q ) ^ * | - | Y ^ * | } { | V ( t , q ) ^ * | } = \\frac { ( q ^ n - 1 ) - ( q ^ { d } - 1 ) } { q ^ t - 1 } \\geq q ^ { d } , \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { \\alpha \\in A _ r } s ^ { \\lambda , n } _ { \\alpha } I ^ { \\alpha } = \\sum _ { \\alpha \\in A _ r } s ^ { \\lambda , n - 1 } _ { \\alpha } I ^ { \\alpha } \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} C _ n ( y _ { n - 1 } ) = \\sum _ { y _ n \\in \\{ 0 , 1 \\} } \\log \\Big ( \\frac { q _ n ( y _ n | x _ n , y _ { n - 1 } ) } { \\nu ^ { \\pi ^ * } _ n ( y _ n | y _ { n - 1 } ) } \\Big ) q _ n ( y _ n | x _ n , y _ { n - 1 } ) , ~ \\forall { x _ n } . \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} \\norm { m + H _ x } = \\inf _ W \\sup _ { y \\in W } \\norm { \\tilde { \\pi } _ y ( m ) } , \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} f \\left ( \\sigma , \\theta \\right ) = f \\left ( \\lambda , \\theta \\right ) e ^ { \\int _ { \\lambda } ^ r \\varphi \\left ( \\sigma , \\theta \\right ) \\ , d \\sigma } \\leq f \\left ( \\lambda , \\theta \\right ) e ^ { 2 \\rho \\left ( 2 \\alpha \\rho + C \\right ) } , \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{gather*} \\partial _ { t _ { k } } \\Psi ( z ; t ) = B _ { k } ( \\partial _ { x } ) \\Psi ( z ; t ) , \\end{gather*}"}
-{"id": "9012.png", "formula": "\\begin{align*} s _ a ( x , \\xi ) = s _ a ^ 1 ( x , \\xi ) + s _ a ^ 2 ( x , \\xi ) , \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} P ( D ) = \\prod _ { ( i , j ) \\in A ( D ) } \\mathrm { p _ a } ( i , j ) \\times \\prod _ { ( i , j ) \\notin A ( D ) } ( 1 - \\mathrm { p _ a } ( i , j ) ) \\ D \\in \\mathcal { D } _ n . \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ { K } ( x _ k + y _ k - 1 ) = S + z _ { 1 , 2 } ^ { ( 3 ) } z _ { 3 , 4 } ^ { ( 3 ) } \\cdots z _ { K - 1 , K } ^ { ( 3 ) } \\geq 0 + \\prod _ { k = 1 } ^ { K } x _ k + \\prod _ { k = 1 } ^ { K } y _ k - 1 , \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} \\tilde { x } _ { i j } = - \\tilde { g } _ { i j } \\tilde { x } + \\tilde { h } _ { i j } x , \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} u v = \\Delta ( \\Omega ) \\cdot \\Theta _ 2 \\left ( \\zeta _ 1 - \\frac { \\tau } { 2 } , \\zeta _ 2 - \\frac { \\rho } { 2 } ; \\Omega \\right ) \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} \\left [ - \\frac { d ^ { \\alpha } } { d x ^ { \\alpha } } + x ^ { 2 } - \\frac { \\alpha } { 2 } \\frac { d ^ { \\alpha / 2 - 1 } } { d x ^ { \\alpha / 2 - 1 } } \\right ] \\psi _ 0 ^ { ( \\alpha ) } ( x ) = 0 . \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} P S ( k ) = ( \\sum \\limits _ { j = 0 } ^ { m - 1 } { x _ j e ^ { \\frac { { - i 2 \\pi j k } } { l } } ) ^ * } ( \\sum \\limits _ { j = 0 } ^ { m - 1 } { x _ j e ^ { \\frac { { - i 2 \\pi j k } } { l } } } ) \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 2 } \\ [ \\bar { e } _ { i , k _ { \\pi ( 1 ) } } , [ \\bar { e } _ { i , k _ { \\pi ( 2 ) } } , \\bar { e } _ { i \\pm 1 , l } ] ] = 0 \\ \\mathrm { a n d } \\ [ \\bar { e } _ { i , k } , \\bar { e } _ { j , l } ] = 0 \\ \\mathrm { f o r } \\ j \\ne i , i \\pm 1 , \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} P _ { \\mu \\nu } Q ^ { \\nu \\lambda } = \\left ( \\mathbf { F } \\cdot \\mathbf { G } \\right ) \\delta _ { \\mu } ^ { \\lambda } = \\frac { 1 } { 4 } \\left ( P _ { \\sigma \\tau } Q ^ { \\tau \\sigma } \\right ) \\delta _ { \\mu } ^ { \\lambda } , \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} \\begin{cases} { \\rm R i c } ^ N ( \\xi , \\xi ) = | A | ^ { 2 } , \\\\ ( { \\rm R i c } ^ N \\ , ( \\xi ) ) ^ { \\top } = \\frac { m } { 2 } { \\rm g r a d } \\ , H , \\ ; \\ ; { \\rm o r } \\ ; \\ ; H = 0 . \\end{cases} \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} \\varphi ( x ) = | x | ^ 2 / ( 1 + 4 \\gamma ) . \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} u _ \\varepsilon ^ { ( s ) } ( t ) = u _ \\varepsilon ^ { ( m - 1 ) } ( t ) \\otimes g _ { \\varepsilon } ( t ) ^ { \\otimes ( s - m + 1 ) } \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} \\tilde { \\theta } _ t ( \\kappa ) = \\frac { 1 } { 2 \\kappa } R e \\int _ 0 ^ t \\left ( e ^ { 2 i \\theta _ s ( \\kappa ) } - 1 \\right ) a ( s ) F ( X _ s ) d s . \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} J = \\begin{pmatrix} 0 & & & & \\\\ 1 & 0 & & & \\\\ 0 & 1 & \\ddots & & \\\\ \\vdots & \\ddots & \\ddots & \\ddots & 0 \\\\ 0 & \\cdots & 0 & 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} \\frac { \\| z _ k \\| _ { \\pi , n } } { \\| z _ k \\| _ { a h , 2 n } } > \\frac { i _ 1 ^ { 2 n } \\sum _ { j = 2 } ^ { k + 1 } { i _ j ^ { 2 n } } } { i _ 1 ^ { 4 n } k + i _ { k + 1 } ^ { 4 n } } > \\frac { i _ 1 ^ { - 2 n } i _ 2 ^ { 2 n } } { 1 + k ^ { - 1 } i _ 1 ^ { - 4 n } i _ { k + 1 } ^ { 4 n } } \\to \\infty , \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\norm { \\Delta _ { \\zeta } \\zeta ( t ) } + \\norm { \\zeta ( t ) } & \\leq \\norm { \\partial _ t \\zeta ( t ) } + \\norm { v ( t ) \\nabla _ H \\zeta ( t ) } + \\norm { ( w ( t ) \\partial _ z \\zeta ( t ) ) } + \\norm { g ( t ) } + \\norm { \\zeta ( t ) } \\\\ & \\leq ( \\tilde { B } _ { \\partial _ t ( v , \\zeta ) } ( t ) ) ^ { 1 / 2 } + C B _ { H ^ 2 } ^ v ( t ) + C B _ { H ^ 1 } ^ { \\zeta } ( t ) + \\norm { g ( t ) } = : B _ { H ^ 2 } ^ { \\zeta } ( t ) , t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\frac { 1 } { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } M ^ { m } \\vec u = \\vec { u } _ { \\infty } . \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} g ( \\theta , x _ 0 , t , c ) = \\Phi \\left ( \\frac { e ^ { t ^ c } - x _ 0 e ^ { \\theta t } } { \\sqrt { v ( \\theta , t ) } } \\right ) + \\Phi \\left ( \\frac { e ^ { t ^ c } + x _ 0 e ^ { \\theta t } } { \\sqrt { v ( \\theta , t ) } } \\right ) - 1 , \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} B ( x ) = & \\ , B ( x , [ 0 ^ { s + 1 } ] , [ v - s , k - \\frac { s - 1 } { 2 } , \\lambda - \\frac { s - 2 } { 3 } ] ) \\\\ = & \\ , \\sum _ { j = 0 } ^ 2 { 2 \\choose j } P ( - x , 2 - j ) P ( s , j ) \\lambda _ j \\\\ = & \\ , x ( x + 1 ) ( v - s ) - 2 x s ( k - ( s - 1 ) / 2 ) + s ( s - 1 ) ( \\lambda - ( s - 2 ) / 3 ) . \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} \\mathrm { d i a m } \\ , \\big ( T _ { \\xi _ { 0 } } \\circ \\cdots \\circ T _ { \\xi _ { n } } ( X ) \\big ) \\to 0 \\mbox { f o r e v e r y } \\ , \\ , \\xi = \\xi _ 0 \\xi _ 1 \\xi _ 2 \\dots \\in \\Sigma _ k ^ + , \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} - \\left ( \\frac { X _ { z _ { i + 1 } } } { X _ { z _ 0 } } \\right ) _ { z _ 0 } = \\left [ \\left ( \\frac { X _ { z _ 0 , z _ 0 } } { X _ { z _ 0 } } + X _ { z _ 0 } \\right ) _ { z _ 0 } - \\frac { 1 } { 2 } \\left ( \\frac { X _ { z _ 0 , z _ 0 } } { X _ { z _ 0 } } + X _ { z _ 0 } \\right ) ^ 2 \\right ] _ { z _ i } , \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} m _ \\mu = \\sup _ { t \\in [ 0 , 1 ] } \\left | t ( 1 - t ) + \\frac { 4 \\mu ( 1 - 2 t ) - 1 } { 1 6 \\mu ^ 2 } \\right | . \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} \\phi _ 1 ( x , y , t ; , \\xi , \\tau , s , \\theta ) = & ( x - y ) \\cdot \\xi + ( t - s ) \\tau + s | \\xi | \\sin \\theta - \\theta \\varpi / 2 \\pi , \\\\ \\phi _ 2 ( x , y , t ; , \\xi , \\tau , s , r ) = & ( x - y ) \\cdot \\xi + ( t - s ) \\tau + i ( | \\xi | s \\sinh r + \\varpi r / 2 \\pi ) . \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{gather*} s _ { n _ { k + 1 } + n _ { k } + 1 } = s _ { n _ { k + 1 } + n _ { k } + 1 } ^ { ( n _ { k } + 1 ) } + \\left ( \\frac { \\Delta _ { k + 1 } } { t _ { n _ { k } } ^ { 2 } } \\right ) ^ { \\tfrac { 1 } { n _ { k + 1 } - n _ { k } } } \\ , \\end{gather*}"}
-{"id": "8187.png", "formula": "\\begin{align*} P _ t ( \\textbf { x } _ t , f _ t , q _ t ) = H _ t ( \\textbf { x } _ t , f _ t , q _ t ) + W _ t ( \\textbf { x } _ t , f _ t , q _ t ) . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} \\alpha ^ l \\sqrt { \\gamma ^ l } = \\sqrt { \\theta ^ l } . \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} \\widetilde { \\pi } _ { C } ( x , L / \\mathbb { Q } ) & = \\sum _ { \\gamma \\in \\Gamma } \\frac { \\widetilde { \\pi } _ { C _ B ( \\gamma ) } ( x , L / L ^ B ) } { [ \\mathrm { C e n t } _ G ( \\gamma ) : \\mathrm { C e n t } _ B ( \\gamma ) ] } , \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} A '' ( z , v ) = \\frac { 1 } { 1 - z } A ' ( z , v ) + \\frac { 1 } { 1 - z } \\big ( A ( z , v ) \\big ) ^ { 2 } , A ( 0 , v ) = v , A ' ( 0 , v ) = v . \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\tilde { t } ^ j _ { k , \\varphi } ( \\omega , \\lambda ) = t ^ j _ k ( x , Y + \\lambda \\nu _ k + i \\tau d \\varphi _ k ( x ) ) , \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} \\frac { N _ { a , b , c } ( r ) } { N _ { a , b , c } ( 0 ) } = \\binom { a + r - 1 } { a - 1 } \\binom { b + c - r - 1 } { c - 1 } \\binom { b + c - 1 } { c - 1 } ^ { - 1 } , \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} P ( D ) = \\int \\mathbf { 1 } _ { \\{ k ( \\mathbf { x } ) = D \\} } d ( \\mu \\mathbf { x } ) D \\in \\mathcal { D } _ n . \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} \\frac { ( - q ; q ) _ \\infty } { ( q ; q ) _ \\infty } \\sum _ { j \\geq 0 } q ^ { { ( 3 j ^ 2 + j ) } / { 2 } } ( 1 - q ^ { 2 j + 1 } ) = { ( - q ; q ) _ \\infty } \\sum _ { j \\geq 0 } \\frac { q ^ { { ( 3 j ^ 2 + j ) } / { 2 } } } { ( q ; q ) _ { 2 j } ( q ^ { 2 j + 2 } ; q ) _ \\infty } . \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} M ( t ) = X ( t ) - X ( 0 ) - \\int _ 0 ^ t b ( s ) X ( s ) \\ , d s , \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} x _ i & = y _ i , & - N \\leq i \\leq N - 1 , \\\\ x _ { i } & \\in C _ 1 , & N \\leq i \\leq N + a - 1 , \\\\ x _ { i } & \\not \\in C _ 1 , & N + a \\leq i \\leq N + a + b - 1 , \\\\ x _ { i } & = z _ { i - 2 N - a - b } , & N + a + b \\leq i \\leq 3 N + a + b - 1 . \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} d ( f , g , h ) \\deg C ( f , g , h ) = \\dim M _ k ( \\Gamma _ 0 ( N ) ) + g ( \\Gamma _ 0 ( N ) ) - 1 - \\sum \\limits _ { \\mathfrak { a } \\in X ( \\Gamma ) } \\min ( D _ f ( \\mathfrak { a } ) , D _ g ( \\mathfrak { a } ) , D _ h ( \\mathfrak { a } ) ) . \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} \\max _ { j , k } ( \\Re ( \\lambda _ j x ) - \\Re ( \\lambda _ k x ) ) < { - } \\Re ( 2 \\pi i x ) \\mbox { i f } \\arg ( x ) = \\psi = \\theta + \\pi / 2 . \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { l l } \\dot { u } = \\log \\det ( u _ { \\alpha \\bar { \\beta } } ) + f ( t , z , u ) \\ ; \\ ; \\ ; & \\mbox { o n } \\ ; \\Omega \\times ( 0 , T ) , \\\\ u = \\varphi & \\mbox { o n } \\ ; \\partial \\Omega \\times [ 0 , T ) , \\\\ u = u _ 0 & \\mbox { o n } \\ ; \\bar { \\Omega } \\times \\{ 0 \\} . \\\\ \\end{array} \\end{cases} \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} \\frac { \\delta ^ l } { \\alpha ^ l } = \\frac { 1 } { \\theta ^ l } = \\delta _ 0 ^ l \\eta ^ j . \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{gather*} \\begin{bmatrix} 1 & 0 & 0 \\\\ C ( z ) & 1 & 0 \\\\ D ( z ) & E ( z ) & 1 \\end{bmatrix} . \\end{gather*}"}
-{"id": "1568.png", "formula": "\\begin{align*} \\lambda \\cdot ( x _ 0 , \\ldots , x _ n ) = ( \\lambda ^ { a _ 0 } x _ 0 , \\ldots , \\lambda ^ { a _ n } x _ n ) . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{gather*} s _ { n _ { k + 1 } + n _ { k } + 1 } = s _ { n _ { k + 1 } + n _ { k } + 1 } ^ { ( n _ { k } + 1 ) } + ( - 1 ) ^ { { \\tfrac { ( n _ { k + 1 } - n _ { k } - 1 ) } { 2 } } } \\left ( \\frac { t _ { n _ { k + 1 } } } { t _ { n _ { k } } } \\right ) ^ { \\tfrac { 1 } { n _ { k + 1 } - n _ { k } } } \\ , \\end{gather*}"}
-{"id": "8525.png", "formula": "\\begin{align*} M _ 1 ( l , u , v ) = \\sum _ { f \\in H _ { 2 k } ^ { * } ( N ) } ^ { h } \\lambda _ f ( l ) L _ f ( 1 / 2 + u + v ) . \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} F ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } ) = \\sum _ { \\underset { \\delta _ { j } = 0 } { \\delta \\in \\{ 0 , 1 \\} ^ { n } } } u _ { \\delta } ( x _ { j } ) \\prod _ { k \\neq j } f _ { k , x _ { j } } ^ { \\delta _ { k } } ( x _ { k } ) . \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ + _ { I V } } & d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , \\alpha } ( s + k - 1 ) T R \\eta ^ { d - 1 } \\end{aligned} \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} P : = \\{ \\ldots \\to 0 \\to P ^ n \\to \\ldots \\to P ^ j \\to \\ldots \\to P ^ { n + m } \\to 0 \\ldots \\} \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} - y ^ { ^ { \\prime \\prime } } ( x ) + q ( x ) y ( x ) = \\mu ^ { 2 } y ( x ) \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} \\sigma _ E ^ { l ( x ) } ( \\kappa ( x ) ) = \\sigma _ E ^ 0 ( r ( x ) ) = r ( x ) . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} & \\gamma ^ j ( w ) = e _ j \\wedge w + \\iota ( e _ j ) w , & & \\rho ^ j ( w ) = e _ j \\wedge w - \\iota ( e _ j ) w , \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} \\gamma ( \\Omega , a , q _ 1 ) ~ = ~ \\beta \\gamma ( \\Omega , a , q _ 2 ) , \\phantom { m } \\phantom { m } \\gamma ( \\Omega , a , q _ 1 ) ~ = ~ \\lambda \\gamma ( \\Omega , a , q _ 3 ) . \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} e ^ { 2 i \\theta _ 1 } - e ^ { 2 i \\theta _ 2 } = 2 i \\sin ( \\theta _ 1 - \\theta _ 2 ) e ^ { i \\theta _ 1 + i \\theta _ 2 } \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} \\rho _ { \\nu + 1 , b } ( x ) & = x \\rho _ { \\nu - 1 , b } ( x ) + \\frac { \\nu } { b } \\rho _ { \\nu , b } ( x ) , \\\\ \\rho _ { \\nu + 1 , b } ' ( x ) & = - b \\rho _ { \\nu , b } ( x ) , \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\mathrm { V o l } _ N \\sum _ { i \\geqslant 0 } & \\widehat { k _ S \\otimes k _ \\xi } ( \\psi _ i ) | \\mathcal { P } _ H ( \\psi _ i ) | ^ 2 \\\\ & = \\mathrm { v o l } ( [ H ] ) \\Pi _ H k _ S ( 1 ) k _ \\xi ( 1 ) + O \\left ( \\# ( { \\rm s u p p } \\ , k _ S / K _ S ) \\beta ( \\xi ) ( 1 + \\| \\xi \\| ) ^ { - 1 / 4 } N ^ { - 1 / 4 } R ^ A \\| k _ S \\| _ { \\infty } \\right ) , \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} U = \\begin{bmatrix} I & \\Psi \\end{bmatrix} , \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} S N R = \\frac { | \\textbf { w } _ { M S } ^ H \\textbf { H } \\textbf { w } _ { B S } | ^ 2 P } { | | \\textbf { w } _ { M S } ^ H | | ^ 2 \\sigma ^ 2 } \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} \\gamma _ { \\chi } ( s ) = \\Big [ \\pi ^ { - \\tfrac { s } { 2 } } \\Gamma \\Big ( \\frac { s } { 2 } \\Big ) \\Big ] ^ { a ( \\chi ) } \\cdot \\Big [ \\pi ^ { - \\tfrac { s + 1 } { 2 } } \\Gamma \\Big ( \\frac { s + 1 } { 2 } \\Big ) \\Big ] ^ { b ( \\chi ) } . \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} F \\left ( x _ { 1 } , \\dots , x _ { n } \\right ) = \\sum _ { \\delta \\in \\left \\{ 0 , 1 \\right \\} ^ { n } } u _ { \\delta } \\prod f _ { i } \\left ( x _ { i } \\right ) ^ { \\delta _ { i } } \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} \\Omega _ 3 \\mid _ { C _ 0 } = \\frac { d v _ 2 \\wedge d v _ 1 } { f _ { v _ 0 } } , ~ \\Omega _ 3 \\mid _ { C _ 1 } = \\frac { d v _ 0 \\wedge d v _ 2 } { f _ { v _ 1 } } , ~ \\Omega _ 3 \\mid _ { C _ 2 } = \\frac { d v _ 1 \\wedge d v _ 0 } { f _ { v _ 2 } } , \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} f _ { 0 , \\pm } \\left ( v \\right ) = \\left \\{ \\begin{array} [ c ] { c c } \\mu _ { \\pm , + } \\left ( \\frac { 1 } { 2 } v ^ { 2 } \\right ) & v > 0 \\\\ \\mu _ { \\pm , - } \\left ( \\frac { 1 } { 2 } v ^ { 2 } \\right ) & v < 0 \\end{array} \\right . . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} P ( G ) = \\int \\mathbf { 1 } _ { \\{ G ( \\mathbf { x } , \\phi ) = G \\} } d ( \\mu \\mathbf { x } ) G \\in \\mathcal { G } _ n , \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} \\frac { \\left ( q ^ { 2 } , q z , q / z ; q ^ { 2 } \\right ) _ { \\infty } } { ( q ; q ) _ { \\infty } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } ( - 1 ) ^ { n } } { ( q ; q ) _ { n } } \\frac { A _ { q } \\left ( q ^ { - n } z \\right ) } { z ^ { n } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } ( - 1 ) ^ { n } } { ( q ; q ) _ { n } } z ^ { n } A _ { q } \\left ( \\frac { 1 } { q ^ { n } z } \\right ) . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} \\big \\| \\bar { \\mathbf { u } } ( t , \\cdot ) \\big \\| _ { \\mathcal { H } } \\leq L \\int _ { 0 } ^ { t } \\big \\| \\tilde { \\mathbf { u } } _ { 1 } ( s , \\cdot ) - \\tilde { \\mathbf { u } } _ { 2 } ( s , \\cdot ) \\big \\| _ { \\mathcal { H } } \\mathrm { d } s L : = 2 \\tilde { C } C _ { \\mathrm { L i p } } . \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} \\dot { x } = \\omega ( \\lambda ) + O ( y , z ) , \\dot { y } = \\sigma ( \\lambda ) + O ( y , z ) , \\dot { z } = Q ( x , \\lambda ) z + O _ 2 ( y , z ) \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} | W ' _ d | = | \\eta ( W ' _ d ) | \\leq \\binom { | L | } { d } 2 ^ { d } \\sum _ { i \\in ( 0 . 5 \\pm \\epsilon ' ) | R | } \\binom { | R | } { i } 2 ^ { i } , \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} \\theta ^ l \\ge \\Theta : = \\frac { 1 - \\sigma } { \\eta \\zeta _ i + ( 1 - \\sigma ) \\delta _ { \\max } } . \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} b ( \\pi ) = ( d - 1 ) \\sum _ { i = 1 } ^ 3 ( a _ i - 1 ) + \\sum _ { i = 1 } ^ 3 \\begin{cases} a _ i - 1 & a _ i \\mid d ; \\\\ a _ 0 a _ 1 a _ 2 - 1 & a _ i \\nmid d . \\end{cases} \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } | A _ f ( n ) | ^ 2 = c _ f X + O ( X ^ { \\frac { 3 } { 5 } + \\epsilon } ) \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\mu _ { k } ^ i \\left ( B \\right ) & = \\frac { 1 } { Z _ { k } ^ i } \\sum \\limits _ { \\theta \\in B } \\prod \\limits _ { j = 1 } ^ { n } \\mu _ { k - 1 } ^ j \\left ( \\theta \\right ) ^ { a _ { i j } } \\ell ^ i \\left ( s _ { k } ^ i | \\theta \\right ) \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} \\sqrt { \\frac { \\pi } { \\log q ^ { - 2 } } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( u v \\right ) ^ { n } } { \\left ( q ; q \\right ) _ { n } } = \\sum _ { j , k = 0 } ^ { \\infty } q ^ { \\left ( j + k \\right ) / 2 } u ^ { j } v ^ { k } \\int _ { - \\infty } ^ { \\infty } S _ { j } \\left ( q ^ { 2 \\alpha - 1 / 2 } ; q \\right ) S _ { k } \\left ( q ^ { 2 \\alpha - 1 / 2 } ; q \\right ) q ^ { 2 \\alpha ^ { 2 } } d \\alpha , \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} \\varphi _ R = \\mathrm { I d } \\quad Z _ R \\backslash Y _ { [ - R - 1 , R + 1 ] } , \\varphi _ R ( u , y ) = ( \\phi _ R ( u ) , y ) ( u , y ) \\in Y _ { [ - R - 1 , R + 1 ] } . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } v ( t , x ) = \\Delta v ( t , x ) + \\sum _ { k = 1 } ^ { \\infty } \\partial _ { t } ^ { \\tilde { \\beta } } \\int _ { 0 } ^ { t } g ^ { k } ( s , x ) d w _ { s } ^ { k } , \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} | \\varphi | ^ 2 = r | \\psi | ^ 2 + s \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} F \\left ( x _ { 1 } , \\dots , x _ { n } \\right ) = u _ { 0 } + g _ { 1 } \\left ( x _ { 1 } \\right ) u _ { 1 } + \\dots + g _ { n } \\left ( x _ { n } \\right ) u _ { n } \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) & = \\frac { 1 } { 2 } \\int _ { \\Omega } ( \\rho \\dot { u } _ { i } ^ 2 + a \\dot { \\tau } ^ 2 + A _ { i K L j } u _ { i , K } u _ { j , L } + C _ { i I J K L j } u _ { i , J I } u _ { j , L K } ) \\ , \\mathrm { d } x \\\\ & + \\frac 1 2 \\int _ { \\Omega } ( K _ { I J } \\tau _ { , I } \\tau _ { , J } + 2 M _ { i J K L } u _ { i , K J } \\tau _ { , L } ) \\ , \\mathrm { d } x , \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} P _ { T o t } ^ { H B F } = & \\ N _ { M S } ( P _ { L N A } + P _ { S P } + N _ { R F } P _ { P S } ) \\\\ & + N _ { R F } ( P _ C + P _ { R F } + 2 P _ { A D C } ) \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z } J _ { j + m } \\left ( \\alpha ^ { - 1 } q ^ { m } ; q \\right ) J _ { j + n } \\left ( \\alpha ^ { - 1 } q ^ { n } ; q \\right ) = \\frac { \\delta _ { m , n } } { 1 - \\alpha ^ { - 2 } q ^ { 2 m } } , \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{gather*} \\big [ \\psi _ { a } ^ { \\pm } ( z ) , \\psi ^ { \\pm } _ { b } ( w ) \\big ] _ { + } = 0 , \\big [ \\psi _ { a } ^ { + } ( w _ { 1 } ) , \\psi _ { b } ^ { - } ( w _ { 2 } ) \\big ] _ { + } = \\delta _ { a b } \\delta ( w _ { 1 } , w _ { 2 } ) , \\end{gather*}"}
-{"id": "6743.png", "formula": "\\begin{align*} \\alpha ( t , x ) = \\ & \\Phi ( { X } _ T ^ { t , x } ) - \\int ^ T _ t \\beta ( r , X _ r ^ { t , x } ) \\mathrm d { W } _ r \\\\ & + \\int ^ T _ t f ( r , { X } _ r ^ { t , x } , \\alpha ( r , X _ r ^ { t , x } ) , \\beta ( r , X _ r ^ { t , x } ) ) \\mathrm d r \\\\ & - w ( t , x ) - \\int ^ T _ t \\nabla w ( r , { X } _ r ^ { t , x } ) \\mathrm d { W } _ r . \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} - L u & = \\mu \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega \\\\ u & = \\nu \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} ( \\mathcal { S } _ N h ) ( q \\ , | \\ , b ) \\triangleq \\sum \\limits _ { p = 0 } ^ N ( - 1 ) ^ p \\sum \\limits _ { k _ 1 < \\cdots < k _ p = 1 } ^ N h \\bigl ( q + b _ 0 + b _ { k _ 1 } + \\cdots + b _ { k _ p } \\bigr ) . \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} J _ { ( i , 2 ) } = - J _ { ( i , 0 ) } , J _ { ( i , 3 ) } = - J _ { ( i , 1 ) } \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\tilde { x } ^ 0 = \\frac { r } { \\sqrt { 1 - r ^ 2 } } , \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} \\{ f \\in C ^ \\infty ( [ 0 , 1 ] , \\mathbb { R } ^ n ) , \\sum _ { i = 1 } ^ n f _ i ( 0 ) = \\sum _ { i = 1 } ^ n f _ i ( 1 ) \\} \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} \\int _ { \\mathcal { V } } \\phi ( v ) \\mathcal { L } \\phi ( v ) \\ , { \\rm d } \\mu ( v ) = \\frac { 1 } { 2 } \\iint _ { \\mathcal { V } \\times \\mathcal { V } } \\Big ( \\phi ( v ) - \\phi ( w ) \\Big ) ^ 2 \\ , { \\rm d } \\mu ( w ) \\ , { \\rm d } \\mu ( v ) \\ge 0 . \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} P ( \\xi ) = ( \\xi - \\xi _ 0 ) ( \\xi - \\xi _ 1 ) ( \\xi - \\xi _ 2 ) ( \\xi - \\xi _ 3 ) ( \\xi - \\xi _ 4 ) ( \\xi - \\xi _ 5 ) . \\end{align*}"}
-{"id": "9963.png", "formula": "\\begin{align*} x x ^ t = c - 1 & = \\begin{cases} a \\ne - 1 & \\gcd ( q , c ) = 1 , \\\\ - 1 & \\gcd ( q , c ) \\ne 1 . \\end{cases} \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} h ( x , y ) = \\sum _ { i \\geq 1 } \\lambda _ i \\psi _ i ( x ) \\psi _ i ( y ) ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} b ( x , D , \\tau ) ^ \\ast = \\sum _ { j = 0 } ^ m D _ n ^ j b _ j ( x , D ' , \\tau ) ^ \\ast . \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{gather*} \\mathcal { S } = \\Delta + 4 \\mu ^ 2 | x + R t ( 1 - t ) e _ 1 | ^ 2 + 2 \\mu R ( 1 - 2 t ) ( x _ 1 + R t ( 1 - t ) ) \\\\ + ( t ^ 2 - t + \\frac 1 6 ) R ^ 2 - \\frac { R ^ 2 ( 1 - 2 t ) } { 1 6 \\mu } , \\\\ \\mathcal { A } = - 4 \\mu ( x + R t ( 1 - t ) e _ 1 ) \\cdot \\nabla - 2 \\mu n , \\end{gather*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n } { 2 } } \\left ( c x \\right ) ^ { n } } { \\left ( q ; q \\right ) _ { \\infty } } A _ { q } \\left ( c q ^ { n - 1 } \\right ) = \\frac { \\left ( c ; q \\right ) _ { \\infty } } { \\left ( c x ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } - n } \\left ( - c ^ { 2 } \\right ) ^ { n } \\prod _ { k = 0 } ^ { 2 n - 1 } \\left ( x - q ^ { k } \\right ) } { \\left ( q ^ { 2 } , c , c q ; q ^ { 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n + 1 } { 2 } } \\left ( - x \\right ) ^ { n } } { \\left ( q ; q \\right ) _ { n } } A _ { q } \\left ( - q ^ { n } \\right ) = \\frac { \\left ( q ^ { 3 / 2 } ; q \\right ) _ { \\infty } } { \\left ( q ^ { 3 / 2 } x ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\left ( n ^ { 2 } + 3 n \\right ) / 4 } \\prod _ { k = 0 } ^ { n - 1 } \\left ( x - q ^ { k } \\right ) } { \\left ( q ^ { 1 / 2 } , q ^ { 3 / 4 } , - q ^ { 3 / 4 } ; q ^ { 1 / 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} F _ 1 ( z ) = \\frac { 1 } { 2 } \\left ( f ( q ) + f ( q ^ c ) \\right ) , F _ 2 ( z ) = - \\frac { 1 } { 2 } \\left ( f ( q ) - f ( q ^ c ) \\right ) \\j \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} f _ M ( t | a ) = t ^ M \\prod \\limits _ { j = 1 } ^ M ( 1 - e ^ { - a _ j t } ) ^ { - 1 } , \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} \\tilde L U + N ( U ) = 0 \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\gamma _ r ( s ) = \\left \\{ \\begin{array} { r l } 2 & s > r \\\\ \\frac { 2 s } { r } & 0 \\leq s \\leq r \\\\ 0 & s < 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\frac { x y w } { z } = \\frac { x _ 1 z y _ 1 z w } { z } = x _ 1 y _ 1 z w . \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} ( g _ f ( x _ 0 ) ) ^ { 2 Q } & = \\lambda _ k ^ Q \\leq K ^ Q \\lambda _ 1 ^ Q \\leq C K ^ Q ( J _ f ( x _ 0 ) ) ^ 2 x _ 0 \\in B ( x , r _ x ) . \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} q = t ^ n q _ 1 \\dots q _ s \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} q _ 0 ^ * | _ V = \\sum _ { p \\leq 4 , i , j \\leq 3 } ( S ^ * ) ^ p _ { i j } \\ , z _ p \\ , x _ i \\ , y _ j = 0 , \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} \\mu ( \\bigcup _ { i = 1 } ^ { \\infty } T ^ { k _ i } A ) = 1 . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\psi _ \\kappa ( [ x , g , y ] ) = [ \\kappa ( x ) , \\kappa \\circ g \\circ \\kappa ^ { - 1 } , \\kappa ( y ) ] . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} \\sqrt { \\gamma ^ l } - \\sqrt { \\gamma ^ { l - 1 } } = \\sqrt { \\gamma ^ l } - \\sqrt { ( 1 - \\alpha ^ l ) \\gamma ^ l } = \\left ( 1 - \\sqrt { 1 - \\alpha ^ l } \\right ) \\sqrt { \\gamma ^ l } \\ge \\frac { \\alpha ^ l \\sqrt { \\gamma ^ l } } { 2 } . \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\beta _ j \\left ( \\tau \\right ) \\ , d \\tau = \\infty . \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\mathbf { h } ^ T \\mathbf { X } ^ T \\mathbf { X } \\mathbf { h } = \\frac { 1 } { n } \\| \\mathbf { X } \\mathbf { h } \\| _ 2 ^ 2 \\geq \\kappa \\| \\mathbf { h } \\| _ 2 ^ 2 , \\ ; \\ ; \\ ; \\ ; \\forall \\mathbf { h } \\in \\mathbb { V } . \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\mathbf { H } _ 3 = \\mathbf { H } ^ { [ 1 : M ] } _ { [ \\ell + 1 : K ] } . \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\frac { P ( M _ 1 | D ) } { P ( M _ 2 | D ) } = \\frac { P ( D | M _ 1 ) } { P ( D | M _ 2 ) } \\frac { P ( M _ 1 ) } { P ( M _ 2 ) } , \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i & = \\big ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } ) _ { , K } \\big ) _ { , J } - E ( | \\dot { u } | \\nu _ m ) \\dot { u } _ i \\\\ a \\ddot { \\tau } & = - \\beta _ { K i } \\dot { u } _ { i , K } + m _ { I J } q _ { I , J } + M _ { j L K I } u _ { j , L K I } + K _ { I J } \\tau _ { , I J } , \\\\ \\kappa \\dot { q } _ i & = \\dot { \\tau } _ { , i } - q _ { i } \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{gather*} D _ { n } = \\begin{vmatrix} 0 & 0 & & \\ldots 0 & s _ { n } \\\\ 0 & 0 & \\ldots & s _ { n } & s _ { n + 1 } \\\\ \\ldots & \\ldots & \\ldots & \\ldots & \\ldots \\\\ 0 & s _ { n } & \\ldots & s _ { 2 n - 2 } & s _ { 2 n - 1 } \\\\ s _ { n } & s _ { n + 1 } & \\ldots & s _ { 2 n - 1 } & s _ { 2 n } \\\\ \\end{vmatrix} = ( - 1 ) ^ { \\tfrac { n ( n + 1 ) } { 2 } } s _ { n } ^ { n + 1 } \\ . \\end{gather*}"}
-{"id": "5338.png", "formula": "\\begin{align*} { \\mathcal Q } _ { k - 1 } : = \\{ [ c _ 0 : \\cdots : c _ k ] \\in { \\mathbb C } P ^ k : c _ 0 ^ 2 + \\cdots + c _ k ^ 2 = 0 \\} . \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} n ^ + = n - \\left ( \\frac { n - \\lambda + 2 \\mu } { 2 - \\lambda + \\mu } \\right ) \\mbox { a n d } k = 1 - \\lambda + 2 \\mu . \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} d _ 4 \\sigma ^ { - 1 } b _ 4 = 0 , \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{gather*} \\Psi ^ { [ k ^ { \\prime } , \\ell ^ { \\prime } ] ( \\alpha ^ { \\prime } , \\beta ^ { \\prime } ) } = \\Psi ^ { [ k , \\ell ] ( \\alpha , \\beta ) } \\Gamma ^ { [ k ^ { \\prime } , \\ell ^ { \\prime } ] ( \\alpha ^ { \\prime } , \\beta ^ { \\prime } ) } _ { [ k , \\ell ] ( \\alpha , \\beta ) } , \\end{gather*}"}
-{"id": "5424.png", "formula": "\\begin{align*} A _ a : = \\begin{pmatrix} S ^ a _ { \\alpha \\mu } \\end{pmatrix} , B _ a : = \\begin{pmatrix} S ^ a _ { \\alpha p } \\end{pmatrix} , C _ a : = \\begin{pmatrix} S ^ a _ { \\mu p } \\end{pmatrix} . \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} a ^ * _ { 0 , N _ T } = 1 , \\textrm { a n d o t h e r s b e i n g 0 } \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} M ^ { 1 } ( \\Omega ) & : = \\{ u \\in L ^ 2 ( \\Omega ) | \\tilde { Y } _ j u \\in L ^ 2 ( \\Omega ) \\mbox { f o r } j \\in \\{ 1 , \\dots , 2 n \\} \\} , \\\\ M ^ { 2 } ( \\Omega ) & : = \\{ u \\in L ^ 2 ( \\Omega ) | \\tilde { Y } _ j u , \\tilde { Y } _ { k } \\tilde { Y } _ \\ell u \\in L ^ 2 ( \\Omega ) j , k , \\ell \\in \\{ 1 , \\ldots , 2 n \\} \\} . \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} \\begin{array} { l l } \\int _ { [ 0 , \\Lambda ] } D \\phi \\ , \\ , e x p \\left \\{ - \\langle { \\phi } , L ( \\Lambda , \\phi , g ) { \\phi } \\rangle \\right \\} = \\int _ { [ 0 , { \\Lambda ' } ] } D \\psi \\ , \\ , e x p \\left \\{ - \\langle { \\psi } , L ( \\Lambda ' , \\psi , g ) { \\psi } \\rangle \\right \\} \\end{array} \\end{align*}"}
-{"id": "10159.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac n { \\mathbb E [ \\mathcal R _ n ] } \\mathbb P ( T _ 0 > n ) = \\gamma \\ , ; \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} ( n + b m ) B _ { m + n } = n \\lambda ^ m B _ n + b m \\lambda ^ n B _ m { \\rm f o r \\ a l l } \\ m , n \\in \\Z . \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\theta _ s \\theta _ { t _ i } \\cong \\begin{cases} \\theta _ { s _ 2 } , & i = 1 ; \\\\ \\theta _ { s _ { i + 1 } } \\oplus \\theta _ { s _ { i - 1 } } , & i > 1 ; \\end{cases} \\theta _ t \\theta _ { s _ i } \\cong \\begin{cases} \\theta _ { t _ 2 } , & i = 1 ; \\\\ \\theta _ { t _ { i + 1 } } \\oplus \\theta _ { t _ { i - 1 } } , & i > 1 ; \\end{cases} \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} \\nu ^ \\alpha V = \\xi _ n ^ \\alpha V \\ . \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} d f = t d x + u d y , \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} \\left \\Vert \\left \\{ \\xi _ { j } - \\sum _ { i = 1 } ^ { m } f _ { i } \\left ( z _ { j } \\right ) \\right \\} _ { j = 1 } ^ { J } \\right \\Vert _ { \\ell ^ { \\infty } } & < \\delta ^ { m } , \\\\ \\left \\Vert \\left \\{ a _ { i } ^ { m } \\right \\} _ { i = 1 } ^ { J } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } , \\left \\Vert f _ { m } \\right \\Vert _ { B _ { 2 } } & \\leq C \\delta ^ { m - 1 } . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} G _ { d , \\ell } \\left ( z ; i t \\right ) = 2 i \\sum _ { a = 0 } ^ { N } \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\sin ( 2 \\pi d z ) } { 1 - \\zeta ^ \\ell } \\right ) \\frac { ( - 2 \\pi t ) ^ a } { a ! } + O \\left ( t ^ { N + 1 } \\right ) . \\\\ \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} c _ { \\rho , \\alpha } \\left ( \\gamma \\right ) = \\left \\{ \\begin{array} [ c ] { l l l } 0 & & \\gamma < \\rho \\\\ \\frac { \\gamma - \\rho } { \\alpha - \\rho } & & \\rho \\leq \\gamma \\leq \\alpha \\\\ 1 & & \\alpha < \\gamma \\end{array} \\right . . \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} \\begin{cases} - d Y _ { t } ^ { i } = f ^ { i } ( t , \\eta _ { t } , Y ^ { i } _ { t } , Z ^ { i } _ { t } , Y ^ { i } _ { t + \\delta ( t ) } ) d t - Z ^ { i } _ { t } d B _ { t } ^ { H } , \\ \\ \\ t \\in [ 0 , T ] ; \\\\ Y _ { t } ^ { i } = g ^ { i } ( \\eta _ { t } ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ t \\in [ T , T + K ] . \\end{cases} \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\rightarrow \\infty } \\textrm { P r } \\left ( \\hat { W } _ { i , j } \\neq W _ { i , j } \\right ) = 0 , ~ ~ ~ \\forall i , \\forall j \\neq i , \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} v _ { { \\mathcal { R } } , { \\mathcal { T } } , p , n } ^ i ( u ) = \\alpha _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { \\tilde { \\mathcal { R } } _ i } ( u ) c _ p ( u ) z _ { { \\mathcal { R } } , { \\mathcal { T } } , n } ^ { \\bar { \\mathcal { R } } _ i } ( u ) , \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} \\mathcal { Z } ^ { ( m , r , l ) } \\mapsto \\mathcal { Z } ^ { T _ { \\gamma } ( m , r , l ) } = \\mathcal { Z } ^ { ( m + l \\gamma , r + d \\alpha _ { \\gamma } ( m ) + l c _ { \\gamma } , l ) } \\end{align*}"}
-{"id": "10033.png", "formula": "\\begin{align*} \\sum _ { i = \\delta } ^ { \\ell } \\alpha _ i ( i - \\delta ) ( i - \\ell ) = 2 y + x - ( \\delta + \\ell ) x + \\delta \\ell z . \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} \\deg D _ f = \\dim M _ k ( \\Gamma _ 0 ( N ) ) + g ( \\Gamma _ 0 ( N ) ) - 1 . \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} \\mu ( B ( z , r ) ) = \\mu ( B ( z ' , r ) ) z \\in V + x z ' \\in V + ( 2 y - x ) . \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} c _ 1 ( \\pi ^ { ( t ) } ) + \\dots + c _ { l ' } ( \\pi ^ { ( t ) } ) = c _ 1 ( \\pi ^ { ( 0 ) } ) + \\dots + c _ { l ' } ( \\pi ^ { ( 0 ) } ) , \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} p ( S _ { t } | y ^ { t } ) & \\stackrel { ( a ) } = B ( \\pi _ { S | Q = q } , y _ t ) \\\\ & \\stackrel { ( b ) } = \\pi _ { S | Q = g ( q , y _ t ) } , \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ { 2 n } ( i _ 1 \\ i _ 2 \\ i _ 3 \\cdots i _ n ) ^ { k \\vee } \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) = { \\bf E } \\bigl [ M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ^ q \\bigr ] , \\ ; \\Re ( q ) < \\tau , \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\omega ^ { \\sharp } ( \\alpha ) = - S _ { a ^ { * } } \\partial _ { a } + S _ { a } \\partial _ { a ^ { * } } - S _ { x ^ { * } } \\partial _ { x } + S _ { x } \\partial _ { x ^ { * } } - S _ { y ^ { * } } \\partial _ { y } + S _ { y } \\partial _ { y ^ { * } } \\ , . \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} & \\lambda ( y ^ { n - 1 } _ { n - J } ) \\\\ & = - \\log \\Big ( \\sum _ { x _ n } \\exp { \\Big \\{ \\sum _ { y _ { n } } \\log \\big ( r _ { n } ( x _ n | y _ n , y ^ { n - 1 } _ { n - M } \\big ) q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) - 1 - s \\gamma _ n ( x _ n , y ^ { n - 1 } _ { n - N } ) \\Big \\} } \\Big ) . \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ m D _ n ^ j a _ j ( x , D ' , \\tau ) = \\sum _ { j = 0 } ^ m \\tilde { a } _ j ( x , D ' , \\tau ) D _ n ^ j , \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} C _ { \\gamma } \\left ( \\overline { z } ^ { m - j } z ^ { n - j } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { j } \\right ) = \\dfrac { - 2 } { z ^ { m - n + 1 } } \\int _ 0 ^ { | z | } r ^ { 2 ( m - j ) + 1 } ( 1 - r ^ 2 ) ^ { \\gamma + j } d r . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} X _ 2 = \\beta _ { 2 , 2 } ^ { - 1 } \\bigl ( 1 , b _ 0 = 2 , \\ , b _ 1 = 1 / 2 , \\ , b _ 2 = 1 / 2 \\bigr ) , \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} \\begin{aligned} & \\forall T > 0 , \\\\ & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , s , k } T R ^ d \\left [ \\alpha + \\frac { y } { \\eta T } + C _ { d , \\alpha } \\left ( \\frac { \\eta } { R } \\right ) ^ { d - 1 } + C _ { d , \\alpha } \\theta ^ { ( d - 1 ) / 2 } \\right ] \\end{aligned} \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} & T _ 1 ( x ) = x + E _ 0 ( x ) , \\\\ & E _ 0 : B _ { 1 / 2 } ^ + \\rightarrow \\R ^ { n + 1 } , \\ | E _ 0 ( x ) | = \\max \\{ \\epsilon _ 0 , c _ { \\ast } \\} O ( | x | ^ { 1 + \\alpha } ) . \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\left \\| \\frac { 1 } { ( 2 L ) ^ d } \\int _ { [ - L , L ] ^ d } V ( x + y ) \\ , d y \\right \\| = 0 \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} \\lim _ { \\nu \\rightarrow \\infty } s _ { j } \\left ( \\nu \\right ) \\cdot \\epsilon ^ { \\nu } = 0 , \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} D ^ F _ { Z _ R } = d ^ F + d ^ { F , * } . \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\pi _ 7 ( [ P , P ] _ p ^ { F N } ) = - \\dfrac 2 3 \\Phi \\wedge \\lambda ^ 2 \\Big ( ( 4 T _ p + \\phi _ \\sigma ( T _ p ) ) ( e _ \\mu ) \\Big ) \\otimes e _ \\mu . \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} \\exp \\ ! \\left ( \\ ! \\hat { q } \\ ! \\sum _ { a > b ( \\ge 1 ) } \\ ! x ^ a x ^ b \\ ! \\right ) \\ ! = \\ ! \\int \\ ! { \\rm D } z \\exp \\left ( \\sum _ { a = 1 } ^ n \\left ( - \\frac { \\hat { q } } { 2 } ( x ^ a ) ^ 2 \\ ! + \\ ! \\sqrt { \\hat { q } } z x ^ a \\ ! \\right ) \\ ! \\right ) \\ ! \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} B F ( e _ { 2 } + \\sp \\{ e _ { 1 } \\} ) = e _ { 2 } + \\sp \\{ e _ { 1 } + e _ { 3 } \\} . \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} { n \\choose i } _ r = f _ { n - i + 1 } \\cdot \\begin{cases} { r + 2 n \\choose r + 2 i - 1 } , & \\mbox { $ n \\ge \\operatorname { m a x } \\{ 0 , \\lceil - r / 2 \\rceil \\} $ , } \\\\ - { - r - 2 i \\choose - r - 2 n - 1 } , & \\mbox { $ 0 \\le n < \\lceil - r / 2 \\rceil $ . } \\end{cases} \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} 0 = & W _ { i k l s } ( W _ { i l k s } + W _ { i k s l } + W _ { i s l k } ) \\\\ = & W _ { i k l s } W _ { i l k s } + W _ { i k l s } W _ { i k s l } + W _ { i k l s } W _ { i s l k } \\\\ = & 2 W _ { i k l s } W _ { i l k s } - | W | ^ 2 \\qquad \\hbox { ~ ~ a t ~ ~ } p . \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} \\lambda \\norm { D u } ^ { - 4 } \\Theta ^ { - 4 } & = - u _ { i j } u ^ i u ^ j \\Theta ^ { - 3 } \\\\ & = v ^ { - 1 } h _ { i j } u ^ i u ^ j \\Theta ^ { - 3 } + \\dot \\vartheta \\vartheta \\sigma _ { i j } u ^ i u ^ j \\Theta ^ { - 3 } . \\\\ \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} d s ^ 2 = \\sum ^ n _ { i , j = 1 } \\tilde { g } _ { i j } ( \\tilde { x } ) d \\tilde { x } ^ i d \\tilde { x } ^ j , \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\mathcal { A } ( x ) = \\frac { \\zeta ( 3 / 2 ) } { \\zeta ( 3 ) } x ^ { 1 / 2 } + \\frac { \\zeta ( 2 / 3 ) } { \\zeta ( 2 ) } x ^ { 1 / 3 } + O ( x ^ { 1 / 6 } \\cdot \\mathrm { e ^ { - c ( \\log ^ { 3 } x / \\log \\log x ) ^ { \\frac { 1 } { 5 } } } } ) \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\Delta f : = f ( \\kappa _ c ) - f ( \\kappa _ d ) , 0 \\le d < c , \\kappa _ x : = \\kappa _ 0 + \\frac x n . \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} \\int f d ( \\mathfrak { T } ^ { n } \\nu ) = \\sum _ { \\xi _ 0 , \\dots , \\xi _ { n - 1 } } p _ { \\xi _ { 0 } } p _ { \\xi _ { 1 } } \\ldots p _ { \\xi _ { n - 1 } } \\ , \\int f \\ , d T _ { \\xi _ { 0 } * } T _ { \\xi _ { 1 } * } \\ldots _ \\ast T _ { \\xi _ { n - 1 } * } \\nu . \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a , b ; q \\right ) _ { n } z ^ { n } } { \\left ( q , c ; q \\right ) _ { n } } = \\frac { \\left ( b ; q \\right ) _ { \\infty } } { \\left ( c , z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( c / b , z ; q \\right ) _ { n } b ^ { n } } { \\left ( q ; q \\right ) _ { n } } \\left ( a z q ^ { n } ; q \\right ) _ { \\infty } \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} { \\hat p } _ n ( e ) = 1 - ( s _ { 1 , n } ( e ) - s _ { 1 , n } ( e _ { \\text f } ) , \\hat { s h } ( e ) , \\hat { s c } ( e ) ) \\ , . \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} \\overline H ( x ) = O \\big ( \\overline F ( x / d ) \\big ) . \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} \\Theta ( T , T ^ * ) = \\mathrm { a r c t a n h } \\tilde { \\Theta } ( T , T ^ * ) , \\rho _ - ( T ) = \\mathrm { a r c t a n h } \\tilde { \\rho } _ - ( T ) , \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} \\tilde { \\pi } _ x ( b ) = g _ { f ( x ) } \\left ( \\tilde { \\pi } _ x ( u ) \\right ) \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\eta ( r ) = \\frac { 1 } { \\sinh r } \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} { \\bf y } _ r = & { \\bf B } _ 1 ( { \\bf s } _ { 1 , 2 } ^ p + { \\bf s } _ { 2 , 1 } ^ p ) + { \\bf B } _ 2 ( { \\bf s } _ { 1 , 3 } ^ p + { \\bf s } _ { 3 , 1 } ^ p ) + { \\bf B } _ 3 ( { \\bf s } _ { 2 , 3 } ^ p + { \\bf s } _ { 3 , 2 } ^ p ) \\\\ & + { \\bf B } _ 4 ( { \\bf s } _ { 1 , 2 } ^ c + { \\bf s } _ { 3 , 1 } ^ c ) + { \\bf B } _ 5 ( { \\bf s } _ { 2 , 3 } ^ c + { \\bf s } _ { 3 , 1 } ^ c ) \\\\ & + { \\bf B } _ 6 { \\bf s } _ { 2 , 3 } ^ r + { \\bf B } _ 7 { \\bf s } _ { 3 , 1 } ^ r + { \\bf n } _ r , \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} K _ { ( 1 : m ) } = K _ m = \\bigoplus _ { t = 1 } ^ { \\frac { m - 1 } { 2 } } N _ t \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} \\mathcal { B } _ { I I } = \\left \\{ \\begin{aligned} & ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in [ 0 , \\infty ) \\times \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } \\textnormal { s u c h t h a t } \\\\ & \\left | \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime ( \\tau ; 0 ) \\right ) \\right | \\leq ( \\sin \\alpha ) \\left | v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime ( \\tau ; 0 ) \\right | \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "9850.png", "formula": "\\begin{align*} \\psi ( z ) = \\sum _ { k \\in \\mathbb { Z } - \\frac { 1 } { 2 } } \\psi _ { - k } z ^ { k - \\frac { 1 } { 2 } } & \\psi ^ * ( z ) = \\sum _ { k \\in \\mathbb { Z } - \\frac { 1 } { 2 } } \\psi _ { k } ^ * z ^ { k + \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} { \\lim _ { n \\rightarrow \\infty } \\{ ( p _ { n + 1 } ) ^ { a } - ( p _ { n } ) ^ { a } } \\} < { \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } = 0 } \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\sum _ { k \\in I , k \\neq i } \\pi _ { V _ { i } } S _ { W } ^ { - 1 } \\pi _ { W _ { k } } f = \\sum _ { i \\in I } \\sum _ { k \\in I , k \\neq i } \\pi _ { V _ { i } } S _ { W } ^ { - 1 } \\pi _ { S _ { W } ^ { - 1 } V _ { i } } \\pi _ { W _ { k } } f = 0 . \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} z ( x ^ p ) = \\tilde A ( x ) z ( x ) , \\ \\ z ( x ^ q ) = \\tilde B ( x ) z ( x ) , \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\lambda ^ G ( \\cdot ) : = \\sum _ { j = 1 } ^ { n + m + 1 } \\alpha _ j ( \\cdot ) \\beta ^ j ( \\cdot ) , \\lambda ^ H ( \\cdot ) : = \\sum _ { j = 1 } ^ { n + m + 1 } \\alpha _ j ( \\cdot ) \\gamma ^ j ( \\cdot ) , \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\Delta \\varphi _ 0 & = 0 & & \\mathbb R ^ n _ + \\\\ \\frac { \\partial \\varphi _ 0 } { \\partial x _ n } & = \\alpha _ n \\frac { n - 2 } { 2 } \\sum _ { i = 1 } ^ { n - 1 } \\frac { k _ i x _ i ^ 2 } { ( 1 + | x | ^ 2 ) ^ { n / 2 } } & & \\partial R ^ n _ + \\\\ \\varphi _ 0 & \\to 0 & & | x | \\to \\infty \\end{aligned} \\right . \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} f ^ \\prime ( x _ 1 , \\dotsc , x _ { n + 1 } ) = f ( 0 , x _ 1 , \\dotsc , x _ { n + 1 } ) , q ^ \\prime ( x _ 1 , \\dotsc , x _ { n + 1 } ) = q ( 0 , x _ 1 , \\dotsc , x _ { n + 1 } ) . \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} \\bar x _ { - i } = \\sum _ { j = 1 , j \\neq i } ^ N x _ j . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} 2 ( \\Delta u ) ' ( 1 ) u ' ( 1 ) + ( 1 - \\sigma ) ( 1 + \\sigma ) ( u ' ( 1 ) ) ^ 2 = - \\dfrac { p + 3 } { p + 1 } \\dfrac { 1 } { \\pi } \\int _ B u ^ { p + 1 } \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 d s } } f _ \\infty ^ { ( s ) } ( t ) d Z _ s = 1 \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c t + a ) ^ 2 - t ^ 2 } , a = c o n s t , \\ ; c = c o n s t \\neq 0 , \\ ; c ^ 2 \\neq \\kappa ^ 2 , \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} \\begin{aligned} \\left | \\phi _ N ^ { ( s ) } ( t , Z _ s ) \\right | \\leq & \\left | \\phi _ N ^ { ( s ) } ( t , Z _ s ^ * ) \\right | + N \\left | \\phi _ N ^ { ( s - 1 ) } ( t , Z _ s ^ { ( i ) } ) \\right | + N \\left | \\phi _ N ^ { ( s - 1 ) } ( t , Z _ s ^ { ( j ) } ) \\right | + \\\\ & + N \\left | \\phi _ N ^ { ( s - 1 ) } ( t , ( Z _ s ^ * ) ^ { ( i ) } ) \\right | + N \\left | \\phi _ N ^ { ( s - 1 ) } ( t , ( Z _ s ^ * ) ^ { ( j ) } ) \\right | \\end{aligned} \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} \\tilde { \\omega } ( x \\mu ) = \\omega ( x ) \\mu ( 1 ) . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} \\omega _ { 1 , \\infty } ^ + = 0 , \\omega _ { 2 , \\infty } ^ - = 0 . \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} u ( x , t ) = \\frac { 1 } { 2 \\sqrt { \\pi t } } \\int \\limits _ { - \\infty } ^ { + \\infty } \\Lambda ( s ) \\exp \\left \\{ - \\frac { ( s - x ) ^ 2 } { 4 t } \\right \\} \\ , d s . \\end{align*}"}
-{"id": "9982.png", "formula": "\\begin{align*} \\int _ A e ^ { - d ( a , b ) } \\ d \\mu ( b ) = 1 \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} \\tilde { l } _ \\alpha ( r ) = \\left \\{ \\begin{array} { l l } l ( 0 ) & \\textrm { i f } r \\leq \\alpha \\sqrt { d } / 2 \\\\ l ( r - \\alpha \\sqrt { d } / 2 ) & \\textrm { i f } r \\geq \\alpha \\sqrt { d } / 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} \\left \\{ \\mathbf { \\bar { v } } _ { q p } ^ { [ { \\sf b s } ] ( a ) } \\right \\} _ { q \\in [ 1 : K ] , p \\in [ 1 : N ] , a \\in [ 1 : \\frac { \\lambda _ 2 } { \\lambda _ 1 } T ^ { K N ( K M + K N ) } ] ~ \\textrm { e x c e p t $ q = i $ a n d $ p = j $ } } \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} \\lambda _ 1 ( g _ { i j } ) = \\inf \\Big \\{ \\mathcal { F } ( g _ { i j } , f ) : f \\in C _ c ^ \\infty ( M ) , \\int _ M e ^ { - f } d \\mu = 1 \\Big \\} , \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} \\left < \\varphi , f \\right > = \\int _ { \\mathbb { R } ^ { 2 d } } \\varphi ( x , v ) f ( x , v ) d x d v \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} \\Phi _ 2 ( a , b ; c ; x , y ) = \\sum _ { m = 0 } ^ \\infty \\frac { ( a ) _ m } { ( c ) _ m } \\ , { } _ 2 F _ 1 \\left [ \\begin{array} { c } - m , \\ , b \\\\ 1 - a - m \\end{array} ; \\ , \\frac { y } { x } \\right ] \\ , \\frac { x ^ m } { m ! } . \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} F _ { k \\ell } ( v , y ) = - \\tilde { a } ^ { k \\ell } ( y ) J ( v ) . \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} & \\textbf { C } = [ \\textbf { C } _ { - 1 } , \\textbf { C } _ 0 , \\ldots , \\textbf { C } _ { N _ h + 1 } ] ^ T , \\\\ & \\textbf { C } _ l = [ c _ { 0 , 0 , l } , c _ { 0 , 1 , l } , \\ldots , c _ { 0 , 2 M , l } , c _ { 1 , 0 , l } , \\ldots , c _ { 1 , 2 M , l } , \\ldots , c _ { 2 ^ k - 1 , 2 M , l } ] , \\\\ & \\textbf { H } ( x ) = [ B _ { - 1 } ( x ) , B _ { 0 } ( x ) , \\ldots , B _ { N _ h + 1 } ( x ) ] , \\\\ \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} \\Psi = \\left [ \\begin{array} { c c } 0 & - e ^ { \\j \\alpha } \\\\ - e ^ { - \\j \\alpha } & 2 \\cos \\beta \\end{array} \\right ] \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { M , M - 1 } ( a , b ) ^ q \\bigr ] = \\eta _ { M , M - 1 } ( q \\ , | a , \\ , b ) . \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} 2 g _ 2 ^ { t r } g _ 2 + 2 b _ 2 b _ 2 ^ { t r } + 2 f _ 2 f _ 2 ^ { t r } = I \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} \\frac { \\partial \\lambda } { \\partial \\rho } = - \\frac { 1 } { 2 } \\zeta ' ( 0 , D ) \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} d e _ \\alpha ^ j & = 0 \\\\ d s ^ { i j } & = \\sum _ { \\alpha , \\beta } g ^ { \\alpha \\beta } e _ \\alpha ^ i e _ \\beta ^ j \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\varphi _ i = \\vartheta ^ { - 1 } u _ i , \\varphi _ { i j } = \\vartheta ^ { - 1 } u _ { i j } - \\cosh u \\ , \\vartheta ^ { - 2 } \\ , u _ i \\ , u _ j , \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} x _ a \\ , x _ { b t t } \\ , + \\ , y _ a \\ , y _ { b t t } \\ , = \\ , x _ b \\ , x _ { a t t } \\ , + \\ , y _ b \\ , y _ { a t t } , \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\overline { s } _ e = \\begin{cases} \\widetilde { \\Phi } ( s _ e ) & \\\\ t _ { e ( w ( v ) ) } & . \\end{cases} \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\lim _ { p _ { k } \\rightarrow \\infty } \\frac { p _ { k + 1 } - p _ { k } } { ( \\log p _ { k } ) ^ { 2 } } = 1 \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} ( \\mu \\bullet f ) \\bullet g - \\mu \\bullet ( f \\bullet g ) = ( - 1 ) ^ { ( n - 1 ) m } ( \\mu \\bullet _ 1 f ) \\bullet _ { m + 1 } g + ( - 1 ) ^ { m - 1 } ( \\mu \\bullet _ 2 f ) \\bullet _ { 1 } g . \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} K ( h ^ { \\frac { 1 } { 2 ^ { n } } } , 2 ) ^ { r _ { n } } H _ { \\nu } ( A , B ) & \\leqslant A \\nabla B - \\sum _ { k = 0 } ^ { n - 1 } r _ { k } [ H _ { \\frac { m _ k } { 2 ^ k } } ( A , B ) - 2 H _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } ( A , B ) + H _ { { \\frac { m _ k + 1 } { 2 ^ k } } } ( A , B ) ] \\\\ & \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ { n } } } , 2 ) ^ { R _ { n } } H _ { \\nu } ( A , B ) \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} ^ { \\partial ( h _ 1 ) } h _ 2 = h _ 1 h _ 2 h _ 1 ^ { - 1 } . \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} j ' e ^ { \\rho \\log \\rho + O ( \\rho ) } + e ^ { - \\Theta ( j ' ) } = e ^ { ( \\log \\rho ) ^ 2 + \\rho \\log \\rho + O ( \\rho ) } = e ^ { \\rho \\log \\rho + O ( \\rho ) } , \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} & T _ { \\xi _ { 1 } } \\circ \\dots \\circ T _ { \\xi _ { \\ell } } ( I ) \\cap T _ { \\omega _ { 1 } } \\circ \\dots \\circ T _ { \\omega _ { s } } ( I ) = \\emptyset \\mbox { a n d } \\\\ & T _ { \\xi _ { 1 } } \\circ \\dots \\circ T _ { \\xi _ { \\ell } } ( I ) \\cup T _ { \\omega _ { 1 } } \\circ \\dots \\circ T _ { \\omega _ { s } } ( I ) \\subset J . \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } a _ g ( n ) n ^ { 1 - k } q ^ n = \\sum _ { n = 0 } ^ N \\frac { ( - x ) ^ n } { n ! } L _ g \\left ( e ^ { \\frac { 2 \\pi i \\ell } { m } } ; k - 1 - n \\right ) + O \\left ( x ^ { N + 1 } \\right ) \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} \\epsilon ( b ) a = a \\circ ( b _ 1 S ( b _ 2 ) ) = ( a _ 1 \\circ b _ 1 ) S ( a _ 2 ) ( a _ 3 \\circ S ( b _ 2 ) ) \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} C _ X ( s ' , s ' + u ) = ( 1 - u ) ^ + \\quad , ~ 0 \\leq s ' \\leq s ' + u , \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} G _ \\pm ( t ) : = ( \\frac { d } { i d t } + H ) E _ \\pm ( t ) = ( H J _ a - J _ a H _ 0 ) E _ \\pm ( t ) , \\end{align*}"}
-{"id": "9979.png", "formula": "\\begin{align*} d _ H ( A , B ) = \\max \\bigl \\{ \\sup _ { a \\in A } d ( a , B ) , \\sup _ { b \\in B } d ( b , A ) \\bigr \\} . \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} h ( x , \\nabla _ x \\phi _ \\pm ( x , \\xi ) ) = h _ 0 ( \\xi ) \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} f _ { | h _ 1 | ^ 2 , | h _ 2 | ^ 2 } ( x _ 1 , x _ 2 ) = \\frac { 2 } { \\beta ^ 2 } e ^ { - \\frac { x _ 1 + x _ 2 } { \\beta } } . \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} \\dot { x } = - \\tilde { F } ^ { - 1 } \\nu \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 d } } v f _ N ^ { ( 1 ) } ( 0 , x , v ) d x d v = \\int _ { \\mathbb { R } ^ { 2 d } } v f _ 0 ( x , v ) d x d v \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\tau ^ { - \\frac { q } { \\tau } } \\frac { \\Gamma _ 2 ( 1 - q \\ , | \\tau ) \\Gamma _ 2 ( \\tau \\ , | \\tau ) } { \\Gamma _ 2 ( 1 \\ , | \\tau ) \\Gamma _ 2 ( \\tau - q \\ , | \\tau ) } = \\frac { \\Gamma ( 1 - q ) } { \\Gamma ( 1 - \\frac { q } { \\tau } ) } , \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} e _ { 3 } ^ \\prime & = A _ { 1 , 1 } \\sum \\limits _ { i = 3 } ^ { n - 2 k } A _ { 2 , i - 1 } e _ i + A _ { 2 , 2 } \\sum \\limits _ { i = 1 } ^ { k } B _ { 1 , i } f _ { k + i } , \\\\ e _ { i } ^ \\prime & = A _ { 1 , 1 } ^ { i - 2 } \\sum \\limits _ { j = i } ^ { n - 2 k } A _ { 2 , j - i + 2 } e _ j , \\ \\ 4 \\leq i \\leq n - 2 k . \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} \\sum _ { m \\geq j } 2 p ^ m m \\frac { \\Gamma ( m + \\rho ) } { \\Gamma ( m + 1 ) } = \\sum _ { m = j } ^ { j ' } 2 p ^ m m \\frac { \\Gamma ( m + \\rho ) } { \\Gamma ( m + 1 ) } + \\sum _ { m = j ' + 1 } ^ { \\infty } 2 p ^ m m \\frac { \\Gamma ( m + \\rho ) } { \\Gamma ( m + 1 ) } . \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} & \\int \\ln \\left ( 1 + | y | \\right ) r _ b ( y ) \\ , d y \\\\ & = \\int \\int \\ln \\left ( 1 + | y | \\right ) s _ b ( y - x ) \\ , d F _ X ( x ) \\ , d y \\\\ & = \\int \\int \\ln \\left ( 1 + | y | \\right ) s _ b ( y - x ) \\ , d y \\ , d F _ X ( x ) \\\\ & \\leq \\int \\int \\left ( \\ln ( 1 + | x | ) + \\ln ( 1 + | y | ) \\right ) s _ b ( y ) \\ , d y \\ , d F _ X ( x ) \\\\ & = S _ b \\int \\ln ( 1 + | x | ) d F _ X ( x ) + L _ b \\\\ & < \\infty , \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} { u _ 1 } _ { | S } = { u _ 2 } _ { | S } \\sum _ { 1 \\leq j \\leq n } a _ { n j } ^ { ( 1 ) } ( x ) { D _ j u _ 1 } _ { | S } = \\sum _ { 1 \\leq j \\leq n } a _ { n j } ^ { ( 2 ) } ( x ) { D _ j u _ 2 } _ { | S } , \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { k } \\bigr ) ( q \\ , | \\ , b ) & = 0 , \\ ; k < M - 1 , \\\\ \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M - 1 } \\bigr ) ( q \\ , | \\ , b ) & = \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M - 1 } \\bigr ) ( 0 \\ , | \\ , b ) , \\\\ \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M } \\bigr ) ( q \\ , | \\ , b ) & - \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M } \\bigr ) ( 0 \\ , | \\ , b ) = - q f ( 0 ) \\ , M ! \\prod _ { j = 1 } ^ { M - 1 } b _ j . \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( 1 + \\frac { x } { i } + \\frac { x ^ 2 } { 4 \\pi ^ 2 } \\right ) & \\left ( 1 + \\frac { x } { i } + \\frac { x ^ 2 } { 1 6 \\pi ^ 2 } \\right ) \\\\ & \\left ( 1 + \\frac { x } { i } + \\frac { x ^ 2 } { 3 6 \\pi ^ 2 } \\right ) \\left ( 1 + \\frac { x } { i } + \\frac { x ^ 2 } { 6 4 \\pi ^ 2 } \\right ) \\cdots , \\end{aligned} \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v + \\nabla _ H p - \\Delta _ H v + f _ 0 k \\times v = 0 , \\\\ \\partial _ z p + T = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\\\ \\partial _ t T + v \\cdot \\nabla _ H T + w \\left ( \\partial _ z T + \\frac { 1 } { h } \\right ) - \\partial _ z ^ 2 T = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} = \\frac { t ^ { p / 2 } } { \\sqrt { \\pi } } \\sum \\limits _ { n = 0 } ^ { N - 1 } 2 ^ { p - n } t ^ { - n / 2 } \\int \\limits _ { \\mu } ^ { + \\infty } z ^ { p - n } e ^ { - ( z - \\eta _ 1 ) ^ 2 } d z \\times \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} a ^ { G _ 3 } ( S , u _ 3 ( \\alpha ) ) = c ( u _ 3 ) \\times \\big \\{ \\mathfrak { c } ( S ) + \\chi _ S ( \\alpha ) \\ , L ^ S ( 1 , \\chi ) \\big \\} \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} { } _ { 1 } \\tilde { \\phi } _ { 1 } ( 0 ; a ; q , z ) = { } _ { 1 } \\tilde { \\phi } _ { 1 } ( 0 ; z ; q , a ) , \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} u ( t , \\xi ) = \\sup _ { M ( t ) } u . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} \\widehat { \\theta } _ 0 ( x _ a ( z ) ) = x _ a ( z ) + \\sum _ { \\beta , \\gamma \\in \\Delta _ + } c _ { a , \\beta } ^ { \\gamma } : \\psi _ { \\gamma } ( z ) \\psi _ { \\beta } ^ * ( z ) : . \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} g ( z ) = e ^ { - \\eta ( \\omega _ 0 ) z } \\frac { \\prod _ { k = 1 } ^ 4 \\sigma ( z - \\bar b _ k - \\frac { 1 } { 2 } ) } { \\prod _ { k = 1 } ^ 4 \\sigma ( z - b _ k ) } , \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} { n \\brace i } = \\frac { 4 ^ { n - i + 1 } - 1 } { n - i + 1 } B _ { 2 ( n - i + 1 ) } { 2 n + 1 \\choose 2 i } , \\quad 0 \\leq i \\leq n . \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} - \\varepsilon u _ { \\varepsilon } ^ { \\prime \\prime } + u _ { \\varepsilon } ^ { \\prime } + \\partial \\phi ( u _ { \\varepsilon } ) & \\ni f ( u _ { \\varepsilon } ) ( 0 , T ) , \\\\ u _ { \\varepsilon } ( 0 ) & = u _ { 0 } u _ { \\varepsilon } ^ { \\prime } ( T ) = 0 \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} C = \\frac { - 1 } { 2 \\bigl ( u ( o ) - m _ u \\bigr ) } > 0 . \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} ( S _ { \\tilde { 1 } } - \\iota _ \\lambda S _ { \\tilde { 0 } } ) \\cdot ( x , y , z ) ^ { t r } = 0 \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} I _ { \\mu } ( z ) & \\sim \\left ( \\frac { z } { 2 } \\right ) ^ \\mu / \\Gamma ( \\mu + 1 ) , z \\to 0 , \\mu > - 1 , \\\\ I _ { \\mu } ( z ) & = \\frac { e ^ z } { \\sqrt { 2 \\pi z } } \\left ( 1 + O \\left ( \\frac { 1 } { z } \\right ) \\right ) , z \\to \\infty , | \\arg z | < \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} \\frac { \\partial w } { \\partial x _ h } ( x ) = \\int _ { \\Omega } \\frac 1 { | x - y | ^ { n } } f \\Big ( x , \\frac { y - x } { | y - x | } \\Big ) v ( y ) d y + C v ( x ) \\hbox { f o r a . e . $ x \\in \\Omega $ , } \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} \\sum _ { a = 0 } ^ 8 p ^ * _ a q ^ * _ a = 0 , \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} \\pi _ j ^ i ( t ) : = \\pi ^ j ( \\pi ^ i ( t ) ) . \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} \\begin{aligned} & b _ { s , s + k + 1 } \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k , 0 ; v _ { s + 1 } , \\dots , v _ { s + k } , v _ { s + k + 1 } ; \\right . \\\\ & \\qquad \\qquad \\qquad \\qquad \\qquad \\left . \\omega _ 1 , \\dots , \\omega _ k , \\omega _ { k + 1 } ; i _ 1 , \\dots , i _ k , i _ { k + 1 } \\right ] = 0 \\end{aligned} \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} e ^ { \\mu \\nu \\sigma \\tau } F _ { \\mu \\nu } F _ { \\sigma \\tau } = 8 \\mathbf { E } \\cdot \\mathbf { B } , e _ { \\mu \\nu \\sigma \\tau } R ^ { \\mu \\nu } R ^ { \\sigma \\tau } = 8 \\mathbf { H } \\cdot \\mathbf { D } . \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} a = ( \\hat \\varphi \\otimes \\iota ) ( [ ( T \\otimes \\iota ) \\hat W ] \\hat W ^ * ) . \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} F [ X ] \\to F ' [ X ] : = F [ X + 1 ] - F [ X ] \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} \\gamma Z = ( A Z + B ) ( C Z + D ) ^ { - 1 } , J ( \\gamma , Z ) = C Z + D \\in G L _ 2 ( \\C ) , \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} P _ { n } \\stackrel { ( d ) } { = } \\boldsymbol { 1 } _ { \\{ B _ { n } = 1 \\} } \\cdot \\left ( P _ { U _ { n } } ' + P _ { n - U _ { n } } '' \\right ) + \\boldsymbol { 1 } _ { \\{ B _ { n } = 0 \\} } \\cdot \\left ( P _ { U _ { n } } ' \\cdot P _ { n - U _ { n } } '' \\right ) , , P _ { 1 } = 1 , \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} a ( \\bar { x } , x ) = \\bigwedge _ { i \\in I } a ( x _ i , x ) , \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} d _ \\ell = 2 a _ 0 + 2 \\sum _ { k ' = 1 } ^ { n _ 1 - 1 } ( \\omega ^ { n _ 2 } ) ^ { k ' \\ell } a _ { n _ 2 k ' } + 2 \\sum _ { k ' = 1 } ^ { n _ 2 - 1 } ( \\omega ^ { n _ 1 } ) ^ { k ' \\ell } a _ { n _ 1 k ' } + 2 \\sum _ { \\substack { 1 \\leq k < n \\\\ ( k , n ) = 1 } } \\omega ^ { k \\ell } a _ k . \\end{align*}"}
-{"id": "10137.png", "formula": "\\begin{align*} p + q + \\gcd ( 2 q , - p ) + \\gcd ( q , - 2 p ) = 0 , \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} F _ { d , \\ell } ( z ; \\tau ) = F _ { d ' , \\ell } ( z ; \\tau ) + \\sum _ { a \\geq 0 } \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\zeta ^ { d } \\left ( 1 - \\zeta ^ { \\ell r } \\right ) } { 1 - \\zeta ^ { \\ell } } \\right ) \\frac { ( 2 \\pi i \\tau ) ^ a } { a ! } . \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} ( x ^ 2 + x ) ^ { m } = \\left [ \\frac { ( 2 x + 1 ) ^ 2 - 1 } 4 \\right ] ^ m \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\sigma _ j ( w ) = B _ j ( x ) w , \\ j = 1 , 2 \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} g ^ * = \\begin{cases} g _ 1 & \\ p > \\frac { 3 \\beta + 1 } { 7 + 5 \\beta } \\\\ g _ 2 & \\ p < \\frac { 3 \\beta + 1 } { 7 + 5 \\beta } \\\\ \\big \\{ ( q , 1 - q ) : 0 \\leq q \\leq 1 \\big \\} & \\ p = \\frac { 3 \\beta + 1 } { 7 + 5 \\beta } . \\end{cases} \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} \\Sigma _ 1 : = \\sum _ b \\Phi _ 0 ( b ) ^ 2 , \\Sigma _ 2 & : = \\sum _ b \\Phi _ 0 ( b ) \\Phi ( b ) , \\Sigma _ 3 : = \\sum _ b \\Phi ( b ) ^ 2 . \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} d \\bar { s } ^ 2 = \\frac { 1 } { ( 1 - r ^ 2 ) ^ 2 } \\ , d r ^ 2 + \\frac { r ^ 2 } { 1 - r ^ 2 } \\sigma _ { i j } d \\xi ^ i d \\xi ^ j . \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { m - 1 } ( - 1 ) ^ i { m - 1 \\choose i } \\gamma _ { m - r + i } = 0 . \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} ( \\sigma \\otimes \\theta _ n ) J _ n ( x ) = ( \\sigma \\otimes \\bigotimes ^ n _ { j = 1 } \\psi _ 0 ) J _ n ( x ) = \\sigma \\circ T ^ n _ { \\psi _ 0 } \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} F = - T \\ln Z , \\langle E \\rangle = T ^ 2 \\frac { \\partial \\ln Z } { \\partial T } , S = \\frac { \\partial } { \\partial T } ( T \\ln Z ) , \\frac { \\partial S } { \\partial T } = \\frac { \\sigma } { T ^ 3 } \\geq 0 \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} A = 1 0 ^ 6 , \\delta \\in [ \\frac { 1 } { A \\log ^ { 1 0 0 } N } , \\frac { 1 } { A \\log ^ 2 N } ] , s _ { - \\delta } = s ( 1 - \\delta ) , s _ { \\delta } = s ( 1 + \\delta ) . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ n \\ell ^ h \\ , \\vartheta _ { \\mathcal L _ { \\Gamma } } ^ { ( \\ell ) } ( z ) = \\sum _ { \\ell = 0 } ^ n \\ell ^ h \\ , \\vartheta _ { \\mathcal L _ { \\Gamma ' } } ^ { ( \\ell ) } ( z ) \\qquad 0 \\leq h \\leq p _ 0 , \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} | \\eta ( t , s ) - \\xi | & = | p ( s , t ; y ( s , t ) , \\xi ) - \\xi | \\\\ & = \\left | \\int _ s ^ t \\nabla _ x V _ \\rho ( \\tau , q ( \\tau , t ; y ( s , t ) , \\xi ) ) d \\tau \\right | \\\\ & \\leq C \\rho ^ { \\varepsilon _ 0 } \\langle s \\rangle ^ { - \\varepsilon _ 1 } . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} x ^ { - 1 } z ^ 2 x & = z ^ { - 2 } & z ^ { - 1 } x ^ 2 z & = x ^ { - 2 } \\\\ y ^ { - 1 } z ^ 2 y & = z ^ { - 2 } & z ^ { - 1 } y ^ 2 z & = y ^ { - 2 } \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\norm { b ( y ) } = \\inf _ { W } \\sup _ { x \\in W \\cap X _ A } \\norm { b ( x ) } , \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\Delta ( \\log p _ t ( x _ 1 , 1 _ 1 ) ) + \\frac { m } { 2 t } = \\frac { m - 1 } { 2 t } - \\frac { ( n - 2 ) e ^ { x _ 1 / t } } { t ^ 2 ( n - 2 - n e ^ { x _ 1 / t } ) ^ 2 } \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\pi : Y = ( M \\times E G ) / G \\to M / G = X . \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} z _ 0 ( z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 ) = z _ 0 , \\ \\ z _ 1 ^ 2 z _ 2 = z _ 1 , \\ \\ ( z _ 0 ^ 2 - z _ 2 ^ 2 ) z _ 2 = z _ 2 . \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} \\log | \\varphi | ^ 2 = \\log \\left ( r | \\psi | ^ 2 + s \\right ) \\ , . \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{gather*} p _ { n _ { k } } ( x ) : = P _ { n _ { k } } ( x ) / D _ { n _ { k } - 1 } \\ , \\ 0 \\leq k < m + 1 \\ , \\ \\ \\ \\ D _ { - 1 } : = 1 \\ , \\ \\end{gather*}"}
-{"id": "2404.png", "formula": "\\begin{align*} T _ { ( k ) } & \\stackrel { d } { = } Y _ { n - k + 1 } + Y _ { n - k + 2 } + \\ldots + Y _ { n } \\\\ & \\stackrel { d } { = } W _ { 1 } + W _ { 2 } + \\ldots + W _ { k } , \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} & [ a _ { ( m ) } , b _ { ( n ) } ] = \\sum _ { i \\geq 0 } \\begin{pmatrix} m \\\\ i \\end{pmatrix} ( a _ { ( i ) } b ) _ { ( m + n - i ) } , \\\\ & ( a _ { ( m ) } b ) _ { ( n ) } = \\sum _ { j \\geq 0 } ( - 1 ) ^ j \\begin{pmatrix} m \\\\ j \\end{pmatrix} ( a _ { ( m - j ) } b _ { ( n + j ) } - ( - 1 ) ^ m b _ { ( m + n - j ) } a _ { ( j ) } ) . \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} T _ F = T _ E \\frac { B } { C _ F } , \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} \\int _ { \\vert z \\vert \\leq R } J ( z ) \\ , d z = \\vert S _ { N - 1 } \\vert \\int _ 0 ^ R r ^ { N - 1 } J ( r ) \\ , d r , \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} \\tfrac { 1 } { ( g ! ) ^ { 2 } } \\int _ { X ^ { g } } g ( P _ { k } , Q ) \\Phi ^ { * } \\nu ^ { g } ( P _ { 1 } , \\dots , P _ { g } ) = \\int _ { X } g ( P _ { k } , Q ) \\mu ( P _ { k } ) = 0 . \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} [ [ x _ 1 , x _ 2 , x _ 3 ] , y _ 2 , y _ 3 ] = [ [ x _ 1 , y _ 2 , y _ 3 ] , x _ 2 , x _ 3 ] + [ [ x _ 2 , y _ 2 , y _ 3 ] , x _ 3 , x _ 1 ] + [ [ x _ 3 , y _ 2 , y _ 3 ] , x _ 1 , x _ 2 ] . \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} \\mathbf { F } = \\mathbf { F } _ { \\parallel } + \\mathbf { F } _ { \\perp } , \\qquad \\mathbf { G } = \\mathbf { G } _ { \\parallel } + \\mathbf { G } _ { \\perp } , \\end{align*}"}
-{"id": "10015.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } n _ { d _ i } ( q ^ { d _ i } - 1 ) = q ^ n - 1 , \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} ( \\partial _ { y _ i y _ j } v ) _ { ( n - 1 ) \\times ( n - 1 ) } & = ( \\partial _ { x _ i x _ j } w ) _ { ( n - 1 ) \\times ( n - 1 ) } - A ( w ) ^ t H ( w ) ^ { - 1 } A ( w ) , \\\\ ( \\partial _ { x _ i x _ j } w ) _ { ( n - 1 ) \\times ( n - 1 ) } & = ( \\partial _ { y _ i y _ j } v ) _ { ( n - 1 ) \\times ( n - 1 ) } - A ( v ) ^ t H ( v ) ^ { - 1 } A ( v ) . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} \\langle d \\phi , g d \\phi \\rangle = - \\langle \\phi , g \\Delta \\phi \\rangle + \\frac { 1 } { 2 } \\langle \\phi , { \\Delta g } \\phi \\rangle \\end{align*}"}
-{"id": "9993.png", "formula": "\\begin{align*} f _ R ( x ) = \\begin{cases} 1 & \\norm { x } _ 2 \\le R , \\\\ e ^ R e ^ { - \\norm { x } _ 2 } & \\norm { x } _ 2 \\ge \\sqrt { R } \\end{cases} \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n P _ i = \\frac { 1 } { Z } \\sum _ { i = 1 } ^ n e ^ { - \\beta E _ i } = 1 \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{gather*} \\Delta _ { \\hat { S } ( \\gg ^ * ) } ( \\partial ^ \\mu ) = \\tilde { \\mathcal { F } } ^ { - 1 } _ R \\Delta _ 0 ( \\partial ^ \\mu ) \\tilde { \\mathcal { F } } _ r . \\end{gather*}"}
-{"id": "8381.png", "formula": "\\begin{align*} f ( 4 m _ 0 k + 2 ) = \\frac { - a } { 1 + 2 a } , f ( 4 m _ 0 k + 3 ) = - \\frac { 1 + a } { 1 + 2 a } , \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} H _ n = \\begin{cases} \\log _ { 1 / p } n + \\frac { 1 } { 2 } \\log _ { p / q } \\log n + o ( \\log \\log n ) & p > q \\\\ \\log _ { 2 } n + \\sqrt { 2 \\log _ 2 n } + o ( \\sqrt { \\log n } ) & p = q \\end{cases} \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\frac { d x \\pm d x ^ 2 } { d x } = 1 \\pm d x = 1 , \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} n _ 1 = n ( v ) ; n _ 2 = g ' ( u ) \\ , l ( v ) + f ' ( u ) \\ , e _ 4 . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} & D _ { 1 } ( L _ { n } ) = n L _ n , & & D _ { 1 } ( G _ n ) = n G _ n , & D _ { 1 } ( 1 ) = 0 , \\\\ & D _ { 2 } ( L _ { n } ) = 0 , & & D _ { 2 } ( G _ n ) = G _ n , & D _ { 2 } ( 1 ) = 0 , \\\\ & D _ { 3 } ( L _ { n } ) = n G _ { n - 1 } , & & D _ { 3 } ( G _ n ) = 0 , & D _ { 3 } ( 1 ) = 0 , \\\\ & D _ { 4 } ( L _ { n } ) = 0 , & & D _ { 4 } ( G _ n ) = L _ { n + 1 } , & D _ { 4 } ( 1 ) = 0 . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} H ( x ) = \\chi _ { ( - 1 / 2 , 1 / 2 ) } ( x ) = \\begin{cases} 1 , & x \\in ( - 1 / 2 , 1 / 2 ) , \\\\ 0 , & | x | \\geq 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 5 } \\nu _ { j } ( \\alpha ) \\ , n ^ { j } \\geq 0 \\ , , n \\geq 2 \\ , , \\ \\ \\alpha > - 1 \\ , . \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} c _ 0 = c _ 0 ( \\alpha , \\beta ) = \\frac { ( 2 \\beta - 1 ) _ + } { \\alpha } , c ' _ 0 = c ' _ 0 ( \\kappa ) = c _ 0 + \\kappa 1 _ { \\beta = 1 / 2 } . \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} Y _ 1 = I ( Y _ 0 , \\Phi ( Y _ 0 ) - Y _ 0 ) . \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} ^ { R \\ ! } D _ { a + } ^ \\alpha x ( t ) : = ( D ^ m I _ { a + } ^ { m - \\alpha } x ) ( t ) \\qquad \\hbox { f o r } t \\in ( a , b ] , \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\left | A ( z , \\zeta ) \\right | ^ { 1 + \\mu _ { 0 } } \\delta _ { \\Omega } ^ { - \\gamma \\mu _ { 0 } - \\varepsilon } ( \\zeta ) \\lesssim \\left ( 2 ^ { i } \\delta _ { \\Omega } ( z ) \\right ) ^ { - \\mu _ { 0 } ( \\gamma + n ) - \\varepsilon } \\prod _ { j = 1 } ^ { n - 1 } \\tau _ { j } ^ { 2 } \\left ( z , 2 ^ { i } \\delta _ { \\Omega } ( z ) \\right ) \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} D _ { L } ( L - N , B ( 0 , \\rho ) ) = 0 . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} P _ { \\mu \\nu } Q ^ { \\mu \\nu } = 2 F _ { \\mu \\nu } R ^ { \\mu \\nu } - \\frac { i } { 2 } \\left ( e ^ { \\mu \\nu \\sigma \\tau } F _ { \\mu \\nu } F _ { \\sigma \\tau } + e _ { \\mu \\nu \\sigma \\tau } R ^ { \\mu \\nu } R ^ { \\sigma \\tau } \\right ) . \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} C U B ( \\alpha _ { k , K } ) \\triangleq \\frac { 1 } { { \\binom { k - 1 } { l - 1 } } } \\sum _ { \\{ L _ i \\subseteq K , \\ ; | L _ i | = l \\} } \\alpha _ { l , L _ i } \\geq \\alpha _ { k , K } . \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} \\mathbb { E } ( D _ { n } ) = \\frac { 1 + \\sqrt { 2 } } { 2 } \\binom { n + \\sqrt { 2 } - 2 } { n - 1 } - \\frac { \\sqrt { 2 } - 1 } { 2 } \\binom { n - \\sqrt { 2 } - 2 } { n - 1 } \\sim \\frac { 1 + \\sqrt { 2 } } { 2 } \\frac { n ^ { \\sqrt { 2 } - 1 } } { \\Gamma ( \\sqrt { 2 } ) } . \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} r ^ { \\pi } _ t ( x _ t | y _ { t - M } ^ { t - 1 } , y _ t ) = \\Big ( \\frac { q _ t ( y _ t | y _ { t - M } ^ { t - 1 } , x _ t ) } { \\nu ^ { \\pi } _ { t } ( y _ t | y _ { t - J } ^ { t - 1 } ) } \\Big ) { \\pi } _ { t } ( x _ t | y _ { t - J } ^ { t - 1 } ) , ~ t \\in \\mathbb { N } _ 0 ^ n . \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} \\mathbb E X ^ s = \\frac 1 { E _ { \\alpha , \\beta } ^ \\gamma ( t ^ \\alpha ) } \\sum _ { j = 0 } ^ s ( \\gamma ) _ j \\ , { s \\brace j } \\ , t ^ { \\alpha j } \\ , E _ { \\alpha , \\alpha j + \\beta } ^ { \\gamma + j } ( t ^ \\alpha ) , s \\in \\mathbb N _ 0 , \\ , t \\geq 0 . \\end{align*}"}
-{"id": "9940.png", "formula": "\\begin{align*} V ^ { \\sigma } ( A ) = \\bigoplus _ { \\delta _ 1 + \\cdots + \\delta _ k = \\sigma } V ( \\delta _ 1 , \\dots , \\delta _ k ) . \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} h ( v ) = \\beta ( d - v ) ( v - p ) , \\ ; \\ ; d , p , \\beta > 0 \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} A _ { t } = A _ { - } \\oplus 0 + 0 \\oplus ( A _ { + } ) _ { t } , \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} d _ 2 ( c _ 2 ) ^ k b _ 2 = 0 , k = 0 , 1 , 2 , 3 , \\cdots \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} \\hat v _ i ^ k = v ^ k _ i , x _ i ^ { k + 1 } = x _ i ^ { k } , \\quad \\hbox { a n d } v _ i ^ { k + 1 } = v _ i ^ { k } \\qquad \\hbox { f o r $ i \\not \\in \\{ I ^ k , J ^ k \\} $ } . \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v - \\Delta v + \\nabla _ H p + f _ 0 k \\times v = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\\\ \\partial _ z p = T , \\\\ \\partial _ t T + v \\cdot \\nabla _ H T + w \\left ( \\partial _ z T + \\frac 1 h \\right ) - \\partial _ z ^ 2 T = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} T ( S x ) = T ( 4 x + 1 ) = ( 3 x + 1 ) 2 ^ { 2 - \\nu _ 2 ( 1 2 x + 4 ) } . \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} & \\textbf { g } _ \\nu = [ g _ \\nu ( t _ 1 ) , g _ \\nu ( t _ 2 ) , \\cdots , g _ \\nu ( t _ { N _ t } ) ] ^ T , \\nu = 1 , 2 , \\\\ & \\textbf { Z } _ 1 = \\bigg [ \\frac { s - p h } { 2 ( p h c - s ) } , 1 , \\frac { s - p h } { 2 ( p h c - s ) } , 0 , \\ldots , 0 \\bigg ] _ { 1 \\times ( N _ h + 3 ) } , \\\\ & \\textbf { Z } _ 2 = \\bigg [ 0 , \\ldots , 0 , \\frac { s - p h } { 2 ( p h c - s ) } , 1 , \\frac { s - p h } { 2 ( p h c - s ) } \\bigg ] _ { 1 \\times ( N _ h + 3 ) } . \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} \\sum _ { j = - \\infty } ^ \\infty | b _ j | < \\infty . \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} \\bar t : = \\begin{cases} \\frac { ( p - 1 ) n } { n - s p } , & 1 < p < \\frac n s , \\\\ + \\infty , & p \\geq \\frac { n } s . \\end{cases} \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} d x _ { \\hat i } = \\sum _ { i = 1 } ^ n X _ { z _ i } d z _ i , X _ { z _ i } = \\frac { \\partial X } { \\partial z _ i } \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} f ( p , q ) \\cdot _ q g ( p , q ) : = ( f ( p , \\cdot ) \\cdot g ( p , \\cdot ) ) ( q ) . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} M _ g ( x ) \\sim x \\exp \\left ( \\sum _ { p \\leq x } \\frac { g ( p ) - 1 } { p } \\right ) = : A x \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} ( F ( x , u ) - F ( \\tilde x , u ) ) ^ T ( x - \\tilde x ) & = \\sum _ { l = 1 } ^ { \\mathcal { L } } \\sum _ { i = 1 } ^ N ( c _ { i l } ' ( g _ { i l } ) - c _ { i l } ' ( \\tilde g _ { i l } ) ) ( g _ { i l } - \\tilde g _ { i l } ) - \\sum _ { l = 1 } ^ { \\mathcal { L } } \\sum _ { i = 1 } ^ N p _ { l } ' ( u _ l ) ( s _ { i l } - \\tilde s _ { i l } ) ^ 2 . \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} S ( u , v ) = ( u - \\frac { 1 } { 3 } u ^ 3 + u v ^ 2 , - v + \\frac { 1 } { 3 } v ^ 3 - v u ^ 2 , u ^ 2 - v ^ 2 ) \\ ; , \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} \\Delta ( x \\rightharpoonup a ) & = x _ 1 \\rightharpoonup a _ 1 \\otimes x _ 2 \\rightharpoonup a _ 2 = x \\rightharpoonup a \\otimes 1 + 1 \\otimes x \\rightharpoonup a . \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\mathcal { B } _ n = \\{ z \\in I n t ( D _ n ) : \\varphi _ n \\ \\ z \\ \\ d d ^ c \\varphi _ n ( z ) > 0 \\} . \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} h | M \\setminus \\big ( 2 \\Omega \\cup B ( o , R _ 1 ) \\big ) = 1 \\end{align*}"}
-{"id": "10126.png", "formula": "\\begin{align*} \\omega = p v d u + q u d v C = \\{ v ^ q - u ^ { - p } = 0 \\} . \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\sigma _ { c , R } : = \\delta _ { c } \\ast \\frac { 1 } { \\ , R } \\ , \\sum _ { 0 \\le k _ 1 \\le a _ 1 R - 1 } \\ , \\dots \\sum _ { 0 \\le k _ s \\le a _ s R - 1 } \\ , \\delta _ { ( - k _ 1 / a _ 1 , \\dots , - k _ 1 / a _ s ) } \\ast \\nu _ \\mu \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} s ( p ( y ) ) = [ y , \\sigma ^ s ( y ) ] , \\forall y \\in P . \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} \\mu = \\mu _ 0 + w \\bigl ( \\Phi ( \\mu _ 0 ) , \\mu _ 0 \\bigr ) \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} z _ { l , q } = \\sum \\limits _ { t = 0 } ^ { l - 1 } { y _ t y _ { t + q } } \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} i = 1 \\begin{array} { | c | c | c | c | c | } \\hline & & & & \\\\ \\hline j = 0 & 3 & 2 & 1 & 0 \\\\ \\hline j = 1 & 0 & 1 & 2 & 3 \\\\ \\hline j = 2 & 2 & 3 & 0 & 1 \\\\ \\hline j = 3 & 1 & 0 & 3 & 2 \\\\ \\hline \\end{array} \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\partial _ t \\norm { D _ { \\eta } \\zeta } ^ 2 + \\norm { \\nabla D _ { \\eta } \\zeta } ^ 2 + \\alpha \\norm { D _ { \\eta } \\tau } ^ 2 _ { L ^ 2 ( \\Gamma _ u ) } = - \\int _ { \\Omega } D _ { \\eta } ( v \\cdot \\nabla _ H \\zeta + w \\partial _ z \\zeta ) D _ { \\eta } \\zeta + \\int _ { \\Omega } D _ { \\eta } g D _ { \\eta } \\zeta . \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} \\begin{aligned} h ( c ( y _ k ) + \\nabla c ( y _ k ) ( x _ k - y _ k ) ) + g ( x _ k ) & \\leq h ( c ( y _ k ) + \\nabla c ( y _ k ) ( w _ k - y _ k ) ) \\\\ & + \\frac { \\tilde \\mu } { 2 } ( \\| w _ k - y _ k \\| ^ 2 - \\| w _ k - x _ k \\| ^ 2 - \\| x _ k - y _ k \\| ^ 2 ) \\\\ & + a _ k g ( v _ k ) + ( 1 - a _ k ) g ( x _ { k - 1 } ) . \\end{aligned} \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} & W = \\Sigma _ X ^ { - 1 } \\Sigma _ { X Y } = S \\Sigma _ Y ^ { 1 \\over 2 } , W , \\\\ & W _ { i j } = \\left \\{ \\begin{array} { l l } S _ { i i } ( 1 - S _ { i i } ^ 2 ) ^ { - { 1 \\over 2 } } , & i = j \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} \\L ^ { ( d e ) } _ { ( X / Y , E / a ) } & : = { \\iota _ { Y ^ { d e } } } _ * \\L ^ { ( d e ) } _ { ( U / Y , E | _ U / a ) } \\textup { , a n d } \\\\ \\phi ^ { ( d e ) } _ { ( X / Y , E / a ) } & : = { \\iota _ { Y ^ { d e } } } _ * ( \\phi ^ { ( d e ) } _ { ( U / Y , E | _ U / a ) } ) : { F ^ { ( d e ) } _ { X / Y } } _ * \\L ^ { ( d e ) } _ { ( X / Y , E / a ) } \\to \\O _ { X _ { Y ^ { d e } } } , \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\widehat u _ \\nu ( k ) = \\frac { 2 ^ { 1 - \\nu } } { \\Gamma ( \\nu ) } | k | ^ { \\nu - \\frac { 1 } { 2 } } K _ { \\nu - \\frac { 1 } { 2 } } ( | k | ) . \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( N ( Q _ { j _ 1 , j _ 2 , \\dots , j _ { k - 1 } , j _ k } ) \\leq \\max ( \\frac { N ( Q ) } { ( 1 + c ) ^ { l } } , N _ 0 ) \\right ) \\geq \\sum \\limits _ { i = l } ^ { k } C ^ { i } _ k \\left ( \\frac { 1 } { 2 A } \\right ) ^ { k - i } \\left ( 1 - \\frac { 1 } { 2 A } \\right ) ^ i \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{gather*} D _ { n _ { k + 1 } } = ( - 1 ) ^ { { \\tfrac { ( n _ { k + 1 } - n _ { k } - 1 ) ( n _ { k + 1 } - n _ { k } ) } { 2 } } } \\left ( s _ { n _ { k + 1 } + n _ { k } + 1 } - s _ { n _ { k + 1 } + n _ { k } + 1 } ^ { ( n _ { k } + 1 ) } \\right ) ^ { n _ { k + 1 } - n _ { k } } t _ { n _ { k } } \\ . \\end{gather*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\omega = \\omega ^ \\mathrm { z m } + \\omega ^ \\mathrm { n z } = \\omega ^ \\mathrm { z m } + \\omega ^ - + \\omega ^ + . \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} \\bigcup _ { j = 1 } ^ { 2 ^ { n ( k - 1 ) } } K _ k ^ j = \\frac 1 2 K _ 0 . \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} B _ { g ( t ) } ( x , r ) \\subset \\Omega _ t ( x , a ) = \\{ y | u ( y , t ) \\geq a \\} \\subset B _ { g ( t ) } ( x , \\rho ) . \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\begin{aligned} & \\limsup _ { N \\rightarrow \\infty } \\left \\Vert \\left ( f _ N ^ { ( s ) } ( t ) - \\int _ { \\mathcal { P } ( \\mathbb { R } ^ { 2 d } ) } h _ N ^ { ( s ) } ( t ; h _ 0 ) d \\pi ( h _ 0 ) \\right ) \\mathbf { 1 } _ { Z _ s \\in \\mathcal { G } _ s \\cap \\hat { \\mathcal { U } } _ s ^ { \\eta ( \\varepsilon ) } } \\mathbf { 1 } _ { E _ s ( Z _ s ) \\leq R ^ 2 } \\right \\Vert _ { L ^ \\infty _ { Z _ s } } \\\\ & = 0 \\end{aligned} \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} S ^ { \\prime } \\left [ \\xi ^ { \\prime } \\right ] \\left ( z _ { 0 } \\right ) & = S \\xi \\left ( z _ { 0 } \\right ) + \\left ( \\xi _ { 0 } - S \\xi \\left ( z _ { 0 } \\right ) \\right ) \\left \\{ 1 - S \\left [ \\varphi _ { z _ { 0 } } \\mid _ { Z } \\right ] \\left ( z _ { 0 } \\right ) \\right \\} \\\\ & = \\xi _ { 0 } - S \\left [ \\varphi _ { z _ { 0 } } \\mid _ { Z } \\right ] \\left ( z _ { 0 } \\right ) \\left ( \\xi _ { 0 } - S \\xi \\left ( z _ { 0 } \\right ) \\right ) . \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} y _ { e , l } = \\mathbf { H } _ { e , l } ^ { H } \\mathbf { x } + \\mathbf { n } _ { e a , l } , ~ l = 1 , 2 , . . . , L \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} h ^ 1 ( L ( 1 ) ) = h ^ 1 ( L ( 3 ) ) = 0 , \\ \\ \\ h ^ 1 ( ( L ^ { - 1 } \\otimes \\omega _ C ) ( 1 ) ) = h ^ 1 ( ( L ^ { - 1 } \\otimes \\omega _ C ) ( 3 ) ) = 0 , \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} F _ \\gamma ( { \\cal P } _ \\gamma ( x ) ) - F _ \\gamma ( \\bar x ) & = F _ \\gamma ( { \\cal P } _ \\gamma ( x ) ) - F _ \\gamma ( y ) \\le \\langle \\nabla F _ \\gamma ( y ) , { \\cal P } _ \\gamma ( x ) - y \\rangle + \\frac { L _ f } 2 \\| { \\cal P } _ \\gamma ( x ) - y \\| ^ 2 \\\\ & = \\frac { L _ f } 2 \\| { \\cal P } _ \\gamma ( x ) - y \\| ^ 2 = \\frac { L _ f } 2 [ { \\rm d i s t } ( { \\cal P } _ \\gamma ( x ) , { \\cal X } ) ] ^ 2 , \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} & \\sum \\limits _ { i , j = 1 } ^ { n + 1 } [ d _ G ( y , y _ 0 ) ^ { - 1 - 2 \\alpha + \\epsilon } Y _ i Y _ j \\tilde { v } ] _ { C ^ { 0 , \\epsilon } _ \\ast ( \\mathcal { N } _ G ( y _ 0 ) ) } \\\\ & + \\sum \\limits _ { i , j = 1 } ^ { n + 1 } \\| d _ G ( y , y _ 0 ) ^ { - 1 - 2 \\alpha } Y _ i Y _ j \\tilde { v } \\| _ { L ^ { \\infty } ( \\mathcal { N } _ G ( y _ 0 ) ) } \\leq C \\| f \\| _ { Y _ { \\alpha , \\epsilon } } . \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} C = ( a , b , T _ { 1 } , C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , Q , n , L ) . \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} d ( Y , t \\bigm | _ Y ) = d ( Y ' , t \\bigm | _ { Y ' } ) \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} & \\beta _ { x x } + 1 = \\frac { q _ 0 } { M } , \\\\ & \\left ( ( \\beta _ t + \\beta \\beta _ x ) _ x + 3 \\beta \\right ) _ x = 0 \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\begin{aligned} & \\biggl ( \\frac { 1 } { m } U _ m \\biggr ) \\hat { \\Delta } = - \\frac { 1 } { \\sqrt { m } } y _ m + \\mathit { r e s t } _ 1 , \\\\ & \\| \\mathit { r e s t } _ 1 \\| \\leq \\| \\hat { \\Delta } \\| \\cdot \\| \\hat { X } _ { \\mathit { t l s } } - X _ 0 \\| \\cdot O _ p ( 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} L \\left ( t \\right ) = L _ 0 e ^ { \\int _ 0 ^ t r - R _ { \\max } \\ , d \\tau } , \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} L _ { 0 } = \\mathcal { L } _ { 0 } - \\frac { 4 \\pi } { c } j ^ { \\nu } A _ { \\nu } , L _ { 1 } = \\mathcal { L } _ { 1 } + \\frac { 4 \\pi } { c } j ^ { \\nu } A _ { \\nu } , \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} \\mu ( t ) = \\mu _ 0 - \\left ( 1 - \\left [ 1 + ( T - t ) \\right ] ^ { - ( d - 1 ) } \\right ) \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} d { x _ 1 } & = \\alpha ( x , t , T ) \\left ( d x - \\beta ( x , t , T ) d t - \\epsilon ( x , t , T ) d T \\right ) , \\\\ x _ 2 & = t , x _ 3 = T \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} \\phi ( x ) = \\left ( 1 - r ^ { - 2 } d ^ 2 ( x ) + C \\bigl ( u ( x ) - m _ u \\bigr ) \\right ) ^ { + } . \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} \\delta _ { n , e } : = \\left \\{ \\begin{array} { l l } 1 , & n \\\\ 0 , & \\end{array} \\right . \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} \\langle u , [ V , \\i A ] v \\rangle = - \\sum _ { i = 1 } ^ d \\langle u , \\big [ ( N _ i - 2 ^ { - 1 } ) ( V - \\tau _ i V ) S _ i + ( N _ i - 2 ^ { - 1 } ) ( V - \\tau _ i ^ * V ) S _ i ^ * \\big ] v \\rangle . \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} \\abs { \\widetilde { E } - E } = c ( M ) \\mathcal { O } ( \\mp ) E , \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} u _ k ( y ) : = \\frac { ( u - p _ k ) ( r _ 0 ^ { 2 k } y '' , r _ 0 ^ k y _ n , r _ 0 ^ k y _ { n + 1 } ) } { r _ 0 ^ { k ( 3 + { 2 \\alpha } ) } } . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} u _ { d _ { i } } ( \\{ j : p ( \\sigma _ { i } ( g ) ( j ) ) \\ne \\sigma _ { i } ( g ) ( p ( j ) ) \\} ) & \\leq 6 \\varepsilon + \\sum _ { r = 1 } ^ { L } | T _ { r } \\setminus g ^ { - 1 } T _ { r } | \\mu ( \\widetilde { V } _ { r } ) \\\\ & \\leq 6 \\varepsilon + \\varepsilon \\sum _ { r = 1 } ^ { L } | T _ { r } | \\mu ( \\widetilde { V } _ { r } ) \\\\ & \\leq 6 \\varepsilon + \\varepsilon \\mu \\left ( \\bigcup _ { r = 1 } ^ { L } T _ { r } \\widetilde { V } _ { r } \\right ) \\\\ & \\leq 7 \\varepsilon , \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\begin{aligned} & \\left | \\tilde { f } _ { N , n , R } ^ { ( s ) } ( t , Z _ s ) - f _ { N , n , R } ^ { ( s ) } ( t , Z _ s ) \\right | \\leq \\\\ & \\leq \\left [ 1 - \\left ( 1 - \\frac { n } { N } \\right ) ^ n \\right ] e ^ { - \\mu _ 0 s } \\exp \\left [ C _ d \\ell ^ { - 1 } n R ^ { d + 1 } e ^ { - \\mu _ 0 } t \\right ] \\end{aligned} \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} y _ n = \\begin{cases} n \\mu ( n ) / \\phi ( n ) & n \\leq N ^ { \\Omega } p \\not { | } n , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} \\sum _ { 1 < a < q / 2 \\atop ( a , q ) = 1 } \\overline { \\chi } ( a ) \\log \\xi _ a , \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} x _ l ^ * \\geq \\frac { \\sigma ( M / 4 ) } { \\alpha + \\sigma ( M / 4 ) } = \\frac { 4 - M } { ( \\alpha - 2 ) M + 4 } \\approx 0 . 1 6 9 1 . \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} h ( z ) = \\sqrt [ q - 1 ] { - T } \\ , M \\big ( E _ u ( z ) ^ { - 1 } , E _ v ( z ) ^ { - 1 } \\big ) ^ { - 1 } . \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} V _ N \\triangleq \\max \\{ V _ { \\varepsilon } ( \\psi _ j ) + 2 \\alpha \\log | 1 + e ^ { i \\psi _ j } | , \\ , j = - N / 2 \\cdots N / 2 \\} \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} A \\nabla _ { \\nu } B \\geqslant A \\sharp _ { \\nu } B + \\sum _ { k = 0 } ^ { \\infty } r _ { k } [ A \\sharp _ { \\frac { m _ k } { 2 ^ k } } B - 2 A \\sharp _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } B + A \\sharp _ { { \\frac { m _ k + 1 } { 2 ^ k } } } B ] . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\| e ^ { i \\sum _ { j = 1 } ^ d t _ j ( s \\alpha _ j + \\beta _ j ) } \\widehat { f } ( s ) \\| \\leq C ( 1 + | s | ^ 2 ) ^ { - 1 } \\| \\widehat { g } ( s ) \\| . \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} E = \\left \\{ [ 1 , z _ 2 , z _ 3 ] \\in \\mathbb { C } P ^ 2 : ~ g _ { z _ 3 } = 0 \\right \\} . \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 d } } | v | ^ 2 f _ N ^ { ( 1 ) } ( t , x , v ) d x d v = \\int _ { \\mathbb { R } ^ { 2 d } } | v | ^ 2 f _ N ^ { ( 1 ) } ( 0 , x , v ) d x d v \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} \\mathbf { P _ * } ( | | \\tilde { Z } _ n ^ * | | & > C _ 1 . n ^ { - 1 / 2 } ( l o g n ) ^ { - 1 } ) \\\\ & \\leq \\mathbf { P _ * } ( | | L _ n ^ * - \\mathbf { E _ * } L _ n ^ * | | > C _ 1 . n ^ { - 1 / 4 } ( l o g n ) ^ { - 1 / 2 } ) \\\\ & = o _ p ( n ^ { - 1 / 2 } ) \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} \\rho ^ Z _ t ( P _ A S _ \\gamma P _ A ) = \\rho ^ A _ { 2 t } ( P _ A S _ \\gamma P _ A ) . \\end{align*}"}
-{"id": "9924.png", "formula": "\\begin{align*} \\lambda ^ { \\min } ( u ^ { - } ( r ^ { - 1 } ) v ) = \\lambda ^ { \\max } ( u ^ { - } ( r ^ { - 1 } ) v ) = \\lambda ^ { \\max } ( v ) . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} S _ { g } ( X ) = \\tfrac { g ( g + 1 ) } { g - 1 } \\log 2 \\pi + \\tfrac { g } { 4 n ( g - 1 ) } \\log \\| \\Delta _ { g } \\| ( X ) + \\tfrac { g + 1 } { 8 ( g - 1 ) } \\delta ( X ) . \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} C ^ { \\pi } _ t ( y _ { t - J } ^ { t - 1 } ) = & \\sup _ { \\big \\{ r _ i ( x _ i | y _ { i - M } ^ { i - 1 } , y _ i ) : ~ i = t , t + 1 , \\ldots , n \\big \\} } { \\bf E } ^ { \\pi } \\bigg \\{ \\sum _ { i = t } ^ n \\log \\Big ( \\frac { r _ i ( X _ i | y _ { i - M } ^ { i - 1 } , Y _ i ) } { { \\pi } _ { i } ( X _ i | y _ { i - J } ^ { i - 1 } ) } \\Big ) \\Big { | } Y _ { t - J } ^ { t - 1 } = y _ { t - J } ^ { t - 1 } \\bigg \\} , ~ t \\in \\mathbb { N } _ 0 ^ n \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} d [ t r \\omega _ R ( z ) ] & = t r [ d \\omega _ R ( z ) ] \\\\ & = - t r \\left ( R ^ { - 1 } ( z ) a R ^ { - 1 } ( z ) t - R ^ { - 1 } ( z ) t R ^ { - 1 } ( z ) a \\right ) d z _ 1 \\wedge d z _ 2 = 0 , \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} \\partial ^ * _ t = \\big ( h ^ { W ^ \\bullet } _ t \\big ) ^ { - 1 } \\overline { \\partial } h ^ { W ^ \\bullet } _ t . \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} \\mathbf { p } _ Y ^ { - 1 } ( t ) = \\mathbb { P } ( \\mathrm { H o m } ( F , L _ t ( 2 ) ) ) , \\ \\ \\ \\mathrm { w h e r e } \\ L _ t : = \\mathbf { L } | _ { C _ t } , \\ \\ \\ t \\in Y . \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} x ( t ) = \\left \\lbrace \\begin{array} { r c l } \\left ( \\frac { 2 } { 3 } \\left ( - t + \\frac { 3 } { 2 } x _ 0 ^ { \\frac { 2 } { 3 } } \\right ) \\right ) ^ { \\frac { 3 } { 2 } } & \\mbox { f o r } & \\left [ 0 , \\frac { 3 } { 2 } x _ 0 ^ { \\frac { 2 } { 3 } } \\right ] \\\\ 0 & \\mbox { f o r } & \\left ( \\frac { 3 } { 2 } x _ 0 ^ { \\frac { 2 } { 3 } } , \\infty \\right ) \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{gather*} \\big ( \\tau _ k ^ { ( \\alpha ) } \\big ) ^ 2 = \\tau _ { k } ^ { ( \\alpha - 1 ) } \\tau _ { k } ^ { ( \\alpha + 1 ) } - \\tau _ { k + 1 } ^ { ( \\alpha - 1 ) } \\tau _ { k - 1 } ^ { ( \\alpha + 1 ) } , \\end{gather*}"}
-{"id": "2342.png", "formula": "\\begin{align*} \\Gamma ( \\delta _ \\lambda ( x ) ; \\delta _ \\lambda ( y ) ) = \\lambda ^ { 2 - q } \\ , \\Gamma ( x ; y ) , . \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} \\mathcal { B } _ { V } ^ + = \\left \\{ ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { A } ^ + \\textnormal { s u c h t h a t } \\left | v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime ( \\tau ; 0 ) \\right | \\leq \\eta \\right \\} \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} \\tilde h ( x ) = \\min \\Bigl ( 1 , \\max \\bigl ( 2 - 2 \\rho ( x ) , L \\sphericalangle ( v _ { 0 } , \\dot \\gamma ^ { o , x } _ { 0 } ) \\bigr ) \\Bigr ) . \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} \\left \\Vert \\Phi _ N \\right \\Vert _ { \\mathcal { L } ^ 1 _ { \\beta , \\mu } } = \\sum _ { s = 1 } ^ N \\frac { 1 } { s ! } \\int _ { \\mathcal { D } _ s } \\left | \\phi _ N ^ { ( s ) } ( Z _ s ) \\right | e ^ { - \\beta E _ s ( Z _ s ) } e ^ { - \\mu s } d Z _ s \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} H = H _ a + H _ c + H _ { a c } = \\frac { \\hbar \\ , \\omega _ { e g } } { 2 } \\sigma _ z + \\hbar \\omega _ c N + ( - i \\hbar \\frac { \\Omega _ 0 } { 2 } ) \\big [ a \\sigma _ + - a ^ \\dagger \\sigma _ - \\big ] \\ , . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} x _ { 1 } & = 0 \\\\ 9 \\ , x _ { 1 } + x _ { 2 } & = 0 \\\\ 3 4 4 \\ , x _ { 1 } + x _ { 7 } & = 0 \\\\ 3 0 9 6 \\ , x _ { 1 } + 3 4 4 \\ , x _ { 2 } + 9 \\ , x _ { 7 } + x _ { 1 4 } & = 0 \\end{align*}"}
-{"id": "10000.png", "formula": "\\begin{align*} S = \\{ x \\in { \\mathcal M } _ \\rho : A ( x - x ^ \\dag ) = 0 \\} . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{gather*} g _ { - } = \\sum _ { a , b = 0 } ^ { n - 1 } g _ { a b } ( z ) E _ { a b } , \\end{gather*}"}
-{"id": "2555.png", "formula": "\\begin{align*} N ^ \\alpha _ t = \\int _ 0 ^ t ( n - N ^ \\alpha _ s ) \\lambda ^ \\top X _ { \\alpha s } \\ , \\dd s + m _ t . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\Psi Q = ( V ^ { \\top } _ * V _ 0 ) Q = \\begin{bmatrix} \\widehat { \\Psi } & 0 _ { r - q } \\end{bmatrix} , \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { M ^ { t } \\vec { v } } { h _ { \\vec v } ( t ) } = \\vec { v } _ { \\infty } \\ , , \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} S _ { \\tt A _ n } ( \\lambda ) = S _ { \\tt A _ n } ( \\lambda _ 1 ) + S _ { \\tt A _ n } ( \\lambda _ 2 ) S _ { \\tt C _ n } ( \\lambda ) = S _ { \\tt C _ n } ( \\lambda _ 1 ) + S _ { \\tt C _ n } ( \\lambda _ 2 ) \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} G _ 1 = \\left [ \\begin{array} { c c } I _ { m - 1 } & 0 \\end{array} \\right ] , F _ 1 = \\left [ \\begin{array} { c c } 0 & I _ { m - 1 } \\end{array} \\right ] , G _ 2 = \\left [ \\begin{array} { c c } 0 & I _ { n - 1 } \\end{array} \\right ] , F _ 2 = \\left [ \\begin{array} { c c } I _ { n - 1 } & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\abs { \\nabla u } a ^ { i i } u _ { 1 ; i i } = \\abs { \\nabla u } a ^ { i i } u _ { i ; i 1 } + \\abs { \\nabla u } ^ 2 a ^ { i i } R ^ { 1 } _ { 1 i i } . \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} L ( s , \\chi ) = \\sum _ { \\mathfrak { n } } \\chi ( \\mathfrak { n } ) \\N \\mathfrak { n } ^ { - s } = \\prod _ { \\mathfrak { p } } \\Big ( 1 - \\frac { \\chi ( \\mathfrak { p } ) } { \\N \\mathfrak { p } ^ { s } } \\Big ) ^ { - 1 } \\end{align*}"}
-{"id": "10010.png", "formula": "\\begin{align*} N ( \\underline { x } , \\underline { y } ) = \\begin{cases} s u p \\{ n : x _ i = y _ i \\ \\forall \\ | i | < n \\} & \\underline { x } , \\underline { y } \\in \\Sigma _ A \\\\ s u p \\{ n : x _ i = y _ i \\ \\forall \\ 0 \\leq i < n \\} & \\underline { x } , \\underline { y } \\in \\Sigma _ A ^ + . \\end{cases} \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} \\rho = { \\bf 1 } \\oplus \\bigoplus _ { n = 0 } ^ { \\infty } \\rho _ n ^ { \\perp } . \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} p ^ { \\omega } _ { K , \\mathbf { a } , f } \\left ( \\phi ( t ) - \\phi _ n ( t ) \\right ) \\le \\sum _ { k = n + 1 } ^ { \\infty } ( M ( t ) ) ^ k M _ f . \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} p _ { 0 , 0 } = \\left [ \\dfrac { \\xi } { \\gamma A } + e ^ { \\frac { \\lambda } { \\xi } } \\left \\lbrace - \\frac { \\gamma } { \\xi } B ( 1 ) + \\frac { ( \\mu - \\xi ) \\gamma } { \\xi \\lambda } C ( 1 ) + \\frac { D ( 1 ) } { A } \\right \\rbrace \\right ] ^ { - 1 } . \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} & x B _ { n } ^ { s } = B _ { n + 1 } ^ { s } + \\sum _ { \\substack { k = 1 \\\\ 1 \\leq i _ { 1 } \\leq \\dots \\leq i _ { k } \\leq d - k + 2 } } ^ { d + 1 } \\rho _ { ( n - 1 ) ( d + 1 ) + i _ { 1 } + 1 + s } \\dots \\rho _ { ( n - k ) ( d + 1 ) + i _ { k } + k + s } B _ { n + 1 - k } ^ { s } , \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} ( \\Delta f ) \\tau ( \\phi ) + 2 \\nabla ^ { \\phi } _ { { \\rm g r a d } \\ , f } \\tau ( \\phi ) = 0 \\ ; \\ ; \\forall \\ ; f : ( M ^ m , g ) \\longrightarrow ( 0 , \\infty ) . \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} f _ { n } = z ^ { n } \\sum _ { k = 0 } ^ { \\infty } \\left ( q ^ { - k } z \\alpha ; q \\right ) _ { - n } \\left ( q ^ { k + 1 } z ^ { - 1 } \\alpha ^ { - 1 } ; q \\right ) _ { \\ ! \\infty } \\frac { q ^ { \\frac { 1 } { 2 } k ( k + 1 ) } } { ( q ; q ) _ { k } } ( - 1 ) ^ { k } \\alpha ^ { - k } z ^ { k } . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} P _ { 1 } ( z ) = p _ { 0 , 0 } e ^ { \\frac { \\lambda } { \\xi } z } z ^ { - \\left ( \\frac { \\mu } { \\xi } - 1 \\right ) } \\left [ - \\frac { \\gamma } { \\xi } B ( z ) + \\frac { ( \\mu - \\xi ) \\gamma } { \\xi \\lambda } C ( z ) + \\frac { D ( z ) } { A } \\right ] , \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} H = H _ 0 - \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ i | a _ i \\rangle \\langle a _ i | \\ ; , \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\sup _ { y \\in [ 0 , \\ , u - \\varepsilon ] } \\left | \\frac { h ( L ^ { \\leftarrow } ( t u ) - L ^ { \\leftarrow } ( t y ) ) } { h ( L ^ { \\leftarrow } ( t ) ) } - u ^ { \\alpha } \\right | = 0 . \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\psi _ { m , n } ^ { \\nu } ( z , \\bar z ) : = \\nabla ^ { \\nu } _ { m } \\left ( z ^ n ( 1 - | z | ^ 2 ) ^ { \\nu - m } \\right ) \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} \\widehat { \\omega } ( t , k , \\eta ) & = \\widehat { \\omega } _ { \\rm i n } ( k , \\eta + k t ) \\exp \\left [ - \\nu \\int _ 0 ^ t k ^ 2 + \\abs { \\eta - k ( \\tau - t ) } ^ 2 d \\tau \\right ] . \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\varphi _ 0 ( f _ 1 + f _ 2 ) = \\log m ( e ^ { T _ 1 + T _ 2 } ) = \\log m ( e ^ { T _ 1 } e ^ { T _ 2 } ) = \\log \\big ( m ( e ^ { T _ 1 } ) m ( e ^ { T _ 2 } ) \\big ) = \\varphi _ 0 ( f _ 1 ) + \\varphi _ 0 ( f _ 2 ) , \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} \\mathcal { M } ' _ b : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J , \\end{align*}"}
-{"id": "9907.png", "formula": "\\begin{align*} \\psi ( s ) = ( \\varphi _ 1 ( s ) - \\varphi _ 1 ( s _ 0 ) ) ^ { - 1 } ( \\varphi _ 2 ( s ) - \\varphi _ 2 ( s _ 0 ) ) \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} P _ { 0 } ( 1 ) = p _ { 0 , 0 } \\frac { \\xi } { \\gamma A } . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} \\left [ T _ { ( A , i ) } , T _ { ( B , j ) } \\right ] & = \\left ( K _ { i j } { } ^ { k } - K _ { i j } { } ^ { k + n } \\right ) C _ { A B } { } ^ { C } T _ { ( C , k ) } , \\quad \\\\ A , B , C & = 1 , \\ldots , \\dim \\mathfrak { g } , i , j , k = 0 , \\cdot \\cdot \\cdot , n - 1 . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} \\mathcal { F } = \\{ F \\in \\mathcal { B } : \\mbox { t h e r e e x i s t s $ i $ s u c h t h a t e i t h e r } F \\sim F _ i \\mbox { o r } F \\sim F _ i ^ c \\} . \\end{align*}"}
-{"id": "10136.png", "formula": "\\begin{gather*} ( p , q ) = ( 2 - 6 u , 1 + 6 u ) , ( p , q ) = ( 4 - 6 u , - 1 + 6 u ) , \\\\ ( p , q ) = ( 1 - 6 u , 2 + 6 u ) , ( p , q ) = ( 5 - 6 u , - 2 + 6 u ) , \\\\ ( p , q ) = ( 1 - 2 u , 1 + 2 u ) , \\end{gather*}"}
-{"id": "2047.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c } ( \\lambda I - A ) ^ { - 1 } B u \\\\ u \\end{array} \\right ] \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} \\begin{array} { l } \\det ( I + B ) = 0 , \\\\ \\det ( I \\wedge I + B \\wedge I + I \\wedge B ) = 0 , \\\\ \\det ( I \\wedge I \\wedge I + B \\wedge I \\wedge I + I \\wedge B \\wedge I + I \\wedge I \\wedge B ) = 0 , \\end{array} \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} J _ n : = [ 2 ^ n - 1 , 2 ^ n ] \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } ^ 0 \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & = \\left ( X _ { s + k } ^ \\prime , V _ { s + k } ^ \\prime \\right ) \\in \\mathbb { R } ^ { 2 d ( s + k ) } \\end{aligned} \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} \\beta _ { j , p _ 1 , p _ 2 } [ t ] = \\left \\{ \\begin{array} { l l l } & g _ { i k , l q } [ t ] \\quad ~ ~ \\\\ & g _ { i k , l q } [ t ] \\quad ~ ~ \\\\ & b _ { r s , l q } [ t ] \\quad ~ ~ \\\\ & 1 \\quad \\quad \\quad ~ ~ ~ \\end{array} \\right . \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P } } ( \\mu , r ) = \\lim _ { P \\rightarrow \\infty } \\limsup _ { L \\rightarrow \\infty } \\frac { T } { L / \\log P } . \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} & \\mathcal { I } _ { 2 , 1 } = { \\mathrm { E } } \\left [ A _ { 1 } A _ { 2 } 1 \\left \\{ A _ { 1 } A _ { 2 } > \\mathcal { C } \\left ( n R _ { n } + \\sqrt { n \\log ^ { r + 1 } n } \\right ) \\right \\} \\right ] , \\\\ & \\mathcal { I } _ { 2 , 2 } = \\mathcal { C } \\left ( n R _ { n } + \\sqrt { n \\log ^ { r + 1 } n } \\right ) \\Pr \\left ( \\bar c \\Vert \\widehat { \\beta } - \\beta \\Vert _ { 2 } > \\Vert \\widehat { \\beta } - \\beta \\Vert _ { 2 , n } \\right ) , \\end{align*}"}
-{"id": "9953.png", "formula": "\\begin{align*} 0 \\geq \\sigma ( c _ { 1 } \\vect { e } _ { 1 } + \\cdots + c _ { i - 1 } \\vect { e } _ { i - 1 } + ( c _ { i } + j ) \\vect { e } _ { i } ) = ( \\lambda - c _ { i } + j ) ( - ( k + 1 ) ) . \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} \\nabla F _ \\gamma ( x ) = \\gamma ^ { - 1 } ( I - \\gamma \\nabla ^ 2 f ( x ) ) ( x - { \\rm p r o x } _ { \\gamma P } ( x - \\gamma \\nabla f ( x ) ) ) . \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } '' _ b : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J , \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\lambda _ k ( q _ 1 ) = \\sum _ { k = 1 } ^ n \\sum _ { m = 1 } ^ \\infty ( a _ m ^ k ) ^ 2 \\lambda _ m ( q _ 1 ) = \\sum _ { m = 1 } ^ \\infty \\left ( \\sum _ { k = 1 } ^ n ( a _ m ^ k ) ^ 2 \\right ) \\lambda _ m ( q _ 1 ) . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} x ( t ) = \\begin{cases} e ^ { i t A _ 1 } \\left ( h - i \\int _ 0 ^ { t } e ^ { - i s A _ 1 } \\Phi ^ * u ( s ) d s \\right ) , & t > 0 ; \\\\ e ^ { i t A _ 1 ^ * } \\left ( h + i \\int _ t ^ { 0 } e ^ { - i s A _ 1 ^ * } \\Phi ^ * y ( s ) d s \\right ) , & t < 0 ; \\\\ \\end{cases} \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\rho = - \\operatorname { d i v } \\mathbf { p } , \\qquad \\mathbf { j } = \\frac { \\partial \\mathbf { p } } { \\partial t } + c \\operatorname { c u r l } \\mathbf { m } \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} r _ { ( P / A ) ^ * } ( X ) & = ( r _ { P / A } ) ( ( E - A ) - X ) + | | X | | _ { r _ { P / A } } - ( r _ { P / A } ) ( E - A ) \\\\ & = r _ P ( E - X ) - r _ P ( A ) + | | X | | _ { r _ { P / A } } - r _ P ( E ) + r _ P ( A ) \\\\ & = r _ P ( E - X ) - r _ P ( E ) + \\sum \\limits _ { a \\in X } ( r _ { P / A } ) ( \\{ a \\} ) \\\\ & = r _ P ( E - X ) - r _ P ( E ) + \\sum \\limits _ { a \\in X } [ r _ P ( A \\cup \\{ a \\} ) - r _ P ( A ) ] . \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} u = \\exp \\left ( - c \\left ( t \\right ) \\right ) h . \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} \\exp ( x ) \\ , \\ , { } _ 1 F _ 1 \\left [ \\begin{array} { c } c - b \\\\ c \\end{array} ; \\ , - x \\right ] \\ , = { } _ 1 F _ 1 \\left [ \\begin{array} { c } b \\\\ c \\end{array} ; \\ , x \\right ] \\ , . \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} \\beta ( x ) \\beta ^ { t r } ( x ) + 2 d ( x ) d ( x ) ^ { t r } = I . \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} ( - i \\nabla + \\mathbf { A } ) ^ 2 u _ n = \\varphi _ n ( - i \\nabla + \\mathbf { A } ) ^ 2 u - 2 i \\nabla \\varphi _ n \\cdot ( - i \\nabla + \\mathbf { A } ) u - ( \\Delta \\varphi _ n ) u \\ , , \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} F P S ( l / k ) = F P S ( l / ( l - k ) , ) 1 \\leqslant k \\leqslant l - 1 \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\mathcal { T } ( P ) = \\beta ( P ) \\log \\sqrt { \\frac { 2 P } { \\pi e } } + H ( \\beta ( P ) ) \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { M , N } ( a , b ) ^ q \\bigr ] = \\eta _ { M , N } ( q \\ , | a , \\ , b ) , \\ ; \\Re ( q ) > - b _ 0 . \\end{align*}"}
-{"id": "10101.png", "formula": "\\begin{align*} \\omega & = - z ( 3 a x ^ 2 + 3 y ^ 2 + c z ^ 2 ) d x + 4 x y z d y - x ( 3 a x ^ 2 + 3 y ^ 2 - c z ^ 2 ) d z , \\\\ \\eta & = z ^ 2 x d x - x ^ 2 z d z , \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} & S _ i b = \\alpha _ { i } ( b ) S _ i , & & S _ i S _ j = \\theta _ { i j } S _ j S _ i , & & S _ i ^ * = S _ i ^ { - 1 } . \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} j ^ 1 _ { ( 0 , 1 ) } \\rho _ 1 = 0 , \\ j ^ 1 _ { ( 0 , 1 ) } \\rho _ 2 = 0 , \\ ; \\ ; \\partial _ { y y } \\rho _ 2 ( 0 , 1 ) = 0 . \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { l - 1 } C ^ { i } _ k p ^ { k - i } \\left ( 1 - p \\right ) ^ i \\leq p ^ { k ( 1 - \\varepsilon ) } \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} f ^ { ( j ) } _ { n , k } ( s ) = ( - 1 ) ^ { j } \\mathbb { E } \\left ( e ^ { - s T _ { ( k ) } } T _ { ( k ) } ^ { j } \\right ) , \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} u ^ { \\prime } + \\mathrm { D } \\phi ( u ) & \\ni 0 \\qquad ( 0 , T ) , \\\\ u ( 0 ) & = u _ { 0 } \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} d ( Y ( G ) , s ) = \\max \\left \\{ \\dfrac { c _ { 1 } ( t ) ^ { 2 } + \\vert V ( G ) \\vert } { 4 } \\left | t \\in { \\rm S p i n } ^ { c } ( X ) , t \\bigm | _ { Y ( G ) } = s \\right . \\right \\} , \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} \\left ( \\left ( \\mathbf { I } + \\bar { K } ( \\nu , 0 ) \\right ) r _ { 0 } , r _ { 0 } \\right ) & = \\left ( \\left ( \\mathbf { I } + \\bar { K } ( \\nu , 0 ) \\right ) r _ { 0 } , r _ { 0 } \\right ) \\\\ & = 1 - F \\left ( i \\lambda \\right ) \\backsim a _ { 0 } \\lambda ^ { 2 } = a _ { 0 } \\nu \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ { N } a _ { i j } ( x ) \\xi _ i \\xi _ j \\geq \\alpha \\sum _ { i = 1 } ^ N \\xi _ i ^ 2 , \\ , \\ , \\forall \\xi = ( \\xi _ 1 , \\xi _ 2 , \\cdots , \\xi _ N ) \\in \\mathbb { R } ^ N \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} & a ^ * _ { 1 , 1 } = a ^ * _ { 0 , 1 } = a ^ * _ { 0 , 2 } = 0 , \\\\ & a ^ * _ { 3 , 0 } + 3 a ^ * _ { 3 , 1 } + 3 a ^ * _ { 3 , 2 } + a ^ * _ { 3 , 3 } + 6 a ^ * _ { 2 , 1 } + 6 a ^ * _ { 2 , 2 } + 2 a ^ * _ { 2 , 3 } + 3 a ^ * _ { 1 , 2 } + a ^ * _ { 1 , 3 } = \\mu _ R . \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} u _ t = A u _ { b + 4 } & + B u _ { b + 3 } u _ { b + 1 } + C u _ { b + 2 } ^ 2 + E u _ { b + 2 } u _ { b + 1 } ^ 2 + G u _ { b + 1 } ^ 4 \\\\ & + H u _ { b + 3 } + I u _ { b + 2 } u _ { b + 1 } + J u _ { b + 1 } ^ 3 + K u _ { b + 2 } + L u _ { b + 1 } ^ 2 \\\\ & + M u _ { b + 1 } + N . \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{gather*} \\frac { \\prod \\limits _ { 1 \\le i < j \\le m } ( w _ i - w _ j ) \\prod \\limits _ { 1 \\le i < j \\le n } ( y _ i - y _ j ) } { \\prod \\limits _ { i = 1 } ^ { m } \\prod \\limits _ { j = 1 } ^ { n } ( w _ i - y _ j ) } . \\end{gather*}"}
-{"id": "8358.png", "formula": "\\begin{align*} ( T _ 2 ) _ { i j , i } ( r ^ { 6 - n } ) _ { , j } = & ( n - 1 0 ) \\sigma _ 1 ( A ) _ { , j } ( 6 - n ) r ^ { 4 - n } x ^ j \\\\ = & ( n - 1 0 ) ( 6 - n ) \\sigma _ 1 ( A ) _ { , j } x ^ j r ^ { 4 - n } \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} A ^ { k } ( i _ { 1 } , \\dots , i _ { l } ) : = \\sum _ { \\substack { | \\delta | = k , \\\\ \\delta _ { i _ { 1 } } = \\cdots = \\delta _ { i _ { l } } = 1 } } u _ { \\delta } , \\ ; \\ ; \\ ; A ^ { k } : = \\sum _ { | \\delta | = k } u _ { \\delta } . \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} S _ c : = \\begin{pmatrix} c _ 0 I & A _ c & B _ c \\\\ A _ c ^ { t r } & - c _ 0 I & C _ c \\\\ B _ c ^ { t r } & C _ c ^ { t r } & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} A : = A ( 1 ) = \\int _ { 0 } ^ { 1 } e ^ { - \\frac { \\lambda } { \\xi } s } ( 1 - s ) ^ { \\frac { \\gamma } { \\xi } - 1 } d s . \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} & A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\psi _ { n - 4 } ^ { ( 1 ) } \\\\ = & \\Big [ A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\alpha _ { n - 4 } ^ { ( 1 ) } + 2 ( B _ { 6 - n } A _ { 4 - n } A _ { 2 - n } + A _ { 6 - n } B _ { 4 - n } A _ { 2 - n } + A _ { 6 - n } A _ { 4 - n } B _ { 2 - n } ) \\beta _ { n - 4 } ^ { ( 1 ) } \\Big ] \\log r \\\\ & + A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\beta _ { n - 4 } ^ { ( 1 ) } \\log ^ 2 r + O ( r ^ { n - 4 } ) . \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} g ( s , v ; 1 ) = \\zeta ( s + v ) . \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} & \\textbf { H } ( x _ 0 ) \\otimes \\mathbf { \\Psi } ( t ) \\textbf { J } ^ { 1 , T } _ { N _ t \\times N _ t } \\cdot \\textbf { C } + \\varphi ( x _ 0 ) = g _ 1 ( t ) , \\\\ & \\textbf { H } ( x _ { N _ h } ) \\otimes \\mathbf { \\Psi } ( t ) \\textbf { J } ^ { 1 , T } _ { N _ t \\times N _ t } \\cdot \\textbf { C } + \\varphi ( x _ { N _ h } ) = g _ 2 ( t ) . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} f ( z ) = h ( z ) + \\overline { g ( z ) } = z + \\sum _ { n = 2 } ^ { \\infty } a _ n z ^ n + \\overline { \\sum _ { n = 1 } ^ { \\infty } b _ n z ^ n } . \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\| \\varphi \\left ( \\omega q ^ { 2 n } \\right ) \\| ^ { 2 } = \\omega ^ { - 4 n } q ^ { - 2 n ( 2 n - 1 ) } \\| \\varphi \\left ( \\omega \\right ) \\| ^ { 2 } \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{gather*} T _ { 2 } ^ { k } T _ { 1 } ^ { \\ell } = ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } + \\frac { \\ell ( \\ell - 1 ) } { 2 } } Q _ { 2 } ^ { k } Q _ { 1 } ^ { \\ell - k } Q _ { 0 } ^ { - \\ell } . \\end{gather*}"}
-{"id": "2796.png", "formula": "\\begin{align*} f ^ { \\lambda } = \\lambda \\left . \\frac { \\partial F } { \\partial \\lambda } F ^ { - 1 } \\right | _ { \\l \\in \\R _ { + } } . \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{gather*} { \\rm { r a n k } } \\ , \\big ( \\ , s _ { i + j } ^ { ( r ) } \\ , \\big ) _ { i , j = 0 } ^ { \\infty } = r \\ . \\end{gather*}"}
-{"id": "9706.png", "formula": "\\begin{align*} H ^ { 1 , 2 } ( D _ T ) : = H ^ 1 ( 0 , T ; L ^ 2 ( D ) ) \\cap L ^ 2 ( 0 , T ; H ^ 2 ( D ) ) . \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} \\mathcal { E } ( f , g ) = \\sum _ { i = 1 } ^ { + \\infty } a _ i \\int _ 0 ^ 1 f ' _ i ( x ) g _ i ' ( x ) d x \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } g _ { i j } ( x , t ) = - 2 h _ { i j } ( x , t ) , \\ \\ \\ \\ ( x , t ) \\in M \\times [ 0 , T ] \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} \\mathfrak { S } = { \\rm V o l } _ N \\sum _ { \\| \\Re \\xi _ i - \\xi \\| > Q } | \\widehat { \\omega _ S } ( \\psi _ i ) | ^ 2 | \\mathcal { P } _ H ( \\psi _ i ) | ^ 2 h _ \\xi ( \\xi _ i ) \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} e ^ { \\mu \\nu \\sigma \\tau } A _ { \\nu \\sigma \\tau } = 6 B ^ { \\mu } , A _ { \\mu \\nu \\lambda } = e _ { \\mu \\nu \\lambda \\sigma } B ^ { \\sigma } . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} F _ { \\beta _ j } F _ { \\beta _ i } - q ^ { - ( \\beta _ i , \\beta _ j ) } F _ { \\beta _ i } F _ { \\beta _ j } = \\sum _ { n _ { i + 1 } , \\ldots , n _ { j - 1 } \\geq 0 } c ( n _ { i + 1 } , \\ldots , n _ { j - 1 } ) F _ { \\beta _ { i + 1 } } ^ { n _ { i + 1 } } \\ldots F _ { \\beta _ { j - 1 } } ^ { n _ { j - 1 } } , \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} | J ^ G _ M ( z , h _ P ) | = \\mathrm { ( c o n s t a n t ) } \\times \\int _ { \\prod _ { v | N } N _ P ( \\Q _ v ) } f _ { z ' _ \\alpha K ( N ) } ( n ) d n \\leq \\mathrm { ( c o n s t a n t ) } \\times N ^ { - 3 } . \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} \\int _ { C ^ \\infty ( \\Sigma , M ) \\times \\mathcal { M } } d \\rho ( \\phi ; g ) = \\int _ { \\mathcal { R G } _ t ( C ^ \\infty ( \\Sigma , M ) \\times \\mathcal { M } ) } \\mathcal { R G } ^ * _ t ( d \\rho ( \\phi ; g ) ) \\end{align*}"}
-{"id": "9891.png", "formula": "\\begin{align*} \\mathcal { N } = \\left [ \\sum _ { k = 0 } ^ m ( - 1 ) ^ k \\ , 2 ^ { m + 1 - k } \\ , { m \\choose k } \\ , \\sum _ { l = 0 } ^ \\infty \\frac { \\sum _ { j = 0 } ^ l b _ j \\ , b _ { l - j } ^ * } { l + m + k + 1 } \\right ] ^ { - 1 / 2 } \\ , , \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{gather*} g ^ { [ k ] ( \\alpha ) } = T ^ { - k } g ^ { ( \\alpha ) } _ { C } , \\end{gather*}"}
-{"id": "5063.png", "formula": "\\begin{align*} \\rho ' ( x ) = ( \\rho \\circ f ) ( x ) g _ f ( x ) . \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} B ^ s _ { p , q } ( \\mathbb { R } ^ d ) = & \\left \\{ f \\in \\mathcal { S } ' ( \\mathbb { R } ^ d ) \\ , \\ , | \\ , \\ , f \\stackrel { \\mathcal { S } ' } { = } \\sum _ { j = 0 } ^ { \\infty } a _ j ( x ) ; \\ , \\ , \\mathrm { s u p p } F a _ j \\subset M _ j ; \\right . \\\\ & \\left . \\| \\{ a _ j \\} \\| _ { l ^ s _ q ( L _ p ) } = \\left [ \\sum _ { j = 0 } ^ { \\infty } ( 2 ^ { s j } \\| a _ j \\| _ { L _ p ( \\mathbb { R } ^ d ) } ) ^ q \\right ] ^ { 1 / q } < \\infty \\right \\} , \\ , \\ , \\ , \\ , \\ , \\ , q \\in [ 1 , \\infty ) \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} F _ \\gamma ( x ) & = \\inf _ y \\left \\{ f ( y ) + P ( y ) + D _ \\phi ( y , x ) \\right \\} \\ge \\inf _ y \\left \\{ f ( y ) + P ( y ) + \\frac 1 2 \\left ( \\frac { 1 } { \\gamma } - L \\right ) \\| y - x \\| ^ 2 \\right \\} . \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} b = c = q = r = 0 , \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} ( S + T ) ( S + T ) ^ H = ( S - T ) ( S - T ) ^ H . \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} a ( u , v ) : = \\frac { c _ { N , \\beta } } { 2 } \\int _ { \\mathbb { R } ^ { N } } \\int _ { \\mathbb { R } ^ { N } } \\frac { ( u ( x , t ) - u ( y , t ) ) ( \\eta ( x , t ) - \\eta ( y , t ) ) } { | x - y | ^ { N + 2 \\beta } } d x d y . \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} P U _ { \\delta , \\xi } ( x ) = U _ { \\delta , \\xi } ( x ) - \\delta ^ { \\frac { 4 - n } { 2 } } \\varphi _ 0 \\ ( \\frac { x - \\xi } { \\delta } \\ ) + O \\ ( \\delta ^ { \\frac { 6 - n } { 2 } } \\ ) \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} | X | + | Y _ 1 | + | Y _ 2 | - 1 = | V ( B ' ) | - 1 = | E ( B ' ) | = | Y _ 1 | + \\sum _ { v \\in Y _ 2 } d _ { B ' } ( v ) \\ge | Y _ 1 | + 2 | Y _ 2 | , \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} \\{ a _ \\lambda b \\} = \\{ a , b \\} \\quad a , b \\in R \\subset J R , \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\Psi _ { v _ 1 , \\ldots , v _ k } : = \\sum _ { a _ 1 , \\ldots , a _ k } w \\left ( \\frac { a _ 1 \\cdots a _ k } x \\right ) v _ 1 ( a _ 1 ) \\cdots v _ k ( a _ k ) d ( a _ 1 \\cdots a _ k + h ) , \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} F _ { d , \\ell } ( z ; i t ) & = \\sum _ { a = 0 } ^ N \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\zeta ^ d } { 1 - \\zeta ^ \\ell } \\right ) \\frac { ( - 2 \\pi t ) ^ a } { a ! } + O \\left ( t ^ { N + 1 } \\right ) , \\\\ G _ { d , \\ell } ( z ; i t ) & = 2 i \\sum _ { a = 0 } ^ N \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\sin ( 2 \\pi d z ) } { 1 - \\zeta ^ \\ell } \\right ) \\frac { ( - 2 \\pi t ) ^ a } { a ! } + O \\left ( t ^ { N + 1 } \\right ) . \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} \\tau _ 0 = - \\log \\Theta ( 0 , T ^ * ) , | D \\tilde { A } | ^ 2 = \\Theta ^ 2 g ^ { i j } h ^ k _ { l ; i } \\Theta h ^ l _ { k ; j } \\Theta . \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} B ( G _ { n , k , b } ) \\leq k \\binom { b } { k } . \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} c _ x f '' ( x ) + b _ x \\mathbf n _ x f = 0 . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} \\mu _ d ( z ) : = \\left \\{ \\begin{array} { l l } $ $ 1 $ $ , & \\hbox { $ d = 1 $ ; } \\\\ $ $ 1 + | \\log z | $ $ , & \\hbox { $ d = 2 $ ; } \\\\ $ $ z ^ { 2 - d } $ $ , & \\hbox { $ d \\geq 3 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} y _ { 1 } = h _ d x _ 1 + h _ c x _ { 2 } + z _ 1 ; y _ 2 = h _ d x _ 2 + z _ 2 , \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{gather*} C ^ { ( \\alpha ) } ( z ) = ( - 1 ) ^ \\alpha z ^ { \\alpha } C ( z ) = \\sum _ { i \\in \\mathbb { Z } } ( - 1 ) ^ \\alpha c _ { i + \\alpha } z ^ { - i - 1 } . \\end{gather*}"}
-{"id": "7226.png", "formula": "\\begin{align*} \\rho = { \\bf 1 } \\oplus \\mu \\oplus \\bigoplus _ { \\theta = \\frac { 2 \\pi j } { 2 ^ n } } \\rho _ { \\theta } , \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} F ( x , y ; z ) = x - \\frac { z ( t ^ 2 - 2 \\Delta t + 1 ) } { ( z - 1 ) ( t ^ 2 - 2 \\Delta t + z ) } y - r ( z ) . \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} X _ s X _ { s ' } + X _ { s ' } X _ s & = 2 Q _ { s , s ' } I s , s ' \\in S , \\\\ Y _ t Y _ { t ' } + Y _ { t ' } Y _ t & = 2 R _ { t , t ' } I t , t ' \\in T . \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} \\xi \\eta \\xi ^ { - 1 } J _ { \\xi \\eta ^ { - 1 } \\xi ^ { - 1 } } = \\Gamma - \\left ( \\{ \\xi \\} \\cup J _ { \\{ \\xi ^ 2 , \\xi \\eta ^ { - 1 } \\xi ^ { - 1 } , \\xi \\eta ^ { - 1 } \\xi , \\xi \\eta ^ { - 2 } , \\xi \\eta ^ 2 , \\xi \\eta \\xi ^ { - 1 } , \\xi \\eta \\xi \\} } \\right ) \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} L ^ { 2 } ( q ^ { \\alpha } ) = 1 + \\sum _ { n = 1 } ^ { \\infty } \\bigl ( 2 4 0 \\ , \\sigma _ { 3 } ( \\frac { n } { \\alpha } ) - 2 8 8 \\ , \\frac { n } { \\alpha } \\ , \\sigma ( \\frac { n } { \\alpha } ) \\bigr ) q ^ { n } \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} \\rho _ n ( z + \\frac { u } { \\sqrt { n } } , z + \\frac { v } { \\sqrt { n } } ) = \\exp ( - \\frac 1 2 \\sum _ { j = 1 } ^ m \\lambda _ j ^ n | u _ j - v _ j | ^ 2 ) ( 1 + \\tau _ n ( u , v ) ) \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} \\mathcal { R } : = \\frac { e ( x ) } { | x | } \\cdot A \\nabla = \\sum _ { j = 1 } ^ { n } \\nu _ { j } \\frac { e _ { j } ( x ) } { | x | } X _ { j } , \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} F = \\left [ \\begin{array} { c c c c c } 0 & I & 0 & \\cdots & 0 \\\\ 0 & 0 & I & \\cdots & 0 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & I \\end{array} \\right ] , G = \\left [ \\begin{array} { c c c c c } I & 0 & \\cdots & 0 & 0 \\\\ 0 & I & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & I & 0 \\end{array} \\right ] , \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\partial ^ * \\star f = - ( f ' _ i ) _ { i \\ge 1 } , \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { n } \\binom { n } { m } \\sum _ { j = 0 } ^ { m } ( - 1 ) ^ { j } \\binom { m } { j } \\left ( \\frac { s } { s + n - m + j } \\right ) ^ { 2 } & = \\frac { n ( n + 2 s ) } { ( s + n ) ^ 2 } , \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} P _ { n } = a ( n + 1 ) - b + o ( 1 ) = a n + O ( 1 ) , \\mbox { a s } n \\to \\infty . \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} - \\Delta u & \\leq \\max _ { t = 0 } \\left ( - \\Delta u - \\frac { R } { 4 } - | \\nabla u | ^ 2 \\right ) + \\frac { R } { 4 } + | \\nabla u | ^ 2 \\leq \\max _ { t = 0 } \\{ - \\Delta u - | \\nabla u | ^ 2 \\} + \\max _ { t = 0 } | \\nabla u | ^ 2 + \\frac { R + 1 } { 4 } \\\\ & \\leq \\max _ { t = 0 } \\left ( - \\Delta u \\right ) + \\max _ { t = 0 } | \\nabla u | ^ 2 + \\frac { R + 1 } { 4 } . \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} \\theta ( \\Delta ) B _ W \\theta ( \\Delta ) = U \\theta ( \\Delta ) \\left ( \\tilde { W } ( S ^ * - S ) - ( S ^ * - S ) \\tilde { W } \\right ) \\theta ( \\Delta ) + \\ . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} \\Q ( q , t ) \\cap \\Q ( ( q ) ) [ [ t ] ] \\cap \\Q ( ( t ) ) [ [ q ] ] \\cap \\Q [ q , t , q ^ { - 1 } , t ^ { - 1 } ] = \\Q [ q , t ] . \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} | v _ { 1 2 } | = | v _ { 2 1 } | \\geq \\frac { 1 } { \\sqrt { 1 0 } } \\left ( \\frac { \\alpha z _ { 1 2 } + \\beta z _ { 2 1 } } { \\alpha ^ 2 - \\beta ^ 2 - z _ { 1 2 } ^ 2 \\wedge z _ { 2 1 } ^ 2 } \\wedge 1 \\right ) . \\\\ \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} C ^ { - 1 } \\left \\Vert \\left \\{ a _ { j } \\right \\} _ { j = 1 } ^ { \\infty } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } ^ { 2 } \\leq \\left \\Vert \\sum \\nolimits _ { j = 1 } ^ { \\infty } a _ { j } \\varphi _ { z _ { j } } \\right \\Vert _ { B _ { 2 } } ^ { 2 } \\leq C \\left \\Vert \\left \\{ a _ { j } \\right \\} _ { j = 1 } ^ { \\infty } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } ^ { 2 } \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} 2 R e \\left \\{ \\left ( e ^ { i \\theta } \\overline p \\sqrt { 1 + | m | ^ 2 } - m \\overline n \\right ) \\varphi \\right \\} = | n | ^ 2 - | p | ^ 2 - 1 \\ , . \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ n q _ i y _ i = \\delta . \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} R _ g = - n ( n + 1 ) + O ( r ^ { n + 2 } \\log r ) . \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\mbox { a n d } \\sum _ { k = 0 } ^ \\infty \\alpha _ k ( \\phi ( x ^ k ) - \\phi ( x ^ * ) ) ^ T ( x ^ k - x ^ * ) < \\infty . \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} d ^ W _ p ( \\mu _ 1 , \\mu _ 2 ) = \\inf \\norm { d } _ { L ^ p ( \\pi ) } \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} C _ { t , p } = \\mathbb { E } \\left [ \\vartheta _ t ( X _ t ) ^ q \\right ] ^ { 1 / q } . \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} & \\frac { \\# \\{ n , \\ , \\ , \\lambda _ n \\ , ( { \\rm m o d } \\ , a ^ { - 1 } ) \\in I , \\ , \\ , M \\le n \\le M + N \\} } { N } \\\\ & = \\frac { \\# \\{ \\lambda \\in \\Lambda , \\ , \\ , \\lambda _ M \\le \\lambda \\le \\lambda _ { M + N - 1 } , \\ , \\ , \\lambda \\ , ( { \\rm m o d } \\ , a ^ { - 1 } ) \\in I \\} } { \\lambda _ { M + N - 1 } - \\lambda _ M } \\ , \\left ( \\frac { \\lambda _ { M + N - 1 } - \\lambda _ M } { \\# A _ { M , N } } \\right ) \\end{align*}"}
-{"id": "10138.png", "formula": "\\begin{gather*} ( p , q ) = ( - 4 - 6 u , 1 + 6 u ) , ( p , q ) = ( - 3 - 6 u , - 1 + 6 u ) , \\\\ ( p , q ) = ( - 5 - 6 u , 2 + 6 u ) , ( p , q ) = ( - 1 - 6 u , - 2 + 6 u ) , \\\\ ( p , q ) = ( - 3 - 2 u , 1 + 2 u ) , \\end{gather*}"}
-{"id": "5543.png", "formula": "\\begin{align*} u ( x _ 1 , x _ 2 , t ) = \\frac { 1 } { 4 \\pi t } \\int \\limits _ { - \\infty } ^ { + \\infty } \\int \\limits _ { - \\infty } ^ { + \\infty } \\Lambda ( s _ 1 , s _ 2 ) \\exp \\left \\{ - \\frac { ( s _ 1 - x _ 1 ) ^ 2 + ( s _ 2 - x _ 2 ) ^ 2 } { 4 t } \\right \\} \\ , d s _ 1 d s _ 2 . \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{gather*} q _ { n , n - 1 - m } = \\sum _ { k = 0 } ^ { m } p _ { n , n - ( m - k ) } s _ { k } \\ , \\ \\ \\ 0 \\leq m \\leq n - 1 \\ , \\ \\ \\ n \\geq 1 \\ . \\end{gather*}"}
-{"id": "6907.png", "formula": "\\begin{align*} ( \\Lambda ( e _ j , 0 ) ( \\pi ( t _ j e _ j ) f ) ) ( \\tau ) = ( \\Lambda ( e _ j , t _ j e _ j ) f ) ( \\tau ) = ( \\Lambda ( e _ j , 0 ) f ) ( \\tau + t _ j ) . \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} \\sum _ { j = - \\infty } ^ { \\infty } f _ { j } ^ { 2 } ( x ) = \\frac { x ^ { 2 } } { 1 - x ^ { 2 } } A ^ { - 1 } W ' ( f ( x ) , g ( x ) ) , \\mbox { f o r } 0 < | x | < 1 . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} H ^ { s , p } _ { p e r } ( \\Omega ) : = \\overline { C ^ { \\infty } _ { p e r } ( \\Omega ) } ^ { \\norm { \\cdot } _ { H ^ { s , p } ( \\Omega ) } } , H ^ { s , p } _ { p e r } ( G ) : = \\overline { C ^ { \\infty } _ { p e r } ( G ) } ^ { \\norm { \\cdot } _ { H ^ { s , p } ( G ) } } , \\end{align*}"}
-{"id": "9926.png", "formula": "\\begin{align*} ( u ( r ) v ) ^ { \\max } = \\sigma ( r ) ( u ( - r ) u ^ { - } ( r ^ { - 1 } ) v ) ^ { \\min } = \\sigma ( r ) ( u ^ { - } ( r ^ { - 1 } ) v ) ^ { \\min } = \\sigma ( r ) ( u ^ { - } ( r ^ { - 1 } ) v ) ^ { \\max } = \\sigma ( r ) v ^ { \\max } . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} { } ^ { c _ i } ( { } ^ { c _ 1 ^ { \\delta _ 1 } \\ldots c _ { i - 1 } ^ { \\delta _ { i - 1 } } c _ { i + 1 } ^ { \\delta _ { i + 1 } } \\ldots c _ r ^ { \\delta _ r } } H ) = { } ^ { c _ 1 ^ { \\delta _ 1 } \\ldots c _ { i - 1 } ^ { \\delta _ { i - 1 } } c _ i c _ { i + 1 } ^ { \\delta _ { i + 1 } } \\ldots c _ r ^ { \\delta _ r } } H = { } ^ { c _ 1 ^ { \\delta _ 1 } \\ldots c _ { i - 1 } ^ { \\delta _ { i - 1 } } c _ { i + 1 } ^ { \\delta _ { i + 1 } } \\ldots c _ r ^ { \\delta _ r } c _ i } H . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = \\omega + \\xi ( z , \\omega , \\mu ) + f ( x , z , \\omega , \\mu ) , \\\\ \\dot { z } & = Q ( \\omega , \\mu ) z + \\zeta ( z , \\omega , \\mu ) + h ( x , z , \\omega , \\mu ) \\end{aligned} \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu } g ( x / \\mu ) = \\mu ^ { - ( \\alpha + 2 ) / 2 } e ^ { i t } g ( x ) = \\lambda _ { \\alpha } g ( x ) . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} X ( t ) = \\mu + \\psi \\Xi + \\int _ 0 ^ t b ( s ) X ( s ) \\ , d s + \\int _ 0 ^ t \\sigma ( s ) \\ , d W ( s ) . \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} | \\tilde \\partial _ x ^ \\alpha V \\left [ x \\right ] | \\leq C _ \\alpha ^ \\prime \\langle x \\rangle ^ { - | \\alpha | - \\varepsilon } , & x \\in \\mathbb { R } ^ d , \\ \\alpha \\in \\mathbb { Z } _ { + } ^ d , \\end{align*}"}
-{"id": "10095.png", "formula": "\\begin{align*} d f _ q ( X ( q ) ) = 0 , \\quad \\forall q \\in C . \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} \\tilde { F } _ j ( x , z ) : = & \\frac { F ( x _ j + r _ j x , \\xi _ j + \\lambda _ j z ) - F ( x _ j + r _ j x , \\xi _ j ) - \\lambda _ j F _ { z } ( x _ j , \\xi _ j ) [ z ] } { \\lambda _ j ^ 2 } ; \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\max \\left \\{ \\beta _ 1 , \\beta _ 2 \\right \\} \\ , d \\tau = \\infty , \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{gather*} ( - 1 ) ^ { \\ell + 1 } h _ { \\underline { k } , \\underline { \\ell } } ^ { ( \\alpha , \\beta ) } = ( - 1 ) ^ { \\ell + 1 } h _ { \\underline { k } , { \\ell } } ^ { ( \\alpha , \\beta ) } h _ { { k + 1 } , { \\underline { \\ell } } } ^ { ( \\alpha - 1 , \\beta ) } + ( - 1 ) ^ { \\ell + 1 } h _ { \\underline { k + 1 } , { \\underline { \\ell } } } ^ { ( \\alpha - 1 , \\beta ) } . \\end{gather*}"}
-{"id": "6400.png", "formula": "\\begin{align*} \\langle \\mathbf { D } , \\hat { \\mathbf { D } } \\rangle _ { \\mathbb { R } ^ { k \\times d } } \\equiv \\mathbf { D } : \\hat { \\mathbf { D } } : = \\sum _ { i = 1 } ^ { k } \\sum _ { j = 1 } ^ { d } D _ { i j } \\hat { D } _ { i j } \\mathbf { D } , \\hat { \\mathbf { D } } \\in \\mathbb { R } ^ { k \\times d } . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ I } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , s , k } R ^ d \\eta ^ { - 1 } y \\end{aligned} \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} - \\beta _ { x x } & = \\int _ { v > 0 } \\mu _ { + , + } ( e _ { + } ) d v + \\int _ { v < 0 } \\mu _ { + , - } ( e _ { + } ) d v \\\\ & - \\int _ { v > 0 } \\mu _ { - , + } ( e _ { - } ) d v - \\int _ { v < 0 } \\mu _ { - , - } ( e _ { - } ) d v \\equiv h \\left ( \\beta \\right ) . \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} \\frac { g ^ { k } _ { \\beta , \\epsilon } } { K } \\leq g \\leq K g ^ { k } _ { \\beta , \\epsilon } , \\ g ^ { k } _ { \\beta , \\epsilon } = \\Sigma _ { j = 1 } ^ { k } \\frac { \\beta _ { j } ^ { 2 } } { ( | z _ { j } | ^ 2 + \\epsilon ^ 2 ) ^ { 1 - \\beta _ { j } } } d z _ { j } \\otimes d \\bar { z } _ { j } + \\Sigma _ { j = k + 1 } ^ { n } d z _ { j } \\otimes d \\bar { z } _ { j } , \\ \\epsilon > 0 . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} \\begin{aligned} d X & = \\frac { 1 } { P } \\ , d z _ 0 + \\frac { \\Omega _ { [ 1 ] } } { 2 } \\ , d z _ 1 - \\frac { \\Delta } { P } \\ , d z _ { n + 1 } , \\\\ Y & = z _ 1 , \\\\ T & = z _ { n + 1 } , \\end{aligned} \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\frac { \\partial S } { \\partial \\beta } = - \\beta \\frac { \\partial ^ 2 } { \\partial \\beta ^ 2 } \\ln Z = - \\beta \\sigma \\leq 0 \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} \\sum _ { e : v \\in e } B ( v , e ) = 1 , & ~ ~ ~ v \\in V ( B _ m ^ L ( 2 ) ) , \\\\ [ 2 m m ] \\prod _ { v : v \\in e } B ( v , e ) = \\alpha , & ~ ~ ~ e \\neq e _ { 2 } . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} \\omega ^ { 2 n - 1 } ( \\Phi ) \\omega ^ { 2 n - 1 } ( \\Psi ) & + \\omega ^ { 2 n } ( \\Phi ) \\omega ^ { 2 n } ( \\Psi ) = \\\\ & = \\sin \\phi ^ 1 \\sin \\phi ^ 2 \\dots \\sin \\phi ^ { 2 n - 2 } \\sin \\psi ^ 1 \\sin \\psi ^ 2 \\dots \\sin \\psi ^ { 2 n - 2 } \\cos ( \\phi ^ { 2 n - 1 } - \\psi ^ { 2 n - 1 } ) \\\\ & \\ge \\frac { 1 } { \\sqrt 2 } \\sin \\phi ^ 1 \\sin \\phi ^ 2 \\dots \\sin \\phi ^ { 2 n - 2 } \\sin \\psi ^ 1 \\sin \\psi ^ 2 \\dots \\sin \\psi ^ { 2 n - 2 } , \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} \\Lambda ( x _ 1 , x _ 2 ) = 0 , x _ 1 < 0 , \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} X \\ ; : \\ ; F ( x _ 0 , \\dotsc , x _ { n + 1 } ) = 0 . \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} \\sum _ { p \\in [ N _ T ] } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { [ N _ T ] } , p } ^ i ( u ) = 0 , \\forall q \\in \\bar { \\mathcal { R } } _ i , \\forall u \\in [ \\rho ] \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} \\partial _ { \\nu } \\left ( \\frac { \\partial L } { \\partial \\left ( \\partial _ { \\nu } A _ { \\mu } \\right ) } \\right ) - \\frac { \\partial L } { \\partial A _ { \\mu } } = 0 . \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} g _ 0 & : = g ( \\cdot , \\cdot , \\cdot , 0 , 0 , 0 ) \\in \\mathcal { M } ^ { 0 , p } ( Q ) \\cap \\mathcal { M } ^ { 0 , 2 } ( Q ) \\\\ f _ 0 & : = f ( \\cdot , \\cdot , \\cdot , 0 , 0 , 0 ) \\in \\mathcal { M } ^ { 0 , \\frac { p ( n + 2 ) } { p + n + 2 } } ( Q ) \\cap \\mathcal { M } ^ { 0 , 2 } ( Q ) . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\Phi ( x , y ) : = \\frac { | \\phi ( x ) - \\phi ( y ) | ^ p } { | x - y | ^ { n + s p } } , \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} \\lambda _ X = 0 , \\lambda _ T - \\lambda ^ n \\lambda _ Y = 0 . \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } V _ i = \\sum _ { i = 1 } ^ { k - 1 } \\left ( B _ i + C _ i \\right ) . \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} u _ { x y } - \\sin u = 0 . \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} \\sum \\limits _ { \\delta | N } \\delta r _ \\delta & = p _ 1 ^ { n _ 1 - 1 } p _ 2 ^ { n _ 2 - 1 } r ( 1 + p _ 1 ^ 2 p _ 2 ^ 2 - p _ 1 ^ 2 - p _ 2 ^ 2 ) \\\\ & = p _ 1 ^ { n _ 1 - 1 } p _ 2 ^ { n _ 2 - 1 } \\frac { 2 4 } { ( p _ 1 - 1 ) ( p _ 2 - 1 ) } ( p _ 1 + 1 ) ( p _ 1 - 1 ) ( p _ 2 + 1 ) ( p _ 2 - 1 ) \\\\ & = 2 4 p _ 1 ^ { n _ 1 - 1 } p _ 2 ^ { n _ 2 - 1 } ( p _ 1 + 1 ) ( p _ 2 + 1 ) . \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} \\omega _ z ( u _ { i j } ^ { k } ) & = \\varphi ( \\pi _ z ( u _ { i j } ^ { k } ) 1 ) = \\varphi ( ( u _ { i j } ^ { k } ) _ { ( 1 ) } ( f _ { 1 - z } \\star S ( ( u _ { i j } ^ { k } ) _ { ( 2 ) } ) ) ) \\\\ & = \\frac { \\delta _ { i j } \\mu _ k ( | q | ^ { \\bar z } + | q | ^ { - \\bar z } ) } { \\mu _ { k } ( | q | + | q | ^ { - 1 } ) } . \\end{align*}"}
-{"id": "9950.png", "formula": "\\begin{align*} \\pi _ { p } ( v _ { i - 1 } ) = \\pi _ { p } ( ( v _ { i } ) _ { i - 1 } ) = \\pi _ { p } ( v _ { i } ) _ { i - 1 } . \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} \\begin{array} { c } h c h ^ { - 1 } C = h \\gamma ^ { \\infty } h ^ { - 1 } C = h \\gamma ^ { \\lambda _ 0 } h ^ { - 1 } C \\subset e C \\ , , \\end{array} \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} \\phi \\big ( M _ 1 ( D _ { i ( 1 ) } ) ^ \\circ \\cdots M _ t ( D _ { i ( t ) } ) ^ \\circ \\big ) = 0 . \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { + \\infty } a _ i < + \\infty . \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} \\frac { z ^ 1 z ^ 2 \\cdots z ^ n } { z ^ i } \\partial h ^ i = \\frac { z ^ 1 z ^ 2 \\cdots z ^ n } { z ^ k } \\partial h ^ k . \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} \\left ( { v - u } \\right ) \\left ( { w - z } \\right ) F _ { t s } \\left ( { \\eta _ 1 , \\eta _ 2 } \\right ) = F \\left ( { v , w } \\right ) - F \\left ( { v , z } \\right ) - F \\left ( { u , w } \\right ) + F \\left ( { u , z } \\right ) \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} \\sigma = \\min \\left \\{ \\frac { 1 - s } { 2 s } , \\frac { p - 1 } { 2 } , \\frac { 1 } { 2 } \\right \\} , \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} \\overrightarrow { H } = \\frac { 1 } { 2 f _ { 2 } ( u ) } \\left \\{ \\kappa _ { \\rho } ( v ) N _ { 1 } + \\left ( - \\kappa _ { \\gamma } f _ { 2 } ( u ) + f _ { 1 } ^ { \\prime } ( u ) \\right ) N _ { 2 } \\right \\} . \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} \\begin{array} { l } ( \\cup _ { \\alpha \\in J } f _ { \\alpha } ) ( A ) : = \\cup _ { \\alpha \\in J } ( f _ { \\alpha } A ) , { } \\\\ ( \\cap _ { \\alpha \\in J } f _ { \\alpha } ) ( A ) : = \\cap _ { \\alpha \\in J } ( f _ { \\alpha } A ) , { } A \\subset X \\ , . \\end{array} \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} f ( y ) = P _ { y _ 0 } ( y ) + O ( d _ G ( y , y _ 0 ) ^ { k + 2 \\alpha } ) , y \\rightarrow y _ 0 . \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} \\lambda x _ { k } ^ { r - 1 } = \\sum _ { i _ { 2 } , \\ldots , i _ { r } } a _ { k , i _ { 2 } , \\ldots , i _ { r } } x _ { i _ { 2 } } \\cdots x _ { i _ { r } } \\ \\ \\ \\ k = 1 , \\ldots , n , \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{gather*} S \\circ \\beta = \\alpha , \\\\ \\big ( S ^ { - 1 } h _ { ( 2 ) } \\big ) _ { ( 1 ) } \\otimes _ A \\big ( S ^ { - 1 } h _ { ( 2 ) } \\big ) _ { ( 2 ) } h _ { ( 1 ) } = S ^ { - 1 } h \\otimes _ A 1 _ H , \\\\ \\big ( S h _ { ( 1 ) } \\big ) _ { ( 1 ) } h _ { ( 2 ) } \\otimes _ A \\big ( S h _ { ( 1 ) } \\big ) _ { ( 2 ) } = 1 _ H \\otimes _ A S h . \\end{gather*}"}
-{"id": "4895.png", "formula": "\\begin{align*} & k \\cdot \\left ( ( 2 - 2 g ) + ( 1 - m ) ( g - 1 - m ) + ( g - m ) m + ( k - 1 - m ) + ( m - 1 ) \\right ) \\\\ = & - k ( g - k + 1 ) . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} = \\theta ( q \\ , T _ p ( p a p ) q ) = \\theta ( T _ p ( q p a p q ) ) = \\theta ( T _ p ( q a q ) ) = \\theta ( q a q ) = \\theta _ q ( a ) \\ , . \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} & \\left \\| \\frac { 1 } { \\sqrt { p } } \\nabla _ p s _ { f } \\right \\| _ { p - S l } ^ 2 = \\frac { 1 } { p } \\inf _ { g \\in L ^ 2 [ 0 , 1 ] : \\nabla _ p s _ { f } = \\nabla _ p s _ { g } } \\| g \\| ^ 2 _ { L ^ 2 [ 0 , 1 ] } \\\\ & = \\frac { 1 } { 2 p ( 3 p - 1 ) } ( 2 s _ { f } ( p ) + \\delta ( 1 - p ) ) ^ 2 + \\frac { 1 } { 2 p } \\| ( f ( t ) - f ( t - p ) ) _ { t \\in [ p , 1 ] } \\| ^ 2 _ { L ^ 2 [ p , 1 ] } , \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\mathcal { C } _ \\rho = \\left \\{ \\left . u \\in \\mathbb { R } ^ d \\right | \\textnormal { d i s t } \\left ( u , L \\right ) \\leq \\rho \\right \\} \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} \\begin{cases} d u = ( \\Delta u + V ( t , x ) u ) \\ , d t + G ( t , x ) u \\ , d W ( t ) , ( t , x ) \\in ( 0 , 1 ] \\times \\R ^ n , \\\\ u ( 0 ) = u _ 0 , \\end{cases} \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} \\tilde { l } _ \\alpha ( | x _ B | ) \\leq \\frac { 1 } { V o l ( B ) } \\int _ { B } \\tilde { l } _ \\alpha ( | x | - { \\rm d i a m } ( B ) ) d x = \\frac { 1 } { \\alpha ^ d } \\int _ { B } \\tilde { l } _ \\alpha ( | x | - \\alpha \\sqrt { d } ) d x . \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} \\frac { 1 } { Z _ n } \\det \\left [ I _ { \\kappa } ( 2 \\alpha _ i \\sqrt { x _ j } ) \\right ] _ { i , j = 1 } ^ n \\det \\left [ x _ j ^ { \\frac { \\nu + i - 1 } { 2 } } K _ { \\nu - \\kappa + i - 1 } ( 2 \\beta \\sqrt { x _ j } ) \\right ] _ { i , j = 1 } ^ n , \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} \\phi _ { K _ p ' } ( x ) = { \\rm v o l } ( k \\in K _ p ^ { \\rm a d } : [ k , x ] \\in K _ p ' ) \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} n ^ \\beta \\exp \\{ - 9 \\alpha ^ { - 2 ( i + 1 ) } f ( n , p ) ^ 2 \\} & \\geq \\exp \\{ \\beta \\log n - 9 \\alpha ^ { - 2 ( i + 1 ) } \\sqrt { \\log n } \\} \\\\ & = \\exp \\{ \\beta \\log n ( 1 - 9 \\alpha ^ { - 2 ( i + 1 ) } ( \\log ^ { - 1 / 2 } n ) ) \\} \\\\ & = \\omega ( 1 ) , \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} { \\textstyle \\sum \\limits _ { j = 1 } ^ { s } } \\gamma _ { j } = u + \\left ( { \\textstyle \\sum \\limits _ { j = 1 } ^ { s } } n _ { j } \\right ) v _ { k } = w + \\left ( { \\textstyle \\sum \\limits _ { j = 1 } ^ { s } } n _ { j } \\right ) h _ { k } , \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} \\Delta ( f ( x ^ { ( k ) } ) ) = H _ k ( x ^ { ( k ) } ) \\otimes f ( x ^ { ( k ) } ) + f ( x ^ { ( k ) } ) \\otimes 1 . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 u _ k = g ( | \\lambda _ k x | ) u _ k ^ p \\quad & \\mbox { i n } B _ { \\frac { 1 } { \\lambda _ k } } ( 0 ) \\\\ u _ k = \\Delta u _ k - ( 1 - \\sigma ) \\lambda _ k ( u _ k ) _ n = 0 \\quad & \\mbox { o n } \\partial B _ { \\frac { 1 } { \\lambda _ k } } ( 0 ) , \\end{cases} \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} g ( 0 , x _ 0 , t , 0 ) = 1 - \\alpha \\quad g ( 0 , x _ 0 , t , 1 ) = 1 - \\alpha . \\end{align*}"}
-{"id": "9935.png", "formula": "\\begin{align*} \\varphi ( s _ 1 ) = [ \\varphi _ 1 ( s _ 1 ) ; \\vect { 0 } ; \\dots ; \\vect { 0 } ] \\psi ( s _ 1 ) = \\vect { 0 } . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} H ( X ) & = \\tfrac { 1 } { ( g ! ) ^ { 2 } } \\int _ { X ^ { g } } \\log \\| \\theta \\| ( 2 P _ { 1 } + P _ { 2 } + \\dots + P _ { g - 1 } - P _ { g } ) \\Phi ^ { * } \\nu ^ { g } \\\\ & = S _ { g } ( X ) + \\int _ { X } g ( \\sigma ( P ) , P ) \\mu ( P ) + \\left ( \\tfrac { g ( g + 1 ) } { 2 } - 1 \\right ) \\tfrac { 1 } { 2 g ( g - 1 ) } \\varphi ( X ) \\\\ & = S _ { g - 1 } ( X ) + \\tfrac { ( g + 2 ) } { 4 g } \\varphi ( X ) . \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} & \\widetilde { J } _ { \\epsilon } ( d , \\xi ) = a ( \\xi ) \\left [ c _ { 1 } + c _ { 2 } \\epsilon \\log | \\epsilon | + c _ 3 \\epsilon + c _ 4 \\mathcal H _ a ( \\xi ) | \\epsilon | d + c _ 5 \\epsilon \\ln d + o ( \\epsilon ) \\right ] , \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\inf _ b \\ , \\sum _ { i = 1 } ^ n \\left \\vert Y _ i - \\frac { b Z _ i } { \\sqrt { n } } \\right \\vert \\approx \\sum _ { i = 1 } ^ n \\vert Y _ i \\vert - \\frac { \\chi ^ 2 _ 1 } { 4 f ( 0 ) } \\ , . \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} \\oplus ( a _ 1 , a _ 2 , a _ 3 ) \\Leftrightarrow \\exists x _ 1 , x _ 2 , x _ 3 \\in K \\left ( \\bigwedge _ { i = 1 } ^ 3 r v _ \\delta ( x _ i ) = a _ i \\wedge x _ 1 + x _ 2 = x _ 3 \\right ) . \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} 0 = t r ( \\phi _ \\sigma ) = 6 \\dim W _ 8 + c \\dim W _ { 4 8 } , \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\begin{aligned} Z _ { s , s + k + 1 } & \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau , 0 ; v _ { s + 1 } , \\dots , v _ { s + k } , v _ { s + k + 1 } ; \\right . \\\\ & \\left . \\omega _ 1 , \\dots , \\omega _ k , \\omega _ { k + 1 } ; i _ 1 , \\dots , i _ k , i _ { k + 1 } \\right ] \\\\ & \\in \\mathcal { K } _ { s + k + 1 } \\cap \\mathcal { U } _ { s + k + 1 } ^ \\eta \\end{aligned} \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} g ( x + y ) = g ( \\frac { x } { y + 1 } ) + g ( y ) + g ( \\frac { x } { y + 1 } y ) \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\left ( 1 , \\tfrac { 1 } { 2 } + \\nu ; \\tfrac { 1 } { 2 } ; - x ^ 2 \\right ) = ( 1 + x ^ 2 ) ^ { - 1 } { } _ 2 F _ 1 ( 1 , - \\nu ; \\tfrac { 1 } { 2 } ; \\tfrac { x ^ 2 } { 1 + x ^ 2 } ) . \\end{align*}"}
-{"id": "10105.png", "formula": "\\begin{align*} C \\ : : \\ : x ^ p y ^ q ( a x + b y + c z ) ^ r - z ^ { p + q + r } = 0 , \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\min ( p , 2 ) } u _ i \\leq k u _ { i + 1 } = u _ i - 1 i = 1 , 2 , \\ldots , p - 1 u _ p = 2 . \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} \\tilde P _ \\pm ^ * \\tilde P _ \\pm u [ x ] = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( x - y ) \\cdot \\eta } r ( x , y , \\eta ) u \\left [ y \\right ] d \\eta , \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\norm { f } _ { L ^ p ( \\Omega ) ^ 2 } = O ( e ^ { - \\beta _ f t } ) \\hbox { a n d } \\norm { g _ { \\tau } } _ { L ^ { q _ { \\tau } } ( \\Omega ) } = O ( e ^ { - \\beta _ g t } ) , \\hbox { a s } t \\to \\infty , \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} \\xi _ { q } ( z ) : = \\sum _ { n = - \\infty } ^ { \\infty } \\frac { q ^ { n / 2 } } { 1 + z q ^ { n - 1 / 2 } } , \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} F ( x _ k ) = F _ t ( x _ k ; x _ k ) & \\ge F _ t ( x _ { k } ^ * ; x _ k ) + \\tfrac { t } { 2 } \\norm { \\mathcal { G } _ t ( x _ k ) } ^ 2 \\ge F _ t ( x _ { k + 1 } ; x _ k ) - \\varepsilon _ { k + 1 } + \\tfrac { t } { 2 } \\norm { \\mathcal { G } _ t ( x _ k ) } ^ 2 . \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} \\psi ( t , \\alpha , \\beta , \\gamma ) : = \\sum _ { j = 1 } ^ { n + m + 1 } \\alpha _ j \\left [ \\nabla _ { x , u } G ( t , x ^ * ( t ) , u ^ * ( t ) ) \\beta ^ j + \\nabla _ { x , u } H ( t , x ^ * ( t ) , u ^ * ( t ) ) \\gamma ^ j \\right ] \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} \\partial ^ { \\alpha } _ t U = - A U + \\partial ^ { \\beta } _ t \\int ^ t _ 0 G ( s ) d W _ s , \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} \\ln \\det ( \\Delta ) = T r \\ln ( \\Delta ) = - \\int _ 0 ^ \\infty \\frac { d t } { t } T r ( e ^ { - t \\Delta } ) \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} \\mu _ d ( z ) : = \\left \\{ \\begin{array} { l l } $ $ 1 $ $ , & \\hbox { $ d \\leq 3 $ ; } \\\\ $ $ 1 + | \\log z | $ $ , & \\hbox { $ d = 4 $ ; } \\\\ $ $ z ^ { 4 - d } $ $ , & \\hbox { $ d \\geq 5 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} \\phi _ E ( ( x , k , y ) ) = [ ( s _ \\mu s _ \\nu ^ * , y ) ] \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} e i t h e r ~ \\omega ( g ) = 1 ~ a n d ~ \\xi ( T ^ { \\langle { g } \\rangle } ) = 1 ~ o r ~ \\omega ( g ) = - 1 ~ a n d ~ \\xi ( T ^ { \\langle { g } \\rangle } ) = - 1 . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} \\omega = \\omega _ 0 + u ( \\omega _ 0 , \\mu _ 0 ) , \\sigma = v ( \\omega _ 0 , \\mu _ 0 ) , \\mu = \\mu _ 0 + w ( \\omega _ 0 , \\mu _ 0 ) \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} [ \\bar { x } ^ + _ { i , r } , \\bar { x } ^ - _ { j , s } ] = \\delta _ { i , j } \\bar { \\xi } _ { i , r + s } . \\end{align*}"}
-{"id": "10083.png", "formula": "\\begin{align*} z ' = - i z + A z ^ 2 + B z \\bar { z } + C \\bar { z } ^ 2 , \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} A ( k ) & = - 2 k ^ 8 - 1 6 k ^ 7 + 4 0 k ^ 6 - 1 7 6 k ^ 5 + 5 3 2 k ^ 4 - 7 5 2 k ^ 3 + 3 6 0 k ^ 2 + 1 7 6 k + 9 4 , \\\\ B ( k ) & = ( k ^ 4 + 1 2 k ^ 3 - 1 8 k ^ 2 - 4 k - 7 ) ( k ^ 2 - 2 k + 5 ) ^ 2 ( k + 3 ) ^ 4 ( k - 1 ) ^ 4 , \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} \\vartheta _ { m } = \\sum _ { \\gamma \\in M _ { k } } \\mathcal { Z } ^ { T _ { \\gamma } ( m , \\varphi ( m ) , 1 ) } = \\sum _ { \\gamma \\in M _ { k } } \\mathcal { Z } ^ { ( m + \\gamma , \\varphi ( m + \\gamma ) , 1 ) } , \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{gather*} P _ { r - 1 } ( x ) Q _ { r } ( x ) - P _ { r } ( x ) Q _ { r - 1 } ( x ) = D _ { r - 1 } ^ { 2 } \\ , \\end{gather*}"}
-{"id": "1483.png", "formula": "\\begin{align*} & H _ { x _ 1 x _ 1 x _ 2 } + 3 H _ { x _ 2 } H _ { x _ 1 } - \\frac { k + 1 } { 4 } \\frac { H _ { x _ 1 x _ 2 } ^ 2 } { H _ { x _ 2 } } = 0 , \\\\ & \\Omega _ { x _ 1 } = H _ { x _ 2 x _ 3 } \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} I _ 0 = [ 0 , T _ 1 ] , I _ 1 = [ T _ 1 ^ { \\prime } - \\varepsilon _ 1 , T _ 1 ] , I _ 2 = [ T _ 1 ^ { \\prime } , T _ 2 ] , \\ldots , I _ n = [ T _ n ^ { \\prime } , T _ { n + 1 } ] , \\ldots , \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\textbf { q } } = - ( e / m ) \\textbf { A } ( \\textbf { q } , t ) \\\\ \\dot { \\textbf { p } } = ( e / m ) \\textbf { p } \\textbf { A } ' ( \\textbf { q } , t ) - \\nabla f ( \\textbf { q } , t ) \\end{cases} , \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\displaystyle \\frac { d \\psi ( t ) } { d t } = \\frac { 2 } { n } ( \\psi ( t ) ) ^ 2 \\\\ \\displaystyle \\ \\ \\psi ( 0 ) = \\mathcal { H } _ { m i n } ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} H _ { k } = \\frac { ( L _ { 1 1 } ^ { k } g _ { 2 2 } + L _ { 2 2 } ^ { k } g _ { 1 1 } - 2 L _ { 1 2 } ^ { k } g _ { 1 2 } ) } { 2 W ^ { 2 } } , \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} P _ \\omega ( x , y ) : = \\sum _ { \\gamma } \\omega ^ { c ( \\gamma ) } . \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} \\chi ( x ) = \\begin{cases} 0 & | x | \\leq 1 , \\\\ 1 & | x | \\geq 2 , \\end{cases} \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} ( N - 1 ) c _ 1 ^ 2 + m _ { 2 , j } ( c _ 2 ^ 2 - c _ 1 ^ 1 ) & = ( m _ { 1 , j } + m _ { 2 , j } ) c _ 1 ^ 2 + m _ { 2 , j } ( c _ 2 ^ 2 - c _ 1 ^ 2 ) \\\\ & = m _ { 1 , j } c _ 1 ^ 2 + m _ { 2 , j } c _ 2 ^ 2 \\\\ & = m _ { 1 , l } c _ 1 ^ 2 + m _ { 2 , l } c _ 2 ^ 2 \\\\ & = ( m _ { 1 , l } + m _ { 2 , l } ) c _ 1 ^ 2 + m _ { 2 , l } ( c _ 2 ^ 2 - c _ 2 ^ 1 ) \\\\ & = ( N - 1 ) c _ 1 ^ 2 + m _ { 2 , l } ( c _ 2 ^ 2 - c _ 1 ^ 2 ) . \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 0 } ^ n k ^ p f _ n ( k ) \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} \\Bigg ( k \\langle x , y \\rangle / \\begin{pmatrix} x ^ 2 y - y x ^ 2 \\\\ x y ^ 2 - y ^ 2 x \\end{pmatrix} \\Bigg ) _ 2 \\cong k [ a , b , c , d ] / ( a d - b c ) , \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} p _ { 1 , 1 } = \\dfrac { \\xi } { \\mu A } p _ { 0 , 0 } , \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} \\{ \\bar Q , \\{ \\bar Q , a \\} \\} = \\frac { 1 } { 2 } \\{ \\{ \\bar Q , \\bar Q \\} , a \\} = 0 \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} h _ { ( \\sigma _ { i } ) _ { i } , \\mu } ( g \\alpha | \\alpha : g \\alpha \\vee \\alpha , F , \\delta ) & \\leq \\delta \\log | A | + \\limsup _ { i \\to \\infty } \\frac { 1 } { d _ { i } } \\log \\binom { d _ { i } } { \\lfloor { \\delta d _ { i } \\rfloor } } \\\\ & = \\delta \\log | A | - \\delta \\log ( \\delta ) - ( 1 - \\delta ) \\log ( 1 - \\delta ) , \\end{align*}"}
-{"id": "10120.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q } { z ^ { p + 2 q } } , \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} d \\sigma ^ { - 1 } + \\gamma ^ { t r } + g \\sigma ^ { - 1 } \\Delta = 0 . \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} \\varphi ^ { ( k ) } ( A ) = \\frac { \\i ( k ! ) } { 2 \\pi } \\int _ { \\C } \\frac { \\partial \\tilde { \\varphi } _ N } { \\partial \\overline { z } } ( z ) ( z - A ) ^ { - 1 - k } d z \\wedge d \\overline { z } \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} \\sum _ { p \\in \\mathcal { T } } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { \\mathcal { T } } , p , n } ^ i ( u ) = 0 , \\forall q \\in \\bar { \\mathcal { R } } _ i , \\forall n \\in [ t N ^ { ( N _ R - r - t ) ( N _ T - t + 1 ) } ] , \\forall u \\in [ S ] , \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} K _ { \\alpha \\beta \\gamma } { } ^ { \\delta } = K _ { \\beta \\gamma \\alpha } { } ^ { \\delta } = K _ { \\gamma \\alpha \\beta } { } ^ { \\delta } . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} 1 + \\sigma ^ 2 \\sum _ { m = 1 } ^ { M } | h _ { k m } | ^ 2 = 1 + \\sigma ^ 2 G , \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} \\ell = \\prod _ { p \\in S _ { } } p \\prod _ { p \\in \\Omega '' ( K ) } p , \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} A _ { p } ^ { ( \\ell ) } ( z ) & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = 0 } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } \\\\ & \\sum _ { \\beta = 0 } ^ { p - j - 2 t } 2 ^ { p - j - 2 t - \\beta } \\binom { n - \\ell } { \\beta } \\binom { \\ell } { p - j - 2 t - \\beta } \\sum _ { \\alpha = 0 } ^ \\beta \\binom { \\beta } { \\alpha } \\sum _ { i = 0 } ^ { j - 1 } z ^ { p - 2 ( j + t + \\alpha - i ) } . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} [ \\tilde { H } _ { 0 } , ( \\gamma \\cdot x ) ] = [ \\sqrt { - \\Delta + m ^ 2 } , ( \\gamma \\cdot x ) ] = \\frac { - ( \\gamma \\cdot \\nabla ) } { \\sqrt { - \\Delta + m ^ 2 } } . \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\nabla _ x q ( s , t ) = I + \\int _ t ^ s A ( p ( \\tau , t ) ) \\nabla _ x p ( \\tau , t ) d \\tau , \\\\ \\nabla _ x p ( s , t ) = - \\int _ t ^ s \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , t ) ) \\nabla _ x q ( \\tau , t ) d \\tau . \\end{array} \\right . \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} C _ { \\delta _ 1 , \\delta _ 2 } ( h ) = C _ { \\delta _ 1 , \\delta _ 2 } \\gamma _ { \\delta _ 1 , \\delta _ 2 } ( h ) \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} C _ { i j k l } = \\frac { 1 } { 3 } c _ { i j k l } . \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} p _ t ^ { ( d + 2 ) } ( r ) = \\frac { - 1 } { 2 \\pi r } \\frac { d } { d r } p _ t ( r ) , r > 0 . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} z e _ k = \\sum _ { j = 0 } ^ { n - 1 } g _ { n + 1 , j } . y _ k ^ j e _ k . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} { \\bf E } [ \\beta _ { 1 , 0 } ( a , b , \\bar { b } ) ^ q ] = & \\frac { \\Gamma ( \\frac { q + b _ 0 } { a } ) } { \\Gamma ( \\frac { b _ 0 } { a } ) } \\frac { \\Gamma ( 1 - \\frac { q + b _ 0 } { a } ) } { \\Gamma ( 1 - \\frac { b _ 0 } { a } ) } , \\\\ = & \\frac { \\sin ( \\frac { \\pi b _ 0 } { a } ) } { \\sin ( \\frac { \\pi ( q + b _ 0 ) } { a } ) } . \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} t r ( d E ( e ^ { i \\theta } ) ) = d s r E ( e ^ { i \\theta } ) = \\frac { d \\theta } { 2 \\pi } . \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} \\omega ( ( \\hat \\epsilon \\otimes \\iota ) \\hat W ) = \\hat \\epsilon ( \\omega \\otimes \\iota ) W ^ * = \\hat \\epsilon ( \\lambda ( \\omega \\circ S ) ) = \\omega ( S ( 1 ) ) = \\omega ( 1 ) . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} \\bar { Q } = \\sum _ \\alpha ( x _ \\alpha - \\chi ( x _ \\alpha ) ) \\otimes x _ \\alpha ^ \\ast - 1 \\otimes \\dfrac { 1 } { 2 } \\sum _ { \\alpha , \\beta , \\gamma } c _ { \\alpha , \\beta } ^ \\gamma x _ \\alpha ^ \\ast x _ \\beta ^ \\ast x _ \\gamma \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} \\gamma _ { j k } = \\gamma _ { * j k } - i \\sigma _ j \\Phi \\Phi ^ * \\sigma _ k + i \\sigma _ k \\Phi \\Phi ^ * \\sigma _ j . \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} ( f ( 2 n _ 0 + 2 s + 1 ) + 1 ) ( f ( 2 m _ 0 + 2 k ) + 1 ) ( f ( 2 m _ 0 + 2 l ) + 1 ) = 0 . \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} \\frac { a _ { p , q } ( x ) - a _ { p , q } ( x _ n ) } { x - x _ n } & = \\frac { \\pi _ q ( 0 ^ m 1 \\overline { a _ { n , m + 2 } a _ { n , m + 3 } \\cdots } ) } { \\pi _ p ( 0 ^ m 1 \\overline { a _ { n , m + 2 } a _ { n , m + 3 } \\cdots } ) } \\\\ & \\ge \\frac { \\pi _ q ( 0 ^ m 1 0 ^ { \\infty } ) } { \\pi _ p ( 0 ^ m 1 ^ { \\infty } ) } \\\\ & = \\frac { p - 1 } { p } \\left ( \\frac { p } { q } \\right ) ^ { m + 1 } \\end{align*}"}
-{"id": "10052.png", "formula": "\\begin{align*} \\alpha \\gamma \\beta = 2 + \\alpha + \\beta + \\gamma , \\beta \\geq \\alpha \\geq \\gamma \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} 0 \\to E _ 0 \\stackrel { i _ 0 } { \\to } & E _ 1 \\stackrel { f _ 1 } { \\to } S _ 1 \\to 0 \\\\ 0 \\to S _ 1 \\stackrel { i _ 1 } { \\to } & E _ 2 \\stackrel { f _ 2 } { \\to } S _ 2 \\to 0 \\\\ & \\vdots \\\\ 0 \\to S _ { n - 2 } \\stackrel { i _ { n - 2 } } { \\to } & E _ { n - 1 } \\stackrel { f _ { n - 1 } } { \\to } S _ { n - 1 } \\to 0 \\\\ 0 \\to S _ { n - 1 } \\stackrel { i _ { n - 1 } } { \\to } & E _ n \\stackrel { f _ n } { \\to } E _ { n + 1 } \\to 0 . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} \\varphi _ a ( x , \\xi + 2 \\pi m ) & = \\varphi _ a ( x , \\xi ) + 2 \\pi x \\cdot m , m \\in \\mathbb { Z } ^ d , \\\\ | \\partial _ x ^ \\alpha \\partial _ \\xi ^ \\beta \\left [ \\varphi _ a ( x , \\xi ) - x \\cdot \\xi \\right ] | & \\leq C _ { \\alpha \\beta , a } \\langle x \\rangle ^ { 1 - \\varepsilon - | \\alpha | } , \\\\ | { } ^ t \\nabla _ x \\nabla _ \\xi \\varphi _ a ( x , \\xi ) - I | & < \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} & | \\lambda | ^ { n } f ^ 0 ( \\lambda ) \\big | A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\bigl ( 1 - e ^ { i \\lambda \\mu } \\bigr ) ^ { n } g ^ 0 ( \\lambda ) + C ^ { \\mu , 0 } _ { N } \\bigl ( e ^ { i \\lambda } \\bigr ) \\big | \\\\ & = \\alpha _ 1 \\big | 1 - e ^ { i \\lambda \\mu } \\big | ^ { n } \\bigl ( f ^ 0 ( \\lambda ) + \\lambda ^ { 2 n } g ^ 0 ( \\lambda ) \\bigr ) , \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ M ) ( q \\ , | \\ , b ) = f ( 0 ) \\frac { d ^ M } { d t ^ M } | _ { t = 0 } \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) + M ! \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j \\ , B ^ { ( f ) } _ { 1 } ( q + b _ 0 ) . \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} p ( a , b c ) = & | A | p ( a , b ) p ( a , c ) \\\\ p ( a x , b ) = & | A | p ( a , b ) p ( x , b ) \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\pi _ { k , p } = \\begin{cases} 0 & \\quad , \\\\ \\pi _ { k \\varepsilon _ 1 + \\Lambda _ p } & \\quad , \\\\ \\pi _ { k \\varepsilon _ 1 + \\Lambda _ n } \\oplus \\pi _ { k \\varepsilon _ 1 + \\bar \\Lambda _ n } & \\quad . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} \\mathbb { C } _ t ^ { \\pi } ( r _ t , r _ { t + 1 } , \\ldots , r _ n ; y _ { t - J } ^ { t - 1 } ) \\triangleq { \\bf E } ^ { \\pi } \\bigg \\{ \\sum _ { i = t } ^ n \\log \\Big ( \\frac { r _ i ( X _ i | y _ { i - M } ^ { i - 1 } , Y _ i ) } { { \\pi } _ { i } ( X _ i | y _ { i - J } ^ { i - 1 } ) } \\Big ) \\Big { | } Y _ { t - J } ^ { t - 1 } = y _ { t - J } ^ { t - 1 } \\bigg \\} , ~ t \\in \\mathbb { N } _ 0 ^ n \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} f _ i ( x _ 1 ( t ) , x _ 2 ( t ) , \\dots , x _ N ( t ) ) = \\sum _ { m \\geq 0 } \\frac { f _ { i , m } } { m ! } t ^ m \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} T _ 1 ( s ) & = \\prod _ p ( 1 - b _ p p ^ { - 2 s } + ( 2 b _ p - 2 ) p ^ { - 3 s } - b _ p p ^ { - 4 s } + p ^ { - 6 s } ) \\\\ T _ 2 ( s ) & = \\prod _ p \\left ( 1 - \\sum _ { i , j = 1 } ^ 3 \\alpha _ { i , p } ^ { - 1 } \\alpha _ { j , p } ^ { - 1 } p ^ { - 2 s } + \\frac { 2 b _ p - 2 } { p ^ { 3 s } } - \\sum _ { i , j = 1 } ^ 3 \\alpha _ { i , p } \\alpha _ { j , p } p ^ { - 4 s } + p ^ { - 6 s } \\right ) \\\\ b _ p & = \\sum _ { i = 1 } ^ 3 \\sum _ { j = 1 } ^ 3 \\frac { \\alpha _ { i , p } } { \\alpha _ { j , p } } . \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} \\mathcal { E } ( k ) : = \\{ E \\in [ 0 , 4 ] : \\ \\ y \\in \\{ 0 , 1 \\} \\ \\ E = g _ k ( E , y ) \\} . \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} \\tilde { F } ' - a ^ { i j } \\tilde { F } _ { : i j } + b ^ i \\tilde { F } _ i + c \\tilde { F } = 0 \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} & ( \\Delta ) = \\sum _ { c / d \\in \\mathcal { C } _ N } \\frac { N } { d ( d , N / d ) } \\textbf { a } _ { \\frac { c } { d } } , \\\\ & ( \\Delta _ N ) = \\sum _ { c / d \\in \\mathcal { C } _ N } \\frac { d } { ( d , N / d ) } \\textbf { a } _ { \\frac { c } { d } } . \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} | u _ k | \\leq \\sup _ { Q ^ j _ k } | u _ k | \\leq \\sup _ { \\partial _ p Q ^ j _ k } | u ^ * | \\leq \\sup _ { \\Omega _ T } | \\psi | = : M \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} \\begin{cases} v _ i ^ * = v _ i + \\omega \\omega \\cdot \\left ( v _ j - v _ i \\right ) \\\\ v _ j ^ * = v _ j - \\omega \\omega \\cdot \\left ( v _ j - v _ i \\right ) \\end{cases} \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} V ( x ) = V \\big ( \\rho ( x ) \\big ) & = \\left ( \\int _ { \\rho ( x ) } ^ \\infty \\frac { d s } { f _ a ^ { n - 1 } ( s ) } \\right ) \\left ( \\int _ 0 ^ { \\rho ( x ) } a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t \\right ) \\\\ & - \\int _ 0 ^ { \\rho ( x ) } \\left ( \\int _ t ^ \\infty \\frac { d s } { f _ a ^ { n - 1 } ( s ) } \\right ) a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t - H + | | \\varphi | | _ \\infty , \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } v _ { \\delta } \\prod _ { i = 1 } ^ { n } f _ { i } ^ { \\delta _ { i } } ( ( A ^ { - 1 } x ) _ { i } ) \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} \\nabla \\tilde H _ \\lambda = ( - 1 ) ^ { | \\lambda | } q ^ { n ' ( \\lambda ) } t ^ { n ( \\lambda ) } \\ ; \\tilde H _ \\lambda , \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} T _ 1 [ X , Y , u , v ] = u ( 1 - v ) . \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\left \\| u - p \\right \\| _ { L ^ 2 ( \\mathcal { B } _ r ^ + ) } ^ 2 = \\sum \\limits _ { k = 5 } ^ { \\infty } | a _ k | ^ 2 \\| p _ k \\| _ { L ^ 2 ( \\mathcal { B } _ r ^ + ) } ^ 2 & \\leq \\sum \\limits _ { k = 5 } ^ { \\infty } | a _ k | ^ 2 r ^ { 1 0 } | \\mathcal { B } _ r ^ + | \\| p _ k \\| _ { L ^ 2 ( \\mathcal { B } _ 1 ^ + ) } ^ 2 \\\\ & \\leq r ^ { 1 0 + 2 n } C ( \\bar { c } ) , \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} F ( z ) = e ^ { - ( B - 2 \\ell A ) z } \\Big ( \\frac { 1 - e ^ { - A z } } { A z } \\Big ) ^ { 2 \\ell } . \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} \\gamma ( 0 ) = S ( 0 , 0 ) , u ' ( 0 ) = \\cos \\theta , v ' ( 0 ) = \\sin \\theta \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } + V _ s \\cdot \\nabla _ { X _ s } \\right ) f _ \\infty ^ { ( s ) } ( t , Z _ s ) = \\ell ^ { - 1 } C _ { s + 1 } ^ 0 f _ \\infty ^ { ( s + 1 ) } ( t , Z _ s ) \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} \\gamma _ j ^ { ' } \\triangleq \\frac { 1 } { \\binom { l - 1 } { p - 1 } } \\sum _ { \\{ i : \\ ; P _ j \\subset L _ i , \\ ; 1 \\leq i \\leq \\binom { n } { l } \\} } \\gamma _ i . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} W _ { 2 n - 1 } ( p , \\psi ) - t W _ { 2 n - 1 } ( q , \\psi ) = ( - 1 ) ^ { n } q ^ { - n } \\left ( q \\psi _ { 2 n - 1 } - t \\psi _ { 2 n } \\right ) \\ ! . \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\frac { 1 - \\delta } { \\bar { \\gamma } } \\sum _ { i = 0 } ^ k \\norm { x _ { i + 1 } - x _ { i } } _ i ^ 2 & \\leq ( 1 - \\delta ) \\sum _ { i = 0 } ^ k \\frac { \\norm { x _ { i + 1 } - x _ { i } } _ i ^ 2 } { \\gamma _ i } \\leq \\sum _ { i = 0 } ^ k \\big ( ( f + g ) ( x _ { i } ) - ( f + g ) ( x _ { i + 1 } ) \\big ) \\\\ & \\leq ( f + g ) ( x _ 0 ) - \\inf _ { k \\in \\N } ( f + g ) ( x _ k ) < + \\infty . \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\aligned & A _ \\alpha ^ * = A _ \\mu ^ * , \\\\ & ( a , \\mu ) \\ ; \\ ; B _ \\alpha ^ * = - ( a , \\alpha ) \\ ; \\ ; B _ \\mu ^ * , \\\\ & ( p , \\mu ) \\ ; \\ ; C _ \\alpha ^ * = - ( p , \\alpha ) \\ ; \\ ; C _ \\mu ^ * , \\\\ & \\theta ^ i _ j - \\theta ^ { i ' } _ { j ' } = \\sum _ k L ^ i _ { j k } ( \\omega ^ k + \\omega ^ { k ' } ) \\endaligned \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} r ( s ) = h \\left ( \\frac { s } { | Q | } \\right ) s \\ge 0 , \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} \\Phi _ { \\lambda , t } ( z _ \\delta , w _ \\delta , t _ \\delta ) = H ( D _ x \\Phi _ \\lambda ( z _ \\delta , w _ \\delta , t _ \\delta ) , z _ \\delta ) \\dot \\xi _ { t _ \\delta } - H ( - D _ y \\Phi _ \\lambda ( z _ \\delta , w _ \\delta , t _ \\delta ) , w _ \\delta ) \\dot \\zeta _ { t _ \\delta } . \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{gather*} ( f \\otimes P ) ( g \\otimes Q ) = \\sum f \\cdot g _ { ( 1 ) } \\otimes { \\boldsymbol \\delta } ( g _ { ( 2 ) } ) ( P ) \\cdot Q f , g \\in S ( \\gg ) , P , Q \\in \\hat { S } ( \\gg ^ * ) . \\end{gather*}"}
-{"id": "1044.png", "formula": "\\begin{align*} q ( x ) = \\sum _ { \\gamma \\in \\Gamma ( k \\pm ) } q _ { \\gamma } e ^ { i \\left \\langle \\gamma , x \\right \\rangle } \\end{align*}"}
-{"id": "10147.png", "formula": "\\begin{align*} d ^ 2 ( x _ { k + 1 } , \\mathrm { A r g } \\min f ) & = \\| x _ { k + 1 } - x _ { k + 1 } ^ \\prime \\| ^ 2 \\leq \\| x _ { k + 1 } - x _ k ^ \\prime \\| ^ 2 \\\\ & = \\| x _ k - h \\cdot \\nabla f ( x _ k ) - x _ k ^ \\prime \\| ^ 2 \\\\ & = d ^ 2 ( x _ k , \\mathrm { A r g } \\min f ) - 2 h \\langle \\nabla f ( x _ k ) , x _ k - x _ k ^ \\prime \\rangle + h ^ 2 \\| \\nabla f ( x _ k ) \\| ^ 2 , \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} A = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} , a , b , c , d \\geq 0 , a ^ 2 > d ^ 2 + b ^ 2 + c ^ 2 . \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} z ' + \\frac { 1 } { t } \\ , z = \\pm \\frac { a } { t } . \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} p _ { n + 1 - r } ( \\sigma , \\varphi ) = p _ { r } ( \\sigma , \\varphi ) - 1 . \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a ; q \\right ) _ { n } q ^ { n ^ { 2 } / 2 } } { \\left ( q , c ; q \\right ) _ { n } } \\left ( - \\frac { c } { a q ^ { 1 / 2 } } \\right ) ^ { n } = \\frac { \\left ( c / a ; q \\right ) _ { \\infty } } { \\left ( c ; q \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "9944.png", "formula": "\\begin{align*} a _ i ( t ) w _ { r + j } = e ^ { ( \\delta _ i - 2 j ) t } w _ { { r + j } } \\delta _ { i } - 2 j \\leq \\delta _ { i } , \\forall \\ , j = 0 , \\dots , l - r . \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot { \\tilde { V } } & = & \\dfrac { \\gamma ^ { - 1 } } { ( 1 - \\gamma ^ { - 1 } V ( x ) ) } \\dot { V } ( x ) \\end{array} \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} X ^ * = \\bigcup _ { n = 0 } ^ { \\infty } X ^ n , \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} C ^ { \\textrm { m a i n } } _ { ( c p + a , d q + b ) } = & \\frac { ( p c ^ 2 + 2 a c - p c ) C _ \\tau } { 2 } + \\left ( c + \\frac { a } { p } \\right ) d C _ \\sigma + \\frac { ( q d ^ 2 + 2 b d - q d ) C _ \\rho } { 2 } \\\\ & + d \\sum _ { l = 0 } ^ { q - 1 } l C _ { \\rho , ( 0 , - l - 1 ) } + c \\sum _ { k = 0 } ^ { p - 1 } k C _ { \\tau , ( - k - 1 , b ) } + c p \\sum _ { l = 0 } ^ { b - 1 } C _ { \\sigma , ( 0 , l ) } . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\tau ( \\mathbf { x } , t ) : = \\int _ { t _ { 0 } } ^ { t } T ( \\mathbf { x } , s ) \\mathrm { d } s . \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} \\bar { Z } _ n ^ * = & 2 ^ { - 1 } \\big ( \\bar { \\tau } _ n \\bar { s } _ n \\big ) ^ { - 2 } \\Bigg [ 2 \\bar { \\tau } _ n \\bar { s } _ n ^ 2 \\Big ( n ^ { - 1 } \\sum _ { i = 1 } ^ { n } \\psi ' ( \\bar { \\epsilon } _ i ) ( G _ i ^ * - \\mu _ { G ^ * } ) \\Big ) \\\\ & - \\bar { \\tau } _ n ^ 2 \\Big ( n ^ { - 1 } \\sum _ { i = 1 } ^ { n } \\psi ^ 2 ( \\bar { \\epsilon } _ i ) [ ( G _ i ^ * - \\mu _ { G ^ * } ) ^ 2 - \\sigma _ { G ^ * } ^ 2 ] \\Big ) \\Bigg ] \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} d ^ m ( f ) = - ( f \\bullet \\mu - ( - 1 ) ^ { m - 1 } \\mu \\bullet f ) . \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} ^ { C \\ ! } D _ { a + } ^ \\alpha x ( t ) : = ( I _ { a + } ^ { m - \\alpha } D ^ m x ) ( t ) , \\hbox { f o r } t \\in ( a , b ] . \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} F ( x ) = \\left [ F ( x ) - F ( y _ { 0 } ) \\right ] + \\left [ F ( y _ { 0 } ) - F ( y _ { 1 } ) \\right ] + \\dots + \\left [ F ( y _ { k - 2 } ) - F ( y _ { k - 1 } ) \\right ] + T ( y _ { k - 1 } ) , \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} \\langle W ^ { \\delta } _ i \\rangle _ t = \\sum _ { k = 1 } ^ { \\infty } ( ( t \\wedge \\tau _ k ^ \\delta ) - ( t \\wedge \\sigma _ k ^ \\delta ) ) , ~ t \\geq 0 , \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} 0 & = \\langle N _ i , D F ( x ) E _ 1 \\rangle + \\langle D _ v N _ i , F ( x ) E _ 1 \\rangle - \\langle F ( x ) N _ i , D _ v E _ 1 \\rangle \\\\ & = \\langle N _ i , D F ( x ) E _ 1 \\rangle - \\langle N _ i , D _ v E _ 2 \\rangle - \\sum _ { j = 1 , 2 } \\langle F ( x ) N _ i , N _ j \\rangle \\langle N _ j , D _ v E _ 1 \\rangle \\\\ & = \\langle N _ i , D F ( x ) E _ 1 \\rangle - \\left ( w _ 2 ^ { i + 2 } ( v ) + \\sum _ { j = 1 , 2 } \\langle F ( x ) N _ i , N _ j \\rangle w _ 1 ^ { j + 2 } ( v ) \\right ) \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} \\mathbf { H ^ * _ n } = \\mathbf { F _ n ^ * } - \\sqrt { n } \\hat { \\sigma } _ n \\mathbf { \\bar { \\Sigma } _ n ^ { - 1 / 2 } } Z ^ * _ n ( ( \\mathbf { E _ * } L _ n ^ * ) ^ { - 1 } \\Delta _ n ^ * ) + R ^ * _ { 4 n } \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} J _ { k _ L } ( n , s ) = O ( p ^ { - \\frac { \\epsilon } { 2 } ( \\log _ { p / q } \\log n ) ^ 2 + o ( ( \\log \\log n ) ^ 2 ) } ) , \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} \\delta ^ { ' } _ { \\mathsf { A c h } } ( \\mu , r ) = ( \\mu M ) \\delta _ { \\mathsf { C a - I A } } + ( 1 - \\mu M ) \\delta _ { \\mathsf { C l - S f } } , \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} H ^ 0 ( V ) : = \\sqrt { - d ^ 2 / d x ^ 2 } + V ( x ) , \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} C _ m = 1 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot \\ldots \\cdot ( 2 m + 1 ) \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} \\zeta ( s , f , D ) = T r ( f D ^ { - s } ) \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} \\left [ T _ { ( A , \\alpha ) } , T _ { ( B , \\beta ) } \\right ] = C _ { ( A , \\alpha ) ( B , \\beta ) } ^ { \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ ( C , \\gamma ) } T _ { ( C , \\gamma ) } . \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} k ( q ) = \\sqrt { 1 - ( k ' ( q ) ) ^ { 2 } } = \\frac { \\vartheta _ { 2 } ^ { 2 } \\left ( 0 \\mid q \\right ) } { \\vartheta _ { 3 } ^ { 2 } \\left ( 0 \\mid q \\right ) } , \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} c _ { \\sigma } ^ { + } ( M ( \\chi ) ) \\sim ( 2 \\pi i ) ^ { - \\lceil n / 2 \\rceil w ( \\chi ) } G _ { \\sigma } ( \\chi ) ^ { \\mathbf { r } _ { \\sigma } } \\delta _ { \\sigma } ( M ) a _ { \\sigma } ^ { * } ( \\chi ) Q _ { \\sigma } ( \\chi ) ^ { \\mathbf { r } _ { \\sigma } - \\lceil n / 2 \\rceil } \\prod _ { j = 1 } ^ { \\mathbf { s } _ { \\sigma } } Q _ { j , \\sigma } , \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} P ( E _ 2 ) = p _ a ^ 8 = ( p _ a ^ 2 ) ^ 4 = ( P ( E _ 1 ) ) ^ 4 P ( E _ 4 ) = ( 1 - p _ a ) ^ 8 = ( ( 1 - p _ a ) ^ 2 ) ^ 4 = ( P ( E _ 3 ) ) ^ 4 . \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} w ( T - t ) - w ^ n ( T - t ) & = \\int _ 0 ^ t P _ p ( r ) \\big ( { \\nabla u ( r + T - t ) b ( r + T - t ) } \\\\ & - { \\nabla u ^ n ( r + T - t ) b ^ n ( r + T - t ) } \\big ) \\d r \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} & { \\bf U } _ { 2 , 1 } ^ p { \\bf H } _ { r , 1 } = { \\bf U } _ { 1 , 2 } ^ p { \\bf H } _ { r , 2 } , \\\\ & { \\bf U } _ { 3 , 1 } ^ p { \\bf H } _ { r , 1 } = { \\bf U } _ { 1 , 3 } ^ p { \\bf H } _ { r , 3 } , \\\\ & { \\bf U } _ { 3 , 2 } ^ p { \\bf H } _ { r , 2 } = { \\bf U } _ { 2 , 3 } ^ p { \\bf H } _ { r , 3 } , \\\\ & { \\bf U } _ { 3 , 1 } ^ { c } { \\bf H } _ { r , 1 } + { \\bf U } _ { 1 , 2 } ^ c { \\bf H } _ { r , 2 } = { \\bf U } _ { 2 , 3 } ^ { c } { \\bf H } _ { r , 3 } . \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} \\nu _ k ^ { \\rm C G } & \\equiv \\frac { \\alpha _ { k + 1 } \\beta _ { k + 1 } } { \\alpha _ k } , & & k = 1 , 2 , \\ldots , \\\\ \\pi _ k ^ { \\rm C G } & \\equiv \\alpha _ { k + 1 } , & & k = 0 , 1 , 2 \\ldots \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} S ( \\psi , g ) = \\langle { \\psi } , L ( \\Lambda ' , \\psi , g ) { \\psi } \\rangle \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} ( i , 0 ) ( j , 1 ) ( i , 2 ) ( j , 3 ) \\ldots ( i , n - 3 ) ( j , n - 2 ) ( k , n - 1 ) , 0 \\leq i , j \\leq w - 1 , k = i \\circ j . \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} A = \\left [ \\begin{array} { c c c c } A _ { 1 1 } & A _ { 1 2 } & \\cdots & A _ { 1 n } \\\\ A _ { 2 1 } & A _ { 2 2 } & \\cdots & A _ { 2 n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ A _ { n 1 } & A _ { n 2 } & \\cdots & A _ { n n } \\end{array} \\right ] . \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} \\mathcal { J } ( \\mathbf { u } ) = \\frac { 1 } { 2 } \\int _ { G } \\big | \\tilde { \\mathbf { u } } ^ { 0 } - \\mathbf { u } \\big | ^ { 2 } \\mathrm { d } \\mathbf { x } + \\lambda \\mathcal { P } ( \\mathbf { u } ) \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} ( \\mathcal { G } _ { D } u _ { 0 } ) _ t ( x ) & = \\int _ { B _ R ( 0 ) } u _ 0 ( y ) p _ D ( t , x , y ) \\ , \\d y \\\\ & \\geq \\int _ { B _ { R - \\epsilon } ( 0 ) } u _ 0 ( y ) p _ D ( t , x , y ) \\ , \\d y \\\\ & \\geq c _ 1 e ^ { - \\mu _ 1 t } , \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} \\mbox { p r o x } _ { \\phi } ( y ; \\lambda , a ) : = \\arg \\min _ { x \\in \\mathbb { R } } \\left \\lbrace \\frac { 1 } { 2 } ( y - x ) ^ 2 + \\lambda \\phi ( x ; a ) \\right \\rbrace . \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} \\Delta _ 0 ( r ^ \\alpha \\varphi ) & = r ^ { \\alpha - 2 } A _ \\alpha \\varphi , \\\\ \\Delta _ 0 ^ 2 ( r ^ \\alpha \\varphi ) & = \\Delta _ 0 ( r ^ { \\alpha - 2 } A _ \\alpha \\varphi ) = r ^ { \\alpha - 4 } A _ { \\alpha - 2 } A _ \\alpha \\varphi , \\\\ \\Delta _ 0 ^ 3 ( r ^ \\alpha \\varphi ) & = r ^ { \\alpha - 6 } A _ { \\alpha - 4 } A _ { \\alpha - 2 } A _ \\alpha \\varphi . \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} \\mu ^ { \\# } & = g \\circ u - ( \\lambda _ { i n t } - \\mu _ { i n t } ) \\ , , \\\\ \\nu ^ { \\# } & = \\nu - ( \\lambda _ { b d } - \\tau _ { b d } ) . \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} b _ 1 ( \\pi ) + \\dots + b _ { i + 1 } ( \\pi ) = b _ 1 ( \\pi ' ) + \\dots + b _ { i + 1 } ( \\pi ' ) , \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} \\lambda \\mathbf { \\centerdot } a : = a . \\overline { \\lambda } \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} M _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\triangleq \\lim \\limits _ { \\tau \\downarrow 1 } \\frac { \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) } { 2 \\pi } \\ , M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} - L u + g \\circ u & = \\mu _ n \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega \\\\ u & = \\nu _ n \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} 1 & = \\norm { f _ j } = \\norm { ( f _ j - h ) ^ + + h - ( f _ j - h ) ^ - } \\\\ & \\ge \\norm { ( f _ j - h ) ^ + + h } - \\norm { ( f _ j - h ) ^ - } = \\norm { ( f _ j - h ) ^ + } + \\norm { h } - \\norm { ( f _ j - h ) ^ - } , \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t \\mathcal U + ( \\mathcal U \\cdot \\nabla ) \\mathcal U + \\nabla p - \\nu _ h \\Delta _ H \\mathcal U - \\nu _ z \\partial _ z ^ 2 \\mathcal U + f _ 0 k \\times \\mathcal U = \\theta e _ 3 , \\\\ \\nabla \\cdot \\mathcal U = 0 , \\\\ \\partial _ t \\theta + \\mathcal U \\cdot \\nabla \\theta - \\kappa _ h \\Delta _ H \\theta - \\kappa _ z \\partial _ z ^ 2 \\theta = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} f _ N ^ { ( s ) } ( t , Z _ s ) = \\int _ { \\mathbb { R } ^ { 2 d ( N - s ) } } f _ N ( t , Z _ N ) d z _ { s + 1 } \\dots d z _ N \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} \\partial _ { t } u & = \\mathrm { d i v } \\ , \\big ( \\mathbf { g } ( \\nabla _ { \\sigma } u ) \\nabla u \\big ) ( 0 , \\infty ) \\times G , \\\\ \\mathbf { g } ( \\nabla _ { \\sigma } u ) \\cdot \\mathbf { n } & = 0 ( 0 , \\infty ) \\times \\Gamma , \\\\ u ( 0 , \\cdot ) & = \\tilde { u } ^ { 0 } \\Omega , \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { 8 } f _ t ( p ' ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ { p ' } | } . \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} J = \\int _ 0 ^ { \\pi / 2 } + \\int _ { \\pi / 2 } ^ { 3 \\pi / 2 } + \\int _ { 3 \\pi / 2 } ^ { 2 \\pi } = J _ 1 + J _ 2 + J _ 3 , \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} u _ t = & A _ 1 u _ 3 + A _ 2 , \\\\ u _ t = & ( A _ 1 u _ 3 + A _ 2 ) ^ { - 2 } + A _ 3 , \\\\ u _ t = & ( 2 A _ 1 u _ 3 + A _ 2 ) ( A _ 1 u _ 3 ^ 2 + A _ 2 u _ 3 + A _ 3 ) ^ { - 1 / 2 } + A _ 4 . \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} f _ { m + 1 } ^ q = f _ m , g _ { m + 1 } ^ q = g _ m , f _ 1 - g _ 1 \\in A \\setminus \\mathfrak { m } A . \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} P = \\bar { P } + \\sigma ^ 2 . \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} Z ( t , g ) = T r ( g e ^ { - t D } ) \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} \\| T f \\| _ { L ^ p } & = \\left ( \\int _ { \\R ^ n } \\Big ( | T f | ^ { p _ 0 } R ( h ) ^ { - 1 } R ( h ) \\Big ) ^ { \\frac { p } { p _ 0 } } \\ , d x \\right ) ^ { \\frac { 1 } { p } } \\\\ & \\leq \\left ( \\int _ { \\R ^ n } | T f | ^ { p _ 0 } R ( h ) ^ { - 1 } \\ , d x \\right ) ^ { \\frac { 1 } { p _ 0 } } \\left ( \\int _ { \\R ^ n } R ( h ) ^ { \\frac { p } { p _ 0 } \\left ( \\frac { p _ 0 } { p } \\right ) ' } \\ , d x \\right ) ^ { \\frac { 1 } { p ( \\frac { p _ 0 } { p } ) ' } } \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\sum _ { y _ { t } } \\Big ( \\log \\Big ( \\frac { r _ t ( x _ t | y ^ { t - 1 } _ { t - M } , y _ t ) } { { \\pi } _ { t } ( x _ t | y ^ { t - 1 } _ { t - J } ) } \\Big ) + C _ { t + 1 } ( y ^ t _ { t + 1 - J } ) \\Big ) q _ t ( y _ t | y ^ { t - 1 } _ { t - M } , x _ t ) - s \\gamma _ t ( x _ t , y ^ { t - 1 } _ { t - N } ) = 1 - \\lambda _ t ( y ^ { t - 1 } _ { t - J } ) , ~ \\forall { x _ t } \\in { \\cal X } _ t . \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} \\widetilde { V } _ p ( u , v , \\lambda ; q ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { 1 / 2 + u + v } } K l ( l , n p ; q ) J _ { 2 \\lambda - 1 } \\left ( 4 \\pi \\frac { \\sqrt { l n p } } { q } \\right ) . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\psi } g ( x ) : = \\sum _ { y \\in \\theta ^ { - 1 } \\left \\{ x \\right \\} } \\exp \\left ( \\psi ( y ) \\right ) \\ , g ( y ) . \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} \\begin{cases} L = \\pi \\sqrt { \\dfrac { ( 1 - a ^ 2 b ) \\alpha ( k , l , m , n , s ) } { 3 r } } , \\\\ \\\\ \\xi _ 0 = - \\dfrac { \\pi } { 3 } ( 5 k + 4 l + 3 m + 2 n + s ) , \\\\ \\\\ p = \\sqrt { \\dfrac { ( 1 - a ^ 2 b ) \\xi _ 0 \\xi _ 1 \\xi _ 2 \\xi _ 3 \\xi _ 4 \\xi _ 5 } { c } } , \\end{cases} \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} C _ { \\left ( A , i \\right ) \\left ( B , j \\right ) } { } ^ { \\left ( C , k \\right ) } & = \\left ( K _ { i j } { } ^ { k } - K _ { i j } { } ^ { k + n } \\right ) C _ { A B } { } ^ { C } \\\\ & = \\left ( K _ { j i } { } ^ { k } - K _ { j i } { } ^ { k + n } \\right ) C _ { A B } { } ^ { C } \\\\ & = - \\left ( K _ { j i } { } ^ { k } - K _ { j i } { } ^ { k + n } \\right ) C _ { B A } { } ^ { C } \\\\ & = - C _ { \\left ( B , j \\right ) \\left ( A , i \\right ) } { } ^ { \\left ( C , k \\right ) } , \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} p _ \\alpha = p _ \\alpha ^ \\top + p _ \\alpha ^ \\bot , \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} f _ { n } = f _ { n } ( z ) : = ( - 1 ) ^ { n } \\alpha ^ { - n } q ^ { \\frac { 1 } { 2 } n ( n + 1 ) } \\ , _ { 1 } \\tilde { \\phi } _ { 1 } ( 0 ; z ^ { - 1 } \\alpha ^ { - 1 } q ^ { n + 1 } ; q , z \\alpha ^ { - 1 } q ^ { n + 1 } ) \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\Gamma ( \\alpha ) : = \\int _ 0 ^ \\infty \\tau ^ { \\alpha - 1 } \\exp ( - \\tau ) \\ ; d \\tau , \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} T ^ 1 _ 1 = T _ 2 ^ 1 = 1 , T ^ 2 _ 1 = - T _ 2 ^ 2 = 1 , \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} - F _ { \\ell - d , \\ell } ( - z ; \\tau ) = \\sum _ { a \\geq 0 } \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\zeta ^ { d } } { 1 - \\zeta ^ { \\ell } } \\right ) \\frac { ( 2 \\pi i \\tau ) ^ a } { a ! } . \\end{align*}"}
-{"id": "9871.png", "formula": "\\begin{align*} { \\rm E } \\left [ { \\sum \\limits _ { i = p } ^ r \\sum \\limits _ { i = 1 } ^ m { \\sum \\limits _ { j = 1 } ^ n { \\sum \\limits _ { k = 1 } ^ l { { \\xi ^ { t p } _ { i j k } } } { x ^ { p } _ { i j k } } } } } \\right ] = \\sum \\limits _ { i = p } ^ r \\sum \\limits _ { i = 1 } ^ m { \\sum \\limits _ { j = 1 } ^ n { \\sum \\limits _ { k = 1 } ^ l { { x ^ { p } _ { i j k } } } } } { \\rm E } \\left [ { { \\xi ^ { t p } _ { i j k } } } \\right ] , t = 1 , 2 , . . . , K . \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} H ( u ) : = h \\left ( f ( u ) \\right ) + c g ( u ) = 0 . \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{gather*} Z _ T ( s , f , \\chi ) = \\nu _ T ( \\overline { f } , \\chi ) + \\sigma _ T ( \\overline { f } , \\chi ) \\frac { ( 1 - q ^ { - 1 } ) q ^ { - s } } { ( 1 - q ^ { - 1 - s } ) } \\\\ + \\int \\limits _ { S _ T ( f ) } \\chi ( a c \\ f ( x , y ) ) \\ | f ( x , y ) | ^ s \\ | d x d y | , \\end{gather*}"}
-{"id": "3444.png", "formula": "\\begin{align*} \\theta ^ { k - l } ( i s ) \\theta ^ l ( s t ) \\theta ^ { k - l } ( i s ) & = \\theta ^ k ( i t ) \\\\ ( j t ) \\theta ^ k ( i t ) ( j t ) & = \\theta ^ k ( i j ) , \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} \\tilde { u } ( 0 ) - \\tilde { y } ( 0 ) = i \\Phi h , \\tilde { u } ^ { \\prime } ( 0 ) - \\tilde { y } ^ { \\prime } ( 0 ) = \\Phi \\Phi ^ * \\tilde { u } ( 0 ) - \\Phi A _ 1 h = \\Phi \\Phi ^ * \\tilde { y } ( 0 ) - \\Phi A _ 1 ^ * h . \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} L _ t \\cdot L _ u \\subseteq L _ { t u } , L _ t ^ { - 1 } = L _ { t ^ * } , \\bigcup _ { t \\in S } L _ t = L ^ 1 . \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} b ^ q = \\det ( \\overline { \\alpha } _ i ^ { q ^ j } ) = ( - 1 ) ^ { n - 1 } b . \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\star \\partial f = ( f ' _ 1 , \\cdots , f ' _ n ) . \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} \\xi ^ { ( n ) } ( m , \\mu ) = ( 1 - B _ { \\mu } ) ^ n \\xi ( m ) = \\sum _ { l = 0 } ^ n ( - 1 ) ^ l { { n \\choose l } } \\xi ( m - l \\mu ) , \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{gather*} P _ { n _ { k + 1 } - 1 } ( x ) = \\gamma _ { k } P _ { n _ { k } } ( x ) \\ , \\end{gather*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\chi _ { j , R } ( u ) = \\psi \\big ( ( - 1 ) ^ j u / R \\big ) , \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} \\gamma v ^ \\gamma + \\abs { p + D _ y v ^ \\gamma } + { f } ( y ) \\cdot \\xi = 0 . \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} f _ 3 = \\sin ( n x ) \\sin ( n y ) , f _ 4 = \\cos ( n x ) \\cos ( n y ) , f _ 6 = \\cos ( n x ) \\sin ( n y ) \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{gather*} \\big \\langle Q ^ { m - n } v _ 0 , \\psi ^ { + } ( w _ 1 ) \\cdots \\psi ^ { + } ( w _ m ) \\psi ^ { - } ( y _ 1 ) \\cdots \\psi ^ { - } ( y _ n ) v _ 0 \\big \\rangle = \\sum _ { \\sigma \\in \\mathfrak { S } _ { n } } ( - 1 ) ^ { \\abs { \\sigma } } \\frac { y _ { \\sigma ( n ) } ^ { 0 } y _ { \\sigma ( n - 1 ) } \\cdots y _ { \\sigma ( m + 1 ) } ^ { n - m - 1 } } { \\prod \\limits _ { i = 0 } ^ { m - 1 } ( w _ { m - i } - y _ { \\sigma ( i + 1 ) } ) } . \\end{gather*}"}
-{"id": "5395.png", "formula": "\\begin{align*} - 2 s z + t q = 0 , ( - 2 s u + t l ) ^ 2 = 1 , w = 0 . \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} c _ 1 ( \\pi ) + \\dots + c _ { i - 1 } ( \\pi ) = c _ 1 ( \\pi ' ) + \\dots + c _ { i - 1 } ( \\pi ' ) \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} \\phi _ s ( i ) = \\left \\lbrace \\begin{array} { l c l } i + 1 & & i \\leq ( s - 3 ) , \\ , i \\\\ i - 1 & & i \\leq ( s - 3 ) , \\ , i \\\\ i + 1 & & i = s - 2 \\\\ i - 2 & & i = s - 1 , \\ , s \\\\ i - 1 & & i = s - 1 , \\ , s \\\\ i & & s \\leq i \\leq x - 1 \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} T _ F = T _ E \\frac { B \\min \\{ M , K \\} } { M C _ F } . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} - d _ 1 \\rho _ { j } \\Delta w _ { \\rho _ { j } } = & \\rho _ { j } w _ { \\rho _ { j } } g \\left ( \\rho _ { j } w _ { \\rho _ { j } } \\right ) \\left [ f \\left ( \\rho _ { j } w _ { \\rho _ { j } } \\right ) - v _ { \\rho _ { j } } \\right ] \\\\ \\ge & \\rho _ { j } w _ { \\rho _ { j } } g \\left ( \\rho _ { j } w _ { \\rho _ { j } } \\right ) \\left [ f \\left ( \\rho _ { j } w _ { \\rho _ { j } } \\right ) - v _ \\epsilon \\right ] \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} \\int _ M K = 2 \\pi \\left ( \\chi ( M ) + \\sum _ { p \\in B } m ( p ) \\right ) , \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} \\nu _ j ( n , s ) = ~ & ( j - \\psi ( n ) ) ^ 2 / 2 + ( j - \\psi ( n ) ) ( s + \\log _ { 1 / p } ( 1 + ( p / q ) ^ s ) + \\psi ( n ) + 1 ) \\\\ & - \\log _ { 1 / p } n \\log _ { 1 / p } ( 1 + ( p / q ) ^ s ) + \\psi ( n ) ^ 2 / 2 + o ( \\psi ( n ) ^ 2 ) . \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} H ^ { i } \\big ( \\pi _ { 1 } ( X ) ; \\ , ^ { \\omega } \\ ! \\Lambda \\big ) = H ^ { i } ( \\widetilde { X } ; \\ , ^ { \\omega } \\ ! \\Lambda ) \\cong H _ { n - i } ( \\widetilde { X } ; \\Lambda ) = 0 \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} g ( x ( t ) ) = g ( x ( \\tau ) ) + \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { m ! } \\frac { d ^ m g ( x ( t ) ) } { d t ^ m } | _ { t = \\tau } ( t - \\tau ) ^ m \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} M _ { n } = ( { \\mathrm { E } } [ \\Vert X \\Vert _ { \\infty } ^ { q } ] ) ^ { 1 / q } . \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} \\beta _ { M , N } ( a , \\ , b ) \\overset { { \\rm i n \\ , l a w } } { = } & \\prod \\limits _ { k = 0 } ^ \\infty \\beta _ { M - 1 , N } ( \\hat { a } _ i , \\ , b _ 0 + k a _ i , \\ , \\ , b _ 1 , \\cdots , b _ N ) , \\\\ \\beta _ { M , N } ( a , \\ , b ) \\overset { { \\rm i n \\ , l a w } } { = } & \\prod \\limits _ { n _ 1 , \\cdots , n _ M = 0 } ^ \\infty \\beta _ { 0 , N } ( b _ 0 + \\Omega , \\ , \\ , b _ 1 , \\cdots , b _ N ) . \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{gather*} e = \\frac { p } { 2 ( p h c - s ) } , a = \\frac { p h c } { p h c - s } , b = \\frac { p } { 2 } \\bigg [ \\frac { c ( c - 1 ) + s ^ 2 } { ( p h c - s ) ( 1 - c ) } \\bigg ] , \\\\ c = \\frac { 1 } { 4 } \\bigg [ \\frac { \\exp ( - p h ) ( 1 - c ) + s ( \\exp ( - p h ) - 1 ) } { ( p h c - s ) ( 1 - c ) } \\bigg ] , d = \\frac { 1 } { 4 } \\bigg [ \\frac { \\exp ( p h ) ( c - 1 ) + s ( \\exp ( p h ) - 1 ) } { ( p h c - s ) ( 1 - c ) } \\bigg ] . \\end{gather*}"}
-{"id": "7703.png", "formula": "\\begin{align*} \\Gamma _ w \\cap B _ { 1 } ' : = \\Gamma _ { 3 / 2 } ( w ) \\cup \\bigcup \\limits _ { \\kappa \\geq 2 } \\Gamma _ { \\kappa } ( w ) , \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} \\norm { \\mathbf { u } } ^ 2 _ { m , s , \\tau } = \\sum _ { j = 0 } ^ m \\norm { u _ j } _ { m - j + s , \\tau } ^ 2 , \\mathbf { u } = ( u _ 0 , \\dots , u _ m ) . \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} ( \\iota \\otimes \\alpha ) \\alpha = ( \\Delta \\otimes \\iota ) \\alpha . \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} S ^ \\nu _ f ( n ) \\overline { S ^ \\nu _ g ( n ) } = \\frac { 1 } { 2 } \\left ( | S ^ \\nu _ { h _ 1 } ( n ) | ^ 2 + i | S ^ \\nu _ { h _ 2 } ( n ) | ^ 2 - ( 1 + i ) \\left ( | S ^ \\nu _ f ( n ) | ^ 2 + | S ^ \\nu _ g ( n ) | ^ 2 \\right ) \\right ) . \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} \\theta _ { s , a ^ 2 } ^ 2 = \\big ( \\bar { r } ^ 2 ( s , a ^ 2 ) - \\bar { r } ^ 2 ( s , a _ s ^ 2 ) \\big ) - \\beta \\left ( \\sum _ { s ' \\in S } p _ { s ' } \\bar { r } ^ 2 ( s ' , a _ { s ' } ^ 2 ) - \\sum _ { s ' \\in S } p ^ 2 ( s ' | s , a ^ 2 ) \\bar { r } ^ 2 ( s ' , a _ { s ' } ^ 2 ) \\right ) , \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { L _ { p } ( \\mathcal { O } ; F ) } : = \\left ( \\int _ { \\mathcal { O } } \\| u ( x ) \\| _ { F } ^ { p } d x \\right ) ^ { 1 / p } < \\infty . \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\ell = [ x _ { 0 } : x _ { 1 } ] \\\\ \\ell ' = [ x _ { 2 } : x _ { 3 } ] \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} - \\mathrm { s i g n } ( x _ { k } ^ { r } ) = \\mathrm { s i g n } ( x _ { k } x _ { i _ { 2 } } \\cdots x _ { i _ { r } } ) \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} q _ 0 = ( 3 / 2 ) \\sqrt { 3 } a ( 1 - a ^ 2 ) , q _ 1 = ( 3 / 2 ) \\sqrt { 3 } ( 1 - a ^ 2 ) ( 1 - 3 a ^ 2 ) . \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} \\Re \\Big \\{ \\sum _ { n = 1 } ^ { \\infty } z _ n ^ m \\Big \\} \\leq \\frac { 1 } { \\alpha ^ m } - \\frac { 1 } { ( \\alpha + 1 - \\beta _ 1 ) ^ { 2 m } } + \\Re \\Big \\{ \\frac { \\delta ( \\chi ) } { ( \\alpha + i \\gamma ' ) ^ { 2 m } } - \\frac { \\delta ( \\chi ) } { ( \\alpha + 1 + i \\gamma ' - \\beta _ 1 ) ^ { 2 m } } \\Big \\} \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} F _ 1 = F U \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; F _ 2 = F V , \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} \\partial _ t \\tilde { h } & = \\partial _ t h + \\partial _ z h \\left ( - \\bar { U } - t \\partial _ t \\bar { U } \\right ) + \\partial _ v h \\partial _ t \\bar { U } \\\\ & = \\partial _ t h - v \\partial _ z h + \\nu b ( \\partial _ v - t \\partial _ z ) h . \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} \\hat { W } ^ { [ { \\sf d } ] } _ { k i } = \\psi _ { k i } ( y _ { k i } ^ { [ { \\sf d } ] } [ 1 ] , \\cdots , y _ { k i } ^ { [ { \\sf d } ] } [ n ] ) , \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} G ( z ) & = z ^ { 2 \\gamma - p } \\left ( \\sum _ { h \\geq 0 } z ^ h \\sum _ { s = 0 } ^ { \\lfloor \\frac { h } { 2 } \\rfloor } N _ { \\mathcal L } ( h - 2 s , \\ell ) \\binom { s + n - 2 } { n - 2 } - \\Upsilon _ { p - 2 \\gamma - 1 } ^ { ( \\ell ) } ( z ) \\right ) \\\\ & = z ^ { 2 \\gamma - p } \\left ( \\frac { \\vartheta _ { \\mathcal L } ^ { ( \\ell ) } ( z ) } { ( 1 - z ^ 2 ) ^ { n - 1 } } - \\Upsilon _ { p - 2 \\gamma - 1 } ^ { ( \\ell ) } ( z ) \\right ) . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} \\psi _ { x t } = A _ 1 \\left ( - \\beta \\psi _ { x x } + \\left ( \\beta _ { x x } - \\frac { 1 } { M } \\right ) \\psi \\right ) + A _ 2 \\left ( - \\beta \\psi _ { x x } - 2 \\beta _ x \\psi _ x - ( 1 + \\beta _ { x x } ) \\right ) , \\\\ \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} M = \\mathbb { S } ^ 1 \\times \\left [ - \\rho , \\rho \\right ] , \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} \\alpha \\circ \\mu ( f ) = \\int _ { L ^ 0 } \\int _ { H ^ { F ^ 0 ( x ) } } f ( h , x ) \\ , \\dd \\mu ^ { F ^ 0 ( x ) } ( h ) \\ , \\dd \\alpha ( x ) \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} \\Gamma ^ { 3 , \\varepsilon } _ { \\alpha 3 } = \\Gamma ^ { p , \\varepsilon } _ { 3 3 } & = 0 \\ \\textrm { i n } \\ \\bar { \\Omega } ^ \\varepsilon , \\\\ A ^ { \\alpha \\beta \\sigma 3 , \\varepsilon } = A ^ { \\alpha 3 3 3 , \\varepsilon } = B ^ { \\alpha \\beta \\sigma 3 , \\varepsilon } & = B ^ { \\alpha 3 3 3 , \\varepsilon } = 0 \\ \\textrm { i n } \\ \\bar { \\Omega } ^ \\varepsilon , \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} \\norm { T ^ n f - h } & \\le \\norm { T ^ n f _ n - h } + M \\varepsilon = \\norm { h \\land T ^ n f - h } + M \\varepsilon \\\\ & = \\norm { ( T ^ n f - h ) ^ - } + M \\varepsilon \\le ( M + 1 ) \\varepsilon . \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} F \\left ( x \\right ) - F \\left ( 0 \\right ) = \\sum _ { i = 1 } ^ { 3 } f _ { i } \\left ( \\alpha _ { i } \\right ) w _ { i } . \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} J _ { ( i , 2 ) } = - J _ { ( i , 0 ) } , J _ { ( i , 3 ) } = - J _ { ( i , 1 ) } \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\nu = \\frac { ( \\nabla u , - 1 ) } { \\sqrt { 1 + | \\nabla u | ^ 2 } } . \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} I ^ { \\alpha } _ t I ^ { \\beta } _ t \\varphi = I ^ { \\alpha + \\beta } _ t \\varphi , \\ , \\ , t \\leq T . \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} u ( x ) = \\sum a _ n \\phi _ n ( x ) \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\varphi = \\{ \\log \\sinh ( \\tfrac { r } { 2 } ) - \\log \\cosh ( \\tfrac { r } { 2 } ) \\} \\big | _ { r _ 2 } ^ u , \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} \\norm { c ( x ) } = \\norm { f ( x ) b ( x ) } = \\abs { f ( x ) } \\norm { b ( x ) } = f ( x ) \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} \\deg _ { H ( u ) } ( v ) = \\deg _ { H } ( u , v ) . \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\overline G ( x ) = o \\big ( \\overline H ( b x ) \\big ) \\ \\ b > 0 . \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c } ( \\lambda E - A ) ^ { - 1 } B u \\\\ u \\end{array} \\right ] \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} \\theta _ { 1 2 } ( 1 ) ( h \\wedge h ^ \\prime ) = 2 h _ 0 h _ 3 - 2 h _ 1 h _ 2 \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} H ( D _ y v , y ) \\xi _ 1 + f ( y ) \\xi _ 2 = \\lambda . \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} \\Re h _ \\mu ^ 0 ( \\lambda ) & = \\sum _ { w \\in W _ \\R } \\Re h _ 0 ( w \\lambda - \\mu ) \\\\ & \\geqslant \\Re h _ 0 ( \\lambda - \\mu ) - C \\delta \\geqslant 1 / 4 . \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} & { \\bf H } _ { 1 , r } { \\bf V } _ { 1 , 2 } ^ p = { \\bf H } _ { 2 , r } { \\bf V } _ { 2 , 1 } ^ p \\triangleq { \\bf B } _ 1 , \\\\ & { \\bf H } _ { 1 , r } { \\bf V } _ { 1 , 3 } ^ p = { \\bf H } _ { 3 , r } { \\bf V } _ { 3 , 1 } ^ p \\triangleq { \\bf B } _ 2 , \\\\ & { \\bf H } _ { 2 , r } { \\bf V } _ { 2 , 3 } ^ p = { \\bf H } _ { 3 , r } { \\bf V } _ { 3 , 2 } ^ p \\triangleq { \\bf B } _ 3 , \\\\ & { \\bf H } _ { 1 , r } { \\bf V } _ { 1 , 2 } ^ { c } + { \\bf H } _ { 2 , r } { \\bf V } _ { 2 , 3 } ^ { c } = { \\bf H } _ { 3 , r } { \\bf V } _ { 3 , 1 } ^ { c } , \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} A _ \\alpha ^ * = - \\sqrt { 2 } \\begin{pmatrix} S ^ a _ { \\alpha p } \\end{pmatrix} , B _ \\alpha ^ * = - 1 / \\sqrt { 2 } \\begin{pmatrix} S ^ a _ { \\alpha \\mu } \\end{pmatrix} , C _ \\alpha ^ * = - 1 / \\sqrt { 2 } \\begin{pmatrix} S ^ p _ { \\alpha \\mu } \\end{pmatrix} . \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} \\theta _ { q ^ { 2 } } \\left ( e ^ { - y - s } \\right ) = 0 , \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i = \\big ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } - B _ { i J K j } \\dot { u } _ { j } ) _ { , K } \\big ) _ { , J } - E _ { i j } \\dot { u } _ { j } . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} \\delta ^ l = \\frac { 1 } { ( 1 - \\sigma ) } \\frac { 2 \\zeta _ i } { l } \\mbox { a n d } \\alpha ^ l = \\frac { 2 } { l + 1 } \\in ( 0 , 1 ] , \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} h : = { \\mathcal { P } } ( \\tilde { h } ) \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} u _ { \\max } = - u ^ * _ { \\min } \\forall t \\in [ t _ \\delta , T ^ * ) , \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} f _ { 1 } ( u , v ) & = A u \\left ( 1 - \\frac { u } { K } \\right ) - \\frac { B u v } { 1 + E u } , \\\\ f _ { 2 } ( u , v ) & = \\frac { C u v } { 1 + E u } - D v , \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} Y _ A \\ ; : \\ ; \\begin{cases} F ( x _ 0 , \\dotsc , x _ n ) = 0 \\\\ \\nabla { F } ( A ) \\cdot ( x _ 0 , \\dotsc , x _ n ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} p _ { 2 n - 1 } = q _ { 2 n } = 0 , p _ { 2 n } = q _ { 2 n + 1 } = ( - q ) ^ { n } , \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { i = 0 \\\\ \\mbox { \\scriptsize $ m + i $ e v e n } } } ^ m ( - 1 ) ^ i { m \\choose i } \\sum \\limits _ j { \\frac { m + i } { 2 } \\brack j } x ^ { 2 j - 1 } + \\sum \\limits _ { \\substack { i = 0 \\\\ \\mbox { \\scriptsize $ m + i $ o d d } } } ^ m ( - 1 ) ^ i { m \\choose i } x ^ { m + i } = 0 , \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} f _ { n } & = z ^ { n } \\left [ \\frac { \\theta _ { q } ( \\alpha z ) } { ( z ^ { 2 } ; q ) _ { \\infty } } + o ( 1 ) \\right ] \\ ! , \\\\ g _ { n } & = z ^ { - n } \\left [ ( q z ^ { 2 } ; q ) _ { \\infty } + o ( 1 ) \\right ] \\ ! , \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} \\Delta ( \\sum _ i ( H _ i , P _ i ) ) = \\sum _ i \\Delta ( H _ i ) \\star \\Delta ( P _ i ) , \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} N _ { a , b , c } ( 0 ) & = M _ { a , b , c - 1 } , & N _ { a , b , c } ( b ) & = M _ { a - 1 , b , c } . \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} \\begin{gathered} \\sup _ { 0 \\leq s \\leq t } | X _ s | \\leq \\left ( 1 + t ^ H \\log ^ 2 t \\right ) \\zeta . \\end{gathered} \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} g ^ * = \\begin{cases} g _ 1 & \\ p > \\frac { 1 } { 3 } \\\\ g _ 2 & \\ p < \\frac { 1 } { 3 } \\\\ \\big \\{ ( q , 1 - q ) : 0 \\leq q \\leq 1 \\big \\} & \\ p = \\frac { 1 } { 3 } . \\end{cases} \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} p _ 1 [ G ] = G , p _ m [ p _ n [ G ] ] = p _ { m n } [ G ] \\quad ( G \\in \\Lambda , \\ ; m , n \\in \\Z _ { > 0 } ) . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} ^ { \\mathfrak { A } } ( M \\cup Y ) = \\{ f ( y ) \\ | \\ y \\in [ Y ] ^ { < \\omega } f \\in M \\cap { } ^ { [ \\eta ] ^ { < \\omega } } H _ \\theta \\} . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} \\begin{aligned} & \\limsup _ { N \\rightarrow \\infty } \\left \\Vert \\left ( f _ N ^ { ( s ) } ( 0 ) - \\int _ { \\mathcal { P } ( \\mathbb { R } ^ { 2 d } ) } h _ N ^ { ( s ) } ( 0 ; h _ 0 ) d \\pi ( h _ 0 ) \\right ) \\mathbf { 1 } _ { Z _ s \\in \\mathcal { G } _ s \\cap \\hat { \\mathcal { U } } _ s ^ { \\eta ( \\varepsilon ) } } \\mathbf { 1 } _ { E _ s ( Z _ s ) \\leq R ^ 2 } \\right \\Vert _ { L ^ \\infty _ { Z _ s } } \\\\ & = 0 \\end{aligned} \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} \\frac { d } { d t } b ( t ) = \\mathcal { Y } _ s ( \\psi _ s ^ t Z _ s ) - 2 t E _ s ( Z _ s ) - \\mathcal { Y } _ s ( Z _ s ) \\geq 0 \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} \\Gamma _ { 1 } : = ( 0 , \\pi ) \\times \\{ 0 , \\pi \\} \\Gamma _ { 2 } : = \\{ 0 , \\pi \\} \\times ( 0 , \\pi ) , \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} H = - { \\hbar ^ 2 \\over 2 m } { d ^ 2 \\over d x ^ 2 } - \\sum _ { i = 1 } ^ { n + 1 } \\lambda _ i \\delta ( x - a _ i ) \\psi ( x ) \\ ; . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} K _ { n } & : = \\left ( { \\mathrm { E } } \\left [ \\max _ { 1 \\leq i \\leq n } \\max _ { 1 \\leq j \\leq p } | X _ { i j } | ^ { 2 } \\right ] \\right ) ^ { 1 / 2 } \\\\ & \\leq \\left ( { \\mathrm { E } } \\left [ \\max _ { 1 \\leq i \\leq n } \\max _ { 1 \\leq j \\leq p } | X _ { i j } | ^ { q } \\right ] \\right ) ^ { 1 / q } \\leq n ^ { 1 / q } M _ { n } . \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} \\tau & \\ge 1 - \\mu _ R + \\sum _ { \\{ ( r , t ) : r + t < N _ R \\} } \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } \\left ( \\frac { 1 } { d _ { r , t } } - 1 \\right ) a _ { r , t } \\\\ & \\ge 1 - \\mu _ R , \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} \\Pi _ m ( \\cdot ) : = \\int _ 0 ^ T \\int _ { \\Omega } E ^ 2 ( | \\dot { u } ^ { ( m ) } | ) | \\dot { u } ^ { ( m ) } | ^ 2 + | \\dot { u } ^ { ( m ) } | ^ 2 + | q ^ { ( m ) } | ^ 2 \\ , \\mathrm { d } x \\mathrm { d } t . \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} \\inf \\left \\{ \\sum \\limits _ { \\beta \\in \\mathcal { T } } \\left \\vert \\bigtriangleup f \\left ( \\beta \\right ) \\right \\vert ^ { 2 } : f \\left ( \\alpha \\right ) = 1 , f \\left ( \\gamma \\right ) = 0 \\forall \\gamma \\in Z \\setminus \\left \\{ \\alpha \\right \\} \\right \\} \\leq \\frac { C } { d ( \\alpha ) } . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} L '^ { ( k ) } _ v [ X , Y ] = L ^ { ( k ) } _ v [ X , Y + 1 ] - L ^ { ( k ) } _ v [ X , Y ] . \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} E _ { \\alpha , \\beta } ^ \\gamma ( z ) = \\sum _ { k \\geq 0 } \\frac { ( \\gamma ) _ k } { \\Gamma ( \\alpha k + \\beta ) } \\frac { z ^ k } { k ! } , \\min \\{ \\Re ( \\alpha ) , \\Re ( \\beta ) , \\Re ( \\gamma ) \\} > 0 ; \\ , z \\in \\mathbb C . \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} Q _ { 3 4 5 } \\cap Q _ { 1 4 5 } \\cap Q _ { 1 2 5 } = L _ 5 \\cup \\Gamma _ 4 \\cup \\Gamma _ 1 \\ , , \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} & X _ 1 : = A / ( x - y + z ) A , X _ 2 : = A / ( x - y - z ) A , \\\\ & X _ 3 : = A / ( x + y + i z ) A , X _ 4 : = A / ( x + y - i z ) A \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} v _ D = \\widehat v _ D = \\widehat H _ g \\leq H _ g \\leq v _ D D , \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} \\mathbf { m } _ e = \\sum _ { d \\in I n ( v ) } \\mathbf { m } _ d \\mathbf { K } _ { d , e } \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} ( A - \\Delta A ) X = B - \\Delta B . \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\begin{aligned} & b _ { s , s + k } ^ 0 \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & \\ ; \\ ; = b _ { s , s + k } ^ 0 \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] \\end{aligned} \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} G ( \\vec { X } ) = G _ 1 ( \\vec { X } ) + G _ 2 ( \\vec { X } ) , \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} & \\| e ^ { i t H _ 0 } J _ a ^ * \\left [ v _ t - E _ + ( t - s ) v _ s \\right ] \\| \\\\ & \\leq \\| e ^ { i t H _ 0 } J _ a ^ * \\| \\| v _ t - E _ + ( t - s ) v _ s \\| \\\\ & = \\| J _ a ^ * \\| \\| e ^ { i ( t - s ) H } ( v _ t - E _ + ( t - s ) v _ s ) \\| \\\\ & = \\| J _ a ^ * \\| \\| v _ s - e ^ { i ( t - s ) H } E _ + ( t - s ) v _ s \\| . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} E ( X _ 1 , \\ldots , X _ n ) = \\underset { i , j \\in [ n ] } { \\max } | \\langle X _ i , X _ j \\rangle - M _ { i , j } | . \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} \\begin{aligned} F ( x _ k ) \\leq a _ k F ( x ) + & ( 1 - a _ k ) F ( x _ { k - 1 } ) + \\frac { \\tilde \\mu a _ k ^ 2 } { 2 } ( \\| x - v _ { k - 1 } \\| ^ 2 - \\| x - v _ k \\| ^ 2 ) \\\\ & - \\frac { \\tilde \\mu - \\mu } { 2 } \\| y _ k - x _ k \\| ^ 2 + \\rho a _ k \\| x - x _ { k - 1 } \\| ^ 2 + \\frac { r a _ k ^ 2 } { 2 } \\| x - v _ { k - 1 } \\| ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} D H ( v ) u = P \\sum _ { | \\alpha | \\le 1 } b _ \\alpha \\partial ^ \\alpha u + \\lambda _ 0 P ( u \\cdot \\nabla ) v , u , v \\in \\mathbb { E } , \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} & z ^ { 2 } ( \\log F ( z ) ) _ { z z } + \\overline { z } ^ { 2 } ( \\log F ( z ) ) _ { \\overline { z } \\overline { z } } \\\\ = & 2 \\sum _ { k = 3 } ^ { p } ( k - 1 ) ( k - 2 ) | z | ^ { 2 ( k - 1 ) } \\log G _ { k } ( z ) + 2 \\sum _ { k = 2 } ^ { p } ( k - 1 ) | z | ^ { 2 ( k - 1 ) } \\mathfrak { L } [ \\log G _ { k } ( z ) ] \\\\ & + \\sum _ { k = 1 } ^ { p } | z | ^ { 2 ( k - 1 ) } \\big ( z ^ { 2 } ( \\log G _ { k } ( z ) ) _ { z z } + \\overline { z } ^ { 2 } ( \\log G _ { k } ( z ) ) _ { \\overline { z } \\overline { z } } \\big ) . \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} K _ { N } ^ { 1 } ( z , \\zeta ) = \\sum _ { k = n - r } ^ { n - 1 } C ' _ { k } \\frac { \\rho ( \\zeta ) ^ { k + N } s \\wedge \\left ( \\partial _ { \\bar { \\zeta } } Q \\right ) ^ { n - r } \\wedge \\left ( \\partial _ { \\bar { z } } Q \\right ) ^ { k + r - n } \\wedge \\left ( \\partial _ { \\bar { z } } s \\right ) ^ { n - k - 1 } } { \\left | z - \\zeta \\right | ^ { 2 \\left ( n - k \\right ) } \\left ( \\frac { 1 } { K _ { 0 } } S ( z , \\zeta ) + \\rho ( \\zeta ) \\right ) ^ { k + N } } , \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} x Q _ n ( x ) = a _ { 2 , n } Q _ { n + 2 } ( x ) + a _ { 1 , n } Q _ { n + 1 } ( x ) + a _ { 0 , n } Q _ n ( x ) + a _ { - 1 , n } Q _ { n - 1 } ( x ) + a _ { - 2 , n } Q _ { n - 2 } ( x ) \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} \\gamma _ { 3 } + ( 1 + \\gamma _ { 5 } ) d - x _ { 1 } = x _ { 1 } ( \\beta _ { 5 } d + \\beta _ { 3 } ) , \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} \\mathcal { L } ( \\vec { x } , \\vec { y } ) = & - \\sum _ { i = 1 } ^ M \\alpha _ i \\log _ 2 \\left ( 1 + \\frac { h _ { s s } ^ i E _ s ^ i } { \\alpha _ i ( \\sigma _ s ^ 2 + h _ { p s } ^ i P _ p ) } \\right ) \\\\ & + \\sum _ { i = 1 } ^ M \\lambda _ i \\left [ \\sum _ { j = 1 } ^ i E _ s ^ j - \\sum _ { j = 1 } ^ i ( 1 - \\alpha _ j ) \\eta P _ p \\right ] \\\\ & + \\sum _ { i = 1 } ^ M \\gamma _ i [ h _ { s p } ^ i E _ s ^ i - P _ { i n t } \\alpha _ i ] + \\sum _ { i = 1 } ^ M \\mu _ i ( \\alpha _ i - 1 ) \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} C _ { n - 1 } ( 0 ) & = \\mu _ 0 ( n - 1 ) ( \\alpha _ { n - 1 } - 1 ) + C _ n ( 0 ) + \\log ( 1 + 2 ^ { \\mu _ 0 ( n - 1 ) + \\Delta { C } _ n } ) - H ( \\alpha _ { n - 1 } ) , \\\\ ~ C _ { n - 1 } ( 1 ) & = \\mu _ 1 ( n - 1 ) ( \\beta _ { n - 1 } - 1 ) + { C } _ n ( 0 ) + \\log ( 1 + 2 ^ { \\mu _ 1 ( n - 1 ) + \\Delta { C } _ n } ) - H ( \\beta _ { n - 1 } ) . \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\min _ { x } ~ F ( x ) : = \\frac { 1 } { m } \\sum _ { i = 1 } ^ m h _ i ( c _ i ( x ) ) + g ( x ) \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x ^ { \\nu } } \\left ( \\frac { \\partial \\mathcal { L } _ { 0 } } { \\partial P _ { \\nu \\mu } } \\right ) = - \\frac { 4 \\pi } { c } j ^ { \\mu } , \\qquad \\frac { \\partial } { \\partial x ^ { \\nu } } \\left ( \\frac { \\partial \\mathcal { L } _ { 1 } } { \\partial P _ { \\nu \\mu } } \\right ) = \\frac { 4 \\pi } { c } j ^ { \\mu } \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} G _ w = \\langle { a ' , b ' } , c ' \\mid ( a ' ) ^ 2 = ( b ' ) ^ 2 = ( c ' ) ^ 2 = 1 , ~ c ' = a ' b ' \\rangle . \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\phi ( u ) = \\int \\sqrt { 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) } a ( u ) d u , \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} & \\bullet \\gamma _ { j k } = - \\gamma _ { k j } , \\gamma _ { * j k } = - \\gamma _ { * k j } , \\\\ & \\bullet \\sigma _ j \\Phi A _ k ^ * - \\sigma _ k \\Phi A _ j ^ * = \\gamma _ { j k } \\Phi , \\\\ & \\bullet \\sigma _ j \\Phi A _ k - \\sigma _ k \\Phi A _ j = \\gamma _ { * j k } \\Phi , \\\\ & \\bullet \\gamma _ { * j k } - \\gamma _ { j k } = i \\left ( \\sigma _ j \\Phi \\Phi ^ * \\sigma _ k - \\sigma _ k \\Phi \\Phi ^ * \\sigma _ j \\right ) . \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} \\frac { G _ 1 ' } { F _ 1 ' } = m \\ , \\frac { F _ 2 ' } { F _ 1 ' } + n \\ , . \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} \\left \\langle J _ { a b } , Z _ { c d } , \\bar { P } _ { e } \\right \\rangle & = \\frac { 1 } { \\sqrt { 2 } } \\left \\langle J _ { a b } , Z _ { c d } , P _ { e } \\right \\rangle + \\frac { 1 } { \\sqrt { 2 } } \\left \\langle J _ { a b } , Z _ { c d } , Z _ { e } \\right \\rangle , \\\\ & = \\left ( \\alpha _ { 1 } + \\alpha _ { 2 } \\right ) \\ , \\varepsilon _ { a b c d e } , \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} N = N _ { \\chi } ( A r ; 1 + i \\tau ) = \\# \\{ \\rho : L ( \\rho , \\chi ) = 0 , | 1 + i \\tau - \\rho | \\leq A r \\} . \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\gamma _ { 2 n + 1 } = \\sum \\limits _ { i = - k } ^ { n } { n + k \\brace i + k } \\gamma _ { 2 i } , \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{gather*} \\Lambda ^ r V \\otimes \\Lambda ^ s V ^ * = V _ { \\underbrace { { \\scriptstyle { 1 , \\dots , 1 } } } _ { \\scriptstyle { r } } } ^ { \\overbrace { \\scriptstyle 1 , \\dotsc , 1 } ^ { \\scriptstyle { s } } } + V _ { \\underbrace { \\scriptstyle 1 , \\dots , 1 } _ { \\scriptstyle { r - 1 } } } ^ { \\overbrace { \\scriptstyle 1 , \\dotsc , 1 } ^ { \\scriptstyle { s - 1 } } } + \\dots + V ^ { \\overbrace { \\scriptstyle 1 , \\dotsc , 1 } ^ { \\scriptstyle { s - r } } } ; \\end{gather*}"}
-{"id": "7701.png", "formula": "\\begin{gather*} \\zeta i _ 2 = i _ 2 f g = F i _ 1 g = F G i _ 2 \\end{gather*}"}
-{"id": "10157.png", "formula": "\\begin{align*} \\langle G ( y _ k ) , z _ k - x ^ * \\rangle = \\frac { 1 } { 2 \\gamma } ( \\| z _ k - x ^ * \\| ^ 2 - \\| z _ { k + 1 } - x ^ * \\| ^ 2 ) + \\frac { \\gamma } { 2 } \\| G ( y _ k ) \\| ^ 2 . \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} 1 - | \\lambda | ^ 2 = \\Big | \\frac { F _ 2 ' } { F _ 1 ' } \\Big | ^ 2 - \\Big | \\frac { G _ 2 ' } { F _ 1 ' } \\Big | ^ 2 \\ , . \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} v _ r = u _ r + \\lambda \\delta _ c , \\ \\ 0 \\leq r \\leq d - 1 . \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} C ( t ) = \\frac { M ^ 2 \\sum _ { j = 1 } ^ { + \\infty } \\frac { 1 } { \\lambda _ j } e ^ { - \\lambda _ j ( t - 2 t _ 0 ) } } { 1 - M ^ 2 \\sum _ { j = 1 } ^ { + \\infty } e ^ { - \\lambda _ j ( t - 2 t _ 0 ) } } . \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} ( \\deg ( _ \\infty ( g / f ) ) , \\deg ( _ \\infty ( h / f ) ) ) = 0 . \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} a _ { 3 } = \\alpha _ { 3 } a _ { 1 } + \\beta _ { 3 } a _ { 2 } + \\gamma _ { 3 } a _ { 4 } , \\ ; \\ ; a _ { 5 } = \\alpha _ { 5 } a _ { 1 } + \\beta _ { 5 } a _ { 2 } + \\gamma _ { 5 } a _ { 4 } \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } x _ i ^ { k + 1 } : = \\Pi _ { K _ i } [ x _ i ^ k - \\alpha _ { k , i } F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) ] \\cr v _ i ^ { k + 1 } : = \\hat v _ i ^ k + x _ i ^ { k + 1 } - x _ i ^ k \\end{array} \\right \\} \\mbox { f o r } i \\in \\{ I ^ k , J ^ k \\} , \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} ( u - v \\mp a ) \\xi _ i ( u ) x _ j ^ { \\pm } ( v ) = ( u - v \\pm a ) x _ j ^ { \\pm } ( v ) \\xi _ i ( v ) \\mp 2 a x _ { j } ^ { \\pm } ( u \\mp a ) \\xi _ { i } ( u ) \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} B _ { 2 , Z } & = \\left \\{ \\varphi = \\sum _ { j = 1 } ^ { \\infty } a _ { j } \\varphi _ { \\zeta _ { j } } : \\sum _ { j = 1 } ^ { \\infty } \\left \\vert a _ { j } \\right \\vert ^ { 2 } \\mu \\left ( \\zeta _ { j } \\right ) < \\infty \\right \\} , \\\\ \\left \\Vert \\varphi \\right \\Vert _ { B _ { 2 , Z } } & \\approx \\left \\Vert \\left \\{ a _ { j } \\right \\} _ { j = 1 } ^ { \\infty } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } . \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} F _ { r _ { B ( U ) , U } } ( r ) = \\Pr \\{ r _ { B ( i ) , i } \\le r \\} = 1 - \\exp \\left ( - \\lambda _ { C } \\pi r ^ 2 \\right ) , \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} a ^ * _ { 3 , 0 } = \\mu _ R , a ^ * _ { 0 , 3 } = 1 - \\mu _ R , \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} p ( y ) = y _ n \\left ( a _ 0 + \\sum _ { i = 1 } ^ { n - 1 } a _ i y _ i + b ( y _ n ^ 2 - 3 y _ { n + 1 } ^ 2 ) \\right ) , \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} \\xi _ { n + 1 } = \\dfrac { p _ { n + 1 } } { q q _ { n + 1 } } + \\theta \\pounds \\theta q q _ { n + 1 } \\cong 0 \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } S _ f ^ \\nu ( n ) \\overline { S _ g ^ \\nu ( n ) } = c _ { f , g } X ^ { 2 \\kappa ( f ) + \\frac { 3 } { 2 } - 2 \\nu } + O ( X ^ { 2 \\kappa ( f ) + 1 - 2 \\nu } \\log ^ 2 X ) \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\frac { 1 } { \\lambda _ u ^ { s } \\cdot { s } ^ { d _ u } } M ^ { s } \\vec { u } _ j = \\vec u _ \\infty ^ j \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} \\mathcal { T } ( P ) = \\log \\left ( 1 + \\sqrt { \\frac { 2 P } { \\pi e } } \\right ) \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} 0 = G _ i = \\norm { D u } ^ { - 2 } u _ { i j } u ^ j + \\lambda u _ i \\Theta ^ { - 1 } , \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} \\varphi _ { n } ( x ) = ( - 1 ; q ) _ { \\infty } x ^ { n } q ^ { n ( n - 1 ) / 2 } A _ { q ^ { 2 } } \\left ( q ^ { - 2 n + 2 } x ^ { - 2 } \\right ) \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} M ' : = ^ { \\mathfrak { A } } ( M \\cup \\{ \\zeta \\} ) . \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} \\beta _ { s , a ^ 1 } ^ 1 = \\frac { \\left [ r ^ 1 ( s , a ^ 1 , a _ s ^ 2 ) - r ^ 1 ( s , a _ s ^ 1 , a _ s ^ 2 ) \\right ] } { \\left [ r ^ 1 ( s , a ^ 1 , a _ { s } ^ 2 ) - \\sum _ { s ' \\in S } p ( s ' | s , a ^ 1 , a _ s ^ 2 ) r ^ 1 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) \\right ] } . \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{gather*} \\tau _ { k } ^ { ( \\alpha ) } \\tau _ { k - 2 } ^ { ( \\alpha + 2 ) } = \\tau _ { k - 1 } ^ { ( \\alpha + 2 ) } \\tau _ { k - 1 } ^ { ( \\alpha ) } - \\big ( \\tau _ { k - 1 } ^ { ( \\alpha + 1 ) } \\big ) ^ 2 . \\end{gather*}"}
-{"id": "3277.png", "formula": "\\begin{gather*} R ^ { \\prime } = \\frac { W } { \\prod \\limits _ { 1 \\le j \\le m } ( z - w _ { j } ) } = ( - 1 ) ^ { m + 1 } \\sum _ { i = 1 } ^ { m } \\frac { W _ { i } } { z - w _ { i } } . \\end{gather*}"}
-{"id": "839.png", "formula": "\\begin{align*} \\sum _ { n \\in \\mathbb { Z } , \\ ; \\ell = 3 \\cdot 2 ^ n } \\| T _ { \\varphi , \\ell } ( f \\mu ) \\| _ { L ^ 2 ( \\mu ) } ^ 2 \\leq C ( M ) \\| f \\| ^ 2 _ { L ^ 2 ( \\mu ) } f \\in L ^ 2 ( \\mu ) . \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} - P _ g ^ 6 = \\Delta _ g ^ 3 + \\Delta _ g \\delta T _ 2 d + \\delta T _ 2 d \\Delta _ g + \\frac { n - 2 } { 2 } \\Delta _ g ( \\sigma _ 1 ( A _ g ) \\Delta _ g ) + \\delta T _ 4 d - \\frac { n - 6 } { 2 } Q _ g ^ 6 , \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\delta ( X ) = \\tfrac { 4 ( g - 1 ) } { g ^ { 2 } } S _ { 1 } ( X ) - \\tfrac { 3 g - 1 } { 2 g } \\tbinom { 2 g } { g - 1 } ^ { - 1 } \\log \\| \\Delta _ { g } \\| ( X ) - 8 g \\log 2 \\pi , \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} S ( u , v ) = ( ( 2 + \\cos u ) \\cos v , ( 2 + \\cos u ) \\sin v , u - \\sin u ) . \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} \\varrho = \\bigotimes _ { j = 0 } ^ m \\big ( ( \\wedge _ k \\partial s _ { j - 1 , k } ) \\wedge ( \\wedge _ k s _ { j , k } ) \\big ) ^ { ( - 1 ) ^ j } \\in \\det V ^ \\bullet . \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} M _ { X , p u r e } = \\sum _ { i \\leq k } w _ i \\mu _ i \\otimes \\mu _ i \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} \\prod _ { j \\in N } a _ j ( i ) = b ( i ) , \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} - \\psi ( n ) - s = - O ( s ) . \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} & A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\psi _ { n - 6 } ^ { ( 0 ) } \\\\ = & A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\alpha _ { n - 6 } ^ { ( 0 ) } + ( B _ { 2 - n } A _ { 4 - n } A _ { 6 - n } + A _ { 2 - n } B _ { 4 - n } A _ { 6 - n } + A _ { 2 - n } A _ { 4 - n } B _ { 6 - n } ) \\beta _ { n - 6 } ^ { ( 0 ) } . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} \\begin{cases} y _ 0 = \\frac { x _ 0 } { x _ 1 } = \\frac { 1 } { z _ 1 } \\\\ y _ 2 = \\frac { x _ 2 } { x _ 1 } = \\frac { z _ 2 } { z _ 1 } \\\\ y _ 3 = \\frac { x _ 3 } { x _ 1 } = \\frac { z _ 3 } { z _ 1 } \\end{cases} \\begin{cases} z _ 1 = \\frac { 1 } { y _ 0 } \\\\ z _ 2 = \\frac { y _ 2 } { y _ 0 } \\\\ z _ 3 = \\frac { y _ 3 } { y _ 0 } \\end{cases} \\begin{cases} d y _ 2 = \\frac { z _ 1 d z _ 2 - z _ 2 d z _ 1 } { z _ 1 ^ 2 } \\\\ d y _ 3 = \\frac { z _ 1 d z _ 3 - z _ 3 d z _ 1 } { z _ 1 ^ 2 } , \\end{cases} \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\lambda ( t ) = \\mathrm { R e } \\ , h ^ { W ^ \\bullet } _ t \\left ( w ( t ) , \\Big ( \\frac { \\partial } { \\partial t } D _ t \\Big ) w ( t ) \\right ) . \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} \\iota _ r = \\begin{pmatrix} 1 & 0 \\\\ 0 & r ^ { - 1 } \\end{pmatrix} \\iota _ 0 \\begin{pmatrix} 1 & 0 \\\\ 0 & r \\end{pmatrix} = a ( r ) \\iota _ 0 a ( r ^ { - 1 } ) . \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\delta _ F ^ { ( 1 ) } = 0 ; ~ ~ ~ ~ ~ \\delta _ E ^ { ( 1 ) } = \\delta _ { \\mathsf { C a - I A } } = \\frac { M + K - 1 } { M } ; \\end{align*}"}
-{"id": "10020.png", "formula": "\\begin{align*} b _ { H , a } = 0 \\ ; \\Rightarrow \\ ; b _ { H , t } = 1 . \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\phi _ N ^ { ( s ) } ( t , Z _ s ^ * ) } { N - s + 1 } & + \\phi _ N ^ { ( s - 1 ) } ( t , ( Z _ s ^ * ) ^ { ( i ) } ) + \\phi _ N ^ { ( s - 1 ) } ( t , ( Z _ s ^ * ) ^ { ( j ) } ) = \\\\ & = \\frac { \\phi _ N ^ { ( s ) } ( t , Z _ s ) } { N - s + 1 } + \\phi _ N ^ { ( s - 1 ) } ( t , Z _ s ^ { ( i ) } ) + \\phi _ N ^ { ( s - 1 ) } ( t , Z _ s ^ { ( j ) } ) \\end{aligned} \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} F _ n ( T _ { ( x y ) } ( i , \\alpha ) ) = F _ n ( T _ x ( i ) \\otimes T _ y ( \\alpha ) ) = F _ n ( T _ x ( i ) ) \\otimes F _ n ( T _ y ( \\alpha ) ) \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} \\displaystyle A \\left ( \\begin{array} { c } u \\\\ v \\end{array} \\right ) = - \\left ( \\begin{array} { c c } \\displaystyle \\partial _ { x x x } & a \\partial _ { x x x } \\\\ \\displaystyle \\frac { a b } { c } \\partial _ { x x x } & \\frac { r } { c } \\partial _ { x } + \\frac { 1 } { c } \\partial _ { x x x } \\end{array} \\right ) \\left ( \\begin{array} { c } u \\\\ v \\end{array} \\right ) \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ { k + 1 } f _ \\alpha ( x , y ) z ^ \\alpha \\in H _ 3 . \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} \\| E a _ k \\| ^ 2 + a _ k ^ H Z a _ k = 0 , k = 1 , \\ldots , r . \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} q ^ { n ^ { 2 } / 2 } S _ { n } \\left ( x q ^ { - n - 1 / 2 } ; q \\right ) = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( x e ^ { i y } ; q \\right ) _ { n } } { \\left ( q ; q \\right ) _ { n } } \\exp \\left ( \\frac { y ^ { 2 } } { \\log q ^ { 2 } } - i n y \\right ) d y . \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda ^ t } M ^ t \\vec v & = \\frac { 1 } { \\lambda ^ t } ( \\lambda ^ t \\vec v + \\sum _ { m = 0 } ^ { t - 1 } \\lambda ^ { t - m - 1 } M ^ { m } \\vec u ) \\\\ & = \\vec v + \\frac { 1 } { \\lambda } \\sum _ { m = 0 } ^ { t - 1 } \\frac { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } { \\lambda ^ { m } } \\frac { 1 } { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } M ^ { m } \\vec u \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} R i c ( \\omega _ { \\infty } ) - \\omega _ { \\infty } = - L _ { v } ( \\omega _ { \\infty } ) , \\textrm { o n t h e r e g u l a r p a r t o f } \\ ; M _ { \\infty } , \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} \\begin{aligned} & C _ { i , s + 1 } ^ - f _ N ^ { ( s + 1 ) } ( t , Z _ s ) = \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { Z _ { s + 1 } \\in \\partial \\mathcal { D } _ { s + 1 } } \\left [ \\omega \\cdot ( v _ { s + 1 } - v _ i ) \\right ] _ { - } \\times \\\\ & \\ ; \\ ; \\times f _ N ^ { ( s + 1 ) } ( t , x _ 1 , v _ 1 , \\dots , x _ i , v _ i , \\dots , x _ s , v _ s , x _ i + \\varepsilon \\omega , v _ { s + 1 } ) d \\omega d v _ { s + 1 } \\end{aligned} \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } D \\tilde f _ { n } ( x ) = D \\tilde f ( x ) , \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} \\ell ( n p ) = \\begin{cases} 1 , 1 , 1 , 2 , 3 , 4 , 5 , \\ldots \\\\ 1 , 1 , 2 , 2 , 3 , 4 , 5 , \\ldots \\\\ 1 , 1 , 2 , 3 , 3 , 4 , 5 , \\ldots . \\end{cases} \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} \\Theta ^ { ( n ) } _ { \\lambda } ( u ) : = \\Theta _ { n u ^ { \\gamma } } ( \\lambda ) , \\Theta ^ { ( n ) } _ { \\lambda , \\lambda ' } ( u ) : = \\Theta ^ { ( n ) } _ { \\lambda ' } ( u ) - \\Theta ^ { ( n ) } _ { \\lambda } ( u ) . \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} D _ n : = \\{ z \\in \\C : \\varphi _ n ^ e ( z ) = \\varphi _ n ( z ) \\} . \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} \\mathcal H = \\bigg \\{ & v \\in L ^ 2 ( \\Omega ) \\bigg | v \\mbox { i s p e r i o d i c i n } ( x , y , z ) , \\mbox { e v e n i n } z , \\\\ & \\mbox { a n d s a t i s f i e s } \\nabla _ H \\cdot \\left ( \\int _ { - h } ^ h v ( x , y , z ) d z \\right ) = 0 \\bigg \\} . \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} G ( s ) : = \\int _ 0 ^ s g ( r ) \\ , d r \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\mathcal { B } ^ + _ V = \\left \\{ \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { A } ^ + \\left | \\left | v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime \\right | \\leq \\eta \\right . \\right \\} \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } q ^ { \\beta ^ { 2 } } \\left ( q ^ { - \\beta + n + 1 } ; q \\right ) _ { \\infty } L _ { n } ^ { ( - \\beta ) } \\left ( - 1 ; q \\right ) A _ { q } \\left ( - q ^ { \\beta + n } \\right ) d \\beta = \\frac { 1 } { \\log q ^ { - 1 } \\left ( q ; q \\right ) _ { n } } \\int _ { - \\infty } ^ { \\infty } \\exp \\left ( \\frac { y ^ { 2 } } { \\log q } \\right ) d y , \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} p ' _ { ( i , a ) , ( j , b ) } = p _ { i j } \\nu _ j ( b ) . \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} p ^ { \\prime } : = \\max \\{ p , 2 \\} \\hbox { a n d } q ^ { \\prime } : = \\max \\{ q , 2 \\} . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} \\lambda _ \\mu = \\lambda _ { \\mu - 1 } + \\delta \\textrm { s i g n } ( T - T _ { \\mu - 1 } ) , \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} \\tilde \\alpha _ 1 & = \\frac { - \\sqrt { - t x _ i } } { ( - 1 ) ^ i x _ i ( t + 1 ) } & & \\tilde \\alpha _ 2 = \\sqrt { x _ i ^ { - 1 } } . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} \\mathbb { P } ( X _ { r } > T _ { ( k ) } ) & = \\mathbb { E } \\left [ \\mathbb { P } ( X _ { r } > T _ { ( k ) } ) | T _ { ( k ) } \\right ] \\\\ & = \\sum _ { j = 0 } ^ { r - 1 } \\frac { s ^ { j } } { j ! } \\mathbb { E } \\left ( e ^ { - s T _ { ( k ) } } T _ { ( k ) } ^ { j } \\right ) . \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k + 1 } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau , 0 ; v _ { s + 1 } , \\dots , v _ { s + k } , v _ { s + k + 1 } ; \\right . \\\\ & \\left . \\qquad \\omega _ 1 , \\dots , \\omega _ k , \\omega _ { k + 1 } ; i _ 1 , \\dots , i _ k , i _ { k + 1 } \\right ] \\\\ & \\in \\mathcal { G } _ { s + k + 1 } \\cap \\hat { \\mathcal { U } } _ { s + k + 1 } ^ \\eta \\end{aligned} \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{gather*} s _ { 0 } = s _ { 1 } = \\ \\ldots \\ = s _ { n _ { 0 } - 1 } = 0 \\ , \\ s _ { n _ { 0 } } = \\Delta _ { 0 } ^ { \\tfrac { 1 } { n _ { 0 } + 1 } } \\ , \\end{gather*}"}
-{"id": "4587.png", "formula": "\\begin{align*} h ( z _ i , \\ldots , z _ m ) : = \\frac { 1 } { m } \\sum _ { i = 1 } ^ m h _ i ( z _ i ) \\textrm { a n d } c ( x ) : = ( c _ 1 ( x ) , \\ldots , c _ m ( x ) ) . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} \\partial _ \\xi ^ \\beta s _ a ^ 2 ( x , \\xi ) = \\begin{cases} 0 , \\ | \\cos ( x , v ( \\xi ) ) | \\geq \\frac { 1 } { 2 } , \\\\ \\mathcal { O } ( \\langle x \\rangle ^ { - \\varepsilon } ) , \\ | \\cos ( x , v ( \\xi ) ) | \\leq \\frac { 1 } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} E = \\bigoplus _ { j = 0 } ^ { n - 1 } E _ j \\otimes \\rho ^ { \\otimes j } \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} & x B _ n ^ d = B _ { n + 1 } ^ i + \\\\ & \\sum \\limits _ { \\substack { t = 0 \\\\ 1 \\leq j _ 0 < j _ 1 < \\dots < j _ t \\leq i + 1 } } ^ { i } \\rho _ { ( d + 1 ) n + j _ 0 } \\rho _ { ( d + 1 ) ( n - 1 ) + j _ 1 } \\dots \\rho _ { ( d + 1 ) ( n - t ) + j _ t } \\ B _ { n - t } ^ { i } , \\ \\ 0 \\leq i \\leq d . \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} \\Delta V & = V '' \\big ( \\rho \\big ) + \\Delta \\rho V ' \\big ( \\rho \\big ) \\\\ & \\le V '' ( \\rho ) + ( n - 1 ) \\frac { f _ a ^ \\prime ( \\rho ) } { f _ a ( \\rho ) } V ' ( \\rho ) \\\\ & = ( n - 1 ) \\frac { f _ a ^ \\prime ( \\rho ) } { f _ a ^ n ( \\rho ) } \\int _ 0 ^ \\rho a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t - a _ 0 ( \\rho ) - ( n - 1 ) \\frac { f _ a ^ \\prime ( \\rho ) } { f _ a ^ n ( \\rho ) } \\int _ 0 ^ \\rho a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t \\\\ & = - a _ 0 ( \\rho ) , \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} A ( b ) f = ( \\mid b + t \\mid ^ { 2 } I + \\Delta ) ^ { - 1 } q f \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\alpha _ i ^ * = \\max \\left \\{ \\frac { \\zeta _ i } { \\zeta _ i + z _ i ^ * - 1 } , \\frac { h _ { s p } ^ i \\eta P _ p } { P _ { i n t } + h _ { s p } ^ i \\eta P _ p } \\right \\} \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\overline F _ \\alpha ( \\overline X , \\overline Y , \\overline Z ) = F _ \\alpha ( X , Y , Z ) + F _ \\alpha ^ \\prime ( X ^ \\prime , Y ^ \\prime , Z ^ \\prime ) , \\alpha = 1 , 2 , 3 , \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\partial ^ * f = \\begin{cases} f ' ( x ) & x \\in I ^ \\circ \\\\ - \\frac { b _ 0 } { c _ 0 } f ( 0 ) & x = 0 \\\\ \\frac { b _ 1 } { c _ 1 } f ( 1 ) & x = 1 . \\end{cases} \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} ( \\widetilde { \\phi } ( x ) ) ( 1 , d _ i , 1 ) = \\phi ( ( 1 , d _ i , 1 ) x ) = \\begin{cases} x & i = 1 , \\\\ 0 & i \\neq 1 . \\end{cases} \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} I ( G ^ \\mathcal { C } \\star H ^ U ) = ( I ( H , x ) ) ^ k I \\Big ( G , \\frac { x I ( H - U , x ) } { I ( H , x ) } \\Big ) , \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\square \\Delta u = 2 R _ { i j } u _ { i j } , \\square | \\nabla u | ^ 2 = - 2 | u _ { i j } | ^ 2 , \\square R = 2 | R _ { i j } | ^ 2 . \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\| \\phi _ j \\phi _ j ^ * - \\phi _ l \\phi _ l ^ * \\| _ { H . S . } ^ 2 & = 2 \\left ( 1 - t r ( \\phi _ j \\phi _ j ^ * \\phi _ l \\phi _ l ^ * ) \\right ) \\\\ & = 2 \\left ( 1 - | \\langle \\phi _ j , \\phi _ l \\rangle | ^ 2 ) \\right ) . \\\\ \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} \\frac { d g _ { i j } } { d t } = - 2 \\alpha ' R _ { i j } + O ( \\alpha '^ 2 ) \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\begin{array} { l l } ( \\nabla _ X J _ 1 ) Y = \\frac { 1 } { 2 ( 2 n - 1 ) } \\left [ g ( X , Y ) p ^ \\top _ 1 - \\overline \\theta _ 1 ( Y ) X + g ( X , J _ 1 Y ) J _ 1 ( p ^ \\top _ 1 ) \\right . \\\\ \\\\ \\qquad \\qquad \\quad \\left . - \\overline \\theta _ 1 ( J _ 1 Y ) J _ 1 X \\right ] , \\end{array} \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} \\nabla _ \\xi p ( t , s ; x , \\eta ( t , s ) ) \\nabla _ \\xi \\eta ( t , s ) = I . \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} 1 6 \\pi X _ { \\mu } = Z _ { \\mu } + Z _ { \\mu } ^ { \\ast } , \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{gather*} \\delta ( z , w ) = \\sum _ { k \\in \\mathbb { Z } } z ^ { k } w ^ { - k - 1 } . \\end{gather*}"}
-{"id": "9868.png", "formula": "\\begin{align*} h _ k [ x _ 1 , \\ldots , x _ n \\mid y _ 1 , \\ldots , y _ m ] : = \\sum _ { i = 0 } ^ k h _ i [ x _ 1 , \\ldots , x _ n ] e _ { k - i } [ y _ 1 , \\ldots , y _ m ] \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\nu _ k \\ge \\theta _ k \\ge 0 \\ > ( k = 0 , 1 , \\ldots ) ; \\nu _ k > 0 \\ > ( k = 1 , 2 , \\ldots ) ; \\pi _ k > 0 \\ > ( k = 0 , 1 , \\ldots ) . \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} \\eta _ { n } = \\frac { ( n ! ) ^ { 2 n } } { 2 } \\xi _ n \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} I _ { a + } ^ { \\alpha } x ( t ) : = \\frac { 1 } { \\Gamma ( \\alpha ) } \\int _ a ^ t ( t - \\tau ) ^ { \\alpha - 1 } x ( \\tau ) \\ ; d \\tau \\hbox { f o r } t \\in ( a , b ] , \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} \\lambda _ f ( m ) \\lambda _ f ( n ) = \\sum _ { \\substack { d | ( m , n ) \\\\ ( d , N ) = 1 } } \\lambda _ { f } \\left ( \\frac { m n } { d ^ 2 } \\right ) . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} \\pi ^ { } _ n ( x _ n | y ^ { n - 1 } _ { n - J } ) & = \\frac { \\exp { \\Big \\{ \\sum _ { y _ n } \\log \\big ( r _ n ( x _ n | y ^ { n - 1 } _ { n - M } , y _ n ) \\big ) q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) - s \\gamma _ n ( x _ n , y ^ { n - 1 } _ { n - N } ) \\Big \\} } } { \\sum _ { x _ n } \\exp { \\Big \\{ \\sum _ { y _ n } \\log \\big ( r _ n ( x _ n | y ^ { n - 1 } _ { n - M } , y _ n ) \\big ) q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) - s \\gamma _ n ( x _ n , y ^ { n - 1 } _ { n - N } ) \\Big \\} } } , ~ \\forall { x _ n } \\in { \\cal X } _ n . \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} N _ \\lambda ( u ; q , t ) = \\prod _ { s \\in \\lambda } ( q ^ { a ( s ) } - u t ^ { l ( s ) + 1 } ) ( q ^ { a ( s ) + 1 } - u ^ { - 1 } t ^ { l ( s ) } ) . \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} S = \\ln Z - \\beta \\frac { \\partial } { \\partial \\beta } \\ln Z \\ , \\propto \\ , \\ln Z \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} v _ { i , j , 1 } = x _ { i , j } , v _ { i , j , 2 } = x _ { i , j } ^ 2 , v _ { i , j , 3 } = x _ { i , j } ^ 4 , \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} T _ t f _ 0 = T _ t T _ { n t _ 0 } f _ 0 = T _ { n t _ 0 } T _ t f _ 0 \\to \\langle \\varphi , T _ t f _ 0 \\rangle f _ 0 n \\to \\infty , \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} u ( b ( t ) , z ) & = u ( 0 , z ) + \\int _ 0 ^ { b ( t ) } [ \\Delta u + V u ] ( s , z ) \\ , d s + \\int _ 0 ^ { b ( t ) } G u ( s , z ) \\ , d W ( s ) \\\\ & = u ( 0 , z ) + \\int _ 0 ^ t [ \\Delta u + V u ] ( b ( s ) , z ) b ' ( s ) \\ , d s + \\int _ 0 ^ t G u ( b ( s ) , z ) \\ , d W ( b ( s ) ) , \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ \\gamma ( z , \\bar z ) : & = ( \\gamma + m + 1 ) _ n ( 1 - | z | ^ 2 ) ^ { - \\nu + m } \\nabla ^ { \\nu } _ { m } \\left ( z ^ n ( 1 - | z | ^ 2 ) ^ { \\nu - m } \\right ) \\\\ & = ( - 1 ) ^ { m + n } \\left ( 1 - \\mid z \\mid ^ { 2 } \\right ) ^ { - \\gamma } \\dfrac { \\partial ^ { m + n } } { \\partial z ^ { m } \\partial \\overline { z } ^ { n } } \\left ( \\left ( 1 - | z | ^ { 2 } \\right ) ^ { \\gamma + m + n } \\right ) , \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} G _ { k , l } \\left ( \\mu ( z ) \\right ) = \\frac { 1 } { W ( f , g ) } \\begin{cases} g _ { k } ( z ) f _ { l } ( z ) , & \\ ; k \\leq l , \\\\ g _ { l } ( z ) f _ { k } ( z ) , & \\ ; k \\geq l , \\end{cases} \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} X _ { k k } = \\sum _ { i = 1 } ^ { \\epsilon _ k + 1 } \\left [ \\begin{array} { c } 1 \\\\ \\lambda _ { k i } \\\\ \\lambda _ { k i } ^ 2 \\\\ \\vdots \\\\ \\lambda _ { k i } ^ { \\epsilon _ k } \\end{array} \\right ] \\left [ \\begin{array} { c } 1 \\\\ \\lambda _ { k i } \\\\ \\lambda _ { k i } ^ 2 \\\\ \\vdots \\\\ \\lambda _ { k i } ^ { \\epsilon _ k } \\end{array} \\right ] ^ H \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} a ^ { G _ 2 } ( S , u _ 2 ( \\alpha ) ) = c ( u _ 2 ) \\times \\big \\{ \\mathfrak { c } _ E ( S ) + \\sum _ \\chi \\chi _ { S _ E } ( \\alpha ) \\ , L _ E ^ S ( 1 , \\chi ) \\big \\} \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} u _ { 0 . 6 } ^ 2 ( g _ 1 ) = [ I - 0 . 6 P ( g _ 1 ) ] ^ { - 1 } \\bar { r } ^ 2 ( g _ 1 ) = ( 6 . 6 , 6 ) . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} r _ k ( \\underline \\pi ) \\equiv \\prod \\limits _ { j = 0 } ^ { a - 1 } \\bigg ( \\det \\left [ \\begin{array} { c c c c c c c c c c } 0 & 1 & \\cdots & k - 1 \\\\ 0 & 1 & \\cdots & k - 1 \\end{array} \\right ] _ { M _ j } \\bigg ) \\bmod \\ { I ^ { \\lceil \\frac { a k ( k - 1 ) ( p - 1 ) + ( p - 1 ) } { 2 d } \\rceil } } . \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} \\tau _ 2 \\ge 1 - \\mu _ R + 1 - \\mu _ R - \\mu _ T = 2 ( 1 - \\mu _ R ) - \\mu _ T . \\end{align*}"}
-{"id": "10059.png", "formula": "\\begin{align*} q + r = m + n + l + k \\leq 2 p + q , \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} \\bar M = M / F ^ 1 M ( = M / C _ 2 ( M ) ) , \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} \\kappa ( x ) = \\begin{cases} x & \\\\ r ( x ) & \\\\ \\mu & . \\end{cases} \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} \\rho \\left ( x , \\varphi \\right ) = \\frac { 1 - c _ { o } } { \\mu \\left ( \\varphi \\right ) } \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} g = \\sum _ { i = w } ^ { \\infty } g _ i t ^ { - i } , \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} \\gamma ^ a \\gamma ^ b + \\gamma ^ b \\gamma ^ a = 0 , \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} F ( \\kappa _ 1 , \\cdots , \\kappa _ n ) = \\varphi ( f ^ 1 ( \\kappa _ 1 , \\cdots , \\kappa _ n ) , \\cdots , f ^ k ( \\kappa _ 1 , \\cdots , \\kappa _ n ) ) \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} \\ell ( n p ) = \\begin{cases} 1 & n = 0 \\\\ n & n \\geqslant 1 . \\end{cases} \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) = { \\bf E } \\bigl [ M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ^ q \\bigr ] , \\ ; \\Re ( q ) < \\tau , \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} V _ s ^ { t , x } = & \\ v + ( \\lambda + 1 ) \\int _ t ^ s \\xi ( r , \\psi ( r , V _ r ^ { t , x } ) ) \\mathrm d r \\\\ & + \\int _ t ^ s ( \\nabla \\xi ( r , \\psi ( r , V _ r ^ { t , x } ) ) + \\mathrm { I } _ d ) \\mathrm d W _ r , \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} \\sigma _ { t , R } = \\delta _ { ( - t , \\dots , - t ) } \\ast \\frac { 1 } { a _ 1 \\ , \\dots \\ , a _ s \\ , R } \\ , \\sum _ { 0 \\le k _ 1 \\le a _ 1 R - 1 } \\ , \\dots \\ , \\sum _ { 0 \\le k _ s \\le a _ s R - 1 } \\ , \\delta _ { ( - k _ 1 / a _ 1 , \\dots , - k _ 1 / a _ s ) } \\ast \\nu _ \\mu , \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\Delta ( \\mu , C _ F , P ) & = \\limsup _ { L \\rightarrow \\infty } \\frac { T _ F + T _ E } { L } . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} | \\gamma ' | ^ 2 & = c \\\\ \\frac { 1 } { | n | } \\langle \\gamma '' , \\gamma ' \\times n \\rangle & = \\kappa | \\gamma ' | ^ 2 , \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} { \\bf E } [ M ^ { - n } _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } ] = ( 2 \\pi ) ^ { - n } \\prod \\limits _ { j = 0 } ^ { n - 1 } \\frac { \\Gamma ( 1 + \\lambda _ 1 + \\frac { ( j + 1 ) } { \\tau } ) \\ , \\Gamma ( 1 + \\lambda _ 2 + \\frac { ( j + 1 ) } { \\tau } ) \\Gamma ( 1 - \\frac { 1 } { \\tau } ) } { \\Gamma ( 1 + \\lambda _ 1 + \\lambda _ 2 + \\frac { ( j + 1 ) } { \\tau } ) \\ , \\Gamma ( 1 + \\frac { j } { \\tau } ) } . \\end{align*}"}
-{"id": "9888.png", "formula": "\\begin{align*} z ^ { ( i + 1 ) } = z ^ { ( i ) } - \\frac { \\mathcal { W } _ \\sigma ^ \\prime ( z ^ { ( i ) } ) \\pm \\left [ \\left ( \\mathcal { W } _ \\sigma ^ \\prime ( z ^ { ( i ) } ) \\right ) ^ 2 - 2 \\mathcal { W } _ \\sigma ( z ^ { ( i ) } ) \\ , \\mathcal { W } _ \\sigma ^ { \\prime \\prime } ( z ^ { ( i ) } ) \\right ] ^ { 1 / 2 } } { \\mathcal { W } _ \\sigma ^ { \\prime \\prime } ( z ^ { ( i ) } ) } \\ , , \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} f = & \\frac { 1 } { 2 h } \\int _ { - h } ^ h \\left ( \\int _ { - h } ^ z \\nabla _ H \\cdot ( v T ) d \\xi + w T + \\nabla _ H \\cdot \\big ( \\nabla _ H \\cdot ( v \\otimes v ) + f _ 0 k \\times v \\big ) \\right ) d z . \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} \\eta _ g : x \\to f _ { g } ^ { - 1 } x f _ g , { } \\phi _ { g } : = \\eta _ { f _ g ^ { - 1 } } : x \\to f _ g x f _ g ^ { - 1 } \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} ( \\mathcal { G } _ { D } u _ 0 ) _ t ( x ) = \\int _ { B _ R ( 0 ) } u _ 0 ( y ) p _ D ( t , x , y ) \\d y , \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\left \\vert x _ { i } - \\dfrac { p _ { i } } { q } \\right \\vert \\leq \\varepsilon t _ { i } \\\\ \\varepsilon q \\leq t _ { i } \\end{array} \\right \\vert i = 1 , 2 , . . . , n \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} \\binom { n z - 1 } { k - 2 } \\le \\frac { z ^ { k - 2 } } { ( k - 2 ) ! } n ^ { k - 2 } . \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\nu _ u : = c _ m \\ , \\Delta u \\in \\mathcal M ^ + ( \\mathcal O ) , c _ m : = \\frac { \\Gamma ( m / 2 ) } { 2 \\pi ^ { m / 2 } \\max \\bigl \\{ 1 , ( m - 2 ) \\bigr \\} } \\ , , \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} \\xi ^ { i k + j } & = 1 & \\Longleftrightarrow & & i k & \\equiv - j \\mod n , \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\begin{aligned} & \\max _ { f \\in \\mathcal { M } ( T , X ) } \\int _ T \\varphi ( t , f ( t ) ) d \\mu \\\\ & \\int _ T f ( t ) d \\mu - \\int _ T \\omega ( t ) d \\mu \\in W \\end{aligned} \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} - \\log _ 2 n + \\psi ( n ) ^ 2 / 2 ( 1 + o ( 1 ) ) = 0 \\implies \\psi ( n ) \\sim \\sqrt { 2 \\log _ { 2 } n } , \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} \\psi _ 1 ( x ) & = 1 , & \\psi _ 2 ( y ) & = 1 , & \\forall z \\in A \\ , . \\ , \\psi _ 1 ( z ) = 0 \\psi _ 2 ( z ) = 0 . \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\left \\vert \\dfrac { R T _ { i , i _ { 0 } } } { R l _ { i _ { 0 } } t _ { i _ { 0 } } } - \\dfrac { H _ { i } } { K } \\right \\vert = \\left \\vert \\dfrac { R T _ { i , i _ { 0 } } K - H _ { i } R l _ { i _ { 0 } } t _ { i _ { 0 } } } { K R l _ { i _ { 0 } } t _ { i _ { 0 } } } \\right \\vert = \\epsilon \\pounds \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} ( \\bar { U } ( t , y ) , 0 ) = ( e ^ { \\nu t \\partial _ { y y } } U ( y ) , 0 ) . \\end{align*}"}
-{"id": "10162.png", "formula": "\\begin{align*} K : = \\limsup _ { n \\rightarrow + \\infty } n \\mathbb P ( - Z _ 1 > b _ n ) < + \\infty \\ , , \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} v _ \\beta ^ 2 ( g _ 2 ) = [ I - \\beta P ( g _ 2 ) ] ^ { - 1 } \\tilde { r } ( g _ 2 ) = \\left [ \\frac { 5 - 2 p + 7 \\beta } { 1 - \\beta ^ 2 } , \\frac { ( 5 - 2 p ) \\beta + 7 } { 1 - \\beta ^ 2 } \\right ] ^ T . \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} g ( x ( 0 ) ) = 0 \\implies \\forall t \\geq 0 : g ( x ( t ) ) = 0 \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\log \\Gamma _ M ( w \\ , | \\ , a ) = & - \\frac { 1 } { M ! } B _ { M , M } ( w \\ , | \\ , a ) \\ , \\log ( w ) + \\sum \\limits _ { k = 0 } ^ M \\frac { B _ { M , k } ( 0 \\ , | \\ , a ) ( - w ) ^ { M - k } } { k ! ( M - k ) ! } \\sum \\limits _ { l = 1 } ^ { M - k } \\frac { 1 } { l } + \\\\ & + R _ M ( w \\ , | \\ , a ) , \\\\ R _ M ( w \\ , | \\ , a ) = & O ( w ^ { - 1 } ) , \\ , | w | \\rightarrow \\infty , \\ , | \\arg ( w ) | < \\pi . \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\| \\phi \\circ T - \\phi \\| \\le \\lim _ { N \\to \\infty } \\| \\frac { 1 } { N } ( \\theta - \\theta \\circ T ^ N ) \\| = 0 , \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} f ' _ \\beta = \\begin{cases} f _ \\beta & \\beta \\in \\mathcal { B } ; \\\\ 0 & \\beta \\in \\mathcal { D } \\setminus \\mathcal { B } . \\end{cases} \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} d ( I - ( v ' \\sigma ^ { - 1 } \\gamma + w ' \\sigma ^ { - 1 } c ) + ( v \\sigma ^ { - 1 } \\gamma + w \\sigma ^ { - 1 } c ) ^ 2 - \\cdots ) \\sigma ^ { - 1 } ( v ' \\beta + w ' b ) = 0 , \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} c _ { f , \\overline { g } } = \\frac { 1 } { 4 \\pi ^ 2 ( 2 \\kappa + \\frac { 3 } { 2 } - 2 \\nu ) } \\sum _ { n \\geq 1 } \\frac { a _ f ( n ) a _ g ( n ) } { n ^ { 2 \\kappa + \\frac { 3 } { 2 } - 2 \\nu } } . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ + _ { V I I } } & d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , \\alpha } \\left ( s + k - 1 \\right ) T R ^ d \\theta ^ { ( d - 1 ) / 2 } \\end{aligned} \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} \\phi _ { \\nu } ( N ) = \\left \\{ \\begin{array} { l l } 1 - p ^ { - 1 } , & \\hbox { i f $ N = p ^ { \\nu } , \\ , \\nu \\geq 3 $ ; } \\\\ 1 - ( p - p ^ { - 1 } ) ^ { - 1 } , & \\hbox { i f $ N = p ^ { \\nu } , \\ , \\nu = 2 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\phi _ v ( g ) = \\begin{cases} g v & g \\in J , \\\\ 0 & g \\notin J . \\end{cases} \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} H ( x , 2 \\rho ( 1 + \\varepsilon _ 1 ) ) \\geq C _ 9 \\rho ^ { n - 1 } \\sup \\limits _ { B ( x , 2 \\rho ) } | u | ^ 2 = C _ 9 2 ^ { 2 N ( x , \\rho ) } \\rho ^ { n - 1 } \\sup \\limits _ { B ( x , \\rho ) } | u | ^ 2 . \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} \\frac { ( 4 - 7 \\delta _ 2 ) \\varepsilon } { 7 } - \\frac { 7 \\delta _ 2 } { 2 } & < \\frac { 4 - 7 \\delta _ 2 } { 7 } \\cdot \\frac { 4 9 \\delta _ 2 } { 8 - 1 4 \\delta _ 2 } - \\frac { 7 \\delta _ 2 } { 2 } \\\\ & = \\frac { 7 \\delta _ 2 } { 2 } - \\frac { 7 \\delta _ 2 } { 2 } = 0 , \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} \\varphi ( x / ( x - 1 ) ) = ( 1 - x ) ^ r \\varphi ( x ) , \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} T ^ { ( p - q ) } ( \\vartheta L , \\vartheta B ) = \\vartheta T ^ { ( p - q ) } ( L , B ) \\quad \\mbox { f o r } \\vartheta \\in { \\rm S O } ( 2 ) . \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} F ( v _ { k } + \\sp \\{ v _ { 1 } , \\dots , v _ { k - 1 } \\} ) = F ( v _ { k } ) + \\sp \\{ F ( v _ { 1 } ) , \\dots , F ( v _ { k - 1 } ) \\} \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} \\delta ( A , \\Theta ) & = 2 ( g - 7 ) H ( A , \\Theta ) - 2 \\Lambda ( A , \\Theta ) - 4 g \\log 2 \\pi , \\\\ \\varphi ( A , \\Theta ) & = ( g + 5 ) H ( A , \\Theta ) - \\Lambda ( A , \\Theta ) + 2 g \\log 2 \\pi . \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} G = \\begin{bmatrix} B & 0 & 0 & 0 \\\\ 0 & B & 0 & 0 \\\\ 0 & 0 & B & 0 \\\\ 0 & 0 & 0 & B \\end{bmatrix} \\begin{bmatrix} H _ { 1 1 } & H _ { 1 2 } & H _ { 1 3 } & H _ { 1 4 } \\\\ H _ { 2 1 } & H _ { 2 2 } & H _ { 2 3 } & H _ { 2 4 } \\\\ H _ { 3 1 } & H _ { 3 2 } & H _ { 3 3 } & H _ { 3 4 } \\\\ H _ { 4 1 } & H _ { 4 2 } & H _ { 4 3 } & H _ { 4 4 } \\end{bmatrix} \\begin{bmatrix} B ^ \\top & 0 & 0 & 0 \\\\ 0 & B ^ \\top & 0 & 0 \\\\ 0 & 0 & B ^ \\top & 0 \\\\ 0 & 0 & 0 & B ^ \\top \\end{bmatrix} . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} \\mathbf { L } | _ { C _ t } \\ \\textit { i s l o c a l l y f r e e } , \\ \\ \\ t \\in Y ^ * : = Y \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { q ^ { n ^ 2 + 2 ( i - 1 ) n } ( - q ; q ^ 2 ) _ n } { ( q ^ 2 ; q ^ 2 ) _ n } = \\frac { 1 } { ( q ^ { 2 i - 1 } ; q ^ 8 ) _ \\infty ( q ^ 4 ; q ^ 8 ) _ \\infty ( q ^ { 9 - 2 i } ; q ^ 8 ) _ \\infty } . \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{gather*} Z ( s , f , \\chi , \\Delta _ 5 ) = \\sum _ { n = 1 } ^ { \\infty } \\int \\limits _ { \\pi ^ { 3 n } O _ v ^ { \\times } \\times \\pi ^ { 2 n } O _ v ^ { \\times } } \\chi ( a c \\ f ( x , y ) ) \\ | f ( x , y ) | ^ s | d x d y | \\\\ = \\sum _ { n = 1 } ^ { \\infty } q ^ { - 5 n - 1 2 n s } \\int \\limits _ { O _ v ^ { \\times 2 } } \\chi ( a c ( ( y ^ 3 - x ^ 2 ) ^ 2 + \\pi ^ { 8 n } x ^ 4 y ^ 4 ) ) | ( y ^ 3 - x ^ 2 ) ^ 2 + \\pi ^ { 8 n } x ^ 4 y ^ 4 | ^ s \\ | d x d y | . \\end{gather*}"}
-{"id": "1028.png", "formula": "\\begin{align*} e _ n ^ j ( z _ { 1 , 1 } , \\dots , z _ { 1 , n } ) & = e _ n ^ j ( z _ { i , 1 } , \\dots , z _ { i , n } ) , & 3 \\leq i \\leq m , \\ 1 \\leq j \\leq n - h - 1 ; \\\\ z _ { 1 , j } & = z _ { 2 , j } , & 1 \\leq j \\leq n ; \\\\ z _ { 1 , n - h - 2 } & = z _ { 1 , j } , & n - h - 1 \\leq j \\leq n . \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} { \\partial } _ { t } u - \\frac { 1 } { } \\nabla \\cdot \\left ( \\frac { 1 } { a } \\nabla u \\right ) = 0 \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} \\mathcal { M } = \\{ ( \\mathbf { w } , \\mathbf { L } ) : & \\\\ & F F P _ { \\mathbf { w } } \\mathcal { E } \\} . \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} P _ 1 u _ 1 = f _ 1 , { u _ 1 } _ { | S } = ( g _ 1 + g _ 2 ) / 2 , \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} \\mathcal { D } = \\{ D \\in \\mathcal { B } : \\mbox { t h e r e e x i s t s $ m \\in \\natural $ s u c h t h a t $ \\bigcup _ { i = m } ^ { \\infty } E _ i \\subset D $ } \\} . \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} \\hat \\theta _ T ^ { ( 1 ) } = - \\left ( \\frac { 1 } { H \\Gamma ( 2 H ) T } \\int _ 0 ^ T X _ t ^ 2 d t \\right ) ^ { - \\frac { 1 } { 2 H } } \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} \\tilde { k } _ m ^ { v , \\infty } : = \\sup _ { s \\in ( 0 , \\infty ) } e ^ { \\tilde { \\beta } _ v s } \\norm { v _ m ( s ) } _ { V _ { \\gamma } } , \\tilde { k } _ m ^ { \\tau , \\infty } : = \\sup _ { s \\in ( 0 , \\infty ) } e ^ { \\tilde { \\beta } _ { \\tau } s } \\norm { \\tau _ m ( s ) } _ { \\hat { V } _ { \\gamma } } , m \\in \\N _ 0 . \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} | S ^ \\nu _ { h _ 2 } ( n ) | ^ 2 = | S ^ \\nu _ f ( n ) | ^ 2 + | S ^ \\nu _ g ( n ) | ^ 2 + 2 \\Im \\left ( S ^ \\nu _ f ( n ) \\overline { S ^ \\nu _ g ( n ) } \\right ) . \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} \\sum \\limits _ j { \\frac { m + i } { 2 } - 1 \\brace j - 1 } \\gamma _ { 2 j - 2 } = \\sum \\limits _ j { \\frac { m + i } { 2 } - 1 \\brace j } \\gamma _ { 2 j } = \\gamma _ { m + i - 1 } , \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} \\epsilon ^ 2 \\int _ M F ^ { i j } & h _ { k i } h ^ k _ j f ^ p \\leq \\{ \\delta ^ { - 1 } c ( p - 1 ) + c \\} \\int _ M F ^ { i j } f _ i f _ j f ^ { p - 2 } \\\\ & + \\{ \\delta c ( p - 1 ) + c \\} \\int _ M \\abs { D A } ^ 2 F ^ { - \\alpha } f ^ { p - 1 } + 2 C \\int _ M f ^ p . \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} | D \\tilde { u } | ^ 2 = \\sigma ^ { i j } \\tilde { u } _ i \\tilde { u } _ j . \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} \\Big | \\int _ { B _ { 2 \\rho } \\setminus \\overline { B _ \\rho } } T _ 2 ( \\nabla \\varphi , \\nabla \\Delta \\varphi ) \\Big | \\leq & C \\int _ { B _ { 2 \\rho } \\setminus \\overline { B _ \\rho } } | \\nabla \\varphi | | \\nabla \\Delta \\varphi | d \\mu _ g \\\\ \\leq & C \\int _ { B _ { 2 \\rho } \\setminus \\overline { B _ \\rho } } | u _ \\epsilon ' | | ( \\Delta u _ \\epsilon ) ' | d x + O ( \\epsilon ^ { n - 6 } ) = O ( \\epsilon ^ { n - 6 } ) . \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} n ^ { 2 ( p + q ) + 4 } \\cdot \\theta ^ { | \\mathbf { J _ n } | - ( p + q + 4 ) } = o ( n ^ { - 1 / 2 } ) \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} S _ 0 - \\sum _ { i = 1 } ^ { k + 1 } S _ i = k ^ 2 ( k + 1 ) \\int _ X u \\ , \\sigma _ { k } ( D ^ 2 w ^ 1 , \\dotsc , D ^ 2 w ^ { k } ) d x + \\sum _ { i = 1 } ^ { 2 k + 1 } U _ i \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} ( \\Delta \\otimes \\iota ) U _ \\pi = U _ { \\pi , 1 3 } U _ { \\pi , 2 3 } . \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} p _ { 0 , 0 } = \\left [ \\frac { \\xi } { A \\gamma } + e ^ { \\frac { \\lambda } { \\xi } } \\left ( - \\frac { \\gamma } { \\xi } B + \\left ( \\frac { 1 } { A } - \\frac { \\xi } { A \\mu } + \\frac { \\gamma } { \\mu } \\right ) E \\right ) \\right ] ^ { - 1 } \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} \\Psi _ n ( z , \\eta ) = z ^ { - n } \\left [ e ^ { - ( z - \\eta ) ^ 2 } - e ^ { - \\eta ^ 2 } \\sum \\limits _ { r = 0 } ^ { n - 1 } z ^ { r } P _ { r } ( \\eta ) \\right ] , \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} I m \\ , \\big ( F ' _ t \\ , \\overline { F ' } \\ , - \\ , G ' \\ , \\overline { G ' _ t } \\big ) _ t = 0 \\ , . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} \\min _ { j = 0 , \\hdots , N - 1 } \\norm { \\mathcal { G } _ t ( x _ j ) } ^ 2 \\le \\frac { 1 } { N } \\sum _ { j = 0 } ^ { N - 1 } \\norm { \\mathcal { G } _ t ( x _ j ) } ^ 2 & \\le \\frac { 2 t ^ { - 1 } \\left ( \\sum _ { j = 0 } ^ { N - 1 } F ( x _ j ) - F ( x _ { j + 1 } ) + \\sum _ { j = 0 } ^ { N - 1 } \\varepsilon _ { j + 1 } \\right ) } { N } \\\\ & \\le \\frac { 2 t ^ { - 1 } \\big ( F ( x _ 0 ) - F ^ * + \\sum _ { j = 0 } ^ { N - 1 } \\varepsilon _ { j + 1 } \\big ) } { N } . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} & r ^ * ( X ) + r ^ * ( Y ) \\\\ = & r ( E - X ) + | | X | | _ r - r ( E ) + r ( E - Y ) + | | Y | | _ r - r ( E ) \\\\ \\geq & r ( E - ( X \\cup Y ) ) + r ( E - ( X \\cap Y ) ) + | | X \\cup Y | | _ r + | | X \\cap Y | | _ r - 2 r ( E ) \\\\ = & r ^ * ( X \\cup Y ) + r ^ * ( X \\cap Y ) . \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} \\dot { \\eta } = - \\frac { \\bar { H } } { n } \\eta , \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} \\mathcal J ( \\varphi | _ { H _ 0 \\cap H ' } ) = \\mathcal J ( \\varphi | _ { H ' } ) | _ { H _ 0 \\cap H ' } = \\mathcal J ( \\varphi ) | _ { H _ 0 \\cap H ' } \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} \\left \\vert \\dfrac { R T _ { i , i _ { 0 } } } { R l _ { i _ { 0 } } t _ { i _ { 0 } } } - x _ { i } \\right \\vert = \\varepsilon \\pounds \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} \\Sigma _ { 1 } ( \\xi _ 1 , \\xi _ 2 ) = \\frac 1 { 2 \\pi i } \\int _ { ( \\sigma ) } \\ ! \\hat f _ 1 ( s ) Z _ 1 ( s ) \\ , d s , \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\tan \\left ( \\frac { \\pi s } { 2 p } + \\frac { \\pi r } { 2 } \\right ) \\tan \\left ( \\frac { \\pi } { 2 p } \\right ) = \\tan \\left ( \\frac { \\pi ( 2 k + 1 ) s } { 2 p } + \\frac { \\pi r } { 2 } \\right ) \\tan \\left ( \\frac { \\pi ( 2 k + 1 ) } { 2 p } \\right ) . \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} c ( \\gamma , \\delta ) = \\frac { 1 } { d ( \\gamma , \\delta ) } \\left ( q _ { \\delta } + \\sum _ { j = 1 } ^ { p - 1 } \\sum _ { \\gamma _ { 1 } , \\gamma _ { 2 } , . . . , \\gamma _ { j } } \\frac { q _ { \\gamma _ { 1 } } q _ { \\gamma _ { 2 } } . . . q _ { \\gamma _ { j } } q _ { \\delta - \\gamma ( j ) } } { d ( \\gamma , \\delta - \\gamma _ { 1 } ) d ( \\gamma , \\delta - \\gamma ( 2 ) ) . . . d ( \\gamma , \\delta - \\gamma ( j ) ) } \\right ) \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{gather*} W _ 1 = \\Lambda ^ 3 V ^ * , W _ 2 = V ^ { 2 , 1 } , W _ 3 = V _ { 1 , 1 } ^ 1 , W _ 4 = V , \\\\ W _ 5 = \\Lambda ^ 3 V , W _ 6 = V _ { 2 , 1 } , W _ 7 = V ^ { 1 , 1 } _ 1 , W _ 8 = V ^ * . \\end{gather*}"}
-{"id": "9278.png", "formula": "\\begin{align*} p _ { i } ( \\sigma , 1 ) = a _ { \\sigma , i } + n - i \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ { \\infty } \\varphi _ { k } \\left ( \\omega q ^ { 2 m } \\right ) \\varphi _ { k } \\left ( \\omega q ^ { 2 n } \\right ) = \\omega ^ { - 4 n } q ^ { - 2 n ( 2 n - 1 ) } \\| \\varphi \\left ( \\omega \\right ) \\| ^ { 2 } \\ , \\delta _ { m , n } \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} & H ' ( u ( t ) ) \\frac { d } { d t } ( k * u ) ( t ) = \\frac { d } { d t } ( k * H ( u ) ) ( t ) + ( - H ( u ( t ) ) + H ' ( u ( t ) ) u ( t ) ) k ( t ) \\\\ & + \\int _ { 0 } ^ { t } ( H ( u ( t - s ) ) - H ( u ( t ) ) - H ' ( u ( t ) ) [ u ( t - s ) - u ( t ) ] ) \\left ( - \\frac { d k ( s ) } { d s } \\right ) d s , \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\mu _ i \\\\ \\nu _ i \\end{array} \\right ] ^ H \\Phi \\left [ \\begin{array} { c } \\mu _ i \\\\ \\nu _ i \\end{array} \\right ] = \\left [ \\begin{array} { c } s _ i \\\\ t _ i \\end{array} \\right ] ^ H \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ] \\left [ \\begin{array} { c } s _ i \\\\ t _ i \\end{array} \\right ] = \\bar s _ i t _ i + s _ i \\bar t _ i = 0 \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} H _ { \\alpha } \\psi ( x ) \\equiv \\left [ - \\frac { d ^ { \\alpha } } { d x ^ { \\alpha } } + x ^ { 2 } \\right ] \\psi ( x ) = E \\psi ( x ) , 1 < \\alpha \\leq 2 . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} V _ N ( 0 , v , k ) \\ll \\begin{cases} \\frac { l ^ { 1 / 2 } ( l T ) ^ { \\epsilon } } { N } \\max { \\left ( \\frac { \\sqrt { T } } { k } , \\frac { 1 } { \\sqrt { k } } \\right ) } & 4 \\pi e l T \\geq N k , \\\\ \\frac { 1 } { \\sqrt { l T } } \\left ( 2 \\pi e \\frac { l T } { N k } \\right ) ^ k & 4 \\pi e l T < N k . \\end{cases} \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { ( 3 n ^ { 2 } - n ) / 2 + 2 m n } } { \\left ( q , q ^ { m + 1 / 2 } , - q ^ { m + 1 / 2 } ; q \\right ) _ { n } } = \\frac { ( - 1 ) ^ { m } q ^ { - m ( m - 1 ) } } { \\left ( q ^ { 2 m + 1 } ; q ^ { 2 } \\right ) _ { \\infty } } \\left \\{ \\frac { a _ { m } ( q ^ { 2 } ) } { ( q ^ { 2 } , q ^ { 8 } ; q ^ { 1 0 } ) _ { \\infty } } - \\frac { b _ { m } ( q ^ { 2 } ) } { ( q ^ { 4 } , q ^ { 6 } ; q ^ { 1 0 } ) _ { \\infty } } \\right \\} . \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\hat { h } _ i ^ j = g _ { i k } \\hat { h } ^ { k j } , \\tilde { h } _ i ^ j = \\tilde { g } ^ { k j } \\tilde { h } _ { k i } , \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} \\langle \\varphi , f ( t ) v \\rangle = \\langle A \\varphi , f ( t ) w \\rangle = 0 . \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ d t ^ { - a } } { ( 1 - t ) ^ d } \\varphi ( 1 - t ) = H ( t ^ { - 1 } ) = \\frac { 1 } { ( 1 - t ^ { - 1 } ) ^ d } \\varphi ( 1 - t ^ { - 1 } ) = \\frac { ( - t ) ^ d } { ( 1 - t ) ^ d } \\varphi ( ( t - 1 ) / t ) . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} C _ { } & \\leq \\sup _ { \\{ p ( x _ t | s _ { t - 1 } , q _ { t - 1 } ) \\} _ { t \\geq 1 } } \\liminf _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N I ( X _ i , S _ { i - 1 } ; Y _ i | Q _ { i - 1 } ) , \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} h ( z ) : = \\sum _ { \\gamma \\in H \\backslash \\Gamma } \\det ( \\gamma ) ( c z + d ) ^ { - q - 1 } t ( \\gamma z ) , \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} D _ { r , \\xi , z } Y ( s , y ) = G ( t - s , x - y ) \\left [ \\sigma ( u _ n ( s , y ) + D _ { r , \\xi , z } u _ n ( s , y ) ) - \\sigma ( u _ n ( s , y ) ) \\right ] 1 _ { [ 0 , t ] } ( s ) . \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} \\dd \\hat { \\zeta } _ t = \\dd \\nu _ t \\otimes \\hat { X } _ { t - } + \\nu _ { t - } \\otimes \\dd \\hat { X } _ t + \\dd [ \\nu , \\hat { X } ] ^ \\otimes _ t . \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} & \\| h \\| _ { C ^ { 0 , \\alpha } } : = \\| h \\| _ { L ^ \\infty } + \\sup _ { x \\neq y \\in \\mathbb R ^ d } \\frac { | h ( x ) - h ( y ) | } { | x - y | ^ \\alpha } \\\\ & \\| h \\| _ { C ^ { 1 , \\alpha } } : = \\| h \\| _ { L ^ \\infty } + \\| \\nabla h \\| _ { L ^ \\infty } + \\sup _ { x \\neq y \\in \\mathbb R ^ d } \\frac { | \\nabla h ( x ) - \\nabla h ( y ) | } { | x - y | ^ \\alpha } , \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} J & = \\frac { f ^ p } { 1 - \\alpha p } - \\left ( \\frac { p } { p - 1 } \\right ) ^ q ( 1 - \\alpha ) ^ q \\frac { f ^ p } { 1 - \\alpha p } = \\\\ & = - \\frac { f ^ p } { 1 - \\alpha p } \\left [ \\left ( \\frac { p } { p - 1 } \\right ) ^ q ( 1 - \\alpha ) ^ q - 1 \\right ] = - f ^ p \\ , G ( \\alpha ) , \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} \\norm { \\zeta ( t ) } _ { H ^ 1 } ^ 2 + \\int _ 0 ^ t \\norm { \\Delta \\zeta ( s ) } ^ 2 d s & \\leq \\norm { b } _ { H ^ 1 } + B _ { L ^ 2 } ^ { \\zeta } ( t ) + C \\tilde { B } _ { H ^ 1 } ^ { \\zeta } ( t ) \\int _ 0 ^ t \\norm { \\Delta v ( s ) } ^ 2 d s + 3 \\int _ 0 ^ t \\norm { g ( s ) } ^ 2 d s \\\\ & \\leq \\norm { b } _ { H ^ 1 } + B _ { L ^ 2 } ^ { \\zeta } ( t ) + C \\tilde { B } _ { H ^ 1 } ^ { \\zeta } ( t ) B _ { H ^ 1 } ^ { v } ( t ) + 3 \\int _ 0 ^ t \\norm { g ( s ) } ^ 2 d s = : B _ { H ^ 1 } ^ { \\zeta } ( t ) . \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} \\theta _ { q } ( x ) : = \\left ( x , q x ^ { - 1 } ; q \\right ) _ { \\infty } = \\frac { 1 } { ( q ; q ) _ \\infty } \\sum _ { n = - \\infty } ^ \\infty q ^ { \\binom { n } { 2 } } ( - x ) ^ n , x \\in \\C , \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} p _ a ^ 8 = \\sum _ { i \\geq 1 } \\lambda _ i ^ 4 . \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} b _ { 2 , 1 } ^ { \\ast } & = b _ { 2 , 1 } \\left ( z _ { 3 } \\right ) , \\\\ b _ { 2 , 2 } ^ { \\ast } & = b _ { 2 , 2 } \\left ( z _ { 3 } \\right ) , \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} \\mu _ i ( G ) + \\mu _ { n - i + 2 } ( \\overline { G } ) \\le \\mu _ 2 ( K _ n ) = - 1 , \\mbox { f o r } i \\ge 2 . \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} \\tilde { \\Delta } _ t = \\partial _ { z } ^ 2 + a ^ 2 \\partial _ { v v } ^ L = \\Delta _ L + ( a ^ 2 - 1 ) \\partial _ { v v } ^ L . \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\begin{array} { c c l } K _ { ( v : m ) } & = & K _ { ( v : m ) } \\otimes K _ { ( 1 : m ) } \\\\ & = & K _ { ( v : m ) } \\otimes \\left ( \\bigoplus _ { t = 1 } ^ { \\frac { m - 1 } { 2 } } N _ t \\right ) \\\\ & = & \\bigoplus _ { t = 1 } ^ { \\frac { m - 1 } { 2 } } \\left ( K _ { ( v : m ) } \\otimes N _ t \\right ) \\\\ \\end{array} \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} u ( t ) = \\sum _ { n \\in \\N ^ d } a ( n ) t ^ n . \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} H ^ { d + 3 } _ { } ( Y , \\mathbb { Q } / \\mathbb { Z } ( d + 2 ) ) & = ( H ^ { d + 3 } ( Y , \\mathbb { Q } / \\mathbb { Z } ( d + 2 ) ) / ( H ^ { d + 3 } ( K _ 0 , \\mathbb { Q } / \\mathbb { Z } ( d + 2 ) ) ) \\\\ & \\rightarrow \\prod _ { v \\in X _ 0 ^ { ( 1 ) } } H ^ { d + 3 } ( Y _ { K _ { 0 , v } } , \\mathbb { Q } / \\mathbb { Z } ( d + 2 ) ) / ( H ^ { d + 3 } ( K _ { 0 , v } , \\mathbb { Q } / \\mathbb { Z } ( d + 2 ) ) ) \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{gather*} v _ { k } ^ { ( \\alpha ) } = \\frac { h _ { k + 1 } ^ { ( \\alpha ) } } { h _ { k } ^ { ( \\alpha + 1 ) } } , w _ { k } ^ { ( \\alpha ) } = \\frac { h _ { k } ^ { ( \\alpha + 1 ) } } { h _ { k } ^ { ( \\alpha ) } } , v _ { - 1 } ^ { ( \\alpha ) } = \\frac 1 { h _ { - 1 } ^ { ( \\alpha ) } } = 0 . \\end{gather*}"}
-{"id": "2945.png", "formula": "\\begin{align*} & \\widetilde { g _ n ^ { ( t , x ) } } ( t _ 1 , x _ 1 , \\ldots , t _ n , x _ n , s , y ) \\\\ & = \\frac { 1 } { n + 1 } \\Big [ g _ n ^ { ( t , x ) } ( t _ 1 , x _ 1 , \\ldots , t _ n , x _ n , s , y ) \\\\ & + \\sum _ { i = 1 } ^ { n } g _ n ^ { ( t , x ) } ( t _ 1 , x _ 1 , \\ldots , t _ { i - 1 } , x _ { i - 1 } , s , y , t _ { i + 1 } , x _ { i + 1 } , \\ldots , t _ n , x _ n , t _ i , x _ i ) \\Big ] \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} \\left ( \\mu ( x ) - \\mu ( y ) \\right ) \\sum _ { j = - \\infty } ^ { n } g _ { j } ( x ) g _ { j } ( y ) = W _ { n } ( g ( x ) , g ( y ) ) . \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} & T _ { \\xi _ { 0 } } \\circ \\cdots \\circ T _ { \\xi _ { N - 1 } } ( I ) \\cap T _ { \\omega _ { 0 } } \\circ \\cdots \\circ T _ { \\omega _ { N - 1 } } ( I ) = \\emptyset \\mbox { a n d } \\\\ & T _ { \\xi _ { 0 } } \\circ \\cdots \\circ T _ { \\xi _ { N - 1 } } ( I ) \\cup T _ { \\omega _ { 0 } } \\circ \\cdots \\circ T _ { \\omega _ { N - 1 } } ( I ) \\subset J . \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} h _ \\rho ( 0 , x , p ( 0 , t ; y ( 0 , t ; x , \\xi ) , \\xi ) ) & = E ( 0 ) = E ( t ) \\\\ & = h _ \\rho ( t , y ( 0 , t ; x , \\xi ) , \\xi ) . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { \\int _ 0 ^ T X _ t ^ 2 \\ , d t } { e ^ { 2 \\theta T } } = \\lim _ { T \\to \\infty } \\frac { X _ T ^ 2 } { 2 \\theta e ^ { 2 \\theta T } } = \\frac { \\xi _ \\theta ^ 2 } { 2 \\theta } . \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} D ^ { F , 2 } _ { X } = d ^ { F } d ^ { F , * } + d ^ { F , * } d ^ { F } : \\ ; \\Omega ^ p ( X , F ) \\rightarrow \\Omega ^ p ( X , F ) . \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} w ( \\Gamma ) = \\frac { 1 } { \\prod _ { x \\in V \\cup E } m ( x ) } \\prod _ { v \\in V } \\left ( p _ { m ( v ) } [ ] , p _ { \\lambda _ E ^ { ( v ) } } [ X ] \\right ) _ X , \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} u ^ { 2 * } ( s ) = \\bar { r } ^ 2 ( s , a _ s ^ 2 ) + \\beta \\sum _ { s ' \\in S } p ^ 2 ( s ' | s , a _ s ^ 2 ) u ^ { 2 * } ( s ' ) , \\ \\forall \\ s \\in S . \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} P _ e ^ { ( k ) } = \\P \\{ \\hat { f } _ \\ell ( X ^ k , Y ^ k ) \\neq f _ \\ell ( X ^ k , Y ^ k ) \\ell \\in [ 1 : L ] \\} . \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} c _ j : = \\ ; { \\mathcal L } _ j \\ ; \\ ; { \\mathcal Q } _ { k - 1 } . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} G _ 0 ( x ^ p ) = A ( x ) G _ 0 ( x ) A _ 0 ^ { - 1 } . \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} \\mathcal { H } : = L ^ { 2 } ( G , \\mathbb { R } ^ { k } ) / \\{ \\mathbf { 1 } \\} \\equiv \\Big \\{ \\mathbf { u } \\in L ^ { 2 } ( G , \\mathbb { R } ^ { k } ) \\ , \\big | \\ , \\int _ { G } \\mathbf { u } \\ , \\mathrm { d } \\mathbf { x } = \\mathbf { 0 } \\Big \\} , \\mathcal { V } : = H ^ { 1 } ( G , \\mathbb { R } ^ { k } ) \\cap \\mathcal { H } . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} R _ i = \\ker ( V _ i \\otimes _ k V _ { i + 1 } \\otimes _ k V _ { i + 2 } \\to A _ { i , i + 3 } ) , \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ \\gamma ( z , \\bar z ) = ( - 1 ) ^ { m } ( m \\wedge n ) ! ( \\gamma + m + 1 ) _ n | z | ^ { | m - n | } e ^ { i [ ( n - m ) \\arg z ] } \\mathrm { P } ^ { ( | m - n | , \\gamma ) } _ { m \\wedge n } ( 1 - 2 | z | ^ 2 ) . \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } & f _ i \\left ( x _ i , \\sum _ { i = 1 } ^ N h _ i ( x _ i ) \\right ) \\cr \\mbox { s u b j e c t t o } & x _ i \\in K _ i , \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\pi i } \\partial \\overline { \\partial } \\varphi = \\int _ { q _ 2 } h ^ 3 - e ^ A _ 1 \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} D _ i & = ( r + ( i - 1 ) \\gamma , r + ( i - 1 ) \\gamma + \\beta ) , \\\\ E _ i & = ( i \\gamma , i \\gamma + \\beta ) . \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} ( H ^ 1 ( \\mathbb { R } ^ d ) , B ^ { 1 + s } _ { 2 , q } ( \\mathbb { R } ^ d ) ) _ { t , 2 } = H ^ { 1 + t s } ( \\mathbb { R } ^ d ) , \\ , \\ , \\ , \\ , \\ , & ( H ^ { - 1 } ( \\mathbb { R } ^ d ) , B ^ { - 1 + s } _ { 2 , q } ( \\mathbb { R } ^ d ) ) _ { t , 2 } = H ^ { 1 + t s } ( \\mathbb { R } ^ d ) , \\\\ ( B ^ { - s _ 1 } _ { 2 , q } ( \\mathbb { R } ^ d ) , B ^ { - s _ 2 } _ { 2 , q } ( \\mathbb { R } ^ d ) ) _ { t , 2 } = & H ^ { - ( 1 - t ) s _ 1 - t s _ 2 } ( \\mathbb { R } ^ d ) . \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{gather*} \\prod _ { i = 1 } ^ { k } c ^ { ( \\alpha ) } _ { i } ( \\sigma f ( z _ { 1 } , z _ { 2 } , \\dots , z _ { k } ) ) = \\prod _ { i = 1 } ^ { k } c ^ { ( \\alpha ) } _ { i } ( f ( z _ { 1 } , z _ { 2 } , \\dots , z _ { k } ) ) . \\end{gather*}"}
-{"id": "93.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { - \\frac { 1 } { 2 } - i \\infty } ^ { - \\frac { 1 } { 2 } + i \\infty } \\zeta _ K ( s + 1 ) \\frac { x ^ s } { s } \\frac { n _ K ! } { \\prod _ { j = 1 } ^ { n _ K } ( s + j ) } d s = \\frac { n _ K ! } { 2 \\pi i } \\int _ { - \\frac { 1 } { 2 } - i \\infty } ^ { - \\frac { 1 } { 2 } + i \\infty } \\zeta _ K ( s + 1 ) \\frac { \\Gamma ( s ) } { \\Gamma ( n _ K + 1 + s ) } x ^ s d s . \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} S ^ a _ { \\alpha p } = 0 , 1 \\leq \\alpha \\leq 4 , 1 \\leq p \\leq 3 , \\forall a = 1 , \\cdots , 7 . \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} H ( X ) & = \\tfrac { 1 } { ( g ! ) ^ { 2 } } \\int _ { X ^ { g } } \\log \\| \\theta \\| ( P _ { 1 } + \\dots + P _ { g } - Q ) \\Phi ^ { * } \\nu ^ { g } = S _ { g } ( X ) + \\tfrac { 1 } { 4 } \\varphi ( X ) , \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} W ^ { ( i ) } = W ^ { ( i ) } _ { \\alpha } | _ { \\alpha = k + n - 1 } . \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} \\textbf { Z } _ 1 \\otimes \\textbf { P } \\textbf { C } & = \\textbf { g } _ { 1 } - \\varphi ( x _ 0 ) \\textbf { I } _ { N _ t \\times 1 } , \\\\ \\textbf { Z } _ 2 \\otimes \\textbf { P } \\textbf { C } & = \\textbf { g } _ { 2 } - \\varphi ( x _ { N _ h } ) \\textbf { I } _ { N _ t \\times 1 } , \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\mathcal { N } ( x , y , l ) = \\binom { x } { l } \\binom { y } { l } , \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } E ( Z _ n ) = \\lim _ { n \\rightarrow \\infty } ( \\sum _ { j = 1 } ^ { n } \\frac { 1 } { j } - \\ln ( n ) ) = \\gamma . \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 x y : n ) } = \\bigoplus _ { ( i , \\alpha , \\gamma ) } H _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) \\varphi ( i , \\alpha , \\gamma ) . \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} d ( \\gamma , \\delta ) = \\mid \\gamma + t \\mid ^ { 2 } - \\mid \\gamma + \\delta + t \\mid ^ { 2 } \\neq 0 , \\forall \\delta \\neq 0 . \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} q _ Y ( y ) = \\left \\{ \\begin{array} { c c } \\frac { \\beta } { 2 A } & | y | \\leq A \\\\ \\frac { 1 - \\beta } { \\sqrt { 2 \\pi \\sigma ^ 2 } } e ^ { - \\frac { ( | y | - A ) ^ 2 } { 2 \\sigma ^ 2 } } & | y | > A \\end{array} \\right . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{gather*} \\exp ( x _ \\rho \\otimes \\partial ^ \\rho ) ( 1 \\otimes x _ \\mu ) \\exp ( - x _ \\sigma \\otimes \\partial ^ \\sigma ) = 1 \\otimes x _ \\mu + x _ \\mu \\otimes 1 , \\\\ \\exp ( x _ \\rho \\otimes \\partial ^ \\rho ) ( x _ \\mu \\otimes 1 ) \\exp ( - x _ \\sigma \\otimes \\partial ^ \\sigma ) = x _ \\mu \\otimes 1 . \\end{gather*}"}
-{"id": "2618.png", "formula": "\\begin{align*} \\mathbf { \\mathcal { W } } ( u ) = ( \\pm 2 ^ { \\frac { n + 1 } { 2 } } H ^ { ( r ) } _ { 2 ^ { p - 2 } } , \\textbf { 0 } _ { 2 ^ { p - 2 } } ) \\quad \\mbox { o r } \\mathbf { \\mathcal { W } } ( u ) = ( \\textbf { 0 } _ { 2 ^ { p - 2 } } , \\pm 2 ^ { \\frac { n + 1 } { 2 } } H ^ { ( r ) } _ { 2 ^ { p - 2 } } ) \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} q _ t ( d y _ t | y _ { t - 1 } , x _ t ) = \\bordermatrix { & 0 , 0 & e , 0 & 1 , 0 & 0 , 1 & e , 1 & 1 , 1 \\cr 0 & \\alpha _ t & \\gamma _ t & \\beta _ t & 0 & 0 & 0 \\cr e & 1 - \\alpha _ t & 1 - \\gamma _ t & 1 - \\beta _ t & 1 - \\alpha _ t & 1 - \\gamma _ t & 1 - \\beta _ t \\cr 1 & 0 & 0 & 0 & \\alpha _ t & \\gamma _ t & \\beta _ t \\cr } , ~ \\alpha _ t , \\beta _ t , \\gamma _ t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} r ^ * ( E - \\{ e \\} ) & = r ( \\{ e \\} ) + | | E - \\{ e \\} | | _ r - r ( E ) \\\\ & = | | E | | _ r - r ( E ) \\\\ & = r ( \\emptyset ) + | | E | | _ r - r ( E ) \\\\ & = r ^ * ( E ) . \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} d \\omega ^ { ( C , k ) } + \\frac { 1 } { 2 } K _ { \\alpha \\beta } ^ { k } C _ { A B } ^ { C } \\omega ^ { ( A , \\alpha ) } \\omega ^ { ( B , \\beta ) } = 0 . \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} ( Z _ 0 ) _ { r e d } = \\ell _ 1 \\sqcup \\ell _ 2 \\sqcup ( \\ell _ 3 \\cup \\ell _ 4 ) , \\ \\ \\ q = \\ell _ 3 \\cap \\ell _ 4 = \\{ \\mathrm { p t } \\} . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} P ( C , L / F ) : = \\min \\{ \\mathrm { N } _ { F / \\mathbb { Q } } ~ \\mathfrak { p } \\colon \\textup { $ \\mathfrak { p } $ u n r a m i f i e d i n $ L $ , $ [ \\tfrac { L / F } { \\mathfrak { p } } ] = C $ , $ \\mathrm { N } _ { F / \\mathbb { Q } } ~ \\mathfrak { p } $ a r a t i o n a l p r i m e } \\} . \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\left ( \\alpha _ { 0 } , \\alpha _ { 1 } , \\alpha _ { 2 } , \\alpha _ { 3 } \\right ) = \\alpha _ { 0 } \\left ( 1 , - 1 , - 1 , - 1 \\right ) . \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} \\phi _ \\sigma : L i n ( W _ 8 , W _ 7 ) & \\longrightarrow L i n ( W _ 8 , W _ 7 ) \\\\ \\phi _ \\sigma ( A ) ( a ) & : = \\sigma ( a , A ( e _ \\mu ) , e _ \\mu , e _ i ) \\ ; e _ i . \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} ( f \\otimes u ) ( x ) = f ( x ) \\wedge u , \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} \\mu _ { G , p } ^ { \\rm c a n } ( k \\in K _ p : k ^ { - 1 } x k \\in K _ p ' ) = \\mu _ { G , p } ^ { \\rm c a n } ( k \\in K _ p : [ k , x ] \\in K _ p ' ) , \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} F ( X , x ) : = \\sum _ { i , j , k = 1 } ^ d \\sigma _ { i k } ( x ) \\sigma _ { j k } ( x ) X _ { i j } , \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{gather*} g ( x , y ) = \\gamma _ { i + 1 } ( x , y ) y ^ { e _ { i + 1 , \\theta } } + \\pi ^ { m ( D _ { i + 1 } - D _ i ) } ( ) , \\\\ g _ r ( x , y ) = \\gamma _ { r + 1 } ( x , y ) y ^ { e _ { r + 1 , \\theta } } + \\pi ^ { m ( D _ { r + 1 } - D _ i ) } ( ) , \\\\ \\intertext { a n d } G ( x , y ) = \\sum _ { \\widetilde { w } _ { i , \\theta } ( \\mathcal { V } _ i ) = 0 } \\gamma _ i ( x , y ) y ^ { e _ { i , \\theta } } , \\end{gather*}"}
-{"id": "9401.png", "formula": "\\begin{align*} [ L ^ p _ { \\overline { \\sigma } } ( \\Omega ) , D ( A _ p ) ] _ { \\theta } & = [ L ^ p ( \\Omega ) \\cap L ^ p _ { \\overline { \\sigma } } ( \\Omega ) , H _ { p e r , b . c . } ^ { 2 , p } ( \\Omega ) \\cap L ^ p _ { \\overline { \\sigma } } ( \\Omega ) ] _ { \\theta } \\\\ & = [ L ^ p ( \\Omega ) , H _ { p e r , b . c . } ^ { 2 , p } ( \\Omega ) ] _ { \\theta } \\cap L ^ p _ { \\overline { \\sigma } } ( \\Omega ) . \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} P ^ x ( X ( \\tau _ D ) \\in A ) = \\int _ A \\int _ D G _ D ( x , y ) \\nu ( y - z ) \\ , d y \\ , d z , x \\in D . \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} \\zeta _ { \\mathrm { c l } } ^ { - 1 } \\left ( z \\right ) = \\left ( 1 - z \\right ) \\cdot G \\left ( z \\right ) \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} ( v , p ) = ( V , p _ 0 ) , \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} g _ f ( x ) ^ Q = \\| D f ( x ) \\| ^ { Q } = | | g ^ { - 1 } f ^ * h ( x ) | | ^ { Q / 2 } \\leq K \\det ( \\overline { g } ^ { - 1 } \\overline { f ^ * h } ) ^ { 1 / 2 } = K J _ f ( x ) \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} P _ { 0 } ( z ) = p _ { 0 , 0 } e ^ { \\frac { \\lambda } { \\xi } z } ( 1 - z ) ^ { - \\frac { c \\gamma } { \\xi } } \\left [ 1 - \\dfrac { A _ { c } ( z ) } { A _ { c } } \\right ] , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\omega ^ 1 ( \\Phi ) = \\cos \\phi ^ 1 \\\\ \\omega ^ 2 ( \\Phi ) = \\sin \\phi ^ 1 \\cos \\phi ^ 2 \\\\ \\dots \\dots \\\\ \\omega ^ { 2 n - 1 } ( \\Phi ) = \\sin \\phi ^ 1 \\sin \\phi ^ 2 \\dots \\sin \\phi ^ { 2 n - 2 } \\cos \\phi ^ { 2 n - 1 } \\\\ \\omega ^ { 2 n } ( \\Phi ) = \\sin \\phi ^ 1 \\sin \\phi ^ 2 \\dots \\sin \\phi ^ { 2 n - 2 } \\sin \\phi ^ { 2 n - 1 } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} \\bar { f } ^ { i } : = f ^ { i } + \\sum _ { j = 1 } ^ { d } \\left ( a ^ { i j } - a _ { 0 } ^ { i j } \\right ) u _ { x ^ { j } } . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} & r _ 1 = r _ p = \\dots = r _ { p ^ { n - 2 } } = 0 \\\\ & r _ { p ^ { n - 1 } } = \\frac { - 2 4 } { p - 1 } \\\\ & r _ { p ^ n } = \\frac { 2 4 p } { p - 1 } \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} D _ j : = A _ j \\setminus \\bigcup _ { i \\in J } A _ i . \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} | G _ i | = \\begin{cases} d & a _ i \\mid d ; \\\\ d - 1 & a _ i \\nmid d . \\end{cases} \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} C _ { A B } { } ^ { C } C _ { C D } { } ^ { E } + C _ { B D } { } ^ { C } C _ { C A } { } ^ { E } + C _ { D A } { } ^ { C } C _ { C B } { } ^ { E } = 0 . \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} D u _ { i _ j } = V _ g ^ { - 1 } ( V _ g ( D u _ { i _ j } ) ) \\to V _ g ^ { - 1 } ( w ) = : v \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} b _ F ( p ) = \\mu _ F ( p ) = p ^ { - 1 } \\lambda ' _ { F , t _ 2 } + p ^ { - 2 } . \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} ( 1 - s ) x _ i & = ( 1 - \\theta ^ { \\nu _ i } ) x _ i , & \\forall i , \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} \\mathcal { F } _ { r , t } = \\left \\{ X \\in \\mathbb { R } ^ { p _ 1 \\times p _ 2 } : \\sigma _ r ( X ) \\geq t \\right \\} . \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\varphi _ { j k } = & \\sum _ { i = 1 } ^ n F _ { i j k } \\abs { A } ^ 2 + \\sum _ { i = 1 } ^ n 2 F _ { i j } \\kappa _ k + \\sum _ { i = 1 } ^ n 2 F _ { i k } \\kappa _ j \\\\ & + 2 \\delta _ { j k } \\sum _ { i = 1 } ^ n F _ i - F _ { j k } H - F _ j - F _ k . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} ( a , b ) ( 1 , c ) ( a , b ) ^ t & = ( a , b ) ( 1 , c ) ( \\bar { a } , \\bar { b } ) = ( 1 , b c \\bar { b } ) = ( 1 , c ) \\forall c \\in \\operatorname { S p } ( 1 ) , \\bar { c } = - c \\\\ \\Leftrightarrow c b & = b c \\forall c \\in \\operatorname { S p } ( 1 ) , \\bar { c } = - c . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} ( - 1 ) ^ { m + n - 2 } \\left ( ( - 1 ) ^ { m - 1 } g \\smile f - ( - 1 ) ^ { ( n - 1 ) m + 1 } f \\smile g \\right ) = ( - 1 ) ^ { n - 1 } \\left ( g \\smile f - ( - 1 ) ^ { n m } f \\smile g \\right ) \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} \\int _ 0 ^ t u ( s , x ) d s & = \\sum _ { k = 1 } ^ \\infty \\int _ { \\mathbb { R } ^ { d } } \\int _ { 0 } ^ { t } I _ s ^ { \\alpha - \\beta } p ( s , x - y ) \\int _ 0 ^ { t - s } g ^ k ( r , y ) d w ^ k _ r d s d y \\\\ & = \\sum _ { k = 1 } ^ \\infty \\int _ { \\mathbb { R } ^ { d } } \\int _ { 0 } ^ { t } p ( t - s , x - y ) I _ s ^ { \\alpha - \\beta } \\int _ 0 ^ s g ^ { k } ( r , y ) d w _ { r } ^ { k } d y d s \\\\ & = \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { d } } p ( t - s , x - y ) w ( s , y ) d y d s . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} d _ { } ( \\mathbf { G _ 1 } , \\mathbf { G _ 2 } ) = \\frac { 1 } { 2 } \\sum _ { G \\in \\mathcal { G } _ n } | P _ 1 ( G ) - P _ 2 ( G ) | . \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} \\left \\| d _ G ( \\cdot , y _ 0 ) ^ { - ( 1 + 4 \\alpha ) } E ^ { y _ 0 , i j } _ 1 \\right \\| _ { L ^ \\infty ( \\mathcal { B } _ { 1 } ^ + ( y _ 0 ) ) } + \\left [ d _ G ( \\cdot , y _ 0 ) ^ { - ( 1 + 4 \\alpha - \\epsilon ) } E ^ { y _ 0 , i j } _ 1 \\right ] _ { \\dot { C } ^ { 0 , \\epsilon } _ \\ast ( \\mathcal { B } _ 1 ^ + ( y _ 0 ) ) } \\leq C . \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} \\Psi ^ l = \\Psi ^ l \\left ( z _ i , X , X _ { z _ { i } } , X _ { z _ { i _ 1 } ^ { j _ 1 } , z _ { i _ 2 } ^ { j _ 2 } } , \\dots , X _ { z _ { i _ 1 } ^ { j _ 1 } , z _ { i _ 2 } ^ { j _ 2 } , z _ { i _ 3 } ^ { j _ 3 } , \\dots , z _ { i _ n } ^ { j _ n } } \\right ) \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ m \\alpha _ i \\ , \\sigma ( v _ i ) = v = \\beta _ 1 v _ 1 + \\sum \\limits _ { i = 2 } ^ { 2 d - m } \\beta _ i \\ , w _ i \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} \\delta ^ { ( a _ 1 \\cdots a _ r ) } _ { b _ 1 \\cdots b _ r } + \\sum _ { t = 1 } ^ s \\sum _ { r - s + 1 \\leqslant i _ 1 < \\cdots < i _ t \\leqslant r } \\varepsilon ^ t \\frac { ( k + r - 1 ) ! } { ( k + r - t - 1 ) ! } \\delta ^ { ( a _ 1 \\cdots \\hat { a } _ { i _ 1 } \\hat { a } _ { i _ 2 } \\cdots \\hat { a } _ { i _ t } \\cdots a _ r ) } _ { b _ 1 \\cdots \\hat { b } _ { i _ 1 } \\hat { b } _ { i _ 2 } \\cdots \\hat { b } _ { i _ t } \\cdots b _ r } \\delta ^ { a _ { i _ 1 } } _ { b _ { i _ 1 } } \\cdots \\delta ^ { a _ { i _ t } } _ { b _ { i _ t } } . \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} e _ { 1 } & = \\left \\{ v , a _ { 1 } , \\cdots , a _ { k - 2 } , a \\right \\} , \\quad ~ e _ { 2 } = \\left \\{ v , b _ { 1 } , \\cdots , b _ { k - 2 } , b \\right \\} , \\\\ e _ { 1 } ^ { \\prime } & = \\left \\{ v , a , b , a _ { 2 } , \\cdots , a _ { k - 2 } \\right \\} , ~ ~ e _ { 2 } ^ { \\prime } = \\left \\{ v , a _ { 1 } , b _ { 1 } , \\cdots , b _ { k - 2 } \\right \\} . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} \\begin{aligned} v ^ * & = v + \\omega \\omega \\cdot ( v _ 2 - v ) \\\\ v _ 2 ^ * & = v _ 2 - \\omega \\omega \\cdot ( v _ 2 - v ) \\end{aligned} \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} \\min _ x ~ f ^ p ( x ) : = \\frac { 1 } { m } \\sum _ { i = 1 } ^ m f _ i ( x ) + p ( x ) , \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} f \\tau _ 2 ( \\phi ) + ( \\Delta f ) \\tau ( \\phi ) + 2 \\nabla ^ { \\phi } _ { { \\rm g r a d } \\ , f } \\tau ( \\phi ) = 0 \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} \\mathbf { m } _ \\ell \\mathbf { m } _ j \\cdot e _ \\ell \\wedge e _ j = e _ { i _ { \\sigma ( \\ell ) } } \\wedge e _ { i _ { \\sigma ( j ) } } + . \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} F = P ( x _ { r - 1 } , x _ { r } ) + \\sum _ { j = 0 } ^ { d - 2 } x _ { j } G _ { j } \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} x ^ r \\left ( \\frac { x } { x - a } \\right ) ^ k & = x ^ r \\left ( 1 - \\frac { a } { x } \\right ) ^ { - k } \\\\ & = x ^ r \\sum \\limits _ { i = 0 } ^ \\infty { - k \\choose i } \\left ( - \\frac { a } { x } \\right ) ^ i \\\\ & = \\sum \\limits _ { i = 0 } ^ \\infty { - k \\choose i } ( - a ) ^ i x ^ { r - i } . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} \\mathcal { F } : = \\biguplus _ { n \\in I } K _ n \\cup F . \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } A _ f ( n ) = O \\left ( X ^ { \\frac { 1 } { 4 } + ( \\frac { 1 } { 2 } + \\epsilon ) \\eta } \\right ) + O \\left ( \\sum _ { X \\leq n \\leq X ' } | A _ f ( n ) | \\right ) , \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} c = a u - p r \\ \\ \\ \\ { \\rm a n d } \\ \\ \\ d = u ^ 2 b - q r u a - r ^ 2 . \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} \\tilde { F } ' = \\dot { F } \\Theta ^ 2 \\frac { \\sinh \\Theta } { \\cosh \\Theta } - \\tilde { F } , \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} \\| \\nabla u \\| _ 2 ^ 2 & = \\int _ { \\mathbb { R } ^ n } | \\xi | ^ 2 | \\hat { u } ( \\xi ) | ^ 2 d \\xi \\leq \\| | \\xi | ^ 2 \\hat { u } ( \\xi ) \\| _ 2 \\| \\hat { u } \\| _ 2 \\\\ & = \\| \\Delta u \\| _ 2 \\| u \\| _ 2 \\leq \\frac { \\varepsilon ^ 2 } { 2 } \\| \\Delta u \\| _ 2 ^ 2 + \\frac { 1 } { 2 \\varepsilon ^ 2 } \\| u \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} k ' ( q ^ { 2 } ) = \\frac { \\vartheta _ { 4 } ^ { 2 } \\left ( 0 \\mid q ^ { 2 } \\right ) } { \\vartheta _ { 3 } ^ { 2 } \\left ( 0 \\mid q ^ { 2 } \\right ) } , \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} - F ^ { i j } f _ { i j } + & 2 \\epsilon ^ 2 F ^ { i j } h _ { k i } h ^ k _ j f \\leq \\alpha F ^ { - 1 } F ^ { i j } F _ { ; i j } f + 2 ( \\alpha - 1 ) F ^ { - 1 } F ^ { i j } F _ i f _ j \\\\ & - 2 \\{ h ^ { i j } - F n F ^ { i j } \\} F ^ { - \\alpha } F _ { ; i j } - 2 \\epsilon ^ 2 \\abs { D A } ^ 2 F ^ { - \\alpha } + 2 C f . \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} h _ m ( C _ { u _ 0 } ) \\subset I _ { x _ 1 } ( \\overline { B _ 1 ( 0 ) } ) \\subset \\R ^ n \\setminus B _ { \\theta _ 1 } ( x _ 1 ) \\theta _ 1 = \\frac { 1 } { 1 + \\abs { x _ 1 } } \\\\ \\hat { k } _ m ( C _ { v _ 0 } ) \\subset I _ { x _ 2 } ( \\overline { B _ 1 ( 0 ) } ) \\subset \\R ^ n \\setminus B _ { \\theta _ 2 } ( x _ 2 ) \\theta _ 2 = \\frac { 1 } { 1 + \\abs { x _ 2 } } \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} K ' = K \\setminus \\{ \\tau \\in K \\mid \\tau \\cap \\partial X _ 0 \\ne \\emptyset \\} . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} \\mathbf { H } ( t + \\tau , \\mathbf { x } ) = \\mathbf { F } \\big ( \\nabla \\mathbf { u } ( t , \\mathbf { x } ) \\big ) ( t , \\mathbf { x } ) \\in ( 0 , \\infty ) \\times G . \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\lambda s \\partial _ s U ( s , x ) = L U ( s , x ) + N ( U ( s , x ) ) , \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P - F S } } & = \\left ( \\frac { \\mu _ 2 - \\mu } { \\mu _ 2 - \\mu _ 1 } \\right ) ^ + \\frac { ( 1 - \\mu _ 1 M ) K } { M r } + \\left ( 1 - \\left ( \\frac { \\mu _ 2 - \\mu } { \\mu _ 2 - \\mu _ 1 } \\right ) ^ + \\right ) \\frac { ( 1 - \\mu _ 2 ) K } { M r } \\\\ & = \\frac { K } { M r } \\left [ 1 - \\mu _ 2 - \\left [ \\mu _ 1 M - \\mu _ 2 \\right ] \\left ( \\frac { \\mu _ 2 - \\mu } { \\mu _ 2 - \\mu _ 1 } \\right ) ^ + \\right ] . \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} d _ X ( x _ m , x _ n ) & \\leq d _ X ( x _ m , x _ { m - 1 } ) + d _ X ( x _ { m - 1 } , x _ { m - 2 } ) + . . . + d _ X ( x _ { n + 1 } , x _ n ) \\\\ & \\leq { \\delta _ 1 } ^ { m - 1 } \\ d _ X ( x _ 1 , x _ 0 ) + { \\delta _ 1 } ^ { m - 2 } \\ d _ X ( x _ 1 , x _ 0 ) + . . . + { \\delta _ 1 } ^ n \\ d _ X ( x _ 1 , x _ 0 ) \\\\ & = { \\delta _ 1 } ^ n \\Big ( 1 + \\delta _ 1 + . . . + { \\delta _ 1 } ^ { m - n - 1 } \\Big ) \\ d _ X ( x _ 1 , x _ 0 ) \\\\ & = \\frac { { \\delta _ 1 } ^ n } { 1 - \\delta _ 1 } \\ d _ X ( x _ 1 , x _ 0 ) \\\\ & \\rightarrow 0 \\ \\ n \\rightarrow \\infty . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} T _ s T _ { t _ 0 } f - h _ { n + 1 } & = T _ s \\big ( g _ n + h _ n - e _ n \\big ) - h _ { n + 1 } \\\\ & = \\big ( T _ s g _ n - a _ n \\big ) ^ + - \\big ( T _ s g _ n - a _ n \\big ) ^ - - T _ s e _ n , \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{gather*} P _ { 0 } \\equiv 1 \\ , \\ P _ { 1 } \\equiv 0 \\ , \\ \\ \\ldots \\ , \\ P _ { n _ { 1 } - 2 } \\equiv 0 \\ , \\ \\ P _ { n _ { 1 } - 1 } \\equiv \\gamma _ { 0 } \\ , \\ \\ \\deg P _ { n _ { 1 } } = n _ { 1 } \\ , \\end{gather*}"}
-{"id": "3147.png", "formula": "\\begin{gather*} S ^ { \\alpha } \\colon \\ B \\to B , S ^ { \\alpha } ( 1 ) = 0 , S ^ { \\alpha } ( c _ { k } ) = c _ { k + \\alpha } , \\alpha \\in \\mathbb { Z } . \\end{gather*}"}
-{"id": "6119.png", "formula": "\\begin{align*} d ^ F F _ { Z _ { j , R } } ( \\omega , \\hat { \\omega } ) = 0 . \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} \\tilde { M } ( t ) = \\textrm { g r a p h } \\ , \\tilde { u } | _ { \\mathbb { S } ^ n } , \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} T r ( f e ^ { - t D } ) = \\int _ M d V \\ , K ( t ; x , x ; D ) f ( x ) = K ( t , f , D ) \\end{align*}"}
-{"id": "9912.png", "formula": "\\begin{align*} v : = \\lim _ { i \\rightarrow \\infty } g \\gamma _ i v _ j / { \\norm { g \\gamma _ i v _ j } } , \\norm { v } = 1 \\lim _ { i \\to \\infty } \\norm { g \\gamma _ i v _ j } = \\infty . \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} \\Delta _ { n } ^ { \\left ( r \\right ) } = \\left ( - 1 \\right ) ^ { \\left ( d + 1 \\right ) n } \\prod \\limits _ { i = 1 } ^ { n } \\gamma _ { i + r } ^ { 0 } \\Delta _ { 0 } ^ { \\left ( r \\right ) } = 1 . \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} \\lambda ( \\Lambda ' ) = \\lambda ( \\Lambda ) - \\int _ { \\Lambda ^ { - 2 } } ^ { \\Lambda '^ { - 2 } } \\frac { d t } { t } K ( t , g , D ) \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} \\norm { \\nabla _ H \\pi _ s ( t ) } \\leq \\norm { f ( t ) } + C ( B _ { H ^ 1 } ^ { \\zeta } ( t ) ) ^ { 1 / 2 } + C B _ { H ^ 2 } ^ v ( t ) = : B _ { H ^ 1 } ^ { \\pi _ s } ( t ) \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} \\partial _ t u = - ( - \\Delta ) ^ { s / 2 } u + u ^ { 1 + p } ( 0 , \\infty ) \\times \\R ^ { N } , 0 < s \\leq 2 , \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\langle f , g \\rangle _ N : = \\int _ { F _ 0 ( N ) } f ( z ) \\overline { g ( z ) } y ^ { k } \\frac { d x d y } { y ^ 2 } . \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} \\rho ( t ) ( y , h , u ) = ( y ^ { \\dagger } ( \\cdot , t _ 2 , \\ldots , t _ d ) | _ { < t _ 1 } , x ( t ) , u ^ { \\dagger } ( \\cdot , t _ 2 , \\ldots , t _ d ) | _ { > t _ 1 } ) . \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P , A c h } } = \\max \\left ( \\alpha \\delta _ F ^ { ( 1 ) } + ( 1 - \\alpha ) \\delta _ F ^ { ( 2 ) } , \\alpha \\delta _ E ^ { ( 1 ) } + ( 1 - \\alpha ) \\delta _ E ^ { ( 2 ) } \\right ) . \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\dot W ^ { 1 , \\epsilon } _ t : = \\begin{cases} \\mu _ \\eta & t \\in ( 4 k \\eta , ( 4 k + 2 ) \\eta ) , \\\\ - \\mu _ \\eta & t \\in ( ( 4 k + 2 ) \\eta , ( 4 k + 4 ) \\eta ) , \\end{cases} \\dot W ^ { 2 , \\epsilon } _ t : = \\begin{cases} \\mu _ \\eta & t \\in ( 2 k \\eta , ( 2 k + 1 ) \\eta ) , \\\\ - \\mu _ \\eta & t \\in ( ( 2 k + 1 ) \\eta , ( 2 k + 2 ) \\eta ) , \\end{cases} \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} \\frac { ( 2 m + r - 2 ) ! } { m ! ( m - 1 ) ! } \\sum \\limits _ { i = 0 } ^ { m } ( - 1 ) ^ i \\frac { ( m + i ) ! } { ( m + i + r - 2 ) ! m } { m \\choose i } \\gamma _ { m + i - 1 } = 0 . \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} \\Big ( g ^ { T Z _ R } _ { \\epsilon , t } \\Big ) _ { ( u , y ) } = \\Big ( g ^ { T Z _ R } \\Big ) _ { ( u , y ) } + \\psi \\Big ( \\frac { R - | u | } { \\epsilon } \\Big ) \\frac { 2 t \\ , d u ^ 2 } { R } , \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} \\sum _ { \\substack { | \\delta | = k , \\\\ \\delta _ { 1 } = \\delta _ { 2 } = \\cdots = \\delta _ { k - 2 } = 1 } } u _ { \\delta } = 0 . \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} \\Lambda _ { \\ \\nu } ^ { \\mu } = \\left ( \\begin{array} [ c ] { c c c c } \\gamma & - \\beta \\gamma & 0 & 0 \\\\ - \\beta \\gamma & \\gamma & 0 & 0 \\\\ 0 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 1 \\end{array} \\right ) , \\qquad \\quad \\beta = \\frac { v } { c } , \\quad \\gamma = \\frac { 1 } { \\sqrt { 1 - \\beta ^ { 2 } } } \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} H ( R \\theta ) \\ ; = \\ ; R \\gamma \\left ( \\theta \\right ) , \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} Q ' : = \\Q { k - 1 } { l - 1 } { m - 1 } { a + 1 } { b + 1 } { c + 1 } { x } , R ' : = \\R { k - 1 } { l - 1 } { m - 1 } { a + 1 } { b + 1 } { c + 1 } { x } . \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} P ^ z \\left ( F _ { \\gamma ^ z } ( \\omega ) = W ( \\cdot , \\omega ) \\right ) = 1 . \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} a ^ { 2 } \\nabla _ { \\nu } b ^ { 2 } \\geqslant a ^ { 2 } \\sharp _ { \\nu } b ^ { 2 } + \\sum _ { k = 0 } ^ { \\infty } r _ { k } \\big [ a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } - a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} \\delta ( H ' ) \\ge \\delta ( H ) - \\binom { n - 1 - \\delta ( G _ { 1 / 3 } ) } 2 & \\ge \\delta ( H ) - \\binom { n - 1 - g _ { . 8 } ( 1 / 3 ) ( n - 1 ) } 2 \\\\ & \\ge . 8 \\binom { n - 1 } 2 - \\binom { . 3 ( n - 1 ) } 2 \\ge . 7 0 9 \\frac { n ^ 2 } 2 , \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} \\phi _ { 6 , u } ( x , y ) = \\left ( \\begin{smallmatrix} 1 & u \\\\ 0 & 1 \\end{smallmatrix} \\right ) \\cdot \\phi _ 6 = x ( u x + y ) ( x ^ 4 - ( u x + y ) ^ 4 ) . \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} Q _ \\infty [ w f , w g ] - \\lambda \\int _ \\Omega U \\ , f g \\ , w ^ 2 d x \\ , & \\ \\leq Q _ \\infty [ w f , w g ] - \\lambda \\int _ \\Omega \\frac { 1 } { 4 | x | ^ 2 } \\left ( \\log \\frac { | x | } { \\rho } + \\beta \\right ) ^ { - 2 } f g \\ , w ^ 2 d x \\\\ & \\ = \\widehat Q _ \\infty [ f , g ] . \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} f ^ * = \\begin{cases} f _ 1 & \\ q < \\frac { 2 } { 3 } \\\\ f _ 2 & \\ q > \\frac { 2 } { 3 } \\\\ \\big \\{ ( p , 1 - p ) : 0 \\leq p \\leq 1 \\big \\} & \\ q = \\frac { 2 } { 3 } . \\end{cases} \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} \\mbox { p r o x } _ { \\phi } ( y ; \\lambda , a ) = 0 , \\forall | y | < \\lambda , \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } z } \\left ( \\det \\left ( \\mathrm { i d } - z \\cdot M \\right ) \\right ) = \\frac { \\mathrm { d } } { \\mathrm { d } z } \\left ( \\left ( 1 - z \\right ) \\cdot G \\left ( z \\right ) \\right ) = - G \\left ( z \\right ) + \\left ( 1 - z \\right ) \\cdot G ' \\left ( z \\right ) . \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ { \\gamma } ( z , \\overline { z } ) = { ( - 1 ) ^ { m } ( \\gamma + m + 1 ) _ { n } } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { - \\gamma } \\sum _ { j = 0 } ^ { + \\infty } \\dfrac { ( - \\gamma - m ) _ { j } } { j ! } \\dfrac { \\partial ^ { m } } { \\partial z ^ { m } } ( z ^ { j + n } ) { \\overline { z } ^ { j } } . \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} \\Theta = ( \\eta ^ \\prime \\eta ) ^ { - 1 } \\eta ^ \\prime ( X ( 0 ) - \\mu ) , V ( t ) = \\int _ 0 ^ t ( \\tau ( s ) ^ \\prime \\tau ( s ) ) ^ { - 1 } \\tau ( s ) ^ \\prime d M ( s ) , \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} { \\bf { C o v } } \\left [ V _ { \\varepsilon } ( \\psi ) , \\ , V _ { \\varepsilon } ( \\xi ) \\right ] = & \\begin{cases} - 2 \\ , \\log | e ^ { i \\psi } - e ^ { i \\xi } | , \\ , | \\xi - \\psi | \\gg \\varepsilon , \\\\ 2 \\left ( \\kappa - \\log \\varepsilon \\right ) , \\psi = \\xi , \\end{cases} \\\\ & + O ( \\varepsilon ) , \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{gather*} \\left \\{ m \\in M \\ \\middle | \\ \\sum _ { j = 1 } ^ { d - 1 } u ^ j m = 0 \\right \\} \\Big / \\Big \\{ u m - m \\ | \\ m \\in M \\Big \\} \\to H ^ 1 ( U , M ) \\\\ \\bar { m } \\mapsto \\left [ c ( m ) _ { u ^ i } = \\sum _ { j = 0 } ^ { i - 1 } u ^ j \\cdot m \\right ] . \\end{gather*}"}
-{"id": "4687.png", "formula": "\\begin{align*} \\nu _ \\chi ( x ) = \\chi ( \\overline { \\zeta } { } _ x ^ { p ^ { - e } } ) , \\phi ( x ) = ( - 1 ) ^ { ( n - 1 ) v _ E ( x ) } . \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} \\gamma _ { j k } = \\sigma _ j \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } - \\sigma _ k \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 j } . \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} c _ { o } ^ { s } = \\frac { \\mu _ { \\nu } } { \\mu _ { w } } \\cdot c _ { o } ^ { 1 - n / p } . \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} R g = \\sum _ { k = 0 } ^ \\infty \\frac { M ^ k g } { 2 ^ k \\| M \\| _ { L ^ q } ^ k } \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} \\bar { \\mathbf { x } } ^ { [ \\sf b s ] } _ { i } [ m ] = \\gamma ^ { [ \\sf d ] } _ { i } \\sum _ { j = 1 } ^ N \\sum _ { a = 1 } ^ { \\frac { \\lambda _ 2 } { \\lambda _ 1 } T ^ { K N ( K M + K N ) } } \\mathbf { \\bar { v } } _ { i j } ^ { [ { \\sf b s } ] ( a ) } [ m ] c ^ { [ { \\sf d } ] ( a ) } _ { i j } [ m ] , \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} \\Phi ^ { ( \\rho , \\sigma , \\kappa ) } _ { \\lambda , \\mu ; \\nu } ( z , s , a ) = \\sum _ { n \\geq 0 } \\frac { ( \\lambda ) _ { \\rho n } \\ , ( \\mu ) _ { \\sigma n } } { n ! \\ , ( \\nu ) _ { \\kappa n } } \\ , \\dfrac { z ^ n } { ( n + a ) ^ s } \\ , , \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} N ^ s _ p ( M ) = B ^ s _ { p , \\infty } ( M ) , \\ , \\ , \\ , \\ , \\tilde { N } ^ s _ p ( \\Omega ) = \\tilde { B } ^ s _ { p , \\infty } ( \\Omega ) , \\\\ W ^ s _ p ( M ) = B ^ s _ { p , p } ( M ) , \\ , \\ , \\ , \\ , \\tilde { W } ^ s _ p ( \\Omega ) = \\tilde { B } ^ s _ { p , p } ( \\Omega ) , \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} \\sigma ( g x ) = R _ g \\sigma ( x ) \\text . \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} X ( k n ) = X ^ * ( ( l - k ) n ) , 1 \\leqslant k \\leqslant l - 1 \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} \\ell ( n D ) - \\ell ( \\kappa - n D ) = \\deg D + 1 - g . \\end{align*}"}
-{"id": "10061.png", "formula": "\\begin{align*} q + p = l + 2 . \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} | \\partial \\mathbf { B } | _ { g ( 0 ) } \\leq 2 ^ { m - 1 } \\sum _ { j = 0 } ^ { N } \\left | \\partial \\mathbf { B } _ j \\cap B _ { g ( j \\xi ^ 2 ) } ( x _ 0 , \\rho ) \\right | _ { g ( j \\xi ^ 2 ) } , \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} \\bigoplus _ { ( i , \\alpha , \\gamma ) } T _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) = \\bigoplus _ { ( i , \\alpha , \\gamma ) } H _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) \\varphi ( i , \\alpha , \\gamma ) \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} & \\mathfrak { L } [ \\log F ( z ) ] - 2 | z | ^ { 2 } ( \\log F ( z ) ) _ { z \\overline { z } } \\\\ = & 2 \\sum _ { k = 3 } ^ { p } ( 3 k - k ^ { 2 } - 2 ) | z | ^ { 2 ( k - 1 ) } \\log G _ { k } ( z ) + \\sum _ { k = 1 } ^ { p } ( 3 - 2 k ) | z | ^ { 2 ( k - 1 ) } \\mathfrak { L } [ \\log G _ { k } ( z ) ] . \\\\ \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} 0 = \\sum _ { k = 0 } ^ n a _ k \\tau _ { k y ^ * } f ( x ) = \\sum _ { k = 0 } ^ n a _ k \\tau _ { k y - k h _ 1 } f ( x ) y , h _ 1 \\in B _ { d } ( \\delta / 2 ) . \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} D _ t ^ j L _ \\alpha ^ 0 = 0 ; ( j = 0 , \\dots , 2 k - s - 2 ) \\ . \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} e ^ { S _ { w } p } = \\prod _ { l = 1 } ^ { k - 1 } p _ { w _ { l } w _ { l + 1 } } \\cdot \\sup _ { s \\in S } p _ { w _ { k } s } = \\frac { \\sup _ { s \\in S } p _ { w _ { k } s } } { \\mu \\left ( \\left [ w _ { 1 } \\right ] \\right ) } \\cdot \\mu \\left ( \\left [ w \\right ] \\right ) , \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} f _ k ( 0 ) & = 0 , \\\\ f _ k ' ( 0 ) & = 1 , \\\\ f _ k '' & = k ^ 2 f _ k . \\end{aligned} \\right . \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} A _ { a , b , c } = A _ n M _ { a , b , c } , n = a + b + c , \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{gather*} \\hat { y } ^ \\phi _ \\alpha = \\hat { x } ^ \\phi _ \\beta \\mathcal { O } ^ { - 1 \\beta } _ \\alpha , \\end{gather*}"}
-{"id": "6761.png", "formula": "\\begin{align*} Z _ N = \\sum _ { \\mathcal { C } } W ( \\mathcal { C } ) , W ( \\mathcal { C } ) = a ^ { n _ 1 + n _ 2 } b ^ { n _ 3 + n _ 4 } c ^ { n _ 5 + n _ 6 } . \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} \\Gamma : \\{ ( x , y ) \\in \\R ^ n \\times \\R ^ n : x \\neq y \\} \\longrightarrow \\R , \\Gamma ( x ; y ) = \\int _ 0 ^ \\infty p _ t ( x ; y ) \\ , \\d t , \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} & \\frac { n - 2 } { 4 n } \\int _ 0 ^ \\infty \\frac { ( n + 4 t ) ^ 2 } { ( 1 + t ) ^ { n - 2 } } t ^ { \\tfrac { n } { 2 } } d t \\\\ = & \\frac { n - 2 } { 4 n } \\int _ 0 ^ \\infty \\frac { ( n - 4 ) ^ 2 + 8 ( n - 4 ) ( 1 + t ) + 1 6 ( 1 + t ) ^ 2 } { ( 1 + t ) ^ { n - 2 } } t ^ { \\tfrac { n } { 2 } } d t \\\\ = & \\frac { n - 2 } { 4 n } \\Big [ ( n - 4 ) ^ 2 B ( \\tfrac { n } { 2 } + 1 , \\tfrac { n } { 2 } - 3 ) + 8 ( n - 4 ) B ( \\tfrac { n } { 2 } + 1 , \\tfrac { n } { 2 } - 4 ) + 1 6 B ( \\tfrac { n } { 2 } + 1 , \\tfrac { n } { 2 } - 5 ) \\Big ] , \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} \\mathcal { V } = \\R ^ d ; \\mathcal { V } = \\S ^ { d - 1 } ; \\mathcal { V } = B ( 0 , l ) : = \\Big \\{ v \\in \\R ^ d \\ , \\mbox { s . t . } | v | \\le l \\Big \\} . \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} \\langle T _ \\mu ^ \\mu \\rangle = - \\frac { 1 } { 1 6 \\pi ^ 2 } ( a E _ 4 - c W ^ 2 ) \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { l l } v ( . , t ) \\in S P S H ( \\bar { \\Omega } ) , \\\\ \\dot { v } = \\log \\det ( v _ { \\alpha \\bar { \\beta } } ) - A v + g ( z , t ) \\ ; \\ ; \\ ; & \\mbox { o n } \\ ; \\Omega \\times ( 0 , T ) , \\\\ v = \\psi & \\mbox { o n } \\ ; \\partial \\Omega \\times [ 0 , T ) . \\\\ \\end{array} \\end{cases} \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} \\lambda ( \\Lambda ' ) = \\lambda ( \\Lambda ) - \\sum _ { k \\geq 0 } a _ k ( g , D ) \\int _ { \\Lambda ^ { - 2 } } ^ { \\Lambda '^ { - 2 } } t ^ { ( k - n - 2 ) / 2 } { d t } \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} & a b = a _ 1 \\circ ( T ( a _ 2 ) \\rightharpoonup b ) , & & S ( a ) = a _ 1 \\rightharpoonup T ( a _ 2 ) , & & a , b \\in A . \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} ( [ F ] , C _ 0 , [ L : = \\mathbf { L } | _ { C _ 0 } ] ) \\ \\textit { s a t i s f i e s t h e s t a t e m e n t s ( i i i ) a n d ( i v ) o f L e m m a \\ref { L e m m a 4 . 2 } } . \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} e ( x ) : = \\exp ( 2 \\pi i x ) . \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} \\omega ^ { \\sharp } ( \\alpha ) = - S _ { a ^ { * } } \\partial _ { a } + S _ { a } \\partial _ { a ^ { * } } \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\mu = O \\left ( t ^ { \\frac { 2 p - \\alpha } { 2 \\alpha } } \\right ) , \\sigma = O \\left ( t ^ { p / \\alpha } \\right ) , \\mu = O \\left ( \\sigma ^ { - \\gamma } \\right ) \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\abs { K ^ \\perp } & = 2 \\abs { A ^ \\circ _ { 1 1 } \\wedge A ^ \\circ _ { 1 2 } } \\abs { N _ 1 \\wedge N _ 2 } = 2 \\left ( \\abs { A ^ \\circ _ { 1 1 } } ^ 2 \\abs { A ^ \\circ _ { 1 2 } } ^ 2 - \\langle A ^ \\circ _ { 1 1 } , A ^ \\circ _ { 1 2 } \\rangle ^ 2 \\right ) ^ { \\frac 1 2 } \\\\ & \\le 2 \\abs { A ^ \\circ _ { 1 1 } } \\abs { A ^ \\circ _ { 1 2 } } \\le \\abs { A ^ \\circ _ { 1 1 } } ^ 2 + \\abs { A ^ \\circ _ { 1 2 } } ^ 2 = \\frac 1 2 \\abs { A ^ \\circ } ^ 2 \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} A = a ^ { i i } ( u _ { 1 ; i } ) ^ 2 W ^ { - 2 } + a ^ { i i } \\sum _ { k \\ne 1 } ( u _ { k ; i } ) ^ 2 \\ge 0 . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} { \\rm q d i m } [ M ^ \\varepsilon _ { r , s } ] = ( - 1 ) ^ { ( s + 1 ) + p ( r + 1 ) } s , \\ \\ { \\rm q d i m } [ F ^ \\varepsilon _ \\lambda ] = e ^ { \\frac { 2 \\pi i k \\lambda } { \\sqrt { 2 p } } } p . \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} I ( X , S ; Y | Q = q _ 1 ) & = I ( X , S ; Y | Q = q _ 4 ) = z H _ 2 ( p ) \\\\ I ( X , S ; Y | Q = q _ 2 ) & = I ( X , S ; Y | Q = q _ 3 ) = H _ 2 ( \\alpha ) - ( 1 - \\alpha ) z H _ 2 ( p ) . \\end{align*}"}
-{"id": "9995.png", "formula": "\\begin{align*} ( I - \\Delta ) ^ { ( n + 1 ) / 2 } h ( x ) = 0 \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} f ( 1 , z _ 1 , z _ 2 , z _ 3 ) = 0 ~ ~ \\widetilde { U } _ 0 . \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} u ^ { \\frac { n + 6 } { n - 6 } } P _ { \\tilde g } ^ 6 \\varphi = P _ g ^ 6 ( u \\varphi ) \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} P _ s \\sim B _ { s - 1 } = \\left ( \\begin{array} { c | c } D _ { s - 1 } & \\\\ \\hline \\mathbf b _ { s - 1 } & \\\\ 0 & \\\\ \\vdots & E _ { n - s + 2 } \\\\ 0 & \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} v _ i ^ { k + 1 } & = \\sum _ { j = 1 } ^ { N } w _ { i j } ( k ) v _ j ^ k + x _ i ^ { k + 1 } - x _ i ^ k , \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} \\Gamma _ i ^ k = \\sum _ { l = 1 } ^ { l _ i ^ k } \\frac { 1 } { \\delta ^ l } = \\frac { l _ i ^ k } { \\delta _ { \\min , i } } , \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} \\mathcal { D } = \\{ D \\in \\mathcal { B } : \\mbox { t h e r e e x i s t s $ h \\in \\natural $ s u c h t h a t $ D _ h \\subset D $ } \\} . \\ \\Box \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} R ^ { ( 1 , k ) } = R \\otimes _ k k \\simeq R \\otimes _ E E ^ { ( 1 , k ) } \\end{align*}"}
-{"id": "10071.png", "formula": "\\begin{align*} \\alpha \\beta \\gamma = \\alpha \\gamma + \\alpha \\beta + \\beta \\gamma . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} H ^ * ( B Q _ { 4 m } ) = H ^ * ( B G ) \\oplus H ^ * ( B G ) [ - 1 ] \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} \\P ( Z ( t ) \\le c ) & = \\P \\left ( \\abs { X _ t } \\le e ^ { t ^ c } \\right ) = \\Phi \\left ( \\frac { e ^ { t ^ c } - x _ 0 e ^ { \\theta t } } { \\sqrt { v ( \\theta , t ) } } \\right ) - \\Phi \\left ( \\frac { - e ^ { t ^ c } - x _ 0 e ^ { \\theta t } } { \\sqrt { v ( \\theta , t ) } } \\right ) \\\\ & = \\Phi \\left ( \\frac { e ^ { t ^ c } - x _ 0 e ^ { \\theta t } } { \\sqrt { v ( \\theta , t ) } } \\right ) + \\Phi \\left ( \\frac { e ^ { t ^ c } + x _ 0 e ^ { \\theta t } } { \\sqrt { v ( \\theta , t ) } } \\right ) - 1 . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} u _ i = 0 , u _ { i , J } = 0 , \\tau = 0 \\Gamma \\times ( 0 , \\infty ) \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{gather*} \\omega = w _ { 0 } \\wedge w _ { 1 } \\wedge w _ { 2 } \\wedge \\cdots , w _ { i } \\in H ^ { ( n ) } , \\end{gather*}"}
-{"id": "1756.png", "formula": "\\begin{align*} \\rho \\leq 2 \\tilde { \\rho } = 4 c \\tilde { \\rho } _ - \\leq 4 c \\rho _ - \\end{align*}"}
-{"id": "9952.png", "formula": "\\begin{align*} u ( \\varphi ( s _ { i } ) ) \\pi _ { p } ( v _ { i } ) = u ( \\varphi ( s _ { i } ) ) \\pi _ { p } ( v ) = \\pi _ { p } ( u ( \\varphi ( s _ { i } ) ) v ) \\in V ^ { 0 - } ( A ) . \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\int _ M 2 h ^ { i j } \\nabla _ i ( Z \\nabla _ j u ) u d \\mu & = - 2 \\int _ M \\nabla _ i ( h ^ { i j } u ) Z \\nabla _ j u u d \\mu \\\\ \\displaystyle & = - 2 \\int _ M | \\nabla u | ^ { p - 2 } h ^ { i j } \\nabla _ i u \\nabla _ j u d \\mu - 2 \\int _ M Z \\langle d i v h , \\nabla u \\rangle u d \\mu . \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} \\displaystyle J _ 1 = \\int _ { 0 } ^ { t } \\| [ \\{ { I _ { \\varepsilon } } | E _ { \\lambda } | ^ { 2 } \\} E _ { \\lambda } - \\{ { I _ { \\varepsilon } } | E _ { \\infty } | ^ { 2 } \\} E _ { \\infty } ] ( s ) \\| _ { H _ { x } ^ { m } } d s \\end{align*}"}
-{"id": "9918.png", "formula": "\\begin{align*} u ( r ) = \\sigma ( r ) u ( - r ) u ^ { - } ( r ^ { - 1 } ) , \\ , \\forall r \\neq 0 . \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} ( a \\nabla _ { \\nu } b ) ^ { 2 } \\leqslant ( a \\sharp _ { \\nu } b ) ^ { 2 } + ( 1 - r _ { 0 } ) ^ { 2 } ( a - b ) ^ { 2 } - \\sum _ { k = 1 } ^ { \\infty } r _ { k } \\big [ a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } - a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } . \\end{align*}"}
-{"id": "10114.png", "formula": "\\begin{align*} C \\ : : \\ : x ^ p y ^ q ( b y + c z ) ^ r - z ^ { p + q + r } = 0 \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{gather*} \\Psi ^ { [ k ] ( \\alpha ) } ( z ) = \\frac { ( - 1 ) ^ k } { \\tau _ { k } ^ { ( \\alpha ) } } \\begin{bmatrix} z ^ { k } & 0 \\\\ 0 & z ^ { - k } \\end{bmatrix} \\begin{bmatrix} S ^ { + } ( z ) & 0 \\\\ 0 & S ^ { - } ( z ) \\end{bmatrix} \\begin{bmatrix} \\tau _ { k } ^ { ( \\alpha ) } & \\tau _ { k - 1 } ^ { ( \\alpha ) } / z \\\\ \\tau _ { k + 1 } ^ { ( \\alpha ) } / z & \\tau _ { k } ^ { ( \\alpha ) } \\end{bmatrix} , \\end{gather*}"}
-{"id": "2819.png", "formula": "\\begin{align*} M ^ 2 \\equiv [ M ( q ) ] ^ 2 : = - \\frac { 2 q ^ 2 } { \\rho ( q ^ 2 ) } \\rho ' ( q ^ 2 ) \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u ( t , x ) = f ( t , x ) + \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } g ^ { k } ( s , x ) d w _ { s } ^ { k } , t \\in ( 0 , T ] , \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} q ^ { - n + 1 } \\psi _ { n - 1 } + q ^ { - n } \\psi _ { n + 1 } = 0 , n \\in \\N , \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} \\mathbb { P } ( X _ { r } > T _ { ( k ) } ) & = \\sum _ { j = 0 } ^ { r - 1 } ( - 1 ) ^ { j } \\frac { s ^ { j } } { j ! } f ^ { ( j ) } _ { n , k } ( s ) \\\\ & = \\sum _ { j = 0 } ^ { r - 1 } ( - 1 ) ^ { j } \\frac { s ^ { j } } { j ! } g ^ { ( k ) } _ { n , k } ( s ) , \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} E _ \\beta = \\inf \\left \\{ E ( u _ 1 , u _ 2 ) : ( u _ 1 , u _ 2 ) \\in \\mathcal { S } _ \\beta \\right \\} . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} \\left ( \\dfrac { D } { p } \\right ) = 1 \\Leftrightarrow \\pm \\left ( \\dfrac { p } { q _ 1 } \\right ) \\left ( \\dfrac { p } { q _ 2 } \\right ) \\ldots \\left ( \\dfrac { p } { q _ m } \\right ) = 1 , \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} | x _ 0 : x _ 1 : x _ 2 | = \\begin{cases} x _ 0 \\cdot | x _ 0 : x _ 1 : x _ 2 | = | x _ 0 ^ 2 : x _ 0 x _ 1 : x _ 0 ^ 2 x _ 2 | = | y _ 0 : y _ 1 : y _ 0 y _ 3 | ; \\\\ x _ 1 \\cdot | x _ 0 : x _ 1 : x _ 2 | = | x _ 0 x _ 1 : x _ 1 ^ 2 : x _ 1 ^ 2 x _ 2 | = | y _ 1 : y _ 2 : y _ 2 y _ 3 | , \\end{cases} \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} d \\check \\mu _ { \\mathbf { v } } = \\sum _ { i , j = 1 } ^ N \\ , d \\mu _ { i , j } \\ , v _ i \\ , \\overline { v _ j } = \\big | \\sum _ { j = 1 } ^ N \\ , v _ j \\ , e ^ { - 2 \\pi i x _ j \\cdot \\xi } \\big | ^ 2 d \\mu ( \\xi ) , \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} ( \\lambda _ 1 \\theta _ 1 + \\lambda _ 2 \\theta _ 2 ) \\circ T ^ n ( p ) > ( \\lambda _ 1 \\| \\theta _ 1 \\| + \\lambda _ 2 \\| \\theta _ 2 \\| ) \\ , ( 1 - \\epsilon ) = \\| \\lambda _ 1 \\theta _ 1 + \\lambda _ 2 \\theta _ 2 \\| \\ , ( 1 - \\epsilon ) , \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} \\sqrt { D } = ( 1 / \\sqrt { 2 } , 1 / \\sqrt { 2 } , b , b ) , \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\phi ( x , y ) \\phi ( y , x ) = a b ( 1 - \\phi ( x , y ) ) ( 1 - \\phi ( y , x ) ) = ( 1 - a ) ( 1 - b ) ~ ~ ~ \\nu ^ 2 \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} F ^ p V _ { \\Delta } = G _ { \\Delta - p } V _ { \\Delta } , \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} & \\Vert Y _ 1 - X _ * \\Vert _ F ^ 2 \\leq \\delta ^ 2 \\Vert Y _ 0 - X _ * \\Vert _ F ^ 2 + \\\\ + \\Big ( \\delta ^ 2 & \\Vert Y _ 0 - X _ * \\Vert _ F ^ 2 - \\sum _ { k = 1 } ^ r \\limits s _ k ^ 2 \\sin ^ 2 \\phi _ { R 1 , k } \\Big ) \\frac { \\sum _ { k = 1 } ^ r \\limits s _ k ^ 2 \\sin ^ 2 \\phi _ { R 0 , k } } { s _ r - \\sum _ { k = 1 } ^ r \\limits s _ k ^ 2 \\sin ^ 2 \\phi _ { R 0 , k } - \\sum _ { k = 1 } ^ r \\limits s _ k ^ 2 \\sin ^ 2 \\phi _ { R 1 , k } } . \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a t ) ^ 2 + t ^ 2 } , c = c o n s t , \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} c _ 0 x + A _ c y + B _ c z = 0 , A _ c ^ { t r } x - c _ 0 y + C _ c z = 0 , B _ c ^ { t r } x + C _ c ^ { t r } y = 0 . \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\zeta ^ k _ { t | s } = \\sum _ { j = 0 } ^ k { n - j \\choose k - j } \\sum _ { i = 0 } ^ k ( - 1 ) ^ { k - i } { k - j \\choose k - i } \\exp ( Q _ { ( n - i ) \\lambda } ( t - s ) ) \\zeta ^ j _ s . \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} h ( z ) = \\bar \\pi ^ { - 1 } \\left ( \\frac { d } { d z } - ( q - 1 ) \\frac { \\Delta ' ( z ) } { \\Delta ( z ) } \\right ) g ( z ) , \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} \\eta _ i = \\left ( \\sum _ l \\kappa _ l ^ r \\right ) ^ { - 1 } \\kappa _ j ^ { r - 1 } \\eta ^ j \\kappa _ i . \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} a = ( 1 , \\ , e ^ { \\j \\omega } , \\ , e ^ { \\j 2 \\omega } , \\ , \\ldots , \\ , e ^ { \\j ( k - 1 ) \\omega } ) \\otimes c , \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} d _ X ( F ^ 2 ( x _ 0 , y _ 0 ) , F ( x _ 0 , y _ 0 ) ) = & d _ X ( F ( F ( x _ 0 , y _ 0 ) , G ( y _ 0 , x _ 0 ) ) , F ( x _ 0 , y _ 0 ) ) \\\\ \\leq & k \\ d _ X ( F ( x _ 0 , y _ 0 ) , x _ 0 ) + l \\ d _ Y ( G ( y _ 0 , x _ 0 ) , y _ 0 ) \\\\ \\leq & ( k + l ) \\ [ d _ X ( x _ 1 , x _ 0 ) + d _ Y ( y _ 1 , y _ 0 ) ] \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} B ( z ) = \\int _ { 0 } ^ { z } s ^ { \\frac { \\mu } { \\xi } - 1 } ( 1 - s ) ^ { - \\left ( \\frac { \\gamma } { \\xi } + 1 \\right ) } \\left ( 1 - \\frac { A ( s ) } { A } \\right ) d s , \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{gather*} \\big ( \\tau _ { k + 1 } ^ { ( \\alpha ) } \\big ) ^ 2 = \\tau _ { k + 1 } ^ { ( \\alpha - 1 ) } \\tau _ { k + 1 } ^ { ( \\alpha + 1 ) } - \\tau _ { k + 2 } ^ { ( \\alpha - 1 ) } \\tau _ k ^ { ( \\alpha + 1 ) } . \\end{gather*}"}
-{"id": "702.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x ^ { \\nu } } \\left ( \\frac { \\partial L _ { 0 , 1 } } { \\partial P _ { \\nu \\mu } } \\right ) - \\frac { \\partial L _ { 0 , 1 } } { \\partial A _ { \\mu } } = 0 . \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} Q _ i & : = \\exp 2 \\pi i \\tau _ i , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ Q _ { i , ( a _ 1 , \\ldots , a _ l ) } : = \\exp 2 \\pi i \\tau _ { i , ( a _ 1 , \\ldots , a _ l ) } \\\\ Q _ { ( i , j ) } & : = \\exp 2 \\pi i \\sigma _ { ( i , j ) } , \\ \\ \\ Q _ { ( i , j ) , ( a _ 1 , \\ldots , a _ l ) } : = \\exp 2 \\pi i \\sigma _ { ( i , j ) , ( a _ 1 , \\ldots , a _ l ) } . \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } q ^ { n ^ { 2 } } S _ { n } \\left ( x q ^ { n } ; q \\right ) t ^ { n } = \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { 3 k ^ { 2 } } \\left ( - x t \\right ) ^ { k } } { \\left ( q ; q \\right ) _ { k } } A _ { q } \\left ( - t q ^ { 3 k } \\right ) . \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} \\int _ I f \\Delta g \\ d x = \\sum _ { x \\in \\partial I } ( g ( x ) \\mathbf n _ x f - f ( x ) \\mathbf n _ x g ) + \\int _ I g \\Delta f \\ d x \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} \\Phi _ a ( x , \\xi , z ) & = \\varphi _ a ( x - z , \\xi ) - \\varphi _ a ( x , \\xi ) + z \\cdot \\nabla _ x \\varphi _ a ( x , \\xi ) \\\\ & = z \\cdot \\left ( \\int _ 0 ^ 1 \\theta _ 1 \\int _ 0 ^ 1 \\nabla _ x ^ 2 \\varphi _ a ( x - \\theta _ 1 \\theta _ 2 z , \\xi ) d \\theta _ 2 d \\theta _ 1 \\right ) z . \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} \\psi \\ge A = 2 \\max _ { \\bar M } | \\varphi | \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} \\bigoplus _ { i = 0 } ^ { x - 1 } T _ x ( i ) \\bigoplus _ { i = 0 } ^ { x - 1 } F _ n ( T _ x ( i ) ) \\bigoplus _ { i = 0 } ^ { x - 1 } F _ n ( T _ x ( \\phi ( i ) ) ) & = \\bigoplus _ { i = 0 } ^ { x - 1 } T _ x ( i ) \\bigoplus _ { i = 0 } ^ { x - 1 } F _ n ( T _ x ( i ) ) \\bigoplus _ { i = 0 } ^ { x - 1 } F _ n ( T _ x ( i ) ) \\\\ & = \\bigoplus _ { i = 0 } ^ { x - 1 } T _ x ( i ) \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} e _ { \\alpha = j } : = - Q ( e _ { \\mu = j } ) , 1 \\leq j \\leq 8 , \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} H ^ i ( Y , R ^ j f _ * ( K _ X \\otimes F \\otimes \\mathcal J ( h ) \\otimes N ) \\otimes H ^ { \\otimes m } ) = 0 \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{gather*} \\tau ( \\hat { g } ) = \\langle v _ { 0 } , \\hat g v _ { 0 } \\rangle . \\end{gather*}"}
-{"id": "9649.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } q ^ { \\beta ^ { 2 } } \\left ( q ^ { - \\beta + n + 1 } ; q \\right ) _ { \\infty } L _ { n } ^ { ( - \\beta ) } \\left ( - 1 ; q \\right ) A _ { q } \\left ( - q ^ { \\beta + n } \\right ) d \\beta = \\frac { \\sqrt { \\pi / \\log q ^ { - 1 } } } { \\left ( q ; q \\right ) _ { n } } , \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} \\sum _ { a \\in A } X _ s ^ a = \\sum _ { b \\in B } Y _ t ^ b = I s \\in S , t \\in T , \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} p ( y ) = y _ n \\left ( a _ 0 + \\sum _ { i = 1 } ^ { n - 1 } a _ i y _ i + b ( y _ n ^ 2 - 3 y _ { n + 1 } ^ 2 ) \\right ) , \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} v \\partial _ { x } \\mp \\beta _ { x } \\partial _ { v } = \\omega _ { \\pm } ( I _ { \\pm } ) \\partial _ { \\theta _ { \\pm } } \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} \\partial _ { t } \\bar { \\mathbf { u } } _ { n } + \\mathcal { A } \\big ( \\mathbf { H } ( \\tilde { \\mathbf { u } } ) \\big ) \\bar { \\mathbf { u } } _ { n } = \\mathbf { f } _ { n } ( 0 , T ) , \\mathbf { u } ( 0 , \\cdot ) = 0 \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} H ( x ) = \\Delta \\rho ( x ) \\ge ( n - 1 ) \\frac { f _ a ^ \\prime \\big ( \\rho ( x ) \\big ) } { f _ a \\big ( \\rho ( x ) \\big ) } \\ge \\sup _ { \\partial B ( o , \\rho ( x ) ) \\times \\R } \\abs { \\bar { \\nabla } f } . \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} P \\left ( x \\right ) = \\left ( 1 - \\rho \\right ) \\delta \\left ( x \\right ) + \\rho \\tilde { P } \\left ( x \\right ) , \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} Q ^ + ( f , f ) = \\int _ { \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } } d \\omega d v _ 2 \\left | \\omega \\cdot ( v - v _ 2 ) \\right | f ( t , x , v ^ * ) f ( t , x , v _ 2 ^ * ) \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} \\big \\langle s , s ' \\big \\rangle _ X = \\int _ X \\big \\langle s ( x ) , s ' ( x ) \\big \\rangle _ { \\Lambda ^ \\bullet ( T ^ { * } X ) \\otimes F } d v _ { X } . \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} - C s - \\frac { ( p / q ) ^ s \\log ( p / q ) } { \\log ( 1 / p ) } \\log _ { 1 / p } n = 0 . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} \\tilde { \\mathcal { A } } ( t ) : = \\mathcal { A } \\big ( \\mathbf { H } ( t , \\cdot ) \\big ) t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} \\mathbb E X ^ s = \\frac { \\gamma \\ , t ^ \\alpha } { E _ { \\alpha , \\beta } ^ \\gamma ( t ^ \\alpha ) \\ , \\Gamma ( \\alpha + \\beta ) } \\ , \\Phi ^ { ( 1 , \\alpha ) } _ { \\gamma + 1 ; \\alpha + \\beta } ( t ^ \\alpha , 1 - s , 1 ) \\ , . \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} \\left [ \\operatorname { c u r l } \\left ( \\mathbf { A \\times B } \\right ) \\right ] _ { p } = \\frac { \\partial } { \\partial x _ { q } } \\left ( A _ { p } B _ { q } - A _ { q } B _ { p } \\right ) \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( Z ) & = & \\tilde { A } Z \\\\ \\sigma ( Z ) & = & \\tilde { B } Z \\end{aligned} \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} ( \\widetilde { \\phi } ( x _ j ) ) ( 1 , d _ i , 1 ) = \\phi ( ( 1 , d _ i d _ j ^ { - 1 } , 1 ) x ) = \\begin{cases} x & i = j , \\\\ 0 & i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} x ^ n + b _ 1 x ^ { n - 1 } + \\cdots + b _ n = ( b _ 1 - a _ 1 ) x ^ { n - 1 } + \\cdots + ( b _ n - a _ n ) . \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} h ( t ) = u ( s ) \\Bigg ( \\frac { | g ( s ) | } { | g ( t ) | } \\Bigg ) ^ { \\frac { 1 } { 2 ( p - 2 ) } } \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{gather*} s _ { 2 r } ^ { ( r ) } \\ ! = \\ ! s _ { r } d _ { 0 } ^ { ( r ) } \\ ! + \\ ! s _ { r + 1 } d _ { 1 } ^ { ( r ) } \\ ! + \\ ! \\ldots \\ ! + \\ ! s _ { 2 r - 2 } d _ { r - 2 } ^ { ( r ) } \\ ! + \\ ! s _ { 2 r - 1 } d _ { r - 1 } ^ { ( r ) } \\ , , \\ s _ { 2 r + m } ^ { ( r ) } \\ ! = \\ ! \\sum \\limits _ { k = 0 } ^ { r - 1 } \\ s _ { r + m + k } ^ { ( r ) } d _ { k } ^ { ( r ) } \\ , , \\ m \\geq 1 \\ , . \\end{gather*}"}
-{"id": "8313.png", "formula": "\\begin{align*} Q _ { S ^ n } ^ 6 = & \\tfrac { n ( n ^ 4 - 2 0 n ^ 2 + 6 4 ) } { 3 2 } , \\\\ P _ { S ^ n } ^ 6 = & - \\Delta _ { S ^ n } ^ 3 - \\tfrac { - 3 n ^ 2 + 6 n + 3 2 } { 4 } \\Delta _ { S ^ n } ^ 2 - \\tfrac { 3 n ^ 4 - 1 2 n ^ 3 - 5 2 n ^ 2 + 1 2 8 n + 1 9 2 } { 1 6 } \\Delta _ { S ^ n } + \\tfrac { n - 6 } { 2 } Q _ { S ^ n } ^ 6 \\\\ = & \\Big ( - \\Delta _ { S ^ n } + \\tfrac { ( n - 6 ) ( n + 4 ) } { 4 } \\Big ) \\Big ( - \\Delta _ { S ^ n } + \\tfrac { ( n - 4 ) ( n + 2 ) } { 4 } \\Big ) \\Big ( - \\Delta _ { S ^ n } + \\tfrac { n ( n - 2 ) } { 4 } \\Big ) . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} \\begin{aligned} & r ^ 2 \\int _ y ^ x \\Big | \\sum _ { \\substack { y \\leq \\N \\mathfrak { p } < u } } \\frac { \\chi ^ * ( \\mathfrak { p } ) \\log \\N \\mathfrak { p } } { \\N \\mathfrak { p } ^ { 1 + i \\tau } } \\Big | \\frac { d u } { u } \\geq \\Big ( \\frac { \\alpha } { 4 e ( 1 + \\alpha ) } \\Big ) ^ { 2 \\phi A \\lambda + 8 } \\cdot \\frac { 1 } { 2 ^ { k + 1 } } \\big \\{ 1 - J ( \\lambda ) \\big \\} , \\end{aligned} \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} W _ \\pm \\big ( D ^ F _ { X _ \\infty } , D ^ F _ { Y _ { \\R _ + } } \\big ) = \\lim _ { t \\rightarrow \\pm \\infty } e ^ { i t D ^ F _ { X _ \\infty } } J e ^ { - i t D ^ F _ { Y _ { \\R _ + } } } : \\ ; L ^ 2 \\big ( \\Omega ^ \\bullet ( Y _ { \\R _ + } , F ) \\big ) \\rightarrow L ^ 2 \\big ( \\Omega ^ \\bullet ( X _ \\infty , F ) \\big ) . \\end{align*}"}
-{"id": "9860.png", "formula": "\\begin{align*} \\psi _ { - 1 / 2 } \\left | \\lambda \\right > & = ( - 1 ) ^ n \\psi _ { \\lambda _ 1 + n - 1 / 2 } ^ \\ast \\psi _ { \\lambda _ 2 + n - 3 / 2 } ^ \\ast \\cdots \\psi _ { \\lambda _ n + 1 / 2 } ^ \\ast \\psi _ { - 1 / 2 } \\left | 0 \\right > \\\\ & = ( - 1 ) ^ n \\psi ^ \\ast ( z _ 1 ) \\psi ^ \\ast ( z _ 2 ) \\cdots \\psi ^ \\ast ( z _ n ) \\left | - 1 \\right > \\ , \\Big \\vert _ { \\boldsymbol { z } ^ { \\lambda } } . \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} \\rho ( x ) = ( x - 2 ) \\varphi ( x ) = \\frac { x - 2 } { x } \\int \\limits _ 0 ^ x \\frac { 1 } { 1 - \\xi } \\chi ( \\xi ) \\ , d \\xi . \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} D _ 1 ( s ) & : = \\sum _ { n \\geq 1 } \\frac { A ( 1 , n ) \\overline { A ( 1 , n ) } } { n ^ s } \\\\ D _ 2 ( s ) & : = \\sum _ { n \\geq 1 } \\frac { A ( 1 , n ) ^ 2 } { n ^ s } . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} P _ { 0 } ( 1 ) + P _ { 1 } ( 1 ) = 1 . \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} h ( x , \\xi ) = & h _ 0 ( \\xi ) + V ( x ) , \\\\ V _ \\rho ( t , x ) = & V ( x ) \\chi ( \\rho x ) \\chi \\left ( \\frac { \\langle \\log \\langle t \\rangle \\rangle x } { \\langle t \\rangle } \\right ) , \\\\ h _ \\rho ( t , x , \\xi ) = & h _ 0 ( \\xi ) + V _ \\rho ( t , x ) , \\\\ \\nabla _ x ^ 2 V _ \\rho ( t , x ) = & { } ^ t \\nabla _ x \\nabla _ x V _ \\rho ( t , x ) , \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} \\frac { c } { n \\lambda } \\log \\left [ \\frac { 1 } { c } \\sum _ { i = 1 } ^ c 2 ^ { \\lambda \\log c } \\right ] = \\frac { c \\log c } { n } , \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} \\prod _ { j \\in S } ( a _ j + b _ j ) = \\sum _ { I \\subset S } \\left ( \\prod _ { j \\in I } a _ j \\right ) \\left ( \\prod _ { J \\in S \\setminus I } b _ j \\right ) . \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} \\vec p _ t ( x , y ) = \\sum _ { c \\in C ( x , y ) } \\vec S ( c ) g _ t ( d _ c ( x , y ) ) , \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} \\Delta ( G _ v g ) = \\Sigma _ { o ( e ) = v } ( \\Sigma _ { G _ e h \\subset { G _ v } } G _ e h g ) - \\Sigma _ { t ( e ) = v } ( \\Sigma _ { G _ e h \\subset { G _ v } } G _ e h g ) . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} [ \\bar { h } _ { i , k } , \\bar { h } _ { i , l } ] = 2 k \\delta _ { k , - l } \\bar { c } , \\ [ \\bar { h } _ { i , k } , \\bar { h } _ { i + 1 , l } ] = - k ( d ^ k + d ^ { - k } ) \\delta _ { k , - l } \\bar { c } , \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} | \\alpha _ { j } ( \\chi ) | = q ^ { 1 / 2 } \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} \\partial _ t \\tilde v + \\tilde v \\cdot \\nabla \\tilde v - \\nu \\Delta \\tilde v + \\nabla \\tilde { p } & = 0 , \\nabla \\cdot \\tilde v = 0 , \\\\ \\tilde v ( t = 0 ) & = \\tilde v _ { \\rm i n } \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} ( f * g ) ( q ) : = \\sum _ { n } \\bigg ( \\sum _ { k + h = n } a _ k b _ h \\bigg ) q ^ n . \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} J = \\bigcup _ { \\substack { k = 0 \\\\ k \\neq i } } ^ n \\left ( p _ i ^ { a _ k } x _ k ^ { a _ i } - p _ k ^ { a _ i } x _ i ^ { a _ k } \\right ) \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} J ( X ) = A _ 1 \\oplus \\dots \\oplus A _ n , \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} x \\theta _ { q ^ { 4 } } \\left ( q ^ { 2 } x ^ { 2 } \\right ) + t \\theta _ { q ^ { 4 } } \\left ( x ^ { 2 } \\right ) = 0 , \\mbox { f o r } t \\in \\R , \\ ; \\mbox { a n d } \\ ; \\theta _ { q ^ { 4 } } \\left ( x ^ { 2 } \\right ) = 0 , \\mbox { f o r } t = \\infty . \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} f \\leq c _ 0 F ^ { - \\delta } \\leq c \\Theta ^ { \\delta } = c e ^ { - \\delta \\tau } \\forall \\tau \\geq \\tau _ 0 , \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} \\ \\hat I ( z ) = \\bar z + \\tfrac { 1 } { 2 } \\ \\ \\hat I ( z ) = - \\bar z + \\tfrac { \\tau } { 2 } . \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 u = g ( x ) | u | ^ { p - 1 } u \\quad & \\mbox { i n } B , \\\\ u = \\Delta u - ( 1 - \\sigma ) u _ n = 0 \\quad & \\mbox { o n } \\partial B , \\end{cases} \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} & B \\dot { u } ( t ) + A u ( t ) = f ( t ) \\ae , \\\\ & u ( 0 ) = u _ 0 . \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} R _ M ( x ) : = \\max ( R _ 1 ( x ) , \\ldots , R _ d ( x ) ) . \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} u _ { \\beta } ^ 2 ( f , g _ 2 ) = [ I - \\beta P ( g _ 2 ) ] ^ { - 1 } \\bar { r } ^ 2 ( g _ 2 ) = \\beta \\left [ \\frac { 5 - 2 p + 7 \\beta } { 1 - \\beta ^ 2 } , \\frac { ( 5 - 2 p ) \\beta + 7 } { 1 - \\beta ^ 2 } \\right ] ^ T . \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} \\| f _ n \\| _ { \\varphi _ n } ^ 2 : = \\int _ { \\C } | f _ n ( z ) | ^ 2 e ^ { - 2 n \\varphi _ n } d V _ m ( z ) \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { K } n _ { k } \\Vert \\widehat { \\beta } _ { - k } ( \\widehat { \\lambda } ) - \\beta \\Vert _ { 2 , n , k } ^ { 2 } & \\leq \\sum _ { k = 1 } ^ { K } n _ { k } \\Vert \\widehat { \\beta } _ { - k } ( \\bar { \\lambda } _ { 0 } ) - \\beta \\Vert _ { 2 , n , k } ^ { 2 } \\\\ & + 2 \\sum _ { k = 1 } ^ { K } \\sum _ { i \\in I _ { k } } \\varepsilon _ { i } X _ { i } ^ { \\prime } ( \\widehat { \\beta } _ { - k } ( \\widehat { \\lambda } ) - \\widehat { \\beta } _ { - k } ( \\bar { \\lambda } _ { 0 } ) ) . \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\partial _ t \\norm { \\zeta } ^ 2 + \\norm { \\nabla \\zeta } ^ 2 + \\alpha \\norm { \\tau } ^ 2 _ { L ^ 2 ( \\Gamma _ u ) } = \\int _ { \\Omega } g \\cdot \\zeta . \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} a \\circ b : = \\sum _ { i \\geq 0 } { \\Delta _ a \\choose i } a _ { ( i - 2 ) } b \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta \\psi = - r ' \\mbox { i n } M \\times \\left ( 0 , T \\right ) \\\\ \\displaystyle \\frac { \\partial \\psi } { \\partial \\eta _ g } = - k _ g R \\quad \\mbox { o n } \\quad \\partial M \\times \\left ( 0 , T \\right ) . \\end{array} \\right . \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} S & : v \\mapsto u = \\arg \\min _ { w } I _ { \\varepsilon , v } ( w ) , \\\\ I _ { \\varepsilon , v } ( w ) & = \\int _ { 0 } ^ { T } \\exp ( - t / \\varepsilon ) \\left ( \\frac { \\varepsilon } { 2 } | w ^ { \\prime } | ^ { 2 } + \\phi ( w ) - ( f ( v ) , w ) \\right ) \\mathrm { d } t , \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { 1 } { \\lambda _ u ^ t \\cdot t ^ { d _ u + 1 } } M ^ t \\vec v _ 0 = \\lambda ' \\lambda _ 0 \\vec u _ \\infty , \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{align*} \\omega ( ( \\epsilon \\otimes \\iota ) \\Delta x ) = \\epsilon ( ( \\iota \\otimes \\omega ) \\Delta x ) = \\lim _ i \\omega _ i ( ( \\iota \\otimes \\omega ) \\Delta x ) = \\lim _ i \\omega _ i \\star \\omega ( x ) = \\omega ( x ) . \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} \\sqrt { - \\frac { d ^ 2 } { d x ^ 2 } } u _ \\nu + V _ \\nu u _ \\nu = 0 , \\end{align*}"}
-{"id": "10070.png", "formula": "\\begin{align*} p = n \\gamma , p = m \\beta , p = l \\alpha \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } '' _ a : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} \\alpha : = \\alpha ( k , l , m , n , s ) = & 5 k ^ 2 + 8 l ^ 2 + 9 m ^ 2 + 8 n ^ 2 + 5 s ^ 2 + 8 k l + 6 k m \\\\ + & 4 k n + 2 k s + 1 2 m l + 8 l n + 3 l s + 1 2 m n + 6 m s + 8 n s . \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} n _ { \\ref { m a i n } } = \\max \\left ( \\frac { n _ { \\ref { c o n n } } } { \\gamma ^ 2 / 4 - 1 4 \\gamma ^ 3 } , \\ ; n _ { \\ref { a b s 5 9 } } , \\ ; \\frac { n _ { \\ref { r e s } } } { 1 - \\gamma } , \\ ; \\frac { n _ { \\ref { c l i c } } } { 1 - \\gamma - \\gamma ^ 2 / 2 } , \\ ; 1 0 ^ { 1 2 } , \\ ; \\frac 2 { \\gamma ^ 3 } \\right ) . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} & \\mathbb { E } \\left [ C _ 1 ^ ( a _ ) \\right ] = \\int _ 0 ^ \\infty \\int _ 0 ^ { x _ 2 } \\frac { 2 } { \\beta ^ 2 } e ^ { - \\frac { x _ 1 + x _ 2 } { \\beta } } \\\\ & \\cdot [ \\log _ 2 ( 1 + \\xi x _ 1 ) - \\log _ 2 \\left ( 1 + ( \\sqrt { 1 + \\xi x _ 2 } - 1 ) \\frac { x _ 1 } { x _ 2 } \\right ) ] d x _ 1 d x _ 2 . \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} \\mathbf { B } _ { j k } = \\left [ \\begin{matrix} \\mathbf { G } _ 1 ^ { 7 j } & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\mathbf { G } _ 2 ^ { 3 k } \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} \\Lambda _ f ( s ) = \\epsilon _ f \\Lambda _ f ( 1 - s ) , \\epsilon _ f = \\pm 1 \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} K ( x , y ) = \\int _ { 1 } ^ \\infty \\varphi ( x + ( y - x ) r ) r ^ { n - 1 } \\ , d r \\quad ( x , y ) \\in \\Omega \\times \\Omega , \\ , x \\neq y . \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} \\tau _ U ( \\hat { \\mu _ R } , 1 / N _ T ) & \\le \\frac { N _ T - 1 + \\frac { N _ R } { N _ R \\hat { \\mu _ R } + 1 } } { N _ T } ( 1 - \\hat { \\mu _ R ) } \\\\ & \\le \\frac { N _ T + \\frac { N _ R } { N _ R \\hat { \\mu _ R } + 1 } } { N _ T } \\\\ & \\le \\frac { N _ T + \\frac { N _ R } { N _ R \\mu _ R } } { N _ T } = 1 + \\frac { 1 } { N _ T \\mu _ R } , \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} L ( \\phi , g , \\gamma ) = g ( - \\Delta + V ) = g D ( \\phi , g , \\gamma ) \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} & \\mathbb { E } [ C _ 2 ^ ( a _ ) ] = \\frac { e ^ { \\frac { 2 } { \\beta \\xi } } } { \\ln ( 4 ) } E _ 1 \\left ( \\frac { 2 } { \\beta \\xi } \\right ) + \\int _ 0 ^ \\infty \\frac { 2 } { \\beta \\ln ( 2 ) } \\\\ & \\cdot \\exp \\left ( - \\frac { x } { \\beta } \\left ( \\frac { \\sqrt { 1 + \\xi x } - 2 } { \\sqrt { 1 + \\xi x } - 1 } \\right ) \\right ) E _ 1 \\left ( \\frac { x \\sqrt { 1 + \\xi x } } { \\beta ( \\sqrt { 1 + \\xi x } - 1 ) } \\right ) d x . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} \\varphi _ 1 ( x ) = \\sum _ { i = 0 } ^ { - r } \\gamma _ i x ^ i = \\sum _ { i = 0 } ^ { \\lfloor - r / 2 \\rfloor } \\delta _ i \\ , x ^ { 2 i } \\ , ( x - 2 ) ^ { - r - 2 i } \\in \\mathcal { F } _ r \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} & M T _ { \\sf d } ^ { K M ( K N + K M ) } + N ( T _ { \\sf d } + 1 ) ^ { K M ( K N + K M ) } + ( T + 1 ) ^ { K N ( K N + K M ) } \\\\ & \\leq ( M + N ) ( T _ { \\sf d } + 1 ) ^ { K M ( K N + K M ) } + ( T + 1 ) ^ { K N ( K N + K M ) } \\\\ & = \\frac { \\lambda _ 2 } { \\lambda _ 1 } \\left ( 1 + \\frac { N } { M } \\right ) T ^ { K N ( K N + K M ) } + ( T + 1 ) ^ { K N ( K N + K M ) } \\\\ & \\leq \\frac { 1 } { \\lambda _ 1 } ( T + 1 ) ^ { K N ( K M + K N ) } , \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} { \\mathrm { E } } \\left [ \\left \\Vert \\sum _ { i = 1 } ^ { n } \\psi _ { i } X _ { i } \\right \\Vert _ { \\infty } ^ { 4 } \\right ] \\leq C n ^ 2 \\log ^ 2 p . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} P ^ x ( \\tau _ D \\in A , X ( \\tau _ D ) \\in B ) = \\int _ A \\int _ B h _ D ( x , s , z ) \\ , d z \\ , d s , x \\in D . \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} \\nabla ^ 2 G _ 2 = \\alpha _ 2 \\vec { I } + \\lambda _ 1 \\cdot \\mathrm { d i a g } \\bigl ( s '' ( x _ 1 ; a _ 1 ) , \\hdots , s '' ( x _ { m n } ; a _ 1 ) \\bigr ) , \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{gather*} \\gamma = \\gamma _ { - } \\gamma _ { 0 + } , \\end{gather*}"}
-{"id": "1792.png", "formula": "\\begin{align*} \\tilde { u } ' = - \\varPhi + \\tilde { u } \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} \\ln \\det ( g D ) | _ { \\Lambda ' } ^ \\Lambda \\sim - 2 \\ln ( \\Lambda / \\Lambda ' ) a _ n ( g , D ) = - 2 \\ln ( \\Lambda / \\Lambda ' ) \\zeta ( 0 , g , D ) \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} { \\bf P } _ e ^ { ( n ) } \\triangleq \\frac { 1 } { M _ n } \\sum _ { w \\in { \\cal M } _ n } { \\bf P } ^ g \\Big \\{ d _ { 0 , n } ( Y ^ { n } ) \\neq w | W = w \\Big \\} \\equiv { \\bf P } ^ g \\Big \\{ d _ { 0 , n } ( Y ^ n ) \\neq W \\Big \\} \\leq \\epsilon _ n \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} U \\theta ( \\Delta ) = \\theta ( \\Delta ) U + \\frac { \\i } { 2 \\pi } \\int _ { \\C } \\frac { \\partial \\tilde { \\theta } } { \\partial \\overline { z } } ( z - \\Delta ) ^ { - 1 } ( \\delta _ { 0 } S ^ * + \\delta _ { 1 } S ^ * - \\delta _ { 0 } S - \\delta _ { - 1 } S ) ( z - \\Delta ) ^ { - 1 } d z \\wedge d \\overline { z } \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\sum _ { k = j } ^ n \\beta ^ k { n \\choose k } { k \\choose j } = { n \\choose j } \\beta ^ j ( 1 + \\beta ) ^ { n - j } \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} h _ E ( x ) ( n ^ * d n ) = h _ E ( \\alpha _ n ( x ) ) ( d ) h _ E ( x ) ( n ^ * n ) . \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} & \\sum _ { x _ i = 1 } ^ \\ell \\sin ( x _ i k _ i ) ^ 2 \\sin ( x _ i k ' _ i ) \\cos ( x _ i k ' _ i ) = \\sum _ { x _ i = 1 } ^ \\ell \\left [ \\frac { \\sin ( 2 x _ i ( k _ i - k ' _ i ) ) - \\sin ( 2 x _ i ( k _ i + k ' _ i ) ) } { 4 } + \\frac { \\sin ( 2 x _ i k ' _ i ) } { 2 } \\right ] . \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ n B _ { \\Gamma , p } ^ { ( \\ell ) } ( z ) & = \\frac { ( - 1 ) ^ { p - 1 } } { z ^ p } \\sum _ { g = 0 } ^ { \\lfloor \\frac { p - 1 } { 2 } \\rfloor } z ^ { 2 g } ( - 1 ) ^ { g } \\binom { n - 1 } { g } \\ ; \\sum _ { h = 0 } ^ { \\lfloor \\frac { p - 1 } { 2 } \\rfloor - g } z ^ { 2 h } \\binom { h + n - 2 } { n - 2 } \\\\ & = \\frac { ( - 1 ) ^ { p - 1 } } { z ^ p } \\sum _ { m = 0 } ^ { \\lfloor \\frac { p - 1 } { 2 } \\rfloor } z ^ { 2 m } \\sum _ { g = 0 } ^ { m } ( - 1 ) ^ { g } \\binom { n - 1 } { g } \\binom { m - g + n - 2 } { n - 2 } . \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} \\Delta \\bar { C } _ t = & \\big ( \\mu _ 1 ( \\beta - 1 ) - \\mu _ 0 ( \\alpha - 1 ) \\big ) + H ( \\alpha ) - H ( \\beta ) + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu _ 1 + \\Delta \\bar { C } _ { t - 1 } } } { 1 + 2 ^ { \\mu _ 0 + \\Delta { C } _ { t - 1 } } } \\Big ) , ~ \\Delta \\bar { C } _ { - 1 } = 0 , ~ t \\in \\mathbb { N } _ 0 ^ n . \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} V _ { n , k , b } = \\left \\{ X \\in \\binom { [ 0 , n ] } { k } : \\overline { X } - \\underline { X } \\leq b \\right \\} \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} ( d - \\sum _ { k = 1 } ^ { j } L _ { k } ) c _ j \\leq \\sum _ { k = j + 1 } ^ K L _ { k } c _ k \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} R _ { \\mathsf { u } } & > N \\sum _ { j = 1 } ^ { L } \\binom { L } { j } \\left ( \\frac { 1 } { N } \\right ) ^ { j } \\left ( 1 - \\frac { 1 } { N } \\right ) ^ { L - j } \\\\ & = N \\sum _ { j = 0 } ^ { L } \\binom { L } { j } \\left ( \\frac { 1 } { N } \\right ) ^ { j } \\left ( 1 - \\frac { 1 } { N } \\right ) ^ { L - j } - N \\left ( 1 - \\frac { 1 } { N } \\right ) ^ { L } \\\\ & = N ( 1 - ( 1 - 1 / N ) ^ L ) . \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{gather*} g _ { a b } ( z ) = \\sum _ { k \\in \\mathbb { Z } } g _ { a b } ^ { ( k ) } z ^ { - k - 1 } , g _ { a b } ^ { ( - 1 ) } = \\delta _ { a b } , g _ { a b } ^ { ( l ) } = 0 , \\ \\ l < - 1 . \\end{gather*}"}
-{"id": "4237.png", "formula": "\\begin{align*} \\overline { \\chi } _ e ( H ) = \\overline { \\chi } _ v ( \\tilde { G } _ H ) . \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} e _ n ^ j ( z _ { 1 , 1 } , \\dots , z _ { 1 , n } ) = e _ n ^ j ( z _ { i , 1 } , \\dots , z _ { i , n } ) \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} W ( k ) = \\mathbb { I } - \\frac { 1 } { 2 } ( e _ { I ^ { k } } - e _ { J ^ { k } } ) ( e _ { I ^ { k } } - e _ { J ^ { k } } ) ^ T , \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} \\lambda u _ 1 = L u _ 1 + D N ( u _ 0 ) u _ 1 . \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} ( \\Delta + \\lambda ) \\varphi = q \\varphi , \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} x \\sim _ n y \\Leftrightarrow T ^ n x = T ^ n y . \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} \\mu _ 0 ( \\alpha _ t , \\gamma _ t ) \\longmapsto \\mu _ 0 ( \\alpha , \\gamma ) = \\frac { H ( \\gamma ) - H ( \\alpha ) } { \\gamma - \\alpha } \\equiv { \\mu } _ 0 , ~ ~ ~ \\mu _ 1 ( \\beta _ t , \\delta _ t ) \\longmapsto \\mu _ 1 ( \\beta , \\delta ) = \\frac { H ( \\beta ) - H ( \\delta ) } { \\beta - \\delta } \\equiv \\mu _ 1 , ~ \\forall { t } . \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} v _ 0 = \\begin{cases} \\max \\{ v \\ge 0 : h ( v ) = 0 \\} , & \\{ v \\ge 0 : h ( v ) = 0 \\} \\ne \\emptyset , \\\\ 0 , & \\{ v \\ge 0 : h ( v ) = 0 \\} = \\emptyset . \\end{cases} \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} B = \\frac { \\theta _ { q } \\ ! \\left ( z ^ { - 1 } \\xi \\right ) } { \\left ( z ^ { 2 } ; q \\right ) _ { \\infty } } . \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} g _ \\varepsilon ^ { ( s ) } ( t ) = g _ \\varepsilon ^ { ( m - 1 ) } ( t ) \\otimes g _ { \\varepsilon } ( t ) ^ { \\otimes ( s - m + 1 ) } \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} \\epsilon _ i \\leq \\rho _ i = \\mathbf { s c r } ( x _ i , t _ i ) \\leq 2 \\epsilon _ i . \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} N _ 0 \\cap N _ 1 = 0 \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} H ^ r ( X _ { m , n , h } , \\Q _ \\ell ) = \\begin{cases} \\Q _ \\ell ( - r / 2 ) & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta _ g f ( p ) = 0 , & \\hbox { ~ ~ i f ~ ~ } n = 1 0 , \\\\ \\nabla ^ k f ( p ) = 0 , k = 2 , 3 , 4 , & \\hbox { ~ ~ i f ~ ~ } n \\geq 1 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} \\mathbf { Q } _ 1 = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 1 \\\\ 0 & 0 \\\\ 0 & 0 \\end{array} \\right ) , \\mathbf { Q } _ 2 = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 0 \\\\ 1 & 0 \\\\ 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} \\xi ( 1 - z ) P ^ { ' } _ { 0 } ( z ) = \\left [ \\lambda ( 1 - z ) + c \\gamma \\right ] P _ { 0 } ( z ) - \\mu p _ { 1 , 1 } , \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} \\varkappa ( z ) = \\exp ( 2 \\pi \\sqrt { - 1 } \\left \\{ T r _ { L _ v / \\mathbb { Q } _ { p } } ( \\pi ^ { - d } z ) \\right \\} _ { p } ) , \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\tilde { \\kappa } _ i = \\kappa _ i ^ { - 1 } . \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} ( X ^ k , Y ) & \\sim \\prod _ { i = 1 } ^ k p ( x _ { i } ) p ( y ) \\\\ & = \\left ( \\prod _ { i = 1 } ^ k \\prod _ { n = 1 } ^ N p ( x ^ { ( n ) } _ { i } ) \\right ) \\prod _ { \\ell = 1 } ^ L p ( y _ { \\ell } ) , \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} A _ p ^ \\# : = \\begin{pmatrix} S ^ p _ { \\alpha \\mu } \\end{pmatrix} , B _ p ^ { \\# } = \\begin{pmatrix} S ^ a _ { \\alpha p } \\end{pmatrix} , C _ p ^ { \\# } = - \\begin{pmatrix} S ^ a _ { \\mu p } \\end{pmatrix} . \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} Z ( s , f , \\chi , \\Delta ) = \\sum \\limits _ { m = 1 } ^ { \\infty } \\int \\limits _ { \\pi ^ { a m } O _ v ^ \\times \\times \\pi ^ { b m } O _ v ^ \\times } \\chi ( a c ( f ( x , y ) ) \\ | f ( x , y ) | ^ s \\ | d x d y | \\\\ = \\sum \\limits _ { m = 1 } ^ { \\infty } q ^ { - ( a + b ) m - d _ 0 m s } \\int \\limits _ { O _ v ^ { \\times 2 } } \\chi ( a c \\ ( F ^ { ( m ) } ( x , y ) ) ) \\ | F ^ { ( m ) } ( x , y ) | ^ s \\ | d x d y | , \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\sup _ { y \\in [ 0 , \\ , u - \\varepsilon ] } \\left | \\frac { L ^ { \\leftarrow } ( t u ) - L ^ { \\leftarrow } ( t y ) } { L ^ { \\leftarrow } ( t u ) } - 1 \\right | = 0 , u > 0 , \\varepsilon \\in ( 0 , u ) \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} \\Sigma : \\dot { g } ( t ) = \\mathcal { X } ( g ( t ) ) + \\sum _ { j = 1 } ^ { m } u _ { j } ( t ) X ^ { j } ( g ( t ) ) , \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} y _ i = \\begin{cases} y _ i \\sim _ \\rho x _ i & \\\\ y _ i \\sim _ 0 x _ i & \\end{cases} \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\sum _ { p \\in \\Omega } t _ p \\log p ~ + ~ \\sum _ { \\ell | q } r _ { \\ell } \\log \\ell ~ = ~ 0 , \\end{align*}"}
-{"id": "10123.png", "formula": "\\begin{align*} C \\ : : \\ : ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q - z ^ { p + 2 q } x ^ { - p } = 0 . \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 3 n ^ { 2 } + ( 2 m + 1 ) n } \\left ( - 1 \\right ) ^ { n } } { \\left ( q ^ { 2 } , - q ^ { m + 1 } , - c q ^ { m + 2 } ; q ^ { 2 } \\right ) _ { n } } = \\frac { ( - 1 ) ^ { m } q ^ { - \\binom { m } { 2 } } a _ { m } ( q ) } { ( q , q ^ { 4 } ; q ^ { 5 } ) _ { \\infty } } - \\frac { ( - 1 ) ^ { m } q ^ { - \\binom { m } { 2 } } b _ { m } ( q ) } { ( q ^ { 2 } , q ^ { 3 } ; q ^ { 5 } ) _ { \\infty } } , \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} n ^ + = n - \\left ( \\frac { 8 ( n - 1 ) } { 8 + n } \\right ) , \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} \\Phi _ 3 ( b ; \\ , c ; \\ , x , \\ , y ) = \\sum _ { n , k = 0 } ^ \\infty \\frac { ( b ) _ n } { ( c ) _ { n + k } } \\frac { x ^ n \\ ; y ^ k } { n ! \\ ; k ! } \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} & d _ { \\Sigma , H D } = \\left \\{ \\begin{array} { l l l } \\frac { K M N } { M + N } { ~ ~ } \\\\ M \\quad \\\\ K N ~ ~ ~ ~ \\end{array} \\right . \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\sum _ { S \\in \\N ( { } ^ { k } H _ { [ i ] } ) } a _ { S } = 0 \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} B ^ { s } _ { 2 , q } ( \\mathbb { R } ^ d ) = ( L _ 2 ( \\mathbb { R } ^ d ) , H ^ 1 ( \\mathbb { R } ^ d ) ) _ { s , q } , \\ , \\ , \\ , \\ , & B ^ { 1 + s } _ { 2 , q } ( \\mathbb { R } ^ d ) = ( H ^ 1 ( \\mathbb { R } ^ d ) , H ^ 2 ( \\mathbb { R } ^ d ) ) _ { s , q } , \\\\ B ^ { - s } _ { 2 , q } ( \\mathbb { R } ^ d ) = & ( L _ 2 ( \\mathbb { R } ^ d ) , H ^ { - 1 } ( \\mathbb { R } ^ d ) ) _ { s , q } . \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} { \\mathrm { E } } [ \\Vert \\widehat { \\beta } ( \\lambda ) \\Vert _ { 0 } \\mid X _ { 1 } ^ { n } ] = \\sum _ { i = 1 } ^ { n } { \\mathrm { E } } [ \\psi _ { i } X _ { i } ^ { \\prime } ( \\widehat { \\beta } ( \\lambda ) - \\beta ) \\mid X _ { 1 } ^ { n } ] , \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} a ^ { i j } ( 0 ) = \\delta ^ { i j } , \\ a ^ { 1 2 } ( x _ 1 , 0 ) = a ^ { 2 1 } ( x _ 1 , 0 ) = 0 , \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} \\frac { c } { n \\lambda } \\log _ 2 \\left [ \\frac { 1 } { c } \\sum _ { k = 1 } ^ { c ( u ^ n ) } \\exp _ 2 \\{ ( \\lambda + 1 ) \\log c _ k ( x ^ n | u ^ n ) \\} \\right ] , \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} & - 2 \\int _ { B _ \\rho } T _ 2 ( \\nabla \\varphi , \\nabla \\Delta \\varphi ) d \\mu _ g \\\\ = & - \\frac { ( n ^ 2 - 2 8 ) ( n - 4 ) ( n - 6 ) ^ 2 } { 1 2 n ( n - 1 ) ( n + 2 ) } | W ( p ) | ^ 2 \\omega _ { n - 1 } \\int _ 0 ^ \\rho r ^ { n + 3 } \\frac { u _ \\epsilon ^ 2 } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 4 } [ ( n + 2 ) \\epsilon ^ 2 + 4 r ^ 2 ] d r \\\\ & + \\int _ { B _ \\rho } \\frac { O ( r ^ 3 ) u _ \\epsilon ^ 2 } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 2 } d x . \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} K ( x , y ) = \\frac { 1 } { | x - y | ^ n } \\int _ { | x - y | } ^ \\infty \\varphi \\bigg ( x + \\frac { y - x } { | y - x | } r \\bigg ) r ^ { n - 1 } \\ , d r \\quad ( x , y ) \\in \\Omega \\times \\Omega , \\ , x \\neq y , \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } \\lvert S _ f ^ \\nu ( n ) \\rvert ^ 2 = c X ^ { 2 \\kappa ( f ) + \\frac { 3 } { 2 } - 2 \\nu } + O ( X ^ { 2 \\kappa ( f ) + 1 - 2 \\nu } \\log ^ 2 X ) \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} K \\min _ { k \\in \\{ 0 , 1 , \\ldots , K - 1 \\} } \\triangle \\ell ^ k & \\leq \\sum _ { k = 0 } ^ { K - 1 } \\triangle \\ell ^ k \\leq ( { \\epsilon } / { ( D ^ p \\rho ) } ) ^ { - \\frac { 1 } { p - 1 } } \\left ( \\Phi ( x ^ { 0 } ) - \\Phi ( x ^ { K } ) \\right ) + \\frac { \\epsilon } { 2 } K \\\\ & \\leq ( { \\epsilon } / { ( D ^ p \\rho ) } ) ^ { - \\frac { 1 } { p - 1 } } ( \\Phi ( x ^ { 0 } ) - \\Phi ^ * ) + \\frac { \\epsilon } { 2 } K , \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } | \\nabla _ { E } \\psi _ { \\epsilon } | _ { L ^ { 2 } ( B ( \\frac { 1 } { 2 } ) ) } = 0 . \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} \\min _ x ~ F ( x ) : = g ( x ) + h ( c ( x ) ) . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} Z _ { s , s } \\left [ Z _ s , t \\right ] = \\psi _ s ^ { - t } Z _ s \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\mathcal { K } = \\{ K \\subset \\mathbb { R } : K \\} . \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\lambda _ n | = \\infty \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} - y ^ { ^ { \\prime \\prime } } ( x ) + q ( x ) y ( x ) = \\lambda y ( x ) , y ( 1 ) = e ^ { i t } y ( 0 ) , y ^ { ^ { \\prime } } ( 1 ) = e ^ { i t } y ^ { ^ { \\prime } } ( 0 ) . \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} & \\hat { A } = \\hat { \\Sigma } _ X ^ { - { 1 \\over 2 } } \\hat U _ { [ : , 1 : r ] } , \\hat A = [ \\hat { \\alpha } ^ { ( 1 ) } , \\hat { \\alpha } ^ { ( 2 ) } , \\cdots , \\hat { \\alpha } ^ { ( r ) } ] , \\\\ & \\hat { B } = \\hat { \\Sigma } _ Y ^ { - { 1 \\over 2 } } \\hat V _ { [ : , 1 : r ] } , \\hat B = [ \\hat { \\beta } ^ { ( 1 ) } , \\hat { \\beta } ^ { ( 2 ) } , \\cdots , \\hat { \\beta } ^ { ( r ) } ] . \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} { \\cal P } _ { \\lambda _ 1 } \\subset { \\cal P } _ { \\lambda _ 2 } , P _ { \\lambda _ 1 } \\cap { \\cal P } _ { \\lambda _ 1 , \\lambda _ 2 } = \\emptyset \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\mathbb { E } [ f ^ p ( z _ i ) - f ^ p ( z ^ * ) ] \\le \\gamma \\left ( 1 - \\tau \\right ) ^ i ( f ^ p ( z _ { 0 } ) - f ^ p ( z ^ * ) ) \\textrm { f o r } i = 1 , \\ldots , \\infty , \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} H _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) ( j , \\beta , \\delta ) & = H _ { ( 2 x ) } ( i , \\alpha ) ( i , \\alpha ) \\otimes H _ { y } ( \\gamma , \\delta ) \\\\ & = T _ x ( i ) \\otimes \\Gamma ( \\alpha ) \\otimes H _ { y } ( \\gamma , \\delta ) \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} \\omega _ p & = d p - p _ t d t - p _ x d x - p _ y d y - p _ z d z , \\\\ \\omega _ q & = d q - q _ t d t - q _ x d x - q _ y d y - q _ z d z , \\\\ \\omega _ r & = d r - r _ t d t - r _ x d x - r _ y d y - r _ z d z \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} S = \\begin{cases} \\sum z _ { 1 , 2 } ^ { ( i _ 1 ) } z _ { 3 , 4 } ^ { ( i _ 3 ) } \\cdots z _ { K - 1 , K } ^ { ( i _ { K - 1 } ) } & \\\\ \\sum z _ { 1 , 2 } ^ { ( i _ 1 ) } z _ { 3 , 4 } ^ { ( i _ 3 ) } \\cdots z _ { K - 2 , K - 1 } ^ { ( i _ { K - 2 } ) } & \\end{cases} \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} Q ( z , \\zeta ) = \\frac { 1 } { K _ { 0 } \\rho ( \\zeta ) } \\sum _ { i = 1 } ^ { n } Q _ { i } ( z , \\zeta ) d \\left ( \\zeta _ { i } - z _ { i } \\right ) \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} \\Lambda : = \\left ( \\frac { \\lambda } { \\theta } - \\frac { \\lambda + 2 \\mu } { \\theta + \\rho } \\right ) , k : = \\frac { \\lambda + 2 \\mu } { \\theta + \\rho } . \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} & f ( \\overline u ) - \\underline v \\le 0 , \\ ; \\ ; h ( \\overline v ) + c g ( \\overline u ) \\le 0 , \\\\ & f ( \\underline u ) - \\overline v \\ge 0 , \\ ; \\ ; h ( \\underline v ) + c g ( \\underline u ) \\ge 0 . \\\\ \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} A _ j = \\begin{pmatrix} \\alpha _ j & \\beta _ j \\\\ \\gamma _ j & \\delta _ j \\end{pmatrix} , A _ 1 = \\begin{pmatrix} I & 0 \\\\ 0 & 0 \\end{pmatrix} , B _ 1 = C _ 1 = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 / \\sqrt { 2 } \\end{pmatrix} , \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} A _ j ^ T = \\rho ( T e _ j ) , \\ , \\sigma _ j ^ T = \\sigma ( T e _ j ) , \\ , \\gamma _ { j k } ^ T = \\gamma ( \\wedge ^ 2 ( T ) e _ j \\wedge e _ k ) . \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} u ( t ) = I _ { t } ^ { \\alpha } f ( t ) , ( a . e . ) \\ , \\ , t \\leq T . \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\langle u , [ V , \\i A ] v \\rangle = - \\langle u , \\big [ ( N - 2 ^ { - 1 } ) ( V - \\tau V ) S + ( N - 2 ^ { - 1 } ) ( V - \\tau ^ * V ) S ^ * \\big ] v \\rangle . \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} m _ 1 A _ { n _ 1 } ~ + ~ \\cdots ~ + ~ m _ k A _ { n _ k } = 0 \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} q _ 1 ( \\lambda ) = c _ 1 \\tilde { t } _ { 1 , \\varphi } ^ 1 ( \\omega , \\lambda ) + c _ 2 \\tilde { t } _ { 1 , \\varphi } ^ 2 ( \\omega , \\lambda ) + U _ 1 ( \\lambda ) \\kappa _ { 1 , \\varphi } ( \\omega , \\lambda ) , \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} S ( a \\rightharpoonup b ) & = S \\left ( S ( a _ 1 ) ( a _ 2 \\circ b ) \\right ) = S ( a _ 1 \\circ b ) a _ 2 = S ( a _ 1 ) ( a _ 2 \\circ S ( b ) ) = a \\rightharpoonup S ( b ) \\end{align*}"}
-{"id": "10005.png", "formula": "\\begin{align*} \\mathcal { J } ( x ) & = \\{ C _ { x ^ i _ { \\alpha } } : x \\in C _ { x ^ i _ { \\alpha } } \\} \\\\ \\mathcal { J } ^ { * } ( x ) & = \\{ C _ { x ^ i _ { \\beta } } : \\exists C _ { x ^ i _ { \\alpha } } \\in \\mathcal { J } ( x ) , C _ { x ^ i _ { \\alpha } } \\cap C _ { x ^ i _ { \\beta } } \\neq \\emptyset \\} \\\\ P ( x ) & = \\bigcap \\{ i n t \\ , R _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ m : C _ { x ^ i _ { \\alpha } } \\in \\mathcal { J } ( x ) , C _ { x ^ i _ { \\beta } } \\in \\mathcal { J } ^ { * } ( x ) , x \\in C _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ m \\} . \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} \\exp ^ x ( q ) = e ^ { x q } . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} v & = 2 ( a + 1 ) ( 2 s ^ 3 ( a + 1 ) ^ 2 - 3 s ^ 2 ( a + 1 ) ^ 3 + 2 s t ( a + 1 ) \\\\ & + s ( a ^ 4 + 8 a ^ 3 + 1 0 a ^ 2 + 8 a + 1 ) - t ( a + 1 ) ^ 2 - 2 a ( a + 1 ) ( a ^ 2 + a + 1 ) ) . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} S ^ { \\prime } \\left [ \\xi ^ { \\prime } \\right ] \\left ( z _ { j } \\right ) & = \\xi _ { j } + \\left ( \\xi _ { 0 } - S \\xi \\left ( z _ { 0 } \\right ) \\right ) \\left \\{ \\varphi _ { z _ { 0 } } \\left ( z _ { j } \\right ) - S \\left [ \\varphi _ { z _ { 0 } } \\mid _ { Z } \\right ] \\left ( z _ { j } \\right ) \\right \\} \\\\ & = \\xi _ { j } + \\left ( \\xi _ { 0 } - S \\xi \\left ( z _ { 0 } \\right ) \\right ) \\left \\{ 0 \\right \\} = \\xi _ { j } , \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} B ( x , M , \\Lambda ) = \\sum _ { j = 0 } ^ t { t \\choose j } P ( - x , t - j ) \\Big ( P ( s , j ) \\lambda _ j - \\sum _ { i = j } ^ s P ( i , j ) m _ i \\Big ) . \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} p ( x = 0 | q _ 1 , s = 0 ) & = p ( x = 1 | q _ 2 , s = 1 ) = a \\\\ p ( x = 1 | q _ 3 , s = 0 ) & = p ( x = 0 | q _ 3 , s = 1 ) = p . \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} { { \\mathcal P } } ^ { - 1 } ( V ) \\overset { \\eqref { d f : c m } } { = } c _ m \\ , \\Delta V \\Bigm | _ { D \\setminus \\{ x _ 0 \\} } + \\left ( 1 - \\limsup \\limits _ { x _ 0 \\neq x \\to x _ 0 } \\dfrac { V ( x ) } { | h _ m ( x - x _ 0 ) | } \\right ) \\cdot { \\delta } _ { x _ 0 } \\ , , V \\in P J _ { x _ 0 } ( D ) , \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } p ( t , x ) = \\Delta p ( t , x ) , \\frac { \\partial p ( t , x ) } { \\partial t } = \\Delta q ( t , x ) , \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} \\Gamma _ M ( k w \\ , | \\ , a ) = k ^ { - B _ { M , M } ( k w \\ , | \\ , a ) / M ! } \\ , \\prod \\limits _ { p _ 1 , \\cdots , p _ M = 0 } ^ { k - 1 } \\Gamma _ M \\Bigl ( w + \\frac { \\sum _ { j = 1 } ^ M p _ j a _ j } { k } \\ , \\Big | \\ , a \\Bigr ) . \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} \\Psi & = - B _ { 1 2 } B _ { 2 2 } ^ { - 1 } \\geq 0 , \\\\ Y & = ( C _ { 1 1 } + \\Psi C _ { 2 1 } ) S ^ { - 1 } \\geq 0 , S : = B _ { 1 1 } + \\Psi B _ { 2 1 } \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} v ( y ) = w ( x ) - x _ { n } y _ n - x _ { n + 1 } y _ { n + 1 } , x = T ^ { - 1 } ( y ) . \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} & g ( x ) : = 2 x - \\sin ( 2 x ) , h ( x ) : = \\frac { 1 } { 1 + g ( x ) ^ 2 } \\\\ & f ( x ) : = \\left ( \\sqrt { ( p + 1 ) ^ 2 + m ^ 2 } + \\sqrt { ( p - 1 ) ^ 2 + m ^ 2 } \\right ) h ( x ) \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} W _ N ( Z _ N ) = W _ N ^ { \\left ( \\tilde { i } _ N ( Z _ N ) , \\tilde { j } _ N ( Z _ N ) \\right ) } ( Z _ N ) \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} \\frac { d } { d z } \\left [ e ^ { - \\frac { \\lambda } { \\xi } z } z ^ { \\frac { \\mu } { \\xi } - 1 } P _ { 1 } ( z ) \\right ] = e ^ { - \\frac { \\lambda } { \\xi } z } z ^ { \\frac { \\mu } { \\xi } - 1 } \\left [ - \\frac { \\gamma } { \\xi ( 1 - z ) } P _ { 0 } ( z ) + p _ { 0 , 0 } \\left ( \\dfrac { ( \\mu - \\xi ) \\gamma } { \\xi \\lambda z } + \\frac { 1 } { A ( 1 ) ( 1 - z ) } \\right ) \\right ] , \\end{align*}"}
-{"id": "10038.png", "formula": "\\begin{align*} G _ k ( z ) = \\tfrac { \\displaystyle g _ k ( z ) } { \\displaystyle z ^ { k - 1 } } = z + \\sum ^ \\infty _ { n = 2 } B _ n z ^ n \\in \\mathcal { S ^ { * } } \\subset \\mathcal { S } . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} f \\ \\in Z ^ 2 ( K , K ) \\Leftrightarrow \\left \\{ \\begin{array} { l l } [ \\delta , \\widetilde { f } ] ( x , y , z ) = 0 , \\\\ \\big [ \\varphi , \\widetilde { f } \\big ] ( x , y , z ) = 0 , \\\\ \\big [ \\lambda , \\widetilde { f } \\big ] ( x , y , v ) = 0 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 \\frac { d y } { ( y ^ 2 + z ^ 2 ) ^ { \\frac { 2 - \\varepsilon } { 2 } } } & = \\frac { 1 } { z ^ { 1 - \\varepsilon } } \\int _ { - 1 / z } ^ { 1 / z } \\frac { d y } { ( 1 + y ^ 2 ) ^ { \\frac { 2 - \\varepsilon } { 2 } } } \\\\ & = \\frac { 2 } { z ^ { 1 - \\varepsilon } } ( \\int _ 0 ^ 1 \\frac { d y } { ( 1 + y ^ 2 ) ^ { \\frac { 2 - \\varepsilon } { 2 } } } + \\int _ 1 ^ { 1 / z } \\frac { d y } { ( 1 + y ^ 2 ) ^ { \\frac { 2 - \\varepsilon } { 2 } } } ) \\\\ & \\sim \\frac { 1 } { z ^ { 1 - \\varepsilon } } \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\frac { \\gamma ^ { j \\alpha } - 1 } { j \\alpha } B _ { j \\alpha } & = \\sum _ { i = 0 } ^ { p - 2 } ( \\gamma ^ \\alpha ) ^ { j i } \\gamma ^ { - i } f _ \\gamma ( \\gamma ^ i ) = \\sum _ { k = 0 } ^ { N - 1 } ( \\gamma ^ \\alpha ) ^ { j k } \\sum _ { i = k \\bmod N } \\gamma ^ { - i } f _ \\gamma ( \\gamma ^ i ) \\pmod p . \\end{align*}"}
-{"id": "10008.png", "formula": "\\begin{align*} W ^ { \\sigma } ( A ; \\gamma ) & : = \\cup _ { x \\in A } W ^ { \\sigma } ( x ; \\gamma ) , \\sigma \\neq c \\\\ s a t ^ c ( A ) & : = \\cup _ { x \\in A } W ^ { c } ( x ; \\gamma ) . \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\xi _ j \\langle \\sigma _ j u , u \\rangle \\geq \\epsilon \\| u \\| ^ 2 . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} & C _ t ( y ^ { t - 1 } _ { t - M } ) = \\sup _ { \\big \\{ { \\pi } _ i ( d x _ i | y _ { i - M } ^ { i - 1 } ) : ~ i = t , t + 1 , \\ldots , n \\big \\} } { \\bf E } ^ { \\pi } \\bigg \\{ \\sum _ { i = t } ^ n \\log \\Big ( \\frac { d { q } _ i ( \\cdot | y ^ { i - 1 } _ { i - M } , x _ i ) } { d \\nu ^ \\pi _ t ( \\cdot | y ^ { i - 1 } _ { i - M } ) } ( Y _ i ) \\Big ) \\Big { | } Y ^ { t - 1 } _ { t - M } = y ^ { t - 1 } _ { t - M } \\bigg \\} . \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} \\abs { \\widetilde { S } _ k - S _ k } \\leq \\sum _ { h = 1 } ^ k \\abs { \\mathcal { F } ^ { h - 1 } \\widetilde { \\mathcal { F } } ( \\widetilde { S } _ { k - h } ) - \\mathcal { F } ^ { h - 1 } \\mathcal { F } ( \\widetilde { S } _ { k - h } ) } . \\end{align*}"}
-{"id": "9877.png", "formula": "\\begin{align*} u _ { \\rm r e g } ( t ) = \\sum _ { j = 0 } ^ \\infty b _ j \\ , t ^ j \\ , , \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{gather*} \\Delta _ { \\hat { S } ( \\gg ^ * ) } ( \\partial ^ \\mu ) = \\tilde { \\mathcal { F } } ^ { - 1 } _ L \\Delta _ 0 ( \\partial ^ \\mu ) \\tilde { \\mathcal { F } } _ l , \\end{gather*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\delta _ E = \\lim _ { \\substack { P \\rightarrow \\infty } } \\frac { T _ E \\log ( P ) } { L } . \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} G _ 2 = \\left [ \\begin{array} { c c } I & 0 \\end{array} \\right ] , F _ 2 = \\left [ \\begin{array} { c c } A & B \\end{array} \\right ] , E _ 1 = I , E _ 2 = \\left [ \\begin{array} { c c } 0 & I \\end{array} \\right ] . \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} H ^ * _ G ( M , \\Z ) = H ^ * ( M , \\Z ) \\otimes H ^ * ( B G , \\Z ) \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} M _ { ( \\tau , \\alpha , \\alpha ) } \\overset { { \\rm i n \\ , l a w } } { = } \\int _ { - \\pi } ^ { \\pi } | 1 + e ^ { i \\psi } | ^ { 2 \\alpha } \\ , d M _ { \\beta } ( \\psi ) , \\ ; \\tau = 1 / \\beta ^ 2 > 1 . \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\gamma _ { j k } = \\gamma _ { 1 k } \\sigma _ 1 ^ { - 1 } \\sigma _ j - \\gamma _ { 1 j } \\sigma _ 1 ^ { - 1 } \\sigma _ k . \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( b c ; q \\right ) _ { n } S _ { n } \\left ( x ; q \\right ) z ^ { n } } { \\left ( c ; q \\right ) _ { n } } = \\frac { \\left ( b ^ { - 1 } , b c z ; q \\right ) _ { \\infty } } { \\left ( c , z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( b c ; q \\right ) _ { n } S _ { n } \\left ( x b z \\right ) , } { \\left ( b c z ; q \\right ) _ { n } b ^ { n } } . \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} y _ e = \\begin{cases} x _ e , & \\frac { 1 + \\rho } { 2 } \\ , , \\\\ 1 - x _ e , & \\frac { 1 - \\rho } { 2 } \\ , . \\end{cases} \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\begin{cases} \\Delta H - H | A | ^ { 2 } + H { \\rm R i c } ^ N ( \\xi , \\xi ) + H ( \\Delta f ) / f + 2 ( { \\rm g r a d } \\ln f ) H = 0 , \\\\ A \\ , ( { \\rm g r a d } \\ , H ) + \\frac { m } { 2 } H { \\rm g r a d } \\ , H - \\ , H \\ , ( { \\rm R i c } ^ N \\ , ( \\xi ) ) ^ { \\top } + H A \\ , ( { \\rm g r a d } \\ , \\ln f ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} \\sum _ { b \\in \\mathfrak { L } } \\eta \\left ( b \\right ) = 2 k \\pi , k \\in \\mathbb { Z } \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} F _ { t , \\varsigma } \\big ( \\sqrt { t } D ^ F _ { Z _ { j , R } } \\big ) ( x , x ) & = F _ { t , \\varsigma } \\big ( \\sqrt { t } D ^ F _ { Z ^ \\mathrm { d b } _ { j , R } } \\big ) ( x , x ) + ( - 1 ) ^ j F _ { t , \\varsigma } \\big ( \\sqrt { t } D ^ F _ { Z ^ \\mathrm { d b } _ { j , R } } \\big ) ( x , \\iota x ) \\\\ & = F _ { t , \\varsigma } \\big ( \\sqrt { t } D ^ F _ { Y _ \\R } \\big ) ( x , x ) + ( - 1 ) ^ j F _ { t , \\varsigma } \\big ( \\sqrt { t } D ^ F _ { Y _ \\R } \\big ) ( x , \\iota x ) . \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} \\begin{cases} & | \\nabla u | ^ 2 \\leq \\max _ { t = s _ 0 } | \\nabla u | ^ 2 , \\\\ & | \\dot { u } | = | \\Delta u | \\leq \\max _ { t = s _ 0 } | \\Delta u | + \\max _ { t = s _ 0 } | \\nabla u | ^ 2 + \\frac { R + 1 } { 4 } , \\end{cases} \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} F _ \\Gamma ^ { p - 1 } ( z ) = \\frac { 1 } { ( 1 - z ^ 2 ) ^ { n - 1 } } \\sum _ { \\ell = 0 } ^ { n } \\vartheta _ { \\mathcal L _ \\Gamma } ^ { ( \\ell ) } ( z ) \\ ; A _ { p } ^ { ( \\ell ) } ( z ) + \\frac { ( - 1 ) ^ p } { z ^ p } \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} k ( \\mathbf { x } ) = D \\ \\Leftrightarrow \\ k ( \\mathbf { y } ) = D ' . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} & \\deg \\left ( _ \\infty \\left ( \\frac { \\Delta } { \\Delta _ { N , 1 2 } } \\right ) \\right ) = 2 ^ { n _ 1 - 1 } 3 ^ { n _ 2 - 1 } 7 ^ { n _ 3 - 1 } \\cdot 3 \\cdot 4 \\cdot 8 - 1 \\\\ & \\deg \\left ( _ \\infty \\left ( \\frac { \\Delta _ N } { \\Delta _ { N , 1 2 } } \\right ) \\right ) = 2 ^ { n _ 1 - 1 } 3 ^ { n _ 2 - 1 } 7 ^ { n _ 3 - 1 } ( 3 \\cdot 4 \\cdot 8 - 2 \\cdot 3 \\cdot 7 ) = 2 ^ { n _ 1 - 1 } 3 ^ { n _ 2 - 1 } 7 ^ { n _ 3 - 1 } \\cdot 5 4 . \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} p _ i = \\sum _ { j = 0 } ^ { h ( i ) } \\omega ^ { a _ j ( i ) } \\left ( \\frac { z } { 2 } \\right ) ^ j . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} \\textbf { a } _ { B S } = \\dfrac { 1 } { \\sqrt { N _ { B S } } } [ 1 , e ^ { j { \\pi } \\sin ( \\theta ) } , . . . , e ^ { j ( N _ { B S } - 1 ) { \\pi } \\sin ( \\theta ) } ] ^ T \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} \\mathcal { E } _ 2 ( k ) = [ \\lambda _ { \\ell } ( k ) , \\lambda _ r ( k ) ] = E _ { k ; - ; - } \\cup E _ { k ; + ; + } \\cup E _ { k ; - ; + } \\cup E _ { k ; + ; - } . \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ { 2 } } } { \\left ( q ^ { 2 } , q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } } = \\frac { 1 } { \\left ( q ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } } { \\left ( q ; q \\right ) _ { n } } A _ { q } ^ { 2 } \\left ( q ^ { n } \\right ) , \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\left ( H _ { x _ 1 , x _ 1 , x _ 2 } + 3 H _ { x _ 2 } H _ { x _ 1 } + n _ 0 \\frac { H _ { x _ 1 x _ 2 } ^ 2 } { H _ { x _ 2 } } \\right ) _ { x _ 1 } = H _ { x _ 2 x _ 3 } \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} Z _ i : = \\d T ( J _ i ) , . \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} \\sum _ { \\alpha = 1 } ^ { 8 - j } x _ \\alpha ^ 2 = 0 , \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} \\phi & = \\begin{pmatrix} 1 & 0 \\\\ h & 1 \\end{pmatrix} : R \\ltimes _ f R \\to R \\ltimes _ g R \\\\ \\psi & = \\begin{pmatrix} 1 & 0 \\\\ - h & 1 \\end{pmatrix} : R \\ltimes _ g R \\to R \\ltimes _ f R . \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} \\Vert 1 _ A - P _ t 1 _ A \\Vert _ 1 = & 2 \\left ( \\mu ( A ) - \\Vert P _ \\frac { t } { 2 } ( 1 _ A ) \\Vert _ 2 ^ 2 \\right ) . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} \\kappa _ E ^ { - 1 } ( \\mathcal { Z } _ E ( \\mu ) ) = \\kappa _ E ^ { - 1 } ( \\{ e _ 1 \\cdots e _ { i _ 0 - 1 } e _ { i _ 0 } ^ \\infty \\} ) = \\{ e _ 1 \\cdots e _ { i _ 0 - 1 } \\} = \\mathcal { Z } _ { E _ \\curlyvee } ( e _ 1 \\cdots e _ { i _ 0 - 1 } ) \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} f ( m ) ( f ( m ) + 1 ) = 0 , ~ ~ \\forall m \\in \\mathbb Z , m \\neq 0 , 1 . \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} r \\iota _ i ( 1 ) = \\iota _ i ( 1 ) r , \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} [ F _ n ] = \\{ ( G \\circ K _ 1 ) + K _ 1 : | G | = n \\} . \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} ( \\min _ { j } \\beta _ { j } ^ { 2 } ) g _ { E } \\leq g ^ { k } _ { \\beta , \\epsilon } \\leq g _ { E } , \\ g _ { E } = \\Sigma _ { j = 1 } ^ { k } ( d s _ { j } ^ 2 + s _ { j } ^ 2 d \\theta _ { j } ^ 2 ) + \\Sigma _ { j = k + 1 } ^ { n } d z _ { j } \\otimes d \\bar { z } _ { j } . \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} h _ D ( x , s , z ) = \\int _ D p _ D ( s , x , y ) \\nu ( y - z ) \\ , d y . \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} r ' & = \\inf \\{ s \\mid \\mu ( s , T _ + ) \\le \\mu ( r , T ) \\} \\\\ r '' & = \\inf \\{ s \\mid \\mu ( s , T _ - ) \\le \\mu ( r , T ) \\} . \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} x _ { k j } [ t ] = \\varphi _ t ( W _ { k j } ^ { [ { \\sf u } ] } ) , \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , M } ( x \\ , | \\ , a ) \\ , \\log ( x ) \\bigr ) ( q | b ) = \\log ( q ) \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , M } ( x \\ , | \\ , a ) \\ , \\bigr ) ( q | b ) + O ( q ) , \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} Z _ n ^ * = & ( 2 s ^ 3 _ n | \\tau _ n | ) ^ { - 1 } \\Big [ 2 \\tau _ n s _ n ^ 2 \\Big ( \\dfrac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\psi '' ( \\bar { \\epsilon } _ i ) [ \\mathbf { x _ i } ' ( ( \\mathbf { E _ * } L _ n ^ * ) ^ { - 1 } \\Delta _ n ^ * ) ] \\Big ) \\\\ & - \\tau ^ 2 _ n \\Big ( \\dfrac { 2 } { n } \\sum _ { i = 1 } ^ { n } \\psi ( \\bar { \\epsilon } _ i ) \\psi ' ( \\bar { \\epsilon } _ i ) [ \\mathbf { x _ i } ' ( ( \\mathbf { E _ * } L _ n ^ * ) ^ { - 1 } \\Delta _ n ^ * ) ] \\Big ) \\Big ] \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} e _ { n , j } ^ d = e _ n ^ d - x _ j e _ { n , j } ^ { d - 1 } = e _ n ^ d - x _ j e _ n ^ { d - 1 } + x _ j ^ 2 e _ { n , j } ^ { d - 2 } = \\dots = \\sum _ { c = 0 } ^ k ( - x _ j ) ^ d e _ n ^ { d - c } . \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} [ \\phi , \\psi ] & = \\phi \\circ \\psi + ( - 1 ) ^ { | \\psi | | \\phi | } \\psi \\circ \\phi . \\\\ \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} J _ { 2 \\lambda - 1 } ( y ) = \\frac { 1 } { 4 \\pi i } \\int \\limits _ { \\Re s = \\Delta } \\frac { \\Gamma ( \\lambda - 1 / 2 + s / 2 ) } { \\Gamma ( \\lambda + 1 / 2 - s / 2 ) } \\left ( \\frac { y } { 2 } \\right ) ^ { - s } d s \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} \\| \\partial e ^ { t \\Delta } f \\| _ { \\mathcal { H } _ x } \\le M ^ 2 \\int _ { X } \\| \\partial f \\| _ { \\mathcal { H } _ y } d \\mu ( y ) \\sum _ { j = 1 } ^ { + \\infty } \\frac { 1 } { \\lambda _ j } e ^ { - \\lambda _ j ( t - 2 t _ 0 ) } . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} N _ { \\mathcal L } ( a q + r , \\ell ) = \\sum _ { s = 0 } ^ { n - \\ell } 2 ^ s \\binom { \\ell + s } { s } \\sum _ { t = s } ^ { a } \\binom { t - s + n - \\ell - 1 } { n - \\ell - 1 } \\ , N _ { \\mathcal L } ^ { \\mathrm { r e d } } ( ( a - t ) q + r , \\ell + s ) . \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} \\pi _ * \\mu _ { G , p } ^ { \\rm c a n } ( V \\cap \\pi ( K _ p ) ) = [ K ^ { \\rm a d } _ p : \\pi ( K _ p ) ] { \\rm v o l } ( V \\cap \\pi ( K _ p ) ) \\leqslant [ K ^ { \\rm a d } _ p : \\pi ( K _ p ) ] { \\rm v o l } ( V ) . \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} g ( x ) = \\inf { \\{ \\sum _ { k = 1 } ^ r \\theta _ k \\mid x = \\sum _ { k = 1 } ^ r \\theta _ k x _ k , \\ ; \\theta _ k \\geq 0 , \\ ; x _ k \\in C , \\ ; k = 1 , \\ldots , r \\} } . \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} C _ { A B } { } ^ { C } C _ { C D } { } ^ { E } + C _ { B D } { } ^ { C } C _ { C A } { } ^ { E } + C _ { D A } { } ^ { C } C _ { C B } { } ^ { E } = 0 , \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} U ( m ) \\alpha _ { i ^ * } ^ m = Q _ { j ^ * , \\ell ^ * + e ^ * } ( n ) \\beta _ { j ^ * } ^ n \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} & A ^ { i j k l , * } : = \\lambda \\delta ^ { i j } \\delta ^ { k l } + \\mu \\left ( \\delta ^ { i k } \\delta ^ { j l } + \\delta ^ { i l } \\delta ^ { j k } \\right ) , \\\\ & B ^ { i j k l , * } : = \\theta \\delta ^ { i j } \\delta ^ { k l } + \\frac { \\rho } { 2 } \\left ( \\delta ^ { i k } \\delta ^ { j l } + \\delta ^ { i l } \\delta ^ { j k } \\right ) , \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\begin{cases} ^ { C \\ ! } D _ { 0 + } ^ { \\alpha } x _ 1 ( t ) = & f _ 1 ( t , x _ 1 ( t ) ) , \\\\ ^ { C \\ ! } D _ { 0 + } ^ { \\alpha } x _ 2 ( t ) = & f _ 2 ( t , x _ 1 ( t ) , x _ 2 ( t ) ) , \\\\ \\cdots & \\cdots \\\\ ^ { C \\ ! } D _ { 0 + } ^ { \\alpha } x _ d ( t ) = & f _ d ( t , x _ 1 ( t ) , x _ 2 ( t ) , \\cdots , x _ d ( t ) ) , \\end{cases} \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} & \\mathbb { E } [ \\| x ^ { k + 1 } _ i - x ^ * _ i \\| ^ 2 \\mid \\mathcal { F } _ k ] \\leq ( 1 + 2 p _ { \\max } \\max _ i L _ i ( \\alpha _ { \\max } - \\alpha _ { \\min } ) ) \\| x _ i ^ k - x ^ * _ i \\| ^ 2 + 4 p _ { \\max } \\alpha _ { \\max } ^ 2 C ^ 2 \\cr & + 2 p _ { \\max } \\alpha _ { \\max } B \\sum _ { j = 1 } ^ N \\mathbb { E } [ W ( k ) ] _ { i j } \\| v ^ k _ j - y ^ k \\| - 2 p _ { \\min } \\alpha _ { \\min } ( F _ i ( x _ i ^ k , \\bar x ^ k ) - F _ i ( x ^ * _ i , \\bar x ^ * ) ) ^ T ( x ^ k _ i - x ^ * _ i ) , \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { R G } } _ { \\Lambda \\Lambda ' } S ( \\phi _ \\Lambda ; g ( \\Lambda ) ) = S ( \\mathcal { R G } _ { \\Lambda \\Lambda ' } ( \\phi _ \\Lambda ; g ( \\Lambda ) ) ) \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} W _ { 1 2 } W _ { 1 3 } W _ { 2 3 } = W _ { 2 3 } W _ { 1 2 } . \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} q _ \\Phi ( \\mu , \\nu ) = \\left [ \\begin{array} { c } \\mu \\\\ \\nu \\end{array} \\right ] ^ H \\Phi \\left [ \\begin{array} { c } \\mu \\\\ \\nu \\end{array} \\right ] = 0 , \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} \\xi _ \\chi ( x ) = \\chi ( \\overline { \\zeta } _ x ) \\psi ( \\overline { u } _ x ) ^ n , \\mu ( x ) = \\lambda ^ { v _ E ( x ) } \\delta ( \\overline { \\zeta } _ x ) ^ { n - 1 } , \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} \\epsilon ^ 2 = \\Big \\lVert e ^ { 4 i R \\lambda _ 0 } C _ { 1 2 } ( \\lambda _ 0 ) v - v \\Big \\rVert _ Y ^ 2 = \\sum _ { j = 1 } ^ m \\Big | e ^ { 4 i R \\lambda _ 0 + i \\theta _ j ( \\lambda _ 0 ) } - 1 \\Big | ^ 2 \\big \\lVert v _ j \\big \\rVert _ Y ^ 2 . \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\varepsilon \\sup _ { s \\in \\left [ 0 , \\varepsilon ^ { - 2 } t \\right ] } \\left \\vert e _ { \\alpha } ^ { \\varepsilon } \\left ( s \\right ) \\right \\vert = 0 \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} f _ { n } & = ( - 1 ) ^ { n } \\alpha ^ { - n } q ^ { \\frac { 1 } { 2 } n ( n + 1 ) } \\left [ 1 + o ( 1 ) \\right ] \\ ! , \\\\ g _ { n } & = ( - 1 ) ^ { n } \\alpha ^ { n } q ^ { - \\frac { 1 } { 2 } n ( n - 1 ) } \\left [ \\theta _ { q } \\ ! \\left ( z ^ { - 1 } \\alpha ^ { - 1 } \\right ) + o ( 1 ) \\right ] \\ ! , \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} \\tau ^ * ( \\mu _ R , \\mu _ T ) \\le \\tau _ U = \\left \\{ \\begin{array} { l l } 1 - \\mu _ R , & ( \\mu _ R , \\mu _ T ) \\in \\mathcal { R } ^ 1 _ { 2 2 } \\\\ 2 - 2 \\mu _ R - \\mu _ T , & ( \\mu _ R , \\mu _ T ) \\in \\mathcal { R } ^ 2 _ { 2 2 } \\end{array} , \\right . \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} Q _ j : = Q ( \\rho _ j , \\theta _ j ) . \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i { 2 m + 2 k - 1 \\choose m - i } { m + i \\choose i } \\gamma _ { m + i - 1 } = 0 \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } | A B | ^ 2 \\sin ( \\angle B A C ) + \\frac { ( 1 - r ) ^ 2 } { 2 } ( \\angle B O C + \\sin ( \\angle B O C ) ) = \\pi r ^ 2 . \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} \\begin{cases} s \\hat { v } ( x , s ) + a \\hat { v } _ { x x x } ( x , s ) = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ \\hat { v } ( 0 , s ) = \\hat { h } _ 0 ( s ) , \\ , \\ , \\hat { v } ( L , s ) = \\hat { h } _ 1 ( s ) , \\ , \\ , \\hat { v } _ { x } ( L , s ) = \\hat { h } _ 2 ( s ) , & \\ , \\ , ( 0 , T ) , \\\\ \\hat { v } ( x , 0 ) = 0 , & \\ , \\ , ( 0 , L ) . \\end{cases} \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } ( - a b q ^ { - \\alpha } ; q ) _ { \\infty } = \\frac { 1 } { \\sqrt { 2 \\pi \\log q ^ { - 1 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i \\alpha x \\right ) } { \\left ( a b q ^ { - 1 / 2 } e ^ { - i x } ; q \\right ) _ { \\infty } } d x \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\langle { \\Lambda _ \\varphi ( x ) | \\Lambda _ \\varphi ( y ) } \\rangle = \\varphi ( y ^ * x ) \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} - \\log ( 1 - x ) = \\sum _ { n = 1 } ^ \\infty \\frac { x ^ n } { n } < \\sum _ { n = 1 } ^ \\infty x ^ n = \\frac { x } { 1 - x } . \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} p _ l = \\sum _ { \\alpha = 1 , p = 1 } ^ { 8 - j , 7 - j } ( S ^ l _ { \\alpha p } + \\pm \\sqrt { - 1 } T ^ l _ { \\alpha p } ) x _ \\alpha z _ p + \\ ; { \\rm o t h e r \\ ; t e r m s } , \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} \\begin{array} { c } \\{ Y _ { \\alpha } \\mid \\alpha \\in J \\} = \\{ Y \\subset X \\mid | Y | = 2 , \\ , \\ , Y = \\{ y _ 1 , y _ 2 \\} , \\\\ { } y _ 1 { } y _ 2 { } \\} \\ , . \\end{array} \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} \\tilde { b } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} u ( x _ 1 , x _ 2 , t ) = U _ 0 ( x _ 1 , x _ 2 , t ) + U _ 1 ( x _ 1 , x _ 2 , t ) , \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{gather*} \\big ( \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta ) } \\big ) ^ { 2 } = \\tau _ { k , \\ell } ^ { ( \\alpha + 1 , \\beta ) } \\tau _ { k , \\ell } ^ { ( \\alpha - 1 , \\beta ) } + \\tau _ { k + 1 , \\ell + 1 } ^ { ( \\alpha - 1 , \\beta ) } \\tau _ { k - 1 , \\ell - 1 } ^ { ( \\alpha + 1 , \\beta ) } - \\tau _ { k + 1 , \\ell } ^ { ( \\alpha - 1 , \\beta ) } \\tau _ { k - 1 , \\ell } ^ { ( \\alpha + 1 , \\beta ) } , \\end{gather*}"}
-{"id": "6731.png", "formula": "\\begin{align*} & \\sup _ { x \\in \\mathbb R } | \\nabla u ( t , x ) | \\leq \\| u ( t ) \\| _ { C ^ { 1 , \\alpha } } \\leq \\| u ( t ) \\| _ { H ^ { 1 + \\delta } _ p } \\\\ \\leq & \\sup _ { t \\in [ 0 , T ] } \\| u ( t ) \\| _ { H ^ { 1 + \\delta } _ p } = \\| u \\| _ { C ( [ 0 , T ] ; H ^ { 1 + \\delta } _ p ) } = : C , \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} \\phi ( t , x , u ) : = A x + B u + c , \\Upsilon ( t , x , u ) : = C x + D u + q \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} \\tau = \\{ ( k _ i , \\lambda ^ { ( i ) } , \\mu ^ { ( i ) } ) \\} _ { i = 1 } ^ r : \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} M : = \\left | \\gamma _ { 2 } + \\Re \\mu + | \\Im \\mu | \\right | + 1 \\in [ 1 , + \\infty ) \\ , . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} \\frac { 1 } { \\| \\rho \\| } \\rho \\circ T ^ n ( q ) > \\frac { 1 } { \\| \\rho \\| } \\ , \\big ( 1 - \\| \\rho \\| \\epsilon - ( 1 - \\| \\rho \\| ) \\big ) = 1 - \\epsilon , \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{gather*} \\epsilon _ { a b } ^ { [ k ] ( \\alpha ) } ( z ) = \\frac { ( - 1 ) ^ { k } z ^ { k ( 2 a - 1 ) } } { \\ell ! } ( - 1 ) ^ { \\frac { \\ell ( \\ell - 1 ) } 2 } ( - 1 ) ^ { \\frac { k ( k - 1 ) } { 2 } } ( - 1 ) ^ { k b } . \\end{gather*}"}
-{"id": "2398.png", "formula": "\\begin{align*} F _ { k } ( x ) = P ( X _ { ( k ) } < x ) = \\sum _ { m = k } ^ { n } \\binom { n } { m } F ^ { m } ( x ) ( 1 - F ( x ) ) ^ { n - m } , \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} \\tilde { M } = \\mathrm { g r a p h } \\ , \\tilde { u } = \\{ \\ \\tilde { \\tau } = \\tilde { u } ( x ) : x \\in \\mathbb { S } ^ n \\} . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} h ( \\sigma ^ a _ b ) = \\sigma ^ a _ b + \\varepsilon ( \\mathrm { T r } \\sigma ) \\delta ^ a _ b , \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\psi ( X ; H ) : = \\sum _ { X - \\frac { H } { 2 } \\leq n \\leq X + \\frac { H } { 2 } } \\Lambda ( n ) \\sim H . \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} \\frac { a _ m ^ k } { a _ n ^ k } = \\frac { a _ m ^ j } { a _ n ^ j } \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} \\xi _ { k } ( u ) = 1 + \\hbar \\sum _ { r \\geq 0 } \\xi _ { k , r } u ^ { - r - 1 } \\ , \\ \\ , \\ x _ k ^ { \\pm } ( u ) = \\hbar \\sum _ { r \\geq 0 } x _ { k , r } ^ { \\pm } u ^ { - r - 1 } . \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} H ( X ) & = \\tfrac { 1 } { 4 ( g ! ) ^ { 2 } } \\int _ { X ^ { g } } \\log \\| \\theta \\| ( 2 P _ { 1 } + P _ { 2 } + \\dots + P _ { g - 1 } - P _ g ) \\Psi ^ { * } \\nu ^ { g } \\\\ & = \\tfrac { 1 } { ( g ! ) ^ { 2 } } \\int _ { X ^ { g } } \\log \\| \\theta \\| ( 2 P _ { 1 } + P _ { 2 } + \\dots + P _ { g - 1 } - P _ { g } ) \\Phi ^ { * } \\nu ^ { g } , \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} \\theta _ \\alpha \\circ J _ \\alpha = - \\frac { 2 n } { 2 n - 1 } \\theta _ 1 \\circ J _ 1 , \\alpha = 2 , 3 . \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} \\left \\langle J _ { ( a b , \\alpha ) } , J _ { ( c d , \\beta ) } , P _ { ( e , \\gamma ) } \\right \\rangle & = \\alpha _ { \\delta } K _ { \\alpha \\beta \\gamma } ^ { \\delta } \\left \\langle \\tilde { J } _ { a b } , \\tilde { J } _ { c d } , \\tilde { P } _ { e } \\right \\rangle , \\\\ & = \\frac { 1 } { 8 } \\left ( \\alpha _ { 0 } K _ { \\alpha \\beta \\gamma } ^ { 0 } + \\alpha _ { 1 } K _ { \\alpha \\beta \\gamma } ^ { 1 } \\right ) \\varepsilon _ { a b c d e } . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l l l l } $ u ( 0 , t ) = h _ 0 ( t ) $ , & $ u ( L , t ) = h _ 1 ( t ) $ , & $ u _ { x } ( L , t ) = h _ 2 ( t ) $ & i n $ ( 0 , T ) $ , \\\\ $ v ( 0 , t ) = 0 $ , & $ v ( L , t ) = 0 $ , & $ v _ { x } ( L , t ) = g _ 2 ( t ) $ & i n $ ( 0 , T ) $ . \\end{tabular} \\right . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i & = \\left ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } ) _ { , K } \\right ) _ { , J } - E ( | \\dot { u } | ) \\dot { u } _ i , \\\\ a \\ddot { \\tau } & = - \\beta _ { K i } \\dot { u } _ { i , K } + m _ { I J } q _ { I , J } + M _ { j L K I } u _ { j , L K I } + K _ { I J } \\tau _ { , I J } , \\\\ \\kappa \\dot { q } _ i & = \\dot { \\tau } _ { , i } - q _ { i } , \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} \\int _ M [ | \\nabla f | ^ { p - 2 } \\langle \\nabla f , \\nabla \\phi \\rangle - \\lambda | f | ^ { p - 2 } \\langle f , \\phi \\rangle ] d \\mu _ g = 0 \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} \\omega _ { s } = \\frac { - \\pi \\sqrt { - 1 } } { \\log | s | } \\sum _ { i j } Z ^ { i j } d w _ { i } \\wedge d \\bar w _ { j } , \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} f _ { n } ( z ) = A ( z ) g _ { n } ( z ) + B ( z ) \\tilde { g } _ { n } ( z ) , \\forall n \\in \\Z . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} b _ k ( x , \\xi ' ) = \\big ( a _ { n n } ^ { ( k ) } \\big ) ^ { - 2 } \\sum _ { i , j = 1 } ^ { n - 1 } \\big ( a _ { i j } ^ { ( k ) } a _ { n n } ^ { ( k ) } - a _ { n i } ^ { ( k ) } a _ { n j } ^ { ( k ) } \\big ) \\xi _ i \\xi _ j , \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { I } \\ \\ \\mathcal { I } = [ 0 , \\lambda ) \\ \\ \\ \\ \\mathcal { I } = ( \\lambda , 4 ] , \\ \\\\ \\mathcal { I } \\cap g _ k ( \\mathcal { I } , y ) = \\emptyset \\ \\ \\ \\ y \\in \\{ 0 , 1 \\} . \\end{cases} \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} \\alpha = \\begin{cases} \\frac { 1 } { 4 } + \\frac { 1 + \\sqrt { 8 \\gamma ^ 2 + 1 } } { 1 6 \\gamma ^ 2 } , & \\mbox { i f } ~ \\gamma \\ge \\frac { 1 + \\sqrt { 2 } } { 2 } , \\\\ \\frac { 2 \\gamma + 1 } { 8 \\gamma ^ 2 } + \\frac { \\sqrt { 8 \\gamma + 3 } } { 1 6 \\gamma ^ 2 } , & \\mbox { i f } ~ \\frac { 1 } { 2 } < \\gamma < \\frac { 1 + \\sqrt { 2 } } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} \\tilde v _ k ( t ) = \\| \\tilde v _ k ( t ) \\| \\omega ( \\tilde \\Phi _ k ( t ) ) , \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} u _ { k ; i j } = u _ { i ; k j } = u _ { i ; j k } + R ^ { \\ell } _ { k j i } u _ { \\ell } , \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} 8 = m _ { - } \\geq 2 k + 1 - j - c _ j , \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} a \\rightharpoonup 1 & = S ( a _ 1 ) ( a _ 2 \\circ 1 ) = S ( a _ 1 ) a _ 2 = \\epsilon ( a ) 1 & & a \\in A . \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} A ( k , \\xi ) - A ( k , \\xi - \\eta ) & = M ( k , \\xi ) \\left ( ( 1 + k ^ 2 + \\xi ^ 2 ) ^ { N / 2 } - ( 1 + k ^ 2 + ( \\xi - \\eta ) ^ 2 ) ^ { N / 2 } \\right ) \\\\ & + ( M ( k , \\xi ) - M ( k , \\xi - \\eta ) ) ( 1 + k ^ 2 + ( \\xi - \\eta ) ^ 2 ) ^ { N / 2 } \\\\ & = { \\rm c o m _ 1 + c o m _ 2 } . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ { \\infty } ( - 1 ) ^ { k } q ^ { k ( k - 1 ) / 2 } A _ { q } \\left ( z q ^ { k + \\ell } \\right ) A _ { q } \\left ( z ^ { - 1 } q ^ { k - \\ell } \\right ) = 0 . \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} H _ { ( x y ) } ( i , \\alpha ) ( j , \\beta ) = H _ x ( i , j ) \\otimes H _ y ( \\alpha , \\beta ) \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } E _ { 1 , j } = 0 \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} P ( t ) = B _ s ( P ( \\cdot ) ) ( t ) + R ( t ) , \\end{align*}"}
-{"id": "9894.png", "formula": "\\begin{align*} ( \\mathbf n _ V - ( \\mathbf n _ V \\cdot \\nu _ { \\partial \\Omega } ) \\nu _ { \\partial \\Omega } ) \\| \\delta V \\| = - \\sigma \\mathbf n _ { B ^ + } \\mathcal H ^ { n - 1 } \\lfloor _ { \\partial ^ * B ^ + } \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\mu ^ { ( m ) } = \\tfrac { 1 } { 4 } \\delta _ { - 2 ^ { 1 - m } t , 2 ^ { 1 - m } t } + \\tfrac { 3 } { 4 } \\mu ^ { ( m - 1 ) } . \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} v ( P ( x , f ( x ) ) ) = v \\left ( \\sum _ { ( i , j ) \\in I } b ( i , j ) x ^ i ( f ( x ) ) ^ j \\right ) \\neq \\infty . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} E ^ r = \\oplus _ { \\lambda \\in \\mathcal { P } ( k , r ) } E _ \\lambda , \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} [ \\sqrt { - \\Delta + m ^ 2 } , e ^ { i \\gamma \\cdot x } ] = e ^ { \\gamma \\cdot x } \\sqrt { - ( \\nabla - i \\gamma ) ^ 2 + m ^ 2 } - e ^ { \\gamma \\cdot x } \\sqrt { - \\Delta + m ^ 2 } . \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} \\mathbf { u } ( 0 , \\cdot ) = \\tilde { \\mathbf { u } } ^ { 0 } - \\frac { 1 } { | G | } \\int _ { G } \\tilde { \\mathbf { u } } ^ { 0 } \\mathrm { d } \\mathbf { x } , \\mathbf { H } ( 0 , \\cdot ) = \\tilde { \\mathbf { H } } ^ { 0 } G \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} \\Gamma & \\le C _ 1 \\int _ 0 ^ 1 ( 1 + \\| \\alpha _ t ( \\beta ) \\| ) \\| \\nabla v ( t , X _ t ) + \\beta _ t \\| \\ , d t \\\\ & = C _ 1 \\sum _ { k = 0 } ^ { N - 1 } \\int _ { k \\delta } ^ { ( k + 1 ) \\delta } ( 1 + \\| \\alpha _ t ( \\beta ) \\| ) \\| \\nabla v ( t , X _ t ) - \\nabla v ( k \\delta , X _ { k \\delta } ) \\| \\ , d t \\\\ & \\le C _ 2 \\sum _ { k = 0 } ^ { N - 1 } \\int _ { k \\delta } ^ { ( k + 1 ) \\delta } ( 1 + \\| \\alpha _ t ( \\beta ) \\| ) ( \\delta + \\| X _ t - X _ { k \\delta } \\| ) \\ , d t \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} \\widetilde { P } = P + R R = \\sum _ { \\beta \\neq 0 } r _ { \\alpha , \\beta } ( x , \\xi ) \\ , D ^ \\alpha _ x D ^ \\beta _ { \\xi } , \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} \\gamma _ { * j k } = \\sigma _ j \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } - \\sigma _ k \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 j } i \\sigma _ j \\Phi \\Phi ^ * \\sigma _ k - i \\sigma _ k \\Phi \\Phi ^ * \\sigma _ j . \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 2 } \\left ( \\sqrt { b } , - \\sqrt { b } ; \\sqrt { c } q ^ { 1 / 4 } , - \\sqrt { c } q ^ { 1 / 4 } ; \\sqrt { q } , - \\frac { c } { b } \\right ) = \\frac { \\left ( c q ^ { 1 / 2 } / b ; q \\right ) _ { \\infty } } { \\left ( c q ^ { 1 / 2 } ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n } { 2 } } } { \\left ( q ; q \\right ) _ { n } } \\left ( - \\frac { c } { b } \\right ) ^ { n } A _ { q } \\left ( - c q ^ { n - 1 } \\right ) \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} \\frac { ( - q ; q ) _ \\infty } { ( q ; q ) _ \\infty } \\left ( 1 + \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { 2 n - 1 } } { ( - q ; q ^ 2 ) _ { n } } \\right ) = \\sum _ { \\pi \\in \\mathcal { U } } { \\omega _ 2 ' } ( \\pi ) q ^ { | \\pi | } . \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} { \\rm d i m } _ { k } { \\rm H o m } _ { { \\rm r e p } ( Q ) } ( X , Y ) = \\left \\{ \\begin{array} { l l } 2 & \\mbox { $ Y \\in \\mathcal { W } ( Z _ { 1 } ) \\cap \\mathcal { W } ( Z _ { 2 } ) ; $ } \\\\ 1 & \\mbox { $ Y \\in ( \\mathcal { W } ( Z _ { 1 } ) \\cup \\mathcal { W } ( Z _ { 2 } ) ) \\backslash ( \\mathcal { W } ( Z _ { 1 } ) \\cap \\mathcal { W } ( Z _ { 2 } ) ) ; $ } \\\\ 0 & \\mbox { $ Y \\notin \\mathcal { W } ( Z _ { 1 } ) \\cup \\mathcal { W } ( Z _ { 2 } ) . $ } \\end{array} \\right . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} \\| P ( \\cdot ) \\| _ 1 = \\| ( 1 - B _ s ) ^ { - 1 } R \\| _ 1 \\leq 2 C \\rho ^ { \\varepsilon _ 0 } \\langle s \\rangle ^ { - 1 - \\varepsilon _ 1 } , \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} X ^ a _ s = { I + ( - 1 ) ^ a X _ s \\over 2 } , \\ Y ^ b _ t = { I + ( - 1 ) ^ b Y _ t \\over 2 } \\ a , b \\in \\{ 0 , 1 \\} . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} \\zeta _ { \\Delta } ( s ) = \\frac { 1 } { \\Gamma ( s ) } \\int _ 0 ^ \\infty t ^ { s - 1 } K ( t , \\Delta ) d t = \\frac { 1 } { \\Gamma ( s ) } \\int _ 0 ^ \\infty t ^ { s - 1 } T r ( e ^ { - t \\Delta } ) d t \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} s _ \\sigma ( x ) = [ y , \\sigma ( y ) ] = q ( y , \\sigma ( y ) ) . \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} I _ { m } ^ { ( 2 ) } \\left ( 2 q ^ { - \\ell / 2 } ; q \\right ) = I _ { | m | } ^ { ( 2 ) } \\left ( 2 q ^ { - \\ell / 2 } ; q \\right ) = \\frac { q ^ { \\ell | m | / 2 } S _ { \\ell } \\left ( - q ^ { - | m | - \\ell } ; q \\right ) } { \\left ( q ^ { \\ell + 1 } ; q \\right ) _ { \\infty } } . \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} P ( \\{ ( 1 , 2 ) , \\dots , ( 1 , m ) \\} \\subset A ( \\mathbf { D } ) ) \\geq \\prod _ { i = 2 } ^ m P ( ( 1 , i ) \\in A ( \\mathbf { D } ) ) = P ( ( 1 , 2 ) \\in A ( \\mathbf { D } ) ) ^ { m - 1 } \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} F _ { \\Gamma } ^ { p } ( z ) - F _ { \\Gamma ' } ^ { p } ( z ) = \\frac { 1 } { ( 1 - z ^ 2 ) ^ { n - 1 } } \\sum _ { \\ell = 0 } ^ { n } \\left ( \\vartheta _ { \\mathcal L } ^ { ( \\ell ) } ( z ) - \\vartheta _ { \\mathcal L ' } ^ { ( \\ell ) } ( z ) \\right ) A _ { p + 1 , \\ell } ( z ) \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} \\begin{aligned} u _ y & = - \\left ( u \\omega _ { [ 1 ] } \\right ) _ x , \\\\ ( v _ { [ i ] } ) _ { x x x } - ( v _ { [ i ] } ) _ x & = - \\left ( u \\omega _ { [ i + 1 ] } \\right ) _ x , \\\\ u _ t & = ( v _ { [ n ] } ) _ { x x x } - ( v _ { [ n ] } ) _ x = \\delta _ x , \\end{aligned} i = 1 , \\dots , n - 1 , \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} c _ { \\Gamma '' } [ S ] \\in \\Lambda , c _ \\Gamma [ S ] = S \\ , \\frac { p _ { m ( e ) } [ S ] } { p _ { m ( v ) } [ S ] } p _ g \\left [ \\frac { c _ { \\Gamma '' } [ S ] } { S } \\right ] = \\frac { S \\ , p _ { m ( e ) } [ S ] } { p _ { m ( v ) } [ S ] p _ g [ S ] } p _ g [ c _ { \\Gamma '' } [ S ] ] . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { p = 1 , 2 } | \\alpha ( g _ p ) f - f | _ { \\infty } ^ - & \\ge C \\left | \\sum _ { \\underset { p = 1 , 2 } { g \\in { \\mathbb F } _ 2 } } ( \\alpha ( g _ p ) f - f ) ( g ) H _ p ( g ) \\right | \\\\ & = C \\left | \\sum _ { \\underset { p = 1 , 2 } { g \\in { \\mathbb F } _ 2 } } ( f ( g ) ( \\alpha ( g _ p ^ { - 1 } ) H _ p - H _ p ) ( g ) \\right | \\\\ & = C | f ( e ) | = C \\end{aligned} \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} v _ { y _ 0 } ( y ) & = - \\frac { 4 } { 2 7 a ^ 2 ( x _ 0 ) } \\left ( \\left ( \\frac { \\nu _ { x _ 0 } \\cdot A ( x _ 0 ) \\nu _ { x _ 0 } } { ( \\nu _ { x _ 0 } ) _ n } y _ n \\right ) ^ 3 \\right . \\\\ & \\left . - 3 \\left ( \\frac { \\nu _ { x _ 0 } \\cdot A ( x _ 0 ) \\nu _ { x _ 0 } } { ( \\nu _ { x _ 0 } ) _ n } ( a ^ { n + 1 , n + 1 } ( x _ 0 ) ) \\right ) y _ n y _ { n + 1 } ^ 2 \\right ) \\\\ & - g ( y _ 0 ) y _ n + y _ { n } \\frac { ( y '' - y _ 0 ) \\cdot \\nu _ { x _ 0 } '' } { ( \\nu _ { x _ 0 } ) _ n } , \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} \\partial _ t u _ t ( x ) = \\mathcal { L } u _ t ( x ) + \\xi \\sigma ( u _ t ( x ) ) \\dot F ( t , x ) , \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} v _ { m , j } = f _ { j } \\left ( \\alpha ^ { - 1 } q ^ { m } \\right ) \\ ! , \\forall j \\in \\Z , \\ , m \\in \\N . \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} & ( n , q ) ( n ' , q ' ) = ( n n ' , q q ' ) , & & ( n , q ) \\circ ( n ' q ' ) = ( n ( q n q ^ { - 1 } ) , q q ' ) , \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P } } ^ * ( \\mu , r ) = \\delta _ { \\mathsf { P - Z F } } , ~ ~ ~ ~ ~ ~ r \\geq \\frac { ( 1 - \\mu ) \\min \\{ M , K \\} } { M } . \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} \\gamma _ { 2 n + 1 } = \\sum \\limits _ { i = 0 } ^ n - { - k - i \\brack - k - n } \\gamma _ { 2 i } . \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} S = \\left \\{ l _ { 1 } , l _ { 2 } , . . . , l _ { n } \\right \\} \\subset \\left [ 0 , 1 \\right ] \\end{align*}"}
-{"id": "9966.png", "formula": "\\begin{align*} J _ { K , N } \\left ( e ^ { \\frac { 2 \\pi \\sqrt { - 1 } } { 3 } } \\right ) = \\left \\{ \\begin{array} { r l } 0 & ( N = 6 l ) \\\\ 1 & ( N = 6 l + 1 ) \\\\ 1 & ( N = 6 l + 2 ) \\\\ 0 & ( N = 6 l + 3 ) \\\\ - 1 & ( N = 6 l + 4 ) \\\\ - 1 & ( N = 6 l + 5 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} b _ x f ( x ) + c _ x \\mathbf n _ x f = 0 . \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} \\varphi _ { \\sigma } = - \\dfrac { R } { 2 } - \\varphi ^ 2 \\leq \\dfrac { \\alpha } { 2 } , \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} L ( t ) = c A ( t ) , \\ t \\ge 0 , ~ ~ P ^ z \\textrm { - } \\hbox { a . s . , } \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} \\tilde { w } = \\min _ { i \\leq k } w _ i . \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\dd N _ t = \\lambda ^ \\top X _ t \\ , \\dd t + \\dd m _ t . \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} [ d , f ] ( x , y ) & = \\Big ( [ \\delta , \\widetilde { f } ] + [ \\delta + \\lambda , v ] + [ \\varphi , \\widehat { f } + \\widehat { v } ] \\Big ) ( x , y ) , \\\\ [ d , f ] ( x , v ) & = \\Big ( [ \\delta , \\widehat { f } ] + [ \\lambda , \\widetilde { f } + \\widehat { f } + \\widehat { v } ] + [ \\varphi , \\widehat { f } ] \\Big ) ( x , v ) , \\\\ [ d , f ] ( u , v ) & = 0 . \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} R ^ 2 A - R B + C = 0 ; \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} v _ m ( x , t ) = v _ m ^ { + } ( x , t ) + v _ m ^ { - } ( x , t ) , \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} \\lim _ { \\gamma \\to + \\infty } T _ \\gamma = \\sup _ \\gamma T _ \\gamma = T _ \\infty , \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} C = \\left \\{ [ 1 , z _ 2 , z _ 3 ] \\in \\mathbb { C } P ^ 2 : ~ g ( 1 , z _ 2 , z _ 3 ) = 0 \\right \\} , \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} B _ { n , 2 } ( x ) = ( - 1 ) ^ { n } \\frac { 2 ( b ^ 2 - a ^ 2 ) ^ { \\mu + \\nu + 1 } } { a ^ \\mu b ^ { \\nu - 1 } n ! } \\sum _ { i = 0 } ^ { \\lfloor \\frac { n - 1 } { 2 } \\rfloor } \\tilde b _ { i , n } x ^ i , \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} \\min _ x ~ f ^ p ( x ) : = f ( x ) + p ( x ) , \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} \\lambda _ { k , p } = \\begin{cases} 0 & \\quad , \\\\ ( k + p ) ( k + 2 n - 2 - p ) & \\quad \\end{cases} \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{align*} S ^ + _ x = \\sqrt { 2 S } a ^ * _ x \\left [ 1 - \\frac { a ^ * _ x a _ x } { 2 S } \\right ] ^ { 1 / 2 } , S ^ - _ x = \\sqrt { 2 S } \\left [ 1 - \\frac { a ^ * _ x a _ x } { 2 S } \\right ] ^ { 1 / 2 } a _ x , S ^ 3 _ x = a ^ * _ x a _ x - S . \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Q } \\int _ { \\Omega } z : \\nabla \\varphi = \\int _ { 0 } ^ { T } \\int _ { \\Q } f \\cdot \\varphi \\qquad \\forall \\varphi \\in L ^ { 2 } ( 0 , T ; H _ { 0 } ^ { 1 } ( \\Q ) ) \\ , . \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} \\bigoplus _ { i = 0 } ^ { x - 1 } F _ n ( T _ x ( \\phi ( i ) ) ) & = \\bigoplus _ { i = 0 } ^ { x - 1 } F _ n ( T _ x ( i ) ) \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} \\mu _ { k } ^ i \\left ( B \\right ) & = \\frac { 1 } { Z _ { k } ^ i } \\int \\limits _ { \\theta \\in B } \\prod \\limits _ { j = 1 } ^ { n } \\left ( \\frac { d \\mu _ { k - 1 } ^ j \\left ( \\theta \\right ) } { d \\lambda ( \\theta ) } \\right ) ^ { a _ { i j } } \\ell ^ i ( s _ { k } ^ i | \\theta ) d \\lambda \\left ( \\theta \\right ) \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} p _ 2 ( u , v , w ) = \\frac 1 2 \\left ( P ( u , v ) + P ( u , - i w ) - P ( u , v - i w ) \\right ) . \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} x u _ { n } = u _ { n - 1 } + \\beta _ { n } u _ { n } + \\sum \\nolimits _ { \\nu = 0 } ^ { d - 1 } \\gamma _ { n + 1 } ^ { d - 1 - \\nu } u _ { n + 1 + \\nu } , n \\geq 0 \\qquad ( u _ { - 1 } = 0 ) . \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} v = W _ { J } ^ + ( \\Gamma ) \\Omega ^ a v \\in \\operatorname { R a n } W _ J ^ + ( \\Gamma ) . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} \\frac { \\delta _ { \\Omega } ( \\zeta ) } { 2 ^ { i } \\delta _ { \\Omega } ( \\zeta ) } \\frac { 1 } { \\prod _ { j = 1 } ^ { n } \\tau _ { j } ^ { 2 } \\left ( \\zeta , \\delta _ { \\Omega } ( \\zeta ) \\right ) } \\tau _ { m } \\left ( \\zeta , \\delta _ { \\Omega } ( \\zeta ) \\right ) \\tau _ { l } \\left ( \\zeta , \\delta _ { \\Omega } ( \\zeta ) \\right ) \\frac { 1 } { \\left | z - \\zeta \\right | } , \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} H ^ j ( L _ { 2 n - 1 } ( m ) ) = \\left \\{ \\begin{array} { l l } \\Z / m , & 2 \\le j \\le 2 n - 2 \\ ; \\\\ \\Z , & j = 0 , 2 n - 1 \\\\ 0 , & \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} C = \\{ a a ^ H \\mid a \\in \\mathcal A , \\ ; \\| E a \\| \\leq 1 \\} \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} P ( E _ 2 ) = p _ a ^ 8 . \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} \\begin{cases} a \\varphi _ { n , x x } ( 0 , \\cdot ) + \\frac { 1 } { c } \\psi _ { n , x x } ( 0 , \\cdot ) \\rightarrow 0 & H ^ { - \\frac 1 3 } ( 0 , T ) , \\\\ a \\varphi _ { n , x x } ( L , \\cdot ) + \\frac { 1 } { c } \\psi _ { n , x x } ( L , \\cdot ) \\rightarrow 0 & H ^ { - \\frac 1 3 } ( 0 , T ) , \\\\ \\varphi _ { n , x } ( L , \\cdot ) \\rightarrow 0 & L ^ 2 ( 0 , T ) , \\\\ \\psi _ { n , x } ( L , \\cdot ) \\rightarrow 0 & L ^ 2 ( 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} X = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} , Y = \\begin{pmatrix} 0 & - \\mathbf { i } \\\\ \\mathbf { i } & 0 \\end{pmatrix} , Z = \\begin{pmatrix} 1 & 0 \\\\ 0 & - 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} \\nabla _ { \\gamma ^ \\prime } \\gamma ^ \\prime = Z ( \\gamma ^ \\prime ) . \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{gather*} \\hat { g } _ { 0 + } v _ { 0 } = \\tau ( \\hat g ) v _ { 0 } . \\end{gather*}"}
-{"id": "5696.png", "formula": "\\begin{gather*} P _ { r - 1 } ( \\lambda ) Q _ { r } ( \\lambda ) = D _ { r - 1 } ^ { 2 } \\ . \\end{gather*}"}
-{"id": "1266.png", "formula": "\\begin{align*} B ( { { z } } ( t ) ) - B ( { z } ( 0 ) ) + \\int _ { 0 } ^ { t } A { z } ( s ) d s = f ( t ) - f ( 0 ) . \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} I _ m ( z ) = \\bar { z } + \\frac 1 2 I _ m ( z ) = - \\bar { z } + i \\frac { b _ m } { 2 } . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} \\delta = a + p v _ { k } \\in \\Gamma ( k , p ) , \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} [ \\bar { e } _ { i , k + 1 } , \\bar { e } _ { i , l } ] = [ \\bar { e } _ { i , k } , \\bar { e } _ { i , l + 1 } ] , \\ [ \\bar { e } _ { i , k + 2 } , \\bar { e } _ { i + 1 , l } ] - ( d + d ^ { - 1 } ) [ \\bar { e } _ { i , k + 1 } , \\bar { e } _ { i + 1 , l + 1 } ] + [ \\bar { e } _ { i , k } , \\bar { e } _ { i + 1 , l + 2 } ] = 0 , \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} h _ { \\pi ^ { ( 0 ) } ( 1 ) } ^ { ( 0 ) } + \\dots + & h _ { \\pi ^ { ( 0 ) } ( l ' ) } ^ { ( 0 ) } + h _ 1 ^ { ( 0 ) } \\\\ & > c _ 1 ( \\pi ^ { ( 0 ) } ) + \\dots + c _ { l ' } ( \\pi ^ { ( 0 ) } ) + ( d - v ) \\\\ & = c _ 1 ( \\pi ' ) + \\dots + c _ { l ' + 1 } ( \\pi ' ) , \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} \\tilde g ( x ) = A ( x ) ^ { - 1 } \\tilde g ( x ^ p ) - r ( x ) . \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} A _ { a , b , c } ( r ) = \\sum _ { s = 1 } ^ { r } A _ n ( s ) N _ { c , b , a } ( r - s ) , r \\in \\{ 1 , \\dots , a + 2 b + c \\} , \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} \\langle d \\phi , d \\phi \\rangle _ g = - \\langle \\phi , \\Delta \\phi \\rangle _ g + \\frac { 1 } { 2 } \\langle \\phi , \\phi \\rangle _ { \\Delta g } \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} I _ s ( ( X _ s - ( T - t ) V _ s , V _ s ) ) = I _ s ( Z _ s ) - ( T - t ) \\mathcal { Y } _ s ( Z _ s ) + ( T - t ) ^ 2 E _ s ( Z _ s ) \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\Phi ( w ) ( g , h ) = ( g , h _ g ) \\phi ( h _ g ^ { - 1 } h w ) ( w \\in W ) . \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a \\ , t ^ 2 ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} f r ( I _ { B , A } ) = \\varepsilon _ B . \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { m _ 1 } \\tau _ { \\alpha _ j } ( D _ j ) \\prod _ { j = 1 } ^ { m _ 2 } \\tau _ { \\beta _ j } ( 1 ) , \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\tau = \\sum _ { k \\in \\omega } \\omega ^ { \\beta _ k } \\cdot p _ k \\le \\omega ^ { \\widetilde { \\alpha } } \\cdot \\left ( \\sum _ { k \\in \\omega } p _ k \\right ) = \\omega ^ { \\widetilde { \\alpha } } \\cdot \\omega = \\omega ^ { \\widetilde { \\alpha } + 1 } = \\omega ^ { \\alpha } . \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} V _ { s + 1 } ^ { 0 } \\left ( \\tau \\right ) = T _ { s } ^ { 0 } \\left ( - \\tau \\right ) C _ { s + 1 } ^ 0 T _ { s + 1 } ^ { 0 } \\left ( \\tau \\right ) \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} L ( \\phi , g , \\gamma ) = - g \\Delta + \\frac { 1 } { 2 } { \\Delta g } = - g _ { i j } ( \\phi ) \\gamma ^ { \\mu \\nu } \\partial _ \\mu \\partial _ \\nu + \\frac { 1 } { 2 } \\gamma ^ { \\mu \\nu } \\partial _ \\mu \\partial _ \\nu g _ { i j } ( \\phi ) \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} 2 g _ R - 2 = \\deg f ( 2 g _ S - 2 ) - b ( f ) \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } N \\left ( n \\right ) = \\infty , \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} \\mu _ + ( A ) = \\lim \\inf _ { \\varepsilon \\to 0 } \\frac { 1 } { \\varepsilon } \\left ( \\mu ( A _ \\varepsilon ) - \\mu ( A ) \\right ) , \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{gather*} ( - 1 ) ^ { \\abs { \\gamma } } \\frac { y _ { \\gamma ( n ) } ^ { 0 } y _ { \\gamma ( n - 1 ) } \\cdots y _ { \\gamma ( m + 1 ) } ^ { n - m - 1 } } { \\prod \\limits _ { i = 0 } ^ { m - 1 } ( w _ { m - i } - y _ { \\gamma ( i + 1 ) } ) } , \\end{gather*}"}
-{"id": "1297.png", "formula": "\\begin{align*} & \\ \\Phi ( x , y ) \\\\ = & \\ \\sum _ { \\alpha \\in F , \\beta \\in F } \\psi ( \\alpha x + \\beta y ) \\left ( \\int \\int \\Phi ( u , v ) \\psi ( - \\alpha u - \\beta v ) d u d v \\right ) \\ , . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} \\langle S _ u , S _ v \\rangle = 0 , | S _ u | ^ 2 = | S _ v | ^ 2 = f ( u , v ) \\geq 0 , \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} ( 1 + \\Pi \\mathcal { O } _ D ) ^ 1 & \\subset J ^ 1 _ { \\mathrm { s } } \\cap ( \\mathcal { O } _ F ^ \\times \\Pi ^ \\Z ( 1 + \\Pi \\mathcal { O } _ D ) / \\varpi ^ \\Z ) \\\\ & \\subset J ^ 1 \\cap ( \\mathcal { O } _ F ^ \\times \\Pi ^ \\Z ( 1 + \\Pi \\mathcal { O } _ D ) / \\varpi ^ \\Z ) = \\mu ( 1 + \\Pi \\mathcal { O } _ D ) ^ 1 . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} A ( k ) & = - 2 ( k ^ 8 - 1 6 k ^ 7 + 7 6 k ^ 6 - 1 6 k ^ 5 - 1 2 2 6 k ^ 4 + 5 4 5 6 k ^ 3 - 1 1 3 4 8 k ^ 2 + 1 1 9 8 4 k \\\\ & \\quad - 5 1 6 7 ) , \\\\ B ( k ) & = ( k ^ 2 - 8 k + 1 1 ) ^ 4 ( k ^ 2 - 5 ) ^ 4 , \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} & \\tilde f ( x ) = \\big ( \\sqrt { ( p + 1 ) ^ 2 + m ^ 2 } + \\sqrt { ( p - 1 ) ^ 2 + m ^ 2 } \\big ) \\tilde h ( x ) , \\\\ & \\tilde u ( x ) = \\tilde f ( x ) \\sin x , \\\\ & \\tilde V ( x ) = \\lambda - \\frac { 1 } { \\tilde u ( x ) } ( \\sqrt { p ^ 2 + m ^ 2 } - m ) \\tilde f ( x ) , \\\\ & \\tilde H = \\sqrt { p ^ 2 + m ^ 2 } - m + \\tilde V ( x ) \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} \\partial _ { t } p _ { \\delta , n } = \\partial \\Upsilon _ { n } ^ { \\delta } \\left ( C _ { n } ^ { - 1 } \\left ( \\xi + V _ { \\xi } ( p _ { \\delta , n } ) ^ s - p _ { \\delta , n } \\right ) - B _ { n } \\ , p _ { \\delta , n } \\right ) \\ , . \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} e ( X ) = \\pi ^ * _ K ( e ( K , L ) ) . \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} \\lim _ { y \\to 0 } \\frac { d _ h ( D f ( 0 ) y , f ( y ) ) } { \\| y \\| } = 0 \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} \\Phi ( y ) = y ' . \\sigma ^ \\Phi ( y , y ' ) . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} ( a \\otimes x ) ( b \\otimes y ) = a ( x _ 1 \\rightharpoonup b _ 1 ) \\otimes ( x _ 2 \\leftharpoonup b _ 2 ) y , & & a , b \\in H , \\ , x , y \\in K . \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} F _ { \\mu \\nu } R ^ { \\mu \\nu } = 2 \\left ( \\mathbf { H } \\cdot \\mathbf { B } - \\mathbf { E } \\cdot \\mathbf { D } \\right ) , \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{gather*} \\begin{aligned} & t _ i = \\frac { 2 i - 1 } { 2 ^ k ( 2 M + 1 ) } , i = 1 , 2 , \\ldots , N _ t , \\\\ & x _ j = j h , j = 0 , 1 , \\ldots , N _ h , \\end{aligned} \\end{gather*}"}
-{"id": "8479.png", "formula": "\\begin{align*} \\int _ { P ( \\zeta , \\delta ) } \\frac { \\delta _ { \\Omega } ^ { \\alpha - 1 } ( z ) } { \\left | z - \\zeta \\right | } d \\lambda ( z ) \\lesssim \\frac { \\delta ^ { \\alpha - 1 } } { \\alpha } \\tau _ { n } ( \\zeta , \\delta ) \\prod _ { j = 1 } ^ { n - 1 } \\tau _ { j } ^ { 2 } ( \\zeta , \\delta ) . \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} R i c ( \\hat { g } ) - \\hat { g } = - L _ { \\nabla \\hat { f } } ( \\hat { g } ) . \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\frac { \\sharp \\Gamma _ 1 \\cap F _ t } { \\lambda ( F _ t ) } \\geq \\limsup _ { t \\to \\infty } \\frac { \\sharp S \\cap F _ t } { \\lambda ( F _ t ) } = \\nu _ S > \\frac { \\nu _ U } { 2 } \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} \\frac { \\Phi _ { p r i m } ^ { o d d } ( Q ) } { \\Phi ( Q ) } = 1 + O ( \\frac { 1 } { q } ) \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} | S ^ \\nu _ { h _ 1 } ( n ) | ^ 2 = | S ^ \\nu _ f ( n ) | ^ 2 + | S ^ \\nu _ g ( n ) | ^ 2 + 2 \\Re \\left ( S ^ \\nu _ f ( n ) \\overline { S ^ \\nu _ g ( n ) } \\right ) \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} h ^ r ( \\kappa _ { ( r ) } ) = ( r ! ) ^ { - 1 } h ^ { r - s } ( Y _ s ) , \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} \\begin{aligned} F _ { t } ( z ; x ) & \\geq ( F ( z ) + \\tfrac { \\mu + t ^ { - 1 } } { 2 } \\| z - x \\| ^ 2 ) - \\mu \\| z - x \\| ^ 2 \\\\ & \\geq F ( \\hat x ) + \\tfrac { \\mu + t ^ { - 1 } } { 2 } \\| \\hat x - x \\| ^ 2 + \\tfrac { 1 } { 2 t } \\| \\hat x - z \\| ^ 2 - \\mu \\| z - x \\| ^ 2 , \\end{aligned} \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} S ^ a _ { \\alpha p } = S ^ a _ { \\mu p } = 0 , 1 \\leq \\alpha , \\mu \\leq 8 - r , 1 \\leq a \\leq 7 - r , \\forall p = 1 , \\cdots , 7 . \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} h ( X , J _ 1 Y ) - J _ 1 h ( X , Y ) = \\frac { 1 } { 2 ( 2 n - 1 ) } \\left [ g ( X , Y ) p ^ \\bot _ 1 + g ( X , J _ 1 Y ) J _ 1 ( p ^ \\bot _ 1 ) \\right ] . \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} \\tilde { I } ( \\zeta ) = \\left ( A \\alpha \\left ( - \\frac { 1 } { \\bar { z } } \\right ) + B \\beta \\left ( - \\frac { 1 } { \\bar { z } } \\right ) , \\bar { B } \\alpha \\left ( - \\frac { 1 } { \\bar { z } } \\right ) - \\bar { A } \\beta \\left ( - \\frac { 1 } { \\bar { z } } \\right ) \\right ) \\ , . \\end{align*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\tilde { \\kappa } _ i = \\kappa _ i ^ { - 1 } , \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 2 } \\ [ \\bar { f } _ { i , k _ { \\pi ( 1 ) } } , [ \\bar { f } _ { i , k _ { \\pi ( 2 ) } } , \\bar { f } _ { i \\pm 1 , l } ] ] = 0 \\ \\mathrm { a n d } \\ [ \\bar { f } _ { i , k } , \\bar { f } _ { j , l } ] = 0 \\ \\mathrm { f o r } \\ j \\ne i , i \\pm 1 . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} \\sum _ { y _ { n } } \\log \\Big ( \\frac { r _ { n } ( x _ n | y ^ { n - 1 } _ { n - M } , y _ n ) } { \\pi _ { n } ( x _ n | y ^ { n - 1 } _ { n - J } ) } \\Big ) q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) - s \\gamma _ n ( x _ n , y ^ { n - 1 } _ { n - N } ) = 1 - \\lambda _ n ( y ^ { n - 1 } _ { n - J } ) , ~ \\forall { x _ n \\in { \\cal X } _ n } . \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } - \\eta _ { t } \\left [ g _ { 1 - \\alpha } * ( u - u _ { 0 } ) \\right ] d x d t + \\int _ { 0 } ^ { T } a ( u , \\eta ) d t \\geq \\int _ { 0 } ^ { T } \\int _ { \\Omega } f \\eta d x d t . \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} S ( g ; \\phi ) = \\int _ \\Sigma d v \\ , t r _ { \\gamma } ( \\phi ^ * g ) \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} w _ 1 ^ j + w _ 2 ^ j + \\dots + w _ n ^ j = d _ j \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} | | P _ t \\mathbf 1 _ E - \\mathbf 1 _ E | | _ { 1 } = 2 \\left ( \\mu ( E ) - \\int \\left ( P _ { t / 2 } \\mathbf 1 _ E \\right ) ^ 2 d \\mu \\right ) . \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\frac { w ^ * K + 1 } { w ^ * ( K + 1 ) } \\cdot \\left ( \\frac { 1 - w ^ * } { w ^ * K + 1 } \\right ) ^ \\frac { 1 } { K + 1 } \\cdot P = 1 . \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ \\xi ^ \\beta Q ^ \\prime ( t ) = \\int _ s ^ t A ( p ( \\tau , s ) ) \\partial _ \\xi ^ \\beta P ^ \\prime ( \\tau ) d \\tau + R ^ \\prime _ { 1 1 } ( t ) + R ^ \\prime _ { 1 2 } ( t ) , \\\\ \\partial _ \\xi ^ \\beta P ^ \\prime ( t ) = - \\int _ s ^ t \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , s ) ) \\partial _ \\xi ^ \\beta Q ^ \\prime ( \\tau ) d \\tau + R ^ \\prime _ { 2 1 } ( t ) + R ^ \\prime _ { 2 2 } ( t ) , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} u ( \\cdot , - s ) = g ( \\cdot , s ) s \\in [ 0 , \\infty ) . \\end{align*}"}
-{"id": "9972.png", "formula": "\\begin{align*} \\langle n _ i ^ { \\pm } S , g \\rangle = \\int _ { B } \\partial _ i ^ { \\mp , h } g d V ^ h = \\int _ { B } \\partial _ i ^ { \\mp , h } f d V ^ h . \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} * \\varphi = x ^ { 4 5 6 7 } + x ^ { 2 3 6 7 } - x ^ { 2 3 4 5 } + x ^ { 1 3 5 7 } + x ^ { 1 3 4 6 } + x ^ { 1 2 5 6 } - x ^ { 1 2 4 7 } \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} h \\sigma ^ { - 1 } + \\sigma ^ { - 1 } h ^ { t r } = 0 . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( x y : n ) } = \\bigoplus _ { ( i , \\alpha ) } T _ { ( x , y ) } ( i , \\alpha ) \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\Delta _ g | \\nabla ^ 2 u | _ g ^ 2 \\\\ = & | \\nabla ^ 3 u | _ g ^ 2 + \\langle \\nabla \\Delta \\nabla _ i u , \\nabla \\nabla ^ i u \\rangle _ g + u _ { , i j } ( R ^ l _ { i j k } u _ { , l k } + R _ { j } ^ l u _ { , i l } + R _ { i j k , k } ^ l u _ { , l } + R _ { i j k } ^ l u _ { , l k } ) . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\lambda _ { \\ell + r m + 1 } & = & ( \\varepsilon _ { 2 } / \\varepsilon _ { 1 } ) \\varepsilon _ { 2 } ^ { ( - 1 ) ^ { m + 1 } } \\lambda _ { m + 1 } ^ { r } , \\\\ \\lambda _ { \\ell + r m + i } & = & ( \\varepsilon _ { 1 } / \\varepsilon _ { 2 } ) ^ { ( - 1 ) ^ { i } } 2 \\leqslant i \\leqslant r . \\end{array} \\right . \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} g \\le \\frac { N _ T + \\frac { 1 } { \\mu _ R } } { \\frac { 1 } { 4 \\mu _ R } - 2 \\mu _ R } = \\frac { 2 \\mu _ R + 1 } { 1 / 4 - 2 \\mu _ R ^ 2 } < \\frac { 2 \\times 1 / 4 + 1 } { 1 / 4 - 2 \\times ( 1 / 4 ) ^ 2 } = 1 2 . \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } + n } \\left ( - c ^ { 2 } z \\right ) ^ { n } } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } } A _ { q ^ { 2 } } \\left ( - c ^ { 2 } q ^ { 2 n } \\right ) = \\frac { \\left ( c ^ { 2 } q ; q ^ { 2 } \\right ) _ { \\infty } } { \\left ( c ^ { 2 } z q ; q ^ { 2 } \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n + 1 } { 2 } } \\prod _ { j = 0 } ^ { n - 1 } \\left ( z - q ^ { 2 j } \\right ) \\left ( - c ^ { 2 } \\right ) ^ { n } } { \\left ( q , c q ^ { 1 / 2 } , - c q ^ { 1 / 2 } ; q \\right ) _ { n } } \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} \\partial _ \\alpha ( A _ { i j } ^ { \\alpha \\beta } ( x ) \\partial _ \\beta u ^ j ) = h _ i + \\partial _ \\beta f ^ \\beta _ i , \\mbox { i n } \\ B _ 4 , i = 1 , \\cdots , N , \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} S _ { k } ( X ) = S _ { g } ( X ) + \\tfrac { ( g - k ) ( g - k + 1 ) } { 2 } \\int _ { X } g ( \\sigma ( P ) , P ) \\mu ( P ) . \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} \\rho ^ { Z , f \\oplus 0 } _ t ( S _ c S _ d ) & = \\exp { ( 2 \\pi \\sqrt { - 1 } t f ) } S _ c S _ d = \\rho ^ { A , f } _ t ( S _ { a } ) , \\\\ \\rho ^ { Z , f \\oplus 0 } _ t ( S _ d S _ c ) & = S _ d \\exp { ( 2 \\pi \\sqrt { - 1 } t f ) } S _ c = S _ d \\exp { ( 2 \\pi \\sqrt { - 1 } t f ) } S _ d ^ * S _ { b } . \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} \\Psi ' ( o ) = \\sum _ { x \\in \\eta } l ' ( | x | ) \\geq \\sum _ { k = K } ^ \\infty \\frac { \\kappa _ d } { 2 } \\int _ { k + 2 } ^ { k + 3 } r ^ { d - 1 } l ( r ) d r = \\frac { \\kappa _ d } { 2 } \\int _ { K + 2 } ^ { \\infty } r ^ { d - 1 } l ( r ) d r = \\infty . \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} & i \\left ( \\mathbf { F \\times F } ^ { \\ast } \\right ) = 2 \\left ( \\mathbf { E \\times H } \\right ) , i \\left ( \\mathbf { G \\times G } ^ { \\ast } \\right ) = 2 \\left ( \\mathbf { D \\times B } \\right ) , \\\\ & \\quad \\mathbf { F } \\cdot \\mathbf { G } ^ { \\ast } + \\mathbf { F } ^ { \\ast } \\cdot \\mathbf { G } = 2 \\left ( \\mathbf { E } \\cdot \\mathbf { D } + \\mathbf { H } \\cdot \\mathbf { B } \\right ) \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} x \\left ( y + z \\right ) & = \\left ( p + \\oslash \\right ) \\left ( q + \\oslash + r + \\oslash \\right ) = \\left ( p + \\oslash \\right ) \\left ( q + r + \\oslash \\right ) \\\\ & = p \\left ( q + r \\right ) + p \\oslash + \\left ( q + r \\right ) \\oslash + \\oslash \\oslash \\\\ & = p \\left ( q + r \\right ) + \\oslash . \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} L ( x ) = \\frac 1 2 \\left ( 1 + \\frac { 1 } { \\sqrt { 1 - 4 x } } \\right ) . \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} T _ 1 ( x ) - T _ 1 ( x _ 0 ) & = ( x - x _ 0 ) + E _ { x _ 0 } ( x ) , x \\in B _ { 1 / 2 } ^ + ( x _ 0 ) \\\\ | E _ { x _ 0 } ( x ) | & \\lesssim \\max \\{ \\epsilon _ 0 , c _ { \\ast } \\} \\left ( | x - x _ 0 | | x _ 0 | ^ \\alpha + | x - x _ 0 | ^ { 1 + \\alpha } \\right ) . \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { m , n } ^ { \\gamma } ) \\right ] ( z ) & = \\dfrac { ( - 1 ) ^ { n } ( \\gamma + 1 ) _ { m + n } ( \\gamma + m + 2 ) _ { n - 1 } ( - m - 1 ) _ { n } } { ( \\gamma + 1 ) _ { n } ( m - n + 2 ) _ { n - 1 } ( m + 1 ) } \\\\ & \\times z ^ { n - 1 } \\overline { z } ^ { m } \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + 1 } { _ 2 F _ 1 } \\left ( \\begin{array} { c } - ( n - 1 ) , - m \\\\ - ( \\gamma + 1 ) - m - ( n - 1 ) \\end{array} \\bigg | \\frac { 1 } { | z | ^ 2 } \\right ) . \\end{align*}"}
-{"id": "10077.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 6 , 6 u + 3 , 6 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 6 , 6 u + 3 , 6 v + 5 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "2070.png", "formula": "\\begin{align*} q _ \\Theta ( \\mu , \\nu ) = \\left [ \\begin{array} { c } \\mu \\\\ \\nu \\end{array} \\right ] ^ H \\left [ \\begin{array} { c c } \\Theta _ { 1 1 } & \\Theta _ { 1 2 } \\\\ \\Theta _ { 2 1 } & \\Theta _ { 2 2 } \\end{array} \\right ] \\left [ \\begin{array} { c } \\mu \\\\ \\nu \\end{array} \\right ] \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} \\left ( \\alpha _ { 0 } , \\alpha _ { 1 } , \\alpha _ { 2 } , \\alpha _ { 3 } \\right ) = \\alpha _ { 0 } \\left ( 1 , - 1 , - 1 , - 1 \\right ) . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } P _ { n } t ^ { n } = \\sum _ { l = 0 } ^ { \\infty } \\frac { t ^ { l } } { ( q ; q ) _ { l } } \\sum _ { k = 0 } ^ { \\infty } \\left ( q z \\xi ^ { - 1 } ; q \\right ) _ { k } z ^ { - 2 k } t ^ { k } = \\frac { 1 } { ( t ; q ) _ { \\infty } } \\ , _ { 2 } \\phi _ { 1 } \\left ( q z \\xi ^ { - 1 } , q ; 0 ; q , z ^ { - 2 } t \\right ) \\end{align*}"}
-{"id": "9911.png", "formula": "\\begin{align*} \\mathcal { V } ^ { + } ( D ) = \\bigoplus _ { \\lambda > 0 } \\mathcal { V } ^ { \\lambda } ( D ) , \\mathcal { V } ^ { - } ( D ) = \\bigoplus _ { \\lambda < 0 } \\mathcal { V } ^ { \\lambda } ( D ) , \\mathcal { V } ^ { \\pm 0 } ( D ) = \\mathcal { V } ^ { \\pm } ( D ) + \\mathcal { V } ^ 0 ( D ) . \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} a _ 1 S ( a _ 2 ) & = a _ 1 ( a _ 2 \\rightharpoonup T ( a _ 3 ) ) = a _ 1 \\circ ( T ( a _ 2 ) \\rightharpoonup ( a _ 3 \\rightharpoonup T ( a _ 4 ) ) \\\\ & = a _ 1 \\circ ( ( T ( a _ 2 ) \\circ a _ 3 ) \\rightharpoonup T ( a _ 4 ) ) = a _ 1 \\circ T ( a _ 2 ) = \\epsilon ( a ) 1 \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} \\varphi ( X ) = \\int _ { X ^ { 2 } } g ( P _ { 1 } , P _ { 2 } ) h ^ { 2 } _ { \\Delta } ( P _ { 1 } , P _ { 2 } ) , \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} A ( \\theta ) : = c _ 1 ( \\theta ) B _ { \\tilde { n } _ 1 } + c _ 2 ( \\theta ) B _ { \\tilde { n } _ 2 } \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} u _ { : i j } = - v h _ { i j } + 2 \\vartheta ^ { - 1 } \\dot \\vartheta u _ i u _ j - \\vartheta \\dot \\vartheta \\sigma _ { i j } . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} Q _ \\infty [ u , v ] - \\lambda \\int _ \\Omega U \\ , \\overline { u } \\ , v \\ , d x = \\int _ { M } \\ , \\nabla \\overline { u } \\cdot \\nabla v \\ , d x - \\lambda \\int _ \\Omega U \\ , \\overline { u } \\ , v \\ , d x \\ , u , v \\in H ^ 1 _ 0 ( M ) \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} \\det \\textbf { ( I + B ) } = 0 . \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} h ( \\kappa _ { ( r ) } ) = \\sum _ { j = 0 } ^ { k + r - n - 1 } \\varepsilon ^ j { k + r - n - 1 \\choose j } \\kappa _ { ( 1 ) } ^ j \\tilde { \\kappa } _ { ( r - j ) } . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u & = [ a ^ { i j } u _ { x ^ { i } x ^ { j } } + b ^ { i } u _ { x ^ { i } } + c u + f ( u ) ] \\\\ & \\qquad + \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } [ \\sigma ^ { i j k } u _ { x ^ { i } x ^ { j } } + \\mu ^ { i k } u _ { x ^ { i } } + \\nu ^ { k } u + g ^ { k } ( u ) ] d w _ { s } ^ { k } \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} v \\partial _ { x } g _ { \\pm } \\mp \\beta _ { x } \\partial _ { v } g _ { \\pm } = \\omega _ { \\pm } ( I _ { \\pm } ) \\partial _ { \\theta _ { \\pm } } g _ { \\pm } , \\end{align*}"}
-{"id": "9967.png", "formula": "\\begin{align*} J _ { K , N } \\Bigl ( e ^ { \\frac { 2 \\pi \\sqrt { - 1 } } { 3 } } \\Bigr ) = \\left \\{ \\begin{array} { r l } 0 & ( N = 6 l ) , \\\\ 1 & ( N = 6 l + 1 ) , \\\\ 1 & ( N = 6 l + 2 ) , \\\\ 0 & ( N = 6 l + 3 ) , \\\\ - 1 & ( N = 6 l + 4 ) , \\\\ - 1 & ( N = 6 l + 5 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\delta ^ { ( \\alpha ) } _ { \\min } ( N ) \\ll N ^ { 1 - \\frac { 4 - \\eta } { 2 } + \\varepsilon } = N ^ { - 1 + \\eta / 2 + \\varepsilon } \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{gather*} c ^ { ( \\alpha ) } ( f ( z ) ) \\cdot c ^ { ( \\alpha ) } ( g ( z ) ) = c ^ { ( \\alpha ) } _ { 1 } ( f ( z _ { 1 } ) ) \\cdot c ^ { ( \\alpha ) } _ { 2 } ( g ( z _ { 2 } ) ) \\end{gather*}"}
-{"id": "6656.png", "formula": "\\begin{align*} { \\bf { C o v } } \\left [ V _ { \\varepsilon } ( u ) , \\ , V _ { \\varepsilon } ( v ) \\right ] & = \\begin{cases} - \\ , 2 \\log | u - v | , \\ , \\varepsilon \\leq | u - v | \\leq 1 , \\\\ 2 \\left ( 1 - \\log \\varepsilon - \\frac { | u - v | } { \\varepsilon } \\right ) , \\ , | u - v | \\leq \\varepsilon . \\end{cases} \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} \\varphi : ( M ^ { 2 } , g = \\lambda ^ 2 { \\bar g } ) \\longrightarrow ( N ^ 3 , h ) \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} \\sum _ { t = 2 } ^ { 8 } f _ t ( p ' ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { 8 } . \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} \\frac { c ^ r - 1 } { r } B _ { r } = \\sum _ { x = 1 } ^ { p - 1 } x ^ { r - 1 } f _ c ( x ) \\pmod p . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} L _ f ( t _ 2 , \\ldots , t _ d ) ( \\cdot ) = ( \\pi ( \\cdot , t _ 2 , \\ldots , t _ d ) f ) ( 0 ) . \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { m = - \\infty } ^ { \\infty } \\left ( a ; q \\right ) _ { m } I _ { \\nu + m } ^ { ( 3 ) } ( 2 z ; q ) \\left ( - q z \\right ) ^ { m } & = \\frac { ( - a q z ^ { 2 } , - 1 / ( a z ^ { 2 } ) ; q ) _ { \\infty } z ^ { \\nu } } { ( q / a , - q ^ { \\nu } / ( a z ^ { 2 } ) ; q ) _ { \\infty } } . \\end{aligned} \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} \\delta \\ , \\sum _ { k \\in \\mathbb { Z } } \\ , \\sup _ { | \\gamma | \\le \\delta / 2 } \\ , g ( \\xi - k \\delta - \\gamma ) & \\le \\left ( \\delta + \\sum _ { k \\in \\mathbb { Z } \\setminus \\{ 0 \\} } \\ , \\frac { \\delta } { 1 + \\delta ^ 2 \\ , ( | k | - 1 ) ^ 2 } \\right ) \\\\ & = 3 \\ , \\delta + 2 \\ , \\sum _ { n = 1 } ^ \\infty \\ , \\frac { \\delta } { 1 + \\delta ^ 2 \\ , n ^ 2 } \\le 3 + 2 \\ , \\int _ 0 ^ \\infty \\ , \\frac { 1 } { 1 + x ^ 2 } \\ , d x = c < \\infty . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} ( i , j ) \\in A ( D ) \\phi ( x _ i , x _ j ) = 1 . \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} I _ n ( u ) & \\geq c _ 1 \\| \\nabla u \\| _ 2 ^ 2 + \\int _ \\Omega ( G _ n \\circ u ) \\ , d x - \\| u \\| _ { 1 } \\cdot \\| f \\| _ \\infty \\\\ & \\geq c _ 1 \\| \\nabla u \\| _ 2 ^ 2 + \\int _ \\Omega ( G _ n \\circ u ) \\ , d x - c _ 2 \\| \\nabla u \\| _ { 2 } \\cdot \\| f \\| _ \\infty \\\\ & = \\left ( c _ 1 \\| \\nabla u \\| _ { 2 } - c _ 2 \\| f \\| _ \\infty \\right ) \\| \\nabla u \\| _ { 2 } + \\int _ \\Omega ( G _ n \\circ u ) \\ , d x , \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} \\left [ H _ 0 - E - \\sum _ { j = 1 } ^ { n + 1 } \\lambda _ j | a _ j \\rangle \\langle a _ j | \\right ] | \\psi \\rangle = | \\chi \\rangle \\ ; , \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} ( \\mu \\bullet _ 1 f ) \\bullet _ { m + 1 } g & = f \\smile g \\\\ ( \\mu \\bullet _ 2 f ) \\bullet _ { 1 } g & = ( \\mu \\bullet _ 1 g ) \\bullet _ { n + 1 } g = g \\smile f \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} \\mathbf { y } _ { 1 } = \\mathbf { D } ^ { q - m } \\mathbf { x } _ { 1 } \\oplus \\mathbf { D } ^ { q - n } \\mathbf { x } _ { 2 } ; \\mathbf { y } _ { 2 } = \\mathbf { D } ^ { q - m } \\mathbf { x } _ { 2 } , \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ { 2 } } \\left ( q ; q ^ { 2 } \\right ) _ { n } S _ { 2 n } \\left ( x q ^ { - 2 n - 1 / 2 } ; q \\right ) } { \\left ( c ; q ^ { 2 } \\right ) _ { n } } \\left ( \\frac { c } { x ^ { 2 } q } \\right ) ^ { n } = \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( c e ^ { i y } / \\left ( q x \\right ) ; q \\right ) _ { \\infty } } { \\left ( c , c e ^ { 2 i y } / \\left ( x ^ { 2 } q \\right ) ; q ^ { 2 } \\right ) _ { \\infty } } \\frac { \\exp \\left ( \\frac { y ^ { 2 } } { \\log q ^ { 2 } } \\right ) d y } { \\sqrt { \\pi \\log q ^ { - 2 } } } . \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { H _ n } { \\log n } \\leq \\limsup _ { r \\to \\infty } \\frac { H _ { n _ { r , t + 1 } } } { \\log n _ { r , t + 1 } } \\frac { \\log n _ { r , t + 1 } } { \\log n _ { r , t } } = \\frac { 1 } { \\log ( 1 / p ) } \\cdot \\frac { ( t + 1 ) ^ 2 } { t ^ 2 } . \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\lim _ { | x | \\to \\infty } | x | V _ \\nu ( x ) = C \\lim _ { z \\uparrow 1 } { } _ 2 F _ 1 ( \\tfrac { 1 } { 2 } + \\nu , - \\tfrac { 1 } { 2 } ; \\tfrac { 1 } { 2 } ; z ) . \\end{align*}"}
-{"id": "10026.png", "formula": "\\begin{align*} n _ { d } ( q ^ { d } - 1 ) + c _ 0 q ^ { d } - 1 = q ^ { n } - 1 \\Rightarrow n _ { d } ( q ^ { d } - 1 ) = q ^ { d } ( q ^ { n - { d } } - c _ 0 ) . \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} \\Sigma _ 2 ^ * + \\Sigma _ 3 ^ * + \\Sigma _ 4 ^ * \\leq M = 1 - 2 N \\approx 0 . 3 6 6 0 . \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} P _ { n } ^ { \\left ( r + 1 \\right ) } \\left ( x \\right ) = \\left ( u _ { 0 } ^ { \\left ( r \\right ) } \\theta _ { 0 } P _ { n + 1 } ^ { \\left ( r \\right ) } \\right ) \\left ( x \\right ) ; u _ { n } ^ { \\left ( r + 1 \\right ) } = \\left ( x u _ { n + 1 } ^ { \\left ( r \\right ) } \\right ) \\left ( u _ { 0 } ^ { \\left ( r \\right ) } \\right ) ^ { - 1 } , \\ n , r \\geq 0 . \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} Z _ { \\lambda _ 1 , \\lambda _ 2 , \\varepsilon } ( \\beta ) \\triangleq \\sum \\limits _ { i = 1 } ^ N x _ i ^ { \\beta \\lambda _ 1 } ( 1 - x _ i ) ^ { \\beta \\lambda _ 2 } e ^ { \\beta V _ \\varepsilon ( x _ i ) } . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} \\int _ \\mathbb { R } e ^ { y q } \\frac { d } { d y } \\exp \\Bigl ( - e ^ { - \\beta y } X \\Bigr ) d y = X ^ { \\frac { q } { \\beta } } \\ , \\Gamma ( 1 - \\frac { q } { \\beta } ) , \\ ; \\Re ( q ) < 0 , \\ , X > 0 . \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { [ n t _ 1 ] } r ( k / n ) Y _ k \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P , A c h } } ( \\mu , r ) \\leq \\delta _ { \\mathsf { P - I A } } = \\frac { ( 1 - \\mu M ) K } { M r } , \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\| T \\| _ { \\mu \\rightarrow \\mu ' } = \\sup _ { m } \\dfrac { \\sum _ { n } | T _ { n , m } | W _ { n } ^ \\mu } { W _ m ^ { \\mu ' } } . \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} \\Lambda _ S = \\Lambda [ S ^ { - 1 } , p _ 2 [ S ] ^ { - 1 } , p _ 3 [ S ] ^ { - 1 } , \\cdots ] . \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} t _ { 2 } + \\frac { 3 } { 4 } t _ { 3 } = d . \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} W a ^ { i j } u _ { i ; j } = W a ^ { i i } u _ { i ; i } = \\langle \\bar { \\nabla } f , ( \\nabla u , - 1 ) \\rangle = f _ { i } u ^ { i } - f _ t . \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} d v & = \\partial _ t \\varphi v \\ , d t + [ - \\Delta \\varphi v - 2 \\nabla \\varphi \\cdot \\nabla v + \\Delta v + | \\nabla \\varphi | ^ 2 v + V v ] \\ , d t + G v \\ , d W ( t ) . \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} \\det B _ { i _ 1 , \\dots , i _ s } ( x _ 1 , \\dots , x _ r ) = \\prod _ { 1 \\leq \\alpha < \\beta \\leq s } ( x _ { i _ \\alpha } - x _ { i _ \\beta } ) . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} \\kappa = { \\lambda _ { \\sf m i n } ( ( m + 1 ) s ) } \\left ( 1 - 3 \\sqrt { \\frac { \\lambda _ { \\sf m a x } ( m s ) } { m \\lambda _ { \\sf m i n } \\left ( ( m + 1 ) s \\right ) } } \\right ) ^ 2 . \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} \\dd Y _ t = \\lambda _ t ( 1 - Y _ t ) \\ , \\dd t + \\dd m _ t , \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ) ^ 3 } g ( x , y ) g ( y , z ) g ( z , x ) ~ d x d y d z = \\frac { 1 } { 8 } ( ( 2 p _ d - 1 ) ^ 3 + 1 ) + \\frac { 1 } { 8 } ( 3 ( 2 p _ d - 1 ) ^ 2 + 3 ( 2 p _ d - 1 ) ) = p _ d ^ 3 . \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} \\bigg ( \\int _ B \\Delta ^ 2 u - ( 1 - \\sigma ) \\int _ B w \\Delta ^ 2 u \\bigg ) \\int _ B w \\Delta ^ 2 u = \\dfrac { p + 3 } { p + 1 } ( 1 + \\sigma ) \\pi \\int _ B u ^ { p + 1 } . \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} T _ 5 = \\left ( \\begin{array} { c | c | c | c | c } I _ { s - 1 } & & & & \\\\ \\hline & 1 & & 1 & \\\\ \\hline \\mathbf a '' & & 1 & & \\\\ \\hline & & & 1 & \\\\ \\hline & & & & I _ { n - s - 1 } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u _ t ( x ) = \\mathcal { L } u _ t ( x ) + \\xi \\sigma ( u _ t ( x ) ) \\dot W ( t , x ) , \\ \\ \\ \\ x \\in B _ R ( 0 ) , \\ \\ \\ t > 0 \\\\ u _ t ( x ) = 0 , \\ \\ \\ x \\in B _ R ( 0 ) ^ c . \\end{cases} \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} L _ t \\cap L _ u = \\bigcup _ { v \\in S , v \\le t , u } L _ v . \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} \\lim _ { n \\longrightarrow \\infty } { \\bf P } _ { X ^ n , Y ^ n } ^ { { \\bf P } ^ * } \\Big \\{ ( X ^ n , Y ^ n ) \\in { \\cal X } ^ n \\times { \\cal Y } ^ n : ~ \\frac { 1 } { n + 1 } \\left | { \\bf E } ^ { { \\bf P } ^ * } \\{ i ^ { { \\bf P } ^ { * } } ( X ^ n , Y ^ n ) \\} - i ^ { { \\bf P } ^ { * } } ( X ^ n , Y ^ n ) \\right | > \\epsilon \\Big \\} = 0 \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} V _ { r - 1 } : = \\sum _ { n _ { i } } A _ { i } + B _ { 1 , 2 } + \\dots \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} S _ 1 ( l , 0 , v ; p ) = \\frac { 1 } { l ^ { 1 / 2 + v } } + 2 \\pi i ^ { 2 k } V _ p ( 0 , v , k ) \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} \\sigma ^ 2 = 2 ^ { - B } P . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} \\pi ^ i ( t ) & = { n \\choose n - i } \\sum _ { k = 0 } ^ i ( - 1 ) ^ { i - k } { i \\choose i - k } \\exp ( Q _ { ( n - k ) \\lambda } t ) x ( 0 ) \\\\ & = { n \\choose i } \\sum _ { k = 0 } ^ i ( - 1 ) ^ { i - k } { i \\choose k } \\exp ( Q _ { ( n - k ) \\lambda } t ) x ( 0 ) . \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} & u ( x ) : = f ( x ) \\sin x \\\\ & V ( x ) : = \\lambda - \\frac { 1 } { u ( x ) } \\left ( \\sqrt { p ^ 2 + m ^ 2 } - m \\right ) u ( x ) , \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} \\tilde { b } _ { s , s } \\left [ Z _ s , t \\right ] = 1 \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} | f _ { l e x } ( Y ) - f _ { l e x } ( X ) | \\leq k \\binom { b } { k } , \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} C _ { \\gamma } \\left ( \\overline { z } ^ { m - j } z ^ { n - j } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { j } \\right ) = \\dfrac { - z ^ { n - j } \\overline { z } ^ { m - j + 1 } } { m - j + 1 } \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + j + 1 } { _ 2 F _ 1 } \\left ( \\begin{array} { c } 1 , \\gamma + m + 2 \\\\ m + 2 - j \\end{array} \\bigg | | z | ^ 2 \\right ) . \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} D _ p ( t ) = \\begin{pmatrix} 2 ^ t \\\\ & \\ddots \\\\ & & 2 ^ t \\end{pmatrix} . \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} F ( \\eta ) _ { X \\boxtimes h } = \\alpha ^ h ( \\tilde \\eta _ { \\alpha ^ { h ^ { - 1 } } ( X ) } ) \\boxtimes \\iota \\colon ( X \\otimes \\alpha ^ h ( U ) ) \\boxtimes h g \\to ( X \\otimes \\alpha ^ h ( V ) ) \\boxtimes h g \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} x _ { n + 1 } & = t _ n T x _ n + ( 1 - t _ n ) x _ n \\\\ & = T _ { t _ n } x _ n \\\\ & = T _ { t ' _ n \\cdot \\frac { 1 - k } { d } } x _ n \\\\ & = T _ { t ' _ n } ( T _ { \\frac { 1 - k } { d } } x _ n ) \\\\ & = t ' _ n T _ { \\frac { 1 - k } { d } } x _ n + ( 1 - t ' _ n ) x _ n . \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} ( 1 - \\varepsilon ) ^ { k - 2 } \\frac { ( n z ) ^ { k - 2 } } { ( k - 2 ) ! } \\le \\left ( 1 - \\frac { k - 2 } { n z } \\right ) ^ { k - 2 } \\frac { ( n z ) ^ { k - 2 } } { ( k - 2 ) ! } \\le \\binom { n z - 1 } { k - 2 } \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} \\mathcal { I } \\cap \\big \\{ \\sum _ { 1 \\leqslant i \\leqslant d } g _ k ( x _ i , y _ i ) : ( x _ 1 , . . . , x _ d ) \\in R \\ \\ \\ \\ ( y _ 1 , . . . , y _ d ) \\in \\{ 0 , 1 \\} ^ d \\big \\} = \\emptyset \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} t = s ^ { \\gamma } , \\gamma : = \\frac { 1 } { 1 - 2 \\alpha } , s \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} \\det E ^ \\bullet = \\bigotimes _ { k = 0 } ^ m \\left ( \\det E ^ k \\right ) ^ { ( - 1 ) ^ k } . \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} g ( x ) = g ( x _ i , \\psi _ i ( x _ { - i } ) ) . \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} & \\deg \\left ( _ \\infty \\left ( \\frac { \\Delta } { \\Delta _ { 5 6 , 1 2 } } \\right ) \\right ) = 9 5 \\\\ & \\deg \\left ( _ \\infty \\left ( \\frac { \\Delta _ { 5 6 } } { \\Delta _ { 5 6 , 1 2 } } \\right ) \\right ) = 4 0 . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} u _ { 0 . 6 7 } ^ 2 ( g _ 2 ) = [ I - 0 . 6 7 P ^ 2 ( g _ 2 ) ] ^ { - 1 } \\bar { r } ^ 2 ( g _ 2 ) = ( 7 . 3 3 , 8 ) . \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } S _ { n } \\left ( x q ^ { \\alpha - 1 / 2 } ; q \\right ) = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( x e ^ { i u } ; q \\right ) _ { n } } { \\left ( q ; q \\right ) _ { n } } \\exp \\left ( \\frac { u ^ { 2 } } { \\log q ^ { 2 } } + i \\alpha u \\right ) d u \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} S _ G = \\sum _ { \\theta \\in \\mbox { e v } ( G ) } \\theta E _ \\theta , \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} \\Pi \\tilde L _ a f ( \\xi ) = & ( 2 \\pi ) ^ { - d } \\sum _ { m \\in \\mathbb { Z } ^ d } \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { T } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi + 2 \\pi m ) - \\varphi _ a ( x , \\eta ) ) } f ( \\eta ) d \\eta d x \\\\ = & ( 2 \\pi ) ^ { - d } \\sum _ { m \\in \\mathbb { Z } ^ d } \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { T } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) + 2 \\pi x \\cdot m - \\varphi _ a ( x , \\eta ) ) } f ( \\eta ) d \\eta d x . \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} \\mbox { e v } ( G ' ) = \\{ \\pm \\sqrt { v } , \\pm 1 \\} , \\beta _ { \\pm \\sqrt { v } } = 0 , \\beta _ { \\pm 1 } = 1 / \\sqrt { 2 } . \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\psi } ^ { * } \\mu = \\lambda \\mu , \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} S _ j ^ \\ell u [ x ] & : = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - y \\cdot \\xi ) } s _ j ^ \\ell ( x , \\xi ) u [ y ] d \\xi , \\\\ P _ j ^ \\ell ( t ) u [ x ] & : = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( x \\cdot \\xi - \\varphi _ a ( y , \\xi ) ) } p _ j ^ \\ell ( y , \\xi ; t ) u [ y ] d \\xi , \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} y _ q ( u ) & = \\sum _ { \\mathcal { R } : | \\mathcal { R } | = r + 1 , \\mathcal { R } \\ni q } \\tilde { h } _ { q , \\mathcal { T } } ^ { \\bar { \\mathcal { R } } } ( u ) x _ { { \\mathcal { R } } , { \\mathcal { T } } } , \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} T _ F = T _ E \\frac { B K } { M C _ F } , \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\deg \\mu _ { 1 } = 2 \\sum _ { a \\geq b > 1 } ^ { } n _ { a , b , 1 } + 4 n _ { d - 2 , 1 , 1 } \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} \\frac { 1 } { N } n _ { \\Delta } ( t ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\mathbf { 1 } _ { ( x _ i ( t ) , v _ i ( t ) ) \\in \\Delta } \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial r } + i \\frac { \\partial F } { \\partial s } = 0 , \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} \\mathcal { L } ^ { \\langle 2 \\rangle } \\cong \\left ( \\bigotimes _ { j = 1 } ^ g p r _ j ^ * T \\otimes \\bigotimes _ { j = 1 } ^ g p r _ { j , g + 1 } ^ * \\O ( \\Delta ) ^ { \\vee } \\otimes \\bigotimes _ { j < k } ^ { g } p r _ { j , k } ^ * \\O ( \\Delta ) \\right ) ^ { \\otimes 8 } \\otimes \\left ( p r _ { g + 2 } ^ * T \\right ) ^ { \\langle 2 \\rangle } , \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ { I I I } ^ - } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq C _ { d , s , k } T \\eta ^ d \\end{aligned} \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} \\{ x ^ k + \\alpha d ^ k : \\ ; \\alpha \\in [ 0 , 1 ] , k = 0 , 1 , \\ldots \\} \\subseteq B ( 0 , R ) , \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} h ^ r ( \\kappa _ { ( r ) } ) = ( r ! ) ^ { - 1 } Y _ r . \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} Q = \\nu _ { 1 } + \\cdots + \\nu _ { n } , \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} Q _ { 0 } : = [ - 2 ^ { \\frac { 2 } { \\alpha } } , 0 ] \\times [ - 1 , 1 ] ^ { d } . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} T _ { \\mu + 1 } = \\frac { \\gamma \\left ( 1 - \\gamma ^ \\mu \\right ) T _ \\mu + ( 1 - \\gamma ) \\delta _ { y _ { \\mu + 1 } , + 1 } } { 1 - \\gamma ^ { \\mu + 1 } } , \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} \\mathcal { E } ( f , g ) = \\sum _ { i = 1 } ^ n \\int _ 0 ^ 1 f ' _ i ( x ) g _ i ' ( x ) d x \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( b ; q \\right ) _ { n } q ^ { n ^ { 2 } / 2 } S _ { n } \\left ( a q ^ { - n - 1 / 2 } ; q \\right ) } { \\left ( c ; q \\right ) _ { n } } \\left ( \\frac { c } { a b } \\right ) ^ { n } = \\frac { \\left ( c / b ; q \\right ) _ { \\infty } } { \\left ( c ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } / 2 } } { \\left ( q ; q \\right ) _ { n } } \\left ( \\frac { c } { a b } \\right ) ^ { n } A _ { q } \\left ( \\frac { c q ^ { n - 1 / 2 } } { a } \\right ) \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} | a ^ { i j } - a _ { 0 } ^ { i j } | _ { B ^ { | \\gamma | } } \\leq \\frac { 1 } { 2 N ( d , \\gamma ) N _ { 0 } } = : \\varepsilon _ 2 . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} \\theta _ { q } \\left ( q ^ { k } x \\right ) = ( - 1 ) ^ { k } x ^ { - k } q ^ { - k ( k - 1 ) / 2 } \\theta _ { q } \\left ( x \\right ) , k \\in \\Z , \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\int \\limits _ { - \\infty } ^ { \\infty } \\frac { \\Pi _ n ( x , x ) d x } { 1 + x ^ 2 } = \\int \\limits _ { - \\infty } ^ { \\infty } \\frac { \\Pi ( x , x ) d x } { 1 + x ^ 2 } . \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\overrightarrow { B _ i B _ { i + 1 } } = \\overrightarrow { C _ i C _ { i + 1 } } = ( \\beta , \\beta ) . \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} a ( x \\otimes u , y ) = \\hom ( u , a ( x , y ) ) , \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} \\widetilde { T } _ { p q } = \\frac { 1 } { 4 \\pi } \\left ( E _ { p } D _ { q } + H _ { p } B _ { q } \\right ) - \\frac { 1 } { 8 \\pi } \\delta _ { p q } \\left ( \\mathbf { E } \\cdot \\mathbf { D } + \\mathbf { H } \\cdot \\mathbf { B } \\right ) , \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} \\lambda p _ { 1 , 0 } = \\gamma p _ { 0 , 0 } . \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} \\omega _ i ^ \\mathrm { z m } \\big | _ { \\partial Z _ { 2 , 0 } } = \\omega _ i ^ \\mathrm { z m } \\big | _ { \\partial Z _ { 1 , 0 } } . \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { p } } \\nabla _ p s _ f ( t ) = \\frac { 1 } { \\sqrt { p } } s _ f ( p ) + s _ { \\frac { 1 } { \\sqrt { p } } \\nabla _ p f } ( t ) , t \\in [ p , 1 ] . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} V _ N \\triangleq \\max \\{ V _ { \\varepsilon } ( x _ i ) + \\lambda _ 1 \\ , \\log ( x _ i ) + \\lambda _ 2 \\ , \\log ( 1 - x _ i ) , \\ , i = 1 \\cdots N \\} \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} ( \\sum _ i a _ i \\otimes b _ i ) ^ p = \\sum _ i a _ i ^ p \\otimes b _ i ^ p = 1 \\otimes ( \\sum _ i a _ i ^ p b _ i ^ p ) = 1 \\otimes \\delta ( \\sum _ i a _ i \\otimes b _ i ) ^ p a _ i , b _ i \\in L \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} \\left \\langle A _ { N _ { k } } X _ { j } , X _ { i } \\right \\rangle = \\left \\langle h ( X _ { i } , X _ { j } ) , N _ { k } \\right \\rangle = L _ { i j } ^ { k } 1 \\leq i , j \\leq 2 ; 1 \\leq k \\leq n - 2 \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} E _ T ( n ) \\sim \\sum _ { k = 1 } ^ \\infty \\frac { 3 } { 4 ^ k + 2 } \\approx 0 . 7 2 7 6 4 9 . \\end{align*}"}
-{"id": "9955.png", "formula": "\\begin{align*} \\pi _ { p } ( v _ { i } ) = u ^ { - } ( \\vect { 0 } , \\dots , - \\varphi ( s _ { i } ) ^ { - 1 } , \\dots , \\vect { 0 } ) \\pi _ { p } ( v _ { i - 1 } ) = \\pi _ { p } ( u ^ { - } ( 0 , \\dots , - \\varphi ( s _ { i } ) ^ { - 1 } , \\dots , 0 ) v _ { i - 1 } ) , \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } - V _ s \\cdot \\nabla _ { X _ s } \\right ) \\left | \\phi _ N ^ { ( s ) } ( t , Z _ s ) \\right | = 0 \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} \\alpha _ p \\left ( \\big [ F _ { Z _ { 1 , R } } ( \\omega , 0 ) \\big ] \\right ) = \\big [ F _ { Z _ R } ( \\omega , 0 , 0 ) \\big ] \\in H ^ p ( Z , F ) . \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} { \\bf P } ^ { \\pi } ( d y ^ t , d x ^ t ) = & \\prod _ { i = 0 } ^ t q _ i ( d y _ i | y _ { i - M } ^ { i - 1 } , x _ i ) \\pi _ i ( d x _ i | y ^ { i - 1 } ) , \\\\ \\nu _ t ^ { \\pi } ( d y _ t | y ^ { t - 1 } ) = & \\int _ { { \\cal X } _ t } q _ t ( d y _ t | y _ { t - M } ^ { t - 1 } , x _ t ) \\pi _ t ( d x _ t | y ^ { t - 1 } ) , ~ t \\in \\mathbb { N } _ 0 ^ { n } . \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { m , n } ^ { \\gamma } ) \\right ] ( z ) = \\overline { \\left [ C _ { \\gamma } ( \\overline { \\mathcal { Z } _ { m , n } ^ { \\gamma } } ) \\right ] ( \\overline { z } ) } = \\overline { \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { n , m } ^ { \\gamma } ) \\right ] ( \\overline { z } ) } . \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\tilde H \\tilde u = \\lambda \\tilde u , \\tilde u \\in D ( \\tilde H ) . \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} p _ F = \\begin{cases} \\frac \\alpha N - 1 & N < \\alpha \\leq N + 2 \\\\ \\frac 2 N & \\alpha > N + 2 , \\end{cases} \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} \\alpha ( P _ { 1 } ) = \\log \\| \\theta \\| ( P _ { 1 } + \\dots + P _ { g } - Q ) - \\sum _ { j = 1 } ^ { g } g ( P _ { j } , Q ) - \\sum _ { k < l } g ( \\sigma ( P _ { k } ) , P _ { l } ) \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} c _ { \\sigma } ^ { + } ( M ( \\chi ^ { ( r + 1 , \\varphi ) } ) ) _ { \\varphi , 1 } \\sim \\delta _ { \\sigma } ( M ) _ { \\varphi } P _ { \\sigma } ( \\chi ^ { ( r + 1 , \\varphi ) } ) _ { 1 } \\prod _ { i = 1 } ^ { j - 1 } Q _ { i , \\sigma , \\varphi } . \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} L _ v ^ { ( 1 ) } [ X , Y ] = X Y ( 1 - v ) . \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} \\nabla \\cdot \\left ( f \\mathbf { A } \\right ) = \\left ( \\nabla f \\right ) \\cdot \\mathbf { A } + f \\left ( \\nabla \\cdot \\mathbf { A } \\right ) . \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{gather*} \\big ( \\tau _ { 0 , \\ell } ^ { ( \\alpha , \\beta ) } \\big ) ^ { 2 } = \\tau _ { 0 , \\ell } ^ { ( \\alpha + 1 , \\beta ) } \\tau _ { 0 , \\ell } ^ { ( \\alpha - 1 , \\beta ) } . \\end{gather*}"}
-{"id": "6169.png", "formula": "\\begin{align*} h _ N ( z ) : = \\sum _ { r = 1 } ^ N H _ N ^ { ( r ) } z ^ { r - 1 } = \\langle \\ , \\sum _ { r = 1 } ^ N \\chi _ { e _ { ( r , 3 / 2 ) } } z ^ { r - 1 } \\ , \\rangle . \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} \\tilde { \\varphi } _ n ( \\zeta ) : = \\varphi _ n ( \\zeta ) - R e ( g _ n ( \\zeta ) ) = \\sum _ { j = 1 } ^ m \\lambda _ j ^ n | \\zeta _ j | ^ 2 + O ( | \\zeta | ^ 3 ) \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{gather*} Q _ { n } ( x ) : = L _ t \\left ( \\frac { P _ { n } ( x ) - P _ { n } ( t ) } { x - t } \\right ) \\ , \\ \\ \\ \\ n \\geq 0 \\ , \\end{gather*}"}
-{"id": "5276.png", "formula": "\\begin{align*} v _ \\beta ^ 2 ( g _ 1 ) - v _ \\beta ^ 2 ( g _ 2 ) = \\left [ \\frac { p ( 7 + 5 \\beta ) - ( 3 \\beta + 1 ) } { 1 - \\beta ^ 2 } , \\frac { \\beta ( p ( 7 + 5 \\beta ) - ( 3 \\beta + 1 ) ) } { 1 - \\beta ^ 2 } \\right ] ^ T . \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} d ( x , y ) = | x _ n - y _ n | + | x _ { n + 1 } - y _ { n + 1 } | + \\frac { | x '' - y '' | } { | x _ n | + | x _ { n + 1 } | + | y _ n | + | y _ { n + 1 } | + | x '' - y '' | ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "10066.png", "formula": "\\begin{align*} 2 q < r - p + 2 q = \\alpha ( p - q ) - p + 2 q = 1 + n + k \\leq 1 + n + q \\leq 1 + \\dfrac { p } { 2 } + q , \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ n B _ { \\Gamma , p } ^ { ( \\ell ) } ( z ) = \\sum _ { g = 0 } ^ { \\lfloor \\frac { p - 1 } { 2 } \\rfloor } z ^ { 2 g - p } \\sum _ { \\ell = n + 1 + 2 g - p } ^ { n } \\Upsilon _ { p - 1 - 2 g } ^ { ( \\ell ) } ( z ) \\ ; C _ { p , g } ^ { ( \\ell ) } . \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} z ' + \\frac { 1 } { t } \\ , z = \\frac { c } { t } , \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} n ^ + = { p - 1 \\choose k } \\mbox { ; } n ^ 0 = 0 \\mbox { ; } n ^ - = { p - 1 \\choose k - 1 } . \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} Q = \\frac { a b ( c ) _ { m } } { c ( a ) _ { k } ( b ) _ { l } } q , \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} \\tau _ h ( f ) \\{ \\phi \\} = f \\{ \\tau _ { - h } ( \\phi ) \\} , h \\in \\mathbb { R } ^ d \\phi \\in \\mathcal { D } ( \\mathbb { R } ^ d ) \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} \\theta \\circ T ^ n ( p ) = \\theta _ \\delta \\circ T ^ n ( p ) + ( \\theta - \\theta _ \\delta ) \\circ T ^ n ( p ) > ( 1 - \\delta ) \\ , ( 1 - \\epsilon ) - \\delta \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} v _ { \\delta ^ { 1 } } g _ { 1 } ( x _ { 1 } ) + u _ { \\delta ^ { 1 , n } } g _ { 1 } ( x _ { 1 } ) g _ { n } ( x _ { n } ) = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } v _ { \\delta } [ g _ { 1 } ( x _ { 1 } - x _ { n } ) ^ { \\delta _ { 1 } } - g _ { 1 } ( - x _ { n } ) ^ { \\delta _ { 1 } } ] f _ { n } ^ { \\delta _ { n } } ( x _ { n } ) \\prod _ { i = 2 } ^ { n - 1 } g _ { i } ^ { \\delta _ { i } } ( - x _ { n } ) \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} \\beta _ { k _ n } & : = \\max \\{ \\beta _ k : k = 0 , 1 , \\ldots , n \\} , \\\\ p _ { k _ n } & : = \\max \\{ p _ k : k = 0 , 1 , \\ldots , n \\} . \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} K _ { Y } + \\tilde { D } + \\sum a _ { i } E _ { i } = \\pi ^ { * } ( K _ { X } + D ) \\sim 0 \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} z _ i \\ln z _ i - z _ i - \\zeta _ i + 1 = 0 \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} a ^ { i j } ( t , x ) = \\sum _ { n = 1 } ^ { M _ 0 } a _ { n } ^ { i j } ( t , x ) 1 _ { ( \\tau _ { n - 1 } , \\tau _ n ] } ( t ) \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} p _ { k + 1 } \\leq s p _ k + \\frac { ( s p _ k - q _ { k + 1 } ) q _ k } { 1 - q _ k - q _ { k + 1 } } , 0 \\leq q _ { k + 1 } \\leq s p _ k . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} \\gamma ( \\Omega , q , q ) = \\frac { 1 } { q } ( \\gamma ( \\Omega ) - \\delta _ { \\Omega } \\log q ) , \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} \\begin{array} { l l } ( \\overline \\nabla _ X J _ \\alpha ) Y = \\frac { 1 } { 4 n } \\left [ g ( X , Y ) p _ \\alpha + \\overline \\theta _ \\alpha ( Y ) X + g ( X , J _ \\alpha Y ) J _ \\alpha p _ \\alpha \\right . \\\\ \\qquad \\quad \\left . + \\overline \\theta _ \\alpha ( J _ \\alpha Y ) J _ \\alpha X \\right ] . \\end{array} \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} h _ { N , N } ( z _ 1 , \\dots , z _ N ) \\Big | _ { z _ N = \\frac { 2 t \\Delta z _ 1 - 1 } { t ^ 2 z _ 1 } } \\propto z _ 1 , z _ 1 \\to 0 , \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} \\sup _ N \\frac { 1 } { N } \\int _ { \\mathcal { D } _ N } \\sum _ { i = 1 } ^ N \\left ( | x _ i | ^ 2 + | v _ i | ^ 2 \\right ) f _ N ( 0 , Z _ N ) d Z _ N < \\infty \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} f _ { \\nu } ' \\left ( z \\right ) = g _ { 1 } ' \\left ( z \\right ) + \\mu _ { \\nu } \\cdot g _ { 2 , \\nu } ' \\left ( z \\right ) . \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} s x _ i & = x _ i - ( \\alpha ^ * _ s , x _ i ) \\alpha _ s , & \\forall i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} & | ( P _ - v _ s , E _ - ( - s ) ^ * v ) | \\\\ & = | ( P _ - v _ s , E _ - ( - s ) ^ * ( \\chi _ { \\{ | x | \\geq R \\} } + \\chi _ { \\{ | x | < R \\} } ) v ) | \\\\ & \\leq \\| E _ - ( - s ) P _ - v _ s \\| \\| \\chi _ { \\{ | x | \\geq R \\} } v \\| + \\| P _ - v _ s \\| \\| \\chi _ { \\{ | x | < R \\} } E _ - ( - s ) \\| \\| v \\| \\\\ & \\leq C _ v ( \\| \\chi _ { \\{ | x | \\geq R \\} } v \\| + \\| \\chi _ { \\{ | x | < R \\} } E _ - ( - s ) \\| ) \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{gather*} D ^ 1 _ { s h } ( V , \\hat { \\mathcal { T } } ) = \\left ( H ^ 1 ( V , \\hat { \\mathcal { T } } ) \\rightarrow \\prod _ { v \\in X _ 0 ^ { ( 1 ) } } H ^ 1 ( K _ { 0 , v } , \\hat { T } ) \\right ) , \\\\ \\mathcal { D } ^ 2 ( V , \\mathcal { T } ) = \\left ( H ^ 2 _ c ( V , \\mathcal { T } ) \\rightarrow H ^ 2 ( K _ 0 , T ) \\right ) , \\end{gather*}"}
-{"id": "1496.png", "formula": "\\begin{align*} & \\psi _ { x x x } + \\frac { M _ x } { M } \\psi _ { x x } + \\frac { M _ { x x } } { M } \\psi _ x - \\frac { \\lambda } { M } \\psi = 0 , \\\\ & \\lambda \\psi _ t + M \\psi _ { x x } + ( \\lambda \\beta + M _ x ) \\psi _ x + \\lambda \\beta _ x \\psi = 0 . \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} ( \\varphi \\otimes \\psi _ n ) \\big ( J _ n ( p _ \\xi ) \\cdot J _ n ( a ) \\cdot J _ n ( p _ \\xi ) \\big ) = ( \\varphi \\otimes \\psi _ n ) \\big ( J _ n ( p _ \\xi a p _ \\xi ) \\big ) = \\varphi ( p _ \\xi a p _ \\xi ) = \\omega _ \\xi ( a ) \\ , \\varphi ( p _ \\xi ) , \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} & y ( s , t ) - x - ( t - s ) v ( \\xi ) \\\\ & = q ( t , s ; x , \\eta ( t , s ) ) - x - ( t - s ) v ( p ( t , s ; x , \\eta ( t , s ) ) ) \\\\ & = \\int _ s ^ t \\left [ v \\left ( \\xi + \\int _ \\tau ^ t \\nabla _ x V _ \\rho ( \\sigma , q ( \\sigma , s ; x , \\eta ( t , s ) ) ) d \\sigma \\right ) - v ( \\xi ) \\right ] d \\tau . \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} \\varphi _ j = \\sum _ { i = 1 } ^ n F _ { i j } \\abs { A } ^ 2 + \\sum _ { i = 1 } ^ n 2 F _ i \\kappa _ j - F _ j H - F , \\\\ \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} c = & ( u '^ 2 + v '^ 2 ) f , \\\\ c \\kappa = & ( u '' v ' - v '' u ' ) f - u '^ 3 \\langle S _ v , S _ { u u } \\rangle + v '^ 3 \\langle S _ u , S _ { v v } \\rangle - \\frac { 1 } { 2 } u '^ 2 v ' f _ u + \\frac { 1 } { 2 } v '^ 2 u ' f _ v . \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} \\mathcal { E } ( f , g ) = \\int _ \\mathbb { M } \\langle d f , d g \\rangle _ { T ^ * \\mathbb { M } } d \\mu , f , g \\in W ^ { 1 , 2 } ( \\mathbb { M } ) \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} s ( x , y ) = \\sum _ { i \\geq 1 } \\lambda _ i \\psi _ i ( x ) \\overline { \\psi _ i ( y ) } ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} \\underset { 0 \\leq x \\leq L } { \\sup } \\| ( \\partial ^ k _ x \\varphi ( x , \\cdot ) , \\partial ^ k _ x \\psi ( x , \\cdot ) ) \\| _ { ( H ^ { \\frac { 1 - k } 3 } ( 0 , T ) ) ^ 2 } \\le C _ T \\| ( \\varphi ^ 0 , \\psi ^ 0 ) \\| _ { ( L ^ 2 ( 0 , L ) ) ^ 2 } , \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} D ( l , m , n ) : = \\begin{vmatrix} ( - 1 ) ^ l & ( - 1 ) ^ m & ( - 1 ) ^ n \\\\ 1 & 1 & 1 \\\\ l & m & n \\\\ \\end{vmatrix} . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} R ^ { + } _ 1 ( x ) & = 1 , R ^ { + } _ 2 ( x ) = 1 + 3 x , R ^ { + } _ 3 ( x ) = 1 + 1 6 x , R ^ { + } _ 4 ( x ) = 1 + 6 1 x + 4 1 x ^ 2 ; \\\\ R ^ { - } _ 1 ( x ) & = x , R ^ { - } _ 2 ( x ) = 3 x , R ^ { - } _ 3 ( x ) = 7 x + 9 x ^ 2 , R ^ { - } _ 4 ( x ) = 1 5 x + 8 0 x ^ 2 ; \\\\ R _ 1 ( x ) & = 1 + x , R _ 2 ( x ) = 1 + 6 x , R _ 3 ( x ) = 1 + 2 3 x + 9 x ^ 2 , R _ 4 ( x ) = 1 + 7 6 x + 1 2 1 x ^ 2 . \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} g ( J _ 1 X , J _ 1 Y ) = - g ( J _ 2 X , J _ 2 Y ) = - g ( J _ 3 X , J _ 3 Y ) = g ( X , Y ) , X , Y \\in T M . \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} S ( l , u , v ; p ) = S _ 1 ( l , u , v ; p ) - \\frac { 1 } { p ^ { 1 / 2 + u + v } } S _ 2 ( l , u , v ; p ) . \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} I ^ \\psi _ { j } & = \\sum _ { u \\in \\psi _ { j } } g _ { u , j } \\ell ( r _ { u , j } ) P _ B , \\\\ I ^ \\varphi _ { j } & = \\sum _ { v \\in \\varphi _ { j } } g _ { v , j } \\ell ( r _ { v , j } ) P _ M , \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} ( 1 \\otimes _ \\nabla X _ j ) ( \\delta _ k \\otimes c \\otimes \\delta _ n \\otimes b ) & = \\delta _ k \\otimes ( c + [ X _ j , c ] ) \\otimes \\delta _ n \\otimes b \\\\ & \\mapsto \\alpha _ d ^ k ( c + [ X _ j , c ] ) \\cdot \\delta _ { ( n , k ) } \\otimes b \\\\ & = \\alpha _ d ^ k ( X _ j c ) \\cdot \\delta _ { ( n , k ) } \\otimes b \\\\ & = X _ j \\left ( \\alpha _ d ^ k ( c ) \\cdot \\delta _ { ( n , k ) } \\otimes b \\right ) \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} \\| f \\| _ { Y _ { \\alpha , \\epsilon } ( \\mathcal { B } _ R ^ + ) } : = \\sup _ { \\bar y \\in P \\cap \\mathcal { B } _ R } \\left ( \\| d _ G ( \\cdot , \\bar y ) ^ { - ( 1 + 2 \\alpha ) } ( f - P _ { \\bar y } ) \\| _ { L ^ { \\infty } ( \\mathcal { B } _ 3 ^ + ( \\bar y ) \\cap \\mathcal { B } _ R ^ + ) } \\right . \\\\ \\left . + [ d _ G ( \\cdot , \\bar y ) ^ { - ( 1 + 2 \\alpha ) } ( f - P _ { \\bar y } ) ] _ { \\dot { C } ^ { 0 , \\epsilon } _ \\ast ( \\mathcal { B } _ 3 ^ + ( \\bar y ) \\cap \\mathcal { B } _ R ^ + ) } + | P _ { \\bar y } | _ 1 \\right ) , \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\zeta ( a _ 1 a _ 2 \\ldots a _ { n } ) = x _ 1 x _ 2 \\ldots x _ { | \\zeta ( a _ 1 ) | } x _ { | \\zeta ( a _ 1 ) | + 1 } \\ldots x _ { | \\zeta ( w ) | } . \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} \\zeta ( \\alpha , \\beta ; s ) = \\sum _ { n + \\alpha > 0 } \\frac { e ( n \\beta ) } { ( n + \\alpha ) ^ s } \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} - \\Re \\Big \\{ \\frac { L ' } { L } ( s , \\chi ) \\Big \\} = \\frac { 1 } { 2 } \\log D _ { \\chi } + \\Re \\Big \\{ \\frac { \\delta ( \\chi ) } { s - 1 } + \\frac { \\delta ( \\chi ) } { s } \\Big \\} - \\sum _ { \\rho } \\Re \\Big \\{ \\frac { 1 } { s - \\rho } \\Big \\} + \\Re \\Big \\{ \\frac { \\gamma _ { \\chi } ' } { \\gamma _ { \\chi } } ( s ) \\Big \\} . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} T ^ \\mu _ \\mu ( x ) = a _ 4 ( x , D ) = \\frac { 1 } { 2 8 8 0 \\pi ^ 2 } ( R ^ { i j k l } R _ { i j k l } - R ^ { i j } R _ { i j } - 3 0 \\Delta R + 5 R ^ 2 ) \\end{align*}"}
-{"id": "9959.png", "formula": "\\begin{align*} { G R S } _ { n , k } ( \\gamma , w ) : = \\left \\{ ( \\gamma _ { 1 } f ( w _ { 1 } ) , \\gamma _ 2 f ( w _ 2 ) , \\ldots , \\gamma _ { n } f ( v _ { n } ) ) \\mid f ( X ) \\in \\mathbb { F } _ \\ell [ X ] _ { k } \\right \\} \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} \\Phi _ 3 ( b ; \\ , c ; \\ , x , \\ , y ) = \\exp \\left ( x + \\frac { y } { x } \\right ) \\ , \\sum _ { k = 0 } ^ \\infty \\ , \\frac { 1 } { k ! } \\ , \\left ( - \\frac { y } { x } \\right ) ^ k { } _ 2 F _ 1 \\left [ \\begin{array} { c } - k , \\ , - k - b + 1 \\\\ c \\end{array} ; \\ , \\frac { x ^ 2 } { y } \\right ] \\ , . \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} { \\cal D } ^ { * } = & \\Big \\{ ( d _ { 1 , 2 } , d _ { 1 , 3 } , d _ { 2 , 1 } , d _ { 2 , 3 } , d _ { 3 , 1 } , d _ { 3 , 2 } ) \\in \\mathbb { R } _ { + } ^ 6 : \\\\ & ~ ~ ~ d _ { p _ 1 , p _ 2 } + d _ { p _ 1 , p _ 3 } \\leq M _ { p _ 1 } , ~ \\forall { \\bf p } \\\\ & ~ ~ ~ d _ { p _ 2 , p _ 1 } + d _ { p _ 3 , p _ 1 } \\leq M _ { p _ 1 } , ~ \\forall { \\bf p } \\\\ & ~ ~ ~ d _ { p _ 1 , p _ 2 } + d _ { p _ 1 , p _ 3 } + d _ { p _ 2 , p _ 3 } \\leq N , ~ \\forall { \\bf p } \\Big \\} , \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} z e _ k & = \\sum _ { j = 0 } ^ { k - 2 } g _ { k , j } . y _ k ^ j e _ k \\\\ & = \\sum _ { j = 0 } ^ { k - 2 } g _ { k + 1 , j } . y _ k ^ j e _ k + \\sum _ { j = 0 } ^ { k - 2 } ( - 1 ) ^ { k - j } g _ { k + 1 , k - 1 } E _ { k - 1 - j } ( x _ 1 , \\ldots , x _ { k - 1 } ) . y _ k ^ j e _ k \\\\ & = \\sum _ { j = 0 } ^ { k - 1 } g _ { k + 1 , j } . y _ k ^ j e _ k . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} & \\mathsf { P r e } _ { t _ 1 , t _ 2 } ( p , q ) : = \\\\ & ~ ~ ~ ~ ~ ~ ~ \\forall v ' \\left ( ( [ \\dot { t _ 1 } ( v ' ) \\wedge p ( v ' ) ] \\vee [ \\dot { t _ 2 } ( v ' ) \\wedge R ( v ' , q ) ] ) \\to ( v ' \\in q ) \\right ) ~ . \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} \\tau _ { m } ( v _ { m } ^ { * } ) \\left \\{ \\begin{array} { l l } \\leq \\lambda _ { m } & \\mbox { i f } v ^ { * } _ { m } = 0 , \\\\ = \\lambda _ { m } & \\mbox { i f } v ^ { * } _ { m } \\in ( 0 , \\gamma _ { m } ) ; \\\\ \\geq \\lambda _ { m } & \\mbox { i f } v ^ { * } _ { m } = \\gamma _ { m } ; \\end{array} \\right . \\forall p \\in \\mathcal { P } _ { m } . \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\begin{array} { l } a _ 0 = V o l ( M , g ) \\\\ a _ 2 = \\frac { 1 } { 6 } \\int _ M d V \\ , R \\\\ a _ 4 = \\frac { 1 } { 3 6 0 } \\int _ M d V \\ , ( - 1 2 \\Delta R + 5 R ^ 2 - 2 R ^ { i j } R _ { i j } + 2 R ^ { i j k l } R _ { i j k l } ) \\\\ \\end{array} \\end{align*}"}
-{"id": "9666.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } A _ { q } ( - q ^ { \\alpha + n } ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ { \\infty } \\frac { ( - q ^ { 1 / 2 + n } e ^ { i x } ; q ) _ { \\infty } \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i \\alpha x \\right ) } { \\sqrt { \\log q ^ { - 1 } } } d x , \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} b c = \\frac { ( 1 - w ) ^ 2 } { 4 w } \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\min \\{ b _ 1 ( \\pi ) + \\dots + & b _ { i + 1 } ( \\pi ) , \\mathcal { M } \\} = \\\\ & \\min \\{ b _ 1 ( \\pi ' ) + \\dots + b _ { i + 1 } ( \\pi ' ) , \\mathcal { M } \\} . \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} ( \\Psi f ) ( t ) = \\begin{cases} \\frac 1 t \\int _ t ^ { 1 - t } f ( s ) \\ , d s , & 0 < t < \\frac 1 2 , \\\\ [ 1 e x ] 0 , & \\frac 1 2 \\le t \\le 1 . \\end{cases} \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} \\begin{cases} - d _ 1 \\Delta w = w q ( \\rho w ) \\left [ f ( \\rho w ) - v \\right ] , & x \\in \\Omega , \\\\ - d _ 2 \\Delta v = v \\left [ h ( v ) + q ( \\rho w ) w \\right ] , & x \\in \\Omega , \\\\ \\partial _ \\nu u = \\partial _ \\nu v = 0 , & x \\in \\partial \\Omega . \\\\ \\end{cases} \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} E ( m T ) \\le S ( m ) m \\in \\mathbb { N } \\lim _ { t \\to \\infty } S ( t ) = 0 . \\end{align*}"}
-{"id": "10129.png", "formula": "\\begin{align*} \\omega = ( a ( p + 2 q ) x ^ 2 + p y ^ 2 + b ( p + q ) x ) d x + 2 q x y d y . \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ u ^ t \\cdot t ^ { d _ u + 1 } } M ^ t \\vec v & = \\frac { 1 } { \\lambda _ u ^ t \\cdot t ^ { d _ u + 1 } } \\left ( \\lambda ^ t \\vec v + \\sum _ { j = 0 } ^ { t - 1 } \\lambda ^ { j } M ^ { t - j - 1 } \\vec u _ { } \\right ) \\\\ & = \\frac { 1 } { t ^ { d _ u + 1 } } \\vec v + \\frac { 1 } { \\lambda } \\sum _ { j = 0 } ^ { t - 1 } \\frac { ( { t - j - 1 } ) ^ { d _ u } } { t ^ { d _ u + 1 } } \\frac { 1 } { \\lambda ^ { t - j - 1 } \\cdot ( { t - j - 1 } ) ^ { d _ u } } M ^ { t - j - 1 } \\vec u _ { } \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , T ] } \\| w ( t ) - w ^ n ( t ) \\| _ { H ^ { 1 + \\delta } _ p } & = \\sup _ { t \\in [ 0 , T ] } \\| w ( T - t ) - w ^ n ( T - t ) \\| _ { H ^ { 1 + \\delta } _ p } \\\\ & \\leq C \\| b - b ^ n \\| _ { \\infty , H ^ { - \\beta } _ q } . \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} \\dim \\mathcal O _ k & = \\dim \\left ( J _ 0 ^ { k + 1 } ( \\mathbb R ^ 2 , \\mathbb R ^ 4 ) \\times J _ 0 ^ k ( \\mathbb R ^ 2 , \\mathbb R ) \\right ) \\\\ & = 4 \\dim \\left ( J _ 0 ^ { k + 1 } ( \\mathbb R ^ 2 , \\mathbb R ) \\right ) + \\dim \\left ( J _ 0 ^ k ( \\mathbb R ^ 2 , \\mathbb R ) \\right ) \\\\ & = 4 \\binom { k + 3 } { 2 } + \\binom { k + 2 } { 2 } = \\frac { ( k + 2 ) ( 5 k + 1 3 ) } { 2 } . \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} \\mathcal { B } _ r ( \\theta ^ * ) & = \\left \\lbrace \\theta \\in \\Theta \\left | \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } D _ { K L } \\left ( \\ell ^ i \\left ( \\cdot | \\theta ^ * \\right ) , \\ell ^ i \\left ( \\cdot | \\theta \\right ) \\right ) \\right . \\leq r \\right \\rbrace \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} P _ C ( t ) = 1 + t + t ^ 2 + t ^ 3 + 2 t ^ 4 + 3 t ^ 5 + \\ldots = 1 + t + t ^ 2 + \\sum _ { m \\geqslant 4 } ( m - 2 ) t ^ m . \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} [ x _ 0 ^ 2 : x _ 0 x _ 1 : x _ 1 ^ 2 : x _ 2 ] & = \\varphi ( | x _ 0 : x _ 1 : x _ 2 | ) \\\\ & = \\varphi \\left ( \\lambda ^ { \\frac 1 2 } \\cdot | x _ 0 : x _ 1 : x _ 2 | \\right ) \\\\ & = \\varphi \\left ( | \\lambda ^ { \\frac 1 2 } x _ 0 : \\lambda ^ { \\frac 1 2 } x _ 1 : \\lambda x _ 2 | \\right ) = [ \\lambda x _ 0 ^ 2 : \\lambda x _ 0 x _ 1 : \\lambda x _ 1 ^ 2 : \\lambda x _ 2 ] . \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\frac { C _ n ^ 2 } { 2 } f '' - W _ + ( r ) f ' = - 1 , f ( \\infty ) = 0 , \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} i ^ 2 = a , \\ j ^ 2 = b , \\ i j = - j i , \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} R ( n ) = \\left \\{ \\begin{array} { c c } e _ { n } + e _ { n + 1 } & \\mbox { i f $ n $ e v e n } \\\\ e _ { 2 c _ { ( n - 1 ) / 2 } } & \\mbox { i f $ n $ o d d } \\end{array} \\right . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} a _ 1 & = \\sum _ { k = 3 } ^ { g + 1 } \\tbinom { k - 1 } { 2 } \\tfrac { k ! ( k - 1 ) ! } { 1 2 } ( - 1 ) ^ { g + 1 - k } ( g + 1 - k ) ! \\tfrac { g ! } { ( k - 1 ) ! } \\tbinom { g } { k - 1 } \\tbinom { g + 1 } { k } \\\\ & = \\tfrac { g ! ( g + 1 ) ! } { 1 2 } ( - 1 ) ^ g \\sum _ { k = 2 } ^ { g } ( - 1 ) ^ { k } \\tbinom { k } { 2 } \\tbinom { g } { k } = 0 , \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} h ( X , J _ \\alpha Y ) - J _ \\alpha h ( X , Y ) = \\frac { 1 } { 4 n } \\left [ g ( X , Y ) p _ \\alpha ^ \\bot + g ( X , J _ \\alpha Y ) J _ \\alpha ( p _ \\alpha ^ \\bot ) \\right ] , \\ \\alpha = 2 , 3 . \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} \\log _ 2 \\Pr _ { a _ L \\sim _ { \\rho } b _ L } [ a _ { L } \\in A _ { L } , b _ L \\in B _ { L } ] & \\geq - | L | \\left ( \\frac { ( 1 - \\tfrac { \\pi } { \\lambda } ) + \\tfrac { \\epsilon } { \\lambda } + o ( 1 ) + 2 \\rho \\sqrt { ( 1 - \\tfrac { \\pi } { \\lambda } ) ( \\tfrac { \\epsilon } { \\lambda } + o ( 1 ) ) } } { 1 - \\rho ^ 2 } \\right ) , \\\\ & = n \\left ( \\frac { \\pi - \\lambda - \\epsilon - 2 \\rho \\sqrt { \\epsilon \\lambda - \\epsilon \\pi } } { 1 - \\rho ^ 2 } - o ( 1 ) \\right ) . \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\sup _ { s \\in \\left [ 0 , \\varepsilon ^ { - 2 } t \\right ] } \\varepsilon \\left \\vert Y ^ { \\varepsilon } \\left ( s \\right ) - \\mathcal { M } _ { { } } ^ { \\varepsilon } \\left ( s \\right ) \\right \\vert = 0 \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} I ( z ) - z = \\bar z - z + \\frac { 1 } { 2 } = - 2 i \\Im ( z ) + \\frac { 1 } { 2 } = n + m \\Re ( \\tau ) + m i \\Im ( \\tau ) \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mbox { m i n i m i z e } & f ( R ) + n \\sum \\limits _ { k = 1 } ^ r | c _ k | ^ 2 \\\\ \\mbox { s u b j e c t t o } & R = \\sigma ^ 2 I + \\sum \\limits _ { k = 1 } ^ r | c _ k | ^ 2 \\left [ \\begin{array} { c } 1 \\\\ e ^ { \\j \\omega _ k } \\\\ \\vdots \\\\ e ^ { \\j ( n - 1 ) \\omega _ k } \\end{array} \\right ] \\left [ \\begin{array} { c } 1 \\\\ e ^ { \\j \\omega _ k } \\\\ \\vdots \\\\ e ^ { \\j ( n - 1 ) \\omega _ k } \\end{array} \\right ] ^ H , \\end{array} \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} \\varrho = \\textrm { a r c c o s h } \\ , x ^ 0 , \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} b _ x \\mathbf n _ x f + c _ x f '' ( x ) = 0 \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} \\sum _ { e : v \\in e } B ( v , e ) = 1 , & ~ ~ ~ v \\in V ( B _ m ^ L ( 2 ) ) , \\\\ \\prod _ { v : v \\in e } B ( v , e ) = \\alpha , & ~ ~ ~ e \\neq e _ { m } . \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} P _ { m + d + 1 } ^ { \\left ( r \\right ) } \\left ( x \\right ) = \\left ( x - \\beta _ { m + d + r } \\right ) P _ { m + d } ^ { \\left ( r \\right ) } \\left ( x \\right ) - \\sum \\nolimits _ { \\nu = 0 } ^ { d - 1 } \\gamma _ { m + d + r - \\nu } ^ { d - 1 - \\nu } P _ { m + d - 1 - \\nu } ^ { \\left ( r \\right ) } \\left ( x \\right ) m \\geq 0 , \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} h ^ i _ j = v ^ { - 1 } \\vartheta ^ { - 1 } \\{ - ( \\sigma ^ { i k } - v ^ { - 2 } \\varphi ^ i \\varphi ^ k ) \\varphi _ { j k } + \\dot { \\vartheta } \\delta ^ i _ j \\} , \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} \\mathfrak { L } _ \\nu = \\nabla _ { \\nu } ^ { * } \\nabla _ { \\nu } - \\nu = \\nabla _ { \\nu - 1 } \\nabla _ { \\nu - 1 } ^ { * } + \\nu . \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} \\mathcal { C } _ { r , s } ( \\varepsilon ; \\tau ) : = \\eta ( \\tau ) \\operatorname { c h } \\left [ M _ { r , s } ^ \\varepsilon \\right ] ( \\tau ) . \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} | \\eta | | q _ { 5 } | \\lesssim | \\eta | | q _ { 4 } | \\sim | q _ { 3 } | \\lesssim | q _ { 2 } | = | \\eta | ^ { - 1 } | q _ { 1 } | \\lesssim \\left \\{ \\begin{array} { r l l } & \\displaystyle | \\eta | & \\ , \\ , \\eta \\in I _ 1 ( t ) , \\\\ & \\displaystyle ( \\lambda t ) ^ { - 2 } | \\eta | ^ { - 1 } & \\ , \\ , \\eta \\in I _ 2 ( t ) , \\\\ & \\displaystyle ( \\lambda t ) ^ { - 2 } | \\eta | ^ { - 3 } & \\ , \\ , \\eta \\in I _ 3 ( t ) . \\end{array} \\right . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} \\nu ^ { \\pi ^ * } _ { n - 1 } ( 0 | 0 ) & = \\sum _ { x _ { n - 1 } \\in \\{ 0 , 1 \\} } q _ { n - 1 } ( 0 | x _ { n - 1 } , 0 ) \\pi ^ * _ { n - 1 } ( x _ { n - 1 } | 0 ) = q _ { n - 1 } ( 0 | 0 , 0 ) \\pi ^ * _ { n - 1 } ( 0 | 0 ) + q _ { n - 1 } ( 0 | 1 , 0 ) \\pi ^ * _ { n - 1 } ( 1 | 0 ) \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} Y _ s = \\xi + \\int _ s ^ T g ( r , Y _ r , Z _ r ) \\mathrm d r - \\int _ s ^ T Z _ r \\mathrm d W _ r \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} F _ \\nu \\left ( \\varphi \\right ) = \\frac { \\rho ^ { - 1 } \\left ( A _ \\nu , \\varphi \\right ) \\cdot \\mu _ { \\nu } - \\rho ^ { - 1 } \\left ( x , \\varphi \\right ) } { \\nu \\cdot \\mu _ { \\nu } } \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} \\frac { 1 } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\sum _ { n = 0 } ^ { \\infty } q ^ { ( 2 n + 1 ) n } ( - q ^ { 2 n + 2 } ; q ) _ \\infty = \\frac { 1 } { ( q ; q ) _ \\infty } \\sum _ { j = 0 } ^ \\infty q ^ { ( 3 j ^ 2 + j ) / 2 } ( 1 - q ^ { 2 j + 1 } ) . \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} S ( l , 0 , v ; p ) = \\frac { 1 } { l ^ { 1 / 2 + v } } + \\frac { 2 \\pi i ^ { 2 k } } { ( 2 \\pi ) ^ { 1 - 2 v } } \\frac { \\Gamma ( k - v ) } { \\Gamma ( k + v ) } \\frac { l ^ { v - 1 / 2 } } { p ^ { 1 + 2 v } } + O ( V _ p ( 0 , v , k ) ) . \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} s _ q ^ + ( t ) : = \\frac { 1 } { q } \\sum _ { j = 1 } ^ n ( 1 - ( - t _ j ) ^ { q } ) x _ j ^ q s _ q ^ - ( t ) : = \\frac { 1 } { q } \\sum _ { j = 1 } ^ n ( 1 - ( - t _ j ) ^ { q } ) x _ j ^ { - q } . \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\alpha _ k \\leq \\frac { 1 } { { \\binom { k - 1 } { l - 1 } } } \\sum _ { i = 1 } ^ { \\binom { k } { l } } \\alpha _ { l , L _ i } , \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{gather*} L \\left ( P _ n ( x ) ^ { 2 } \\right ) = D _ { n } D _ { n - 1 } \\ , \\ \\ n \\geq 0 \\ , \\ \\ D _ { - 1 } : = 1 \\ . \\end{gather*}"}
-{"id": "9227.png", "formula": "\\begin{align*} H ( X ) = \\frac { 1 } { N } H ( X _ 1 , \\ldots , X _ N ) = \\sum _ { i \\leq k } w _ i H ( \\mu _ i ) \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} \\displaystyle \\det ( \\wedge ^ k I + \\sum _ { j = 0 } ^ { k - 1 } ( \\wedge ^ j I ) \\wedge B \\wedge ( \\wedge ^ { k - 1 - j } I ) ) = 0 . \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { m = - \\infty } ^ { \\infty } ( q ^ { - \\ell } ; q ) _ { m } 2 ^ { m } q ^ { \\ell ( | m | - m ) / 2 } S _ { \\ell } \\left ( - q ^ { - | m | - \\ell } ; q \\right ) \\\\ & = \\left ( - \\frac { 1 } { 2 } \\right ) ^ { \\ell } q ^ { \\binom { \\ell + 1 } { 2 } } \\frac { \\left ( q ^ { \\ell } / 2 ; q \\right ) _ { \\infty } } { \\left ( q , q / 2 ; q \\right ) _ { \\ell } } . \\end{aligned} \\end{align*}"}
-{"id": "9910.png", "formula": "\\begin{align*} \\left | \\int _ { \\cup _ { i = 1 } ^ l B _ i } f ( a ( t ) u ( \\varphi ( s ) ) x ) \\dd s \\right | \\leq \\epsilon | I | . \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} \\sigma _ 1 \\sigma _ 2 ^ { - 1 } \\sigma _ k = \\sigma _ k \\sigma _ 2 ^ { - 1 } \\sigma _ 1 . \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} L _ a f ( \\xi ) & = F J _ a ^ * J _ a F ^ * f ( \\xi ) \\\\ & = ( 2 \\pi ) ^ { - d } \\sum _ { x \\in \\mathbb { Z } ^ d } \\int _ { \\mathbb { T } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - \\varphi _ a ( x , \\eta ) ) } f ( \\eta ) d \\eta , \\\\ \\tilde L _ a f ( \\xi ) & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { T } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - \\varphi _ a ( x , \\eta ) ) } f ( \\eta ) d \\eta d x . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} - a _ { 2 1 } M \\hat { f } ^ * L _ 1 | _ { \\hat { C } ' _ 1 } & \\sim _ { \\mathbb { Q } } a _ { 2 1 } ( M m ( K _ { \\hat { X } } + \\hat { B } ) - M \\hat { f } ^ * L _ 1 ) | _ { \\hat { C } ' _ 1 } \\\\ & \\sim _ { \\mathbb { Q } } a _ { 2 1 } \\hat { \\Gamma } _ 1 | _ { \\hat { C } ' _ 1 } \\sim _ { \\mathbb { Q } } a _ { 1 1 } a _ { 2 1 } \\hat { G } _ 1 | _ { \\hat { C } ' _ 1 } \\\\ & \\sim _ { \\mathbb { Q } } a _ { 1 1 } \\hat { \\Gamma } _ 2 | _ { \\hat { C } ' _ 1 } \\sim _ { \\mathbb { Q } } - a _ { 1 1 } M \\hat { f } ^ * L _ 2 | _ { \\hat { C } ' _ 1 } \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} | a ^ 0 _ k - a ^ 0 _ { k - 1 } | & \\leq C r _ 0 ^ { k ( 2 + { 2 \\alpha } ) } , \\\\ | a _ k ^ j - a _ { k - 1 } ^ j | & \\leq C r _ 0 ^ { { 2 k \\alpha } } , j = 1 , \\dots , n - 1 , \\\\ | b _ k - b _ { k - 1 } | & \\leq C r _ 0 ^ { { 2 k \\alpha } } . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} T = \\frac { 1 } { \\sqrt 2 } \\left [ \\begin{array} { c c } 1 & 1 \\\\ - 1 & 1 \\end{array} \\right ] R , \\epsilon = \\frac { 1 } { 2 } ( \\alpha + \\gamma ) , \\delta = \\frac { 1 } { 2 } ( \\alpha - \\gamma ) , \\eta = \\beta . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} C ^ { \\infty } _ { p e r } ( \\Omega ) = \\{ \\varphi \\in C ^ { \\infty } ( \\overline { \\Omega } ) \\mid \\hbox { $ \\varphi $ p e r i o d i c o f o r d e r $ m $ o n $ \\Gamma _ l $ f o r a l l $ m \\in \\N $ } \\} , \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} M _ { a , b , c } = \\prod _ { j = 0 } ^ { a - 1 } \\prod _ { k = 0 } ^ { b - 1 } \\prod _ { l = 0 } ^ { c - 1 } \\frac { j + k + l + 2 } { j + k + l + 1 } = \\prod _ { j = 0 } ^ { b - 1 } \\frac { j ! ( j + a + c ) ! } { ( j + a ) ! ( j + c ) ! } , \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} n ^ + = n - \\left ( \\frac { n + 2 \\mu } { 2 + \\mu } \\right ) \\mbox { a n d } k = 1 + 2 \\mu . \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( - u L ^ { * } \\varphi + ( g \\circ u ) \\varphi ) d x & = \\int _ \\Omega \\varphi d \\mu - \\int _ { \\partial \\Omega } \\frac { \\partial \\varphi } { \\partial { \\bf { n } } _ { L ^ { * } } } d \\nu \\ , , \\ , \\ , \\ , \\forall \\ , \\varphi \\in C ^ { 2 , L } _ c ( \\bar { \\Omega } ) \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} ( a , a ^ { \\prime } ) [ ( b , b ^ { \\prime } ) ( c , c ^ { \\prime } ) ] & = ( a , a ^ { \\prime } ) ( b c , b ^ { \\prime } c ^ { \\prime } ) , \\\\ & = \\left ( a ( b c ) , a ^ { \\prime } ( b ^ { \\prime } c ^ { \\prime } ) \\right ) , \\\\ & = \\left ( ( a b ) c , ( a ^ { \\prime } b ^ { \\prime } ) c ^ { \\prime } \\right ) , \\\\ & = ( a b , a ^ { \\prime } b ^ { \\prime } ) ( c , c ^ { \\prime } ) , \\\\ & = [ ( a , a ^ { \\prime } ) ( b , b ^ { \\prime } ) ] ( c , c ^ { \\prime } ) . \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d t ^ 2 } | _ { t = 0 } \\int _ { L } \\varphi ( \\xi _ t ) ^ 2 \\ , d v o l _ x = 2 \\frac { d } { d t } _ { | t = 0 } \\int _ { L } \\varphi ( \\xi _ t ) \\cdot \\frac { d } { d t } ( \\exp t V ) ^ * \\varphi ( \\xi ( x ) ) \\ , d v o l _ x . \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} \\| \\varphi _ { g } \\| ( X ) = \\prod _ { \\lbrace j _ { 1 } , \\dots , j _ { g + 1 } \\rbrace \\in \\mathcal { T } } \\| \\theta \\| ( W _ { j _ { 1 } } + \\dots + W _ { j _ { g } } - W _ { j _ { g + 1 } } ) ^ { 8 } . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} & C _ 1 ^ ( a ) = \\log _ 2 \\left ( 1 + \\dfrac { ( 1 - a ) \\xi | h _ 1 | ^ 2 } { a \\xi | h _ 1 | ^ 2 + 1 } \\right ) \\\\ & C _ 2 ^ ( a ) = \\log _ 2 \\left ( 1 + a \\xi | h _ 2 | ^ 2 \\right ) . \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} \\theta _ { s , a ^ 1 } ^ 1 = \\big ( \\bar { r } ^ 1 ( s , a ^ 1 ) - \\bar { r } ^ 1 ( s , a _ s ^ 1 ) \\big ) - \\beta \\left ( \\sum _ { s ' \\in S } p _ { s ' } \\bar { r } ^ 1 ( s ' , a _ { s ' } ^ 1 ) - \\sum _ { s ' \\in S } p ^ 1 ( s ' | s , a ^ 1 ) \\bar { r } ^ 1 ( s ' , a _ { s ' } ^ 1 ) \\right ) , \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ u } \\sum _ { j = 0 } ^ { t _ 0 - 1 } \\left ( \\frac { \\lambda } { \\lambda _ u } \\right ) ^ j \\left ( \\frac { { t - j - 1 } } { t } \\right ) ^ { d _ u } \\frac { 1 } { \\lambda _ u ^ { t - j - 1 } \\cdot ( { t - j - 1 } ) ^ { d _ u } } M ^ { t - j - 1 } \\vec u _ { j + 1 } \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} \\lim \\limits _ { l \\rightarrow \\infty } \\mu ( \\mathbf { z } ^ { l } ) = \\mu ^ { * } ; \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{gather*} d _ { r } \\ = \\ \\inf \\left \\{ \\ \\ m \\geq 0 \\ \\ \\big | \\ \\ D _ { r + m } \\neq 0 \\ \\ \\right \\} \\ . \\end{gather*}"}
-{"id": "502.png", "formula": "\\begin{align*} ( F ( x ) ) _ { i } = f _ { i } ( x _ { i } ) \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} y _ k ^ n e _ k = y _ k ^ { k - 1 } ( \\psi _ k \\cdots \\psi _ n ) ( \\psi _ n \\cdots \\psi _ k ) e _ k + \\sum _ { j = 1 } ^ { n - k + 1 } ( - 1 ) ^ { j + 1 } E _ j ( x _ k , \\ldots , x _ n ) . ( y _ k ^ { n - j } e _ k ) . \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} \\lim _ { l \\to e } \\gamma ^ l = \\mathrm { p } _ M \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} \\Delta ^ 2 u : W ^ { 2 , q ' } _ 0 ( \\Omega ) \\ni \\varphi \\mapsto < \\Delta ^ 2 u , \\varphi > : = \\int _ \\Omega g ( x ) | u | ^ { p - 1 } u \\varphi . \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} q _ { 2 n } + { \\textstyle \\sum \\limits _ { p = 1 } ^ { 2 n - 1 } } c _ { p } q _ { 2 n - p } = 0 , \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} h ( n ) = h _ 1 ^ 2 ( n ) \\ , . \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ i { i \\choose r } ( k x ) ^ r . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} \\tau \\left ( \\zeta , v , \\varepsilon \\right ) = \\sup \\left \\{ c \\mbox { s u c h t h a t } \\left | \\rho \\left ( \\zeta + \\lambda v \\right ) - \\rho ( \\zeta ) \\right | < \\varepsilon , \\ , \\forall \\lambda \\in \\mathbb { C } , \\ , \\left | \\lambda \\right | < c \\right \\} . \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 : n ) } = \\bigoplus _ { j = 0 } ^ 3 \\Gamma ( j ) = \\bigoplus _ { j = 0 } ^ 3 \\left ( \\Gamma ( j ) \\oplus F _ n ( \\Gamma ( j ) ) \\oplus F _ n ( \\Gamma ( \\pi ( j ) ) \\right ) = \\bigoplus _ { j = 0 } ^ 3 \\Lambda ( j , \\pi ( j ) ) \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} 0 = \\sum _ { r = 1 } ^ { N } m _ { t , r } \\left ( C _ { t , t } \\left ( 1 \\right ) - C _ { t , r } \\left ( 1 \\right ) \\right ) . \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{align*} \\norm { v } { H ^ { k , 2 k } ( D _ T ) } : = \\sum _ { \\ell = 0 } ^ k \\norm { \\partial _ t ^ \\ell v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 \\ell } ( D ) ) } . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} a _ { 0 1 } a _ { 1 0 } = a _ 0 + k e _ 0 \\qquad a _ { 1 0 } a _ { 0 1 } = a _ 1 + 2 e _ 1 \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} \\prod _ { m = 1 } ^ { \\infty } ( 1 - \\frac { 1 } { \\sqrt { r _ m } } ) > 0 . \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} \\frac { 1 } { \\rho } \\left ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } F ^ 3 - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } - B _ { i J K j } F ^ 1 _ { j } ) _ { , K } \\right ) _ { , J } & = F ^ 2 _ i , \\\\ \\frac { 1 } { a } \\left ( - \\beta _ { K i } F ^ 1 _ { i , K } + ( m _ { I J } F ^ 3 _ { , J } + M _ { j L K I } u _ { j , L K } + K _ { I J } \\tau _ { , J } ) _ { , I } \\right ) & = F ^ 4 . \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} \\Omega _ t = - \\beta \\Omega _ x - k \\Omega \\beta _ x + \\alpha ^ { k - 2 } \\left ( - k \\beta _ { x x x } + ( k - 2 ) \\beta _ { x x } \\frac { \\alpha _ x } { \\alpha } + 3 k \\alpha ^ k \\alpha _ x \\right ) . \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} w _ k ^ 2 L _ k & = w _ k ^ 2 ( \\pi _ { W _ k } ) \\leq \\sum _ { k ' = 1 } ^ K w _ { k ' } ^ 2 ( \\pi _ { W _ { k ' } } \\pi _ { W _ { k } } ) = ( \\sum _ { k ' = 1 } ^ K w _ { k ' } ^ 2 \\pi _ { W _ { k ' } } \\pi _ { W _ { k } } ) \\\\ & = ( S _ { \\mathbf { W } , \\mathbf { w } } \\pi _ { W _ { k } } ) = ( \\alpha I _ { \\mathcal { H } } \\pi _ { W _ { k } } ) = \\alpha ( \\pi _ { W _ { k } } ) = \\alpha L _ { k } \\end{align*}"}
-{"id": "10103.png", "formula": "\\begin{align*} B _ { \\pm } ( \\alpha ) = \\left [ \\frac { - \\alpha \\pm \\sqrt { \\alpha ^ 2 + 1 2 a c } } { 6 a } : 0 : 1 \\right ] . \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\exp ( t X ) ( x _ 0 ) ) & = X ( t , \\exp ( t X ) ( x _ 0 ) ) , \\qquad \\mbox { f o r a l m o s t e v e r y } t \\in \\mathbb { T } _ { x _ 0 } \\\\ \\exp ( 0 X ) ( x _ 0 ) & = x _ 0 . \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} w ( \\Gamma _ 4 ) = 4 A _ { ( 1 , 1 ) } B _ { ( 1 , 1 ) } , \\ ; w ( \\Gamma _ 5 ) = 2 A _ { ( 1 , 1 ) } B _ { ( 1 ) } ^ 2 , \\ ; w ( \\Gamma _ 6 ) = 2 A _ { ( 1 ) } ^ 2 B _ { ( 1 , 1 ) } . \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} g = x . x ^ { 2 \\alpha } , \\ , h = x . ( x ^ { 2 \\alpha } y ^ 2 z ^ 2 ) g = x . ( x ^ { 2 \\alpha } z ^ 2 ) , \\ , h = x . ( x ^ { 2 \\alpha } y ^ 2 ) , \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} u _ i = 0 , u _ { i , J } = 0 , \\tau = 0 \\Gamma \\times ( 0 , \\infty ) \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} d ( y , \\alpha ) & \\geq d ( y , z ) - 6 C _ \\eta & \\\\ & = d ( o , y ) - d ( o , z ) - 6 C _ \\eta \\\\ & \\geq d ( o , x ) - d ( o , z ) - 6 C _ \\eta - 2 C _ \\zeta \\\\ & \\geq ( \\xi \\cdot \\zeta ) _ o - ( \\zeta \\cdot \\eta ) _ o - K _ \\eta - K _ \\zeta - 6 C _ \\eta - 2 C _ \\zeta & \\\\ & \\geq R _ \\zeta - ( \\zeta \\cdot \\eta ) _ o - K _ \\eta - K _ \\zeta - 6 C _ \\eta - 2 C _ \\zeta & \\xi \\in U ( \\zeta , R _ \\zeta ) \\\\ & = 2 C _ \\zeta = d ( x , y ) \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} K = C _ 1 \\left ( 1 - \\frac { C C _ T \\beta L } { T } \\right ) ^ { - 1 } > 0 . \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\zeta ( s , f , D ) = \\frac { 1 } { \\Gamma ( s ) } \\int _ 0 ^ \\infty d t \\ , t ^ { s - 1 } T r ( f e ^ { - t D } ) \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} Z _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\in \\mathcal { K } _ { s + k } \\cap \\mathcal { U } _ { s + k } ^ \\eta \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} Z ( t ) = X ( t ) + Y ( G _ i ) = X ( t ) - X ( G _ i ) , t \\in [ G _ i , ( G _ i + \\epsilon ) \\wedge D _ i ] . \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\sum _ { \\theta \\in \\mathcal { B } _ { r } ^ c ( \\theta ^ * ) } \\exp \\left ( - k \\gamma ^ 2 ( \\theta ) \\right ) & = \\sum \\limits _ { l \\geq 1 } \\sum _ { \\theta \\in \\mathcal { B } _ { r _ { l + 1 } } \\backslash \\mathcal { B } _ { r _ { l } } } \\exp \\left ( - k \\gamma ^ 2 ( \\theta ) \\right ) \\\\ & \\leq \\sum \\limits _ { l \\geq 1 } \\mathcal { N } _ { r _ l } \\exp \\left ( - k r _ { l } ^ 2 \\right ) \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} x _ i ' = \\frac { x _ i } { z _ i n } y _ i ' = \\frac { y _ i } { z _ i n } k _ i = \\frac { v z _ i n } { x _ i y _ i } \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} ( \\Lambda _ { N } ( t ) - \\mid \\gamma + t \\mid ^ { 2 } ) ( \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) = \\sum _ { \\gamma _ { 1 } \\in \\Gamma ( k + ) } q _ { \\gamma _ { 1 } } ( \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma - \\gamma _ { 1 } + t , x \\right \\rangle } ) . \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\min _ z \\ h ( z ) : = \\frac 1 2 \\| A z - b \\| ^ 2 + \\mu ( \\| z \\| _ 1 - \\| z \\| ) , \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} w = ( ( z - \\alpha _ { 1 } ) ^ { n _ { 1 } - 1 } , 0 , 0 , \\dots ) . \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} \\left ( \\frac { x _ { z _ { i + 1 } } } { x _ { z _ 0 } } + \\frac { x _ { z _ i , z _ 0 , z _ 0 } } { x _ { z _ 0 } } \\right ) _ { z _ 0 } = \\left ( \\frac { x _ { z _ 0 } ^ 2 } { 2 } \\right ) _ { z _ i } , i = 1 , \\dots , n . \\end{align*}"}
-{"id": "10119.png", "formula": "\\begin{align*} ( p , q , r ) = ( p _ 0 , q _ 0 + p _ 0 u , r _ 0 + p _ 0 v ) , \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\mathbf { X } = \\left [ \\begin{array} { c c c c } x _ { n - 1 } & x _ { n - 2 } & \\cdots & x _ { n - p } \\\\ x _ { n - 2 } & x _ { n - 3 } & \\cdots & x _ { n - p - 1 } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ x _ { 0 } & x _ { - 1 } & \\cdots & x _ { - p + 1 } \\end{array} \\right ] . \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{gather*} \\mathcal { F } _ c = \\sum _ { i _ 1 , \\dots , i _ n = 0 } ^ \\infty \\frac { x _ 1 ^ { i _ 1 } \\cdots x _ n ^ { i _ n } \\otimes 1 } { i _ 1 ! \\cdots i _ n ! } \\prod _ { \\nu = 1 } ^ n \\big [ \\big ( \\Delta _ { \\hat { S } ( \\gg ) } - \\Delta _ 0 \\big ) ( \\partial ^ \\nu ) \\big ] ^ { i _ \\nu } \\end{gather*}"}
-{"id": "5576.png", "formula": "\\begin{align*} M = ( \\lambda s \\partial _ s - L ) ^ { - 1 } . \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} \\int _ { \\frac { i } { k } } ^ { \\frac { i + 1 } { k } } r | u _ { \\varepsilon , i } | ^ p \\ , d r = k ^ { - 2 } \\int _ 0 ^ 1 ( s + i ) | v _ \\varepsilon | ^ p \\ , d s > k ^ { - 2 } \\int _ \\varepsilon ^ { 1 - \\varepsilon } ( s + i ) \\ , d s = \\frac { k ^ { - 2 } } { 2 } ( 2 i + 1 ) ( 1 - 2 \\varepsilon ) . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} v ' _ \\theta ( \\theta , t ) = H \\int _ 0 ^ t s ^ { 2 H - 1 } \\left ( s e ^ { \\theta s } + ( 2 t - s ) e ^ { \\theta ( 2 t - s ) } \\right ) d s > 0 , \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} \\alpha = \\left ( \\begin{array} { c } \\sigma _ 4 \\\\ \\sigma _ 3 \\\\ - \\sigma _ 2 \\\\ - \\sigma _ 1 \\\\ \\end{array} \\right ) ~ ~ { \\rm a n d } ~ ~ \\beta = \\left ( \\begin{array} { c c c c } \\sigma _ 1 & \\sigma _ 2 & \\sigma _ 3 & \\sigma _ 4 \\end{array} \\right ) \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in E _ Z } S _ \\gamma P _ A S _ \\gamma ^ * = \\sum _ { \\gamma \\in E _ Z } P _ B S _ \\gamma S _ \\gamma ^ * = P _ B . \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} P _ n ( x ) = Q _ { n } ( x ) + \\sum _ { i = 1 } ^ { r } a _ { n , i } Q _ { n - i } ( x ) , \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} \\omega = \\int _ { - \\delta _ Y } ^ { \\delta _ Y } E _ 0 ( \\phi _ \\lambda , \\lambda ) d \\lambda \\in L ^ 2 \\big ( \\Omega ^ \\bullet ( Y _ { \\R _ + } , F ) \\big ) , \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} d _ { r , t } = \\left \\{ \\begin{array} { l l } 1 , & r + t \\ge N _ R \\\\ \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } t } { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } t + 1 } , & r + t = N _ R - 1 \\\\ \\max \\left \\{ d ' _ { r , t } , \\frac { r + t } { N _ R } \\right \\} , & r + t \\le N _ R - 2 \\end{array} , \\right . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} \\hat { \\psi } _ s ^ { t ^ \\prime } Z _ s ^ 0 = ( X _ s ^ 0 + V _ s ^ 0 t ^ \\prime , V _ s ^ 0 ) \\end{align*}"}
-{"id": "10109.png", "formula": "\\begin{align*} p + q + r = \\gcd ( q , p + r ) + \\gcd ( p , q + r ) + \\gcd ( r , p + q ) . \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} \\rho _ j = \\left ( 1 + 2 ^ { - j } \\right ) \\frac \\rho 2 , \\theta _ j = \\left ( 1 + 2 ^ { - j } \\right ) \\frac \\theta 2 , k _ j = \\left ( 1 - 2 ^ { - j } \\right ) k \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} K _ { a , \\psi } f ( \\xi ) = & \\psi \\circ ( \\Pi \\tilde L _ a - \\tilde L _ a ) f ( \\xi ) \\\\ = & \\sum _ { m \\in \\mathbb { Z } ^ d \\backslash \\{ 0 \\} } \\psi ( \\xi ) \\int _ { \\mathbb { R } ^ d } \\int _ { \\mathbb { T } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi + 2 \\pi m ) - \\varphi _ a ( x , \\eta ) ) } f ( \\eta ) d \\eta d x \\\\ = & \\int _ { \\mathbb { T } ^ d } k _ { a , \\psi } ( \\xi , \\eta ) f ( \\eta ) d \\eta , \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} G ^ { ( n + 1 ) } ( x , y ) = G ^ { ( n ) } ( x , y ) + { \\lambda _ { n + 1 } G ^ { ( n ) } ( x , a _ { n + 1 } ) G ^ { ( n ) } ( a _ { n + 1 } , y ) \\over 1 - \\lambda _ { n + 1 } G ^ { ( n ) } ( a _ { n + 1 } , a _ { n + 1 } ) } \\ ; , \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} \\sigma _ E ^ { m ( x ) } ( \\kappa ( \\sigma _ E ( x ) ) ) = \\sigma _ E ^ { l ( x ) } ( \\kappa ( x ) ) \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{gather*} ( \\zeta '' ) ^ \\tau _ { \\alpha _ 1 , \\dots , \\alpha _ s \\beta _ 1 , \\dots , \\beta _ r } = \\frac { \\partial } { \\partial ( \\partial ^ \\lambda ) } \\big ( \\zeta ^ \\tau _ { \\alpha _ 1 , \\dots , \\alpha _ s } \\big ) \\cdot ( \\zeta ' ) ^ \\lambda _ { \\beta _ 1 , \\dots , \\beta _ r } - \\frac { \\partial } { \\partial ( \\partial ^ \\lambda ) } \\big ( ( \\zeta ' ) ^ \\lambda _ { \\beta _ 1 , \\dots , \\beta _ r } \\big ) \\cdot \\zeta ^ \\tau _ { \\alpha _ 1 , \\dots , \\alpha _ s } . \\end{gather*}"}
-{"id": "4265.png", "formula": "\\begin{align*} \\mu ( Q ) = ( q + 1 ) c _ 2 ( \\beta , k ) - \\left ( ( q + 1 ) c _ 2 ( \\beta , k ) - q c _ 1 ( \\beta , k ) \\right ) = q c _ 1 ( \\beta , k ) . \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} p ( s ) = ( c I + r ) ^ { - 1 } \\left ( M s \\right ) s \\ge 0 , \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} p = \\pm \\sqrt { 2 } x , q = \\pm 2 s y , z = \\pm t y , l ^ 2 = t ^ 2 , \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} \\alpha ( t ) : = & x _ { a } ( t ^ { \\theta } ) x _ { b } ( t ) x _ { a + b } ( t ^ { \\theta + 1 } ) x _ { 2 a + b } ( t ^ { 2 \\theta + 1 } ) , \\\\ \\beta ( t ) : = & x _ { a + b } ( t ^ { \\theta } ) x _ { 3 a + b } ( t ) , \\\\ \\gamma ( t ) : = & x _ { 2 a + b } ( t ^ { \\theta } ) x _ { 3 a + 2 b } ( t ) , \\\\ \\tau : = \\tau ( 1 ) = & \\pi _ { a + b } ( 1 ) \\pi _ { 3 a + b } ( 1 ) , \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} \\begin{dcases} d X = - \\sum _ { i = 1 } ^ m D _ p H ^ i ( P , X ) \\cdot d W ^ i , & X ( x , p , t _ 0 ) = x , \\\\ d P = \\sum _ { i = 1 } ^ m D _ x H ^ i ( P , X ) \\cdot d W ^ i , & P ( x , p , t _ 0 ) = p . \\end{dcases} \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\begin{array} { l c } F _ 1 ( X , Y , Z ) = \\frac { 1 } { 2 ( 2 n - 1 ) } \\left [ g ( X , Y ) \\theta _ 1 ( Z ) - g ( X , Z ) \\theta _ 1 ( Y ) \\right . \\\\ \\qquad \\qquad \\qquad \\left . - g ( X , J _ 1 Y ) \\theta _ 1 ( J _ 1 Z ) + g ( X , J _ 1 Z ) \\theta _ 1 ( J _ 1 Y ) \\right ] , \\end{array} \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} \\begin{array} { c c l } K _ { ( v : m ) } \\otimes N _ t & = K _ { ( v : m ) } \\otimes \\left ( \\bigoplus _ { i = 1 } ^ p C ( w _ i ) \\right ) \\\\ & = \\bigoplus _ { i = 1 } ^ p \\left ( K _ { ( v : m ) } \\otimes C ( w _ i ) \\right ) \\\\ \\end{array} \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} a ^ { i j } h _ { i ; j } = a ^ { i i } h _ { i ; i } = a ^ { i i } \\left ( W \\eta _ { i ; i } + 2 \\eta _ { i } W _ { i } + \\eta W _ { i ; i } \\right ) \\le 0 . \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} p _ k ( x , \\xi ) = a _ { n n } ^ { ( k ) } \\Big ( \\big ( \\xi _ n + \\sum _ { j = 1 } ^ { n - 1 } a _ { n j } ^ { ( k ) } / a _ { n n } ^ { ( k ) } \\xi _ j \\big ) ^ 2 + b _ k ( x , \\xi ' ) \\Big ) , \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} \\chi _ { q } ( z ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { n / 2 } } { 1 + z q ^ { n - 1 / 2 } } . \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } ( \\omega _ \\xi \\otimes \\theta _ k ) J _ { n _ k } ( a ) = \\rho ( a ) , \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} \\frac { k ^ 2 + \\abs { \\eta - k t } ^ 2 } { k ^ 2 + | \\xi - k t | ^ 2 } = \\frac { k ^ 2 + \\abs { \\xi - k t + \\eta - \\xi } ^ 2 } { k ^ 2 + | \\xi - k t | ^ 2 } \\lesssim 1 + \\abs { \\eta - \\xi } ^ 2 , \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} f ( a + b ) = f ( a ) + f ( b ) + f ( b ) H ( a ) , \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} P _ { n } = P _ { n } ( z , \\xi ; q ) : = \\sum _ { j = 0 } ^ { n } \\frac { ( q z \\xi ^ { - 1 } ; q ) _ { j } } { ( q ; q ) _ { n - j } } z ^ { - 2 j } . \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} \\int _ { I _ { m , 1 } } B _ { p , q } ' ( x ) \\ d x & = \\int _ { \\frac { 1 } { p ^ { m + 1 } ( p - 1 ) } } ^ { \\frac { 1 } { p ^ { m + 1 } } } B _ { p , q } ' ( x ) \\ d x \\\\ & = B _ { p , q } \\left ( \\frac { 1 } { p ^ { m + 1 } } \\right ) - B _ { p , q } \\left ( \\frac { 1 } { p ^ { m + 1 } ( p - 1 ) } \\right ) \\\\ & = \\frac { 1 } { q ^ { m + 1 } } - \\frac { 1 } { q ^ { m + 1 } ( q - 1 ) } \\\\ & = \\frac { q - 2 } { q ^ { m + 1 } ( q - 1 ) } \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} z _ j = \\frac { a } { b } \\frac { b ( \\lambda _ j ) } { a ( \\lambda _ j ) } , j = 1 , \\ldots , N . \\end{align*}"}
-{"id": "7007.png", "formula": "\\begin{align*} \\Sigma ^ { \\left [ d \\right ] } : = \\overline { \\Sigma \\times _ { \\pi } \\dots \\times _ { \\pi } \\Sigma \\setminus } \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} x _ 1 ( 0 ) = x _ { 1 0 } < x _ { 2 0 } = x _ 2 ( 0 ) . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} I _ 3 = & \\delta T _ 2 d \\Delta ( r ^ { 6 - n } ) \\\\ = & - [ ( T _ 2 ) _ { i j } ( \\Delta r ^ { 6 - n } ) _ { , j } ] _ { , i } \\\\ = & - ( T _ 2 ) _ { i j , i } ( \\Delta r ^ { 6 - n } ) _ { , j } - ( T _ 2 ) _ { i j } ( \\Delta r ^ { 6 - n } ) _ { , j i } . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\ell b ( x , y ) = b ( \\varphi ( x ) , \\varphi ( y ) ) \\ \\forall x , y \\in L ^ * . \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\abs { A ^ \\circ _ { 1 1 } } ^ 2 = \\abs { A ^ \\circ _ { 1 2 } } ^ 2 , \\langle A ^ \\circ _ { 1 1 } , A ^ \\circ _ { 1 2 } \\rangle = 0 \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} ^ { C \\ ! } D _ { 0 + } ^ { \\alpha } x ( t ) = A ( t ) x ( t ) , \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} a = i ( f ) + \\varpi ^ m a _ 1 = c _ f \\bigl ( i ( f ' ) + ( \\varpi ^ m / c _ f ) a _ 1 \\bigr ) . \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} a \\partial _ \\theta U ( x , \\theta , s ) + \\lambda s \\partial _ s U ( x , \\theta , s ) = L U ( x , \\theta , s ) + N ( U ( x , \\theta , s ) ) . \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} v _ { \\mu _ 1 , \\mu _ 2 } : = \\min _ z \\ \\ J ( z ) : = \\frac 1 2 \\| A z - b \\| ^ 2 + \\mu _ 1 H _ 1 ( z ) - \\mu _ 2 H _ 2 ( z ) , \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\int _ { P ( z , 2 ^ { - ( i - 1 ) } \\delta _ { \\Omega } ( z ) ) \\setminus P ( z , 2 ^ { - i } \\delta _ { \\Omega } ( z ) ) } \\frac { d \\lambda ( \\zeta ) } { \\left | z - \\zeta \\right | ^ { 1 + \\eta } } \\lesssim \\left ( 2 ^ { - i } \\right ) ^ { 2 } \\prod _ { j = 1 } ^ { n - 1 } \\tau _ { j } ^ { 2 } \\left ( z , \\delta _ { \\Omega } ( z ) \\right ) \\tau _ { n } ^ { \\frac { 1 - \\eta } { m } } \\left ( z , \\delta _ { \\Omega } ( z ) \\right ) . \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} \\bar { \\alpha } _ p ( R ) ( \\hat { \\omega } ) = \\hat { \\eta } . \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} T ^ { ( p - 2 j ) } ( \\vartheta L , \\vartheta B ) = i _ { \\vartheta L ^ \\perp } ^ * \\vartheta \\pi _ { L ^ \\perp } ^ * T ^ { ( p - 2 j ) } ( L , B ) , \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} D S ( \\hat t , t _ 0 ) \\hat \\eta ( \\hat x ) = D \\zeta ( \\hat x ) D ^ 2 S ( \\hat t , t _ 0 ) \\hat \\eta ( \\hat x ) \\le D ^ 2 \\zeta ( \\hat x ) , \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} \\partial _ t f ( t , x , v ) + v \\cdot \\nabla _ x f ( t , x , v ) = \\sigma ( x ) \\left ( \\int _ { \\mathcal { V } } f ( t , x , w ) \\ , { \\rm d } \\mu ( w ) - f ( t , x , v ) \\right ) \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} & J _ g [ B ^ * B \\alpha + \\tfrac { 1 } { 2 } \\Phi ^ { - 1 } ( c ) \\beta , \\beta ] - \\Phi ^ { - 1 } ( c ) \\\\ & = J _ f ^ { B , c } [ \\alpha , \\beta ] + \\Phi ^ { - 1 } ( c ) \\ , \\mathbf { E } \\bigg [ \\frac { 1 } { 2 } \\int _ 0 ^ 1 e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ t \\| \\beta _ s \\| ^ 2 d s } \\| \\beta _ t \\| ^ 2 \\ , d t + e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\| \\beta _ t \\| ^ 2 d t } - 1 \\bigg ] \\\\ & = J _ f ^ { B , c } [ \\alpha , \\beta ] , \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] \\\\ & \\ ; \\ ; \\in \\mathcal { K } _ { s + k } \\cap \\mathcal { U } _ { s + k } ^ \\eta \\end{aligned} \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} c = \\inf \\{ s \\mid \\lambda ( s , A ) \\le b \\} , d = \\sup \\{ s \\mid \\lambda ( s , A ) \\ge a \\} . \\end{align*}"}
-{"id": "2302.png", "formula": "\\begin{align*} \\mu ( ( T ^ { q } ) ^ { h _ { N } } ( A ^ { * } _ { N } ) \\Delta A ^ { * } _ { N } ) & < 2 \\mu ( A ^ { * } _ { N } \\Delta B ^ { * } _ { N } ) + \\mu ( B ^ { * } _ { N } \\Delta ( T ^ { q } ) ^ { h _ { N } } ( B ^ { * } _ { N } ) ) \\\\ & \\leq ( q - 1 ) ( q - 2 ) \\frac { 1 } { b ^ { N - 1 } } \\mu ( B ^ { * } _ { N } ) + \\frac { q } { b ^ { N - 1 } } \\mu ( B ^ { * } _ { N } ) \\\\ & = \\frac { ( q ^ { 2 } - 2 q + 2 ) } { b ^ { N - 1 } } \\mu ( B ^ { * } _ { N } ) \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} L _ { d n + k } ( x ; c ) = \\sum _ { j = 0 } ^ { n - 1 } P _ j ( c ) \\left \\{ \\sum _ { r = 0 } ^ { d - 1 } \\frac { P _ { d j + r } ( x ) } { \\left \\langle u _ { r } , P _ { d j + r } P _ j \\right \\rangle } \\right \\} + P _ n ( c ) \\left \\{ \\sum _ { r = 0 } ^ { k } \\frac { P _ { d n + r } ( x ) } { \\left \\langle u _ { r } , P _ { d n + r } P _ n \\right \\rangle } \\right \\} . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} \\Omega ( a ) \\ = \\ \\{ \\omega \\in \\Sigma ^ + \\mid \\exists \\textrm { i n f i n i t e l y m a n y } j \\in \\N : \\ \\omega _ { n ^ x _ j } = 1 \\textrm { i f f } a _ x = 1 \\} \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} - L u + g \\circ u & = f \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} d s ^ 2 = d \\sigma ^ 2 + f \\left ( \\theta , \\sigma \\right ) ^ 2 d \\theta ^ 2 \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} \\frac { h _ { n } \\left ( \\sinh \\left ( \\xi + \\eta \\right ) \\vert q \\right ) } { \\left ( e ^ { 2 \\xi } ; q \\right ) _ { n } } = \\sum _ { k = 0 } ^ { n } \\frac { \\left ( q ^ { - n } ; q \\right ) _ { k } q ^ { k ^ { 2 } / 2 } h _ { k } \\left ( \\sinh \\left ( \\log q ^ { \\left ( n - 1 \\right ) / 2 } - \\eta \\right ) \\vert q \\right ) } { \\left ( - e ^ { \\xi + \\eta } \\right ) ^ { n } \\left ( q , e ^ { 2 \\xi } ; q \\right ) _ { k } \\left ( - q ^ { n / 2 } e ^ { 2 \\xi + \\eta } \\right ) ^ { - k } } . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} T _ k [ X , Y ; u , v ] = L '^ { ( k ) } _ v [ X + u , Y ] - L '^ { ( k ) } _ v [ X , Y ] , \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} \\Gamma ( P + t , \\eta + t ) = \\sum _ { j = 0 } ^ q \\Gamma _ { p - j } ( P , \\eta ) \\frac { t ^ j } { j ! } \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} \\left [ ( - 1 ) ^ { \\ell ( \\sigma _ J ) } \\mathbf { z } ^ { \\sigma _ J ( \\mu _ - ) } \\prod _ { 1 \\leq j \\leq n } \\frac { ( 1 + t z _ j x ) } { ( 1 - z _ j x ) } \\right ] \\Big \\vert _ { \\mathbf { z } ^ \\lambda } & = \\left [ ( - 1 ) ^ { \\ell ( \\sigma _ J ) } \\prod _ { 1 \\leq j \\leq n } \\frac { ( 1 + t z _ j x ) } { ( 1 - z _ j x ) } \\right ] \\Big \\vert _ { \\mathbf { z } ^ { \\lambda - \\sigma _ J ( \\mu _ - ) } } \\\\ & = ( - 1 ) ^ { | J | } ( 1 + t ) ^ { s ( \\lambda ; \\mu ) + 1 - | J | } x ^ { | \\lambda | - | \\mu | } , \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} & ( x _ k + y _ k - 1 ) ( x _ { k + 1 } + y _ { k + 1 } - 1 ) \\\\ & = x _ k x _ { k + 1 } + x _ k y _ { k + 1 } - x _ k + y _ k x _ { k + 1 } + y _ k y _ { k + 1 } - y _ k - x _ { k + 1 } - y _ { k + 1 } + 1 \\\\ & = ( x _ k y _ { k + 1 } - x _ k - y _ { k + 1 } + 1 ) + ( y _ k x _ { k + 1 } - x _ { k + 1 } - y _ k + 1 ) + ( x _ k x _ { k + 1 } + y _ k y _ { k + 1 } - 1 ) \\\\ & = ( x _ k - 1 ) ( y _ { k + 1 } - 1 ) + ( y _ k - 1 ) ( x _ { k + 1 } - 1 ) + ( x _ k x _ { k + 1 } + y _ k y _ { k + 1 } - 1 ) . \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} b _ { j } \\in \\Gamma ( k , p _ { j } ) : = \\left \\{ p _ { j } v _ { k } + a : a \\in \\Gamma ( k ) \\right \\} . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} L ( u , \\chi ) = \\prod _ { j = 1 } ^ { \\deg Q - 1 } ( 1 - \\alpha _ { j } ( \\chi ) u ) \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} \\left \\Vert \\varphi \\right \\Vert _ { B _ { 2 } } \\leq C \\left \\Vert \\left \\{ a _ { j } \\right \\} _ { j = 1 } ^ { J } \\right \\Vert _ { \\ell ^ { 2 } \\left ( \\mu \\right ) } . \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} \\begin{aligned} & \\sum _ { m = - \\infty } ^ { \\infty } ( q ^ { - \\ell } ; q ) _ { m } w ^ { m } q ^ { \\ell | m | / 2 } S _ { \\ell } \\left ( - q ^ { - | m | - \\ell } ; q \\right ) \\\\ & = \\frac { \\left ( 1 - q ^ { 3 \\ell / 2 } / w \\right ) } { \\left ( q , w / q ^ { \\ell / 2 } ; q \\right ) _ { \\ell } } \\left ( q ^ { 1 + \\ell / 2 } / w ; q \\right ) _ { \\infty } . \\end{aligned} \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\| x ( t ) \\| ^ 2 = \\| x ( 0 ) \\| ^ 2 + \\int _ 0 ^ t \\langle u ( s ) , u ( s ) \\rangle d s - \\int 0 ^ t \\langle \\tilde { y } ( s ) , \\tilde { y } ( s ) \\rangle d s . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i & = \\big ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } - B _ { i J K j } \\dot { u } _ { j } ) _ { , K } \\big ) _ { , J } \\\\ a \\ddot { \\tau } & = - \\beta _ { K i } \\dot { u } _ { i , K } + m _ { I J } \\dot { \\tau } _ { , I J } + M _ { j L K I } u _ { j , L K I } + K _ { I J } \\tau _ { , I J } . \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} \\# I ( A ; h ) = q ^ { h + 1 } : = H \\ ; . \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} w _ \\sigma ( \\eta ) = \\sum _ { k = 0 } ^ \\infty \\ , a _ { k , \\sigma } \\ , \\eta ^ { k + \\sigma } \\ , , \\sigma = 0 , 1 \\ , , \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} \\beta _ b ( \\tau ) = \\limsup _ { r \\to 1 } \\frac { 1 } { | \\log ( 1 - r ) | } \\cdot \\log \\int _ { | z | = r } \\bigl | e ^ { \\tau b ( z ) } \\bigr | \\ , | d z | , \\tau \\in \\mathbb { C } . \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} a _ 1 ( x ) R ^ + ( x ; t ) + b _ 1 ( x ) R ^ - ( x ; t ) & = F _ 1 ( u ( x , t ) ) , a _ 2 ( x ) R ^ + ( x ; t ) + b _ 2 ( x ) R ^ - ( x ; t ) & = F _ 2 ( u ( x , t ) ) , \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} \\tilde { Q } [ \\psi ] & = - \\frac { C h ' } { W ^ 3 } - \\frac { h ' H } { W } - b \\\\ & < - \\abs { b } \\left ( \\frac { h ' } { W ^ 3 } + \\frac { h ' } { W } - 1 \\right ) \\\\ & \\le 0 . \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } \\bar { u } ( t , x ) = a ^ { i j } \\bar { u } _ { x ^ { i } x ^ { j } } ( t , x ) + a ^ { i j } ( t ) v _ { x ^ { i } x ^ { j } } ( t , x ) - \\Delta v ( t , x ) + f ( t , x ) . \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\lambda _ x = 0 , \\lambda _ t - \\lambda ^ n \\lambda _ y = 0 . \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} V = 0 , M = \\alpha , N ( u ) = 0 \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { L } _ { 0 } } { \\partial P _ { \\alpha \\beta } } = Q ^ { \\beta \\alpha } , \\qquad \\frac { \\partial \\mathcal { L } _ { 0 } } { \\partial P _ { \\alpha \\beta } ^ { \\ast } } = \\overset { \\ast } { \\left . Q ^ { \\beta \\alpha } \\right . } \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } + V _ s \\cdot \\nabla _ { X _ s } \\right ) f _ \\infty ^ { ( s ) } ( t , Z _ s ) = \\ell ^ { - 1 } C _ { s + 1 } ^ 0 f _ \\infty ^ { ( s + 1 ) } ( t , Z _ s ) \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\mathcal { L } ( \\alpha _ i , \\lambda ) = & - \\alpha _ i \\log _ 2 \\left ( 1 + \\frac { 1 - \\alpha _ i } { \\alpha _ i } \\zeta _ i \\right ) + \\lambda \\left ( \\alpha _ i - \\Psi _ i \\right ) \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} 8 \\pi \\frac { \\partial } { \\partial x _ { q } } \\left ( \\widetilde { T } _ { p q } - T _ { p q } \\right ) = \\left [ \\operatorname { c u r l } \\left ( \\mathbf { E \\times D } + \\mathbf { H } \\times \\mathbf { B } \\right ) \\right ] _ { p } . \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} \\tilde { \\varphi } _ N ( x + \\i y ) = 0 \\ \\forall y \\in \\R \\ \\mathrm { w h e n e v e r } \\ \\varphi ( x ) = 0 ; \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} A _ \\alpha ^ * = \\begin{pmatrix} 0 & 0 \\\\ 0 & \\delta _ \\alpha \\end{pmatrix} , B _ 1 ^ * = C _ 1 ^ * = \\begin{pmatrix} 0 & I / \\sqrt { 2 } & 0 \\\\ 0 & 0 & \\sqrt { D } \\end{pmatrix} , \\alpha \\leq 4 , \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} ( E _ \\pm ( 0 ) - P _ \\pm ) u [ x ] = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( x - y ) \\cdot \\eta } r ( x , y , \\eta ) u \\left [ y \\right ] d \\eta , \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} \\theta _ { s , a ^ 2 } ^ 2 = \\bar { r } ^ 2 ( s , a ^ 2 ) + \\beta \\sum _ { s ' \\in S } p ^ 2 ( s ' | s , a ^ 2 ) u ^ { 2 * } ( s ' ) - u ^ { 2 * } ( s ) , \\ \\forall \\ s \\in S , a ^ 2 \\in A ^ 2 ( s ) , a ^ 2 \\neq a _ s ^ 2 . \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\ , | f ( x ) | ^ 2 \\ , d x = \\sum _ { i = 1 } ^ { N } \\ , \\int _ { \\mathbb { R } ^ d } \\ , | \\hat F _ i ( \\lambda ) | ^ 2 \\ , d \\lambda \\hat { f } ( \\xi ) = \\epsilon ^ { d / 2 } \\ , \\sum _ { i = 1 } ^ { N } \\ , \\hat { F _ i } ( \\epsilon \\xi ) \\ , e ^ { - 2 \\pi i x _ i \\cdot \\xi } . \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} u _ { \\epsilon } = G ( u ) , \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} A ^ - ( \\xi ) = \\left ( \\begin{matrix} A _ { 1 1 } & 0 \\\\ A ^ - _ { 2 1 } ( \\xi ^ { - r } ) & A _ { 2 2 } \\end{matrix} \\right ) , \\ \\ B ^ - ( \\xi ) = \\left ( \\begin{matrix} B _ { 1 1 } & 0 \\\\ B _ { 2 1 } ( \\xi ^ { - r } ) & B _ { 2 2 } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} F _ { i , j } ( a , b ) = F ( a e _ { j } + b e _ { i } ) . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\left ( a \\cdot \\imath ( a _ 1 ) \\mid \\imath ( a _ 2 ) \\right ) _ { B } & = \\Phi _ 0 ( ( a a _ 1 ) ^ * a _ 2 ) \\\\ & = \\Phi _ 0 ( a _ 1 a ^ * a _ 2 ) = \\left ( \\imath ( a _ 1 ) \\mid a ^ * \\cdot \\imath ( a _ 2 ) \\right ) _ { B } \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} Z ( u _ 1 ) \\in \\partial S _ 1 \\setminus \\{ 0 \\} , \\ Z ( u _ 2 ) \\in \\partial S _ 2 \\setminus \\{ 0 \\} , \\ \\hbox { a n d } \\ Z ( u ) = 0 . \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} & N T ^ { K N ( K N + K M ) } + \\min \\{ ( K - 1 ) N , M \\} ( T + 1 ) ^ { K N ( K N + K M ) } + M N ( T _ { \\sf d } + 1 ) ^ { K M ( K N + K M ) } \\\\ & = N T ^ { K N ( K N + K M ) } + \\min \\{ ( K - 1 ) N , M \\} ( T + 1 ) ^ { K N ( K N + K M ) } + N \\frac { \\lambda _ 2 } { \\lambda _ 1 } T ^ { K N ( K N + K M ) } \\\\ & \\leq \\frac { M } { \\lambda _ 1 } ( T + 1 ) ^ { K N ( K M + K N ) } , \\end{align*}"}
-{"id": "10075.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 4 , 4 u + 2 , 4 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 4 , 4 u + 2 , 4 v + 3 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "8012.png", "formula": "\\begin{align*} y _ { s , k } = \\mathbf { h } _ { s , k } ^ { H } \\mathbf { x } + n _ { s a , k } , ~ k = 1 , 2 , . . . , K \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} L ^ { p - 1 } _ { s p } ( \\R ^ n ) : = \\Big \\{ f \\in L _ { \\rm l o c } ^ { p - 1 } ( \\R ^ n ) \\ , : \\ , { \\rm T a i l } ( f ; 0 , 1 ) < \\infty \\Big \\} . \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} \\nabla _ x q ( s , t ; y ( s , t ) , \\xi ) \\nabla _ x y ( s , t ) = I . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} & ( T _ 4 ) _ { i j } x ^ i x ^ j \\\\ = & - \\frac { n - 6 } { 1 2 ( n - 1 ) } | W ( p ) | ^ 2 r ^ 2 + \\frac { 3 2 } { 9 ( n - 4 ) ( n - 2 ) } \\sum _ { k , l , s } [ ( W _ { i k l s } ( p ) + W _ { i l k s } ( p ) ) x ^ i ] ^ 2 \\\\ & - \\frac { 4 } { 3 ( n - 4 ) ( n - 2 ) ( n - 1 ) } | W ( p ) | ^ 2 r ^ 2 + \\frac { 1 6 ( 7 n - 8 ) } { ( n - 4 ) ( n - 2 ) } \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j + O ( r ^ 3 ) , \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} ( x ^ 2 + x ) ^ { m } = ( x + 1 ) ^ { m } x ^ { m } = \\sum \\limits _ { i = 0 } ^ m { m \\choose i } x ^ { m + i } \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} \\mathcal { L } _ { 1 } & = - \\mathcal { L } _ { 1 } ^ { \\ast } = P _ { \\sigma \\tau } ^ { \\ast } Q ^ { \\tau \\sigma } = \\frac { 1 } { 2 } \\left ( P _ { \\sigma \\tau } ^ { \\ast } Q ^ { \\tau \\sigma } - P _ { \\sigma \\tau } \\overset { \\ast } { \\left . Q ^ { \\tau \\sigma } \\right . } \\right ) \\\\ & = \\frac { i } { 2 } e ^ { \\sigma \\tau \\kappa \\rho } P _ { \\sigma \\tau } P _ { \\kappa \\rho } ^ { \\ast } = 4 i \\left ( \\mathbf { E } \\cdot \\mathbf { B - H } \\cdot \\mathbf { D } \\right ) . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} { \\mathcal L } _ j : = \\{ \\lambda \\in { \\mathcal Q } _ { k - 1 } : r _ \\lambda = j \\} , \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} \\mathcal { M } ( l , \\hat { l } ) : = \\frac { 1 } { n } \\min _ { \\pi } \\left | \\left \\{ i : l _ i \\neq \\pi ( \\hat { l } _ i ) \\right \\} \\right | . \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\sigma _ E ^ { n _ i } ( \\beta ^ { - 1 } ( \\sigma _ F ^ { | \\mu _ i | } ( \\mu _ i e _ { w _ 1 } ^ \\infty ) ) ) = \\sigma _ E ^ { m _ i } ( \\beta ^ { - 1 } ( \\mu _ i e _ { w _ 1 } ^ \\infty ) ) \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} \\sum _ { \\substack { I \\in S _ \\phi \\\\ I \\neq X } } \\mu ( I ) ( y _ { I ^ \\star } ) ^ p = \\sum _ { I \\in S _ \\phi } ( \\mu ( I ) - a _ I ) y _ I ^ p , \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} E _ 0 \\simeq E _ { ( k ) } . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} | h _ m | \\leq \\sup _ { \\partial _ p U _ m } | h _ m | = \\sup _ { \\partial _ p U _ m } | u _ k | \\leq M \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} J ( \\lambda ) : = \\frac { W _ 1 \\lambda + W _ 2 } { A _ 1 ( 1 + \\eta ) ^ { k _ 0 } } < 1 \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} \\det ( \\bar P _ k ) = \\det \\left [ \\begin{array} { c c c c c c c c c c } 1 & 2 & \\cdots & \\lfloor \\frac { i } { p } \\rfloor + i & \\cdots & k p - p - 1 \\\\ k & k + 1 & \\cdots & k + i - 1 & \\cdots & k p - p \\end{array} \\right ] _ { \\bar P _ k } . \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} \\left ( c _ 1 \\mathrm { I } + | Q | r \\right ) ( s ) = c _ 1 s + | Q | r ( s ) s \\ge 0 . \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} c _ n ( \\alpha ) \\ ; : = \\ ; \\sum _ { k = 1 } ^ { n } \\frac { ( - 1 ) ^ { n - k } k ^ { n - \\alpha } } { k ! ( n - k ) ! } \\ , . \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} A _ { q } \\left ( z \\right ) = \\left ( z q / a ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } S _ { n } \\left ( a q ^ { - n } ; q \\right ) } { \\left ( z q / a ; q \\right ) _ { n } } \\left ( \\frac { z } { a } \\right ) ^ { n } . \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} \\mathcal { D } _ N = \\left \\{ \\left . Z _ N \\in \\mathbb { R } ^ { 2 d N } \\right | \\ ; \\forall 1 \\leq i < j \\leq N , \\ ; | x _ i - x _ j | > 1 \\right \\} \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} \\partial f = \\sum _ { j = 1 } ^ { + \\infty } \\langle f , \\phi _ j \\rangle \\partial \\phi _ j . \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} { \\rm t r } ( T _ 4 ) = & - \\tfrac { 3 n ^ 3 - 1 2 n ^ 2 - 3 6 n + 6 4 } { 4 } \\sigma _ 1 ( A ) ^ 2 + 4 ( n ^ 2 - 4 n - 1 2 ) | A | ^ 2 + n ( n - 6 ) \\Delta \\sigma _ 1 ( A ) \\\\ = & - \\frac { n ( n - 6 ) } { 1 2 ( n - 1 ) } | W ( p ) | ^ 2 + O ( r ) . \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} \\partial _ t g _ { i j } ( t , x ) = - 2 R _ { i j } . \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} B = Q ^ { ( 4 ) } ( A _ { 2 } , A _ { 1 } , \\bar { A } ) + Q ^ { ( 4 ) } ( A , A _ { 2 } , \\bar { A } ) . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} u _ i = \\frac { u _ i ^ 0 } { \\nu _ { m } } , \\dot { u } _ i = \\frac { \\dot { u } _ i ^ 0 } { \\nu _ { m } } , \\tau = \\frac { \\tau ^ 0 } { \\nu _ { m } } , \\dot { \\tau } = \\frac { \\dot { \\tau } ^ 0 } { \\nu _ { m } } , q _ i = \\frac { q ^ 0 _ i } { \\nu _ { m } } \\Omega \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} k = f _ { x x } f _ { y y } - f _ { x y } ^ 2 , \\end{align*}"}
-{"id": "9898.png", "formula": "\\begin{align*} \\| \\nabla _ { \\partial \\Omega } \\nu ( b ) \\| = \\| \\nabla _ { \\partial \\Omega } \\tau ( b ) \\| \\le \\varkappa \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} \\widetilde { \\omega } _ 2 ( \\pi ) = \\prod _ { i = 1 } ^ { \\nu ( \\pi ) } ( 2 \\lambda _ i - 2 \\lambda _ { i + 1 } - 1 ) . \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\begin{array} { r @ { \\ = \\ } l } u _ t + \\lambda _ 0 \\left [ ( u + V ) \\cdot \\nabla \\right ] u + ( M + \\beta | u | ^ 2 ) u - \\Gamma _ 0 \\Delta u + \\Gamma _ 2 \\Delta ^ 2 u + \\nabla q & f + N ( u ) , \\\\ { \\mathrm { d i v \\ , } } u & 0 , \\\\ u ( 0 ) & u _ 0 . \\end{array} \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} \\Phi ( x , y , \\xi ; t ) : = \\varphi _ a ( x , \\xi ) - t h _ 0 ( \\xi ) - \\varphi _ a ( y , \\xi ) . \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} n ^ + ( G ) + n ^ + ( \\overline { G } ) = 2 + f ( G ) + f ( \\overline { G } ) = 2 + f ( G ) + g ( G ) = n + 1 . \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} W ' = \\{ ( i , j ) : 1 \\leq i \\leq r , \\ ; 1 \\leq j \\leq l ( \\mu ^ { ( i ) } ) \\} . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} \\left | \\alpha _ { k , i } - \\frac { 1 } { k p _ i } \\right | \\leq \\frac { 2 } { k ^ 2 p ^ 2 _ { i } } \\frac { k ^ { \\frac { 1 } { 2 } + q } } { N ^ 2 } = \\frac { 2 } { k ^ { \\frac { 3 } { 2 } - q } p ^ 2 _ { i } N ^ 2 } . \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} \\nu _ \\beta ( \\xi ) & = \\sup _ { \\pi } \\sum _ { t = 1 } ^ { \\infty } \\beta ^ t I ( X _ t , S _ { t - 1 } ; Y _ { t } | Q _ { t - 1 } ) . \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} \\left ( \\partial _ t - V _ s \\cdot \\nabla _ { X _ s } \\right ) \\phi _ N ^ { ( s ) } ( t , Z _ s ) = 0 \\left ( Z _ s \\in \\mathcal { D } _ s , \\ ; s = 1 , \\dots , N \\right ) \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} g _ j ( x ) : = \\begin{cases} \\psi _ j ( x ) , & x \\in U , \\\\ [ 0 . 5 e x ] \\min \\{ j , u ( x ) \\} , & x \\in \\R ^ { n } \\setminus U . \\end{cases} \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} ( x \\leftharpoonup ( y _ 1 \\rightharpoonup h _ 1 ) ) \\circ ( y _ 2 \\leftharpoonup h _ 2 ) & = T ( x _ 1 \\circ y _ 1 \\rightharpoonup h _ 1 ) \\circ x _ 2 \\circ y _ 2 \\circ h _ 2 = ( x \\circ y ) \\leftharpoonup h . \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} a ^ * _ { 0 , 1 } = \\frac { 1 - \\mu _ R } { N _ T } , a ^ * _ { N _ R , 0 } = \\mu _ R , \\textrm { a n d o t h e r s b e i n g 0 . } \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} [ \\bar { h } _ { 0 , k } , \\bar { h } _ { 0 , l } ] = k ( 2 - d ^ k - d ^ { - k } ) \\delta _ { k , - l } \\bar { c } , \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} & \\Delta _ { 0 } = ( - 1 ) ^ { \\frac { ( n _ { 0 } + 1 ) ^ { 2 } } { 2 } } s _ { n _ { 0 } } ^ { n _ { 0 } + 1 } > 0 \\ , \\\\ & \\Delta _ { k + 1 } = ( - 1 ) ^ { { \\tfrac { ( n _ { k + 1 } - n _ { k } ) ^ { 2 } } { 2 } } } \\left ( s _ { n _ { k + 1 } + n _ { k } + 1 } - s _ { n _ { k + 1 } + n _ { k } + 1 } ^ { ( n _ { k } + 1 ) } \\right ) ^ { n _ { k + 1 } - n _ { k } } t _ { n _ { k } } ^ { 2 } > 0 \\ , \\ 0 \\leq k < m - 1 \\ , \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} \\frac { d ^ n p _ { N } } { d u ^ n } \\left ( u \\right ) = \\frac { ( - i ) ^ n } { 2 \\pi } \\int \\omega ^ n \\phi _ { N } ( \\omega ) e ^ { - i \\omega u } \\ , d \\omega , \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} H _ { j _ i } ( s ) & : = \\left ( 2 ^ { j _ i } X _ 3 \\right ) ^ s \\int _ \\frac 1 2 ^ 2 \\ ! u _ 0 ( \\eta ) \\eta ^ { s - 1 } \\ , d \\eta , j _ i \\geq 1 , \\\\ H _ 0 ( s ) & : = \\zeta ( 1 - s ) { d _ i } ^ s \\frac { \\psi _ { - s } ( c _ i ) } { \\psi _ 0 ( c _ i ) } - \\frac { { X _ 3 } ^ s } s \\int _ 1 ^ 2 \\ ! u _ 0 ' ( \\eta ) \\eta ^ s \\ , d \\eta , \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} p _ t ( x , y ) = 1 + \\sum _ { j = 1 } ^ { + \\infty } e ^ { - \\lambda _ j t } \\phi _ j ( x ) \\phi _ j ( y ) . \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\frac { g ^ { k } _ { \\beta } } { K } \\leq g \\leq K g ^ { k } _ { \\beta } \\ ( \\textrm { q u a s i - i s o m e t r i c } ) , \\ g ^ { k } _ { \\beta } = \\Sigma _ { j = 1 } ^ { k } \\frac { \\beta _ { j } ^ { 2 } } { | z _ { j } | ^ { 2 - 2 \\beta _ { j } } } d z _ { j } \\otimes d \\bar { z } _ { j } + \\Sigma _ { j = k + 1 } ^ { n } d z _ { j } \\otimes d \\bar { z } _ { j } . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} f _ { A B } { } ^ { C } & = C _ { A B } { } ^ { C } , \\\\ f _ { A B } { } ^ { C + m } & = 0 , \\\\ f _ { A ( B + m ) } { } ^ { C } = f _ { ( A + m ) B } { } ^ { C } & = 0 , \\\\ f _ { A ( B + m ) } { } ^ { C + m } = f _ { ( A + m ) B } ^ { C + m } & = C _ { A B } { } ^ { C } , \\\\ f _ { ( A + m ) ( B + m ) } { } ^ { C } & = - C _ { A B } { } ^ { C } , \\\\ f _ { ( A + m ) ( B + m ) } { } ^ { C + m } & = 0 . \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( b ; q \\right ) _ { n } S _ { n } \\left ( x ; q \\right ) z ^ { n } } { \\left ( c ; q \\right ) _ { n } } = \\frac { \\left ( c / b , b z ; q \\right ) _ { \\infty } } { \\left ( c , z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( b ; q \\right ) _ { n } \\left ( \\frac { c } { b } \\right ) ^ { n } } { \\left ( b z ; q \\right ) _ { n } } S _ { n } \\left ( \\frac { x b z } { c } \\right ) , \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 1 } \\left ( a , b ; c ; q , z \\right ) = \\frac { \\left ( a z ; q \\right ) _ { \\infty } } { \\left ( z ; q \\right ) _ { \\infty } } { } _ { 2 } \\phi _ { 2 } \\left ( a , c / b ; c , a z ; q , b z \\right ) , \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} \\partial e ^ { t \\Delta } f = e ^ { t \\vec { \\Delta } } \\partial f , t \\ge 0 . \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l l l l } $ u ( 0 , t ) = 0 $ & $ u ( L , t ) = 0 $ & $ u _ { x } ( L , t ) = 0 $ , & i n $ ( 0 , T ) $ , \\\\ $ v ( 0 , t ) = 0 , $ & $ v ( L , t ) = 0 , $ & $ v _ { x } ( L , t ) = g _ 2 ( t ) $ , & i n $ ( 0 , T ) $ . \\end{tabular} \\right . \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} \\varphi ( m ) = \\overline { \\varphi } ( m ) , \\ \\ \\ \\ { \\rm f o r \\ \\ a n y } \\ m \\in M . \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} \\overline { d } _ { \\mathbb { Z } / 2 } ( X ) = \\underline { d } _ { \\mathbb { Z } / 2 } ( X ) \\pm 1 , \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} \\left \\{ ( z , a _ { 0 } , \\dots , a _ { n - 2 } ) \\mid z ^ { n } + a _ { n - 2 } z ^ { n - 2 } + \\dots a _ { 0 } = 0 \\right \\} \\to ( a _ { 0 } , \\dots , a _ { n - 2 } ) \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} \\widetilde G ( s ) : = \\sup _ { r \\geq 0 } \\ , \\big ( s r - G ( r ) \\big ) . \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} \\log _ { 1 / p } ( 1 + ( p / q ) ^ s ) = \\frac { ( p / q ) ^ s } { \\log ( 1 / p ) } + O ( ( p / q ) ^ 2 ) , \\end{align*}"}
-{"id": "9941.png", "formula": "\\begin{align*} V = \\bigoplus _ { p } V _ { p } . \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} H u = \\lambda u , u \\in D ( H ) . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} \\delta = \\frac { 1 } { 1 0 ^ 8 n ^ 2 \\log ^ { 2 } N } . \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} \\rho ^ { ( - 1 ) } _ t = & ( a ^ { - 1 } ) _ b D ^ b F , \\\\ \\rho ^ { ( 1 ) } _ t = & 2 P ^ { ( 1 ) } u _ { b + 1 } D ^ { b + 1 } F + P ^ { ( 1 ) } _ b u _ { b + 1 } ^ 2 D ^ b F , \\\\ \\rho ^ { ( 3 ) } _ t = & 2 P ^ { ( 3 ) } u _ { b + 2 } D ^ { b + 2 } F + 4 Q ^ { ( 3 ) } u _ { b + 1 } ^ 3 D ^ { b + 1 } F \\\\ & \\quad \\quad + ( P ^ { ( 3 ) } _ b u _ { b + 2 } ^ 2 + Q ^ { ( 3 ) } _ b u _ { b + 1 } ^ 4 ) D ^ { b } F . \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} F ( x _ k ) = F _ t ( x _ k ; x _ k ) & \\ge F _ t ( x _ { k + 1 } ; x _ k ) + \\tfrac { t } { 2 } \\norm { \\mathcal { G } _ t ( x _ k ) } ^ 2 \\ge F ( x _ { k + 1 } ) + \\tfrac { t } { 2 } \\norm { \\mathcal { G } _ t ( x _ k ) } ^ 2 . \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} \\big \\lVert \\omega _ { 1 , \\infty } \\big \\rVert ^ 2 _ { Z _ { 1 , 0 } } + \\big \\lVert \\omega _ { 2 , \\infty } \\big \\rVert ^ 2 _ { Z _ { 2 , 0 } } = 1 , D ^ F _ { Z _ { j , \\infty } } \\omega _ { j , \\infty } = \\lambda _ \\infty \\omega _ { j , \\infty } , \\omega _ { j , \\infty } ^ \\mathrm { z m } = 0 , j = 1 , 2 . \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} \\pi = { \\bf 1 } \\oplus \\bigoplus _ { n = 0 } ^ { \\infty } \\pi _ n ^ { \\perp } , \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} T _ 1 = \\left ( \\begin{array} { c | c | c | c } I _ { s - 1 } & & & \\\\ \\hline - \\mathbf a '' & 1 & & \\\\ \\hline & & 1 & \\\\ \\hline & & & I _ { n - s } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} h ( t ) : = \\frac { f ( t ) } { ( 2 \\pi ) ^ { N } } = \\int _ { \\R ^ { N } } \\left ( e ^ { - t } \\delta _ 0 + \\psi ( t , x ) \\right ) u _ 0 ( x ) d x = e ^ { - t } u _ 0 ( 0 ) + \\int _ { \\R ^ N } \\psi ( t , x ) u _ 0 ( x ) d x , \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} 0 & \\to \\left ( \\mathcal O _ { H _ { i _ 1 \\cdots i _ { l - 1 } } } / \\mathcal J ( \\varphi ) | _ { H _ { i _ 1 \\cdots i _ { l - 1 } } } \\right ) \\otimes \\mathcal O _ { H _ { i _ 1 \\cdots i _ { l - 1 } } } ( - H _ { i _ l } ) \\\\ & \\to \\mathcal O _ { H _ { i _ 1 \\cdots i _ { l - 1 } } } / \\mathcal J ( \\varphi ) | _ { H _ { i _ 1 \\cdots i _ { l - 1 } } } \\to \\mathcal O _ { H _ { i _ 1 \\cdots i _ l } } / \\mathcal J ( \\varphi ) | _ { H _ { i _ 1 \\cdots i _ l } } \\to 0 \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x & = & \\dfrac { P _ { x } } { Q } + \\varepsilon \\phi \\\\ \\varepsilon Q & \\cong & 0 \\end{array} \\right . \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} d _ X ( F ( x , y ) , x ) & \\leq d _ X ( F ( x , y ) , F ^ { n + 1 } ( x _ 0 , y _ 0 ) ) + d _ X ( F ^ { n + 1 } ( x _ 0 , y _ 0 ) , x ) \\\\ & = d _ X ( F ( x , y ) , F ( F ^ n ( x _ 0 , y _ 0 ) , G ^ n ( y _ 0 , x _ 0 ) ) ) + d _ X ( F ^ { n + 1 } ( x _ 0 , y _ 0 ) , x ) \\\\ & \\leq k \\ d _ X ( x , F ( ( F ^ n ( x _ 0 , y _ 0 ) , G ^ n ( y _ 0 , x _ 0 ) ) ) + l \\ d _ X ( F ^ n ( x _ 0 , y _ 0 ) , F ( x , y ) ) \\\\ & \\ \\ \\ + d _ X ( F ^ { n + 1 } ( x _ 0 , y _ 0 ) , x ) \\\\ & = k \\ d _ X ( x , F ^ { n + 1 } ( x _ 0 , y _ 0 ) ) + l \\ d _ X ( F ^ n ( x _ 0 , y _ 0 ) , F ( x , y ) ) \\\\ & \\ \\ \\ + d _ X ( F ^ { n + 1 } ( x _ 0 , y _ 0 ) , x ) \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} v _ i = \\frac { u _ i ^ 0 } { \\nu _ { m } } , \\dot { v } _ i = \\frac { \\dot { u } _ i ^ 0 } { \\nu _ { m } } , \\omega = \\frac { \\tau ^ 0 } { \\nu _ { m } } , \\dot { \\omega } = \\frac { \\dot { \\tau } ^ 0 } { \\nu _ { m } } , r _ i = \\frac { q ^ 0 _ i } { \\nu _ { m } } \\Omega \\times \\{ 0 \\} . \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\mathbf { m } \\cdot e _ 1 \\wedge \\ldots \\wedge e _ k = e _ { i _ 1 } \\wedge \\ldots \\wedge e _ { i _ k } + . \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} g \\cdot y = ( \\sigma _ { a _ 0 } , \\sigma _ { a _ 1 } , \\sigma _ { a _ 2 } ) \\cdot [ y _ 0 : y _ 1 : y _ 2 ] = [ \\sigma _ { a _ 0 } y _ 0 : \\sigma _ { a _ 1 } y _ 1 : \\sigma _ { a _ 2 } y _ 2 ] . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{gather*} \\colon \\psi _ { a } ^ { + } ( z ) \\psi _ { b } ^ { - } ( w ) \\colon = \\psi _ { a } ^ { + } ( z ) \\psi _ { b } ^ { - } ( w ) . \\end{gather*}"}
-{"id": "1163.png", "formula": "\\begin{align*} X _ * V _ 0 = U _ 1 U ^ { \\top } _ 1 X _ * V _ 0 + U _ 1 U ^ { \\top } _ 1 H V _ 0 - H V _ 0 = U _ * S _ * V ^ { \\top } _ * V _ 0 = U _ * \\begin{bmatrix} \\widehat { \\Psi } & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} \\tilde { Q } [ w + \\varphi ] \\le 0 = \\tilde { Q } [ u ] , \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} S _ { m , n } ( c , \\eta ) = ( 1 - \\eta ^ 2 ) ^ { m / 2 } \\ , w ( \\eta ) \\ , , \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} F _ \\gamma ( \\omega ( T + \\cdot ) ) = F _ \\gamma ( \\omega ) ( T + \\cdot ) - F _ \\gamma ( \\omega ) ( T ) . \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 2 } \\ [ \\bar { x } ^ { \\pm } _ { i , r _ { \\pi ( 1 ) } } , [ \\bar { x } ^ { \\pm } _ { i , r _ { \\pi ( 2 ) } } , \\bar { x } ^ { \\pm } _ { i \\pm 1 , s } ] ] = 0 \\ \\mathrm { a n d } \\ [ \\bar { x } ^ { \\pm } _ { i , r } , \\bar { x } ^ { \\pm } _ { j , s } ] = 0 \\ \\mathrm { f o r } \\ j \\ne i , i \\pm 1 . \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\dim \\delta H _ 3 ^ g = & \\ ; 6 + 9 k + ( k - 1 ) k + { 3 ( k - 1 ) ^ 2 } + ( k - 2 ) ( k - 1 ) ^ 2 / 2 \\\\ & \\ ; - \\left ( { ( k + 1 ) ^ 2 ( k + 2 ) / 2 + k + 3 } \\right ) \\\\ = & \\ ; k + 4 , \\\\ \\dim \\delta E _ 3 ^ g = & \\ ; I _ M ( V _ 3 ^ g \\times W _ 3 ^ g ) = \\ ; k + 2 , \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} Z = \\bigcup _ { \\substack { 1 \\leq r \\leq s \\\\ \\alpha _ 1 + \\dots + \\alpha _ r = n \\\\ 1 \\leq i _ 1 < \\dots < i _ r \\leq n } } Z _ { r ; \\alpha _ 1 , \\dots , \\alpha _ r } ^ { i _ 1 , \\dots , i _ r } , \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} \\frac { d ^ { 2 } } { d t ^ { 2 } } ( [ A ^ { \\ast } ( t ) A ( t ) ] ^ { - 1 } ) | _ { t = 0 } = D . \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} I _ n ( f ) = I _ n ( \\widetilde { f } ) \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} & a ( x ) + x b ( x ) - 2 x ( 1 - 2 x ) \\frac { d } { d x } a ( x ) = 0 , \\\\ & 2 x b ( x ) + a ( x ) - 2 x ( 1 - 2 x ) \\frac { d } { d x } b ( x ) = 0 . \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\left [ J _ { i } , J _ { j } \\right ] = \\varepsilon _ { i j } ^ { k } J _ { k } , \\quad \\left [ J _ { i } , K _ { j } \\right ] = \\varepsilon _ { i j } ^ { k } K _ { k } , \\quad \\left [ K _ { i } , K _ { j } \\right ] = - \\varepsilon _ { i j } ^ { k } J _ { k } . \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} \\lim _ { \\| \\eta \\| \\to \\infty } \\Psi _ { x , y } ( \\eta ) = \\infty . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} \\omega _ 1 ^ { K [ G ] } = \\omega _ 1 ^ K \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} \\theta _ { C , R } ( s ) = - \\sum _ { \\lambda \\in \\Lambda ^ * _ R ( C ) } \\lambda ^ { - 2 s } . \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} \\tau = \\frac { N _ R ( 1 - \\mu _ R ) } { \\min \\{ N _ R , N _ R \\mu _ R + N _ T \\mu _ T \\} } , \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{gather*} \\big ( \\tau _ k ^ { ( \\alpha ) } \\big ) ^ 2 = \\tau _ { k } ^ { ( \\alpha - 1 ) } \\tau _ { k } ^ { ( \\alpha + 1 ) } - \\tau _ { k + 1 } ^ { ( \\alpha - 1 ) } \\tau _ { k - 1 } ^ { ( \\alpha + 1 ) } , \\alpha \\in \\mathbb { Z } , k = 0 , 1 , \\dots . \\end{gather*}"}
-{"id": "6719.png", "formula": "\\begin{align*} g h : = \\lim _ { j \\to \\infty } S ^ j g S ^ j h , \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\sum _ { m = k } ^ { n } \\binom { n } { m } \\sum _ { j = 0 } ^ { m } ( - 1 ) ^ { j } \\binom { m } { j } \\left ( \\frac { s } { s + n - m + j } \\right ) ^ { r } & = \\sum _ { j = 0 } ^ { r - 1 } ( - 1 ) ^ { j } \\frac { s ^ { j } } { j ! } f ^ { ( j ) } _ { n , k } ( s ) \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{gather*} D _ { n _ { k } + 1 } = 0 = t _ { n _ { k } + 1 } \\ , . . . \\ , D _ { n _ { k + 1 } - 1 } = 0 = t _ { n _ { k + 1 } - 1 } \\ , \\end{gather*}"}
-{"id": "2994.png", "formula": "\\begin{align*} \\Phi _ { s , t } : = \\Phi _ { 0 , t } \\circ \\Phi ^ { - 1 } _ { 0 , s } \\qquad \\hbox { f o r a l l } s , t \\in J , \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} F ( \\vec { X } ) : = \\dfrac { 1 } { 2 } \\| \\vec { Y - X } \\| _ F ^ 2 & + \\lambda _ 0 \\sum _ { i = 1 } ^ { m } \\phi ( \\sigma _ i ( \\vec { X } ) ; a _ 0 ) \\\\ & + \\lambda _ 1 \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } \\phi ( \\vec { X } _ { i , j } ; a _ 1 ) , \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} V _ { \\infty } : = \\liminf _ { \\abs { x } \\to + \\infty } \\abs { V ( x ) } \\ , , \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & i \\partial _ t \\Psi _ 1 + \\left ( - \\Delta \\right ) ^ \\alpha \\Psi _ 1 = f _ 1 \\left ( \\Psi _ 1 \\right ) + \\partial _ 1 F ( \\Psi _ 1 , \\Psi _ 2 ) , \\ x \\in \\mathbb { R } ^ N , \\ t > 0 , \\\\ & i \\partial _ t \\Psi _ 2 + \\left ( - \\Delta \\right ) ^ \\alpha \\Psi _ 2 = f _ 2 \\left ( \\Psi _ 2 \\right ) + \\partial _ 2 F ( \\Psi _ 1 , \\Psi _ 2 ) , \\ x \\in \\mathbb { R } ^ N , \\ t > 0 , \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} A ' & = \\{ x \\in \\textstyle \\prod X _ { n , k } \\colon ( \\exists m ) ( \\forall n > m ) ( \\forall j ) ( \\exists k > j ) ( x ( n , k ) \\in A _ { n , k } ) \\} , \\\\ B ' & = \\{ x \\in \\textstyle \\prod X _ { n , k } \\colon ( \\exists m ) ( \\forall n > m ) ( \\forall j ) ( \\exists k > j ) ( x ( n , k ) \\in B _ { n , k } ) \\} . \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} \\det ( t I + B '' ) & = ( t + \\varepsilon _ 2 ) \\det ( ( t + \\varepsilon _ 2 ) I + ( 1 + ( k + 1 ) \\varepsilon _ 2 ) B ) , \\\\ & = \\sum _ { i = 0 } ^ n ( t + \\varepsilon _ 2 ) ^ { n + 1 - i } I _ i ( B ) ( 1 + ( k + 1 ) \\varepsilon _ 2 ) ^ i , \\\\ & = \\sum _ { m = 0 } ^ { n + 1 } t ^ { n + 1 - m } \\sum _ { i = 0 } ^ { n } \\varepsilon _ 2 ^ { m - i } { n + 1 - i \\choose n + 1 - m } I _ i ( B ) ( 1 + ( k + 1 ) \\varepsilon _ 2 ) ^ i , \\\\ \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} S ( t , t _ 0 ) \\phi _ 1 ( x ) \\le S ( t , t _ 0 ) ( \\tilde \\phi _ 2 + \\epsilon ) ( x ) = S ( t , t _ 0 ) ( \\phi _ 2 + \\epsilon ) ( x ) = S ( t , t _ 0 ) \\phi _ 2 ( x ) + \\epsilon . \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} M _ \\lambda \\ ; v = 0 \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} \\begin{aligned} d _ X ( F ( x , y ) , F ( u , v ) ) \\leq a \\ d _ X ( x , F ( u , v ) ) + & b \\ d _ X ( u , F ( x , y ) ) + c \\ d _ X ( x , u ) ; \\\\ & \\forall x \\geq _ { P _ 1 } u , \\ y \\leq _ { P _ 2 } v ; \\ 2 b + c < 1 \\end{aligned} \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i Q _ i ^ { m - 1 } n ^ { m - i } \\sum \\limits _ { k = 0 } ^ n k ^ { m + i - 1 } f _ n ( k ) = 0 . \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} b _ { l , j } = a _ { l , j } + n _ { l } v _ { k } , a _ { l , j } \\in \\Gamma ( k ) . \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} \\mathbf { Q } _ 5 = \\left ( \\begin{array} { c c } \\underline { a } _ 1 & 0 \\\\ \\underline { a } _ 2 & 0 \\\\ 0 & \\underline { b } _ 1 \\\\ 0 & \\underline { b } _ 2 \\end{array} \\right ) , \\mathbf { Q } _ 6 = \\left ( \\begin{array} { c c } a _ 1 + \\underline { a } _ 1 & 0 \\\\ a _ 2 + \\underline { a } _ 2 & 0 \\\\ 0 & b _ 1 + \\underline { b } _ 1 \\\\ 0 & b _ 2 + \\underline { b } _ 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "10176.png", "formula": "\\begin{align*} \\quad \\lim _ { n \\rightarrow + \\infty } n ^ { \\frac 1 4 } \\mathbb P ( T _ 0 ^ { ( 1 ) } > n ) = \\frac 3 2 K _ p \\ , \\mathbb E \\left [ \\sup _ { t \\in [ 0 , 1 ] } \\Delta ^ { ( 0 ) } _ t \\right ] . \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} o ( 1 ) = & \\int _ M | \\nabla \\Delta ( u - u _ k ) | _ g ^ 2 d \\mu _ g - \\int _ M f | u - u _ k | ^ { 2 ^ \\sharp } d \\mu _ g \\\\ \\geq & \\int _ M | \\nabla \\Delta ( u - u _ k ) | _ g ^ 2 d \\mu _ g \\Big [ 1 - ( \\max _ M f ) ( 1 + \\epsilon ) ^ { \\frac { 2 ^ \\sharp } { 2 } } Y _ 6 ( S ^ n ) ^ { - \\frac { 2 ^ \\sharp } { 2 } } \\big ( \\int _ M | \\nabla \\Delta ( u - u _ k ) | _ g ^ 2 d \\mu _ g \\big ) ^ { \\frac { 6 } { n - 6 } } \\Big ] . \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} c _ { k o } = c _ { o } ^ { k } = \\left ( p _ { x _ { 1 } , x _ { 2 } } \\cdots p _ { x _ { p } , x _ { p + 1 } } \\right ) ^ { k } \\quad \\textrm { f o r } 0 \\leq k \\cdot o < k _ { 0 } \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\| u \\| _ { H _ \\sigma ( \\Omega ) } : = \\bigg ( \\int _ \\Omega ( \\Delta u ) ^ 2 - 2 ( 1 - \\sigma ) \\int _ \\Omega d e t ( \\nabla ^ 2 u ) \\bigg ) ^ { \\frac { 1 } { 2 } } \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} \\nu ^ { \\pi ^ * } _ { n - 1 } ( 0 | 0 ) \\equiv { { c } _ 0 ( n - 1 ) = \\frac { 1 } { 1 + 2 ^ { \\mu _ 0 ( n - 1 ) + \\Delta { C } _ n } } } , ~ \\mu _ 0 ( n - 1 ) \\triangleq \\frac { H ( \\gamma _ { n - 1 } ) - H ( \\alpha _ { n - 1 } ) } { \\gamma _ { n - 1 } - \\alpha _ { n - 1 } } . \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} ( \\mathcal { T } x ^ { k - 1 } ) _ { i } = \\sum _ { i _ { 2 } , \\cdots , i _ { k } = 1 } ^ { n } \\mathcal { T } _ { i i _ { 2 } \\cdots i _ { k } } x _ { i _ { 2 } } \\cdots x _ { i _ { k } } , ( i = 1 , \\cdots , n ) . \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} G _ j ^ { \\ast } \\Psi _ j ^ { \\ast } A _ j = \\Psi _ 1 ^ { \\ast } A _ 1 + i \\delta \\xi _ j + i \\nu _ j . \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} f _ \\ell ( \\Sigma ) = f _ \\ell ( \\Delta ) = \\mathcal { O } ( n ^ { \\ell + 1 - 3 ^ { - \\ell } } ) . \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} u ^ { 1 * } ( s ) = \\bar { r } ^ 1 ( s , a _ s ^ 1 ) + \\beta \\sum _ { s ' \\in S } p ^ 1 ( s ' | s , a _ s ^ 1 ) u ^ { 1 * } ( s ' ) , \\ \\forall \\ s \\in S , \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} \\int _ U \\Big ( \\partial _ s ( g _ { a b } ) \\partial _ r ( g ^ { r s } \\sqrt { g } \\ , \\eta ) - f _ { a b } \\sqrt { g } \\ , \\eta \\Big ) d x = 0 , \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} \\phi _ { k } ^ { \\pm , \\varepsilon } ( I _ { \\pm } ) & = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\phi ( x \\left ( I _ { \\pm } , \\theta _ { \\pm } \\right ) ) e ^ { - i k \\theta _ { \\pm } } d \\theta _ { \\pm } , \\\\ \\Psi _ { k } ^ { \\pm , \\varepsilon } ( I _ { \\pm } ) & = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } \\Psi ( x \\left ( I _ { \\pm } , \\theta _ { \\pm } \\right ) ) e ^ { - i k \\theta _ { \\pm } } d \\theta _ { \\pm } . \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} | \\tau | = \\sum _ { i = 1 } ^ r k _ i | \\lambda ^ { ( i ) } | . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{gather*} \\tau _ { k } ^ { ( \\alpha ) } = \\big \\langle T ^ { k } v _ { 0 } , g _ { C } ^ { ( \\alpha ) } v _ { 0 } \\big \\rangle , \\end{gather*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u _ t ( t , x ) + L ^ b u ( t , x ) + f ( t , x , u ( t , x ) , \\nabla u ( t , x ) ) = 0 , \\\\ u ( T , x ) = \\Phi ( x ) , \\\\ \\forall ( t , x ) \\in [ 0 , T ] \\times \\mathbb R ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { t ^ 2 - ( c \\pm a \\ , t ^ 2 ) ^ 2 } , a = c o n s t \\neq 0 , c = c o n s t , \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} A B \\ , \\leq \\ , \\frac { \\delta ^ r } { r } \\ , A ^ r + \\frac { \\delta ^ { - r ' } } { r ' } \\ B ^ { r ' } , A , B > 0 , \\frac 1 r + \\frac { 1 } { r ' } = 1 , \\delta > 0 , \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} \\sum _ k \\left | \\omega _ k \\cdot \\left ( v _ { j _ k } ( t _ k ^ - ) - v _ { i _ k } ( t _ k ^ - ) \\right ) \\right | \\leq 2 \\lambda ^ { - 1 } \\sum _ { i = 1 } ^ N \\left ( \\lambda ^ 2 | x _ i | ^ 2 + | v _ i | ^ 2 \\right ) \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} \\lambda _ i ( x ) = \\sum \\limits _ { j = 1 } ^ { n _ 0 } l _ { i j } x _ j + l _ { i ( n _ 0 + 1 ) } , ~ i = 1 , \\dots , ( n _ 0 + 1 ) . \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} \\tau ' ( p ) = \\frac { 1 } { \\tau ' ( q ) } . \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} \\widetilde { Q } _ { \\sigma , \\lambda } [ u , v ] = \\int _ { M } \\ , \\nabla \\overline { u } \\cdot \\nabla v \\ , d x - \\lambda \\int _ M \\ , U \\overline { u } v \\ , d x + \\int _ { \\partial M } \\sigma \\ , \\overline { u } \\ , v \\ , d S \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ { s _ i } \\lambda _ j ^ i \\cdot F _ \\phi ( P _ j ^ i ) = 0 . \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial \\rho } = \\frac { 1 } { 2 } \\zeta ' ( 0 , D ) - \\rho \\zeta ( 0 , D ) \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} C _ { t , r } \\left ( 1 \\right ) = C _ { t , t } \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} U _ s ( y ) = U _ s '' ( a ) \\frac { ( y - a ) ^ 2 } { 2 } , U _ s '' ( a ) < 0 , \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} x \\frac { d B ( x ) } { d x } + B ( x ) A ( x ) = A ( q x ) B ( x ) . \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} x ^ * = { \\rm p r o x } _ { \\gamma P } ( x ^ * - \\gamma \\nabla f ( x ^ * ) ) \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} D ^ F _ { Z _ { R _ i } } \\omega _ i = \\lambda _ i \\omega _ i , \\end{align*}"}
-{"id": "9869.png", "formula": "\\begin{align*} & \\langle 0 \\mid e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast _ { k + 1 / 2 } \\mid - 1 \\rangle = \\langle 0 \\mid e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast ( z ) \\mid - 1 \\rangle \\mid _ { z ^ { k } } \\\\ & = \\left [ e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\langle 0 \\mid \\psi ^ \\ast ( z ) \\mid - 1 \\rangle \\right ] \\mid _ { z ^ { k } } = \\left [ e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\right ] \\mid _ { z ^ { k + 1 } } = : h _ { k + 1 } [ \\boldsymbol { t } ] . \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} \\tau & = \\sum _ { k \\in \\omega } \\omega ^ { \\beta _ k } \\cdot p _ k \\ge \\omega ^ { \\widetilde { \\alpha } } \\cdot \\sup \\{ m _ n : n \\in \\omega \\} \\\\ & = \\omega ^ { \\widetilde { \\alpha } } \\cdot \\omega = \\omega ^ { \\widetilde { \\alpha } + 1 } = \\omega ^ \\alpha . \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} [ \\bar { h } _ { i , k } , \\bar { h } _ { j , l } ] = k a _ { i , j } d ^ { - k m _ { i , j } } \\delta _ { k , - l } \\bar { c } , \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} G = ( q _ \\alpha ( \\beta _ { k } v _ { i } t , \\beta _ { m } v _ { j } t ) ) _ { { \\tiny \\substack { 1 \\leq k , m \\leq n \\\\ 1 \\leq i , j \\leq 4 } } } . \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\big | \\lambda _ k ( R _ { \\epsilon , t } ) \\big | \\Big | _ { t = 0 } \\geqslant - R ^ { - 1 } \\big | \\lambda _ k ( R ) \\big | , k = 1 , \\cdots , m . \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} \\sum _ k ( S ^ * ) ^ b _ { i j ; k } \\ , \\omega ^ k = d ( S ^ * ) ^ b _ { i j } - \\sum _ t ( S ^ * ) ^ b _ { t j } \\ , \\theta ^ t _ i - \\sum _ t ( S ^ * ) ^ b _ { i t } \\ , \\theta ^ t _ j , \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} \\| u _ 1 - u _ 2 \\| _ { L ^ 1 ( \\Omega ) } + & \\| g \\circ u _ 1 - g \\circ u _ 2 \\| _ { L ^ 1 ( \\Omega , \\rho ) } \\leq \\\\ & C ( \\| f _ 1 - f _ 2 \\| _ { L ^ 1 ( \\Omega , \\rho ) } + \\| \\eta _ 1 - \\eta _ 2 \\| _ { L ^ 1 ( \\partial \\Omega ) } ) \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} \\tau = \\textrm { a r c s i n h } \\ , x ^ 0 , \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} z ^ 2 I + b b ^ { t r } = I / 2 . \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} g _ s ( x ) = ( x - 2 ) ^ { - r - 2 s } x ^ { 2 s } , 0 \\le s \\le - k - 1 \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} \\Gamma ( x ; y ) = \\Gamma ( y ; x ) \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\nabla _ r X _ { i r } - ( \\partial _ r f ) X _ { i r } = \\frac { 1 } { 2 } \\partial _ { i } S , \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{gather*} \\mathcal { F } ^ { - 1 } \\Delta _ 0 ( h _ 1 h _ 2 ) \\mathcal { F } = \\mathcal { F } ^ { - 1 } \\Delta _ 0 ( h _ 1 ) \\mathcal { F } \\mathcal { F } ^ { - 1 } \\Delta _ 0 ( h _ 2 ) \\mathcal { F } = \\Delta _ H ( h _ 1 ) \\Delta _ H ( h _ 2 ) = \\Delta _ H ( h _ 1 h _ 2 ) , \\end{gather*}"}
-{"id": "6646.png", "formula": "\\begin{align*} \\beta ^ { \\kappa } _ { M , M - 1 } ( \\kappa \\ , a , \\kappa \\ , b , \\kappa \\ , \\bar { b } ) \\overset { { \\rm i n \\ , l a w } } { = } \\beta _ { M , M - 1 } ( a , \\ , b , \\bar { b } ) . \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} u '' + a ( t ) u ^ { p } = 0 , \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} y = \\frac { 1 } { a } x ^ { ( 1 - a ) } - ( \\log x ) ^ { 2 } \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} \\frac { d } { d t } [ \\log H ( t ) ] = \\frac { \\dot { H } ( t ) } { H ( t ) } = \\frac { 2 D ( t ) } { H ( t ) } - \\dot { F } , \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} \\partial ^ { \\alpha } _ t u = a ^ { i j } u _ { x ^ i x ^ j } + f \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} x ^ q y ^ { ( p - 1 ) q } = a x ^ { p q } + b y ^ { p q } + c x y ( x - y ) . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} h _ \\alpha k _ \\alpha ^ { t r } = 0 , h _ \\alpha h _ \\alpha ^ { t r } = I / 2 , \\alpha \\leq 4 , \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ 3 } : = & \\Big ( \\int _ { M } ( | \\nabla \\Delta u | _ g ^ 2 d \\mu _ g + | \\nabla ^ 2 u | _ g ^ 2 + | \\nabla u | _ g ^ 2 + u ^ 2 ) d \\mu _ g \\Big ) ^ { \\frac { 1 } { 2 } } \\\\ \\approx & \\Big ( \\int _ { M } ( | \\nabla ^ 3 u | _ g ^ 2 d \\mu _ g + | \\nabla ^ 2 u | _ g ^ 2 + | \\nabla u | _ g ^ 2 + u ^ 2 ) d \\mu _ g \\Big ) ^ { \\frac { 1 } { 2 } } , ~ ~ u \\in H ^ 3 ( M , g ) . \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} \\sum _ { 1 \\le a < q \\atop ( a , q ) = 1 } \\alpha _ a \\gamma ( \\Omega , a , q ) = 0 \\phantom { m } \\phantom { m } \\sum _ { 1 \\le b < r \\atop ( b , r ) = 1 } \\beta _ b \\gamma ( \\Omega , b , r ) = 0 . \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} a _ { r s } = \\cos ( ( r - l ) \\frac { { 2 k \\pi } } { l } ) \\cos ( ( s - l ) \\frac { { 2 k \\pi } } { l } ) + \\sin ( ( r - l ) \\frac { { 2 k \\pi } } { l } ) \\sin ( ( s - l ) \\frac { { 2 k \\pi } } { l } ) \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\lambda _ i ( \\tau _ { P y } ) = \\lambda _ { \\sigma ( i ) } ( \\tau _ y ) , \\ i = 1 , \\cdots , N . \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} 0 = \\big \\langle \\omega , \\mu _ 1 ^ \\mathrm { p p } \\big \\rangle _ { Z _ { 1 , \\infty } } = \\big \\langle \\omega , \\mu _ 1 ^ \\mathrm { p p } \\big \\rangle _ { Z _ { 1 , R } } + \\big \\langle \\omega , \\mu _ 1 ^ \\mathrm { p p } \\big \\rangle _ { Y _ { [ R , \\infty ) } } . \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} \\partial _ { t } f + v \\partial _ { x } f - E \\partial _ { v } f = 0 , \\ \\ \\ \\ \\ \\ \\ \\ E _ { x } = - \\int _ { - \\infty } ^ { + \\infty } f \\ d v + 1 , \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} \\| A ^ { 1 - \\nu } X B ^ { \\nu } \\| _ { 2 } ^ { 2 } + r _ { 0 } ^ { 2 } \\| A X - X B \\| _ { 2 } ^ { 2 } & + \\sum _ { k = 1 } ^ { \\infty } r _ { k } \\| A ^ { 1 - \\frac { m _ { k } } { 2 ^ { k } } } X B ^ { \\frac { m _ { k } } { 2 ^ { k } } } - A ^ { 1 - \\frac { m _ { k } + 1 } { 2 ^ { k } } } X B ^ { \\frac { m _ { k } + 1 } { 2 ^ { k } } } \\| _ { 2 } ^ { 2 } \\\\ & \\leqslant \\| ( 1 - \\nu ) A X - \\nu X B \\| _ { 2 } ^ { 2 } . \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} - \\dfrac { \\tilde { A } '' ( \\tilde { t } ) } { \\tilde { A } ' ( \\tilde { t } ) } = \\frac { 1 } { 2 } \\tilde { R } _ { \\min } . \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\sin \\left ( \\frac { u _ { 1 2 } - u _ 1 - u _ 2 + u } { 4 } \\right ) = \\frac { p q } { 4 } \\sin \\left ( \\frac { u _ { 1 2 } + u _ 1 + u _ 2 + u } { 4 } \\right ) . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} 2 \\frac { ( d - 1 ) ( d - 2 ) } { 2 } - 2 = a _ 0 a _ 1 a _ 2 ( 2 g _ C - 2 ) + b ( \\pi ) . \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} ( \\min _ { j } \\beta _ { j } ^ { 2 } ) g _ { E } \\leq g ^ { k } _ { \\beta } \\leq g _ { E } , \\ g _ { E } = \\Sigma _ { j = 1 } ^ { k } ( d r _ { j } ^ 2 + r _ { j } ^ 2 d \\theta _ { j } ^ 2 ) + \\Sigma _ { j = k + 1 } ^ { n } d z _ { j } \\otimes d \\bar { z } _ { j } . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} x ^ p + y ^ q = z ^ r , 1 / p + 1 / q + 1 / r < 1 , x , y , z \\in \\Z , x y z \\ne 0 , \\gcd ( x , y , z ) = 1 , \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} D = \\begin{pmatrix} \\alpha \\\\ & 1 \\\\ & & \\alpha \\end{pmatrix} \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} { \\overleftarrow { P } } _ { 0 , n } ( d x ^ n | y ^ { n - 1 } ) & \\triangleq \\otimes _ { t = 0 } ^ n { p } _ { t } ( d x _ t | x ^ { t - 1 } , y ^ { y - 1 } ) \\in { \\cal M } ( { \\cal X } ^ n ) \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} \\tilde \\Lambda _ { 1 1 } = \\left [ \\begin{array} { c c } Q _ 1 & Q _ 2 \\end{array} \\right ] \\left [ \\begin{array} { c c } \\Delta _ 1 & 0 \\\\ 0 & \\Delta _ 2 \\end{array} \\right ] \\left [ \\begin{array} { c c } Q _ 1 & Q _ 2 \\end{array} \\right ] ^ H \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} Z \\ ; : \\ ; f ( x _ 0 , \\dotsc , x _ { n + 1 } ) = q ( x _ 0 , \\dotsc , x _ { n + 1 } ) = l ( x _ 0 , \\dotsc , x _ { n + 1 } ) = 0 \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} \\mathbf { \\bar { U } _ n ^ { * } } = \\mathbf { \\bar { M } _ { 0 n } ^ * } \\mathbf { \\bar { S } _ n ^ * } + ( \\mathbf { { \\bar { S } ^ { * ' } _ n } } \\mathbf { \\bar { M } _ { 1 n } ^ * } \\mathbf { \\bar { S } _ n ^ * } , \\ldots , \\mathbf { { \\bar { S } ^ { * ' } _ n } } \\mathbf { \\bar { M } _ { p n } ^ * \\bar { S } _ n ^ * } ) ' \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ { \\gamma } ( z , \\overline { z } ) = \\frac { ( - 1 ) ^ { m } ( \\gamma + m + 1 ) _ { n } m ! } { 2 \\pi i } \\left ( 1 - \\mid { z } \\mid ^ { 2 } \\right ) ^ { - \\gamma } \\oint _ { \\mid t \\mid = 1 } \\dfrac { t ^ { n } } { ( t - z ) ^ { m + 1 } } \\left ( \\sum _ { j = 0 } ^ { + \\infty } \\dfrac { ( - \\gamma - m ) _ { j } } { j ! } { t ^ { j } \\overline { z } ^ { j } } \\right ) d t , \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} \\omega _ 1 ( \\pi ) : = \\frac { \\lambda _ { \\nu ( \\pi ) } + \\delta _ { \\lambda _ { \\nu ( \\pi ) } , o } } { 2 } \\cdot \\prod _ { i = 1 } ^ { \\nu ( \\pi ) - 1 } \\frac { \\lambda _ i - \\lambda _ { i + 1 } - \\delta _ { \\lambda _ i , e } - \\delta _ { \\lambda _ { i + 1 } , e } } { 2 } \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} \\| \\Delta ^ 2 u \\| _ { W ^ { - 2 , q } ( \\Omega ) } = \\sup _ { 0 \\neq \\varphi \\in W ^ { 2 , q ' } _ 0 ( \\Omega ) } \\dfrac { \\bigg | \\int _ \\Omega g ( x ) | u | ^ { p - 1 } u \\varphi \\bigg | } { \\| \\varphi \\| _ { W ^ { 2 , q ' } _ 0 ( \\Omega ) } } \\leq C ( p , q , \\Omega ) \\| g \\| _ 1 \\| u \\| _ { H ^ 2 ( \\Omega ) } ^ p , \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} \\prod _ { k = 1 } ^ K x _ k + \\prod _ { k = 1 } ^ K y _ k - 1 \\leq \\prod _ { k = 1 } ^ K ( x _ k + y _ k - 1 ) . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } { \\rm I _ 2 } = 0 \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} S ( g ; \\phi ) = \\frac { T } { 2 } \\int _ \\Sigma d v \\ , \\ , \\gamma ^ { \\mu \\nu } \\partial _ \\mu \\phi ^ i ( x ) \\partial _ \\nu \\phi ^ j ( x ) g _ { i j } ( \\phi ( x ) ) \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} { \\bf W } = \\left ( [ { \\bf B } _ 1 ~ { \\bf B } _ 2 ~ { \\bf B } _ 3 ~ { \\bf B } _ 4 ~ { \\bf B } _ 5 ~ { \\bf B } _ 6 ~ { \\bf B } _ 7 ] \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} a ^ \\omega = 1 + k \\omega . \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} T _ { ( A , i + n ) } = - T _ { ( A , i ) } . \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} \\phi \\big ( M ( D _ i , D _ j ) \\big ) & = \\prod _ { k = 1 } ^ K \\sum _ { \\pi \\in N _ { o } ^ { ( k ) } } \\kappa _ { \\pi } [ a _ { k ; i } ^ { n _ 1 } , a _ { k ; j } ^ { n _ 2 } , \\cdots , a _ { k ; i } ^ { n _ { 2 t - 1 } } , a _ { k ; j } ^ { n _ { 2 t } } ] . \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} \\begin{array} { c } \\displaystyle \\sum \\limits _ { i = 1 } ^ { n } \\left [ \\langle { \\mathbf { g } _ { i } ( \\mathbf { x } ^ { * } ) , \\mathbf { y } _ { i } - \\mathbf { x } ^ { * } _ { i } } \\rangle + h _ { i } ( \\mathbf { y } _ { i } ) - h _ { i } ( \\mathbf { x } ^ { * } _ { i } ) \\right ] \\geq 0 \\\\ \\displaystyle \\forall \\mathbf { y } _ { i } \\in X _ { i } , \\mbox { f o r } \\ i = 1 , \\dots , n . \\end{array} \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} \\frac { \\partial g _ { i j } } { \\partial t } = - R _ { i j } \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\lambda _ { \\alpha _ { 1 } } \\cdots \\lambda _ { \\alpha _ { r } } = \\lambda _ { \\rho ( \\alpha _ { 1 } , \\ldots , \\alpha _ { r } ) } = K _ { \\alpha _ { 1 } \\ldots \\alpha _ { r } } ^ { \\gamma } \\lambda _ { \\gamma } . \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { T _ 1 } ^ t \\left < \\Phi ( \\tau ) , C _ 2 f _ N ^ { ( 2 ) } ( \\tau ) \\right > d \\tau = \\\\ & \\int _ { T _ 1 } ^ t \\int d \\tau d \\omega d x _ 1 d v _ 1 d v _ 2 \\ ; \\omega \\cdot ( v _ 2 - v _ 1 ) \\Phi ( \\tau , x _ 1 , v _ 1 ) f _ N ^ { ( 2 ) } ( \\tau , x _ 1 , v _ 1 , x _ 1 + \\varepsilon \\omega , v _ 2 ) \\end{aligned} \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} f _ \\ell ( X ^ k , Y ^ k ) = \\left [ X _ 1 ^ { ( Y _ { \\ell 1 } ) } , \\ldots , X _ k ^ { ( Y _ { \\ell k } ) } \\right ] . \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} Y _ { i , k } & = X _ k + Z _ { i , k } , i = 1 , 2 , \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} \\begin{array} { c } e C \\supset \\cup _ { \\alpha \\in J } h ( \\psi _ { \\alpha } d _ { \\alpha } \\psi _ { \\alpha } ^ { - 1 } ) h ^ { - 1 } C = h ( \\cup _ { \\alpha \\in J } \\psi _ { \\alpha } d _ { \\alpha } \\psi _ { \\alpha } ^ { - 1 } ) h ^ { - 1 } C \\ , , \\end{array} \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} \\mathcal { Z } _ N = \\int _ { \\mathbb { R } ^ { 2 d N } } f _ 0 ^ { \\otimes N } ( Z _ N ) \\mathbf { 1 } _ { Z _ N \\in \\mathcal { D } _ N } d Z _ N \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } A _ { q } \\left ( q ^ { \\alpha } z \\right ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( z q ^ { 1 / 2 } e ^ { i x } ; q \\right ) _ { \\infty } \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i \\alpha x \\right ) } { \\sqrt { \\log q ^ { - 1 } } } d x , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} Q _ { d / 2 , i } = | q | ^ { 2 i } \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{gather*} { \\rm { r a n k } } \\ , \\big ( s _ { i + j } ^ { ( r ) } \\big ) _ { i , j = 0 } ^ { \\infty } = r \\ , \\end{gather*}"}
-{"id": "9555.png", "formula": "\\begin{align*} S _ { 2 n + 1 } \\left ( q ^ { - 2 n - 1 } ; q \\right ) = 0 , S _ { 2 n } \\left ( q ^ { - 2 n } ; q \\right ) = \\frac { \\left ( - 1 \\right ) ^ { n } q ^ { n - n ^ { 2 } } } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} \\int _ { \\mathbb S ^ { n - 1 } } e ^ { - 2 \\pi \\theta \\cdot \\xi } d \\theta = & \\frac { 2 \\pi } { | \\xi | ^ { \\frac { n - 2 } 2 } } J _ { \\frac { n - 2 } 2 } ( 2 \\pi | \\xi | ) \\ , , \\\\ \\int _ 0 ^ 1 J _ { \\mu } ( t s ) s ^ { \\mu + 1 } ( 1 - s ^ 2 ) ^ \\nu d s = & \\frac { \\Gamma ( \\nu + 1 ) 2 ^ \\nu } { t ^ { \\nu + 1 } } J _ { \\mu + \\nu + 1 } ( t ) \\ , . \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\mu ( P ) = \\frac { 1 } { ( k - 2 ) ! } \\iint _ P ( y - x ) ^ { k - 2 } \\ , d x \\ , d y . \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\mathrm { R S E } : = \\dfrac { \\| \\vec { X } _ { \\mathrm { e s t } } - \\vec { X } _ { \\mathrm { o r g } } \\| _ F } { \\| \\vec { X } _ { \\mathrm { o r g } } \\| _ F } , \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} y _ i = f ( z _ i , x _ i ) , ~ ~ ~ z _ { i + 1 } = g ( z _ i , x _ i ) , ~ ~ ~ i = 1 , 2 , . . . \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} B \\dot { z } ( t ) + A { z } ( t ) = \\dot { f } ( t ) , \\ae , \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} \\mathbf { U _ n ^ { * } } = \\mathbf { M _ { 0 n } ^ * S _ n ^ * } + ( \\mathbf { { S ^ * _ n } } ' \\mathbf { M _ { 1 n } ^ * S _ n ^ * } , \\ldots , \\mathbf { { S ^ * _ n } } ' \\mathbf { M _ { p n } ^ * S _ n ^ * } ) ' \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { l l } \\dot { u } _ k = \\log \\det ( u _ k ) _ { \\alpha \\bar { \\beta } } + f ( z , t ) \\ ; \\ ; \\ ; & \\mbox { o n } \\ ; \\Omega \\times ( 0 , T ) , \\\\ u _ k = \\varphi _ k & \\mbox { o n } \\ ; \\partial \\Omega \\times [ 0 , T ) , \\\\ u _ k = u _ { 0 , k } & \\mbox { o n } \\ ; \\bar { \\Omega } \\times \\{ 0 \\} . \\\\ \\end{array} \\end{cases} \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{gather*} \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M } ( x ) \\bigr ) ( q \\ , | \\ , b ) - \\bigl ( \\mathcal { S } _ { M - 1 } B ^ { ( f ) } _ { M } ( x ) \\bigr ) ( 0 \\ , | \\ , b ) = \\\\ M ! \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j \\Bigl ( B ^ { ( f ) } _ { 1 } ( q + b _ 0 ) - B ^ { ( f ) } _ { 1 } ( b _ 0 ) \\Bigr ) . \\end{gather*}"}
-{"id": "4446.png", "formula": "\\begin{align*} g _ \\varepsilon ^ { ( s ) } ( t ) = \\tilde { T } _ s ( t ) g _ \\varepsilon ^ { ( s ) } ( 0 ) + \\ell ^ { - 1 } \\int _ 0 ^ t \\tilde { T } _ s ( t - \\tau ) \\tilde { C } _ { s + 1 } g _ \\varepsilon ^ { ( s + 1 ) } ( \\tau ) d \\tau ( s \\geq m - 1 ) \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} \\left \\vert \\mathbb { T } _ { p } ^ { \\left ( 1 \\right ) } \\ \\mathbb { T } _ { p } ^ { \\left ( 2 \\right ) } . . . \\mathbb { T } _ { p } ^ { \\left ( d + 1 \\right ) } \\right \\vert ^ { T } = \\Delta _ { n - d } ^ { - 1 } \\Delta _ { n + p } \\neq 0 . \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} & \\mathbb { E } [ \\| x ^ { k + 1 } - x ^ * \\| ^ 2 \\mid \\mathcal { F } _ k ] \\leq ( 1 + 2 p _ { \\max } \\max _ i L _ i ( \\alpha _ { \\max } - \\alpha _ { \\min } ) ) \\| x ^ k - x ^ * \\| ^ 2 + 4 p _ { \\max } \\alpha _ { \\max } ^ 2 C ^ 2 N \\cr & + 2 p _ { \\max } \\alpha _ { \\max } B \\sum _ { i = 1 } ^ N \\sum _ { j = 1 } ^ N \\mathbb { E } [ W ( k ) ] _ { i j } \\| v ^ k _ j - y ^ k \\| - 2 p _ { \\min } \\alpha _ { \\min } ( \\phi ( x ^ k ) - \\phi ( x ^ * ) ) ^ T ( x ^ k - x ^ * ) , \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} M = \\begin{pmatrix} ( \\bar { a } \\partial _ \\theta - L ) ^ { - 1 } & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "10025.png", "formula": "\\begin{align*} \\sum _ { i = a } ^ t n _ i ( q ^ i - 1 ) = c _ 0 q ^ { d } - 1 , \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} | \\textbf { n } _ 1 | = | \\textbf { n } _ 2 | + 1 , \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} R _ { \\epsilon , t } = \\int _ 0 ^ R \\sqrt { 1 + \\psi \\Big ( \\frac { R - u } { \\epsilon } \\Big ) \\frac { 2 t } { R } } \\ ; d u . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} B '' _ j ( x _ k ) = \\left \\{ \\begin{aligned} & \\frac { - p ^ 2 s } { p h c - s } , & \\textrm { i f } \\ \\ k = j , \\\\ & \\frac { p ^ 2 s } { 2 ( p h c - s ) } , & \\textrm { i f } \\ \\ k = j \\pm 1 , \\\\ & 0 , & \\textrm { i f } \\ \\ k = j \\pm 2 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} \\psi _ { n , m } ( t ) \\cong \\sum _ { i = 0 } ^ { N _ t - 1 } Q _ i b _ i ( t ) , Q _ i = N _ t \\int _ 0 ^ 1 b _ i ( \\xi ) \\psi _ { n , m } ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} g ( x w ) = y h ( w ) \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { u } + \\mathcal { A } ( \\mathbf { H } ( \\mathbf { u } ) ) \\mathbf { u } = 0 ( 0 , T ) , \\mathbf { u } ( 0 , \\cdot ) = \\mathbf { u } ^ { 0 } \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} a ^ * _ { N _ R , 0 } = \\mu _ R , a ^ * _ { 0 , N _ R } = \\frac { 1 - \\mu _ R } { \\binom { N _ T } { N _ R } } , \\textrm { a n d o t h e r s b e i n g 0 } . \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{align*} \\mu \\times \\mu ( \\bigcup _ { i = 1 } ^ { \\infty } ( R \\times S ) ^ { k _ i } A ) > ( 1 - \\epsilon ) \\mu ( \\bigcup _ { i = 1 } ^ { \\infty } R ^ { k _ i } F ) = 1 - \\epsilon . \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} P _ { C , 2 p } & = 1 + \\ell ( 2 p ) t + \\ell ( 4 p ) t ^ 2 + \\ldots = \\frac { 1 - t ^ 4 } { ( 1 - t ) ( 1 - t ) ( 1 - t ^ 2 ) } \\\\ P _ { C , 3 p } & = 1 + \\ell ( 3 p ) t + \\ell ( 6 p ) t ^ 2 + \\ldots = \\frac { 1 - t ^ 3 } { ( 1 - t ) ( 1 - t ) ( 1 - t ) } \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} \\omega _ { \\mu + m , a } ( x ) = a ^ { - m } r _ { m , \\mu } ( a ^ 2 x ) \\omega _ { \\mu , a } ( x ) + a ^ { 1 - m } s _ { m , \\mu } ( a ^ 2 x ) \\omega _ { \\mu + 1 , a } ( x ) \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} h \\cdot ( g _ 1 \\cdot g _ 2 ) = ( h \\cdot g _ 1 ) \\cdot ( h \\cdot g _ 2 ) \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } \\overset { { \\rm i n \\ , l a w } } { = } 2 \\pi \\ , 2 ^ { - \\bigl [ 3 ( 1 + \\tau ) + 2 \\tau ( \\lambda _ 1 + \\lambda _ 2 ) \\bigr ] / \\tau } \\ , \\Gamma \\bigl ( 1 - 1 / \\tau \\bigr ) ^ { - 1 } \\ , L \\ , X _ 1 \\ , X _ 2 \\ , X _ 3 \\ , Y . \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} x ^ 0 ( t , \\xi ) & = \\Theta ( t ) , \\\\ x ^ i ( t , \\xi ) & = x ^ i ( 0 , \\xi ) . \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} \\delta _ m \\left ( \\sum _ { j = 1 } ^ { \\infty } p ^ { ( i ) } _ j t ^ { - j } \\right ) = \\left \\{ \\sum _ { j = 1 } ^ { \\infty } p ^ { ( i ) } _ j t ^ { m - 1 - j } \\right \\} = \\sum _ { j = 1 } ^ { \\infty } p ^ { ( i ) } _ { m - 1 + j } t ^ { - j } . \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} m _ 1 m _ 2 - d & = \\frac { 2 ^ { 9 } - 1 } { 7 } \\frac { 2 ^ { 6 l - 8 } - 1 } { 3 } - \\left \\lceil \\frac { 2 ^ { 6 l + 1 } - 1 } { 2 2 } \\right \\rceil \\\\ & > \\frac { 2 ^ { 6 l + 1 } } { 2 1 \\cdot 2 2 } - \\frac { 2 ^ 9 + 2 ^ { 6 l - 8 } - 1 } { 2 1 } - 1 > 0 , \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} F _ \\Gamma ^ { p - 1 } ( z ) & = \\sum _ { \\ell = 0 } ^ n \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = 0 } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } \\\\ & \\sum _ { \\beta = 0 } ^ { p - j - 2 t } 2 ^ { p - j - 2 t - \\beta } \\binom { n - \\ell } { \\beta } \\binom { \\ell } { p - j - 2 t - \\beta } \\sum _ { \\alpha = 0 } ^ \\beta \\binom { \\beta } { \\alpha } \\sum _ { i = 0 } ^ { j - 1 } G ( z ) , \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} \\int _ { \\mathbb { G } } f ( x ) d x = \\int _ { 0 } ^ { \\infty } \\int _ { \\wp } f ( r y ) r ^ { Q - 1 } d \\sigma ( y ) d r . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\epsilon ^ 1 _ { j 1 } = \\epsilon ^ 2 _ { j 2 } = \\epsilon ^ 3 _ { j 3 } = \\epsilon ^ 4 _ { j 4 } , 1 \\leq j \\leq 4 . \\end{align*}"}
-{"id": "10058.png", "formula": "\\begin{align*} q + r = m + n + l + k < q + r , \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} | V _ { Z _ 1 } ( t , x ; \\xi ) | \\leq \\begin{cases} C t ^ { ( \\gamma - 1 ) \\frac \\alpha 2 } p ( t , x - \\xi ) \\ , , & \\ d = 1 \\ , ; \\\\ C t ^ { \\nu _ 0 \\alpha - 1 } | x - \\xi | ^ { - d + \\gamma - \\nu _ 1 + 2 - \\nu _ 0 } p ( t , x - \\xi ) \\ , , & \\ d \\geq 2 \\\\ \\end{cases} \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c | c } & & \\\\ & D & & b \\\\ & & \\\\ \\hline 0 & \\cdots & 0 & 1 \\end{array} \\right ) \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} \\nabla \\times \\left ( f \\mathbf { A } \\right ) = \\left ( \\nabla f \\right ) \\times \\mathbf { A } + f \\left ( \\nabla \\times \\mathbf { A } \\right ) . \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} S ( u , v ) = ( \\cosh u \\cos v , \\cosh u \\sin v , u ) \\ ; , \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} S = - \\ , \\rho \\ln \\rho = - \\langle \\ln \\rho \\rangle \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} \\| g ^ { - 1 } f ^ * h ( x ) \\| ^ k & = \\| D _ H f ( x ) \\| ^ { 2 k } \\leq K ^ { 2 n / ( n + 1 ) } J ( x , f ) ^ { 2 n / ( n + 1 ) } \\\\ & = K ^ { 2 n / ( n + 1 ) } H J ( x , f ) ^ 2 = K ^ { 2 n / ( n + 1 ) } \\det ( g ^ { - 1 } f ^ * h ( x ) ) . \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} e _ { \\mu \\nu \\sigma \\tau } Q ^ { \\sigma \\tau } = 2 i Q _ { \\mu \\nu } , \\qquad 2 i Q ^ { \\mu \\nu } = e ^ { \\mu \\nu \\sigma \\tau } Q _ { \\sigma \\tau } \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} g _ { n } = ( - 1 ) ^ { n } \\alpha ^ { n } q ^ { - \\frac { 1 } { 2 } n ( n - 1 ) } \\sum _ { k = 0 } ^ { \\infty } \\left ( q ^ { - k } z ^ { - 1 } \\alpha ^ { - 1 } ; q \\right ) _ { n } \\left ( q ^ { k + 1 } z \\alpha ; q \\right ) _ { \\ ! \\infty } \\frac { q ^ { \\frac { 1 } { 2 } k ( k + 1 ) + k n } } { ( q ; q ) _ { k } } ( - 1 ) ^ { k } z ^ { 2 k } \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} | f | _ { \\Phi } = \\Phi ( f ^ * ( 1 ) \\ge f ^ * ( 2 ) \\ge \\dots ) \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} \\Phi = T ^ H \\Phi _ \\mathrm u T , \\Psi = T ^ H \\left [ \\begin{array} { c c } \\epsilon + \\eta & - \\delta \\\\ - \\delta & \\epsilon - \\eta \\end{array} \\right ] T . \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\begin{cases} \\Delta H - H [ | A | ^ { 2 } - { \\rm R i c } ^ N ( \\xi , \\xi ) - ( \\Delta \\lambda ^ 2 ) / \\lambda ^ 2 ] + 4 g ( { \\rm g r a d } \\ln \\lambda ) , { \\rm g r a d } H ) = 0 , \\\\ A \\ , ( { \\rm g r a d } \\ , H ) + H [ { \\rm g r a d } \\ , H - ( { \\rm R i c } ^ N \\ , ( \\xi ) ) ^ { \\top } ] + 2 A \\ , ( { \\rm g r a d } \\ , \\ln \\lambda ) ] = 0 , \\end{cases} \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} E _ \\mathfrak { c } ( z ; s ) = \\delta _ { \\mathfrak { c } \\infty } y ^ s & + \\pi ^ \\frac 1 2 \\frac { \\Gamma \\left ( s - \\frac 1 2 \\right ) } { \\Gamma ( s ) } \\varphi _ { \\mathfrak { c } , 0 } ( s ) y ^ { 1 - s } \\\\ & + 2 y ^ \\frac 1 2 \\frac { \\pi ^ s } { \\Gamma ( s ) } \\sum _ { n \\neq 0 } | n | ^ { s - \\frac 1 2 } \\varphi _ { \\mathfrak { c } , n } ( s ) K _ { s - \\frac 1 2 } ( 2 \\pi | n | y ) e ( n x ) . \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} \\begin{aligned} & \\{ ( \\sigma - \\omega ) ^ { - 2 } : \\omega \\neq \\beta _ 1 \\} , \\\\ & \\{ ( \\sigma + i \\gamma ' - \\omega ) ^ { - 2 } : \\omega \\neq \\beta _ 1 \\} . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} k _ 1 ( c _ i ^ a v ^ i _ b + c _ i ^ d B ^ i _ j v ^ j _ d \\delta _ b ^ a ) + k _ 2 \\delta _ b ^ a ( c _ i ^ d v ^ i _ d + k c _ i ^ d B ^ i _ j v ^ j _ d \\delta _ b ^ a ) = c _ i ^ a v ^ i _ b + c _ i ^ d \\tilde { B } ^ i _ j v ^ j _ d \\delta _ b ^ a . \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} h _ { \\mu , f } \\bigl ( f ^ 0 , g ^ 0 \\bigr ) = \\frac { | A _ N ( e ^ { i \\lambda } ) ( 1 - e ^ { i \\lambda \\mu } ) ^ { n } \\lambda ^ { 2 n } g ^ 0 ( \\lambda ) + \\lambda ^ { 2 n } C ^ { \\mu , 0 } _ { N } ( e ^ { i \\lambda } ) | } { | \\lambda | ^ { n } | 1 - e ^ { i \\lambda \\mu } | ^ { n } p ^ 0 ( \\lambda ) } , \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} S _ 1 ( l , u , v ; N ) = \\frac { 1 } { l ^ { 1 / 2 + u + v } } + 2 \\pi i ^ { 2 k } V _ N ( u , v , k ) , \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} y _ { e , l } = \\sqrt { \\rho _ { e , l } } ( \\mathbf { H } _ { e a , l } ^ { H } \\mathbf { x } + \\mathbf { n _ { e a , l } } ) + \\mathbf { n } _ { e p , l } , ~ \\forall l \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} \\Delta ( \\log p _ t ^ K ( x _ 1 , 1 _ 1 ) ) + \\frac { m } { 2 t } = \\frac { m - 1 } { 2 t } - \\frac { ( n - 2 ) e ^ { x _ 1 / t } } { t ^ 2 ( n - 2 - n e ^ { x _ 1 / t } ) ^ 2 } \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\left ( { v - u } \\right ) \\left ( { w - z } \\right ) F _ { t s } \\left ( { \\eta _ 1 , \\eta _ 2 } \\right ) = F \\left ( { v , w } \\right ) - F \\left ( { v , z } \\right ) - F \\left ( { u , w } \\right ) + F \\left ( { u , z } \\right ) . \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} | u \\circ v | = | u | | v | , \\forall u , v . \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} F = - \\frac { 1 } { \\beta } \\ln Z \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} P _ g u = f u ^ { \\frac { n + 6 } { n - 6 } } , u > 0 \\hbox { ~ ~ i n ~ ~ } M . \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\int _ M f \\varphi ^ { \\frac { 2 n } { n - 6 } } d \\mu _ g = & \\int _ { B _ \\rho } \\Big [ f ( p ) + \\sum _ { k = 2 } ^ 4 \\frac { 1 } { k ! } \\partial _ { i _ 1 \\cdots i _ k } f ( p ) x ^ { i _ 1 } \\cdots x ^ { i _ k } + O ( | x | ^ 5 ) \\Big ] u _ \\epsilon ^ { 2 ^ \\sharp } d x + O ( \\epsilon ^ n ) \\\\ = & f ( p ) \\int _ { \\mathbb { R } ^ n } u _ \\epsilon ^ { \\frac { 2 n } { n - 6 } } d x + \\left \\{ \\begin{array} { l l } O ( \\epsilon ^ 4 ) & \\hbox { ~ ~ i f ~ ~ } n = 1 0 , \\\\ o ( \\epsilon ^ 4 ) & \\hbox { ~ ~ i f ~ ~ } n \\geq 1 1 , \\end{array} \\right . \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\dd N _ t = \\lambda ^ \\top X _ t ( n - N _ t ) \\ , \\dd t + \\dd m _ t . \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} 6 i \\frac { \\partial Q ^ { \\mu \\nu } } { \\partial x ^ { \\nu } } = e ^ { \\mu \\nu \\lambda \\sigma } \\left ( \\frac { \\partial P _ { \\lambda \\sigma } } { \\partial x ^ { \\nu } } + \\frac { \\partial P _ { \\nu \\lambda } } { \\partial x ^ { \\sigma } } + \\frac { \\partial P _ { \\sigma \\nu } } { \\partial x ^ { \\lambda } } \\right ) , \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} P _ { \\mu \\lambda } ^ { \\ast } \\frac { \\partial Q ^ { \\lambda \\nu } } { \\partial x ^ { \\nu } } + P _ { \\mu \\lambda } \\frac { \\partial \\overset { \\ast } { \\left . Q ^ { \\lambda \\nu } \\right . } } { \\partial x ^ { \\nu } } = - \\frac { 8 \\pi } { c } F _ { \\mu \\lambda } j ^ { \\lambda } . \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} P ^ 2 ( g ^ * ) = \\frac { Q ( f ^ * , g ^ * ) } { | | \\mu | | } + I . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} & \\delta _ { \\mathsf { C a - I A } } = \\frac { M + K - 1 } { M } \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} \\frac { d E } { d \\tau } ( \\tau ) & = \\partial _ t h _ \\rho ( \\tau , q ( \\tau , 0 ; x , \\eta ( t , 0 ; x , \\xi ) ) , p ( \\tau , 0 ; x , \\eta ( t , 0 ; x , \\xi ) ) ) \\\\ & = \\partial _ t V _ \\rho ( \\tau , q ( \\tau , 0 ; x , \\eta ( t , 0 ; x , \\xi ) ) ) . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} B v ( t ) = f ( t ) - A u ( t ) , \\ae , \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\mathbb { E } ( L _ { n } ) = n \\left ( \\frac { 3 + \\sqrt { 5 } } { 2 \\sqrt { 5 } } \\binom { n + \\frac { \\sqrt { 5 } } { 2 } - \\frac { 3 } { 2 } } { n } - \\frac { 3 - \\sqrt { 5 } } { 2 \\sqrt { 5 } } \\binom { n - \\frac { \\sqrt { 5 } } { 2 } - \\frac { 3 } { 2 } } { n } \\right ) \\sim \\frac { 1 + \\sqrt { 5 } } { 2 \\sqrt { 5 } } \\frac { n ^ { \\frac { \\sqrt { 5 } - 1 } { 2 } } } { \\Gamma ( \\frac { \\sqrt { 5 } - 1 } { 2 } ) } . \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} \\exists \\tau ( e \\left ( \\tau \\right ) = 0 \\wedge \\forall x ( x \\in A \\leftrightarrow x < \\tau ) . \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} A = { \\rm d i a g } ( \\nu _ { 1 } , \\ldots , \\nu _ { n } ) , \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} \\left \\langle J _ { ( a b , i ) } , P _ { ( c , j ) } \\right \\rangle = \\frac { 1 } { 2 } \\left ( \\alpha _ { 0 } K _ { i j } { } ^ { 0 } + \\alpha _ { 1 } K _ { i j } { } ^ { 1 } + \\alpha _ { 2 } K _ { i j } { } ^ { 2 } + \\alpha _ { 3 } K _ { i j } { } ^ { 3 } \\right ) \\varepsilon _ { a b c } , \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} \\widetilde { \\ ! \\widetilde A \\ , } = A \\ , \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { t ^ 2 - ( c \\pm a t ) ^ 2 } , c = c o n s t , \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} f ^ * = \\begin{cases} f _ 1 & \\ q < \\frac { 2 } { 3 } \\\\ f _ 2 & \\ q > \\frac { 2 } { 3 } \\\\ \\big \\{ ( p , 1 - p ) : 0 \\leq p \\leq 1 \\big \\} & \\ q = \\frac { 2 } { 3 } . \\end{cases} \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} \\frac { f ( a + b ) - f ( b ) - f ( a ) } { f ( a ) } = H ( b ) , \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\mathbf { F } ( \\mathbf { D } ) = \\mathbb { P } _ { \\hat { \\mathbf { D } } ^ { \\perp } } ( \\mathbf { D } ) . \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} W _ t ( \\textbf { x } _ t , f _ t , q _ t ) & = w _ { t + K - 1 } \\mathrm { E } [ ( q _ t - A _ { K - 1 , t } - D _ { t + K - 1 } ) ^ + | f _ t ] \\\\ & = w _ { t + K - 1 } \\sum \\limits _ { u = 1 } ^ { q _ t } \\mathrm { P } ( A _ { K - 1 , t } + D _ { t + K - 1 } < u | f _ t ) \\\\ & = w _ { t + K - 1 } \\sum \\limits _ { u = 1 } ^ { q _ t } R _ { K , t } ( u ) . \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} c _ { j } = 0 \\quad \\textrm { f o r } j \\notin o \\mathbb { N } _ { 0 } , \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} \\mathbf { \\Psi } ( t ) = [ \\psi _ { 0 , 0 } , \\psi _ { 0 , 1 } , \\ldots , \\psi _ { 0 , 2 M } , \\psi _ { 1 , 0 } , \\ldots , \\psi _ { 1 , 2 M } , \\ldots , \\psi _ { 2 ^ k - 1 , 2 M } ] , \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} \\dot { x } = - F \\nu \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} \\left ( \\mu ( x ) - \\mu ( y ) \\right ) \\sum _ { j = m + 1 } ^ { n } g _ { j } ( x ) g _ { j } ( y ) = W _ { n } ( g ( x ) , g ( y ) ) - W _ { m } ( g ( x ) , g ( y ) ) . \\end{align*}"}
-{"id": "10108.png", "formula": "\\begin{align*} C \\ : : \\ : x ^ p y ^ q ( a x + b y + c z ) ^ r - z ^ { p + q + r } = 0 \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} \\begin{array} { c } e C \\supset g _ { \\alpha } d _ { \\alpha } g _ { \\alpha } ^ { - 1 } C = ( U g _ { \\alpha } ) d _ { \\alpha } ( U g _ { \\alpha } ) ^ { - 1 } C = \\\\ ( h \\psi _ { \\alpha } ) d _ { \\alpha } ( h \\psi _ { \\alpha } ) ^ { - 1 } C = h ( \\psi _ { \\alpha } d _ { \\alpha } \\psi _ { \\alpha } ^ { - 1 } ) h ^ { - 1 } C \\ , . \\end{array} \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau ( \\zeta , \\gamma , \\varepsilon ) } \\simeq \\sum _ { j = 1 } ^ { n } \\frac { \\left | a _ { j } \\right | } { \\tau _ { j } ( \\zeta , \\varepsilon ) } . \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} \\lim \\limits _ { n \\to \\infty } \\frac { \\rho ^ { \\Pi _ n } ( p ) } { \\rho ^ { \\Pi _ n } ( q ) } = \\lim \\limits _ { n \\to \\infty } \\frac { ( 1 - ( 1 - p / 2 n ^ 2 ) ) ^ s } { ( 1 - ( 1 - q / 2 n ^ 2 ) ) ^ s } = \\frac { p ^ s } { q ^ s } = \\frac { \\rho ^ { J _ s } ( p ) } { \\rho ^ { J _ s } ( q ) } . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} \\dot { \\tilde { x } } = \\varPhi \\tilde { \\nu } + \\varPhi ^ m \\tilde { h } _ m ^ k \\tilde { x } _ k . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} \\begin{cases} \\varphi _ n ( 0 , \\cdot ) \\rightarrow \\varphi ( 0 , \\cdot ) , \\varphi _ n ( L , \\cdot ) \\rightarrow \\varphi ( L , \\cdot ) L ^ 2 ( 0 , T ) , \\\\ \\psi _ n ( 0 , \\cdot ) \\rightarrow \\psi ( 0 , \\cdot ) , \\psi _ n ( L , \\cdot ) \\rightarrow \\psi ( L , \\cdot ) L ^ 2 ( 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} h ^ { W ^ \\bullet } _ t \\big ( w ( t ) , w ( t ) \\big ) = 1 . \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\langle T _ \\mu ^ \\mu \\rangle = - \\frac { a } { 1 6 \\pi ^ 2 } E _ 4 \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\varepsilon < < 1 , \\ K ( \\lambda , \\varepsilon ) \\ \\lambda = 0 . \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} r ( [ x , h , y ] ) = x s ( [ x , h , y ] ) = y . \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha } \\frac { | t | } { \\eta ^ { 1 + \\frac { 2 } { \\alpha } } } \\left | \\frac { d p _ N } { d u } \\right | _ { u = \\frac { t } { \\sqrt [ \\alpha ] { \\eta } } } \\leq \\frac { 1 } { \\alpha } \\frac { \\kappa _ 1 } { | t | ^ { 1 + \\alpha } } \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} w ( t ) & = { P _ p ( { T - t } ) w ( T ) } + \\int ^ T _ t P _ p ( r - t ) { \\nabla u ( r ) b ( r ) } \\d r \\\\ & = \\int ^ T _ t P _ p ( r - t ) { \\nabla u ( r ) b ( r ) } \\d r . \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} S ( \\alpha ) = \\left \\{ \\beta \\in \\mathcal { T } : \\beta \\geq \\alpha \\right \\} . \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\vec u _ t } { h ( t ) } = \\vec v _ U \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} z _ N = \\frac { 2 t \\Delta z _ 1 - 1 } { t ^ 2 z _ 1 } , \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} F P S ( l / k ) = ( \\sum \\limits _ { t = 0 } ^ { l - 1 } { y _ t \\cos ( 2 \\pi t k / l ) ) ^ 2 + ( } \\sum \\limits _ { t = 0 } ^ { l - 1 } { y _ t \\sin ( 2 \\pi t k / l ) } ) ^ 2 \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} G = \\langle \\sigma _ { 0 } \\rangle \\times \\left ( \\langle \\sigma _ { 1 } , \\sigma _ { 2 } \\rangle \\ltimes \\left ( \\langle \\sigma _ { 3 } \\rangle \\times \\langle \\sigma _ { 4 } \\rangle \\times \\langle \\sigma _ { 5 } \\rangle \\right ) \\right ) \\cong \\mathbb { Z } / 2 \\mathbb { Z } \\times \\left ( S _ { 3 } \\ltimes \\left ( \\mathbb { Z } / 2 \\mathbb { Z } \\right ) ^ { 3 } \\right ) , \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} \\int _ { - \\delta } ^ \\delta e ^ { - i \\lambda u _ 1 } \\phi _ \\lambda d \\lambda = \\int _ { - \\delta } ^ \\delta e ^ { i \\lambda u _ 1 } C _ 1 ( \\lambda ) \\phi _ \\lambda d \\lambda = 0 . \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\tau _ 0 = \\inf \\{ t > 0 : Z ( t ) = 0 \\} . \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} ( \\Lambda _ { N } ( t ) - \\mid \\gamma + t \\mid ^ { 2 } ) ( \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) = \\sum _ { \\gamma _ { 1 } , \\gamma _ { 2 } , . . . , \\gamma _ { m } } \\frac { q _ { \\gamma _ { 1 } } q _ { \\gamma _ { 2 } } . . . q _ { \\gamma _ { m } } ( q \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma + t - \\gamma ( m ) , x \\right \\rangle } ) } { { \\textstyle \\prod \\limits _ { s = 1 , 2 , . . . , m } } [ \\Lambda _ { N } ( t ) - \\left \\vert \\gamma + t - \\gamma ( s ) \\right \\vert ^ { 2 } ] } , \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} P _ s P _ t = \\lim _ { n \\to \\infty } T _ { s n } \\lim _ { n \\to \\infty } T _ { t n } = \\lim _ { n \\to \\infty } T _ { s n } T _ { t n } = \\lim _ { n \\to \\infty } T _ { ( s + t ) n } = P _ { s + t } , \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} \\omega _ { 1 , \\infty } ^ \\mathrm { a c } = \\int _ { - \\delta } ^ \\delta E ( \\phi _ \\lambda , \\lambda ) d \\lambda . \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{gather*} V _ { k } ^ { ( \\alpha ) } \\big ( W _ { k + 1 } ^ { ( \\alpha ) } \\big ) ^ { - 1 } = \\big ( W _ { k + 1 } ^ { ( \\alpha - 1 ) } \\big ) ^ { - 1 } V _ { k + 1 } ^ { ( \\alpha - 1 ) } . \\end{gather*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\begin{array} { l } g ( e _ 1 , e _ 1 ) = g ( e _ 2 , e _ 2 ) = - g ( e _ 3 , e _ 3 ) = - g ( e _ 4 , e _ 4 ) \\\\ \\phantom { g ( e _ 1 , e _ 1 ) } = - g ( e _ 5 , e _ 5 ) = g ( e _ 6 , e _ 6 ) = g ( e _ 7 , e _ 7 ) = 1 , \\\\ g ( e _ i , e _ j ) = 0 , \\ ; i \\neq j . \\end{array} \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} \\Gamma _ { M } ( w \\ , | \\ , a ) = \\Gamma _ { M - 1 } ( w \\ , | \\ , \\hat { a } _ i ) \\ , \\Gamma _ M \\bigl ( w + a _ i \\ , | \\ , a \\bigr ) , \\ , i = 1 \\cdots M , \\ , \\ , M \\in \\mathbb { N } , \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l l } $ u _ t + u _ { x x x } + a v _ { x x x } = 0 $ & i n $ ( 0 , L ) \\times ( 0 , T ) $ , \\\\ $ v _ t + \\frac { r } { c } v _ x + \\frac { a b } { c } u _ { x x x } + \\frac { 1 } { c } v _ { x x x } = 0 $ & i n $ ( 0 , L ) \\times ( 0 , T ) $ , \\\\ $ u ( x , 0 ) = u ^ 0 ( x ) , v ( x , 0 ) = v ^ 0 ( x ) $ , & i n $ ( 0 , L ) $ \\end{tabular} \\right . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} \\big ( \\delta ^ * _ \\lambda ( \\xi ) \\big ) ^ \\beta = \\lambda ^ { | \\beta | _ * } \\ , \\xi ^ \\beta . \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} V _ { i } = \\left ( \\dfrac { T _ { 1 , i } } { l _ { i } t _ { i } } , \\dfrac { T _ { 2 , i } } { l _ { i } t _ { i } } , . . . , \\dfrac { T _ { n + 1 , i } } { l _ { i } t _ { i } } \\right ) ^ { T } ; i = 1 , 2 , . . . , e \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} \\widetilde { F } \\left ( 1 , 0 , \\frac { - 1 } { \\alpha - \\beta } \\right ) = \\widetilde { F } \\left ( \\frac { \\alpha } { \\alpha - \\beta } + 1 , \\frac { \\beta } { \\alpha - \\beta } , \\frac { - 1 } { \\alpha - \\beta } \\right ) \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} \\tilde { F } ' + \\tilde { F } - \\{ F ^ { i j } F _ { ; i j } + F ^ { i j } h _ { i k } h ^ k _ j F + K _ N F ^ { i j } g _ { i j } F \\} \\Theta ^ 2 \\frac { \\sinh \\Theta } { \\cosh \\Theta } = 0 . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} e ^ { - W } = \\int [ D \\phi ] e ^ { - S ( g ; \\phi ) } \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} u _ { \\beta } ^ 2 ( f , g _ 1 ) - u _ { \\beta } ^ 2 ( f , g _ 2 ) = \\beta \\left [ \\frac { p ( 7 + 5 \\beta ) - ( 3 \\beta + 1 ) } { 1 - \\beta ^ 2 } , \\frac { \\beta ( p ( 7 + 5 \\beta ) - ( 3 \\beta + 1 ) ) } { 1 - \\beta ^ 2 } \\right ] ^ T . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} = \\sum \\limits _ { n = 0 } ^ { N - 1 } \\frac { \\Lambda ^ + _ n } { \\sqrt { \\pi } 2 ^ n } t ^ { - n / 2 } \\int \\limits _ { \\mu } ^ { + \\infty } z ^ { - n } e ^ { - ( z - \\eta ) ^ 2 } d z + \\int \\limits _ { \\mu } ^ { + \\infty } R _ N ( z \\sqrt { t } ) e ^ { - ( z - \\eta ) ^ 2 } d z , \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( d v + \\sum _ { j = 1 } ^ N \\frac { 1 } { 2 } ( b _ j + c _ j ) \\frac { \\partial v } { \\partial x _ j } \\right ) d x \\geq 0 , ~ ~ \\forall ~ v \\in C _ c ^ 1 ( \\Omega ) , v \\geq 0 . \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q \\ , | \\ , b ) = \\exp \\Bigl ( \\bigl ( \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j / \\prod \\limits _ { i = 1 } ^ M a _ i \\bigr ) q \\log ( q ) + O ( q ) \\Bigr ) , \\ ; q \\rightarrow \\infty . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} F _ { t , \\varsigma } \\big ( \\sqrt { t } D ^ F _ { Z _ { j , R } } \\big ) ( x , x ) = F _ { t , \\varsigma } \\big ( \\sqrt { t } D ^ F _ { Z _ R } \\big ) ( x , x ) . \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} { \\rm q d i m } [ M ^ \\varepsilon _ { r , s } ] = ( - 1 ) ^ { k ( r + 1 ) } \\frac { \\sin \\left ( \\frac { \\pi k s } { p } \\right ) } { \\sin \\left ( \\frac { \\pi k } { p } \\right ) } , \\ \\ \\ { \\rm q d i m } [ F ^ \\varepsilon _ \\lambda ] = 0 . \\end{align*}"}
-{"id": "10173.png", "formula": "\\begin{align*} \\mathbb P \\left ( \\Delta _ 1 \\le - 2 a , \\ \\max _ { k = 1 , . . . , n } ( \\Delta _ { k + 1 } - \\Delta _ 1 ) \\le a \\right ) \\ge c _ a \\mathbb P \\left ( \\max _ { k = 1 , . . . , n } ( \\Delta _ { k + 1 } - \\Delta _ 1 ) \\le a \\right ) \\ , . \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } F ( t , z ) \\\\ G ( t , z ) \\end{array} \\right ) = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & e ^ { i \\xi _ 0 t } \\end{array} \\right ) \\left ( \\begin{array} { c c } e ^ { i \\nu _ 0 t } & 0 \\\\ 0 & e ^ { - i \\nu _ 0 t } \\end{array} \\right ) \\left ( \\begin{array} { c } F _ 0 ( z ) \\\\ G _ 0 ( z ) \\end{array} \\right ) + \\left ( \\begin{array} { c } \\mu ( t ) \\\\ \\nu ( t ) \\end{array} \\right ) \\ , , \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} Z ( \\mu , \\nu ) : = \\{ ( \\mu x , | \\mu | - | \\nu | , \\nu x ) \\ \\vert \\ x \\in X , r ( \\mu ) = s ( x ) \\} , \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{gather*} s _ { n } ^ { ( r ) } = s _ { n } \\ , \\ 0 \\leq n \\leq 2 r - 1 \\ . \\end{gather*}"}
-{"id": "9832.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a \\ , t ^ 2 ) ^ 2 + t ^ 2 } , a = c o n s t \\neq 0 , c = c o n s t , \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} \\widehat { g _ 1 } ( \\eta ) = \\omega \\cdot \\widehat { f _ 1 } ( \\eta \\omega ) \\eta ^ { d - 1 } / \\sqrt { 1 + \\varepsilon ^ 2 \\eta ^ 2 } . \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} f ( x _ 0 + z ) & = f ( x _ 0 ) + B z + g ( z ) , g ( 0 ) = 0 , \\\\ \\| g ( z ) \\| & = O ( \\| z \\| _ { X ^ \\gamma } ^ s ) , \\quad z \\to 0 X ^ \\gamma , \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} h ( u , v ) = \\frac { \\rho u v } { 1 + ( u ) ^ { + } + \\delta u ^ { 2 } } . \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} T _ { \\psi _ \\lambda } ( x ) = \\lambda \\big ( ( a ^ + ) ^ * x a ^ + + s ^ * b ^ * \\ , x \\ , b s \\big ) + ( 1 - \\lambda ) \\big ( ( b ^ + ) ^ * s \\ , x \\ , s ^ * b ^ + + a ^ * x a \\big ) \\ , . \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} \\delta _ Y = \\mathrm { m i n } \\big \\{ \\abs { \\mu } \\ ; : \\ ; 0 \\neq \\mu \\in \\mathrm { S p } \\big ( D ^ { F } _ { Y } \\big ) \\big \\} . \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} u ^ { \\prime } + \\partial \\varphi _ { 1 } \\left ( u \\right ) - \\partial \\varphi _ { 2 } \\left ( u \\right ) & \\ni f ( u ) \\qquad ( 0 , T ) , \\\\ u ( 0 ) & = u _ { 0 } . \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} \\lim _ { \\Omega \\ni y \\to x } h ( y ) = g ( x ) x \\in \\partial \\Omega h = g \\R ^ n \\setminus \\Omega \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} & q ( t , s ; x , \\xi ) - x - ( t - s ) v ( p ( t , s ) ) \\\\ & = \\int _ s ^ t \\left [ v \\left ( p ( t , s ) + \\int _ \\tau ^ t \\nabla _ x V _ \\rho ( \\sigma , q ( \\sigma , s ) ) d \\sigma \\right ) - v ( p ( t , s ) ) \\right ] d \\tau . \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} A _ j = \\frac 1 { \\sqrt { m } } \\cos ( 2 \\pi j w / F ) , ~ j = 1 , . . . , n , \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} D T = \\begin{pmatrix} I _ { n - 1 } & 0 \\\\ A ( w ) & H ( w ) \\end{pmatrix} , D T ^ { - 1 } = \\begin{pmatrix} I _ { n - 1 } & 0 \\\\ A ( v ) & H ( v ) \\end{pmatrix} \\mbox { i n } U \\setminus P , \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} F _ A \\left ( x , v \\right ) = \\left ( f ( x ) , A ( x ) v \\right ) . \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n = 1 } ^ { \\infty } a _ f ( n ) e ^ { 2 \\pi i n z } \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} u _ i = 0 , u _ { i , J } = 0 , \\tau = 0 \\Gamma \\times ( 0 , \\infty ) \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 n } \\omega ^ i ( \\Phi ) \\omega ^ i ( \\Psi ) = & \\cos \\phi ^ 1 \\cos \\psi ^ 1 + \\\\ & + \\sin \\phi ^ 1 \\cos \\phi ^ 2 \\sin \\psi ^ 1 \\cos \\psi ^ 2 + \\\\ & + \\dots + \\\\ & + \\sin \\phi ^ 1 \\sin \\phi ^ 2 \\dots \\sin \\phi ^ { 2 n - 2 } \\cos \\phi ^ { 2 n - 1 } \\sin \\psi ^ 1 \\sin \\psi ^ 2 \\dots \\sin \\psi ^ { 2 n - 2 } \\cos \\psi ^ { 2 n - 1 } + \\\\ & + \\sin \\phi ^ 1 \\sin \\phi ^ 2 \\dots \\sin \\phi ^ { 2 n - 2 } \\sin \\phi ^ { 2 n - 1 } \\sin \\psi ^ 1 \\sin \\psi ^ 2 \\dots \\sin \\psi ^ { 2 n - 2 } \\sin \\psi ^ { 2 n - 1 } ; \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} \\{ x \\} & = ( x - \\epsilon , x + \\epsilon ) \\cap \\left ( \\biguplus _ { m \\in \\omega } K _ m ^ { ( \\beta ) } \\uplus \\{ b \\} \\right ) \\\\ [ 3 p t ] & = ( x - \\epsilon , x + \\epsilon ) \\cap K ^ { ( \\beta ) } , \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( b ; q \\right ) _ { n } S _ { n } \\left ( x ; q \\right ) z ^ { n } } { \\left ( c ; q \\right ) _ { n } } = \\frac { \\left ( b ; q \\right ) _ { \\infty } } { \\left ( c , z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( c / b , z ; q \\right ) _ { n } b ^ { n } } { \\left ( q ; q \\right ) _ { n } } A _ { q } \\left ( x z q ^ { n } \\right ) . \\end{align*}"}
-{"id": "9997.png", "formula": "\\begin{align*} ( I - \\Delta ) ^ m f ( x ) = 0 \\R ^ n \\setminus A \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} M ( t ) = \\mathrm { g r a p h \\ , } u ( t , \\cdot ) \\forall t _ \\delta \\leq t \\leq t _ 0 , \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial x ^ { \\nu } } \\left ( x _ { \\lambda } T _ { \\mu } { } ^ { \\nu } - x _ { \\mu } T _ { \\lambda } { } ^ { \\nu } \\right ) = \\left ( T _ { \\mu \\lambda } - T _ { \\lambda \\mu } \\right ) \\\\ & - \\left ( x _ { \\lambda } X _ { \\mu } - x _ { \\mu } X _ { \\lambda } \\right ) - \\frac { 1 } { c } \\left ( x _ { \\lambda } F _ { \\mu \\nu } - x _ { \\mu } F _ { \\lambda \\nu } \\right ) j ^ { \\nu } \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} v _ \\beta ^ 2 ( g _ 1 ) = [ I - \\beta P ( g _ 1 ) ] ^ { - 1 } \\tilde { r } ( g _ 1 ) = \\left [ \\frac { 4 + 5 p } { 1 - \\beta } , \\frac { ( 4 + 5 p ) \\beta } { 1 - \\beta } + 7 \\right ] ^ T . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} \\min _ { j = 1 , \\hdots N } ~ \\norm { x _ j - y _ j } ^ 2 \\le \\frac { 2 4 } { \\tilde { \\mu } - \\mu } \\left ( \\frac { \\tilde { \\mu } \\norm { x ^ * - v _ 0 } ^ 2 } { N ( N + 1 ) ( 2 N + 1 ) } + \\frac { M ^ 2 ( r + \\frac { \\rho } { 2 } ( N + 3 ) } { ( N + 1 ) ( 2 N + 1 ) } \\right ) , \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} ( P \\cap H ) + * R = ( Q \\cap H ) + R . \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} a l _ 3 = a p \\ \\ \\ { \\rm a n d } \\ \\ l _ 3 ^ 2 = l _ 3 p . \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} \\varphi ( A ) = \\frac { \\i } { 2 \\pi } \\int _ { \\C } \\frac { \\partial \\tilde { \\varphi } _ N } { \\partial \\overline { z } } ( z ) ( z - A ) ^ { - 1 } d z \\wedge d \\overline { z } . \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\mathcal G _ { n , \\nu } ( X , Y ) \\rightrightarrows ( Y , Y ^ 2 + \\nu ) = : \\mathcal G _ \\nu ( X , Y ) \\mbox { f o r } ( X , Y ) \\in K , | \\nu | \\le 1 \\ ( n \\to \\infty ) . \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} 2 ^ { - \\lfloor ( k + 2 ) / 2 \\rfloor } A _ i ^ { k + 1 } = { 2 k + 3 \\choose 2 i } - \\sum \\limits _ { j = i } ^ k { 2 k + 3 \\choose 2 j + 1 } 2 ^ { - \\lfloor ( j + 3 ) / 2 \\rfloor } A _ i ^ j , \\quad 0 \\leq i \\leq k \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} R ^ { \\bot } ( X _ { i } , X _ { j } ) N _ { \\alpha } = h ( X _ { i } , A _ { N _ { \\alpha } } X _ { j } ) - h ( X _ { j } , A _ { N _ { \\alpha } } X _ { i } ) , \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\left ( A ( \\gamma ) \\right ) ^ { n } e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } = \\sum _ { \\gamma _ { 1 } , \\gamma _ { 2 } , . . . , \\gamma _ { m } } \\frac { q _ { \\gamma _ { 1 } } q _ { \\gamma _ { 2 } } . . . q _ { \\gamma _ { n } } e ^ { i \\left \\langle \\gamma + t + \\gamma ( n ) , x \\right \\rangle } } { { \\textstyle \\prod \\limits _ { s = 1 , 2 , . . . , n } } \\left ( \\mid \\gamma + t \\mid ^ { 2 } - \\left \\vert \\gamma + t + \\gamma ( s ) \\right \\vert ^ { 2 } \\right ) } . \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} \\begin{aligned} u _ y & = j v _ { [ 1 ] } , \\\\ j v _ { [ i + 1 ] } & = k v _ { [ i ] } , \\\\ ( \\omega _ { [ i ] } ) _ x & = u ( v _ { [ i ] } ) _ x , \\\\ u _ t & = k v _ { [ n ] } , \\end{aligned} i = 1 , \\dots , n - 1 , \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , k } ( x | a ) \\bigr ) ( q \\ , | \\ , b ) & - \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , k } ( x | a ) \\bigr ) ( 0 \\ , | \\ , b ) = 0 . \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} \\sqrt { - d ^ 2 / d x ^ 2 } \\ , u _ \\nu ( x ) & = \\widehat { ( | k | \\hat u _ \\nu ) } ( x ) = \\frac { 2 \\Gamma ( \\tfrac { 1 } { 2 } + \\nu ) } { \\sqrt { \\pi } \\Gamma ( \\nu ) } \\ ; { } _ 2 F _ 1 \\left ( 1 , \\tfrac { 1 } { 2 } + \\nu ; \\tfrac { 1 } { 2 } ; - x ^ 2 \\right ) . \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} \\sum _ { n = - \\infty } ^ { \\infty } \\omega ^ { 2 n } q ^ { n ( n - 1 ) } \\left [ \\varphi _ { k } \\left ( \\omega q ^ { n } \\right ) \\varphi _ { \\ell } \\left ( \\omega q ^ { n } \\right ) + \\omega ^ { 2 } q ^ { 2 n } \\varphi _ { k } \\left ( - \\omega ^ { - 1 } q ^ { - n } \\right ) \\varphi _ { \\ell } \\left ( - \\omega ^ { - 1 } q ^ { - n } \\right ) \\right ] = 2 \\| \\varphi \\left ( \\omega \\right ) \\ ! \\| ^ { 2 } \\delta _ { k , \\ell } , \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} \\mathrm { T r } ^ N _ 1 ( ( g E _ { \\lambda , N } ) ^ \\mu ) = c _ 1 \\sum _ { i = 1 } ^ { r } \\frac { D ( k \\mu - 1 , f _ i , g ^ \\mu E _ { \\lambda , N } ^ { \\mu - 1 } ) } { \\pi ^ { k \\mu } \\langle f _ i , f _ i \\rangle } f _ i , \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} \\delta _ { n , o } : = 1 - \\delta _ { n , e } . \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} \\Delta { C } ^ \\infty = \\log \\Big ( ( 2 ^ { \\ell _ 1 } - 1 ) + \\sqrt { ( 1 - 2 ^ { \\ell _ 1 } ) ^ 2 + 2 ^ { \\ell _ 0 + 2 } } \\Big ) - \\mu _ 0 - 1 \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} v '' + ( \\lambda + q ( t ) ) v = 0 , \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} H ^ { n - 1 } ( \\{ u = 0 \\} \\cap Q ) \\leq C d i a m ^ { n - 1 } ( Q ) N _ u ^ \\alpha ( Q ) , \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} F \\left ( x _ { 1 } , \\dots , x _ { 2 n } \\right ) = \\left ( x _ { 1 } , x _ { 2 } , \\dots , x _ { 2 n - 1 } , x _ { 2 n } + \\sum _ { \\substack { \\delta \\in \\left \\{ 0 , 1 \\right \\} ^ { 2 n } \\\\ \\left | \\delta \\right | = n } } \\alpha _ { n } \\prod _ { i = 1 } ^ { 2 n } x _ { i } ^ { \\delta _ { i } } \\right ) , \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\Delta _ g u = - \\lambda u . \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} N _ { ( 1 , 9 ) } ( n ) & = 8 \\ , \\sigma ( n ) - 3 2 \\ , \\sigma ( \\frac { n } { 4 } ) + 8 \\ , \\sigma ( \\frac { n } { 9 } ) - 3 2 \\ , \\sigma ( \\frac { n } { 3 6 } ) \\\\ & + 6 4 \\ , W _ { ( 1 , 9 ) } ( n ) + 1 0 2 4 \\ , W _ { ( 1 , 9 ) } ( \\frac { n } { 4 } ) - 2 5 6 \\ , \\biggl ( W _ { ( 4 , 9 ) } ( n ) + W _ { ( 1 , 3 6 ) } ( n ) \\biggr ) . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} x _ i ^ { k + 1 } & = \\Pi _ { K _ i } [ x _ i ^ k - \\alpha _ k ( F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) + \\epsilon _ { k , i } x _ i ^ k ) ] , \\\\ v _ i ^ { k + 1 } & = { \\hat v _ i ^ k } + x _ i ^ { k + 1 } - x _ i ^ k , \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} \\left ( Q ^ { \\mu \\nu } + \\overset { \\ast } { \\left . Q ^ { \\mu \\nu } \\right . } \\right ) = \\epsilon ^ { \\mu \\nu \\sigma \\tau } \\left ( P _ { \\sigma \\tau } + \\overset { \\ast } { \\left . P _ { \\sigma \\tau } \\right . } \\right ) . \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 d } } f _ 0 ( x , v ) d x d v = 1 \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} L v + A v = \\lambda v , \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} X _ P \\ ; : \\ ; \\begin{cases} x _ { n + 1 } = 0 , \\\\ q ^ \\prime ( x _ 1 , \\dotsc , x _ n ) + c ^ \\prime ( x _ 1 , \\dotsc , x _ n ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} \\exp ( X ) ( t , x ) = \\exp ( t X ) ( x ) , \\qquad \\forall ( t , x ) \\in \\R \\times \\R ^ n . \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} D ( n , k ) - T ( n , k ) = ( - 1 ) ^ k \\binom { n - k + 1 } { k } , \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} D _ { i , j } & = 0 , & & 1 \\leq j < i \\leq k , \\\\ D _ { i , j } & = D _ { 1 , j - i + 1 } , & & 1 \\leq i \\leq j \\leq k , \\\\ C _ { i , n - 2 k } & = 0 , & & 1 \\leq i \\leq k - 1 , \\\\ D _ { i , j } E _ 2 & = - D _ { i , k + j } - D _ { i + 1 , k + j } , & & 1 \\leq i \\leq k - 1 , \\ \\ 1 \\leq j \\leq k . \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} \\phi ( ( u _ 1 , u _ 1 ' ) , ( u _ 2 , u _ 2 ' ) ) \\phi ( ( u _ 2 , u _ 2 ' ) , ( u _ 1 , u _ 1 ' ) ) & = f ( u _ 1 , u _ 2 ) ( 2 p _ d - 1 ) , \\\\ \\phi ( ( u _ 1 , u _ 1 ' ) , ( u _ 2 , u _ 2 ' ) ) ( 1 - \\phi ( ( u _ 2 , u _ 2 ' ) , ( u _ 1 , u _ 1 ' ) ) ) & = f ( u _ 1 , u _ 2 ) ( g ( u _ 1 ' , u _ 2 ' ) - ( 2 p _ d - 1 ) ) , \\\\ ( 1 - \\phi ( ( u _ 1 , u _ 1 ' ) , ( u _ 2 , u _ 2 ' ) ) ) ( 1 - \\phi ( ( u _ 2 , u _ 2 ' ) , ( u _ 1 , u _ 1 ' ) ) ) & = 1 - f ( u _ 1 , u _ 2 ) . \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} \\min _ { \\theta \\in \\Theta } F ( \\theta ) & \\triangleq D _ { K L } \\left ( \\boldsymbol { f } \\| \\boldsymbol { \\ell } \\left ( \\cdot | \\theta \\right ) \\right ) \\ \\ \\ \\\\ & = \\sum \\limits _ { i = 1 } ^ n D _ { K L } \\left ( f ^ i \\| \\ell ^ i \\left ( \\cdot | \\theta \\right ) \\right ) \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} \\begin{aligned} | f ( \\rho ( h ) ) - f ( \\rho ( g _ p h ) ) | & \\le \\max _ { t \\in K ^ M } | f ( \\rho ( h ) ) - f ( t \\rho ( h ) ) | \\\\ & \\le \\left ( \\left ( \\sum _ { t \\in K ^ M } | \\alpha ( t ) f - f | \\right ) \\circ \\rho \\right ) ( h ) . \\end{aligned} \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} w = x _ 0 x _ 1 y _ 2 y _ 3 - y _ 0 x _ 1 x _ 2 y _ 3 - x _ 0 y _ 1 y _ 2 x _ 3 + y _ 0 y _ 1 x _ 2 x _ 3 . \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} F ^ { ( m ) } ( x , \\theta + \\pi ^ k y ) = c _ j \\pi ^ { ( d _ j - d _ 0 ) m + k e _ { j , \\theta } } \\gamma _ j ( x , y ) y ^ { e _ { j , \\theta } } , \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\mathcal { T } g ( t , x ) & : = \\left [ \\int _ { - \\infty } ^ { t } \\left | ( - \\Delta ) ^ { { c _ 1 } / 2 } T _ { t - s } ^ { \\alpha , \\beta } g ( s , \\cdot ) ( x ) \\right | _ { H } ^ { 2 } d s \\right ] ^ { 1 / 2 } , \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} \\nu ^ { { \\pi ^ { * , \\infty } } } ( 0 ) = \\frac { 1 + 2 ^ { \\mu ^ s _ 0 + \\Delta { K } ^ { s , \\infty } } } { 1 + 2 ^ { \\mu ^ s _ 0 + \\mu ^ s _ 1 + 2 \\Delta { K } ^ { s , \\infty } } + 2 ^ { \\mu ^ s _ 0 + 1 + \\Delta { K } ^ { s , \\infty } } } , ~ \\nu ^ { { \\pi ^ { * , \\infty } } } ( 1 ) = \\frac { 2 ^ { \\mu ^ s _ 0 + \\Delta { K } ^ { s , \\infty } } ( 1 + 2 ^ { \\mu ^ s _ 1 + \\Delta { K } ^ { s , \\infty } } ) } { 1 + 2 ^ { \\mu ^ s _ 0 + \\mu ^ s _ 1 + 2 \\Delta { K } ^ { s , \\infty } } + 2 ^ { \\mu ^ s _ 0 + 1 + \\Delta { K } ^ { s , \\infty } } } . \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} \\beta \\beta ^ { t r } = \\gamma ^ { t r } \\gamma = 1 , X X ^ { t r } = Y Y ^ { t r } = 1 . \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} x ^ j y ^ { p q + 1 - j } & = x ^ { q + 1 } y ^ { p q - q } - \\sum _ { i = 1 } ^ { j - 2 } x ^ { q + i } y ^ { p q - q - i } ( y - x ) - x ^ j y ^ { p q - q + 2 - j } ( x ^ { q - 1 } - y ^ { q - 1 } ) \\\\ & = x ^ { q + 1 } y ^ { p q - q } - \\sum _ { i = 1 } ^ { j - 2 } t ^ i x ^ { q } y ^ { p q - q } ( y - x ) - t ^ { j - 1 } x y ^ { p q - q + 1 } ( x ^ { q - 1 } - y ^ { q - 1 } ) \\\\ & = x ^ { q + 1 } y ^ { p q - q } - \\sum _ { i = 1 } ^ { j - 2 } t ^ i x ^ { q } y ^ { p q - q } ( y - x ) - t ^ { j - 1 } x ^ q y ^ { p q - q + 1 } + t ^ { j - 1 } x y ^ { p q } \\\\ & \\in ( x ^ q , y ^ q ) ^ p = I ^ { [ q ] } . \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} y _ i = \\beta x _ i + \\epsilon _ i \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} Q ^ - ( f , f ) = \\int _ { \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } } d \\omega d v _ 2 \\left | \\omega \\cdot ( v - v _ 2 ) \\right | f ( t , x , v ) f ( t , x , v _ 2 ) \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} X ( t , \\omega ) = W ( t , \\omega ) + \\omega ( 0 ) , \\ t \\ge 0 , \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ 2 } ^ 2 = \\int _ { - \\infty } ^ { \\infty } \\| f ( t ) \\| ^ 2 d t < \\infty . \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} N = \\{ ( x ^ 0 , r , \\xi ^ i ) : r = \\sqrt { 1 + \\abs { x ^ 0 } ^ 2 } , x ^ 0 \\in \\mathbb { R } , \\xi \\in \\mathbb { S } ^ n \\} , \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} \\omega ( a , b , c ) \\omega ( a , b c , d ) \\omega ( b , c , d ) = \\omega ( a b , c , d ) \\omega ( a , b , c d ) . \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} \\frac { 1 } { t ^ { d _ u + 1 } } \\sum _ { k = 0 } ^ { t - 1 } { k } ^ { d _ u } \\le 1 \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} S = \\left \\{ e _ { i } \\ , : \\ , i \\in \\left \\{ 1 , \\dots , n \\right \\} \\right \\} \\cup \\left \\{ e _ { i } + e _ { j } \\ , : \\ , i , j \\in \\left \\{ 1 , \\dots , n \\right \\} \\right \\} \\cup \\left \\{ e _ { 1 } + \\dots + e _ { n } \\right \\} \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u - \\div \\bigl ( \\alpha _ \\varepsilon ( u ) \\nabla u \\bigr ) = 0 & \\Q \\times ( 0 , \\infty ) , \\\\ u & \\\\ u ( 0 ) = u _ 0 & \\Q , \\end{cases} \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} g _ { u , v } ( \\eta ^ { ( i ) } ) - g _ { u , v } ( \\beta ) = a _ i ( \\beta ) u ^ 2 + b _ i ( \\beta ) v ^ 2 - c ( \\beta ) u ^ 2 v ^ 2 > 0 . \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} \\nabla _ x \\phi _ \\pm ( x , \\xi ) = \\lim _ { t \\to \\pm \\infty } \\eta ( t , 0 ; x , \\xi ) = \\lim _ { t \\to \\pm \\infty } p ( 0 , t ; y ( 0 , t ; x , \\xi ) , \\xi ) . \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} & n - 1 0 + \\frac { 1 } { 1 6 n ( n + 2 ) ( n - 3 ) } \\Big \\{ ( n - 2 ) ( n - 8 ) ( n - 1 0 ) ( 5 n ^ 2 - 2 n - 1 2 0 ) \\\\ & + 8 ( n - 3 ) [ ( n - 1 0 ) ( - n ^ 3 + 8 n ^ 2 + 2 8 n - 1 7 6 ) + 8 ( - n ^ 3 + 6 n ^ 2 + 3 0 n - 1 1 6 ) ] \\Big \\} \\\\ = & n - 1 0 + \\frac { 1 } { 1 6 n ( n + 2 ) ( n - 3 ) } \\Big [ - 3 n ^ 5 + 2 n ^ 4 + 2 2 8 n ^ 3 - 2 6 4 n ^ 2 - 1 7 6 0 n - 7 6 8 \\Big ] \\\\ = & \\frac { - 3 n ^ 5 + 1 8 n ^ 4 + 5 2 n ^ 3 - 2 0 0 n ^ 2 - 8 0 0 n - 7 6 8 } { 1 6 n ( n + 2 ) ( n - 3 ) } < 0 . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} w ( \\Gamma _ 1 ) = 2 A _ { ( 2 ) } B _ { ( 2 ) } , \\ ; w ( \\Gamma _ 2 ) = A _ { ( 2 ) } p _ 2 [ B _ { ( 1 ) } ] , \\ ; w ( \\Gamma _ 3 ) = p _ 2 [ A _ { ( 1 ) } ] B _ { ( 2 ) } , \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\mathbb { E } ( P _ { n } ) & = \\frac { 2 p } { n - 1 } \\sum _ { k = 1 } ^ { n - 1 } \\mathbb { E } ( P _ { k } ) + \\frac { 1 - p } { n - 1 } \\sum _ { k = 1 } ^ { n - 1 } \\mathbb { E } ( P _ { k } ) \\mathbb { E } ( P _ { n - k } ) , n \\ge 2 , \\mathbb { E } ( P _ { 1 } ) = 1 . \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} L = \\left \\{ n \\in \\mathbb { N } ^ { \\ast } : n \\leq \\left \\vert S \\right \\vert \\forall s \\in \\left \\{ 1 . . . n \\right \\} \\forall Z = \\left \\{ x _ { 1 } , x _ { 2 } , . . . , x _ { s } \\right \\} \\subset S : B ( Z ) \\right \\} \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { u } + \\mathrm { d i v } \\ , \\mathbf { J } = \\mathbf { 0 } ( 0 , \\infty ) \\times G , \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} & \\delta ^ { k } _ T ( f ) ( x _ { 0 } , \\dots , x _ { k } ) = \\\\ & \\sum _ { 0 \\leq s < t \\leq k } ( - 1 ) ^ { t + | x _ { t } | ( | x _ { s + 1 } | + \\dots + | x _ { t - 1 } | ) } f \\Big ( \\alpha ( x _ { 0 } ) , \\dots , \\alpha ( x _ { s - 1 } ) , [ x _ { s } , x _ { t } ] , \\alpha ( x _ { s + 1 } ) , \\dots , \\widehat { x _ { t } } , \\dots , \\alpha ( x _ { k } ) \\Big ) . \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{gather*} g ^ { [ k ] ( \\alpha ) } _ { a b } ( z ) = \\frac { ( - 1 ) ^ { ( a + 1 ) \\ell } \\epsilon _ { a b } ^ { [ k ] ( \\alpha ) } ( z ) } { \\tau _ { k } ^ { ( \\alpha ) } } \\prod _ { i = 1 } ^ { \\ell } c _ { i } ^ { ( \\alpha ) } \\left ( \\det \\big ( V ^ { ( \\ell ) } _ { \\{ z _ { i } \\} } \\big ) ^ { 2 } \\prod _ { j = 1 } ^ \\ell ( z - { z } _ { j } ) ^ { 1 - 2 a } \\right ) . \\end{gather*}"}
-{"id": "225.png", "formula": "\\begin{align*} P S ( k n ) = X ^ * ( k n ) X ( k n ) = X ^ * ( k m / l ) X ( k m / l ) = Y ^ * ( k ) Y ( k ) . \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} c _ 1 ( \\pi ) + \\dots + c _ i ( \\pi ) = c _ 1 ( \\pi ' ) + \\dots + c _ i ( \\pi ' ) = \\mathcal { M } \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} Z ( s , f , \\chi ) = \\overline { \\chi } ^ 4 ( \\overline { y } _ 0 ) \\overline { \\chi } ^ 2 ( \\overline { y } _ 0 - 1 ) \\frac { ( q - 2 ) ( 1 - q ^ { - 1 } ) q ^ { - 6 - 1 2 s } } { ( 1 - q ^ { - 5 - 1 2 s } ) } , \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} \\Gamma ( k \\pm ) = \\left \\{ u \\pm n v _ { k } : u \\in \\Gamma ( k ) , n \\in \\mathbb { N } \\right \\} , \\mathbb { N = } \\left \\{ 1 , 2 , . . . \\right \\} . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} U _ \\infty ( x ) : = \\frac { 1 } { 4 | x | ^ 2 } \\left ( \\log \\frac { | x | } { \\rho } \\right ) ^ { - 2 } . \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{gather*} \\sum \\limits _ { k = 0 } ^ { r } \\ p _ { r , k } s _ { k + m } ^ { ( r ) } = 0 \\ , , \\ \\ \\ \\ m \\geq 0 \\ , , \\end{gather*}"}
-{"id": "7105.png", "formula": "\\begin{align*} h _ { ( \\sigma _ { i } ) _ { i } , \\mu } ( \\mathcal { G } _ { A } | \\mathcal { F } : \\mathcal { X } ) & \\leq \\liminf _ { n \\to \\infty } h _ { ( \\sigma _ { i } ) _ { i } , \\mu } \\left ( \\bigvee _ { j = 1 } ^ { n } \\mathcal { G } _ { A _ { j } } | \\mathcal { F } : \\mathcal { X } \\right ) \\\\ & \\leq \\liminf _ { n \\to \\infty } \\sum _ { j = 1 } ^ { n } h _ { ( \\sigma _ { i } ) _ { i } , \\mu } ( \\mathcal { G } _ { A _ { j } } | \\mathcal { F } : \\mathcal { X } ) \\\\ & \\leq 0 . \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x _ { i } = \\dfrac { p _ { i } k _ { 1 } } { q k _ { 1 } } + \\varepsilon \\phi = \\dfrac { P _ { i } } { Q } + \\varepsilon \\phi & ; & i = 1 , 2 , . . . , n \\\\ x _ { n + 1 } = \\dfrac { h _ { 1 } q } { k _ { 1 } q } + \\varepsilon . 0 = \\dfrac { P _ { n + 1 } } { Q } + \\varepsilon \\phi & & \\\\ \\varepsilon Q = \\varepsilon q k _ { 1 } \\cong 0 & & \\end{array} \\right . \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} & \\pi ( a _ 1 ) ( a _ 2 \\rightharpoonup \\pi ( b ) ) = a _ 1 S ( a _ 2 ) ( a _ 3 \\circ b ) = a \\circ b = \\pi ( a \\circ b ) & a , b \\in A . \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} F ( x ) - F \\left ( 0 \\right ) = \\sum _ { i = 1 } ^ { n } f _ { i } ( \\alpha _ { i } ) w _ { i } . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} U = \\{ k \\in K _ p : [ k , x ] \\in K _ p ' \\} \\quad V = \\{ k \\in K _ p ^ { \\rm a d } : [ k , \\pi ( x ) ] \\in K _ p ' \\} . \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} \\Delta { K } ^ { s , \\infty } = \\log \\Big ( ( 2 ^ { \\ell _ 1 } - 1 ) + \\sqrt { ( 1 - 2 ^ { \\ell _ 1 } ) ^ 2 + 2 ^ { \\ell _ 0 + 2 } } \\Big ) - \\mu _ 0 - 1 \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} \\begin{array} { c l } P _ { m + 1 } ^ { \\left ( i \\right ) } & = P _ { m + 1 } ^ { \\left ( i + 1 \\right ) } + l _ { ( d + 1 ) m + i + 2 } ^ { ( i ) } P _ { m } ^ { \\left ( i + 1 \\right ) } , \\ \\ m = 0 , 1 , . . . \\\\ P _ { 0 } ^ { \\left ( i + 1 \\right ) } & = 1 , \\end{array} \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} \\frac { g ( x + y ) - g ( y ) } { g ( x ) } = g ( \\frac { 1 } { 1 + y } ) + g ( \\frac { y } { 1 + y } ) . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\lambda ^ { - 1 - 2 \\alpha + \\epsilon } \\sum \\limits _ { i , j = 1 } ^ { n + 1 } [ Y _ i Y _ j \\tilde { v } ] _ { C ^ { 0 , \\epsilon } _ \\ast ( \\mathcal { B } _ { \\lambda / 4 } ( y ) ) } + \\lambda ^ { - 1 - 2 \\alpha } \\sum \\limits _ { i , j = 1 } ^ { n + 1 } \\| Y _ i Y _ j \\tilde { v } \\| _ { L ^ { \\infty } ( \\mathcal { B } _ { \\lambda / 4 } ( y ) ) } \\leq C \\| f \\| _ { Y _ { \\alpha , \\epsilon } } . \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u = D _ { x ^ { i } } ( a _ { 0 } ^ { i j } u _ { x ^ { i } } + \\bar { f } ^ { i } ) + h \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} \\lambda \\leq \\bar \\lambda _ { ( i ) } ( \\mathbf { x } ) = 2 ( 1 - \\beta ) \\varphi _ { i } ( \\mathbf { x } ) / ( \\| \\mathbf { d } _ { i } \\| ^ { 2 } L _ { i } ) . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} ( \\ ; + \\ ; ) = k , \\geq 1 , \\ ; \\leq k - 1 . \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} { } _ 2 F _ 1 \\left ( 2 , \\tfrac { 1 } { 2 } + \\nu ; \\tfrac { 3 } { 2 } ; - x ^ 2 \\right ) & = ( 1 + x ^ 2 ) ^ { - 2 } \\ , { } _ 2 F _ 1 \\left ( 2 , 1 - \\nu ; \\tfrac { 3 } { 2 } ; \\tfrac { x ^ 2 } { 1 + x ^ 2 } \\right ) \\\\ & = ( 1 + x ^ 2 ) ^ { - ( \\frac { 1 } { 2 } + \\nu ) } \\ , { } _ 2 F _ 1 \\left ( \\tfrac { 1 } { 2 } + \\nu , - \\tfrac { 1 } { 2 } ; \\tfrac { 3 } { 2 } ; \\tfrac { x ^ 2 } { 1 + x ^ 2 } \\right ) . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{gather*} [ \\hat { x } ^ \\phi _ \\alpha , \\hat { y } ^ \\phi _ \\beta ] = 0 . \\end{gather*}"}
-{"id": "9266.png", "formula": "\\begin{align*} E _ j ( x _ 2 , \\ldots , x _ n ) = ( - 1 ) ^ j x _ { 1 } ^ j + ( - 1 ) ^ { j - 1 } x _ { 1 } ^ { j - 1 } E _ 1 ( x _ 1 , \\ldots , x _ n ) + \\cdots + ( - 1 ) ^ 0 x _ { 1 } ^ 0 E _ j ( x _ 1 , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} B _ { \\alpha } A _ { \\alpha } = - \\frac { d ^ { \\alpha } } { d x ^ { \\alpha } } - \\frac { \\alpha } { 2 } \\frac { d ^ { \\alpha / 2 - 1 } } { d x ^ { \\alpha / 2 - 1 } } + x ^ { 2 } . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} e ^ { \\mu \\nu \\sigma \\tau } e _ { \\mu \\nu \\lambda \\rho } = - 2 \\left \\vert \\begin{array} [ c ] { c c } \\delta _ { \\lambda } ^ { \\sigma } & \\delta _ { \\rho } ^ { \\sigma } \\\\ \\delta _ { \\lambda } ^ { \\tau } & \\delta _ { \\rho } ^ { \\tau } \\end{array} \\right \\vert = - 2 \\left ( \\delta _ { \\lambda } ^ { \\sigma } \\delta _ { \\rho } ^ { \\tau } - \\delta _ { \\rho } ^ { \\sigma } \\delta _ { \\lambda } ^ { \\tau } \\right ) , \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} d ( X ) = \\mathrm { m i n } \\{ r \\mid \\exists \\ , x \\in \\tilde { H } ^ r _ { S ^ 1 } ( X ) , U ^ l x \\neq 0 \\ ; \\mathrm { f o r \\ ; a l l } \\ ; l \\geq 0 \\} . \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} \\epsilon R = \\epsilon R ( \\iota \\otimes \\epsilon ) \\Delta = \\epsilon ( \\epsilon R \\otimes \\iota ) \\Delta = \\epsilon \\implies ( \\iota \\otimes \\epsilon ) \\Delta = ( \\epsilon \\otimes \\iota ) \\Delta = \\iota . \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} & \\langle \\mathcal { L } _ 0 , \\dots , \\mathcal { L } _ q \\rangle ( \\mathcal { X } _ g ^ { q + 1 } / \\mathcal { X } _ g ) \\\\ = & \\langle \\mathcal { L } _ 0 , \\dots , \\mathcal { L } _ { q - 2 } , \\langle \\mathcal { L } _ { q - 1 } , \\mathcal { L } _ q \\rangle ( \\mathcal { X } _ g ^ { q + 1 } / \\mathcal { X } ^ q _ g ) \\rangle ( \\mathcal { X } _ g ^ q / \\mathcal { X } _ g ) , \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} v ( y ) : = u ( x ) - x _ n y _ n - x _ { n + 1 } y _ { n + 1 } , \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\sup _ { y \\in [ 0 , u - \\varepsilon ] } \\left | \\frac { h ( L ^ { \\leftarrow } ( t u ) - L ^ { \\leftarrow } ( t y ) ) } { h ( L ^ { \\leftarrow } ( t u ) ) } - 1 \\right | = 0 \\end{align*}"}
-{"id": "9988.png", "formula": "\\begin{align*} \\norm { f } _ { \\mathcal { H } } ^ 2 = \\int _ { \\R ^ n } \\frac { 1 } { \\widehat { F _ p } ( x ) } \\abs { \\widehat { f } ( x ) } ^ 2 \\ d x . \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\delta _ T : = { 2 + \\gamma \\over 2 \\gamma } \\Bigl ( T - T _ \\gamma \\Bigr ) \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} d z = - \\frac { A } { C } d x - \\frac { B } { C } d y , \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} & \\delta ^ * ( \\mu , r ) = 2 - \\mu . \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} \\sum _ { \\ell = 0 } ^ { N } R ^ { \\eta ^ { - \\ell } } & \\leq \\sum _ { \\ell \\geq 0 : R ^ { \\eta ^ { - \\ell } } > 2 } R ^ { \\eta ^ { - \\ell } } + \\sum _ { \\ell \\leq N : R ^ { \\eta ^ { - \\ell } } \\leq 2 } R ^ { \\eta ^ { - \\ell } } \\\\ & \\leq R \\sum _ { j = 0 } ^ { \\infty } \\left ( 2 ^ { 1 - \\eta } \\right ) ^ { j } + 2 \\left ( N + 1 \\right ) . \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} F ( t _ { 0 } , s _ { 0 } ) = f ( t _ { 0 } ) e _ { 1 } + g ( s _ { 0 } ) e _ { 2 } + f ( t _ { 0 } ) g ( s _ { 0 } ) e _ { 3 } \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} \\| b \\| _ { \\mathcal B ( \\mathbb { H } ) } = \\sup _ { z \\in \\mathbb { H } } \\ , 2 y \\cdot | b ' ( z ) | < \\infty . \\end{align*}"}
-{"id": "10055.png", "formula": "\\begin{align*} q + r = \\gcd ( p , q ) + \\gcd ( p , r ) + \\gcd ( q , q + r - p ) + \\gcd ( r , q + r - p ) . \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda \\ell } \\log _ 2 \\left [ \\frac { \\ell } { n } \\sum _ { t = 0 } ^ { n / \\ell - 1 } 2 ^ { \\lambda L ( y _ { t \\ell + 1 } ^ { t \\ell + \\ell } ) } \\right ] \\ge \\hat { H } _ \\lambda ^ \\ell ( x ^ n ) - \\frac { \\gamma ( s , \\ell ) } { \\ell } , \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} 0 = H _ 1 ( X , \\Z ) \\to H _ 1 ^ { B M } ( X ^ \\circ , \\Z ) \\to H _ 0 ( \\{ P _ 1 , \\dots , P _ r \\} , \\Z ) = \\Z ^ r \\end{align*}"}
-{"id": "10145.png", "formula": "\\begin{align*} T ( F ( u , c v ) ) = X _ { \\tilde u , \\ , \\tilde v c / a ^ { 4 m } } ( a ^ { 2 m } ) = Y _ { \\tilde u c / a ^ { 4 m } , \\ , \\tilde v } ( a ^ { 2 m } ) = T ( S ( u c , v ) ) . \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} g ( P , Q ) = \\tfrac { 1 } { g ! } \\int _ { \\Theta + P - Q } \\log \\| \\theta \\| \\nu ^ { g - 1 } + A ( X ) . \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} S ( l , 0 , v ; p ^ 2 ) = S _ 1 ( l , 0 , v ; p ^ 2 ) = \\frac { 1 } { l ^ { 1 / 2 + v } } + 2 \\pi i ^ { 2 k } V _ { p ^ 2 } ( 0 , v , k ) , \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} X ^ 2 V - X D + Q = 0 . \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} f ( \\eta \\cdot \\lambda ) = & f \\begin{pmatrix} r \\lambda _ 1 + f \\lambda _ 3 & r \\lambda _ 2 + f \\lambda _ 4 \\\\ a \\lambda _ 1 + \\sigma \\lambda _ 3 & a \\lambda _ 2 + \\sigma \\lambda _ 4 \\\\ \\end{pmatrix} \\\\ = & \\phi ( r \\lambda _ 2 + f \\lambda _ 4 ) + a ( \\lambda _ 1 ) + \\sigma ( \\lambda _ 3 ) \\\\ = & \\phi ( r \\lambda _ 2 ) + \\phi ( f \\lambda _ 4 ) + a ( \\lambda _ 1 ) + \\sigma ( \\lambda _ 3 ) . \\\\ \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} \\begin{aligned} \\prod _ { \\mathclap { \\substack { j \\in N \\\\ \\ell _ i \\le j \\le r _ i } } } a _ j ( i ) = \\prod _ { \\mathclap { \\substack { \\ell _ i \\le j \\le r _ i } } } a _ j ( i ) . \\end{aligned} \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{gather*} e \\big ( e _ { a } ^ { k } \\big ) \\alpha = e _ { a } ^ { k } \\wedge \\alpha , i \\big ( e _ { a } ^ { k } \\big ) \\alpha = \\beta , \\alpha = e _ { a } ^ { k } \\wedge \\beta . \\end{gather*}"}
-{"id": "6070.png", "formula": "\\begin{align*} d ^ F F _ { Z _ R } ( \\omega _ 1 , \\omega _ 2 , \\hat { \\omega } ) = d ^ { F , * } G _ { Z _ R } ( \\omega _ 1 , \\omega _ 2 , \\hat { \\omega } ) = 0 . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { N _ R - r } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { \\mathcal { T } } , p } ( u ) = 0 , \\forall q \\in \\bar { \\mathcal { R } } , \\forall u \\in [ \\rho ] . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} P _ { v a c } = \\mathbb { P } ( J = 0 ) & = \\sum _ { n = 0 } ^ { \\infty } \\mathbb { P } ( J = 0 , N = n ) \\\\ & = \\sum _ { n = 0 } ^ { \\infty } p _ { 0 , n } \\\\ & = P _ { 0 } ( 1 ) = p _ { 0 , 0 } \\frac { \\xi } { \\gamma A } , \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} \\pi _ { 1 } = [ a \\partial _ { a ^ { * } } , \\partial _ { a } ] + [ a ^ { * } \\partial _ { a ^ { * } } , \\partial _ { a ^ { * } } ] \\ , . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} B _ { \\Gamma , p } ^ { ( \\ell ) } ( z ) & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = 0 } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } \\sum _ { \\beta = 0 } ^ { p - j - 2 t } 2 ^ { p - j - 2 t - \\beta } \\binom { n - \\ell } { \\beta } \\\\ & \\binom { \\ell } { p - j - 2 t - \\beta } \\sum _ { g = 0 } ^ { p - 1 } \\sum _ { i = 0 } ^ { j - 1 } \\binom { \\beta } { p + i - j - t - g } z ^ { 2 g - p } \\ ; \\Upsilon _ { p - 1 - 2 g } ^ { ( \\ell ) } ( z ) . \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} ( \\star ) A _ { n + 8 r } = t ^ { 8 r } A _ { n } + t ^ { 2 r } A _ { n + 2 r } . \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} \\frac { 1 - t ^ { d _ 1 } - t ^ { d _ 2 } + t ^ { d _ 1 + d _ 2 } } { \\prod ( 1 - t ^ { a _ i } ) } = \\frac { ( 1 - t ^ { d _ 1 } ) ( 1 - t ^ { d _ 2 } ) } { \\prod ( 1 - t ^ { a _ i } ) } . \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} L _ k ( x , y ) = \\sum _ { n = 0 } ^ \\infty \\left ( e ^ { - \\Delta / 2 , \\cdot } E _ n ( x , \\cdot ) \\right ) ( y ) , \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} N = 1 / \\varepsilon \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} \\Delta ( P ) \\star \\Delta ( H _ 1 H _ 2 ) = & \\Delta ( P ) \\star \\Delta ( H _ 1 ) \\star \\Delta ( H _ 2 ) = \\Delta ( P \\star H _ 1 ) \\star \\Delta ( H _ 2 ) \\\\ = & \\Delta ( H _ 1 \\star H _ 1 ( P ) ) \\star \\Delta ( H _ 2 ) = \\Delta ( H _ 1 ) \\star \\Delta ( H _ 1 ( P ) ) \\star \\Delta ( H _ 2 ) \\\\ = & \\Delta ( H _ 1 ) \\star \\Delta ( H _ 1 ( P ) \\star H _ 2 ) = \\Delta ( H _ 1 ) \\star \\Delta ( H _ 2 \\star H _ 2 H _ 1 ( P ) ) \\\\ = & \\Delta ( H _ 1 H _ 2 ) \\star \\Delta ( ( H _ 1 H _ 2 ) ( P ) ) . \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} \\Delta ( \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j r ^ 2 ) = & \\Delta ( \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j ) r ^ 2 + 2 \\nabla _ s ( \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j ) \\nabla _ s r ^ 2 \\\\ & + ( \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j ) \\Delta r ^ 2 \\\\ = & 2 \\Delta \\sigma _ 1 ( A ) ( p ) r ^ 2 + 8 \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j + 2 n \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j + O ( r ^ 3 ) \\\\ = & - \\frac { 1 } { 6 ( n - 1 ) } | W ( p ) | ^ 2 r ^ 2 + 2 ( n + 4 ) \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j + O ( r ^ 3 ) \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} F _ f ( \\sigma ) = \\rho ( \\sigma ) \\prod _ { \\alpha \\in \\Delta ^ + } ( 1 - \\alpha ( \\sigma ) ^ { - 1 } ) \\int _ { T \\backslash G } f ( x ^ { - 1 } \\sigma x ) d x . \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} \\sigma _ E ^ { m ( x ) } ( \\kappa ( \\sigma _ E ( x ) ) ) = \\sigma _ E ^ { | \\mu _ { r ( x ) } | } ( \\kappa ( r ( x ) ) ) = \\sigma _ E ^ { | \\mu _ { r ( x ) } | } ( \\mu _ { r ( x ) } ) = r ( x ) \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} M _ 1 ( t , k , \\xi ) = \\exp { \\left ( - \\int _ 0 ^ t \\frac { \\abs { k } } { k ^ 2 + | \\xi - k s | ^ 2 } \\ , d s \\right ) } , \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} F _ i = \\bigcup _ { m = 0 } ^ { \\infty } \\bigcup _ { j = 1 } ^ { 2 ^ i } C _ { m 2 ^ { i + 1 } + j } . \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} \\rho ( e _ j ) = A _ j , \\ , \\sigma ( e _ j ) = \\sigma _ j , \\ , \\gamma ( e _ j \\wedge e _ k ) = \\gamma _ { j k } . \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 x y : n ) } = \\bigoplus _ { ( i , \\alpha , \\gamma ) } H _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) \\varphi ( i , \\alpha , \\gamma ) \\end{align*}"}
-{"id": "9948.png", "formula": "\\begin{align*} v ' _ i : = \\sum _ { \\vect { \\delta } \\in E _ i } q ( \\vect { \\delta } ) ( v ' ) . \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} 4 0 0 = d \\cdot N . \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} \\mathcal { S } e _ { j } = \\sum _ { i = 1 } ^ { \\infty } b _ { i } \\varphi _ { z _ { i } } \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\int _ { \\vert y \\vert \\geq m \\tau ^ { 1 / \\beta } } \\psi ( T , x - y ) d y \\leq e ^ { - T } \\sum _ { k = 1 } ^ { + \\infty } \\frac { T ^ { k } } { k ! } k \\frac { C ' } { \\tau } = C ' \\frac T \\tau . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\dim ( S ^ 5 \\cap { \\mathcal C } ) \\geq 5 + 8 - 1 0 = 3 , \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\delta _ F + \\delta _ E = \\frac { K } { \\min \\{ M , K \\} } + \\frac { K } { r } . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} ( d ^ { [ { \\sf d } ] } _ { 1 1 } , \\cdots , d _ { K N } ^ { [ { \\sf d } ] } , d ^ { [ { \\sf u } ] } _ { 1 1 } , \\cdots , d _ { K N } ^ { [ { \\sf u } ] } ) = \\lim _ { P \\to \\infty } \\left ( \\frac { R ^ { [ { \\sf d } ] } _ { 1 1 } } { \\frac { 1 } { 2 } \\log P } , \\cdots , \\frac { R ^ { [ { \\sf d } ] } _ { K N } } { \\frac { 1 } { 2 } \\log P } , \\frac { R ^ { [ { \\sf u } ] } _ { 1 1 } } { \\frac { 1 } { 2 } \\log P } , \\cdots , \\frac { R ^ { [ { \\sf u } ] } _ { K N } } { \\frac { 1 } { 2 } \\log P } \\right ) \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q | a , b ) = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty \\frac { d t } { t } \\Bigl [ ( e ^ { - t q } - 1 ) e ^ { - b _ 0 t } \\frac { \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } + q e ^ { - t } \\frac { \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j } { \\prod \\limits _ { i = 1 } ^ M a _ i } \\Bigr ] \\Bigr ) . \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} & = \\sum \\limits _ { s = 1 } ^ { n - j + 1 } { 2 n \\choose 2 s + 2 j - 2 } { 2 s + 2 j - 2 \\choose 2 j - 1 } \\frac { 4 ^ s - 1 } { s } B _ { 2 s } B _ { 2 ( n - s - j + 1 ) } \\\\ & = \\sum \\limits _ { s = 1 } ^ { n - j + 1 } { 2 n \\choose 2 ( s + j - 1 ) } { s + j - 1 \\brack j } B _ { 2 ( n - s - j + 1 ) } , \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} f _ { z _ 1 } d z _ 1 + f _ { z _ 2 } d z _ 2 + f _ { z _ 3 } d z _ 3 = 0 ~ ~ \\widetilde { U } _ 0 , \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} \\mathbb { \\tilde { E } } \\left [ M _ { \\alpha } ^ { \\varepsilon } \\left ( t \\right ) M _ { \\beta } ^ { \\varepsilon } \\left ( t \\right ) \\right ] = 2 t \\mathbb { \\tilde { E } } \\tilde { V } ^ { ^ { \\prime \\prime } } \\left ( \\hat { \\eta } _ { t } ^ { \\varepsilon } \\left ( 0 , \\varepsilon e _ { \\alpha } \\right ) \\right ) \\delta _ { \\alpha \\beta } . \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} \\Psi ( y ) : = \\sum _ { x \\in \\eta } l ( | x - y | ) . \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} \\aligned & B _ a = C _ a , \\ ; \\forall a , \\\\ & A _ a \\ ; , \\ ; \\forall a , \\\\ & A _ a ^ \\# \\ ; , \\ ; \\forall a . \\endaligned \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} D _ 1 ( s ) & = \\sum _ { n \\geq 1 } \\frac { A ( 1 , n ) \\overline { A ( 1 , n ) } } { n ^ s } = L ( s , f \\times f ) T _ 1 ( s ) \\\\ D _ 2 ( s ) & = \\sum _ { n \\geq 1 } \\frac { A ( 1 , n ) ^ 2 } { n ^ s } = L ( s , f \\times \\overline { f } ) T _ 2 ( s ) , \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} \\omega _ { i j } = ( - 1 ) ^ { n - 1 } \\omega _ { j i } & \\qquad ( ) \\\\ \\omega _ { i j } \\omega _ { j k } + \\omega _ { k i } \\omega _ { i j } + \\omega _ { j k } \\omega _ { k i } = 0 & ( ) \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} \\mathcal { W } _ { x _ 0 } ( x ) = a ( x _ 0 ) w _ { 3 / 2 } \\left ( \\frac { ( x - x _ 0 ) \\cdot \\nu _ { x _ 0 } } { ( \\nu _ { x _ 0 } \\cdot A ( x _ 0 ) \\nu _ { x _ 0 } ) ^ { 1 / 2 } } , \\frac { x _ { n + 1 } } { ( a ^ { n + 1 , n + 1 } ( x _ 0 ) ) ^ { 1 / 2 } } \\right ) . \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\{ x \\} & = ( x - \\epsilon , x + \\epsilon ) \\cap \\left ( \\biguplus _ { m \\in \\omega } K _ m ^ { ( \\rho ) } \\uplus \\{ b \\} \\right ) \\\\ [ 3 p t ] & = ( x - \\epsilon , x + \\epsilon ) \\cap K ^ { ( \\rho ) } , \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} f _ { \\gamma } ( T ) : = \\lim _ { i \\to \\infty } { y _ 0 - y _ i ^ T \\over i } \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} V _ 1 ( \\mu , \\eta _ 1 , \\eta _ 2 , t ) = \\exp \\left \\{ - ( \\eta _ 1 ^ 2 + \\eta _ 2 ^ 2 ) \\right \\} \\sum \\limits _ { r ^ 2 _ s + q ^ 2 _ s \\neq 0 } a ' _ s \\eta _ 1 ^ { m _ s } \\eta _ 2 ^ { n _ s } t ^ { l _ s } \\mu ^ { r _ s } \\ln ^ { q _ s } \\mu , \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} B _ { p , q } ' ( x ) = \\frac { q - 2 } { q ^ { m + 1 } ( q - 1 ) } \\cdot \\frac { p ^ { m + 1 } ( p - 1 ) } { p - 2 } \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} \\alpha _ { i _ s } \\alpha _ { i _ t } + \\alpha _ { i _ t } \\alpha _ { i _ s } = - 2 \\delta _ { s t } I . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} \\tilde { g } ^ { i k } \\varphi _ { j k } = - v \\vartheta h ^ i _ j + \\dot { \\vartheta } \\delta ^ i _ j , \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\phi ( w ) = \\frac { w - c - \\sqrt { ( w - c ) ^ 2 - \\rho ^ 2 } } { \\rho R } , \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\aligned \\gamma _ 2 \\gamma _ 2 ^ { t r } & = ( - d \\sigma ^ { - 1 } ( I - \\Delta ^ { t r } \\Delta ) + \\theta \\Delta ) ^ { t r } ( - d \\sigma ^ { - 1 } ( I - \\Delta ^ { t r } \\Delta ) + \\theta \\Delta ) \\\\ & + ( \\mu \\Delta ) ^ { t r } ( \\mu \\Delta ) \\endaligned \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} \\left ( z , z t ; q \\right ) _ { \\infty } = \\sum _ { k = 0 } ^ { \\infty } \\left ( - z \\right ) ^ { k } q ^ { \\binom { k } { 2 } } S _ { k } \\left ( - t q ^ { - k } ; q \\right ) . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} J _ k ( n , s ) : = \\sum _ { j = 0 } ^ k n ^ { - s } T ( s ) ^ { k - j } \\sum _ { m \\geq j } T ( - m ) ( \\mu _ { m , j } - \\mu _ { m , j - 1 } ) \\frac { \\Gamma ( m + s ) } { \\Gamma ( m + 1 ) } , \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\langle \\sqrt { \\epsilon } f , \\frac { 1 } { \\sqrt { \\epsilon } } g \\rangle \\leq \\frac { 1 } { 2 } \\left [ \\epsilon \\| f \\| ^ 2 + \\frac { 1 } { \\epsilon } \\| g \\| ^ 2 \\right ] . \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} & ( h \\# k ) ( h ' \\# k ' ) = h h ' \\# k k ' , & & ( h \\# k ) \\circ ( h ' \\# k ' ) = h ( k _ 1 \\rightharpoonup h ' ) \\# k _ 2 k ' , \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} f ( a + b ) = f ( a ) + f ( b ) + f ( a ) H ( b ) . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} B _ { 2 , 2 } ( x \\ , | \\ , a ) = \\frac { x ^ 2 } { a _ 1 a _ 2 } - \\frac { x ( a _ 1 + a _ 2 ) } { a _ 1 a _ 2 } + \\frac { a _ 1 ^ 2 + 3 a _ 1 a _ 2 + a _ 2 ^ 2 } { 6 a _ 1 a _ 2 } , \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } } { \\left ( q ; q \\right ) _ { n } } A _ { q } ^ { 2 } \\left ( q ^ { n } \\right ) = \\frac { 1 } { \\left ( - q ; q \\right ) _ { \\infty } } . \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} \\mathcal F ( \\lambda , Z ) = \\left [ \\begin{array} { c } ( \\lambda I - A ) ^ { - 1 } B \\\\ I \\end{array} \\right ] ^ H \\left [ \\begin{array} { c c } M _ { 1 1 } + Z _ { 1 1 } & M _ { 1 2 } + Z _ { 1 2 } \\\\ M _ { 2 1 } + Z _ { 2 1 } & M _ { 2 2 } + Z _ { 2 2 } \\end{array} \\right ] \\left [ \\begin{array} { c } ( \\lambda I - A ) ^ { - 1 } B \\\\ I \\end{array} \\right ] \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} N \\cap W = M N : = ^ { ( H _ { \\omega _ 3 } , \\in , \\Delta ) } \\big ( M \\cup \\{ W \\cap \\omega _ 2 \\} \\big ) . \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} J _ { \\mu + m } ( x ) = h _ { m , \\mu } ( 1 / x ) J _ { \\mu } ( x ) - h _ { m - 1 , \\mu + 1 } ( 1 / x ) J _ { \\mu - 1 } ( x ) \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} R _ \\mathrm m = \\frac { 1 } { N - n + 1 } Y Y ^ H \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N } v ^ k _ j = \\sum _ { j = 1 } ^ { N } x ^ k _ j . \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} \\delta ^ * ( 0 , r ) = \\delta _ { \\mathsf { C l - S f } } \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} ( \\Delta + \\mid b _ { 2 , j } + t \\mid ^ { 2 } ) \\varphi _ { 2 , j } = q \\varphi _ { 2 , l } + \\Psi , \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\det M _ \\lambda = 0 . \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\begin{array} { l } a _ 0 ( f , D ) = \\int _ M d V ~ T r \\{ f \\} \\\\ a _ 2 ( f , D ) = \\frac { 1 } { 6 } \\int _ M d V \\ , T r \\{ f ( 6 V + R ) \\} \\\\ a _ 4 ( f , D ) = \\frac { 1 } { 3 6 0 } \\int _ M d V \\ , T r \\{ f ( 6 0 \\Delta V + 6 0 R V + 1 8 0 V ^ 2 \\\\ + 1 2 \\Delta R + 5 R ^ 2 - 2 R ^ { i j } R _ { i j } + 2 R ^ { i j k l } R _ { i j k l } ) \\} \\end{array} \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & \\ ; \\ ; = \\psi _ { s + k } ^ { - \\tau } Z _ { s , s + k } \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] \\end{aligned} \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} \\sinh ^ 2 \\tfrac { 1 } { 2 } d _ { \\xi } z = \\sinh ^ 2 ( \\tfrac { 1 } { 2 } T _ { \\xi } ) \\cosh ^ 2 d _ z \\mathcal { A } + \\sin ^ 2 \\theta \\sinh ^ 2 d _ z \\mathcal { A } \\\\ \\leq \\left ( \\sinh ^ 2 ( \\tfrac { 1 } { 2 } T _ { \\xi } ) + \\sin ^ 2 \\theta \\right ) \\cosh ^ 2 d _ z \\mathcal { A } , \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} \\rho ( x ) - \\frac { { \\rm d } } { { \\rm d } x } \\left ( \\frac { \\mathfrak { D } } { \\sigma ^ * } \\frac { { \\rm d } \\rho } { { \\rm d } x } \\right ) = g ( x ) \\mbox { f o r } x \\in ( - \\ell , + \\ell ) . \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} S _ { n } \\left ( e ^ { 2 \\xi } q ^ { - 1 } \\right ) = \\frac { q ^ { \\binom { n + 1 } { 2 } } \\left ( - e ^ { \\xi } t \\right ) ^ { n } h _ { n } \\left ( \\sinh \\left ( \\xi + \\log q ^ { \\left ( n - 1 \\right ) / 2 } \\right ) \\vert q \\right ) } { \\left ( q ; q \\right ) _ { n } } , \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\int _ { x \\in G } \\psi ( g x ) \\ , d \\mu = \\int _ { x \\in G } \\psi ( x g ) = \\int _ { x \\in G } \\psi ( x ^ { - 1 } ) \\ , d \\mu = \\int _ { x \\in G } \\psi ( x ) \\ , d \\mu \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\overline \\nabla _ X N = - A _ N X + D _ X N , \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} \\mathcal { T } _ 0 = - \\iint \\left ( A ( u _ 0 ^ z \\partial _ z \\ne { f } ) - u _ 0 ^ z \\partial _ z A \\ne { f } \\right ) A \\ne { f } \\ , d V d t . \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} \\liminf _ { ( y , s ) \\to ( x , t ) } u ( y , s ) = \\liminf _ { ( y , s ) \\to ( x , t ) } w ( y , s ) \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { f ( t ) } { g ( t ) } = C \\ , . \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} \\partial _ { t } p _ { \\delta , n } = \\partial \\Upsilon _ { n } ^ { \\delta } \\left ( z _ { \\delta , n } - B _ { n } \\ , p _ { \\delta , n } \\right ) \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\lambda _ k ( \\Omega ; p ) : = \\inf \\limits _ { \\mathcal { A } \\in \\mathcal { F } _ k } \\sup \\limits _ { u \\in \\mathcal { A } } \\int _ { \\Omega } | \\nabla u | ^ p \\ , d x . \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} c F _ \\gamma ( x ^ k ) ^ { \\theta } = c | F _ \\gamma ( x ^ k ) | ^ { \\theta } \\le \\| \\nabla F _ \\gamma ( x ^ k ) \\| \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} H ( \\rho , x ) = \\sum _ { j \\in \\Z } A ( \\rho + i t _ j ) \\Gamma ( \\rho + 1 + i t _ j ) e ^ { - 2 j \\pi i x } , \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\int _ { P ( z , \\delta ) } \\frac { d \\lambda ( \\zeta ) } { \\left | z - \\zeta \\right | ^ { 1 + \\mu } } \\lesssim \\tau _ { n } ( z , \\delta ) ^ { 1 - \\mu } \\prod _ { j = 1 } ^ { n - 1 } \\tau _ { j } ^ { 2 } ( z , \\delta ) , \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} Y _ 0 & = \\begin{pmatrix} \\cos \\phi _ { L 0 } & \\sin \\phi _ { L 0 } \\end{pmatrix} s _ 0 \\begin{pmatrix} \\cos \\phi _ { R 0 } \\\\ \\sin \\phi _ { R 0 } \\end{pmatrix} \\in \\pi ^ { - 1 } ( \\Omega ) , \\\\ Y _ 1 & = I ( Y _ 0 , \\Phi ( Y _ 0 ) - Y _ 0 ) = \\begin{pmatrix} \\cos \\phi _ { L 1 } & \\sin \\phi _ { L 1 } \\end{pmatrix} s _ 0 \\begin{pmatrix} \\cos \\phi _ { R 1 } \\\\ \\sin \\phi _ { R 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\varphi _ { 2 n + 1 } = \\sum _ { i = 0 } ^ n { n \\choose i } _ r \\varphi _ { 2 i } , \\quad \\psi _ { 2 n + 1 } = \\sum _ { i = 0 } ^ n { n \\choose i } _ s \\psi _ { 2 i } . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} \\int _ { \\Omega } v \\nabla _ H v \\cdot v + w \\partial _ z v \\cdot v = 0 . \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} \\Phi _ 0 = & d x ^ { 1 2 3 4 } + d x ^ { 1 2 5 6 } - d x ^ { 1 2 7 8 } + d x ^ { 1 3 5 7 } + d x ^ { 1 3 6 8 } + d x ^ { 1 4 5 8 } - d x ^ { 1 4 6 7 } \\\\ & - d x ^ { 2 3 5 8 } + d x ^ { 2 3 6 7 } + d x ^ { 2 4 5 7 } + d x ^ { 2 4 6 8 } - d x ^ { 3 4 5 6 } + d x ^ { 3 4 7 8 } + d x ^ { 5 6 7 8 } , \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} k = \\frac { m z } { x y } x ' = \\frac { x } { z w } y ' = \\frac { y } { z w } m ' = \\frac { m } { w } = \\frac { x y k } { z w } = x ' y ' z w k \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} \\sum _ { m > N } \\left ( \\frac { t } { m } - \\log \\left ( 1 + \\frac { t } { m } \\right ) \\right ) = \\sum _ { l \\geq 2 } \\frac { ( - 1 ) ^ l t ^ l } { l } \\sum _ { m > N } \\frac { 1 } { m ^ l } . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial x ^ { \\nu } } \\left ( e ^ { \\mu \\lambda \\sigma \\tau } x _ { \\sigma } T _ { \\tau } { } ^ { \\nu } \\right ) + e ^ { \\mu \\lambda \\sigma \\tau } T _ { \\sigma \\tau } \\\\ & \\ + e ^ { \\mu \\lambda \\sigma \\tau } x _ { \\sigma } X _ { \\tau } { } + \\frac { 1 } { c } e ^ { \\mu \\lambda \\sigma \\tau } x _ { \\sigma } F _ { \\tau \\nu } j ^ { \\nu } = 0 ^ { \\mu \\lambda } . \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} a ^ { i j } \\partial _ { i j } w + ( \\partial _ i a ^ { i j } ) \\partial _ j w = 0 . \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} \\delta _ { 1 } & = - a _ 1 a _ { n - 2 k } , \\\\ b _ { 2 } & = \\frac { ( - 1 ) ^ { k - 1 } ( a _ 2 + 1 ) ^ k \\theta _ { n } } { a _ 1 } , \\\\ \\theta _ { n - k + i } & = ( - 1 ) ^ { k - i } ( a _ 2 + 1 ) ^ { k - i } \\theta _ { n } , 1 \\leq i \\leq k - 1 . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} u ( x _ 0 ) + \\int _ { D \\setminus \\{ x _ 0 \\} } V _ { \\mu } \\ , d { \\nu } _ u = \\int _ { D } u \\ , d \\mu . \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{gather*} { \\boldsymbol \\delta } ( f ) ( P \\cdot Q ) = { \\boldsymbol \\delta } ( f _ { ( 1 ) } ) ( P ) { \\boldsymbol \\delta } ( f _ { ( 2 ) } ) ( Q ) { \\boldsymbol \\delta } ( f ) ( 1 ) = \\epsilon ( f ) 1 , \\end{gather*}"}
-{"id": "6830.png", "formula": "\\begin{align*} I \\left ( X _ i ; \\bar { X } _ i \\right ) & ~ ~ = ~ ~ \\log _ 2 \\left ( 1 + \\frac { \\bar { P } } { \\sigma ^ 2 } \\right ) = B \\\\ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\sigma ^ 2 & ~ ~ = ~ ~ \\frac { \\bar { P } } { 2 ^ B - 1 } . \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\textit { T h e r e e x i s t s $ c \\in X ^ \\sigma $ a n d $ d \\in X \\setminus X ^ \\sigma $ s u c h t h a t $ [ c , d ] = 1 $ . } \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} \\{ u ^ * > \\psi \\} = \\bigcup _ { k = 0 } ^ \\infty \\{ \\varphi _ k > \\psi \\} . \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} L ^ { 2 } ( q ) = 1 + \\sum _ { n = 1 } ^ { \\infty } \\biggl ( 2 4 0 \\ , \\sigma _ { 3 } ( n ) - 2 8 8 \\ , n \\ , \\sigma ( n ) \\biggr ) q ^ { n } \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ \\infty T ( p ^ n ) t ^ n = \\frac { P _ { p } ( t ) } { Q _ { p } ( t ) } , \\ P _ { p } ( t ) = 1 - p ^ 2 R _ p t ^ 2 , \\\\ & Q _ { p } ( t ) = 1 - T ( p ) t + \\{ T ( p ) ^ 2 - T ( p ^ 2 ) - p ^ 2 R _ p \\} t ^ 2 - p ^ 3 R _ p T ( p ) t ^ 3 + p ^ { 6 } R _ { p ^ 2 } t ^ 4 . \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} \\vec { v } _ { \\infty } = \\lim _ { t \\to \\infty } \\frac { 1 } { \\| M ^ t \\vec v \\| } M ^ t \\vec v \\neq \\vec 0 \\ , , \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} h ^ { \\prime } ( \\theta ) & = \\frac { \\pi } { 2 } \\sum _ { i = 1 } ^ { n } F _ { i } ( \\tilde e \\cos ( \\pi \\theta / 2 ) + e \\sin ( \\pi \\theta / 2 ) ) ( e _ { j } \\cos ( \\pi \\theta / 2 ) - \\tilde e _ { j } \\sin ( \\pi \\theta / 2 ) ) \\\\ & = \\frac { \\pi } { 2 } ( \\nabla F ( \\widetilde { W } _ { \\theta } ) , W _ { \\theta } ) , \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} \\abs { \\mathbf { B } ( x ) } : = \\sqrt { \\sum _ { j , k = 1 } ^ d B _ { j k } ( x ) ^ 2 } \\ , , \\abs { \\nabla \\mathbf { B } ( x ) } : = \\sqrt { \\sum _ { j , k = 1 } ^ d \\abs { \\nabla B _ { j k } ( x ) } ^ 2 } \\ , , \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} e ^ { \\mu \\nu \\sigma \\tau } e _ { \\mu \\kappa \\lambda \\rho } = - \\left \\vert \\begin{array} [ c ] { c c c } \\delta _ { \\kappa } ^ { \\nu } & \\delta _ { \\lambda } ^ { \\nu } & \\delta _ { \\rho } ^ { \\nu } \\\\ \\delta _ { \\kappa } ^ { \\sigma } & \\delta _ { \\lambda } ^ { \\sigma } & \\delta _ { \\rho } ^ { \\sigma } \\\\ \\delta _ { \\kappa } ^ { \\tau } & \\delta _ { \\lambda } ^ { \\tau } & \\delta _ { \\rho } ^ { \\tau } \\end{array} \\right \\vert , \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} { f ( x ) = \\sum _ { i , j = 1 } ^ t r _ { i j } ( x ) x ^ { \\alpha _ i } \\log ( x ) ^ j } \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} & \\partial _ t f + u \\cdot \\nabla _ L f = \\nu \\Delta _ L f \\\\ & u = - \\nabla ^ { \\perp } _ L ( - \\Delta _ L ) ^ { - 1 } f , \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} \\beta _ { s , a ^ 1 } ^ 1 = \\frac { \\left [ r ^ 1 ( s , a ^ 1 , a _ s ^ 2 ) - r ^ 1 ( s , a _ s ^ 1 , a _ s ^ 2 ) \\right ] } { \\left [ \\sum _ { s ' \\in S } p _ { s ' } \\ r ^ 1 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) - \\sum _ { s ' \\in S } p ( s ' | s , a ^ 1 , a _ s ^ 2 ) r ^ 1 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) \\right ] } . \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} \\varphi = x ^ { 1 2 3 } + x ^ { 1 4 5 } - x ^ { 1 6 7 } + ^ { 2 4 6 } + x ^ { 2 5 7 } + x ^ { 3 4 7 } - x ^ { 3 5 6 } . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) = 1 \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} R _ { \\mathsf { u } } & > \\sum _ { n = 1 } ^ N \\sum _ { j = 1 } ^ { L - r _ n } \\frac { j } { j + r _ n } \\binom { L - r _ n } { j } p _ n ^ { j } ( 1 - p _ n ) ^ { L - r _ n - j } \\\\ & = N \\sum _ { j = 1 } ^ { L - r } \\frac { j } { j + r } \\binom { L - r } { j } \\left ( \\frac { 1 } { N } \\right ) ^ { j } \\left ( 1 - \\frac { 1 } { N } \\right ) ^ { L - r - j } . \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} G _ \\pm ( t ) u [ x ] & = ( H J _ a - J _ a H _ 0 ) e ^ { - i t H _ 0 } \\tilde P _ \\pm u [ x ] \\\\ & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i \\Phi ( x , y , \\xi ; t ) } s _ a ( x , \\xi ) p _ \\pm ( y , \\xi ) u [ y ] d \\xi . \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} \\frac { d a ( x ) } { d x } = - \\dfrac { \\frac { 1 } { 2 } \\xi x + 1 - ( 1 + \\xi x ) ^ { 1 / 2 } } { \\xi x ^ 2 ( 1 + \\xi x ) ^ { 1 / 2 } } \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} k [ x , y ] ^ { ( 2 ) } = k [ x ^ 2 , x y , y ^ 2 ] \\cong \\frac { k [ u , v , w ] } { ( u w - v ^ 2 ) } . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} \\chi _ { q } ( z ) & = \\sum _ { n = 1 } ^ { \\infty } q ^ { n / 2 } \\sum _ { k = 0 } ^ { \\infty } \\left ( - z \\right ) ^ { k } q ^ { ( 2 n - 1 ) k / 2 } = \\sum _ { k = 0 } ^ { \\infty } q ^ { - k / 2 } \\left ( - z \\right ) ^ { k } \\sum _ { n = 1 } ^ { \\infty } q ^ { ( 2 k + 1 ) n / 2 } \\\\ & = \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { ( k + 1 ) / 2 } } { 1 - q ^ { k + 1 / 2 } } \\left ( - z \\right ) ^ { k } \\ ! , \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} \\sum _ { y _ { n } } \\log \\Big ( \\frac { q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) } { \\nu ^ { \\pi } _ { n } ( y _ n | y ^ { n - 1 } _ { n - J } ) } \\Big ) q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) - s \\gamma _ n ( x _ n , y ^ { n - 1 } _ { n - N } ) = 1 - \\lambda _ n ( y ^ { n - 1 } _ { n - J } ) , ~ \\forall { x _ n \\in { \\cal X } _ n } . \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} x ^ p + y ^ p = z ^ r \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} \\mathcal { C } _ N = \\left \\lbrace \\frac { a } { d } : d | N , ( a , d ) = 1 , a \\in ( \\mathbb { Z } / k \\mathbb { Z } ) ^ * k = ( d , N / d ) \\right \\rbrace . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} \\mathcal { M } _ q [ N ] = \\left \\{ \\prod _ { \\substack { \\mathcal { R } , \\bar { \\mathcal { R } } _ i , \\mathcal { T } : \\\\ \\mathcal { R } \\cup \\bar { \\mathcal { R } } _ i \\not \\ni q } } \\left [ \\alpha _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { \\bar { \\mathcal { R } } _ i } \\tilde { h } _ { q , \\mathcal { T } } ^ { \\bar { \\mathcal { R } } _ i } \\right ] ^ { s _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { i } } : 1 \\le s _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { i } \\le N \\right \\} , \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} W _ { \\nu } ( z ) = w _ { \\nu } ^ { \\prime } ( - 4 z ) = \\frac { \\sqrt { \\pi } } { 2 } \\sum _ { n \\geq 0 } \\frac { \\left ( n + 1 \\right ) z ^ { n } } { \\Gamma \\left ( n + \\frac { 3 } { 2 } \\right ) \\left ( \\nu + \\frac { 3 } { 2 } \\right ) _ { n } } = \\frac { \\sqrt { \\pi } } { 2 } \\sum _ { n \\geq 0 } \\frac { \\left ( n + 1 \\right ) ! } { \\Gamma \\left ( n + \\frac { 3 } { 2 } \\right ) \\left ( \\nu + \\frac { 3 } { 2 } \\right ) _ { n } } \\cdot \\frac { z ^ n } { n ! } . \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} \\pi _ C ( x , L / F ) : = \\# \\{ \\mathfrak { p } : \\textup { $ \\mathfrak { p } $ i s u n r a m i f i e d i n $ L $ , $ [ \\tfrac { L / F } { \\mathfrak { p } } ] = C $ , $ \\mathrm { N } _ { F / \\mathbb { Q } } ~ \\mathfrak { p } \\leq x $ } \\} . \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{gather*} \\mu _ A ( ( ( \\beta ( c ) h ) \\blacktriangleright a ) \\otimes ( h ' \\blacktriangleright b ) ) = \\mu _ A ( ( h \\blacktriangleright a ) \\otimes ( ( \\alpha ( c ) h ' ) \\blacktriangleright b ) ) . \\end{gather*}"}
-{"id": "3546.png", "formula": "\\begin{align*} & a ^ * _ { 1 , 1 } = \\frac { 1 } { 3 } - \\frac { \\mu _ R } { 3 } - \\frac { \\mu _ T } { 3 } , a ^ * _ { 3 , 0 } = 2 \\mu _ R + \\mu _ T - 1 , a ^ * _ { 0 , 3 } = \\mu _ R + 2 \\mu _ T - 1 , \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} \\rho _ E ^ \\lambda ( x ^ n ) = \\frac { c } { n \\lambda } \\log \\left [ \\frac { 1 } { c } \\sum _ { i = 1 } ^ c 2 ^ { \\lambda L ( y _ { n _ { i - 1 } + 1 } ^ { n _ i } ) } \\right ] , ~ ~ ~ ~ n _ 0 \\equiv 0 , ~ ~ n _ c \\equiv n , \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} & f _ n ( t _ 1 , x _ 1 , \\ldots , t _ n , x _ n , t , x ) \\\\ & = \\lambda ^ n G ( t - t _ n , x - x _ n ) \\ldots G ( t _ 2 - t _ 1 , x _ 2 - x _ 1 ) \\ , \\eta 1 _ { \\{ 0 < t _ 1 < \\ldots < t _ n < t \\} } , \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} L _ { t } ( q ) \\Psi _ { N , t } ( x ) = \\Lambda _ { N } ( t ) \\Psi _ { N , t } ( x ) . \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} & \\quad \\quad \\quad \\int _ { 0 } ^ { T } a ( u , u ^ { - } ) d t = \\int _ { 0 } ^ { T } a ( u ^ { + } , u ^ { - } ) d t - \\int _ { 0 } ^ { T } a ( u ^ { - } , u ^ { - } ) d t , \\\\ & \\int _ { 0 } ^ { T } a ( u ^ { - } , u ^ { - } ) d t = \\frac { c _ { N , \\beta } } { 2 } \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { N } } \\int _ { \\mathbb { R } ^ { N } } \\frac { ( u ^ { - } ( x , t ) - u ^ { - } ( y , t ) ) ^ { 2 } } { | x - y | ^ { N + 2 \\beta } } d x d y d t > 0 , \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ n \\log \\Big ( \\frac { d { \\bf P } _ { Y _ t | Y ^ { t - 1 } , X ^ t } ( \\cdot | y ^ { t - 1 } , x ^ t ) } { d { \\bf P } _ { Y _ t | Y ^ { t - 1 } } ^ { { \\bf P } ^ * } ( \\cdot | y ^ { t - 1 } ) } ( Y _ t ) \\Big ) . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} \\ell : = \\zeta _ d - \\frac { \\alpha } { 4 } \\kappa + \\frac { \\alpha } { 2 } \\kappa _ d + \\frac { \\alpha } { 2 } d \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ + } \\ ; d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ d \\left ( s + k \\right ) T R ^ d \\left [ \\alpha + \\frac { y } { \\eta T } + C _ { d , \\alpha } \\left ( \\frac { \\eta } { R } \\right ) ^ { d - 1 } + C _ { d , \\alpha } \\theta ^ { ( d - 1 ) / 2 } \\right ] \\end{aligned} \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} S _ z = \\{ \\mathfrak { n } \\colon \\mathfrak { p } \\mid \\mathfrak { n } \\implies \\N \\mathfrak { p } > z \\} V ( z ) = \\sum _ { \\N \\mathfrak { n } \\leq z } \\frac { 1 } { \\N \\mathfrak { n } } . \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} 0 = \\sum _ { k = 0 } ^ n a _ k \\tau _ { n h _ 1 } \\tau _ { k y - k h _ 1 } f ( x ) = \\sum _ { k = 0 } ^ { n } a _ k \\tau _ { ( n - k ) h _ 1 } ( \\tau _ { k y } f ) ( x ) y , h _ 1 \\in B _ { d } ( \\delta / 2 ) . \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} { \\cal P } _ { 0 , n } ( \\kappa ) \\triangleq \\Big \\{ { \\bf P } _ { X _ t | X ^ { t - 1 } , Y ^ { t - 1 } } , t = 0 , \\ldots , n : \\frac { 1 } { n + 1 } { \\bf E } \\Big \\{ c _ { 0 , n } ( X ^ n , Y ^ { n - 1 } ) \\Big \\} \\leq \\kappa \\Big \\} , ~ \\kappa \\in [ 0 , \\infty ) \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} | \\phi _ k ( a _ { k ; i } ^ m ) | ^ 2 | \\phi _ l ( a _ { l ; j } ^ n ) | ^ 2 \\big ( 1 - | \\phi _ l ( a _ { l ; i } ^ m ) | ^ 2 \\big ) \\big ( 1 - | \\phi _ k ( a _ { k ; j } ^ n ) | ^ 2 \\big ) = 0 . \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} f f '' + ( f ' ) ^ 2 - 1 = \\pm 2 a f \\sqrt { f '^ 2 - 1 } , a = c o n s t \\neq 0 . \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} [ - 1 , 1 ] ^ n = \\underbrace { [ - 1 , 1 ] \\times [ - 1 , 1 ] \\times \\ldots \\times [ - 1 , 1 ] } _ { n \\small \\mbox { t i m e s } } . \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} \\begin{array} { l } g ( \\{ x , y \\} , z ) = g ( \\nabla _ x y + \\nabla _ y x , z ) \\\\ \\phantom { g ( \\{ x , y \\} , z ) } = x g ( y , z ) + y g ( x , z ) - z g ( x , y ) - g ( [ y , z ] , x ) + g ( [ z , x ] , y ) . \\end{array} \\end{align*}"}
-{"id": "9905.png", "formula": "\\begin{align*} \\varepsilon _ 3 ( x ) = \\langle \\tau ( x ) , \\nabla _ { \\partial \\Omega } ( \\tau - \\nu ) ( x ) , x - a \\rangle . \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} \\nu ( \\alpha ) ^ { - \\frac { k _ 1 + k _ 2 - 3 } { 2 } } T _ \\alpha F ( Z ) = \\nu ( \\alpha ) ^ { - \\frac { 3 } { 2 } } \\widetilde { T } _ { \\alpha ^ { - 1 } } \\phi ( g ) , \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} S ( u , v ) = \\int ^ { z = u + i v } ( 1 + g ^ 2 , i - i g ^ 2 , 2 g ) \\eta \\ ; , \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\phi _ { A , 0 } \\circ q ^ * = \\phi _ { I , 0 } \\circ A \\circ q ^ * = \\phi _ { I , 0 } \\circ q ^ * _ A \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} g _ 1 ( X , Y ) = g ( J _ 1 X , Y ) , g _ 2 ( X , Y ) = g ( J _ 2 X , Y ) , g _ 3 ( X , Y ) = g ( J _ 3 X , Y ) . \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} d \\bar { s } ^ 2 = - { d x ^ 0 } ^ 2 + d r ^ 2 + r ^ 2 \\sigma _ { i j } d \\xi ^ i d \\xi ^ j , \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} \\langle \\alpha , x y \\rangle \\cap \\langle \\beta , x y \\rangle = x y . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\Gamma ^ { - 1 } _ M ( w \\ , | \\ , a ) = e ^ { P ( w \\ , | \\ , a ) } \\ , w \\prod \\limits _ { n _ 1 , \\cdots , n _ M = 0 } ^ \\infty { } ' \\Bigl ( 1 + \\frac { w } { \\Omega } \\Bigr ) \\exp \\Bigl ( \\sum _ { k = 1 } ^ M \\frac { ( - 1 ) ^ k } { k } \\frac { w ^ k } { \\Omega ^ k } \\Bigr ) , \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} \\int _ { \\widetilde { M } } | D f ^ { - 1 } ( w ) | ^ { 2 } \\ , \\mathrm { d } w \\ = \\ \\int _ { M } K _ { f } ( z ) \\ , \\mathrm { d } z \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{gather*} \\mathcal { D } ^ 1 ( U , \\tilde { \\mathcal { G } } ) = \\left ( H ^ 1 _ c ( U , \\tilde { \\mathcal { G } } ) \\rightarrow H ^ 1 ( K _ 0 , \\tilde { G } ) \\right ) , \\\\ D _ { s h } ^ 1 ( U , \\mathcal { G } ) = \\left ( H ^ 1 ( U , \\mathcal { G } ) \\rightarrow \\prod _ { v \\in X _ 0 ^ { ( 1 ) } } H ^ 1 ( K _ { 0 , v } , G ) \\right ) . \\end{gather*}"}
-{"id": "6000.png", "formula": "\\begin{align*} C _ { \\alpha , 2 j } : = ( - 1 ) ^ j 2 ^ { 2 j - \\alpha } \\int _ 0 ^ \\infty \\frac { \\sin ^ { 2 j } u } { u ^ { 1 + \\alpha } } \\ , d u . \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} [ \\bar { f } _ { 0 , k + 3 } , \\bar { f } _ { 0 , l } ] - ( 1 + d + d ^ { - 1 } ) [ \\bar { f } _ { 0 , k + 2 } , \\bar { f } _ { 0 , l + 1 } ] + ( 1 + d + d ^ { - 1 } ) [ \\bar { f } _ { 0 , k + 1 } , \\bar { f } _ { 0 , l + 2 } ] - [ \\bar { f } _ { 0 , k } , \\bar { f } _ { 0 , l + 3 } ] = 0 , \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} \\| x ^ { k + 1 } _ i - x ^ * _ i \\| ^ 2 & \\leq \\| x _ i ^ k - \\alpha _ { k , i } F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) - x ^ k _ i - x ^ * _ i + \\alpha _ { k , i } F _ i ( x ^ * _ i , \\bar x ^ * ) \\| ^ 2 \\\\ & = \\| x _ i ^ k - x ^ * _ i \\| ^ 2 + \\alpha _ { k , i } ^ 2 \\| F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) - F _ i ( x ^ * _ i , \\bar x ^ * ) \\| ^ 2 \\cr & - 2 \\alpha _ { k , i } ( F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) - F _ i ( x ^ * _ i , \\bar x ^ * ) ) ^ T ( x ^ k _ i - x ^ * _ i ) . \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} \\mu = \\begin{pmatrix} 0 & A \\end{pmatrix} , A A ^ { t r } = I , \\quad \\theta = \\begin{pmatrix} \\tau & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "9893.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } g \\cdot { \\mathbf n } _ V \\ , d \\| \\delta V \\| + \\sigma \\int _ { \\partial ^ * B ^ + } g \\cdot \\mathbf n _ { B ^ + } \\ , d \\mathcal H ^ { n - 1 } = 0 . \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} \\left ( x , v \\right ) \\in \\Omega _ { - } = \\left \\{ e _ { \\pm } > - \\min \\beta , v < 0 \\right \\} , \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} F _ { d , \\ell } ( z ; \\tau ) = e ^ { \\frac { 2 \\pi i d c } { h \\ell } } \\sum _ { j = 0 } ^ { h - 1 } e ^ { \\frac { 2 \\pi i c j } { h } } F _ { \\ell j + d , \\ell h } \\left ( \\frac { w _ 0 } { \\ell } ; \\tau \\right ) . \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\textbf { p } ( t ) \\frac { \\partial \\bar { \\textbf { g } } } { \\partial \\textbf { x } } ( t , t _ 0 , \\textbf { x } ) + \\frac { \\partial \\bar { c } } { \\partial \\textbf { x } } ( t , t _ 0 , \\textbf { x } ) = \\textbf { p } ( t _ 0 ) \\frac { \\partial \\bar { \\textbf { g } } } { \\partial \\textbf { x } } ( t _ 0 , t _ 0 , \\textbf { x } ) + \\frac { \\partial \\bar { c } } { \\partial \\textbf { x } } ( t _ 0 , t _ 0 , \\textbf { x } ) . \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} \\mu _ { a _ i } ( { \\rm C y l } _ w ) = \\lim _ { t \\to \\infty } \\frac { | \\zeta ^ { t } ( a _ i ) | _ w } { | \\zeta ^ { t } ( a _ i ) | } . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} \\mathbb { E } [ \\| D ( k ) ( v ^ k ( \\ell ) - [ y ^ { k } ] _ \\ell \\mathbf { 1 } ) \\| ] = \\mathbb { E } \\left [ \\mathbb { E } [ \\| D ( k ) ( v ^ k ( \\ell ) - [ y ^ { k } ] _ \\ell \\mathbf { 1 } ) \\| \\mid \\mathcal { F } _ k ] \\right ] \\le \\sqrt { \\lambda } \\ , \\mathbb { E } [ \\| v ^ k ( \\ell ) - [ y ^ { k } ] _ \\ell \\mathbf { 1 } \\| ] . \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } g _ { i j } ( x , t ) = ( r ( t ) - R ( x , t ) ) g _ { i j } ( x , t ) , \\ \\ ( x , t ) \\in M \\times [ 0 , \\infty ) , \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { 3 p - 1 } ( s _ { g _ 0 } ( p ) - ( 2 p - 1 ) \\delta _ 0 ) . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\theta \\circ T ^ n = \\omega _ \\xi \\ , . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} - u '' ( x ) + x ^ 2 u ( x ) = \\lambda u ( x ) , \\ ; \\ ; x \\in \\R , \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} d \\tilde { R } ( t ) = \\frac 1 2 \\left ( \\lambda \\frac { \\beta } { 4 } e ^ { - \\frac { \\beta } { 4 } t } \\cosh \\tilde { R } ( t ) + \\tanh \\tilde { R } ( t ) \\right ) d t + d B _ t . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} g \\circ I = - \\frac { 1 } { \\bar g } \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( \\sum _ { i , j = 1 } ^ N a _ { i j } \\frac { \\partial u _ n } { \\partial x _ j } \\frac { \\partial v } { \\partial x _ i } + \\sum _ { i = 1 } ^ N \\left ( b _ i \\frac { \\partial u _ n } { \\partial x _ i } v + c _ i \\frac { \\partial v } { \\partial x _ i } u _ n \\right ) + d u _ n v \\right ) + \\int _ { \\Omega } ( g _ n \\circ u _ n ) v = \\int _ { \\Omega } f v , \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} \\beta ( t ) = \\alpha _ { 3 3 } ( t ) - \\tau _ { 3 2 } ( t ) \\tau _ { 3 2 } ( t ) ^ \\prime . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} A _ k - A _ k ^ * = i \\Phi ^ * \\sigma _ k \\Phi . \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} \\big \\langle \\omega , \\omega ' \\big \\rangle _ { Y _ I } = \\int _ I \\big \\langle \\omega _ u , \\omega ' _ u \\big \\rangle _ Y d u . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} \\begin{aligned} v _ 1 & : = y _ 1 \\cdots y _ { j _ 1 } , & & \\\\ v _ r & : = y _ { j _ { r - 1 } + 1 } \\cdots y _ { j _ r } & & r \\in \\{ 2 , \\ldots , m \\} , \\\\ v _ { m + 1 } & : = y _ { j _ m + 1 } \\cdots y _ \\ell . & & \\end{aligned} \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} \\lambda _ { k } = \\lambda _ { ( s ) } ( \\mathbf { x } ^ { k } ) = 2 ( 1 - \\beta ) \\varphi _ { s } ( \\mathbf { x } ^ { k } ) / ( \\| \\mathbf { d } ^ { k } _ { s } \\| ^ { 2 } L _ { s } ) \\geq 2 ( 1 - \\beta ) \\varphi _ { s } ( \\mathbf { x } ^ { k } ) / ( \\rho ^ { 2 } L ) , \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} g _ \\infty ( z ) = \\begin{cases} \\displaystyle \\Biggl ( \\int _ 0 ^ 1 { 1 \\over z + W ( s ) } d s \\Biggl ) ^ { - 1 } & \\displaystyle { 1 \\over z + W ( s ) } \\cr \\cr 0 & . \\end{cases} \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} W = U \\oplus V _ 1 \\oplus \\ldots \\oplus V _ { p - 1 } \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} b = P \\mu + b _ 0 , \\| \\mu \\| _ \\infty \\le C \\| b \\| _ { \\mathcal B } , b _ 0 \\in \\mathcal B _ 0 . \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\left . h _ { N , s } ( z _ 1 , \\dots , z _ s ) \\right \\vert _ { z _ s = 1 } = h _ { N , s - 1 } ( z _ 1 , \\dots , z _ { s - 1 } ) . \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} \\partial ^ { \\tau } A _ { \\tau } + \\kappa u ^ { \\nu } u ^ { \\tau } \\partial _ { \\nu } A _ { \\tau } = \\left ( g ^ { \\nu \\tau } + \\kappa u ^ { \\nu } u ^ { \\tau } \\right ) \\partial _ { \\nu } A _ { \\tau } = 0 , \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} T = T ^ w : B _ 1 ^ + \\rightarrow \\R ^ { n + 1 } , y = T ( x ) = ( x '' , \\partial _ { n } w ( x ) , \\partial _ { n + 1 } w ( x ) ) . \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} T _ k ( \\xi ) = \\psi ^ { - 1 } ( 0 , \\ \\cdots , \\ 0 , \\ \\xi , \\ 0 , \\ \\cdots , \\ 0 ) , \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} \\begin{cases} \\sigma _ k ( D ^ 2 u ) = 0 , & , \\\\ u = f , & , \\end{cases} \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\delta ( t , x ) = \\int _ 0 ^ \\infty ( 1 - \\frac { u ( t , x , y ) } { U ( t , x ) } ) d y . \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} x \\oplus y = \\begin{cases} x + y , & x + y < 1 \\\\ x + y - 1 , & x + y \\geq 1 \\end{cases} \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} L ( s , S _ f ^ \\nu ) & = \\sum _ { n \\geq 1 } \\sum _ { h \\geq 0 } \\frac { a _ f ( n ) } { n ^ \\nu ( n + h ) ^ s } \\\\ & = L ^ \\nu ( s , f ) + \\frac { 1 } { 2 \\pi i } \\int _ { ( \\gamma ) } L ^ \\nu ( s - w , f ) \\zeta ( w ) \\frac { \\Gamma ( w ) \\Gamma ( s - w ) } { \\Gamma ( s ) } d w , \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} H _ { a , b , c } ^ { ( r ) } : = \\frac { A _ { a , b , c } ( r ) } { A _ { a , b , c } } . \\end{align*}"}
-{"id": "9647.png", "formula": "\\begin{align*} \\frac { \\left ( - b ; q ^ { 2 } \\right ) _ { \\infty } } { \\left ( b ; q \\right ) _ { \\infty } } = \\sqrt { \\frac { \\log q ^ { - 1 } } { \\pi } } \\int _ { - \\infty } ^ { \\infty } { } _ { 1 } \\phi _ { 1 } \\left ( \\begin{array} { c c c } \\begin{array} { c } a \\\\ b \\end{array} & \\vert & q , - b q ^ { \\alpha } \\end{array} \\right ) \\left ( - a b q ^ { - \\alpha } ; q \\right ) _ { \\infty } q ^ { \\alpha ^ { 2 } } d \\alpha . \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} K ^ { ( \\beta ) } = \\biguplus _ { m \\in \\omega } K _ m ^ { ( \\beta ) } \\uplus \\{ b \\} . \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\gamma _ { 1 j } \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } - \\gamma _ { 1 k } \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 j } = \\Phi A _ 1 \\Phi ^ * \\left ( \\sigma _ j \\sigma _ 1 ^ { - 1 } \\sigma _ k - \\sigma _ k \\sigma _ 1 ^ { - 1 } \\sigma _ j \\right ) \\Phi A _ 1 ^ * \\Phi ^ * . \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} \\begin{aligned} & g _ \\varepsilon ^ { ( s ) } ( t ) = \\sum _ { k = 0 } ^ \\infty \\ell ^ { - k } \\times \\\\ & \\times \\int _ 0 ^ t \\int _ 0 ^ { t _ 1 } \\dots \\int _ 0 ^ { t _ { k - 1 } } \\tilde { T } _ s ( t - t _ 1 ) \\tilde { C } _ { s + 1 } \\dots \\tilde { T } _ { s + k } ( t _ k ) g _ \\varepsilon ^ { ( s + k ) } ( 0 ) d t _ k \\dots d t _ 1 \\\\ & \\textnormal { ( i f } s \\geq m - 1 \\textnormal { ) } \\end{aligned} \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} { Z } ' _ t ( \\omega ) = \\left \\{ \\begin{array} { l l } \\underset { s \\in { \\mathbb Q } , s \\downarrow t } \\lim { Z } _ s ( \\omega ) \\quad & \\omega \\in \\Omega '' , \\\\ 0 \\quad & \\omega \\notin \\Omega '' . \\end{array} \\right . \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} f _ j ( x , y ) = c _ j x ^ { u _ j } y ^ { v _ j } \\prod _ { i = 1 } ^ { l _ j } ( y ^ a - \\alpha _ { i , j } x ^ b ) ^ { e _ { i , j } } , c _ j \\in L _ { v } ^ { \\times } . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} & \\mathfrak { a } _ { \\Omega , 4 m , a , b , q } ( f _ { K , \\Omega , j } , f _ { K , \\Omega , j } ) ) - \\lambda \\ , \\mathfrak { b } _ { \\Omega , 2 m , a , b , q } ( f _ { K , \\Omega , j } , f _ { K , \\Omega , j } ) \\\\ & = ( \\lambda _ { K , \\Omega , j } - \\lambda ) \\| f _ { K , \\Omega , j } \\| _ { L ^ 2 ( \\Omega ) } ^ 2 < 0 , \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} 4 \\pi T _ { \\mu } { } ^ { \\nu } = F _ { \\mu \\lambda } \\epsilon ^ { \\lambda \\nu \\sigma \\tau } F _ { \\sigma \\tau } + \\frac { 1 } { 4 } \\delta _ { \\mu } ^ { \\nu } F _ { \\sigma \\tau } \\epsilon ^ { \\sigma \\tau \\lambda \\rho } F _ { \\lambda \\rho } \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} h _ 0 ( \\nabla _ x \\varphi ( x , \\xi ) ) + V ( x ) = h _ 0 ( \\xi ) \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\Sigma _ s ( i , j ) = \\left \\{ X _ s \\in \\mathbb { R } ^ { d s } \\left | | x _ i - x _ j | = \\varepsilon \\right . \\right \\} \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} \\varphi ( x _ 1 ) = \\chi ( x _ 1 ) , \\ \\ldots , \\ \\varphi ( x _ i ) = \\chi ( x _ i ) \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} K _ { \\alpha _ { 1 } \\ldots \\alpha _ { r } } ^ { \\gamma } = K _ { \\alpha _ { 1 } \\alpha _ { 2 } } ^ { \\beta _ { 1 } } K _ { \\beta _ { 1 } \\alpha _ { 3 } } ^ { \\beta _ { 2 } } \\cdots K _ { \\beta _ { r - 3 } \\alpha _ { r - 1 } } ^ { \\beta _ { r - 2 } } K _ { \\beta _ { r - 2 } \\alpha _ { r } } ^ { \\gamma } . \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} H = \\frac { \\kappa } { 2 f } \\ , n _ 1 + \\frac { f f '' + ( f ' ) ^ 2 - 1 } { 2 f \\sqrt { 1 - f '^ 2 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} D _ 2 = S _ 2 \\times S _ 2 = G ( 2 , 2 , 2 ) , & & B _ 2 = G ( 2 , 1 , 2 ) \\cong D i h _ 4 = I _ 2 ( 4 ) = G ( 4 , 4 , 2 ) . \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} f ^ * ( t ) = & \\left ( \\frac { 1 } { 3 p - 1 } ( s _ { f } ( p ) + \\frac { 1 - p } { 2 } \\delta ) + \\frac { 1 } { 2 } ( - f ( t + p ) + f ( t ) ) \\right ) { \\bf 1 } _ { [ 0 , 1 - p ] } ( t ) \\\\ & + \\frac { 1 } { 2 p - 1 } ( s _ f ( p ) - \\frac { 1 - p } { 3 p - 1 } ( s _ { f } ( p ) - ( 2 p - 1 ) \\delta ) ) { \\bf 1 } _ { ( 1 - p , p ) } ( t ) \\\\ & + \\left ( \\frac { 1 } { 3 p - 1 } ( s _ { f } ( p ) + \\frac { 1 - p } { 2 } \\delta ) + \\frac { 1 } { 2 } ( f ( t + p ) - f ( t ) ) \\right ) { \\bf 1 } _ { [ p , 1 ] } ( t ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} g _ { k } ^ { + } ( I _ { + } ) = \\frac { \\mu _ { + , \\pm } ^ { \\prime } ( e _ { + } ) \\omega _ { + } \\left ( I _ { + } \\right ) \\phi _ { k } ^ { + } ( I _ { + } ) } { \\omega _ { + } \\pm \\lambda / i k } \\pm v > 0 , \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} P _ t f ( x , y ) = ( P _ t ^ 1 P _ t ^ 2 f ) ( x , y ) . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\dim \\xi = \\dim \\rho _ { a , b } = \\frac { ( a - b + 1 ) ( a + b + 3 ) ( a + 2 ) ( b + 1 ) } { 6 } . \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} E _ c : = \\left \\{ ( x , y ) \\in \\R ^ { d } \\times \\R ^ r \\mid | x | \\in [ 1 , 2 ] , | x | ^ { d / r } y _ j \\in [ 0 , c ] j = 1 , \\dots , r \\right \\} \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} \\begin{array} { r l } v _ t + \\lambda _ 0 v \\cdot \\nabla v & = f - \\nabla p + \\lambda _ 1 \\nabla | v | ^ 2 - ( \\alpha + \\beta | v | ^ 2 ) v + \\Gamma _ 0 \\Delta v - \\Gamma _ 2 \\Delta ^ 2 v \\\\ { \\mathrm { d i v \\ , } } v & = 0 \\\\ v ( 0 ) & = v _ 0 \\end{array} \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} \\tilde { h } _ { i j } \\tilde { v } = ( 1 - r ^ 2 ) h _ { i j } v , \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} \\mathcal { B } _ { I V } ^ + = \\left \\{ \\begin{aligned} & ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { A } ^ + \\backslash \\mathcal { B } _ { I I } \\textnormal { s u c h t h a t } \\\\ & \\exists \\ ; i \\in \\left \\{ 1 , 2 , \\dots , s + k \\right \\} \\backslash \\left \\{ i _ { k + 1 } \\right \\} , \\ ; t ^ \\prime \\geq 0 \\ ; : \\ ; \\left | v _ { i _ { k + 1 } } ^ { \\prime * } - v _ i ^ \\prime ( \\tau ; t ^ \\prime ) \\right | \\leq \\eta \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} R ^ * \\psi & = \\sum _ { \\beta \\neq 0 } ( - 1 ) ^ { | \\alpha | + | \\beta | } \\ , D ^ \\alpha _ x \\Big ( r _ { \\alpha , \\beta } ( x ) \\ , D ^ \\beta _ { \\xi } \\psi ( x , \\xi ) \\Big ) = : \\sum _ { \\beta \\neq 0 } r ^ * _ { \\alpha , \\beta } ( x ) \\ , D ^ \\alpha _ x D ^ \\beta _ { \\xi } \\psi ( x , \\xi ) . \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} K = \\frac { 1 } { W ^ { 2 } } \\sum \\limits _ { k = 1 } ^ { n - 2 } ( L _ { 1 1 } ^ { k } L _ { 2 2 } ^ { k } - ( L _ { 1 2 } ^ { k } ) ^ { 2 } ) . \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} & a _ { ( n ) } \\Gamma ^ q M \\subset \\Gamma ^ { p + q - n - 1 } M a \\in F ^ p V , \\ \\forall n \\in \\Z , \\\\ & a _ { ( n ) } \\Gamma ^ q M \\subset \\Gamma ^ { p + q - n } M a \\in F ^ p V , \\ n \\geq 0 , \\\\ & H . \\Gamma ^ p M \\subset \\Gamma ^ p M p \\geq 0 , \\\\ & \\bigcap _ p \\Gamma ^ p M = 0 . \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} k _ L = \\log _ { 1 / p } n + ( 1 - \\epsilon ) \\psi ( n ) \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} \\dot { \\tilde { x } } = - \\varPhi \\tilde { \\nu } , \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{gather*} \\int \\limits _ { O _ v ^ { \\times 2 } } \\chi ( a c ( F ^ { ( m ) } ( x , \\theta + \\pi ^ { k } y ) ) ) \\ | F ^ { ( m ) } ( x , \\theta + \\pi ^ { k } y ) | ^ s \\ | d x d y | \\\\ = q ^ { - ( D _ { i + 1 } - d _ 0 ) m s - \\varepsilon _ { i + 1 } k s } \\int \\limits _ { O _ v ^ { \\times 2 } } \\chi ( a c ( g ( x , y ) ) \\ | g ( x , y ) | ^ s \\ | d x d y | . \\end{gather*}"}
-{"id": "3349.png", "formula": "\\begin{align*} \\dim \\delta H _ 3 ^ g = & \\ ; 8 + 1 2 k + 6 ( k - 1 ) ^ 2 + { ( k - 1 ) ^ 3 } - \\left ( { ( k + 1 ) ^ 3 + 3 } \\right ) \\\\ = & \\ ; 9 , \\\\ \\dim \\delta E _ 3 ^ g = & \\ ; I _ M ( V _ 3 ^ g \\times W _ 3 ^ g ) = \\ ; 6 , \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{gather*} { Q _ { r } ( x ) } / { P _ { r } ( x ) } = \\sum \\nolimits _ { k \\geq 0 } \\ { s _ { k } } { x ^ { - k - 1 } } \\ \\ . \\end{gather*}"}
-{"id": "6137.png", "formula": "\\begin{align*} C _ j ^ p = \\left ( \\begin{array} { c c } S _ j ^ p & 0 \\\\ 0 & - S _ j ^ { p - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} H ^ { s } _ { 0 } ( \\Omega ) : = \\mathrm { c l o s } \\big ( C _ { 0 } ^ { \\infty } ( \\Omega ) , \\| \\cdot \\| _ { H ^ { s } } \\big ) s \\in \\mathbb { N } , \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} g _ { k } ^ { - } ( I _ { - } ) = - \\frac { \\mu _ { - , \\pm } ^ { \\prime } ( e _ { - } ) \\omega _ { - } \\left ( I _ { - } \\right ) \\phi _ { k } ^ { - } ( I _ { - } ) } { \\omega _ { + } \\pm \\lambda / i k } \\pm v > 0 . \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} P ^ 0 ( T _ { \\partial S _ i } ( t ) < \\infty ) = E ^ 0 \\left [ P ^ 0 ( T _ { \\partial S _ i } ( t ) < \\infty | Z ( t ) ) \\right ] = E ^ 0 \\left [ P ^ { Z ( t ) } ( T _ { \\partial S _ i } < \\infty ) \\right ] . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial \\rho } = \\frac { 1 } { 2 } \\zeta ' ( 0 , D ) - \\rho \\zeta ( 0 , D ) \\end{align*}"}
-{"id": "10100.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) z ^ 2 } { x ^ { 4 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z } { x ^ { 3 } } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 2 z } { x ^ { 5 } } . \\\\ \\end{gather*}"}
-{"id": "4101.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\log ( \\mu ( u , e ^ { T _ 0 } e ^ { S _ 0 } ) ) \\ , d u & = \\log ( \\Delta _ \\tau ( e ^ { T _ 0 } e ^ { S _ 0 } ) ) \\\\ & = \\log ( \\Delta _ \\tau ( e ^ { T _ 0 } ) ) + \\log ( \\Delta _ \\tau ( e ^ { S _ 0 } ) ) = \\tau ( T _ 0 ) + \\tau ( S _ 0 ) . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} x _ l = \\sum _ { l m \\leq N ^ { \\Omega } } \\frac { \\mu * \\mu ( m ) } { m } y _ { l m } , \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x ^ { \\nu } } T _ { \\mu } { } ^ { \\nu } + X _ { \\mu } = - \\frac { 1 } { c } F _ { \\mu \\lambda } j ^ { \\lambda } , \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} R ^ - ( n , k ) = 2 k R ^ - ( n - 1 , k ) + ( 2 n - 4 k + 3 ) R ^ - ( n - 1 , k - 1 ) + R ( n - 1 , k - 1 ) , \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} V _ N = 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\log X + \\log Y + \\log Y ' + o ( 1 ) . \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} U _ { ( x , n _ 1 , n _ 2 ) } = \\left ( h _ E ( x ) ( n _ 1 ^ * n _ 1 n _ 2 ^ * n _ 2 ) \\right ) ^ { - 1 / 2 } p _ x n _ 1 ^ * n _ 2 p _ x \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\partial ^ { 2 } _ { P _ i } : ( \\Z / m ) ^ r \\to H ^ 3 ( M / G , \\Z ) \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { k \\in I _ j } w _ { k } ^ { 2 } \\sum _ { l = 1 } ^ { L _ k } \\overline { z } _ { k , l } f _ { k , l } = 0 . \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} H ^ { 2 n - 1 } ( X ^ \\circ , \\Z ) = H _ 1 ^ { B M } ( X ^ \\circ , \\Z ) \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} \\dot { u } _ i & = v _ i , \\\\ \\dot { v } _ i & = \\frac { 1 } { \\rho } \\big ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\theta - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } - B _ { i J K j } v _ { j } ) _ { , K } \\big ) _ { , J } , \\\\ \\dot { \\tau } & = \\theta , \\\\ \\dot { \\theta } & = \\frac { 1 } { a } \\big ( - \\beta _ { K i } v _ { i , K } + ( m _ { I J } \\theta _ { , J } + M _ { j L K I } u _ { j , L K } + K _ { I J } \\tau _ { , J } ) _ { , I } \\big ) . \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\left ( g _ { \\lambda \\rho } - \\frac { \\kappa } { 1 + \\kappa } u _ { \\lambda } u _ { \\rho } \\right ) \\left ( g ^ { \\lambda \\sigma } + \\kappa u ^ { \\lambda } u ^ { \\sigma } \\right ) = \\delta _ { \\rho } ^ { \\sigma } , \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{gather*} \\psi _ { a } ^ { \\pm } ( w ) = \\sum _ { k \\in \\mathbb { Z } } { } _ { a } \\psi _ { ( k ) } ^ { \\pm } w ^ { - k - 1 } , a = 0 , 1 , \\dots , n - 1 , \\end{gather*}"}
-{"id": "3618.png", "formula": "\\begin{align*} \\phi \\big ( M ( D _ i : i \\in I ) \\big ) = \\phi _ 1 \\big ( M ( a _ { 1 ; i } : i \\in I ) \\big ) \\cdots \\phi _ K \\big ( M ( a _ { K ; i } : i \\in I ) \\big ) . \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} u _ i ( \\cdot , 0 ) = u _ i ^ 0 , \\dot { u } _ i ( \\cdot , 0 ) = \\dot { u } _ i ^ 0 , \\tau ( \\cdot , 0 ) = \\tau ^ 0 , \\dot { \\tau } ( \\cdot , 0 ) = \\dot { \\tau } ^ 0 \\Omega , \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} Z ( \\gamma ' ) = \\kappa J ^ { 9 0 } _ \\gamma ( \\gamma ' ) , \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} \\alpha _ 0 = \\min \\{ \\alpha _ 0 ^ 1 , \\alpha _ 0 ^ 2 \\} , \\end{align*}"}
-{"id": "9928.png", "formula": "\\begin{align*} V ^ { \\lambda } ( A ) = \\bigoplus _ { n \\delta _ 1 + \\delta _ 2 = \\lambda } V ^ { \\delta _ 1 , \\delta _ 2 } . \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} \\tilde { u } _ { \\nu } = u _ { \\nu } \\left [ \\lambda \\delta + \\bar { \\lambda } \\left ( 1 - \\beta _ { 0 } x ^ { - 1 } \\right ) u _ { 0 } - \\bar { \\lambda } \\sum \\nolimits _ { \\nu = 0 } ^ { d - 2 } \\gamma _ { 1 } ^ { d - 1 - \\nu } \\left ( x ^ { - 1 } u _ { \\nu + 1 } \\right ) \\right ] ^ { - 1 } \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} \\begin{cases} U _ t = A U , \\\\ U ( 0 ) = U _ 0 , \\end{cases} \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} f _ w ( a ) : = \\lim _ { t \\to \\infty } \\frac { | \\zeta ^ { t } ( a ) | _ w } { | \\zeta ^ { t } ( a ) | } \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} a ( \\chi ) + b ( \\chi ) = n _ K . \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } ^ 0 \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & \\ ; \\ ; = \\left ( X _ { s + k } ^ \\prime , V _ { s + k } ^ \\prime \\right ) \\in \\mathbb { R } ^ { 2 d ( s + k ) } \\end{aligned} \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} \\frac { \\mu + n \\xi } { \\beta _ n } \\mathbb { E } ( S \\mid X _ { 0 } = ( 1 , n ) ) & = \\frac { \\mu } { \\beta _ { n } } \\mathbb { E } ( S _ { 1 , n - 1 } ) + \\frac { n \\xi } { \\beta _ { n } } \\mathbb { E } ( S _ { 1 , n - 1 } ) \\\\ & = \\frac { \\mu } { \\beta _ { n } } \\mathbb { E } ( S _ { 1 , n - 1 } ) + \\frac { n \\xi } { \\beta _ { n } } \\left ( \\frac { 1 } { n } \\times 0 + \\frac { n - 1 } { n } \\mathbb { E } ( S _ { 1 , n - 1 } ) \\right ) , \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} v ( x ) = v _ 0 , p ( x ) = p _ 0 - ( \\alpha + \\beta | v _ 0 | ^ 2 ) v _ 0 \\cdot x , x \\in \\Omega , \\ , p _ 0 \\in \\mathbb { R } , \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} \\Phi ( \\tau ) = \\mathcal { T } ( - ( t - \\tau ) ) \\varphi \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} m _ { z _ 0 } = \\frac { x _ { z _ 0 } ^ 2 } { 2 } , m _ { z _ i } = \\frac { x _ { z _ { i + 1 } } } { x _ { z _ 0 } } + \\frac { x _ { z _ i , z _ 0 , z _ 0 } } { x _ { z _ 0 } } , i = 1 , \\dots , n . \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} ( f ( 2 m _ 0 + 2 s ) + 1 ) ( f ( 2 n _ 0 + 2 k + 1 ) + 1 ) ( f ( 2 n _ 0 + 2 l + 1 ) + 1 ) = 0 , \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} & 0 \\to L ( 1 ) \\to F \\otimes L ( 2 ) \\to L ( 3 ) \\to 0 , \\\\ & 0 \\to ( L ^ { - 1 } \\otimes \\omega _ C ) ( 1 ) \\to F \\otimes ( L ^ { - 1 } \\otimes \\omega _ C ) ( 2 ) \\to ( L ^ { - 1 } \\otimes \\omega _ C ) ( 3 ) \\to 0 . \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} E _ f [ u _ k ] - E _ f [ u ] = & \\frac { 1 } { 2 } \\int _ M | \\nabla \\Delta ( u - u _ k ) | _ g ^ 2 - \\frac { 1 } { 2 ^ \\sharp } \\int _ M f | u - u _ k | ^ { 2 ^ \\sharp } d \\mu _ g + o ( 1 ) \\\\ = & E _ f [ u - u _ k ] + o ( 1 ) . \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} \\left \\langle \\partial \\varphi , \\partial \\varphi \\right \\rangle = \\int _ { D } \\sum _ { \\alpha = 1 } ^ { d } \\left ( \\frac { \\partial \\varphi } { \\partial x _ { \\alpha } } \\right ) ^ { 2 } d x . \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} \\begin{aligned} h \\big ( c ( y _ k ) + \\nabla c ( y _ k ) ( x _ k - y _ k ) \\big ) + g ( x _ k ) & \\le h \\big ( c ( y _ k ) + \\nabla c ( y _ k ) ( w _ k - y _ k ) \\big ) \\\\ & + \\frac { \\tilde { \\mu } _ k } { 2 } \\left ( \\norm { w _ k - y _ k } ^ 2 - \\norm { w _ k - x _ k } ^ 2 - \\norm { x _ k - y _ k } ^ 2 \\right ) \\\\ & + a _ k g ( v _ k ) + ( 1 - a _ k ) g ( x _ { k - 1 } ) \\end{aligned} \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} \\dot { S } _ { i j } - F ^ { k l } S _ { i j ; k l } = & \\ , F ^ { k l } h _ { r k } h ^ r _ l h _ { i j } - 2 F h _ i ^ k h _ { k j } \\\\ & + K _ N \\{ 2 F g _ { i j } - F ^ { k l } g _ { k l } h _ { i j } \\} + 2 F h _ { i j } + F ^ { k l , r s } h _ { k l ; i } h _ { r s ; j } \\\\ \\equiv & \\ , N _ { i j } + \\tilde { N } _ { i j } , \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} 0 \\leq \\underset { k \\in \\N } { \\sup } \\frac { \\beta _ k ^ p } { \\alpha _ k ^ 2 } = \\Lambda < \\infty \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\mathbb { E } [ \\| v _ i ^ { k } - y ^ { k } \\| ] \\le \\sqrt { N } \\sqrt { \\sum _ { i = 1 } ^ N \\mathbb { E } [ \\| v _ i ^ { k } - y ^ { k } \\| ^ 2 ] } . \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} c _ { p } = \\frac { 1 } { 4 \\pi ^ { 2 } p ( 2 n - p ) } \\left ( q _ { p } + \\sum _ { k = 1 } ^ { p - 1 } \\sum _ { n _ { 1 } , n _ { 2 } , . . . , n _ { k } } \\frac { q _ { n _ { 1 } } q _ { n _ { 2 } } . . . q _ { n _ { k } } q _ { p - n ( k ) } } { b ( n , p , k ) } \\right ) , \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} \\gamma ( t ) = x _ 0 + t ^ k \\mathbf { v } + \\mathcal { O } ( t ^ { k + 1 } ) . \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} D _ h = ( B \\setminus C _ h ) \\cup ( T ^ h B \\setminus C _ h ) . \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Upsilon _ { n } ^ { \\delta } ( z _ { n } ) = \\Upsilon ^ { \\delta } ( z ) \\ , . \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} a ( u ) = \\left ( a _ { 1 } ( u ) , . . . , a _ { n } ( u ) ; 0 \\right ) , \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} I _ 3 = & - 4 ( n - 6 ) ( n - 4 ) \\Big [ ( n - 6 ) \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j r ^ { 2 - n } + 4 ( n - 2 ) r ^ { - n } ( A _ { i j , k l } ( p ) x ^ i x ^ j x ^ k x ^ l ) \\Big ] \\\\ & - 4 ( n - 6 ) ( n - 4 ) ( n - 1 0 ) r ^ { 2 - n } \\sigma _ 1 ( A ) _ { , i } x ^ i + O ( r ^ { 5 - n } ) \\\\ = & - 8 ( n - 8 ) ( n - 6 ) ( n - 4 ) \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j r ^ { 2 - n } \\\\ & - 1 6 ( n - 6 ) ( n - 4 ) ( n - 2 ) r ^ { - n } ( A _ { i j , k l } ( p ) x ^ i x ^ j x ^ k x ^ l ) + O ( r ^ { 5 - n } ) . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\dd N _ t = \\hat \\lambda _ t \\ , \\dd t + \\dd \\hat m _ t , \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} \\Delta _ L = \\Delta _ t - ( a ^ 2 - 1 ) \\partial _ { v v } ^ L - b \\partial _ { v } ^ L , \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} z _ { n + 1 } = q ^ \\prime ( z _ 1 , \\dotsc , z _ n ) + ( ) . \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{gather*} \\big ( D _ { x } D _ { t } + D _ { x } ^ { 4 } \\big ) \\tau \\cdot \\tau = 0 , \\end{gather*}"}
-{"id": "7489.png", "formula": "\\begin{align*} V ' ( r ) & = - \\frac { 1 } { f _ a ^ { n - 1 } ( r ) } \\int _ 0 ^ r a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t < 0 , \\\\ V '' ( r ) & = ( n - 1 ) \\frac { f _ a ^ \\prime ( r ) } { f _ a ^ n ( r ) } \\int _ 0 ^ r a _ 0 ( t ) f _ a ^ { n - 1 } ( t ) d t - a _ 0 ( r ) , \\\\ \\noalign { a n d } \\left | \\nabla V \\big ( \\rho ( x ) \\big ) \\right | & = \\left | V ' \\big ( \\rho ( x ) \\big ) \\nabla \\rho ( x ) \\right | = \\left | V ' \\big ( \\rho ( x ) \\big ) \\right | . \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} \\Phi _ A ( \\psi ) = \\sup _ { x \\in A } \\psi ( x ) . \\end{align*}"}
-{"id": "10104.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } y ^ { q } ( a x + b y + c z ) ^ r } { z ^ { p + q + r } } , \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\begin{cases} \\dot { u } = \\log \\det ( u _ { \\alpha \\bar { \\beta } } ) + f ( z , t ) \\ ; \\ ; \\mbox { o n } \\ ; \\ ; \\Omega \\times ( 0 , T ) , \\\\ u = \\varphi \\ ; \\ ; \\ ; \\mbox { o n } \\ ; \\ ; \\partial \\Omega \\times [ 0 , T ) . \\end{cases} \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} W ( f , g ) = \\lim _ { n \\rightarrow \\infty } \\left ( f _ { n + 1 } g _ { n } - f _ { n } g _ { n + 1 } \\right ) \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} \\left ( p q , ( p q ) ^ { 1 / 2 } z , ( p q ) ^ { 1 / 2 } / z ; p q \\right ) _ { \\infty } = \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( q , q ^ { 1 / 2 } / ( z e ^ { i x } ) , z q ^ { 1 / 2 } e ^ { i x } ; q \\right ) _ { \\infty } \\exp \\left ( \\frac { x ^ { 2 } } { \\log p ^ { 2 } } \\right ) } { \\sqrt { \\pi \\log p ^ { - 2 } } } d x . \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} h _ { N , N } ( z _ 1 , \\dots , z _ N ) = \\frac { Z _ N ( \\boldsymbol { \\lambda } ) } { Z _ N } \\prod _ { j = 1 } ^ { N } \\left [ \\frac { a } { a ( \\lambda _ j ) } \\right ] ^ { N - 1 } , \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { q ^ { ( n ^ 2 + n ) / { 2 } } ( - q ; q ) _ n } { ( q ; q ) ^ 2 _ n } = \\lim _ { b \\rightarrow - 1 } \\lim _ { \\rho \\rightarrow \\infty } \\frac { ( b ; q ) _ \\infty ( q ^ 2 b ^ 2 / \\rho ; q ) _ \\infty } { ( q b ^ 2 ; q ) _ \\infty ( q b / \\rho ; q ) _ \\infty } \\sum _ { n \\geq 0 } \\frac { ( q b ; q ) _ n } { ( q ^ 2 b ^ 2 / \\rho ; q ) _ n } b ^ n . \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} _ { 1 } \\phi _ { 2 } \\left ( \\begin{array} { c c c } \\begin{array} { c } a \\\\ b \\sqrt { q } , - b \\sqrt { q } \\end{array} & \\vert & q , z q \\end{array} \\right ) = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 4 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( a z e ^ { i x } ; q \\right ) _ { \\infty } \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 4 } } \\right ) d x } { \\left ( z e ^ { i x } ; q \\right ) _ { \\infty } \\left ( b ^ { 2 } q , - b ^ { 2 } e ^ { i x } ; q ^ { 2 } \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} x _ k = \\theta _ 1 x _ { k - 1 } + \\theta _ 2 x _ { k - 2 } + \\cdots + \\theta _ p x _ { k - p } + w _ k = \\boldsymbol { \\theta } ^ T \\mathbf { x } _ { k - p } ^ { k - 1 } + w _ k , \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } ( p ) & = ( \\varphi ^ { - 1 } \\circ \\psi ^ { - 1 } ) ( | p _ 0 : p _ 1 : p _ 2 | ) \\\\ & = \\{ [ \\sigma _ 0 p _ 0 ^ { 1 / a _ 0 } : \\sigma _ 1 p _ 1 ^ { 1 / a _ 1 } : \\sigma _ 2 p _ 2 ^ { 1 / a _ 2 } ] \\mid \\sigma _ i \\in \\mu ^ { a _ i } \\} \\\\ & = \\{ g \\cdot [ p _ 0 ^ { 1 / a _ 0 } : p _ 1 ^ { 1 / a _ 1 } : p _ 2 ^ { 1 / a _ 2 } ] \\mid g \\in G \\} \\end{align*}"}
-{"id": "3266.png", "formula": "\\begin{gather*} \\det \\big ( V ^ { k } _ { \\{ z _ { i } \\} } \\big ) = \\prod _ { k \\ge \\alpha > \\beta \\ge 1 } ( z _ { \\alpha } - z _ { \\beta } ) . \\end{gather*}"}
-{"id": "8965.png", "formula": "\\begin{align*} P _ 0 ( s ) = B _ t ( P _ 0 ( \\cdot ) ) ( s ) + R _ 0 ( s ) , \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} p _ T ( y ) = \\sum _ { 1 \\le i _ 1 \\le \\dots \\le i _ r \\le n } t _ { i _ 1 \\dots i _ r } y _ { i _ 1 } \\cdots y _ { i _ r } , y = \\sum _ { i = 1 } ^ n y _ i e _ i . \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} \\Delta _ k ( L , B ) = \\sum _ { j = 0 } ^ { \\lfloor p / 2 \\rfloor } Q _ L ^ j \\pi _ { L ^ \\perp } ^ * T ^ { ( p - 2 j ) } ( L , B ) \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} r _ { Z _ N } ( t ) - r _ { Z _ N } ( 0 ) = \\sum _ k \\left | \\omega _ k \\cdot \\left ( v _ { j _ k } ( t _ k ^ - ) - v _ { i _ k } ( t _ k ^ - ) \\right ) \\right | \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\gamma _ { j k } = \\sigma _ j \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } - \\sigma _ k \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 j } . \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} ( T _ \\psi ) ^ n = P _ { \\psi _ n } J _ n \\ , , \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} \\langle \\Pi _ x , z \\rangle ^ { \\gamma } = \\langle \\Pi _ x , \\rho \\rangle ^ { \\gamma } = \\langle \\Pi _ x ^ { \\gamma } , \\rho ^ { \\gamma } \\rangle = \\langle \\Pi _ x , \\rho \\rangle , \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} W ( p ) \\le C _ 2 = 1 + \\frac { 1 6 \\bigl ( m _ u - u ( o ) \\bigr ) } { r } . \\end{align*}"}
-{"id": "9954.png", "formula": "\\begin{align*} \\pi _ { p } ( v _ { i } ) _ { i - 1 } = \\pi _ { p } ( ( v _ { i } ) _ { i - 1 } ) = \\pi _ { p } ( v _ { i - 1 } ) . \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} h _ { \\pi ( 1 ) } + \\dots + h _ { \\pi ( i ) } & = h _ { \\pi ' ( 1 ) } + \\dots + h _ { \\pi ' ( i ) } \\\\ & \\leq c _ 1 ( \\pi ' ) + \\dots + c _ i ( \\pi ' ) . \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\int f _ { 0 } \\left ( v \\right ) d v = 1 , \\ \\int \\frac { f _ { 0 } ^ { \\prime } \\left ( v \\right ) } { v } = \\left ( \\frac { 2 \\pi } { P _ { 0 } } \\right ) ^ { 2 } , \\ \\ \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} \\mathcal H : = & \\{ ( v , w , p , T ) | v , w , p \\mbox { a n d } T \\mbox { a r e s p a t i a l l y p e r i o d i c i n a l l t h r e e v a r i a b l e s } \\\\ & \\mbox { a n d a r e e v e n , o d d , e v e n a n d o d d w i t h r e s p e c t t o } z \\mbox { v a r i a b l e } , \\mbox { r e s p e c t i v e l y } \\} , \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( q ; q \\right ) _ { n } q ^ { n ^ { 2 } } c ^ { n } S _ { n } \\left ( x q ^ { - n } ; q \\right ) S _ { n } \\left ( y q ^ { - n } ; q \\right ) } { \\left ( c x y q ; q \\right ) _ { n } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } c ^ { n } A _ { q } \\left ( c x q ^ { n } \\right ) A _ { q } \\left ( c y q ^ { n } \\right ) } { \\left ( q ; q \\right ) _ { n } \\left ( c x y q ; q \\right ) _ { \\infty } } . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} d S _ { \\pm } = \\left ( \\lambda ( 1 \\pm \\epsilon ) \\cosh _ { \\pm } ( S _ { \\pm } ) + \\frac { C _ n ^ 2 } { 2 } \\tanh _ { \\pm , \\epsilon } ( S _ { \\pm } ) \\right ) d t + C _ n d W _ t . \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} X + \\langle Y - \\bar y \\rangle = X + Y - \\bar y . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\dim \\mathrm { E x t } ^ 1 ( E ( \\mathbf { w } ) , E ( \\mathbf { w } ) ) = \\dim W ( d _ 1 , d _ 2 , c ) . \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} [ \\varphi ( \\xi ) ] ^ 2 + | \\Psi ( \\xi ) | ^ 2 = | \\xi | ^ 2 \\xi \\in G r _ k ( T _ x M ^ n ) . \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\gamma _ { 1 j } \\Phi = \\sigma _ 1 \\Phi A _ j ^ * - \\sigma _ j \\Phi A _ 1 ^ * . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} \\frac { d ^ 3 } { d x ^ 3 } f _ A ( x ) = \\frac { d ^ 3 } { d u ^ 3 } g ( A + x ) - \\frac { d ^ 3 } { d x ^ 3 } g ( A - x ) \\geq 0 \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( f \\ , H ) - ( f \\ , H ) [ m H ^ { 2 } - \\lambda ] = 0 , \\\\ H { \\rm g r a d } ( f \\ , H ) + \\frac { m } { 2 } ( f \\ , H ) { \\rm g r a d } \\ , H = 0 . \\end{cases} \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} i \\ , \\R \\cap \\Gamma = \\{ 2 m i \\Im ( \\tau ) : m \\in \\Z \\} . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} \\Vert \\widehat { \\beta } \\Vert _ { 0 } = | \\widehat T | = \\sum _ { i = 1 } ^ { n } \\frac { \\partial ( X _ { i } ^ { \\prime } ( \\widehat { \\beta } - \\beta ) ) } { \\partial \\varepsilon _ { i } } . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} \\tilde \\theta _ { T , H } = \\frac { X _ T ^ 2 - X _ 0 ^ 2 } { 2 \\int _ 0 ^ T X _ t ^ 2 d t } - \\left ( \\frac { 1 } { H \\Gamma ( 2 H ) T } \\int _ 0 ^ T X _ t ^ 2 d t \\right ) ^ { - \\frac { 1 } { 2 H } } . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} & [ L _ n + z , L _ m + z ' ] = ( \\{ m \\} - \\{ n \\} ) L _ { n + m } & & \\\\ & [ L _ n + z , G _ m + z ' ] = ( \\{ m + 1 \\} - \\{ n \\} ) G _ { n + m } + c \\frac { 1 } { q ^ { n - 2 } } \\frac { 1 + q ^ { 2 } } { 1 + q ^ { n } } \\frac { \\{ n + 1 \\} \\{ n \\} \\{ n - 1 \\} } { \\{ 1 2 \\} } \\delta _ { n + m , 0 } , \\ \\forall n > 0 \\\\ & [ G _ n + z , G _ m + z ' ] = 0 . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} \\sum _ { 0 \\leq k < n } \\frac { 2 a _ k } { ( k - n ) } t ^ { ( k - n ) / 2 } | _ { \\Lambda ^ { - 2 } } ^ { \\Lambda '^ { - 2 } } = \\sum _ { 0 \\leq k < n } \\frac { 2 a _ k } { ( k - n ) } { \\Lambda } ^ { n - k } ( e ^ { - \\tau ( n - k ) / 2 } - 1 ) \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} \\| u \\| _ { H _ \\sigma ( \\Omega ) } ^ 2 & = \\int _ \\Omega u _ { x x } ^ 2 + u _ { y y } ^ 2 + 2 ( 1 - \\sigma ) u _ { x y } ^ 2 + 2 \\sigma u _ { x x } u _ { y y } \\\\ & \\geq \\int _ \\Omega u _ { x x } ^ 2 + u _ { y y } ^ 2 + 2 ( 1 - \\sigma ) u _ { x y } ^ 2 - | \\sigma | ( u _ { x x } ^ 2 + u _ { y y } ^ 2 ) \\geq ( 1 - | \\sigma | ) \\| | \\nabla ^ 2 u | \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ s } \\ , \\varphi ( \\xi _ 1 , \\dots , \\xi _ s ) \\ , d \\nu _ \\mu ( \\xi _ 1 , \\dots , \\xi _ s ) = \\int _ { \\mathbb { R } } \\ , \\varphi ( \\lambda , \\dots , \\lambda ) \\ , d \\mu ( \\lambda ) , \\varphi \\in C _ c ( \\mathbb { R } ^ s ) , \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} h _ { N , s } ( z _ 1 , \\dots , z _ s ) \\Big | _ { z _ s = \\frac { 2 \\Delta t z _ j - 1 } { t ^ 2 z _ j } } \\propto z _ j , z _ j \\to 0 , j = 1 , \\dots , s - 1 , \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} \\sigma _ E ^ { m ( x ) } ( \\kappa ( \\sigma _ E ( x ) ) ) = \\sigma _ E ^ { l ( x ) } ( \\kappa ( x ) ) \\sigma _ E ^ { m ' ( x ) } ( \\kappa ^ { - 1 } ( \\sigma _ E ( x ) ) ) = \\sigma _ E ^ { l ' ( x ) } ( \\kappa ^ { - 1 } ( x ) ) \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\min _ x ~ \\| c ( x ) \\| \\textrm { s u b j e c t t o } l _ i \\leq x _ i \\leq u _ i \\quad \\textrm { f o r } i = 1 , \\ldots , m . \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} \\Phi _ { i j } ( \\epsilon ) = \\begin{cases} { 1 \\over \\lambda _ i ( \\epsilon ) } - \\langle a _ { i } ^ { \\epsilon } | G ^ { ( n ) } _ { 0 } | a _ { i } ^ { \\epsilon } \\rangle & \\textrm { i f $ i = j $ } \\\\ - \\ ; \\langle a _ { i } ^ { \\epsilon } | G ^ { ( n ) } _ { 0 } | a _ { j } ^ { \\epsilon } \\rangle & \\textrm { i f $ i \\neq j $ } . \\end{cases} \\ ; . \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} & \\left \\| \\sin \\Theta ( V , W ) \\right \\| _ F ^ 2 \\overset { \\eqref { e q : d e f _ s i n _ T h e t a _ F r o b e n i u s } } { = } \\| W ^ { \\intercal } V _ { \\perp } \\| _ F ^ 2 \\\\ = & \\| [ \\alpha ^ { ( r + 1 ) } ~ \\cdots ~ \\alpha ^ { ( p _ 2 ) } ] \\| _ F ^ 2 \\leq \\frac { \\| \\tilde { U } ^ { \\intercal } A W _ { \\perp } \\| _ F ^ 2 \\sigma _ { r } ^ 2 ( A W ) } { \\left ( \\sigma _ { r } ^ 2 ( A W ) - \\sigma ^ 2 _ { r + 1 } ( A ) \\right ) ^ 2 } . \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} \\bigoplus _ { i = 0 } ^ { x - 1 } T _ x ( i ) = \\bigoplus _ { i = 0 } ^ { x - 1 } H _ x ( i , \\phi ( i ) ) \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} ( \\mu \\bullet f ) \\bullet g - \\mu \\bullet ( f \\bullet g ) = \\sum _ { ( 1 \\leq j \\leq i - 1 ) \\vee ( m + i \\leq j \\leq m + 1 ) } ( - 1 ) ^ { ( m - 1 ) ( i - 1 ) + ( n - 1 ) ( j - 1 ) } ( \\mu \\bullet _ i f ) \\bullet _ j g \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} \\mu = \\frac { \\langle \\hat { v } , g _ s ^ { - 1 } \\hat { v } \\rangle } { \\langle \\hat { v } , h _ s ^ { - 1 } \\hat { v } \\rangle } = \\frac { \\langle b \\hat { v } , A _ g b \\hat { v } \\rangle } { \\langle b \\hat { v } , b \\hat { v } \\rangle } = \\frac { \\langle b \\hat { v } , A _ g A _ h ^ { - 1 } b \\hat { v } \\rangle } { \\langle b \\hat { v } , b \\hat { v } \\rangle } . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\hat { M } ( t ) \\subset B _ { \\tilde { \\rho } } ( 0 ) , \\tilde { \\rho } ( t ) = 2 c \\tilde { \\rho } _ - ( t ) . \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} d u ( b ( t ) , a ( t ) x ) = d u ( b ( t ) , z ) \\big | _ { z = a ( t ) x } + d u ( \\tau , a ( t ) , x ) \\big | _ { \\tau = b ( t ) } . \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\max \\ \\to \\ \\sum \\limits _ { i = 1 } ^ { n } \\langle \\mathbf { c } _ { i } , \\mathbf { x } _ { i } \\rangle \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} P _ \\pm u \\left [ x \\right ] = & ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } L ^ k \\left ( e ^ { i ( x - y ) \\cdot \\xi } \\right ) p _ \\pm ( y , \\xi ) u \\left [ y \\right ] d \\xi \\\\ = & ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( x - y ) \\cdot \\xi } ( L ^ * ) ^ k \\left ( p _ \\pm ( y , \\xi ) \\right ) u \\left [ y \\right ] d \\xi \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} d = \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } \\binom { N _ R - r - 1 } { t - 1 } t } { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } \\binom { N _ R - r - 1 } { t - 1 } t + \\binom { N _ R - 1 } { r + 1 } \\binom { N _ R - r - 2 } { t - 1 } \\binom { N _ T } { t - 1 } } \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} M _ { p q } = e _ { p r s } x _ { r } T _ { s q } . \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{gather*} N _ { a } = \\psi _ { a } ^ { + } ( z ^ { a } _ { 1 } ) \\psi _ { a } ^ { + } ( z ^ { a } _ { 2 } ) \\psi ^ { - } _ { a } ( z ^ { a } _ { 3 } ) , \\end{gather*}"}
-{"id": "6661.png", "formula": "\\begin{align*} N = 2 \\pi / \\varepsilon \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\mathcal { S ' } _ { \\chi } ( m ) : = \\sum _ { \\substack { 2 j + 3 l + 6 k + 3 i = m \\\\ 0 \\leq j , l \\leq n - h - 2 \\\\ 0 \\leq k , i } } ( - 1 ) ^ { i } q ^ { \\frac { j + k + l } { 2 } } \\Lambda _ { j } ( \\chi ^ { 2 } ) \\Lambda _ { l } ( \\chi ^ { 3 } ) S y m ^ { k } ( \\chi ^ { 6 } ) \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} ( a b ) c - ( - 1 ) ^ { j k } ( a c ) b = a ( b c - ( - 1 ) ^ { j k } c b ) . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} t _ { D e l , R e d } = \\frac { t _ { D e l } } { k } \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} U ^ * ( x , t ) : = \\limsup _ { ( y , s ) \\to ( x , t ) } U ( y , s ) U _ * ( x , t ) : = \\liminf _ { ( y , s ) \\to ( x , t ) } U ( y , s ) . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u _ t ( x ) = m u _ t ( x ) - ( m ^ { \\frac { 2 } { \\alpha } } - \\Delta ) ^ { \\frac { \\alpha } { 2 } } u _ t ( x ) + \\xi \\sigma ( u _ t ( x ) ) \\dot F ( t , x ) , \\ \\ \\ x \\in B _ R ( 0 ) , \\ \\ t > 0 \\\\ u _ t ( x ) = 0 , \\ \\ \\ x \\in { B _ R ( 0 ) ^ c } . \\end{cases} \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} c \\lambda = ( c \\lambda _ 1 , c \\lambda _ 2 , \\ldots , c \\lambda _ { l ( \\lambda ) } ) . \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} F _ 1 ( x ) & = - \\frac { p ( ( \\tan ^ 2 ( x ) + 1 ) / 2 ) } { \\sqrt { 2 } \\sin ( x ) } = \\frac { - 1 } { \\sqrt { 2 } \\sin ( x ) } \\left ( \\frac { - \\sin ( x ) } { 1 + \\cos ( x ) } \\right ) ^ { \\frac { - 1 } { \\sqrt { 2 } } } , \\\\ F _ 2 ( x ) & = - \\frac { p ( ( \\tan ^ 2 ( x ) + 1 ) / 2 ) } { \\sqrt { 2 } \\sin ( x ) } = \\frac { - 1 } { \\sqrt { 2 } \\sin ( x ) } \\left ( \\frac { - \\sin ( x ) } { 1 + \\cos ( x ) } \\right ) ^ { \\frac { 1 } { \\sqrt { 2 } } } . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} J ^ * ( v ) = \\sup _ { x \\in R ^ { n } } \\left \\{ \\langle v , x \\rangle - J ( x ) \\right \\} , \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} f _ { k + 1 } = & A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\Big ( \\sum _ { i = 1 } ^ { k + 1 } \\psi _ i \\Big ) + K _ { 6 - n } \\Big ( \\sum _ { i = 1 } ^ { k + 1 } \\psi _ i \\Big ) + f \\\\ = & f _ k + A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\psi _ { k + 1 } + K _ { 6 - n } \\psi _ { k + 1 } \\\\ = & O ( r ^ { k + 2 } ) . \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} D _ { a , b } ( { \\cal B } ) \\leq w ( e ( a , b ) ) = D _ { a , b } , \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} \\mathbf { H } ( t , \\cdot ) = \\exp \\big ( - t / \\tau \\big ) \\mathbf { H } ^ { 0 } + \\int _ { 0 } ^ { t } \\exp \\big ( - ( t - s ) / \\tau \\big ) \\mathbf { F } \\big ( \\nabla \\mathbf { u } ( s , \\cdot ) \\big ) \\mathrm { d } \\mathbf { s } . \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} R ( \\vec { z } ) = \\lambda _ { \\tau } ( \\vec { z } ) \\ , ( \\tau Q ) ( \\vec { z } ) . \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x _ { i } & = & \\dfrac { P _ { i } } { Q } + \\varepsilon \\phi \\\\ \\varepsilon Q & \\cong & 0 \\end{array} \\right . ; i = 1 , 2 , . . . , n \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } \\Upsilon _ { n } ^ { \\delta } ( z _ { n } ) = \\lim _ { n \\rightarrow \\infty } \\int _ { \\Omega } \\Psi ^ { \\delta } ( z _ { n } ) = \\int _ { \\Omega } \\Psi ^ { \\delta } ( z ) \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} R = a ^ 6 D ^ 6 & + R ^ { ( 5 ) } D ^ 5 + R ^ { ( 4 ) } D ^ 4 + R ^ { ( 3 ) } D ^ 3 + R ^ { ( 2 ) } D ^ 2 + R ^ { ( 1 ) } D + R ^ { ( 0 ) } \\\\ & + \\sigma ^ { ( 1 ) } D ^ { - 1 } \\gamma ^ { ( 1 ) } + \\sigma ^ { ( 2 ) } D ^ { - 1 } \\gamma ^ { ( 2 ) } , \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} K ( N ) = \\prod _ { v \\nmid D \\infty } K _ v ( N ) \\prod _ { v \\mid D } K _ v \\qquad K ^ S ( N ) = \\prod _ { \\substack { v \\nmid D \\infty \\\\ v \\notin S } } K _ v ( N ) \\prod _ { v \\mid D } K _ v . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} N : = \\frac { R } { r } = \\frac { s } { 4 a b c d } \\sqrt { ( a b + c d ) ( a c + b d ) ( a d + b c ) } \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} F = p ( x _ { 0 } , x _ { 1 } ) + q ( x _ { 2 } , x _ { 3 } ) + \\sum _ { i = 0 , 1 ; j = 2 , 3 } ^ { } x _ { i } x _ { j } A _ { i , j } + \\sum _ { i = 4 } ^ { r } x _ { i } G _ { i } \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} & A ! _ { \\nu } B \\\\ & \\leqslant \\{ A ^ { - 1 } \\sharp _ { \\nu } B ^ { - 1 } + \\sum _ { k = 0 } ^ { n } r _ { k } ( A ^ { - 1 } \\sharp _ { \\frac { m _ k } { 2 ^ k } } B ^ { - 1 } - 2 A ^ { - 1 } \\sharp _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } B ^ { - 1 } + A ^ { - 1 } \\sharp _ { \\frac { m _ k + 1 } { 2 ^ k } } B ^ { - 1 } ) \\} ^ { - 1 } \\\\ & \\leqslant A \\sharp _ { \\nu } B . \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} s _ { 2 i + 1 } = p _ { 2 i + 1 } + \\frac { \\omega ^ { a _ { h ( 2 i + 1 ) } ( 2 i + 1 ) } } { 2 ^ { h ( 2 i + 1 ) + 1 } } ( 1 + z + \\dots + z ^ { k - h ( 2 i + 1 ) - 1 } ) ( z ^ { h ( 2 i + 1 ) + 1 } - 1 ) . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} C ^ 3 d t = ( A \\frac { \\partial C } { \\partial p } - C \\frac { \\partial A } { \\partial p } ) ( b ' d x - a ' d y ) + ( C \\frac { \\partial A } { \\partial q } - A \\frac { \\partial C } { \\partial q } ) ( b d x - a d y ) , \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} \\tau = \\ln ( \\Lambda '^ { - 2 } / \\Lambda ^ { - 2 } ) , i . e . \\Lambda '^ { - 2 } = e ^ \\tau \\Lambda ^ { - 2 } \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} b _ { i , j } = \\left [ b _ { i - 1 , j } ^ { \\ast } - b _ { i - 1 , j } \\right ] + \\left ( 1 - \\beta _ { i , i } \\right ) b _ { i - 1 , j } + \\beta _ { i , i } \\delta _ { i , j } . \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { m , n } ^ { \\gamma } ) \\right ] ( z ) = \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { \\gamma + 1 } \\mathcal { Z } _ { m , n - 1 } ^ { \\gamma + 1 } ( z , \\overline { z } ) . \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} \\left | r _ { Z _ N } ( t ) \\right | \\leq \\frac { 1 } { 2 } \\lambda ^ { - 1 } \\sum _ { i = 1 } ^ N \\left ( \\lambda ^ 2 | x _ i | ^ 2 + | v _ i | ^ 2 \\right ) \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} \\alpha : = \\alpha ( k , l , m , n , s ) = & 5 k ^ 2 + 8 l ^ 2 + 9 m ^ 2 + 8 n ^ 2 + 5 s ^ 2 + 8 k l + 6 k m \\\\ + & 4 k n + 2 k s + 1 2 m l + 8 l n + 3 l s + 1 2 m n + 6 m s + 8 n s . \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} _ 2 F _ 1 \\left [ \\begin{array} { c } a , \\ , b \\\\ c \\end{array} ; 1 \\right ] = \\frac { \\Gamma ( c ) \\ ; \\Gamma ( c - a - b ) } { \\Gamma ( c - a ) \\ ; \\Gamma ( c - b ) } \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} \\frac { 1 } { \\Phi ( Q ) } \\sum _ { A \\mod Q } \\bar { \\chi _ { 1 } } ( A ) \\chi _ { 2 } ( A ) = \\begin{cases} 1 & \\chi _ { 1 } = \\chi _ { 2 } \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} \\mathcal { A } - z \\mathcal { E } = \\begin{bmatrix} D & - T ^ \\top \\\\ I _ n & 0 \\end{bmatrix} - z \\begin{bmatrix} V & 0 \\\\ 0 & I _ n \\end{bmatrix} . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} \\delta _ { n } - \\delta _ { n - 1 } & = \\psi _ { \\beta } \\left ( \\delta _ { n - 1 } \\right ) - \\psi _ { \\beta } \\left ( \\delta _ { n - 2 } \\right ) = \\left ( 1 - 3 \\sqrt { \\beta } \\right ) \\left ( \\delta _ { n - 1 } - \\delta _ { n - 2 } \\right ) \\\\ & = \\left ( 1 - 3 \\sqrt { \\beta } \\right ) ^ { n - 1 } \\left ( \\delta _ { 1 } - \\delta _ { 0 } \\right ) = \\frac { \\beta } { 2 } \\left ( 1 - 3 \\sqrt { \\beta } \\right ) ^ { n - 1 } , \\end{align*}"}
-{"id": "10085.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ 2 } { x z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ 2 } { x ^ 3 z } \\\\ \\end{gather*}"}
-{"id": "495.png", "formula": "\\begin{align*} F ' ( x _ { 1 } , . . . . , x _ { n } ) = \\sum _ { i = 1 } ^ { n } f _ { i } ( x _ { i } ) e _ { i } , \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\mathbf { G } = \\left ( \\begin{array} { c c c c c c } 1 & 0 & 0 & 0 & a _ 1 & \\underline { a _ 1 } \\\\ 0 & 1 & 0 & 0 & a _ 2 & \\underline { a _ 2 } \\\\ 0 & 0 & 1 & 0 & b _ 1 & \\underline { b _ 1 } \\\\ 0 & 0 & 0 & 1 & b _ 2 & \\underline { b _ 2 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { q ^ { ( n ^ 2 + n ) / { 2 } } ( - q ; q ) _ n } { ( q ; q ) ^ 2 _ n } = \\lim _ { b \\rightarrow - 1 } \\frac { ( b q ; q ) _ \\infty } { ( q b ^ 2 ; q ) _ \\infty } ( 1 - b ) F ( b , 0 ; b ) , \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} v _ { \\delta ^ { 1 } } g _ { 1 } ( x _ { 1 } ) + u _ { \\delta ^ { n } } g _ { n } ( x _ { n } ) + u _ { \\delta ^ { 1 , n } } g _ { 1 } ( x _ { 1 } ) g _ { n } ( x _ { n } ) = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } v _ { \\delta } g _ { 1 } ( x _ { 1 } - x _ { n } ) ^ { \\delta _ { 1 } } f _ { n } ^ { \\delta _ { n } } ( x _ { n } ) \\prod _ { i = 2 } ^ { n - 1 } g _ { i } ^ { \\delta _ { i } } ( - x _ { n } ) . \\end{align*}"}
-{"id": "10079.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { x y ( a x + b y + c z ) } { z ^ 3 } , f ( x , y , z ) = \\dfrac { x y ( a x + b y + c z ) ^ 2 } { z ^ 4 } , \\\\ f ( x , y , z ) = \\dfrac { x y ^ 2 ( a x + b y + c z ) ^ 3 } { z ^ 6 } , \\end{gather*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\big ( F _ { Z _ R } - G _ { Z _ R } \\big ) ( \\omega _ 1 , \\omega _ 2 , \\hat { \\omega } ) = \\mu _ 0 + \\mu _ 1 - \\mu _ 2 . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} d \\mu = Z ^ { - 1 } \\exp \\left [ - \\sum _ { e = \\left ( i , j \\right ) \\in D ^ { \\ast } } V _ { e } \\left ( \\theta \\left ( i \\right ) - \\theta \\left ( j \\right ) \\right ) \\right ] \\prod _ { i \\in D \\backslash \\partial D } d \\theta \\left ( i \\right ) \\prod _ { i \\in \\partial D } \\delta _ { 0 } \\left ( d \\theta \\left ( i \\right ) \\right ) . \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} [ \\alpha _ p ( \\omega ) ] = [ \\omega ' ] \\in H ^ p ( Z , F ) . \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\epsilon _ { \\alpha } = \\frac { \\alpha } { 2 } \\frac { d ^ { \\alpha / 2 - 1 } } { d x ^ { \\alpha / 2 - 1 } } . \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{gather*} [ i , \\lambda ] \\cdot [ j , \\mu ] : = \\begin{cases} [ i , \\mu ] & P ( \\lambda , j ) = 1 , \\\\ 0 & P ( \\lambda , j ) = 0 , \\end{cases} \\\\ 0 \\cdot [ i , \\lambda ] : = [ i , \\lambda ] \\cdot 0 : = 0 \\cdot 0 : = 0 . \\end{gather*}"}
-{"id": "7148.png", "formula": "\\begin{align*} ( S \\otimes \\iota ) U = U ^ * \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { i = 1 } ^ { n } f _ { i } ( x _ { i } ) e _ { i } \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} \\epsilon : = \\Big \\lVert e ^ { 4 i R \\lambda _ 0 } C _ { 1 2 } ( \\lambda _ 0 ) v - v \\Big \\rVert _ Y < \\big \\lVert v \\big \\rVert _ Y , \\end{align*}"}
-{"id": "9676.png", "formula": "\\begin{align*} \\frac { \\left ( - q z ^ { 2 } ; q \\right ) _ { \\infty } } { ( q ; q ) _ { \\infty } } z ^ { \\nu } = \\sum _ { m = 0 } ^ { \\infty } \\frac { q ^ { m ^ { 2 } } } { \\left ( q ; q \\right ) _ { m } } I _ { \\nu + m } ^ { ( 3 ) } ( 2 z q ^ { m / 2 } ; q ) \\left ( \\frac { q ^ { \\nu } } { z } \\right ) ^ { m } , \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\widehat { \\Theta } _ t ( \\lambda ) = \\pi \\int _ { [ 0 , t ] \\times [ 0 , \\lambda ] } \\widehat { P } ( d s d \\lambda ' ) \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} z ' _ \\alpha = \\left \\{ \\begin{array} { c c } z _ \\alpha \\cdot E _ 4 & \\ { \\rm i f } \\ \\nu ( \\alpha ) \\in ( \\Q ^ \\times ) ^ 2 \\\\ E _ 4 & \\ { \\rm o t h e r w i s e } \\end{array} \\right . . \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} C _ h = \\bigcup _ { x \\in \\hat { E } } \\{ T ^ i x : 0 \\leq i \\leq i _ x + h \\} , \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} X _ s ^ { t , x } = \\ x + \\int _ t ^ s b ( r , X _ r ^ { t , x } ) \\mathrm d r + \\int _ t ^ s \\mathrm d W _ r , s \\in [ t , T ] \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} A _ { q } \\left ( q ^ { 2 \\alpha } u \\right ) = \\left ( u q ^ { 1 / 2 } ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } S _ { n } \\left ( q ^ { 2 \\alpha - 1 / 2 } ; q \\right ) q ^ { n / 2 } u ^ { n } \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\dd \\hat { X } _ t = Q \\hat { X } _ t \\ , \\dd t + \\hat { \\lambda } _ { t - } ^ + ( \\widehat { X \\lambda } _ { t - } - \\hat { X } _ { t - } \\hat { \\lambda } _ { t - } ) ( \\dd N _ t - \\hat \\lambda _ t \\ , \\dd t ) , \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{gather*} h _ { k } ^ { ( \\alpha ) } = \\frac { \\tau _ { k + 1 } ^ { ( \\alpha ) } } { \\tau _ { k } ^ { ( \\alpha ) } } , \\end{gather*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\mathbf { C V = V B } , \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} \\Sigma \\cap Q = L _ 1 + L _ 2 + L _ 3 + L _ 1 ^ \\prime + L _ 2 ^ \\prime + L _ 3 ^ \\prime \\end{align*}"}
-{"id": "9899.png", "formula": "\\begin{align*} \\nabla g ( x ) = \\nabla ( \\gamma ( r ) ( x - a ) ) + \\nabla ( \\gamma ( \\tilde { r } ) ( i _ x ( \\tilde { x } - a ) ) ) . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} W : = \\frac { 1 } { 2 } \\ln \\det ( \\Delta ) = - \\frac { 1 } { 2 } \\int _ 0 ^ \\infty \\frac { d t } { t } T r ( e ^ { - t \\Delta } ) \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} \\xi ( s , \\chi ) = e ^ { A ( \\chi ) + B ( \\chi ) s } \\prod _ { \\rho } \\left ( 1 - \\frac { s } { \\rho } \\right ) e ^ { s / \\rho } . \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} [ \\bar { h } _ { i , k } , \\bar { f } _ { i , l } ] = - 2 \\bar { f } _ { i , l + k } , \\ [ \\bar { h } _ { i , k } , \\bar { f } _ { i + 1 , l } ] = ( d ^ k + d ^ { - k } ) \\bar { f } _ { i + 1 , l + k } , \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} \\sigma _ { [ i _ 1 , \\ldots , i _ l ] } = \\sigma _ { [ j _ 1 , \\ldots , i _ j ] } \\Leftrightarrow [ i _ 1 , \\ldots , i _ l ] = [ j _ 1 , \\ldots , j _ l ] . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} H ( u ) : = \\beta P | u | ^ 2 u + \\lambda _ 0 P ( u \\cdot \\nabla ) u - P N ( u ) \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\overline F \\big ( ( x - 1 ) / a ( x ) \\big ) \\overline G \\big ( a ( x - 1 ) \\big ) + \\overline F \\big ( ( x - 1 ) / a ( x - 1 ) \\big ) \\overline G \\big ( ( x - 1 ) a ( x ) / x \\big ) = o \\big ( \\overline { H } ( x - 1 ) \\big ) . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} \\prod _ { p | q _ 1 q _ 2 } p ^ { c _ p } ~ = ~ \\prod _ { 1 < a < q _ 1 / 2 \\atop ( a , q _ 1 ) = 1 } { \\xi } _ a ^ { d _ a } \\prod _ { 1 < b < q _ 2 / 2 \\atop ( b , q _ 2 ) = 1 } { \\xi } _ b ^ { e _ b } \\prod _ { 1 < c < q _ 3 / 2 \\atop ( c , q _ 3 ) = 1 } { \\xi } _ c ^ { f _ c } \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} Y = \\left \\{ \\begin{array} { l l l } A _ { 2 , 2 t - 3 } & \\mbox { $ t = 3 , 4 ; $ } \\\\ B _ { t - 3 , t } & \\mbox { $ t \\geq 5 , t ~ { \\rm o d d } ; $ } \\\\ B _ { t - 4 , t + 1 } & \\mbox { $ t \\geq 6 , t ~ { \\rm e v e n } . $ } \\end{array} \\right . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} S = \\{ i : F _ i ^ { \\prime } \\sim F _ i ^ { \\prime \\prime } , 1 \\leq i \\leq k \\} . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} { \\mathbf { y } } ^ { [ \\sf b s ] } _ { i } [ t ] & = \\sum _ { j = 1 } ^ { K } \\sum _ { k = 1 } ^ { N } \\mathbf { { f } } _ { i , j k } [ t ] x ^ { [ \\sf u ] } _ { j k } [ t ] + \\sum _ { j = 1 } ^ K \\mathbf { { B } } _ { i j } [ t ] { \\mathbf { x } } ^ { [ \\sf b s ] } _ { j } [ t ] + \\mathbf { { z } } ^ { [ \\sf b s ] } _ i [ t ] , \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} \\Delta u = \\alpha u \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} \\Delta u = \\frac { d } { d t } u \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} v ( y ) = - ( y _ 1 ^ 3 - 3 y _ 1 y _ 2 ^ 2 ) + c _ 1 y _ 1 ^ 4 + c _ 2 y _ 1 ^ 2 y _ 2 ^ 2 + c _ 3 y _ 1 y _ 2 ^ 3 + c _ 4 y _ 1 ^ 5 + c _ 5 y _ 1 ^ 3 y _ 1 ^ 2 + c _ 6 y _ 1 ^ 2 y _ 2 ^ 3 + c _ 7 y _ 1 y _ 2 ^ 4 + h . o . t , \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} L _ f ( s ) = \\sum _ { n \\geq 1 } \\frac { \\lambda _ f ( n ) } { n ^ s } , \\Re { s } > 1 \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} \\psi _ i ( \\vect { x } _ 1 ) = \\psi _ i ( \\vect { x } _ 2 ) = 0 , i = 1 , \\dots , K \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} \\| m _ { \\mathbf { B } , V } ^ \\frac 1 2 ( - i \\nabla + \\mathbf { A } ) u \\| ^ 2 = \\langle m _ { \\mathbf { B } , V } ( - i \\nabla + \\mathbf { A } ) u , ( - i \\nabla + \\mathbf { A } ) u \\rangle \\ , , \\end{align*}"}
-{"id": "10132.png", "formula": "\\begin{align*} \\gcd ( 2 q , - p ) + 2 = p + 2 q , \\end{align*} % \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} f _ i ( 1 ) = f _ i ( 0 ) = f _ j ( 0 ) = f _ j ( 1 ) . \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} | \\mathbb { E } ( | w _ j | ^ k ) | \\leq ( { \\tilde { c } \\sigma _ { \\sf w } } ) ^ k k ! , \\ k = 2 , 3 , \\cdots , \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} p _ n = \\frac { \\displaystyle \\xi _ n } { \\displaystyle \\left ( n ! \\right ) ^ { 2 n } - 1 } , \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} H ^ j _ { c , G } ( M , \\Z ) : = H ^ j _ { c } ( ( M \\times E G ) / G , \\Z ) . \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} b _ 1 \\leqslant R ^ { - 2 } , b _ l \\geqslant 2 \\delta / 3 , b _ { j + 1 } \\leqslant b _ j + e ^ { - a R / 2 } , j = 1 , \\cdots , l - 1 , \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} r ^ * ( \\emptyset ) = r ( E ) + | | \\emptyset | | _ r - r ( E ) = 0 , \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} P _ { 0 } ( z ) = e ^ { \\frac { \\lambda } { \\xi } z } ( 1 - z ) ^ { - \\frac { \\gamma } { \\xi } } \\left [ P _ { 0 } ( 0 ) - p _ { 1 , 1 } \\frac { \\mu } { \\xi } \\int _ { 0 } ^ { z } e ^ { - \\frac { \\lambda } { \\xi } s } ( 1 - s ) ^ { \\frac { \\gamma } { \\xi } - 1 } d s \\right ] . \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} \\breve { \\{ } 2 ( e _ { \\pm } \\mp \\beta ( x ^ { \\prime } ) ) \\} ^ { - 1 / 2 } = \\{ 2 e _ { \\pm } \\} ^ { - 1 / 2 } \\{ 1 \\mp \\frac { \\varepsilon } { 2 e _ { \\pm } } \\cos \\frac { 2 \\pi x ^ { \\prime } } { P _ { \\beta } } + O \\left ( \\left ( \\frac { \\varepsilon } { e _ { \\pm } } \\right ) ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} Q ^ { \\mu \\nu } & = R ^ { \\mu \\nu } + i S ^ { \\mu \\nu } = R ^ { \\mu \\nu } - \\frac { i } { 2 } e ^ { \\mu \\nu \\sigma \\tau } F _ { \\sigma \\tau } , \\\\ P _ { \\mu \\nu } & = F _ { \\mu \\nu } + i G _ { \\mu \\nu } = F _ { \\mu \\nu } - \\frac { i } { 2 } e _ { \\mu \\nu \\sigma \\tau } R ^ { \\sigma \\tau } . \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\beta \\in Z , \\beta \\geq \\alpha \\\\ \\mu \\left ( \\gamma \\right ) = 0 \\alpha < \\gamma < \\beta } } \\mu \\left ( \\beta \\right ) \\leq C d \\left ( \\alpha \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} \\ell = \\prod _ { p \\in S _ { } } p . \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} | G | - | C ( G ) | = \\sum _ { k \\in \\pi _ e ( G ) } c _ k ( \\phi ( k ) - 1 ) . \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} \\tilde { u } ( x ) : = \\frac { w ( x _ 0 + \\lambda x ) - \\mathcal { W } _ { x _ 0 } ( x _ 0 + \\lambda x ) } { \\lambda ^ { \\frac { 3 } { 2 } + \\alpha } } , \\ 0 < \\lambda < 1 / 4 . \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} [ \\bar { e } _ { i , k } , \\bar { f } _ { j , l } ] = \\delta _ { i , j } \\bar { h } _ { i , k + l } + k \\delta _ { i , j } \\delta _ { k , - l } \\bar { c } , \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} \\| b _ i \\| _ { M A ( G ) } = \\sup _ { \\| a \\| _ { A ( G ) } = 1 } \\| b _ i a \\| _ { A ( G ) } . \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} ( g ' , d ' , 1 ) = ( 1 , \\zeta , 1 ) ( a , a , 1 ) ( \\varphi , \\Pi , 1 ) ^ { - m } ( g '' , d '' , 1 ) \\in \\zeta J ^ 1 _ { \\mathrm { s } } \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} \\O = B ( \\psi _ J ) = \\bigcap _ { I \\in P } B ( \\psi _ I ) \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} ( \\log F ( z ) ) _ { \\overline { z } \\overline { z } } = & \\sum _ { k = 3 } ^ { p } ( k - 1 ) ( k - 2 ) | z | ^ { 2 ( k - 3 ) } z ^ { 2 } \\log G _ { k } ( z ) + 2 \\sum _ { k = 2 } ^ { p } ( k - 1 ) | z | ^ { 2 ( k - 2 ) } z ( \\log G _ { k } ( z ) ) _ { \\overline { z } } \\\\ & + \\sum _ { k = 1 } ^ { p } | z | ^ { 2 ( k - 1 ) } ( \\log G _ { k } ( z ) ) _ { \\overline { z } \\overline { z } } . \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} F ( a e _ { i } ) = f _ { i } ( a ) e _ { i } . \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} g _ { n } = g _ { n } ( z ) : = z ^ { - n } \\ , _ { 1 } \\tilde { \\phi } _ { 1 } ( 0 ; z \\alpha q ^ { 1 - n } ; q , q z ^ { 2 } ) . \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} \\begin{array} { c } f _ { 1 } ( u ) = \\int \\sqrt { 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) } d u , \\\\ f _ { 2 } ( u ) = \\lambda \\cos \\left ( \\frac { u } { c } \\right ) . \\end{array} \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{gather*} \\omega _ { \\underline k } = M ( \\underline k ) v _ { 0 } ^ { ( 1 ) } = M ^ { + } ( \\underline k ^ { + } ) M ^ { - } ( \\underline k ^ { - } ) v _ { 0 } ^ { ( 1 ) } , \\end{gather*}"}
-{"id": "1224.png", "formula": "\\begin{align*} \\varphi = \\frac { f _ { \\sigma } \\left ( \\sigma , \\theta \\right ) } { f \\left ( \\sigma , \\theta \\right ) } = - k \\left ( \\sigma , \\theta \\right ) , \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} e ^ { \\mu \\nu \\sigma \\tau } A _ { \\sigma \\tau } = 2 B ^ { \\mu \\nu } , e _ { \\mu \\nu \\sigma \\tau } B ^ { \\sigma \\tau } = A _ { \\nu \\mu } - A _ { \\mu \\nu } . \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{gather*} U _ { k } ^ { ( \\alpha ) } = V _ { k } ^ { ( \\alpha ) } \\big ( W _ { k + 1 } ^ { ( \\alpha ) } \\big ) ^ { - 1 } = \\big ( W _ { k + 1 } ^ { ( \\alpha - 1 ) } \\big ) ^ { - 1 } V _ { k + 1 } ^ { ( \\alpha - 1 ) } . \\end{gather*}"}
-{"id": "7683.png", "formula": "\\begin{align*} Z _ { \\mathrm { a } } = \\overline { U _ 2 \\cdot [ \\phi _ 6 ] } \\subset Y \\subset \\P ( M _ 6 ) \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} S ( l , u , v ; N ) = \\sum _ { \\substack { n = 1 \\\\ ( n , p ) = 1 } } ^ { \\infty } \\frac { 1 } { n ^ { 1 / 2 + u + v } } \\Delta _ { 2 k , N } ( l , n ) . \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} p ( z ) + \\sum _ { j = 0 } ^ { r - 2 } \\left ( c _ { r - 1 , j } z + c _ { r , j } \\right ) g _ { j } ( z ) . \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} \\| F _ i ( x _ i , u ) - F _ i ( x _ i , z ) \\| & = \\sqrt { \\sum _ { l = 1 } ^ \\mathcal { L } \\| F _ { i l } ( x _ { i l } , u _ l ) - F _ { i l } ( x _ { i l } , z _ l ) \\| ^ 2 } \\leq \\sqrt { 2 } \\sqrt { \\sum _ { l = 1 } ^ \\mathcal { L } ( C _ l ^ 2 + M _ l ^ 2 { \\rm c a p } _ { i l } ^ 2 ) } \\ , \\| u - z \\| . \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} \\dot { z } _ 1 = z _ 2 + z _ 2 \\psi _ 1 ( z _ 1 , z _ 2 ^ 2 ) , \\dot { z } _ 2 = \\mu z _ 1 + \\psi _ 2 ( z _ 1 , z _ 2 ^ 2 ) \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} \\widetilde { F } ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) = a _ { 1 } [ x _ { 1 } + \\alpha x _ { 3 } ( x _ { 1 } - x _ { 2 } ) ] + a _ { 2 } [ x _ { 2 } + \\beta x _ { 3 } ( x _ { 1 } - x _ { 2 } ) ] + a _ { 3 } x _ { 3 } [ 1 + \\gamma ( x _ { 1 } - x _ { 2 } ) ] . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} g _ \\varepsilon ^ { ( s ) } ( t , Z _ s ^ * ) = g _ \\varepsilon ^ { ( s ) } ( t , Z _ s ) \\textnormal { a . e . } ( t , Z _ s ) \\in [ 0 , T ) \\times \\partial \\tilde { \\mathcal { D } } _ s \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\epsilon & < d ( g _ { N + 1 } x _ j , g _ { N + 1 } x _ k ) = d ( \\tilde { r } _ { N + 1 } ( \\omega _ j ) , \\tilde { r } _ { N + 1 } ( \\omega _ k ) ) \\leq 8 \\delta + d ( r ( \\omega _ j ) , r ( \\omega _ k ) ) \\\\ & \\leq 1 6 \\delta + d ( \\tilde { r } _ N ( \\omega _ j ) , \\tilde { r } _ N ( \\omega _ k ) ) = \\epsilon + d ( g _ N x _ i , g _ N x _ i ) = \\epsilon , \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} \\tfrac { ( \\alpha ^ { - 1 } - 1 ) \\alpha } { 2 \\tilde { \\mu } _ 0 } \\cdot \\left ( \\sum _ { j = 1 } ^ N \\tfrac { 1 } { a _ j ^ 2 } \\right ) \\min _ { j = 1 , \\hdots , N } \\norm { \\tilde { \\mu } _ j ( x _ j - y _ j ) } ^ 2 & \\le \\tfrac { \\tilde { \\mu } _ { \\max } } { \\tilde { \\mu } _ 0 } \\left ( \\tfrac { \\tilde { \\mu } _ 0 } { 2 } \\norm { x ^ * - v _ 0 } ^ 2 + \\rho M ^ 2 \\left ( \\sum _ { j = 1 } ^ N \\tfrac { 1 } { a _ j } \\right ) + \\tfrac { N r M ^ 2 } { 2 } \\right ) . \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} q ^ { - \\dim ( G / P _ \\lambda ) } \\# ( G / P _ \\lambda ) ( k ) = q ^ { \\dim M _ \\lambda - \\dim G } \\# G ( k ) / \\# M _ \\lambda ( k ) . \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} \\textbf { A } \\otimes \\textbf { P } ^ \\alpha \\textbf { C } + \\textrm { d i a g } ( \\textbf { a } ) \\textbf { K } _ 1 \\otimes \\textbf { P } \\textbf { C } + \\textrm { d i a g } ( \\textbf { b } ) \\textbf { K } _ 2 \\otimes \\textbf { P } \\textbf { C } = \\textbf { F } \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} \\int _ M \\varphi P _ g \\varphi d \\mu _ g = & \\int _ M | \\nabla \\Delta \\varphi | _ g ^ 2 d \\mu _ g - 2 \\int _ M T _ 2 ( \\nabla \\varphi , \\nabla \\Delta \\varphi ) d \\mu _ g - \\frac { n - 2 } { 2 } \\int _ M \\sigma _ 1 ( A ) ( \\Delta \\varphi ) ^ 2 d \\mu _ g \\\\ & - \\int _ M T _ 4 ( \\nabla \\varphi , \\nabla \\varphi ) d \\mu _ g + \\frac { n - 6 } { 2 } \\int _ M Q _ g \\varphi ^ 2 d \\mu _ g . \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} \\Lambda _ { n , k } ( X _ { 1 } ^ { n } , T ) = \\left \\{ \\lambda \\in \\Lambda _ { n } \\colon { \\mathrm { E } } [ \\Vert \\widehat { \\beta } _ { - k } ( \\lambda ) - \\beta \\Vert _ { 2 , n , - k } \\mid X _ { 1 } ^ { n } ] \\leq T \\right \\} , T > 0 . \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} | v _ k | \\leq \\sup _ { Q ^ k } | v _ k | \\leq \\sup _ { \\partial _ p Q ^ k } | v _ k | = \\sup _ { \\partial _ p Q ^ k } | \\varphi _ k | \\leq \\sup _ { \\Omega _ T } | \\psi | \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\frac { 1 } { 2 } | \\gamma ' | ^ 2 = \\langle \\nabla _ { \\gamma ^ \\prime } \\gamma ^ \\prime , \\gamma ^ \\prime \\rangle = \\langle Z ( \\gamma ' ) , \\gamma ^ \\prime \\rangle = \\Omega ( \\gamma ^ \\prime , \\gamma ^ \\prime ) = 0 \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} \\tilde Y ( t \\vee \\tilde \\tau _ 0 ) = \\tilde Y ( \\tilde \\tau _ 0 ) , t \\ge 0 , \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} W = \\{ ( i , j ) : 1 \\leq i \\leq r , \\ ; 1 \\leq j \\leq l ( \\lambda ^ { ( i ) } ) \\} \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} c _ { f , g } = \\frac { 1 } { 4 \\pi ^ 2 ( 2 \\kappa + \\frac { 3 } { 2 } - 2 \\nu ) } \\sum _ { n \\geq 1 } \\frac { a _ f ( n ) \\overline { a _ g ( n ) } } { n ^ { 2 \\kappa + \\frac { 3 } { 2 } - 2 \\nu } } , \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z } J _ { j } ^ { 2 } \\left ( x ; q \\right ) = \\frac { 1 } { 1 - x ^ { 2 } } \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} u _ { \\delta ^ { n } } g _ { n } ( x _ { n } ) = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } v _ { \\delta } f _ { n } ^ { \\delta _ { n } } ( x _ { n } ) \\prod _ { i = 1 } ^ { n - 1 } g _ { i } ^ { \\delta _ { i } } ( - x _ { n } ) . \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} S ( g ; \\phi ) = - \\int _ \\Sigma d v \\ , g _ { i j } \\phi ^ i \\partial ^ \\mu \\partial _ \\mu \\phi ^ j + \\frac { 1 } { 2 } \\int _ \\Sigma d v \\ , ( \\partial ^ \\mu \\partial _ \\mu g _ { i j } ) \\phi ^ i \\phi ^ j \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} c _ \\pi = \\pm 1 . \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} X = \\tau _ { a } Y = \\tau _ { a ^ { * } } v = \\begin{pmatrix} - \\tau _ { x } & \\tau _ { y ^ { * } } \\end{pmatrix} w = \\begin{pmatrix} \\tau _ { x ^ { * } } \\\\ \\tau _ { y } \\end{pmatrix} \\ , . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} & \\Big ( \\Big \\{ { \\cal X } _ t : ~ t = 0 , \\ldots , n \\Big \\} , \\Big \\{ { \\cal Y } _ t : ~ t = 0 , \\ldots , n \\Big \\} , { \\cal C } _ { 0 , n } \\triangleq \\Big \\{ { \\bf P } _ { Y _ t | Y ^ { t - 1 } , X ^ t } : ~ t = 0 , \\ldots , n \\Big \\} , \\\\ & \\qquad \\qquad { \\cal P } _ { 0 , n } \\triangleq \\Big \\{ { \\bf P } _ { X _ t | X ^ { t - 1 } , Y ^ { t - 1 } } : ~ t = 0 , \\ldots , n \\Big \\} \\Big ) \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\frac { \\tilde { \\mu } - \\mu } { 2 } \\sum _ { j = 1 } ^ N \\frac { \\norm { x _ j - y _ j } ^ 2 } { a _ j ^ 2 } & \\le \\frac { \\tilde { \\mu } } { 2 } \\norm { x ^ * - v _ 0 } ^ 2 + \\rho M ^ 2 \\left ( \\sum _ { j = 1 } ^ N \\frac { 1 } { a _ j } \\right ) + \\frac { N r M ^ 2 } { 2 } \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} \\mathcal { P } _ k ^ { h o m } = & \\{ p _ k ( y ) | \\ p _ k ( y ) = \\sum _ { | \\beta | \\leq k } a _ \\beta y ^ { \\beta } , \\\\ & a _ \\beta = 0 \\sum _ { i = 1 } ^ { n - 1 } 2 \\beta _ i + \\beta _ n + \\beta _ { n + 1 } \\neq k \\} . \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} ( \\star \\star ) A _ { n + 1 6 r } = t ^ { 1 6 r } A _ { n } + t ^ { 4 r } A _ { n + 4 r } + t ^ { 2 r } A _ { n + 2 r } . \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} B _ { ( d + 1 ) n + s } = x ^ s \\sum _ { p = 0 } ^ { n } a _ { ( d + 1 ) n + s , ( d + 1 ) p + s } x ^ { ( d + 1 ) p } , \\ \\ 0 \\leq s \\leq d , \\ \\ n \\geq 0 . \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} V ( x ) > 0 \\ \\forall x \\in \\left \\lbrace x | R _ i ( x ) \\leq \\gamma _ i , i = 1 , \\ldots , d \\right \\rbrace \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ { \\gamma } ( z , \\overline { z } ) = ( - 1 ) ^ { m } ( \\gamma + m + 1 ) _ { n } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { - \\gamma } \\dfrac { \\partial ^ { m } } { \\partial z ^ { m } } \\left ( z ^ { n } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { \\gamma + m } \\right ) \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} ( u _ { j } ) _ t = \\sum _ { l = 1 } ^ k v ^ { j } _ { l } ( u ) ( u _ l ) _ { x _ i } , \\forall 1 \\leq i \\leq n , j = 1 , \\dots , k \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\dd \\nu ^ k _ { t } & = ( \\nu ^ { k - 1 } _ { t - } - \\nu ^ k _ { t - } ) ( \\lambda ^ \\top X _ { t - } ( n - N _ t ) \\ , \\dd t + \\dd m _ t ) \\\\ & = \\lambda ^ \\top X _ t ( ( n - k + 1 ) \\nu ^ { k - 1 } _ t - ( n - k ) \\nu ^ k _ t ) \\ , \\dd t + \\dd M _ t . \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\gamma _ { 1 k } = \\sigma _ 1 \\sigma _ 2 ^ { - 1 } \\gamma _ { 2 k } - \\sigma _ k \\sigma _ 2 ^ { - 1 } \\gamma _ { 2 1 } . \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} \\begin{array} { l l } S ( g ; \\phi ) & = - \\int _ \\Sigma d v \\ , \\phi ^ i \\Delta _ g \\phi ^ j + \\frac { 1 } { 2 } \\int _ \\Sigma d v \\ , \\phi ^ i \\Delta g _ { i j } ( \\phi ) \\phi ^ j \\end{array} \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } P _ { n } t ^ { n } = \\frac { 1 } { ( 1 - z ^ { - 2 } t ) ( t ; q ) _ { \\infty } } \\ , _ { 1 } \\phi _ { 1 } \\left ( q ; q z ^ { - 2 } t ; q , q z ^ { - 1 } \\xi ^ { - 1 } t \\right ) = : \\Psi ( t ) . \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} \\abs { I _ { m , k } } = \\frac { p - 2 } { p ^ { m + 1 } ( p - 1 ) } \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} \\Phi _ { \\nu } ( z ) = \\phi _ { \\nu } ^ { \\prime } ( - 4 z ) = \\sum _ { n \\geq 0 } \\frac { n + 1 } { \\left ( \\nu + 1 \\right ) _ { n } } \\cdot \\frac { z ^ n } { n ! } . \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{align*} \\left | g \\circ \\pi ( \\omega ) - \\int g \\ , d T _ { \\omega _ { 0 } * } \\ldots _ \\ast T _ { \\omega _ { n } * } \\mu _ { n } \\right | & = \\left | \\int g \\circ \\pi ( \\omega ) \\ , d \\mu _ { n } - \\int g \\circ T _ { \\omega _ { 0 } } \\circ \\cdots \\circ T _ { \\omega _ { n } } ( x ) \\ , d \\mu _ { n } \\right | \\\\ & \\leq \\int | g \\circ \\pi ( \\omega ) - g \\circ T _ { \\omega _ { 0 } } \\circ \\cdots \\circ T _ { \\omega _ { n } } ( x ) | \\ , d \\mu _ { n } \\le \\epsilon . \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} \\mathbb { P } ( \\sum _ { i = 1 } ^ k y _ i \\geq l ) \\geq \\sum \\limits _ { i = l } ^ { k } C ^ { i } _ k p ^ { k - i } \\left ( 1 - p \\right ) ^ i . \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{gather*} \\psi _ { r } ( z ) = \\sum _ { n \\geq 0 } a _ { n } z ^ { n } \\ , \\ \\ \\ \\ | z | < 1 / \\rho _ { r } \\ , \\end{gather*}"}
-{"id": "9328.png", "formula": "\\begin{align*} ( 1 - 2 x t ) \\frac { d } { d t } M _ { a _ j , b _ j } ( x ; t ) & = 2 x ( 1 - 2 x ) \\frac { d } { d x } M _ { a _ j , b _ j } ( x ; t ) = 0 , \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} { } - [ C r , \\chi ] ^ { F N } _ p = [ ( \\imath _ { e _ i } \\varphi ) \\otimes e _ i , ( \\imath _ { e _ j } \\ast \\varphi ) \\otimes e _ j ] ^ { F N } _ p = : \\beta _ j \\otimes e _ j , \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} x _ i ^ { k + 1 } & : = \\Pi _ { K _ i } [ x _ i ^ k - \\alpha _ k F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) ] , \\\\ v _ i ^ { k + 1 } & : = { \\hat v _ i ^ k } + x _ i ^ { k + 1 } - x _ i ^ k , \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} A & = - 2 k ^ 8 + 1 6 k ^ 7 - 4 8 k ^ 6 + 4 8 k ^ 5 + 1 1 6 k ^ 4 - 5 2 8 k ^ 3 + 9 1 2 k ^ 2 - 7 8 4 k + 2 7 4 , \\\\ B & = ( k ^ 2 - 1 ) ^ 4 ( k ^ 2 - 4 k + 5 ) ^ 4 . \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} R _ { j , t } ( a , b ) = \\mu _ j ( a ) \\mu _ j ( b ) - \\frac { 1 } { | A _ j ^ t | } \\sum _ { ( i , i + 1 ) \\in A _ j ^ t } \\delta ( X _ i = a ) \\cdot \\delta ( X _ { i + 1 } = b ) \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} \\gamma = \\min \\{ \\gamma _ 0 , 1 0 ^ { - 6 } / 3 \\} . \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} S _ d ( \\delta _ k \\otimes c \\otimes \\delta _ n \\otimes b ) & = \\delta _ { k + 1 } \\otimes c \\otimes \\delta _ n \\otimes b \\\\ & \\mapsto \\alpha _ d ^ { k + 1 } ( c ) \\cdot \\delta _ { ( n , k + 1 ) } \\otimes b \\\\ & = S _ d \\alpha _ d ^ k ( c ) S _ d ^ * \\cdot \\delta _ { ( n , k + 1 ) } \\otimes b \\\\ & = S _ d \\left ( \\alpha _ d ^ k ( c ) \\cdot \\delta _ { ( n , k ) } \\otimes b \\right ) \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} J _ \\sigma ( u ) : = \\int _ \\Omega \\dfrac { ( \\Delta u ) ^ 2 } { 2 } - ( 1 - \\sigma ) \\int _ \\Omega d e t ( \\nabla ^ 2 u ) - \\int _ \\Omega \\dfrac { g ( x ) | u | ^ { p + 1 } } { p + 1 } . \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} C _ m S _ { n , 2 m } = \\sum \\limits _ { i = 0 } ^ m G _ i ^ m S _ { n , 2 i + 1 } , \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\| \\vec { Y } \\| _ * : = \\sum _ { i = 1 } ^ { k } \\sigma _ i ( \\vec { Y } ) , \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} { t } = \\bigg { \\{ } 0 , \\ 2 \\pi ( \\frac { 1 } { 2 ^ q } ) , \\ 2 \\pi ( \\frac { 2 } { 2 ^ q } ) , \\ . . . , \\ 2 \\pi ( \\frac { 2 ^ q - 1 } { 2 ^ q } ) \\bigg { \\} } \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} Q ^ { \\mu \\nu } = F ^ { \\mu \\nu } - \\frac { i } { 2 } e ^ { \\mu \\nu \\sigma \\tau } F _ { \\sigma \\tau } , F ^ { \\mu \\nu } = g ^ { \\mu \\sigma } g ^ { \\nu \\tau } F _ { \\sigma \\tau } , g _ { \\mu \\sigma } g _ { \\nu \\tau } Q ^ { \\sigma \\tau } = Q _ { \\mu \\nu } . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ { \\gamma } ( z , \\bar { z } ) = \\frac { \\left ( ( \\gamma + 1 ) _ { m + n } \\right ) ^ 2 } { ( \\gamma + 1 ) _ { m } ( \\gamma + 1 ) _ { n } } \\overline { z } ^ m z ^ n { _ 2 F _ 1 } \\left ( \\begin{array} { c } - m , - n \\\\ - \\gamma - m - n \\end{array} \\bigg | \\frac 1 { | z | ^ 2 } \\right ) = ( \\gamma + 1 ) _ { m + n } \\overline { R _ { m , n } ^ { ( \\gamma ) } ( z ) } . \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { t } & ( t - s _ { 1 } ) ^ { \\theta - 1 } \\int _ { 0 } ^ { s _ { 1 } } ( s _ { 1 } - s _ { 2 } ) ^ { \\theta - 1 } \\cdots \\int _ { 0 } ^ { s _ { n - 1 } } ( s _ { n - 1 } - s _ { n } ) ^ { \\theta - 1 } d s _ { n } \\cdots d s _ { 1 } \\\\ & = \\frac { \\left \\{ \\Gamma ( \\theta ) \\right \\} ^ { n } } { \\Gamma ( n \\theta + 1 ) } t ^ { n \\theta } , \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} \\frac { d } { d t } ( [ A ^ { \\ast } ( t ) A ( t ) ] ^ { - 1 } ) | _ { t = 0 } = 0 \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} \\eta _ { M , M - 1 } ( q | a , b ) \\triangleq \\exp \\Bigl ( \\bigl ( \\mathcal { S } _ { M - 1 } \\log \\Gamma _ M \\bigr ) ( q \\ , | a , \\ , b ) - \\bigl ( \\mathcal { S } _ { M - 1 } \\log \\Gamma _ M \\bigr ) ( 0 \\ , | a , \\ , b ) \\Bigr ) . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} 6 i \\frac { \\partial Q ^ { \\mu \\nu } } { \\partial x ^ { \\nu } } = e ^ { \\mu \\nu \\lambda \\sigma } \\left ( \\frac { \\partial P _ { \\lambda \\sigma } } { \\partial x ^ { \\nu } } + \\frac { \\partial P _ { \\nu \\lambda } } { \\partial x ^ { \\sigma } } + \\frac { \\partial P _ { \\sigma \\nu } } { \\partial x ^ { \\lambda } } \\right ) \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} { n \\brack i } _ r : = \\begin{cases} \\frac { 2 } { 2 n + r } { 2 n + r \\choose 2 i + r } B _ { 2 ( n - i ) } , & n \\ge \\operatorname { m a x } \\{ 0 , \\lfloor - r / 2 \\rfloor + 1 \\} , \\\\ \\frac { 2 } { 2 i + r } { - 2 i - r \\choose - 2 n - r } B _ { 2 ( n - i ) } , & 0 \\le n \\le \\lfloor - ( r + 1 ) / 2 \\rfloor . \\end{cases} \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} & \\left ( \\frac { 1 } { 2 \\pi i } \\right ) ^ 2 \\iint \\limits _ { ( \\sigma ) ( \\gamma ) } L ^ \\nu ( s - w ) \\zeta ( w ) \\frac { \\Gamma ( w ) \\Gamma ( s - w ) } { \\Gamma ( s ) } V ( s ) X ^ s d w d s \\\\ & = \\left ( \\frac { 1 } { 2 \\pi i } \\right ) ^ 2 \\iint \\limits _ { ( \\epsilon ) ( \\gamma ) } L ^ \\nu ( s ) \\zeta ( w ) \\frac { \\Gamma ( w ) \\Gamma ( s ) } { \\Gamma ( s + w ) } V ( s + w ) X ^ { s + w } d w d s . \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} Y _ s ^ { t , x } = \\alpha ( s , X _ s ^ { t , x } ) Z _ s ^ { t , x } = \\beta ( s , X _ s ^ { t , x } ) \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} X ^ { t , x } _ { s } = \\ & x + \\xi ( t , x ) - \\xi ( s , X ^ { t , x } _ { s } ) + ( \\lambda + 1 ) \\int _ { t } ^ { s } \\xi ( r , X ^ { t , x } _ { r } ) \\mathrm { d } r \\\\ & + \\int _ { t } ^ { s } { ( \\nabla \\xi ( r , X ^ { t , x } _ { r } ) + \\mathrm I _ d ) } \\mathrm { d } W _ { r } , \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} S z ( L _ p ( C ( K ) ) ) & = \\max \\{ S z ( L _ p ( C _ 0 ( K ) ) ) , S z ( L _ p ) \\} = \\max \\{ S z ( L _ p ( C _ 0 ( K ) ) ) , \\omega \\} \\\\ & = S z ( L _ p ( C _ 0 ( K ) ) ) . \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} H _ g = \\widehat H _ g = \\widehat v _ k \\geq \\widehat v = v \\Omega \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} \\theta _ j ( \\xi ) : = \\theta ( \\delta ^ * _ { 2 ^ { - j } } ( \\xi ) ) . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{gather*} \\big ( \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta ) } \\big ) ^ { 2 } = \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta + 1 ) } \\tau _ { k , \\ell } ^ { ( \\alpha , \\beta - 1 ) } - \\tau _ { k , \\ell + 1 } ^ { ( \\alpha , \\beta - 1 ) } \\tau _ { k , \\ell - 1 } ^ { ( \\alpha , \\beta + 1 ) } - \\tau _ { k - 1 , \\ell } ^ { ( \\alpha , \\beta - 1 ) } \\tau _ { k + 1 , \\ell } ^ { ( \\alpha , \\beta + 1 ) } , \\end{gather*}"}
-{"id": "8028.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ { i } & = \\big ( A _ { i J R s } u _ { s , R } - \\beta _ { J i } T - ( C _ { i J K S R l } u _ { l , R S } + M _ { i J K R } \\tau _ { , R } ) _ { , K } \\big ) _ { , J } + \\rho f _ { i } , \\\\ a \\ddot { \\tau } & = - \\beta _ { K i } \\dot { u } _ { i , K } - M _ { j L K I } u _ { j , L K I } + K _ { I J } \\tau _ { , I J } + \\rho T _ { 0 } ^ { - 1 } R \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} V X = \\{ A \\subseteq X \\mid A \\} \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} G _ { [ 1 : p _ 1 , 1 : r ] } = \\hat { U } _ X \\hat { \\Sigma } _ X \\hat { U } _ X ^ { \\intercal } W _ { [ : , 1 : r ] } , W _ { [ : , 1 : r ] } = \\begin{bmatrix} W _ { 1 1 } & & \\\\ & \\ddots & \\\\ & & W _ { r r } \\\\ & 0 & \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} \\Lambda _ { n } ( X _ { 1 } ^ { n } , T ) = \\cap _ { k = 1 } ^ { K } \\Lambda _ { n , k } ( X _ { 1 } ^ { n } , T ) , T > 0 , \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} \\begin{cases} H _ 0 u [ x ] = \\displaystyle \\sum _ { y \\in \\mathbb { Z } ^ d } f [ y ] u [ x - y ] , \\\\ H u [ x ] = H _ 0 u [ x ] + V [ x ] u [ x ] , \\end{cases} \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} \\varphi ^ { \\det } _ j : \\det H ^ \\bullet _ \\mathrm { b d } ( Z _ { j , R } , F ) \\rightarrow \\det H ^ \\bullet _ \\mathrm { b d } ( Z _ { j , R ' } , F ) , j = 0 , 1 , 2 \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P , A c h } } = \\max \\left ( \\alpha \\delta _ F ^ { ( 1 ) } + ( 1 - \\alpha ) \\delta _ F ^ { ( 2 ) } , \\alpha \\delta _ E ^ { ( 1 ) } + ( 1 - \\alpha ) \\delta _ E ^ { ( 2 ) } \\right ) . \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} \\varphi _ \\sigma ( f ) = \\begin{cases} d _ \\sigma & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "9883.png", "formula": "\\begin{align*} W \\left [ w _ \\sigma , \\ , w _ { \\rm r e g } \\right ] ( 1 / 2 ) = - \\sum _ { l = 0 } ^ \\infty 2 ^ { - l } \\ , ( l + 1 ) \\left ( \\sum _ { j = 0 } ^ { l + 1 - \\sigma } a _ { l + 1 - \\sigma - j , \\sigma } \\ , b _ j \\right ) \\ , . \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ n \\frac { 2 } { 2 i + 2 k + 1 } { - 2 i - 2 k - 1 \\choose - 2 n - 2 k - 1 } B _ { 2 n - 2 i } \\sum \\limits _ { j = 0 } ^ i - { - k - j - 1 \\brace - k - i - 1 } \\gamma _ { 2 j } . \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} ( n - k + 1 ) ( T _ { ( k ) } - T _ { ( k - 1 ) } ) \\stackrel { d } { = } T _ { 1 } , ~ ~ 1 \\leq k \\leq n , \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} | A | = \\sum _ { l = n _ 1 + 1 } ^ { N } l k _ { B , l } + k _ 1 = q + 1 . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} ( w _ x ^ 2 + w _ y ^ 2 + w _ z ^ 2 ) k = h ( w _ x , w _ y , w _ z , w _ { x x } , w _ { x y } , w _ { y y } , w _ { x z } , w _ { y z } , w _ { z z } ) , \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} | | X | | _ \\lambda = \\sum _ { x \\in X } \\lambda ( \\{ x \\} ) . \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} \\Psi _ { b _ { j } + t } ( x ) = e ^ { i \\left \\langle b _ { j } + t , x \\right \\rangle } + \\left ( A ( b _ { j } ) \\right ) e ^ { i \\left \\langle b _ { j } + t , x \\right \\rangle } + \\left ( A ( b _ { j } ) \\right ) ^ { 2 } e ^ { i \\left \\langle b _ { j } + t , x \\right \\rangle } + . . . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} ( 1 + \\abs { \\nabla \\psi } ^ 2 ) \\Delta \\psi + ( 1 + \\abs { \\nabla \\psi } ^ 2 ) ^ { 3 / 2 } F & \\le - c \\left ( \\frac { \\rho f _ a ^ { \\prime } \\circ \\rho } { f _ a \\circ \\rho } \\right ) \\rho ^ { - \\delta - 2 } + ( 1 + c \\rho ^ { - 2 } ) F \\\\ & \\le - c \\left ( \\frac { \\rho f _ a ^ { \\prime } \\circ \\rho } { f _ a \\circ \\rho } \\right ) \\rho ^ { - \\delta - 2 } \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} A ( s ) = \\sum _ { j = 0 } ^ \\infty T ( s ) ^ { - j } \\sum _ { n = j } ^ \\infty T ( - n ) ( \\mu _ { n , j } - \\mu _ { n , j - 1 } ) \\frac { \\phi _ n ( s ) } { n ! } , \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} - L w + g \\circ w & = \\mu _ n \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ w & = \\nu _ n \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} { { m } _ { m a x } } = \\left \\{ \\begin{matrix} N - ( M + 1 ) / 2 M \\ i s \\ o d d \\\\ N - M / 2 \\ \\ \\ M \\ i s \\ e v e n \\\\ \\end{matrix} \\right . \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} \\mathbf { P _ * } ( | | R ^ * _ { 5 n } | | = o ( n ^ { - 1 / 2 } ) ) = 1 - o ( n ^ { - 1 / 2 } ) \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n ( \\lambda _ k ) ^ { - s } \\geq \\sum _ { k = 1 } ^ n \\left ( ( s + 1 ) ( a _ k ) ^ { - s } - s ( a _ k ) ^ { - s - 1 } b _ k \\right ) . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} \\bar { x } _ 1 = \\lambda _ 1 \\hat { x } _ 1 + \\lambda _ 2 \\hat { x } _ 2 + \\lambda _ 3 \\hat { x } _ 4 , \\ \\bar { x } _ 2 = \\lambda _ 2 \\hat { x } _ 3 + \\lambda _ 3 \\hat { x } _ 5 + \\lambda _ 4 \\hat { x } _ 7 , \\ \\bar { x } _ 3 = \\lambda _ 3 \\hat { x } _ 6 + \\lambda _ 4 \\hat { x } _ 8 + \\lambda _ 5 \\hat { x } _ 9 . \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} X _ A = \\{ ( x _ n ) _ { n \\in \\N } \\in \\{ 1 , \\dots , N \\} ^ { \\N } \\mid A ( x _ n , x _ { n + 1 } ) = 1 n \\in { \\N } \\} . \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} \\alpha ( G [ \\cup _ { i = 1 } ^ j V _ i ] ) = \\max _ { v \\in V _ j } \\{ \\alpha ( G [ ( \\cup _ { i = 1 } ^ { j - 1 } V _ i ) \\cup \\{ v \\} ] ) \\} . \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} N = \\langle [ h , h ' ] \\mid h ' \\in H \\rangle ^ H = \\pi \\mathbb { Z } [ \\omega ] \\oplus \\{ 0 \\} \\oplus \\{ 0 \\} \\oplus \\{ 0 \\} \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\zeta ^ d } { 1 - \\zeta ^ \\ell } \\right ) = - \\ell ^ { 2 a } \\left ( \\frac { ( 2 a ) ! } { \\left ( 2 \\pi i \\ell z \\right ) ^ { 2 a + 1 } } + \\sum _ { b \\geq 0 } \\frac { \\left ( 2 \\pi i \\ell z \\right ) ^ { b } B _ { 2 a + b + 1 } \\left ( \\frac { d } { \\ell } \\right ) } { b ! ( 2 a + b + 1 ) } \\right ) . \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { p - 1 } ( \\epsilon _ { a , k } p + k ) C _ { \\tau , ( a - 1 - k , b ) } & = - p C _ \\tau + \\sum _ { k = 0 } ^ { p - 1 } k C _ { \\tau , ( a - 1 - k , b ) } + p \\sum _ { k = 0 } ^ { a - 1 } C _ { \\tau , ( a - 1 - k , b ) } \\\\ & = - p C _ \\tau + \\sum _ { k = - a } ^ { p - a - 1 } ( k + a ) C _ { \\tau , ( - k - 1 , b ) } + p \\sum _ { k = p - a } ^ { p - 1 } C _ { \\tau , ( - k - 1 , b ) } \\\\ & = ( - p + a ) C _ \\tau + \\sum _ { k = 0 } ^ { p - 1 } k C _ { \\tau , ( - k - 1 , b ) } . \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} T _ P { X } \\ ; : \\ ; \\nabla { F } ( P ) \\cdot ( x _ 0 , \\dotsc , x _ { n + 1 } ) = 0 . \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l l l l } $ u ( 0 , t ) = 0 $ , & $ u ( L , t ) = h _ 1 ( t ) $ , & $ u _ { x } ( L , t ) = h _ 2 ( t ) $ & i n $ ( 0 , T ) $ , \\\\ $ v ( 0 , t ) = 0 $ , & $ v ( L , t ) = g _ 1 ( t ) $ , & $ v _ { x } ( L , t ) = g _ 2 ( t ) $ & i n $ ( 0 , T ) $ , \\end{tabular} \\right . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\frac { t _ h ^ 2 / 2 ^ { \\kappa _ h } } { 1 2 q ^ { - h } N + \\frac { 2 } { 3 } t _ h } \\geq \\frac { C _ 1 ^ 2 N d h q ^ { - h } } { 1 2 q ^ { - h } N + \\frac { 2 } { 3 } C _ 1 \\sqrt { c ( q ) } q ^ { - h } N } = \\dfrac { C _ 1 ^ 2 d h } { 1 2 + \\frac { 2 } { 3 } C _ 1 \\sqrt { c ( q ) } } . \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} Z ( ( \\mu , \\nu ) \\setminus F ) : = Z ( \\mu , \\nu ) \\cap \\big ( G _ E \\setminus \\Big ( \\bigcup _ { \\alpha \\in F } Z ( \\mu \\alpha , \\nu \\alpha ) \\Big ) \\Big ) . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} \\bar { v } _ { k } ( \\theta ) & = \\sum \\limits _ { { t = 1 } } ^ { k } \\frac { 1 } { n } \\sum \\limits _ { i = 1 } ^ { n } \\log \\frac { \\ell ^ i ( S _ { t } ^ i | \\theta ) } { \\ell ^ i ( S _ { t } ^ i | \\theta ^ * ) } \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} \\exp ^ x _ p ( q ) - \\exp ^ x ( p + q ) = \\frac { x ^ 2 } { 2 } ( p q - q p ) + \\sum _ { n \\ge 3 } \\frac { x ^ n } { n ! } \\big ( ( p + q ) ^ { \\cdot _ q n } - ( p + q ) ^ n \\big ) , \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} W ^ { 1 , A } _ 0 ( \\Omega ) = \\{ u \\in W ^ { 1 , A } ( \\Omega ) : \\ , \\ , & \\hbox { t h e c o n t i n u a t i o n o f $ u $ b y $ 0 $ o u t s i d e $ \\Omega $ } \\\\ & \\hbox { i s w e a k l y d i f f e r e n t i a b l e i n $ \\R ^ n $ } \\} . \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} \\sum _ { ( i , j ) \\in A ( D ) } P ( D ) = \\sum _ { ( k , l ) \\in A ( D ' ) } P ( D ' ) . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} ( a \\star b , c ) = ( a \\otimes b , \\Delta ( c ) ) ( c , a \\star b ) = ( \\Delta ( c ) , a \\otimes b ) . \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} & \\sum _ { \\alpha \\in \\Q ^ \\times / ( N _ { E / \\Q } ( E ^ \\times _ { S _ E } ) \\cap \\Q ^ \\times ) } a ^ G ( S , \\delta _ 3 u _ 3 ( \\alpha ) ) \\ , J _ G ( \\delta _ 3 u _ 3 ( \\alpha ) , f _ \\xi ) \\ , J _ G ( \\delta _ 3 u _ 3 ( \\alpha ) , h ) \\\\ & = 2 c _ 3 ' \\times ( - e ^ { i l _ 2 \\theta } + e ^ { i l _ 1 \\theta } ) L ^ S ( 1 , \\chi ) \\sum _ { \\alpha \\in \\Q ^ \\times / ( N _ { E / \\Q } ( E ^ \\times _ { ( S _ 0 ) _ E } ) \\cap \\Q ^ \\times ) } \\chi _ { S _ 0 } ( \\alpha ) \\ , J _ G ( \\delta _ 3 u _ 3 ( \\alpha ) , h ) \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} ( f ( 0 ) + f ( 1 ) + 1 ) f ( m ) ( f ( m ) + 1 ) = 0 . \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} \\norm { D u } ^ 2 \\equiv g ^ { i j } u _ i u _ j = v ^ { - 2 } \\bar { g } ^ { i j } u _ i u _ j \\equiv v ^ { - 2 } \\abs { D u } ^ 2 , \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} f = \\left ( \\frac { \\varphi ^ 1 \\bar g - \\bar \\psi ^ 1 } { 1 + | g | ^ 2 } , \\frac { \\bar \\psi ^ 1 g + \\varphi ^ 1 } { 1 + | g | ^ 2 } \\right ) . \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} R ( L _ { m } ) = f ( m + k ) L _ { m + k } , ~ ~ \\forall m \\in \\mathbb Z , \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { i = 1 } ^ { n } f _ { i } ( x _ { i } ) e _ { i } , \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} [ \\bar { \\xi } _ { i , r } , \\bar { \\xi } _ { j , s } ] = 0 , \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} \\frac { q ^ { { ( n ^ 2 + n ) } / { 2 } } ( - 1 ; q ) _ n } { ( q ; q ) ^ 2 _ n } = \\frac { q ^ { { ( n ^ 2 + n ) } / { 2 } } } { ( q ; q ) _ n } \\frac { 2 } { 1 - q ^ n } \\frac { ( - q ; q ) _ { n - 1 } } { ( q ; q ) _ { n - 1 } } . \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ d \\sum _ { j = 0 } ^ i ( - 1 ) ^ i q ^ { \\frac { d - i } { 2 } } t ^ { \\frac { j } { 2 } } \\dim \\left ( W _ { i + j } H ^ i ( X , \\C ) / W _ { i + j - 1 } H ^ i ( X , \\C ) \\right ) \\in \\Z [ q ^ { \\frac 1 2 } , t ^ { \\frac 1 2 } ] . \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} \\begin{cases} u ( 0 , t ) = h _ 0 ( t ) , \\ , \\ , u ( L , t ) = h _ 1 ( t ) , \\ , \\ , u _ { x } ( L , t ) = h _ 2 ( t ) , \\\\ v ( 0 , t ) = g _ 0 ( t ) , \\ , \\ , v ( L , t ) = g _ 1 ( t ) , \\ , \\ , v _ { x } ( L , t ) = g _ 2 ( t ) . \\end{cases} \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} & c ( x \\otimes y ) = ( x _ 1 \\rightharpoonup y _ 1 ) \\otimes ( x _ 2 \\leftharpoonup y _ 2 ) , & & x , y \\in A , \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\varphi } \\left ( p , \\mathcal { C } \\right ) = - \\rho \\left ( A , \\varphi \\right ) . \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} \\overline G ( x / d ) - \\overline G \\big ( ( x + 1 ) / d \\big ) = o \\big ( \\overline H ( x ) \\big ) \\ \\ d \\in D [ F ] . \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} U _ 1 ( x _ 1 , x _ 2 , t ) = t ^ { p / 2 } \\sum \\limits _ { n = 1 } ^ { N } t ^ { - n / 2 } \\widetilde { S } _ n ( \\eta _ 1 , \\eta _ 2 ) + V _ 1 ( \\mu , \\eta _ 1 , \\eta _ 2 , t ) + O ( \\sigma ^ { - \\rho _ 4 N } ) , \\quad \\rho _ 4 > 0 , \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} \\aligned 1 6 ( \\sum _ { a = 0 } ^ 8 ( q ^ * _ a | _ V ) ^ 2 ) & = 1 6 G | _ V ( | x | ^ 2 + | y | ^ 2 + | z | ^ 2 ) - \\langle \\nabla ( G | _ V ) , \\nabla ( G | _ V ) \\rangle \\\\ & - 4 c ^ 2 ( \\sum _ { a = 1 } ^ 4 p _ a ^ * | _ V \\ , z _ a ) ^ 2 , \\endaligned \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} - L u + g \\circ u & = f \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = \\eta \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} a \\circ b = \\pi ( \\pi ^ { - 1 } ( a ) \\pi ^ { - 1 } ( b ) ) , a , b \\in A . \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} \\frac { \\frac { \\theta _ i E _ s ^ { i * } } { \\alpha _ i ^ * } } { \\ln 2 \\cdot \\left ( 1 + \\frac { \\theta _ i E _ s ^ { i * } } { \\alpha _ i ^ * } \\right ) } - \\log _ 2 \\left ( 1 + \\frac { \\theta _ i E _ s ^ { i * } } { \\alpha _ i ^ * } \\right ) + \\sum _ { j = i } ^ M \\eta P _ p \\lambda _ j ^ * \\\\ - P _ { i n t } \\gamma _ i ^ * + \\mu _ i ^ * & = 0 \\\\ - \\frac { \\theta _ i } { \\ln 2 \\left ( 1 + \\frac { \\theta _ i E _ s ^ { i * } } { \\alpha _ i ^ * } \\right ) } + \\sum _ { j = i } ^ M \\lambda _ j ^ * + \\gamma _ i ^ * h _ { s p } ^ i & = 0 \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i { m \\choose i } \\widetilde { \\gamma } _ { m + i + 1 } = 0 , \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} j ( A _ N ) = \\{ \\alpha < \\kappa \\ | \\ \\forall f \\in j ( N ) = j [ N ] \\ \\ f ( \\alpha ) < \\alpha \\implies f ( \\alpha ) \\in j ( N ) = j [ N ] \\} . \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} C - 1 \\geq r + i - D - 1 + m - A = B + C + D - i - 1 , \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} \\chi ^ { v i r t } ( \\pi ) = \\chi ( F ( r ) ) / | C | = \\frac { 1 - r } 4 . \\end{align*}"}
-{"id": "10172.png", "formula": "\\begin{align*} \\mathbb P ( Z _ n ^ * \\le - 1 ) = \\frac 1 2 \\mathbb P ( T _ 0 > n ) \\sim \\frac { \\gamma a _ n } { n } \\mathbb E \\left [ \\sup _ { t \\in [ 0 , 1 ] } \\Delta _ t \\right ] , \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} \\begin{bmatrix} \\ddots & \\ddots \\\\ \\ddots & - B & C \\\\ & A & - B & C \\\\ & & A & - B & \\ddots \\\\ & & & \\ddots & \\ddots & \\end{bmatrix} \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} e = ( g , e ) ( e , h ) ( g , e ) ^ { - 1 } ( e , h ) ^ { - 1 } , g , h \\neq e , \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} | | X | | _ r = \\sum _ { x \\in X } ( r ( \\{ x \\} ) ) . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} c _ 0 = \\tfrac { 1 } { 4 } c , \\sigma \\leq \\min ( \\epsilon ^ 3 p ^ { - 1 / 2 } { c _ 0 } ^ { - 1 } , \\sigma _ 0 ) , \\delta = \\epsilon p ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} \\Psi ^ b ( c ) ( y ) = \\Psi _ { y } ( c ( \\psi ^ * ( y ) ) ) \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u & = \\left [ D _ { x ^ { i } } \\left ( a ^ { i j } u _ { x ^ { j } } + b ^ { i } u + f ^ { i } ( u ) \\right ) + c u + h ( u ) \\right ] \\\\ & \\qquad + \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } [ \\sigma ^ { i j k } u _ { x ^ { i } x ^ { j } } + \\mu ^ { i k } u _ { x ^ { i } } + \\nu ^ { k } u + g ^ { k } ( u ) ] d w _ { s } ^ { k } . \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} \\frac { q _ 1 } { p _ 1 } \\leq \\dfrac { q / \\delta ^ 4 d _ * ^ 2 } { p / \\delta ^ 4 d _ * ^ 2 } = \\frac { q } { p } \\leq s . \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} X ( u , v ) = ( f _ { 1 } ( u ) , f _ { 2 } ( u ) , \\lambda \\cos \\left ( \\frac { u } { c } \\right ) \\cos v , \\lambda \\cos \\left ( \\frac { u } { c } \\right ) \\sin v ) , \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\left ( \\left \\{ \\mathbf { \\bar { G } } _ { i j , k l } \\mathbf { \\bar { v } } _ { k p , l } ^ { [ \\sf b s ] ( \\mathbf { s } ^ { [ \\sf d ] } ) } \\right \\} _ { p \\in [ 1 : N ] , k \\in [ 1 : K ] , l \\in [ 1 : M ] , \\mathbf { s } ^ { [ \\sf d ] } \\in \\mathcal { S } _ { T _ { \\sf d } } ^ { [ \\sf d ] } ~ } \\right ) \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} \\frac { F ( x _ N ) - F ( x ^ * ) } { a _ N ^ 2 } + \\frac { \\tilde { \\mu } } { 2 } \\| x ^ * - v _ N \\| ^ 2 \\le & \\frac { \\tilde { \\mu } } { 2 } \\norm { x ^ * - v _ 0 } ^ 2 + \\rho M ^ 2 \\left ( \\sum _ { j = 1 } ^ N \\frac { 1 } { a _ j } \\right ) \\\\ & + \\frac { N r M ^ 2 } { 2 } - \\frac { \\tilde { \\mu } - \\mu } { 2 } \\sum _ { j = 1 } ^ N \\frac { \\norm { x _ j - y _ j } ^ 2 } { a _ j ^ 2 } + 2 L \\sum _ { j = 1 } ^ N \\frac { \\varepsilon _ j + \\delta _ j } { a _ j ^ 2 } . \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} x \\mapsto z & = x - t \\bar { U } ( t , y ) \\\\ y \\mapsto v & = \\bar { U } ( t , y ) . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} D ( n , k ) & = \\frac { 1 } { 2 } R ( n , k ) + \\frac { 1 } { 2 } \\binom { n - k + 1 } { k } ( - 1 ) ^ k ; \\\\ T ( n , k ) & = \\frac { 1 } { 2 } R ( n , k ) - \\frac { 1 } { 2 } \\binom { n - k + 1 } { k } ( - 1 ) ^ k . \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\xi & = & \\dfrac { p _ { i } } { q } + \\varepsilon l \\\\ \\varepsilon q & \\cong & 0 \\end{array} \\right . \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} F ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } ) = \\sum _ { \\underset { \\delta _ { j } = 0 } { \\delta \\in \\{ 0 , 1 \\} ^ { n } } } u _ { \\delta } ( x _ { j } ) \\left [ \\prod _ { k \\neq i , j } f _ { k } ^ { \\delta _ { k } } ( x _ { k } ) \\right ] f _ { i } ^ { \\delta _ { i } } ( x _ { i } ) , \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} Q ( t ) & : = \\nabla _ x q ( t , s ) - I , \\\\ P ( t ) & : = \\nabla _ x p ( t , s ) , \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} P ( \\{ ( 1 , 2 ) , ( 1 , 3 ) \\} \\subset A ( \\mathbf { D } ) ) = ( p _ e p _ d ) ^ 2 = \\left ( P ( ( 1 , 2 ) \\in A ( \\mathbf { D } ) ) \\right ) ^ 2 . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ \\infty \\lambda _ F ( p ^ n ) t ^ n = \\frac { P _ { F , p } ( t ) } { Q _ { F , p } ( t ) } , \\ P _ { F , p } ( t ) = 1 - p ^ { \\mu - 1 } \\chi ( p ) t ^ 2 , \\\\ & Q _ { F , p } ( t ) = 1 - \\lambda _ F ( p ) T + \\{ \\lambda _ F ( p ) ^ 2 - \\lambda _ F ( p ^ 2 ) - p ^ { \\mu - 1 } \\chi ( p ) \\} t ^ 2 - \\chi ( p ) p ^ { \\mu } \\lambda _ F ( p ) t ^ 3 + \\chi ( p ) ^ 2 p ^ { 2 \\mu } t ^ 4 \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} C _ Y ( I ) > C _ Y ( H ) \\Leftrightarrow \\bigg [ \\Big [ k = 1 C _ X ( I ' ) > C _ X ( H ' ) \\Big ] \\Big [ k > 1 C _ X ( I ' ) > 1 \\Big ] \\bigg ] . \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} \\frac { ( - q ; q ^ 2 ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\sum _ { j \\geq 0 } q ^ { 3 j ^ 2 + 2 j } ( 1 - q ^ { 2 j + 1 } ) = \\sum _ { j \\geq 0 } q ^ { 3 j ^ 2 + 2 j } \\frac { ( - q ; q ^ 2 ) _ { j } } { ( q ^ 2 ; q ^ 2 ) _ { 2 j } } \\frac { ( - q ^ { 2 j + 3 } ; q ^ 2 ) _ \\infty } { ( q ^ { 4 j + 4 } ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} ( D _ { k , i } + D _ { k , i + 1 } ) H & = D _ { k , i + 1 } , & & 1 \\leq i \\leq k - 1 , \\\\ D _ { k , 1 } H & = D _ { k , 1 } , \\\\ D _ { k , i } E _ 2 & = - D _ { k , k + i } , & & 1 \\leq i \\leq k , \\\\ C _ { k , n - 2 k } & = 0 . \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} ( U _ { n - 1 } ( x / 2 ) , U _ { m - 1 } ( x / 2 ) ) = U _ { ( n , m ) - 1 } ( x / 2 ) \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} \\check { \\tilde { u } } ^ i = \\frac { 1 } { \\cosh ^ 2 \\tilde { u } } \\sigma ^ { i j } \\tilde { u } _ j . \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} ( g \\phi ) ( w ) & = \\bigl ( ( g \\Phi ) ( w ) \\bigr ) ( 1 , 1 ) = \\bigl ( ( g , 1 ) \\Phi ( w ) \\bigr ) ( 1 , 1 ) = \\bigl ( ( g , h _ g ) ( 1 , h _ g ^ { - 1 } ) \\Phi ( w ) \\bigr ) ( 1 , 1 ) \\\\ & = \\bigl ( ( g , h _ g ) \\Phi ( h _ g ^ { - 1 } w ) \\bigr ) ( 1 , 1 ) = \\Phi ( h _ g ^ { - 1 } w ) ( g , h _ g ) = ( g , h _ g ) \\bigl ( \\Phi ( h _ g ^ { - 1 } w ) ( 1 , 1 ) \\bigr ) \\\\ & = ( g , h _ g ) \\phi ( h _ g ^ { - 1 } w ) . \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} k _ r ( x , y ) = \\int _ 0 ^ \\infty g _ r ( s ) \\partial _ x p _ s ( x , y ) \\ d s \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} P _ g ^ 6 = \\Big ( - \\Delta _ g + \\frac { ( n - 6 ) ( n + 4 ) } { 4 n ( n - 1 ) } R _ g \\Big ) \\Big ( - \\Delta _ g + \\frac { ( n - 4 ) ( n + 2 ) } { 4 n ( n - 1 ) } R _ g \\Big ) \\Big ( - \\Delta _ g + \\frac { n - 2 } { 4 ( n - 1 ) } R _ g \\Big ) . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} f _ 1 ( x , y ) & = x ^ { 1 2 } ( y _ 0 + \\pi ^ { j + 1 } y ) ^ 4 ( ( y _ 0 - 1 + \\pi ^ { j + 1 } y ) ^ 2 + \\pi ^ { 8 n } x ^ 8 ( y _ 0 + \\pi ^ { j + 1 } y ) ^ 4 ) , \\\\ f _ 2 ( x , y ) & = x ^ { 1 2 } ( 1 + \\pi ^ { j + 1 } y ) ^ 4 ( y ^ 2 + \\pi ^ { 8 n - ( 2 + 2 j ) } x ^ 8 ( 1 + \\pi ^ { j + 1 } y ) ^ 4 ) , \\\\ f _ 3 ( x , y ) & = x ^ { 1 2 } ( 1 + \\pi ^ { j + 1 } y ) ^ 4 ( y ^ 2 + x ^ 8 ( 1 + \\pi ^ { j + 1 } y ) ^ 4 ) , \\\\ \\intertext { a n d } f _ 4 ( x , y ) & = x ^ { 1 2 } ( 1 + \\pi ^ { j + 1 } y ) ^ 4 ( \\pi ^ { 2 + 2 j - 8 n } y ^ 2 + x ^ 8 ( 1 + \\pi ^ { j + 1 } y ) ^ 4 ) . \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} \\| W _ J ^ \\pm ( \\Gamma ) u \\| ^ 2 = \\lim _ { t \\to \\pm \\infty } \\| J _ a u _ t \\| ^ 2 = \\lim _ { t \\to \\pm \\infty } ( ( J _ a ^ * J _ a - I ) u _ t , u _ t ) + \\| u \\| ^ 2 . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} C _ { t , r } \\left ( 1 \\right ) = C _ { t , t } \\left ( 1 \\right ) \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} d \\mu _ t = \\mu _ t \\ , d x \\ , \\textrm { o n } \\ , M _ t , \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { u } + \\mathcal { A } \\big ( \\mathbf { H } ( \\mathbf { u } ) \\big ) \\mathbf { u } = 0 ( 0 , T ) , \\mathbf { u } ( 0 , \\cdot ) = \\mathbf { u } ^ { 0 } \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} y _ { k q + i } ^ T = y ^ T _ i + k p \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} \\omega ^ { L C } = \\omega + \\tau + \\tilde \\lambda , \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} \\Delta _ i ^ + = \\partial _ { i + 1 } \\delta _ i , \\qquad \\Delta _ i ^ - = \\delta _ { i - 1 } \\partial _ i , \\qquad \\Delta _ i = \\Delta _ i ^ + + \\Delta _ i ^ - \\ , , \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} b ( f ( x ) , y ) = a ( x , g ( y ) ) , \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} s _ f ( p ) = \\alpha ( 1 - p ) + \\beta ( 2 p - 1 ) , f ( t ) - f ( t - p ) = \\gamma - \\alpha + c ( t ) - a ( t ) , t \\in [ p , 1 ] . \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} \\frac { \\partial ^ \\alpha y ( x , t ) } { \\partial t ^ \\alpha } = \\frac { 1 } { \\Gamma ( 1 - \\alpha ) } \\int ^ t _ 0 \\frac { \\partial y ( x , \\xi ) } { \\partial \\xi } \\frac { d \\xi } { ( t - \\xi ) ^ \\alpha } , \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} J _ t = 0 \\ , . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} P _ { X , D } ( t ) = \\sum _ { m \\geqslant 0 } \\ell ( m D ) t ^ m . \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} \\begin{array} { l l l l l l l l } T = T _ 0 , & \\lambda _ 0 = 1 , & \\alpha = 0 . 1 5 4 9 , & \\eta = 0 . 0 5 7 2 2 , \\\\ \\epsilon = 1 0 ^ { - 5 } , & \\upsilon = 0 . 1 , & \\xi = 1 . 0 0 3 0 , & \\omega = 0 . 0 2 0 7 4 . \\end{array} \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} { \\mathrm { E } } \\left [ \\frac { \\partial ( X _ { i } ^ { \\prime } ( \\widehat { \\beta } - \\beta ) ) } { \\partial \\varepsilon _ { i } } \\right ] = { \\mathrm { E } } [ \\psi _ { i } X _ { i } ^ { \\prime } ( \\widehat { \\beta } - \\beta ) ] , i = 1 , \\dots , n . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} \\widetilde { \\omega } = \\sum _ { j = 1 } ^ m J _ { A , Z _ R } ( \\widetilde { \\omega } _ { 1 , j } , \\widetilde { \\omega } _ { 1 , j } ^ \\mathrm { z m } , \\widetilde { \\omega } _ { 2 , j } , \\widetilde { \\omega } _ { 2 , j } ^ \\mathrm { z m } ) . \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} Y _ { + } ( x ) = Y _ { - } ( x ) \\begin{pmatrix} I _ 2 & W ( x ) \\\\ 0 & I _ 2 \\end{pmatrix} , x \\in \\mathbb ( 0 , + \\infty ) , \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} \\mathcal { R } _ { \\sf e s t } ( \\widehat { \\boldsymbol { \\theta } } ) : = \\sup _ { \\mathcal { H } } \\left ( \\mathbb { E } \\left [ \\| \\widehat { \\boldsymbol { \\theta } } - \\boldsymbol { \\theta } \\| _ 2 ^ 2 \\right ] \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial t } = \\Delta _ { g } f + r f + \\psi . \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} \\sum _ { a = 5 } ^ 8 \\langle ( e _ b e _ c - e _ c e _ b ) e _ p , f _ a \\rangle \\ , S ^ a _ { b , c ' } = 0 , \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} C _ { } & = \\max _ { 0 \\leq p \\leq 0 . 5 } \\frac { H _ 2 ( p ) } { \\frac { 1 } { 1 - \\epsilon } + p } . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} u '' + \\lambda a ( t ) \\biggl { ( } \\dfrac { u ^ { \\gamma } } { 1 + u ^ { \\sigma } } \\biggr { ) } = 0 , \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} \\operatorname { e r f } ( w ) & = e ^ { - w ^ 2 } \\sum _ { n \\geq 1 } \\frac { w ^ { 2 n - 1 } } { \\Gamma \\left ( n + \\frac { 1 } { 2 } \\right ) } , \\\\ ( w ) & = \\frac { e ^ { - w ^ 2 } } { \\sqrt { \\pi } } \\sum _ { m = 0 } ^ N \\frac { ( - 1 ) ^ m \\left ( \\frac { 1 } { 2 } \\right ) _ m } { w ^ { 2 m + 1 } } + O \\left ( w ^ { - 2 N - 3 } \\right ) . \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} E _ 4 = R _ { \\mu \\nu \\rho \\sigma } ^ 2 - 4 R _ { \\mu \\nu } ^ 2 + R ^ 2 , W ^ 2 _ { \\mu \\nu \\rho \\sigma } = R _ { \\mu \\nu \\rho \\sigma } ^ 2 - 2 R _ { \\mu \\nu } ^ 2 + \\frac { 1 } { 3 } R ^ 2 \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} r _ { 4 } ( n ) = 8 \\sigma ( n ) - 3 2 \\sigma ( \\frac { n } { 4 } ) . \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} h ( X _ { i } , X _ { _ { j } } ) = \\widetilde { \\nabla } _ { X _ { _ { i } } } X _ { _ { j } } - \\nabla _ { X _ { _ { i } } } X _ { _ { j } } 1 \\leq i , j \\leq 2 \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} \\xi ( s , \\chi ) = w ( \\chi ) \\xi ( 1 - s , \\bar { \\chi } ) \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} [ v _ { \\delta ^ { 1 } } + u _ { \\delta ^ { 1 , n } } g _ { n } ( x _ { n } ) ] = \\frac { [ g _ { 1 } ( x _ { 1 } - x _ { n } ) - g _ { 1 } ( - x _ { n } ) ] } { g _ { 1 } ( x _ { 1 } ) } \\sum _ { \\substack { \\delta \\in \\{ 0 , 1 \\} ^ { n - 1 } , \\\\ \\delta _ { 1 } = 1 } } v _ { \\delta } f _ { n } ^ { \\delta _ { n } } ( x _ { n } ) \\prod _ { i = 2 } ^ { n - 1 } g _ { i } ^ { \\delta _ { i } } ( - x _ { n } ) . \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} Y _ l = \\prod _ { i \\in I '' } { } ^ { i } ( { } ^ { j _ l } X ) . \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} C ^ { F B , A . 1 } = ( 1 - \\nu _ e ) \\log ( 1 + 2 ^ { \\Delta { C } ^ { 1 , \\infty } } ) - \\nu _ 0 \\Delta { C } ^ { 1 , \\infty } \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} w ( \\sigma ) = \\prod _ { i = 1 } ^ 6 w _ i ^ { n _ i ( \\sigma ) } , \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\tilde { g } ^ { i j } = \\sigma ^ { i j } - v ^ { - 2 } \\varphi ^ i \\varphi ^ j , \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} \\phi \\big ( ( D _ i D _ i ^ * ) M ( D _ j ) ( D _ i D _ i ^ * ) M ( D _ j ) ^ * \\big ) = \\phi _ 1 \\big ( ( a _ { 1 ; i } a _ { 1 ; i } ^ * ) M ( a _ { 1 ; j } ) ( a _ { 1 ; i } a _ { 1 ; i } ^ * ) M ( a _ { 1 ; j } ) ^ * \\big ) . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} W = \\left ( \\overline { \\zeta } _ { m } - \\overline { z } _ { m } \\right ) \\rho ^ { N } ( \\zeta ) \\prod _ { k = 1 } ^ { n - 1 } \\frac { \\partial Q _ { i _ { k } } ( z , \\zeta ) } { \\partial \\overline { \\zeta } _ { j _ { k } } } \\bigwedge _ { i = 1 } ^ { n } \\left ( d \\zeta _ { i } \\wedge d \\overline { \\zeta _ { i } } \\right ) \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} \\theta _ { u ^ * ( j ) t } ( x , y ) = \\theta ' _ j ( v ( x ) , v ( y ) ) , \\ \\forall t \\in A \\mbox { a n d } j \\in H . \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} u _ { t t } - c ^ { 2 } \\triangle u + \\alpha \\triangle ^ { 2 } u + c ^ { 2 } \\gamma ( - \\triangle ) ^ { 1 / 2 } \\theta & = \\chi _ { \\omega } u _ { 1 } , \\\\ \\theta _ { t } - \\triangle \\theta - \\gamma ( - \\triangle ) ^ { 1 / 2 } u _ { t } & = \\chi _ { \\omega } u _ { 2 } \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} y _ { s , k } = \\sqrt { \\rho _ { s , k } } ( \\mathbf { h } _ { s a , k } ^ { H } \\mathbf { x } + n _ { s a , k } ) + n _ { s p , k } , ~ \\forall k \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} \\tilde { u } ( \\tau , \\xi ) = u ( t , \\xi ) \\Theta ( t , T ^ * ) ^ { - 1 } , \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} ( H J _ a - J _ a H _ 0 ) u \\left [ x \\right ] = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - y \\cdot \\xi ) } s _ a ( x , \\xi ) u \\left [ y \\right ] d \\xi , \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} p _ k & \\leq \\frac { p _ 0 } { c _ * ( s , p _ 0 , q _ 0 ) } s ^ k , 0 < c _ * ( s , p _ 0 , q _ 0 ) \\leq 1 - \\sum _ { j = 0 } ^ k \\limits q _ k \\leq s p _ { k - 1 } + q _ { k - 1 } , \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} X ( f ) = \\sum _ { l \\leq L } \\frac { x _ l \\lambda _ f ( l ) } { \\sqrt { l } } , \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} \\Delta f = \\Delta ^ S f _ { \\mathfrak { L } } + \\Delta ^ D f _ { \\mathfrak { L } ^ \\perp } . \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} \\varphi ( i , \\alpha ) = ( \\psi _ { \\alpha } ( i ) , \\sigma _ i ( \\alpha ) ) \\end{align*}"}
-{"id": "8881.png", "formula": "\\begin{align*} ( a \\nabla _ { \\nu } b ) ^ { 2 } \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ { n - 1 } } } , 2 ) ^ { - r _ { n } } ( a \\sharp _ { \\nu } b ) ^ { 2 } + R _ { 0 } ^ { 2 } ( a - b ) ^ { 2 } - \\sum _ { k = 1 } ^ { n - 1 } r _ { k } \\big [ a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } - a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } , \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} \\theta ( x _ { j , i } ) ^ 2 = \\theta ( v _ { j , i } ) ^ 2 . \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} A _ 0 : = - \\i \\left ( 2 ^ { - 1 } ( S ^ * + S ) + N ( S ^ * - S ) \\right ) = \\i \\left ( 2 ^ { - 1 } ( S ^ * + S ) - ( S ^ * - S ) N \\right ) , \\mathcal { D } ( A _ 0 ) = \\ell _ 0 ( \\Z ) . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} \\Phi _ s = \\frac 1 { E _ { \\alpha , \\beta } ^ \\gamma ( t ^ \\alpha ) } \\sum _ { r = 1 } ^ s ( - 1 ) ^ r \\ , e _ r \\ , \\sum _ { j = 0 } ^ r ( \\gamma ) _ j \\ , { r \\brace j } \\ , t ^ { \\alpha j } \\ , E _ { \\alpha , \\alpha j + \\beta } ^ { \\gamma + j } ( t ^ \\alpha ) \\ , . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} J _ { k _ U } ( n , s ) = O ( p ^ { \\frac { \\epsilon } { 2 } ( \\log _ { p / q } \\log n ) ^ 2 + o ( ( \\log \\log n ) ^ 2 ) } ) \\to 0 . \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} { C } _ { X ^ n \\rightarrow Y ^ n } ^ { F B , A . M } = \\sup _ { { \\cal P } ^ { A . M } _ { 0 , n } } \\sum _ { t = 0 } ^ n { \\bf E } ^ { \\pi } \\left \\{ \\log \\Big ( \\frac { q _ t ( \\cdot | Y _ { t - M } ^ { t - 1 } , X _ t ) } { \\nu _ t ^ { \\pi } ( \\cdot | Y _ { t - M } ^ { t - 1 } ) } ( Y _ t ) \\Big ) \\right \\} \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} \\langle u , [ T , A ] v \\rangle : = \\langle T u , A v \\rangle - \\langle A u , T v \\rangle . \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} \\left | \\left ( \\delta _ { a _ 1 ^ { - 1 } e _ 1 } - \\delta _ 0 \\right ) * \\sigma _ j \\right | ( [ - r , r ] ^ s ) & = \\left | \\left ( \\delta _ { N _ j a _ 1 ^ { - 1 } e _ 1 } - \\delta _ 0 \\right ) * \\rho _ j \\right | ( [ - r , r ] ^ s ) \\\\ & \\le \\left ( \\delta _ { N _ j a _ 1 ^ { - 1 } e _ 1 } + \\delta _ 0 \\right ) * \\rho _ j ( [ - r , r ] ^ s ) \\to 0 , \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} \\mathcal { L } _ { , \\mathfrak { C } _ { 5 } } ( A ) = \\kappa \\ , Q ^ { ( 5 ) } ( A ) = \\kappa \\ , Q ^ { ( 5 ) } ( A , \\bar { A } = 0 ) \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} q _ { 2 i } & = p _ { 2 i } + \\omega ^ { a _ { h ( 2 i ) } ( 2 i ) } \\left ( \\frac { z } { 2 } \\right ) ^ { h ( 2 i ) } ( z ^ 1 + z ^ 2 + \\dots + z ^ { k - h ( 2 i ) } ) \\\\ q _ { 2 i + 1 } & = p _ { 2 i + 1 } + \\omega ^ { a _ { h ( 2 i + 1 ) } ( 2 i + 1 ) } \\left ( \\frac { z } { 2 } \\right ) ^ { h ( 2 i + 1 ) } ( z ^ 1 + z ^ 2 + \\dots + z ^ { k - h ( 2 i + 1 ) } ) \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} \\left ( \\mathcal { V } G _ \\infty \\right ) ( t ) = \\ell ^ { - 1 } \\int _ 0 ^ t V ^ 0 ( \\tau ) G _ \\infty ( \\tau ) d \\tau \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} C ^ { F B , A . J } _ { X ^ n \\rightarrow { Y ^ n } } ( \\kappa ) = \\inf _ { s \\geq { 0 } } \\sum _ { y ^ { - 1 } _ { - J } } C _ 0 ( y ^ { - 1 } _ { - J } ) \\mu ( y ^ { - 1 } _ { - J } ) . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} G _ 2 ( \\vec { X } ) : = \\dfrac { \\alpha _ 2 } { 2 } \\| \\vec { Y } - \\vec { X } \\| _ F ^ 2 + \\lambda _ 1 \\sum _ { i = 1 } ^ { m } \\sum _ { j = 1 } ^ { n } s ( \\vec { X } _ { i j } ; a _ 1 ) , \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} U _ a H U _ a ^ { - 1 } = \\frac { 1 } { a } \\left ( \\sqrt { p ^ 2 + ( a m ) ^ 2 } - a m + a V ( a x ) \\right ) . \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i & = \\big ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } - B _ { i J K j } \\dot { u } _ { j } ) _ { , K } \\big ) _ { , J } \\\\ a \\ddot { \\tau } & = - \\beta _ { K i } \\dot { u } _ { i , K } + m _ { I J } \\dot { \\tau } _ { , I J } + M _ { j L K I } u _ { j , L K I } + K _ { I J } \\tau _ { , I J } \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} K ( h ^ { \\frac { 1 } { 2 ^ n } } , 2 ) ^ { r _ n } a \\sharp _ { \\nu } b & \\leqslant a \\nabla _ { \\nu } b - \\sum _ { k = 0 } ^ { n - 1 } r _ { k } \\big [ \\big ( a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } - \\big ( a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ] ^ { 2 } \\\\ & \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ n } } , 2 ) ^ { R _ n } a \\sharp _ { \\nu } b , \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} - 3 \\lambda \\begin{pmatrix} I _ d & 0 \\\\ 0 & I _ d \\end{pmatrix} \\le \\begin{pmatrix} X _ \\lambda & 0 \\\\ 0 & - Y _ \\lambda \\end{pmatrix} \\le 3 \\lambda \\begin{pmatrix} I _ d & - I _ d \\\\ - I _ d & I _ d \\end{pmatrix} \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} \\omega ( G ) \\le \\chi _ v ( G ) \\le \\theta ( \\overline { G } ) \\le \\chi _ f ( G ) \\le \\chi _ c ( G ) \\le \\lceil \\chi _ c ( G ) \\rceil = \\chi ( G ) , \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} \\min \\limits _ { \\mathbf { y } \\in X _ { 1 } \\times \\dots \\times X _ { n } } \\to \\sum \\limits _ { i = 1 } ^ { n } \\Phi _ { i } ( \\mathbf { x } , \\mathbf { y } _ { i } ) , \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} r _ M ( X ) + r _ { M ^ * } ( X ) = \\lambda ( X ) + | X | . \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} \\| \\sin \\Theta ( \\hat V , V ) \\| _ F = \\sqrt { r - \\| V ^ { \\intercal } \\hat V \\| _ F ^ 2 } = \\| \\hat V ^ { \\intercal } V _ { \\perp } \\| _ F . \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} \\rho _ { \\nu + m , b } ( x ) = ( - b ) ^ { - m } r _ { m , \\nu } ( b ^ 2 x ) \\rho _ { \\nu , b } ( x ) + ( - b ) ^ { 1 - m } s _ { m , \\nu } ( b ^ 2 x ) \\rho _ { \\nu + 1 , b } ( x ) , \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} \\theta _ { u ^ * ( j ) t } ( x , y ) = \\theta _ { u ^ * ( j ) } ( x , y ) \\ \\forall t \\in A \\mbox { a n d } j \\in H . \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\nu ^ { \\pi ^ * } _ n ( 1 | 1 ) \\equiv { { c } _ 1 ( n ) = \\frac { 2 ^ { \\mu _ 1 ( n ) } } { 1 + 2 ^ { \\mu _ 1 ( n ) } } } , ~ \\pi ^ * _ n ( 1 | 1 ) \\equiv { d _ 1 ( n ) } = \\frac { \\beta _ n ( 1 + 2 ^ { \\mu _ 1 ( n ) } ) - 1 } { ( \\beta _ n - \\delta _ n ) ( 1 + 2 ^ { \\mu _ 1 ( n ) } ) } , ~ \\mu _ 1 ( n ) \\triangleq \\frac { H ( \\beta _ n ) - H ( \\delta _ n ) } { \\beta _ n - \\delta _ n } . \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} P = \\frac { 1 } { \\varepsilon } ( q - \\alpha \\theta - \\hat q ) ^ + , \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} S _ 1 ( l , u , v ; N ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { 1 } { n ^ { 1 / 2 + u + v } } \\Delta _ { 2 k , N } ( l , n ) \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} \\pi ^ * _ n ( 0 | 0 ) \\equiv { { d _ 0 } ( n ) = \\frac { 1 - \\gamma _ n ( 1 + 2 ^ { \\mu _ 0 ( n ) } ) } { ( \\alpha _ n - \\gamma _ n ) ( 1 + 2 ^ { \\mu _ 0 ( n ) } ) } } . \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\left [ ( \\sigma _ k ^ { \\delta ( m ) } \\wedge T ) - ( \\tau _ { k - 1 } ^ { \\delta ( m ) } \\wedge T ) \\right ] = \\int _ 0 ^ T \\sum _ { k = 1 } ^ { \\infty } 1 \\{ t \\in [ \\tau _ { k - 1 } ^ { \\delta ( m ) } , \\sigma _ { k } ^ { \\delta ( m ) } ] \\} d t \\leq \\int _ 0 ^ T 1 \\{ Z ( t ) \\in S \\backslash S _ { 2 \\delta ( m ) } \\} d t , \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} \\begin{cases} \\varphi _ t + \\varphi _ { x x x } + \\frac { a b } { c } \\psi _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ \\psi _ t + \\frac { r } { c } \\psi _ x + a \\varphi _ { x x x } + \\frac { 1 } { c } \\psi _ { x x x } = 0 , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\end{cases} \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\left | \\Pi _ n \\right | = \\left ( n ! \\right ) ^ { 2 n } \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} g \\le \\frac { 4 } { 3 } \\left ( 1 + \\frac { 1 } { N _ T / 4 } \\right ) = 4 . \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} \\Delta _ 1 ( L , B ) = \\sum _ { j = 0 } ^ p v _ L ^ j \\pi _ { L ^ \\perp } ^ * T ^ { ( p - j ) } ( L , B ) \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} d = \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } \\binom { N _ R - r - 1 } { t - 1 } t N ^ { ( N _ R - r - t ) ( N _ T - t + 1 ) } } { S } \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} y ( x ^ p ) = A ( x ) y ( x ) , \\ \\ \\ y ( x ^ q ) = B ( x ) y ( x ) \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} M _ i = \\begin{pmatrix} 1 & - i / q \\\\ 1 & 1 - i / q \\end{pmatrix} M _ i ^ { - 1 } = \\begin{pmatrix} 1 - i / q & i / q \\\\ - 1 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } g = - 2 R i c \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{gather*} \\mu _ A \\mathcal { F } ( \\blacktriangleright \\otimes \\blacktriangleright ) ( a \\otimes b ) : = ( \\mu _ A \\circ ( \\blacktriangleright \\otimes \\blacktriangleright ) ) ( \\mathcal { F } \\otimes ( a \\otimes b ) ) . \\end{gather*}"}
-{"id": "3845.png", "formula": "\\begin{align*} K _ 0 ( z ) \\leq e ^ { - z } \\int _ 0 ^ \\infty \\frac { e ^ { - s } } { \\sqrt { 2 s z } } d s = \\frac { e ^ { - z } } { \\sqrt { 2 z } } \\sqrt { \\pi } . \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} g _ l ( \\check { q } ) : = \\sum _ { d } \\frac { ( - 1 ) ^ { ( D _ l \\cdot d ) } ( - ( D _ l \\cdot d ) - 1 ) ! } { \\prod _ { p \\neq l } ( D _ p \\cdot d ) ! } \\check { q } ^ d , \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} \\hat v _ i ^ k = \\sum _ { j = 1 } ^ { N } w _ { i j } ( k ) v _ j ^ k \\mbox { w i t h } v _ j ^ 0 = x _ j ^ 0 \\mbox { f o r a l l } j = 1 , \\ldots , N . \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} \\lVert J _ k ( x _ k , \\gamma _ k , \\tilde { \\lambda } _ k ) - x _ k \\rVert _ k = \\frac { \\tilde { \\lambda } _ k } { \\lambda _ k } \\norm { J _ k ( x _ k , \\gamma _ k , \\lambda _ k ) - x _ k } _ k = \\frac { 1 } { \\theta } \\norm { x _ { k + 1 } - x _ k } _ k \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} r _ { \\alpha , \\beta } ( x , \\xi ) = \\sum _ { | \\gamma | _ { * } \\leq | \\beta | _ { * } - 1 } c _ { \\alpha , \\beta , \\gamma } ( x ) \\ , \\xi ^ \\gamma , \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} M \\vec v = \\lambda \\vec v + \\vec u . \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} R ^ { * } \\ ! = \\ ! \\max _ { t } \\log _ { 2 } ( 1 \\ ! + \\ ! f ( t ) ) \\ ! + \\ ! \\log _ { 2 } ( t ) , ~ s . t . ~ t _ { \\min } \\ ! \\leq \\ ! t \\ ! \\leq \\ ! 1 , \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} L _ { e } ^ { p } ( \\Omega ) : = \\left \\{ u \\in L _ { e } ^ { p } ( \\mathbb { R } ^ { N } ) \\ , : \\ , u = 0 \\mathbb { R } ^ { N } \\backslash \\Omega \\right \\} . \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} F ( x _ { 1 } , x _ { 2 } , . . . , x _ { n } ) = \\sum _ { \\underset { \\delta _ { i } = 0 } { \\delta \\in \\{ 0 , 1 \\} ^ { n } } } v _ { \\delta } ( x _ { i } ) \\left [ \\prod _ { k \\neq i , j } f _ { k } ^ { \\delta _ { k } } ( x _ { k } ) \\right ] g _ { j } ^ { \\delta _ { j } } ( x _ { j } ) . \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} [ \\bar { x } ^ { \\pm } _ { i , r + 1 } , \\bar { x } ^ { \\pm } _ { j , s } ] - [ \\bar { x } ^ { \\pm } _ { i , r } , \\bar { x } ^ { \\pm } _ { j , s + 1 } ] = - m _ { i , j } \\beta [ \\bar { x } ^ { \\pm } _ { i , r } , \\bar { x } ^ { \\pm } _ { j , s } ] , \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{gather*} v _ m ^ + ( x , t ) = \\frac { 1 } { 2 \\pi i } \\sum _ { j = 0 } ^ { 2 } \\int _ { 0 } ^ { i \\infty } e ^ { s t } \\frac { \\Delta _ { j , m } ( s ) } { \\Delta ( s ) } e ^ { \\lambda _ j ( s ) x } \\hat { h } _ m ( s ) d s , \\\\ v _ m ^ - ( x , t ) = \\frac { 1 } { 2 \\pi i } \\sum _ { j = 0 } ^ { 2 } \\int ^ { 0 } _ { - i \\infty } e ^ { s t } \\frac { \\Delta _ { j , m } ( s ) } { \\Delta ( s ) } e ^ { \\lambda _ j ( s ) x } \\hat { h } _ m ( s ) d s . \\end{gather*}"}
-{"id": "2939.png", "formula": "\\begin{align*} F = f ( X ( \\varphi _ 1 ) , \\ldots , X ( \\varphi _ n ) ) , \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} \\big ( \\mathbf { H } ( \\mathbf { u } ) \\big ) ( t , \\cdot ) = \\exp ( - t / \\tau ) \\mathbf { H } _ { 0 } + \\int _ { 0 } ^ { t } \\exp \\big ( - ( t - s ) / \\tau \\big ) \\mathbf { F } \\big ( \\nabla _ { \\sigma } \\mathbf { u } ( s , \\cdot ) \\big ) \\mathrm { d } s t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} A _ \\alpha ^ * = \\begin{pmatrix} 0 & 0 \\\\ 0 & \\delta _ \\alpha \\end{pmatrix} , 1 \\leq \\alpha \\leq 4 , \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} \\big \\lVert \\ , \\cdot \\ , \\big \\rVert ^ \\mathrm { R S } _ { \\det H ^ { \\bullet } _ \\mathrm { b d } ( X , F ) } = \\big \\lVert \\ , \\cdot \\ , \\big \\rVert ^ { L ^ { 2 } } _ { \\det H ^ { \\bullet } _ \\mathrm { b d } ( X , F ) } T _ \\mathrm { b d } ( X , g ^ { T X } , h ^ { F } ) . \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} \\gamma _ \\mu = 2 ( ( \\imath _ { e _ \\nu } \\Phi ) \\wedge \\imath _ { e _ \\mu } ( \\nabla _ { e _ \\nu } \\Phi ) + ( \\imath _ { e _ \\mu } \\imath _ { e _ \\nu } \\Phi ) \\wedge e ^ \\rho \\wedge ( \\imath _ { e _ \\nu } \\nabla _ { e _ \\rho } \\Phi ) ) . \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} H = \\frac { \\kappa } { 2 f } \\ , n _ 1 + \\frac { f f '' + ( f ' ) ^ 2 - 1 } { 2 f \\sqrt { 1 - f '^ 2 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} D = - \\Delta + \\frac { 1 } { 2 } ( \\partial ^ \\mu \\phi \\partial _ \\mu \\phi + \\Delta \\phi ) \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} f ( p ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ p | } , \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} \\mathbf { F } _ { \\parallel } ^ { \\prime } = \\mathbf { F } _ { \\parallel } , \\qquad \\mathbf { G } _ { \\parallel } ^ { \\prime } = \\mathbf { G } _ { \\parallel } \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} V ( x , \\theta ) = \\dfrac { V _ N ( x , \\theta ) } { \\gamma - R ( x ) } \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l } $ u = 2 a \\widetilde { u } + 2 a \\widetilde { v } $ , \\\\ $ v = \\left ( \\left ( \\frac { 1 } { c } - 1 \\right ) + \\lambda \\right ) \\widetilde { u } + \\left ( \\left ( \\frac { 1 } { c } - 1 \\right ) - \\lambda \\right ) \\widetilde { v } $ \\end{tabular} \\right . \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} u _ { 0 . 6 } ^ 1 ( f _ 2 ) = [ I - 0 . 6 P ( f _ 2 ) ] ^ { - 1 } \\bar { r } ^ 1 ( f _ 2 ) = ( 5 . 7 , 4 . 5 ) . \\end{align*}"}
-{"id": "10073.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 3 , 3 u + 1 , 3 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 3 , 3 u + 2 , 3 v + 2 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "3510.png", "formula": "\\begin{align*} & \\left \\{ \\alpha _ { { \\mathcal { R } } , { [ N _ T ] } } ^ { \\bar { \\mathcal { R } } _ i } \\sum _ { p = 1 } ^ { N _ T } h _ { q p } c _ p \\right \\} _ { \\mathcal { R } , i : \\mathcal { R } \\ni q , | \\mathcal { R } | = r + 1 , i \\in [ \\rho ] } \\cup \\left \\{ \\alpha _ { { \\mathcal { R } } , { [ N _ T ] } } ^ { \\bar { \\mathcal { R } } _ i } \\sum _ { p = 1 } ^ { N _ T } h _ { q p } c _ p \\right \\} _ { \\mathcal { R } , \\bar { \\mathcal { R } } _ i : \\mathcal { R } \\cup \\bar { \\mathcal { R } } _ i \\not \\ni q } . \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} \\begin{array} { l } x ( n ) = \\sin ( 2 \\pi \\frac { n } { { 3 . 7 } } + \\frac { \\pi } { 4 } ) + \\cos ( 2 \\pi \\frac { n } { { 5 . 6 } } + \\frac { { 3 \\pi } } { 4 } ) + n o i s e \\\\ n = 1 , 2 , \\cdots , 3 0 0 \\\\ \\end{array} \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} B _ { f _ { s p o } } ( G _ { n , k , b } ) = \\left \\lceil \\frac { u + | C | - 1 + | V _ { n , k , b } | - u - 1 } { 2 } \\right \\rceil = \\left \\lceil \\frac { | V _ { n , k , b } | + | C | - 2 } { 2 } \\right \\rceil . \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} & T _ { p q } = \\frac { 1 } { 1 6 \\pi } \\left [ F _ { p } G _ { q } ^ { \\ast } + F _ { p } ^ { \\ast } G _ { q } + F _ { q } G _ { p } ^ { \\ast } + F _ { q } ^ { \\ast } G _ { p } \\right . \\\\ & \\qquad \\quad - \\left . \\delta _ { p q } \\left ( \\mathbf { F } \\cdot \\mathbf { G } ^ { \\ast } + \\mathbf { F } ^ { \\ast } \\cdot \\mathbf { G } \\right ) \\right ] = T _ { q p } \\qquad \\left ( p , q = 1 , 2 , 3 \\right ) \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} \\lim \\limits _ { p \\to 1 + } \\lambda _ 1 ( \\Omega ; p ) = h ( \\Omega ) , \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { f ( t ) } { g ( t ) } = 0 \\ , . \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} 2 - 2 \\cos ( \\phi ) = 2 - 2 \\cos ( \\phi + k ) , 2 - 2 \\cos ( \\phi ) = 2 - 2 \\cos ( \\phi - k ) \\end{align*}"}
-{"id": "9931.png", "formula": "\\begin{align*} ( u ' ( \\psi ( s ) ) v ) ^ { \\mu _ 0 } ( A ) = u ' ( \\psi ( s ) ) v ^ { \\mu _ 0 } ( A ) . \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} { \\omega _ 2 ' } ( \\pi ) = 2 ^ { \\nu _ { d e } ( \\pi ) } \\left ( 1 + \\sum _ { i \\geq 0 } \\chi ( ( 4 i + 3 ) \\in \\pi ) 2 ^ { \\mu _ { 4 i + 3 , o } ( \\pi ) } \\right ) . \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} \\varphi \\left ( z \\right ) = \\mathcal { S } \\xi \\left ( z \\right ) = \\sum _ { j = 1 } ^ { J } a _ { j } \\varphi _ { z _ { j } } \\left ( z \\right ) , \\ ; \\ ; \\ ; \\ ; \\ ; z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\sum _ { m = 1 } ^ \\infty & \\lambda _ k ( a ^ k _ m ) ^ 2 - \\sum _ { k = 1 } ^ n \\sum _ { m = 1 } ^ \\infty \\lambda _ m ( a ^ k _ m ) ^ 2 = \\\\ & \\sum _ { k = 1 } ^ n \\sum _ { m = n + 1 } ^ \\infty \\lambda _ k \\left ( ( a ^ m _ k ) ^ 2 - ( a ^ k _ m ) ^ 2 \\right ) + \\sum _ { k = 1 } ^ n \\sum _ { m = n + 1 } ^ \\infty ( \\lambda _ k - \\lambda _ m ) ( a ^ k _ m ) ^ 2 . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} ( L \\otimes \\iota ) \\hat W & = ( \\iota \\otimes \\iota \\otimes \\omega ) ( U _ { 1 3 } \\hat W _ { 1 2 } U ^ * _ { 1 3 } ) . \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} | \\mathbb { Q } | = \\{ \\omega _ 2 , 2 ^ \\omega \\} = \\omega _ 2 \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} t v ( \\theta , t ) - \\tfrac 1 2 v ' _ \\theta ( \\theta , t ) = H \\int _ 0 ^ t s ^ { 2 H - 1 } \\left ( \\left ( t - \\tfrac 1 2 s \\right ) e ^ { \\theta s } + \\tfrac 1 2 s e ^ { \\theta ( 2 t - s ) } \\right ) d s > 0 . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} x P _ { n } & = P _ { n + 1 } + \\left ( l _ { n n } + m _ { n + 1 } \\right ) P _ { n } + \\sum \\nolimits _ { i = 0 } ^ { d - 2 } \\left ( l _ { n , n - 1 - i } + l _ { n , n - i } m _ { n - i } \\right ) P _ { n - i - 1 } \\\\ & + l _ { n , n - d + 1 } m _ { n - d + 1 } P _ { n - d } \\end{align*}"}
-{"id": "9128.png", "formula": "\\begin{align*} w _ a \\Big ( e _ { \\Lambda _ z } \\big ( \\bar \\pi a ^ { - 1 } ( u _ 1 z + u _ 2 ) \\big ) , e _ { \\Lambda _ z } \\big ( \\bar \\pi a ^ { - 1 } ( v _ 1 z + v _ 2 ) \\big ) \\Big ) = \\bar \\pi e _ A \\big ( a ^ { - 1 } ( u _ 1 v _ 2 - u _ 2 v _ 1 ) \\big ) h ( z ) ^ { - 1 } . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} - \\Delta + \\sum _ { j = 1 } ^ N \\alpha _ j \\delta ( x - x _ j ) , \\alpha _ 1 , \\cdots , \\alpha _ N \\in \\R \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} I ^ \\psi _ { B ( i ) } = \\sum _ { u \\in \\psi _ { B ( i ) } } g _ { u , B ( i ) } \\ell ( r _ { u , B ( i ) } ) P _ B , \\\\ I ^ \\varphi _ { B ( i ) } = \\sum _ { v \\in \\varphi _ { B ( i ) } } g _ { v , B ( i ) } \\ell ( r _ { v , B ( i ) } ) P _ M , \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} \\frac { 1 } { f ( x ) } \\left | \\left ( \\sqrt { ( p + 1 ) ^ 2 + m ^ 2 } - \\sqrt { 1 + m ^ 2 } \\right ) f ( x ) \\right | = O ( | x | ^ { - 1 } ) . \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} \\| ( 1 - \\nu ) A X - \\nu X B \\| _ { 2 } ^ { 2 } & \\leqslant \\underline { K _ { t } } ^ { - 1 } \\| A ^ { 1 - \\nu } X B ^ { \\nu } \\| _ { 2 } ^ { 2 } + R _ { 0 } ^ { 2 } \\| A X - X B \\| _ { 2 } ^ { 2 } \\\\ & - \\sum _ { k = 1 } ^ { \\infty } r _ { k } \\| A ^ { 1 - \\frac { m _ { k } } { 2 ^ { k } } } X B ^ { \\frac { m _ { k } } { 2 ^ { k } } } - A ^ { 1 - \\frac { m _ { k } + 1 } { 2 ^ { k } } } X B ^ { \\frac { m _ { k } + 1 } { 2 ^ { k } } } \\| _ { 2 } ^ { 2 } . \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} \\frac { \\left ( \\frac { 2 } { n } \\sum _ { i = 1 } ^ n \\rho ^ { ( 2 ) } \\left ( \\frac { r _ i ( e ) } { \\sigma _ n } \\right ) \\right ) ( s _ { \\rho , n } ( e ) - s _ { \\rho , n } ( e _ { \\text f } ) ) } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\rho ^ { ( 1 ) } \\left ( \\frac { r _ i ( e ) } { \\sigma _ n } \\right ) ^ 2 } \\le \\chi ^ 2 _ { k - k ( e ) } \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} \\overline { \\left \\{ \\sum \\limits _ \\mathrm { f i n i t e } ( a _ i \\otimes 1 ) \\delta ( b _ i ) : a _ i , b _ i \\in A \\right \\} } = A \\otimes H . \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} Y ( a , z ) = e ^ { z T } a \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} c _ 2 = - ( c _ 2 ) ^ { t r } , \\ ; \\ ; f _ 2 = - ( f _ 2 ) ^ { t r } , \\ ; \\ ; \\delta _ 2 = - ( \\delta _ 2 ) ^ { t r } , \\ ; \\ ; \\gamma _ 2 ^ { t r } = - \\sqrt { 2 } d _ 2 , \\ ; \\ ; \\beta _ 2 = - \\sqrt { 2 } g _ 2 , \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} x \\cdot x & = 0 , & y \\cdot x & = ( 1 - \\alpha ^ { - 1 } ) z , & z \\cdot x & = ( \\alpha + 1 ) x , \\\\ x \\cdot y & = - ( 1 - \\alpha ^ { - 1 } ) z , & y \\cdot y & = 0 , & z \\cdot y & = ( \\alpha - 1 ) y , \\\\ x \\cdot z & = \\alpha x , & y \\cdot z & = \\alpha z , & z \\cdot z & = \\alpha z , \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} ( a _ 1 \\circ b ) S ( a _ 2 ) ( a _ 3 \\circ c ) & = \\pi ( \\pi ^ { - 1 } ( a _ 1 ) \\pi ^ { - 1 } ( b ) ) S ( a _ 2 ) \\pi ( \\pi ^ { - 1 } ( a _ 3 ) \\pi ^ { - 1 } ( c ) ) \\\\ & = a _ 1 ( \\pi ^ { - 1 } ( a _ 2 ) \\rightharpoonup b ) S ( a _ 3 ) a _ 4 ( \\pi ^ { - 1 } ( a _ 5 ) \\rightharpoonup c ) \\\\ & = a _ 1 ( \\pi ^ { - 1 } ( a _ 2 ) \\rightharpoonup b ) ( \\pi ^ { - 1 } ( a _ 3 ) \\rightharpoonup c ) = a _ 1 ( \\pi ^ { - 1 } ( a _ 2 ) \\rightharpoonup ( b c ) ) . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} E ^ { 1 , 2 } _ 2 = H ^ 1 ( G , H ^ 2 ( M , \\Z ) ) \\to H ^ 3 _ G ( M , \\Z ) = E ^ { 1 , 2 } _ \\infty \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} H ^ 1 ( G , H ^ 0 ( Z ) ) = H ^ 1 ( H , \\Z ) = 0 . \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} g _ t ( u ) = \\frac { 1 } { 2 \\pi i } \\int _ { 3 - i \\infty } ^ { 3 + i \\infty } G _ t ( w ) u ^ { - w } d w \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} \\Lambda ^ \\nu ( s ) : = \\frac { q ( f ) ^ { s / 2 } } { \\pi ^ s } \\Gamma ( s + \\nu ) L ^ \\nu ( s , f ) = \\varepsilon ( f ) \\Lambda ^ \\nu ( 2 \\kappa ( f ) + 1 - 2 \\nu - s , \\widetilde { f } ) , \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} A \\sharp _ { \\nu } B \\leqslant A \\sharp _ { \\nu } B + \\sum _ { k = 0 } ^ { \\infty } r _ { k } ( A \\sharp _ { \\frac { m _ k } { 2 ^ k } } B - 2 A \\sharp _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } B + A \\sharp _ { \\frac { m _ k + 1 } { 2 ^ k } } B ) \\leqslant A \\nabla _ { \\nu } B . \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} \\Delta \\log \\left ( r | \\psi | ^ 2 + s \\right ) = 4 r s \\cdot \\frac { | \\psi ' | ^ 2 } { ( r | \\psi | ^ 2 + s ) ^ 2 } \\ , , \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} K _ { a , \\psi } : = \\psi \\circ ( L _ a - \\tilde L _ a ) \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} z & = \\frac { \\xi ( 1 - \\gamma - \\xi ) } { ( \\beta - \\xi ) ( \\gamma + \\xi ) } , & z & = \\frac { \\eta ( 2 - \\eta ) } { 1 - \\eta ^ 2 } , \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} \\mathrm { c l } _ { K } ( F ) = \\mathrm { c l } _ W ( F ) \\supsetneq F , \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} \\mathcal J ( H , D | _ H ) = \\mathcal J ( X , D ) | _ H \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} P ( D ) = P _ { \\mathbf { G } } ( U ( D ) ) ( 1 - p _ d ) ^ { n _ { a s } } ( 2 p _ d - 1 ) ^ { n _ s } D \\in \\mathcal { D } _ n , \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} \\gamma ( H _ 0 ) - P _ + - P _ - = \\gamma ( H _ 0 ) ( 1 - \\chi ) , \\end{align*}"}
-{"id": "10064.png", "formula": "\\begin{align*} 3 p - 2 q \\leq ( \\alpha - 4 ) ( p - q ) + 3 p - 2 q = 1 + n + k \\leq 1 + \\dfrac { p } { 2 } + q , \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} \\dot w ( t ) + A w ( t ) = f ( w ( t ) ) , \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} \\frac { 1 } { \\phi ( 2 N + 1 ) } \\left ( \\frac { 1 } { | S _ { m , N } | } \\sum _ { k \\in S _ { m , N } } | \\xi ^ { m _ 0 , N _ 0 } _ k | ^ p \\right ) ^ \\frac 1 p & \\le \\frac { 1 } { \\phi ( 2 N + 1 ) } \\left ( \\frac { 1 } { | S _ { m _ 0 , N } | } \\sum _ { k \\in S _ { m _ 0 , N } } | \\xi ^ { m _ 0 , N _ 0 } _ k | ^ p \\right ) ^ \\frac 1 p \\\\ & = \\frac { 1 } { \\phi ( 2 N + 1 ) } \\left ( \\frac { | S _ { m _ 0 , N } | } { | S _ { m _ 0 , N } | } \\right ) ^ \\frac 1 p \\le \\frac { 1 } { C _ 1 \\phi ( 2 N _ 0 + 1 ) } . \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x & = & \\dfrac { P _ { x } } { Q } + \\varepsilon \\phi \\\\ \\varepsilon Q & \\cong & 0 \\end{array} \\right . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} \\mathbf { F } _ { \\perp } ^ { \\prime } = \\frac { \\mathbf { F } _ { \\perp } - \\dfrac { i } { c } \\left ( \\mathbf { v \\times G } \\right ) } { \\sqrt { 1 - v ^ { 2 } / c ^ { 2 } } } , \\qquad \\mathbf { G } _ { \\perp } ^ { \\prime } = \\frac { \\mathbf { G } _ { \\perp } - \\dfrac { i } { c } \\left ( \\mathbf { v \\times F } \\right ) } { \\sqrt { 1 - v ^ { 2 } / c ^ { 2 } } } . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} \\widetilde { F } ( 0 , \\frac { 1 } { \\gamma } , 1 ) = \\widetilde { F } ( x _ { 1 } , x _ { 2 } , 0 ) \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} 2 ( R c - \\mathcal { S } ) ( \\nabla u , \\nabla u ) = 2 \\alpha | \\nabla \\varphi | ^ 2 | \\nabla u | ^ 2 . \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} \\Delta = \\lfloor \\log _ { q } \\alpha \\rfloor \\mbox { a n d } \\beta = \\alpha q ^ { - \\lfloor \\log _ { q } \\alpha \\rfloor } . \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} u ( x ) = \\int \\limits _ { \\R ^ { n + 1 } } k ( x , y ) f ( y ) d y . \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} I _ 2 ^ { ( 5 ) } = & r ^ { 2 - n } \\Big [ \\frac { ( n - 6 ) ( n - 4 ) } { 1 2 ( n - 1 ) } | W ( p ) | ^ 2 r ^ 2 - \\frac { 3 2 } { 9 ( n - 2 ) } \\sum _ { k , l , s } \\big ( ( W _ { i k l s } ( p ) + W _ { i l k s } ( p ) ) x ^ i \\big ) ^ 2 \\\\ & + \\frac { 4 } { 3 ( n - 2 ) ( n - 1 ) } | W ( p ) | ^ 2 r ^ 2 - \\frac { 1 6 ( 7 n - 8 ) } { n - 2 } \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j \\Big ] + O ( r ^ { 5 - n } ) . \\end{align*}"}
-{"id": "10056.png", "formula": "\\begin{align*} q + r = m + n + l + k . \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} A _ { \\mathcal { S } , n } & = \\prod _ { j \\in S } \\mathbb { 1 } \\{ Y _ j = n Q _ j ^ { ( n ) } = 1 \\} \\prod _ { j \\in S ^ c } \\mathbb { 1 } \\{ Y _ j \\neq n Q _ j ^ { ( n ) } = 0 \\} . \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} \\lim _ { \\nu \\rightarrow \\infty } \\mu _ { \\nu } \\cdot \\sup _ { 1 \\leq z \\leq 1 + \\gamma \\mu _ { \\nu } } \\left | g _ { 2 , \\nu } ' \\left ( z \\right ) \\right | = 0 . \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } w ^ n _ t + \\frac 1 2 \\Delta w ^ n = { \\nabla u ^ n \\ , b ^ n } , \\\\ w ^ n ( T , x ) = 0 , \\\\ { \\forall } ( t , x ) \\in [ 0 , T ] \\times \\mathbb R ^ d . \\end{array} \\right . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} \\kappa = \\sqrt { \\frac { \\alpha } { \\beta } } - \\sqrt { \\frac { \\beta } { \\alpha } } , a ( t ) = \\frac { \\sqrt { \\alpha \\beta } } { \\alpha ( 1 - t ) + \\beta t } \\quad b ( t ) = \\sqrt { \\frac { \\beta } { \\alpha } } a ( t ) t . \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\neq 0 } A _ f ( n ) \\sqrt { y } K _ { i t _ j } ( 2 \\pi i \\lvert y \\rvert n ) e ^ { 2 \\pi i n x } , \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} \\begin{array} [ c ] { c } \\oslash = \\operatorname * { z u p } \\Phi \\\\ \\pounds = \\operatorname * { w i n f } \\Psi \\end{array} . \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\int g ( f - P _ t f ) d \\mu & = \\int _ 0 ^ t \\int g \\frac { \\partial P _ s f } { \\partial s } d \\mu d s = \\int _ 0 ^ t \\int g \\Delta P _ s f d \\mu d s = - \\int _ 0 ^ t \\int \\Gamma ( P _ s g , f ) d \\mu d s \\\\ & \\le \\int _ 0 ^ t \\| \\sqrt { \\Gamma ( P _ s g ) } \\| _ { \\infty } \\int \\sqrt { \\Gamma ( f ) } d \\mu d s \\le C _ 1 \\sqrt { 2 t } \\int \\sqrt { \\Gamma ( f ) } d \\mu . \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} \\pi _ { l } ( x ) Q _ n ( x ) & = \\sum _ { v = n - \\rho } ^ { n + l } a _ { n , v } P _ v ( x ) . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} \\det ( D T ( x ) ) = - c | x - x _ 0 | ^ { - 1 } + \\max \\{ \\epsilon _ 0 , c _ { \\ast } \\} O ( | x - x _ 0 | ^ { - 1 + \\alpha } ) < 0 . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} { \\cal P } _ { 0 , n } ( \\kappa ) \\triangleq \\Big \\{ p _ t ( d x _ t | x ^ { t - 1 } , y ^ { t - 1 } ) , t = 0 , 1 , \\ldots , n : \\frac { 1 } { n + 1 } { \\bf E } ^ { p } \\Big ( c _ { 0 , n } ( X ^ n , Y ^ { n - 1 } ) \\Big ) \\leq \\kappa \\Big \\} \\subset { \\cal P } _ { 0 , n } , ~ \\kappa \\in [ 0 , \\infty ) \\end{align*}"}
-{"id": "9960.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c c } \\gamma _ { 1 } & \\gamma _ { 2 } & \\ldots & \\gamma _ { n } \\\\ \\gamma _ { 1 } w _ { 1 } & \\gamma _ { 2 } w _ { 2 } & \\ldots & \\gamma _ { n } w _ { n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\gamma _ { 1 } w _ { 1 } ^ { k - 1 } & \\gamma _ { 2 } w _ { 2 } ^ { k - 1 } & \\ldots & \\gamma _ { n } w _ { n } ^ { k - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\left ( \\delta _ { \\min } ( N ) \\geq \\frac { ( \\log \\log N ) ^ 2 } { N } \\right ) = 0 . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} d = \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } t N ^ { \\binom { N _ R - 1 } { r + 1 } \\binom { N _ T } { t } } } { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } t N ^ { \\binom { N _ R - 1 } { r + 1 } \\binom { N _ T } { t } } + ( N + 1 ) ^ { \\binom { N _ R - 1 } { r + 1 } \\binom { N _ T } { t } } } \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} \\limsup _ { k \\to \\infty } \\sqrt { \\sum _ { i = 1 } ^ N \\mathbb { E } [ \\| v _ i ^ { k } - y ^ { k } \\| ^ 2 ] } \\le \\frac { \\sqrt { 2 n } \\ , C \\alpha _ { \\max } } { 1 - \\sqrt { \\lambda } } . \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} \\cos ^ 2 \\alpha _ 1 + \\cos ^ 2 \\alpha _ 2 + \\cos ^ 2 \\alpha _ 3 = 1 . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} p _ e p _ d = P ( ( 1 , 2 ) \\in A ( \\mathbf { D } ) ) = \\int \\int \\phi ( x _ 1 , x _ 2 ) d ( \\mu x _ 2 ) d ( \\mu x _ 1 ) = \\int c \\ d ( \\mu x _ 1 ) = c , \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} I \\subseteq I ' \\subseteq m ' ( I ) \\implies m ' ( I ' ) = m ' ( I ) \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { x } u ^ { a - 1 } ( 1 - u ) ^ { b - 1 } d u = \\dfrac { 1 } { a } x ^ { a } \\left ( 1 - x \\right ) ^ { b } { _ 2 F _ 1 } \\left ( \\begin{array} { c } 1 , a + b \\\\ a + 1 \\end{array} \\bigg | x \\right ) , \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} r = \\tanh u . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} Q \\cap Q ^ \\prime = L _ 2 \\cup L _ 3 \\cup X \\ , . \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} F _ { t } ( A _ { 2 } , A _ { 1 } ) & = d A _ { t } ( A _ { 2 } , A _ { 1 } ) + \\frac { 1 } { 2 } \\left [ A _ { t } ( A _ { 2 } , A _ { 1 } ) , A _ { t } ( A _ { 2 } , A _ { 1 } ) \\right ] , \\\\ & = d \\omega + t d e + \\frac { 1 } { 2 } \\left [ \\omega + t e , \\omega + t e \\right ] , \\\\ & = d \\omega + t d e + \\frac { 1 } { 2 } \\left [ \\omega , \\omega \\right ] + \\frac { t } { 2 } \\left [ \\omega , e \\right ] + \\frac { t } { 2 } \\left [ e , \\omega \\right ] + \\frac { t ^ { 2 } } { 2 } \\left [ e , e \\right ] , \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} L ( T _ k ) = k + 1 . \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} \\frac { W ( a , b , c \\ , ; x ) } { W ( a + k , b + l , c + m \\ , ; x ) } = \\frac { ( - 1 ) ^ { k + l - m } ( c - a ) _ { m - k } ( c - b ) _ { m - l } } { ( a ) _ { k } ( b ) _ { l } } x ^ { m } ( 1 - x ) ^ { k + l - m } . \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i , j = 0 } ^ I r _ { i j } ( x ) ( \\log ( x ) ) ^ { \\alpha _ i } \\log ( \\log ( x ) ) ^ j \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} \\tilde { w } = \\left [ \\begin{matrix} \\operatorname { R e } ( w ) \\\\ \\operatorname { I m } ( w ) \\\\ \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} G \\left ( \\sum _ { i = 1 } ^ { n } x _ { i } e _ { i } \\right ) = \\sum _ { i = 1 } ^ { n } g _ { i } ( x _ { i } ) e _ { i } : = \\sum _ { i = 1 } ^ { n } \\frac { f ( x _ { i } + \\alpha _ { i } ) - \\alpha _ { i } } { f ( 1 + \\alpha _ { i } ) - \\alpha _ { i } } e _ { i } . \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\frac { \\partial \\log F ( r e ^ { i t } ) } { \\partial t } = & i z \\left ( \\sum _ { k = 2 } ^ { p } ( k - 1 ) | z | ^ { 2 ( k - 2 ) } \\overline { z } \\log G _ { k } ( z ) + \\sum _ { k = 1 } ^ { p } | z | ^ { 2 ( k - 1 ) } ( \\log G _ { k } ( z ) ) _ { z } \\right ) \\\\ & - i \\overline { z } \\left ( \\sum _ { k = 2 } ^ { p } ( k - 1 ) | z | ^ { 2 ( k - 2 ) } z \\log G _ { k } ( z ) + \\sum _ { k = 1 } ^ { p } | z | ^ { 2 ( k - 1 ) } ( \\log G _ { k } ( z ) ) _ { \\overline { z } } \\right ) \\\\ = & i \\sum _ { k = 1 } ^ { p } | z | ^ { 2 ( k - 1 ) } L [ \\log G _ { k } ( z ) ] , \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} S ( g _ j , g _ k ) = \\sum _ { i = 1 } ^ n a _ { i j k } g _ i , \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{gather*} T ^ { k } g _ { C } = ( - 1 ) ^ { k } \\begin{bmatrix} z ^ { k } & 0 \\\\ z ^ { - k } C ( z ) & z ^ { - k } \\end{bmatrix} = \\begin{bmatrix} z ^ { k } & 0 \\\\ \\big ( z ^ { - k } C ( z ) \\big ) _ { - } & z ^ { - k } \\end{bmatrix} \\left ( ( - 1 ) ^ { k } \\begin{bmatrix} 1 & 0 \\\\ \\big ( z ^ { - k } C ( z ) \\big ) _ { 0 + } z ^ { k } & 1 \\end{bmatrix} \\right ) . \\end{gather*}"}
-{"id": "8460.png", "formula": "\\begin{align*} F ( \\xi _ 1 , \\xi _ 2 , \\eta ) : = \\frac 1 { \\eta ^ 2 } v _ 1 \\left ( \\frac { \\xi _ 1 } { \\eta } \\right ) v _ 1 \\left ( \\frac { \\xi _ 2 } { \\eta } \\right ) w \\left ( \\frac { \\xi _ 1 } x \\right ) w \\left ( \\frac { \\xi _ 2 } x \\right ) . \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} M = \\begin{pmatrix} ( L + c \\partial _ x ) ^ { - 1 } & 0 \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\tau } \\lambda ( \\tau ) = - a _ n ( g , D ) \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\mathfrak { G } _ { R } = { \\textstyle \\bigoplus _ { p \\in I } } W _ { p } \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { N _ T } \\binom { N _ T } { t } a _ { 0 , t } = 1 . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} P ( D ) = \\int P _ { \\mathbf { x } } ( D ) \\mu ( d \\mathbf { x } ) = \\int P _ { \\mathbf { y } } ( D ' ) \\mu ( d \\mathbf { y } ) . \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} T ( u ) = u + M F ( u ) = u . \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} ( W ' u ) ( n ) : = \\left ( \\prod _ { i = 1 } ^ d \\frac { q _ i \\sin ( k _ i n _ i ) } { n _ i } \\right ) u ( n ) , \\ n \\in \\Z ^ d \\ \\ u \\in \\mathcal { H } , \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} \\frac { d \\mathbb { P } ^ p } { d \\mathbb { P } ^ q } ( X ) = \\frac { \\frac { d \\xi \\mathbb { P } _ { ( \\cdot | X ; C ) } } { d \\xi \\mathbb { P } } ( p ) } { \\frac { d \\xi \\mathbb { P } _ { ( \\cdot | X ; C ) } } { d \\xi \\mathbb { P } } ( q ) } \\cdot \\frac { d \\mathbb { P } _ { ( \\cdot | X ; C ) } ^ p } { d \\mathbb { P } _ { ( \\cdot | X ; C ) } ^ q } ( X | _ { E \\backslash C } ) . \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} \\phi _ { 1 } = \\frac { 1 + \\sqrt { 1 + 4 C _ { 1 } ^ { 2 } } } { 2 } > 1 , \\quad \\quad \\delta _ { 1 } = \\min \\left \\lbrace C _ { 4 } , \\frac { - 1 + ( n - 1 ) \\phi _ { 1 } } { 1 + ( n - 1 ) \\phi _ { 1 } } \\right \\rbrace \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\int _ M u \\Delta _ p u d \\mu = - p \\int _ M | \\nabla u | ^ { p - 2 } h ^ { i j } \\nabla _ i u \\nabla _ j u d \\mu - \\int _ M Z \\langle \\nabla \\mathcal { H } , \\nabla u \\rangle u d \\mu . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} \\lambda _ { \\tau } ( \\vec { z } ) : = \\frac { c ( c + 1 ) } { ( a + 1 ) ( b + 1 ) x ( 1 - x ) } , \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} \\left . \\frac { d } { d t } r _ { Z _ N } ( t ) \\right | _ { t = t _ 0 } = 0 \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} V _ \\nu ( x ) & : = - \\frac { 2 } { \\sqrt { \\pi } } \\frac { \\Gamma ( \\tfrac { 1 } { 2 } + \\nu ) } { \\Gamma ( \\nu ) } ( 1 + x ^ 2 ) ^ { - 1 / 2 } { } _ 2 F _ 1 ( 1 , \\tfrac { 1 } { 2 } + \\nu , \\tfrac { 1 } { 2 } , - x ^ 2 ) \\\\ u _ \\nu ( x ) & : = \\frac { 1 } { ( 1 + x ^ 2 ) ^ { \\nu } } . \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} H ^ j _ { c , G } ( V ) = H ^ { j - 2 n } ( B G ) = \\left \\{ \\begin{array} { l l } \\Z , & j = 2 n \\\\ \\Z / m , & j > 2 n \\ ; e v e n \\\\ 0 , & \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} F ( \\alpha , w ) = \\left ( \\begin{array} { c c } ( \\bar { a } + \\alpha ) \\partial _ \\theta ( \\bar { v } + w ) - L ( \\bar { v } + w ) - N ( \\bar { v } + w ) \\\\ \\langle \\partial _ \\theta \\bar { v } , w \\rangle \\end{array} \\right ) = 0 . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} \\rho ( t ) = \\mathrm { a r c t a n h } \\ , \\tilde { \\rho } ( t ) \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\alpha _ k \\left ( \\sum _ { s = 1 } ^ { k } \\beta ^ { k - s } \\alpha _ { s - 1 } \\right ) = \\sum _ { k = 1 } ^ \\infty \\left ( \\sum _ { s = 1 } ^ { k } \\beta ^ { k - s } \\alpha _ k \\alpha _ { s - 1 } \\right ) \\leq \\sum _ { k = 1 } ^ \\infty \\left ( \\sum _ { s = 1 } ^ { k } \\beta ^ { k - s } \\alpha _ { s - 1 } ^ 2 \\right ) . \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } - V _ s \\cdot \\nabla _ { X _ s } \\right ) I _ s ( ( X _ s - ( T - t ) V _ s , V _ s ) ) = 0 \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\exp _ 2 \\{ \\lambda L ( y _ { n _ { i - 1 } + 1 } ^ { n _ i } ) \\} \\ge \\frac { s ^ 2 \\left [ \\exp _ 2 \\left \\{ ( \\lambda + 1 ) \\log \\left ( \\frac { c + s ^ 2 } { 2 s ^ 2 } \\right ) \\right \\} - 1 \\right ] } { 2 ^ { \\lambda + 1 } - 1 } . \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{gather*} p _ { n _ { k + 1 } } ( x ) = a _ { k } ( x ) p _ { n _ { k } } ( x ) - \\beta _ { k } p _ { n _ { k - 1 } } ( x ) \\ , \\ \\ 0 \\leq k < m \\ , \\ \\ \\ \\ p _ { n _ { - 1 } } : = 0 \\ . \\end{gather*}"}
-{"id": "422.png", "formula": "\\begin{align*} u _ { 0 } ( 0 ) + u _ { \\delta ^ { p } } ( 0 ) f _ { p , 0 } ( x _ { p } ) = v _ { 0 } ( 0 ) + v _ { \\delta ^ { p } } ( 0 ) g _ { p , 0 } ( x _ { p } ) \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\dot { z } _ 1 = z _ 2 + \\psi _ 1 ( z _ 1 ^ 2 , z _ 2 ) , \\dot { z } _ 2 = \\mu z _ 1 + z _ 1 \\psi _ 2 ( z _ 1 ^ 2 , z _ 2 ) \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} \\sum \\limits _ { i \\in \\left [ k \\right ] } \\left \\Vert \\mathbf { y } ^ { ( i ) } \\right \\Vert ^ { 2 } = \\sum \\limits _ { i \\in \\left [ k \\right ] } \\left ( \\left ( k - 1 \\right ) ^ { 2 } \\left \\vert V _ { i } \\right \\vert + n - \\left \\vert V _ { i } \\right \\vert \\right ) = \\left ( k ^ { 2 } - k \\right ) n . \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} \\begin{aligned} & \\max _ { \\lambda \\in \\mathcal { R } ( T , X ) } \\int _ T \\int _ X \\varphi ( t , x ) \\lambda ( t , d x ) d \\mu \\\\ & \\int _ T \\int _ X \\imath _ X ( x ) \\lambda ( t , d x ) d \\mu - \\int _ T \\omega ( t ) d \\mu \\in W . \\end{aligned} \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} P ( z ) : = V z ^ 2 - D z + Q . \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} \\beta = ( 1 , 0 ) . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} \\beta ( g ) : = \\Lambda \\frac { \\partial g } { \\partial \\Lambda } = \\frac { \\partial g } { \\partial \\ln \\Lambda } \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} \\sum _ { \\substack { | \\delta | = k , \\\\ \\delta _ { i _ { 1 } } = \\cdots = \\delta _ { i _ { l } } = 1 } } u _ { \\delta } = 0 . \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} O \\cdot \\tilde { I } ( u , v ) & = \\tilde { I } ( O \\cdot ( u , v ) ) & & \\forall ( u , v ) \\in H \\oplus H , O \\in S O ( 4 ) \\ \\ \\\\ \\frac { - 1 } { \\overline { m ( z ) } } & = m \\left ( \\frac { - 1 } { \\bar { z } } \\right ) & & \\forall z \\in \\C , m \\in G \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} r _ N \\left ( t , Z _ N \\right ) = \\sum _ { i = 1 } ^ N \\left ( x _ i \\cdot v _ i - | v _ i | ^ 2 t \\right ) \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\prod _ { \\lbrace j _ { 1 } , \\dots , j _ { g } \\rbrace \\in \\mathcal { U } _ { g } } \\| J \\| ( W _ { j _ { 1 } } , \\dots , W _ { j _ { g } } ) = \\pi ^ { \\binom { 2 g + 2 } { g } g } \\cdot \\| \\varphi _ { g } \\| ( X ) ^ { ( g + 1 ) / 4 } , \\end{align*}"}
-{"id": "10158.png", "formula": "\\begin{align*} d ( A ^ T x , \\partial g ( \\bar { y } ) ) & \\geq \\gamma \\cdot d ( A ^ T x , \\partial g ( \\bar { y } ) \\cap R ( A ^ T ) ) = \\gamma \\cdot \\min _ { A ^ T u \\in \\partial g ( \\bar { y } ) } \\| A ^ T x - A ^ T u \\| \\\\ & = \\gamma \\cdot \\min _ { y \\in A ^ T X _ 0 } \\| A ^ T x - y \\| \\geq \\gamma \\cdot \\min _ { y \\in V _ 0 } \\| A ^ T x - y \\| = \\gamma \\cdot \\| A ^ T x - \\hat { y } \\| , \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} v \\left ( \\sum _ { ( i , j ) \\in I } a ( i , j ) x ^ i ( f ( x ) ) ^ j \\right ) = v \\left ( \\sum _ { ( i , j ) \\in I } b ( i , j ) x ^ i ( f ( x ) ) ^ j \\right ) < v \\left ( \\sum _ { ( i , j ) \\in J } a ( i , j ) x ^ i ( f ( x ) ) ^ j \\right ) . \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} { G ( x ^ p ) = A ( x ) G ( x ) A _ 0 ^ { - 1 } , \\ \\ G ( x ^ q ) = B ( x ) G ( x ) B _ 0 ^ { - 1 } } \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} T ^ { - 1 } ( u ) & = T ^ { - 1 } ( d _ { 1 / \\lambda } ( d _ \\lambda ( u ) ) ) \\stackrel { \\eqref { s e c . t w o _ 1 : e q _ h o m p r o p i n v T } } { = } D _ { 1 / \\lambda } ( T ^ { - 1 } ( d _ \\lambda ( u ) ) ) \\\\ & = ( D _ { 1 / \\lambda } \\circ ( T ^ { - 1 } ) | _ { \\mathcal { W } } \\circ d _ \\lambda ) ( u ) . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\tilde { C } _ { s + 1 } = \\sum _ { i = 1 } ^ s \\left ( \\tilde { C } _ { i , s + 1 } ^ + - \\tilde { C } _ { i , s + 1 } ^ - \\right ) \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} A \\cdot \\vec { u } _ \\ell = 0 , \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} f ( z ) = \\frac { 2 } { \\pi } \\int _ { \\mathbb { D } } \\frac { f ( w ) ( 1 - | w | ^ 2 ) } { ( 1 - z \\overline { w } ) ^ 3 } | d w | ^ 2 \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} \\begin{cases} - d Y _ { t } = f ( t , \\eta _ { t } , Y _ { t } , Z _ { t } , Y _ { t + \\delta ( t ) } , Z _ { t + \\zeta ( t ) } ) d t - Z _ { t } d B _ { t } ^ { H } , \\ \\ \\ t \\in [ 0 , T ] ; \\\\ Y _ { t } = g ( \\eta _ { t } ) , \\ \\ Z _ { t } = h ( \\eta _ { t } ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ t \\in [ T , T + K ] , \\end{cases} \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } A _ { q } \\left ( q ^ { \\alpha } z \\right ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( z q ^ { 1 / 2 } e ^ { i x } ; q \\right ) _ { \\infty } \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i \\alpha x \\right ) } { \\sqrt { \\log q ^ { - 1 } } } d x \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} x \\cdot x & = 0 , & x \\cdot y & = 0 , & y \\cdot x & = x , & y \\cdot y & = \\alpha y , \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\Sigma _ 3 } \\ [ \\bar { x } ^ { \\pm } _ { 0 , r _ { \\pi ( 1 ) } } , [ \\bar { x } ^ { \\pm } _ { 0 , r _ { \\pi ( 2 ) } } , \\bar { x } ^ { \\pm } _ { 0 , r _ { \\pi ( 3 ) } + 1 } ] ] = 0 . \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} \\begin{cases} - d _ 1 \\Delta u = g ( u ) \\left ( f ( u ) - d \\right ) , & x \\in \\Omega , \\\\ \\partial _ \\nu u = 0 , & x \\in \\partial \\Omega . \\\\ \\end{cases} \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} g \\ : = \\ : & d r ^ 2 + r ^ 2 h + h _ { 0 } + r ^ { - 2 } h _ { 2 } + \\cdots + r ^ { - 2 i } h _ { 2 i } + \\cdots \\\\ f \\ : = \\ : & - \\frac 1 4 r ^ 2 + f _ { 0 } + r ^ { - 2 } f _ { 2 } + \\cdots + r ^ { - 2 i } f _ { 2 i } + \\cdots , \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} \\lim _ { \\nu \\rightarrow \\infty } s _ { j } \\left ( \\nu \\right ) \\cdot \\mu _ { \\nu } = 0 . \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} \\partial _ t ( | \\nabla f | ^ 2 ) & = \\partial _ t ( g ^ { i j } \\partial _ i f \\partial _ j f ) \\\\ \\displaystyle & = ( \\partial _ t g ^ { i j } ) \\partial _ i f \\partial _ j f + 2 g ^ { i j } \\partial _ i f \\partial _ j f _ t \\\\ \\displaystyle & = 2 h ^ { i j } \\partial _ i f \\partial _ j f + 2 g ^ { i j } \\partial _ i f \\partial _ j f _ t \\\\ \\displaystyle & = 2 h _ { i j } f _ i f _ j + 2 f _ i f _ { t , j } , \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} \\mbox { \\bf T e r m 1 } \\le 2 \\| F _ i ( x _ i ^ k , N \\hat v _ i ^ k ) \\| ^ 2 + 2 \\| F _ i ( x ^ * _ i , \\bar { x } ^ * ) \\| ^ 2 \\le \\tilde C \\qquad \\hbox { w i t h } \\tilde C = 2 C ^ 2 + 2 \\max _ { ( x _ i , \\bar x ) \\in K _ i \\times \\bar K } \\| F _ i ( x _ i , \\bar { x } ) \\| ^ 2 , \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} V X = \\{ A \\subseteq X \\mid A \\} \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} t ^ 2 & = ( a + 1 ) ^ 2 s ^ 4 - 2 ( a + 1 ) ^ 3 s ^ 3 + ( a ^ 4 + 8 a ^ 3 + 1 0 a ^ 2 + 8 a + 1 ) s ^ 2 \\\\ & - 4 a ( a + 1 ) ( a ^ 2 + a + 1 ) s + 4 a ^ 2 ( a ^ 2 + 1 ) . \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| Q _ n J _ n ( a ) - J _ n ( a ) \\| _ { \\varphi \\otimes \\psi _ n } = 0 \\ , , \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} \\pi & = ( 2 ^ { f _ 2 } , 4 ^ { f _ 4 } , 6 ^ { f _ 6 } , \\dots , ( 4 j - 2 ) ^ { f _ { 4 j - 2 } } , ( 4 j - 1 ) ^ { 1 } , \\dots ) , \\\\ \\intertext { w h e r e $ f _ 2 , \\ f _ 4 , \\ \\dots , \\ f _ { 4 j - 2 } $ a r e a l l p o s i t i v e , o r a s } \\pi & = ( 2 ^ { f _ 2 } , 4 ^ { f _ 4 } , 6 ^ { f _ 6 } , \\dots , ( 4 j ) ^ { f _ { 4 j } } , ( 4 j + 1 ) ^ { 0 } , ( 4 j + 2 ) ^ { 0 } , \\dots ) , \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} G '' ( z , v ) = G ' ( z , v ) G ( z , v ) + \\frac { 1 } { 1 - z } G ' ( z , v ) , G ( 0 , v ) = v , G ' ( 0 , v ) = v . \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\beta } \\left ( \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { r } g ^ { k } ( s , x ) d w _ { s } ^ { k } \\right ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { t } ( t - s ) ^ { - \\beta } g ^ { k } ( s , x ) d w _ { s } ^ { k } , \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} I _ 2 = & \\Delta \\delta T _ 2 d ( r ^ { 6 - n } ) \\\\ = & - \\Delta [ ( T _ 2 ) _ { i j } ( r ^ { 6 - n } ) _ { , j } ] _ { , i } \\\\ = & - \\Delta [ ( T _ 2 ) _ { i j , i } ( r ^ { 6 - n } ) _ { , j } + ( T _ 2 ) _ { i j } ( r ^ { 6 - n } ) _ { , j i } ] . \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} \\min \\limits _ { \\mathbf { x } \\in X _ { 1 } \\times \\dots \\times X _ { n } } \\to \\mu ( \\mathbf { x } ) = \\left \\{ f ( \\mathbf { x } ) + \\sum \\limits ^ { n } _ { i = 1 } h _ { i } ( \\mathbf { x } _ { i } ) \\right \\} . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} q _ { i j } & = \\xi ^ i , & \\forall i \\in I , \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} H ^ i ( B , H \\otimes p ^ * \\mathcal F ^ j ) = 0 \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} ( \\iota \\otimes \\omega _ { \\pi ( \\omega _ 1 ) \\xi , \\pi ( \\omega _ 2 ) \\xi } ) U = \\omega _ 1 \\star a \\star \\omega _ 2 ^ \\sharp . \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} H _ { [ i _ 1 , \\ldots , i _ l ] } = \\{ ( x _ 1 , \\dots , x _ r ) \\in H | x _ { i _ 1 } = \\ldots = x _ { i _ l } = 0 \\} . \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} e _ 1 = - \\frac { i } { 2 } \\widetilde F \\sigma _ 1 \\widetilde F ^ { - 1 } , \\ ; \\ ; e _ 2 = - \\frac { i } { 2 } \\widetilde F \\sigma _ 2 \\widetilde F ^ { - 1 } \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; e _ 3 = - \\frac { i } { 2 } \\widetilde F \\sigma _ 3 \\widetilde F ^ { - 1 } . \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} F ^ p V = \\sum _ { j \\geq 0 } ( F ^ 0 V ) _ { ( - j - 1 ) } F ^ { p - j } V . \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} F = ( \\tfrac { 1 } { n } \\sum _ { i } \\kappa _ i ^ r ) ^ { 1 / r } - 1 \\leq r < 1 \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} F _ x = F U \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; F _ y = F V , \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} \\det ( \\mathcal { J } _ { \\displaystyle \\Psi _ { x , y } } ( \\eta ) ) = 1 , . \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} r ( x , m n ) = \\frac { \\gamma _ x ( u ^ * ( m ) , u ^ * ( n ) ) } { \\gamma ' _ { v ( x ) } ( m , n ) } \\ , r ( x , m ) r ( x , n ) . \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} ( \\Lambda _ { N } ( t ) - \\mid \\gamma + t \\mid ^ { 2 } ) ( \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) = ( q \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} \\frac { v ( \\theta , t ) } { e ^ { 2 \\theta t } } = H e ^ { - 2 \\theta t } \\int _ 0 ^ t s ^ { 2 H - 1 } e ^ { \\theta s } d s + H \\int _ 0 ^ t s ^ { 2 H - 1 } e ^ { - \\theta s } d s \\to \\frac { H \\Gamma ( 2 H ) } { \\theta ^ { 2 H } } , \\quad t \\to \\infty . \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} Z = \\sum _ { i = 1 } ^ n e ^ { - \\beta E _ i } \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} F _ i ( x _ i , \\bar x ) = \\nabla _ { x _ i } f _ i ( x _ i , \\bar x ) \\qquad \\hbox { f o r a l l } i = 1 , \\ldots , N . \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} a = \\lceil N \\alpha \\rceil , b = \\lceil N \\beta \\rceil , c = \\lceil N \\gamma \\rceil , r - s = \\lfloor N \\xi \\rfloor , s = \\lceil N \\eta \\rceil , \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} U _ { \\tau _ 0 } w = f + ( U _ { \\tau _ 0 } - U _ \\tau ) u \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} b : = \\frac { f ( y _ 1 ) + f ( y _ 2 ) } { 2 } , \\xi ^ * : = \\frac { f ( y _ 1 ) - f ( y _ 2 ) } { | f ( y _ 1 ) - f ( y _ 2 ) | } , \\mu : = \\frac { | f ( y _ 1 ) - f ( y _ 2 ) | } { 2 } . \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} \\frac { \\lambda ' } { \\lambda } \\sum _ { m = 0 } ^ { s } \\frac { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } { \\lambda ^ { m } } \\frac { 1 } { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } M ^ { m } \\vec u \\ , , \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} ( u _ 0 , v _ 0 , T _ { e , 0 } , q _ { e , 0 } ) \\in H ^ 1 ( \\mathbb R ^ 2 ) , \\nabla \\cdot u _ 0 = 0 , q _ { e , 0 } \\leq 0 , \\mbox { a . e . ~ o n } \\mathbb R ^ 2 . \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} \\alpha & = \\frac { 5 } { 1 4 } + \\epsilon , \\sigma _ 0 = 0 , \\beta = \\frac { 3 } { 2 } + \\epsilon , \\\\ \\gamma _ { 1 } & = 1 , R _ 1 \\sim \\langle f , f \\rangle , \\sigma _ 1 = \\frac { 6 } { 7 } + \\epsilon , \\eta _ { 1 } = \\frac { 9 } { 1 4 } + \\epsilon , \\\\ \\gamma _ { 2 } & = 1 , R _ 2 \\sim \\langle f , \\overline { f } \\rangle , \\sigma _ 2 = \\frac { 6 } { 7 } + \\epsilon , \\eta _ { 2 } = \\frac { 9 } { 1 4 } + \\epsilon . \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\textbf { B } ( r , \\theta , \\phi ) = \\frac { B _ 0 R } { R + r \\cos ( \\theta ) } \\left ( \\hat { \\textbf { e } } _ \\phi + \\frac { r } { Q R } \\hat { \\textbf { e } } _ \\theta \\right ) , \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} \\varphi ( \\lambda , \\nu ) = \\frac { c } { a ( \\lambda , \\nu ) b ( \\lambda , \\nu ) } , d ( \\lambda , \\nu ) = \\sin ( \\lambda - \\nu ) , e ( \\lambda , \\nu ) = \\sin ( \\lambda - \\nu + 2 \\eta ) . \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\omega ( u _ { i j } ^ \\alpha ) = \\tilde \\omega ( \\varphi u _ { i j } ^ \\alpha \\varphi ) = \\tilde \\omega \\Big ( \\frac { \\delta _ { i j } \\sigma _ { - i / 2 } ( \\chi _ \\alpha ) \\varphi } { \\dim _ q ( \\alpha ) } \\Big ) . \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} \\mathbf { H } ( t , \\cdot ) \\in L ^ { \\infty } \\big ( G , \\mathcal { S } _ { \\geq \\kappa } ( \\mathbb { R } ^ { k \\times d } ) \\big ) t \\in [ 0 , T ] \\kappa : = \\alpha \\exp ( - 1 / \\tau ) . \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} { \\bf P } _ { Y _ t | Y ^ { t - 1 } , X ^ t , W } ( d y _ t | y ^ { t - 1 } , x ^ t , w ) = { \\bf P } _ { Y _ t | Y ^ { t - 1 } , X ^ t } ( d y _ t | y ^ { t - 1 } , x ^ t ) , ~ t \\in \\mathbb { N } ^ n _ 0 . \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} ( x , k , y ) ( y , l , z ) : = ( x , k + l , z ) \\quad ( x , k , y ) ^ { - 1 } : = ( y , - k , x ) . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } A = \\omega + e + k + h , \\\\ A _ { 2 } = \\omega + e , \\\\ A _ { 1 } = \\omega , \\\\ \\bar { A } = 0 , \\end{array} \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} & \\dot { x } = - F \\nu , \\\\ & x ( 0 ) = M _ 0 \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} & R _ 1 \\leq m , R _ 2 = 0 . \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} c _ \\Gamma [ S ] = S \\ , \\frac { \\prod _ { e \\in E } p _ { m ( e ) } [ S ] } { \\prod _ { v \\in V } p _ { m ( v ) } [ S ] } \\in S ^ { b ( \\Gamma ) } \\Lambda . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\pi i } \\partial \\overline \\partial \\log \\| \\theta _ { \\alpha } \\| = \\omega _ 0 + \\tfrac { 1 } { 2 } \\omega _ { \\mathrm { H d g } } - \\delta _ { \\Theta _ { \\alpha } } . \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n + 1 } { 2 } } } { \\left ( q ; q \\right ) _ { n } } \\left ( - x \\right ) ^ { n } A _ { q } \\left ( - q ^ { n } \\right ) = \\frac { \\left ( q ^ { 1 / 2 } ; q \\right ) _ { \\infty } } { \\left ( q ^ { 1 / 2 } x ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\left ( n ^ { 2 } + n \\right ) / 4 } \\prod _ { k = 0 } ^ { n - 1 } \\left ( x - q ^ { k } \\right ) } { \\left ( q ^ { 1 / 2 } , q ^ { 1 / 4 } , - q ^ { 1 / 4 } ; q ^ { 1 / 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} \\tilde v _ k = \\frac { \\| E _ 2 w _ k \\| ^ { 1 / 2 } } { \\| E _ 1 v _ k \\| ^ { 1 / 2 } } v _ k , \\tilde w _ k = \\frac { \\| E _ 1 v _ k \\| ^ { 1 / 2 } } { \\| E _ 2 w _ k \\| ^ { 1 / 2 } } w _ k \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} \\mu ' ( \\mathbf { x } ; \\mathbf { d } ) = \\langle { \\mathbf { g } ( \\mathbf { x } ) , \\mathbf { d } } \\rangle + h ' ( \\mathbf { x } ; \\mathbf { d } ) , \\ \\mbox { w i t h } \\ h ' ( \\mathbf { x } ; \\mathbf { d } ) = \\sum \\limits ^ { n } _ { i = 1 } \\max _ { \\mathbf { b } _ { i } \\in \\partial h _ { i } ( \\mathbf { x } _ { i } ) } \\langle { \\mathbf { b } _ { i } , \\mathbf { d } _ { i } } \\rangle ; \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} \\hat v ^ k _ i = \\sum _ { \\ell = 1 } ^ { N } [ \\Phi ( k , 0 ) ] _ { i \\ell } v _ \\ell ^ 0 + \\sum _ { s = 1 } ^ { k } \\left ( \\sum _ { j = 1 } ^ { N } [ \\Phi ( k , s ) ] _ { i j } ( x _ j ^ s - x _ j ^ { s - 1 } ) \\right ) . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} H ^ 2 ( M ' ) = H ^ 2 ( M ) \\oplus H ^ 0 ( Z ) . \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i = \\big ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } - B _ { i J K j } \\dot { u } _ { j } ) _ { , K } \\big ) _ { , J } + E _ { i j } \\dot { u } _ { j } \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} W _ { i j } = S _ { i j } + \\epsilon g _ { i j } . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} \\tilde { \\phi } _ N ^ { ( s ) } ( t , Z _ s ) = \\phi _ N ^ { ( s ) } ( t , Z _ s ) e ^ { - \\beta ( t ) I _ s ( ( X _ s - ( T - t ) V _ s , V _ s ) ) } \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} & \\big | A _ N \\bigl ( e ^ { i \\lambda } \\bigr ) \\bigl ( 1 - e ^ { i \\lambda \\mu } \\bigr ) ^ { n } \\lambda ^ { 2 n } g ^ 0 ( \\lambda ) + \\lambda ^ { 2 n } C ^ { \\mu , 0 } _ { N } \\big ( e ^ { i \\lambda } \\big ) \\big | ^ 2 \\\\ & = \\alpha _ 1 | \\lambda | ^ { 2 n } \\big | 1 - e ^ { i \\lambda \\mu } \\big | ^ { 2 n } \\bigl ( f ^ 0 ( \\lambda ) - f _ 1 ( \\lambda ) \\bigr ) \\bigl ( f ^ 0 ( \\lambda ) + \\lambda ^ { 2 n } g ^ 0 ( \\lambda ) \\bigr ) ^ 2 , \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} ( \\lambda + \\gamma ) p _ { 0 , 0 } = \\xi p _ { 0 , 1 } + \\mu p _ { 1 , 1 } . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} A _ 1 : = A \\cap [ 1 , v _ 1 ' ] , A _ 2 : = A \\cap [ v _ 1 ' + 1 , v _ 1 ' + v _ 2 ' ] , B _ 1 : = B \\cap [ 1 , v _ 1 ' ] , B _ 2 : = B \\cap [ v _ 1 ' + 1 , v _ 1 ' + v _ 2 ' ] . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} \\alpha ( L ) : = \\{ \\alpha _ 0 ( L ) , \\alpha _ 1 ( L ) , \\ldots , \\alpha _ { g - 1 } ( L ) \\} \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} & \\delta _ { \\mathsf { P } } ^ * ( \\mu , r ) = 1 , ~ ~ ~ ~ ~ ~ ~ \\mu \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{gather*} \\begin{bmatrix} \\dot { x } \\\\ \\dot { y } \\\\ \\dot { \\omega } \\end{bmatrix} = \\begin{bmatrix} r \\dot { \\cos ( \\theta ) } \\\\ r \\dot { \\sin ( \\theta ) } \\\\ - \\frac { g \\cos ( \\theta ) } { r } \\end{bmatrix} = \\begin{bmatrix} - r \\sin ( \\theta ) \\dot { \\theta } \\\\ r \\cos ( \\theta ) \\dot { \\theta } \\\\ - \\frac { g ( r \\cos ( \\theta ) ) } { r ^ 2 } \\end{bmatrix} = \\begin{bmatrix} - y \\omega \\\\ x \\omega \\\\ - \\frac { g x } { r ^ 2 } \\end{bmatrix} \\end{gather*}"}
-{"id": "9568.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 1 } \\left ( a , b ; c ; q , c / \\left ( a b \\right ) \\right ) = \\frac { \\left ( c / a , c / b ; q \\right ) _ { \\infty } } { \\left ( c , c / \\left ( a b \\right ) ; q \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\Delta _ { g _ { t } } u + f \\ \\textrm { o v e r } \\ C _ { r } ( Y ) . \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} \\sum \\limits _ { i = j } ^ n { 2 n \\choose 2 i } B _ { 2 n - 2 i } { i \\brack j } = \\begin{cases} 0 , & j < n , \\\\ n , & j = n . \\end{cases} \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{gather*} \\Psi ^ { [ k ] ( \\alpha ) } = T ^ { k } Q _ 0 ^ { - \\alpha } g _ { - } ^ { [ k ] ( \\alpha ) } . \\end{gather*}"}
-{"id": "2026.png", "formula": "\\begin{gather*} \\int \\limits _ { O _ v ^ { \\times 2 } } \\chi ( a c \\ ( F ^ { ( m ) } ( x , y ) ) ) \\ | F ^ { ( m ) } ( x , y ) | ^ s \\ | d x d y | = \\frac { U _ 0 ( q ^ { - s } , \\chi ) } { 1 - q ^ { - 1 - s } } \\\\ + \\sum _ { \\{ \\theta \\in O _ v | f _ 0 ( 1 , \\theta ^ a ) = 0 \\} } J _ \\theta ( s , m , \\chi ) , \\end{gather*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\big \\lVert \\omega ^ \\mathrm { z m } \\big \\rVert _ Y = \\big \\lVert \\omega ^ \\mathrm { z m } _ u \\big \\rVert _ Y . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} \\alpha _ 2 \\alpha _ 2 ^ { t r } + 2 g _ 2 g _ 2 ^ { t r } + 2 d _ 2 d _ 2 ^ { t r } = I \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} P _ { n } ^ { \\left ( 1 \\right ) } \\left ( x \\right ) = \\left \\langle u _ { 0 } , \\frac { P _ { n + 1 } \\left ( x \\right ) - P _ { n + 1 } \\left ( \\xi \\right ) } { x - \\xi } \\right \\rangle , n \\geq 0 . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} - L u + g _ n \\circ u & = f \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega , \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} & \\min _ { \\mathbf { W } _ { k } \\in \\mathbb { H } ^ { N } , k = 1 , . . . , K } ~ \\sum _ { k = 1 } ^ { K } \\textrm { t r } ( \\mathbf { A } _ { k } \\mathbf { W } _ { k } ) \\\\ & s . t . ~ \\sum _ { k = 1 } ^ { K } \\textrm { t r } ( \\mathbf { B } _ { m , k } \\mathbf { W } _ { k } ) \\unrhd _ { m } b _ { m } , m \\ ! = \\ ! 1 , . . . , M , \\mathbf { W } _ { k } \\succeq 0 , k \\ ! = \\ ! 1 , . . . , K , \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} \\hat \\theta _ n ^ { ( 3 ) } ( m ) = - \\left ( \\frac { 1 } { H \\Gamma ( 2 H ) n ^ m } \\sum _ { k = 0 } ^ { n ^ m - 1 } X _ { k / n } ^ 2 \\right ) ^ { - \\frac 1 { 2 H } } \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} i _ { v , w } \\Gamma _ v & = \\Gamma _ w i _ { v , w } \\\\ i _ { v , w } ( T _ v ( H ) ) & = T _ w ( i _ { v , w } ( H ) ) , \\ , \\ , \\forall \\ , H \\in I ( X ) \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} P = \\frac { 1 } { Z } e ^ { - \\beta E } \\ , Z = e ^ { - \\beta F } \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\left ( \\left \\{ \\mathbf { \\bar { G } } _ { i j , k l } \\mathbf { \\bar { v } } _ { k p , l } ^ { [ \\sf b s ] ( \\mathbf { s } ^ { [ \\sf d ] } ) } \\right \\} _ { k \\in [ 1 : K ] , l \\in [ 1 : M ] , \\mathbf { s } ^ { [ \\sf d ] } \\in \\mathcal { S } _ { T _ { \\sf d } } ^ { [ \\sf d ] } ~ } \\right ) \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} & \\mathbb { E } [ | \\pi | ^ q ( X _ 1 ) ] = \\mathbb { E } [ | W _ 1 | ^ q ] = \\frac { 2 ^ { ( q + n + 1 ) / 2 } \\pi ^ { ( n - 1 ) / 2 } } { \\sqrt { n } } \\frac { \\Gamma ( \\frac { n + q } { 2 } ) } { \\Gamma ( \\frac { n } { 2 } ) } \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} \\int _ \\Omega \\frac { \\mathfrak { D } } { \\bar { \\sigma } ( x ) } \\nabla \\rho ( x ) \\cdot \\nabla \\Psi ( x ) \\ , { \\rm d } x + \\int _ \\Omega \\rho ( x ) \\Psi ( x ) \\ , { \\rm d } x = \\int _ \\Omega g ( x ) \\Psi ( x ) \\ , { \\rm d } x , \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} ( 2 \\pi n ) ^ { 2 } - ( 2 \\pi ( n + p ) ) ^ { 2 } ) ( \\Phi , e ^ { i 2 \\pi ( n + p ) x } ) = \\sum _ { m \\in \\mathbb { N } } q _ { m } ( \\Phi , e ^ { i 2 \\pi ( n + p - m ) x } ) + c ( \\Psi , e ^ { i 2 \\pi ( n + p ) x } ) \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} \\Delta ( z ) & = q - 2 4 q ^ 2 + 2 5 2 q ^ 3 - 1 4 2 7 q ^ 4 + . . . \\\\ \\Delta _ p ( z ) & = q ^ p - 2 4 q ^ { 2 p } + 2 5 2 q ^ { 3 p } - 1 4 2 7 q ^ { 4 p } + . . . \\\\ \\eta ( z ) ^ { 1 2 } \\eta ( p z ) ^ { 1 2 } & = q ^ { \\frac { p + 1 } { 2 } } - 1 2 q ^ { \\frac { p + 1 } { 2 } + 1 } + 5 4 q ^ { \\frac { p + 1 } { 2 } + 2 } - 8 8 q ^ { \\frac { p + 1 } { 2 } + 3 } - 9 9 q ^ { \\frac { p + 1 } { 2 } + 4 } + . . . \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { u } - \\mathrm { d i v } \\bigg ( \\Big ( \\int _ { 0 } ^ { \\cdot } \\exp \\big ( - ( \\cdot - s ) / \\tau \\big ) \\big ( \\mathbf { F } ( \\nabla \\mathbf { u } ) \\big ) ( s ) \\mathrm { d } s \\Big ) \\nabla \\mathbf { u } \\bigg ) = \\mathbf { 0 } ( 0 , \\infty ) \\times G , \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} X _ { \\mathbf { u } } = X _ 0 + \\sum _ { i = 1 } ^ { m } u _ i X _ i . \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} \\dim \\delta H _ 3 ^ g = & \\ ; 6 + 9 ( k + 2 ) + ( k + 1 ) ( k + 2 ) + 3 \\ , k ( k + 1 ) / 2 \\\\ & \\ ; + { ( k - 1 ) k ( k + 1 ) } / { 6 } - { ( k + 4 ) ( k + 5 ) ( k + 6 ) } / { 6 } \\\\ = & \\ ; k + 6 , \\\\ \\dim \\delta E _ 3 ^ g = & \\ ; I _ M ( V _ 3 ^ g \\times W _ 3 ^ g ) = \\ ; k + 3 , \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} d _ { f _ { s p o } } ( X , Y ) = | f _ { s p o } ( X ) - f _ { s p o } ( Y ) | . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\dim H _ 3 - \\dim E _ 3 + \\dim V _ 3 - \\dim W _ 3 = 1 . \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\Delta { K } ^ { s , \\infty } = & \\big ( \\mu ^ s _ 1 ( \\beta - 1 ) - \\mu ^ s _ 0 ( \\alpha - 1 ) \\big ) + H ( \\alpha ) - H ( \\beta ) + s + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu ^ s _ 1 + \\Delta { K } ^ { s , \\infty } } } { 1 + 2 ^ { \\mu ^ s _ 0 + \\Delta { K } ^ { s , \\infty } } } \\Big ) . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} \\sum _ { k \\in \\mathcal { R } } \\left ( 1 - \\left \\vert z _ { j } \\right \\vert ^ { 2 } \\right ) ^ { 2 } \\mu \\left ( k \\right ) & \\leq \\left ( 1 - R ^ { 2 } \\right ) ^ { 2 } \\sum _ { i = 1 } ^ { N } \\sum _ { z _ { k } \\in B _ { i } : k \\in \\mathcal { R } } \\mu \\left ( k \\right ) \\\\ & \\leq C \\left ( 1 - R ^ { 2 } \\right ) ^ { 2 } N \\\\ & \\leq C \\left ( 1 - R ^ { 2 } \\right ) < C \\varepsilon , \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} K ( h ^ { \\frac { 1 } { 2 ^ { t - 1 } } } , 2 ) ^ { r _ { t - 1 } } a \\sharp _ { \\nu } b & = a \\nabla _ { \\nu } b - \\sum _ { k = 0 } ^ { t - 2 } r _ { k } \\big [ \\big ( a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } - \\big ( a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ] ^ { 2 } \\\\ & = K ( h ^ { \\frac { 1 } { 2 ^ { t - 1 } } } , 2 ) ^ { R _ { t - 1 } } a \\sharp _ { \\nu } b , \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} F _ { d , \\ell } ( z ; \\tau ) = \\mathcal { M } _ { d , \\ell } ( z ; \\tau ) - F _ { \\ell - d , \\ell } ( - z ; \\tau ) . \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} [ W ] _ { i j } = \\left \\{ \\begin{array} { l l } 0 & \\mbox { i f } A _ { i j } = 0 \\\\ \\delta & \\mbox { i f $ A _ { i j } = 1 $ a n d $ i \\ne j $ } \\\\ 1 - \\delta d ( i ) & \\mbox { i f } i = j , \\end{array} \\right . \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} & \\delta _ { \\mathsf { A c h } } ( \\mu , r ) = \\frac { K } { \\min \\{ M , K \\} } \\left ( \\frac { \\mu M - 1 } { M - 1 } \\right ) + ( 1 - \\mu ) \\frac { M + K - 1 } { M - 1 } ; \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} \\alpha _ \\gamma = \\begin{cases} \\frac { 1 } { 4 } + \\frac { 1 + \\sqrt { 8 \\gamma ^ 2 + 1 } } { 1 6 \\gamma ^ 2 } , & \\mbox { i f } ~ \\gamma \\ge \\frac { 1 + \\sqrt { 2 } } { 2 } , \\\\ \\frac { 2 \\gamma + 1 } { 8 \\gamma ^ 2 } + \\frac { \\sqrt { 8 \\gamma + 3 } } { 1 6 \\gamma ^ 2 } , & \\mbox { i f } ~ \\frac { 1 } { 2 } < \\gamma < \\frac { 1 + \\sqrt { 2 } } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} \\dim \\delta H _ 2 ^ g = 4 ( k + 1 ) + ( k - 1 ) ^ 2 - \\left ( ( k + 1 ) ^ 2 + 2 \\right ) = 2 . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} G = \\tfrac { 1 } { 2 } \\log ( \\norm { D u } ^ 2 \\Theta ^ { - 2 } ) + \\lambda u \\Theta ^ { - 1 } . \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} ( m - 4 ) y ^ 4 - ( m - 1 ) y ^ 3 - y + 1 = 0 . \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} f ( y ) = y _ 0 ^ 4 f ( z ) , \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\displaystyle J _ 2 = \\int _ { 0 } ^ { t } \\| ( Q _ { \\lambda } E _ { \\lambda } ) ( s ) \\| _ { H _ { x } ^ { m } } d s . \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} 0 = G _ { - 1 } V \\subset G _ 0 V \\subset \\dots G _ 1 V \\subset \\dots , V = \\bigcup _ p G _ p V . \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} X ( u , v ) = \\phi ( u ) \\overrightarrow { e _ { 1 } } + \\lambda \\cos \\left ( \\frac { u } { c } \\right ) \\rho ( v ) , \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} e _ 1 ^ \\prime = e _ 1 , e _ 2 ^ \\prime = e _ 1 - f _ { k + 1 } , e _ { i + 1 } ^ \\prime = e _ { i } , \\ 2 \\leq i \\leq n - 2 k - 1 , f _ { j } ^ \\prime = f _ { j } , f _ { k + j } ^ \\prime = f _ { k + 1 + j } , \\ 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} | \\mathcal { T } _ { \\neq } | & = \\left | \\iint A ( \\nabla _ L ^ { \\perp } \\Delta _ L ^ { - 1 } \\ne { f } \\cdot \\nabla _ L f ) A f \\ , d V d t \\right | \\\\ & \\lesssim \\Vert \\nabla _ L ^ { \\perp } \\Delta _ L ^ { - 1 } \\ne { f } \\Vert _ { { L ^ 2 } H ^ { N } } \\Vert \\nabla _ L f \\Vert _ { L ^ { 2 } H ^ { N } } \\Vert A f \\Vert _ { L ^ { \\infty } L ^ { 2 } } . \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} ( \\star \\omega ) _ i = \\star \\omega _ i . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { + \\infty } x ^ { \\alpha - 1 } \\ , e ^ { - \\left ( p \\ , x + \\frac { q } { x } \\right ) } \\ , { \\rm d } x = 2 \\left ( \\frac { q } { p } \\right ) ^ { { \\alpha } / { 2 } } K _ \\alpha ( 2 \\sqrt { p \\ , q } ) \\ : . \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} \\left ( F R + G S \\right ) = 0 , \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} q ( s ) = s - ( I + p ) ^ { - 1 } \\left ( s \\right ) s \\ge 0 . \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} X = Z _ { \\lambda _ 1 , \\lambda _ 2 , \\varepsilon } ( \\beta ) \\frac { e ^ { \\kappa } \\ , \\Gamma ( 1 - \\beta ^ 2 ) } { 2 \\pi } \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} f _ { k } ( t ) = \\frac { 1 } { B ( k , n - k + 1 ) } \\left ( 1 - e ^ { - t } \\right ) ^ { k - 1 } e ^ { - ( n - k + 1 ) t } , ~ ~ t > 0 . \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} Y ( z ) = \\begin{pmatrix} Y _ { 1 , 1 } ( z ) & Y _ { 1 , 2 } ( z ) \\\\ Y _ { 2 , 1 } ( z ) & Y _ { 2 , 2 } ( z ) \\end{pmatrix} , \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} & \\phi \\circ \\psi ( X , Y , Z ) = ( - 1 ) ^ { | X | | Z | } \\circlearrowleft _ { X , Y , Z } ( - 1 ) ^ { | X | ( | \\psi | + | Z | ) } \\phi ( \\gamma ( X ) , \\psi ( Y , Z ) ) , \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} \\dim \\delta H _ 3 ^ g = & \\ ; 8 + 1 2 ( k + 2 ) + 6 \\ , k ( k + 1 ) / 2 + ( k - 2 ) ( k - 1 ) k / 6 \\\\ & \\ ; - ( k + 4 ) ( k + 5 ) ( k + 6 ) / 6 \\\\ = & \\ ; 3 ( k + 4 ) , \\\\ \\dim \\delta E _ 3 ^ g = & \\ ; I _ M ( V _ 3 ^ g \\times W _ 3 ^ g ) = \\ ; 3 ( k + 2 ) , \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} F _ { Z _ R } ( \\omega _ 1 , \\omega _ 2 , \\hat { \\omega } ) - \\varphi ^ * _ { R , R ' } F _ { Z _ { R ' } } ( \\omega _ 1 , \\omega _ 2 , \\hat { \\omega } ) = d ^ F \\mu , \\end{align*}"}
-{"id": "9701.png", "formula": "\\begin{align*} \\big ( ( \\mathfrak { S } ^ { n } \\widehat { \\mu } ) _ { 1 } ( X ) , \\dots , ( \\mathfrak { S } ^ { n } \\widehat { \\mu } ) _ { k } ( X ) \\big ) = \\widehat { p } \\ , P ^ { n } . \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} A _ { x , y } = \\begin{cases} 1 , & \\textup { i f $ ( x , y ) \\in E $ } , \\\\ 0 , & \\textup { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} \\mathbf { F } ^ { \\prime } = \\gamma \\left ( \\mathbf { F } - \\dfrac { i } { c } \\mathbf { v \\times G } \\right ) - \\left ( \\gamma - 1 \\right ) \\frac { \\mathbf { v } \\cdot \\mathbf { F } } { v ^ { 2 } } \\mathbf { v } , \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 2 } \\left ( b , b q ; c , c q ; q ^ { 2 } , \\frac { c ^ { 2 } } { b ^ { 2 } } \\right ) = \\frac { \\left ( c / b ; q \\right ) _ { \\infty } } { \\left ( c ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n } { 2 } } } { \\left ( q ; q \\right ) _ { n } } \\left ( \\frac { c } { b } \\right ) ^ { n } A _ { q } \\left ( c q ^ { n - 1 } \\right ) \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} \\frac { H ^ { n - 1 } ( \\{ u = 0 \\} \\cap B _ g ( x , \\rho ) ) } { \\rho ^ { n - 1 } } \\geq \\frac { c _ 1 } { ( \\beta ( x , \\rho / 2 ) ) ^ { n - 1 } } \\geq \\frac { c _ 2 } { N ^ { n - 1 } } . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} P _ { n - k } ^ { \\left ( k \\right ) } = P _ { r } ^ { \\left ( k \\right ) } P _ { n - \\left ( k + r \\right ) } ^ { \\left ( k + r \\right ) } - q _ { 1 , r - 1 } P _ { n - \\left ( k + r + 1 \\right ) } ^ { \\left ( k + r + 1 \\right ) } - . . . - q _ { d , r - 1 } P _ { n - \\left ( k + r + d \\right ) } ^ { \\left ( k + r + d \\right ) } , \\end{align*}"}
-{"id": "9984.png", "formula": "\\begin{align*} w _ { a _ i } = \\frac { 1 } { 2 } \\left ( \\tanh \\frac { a _ i - a _ { i - 1 } } { 2 } + \\tanh \\frac { a _ { i + 1 } - a _ i } { 2 } \\right ) \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} \\sqrt { - d ^ 2 / d x ^ 2 } \\ , u _ \\nu ( x ) = - V _ \\nu ( x ) u _ \\nu ( x ) . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} e _ { + } = \\frac { 1 } { \\sqrt { 1 + | a | ^ 2 \\l ^ { 2 } } } \\begin{pmatrix} e ^ { i \\theta } & a \\l \\\\ - \\bar a \\l & e ^ { - i \\theta } \\end{pmatrix} , \\ ; e _ { - } = \\frac { 1 } { \\sqrt { 1 + | b | ^ 2 \\l ^ { - 2 } } } \\begin{pmatrix} e ^ { i \\kappa } & b \\l ^ { - 1 } \\\\ - \\bar b \\l ^ { - 1 } & e ^ { - i \\kappa } \\end{pmatrix} , \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } v ( t , x ) = \\Delta v ( t , x ) + \\sum _ { k = 1 } ^ { \\infty } \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } ( \\sigma ^ { i j k } u _ { x ^ i x ^ j } + g ^ { k } ) d w _ { s } ^ { k } \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\aligned & d _ 1 ( x ) d _ 1 ( x ) ^ { t r } = s ^ 2 ( x _ 3 ^ 2 + x _ 4 ^ 2 ) , d _ 2 ( x ) d _ 2 ( x ) ^ { t r } = s ^ 2 ( x _ 1 ^ 2 + x _ 2 ^ 2 ) . \\\\ & \\tau d _ 1 ( x ) J g _ 1 ( x ) ^ { t r } = 0 = \\tau d _ 2 ( x ) J g _ 2 ( x ) ^ { t r } . \\endaligned \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\rho ( t ) ( y , h , u ) = ( \\tilde { y } _ t | _ { \\R _ { < 0 } } , x ( t ) , \\tilde { u } _ t | _ { \\R _ { > 0 } } ) . \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} b _ j ^ { t r } ( c _ j - f _ j ) = 0 , d _ j ^ { t r } d _ j + c _ j ^ { t r } c _ j = g _ j ^ { t r } g _ j + f _ j ^ { t r } f _ j . \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} \\langle h \\rangle ^ G = \\langle g \\rangle ^ G = \\langle x ^ { 2 \\alpha } y ^ { 2 \\beta } , x ^ { 4 \\alpha } , y ^ { 4 \\beta } \\rangle . \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\tilde { F } ( v , y ) = g ( y ) . \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\theta _ 1 ( Z ) = \\frac { 2 m - 1 } { 2 n - 1 } \\overline \\theta _ 1 ( Z ) , \\theta _ \\alpha ( Z ) = \\frac { m } { n } \\overline \\theta _ \\alpha ( Z ) , \\alpha = 2 , 3 . \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} t _ h = C _ 1 \\sqrt { N d h q ^ { - h } 2 ^ { \\kappa _ h } } \\leq C _ 1 \\sqrt { c ( q ) } q ^ { - h } N . \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} - \\varepsilon u ^ { \\prime \\prime } + u ^ { \\prime } + \\xi - f ( \\alpha u ) & = 0 ( 0 , T ) , \\\\ \\xi & \\in \\partial \\phi ( u ) ( 0 , T ) , \\\\ u ^ { \\prime } ( T ) & = 0 , \\\\ u ( 0 ) & = u _ { 0 } . \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} a ^ { i j } ( t , x ) = \\sum _ { n = 1 } ^ { M _ 0 } a _ { n } ^ { i j } ( t , x ) 1 _ { ( \\tau _ { n - 1 } , \\tau _ n ] } ( t ) , \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} \\frac { T } { 2 } \\int _ 0 ^ L \\left ( \\varphi _ 0 ^ 2 ( x ) + \\frac { b } { c } \\psi _ 0 ^ 2 ( x ) \\right ) d x = & \\frac { 1 } { 2 } \\int _ 0 ^ T \\int _ 0 ^ L \\left ( \\varphi ^ 2 ( x , t ) + \\frac { b } { c } \\psi ^ 2 ( x , t ) \\right ) d x d t \\\\ + & \\frac 1 2 \\int _ 0 ^ T ( T - t ) \\left [ \\varphi _ x ^ 2 ( 0 , t ) + \\frac { 2 a b } { c } \\psi _ x ( 0 , t ) \\varphi _ x ( 0 , t ) + \\frac { b } { c ^ 2 } \\psi _ x ^ 2 ( 0 , t ) \\right ] d t . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\Delta _ L ( t ) \\equiv \\left [ \\frac { \\partial } { \\partial x } E _ { \\alpha , \\beta } ^ \\gamma \\left ( t { \\rm e } ^ { { \\rm i } x } \\right ) \\Big | _ { x = 0 } \\right ] ^ 2 - E _ { \\alpha , \\beta } ^ \\gamma ( t ) \\cdot \\frac { \\partial ^ 2 } { \\partial x ^ 2 } E _ { \\alpha , \\beta } ^ \\gamma \\left ( t { \\rm e } ^ { { \\rm i } x } \\right ) \\Big | _ { x = 0 } \\geq 0 , \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} \\dim \\delta H _ 2 ^ g = 4 ( k + 2 ) + ( k - 1 ) k / 2 - ( k + 3 ) ( k + 4 ) / 2 = 2 . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c 2 ^ { \\lambda L ( y _ { n _ { i - 1 } + 1 } ^ { n _ i } ) } \\ge \\frac { s ^ 2 \\left [ \\exp _ 2 \\left \\{ ( \\lambda + 1 ) \\log \\left ( \\frac { c + s ^ 2 } { 2 s ^ 2 } \\right ) \\right \\} - 1 \\right ] } { 2 ^ { \\lambda + 1 } - 1 } , \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} \\iota ( t _ 1 , t _ 2 ) = g _ 0 ^ { - 1 } \\begin{pmatrix} t _ 1 & \\\\ & t _ 2 \\end{pmatrix} g _ 0 , \\forall t _ 1 , t _ 2 \\in F , \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} D _ { v } e _ i : = w _ i ^ j ( v ) e _ j v \\in T _ x M \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} g ( 0 , x _ 0 , t , c _ t ) = 1 - \\alpha . \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} e _ { r e l } ( x ) : = 1 0 0 \\left ( \\frac { \\hat v _ \\bold { 0 } ^ { \\Pi } ( 0 , x ) } { v _ { \\bold { 0 } } ^ { \\Pi } ( 0 , x , 0 , 0 ) } - 1 \\right ) , \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} d \\mu ^ { \\varepsilon } = Z ^ { - 1 } \\exp \\left [ - \\beta \\left ( \\varepsilon \\right ) \\sum _ { b \\in \\left ( \\varepsilon \\mathbb { Z } ^ { d } \\right ) ^ { \\ast } } V \\left ( \\eta ^ { \\varepsilon } \\left ( b \\right ) \\right ) \\right ] \\prod _ { i \\in \\varepsilon \\mathbb { Z } ^ { d } } d \\theta ^ { \\varepsilon } \\left ( i \\right ) , \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} [ X _ i , Y _ i ] = - \\xi \\forall \\ i = 1 , \\ldots , n \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{gather*} \\big ( E _ { a b } z ^ { k } \\big ) \\cdot \\big ( e _ { c } z ^ { m } \\big ) = \\delta _ { b c } e _ { a } z ^ { k + m } . \\end{gather*}"}
-{"id": "1368.png", "formula": "\\begin{align*} L ^ { 2 } ( q ^ { \\beta } ) = 1 + \\sum _ { n = 1 } ^ { \\infty } \\bigl ( 2 4 0 \\ , \\sigma _ { 3 } ( \\frac { n } { \\beta } ) - 2 8 8 \\ , \\frac { n } { \\beta } \\ , \\sigma ( \\frac { n } { \\beta } ) \\bigr ) q ^ { n } . \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\begin{aligned} v _ i ^ * & = v _ i + \\omega \\omega \\cdot ( v _ { s + 1 } - v _ i ) \\\\ v _ { s + 1 } ^ * & = v _ { s + 1 } - \\omega \\omega \\cdot ( v _ { s + 1 } - v _ i ) \\end{aligned} \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} \\vartheta _ { \\mathcal L } ^ { ( \\ell ) } ( z ) & = \\sum _ { s = 0 } ^ { n - \\ell } 2 ^ s \\binom { \\ell + s } { s } z ^ { s q } \\sum _ { r = 0 } ^ { q - 1 } z ^ r \\left ( \\frac { 1 } { ( 1 - z ^ q ) ^ { n - \\ell } } \\sum _ { h \\geq 0 } N _ { \\mathcal L } ^ { \\mathrm { r e d } } ( h q + r , \\ell + s ) z ^ { h q } \\right ) \\\\ & = \\frac { 1 } { ( 1 - z ^ q ) ^ { n - \\ell } } \\sum _ { s = 0 } ^ { n - \\ell } 2 ^ s \\binom { \\ell + s } { s } z ^ { s q } \\sum _ { k \\geq 0 } N _ { \\mathcal L } ^ { \\mathrm { r e d } } ( k , \\ell + s ) z ^ { k } , \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} G _ \\infty = G _ \\infty ( 0 ) + \\mathcal { V } G _ \\infty \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} ( \\overline \\nabla _ X J _ \\alpha ) Y = ( \\nabla _ X J _ \\alpha ) Y + h ( X , J _ \\alpha Y ) - J _ \\alpha h ( X , Y ) , \\alpha = 1 , 2 , 3 . \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} \\begin{array} { c } h \\gamma ^ { \\kappa + 1 } h ^ { - 1 } C = h \\gamma \\ , { } \\ , \\gamma ^ { \\kappa } h ^ { - 1 } C \\subset \\\\ h \\gamma h ^ { - 1 } ( h \\gamma ^ { \\kappa } h ^ { - 1 } C ) \\subset h \\gamma h ^ { - 1 } ( e C ) \\subset e C \\ , . \\end{array} \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} h \\sum _ { i = 1 } ^ s b _ i l _ i ( \\textbf { q } ( t ) ) = z ( t + h ) - z ( t ) + \\mathcal { O } ( h ^ { p + 1 } ) = \\int _ t ^ { t + h } f ( \\textbf { q } ( \\tau ) , \\tau ) d \\tau + \\mathcal { O } ( h ^ { p + 1 } ) , \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} f _ N ^ { ( 2 ) } ( \\tau , x _ 1 , v _ 1 ^ * , x _ 1 + \\varepsilon \\omega , v _ 2 ^ * ) = f _ N ^ { ( 2 ) } ( \\tau , x _ 1 , v _ 1 , x _ 1 + \\varepsilon \\omega , v _ 2 ) \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} ( \\tilde H _ \\lambda , \\tilde H _ \\lambda ) ^ { S _ { q , t } } = \\prod _ { s \\in \\lambda } ( q ^ { a ( s ) } - t ^ { l ( s ) + 1 } ) ( q ^ { a ( s ) + 1 } - t ^ { l ( s ) } ) \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} \\Delta _ { j } = \\dfrac { R T _ { n + 1 , i _ { 0 } } + j H _ { n + 1 } } { R l _ { i _ { 0 } } t _ { i _ { 0 } } + j K } - \\dfrac { H _ { n + 1 } } { K } \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} C ^ { \\textrm { m a i n } } _ { ( c p + a , d q + b ) } = & \\frac { ( p c ^ 2 + 2 a c - p c ) C _ \\tau } { 2 } + \\left ( c + \\frac { a } { p } \\right ) \\left ( d + \\frac { b } { q } \\right ) C _ \\sigma - \\frac { a b } { p q } C _ \\sigma + \\frac { ( q d ^ 2 + 2 b d - q d ) C _ \\rho } { 2 } \\\\ & + d \\sum _ { l = 0 } ^ { q - 1 } l C _ { \\rho , ( 0 , - l - 1 ) } + c \\sum _ { k = 0 } ^ { p - 1 } k C _ { \\tau , ( - k - 1 , 0 ) } . \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\hat { \\mathbf { D } } ^ { \\perp } } ( \\mathbf { D } ) = \\mathbf { D } - \\frac { ( \\mathbf { D } : \\hat { \\mathbf { D } } ) \\hat { \\mathbf { D } } } { \\hat { \\mathbf { D } } : \\hat { \\mathbf { D } } } \\mathbf { D } \\in \\mathbb { R } ^ { k \\times d } . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} W _ s : = - \\frac { 1 } { 2 } \\tilde { \\mu } ^ { 2 s } \\Gamma ( s ) \\zeta _ { \\Delta } ( s ) = - \\frac { 1 } { 2 } \\tilde { \\mu } ^ { 2 s } \\int _ 0 ^ \\infty t ^ { s - 1 } T r ( e ^ { - t \\Delta } ) d t \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( \\lambda ^ 2 H ) - ( \\lambda ^ 2 H ) ( | A | ^ 2 - 2 C ) = 0 , \\\\ A ( { \\rm g r a d } ( \\lambda ^ 2 H ) ) + ( \\lambda ^ 2 H ) { \\rm g r a d } H = 0 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} F ( t ^ { d ^ r } , t ^ { d ^ { r - 1 } } , \\ldots , t ^ r , t ) = \\sum a _ I t ^ { i _ 1 d ^ r + \\ldots + i _ n r + \\ldots + i _ { n + 1 } } \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} \\dot { x } = f ( x ) \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{gather*} \\langle \\omega _ { n - 1 } \\otimes \\omega _ { n - 2 } \\otimes \\dots \\otimes \\omega _ { 0 } , \\tilde \\omega _ { n - 1 } \\otimes \\tilde \\omega _ { n - 2 } \\otimes \\dots \\otimes \\tilde \\omega _ { 0 } \\rangle _ { F ^ { \\otimes n } } = \\prod \\langle \\omega _ { a } , \\tilde \\omega _ { a } \\rangle _ { F } . \\end{gather*}"}
-{"id": "8252.png", "formula": "\\begin{align*} A _ { J _ 1 N } X = J _ 1 ( A _ N X ) + \\frac { 1 } { 2 ( 2 n - 1 ) } \\left [ \\overline \\theta _ 1 ( N ) X + \\overline \\theta _ 1 ( J _ 1 N ) J _ 1 X \\right ] , \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} \\int _ { p r _ { g + 1 } } \\delta _ { \\lbrace P _ g \\in \\sigma ( P _ 1 + \\dots + P _ { g - 1 } ) \\rbrace } \\gamma ^ * \\omega _ 0 ^ { g } = \\sum _ { j = 1 } ^ { g - 1 } \\int _ { p r _ { g + 1 } } \\delta _ { \\lbrace P _ j = P _ g \\rbrace } \\gamma _ { \\sigma } ^ * \\omega _ 0 ^ g . \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} \\delta ^ * ( \\mu , r ) \\geq \\delta _ E + \\delta _ F = 1 + \\frac { K \\left ( 1 - \\mu M \\right ) } { M r } . \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\hat { f } ( t , x ) = \\begin{cases} f ( t , x ) , & \\quad \\hbox { i f } | x | \\leq M _ 5 \\\\ f ( t , M _ 5 \\frac { x } { | x | } ) , & \\hbox { i f } | x | > M _ 5 , \\end{cases} \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{gather*} \\deg P _ { n _ { k } } = n _ { k } \\ , \\ \\ P _ { n _ { k + 1 } - 1 } = \\gamma _ { k } P _ { n _ { k } } \\ , \\ \\ \\deg P _ { n _ { k + 1 } } = n _ { k + 1 } \\ , \\end{gather*}"}
-{"id": "9784.png", "formula": "\\begin{align*} \\mathcal { M } '' : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} k _ { \\Phi } ( \\tau ) = \\liminf _ { A \\in R _ 1 ^ + ( { \\mathcal H } ) } \\max _ { 1 \\le j \\le n } | [ T _ j , A ] | _ { \\Phi } \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\| ( u - d - \\textrm { e s s s u p } _ { ( \\omega , t , x ) \\in \\Omega \\times \\partial _ p \\mathcal { O } _ t } u ^ + - \\textrm { e s s s u p } _ { ( \\omega , t , x ) \\in \\Omega \\times \\mathcal { O } _ t } \\hat { \\xi } ^ + ) ^ + \\| _ { \\mathcal { V } _ 2 ( \\mathcal { O } _ t ) } = 0 , \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} \\mathcal { H } _ { p - } \\ ; & = \\ ; \\left \\{ \\frac { 1 } { \\sqrt { p } } \\nabla _ p s _ f : [ p , 1 ] \\to \\R \\ ; | \\ ; f \\in L ^ 2 [ 0 , 1 ] \\right \\} \\\\ & = \\left \\{ c + s _ g : [ p , 1 ] \\to \\R \\ ; | \\ ; c \\in \\R \\ ; \\ ; g \\in L ^ 2 [ p , 1 ] \\right \\} . \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} L ( h ) = d i v ( \\sqrt { g } ( g ^ { i j } ) \\nabla h ) = 0 \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k | \\alpha _ i | & \\leq | \\beta | . \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} & \\lim _ { i \\to \\infty } \\mathbf { c r } ( M _ i , 0 ) = \\infty , \\\\ & \\lim _ { i \\to \\infty } \\mathbf { s c r } ( M _ i , 0 ) = \\infty . \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} \\lambda _ { \\alpha } \\lambda _ { \\beta } & = K _ { \\alpha \\beta } ^ { \\gamma } \\lambda _ { \\gamma } , \\\\ \\ \\quad \\lambda _ { \\alpha } \\lambda _ { \\beta } \\lambda _ { \\delta } & = K _ { \\alpha \\beta } ^ { \\gamma } \\lambda _ { \\gamma } \\lambda _ { \\delta } , \\\\ K _ { \\alpha \\beta \\delta } ^ { \\varepsilon } \\lambda _ { \\varepsilon } & = K _ { \\alpha \\beta } ^ { \\gamma } K _ { \\gamma \\delta } ^ { \\varepsilon } \\lambda _ { \\varepsilon } , \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} x \\otimes 1 - 1 \\otimes x = z \\otimes 1 - 1 \\otimes z = ( y _ n \\otimes 1 - 1 \\otimes y _ n ) ^ { p ^ n } \\in J ^ { p ^ n } \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} J _ i ( \\pi ) ( T ^ { - 1 } ( x , \\xi ) ) = X _ i ( \\pi ( T ^ { - 1 } ( x , \\xi ) ) ) . \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} \\mathbb { E } ( S _ { 0 , 0 } ) & = \\frac { \\gamma } { \\lambda + \\mu + \\xi } \\mathbb { E } ( S \\mid X _ { 0 } = ( 0 , 1 ) , X _ { 1 } = ( 1 , 1 ) ) + \\frac { \\lambda } { \\lambda + \\mu + \\xi } \\mathbb { E } ( S \\mid X _ { 0 } = ( 0 , 1 ) , X _ { 1 } = ( 0 , 2 ) ) \\\\ & \\qquad + \\frac { \\xi } { \\lambda + \\mu + \\xi } \\mathbb { E } ( S \\mid X _ { 0 } = ( 0 , 1 ) , X _ { 1 } = ( 0 , 0 ) ) , \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} A _ { \\mathrm { t } } = \\bigcup _ { \\xi \\in S _ \\mathrm { t } } I _ \\xi . \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} g _ \\varepsilon ^ { ( s ) } ( 0 ) = g _ \\varepsilon ^ { ( m - 1 ) } ( 0 ) \\otimes g _ { \\varepsilon } ( 0 ) ^ { \\otimes ( s - m + 1 ) } \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{gather*} \\hat { g } = \\hat { g } _ { - } \\hat { g } _ { 0 + } . \\end{gather*}"}
-{"id": "8484.png", "formula": "\\begin{align*} \\left | \\nabla _ { z } K _ { N } ^ { 1 } ( z , \\zeta ) \\right | \\lesssim \\frac { 1 } { \\prod _ { j = 1 } ^ { n - 1 } \\tau _ { j } ^ { 2 } \\left ( z , \\delta _ { \\Omega } ( z ) \\right ) } \\frac { 1 } { \\left | z - \\zeta \\right | } \\frac { 1 } { 2 ^ { - i } \\delta _ { \\Omega } ( z ) } , \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { R } ( ( x w ^ * ) * R ( w w ^ * ) ^ * ) & = \\mathcal { R } ( ( w w ^ * ) * R ( w w ^ * ) ^ * ) , \\\\ \\mathcal { R } ( \\mathcal { F } ( ( x w ^ * ) * R ( w w ^ * ) ^ * ) ) & = \\mathcal { R } ( \\mathcal { F } ( ( w w ^ * ) * R ( w w ^ * ) ^ * ) ) . \\end{aligned} \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} \\begin{aligned} h \\big ( c ( y _ k ) + a _ k \\nabla c ( y _ k ) ( v _ k - v _ { k - 1 } ) \\big ) + & a _ k g ( v _ k ) \\le h \\big ( c ( y _ k ) + a _ k \\nabla c ( y _ k ) ( x - v _ { k - 1 } ) \\big ) + a _ k g ( x ) \\\\ & + \\frac { \\tilde { \\mu } a _ k ^ 2 } { 2 } \\left ( \\norm { x - v _ { k - 1 } } ^ 2 - \\norm { x - v _ k } ^ 2 - \\norm { v _ k - v _ { k - 1 } } ^ 2 \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "9949.png", "formula": "\\begin{align*} \\pi _ { p } ( v ' ) _ { j } = \\pi _ { p } ( v ' _ { j } ) , \\quad . \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} & \\pi ^ * I _ 0 + \\pi ^ * I _ 1 . s + \\cdots + \\pi ^ * I _ { k - 1 } . s ^ { k - 1 } + ( s ^ k ) , \\\\ & I _ 0 \\subset I _ 1 \\subset \\cdots \\subset I _ { k - 1 } \\subset \\O _ { Z \\times S } , \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} F ( x + \\sp \\{ v _ { i } \\} ) = F ( x ) + \\sp \\{ F ( v _ { i } ) \\} \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} f _ { \\pm } ^ { \\beta } \\left ( x , v \\right ) = \\left \\{ \\begin{array} [ c ] { c c } \\mu _ { \\pm , + } \\left ( e _ { \\pm } \\right ) & v > 0 \\\\ \\mu _ { \\pm , - } \\left ( e _ { \\pm } \\right ) & v < 0 \\end{array} \\right . , \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{gather*} \\int \\limits _ { a + \\pi ^ e O _ v } \\chi ( a c ( x ) ) ^ N | x | ^ { s N + n - 1 } \\ d x \\\\ = \\begin{cases} \\frac { ( 1 - q ^ { - 1 } ) ( q ^ { - e n - e N s } ) } { ( 1 - q ^ { - n - N s } ) } & a \\in \\pi ^ e O _ v , \\chi ^ N = \\chi _ { t r i v } \\\\ \\\\ q ^ { - e } \\chi ( a c ( a ) ) ^ N | a | ^ { s N + n - 1 } & a \\notin \\pi ^ e O _ v , \\chi ^ N | _ { 1 + \\pi ^ e a ^ { - 1 } O _ v } = \\chi _ { t r i v } \\\\ \\\\ 0 & \\textit { a l l o t h e r c a s e s } . \\end{cases} \\end{gather*}"}
-{"id": "7648.png", "formula": "\\begin{align*} x ' ( t ) = - \\gamma \\ , f _ \\gamma ( h ^ \\prime ( x ( t ) ) ) \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\tilde { Z } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} \\max \\Big \\{ 0 : X \\in \\mathcal S _ + ^ { S \\cup T } , \\ , X _ { k , k } = 1 k \\in S \\cup T , \\ , X _ { s , t } = C _ { s , t } s \\in S , t \\in T \\Big \\} , \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} \\mu & ( B ( x + 2 m k z , r ) ) - \\mu ( B ( x , r ) ) \\\\ & = \\sum _ { j = 1 } ^ m \\bigl [ \\mu ( B ( x + 2 j k z , r ) ) - \\mu ( B ( x + 2 ( j - 1 ) k z , r ) ) \\bigl ] . \\end{align*}"}
-{"id": "9901.png", "formula": "\\begin{align*} \\dfrac { \\partial } { \\partial x _ i } ( \\gamma ( \\tilde { r } ) ) = & \\ , \\gamma ^ { \\prime } ( \\tilde { r } ) \\dfrac { \\partial } { \\partial x _ i } \\sqrt { ( 2 \\xi _ 1 ( x ) - x _ 1 - a _ 1 ) ^ 2 + \\cdots + ( 2 \\xi _ { n + 1 } ( x ) - x _ { n + 1 } - a _ { n + 1 } ) ^ 2 } \\\\ = & \\ , \\dfrac { \\gamma ^ { \\prime } ( \\tilde { r } ) } { \\tilde { r } } \\left \\{ \\left ( \\sum _ { k } 2 \\dfrac { \\partial \\xi _ k ( x ) } { \\partial x _ i } ( \\tilde { x } - a ) _ k \\right ) - ( \\tilde { x } - a ) _ i \\right \\} . \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} x _ s = \\bigoplus _ i \\sqrt { \\lambda _ i } \\begin{pmatrix} \\Re ( X _ s \\psi _ i ) \\\\ \\Im ( X _ s \\psi _ i ) \\end{pmatrix} y _ t = \\bigoplus _ i \\sqrt { \\lambda _ i } \\begin{pmatrix} \\Re ( Y _ t \\psi _ i ) \\\\ \\Im ( Y _ t \\psi _ i ) \\end{pmatrix} \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} v _ { s + k + 1 } ^ * = v _ { s + k + 1 } - \\omega _ { k + 1 } \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime \\right ) \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} p ( \\varepsilon ) = & \\sum _ { g \\in G } p ( e _ g ) \\\\ = & \\sum _ { g \\in G } \\sum _ { h \\in H } p ( g , h ) h \\\\ = & 1 . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} p _ { n + 1 } < ( ( p _ { n } ) ^ { \\frac { 1 } { k } } + \\frac { 2 } { k } ) ^ { k } = p _ { n } + 2 ( p _ { n } ) ^ { ( \\frac { k - 1 } { k } ) } + . . . + ( \\frac { 2 } { k } ) ^ { k } \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} \\frac { R ' _ { \\sf u p p e r } ( \\omega _ \\ell ) } { R _ { \\sf l o w e r } ( \\omega _ \\ell ) } & \\le \\left . e ^ z \\left ( 1 + \\frac { z } { e ^ z - 1 } \\right ) ^ 2 \\right | _ { z = - \\frac { 5 } { 4 } \\ln ( 1 - 1 / 5 ) } \\\\ & = \\left . \\nu ^ \\nu \\left ( 1 + \\frac { \\nu \\ln \\nu } { \\nu ^ \\nu - 1 } \\right ) ^ 2 \\right | _ { \\nu = \\frac { 5 } { 4 } } \\\\ & \\approx 4 . 6 0 7 . \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} S _ { n , 2 m } = \\frac { 2 } { 2 m + 1 } \\sum \\limits _ { i = 0 } ^ m { 2 m + 1 \\choose 2 i + 1 } B _ { 2 m - 2 i } S _ { n , 2 i + 1 } . \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} d u = ( \\Delta u + V ( t , x ) u ) \\ , d t + G ( t , x ) u \\ , d W ( t ) , ( t , x ) \\in [ 0 , 1 ] \\times \\R ^ n . \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} F ^ { \\ , \\bf a } _ k = b _ 1 F ^ { \\bf a } _ { k - 1 } + \\cdots + b _ n F ^ { \\bf a } _ { k - n } , \\ , \\ , \\ , k > 0 , \\ , \\ , \\ , \\ , \\ , \\ , F ^ { \\bf a } _ k \\equiv a _ k , \\ , \\ , \\ , k = - n + 1 , \\dots , 0 . \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{gather*} D _ { n _ { k + 1 } } = \\left ( s _ { 2 n _ { k + 1 } } - s _ { 2 n _ { k + 1 } } ^ { ( n _ { k } + 1 ) } \\right ) t _ { n _ { k } } \\ , \\end{gather*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\log \\left ( \\frac { \\Gamma ( t + 1 ) } { \\Gamma ( r + 1 ) } \\right ) & = - \\gamma ( t - r ) + ( t - r ) \\sum _ { l \\geq 2 } \\frac { ( - 1 ) ^ l } { l } \\zeta ( l ) \\left ( \\sum _ { 0 \\leq k \\leq l - 1 } t ^ k r ^ { l - 1 - k } \\right ) \\\\ & = ( t - r ) \\left ( - \\gamma + \\sum _ { l \\geq 2 } \\frac { ( - 1 ) ^ l } { l } \\left ( \\sum _ { 0 \\leq k \\leq l - 1 } t ^ k r ^ { l - 1 - k } \\right ) \\zeta ( l ) \\right ) . \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} x _ { 1 j } \\cdot x _ { d - 1 w } = m = x _ { d - 1 w } \\cdot x _ { 1 \\sigma ( j ) } \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} \\varphi ( y ) = \\inf _ { \\psi \\in C X } \\hom ( \\psi ( y ) , \\Phi ( \\psi ) ( x ) ) , \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} \\bar { r } ^ 1 ( s , f ^ * ) = \\max _ { f \\in F _ S } \\bar { r } ^ 1 ( s , f ) = \\frac { 1 } { 1 + \\alpha } \\max _ { a ^ 1 \\in A ^ 1 ( s ) } [ R ^ 1 ( s ) g ( s ) ] _ { a ^ 1 } . \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} q ^ { \\alpha ^ { 2 } / 2 } \\left ( - z q ^ { \\alpha + 1 / 2 } ; q \\right ) _ { \\infty } = \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( x ^ { 2 } / \\log q ^ { 2 } + i \\alpha x \\right ) d x } { \\left ( z e ^ { i x } ; q \\right ) _ { \\infty } \\sqrt { \\pi \\log q ^ { - 2 } } } , \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i , j = 0 } ^ I r _ { i j } ( x ) x ^ { \\alpha _ i } \\log ( x ) ^ j \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} F ' _ t \\ , \\overline { F ' } \\ , - \\ , G ' \\ , \\overline { G ' _ t } = i \\ , \\nu ( z , \\ , \\bar { z } ) \\ , \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} { } ^ k H _ { [ i ] } = { } ^ { c _ 1 ^ { \\delta _ 1 } c _ 2 ^ { \\delta _ 2 } \\ldots c _ r ^ { \\delta _ r } } H _ { [ i ] } < { } ^ { c _ 1 ^ { \\delta _ 1 } c _ 2 ^ { \\delta _ 2 } \\ldots c _ r ^ { \\delta _ r } } H . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} \\inf _ { \\beta \\in \\mathbb C ^ 2 , \\| \\beta \\| = 1 } \\Big \\{ 1 - u ^ 2 \\| A _ { 1 } ^ * \\beta \\| ^ 2 - v ^ 2 \\| A _ { 2 } ^ * \\beta \\| ^ 2 \\Big \\} \\geq 0 . \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} G _ 2 ( M ) = G _ 2 ( M , \\varphi ) \\longrightarrow M , \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} \\nu = 3 , p \\rightarrow \\infty & & \\Omega = 1 / 1 2 - \\epsilon & & P N \\geq 1 / 1 4 - \\epsilon ; \\\\ \\nu = 4 , p \\rightarrow \\infty & & \\Omega = 1 / 8 - \\epsilon & & P N \\geq 1 / 1 0 - \\epsilon . \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} | g ( y ) | \\leq \\left \\{ \\begin{array} { l l } \\ C ( y _ n ^ 2 + y _ { n + 1 } ^ 2 ) ^ { \\frac { 3 } { 2 } - \\frac { n + 1 } { p } } & p \\in ( n + 1 , \\infty ] f = 0 , \\\\ C ( y _ n ^ 2 + y _ { n + 1 } ^ 2 ) ^ { 1 - \\frac { n + 1 } { p } } & p \\in ( 2 ( n + 1 ) , \\infty ] . \\end{array} \\right . \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { q ^ { ( n ^ 2 + n ) / { 2 } } ( - 1 ; q ) _ n } { ( q ; q ) ^ 2 _ n } = \\sum _ { \\pi \\in \\mathcal { D } } \\widetilde { \\omega } _ 1 ( \\pi ) q ^ { | \\pi | } , \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} v _ i ^ 0 = h _ i ( x _ i ^ 0 ) \\mbox { f o r a l l } i = 1 , \\ldots , N , \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} P ( z ) = P _ 0 + P _ 1 z + \\dots + P _ g z ^ g \\in \\mathbb { C } [ z ] ^ { n \\times n } , \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\begin{gathered} \\sum _ { i = 1 } ^ { m _ 1 } { } \\beta _ i p _ i \\ + \\sum _ { v \\in \\{ 0 , 1 \\} ^ { m _ 2 } } \\big ( \\sum _ t b _ { t v } \\big ) ^ 2 \\cdot \\prod _ { j = 1 } ^ { m _ 2 } q _ j ^ { v _ j } + \\prod _ { k = 1 } ^ { m _ 3 } r _ k ^ { d _ k } \\\\ + \\sum _ { v \\in \\{ 0 , 1 \\} ^ { m _ 3 } } \\big ( \\sum _ t c _ { t v } \\big ) ^ 2 \\cdot \\prod _ { k = 1 } ^ { m _ 3 } r _ k ^ { v _ k } + \\sum _ w s _ w ^ 2 = 0 \\end{gathered} \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} c _ { i j k l } : = \\frac { \\partial ^ 2 g _ { i j } } { \\partial x ^ k \\partial x ^ l } ( 0 ) , \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} \\left \\vert \\left ( A ( \\gamma ) \\right ) ^ { n } e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } \\right \\vert < \\frac { c } { 2 ^ { n } } , \\forall n = 1 , 2 , . . . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } U & \\tilde U \\end{array} \\right ] \\left [ \\begin{array} { c c } V & 0 \\end{array} \\right ] ^ H + \\left [ \\begin{array} { c c } V & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } U & \\tilde U \\end{array} \\right ] ^ H = 0 . \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } q ^ { n ^ { 2 } / 2 } S _ { n } \\left ( x q ^ { n } ; q \\right ) t ^ { n } = \\left ( - t q ^ { 1 / 2 } ; q \\right ) _ { \\infty } \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { 5 k ^ { 2 } / 2 } \\left ( - x t \\right ) ^ { k } } { \\left ( q ; q \\right ) _ { k } \\left ( - t q ^ { 1 / 2 } ; q \\right ) _ { 2 k } } \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} 4 t _ { 2 } + 3 t _ { 3 } + t _ { 4 } = 4 d + \\sum _ { r \\geq 5 } ( 2 r - 9 ) t _ { r } . \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P - I A } } & = \\max \\left \\{ \\frac { ( 1 - \\mu M ) K } { M r } ~ , ~ \\frac { ( 1 - \\mu M ) K } { \\min \\{ M , K \\} } + \\mu ( M + K - 1 ) \\right \\} , \\\\ & = \\frac { ( 1 - \\mu M ) K } { M r } , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\mu \\leq \\mu _ 1 . \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} \\left \\langle \\varphi ( \\omega q ^ { 2 m } ) , \\varphi ( \\omega q ^ { 2 n } ) \\right \\rangle = \\| \\varphi \\left ( \\omega q ^ { 2 n } \\right ) \\| ^ { 2 } \\ , \\delta _ { m , n } \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} C _ j ( \\lambda ) \\big | _ { \\lambda \\in [ - a , a ] } = \\sum _ { k \\in \\Z } \\exp \\big ( i \\pi a ^ { - 1 } k \\lambda \\big ) C _ { j , a , k } \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} f _ { n , k } ( s ) - s f ^ { ' } _ { n , k } ( s ) = g _ { n , k } ( s ) - s g ^ { ' } _ { n , k } ( s ) \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} | | u | | _ { W ^ { 1 , G } ( \\Omega ) } : = | | u | | _ { L ^ G ( \\Omega ) } + | | D u | | _ { L ^ G ( \\Omega ) } . \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} ( f _ { 1 } ^ { \\prime } ) ^ { 2 } + . . . + ( f _ { n } ^ { \\prime } ) ^ { 2 } = 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) . \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } \\varphi = D _ { t } ^ { \\alpha } \\left ( \\varphi ( t ) - \\sum _ { k = 0 } ^ { n - 1 } \\frac { t ^ { k } } { k ! } \\varphi ^ { ( k ) } ( 0 ) \\right ) . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} \\theta \\mu ^ { t r } = 0 , \\quad \\mu \\mu ^ { t r } = I . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} g _ 0 = d r ^ 2 + f \\left ( \\theta \\right ) ^ 2 d \\theta ^ 2 , \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} \\textrm { d o m } \\ , f & : = \\{ x \\in \\R ^ d : f ( x ) < + \\infty \\} , \\textrm { e p i } \\ , f : = \\{ ( x , r ) \\in \\R ^ d \\times \\R : f ( x ) \\leq r \\} , \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} \\omega ( \\pi ) : = \\lambda _ { \\nu ( \\pi ) } \\cdot \\prod _ { i = 1 } ^ { \\nu ( \\pi ) - 1 } ( \\lambda _ { i } - \\lambda _ { i + 1 } - 1 ) , \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} \\alpha '' _ k ( t ) - \\lambda _ k ^ 2 \\alpha _ k ( t ) = 0 . \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} \\bigl ( \\mathcal { S } _ { M - 1 } B _ { M , M } ( x \\ , | \\ , a ) \\ , \\log ( x ) \\bigr ) ( q | b ) = - M ! \\bigl ( \\prod \\limits _ { j = 1 } ^ { M - 1 } b _ j / \\prod \\limits _ { i = 1 } ^ M a _ i \\bigr ) q \\log ( q ) + O ( q ) . \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} K ( x , y ) = \\sum _ { \\gamma \\in G ( F ) } \\phi ( x ^ { - 1 } \\gamma y ) \\qquad K _ H ( x , y ) = \\sum _ { \\gamma \\in H ( F ) } \\phi ( x ^ { - 1 } \\gamma y ) . \\end{align*}"}
-{"id": "10170.png", "formula": "\\begin{align*} a _ n : = \\left \\{ \\begin{array} { l l l } n ^ { 1 - \\frac { 1 } { 2 \\alpha } } & & S d = 1 , \\ \\alpha \\in ( 1 , 2 ] . \\\\ \\sqrt { n \\log n } & & S \\alpha = d \\in \\{ 1 , 2 \\} . \\\\ \\sqrt { n } & & S . \\end{array} \\right . \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} M _ 0 : = \\iiint \\ ! \\Delta _ { \\delta _ 1 } ( \\xi _ 1 + h ) \\Delta _ { \\delta _ 2 } ( \\xi _ 2 + h ) C _ { \\delta _ 1 , \\delta _ 2 } \\gamma _ { \\delta _ 1 , \\delta _ 2 } ( h ) F ( \\xi _ 1 , \\xi _ 2 , \\eta ) \\ , d \\xi _ 1 d \\xi _ 2 d \\eta , \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} \\Theta _ { l _ 1 , l _ 2 } ^ { { \\rm h o l } } ( z g ) = \\chi _ \\xi ( z ) ^ { - 1 } \\times \\{ \\Theta _ { l _ 1 , l _ 2 } ( g ) + \\overline { \\Theta _ { l _ 1 , l _ 2 } ( g ) } \\} ( z \\in A _ { G , \\infty } , \\ ; \\ ; g \\in S p _ 4 ( \\R ) ) . \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} \\widetilde { \\omega } _ 1 ( \\pi ) = \\prod _ { i = 1 } ^ { \\nu ( \\pi ) } ( 2 \\lambda _ i - 2 \\lambda _ { i + 1 } - 1 ) . \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} W ^ { i , \\epsilon } _ 0 : = 0 W ^ { i , \\epsilon } _ t : = W ^ { i , \\epsilon } _ { k \\eta } + \\frac { t - k \\eta } { \\eta ^ { 1 / 2 } } X ^ i _ k . \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} M _ { X } ( a , b ) = \\phi _ a \\cdot U \\cdot \\chi _ b , \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} T ^ { ( p - 2 j ) } ( L , \\theta B ) = \\theta T ^ { ( p - 2 j ) } ( L , B ) , \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} P _ { i , 1 } ( m ) \\alpha _ { i } ^ m - w _ { \\mu } = \\sum _ { j = 1 } ^ { q } \\left ( \\sum _ { \\ell = 1 } ^ { n _ j } c _ { j , \\ell } Q _ { j , \\ell } ( n ) \\right ) \\beta _ j ^ n = \\sum _ { j = 1 } ^ q Q _ j ( n ) \\beta _ j ^ n \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} 0 = C l _ { - 1 } \\subset C l _ 0 \\subset C l _ 1 \\cdots \\subset C l _ { N } = C l , \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} H ( Y _ i | Y ^ { i - 1 } ) & = H ( Y _ i | Y ^ { i - 1 } , \\Phi _ { i - 1 } ( Y ^ { i - 1 } ) ) \\\\ & \\leq H ( Y _ i | \\Phi _ { i - 1 } ( Y ^ { i - 1 } ) ) . \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} \\begin{array} { r @ { \\ = \\ } l l } u _ t + \\lambda _ 0 ( V \\cdot \\nabla ) u + M u - \\Gamma _ 0 \\Delta u + \\Gamma _ 2 \\Delta ^ 2 u + \\nabla q & f & ( 0 , \\infty ) \\times \\mathbb { R } ^ n , \\\\ { \\mathrm { d i v \\ , } } u & 0 & ( 0 , \\infty ) \\times \\mathbb { R } ^ n , \\\\ u ( 0 ) & u _ 0 & \\mathbb { R } ^ n . \\end{array} \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} & A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\psi _ { n - 5 } ^ { ( 0 ) } \\\\ = & A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\alpha _ { n - 5 } ^ { ( 0 ) } + ( B _ { 2 - n } A _ { 4 - n } A _ { 6 - n } + A _ { 2 - n } B _ { 4 - n } A _ { 6 - n } + A _ { 2 - n } A _ { 4 - n } B _ { 6 - n } ) \\beta _ { n - 5 } ^ { ( 0 ) } . \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} v _ m ^ + ( x , t ) = \\frac { 3 a L ^ 3 } { 2 \\pi } \\sum _ { j = 0 } ^ { 2 } \\int _ { 0 } ^ { \\infty } e ^ { i a \\rho ^ 3 L ^ 3 t } \\frac { \\Delta _ { j , m } ^ + ( \\rho ) } { \\Delta ^ + ( \\rho ) } e ^ { \\lambda _ j ^ + ( \\rho ) x } \\hat { h } ^ + _ m ( \\rho ) \\rho ^ 2 d \\rho v _ m ^ - ( x , t ) = \\overline { v _ m ^ + ( x , t ) } , \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} ( L _ { \\lambda ^ \\prime } \\mathbb { E } ) _ t = \\bigoplus _ { | \\nu | = r - t } K _ { \\lambda ^ \\prime / \\nu } F _ 1 \\otimes K _ { \\nu ^ \\prime } F _ 0 . \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} \\gamma ^ l = \\frac { 1 } { \\delta ^ 1 } \\prod _ { j = 2 } ^ l ( 1 - \\alpha ^ j ) ^ { - 1 } = \\frac { 1 } { \\delta ^ 1 } \\prod _ { j = 2 } ^ l ( \\Lambda ^ { j } / \\Lambda ^ { j - 1 } ) = \\frac { 1 } { \\delta ^ 1 } \\frac { \\Lambda ^ l } { \\Lambda ^ 1 } = \\Lambda ^ l . \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} k = \\frac { T V - U ^ 2 } { ( 1 + t ^ 2 + u ^ 2 ) } , \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} I _ { \\mu - 1 } ( z ) - I _ { \\mu + 1 } ( z ) & = \\frac { 2 \\mu } { z } I _ { \\mu } ( z ) , \\\\ I _ { \\mu - 1 } ( z ) + I _ { \\mu + 1 } ( z ) & = 2 I _ { \\mu } ' ( z ) ; \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{gather*} i _ 2 f = F i _ 1 \\\\ i _ 1 g = G i _ 2 \\end{gather*}"}
-{"id": "5161.png", "formula": "\\begin{align*} \\mu ^ { \\# } & = g \\circ u - ( \\lambda _ { i n t } - \\mu _ { i n t } ) \\ , , \\\\ \\nu ^ { \\# } & = \\nu - \\frac { ( \\lambda _ { b d } - \\tau _ { b d } ) } { { \\bf { n } } \\cdot { \\bf { n } } A ^ T } . \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} \\mathbf { B } = ( B _ { j k } ) _ { j , k = 1 } ^ d \\ , , B _ { j k } : = \\partial _ j A _ k - \\partial _ k A _ j = i [ P _ j , P _ k ] \\ , , \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\Box _ d : = \\prod _ { i : ( D _ i , d ) > 0 } \\prod _ { k = 0 } ^ { ( D _ i , d ) - 1 } ( \\widehat { D } _ i - k z ) - y ^ d \\prod _ { i : ( D _ i , d ) < 0 } \\prod _ { k = 0 } ^ { - ( D _ i , d ) - 1 } ( \\widehat { D } _ i - k z ) . \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} \\mathcal { L } ( \\boldsymbol { \\gamma } ) = \\int _ { 0 } ^ { 1 } \\sqrt { \\tau } ( R + | \\dot { \\gamma } | ^ 2 ) d \\tau = \\int _ { 0 } ^ { 1 } \\sqrt { \\tau } | \\dot { \\gamma } | ^ 2 d \\tau \\leq \\frac { 2 } { 3 } \\cdot 9 d ^ 2 = 6 d ^ 2 , \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} N ^ { 1 , p } ( M , \\mathbb { V } ) : = \\tilde { N } ^ { 1 , p } ( M , \\mathbb { V } ) / \\{ f \\in \\tilde { N } ^ { 1 , p } ( M , \\mathbb { V } ) : \\| f \\| _ { \\tilde { N } ^ { 1 , p } ( M , \\mathbb { V } ) } = 0 \\} . \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} K ^ \\perp ( x ) = R ^ X ( E _ 1 , E _ 2 , N _ 2 , N _ 1 ) + 2 \\langle A ^ \\circ _ { 1 1 } \\wedge A ^ \\circ _ { 1 2 } , N _ 1 \\wedge N _ 2 \\rangle . \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\Delta { C } _ t = & \\big ( \\mu _ 1 ( \\beta - 1 ) - \\mu _ 0 ( \\alpha - 1 ) \\big ) + H ( \\alpha ) - H ( \\beta ) + \\log \\Big ( \\frac { 1 + 2 ^ { \\mu _ 1 + \\Delta { C } _ { t + 1 } } } { 1 + 2 ^ { \\mu _ 0 + \\Delta { C } _ { t + 1 } } } \\Big ) , ~ \\Delta { C } _ { n + 1 } = 0 , \\\\ = & f ( \\alpha , \\beta , \\mu _ 0 , \\mu _ 1 , \\Delta { C } _ { t + 1 } ) , ~ t \\in \\{ n , \\ldots , 0 \\} \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u ^ n _ t ( t , x ) + L ^ { b ^ n } u ^ n ( t , x ) + f ( t , x , u ^ n ( t , x ) , \\nabla u ^ n ( t , x ) ) = 0 , \\\\ u ^ n ( T , x ) = \\Phi ( x ) , \\\\ \\forall ( t , x ) \\in [ 0 , T ] \\times \\R ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ N S _ j S _ j ^ * = 1 , S _ i ^ * S _ i = \\sum _ { j = 1 } ^ N A ( i , j ) S _ j S _ j ^ * , i = 1 , \\dots , N . \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} \\Phi ( x , t ) = \\phi ( x ) + \\sum _ { i = 1 } ^ m \\int _ { t _ 0 } ^ t H ^ i ( D \\Phi ( x , s ) , x ) \\cdot d W ^ i _ s . \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} \\xi _ { s p } & = r ( z ) , & z & : = \\frac { \\xi _ { s p } } { \\xi _ { s p } - \\eta _ { s p } } . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\{ f _ i \\in C ^ \\infty ( [ 0 , 1 ] , \\mathbb { R } ) , \\sum _ { i = 1 } ^ { + \\infty } a _ i f _ i ( 0 ) = \\sum _ { i = 1 } ^ { + \\infty } a _ i f _ i ( 1 ) \\} \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} 0 = \\Phi _ A ( \\psi _ 1 \\otimes \\psi _ 2 ) = \\Phi _ A ( \\psi _ 1 ) \\otimes \\Phi _ A ( \\psi _ 2 ) . \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} H _ { \\alpha } = B _ { \\alpha } A _ { \\alpha } + \\epsilon _ { \\alpha } , \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} V _ \\ell = \\left ( X ^ { ( n ) } Q _ \\ell ^ { ( n ) } : n \\in [ 1 : N ] \\right ) . \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} V ^ { \\intercal } V = I _ r , \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} \\left \\vert \\phi _ { k } ^ { \\pm } \\left ( I _ { \\pm } \\right ) \\right \\vert = \\left \\vert \\frac { 1 } { 2 \\pi k i } \\int _ { 0 } ^ { 2 \\pi } e ^ { - i k \\theta _ { \\pm } } \\phi ^ { \\prime } ( x ) \\frac { \\partial x } { \\partial \\theta _ { \\pm } } d \\theta _ { \\pm } \\right \\vert \\leq \\frac { 1 } { 2 \\pi k } | | \\phi | | _ { W ^ { 1 , 1 } } \\lesssim \\frac { 1 } { k } | | \\phi | | _ { H ^ { 1 } } . \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\nabla \\cdot \\left ( \\mathbf { A } \\times \\mathbf { B } \\right ) = \\mathbf { B } \\cdot \\left ( \\nabla \\times \\mathbf { A } \\right ) - \\mathbf { A } \\cdot \\left ( \\nabla \\times \\mathbf { B } \\right ) . \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} \\varphi _ P ( f ) = \\sum _ { I \\in P } m ( I ) \\varphi _ I ( f ) = \\sum _ { I \\in P } \\sum _ { I ' \\in P ' } m ( I \\cap I ' ) \\varphi _ { I \\cap I ' } ( f ) = \\sum _ { I ' \\in \\P ' } m ( I ' ) \\varphi _ { I ' } ( f ) = \\varphi _ { P ' } ( f ) \\ , . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( \\lambda ^ 2 H ) - ( \\lambda ^ 2 H ) ( | A | ^ 2 - 2 ) = 0 , \\\\ A ( { \\rm g r a d } ( \\lambda ^ 2 H ) ) + ( \\lambda ^ 2 H ) { \\rm g r a d } H = 0 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} \\frac { ( \\gamma + \\eta ) ( \\xi - \\eta ) } { \\eta ( \\alpha + \\beta - \\eta ) } = 1 \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} J _ { D f ( p _ 0 ) } ( 0 ) = J _ f ( p _ 0 ) . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} Q _ 1 & = \\big \\{ ( x , t ) \\in \\Omega \\times ( 0 , T ) \\ , | \\ , \\big | \\dot { u } ( x , t ) \\big | > 1 \\big \\} , \\\\ Q _ 2 & = \\big \\{ ( x , t ) \\in \\Omega \\times ( 0 , T ) \\ , | \\ , \\big | \\dot { u } ( x , t ) \\big | \\le 1 \\big \\} . \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} \\texttt { \\rm d e g } _ B ( f _ \\varepsilon - p _ 0 , B _ { \\R ^ { 2 n } } ( \\widetilde \\eta , R _ 0 ) , 0 ) = 1 , \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} h ( X , Y ) = \\frac { 1 } { 2 ( 2 n - 1 ) } g ( X , Y ) J _ 1 ( p ^ \\bot _ 1 ) = \\frac { 1 } { 4 n } g ( X , Y ) J _ \\alpha ( p _ \\alpha ^ \\bot ) , \\alpha = 2 , 3 , \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } z ^ { - n } f _ { n } = \\sum _ { k = 0 } ^ { \\infty } \\theta _ { q } \\left ( q ^ { - k } z \\alpha \\right ) \\frac { q ^ { \\frac { 1 } { 2 } k ( k + 1 ) } } { ( q ; q ) _ { k } } ( - 1 ) ^ { k } \\alpha ^ { - k } z ^ { k } . \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} \\gamma _ { 1 } | V ( x ) | - \\Re \\mu - | \\Im \\mu | - \\gamma _ { 2 } \\geq \\gamma _ { 1 } \\check V _ { \\infty } - \\Re \\mu - | \\Im \\mu | - \\gamma _ { 2 } = : \\gamma > 0 \\ , . \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\zeta _ n ( w ) = ( x _ 1 \\ldots x _ { n } ) ( x _ 2 \\ldots x _ { n + 1 } ) \\ldots ( x _ { | \\zeta ( a _ 1 ) | } \\ldots x _ { | \\zeta ( a _ 1 ) | + n - 1 } ) . \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} \\beta \\partial _ t u - u \\Delta u & = 0 \\quad D _ T , \\\\ \\partial _ n u & = 0 \\quad \\Gamma _ T \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} i = 2 \\begin{array} { | c | c | c | c | c | } \\hline & & & & \\\\ \\hline j = 0 & 1 & 0 & 3 & 2 \\\\ \\hline j = 1 & 2 & 3 & 0 & 1 \\\\ \\hline j = 2 & 0 & 1 & 2 & 3 \\\\ \\hline j = 3 & 3 & 2 & 1 & 0 \\\\ \\hline \\end{array} \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\lim _ { P \\rightarrow \\infty } \\lim _ { L \\rightarrow \\infty } \\frac { ( T _ E + T _ F ) \\log ( P ) } { L } & = \\left ( 1 + \\frac { 1 } { r } \\right ) \\lim _ { P \\rightarrow \\infty } \\lim _ { L \\rightarrow \\infty } \\frac { T _ E \\log ( P ) } { L } \\\\ & = \\left ( 1 + \\frac { 1 } { r } \\right ) \\frac { K } { \\min \\{ M , K \\} } , \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} \\sum _ { n = m + 1 } ^ { \\infty } \\mu ( \\hat { C } _ { h _ n } \\cup \\hat { D } _ { h _ n } ) < \\frac { \\epsilon _ { h _ m } } { 4 } . \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} f _ { 0 } \\left ( v \\right ) = \\left \\{ \\begin{array} [ c ] { c c } \\mu _ { - , + } \\left ( \\frac { 1 } { 2 } v ^ { 2 } \\right ) & v > 0 \\\\ \\mu _ { - , - } \\left ( \\frac { 1 } { 2 } v ^ { 2 } \\right ) & v < 0 \\end{array} \\right . . \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} \\psi _ { k , n } ( t ) = | a _ 0 | ^ { \\frac { k } { 2 } } \\psi ( a _ 0 ^ { k } t - n b _ 0 ) , \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} 8 = m _ { - } \\geq 1 3 - j - c _ j . \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} B ( J ) = \\alpha _ { j , J } + { \\sum _ { i = 1 } ^ { t } \\alpha _ { 1 , \\{ i + m a x ( J ) \\} } } , \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\phi ( t , \\xi ) = H _ { l + 1 } ( W _ { i j } ( t , \\xi ) ) + \\frac { H _ { l + 2 } ( W _ { i j } ( t , \\xi ) ) } { H _ { l + 1 } ( W _ { i j } ( t , \\xi ) ) } , \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} H _ 0 = \\left \\{ e \\in H \\colon \\binom e 2 \\cap G _ { 1 / 3 } = \\emptyset \\right \\} \\quad \\mbox { a n d } H ' = H \\setminus H _ 0 , \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} h ( z ) & = \\exp \\left ( + 4 \\eta ( 1 ) \\bar z + \\eta ( 1 ) - \\frac { m } { 2 } \\eta ( \\tau ) - 4 \\eta ( 1 ) \\bar z + \\eta ( 1 ) \\sum _ { k = 1 } ^ 4 \\Re ( b _ k ) + \\eta ( 1 ) \\frac { m } { 2 } \\tau \\right ) \\\\ & = \\exp \\left ( \\eta ( 1 ) \\left ( 1 + \\sum _ { k = 1 } ^ 4 \\Re ( b _ k ) \\right ) \\right ) \\exp \\left ( m \\pi i \\right ) , \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} 2 \\sum \\limits _ { j = 1 } ^ n \\gamma _ { 2 j } \\sum \\limits _ { i = j } ^ n \\frac { { 2 i + 2 \\choose 2 } ( 2 n ) ! ( 2 j + 1 ) } { { 2 j + 1 \\choose 2 } ( 2 i + 2 ) ! ( 2 n - 2 i ) ! } B _ { 2 n - 2 i } { i \\brack j } = \\sum \\limits _ { j = 1 } ^ n \\frac { \\gamma _ { 2 j } } { j } \\sum \\limits _ { i = j } ^ n { 2 n \\choose 2 i } B _ { 2 n - 2 i } { i \\brack j } . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} K ^ \\perp ( x ) & = R ^ X ( E _ 1 , E _ 2 , N _ 2 , N _ 1 ) + \\langle ( D _ { E _ 1 } N _ 1 ) ^ \\top , ( D _ { E _ 2 } N _ 2 ) ^ \\top \\rangle - \\langle ( D _ { E _ 1 } N _ 2 ) ^ \\top , ( D _ { E _ 2 } N _ 1 ) ^ \\top \\rangle \\\\ & = R ^ X ( E _ 1 , E _ 2 , N _ 2 , N _ 1 ) + \\left ( \\sum _ { j = 1 , 2 } \\langle A _ { 1 j } , N _ 1 \\rangle \\langle A _ { 2 j } , N _ 2 \\rangle - \\langle A _ { 1 j } , N _ 2 \\rangle \\langle A _ { 2 j } , N _ 1 \\rangle \\right ) . \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} \\dot { x } = - F \\nu , \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} L _ \\epsilon ( \\lambda ) = \\left [ \\begin{array} { c c c c c c } \\lambda & - 1 & 0 & \\cdots & 0 & 0 \\\\ 0 & \\lambda & - 1 & \\cdots & 0 & 0 \\\\ \\vdots & \\vdots & \\vdots & & 0 & 0 \\\\ 0 & 0 & 0 & \\cdots & - 1 & 0 \\\\ 0 & 0 & 0 & \\cdots & \\lambda & - 1 \\end{array} \\right ] , \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} S = S _ 0 \\subseteq S _ 1 \\subseteq S _ 2 \\subseteq \\cdots \\subseteq X ( \\Q ) \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\mathcal { I } ^ 2 \\ll \\frac { 1 } { T ^ 2 } \\int _ { | y | > M ^ { \\varepsilon } } \\Big | \\widehat { w _ 2 } \\left ( \\frac { y } { T } \\right ) \\Big | ^ 2 \\Big | \\Sigma ( M ^ { \\frac \\eta 4 } , y ) \\Big | ^ 8 \\Big | \\Sigma ( M ^ { 1 - \\frac \\eta 4 } , y ) \\Big | ^ 8 d y . \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } _ { V I } ^ + } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq C _ { d , s , k } C _ { d , \\alpha } T R ^ d \\theta ^ { ( d - 1 ) / 2 } \\end{aligned} \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{gather*} P _ { 0 } \\equiv 1 \\ , \\ \\ P _ { 1 } \\equiv \\gamma _ { 0 } \\ , \\ \\ \\deg P _ { n _ { 1 } } = n _ { 1 } = 2 \\ , \\end{gather*}"}
-{"id": "3336.png", "formula": "\\begin{align*} { \\rm d i s t } _ { { \\rm b d } \\ , X ( t ' ) } ( x ^ * ( t ' ) ) = \\| x ^ * ( t ' ) - \\bar { x } ^ * ( t ' ) \\| \\leq \\epsilon , \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} { } [ \\chi , \\chi ] ^ { F N } _ p = c _ { i j } \\ast e ^ i \\otimes e _ j , \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} \\frac { g _ { 1 } ( a + b ) - g _ { 1 } ( b ) } { g _ { 1 } ( a ) } = G ( b ) , \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{gather*} D _ { r - 1 } ^ { 2 } \\sum _ { k = 0 } ^ { r - 1 } \\frac { P _ { k } ( x ) P _ { k } ( y ) } { D _ { k } D _ { k - 1 } } = \\frac { P _ { r } ( x ) P _ { r - 1 } ( y ) - P _ { r } ( y ) P _ { r - 1 } ( x ) } { x - y } \\ , \\ \\ D _ { - 1 } : = 1 \\ , \\ x \\neq y \\ , \\end{gather*}"}
-{"id": "8351.png", "formula": "\\begin{align*} \\int _ M | \\nabla \\Delta u _ k | _ g ^ 2 d \\mu _ g = & \\int _ M | \\nabla \\Delta u | _ g ^ 2 d \\mu _ g + \\int _ M | \\nabla \\Delta ( u - u _ k ) | _ g ^ 2 d \\mu _ g + o ( 1 ) , \\\\ \\int _ M f | u _ k | ^ { 2 ^ \\sharp } d \\mu _ g = & \\int _ M f | u | ^ { 2 ^ \\sharp } d \\mu _ g + \\int _ M f | u - u _ k | ^ { 2 ^ \\sharp } d \\mu _ g + o ( 1 ) , \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} Z ( I , X ) ^ { - 1 } \\prod \\limits _ { 1 \\leq i < j \\leq \\# _ I ( X ) } ( t _ i - t _ j ) ^ 2 \\prod \\limits _ { i = 1 } ^ { \\# _ I ( X ) } \\rho ^ { \\Pi } _ { I , X } ( t _ i ) , \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} C = \\frac { 1 } { 2 ( 2 n - 1 ) } J _ 1 ( p ^ \\bot _ 1 ) = \\frac { 1 } { 4 n } J _ \\alpha ( p _ \\alpha ^ \\bot ) , \\alpha = 2 , 3 . \\end{align*}"}
-{"id": "10110.png", "formula": "\\begin{align*} C \\ : : \\ : y ^ q ( a x + b y + c z ) ^ r - x ^ { - p } z ^ { p + q + r } = 0 \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} \\alpha ( y _ 1 \\cdots y _ k ) = 0 \\quad \\alpha ( z _ 1 \\cdots z _ \\ell ) = 0 . \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} K & ( x , y , \\Psi _ { x , y } ^ { - 1 } ( u ) ) \\stackrel { \\eqref { s e c . t w o _ 2 : e q _ d e f K e r n e l K } } { \\geq } \\sum _ { k = 1 } ^ p | F _ { n + k } ( x , y , \\Psi _ { x , y } ^ { - 1 } ( u ) ) | ^ { { 1 } / { \\sigma ^ * _ k } } \\\\ & \\stackrel { \\eqref { s e c . t w o _ 2 : d e f _ m a p P s i x y } } { = } \\sum _ { k = 1 } ^ p | \\Psi _ { x , y } ( \\Psi _ { x , y } ^ { - 1 } ( u ) ) | ^ { { 1 } / { \\sigma ^ * _ k } } = \\sum _ { k = 1 } ^ p | u _ k | ^ { { 1 } / { \\sigma ^ * _ k } } { = } N ( u ) . \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} \\dfrac { d x } { d s } = i \\xi A x - i \\Phi ^ * \\sigma ( \\xi ) ( \\Lambda ( \\xi , \\eta ) \\tilde { u } ) . \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} \\nabla _ { \\sigma } \\mathbf { u } : = \\nabla \\big ( \\mathbf { u } \\ast \\rho _ { \\sigma } \\big ) \\rho _ { \\sigma } ( \\mathbf { x } ) : = \\tfrac { 1 } { \\sigma ^ { d } } \\rho \\big ( \\tfrac { \\mathbf { x } } { \\sigma } \\big ) \\mathbf { x } \\in G \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} [ G , G ] = \\left \\{ { \\textstyle \\sum _ { i \\in I } b _ i ( \\zeta ) \\ , e _ i } \\in B \\mid b _ \\infty ( 1 ) = b _ 0 ( 1 ) = \\ldots = b _ { p - 1 } ( 1 ) \\right \\} . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} B _ { ( d + 1 ) n + s } ( x ) = x ^ s B _ n ^ s ( x ^ { d + 1 } ) , \\ \\ 0 \\leq s \\leq d , \\ \\ n \\geq 0 . \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\Delta _ 1 ( L , B ) ( v _ L , \\dots , v _ L , x _ 1 , \\dots , x _ { p - q } ) = \\binom { p } { q } ^ { - 1 } T ^ { ( p - q ) } ( L , B ) ( x _ 1 , \\dots , x _ { p - q } ) . \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\kappa ^ l _ g Z ^ G _ { t , g } \\leq x ^ { G + } _ { t , g } \\leq \\kappa ^ u _ g Z ^ G _ { t , g } , ~ \\mbox { f o r } t = 0 , \\dots , T . \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} G _ 1 ( p t ) = \\bar A ( t ) G _ 1 ( t ) A _ 1 ^ { - 1 } \\mbox { f o r s m a l l } t . \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} { \\mathcal T } : = \\sqrt { h ^ { - 1 } } ( u ' v '' - v ' u '' ) + \\sqrt { h } R ( v ' f _ u - u ' f _ v ) \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} \\left ( c ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( - \\sqrt { q } ; \\sqrt { q } \\right ) _ { n } q ^ { \\binom { n } { 2 } } c ^ { n } } { \\left ( \\sqrt { q } , \\sqrt { c } , - \\sqrt { c } ; \\sqrt { q } \\right ) _ { n } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } c ^ { n } A _ { q } ^ { 2 } \\left ( - c q ^ { n - 1 / 2 } \\right ) } { \\left ( q ; q \\right ) _ { n } } \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} \\mathcal { A } ^ - = \\left \\{ \\begin{aligned} & ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in [ 0 , \\infty ) \\times \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } \\textnormal { s u c h t h a t } \\\\ & \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime ( \\tau ; 0 ) \\right ) \\leq 0 \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} \\omega _ { J ^ p } = \\cos \\alpha _ p d V _ { \\phi ^ * h } . \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } | A _ f ( n ) | ^ 2 & = \\sum _ { n \\leq X } \\sum _ { d ^ 2 | n } \\mu _ N ( d ) C _ f ( n / d ^ 2 ) = \\sum _ { d ^ 2 \\leq X } \\mu _ N ( d ) \\sum _ { v \\leq X / d ^ 2 } C _ f ( v ) \\\\ & = \\sum _ { d \\leq \\sqrt { X } } \\mu _ N ( d ) \\left ( R _ 1 \\frac { X } { d ^ 2 } + O \\left ( \\frac { X ^ { 3 / 5 } } { d ^ { 6 / 5 } } \\right ) \\right ) . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} F _ n ( T _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) ) = F _ n ( T _ { ( 2 x ) } ( i , \\alpha ) \\otimes T _ y ( \\gamma ) ) = F _ n ( T _ { ( 2 x ) } ( i , \\alpha ) ) \\otimes F _ n ( T _ y ( \\gamma ) ) \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{gather*} T _ { a b } = Q _ { a } Q _ { b } ^ { - 1 } , \\end{gather*}"}
-{"id": "1401.png", "formula": "\\begin{align*} \\kappa ( r , p ) & = \\sum _ { j = 1 } ^ { L - r } \\frac { j } { j + r } \\binom { L - r } { j } p ^ j ( 1 - p ) ^ { L - r - j } . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} h ( n ) \\geq \\left ( \\frac 5 9 + o ( 1 ) \\right ) \\binom { n - 1 } 2 \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{gather*} \\big \\langle Q ^ { m - n } v _ 0 , \\psi ^ { + } ( w _ 1 ) \\cdots \\psi ^ { + } ( w _ m ) \\psi ^ { - } ( y _ 1 ) \\cdots \\psi ^ { - } ( y _ n ) v _ 0 \\big \\rangle = \\frac { \\prod \\limits _ { 1 \\le i < j \\le m } ( w _ i - w _ j ) \\prod \\limits _ { 1 \\le i < j \\le n } ( y _ i - y _ j ) } { \\prod \\limits _ { i = 1 } ^ { m } \\prod \\limits _ { j = 1 } ^ { n } ( w _ i - y _ j ) } . \\end{gather*}"}
-{"id": "5719.png", "formula": "\\begin{gather*} \\left ( \\sum _ { m = 0 } ^ { r } p _ { r , m } x ^ { m } \\right ) \\sum _ { m \\geq 0 } \\frac { a _ { m } } { x ^ { m + 1 } } = \\sum _ { m \\geq 0 } \\frac { \\begin{displaystyle} \\sum \\nolimits _ { k = 0 } ^ { r } \\end{displaystyle} \\ p _ { r , k } a _ { m + k } } { x ^ { m + 1 } } + \\sum _ { m = 0 } ^ { r - 1 } x ^ { m } \\sum _ { k = m } ^ { r - 1 } p _ { r , k + 1 } a _ { k - m } \\ , \\end{gather*}"}
-{"id": "3772.png", "formula": "\\begin{align*} ( F _ i ( x _ i ^ k , N \\hat v ^ k _ i ) - F _ i ( x ^ * _ i , \\bar x ^ * ) ) ^ T ( x _ i ^ k - x _ i ^ * ) & \\ge - \\bar L _ i N M \\sum _ { j = 1 } ^ N [ W ( k ) ] _ { i j } \\| v ^ k _ j - y ^ k \\| \\cr & + ( F _ i ( x ^ k _ i , \\bar x ^ k ) - F _ i ( x ^ * _ i , \\bar x ^ * ) ) ^ T ( x _ i ^ k - x _ i ^ * ) . \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} R ( n , k ) = ( 2 k + 1 ) R ( n - 1 , k ) + ( 2 n - 4 k + 3 ) R ( n - 1 , k - 1 ) + R ^ - ( n - 1 , k - 1 ) . \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} \\Q _ \\sigma [ u , v ] = \\int _ 0 ^ \\infty \\overline { u ' } v ' \\ , d x + \\sigma \\ , \\overline { u ( 0 ) } \\ , v ( 0 ) , u , v \\in H ^ 1 ( \\R _ + ) , \\sigma > 0 \\ , . \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} \\frac { q ^ { n \\left ( n + \\nu \\right ) / 2 } h _ { n } \\left ( i \\cosh \\left ( \\log q ^ { \\nu / 2 } \\right ) \\vert q \\right ) } { i ^ { n - \\nu } \\left ( q ; q \\right ) _ { \\infty } } = q ^ { \\nu \\left ( \\nu + 1 / 2 \\right ) } J _ { \\nu } ^ { ( 2 ) } \\left ( \\frac { 2 i } { q ^ { \\nu + \\left ( n + 1 \\right ) / 2 } } ; q \\right ) \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{gather*} H _ \\gg = U ( \\gg ) \\sharp _ { { \\boldsymbol \\phi } } \\hat { S } ( \\gg ^ * ) \\end{gather*}"}
-{"id": "8391.png", "formula": "\\begin{align*} \\mathcal E ( w ( \\lambda ) ) = \\varepsilon ( w ) \\mathcal E ( \\lambda ) . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\tilde { Q } [ w + \\varphi ] = a ^ { i j } ( w _ { i ; j } + \\varphi _ { i ; j } ) - b \\le a ^ { i j } w _ { i ; j } + \\Lambda \\norm { \\varphi } _ { C ^ 2 } - b , \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} \\lim _ { \\epsilon \\rightarrow 0 } 1 0 0 \\int ^ { t _ { 2 } } _ { t _ { 1 } } \\int _ { \\C ^ { n } } u ^ 2 \\eta ^ { 2 } | \\nabla _ { g } \\psi _ { \\epsilon } | ^ 2 d V _ { g } d s = 0 . \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} \\left | \\alpha _ { k , i } - \\frac { 1 } { k p _ i } \\right | = \\frac { 1 } { k p _ i } \\frac { 1 } { \\Gamma _ k ( i ) } | k p _ i - \\Gamma _ k ( i ) | \\leq \\frac { 2 } { k ^ 2 p _ i ^ 2 } | k p _ i - \\Gamma _ k ( i ) | , \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} ( 1 - \\phi ( x , y ) ) ( 1 - \\phi ( y , x ) ) = ( 1 - p _ a ) ^ 2 ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} F _ { t , \\varsigma } ( z ) & = \\int _ { - \\infty } ^ { + \\infty } e ^ { i \\sqrt { 2 } v z } e ^ { - \\frac { 1 } { 2 } v ^ 2 } f \\big ( \\sqrt { \\varsigma t } v \\big ) \\frac { d v } { \\sqrt { 2 \\pi } } , \\\\ G _ { t , \\varsigma } ( z ) & = \\int _ { - \\infty } ^ { + \\infty } e ^ { i \\sqrt { 2 } v z } e ^ { - \\frac { 1 } { 2 } v ^ 2 / t } \\left ( 1 - f \\big ( \\sqrt { \\varsigma } v \\big ) \\right ) \\frac { d v } { \\sqrt { 2 \\pi t } } . \\\\ \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} \\sqrt { - d ^ 2 / d x ^ 2 } \\ , v _ { 3 / 2 } = - \\frac { 2 } { \\pi } \\frac { d } { d x } \\bigg ( \\frac { 1 } { x ^ 2 + 1 } - \\frac { x \\sinh ^ { - 1 } x } { \\left ( x ^ 2 + 1 \\right ) ^ { 3 / 2 } } \\bigg ) . \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} \\| ( 1 - \\nu ) A X - \\nu X B \\| _ { 2 } ^ { 2 } & \\leqslant \\| A ^ { 1 - \\nu } X B ^ { \\nu } \\| _ { 2 } ^ { 2 } + ( 1 - r _ { 0 } ) ^ { 2 } \\| A X - X B \\| _ { 2 } ^ { 2 } \\\\ & - \\sum _ { k = 1 } ^ { \\infty } r _ { k } \\| A ^ { 1 - \\frac { m _ { k } } { 2 ^ { k } } } X B ^ { \\frac { m _ { k } } { 2 ^ { k } } } - A ^ { 1 - \\frac { m _ { k } + 1 } { 2 ^ { k } } } X B ^ { \\frac { m _ { k } + 1 } { 2 ^ { k } } } \\| _ { 2 } ^ { 2 } . \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} f ( x _ 0 , \\dotsc , x _ { n + 1 } ) = \\nabla { F } ( x _ 0 , \\dotsc , x _ { n + 1 } ) \\cdot D ^ \\prime , g = \\nabla { F } ( B ) \\cdot ( x _ 0 , \\dotsc , x _ { n + 1 } ) . \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} \\circlearrowleft _ { x , y , z } ( - 1 ) ^ { | x | | z | } \\sum _ { s \\geq 0 } t ^ s \\Big ( \\sum _ { k = 0 } ^ { s } \\sum _ { i = 0 } ^ { s - k } [ \\alpha _ { i } ( x ) , [ y , z ] ' _ { k } ] ' _ { s - i - k } \\Big ) = 0 . \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} \\left ( - z q ^ { 1 / 2 } , - z q ; q \\right ) _ { \\infty } = \\left ( - z q ^ { 1 / 2 } ; q ^ { 1 / 2 } \\right ) _ { \\infty } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ( n + 1 ) / 4 } z ^ { n } } { \\left ( q ^ { 1 / 2 } ; q ^ { 1 / 2 } \\right ) _ { n } } = \\sum _ { k = 0 } ^ { \\infty } z ^ { k } q ^ { k ^ { 2 } / 2 } S _ { k } \\left ( - q ^ { - k + 1 / 2 } ; q \\right ) \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { 2 , 1 } ( a , b ) ^ q \\bigr ] = \\frac { \\Gamma _ 2 ( q + b _ 0 \\ , | \\ , a ) } { \\Gamma _ 2 ( b _ 0 \\ , | \\ , a ) } \\frac { \\Gamma _ 2 ( b _ 0 + b _ 1 \\ , | \\ , a ) } { \\Gamma _ 2 ( q + b _ 0 + b _ 1 \\ , | \\ , a ) } . \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda } \\log \\left [ \\frac { 1 } { n } \\sum _ { t = 1 } ^ n 2 ^ { \\lambda l ( y _ t ) } \\right ] , \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} c _ 1 = 1 + \\frac 1 2 \\sum _ { j = 1 } ^ k z ^ j . \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} L ( x ^ n ) = s _ n , n \\ge 0 . \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v + \\nabla _ H p ( x , y , t ) - \\Delta v + f _ 0 k \\times v = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} d _ X ( F ^ 2 ( x _ 0 , y _ 0 ) , F ( x _ 0 , y _ 0 ) ) & = d _ X ( F ( F ( x _ 0 , y _ 0 ) , G ( y _ 0 , x _ 0 ) ) , F ( x _ 0 , y _ 0 ) ) \\\\ & \\leq a \\ d _ X ( F ( x _ 0 , y _ 0 ) , F ^ 2 ( x _ 0 , y _ 0 ) ) + b \\ d _ X ( x _ 0 , F ( x _ 0 , y _ 0 ) ) + \\\\ & \\ \\ \\ \\ \\ c \\ d _ X ( F ( x _ 0 , y _ 0 ) , x _ 0 ) \\\\ i e , \\ d _ X ( F ^ 2 ( x _ 0 , y _ 0 ) , F ( x _ 0 , y _ 0 ) ) & \\leq \\dfrac { b + c } { 1 - a } \\ d _ X ( x _ 0 , x _ 1 ) \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\mathbf { Q ' } _ 1 = [ \\mathbf { Q } _ { x _ 1 } \\mathbf { b } _ 1 , \\dots , \\mathbf { Q } _ { x _ d } \\mathbf { b } _ d ] \\mathbf { Z } , \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} Z _ i ( x , \\xi ) & \\ , \\ , \\ , = \\d T ( J _ i ) ( x , \\xi ) = J _ i ( T ) ( T ^ { - 1 } ( x , \\xi ) ) \\\\ & \\stackrel { \\eqref { s e c . t w o _ 1 : d e f _ m a p T } } { = } \\Big ( J _ i ( \\pi ) , J _ i ( a \\mapsto a _ { j _ 1 } ) , \\ldots , J _ i ( a \\mapsto a _ { j _ p } ) \\Big ) ( T ^ { - 1 } ( x , \\xi ) ) . \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} R _ i ( \\alpha _ i , P _ s ^ i ) = \\alpha _ i \\log _ 2 \\left ( 1 + \\frac { h _ { s s } ^ i P _ s ^ i } { \\sigma _ s ^ 2 + h _ { p s } ^ i P _ p } \\right ) , \\ ; i = 1 , \\ldots , M . \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} a _ 0 ( r ) = O \\left ( \\frac { 1 } { r ( \\log r ) ^ \\alpha } \\right ) , \\ \\alpha > 1 , \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( \\mathcal { Z } _ { m , n } ^ \\gamma ) \\right ] ( z ) & = \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + 1 } \\mathcal { Z } _ { m , n - 1 } ^ { \\gamma + 1 } ( z , \\bar { z } ) ; m \\geq n . \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} \\textbf { w } ^ i = \\dfrac { 1 } { \\sqrt { N } } [ 1 , \\ e ^ { j \\vartheta _ i } , \\ e ^ { j 2 \\vartheta _ i } , \\ . . . , \\ e ^ { j ( N - 1 ) \\vartheta _ i } ] ^ T \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} D : = D _ 1 \\hat \\otimes 1 + 1 \\hat \\otimes _ \\nabla D _ 2 \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 2 n - m + 1 } ^ m { m \\choose i } \\frac { ( m + i ) ! } { ( m + i - 2 n - 1 ) ! } \\ : ( - 2 ) ^ { - i } = 0 . \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\hat \\theta _ n ^ { ( 4 ) } ( m ) = \\frac { X _ { n ^ { m - 1 } } ^ 2 } { 2 \\int _ 0 ^ { n ^ { m - 1 } } X ^ 2 _ t \\ , d t + 2 \\vartheta _ n } = \\left ( \\frac { 1 } { \\hat \\theta _ { n ^ { m - 1 } } ^ { ( 2 ) } } + 2 \\cdot \\frac { e ^ { 2 \\theta n ^ { m - 1 } } } { X _ { n ^ { m - 1 } } ^ 2 } \\cdot \\frac { \\vartheta _ n } { e ^ { 2 \\theta n ^ { m - 1 } } } \\right ) ^ { - 1 } \\to \\theta \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} T _ r : = \\inf \\left \\{ s \\ ; \\middle | \\ ; R ^ { ( n ) } ( s ) = r \\right \\} \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} \\inf \\left \\{ \\max _ { k \\in K } | \\alpha ( k ) f - f | _ { \\infty } ^ - | \\mid f \\in { \\mathcal F } ( G ) , f ( e ) = 1 \\right \\} = 0 \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} \\mathcal { R } ^ 2 _ { \\sf p r e } ( \\widehat { \\boldsymbol { \\theta } } ) : = \\sup _ { \\mathcal { H } } \\mathbb { E } \\left [ \\left ( x _ { k } - \\widehat { \\boldsymbol { \\theta } } ' \\mathbf { x } _ { k - p } ^ { k - 1 } \\right ) ^ 2 \\right ] . \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\varphi ( x ) = \\inf _ { \\psi \\in C X } \\hom ( \\psi ( x ) , \\Phi ( \\psi ) ) . \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} \\psi ( x , y ) \\psi ( y , x ) = a b ( 1 - \\psi ( x , y ) ) ( 1 - \\psi ( y , x ) ) = ( 1 - a ) ( 1 - b ) x \\neq y . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} a _ { \\ell _ i } a _ { r _ i } ( i ) = 0 \\quad a _ { \\ell _ i + 1 } ( i ) = \\ldots = a _ { r _ i - 1 } ( i ) = 1 . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\begin{aligned} & \\left < \\varphi , f _ N ^ { ( 1 ) } ( t ) - \\mathcal { T } ( t - T _ 1 ) f _ N ^ { ( 1 ) } ( T _ 1 ) \\right > = \\int _ { T _ 1 } ^ t \\left < \\varphi , \\mathcal { T } ( t - \\tau ) C _ 2 f _ N ^ { ( 2 ) } ( \\tau ) \\right > d \\tau \\end{aligned} \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\dot { \\tilde { V } } & = & - \\dfrac { 1 } { V ( x ) ( 1 - \\gamma ^ { - 1 } D ( x ) ) } \\left [ \\dot { V } ( x ) \\left ( \\gamma - D ( x ) \\right ) + V ( x ) \\dot { D } ( x ) \\right ] \\\\ & = & - \\dfrac { 1 } { V ( x ) ( 1 - \\gamma ^ { - 1 } D ( x ) ) } \\left [ \\gamma \\dot { V } ( x ) - \\dot { V } ( x ) D ( x ) + V ( x ) \\dot { D } ( x ) \\right ] \\end{array} \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} d s ^ 2 = \\sum _ { j = 1 } ^ { n - 1 } \\frac { d y _ j ^ 2 } { y _ n ^ 2 + y _ { n + 1 } ^ 2 } + d y _ n ^ 2 + d y _ { n + 1 } ^ 2 . \\end{align*}"}
-{"id": "10107.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p - q - r } y ^ { q } ( c x + b y ) ^ { r } } { z ^ { p } } \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} \\xi _ { q } ( z ) = \\frac { q ^ { 1 / 2 } } { 1 - q ^ { 1 / 2 } } { } _ { 1 } \\psi _ { 1 } \\left ( q ^ { 1 / 2 } ; q ^ { 3 / 2 } ; q , - q ^ { 1 / 2 } z \\right ) = q ^ { 1 / 2 } \\frac { \\left ( q , q , - q z , - z ^ { - 1 } ; q \\right ) _ { \\infty } } { \\left ( q ^ { 1 / 2 } , q ^ { 1 / 2 } , - q ^ { 1 / 2 } z , - q ^ { 1 / 2 } z ^ { - 1 } ; q \\right ) _ { \\infty } } . \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{gather*} D _ { r - 1 } s _ { 2 r + m } ^ { ( r ) } \\ ! + \\ ! \\sum \\limits _ { k = 0 } ^ { r - 1 } \\ p _ { r , k } s _ { r + m + k } ^ { ( r ) } = 0 \\ , , \\ \\ \\ \\ m \\geq 0 \\ , , \\end{gather*}"}
-{"id": "8483.png", "formula": "\\begin{align*} \\int _ { P ^ { i } ( \\zeta ) } \\left | \\frac { \\delta _ { \\Omega } ( z ) } { \\delta _ { \\Omega } ( \\zeta ) } \\right | ^ { \\gamma ' - \\varepsilon } \\frac { d \\lambda ( z ) } { \\left | z - \\zeta \\right | } \\lesssim _ { \\gamma ' - \\epsilon } \\tau _ { n } \\left ( \\zeta , 2 ^ { i } \\delta _ { \\Omega } ( \\zeta ) \\right ) \\prod _ { j = 1 } ^ { n - 1 } \\tau _ { j } ^ { i } \\left ( \\zeta , 2 ^ { i } \\delta _ { \\Omega } ( \\zeta ) \\right ) , \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} A ( y + 1 ) & = { \\rm a r g m i n } \\Bigl \\{ T z + W ( z ) + { 1 \\over 2 } \\gamma ( z - y - 1 ) ^ 2 \\Bigr \\} \\\\ & = 1 + { \\rm a r g m i n } \\Bigl \\{ T ( z - 1 ) + W ( z - 1 ) + { 1 \\over 2 } \\gamma ( z - y ) ^ 2 \\Bigr \\} \\\\ & = 1 + { \\rm a r g m i n } \\Bigl \\{ T z + W ( z ) + { 1 \\over 2 } \\gamma ( z - y ) ^ 2 \\Bigr \\} \\\\ & = 1 + A ( y ) . \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} X ( u , v ) = ( \\phi ( u ) , \\lambda \\cos \\left ( \\frac { u } { c } \\right ) \\cos v , \\lambda \\cos \\left ( \\frac { u } { c } \\right ) \\sin v ) , \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} & 2 M A _ 1 \\left ( \\beta _ { x x } + a _ 0 \\beta - \\frac { 1 } { M } \\right ) _ x - A _ 2 \\left ( M \\beta _ { x x } + a _ 0 \\beta + M \\right ) _ x = 0 , \\\\ & M _ t = 3 M \\beta _ x - \\beta M _ x . \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\| v ^ { k + 1 } ( \\ell ) - [ y ^ { k + 1 } ] _ \\ell \\mathbf { 1 } \\| \\right ] & \\le \\left ( \\sqrt { \\lambda } \\right ) ^ { k + 1 } \\mathbb { E } \\left [ \\| v ^ 0 ( \\ell ) - [ y ^ { 0 } ] _ \\ell \\mathbf { 1 } \\| \\right ] + \\sqrt { 2 } C \\alpha _ { \\max } \\sum _ { s = 0 } ^ k ( \\sqrt { \\lambda } ) ^ s \\qquad \\hbox { f o r a l l } k \\ge 0 . \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} \\partial _ t u = J * u - u + u ^ { 1 + p } ( 0 , \\infty ) \\times \\R ^ { N } , \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } w _ t + \\frac 1 2 \\Delta w = { \\nabla u \\ , b } , \\\\ w ( T , x ) = 0 , \\\\ { \\forall } ( t , x ) \\in [ 0 , T ] \\times \\mathbb R ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} - L u _ n + g _ n \\circ u _ n & = f \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u _ n & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} f ( a ) H ( b ) = f ( b ) H ( a ) \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} f ^ \\infty _ { i j } ( t ) = { n - j \\choose n - i } \\exp ( - ( n - i ) \\lambda _ \\infty t ) ( 1 - \\exp ( - \\lambda _ \\infty t ) ) ^ { i - j } . \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\alpha = H \\left ( 1 \\right ) = g \\left ( 1 / 2 \\right ) + g \\left ( 1 / 2 \\right ) - 1 = \\frac { 1 } { 2 } g \\left ( 1 \\right ) + \\frac { 1 } { 2 } g \\left ( 1 \\right ) - 1 = 0 , \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} | a _ { i } - b _ { i } | = 1 , i \\in I _ { 2 n } , \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} [ z ] \\hat \\otimes _ A [ \\lambda ^ { ( d ) } ] & = ( - 1 ) ^ { d - 1 } [ z ] \\hat \\otimes _ A \\big ( [ \\mathrm { e x t } ] \\hat \\otimes _ { C } [ \\lambda ^ { ( d - 1 ) } ] \\big ) \\\\ & = ( - 1 ) ^ { d - 1 } \\big ( [ z ] \\hat \\otimes _ A [ \\mathrm { e x t } ] \\big ) \\hat \\otimes _ { C } [ \\lambda ^ { ( d - 1 ) } ] \\\\ & = ( - 1 ) ^ { d - 1 } \\partial [ z ] \\hat \\otimes _ { C } [ \\lambda ^ { ( d - 1 ) } ] \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} \\begin{gathered} \\exists k \\in \\R \\backslash \\{ 0 \\} : \\ k \\psi _ i ( \\vect { x } _ 1 ) = \\psi _ i ( \\vect { x } _ 2 ) , i = 1 , \\dots , K \\end{gathered} \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} L _ v ^ { ( 2 ) } [ X , Y ] = v ( v - 1 ) e _ 2 [ X ] e _ 2 [ Y ] . \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} x ( t ) = a x _ 1 ( t ) + b x _ 2 ( t ) \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\prod _ { k = 1 } ^ n ( z - x _ k ) \\right ] = \\det Y _ { 1 , 1 } ( z ) , \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} F ( s e _ { 1 } + t e _ { 2 } ) - F \\left ( 0 \\right ) = f \\left ( s \\right ) u _ { 1 } + f \\left ( t \\right ) u _ { 2 } , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\mathfrak { M } \\bigl ( \\frac { q } { \\tau } \\ , | \\ , \\frac { 1 } { \\tau } , \\tau \\lambda _ 1 , \\tau \\lambda _ 2 \\bigr ) ( 2 \\pi ) ^ { - \\frac { q } { \\tau } } \\ , \\Gamma ^ { \\frac { q } { \\tau } } ( 1 - \\tau ) \\Gamma ( 1 - \\frac { q } { \\tau } ) = & \\mathfrak { M } ( q \\ , | \\ , \\tau , \\lambda _ 1 , \\lambda _ 2 ) ( 2 \\pi ) ^ { - q } \\times \\\\ & \\times \\Gamma ^ { q } ( 1 - \\frac { 1 } { \\tau } ) \\Gamma ( 1 - q ) . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{gather*} g _ { - } = \\sum _ { a , b = 0 } ^ { n - 1 } g _ { a b } ( z ) E _ { a b } , \\end{gather*}"}
-{"id": "1732.png", "formula": "\\begin{align*} N _ { i j } \\eta ^ i \\eta ^ j = & \\ , F ^ { k l } h _ { r k } h ^ r _ l ( 1 - \\epsilon n ) \\abs { \\eta } ^ 2 + 2 F h _ { i j } ( \\epsilon n - 1 ) \\eta ^ i \\eta ^ j \\\\ & + \\{ F ^ { k l } g _ { k l } - 2 F \\} ( 1 - \\epsilon n ) \\abs { \\eta } ^ 2 - 2 F ( \\epsilon n - 1 ) h _ { i j } \\eta ^ i \\eta ^ j \\\\ = & \\ , ( 1 - \\epsilon n ) \\sum _ { i } F _ i ( \\kappa _ i ^ 2 - 2 \\kappa _ i + 1 ) \\abs { \\eta } ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\tilde \\chi _ 0 ( \\tau ) = & | \\tau | ^ \\beta \\chi _ 0 ( \\tau ) \\\\ \\tilde { \\tilde { \\chi } } ( s ) = & \\tilde \\chi ( s ) s ^ { - \\frac 1 2 } \\ , . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\cosh \\Theta = ( \\cosh r _ 0 ) e ^ { - t } . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} u '' + h ( t , u ) = 0 \\end{align*}"}
-{"id": "9951.png", "formula": "\\begin{align*} \\pi _ { p } ( v _ { i } ) = \\pi _ { p } ( v ) _ { i } = \\pi _ { p } ( v ) . \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\inf _ { \\vert x \\vert \\leq R } u ( t , x ) = 1 , R \\geq 0 . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} d ^ { m + n - 1 } ( f \\bullet g ) = ( - 1 ) ^ { n } \\left ( g \\smile f - ( - 1 ) ^ { m n } f \\smile g \\right ) . \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma _ 1 ( B _ 2 ) B _ 1 = \\sigma _ 2 ( B _ 1 ) B _ 2 . \\end{aligned} \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} \\delta _ f & = | \\{ v \\in S ~ : ~ d _ G ( v ) \\neq d _ { G ' } ( f ( v ) ) \\} | + | V \\setminus S | \\\\ & \\qquad + | \\{ v ' \\in S ' ~ : ~ d _ { G ' } ( v ' ) \\neq d _ G ( f ^ { - 1 } ( v ' ) ) \\} | + | V ' \\setminus S ' | . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} ( \\gamma ^ { \\prime \\prime } ) ^ k = - \\sum _ { i , j = 1 } ^ 2 \\Gamma ^ k _ { i j } ( \\gamma ^ \\prime ) ^ i ( \\gamma ^ \\prime ) ^ j - \\kappa ( J ^ { 9 0 } _ \\gamma ( \\gamma ' ) ) ^ k , k = 1 , 2 . \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} X = \\{ | x _ 0 : 1 : 1 | \\} \\cup \\{ | 1 : 0 : 0 | \\} \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\begin{aligned} Y = Y _ { \\lambda } = \\frac { 1 } { e A _ 1 } \\cdot \\frac { 1 } { \\alpha } \\Big \\{ 2 \\phi A + \\frac { 8 } { \\lambda } \\Big \\} , X = X _ { \\lambda } = \\frac { 2 \\log \\big ( \\frac { 2 A _ 1 } { 1 - \\omega } \\big ) } { ( 1 - \\omega ) } \\cdot \\frac { 1 + \\alpha } { \\alpha } \\Big \\{ 2 \\phi A + \\frac { 8 } { \\lambda } \\Big \\} , \\end{aligned} \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} ( 1 - 2 x t ) \\frac { d } { d t } R ^ + ( x ; t ) & = 2 x ( 1 - 2 x ) \\frac { d } { d x } R ^ + ( x ; t ) + R ^ + ( x ; t ) + R ^ - ( x ; t ) , \\\\ ( 1 - 2 x t ) \\frac { d } { d t } R ^ - ( x ; t ) & = 2 x ( 1 - 2 x ) \\frac { d } { d x } R ^ - ( x ; t ) + x R ^ + ( x ; t ) + 2 x R ^ - ( x ; t ) . \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} H _ { k } ( w ) g H _ k ( w ) ^ { - 1 } : = g \\widehat { \\Phi _ k } ( w , v ) . \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\tilde { r } = \\left ( \\frac { ( ( x - x _ 0 ) \\cdot \\nu _ { x _ 0 } ) ^ 2 } { ( \\nu _ { x _ 0 } \\cdot A ( x _ 0 ) \\nu _ { x _ 0 } ) } + \\frac { x _ { n + 1 } ^ 2 } { a ^ { n + 1 , n + 1 } ( x _ 0 ) } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} \\| \\vec { Y } \\| _ F ^ 2 : = \\sum _ { i , j } | \\vec { Y } _ { i , j } | ^ 2 , \\| \\vec { Y } \\| _ 1 : = \\sum _ { i , j } | \\vec { Y } _ { i , j } | . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} \\Psi _ 2 ( a ; \\ , b , \\ , c ; \\ , x , \\ ; y ) = \\sum _ { k = 0 } ^ \\infty \\frac { ( a ) _ k } { ( b ) _ k } \\ , { } _ 2 F _ 1 \\left [ \\begin{array} { c } - k , \\ , - k - b + 1 \\\\ c \\end{array} ; \\ , \\frac { y } { x } \\right ] \\ , \\frac { x ^ k } { k ! } \\ , . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\mathcal { X } _ i = \\mathcal { X } \\xrightarrow { g _ i h g _ { j _ i } ^ { - 1 } } \\mathcal { X } = \\mathcal { X } _ { j _ i } , \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} \\delta _ v \\circ ( T _ v ^ { - 1 } \\iota _ v T _ v ) = \\iota _ r . \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} ( a ) & \\ \\ P ( R ) = \\bigcup _ { k = 0 } ^ { n - 1 } \\{ z \\in \\C ^ 3 : \\ z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta _ k = 0 \\} . \\\\ ( b ) & \\ \\ t r R ^ { - 1 } ( z ) = \\frac { 1 } { n } \\sum _ 0 ^ { n - 1 } \\frac { z _ 0 } { z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta _ k } . \\\\ ( c ) & \\ \\ t r \\omega _ R ( z ) = \\partial \\big ( \\frac { 1 } { 2 n } \\sum _ { k = 0 } ^ { n - 1 } \\log ( z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 - 2 z _ 1 z _ 2 \\cos \\theta _ k ) \\big ) . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} H : = \\Delta + W + V \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} g _ \\varepsilon ^ { ( s ) } ( t ) = g _ \\varepsilon ^ { ( m - 1 ) } ( t ) \\otimes g _ { \\varepsilon } ( t ) ^ { \\otimes ( s - m + 1 ) } \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} \\begin{cases} \\lim _ { \\epsilon \\to 0 } \\eta ( \\epsilon ) = 0 , & \\\\ [ 1 m m ] \\lim _ { \\epsilon \\to 0 } \\frac { \\epsilon ^ \\beta } { \\eta ( \\epsilon ) } = 0 & \\\\ [ 1 m m ] \\lim _ { \\epsilon \\to 0 } \\frac { \\abs { \\log \\epsilon } } { \\eta ( \\epsilon ) } = 0 & \\end{cases} \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} d _ { \\Sigma , 2 } = \\max _ { \\substack { \\lambda _ 1 , \\lambda _ 2 \\in ( 0 , 1 ] \\\\ \\lambda _ 1 ( N + \\min \\{ M , K ( N - 1 ) \\} \\leq M \\\\ \\lambda _ 1 ( K - 1 ) N + \\lambda _ 2 K N \\leq M \\\\ \\lambda _ 1 + \\lambda _ 2 \\leq 1 \\\\ } } K N ( \\lambda _ 1 + \\lambda _ 2 ) . \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} \\left | F _ N \\right | _ { L ^ \\infty _ { \\beta , \\mu } } = \\sup _ { 1 \\leq s \\leq N } \\sup _ { Z _ s \\in \\mathcal { D } _ s } \\left | f _ N ^ { ( s ) } ( Z _ s ) \\right | e ^ { \\beta E _ s ( Z _ s ) } e ^ { \\mu s } \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} \\varphi = F ^ { k l } g _ { k l } \\abs { A } ^ 2 - F H . \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} Z ( u ) = \\prod _ { P } ( 1 - u ^ { \\deg P } ) ^ { - 1 } = \\frac { 1 } { 1 - q u } \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} X ^ 2 U V - X U D + Q = 0 . \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} f _ N ( 0 , Z _ N ) = \\mathcal { Z } _ N ^ { - 1 } f _ 0 ^ { \\otimes N } ( Z _ N ) \\mathbf { 1 } _ { Z _ N \\in \\mathcal { D } _ N } \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} \\begin{cases} \\partial u _ t ( x ) = - { \\nu } ( - \\Delta ) ^ { \\frac { \\alpha } { 2 } } u _ t ( x ) - a ^ \\beta ( - \\Delta ) ^ { \\frac { \\beta } { 2 } } u _ t ( x ) + \\xi \\sigma ( u _ t ( x ) ) \\dot F ( t , x ) , \\ \\ \\ x \\in B _ R ( 0 ) , \\ \\ t > 0 \\\\ u _ t ( x ) = 0 , \\ \\ \\ x \\in { B _ R ( 0 ) ^ c } . \\end{cases} \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} S _ { i _ \\ell } & = \\{ i _ 1 , i _ 2 , \\ldots , i _ { \\ell - 1 } , i _ \\ell - 1 , i _ { \\ell + 1 } - 1 , i _ { \\ell + 2 } - 1 , \\ldots , i _ { s } - 1 \\} , \\\\ \\widehat { S } _ { i _ \\ell } & = \\{ i _ 1 , i _ 2 , \\ldots , i _ { \\ell - 1 } , \\widehat { i _ \\ell } , i _ { \\ell + 1 } - 1 , i _ { \\ell + 2 } - 1 , \\ldots , i _ { s } - 1 \\} , \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} P _ { \\phi } = \\int _ { \\mathcal { P } _ { \\phi } } P _ { \\theta } d m ^ { ( \\phi ) } . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} t _ 1 \\cdot B _ p : = t _ 1 ^ { m _ { h _ p } } B _ p , \\ , \\ t _ 2 \\cdot B _ p ^ * : = t _ 2 ^ { m _ { h _ p ^ * } } B _ p ^ * . \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} q ^ * : = \\displaystyle { \\frac { n q } { n - q } } \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} J _ { \\nu } ^ { ( 1 ) } \\left ( z ; q \\right ) \\left ( \\frac { z } { 2 } \\right ) ^ { - \\nu } = \\frac { \\sqrt { \\log q ^ { - 1 } / \\left ( 2 \\pi \\right ) } } { \\left ( q ; q \\right ) _ { \\infty } } \\int _ { - \\infty } ^ { \\infty } q ^ { \\alpha ^ { 2 } / 2 } \\left ( \\frac { z ^ { 2 } q ^ { 1 / 2 } } { 4 q ^ { \\alpha } } , q \\right ) _ { \\infty } A _ { q } \\left ( q ^ { \\alpha + \\nu + 1 / 2 } \\right ) d \\alpha . \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} m _ { \\pi _ { k , p } } ( \\mu ) & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = 0 } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } \\sum _ { \\beta = 0 } ^ { p - j - 2 t } 2 ^ { p - j - 2 t - \\beta } \\binom { n - Z ( \\mu ) } { \\beta } \\binom { Z ( \\mu ) } { p - j - 2 t - \\beta } \\\\ & \\sum _ { \\alpha = 0 } ^ \\beta \\binom { \\beta } { \\alpha } \\sum _ { i = 0 } ^ { j - 1 } \\binom { r ( \\mu ) - i - p + \\alpha + t + j + n - 2 } { n - 2 } , \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} \\mathcal { F } \\{ D _ t ^ \\sigma q _ { \\alpha , \\beta } ( t , \\cdot ) \\} ( \\xi ) = t ^ { \\alpha - \\beta - \\sigma } E _ { \\alpha , 1 + \\alpha - \\beta - \\sigma } ( - | \\xi | ^ { 2 } t ^ { \\alpha } ) , \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} T _ n ( x ) ^ 2 - ( x ^ 2 - 1 ) U _ { n - 1 } ( x ) ^ 2 = 1 . \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} \\Theta = \\mathrm { a r c c o s h } \\ , e ^ { ( T ^ * - t ) } . \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\lim _ { b \\to + \\infty , \\ ; b - a = c } \\sqrt { \\frac { 4 b } { \\pi } } e ^ { 2 b x } P _ n ( x ^ 2 ) = \\frac { 2 ( b ^ 2 - a ^ 2 ) ^ { \\mu + \\nu + 1 } } { a ^ \\mu b ^ \\nu \\Gamma ( \\mu + \\nu + 1 + n ) n ! } x ^ { \\nu - \\frac { 1 } { 2 } } L _ n ^ { ( \\mu + \\nu ) } ( 2 c x ) , \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} \\mathbf { H } ( t , \\cdot ) \\in L ^ { \\infty } \\big ( G , \\mathcal { S } _ { \\geq \\kappa } ( \\mathbb { R } ^ { k \\times d } ) \\big ) t \\in [ 0 , T ] \\kappa : = \\alpha \\exp ( - T / \\tau ) . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} ( \\mu _ A ( f ) a ) ( x ) = f ( x ) a ( x ) \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} X _ { A _ { 0 } \\cdots A _ { r } } ^ { \\left ( p \\right ) } = C _ { A _ { 0 } A _ { p } } ^ { B } \\left \\langle \\boldsymbol { T } _ { A _ { 1 } } \\cdots \\boldsymbol { T } _ { A _ { p - 1 } } \\boldsymbol { T } _ { B } \\boldsymbol { T } _ { A _ { p + 1 } } \\cdots \\boldsymbol { T } _ { A _ { r } } \\right \\rangle . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} \\varphi ( x , t ) \\ ; = \\ ; & \\left ( J ^ * + t H \\right ) ^ * ( x ) \\\\ \\ ; = \\ ; & - \\min _ { v \\in R ^ { n } } \\left \\{ J ^ * ( v ) + t H ( v ) - \\langle x , v \\rangle \\right \\} \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} q ^ n - 1 + \\gamma - r & = \\sum _ { i \\in [ 0 , n - 1 ] } ( q - 1 ) q ^ i + \\sum _ { i \\in G } ( 1 - r _ i ) q ^ i - \\sum _ { i \\in E \\setminus G } r _ i q ^ i \\\\ & = \\sum _ { i \\in G } ( q - r _ i ) q ^ i + \\sum _ { i \\in E \\setminus G } ( q - 1 - r _ i ) q ^ i + \\sum _ { i \\in [ 0 , n - 1 ] \\setminus E } ( q - 1 ) q ^ i . \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} X _ { k + 1 } = \\Phi ( X _ k ) , k = 0 , \\ldots \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\rho \\left ( \\frac { x } { x - 1 } \\right ) = \\rho ( x ) . \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } P _ n ( x ) = ( - 1 ) ^ n \\frac { ( b ^ 2 - a ^ 2 ) ^ { \\mu + \\nu + 1 } \\Gamma ( \\nu ) } { a ^ \\mu b ^ { 2 \\nu } \\Gamma ( \\mu + \\nu + 1 ) n ! } { \\ ; } _ 2 F _ 1 \\left ( { - n , \\nu \\atop \\mu + \\nu + 1 } \\Big { | } 1 - \\frac { a ^ 2 } { b ^ 2 } \\right ) \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} V _ q ( L ^ { - 1 } , [ 0 , a ] ) = \\sum _ { i : L ( G _ i ) \\le a } ( D _ i - G _ i ) ^ q \\ \\ \\hbox { a n d } \\ \\ V _ q ( L ^ { - 1 } , [ 0 , a ) ) = \\sum _ { i : L ( G _ i ) < a } ( D _ i - G _ i ) ^ q , ~ a \\geq 0 . \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} V : = V _ { + } ^ * \\oplus V _ { - } ^ * \\oplus V _ 0 ^ * \\subset E _ { + } ^ * \\oplus E _ { - } ^ * \\oplus E _ 0 ^ * : = E . \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\mathbf { Q } ^ V _ i = Q _ { ( n - i ) \\lambda } . \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{gather*} \\left \\{ e _ { 0 } = \\begin{bmatrix} 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{bmatrix} , e _ { 1 } = \\begin{bmatrix} 0 \\\\ 1 \\\\ \\vdots \\\\ 0 \\end{bmatrix} , \\dots , e _ { n - 1 } = \\begin{bmatrix} 0 \\\\ \\vdots \\\\ 0 \\\\ 1 \\end{bmatrix} \\right \\} \\end{gather*}"}
-{"id": "7663.png", "formula": "\\begin{align*} c ( g , g ' ) = A ( g g ' , z ) - A ( g , g ' z ) - A ( g ' , z ) \\quad . \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} L _ { \\mathrm { C h S } } ^ { ( 5 ) } = \\alpha _ { 1 } l ^ { 2 } \\varepsilon _ { a b c d e } R ^ { a b } R ^ { c d } e ^ { e } + \\alpha _ { 3 } \\varepsilon _ { a b c d e } \\left ( \\frac { 2 } { 3 } R ^ { a b } e ^ { c } e ^ { d } e ^ { e } + 2 l ^ { 2 } k ^ { a b } R ^ { c d } T ^ { e } + l ^ { 2 } R ^ { a b } R ^ { c d } h ^ { e } \\right ) , \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} p _ l = \\sum _ { \\alpha = 1 , p = 1 } ^ { m _ { - } \\ , - r , \\ , m _ { + } \\ , - r } ( S ^ l _ { \\alpha p } + \\iota T ^ l _ { \\alpha p } ) x _ \\alpha z _ p + \\ ; { \\rm o t h e r \\ ; t e r m s } , \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} ^ { C \\ ! } D _ { 0 + } ^ { \\alpha } x ( t ) = f ( t , x ( t ) ) , \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} \\pi _ p ( c ) : = \\sum _ { i = 1 } ^ { \\infty } \\frac { c _ i } { p ^ i } = x . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\begin{bmatrix} I & - M _ { 1 2 } M _ { 2 2 } ^ { - 1 } \\\\ 0 & I \\end{bmatrix} . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} \\Phi _ { i j } = \\begin{cases} { 1 \\over \\lambda _ i } - G ^ { ( n ) } _ { 0 } ( a _ { i } , a _ { i } ) & \\textrm { i f $ i = j $ } \\\\ - \\ ; G ^ { ( n ) } _ { 0 } ( a _ { i } , a _ { j } ) & \\textrm { i f $ i \\neq j $ } . \\end{cases} \\ ; . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} H _ { [ i _ 1 , \\ldots , i _ l ] } = H _ { [ j _ 1 , \\ldots , j _ l ] } \\Leftrightarrow \\{ i _ 1 , \\ldots , i _ l \\} = \\{ j _ 1 , \\ldots , j _ l \\} , \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} y _ { 2 } y _ { 3 } & = e ^ { 4 \\pi i / 6 } e ^ { 6 \\pi i / 6 } = - y _ { 1 } ^ { 2 } , y _ { 3 } y _ { 4 } = e ^ { 6 \\pi i / 6 } e ^ { 8 \\pi i / 6 } = - y _ { 2 } ^ { 2 } , y _ { 4 } y _ { 5 } = e ^ { 8 \\pi i / 6 } e ^ { 1 0 \\pi i / 6 } = - y _ { 3 } ^ { 2 } , \\\\ y _ { 5 } y _ { 6 } & = e ^ { 1 0 \\pi i / 6 } e ^ { 1 2 \\pi i / 6 } = - y _ { 4 } ^ { 2 } , y _ { 6 } y _ { 1 } = e ^ { 1 2 \\pi i / 6 } e ^ { 2 \\pi i / 6 } = - y _ { 5 } ^ { 2 } , y _ { 1 } y _ { 2 } = e ^ { 2 \\pi i / 6 } e ^ { 4 \\pi i / 6 } = - y _ { 6 } ^ { 2 } . \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} \\frac { ( n ! ) ^ { 2 n } } { n ^ { n ^ 2 } } \\le n ^ 2 \\binom { n ^ 2 } { k } \\frac { n ! ^ { 2 n - \\frac { k } { n } } e ^ { n ( 3 + \\frac { \\ln ( 2 \\pi n ) ^ 2 } { 4 } ) } } { ( n - \\frac { k } { n } ) ! ^ { 2 n } e ^ k } . \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} \\mathcal { A } _ t = \\{ \\alpha \\in L ^ \\infty ( [ 0 , t ] ; \\R ^ d ) \\ ; : \\ ; \\| \\alpha \\| _ \\infty \\leq 1 \\} \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ I r _ i ( x ) e ^ { \\alpha _ i x } \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} v _ m = \\left \\{ \\begin{array} { l l } h _ m & \\textrm { i n } \\ , \\ , U _ m \\\\ u _ k & \\textrm { i n } \\ , \\ , \\widetilde Q \\setminus U _ m \\\\ \\end{array} \\right . , \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} ( \\lambda + \\mu ) p _ { 1 , 1 } = ( \\mu + \\xi ) p _ { 1 , 2 } + \\gamma p _ { 0 , 1 } . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} \\Lambda ( X ) & = \\tfrac { 1 } { ( g - 1 ) ! g ! } \\int _ { X ^ { g - 1 } } \\log \\| \\eta \\| ( P _ 1 + \\dots + P _ { g - 1 } ) \\Phi _ { \\Theta } ^ * \\nu ^ { g - 1 } \\\\ & = \\tfrac { 1 } { ( g ! ) ^ 2 } \\int _ { X ^ g } \\log \\| \\eta \\| ( P _ 1 + \\dots + P _ { g - 1 } ) \\Phi ^ * \\nu ^ g , \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} T _ { \\pm } ( e _ { \\pm } ) = \\int _ { 0 } ^ { P _ { \\beta } } \\frac { d x ^ { \\prime } } { \\sqrt { 2 ( e _ { \\pm } \\mp \\beta ( x ^ { \\prime } ) ) } } = \\frac { P _ { \\beta } } { \\sqrt { 2 e _ { \\pm } } } + O ( \\frac { \\varepsilon } { \\left ( e _ { \\pm } \\right ) ^ { \\frac { 3 } { 2 } } } ) , \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} H ^ i _ c ( M ^ G ) \\to H ^ { i } _ { c , G } ( M ^ G ) = H ^ i _ { c } ( M ^ G \\times B G ) \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\sum ^ { N - 1 } _ { n = 0 } \\theta \\circ T ^ n ( p _ \\xi ) = \\omega _ \\xi ( p _ \\xi ) = 1 \\ , . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} V ( M ) : = G _ 2 ( M ) \\times _ { G _ 2 } V \\longrightarrow M \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} g _ { k ; \\pm } ( x ) : = 2 + ( x - 2 ) \\cos ( k ) \\pm \\sin ( k ) \\sqrt { x ( 4 - x ) } h _ { k ; \\pm } ( x ) : = g _ { k ; \\pm } ( x ) - x . \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} 0 = \\mathcal { S } e _ { j } \\left ( z _ { j _ { \\ell } } \\right ) = \\sum _ { k = 0 } ^ { \\ell } b _ { j _ { k } } \\varphi _ { z _ { j _ { k } } } \\left ( z _ { j _ { \\ell } } \\right ) + \\sum _ { i \\notin \\left \\{ j _ { 0 } , j _ { 1 } , . . . , j _ { \\ell } \\right \\} } b _ { i } \\varphi _ { z _ { i } } \\left ( z _ { j _ { \\ell } } \\right ) . \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} \\mathcal J _ k ( r ) = & \\frac 1 { 2 \\pi } \\int ^ { \\pi } _ { - \\pi } e ^ { i r \\sin \\theta } e ^ { - i \\theta k } d \\theta - \\frac { \\sin ( k \\pi ) } { \\pi } \\int _ 0 ^ \\infty e ^ { - ( r \\sinh ( s ) + k s ) } d s \\\\ : = & \\tilde J _ k ( r ) - E _ k ( r ) \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} F ( u ) = S ( \\xi _ { s p } , \\eta _ { s p } ; u ) , \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} \\tilde Y ( 0 ) = 0 , \\ \\tilde Z ( t ) \\in S \\ \\hbox { f o r e a c h } \\ t \\ge 0 , \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} u _ { \\hat { \\delta } } ( x _ { j } ) = \\hat { v } _ { 1 } + \\hat { v } _ { 2 } g _ { j } ( x _ { j } ) \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\Pi _ H k _ S ( g ) = \\int _ { H ( F _ S ) } k _ S ( g h ) d h . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} \\begin{aligned} \\norm { x _ { k + 1 } - x } _ k ^ 2 \\leq \\norm { x _ k - x } _ k ^ 2 & + 2 \\gamma _ k \\lambda _ k \\big ( ( f + g ) ( x ) - ( f + g ) ( x _ { k + 1 } ) \\big ) \\\\ [ 0 . 5 e x ] & + \\frac { 2 \\gamma _ k } { 1 - \\delta } \\big ( ( f + g ) ( x _ { k } ) - ( f + g ) ( x _ { k + 1 } ) \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{gather*} D _ { n } = ( - 1 ) ^ { \\frac { n ( n + 1 ) } { 2 } } s _ { n } ^ { n + 1 } \\ \\ \\ . \\end{gather*}"}
-{"id": "6766.png", "formula": "\\begin{align*} \\left . G _ { N , s } ^ { ( r _ 1 , \\dots , r _ s ) } \\right \\vert _ { r _ s = N } = G _ { N , s - 1 } ^ { ( r _ 1 , \\dots , r _ { s - 1 } ) } , \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} U _ 0 = P _ C , U _ k & = S _ { c ( k ) } S _ { d _ k } S _ { d _ k } ^ * k = 1 , \\dots , N _ D , \\\\ T _ 0 = P _ D , T _ l & = S _ { d ( l ) } S _ { c _ l } S _ { c _ l } ^ * l = 1 , \\dots , N _ C . \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} A _ i = \\frac { 1 } { \\sqrt { \\alpha _ i } } \\sum _ { s = 1 } ^ m ( v _ i ) _ s X _ s i \\in [ r ] . \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} \\beta ( g ) = \\alpha ( m g ^ { - 1 } , g ) \\alpha ( m g m ^ { - 1 } , m g ^ { - 1 } ) ^ { - 1 } \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( w ) } ^ { p _ 0 } = \\left ( \\int _ { \\R ^ n } f ^ { p _ 0 \\frac { p } { p _ 0 } } w ^ { \\frac { p _ 0 } { p } \\frac { p } { p _ 0 } } \\ , d x \\right ) ^ { \\frac { p _ 0 } { p } } \\\\ = \\int _ { \\R ^ n } f ^ { p _ 0 } w ^ { \\frac { p _ 0 } { p } } h \\ , d x . \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} P _ 1 P _ 2 P _ 1 ( P _ 1 y ) = P _ 1 P _ 2 P _ 1 ( P _ 1 P _ 2 y ) = P _ 1 ( P _ 2 P _ 1 P _ 2 y ) = \\lambda P _ 1 y . \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} A _ { \\alpha , g } = & A _ \\alpha + r ^ 2 ( \\Delta _ g - \\Delta _ 0 ) = A _ \\alpha + r ^ 2 \\partial _ i ( ( g ^ { i j } - \\delta ^ { i j } ) \\partial _ j ) , \\\\ - P _ g ( r ^ \\alpha \\varphi ) = & r ^ { \\alpha - 6 } ( A _ { \\alpha - 4 } A _ { \\alpha - 2 } A _ \\alpha \\varphi + K _ \\alpha \\varphi ) , \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} ( 1 - \\nu ) I + \\nu X \\geqslant K ( h ^ { \\frac { 1 } { 2 ^ { n } } } , 2 ) ^ { r _ { n } } X ^ { \\nu } + \\sum _ { k = 0 } ^ { n - 1 } r _ { k } [ X ^ { \\frac { m _ k } { 2 ^ k } } - 2 X ^ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } + X ^ { \\frac { m _ k + 1 } { 2 ^ k } } ] . \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z } q ^ { j } J _ { j } ^ { 2 } \\left ( z ; q \\right ) = 1 , | z | < 1 , \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} & y _ { 1 } ( \\beta _ { 3 } - \\gamma _ { 5 } ) + y _ { 2 } ( \\alpha _ { 3 } + ( 1 + \\gamma _ { 5 } ) ) = \\gamma _ { 3 } - \\alpha _ { 3 } \\beta _ { 3 } d - \\gamma _ { 5 } \\alpha _ { 3 } x _ { 3 } + ( 1 + \\gamma _ { 5 } ) \\beta _ { 3 } x _ { 3 } . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} - z _ 1 ^ 3 f _ { y _ 0 } = z _ 1 f _ { z _ 1 } + z _ 2 f _ { z _ 2 } + z _ 3 f _ { z _ 3 } . \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} \\norm { J _ k ( x _ k , \\tilde { \\gamma } _ k , \\lambda _ k ) - x _ k } _ k \\leq \\frac { \\tilde { \\gamma } _ k } { \\gamma _ k } \\norm { J _ k ( x _ k , \\gamma _ k , \\lambda _ k ) - x _ k } _ k = \\frac { 1 } { \\theta } \\norm { x _ { k + 1 } - x _ k } _ k \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{gather*} \\frac { Q _ { m } ( x ) } { P _ { m } ( x ) } = \\sum _ { k \\geq 0 } \\ \\ , \\frac { s _ { k } ^ { ( m ) } } { x ^ { k + 1 } } \\ , \\ \\ \\ \\ \\ | x | > \\max \\{ \\ | z | \\ | \\ P _ { m } ( z ) = 0 \\ \\} \\ . \\end{gather*}"}
-{"id": "7318.png", "formula": "\\begin{align*} - s \\log ( p / q ) \\cdot e ^ { - s \\log ( p / q ) } = \\Theta ( \\log n ) . \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} Y _ { B , D } \\ ; : \\ ; \\begin{cases} F ( x _ 0 , \\dotsc , x _ { n + 1 } ) = 0 \\\\ \\nabla { F } ( x _ 0 , \\dotsc , x _ { n + 1 } ) \\cdot D = 0 \\\\ \\nabla { F } ( B ) \\cdot ( x _ 0 , \\dotsc , x _ { n + 1 } ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} 0 = \\int \\limits _ 0 ^ { 2 \\pi } \\int \\limits _ 0 ^ { 2 \\pi } \\left [ \\frac { \\partial f } { \\partial x _ 1 } X ( g ) - \\frac { \\partial g } { \\partial x _ 1 } X ( f ) + f Y ( g ) - g Y ( f ) \\right ] d x _ 4 d x _ 5 . \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} q ( p ^ { - 1 } ( T _ 2 \\cdot [ \\phi _ { 6 , u } ] ) ) & = ( T _ 2 \\cdot [ x ( u x + y ) ] ) \\cup ( T _ 2 \\cdot [ x ^ 2 - ( u x + y ) ^ 2 ] ) \\cup ( T _ 2 \\cdot [ ( x ^ 2 + ( u x + y ) ^ 2 ] ) = \\\\ & = \\{ x ( u x + t y ) \\} \\cup \\{ x ^ 2 - ( u x + t y ) ^ 2 \\} \\cup \\{ x ^ 2 + ( u x + t y ) ^ 2 \\} , \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int g \\ , d ( \\mathfrak { S } ^ { n } ( { \\widehat { \\mu } } ) ) _ { j } = \\int _ { [ j ] } ( g \\circ \\pi ) \\ , d \\ , \\mathbb { P } ^ { - } \\mbox { f o r e v e r y $ g \\in C ^ { 0 } ( X ) $ } . \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u + u \\partial _ x u + v \\partial _ y u + \\partial _ x P - \\partial _ y ^ 2 u = 0 , & \\\\ \\partial _ x u + \\partial _ y v = 0 , & { \\mbox i n } \\quad \\Omega , \\\\ ( u , v ) | _ { y = 0 } = 0 , \\lim \\limits _ { y \\to + \\infty } u = U ( t , x ) , \\end{cases} \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} e ( \\nu ) = \\frac { 1 } { 2 \\pi } \\int _ { M } K ^ \\perp . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} a e _ { 2 } + t ( e _ { 1 } + a e _ { 3 } ) = ( 1 - a ) ( t e _ { 1 } ) + a ( t e _ { 1 } + e _ { 2 } + t e _ { 3 } ) , \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v + \\nabla _ H p ( \\textbf { x } ^ H , t ) - \\Delta v + f _ 0 k \\times v = 0 , \\\\ \\nabla _ H \\cdot \\left ( \\int _ { - h } ^ h v ( x , y , z , t ) d z \\right ) = 0 , \\\\ w ( x , y , z , t ) = - \\nabla _ H \\cdot \\left ( \\int _ 0 ^ z v ( x , y , \\xi , t ) d \\xi \\right ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} \\begin{cases} \\mathcal { L } u = 0 & \\ \\Omega \\\\ [ 0 . 5 e x ] u = g & \\ \\Omega , \\end{cases} \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} [ ( n , x ) ] ^ { - 1 } = [ ( n ^ * , \\alpha _ n ( x ) ) ] . \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} X = \\begin{bmatrix} U & U _ { \\perp } \\\\ \\end{bmatrix} \\cdot \\begin{bmatrix} \\Sigma _ 1 & 0 \\\\ 0 & \\Sigma _ 2 \\\\ \\end{bmatrix} \\cdot \\begin{bmatrix} V ^ { \\intercal } \\\\ V _ { \\perp } ^ { \\intercal } \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} C _ n ( 0 ) = \\mu _ 0 ( n ) ( \\alpha _ n - 1 ) + \\log ( 1 + 2 ^ { \\mu _ 0 ( n ) } ) - H ( \\alpha _ n ) , ~ C _ n ( 1 ) = \\mu _ 1 ( n ) ( \\beta _ n - 1 ) + \\log ( 1 + 2 ^ { \\mu _ 1 ( n ) } ) - H ( \\beta _ n ) . \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} \\dot { x } = F ( \\sigma , \\mu ) + \\xi ( y , \\sigma , \\mu ) , \\dot { y } = \\sigma + \\eta ( y , \\sigma , \\mu ) \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} \\| \\psi _ i \\| _ 2 = 1 , ( \\Delta + \\lambda _ i ^ 2 ) \\psi _ i = 0 , \\quad \\| \\psi _ i \\| _ \\infty \\gg \\lambda _ i ^ \\delta . \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} \\delta _ F + \\delta _ E = \\frac { M + K - 1 } { M } + \\frac { K } { M r } , \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} \\lambda : = [ c _ 0 : \\cdots : c _ { k } ] \\in { \\mathbb C } P ^ { k } . \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\tilde { \\kappa } _ { ( r ) , B } = \\sum _ { i = 0 } ^ { r } I _ i ( B ) \\ \\kappa _ { ( r - i ) } \\cdot \\kappa _ { ( 1 ) } ^ i . \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} d _ 2 ( f \\otimes e _ 1 \\wedge \\ldots \\wedge e _ n ) = { \\textstyle \\sum } _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } f ( \\phi ( e _ i ) ) e _ 1 \\wedge \\ldots \\wedge { \\widehat { e _ i } } \\wedge \\cdots \\wedge e _ n \\ , . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} \\sum _ { m , s , j \\atop 2 m + 2 j + s = p - r } \\overline a _ { m r s j } Q ^ m Q _ { L ( F ) } ^ j u ^ s + \\sum _ { m , s , j \\atop 2 m + 2 j + s + 2 = p - r } b _ { m r s j } Q ^ m v _ F ^ { 2 j + 1 } ( v _ F \\times u ) u ^ s = 0 \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} W _ i = V _ i \\otimes _ k R _ { i + 1 } \\cap R _ i \\otimes _ k V _ { i + 3 } \\subset V _ i \\otimes _ k V _ { i + 1 } \\otimes _ k V _ { i + 2 } \\otimes _ k V _ { i + 3 } , \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} w _ K = 2 h _ K . \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} K ( h ^ { \\frac { 1 } { 2 ^ { n } } } , 2 ) ^ { r _ { n } } A \\sharp _ { \\nu } B & \\leqslant A \\nabla _ { \\nu } B - \\sum _ { k = 0 } ^ { n - 1 } r _ { k } [ A \\sharp _ { \\frac { m _ k } { 2 ^ k } } B - 2 A \\sharp _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } B + A \\sharp _ { { \\frac { m _ k + 1 } { 2 ^ k } } } B ] \\\\ & \\leqslant K ( h ^ { \\frac { 1 } { 2 ^ { n } } } , 2 ) ^ { R _ { n } } A \\sharp _ { \\nu } B , \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\sqrt { d e t g _ { i j } } \\frac { \\partial u } { \\partial t } - \\frac { \\partial } { \\partial x _ { i } } ( g ^ { i j } \\sqrt { d e t g _ { i j } } \\frac { \\partial u } { \\partial x _ { j } } ) = F = f \\sqrt { d e t g _ { i j } } . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\dim \\mathbb { P } \\mathrm { H o m } ( F , C , L ) _ e = 4 d _ 1 d _ 2 - 1 . \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} E _ { s , k } = \\eta _ { s , k } ( 1 - \\rho _ { s , k } ) \\bigg [ \\mathbf { h } _ { s , k } ^ { H } \\mathbf { q } \\mathbf { q } ^ { H } \\mathbf { h } _ { s , k } + \\mathbf { h } _ { s , k } ^ { H } \\mathbf { V } \\mathbf { h } _ { s , k } + \\sigma _ { s a , k } ^ { 2 } \\bigg ] \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} \\norm { \\partial _ t ^ i D ^ j v } { H ^ { k - m , 2 k - 2 m } ( D _ T ) } & \\le \\sum _ { \\ell = 1 } ^ { k } \\norm { \\partial _ t ^ { \\ell } v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 m - 2 \\ell + 2 i + j } ( D ) ) } \\\\ & \\le \\sum _ { \\ell = 1 } ^ { k } \\norm { \\partial _ t ^ { \\ell } v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 \\ell } ( D ) ) } \\le | v | _ { H ^ { k , 2 k } ( D _ T ) } . \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} \\Phi = \\Phi _ \\mathrm r , \\Psi = 0 , G = \\left [ \\begin{array} { c c } I _ { n - 1 } & 0 \\end{array} \\right ] , F = \\left [ \\begin{array} { c c } J & \\beta _ { n - 1 } e _ { n - 1 } \\end{array} \\right ] , \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} \\Lambda = 1 , M = 3 2 , \\Lambda _ r = \\begin{cases} 0 . 2 8 6 + 0 . 0 0 1 r & 1 \\leq r \\leq 1 4 , \\\\ 0 . 3 0 0 + 0 . 0 2 5 ( r - 1 4 ) & 1 5 \\leq r \\leq 2 2 , \\\\ 0 . 5 + 0 . 0 5 ( r - 2 2 ) & 2 3 \\leq r \\leq 3 2 , \\end{cases} \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} G ( x _ 0 ) = \\sup _ { \\mathbb { S } ^ n } G , \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} \\| x ( t ) \\| ^ 2 - \\| h \\| ^ 2 = \\int _ 0 ^ t \\langle \\tilde { u } , \\tilde { u } \\rangle - \\int _ 0 ^ t \\langle \\tilde { y } , \\tilde { y } \\rangle . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} d _ j ^ { t r } d _ j = g _ j ^ { t r } g _ j , 2 \\leq j \\leq 4 . \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} \\left \\langle Z _ { a b } , Z _ { c d } , \\bar { Z } _ { e } \\right \\rangle & = \\frac { 1 } { \\sqrt { 2 } } \\left \\langle Z _ { a b } , Z _ { c d } , P _ { e } \\right \\rangle - \\frac { 1 } { \\sqrt { 2 } } \\left \\langle Z _ { a b } , Z _ { c d } , Z _ { e } \\right \\rangle , \\\\ & = \\left ( \\alpha _ { 2 } - \\alpha _ { 3 } \\right ) \\ , \\varepsilon _ { a b c d e } , \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} a = \\lim _ { t \\to 1 } ( t - 1 ) ^ { 2 } \\Psi ( t ) = \\frac { 1 } { \\left ( q ; q \\right ) _ { \\infty } } \\ , _ { 0 } \\phi _ { 0 } \\left ( - ; - ; q , q \\xi ^ { - 1 } \\right ) = \\frac { \\left ( q \\xi ^ { - 1 } ; q \\right ) _ { \\infty } } { \\left ( q ; q \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} K ^ \\perp & = 2 \\langle A ^ \\circ _ { 1 1 } \\wedge A ^ \\circ _ { 1 2 } , N _ 1 \\wedge N _ 2 \\rangle = 2 \\left ( \\langle A ^ \\circ _ { 1 1 } , N _ 1 \\rangle \\langle A ^ \\circ _ { 1 2 } , N _ 2 \\rangle - \\langle A ^ \\circ _ { 1 1 } , N _ 2 \\rangle \\langle A ^ \\circ _ { 1 2 } , N _ 1 \\rangle \\right ) \\\\ & = - 2 \\abs { A ^ \\circ _ { 1 1 } } ^ 2 = - 2 \\abs { A ^ \\circ _ { 1 1 } } \\abs { A ^ \\circ _ { 1 2 } } . \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} f \\smile g = ( \\mu \\bullet _ 1 f ) \\bullet _ { m + 1 } g . \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} \\overline { 0 } = ( - 1 ) ^ { n } \\left ( \\overline { g } \\smile \\overline { f } - ( - 1 ) ^ { m n } \\overline { f } \\smile \\overline { g } \\right ) , \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} \\mathcal { A } _ { c } = \\{ \\phi \\in [ 0 , \\pi ] \\mid 2 \\cos \\phi \\in [ - 2 , 2 ] \\cap \\mathcal { A } \\} \\quad \\mbox { a n d } \\mathcal { A } _ { d } = \\{ m > \\lfloor \\log | \\alpha | \\rfloor \\mid \\alpha ^ { - 1 } q ^ { m } + \\alpha q ^ { - m } \\in \\mathcal { A } \\} . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} z = f ( x , y ) , \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} \\ln Z _ { C F T } = - \\frac { 1 } { 2 } \\ln \\det D = \\frac { 1 } { 2 } [ \\zeta ' ( 0 , D ) + \\tau \\zeta ( 0 , D ) ] \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} & A _ n ( p , q ) + a A _ m ( p , q ) + b A _ k ( p , q ) = 0 , \\\\ & c = - B _ n ( p , q ) - a B _ m ( p , q ) - b B _ k ( p , q ) . \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} V _ N ( 0 , i t , k ) \\ll \\begin{cases} \\frac { l ^ { 1 / 2 } } { N } d _ { 0 } ^ { \\epsilon } \\max { \\left ( \\frac { \\sqrt { T } } { k } , \\frac { 1 } { \\sqrt { k } } \\right ) } , & d _ 0 \\geq 1 \\\\ \\frac { 1 } { \\sqrt { l T } } \\left ( \\frac { d _ 0 } { 2 } \\right ) ^ k , & d _ 0 < 1 . \\end{cases} \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} f ( 2 m _ 0 k ) = f ( - 2 m _ 0 k ) . \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} \\Psi ( x + \\omega ) = e ^ { i \\left \\langle t , \\omega \\right \\rangle } \\Psi ( x ) , \\ \\forall \\omega \\in \\Omega , \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} u \\left ( t \\right ) = & P _ p ( T - t ) \\Phi + \\int _ { t } ^ { T } P _ p ( r - t ) \\left ( \\nabla u \\left ( r \\right ) b \\left ( r \\right ) \\right ) d r \\\\ & + \\int _ { t } ^ { T } P _ p \\left ( r - t \\right ) f \\left ( r , u ( r ) , \\nabla u ( r ) \\right ) d r , \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} { \\mathcal H _ a ( \\xi ) } : = { 2 \\over n - 1 } { \\partial _ \\nu a ( \\xi ) \\over a ( \\xi ) } - H ( \\xi ) . \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} Z ( t ) = \\frac { \\log ^ + \\log \\abs { X _ t } } { \\log t } , t > 1 . \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} - { \\hbar ^ 2 \\over 2 m } \\nabla _ { g } ^ 2 \\psi ( x ) - \\sum _ { j = 1 } ^ { N } \\lambda _ j \\delta _ { g } ( x , a _ j ) \\psi ( x ) = E \\psi ( x ) \\ ; , \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} n e _ n & = \\sum \\limits ^ d _ { i = 1 } \\sum \\limits ^ \\infty _ { r = 0 } i \\cdot e _ { n - i p ^ r } a _ i ^ { p ^ r } \\big ( \\sum \\limits _ { j = 1 } ^ \\ell ( c _ j \\pi _ j ) ^ { p ^ r } \\big ) \\\\ & \\equiv \\sum \\limits ^ d _ { i = 1 } \\sum \\limits ^ \\infty _ { r = 0 } i \\cdot e _ { n - i p ^ r } a _ i ^ { p ^ r } \\big ( \\sum \\limits _ { j = 1 } ^ \\ell c _ j \\pi _ j \\big ) ^ { p ^ r } \\pmod { p } . \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{gather*} \\mathcal { F } _ c = { : } \\exp \\left ( \\sum _ { \\mu = 1 } ^ n ( x _ \\mu \\otimes 1 ) \\big ( \\Delta _ { \\hat { S } ( \\gg ^ * ) } - \\Delta _ 0 \\big ) ( \\partial ^ \\mu ) \\right ) { : } . \\end{gather*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\begin{cases} ( n ) & q \\geq n , \\\\ ( q , q , \\dots , q , s ) ( 0 \\leq s \\leq n - 1 ) & q < n . \\end{cases} \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} C _ { } & = \\lim _ { N \\rightarrow \\infty } \\sup _ { \\{ p ( x _ t | s _ { t - 1 } , y ^ { t - 1 } ) \\} _ { t = 1 } ^ N } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N I ( X _ i , S _ { i - 1 } ; Y _ i | Y ^ { i - 1 } ) . \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} \\tilde { f } ( y ) & = - \\tilde { a } ^ { i j } ( y _ 0 ) \\tilde { E } ^ { y _ 0 , i j } _ 1 ( y ) - \\hat E ^ { y _ 0 , i j } ( y ) G ^ { i j } ( v _ { y _ 0 } ) + g ( y ) . \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} A = A _ { 1 } \\sqcup A _ { 2 } , B = B _ { 1 } \\sqcup B _ { 2 } , \\ , \\ A _ 1 \\sqcup B _ 1 = [ 1 , v _ 1 ' ] , A _ 2 \\sqcup B _ 2 = [ v _ 1 ' + 1 , v _ 1 ' + v _ 2 ' ] . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} \\deg \\alpha = 2 \\sum _ { a \\geq b > 1 } ^ { } n _ { ( a , b , 1 ) } + 4 n _ { d - 2 , 1 , 1 } . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} v _ { e a } ^ 2 ( g _ 1 ) - v _ { e a } ^ 2 ( g _ 2 ) = \\left [ 6 p - 2 , 6 p - 2 \\right ] ^ T . \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} J u \\left [ x \\right ] = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( \\varphi ( x , \\xi ) - y \\cdot \\xi ) } u [ y ] d \\xi , \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} \\phi _ { 0 } ^ { ( \\alpha ) } ( k ) = \\exp \\left ( - \\frac { 2 | k | ^ { \\alpha / 2 + 1 } } { \\alpha + 2 } \\right ) \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} \\alpha _ k = \\alpha _ { k , K ^ { * } } & \\leq \\frac { 1 } { \\binom { k - 1 } { l - 1 } } \\sum _ { \\{ L _ i \\subseteq K ^ { * } , \\ ; | L _ i | = l \\} } \\alpha _ { l , L _ i } \\\\ & \\leq \\frac { \\binom { k } { l } } { \\binom { k - 1 } { l - 1 } } \\alpha _ l = \\alpha _ l \\cdot \\frac { k } { l } . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} \\phi _ 7 = ( 0 1 ) ( 2 3 ) ( 4 5 6 ) ( 7 ) ( 8 ) ( 9 ) ( 1 0 ) \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } v ( t , x _ 0 ) = 1 . \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} u _ { x ^ { i } } = \\frac { \\partial u } { \\partial x ^ { i } } = D _ { ^ { i } } u , D _ { x } ^ { \\mathfrak { a } } u = D _ { 1 } ^ { \\mathfrak { a } _ { 1 } } \\cdots D _ { d } ^ { \\mathfrak { a } _ { d } } u , \\quad | \\mathfrak { a } | = \\mathfrak { a } _ { 1 } + \\cdots + \\mathfrak { a } _ { d } . \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} \\tilde { P } ^ x \\left ( \\tilde { X } ( \\tau _ { D _ + } ) \\in B \\right ) = \\int _ { D _ + } \\tilde { G } _ { D _ + } ( x , y ) \\int _ B \\tilde { \\nu } ( y , z ) \\ , d z \\ , d y . \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} B \\dot { w } ( t ) + A ( { w } ( t ) - { u } _ 0 ) = f ( t ) - A u _ 0 , \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} A ( \\gamma ) e ^ { i \\left \\langle b + t , x \\right \\rangle } = \\sum _ { \\gamma _ { 1 } \\in \\Gamma ( k + ) } \\frac { q _ { \\gamma _ { 1 } } e ^ { i \\left \\langle b + \\gamma _ { 1 } + t , x \\right \\rangle } } { \\mid \\gamma + t \\mid ^ { 2 } - \\mid b + \\gamma _ { 1 } + t \\mid ^ { 2 } } . \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} P ( D ) = \\int \\mathbf { 1 } _ { \\{ D ( \\mathbf { x } , \\phi ) = D \\} } d ( \\mu \\mathbf { x } ) D \\in \\mathcal { D } _ n . \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{gather*} \\omega _ { ( \\underline k _ { n - 1 } , \\underline k _ { n - 2 } , \\dots , \\underline k _ { 0 } ) } = M _ { n - 1 } ( \\underline k _ { n - 1 } ) M _ { n - 2 } ( \\underline k _ { n - 2 } ) \\cdots M _ { 0 } ( \\underline k _ { 0 } ) v _ { 0 } ^ { ( n ) } . \\end{gather*}"}
-{"id": "10034.png", "formula": "\\begin{align*} ( i - \\delta ) ( i - \\ell ) \\begin{cases} = 0 \\ ; & \\mbox { i f } i = \\delta , \\\\ < 0 \\ ; & \\mbox { i f } \\delta < i < \\ell , \\\\ = 0 \\ ; & \\mbox { i f } i = \\ell . \\end{cases} \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} \\{ f ^ * A _ 2 , \\cdots , f ^ * A _ l , H : = f ^ * A _ 0 \\} \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} \\mathbf { \\tilde { H } ^ * _ n } = \\mathbf { F _ n ^ * } - \\sqrt { n } \\hat { \\sigma } _ n \\mathbf { \\bar { \\Sigma } _ n ^ { - 1 / 2 } } ( Z _ { n } ^ * - \\bar { Z } _ { n } ^ * ) ( ( \\mathbf { E _ * } L _ n ^ * ) ^ { - 1 } \\Delta _ n ^ * ) + R ^ * _ { 5 n } \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} u ( x , T ) = u ^ 1 ( x ) v ( x , T ) = v ^ 1 ( x ) ? \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} p ( \\sigma _ { i } ( g ) \\sigma _ { i } ( h ) ( j ) ) = p ( \\sigma _ { i } ( g h ) ( j ) ) = p _ { g h , r } ( \\sigma _ { i } ( g h ) ( j ) ) = \\sigma _ { i } ( g h ) p _ { e , r } ( j ) & = \\sigma _ { i } ( g ) \\sigma _ { i } ( h ) p _ { e , r } ( j ) \\\\ & = \\sigma _ { i } ( g ) p _ { h , r } ( \\sigma _ { i } ( h ) ( j ) ) . \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} q = \\sum _ { \\alpha = 0 } ^ { 2 k + 3 } f _ \\alpha ( x , y ) z ^ \\alpha . \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} D _ t w ( t ) = \\lambda ( t ) w ( t ) , \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} ( \\mathcal { F } \\Delta \\mathcal { F } ^ { - 1 } f ) ( \\xi ) = f ( \\xi ) \\sum \\limits _ { i = 1 } ^ d ( 2 - 2 \\cos ( \\xi _ i ) ) , \\ ( \\mathcal { F } u ) ( \\xi ) : = \\sum \\limits _ { n \\in \\Z ^ d } u ( n ) e ^ { \\i n \\cdot \\xi } ( 2 \\pi ) ^ { - d / 2 } . \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} \\psi ( b c _ 1 b ^ * c _ 2 b c _ 1 ^ * b ^ * c _ 2 ^ * ) = | \\psi ( c _ 1 ) | ^ 2 \\psi ( c _ 2 c _ 2 ^ * ) + | \\psi ( c _ 2 ) | ^ 2 \\psi ( c _ 1 c _ 1 ^ * ) - | \\psi ( c _ 1 ) | ^ 2 | \\psi ( c _ 2 ) | ^ 2 . \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} G : = \\{ g , x ^ n , x ^ { n - 1 } y ^ 3 , x ^ { n - 2 } y ^ 4 , \\cdots , x ^ 3 y ^ { n - 1 } , y ^ { n } \\} \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} \\sum _ { t ^ 2 \\le 4 n } H ( 4 n - t ^ 2 ) \\ = \\sum _ { n = a d \\atop a , \\ , d > 0 } \\max ( a , d ) \\ ; . \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{gather*} \\partial _ \\nu \\big ( \\phi ^ { - 1 \\gamma } _ \\mu \\partial _ \\rho \\phi ^ \\rho _ \\gamma \\big ) = \\partial _ \\mu \\big ( \\phi ^ { - 1 \\gamma } _ \\nu \\partial _ \\rho \\phi ^ \\rho _ \\gamma \\big ) . \\end{gather*}"}
-{"id": "729.png", "formula": "\\begin{align*} P Q & = Q P = \\left ( \\mathbf { F } \\cdot \\mathbf { G } \\right ) I , \\\\ \\det P & = \\det Q = - \\left ( \\mathbf { F } \\cdot \\mathbf { G } \\right ) ^ { 2 } \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} a _ { n + 1 } = b a _ n + d a _ { n - 1 } , ( b , d ) = 1 , a _ 0 = 0 , a _ 1 = 1 \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\Phi _ \\lambda ( x , y , t ) : = \\frac { \\lambda } { 2 } | x - y | ^ 2 \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} & \\ \\Phi ( x , y ) \\\\ = & \\ \\int \\int \\Phi ( u , v ) d u d v \\\\ + & \\ \\sum _ { \\alpha \\in F ^ \\times } \\psi ( \\alpha x ) \\left ( \\int \\int \\Phi ( u , v ) \\psi ( - \\alpha u ) d u d v \\right ) \\\\ + & \\ \\sum _ { \\beta \\in F ^ \\times } \\psi ( \\beta y ) \\left ( \\int \\int \\Phi ( u , v ) \\psi ( - \\beta v ) d u d v \\right ) \\ , . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { C l - S f } } = \\frac { K } { \\min \\{ M , K \\} } + \\frac { K } { M r } , \\end{align*}"}
-{"id": "10148.png", "formula": "\\begin{align*} d ( x _ k , \\mathrm { A r g } \\min f ) & \\leq \\| x _ k - u _ { k + 1 } \\| \\leq \\| x _ { k + 1 } - u _ { k + 1 } \\| + \\| x _ { k + 1 } - x _ k \\| \\\\ & = d ( x _ { k + 1 } , \\mathrm { A r g } \\min f ) + h \\| \\nabla f ( x _ k ) \\| , ~ k \\geq 0 . \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} T _ { ( A , 2 ) } = - T _ { ( A , 0 ) } , T _ { ( A , 3 ) } = - T _ { ( A , 1 ) } \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} \\rho ( x , y ) = \\rho _ { \\tilde h } ( \\tau ^ { - 1 } _ i ( x ) , p _ 2 ) + \\rho _ { \\tilde h } ( p _ 2 , \\tau ^ { - 1 } _ j ( y ) ) + ( j - i ) \\rho _ { \\tilde h } ( p _ 1 , p _ 2 ) . \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} a _ { i _ { 1 } , \\ldots , i _ { r } } : = \\left \\{ \\begin{array} [ c ] { l l } 1 , & \\left \\{ i _ { 1 } , \\ldots , i _ { r } \\right \\} \\in E \\left ( G \\right ) \\\\ 0 , & \\end{array} \\right . \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} \\phi _ a ( x ^ { b ^ n } ) = \\phi _ a ( x ) - n \\ , \\phi _ c ( x ) \\phi _ c ( x ^ { b ^ n } ) = \\phi _ c ( x ) \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} \\prod _ { i < j } ( x _ i - x _ j ) ^ { - 1 } \\left [ \\prod _ { k = 1 } ^ n ( z _ k ^ { - 1 } + t _ n x _ n ) \\det _ { 1 \\leq i , j \\leq n } \\left \\{ ( 1 - x _ i z _ j ) ^ { - 1 } \\right \\} \\right ] \\ , \\Big \\vert _ { \\boldsymbol { z } ^ { \\lambda } } = \\prod _ { k = 1 } ^ n ( x _ k + t _ n x _ n ) s _ { \\lambda - \\rho } ( \\boldsymbol { x } ) . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} u _ { i } ( \\cdot , 0 ) = u _ { i } ^ { 0 } , \\dot { u } _ { i } ( \\cdot , 0 ) = u _ { i } ^ { 1 } , \\tau ( \\cdot , 0 ) = \\tau ^ { 0 } , T ( \\cdot , 0 ) = T ^ { 0 } \\Omega . \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} \\mathbf { r } ^ { \\prime } = \\mathbf { r } + \\left [ \\left ( \\gamma - 1 \\right ) \\frac { \\mathbf { v } \\cdot \\mathbf { r } } { v ^ { 2 } } - \\gamma t \\right ] \\mathbf { v } , t ^ { \\prime } = \\gamma \\left ( t - \\frac { \\mathbf { v } \\cdot \\mathbf { r } } { c ^ { 2 } } \\right ) , \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} D ( G _ 1 + G _ 2 , x ) = \\Big ( ( 1 + x ) ^ { n _ 1 } - 1 \\Big ) \\Big ( ( 1 + x ) ^ { n _ 2 } - 1 \\Big ) + D ( G _ 1 , x ) + D ( G _ 2 , x ) . \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} X & : = E ^ \\infty \\cup \\{ \\mu \\in { \\rm P a t h } ( E ) \\mid r ( \\mu ) \\} \\cup \\{ \\mu \\in { \\rm P a t h } ( E ) \\ \\vert \\ r ( \\mu ) \\in \\operatorname { I n f } ( E ) \\} \\\\ G _ E & : = \\{ ( \\alpha x , | \\alpha | - | \\beta | , \\beta x ) \\ \\vert \\ \\alpha , \\beta \\in { \\rm P a t h } ( E ) , x \\in X , r ( \\alpha ) = r ( \\beta ) = s ( x ) \\} . \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} w ( x , y , z , t ) = - \\int _ { - h } ^ z \\nabla _ H \\cdot v ( x , y , z ' , t ) d z ' , \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} F _ { ; i j } = F _ { : i j } - \\{ \\varGamma _ { i j } ^ k - \\tilde { \\varGamma } _ { i j } ^ k \\} F _ k . \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} t ' _ k = 2 ( \\langle y ' , x ' _ k \\rangle - \\langle x ' , y ' _ k \\rangle ) = t _ k + 2 ( \\langle y _ k , x ' _ k \\rangle - \\langle x _ k , y ' _ k \\rangle ) . \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\frac { d \\tilde { u } } { d t } = \\varPhi \\tilde { v } ^ { - 1 } + \\varPhi ^ m \\tilde { h } _ m ^ { k } \\tilde { u } _ k , \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} \\sum _ { \\substack { I \\in S _ \\phi \\\\ I \\neq X } } \\mu ( I ) y _ I ^ p = \\sum _ { I \\in S _ \\phi } \\mu ( I ) y _ I ^ p - y _ X ^ p , \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} [ H _ 3 , \\bar { x } ^ + _ { i , r } ] = \\bar { x } ^ + _ { i , r + 1 } , \\ [ H _ 4 , \\bar { x } ^ + _ { i , r } ] = \\bar { x } ^ + _ { i , r + 2 } , \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} K _ { \\lambda ^ \\prime / \\nu } F = \\displaystyle \\bigoplus _ { | \\mu | = | \\lambda ^ \\prime | - | \\nu | } c ^ { \\lambda ^ \\prime } _ { \\mu \\nu } K _ \\mu F , \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} u ( t ; x , \\eta ) : = x \\cdot \\eta + \\int _ 0 ^ t \\{ h _ \\rho - x \\cdot \\nabla _ x h _ \\rho \\} ( \\tau , q ( \\tau , 0 ; x , \\eta ) , p ( \\tau , 0 ; x , \\eta ) ) d \\tau . \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} \\overline { \\gamma } _ { 2 n + 1 } = \\sum \\limits _ { i = 0 } ^ { n } { n \\brace i } \\overline { \\gamma } _ { 2 i } , \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} M \\xi \\left ( z \\right ) = \\sum _ { j = 1 } ^ { J } \\xi _ { j } \\varphi _ { z _ { j } } \\left ( z \\right ) , \\ ; \\ ; \\ ; \\ ; \\ ; z \\in \\mathbb { D } , \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\delta \\in \\{ 0 , 1 \\} ^ { n } , \\\\ \\delta _ { k } = 0 , \\ ; \\forall k \\neq p _ { 1 } , \\ldots p _ { N + 1 } } } u _ { \\delta } ( x _ { j } ) = w _ { 1 } + w _ { 2 } g _ { j } ( x _ { j } ) \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} 0 < c _ { - } \\leq V ^ { \\prime \\prime } \\left ( x \\right ) \\leq c _ { + } < \\infty V ^ { \\prime \\prime } V ^ { \\prime \\prime } \\left ( 0 \\right ) = 1 . \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} L _ f ( x ) = \\limsup _ { r \\to 0 } \\frac { L _ f ( x , r ) } { r } l _ f ( x ) = \\liminf _ { r \\to 0 } \\frac { l _ f ( x , r ) } { r } . \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} \\begin{aligned} f _ \\infty ^ { ( s ) } ( t ) & = \\sum _ { k = 0 } ^ \\infty \\ell ^ { - k } \\\\ & \\times \\int _ 0 ^ t \\int _ 0 ^ { t _ 1 } \\dots \\int _ 0 ^ { t _ { k - 1 } } T _ s ^ 0 ( t - t _ 1 ) C _ { s + 1 } ^ 0 \\dots T _ { s + k } ^ 0 ( t _ k ) f _ \\infty ^ { ( s + k ) } ( 0 ) d t _ k \\dots d t _ 1 \\end{aligned} \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } u _ j ( x ) = u ( x ) x \\in \\R ^ n . \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} Q _ g ^ 6 = & \\frac { n ^ 4 - 2 0 n ^ 2 + 6 4 } { 3 2 n ^ 2 ( n - 1 ) ^ 3 } R _ g ^ 3 , \\\\ - P _ g ^ 6 = & \\Delta _ g ^ 3 + \\frac { - 3 n ^ 2 + 6 n + 3 2 } { 4 n ( n - 1 ) } R _ g \\Delta _ g ^ 2 + \\frac { 3 n ^ 4 - 1 2 n ^ 3 - 5 2 n ^ 2 + 1 2 8 n + 1 9 2 } { 1 6 n ^ 2 ( n - 1 ) ^ 2 } R _ g ^ 2 \\Delta _ g - \\frac { n - 6 } { 2 } Q _ g ^ 6 . \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{gather*} \\pi ( g _ { C } ) = \\begin{bmatrix} 1 & 0 \\\\ C ( z ) & 1 \\end{bmatrix} , \\end{gather*}"}
-{"id": "2779.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N _ D } U _ k ^ * U _ k & = \\sum _ { k = 1 } ^ { N _ D } S _ { d _ k } S _ { d _ k } ^ * S _ { c ( k ) } ^ * S _ { c ( k ) } S _ { d _ k } S _ { d _ k } ^ * = \\sum _ { k = 1 } ^ { N _ D } S _ { d _ k } S _ { d _ k } ^ * = P _ D , \\\\ \\sum _ { k = 1 } ^ { N _ C } T _ l ^ * T _ l & = \\sum _ { l = 1 } ^ { N _ C } S _ { c _ l } S _ { c _ l } ^ * S _ { d ( l ) } ^ * S _ { d ( l ) } S _ { c _ l } S _ { c _ l } ^ * = \\sum _ { l = 1 } ^ { N _ C } S _ { c _ l } S _ { c _ l } ^ * = P _ C . \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} \\Omega _ x = - k \\alpha ^ { k + 1 } \\epsilon _ x , \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} \\frac { t } { 2 } \\ , ( \\varphi ^ 2 ) ' + \\varphi ^ 2 + 1 = c \\sqrt { \\varphi ^ 2 + 1 } . \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l } D ( S _ { \\psi } ) & = \\ \\left \\{ x \\in { \\cal H } \\ ; \\ \\lim _ { n \\rightarrow \\infty } \\sum _ { k = 0 } ^ { n } ( x | \\psi _ { k } ) \\psi _ { k } \\ { \\rm e x i s t s } \\ { \\rm i n } \\ { \\cal H } \\right \\} \\\\ \\\\ S _ { \\psi } x & = \\ \\sum _ { n = 0 } ^ { \\infty } ( x | \\psi _ { n } ) \\psi _ { n } , \\ ; \\ ; \\ ; x \\in D ( S _ { \\psi } ) . \\\\ \\end{array} \\right . \\\\ \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} \\lambda _ { 0 } v ( t ) f ( u ( t ) ) & = - a ( t ) f ( u ( t ) ) f ' ( u ( t ) ) v ( t ) - v '' ( t ) f ( u ( t ) ) \\\\ & = u '' ( t ) f ' ( u ( t ) ) v ( t ) - v '' ( t ) f ( u ( t ) ) , \\forall \\ , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} h ( z ) \\ = \\ \\frac { z } { | z | } \\rho ( | z | ) \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ { k + 1 } r ^ { k + 1 } } { k ! } \\cdot F ^ { ( k ) } ( z ) = \\sum _ { \\mathfrak { n } } \\frac { \\Lambda _ K ( \\mathfrak { n } ) \\chi ^ * ( \\mathfrak { n } ) } { \\N \\mathfrak { n } ^ { 1 + i \\tau } } \\cdot r E _ k ( r \\log \\N \\mathfrak { n } ) \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} \\Psi ( z ) = \\exp ( 2 \\pi \\sqrt { - 1 } \\left \\{ T r _ { L _ v / \\mathbb { Q } _ { p } } ( z ) \\right \\} _ { p } ) , \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} \\int _ T \\Gamma ( t ) d \\mu = \\int _ T \\mathrm { e x } \\ , \\overline { \\mathrm { c o } } ^ { \\ , \\mathit { w } ^ * } \\Gamma ( t ) d \\mu . \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} H _ { \\rm r } = \\sqrt { - \\Delta + m ^ 2 } - m + W ( x ) , \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} \\sigma _ { k } ( \\overbrace { B , \\dotsc , B } ^ { l } , C , \\dotsc , C ) & : = \\sigma _ { k } ( \\overbrace { B , \\dotsc , B } ^ { l } , \\overbrace { C , \\dotsc , C } ^ { k - l } ) , \\\\ T _ { k } ( \\overbrace { B , \\dotsc , B } ^ { l } , C , \\dotsc , C ) _ { i j } & : = T _ { k } ( \\overbrace { B , \\dotsc , B } ^ { l } , \\overbrace { C , \\dotsc , C } ^ { k - l } ) _ { i j } . \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} \\eta ( t , 0 ; x , \\xi + 2 \\pi m ) & = \\eta ( t , 0 ; x , \\xi ) + 2 \\pi m , \\\\ q ( t , 0 ; x , \\xi + 2 \\pi m ) & = q ( t , 0 ; x , \\xi ) \\end{align*}"}
-{"id": "10067.png", "formula": "\\begin{align*} 5 q < 5 + \\dfrac { 5 } { 2 } p \\leq 5 + ( 1 + 3 q ) = 6 + 3 q , \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} b _ { i j } = \\ast ( ( \\imath _ { e _ i } \\varphi ) \\wedge \\beta _ j ) . \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} L ( s , S _ f ) & : = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) } { n ^ s } , \\\\ L ( s , S _ f \\times S _ f ) : = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ f ( n ) } } { n ^ s } , & L ( s , S _ f \\times \\overline { S _ f } ) : = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) S _ f ( n ) } { n ^ s } , \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } u ( \\ell + 1 + r ( k m + i ) ) & = & \\alpha _ { i + 1 } ( u ( k m + i + 1 ) ) ^ { \\gamma } , \\\\ u ( \\ell + 1 + r ( k m + i ) + j ) & = & \\beta _ { i , j } ( 1 \\leqslant j < r ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} o s c _ { C _ { r } } u \\leq ( 1 - b ) o s c _ { C _ { 2 r } } u + 4 r ^ { 2 } | f | _ { 0 , C _ { 2 r } } \\ , \\ b = b ( n , \\beta , K ) > 0 . \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{gather*} \\det \\big ( T ^ { ( \\alpha ) } _ { k } \\big ) = \\frac 1 { k ! } \\prod _ { i = 1 } ^ { k } c ^ { ( \\alpha ) } _ { i } \\left ( \\sum _ { \\sigma } \\sigma \\det \\big ( z _ { i } ^ { i + j - 2 } \\big ) \\right ) , \\end{gather*}"}
-{"id": "6267.png", "formula": "\\begin{align*} \\beta _ j = & \\ ; ( \\imath _ { e _ k } \\varphi ) \\wedge ( \\imath _ { e _ j } \\nabla _ { e _ k } \\ast \\varphi ) - ( \\imath _ { e _ j } \\imath _ { e _ k } \\varphi ) \\wedge e ^ l \\wedge ( \\imath _ { e _ k } \\nabla _ { e _ l } \\ast \\varphi ) \\\\ & - ( \\imath _ { e _ j } \\nabla _ { e _ k } \\varphi ) \\wedge ( \\imath _ { e _ k } \\ast \\varphi ) - e ^ l \\wedge ( \\imath _ { e _ k } \\nabla _ { e _ l } \\varphi ) \\wedge ( \\imath _ { e _ j } \\imath _ { e _ k } \\ast \\varphi ) . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\inf \\left \\{ \\max _ { p = 1 , 2 } | \\alpha ( g _ p ) f - f | _ { \\infty } ^ - \\mid f \\in { \\mathcal F } ( { \\mathbb F } _ 2 ) , f ( e ) = 1 \\right \\} = 0 . \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} ( \\mathcal { A } ( G ^ { k , s } ) y ) _ { w } & = [ ( x ^ { e } ) ^ { \\frac { t } { k } } ] ^ { s } [ ( \\mu ^ { - 1 } x ^ { e } ) ^ { \\frac { 1 } { k } } ] ^ { k - t s - 1 } \\\\ & = \\mu ^ { \\frac { t s } { k } } ( \\mu ^ { - 1 } x ^ { e } ) ^ { \\frac { k - 1 } { k } } \\\\ & = \\mu ^ { \\frac { t s } { k } } y _ { w } ^ { k - 1 } . \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} \\frac { c _ 2 ( \\beta , k ) } { c _ 3 ( \\beta , k ) } = \\left ( \\frac { \\beta } { q \\beta - q r } \\right ) ^ k \\frac { q } { \\beta } ( k - ( k - 1 ) \\beta ) \\ge \\left ( \\frac { \\beta } { q \\beta - q r } \\right ) ^ k \\frac { k ( q - 1 ) + 1 } { \\beta } \\ge 6 \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} P ( D ) = \\begin{cases} \\frac { 1 } { { n ( n - 1 ) \\choose m } } , & | A ( D ) | = m \\\\ 0 , & { \\rm o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} s = \\delta ^ 2 , p _ k = \\dfrac { \\Vert Y _ k - X _ * \\Vert ^ 2 _ F } { s _ r ^ 2 } , q _ k = \\dfrac { 1 } { s _ r ^ 2 } \\sum _ { k = 1 } ^ r \\limits s _ k ^ 2 \\sin ^ 2 \\phi _ { R k } , \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} \\omega _ k = \\frac { u _ k } { \\| u _ k \\| } , \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} \\int _ { \\mathcal { V } } \\psi ( v ) \\mathcal { L } \\phi ( v ) \\ , { \\rm d } \\mu ( v ) = \\int _ { \\mathcal { V } } \\phi ( v ) \\mathcal { L } \\psi ( v ) \\ , { \\rm d } \\mu ( v ) \\mbox { f o r a l l } \\phi , \\psi \\in L ^ 2 ( \\mathcal { V } ; { \\rm d } \\mu ) . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\begin{aligned} & I _ s ( ( X _ s - ( T - t ) V _ s , V _ s ) ) + E _ s ( Z _ s ) \\geq \\\\ & \\geq \\frac { 1 } { 2 } \\left ( \\left | x _ i - ( T - t ) v _ i \\right | ^ 2 + | v _ i | ^ 2 \\right ) + E _ { s - 1 } ( Z _ s ^ { ( i ) } ) \\end{aligned} \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} A _ r ^ s ( x _ 1 , \\dots , x _ r ) : = \\begin{pmatrix} e _ { r , 1 } ^ 0 & e _ { r , 2 } ^ 0 & \\cdots & e _ { r , r } ^ 0 \\\\ e _ { r , 1 } ^ 1 & e _ { r , 2 } ^ 1 & \\cdots & e _ { r , r } ^ 1 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ e _ { r , 1 } ^ { s - 1 } & e _ { r , 2 } ^ { s - 1 } & \\cdots & e _ { r , r } ^ { s - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\begin{bmatrix} A _ 0 + I - B _ 0 & C _ 0 \\\\ A _ 0 & I - B _ 0 & C _ 0 \\\\ & \\ddots & \\ddots & \\ddots \\\\ & & A _ 0 & I - B _ 0 & C _ 0 \\\\ & & & A _ 0 & I - B _ 0 + C _ 0 \\end{bmatrix} \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} \\kappa ( x _ 0 ) : = \\lim \\limits _ { r \\rightarrow 0 _ + } \\frac { \\ln \\left ( r ^ { - \\frac { n + 1 } { 2 } } \\left \\| w \\right \\| _ { L ^ 2 ( B _ r ^ + ( x _ 0 ) ) } \\right ) } { \\ln ( r ) } . \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { p = 1 , 2 } | \\alpha ( g _ p ) f _ n - f _ n | ^ - _ { \\infty } = 0 . \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} ( x - y ) \\sum _ { k = m + 1 } ^ { n } \\varphi _ { k } ( x ) \\varphi _ { k } ( y ) = W _ { n } \\left ( \\varphi ( x ) , \\varphi ( y ) \\right ) - W _ { m } \\left ( \\varphi ( x ) , \\varphi ( y ) \\right ) \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( \\| E _ 1 \\tilde v _ k \\| ^ 2 + \\| E _ 2 \\tilde w _ k \\| ^ 2 ) = \\| E _ 1 v _ k \\| \\| E _ 2 w _ k \\| \\leq \\frac { 1 } { 2 } ( \\| E _ 1 v _ k \\| ^ 2 + \\| E _ 2 w _ k \\| ^ 2 ) . \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\norm { \\Delta v ( t ) } \\leq C ( \\tilde { B } _ { \\partial _ t ( v , \\zeta ) } ( t ) ) ^ { 1 / 2 } + \\frac { C } { \\varepsilon } B _ { H ^ 1 } ^ v ( t ) + \\norm { f } + C ( B _ { H ^ 1 } ^ { \\zeta } ( t ) ) ^ { 1 / 2 } = : B _ { H ^ 2 } ^ v ( t ) , \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} \\displaystyle A ^ * \\left ( \\begin{array} { c c } \\varphi \\\\ \\psi \\end{array} \\right ) = \\left ( \\begin{array} { c c } \\displaystyle \\partial _ { x x x } & \\frac { a b } { c } \\partial _ { x x x } \\\\ \\displaystyle a \\partial _ { x x x } & \\displaystyle \\frac { r } { c } \\partial _ { x } + \\displaystyle \\frac { 1 } { c } \\partial _ { x x x } \\end{array} \\right ) \\left ( \\begin{array} { c } \\varphi \\\\ \\psi \\end{array} \\right ) \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} e ^ { i t H _ 0 } J _ a ^ * E _ \\pm ( t - s ) & = e ^ { i t H _ 0 } J _ a ^ * J _ a e ^ { - i ( t - s ) H _ 0 } \\tilde P _ \\pm \\\\ & = e ^ { i t H _ 0 } ( J _ a ^ * J _ a - I ) e ^ { - i t H _ 0 } e ^ { i s H _ 0 } \\tilde P _ \\pm + e ^ { i s H _ 0 } \\tilde P _ \\pm . \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} T _ { n } = C \\left ( \\frac { s \\log ( p n ) } { n } + \\frac { \\log ^ { r + 1 } n } { n } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} \\sup \\left \\{ h _ \\mu ( f ) : \\mu \\in M ( X , f ) \\right \\} = \\sup \\left \\{ h _ { \\widetilde { \\mu } } ( \\widetilde { f } ) : \\widetilde { \\mu } \\in M ( \\widetilde { X } , \\widetilde { f } ) \\right \\} \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} R = J K ^ { - 1 } , K = \\partial _ { X X X } - \\partial _ X , J = - \\frac { 1 } { 2 } \\left ( \\partial _ X U + U \\partial _ X \\right ) . \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} 0 & = [ K ( \\lambda , \\varepsilon ) - K ( \\lambda , 0 ) ] \\left ( Z ^ { \\perp } ( \\lambda , \\varepsilon ) + \\mathbf { I } \\right ) \\left ( a _ { 1 } r _ { 1 } + a _ { 2 } r _ { 2 } \\right ) \\\\ & \\ \\ + \\left ( \\mathbf { I } + K ( \\lambda , 0 ) \\right ) \\left ( a _ { 1 } r _ { 1 } + a _ { 2 } r _ { 2 } \\right ) + \\left ( \\mathbf { I } + K ( \\lambda , 0 ) \\right ) \\left ( \\left ( \\mathbf { I } - \\mathbf { P } \\right ) r \\right ) . \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} A _ \\alpha = & r ^ 2 \\Delta _ 0 + 2 \\alpha r \\partial _ r + \\alpha ( \\alpha + n - 2 ) , \\\\ A _ { \\alpha , g } = & r ^ 2 \\Delta _ g + 2 \\alpha r \\partial _ r + \\alpha ( \\alpha + n - 2 ) , \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} d X _ t & = a ( t , X _ t ) d t + \\sigma ( t , X _ t ) d W _ t , t \\in [ 0 , T ] , \\\\ X _ 0 & = x _ 0 , \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} a \\cdot f \\otimes g = ( a f ) \\otimes g - a \\otimes ( f g ) \\quad f \\otimes g \\cdot a = f \\otimes a g \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} J E _ { H _ 0 } ( \\Gamma ) u = J _ a u \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} \\hom ( x , y ) = \\begin{cases} 1 & \\\\ a _ i + ( b _ i - a _ i ) \\cdot \\hom _ i \\left ( \\frac { x - a _ i } { b _ i - a _ i } , \\frac { y - a _ i } { b _ i - a _ i } \\right ) & \\\\ y & \\end{cases} \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} \\tilde { \\rho } ( t ) = 2 c \\tilde { \\rho } _ - ( t ) \\leq 2 c \\tilde { \\Theta } . \\end{align*}"}
-{"id": "9947.png", "formula": "\\begin{align*} E _ { 0 } = \\{ \\vect { 0 } \\} E _ i : = \\{ c _ 1 \\vect { e } _ 1 + \\cdots + c _ i \\vect { e } _ i : c _ 1 , \\dots , c _ i \\in \\Z _ { \\geq 0 } \\} , \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\psi ( t , x ) d x = 1 - e ^ { - t } . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} ( 2 \\alpha ) ^ { \\lambda } \\sum _ { k = 1 } ^ { c ( u ^ n ) } \\exp _ 2 \\{ ( \\lambda + 1 ) \\log c _ k ( x ^ n | u ^ n ) \\} , \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} P _ { 0 } ( 1 ) = \\frac { C _ { \\lambda , \\mu } } { \\gamma } . \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} r = 0 , - t u - s l = 0 , \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} \\varphi _ { \\zeta } ( t ) : = \\norm { \\nabla _ H \\zeta } ^ 4 + \\norm { \\nabla \\zeta } _ { L _ z ^ 2 L _ { x y } ^ 3 } ^ { 6 } + 1 \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} \\zeta ( z ; \\alpha , \\beta , \\gamma ) = \\frac { 3 - \\alpha } { 2 } - \\frac { 2 z - 1 } { 2 \\sqrt { z ^ 2 - z + 1 } } - \\frac { ( 1 - \\alpha ) ^ 2 z + \\alpha \\gamma - \\beta } { 2 \\sqrt { [ ( \\gamma - \\beta ) z + \\alpha + \\beta ] ^ 2 - 4 \\gamma \\beta z ( 1 - z ) } } \\end{align*}"}
-{"id": "10097.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ 2 } { x z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ 2 } { x ^ 3 z } \\\\ \\end{gather*}"}
-{"id": "3486.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { N _ R - r } a _ { p } c _ { p } = \\det ( \\mathbf { H } _ { \\bar { \\mathcal { R } } } ) . \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} F _ { \\beta _ { t } } F _ { \\beta _ { l + 1 } } - q ^ { - ( \\beta _ t , \\beta _ { l + 1 } ) } F _ { \\beta _ { l + 1 } } F _ { \\beta _ { t } } = \\sum _ { n _ { l + 2 } , \\ldots , n _ { t - 1 } \\geq 0 } c ( n _ { l + 2 } , \\ldots , n _ { t - 1 } ) F _ { \\beta _ { l + 2 } } ^ { n _ { l + 2 } } \\ldots F _ { \\beta _ { t - 1 } } ^ { n _ { t - 1 } } . \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} M _ X = \\mu \\otimes \\mu . \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} f ^ * h = c g . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} ( \\Lambda ( \\xi , \\eta ) \\tilde { y } ) ( s ) = ( \\Lambda ( \\xi , \\eta ) ( \\tilde { u } ) ( s ) + i \\Phi x ( \\xi s + \\eta ) . \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} \\mathbf { A } _ 0 = \\left [ \\begin{matrix} \\mathbf { G } _ 1 & \\mathbf { 0 } \\\\ \\mathbf { 0 } & \\mathbf { G } _ 2 \\end{matrix} \\right ] . \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} \\bigcap _ n m _ 1 ^ n \\subseteq \\bigcap _ l ( m ^ l \\otimes _ k k ) = ( \\bigcap m ^ l ) \\otimes _ k k = 0 \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} & \\frac { K N T ^ { K N ( K N + K M ) } + K N M T _ { \\sf d } ^ { K M ( K N + K M ) } } { \\frac { 1 } { \\lambda _ 1 } ( T + 1 ) ^ { K N ( K M + K N ) } } \\\\ & = \\frac { K N T ^ { K N ( K N + K M ) } + K N M \\left ( \\left ( \\frac { \\lambda _ 2 } { \\lambda _ 1 M } ^ { \\frac { 1 } { K M ( K N + K M ) } } \\right ) T ^ { \\frac { N } { M } } - 1 \\right ) ^ { K M ( K N + K M ) } } { \\frac { 1 } { \\lambda _ 1 } ( T + 1 ) ^ { K N ( K M + K N ) } } \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} \\iota ^ * : \\tilde { H } ^ { * } _ H ( X ) \\to \\tilde { H } ^ * _ H ( X ^ { S ^ 1 } ) = H ^ { * + s } ( B H ) . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} f _ { z _ 2 } = f _ { z _ 3 } = 0 , ~ ~ \\phi ( S ^ 2 ) \\cap V _ 1 . \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} W = \\{ ( i , j ) : 1 \\leq i \\leq r , \\ ; 1 \\leq j \\leq l ( \\lambda ^ { ( i ) } ) \\} \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} t \\left ( \\tilde { t } \\right ) = \\int _ 0 ^ { \\tilde { t } } \\phi \\left ( \\tau \\right ) \\ , d \\tau . \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\prod _ { v \\in S _ N } O _ \\gamma ( { \\bf 1 } _ { K _ v ( N ) K _ { H _ v } } ) \\prod _ { \\substack { v \\in S _ \\gamma \\\\ v \\notin S _ N } } O _ \\gamma ( { \\bf 1 } _ { K _ v } ) & \\ll N ^ { - \\delta } D _ { S _ N } ( \\gamma ) ^ { - C } \\prod _ { v \\in S _ N } O _ \\gamma ( { \\bf 1 } _ { K _ v } ) \\prod _ { \\substack { v \\in S _ \\gamma \\\\ v \\notin S _ N } } O _ \\gamma ( { \\bf 1 } _ { K _ v } ) \\\\ & = N ^ { - \\delta } D _ { S _ N } ( \\gamma ) ^ { - C } \\prod _ { v \\in S _ \\gamma } O _ \\gamma ( { \\bf 1 } _ { K _ v } ) . \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} H _ 1 = \\bigoplus _ { 0 \\leqslant \\lambda \\leqslant 1 } H _ 1 ^ \\lambda , H _ 2 = \\bigoplus _ { 0 \\leqslant \\lambda \\leqslant 1 } H _ 2 ^ \\lambda \\end{align*}"}
-{"id": "10091.png", "formula": "\\begin{gather*} ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 3 } { x ^ 2 z ^ 2 } , { f ( x , y , z ) = \\dfrac { y ^ 2 ( a x + b y + c z ) ^ 2 } { x z ^ { 3 } } } , \\\\ ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 4 } { x ^ 2 z ^ { 3 } } , ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ^ 2 ( a x + b y + c z ) ^ 3 } { x z ^ { 4 } } , \\\\ ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 6 } { x ^ 3 z ^ { 4 } } , ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ^ 3 ( a x + b y + c z ) ^ 4 } { x z ^ { 6 } } , \\end{gather*}"}
-{"id": "6771.png", "formula": "\\begin{align*} K _ n ( x ) = ( - 1 ) ^ n n ! \\varphi ^ { n + 1 } \\frac { \\det \\left [ \\begin{cases} x ^ { j - 1 } & ( k = 1 ) \\\\ \\partial _ \\lambda ^ { j + k - 3 } \\varphi & ( k \\geq 2 ) \\end{cases} \\right ] _ { j , k = 1 , \\dots , n + 1 } } { \\det \\left [ \\partial _ \\lambda ^ { j + k - 2 } \\varphi \\right ] _ { j , k = 1 , \\dots , n + 1 } } . \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} a _ 3 x + a _ 4 y + a _ 5 z = 0 , b _ 3 x + b _ 4 y + b _ 5 z = 0 . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} \\prod _ { 1 < a < q _ 1 / 2 \\atop ( a , q _ 1 ) = 1 } { \\xi } _ a ^ { d _ a } ~ = ~ \\prod _ { 1 < b < q _ 2 / 2 \\atop ( b , q _ 2 ) = 1 } { \\xi } _ b ^ { - e _ b } \\prod _ { 1 < c < q _ 3 / 2 \\atop ( c , q _ 3 ) = 1 } { \\xi } _ c ^ { - f _ c } \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} \\frac { 1 } { f ( 2 m _ { 0 } k ) } + \\frac { 1 } { f ( - 2 m _ { 0 } k ) } + \\frac { 1 } { f ( 2 m _ { 0 } k ) f ( - 2 m _ { 0 } k ) } = \\frac { 1 + 2 a } { a ^ { 2 } } , \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i & = \\big ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } , \\\\ a \\ddot { \\tau } & = - \\beta _ { K i } \\dot { u } _ { i , K } + m _ { I J } \\dot { \\tau } _ { , I J } + M _ { j L K I } u _ { j , L K I } + K _ { I J } \\tau _ { , I J } . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} F _ { k \\ell } ( M , P , y ) & : = \\frac { \\partial F ( M , P , y ) } { \\partial m _ { k \\ell } } , M = ( m _ { k \\ell } ) \\in \\R ^ { ( n + 1 ) \\times ( n + 1 ) } _ { s y m } , \\ P \\in \\R ^ { n + 1 } , \\ y \\in \\R ^ { n + 1 } , \\\\ F _ k ( M , P , y ) & : = \\frac { \\partial F ( M , P , y ) } { \\partial p _ { k } } . \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} | C | & = n - \\theta ( ( n p ) ^ 3 ) \\\\ & = n - o ( n ) , \\\\ | D | & \\leq ( n - 1 ) p ( 1 + \\delta ) + 1 \\\\ & = o ( \\sqrt [ 3 ] { n } ) . \\\\ \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} k = f _ { x x } f _ { y y } , \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} m _ { \\cdot } : h ^ A ( R ) \\times h ^ A ( R ) & \\to h ^ A ( R ) \\\\ ( f _ 1 , f _ 2 ) & \\mapsto f _ 1 \\cdot f _ 2 : = ( f _ 1 , f _ 2 ) \\circ \\Delta _ \\cdot , \\\\ m _ { \\circ } : h ^ A ( R ) \\times h ^ A ( R ) & \\to h ^ A ( R ) \\\\ ( f _ 1 , f _ 2 ) & \\mapsto f _ 1 \\cdot f _ 2 : = ( f _ 1 , f _ 2 ) \\circ \\Delta _ \\circ . \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} \\tau _ { 2 } ( \\phi ) : = { \\rm T r a c e } _ { g } ( \\nabla ^ { \\phi } \\nabla ^ { \\phi } - \\nabla ^ { \\phi } _ { \\nabla ^ { M } } ) \\tau ( \\phi ) - { \\rm T r a c e } _ { g } R ^ { N } ( { \\rm d } \\phi , \\tau ( \\phi ) ) { \\rm d } \\phi = 0 , \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} R ^ - ( n , k ) & = 2 k R ^ - ( n - 1 , k ) + ( 2 n - 4 k + 4 ) R ^ - ( n - 1 , k - 1 ) + R ^ + ( n - 1 , k - 1 ) \\\\ & = 2 k R ^ - ( n - 1 , k ) + ( 2 n - 4 k + 3 ) R ^ - ( n - 1 , k - 1 ) + R ( n - 1 , k - 1 ) . \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} F _ { d , \\ell } ( z ; \\tau ) & = \\sum _ { a \\geq 0 } \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\zeta ^ d } { 1 - \\zeta ^ \\ell } \\right ) \\frac { \\left ( 2 \\pi i \\tau \\right ) ^ a } { a ! } , \\\\ G _ { d , \\ell } ( z ; \\tau ) & = 2 i \\sum _ { a \\geq 0 } \\mathcal { D } _ z ^ { 2 a } \\left ( \\frac { \\sin \\left ( 2 \\pi d z \\right ) } { 1 - \\zeta ^ \\ell } \\right ) \\frac { \\left ( 2 \\pi i \\tau \\right ) ^ a } { a ! } . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} L ( a , b ) : = | a - b | ^ { p - 2 } ( a - b ) , a , b \\in \\R . \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} \\mu ( \\bigcup _ { i = 1 } ^ { \\infty } T ^ { k _ i } A ) = 1 . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} { \\bf E } [ \\beta _ { 2 , 1 } ( a , b ) ^ q ] = & \\frac { \\Gamma _ 2 ( q + b _ 0 \\ , | \\ , a ) } { \\Gamma _ 2 ( b _ 0 \\ , | \\ , a ) } \\frac { \\Gamma _ 2 ( b _ 0 + b _ 1 \\ , | \\ , a ) } { \\Gamma _ 2 ( q + b _ 0 + b _ 1 \\ , | \\ , a ) } , \\\\ { \\bf E } [ \\beta _ { 2 , 1 } ( a , b , \\bar { b } ) ^ q ] = & \\frac { S _ 2 ( b _ 0 \\ , | \\ , a ) } { S _ 2 ( q + b _ 0 \\ , | \\ , a ) } \\frac { S _ 2 ( q + b _ 0 + b _ 1 \\ , | \\ , a ) } { S _ 2 ( b _ 0 + b _ 1 \\ , | \\ , a ) } . \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} \\sum _ { k \\geq 0 } \\frac { | A ( 1 , p ^ k ) | ^ 2 } { p ^ { k s } } = & \\frac { 1 - b _ p p ^ { - 2 s } + ( 2 b _ p - 2 ) p ^ { - 3 s } - b _ p p ^ { - 4 s } + p ^ { - 6 s } } { \\prod _ { i = 1 } ^ 3 \\prod _ { j = 1 } ^ 3 ( 1 - \\frac { \\alpha _ { i , p } } { \\alpha _ { j , p } } p ^ { - s } ) } , \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} \\sigma = \\sum \\limits _ { i = 1 } ^ { n _ 0 + 1 } l ' _ { i ( n _ 0 + 1 ) } \\ge - \\sum \\limits _ { i = 1 } ^ { n _ 0 + 1 } z _ i . \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} K _ { W ' } : = K _ { W _ 1 ' } \\otimes W _ 2 ' \\otimes . . . \\otimes W _ d ' \\ + \\ W _ 1 ' \\otimes K _ { W _ 2 ' } \\otimes . . . \\otimes W _ d ' \\ + \\ . . . \\ + \\ W _ 1 ' \\otimes . . . \\otimes W _ { d - 1 } ' \\otimes K _ { W _ d ' } , \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} F _ { n } ^ { \\left ( r \\right ) } = \\left \\vert \\mathbf { P } _ { n } ^ { r } \\ \\mathbf { P } _ { m _ { 1 } } ^ { r } . . . \\mathbf { P } _ { m _ { d } } ^ { r } \\right \\vert ^ { T } = \\Delta _ { n } ^ { \\left ( r \\right ) } \\left \\vert \\mathbf { P } _ { m _ { 1 } - n - 1 , - 1 } ^ { r + n + 1 } \\ \\mathbf { P } _ { m _ { 2 } - n - 1 , - 1 } ^ { r + n + 1 } . . . \\mathbf { P } _ { m _ { d } - n - d , - 1 } ^ { r + n + 1 } \\right \\vert ^ { T } \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\begin{array} { l l } \\pi \\circ f _ B - f _ { B _ 1 } \\circ \\pi & = c ^ a _ n v ^ n _ b + \\sum _ { j = 1 } ^ { n - 1 } c ^ d _ n B ^ n _ j v ^ j _ d \\delta ^ a _ b + \\sum _ { i = 1 } ^ { n } c ^ d _ i B ^ i _ n v ^ n _ d \\delta ^ a _ b , \\\\ & = 0 . \\end{array} \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} \\det ( A ( m , n ) ) & = \\det ( A ( m , n - m ) ) = \\pm \\det ( A ( n - m , 2 m - n ) ) \\\\ & = \\pm \\det ( A ( n - m , 3 m - 2 n ) ) = \\cdots \\\\ & = \\pm \\det ( A ( n - m , k m - ( k - 1 ) n ) ) . \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} ( g r ) ( \\omega ) = g ( \\omega ) ( r ( \\omega ) ) \\omega . \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} v _ { { \\mathcal { R } } , { [ N _ T ] } , p } ^ i ( u ) = \\alpha _ { { \\mathcal { R } } , { [ N _ T ] } } ^ { \\bar { \\mathcal { R } } _ i } ( u ) c _ p ( u ) , \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} a ( \\mathbf { u } , \\mathbf { v } ; t ) : = \\int _ { G } ( \\mathbf { H } \\nabla \\mathbf { u } ) : ( \\nabla \\mathbf { v } ) \\mathrm { d } \\mathbf { x } . \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{gather*} D _ { r + d } = ( - 1 ) ^ { { \\tfrac { d ( d + 1 ) } { 2 } } } \\left ( s _ { 2 r + d } - s _ { 2 r + d } ^ { ( r ) } \\right ) ^ { d + 1 } D _ { r - 1 } \\ . \\end{gather*}"}
-{"id": "9347.png", "formula": "\\begin{align*} Y ^ S _ j ( x ) = \\sum _ { i = 1 } ^ m Y ^ T _ i C ^ + _ { i j } \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} E _ \\tau = \\lim _ { n \\to \\infty } E \\left ( u _ 1 ^ n , u _ 2 ^ n \\right ) \\geq \\liminf _ { n \\to \\infty } e _ 2 \\left ( u _ 2 ^ n \\right ) . \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} \\langle h \\rangle ^ G = \\langle g \\rangle ^ G = \\langle x ^ { 2 \\alpha } y ^ { 2 \\beta } z ^ { 2 \\gamma } , x ^ { 4 \\alpha } , y ^ { 4 \\beta } , z ^ { 4 \\gamma } \\rangle . \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } U & \\tilde U \\end{array} \\right ] \\left [ \\begin{array} { c c } U & \\tilde U \\end{array} \\right ] ^ H = \\left [ \\begin{array} { c c } V & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } V & 0 \\end{array} \\right ] ^ H \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} f f '' + ( f ' ) ^ 2 + 1 = \\pm 2 a f \\sqrt { f '^ 2 + 1 } , a = c o n s t \\neq 0 . \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} \\partial ( \\overline A \\partial w _ k ) = 0 , \\ \\frac 3 { 4 ^ { k + 1 } } B _ 1 , \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} C = \\{ \\textup { $ A \\in G $ : $ \\mathrm { t r } ( A ) \\equiv a \\pmod { \\ell } $ , $ \\mathrm { t r } ( A ) ^ 2 - 4 \\det ( A ) $ i s a s q u a r e i n $ \\mathbb { F } _ { \\ell } ^ \\times $ } \\} \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\min _ { j = 0 , \\hdots , N - 1 } \\norm { \\mathcal { G } _ t ( x _ j ) } ^ 2 \\le \\frac { 2 t ^ { - 1 } \\big ( F ( x _ 0 ) - F ^ * \\big ) } { N } , \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} \\alpha _ { s ^ k _ i } ^ * & = y _ i , & \\alpha _ { s _ i ^ k } & : = ( 1 - \\theta ^ k ) x _ i . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} x K _ { n } ( x ) = P _ { n + 1 } ( x ) - \\frac { P _ { n + 1 } ( 0 ) } { P _ n ( 0 ) } P _ n ( x ) , \\ \\ n \\geq 0 . \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\begin{matrix} \\Phi ( 0 ) = \\mathbf { 0 } , \\\\ \\Phi ( \\gamma ^ k ) = \\mathbf { C } _ p ^ k , ~ 0 \\leq k \\leq q ^ L - 2 , \\end{matrix} \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} \\frac { q ^ { n ^ { 2 } / 2 } } { \\left ( a ; q \\right ) _ { n } } = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i n x \\right ) } { \\left ( a , - a q ^ { - 1 / 2 } e ^ { i x } ; q \\right ) _ { \\infty } } d x \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} I ( t ) & = 8 \\gamma \\int _ { \\R ^ n } | \\nabla f | ^ 2 \\ , d x - 2 \\int _ { \\R ^ n } | \\nabla f | ^ 2 | G | ^ 2 \\ , d x \\\\ & + 3 2 \\gamma ^ 3 \\int _ { \\R ^ n } | x | ^ 2 | f | ^ 2 \\ , d x - 4 \\gamma ^ 2 \\| G \\| ^ 2 _ \\infty \\int _ { \\R ^ n } | x | ^ 2 | f | ^ 2 \\ , d x \\\\ & \\ge 2 ( 4 \\gamma - \\| G \\| ^ 2 _ \\infty ) \\int _ { \\R ^ n } | \\nabla f | ^ 2 \\ , d x + 4 \\gamma ^ 2 ( 8 \\gamma - \\| G \\| ^ 2 _ \\infty ) \\int _ { \\R ^ n } | x | ^ 2 | f | ^ 2 \\ , d x \\\\ & \\ge 0 , \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} \\min _ { z } ~ F _ t ( z ; x ) : = \\frac { 1 } { m } \\sum _ { i = 1 } ^ m h _ i \\Big ( c _ i ( x ) + \\langle \\nabla c _ i ( x ) , z - x \\rangle \\Big ) + g ( z ) + \\frac { 1 } { 2 t } \\| z - x \\| ^ 2 . \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} G \\left ( 1 \\right ) = \\sum _ { t = 1 } ^ { N } \\sum _ { r = 1 } ^ { N } m _ { t , r } \\cdot C _ { t , r } \\left ( 1 \\right ) = \\sum _ { t = 1 } ^ { N } C _ { t , t } \\left ( 1 \\right ) . \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} u ( x ) = w ( x ) \\ , f ( x ) , \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n + 1 } { 2 } } } { \\left ( q ; q \\right ) _ { n } } \\left ( - x \\right ) ^ { n } A _ { q } \\left ( - q ^ { n } \\right ) = \\frac { \\left ( - q ; q \\right ) _ { \\infty } } { \\left ( - q x ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } + n } \\left ( - 1 \\right ) ^ { n } \\prod _ { k = 0 } ^ { 2 n - 1 } \\left ( x - q ^ { k } \\right ) } { \\left ( q ^ { 2 } , - q , - q ^ { 2 } ; q ^ { 2 } \\right ) _ { k } } , \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} \\tau ( 1 ) = 0 \\qquad \\Pi _ H \\tau ( 1 ) \\gg 1 , \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} Q _ \\sigma [ u , u ] & - C \\int _ M \\frac { | u ( x ) | ^ 2 } { | x | ^ 2 \\left ( \\log \\frac { | x | } { \\rho } + \\frac { 1 } { 2 \\rho \\ , \\sigma _ 0 } \\right ) ^ { 2 } } \\ d x = 2 \\pi \\int _ 1 ^ \\infty ( f ' ( r ) ) ^ 2 \\ r \\ , \\left ( \\log r + \\frac { 1 } { 2 \\rho \\ , \\sigma _ 0 } \\right ) \\ , d r \\\\ & \\qquad \\qquad - 2 \\pi \\left ( C - \\frac 1 4 \\right ) \\int _ 1 ^ \\infty f ^ 2 ( r ) \\ , r ^ { - 1 } \\ , \\left ( \\log r + \\frac { 1 } { 2 \\rho \\ , \\sigma _ 0 } \\right ) ^ { - 1 } \\ , d r . \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} a \\nabla _ { \\nu } b \\geqslant a \\sharp _ { \\nu } b + \\sum _ { k = 0 } ^ { n } r _ { k } \\big [ \\big ( a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } - \\big ( a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ) ^ { \\frac { 1 } { 2 } } \\big ] ^ { 2 } . \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ N y _ j = 0 . \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\frac { d } { d t } \\lambda ( t ) \\int _ M \\varphi ^ 2 \\ d \\mu = - \\int _ M \\varphi \\Big ( \\frac { \\partial } { \\partial t } \\Delta \\Big ) \\varphi \\ d \\mu . \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} S ( 0 ) : = \\{ x | \\lim _ { t \\rightarrow \\infty } \\phi ( t , x ) = 0 \\} \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} \\frac { \\psi ' } { W } & \\ge c _ 1 = c _ 1 \\big ( \\max _ { \\bar { \\Omega } } | \\nabla \\varphi | \\big ) > 0 \\\\ \\noalign { a n d } W - \\psi ' & \\le c _ 2 = c _ 2 \\big ( \\max _ { \\bar { \\Omega } } | \\nabla \\varphi | \\big ) \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} f ( x ) = f _ 1 ( x ) f _ 2 ( x ) \\cdots f _ k ( x ) , \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} J _ a ^ * J _ a - I = F ^ * ( L _ a - I ) F = F ^ * \\sum _ { j = 1 } ^ J \\left ( K _ { a , \\psi _ j } + \\psi _ j \\circ ( \\tilde L _ a - I ) \\right ) F \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} N _ { t } ^ { \\varphi } = \\min \\left \\{ n \\in \\mathbb { N } \\middle | \\ ; S _ { n } \\varphi > t \\right \\} \\leq \\min \\left \\{ n \\in \\mathbb { N } \\middle | \\ ; S _ { n } \\psi - c > t \\right \\} = N _ { t + c } ^ { \\psi } , \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} T _ 3 = \\left ( \\begin{array} { c | c | c | c | c } I _ { s - 1 } & & & & \\\\ \\hline & 1 & & & \\\\ \\hline & - 1 & 1 & 1 & \\\\ \\hline & & & 1 & \\\\ \\hline & & & & I _ { n - s - 1 } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\gamma _ 0 = \\frac { 1 } { | G | } \\quad \\quad \\mbox { a n d } \\quad \\gamma _ 1 = \\frac { p } { 2 | G | } , \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} K _ { \\nu - 1 } ( z ) - K _ { \\nu + 1 } ( z ) & = - \\frac { 2 \\nu } { z } K _ { \\nu } ( z ) , \\\\ K _ { \\nu - 1 } ( z ) + K _ { \\nu + 1 } ( z ) & = - 2 K _ { \\nu } ' ( z ) ; \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} ( X , \\begin{bmatrix} I & \\Psi \\end{bmatrix} ) , X = h ^ { - 1 } ( Y - I ) \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} \\psi = \\frac { G _ 2 ' } { F _ 1 ' } = e ^ { i \\theta } \\sqrt { 1 + | m | ^ 2 } \\ , \\frac { F _ 2 ' } { F _ 1 ' } + e ^ { i s } m \\ , . \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} P ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) = ( x _ { 1 } + x _ { 3 } ( x _ { 1 } - x _ { 2 } ) , x _ { 2 } + x _ { 3 } ( x _ { 1 } - x _ { 2 } ) , x _ { 3 } ) . \\end{align*}"}
-{"id": "10130.png", "formula": "\\begin{align*} \\omega '' = y p d s + 2 ( p + q ) s d y , C '' \\ : : \\ : y ^ { 2 ( p + q ) } - s ^ { - p } = 0 , \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} \\left \\vert K \\left ( k , j \\right ) \\right \\vert = \\left \\vert \\varphi _ { z _ { j } } \\left ( z _ { k } \\right ) - \\varphi _ { z _ { j } } \\left ( z _ { j } \\right ) \\right \\vert \\leq \\left ( \\log \\frac { 1 } { 1 - \\left \\vert z _ { j } \\right \\vert ^ { 2 } } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} P _ n ( x ) & = Q _ { n } ( x ) + a _ { n , 1 } Q _ { n - 1 } ( x ) + . . . + a _ { n , ( l - 1 ) d } Q _ { n - ( l - 1 ) d } ( x ) \\\\ & + \\sum _ { i = 1 } ^ { r } a _ { n , ( l - 1 ) d + i } Q _ { n - ( l - 1 ) d - i } ( x ) , \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} g ( 1 , z _ 2 , z _ 3 ) = 0 , ~ g _ { z _ 2 } = f _ { z _ 2 } = 0 , ~ g _ { z _ 3 } = f _ { z _ 3 } = 0 , ~ ~ \\phi ( S ^ 2 ) \\cap V _ 1 . \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} f _ { k } ( x ) = n \\binom { n - 1 } { k - 1 } F ^ { k - 1 } ( x ) ( 1 - F ( x ) ) ^ { n - k } f ( x ) , \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} m ( L _ x f ) = m ( f ) ( f \\in L ^ \\infty ( G ) , \\ g \\in G ) , \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} P ( t ) \\ = \\ \\begin{cases} t ^ { 2 } , & 0 \\ \\le \\ t \\ \\le \\ 1 \\\\ \\frac { t ^ { 2 } } { \\mathcal { A } ^ { - 1 } ( l o g t ^ { 2 } ) } , & t \\ \\ge \\ 1 . \\end{cases} \\end{align*}"}
-{"id": "9932.png", "formula": "\\begin{align*} p _ W ( u ' ( \\psi ( s ) ) v ^ { \\mu _ 0 } ( A ) ) = a _ i w _ i , 0 \\leq i \\leq r , \\ 0 \\neq a _ { i } \\in \\R ; \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} K _ 0 ( z ) = \\int _ 0 ^ \\infty \\exp ( - z \\cosh t ) d t = e ^ { - z } \\int _ 0 ^ \\infty \\frac { e ^ { - s } } { \\sqrt { s ( s + 2 z ) } } d s \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} Q _ { n } \\left ( x \\right ) = P _ { n } \\left ( x \\right ) + a _ { n } ^ { \\left ( 1 \\right ) } P _ { n - 1 } \\left ( x \\right ) + . . . + a _ { n } ^ { \\left ( r \\right ) } P _ { n - r } \\left ( x \\right ) , \\ \\ n \\geq 1 . \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} h ( X , Y ) = g ( X , Y ) C , X , Y \\in T M . \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{gather*} \\mu _ { k - 1 } \\left ( r ( \\blacktriangleright _ \\gg \\otimes \\cdots \\otimes \\blacktriangleright _ \\gg ) ( a _ 1 \\otimes \\cdots \\otimes a _ k ) \\right ) = 0 a _ 1 , \\dots , a _ k \\in H . \\end{gather*}"}
-{"id": "5302.png", "formula": "\\begin{align*} u _ { \\beta } ^ 2 ( f , g _ 1 ) = [ I - \\beta P ( g _ 1 ) ] ^ { - 1 } \\bar { r } ^ 2 ( g _ 1 ) = \\beta \\left [ \\frac { 4 + 5 p } { 1 - \\beta } , \\frac { ( 4 + 5 p ) \\beta } { 1 - \\beta } + 7 \\right ] ^ T . \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} \\sum _ { p \\in \\mathcal { T } } h _ { q p } ( u ) v _ { { \\mathcal { R } } , { \\mathcal { T } } , p , n } ^ i ( u ) = 0 , \\forall q \\in \\bar { \\mathcal { R } } _ i , \\forall n \\in [ N ^ { \\binom { N _ R - 1 } { r + 1 } \\binom { N _ T } { t } } ] , \\forall u \\in [ S ] \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} v _ { k } = u _ { k } + h _ { k } , u _ { k } \\in P ( k ) , h _ { k } \\perp P ( k ) = 0 , h _ { k } \\neq 0 . \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} S _ f ( X ) = & \\sum _ { n \\leq X } a ( n ) = Q ( X ) + O ( X ^ { \\frac { \\delta } { 2 } - \\frac { 1 } { 4 A } + 2 A ( w - \\frac { \\delta } { 2 } - \\frac { 1 } { 4 A } ) \\eta + \\epsilon } ) \\\\ & + O ( X ^ { q - \\frac { 1 } { 2 A } - \\eta } \\log ( X ) ^ { r - 1 } ) + O \\bigg ( \\sum _ { X \\leq n \\leq X ' } | a ( n ) | \\bigg ) \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\mu ' ( x ) \\sum _ { j = - \\infty } ^ { n } g _ { j } ^ { 2 } ( x ) = W _ { n } ( g ' ( x ) , g ( x ) ) , \\mbox { f o r } 0 < | x | < 1 , \\end{align*}"}
-{"id": "10121.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q } { z ^ { p + 2 q } x ^ { - p } } , \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} U _ 1 S _ 1 V _ 1 ^ { \\top } = \\begin{pmatrix} \\cos \\phi _ { R 0 } \\\\ \\delta d _ 0 \\sin \\phi _ { R 0 } \\end{pmatrix} \\frac { 1 } { 1 + ( \\delta ^ 2 d _ 0 ^ 2 - 1 ) \\sin ^ 2 \\phi _ { R 0 } } \\begin{pmatrix} \\cos \\phi _ { R 0 } & \\delta ^ 2 d _ 0 ^ 2 \\sin \\phi _ { R 0 } \\end{pmatrix} \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} s _ { * x } ( v ) = q _ { * ( y , \\sigma ( y ) ) } ( w , \\sigma _ { * y } ( w ) ) . \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} \\Delta \\psi = A R _ 3 ^ \\delta \\big ( \\delta ( \\delta + 1 ) \\rho ^ { - \\delta - 2 } - \\delta \\rho ^ { - \\delta - 1 } \\Delta \\rho \\big ) + A \\Delta h . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} ( m _ { i j } ) : = \\left ( \\begin{array} { c c } \\delta & 0 \\\\ 0 & \\delta \\\\ \\end{array} \\right ) \\delta > 0 . \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{align*} f _ { n - 4 } ^ { ( 1 ) } = & f _ { n - 5 } + A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\psi _ { n - 4 } ^ { ( 1 ) } + K _ { 6 - n } \\psi _ { n - 4 } ^ { ( 1 ) } \\\\ = & O ( r ^ { n - 4 } ) + O ( r ^ { n - 3 } ) \\log r + O ( r ^ { n - 2 } ) \\log ^ 2 r \\\\ : = & b _ { n - 4 } ^ { ( 0 ) } + O ( r ^ { n - 3 } ) \\log r + O ( r ^ { n - 3 } ) + O ( r ^ { n - 2 } ) \\log ^ 2 r . \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} T f ( z ) = \\int _ { \\Omega } f ( \\zeta ) \\wedge K _ { N } ^ { 1 } ( z , \\zeta ) - \\overline { \\partial } ^ { * } \\mathcal { N } \\left ( \\int _ { \\Omega } f ( \\zeta ) \\wedge P _ { N } ( z , \\zeta ) \\right ) , \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} b ^ { q ^ n } = ( - 1 ) ^ { ( n - 1 ) ( 1 + q + \\cdots + q ^ { n - 1 } ) } b = b , \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} H ^ 2 ( g , h ) = \\frac { \\lambda _ k ^ k } { \\lambda _ 1 \\cdots \\lambda _ k } = \\frac { \\lambda _ k ^ { k + 2 n _ 2 + 3 n _ 3 + \\cdots + m n _ m } } { \\lambda _ 1 \\cdots \\lambda _ k \\lambda _ k ^ { 2 n _ 2 + 3 n _ 3 + \\cdots + m n _ m } } \\leq \\frac { \\lambda _ k ^ Q } { \\lambda _ 1 \\cdots \\lambda _ k \\mu _ { k + 1 } \\mu _ { k + 2 } \\cdots \\mu _ n } . \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} \\frac { i - 1 } { i } = 1 , \\ ; \\frac { i - 1 } { 2 i } = \\frac { 1 } { 2 } , \\ ; \\frac { i - 2 } { 3 i } = \\frac { 1 } { 3 } , \\ldots \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} K _ { \\alpha \\beta \\gamma } { } ^ { \\delta } = K _ { \\alpha \\beta } { } ^ { \\varepsilon } K _ { \\varepsilon \\gamma } { } ^ { \\delta } \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} 2 \\log ( | x | + \\sqrt { t } \\ , ) & = \\log ( | x | ^ 2 + 2 | x | \\ , \\sqrt { t } + t ) \\geq \\log ( | x | + | x | \\ , \\sqrt { t } \\ , ) \\ , , \\\\ & = \\log ( 1 + \\sqrt { t } \\ , ) + \\log | x | , \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} y ( t , x ) = [ a ( t ) ] ^ { \\frac { n } { 2 } } u ( b ( t ) , a ( t ) x ) e ^ { \\frac { a ( t ) \\kappa | x | ^ 2 } { 4 } } , \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} Q _ { d n + k } ( x ) = P _ { d n + k } ( x ) - \\lambda Q _ { d n + k } ( c ) L _ { d n + k - 1 } ( x ; c ) , \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} & a ^ * _ { 0 , 1 } = 1 - \\mu _ R - \\mu _ T , a ^ * _ { 2 , 0 } = \\mu _ R , a ^ * _ { 0 , 2 } = 2 \\mu _ T - ( 1 - \\mu _ R ) \\textrm { a n d o t h e r r a t i o s a r e 0 } . \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} \\sigma ^ { 2 } ( P ) = 1 + \\frac { 2 g ( 2 \\sqrt { P } ) } { 1 + 2 Q ( 2 \\sqrt { P } ) } , \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} P + * P = Q + * Q . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } X _ s ^ { t , x } = \\ x + \\int _ t ^ s b ( r , X _ r ^ { t , x } ) \\mathrm d r + \\int _ t ^ s \\mathrm d W _ r , \\\\ Y _ s ^ { t , x } = \\Phi ( X _ T ^ { t , x } ) - \\int _ s ^ T Z _ r ^ { t , x } \\mathrm d W _ r + \\int _ s ^ T f ( r , X _ r ^ { t , x } , Y _ r ^ { t , x } , Z _ r ^ { t , x } ) \\mathrm d r , \\\\ \\forall s \\in [ t , T ] . \\end{array} \\right . \\end{align*}"}
-{"id": "10127.png", "formula": "\\begin{align*} C \\ : : \\ : ( y ^ 2 + a x ^ 2 + b x z ) ^ q - z ^ { p + 2 q } x ^ { - p } = 0 . \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} \\Gamma ( x ; y ) = \\frac { c \\ , \\sqrt { 2 } } { \\sqrt [ 4 ] { ( x _ 1 ^ 2 + y _ 1 ^ 2 ) ^ 2 + 4 \\ , ( x _ 2 - y _ 2 ) ^ 2 } } \\cdot \\mathrm { K } \\left ( \\frac { 1 } { 2 } + \\frac { x _ 1 y _ 1 } { \\sqrt [ 4 ] { ( x _ 1 ^ 2 + y _ 1 ^ 2 ) ^ 2 + 4 \\ , ( x _ 2 - y _ 2 ) ^ 2 } } \\right ) , \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\Delta _ { g } f = R - r \\quad \\mbox { i n } M \\\\ \\dfrac { \\partial f } { \\partial \\eta _ { g } } = 0 , \\mbox { o n } \\partial M , \\end{array} \\right . \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} Q ( u ) \\ , = \\ , { \\varepsilon ^ 2 \\int _ { \\mathcal { D } } | \\nabla u | ^ 2 + \\int _ { \\mathcal { D } } | u | ^ 2 \\over ( \\ , \\int _ { \\mathcal { D } } | u | ^ { q } ) ^ { 2 \\over q } } , u \\in H ^ 1 ( { \\mathcal { D } } ) \\setminus \\{ 0 \\} , \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\alpha = \\alpha ( x ; a , b , \\varepsilon , r ) = r \\cdot \\left ( \\frac { x + \\sqrt { x ^ 2 + 4 \\varepsilon } } { 2 } \\right ) ^ a \\cdot \\left ( \\sqrt { x ^ 2 + 4 \\varepsilon } \\right ) ^ b \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} \\frac { 1 } { \\alpha } \\frac { | t | } { \\eta ^ { 1 + \\frac { 2 } { \\alpha } } } \\left | \\frac { d p _ N } { d u } \\right | _ { u = \\frac { t } { \\sqrt [ \\alpha ] { \\eta } } } \\leq \\frac { 1 } { \\pi \\alpha ^ 2 } \\frac { | t | } { b ^ { 1 + \\frac { 2 } { \\alpha } } } \\Gamma \\left ( \\frac { 2 } { \\alpha } \\right ) , \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\sigma _ j \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } - \\sigma _ k \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 j } - \\gamma _ { j k } = \\left ( \\sigma _ j \\sigma _ 1 ^ { - 1 } \\sigma _ k - \\sigma _ k \\sigma _ 1 ^ { - 1 } \\sigma _ j \\right ) \\Phi A _ 1 ^ * \\Phi ^ * . \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} \\partial e ^ { t \\Delta } = e ^ { t \\vec { \\Delta } } \\partial . \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} F _ j ( x , z ) : = & \\frac { F ( x _ j + r _ j x , \\xi _ j + \\lambda _ j z ) - F ( x _ j + r _ j x , \\xi _ j ) - \\lambda _ j F _ { z } ( x _ j + r _ j x , \\xi _ j ) [ z ] } { \\lambda _ j ^ 2 } \\\\ = & \\int _ 0 ^ 1 ( 1 - t ) F _ { z z } ( x _ j + r _ j x , \\xi _ j + t \\lambda _ j z ) [ z , z ] ; \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} - ( a ( x ) \\nabla u ) + a ( x ) u = a ( x ) u ^ { q - 1 } \\ \\ \\Omega , u > 0 \\ \\ \\Omega , \\partial _ \\nu u = 0 \\ \\ \\partial \\Omega , \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} \\omega ^ { ( A , i + n ) } \\overset { ! } { = } - \\omega ^ { ( A , i ) } ; i = 1 , \\ldots , n - 1 \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} M _ t = 3 M \\beta _ x - \\beta M _ x , M _ T = 3 M \\epsilon _ x - & \\epsilon M _ x , \\\\ & \\beta _ T - \\epsilon _ t + \\epsilon \\beta _ x - \\epsilon \\beta _ x = 0 . \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} - L u _ n + g _ n \\circ u _ n & = f \\leq | f | = - L \\tilde { u } _ n + g _ n \\circ \\tilde { u } _ n \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u _ n & = 0 \\ , \\ , \\ , \\ , \\ , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\ , \\tilde { u } _ n = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} + \\Lambda ^ + _ { n } \\ln \\sigma + \\sum \\limits _ { m = 1 } ^ { n } \\frac { \\Lambda ^ + _ { n - m } } { m } \\sigma ^ { m } - \\int \\limits _ { \\sigma } ^ { + \\infty } \\Phi ^ + _ n ( s ) d s , \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\tfrac { 1 } { \\pi i } \\partial \\overline { \\partial } H ( X ) = \\tfrac { 1 } { 2 } \\omega _ { \\mathrm { H d g } } + \\tfrac { 1 } { ( g ! ) ^ 2 } \\left ( \\int _ { p r _ { g + 1 } } \\gamma ^ * \\omega _ 0 ^ { g + 1 } - \\int _ { p r _ { g + 1 } } \\gamma ^ * \\left ( \\delta _ { \\Theta _ { \\alpha } } \\right ) \\gamma ^ * \\omega _ 0 ^ { g } \\right ) . \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} i x ^ { \\prime } ( t ) + A _ 1 ^ * x ( t ) = \\Phi ^ * y ( t ) . \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} \\Pi _ { d , \\xi } \\left \\{ P U _ { \\delta , \\xi } + \\phi - i ^ { \\ast } \\left [ f _ { \\epsilon } \\ ( P U _ { \\delta , \\xi } + \\phi \\ ) \\right ] \\right \\} = 0 . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} ( \\sigma _ { a _ 0 } , \\sigma _ { a _ 1 } , \\sigma _ { a _ 2 } ) \\cdot f ( y _ 0 , y _ 1 , y _ 2 ) = f ( \\sigma _ { a _ 0 } y _ 0 , \\sigma _ { a _ 1 } y _ 1 , \\sigma _ { a _ 2 } y _ 2 ) . \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} \\left ( \\partial _ t + v \\cdot \\nabla _ x \\right ) f ( t , x , v ) = Q ^ + ( f , f ) ( t , x , v ) - Q ^ - ( f , f ) ( t , x , v ) \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} P ^ z \\left ( \\left ( Z \\left ( \\left ( V _ k ^ n + \\cdot \\right ) \\wedge T _ k ^ n \\right ) , Z \\left ( \\left ( V _ k ^ n + \\cdot \\right ) \\wedge T _ k ^ n \\right ) - X \\left ( \\left ( V _ k ^ n + \\cdot \\right ) \\wedge T _ k ^ n \\right ) \\right ) \\in { \\mathcal A } \\right ) = 1 . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} & y ( x , 0 ) = \\varphi ( x ) , 0 \\leq x \\leq \\ell , \\\\ & y ( 0 , t ) = g _ 1 ( t ) , y ( \\ell , t ) = g _ 2 ( t ) , 0 < t \\leq 1 , \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} A ^ { k } ( i _ { 1 } , \\dots , i _ { l } ) = \\frac { 1 } { \\binom { k - l } { k - 2 - l } } \\sum _ { p = 1 } ^ { \\binom { n - l } { k - 2 - l } } A ^ { k } ( i _ { 1 } , \\dots , i _ { l } , i _ { l + 1 } ^ { p } , \\dots , i _ { k - 2 } ^ { p } ) = 0 . \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} \\sum _ { i \\geq 1 } \\lambda _ i ^ 2 \\psi _ i ( x ) \\overline { \\psi _ i ( z ) } = 0 ~ ~ ~ \\mu ^ 2 \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} D : = \\left ( \\bigcup _ { i } \\sigma _ { i } \\right ) \\cap \\P ^ { 1 } _ { b } . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} \\hat { Q } _ { ( 0 ) } ^ 2 = 0 . \\end{align*}"}
-{"id": "10089.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } y ^ { q } ( a x + b y + c z ) ^ r } { z ^ { p + q + r } } , \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} \\frac { 2 i + 1 } { 2 n + 2 } { n + 1 \\brack i + 1 } = { n \\brace i } , \\quad 0 \\leq i \\leq n , \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\theta ( x _ { j , i } ) ^ { k _ i } = \\left ( \\theta ( v _ { j , i } ) ^ { x _ i ^ { k _ i - 1 } } \\right ) \\cdots \\left ( \\theta ( v _ { j , i } ) ^ { x _ i ^ 2 } \\right ) \\left ( \\theta ( v _ { j , i } ) ^ { x _ i } \\right ) \\left ( \\theta ( v _ { j , i } ) ^ 1 \\right ) . \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} p _ { 2 i + 1 } - p _ i = \\frac 1 2 ( p _ i - p _ { \\lfloor i / 2 \\rfloor } ) \\omega z . \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} y ^ T _ { h } = \\Bigl ( { \\gamma \\over 2 + \\gamma } \\Bigr ) ^ h \\Bigl ( y _ 0 + { T \\over 2 } \\Bigr ) - { T \\over 2 } \\ , . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { n - 1 } \\frac { b _ i } { p ^ i } \\right ) + \\frac { 1 } { p ^ n } \\le x , \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} & C _ t ( y ^ { t - 1 } _ { t - J } ) = \\sup _ { \\big \\{ \\pi _ i ( d x _ i | y ^ { i - 1 } _ { i - J } ) : ~ i = t , t + 1 , \\ldots , n \\big \\} } { \\bf E } ^ { \\pi } \\bigg \\{ \\sum _ { i = t } ^ n \\log \\Big ( \\frac { d q _ i ( \\cdot | y ^ { i - 1 } _ { i - M } , X _ i ) } { d { \\nu } ^ { \\pi } _ { i } ( \\cdot | y ^ { i - 1 } _ { i - J } ) } ( Y _ i ) \\Big ) \\\\ & \\qquad - s \\Big ( \\sum _ { i = t } ^ n \\gamma _ i ( x _ i , y ^ { i - 1 } _ { i - N } ) - ( n + 1 ) \\kappa \\Big ) \\Big { | } Y ^ { t - 1 } _ { t - J } = y ^ { t - 1 } _ { t - J } \\bigg \\} \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} C \\big ( D ^ F _ { X _ \\infty } , D ^ F _ { Y _ { \\R _ + } } \\big ) = W ^ * _ + \\big ( D ^ F _ { X _ \\infty } , D ^ F _ { Y _ { \\R _ + } } \\big ) \\circ W _ - \\big ( D ^ F _ { X _ \\infty } , D ^ F _ { Y _ { \\R _ + } } \\big ) \\in \\mathrm { E n d } \\Big ( L ^ 2 \\big ( \\Omega ^ \\bullet ( Y _ { \\R _ + } , F ) \\big ) \\Big ) . \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} I ( \\gamma , x , y ) : = \\int _ 0 ^ 1 H ^ * ( - \\dot \\gamma _ s + x - y , \\gamma _ s + x + s ( y - x ) ) \\ ; d s . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} G \\otimes \\left ( H _ 1 \\oplus H _ 2 \\right ) = \\left ( G \\otimes H _ 1 \\right ) \\oplus \\left ( G \\otimes H _ 2 \\right ) , \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} P _ { \\tilde { i } , d _ { \\tilde { i } } + 1 } ( m ) \\alpha _ { \\tilde { i } } ^ m = R ( n ) \\beta _ { j _ * } ^ n \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k + 1 } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau , 0 ; v _ { s + 1 } , \\dots , v _ { s + k } , v _ { s + k + 1 } ; \\right . \\\\ & \\left . \\qquad \\omega _ 1 , \\dots , \\omega _ k , \\omega _ { k + 1 } ; i _ 1 , \\dots , i _ k , i _ { k + 1 } \\right ] \\\\ & \\in \\mathcal { G } _ { s + k + 1 } \\cap \\hat { \\mathcal { U } } _ { s + k + 1 } ^ \\eta \\end{aligned} \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} \\nabla ^ A F _ n = ( A - B _ n ) F _ n \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} & \\underset { z } { } \\ ; \\ ; \\| z _ L \\| _ { 1 } \\\\ & \\ ; \\ ; \\| z \\| _ { 1 } \\leq 1 , \\ ; A z = 0 . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} \\left ( x , v \\right ) \\in \\Omega _ { 0 } = \\left \\{ e _ { \\pm } < \\max \\beta = - \\min \\beta \\right \\} , \\end{align*}"}
-{"id": "9667.png", "formula": "\\begin{align*} q ^ { \\beta ^ { 2 } / 2 } \\left ( q ^ { - \\beta + n + 1 } ; q \\right ) _ { \\infty } L _ { n } ^ { ( - \\beta ) } \\left ( - 1 ; q \\right ) = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { y ^ { 2 } } { \\log q ^ { 2 } } + i \\beta y \\right ) } { \\left ( q ; q \\right ) _ { n } \\left ( - q ^ { 1 / 2 + n } e ^ { - i y } ; q \\right ) _ { \\infty } } d y , \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} u ( t , x ) = U ( e ^ { \\lambda t } , x ) \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} \\delta - \\gamma ( j ) \\neq 0 \\And d ( \\gamma , \\delta - \\gamma ( j ) ) \\neq 0 , \\forall j = 1 , 2 , . . . \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} \\theta _ { s , a ^ 2 } ^ 2 = r ^ 2 ( s , a _ s ^ 1 , a ^ 2 ) + \\beta \\sum _ { s ' \\in S } p ( s ' | s , a _ s ^ 1 , a ^ 2 ) v _ \\beta ^ { 2 * } ( s ' ) - v _ \\beta ^ { 2 * } ( s ) . \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} & \\beta _ { x x } + a _ 0 \\beta = \\frac { 1 } { M } + q _ 0 , \\\\ & ( \\beta _ { x x } + a _ 0 \\beta ) _ t + \\beta \\beta _ { x x x } + 3 \\beta _ x \\beta _ { x x } + 4 a _ 0 \\beta \\beta _ x - 3 q _ 0 \\beta _ x = 0 . \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} & \\Delta \\left ( \\mu = 1 , 0 , P \\right ) = \\frac { K / \\log ( P ) } { { \\min \\{ M , K \\} } } , \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} \\varphi ( x ) = \\frac { 1 } { x } \\int \\limits _ 0 ^ x \\frac { 1 } { 1 - \\xi } \\psi \\left ( \\frac { \\xi ^ 2 } { 1 - \\xi } \\right ) \\ , d \\xi . \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} Z ( s , f , \\chi ) = q ^ { - 1 } ( 1 - q ^ { - 1 } ) + \\frac { q ^ { - 3 - 4 s } ( 1 - q ^ { - 1 } ) } { ( 1 - q ^ { - 2 - 4 s } ) } + \\frac { q ^ { - 2 - 4 s } ( 1 - q ^ { - 1 } ) ^ 2 } { ( 1 - q ^ { - 2 - 4 s } ) } \\\\ + \\frac { q ^ { - 7 - 1 6 s } ( 1 - q ^ { - 1 } ) ^ 2 } { ( 1 - q ^ { - 2 - 4 s } ) ( 1 - q ^ { - 5 - 1 2 s } ) } . \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\big \\langle Y _ i , Y _ j \\big \\rangle = \\big \\langle U ^ * X _ i U , U ^ * X _ j U \\big \\rangle = \\big \\langle P ^ * X _ i P , P ^ * X _ j P \\big \\rangle = \\big \\langle X _ i , X _ j \\big \\rangle \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} D _ y ( x _ 1 \\cdots x _ N ) = \\sum _ { x _ i = y } ( - 1 ) ^ { n _ i m _ i } x _ { i + 1 } \\cdots x _ N x _ { 1 } \\cdots x _ { i - 1 } \\ , , \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} \\omega _ { \\mu + m , 1 } ( x ) = r _ { m , \\mu } ( x ) \\omega _ { \\mu , 1 } ( x ) + s _ { m , \\mu } ( x ) \\omega _ { \\mu + 1 , 1 } ( x ) \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\operatorname { d i v } V = 0 , \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} \\norm { T _ 1 f } _ p ^ p & = \\sum _ { k = 0 } ^ \\infty \\abs { T f ( k ) } ^ p \\mu ( \\{ k \\} ) = \\abs { \\langle \\psi _ 1 - \\alpha _ 1 , f \\rangle } ^ p + \\sum _ { k = 2 } ^ \\infty \\frac { k ^ p } { ( k - 1 ) ^ p } \\abs { f ( k - 1 ) } ^ p \\frac { 1 } { k ^ p } \\\\ & \\le \\langle \\psi , \\abs { f } \\rangle ^ p + \\sum _ { k = 1 } ^ \\infty \\abs { f ( k ) } ^ p \\frac { 1 } { k ^ p } \\le ( \\norm { \\psi } ^ p + 1 ) \\norm { f } _ p ^ p < \\infty . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} & \\left [ \\partial ^ { \\tau } \\partial _ { \\tau } + \\kappa \\left ( u ^ { \\tau } \\partial _ { \\tau } \\right ) ^ { 2 } \\right ] A _ { \\sigma } - \\partial _ { \\sigma } \\left ( \\partial ^ { \\tau } A _ { \\tau } + \\kappa u ^ { \\nu } u ^ { \\tau } \\partial _ { \\nu } A _ { \\tau } \\right ) \\\\ & \\qquad = \\frac { 4 \\pi \\mu } { c } \\left ( g _ { \\sigma \\lambda } - \\frac { \\kappa } { 1 + \\kappa } u _ { \\sigma } u _ { \\lambda } \\right ) j ^ { \\lambda } . \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} & S ( a _ 1 ) ( a _ 2 \\circ b ) = S ( a _ 1 ) a _ 2 ( a _ 3 \\rightharpoonup b ) = a \\rightharpoonup b & a , b \\in A . \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} D _ { a , b } = m i n \\left \\{ \\sum _ { i = a } ^ { b - 1 } D _ { i , i + 1 } , \\sum _ { i = b } ^ { n - 1 } D _ { i , i + 1 } + D _ { 1 , n } + \\sum _ { i = 1 } ^ { a - 1 } D _ { i , i + 1 } \\right \\} . \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} \\| ( \\varphi \\otimes \\psi _ n ) _ { J _ n ( a ) } \\| = \\| J ^ { \\psi _ n } _ n \\varphi _ a \\| = \\| \\varphi _ a \\| \\ , , \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} \\partial ( f \\circ \\Phi ) = \\frac { \\partial f } { \\partial x ^ 1 } \\partial \\phi ^ 1 + \\frac { \\partial f } { \\partial x ^ 2 } \\partial \\phi ^ 2 \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} J _ f ^ { B , c } [ \\alpha , \\beta ] : = \\mathbf { E } \\bigg [ & \\int _ 0 ^ 1 e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ t \\| \\beta _ s \\| ^ 2 d s } \\langle B ^ * B \\alpha _ t , \\beta _ t \\rangle \\ , d t \\\\ & + e ^ { - \\frac { 1 } { 2 } \\int _ 0 ^ 1 \\| \\beta _ t \\| ^ 2 d t } f \\bigg ( B W _ 1 + \\int _ 0 ^ 1 B \\alpha _ t \\ , d t + \\frac { \\Phi ^ { - 1 } ( c ) } { 2 } \\int _ 0 ^ 1 B \\beta _ t \\ , d t \\bigg ) \\bigg ] \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} [ \\Delta , \\i A ] _ { \\circ } = \\Delta ( 4 - \\Delta ) . \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} u > \\psi ( x ) \\otimes \\inf _ { \\psi ' \\in C X } \\hom ( \\psi ' ( x ) , \\Phi ( \\psi ' ) ) = \\inf _ { \\psi ' \\in C X , \\psi ' ( x ) = 1 } \\psi ( x ) \\otimes \\Phi ( \\psi ' ) , \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} \\frac { k \\binom { b } { k } + O ( \\frac { b ^ { k + 1 } } { n } ) } { 1 + O ( \\frac { b } { n } ) } \\ge \\left ( k \\binom { b } { k } + o ( 1 ) \\right ) \\left ( 1 - O ( \\frac { b } { n } ) \\right ) = k \\binom { b } { k } - o ( 1 ) . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} \\sum \\limits _ { \\substack { i = 0 \\\\ \\mbox { \\scriptsize $ m + i $ e v e n } } } ^ m ( - 1 ) ^ i { m \\choose i } ( m + i ) \\sum \\limits _ j { \\frac { m + i } { 2 } - 1 \\brace j - 1 } x ^ { 2 j - 2 } \\\\ \\quad + \\sum \\limits _ { \\substack { i = 0 \\\\ \\mbox { \\scriptsize $ m + i $ o d d } } } ^ m ( - 1 ) ^ i { m \\choose i } ( m + i ) x ^ { m + i - 1 } & = 0 . \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } q ^ { n ^ { 2 } / 2 } \\left ( - t \\right ) ^ { n } S _ { n } \\left ( x q ^ { n } ; q \\right ) = \\left ( t q ^ { 1 / 2 } ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 5 n ^ { 2 } / 2 } \\left ( x t \\right ) ^ { n } } { \\left ( q ; q \\right ) _ { n } \\left ( t q ^ { 1 / 2 } , t q ^ { 3 / 2 } ; q ^ { 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\sum _ { n = 0 } ^ \\infty f ( n ) \\overline { g ( n ) } . \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} \\begin{cases} \\Delta H - H | A | ^ { 2 } + H { \\rm R i c } ^ N ( \\xi , \\xi ) = 0 , \\\\ 2 A \\ , ( { \\rm g r a d } \\ , H ) + \\frac { m } { 2 } { \\rm g r a d } \\ , H ^ 2 - 2 \\ , H \\ , ( { \\rm R i c } ^ N \\ , ( \\xi ) ) ^ { \\top } = 0 , \\end{cases} \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} g _ 0 = d r ^ 2 + f \\left ( \\theta \\right ) ^ 2 d \\theta ^ 2 , \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\mathcal J ( \\varphi | _ { X _ 1 } ) | _ { f ^ { - 1 } ( Q ) } = \\mathcal J ( \\varphi ) | _ { f ^ { - 1 } ( Q ) } \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{gather*} g ^ { ( \\alpha ) } _ { C } = \\exp \\big ( \\Gamma ^ { ( \\alpha ) } _ { C } \\big ) = \\sum _ { \\ell = 0 } ^ { \\infty } \\frac 1 { \\ell ! } \\big ( \\Gamma ^ { ( \\alpha ) } _ { C } \\big ) ^ { \\ell } , \\end{gather*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\tilde { B } ^ i _ j = E T ( B ) ^ i _ j = ( 1 + k \\varepsilon ) B ^ i _ j + \\varepsilon \\delta ^ i _ j , 1 + k \\varepsilon \\neq 0 . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} E _ 3 ^ { y _ 0 } ( y ) : = \\tilde { a } ^ { i j } ( y _ 0 ) E _ 1 ^ { y _ 0 , i j } ( y ) + E ^ { y _ 0 , i j } _ 2 ( y ) G ^ { i j } ( v ) . \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} L _ n ^ { * - 1 } = ( \\mathbf { E _ * } L _ n ^ * ) ^ { - 1 } + W _ n ^ * + \\tilde { Z } _ n ^ * \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} s ( f - x y ^ n ) & = s x y ^ 2 ( x - y ) \\left ( \\sum _ { j = 0 } ^ { m - 1 } x ^ { n - 3 - 2 j } y ^ { 2 j } \\right ) \\\\ & \\equiv x y ( x ^ 2 - y ^ 2 ) \\left ( \\sum _ { j = 0 } ^ { m - 1 } x ^ { n - 3 - 2 j } y ^ { 2 j } \\right ) \\\\ & = x y ( x ^ { 2 m } - y ^ { 2 m } ) = x ^ { n } y - x y ^ { n } . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} \\sum _ { \\substack { | \\delta | = k , \\\\ \\delta _ { i _ { 1 } } = \\cdots = \\delta _ { i _ { l } } = 1 } } u _ { \\delta } = 0 . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} Q ( f , f ) = \\int _ { \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } } \\left [ \\omega \\cdot ( v _ 1 - v ) \\right ] _ + \\left ( f ( x , v ^ * ) f ( x , v _ 1 ^ * ) - f ( x , v ) f ( x , v _ 1 ) \\right ) d \\omega d v _ 1 \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{gather*} D _ { r - 1 } \\neq 0 \\ , \\ \\ \\ \\ D _ { n } = 0 \\ , \\ \\ \\ n \\geq r \\ . \\end{gather*}"}
-{"id": "3403.png", "formula": "\\begin{align*} ( x t ^ n ) \\cdot ( a m ) = ( x _ { ( n ) } a ) m + a ( x t ^ n ) \\cdot m \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} N ( \\sigma , T , \\chi ) = \\# \\{ \\rho : L ( \\rho , \\chi ) = 0 , \\sigma < \\Re \\{ \\rho \\} < 1 , | \\Im ( \\rho ) | \\leq T \\} . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} \\langle \\mathbf { h } \\rangle : = \\int _ { \\mathcal { V } } \\mathbf { h } ( v ) \\ , { \\rm d } \\mu ( v ) \\mbox { f o r a l l } \\mathbf { h } \\in L ^ 1 ( \\mathcal { V } ; { \\rm d } \\mu ) . \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma _ j ( Y ) & = & B _ j \\ , Y , \\ j = 1 , 2 \\end{aligned} \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} g ( \\eta ) = \\lambda \\cdot f ( \\eta ) = f ( \\eta \\cdot \\lambda ) \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} Z ( s , f , \\chi ) = \\frac { q ^ { - 8 - 1 8 s } ( 1 - q ^ { - 1 } ) ^ 2 } { ( 1 - q ^ { - 3 - 6 s } ) ( 1 - q ^ { - 5 - 1 2 s } ) } + \\frac { q ^ { - 3 - 6 s } ( 1 - q ^ { - 1 } ) ^ 2 } { ( 1 - q ^ { - 3 - 6 s } ) } \\\\ + \\frac { q ^ { - 4 - 6 s } ( 1 - q ^ { - 1 } ) } { ( 1 - q ^ { - 3 - 6 s } ) } + q ^ { - 1 } ( 1 - q ^ { - 1 } ) . \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} \\begin{array} { c } ( X , \\mu ) : = \\coprod _ { \\alpha \\in J } ( X _ { \\alpha } , \\mu _ { \\alpha } ) { } \\\\ \\left ( \\coprod _ { \\alpha \\in J } X _ { \\alpha } , \\coprod _ { \\alpha \\in J } \\kappa ( \\mu _ { \\alpha } , k _ { \\alpha } ) \\right ) : = \\coprod _ { \\alpha \\in J } \\left ( X _ { \\alpha } , \\kappa ( \\mu _ { \\alpha } , k _ { \\alpha } ) \\right ) \\end{array} \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\theta _ { q } \\left ( z \\right ) = - z \\theta _ { q } \\left ( z ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} A \\cdot \\Xi \\cdot ( \\beta _ 1 , \\ldots , \\beta _ d ) ^ T = 0 . \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { \\mathbb { R } ^ { 2 d ( N - s ) } } \\mathbf { 1 } _ { Z _ { ( s + 1 ) : N } \\in \\mathcal { D } _ { N - s } } \\mathbf { 1 } _ { | x _ i - x _ j | \\leq \\varepsilon } & f _ 0 ^ { \\otimes ( N - s ) } ( Z _ { ( s + 1 ) : N } ) d Z _ { ( s + 1 ) : N } \\leq \\\\ & \\leq \\mathcal { Z } _ { N - s - 1 } \\varepsilon ^ d | B _ 1 ^ d | \\left \\Vert f _ 0 \\right \\Vert _ { L ^ \\infty _ x L ^ 1 _ v } \\end{aligned} \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\theta = \\begin{pmatrix} 0 & \\tau \\end{pmatrix} , \\quad \\mu = \\begin{pmatrix} 0 & A \\end{pmatrix} , A A ^ { t r } = I , \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} [ \\bar { h } _ { i , k } , \\bar { e } _ { j , l } ] = a _ { i , j } d ^ { - k m _ { i , j } } \\bar { e } _ { j , l + k } , \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} u ( x ) = \\sum a _ n \\phi _ n ( x ) \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} c = & u '^ 2 + v '^ 2 ( \\sqrt { 2 } + \\cos u ) ^ 2 , \\\\ c \\kappa = & ( u '' v ' - v '' u ' ) ( \\sqrt { 2 } + \\cos u ) + v ' \\sin u ( v '^ 2 ( \\sqrt { 2 } + \\cos u ) ^ 2 + 2 u '^ 2 ) . \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} \\overline { \\partial } ( \\omega \\otimes f ) : = \\overline { \\partial } ( \\omega ) \\otimes f + ( - 1 ) ^ q \\omega \\otimes \\overline { \\partial } ( f ) \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} \\Psi _ { \\gamma + t } ( x ) = e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } + A ( \\gamma ) e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } + \\left ( A ( \\gamma ) \\right ) ^ { 2 } e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } + . . . , \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} G = \\left [ \\begin{array} { c c } E & 0 \\end{array} \\right ] , F = \\left [ \\begin{array} { c c } A & B \\end{array} \\right ] , \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} t ( 2 - t ) \\frac { d ^ 2 } { d t ^ 2 } u ( t ) + 2 ( m + 1 ) ( 1 - t ) \\ , \\frac { d } { d t } u ( t ) + \\left [ z - c ^ 2 ( 1 - t ) ^ 2 \\right ] u ( t ) = 0 \\ , . \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\omega \\big | _ { Y _ { [ - R , R ] } } = e ^ { - i \\lambda u } \\phi _ 1 + e ^ { i \\lambda u } \\phi ' _ 1 + \\omega ^ \\mathrm { n z } . \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} A \\ = \\ e ^ { i \\theta _ 1 } \\begin{pmatrix} \\lambda & 0 \\\\ 0 & \\lambda ^ { - 1 } \\end{pmatrix} e ^ { i \\theta _ 2 } , \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} & L = \\limsup _ { R \\to \\infty } \\ , \\sup _ { t \\in \\mathbb { R } } \\ , \\frac { 1 } { R } \\ , \\int _ { \\prod _ { k = 1 } ^ s [ t , t + N _ k ( R ) \\ , \\ , a _ k ^ { - 1 } ) } \\ , \\big | Q ( \\xi ) \\big | ^ 2 \\ , d \\nu _ { \\mu } ( \\xi ) \\\\ & = \\limsup _ { R \\to \\infty } \\ , \\sup _ { t \\in \\mathbb { R } } \\ , \\frac { 1 } { R } \\ , \\sum _ { 0 \\le k _ 1 \\le a _ 1 R - 1 } \\ , \\dots \\ , \\sum _ { 0 \\le k _ s \\le a _ s R - 1 } \\ , \\int _ { I ( t , k _ 1 , \\dots , k _ s ) } \\ , \\big | Q ( \\xi ) \\big | ^ 2 \\ , d \\nu _ { \\mu } ( \\xi ) . \\\\ \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} \\lambda _ i ^ * \\left [ \\sum _ { j = 1 } ^ i E _ s ^ { j * } - \\sum _ { j = 1 } ^ i ( 1 - \\alpha _ j ^ * ) \\eta P _ p \\right ] & = 0 \\\\ \\gamma _ i ^ * [ h _ { s p } ^ i E _ s ^ { i * } - \\alpha _ i ^ * P _ { i n t } ] & = 0 \\\\ \\mu _ i ^ * [ \\alpha _ i ^ * - 1 ] & = 0 \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} \\psi = A ( R _ { 3 } ^ { \\delta } \\rho ^ { - \\delta } + h ) \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} \\frac { t } { 2 } \\ , ( \\varphi ^ 2 ) ' + \\varphi ^ 2 + 1 = \\pm 2 a t \\sqrt { \\varphi ^ 2 + 1 } . \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} | \\widehat { f } _ c ( k ) - \\widehat { h } ( k ) | & = \\frac { 1 } { 2 m } \\left | \\left ( \\sqrt { \\frac { 1 } { c ^ 2 } ( k + 1 ) ^ 2 + m ^ 2 } + \\sqrt { \\frac { 1 } { c ^ 2 } ( k - 1 ) ^ 2 + m ^ 2 } - 2 m \\right ) \\widehat { h } ( k ) \\right | \\\\ & \\leq \\frac { 1 } { 2 m } \\frac { 2 k ^ 2 + 2 } { m c ^ 2 } | \\widehat { h } ( k ) | . \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} a ( x _ { 1 } , \\ldots , x _ { m } ) : = x _ { 1 } ^ { M _ { 1 } - 1 } \\cdots x _ { m } ^ { M _ { m } - 1 } , \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} \\frac { i } { c } \\left ( \\frac { \\partial \\mathbf { G } } { \\partial t } + 4 \\pi \\mathbf { j } \\right ) = \\operatorname { c u r l } \\mathbf { F } , \\qquad \\mathbf { j } = \\mathbf { j } ^ { \\ast } , \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\xi _ { q } ( z ) = \\sum _ { k = - \\infty } ^ { \\infty } \\frac { q ^ { ( k + 1 ) / 2 } } { 1 - q ^ { k + 1 / 2 } } \\left ( - z \\right ) ^ { k } \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} M _ 1 ( l , u , v ) = \\sum _ { f \\in H _ { 2 k } ^ { * } ( N ) } ^ { h } \\lambda _ f ( l ) L _ f ( 1 / 2 + u + v ) , \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\Psi _ L ( x _ 1 ^ { K + 1 } , \\cdots , x _ N ^ { K + 1 } , \\lambda ^ { K + 1 } , x _ N ^ K ) - \\Psi _ L ( x _ 1 ^ 1 , \\cdots , x _ N ^ 1 , \\lambda ^ 1 , x _ N ^ 0 ) \\leq - \\tau \\sum _ { k = 1 } ^ { K } \\sum _ { i = 1 } ^ N \\left \\| x _ i ^ k - x _ i ^ { k + 1 } \\right \\| ^ 2 , \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} P _ { n } ^ { [ m ] } ( x ) = \\sum _ { i = 0 } ^ { d + 1 } \\tilde { \\lambda } _ { n , \\nu } P _ { n - \\nu } ^ { [ m + 1 ] } ( x ) , \\ \\forall n , m \\geq 0 . \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} \\pi _ 7 ( [ P , P ] _ p ^ { F N } ) = \\frac 1 { 1 2 } \\ast ( \\gamma _ \\mu \\wedge \\lambda ^ 2 ( e _ i ) ) \\ ; \\Phi \\wedge \\lambda ^ 2 ( e _ i ) \\otimes e _ \\mu . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} P _ t ( f ^ 2 ) - ( P _ t f ) ^ 2 = 2 \\int _ 0 ^ t P _ s ( \\Gamma ( P _ { t - s } f ) d s \\ge \\frac { 2 } { C _ 1 ^ 2 } t \\Gamma ( P _ t f ) . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} P ( \\{ ( 1 , 2 ) , ( 1 , 3 ) \\} \\subset A ( \\mathbf { D } ) ) = \\frac { k ( k - 1 ) } { ( n - 1 ) ( n - 2 ) } < \\left ( \\frac { k } { n - 1 } \\right ) ^ 2 = ( P ( ( 1 , 2 ) \\in A ( \\mathbf { D } ) ) ) ^ 2 \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} S _ f ^ \\nu ( n ) : = \\sum _ { m \\leq n } \\frac { B _ f ( m ) } { m ^ \\nu } , \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} p _ { \\kappa ( x ) } \\mathcal { D } ( F ) p _ { \\kappa ( x ) } = \\Psi ( p _ x \\mathcal { D } ( E ) p _ x ) , \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} \\pi ^ i ( t ) = { n \\choose i } \\sum _ { k = 0 } ^ i ( - 1 ) ^ { i - k } { i \\choose k } \\exp ( Q _ { ( n - k ) \\lambda } t ) x ( 0 ) . \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} g _ 0 = d r ^ 2 + f \\left ( r \\right ) ^ 2 d \\theta ^ 2 , \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} A _ { q } \\left ( x t \\right ) = \\left ( t ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } S _ { n } \\left ( x ; q \\right ) t ^ { n } . \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} \\wedge ^ t ( \\mathcal { Q } ^ * \\otimes \\mathcal { S } ) = \\bigoplus _ { | \\mu | = t } K _ { \\mu } \\mathcal { Q } ^ * \\otimes K _ { \\mu ^ \\prime } \\mathcal { S } \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} \\left ( - 1 \\right ) ^ { n } q ^ { n ^ { 2 } } = \\sum _ { k = 0 } ^ { n } \\frac { \\left ( q ^ { - n } , - q ^ { - n } ; q \\right ) _ { k } q ^ { 2 n k } } { \\left ( q ; q \\right ) _ { k } q ^ { \\binom { k } { 2 } } } . \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} \\int _ T \\int _ X \\Phi ( t , x ) \\lambda ( t , d x ) d \\mu = \\int _ T \\Phi ( t , f ( t ) ) d \\mu . \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} \\{ y _ 1 , \\ldots , y _ k \\} = G _ 0 \\sqcup G _ 1 \\sqcup G _ 2 \\sqcup \\{ p _ 0 , p _ 1 , p _ 2 \\} . \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} g _ 2 ( f _ 2 ) ^ k b _ 2 = 0 , k = 0 , 1 , 2 , 3 , \\cdots \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} \\frac { d } { d x } \\alpha _ \\varepsilon ( u ) = - 2 \\alpha _ \\varepsilon ^ 2 ( u ) \\partial ^ { 1 - \\varepsilon } u \\frac { d } { d x } \\partial ^ { 1 - \\varepsilon } u . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} b _ 4 = \\begin{pmatrix} s & a & 0 \\\\ 0 & b & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , b _ 5 = \\begin{pmatrix} 0 & - b & 0 \\\\ s & a & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} , b _ 6 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ s & a & 0 \\\\ 0 & \\pm b & 0 \\end{pmatrix} , b _ 7 = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & \\mp b & 0 \\\\ s & a & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} \\theta = k ^ 2 G \\left ( \\frac { k } { \\rho } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} \\oslash \\leq \\oslash 1 \\leq \\oslash \\pounds \\leq 1 \\pounds = \\pounds . \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} d _ 2 ^ { t r } d _ 2 = g _ 2 ^ { t r } g _ 2 , \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\overrightarrow { G } \\otimes \\overrightarrow { H } & = \\left ( \\oplus _ i \\overrightarrow { C } ( i ) \\right ) \\otimes \\left ( \\oplus _ j \\overrightarrow { C } ' ( j ) \\right ) \\\\ & = \\oplus _ i \\oplus _ j \\left ( \\overrightarrow { C } ( i ) \\otimes \\overrightarrow { C } ' ( j ) \\right ) \\\\ \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} | \\nabla _ x y ( s , t ) - I | & = | ( \\nabla _ x q ( s , t ; y ( s , t ) , \\xi ) ) ^ { - 1 } - I | \\\\ & \\leq C | \\nabla _ x q ( s , t ; y ( s , t ) , \\xi ) - I | \\\\ & \\leq C \\rho ^ { \\varepsilon _ 0 } \\langle s \\rangle ^ { - \\varepsilon _ 1 } . \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} I _ { m + \\nu } ^ { ( 2 ) } ( z ; q ) = \\frac { ( z / 2 ) ^ { \\nu } } { ( q ; q ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } + n \\nu } } { ( q ; q ) _ { n } } \\left ( \\frac { z ^ { 2 } } { 4 } \\right ) ^ { n } \\left ( q ^ { n + \\nu + 1 } ; q \\right ) _ { \\infty } \\frac { \\left ( z q ^ { n } / 2 \\right ) ^ { m } } { ( q ^ { n + \\nu + 1 } ; q ) _ { m } } . \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} P ( D ) = p _ a ^ { | A ( D ) | } ( 1 - p _ a ) ^ { n ( n - 1 ) - | A ( D ) | } \\ D \\in \\mathcal { D } _ n . \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} T _ { k } ^ 1 = ( - 1 ) ^ k , T _ k ^ 2 = ( - 1 ) ^ k \\sum _ { 1 \\leq j \\leq n } a _ { n j } ^ { ( k ) } ( x ) D _ j , k = 1 , 2 . \\end{align*}"}
-{"id": "10142.png", "formula": "\\begin{align*} x _ { i j } ( r ) x _ { i j } ( s ) & = x _ { i j } ( r + s ) , & \\\\ [ x _ { i j } ( r ) , \\ , x _ { h k } ( s ) ] & = 1 , & h \\neq j , \\ k \\neq i , \\\\ [ x _ { i j } ( r ) , \\ , x _ { j k } ( s ) ] & = x _ { i k } ( r s ) . & \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\frac { b _ { p , q } ( x ) - b _ { p , q } ( x _ n ) } { x - x _ n } & = \\frac { \\pi _ q ( b _ 1 b _ 2 \\cdots ) - \\pi _ q ( b _ 1 \\cdots b _ { n - 1 } 0 ^ { \\infty } ) } { \\pi _ p ( b _ 1 b _ 2 \\cdots ) - \\pi _ p ( b _ 1 \\cdots b _ { n - 1 } 0 ^ { \\infty } ) } \\\\ & = \\frac { \\pi _ q ( 0 ^ { n - 1 } 1 b _ { n + 1 } \\cdots ) } { \\pi _ p ( 0 ^ { n - 1 } 1 b _ { n + 1 } \\cdots ) } \\\\ & \\ge \\frac { \\pi _ q ( 0 ^ { n - 1 } 1 0 ^ { \\infty } ) } { \\pi _ p ( 0 ^ { n - 1 } 1 ^ { \\infty } ) } \\\\ & = \\frac { p - 1 } { p } \\left ( \\frac { p } { q } \\right ) ^ n \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta _ { M , N } ( a , b ) ^ { k } \\bigr ] & = \\exp \\Bigl ( - \\sum \\limits _ { l = 0 } ^ { k - 1 } \\bigl ( \\mathcal { S } _ N \\log \\Gamma _ { M - 1 } \\bigr ) ( l \\ , | \\ , \\hat { a } _ i , b ) \\Bigr ) , \\\\ { \\bf E } \\bigl [ \\beta _ { M , N } ( a , b ) ^ { - k } \\bigr ] & = \\exp \\Bigl ( \\sum \\limits _ { l = 0 } ^ { k - 1 } \\bigl ( \\mathcal { S } _ N \\log \\Gamma _ { M - 1 } \\bigr ) ( - ( l + 1 ) \\ , | \\ , \\hat { a } _ i , b ) \\Bigr ) , \\ ; k < b _ 0 . \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} ( \\rho _ i - J _ { s _ i } ) X _ { 1 1 , i } ( \\rho _ i - J _ { s _ i } ) ^ H = X _ { 1 1 , i } \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{gather*} \\pi ( g ^ { [ k ] ( \\alpha ) } _ { - } ) = \\begin{bmatrix} S ^ { + } ( z ) \\tau ^ { ( \\alpha ) } _ { k } & S ^ { + } ( z ) \\tau ^ { ( \\alpha ) } _ { k - 1 } / z \\\\ S ^ { - } ( z ) \\tau ^ { ( \\alpha ) } _ { k + 1 } / z & S ^ { - } ( z ) \\tau ^ { ( \\alpha ) } _ { k } \\end{bmatrix} / \\tau ^ { ( \\alpha ) } _ { k } . \\end{gather*}"}
-{"id": "10106.png", "formula": "\\begin{align*} C \\ : : \\ : y ^ q ( a x + b y + c z ) ^ r - x ^ { - p } z ^ { p + q + r } = 0 , \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} \\begin{cases} \\mu ^ { ( 0 ) } & = \\delta _ { t , t } , \\\\ \\mu ^ { ( m ) } & = \\tfrac { 1 } { 3 } \\delta _ { 2 ^ { - m } t , - 2 ^ { - m } t } + \\tfrac { 1 } { 6 } \\delta _ { - 2 ^ { 1 - m } t , 2 ^ { 1 - m } t } + \\tfrac { 1 } { 2 } \\mu ^ { ( m - 1 ) } \\end{cases} \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{gather*} D _ { r } = \\ . . . \\ = D _ { r + d - 1 } = 0 \\ , \\end{gather*}"}
-{"id": "7164.png", "formula": "\\begin{align*} \\dim ( U ^ { d / 2 } ) = d + 1 = \\mu _ d ( 2 ) \\& \\dim _ q ( U ^ { d / 2 } ) = ( Q _ { d / 2 } ) = \\sum _ i | q | ^ { 2 i } = \\mu _ d ( | q | + | q | ^ { - 1 } ) . \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\sigma } ( \\mathbf { u } ) : = \\mathbf { F } ( \\nabla _ { \\sigma } \\mathbf { u } ) \\mathbf { u } \\in L ^ { 2 } ( G , \\mathbb { R } ^ { k } ) . \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} R _ { \\mathsf { H P F } } ( R _ { \\mathsf { c } } ) & > \\sum _ { n = R _ { \\mathsf { c } } + 1 } ^ N ( 1 - ( 1 - p _ n ) ^ { L } ) \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} f \\frac { \\partial g } { \\partial x } = \\frac { 1 } { 2 } \\frac { \\partial } { \\partial x } \\left ( f g \\right ) + \\frac { 1 } { 2 } \\left ( f \\frac { \\partial g } { \\partial x } - \\frac { \\partial f } { \\partial x } g \\right ) \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} ( D _ { \\eta } \\zeta ) ( t ) : = \\tfrac { 1 } { \\eta } \\left ( ( s _ { \\eta } \\zeta ) ( t ) - \\tau ( t ) \\right ) , ( D _ { \\eta } v ) ( t ) : = \\tfrac { 1 } { \\eta } \\left ( ( s _ { \\eta } v ) ( t ) - v ( t ) \\right ) , \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} P = e ^ { \\beta ( F - E ) } \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} & - \\frac { n - 2 } { 2 } \\int _ M \\sigma _ 1 ( A ) ( \\Delta \\varphi ) ^ 2 d \\mu _ g \\\\ = & \\frac { ( n - 6 ) ^ 2 ( n - 2 ) | W ( p ) | ^ 2 } { 4 8 n ( n - 1 ) } \\omega _ { n - 1 } \\int _ 0 ^ \\rho \\frac { ( n \\epsilon ^ 2 + 4 r ^ 2 ) ^ 2 } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 4 } u _ \\epsilon ^ 2 r ^ { n + 1 } d r + \\left \\{ \\begin{array} { l l } O ( \\epsilon ^ 4 ) & \\hbox { ~ ~ i f ~ ~ } n = 1 0 , \\\\ O ( \\epsilon ^ 5 | \\log \\epsilon | ) & \\hbox { ~ ~ i f ~ ~ } n = 1 1 , \\\\ O ( \\epsilon ^ 5 ) & \\hbox { ~ ~ i f ~ ~ } n \\geq 1 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} \\frac { H ^ { n - 1 } ( \\{ u = 0 \\} \\cap Q ) } { d i a m ^ { n - 1 } ( Q ) } > \\frac { 3 } { 4 } F ( N ) \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} I _ 3 : & = E \\left [ \\int _ t ^ T \\langle 2 ( u ( s ) - \\hat { \\xi } ( s ) ) , f ( s , u ( s ) , \\nabla u ( s ) , v ( s ) ) - \\hat { f } ( s ) \\rangle \\ , d s | \\mathcal { F } _ t \\right ] \\\\ & \\leq E \\left [ \\int _ t ^ T \\langle 2 | u ( s ) - \\hat { \\xi } ( s ) | , | f _ 0 | + L | u ( s ) | + L | \\nabla u ( s ) | + L | v ( s ) | + | \\hat { f } ( s ) | \\rangle \\ , d s | \\mathcal { F } _ t \\right ] . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\int _ \\Omega \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ N ( b _ i + c _ i ) \\frac { \\partial v _ m ^ 2 } { \\partial x _ i } = \\int _ { \\Omega } \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ N ( b _ i + c _ i ) \\frac { \\partial u ^ 2 } { \\partial x _ i } . \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} H _ { x _ 1 } = \\frac { 1 } { 3 } \\left ( \\frac { \\Omega } { \\alpha ^ k } - k \\frac { \\alpha _ { x x } } { \\alpha ^ 3 } + ( 2 k - 1 ) \\left ( \\frac { \\alpha _ x } { \\alpha ^ 2 } \\right ) ^ 2 \\right ) \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} \\mathcal { P } ( \\mathbf { u } ) = \\frac { 1 } { 2 } \\int _ { G } \\sum _ { | \\beta | \\leq k } c _ { \\beta } \\Big ( \\big ( \\nabla ^ { \\alpha } \\mathbf { u } \\big ) _ { | \\alpha | \\leq s } \\Big ) ^ { \\beta } \\mathrm { d } \\mathbf { x } \\mathcal { P } ( \\mathbf { u } ) = \\mathrm { T V } ( \\mathbf { u } ) , \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} - L v & = f \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ v & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} J ^ { ( \\mu ) } \\mathbf { \\Psi } ( t ) = \\frac { 1 } { \\Gamma ( \\mu ) } \\int ^ t _ 0 \\frac { \\mathbf { \\Psi } ( \\xi ) d \\xi } { ( t - \\xi ) ^ { 1 - \\mu } } \\cong \\mathbf { \\Psi } ( t ) \\textbf { J } ^ { \\mu , T } _ { N _ t \\times N _ t } , \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{gather*} \\mu _ A ( \\epsilon ( \\beta ( c ) h \\alpha ( a ) ) \\otimes \\epsilon ( h ' \\alpha ( b ) ) ) = \\mu _ A ( \\epsilon ( h \\alpha ( a ) ) \\otimes \\epsilon ( \\alpha ( c ) h ' \\alpha ( b ) ) ) . \\end{gather*}"}
-{"id": "1293.png", "formula": "\\begin{align*} A & = \\{ x \\in \\textstyle \\prod X _ { n , k } \\colon ( \\exists m ) ( \\forall n > m ) ( \\exists j ) ( \\forall k > j ) ( x ( n , k ) \\in A _ { n , k } ) \\} , \\\\ B & = \\{ x \\in \\textstyle \\prod X _ { n , k } \\colon ( \\exists m ) ( \\forall n > m ) ( \\exists j ) ( \\forall k > j ) ( x ( n , k ) \\in B _ { n , k } ) \\} . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} \\nu ( A \\times B ) = \\sum _ { i = 0 } ^ N a _ i \\mu ( T ^ { - i } A \\cap B ) , \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\left [ \\bar { P } _ { a } , \\bar { Z } _ { b } \\right ] & = \\frac { 1 } { 2 } \\left [ P _ { a } + Z _ { a } , P _ { b } - Z _ { b } \\right ] , \\\\ & = \\frac { 1 } { 2 } \\left ( \\left [ P _ { a } , P _ { b } \\right ] - \\left [ P _ { a } , Z _ { b } \\right ] + \\left [ Z _ { a } , P _ { b } \\right ] - \\left [ Z _ { a } , Z _ { b } \\right ] \\right ) , \\\\ & = \\frac { 1 } { 2 } \\left ( J _ { a b } - Z _ { a b } - Z _ { b a } + J _ { a b } \\right ) , \\\\ & = J _ { a b } , \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} F _ n ( T _ { ( 2 x ) } ( i , \\alpha ) ) = F _ n ( T _ x ( i ) \\otimes \\Gamma ( \\alpha ) ) = F _ n ( T _ x ( i ) ) \\otimes F _ n ( \\Gamma ( \\alpha ) ) \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} d u ^ n ( t , x ) = - ( - \\Delta u ^ n ( t , x ) + \\bar { f } ( t , x ) + f ^ n ( t , x ) + \\nabla \\cdot \\hat { g } ( t , x ) ) \\ , d t + ( v ^ r ( t , x ) - \\check { v } ^ { n , r } ( t , x ) ) \\ , d W _ t ^ r . \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} v \\partial _ { x } \\mp \\beta _ { x } \\partial _ { v } & = \\omega _ { \\pm } ( I _ { \\pm } ) \\partial _ { \\theta _ { \\pm } } , v > 0 , \\\\ v \\partial _ { x } \\mp \\beta _ { x } \\partial _ { v } & = - \\omega _ { \\pm } ( I _ { \\pm } ) \\partial _ { \\theta _ { \\pm } } , v < 0 . \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} P ' _ { 0 } ( 1 ) = \\frac { \\lambda } { \\gamma + \\xi } P _ { 0 } ( 1 ) = \\frac { \\lambda } { \\gamma + \\xi } \\frac { \\xi } { \\gamma A } p _ { 0 , 0 } \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} \\underline { K _ { t } } \\| A ^ { 1 - \\nu } X B ^ { \\nu } \\| _ { 2 } ^ { 2 } & \\leqslant \\| ( 1 - \\nu ) A X - \\nu X B \\| _ { 2 } ^ { 2 } - r _ { 0 } ^ { 2 } \\| A X - X B \\| _ { 2 } ^ { 2 } \\\\ & - \\sum _ { k = 1 } ^ { t - 1 } r _ { k } \\| A ^ { 1 - \\frac { m _ { k } } { 2 ^ { k } } } X B ^ { \\frac { m _ { k } } { 2 ^ { k } } } - A ^ { 1 - \\frac { m _ { k } + 1 } { 2 ^ { k } } } X B ^ { \\frac { m _ { k } + 1 } { 2 ^ { k } } } \\| _ { 2 } ^ { 2 } \\\\ & \\leqslant \\overline { K _ { t } } \\| A ^ { 1 - \\nu } X B ^ { \\nu } \\| _ { 2 } ^ { 2 } . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} ( u _ 0 , v _ 0 , T _ { e , 0 } , q _ { e , 0 } ) \\in H ^ 1 ( \\mathbb R ^ 2 ) , \\quad \\mbox { w i t h } \\nabla \\cdot u _ 0 = 0 . \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} & G ^ 0 = G \\\\ & G ^ { \\alpha } = ( G ^ { \\pi _ \\alpha ( n ) } ) ^ * \\mbox { i f } \\alpha > 0 \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } \\frac { b ' ( t ) } { b ( t ) ^ 2 } = 0 \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} \\lambda p _ { 0 , 0 } = \\xi p _ { 0 , 1 } + \\mu p _ { 1 , 1 } . \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} \\lambda _ { p , 1 } ( t ) = \\lambda _ { p , 1 } ( u ( t ) , t ) = - \\int _ M u ( t , x ) \\Delta _ p u ( t , x ) d \\mu _ { g ( t ) } . \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} \\mathcal { S } e _ { j } = \\varphi _ { z _ { j } } - \\sum _ { z _ { i } \\in \\mathcal { C } \\left ( z _ { j } \\right ) } \\varphi _ { z _ { j } } \\left ( z _ { i } \\right ) \\varphi _ { z _ { i } } + f _ { j } , \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} I m ( d _ 3 ) = K e r ( \\phi : \\Z / m \\to H ^ 4 _ G ( M , \\Z ) ) , \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} \\partial _ x ^ k u \\in L ^ { \\infty } _ x ( 0 , L ; H ^ { \\frac { 1 - k } { 3 } } ( 0 , T ) ) , k = 0 , 1 , 2 . \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} 4 \\pi T _ { \\mu } { } ^ { \\nu } = \\left ( \\begin{array} [ c ] { c c } \\left ( \\mathbf { E } \\cdot \\mathbf { D } + \\mathbf { H } \\cdot \\mathbf { B } \\right ) / 2 & \\left ( \\mathbf { E \\times H } \\right ) _ { q } \\bigskip \\\\ - \\left ( \\mathbf { D \\times B } \\right ) _ { p } & E _ { p } D _ { q } + H _ { p } B _ { q } - \\delta _ { p q } \\left ( \\mathbf { E } \\cdot \\mathbf { D } + \\mathbf { H } \\cdot \\mathbf { B } \\right ) / 2 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} b _ { s , s } \\left [ Z _ s , t \\right ] = \\mathbf { 1 } _ { Z _ s \\in \\mathcal { D } _ s } \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} & ( z _ 0 ' ) ^ 2 - ( z _ 1 ' ) ^ 2 \\\\ & = ( z _ 0 ' - z _ 1 ' ) ( z _ 0 ' + z _ 1 ' ) \\\\ & = \\big ( z _ 0 ( z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 ) - z _ 1 ^ 2 z _ 2 \\big ) \\big ( z _ 0 ( z _ 0 ^ 2 - z _ 1 ^ 2 - z _ 2 ^ 2 ) + z _ 1 ^ 2 z _ 2 \\big ) \\\\ & = ( z _ 0 + z _ 2 ) \\big ( z _ 0 ( z _ 0 - z _ 2 ) - z _ 1 ^ 2 \\big ) ( z _ 0 - z _ 2 ) \\big ( z _ 0 ( z _ 0 + z _ 2 ) - z _ 1 ^ 2 \\big ) , \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} \\Delta ^ { 1 , 2 } ( x _ 1 y _ 1 z _ 3 ) + \\Delta ^ { 2 , 3 } ( x _ 1 y _ 2 z _ 2 ) + \\Delta ^ { 3 , 1 } ( x _ 3 y _ 2 z _ 3 ) = 3 x _ 1 y _ 2 z _ 3 + \\tau ( x _ 3 y _ 2 z _ 1 ) \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} \\pi ( [ x , y ] ) & = [ \\pi ( x ) , \\pi ( y ) ] + \\rho ( x ) \\pi ( y ) - \\rho ( y ) \\pi ( x ) , & x , y & \\in \\mathfrak { g } . \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} \\gamma ^ l = \\frac { 1 } { \\delta ^ 1 } \\prod _ { j = 2 } ^ l ( 1 - \\alpha ^ j ) ^ { - 1 } = \\frac { 1 } { \\delta ^ 1 } \\prod _ { j = 2 } ^ l \\frac { j + 1 } { j - 1 } = \\frac { 1 } { \\delta ^ 1 } \\frac { l ( l + 1 ) } { 2 } . \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} & v _ { 2 n - 2 } ^ * v _ { 2 n - 2 } + v _ { 2 n - 1 } ^ * v _ { 2 n - 1 } \\\\ = & v _ { 2 n - 2 } ^ * v _ { 2 n - 2 } + ( 1 \\otimes f _ n - v _ { 2 n - 2 } ^ * v _ { 2 n - 2 } ) w _ n ^ * w _ n ( 1 \\otimes f _ n - v _ { 2 n - 2 } ^ * v _ { 2 n - 2 } ) \\\\ = & v _ { 2 n - 2 } ^ * v _ { 2 n - 2 } + 1 \\otimes f _ n - v _ { 2 n - 2 } ^ * v _ { 2 n - 2 } \\\\ = & 1 \\otimes f _ n . \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} f _ \\infty ^ { ( s ) } ( t , Z _ s ) = \\int _ { \\mathbb { R } ^ { 2 d } } f _ \\infty ^ { ( s + 1 ) } ( t , Z _ { s + 1 } ) d z _ { s + 1 } \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} \\bigoplus _ { ( i , \\alpha ) } T _ { ( 2 x ) } ( i , \\alpha ) = \\bigoplus _ { ( i , \\alpha ) } H _ { ( 2 x ) } ( i , \\alpha ) \\varphi ( i , \\alpha ) \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} S ( u , v ) = ( u + \\frac { 1 } { 3 } u ^ 3 - u v ^ 2 , - v - \\frac { 1 } { 3 } v ^ 3 + v u ^ 2 , v ^ 2 - u ^ 2 ) \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} t _ m = c _ 1 A ^ { - 1 } ( c _ 2 2 ^ m ) \\end{align*}"}
-{"id": "10053.png", "formula": "\\begin{align*} \\beta ^ 2 \\gamma = 2 + 2 \\beta + \\gamma \\iff \\gamma ( \\beta - 1 ) = 2 . \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} \\mathbf { H } = \\mathbf { F } ( \\nabla \\mathbf { u } ) ( 0 , \\infty ) \\times G \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} \\Xi ( \\xi _ 0 ) : = \\{ ( y _ { 1 } , \\dots , y _ { m } , x ^ { \\prime } ) \\in \\partial D : \\left ( \\left \\vert y _ { 1 } \\right \\vert , \\dots , \\left \\vert y _ { m } \\right \\vert , x ^ { \\prime } \\right ) = \\xi _ 0 \\in \\partial \\Omega \\} \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} 3 z + 1 = 2 ^ 4 y = 2 ^ 2 ( 3 \\xi ( T x ) + 1 ) . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} \\frac { b } { [ \\log ( b ) ] ^ { 2 } } \\ = \\ \\log ( s ) , \\ \\mbox { i . e . } \\ \\frac { b ^ { \\frac { 1 } { 2 } } } { \\log ( { b ^ { \\frac { 1 } { 2 } } ) } } \\ = \\ \\sqrt { 4 \\log ( s ) } . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} U _ t ( f ) = \\exp ( { 2 \\pi \\sqrt { - 1 } t f } ) , \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} E _ R ( n ) \\sim \\sum _ { k = 1 } ^ \\infty \\frac { 9 } { 4 ^ { 1 - k } + 4 + 4 ^ k } \\approx 1 . 6 2 2 9 7 . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} \\beta \\beta ^ { t r } + 2 d d ^ { t r } = I d , \\gamma ^ { t r } \\gamma + 2 g g ^ { t r } = I d , \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } e ^ k = \\lim _ { k \\to \\infty } \\psi ( e ^ k ) = 0 . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} \\phi ( 0 ) = \\xi \\ , , \\phi ' ( 0 ) = \\eta \\ , . \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} \\frac { \\partial \\mathcal { L } _ { 1 } } { \\partial P _ { \\alpha \\beta } ^ { \\ast } } = Q ^ { \\beta \\alpha } , \\qquad \\frac { \\partial \\mathcal { L } _ { 1 } ^ { \\ast } } { \\partial P _ { \\alpha \\beta } } = \\overset { \\ast } { \\left . Q ^ { \\beta \\alpha } \\right . } \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} H ^ p ( Y , R ^ q f _ * ( K _ X \\otimes F \\otimes \\mathcal J ( h ) ) \\otimes H ^ { \\otimes m } ) = 0 \\end{align*}"}
-{"id": "9909.png", "formula": "\\begin{align*} \\left | ( \\sum _ { i = 1 } ^ l | B _ i | ) \\int _ { G / \\Gamma } f \\dd \\mu _ G \\right | \\leq \\epsilon | I | , \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\varPhi = \\varPhi ( F ) = \\varPhi ( \\tilde { F } ^ { - 1 } ) . \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} \\left ( - z q ^ { 1 / 2 } , - z ; q \\right ) _ { \\infty } = \\left ( - z ; q ^ { 1 / 2 } \\right ) _ { \\infty } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ( n - 1 ) / 4 } z ^ { n } } { \\left ( q ^ { 1 / 2 } ; q ^ { 1 / 2 } \\right ) _ { n } } = \\sum _ { k = 0 } ^ { \\infty } z ^ { k } q ^ { k ^ { 2 } / 2 } S _ { k } \\left ( - q ^ { - k - 1 / 2 } ; q \\right ) , \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } X _ s ^ { n } = x + \\int _ t ^ s b ^ n ( r , X _ r ^ { n } ) \\mathrm d r + \\int _ t ^ s \\mathrm d W _ r , \\\\ Y _ s ^ { n } = \\Phi ( X _ T ^ { n } ) - \\int _ s ^ T Z ^ n _ r \\mathrm d W _ r + \\int _ s ^ T f ( r , X ^ n _ r , Y _ r ^ { n } , Z _ r ^ n ) \\mathrm d r \\end{array} \\right . \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} \\begin{array} { c l } P _ { n } ^ { c } \\left ( x \\right ) & = P _ { n } \\left ( x \\right ) - \\sum \\limits _ { i = 1 } ^ { d } A _ { i } \\left ( x \\right ) P _ { n - k - i } ^ { \\left ( k + i \\right ) } \\left ( x \\right ) , d \\geq 1 , n , k \\geq 0 , \\\\ P _ { n } ^ { c } \\left ( x \\right ) & = P _ { n } \\left ( x \\right ) , n \\leq k . \\end{array} \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} \\Lambda _ { K _ h , c } : = \\left \\{ B _ A \\in \\Lambda _ { n - 1 } \\ : \\ \\Delta _ { K _ h } ( \\phi ( \\delta _ m ( B _ A ) ) ) = c \\right \\} . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} & \\alpha ~ = ~ \\frac { \\ell } { M } \\left ( 1 + \\frac { 1 } { r } \\right ) - 1 , ~ ~ ~ \\beta ~ = ~ 1 - \\frac { \\ell - 1 } { M } \\left ( 1 + \\frac { 1 } { r } \\right ) , \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} A _ - + * ( A _ + ) = ( A + * A ) _ - = ( B + * B ) _ - = B _ - + * ( B _ + ) . \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} ( ( 2 \\pi n ) ^ { 2 } - ( 2 \\pi ( n + p ) ) ^ { 2 } ) ( \\Psi , e ^ { i 2 \\pi ( n + p ) x } ) = \\sum _ { m \\in \\mathbb { N } } q _ { m } ( \\Psi , e ^ { i 2 \\pi ( n + p - m ) x } ) \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ { \\mathbb { R } ^ { 2 d } } \\left ( x \\cdot v - | v | ^ 2 t \\right ) f ( t , x , v ) d x d v = 0 \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} E _ 1 = \\begin{pmatrix} C _ 1 & C _ 1 \\\\ C _ 1 & C _ 1 \\end{pmatrix} E _ 2 = \\begin{pmatrix} C _ 2 & C _ 2 \\\\ C _ 2 & C _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} P ^ \\prime ( t ) = B _ s ( P ^ \\prime ( \\cdot ) ) ( t ) + R ^ \\prime ( t ) , \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} \\mu _ 1 = \\mu _ 1 ^ \\mathrm { p p } + \\mu _ 1 ^ \\mathrm { a c } \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} \\rho ^ + ( \\sigma ) + \\rho ^ - ( \\sigma ) = 1 , \\\\ \\rho ^ + ( \\sigma ) = \\begin{cases} 1 , & \\sigma \\geq \\frac { 3 } { 4 } , \\\\ 0 , & \\sigma \\leq \\frac { 1 } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} ( - z \\xi ) ^ { n } q ^ { - \\frac { 1 } { 2 } n ( n + 1 ) } \\left ( z ^ { - 1 } \\xi ; q \\right ) _ { \\infty } \\ ! \\left [ \\sum _ { j = 1 } ^ { n } \\frac { ( q z \\xi ^ { - 1 } ; q ) _ { j } } { ( q ; q ) _ { n - j } } z ^ { - 2 j } + \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { k ( k - 1 ) / 2 } } { \\left ( z ^ { - 1 } \\xi ; q \\right ) _ { k } ( q ; q ) _ { k + n } } ( - z \\xi ) ^ { k } \\right ] \\ ! \\ ! . \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} d z _ 0 = P \\ , d X - \\frac { 1 } { 2 } P \\Omega _ { [ 1 ] } \\ , d Y + \\Delta \\ , d T . \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\left \\langle J _ { a b } , Z _ { c d } , \\bar { Z } _ { e } \\right \\rangle & = \\frac { 1 } { \\sqrt { 2 } } \\left \\langle J _ { a b } , Z _ { c d } , P _ { e } \\right \\rangle - \\frac { 1 } { \\sqrt { 2 } } \\left \\langle J _ { a b } , Z _ { c d } , Z _ { e } \\right \\rangle , \\\\ & = \\left ( \\alpha _ { 1 } - \\alpha _ { 2 } \\right ) \\ , \\varepsilon _ { a b c d e } , \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} \\frac { d } { d t } \\lambda ( t ) \\int _ M \\varphi ^ 2 \\ d \\mu = \\lambda ( t ) \\int _ M \\mathcal { H } \\varphi ^ 2 \\ d \\mu - \\int _ M \\mathcal { H } | \\nabla \\varphi | ^ 2 \\ d \\mu + 2 \\int _ M \\langle h , d \\varphi \\otimes d \\varphi \\rangle \\ d \\mu . \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} K _ { k + n , l } { } ^ { m } & = \\left \\{ \\begin{array} [ c ] { l l } 1 , & m \\equiv k + n + l \\ \\left ( \\operatorname { m o d } 2 n \\right ) \\\\ 0 , & o t h e r w i s e \\end{array} \\right . \\\\ & = \\left \\{ \\begin{array} [ c ] { l l } 1 , & m + n \\equiv k + { 2 n } + l \\ \\left ( \\operatorname { m o d } 2 n \\right ) \\\\ 0 , & o t h e r w i s e \\end{array} \\right . \\\\ & = K _ { k l } { } ^ { m + n } . \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} & \\mathrm { I d } = P _ 1 ( \\lambda ) + \\cdots + P _ m ( \\lambda ) , P _ j ( \\lambda ) P _ k ( \\lambda ) = 0 , j \\neq k , \\\\ & C ( \\lambda ) = e ^ { i \\theta _ 1 ( \\lambda ) } P _ 1 ( \\lambda ) + \\cdots + e ^ { i \\theta _ m ( \\lambda ) } P _ m ( \\lambda ) . \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - & ( r ^ { N - 1 } | u ' | ^ { p - 2 } u ' ) ' = \\mu _ k ( p ) r ^ { N - 1 } | u | ^ { p - 2 } u , & & r \\in ( 0 , 1 ) , \\\\ & u ' ( 0 ) = 0 , ~ ~ u ( 1 ) = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} \\norm { w } _ { L ^ { \\infty } } & \\leq C \\norm { w } _ { H ^ 1 _ z L _ { x y } ^ { \\infty } } = C \\norm { \\partial _ z w } _ { L ^ 2 _ z L _ { x y } ^ { \\infty } } = C \\norm { \\div _ H v } _ { L ^ 2 _ z L _ { x y } ^ { \\infty } } \\\\ & \\leq C \\norm { \\div _ H v } _ { L ^ 2 _ z H ^ 1 _ { x y } } \\leq C \\norm { v } _ { L ^ 2 _ z H ^ 2 _ { x y } } \\leq C \\norm { \\Delta v } , \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} M _ 2 ( l , u , v ) = \\sum _ { f \\in H _ { 2 k } ^ { * } ( N ) } ^ { h } \\lambda _ f ( l ) L _ f ( 1 / 2 + u + v ) L _ f ( 1 / 2 + u - v ) . \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\mathcal { H } _ { g ( t ) } \\ \\ \\geq \\ \\ \\psi ( t ) \\ \\ = \\ \\ \\frac { \\mathcal { H } _ { m i n } ( 0 ) } { 1 - \\frac { 2 } { n } \\mathcal { H } _ { m i n } ( 0 ) t } \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} q ( x ) = \\sum _ { \\gamma \\in Z ^ { + } } q _ { \\gamma } e ^ { i \\left \\langle \\gamma , x \\right \\rangle } , \\sum _ { \\gamma \\in Z ^ { + } } \\mid q _ { \\gamma } \\mid < \\infty , \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} P _ 1 & = ( 4 , 8 7 9 3 6 0 / 1 1 6 2 8 1 ) , \\\\ P _ 2 & = ( 3 1 6 8 4 / 1 1 6 2 8 1 , 1 9 0 7 1 0 6 2 4 0 / 1 3 5 2 1 2 7 0 9 6 1 ) , \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} X _ A = \\mathrm { I m } ( \\phi _ A ) = \\{ x \\in X : J _ x \\neq A \\} . \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} a = \\sigma , \\ b = ( a , \\ c ) , \\ c = ( a , \\ d ) , \\ d = ( 1 , \\ b ) , \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} ( f \\bullet _ i g ) \\bullet _ j h = \\left \\{ \\begin{aligned} & ( f \\bullet _ j h ) \\bullet _ { i + p - 1 } g & & & 1 \\leq j \\leq i - 1 & \\\\ & f \\bullet _ i ( g \\bullet _ { j - i + 1 } h ) & & & i \\leq j \\leq n & \\end{aligned} \\right . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} F ^ { k l } \\phi _ { ; k l } - \\dot { \\phi } \\leq & \\ , \\phi ^ { i j } \\{ - F ^ { k l } h _ { r k } h ^ r _ l W _ { i j } - F ^ { k l } g _ { k l } W _ { i j } + 2 F h _ i ^ k W _ { k j } \\\\ & \\ , + 2 F \\epsilon g _ { i j } - F ^ { k l , r s } W _ { k l ; i } W _ { r s ; j } \\} + F ^ { k l } \\phi ^ { i j , r s } W _ { i j ; k } W _ { r s ; l } + O ( \\phi ) . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} e ^ { - S _ { e f f } [ \\Lambda , g ] } = \\int _ { C ^ { \\infty } ( \\Sigma , M ) [ 0 , \\Lambda ] } D \\phi \\ , \\ , e x p \\left \\{ - \\langle { \\phi } , L ( \\Lambda , \\phi , g ) { \\phi } \\rangle \\right \\} \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} w _ i = 0 , w _ { i , J } = 0 , \\chi = 0 \\Gamma \\times ( 0 , \\infty ) \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\dot { x } = f ( x , \\theta ) , \\theta \\in \\Theta \\subset \\R ^ { n _ p } \\end{align*}"}
-{"id": "10160.png", "formula": "\\begin{align*} \\mathbb P \\left ( Z ^ * _ n \\le - 1 \\right ) = \\frac 1 2 \\mathbb P ( T _ 0 > n ) \\sim \\frac { \\gamma \\ , \\mathbb E [ \\mathcal R _ n ] } { 2 n } \\ , . \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} D _ { L } ( L - N , B ( 0 , r ) ) = _ { B } \\biggl { ( } - \\dfrac { 1 } { T } \\int _ { 0 } ^ { T } \\tilde { f } ( t , \\cdot ) ~ \\ ! d t , \\mathopen { ] } - r , r \\mathclose { [ } , 0 \\biggr { ) } = 1 , \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} \\bar \\Gamma \\ni \\bar I ( z + \\omega ) - \\bar I ( z ) = \\Phi ( z + \\omega ) - \\Phi ( z ) = c \\omega \\ \\ \\ \\ \\forall \\omega \\in \\Gamma , \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} & f _ 0 = f _ Y | _ { \\mathbb { P } \\mathrm { H o m } ( F , C _ 0 , L _ 0 ) _ e } : \\mathbb { P } \\mathrm { H o m } ( F , C _ 0 , L _ 0 ) _ e \\to \\overline { \\mathcal { C } ( d _ 1 , d _ 2 , c ) } : \\\\ & ( [ F ] , C _ 0 , [ L _ 0 ] , \\mathbf { k } \\varphi ) \\mapsto [ \\ker ( \\varphi : F \\twoheadrightarrow L _ 0 ( 2 ) ) ] . \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} S _ { k + 1 } & : = \\frac { ( - 1 ) ^ { k + 1 } k } { ( k - 1 ) ! } \\sum _ { p _ 1 \\neq \\dotsb \\neq p _ k } \\Bigl [ \\oint _ M w ^ { p _ 1 } w ^ { p _ 2 } _ { n } \\dotsm w ^ { p _ { k } } _ n u _ n \\sigma _ { k - 1 } ( L ) d \\mu \\\\ & + \\frac { 1 } { k } \\oint _ M u w ^ { p _ 1 } _ n \\dotsm w ^ { p _ { k } } _ n \\sigma _ { k - 1 } ( L ) d \\mu \\Bigr ] . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} y _ i = \\beta _ 0 + \\beta _ 1 x _ i + \\epsilon _ i \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} Q = \\left [ \\begin{array} { c c } u & \\j v \\\\ \\j v & u \\end{array} \\right ] \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } S _ { j } \\left ( q ^ { 2 \\alpha - 1 / 2 } ; q \\right ) S _ { k } \\left ( q ^ { 2 \\alpha - 1 / 2 } ; q \\right ) q ^ { 2 \\alpha ^ { 2 } } d \\alpha = \\sqrt { \\frac { \\pi } { \\log q ^ { - 2 } } } \\frac { q ^ { - j } \\delta _ { j , k } } { \\left ( q ; q \\right ) _ { j } } . \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} & \\partial _ t f + u \\cdot \\nabla _ t f = b \\partial _ z \\phi + \\nu \\tilde { \\Delta } _ t f \\\\ & \\Delta _ t \\phi = f \\\\ & u = \\nabla ^ { \\perp } _ t \\phi , \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } v = a ^ { i j } v _ { x ^ { i } x ^ { j } } + f ( u ) + \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { t } [ \\sigma ^ { i j k } v _ { x ^ { i } x ^ { j } } + g ^ { k } ( u ) ] d w _ { s } ^ { k } \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} H ^ i ( Y , R ^ j f _ * ( K _ X \\otimes F \\otimes \\mathcal J ( h ) \\otimes N ) ) = 0 \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\vert y _ i - m \\vert - \\inf _ b \\ , \\sum _ { i = 1 } ^ n \\vert y _ i - m - b Z _ i \\vert \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} B _ \\alpha ^ * ( B _ \\alpha ^ * ) ^ { t r } = \\begin{pmatrix} I / 2 & 0 \\\\ 0 & D _ \\alpha \\end{pmatrix} , D _ \\alpha = \\begin{pmatrix} 1 / 2 & 0 \\\\ 0 & e _ \\alpha \\end{pmatrix} , \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\frac { k ^ { 1 / \\nu } + k m ^ { 1 / \\nu - 1 } + o ( k m ^ { - 1 / 2 } ) + O ( m ^ { - 1 / 4 } k ^ { 1 / 2 } \\sqrt { \\log k } ) } { z ( m , k , \\gamma ) } = 0 . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} V _ { i j } = { n - j \\choose n - i } ( - 1 ) ^ { i - j } I . \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} 2 R ^ { \\mu \\nu } = e ^ { \\mu \\nu \\sigma \\tau } G _ { \\sigma \\tau } , \\qquad - 2 S ^ { \\mu \\nu } = e ^ { \\mu \\nu \\sigma \\tau } F _ { \\sigma \\tau } . \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} \\| x ^ { k + 1 } - x ^ * \\| ^ 2 & \\leq \\| x ^ k - x ^ * \\| ^ 2 + N \\tilde C \\alpha _ k ^ 2 + 4 \\alpha _ k M N \\sum _ { i = 1 } ^ N \\bar L _ i \\| \\hat v _ i ^ k - y ^ k \\| \\cr & \\ ; - 2 \\alpha _ k \\left ( \\phi ( x ^ k ) - \\phi ( x ^ * ) \\right ) ^ T ( x ^ k - x ^ * ) , \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} ( \\nabla | u | ^ 2 u ) \\nabla u & = \\sum _ { j , k , \\ell = 1 } ^ n ( \\partial _ k u ^ j ) \\partial _ k ( u ^ \\ell ) ^ 2 u ^ j \\\\ & = \\sum _ { j , k , \\ell = 1 } ^ n ( u ^ \\ell ) ^ 2 ( \\partial _ k u ^ j ) ^ 2 + 2 \\sum _ { k = 1 } ^ n ( u \\cdot \\partial _ k u ) ^ 2 . \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{gather*} \\big ( \\tau _ { 0 } ^ { ( \\alpha ) } \\big ) ^ 2 = \\tau _ { 0 } ^ { ( \\alpha - 1 ) } \\tau _ { 0 } ^ { ( \\alpha + 1 ) } - \\tau _ { 1 } ^ { ( \\alpha - 1 ) } \\tau _ { - 1 } ^ { ( \\alpha + 1 ) } , \\end{gather*}"}
-{"id": "9996.png", "formula": "\\begin{align*} h ( x ) = \\begin{cases} e ^ { x - a } & x < a , \\\\ 1 & a \\le x \\le b , \\\\ e ^ { b - x } & x > b . \\end{cases} \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} g _ s ( x ) = ( x - 2 ) ^ { - r - 2 s } x ^ { 2 s } = \\sum _ { j = 0 } ^ { - r - 2 s } { - r - 2 s \\choose j } ( - 2 ) ^ { - r - 2 s - j } x ^ { 2 s + j } = \\sum _ { i = 0 } ^ { - r } \\gamma _ i x ^ i \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} T \\big ( D _ \\lambda ( a ) \\big ) = d _ \\lambda \\big ( T ( a ) \\big ) , ; \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} \\nu ^ { \\pi ^ * } _ { n - 1 } ( y _ { n - 1 } | y _ { n - 2 } ) = \\sum _ { x _ { n - 1 } \\in \\{ 0 , 1 \\} } q _ { n - 1 } ( y _ { n - 1 } | x _ { n - 1 } , y _ { n - 2 } ) \\pi ^ * _ { n - 1 } ( x _ { n - 1 } | y _ { n - 2 } ) . \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{gather*} \\partial _ \\mu R = F _ \\mu \\end{gather*}"}
-{"id": "6706.png", "formula": "\\begin{align*} g ( t ) \\triangleq f ( t ) e ^ { - q t } \\frac { d ^ r } { d t ^ r } \\bigl [ e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ { M - 1 } ( 1 - e ^ { - b _ j t } ) \\bigr ] . \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} \\star \\partial e ^ { t \\Delta } f = e ^ { t \\Delta } \\star \\partial f , \\end{align*}"}
-{"id": "9886.png", "formula": "\\begin{align*} \\mathcal { W } _ \\sigma ( z ) \\equiv - \\ , W \\left [ w _ \\sigma , \\ , w _ { \\rm r e g } \\right ] ( 0 ) = \\delta _ { \\sigma , 0 } \\ , \\left ( \\sum _ { j = 0 } ^ \\infty j \\ , b _ j ( z ) \\right ) + \\delta _ { \\sigma , 1 } \\ , \\left ( \\sum _ { j = 0 } ^ \\infty b _ j ( z ) \\right ) \\ , , \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} \\phi _ { A _ { \\nu } } \\left ( z \\right ) = 1 + \\sum _ { k = 1 } ^ { s - 1 } c _ { k o } z ^ { k o } = \\frac { 1 - \\left ( c _ { o } z ^ { o } \\right ) ^ { s } } { 1 - c _ { o } z ^ { o } } = \\frac { 1 - \\frac { \\mu _ { \\nu } } { \\mu _ { w } } \\cdot c _ { o } ^ { 1 - n / p } z ^ { o s } } { 1 - c _ { o } z ^ { o } } . \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} b _ { \\mu , N } ( l ) - \\sum _ { m = 0 } ^ { N + \\mu n } T ^ { \\mu } _ { l , m } a _ { \\mu , N } ( m ) = \\sum _ { k = 0 } ^ { N + \\mu n } P _ { l , k } ^ { \\mu } c _ { \\mu } ( k ) , \\quad 0 \\leq l \\leq N , \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\left \\vert x _ { i } - \\dfrac { p _ { i } } { q } \\right \\vert \\leq \\dfrac { 1 } { T q } & & \\left ( i = 1 , 2 , . . . , n \\right ) \\\\ 1 \\leq q < T ^ { n } & & \\end{array} \\right . \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} T ( t , s ) \\big ( ( u _ 0 , v _ 0 ) \\big ) ~ : = ~ ( u , v ) ( t , \\cdot ) , \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} V _ { s + 1 } ^ 0 ( \\tau ) = V _ { s + 1 } ^ { 0 , + } ( \\tau ) - V _ { s + 1 } ^ { 0 , - } ( \\tau ) \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} \\frac { 1 } { | n | } \\langle \\gamma '' , \\gamma ' \\times n \\rangle = \\langle \\gamma '' , J ^ { 9 0 } _ \\gamma ( \\gamma ' ) \\rangle = \\frac { 1 } { \\kappa } \\langle \\gamma '' , \\nabla _ { \\gamma ' } \\gamma ' \\rangle = \\frac { 1 } { \\kappa } | \\nabla _ { \\gamma ' } \\gamma ' | ^ 2 \\ ; . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} \\dim \\delta H _ 3 ^ g = & \\ ; 5 + 8 ( k + 2 ) + 2 ( k + 1 ) ( k + 2 ) + k ( k + 1 ) / 2 \\\\ & \\ ; + { ( k - 1 ) k ( k + 1 ) } / { 6 } - { ( k + 4 ) ( k + 5 ) ( k + 6 ) } / { 6 } \\\\ = & \\ ; 2 \\ , k + 5 , \\\\ \\dim \\delta E _ 3 ^ g = & \\ ; I _ M ( V _ 3 ^ g \\times W _ 3 ^ g ) = \\ ; 3 , \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} \\alpha ( t ) = & P _ p ( { T - t } ) \\Phi + \\int ^ T _ t P _ p ( { r - t } ) \\left ( { \\nabla u ( r ) \\ , b ( r ) } \\right ) \\mathrm d r \\\\ & + \\int ^ T _ t P _ p ( { r - t } ) f ( r , \\alpha ( r ) , \\nabla \\alpha ( r ) ) \\mathrm d r . \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} [ C _ { \\gamma } ( f ) ] ( z ) & = \\dfrac { 1 } { \\pi } \\int _ { D } \\dfrac { f ( w ) } { w - z } d \\mu _ { \\gamma } ( w ) . \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} \\varphi ^ e _ n ( z ) : = \\sup \\{ \\psi ( z ) : \\psi \\in \\mathcal { L } ( \\C ) , \\psi \\leq \\varphi _ n \\ \\ \\C \\} \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } q ^ { 2 n ^ { 2 } - n } S _ { 2 n } \\left ( x q ^ { - 2 n } ; q \\right ) = \\frac { \\left ( x ; q \\right ) _ { \\infty } } { \\left ( q ; q ^ { 2 } \\right ) _ { \\infty } } . \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} A = \\frac { 1 } { 4 \\pi } \\int _ { 0 } ^ { 2 \\pi } { \\int _ { 0 } ^ { \\pi } { { { v } ^ { H } } ( \\theta , \\phi ) v ( \\theta , \\phi ) \\sin \\theta d \\theta d \\phi } } , \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} A = { \\rm d i a g } ( \\nu _ { 1 } , \\ldots , \\nu _ { n } ) . \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} \\hat { D } ( i , l ) = \\begin{cases} 1 & d ( a _ i ) = d ( b _ l ) , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} \\norm { \\zeta ( t ) } ^ 2 + \\int _ 0 ^ t \\norm { \\zeta ( s ) } _ { H ^ 1 ( \\Omega ) ^ 2 } ^ 2 d s \\leq ( 1 + t ) B _ { L ^ 2 , 1 } ^ { \\zeta } ( t ) + B _ { L ^ 2 , 2 } ^ { \\zeta } ( t ) = : B _ { L ^ 2 } ^ { \\zeta } ( t ) . \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} x _ 0 ^ * : = ( x _ 0 + n _ 0 ) / \\sqrt { 2 } , n _ 0 ^ * : = ( x _ 0 - n _ 0 ) / \\sqrt { 2 } . \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} \\rho ( t ) ( f + \\sum _ { i \\leq j } c _ { i j } e _ { i , j } ) = f + \\sum _ { i \\leq j } t ^ { 2 ( i - j + 1 ) } c _ { i j } e _ { i , j } . \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} \\mathcal I _ j ( x , y ) = & \\iint a _ { 1 , j } ( \\tau , s , \\xi ) e ^ { 2 \\pi i \\phi _ 1 ( x , y , t ; , \\xi , \\tau , s , \\theta ) } d \\theta d s d \\tau d \\xi , \\\\ \\mathcal J _ j ( x , y ) = & \\iint a _ { 2 , j } ( \\tau , s , \\xi ) e ^ { 2 \\pi i \\phi _ 2 ( x , y , t ; , \\xi , \\tau , s , r ) } d r d s d \\tau d \\xi , \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} v _ { 0 } ( 0 ) + v _ { \\delta ^ { j } } ( 0 ) g _ { j , 0 } ( x _ { j } ) = u _ { 0 } ( x _ { j } ) \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} \\left \\Vert f ( z ) \\right \\Vert _ { k } = \\sup _ { \\left \\Vert v _ { i } \\right \\Vert = 1 } \\frac { \\left | \\left \\langle f ; v _ { 1 } , \\ldots , v _ { q } \\right \\rangle ( z ) \\right | } { \\sum _ { i = 1 } ^ { q } k \\left ( z , v _ { i } \\right ) } , \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} Q ^ \\prime ( t ) & : = \\nabla _ \\xi q ( t , s ) - ( t - s ) A ( \\xi ) , \\\\ P ^ \\prime ( t ) & : = \\nabla _ \\xi p ( t , s ) - I . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ { \\gamma } ( z , \\bar { z } ) = ( \\gamma + 1 ) _ { m + n } \\overline { z } ^ m z ^ n { _ 2 F _ 1 } \\left ( \\begin{array} { c } - m , - n \\\\ \\gamma + 1 \\end{array} \\bigg | 1 - \\frac 1 { | z | ^ 2 } \\right ) = ( \\gamma + 1 ) _ { m + n } \\overline { P _ { m , n } ^ { \\gamma } ( z , \\bar z ) } \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} f _ n = & A _ { 2 - n } A _ { 4 - n } A _ { 6 - n } \\Big ( \\sum _ { k = 1 } ^ n \\psi _ k \\Big ) + K _ { 6 - n } \\Big ( \\sum _ { k = 1 } ^ n \\psi _ k \\Big ) + f \\\\ = & O ( r ^ { n + 1 } ) ( \\log ^ 3 r + \\log ^ 2 r + \\log r + 1 ) + O ( r ^ { n + 2 } ) \\log ^ 4 r . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\partial _ { \\nu } R ^ { \\mu \\nu } = - \\frac { 4 \\pi } { c } j ^ { \\mu } , \\qquad \\partial _ { \\nu } S ^ { \\mu \\nu } = 0 . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} \\dot { F } = \\mathbf { Q } F . \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} [ W ' , \\i A ] _ { \\circ } & \\approx [ W _ 1 ' , \\i A _ 1 ] _ { \\circ } \\otimes W _ 2 ' \\otimes . . . \\otimes W _ d ' \\ + \\ W _ 1 ' \\otimes [ W _ 2 ' , \\i A _ 2 ] _ { \\circ } \\otimes . . . \\otimes W _ d ' \\\\ & + \\ . . . \\ + \\ W _ 1 ' \\otimes . . . \\otimes W _ { d - 1 } ' \\otimes [ W _ d ' , \\i A _ d ] _ { \\circ } \\\\ & : = K _ { W ' } + B _ { W ' } \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\tilde { \\jmath } ( z ) : = \\frac { g ( z ) ^ { m ( q + 1 ) / ( q - 1 ) } } { h ( z ) ^ m } , \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} f ( v ) = \\begin{cases} 1 & \\mbox { i f $ d ( v ) \\le S ( 2 n - 2 ) $ } \\\\ ( 1 - ( d ( v ) - S ( 2 n - 2 ) ) h _ { 2 n - 1 } ^ { - 1 } ) _ + & \\mbox { i f $ d ( v ) > S ( 2 n - 2 ) $ . } \\end{cases} \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} \\xi ^ l : = \\delta ^ l \\alpha ^ l \\gamma ^ l = \\delta ^ l \\left ( \\frac { 1 / \\delta ^ l } { \\Lambda ^ l } \\right ) \\Lambda ^ l = 1 . \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} x S ^ 3 _ { \\alpha p } + y S ^ 4 _ { \\alpha p } + z S ^ 5 _ { \\alpha p } = 0 . x T ^ 3 _ { \\alpha p } + y T ^ 4 _ { \\alpha p } + z T ^ 5 _ { \\alpha p } = 0 . \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} \\hat { { \\bf s } } _ i = & { \\bf U } _ i { \\bf H } _ { r , i } { \\bf T } { \\bf s } _ r + { \\bf U } _ i { \\bf H } _ { r , i } { \\bf T } { \\bf W } { \\bf n } _ r + { \\bf U } _ i { \\bf n } _ i , \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} W ( g ) = \\frac { 1 } { 2 } \\ln \\det ( g D ) = - \\frac { 1 } { 2 } \\int _ 0 ^ \\infty \\frac { d t } { t } T r ( g e ^ { - t D } ) \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} x ( t ; \\ , a , b ) \\ , + \\ , i \\ , y ( t ; \\ , a , b ) = F ( t , z ) \\ , + \\ , \\overline { G ( t , z ) } , z = a + i b , \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in \\omega q ^ { 2 \\Z } } \\frac { \\varphi _ { k } \\left ( \\lambda \\right ) \\varphi _ { \\ell } \\left ( \\lambda \\right ) } { \\| \\varphi \\left ( \\lambda \\right ) \\| ^ { 2 } } + \\frac { \\varphi _ { k } \\left ( - \\lambda ^ { - 1 } \\right ) \\varphi _ { \\ell } \\left ( - \\lambda ^ { - 1 } \\right ) } { \\| \\varphi \\left ( - \\lambda ^ { - 1 } \\right ) \\| ^ { 2 } } = \\delta _ { k , \\ell } , \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} \\dim Y _ G & = n \\left ( m \\frac { ( n - 1 ) } { 2 } + \\frac { m } { p } - 1 \\right ) . \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} D ^ F _ { Z _ { 1 , \\infty } } \\omega _ { 1 , \\infty } = 0 . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} f ( 1 , z _ 1 , z _ 2 , z _ 3 ) = \\frac { 1 } { x _ 0 ^ 4 } f ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) , ~ f _ { z _ i } = \\frac { \\partial f ( 1 , z _ 1 , z _ 2 , z _ 3 ) } { \\partial z _ i } . \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} \\tilde { \\pi } ( \\{ \\alpha , \\beta \\} _ { \\pi } ) = [ \\tilde { \\pi } ( \\alpha ) , \\tilde { \\pi } ( \\beta ) ] \\ , , \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\widetilde { g } ( s , v ; p ) = - \\zeta ( s + v ) . \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} u ( t , x ) : = \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { t } \\int _ { \\mathbb { R } ^ { d } } q _ { \\alpha , \\beta } ( t - s , x - y ) g ^ { k } ( s , y ) d y d w _ { s } ^ { k } . \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} A P _ { \\varepsilon } ( \\zeta ) = \\left \\{ z = \\zeta + \\sum _ { k = 1 } ^ { n } \\lambda _ { k } v _ { k } \\mbox { s u c h t h a t } \\left | \\lambda _ { k } \\right | \\leq c _ { 0 } A \\tau _ { k } ( \\zeta , \\varepsilon ) \\right \\} , \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\widehat { u } = e ^ { \\sum _ { j = 2 } ^ d i t _ j \\left ( \\alpha _ j \\tau _ 1 + \\beta _ j \\right ) } \\widehat { f } ( \\tau _ 1 ) . \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} \\norm { T _ t f - \\langle \\varphi , f \\rangle f _ 0 } = \\norm { T _ { t - n _ 0 t _ 0 } T _ { n _ 0 t _ 0 } f - \\langle \\varphi , f \\rangle T _ { t - n _ 0 t _ 0 } f _ 0 } \\le \\sup _ { t \\in ( 0 , \\infty ) } \\norm { T _ t } \\cdot \\varepsilon . \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} 0 = \\sum _ { k = 0 } ^ n a _ k \\tau _ { k y } f ( x ) y \\in B _ { d } ( \\delta ) : = \\{ h \\in \\mathbb { R } ^ d : \\| h \\| < \\delta \\} . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} [ \\bar { \\xi } _ { i , r + 1 } , \\bar { x } ^ { \\pm } _ { j , s } ] - [ \\bar { \\xi } _ { i , r } , \\bar { x } ^ { \\pm } _ { j , s + 1 } ] = - m _ { i , j } \\beta [ \\bar { \\xi } _ { i , r } , \\bar { x } ^ { \\pm } _ { j , s } ] , \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\tau _ 1 ( T ) & = 2 \\Big ( T - 2 T ^ \\top - t r ( T ) i d \\Big ) , \\\\ \\tau _ 2 ( T ) & = - 4 ( T + T ^ \\top ) + 6 \\ast ( e ^ i \\wedge \\sigma _ T \\wedge \\varphi ) \\otimes e _ i , \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\rho \\left ( \\frac { y _ i - m } { \\sigma _ n } \\right ) - \\inf _ b \\ , \\sum _ { i = 1 } ^ n \\rho \\left ( \\frac { y _ i - m - b Z _ i } { \\sigma _ n } \\right ) , \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} a ( b + c ) + a = a b + a c \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} R ^ \\star _ { \\sf u } ( R _ { \\sf c } ) & \\le \\sum _ { n = 1 } ^ N \\frac { p _ n ( 1 - r _ n ) } { 1 - \\alpha _ n } \\left [ 1 - \\alpha _ n ^ L \\right ] , \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} v _ m ( x , t ) = \\frac { 1 } { 2 \\pi i } \\sum _ { j = 0 } ^ { 2 } \\int _ { r - i \\infty } ^ { r + i \\infty } e ^ { s t } \\frac { \\Delta _ { j , m } ( s ) } { \\Delta ( s ) } e ^ { \\lambda _ j ( s ) x } \\hat { h } _ m ( s ) d s : = [ W _ { m } ( t ) h _ m ( t ) ] ( x ) \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m s ( a _ i , b _ i ; \\hat { X } _ { \\mathit { t l s } } ) = 0 . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} - \\varepsilon \\int _ { 0 } ^ { t } ( u ^ { \\prime \\prime } , u ^ { \\prime } ) + \\int _ { 0 } ^ { t } | u ^ { \\prime } | ^ { 2 } + \\phi ( u ( t ) ) - \\phi ( u _ { 0 } ) = \\int _ { 0 } ^ { t } ( f ( \\alpha u ) , u ^ { \\prime } ) . \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} - \\Delta u & = \\lambda k ( u ) + \\mu \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "10027.png", "formula": "\\begin{align*} q ^ { t + r _ t + r _ d } - 1 = | W ^ * | & = \\sum _ { X \\in S T } | X ^ * | \\cr & \\leq | V ( t , q ) ^ * | + ( | S T | - 1 ) \\cdot | V ( r _ t + r _ d , q ) ^ * | = q ^ { t + r _ t + r _ d } - 1 . \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} \\mathcal { F } ^ { ( \\beta ) } = \\biguplus _ { n \\in I } K _ n ^ { ( \\beta ) } \\cup F . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} \\| u \\| _ { B ^ s _ { p , q } ( M ) } = \\sum _ { U } \\| \\psi _ U u \\| _ { \\tilde { B } ^ s _ { p , q } ( U ) } . \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} C _ { s + 1 } ^ 0 = \\sum _ { i = 1 } ^ s C _ { i , s + 1 } ^ { 0 } \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { u } - \\mathrm { d i v } \\big ( \\mathbf { F } ( \\nabla \\mathbf { u } ) \\nabla \\mathbf { u } \\big ) = \\mathbf { 0 } ( 0 , \\infty ) \\times G . \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\int _ M u \\Delta _ p u d \\mu & = - p \\int _ M | \\nabla u | ^ { p - 2 } h ^ { i j } \\nabla _ i u \\nabla _ j u d \\mu - \\int _ M Z \\langle \\nabla \\mathcal { H } , \\nabla u \\rangle u d \\mu \\\\ \\displaystyle & \\ \\ \\ + ( p - 1 ) \\int _ M \\Delta _ p u \\ u _ t d \\mu + \\int _ M \\Delta _ p u \\frac { \\partial } { \\partial t } ( u d \\mu ) . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} \\lambda _ { \\sigma ( i ) } ( \\tau _ y ) = \\lambda _ i ( \\tau _ { P y } ) = \\lambda _ i ( \\tau _ y ) \\lambda _ i ( \\tau _ { ( P - I ) y } ) , \\ i = 1 , \\cdots , N . \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{gather*} g _ { - } v _ { 0 } = g _ { - } \\hat { g } _ { 0 + } v _ { 0 } / \\tau ( \\hat g ) = \\hat { g } v _ { 0 } / \\tau ( \\hat g ) . \\end{gather*}"}
-{"id": "9780.png", "formula": "\\begin{align*} n _ 1 = n ( v ) ; n _ 2 = g ' ( u ) \\ , l ( v ) + f ' ( u ) \\ , e _ 4 , \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} F _ { \\varphi } ( x , y ) & = \\chi \\bigl ( b ( \\rho ( y ) ) d ( x , y ) \\bigr ) \\varphi ( y ) , \\\\ { \\mathcal { R } } ( \\varphi ) ( x ) & = \\int _ { M } F _ { \\varphi } ( x , y ) d m ( y ) , \\ \\\\ { \\mathcal { P } } ( \\varphi ) & = \\frac { { \\mathcal { R } } ( \\varphi ) } { { \\mathcal { R } } ( 1 ) } , \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} D g '' ( \\phi ( p ) ) - C _ 0 g ' ( \\phi ( p ) ) - N g ( \\phi ( p ) ) = ( D C _ 1 ^ 2 - C _ 1 C _ 0 - N ) e ^ { C _ 1 \\phi ( p ) } + N > 0 \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} \\chi ( X ) = \\chi ( \\Gamma ) + \\chi ( X ' ) - \\chi ( \\Gamma \\cap X ' ) . \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} F F P _ { \\mathbf { w } } ( \\mathbf { W } ) \\geq \\frac { 1 } { d } \\left ( \\sum _ { k = 1 } ^ { K } w _ k ^ 2 L _ k \\right ) ^ 2 , \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { B } ^ + } d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ { d , s , k } T R ^ d \\left [ \\alpha + \\frac { y } { \\eta T } + C _ { d , \\alpha } \\left ( \\frac { \\eta } { R } \\right ) ^ { d - 1 } + C _ { d , \\alpha } \\theta ^ { ( d - 1 ) / 2 } \\right ] \\end{aligned} \\end{align*}"}
-{"id": "10099.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z ^ 2 } { x ^ { 4 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z } { x ^ { 3 } } , \\\\ \\end{gather*}"}
-{"id": "1820.png", "formula": "\\begin{align*} \\cosh ^ 2 u ^ * = 1 + \\abs { \\tilde { x } ^ 0 } ^ 2 . \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} \\prod \\nolimits _ { \\nu = 0 } ^ { n } \\gamma _ { \\nu d } ^ 0 = \\left \\langle u _ { d - 1 } , x ^ { n } P _ { \\left ( n + 1 \\right ) d - 1 } \\right \\rangle . \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{gather*} C ( z ) = \\sum _ { i \\in \\mathbb { Z } } c _ { i } z ^ { - i - 1 } , \\end{gather*}"}
-{"id": "7974.png", "formula": "\\begin{align*} P ( E _ 2 ) = ( a b ) ^ 4 = ( P ( E _ 1 ) ) ^ 4 P ( E _ 4 ) = ( ( 1 - a ) ( 1 - b ) ) ^ 4 = ( P ( E _ 3 ) ) ^ 4 . \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} \\frac { \\left ( q z ^ { 2 } ; q \\right ) _ { \\infty } } { ( q ; q ) _ { \\infty } } z ^ { \\nu } = \\sum _ { m = 0 } ^ { \\infty } \\frac { q ^ { m ^ { 2 } } } { \\left ( q ; q \\right ) _ { m } } J _ { \\nu + m } ^ { ( 3 ) } ( 2 z q ^ { m / 2 } ; q ) \\left ( \\frac { q ^ { \\nu } } { z } \\right ) ^ { m } . \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} & \\eta f = g \\pi , \\\\ & g ( h \\rightharpoonup a ) = f ( h ) \\rightharpoonup g ( a ) , & & a \\in A , \\ ; h \\in H . \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} \\begin{aligned} \\delta ( Y ) & = & A Y \\\\ \\sigma ( Y ) & = & B Y \\end{aligned} \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\rightarrow 0 } \\sup _ { 1 \\leq j \\leq \\left [ \\varepsilon ^ { - 2 } t \\right ] } \\varepsilon \\left \\vert Y _ { j } ^ { \\varepsilon } - \\mathcal { M } _ { j } ^ { \\varepsilon } \\right \\vert = 0 \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} P ( X , Y ) = \\sum _ { ( i , j ) } a ( i , j ) X ^ i Y ^ j \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} - L w + \\tilde { g } \\circ w & = \\tilde { f } \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ w & = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega , \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} I _ 4 = & \\frac { n - 2 } { 2 } \\Delta ( \\sigma _ 1 ( A ) \\Delta ( r ^ { 6 - n } ) ) \\\\ = & 2 ( n - 2 ) ( 6 - n ) \\Delta ( \\sigma _ 1 ( A ) r ^ { 4 - n } ) + O ( r ^ { N + 4 - n } ) \\\\ = & 2 ( n - 2 ) ( 6 - n ) r ^ { 2 - n } [ \\Delta \\sigma _ 1 ( A ) r ^ 2 + 2 ( 4 - n ) \\sigma _ 1 ( A ) _ { , i } x ^ i + 2 ( 4 - n ) \\sigma _ 1 ( A ) ] + O ( r ^ { N + 2 - n } ) \\\\ = & 2 ( n - 2 ) ( n - 6 ) r ^ { 2 - n } \\Big [ \\frac { 1 } { 1 2 ( n - 1 ) } | W ( p ) | ^ 2 r ^ 2 + 3 ( n - 4 ) \\sigma _ 1 ( A ) _ { , i j } ( p ) x ^ i x ^ j \\Big ] + O ( r ^ { 5 - n } ) . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} 1 = \\prod _ { v \\in T } D _ v ( \\gamma ) \\prod _ { v \\in S _ D \\cup \\infty } D _ v ( \\gamma ) \\prod _ { v \\in S _ \\gamma } D _ v ( \\gamma ) \\ll q _ T ^ { A \\kappa + B } q _ { S _ \\gamma } ^ { - 1 } , \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} \\Lambda ( i , j ) & = \\Gamma ( i ) \\oplus F _ n ( \\Gamma ( i ) ) \\oplus F _ n ( \\Gamma ( j ) ) \\\\ & = \\left ( \\bigoplus _ { t = 0 } ^ { b - 2 } \\gamma _ { 0 , i } ( t ) \\right ) \\oplus \\gamma _ { a , i } ( n ) \\oplus F _ n ( \\Gamma ( i ) ) \\oplus F _ n ( \\Gamma ( j ) ) \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} & x ^ { d + 1 } B _ { n } = B _ { n + d + 1 } + \\\\ & \\sum _ { \\substack { k = 1 \\\\ 1 \\leq i _ { 1 } \\leq \\dots \\leq i _ { k } \\leq d - k + 2 } } ^ { d + 1 } \\rho _ { n - ( d + 1 ) + i _ { 1 } + 1 } \\dots \\rho _ { n - k ( d + 1 ) + i _ { k } + k } B _ { n + ( 1 - k ) ( d + 1 ) } . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { N } C _ { t , t } \\left ( 1 \\right ) \\cdot m _ { t , r } = \\sum _ { t = 1 } ^ { N } C _ { t , r } \\left ( 1 \\right ) \\cdot m _ { t , r } = C _ { r , r } \\left ( 1 \\right ) \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} \\gamma _ { 2 n + 1 } = \\sum \\limits _ { i = 0 } ^ n { n \\choose i } _ r \\gamma _ { 2 i } . \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} x ^ i = \\sum ^ n _ { j = 1 } g _ { i j } x ^ j , \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} Z = \\left [ \\begin{array} { c c } x _ 1 & x _ 2 \\\\ x _ 2 & x _ 1 \\end{array} \\right ] , x _ i \\in \\mathbb { R } , \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} \\bar x \\ , : = \\ , \\sum _ { j = 1 } ^ N x _ j = x _ i + \\bar x _ { - i } , \\bar x \\in \\bar K , \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} H _ n \\colon ( x , y ) \\mapsto ( X , Y ) = ( \\sigma ^ n ( x - 1 ) , \\ \\sigma ^ { 2 n } ( y - \\sigma ^ { - n } ) ) . \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} R = d \\omega + \\frac { 1 } { 2 } \\left [ \\omega , \\omega \\right ] & \\propto J _ { a b } , \\\\ T = d e + \\left [ \\omega , e \\right ] & \\propto P _ { a } , \\\\ \\left [ e , e \\right ] = \\frac { 1 } { \\ell ^ { 2 } } e ^ { a } e ^ { b } \\left [ P _ { a } , P _ { b } \\right ] & \\propto Z _ { a b } . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} q ( y '' , y _ n , y _ { n + 1 } ) = y _ n \\left ( a _ 0 + \\sum _ { j = 1 } ^ { n - 1 } a _ j y _ j \\right ) + b ( y _ n ^ 3 - 3 y _ n y _ { n + 1 } ^ 2 ) , \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} U _ \\mathcal { S } = \\left ( X ^ { ( n ) } A _ { \\mathcal { S } , n } : n \\in [ 1 : N ] \\right ) , \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\Delta _ { j _ { 0 } } = \\varepsilon a + \\varepsilon \\phi \\end{align*}"}
-{"id": "10019.png", "formula": "\\begin{align*} ( i - q ^ { t - a } ) ( i - q ^ t ) \\begin{cases} = 0 \\ ; & \\mbox { i f } i = q ^ { t - a } , \\\\ < 0 \\ ; & \\mbox { i f } q ^ { t - a } < i < q ^ t , \\\\ = 0 \\ ; & \\mbox { i f } i = q ^ t . \\end{cases} \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{gather*} v _ { 0 } = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\wedge \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} \\wedge \\begin{bmatrix} z \\\\ 0 \\end{bmatrix} \\wedge \\begin{bmatrix} 0 \\\\ z \\end{bmatrix} \\wedge \\begin{bmatrix} z ^ { 2 } \\\\ 0 \\end{bmatrix} \\wedge \\begin{bmatrix} 0 \\\\ z ^ { 2 } \\\\ \\end{bmatrix} \\wedge \\cdots , \\end{gather*}"}
-{"id": "3567.png", "formula": "\\begin{align*} x ^ * _ i = \\begin{cases} 1 & { \\rm i f } \\ i = i _ * , \\\\ 0 & { \\rm o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} & a \\circ ( b c ) = ( a _ 1 \\circ b ) S ( a _ 2 ) ( a _ 3 \\circ c ) , & & a , b , c \\in A . \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\partial e ^ { t \\Delta } f = e ^ { t \\vec { \\Delta } } \\partial f . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} \\tilde Y _ z : = \\{ y \\in Y : z \\in X + Y + y \\} \\ \\ \\ \\ Y _ z : = Y \\setminus \\tilde Y _ z . \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{gather*} \\tilde \\phi \\big ( Q ^ { \\alpha _ { n - 1 } } v _ { 0 } ^ { ( 1 ) } \\otimes Q ^ { \\alpha _ { n - 2 } } v _ { 0 } ^ { ( 1 ) } \\otimes \\cdots \\otimes Q ^ { \\alpha _ { 0 } } v _ { 0 } ^ { ( 1 ) } \\big ) = Q _ { n - 1 } ^ { \\alpha _ { n - 1 } } Q _ { n - 2 } ^ { \\alpha _ { n - 2 } } \\cdots Q _ { 0 } ^ { \\alpha _ { 0 } } v _ { 0 } ^ { ( n ) } , \\end{gather*}"}
-{"id": "7764.png", "formula": "\\begin{align*} | g ( y ) | \\leq C d _ G ( y , y _ 0 ) ^ { \\eta _ 0 } , \\eta _ 0 = \\min \\left \\{ 1 + 4 \\alpha , 2 - \\frac { 2 ( n + 1 ) } { p } \\right \\} . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} \\beta _ 0 = \\max _ { s \\in S } \\max _ { a ^ 1 \\in A ^ 1 ( s ) ; a ^ 1 \\neq a ^ 1 _ s } \\max _ { a ^ 2 \\in A ^ 2 ( s ) ; a ^ 2 \\neq a ^ 2 _ s } \\{ 0 , \\beta _ { s , a ^ 1 } ^ 1 , \\beta _ { s , a ^ 2 } ^ 2 \\} \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\pi _ { \\phi } ( x ) = \\sum _ { \\substack { \\textup { $ \\mathfrak { p } $ u n r a m i f i e d i n $ L $ } \\\\ \\mathrm { N } _ { F / \\mathbb { Q } } ~ \\mathfrak { p } \\leq x } } \\phi ( \\mathrm { F r o b } _ { \\mathfrak { p } } ) , \\widetilde { \\pi } _ { \\phi } ( x ) = \\sum _ { \\substack { \\textup { $ \\mathfrak { p } $ u n r a m i f i e d i n $ L $ } \\\\ \\mathrm { N } _ { F / \\mathbb { Q } } ~ \\mathfrak { p } ^ m \\leq x } } \\frac { 1 } { m } \\phi ( \\mathrm { F r o b } _ { \\mathfrak { p } } ^ m ) . \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} \\min _ x \\ , F ( x ) : = g ( x ) + h ( c ( x ) ) , \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } D _ n / \\log n = 1 / \\log ( 1 / q ) , & & \\limsup _ { n \\to \\infty } D _ n / \\log n = 1 / \\log ( 1 / p ) . \\end{align*}"}
-{"id": "10116.png", "formula": "\\begin{align*} y ^ q ( b y + c z ) ^ r - z ^ { p + q + r } = 0 \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} D _ t \\subset \\left \\{ x \\mid \\frac { x } { t } \\in v ( U ) \\right \\} = : D _ t ^ \\prime . \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} \\mathcal { S } ( M , T , \\alpha ) : = \\# \\Bigl \\{ n _ 1 , n _ 2 , n _ 3 , n _ 4 \\asymp M : \\Bigl | \\frac { n _ 1 n _ 2 } { n _ 3 n _ 4 } - \\alpha \\Bigr | \\ll \\frac { 1 } { T } \\Bigr \\} . \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} v : = \\min \\{ u , t \\} - t \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} g _ { B } = \\sum _ { i j } \\frac { \\partial ^ { 2 } \\overline { \\varphi } } { \\partial y _ { i } \\partial y _ { j } } d y _ { i } d y _ { j } , \\ \\ { \\rm a n d } \\ \\ \\det ( \\frac { \\partial ^ { 2 } \\overline { \\varphi } } { \\partial y _ { i } \\partial y _ { j } } ) \\equiv \\det ( Z _ { i j } ) . \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{gather*} ( a _ \\sigma ) _ { i j k } = b _ { i j k } , \\quad ( b _ \\sigma ) _ { i j k } = a _ { i j k } , ( c _ \\sigma ) _ { i j k } = d _ { i j k } , ( d _ \\sigma ) _ { i j k } = c _ { i j k } , \\\\ ( \\lambda _ \\sigma ) _ i = - \\lambda _ { n + i } , ( \\lambda _ \\sigma ) _ { n + i } = - \\lambda _ i ; \\end{gather*}"}
-{"id": "8753.png", "formula": "\\begin{align*} L ^ { S _ { q , t } } _ { \\nabla ^ { - 1 } } [ X , Y ; q , t ] = q \\ , t \\ ; L ^ { S _ { q , t } } _ { \\nabla } [ - X , \\ ; - ( q t ) ^ { - 1 } Y ; \\ ; q ^ { - 1 } , \\ ; t ^ { - 1 } ] . \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} h _ p = x _ p + y _ p - u _ p + v _ p . \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\Big | \\sum _ { \\mathrm { N } \\mathfrak { n } < x } \\frac { 1 } { \\mathrm { N } \\mathfrak { n } } \\Big ( 1 - \\frac { \\mathrm { N } \\mathfrak { n } } { x } \\Big ) ^ { n _ K } - \\kappa _ K \\Big ( \\log x - \\sum _ { j = 1 } ^ { n _ K } \\frac { 1 } { j } \\Big ) - \\kappa _ K \\gamma _ K \\Big | \\ll _ { \\epsilon } \\big ( n _ K ^ { n _ K } D _ K \\big ) ^ { 1 / 4 + \\epsilon } x ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} \\mathbb { E } [ C _ 1 ^ ] = & \\int _ 0 ^ \\infty \\int _ { x _ 1 } ^ \\infty \\frac { 2 } { \\beta ^ 2 } e ^ { - \\frac { x _ 1 + x _ 2 } { \\beta } } \\cdot \\frac { 1 } { 2 } \\log _ 2 ( 1 + \\xi x _ 1 ) d x _ 2 d x _ 1 \\\\ = & \\frac { e ^ { \\frac { 2 } { \\beta \\xi } } } { \\ln ( 4 ) } E _ 1 \\left ( \\frac { 2 } { \\beta \\xi } \\right ) , \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} q ( x ) = { \\textstyle \\sum \\limits _ { n = 1 } ^ { \\infty } } q _ { n } e ^ { i n x } , \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } Q _ 0 ( s ) = \\int _ t ^ s A ( p ( \\tau , t ) ) P _ 0 ( \\tau ) d \\tau , \\\\ P _ 0 ( s ) = - \\int _ t ^ s \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , t ) ) Q _ 0 ( \\tau ) d \\tau - \\int _ t ^ s \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , t ) ) d \\tau . \\end{array} \\right . \\end{align*}"}
-{"id": "10086.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 2 } { x z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ 2 z ^ { 4 } } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ { 4 } z ^ { 2 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ { 5 } z } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 4 } { x ^ { 5 } z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 4 } { x ^ { 7 } z } , \\end{gather*}"}
-{"id": "5151.png", "formula": "\\begin{align*} - L u _ n & = f - g _ n \\circ u _ n \\leq f = - L v \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u _ m & = 0 \\ , \\ , \\ , \\ , \\ , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ v = 0 \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} p ( x , y ) + t f ( x , y ) = U ( x , y , t ) P ( x , y , t ) , \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} f _ \\sigma = F ^ { - \\alpha } ( \\abs { A } ^ 2 - n F ^ 2 ) , \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} \\widetilde { W } _ Y : = \\{ ( [ F ] , C _ t , [ L _ t ] , \\mathbf { k } \\varphi ) \\ | \\ t \\in Y , \\ \\mathbf { k } \\varphi \\in \\mathbb { P } \\mathrm { H o m } ( F , C _ t , L _ t ) _ e \\} . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} \\left ( A ( z ) - B \\left ( z ^ { - 1 } \\right ) \\right ) g _ { n } ( z ) = \\left ( A \\left ( z ^ { - 1 } \\right ) - B ( z ) \\right ) \\tilde { g } _ { n } ( z ) , \\forall n \\in \\Z . \\end{align*}"}
-{"id": "10178.png", "formula": "\\begin{align*} \\varphi _ q ^ k & = \\frac { \\mu ( x _ q ^ k ) } { q } \\left ( 1 + \\frac { \\beta ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) \\right ) \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} F _ { i j } \\xi ^ i \\xi ^ j < 0 \\forall \\xi \\not \\sim \\kappa \\textrm { a n d } \\xi \\not = 0 , \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} H _ c : = \\sqrt { c ^ 2 p ^ 2 + m ^ 2 c ^ 4 } - m c ^ 2 + V _ c ( x ) . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { \\tilde \\mu - \\mu } { 2 } \\sum _ { i = 1 } ^ N \\frac { \\| x _ i - y _ i \\| ^ 2 } { a _ i ^ 2 } \\leq & \\frac { \\tilde \\mu } { 2 } \\| x ^ * - v _ 0 \\| ^ 2 + \\frac { \\rho M ^ 2 N ( N + 3 ) } { 4 } + \\frac { N r M ^ 2 } { 2 } + \\sum ^ N _ { i = 1 } \\frac { \\delta _ i } { a _ i } \\\\ & + 2 \\sum _ { i = 1 } ^ N \\frac { \\varepsilon _ i } { a _ i ^ 2 } + \\sqrt { 2 \\tilde \\mu } \\sum _ { i = 1 } ^ N \\| x ^ * - v _ { i } \\| \\cdot \\sqrt { \\frac { \\delta _ i } { a _ i } } . \\end{aligned} \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} \\vec { M } : = \\vec { A } \\cdot \\vec { B } , \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\frac { 2 } { 2 n + 1 } \\sum \\limits _ { i = 0 } ^ n { 2 n + 1 \\choose 2 i + 1 } B _ { 2 n - 2 i } \\gamma _ { 2 i + 1 } = \\frac { 2 } { 2 n + 1 } \\sum \\limits _ { i = 0 } ^ n { 2 n + 1 \\choose 2 i + 1 } B _ { 2 n - 2 i } \\sum \\limits _ { j = 0 } ^ i { i \\brace j } \\gamma _ { 2 j } . \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} F _ n = \\begin{cases} \\log _ { 1 / q } n - \\log _ { 1 / q } \\log \\log n + o ( \\log \\log \\log n ) & p > q \\\\ \\log _ { 2 } n - \\log _ 2 \\log n + o ( \\log \\log n ) & p = q \\end{cases} \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{gather*} g ^ { [ k , \\ell ] ( \\alpha , \\beta ) } = T ^ { - \\ell } _ { 2 } T _ { 1 } ^ { - k } g ^ { ( \\alpha , \\beta ) } , \\end{gather*}"}
-{"id": "3518.png", "formula": "\\begin{align*} \\tau _ U = \\binom { N _ T } { N _ R } a ^ * _ { 0 , N _ R } = 1 - \\mu _ R , \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} \\partial _ { \\nu } Q ^ { \\mu \\nu } = 0 , \\qquad \\partial ^ { \\nu } Q _ { \\mu \\nu } = 0 , \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} W ( g , \\tilde { g } ) = z ^ { - 1 } \\theta _ { q } \\left ( z ^ { 2 } \\right ) \\ ! . \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} \\frac { N _ { a , b , c } ( r ) } { M _ { a , b , c } } = \\binom { a + r - 1 } { a - 1 } \\binom { b + c - r - 1 } { c - 1 } \\binom { a + b + c - 1 } { b } ^ { - 1 } , \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} \\lim _ { g \\uparrow h } \\theta _ { > } ( g ) \\leq \\lim _ { n \\to \\infty } \\theta _ { > } ( h ) + n ^ { - 1 } = \\theta _ { > } ( h ) . \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} \\zeta ^ { ( n ) } ( \\lambda ) \\to Q _ { \\lambda } : = P o i s s o n \\left ( \\frac { \\lambda } { \\pi } 1 _ { [ 0 , 1 ] } d t \\right ) . \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v - \\Delta v + \\nabla _ H p + f _ 0 k \\times v = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\\\ \\partial _ z p = \\theta , \\\\ \\partial _ t \\theta + v \\cdot \\nabla _ H \\theta + w \\partial _ z \\theta - \\Delta \\theta = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ \\infty \\gamma _ i x ^ i = \\frac { 1 } { x } \\cdot \\frac { d ^ { 2 k - 1 } } { d x ^ { 2 k - 1 } } \\left [ x ^ { 2 k - 2 } \\left ( \\frac { x ^ 2 } { 1 - x } \\right ) \\psi \\left ( \\frac { x ^ 2 } { 1 - x } \\right ) \\right ] , \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ \\infty \\gamma _ i x ^ i = \\frac { 1 } { x } \\cdot \\frac { d ^ { 2 k - 1 } } { d x ^ { 2 k - 1 } } \\left [ \\frac { x ^ { 2 k } } { 1 - x } \\psi \\left ( \\frac { x ^ 2 } { 1 - x } \\right ) \\right ] , \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 g ( \\rho ) \\ , d \\rho = 1 \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ n c ^ G _ i ( E \\otimes \\rho ) = \\sum _ { i = 0 } ^ n c _ i ( E ) ( 1 + c _ 1 ( \\rho ) ) ^ { n - i } . \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} D ( z ) = \\int _ { 0 } ^ { z } e ^ { - \\frac { \\lambda } { \\xi } s } s ^ { \\frac { \\mu } { \\xi } - 1 } ( 1 - s ) ^ { - 1 } d s . \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} & x _ { 2 } = - x _ { 3 } \\left ( \\beta _ { 5 } d + \\beta _ { 3 } \\right ) \\\\ & x _ { 3 } \\left ( \\gamma _ { 3 } + \\left ( 1 + \\gamma _ { 5 } \\right ) d - x _ { 1 } \\right ) = - x _ { 1 } x _ { 2 } \\\\ & d = x _ { 1 } - x _ { 2 } \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} \\inf _ { P \\nu = b + b _ 0 } \\ , \\| \\nu \\| _ \\infty \\ , = \\ , \\sup _ { \\{ g _ n \\} } \\ , \\limsup _ { n \\to \\infty } \\ , \\biggl | \\int _ { \\mathbb { D } } { \\mu } _ b ( z ) \\overline { g _ n ( z ) } \\ , | d z | ^ 2 \\biggr | , \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} \\frac { { x _ 1 f \\left ( { x _ 2 } \\right ) - x _ 2 f \\left ( { x _ 1 } \\right ) } } { { x _ 1 - x _ 2 } } = f \\left ( \\xi \\right ) - \\xi f ' \\left ( \\xi \\right ) . \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { N } { v _ j ^ k } = & \\sum _ { j = 1 } ^ { N } ( \\hat v _ j ^ { k - 1 } + x _ j ^ { k } - x _ j ^ { k - 1 } ) \\\\ = & \\sum _ { j = 1 } ^ N \\sum _ { i = 1 } ^ N [ W ( k - 1 ) ] _ { j i } v _ i ^ { k - 1 } + \\sum _ { j = 1 } ^ { N } ( x _ j ^ { k } - x _ j ^ { k - 1 } ) \\\\ = & \\sum _ { i = 1 } ^ N v _ i ^ { k - 1 } + \\sum _ { j = 1 } ^ { N } ( x _ j ^ { k } - x _ j ^ { k - 1 } ) , \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} F = - \\frac { 1 } { 2 } [ \\tau \\zeta ' ( 0 , D ) + \\tau ^ 2 \\zeta ( 0 , D ) ] \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} \\lim _ { b \\rightarrow - 1 } ( 1 - b ) F ( b , 0 ; b ) & = \\sum _ { j \\geq 0 } q ^ { { ( 3 j ^ 2 + j ) } / { 2 } } ( 1 - q ^ { 2 j + 1 } ) \\\\ & = 1 + \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { 2 n - 1 } } { ( - q ; q ^ 2 ) _ n } \\\\ & = \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ n q ^ { ( n ^ 2 + n ) / 2 } } { ( - q ; q ) _ n } . \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} \\| \\varphi \\left ( x ^ { - 1 } \\right ) \\| ^ { 2 } = x ^ { - 2 } \\| \\varphi ( x ) \\| ^ { 2 } , \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} \\left ( q / a ; q ^ { 2 } \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a q ; q ^ { 2 } \\right ) _ { n } z ^ { n } } { \\left ( q ; q \\right ) _ { n } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } A _ { q } \\left ( q ^ { - 2 n } a z \\right ) } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } \\left ( - a \\right ) ^ { n } } , \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} a _ { k } = \\xi _ { k } - \\omega _ { k } , \\ ; \\ ; \\ ; \\ ; \\ ; k \\geq 1 . \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} u ( \\cdot , T ) = u ^ 1 ( \\cdot ) , v ( \\cdot , T ) = v ^ 1 ( \\cdot ) . \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} \\sum a _ { i } x _ { i } = 0 \\end{align*}"}
-{"id": "9974.png", "formula": "\\begin{align*} \\begin{array} { l l l } I _ 1 ^ h & = & \\displaystyle { \\frac { 1 } { 4 } \\int _ { \\partial B ^ h } \\Big ( \\phi ( z + h ) n _ 1 ^ { + } ( z ) + \\phi ( z - h ) n _ 1 ^ { - } ( z ) + i \\phi ( z + i h ) n _ 2 ^ { + } ( z ) } \\\\ & & + i \\phi ( z - i h ) n _ 2 ^ { - } ( z ) \\Big ) f ^ h ( z ) d S ^ h ( z ) - \\displaystyle { \\int _ { B ^ h } \\phi \\partial _ { \\bar z } ^ h f ^ h d V ^ h } . \\end{array} \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} H ^ { i m } \\left ( \\nabla _ i H _ { m l , r } - \\nabla _ l H _ { i m , r } \\right ) + 2 f _ { , r l } - H ^ { m n } H _ { n l , r } f _ { , m } = 0 , \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} \\big \\langle D ^ F _ { Z _ { 1 , R } } \\mu , \\mu \\big \\rangle _ { Z _ { 1 , R } } - \\big \\langle \\mu , D ^ F _ { Z _ { 1 , R } } \\mu \\big \\rangle _ { Z _ { 1 , R } } = \\big \\langle \\lambda \\mu , \\mu \\big \\rangle _ { Z _ { 1 , R } } - \\big \\langle \\mu , \\lambda \\mu \\big \\rangle _ { Z _ { 1 , R } } = 0 . \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} H _ g \\leq v _ D \\leq u \\widehat H _ g \\leq \\widehat v _ D \\leq \\widehat u = \\widehat H _ g , \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\hat v _ i ^ k = \\frac { v ^ k _ { I ^ { k } } + v ^ k _ { J ^ { k } } } { 2 } \\mbox { f o r } i \\in \\{ I ^ k , J ^ k \\} , \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} \\xi : = \\left \\{ \\begin{array} { l c } \\lceil ( \\ell - k ) / \\beta \\rceil , & \\textrm { i f } k < \\ell \\\\ \\lceil ( k - \\ell ) / \\gamma \\rceil , & \\textrm { i f } k \\ge \\ell \\end{array} \\right . \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} \\left \\| \\nabla \\psi \\left ( t \\right ) \\right \\| _ { \\infty } = \\max _ { p \\in M } \\left \\| \\nabla \\psi \\left ( p , t \\right ) \\right \\| , \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} d \\bar { s } _ { \\mathbb { H } ^ { n + 1 } } ^ 2 = d \\varrho ^ 2 + \\sinh ^ 2 \\varrho \\ , \\sigma _ { i j } \\ , d \\xi ^ i d \\xi ^ j . \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} \\mathbb { P } ( T _ { ( k ) } < X _ { r } ) & = \\sum _ { m = k } ^ { n } \\binom { n } { m } \\sum _ { j = 0 } ^ { m } ( - 1 ) ^ { j } \\binom { m } { j } \\left ( \\frac { s } { s + n - m + j } \\right ) ^ { r } . \\end{align*}"}
-{"id": "10043.png", "formula": "\\begin{align*} | p _ n | = \\Big | \\tfrac { \\displaystyle p ^ { ( n ) } ( 0 ) } { \\displaystyle n ! } \\Big | \\leq | \\varphi ' ( 0 ) | ( n \\in \\mathbb { N } ) . \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} \\ell = \\sum _ { i = 1 } ^ d \\ell _ i = \\zeta _ { d } + \\frac { | H | \\alpha } { 2 } + \\frac { \\kappa _ d \\alpha } { 2 } { \\rm w i t h } | H | = \\sum _ { i = 1 } ^ d H _ i \\ , . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{gather*} e ^ { - \\beta ^ 2 ( \\kappa - \\log \\varepsilon ) } \\int _ a ^ b e ^ { \\beta V _ { \\varepsilon } ( u ) } \\ , d u \\longrightarrow M _ { \\beta } ( a , b ) , \\\\ { \\bf { E } } [ M _ { \\beta } ( a , b ) ] = | b - a | . \\end{gather*}"}
-{"id": "7057.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( x y : n ) } = \\bigoplus _ { ( i , \\alpha ) } H _ { ( x y ) } ( i , \\alpha ) \\psi ( i , \\alpha ) \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} & \\| f ( t , u ) - f ( t , v ) \\| _ { H ^ { 1 + \\delta } _ p } = \\| u e ^ { - t | \\cdot | } - v e ^ { - t | \\cdot | } \\| _ { H ^ { 1 + \\delta } _ p } \\\\ = & \\| ( u - v ) e ^ { - t | \\cdot | } \\| _ { H ^ { 1 + \\delta } _ p } \\leq c \\| u - v \\| _ { H ^ { 1 + \\delta } _ p } \\| e ^ { - t | \\cdot | } \\| _ { H ^ { 1 + \\delta } _ p } . \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} u _ i ( \\cdot , 0 ) & = u _ i ^ 0 , \\dot { u } _ i ( \\cdot , 0 ) = \\dot { u } _ i ^ 0 , \\\\ \\tau ( \\cdot , 0 ) & = \\tau ^ 0 , \\dot { \\tau } ( \\cdot , 0 ) = \\dot { \\tau } ^ 0 , q _ i ( \\cdot , 0 ) = q ^ 0 _ i \\Omega . \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} H _ n : X = \\sigma ^ n ( x - e ) , \\ Y = b \\sigma ^ { 2 n } ( y - \\sigma ^ { - n } ) . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} & \\theta _ { s , a ^ 1 } ^ 1 = r ^ 1 ( s , a ^ 1 , a _ s ^ 2 ) + \\beta \\sum _ { s ' \\in S } p ( s ' | s , a ^ 1 , a _ s ^ 2 ) \\frac { r ^ 1 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) } { 1 - \\beta } - \\frac { r ^ 1 ( s , a _ { s } ^ 1 , a _ { s } ^ 2 ) } { 1 - \\beta } . \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} V \\in B _ { \\alpha , \\beta } , M = 2 \\beta V V ^ t , N ( u ) = - \\beta | u | ^ 2 V - 2 \\beta ( u \\cdot V ) u \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} A _ i = \\cup _ { k = 1 } ^ l C _ { i _ k } , \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} D = \\frac { { { \\left | F ( { { \\theta } _ { 0 } } , { { \\phi } _ { 0 } } ) \\right | } ^ { 2 } } } { \\frac { 1 } { 4 \\pi } \\int _ { 0 } ^ { 2 \\pi } { \\int _ { 0 } ^ { \\pi } { { { \\left | F ( \\theta , \\phi ) \\right | } ^ { 2 } } \\sin \\theta d \\theta d \\phi } } } , \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} \\| \\partial _ x ^ 2 v _ 0 ^ + ( x , \\cdot ) \\| ^ 2 _ { H ^ { - \\frac { 1 } { 3 } } ( 0 , T ) } & \\leq C \\int _ { 0 } ^ { \\infty } \\rho ^ 4 \\left | \\hat { h } _ 0 ^ + ( \\rho ) \\right | ^ 2 d \\rho = C \\int _ { 0 } ^ { \\infty } \\rho ^ 4 \\left | \\hat { h } _ 0 ( i a \\rho ^ 3 L ^ 3 ) \\right | ^ 2 d \\rho \\\\ & = C \\int _ { 0 } ^ { \\infty } \\rho ^ 4 \\left | \\int _ 0 ^ { \\infty } e ^ { - i a \\rho ^ 3 L ^ 3 t } h _ 0 ( t ) d t \\right | ^ 2 d \\rho . \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} - \\partial _ + \\partial _ - p q & = - \\partial _ + ( n \\lambda ^ { 0 , 1 } - ( n - 1 ) f _ 4 ) ) + \\partial _ - ( - n \\lambda ^ { 1 , 0 } + ( n - 1 ) f _ 8 ) \\\\ & = ( - n d \\lambda + ( n - 1 ) ( - d f _ 4 + d f _ 8 ) ) ^ { 1 , 1 } . \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} \\mathbb { E } ( e \\omega ) = \\mathbb { E } ( e \\otimes 1 ) \\omega + ( - 1 ) ^ { \\deg ( e ) } e \\overline { \\partial } ( \\omega ) \\qquad \\mathbb { E } \\circ \\mathbb { E } ( e ) = 0 , \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} ( 1 , 0 ) \\cdot ( y _ 1 , y _ 2 , y _ 3 ) = ( y _ 1 , y _ 2 , y _ 3 - y _ 1 ) + ( - 2 , - 1 , 2 ) , \\\\ ( 0 , 1 ) \\cdot ( y _ 1 , y _ 2 , y _ 3 ) = ( y _ 1 , y _ 2 , y _ 3 - y _ 2 ) + ( - 1 , - 2 , 2 ) . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} \\mu ( [ A , B ] ) = t ( a _ 2 - a _ 1 ) + ( 1 - t ) ( b _ 2 - b _ 1 ) = \\ell . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} c _ 1 c _ 2 c _ 3 c _ 4 c _ 5 c _ 6 = 1 . \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} [ \\bar { f } _ { i , k + 1 } , \\bar { f } _ { i , l } ] = [ \\bar { f } _ { i , k } , \\bar { f } _ { i , l + 1 } ] , \\ [ \\bar { f } _ { i , k + 2 } , \\bar { f } _ { i + 1 , l } ] - ( d + d ^ { - 1 } ) [ \\bar { f } _ { i , k + 1 } , \\bar { f } _ { i + 1 , l + 1 } ] + [ \\bar { f } _ { i , k } , \\bar { f } _ { i + 1 , l + 2 } ] = 0 , \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} \\sum _ { x \\in \\xi _ D } g _ J ( x ) = \\sum _ { j \\in J } \\tilde { \\Psi } _ { \\xi _ D } ( y _ { c } ( B _ j ) ) . \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\{ ( x , \\pm \\sqrt { - 1 } x , z ) : \\sum _ { \\alpha = 1 } ^ 8 ( x _ \\alpha ) ^ 2 = 0 \\} . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} s _ { i j } : = \\begin{cases} 1 & ( i , j ) = ( 1 , 2 ) , \\\\ 0 & \\end{cases} t _ { i j } : = \\begin{cases} 1 & ( i , j ) = ( 2 , 1 ) , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} M = \\begin{bmatrix} 3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\ 1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 \\\\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\\\ 4 & 0 & 0 & 0 & 3 & 1 & 0 & 0 \\\\ 1 & 1 & 0 & 0 & 1 & 1 & 0 & 0 \\\\ 0 & 3 & 1 & 3 & 2 & 3 & 2 & 1 \\\\ 1 & 1 & 2 & 1 & 0 & 4 & 1 & 1 \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ T \\int _ { B _ { 2 R } ^ d } \\int _ { \\mathbb { S } ^ { d - 1 } } \\mathbf { 1 } _ { \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { B } ^ - } & d \\omega _ { k + 1 } d v _ { s + k + 1 } d \\tau \\leq \\\\ & \\leq C _ d \\left ( s + k \\right ) T R ^ d \\left [ \\frac { y } { \\eta T } + \\left ( \\frac { \\eta } { R } \\right ) ^ d + \\theta ^ { d - 1 } \\right ] \\end{aligned} \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} \\left [ Z _ { a b } , \\bar { Z } _ { c } \\right ] & = \\frac { 1 } { \\sqrt { 2 } } \\left [ Z _ { a b } , P _ { c } \\right ] - \\frac { 1 } { \\sqrt { 2 } } \\left [ Z _ { a b } , Z _ { c } \\right ] , \\\\ & = \\frac { 1 } { \\sqrt { 2 } } f _ { a b , c } ^ { d } Z _ { d } + \\frac { 1 } { \\sqrt { 2 } } f _ { a b , c } ^ { d } P _ { d } , \\\\ & = f _ { a b , c } ^ { d } \\bar { P } _ { d } , \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} \\mathcal { B } _ r ( \\theta ^ * ) & = \\left \\lbrace \\theta \\in \\Theta \\left | \\frac { 1 } { \\sqrt { n } } h \\left ( \\boldsymbol { \\ell } \\left ( \\cdot | \\theta ^ * \\right ) , \\boldsymbol { \\ell } \\left ( \\cdot | \\theta \\right ) \\right ) \\right . \\leq r \\right \\rbrace \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} p _ i - p _ { \\lfloor i / 2 \\rfloor } = \\omega ^ { a _ { h ( i ) } ( i ) } \\left ( \\frac { z } { 2 } \\right ) ^ { h ( i ) } . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} & \\sum \\limits _ { i , j = 1 } ^ { n + 1 } [ d _ G ( y , y _ 0 ) ^ { - 1 - 2 \\alpha + \\epsilon } Y _ i Y _ j \\tilde { v } ] _ { C ^ { 0 , \\epsilon } _ \\ast ( \\mathcal { B } _ 1 ^ + ( y _ 0 ) ) } \\\\ & + \\sum \\limits _ { i , j = 1 } ^ { n + 1 } \\| d _ G ( y , y _ 0 ) ^ { - 1 - 2 \\alpha } Y _ i Y _ j \\tilde { v } \\| _ { L ^ { \\infty } ( \\mathcal { B } _ 1 ^ + ( y _ 0 ) ) } \\leq C \\| f \\| _ { Y _ { \\alpha , \\epsilon } } . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} g _ { 1 } ( x _ { 1 } ) [ v _ { \\delta ^ { 1 } } + u _ { \\delta ^ { 1 , n } } g _ { n } ( x _ { n } ) ] = [ g _ { 1 } ( x _ { 1 } - x _ { n } ) - g _ { 1 } ( - x _ { n } ) ] \\sum _ { \\substack { \\delta \\in \\{ 0 , 1 \\} ^ { n - 1 } , \\\\ \\delta _ { 1 } = 1 } } v _ { \\delta } f _ { n } ^ { \\delta _ { n } } ( x _ { n } ) \\prod _ { i = 2 } ^ { n - 1 } g _ { i } ^ { \\delta _ { i } } ( - x _ { n } ) \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} T ^ \\mu _ \\mu ( x ) = a _ n ( x , D ) \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} x ( t ) = \\begin{cases} e ^ { i t A _ 1 } h , & t > 0 \\\\ e ^ { i t A _ 1 ^ * } h , & t < 0 \\end{cases} . \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} \\tilde { v } _ \\lambda ( \\xi ) : = \\frac { ( v - P _ { y _ 0 } ) ( y _ 0 + \\lambda ^ 2 \\xi '' , \\lambda \\xi _ n , \\lambda \\xi _ { n + 1 } ) } { \\lambda ^ { 3 + 2 \\alpha } } , \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} d _ { \\Sigma , 1 } = \\max _ { \\substack { \\lambda _ 1 , \\lambda _ 2 \\in ( 0 , 1 ] \\\\ \\lambda _ 1 ( N + \\min \\{ M , K ( N - 1 ) \\} + \\lambda _ 2 N \\leq M \\\\ \\lambda _ 1 + \\lambda _ 2 \\left ( 1 + \\frac { N } { M } \\right ) \\leq 1 \\\\ } } K N ( \\lambda _ 1 + \\lambda _ 2 ) . \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} f ( p , q ) ^ { \\cdot _ q n } : = \\underbrace { f ( p , q ) \\cdot _ q f ( p , q ) \\cdot _ q \\cdots \\cdot _ q f ( p , q ) } _ { } , \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} \\frac { d } { d t } \\mu _ t = \\frac { d } { d t } \\sqrt { g } = \\tfrac { 1 } { 2 } \\mu _ t g ^ { i j } \\dot { g } _ { i j } = - F H \\mu _ t , \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{gather*} \\mathop { \\vec { \\prod } } \\limits _ { l = 1 } ^ { k } { } _ { a } \\psi _ { ( - l ) } ^ { \\pm } = { } _ { a } \\psi ^ { \\pm } _ { ( - k ) } \\cdots { } _ { a } \\psi _ { ( - 1 ) } ^ { \\pm } . \\end{gather*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\mathcal { A } ^ + = \\left \\{ \\begin{aligned} & \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\subset [ 0 , \\infty ) \\times \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } \\textnormal { s u c h t h a t } \\\\ & \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime \\right ) > 0 \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} \\delta _ 2 ( \\phi _ { A , 0 } \\circ q ^ * ) = \\delta _ 2 ( \\phi _ { I , 0 } \\circ q ^ * _ A ) = 0 . \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ M ^ l _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } \\bigr ] = \\prod _ { k = 0 } ^ { l - 1 } \\frac { \\Gamma ( 1 - ( k + 1 ) / \\tau ) } { \\Gamma ( 1 - 1 / \\tau ) } \\frac { \\Gamma ( 1 + \\lambda _ 1 - k / \\tau ) \\Gamma ( 1 + \\lambda _ 2 - k / \\tau ) } { \\Gamma ( 2 + \\lambda _ 1 + \\lambda _ 2 - ( l + k - 1 ) / \\tau ) } . \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} \\Pr _ { a \\sim ^ L _ { \\rho } b } [ a \\in A , b \\in B ] = 2 ^ { - n } \\sum _ { d = 0 } ^ { | L | } \\Big { ( } \\frac { 1 + \\rho } { 2 } \\Big { ) } ^ { | L | - d } \\Big { ( } \\frac { 1 - \\rho } { 2 } \\Big { ) } ^ { d } 2 ^ { - | R | } W _ { d } \\ , . \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} F P S ( l / k ) = z _ { l , 0 } + 2 \\sum \\limits _ { q = 1 } ^ { \\frac { { l - 1 } } { 2 } } { z _ { l , q } \\cos ( ( q ) } \\frac { { 2 k \\pi } } { l } ) . \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} R ^ + ( n , k ) & = ( 2 k + 1 ) R ^ + ( n - 1 , k ) + ( 2 n - 4 k + 2 ) R ^ + ( n - 1 , k - 1 ) + R ^ - ( n - 1 , k ) \\\\ & = 2 k R ^ + ( n - 1 , k ) + ( 2 n - 4 k + 2 ) R ^ + ( n - 1 , k - 1 ) + R ( n - 1 , k ) . \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} \\hat { \\varphi } ( \\xi ) & = \\frac { \\left ( ( i \\xi ) ^ 3 + r ( i \\xi ) + c \\lambda \\right ) \\left ( \\alpha + \\beta e ^ { - i L \\xi } \\right ) } { ( 1 - a ^ 2 b ) ( i \\xi ) ^ 6 + r ( i \\xi ) ^ 4 + ( c + 1 ) \\lambda ( i \\xi ) ^ 3 + r \\lambda ( i \\xi ) + c \\lambda ^ 2 } . \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} 0 & = \\lim \\limits _ { j \\to \\infty } | q | | \\sin ( k ( j - 1 ) ) - \\sin ( k j ) | + | q | | \\sin ( k ( j + 1 ) ) - \\sin ( k j ) | \\\\ & = \\lim \\limits _ { j \\to \\infty } 2 | q | | \\cos ( k j - k / 2 ) | | \\sin ( k / 2 ) | + 2 | q | | \\cos ( k j + k / 2 ) | | \\sin ( k / 2 ) | . \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} \\mathcal { E } = \\{ E \\in \\mathcal { B } : E \\cap ( C _ m \\cup D _ m ) \\in \\{ \\emptyset , C _ m , D _ m , C _ m \\cup D _ m \\} \\} . \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} | G | = q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) \\leq q ^ 3 ( q ^ 2 - 1 ) ( q + \\sqrt { 3 q } + 1 ) \\cdot \\frac { ( q - \\sqrt { 3 q } + 1 ) r } { | G _ { p } | } = f ( p ) , \\end{align*}"}
-{"id": "9964.png", "formula": "\\begin{align*} x x ^ \\dagger = c - 1 & = \\begin{cases} a \\ne - 1 & \\gcd ( q , c ) = 1 , \\\\ - 1 & \\gcd ( q , c ) \\ne 1 . \\end{cases} \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\gamma ^ a \\eta _ b + \\eta _ b \\gamma ^ a = \\delta _ { a , b } I d . \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\varphi ^ 1 \\circ I = \\frac { \\bar \\psi ^ 1 } { \\bar g } = \\bar \\psi ^ 2 \\ \\ \\ \\ \\psi ^ 1 \\circ I = - \\frac { \\bar \\varphi ^ 1 } { \\bar g } = - \\bar \\varphi ^ 2 . \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} \\frac { \\rho ^ { \\Pi } _ { I , X } ( p ) } { \\rho ^ { \\Pi } _ { I , X } ( q ) } = \\frac { \\rho ^ { \\Pi } ( p ) } { \\rho ^ { \\Pi } ( q ) } \\prod \\limits _ { x \\in X \\setminus I } \\left ( \\frac { x - p } { x - q } \\right ) ^ 2 . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} \\mathcal { Z } _ { m , n } ^ { \\gamma } ( z , \\overline { z } ) = \\dfrac { - ( \\gamma + m + 1 ) _ { n } m ! } { 2 \\pi i } \\left ( 1 - | z | ^ { 2 } \\right ) ^ { - \\gamma } \\oint _ { \\mid t \\mid = 1 } t ^ { n } \\dfrac { \\left ( 1 - { t \\overline { z } } \\right ) ^ { \\gamma + m } } { ( z - t ) ^ { m + 1 } } d t \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\{ x \\in X ; \\ \\theta ( x ) < 0 \\} = \\Omega _ 1 \\cap X , \\{ x \\in X ; \\ \\theta ( x ) = 0 \\} = S \\cap X , \\{ x \\in X ; \\ \\theta ( x ) > 0 \\} = \\Omega _ 2 \\cap X . \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} \\Theta _ { l _ 1 , l _ 2 } ( t _ 4 ( \\theta _ 1 , \\theta _ 2 ) ) = \\frac { - e ^ { i l _ 1 \\theta _ 1 + i l _ 2 \\theta _ 2 } + e ^ { i l _ 2 \\theta _ 1 + i l _ 1 \\theta _ 2 } } { ( e ^ { i \\theta _ 1 } - e ^ { - i \\theta _ 1 } ) ( e ^ { i \\theta _ 2 } - e ^ { - i \\theta _ 2 } ) ( 1 - e ^ { i \\theta _ 1 + i \\theta _ 2 } ) ( e ^ { - i \\theta _ 1 } - e ^ { - i \\theta _ 2 } ) } . \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} \\dd \\nu ^ k _ { t } = ( \\nu ^ { k - 1 } _ { t - } - \\nu ^ k _ { t - } ) \\ , \\dd N _ t . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} \\tilde { u } ^ { \\prime \\prime } ( 0 ) - \\tilde { y } ^ { \\prime \\prime } ( 0 ) = \\Phi \\Phi ^ * \\sigma u ^ { \\prime } ( 0 ) - i A _ 1 ^ 2 h + i A _ 1 \\Phi ^ * \\sigma u ( 0 ) . \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} W ( \\omega ) \\ = \\ C \\cup \\bigcup _ { n \\in \\N : \\omega _ n = 1 } G _ n \\ , \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} \\sum _ { m = s + 1 } ^ { t - 1 } \\frac { \\lambda _ u ^ { m - s } \\cdot { m } ^ { d _ u } } { \\lambda ^ { m - s } } \\frac { 1 } { \\lambda _ u ^ { m } \\cdot { m } ^ { d _ u } } M ^ { m } \\vec u _ { t - m } \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} \\circ ^ { - 1 } ( \\partial _ i \\mathcal { M } ( x , y ) ) = \\left \\{ \\begin{array} { l r } \\partial _ i \\mathcal { M } ( z , y ) \\times \\mathcal { M } ( x , z ) & \\mbox { f o r } i < m \\\\ \\mathcal { M } ( z , y ) \\times \\partial _ { i - m } \\mathcal { M } ( x , z ) & \\mbox { f o r } i > m \\end{array} \\right . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} u ( t _ 1 , \\ldots , t _ d ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int _ { - \\infty } ^ { \\infty } e ^ { \\sum _ { j = 1 } ^ d i t _ j \\left ( \\alpha _ j \\tau _ 1 + \\beta _ j \\right ) } \\widehat { f } ( \\tau _ 1 ) d \\tau _ 1 . \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} G \\left ( 1 \\right ) \\cdot \\vect v = \\left ( C _ { t , t } \\left ( 1 \\right ) \\right ) _ { t = 1 , \\dots , N } = \\left ( C _ { t , r } \\left ( 1 \\right ) \\right ) _ { t = 1 , \\dots , N } , \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\sum _ { f \\in M _ { n } } \\alpha _ { 2 } ( f ) u ^ { n } = \\frac { Z ( u ^ { 2 } ) Z ( u ^ { 3 } ) } { Z ( u ^ { 6 } ) } \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} \\sum _ { w \\in W } \\left [ x ^ { n - w } \\right ] h _ { \\zeta ^ k } ( x ) & = \\sum _ { w \\in W } ( - 1 ) ^ w \\sigma _ w ( \\zeta ^ k ) = \\sum _ { w \\in W } ( - 1 ) ^ w \\mathcal { F } _ { \\zeta } [ \\delta _ w ] ( k ) \\\\ & = \\mathcal { F } _ \\zeta \\left [ \\sum _ { w \\in W } ( - 1 ) ^ w \\delta _ w \\right ] ( k ) . \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} \\left \\{ \\sup _ { y \\in B _ 1 } \\Psi ( y ) \\geq h , \\ldots , \\sup _ { y \\in B _ k } \\Psi ( y ) \\geq h \\right \\} \\subset \\left \\{ \\sum _ { l = 1 } ^ k \\tilde { \\Psi } ( y _ c ( B _ l ) ) \\geq k h \\right \\} . \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} P N = \\sum _ { \\substack { f \\in H _ { 2 k } ^ { * } ( N ) \\\\ L _ f ( 1 / 2 ) \\neq 0 } } ^ { h } 1 \\geq \\frac { p - 1 } { p } \\frac { \\Omega } { 1 + 2 \\Omega } - \\epsilon . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{align*} | \\lambda | = \\sum _ { i = 1 } ^ { l ( \\lambda ) } \\lambda _ i , \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} a = c \\ ; ( 1 , \\frac { \\sin { 2 \\omega } } { \\sin { \\omega } } , \\ , \\frac { \\sin { 3 \\omega } } { \\sin { \\omega } } , \\ , \\ldots , \\ , \\frac { \\sin { n \\omega } } { \\sin { \\omega } } ) , \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} G _ a \\ = \\ \\left ( \\sum _ { n = 1 } ^ { \\vert a \\vert } 2 a _ n 3 ^ { - n } + 3 ^ { - n } , \\sum _ { n = 1 } ^ { \\vert a \\vert } 2 a _ n 3 ^ { - n } + 2 \\cdot 3 ^ { - \\vert a \\vert } \\right ) \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} { } [ K , K ] ^ { F N } _ p = & 2 \\Big ( ( \\imath _ { e _ i } \\Psi ) \\wedge ( \\imath _ { e _ j } \\nabla _ { e _ i } \\Psi ) + ( \\imath _ { e _ j } \\imath _ { e _ i } \\Psi ) \\wedge e ^ k \\wedge \\imath _ { e _ i } \\nabla _ { e _ k } \\Psi \\Big ) \\otimes ( e ^ j ) ^ \\# . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} & \\forall t \\in [ p , 1 ] : \\nabla _ p s _ { f } ( t ) = s _ f ( p ) + \\int _ p ^ t f ( s ) - f ( s - p ) d s = \\nabla _ p s _ { g } ( t ) \\Longleftrightarrow \\\\ & f ( t ) - f ( t - p ) = g ( t ) - g ( t - p ) ~ \\lambda - a . s . \\mbox { f o r a l l } t \\in [ p , 1 ] , ~ ~ s _ f ( p ) = s _ g ( p ) . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} | \\nabla V ( x ) | + | \\nabla \\mathbf B ( x ) | & = o \\left ( ( | V ( x ) | + | \\mathbf B ( x ) | ) ^ \\frac 3 2 + 1 \\right ) , \\\\ ( \\Re V ( x ) ) _ - & = o \\Big ( | V ( x ) | + | \\mathbf B ( x ) | + 1 \\Big ) , \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} h ( v ) = \\beta ( d - v ) , \\ ; \\ ; \\beta , d > 0 , \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} d = \\max \\left \\{ d ' _ { r , t } , \\frac { r + t } { N _ R } \\right \\} , \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} & \\tfrac { 1 } { ( g ! ) ^ 2 } \\int _ { X ^ g } \\log \\| \\theta _ { \\alpha _ X } \\| ( P _ 1 + \\dots + P _ g - Q - \\alpha _ X ) ( ( \\gamma | _ { X ^ { g + 1 } } ) \\circ s _ Q ) ^ * \\nu ^ g \\\\ = & \\tfrac { 1 } { ( g ! ) ^ 2 } \\int _ { p r _ { g + 1 } | _ { X ^ { g + 1 } } } \\log \\| \\theta _ { \\alpha _ X } \\| ( P _ 1 + \\dots + P _ g - P _ { g + 1 } - \\alpha _ X ) ( \\gamma | _ { X ^ { g + 1 } } ) ^ * \\nu ^ g , \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} \\mathcal { T } = \\iint A ( u _ 0 ^ x \\partial _ z f ) A f \\ , d V d t + \\iint A ( \\ne { u } \\cdot \\nabla _ L f ) A f \\ , d V d t = \\mathcal { T } _ 0 + \\mathcal { T } _ { \\neq } . \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} \\cos ( ( q ) \\frac { { 2 k \\pi } } { l } ) = \\cos ( ( q ) \\frac { { 2 ( l - k ) \\pi } } { l } ) , 0 \\leqslant k \\leqslant l - 1 . \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { r _ 1 } ( L ' _ i ) ^ d & = \\sum _ { i = 1 } ^ { r _ 1 } \\big ( \\alpha _ { i , 0 } ( x _ 0 - a ' _ 1 y _ 1 + \\ldots - a ' _ m y _ m ) + \\ldots + \\alpha _ { i , n } x _ n \\big ) ^ d . \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} d ( \\psi _ 1 , \\psi _ 2 ) = \\inf _ { x \\in X } \\hom ( \\psi _ 1 ( x ) , \\psi _ 2 ( x ) ) , \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} \\vartheta _ { 1 } ( u \\mid q ^ { 2 } ) \\vartheta _ { 4 } ( v \\mid q ^ { 2 } ) + \\vartheta _ { 1 } ( v \\mid q ^ { 2 } ) \\vartheta _ { 4 } ( u \\mid q ^ { 2 } ) = \\vartheta _ { 1 } \\left ( \\frac { u + v } { 2 } \\ , \\Big | \\ , q \\right ) \\vartheta _ { 2 } \\left ( \\frac { u - v } { 2 } \\ , \\Big | \\ , q \\right ) \\ ! , \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} \\Phi _ 0 \\left ( T ^ { \\psi } _ { \\varphi _ i ( h ) } u - u , T ^ { \\psi } _ { \\varphi _ i ( h ) } u - u \\right ) = \\Phi _ 0 ( T ^ { \\psi } _ { \\varphi _ i ( h ) } u , T ^ { \\psi } _ { \\varphi _ i ( h ) } u ) - \\Phi _ 0 ( u , u ) + 2 \\Phi _ 0 ( u , T ^ { \\psi } _ { \\varphi _ i ( h ) } u - u ) . \\end{align*}"}
-{"id": "9983.png", "formula": "\\begin{align*} \\mu _ { [ a , b ] } = \\frac { 1 } { 2 } ( \\delta _ a + \\lambda _ { [ a , b ] } + \\delta _ b ) \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} v ( x , t ) = [ W _ { b d r } \\overrightarrow { h } ] ( x , t ) = \\sum _ { i = 0 } ^ 2 [ W _ j ( t ) h _ j ] ( x ) , \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} \\nu ( x _ 1 , \\ldots , x _ n ) = a \\sum _ { 1 \\le i \\le j \\le n } x _ i x _ j \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{gather*} \\det \\big \\{ s _ { i + j } ^ { \\ , \\varepsilon _ { k } } \\big \\} _ { i , j = 0 } ^ { n } \\neq 0 \\ , \\ \\ n , k \\geq 0 \\ . \\end{gather*}"}
-{"id": "3702.png", "formula": "\\begin{align*} \\int _ 0 ^ { 2 \\pi } \\dfrac { e ^ { i ( n - m ) \\theta } } { r e ^ { i \\theta } - z } d \\theta = \\left \\{ \\begin{array} { l l } - 2 \\pi \\dfrac { r ^ { m - n } } { z ^ { m - n + 1 } } & \\mbox { i f } 0 < r < | z | \\\\ 0 & \\mbox { i f } 0 \\leq | z | < r < 1 \\end{array} \\right . . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} C _ n ( 0 ) & = \\sum _ { y _ n \\in \\{ 0 , 1 \\} } \\log \\Big ( \\frac { q _ n ( y _ n | 0 , 0 ) } { \\nu ^ { \\pi ^ * } _ n ( y _ n | 0 ) } \\Big ) q _ n ( y _ n | 0 , 0 ) = \\log \\Big ( \\frac { q _ n ( 0 | 0 , 0 ) } { \\nu ^ { \\pi ^ * } _ n ( 0 | 0 ) } \\Big ) q _ n ( 0 | 0 , 0 ) + \\log \\Big ( \\frac { q _ n ( 1 | 0 , 0 ) } { \\nu ^ { \\pi ^ * } _ n ( 1 | 0 ) } \\Big ) q _ n ( 1 | 0 , 0 ) \\\\ & = \\alpha _ n \\log \\big ( \\frac { 1 - c _ 0 ( n ) } { c _ 0 ( n ) } \\big ) + \\log \\big ( \\frac { 1 } { 1 - c _ 0 ( n ) } \\big ) - H ( \\alpha _ n ) . \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } u _ { \\delta } \\prod _ { i = 1 } ^ { n } x _ { i } ^ { \\delta _ { i } } = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } v _ { \\delta } \\ , x _ { n } ^ { \\delta _ { n } } \\prod _ { i = 1 } ^ { n - 1 } \\left ( x _ { i } - x _ { n } \\right ) ^ { \\delta _ { i } } \\end{align*}"}
-{"id": "10144.png", "formula": "\\begin{align*} ( w ^ t \\cdot v ) \\cdot u = \\sum _ { p < q } u _ { p q } , u _ { p q } = ( e _ p v _ q - e _ q v _ p ) \\cdot ( u _ p w _ q - u _ q w _ p ) \\in { } \\ ! R ^ n , \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} \\phi ( s ) : = \\prod _ { i = 1 } ^ { n } ( s _ { \\sigma ( i + 1 ) } - s _ { \\sigma ( i ) } ) , \\phi ( r ) : = \\prod _ { i = 1 } ^ { n } ( r _ { \\rho ( i + 1 ) } - r _ { \\rho ( i ) } ) , \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} f ( v ) & \\le \\sum _ { k = 0 } ^ q \\omega ^ { \\beta _ k } \\cdot p _ k \\le \\sum _ { k = 0 } ^ { r - 1 } \\omega ^ { \\beta _ k } \\cdot p _ k \\\\ & < \\sum _ { k = 0 } ^ { r - 1 } \\omega ^ { \\beta _ k } \\cdot p _ k + 1 + f _ r ( u ) = f ( u ) , \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} \\widehat { \\mu } ^ { \\rm p l } _ { S ' } ( \\widehat { f } _ { S ' } ) = f _ { S ' } ( 1 ) . \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\eta ( v ) = v \\otimes 1 + a \\cdot 1 \\otimes x + l _ 3 \\cdot v \\otimes x . \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { K } \\Vert \\widehat { \\beta } _ { - k } ( \\widehat { \\lambda } ) - \\beta \\Vert _ { 2 , n , k } ^ { 2 } & \\lesssim \\sum _ { k = 1 } ^ { K } \\Vert \\widehat { \\beta } _ { - k } ( \\bar { \\lambda } _ { 0 } ) - \\beta \\Vert _ { 2 , n , k } ^ { 2 } \\\\ & + \\sqrt { \\frac { \\log ^ { r + 1 } n } { n } } \\sum _ { k = 1 } ^ { K } \\Vert \\widehat { \\beta } _ { - k } ( \\widehat { \\lambda } ) - \\widehat { \\beta } _ { - k } ( \\bar { \\lambda } _ { 0 } ) \\Vert _ { 2 , n , k } , \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} \\overline { \\varphi } ( y ) = \\frac { 1 } { 2 } Z ( y , y ) \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} V _ \\nu ( x ) = - \\frac { 2 \\Gamma ( \\tfrac { 1 } { 2 } + \\nu ) } { \\sqrt { \\pi } \\Gamma ( \\nu ) } ( 1 + x ^ 2 ) ^ { - 1 + \\nu } { } _ 2 F _ 1 ( 1 , - \\nu ; \\tfrac { 1 } { 2 } ; \\tfrac { x ^ 2 } { 1 + x ^ 2 } ) , \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} \\beta _ { s , a ^ 2 } ^ 2 = \\frac { r ^ 2 ( s , a _ s ^ 1 , a ^ 2 ) - r ^ 2 ( s , a _ s ^ 1 , a _ s ^ 2 ) } { r ^ 2 ( s , a _ s ^ 1 , a ^ 2 ) - \\sum _ { s ' \\in S } \\left ( \\frac { \\mu ( s ' , s , a _ s ^ 1 , a ^ 2 ) } { | | \\mu | | } + \\delta ( s , s ' ) \\right ) r ^ 2 ( s ' , a _ { s ' } ^ 1 , a _ { s ' } ^ 2 ) } , \\end{align*}"}
-{"id": "10165.png", "formula": "\\begin{align*} \\mathbb E [ Z ^ * _ { \\lfloor \\varepsilon n \\rfloor + 1 , n + \\lfloor \\varepsilon n \\rfloor } ] = \\mathbb E [ Z _ { \\lfloor \\varepsilon n \\rfloor } ] + \\mathbb E [ Z ^ * _ { \\lfloor \\varepsilon n \\rfloor + 1 , n + \\lfloor \\varepsilon n \\rfloor } - Z _ { \\lfloor \\varepsilon n \\rfloor } ] = 0 + \\mathbb E [ Z _ n ^ * ] = \\mathbb E [ Z _ n ^ * ] \\ , . \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} u [ k | k ; i { , } j ] { = } L C ( u [ k | k { , } i { ; } j ] { , } u [ k | k { , } j { ; } i ] ) { , } k { \\neq } i { \\neq } j , \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} p _ S ( x ) = \\displaystyle \\sum _ { j = 0 } ^ { \\max ( S ) } ( \\Delta ^ j p _ S ) ( k ) \\binom { x - k } { j } \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } = & \\frac { 2 \\pi \\tau ^ { 1 / \\tau } } { \\Gamma ( 1 - 1 / \\tau ) } \\ , \\beta ^ { - 1 } _ { 2 2 } ( \\tau , b _ 0 = \\tau , \\ , b _ 1 = 1 + \\tau \\lambda _ 1 , \\ , b _ 2 = 1 + \\tau \\lambda _ 2 ) \\times \\\\ & \\times \\beta _ { 1 , 0 } ^ { - 1 } ( \\tau , b _ 0 = \\tau ( \\lambda _ 1 + \\lambda _ 2 + 1 ) + 1 ) , \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} \\partial _ { i j } \\varphi _ a ( x ) = \\frac { \\varphi ' _ a ( | x | ) } { | x | } \\delta _ { i j } - a \\frac { x _ i x _ j } { | x | ^ 2 } \\zeta _ a ( | x | ) , \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} r ^ 1 ( s , f ^ * , g ) = \\max _ { f \\in F _ S } r ^ 1 ( s , f , g ) = \\max _ { a ^ 1 \\in A ^ 1 ( s ) } [ R ^ 1 ( s ) g ( s ) ] _ { a ^ 1 } . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} \\dim _ y Y = \\dim _ { f ( y ) } S + \\dim _ y f ^ { - 1 } ( f ( y ) ) \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} A _ { \\eta } ( X ) = \\langle \\eta , \\eta \\rangle X \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} - \\Delta u ( x ) + q ( x ) u ( x ) = \\lambda u ( x ) , \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} \\mathcal { D } _ N = \\left \\{ Z _ N = ( X _ N , V _ N ) \\in \\mathbb { R } ^ { d N } \\times \\mathbb { R } ^ { d N } \\left | \\forall 1 \\leq i < j \\leq N , \\ ; | x _ i - x _ j | > \\varepsilon \\right . \\right \\} \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} & s ( \\tilde { \\gamma } ) + t ( \\tilde { \\gamma } ) = x + y + 1 , \\\\ & t ( \\tilde { \\gamma } ) = 2 c ( \\tilde { \\gamma } ) + 1 . \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} \\inf \\left \\{ \\max _ { g \\in K _ 1 } | \\alpha ( g ) f - f | _ { \\Phi } \\mid f \\in { \\mathcal F } ( G _ 1 ) , f ( e ) = 1 \\right \\} > 0 \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} H ^ 1 ( G , H ^ 2 ( M ) ) = 0 . \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} q ^ * _ 5 | _ V { \\bf 1 } + q ^ * _ 6 | _ V { \\bf i } + q ^ * _ 7 | _ V { \\bf j } + q ^ * _ 8 | _ V { \\bf k } = ( x y - y x ) z . \\end{align*}"}
-{"id": "9922.png", "formula": "\\begin{align*} \\lambda ^ { \\max } ( u ^ { - } ( s ) v ) = \\lambda ^ { \\max } ( v ) . \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} H _ { \\nu + m } ^ { ( 1 ) } ( i y ) = \\left ( \\frac { 2 i } { y } \\right ) ^ { m } r _ { m , \\nu } \\left ( \\frac { y ^ 2 } { 4 } \\right ) H _ { \\nu } ^ { ( 1 ) } ( i y ) + \\left ( \\frac { 2 i } { y } \\right ) ^ { m - 1 } s _ { m , \\nu } \\left ( \\frac { y ^ 2 } { 4 } \\right ) H _ { \\nu + 1 } ^ { ( 1 ) } ( i y ) . \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v + ( v \\cdot \\nabla _ H ) v + w \\partial _ z v + \\nabla _ H p - \\Delta _ H v + f _ 0 k \\times v = 0 , \\\\ \\partial _ z p + T = 0 , \\\\ \\nabla _ H \\cdot v + \\partial _ z w = 0 , \\\\ \\partial _ t T + v \\cdot \\nabla _ H T + w \\left ( \\partial _ z T + \\frac { 1 } { h } \\right ) - \\Delta _ H T = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} & a ^ * _ { N _ R , 0 } = 1 - ( 1 - \\mu _ R ) \\frac { N _ R } { N _ T } , a ^ * _ { N _ R - N _ T , N _ T } = \\frac { 1 - \\mu _ R } { \\binom { N _ R - 1 } { N _ R - N _ T } } , \\textrm { a n d o t h e r s b e i n g 0 } . \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} P ^ - \\sum _ { i = 1 } ^ { \\chi - 1 } U _ i B U _ i ^ * P ^ - - P ^ - \\sum _ { i = 1 } ^ { \\chi - 1 } U _ i C U _ i ^ * P ^ - = C . \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} C \\big ( D ^ F _ { X _ \\infty } , D ^ F _ { Y _ { \\R _ + } } \\big ) \\omega = \\int _ { - \\delta _ Y } ^ { \\delta _ Y } E _ 0 ( C ( \\lambda ) \\phi _ \\lambda , \\lambda ) d \\lambda . \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} \\mathbb { E } [ S _ { } ] = \\frac { e ^ { \\frac { 1 } { \\beta \\xi } } } { \\ln ( 2 ) } E _ 1 \\left ( \\frac { 1 } { \\beta \\xi } \\right ) . \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} \\left \\Vert \\left \\{ a _ { j } \\right \\} _ { j = 1 } ^ { J } \\right \\Vert _ { \\ell ^ { 2 } \\left ( d \\mu \\right ) } \\leq C \\left \\Vert \\left \\{ \\xi _ { j } \\right \\} _ { j = 1 } ^ { J } \\right \\Vert _ { \\ell ^ { 2 } \\left ( d \\mu \\right ) } . \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} P ^ - \\sum _ { i = 1 } ^ { \\chi - 1 } U _ i C U _ i ^ * P ^ - \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} d ^ { A ^ { ( 1 ) } _ { 2 n } } _ { k , l } ( z ) & = D _ { k , l } ( z ) \\times ( z + q ^ { \\mathtt { h } ^ \\vee } ) ^ { \\delta _ { l , k ^ * } } \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\left [ J _ { a b } , \\bar { Z } _ { c } \\right ] & = \\frac { 1 } { \\sqrt { 2 } } \\left [ J _ { a b } , P _ { c } \\right ] - \\frac { 1 } { \\sqrt { 2 } } \\left [ J _ { a b } , Z _ { c } \\right ] , \\\\ & = \\frac { 1 } { \\sqrt { 2 } } f _ { a b , c } ^ { d } P _ { d } - \\frac { 1 } { \\sqrt { 2 } } f _ { a b , c } ^ { d } Z _ { d } , \\\\ & = f _ { a b , c } ^ { d } \\bar { Z } _ { d } , \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} \\operatorname { a d j o i n t } ( g \\circ f ) = \\operatorname { a d j o i n t } ( g ) \\circ \\Phi ^ * f , \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} R ^ \\prime ( t ) : = - \\int _ s ^ t \\nabla _ x ^ 2 V _ \\rho ( \\tau , q ( \\tau , s ) ) \\int _ s ^ \\tau A ( p ( \\sigma , s ) ) d \\sigma d \\tau . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\frac { a _ { p , q } ( x _ n ) - a _ { p , q } ( x ) } { x _ n - x } & = \\frac { \\pi _ q ( 0 ^ m 1 a _ { n , m + 2 } a _ { n , m + 3 } \\cdots ) } { \\pi _ p ( 0 ^ m 1 a _ { n , m + 2 } a _ { n , m + 3 } \\cdots ) } \\ge \\left ( \\frac { p } { q } \\right ) ^ m \\frac { \\pi _ q ( 1 0 ^ { \\infty } ) } { \\pi _ p ( 1 ^ { \\infty } ) } \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} \\xi _ 1 = \\xi _ 0 + \\frac { 2 \\pi } { L } k , \\xi _ 2 = \\xi _ 1 + \\frac { 2 \\pi } { L } l , \\xi _ 3 = \\xi _ 2 + \\frac { 2 \\pi } { L } m , \\xi _ 4 = \\xi _ 3 + \\frac { 2 \\pi } { L } n \\xi _ 5 = \\xi _ 4 + \\frac { 2 \\pi } { L } s , \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} \\lim _ { W ( x ) \\rightarrow - 1 } - \\ln ( W ( x ) + 1 ) = + \\infty \\end{align*}"}
-{"id": "9729.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq 1 } A _ f ( n ) n ^ { \\frac { k - 1 } { 2 } } e ^ { 2 \\pi i n z } . \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} \\Psi ( x , y , t ) : = u ( x , t ) - v ( y , t ) - \\Phi _ \\lambda ( x , y , t ) - \\frac { \\beta } { 2 } ( | x | ^ 2 + | y | ^ 2 ) - \\mu t . \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} \\delta ' ( \\boldsymbol { x } _ 1 , \\ldots , \\boldsymbol { x } _ n ) ^ { q ^ { - m } } = ( \\xi ^ { q ^ { - m } } ) ^ { - ( 1 + q + \\cdots + q ^ { n - 1 } ) } \\delta ( \\boldsymbol { \\xi } \\boldsymbol { x } _ 1 , \\ldots , \\boldsymbol { \\xi } \\boldsymbol { x } _ n ) ^ { q ^ { - m } } \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n } f ( t \\alpha _ { i } ) e _ { i } = F ( t \\alpha ) = s ( t ) F ( \\alpha ) = s ( t ) \\sum _ { i = 1 } ^ { n } f ( \\alpha _ { i } ) e _ { i } \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} \\gamma _ D ( x ) \\ ; = \\ ; \\inf \\left \\{ \\lambda > 0 \\ ; | \\ : x \\in \\lambda D \\right \\} , \\end{align*}"}
-{"id": "10004.png", "formula": "\\begin{align*} R _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ { m } & = c l \\ C _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ m , \\ m = 1 , \\ldots , 4 \\\\ F _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } & = \\cup _ { m = 1 } ^ 4 \\partial C _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ m , \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} { \\rm d i m } _ { k } { \\rm H o m } _ { { \\rm r e p } ( Q ) } ( X , Y ) = \\left \\{ \\begin{array} { l l } 0 & \\mbox { $ Y \\in \\{ Z ~ | ~ Z $ i s n o t a s u c c e s s o r o f $ X \\} \\cup \\{ A _ { l ' } ^ { ( 1 - k ) } ~ | ~ l ' > l , l ' $ ~ { \\rm o d d } $ \\} ; $ } \\\\ 1 & \\mbox { $ Y \\in \\{ Z ~ | ~ Z $ i s a s u c c e s s o r o f $ X \\} \\backslash \\{ A _ { l ' } ^ { ( 1 - k ) } ~ | ~ l ' > l , l ' $ ~ { \\rm o d d } $ \\} . $ } \\end{array} \\right . \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\sigma ^ { i j } w _ i w _ j = \\norm { D u } ^ 2 \\Theta ^ { - 2 } \\theta ^ 2 v ^ 2 \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} P ^ + = \\sum _ { i = 1 } ^ { n ^ + } v _ i v _ i ^ * \\mbox { a n d } P ^ - = \\sum _ { i = n - n ^ - + 1 } ^ n v _ i v _ i ^ * \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} \\beta _ { i , j } = \\varphi _ { P z _ { j } } \\left ( z _ { i } \\right ) . \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} R ^ \\star _ { \\sf u } ( R _ { \\mathsf { c } } ) \\ge \\max _ { \\ell \\in [ 1 : L ] } \\sum _ { n = 1 } ^ N ( s _ n ( \\ell ) - s _ { n + 1 } ( \\ell ) ) \\left ( n - \\ell R _ { \\sf c } \\right ) ^ + , \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} v _ t = \\sum _ { i = 0 } ^ { d - 1 } \\phi _ t ^ i u _ i , \\ 0 \\leq t \\leq d - 1 , \\ \\ \\max \\limits _ { t , i } \\deg ( \\phi _ t ^ i ) = l . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} D _ m = \\det \\left ( \\kappa _ { ( 1 + j - i ) } \\right ) _ { 1 \\leq i , j \\leq m } , \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} \\aligned & S ^ 3 _ { \\alpha p } = a _ 3 S ^ 2 _ { \\alpha p } - b _ 3 T ^ 2 _ { \\alpha p } , T ^ 3 _ { \\alpha p } = b _ 3 S ^ 2 _ { \\alpha p } + a _ 3 T ^ 2 _ { \\alpha p } , \\\\ & S ^ 4 _ { \\alpha p } = a _ 4 S ^ 2 _ { \\alpha p } - b _ 4 T ^ 2 _ { \\alpha p } , T ^ 4 _ { \\alpha p } = b _ 4 S ^ 2 _ { \\alpha p } + a _ 4 T ^ 2 _ { \\alpha p } , \\\\ & S ^ 5 _ { \\alpha p } = a _ 5 S ^ 2 _ { \\alpha p } - b _ 5 T ^ 2 _ { \\alpha p } , T ^ 5 _ { \\alpha p } = b _ 5 S ^ 2 _ { \\alpha p } + a _ 5 T ^ 2 _ { \\alpha p } . \\endaligned \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{gather*} D _ { 0 } = D _ { 1 } = \\ \\ldots \\ = D _ { n _ { 0 } - 1 } = 0 \\ , \\ D _ { n _ { 0 } } = ( - 1 ) ^ { \\frac { n _ { 0 } ( n _ { 0 } + 1 ) } { 2 } } s _ { n _ { 0 } } ^ { n _ { 0 } + 1 } = ( - 1 ) ^ { n _ { 0 } ( n _ { 0 } + 1 ) } t _ { n _ { 0 } } = t _ { n _ { 0 } } \\ . \\end{gather*}"}
-{"id": "2319.png", "formula": "\\begin{align*} \\pi \\big ( D _ \\lambda ( a ) \\big ) = \\delta _ \\lambda \\big ( \\pi ( a ) \\big ) , . \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} \\mu ( u , e ^ T e ^ S ) \\leq \\mu ( \\frac { u } { 2 } , e ^ T ) \\mu ( \\frac { u } { 2 } , e ^ S ) \\leq \\mu ( \\frac { u } { 2 } , e ^ { T _ + } ) \\mu ( \\frac { u } { 2 } , e ^ { S _ + } ) & = e ^ { \\mu ( \\frac { u } { 2 } , T _ + ) + \\mu ( \\frac { u } { 2 } , S _ + ) } \\\\ & \\leq e ^ { \\mu ( \\frac { u } { 2 } , T ) + \\mu ( \\frac { u } { 2 } , S ) } , \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} H = ( - \\Delta + m ^ 2 ) ^ { 1 / 2 } - m + V \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} X \\diamond Y = X + Y + \\frac { 1 } { 2 } [ X , Y ] + \\frac { 1 } { 1 2 } [ X [ X , Y ] ] - \\frac { 1 } { 1 2 } [ Y [ X , Y ] ] + \\cdots , \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} | V _ { n , k } ( P ) | = \\sum _ { \\frac { 1 } { n } ( i , j ) \\in P } \\binom { j - i - 1 } { k - 2 } = \\sum _ { \\frac { 1 } { n } ( i , j ) \\in P } \\binom { n ( j / n - i / n ) - 1 } { k - 2 } . \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} [ \\pi _ A ( a ) , \\pi _ B ( b ) ] = \\pi _ A ( a ) \\pi _ B ( b ) - \\pi _ A ( a ) \\pi _ B ( b ) = 0 a , b \\in \\mathcal C ( r ) . \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} & H _ k ^ - ( x ^ { ( k ) } ) \\star 1 ( - E _ k ( - u ) , E _ k ( v ) ) + F _ k ( u ) \\star E _ k ( v ) ( 1 , H _ k ( x ^ { ( k ) } ) ) \\\\ = & E _ k ( v ) \\star F _ k ( u ) ( 1 , H _ k ( x ^ { ( k ) } ) ) + H _ k ^ + ( x ^ { ( k ) } ) \\star 1 ( E _ k ( v ) , - E _ k ( - u ) ) . \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{gather*} \\sum _ { k = 0 } ^ { r - 1 } \\frac { P _ { k } ( \\lambda ) ^ { 2 } } { D _ { k } D _ { k - 1 } } \\geq \\frac { P _ { r - 1 } ( \\lambda ) ^ { 2 } } { D _ { r - 1 } D _ { r - 2 } } > 0 \\ . \\end{gather*}"}
-{"id": "7609.png", "formula": "\\begin{align*} \\mathcal { D } ( X ) : = 2 ( R _ { i j } - h _ { i j } ) ( X , X ) + 2 \\langle \\nabla \\mathcal { H } - 2 d i v \\ h , X \\rangle + \\partial _ t \\mathcal { H } - \\Delta \\mathcal { H } - 2 | h _ { i j } | ^ 2 , \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} \\aligned 0 & = ( - d \\sigma ^ { - 1 } ( I - \\Delta ^ { t r } \\Delta ) + \\theta \\Delta ) \\Delta ^ { t r } \\mu ^ { t r } - 2 ( d \\Delta - \\theta \\sigma ) \\sigma \\mu ^ { t r } \\\\ & = d ( - \\sigma ^ { - 1 } ( I - \\Delta ^ { t r } \\Delta ) \\Delta ^ { t r } - 2 \\Delta \\sigma ) u ^ { t r } + \\theta ( \\Delta \\Delta ^ { t r } + 2 \\sigma ^ 2 ) u ^ { t r } \\\\ & = d \\ ; ( \\sqrt { 2 } , 0 , 0 ) \\mu ^ { t r } + \\theta \\mu ^ { t r } \\\\ & = ( \\sqrt { 2 } p , 0 , 0 ) \\mu ^ { t r } , \\endaligned \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{gather*} \\left \\langle Q ^ { \\pm k } v _ { 0 } , \\mathop { \\overleftarrow \\prod } \\limits _ { i = 1 } ^ { k } \\psi ^ { \\pm } ( z _ { i } ) v _ { 0 } \\right \\rangle = \\det \\big ( V ^ { ( k ) } _ { \\{ z _ { i } \\} } \\big ) . \\end{gather*}"}
-{"id": "9355.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ I \\sum _ { j = 0 } ^ J \\sum _ { k = 0 } ^ { K } r _ { i , j , k } ( x ) \\log ( x ) ^ j x ^ { \\alpha _ i } g _ { i , k } ( t ) \\zeta ( t ) ^ k \\rho ( \\delta _ i , t ) , \\ t = \\log ( x ) / \\log ( q _ 1 ) , \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} A ^ { ( 1 ) } = u , x _ { 1 } = y , A ^ { ( 2 ) } = - & u \\omega _ { [ 1 ] } , x _ { 2 } = x \\\\ & A ^ { ( 1 ) } = u , x _ { 1 } = t , A ^ { ( 2 ) } = \\delta , x _ { 2 } = x , \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { m } ( - 1 ) ^ i { 2 m + r - 2 \\choose m - i } { m + i \\choose i } \\gamma _ { m + i - 1 } = 0 . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} \\norm { v } _ { S _ T } : = & \\sup _ { 0 \\leq s \\leq T } \\norm { v ( s ) } _ { V _ { \\delta } } + \\sup _ { 0 \\leq s \\leq T } s ^ { 1 - \\gamma } \\norm { v ( s ) } _ { V _ { \\gamma } } , \\\\ \\norm { \\zeta } _ { \\hat { S } _ T } : = & \\sup _ { 0 \\leq s \\leq T } \\norm { \\zeta ( s ) } _ { \\hat { V } _ { \\delta } } + \\sup _ { 0 \\leq s \\leq T } s ^ { 1 - \\gamma } \\norm { \\zeta ( s ) } _ { \\hat { V } _ { \\gamma } } . \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} h : = \\Delta _ { g } f + \\left | \\nabla f \\right | ^ { 2 } , \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} h ^ 1 ( N _ C ) = h ^ 1 ( \\mathcal { O } _ C ( 2 ) ) . \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} \\delta _ \\lambda ( X _ j ( x ) ) = \\lambda \\ , X \\big ( \\delta _ \\lambda ( x ) \\big ) , . \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} \\| \\partial ^ \\mu _ { t , x } b \\| _ { \\ell ^ { 1 + | \\mu | } _ 1 L ^ \\infty _ { t , x } } + \\| \\partial _ { t , x } ^ \\mu c \\| _ { \\ell ^ { 2 + | \\mu | } _ 1 L ^ \\infty _ { t , x } } = \\O ( 1 ) , | \\mu | \\le 2 . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} \\int _ t ^ T \\int _ { \\mathcal { O } } ( u - k ) ^ + \\ , \\mu ( d x d s ) \\leq \\int _ t ^ T \\int _ { \\mathcal { O } } ( u - \\xi ) ^ + \\ , \\mu ( d x d s ) + \\int _ t ^ T \\int _ { \\mathcal { O } } ( \\xi - \\hat { \\xi } ^ + ) ^ + \\ , \\mu ( d x d s ) = 0 . \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\omega _ { n } ^ { s } = \\begin{cases} ~ 1 , & ~ n = 0 \\\\ | n | ^ { s } , & ~ n \\neq 0 . \\end{cases} \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} Q _ { m } ( x ) = P _ { m } ( x ) - \\lambda \\frac { P _ m ( c ) } { 1 + \\lambda L _ { m - 1 } ( c ; c ) } L _ { m - 1 } ( x ; c ) . \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} \\nabla \\tilde { h } ( t , x , y ) = \\begin{pmatrix} \\partial _ x \\tilde h \\\\ \\partial _ y \\tilde h \\end{pmatrix} = \\begin{pmatrix} \\partial _ z h \\\\ a ( \\partial _ v - t \\partial _ z ) h \\end{pmatrix} = \\begin{pmatrix} \\partial ^ t _ z h \\\\ \\partial ^ t _ v h \\end{pmatrix} = \\nabla _ t h , \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} { n - \\beta \\choose \\alpha } _ r + { n - \\alpha \\choose \\beta } _ s = { n \\choose \\alpha + \\beta } _ { r + s } + \\sum _ k \\sum _ { \\substack { \\ell , m \\ge 0 , \\\\ \\ell + m = k - 1 } } { n \\choose k } _ { r + s } { \\ell \\choose \\alpha } _ r { m \\choose \\beta } _ s . \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\exp _ 2 \\{ \\lambda L ( y _ { n _ { i - 1 } + 1 } ^ { n _ i } ) \\} \\ge \\exp _ 2 \\{ ( \\lambda + 1 ) \\log c - \\lambda \\gamma ( s , \\log _ \\alpha c ) \\} . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} & \\left ( \\delta _ { e _ 1 / a _ 1 } - \\delta _ 0 \\right ) * \\sigma _ j \\\\ & = \\delta _ { c _ j } \\ast ( \\delta _ { e _ 1 \\ , N _ 1 / a _ 1 } - \\delta _ 0 ) \\ast \\frac { 1 } { \\ , R _ j } \\ , \\sum _ { 0 \\le k _ 2 \\le a _ 1 R _ j - 1 } \\ , \\dots \\sum _ { 0 \\le k _ s \\le a _ s R _ j - 1 } \\ , \\delta _ { ( 0 , - k _ 2 / a _ 2 , \\dots , - k _ 1 / a _ s ) } \\ast \\nu _ \\mu , \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} \\sum _ { m , r , s , k \\atop 2 m + r + s = p } a _ { k m r s } Q ^ m \\phi _ k ^ { r , s , 0 } + \\sum _ { m , r , s , k \\atop 2 m + r + s + 1 = p } b _ { k m r s } Q ^ m \\widetilde \\phi _ k ^ { r , s } = 0 \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} F ( u ) = L u + N ( u ) = 0 . \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} \\int \\widehat f \\ , d \\mathfrak { S } ^ { n } ( { \\widehat { \\mu } } ) = \\sum _ { j = 1 } ^ { k } \\int f _ { j } \\ , d ( \\mathfrak { S } ^ { n } ( { \\widehat { \\mu } } ) ) _ { j } , \\mathfrak { S } ^ { n } ( { \\widehat { \\mu } } ) = \\big ( ( \\mathfrak { S } ^ { n } ( { \\widehat { \\mu } } ) ) _ { 1 } , \\dots , ( \\mathfrak { S } ^ { n } ( { \\widehat { \\mu } } ) ) _ { k } \\big ) . \\end{align*}"}
-{"id": "9343.png", "formula": "\\begin{align*} \\delta ( Z ) = \\mu ^ N \\sigma ^ N ( A ) Z , \\ \\ \\sigma ( Z ) = \\sigma ^ N ( B ) Z . \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} I = ( T _ 1 , \\dots , T _ \\ell ) = ( \\pi _ 1 , \\dots , \\pi _ \\ell ) . \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} f _ i ( \\Sigma ) / f _ j ( \\Sigma ) \\leq { n \\choose i + 1 } / { n \\choose j + 1 } = \\mathcal { O } ( n ^ { i - j } ) , \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} ( D F ) [ X ] = \\int _ { Y } ^ S K _ D ^ S [ X , Y ] F [ Y ^ * ] . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} & \\{ a _ X \\} + \\{ a ' _ X \\} = \\{ a _ X + a ' _ X \\} , \\\\ & \\{ a _ X \\} \\cdot \\{ a ' _ X \\} = \\{ a _ X \\circ a ' _ X \\} , \\\\ & \\{ a _ X \\} ^ * = \\{ a _ X ^ * \\} , \\\\ & \\alpha \\{ a _ X \\} = \\{ \\alpha a _ X \\} \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} \\dim \\delta H _ 3 ^ g = & \\ ; 4 + 6 ( k + 2 ) + 4 \\ , ( k + 1 ) ( k + 2 ) / 2 + k ( k + 1 ) ( k + 2 ) / 6 \\\\ & \\ ; - ( k + 4 ) ( k + 5 ) ( k + 6 ) / 6 \\\\ = & \\ ; 0 \\\\ \\dim \\delta E _ 3 ^ g = & \\ ; I _ M ( V _ 3 ^ g \\times W _ 3 ^ g ) \\\\ = & \\ ; 0 , \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} & \\left [ ( \\lambda z - \\mu j ) ( 1 - z ) + ( c - j ) \\gamma z + j \\xi ( 1 - z ) \\right ] P _ { j } ( z ) - \\xi z ( 1 - z ) P ^ { ' } _ { j } ( z ) \\\\ & = ( c - j + 1 ) \\gamma z P _ { j - 1 } ( z ) + j \\mu ( 1 - z ) \\sum _ { n = 0 } ^ { j } z ^ { n } p _ { j , n } - \\mu z \\sum _ { n = 1 } ^ { j } n z ^ { n } p _ { j , n } \\\\ & + \\mu \\sum _ { n = 1 } ^ { j + 1 } n z ^ { n } p _ { j + 1 , n } + j \\xi ( 1 - z ) \\sum _ { n = 0 } ^ { j } z ^ { n } p _ { j , n } , \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} N _ { i j } \\eta ^ i \\eta ^ j = \\ , \\{ F ^ { k l } h _ { r k } h _ l ^ r - 2 F + F ^ { k l } g _ { k l } \\} \\abs { \\eta } ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} T _ \\mathrm { b d } ( X , g ^ { T X } , h ^ { F } ) = \\exp \\Big ( \\frac { 1 } { 2 } { \\theta ^ { F } _ { X , \\mathrm { b d } } } ' ( 0 ) \\Big ) . \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} X ( k n ) = Y ( k ) , 0 \\le k \\le l - 1 \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} \\phi \\big ( M ( D _ i , D _ j ) \\big ) & = \\prod _ { k = 1 } ^ K \\sum _ { \\pi \\in N C _ { e , o } ( 2 t ) } \\kappa _ { \\pi } [ a _ { k ; i } ^ { n _ 1 } , a _ { k ; j } ^ { n _ 2 } , \\cdots , a _ { k ; i } ^ { n _ { 2 t - 1 } } , a _ { k ; j } ^ { n _ { 2 t } } ] . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} & a ( t ) \\geq 0 , \\ ; t \\in I ^ { + } _ { i } , a ( t ) \\not \\equiv 0 \\ ; I ^ { + } _ { i } , i = 1 , \\ldots , m ; \\\\ & a ( t ) \\leq 0 , \\ ; t \\in ( \\mathbb { R } / T \\mathbb { Z } ) \\setminus \\bigcup _ { i = 1 } ^ { m } I ^ { + } _ { i } . \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} & r a n k \\left ( \\mathbf { I } - \\mathbf { B } _ { j } \\right ) = L ~ ~ \\forall ~ 1 \\leq j \\leq \\omega , \\\\ & r a n k \\left ( \\mathbf { B } _ { \\omega + 1 } + \\mathbf { M } _ { \\omega } \\mathbf { M } _ { \\omega - 1 } \\cdots \\mathbf { M } _ 1 \\right ) = L \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\forall ~ \\mathbf { M } _ j \\in \\left \\{ \\mathbf { I } , \\mathbf { B } _ { j } \\right \\} , 1 \\leq j \\leq \\omega . \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} \\abs { p _ { k , h } ( z ) - 1 } & = \\biggl \\lvert \\sum _ { j = 0 } ^ { h - 1 } z ^ j + \\frac 1 2 \\sum _ { j = 0 } ^ { k - h - 1 } z ^ j \\sum _ { j = 0 } ^ { h } z ^ j - \\omega \\sum _ { j = 0 } ^ { k - h } z ^ j \\sum _ { j = 0 } ^ { h - 1 } z ^ j \\biggr \\rvert \\cdot \\abs { z - 1 } \\\\ & \\leq \\left ( h + \\frac 1 2 ( k - h ) ( h + 1 ) + ( k - h + 1 ) h \\right ) \\abs { z - 1 } \\\\ & < k ^ 2 \\abs { z - 1 } , \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} \\norm { \\partial _ t ^ i D ^ j v } { H ^ { k - m , 2 k - 2 m } ( D _ T ) } & = \\sum _ { \\ell = 0 } ^ { k - m } \\norm { \\partial _ t ^ { \\ell + i } D ^ j v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 m - 2 \\ell } ( D ) ) } \\\\ & = \\sum _ { \\ell = i } ^ { k - m + i } \\norm { \\partial _ t ^ { \\ell } D ^ j v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 m - 2 \\ell + 2 i } ( D ) ) } . \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} u _ { n } ^ { \\left ( r + 1 \\right ) } = x ^ { r + 1 } \\left ( u _ { n + r + 1 } \\right ) \\left ( u _ { 0 } ^ { \\left ( 0 \\right ) } u _ { 0 } ^ { \\left ( 1 \\right ) } u _ { 0 } ^ { \\left ( 2 \\right ) } . . . u _ { 0 } ^ { \\left ( r \\right ) } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} \\frac { 2 } { 2 n + 1 } \\sum \\limits _ { j = 0 } ^ n \\gamma _ { 2 j } \\sum \\limits _ { i = j } ^ n { 2 n + 1 \\choose 2 i + 1 } B _ { 2 n - 2 i } \\frac { 2 j + 1 } { 2 i + 2 } { i + 1 \\brack j + 1 } . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} & \\delta _ { \\mathsf { A c h } } ( \\mu , r ) = ( M + K - 1 ) \\mu + \\left ( 1 - \\mu M \\right ) \\left [ \\frac { K } { \\min \\{ M , K \\} } + \\frac { K } { M r } \\right ] ; \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} e _ { \\mu \\nu \\sigma \\tau } Q ^ { \\sigma \\tau } = 2 i P _ { \\mu \\nu } , \\qquad 2 i Q ^ { \\mu \\nu } = e ^ { \\mu \\nu \\sigma \\tau } P _ { \\sigma \\tau } . \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} u _ { t } & = D _ { 1 } \\Delta u + f _ { 1 } ( u , v ) \\qquad \\Omega \\times ( 0 , T ) \\\\ v _ { t } & = D _ { 2 } \\Delta v + f _ { 2 } ( u , v ) \\qquad \\Omega \\times ( 0 , T ) \\\\ \\partial _ { n } u & = 0 = \\partial _ { n } v \\qquad \\partial \\Omega \\times ( 0 , T ) \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} g ^ { k m } g _ { m i : j } = \\tilde { g } ^ { k m } \\{ \\varphi _ { m j } \\varphi _ i + \\varphi _ { i j } \\varphi _ m + 2 \\cosh u \\ , \\varphi _ j ( \\varphi _ m \\varphi _ i + \\sigma _ { m i } ) \\} . \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\delta ( \\mu , r ) = \\lim _ { \\substack { P \\rightarrow \\infty } } \\frac { \\Delta ( \\mu , r \\log ( P ) , P ) } { 1 / \\log P } . \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} u '' + \\lambda a ( t ) g ( u ) = 0 \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} \\tilde { \\mathbf { H } } ^ { 0 } ( \\mathbf { x } ) = \\mathrm { C o v } \\big [ \\nabla \\boldsymbol { \\varepsilon } ( \\mathbf { x } ) \\big ] \\mathbf { x } \\in G \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} L = ( m + 1 ) \\max \\{ L _ i \\mid i \\in \\{ 0 , 1 , \\ldots , m \\} \\} , \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} \\gamma _ i = \\begin{cases} 0 , & i < 2 s , \\\\ { - r - 2 s \\choose i - 2 s } ( - 2 ) ^ { - r - i } , & i \\ge 2 s . \\end{cases} \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\{ I _ { k } , I _ { \\ell } \\} _ { m } = 0 & \\{ J _ { \\ell } , I _ { k } \\} _ { m } = ( k + \\ell + m - 2 ) I _ { k + \\ell + m - 2 } & \\{ J _ { k } , J _ { \\ell } \\} _ { m } = ( \\ell - k ) J _ { k + \\ell + m - 2 } \\end{array} \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} \\alpha = \\begin{cases} x _ 1 \\dots x _ { n } & n > 0 \\\\ s ( x ) & . \\end{cases} \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} \\xi _ \\varepsilon = \\xi \\sqrt { 1 + \\varepsilon ^ 2 \\xi ^ 2 } \\quad \\ d = 1 , \\xi _ \\varepsilon = | \\xi | \\sqrt { 1 + \\varepsilon ^ 2 | \\xi | ^ 2 } \\quad \\ d = 2 , 3 . \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} \\Psi ( s _ i \\otimes s _ j ) = \\xi ^ i s _ j \\otimes s _ i . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ { k - 2 } \\frac { \\lambda ^ { l } } { l ! } e ^ { - \\lambda } < U \\leq \\sum _ { l = 0 } ^ { k - 1 } \\frac { \\lambda ^ { l } } { l ! } e ^ { - \\lambda } \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{gather*} E _ { a b } z ^ { k } \\mapsto \\colon \\sum _ { l \\in \\mathbb { Z } } \\big ( e _ { a } z ^ { k + l } \\wedge \\big ) \\big ( i \\big ( e _ { b } z ^ { l } \\big ) \\big ) \\colon = \\sum _ { l \\in \\mathbb { Z } } \\colon { } _ { a } \\psi ^ { + } _ { ( k + l ) } { } _ { b } \\psi ^ { - } _ { ( - l - 1 ) } \\colon . \\end{gather*}"}
-{"id": "6676.png", "formula": "\\begin{align*} M _ { ( \\tau , \\lambda _ 1 , \\lambda _ 2 ) } \\overset { { \\rm i n \\ , l a w } } { = } \\int _ 0 ^ 1 s ^ { \\lambda _ 1 } ( 1 - s ) ^ { \\lambda _ 2 } \\ , M _ \\beta ( d s ) , \\ ; \\tau = 1 / \\beta ^ 2 > 1 . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} \\left \\langle T _ { ( A _ { 1 } , i _ { 1 } ) } , \\ldots , T _ { ( A _ { r } , i _ { r } ) } \\right \\rangle = \\alpha _ { \\gamma } K _ { i _ { 1 } \\ldots i _ { r } } ^ { \\gamma } \\left \\langle T _ { A _ { 1 } } , \\ldots , T _ { A _ { r } } \\right \\rangle , \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{gather*} t _ { n _ { 0 } } = ( - 1 ) ^ { \\tfrac { n _ { 0 } + 1 } { 2 } } \\Delta _ { 0 } \\ , \\ \\Delta _ { 0 } > 0 \\ , \\end{gather*}"}
-{"id": "7809.png", "formula": "\\begin{align*} \\alpha ( G ) \\le n ^ 0 + \\min ( n ^ + , n ^ - ) = \\min ( n ^ + , n ^ - ) . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} N ( \\delta _ k \\otimes c \\otimes \\delta _ n \\otimes b ) & = k \\delta _ k \\otimes c \\otimes \\delta _ n \\otimes b \\\\ & \\mapsto k \\alpha _ d ^ k ( c ) \\cdot \\delta _ { ( n , k ) } \\otimes b = X _ d \\left ( \\alpha _ d ^ k ( c ) \\cdot \\delta _ { ( n , k ) } \\otimes b \\right ) \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} y _ 2 ^ { p ^ \\ell } - y _ 2 + \\frac { y _ 1 ^ { p ^ { \\ell + 1 } } - y _ 1 ^ p - ( y _ 1 ^ { p ^ \\ell } - y _ 1 ) ^ p } { p } = \\frac { f ^ \\sigma ( x ^ p ) - ( f ( x ) ) ^ p } { p } \\bmod p , \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} \\tilde Z ( t ) = \\begin{cases} \\tilde X ( t ) + \\tilde Y ( t ) & \\mbox { i f } t \\le \\tilde \\tau _ 0 , \\\\ 0 & \\mbox { i f } t \\ge \\tilde \\tau _ 0 , \\end{cases} \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} r _ { \\sigma , \\varphi , 1 } ( \\chi ) = n \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} \\sinh ^ 2 \\tfrac { 1 } { 2 } d _ { \\xi } z = \\sinh ^ 2 ( \\tfrac { 1 } { 2 } T _ { \\xi } ) \\cosh ^ 2 d _ z \\mathcal { A } + \\sin ^ 2 \\theta \\sinh ^ 2 d _ z \\mathcal { A } , \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{gather*} \\Delta _ 0 \\partial ^ \\mu = \\exp ( - \\partial ^ \\alpha \\otimes x _ \\alpha ) ( 1 \\otimes \\partial ^ \\mu ) \\exp ( \\partial ^ \\alpha \\otimes x _ \\alpha ) , \\\\ \\Delta _ 0 \\partial ^ \\mu = \\exp ( - x _ \\alpha \\otimes \\partial ^ \\alpha ) ( \\partial ^ \\mu \\otimes 1 ) \\exp ( x _ \\alpha \\otimes \\partial ^ \\alpha ) . \\end{gather*}"}
-{"id": "8625.png", "formula": "\\begin{align*} B _ { g ( 0 ) } \\left ( x _ 0 , \\frac { r } { 2 K ^ 2 } \\right ) \\cap \\mathbf { B } = \\emptyset , \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} \\bar { V } = \\dfrac { V ( x ) } { ( \\gamma - R ( x ) ) } \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} B = { { v } ^ { H } } ( { { \\theta } _ { 0 } } , { { \\phi } _ { 0 } } ) v ( { { \\theta } _ { 0 } } , { { \\phi } _ { 0 } } ) . \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} \\big ( \\mathbf { H } ( \\mathbf { u } ) \\big ) ( t , \\cdot ) = \\exp ( - t / \\tau ) \\mathbf { H } _ { 0 } + \\int _ { 0 } ^ { t } \\exp \\big ( - ( t - s ) / \\tau \\big ) \\mathbf { F } \\big ( \\nabla \\mathbf { u } ( s , \\cdot ) \\big ) \\mathrm { d } s t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} g ( 0 , x _ 0 , t , 0 ) = \\Phi \\left ( \\frac { e - x _ 0 } { t ^ H } \\right ) + \\Phi \\left ( \\frac { e + x _ 0 } { t ^ H } \\right ) - 1 \\to 2 \\Phi ( 0 ) - 1 = 0 , \\quad t \\to \\infty . \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} d \\mu _ { i j } ( \\xi ) = e ^ { - 2 \\pi i ( x _ i - x _ j ) \\cdot \\xi } \\ , d \\mu ( \\xi ) , 1 \\le i , j \\le N . \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} & \\mathcal { L } _ { _ { 3 } , C \\mathfrak { _ { 3 } } } \\\\ & = \\varepsilon _ { a b c } \\left ( \\left ( \\alpha _ { 0 } + \\alpha _ { 1 } \\right ) R ^ { a b } e ^ { c } + \\frac { 1 } { 3 \\ell ^ { 2 } } \\left ( \\alpha _ { 1 } + \\alpha _ { 2 } \\right ) e ^ { a } e ^ { b } e ^ { c } + \\left ( \\alpha _ { 1 } + \\alpha _ { 2 } \\right ) k ^ { a b } T ^ { c } - \\frac { 1 } { 2 } \\left ( \\alpha _ { 1 } - \\alpha _ { 2 } \\right ) k ^ { a b } k _ { d } ^ { c } e ^ { d } \\right ) . \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} \\widetilde { v } : = \\begin{cases} \\max \\{ v , v _ 0 \\} & , \\\\ v _ 0 & , \\end{cases} \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} - ( p - 1 ) \\int _ M | \\nabla u | ^ { p - 1 } g ^ { i j } \\nabla _ i u \\nabla _ j u _ t d \\mu & = ( p - 1 ) \\int _ M g ^ { i j } \\nabla _ i ( | \\nabla u | ^ { p - 1 } \\nabla _ i u ) u _ t \\\\ \\displaystyle & = ( p - 1 ) \\int _ M \\Delta _ p u u _ t d \\mu \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} - g ^ { i j } w _ { : i j } - \\vartheta ^ { - 3 } \\dot \\vartheta \\Theta v ^ { - 2 } | D w | ^ 2 + v H \\Theta ^ { - 1 } + n \\vartheta ^ { - 1 } \\dot \\vartheta \\Theta ^ { - 1 } = 0 . \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} \\epsilon _ A ( [ p ] ) = \\epsilon _ A \\circ \\xi _ * ( [ 1 _ B ] ) = \\Phi _ { { ( D _ n \\cdots D _ 2 D _ 1 ) } ^ t } \\circ \\epsilon _ B ( [ 1 _ B ] ) = \\Phi _ { { ( D _ n \\cdots D _ 2 D _ 1 ) } ^ t } ( [ 1 , 1 , \\dots , 1 ] ^ t ) \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} a ^ { - \\frac { n } { 2 } } e ^ { - \\frac { a \\kappa | x | ^ 2 } { 4 } } \\Delta y = a ^ { 2 } \\Delta u + \\kappa a ^ { 2 } \\nabla u \\cdot x + \\frac { n \\kappa } { 2 } a u + \\frac { \\kappa ^ 2 } { 4 } a ^ 2 | x | ^ 2 u . \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} v '' + h ^ { * } ( t , v ) = 0 . \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} f _ N ( 0 , Z _ N ) = \\mathcal { Z } _ N ^ { - 1 } \\mathbf { 1 } _ { Z _ N \\in \\mathcal { D } _ N } f _ 0 ^ { \\otimes N } ( Z _ N ) \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} I ( A ; h ) = \\{ f : | | f - A | | \\leq q ^ h \\} \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} \\left [ T _ { ( A , i ) } , T _ { ( B , j ) } \\right ] & = K _ { i j } { } ^ { \\gamma } C _ { A B } { } ^ { C } T _ { ( C , \\gamma ) } , \\\\ & = K _ { i j } { } ^ { k } C _ { A B } { } ^ { C } T _ { ( C , k ) } + K _ { i j } { } ^ { k + n } C _ { A B } { } ^ { C } T _ { ( C , k + n ) } \\\\ & = K _ { i j } { } ^ { k } C _ { A B } { } ^ { C } T _ { ( C , k ) } - K _ { i j } { } ^ { k + n } C _ { A B } { } ^ { C } T _ { ( C , k ) } \\\\ & = \\left ( K _ { i j } { } ^ { k } - K _ { i j } { } ^ { k + n } \\right ) C _ { A B } { } ^ { C } T _ { ( C , k ) } , \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} _ { 1 } \\phi _ { 1 } \\left ( \\begin{array} { c c c } \\begin{array} { c } q ^ { n } \\\\ - q ^ { \\alpha + 1 } \\end{array} & \\vert & q , - x q ^ { 1 + \\alpha - \\beta } \\end{array} \\right ) q ^ { \\beta ^ { 2 } / 2 } = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { y ^ { 2 } } { \\log q ^ { 2 } } + i \\beta y \\right ) d y } { \\left ( - q ^ { \\alpha + 1 } , q ^ { \\alpha + 1 / 2 } e ^ { - i y } ; q \\right ) _ { \\infty } \\left ( x q ^ { \\alpha + 1 / 2 } e ^ { - i y } ; q \\right ) _ { n } } \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} T f ( z ) = \\int _ { \\Omega } K ( z , \\zeta ) f ( \\zeta ) d \\mu ( \\zeta ) \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} g \\mapsto \\Lambda _ T g = ( u _ g ( T ) , u ' _ g ( T ) ) \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} & - [ \\alpha ( x _ n ) , g ( x _ m ) ] + [ \\alpha ( x _ m ) , g ( x _ n ) ] + g ( [ x _ n , x _ m ] ) + \\bar { f } ( 1 , \\varphi ( x _ n , x _ m ) ) = 0 . \\\\ & - \\varphi ( \\alpha ( x _ n ) , g ( x _ m ) ) + \\varphi ( \\alpha ( x _ m ) , g ( x _ n ) ) + \\hat { v } ( ( 1 , [ x _ n , x _ m ] ) + \\bar { v } ( 1 , \\varphi ( x _ n , x _ m ) ) = 0 . \\\\ \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} \\phi _ { \\sigma _ { i } } ^ { F _ { 0 } } ( j ) = \\psi _ { \\sigma _ { i } } ^ { F _ { 0 } } ( p _ { e , r } ( j ) ) \\mbox { f o r a l l $ j \\in B _ { e , r } . $ } \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} { } [ \\chi , \\chi ] _ p ^ { F N } = - 4 \\ast ( T _ p + T _ p ^ \\top ) ( e _ i ) \\otimes e _ i + 6 e ^ i \\wedge \\tau _ p \\wedge \\varphi \\otimes e _ i , \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} \\gamma _ 2 \\gamma _ 2 ^ { t r } = \\beta _ 2 ^ { t r } \\beta _ 2 , \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} H = - \\frac { \\kappa } { 2 f } \\ , n _ 1 + \\frac { f f '' + ( f ' ) ^ 2 - 1 } { 2 f \\sqrt { f '^ 2 - 1 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} F P S ( l / k ) = y ^ t A y \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} 2 = C \\cdot ( - K _ X ) = C \\cdot F = \\pi _ 0 ^ * ( \\pi _ 0 ( C ) ) \\cdot \\pi _ 0 ^ * ( D ) = n \\pi _ 0 ( C ) \\cdot D , \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} \\hat \\mu = \\dfrac { z - \\alpha ^ { - 1 } \\varphi ( \\alpha { x } , \\alpha { y } ) } { \\sqrt { 1 + \\abs { \\nabla \\varphi ( \\alpha { x } , \\alpha { y } ) } ^ 2 } } . \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} P _ { 0 } ( z ) = e ^ { \\lambda / \\xi z } ( 1 - z ) ^ { - \\gamma / \\xi } p _ { 0 , 0 } \\left [ 1 - \\frac { A ( z ) } { A ( 1 ) } \\right ] . \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} \\vartheta _ { \\mathcal L _ \\Gamma } ( z ) = \\sum _ { k \\geq 0 } N _ { \\mathcal L _ \\Gamma } ( k ) z ^ k . \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} x ( t _ { 1 } , u _ { 1 } , v _ { 1 } ) x ( t _ { 2 } , u _ { 2 } , v _ { 2 } ) = x ( t _ { 1 } + t _ { 2 } , & u _ { 1 } + u _ { 2 } - t _ { 1 } t _ { 2 } ^ { 3 \\theta } , \\\\ & v _ { 1 } + v _ { 2 } - t _ { 2 } u _ { 1 } + t _ { 1 } t _ { 2 } ^ { 3 \\theta + 1 } - t _ { 1 } ^ { 2 } t _ { 2 } ^ { 3 \\theta } ) . \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} \\lambda _ { \\ell } ( k ) : = \\begin{cases} 4 - 4 \\cos ( k / 2 ) & \\ \\ k \\in ( 0 , \\pi ) \\\\ 4 + 4 \\cos ( k / 2 ) & \\ \\ k \\in ( \\pi , 2 \\pi ) \\end{cases} \\lambda _ { r } ( k ) : = \\begin{cases} 4 + 4 \\cos ( k / 2 ) & \\ \\ k \\in ( 0 , \\pi ) \\\\ 4 - 4 \\cos ( k / 2 ) & \\ \\ k \\in ( \\pi , 2 \\pi ) . \\end{cases} \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} \\sum _ { m , r , s , j , k \\atop 2 m + 2 j + r + s = p } a _ { k m r s j } Q ^ m \\phi _ k ^ { r , s , j } + \\sum _ { m , r , s , j \\atop 2 m + 2 j + r + s + 2 = p } b _ { m r s j } Q ^ m \\widetilde \\phi ^ { r , s , j } = 0 \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} G ( t ) : = \\big \\{ ( x , \\phi ( t , x , u ) , c ( t ) ( u - u ^ * ( t ) ) ) : ( x , u ) \\in S ( t ) \\big \\} \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} P _ \\Phi : = - \\delta _ { g _ \\Phi } ( \\Phi ) , \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} \\theta _ { C , R } { } ' ( 0 ) = r \\log ( 2 R ) + \\log 2 \\cdot \\dim V + \\frac { 1 } { 2 } \\log { \\det } ^ * \\Big ( \\frac { 2 - C ( 0 ) - { C ( 0 ) } ^ { - 1 } } { 4 } \\Big ) . \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} [ \\bar { f } _ { i , k + 1 } , \\bar { f } _ { j , l } ] = d ^ { - m _ { i , j } } [ \\bar { f } _ { i , k } , \\bar { f } _ { j , l + 1 } ] , \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} v : = \\big ( u ( x _ 0 , t _ 0 ) - u \\big ) _ + \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} R ( P ) = \\left \\{ \\begin{array} { c c } \\log \\left ( 1 + \\sqrt { \\frac { 2 P } { \\pi e } } \\right ) & P \\geq 6 . 3 0 3 \\mbox { d B } \\\\ \\beta ( P ) \\log \\sqrt { \\frac { 2 P } { \\pi e } } + H ( \\beta ( P ) ) & \\mbox { o t h e r w i s e } \\end{array} \\right . . \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} \\Phi = R ^ H \\left [ \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ] R , \\Psi = R ^ H \\left [ \\begin{array} { c c } \\alpha & \\beta \\\\ \\beta & \\gamma \\end{array} \\right ] R \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} s _ a ^ 1 ( x , \\xi ) & = \\sum _ { z \\in \\mathbb { Z } ^ d } f \\left [ z \\right ] e ^ { i ( \\varphi _ a ( x - z , \\xi ) - \\varphi _ a ( x , \\xi ) ) } - h _ 0 ( \\nabla _ x \\varphi _ a ( x , \\xi ) ) , \\\\ s _ a ^ 2 ( x , \\xi ) & = h ( x , \\nabla _ x \\varphi _ a ( x , \\xi ) ) - h _ 0 ( \\xi ) . \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } { } _ { 1 } \\phi _ { 1 } \\left ( \\begin{array} { c c c } \\begin{array} { c } - 1 / \\sqrt { b } \\\\ b \\end{array} & \\vert & q , - b q ^ { \\alpha } \\end{array} \\right ) A _ { q } \\left ( \\sqrt { b } q ^ { - \\alpha - 1 } \\right ) q ^ { \\alpha ^ { 2 } } d \\alpha = \\frac { \\sqrt { \\pi / \\log q ^ { - 1 } } } { \\left ( b ; q \\right ) _ { \\infty } } . \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} F = - \\frac { 1 } { \\beta } \\ln Z _ { S ^ 3 } = - T \\ln Z _ { S ^ 3 } \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} \\tau _ { i t } ( r ) = \\sum _ { a b = r } \\left ( \\frac { a } { b } \\right ) ^ { i t } . \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{gather*} \\mu \\mathcal { F } _ l ( \\triangleright \\otimes \\triangleright ) ( g \\otimes f ) = \\mu \\mathcal { F } _ r ( \\triangleright \\otimes \\triangleright ) ( g \\otimes f ) = \\xi ^ { - 1 } ( \\xi ( g ) \\cdot _ { U ( \\gg ) } \\xi ( f ) ) , g , f \\in S ( \\gg ) , \\end{gather*}"}
-{"id": "6398.png", "formula": "\\begin{align*} \\big ( \\mathbf { u } \\ast \\rho \\big ) ( \\mathbf { x } ) : = \\int _ { G } \\rho ( \\mathbf { x } - \\mathbf { y } ) \\mathbf { u } ( \\mathbf { y } ) \\mathrm { d } \\mathbf { y } \\mathbf { x } \\in G . \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\cos \\frac { 2 k \\pi } { i } = 1 - \\frac { 2 k ^ 2 \\pi ^ 2 } { { i } ^ 2 } . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } q ^ { - n } \\left ( \\varphi _ { 2 n + 1 } ( x ) + t \\varphi _ { 2 n } ( x ) \\right ) = 0 \\mbox { a n d } \\lim _ { n \\to \\infty } q ^ { - n } \\left ( q \\varphi _ { 2 n - 1 } ( x ) - t \\varphi _ { 2 n } ( x ) \\right ) = 0 , \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} F _ { d , \\ell } \\left ( z ; i t \\right ) & = \\sum _ { a = 0 } ^ N \\mathcal D _ z ^ { 2 a } \\left ( \\frac { e ^ { 2 \\pi i d z } } { 1 - \\zeta ^ \\ell } \\right ) \\frac { ( - 2 \\pi t ) ^ a } { a ! } + O \\left ( t ^ { N + 1 } \\right ) , \\\\ G _ { d , \\ell } \\left ( z ; i t \\right ) & = 2 i \\sum _ { a = 0 } ^ N \\mathcal D _ z ^ { 2 a } \\left ( \\frac { \\sin ( 2 \\pi d z ) } { 1 - \\zeta ^ \\ell } \\right ) \\frac { ( - 2 \\pi t ) ^ a } { a ! } + O \\left ( t ^ { N + 1 } \\right ) . \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{align*} \\alpha _ 1 & = \\frac { t } { ( - 1 ) ^ i ( t + 1 ) } & & \\alpha _ 2 = x _ i ^ { - 1 } & & \\beta _ 1 = \\frac { ( - 1 ) ^ { i - 1 } } { ( t + 1 ) x _ i ^ { \\lambda _ 1 + 1 } } & & \\beta _ 2 = \\frac { 1 } { ( t + 1 ) x _ i ^ { \\lambda _ 1 } } . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} \\frac { \\partial L } { \\partial \\left ( \\partial _ { \\nu } A _ { \\mu } \\right ) } = \\frac { \\partial L } { \\partial F _ { \\sigma \\tau } } \\frac { \\partial F _ { \\sigma \\tau } } { \\partial \\left ( \\partial _ { \\nu } A _ { \\mu } \\right ) } = R ^ { \\mu \\nu } \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} T ^ * : = \\min \\left \\{ ( 1 / 2 C ^ 2 ) ^ { 1 / ( 1 - \\gamma ) } , T _ v ^ * , T _ { \\zeta } ^ * \\right \\} , \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} - \\phi _ { x x } = \\rho = \\int \\left ( g _ { + } - g _ { - } \\right ) d v , \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} F = \\left ( \\begin{array} { c c c c c c c c c } \\lambda _ 1 & \\lambda _ 2 & 0 & \\lambda _ 3 & 0 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & \\lambda _ 2 & 0 & \\lambda _ 3 & 0 & \\lambda _ 4 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & \\lambda _ 3 & 0 & \\lambda _ 4 & \\lambda _ 5 \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{gather*} \\gamma _ { i + 1 } ( x , y ) y ^ { e _ { i + 1 , \\theta } } + \\pi ^ { m ( D _ { i + 1 } - D _ i ) } ( \\cdots ) \\\\ = \\gamma _ { i + 1 } ( x , y ) y ^ { e _ { i + 1 , \\theta } } + \\pi ^ { m ( D _ { i + 1 } - D _ i ) } ( \\geqslant D _ { i + 1 } ) . \\end{gather*}"}
-{"id": "2099.png", "formula": "\\begin{align*} \\rho _ i X _ { 1 1 , i } J _ { s _ i } ^ T + \\bar \\rho _ i J _ { s _ i } X _ { 1 1 , i } = J _ { s _ i } X _ { 1 1 , i } J _ { s _ i } ^ T . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} [ a _ { ( m ) } , b _ { ( n ) } ] = 0 \\quad \\forall a , b \\in \\Z , \\ m , n \\in \\Z \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} - \\frac { { \\rm d } \\zeta ^ 0 } { { \\rm d } x } = g ( x ) - \\rho ( x ) . \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} \\begin{aligned} \\| u ^ n \\| _ { C ^ { 0 , \\gamma } ( [ 0 , T ] ; H ^ { 1 + \\delta } _ p ) } = & \\sup _ { 0 \\le t \\le T } \\| u ^ n ( t ) \\| _ { H ^ { 1 + \\delta } _ p } \\\\ & + \\sup _ { 0 \\le s < t \\le T } \\frac { \\| u ^ n ( t ) - u ^ n ( s ) \\| _ { H _ p ^ { 1 + \\delta } } } { | t - s | ^ \\gamma } . \\end{aligned} \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} \\omega ( u , v ) = h ( u , \\epsilon v ) \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} \\left [ \\mathcal { T } ( \\tau ) g \\right ] ( x , v ) = g ( x - \\tau v , v ) \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} \\dot { x } = - \\varPhi ( F ) \\nu \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\mu _ i \\\\ \\nu _ i \\end{array} \\right ] = R ^ { - 1 } \\left [ \\begin{array} { c } s _ i \\\\ t _ i \\end{array} \\right ] , i = 1 , \\ldots , r . \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} & F _ { k \\ell } ( v , y ) - F _ { k \\ell } ( v _ { 0 } , 0 ) \\\\ & = - \\left ( \\tilde { a } ^ { k \\ell } ( y ) - \\tilde { a } ^ { k \\ell } ( 0 ) \\right ) J ( v _ { 0 } ) - \\tilde { a } ^ { k \\ell } ( y ) \\left ( J ( v ) - J ( v _ { 0 } ) \\right ) \\\\ & \\lesssim c _ \\ast ( y _ n ^ 2 + y _ { n + 1 } ^ 2 ) + \\delta _ 0 ( y _ n ^ 2 + y _ { n + 1 } ^ 2 ) . \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} L _ { v } \\left ( s - \\frac { n - 1 } { 2 } , \\Pi ^ { \\varphi } \\right ) = L _ { v } ( M , s ) _ { \\varphi } , \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} S _ { n } \\left ( - q ^ { - n + 1 / 2 } ; q \\right ) = \\frac { q ^ { - \\left ( n ^ { 2 } - n \\right ) / 4 } } { \\left ( q ^ { 1 / 2 } ; q ^ { 1 / 2 } \\right ) _ { n } } , \\ S _ { n } \\left ( - q ^ { - n - 1 / 2 } ; q \\right ) = \\frac { q ^ { - \\left ( n ^ { 2 } + n \\right ) / 4 } } { \\left ( q ^ { 1 / 2 } ; q ^ { 1 / 2 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} Q ( N , \\kappa _ h , \\gamma ) : = \\left \\{ n \\in \\{ 1 , \\ldots , N \\} ~ : ~ n \\equiv \\gamma \\pmod { 2 ^ { \\kappa _ h } } \\right \\} . \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} L ( q ) = E _ { 2 } ( q ) = 1 - 2 4 \\ , \\sum _ { n = 1 } ^ { \\infty } \\sigma ( n ) q ^ { n } , \\\\ M ( q ) = E _ { 4 } ( q ) = 1 + 2 4 0 \\ , \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { 3 } ( n ) q ^ { n } . \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial x ^ { \\nu } } Q ^ { \\mu \\nu } = - \\frac { \\partial } { \\partial x ^ { \\nu } } Q ^ { \\nu \\mu } = - \\frac { 4 \\pi } { c } j ^ { \\mu } \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} \\delta ^ * ( 0 , r ) \\geq \\max \\left ( \\frac { K } { M } + \\frac { K } { M r } , 1 + \\frac { K } { M r } \\right ) = \\frac { K } { \\min \\{ M , K \\} } + \\frac { K } { M r } , \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} d \\nu ^ { f } = Z ^ { - 1 } \\exp \\left [ - \\sum _ { \\left ( x , y \\right ) \\in D ^ { \\ast } } V \\left ( \\theta \\left ( y \\right ) - \\theta \\left ( x \\right ) \\right ) \\right ] \\prod _ { x \\in D \\backslash \\partial D } d \\theta \\left ( x \\right ) \\prod _ { x \\in \\partial D } \\delta \\left ( \\theta \\left ( x \\right ) - f \\left ( x \\right ) \\right ) . \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} b _ { K , j } = \\sigma _ j + O ( K ^ { 2 - s } ) . \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} \\left \\lbrace \\begin{tabular} { l l l l } $ u ( 0 , t ) = 0 $ , & $ u ( L , t ) = 0 $ , & $ u _ { x } ( L , t ) = h _ 2 ( t ) $ & i n $ ( 0 , T ) $ , \\\\ $ v ( 0 , t ) = g _ 0 ( t ) $ , & $ v ( L , t ) = g _ 1 ( t ) $ , & $ v _ { x } ( L , t ) = g _ 2 ( t ) $ & i n $ ( 0 , T ) $ . \\end{tabular} \\right . \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} ( U _ 1 U ^ { \\top } _ 1 ) ( X _ * V _ 0 + H V _ 0 ) = X _ * V _ 0 + H V _ 0 . \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } ( 1 , \\ , \\tau ) = ( 1 , \\ , \\frac { 1 } { \\tau } ) \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} U _ S = \\begin{bmatrix} 1 & & \\\\ & \\ddots & \\\\ & & 1 \\\\ 0 & \\cdots & 0 \\end{bmatrix} \\in \\mathbb { O } ^ { p _ 1 , r } \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{gather*} u _ t + u _ { x x x } + 6 u u _ x = 0 \\end{gather*}"}
-{"id": "7915.png", "formula": "\\begin{align*} \\inf \\left \\{ \\max _ { g \\in K _ 2 } | \\alpha ( g ) f - f | _ { \\Phi } \\mid f \\in { \\mathcal F } ( G _ 2 ) , f ( e ) = 1 \\right \\} > 0 . \\end{align*}"}
-{"id": "10169.png", "formula": "\\begin{align*} \\P ( Z _ n ^ * \\leq - a ) & = \\P ( \\forall k \\leq n \\ , : \\ , Z _ k + f ( k ) \\leq - a + f ( k ) ) \\\\ & \\geq \\P ( \\forall k \\leq n \\ , : \\ , Z _ k + f ( k ) \\leq a ) \\\\ & \\geq \\P ( Z _ n ^ * \\leq a ) \\ , \\exp ( - \\sqrt { 2 \\| f \\| _ { \\mathcal H } ^ 2 \\log ( 1 / \\P ( Z _ n ^ * \\leq a ) ) } - \\| f \\| _ { \\mathcal H } ^ 2 / 2 ) \\ , . \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\overline G \\big ( a ( x ) \\big ) = O \\big ( \\overline H ( x ) \\big ) , \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} F _ { Z _ { 2 , R } } ( \\omega , 0 ) \\big | _ { Z _ { 1 , R } } = 0 . \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} - \\frac { \\dot { M _ 1 } } { M _ 1 } & = \\frac { \\abs { k } } { k ^ 2 + | \\xi - k t | ^ 2 } \\\\ M _ 1 ( 0 , k , \\eta ) & = 1 . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} \\sum _ { m \\in \\mathbb { Z } ^ d } e ^ { 2 \\pi i x \\cdot m } = \\sum _ { m \\in \\mathbb { Z } ^ d } \\delta _ { x - m } \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} \\Theta ( z ; \\tau ) = ( - 2 i \\tau ) ^ { - \\frac { 1 } { 2 } } \\sum _ { n \\in 1 + 2 \\mathbb { Z } } e ^ { - \\frac { \\pi i } { 8 \\tau } ( n + 2 z ) ^ 2 } . \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } \\mu ' \\\\ \\nu ' \\end{array} \\right ] ^ H \\left [ \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ] \\left [ \\begin{array} { c } \\mu ' \\\\ \\nu ' \\end{array} \\right ] = 0 , \\left [ \\begin{array} { c } \\mu ' \\\\ \\nu ' \\end{array} \\right ] ^ H \\left [ \\begin{array} { c c } 0 & - \\delta \\\\ - \\delta & 2 \\epsilon \\end{array} \\right ] \\left [ \\begin{array} { c } \\mu ' \\\\ \\nu ' \\end{array} \\right ] \\leq 0 , \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} Q _ { 1 2 5 } \\cap Q _ { 2 3 5 } \\cap Q _ { 3 4 5 } = L _ 5 \\cup \\Gamma _ 2 \\cup \\Gamma _ 3 \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} u '' + \\tilde { f } ( t , u ) = 0 . \\end{align*}"}
-{"id": "10039.png", "formula": "\\begin{align*} F ' ( z ) = f ' ( z ) + z f '' ( z ) ( \\lambda - \\mu + 2 \\lambda \\mu ) + \\lambda \\mu z ^ { 2 } f ''' ( z ) \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( m z w : n ) } = & \\left ( \\oplus _ { i = 1 } ^ { t - 1 } \\overrightarrow { C } _ { ( m : n ) } \\otimes Z _ i \\right ) \\oplus \\left ( \\overrightarrow { C } _ { ( m : n ) } \\otimes Z _ { t } \\right ) \\oplus \\\\ & \\left ( \\overrightarrow { C } _ { ( m : n ) } \\otimes Z _ { t + 1 } \\right ) \\oplus \\left ( \\oplus _ { i = t + 2 } ^ { z w } \\overrightarrow { C } _ { ( m : n ) } \\otimes Z _ i \\right ) \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} b _ { i , j } & = 0 , \\ ; \\ ; \\ ; \\ ; \\ ; i < j , \\\\ b _ { i , i } & = \\beta _ { i , i } , \\\\ b _ { i , j } & = b _ { i - 1 , j } ^ { \\ast } - b _ { i - 1 , j } \\beta _ { i , i } , \\ ; \\ ; \\ ; \\ ; \\ ; i > j , \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} \\Pi _ n ( g \\chi ) = g \\chi - u \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} r : = 2 \\gamma _ 1 / \\gamma _ 0 = - ( a + d ) , \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} s ( x ) & : = \\begin{cases} 2 & x = 1 , \\\\ 3 & \\end{cases} t ( x ) : = \\begin{cases} 1 & x = 2 , \\\\ 3 & \\end{cases} \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} & \\| u \\| _ { L ^ \\infty ( ( 0 , T ) , L ^ 2 ) } ^ 2 + \\| u \\| _ { L ^ 2 ( ( 0 , T ) , H ^ 2 ) } ^ 2 + \\| u \\| _ { L ^ 4 ( ( 0 , T ) , L ^ 4 ) } ^ 4 \\\\ & \\quad \\le C ( 1 + T e ^ { \\omega T } ) \\left ( \\| u _ 0 \\| _ 2 ^ 2 + \\| f \\| _ { L ^ 2 ( ( 0 , T ) , L ^ 2 ) } ^ 2 \\right ) \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} \\langle \\alpha , \\beta \\rangle _ { g _ H ^ * } = \\langle \\sharp ^ H \\alpha , \\sharp ^ H \\beta \\rangle _ { g _ H } , \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} ( \\log F ( z ) ) _ { z z } = & \\sum _ { k = 3 } ^ { p } ( k - 1 ) ( k - 2 ) | z | ^ { 2 ( k - 3 ) } \\overline { z } ^ { 2 } \\log G _ { k } ( z ) + 2 \\sum _ { k = 2 } ^ { p } ( k - 1 ) | z | ^ { 2 ( k - 2 ) } \\overline { z } ( \\log G _ { k } ( z ) ) _ { z } \\\\ & + \\sum _ { k = 1 } ^ { p } | z | ^ { 2 ( k - 1 ) } ( \\log G _ { k } ( z ) ) _ { z z } \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} \\theta ^ { ( n ) } _ d \\left ( q \\to 1 \\ \\mathrm { l i m i t \\ o f } \\ \\frac { \\prod _ { j = 1 } ^ k x _ { i - 1 , j } } { \\prod _ { j = 1 } ^ k x _ { i , j } } F _ { 0 , k } \\right ) = \\mu _ k A _ i ( d ^ k ) \\otimes Z ^ k \\ \\mathrm { f o r \\ a n y } \\ 1 \\leq i \\leq n . \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} | | P _ t \\mathbf 1 _ E - \\mathbf 1 _ E | | _ { 1 } = & \\int _ E ( 1 - P _ t 1 _ E ) d \\mu + \\int _ { E ^ c } P _ t ( 1 _ E ) d \\mu \\\\ = & \\int _ E ( 1 - P _ t 1 _ E ) d \\mu + \\int _ E ( P _ t 1 _ { E ^ c } ) d \\mu \\\\ = & 2 \\left ( \\mu ( E ) - \\int _ E P _ t ( 1 _ E ) d \\mu \\right ) \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\lambda _ { n } = \\begin{cases} 0 & n = 5 N \\mod 6 \\\\ 7 & n = 5 N + 1 \\mod 6 \\\\ 6 & n = 5 N + 2 \\mod 6 \\\\ 3 & n = 5 N + 3 \\mod 6 \\\\ 4 & n = 5 N + 4 \\mod 6 \\\\ 1 1 & n = 5 N + 5 \\mod 6 \\end{cases} \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} \\partial _ { y _ i } v = \\partial _ { x _ i } w , \\ i = 1 , \\ldots , n - 1 , \\partial _ { y _ { n } } v = - x _ n , \\partial _ { y _ { n + 1 } } v = - x _ { n + 1 } . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} \\begin{aligned} \\left | | \\alpha ( t ) f - f | \\right | _ { \\infty } ^ - & = | \\alpha ( t ) f - f | _ { \\infty } ^ - \\\\ & \\le M \\max _ { k \\in K } | \\alpha ( k ) f - f | _ { \\infty } ^ - \\end{aligned} \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{gather*} B = \\mathbb { C } [ c _ { k } ] _ { k \\in \\mathbb { Z } } \\end{gather*}"}
-{"id": "1370.png", "formula": "\\begin{align*} B _ { E , 1 4 } & = \\{ \\ , M ( q ^ { t } ) \\ , \\mid \\ , t 1 4 \\ , \\} , \\\\ B _ { E , 2 2 } & = \\{ \\ , M ( q ^ { t } ) \\ , \\mid \\ , t 2 2 \\ , \\} , \\\\ B _ { E , 2 6 } & = \\{ \\ , M ( q ^ { t } ) \\ , \\mid \\ , t 2 6 \\ , \\} \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} ( 1 - 2 x t ) \\frac { d } { d t } M ( x ; t ) & = 2 x ( 1 - 2 x ) \\frac { d } { d x } M ( x ; t ) \\\\ & + \\left ( a ( x ) + x b ( x ) - 2 x ( 1 - 2 x ) \\frac { d } { d x } a ( x ) \\right ) R ^ + ( x ; t ) \\\\ & + \\left ( 2 x b ( x ) + a ( x ) - 2 x ( 1 - 2 x ) \\frac { d } { d x } b ( x ) \\right ) R ^ - ( x ; t ) . \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} P _ { i , 1 } ( m ) \\alpha _ { i } ^ m = Q ( n ) \\beta _ { j ^ * } ^ n . \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} \\pi _ { 2 } = [ a ^ { 2 } \\partial _ { a ^ { * } } , \\partial _ { a } ] + [ a ^ { * } a \\partial _ { a ^ { * } } , \\partial _ { a ^ { * } } ] + [ a ^ { * } \\partial _ { a ^ { * } } , a \\partial _ { a ^ { * } } ] \\ , . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} P _ { m + d + 1 } ^ { c } \\left ( x \\right ) = \\left ( x - \\beta _ { m + d } \\right ) P _ { m + d } ^ { c } \\left ( x \\right ) - \\sum \\nolimits _ { \\nu = 0 } ^ { d - 1 } \\gamma _ { m + d - \\nu } ^ { d - 1 - \\nu } P _ { m + d - 1 - \\nu } ^ { c } \\left ( x \\right ) , \\ \\ m \\geq k . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} g ^ { ( j ) } _ { \\pi } ( x , y ) = \\phi _ j ( x ) \\cdot y \\oplus d ( x ) , \\ ; \\ ; \\ ; x \\in \\Z _ 2 ^ k , \\ ; \\ ; \\ ; y \\in \\Z _ 2 ^ { k + 1 } , \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} H = - \\frac { \\kappa } { 2 f } \\ , n _ 1 - \\frac { f f '' + ( f ' ) ^ 2 + 1 } { 2 f \\sqrt { f '^ 2 + 1 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} D _ H f ( x ) X _ i = X _ i f ( x ) i = 1 , \\dots , 2 n . \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} \\dim \\delta H _ 3 ^ g = & \\ ; 6 + 9 k + ( k - 1 ) k + { 3 \\ , k ^ 2 } + ( k - 1 ) k ^ 2 / 2 \\\\ & \\ ; - { ( k + 2 ) ^ 2 ( k + 3 ) / 2 } \\\\ = & \\ ; 0 , \\\\ \\dim \\delta E _ 3 ^ g = & \\ ; I _ M ( V _ 3 ^ g \\times W _ 3 ^ g ) = \\ ; 0 . \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} E _ 1 = \\frac { 1 } { \\sqrt n } I , E _ 2 = 1 , F _ 1 = \\left [ \\begin{array} { c c } 0 & I _ { n _ 1 - 1 } \\end{array} \\right ] , G _ 1 = \\left [ \\begin{array} { c c } I _ { n _ 1 - 1 } & 0 \\end{array} \\right ] , \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - d \\hat { \\xi } ( t , x ) & = [ \\partial _ j ( a ^ { i j } \\partial _ i \\hat { \\xi } ( t , x ) + \\sigma ^ { j r } \\hat { v } ^ r ( t , x ) ) + \\hat { f } ( t , x ) + \\nabla \\cdot \\hat { g } ( t , x ) ] d t - \\hat { v } ^ r ( t , x ) d W ^ r _ t , \\\\ & ~ ~ ~ ( t , x ) \\in Q , \\\\ \\hat { \\xi } ( T , x ) & = \\hat { G } ( x ) , ~ ~ ~ x \\in \\mathcal { O } , \\end{array} \\right . \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} T = V \\oplus H , \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} q \\Psi _ { \\gamma + t } = ( \\Delta + \\mid \\gamma + t \\mid ^ { 2 } I ) ( \\Psi _ { \\gamma + t } - e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) = \\Delta \\Psi _ { \\gamma + t } + \\mid \\gamma + t \\mid ^ { 2 } \\Psi _ { \\gamma + t } , \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} a _ n ( f , D ) = \\zeta ( 0 , f , D ) \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} P \\left ( x _ { r - 1 } , x _ { r } \\right ) + \\sum _ { j = 0 } ^ { r - 2 } \\left ( c _ { r - 1 , j } x _ { r - 1 } + c _ { r , j } x _ { r } \\right ) g _ { j } . \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} W _ n ^ { \\mu _ 1 , \\mu _ 2 } = ( 1 + | n | ) ^ { \\mu _ 1 } e ^ { \\mu _ 2 | n | } \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} f _ 0 ( y '' ) = 0 \\mbox { a n d } f _ 1 ( y ) = 0 \\mbox { f o r } y = ( y '' , y _ n , y _ { n + 1 } ) \\mbox { s u c h t h a t } ( y '' , 0 , 0 ) \\in P \\setminus \\mathcal { B } _ 3 ^ + . \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} \\lim _ { M \\rightarrow \\infty } | S _ { m , N } | ^ { \\frac { 1 } { q } - \\frac { 1 } { p } } \\gamma \\left | \\bigcap _ { i = M } ^ { \\infty } \\{ k \\in S _ { m , N } : | x ^ { ( i ) } _ { k } - x ^ { ( j ) } _ { k } | > \\gamma \\} \\right | ^ { \\frac { 1 } { p } } < \\epsilon . \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} f ^ { \\lambda } = \\lambda \\left . \\frac { \\partial F } { \\partial \\lambda } F ^ { - 1 } \\right | _ { \\l \\in \\R _ { + } } . \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\sqrt { \\Gamma ( f ) } ( x ) = \\lim \\sup _ { d ( x , y ) \\to 0 } \\frac { | f ( x ) - f ( y ) | } { d ( x , y ) } . \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\| \\textbf { \\textit { v } } _ { m } \\| _ { \\ell ^ { 2 } ( \\mathbb { Z } ) } = \\frac { | \\alpha | ^ { - m } q ^ { m ( m + 1 ) / 2 } } { \\sqrt { 1 - \\alpha ^ { - 2 } q ^ { 2 m } } } \\ , ( q ; q ) _ { \\infty } , m > \\lfloor \\log _ { q } | \\alpha | \\rfloor . \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} v ( \\zeta ^ { - 1 } \\boldsymbol { x } _ n - 1 ) = v ( ( \\zeta ^ { - 1 } - 1 ) \\boldsymbol { x } _ n + \\boldsymbol { x } _ n - 1 ) = 0 . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} y = T x \\ \\Leftrightarrow \\ 3 x + 1 = y 2 ^ { \\nu _ 2 ( 3 x + 1 ) } . \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} \\tilde P _ \\pm ^ * \\tilde P _ \\pm u [ x ] & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi ) - \\varphi _ a ( y , \\xi ) ) } p _ \\pm ( x , \\xi ) p _ \\pm ( y , \\xi ) u \\left [ y \\right ] d \\xi \\\\ & = ( 2 \\pi ) ^ { - d } \\int _ { \\mathbb { T } ^ d } \\sum _ { y \\in \\mathbb { Z } ^ d } e ^ { i ( x - y ) \\cdot \\eta ( \\xi ; x , y ) } p _ \\pm ( x , \\xi ) p _ \\pm ( y , \\xi ) u \\left [ y \\right ] d \\xi , \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} \\kappa _ E ^ { - 1 } \\left ( \\mathcal { Z } _ E ( \\mu \\setminus F ) \\right ) & = \\kappa _ E ^ { - 1 } ( \\mathcal { Z } _ E ( \\mu ) ) \\setminus \\left ( \\bigcup _ { e \\in F } \\kappa _ E ^ { - 1 } ( \\mathcal { Z } _ E ( \\mu e ) ) \\right ) \\\\ & = \\mathcal { Z } _ { E _ \\curlyvee } ( \\mu ) \\setminus \\left ( \\bigcup _ { e \\in F } \\mathcal { Z } _ { E _ \\curlyvee } ( \\mu e ) \\right ) = \\mathcal { Z } _ { E _ \\curlyvee } ( \\mu \\setminus F ) \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } x _ { i } & = & \\dfrac { P _ { i } } { Q } + \\varepsilon \\phi i = 1 , 2 , . . . , n + 1 \\\\ & \\varepsilon Q \\cong 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} I _ N ( Z _ N ) = \\sum _ { i = 1 } ^ N | x _ i | ^ 2 \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\tau _ { 2 , f } ( \\phi ) \\equiv f \\tau _ 2 ( \\phi ) + ( \\Delta f ) \\tau ( \\phi ) + 2 \\nabla ^ { \\phi } _ { { \\rm g r a d } \\ , f } \\tau ( \\phi ) = 0 , \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} v ^ { - 1 } h _ { i j } & = - v ^ { - 2 } u _ { ; i j } - \\dot \\vartheta \\vartheta \\sigma _ { i j } \\\\ & = - v ^ { - 2 } \\{ u _ { : i j } - \\tfrac { 1 } { 2 } \\bar { g } ^ { k m } \\left ( ( \\vartheta ^ 2 ) _ j \\sigma _ { m i } + ( \\vartheta ^ 2 ) _ i \\sigma _ { m j } - ( \\vartheta ^ 2 ) _ m \\sigma _ { i j } \\right ) u _ k \\} - \\dot \\vartheta \\vartheta \\sigma _ { i j } . \\\\ \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} D \\big ( \\mathcal { A } ( \\mathbf { H } ) \\big ) : = \\big \\{ \\mathbf { u } \\in \\mathcal { V } \\ , | \\ , \\mathcal { A } ( \\mathbf { H } ) \\mathbf { u } \\in \\mathcal { H } \\big \\} \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} F ( t \\beta ) = \\sum _ { i = 1 } ^ { n } f ( t \\beta _ { i } ) e _ { i } = g ( t ) \\sum _ { i = 1 } ^ { n } f ( \\beta _ { i } ) e _ { i } \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} \\begin{aligned} & P _ Y = - \\frac { 1 } { 2 } \\left ( P \\Omega _ { [ 1 ] } \\right ) _ X , \\\\ & ( \\Omega _ { [ i ] } ) _ { X X X } - ( \\Omega _ { [ i ] } ) _ X = - P \\left ( P \\Omega _ { [ i + 1 ] } \\right ) _ X , \\\\ & P _ T = \\frac { ( \\Omega _ { [ n ] } ) _ { X X X } - ( \\Omega _ { [ n ] } ) _ X } { 2 P } = \\Delta _ X . \\end{aligned} i = 1 , \\dots , n - 1 , \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} F R - G S = \\frac { 1 } { 2 } \\left ( F R \\right ) I = \\left ( \\mathbf { E } \\cdot \\mathbf { D } - \\mathbf { H } \\cdot \\mathbf { B } \\right ) I , \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} \\| \\partial _ x ^ 2 v _ 0 ^ + ( x , \\cdot ) \\| ^ 2 _ { H ^ { - \\frac { 1 } { 3 } } ( 0 , T ) } & = C \\int _ { 0 } ^ { \\infty } \\rho ^ 4 \\left | \\int _ 0 ^ { \\infty } e ^ { - i a \\rho ^ 3 L ^ 3 t } h _ 0 ( t ) d t \\right | ^ 2 d \\rho & \\leq C \\int _ { 0 } ^ { \\infty } \\mu ^ { \\frac { 2 } { 3 } } \\left | \\int _ 0 ^ { \\infty } e ^ { - i \\mu t } h _ 0 ( t ) d t \\right | ^ 2 d \\mu \\\\ & \\leq C \\| h _ 0 \\| _ { H ^ { \\frac { 1 } { 3 } } ( \\R ^ + ) } ^ 2 . \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} F ( z , v ) : = \\sum _ { n \\ge 1 } \\sum _ { m \\ge 0 } T _ { n } \\mathbb { P } \\{ D _ { n } = m \\} \\frac { z ^ { n } } { n ! } v ^ { m } , A ( z , v ) : = \\sum _ { n \\ge 0 } \\sum _ { m \\ge 0 } \\tilde { T } _ { n } \\mathbb { P } \\{ \\tilde { D } _ { n } = m \\} \\frac { z ^ { n } } { n ! } v ^ { m } , \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} \\left ( a z q ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } S _ { n } \\left ( a q ^ { - n } ; q \\right ) } { \\left ( a z q ; q \\right ) _ { n } } \\left ( - z \\right ) ^ { n } = \\left ( z q / a ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } S _ { n } \\left ( a q ^ { - n } ; q \\right ) } { \\left ( z q / a ; q \\right ) _ { n } } \\left ( \\frac { z } { a } \\right ) ^ { n } . \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\{ p \\} = \\bigcap _ { n \\geq 0 } T _ { \\xi _ { 0 } } \\circ \\cdots \\circ T _ { \\xi _ { n } } ( X ) \\supset \\bigcap _ { n \\geq 0 } T _ { \\xi _ { 0 } } \\circ \\cdots \\circ T _ { \\xi _ { n } } ( K ) . \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} T _ { ( A , ( j + i ) + n \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } = - T _ { ( A , j + i \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } . \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} X _ s X _ { s ' } + X _ { s ' } X _ s & = 2 Q _ { s , s ' } I s , s ' \\in S , \\\\ Y _ t Y _ { t ' } + Y _ { t ' } Y _ t & = 2 R _ { t , t ' } I t , t ' \\in T . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} A \\ , \\sum _ { i = 1 } ^ { N } \\ , \\int _ { \\mathbb { R } ^ d } \\ , | \\hat { F _ i } ( \\lambda ) | ^ 2 \\ , d \\lambda \\leq \\int _ { \\mathbb { R } ^ d } \\ , \\epsilon ^ d \\ , \\left | \\sum _ { i = 1 } ^ { N } \\ , \\hat { F _ i } ( \\epsilon \\lambda ) \\ , e ^ { - 2 \\pi i x _ i \\cdot \\lambda } \\right | ^ 2 \\ , d \\mu ( \\lambda ) \\leq B \\ , \\sum _ { i = 1 } ^ { N } \\int _ { \\mathbb { R } ^ d } \\ , | \\hat { F _ i } ( \\lambda ) | ^ 2 \\ , d \\lambda . \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{gather*} t _ { n _ { k + 1 } } = ( - 1 ) ^ { { \\tfrac { ( n _ { k + 1 } - n _ { k } - 1 ) ( n _ { k + 1 } - n _ { k } ) } { 2 } } } \\left ( s _ { n _ { k + 1 } + n _ { k } + 1 } - s _ { n _ { k + 1 } + n _ { k } + 1 } ^ { ( n _ { k } + 1 ) } \\right ) ^ { n _ { k + 1 } - n _ { k } } t _ { n _ { k } } \\ . \\end{gather*}"}
-{"id": "8126.png", "formula": "\\begin{align*} b ( \\lambda ) = \\frac { 1 } { K - 1 } \\sum _ { k = 1 } ^ { K } \\frac { n - n _ { k } } { n } \\widehat { \\beta } _ { - k } ( \\lambda ) . \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} d \\bar { s } _ { N } ^ 2 = - d \\tau ^ 2 + \\cosh ^ 2 \\tau \\sigma _ { i j } d \\xi ^ i d \\xi ^ j . \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} \\tilde { Z } _ { s , s } \\left [ Z _ s , t \\right ] = \\tilde { \\psi } _ s ^ { - t } Z _ s \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} & \\limsup _ { R \\to \\infty } \\ , \\sup _ { t \\in \\mathbb { R } } \\ , \\frac { 1 } { R } \\ , \\int _ { [ t , t + R ] ^ s } \\ , \\big | \\sum _ { i = 1 } ^ m \\ , c _ i \\ , e ^ { - 2 \\pi i ( \\sum _ { j = 1 } ^ s \\ , n _ { i j } \\ , a _ j \\ , \\xi _ j ) } \\big | ^ 2 \\ , d \\nu _ { \\mu } ( \\xi _ 1 , \\dots , \\xi _ s ) \\\\ & = \\limsup _ { R \\to \\infty } \\ , \\sup _ { t \\in \\mathbb { R } } \\ , \\frac { 1 } { R } \\int _ { [ t , t + R ] } \\ , \\big | \\sum _ { i = 1 } ^ m \\ , c _ i \\ , e ^ { - 2 \\pi i x _ i \\ , \\lambda } \\big | ^ 2 \\ , d \\mu ( \\lambda ) : = L . \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} h ( w ) : = \\exp \\left ( \\frac { 2 \\pi } { \\log | t | } Z ^ { i j } y _ i y _ j \\right ) = \\exp k \\left ( \\frac { 2 \\pi } { \\log | s | } Z ^ { i j } y _ i y _ j \\right ) \\end{align*}"}
-{"id": "9989.png", "formula": "\\begin{align*} \\norm { \\widehat { \\varphi _ k } } _ \\infty \\le \\norm { \\varphi _ k } _ 1 = \\int _ { \\R ^ n } \\frac { 1 } { \\widehat { F _ p } ( x / k ) } \\abs { \\widehat { f } ( x ) } \\ d x \\le \\int _ { \\R ^ n } \\frac { 1 } { \\widehat { F _ p } ( x ) } \\abs { \\widehat { f } ( x ) } \\ d x < \\infty \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} A = A _ 0 + \\tilde { A } , B = B _ 0 + \\tilde { B } , \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} \\beta ( x , y ) = p ( \\sum _ { g \\in G } \\theta _ g ( x , y ) e _ g ) = p ( \\sum _ { g \\in A } \\theta _ g ( x , y ) e _ g ) . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } ^ 0 \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & = \\left ( X _ { s + k } ^ \\prime - V _ { s + k } ^ \\prime \\tau , V _ { s + k } ^ \\prime \\right ) \\end{aligned} \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} p ( a , b | s , t ) = \\psi ^ * ( X _ s ^ a \\otimes Y _ t ^ b ) \\psi a \\in A , b \\in B , s \\in S , t \\in T . \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} \\beta ( y ) = \\alpha ( y + \\beta ( y ) ) , \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} s _ { e _ \\nu } & = s _ { e _ \\nu } p _ { s _ E ( e _ 1 ) } = s _ { e _ { \\nu } } s _ { e _ 1 } s _ { e _ 1 } ^ * \\\\ & = s _ { e _ { \\nu } } s _ { e _ 1 } p _ { s _ E ( e _ 2 ) } s _ { e _ 1 } ^ * = s _ { e _ { \\nu } } s _ { e _ 1 } s _ { e _ 2 } s _ { e _ 2 } ^ * s _ { e _ 1 } ^ * \\\\ & \\ \\vdots \\\\ & = s _ { e _ { \\nu } } s _ { \\mu } s _ { \\mu } ^ * \\\\ & = s _ \\nu s _ \\mu ^ * \\\\ & = \\Psi ( t _ { \\overline { \\nu } } t _ \\mu ) \\in \\psi ( C ^ * ( F ) ) . \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} x _ 0 ( \\xi s ) = e ^ { i \\xi A } h + i \\int _ { 0 } ^ { s } e ^ { i ( \\xi - w \\xi ) A } \\Phi ^ * \\sigma ( \\xi ) ( \\Lambda ( \\xi ) ( \\tilde { u _ 0 } ) ( w ) d w . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} \\mathbf { Q } _ 3 = \\left ( \\begin{array} { c c } a _ 1 & \\underline { a } _ 1 \\\\ a _ 2 & \\underline { a } _ 2 \\\\ b _ 1 & \\underline { b } _ 1 \\\\ b _ 2 & \\underline { b } _ 2 \\end{array} \\right ) , \\mathbf { Q } _ 4 = \\left ( \\begin{array} { c c } a _ 1 & 0 \\\\ a _ 2 & 0 \\\\ 0 & b _ 1 \\\\ 0 & b _ 2 \\end{array} \\right ) \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} \\gamma ( \\Omega _ 1 , a , q ) ~ = ~ \\beta \\gamma ( \\Omega _ 2 , a , q ) \\phantom { m } \\phantom { m } \\gamma ( \\Omega _ 1 , a , q ) ~ = ~ \\lambda \\gamma ( \\Omega _ 3 , a , q ) . \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} q ( t , s ) & = x + \\int _ s ^ t v ( p ( \\tau , s ) ) d \\tau , \\\\ p ( t , s ) & = \\xi - \\int _ s ^ t \\nabla _ x V _ \\rho ( \\tau , q ( \\tau , s ) ) d \\tau . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} A ^ - ( \\xi ) G ( \\xi ) = G ( \\xi ^ p ) A ^ - ( 0 ) . \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} & \\| \\nabla _ p s _ { g _ 0 } \\| ^ 2 _ { p - S l } = \\frac { 1 } { 3 p - 1 } ( 2 s _ { g _ 0 } ( p ) ^ 2 + 2 ( 1 - p ) s _ { g _ 0 } ( p ) \\delta _ 0 + \\delta _ 0 ^ 2 ( 1 - p ) p ) . \\end{align*}"}
-{"id": "9879.png", "formula": "\\begin{align*} ( k + \\sigma ) ( k - 1 + \\sigma ) \\ , a _ { k , \\sigma } = \\left [ ( k - 1 + 2 m + \\sigma ) ( k - 2 + \\sigma ) - z \\right ] \\ , a _ { k - 2 , \\sigma } \\ , . \\\\ \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\tilde { h } _ i ^ j = \\hat { h } _ i ^ j . \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} W _ s = \\left ( \\begin{array} { c | c } I _ s & \\\\ \\hline & E _ { n - s + 1 } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} y _ \\mu \\left ( \\sum _ { i = 1 } ^ N \\Phi _ { \\mu i } x _ i + \\lambda _ \\mu \\right ) > 0 \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} \\delta ^ d \\ , \\sum _ { k \\in \\mathbb { Z } ^ d } \\ , \\sup _ { \\gamma \\in Q _ \\delta } \\ , | \\psi ( \\xi - \\delta k - \\gamma ) | & \\le \\sum _ { k \\in \\mathbb { Z } ^ d } \\ , C _ 1 \\ , \\prod _ { i = 1 } ^ d \\ , \\delta \\ , \\sup _ { | \\gamma _ i | \\le \\delta / 2 } \\ , g ( \\xi _ i - \\delta k _ i - \\gamma _ i ) \\\\ & = C _ 1 \\ , \\prod _ { i = 1 } ^ d \\ , \\delta \\ , \\sum _ { k _ i \\in \\mathbb { Z } } \\ , \\sup _ { | \\gamma _ i | \\le \\delta / 2 } \\ , g ( \\xi _ i - \\delta k _ i - \\gamma _ i ) \\le C _ 1 \\ , c ^ d = C < \\infty . \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} V ( \\varepsilon ) \\leq \\sum ^ { l ( \\varepsilon ) } _ { l = 1 } V _ { l } , \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} x ' _ k = ( 1 + \\epsilon _ k ) x ' , y ' _ k = ( 1 + \\epsilon _ k ) y ' + A _ k \\epsilon _ k ^ 2 x ' , A _ k = - \\mathrm { s g n } ( t _ k + 2 ( \\langle x ' , y _ k \\rangle - \\langle y ' , x _ k \\rangle ) ) . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} \\left ( \\mathbf { A } \\times \\operatorname { c u r l } \\mathbf { B } \\right ) _ { p } = A _ { q } \\left ( \\frac { \\partial B _ { q } } { \\partial x _ { p } } - \\frac { \\partial B _ { p } } { \\partial x _ { q } } \\right ) . \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} \\widehat { C } _ { \\tau } f ( t ) = e ^ { i w _ 0 t } f ( t ) . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } u _ t ( t , x ) + L ^ b u ( t , x ) + f ( t , x , u ( t , x ) , \\nabla u ( t , x ) ) = 0 , \\\\ u ( T , x ) = \\Phi ( x ) , \\\\ \\forall ( t , x ) \\in [ 0 , T ] \\times \\mathbb R ^ d . \\end{array} \\right . \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} \\eta _ j = \\mathrm { d } f _ j + \\nu _ j . \\end{align*}"}
-{"id": "9937.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\vect { \\delta } = ( \\delta _ 1 , \\dots , \\delta _ k ) } V ( \\vect { \\delta } ) , \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} W \\xi ( s , t ) = \\xi ( s , s ^ { - 1 } t ) ( \\xi \\in L ^ 2 ( G \\times G ) = L ^ 2 ( G ) \\otimes L ^ 2 ( G ) ) , \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} X _ m [ t ] = \\pi ^ { m } _ { \\mathsf { P } , e } \\Big ( S _ m , \\left \\{ U _ m [ 1 ] , U _ m [ 2 ] , \\ldots , U _ m [ t - 1 ] \\right \\} , \\mathbf { D } , \\mathbf { H } \\Big ) , ~ ~ t \\in [ 1 : T ] \\end{align*}"}
-{"id": "10041.png", "formula": "\\begin{align*} z f ' ( z ) + z ^ 2 f '' ( z ) ( \\lambda - \\mu + 2 \\lambda \\mu ) + \\lambda \\mu z ^ 3 f ''' ( z ) = p ( z ) G _ k ( z ) . \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} \\widehat { I } _ { p } = { \\det } ^ \\ast \\big ( \\frac { 2 - S ^ p _ 1 \\circ S ^ p _ 2 - S ^ p _ 2 \\circ S ^ p _ 1 } { 4 } \\big ) ^ { \\frac { 1 } { 4 } } , I _ p = { \\det } ^ \\ast \\big ( \\frac { 2 - C ^ p _ { 1 2 } - ( C ^ p _ { 1 2 } ) ^ { - 1 } } { 4 } \\big ) ^ { \\frac { 1 } { 4 } } . \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} \\left \\langle R ^ { \\bot } ( X _ { i } , X _ { j } ) N _ { \\alpha } , N _ { \\beta } \\right \\rangle = \\left \\langle [ A _ { N _ { \\alpha } } , A _ { N _ { \\beta } } ] X _ { i , } X _ { j } \\right \\rangle , \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\frac { d s _ { j } } { d \\rho _ { j } } = \\frac { \\beta _ { j } } { ( \\rho _ { j } ^ 2 + \\epsilon ^ 2 ) ^ { \\frac { 1 - \\beta _ { j } } { 2 } } } , \\ s _ { j } ( 0 ) = 0 , \\ \\rho _ { j } = | z _ { j } | . \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} \\alpha & \\leq \\Pr ( X _ { 1 } + \\dots + X _ { m } \\geq Q _ { 1 - \\alpha } ( X _ { 1 } + \\dots + X _ { m } ) ) \\\\ & \\leq \\Pr ( X _ { 1 } + \\dots + X _ { m } > Q _ { 1 - \\alpha / ( 2 m ) } ( X _ { 1 } ) + \\dots + Q _ { 1 - \\alpha / ( 2 m ) } ( X _ { m } ) ) \\\\ & \\leq \\Pr ( X _ { 1 } > Q _ { 1 - \\alpha / ( 2 m ) } ( X _ { 1 } ) ) + \\dots + \\Pr ( X _ { m } > Q _ { 1 - \\alpha / ( 2 m ) } ( X _ { m } ) ) \\\\ & \\leq \\alpha / ( 2 m ) + \\dots + \\alpha / ( 2 m ) = \\alpha / 2 , \\end{align*}"}
-{"id": "10118.png", "formula": "\\begin{gather*} \\pi ^ * _ { P _ 3 } C \\ : : \\ : t ^ q ( b t + c ) ^ r - z ^ p = 0 , \\\\ \\pi ^ * _ { P _ 3 } \\omega = - p t ( b t + c ) d z + z ( b t ( q + r ) + q c ) d t . \\end{gather*}"}
-{"id": "5346.png", "formula": "\\begin{align*} S ^ l _ { \\alpha p } : = \\langle S ( X _ \\alpha , Z _ p ) , n _ l \\rangle , T ^ l _ { \\alpha p } : = \\langle S ( Y _ \\alpha , Z _ p ) , n _ l \\rangle \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} m ( \\omega \\otimes \\iota ) \\Delta ( f ) = m ( f ) \\int _ { G } \\omega ( x ) d x = \\omega ( 1 ) m ( f ) ( \\omega \\in L ^ 1 ( G ) , \\ f \\in L ^ \\infty ( G ) ) . \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} \\lim _ { s \\to \\infty } \\limsup _ { t \\to \\infty } \\| e ^ { i t H _ 0 } J _ a ^ * e ^ { - i t H } v - e ^ { i s H _ 0 } \\tilde P _ + v _ s \\| = 0 . \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} E _ 1 = Y _ 1 - X _ * = P _ { \\mathcal { T } ( X ) } ( \\Phi ( Y _ 0 ) - \\Phi ( X _ * ) ) + P _ { \\mathcal { T } ( X ) } ( X _ * ) - X _ * . \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} \\begin{cases} \\tau ~ & = ~ \\big | \\frac { U _ s '' ( a ) } { 2 } \\big | ^ { \\frac { 1 } { 2 } } ~ \\tilde \\tau , \\\\ V ( z ) ~ & = ~ \\big | \\frac { U _ s '' ( a ) } { 2 } \\big | ^ { \\frac { 1 } { 2 } } \\Big [ \\big ( \\tilde \\tau + \\big | \\frac { U _ s '' ( a ) } { 2 } \\big | ^ { \\frac { 1 } { 2 } } z ^ 2 \\big ) W \\big ( \\big | \\frac { U _ s '' ( a ) } { 2 } \\big | ^ { \\frac { 1 } { 4 } } z \\big ) - 1 _ { \\R ^ + } \\big ( \\tilde \\tau + \\big | \\frac { U _ s '' ( a ) } { 2 } \\big | ^ { \\frac { 1 } { 2 } } z ^ 2 \\big ) \\Big ] , \\end{cases} \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} D ^ { ( n ) } ( m , \\mu _ 1 , \\mu _ 2 ) = \\int _ { - \\pi } ^ { \\pi } e ^ { i \\lambda m } \\bigl ( 1 - e ^ { - i \\mu _ 1 \\lambda } \\bigr ) ^ n \\bigl ( 1 - e ^ { i \\mu _ 2 \\lambda } \\bigr ) ^ n \\frac { 1 } { \\lambda ^ { 2 n } } d F ( \\lambda ) , \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} s _ { l + 1 } \\ ; = \\ ; s _ { l } - \\psi ( z , s _ { l } ) \\left ( \\frac { \\partial \\psi } { \\partial t } ( z , s _ { l } ) \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} f & = \\Delta _ t \\phi = ( \\partial _ z ^ 2 + a ^ 2 \\partial ^ L _ { v v } + b \\partial ^ L _ { v } ) \\phi = ( \\partial _ z ^ 2 + a ^ 2 ( \\partial _ { v } - t \\partial _ z ) ^ 2 + b ( \\partial _ { v } - t \\partial _ z ) ) \\phi \\\\ & = \\Delta _ L \\phi + ( ( a ^ 2 - 1 ) \\partial ^ L _ { v v } + b \\partial ^ L _ { v } ) \\phi . \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} F _ \\gamma ( \\omega ) ( \\cdot \\wedge T ) = F _ \\gamma ( \\omega ( \\cdot \\wedge T ) ) . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} \\left ( W ( k ) - \\frac { 1 } { N } \\mathbf { 1 } \\mathbf { 1 } ^ T \\right ) [ y ^ { k } ] _ \\ell \\mathbf { 1 } = 0 . \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} U _ T = R ^ { - n } U _ Y , \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} \\Psi _ 2 ( a ; \\ , b , \\ , c ; \\ , x , \\ , y ) = \\sum _ { n , k = 0 } ^ \\infty \\frac { ( a ) _ { n + k } } { ( b ) _ { n } \\ ; ( c ) _ { k } } \\frac { x ^ n \\ ; y ^ k } { n ! \\ ; k ! } , \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} f \\left ( \\alpha \\right ) \\leq C \\left ( J _ { \\infty } g \\left ( \\alpha \\right ) + \\sum _ { k = 1 } ^ { N \\left ( \\alpha \\right ) } \\left \\{ \\prod _ { k \\leq \\ell \\leq N \\left ( \\alpha \\right ) } \\left ( 1 - \\left | z _ { m _ { \\ell } - 1 } \\right | ^ { 2 } \\right ) ^ { \\sigma } \\right \\} J _ { k } g \\left ( \\alpha \\right ) \\right ) , \\ ; \\ ; \\ ; \\ ; \\ ; \\alpha \\in \\mathcal { Y } . \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} \\Phi _ c ^ { - 1 } ( f _ i ^ s ( x ) ) & : = \\sup _ { y \\in \\mathbb { R } ^ { n _ i } } \\{ \\Phi _ c ^ { - 1 } ( f _ i ( y ) ) - s ^ { - 1 } \\| x - y \\| \\} , \\\\ \\Phi _ c ^ { - 1 } ( h ^ s ( x ) ) & : = \\sup _ { y \\in \\mathbb { R } ^ { n } } \\{ \\Phi _ c ^ { - 1 } ( h ( y ) ) - s ^ { - 1 } \\| x - y \\| \\} . \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} \\mathcal { E } ( t ) \\le \\frac { 1 } { N - \\widetilde { C } } \\mathcal { F } ( t ) \\le \\frac { 1 } { N - \\widetilde { C } } \\mathcal { E } ( 0 ) e ^ { - \\widehat { C } / ( N + \\widetilde { C } ) } t = C \\mathcal { E } ( 0 ) e ^ { - c _ 0 t } . \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} r ^ * ( X ) & = \\lambda ( X ) + | | E - X | | _ \\lambda + 2 | | X | | _ \\lambda - | | E | | _ \\lambda \\\\ & = \\lambda ( X ) + | | X | | _ \\lambda \\\\ & = r ( X ) . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} e ^ { i s ( t _ 1 + h ) } - e ^ { i s t _ 1 } = i s h e ^ { i s c } . \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} \\mathcal { S } ( \\mathbb { R } ^ { k \\times d } ) = \\big \\{ \\mathbf { H } \\in \\mathbb { R } ^ { ( k \\times d ) \\times ( k \\times d ) } \\ , | \\ , H _ { i j I J } = H _ { I J i j } i , I = 1 , \\dots , k j , J = 1 , \\dots , d \\big \\} \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} d \\omega _ { \\pm } & = \\omega _ { \\pm } ^ { \\prime } ( I _ { \\pm } ) d I _ { \\pm } \\backsim d I _ { \\pm } , \\\\ \\frac { d } { d \\omega _ { \\pm } } & = \\frac { d I _ { \\pm } } { d \\omega _ { \\pm } } \\frac { d } { d I _ { \\pm } } = \\frac { 1 } { \\omega _ { \\pm } ^ { \\prime } ( I _ { \\pm } ) } \\frac { d } { d I _ { \\pm } } \\backsim \\frac { d } { d I _ { \\pm } } , \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { n } - { n \\brace i } ( - 2 ) ^ { - 2 i } = ( - 2 ) ^ { - 2 n - 1 } \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} \\tilde { V } ^ + ( f _ 1 \\vee f _ 2 ) + \\tilde { V } ^ + ( f _ 1 \\wedge f _ 2 ) = \\tilde { V } ^ + ( f _ 1 ) + \\tilde { V } ^ + ( f _ 2 ) . \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} f _ { \\nu } \\left ( z \\right ) = f _ { \\nu } \\left ( 1 \\right ) + \\sum _ { l = 1 } ^ { k } \\frac { 1 } { l ! } f _ { \\nu } ^ { \\left ( l \\right ) } \\left ( 1 \\right ) \\cdot \\left ( z - 1 \\right ) ^ { l } + \\frac { 1 } { \\left ( k + 1 \\right ) ! } f _ { \\nu } ^ { \\left ( k + 1 \\right ) } \\left ( \\xi \\right ) \\cdot \\left ( z - 1 \\right ) ^ { k + 1 } \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} \\| \\partial ^ \\mu _ { t , x } g _ { i , { \\alpha \\beta } } \\| _ { \\ell ^ { i + | \\mu | } _ 1 L ^ \\infty _ { t , x } } = \\O ( 1 ) , i = 0 , 1 , | \\mu | \\le 3 . \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} \\alpha _ { l , L } \\triangleq \\underset { \\{ z : \\ ; A z = 0 , \\ ; z \\neq 0 \\} } { } \\frac { \\| z _ L \\| _ { 1 } } { \\| z \\| _ { 1 } } . \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} \\gamma _ { B ( i ) } = \\frac { g _ { i , B ( i ) } \\ell ( r _ { i , B ( i ) } ) P _ M } { I ^ \\psi _ { B ( i ) } + I ^ \\varphi _ { B ( i ) } + \\sigma ^ 2 } . \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} \\mathbf { r } _ { \\parallel } ^ { \\prime } = \\frac { \\mathbf { r } _ { \\parallel } - \\mathbf { v } t } { \\sqrt { 1 - v ^ { 2 } / c ^ { 2 } } } , \\qquad \\mathbf { r } _ { \\perp } ^ { \\prime } = \\mathbf { r } _ { \\perp } , t ^ { \\prime } = \\frac { t - \\left ( \\mathbf { v } \\cdot \\mathbf { r } \\right ) / c ^ { 2 } } { \\sqrt { 1 - v ^ { 2 } / c ^ { 2 } } } , \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} \\sum _ { \\mathfrak { n } } = \\sum _ { \\N \\mathfrak { p } < y } + \\sum _ { y \\leq \\N \\mathfrak { p } < x } + \\sum _ { \\N \\mathfrak { p } \\geq x } + \\sum _ { \\mathfrak { n } } = S _ 1 + S _ 2 + S _ 3 + S _ 4 . \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} \\epsilon _ { m , j } = \\sup \\{ T > t \\geq 0 : \\nu ( u ( . , t ' ) , a _ { m , j } ) > 0 , \\forall 0 \\leq t \\leq t ' \\} . \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} D _ X J _ \\alpha N = J _ \\alpha ( D _ X N ) , \\alpha = 1 , 2 , 3 , \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} p _ t ( x , y ) \\ge 1 - M ^ 2 \\sum _ { j = 1 } ^ { + \\infty } e ^ { - \\lambda _ j ( t - 2 t _ 0 ) } . \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} \\mathcal K _ { g , h } ( \\Omega , \\Omega ' ) = \\Big \\{ u \\in W ^ { s , p } ( \\Omega ' ) \\ , : \\ , u \\geq h \\ , \\Omega , \\ , u = g \\ , \\R ^ n \\setminus \\Omega \\Big \\} , \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} H _ { ( 2 x y ) } ( i , \\alpha , \\gamma ) ( j , \\beta , \\delta ) = H _ { ( 2 x ) } ( i , \\alpha ) ( j , \\beta ) \\otimes H _ y ( \\gamma , \\delta ) \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( f \\ , H ) - ( f \\ , H ) [ | A | ^ { 2 } - { \\rm R i c } ^ N ( \\xi , \\xi ) ] = 0 , \\\\ A \\ , ( { \\rm g r a d } ( f \\ , H ) ) + ( f \\ , H ) [ \\frac { m } { 2 } { \\rm g r a d } \\ , H - ( { \\rm R i c } ^ N \\ , ( \\xi ) ) ^ { \\top } ] = 0 , \\end{cases} \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} \\Psi ( \\mu _ A ( f ) a ) = \\mu _ B ( \\psi ( f ) ) \\Psi ( a ) . \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} d _ X ( F ^ 2 ( x _ 0 , y _ 0 ) , F ( x _ 0 , y _ 0 ) ) & = d _ X ( F ( F ( x _ 0 , y _ 0 ) , G ( y _ 0 , x _ 0 ) ) , F ( x _ 0 , y _ 0 ) ) \\\\ & \\leq k \\ d _ X ( F ( x _ 0 , y _ 0 ) , x _ 0 ) + l \\ d _ Y ( G ( y _ 0 , x _ 0 ) , y _ 0 ) \\\\ & \\leq ( k + l ) \\ [ d _ X ( x _ 1 , x _ 0 ) + d _ Y ( y _ 1 , y _ 0 ) ] \\\\ & < ( m + n ) \\ [ d _ X ( x _ 1 , x _ 0 ) + d _ Y ( y _ 1 , y _ 0 ) ] \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} \\sigma _ s ( \\boldsymbol { \\theta } ) : = \\| \\boldsymbol { \\theta } - \\boldsymbol { \\theta } _ s \\| _ 1 ~ ~ \\varsigma _ s ( \\boldsymbol { \\theta } ) : = \\| \\boldsymbol { \\theta } - \\boldsymbol { \\theta } _ s \\| _ 2 , \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\left ( I ^ { \\alpha } \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { \\cdot } h ^ { k } ( s ) d w _ { s } ^ { k } \\right ) ( t ) & = \\frac { 1 } { \\Gamma ( \\alpha ) } \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { t } ( t - s ) ^ { \\alpha - 1 } h ^ { k } ( s ) d w _ { s } ^ { k } \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} \\lim _ { N ^ \\prime } \\int _ { T _ 1 } ^ { T _ 2 } \\int \\left | \\omega \\cdot ( v _ 2 - v _ 1 ) \\right | ^ 2 f _ { N ^ \\prime } ^ { ( 2 ) } ( t , x _ 1 , v _ 1 , x _ 1 + \\varepsilon \\omega , v _ 2 ) d \\omega d x _ 1 d v _ 1 d v _ 2 d t = 0 \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} \\mathrm { R e } ( c ) = - k , \\mathrm { I m } ( c ) = 0 . \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\gamma ( u ) = \\phi ( u ) + \\lambda \\cos \\left ( \\frac { u } { c } \\right ) e _ { n + 1 } , \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} u _ { \\delta } ( x _ { j } ) = w _ { \\delta } + y _ { \\delta } g _ { j } ( x _ { j } ) . \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 k + 1 } U _ i = \\tilde U _ 1 + \\sum _ { i = 1 } ^ k U _ { 2 i + 1 } + \\sum _ { i = 1 } ^ { 2 k - 2 } W _ i + \\sum _ { i = 1 } ^ { k - 1 } V _ i . \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 n } } \\prod _ { k = 1 } ^ n \\varphi _ { H _ i } ( y _ { i k } - z _ { i k } ) \\Theta _ { n } ( t , y _ { i k } , s ) \\Theta _ { n } ( t , z _ { i k } , s ) d y _ i d z _ i \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} \\frac { d } { d t } m ( t ) = \\int _ { \\vert x \\vert \\leq R } J * u ( t , \\cdot ) ( x ) \\ , d x - m ( t ) + \\int _ { \\vert x \\vert \\leq R } u ^ { 1 + p } ( t , x ) \\ , d x . \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} 2 \\mu \\left ( a ^ { 2 2 } a ^ { 1 1 } - a ^ { 2 1 } a ^ { 1 2 } \\right ) e _ { 1 | | 3 } ^ 0 + \\rho \\left ( a ^ { 2 2 } a ^ { 1 1 } - a ^ { 2 1 } a ^ { 1 2 } \\right ) \\dot { e } _ { 1 | | 3 } ^ 0 = 2 \\mu a e _ { 1 | | 3 } ^ 0 + \\rho a \\dot { e } _ { 1 | | 3 } ^ 0 = 0 , \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} \\hat { \\psi } ( \\xi ) = - \\frac { a c ( i \\xi ) ^ 3 \\left ( \\alpha + \\beta e ^ { - i L \\xi } \\right ) } { ( 1 - a ^ 2 b ) ( i \\xi ) ^ 6 + r ( i \\xi ) ^ 4 + ( c + 1 ) \\lambda ( i \\xi ) ^ 3 + r \\lambda ( i \\xi ) + c \\lambda ^ 2 } . \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} H ^ j _ c ( X ) = \\left \\{ \\begin{array} { l l } \\Z / m , & 3 \\le j \\le 2 n - 1 \\ ; \\\\ \\Z , & j = 2 n \\\\ 0 , & \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} u _ { 0 } ( 0 ) + u _ { \\delta ^ { i } } ( 0 ) f _ { i , 0 } ( x _ { i } ) = v _ { 0 } ( x _ { i } ) . \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} v ( \\theta , \\phi ) = \\left [ \\begin{matrix} { { g } _ { 1 } } ( \\theta , \\phi ) \\exp ( j 2 \\pi { { f } _ { 0 } } { { \\tau } _ { 1 } } ) \\\\ { { g } _ { 2 } } ( \\theta , \\phi ) \\exp ( j 2 \\pi { { f } _ { 0 } } { { \\tau } _ { 2 } } ) \\\\ \\vdots \\\\ { { g } _ { N } } ( \\theta , \\phi ) \\exp ( j 2 \\pi { { f } _ { 0 } } { { \\tau } _ { N } } ) \\\\ \\end{matrix} \\right ] , \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} c \\left ( t \\right ) = \\int _ 0 ^ t \\left ( r + 2 \\left \\| \\nabla \\psi \\right \\| _ { \\infty } \\right ) \\ , d t , \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { B _ j } & = \\sigma _ j ( G ) B _ j G ^ { - 1 } , j = 1 , 2 \\end{aligned} \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} u _ n = \\sum _ { i = 0 } ^ q P _ i ( n ) \\alpha _ i ^ n \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\xi = 1 + 1 0 ^ { - 5 } , \\upsilon = 1 0 ^ { - 5 } , \\eta = 1 0 ^ { - 5 } , \\omega = 1 0 ^ { - 5 } , \\alpha = 0 . 1 5 . \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} R ' _ 0 & = \\{ X \\in R ' : \\ , \\underline { X } + \\overline { X } < n \\} , \\\\ R ' _ 1 & = \\{ X \\in R ' : \\ , \\underline { X } + \\overline { X } > n \\} \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} & [ L _ { n } , L _ { m } ] = ( \\{ m \\} - \\{ n \\} ) L _ { n + m } , \\\\ & [ L _ { n } , G _ { m } ] = ( \\{ m + 1 \\} - \\{ n \\} ) G _ { n + m } , \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} { \\rm q d i m } [ M ^ \\varepsilon _ { r , s } ] = 0 , \\ \\ { \\rm q d i m } [ F ^ \\varepsilon _ \\lambda ] = 0 ; \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} \\vec { Y } = \\vec { X } + \\vec { W } , \\vec { W } \\in \\mathbb { R } ^ { m \\times n } , \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} d z _ { i } = A ^ { ( j ) } d x _ { i } + A ^ { ( j ' ) } d x _ { i ' } , \\forall 1 \\leq i , i ' \\leq n , \\forall 1 \\leq j , j ' \\leq 2 n . \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\gamma = \\frac { \\int \\limits _ { 0 } ^ { 2 \\pi } { \\int \\limits _ { 0 } ^ { \\pi } { \\left | F { { ( \\theta , \\phi ) } ^ { 2 } } \\right | } } \\sin \\theta d \\theta d \\phi } { 4 \\pi \\sum \\limits _ { k = 1 } ^ { N } { { { \\left | { { w } _ { k } } \\right | } ^ { 2 } } } } . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} ( d - \\sum _ { k \\in I _ { J } ^ { c } } L _ { k } - \\sum _ { k \\in \\{ 1 , \\ldots , k _ { 1 } \\} \\cap I _ { J } } L _ { k } ) w _ { k _ { 1 } } ^ 2 & = ( d - \\sum _ { j = 1 } ^ { J - 1 } \\sum _ { k \\in I _ { j } } L _ { k } - \\sum _ { k \\in \\{ 1 , \\ldots , k _ { 1 } \\} \\cap I _ { J } } L _ { k } ) w _ { k _ { 1 } } ^ 2 \\\\ & = ( ( E _ { J } ) - \\sum _ { k \\in \\{ 1 , \\ldots , k _ { 1 } \\} \\cap I _ { J } } L _ { k } ) w _ { k _ { 1 } } ^ 2 \\\\ & \\leq \\sum _ { k \\in I _ J , k > k _ { 1 } } L _ { k } w _ k ^ 2 . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty \\frac { ( i + 1 ) ! } { ( i + 2 k ) ! } \\gamma _ i x ^ { i + 2 k } = \\left ( \\frac { x ^ { 2 k } } { 1 - x } \\right ) \\psi \\left ( \\frac { x ^ 2 } { 1 - x } \\right ) , \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} \\Vert \\partial _ v { ( a ^ 2 - 1 ) } \\Vert _ { L ^ 2 _ t H ^ \\sigma _ y } = \\Vert 2 b \\Vert _ { L ^ 2 _ t H ^ \\sigma _ y } \\lesssim \\Vert \\bar { U } '' \\Vert _ { L ^ 2 _ t H ^ \\sigma _ y } \\lesssim \\delta \\nu ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n & \\sqrt { ( x _ i - x _ { i - 1 } ) ^ 2 + ( f ( x _ i ) - f ( x _ { i - 1 } ) ) ^ 2 } \\\\ & \\qquad \\le \\sum _ { i = 1 } ^ n \\left ( ( x _ i - x _ { i - 1 } ) + ( f ( x _ i ) - f ( x _ { i - 1 } ) ) \\right ) \\\\ & \\qquad = ( b - a ) + ( f ( b ) - f ( a ) ) \\\\ & \\qquad \\le ( b - a ) + ( d - c ) . \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} M : = \\| V \\| _ \\infty , M _ 0 : = \\| G \\| _ \\infty , M _ 1 : = \\| \\nabla G \\| _ \\infty , \\quad ~ \\gamma > M _ 0 ^ 2 / 4 . \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} q \\in L _ { 1 } [ 0 , 1 ] , q ( x + 1 ) = q ( x ) , q _ { n } = 0 , \\forall n = 0 , - 1 , - 2 , . . . , \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} \\Phi : \\C ^ { m } \\times \\C ^ { m _ 1 ^ - } \\times \\C ^ { m _ 2 ^ - } & \\to \\C _ { m _ 1 - 1 } [ \\lambda ] \\times \\C _ { m _ 2 - 1 } [ \\lambda ] , \\\\ ( c , \\gamma _ 1 , \\gamma _ 2 ) & \\mapsto \\Big ( \\sum _ { j = 1 } ^ m c _ j \\tilde { e } _ { 1 } ^ j + \\sum _ { j = 1 } ^ { m _ 1 ^ - } \\gamma _ { 1 , j } \\tilde { e } _ { 1 } ^ { j + m } , \\sum _ { j = 1 } ^ m c _ j \\tilde { e } _ { 2 } ^ j + \\sum _ { j = 1 } ^ { m _ 2 ^ - } \\gamma _ { 2 , j } \\tilde { e } _ { 2 } ^ { j + m } \\Big ) . \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\widetilde \\psi _ j ( x ) : = \\min _ { y \\in \\overline D } \\Big \\{ \\min \\big \\{ j , \\ , u ( y ) \\big \\} + j ^ 2 | x - y | \\Big \\} - \\frac 1 { j } . \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} w ( a , b ) w ( c , d ) = w ( a + c , b + d + ( a \\pi ) c ) . \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\rho = \\frac { k } { G ^ { - 1 } \\left ( \\frac { k ^ 2 } { r ^ 2 } \\right ) } = \\frac { s _ { m ^ * } } { G ^ { - 1 } ( s _ { m ^ * } ^ 2 ) } r \\leq S _ { m ^ * } r \\leq r . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} T = T _ { r _ 1 } \\times T _ { r _ 2 } \\times \\ldots \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} D _ \\lambda ( q , t ) = - 1 - S _ { q , t } B _ \\lambda , \\bar D _ \\lambda ( q , t ) = D _ \\lambda ( q ^ { - 1 } , t ^ { - 1 } ) . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\begin{array} [ c ] { l } \\left \\langle J _ { a b } , J _ { c d } , Z _ { e } \\right \\rangle = \\alpha _ { 0 } \\ , \\varepsilon _ { a b c d e } , \\\\ \\left \\langle J _ { a b } , Z _ { c d } , P _ { e } \\right \\rangle = - \\alpha _ { 0 } \\ , \\varepsilon _ { a b c d e } , \\\\ \\left \\langle Z _ { a b } , Z _ { c d } , P _ { e } \\right \\rangle = - \\alpha _ { 0 } \\ , \\varepsilon _ { a b c d e } . \\end{array} \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{gather*} \\big ( \\tau _ k ^ { ( \\alpha ) } \\big ) ^ 2 = \\tau _ { k } ^ { ( \\alpha - 1 ) } \\tau _ { k } ^ { ( \\alpha + 1 ) } - \\tau _ { k + 1 } ^ { ( \\alpha - 1 ) } \\tau _ { k - 1 } ^ { ( \\alpha + 1 ) } , k = 0 , 1 , \\dots . \\end{gather*}"}
-{"id": "6831.png", "formula": "\\begin{align*} \\bar { P } = P \\left ( 1 - 2 ^ { - B } \\right ) , \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P - Z F } } = \\frac { ( 1 - \\mu _ 2 ) K } { M r } , \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } u _ t = \\Delta u _ t \\\\ u _ 0 = f . \\end{cases} \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\| x ^ { k + 1 } _ i - x ^ * _ i \\| ^ 2 & \\leq \\sum _ { i = 1 } ^ N \\| x ^ k _ i - x ^ * _ i \\| ^ 2 + N \\tilde C \\alpha _ k ^ 2 + 4 \\alpha _ k M N \\sum _ { i = 1 } ^ N \\bar L _ i \\| \\hat v _ i ^ k - y ^ k \\| \\cr & \\ ; - 2 \\alpha _ k \\sum _ { i = 1 } ^ N \\left ( F _ i ( x ^ k _ i , N y ^ k ) - F _ i ( x ^ * _ i , \\bar { x } ^ * ) \\right ) ^ T ( x _ i ^ k - x ^ * _ i ) . \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ { i } = t _ { K i , K } + \\rho f _ { i } , \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} & X ^ 0 = 2 \\\\ & X ^ { \\alpha } = \\prod _ n X ^ { \\pi _ \\alpha ( n ) } \\quad \\mbox { i f } \\alpha > 0 \\end{align*}"}
-{"id": "1287.png", "formula": "\\begin{align*} \\phi ( 0 ) = \\phi _ n \\ , , \\phi ' ( 0 ) = 0 \\ , . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} r ^ { * , \\pi } _ n ( x _ n | y ^ { n - 1 } _ { n - M } , y _ n ) = \\Big ( \\frac { q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) } { \\nu ^ { \\pi } _ { n } ( y _ n | y ^ { n - 1 } _ { n - J } ) } \\Big ) { \\pi } _ { n } ( x _ n | y ^ { n - 1 } _ { n - J } ) . \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} \\left [ \\partial ^ { \\tau } \\partial _ { \\tau } + \\kappa \\left ( u ^ { \\tau } \\partial _ { \\tau } \\right ) ^ { 2 } \\right ] \\partial _ { \\sigma } Z ^ { \\lambda \\sigma } = - \\frac { 4 \\pi \\mu } { c } j ^ { \\lambda } . \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} \\lim _ { r ( x ) \\to \\infty } V ( x ) = | | \\varphi | | _ \\infty . \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} \\hat { \\theta } _ T = \\frac { \\int _ 0 ^ T X _ t d X _ t } { \\int _ 0 ^ T X ^ 2 _ t d t } , \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} S _ { 2 n + 1 } \\left ( q ^ { - 2 n - 1 } ; q \\right ) = 0 , S _ { 2 n } \\left ( q ^ { - 2 n } ; q \\right ) = \\frac { \\left ( - 1 \\right ) ^ { n } q ^ { - n ^ { 2 } } } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } } \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} \\hat X = X + Z = \\begin{bmatrix} \\hat U & \\hat U _ { \\perp } \\\\ \\end{bmatrix} \\cdot \\begin{bmatrix} \\hat \\Sigma _ 1 & 0 \\\\ 0 & \\hat \\Sigma _ 2 \\\\ \\end{bmatrix} \\cdot \\begin{bmatrix} \\hat V ^ { \\intercal } \\\\ \\hat V _ { \\perp } ^ { \\intercal } \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} \\lambda g _ { + } \\pm \\omega _ { + } ( I _ { + } ) \\partial _ { \\theta _ { + } } g _ { + } \\mp \\mu _ { + , \\pm } ^ { \\prime } ( e _ { + } ) \\omega _ { + } ( I _ { + } ) \\partial _ { \\theta _ { + } } \\phi & = 0 , \\pm v > 0 , \\\\ \\lambda g _ { - } \\pm \\omega _ { - } ( I _ { - } ) \\partial _ { \\theta _ { - } } g _ { - } \\pm \\mu _ { - , \\pm } ^ { \\prime } ( e _ { - } ) \\omega _ { - } ( I _ { - } ) \\partial _ { \\theta _ { - } } \\phi & = 0 , \\pm v > 0 , \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} h ( v ) = \\beta ( d - v ) ( v + p ) , \\ ; \\ ; d > p > 0 , \\beta > 0 , \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} \\mathrm { E n d } _ { \\overline { K } } \\left ( J ^ + _ r ( t ) \\right ) = \\mathcal { O } _ K . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} \\Delta = \\{ ( x , y ) \\mid x \\leq y \\} \\cap \\{ ( x , y ) \\mid y \\leq x \\} \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Big ( \\int _ M | u | ^ p d \\mu \\Big ) \\Big | _ { t = s } = 0 , \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\frac { b _ { p , q } ( x ) - b _ { p , q } ( x _ n ) } { x - x _ n } & = \\frac { \\pi _ q ( 0 ^ m 1 \\overline { b _ { n , m + 2 } b _ { n , m + 3 } \\cdots } ) } { \\pi _ p ( 0 ^ m 1 \\overline { b _ { n , m + 2 } b _ { n , m + 3 } \\cdots } ) } \\\\ & \\ge \\frac { \\pi _ q ( 0 ^ m 1 0 ^ { \\infty } ) } { \\pi _ p ( 0 ^ m 1 ^ { \\infty } ) } \\\\ & = \\frac { p - 1 } { p } \\left ( \\frac { p } { q } \\right ) ^ { m + 1 } \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} G _ 0 ( x ^ q ) = B ( x ) G _ 0 ( x ) B _ 0 ^ { - 1 } . \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\left ( a ; q \\right ) _ { 2 n } = \\left ( a ; q ^ { 2 } \\right ) _ { n } \\left ( a q ; q ^ { 2 } \\right ) _ { n } \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} \\begin{cases} P _ 1 u _ 1 = f _ 1 & \\ V _ 1 = \\{ x _ n < 0 \\} , \\\\ P _ 2 u _ 2 = f _ 2 & \\ V _ 2 = \\{ x _ n > 0 \\} , \\\\ T ^ j _ 1 u _ 1 + T _ 2 ^ j u _ 2 = g ^ j , & \\ S , j = 1 , \\dots , m . \\end{cases} \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} \\deg ( \\Gamma ) & = ( D - 1 ) E - D V + \\sum _ { i = 1 } ^ { V } \\deg d e c ( v _ i ) \\\\ & = ( D - 3 ) ( E - V ) + \\sum ^ V _ { i = 1 } \\left ( \\deg d e c ( v _ i ) + ( v _ i ) - 3 \\right ) . \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} \\Big ( N \\delta _ { \\min , k } ( N ) \\leq u _ N \\Big ) = \\begin{cases} 1 , & \\sum _ n u _ n ^ k / n = \\infty , \\\\ \\\\ 0 , & \\sum _ n u _ n ^ k / n < \\infty . \\end{cases} \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} B \\phi ( z _ 1 ) - B \\phi ( z _ 2 ) = A \\left ( \\int _ { 0 } ^ { t } z _ 1 ( s ) d s + u _ 0 \\right ) - A \\left ( \\int _ { 0 } ^ { t } z _ 2 ( s ) d s + u _ 0 \\right ) \\ae . \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} \\| f ( z ) \\| _ { A ^ 2 _ 1 } ^ 2 = \\frac { 2 } { \\pi } \\int _ { \\mathbb { D } } | f ( z ) | ^ 2 ( 1 - | z | ^ 2 ) | d z | ^ 2 , \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} u _ \\omega = | v | ^ { - 1 } \\left ( 2 \\omega \\omega \\cdot v - v \\right ) \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} H _ { \\mathrm { L T } } = \\bigoplus _ { \\rho } H _ { \\mathrm { L T } , \\rho } , \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} B _ a = \\begin{pmatrix} 0 & 0 \\\\ \\beta _ a & \\gamma _ a \\end{pmatrix} , 1 \\leq a \\leq 7 - r , \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} ( s \\cdot r ) ( x , z ) = \\bigvee _ { y \\in Y } r ( x , y ) \\otimes s ( y , z ) , \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} N ^ { s _ 0 } _ { p _ 0 } ( \\mathbb { R } ^ d ) \\hookrightarrow N ^ { s _ 1 } _ { p _ 1 } ( \\mathbb { R } ^ d ) , \\ , \\ , \\ , \\ , s _ 0 - \\frac { d } { p _ 0 } = s _ 1 - \\frac { d } { p _ 1 } ; \\\\ N ^ { s _ 0 } _ { p _ 0 } ( \\mathbb { R } ^ d ) \\hookrightarrow L _ { p _ 2 } ( \\mathbb { R } ^ d ) , \\ , \\ , \\ , \\ , s _ 0 - \\frac { d } { p _ 0 } > - \\frac { d } { p _ 2 } . \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} a _ { j , 1 } ( \\tau , \\xi ) = & | \\xi | ^ { - \\varpi } \\varphi _ j ( \\xi ) | \\tau | ^ \\beta \\int _ { - \\pi } ^ { \\pi } \\ ^ { \\chi } _ 1 ( \\tau - | \\xi | \\sin \\theta ) e ^ { - i \\theta \\varpi } d \\theta , \\\\ a _ { j , 2 } ( \\tau , \\xi ) = & | \\xi | ^ { - \\varpi } \\varphi _ j ( \\xi ) | \\tau | ^ \\beta \\int ^ \\infty _ 0 \\ ^ { \\chi _ 2 } \\bigl ( \\tau - i | \\xi | \\sinh s \\bigr ) e ^ { - \\varpi s } d s . \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} \\theta _ { j , R } ( s ) = \\theta ^ S _ { j , R } ( s ) + \\theta ^ L _ { j , R } ( s ) . \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\in \\Z \\\\ m < 0 } } c ( m ) a ( - m ) = 0 \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} P _ { \\mu \\lambda } Q ^ { \\lambda \\nu } + P _ { \\mu \\lambda } ^ { \\ast } \\overset { \\ast } { \\left . Q ^ { \\lambda \\nu } \\right . } & = \\frac { 1 } { 4 } \\left ( P _ { \\sigma \\tau } Q ^ { \\tau \\sigma } + P _ { \\sigma \\tau } ^ { \\ast } \\overset { \\ast } { \\left . Q ^ { \\tau \\sigma } \\right . } \\right ) \\delta _ { \\mu } ^ { \\nu } \\\\ & = \\frac { 1 } { 2 } \\left ( \\mathbf { E } \\cdot \\mathbf { D } - \\mathbf { H } \\cdot \\mathbf { B } \\right ) \\delta _ { \\mu } ^ { \\nu } . \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} L _ j ( x , t , y ) = \\int ^ \\infty _ 0 d s & \\int ( 1 - \\chi _ { I _ j } ( \\tau ) ) | \\tau | ^ \\beta e ^ { 2 \\pi i t \\tau } e ^ { - \\varpi s } d \\tau \\\\ & \\times \\int _ { \\R ^ n } \\varphi _ j ( \\xi ) \\ ^ { \\chi _ 2 } \\bigl ( \\tau - i | \\xi | \\sinh s \\bigr ) \\frac { e ^ { 2 \\pi i ( x - y ) \\cdot \\xi } } { | \\xi | ^ \\varpi } d \\xi . \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} \\frac { \\partial W ( g ) } { \\partial g } = - \\frac { 1 } { 2 } \\int _ 0 ^ \\infty \\frac { d t } { t } T r ( e ^ { - t D } ) \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} P _ { \\mathbf { x } } ( G ) = \\prod _ { i j \\in E ( G ) } \\phi ( x _ i , x _ j ) \\times \\prod _ { i j \\notin E ( G ) } ( 1 - \\phi ( x _ i , x _ j ) ) . \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} g _ s ^ { - 1 } = b ^ T A _ g b . \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} N _ \\lambda ( u ) = \\left ( \\tau _ { u ^ { - 1 } - q t } \\tilde H _ \\lambda , \\ ; \\tau _ { u - 1 } \\tilde H _ \\lambda \\right ) ^ { S _ { q , t } } . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} \\| \\sin \\Theta ( \\hat V , V ) \\| = \\sqrt { 1 - \\sigma _ { \\min } ^ 2 ( \\hat V ^ { \\intercal } V ) } = \\| \\hat V ^ { \\intercal } V _ { \\perp } \\| , \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} \\sigma _ j \\sigma _ 2 ^ { - 1 } \\sigma _ k = \\sigma _ j \\sigma _ 1 ^ { - 1 } \\sigma _ k \\sigma _ 2 ^ { - 1 } \\sigma _ 1 . \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} d ( - Y , s ) = - d ( Y , s ) \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} H ( p , x ) = \\langle g ( x ) p , p \\rangle ^ { q / 2 } . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} \\left | K _ { N } ^ { 1 } ( z , \\zeta ) \\right | \\lesssim \\frac { 1 } { \\prod _ { i = 1 } ^ { n - 1 } \\tau _ { i } ( z , \\varepsilon ) } \\frac { 1 } { \\left | z - \\zeta \\right | } . \\end{align*}"}
-{"id": "9942.png", "formula": "\\begin{align*} w _ r \\in V ( \\delta _ 1 , \\delta _ 2 , \\cdots , \\delta _ k ) \\delta _ i = l - 2 r . \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} \\bigl ( \\det ( x _ i ^ { q ^ { j - 1 } } ) _ { 1 \\le i , j \\le n } \\bigr ) ^ { q - 1 } = ( - 1 ) ^ { n - 1 } . \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in \\Gamma ( k \\pm ) } \\mid q _ { \\gamma } \\mid = M < \\infty , \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} \\left ( L _ { t } ( q ) - \\lambda I \\right ) \\varphi = \\Psi \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} u ( t , x ) = \\sum _ { n \\geq 0 } I _ n ( f _ n ( \\cdot , t , x ) ) , \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} { \\bf { C o v } } \\left [ V _ { \\varepsilon } ( u ) , \\ , V _ { \\varepsilon } ( v ) \\right ] = & \\begin{cases} - 2 \\ , \\log | u - v | , \\ , \\varepsilon \\ll | u - v | \\leq 1 , \\\\ 2 \\left ( \\kappa - \\log \\varepsilon \\right ) , \\ , u = v , \\end{cases} \\\\ & + O ( \\varepsilon ) , \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} w _ { \\max } = \\sup _ { \\tilde { M } ( \\tau ) } w \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u ( t , x ) = \\Delta u ( t , x ) , u ( 0 ) = u _ 0 , 1 _ { \\alpha > 1 } u ' ( 0 ) = 0 . \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} S ( x , y ) & = \\dfrac { c } { \\sqrt { 1 + \\abs { \\nabla \\varphi ( x , y ) } ^ 2 } } - k \\\\ & = \\dfrac { m _ * ^ 2 - \\abs { \\nabla \\varphi } ^ 2 } { \\sqrt { 1 + \\abs { \\nabla \\varphi ( x , y ) } ^ 2 } \\left ( c + k \\sqrt { 1 + \\abs { \\nabla \\varphi ( x , y ) } ^ 2 } \\right ) } . \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} \\xi ( 1 - z ) P ' _ { 0 } ( z ) - \\left [ \\lambda ( 1 - z ) + \\gamma \\right ] P _ { 0 } ( z ) = - ( \\gamma p _ { 0 , 0 } + \\mu p _ { 1 , 1 } ) . \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} d ( [ x , y ] ) = [ d ( x ) , y ] + [ x , d ( y ) ] . \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} \\bigcup _ { n \\geq 0 } \\theta ^ { - n } \\left ( A \\right ) = X \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} \\square \\left ( \\Delta u - \\frac { R } { 4 } - | \\nabla u | ^ 2 \\right ) = - 2 \\left | u _ { i j } - \\frac { R _ { i j } } { 2 } \\right | ^ 2 , \\square \\left ( - \\Delta u - \\frac { R } { 4 } - | \\nabla u | ^ 2 \\right ) = - 2 \\left | u _ { i j } + \\frac { R _ { i j } } { 2 } \\right | ^ 2 . \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} \\chi _ 2 ( - 1 ) = ( - 1 ) ^ { k _ 1 + k _ 2 } . \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} A \\nabla B \\leqslant H _ { \\nu } ( A , B ) & + ( A - 2 A \\sharp B + B ) \\\\ & - \\sum _ { k = 0 } ^ { \\infty } r _ { k } [ H _ { \\frac { m _ k } { 2 ^ k } } ( A , B ) - 2 H _ { \\frac { 2 m _ k + 1 } { 2 ^ { k + 1 } } } ( A , B ) + H _ { { \\frac { m _ k + 1 } { 2 ^ k } } } ( A , B ) ] . \\end{align*}"}
-{"id": "10134.png", "formula": "\\begin{align*} \\omega = 2 q y z d y - ( ( p + 2 q ) y ^ 2 + b ( p + q ) z ) d z . \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} r _ { \\partial } = \\dfrac { \\int _ { \\partial M } k _ { \\tilde { g } } \\tilde { R } d s _ { \\tilde { g } } } { \\int _ { \\partial M } k _ { \\tilde { g } } d s _ { \\tilde { g } } } , \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} u ^ { 0 } \\in H ^ { 2 , q } _ { \\mathrm { N e u } } : = \\Big \\{ u \\in H ^ { 2 , q } ( G ) \\ , \\big | \\ , \\frac { \\partial u } { \\partial \\mathbf { n } } = 0 \\Gamma \\Big \\} \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{gather*} \\xi _ 0 \\xi _ 1 \\xi _ 2 \\xi _ 3 \\xi _ 4 \\xi _ 5 = \\left ( \\frac { c } { 1 - a ^ 2 b } \\right ) p ^ 2 . \\end{gather*}"}
-{"id": "6558.png", "formula": "\\begin{align*} { n \\brack i } = \\frac { ( 2 n ) ( 2 n - 1 ) } { ( 2 i - 1 ) ( 2 i - 2 ) } { n - 1 \\brack i - 1 } = \\frac { { 2 n \\choose 2 } } { { 2 i - 1 \\choose 2 } } { n - 1 \\brack i - 1 } , \\quad 2 \\leq i \\leq n . \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} k [ x , y ] ^ { ( 2 ) } = k [ x ^ 2 , x y , y ^ 2 ] . \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\overline { \\mu } ( T ^ n A \\cap B _ m ) & \\leq \\frac { 1 } { r _ m } \\\\ & = \\frac { 1 } { r _ m } \\sqrt { r _ m } \\mu ( B _ m ) = \\frac { 1 } { \\sqrt { r _ m } } \\mu ( B _ m ) . \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} \\sigma ^ a _ b = c ^ { a ^ \\prime } _ { i ^ \\prime } ( \\delta ^ a _ { a ^ \\prime } v ^ { i ^ \\prime } _ b + \\delta ^ a _ b B ^ { i ^ \\prime } _ j v ^ j _ { a ^ \\prime } ) . \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} \\langle U , U ^ * \\rangle _ { \\mathcal { H } _ \\kappa } = \\big \\langle ( u , v , \\tau , \\theta ) ^ T , ( u ^ * , v ^ * , \\tau ^ * , \\theta ^ * ) ^ T \\big \\rangle _ { \\mathcal { H } } + \\int _ { \\Omega } \\kappa m _ { i j } q _ i ^ * q _ j \\ , \\mathrm { d } x \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} L _ v [ X , Y ] = \\sum _ { k = 1 } ^ \\infty L ^ { ( k ) } _ v [ X , Y ] , \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} \\Phi _ 3 ( b ; \\ , c ; \\ , x , \\ , x ^ 2 ) = \\exp \\left ( 2 x \\right ) \\ ; \\sum _ { k = 0 } ^ \\infty \\ ; \\frac { ( - x ) ^ k } { k ! } \\ ; \\left ( - \\frac { y } { x } \\right ) ^ k { } _ 2 F _ 1 \\left [ \\begin{array} { c } - k , \\ ; - k - c + b + 1 \\\\ c \\end{array} ; \\ ; 1 \\right ] \\ , . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} F _ { k } ( x ) = \\sum _ { m = k } ^ { n } \\binom { n } { m } \\left ( 1 - e ^ { - t } \\right ) ^ { m } e ^ { - ( n - m ) t } , ~ ~ t > 0 . \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } u _ { \\delta } \\prod _ { i = 1 } ^ { n } g _ { i } ^ { \\delta _ { i } } ( x _ { i } ) = \\sum _ { \\delta \\in \\{ 0 , 1 \\} ^ { n } } v _ { \\delta } f _ { n } ^ { \\delta _ { n } } ( x _ { n } ) \\prod _ { i = 1 } ^ { n - 1 } f _ { i } ^ { \\delta _ { i } } ( x _ { i } - x _ { n } ) . \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} \\tilde { \\kappa } _ { ( r ) } = \\sum _ { i = 0 } ^ { r } I _ i \\ \\kappa _ { ( r - i ) } \\cdot \\kappa _ { ( 1 ) } ^ i . \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} ( r + s i ) ( x , y ) : = ( r x - s y , r y + s x ) \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} H \\ : = \\ : & r ^ 2 h + h _ { 0 } + r ^ { - 2 } h _ { 2 } + \\cdots + r ^ { - 2 i } h _ { 2 i } + \\cdots , \\\\ f \\ : = \\ : & - \\frac 1 4 r ^ 2 + f _ { 0 } + r ^ { - 2 } f _ { 2 } + \\cdots + r ^ { - 2 i } f _ { 2 i } + \\cdots . \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{gather*} \\begin{bmatrix} \\dot { x _ 1 } \\\\ \\dot { v _ 1 } \\\\ \\dot { x _ 2 } \\\\ \\dot { v _ 2 } \\end{bmatrix} = \\begin{bmatrix} v _ 1 \\\\ - \\frac { k _ 1 } { m _ 1 } x _ 1 - \\frac { k _ 2 } { m _ 1 } ( x _ 1 - x _ 2 ) \\\\ v _ 2 \\\\ - \\frac { k _ 2 } { m _ 2 } ( x _ 2 - x _ 1 ) \\end{bmatrix} \\end{gather*}"}
-{"id": "6955.png", "formula": "\\begin{align*} \\int _ { \\mathcal { V } } v \\ , { \\rm d } \\mu ( v ) = 0 . \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} g + 4 = 5 g ' + 2 n . \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} s _ a ^ 1 ( x , \\xi ) & = \\sum _ { z \\in \\mathbb { Z } ^ d } f \\left [ z \\right ] \\left ( e ^ { i ( \\varphi _ a ( x - z , \\xi ) - \\varphi _ a ( x , \\xi ) ) } - e ^ { - i z \\cdot \\nabla _ x \\varphi _ a ( x , \\xi ) } \\right ) \\\\ & = \\sum _ { z \\in \\mathbb { Z } ^ d } f \\left [ z \\right ] e ^ { - i z \\cdot \\nabla _ x \\varphi _ a ( x , \\xi ) } \\left ( e ^ { i \\Phi _ a ( x , \\xi , z ) } - 1 \\right ) , \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} u ( x _ 0 , 0 ) \\geq \\left . \\int _ { M } u \\left ( 4 \\pi K ^ { - 2 } r ^ 2 \\right ) ^ { - \\frac { m } { 2 } } e ^ { - l } d \\mu \\right | _ { t = - K ^ { - 2 } r ^ 2 } = \\left . \\int _ { M } \\varphi \\left ( 4 \\pi K ^ { - 2 } r ^ 2 \\right ) ^ { - \\frac { m } { 2 } } e ^ { - l } d \\mu \\right | _ { t = - K ^ { - 2 } r ^ 2 } . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} \\begin{aligned} & \\limsup _ { N \\rightarrow \\infty } \\left \\Vert \\left ( f _ N ^ { ( s ) } ( t ) - \\int _ { \\mathcal { P } \\left ( \\mathbb { R } ^ { 2 d } \\right ) } h ( t ) ^ { \\otimes s } d \\pi ( h _ 0 ) \\right ) \\mathbf { 1 } _ { Z _ s \\in \\mathcal { K } _ s \\cap \\mathcal { U } _ s ^ { \\eta ( \\varepsilon ) } } \\mathbf { 1 } _ { E _ s ( Z _ s ) \\leq R ^ 2 } \\right \\Vert _ { L ^ \\infty _ { Z _ s } } = 0 \\end{aligned} \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } V a r ( Z _ n ) = \\lim _ { n \\rightarrow \\infty } ( \\sum _ { j = 1 } ^ { n } \\frac { 1 } { j ^ 2 } ) = \\sum _ { j = 1 } ^ { \\infty } \\frac { 1 } { j ^ 2 } = \\frac { \\pi ^ 2 } { 6 } , \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{gather*} b a = q a b , \\ d c = q c d , \\ c a = p a c , \\ d b = p b d , \\ q c b = p b c , \\\\ d a - a d = ( p - q ^ { - 1 } ) b c , \\ ( a d - q ^ { - 1 } b c ) \\delta ^ { - 1 } = 1 = \\delta ^ { - 1 } ( a d - q ^ { - 1 } b c ) . \\end{gather*}"}
-{"id": "2162.png", "formula": "\\begin{align*} \\hat { \\varphi } ( \\xi ) & = \\left ( 1 + \\frac { a ^ 2 b ( i \\xi ) ^ 6 } { ( 1 - a ^ 2 b ) ( i \\xi ) ^ 6 + r ( i \\xi ) ^ 4 + ( c + 1 ) \\lambda ( i \\xi ) ^ 3 + r \\lambda ( i \\xi ) + c \\lambda ^ 2 } \\right ) \\frac { \\left ( \\alpha + \\beta e ^ { - i L \\xi } \\right ) } { ( i \\xi ) ^ 3 + \\lambda } , \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} \\begin{aligned} & \\limsup _ { N \\rightarrow \\infty } \\left \\Vert \\left ( f _ N ^ { ( s ) } ( 0 ) - \\int _ { \\mathcal { P } \\left ( \\mathbb { R } ^ { 2 d } \\right ) } h _ 0 ^ { \\otimes s } d \\pi ( h _ 0 ) \\right ) \\mathbf { 1 } _ { Z _ s \\in \\mathcal { K } _ s \\cap \\mathcal { U } _ s ^ { \\eta ( \\varepsilon ) } } \\mathbf { 1 } _ { E _ s ( Z _ s ) \\leq R ^ 2 } \\right \\Vert _ { L ^ \\infty _ { Z _ s } } = 0 \\end{aligned} \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} F ( s ) & = \\sup _ { r > 0 } \\big ( ( s r ) ^ q - G ( r ) \\big ) ^ { 1 / q } = \\sup _ { \\substack { r > 0 \\\\ ( s r ) ^ q \\geq G ( r ) } } \\left ( ( s r ) ^ q - 2 ^ q \\left ( \\frac 1 2 G ( r ) ^ { 1 / q } \\right ) ^ q \\right ) ^ { 1 / q } \\\\ & \\leq 2 \\sup _ { \\substack { r > 0 \\\\ ( 2 s r ) ^ q \\geq G ( r ) } } \\left ( s r - \\frac 1 2 G ( r ) ^ { 1 / q } \\right ) = \\sup _ { r > 0 } \\left ( s r - G \\left ( \\frac r 2 \\right ) ^ { 1 / q } \\right ) , \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} - \\sum \\limits _ { j = 0 } ^ n \\gamma _ { 2 j } \\sum \\limits _ { i = j } ^ n \\frac { 1 } { i + k } { - 2 i - 2 k \\choose - 2 n - 2 k } { - 2 k - 2 j \\choose - 2 k - 2 i - 1 } \\frac { 4 ^ { i - j + 1 } - 1 } { i - j + 1 } B _ { 2 ( n - i ) } B _ { 2 ( i - j + 1 ) } . \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\frac { F '' ( v ) } { F ' ( v ) } \\| \\nabla v \\| ^ 2 = - \\frac { 1 } { 2 } v \\| \\nabla v \\| ^ 2 + \\frac { 1 } { 2 v } \\| \\nabla v \\| ^ 2 \\\\ & = \\sup _ { a \\in \\mathbb { R } ^ n } \\inf _ { b \\in \\mathbb { R } ^ n } \\bigg \\{ \\langle a + c b , \\nabla v + b \\rangle - \\frac { 1 } { 2 } v \\| b \\| ^ 2 \\bigg \\} + \\sup _ { \\tilde a \\in \\mathbb { R } ^ n } \\bigg \\{ \\langle \\tilde a , \\nabla v \\rangle - \\frac { 1 } { 2 } v \\| \\tilde a \\| ^ 2 \\bigg \\} . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} p _ { k + 1 } & \\leq \\dfrac { s p _ k } { 1 - \\left ( \\dfrac { q _ { k + 1 } } { 1 - q _ k } \\right ) } \\leq \\dfrac { s ^ { k + 1 } p _ 0 } { \\prod \\limits _ { j = 0 } ^ k \\Big ( 1 - \\dfrac { q _ { j + 1 } } { 1 - q _ j } \\Big ) } . \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} L _ { b , \\gamma } = \\Gamma ( \\gamma + 1 ) \\ , \\left ( e ^ { - b } - b \\ , \\int _ b ^ \\infty \\ , s ^ { - 1 } \\ , e ^ { - s } \\ , d s \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} c _ 1 ( \\pi ) + \\dots + c _ { i + 1 } ( \\pi ) = c _ 1 ( \\pi ' ) + \\dots + c _ { i + 1 } ( \\pi ' ) . \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} D _ { t , x } F = \\sum _ { n = 1 } ^ { \\infty } n \\ , I _ { n - 1 } ( f _ n ( \\cdot , t , x ) ) \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} \\partial f = f \\otimes \\mathbf 1 , \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} d \\mu ( t ) = \\frac { 1 } { 2 \\pi } \\sqrt { 4 - t ^ 2 } 1 _ { [ - 2 , 2 ] } d t . \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} \\Vert P _ \\frac { t } { 2 } ( 1 _ A ) \\Vert _ 2 ^ 2 = \\mu ( A ) ^ 2 + \\Vert P _ \\frac { t } { 2 } ( 1 _ A - \\mu ( A ) ) \\Vert _ 2 ^ 2 \\le \\mu ( A ) ^ 2 + e ^ { - \\lambda _ 1 t } \\| 1 _ A - \\mu ( A ) \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} [ W , \\i A ] _ { \\circ } = K _ { W } + B _ { W } , \\ [ W ' , \\i A ] _ { \\circ } = K _ { W ' } + B _ { W ' } . \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} { m + n \\choose l } = \\sum _ { i = 0 } ^ { m } { m \\choose i } { n \\choose l - i } , \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} \\Psi _ 2 ( b ; \\ ; b , \\ ; 2 b ; \\ ; x , \\ ; x ) = \\ , _ 2 F _ 2 \\left [ \\begin{array} { c } \\frac { 3 b } { 2 } , \\ ; \\frac { 3 b - 1 } { 2 } \\\\ 2 b , \\ ; 3 b - 1 \\end{array} ; \\ ; 4 x \\right ] \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} \\widetilde { N } ' : = \\{ f ( y ) \\ | \\ f \\in \\widetilde { N } \\cap { } ^ { \\omega _ 2 } H _ \\theta y \\in N ' \\cap \\omega _ 2 \\} . \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} T _ 2 = & ( n - 2 ) \\sigma _ 1 ( A _ g ) g - 8 A _ g = - \\frac { 8 } { n - 2 } { \\rm R i c } _ g + \\frac { n ^ 2 - 4 n + 1 2 } { 2 ( n - 1 ) ( n - 2 ) } R _ g g ; \\\\ T _ 4 = & - \\frac { 3 n ^ 2 - 1 2 n - 4 } { 4 } \\sigma _ 1 ( A _ g ) ^ 2 g + 4 ( n - 4 ) | A | _ g ^ 2 g + 8 ( n - 2 ) \\sigma _ 1 ( A _ g ) A _ g \\\\ & + ( n - 6 ) \\Delta _ g \\sigma _ 1 ( A _ g ) g - 4 8 A _ g ^ 2 - \\frac { 1 6 } { n - 4 } B _ g ; \\\\ v _ 6 = & - \\frac { 1 } { 8 } \\sigma _ 3 ( A _ g ) - \\frac { 1 } { 2 4 ( n - 4 ) } \\langle B , A \\rangle _ g . \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} \\sum \\limits _ { s = 1 } ^ \\ell { 2 \\ell \\choose 2 s } ( 2 ^ { 2 s } - 1 ) B _ { 2 s } B _ { 2 ( \\ell - s ) } = 0 , \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} q _ 2 ( \\lambda ) = c _ 1 \\tilde { t } _ { 2 , \\varphi } ^ 1 ( \\omega , \\lambda ) + c _ 2 \\tilde { t } _ { 2 , \\varphi } ^ 2 ( \\omega , \\lambda ) + U _ 2 ( \\lambda ) \\kappa _ { 2 , \\varphi } ( \\omega , \\lambda ) . \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} D H ( v ) u & = \\beta P | v | ^ 2 u + 2 \\beta P ( u \\cdot v ) v + \\lambda _ 0 P ( u \\cdot \\nabla ) v \\\\ & + \\lambda _ 0 P ( v \\cdot \\nabla ) u - 2 P \\sum _ { i , k = 1 } ^ n a _ { j k } ( u ^ j v ^ k + u ^ k v ^ j ) . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} \\Gamma _ { 0 } ( N ) & = \\bigl \\{ ~ \\left ( \\begin{smallmatrix} a & b \\\\ c & d \\end{smallmatrix} \\right ) \\in _ { 2 } ( \\mathbb { Z } ) ~ | ~ c \\equiv 0 \\pmod { N } ~ \\bigr \\} . \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} - L u & = f \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = \\eta \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} \\begin{aligned} F ( x _ k ) & \\le h \\big ( c ( y _ k ) + \\nabla c ( y _ k ) ( w _ k - y _ k ) \\big ) + a _ k g ( v _ k ) + ( 1 - a _ k ) g ( x _ { k - 1 } ) \\\\ & + \\frac { \\tilde { \\mu } _ k } { 2 } \\left ( \\norm { w _ k - y _ k } ^ 2 - \\norm { w _ k - x _ k } ^ 2 \\right ) + \\frac { \\left ( 1 - \\alpha ^ { - 1 } \\right ) } { 2 t _ k } \\norm { x _ k - y _ k } ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} \\rho \\left \\vert x _ { k } \\right \\vert ^ { r } = \\left \\vert \\sum _ { i _ { 2 } , \\ldots , i _ { r } } a _ { k , i _ { 2 } , \\ldots , i _ { r } } x _ { k } x _ { i _ { 2 } } \\cdots x _ { i _ { r } } \\right \\vert \\leq \\sum _ { i _ { 2 } , \\ldots , i _ { r } } a _ { k , i _ { 2 } , \\ldots , i _ { r } } \\left \\vert x _ { k } \\right \\vert | x _ { i _ { 2 } } | \\cdots \\left \\vert x _ { i _ { r } } \\right \\vert . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} s _ - ( x , \\xi ) & : = s _ a ( x , \\xi ) \\chi _ { \\{ x \\neq 0 \\} } \\rho ^ - ( \\cos ( x , v ( \\xi ) ) ) , \\\\ s _ + ( x , \\xi ) & : = s _ a ( x , \\xi ) - s _ - ( x , \\xi ) . \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { m } \\langle \\xi _ 1 ( \\eta _ { i + 1 } ) , v _ i \\rangle \\le \\sum _ { i = 0 } ^ { m } \\langle \\xi _ 1 ( \\eta _ { i } ) , v _ i \\rangle . \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} \\textbf { Q } & = \\textbf { q } + h \\sum _ { i = 1 } ^ s b _ i \\textbf { k } _ i ( \\textbf { q } ) \\\\ \\textbf { P } & = \\left ( \\textbf { p } - h \\sum _ { i = 1 } ^ s b _ i \\nabla l _ i ( \\textbf { q } ) \\right ) \\left ( \\textbf { I } + h \\sum _ { i = 1 } ^ { s } b _ i \\textbf { k } ' _ i ( \\textbf { q } ) \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} ( F _ { + } ) _ { 1 } = F _ { + } \\xi _ { + } \\ ; \\ ; \\mbox { a n d } \\ ; \\ ; ( G _ { - } ) _ { 2 } = G _ { - } \\xi _ { - } , \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} c = & u '^ 2 + v '^ 2 \\cos ^ 2 u \\ ; , \\\\ c \\kappa = & ( u '' v ' - v '' u ' ) \\cos u + v '^ 3 \\cos ^ 2 u \\sin u + 2 u '^ 2 v ' \\sin u \\ ; . \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} \\left \\| \\left ( \\mathbb { I } - \\frac { 1 } { N } \\mathbf { 1 } \\mathbf { 1 } ^ T \\right ) \\zeta ^ { k + 1 } ( \\ell ) \\right \\| \\le \\left \\| \\mathbb { I } - \\frac { 1 } { N } \\mathbf { 1 } \\mathbf { 1 } ^ T \\right \\| \\ , \\| \\zeta ^ { k + 1 } ( \\ell ) \\| = \\| \\zeta ^ { k + 1 } ( \\ell ) \\| . \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} \\norm { b ( y ) } = \\inf _ W \\sup _ { x \\in W \\cap X _ A } \\norm { b ( x ) } , \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = \\omega + \\xi ( y , z , \\omega , \\sigma , \\mu ) + f ( x , y , z , \\omega , \\sigma , \\mu ) , \\\\ \\dot { y } & = \\sigma + \\eta ( y , z , \\omega , \\sigma , \\mu ) + g ( x , y , z , \\omega , \\sigma , \\mu ) , \\\\ \\dot { \\sigma } & = \\Lambda y , \\\\ \\dot { z } & = Q ( \\omega , \\mu ) z + \\zeta ( y , z , \\omega , \\sigma , \\mu ) + h ( x , y , z , \\omega , \\sigma , \\mu ) \\end{aligned} \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} \\sup _ { \\theta \\in \\mathcal { S } ^ { p } ( \\ell _ { n } ) } \\left \\vert \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } ( X _ { i } ^ { \\prime } \\theta ) ^ { 2 } - { \\mathrm { E } } [ ( X ^ { \\prime } \\theta ) ^ { 2 } ] \\right \\vert \\leq C n ^ { - c } \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { \\left ( b \\pm \\frac { t } { 2 } \\sqrt { a ^ 2 - 4 c t ^ 2 } \\pm \\frac { a ^ 2 } { 4 \\sqrt { - c } } \\arcsin \\frac { 2 \\sqrt { - c } \\ , t } { a } \\right ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} x _ 1 ^ * + \\Sigma _ 2 ^ * + \\Sigma _ 3 ^ * + \\Sigma _ 4 ^ * \\leq M = 1 - N \\approx 0 . 6 8 3 0 . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} + \\sum \\limits _ { r = 0 } ^ { n - 2 } \\mu ^ { r - n + 1 } e ^ { - \\eta ^ 2 } P _ r ( \\eta ) + \\int \\limits _ { 0 } ^ { 1 } \\Psi _ n ( z , \\eta ) d z - \\int \\limits _ { 0 } ^ { \\mu } \\Psi _ n ( z , \\eta ) d z . \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} \\Big | \\sqrt { c ^ 2 k ^ 2 + m ^ 2 c ^ 4 } - m c ^ 2 - \\frac { k ^ 2 } { 2 m } \\Big | = \\Big | - \\frac { k ^ 4 } { 2 m c ^ 2 } \\frac { 1 } { ( m + \\sqrt { c ^ { - 2 } k ^ 2 + m ^ 2 } ) ^ 2 } \\Big | \\leq \\frac { k ^ 4 } { 4 m ^ 2 c ^ 2 } \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} \\sqrt { - d ^ 2 / d x ^ 2 } \\ , u _ { 3 / 2 } ( x ) = \\frac { 2 } { \\pi } \\bigg ( \\frac { 1 } { x ^ 2 + 1 } - \\frac { x \\sinh ^ { - 1 } x } { \\left ( x ^ 2 + 1 \\right ) ^ { 3 / 2 } } \\bigg ) , \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( 1 - 2 t ) \\dot { H } ( t ) \\ , d t = 2 \\int _ 0 ^ 1 H ( t ) \\ , d t - H ( 0 ) - H ( 1 ) . \\end{align*}"}
-{"id": "10012.png", "formula": "\\begin{align*} h ^ { s } ( \\underline { a } ) = \\bigcap _ { n \\geq 0 } ( \\phi _ { a [ o , n ] } ^ { + } ) ^ { - 1 } ( R _ { a _ n } ) = \\bigcap _ { n \\geq 0 } V _ { a [ 0 , n ] } \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} & a \\circ b = a _ 1 ( a _ 2 \\rightharpoonup b ) , \\\\ & a b = a _ 1 \\circ ( T ( a _ 2 ) \\rightharpoonup b ) \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} \\int _ M | \\nabla \\Delta u | _ g ^ 2 d \\mu _ g = \\int _ M | \\Delta \\nabla _ j u - R _ j ^ k \\nabla _ k u | _ g ^ 2 d \\mu _ g . \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } q ^ { \\alpha ^ { 2 } - 2 \\nu \\alpha } \\left ( i q ^ { n / 2 } \\right ) ^ { - \\alpha } S _ { n } \\left ( - q ^ { 2 \\nu - \\alpha } ; q \\right ) J _ { \\alpha } ^ { ( 2 ) } \\left ( 2 i q ^ { n / 2 } ; q \\right ) d \\alpha = \\sqrt { \\frac { \\pi } { \\log q ^ { - 1 } } } \\frac { q ^ { - \\nu ^ { 2 } } } { \\left ( q ; q \\right ) _ { \\infty } \\left ( q ; q \\right ) _ { n } } . \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\ell = \\prod _ { p \\in S _ { } } p . \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} g ' ( u ) & = 2 u Q ( u ) - \\psi ( u ) \\\\ & > \\frac { u ^ 2 - 1 } { u ^ 2 + 1 } \\psi ( u ) \\\\ & \\geq 0 \\ \\ , \\ \\ \\mbox { f o r } u \\geq 1 \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} y ' = \\frac { ( 1 - a ) } { a } x ^ { - a } - \\frac { 2 } { x } \\log x \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} \\Delta _ 0 u _ \\epsilon = & - ( n - 6 ) \\frac { u _ \\epsilon } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 2 } ( n \\epsilon ^ 2 + 4 r ^ 2 ) , \\\\ ( \\Delta _ 0 u _ \\epsilon ) ' = & ( n - 6 ) ( n - 4 ) \\frac { u _ \\epsilon r } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 3 } \\Big [ ( n + 2 ) \\epsilon ^ 2 + 4 r ^ 2 \\Big ] . \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{gather*} R = \\sum _ { i = 1 } ^ { m } \\frac { W _ { i } } { w _ { i } - z } , \\end{gather*}"}
-{"id": "7652.png", "formula": "\\begin{align*} y _ { 1 , k } = { - T + 2 k \\over 2 + \\gamma } + { \\gamma \\over 2 + \\gamma } y _ 0 \\end{align*}"}
-{"id": "9962.png", "formula": "\\begin{align*} r a n k ( H H ^ \\dagger ) & = r a n k ( \\overline { K } \\ , \\overline { K } ^ \\dagger ) \\\\ & = r a n k ( K { K } ^ \\dagger ) \\\\ & = n - k - m \\\\ & = n - k - \\dim ( H u l l _ h ( C ) ) \\\\ & = n - k - \\dim ( H u l l _ h ( C ^ { \\bot h } ) ) \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} \\partial \\varphi ( \\sigma ) = \\left \\{ \\varepsilon \\in X \\ , | \\ , \\varphi ( \\xi ) \\geq \\varphi ( \\sigma ) + \\varepsilon \\cdot ( \\xi - \\sigma ) \\quad \\forall \\ , \\xi \\in X \\right \\} . \\end{align*}"}
-{"id": "9957.png", "formula": "\\begin{align*} \\{ u ( \\varphi ( s ) ) : s \\in I \\} \\subset P ^ - ( A ) F \\xi ( s _ 0 ) \\cap U ^ + ( A ) = P ^ - ( A ) ( F \\cap H ) \\xi ( s _ 0 ) \\cap U ^ + ( A ) . \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} \\mathcal { S } _ \\psi : = \\{ v \\in \\mathcal { S } : v = \\hat v \\ , \\ , \\textrm { e v e r y w h e r e i n } \\ , \\Omega _ T , v \\geq \\psi \\ , \\ , \\textrm { a l m o s t e v e r y w h e r e i n } \\ , \\Omega _ T \\} . \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} \\begin{array} { l } \\kappa ( \\mu _ 0 , k _ 0 ) = P ( X _ 0 ) \\ , . \\end{array} \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} \\chi ^ * = \\chi _ { \\bar U } = \\chi _ { F ^ { - 1 } U F } = \\chi . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } \\dfrac { 1 } { 2 ( 1 + \\xi x ) ^ { 1 / 2 } } = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} Z _ s : = Y _ s ^ { 0 , 0 , 0 } + \\int _ 0 ^ s \\psi _ 0 ( r , X _ r , 0 ) d r . \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} K = \\{ ( x , y , z ) : 0 < x , \\ ; 0 < y , \\ ; 0 < z < 1 , \\ ; x + y < 1 \\} \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} \\nabla _ \\alpha = - ( 1 - | z | ^ 2 ) \\frac { \\partial } { \\partial z } + \\alpha \\bar z \\mbox { a n d } \\nabla _ \\alpha ^ { * } = ( 1 - | z | ^ 2 ) \\frac { \\partial } { \\partial \\bar z } + ( \\alpha + 1 ) z , \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} f ( x , t ) = m ( x ) + r ( t ) , \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} ( \\tilde H _ \\lambda , \\tilde H _ \\lambda ) ^ { S _ { q , t } } = N _ \\lambda ( 1 ; q , t ) \\in \\Z [ q , t ] . \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} \\mathcal { A } : = \\left \\{ \\max _ { i , j } | \\widehat { R } _ { i j } - R _ { i j } | \\leq \\tau \\right \\} . \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} \\kappa _ { d } = \\begin{cases} 0 , & d = 1 ; \\\\ - d + \\gamma - 2 \\nu _ 0 + \\frac { 2 } { \\alpha } , & d \\geq 2 \\ , . \\end{cases} \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} \\frac { \\left ( q ^ { \\alpha + n + 1 } ; q \\right ) _ { \\infty } L _ { n } ^ { ( \\alpha ) } \\left ( x q ^ { - n - \\alpha } ; q \\right ) q ^ { n ^ { 2 } / 2 } } { \\left ( - x \\right ) ^ { n } } = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( q ^ { 1 / 2 } e ^ { - i y } / x ; q \\right ) _ { n } } { \\left ( q ; q \\right ) _ { n } } \\frac { \\exp \\left ( \\frac { y ^ { 2 } } { \\log q ^ { 2 } } + i n y \\right ) } { \\left ( - q ^ { \\alpha + 1 / 2 } e ^ { i y } ; q \\right ) _ { \\infty } } d y , \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} F _ { \\gamma ^ z } ( \\omega ( \\tau + \\cdot ) ) = F _ { \\gamma ^ z } ( \\omega ) ( \\tau + \\cdot ) - F _ { \\gamma ^ z } ( \\omega ) ( \\tau ) = W ( \\tau + \\cdot ) - W ( \\tau ) \\not = \\underline 0 , \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} \\lambda _ i ( \\tau _ { ( P - I ) y } ) = \\frac { \\lambda _ { \\sigma ( i ) } ( \\tau _ y ) } { \\lambda _ i ( \\tau _ y ) } , i = 1 , \\cdots , N , \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} X _ V = t U _ 0 V ^ { \\intercal } , V \\in \\mathbb { O } _ { p _ 2 , r } , \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\ell ( \\gamma ) : = \\int _ { a } ^ { b } \\sqrt { g _ { \\gamma ( t ) } ( \\dot { \\gamma } ( t ) , \\dot { \\gamma } ( t ) ) } d t . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} \\Psi = \\left [ \\begin{array} { c c } 2 & - ( \\alpha + \\beta ) \\\\ - ( \\alpha + \\beta ) & 2 \\alpha \\beta \\end{array} \\right ] \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} E = \\begin{pmatrix} C & C \\\\ C & C \\end{pmatrix} . \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\vartheta ( r ) = \\sinh r , \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( a , b ; q \\right ) _ { n } z ^ { n } } { \\left ( q , c ; q \\right ) _ { n } } = \\frac { \\left ( a z ; q \\right ) _ { \\infty } } { \\left ( z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { \\binom { n } { 2 } } \\left ( a , c / b ; q \\right ) _ { n } } { \\left ( q , c , a z ; q \\right ) _ { n } } \\left ( - b z \\right ) ^ { n } . \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} a _ { \\mu , N } ( m ) = \\sum _ { l = \\max \\{ [ \\frac { m - N } { \\mu } ] ' , 0 \\} } ^ { \\min \\{ [ \\frac { m } { \\mu } ] , n \\} } ( - 1 ) ^ l { n \\choose l } a ( m - \\mu l ) , \\quad 0 \\leq m \\leq N + \\mu n , \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} M ( L / K ) : = ( [ L : K ] ^ { 2 } n _ K ^ { 1 + \\omega ( D _ L ) } \\mathrm { r a d } ( D _ L ) ^ { 3 } ) ^ { n _ K } \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\chi _ n = \\omega _ n - \\sum _ { k = 0 } ^ n \\frac { 1 } { \\sqrt { 2 k + 1 } } = \\omega _ n - \\left ( 1 - \\frac { 1 } { \\sqrt { 2 } } \\right ) \\sum _ { k = 1 } ^ { n } \\frac { 1 } { \\sqrt { k } } - \\sum _ { k = n + 1 } ^ { 2 n + 1 } \\frac { 1 } { \\sqrt { k } } ; \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} y = \\sqrt { P } \\textbf { w } _ { M S } ^ H \\textbf { H } \\textbf { w } _ { B S } { s } \\ + \\ \\textbf { w } _ { M S } ^ H \\textbf { n } \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} - L u & = f _ 1 - f _ 2 - g \\circ u _ 1 + g \\circ u _ 2 \\ , \\ , \\ , \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ u & = \\eta _ 1 - \\eta _ 2 \\ , \\ , \\ , \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} \\chi _ e ( \\mathcal { C } ) = \\left \\{ \\begin{array} { l } 1 \\quad \\mathrm { i f \\ a r r o w \\ o n } \\ e \\ \\mathrm { p o i n t s \\ l e f t \\ o r \\ d o w n } \\\\ 0 \\quad \\mathrm { i f \\ a r r o w \\ o n } \\ e \\ \\mathrm { p o i n t s \\ r i g h t \\ o r \\ u p . } \\end{array} \\right . \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} r ^ 1 ( s , f ^ * , g ) = \\max _ { f \\in F _ S } r ^ 1 ( s , f , g ) = \\max _ { a ^ 1 \\in A ^ 1 ( s ) } [ R ^ 1 ( s ) g ( s ) ] _ { a ^ 1 } . \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} ( a \\nabla _ { \\nu } b ) ^ { 2 } - r _ { 0 } ^ { 2 } ( a - b ) ^ { 2 } & = a ^ { 2 } \\nabla _ { \\nu } b ^ { 2 } - r _ { 0 } ( a - b ) ^ { 2 } \\\\ & \\geq ( a \\sharp _ { \\nu } b ) ^ { 2 } + \\sum _ { k = 1 } ^ { \\infty } r _ { k } \\big [ a ^ { 1 - \\frac { m _ k } { 2 ^ k } } b ^ { \\frac { m _ k } { 2 ^ k } } - a ^ { 1 - \\frac { m _ k + 1 } { 2 ^ k } } b ^ { \\frac { m _ k + 1 } { 2 ^ k } } \\big ] ^ { 2 } \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} F _ G = \\frac { 1 } { v } J + E , \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} h ( t , z , v ) = \\tilde h ( t , x , y ) , \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} g ( u ) = \\begin{cases} 1 & \\mbox { i f $ d ( u ) \\le S ( 2 n - 1 ) $ } \\\\ 1 - ( d ( u ) - S ( 2 n - 1 ) h _ { 2 n } ^ { - 1 } ) _ + & \\mbox { i f $ d ( u ) > S ( 2 n - 1 ) $ } \\end{cases} \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} g \\le \\frac { 1 } { 1 - \\mu _ R } \\cdot \\min _ { \\{ a _ { r , t } \\} } \\sum _ { r = 0 } ^ { N _ R - 1 } \\sum _ { t = 1 } ^ { N _ T } \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } } { d _ { r , t } } a _ { r , t } , \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} | v _ { 1 2 } | = | v _ { 2 1 } | \\geq \\frac { 1 } { \\sqrt { 2 } } . \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c t + a ) ^ 2 + t ^ 2 } , a = c o n s t , \\ ; c = c o n s t \\neq 0 , \\ ; c ^ 2 \\neq \\kappa ^ 2 , \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} K _ { w _ 0 } ^ { \\alpha } ( w ) = \\frac { k _ { \\alpha } } { ( w - \\overline { w _ 0 } ) ^ { \\alpha + 2 } } , \\end{align*}"}
-{"id": "10174.png", "formula": "\\begin{align*} \\mathbb E [ \\max _ { k = 1 , . . . , n } \\Delta _ k ] = n ^ h \\mathbb E [ \\max _ { k = 1 , . . . , n } \\Delta _ { \\frac k n } ] \\sim n ^ h \\mathbb E [ \\sup _ { [ 0 , 1 ] } \\Delta ] \\ , . \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} P _ { \\mathcal { L } } ( n ) = \\frac { 3 e ( Y _ { n } ^ { \\mathcal { L } } ) - K _ { Y _ { n } ^ { \\mathcal { L } } } ^ { 2 } } { n ^ { d - 3 } } = n ^ { 2 } ( f _ { 0 } - d ) + 2 n ( d - f _ { 1 } + f _ { 0 } ) + 2 f _ { 1 } + f _ { 0 } - d - 4 t _ { 2 } \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} \\nabla b _ k & = \\nabla u _ { j _ k } - \\nabla g _ k = ( \\nabla u _ { j _ k } - v _ k ) + ( v _ k - \\rho _ { l _ k } v _ k ) + \\rho _ { l _ k } ( v _ k - \\nabla \\tilde { g } _ k ) - \\tilde { g } _ k \\otimes \\nabla \\rho _ { l _ k } \\\\ & = ( \\nabla u _ { j _ k } - v _ k ) + ( 1 - \\rho _ { l _ k } ) v _ k + \\rho _ { l _ k } \\tilde { \\sigma } _ k - \\tilde { g } _ k \\otimes \\nabla \\rho _ { l _ k } \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} A _ 0 = \\C [ a _ 0 ] \\qquad A _ 1 = \\C [ a _ 1 ] \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} W \\ = \\ C _ 0 \\cup \\bigcup _ { k \\in \\mathbb { N } } \\left ( I _ k \\cap ( \\inf ( I _ k ) + A _ k ) \\right ) \\ . \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} \\int _ { [ 0 , 1 ) ^ 3 } g ( x , y ) g ( y , z ) ~ d x d y d z = \\frac { 1 } { 4 } \\cdot \\frac { 1 } { 2 } ( ( 2 p _ d - 1 ) ^ 2 + 1 ) + \\frac { 3 } { 4 } \\cdot \\frac { 1 } { 6 } ( ( 2 p _ d - 1 ) ^ 2 + 4 ( 2 p _ d - 1 ) + 1 ) = p _ d ^ 2 . \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} N \\subseteq Q \\sqsubseteq R Q , R \\prec ( H _ \\theta , \\in ) \\ \\implies \\ \\ R \\cap ( Q \\cap \\omega _ 2 ) = Q \\cap \\omega _ 2 . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} & \\big ( z _ 0 + \\frac { z _ 1 + z _ 2 } { 2 } \\big ) ^ 2 - \\big ( \\frac { z _ 1 } { 2 } \\big ) ^ 2 - \\big ( \\frac { z _ 2 } { 2 } \\big ) ^ 2 - \\frac { z _ 1 z _ 2 } { 2 } \\cos \\theta \\\\ & = z _ 0 ^ 2 + z _ 0 ( z _ 1 + z _ 2 ) + \\frac { z _ 1 z _ 2 } { 2 } ( 1 - \\cos \\theta ) \\\\ & = z _ 0 ^ 2 + z _ 0 ( z _ 1 + z _ 2 ) + z _ 1 z _ 2 \\sin ^ 2 \\frac { \\theta } { 2 } = 0 , \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} 0 \\leqslant \\psi ' \\leqslant 3 , \\psi ( u ) = 0 \\quad u \\leqslant 1 / 4 , \\psi ( u ) = 1 \\quad u \\geqslant 3 / 4 . \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} \\otimes _ { t = 0 } ^ n \\big ( s _ t ( d y _ t | y ^ { t - 1 } , x ^ { t - 1 } ) \\otimes { r } _ t ( d x _ t | x ^ { t - 1 } , y ^ t ) \\big ) \\in { \\cal M } ( { { \\cal X } ^ { n } } \\times { \\cal Y } ^ { n } ) . \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} \\mathbb { V } : = \\{ \\mathbf { h } \\in \\mathbb { R } ^ p | \\| \\mathbf { h } _ { S ^ c } \\| _ 1 \\leq 3 \\| \\mathbf { h } _ S \\| _ 1 + 4 \\| \\boldsymbol { \\theta } _ { S ^ c } \\| _ 1 \\} . \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} \\sigma _ E ^ { m ( x ) } ( \\kappa ( \\sigma _ E ( x ) ) ) = \\sigma _ E ^ { l ( x ) } ( \\kappa ( x ) ) \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} \\widetilde { \\omega } \\big | _ { Y _ { [ 0 , R ] } } = e ^ { - i \\lambda u _ 1 } \\phi + e ^ { i \\lambda u _ 1 } C _ 1 ( \\lambda ) \\phi ' + \\widetilde { \\omega } ^ \\mathrm { n z } . \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\mathcal { T } _ 0 = \\iint \\Big ( A ( - \\partial _ v \\partial _ v ^ { - 2 } f _ 0 \\ , \\partial _ z \\ne { f } ) + \\partial _ v \\partial _ v ^ { - 2 } f _ 0 \\ , \\partial _ z A \\ne { f } \\Big ) A \\ne { f } \\ , d V d t . \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } { X } _ s ^ { t , x } = \\ x + \\int _ t ^ s \\mathrm d W _ r , \\\\ { Y } _ s ^ { t , x } = \\ \\Phi ( { X } _ T ^ { t , x } ) - \\int _ s ^ T { Z } _ r ^ { t , x } \\mathrm d { W } _ r + \\int _ s ^ T f ( r , { X } _ r ^ { t , x } , { Y } _ r ^ { t , x } , { Z } _ r ^ { t , x } ) \\mathrm d r \\\\ + \\int ^ T _ s { { Z } _ r ^ { t , x } b ( r , { X } _ r ^ { t , x } ) } \\mathrm { d } r , \\\\ { \\forall } s \\in [ t , T ] , \\end{array} \\right . \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} f ( x ) & = \\sum _ { n = k } ^ { \\infty } 2 ^ { - n } ( c _ n ) f _ n ( x ) \\\\ & \\leq c _ k \\ \\sum _ { n = k } ^ { \\infty } 2 ^ { - n } f _ n ( x ) \\\\ & \\leq c _ k 2 ^ { - k + 1 } \\leq c _ k < \\norm { u ( x ) } , \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} \\sigma = ( x _ 1 ^ 2 + x _ 2 ^ 2 + t ) ^ { \\beta / 2 } , 0 < \\beta < 1 , \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} a _ { i , j + 1 } a _ { i + 1 , j } - a _ { i j } a _ { i + 1 , j + 1 } = 1 . \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { k z } { i } \\right ) ^ i = 1 + \\frac { i } { i } k z + \\frac { i ( i - 1 ) } { i \\cdot 2 i } k ^ 2 z ^ 2 + \\frac { i ( i - 1 ) ( i - 2 ) } { i \\cdot 2 i \\cdot 3 i } k ^ 3 z ^ 3 + \\ldots \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} \\max _ { k = 1 , \\ldots , K } w _ { k } ^ { 2 } \\leq \\frac { 1 } { d } \\sum _ { k = 1 } ^ { K } w _ k ^ { 2 } L _ { k } . \\end{align*}"}
-{"id": "9998.png", "formula": "\\begin{align*} F ( x ) = y , \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} h ( X _ { i } , X _ { j } ) = \\overset { n - 2 } { \\underset { k = 1 } { \\sum } } L _ { i j } ^ { k } N _ { k } , \\ \\ \\ \\ \\ 1 \\leq i , j \\leq 2 \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} { } \\tau _ 1 ( T ) = K _ { \\pi _ 7 ( [ C r , \\chi ] ^ { F N } ) } \\mbox { a n d } \\tau _ 2 ( T ) = L _ { [ \\chi , \\chi ] ^ { F N } } , \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} H _ k ( L _ { 2 n - 1 } ( m ) ) = \\left \\{ \\begin{array} { l l } \\Z / m , & 1 \\le j \\le 2 n - 3 \\ ; \\\\ \\Z , & k = 0 , 2 n - 1 \\\\ 0 , & \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\delta ( \\mu , r ) = 1 + \\frac { K } { M r } \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} w ' = w _ 0 w _ 1 \\ldots w _ k \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} ( P _ \\lambda ) = ( T ) + \\bigoplus _ { \\langle \\alpha , \\lambda \\rangle \\ge 0 } ( G ) _ \\alpha . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\mathcal { A } _ { n , R } = \\sum _ { k = 0 } ^ n C _ d ^ k \\ell ^ { - k } R ^ { k ( d + 1 ) } n ^ k e ^ { - \\mu _ 0 k } T _ L ^ k \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} \\mathcal { E } = \\{ E \\in \\mathcal { B } : E \\cap C _ m \\in \\{ \\emptyset , C _ m \\} \\} . \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} { \\mathcal L } _ { \\theta ' } ( z ' ) & = 4 z _ 1 ' z _ 2 ' \\frac { { \\mathcal L } _ { \\theta } ( z ) { \\mathcal L } _ { - \\theta } ( z ) } { 4 z _ 1 ^ 2 z _ 2 ^ 2 } \\\\ & = ( z _ 0 ^ 2 - z _ 2 ^ 2 ) { \\mathcal L } _ { \\theta } ( z ) { \\mathcal L } _ { - \\theta } ( z ) . \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} \\int _ { \\vert y \\vert \\geq m \\tau ^ { 1 / \\beta } } \\psi ( T , x - y ) d y \\leq \\int _ { \\vert z \\vert \\geq \\frac m 2 \\tau ^ { 1 / \\beta } } \\psi ( T , z ) d z = e ^ { - T } \\sum _ { k = 1 } ^ \\infty \\frac { T ^ { k } } { k ! } \\int _ { \\vert z \\vert \\geq \\frac m 2 \\tau ^ { 1 / \\beta } } J ^ { * ( k ) } ( z ) d z . \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} S _ 2 ( l , u , v ; p ) = 2 \\pi i ^ { 2 k } \\widetilde { V } _ { p } ( u , v , k ) , \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} f _ { \\varepsilon } = q ( x _ { 2 } , x _ { 3 } ) + \\varepsilon \\ell _ { 0 } ( x _ { 2 } a _ { 0 , 2 } + x _ { 3 } a _ { 0 , 3 } ) + \\varepsilon \\ell _ { 1 } ( x _ { 2 } a _ { 1 , 2 } + x _ { 3 } a _ { 1 , 3 } ) + \\sum _ { i = 4 } ^ { r } \\varepsilon \\ell _ { i } g _ { i } . \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} Z = Z _ { 1 1 } + Z _ { 1 2 } + Z _ { 2 1 } + Z _ { 2 2 } , \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} T _ \\psi : = P _ \\psi J \\ , , \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} \\alpha [ ( r - 3 ) \\binom { m _ 1 } { 2 } - m _ 1 r + r - 2 ] + ( \\varepsilon _ 1 + 1 ) [ ( r - 3 ) m _ 1 - r ] + 4 + \\mu _ 1 ( r - 3 ) > 0 \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ d } \\ , \\epsilon ^ d \\ , \\bigg | \\sum _ { i = 1 } ^ { N } \\ , \\hat { F _ i } ( \\epsilon \\lambda ) \\ , e ^ { - 2 \\pi i x _ i \\cdot \\lambda } \\bigg | ^ 2 \\ , d \\mu ( \\lambda ) \\geq \\left ( G ( \\delta , \\epsilon ) - I ( \\delta , \\epsilon ) \\right ) ^ 2 \\\\ & \\ge \\left ( \\sqrt { \\mathcal { D } ^ { - } _ N ( \\check { \\mu } ) - \\rho ' } - \\sqrt { \\rho ' } \\right ) ^ 2 \\ , \\| f \\| _ 2 ^ 2 \\geq ( \\mathcal { D } ^ { - } _ N ( \\check { \\mu } ) - \\rho ) \\ , \\| f \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} ( 2 T _ n ( x / 2 ) , 2 T _ m ( x / 2 ) ) = 2 T _ { ( n , m ) } ( x / 2 ) \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} P _ { T o t } ^ { A B F } = N _ { M S } ( P _ { L N A } + P _ { P S } ) + P _ C + P _ { R F } + 2 P _ { A D C } \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} \\varphi _ 0 = \\psi , \\qquad \\varphi _ { k + 1 } = \\max \\{ \\varphi _ k , v _ k \\} , \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } \\frac { L ^ { \\leftarrow } ( t a ) } { L ^ { \\leftarrow } ( t b ) } = 0 . \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} \\lambda ^ { - { 1 \\over \\alpha } } Y \\circ L ^ { - 1 } ( \\lambda \\cdot ) \\buildrel d \\over = Y \\circ L ^ { - 1 } ( \\cdot ) , \\lambda > 0 . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = \\omega + \\xi ( y , \\sigma , \\mu ) + f ( x , y , \\sigma , \\mu ) , \\\\ \\dot { y } & = \\sigma + \\eta ( y , \\sigma , \\mu ) + g ( x , y , \\sigma , \\mu ) \\end{aligned} \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} \\lim _ { a \\to 0 , \\ ; b / \\sqrt { a } \\to k } Q _ n \\left ( \\frac { x } { a } \\right ) = ( - 1 ) ^ n \\frac { ( \\mu + \\nu + 1 ) _ n } { \\Gamma ( \\mu + 1 ) } { \\ ; } _ 1 F _ 2 \\left ( { - n \\atop \\mu + \\nu + 1 , \\mu + 1 } \\Big { | } k ^ 2 x \\right ) x ^ \\mu , \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} \\psi ( i , \\alpha ) = ( \\phi _ { \\alpha } ( i ) , \\sigma _ i ( \\alpha ) ) \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} - f \\bullet \\mu = - \\sum ^ { m } _ { i = 1 } ( - 1 ) ^ { ( i - 1 ) \\cdot 1 } f \\bullet _ i \\mu = - \\sum ^ { m } _ { i = 1 } ( - 1 ) ^ { i - 1 } f \\circ \\alpha ^ { i - 1 , 1 } _ m \\circ \\mu ^ { i - 1 } _ { m + 1 } \\circ \\alpha ^ { i - 1 , 2 } _ { m + 1 } \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} D _ { t } ^ { \\alpha } \\varphi : = \\left ( \\frac { d } { d t } \\right ) ^ { n } \\left ( I _ { t } ^ { n - \\alpha } \\varphi \\right ) , \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} | d e t _ { t r } x | = \\det x . \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} \\omega _ { \\mu + 1 , a } ( x ) & = x \\omega _ { \\mu - 1 , a } ( x ) - \\frac { \\mu } { a } \\omega _ { \\mu , a } ( x ) , \\\\ \\omega _ { \\mu + 1 , a } ' ( x ) & = a \\omega _ { \\mu , a } ( x ) , \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} T _ { ( A , j + n + i + n \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } & = - T _ { ( A , j + n + i \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } , \\\\ T _ { ( A , j + i \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } & = - T _ { ( A , ( j + i ) + n \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } , \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} | \\psi ' | ^ 2 = | \\varphi ' | ^ 2 + | \\omega _ 1 ' | ^ 2 \\ , . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} | S _ { m , N } | ^ { \\frac { 1 } { q } - \\frac { 1 } { p } } \\gamma & \\left | \\bigcap _ { i = M } ^ { \\infty } \\{ k \\in S _ { m , N } : | x ^ { ( i ) } _ { k } - x ^ { ( j ) } _ { k } | > \\gamma \\} \\right | ^ { \\frac { 1 } { p } } \\\\ & \\leq | S _ { m , N } | ^ { \\frac { 1 } { q } - \\frac { 1 } { p } } \\gamma \\left | \\{ k \\in S _ { m , N } : | x ^ { ( M ) } _ { k } - x ^ { ( j ) } _ { k } | > \\gamma \\} \\right | ^ { \\frac { 1 } { p } } \\\\ & \\leq | | x ^ { ( M ) } - x ^ { ( j ) } | | _ { w \\ell _ { q } ^ { p } } . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} \\dim \\mathcal { S D E } _ k = 4 + 3 \\binom { k + 4 } { 4 } - 3 \\binom { k + 2 } { 4 } = k ^ 3 + \\frac { 9 } { 2 } k ^ 2 + \\frac { 1 3 } { 2 } k + 7 . \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} A _ { N _ { k } } X _ { j } = - ( \\widetilde { \\nabla } _ { X _ { j } } N _ { k } ) ^ { T } , X _ { j } \\in \\chi ( M ) . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} f ( v _ 0 , v _ 1 , v _ 2 , 1 ) = \\frac { 1 } { x _ 3 ^ 4 } f ( x _ 0 , x _ 1 , x _ 2 , x _ 3 ) . \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} q _ { a _ { i } - \\delta + \\left ( n _ { 1 } - n _ { 2 } \\right ) v _ { k } } + { \\textstyle \\sum \\limits _ { m = 1 } ^ { n _ { 1 } - n _ { 2 } - 1 } } \\left ( { \\textstyle \\sum \\limits _ { u \\in \\Gamma ( k ) } } c ( a _ { i } - u , n _ { 1 } - m ) q _ { u + m v _ { k } } \\right ) = 0 , \\forall i = 1 , 2 , . . . , s , \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} \\sum _ { k = - \\infty } ^ { \\infty } \\varphi _ { k } \\left ( \\omega q ^ { 2 m } \\right ) \\varphi _ { k } \\left ( - \\omega ^ { - 1 } q ^ { 2 n } \\right ) = 0 \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} X _ P \\ ; : \\ ; \\begin{cases} F ( x _ 0 , \\dotsc , x _ n ) = 0 , \\\\ \\nabla { F } ( P ) \\cdot ( x _ 0 , \\dotsc , x _ n ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} \\tau = \\sum _ { r = 0 } ^ { N _ R - 1 } \\sum _ { t = 1 } ^ { N _ T } \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t } } { d _ { r , t } } a _ { r , t } . \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} A _ { q } \\left ( z \\right ) = \\left ( a z q ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } S _ { n } \\left ( a q ^ { - n } ; q \\right ) } { \\left ( a z q ; q \\right ) _ { n } } \\left ( - z \\right ) ^ { n } . \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} \\Omega = ( 0 , \\pi ) \\times ( 0 , \\pi ) , \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} c _ 1 ( \\pi ^ { ( t ) } ) + \\dots + c _ { l ' } ( \\pi ^ { ( t ) } ) + & ( d - v ) = \\\\ & c _ 1 ( \\pi ^ { ( 0 ) } ) + \\dots + c _ { l ' + 1 } ( \\pi ^ { ( 0 ) } ) . \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} f : ( x , n ) \\mapsto ( x ^ * , n ^ * ) = ( ( x + n ) / \\sqrt { 2 } , ( x - n ) / \\sqrt { 2 } ) \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} a ^ * _ { 0 , 1 } = 1 / N _ T , \\textrm { a n d o t h e r s b e i n g 0 . } \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} J _ s ( x , y ) = \\frac { \\sqrt { x } J _ { s + 1 } ( \\sqrt { x } ) J _ s ( \\sqrt { y } ) - \\sqrt { y } J _ { s + 1 } ( \\sqrt { y } ) J _ s ( \\sqrt { x } ) } { 2 ( x - y ) } \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\tilde { a } ^ { i j } ( y ) = \\tilde { a } ^ { i j } ( y _ 0 ) + E _ 2 ^ { y _ 0 , i j } ( y ) , \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} Y \\ ; : \\ ; C ( x _ 1 , \\dotsc , x _ k ) = 0 , Q ( x _ 1 , \\dotsc , x _ k ) = 0 \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} \\langle f _ 0 , \\beta ^ R ( \\lambda ) \\beta ^ R ( \\nu ) f _ 1 \\rangle & = \\iint \\langle f _ 0 ( \\omega _ 0 ) , \\beta ( x ) ( \\beta ^ R ( \\nu ) f _ 1 ( \\omega ) ) \\rangle d \\lambda ^ \\omega ( \\omega _ 0 , x ) d \\omega \\\\ & = \\iiint \\langle f _ 0 ( \\omega _ 0 ) , \\beta ( x ) \\beta ( y ) f _ 1 ( \\omega _ 1 ) ) \\rangle d \\nu _ \\omega ( y , \\omega _ 1 ) d \\lambda ^ \\omega ( \\omega _ 0 , x ) d \\omega \\\\ & = \\langle f _ 0 , \\beta ^ R ( \\lambda \\nu ) f _ 1 \\rangle . \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} d ( \\Lambda _ 1 , \\Lambda _ 2 ) \\ , : = \\ , d ^ { } _ { \\mathrm { H } } \\bigl ( j ^ { - 1 } ( \\Lambda _ 1 \\cup \\{ \\infty \\} ) , j ^ { - 1 } ( \\Lambda _ 2 \\cup \\{ \\infty \\} ) \\bigr ) , \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r r r r } \\partial _ z v = 0 , w = 0 , & \\partial _ z \\tau + \\alpha \\tau = 0 , & \\partial _ z \\sigma = 0 & \\hbox { o n } \\Gamma _ u \\times ( 0 , \\infty ) , \\\\ v = 0 , w = 0 , & \\partial _ z \\tau = 0 , & \\partial _ z \\sigma = 0 & \\hbox { o n } \\Gamma _ b \\times ( 0 , \\infty ) , \\\\ v , \\pi , \\tau , \\sigma & \\hbox { a r e p e r i o d i c } & & \\hbox { o n } \\Gamma _ l \\times ( 0 , \\infty ) , \\end{array} \\right . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} \\bigoplus _ { k = 1 } ^ s H ^ j ( Z _ k , \\Z ) \\otimes \\Z / m \\to \\bigoplus _ { k = 1 } ^ s H ^ j ( Z _ k , \\Z ) \\otimes \\Z / m . \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} \\frac { \\partial M _ { p q } } { \\partial x _ { q } } = \\mathcal { T } _ { p } + \\frac { \\partial \\mathcal { L } _ { p } } { \\partial t } , \\qquad \\overrightarrow { \\mathcal { L } } = \\mathbf { r \\times } \\overrightarrow { \\mathcal { G } } , \\quad \\overrightarrow { \\mathcal { T } } = \\mathbf { r \\times } \\overrightarrow { \\mathcal { F } } , \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} \\lim \\limits _ { t \\rightarrow \\infty } t \\Psi ' ( t ) \\ = \\ \\infty . \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} \\mathtt { M } = \\left ( \\begin{array} { c | c | c | c | c } 2 E _ 1 & B _ { 1 2 } & B _ { 1 3 } & \\dots & B _ { 1 n } \\\\ \\hline B _ { 2 1 } & 2 E _ 2 & B _ { 2 3 } & \\dots & B _ { 2 n } \\\\ \\hline B _ { 3 1 } & B _ { 3 2 } & 2 E _ 3 & \\dots & B _ { 3 n } \\\\ \\hline \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\hline B _ { n 1 } & B _ { n 2 } & B _ { n 3 } & \\dots & 2 E _ n \\end{array} \\right ) , \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} L _ { t } ( 0 ) e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } = \\mid \\gamma + t \\mid ^ { 2 } e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } . \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} L _ { d , \\xi } \\phi : = \\phi - \\Pi _ { d , \\xi } ^ { \\perp } i ^ { \\ast } \\left [ f ^ { \\prime } _ \\epsilon ( P U _ { \\delta , \\xi } ) \\phi \\right ] . \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} X ( k ) = \\sum \\limits _ { j = 0 } ^ { m - 1 } { x _ j e ^ { \\frac { { - i 2 \\pi j k } } { l } } } \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\sinh ^ 2 \\tfrac { 1 } { 2 } d _ { \\eta \\xi \\eta ^ { - 1 } } z = \\sinh ^ 2 ( \\tfrac { 1 } { 2 } T _ { \\xi } ) \\cosh ^ 2 d _ z \\mathcal { B } + \\sin ^ 2 \\theta \\sinh ^ 2 d _ z \\mathcal { B } . \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} x \\sim y \\Leftrightarrow \\exists m , n \\ge 0 : \\ T ^ m x = T ^ n y . \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} \\exp ^ x _ p ( q ) = \\exp ^ x ( p + q ) . \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} & \\tau ^ * ( \\mu _ R , \\mu _ T ) \\\\ \\le & \\tau _ U ( \\mu _ R , \\mu _ T ) \\\\ \\le & \\tau _ U ( \\mu _ R , 1 / N _ T ) \\\\ \\le & \\tau _ U ( 1 , 1 / N _ T ) + \\frac { \\tau _ U ( \\mu _ R ^ 0 , 1 / N _ T ) - \\tau _ U ( 1 , 1 / N _ T ) } { 1 - \\mu _ R ^ 0 } ( 1 - \\mu _ R ) \\\\ = & \\frac { \\tau _ U ( \\mu _ R ^ 0 , 1 / N _ T ) } { 1 - \\mu _ R ^ 0 } ( 1 - \\mu _ R ) . \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} \\lambda _ P ( X ) & = r ( X ) + r ( E - X ) - r ( E ) \\\\ & = \\lambda ( X ) + | | X | | _ \\lambda + \\lambda ( E - X ) + | | E - X | | _ \\lambda - | | E | | _ \\lambda \\\\ & = \\lambda ( X ) + \\lambda ( E - X ) \\\\ & = 2 \\lambda ( X ) . \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} { } _ { 1 } \\tilde { \\phi } _ { 1 } ( 0 ; - \\xi q ^ { - n } ; q , - \\xi q ^ { - n } ) = \\xi ^ { n } q ^ { - \\frac { 1 } { 2 } n ( n + 1 ) } \\left [ \\frac { \\theta _ { q } \\left ( - \\xi \\right ) } { \\left ( q ; q \\right ) _ { \\infty } } n + O ( 1 ) \\right ] \\ ! . \\end{align*}"}
-{"id": "9965.png", "formula": "\\begin{align*} g _ k g _ { j + 1 } ^ \\dagger = \\sum _ { t = 0 } ^ { r - 1 } \\left ( \\beta _ t ^ { k - 1 + q j } \\sum _ { m = 0 } ^ { q - 2 } \\omega ^ { m ( k - 1 + q j ) } \\right ) = \\sum _ { t = 0 } ^ { r - 1 } \\left ( \\beta _ t ^ { k - 1 + q j } \\cdot 0 \\right ) = 0 , \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} d _ \\lambda ( x , \\xi ) = ( \\delta _ \\lambda ( x ) , \\delta ^ * _ \\lambda ( \\xi ) ) , \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} \\frac { 1 } { \\prod _ { j = 1 } ^ { n } \\tau _ { j } ^ { 2 } \\left ( \\zeta , \\delta _ { \\Omega } ( \\zeta ) \\right ) } \\tau _ { m } \\left ( \\zeta , \\delta _ { \\Omega } ( \\zeta ) \\right ) \\tau _ { l } \\left ( \\zeta , \\delta _ { \\Omega } ( \\zeta ) \\right ) \\frac { 1 } { \\left | z - \\zeta \\right | } . \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} v ^ { k + 1 } ( \\ell ) - [ y ^ { k + 1 } ] _ \\ell \\mathbf { 1 } = \\left ( W ( k ) - \\frac { 1 } { N } \\mathbf { 1 } \\mathbf { 1 } ^ T W ( k ) \\right ) v ^ k ( \\ell ) + \\left ( \\mathbb { I } - \\frac { 1 } { N } \\mathbf { 1 } \\mathbf { 1 } ^ T \\right ) \\zeta ^ { k + 1 } ( \\ell ) , \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} \\dot { v } & = \\frac { X } { m } - g \\sin ( \\theta ) - q w , & \\dot { w } & = \\frac { Z } { m } + g \\cos ( \\theta ) + q v , \\\\ \\dot { x } & = w \\sin ( \\theta ) + v \\cos ( \\theta ) , & \\dot { z } & = - v \\sin ( \\theta ) + w \\cos ( \\theta ) , \\\\ \\dot { \\theta } & = q , & \\dot { q } & = \\frac { M } { I _ { y y } } , \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} T _ { ( k ) } \\stackrel { d } { = } \\sum _ { j = n - k + 1 } ^ { n } Y _ { j } , \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\left \\Vert \\phi ^ { \\prime } ( u ) \\right \\Vert ^ { 2 } = 1 - \\frac { \\lambda ^ { 2 } } { c ^ { 2 } } \\sin ^ { 2 } \\left ( \\frac { u } { c } \\right ) . \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} \\sup _ { \\{ p ( x _ t | s _ { t - 1 } , q _ { t - 1 } ) \\} _ { t \\geq 1 } } \\liminf _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N I ( X _ i , S _ { i - 1 } ; Y _ i | Q _ { i - 1 } ) & = \\sup _ { P _ { X | S , Q } } \\liminf _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N I ( X _ i , S _ { i - 1 } ; Y _ i | Q _ { i - 1 } ) , \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} \\Psi ' ( o ) = \\sum _ { x \\in \\eta } l ' ( | x | ) \\leq \\inf _ { y \\in B ( o , 1 ) } \\sum _ { x \\in \\eta } l ( | x - y | ) = \\inf _ { y \\in B ( o , 1 ) } \\Psi ( y ) . \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} \\left [ \\bar { P } _ { a } , \\bar { P } _ { b } \\right ] & = \\frac { 1 } { 2 } \\left [ P _ { a } + Z _ { a } , P _ { b } + Z _ { b } \\right ] , \\\\ & = \\frac { 1 } { 2 } \\left ( \\left [ P _ { a } , P _ { b } \\right ] + \\left [ P _ { a } , Z _ { b } \\right ] + \\left [ Z _ { a } , P _ { b } \\right ] + \\left [ Z _ { a } , Z _ { b } \\right ] \\right ) , \\\\ & = \\frac { 1 } { 2 } \\left ( J _ { a b } + Z _ { a b } - Z _ { b a } - J _ { a b } \\right ) , \\\\ & = Z _ { a b } , \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} F _ 1 ^ { ( \\lambda ) } \\cup F _ 2 ^ { ( \\lambda ) } & = \\bigcap _ { \\beta < \\lambda } F _ 1 ^ { ( \\beta ) } \\cup \\bigcap _ { \\beta < \\lambda } F _ 2 ^ { ( \\beta ) } \\\\ & \\subset \\bigcap _ { \\beta < \\lambda } ( F _ 1 ^ { ( \\beta ) } \\cup F _ 2 ^ { ( \\beta ) } ) \\\\ & = \\bigcap _ { \\beta < \\lambda } ( F _ 1 \\cup F _ 2 ) ^ { ( \\beta ) } \\\\ & = ( F _ 1 \\cup F _ 2 ) ^ { ( \\lambda ) } . \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} \\begin{cases} \\begin{array} { l l } \\dot { v } = \\log \\det ( v _ { \\alpha \\bar { \\beta } } ) - n \\log ( A t + 1 ) + f ( z , \\frac { \\log ( A t + 1 ) } { A } ) \\ ; \\ ; & \\mbox { o n } \\ ; \\Omega \\times ( 0 , \\frac { e ^ { A T } - 1 } { A } ) , \\\\ v ( z , 0 ) = u _ 0 ( z ) & \\mbox { o n } \\ ; \\Omega , \\\\ v ( z , t ) = \\varphi ( z , \\frac { \\log ( A t + 1 ) } { A } ) & \\mbox { o n } \\ ; \\partial \\Omega \\times [ 0 , T ) . \\end{array} \\end{cases} \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\bar { V } = \\max _ { i } \\left \\lbrace \\dfrac { V ( x ) } { ( \\gamma _ i - R _ i ( x ) ) } \\right \\rbrace \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} [ A , B ] _ t = ( t a _ 1 - ( 1 - t ) b _ 2 , t a _ 2 - ( 1 - t ) b _ 1 ) \\subset E _ 0 \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} S _ { n , p } : = \\frac { 1 } { n ^ p } \\sum \\limits _ { k = 0 } ^ n k ^ p f _ n ( k ) = \\sum \\limits _ { k = 0 } ^ n \\frac { k ^ p } { n ^ p } f _ n ( k ) , \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\left ( a ; q \\right ) _ { n } S _ { n } \\left ( x ; q \\right ) z ^ { n } = \\frac { \\left ( a z ; q \\right ) _ { \\infty } } { \\left ( z ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } \\left ( a ; q \\right ) _ { n } \\left ( - x z \\right ) ^ { n } } { \\left ( q , a z ; q \\right ) _ { n } } . \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} \\tilde { u } _ { 0 } = u _ { 0 } \\left [ \\lambda \\delta + \\bar { \\lambda } \\left ( 1 - \\beta _ { 0 } x ^ { - 1 } \\right ) u _ { 0 } - \\bar { \\lambda } \\sum \\nolimits _ { \\nu = 0 } ^ { d - 2 } \\gamma _ { 1 } ^ { d - 1 - \\nu } \\left ( x ^ { - 1 } u _ { \\nu + 1 } \\right ) \\right ] ^ { - 1 } . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { c c c c } m _ { 1 } & 1 & & \\\\ & m _ { 2 } & 1 & \\\\ & & \\ddots & \\ddots \\end{array} \\right ) , \\ \\ L = \\left ( \\begin{array} { c c c c c } 1 & & & & \\\\ l _ { 1 1 } & 1 & & & \\\\ \\vdots & \\ddots & \\multicolumn { 1 } { l } { \\ddots } & \\multicolumn { 1 } { l } { } & \\\\ l _ { d 1 } & \\cdots & l _ { d d } & 1 & \\\\ 0 & \\ddots & & & \\ddots \\end{array} \\right ) . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} d _ 1 ( n { , } K { , } 1 { , } K ) { = } \\frac { n ^ 2 } { 1 { + } \\frac { n ( n { - } 1 ) } { d _ 2 ( K { , } 1 { , } K ) } } . \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} L _ { ( d _ 1 d _ 2 ) } = L _ 0 \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} g ( \\theta _ 0 , x _ 0 , t , \\tilde c _ t ) = \\alpha . \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} G _ { N , s } ^ { ( r _ 1 , \\dots , r _ s ) } > 0 , r _ j \\geq j , j = 1 , \\dots , s . \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} v _ A ^ * ( S _ { a _ i } S _ { a _ i } ^ * \\otimes p _ { \\bar { 0 } } ) v _ A & = v _ 1 ^ * ( S _ { a _ i } S _ { a _ i } ^ * \\otimes p _ { \\bar { 0 } } ) v _ 1 \\\\ & = ( P _ C \\otimes s _ { 1 _ 0 , 1 } ) ^ * ( S _ { a _ i } S _ { a _ i } ^ * \\otimes p _ { \\bar { 0 } } ) ( P _ C \\otimes s _ { 1 _ 0 , 1 } ) \\\\ & = S _ { a _ i } S _ { a _ i } ^ * \\otimes s _ { 1 , 1 _ 0 } p _ { \\bar { 0 } } s _ { 1 _ 0 , 1 } . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} \\begin{array} { l } \\kappa ( \\mu , k ) = \\{ \\emptyset \\} \\cup \\{ A \\subset X \\mid 0 \\in A { } \\\\ A \\setminus \\{ 0 \\} { } X \\setminus \\{ 0 \\} = X _ 0 \\} \\ , . \\end{array} \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} \\pi _ { \\Omega } ( y ) \\ ; = \\ ; y - \\bar { s } \\frac { \\nabla _ x \\psi ( y , \\bar { s } ) } { \\| \\nabla _ x \\psi ( y , \\bar { s } ) \\| _ 2 } , \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} F _ { t s } \\left ( { t , s } \\right ) & = \\frac { 1 } { { t s } } H _ { s t } \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) - \\frac { 1 } { t } H _ t \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) - \\frac { 1 } { s } H _ s \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) + H \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) \\\\ & = F _ { s t } \\left ( { t , s } \\right ) . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} f ( \\lambda ^ { a _ 0 } x _ 0 , \\ldots , \\lambda ^ { a _ n } x _ n ) = \\lambda ^ d f ( x _ 0 , \\ldots , x _ n ) . \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} \\bar c \\leq \\inf _ { \\delta \\in \\mathcal S ^ p ( J _ n ( \\lambda ) ) } \\| \\delta \\| _ { 2 , n } \\quad \\max _ { 1 \\leq j \\leq p } \\sqrt { \\frac { 1 } { n } \\sum _ { i = 1 } ^ n X _ { i j } ^ 2 } \\leq \\bar C , \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} a & = \\lceil N \\alpha \\rceil , & b & = \\lceil N \\beta \\rceil , & c & = \\lceil N \\gamma \\rceil , & r & = \\lceil N \\xi \\rceil , & s & = \\lfloor N \\eta \\rfloor , \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} | V _ { Z _ 2 } ( t , x ; \\xi ) | \\leq \\begin{cases} C t ^ { ( \\gamma - 1 ) \\frac \\alpha 2 + 1 } p ( t , x - \\xi ) \\ , , & \\ d = 1 \\ , ; \\\\ C t ^ { \\frac { \\nu _ 0 \\alpha } { 2 } + 1 - \\alpha } | x - \\xi | ^ { - d + \\gamma - \\nu _ 1 + 2 - \\nu _ 0 } p ( t , x - \\xi ) \\ , , & \\ d \\geq 2 \\\\ \\end{cases} \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} \\partial _ { t } u - \\mathrm { d i v } \\ , \\left ( g \\Big ( \\int _ { 0 } ^ { t } \\theta ( t - s ) \\big | \\nabla u ( s , \\cdot ) \\big | ^ { 2 } \\mathrm { d } s \\Big ) \\nabla u \\right ) & = 0 ( 0 , \\infty ) \\times G , \\\\ \\frac { \\partial u } { \\partial \\mathbf { n } } & = 0 ( 0 , \\infty ) \\times \\Gamma , \\\\ u ( 0 , \\cdot ) & = u ^ { 0 } \\Omega , \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} \\left [ T _ { A } , T _ { B } \\right ] = C _ { A B } ^ { \\ \\ \\ C } T _ { C } . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} \\partial _ t u = \\Delta u + u ^ { 1 + p } ( 1 - u ) , \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} T ( \\Gamma _ w ) \\subset \\{ y \\in \\R ^ { n + 1 } | \\ y _ n = y _ { n + 1 } = 0 \\} . \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} p _ n ( e ) \\approx 1 - \\left ( \\frac { \\left ( \\frac { 2 } { n } \\sum _ { i = 1 } ^ n \\rho ^ { ( 2 ) } \\left ( \\frac { r _ i ( e ) } { \\sigma _ n } \\right ) \\right ) ( s _ { \\rho , n } ( e ) - s _ { \\rho , n } ( e _ { \\text f } ) ) } { \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\rho ^ { ( 1 ) } \\left ( \\frac { r _ i ( e ) } { \\sigma _ n } \\right ) ^ 2 } , k - k ( e ) \\right ) \\ , . \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { 8 } f _ t ( p ' ) = ( q - 1 ) ( q ^ 3 + 1 ) r . \\end{align*}"}
-{"id": "9934.png", "formula": "\\begin{align*} \\psi _ i ( s ) = \\varphi ^ { - 1 } _ 1 ( s ) \\varphi _ i ( s ) \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} F _ \\gamma ( \\omega ( ( T _ 1 + \\cdot ) \\wedge T _ 2 ) ) ( t ) = F _ \\gamma ( \\omega ) ( ( T _ 1 + t ) \\wedge T _ 2 ) - F _ \\gamma ( \\omega ) ( T _ 1 ) . \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} u = \\triangle u = 0 \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\left ( \\frac { \\partial \\bar { \\textbf { g } } } { \\partial \\textbf { x } } \\right ) = \\frac { \\partial } { \\partial \\textbf { x } } \\left ( \\frac { \\partial \\bar { \\textbf { g } } } { \\partial t } \\right ) = - \\frac { e } { m } \\textbf { A } ' \\frac { \\partial \\bar { \\textbf { g } } } { \\partial \\textbf { x } } . \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} D ( G _ n ( W ) \\| G _ n ( W ' ) ) & \\le 8 \\int _ { \\ell \\in [ 0 , 1 ] ^ n } \\sum _ { i , j = 1 } ^ n ( W ( { \\ell _ i } , { \\ell _ j } ) - W ' ( { \\ell _ i } , { \\ell _ j } ) ) ^ 2 d \\ell \\\\ & = 8 \\sum _ { i , j = 1 } ^ n \\int _ { \\ell \\in [ 0 , 1 ] ^ n } ( W ( \\ell _ i , \\ell _ j ) - W ' ( \\ell _ i , \\ell _ j ) ) ^ 2 d \\ell \\\\ & \\le 8 n ^ 2 \\int _ { [ 0 , 1 ] ^ 2 } ( W ( x , y ) - W ' ( x , y ) ) ^ 2 d x d y \\\\ & = 8 n ^ 2 \\| W - W ' \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "9956.png", "formula": "\\begin{align*} \\xi ( s ) u ( \\varphi ( s ) ) \\xi ( s _ 0 ) ^ { - 1 } F = \\exp ( w ^ 0 ( s ) ) \\exp ( w ^ - ( s ) ) \\exp ( w ^ + ( s ) ) F , \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} ( x ^ 2 ) + 2 p = ( x ) + ( x ) + p + p = ( x ) + p + ( x ) + p \\geqslant 0 . \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} C ( \\Delta ( r + 2 ) ) - d - 1 + \\left \\lfloor \\frac { 1 7 s - 4 } { 6 } \\right \\rfloor = \\frac { 2 ^ { r + 2 } - 1 } { 3 } - ( 2 ^ { r + 2 } - 1 ) + \\frac { 1 7 \\cdot 2 ^ { r - 1 } - 4 } { 3 } = \\frac { 2 ^ { r - 1 } - 2 } { 3 } \\geq 0 \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} W ( u ) = W _ { \\chi } ( u ; \\tau ) : = \\sum _ { y \\leq \\N \\mathfrak { p } < u } \\frac { \\chi ( \\mathfrak { p } ) \\log \\N \\mathfrak { p } } { \\N \\mathfrak { p } ^ { 1 + i \\tau } } , \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} c = u a + 2 r , \\ \\ \\ \\ \\ \\ \\ d = u ^ 2 b - r u a - r ^ 2 . \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} \\partial _ t u = \\alpha \\partial _ { x } ^ 2 u + \\partial _ x ^ 4 u + u \\partial _ x u , \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} Q & = Q _ { R } ( s , y ) , R > 0 \\\\ & = ( s - R ^ { \\frac { 2 } { \\alpha } } / 2 , s + R ^ { \\frac { 2 } { \\alpha } } / 2 ) \\times ( y ^ { 1 } - R / 2 , y ^ { 1 } + R / 2 ) \\times \\cdots \\times ( y ^ { d } - R / 2 , y ^ { d } + R / 2 ) . \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} & ( p ^ { d e } - 1 ) a ^ { - 1 } ( a m A - ( a K _ { X ^ { d } / Y ^ { d } } + E ) ) + p ^ { d e } N \\\\ = & ( p ^ { d e } - 1 ) ( m - m _ 0 ) A + ( p ^ { d e } - 1 ) a ^ { - 1 } ( a m _ 0 A - ( a K _ { X ^ { d } / Y ^ { d } } + E ) ) + p ^ { d e } N \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} \\sigma _ 2 ^ { - 1 } \\sigma _ k = \\sigma _ 1 ^ { - 1 } \\sigma _ k \\sigma _ 2 ^ { - 1 } \\sigma _ 1 . \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c t + a ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} h _ k [ \\boldsymbol { t } ] & : = \\langle 1 \\mid e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast _ { k + 1 / 2 } \\mid 0 \\rangle = \\langle 1 \\mid e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast ( z ) \\mid 0 \\rangle \\mid _ { z ^ { k } } \\\\ & = \\left [ e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\langle 1 \\mid \\psi ^ \\ast ( z ) \\mid 0 \\rangle \\right ] \\mid _ { z ^ { k } } = \\left [ e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\right ] \\mid _ { z ^ { k } } . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} f ( z ) = \\left ( - 1 \\right ) ^ { q + 1 } \\overline { \\partial _ { z } } \\left ( \\int _ { \\Omega } f ( \\zeta ) \\wedge K _ { N } ^ { 1 } ( z , \\zeta ) \\right ) - \\int _ { \\Omega } f ( \\zeta ) \\wedge P _ { N } ( z , \\zeta ) , \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( \\partial ^ 2 _ t - \\Delta ) & u ( x , t ) = 0 , \\ , ( x , t ) \\in \\R ^ n \\times \\R \\ , , \\\\ u ( x , 0 ) & = f ( x ) \\ , , \\ ; \\partial _ t u ( x , 0 ) = g ( x ) \\ , . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} \\sinh ^ 2 \\tfrac { 1 } { 2 } d _ { \\eta \\xi \\eta ^ { - 1 } } z = \\sinh ^ 2 ( \\tfrac { 1 } { 2 } T _ { \\xi } ) \\cosh ^ 2 d _ z \\mathcal { B } + \\sin ^ 2 \\theta \\sinh ^ 2 d _ z \\mathcal { B } \\\\ \\leq \\left ( \\sinh ^ 2 ( \\tfrac { 1 } { 2 } T _ { \\xi } ) + \\sin ^ 2 \\theta \\right ) \\cosh ^ 2 d _ z \\mathcal { B } . \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} H _ f ( x ) = \\limsup _ { r \\to 0 } \\frac { L _ f ( x , r ) } { l _ f ( x , r ) } = \\lim _ { r \\to 0 } \\frac { L _ f ( x , r ) } { l _ f ( x , r ) } = \\liminf _ { r \\to 0 } \\frac { L _ f ( x , r ) } { l _ f ( x , r ) } = h _ f ( x ) . \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{gather*} \\pi \\big ( g _ { C , D , E } ^ { ( \\alpha , \\beta ) } \\big ) = \\begin{bmatrix} 1 & 0 & 0 \\\\ C ^ { ( \\alpha - \\beta ) } ( z ) & 1 & 0 \\\\ D ^ { ( \\alpha ) } ( z ) & E ^ { ( \\beta ) } ( z ) & 1 \\end{bmatrix} . \\end{gather*}"}
-{"id": "8593.png", "formula": "\\begin{align*} \\tilde U ( X ) & = - X ^ { 1 1 } - X ^ { 1 0 } - X ^ 9 + X ^ 8 - X ^ 7 + X ^ 6 - X ^ 5 - X ^ 4 + X ^ 3 + X ^ 2 + X - 1 , \\\\ \\tilde V ( X ) & = - 5 1 X ^ { 1 1 } + 3 9 X ^ { 1 0 } + 6 3 X ^ 9 + 6 0 X ^ 8 + 4 5 X ^ 7 + 6 2 X ^ 6 \\\\ & \\qquad \\qquad - 1 8 X ^ 5 - 4 7 X ^ 4 + 5 2 X ^ 3 - 2 4 X ^ 2 - 3 2 X - 1 9 . \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} \\int \\partial _ v \\partial _ v ^ { - 2 } f _ 0 \\partial _ z A \\ne { f } A \\ne { f } \\ , d V = 0 , \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} f ( x , t ) = m ( x ) + r ( t ) . \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} \\| \\sin \\Theta ( \\hat V , V ) \\| = \\sin ( \\cos ^ { - 1 } ( \\sigma _ r ) ) = \\sqrt { 1 - \\sigma _ r ^ 2 } = \\sqrt { 1 - \\sigma _ { \\min } ^ 2 ( V ^ { \\intercal } \\hat V ) } , \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} \\left \\Vert G _ \\infty \\right \\Vert = \\sup _ { t \\geq 0 } \\sup _ { s \\in \\mathbb { N } } \\sup _ { Z _ s \\in \\mathbb { R } ^ { 2 d s } } e ^ { \\mu ( t ) s } e ^ { \\beta ( t ) \\left ( E _ s ( Z _ s ) + I _ s ( Z _ s ) \\right ) } \\left | g _ \\infty ^ { ( s ) } ( t , Z _ s ) \\right | \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} ( - 1 ) ^ { k + 1 } \\frac { d ^ k } { d s ^ k } \\frac { L ' } { L } ( s , \\chi ) = \\sum _ { \\mathfrak { p } } \\sum _ { m = 1 } ^ { \\infty } ( \\log \\N \\mathfrak { p } ) \\chi ( \\mathfrak { p } ) \\frac { ( \\log \\N \\mathfrak { p } ^ m ) ^ k } { ( \\N \\mathfrak { p } ^ m ) ^ { s } } = \\frac { \\delta ( \\chi ) k ! } { ( s - 1 ) ^ { k + 1 } } - \\sum _ { \\omega } \\frac { k ! } { ( s - \\omega ) ^ { k + 1 } } \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} p ( t ) = \\exp ( \\Lambda _ t A ) e _ 0 , \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} \\varphi _ 1 ( y ) \\otimes \\psi ( y ) < u & & & & \\psi ( x ) = 1 ; \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} S = \\frac { 2 } { \\rho } = . \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} g ( s , v ; q ) = \\sum _ { n = 1 } ^ { \\infty } \\frac { K l ( l , n ; q ) } { n ^ { s + v } } . \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} \\chi _ b & = \\sum _ { a \\in A } f ( a , b ) e _ a \\\\ & = \\sum _ { a \\in A } \\sum _ { c \\in B } f ( a , b ) p ( a , c ) \\chi _ c . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} \\times \\deg \\pi \\colon H ^ i _ c ( Y , \\Lambda ) = \\bigoplus _ j H ^ i _ c ( Y _ j , \\Lambda ) \\xrightarrow { \\bigoplus _ j ( \\times \\delta _ j ) } \\bigoplus _ j H ^ i _ c ( Y _ j , \\Lambda ) = H ^ i _ c ( Y , \\Lambda ) . \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\widetilde { \\varphi } ( x ) = \\sum _ { i = 0 } ^ \\infty \\frac { ( i + 1 ) ! } { ( i + 2 k ) ! } \\gamma _ i x ^ { i + 2 } = \\left ( \\frac { x ^ 2 } { 1 - x } \\right ) \\psi \\left ( \\frac { x ^ 2 } { 1 - x } \\right ) . \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} \\mathcal { A } \\left ( \\begin{array} [ c ] { c } g _ { + } \\\\ g _ { - } \\end{array} \\right ) = \\left ( \\begin{array} [ c ] { c } - \\left ( v \\partial _ { x } - \\beta _ { x } \\partial _ { v } \\right ) g _ { + } + \\partial _ { x } \\phi \\partial _ { v } \\mu _ { + } \\\\ - \\left ( v \\partial _ { x } + \\beta _ { x } \\partial _ { v } \\right ) g _ { - } - \\partial _ { x } \\phi \\partial _ { v } \\mu _ { - } \\end{array} \\right ) , \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} | V _ Y ( t , x ; \\xi ) | \\leq \\begin{cases} C t ^ { \\alpha - 1 + ( \\gamma - 1 ) \\frac \\alpha 2 } p ( t , x - \\xi ) \\ , , & \\ d = 1 \\ , ; \\\\ C t ^ { \\nu _ 0 \\alpha - 1 } | x - \\xi | ^ { - d + \\gamma - \\nu _ 1 + 2 - \\nu _ 0 } p ( t , x - \\xi ) \\ , , & \\ d \\geq 2 \\\\ \\end{cases} \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\widetilde { g } ( s , v ; q ) = \\sum _ { a , b = 1 } ^ { q } \\delta _ q ( a b - 1 ) e \\left ( \\frac { a l } { q } \\right ) \\zeta ( 0 , b p / q ; s + v ) . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} T _ { n + 1 } ( x ) = 2 x T _ { n } ( x ) - T _ { n - 1 } ( x ) , U _ { n + 1 } ( x ) = 2 x U _ { n } ( x ) - U _ { n - 1 } ( x ) , \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 1 } ^ I \\sum _ { k = 0 } ^ { K } r _ { i , k } ( x ) g _ { i , k } ( x ) \\zeta ( x ) ^ k e ^ { \\alpha _ i x } \\rho ( \\delta _ i , x ) \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} \\prod _ { \\alpha = 1 } ^ { s - 1 } ( - 1 ) ^ { s - \\alpha } x _ { i _ { \\alpha + 1 } } x _ { i _ { \\alpha + 2 } } \\cdots x _ { i _ s } = ( - 1 ) ^ { \\frac { ( s - 1 ) s } { 2 } } x _ { i _ 2 } x _ { i _ 3 } ^ 2 x _ { i _ 4 } ^ 3 \\cdots x _ { i _ s } ^ { s - 1 } . \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\mathbb { Z } _ { n } = \\left \\{ a , a a , a a a , \\ldots , a ^ { n } \\right \\} . \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} \\frac { 1 } { 4 \\pi } \\left ( \\mathbf { E } \\cdot \\frac { \\partial \\mathbf { D } } { \\partial t } + \\mathbf { H } \\cdot \\frac { \\partial \\mathbf { B } } { \\partial t } \\right ) + \\mathbf { j } \\cdot \\mathbf { E } + \\operatorname { d i v } \\left ( \\frac { c } { 4 \\pi } \\mathbf { E } \\times \\mathbf { H } \\right ) = 0 \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} r ^ { * , \\pi } _ t ( x _ t | y ^ { t - 1 } _ { t - M } , y _ t ) = \\Big ( \\frac { q _ t ( y _ t | y ^ { t - 1 } _ { t - M } , x _ t ) } { \\nu ^ { \\pi } _ { t } ( y _ t | y ^ { t - 1 } _ { t - J } ) } \\Big ) { \\pi } _ { t } ( x _ t | y ^ { t - 1 } _ { t - J } ) , ~ \\forall { x _ t } \\in { \\cal X } _ t , ~ t \\in \\mathbb { N } _ 0 ^ { n - 1 } . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} Z _ R = Z _ { 1 , R } \\cup _ Y Z _ { 2 , R } , \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{gather*} W = \\prod _ { 1 \\le j < k \\le m } ( w _ { j } - w _ { k } ) . \\end{gather*}"}
-{"id": "5254.png", "formula": "\\begin{align*} a _ i ^ 2 = 1 a _ i a _ j + a _ j a _ i = 0 i \\neq j . \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\Lambda ( f ) = \\begin{cases} \\deg ( g ) & f = g ^ k , \\ g , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} t _ q = \\frac { 1 } { q } \\sum _ { i = 1 } ^ n x _ i ^ q \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} y ^ 2 & = x ^ 4 - 2 ( a + 1 ) ^ 2 x ^ 3 + ( a ^ 4 + 8 a ^ 3 + 1 0 a ^ 2 + 8 a + 1 ) x ^ 2 \\\\ & - 4 a ( a + 1 ) ^ 2 ( a ^ 2 + a + 1 ) x + 4 a ^ 2 ( a + 1 ) ^ 2 ( a ^ 2 + 1 ) . \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} P _ { C , p } ( t ) = \\sum _ { m \\geqslant 0 } \\ell ( m p ) t ^ m = 1 + t + 2 t ^ 2 + 3 t ^ 3 + \\ldots . \\end{align*}"}
-{"id": "10140.png", "formula": "\\begin{align*} C \\ : : \\ : z ^ { - ( p + 2 q ) } ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q - x ^ { - p } = 0 , \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{gather*} t _ { n _ { k } + 1 } = . . . = t _ { n _ { k + 1 } - 1 } = 0 \\end{gather*}"}
-{"id": "5586.png", "formula": "\\begin{align*} X = \\{ a : \\| a \\| _ s < \\infty \\} , \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} _ { 1 } \\phi _ { 1 } \\left ( \\begin{array} { c c c } \\begin{array} { c } a \\\\ b \\end{array} & \\vert & q , - b q ^ { \\alpha } \\end{array} \\right ) q ^ { \\alpha ^ { 2 } / 2 } = \\frac { 1 } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\left ( a b q ^ { - 1 / 2 } e ^ { i x } ; q \\right ) \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i \\alpha x \\right ) d x } { \\left ( b ; q \\right ) _ { \\infty } \\left ( b e ^ { 2 i x } q ^ { - 1 } ; q ^ { 2 } \\right ) _ { \\infty } } \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} H _ { m , n } ( 0 , 0 ) = \\lim \\limits _ { \\rho \\to + \\infty } \\frac { \\mathcal { Z } _ { m , n } ^ { \\rho ^ 2 } ( 0 , 0 ) } { \\rho ^ { m + n } } = \\lim \\limits _ { \\rho \\to + \\infty } ( - 1 ) ^ m m ! \\delta _ { m , n } \\dfrac { \\Gamma ( \\rho ^ 2 + 2 m + 1 ) } { \\rho ^ { 2 m } \\Gamma ( \\rho ^ 2 + m + 1 ) } = ( - 1 ) ^ m m ! \\delta _ { m , n } . \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} | F _ 1 ' | ^ 2 - | G _ 1 ' | ^ 2 = | F _ 2 ' | ^ 2 - | G _ 2 ' | ^ 2 \\ , , \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} \\phi ^ * g _ { \\mu \\nu } ( x ) = g _ { i j } ( \\phi ( x ) ) \\partial _ \\mu \\phi ^ i ( x ) \\partial _ \\nu \\phi ^ j ( x ) \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} y _ i = f ( z _ i , x _ i , u _ i ) , ~ ~ ~ z _ { i + 1 } = g ( z _ i , x _ i , u _ i ) , ~ ~ ~ i = 1 , 2 , . . . \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} \\partial F ( f ^ 1 , f ^ 2 , \\ldots , f ^ n ) = \\sum _ { i = 1 } ^ n \\frac { \\partial F } { \\partial x ^ i } \\partial f ^ i . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} a ( t , v ( t , y ) ) & = \\bar { U } ' ( t , y ) \\\\ b ( t , v ( t , y ) ) & = \\bar { U } '' ( t , y ) . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{gather*} A _ { n - 1 } = \\mathop { \\oplus } \\limits _ { i = 1 } ^ { n - 1 } \\mathbb { Z } \\alpha _ { i } \\subset \\mathbb { Z } ^ { n } . \\end{gather*}"}
-{"id": "8691.png", "formula": "\\begin{align*} \\Phi _ { i j } = \\begin{cases} \\int _ { 0 } ^ { \\infty } { d t \\over \\hbar } \\ ; K _ t ( a _ i , a _ i ) \\left ( e ^ { - t \\mu _ { i } ^ 2 / \\hbar } - e ^ { t E / \\hbar } \\right ) & \\textrm { i f $ i = j $ } \\\\ - \\int _ { 0 } ^ { \\infty } { d t \\over \\hbar } \\ ; K _ t ( a _ i , a _ j ) e ^ { t E / \\hbar } & \\textrm { i f $ i \\neq j $ } . \\end{cases} \\ ; . \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { \\mathcal { D } _ N } \\left \\{ \\sum _ k W _ N \\left ( \\psi _ N ^ { t _ k } Z _ N \\right ) \\right \\} f _ N ( 0 , Z _ N ) d Z _ N = \\\\ & = \\int _ { \\mathbb { R } } \\int _ { \\partial \\mathcal { D } _ N } \\left [ W _ N \\left ( Z _ N \\right ) \\right ] ^ 2 f _ N ( t , Z _ N ) d \\sigma _ N d V _ N d t \\end{aligned} \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} \\mathfrak { L } = \\left \\{ f \\in L ^ 2 ( X , \\mu ) , t \\ge 0 , e ^ { t \\Delta } f = e ^ { t \\Delta ^ \\perp } f \\right \\} . \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} \\begin{array} { c l } J _ { d } ^ { \\left ( 0 \\right ) } & = J _ { d } , \\\\ J _ { d } ^ { \\left ( i \\right ) } & = L _ { i + 1 } . . . L _ { d } U L _ { 1 } . . . L _ { i } + \\lambda I , i = 1 , 2 , . . , d . \\end{array} \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} \\eta _ { j } = ( \\delta - w _ { j k } ) \\tilde \\varphi \\qquad \\textrm { a n d } \\qquad \\eta _ { k } = ( \\delta + w _ { j k } ) \\tilde \\varphi \\end{align*}"}
-{"id": "9930.png", "formula": "\\begin{align*} a ( t ) w _ i = e ^ { ( ( r - 2 i ) ( m + n ) / 2 + \\delta ) t } w _ i , \\quad 1 \\leq i \\leq r . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} g _ C = \\frac { 1 } { a _ 0 a _ 1 a _ 2 } \\left ( \\frac { ( d - 1 ) ( d - 2 ) } { 2 } - \\left [ \\frac { b ( \\pi ) } { 2 } + 1 - a _ 0 a _ 1 a _ 2 \\right ] \\right ) \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} \\int ^ { c \\in \\mathcal C } \\ ! G ( ? ; c , c ) = \\big ( \\int ^ { c \\in \\mathcal C } \\ ! \\widetilde G ( c , c ) \\ , \\big ) \\ , ( ? ) \\ , . \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} [ a ( z ) , b ( w ) ] = \\sum _ { n = 0 } ^ { N _ { a , b } - 1 } ( a _ { ( n ) } b ) ( w ) \\frac { 1 } { n ! } \\partial _ w ^ n \\delta ( z - w ) , \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} Q = \\sum _ \\alpha ( x _ \\alpha - \\chi ( x _ \\alpha ) ) \\otimes x _ \\alpha ^ \\ast - 1 \\otimes \\dfrac { 1 } { 2 } \\sum _ { \\alpha , \\beta , \\gamma } c _ { \\alpha , \\beta } ^ \\gamma x _ \\alpha ^ \\ast x _ \\beta ^ \\ast x _ \\gamma . \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} \\left | r _ { j _ k } v _ k ( x ) \\right | = \\left | r _ { j _ k } T _ k \\circ \\nabla u _ { j _ k } ( x ) \\right | = \\left | T _ { r _ { j _ k } k } ( r _ { j _ k } \\nabla u _ { j _ k } ( x ) ) \\right | \\leq \\left | T _ k ( r _ { j _ k } \\nabla u _ { j _ k } ) \\right | = \\left | T _ k ( \\nabla \\zeta _ { j _ k } ) \\right | . \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\widetilde \\phi ^ { r , s , j } ( P + t , \\eta + t ) = \\sum _ { i = 0 } ^ r \\binom { r } { i } \\widetilde \\phi ^ { r - i , s , j } ( P , \\eta ) \\ , t ^ i \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} | n | ^ 2 = 1 + | p | ^ 2 . \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} | \\psi \\rangle = G ^ { ( n ) } | \\chi \\rangle + \\lambda _ { n + 1 } G ^ { ( n ) } | a _ { n + 1 } \\rangle \\langle a _ { n + 1 } | \\psi \\rangle \\ ; . \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} ( V \\rfloor \\varphi ) _ { | L } = 0 . \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} \\mathcal { F } \\left ( f \\right ) ( \\xi ) = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } } \\int _ { \\mathbb { R } ^ { d } } e ^ { - i \\xi \\cdot x } f ( x ) d x , \\quad \\mathcal { F } ^ { - 1 } ( g ) ( x ) : = \\frac { 1 } { ( 2 \\pi ) ^ { d / 2 } } \\int _ { \\mathbb { R } ^ { d } } e ^ { i \\xi \\cdot x } g ( \\xi ) d \\xi , \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} \\dim \\{ X _ { i _ 1 } ( x ) , \\ldots , X _ { i _ n } ( x ) \\} = n , . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} S ^ * U = U S ^ * + \\delta _ { \\{ 0 \\} } S ^ * + \\delta _ { \\{ - 1 \\} } S ^ * , S U = U S - \\delta _ { \\{ 0 \\} } S - \\delta _ { \\{ 1 \\} } S , \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} d < 4 - \\frac { 2 ( 2 \\beta - 1 ) _ + } { \\alpha } = : d _ 0 . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} J h ^ s _ { p , z } = \\lim _ { n \\to \\infty } \\frac { J f ^ n _ c ( t ) } { J f ^ n _ c ( h ^ s _ { p , z } ( t ) ) } , \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} & ( a - 1 ) ( b - 1 ) d _ 0 - ( d ' - 1 ) \\frac { p ^ L - 1 } { d ' } \\\\ \\geq & ( a - 1 ) ( b - 1 ) \\frac { ( p ^ 9 - 1 ) ( p ^ { L - 9 } - 1 ) } { a b } - \\frac { b - 2 } { b } ( p ^ L - 1 ) \\\\ = & \\frac { a - b + 1 } { a b } p ^ L - \\frac { a b - a - b + 1 } { a b } ( p ^ 9 + p ^ { L - 9 } - 1 ) + \\frac { b - 2 } { b } \\\\ > & \\frac { 2 } { a b } p ^ L - p ^ 9 - p ^ { L - 9 } > 0 . \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} \\nu _ j ( n , s ) & = - \\frac { ( s + \\log _ { 1 / p } ( 1 + ( p / q ) ^ s ) + \\psi ( n ) + 1 ) ^ 2 } { 2 } \\\\ & ~ ~ ~ ~ - \\log _ { 1 / p } n \\log _ { 1 / p } ( 1 + ( p / q ) ^ s ) + \\psi ( n ) ^ 2 / 2 + o ( \\psi ( n ) ^ 2 ) . \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} C _ { \\gamma } \\left ( \\overline { z } ^ { m - j } z ^ { n - j } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { j } \\right ) = \\dfrac { - 1 } { z ^ { m - n + 1 } } \\int _ 0 ^ { | z | ^ 2 } t ^ { m - j } ( 1 - t ) ^ { \\gamma + j } d t . \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} X ~ i s ~ o r i e n t a b l e , ~ v e r t i c e s ~ o f ~ \\Gamma ~ h a v e ~ v a l e n c e ~ 3 , ~ | V | ~ i s ~ e v e n ~ a n d ~ r = 1 + 2 | V | \\equiv 1 ~ m o d ~ ( 4 ) . \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} W ^ { k + t , p } ( \\Omega , X ) & : = \\left [ W ^ { k , p } ( \\Omega , X ) , W ^ { k + 1 , p } ( \\Omega , X ) \\right ] _ { t } , \\\\ W ^ { k + t } _ p ( \\Omega , X ) & : = \\left ( W ^ { k , p } ( \\Omega , X ) , W ^ { k + 1 , p } ( \\Omega , X ) \\right ) _ { t } \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\Phi _ 0 ( \\gamma _ 1 ) + \\gamma _ 0 = \\gamma _ 1 . \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} \\theta _ 0 ( x _ a ) = x _ a \\ * 1 + 1 \\ * \\sum _ { \\beta , \\gamma \\in \\Delta _ + } c _ { a , \\beta } ^ { \\gamma } x _ { \\gamma } x _ { \\beta } ^ * . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} P _ { n } = K _ { n } + l _ { n , n } K _ { n - 1 } + l _ { n , n - 1 } K _ { n - 2 } + . . . + l _ { n , n - d + 1 } K _ { n - d } , \\ n \\geq 0 \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} H _ F ( P ) = \\left ( \\frac { \\partial ^ 2 F } { \\partial x _ i \\partial x _ j } ( P ) \\right ) _ { i , j = 0 , \\dots , n + 1 } . \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} \\overline g ( ( X , X ^ \\prime ) , ( Y , Y ^ \\prime ) ) = g ( X , Y ) + g ^ \\prime ( X ^ \\prime , Y ^ \\prime ) \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} & ( n - 6 ) \\int _ { B _ \\rho } \\Delta \\sigma _ 1 ( A ) | \\nabla \\varphi | _ g ^ 2 d \\mu _ g \\\\ = & ( n - 6 ) \\int _ { B _ \\rho } ( \\Delta \\sigma _ 1 ( A ) ( p ) + O ( r ) ) ( | \\varphi ' | ^ 2 + O ( r ^ 2 ) | \\varphi | ^ 2 ) d x \\\\ = & - ( n - 6 ) ^ 3 \\frac { | W ( p ) | ^ 2 } { 1 2 ( n - 1 ) } \\int _ { B _ \\rho } \\frac { u _ \\epsilon ^ 2 r ^ 2 } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 2 } d x + \\int _ { B _ \\rho } \\frac { O ( r ^ 3 ) u _ \\epsilon ^ 2 } { ( \\epsilon ^ 2 + r ^ 2 ) ^ 2 } d x . \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} \\partial ^ { \\alpha } _ t u = \\Delta u + \\partial ^ { \\beta } _ t \\int ^ t _ 0 g ^ k d w ^ k _ s . \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} \\beta ^ \\prime ( t ) = \\frac { 1 } { 2 } \\beta _ 0 ( d - 1 ) [ 1 + ( T - t ) ] ^ { - d } \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} f _ N ^ { ( s ) } ( t ) = T _ s ( t ) f _ N ^ { ( s ) } ( 0 ) + ( N - s ) \\varepsilon ^ { d - 1 } \\int _ 0 ^ t T _ s ( t - t _ 1 ) C _ { s + 1 } f _ N ^ { ( s + 1 ) } ( t _ 1 ) d t _ 1 \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} \\frac { 1 } { \\Phi ( Q ) ^ { 2 } } \\sum _ { \\substack { \\chi \\neq \\chi _ { 0 } \\\\ \\chi \\in \\Gamma _ { 6 - p r i m } ^ { 6 - o d d } ( Q ) } } | \\sum _ { \\substack { 2 j + 3 l + 6 k = n \\\\ 0 \\leq j \\leq N \\\\ 0 \\leq l \\leq N \\\\ 0 \\leq k } } q ^ { \\frac { j + k + l } { 2 } } \\Lambda _ { j } ( \\chi ^ { 2 } ) \\Lambda _ { l } ( \\chi ^ { 3 } ) S y m ^ { k } ( \\chi ^ { 6 } ) | ^ 2 \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} \\lambda \\cdot [ x _ 0 ^ 2 : x _ 0 x _ 1 : x _ 1 ^ 2 : x _ 2 ] = [ \\lambda x _ 0 ^ 2 : \\lambda x _ 0 x _ 1 : \\lambda x _ 1 ^ 2 : \\lambda x _ 2 ] = [ x _ 0 ^ 2 : x _ 0 x _ 1 : x _ 1 ^ 2 : x _ 2 ] . \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} \\alpha ^ { \\frac 4 3 } - 4 \\alpha - \\alpha ^ { \\frac 1 3 } + 1 = 0 . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial \\kappa ^ i } = n ^ { - 1 / r } \\left ( \\sum _ { l } \\kappa _ l ^ r \\right ) ^ { \\frac { 1 } { r } - 1 } \\kappa _ i ^ { r - 1 } , \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} F _ { d , \\ell } \\left ( z ; i t \\right ) = \\frac { 1 } { 2 } \\left ( 2 \\ell ^ 2 t \\right ) ^ { - \\frac { 1 } { 2 } } \\zeta ^ d - \\zeta ^ d \\sum _ { a = 0 } ^ { N } \\frac { B _ { 2 a + 1 } \\left ( \\frac { d } { \\ell } \\right ) } { 2 a + 1 } \\frac { \\left ( - 2 \\pi \\ell ^ 2 t \\right ) ^ a } { a ! } + O \\left ( t ^ { N + 1 } \\right ) . \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} ( E _ k ( u ) , H _ l ( x ) ) = 0 , \\ , \\ ( 1 , E _ k ( u ) ) = 0 . \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} n C _ n = \\frac { 1 } { 1 2 n } + \\sum _ { i = 1 } ^ n \\left ( F _ 0 ( e _ i ) - \\frac { 2 i - 1 } { 2 n } \\right ) ^ 2 , \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{gather*} \\sum _ { k \\geq 0 } \\frac { s _ { k } } { x ^ { k + 1 } } = \\frac { Q _ { r } ( x ) } { P _ { r } ( x ) } \\ , \\end{gather*}"}
-{"id": "2984.png", "formula": "\\begin{align*} ^ { C \\ ! } D _ { 0 + } ^ \\alpha x ( t ) = f ( t , x ( t ) ) , \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} 0 < \\tfrac { c } { 2 } \\Theta = \\tfrac { c } { 2 } \\mathrm { a r c t a n h } \\tilde { \\Theta } \\leq c \\tilde { \\Theta } \\leq \\mathrm { a r c t a n h } ( c \\tilde { \\Theta } ) \\leq \\rho _ - ( T ) . \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} b _ 1 + b _ 2 - p _ 1 - p _ 2 = 0 \\in \\Gamma \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i { m \\choose i } \\frac { ( m + i ) ! } { ( m + 2 k + i - 1 ) ! } \\gamma _ { m + i - 1 } = 0 . \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} \\Omega F = \\frac { 1 } { 1 2 } ( l ^ 2 _ 1 + l ^ 2 _ 2 - 5 ) F , \\Delta F = ( l _ 1 l _ 2 ) ^ 2 F . \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} \\begin{array} { c } \\gamma _ 0 = \\{ \\emptyset , \\left [ 0 , 1 \\right ] \\} \\cup \\{ \\left [ 0 , p ) \\mid p \\in ( 0 , 1 \\right ] \\} \\cup \\\\ \\left . \\{ ( q , 1 \\right ] \\mid q \\in \\left [ 0 , 1 ) \\} \\cup \\{ \\left [ 0 , r ) \\cup ( s , 1 \\right ] \\mid r , s \\in ( 0 , 1 ) , \\ , \\ , r \\le s \\} \\ , . \\right . \\end{array} \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} \\lim _ j \\norm { ( f _ j - h ) ^ - } = 0 . \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} h _ { i j ; k } - h _ { i k ; j } = \\bar { R } _ { \\alpha \\beta \\gamma \\delta } \\nu ^ { \\alpha } x _ i ^ { \\beta } x _ j ^ { \\gamma } x _ k ^ { \\delta } , \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} \\rho ^ { \\omega } _ { K , \\mathbf { a } , f } ( Y _ { \\mathbf { u } } ) = \\rho ^ { \\omega } _ { K , \\mathbf { a } , f } ( Y _ 0 + \\sum _ { i = 1 } ^ { m } u _ i Y _ i ) \\le ( m + 1 ) \\max \\{ \\rho ^ { \\omega } _ { K , \\mathbf { a } , f } ( Y _ i ) \\mid i \\in \\{ 0 , 1 , \\ldots , m \\} \\} . \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} f ( \\ell ) = ( a ( \\ell ) t + \\ell ' ) f \\end{align*}"}
-{"id": "4084.png", "formula": "\\begin{align*} E Y ^ { \\intercal } Y = X ^ { \\intercal } X + p _ 1 I _ { p _ 2 } = V \\Sigma ^ 2 V ^ { \\intercal } + p _ 1 I _ { p _ 2 } , E V ^ { \\intercal } Y ^ { \\intercal } Y V = \\Sigma ^ 2 + p _ 1 I _ { p _ 2 } , \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} d \\tilde { s } ^ 2 = d r ^ 2 + r ^ 2 \\sigma _ { i j } d \\xi ^ i d \\xi ^ j \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} U _ * S _ * = U _ 1 \\Psi _ 1 - H V ^ { ( q ) } _ 0 \\widehat { \\Psi } ^ { - 1 } , \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} [ \\psi _ i , \\psi _ j ] _ + = 0 , [ \\psi _ i , \\psi _ j ^ \\ast ] _ + = \\delta _ { i , j } , [ \\psi _ i ^ \\ast , \\psi _ j ^ \\ast ] _ + = 0 , \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} \\phi ^ z _ \\delta ( x ) : = \\delta ^ { - d / p } \\phi ( \\delta ^ { - 1 } ( x - z ) ) . \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\rho : P _ { 1 } \\otimes V _ { r - 1 } \\to \\bigoplus _ { i = 1 } ^ { k } k [ z ] / ( z - \\alpha _ { i } ) ^ { n _ { i } - 1 } \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} K _ \\delta : = \\left \\{ \\ , x \\in M \\ : \\ { \\rm d i s t } ( x , K ) < \\delta \\right \\} \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} \\mathcal { U } _ s ^ \\eta = \\left \\{ Z _ s = \\left ( X _ s , V _ s \\right ) \\in \\overline { \\mathcal { D } _ s } \\left | \\inf _ { 1 \\leq i < j \\leq s } \\left | v _ i - v _ j \\right | > \\eta \\right . \\right \\} \\subset \\mathbb { R } ^ { 2 d s } \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} r ( Y ) - r ( X ) & = \\lambda ( Y ) + | | Y | | _ \\lambda - \\lambda ( X ) - | | X | | _ \\lambda \\\\ & = | | Y - X | | _ \\lambda + \\lambda ( Y ) - \\lambda ( X ) \\\\ & = | | ( E - X ) - ( E - Y ) | | _ \\lambda - ( \\lambda ( E - X ) - \\lambda ( E - Y ) ) . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} ( \\log F ( z ) ) _ { z \\overline { z } } = & \\sum _ { k = 2 } ^ { p } ( k - 1 ) ^ { 2 } | z | ^ { 2 ( k - 2 ) } \\log G _ { k } ( z ) + \\sum _ { k = 2 } ^ { p } ( k - 1 ) | z | ^ { 2 ( k - 2 ) } \\mathfrak { L } [ \\log G _ { k } ( z ) ) ] . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} K ( P _ x ) \\le - a \\big ( \\rho ( x ) \\big ) ^ 2 = - \\frac { \\phi ( \\phi - 1 ) } { \\rho ( x ) ^ 2 } , \\phi > 1 , \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} \\bordermatrix { & \\cr & \\nu ^ { { \\pi ^ { * , \\infty } } } ( 0 ) \\cr & \\nu ^ { { \\pi ^ { * , \\infty } } } ( 1 ) \\cr } = \\bordermatrix { & \\cr & \\nu ^ { \\pi ^ { * , \\infty } } ( 0 | 0 ) & \\nu ^ { \\pi ^ { * , \\infty } } ( 0 | 1 ) \\cr & \\nu ^ { \\pi ^ { * , \\infty } } ( 1 | 0 ) & \\nu ^ { \\pi ^ { * , \\infty } } ( 1 | 1 ) \\cr } \\bordermatrix { & \\cr & \\nu ^ { { \\pi ^ { * , \\infty } } } ( 0 ) \\cr & \\nu ^ { { \\pi ^ { * , \\infty } } } ( 1 ) \\cr } \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} \\varrho = \\mathrm { a r c t a n h } r = \\tfrac { 1 } { 2 } ( \\log ( 1 + r ) - \\log ( 1 - r ) ) , \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{gather*} n _ d + n _ e = \\ell , n _ c + n _ d = k . \\end{gather*}"}
-{"id": "4688.png", "formula": "\\begin{align*} z ^ { p ^ m } - z = y ^ { p ^ e + 1 } - \\frac { 1 } { n ' } \\sum _ { 1 \\le i \\le j \\le n - 2 } y _ i y _ j . \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c } \\begin{array} { l l l l l } x _ { i } & = & \\dfrac { T _ { i , 1 } } { l _ { 1 } t _ { 1 } } & + & \\varepsilon \\pounds \\end{array} & ; & i = 1 , 2 , . . . , n \\\\ \\begin{array} { l l l l l } x _ { n + 1 } & = & \\dfrac { T _ { n + 1 , 1 } } { l _ { 1 } t _ { 1 } } & - & \\lambda _ { 1 } \\end{array} & & \\end{array} \\right . \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} F + ( F \\cap H ) = F _ + + F _ - \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} U _ \\mathcal { S } = \\left ( X ^ { ( n ) } \\cdot A _ { \\mathcal { S } , n } : n \\in [ 1 : N ] \\right ) , \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\Xi = \\begin{bmatrix} \\Xi _ 1 \\\\ \\Xi _ 2 \\end{bmatrix} = ( \\psi ^ \\prime \\psi ) ^ { - 1 } \\psi ^ \\prime ( X ( 0 ) - \\mu ) , \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} \\gamma ^ * \\omega _ 0 = \\tfrac { 1 } { 4 } ( [ 2 ] \\circ \\gamma ) ^ * \\omega _ 0 = - \\tfrac { 1 } { 8 } c _ 1 ( \\mathcal { L } ^ { \\langle 2 \\rangle } ) . \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} g _ 0 = d r ^ 2 + f \\left ( \\theta \\right ) ^ 2 d \\theta ^ 2 . \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} d _ X ( F ( x , y ) , x ) & = \\lim _ { j \\rightarrow \\infty } d _ X ( F ( F ^ j ( x _ 0 , y _ 0 ) , G ^ j ( y _ 0 , x _ 0 ) ) , F ^ j ( x _ 0 , y _ 0 ) ) \\\\ & = \\lim _ { j \\rightarrow \\infty } d _ X ( F ^ { j + 1 } ( x _ 0 , y _ 0 ) , F ^ j ( x _ 0 , y _ 0 ) ) \\\\ & = 0 \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} \\partial e ^ { t \\Delta } = e ^ { t \\vec { \\Delta } } \\partial , \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} - L v _ n & = \\mu _ n \\ , \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega , \\\\ v _ n & = \\nu _ n \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega \\end{align*}"}
-{"id": "9971.png", "formula": "\\begin{align*} S = h ^ { - 2 } s , \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} a \\pm n d x - a = 0 . \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} { y _ i ^ T \\over i } & = { y _ { k m + n } ^ T \\over k m + n } \\le { y _ { k m } ^ T + y _ n ^ T + 1 \\over k m + n } \\\\ & \\le { k y _ { m } ^ T + y _ n ^ T + k \\over k m + n } = { k y _ { m } ^ T \\over k m + n } + { y _ n ^ T \\over k m + n } + { k \\over k m + n } \\ , . \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\frac { d } { d \\eta } h ( Y ) & = - \\int \\frac { d } { d \\eta } \\left ( q _ { \\eta } ( y ) \\ , \\ln q _ { \\eta } ( y ) \\right ) \\ , d y \\\\ & = - \\int \\frac { d q _ { \\eta } } { d \\eta } ( y ) \\ , \\ln q _ { \\eta } ( y ) \\ , d y - \\int \\frac { d q _ { \\eta } } { d \\eta } ( y ) \\ , d y \\\\ & = - \\int \\frac { d q _ { \\eta } } { d \\eta } ( y ) \\ , \\ln q _ { \\eta } ( y ) \\ , d y - \\frac { d } { d \\eta } \\int q _ { \\eta } ( y ) \\ , d y \\\\ & = - \\int \\frac { d q _ { \\eta } } { d \\eta } ( y ) \\ , \\ln q _ { \\eta } ( y ) \\ , d y . \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} ( r ^ * ) ^ * ( X ) & = r ^ * ( E - X ) + | | X | | _ { r ^ * } - r ^ * ( E ) \\\\ & = r ( E - ( E - X ) ) + | | E - X | | _ r - r ( E ) + | | X | | _ { r ^ * } \\\\ & { \\text ~ ~ ~ ~ ~ ~ ~ ~ ~ } - r ( E - E ) - | | E | | _ r + r ( E ) \\\\ & = r ( X ) + \\sum _ { x \\in X } ( \\lambda _ P ( \\{ x \\} ) - r ( \\{ x \\} ) ) \\\\ & = r ^ \\flat ( X ) . \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\mbox { m i n i m i z e } \\sum _ { t = 0 } ^ { n - 1 } | y _ \\mathrm m ( t ) - \\sum _ { k = 1 } ^ r c _ k e ^ { \\j \\omega _ k t } | ^ 2 , \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} C ^ { F B , A . 1 } = \\sum _ { y \\in \\{ 0 , 1 \\} } \\bigg ( \\sum _ { x \\in \\{ 0 , 1 \\} , z \\in \\{ 0 , 1 \\} } \\log \\Big ( \\frac { q ( z | y , x ) } { \\nu ^ { * , \\infty } ( z | y ) } \\Big ) q ( z | y , x ) \\pi ^ { * , \\infty } ( x | y ) \\bigg ) \\nu ^ { { \\pi ^ { * , \\infty } } } ( y ) . \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} E ( k ) : = \\begin{cases} 4 - 4 \\cos ( k / 2 ) & \\ \\ k \\in ( 0 , \\pi ) \\\\ 4 + 4 \\cos ( k / 2 ) & \\ \\ k \\in ( \\pi , 2 \\pi ) \\end{cases} \\mu ( H ) : = [ 0 , E ( k ) ) \\cup ( 4 d - E ( k ) , 4 d ] . \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} \\langle f , \\star \\partial g \\rangle _ 2 = - \\langle \\star \\partial f , g \\rangle _ 2 . \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\partial _ { t } f _ { \\pm } + v \\partial _ { x } f _ { \\pm } \\pm E \\partial _ { v } f _ { \\pm } = 0 , E _ { x } = \\int [ f _ { + } - f _ { - } ] d v \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} h ( \\bot ) = \\bot & & & & h ( x \\vee y ) = h ( x ) \\vee h ( y ) , \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\int _ { p r _ { g + 1 } } \\gamma ^ * ( \\delta _ { \\Theta _ { \\alpha } } ) \\gamma ^ * \\omega _ 0 ^ { g } = & \\sum _ { j = 1 } ^ g \\int _ { p r _ { g + 1 } } \\delta _ { \\lbrace P _ j = P _ { g + 1 } \\rbrace } \\gamma ^ * \\omega _ 0 ^ g \\\\ & + \\int _ { p r _ { g + 1 } } \\delta _ { \\lbrace P _ g \\in \\sigma ( P _ 1 , \\dots , P _ { g - 1 } ) \\rbrace } \\gamma ^ * \\omega _ 0 ^ { g } , \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} \\begin{aligned} x _ i x _ { i + 1 } y _ { i + 2 } - y _ i x _ { i + 1 } x _ { i + 2 } & = 0 \\\\ x _ i y _ { i + 1 } y _ { i + 2 } - y _ i y _ { i + 1 } x _ { i + 2 } & = 0 . \\end{aligned} \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} [ \\bar { h } _ { i , k } , \\bar { e } _ { i , l } ] = 2 \\bar { e } _ { i , l + k } , \\ [ \\bar { h } _ { i , k } , \\bar { e } _ { i + 1 , l } ] = - ( d ^ k + d ^ { - k } ) \\bar { e } _ { i + 1 , l + k } , \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} \\sum _ { m , s \\atop 2 m + s = p - r } \\overline a _ { 1 m r s } Q ^ m u _ F ^ s + \\sum _ { m , s \\atop 2 m + s + 1 = p - r } b _ { 1 m r s } Q ^ m \\ , \\overline { u _ F } \\ , u _ F ^ s = 0 . \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} { \\bf E } \\bigl [ \\beta ^ { - q } _ { 1 , 0 } \\bigl ( ( \\tau , 1 + \\tau ( 1 + \\lambda _ 1 + \\lambda _ 2 ) \\bigr ) \\bigr ] = \\tau ^ { - \\frac { q } { \\tau } } \\frac { \\Gamma ( - \\frac { q } { \\tau } + 1 + \\lambda _ 1 + \\lambda _ 2 + \\frac { 1 } { \\tau } ) } { \\Gamma ( 1 + \\lambda _ 1 + \\lambda _ 2 + \\frac { 1 } { \\tau } ) } , \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} d s ^ 2 = d \\sigma ^ 2 + f \\left ( \\theta , \\sigma \\right ) ^ 2 d \\theta ^ 2 \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} R ^ { ( n ) } ( t ) : = \\widetilde { R } ( n \\tau ( t ) ^ { \\gamma } ) \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} x \\ , \\varphi \\ , y \\iff y \\in \\bigcap _ { \\Phi ( \\psi ) ( x ) = 0 } Z ( \\psi ) . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\mathrm { I I } _ { j } = 0 . \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 d } } | v | ^ 2 f _ N ^ { ( 1 ) } ( 0 , x , v ) d x d v = \\int _ { \\mathbb { R } ^ { 2 d } } | v | ^ 2 f _ 0 ( x , v ) d x d v \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} \\bar { P } _ { l , t } ( Y ) = C _ { l , t } \\int _ { \\mu _ 0 \\in \\mathbb { R } ^ { p } : \\| \\mu _ 0 \\| \\geq 1 / 2 } & \\frac { 1 } { ( 2 \\pi ) ^ { p n / 2 } } \\exp ( - \\| Y - 2 t \\mu _ 0 l ^ { \\intercal } \\| _ F ^ 2 / 2 ) \\\\ & \\cdot \\left ( \\frac { p } { 2 \\pi } \\right ) ^ { p _ 1 r / 2 } \\exp ( - p \\| \\mu _ 0 \\| _ 2 ^ 2 / 2 ) d \\mu _ 0 . \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} Z _ N ( \\boldsymbol { \\lambda } ; \\boldsymbol { \\nu } ) = \\frac { \\prod _ { j , k = 1 } ^ { N } a ( \\lambda _ j , \\nu _ k ) b ( \\lambda _ j , \\nu _ k ) } { \\prod _ { 1 \\leq j < k \\leq N } d ( \\lambda _ k , \\lambda _ j ) d ( \\nu _ j , \\nu _ k ) } \\det [ \\varphi ( \\lambda _ j , \\nu _ k ) ] _ { j , k = 1 , \\dots , N } , \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} H ^ 0 ( Y , R ^ i f _ * ( K _ X \\otimes F \\otimes \\mathcal J ( h ) ) \\otimes H ^ { \\otimes m } ) \\to H ^ 0 ( Y , R ^ i f _ * ( K _ X \\otimes F \\otimes \\mathcal J ( h ) ) \\otimes H ^ { \\otimes m + 1 } ) \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} G _ \\infty = \\left ( \\mathcal { I } - \\mathcal { V } \\right ) ^ { - 1 } G _ \\infty ( 0 ) = \\sum _ { j = 0 } ^ \\infty \\mathcal { V } ^ j G _ \\infty ( 0 ) \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} \\mathbf { P _ * } ( | | R ^ * _ { 4 n } | | = o ( n ^ { - 1 / 2 } ) ) = 1 - o ( n ^ { - 1 / 2 } ) \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\nabla _ \\xi \\varphi _ a ( x , \\xi ) - v ( \\xi ) = 0 . \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{align*} A _ \\lambda ' = \\frac { A _ \\lambda p _ \\lambda [ S ] } S , B _ \\lambda ' = \\frac { B _ \\lambda } { S } . \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} K _ { i j l } ^ { m } & = K _ { i j } { } ^ { k } K _ { k l } { } ^ { m } + K _ { i j } { } ^ { k + n } K _ { k + n , l } { } ^ { m } \\\\ K _ { i j l } ^ { m + n } & = K _ { i j } { } ^ { k } K _ { k l } { } ^ { m + n } + K _ { i j } { } ^ { k + n } K _ { k + n , l } { } ^ { m + n } \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} z ' + \\frac { 1 } { t } \\ , z = \\pm \\frac { \\sqrt { a ^ 2 + 4 c t ^ 2 } } { t } . \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} C = \\max _ { F _ X ( x ) : | X | \\leq A } I ( X ; Y ) \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} \\odot : \\mathcal { I } _ 0 \\times \\mathcal { I } _ 0 \\longrightarrow \\mathcal { I } _ 0 \\ \\ \\ \\ \\ \\ a ^ 0 \\odot b ^ 0 = [ \\mu ( a ^ 0 \\cap b ^ 1 ) ] ^ 0 \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} T _ { ( A , j + n + i \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } & = - T _ { ( A , j + i \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } , \\\\ T _ { ( A , ( j + i ) + n \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } & = - T _ { ( A , j + i \\ \\left ( \\operatorname { m o d } 2 n \\right ) ) } . \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} B \\dot { u } ( t ) + A u ( t ) = f ( t ) \\ae . \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} F _ { Z _ R } ( \\omega , 0 , 0 ) \\big | _ { Z _ { 1 , R } } = F _ { Z _ { 1 , R } } ( \\omega , 0 ) , F _ { Z _ R } ( \\omega , 0 , 0 ) \\big | _ { Z _ { 2 , R } } = 0 . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\tilde { g } ( t ) & : = g ( t ) \\ , \\ , \\ , t \\in H ^ 2 ( X , \\mathbb { Z } ) \\cong v ^ \\perp \\\\ \\tilde { g } ( v ) & : = \\chi ( g ) v . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} X & \\triangleq \\beta _ { 2 , 2 } ^ { - 1 } \\bigl ( \\tau = 1 , b _ 0 = 1 , \\ , b _ 1 = 1 + \\alpha , \\ , b _ 2 = 1 + \\alpha \\bigr ) , \\\\ Y & \\triangleq \\beta _ { 1 , 0 } ^ { - 1 } ( \\tau = 1 , b _ 0 = 2 \\alpha + 2 ) , \\\\ Y ' & \\triangleq \\beta _ { 1 , 0 } ^ { - 1 } \\bigl ( \\tau = 1 , b _ 0 = 1 \\bigr ) \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd t } p ^ { k } ( t ) = p ^ { k - 1 } ( t ) - p ^ { k } ( t ) \\lambda ^ \\top X _ t . \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} \\mu _ { \\nu } = \\frac { \\mu _ w } { c _ { o } } \\cdot c _ { o } ^ { \\nu } , \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\mathcal { J } _ { \\pi } ( 0 ) = ( X _ 1 ( 0 ) \\cdots X _ N ( 0 ) ) , \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} Q _ \\sigma [ u , v ] = \\int _ { M } \\ , \\nabla \\overline { u } \\cdot \\nabla v \\ , d x + \\int _ { \\partial M } \\sigma \\ , \\overline { u } \\ , v \\ , d S , u , v \\in H ^ 1 ( M ) . \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\left [ Z _ { a b } , \\bar { P } _ { c } \\right ] & = \\frac { 1 } { \\sqrt { 2 } } \\left [ Z _ { a b } , P _ { c } \\right ] + \\frac { 1 } { \\sqrt { 2 } } \\left [ Z _ { a b } , Z _ { c } \\right ] , \\\\ & = \\frac { 1 } { \\sqrt { 2 } } f _ { a b , c } ^ { d } Z _ { d } - \\frac { 1 } { \\sqrt { 2 } } f _ { a b , c } ^ { d } P _ { d } , \\\\ & = - f _ { a b , c } ^ { d } \\bar { Z } _ { d } , \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} \\left \\| d _ G ( \\cdot , y _ 0 ) ^ { - 2 ( 1 - \\frac { n + 1 } { p } ) } \\hat E ^ { y _ 0 , i j } \\right \\| _ { L ^ \\infty ( \\mathcal { B } _ { 1 / 2 } ^ + ( y _ 0 ) ) } \\\\ + \\left [ d _ G ( \\cdot , y _ 0 ) ^ { - 2 ( 1 - \\frac { n + 1 } { p } - \\epsilon ) } \\hat E ^ { y _ 0 , i j } \\right ] _ { \\dot { C } ^ { 0 , \\epsilon } _ \\ast ( \\mathcal { B } _ { 1 / 2 } ^ + ( y _ 0 ) ) } \\leq C . \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} U = \\frac { i } { 2 } \\begin{pmatrix} - u _ x & \\l \\\\ \\l & u _ x \\end{pmatrix} , \\ ; \\ ; V = - \\frac { i } { 2 } \\begin{pmatrix} 0 & \\l ^ { - 1 } e ^ { i u } \\\\ \\l ^ { - 1 } e ^ { - i u } & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} g ( t ) = e ^ { C _ 1 t } - 1 \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} h _ { \\Delta } ( P _ { 1 } , P _ { 2 } ) = \\mu ( P _ { 1 } ) + \\mu ( P _ { 2 } ) - \\tfrac { i } { 2 } \\sum _ { k = 1 } ^ { g } \\left ( \\psi _ { k } ( P _ { 1 } ) \\wedge \\bar { \\psi } _ { k } ( P _ { 2 } ) + \\psi _ { k } ( P _ { 2 } ) \\wedge \\bar { \\psi } _ { k } ( P _ { 1 } ) \\right ) , \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} \\frac { \\left ( f f '' + ( f ' ) ^ 2 + 1 \\right ) ^ 2 - \\kappa ^ 2 ( f '^ 2 + 1 ) } { 4 f ^ 2 ( f '^ 2 + 1 ) } = c . \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} \\Phi = \\Phi _ \\mathrm r = \\left [ \\begin{array} { c c } 0 & \\j \\\\ - \\j & 0 \\end{array} \\right ] , \\Psi = \\Phi _ \\mathrm u = \\left [ \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} P _ { 1 } ( 1 ) = p _ { 0 , 0 } e ^ { \\frac { \\lambda } { \\xi } } \\left [ - \\frac { \\gamma } { \\xi } B ( 1 ) + \\left ( \\frac { 1 } { A } - \\frac { \\xi } { A \\mu } + \\frac { \\gamma } { \\mu } \\right ) E ( 1 ) \\right ] . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} \\rho ^ { Z , f \\oplus 0 } _ t ( S _ c S _ d ) = \\rho ^ { A , f } _ t ( S _ { a } ) , & \\rho ^ { Z , f \\oplus 0 } _ t ( S _ d S _ c ) = \\rho ^ { B , \\phi ( f ) } _ t ( S _ { b } ) , \\\\ \\rho ^ { Z , 0 \\oplus g } _ t ( S _ d S _ c ) = \\rho ^ { B , g } _ t ( S _ { b } ) , & \\rho ^ { Z , 0 \\oplus g } _ t ( S _ c S _ d ) = \\rho ^ { A , \\psi ( g ) } _ t ( S _ { a } ) \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} \\left [ C _ { \\gamma } ( f ) \\right ] ( z ) & = \\dfrac { 1 } { \\pi } \\int _ { D } \\dfrac { f ( w ) } { w - z } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { \\gamma } d x d y \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} | \\langle \\phi _ j , \\phi _ l \\rangle | ^ 2 = \\frac { 1 } { M } + \\frac { M - 1 } { M } \\langle y _ j y _ l \\rangle \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} Z ( s , f , \\chi ) = Z ( s , f , \\chi , O _ { v } ^ { \\times 2 } ) + \\sum _ { \\tau \\subset \\Gamma ^ { g e o m } ( f ) } Z ( s , f , \\chi , \\Delta _ \\tau ) . \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\zeta ^ k _ { t | s } = ( e _ k ^ \\top \\otimes I ) V \\begin{pmatrix} \\exp ( Q _ { n \\lambda } ( t - s ) ) & & \\\\ & \\ddots & \\\\ & & \\exp ( Q ( t - s ) ) \\end{pmatrix} V ^ { - 1 } \\zeta _ s . \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} \\exp ^ x ( p + q ) = \\exp ^ x ( p ) \\exp ^ x ( q ) . \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} \\Omega \\mid _ { \\widetilde { U } _ i } = \\Omega _ i . \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{gather*} e ^ { - \\beta ^ 2 ( \\kappa - \\log \\varepsilon ) } \\int _ \\phi ^ \\psi e ^ { \\beta V _ { \\varepsilon } ( \\theta ) } \\ , d \\theta \\longrightarrow M _ { \\beta } ( \\phi , \\psi ) , \\\\ { \\bf { E } } [ M _ { \\beta } ( \\phi , \\psi ) ] = | \\psi - \\phi | . \\end{gather*}"}
-{"id": "3493.png", "formula": "\\begin{align*} y _ q ( u ) = & \\sum _ { \\mathcal { R } : | \\mathcal { R } | = r + 1 , \\mathcal { R } \\ni q } \\sum _ { \\mathcal { T } : | \\mathcal { T } | = t } \\sum _ { i = 1 } ^ t \\left [ \\sum _ { p \\in \\mathcal { T } } h _ { q p } ( u ) \\left ( \\mathbf { v } _ { { \\mathcal { R } } , { \\mathcal { T } } , p } ^ i ( u ) \\right ) ^ T \\right ] \\mathbf { x } _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ i + \\sum _ { m ( u ) \\in \\mathcal { M } _ q [ N + 1 ] ( u ) } m ( u ) x _ { m ( u ) } , \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} \\sup _ { P _ { X | S , Q } } \\liminf _ { N \\rightarrow \\infty } \\frac { 1 } { N } \\sum _ { i = 1 } ^ N I ( X _ i , S _ { i - 1 } ; Y _ i | Q _ { i - 1 } ) & \\leq \\sup _ { P _ { X | S , Q } \\in \\mathcal { P } _ { \\pi } } I ( X , S ; Y | Q ) , \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} f _ N ^ { ( 1 ) } ( t ) = \\mathcal { T } ( t - T _ 1 ) f _ N ^ { ( 1 ) } ( T _ 1 ) + \\int _ { T _ 1 } ^ t \\mathcal { T } ( t - \\tau ) C _ 2 f _ N ^ { ( 2 ) } ( \\tau ) d \\tau \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} \\mathrm { E x t } ^ 3 ( L ( 2 ) , F ) \\simeq \\mathrm { H o m } ( F , L ( - 2 ) ) ^ { \\vee } = 0 . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { 2 k ^ { 2 } - \\left ( 2 n + 1 \\right ) k } } { \\left ( q , q ^ { 2 } ; q ^ { 2 } \\right ) _ { k } } = \\frac { \\left ( - 1 \\right ) ^ { n } \\left ( - q ; q \\right ) _ { n } } { \\left ( q ; q ^ { 2 } \\right ) _ { \\infty } q ^ { \\binom { n + 1 } { 2 } } } . \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} x ( \\xi s + \\eta ) = e ^ { i ( \\xi s + \\eta ) A } \\left ( h + i \\int _ 0 ^ s e ^ { - i ( \\xi w + \\eta ) A } \\Phi ^ * \\sigma ( \\xi ) ( \\Lambda ( \\xi ) \\tilde { u } ) ( w ) d w \\right ) . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} \\| F _ { i l } ( x _ { i l } , u _ l ) - F _ { i l } ( \\tilde x _ { i l } , u _ l ) \\| & = \\sqrt { | c _ { i l } ' ( g _ { i l } ) - c _ { i l } ' ( \\tilde g _ { i l } ) | ^ 2 + | p ' _ l ( u _ l ) ( s _ { i l } - \\tilde s _ { i l } ) | ^ 2 } \\cr & \\le \\sqrt { L _ { i l } ^ 2 | g _ { i l } - \\tilde g _ { i l } | ^ 2 + \\bar p _ l ^ 2 | s _ { i l } - \\tilde s _ { i l } | ^ 2 } . \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } '' : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} \\tilde { u } _ { \\nu + 1 } \\tilde { u } _ { 0 } ^ { - 1 } = u _ { \\nu + 1 } u _ { 0 } ^ { - 1 } , 0 \\leq \\nu \\leq d - 2 , d \\geq 2 , \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d \\mathrm { x } } { d s } ( s ) \\ ; = \\ ; f \\left ( \\beta ( s ) \\right ) & \\mbox { i n } ( t , T ) , \\\\ \\mathrm { x } ( t ) \\ ; = \\ ; x , \\end{cases} \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} S ( \\omega ) = \\frac { 1 } { 2 \\pi } \\frac { \\sigma ^ 2 _ { \\sf w } } { | 1 - \\sum _ { \\ell = 1 } ^ p \\theta _ { \\ell } e ^ { - j \\ell \\omega } | ^ 2 } . \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} \\begin{aligned} & Z _ { s , s + k } \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & = \\left ( X _ { s + k } ^ \\prime , V _ { s + k } ^ \\prime \\right ) \\in \\mathcal { K } _ { s + k } \\cap \\mathcal { U } _ { s + k } ^ \\eta \\end{aligned} \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} \\sup f _ { m n } ( C ) & \\leq f _ { m n } ( x _ C ) + 1 \\\\ & \\leq \\inf f _ { m n } ( B _ C ) + \\frac { m n } { n } + 1 \\\\ & = \\inf f _ { m n } ( B _ C ) + m + 1 . \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} \\Re \\Big \\{ \\sum _ { n = 1 } ^ { \\infty } z _ n ^ m \\Big \\} \\leq \\frac { 1 } { \\alpha ^ m } - \\frac { 1 } { ( \\alpha + 1 - \\beta _ 1 ) ^ { 2 m } } + \\Re \\Big \\{ \\frac { \\delta ( \\chi ) + \\delta ( \\psi \\chi ) } { ( \\alpha + i \\gamma ' ) ^ { 2 m } } - \\frac { \\delta ( \\chi ) + \\delta ( \\psi \\chi ) } { ( \\alpha + 1 + i \\gamma ' - \\beta _ 1 ) ^ { 2 m } } \\Big \\} \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\| y ^ k - \\hat v _ i ^ k \\| \\leq \\theta \\beta ^ { k } \\sum _ { \\ell = 1 } ^ { N } \\| v _ \\ell ^ 0 \\| + \\theta N \\sum _ { s = 1 } ^ { k } \\beta ^ { k - s } \\alpha _ { s - 1 } C \\le \\theta \\beta ^ { k } M + \\theta N C \\sum _ { s = 1 } ^ { k } \\beta ^ { k - s } \\alpha _ { s - 1 } , \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} \\abs { \\ne { \\mathcal { T } } } & \\lesssim \\Vert \\nabla _ t ^ { \\perp } \\Delta _ t ^ { - 1 } \\ne { f } \\Vert _ { L ^ { 2 } H ^ { N } } \\Vert \\nabla _ t f \\Vert _ { L ^ { 2 } H ^ { N } } \\Vert A f \\Vert _ { L ^ { \\infty } L ^ { 2 } } \\\\ & \\lesssim \\Vert \\nabla _ L ^ { \\perp } \\Delta _ t ^ { - 1 } \\ne { f } \\Vert _ { L ^ { 2 } H ^ { N } } \\Vert \\nabla _ L f \\Vert _ { L ^ { 2 } H ^ { N } } \\Vert A f \\Vert _ { L ^ { \\infty } L ^ { 2 } } . \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} | f _ 0 ( P U _ { \\delta , \\xi } ) - f _ 0 ( U _ { \\delta , \\xi } ) | _ { 2 n \\over n + 2 } & = O \\ ( | P U _ { \\delta , \\xi } - U _ { \\delta , \\xi } | ^ p _ { 2 n \\over n - 2 } \\ ) + O \\ ( U _ { \\delta , \\xi } ^ { p - 1 } \\ ( P U _ { \\delta , \\xi } - U _ { \\delta , \\xi } \\ ) | _ { 2 n \\over n + 2 } \\ ) \\\\ & = O \\ ( \\delta \\ ) \\hbox { ( b e c a u s e $ n \\ge 5 $ ) . } \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} - A ( X ) = & \\tfrac { 1 } { ( g - 1 ) ! g ! } \\int _ { X ^ { g - 1 } } \\log \\| \\Lambda \\| ( P _ 1 + \\dots + P _ { g - 1 } ) \\Phi _ { \\Theta } ^ * \\nu ^ { g - 1 } ( P _ 1 , \\dots , P _ { g - 1 } ) \\\\ = & \\tfrac { 1 } { ( g ! ) ^ 2 } \\int _ { X ^ { g } } \\log \\| \\Lambda \\| ( P _ 1 + \\dots + P _ { g - 1 } ) \\Phi ^ * \\nu ^ { g } ( P _ 1 , \\dots , P _ g ) , \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} \\| W ( \\xi _ i \\otimes \\eta ) - ( \\xi _ i \\otimes \\eta ) \\| ^ 2 & = 2 \\| \\eta \\| ^ 2 - 2 ( W ( \\xi _ i \\otimes \\eta ) | \\xi _ i \\otimes \\eta ) \\\\ & = 2 \\| \\eta \\| ^ 2 - 2 ( \\omega _ { \\xi _ i } \\otimes \\omega _ \\eta ) W \\\\ & \\to 2 \\| \\eta \\| ^ 2 - 2 ( \\epsilon \\otimes \\omega _ \\eta ) W \\\\ & = 2 \\| \\eta \\| ^ 2 - 2 \\omega _ \\eta ( 1 ) = 0 . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial x _ 1 } X ( g ) - \\frac { \\partial g } { \\partial x _ 1 } X ( f ) = \\frac { \\partial g } { \\partial x _ 4 } - \\frac { \\partial f } { \\partial x _ 5 } + g Y ( f ) - f Y ( g ) , \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} F _ s \\left ( { t , s } \\right ) = t f \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) - \\frac { t } { s } f _ s \\left ( { \\frac { 1 } { t } , \\frac { 1 } { s } } \\right ) , \\end{align*}"}
-{"id": "9884.png", "formula": "\\begin{align*} \\limsup _ { k \\to \\infty } \\left ( | a _ { k , \\sigma } | \\right ) ^ { 1 / k } = 1 \\ , , \\limsup _ { j \\to \\infty } \\left ( | b _ j | \\right ) ^ { 1 / j } = 2 ^ { - 1 } \\ , , \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} h & = \\lim _ { m \\to \\infty } h _ m = \\lim _ { m \\to \\infty } h _ m \\circ I = h \\circ I , \\\\ \\hat k & = \\lim _ { m \\to \\infty } \\hat k _ m = \\lim _ { m \\to \\infty } \\hat k _ m \\circ I = \\hat k \\circ I , \\end{align*}"}
-{"id": "9913.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\mu ( \\pi ( N ( L , W ) ) ) > 0 & & \\mu ( \\pi ( S ( L , W ) ) ) = 0 . \\end{array} \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} R ( v ) = ( \\log v _ 1 , \\ldots , \\log v _ F ) \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} \\beta ( H _ p ) \\le \\beta _ { \\beta ( C _ p ) } ( C _ p \\times C _ p ) = p ^ 2 + p - 1 . \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} Z ( T ) = \\frac { 1 } { 1 - q T } . \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} x B _ { n } ^ d ( x ) = B _ { n + 1 } ^ 0 ( x ) + \\rho _ { ( d + 1 ) n + 1 } B _ { n } ^ 0 ( x ) . \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } u _ { \\rho } = a _ { \\rho } ^ { i j } ( u _ { \\rho } ) _ { x ^ { i } x ^ { j } } + f _ { \\rho } , t \\leq \\rho ^ { - 2 / { \\alpha } } T . \\end{align*}"}
-{"id": "9992.png", "formula": "\\begin{align*} \\norm { f } _ { H ^ { ( n + 1 ) / 2 } } ^ 2 = \\int _ { \\R ^ n } f ( x ) \\bigl [ ( I - \\Delta ) ^ { ( n + 1 ) / 2 } f \\bigr ] ( x ) \\ d x , \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} F _ \\gamma ( \\omega ) ( t ) - F _ \\gamma ( \\omega ) ( s ) = F _ \\gamma ( \\omega ( s + \\cdot ) ) ( t - s ) , \\ 0 \\le s \\le t . \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} Y _ { \\alpha , \\epsilon } = y _ n C ^ { 0 , \\alpha } + r ^ { 1 + 2 \\alpha - \\epsilon } C _ { \\ast } ^ { 0 , \\epsilon } . \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} [ \\bar { e } _ { 0 , k + 3 } , \\bar { e } _ { 0 , l } ] - ( 1 + d + d ^ { - 1 } ) [ \\bar { e } _ { 0 , k + 2 } , \\bar { e } _ { 0 , l + 1 } ] + ( 1 + d + d ^ { - 1 } ) [ \\bar { e } _ { 0 , k + 1 } , \\bar { e } _ { 0 , l + 2 } ] - [ \\bar { e } _ { 0 , k } , \\bar { e } _ { 0 , l + 3 } ] = 0 , \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} \\mathbf { \\mathcal { W } } ( u ) = ( W _ { h _ 0 } ( u ) , W _ { h _ 1 } ( u ) , \\ldots , W _ { h _ { 2 ^ { p - 1 } - 1 } } ( u ) ) = \\pm 2 ^ { \\frac { n } { 2 } } H _ { 2 ^ { p - 1 } } ^ { ( r ) } \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} & \\delta ^ * ( \\mu , r ) = 1 + \\frac { 1 - \\mu } { r } . \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } z } \\left ( \\det \\left ( \\mathrm { i d } - z \\cdot M \\right ) \\right ) = \\sum _ { t = 1 } ^ { N } \\sum _ { r = 1 } ^ { N } - m _ { t , r } \\cdot C _ { t , r } \\left ( z \\right ) . \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\mathcal { S ' } _ { \\chi } ( 1 ) , \\mathcal { S ' } _ { \\chi } ( - 1 ) , \\mathcal { S ' } _ { \\chi } ( - 2 ) : = 0 . \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ d ( 2 - 2 \\cos ( \\xi _ i ) ) = \\sum _ { i = 1 } ^ d ( 2 - 2 \\cos ( \\xi _ i + k ) ) , \\sum _ { i = 1 } ^ d ( 2 - 2 \\cos ( \\xi _ i ) ) = \\sum _ { i = 1 } ^ d ( 2 - 2 \\cos ( \\xi _ i - k ) ) , \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} T ^ n = T ^ n ( P + Q ) = P + T ^ n Q \\to P t \\to \\infty \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} f _ \\ell ( X ^ k , Y ) = \\left [ X _ 1 ^ { ( Y _ { \\ell } ) } , \\ldots , X _ k ^ { ( Y _ { \\ell } ) } \\right ] . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} g _ k ( x , y ) : = 2 + ( x - 2 ) \\cos ( k ) - \\sin ( k ) \\sqrt { x ( 4 - x ) } ( 2 y - 1 ) \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 3 n ^ { 2 } } \\left ( - w \\right ) ^ { n } } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } \\left ( - w q ^ { 3 } ; q ^ { 4 } \\right ) _ { n } } = \\frac { 1 } { \\left ( 1 + w q ^ { 3 } \\right ) } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 3 n ^ { 2 } } \\left ( - q ^ { 2 } w \\right ) ^ { n } } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } \\left ( - w q ^ { 7 } ; q ^ { 4 } \\right ) _ { n } } . \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} \\frac { F ( x _ { N } ) - F ^ * } { a _ N ^ 2 } + \\frac { \\tilde \\mu } { 2 } \\| x ^ * - v _ N \\| ^ 2 \\leq & \\frac { \\tilde \\mu } { 2 } \\| x ^ * - v _ 0 \\| ^ 2 + \\rho M ^ 2 \\sum _ { i = 1 } ^ N \\frac { 1 } { a _ i } + \\frac { N r M ^ 2 } { 2 } + \\sum ^ N _ { i = 1 } \\frac { \\delta _ i } { a _ i } \\\\ & - \\frac { \\tilde \\mu - \\mu } { 2 } \\sum _ { i = 1 } ^ N \\frac { \\| x _ i - y _ i \\| ^ 2 } { a _ i ^ 2 } + 2 \\sum _ { i = 1 } ^ N \\frac { \\varepsilon _ i } { a _ i ^ 2 } + \\sqrt { 2 \\tilde \\mu } \\sum _ { i = 1 } ^ N \\| x ^ * - v _ { i } \\| \\cdot \\sqrt { \\frac { \\delta _ i } { a _ i } } . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} \\chi ( G ) \\ge 1 + \\frac { n ^ + } { n ^ - } = \\frac { n ^ - + n ^ + } { n ^ - } = \\frac { n - n ^ 0 } { n - n ^ + - n ^ 0 } \\ge \\frac { n } { n - n ^ + } . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} \\dot { x } ( t ) = X _ 0 ( x ( t ) ) + \\sum _ { i = 1 } ^ { m } u _ { i } ( t ) X _ i ( x ( t ) ) , \\mbox { f o r a l m o s t e v e r y } t \\in [ 0 , T ] , \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} \\displaystyle \\min _ { j = 0 , 1 , \\ldots , N - 1 } \\| \\mathcal { G } _ t ( x _ j ) \\| ^ 2 \\leq \\frac { 1 } { N } \\sum ^ { N - 1 } _ { j = 0 } \\| \\mathcal { G } _ t ( x _ j ) \\| ^ 2 \\leq \\frac { 4 t ^ { - 1 } ( F ( x _ 0 ) - F ^ * + 4 L \\sum ^ { N } _ { j = 1 } \\varepsilon _ { j } ) } { N } , \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} \\Vert F \\Vert _ { \\mathcal { A } _ { \\alpha } ^ 2 ( \\Pi ^ + ) } ^ 2 = \\Vert f \\Vert _ { L _ { \\alpha + 1 } ^ 2 } ^ 2 = b _ { \\alpha } \\int _ 0 ^ { \\infty } \\vert f ( t ) \\vert ^ 2 t ^ { - ( \\alpha + 1 ) } \\ , d t , \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} \\sigma ( z + \\omega ) = - e ^ { \\eta ( \\omega ) \\left ( z + \\frac { \\omega } { 2 } \\right ) } \\sigma ( z ) , \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} \\frac { 1 } { \\Phi ( Q ) } \\sum _ { \\chi \\mod Q } \\bar { \\chi } ( A ) \\chi ( N ) = \\begin{cases} 1 & N = A \\mod Q \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} k _ { a , \\psi } ( \\xi , \\eta ) = \\sum _ { m \\in \\mathbb { Z } ^ d \\backslash \\{ 0 \\} } \\psi ( \\xi ) \\int _ { \\mathbb { R } ^ d } e ^ { i ( \\varphi _ a ( x , \\xi + 2 \\pi m ) - \\varphi _ a ( x , \\eta ) ) } d x \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} \\sum _ { x \\in \\eta _ { A _ J ^ c } } g _ J ( x ) = \\sum _ { x \\in \\eta _ { A _ J ^ c } } \\sum _ { j \\in J } \\tilde { l } _ \\alpha ( | y _ { c } ( B _ j ) - x | ) = \\sum _ { j \\in J } \\sum _ { x \\in \\eta _ { A _ J ^ c } } \\tilde { l } _ \\alpha ( | y _ { c } ( B _ j ) - x | ) = \\sum _ { j \\in J } \\tilde { \\Psi } _ { \\eta _ { A _ J ^ c } } ( y _ { c } ( B _ j ) ) , \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} a _ { h ( i ) } ( i ) = a _ { h ( 2 i ) } ( 2 i ) + 1 = a _ { h ( 2 i ) - 1 } ( 2 i ) = a _ { h ( 2 i + 1 ) } ( 2 i + 1 ) - 1 = a _ { h ( 2 i + 1 ) - 1 } ( 2 i + 1 ) . \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} \\partial _ { t } \\mathbf { u } - \\mathrm { d i v } ( \\mathbf { H } \\nabla \\mathbf { u } ) = \\mathbf { 0 } ( 0 , \\infty ) \\times G . \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} \\left ( q ; q \\right ) _ { \\infty } = \\frac { 1 } { \\sqrt { 2 \\pi \\log q ^ { - 1 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } \\right ) d x } { \\left | \\left ( - q ^ { 1 / 2 } e ^ { i x } ; q \\right ) _ { \\infty } \\right | ^ { 2 } } \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} E _ k ( r \\log \\N \\mathfrak { n } ) = ( 2 r ) ^ k ( \\N \\mathfrak { n } ) ^ { 1 / 2 - r } E _ k ( \\tfrac { 1 } { 2 } \\log \\N \\mathfrak { n } ) \\leq ( 2 r ) ^ { k } ( \\N \\mathfrak { n } ) ^ { 1 / 2 - r } . \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { T } \\bigl { ( } v ( t ) f ' ( u ( t ) ) u '' ( t ) + v ' ( t ) f ' ( u ( t ) ) u ' ( t ) \\bigr { ) } ~ \\ ! d t \\\\ & = \\Bigl { [ } v ( t ) f ' ( u ( t ) ) u ' ( t ) \\Bigr { ] } _ { t = 0 } ^ { t = T } - \\int _ { 0 } ^ { T } v ( t ) f '' ( u ( t ) ) u ' ( t ) ^ { 2 } ~ \\ ! d t \\\\ & = - \\int _ { 0 } ^ { T } v ( t ) f '' ( u ( t ) ) u ' ( t ) ^ { 2 } ~ \\ ! d t . \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} \\norm { ( z _ 1 , \\ldots , z _ n ) } = \\sqrt { \\sum _ { j = 1 } ^ n | z _ j | ^ 2 } . \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} \\Psi ( \\mu _ A ( f ) a ) ( y ) = \\left ( \\mu _ B ( \\psi ( f ) ) \\Psi ( a ) \\right ) ( y ) = f ( \\psi ^ * ( y ) ) \\Psi ( a ) ( y ) . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} h = \\inf \\frac { \\mu ( \\partial A ) } { \\mu ( A ) } , \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} X = \\left [ \\begin{array} { c c c c c c } 1 & \\cdots & 0 & 0 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\vdots & & \\vdots \\\\ 0 & \\cdots & 1 & 0 & \\cdots & 0 \\\\ 0 & \\cdots & 0 & X _ { 1 1 } & \\cdots & 0 \\\\ \\vdots & & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 & 0 & \\cdots & X _ { r r } \\end{array} \\right ] \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ m | x _ j | ^ 2 \\leqslant \\Big ( \\sum _ { j = 1 } ^ m | x _ j | \\big \\lVert v _ j - w _ j \\big \\rVert \\Big ) ^ 2 \\leqslant \\Big ( \\sum _ { j = 1 } ^ m | x _ j | ^ 2 \\Big ) \\Big ( \\sum _ { j = 1 } ^ m \\big \\lVert v _ j - w _ j \\big \\rVert ^ 2 \\Big ) . \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} \\left ( c , z ; q \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { S _ { n } \\left ( x ; q \\right ) z ^ { n } } { \\left ( c ; q \\right ) _ { n } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { \\left ( z ; q \\right ) _ { n } q ^ { \\binom { n } { 2 } } \\left ( - c \\right ) ^ { n } } { \\left ( q ; q \\right ) _ { n } } A _ { q } \\left ( x z q ^ { n } \\right ) . \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} d \\bar { s } ^ 2 = d \\varrho ^ 2 + \\sinh ^ 2 \\varrho \\ , \\sigma _ { i j } \\ , d \\xi ^ i d \\xi ^ j . \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} b = e , c ^ { t r } \\sigma + \\sigma c = f ^ { t r } \\sigma + \\sigma f . \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} c _ 1 ( \\delta _ k \\otimes c _ 2 \\otimes \\delta _ n \\otimes b ) & = \\delta _ k \\otimes \\alpha _ d ^ { - k } ( c _ 1 ) c _ 2 \\otimes \\delta _ n \\otimes b \\\\ & \\mapsto \\alpha _ d ^ k \\ ! \\left ( \\alpha _ d ^ { - k } ( c _ 1 ) c _ 2 \\right ) \\cdot \\delta _ { ( n , k ) } \\otimes b \\\\ & = c _ 1 \\left ( \\alpha _ d ^ k ( c _ 2 ) \\cdot \\delta _ { ( n , k ) } \\otimes b \\right ) . \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} H ^ 0 \\left ( G _ \\Sigma , D ^ * _ { \\rho _ { \\pmb { 1 } , 2 } } \\right ) ^ \\vee = H ^ 0 \\left ( G _ \\Sigma , \\hat { R [ [ \\Gamma ] ] } ( \\tau ^ { - 1 } \\varkappa \\kappa ) \\right ) ^ \\vee \\cong \\frac { R [ [ \\Gamma ] ] } { \\left ( \\tau ^ { - 1 } \\varkappa \\kappa ( \\gamma _ 0 ) - 1 \\right ) } . \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} \\frac { \\partial P _ { \\mu \\nu } } { \\partial x ^ { \\lambda } } + \\frac { \\partial P _ { \\nu \\lambda } } { \\partial x ^ { \\mu } } + \\frac { \\partial P _ { \\lambda \\mu } } { \\partial x ^ { \\nu } } = - \\frac { 4 \\pi i } { c } e _ { \\mu \\nu \\lambda \\sigma } j ^ { \\sigma } \\end{align*}"}
-{"id": "10050.png", "formula": "\\begin{align*} p + r = \\alpha q , q + r = \\beta p , p + q = \\gamma r . \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\vartheta = - 2 \\varepsilon , \\ : p > \\max \\{ \\frac { 2 N + 2 } { \\varepsilon } , \\frac { 4 N + 5 } { 2 } \\} , \\ : \\varepsilon > 1 - \\frac { 1 } { 2 p } . \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\mu _ { n , k } = ( p ^ n + q ^ n ) \\mu _ { n , k } + \\sum _ { j = 1 } ^ { n - 1 } { n \\choose j } p ^ j q ^ { n - j } ( \\mu _ { j , k - 1 } + \\mu _ { n - j , k - 1 } ) \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} ( \\Lambda _ { N } ( t ) - \\mid \\gamma + t \\mid ^ { 2 } ) ( \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) = 0 \\And ( \\Psi _ { N , t } , e ^ { i \\left \\langle \\gamma + t , x \\right \\rangle } ) = 0 \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} A _ { q } ( - q ^ { m } ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } + m n } } { ( q ; q ) _ { n } } = \\frac { ( - 1 ) ^ { m } q ^ { - \\binom { m } { 2 } } a _ { m } ( q ) } { ( q , q ^ { 4 } ; q ^ { 5 } ) _ { \\infty } } - \\frac { ( - 1 ) ^ { m } q ^ { - \\binom { m } { 2 } } b _ { m } ( q ) } { ( q ^ { 2 } , q ^ { 3 } ; q ^ { 5 } ) _ { \\infty } } . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} \\alpha = \\nabla _ x \\Psi ( t , x , u ; \\lambda , \\upsilon , \\mu , \\nu ) , \\beta = \\nabla _ u \\Psi ( t , x , u ; \\lambda , \\upsilon , \\mu , \\nu ) + \\zeta . \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} A _ a = \\begin{pmatrix} z _ a & 0 \\\\ 0 & \\cdot \\end{pmatrix} , 2 \\leq a \\leq 7 - r , \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\alpha } c _ n ( \\alpha ) \\ , \\frac { x ^ n \\sin ( n x ) } { \\sin ^ n x } \\ = \\ A _ \\alpha x ^ { 2 \\alpha - 1 } + O ( x ^ { 2 \\alpha + 1 } ) \\qquad \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} Y ( k ) = \\sum \\limits _ { t = 0 } ^ { l - 1 } { y _ t } e ^ { - i 2 \\pi t k / l } , k = 0 , 1 , \\ldots l - 1 \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} \\limsup _ { N \\rightarrow \\infty } \\left \\Vert \\left ( f _ N ^ { ( s ) } ( 0 , Z _ s ) - f _ 0 ^ { \\otimes s } ( Z _ s ) \\right ) \\mathbf { 1 } _ { Z _ s \\in \\mathcal { K } _ s \\cap \\mathcal { U } _ s ^ { \\eta ( \\varepsilon ) } } \\mathbf { 1 } _ { E _ s ( Z _ s ) \\leq R ^ 2 } \\right \\Vert _ { L ^ \\infty _ { Z _ s } } = 0 \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} & Z _ { 1 , R } = Z _ 1 \\cup _ Y Y _ { [ 0 , R ] } , Z _ { 2 , R } = Z _ 2 \\cup _ Y Y _ { [ - R , 0 ] } , \\\\ & Z _ { 1 , \\infty } = Z _ 1 \\cup _ Y Y _ { [ 0 , \\infty ) } , Z _ { 2 , \\infty } = Z _ 2 \\cup _ Y Y _ { ( - \\infty , 0 ] } , \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} s _ i = p _ i + \\frac { \\omega ^ { a _ { h ( i ) } ( i ) } } { 2 ^ { h ( i ) + 1 } } \\left ( \\sum _ { j = 0 } ^ { k - h ( i ) - 1 } z ^ j \\right ) ( z ^ { h ( i ) + 1 } - 1 ) \\qquad ( i = 1 , \\dots , 2 ^ k - 1 ) . \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} D _ n ( x ) - T _ n ( x ) = D ^ + _ { n + 1 } ( x ) - T ^ + _ { n + 1 } ( x ) = \\frac { 1 } { x } \\left ( T ^ - _ { n + 2 } ( x ) - D ^ - _ { n + 2 } ( x ) \\right ) = x ^ { \\frac { n + 1 } { 2 } } U _ { n + 1 } \\left ( \\frac { 1 } { 2 \\sqrt { x } } \\right ) . \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} & \\sum _ { n \\leq X } \\Re a ( n ) ^ 2 + \\Im a ( n ) ^ 2 = c _ { 1 } X ^ { \\gamma _ { 1 } } + O ( X ^ { \\eta _ { 1 } } ) , \\\\ & \\sum _ { n \\leq X } \\Re a ( n ) ^ 2 - \\Im a ( n ) ^ 2 + 2 i \\Re a ( n ) \\Im a ( n ) = c _ { 2 } X ^ { \\gamma _ { 2 } } + O ( X ^ { \\eta _ { 2 } } ) , \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} y ^ k = { \\frac { 1 } { N } \\sum _ { i = 1 } ^ N v _ i ^ k } \\mbox { f o r a l l } k \\geq 0 . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} \\varphi _ { n , k } ( r , \\theta ) = ( a \\cos ( n \\theta ) + b \\sin ( n \\theta ) ) J _ n ( \\alpha _ { n , k } r ) , \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} & \\delta _ { \\mathsf { C a - Z F } } = \\frac { K } { \\min \\{ M , K \\} } \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} I ( X ^ n \\rightarrow { Y } ^ n ) = \\int _ { { \\cal X } _ { 0 , n } \\times { \\cal Y } _ { 0 , n } } \\log \\Big ( \\frac { d { \\overrightarrow Q } _ { 0 , n } ( \\cdot | x ^ n ) } { d \\nu _ { 0 , n } ( \\cdot ) } ( y ^ n ) \\Big ) ( { \\overleftarrow P } _ { 0 , n } \\otimes { \\overrightarrow Q } _ { 0 , n } ) ( d x ^ n , d y ^ n ) \\equiv { \\mathbb { I } } _ { X ^ n \\rightarrow { Y ^ n } } ( { \\overleftarrow P } _ { 0 , n } , { \\overrightarrow Q } _ { 0 , n } ) \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} \\eta _ 1 = ( 1 - w _ { \\varepsilon } ) \\varphi \\qquad \\textrm { a n d } \\qquad \\eta _ 2 = w _ { \\varepsilon } \\varphi \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} | f _ { l e x } ( Y ' ) - f _ { l e x } ( X ' ) | \\leq ( b - k + 1 ) \\binom { b } { k - 1 } = k \\binom { b } { k } . \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} q _ i = q ^ { d _ i } , \\ , \\ , [ n ] _ q ! = \\prod _ { i = 1 } ^ n \\frac { q ^ n - q ^ { - n } } { q - q ^ { - 1 } } , \\ \\ E _ i ^ { ( n ) } = \\frac { E _ i ^ n } { [ n ] _ { q _ i } ! } \\ \\ \\ \\ F _ i ^ { ( n ) } = \\frac { F _ i ^ n } { [ n ] _ { q _ i } ! } . \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} P ' _ { 0 } ( z ) - \\left [ \\dfrac { \\lambda } { \\xi } + \\dfrac { \\gamma } { \\xi ( 1 - z ) } \\right ] P _ { 0 } ( z ) = - \\dfrac { \\mu } { \\xi ( 1 - z ) } p _ { 1 , 1 } . \\end{align*}"}
-{"id": "9861.png", "formula": "\\begin{align*} \\det _ { 1 \\leq i , j \\leq n } \\left \\{ ( 1 - x _ i z _ j ) ^ { - 1 } \\right \\} = \\frac { \\prod _ { i < j } ( x _ i - x _ j ) ( z _ i - z _ j ) } { \\prod _ { 1 \\leq i , j \\leq n } ( 1 - z _ i x _ j ) } \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} S _ j ( f ( x ) ) = \\sigma _ j ^ { m _ j } ( f ( x ) ) + b _ { j , m _ j - 1 } ( x ) \\sigma _ j ^ { m _ j - 1 } ( f ( x ) ) + \\ldots + b _ { j , 0 } ( x ) f ( x ) = 0 , \\ j = 1 , 2 \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} B ( G _ { n , k , b } ) \\sim k \\binom { b } { k } n \\rightarrow \\infty . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} [ h , l ] = \\bigwedge _ { s \\in S } \\hom ( h ( s ) , l ( s ) ) , \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} F _ { \\rm A S M } ( x , y ; z ) = x - \\frac { 1 } { z - 1 } y - r _ { \\rm A S M } ( z ) , z \\in [ 1 , + \\infty ) . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\Big ( \\int _ M | u | ^ p d \\mu \\Big ) = 0 & = \\frac { \\partial } { \\partial t } \\Big ( \\int _ M | u | ^ { p - 1 } u d \\mu \\Big ) \\\\ \\displaystyle & = \\int _ M ( p - 1 ) | u | ^ { p - 2 } u \\frac { \\partial u } { \\partial t } d \\mu + \\int _ M | u | ^ { p - 1 } \\frac { \\partial } { \\partial t } ( u d \\mu ) . \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} \\det R ( z ) & = \\exp \\big ( \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } \\log | 2 z _ 1 z _ 2 | + \\log | \\cos \\theta | d \\theta \\big ) \\\\ & = \\sqrt { 2 | z _ 1 z _ 2 | } \\exp \\big ( \\frac { 1 } { 4 \\pi } \\int _ 0 ^ { 2 \\pi } \\log | \\cos \\theta | d \\theta \\big ) \\\\ & = \\sqrt { 2 | z _ 1 z _ 2 | } \\exp \\big ( \\frac { 1 } { \\pi } \\int _ 0 ^ { \\pi / 2 } \\log \\cos \\theta d \\theta \\big ) \\\\ & = \\sqrt { 2 | z _ 1 z _ 2 | } / \\sqrt { 2 } \\\\ & = \\sqrt { | z _ 1 z _ 2 | } . \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} \\partial _ t ^ \\alpha ( u ^ { n + 1 } - u ^ n ) & = L _ { \\lambda _ 0 } ( u ^ { n + 1 } - u ^ n ) + ( \\lambda - \\lambda _ 0 ) ( L _ 1 - L _ 0 ) ( u ^ n - u ^ { n - 1 } ) \\\\ & ~ ~ ~ + \\partial _ { t } ^ { \\beta } \\int _ { 0 } ^ { t } \\Lambda ^ k _ { \\lambda _ 0 } ( u ^ { n + 1 } - u ^ n ) + ( \\lambda - \\lambda _ 0 ) ( \\Lambda _ { 1 } - \\Lambda _ 0 ) ( u ^ n - u ^ { n - 1 } ) d w _ { s } ^ { k } . \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} \\prod _ { p \\in \\{ \\} } ( a , b ) _ p = - 1 . \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( \\pm a t + c ) ^ 2 - t ^ 2 } , c = c o n s t , \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} \\Delta _ i ^ { s , t } ( f ) : = \\sum _ { k = 0 } ^ { p - 1 } x ^ { ( t ) } _ { i , k } \\partial ^ { ( s ) } _ { i , k } f \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} \\left \\langle J _ { a b } , J _ { c d } , \\bar { P } _ { e } \\right \\rangle & = \\frac { 1 } { \\sqrt { 2 } } \\left \\langle J _ { a b } , J _ { c d } , P _ { e } \\right \\rangle + \\frac { 1 } { \\sqrt { 2 } } \\left \\langle J _ { a b } , J _ { c d } , Z _ { e } \\right \\rangle , \\\\ & = \\left ( \\alpha _ { 0 } + \\alpha _ { 1 } \\right ) \\varepsilon _ { a b c d e } , \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{gather*} \\Delta _ { 0 } \\ ! : = \\ ! ( - 1 ) ^ { \\tfrac { n _ { 0 } \\ ! + \\ ! 1 } { 2 } } t _ { n _ { 0 } } \\ , , \\ \\Delta _ { k + 1 } \\ ! : = \\ ! ( - 1 ) ^ { \\tfrac { n _ { k \\ ! + \\ ! 1 } \\ ! - \\ ! n _ { k } } { 2 } } t _ { n _ { k \\ ! + \\ ! 1 } } t _ { n _ { k } } \\ , , \\ 0 \\ ! \\leq \\ ! k \\ ! < \\ ! m \\ ! - \\ ! 1 \\ , , \\ 2 \\ ! \\leq \\ ! m \\ ! \\leq \\ ! \\infty \\ . \\end{gather*}"}
-{"id": "6464.png", "formula": "\\begin{align*} \\lambda g _ { \\pm } + v \\partial _ { x } g _ { \\pm } \\mp \\beta _ { x } \\partial _ { v } g _ { \\pm } \\mp \\phi _ { x } v \\mu _ { \\pm , + } ^ { \\prime } & = 0 , \\ v > 0 \\\\ \\lambda g _ { \\pm } + v \\partial _ { x } g _ { \\pm } \\mp \\beta _ { x } \\partial _ { v } g _ { \\pm } \\mp \\phi _ { x } v \\mu _ { \\pm , - } ^ { \\prime } & = 0 , \\ v < 0 \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} R _ { n + 1 } ^ + ( x ) & = 2 n x R _ { n } ^ + ( x ) + 2 x ( 1 - 2 x ) \\frac { d } { d x } R _ { n } ^ + ( x ) + R _ { n } ( x ) , \\\\ R _ { n + 1 } ^ - ( x ) & = ( 2 n + 1 ) x R _ { n } ^ - ( x ) + 2 x ( 1 - 2 x ) \\frac { d } { d x } R _ { n } ^ - ( x ) + x R _ { n } ( x ) , \\\\ R _ { n + 1 } ( x ) & = ( 1 + ( 2 n + 1 ) x ) R _ { n } ( x ) + 2 x ( 1 - 2 x ) \\frac { d } { d x } R _ { n } ( x ) + x R _ { n } ^ - ( x ) , \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} W _ { n } \\left ( \\varphi ( x ) , \\varphi ( y ) \\right ) = q ^ { - n } \\left ( \\varphi _ { n + 1 } ( x ) \\varphi _ { n } ( y ) - \\varphi _ { n } ( x ) \\varphi _ { n + 1 } ( y ) \\right ) \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} \\lim _ { L \\to \\infty } \\sup _ { x \\in \\R ^ d } \\left \\| \\frac { 1 } { L ^ d } \\int _ { [ 0 , L ] ^ d } V ( x + y ) \\ , d y \\right \\| = 0 . \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} ( S _ i u ) ( n ) : = u ( n _ 1 , . . . , n _ i - 1 , . . . , n _ d ) , \\ n \\in \\Z ^ d \\ \\ u \\in \\mathcal { H } . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} u _ t ( x ) = ( \\mathcal { G } _ { D } u _ 0 ) _ t ( x ) + \\xi \\int _ { B _ R ( 0 ) } \\int _ { 0 } ^ { t } p _ D ( t - s , x , y ) \\sigma ( u _ s ( y ) ) F ( \\d s { , } \\d y ) , \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} \\tilde { \\Psi } _ { A _ o } ( y ) : = \\sum _ { x \\in \\xi } \\tilde { l } _ \\alpha ( | x - y | ) , \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\chi ( x ) = ( 1 - x ) \\left [ \\frac { x } { x - 2 } \\rho ^ \\prime ( x ) - \\frac { 2 } { ( x - 2 ) ^ 2 } \\rho ( x ) \\right ] , \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{gather*} g _ { a b } ( z ) = \\big \\langle Q _ { b } ^ { - 1 } v _ { 0 } , \\psi _ { a } ^ { - } ( z ) \\hat g v _ { 0 } \\big \\rangle / \\tau ( \\hat { g } ) , \\end{gather*}"}
-{"id": "9817.png", "formula": "\\begin{align*} \\sum _ { { t = 3 } \\atop { t \\neq 7 } } ^ { 8 } f _ t ( p ' ) = q ^ 3 ( q ^ 2 - 1 ) ( q - \\sqrt { 3 q } + 1 ) \\left ( \\frac { ( q + \\sqrt { 3 q } + 1 ) r } { | G _ { p ' } | } - k _ 7 \\right ) . \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} \\begin{cases} - d Y _ { t } = f ( t , \\eta _ { t } , y _ { t } , z _ { t } , y _ { t + \\delta ( t ) } , z _ { t + \\zeta ( t ) } ) d t - Z _ { t } d B _ { t } ^ { H } , \\ \\ \\ t \\in [ 0 , T ] ; \\\\ Y _ { t } = g ( \\eta _ { t } ) , \\ \\ Z _ { t } = h ( \\eta _ { t } ) , \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ t \\in [ T , T + K ] . \\end{cases} \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} V _ N = 2 \\log N - \\frac { 3 } { 2 } \\log \\log N + { \\rm c o n s t } + \\log Y + \\log Y ' + o ( 1 ) , \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} A _ 1 & = \\{ ( n - 2 ) p ( 1 - p ) ( 1 - \\varepsilon _ 1 ) < | X _ 1 | , | X _ 2 | < ( n - 2 ) p ( 1 - p ) ( 1 + \\varepsilon _ 1 ) \\} , \\\\ A _ 2 & = \\{ | Y | < ( n - 2 ) p ^ 2 ( 1 + \\varepsilon _ 2 ) \\} , \\\\ A _ 3 & = \\{ C _ 1 , C _ 2 < s p ( 1 + \\varepsilon _ 3 ) \\} , \\\\ \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} ( A ^ m ) _ { k , \\ell } = 0 , \\textrm { f o r e v e r y } m < \\xi . \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + \\delta + t , x \\right \\rangle } ) = \\frac { 1 } { d ( \\gamma , \\delta ) } \\sum _ { \\gamma _ { 1 } \\in \\Gamma ( k + ) } q _ { \\gamma _ { 1 } } ( \\Psi _ { \\gamma + t } , e ^ { i \\left \\langle \\gamma + \\delta - \\gamma _ { 1 } + t , x \\right \\rangle } ) \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} \\norm { f } _ p = \\left ( \\int _ M f ^ p \\right ) ^ \\frac { 1 } { p } \\leq e ^ { 5 C T ^ * } ( \\abs { M _ 0 } + 1 ) \\sup _ { 0 \\leq \\sigma \\leq 1 / 2 } \\sup _ { M _ 0 } f _ \\sigma . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} { \\bf E } \\Bigl [ \\exp \\bigl ( q \\log \\beta _ { M , N } ( a , b ) \\bigr ) \\Bigr ] = \\exp \\Bigl ( \\int \\limits _ 0 ^ \\infty ( e ^ { - t q } - 1 ) e ^ { - b _ 0 t } \\frac { \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } \\frac { d t } { t } \\Bigr ) . \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} m _ { \\Delta } : h ^ H ( R ) \\times h ^ H ( R ) & \\to h ^ H ( R ) \\\\ ( f _ 1 , f _ 2 ) & \\mapsto ( f 1 * f _ 2 ) ( h ) : = f _ 1 ( h _ 1 ) f _ 2 ( h _ 2 ) . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} q ( n , k ) = \\sum _ { \\overline { A } \\in \\overline { \\mathfrak { B } } _ { n , k } } | \\overline { A } | \\prod _ { i = 0 } ^ { n - 2 } \\left [ \\left ( n - i \\right ) ! \\right ] ^ { \\psi _ i ( \\overline { A } ) } \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} \\sum _ { \\gamma \\in E _ Z } S _ \\gamma P _ A S _ \\gamma ^ * = P _ B , \\sum _ { \\gamma \\in E _ Z } S _ \\gamma P _ B S _ \\gamma ^ * = P _ A . \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} a _ { 0 } & = \\xi _ { 0 } , \\\\ a _ { k } & = \\xi _ { k } - \\omega _ { k } , \\ ; \\ ; \\ ; \\ ; \\ ; k \\geq 1 . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} a ( t ) \\begin{cases} = C _ 1 t ^ { - 1 } & \\\\ \\ge C _ 1 t ^ { - 1 } & \\\\ \\end{cases} \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} \\overline { \\varphi } ( y + \\gamma ) = \\overline { \\varphi } ( y ) + \\alpha _ { \\gamma } ( y ) . \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} \\liminf _ { E \\cap D \\ni y \\to x } w ( y ) \\geq w ( x ) \\geq v ( x ) = \\limsup _ { E \\cap D \\ni y \\to x } v ( y ) \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} \\nu ^ { { \\pi ^ { * , \\infty } } } ( 0 ) = \\frac { 1 + 2 ^ { \\mu _ 0 + \\Delta { C } ^ { \\infty } } } { 1 + 2 ^ { \\mu _ 0 + \\mu _ 1 + 2 \\Delta { C } ^ { \\infty } } + 2 ^ { \\mu _ 0 + 1 + \\Delta { C } ^ { \\infty } } } , ~ \\nu ^ { { \\pi ^ { * , \\infty } } } ( 1 ) = \\frac { 2 ^ { \\mu _ 0 + \\Delta { C } ^ { \\infty } } ( 1 + 2 ^ { \\mu _ 1 + \\Delta { C } ^ { \\infty } } ) } { 1 + 2 ^ { \\mu _ 0 + \\mu _ 1 + 2 \\Delta { C } ^ { \\infty } } + 2 ^ { \\mu _ 0 + 1 + \\Delta { C } ^ { \\infty } } } . \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} \\chi ( x ) = ( 1 - x ) \\frac { d } { d x } \\left [ x \\varphi ( x ) \\right ] , \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} \\sum _ { k } \\int w \\left ( e _ { \\pm } \\right ) \\left \\vert \\phi _ { k } ^ { \\pm } ( I _ { \\pm } ) \\right \\vert ^ { 2 } d I _ { \\pm } & = \\int \\int w \\left ( e _ { \\pm } \\right ) \\left \\vert \\phi \\left ( x \\right ) \\right \\vert ^ { 2 } d x d v \\\\ & \\leq \\sup _ { x } \\int w \\left ( e _ { \\pm } \\right ) d v \\left \\Vert \\phi \\right \\Vert _ { L ^ { 2 } } ^ { 2 } \\lesssim \\left \\Vert \\phi \\right \\Vert _ { L ^ { 2 } } ^ { 2 } . \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ( f \\ , H ) - ( f \\ , H ) [ | A | ^ { 2 } - m C ] = 0 , \\\\ A \\ , ( { \\rm g r a d } ( f \\ , H ) ) + \\frac { m } { 2 } ( f \\ , H ) { \\rm g r a d } \\ , H = 0 . \\end{cases} \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} z ^ 2 = y ^ 3 + a x ^ 4 y + b x ^ 6 . \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} \\sum _ { j \\in \\Z } J _ { j } ^ { 2 } ( z ) = 1 , \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} v _ { e a } ^ 2 ( g _ 1 ) = P ^ * ( g _ 1 ) \\tilde { r } ( g _ 1 ) = \\left [ 4 + 5 p , 4 + 5 p \\right ] ^ T . \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} & \\frac 1 2 \\| u ( t ) \\| _ 2 ^ 2 + \\Gamma _ 2 \\int _ 0 ^ t \\| \\Delta u \\| _ 2 ^ 2 \\ , d s + \\beta \\int _ 0 ^ t \\| u \\| _ 4 ^ 4 \\ , d s = \\frac 1 2 \\| u _ 0 \\| _ 2 ^ 2 + \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ n } f u \\ , d x d s \\\\ & \\qquad \\strut + \\Gamma _ 0 \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ n } u \\Delta u \\ , d x d s - \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ n } u M u \\ , d x d s + \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ n } u N ( u ) \\ , d x d s \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} & \\widetilde { \\mathbf { s c r } } ( x _ i , 0 ) = 2 ; \\\\ & \\widetilde { \\mathbf { s c r } } ( x , t ) \\geq 1 , \\forall \\ ; x \\in M _ i , \\ ; t \\in [ - 4 L , 4 L ] . \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} \\mathbb { P } \\{ D _ { n } = m \\} = [ z ^ { n - 1 } v ^ { m } ] F ' ( z , v ) = \\frac { 1 } { 2 \\pi i } \\oint \\frac { ( 1 - ( 1 - z ) ^ { p } ) ^ { m - 1 } } { z ^ { n } ( 1 - z ) ^ { 1 - p } } d z , \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} b ( 1 + b ) A _ 2 = ( b - 1 ) ( b - 2 ) \\lambda ^ 2 . \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} \\lambda _ { p , 1 } ( u ( t ) , t ) : = - \\int _ M u ( t ) \\Delta _ p u ( t ) d \\mu _ { g ( t ) } = \\int _ M | \\nabla u ( t ) | ^ p d \\mu _ { g ( t ) } , \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} x ( \\mathbb { P } ) _ { n - 1 } = ( J ) _ { n } ( \\mathbb { P } ) _ { n - 1 } \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} j = - ( s + \\log _ { 1 / p } ( 1 + ( p / q ) ^ s ) + 1 ) . \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} \\Psi ( x ) = M ( x ) + R ( x ) , \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} \\theta _ { \\mu } ' ( t ) = \\mu \\dfrac { v ' ( t ) ^ { 2 } - v ( t ) v '' ( t ) } { \\mu ^ { 2 } v ( t ) ^ { 2 } + v ' ( t ) ^ { 2 } } . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} \\chi _ { e } ( \\sigma ) : = \\left \\{ \\begin{array} { l } 1 , \\quad \\mathrm { i f \\ } e \\mathrm { \\ i s \\ t h i c k } , \\\\ 0 , \\quad \\mathrm { i f \\ } e \\mathrm { \\ i s \\ t h i n } , \\end{array} \\right . \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} \\psi ( z ) = \\frac { \\rho } { 2 } \\left ( R z + \\frac { 1 } { R z } \\right ) + c , \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} \\mathcal { M R } _ { a } ( \\mathcal { H } ) : = \\big \\{ u \\in L ^ { 2 } ( 0 , T ; \\mathcal { V } ) \\cap H ^ { 1 } ( 0 , T ; \\mathcal { H } ) \\ , | \\ , \\mathcal { A } u \\in L ^ { 2 } ( 0 , T ; \\mathcal { H } ) \\big \\} . \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} m _ \\lambda [ X + 1 ] - m _ \\lambda [ X ] = \\sum _ { j = 1 } ^ r m _ { \\lambda _ 1 , \\lambda _ 2 , \\ldots , \\widehat { \\lambda _ { i _ j } } , \\ldots , \\lambda _ r } [ X ] , \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} \\frac { ( - q ; q ^ 2 ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\sum _ { n \\geq 0 } \\frac { ( - 1 ) ^ n q ^ { n ^ 2 + n } } { ( - q ; q ^ 2 ) _ { n + 1 } } = \\frac { ( - q ; q ^ 2 ) _ \\infty } { ( q ^ 2 ; q ^ 2 ) _ \\infty } \\left ( \\sum _ { j \\geq 0 } \\frac { q ^ { 4 j ^ 2 + 2 j } ( 1 - q ^ { 4 j + 2 } ) } { ( - q ; q ^ 2 ) _ { 2 j + 1 } } + \\sum _ { j \\geq 1 } \\frac { q ^ { 4 j ^ 2 + 2 j - 1 } } { ( - q ; q ^ 2 ) _ { 2 j } } \\right ) . \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} \\| ( \\sum _ { n = 1 } ^ \\infty | M _ { n , 1 } x | ^ 2 + | M _ { n , 2 } x | ^ 2 ) ^ \\frac 1 2 \\| _ { L ^ p } \\simeq ^ { c _ p } \\| x \\| _ { L ^ p } \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} \\left ( a ; q \\right ) _ { n } = \\frac { \\left ( - a q ^ { - 1 / 2 } \\right ) ^ { n } } { \\sqrt { \\pi \\log q ^ { - 2 } } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q ^ { 2 } } + i n x \\right ) } { \\left ( q / a , - q ^ { 1 / 2 } / \\left ( a e ^ { i x } \\right ) ; q \\right ) _ { \\infty } } d x , \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} & \\frac { \\partial } { \\partial \\pi _ n } \\Big \\{ \\sum _ { x _ n , y _ { n } } \\log \\Big ( \\frac { r _ { n } ( x _ n | y ^ { n - 1 } _ { n - M } , y _ n ) } { \\pi _ { n } ( x _ n | y ^ { n - 1 } _ { n - J } ) } \\Big ) q _ n ( y _ n | y ^ { n - 1 } _ { n - M } , x _ n ) \\pi _ n ( x _ n | y ^ { n - 1 } _ { n - J } ) \\\\ & - s \\sum _ { x _ n } \\gamma _ n ( x _ n , y ^ { n - 1 } _ { n - N } ) \\pi _ n ( x _ n | y ^ { n - 1 } _ { n - J } ) + \\lambda _ n ( y ^ { n - 1 } _ { n - J } ) \\Big { ( } \\sum _ { x _ n } \\pi _ { n } ( x _ n | y ^ { n - 1 } _ { n - J } ) - 1 \\Big { ) } \\Big \\} = 0 , ~ \\forall { x _ n } \\in { \\cal X } _ n \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ { \\lfloor \\lambda \\rfloor } \\frac { \\lambda ^ l } { l ! } e ^ { - \\lambda } + \\sum _ { l = \\lfloor \\lambda \\rfloor + 2 } ^ { k - 1 } \\frac { \\lambda ^ l } { l ! } e ^ { - \\lambda } < U \\leq \\sum _ { l = 0 } ^ { \\lfloor \\lambda \\rfloor } \\frac { \\lambda ^ l } { l ! } e ^ { - \\lambda } + \\sum _ { l = \\lfloor \\lambda \\rfloor + 2 } ^ { k } \\frac { \\lambda ^ l } { l ! } e ^ { - \\lambda } \\end{align*}"}
-{"id": "10054.png", "formula": "\\begin{align*} \\gamma ^ 2 \\beta = 2 + 2 \\gamma + \\beta \\iff \\beta ( \\gamma - 1 ) = 2 . \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} | - 1 + l \\tau | ^ 2 = 1 + l ^ 2 | \\tau | ^ 2 - 2 l \\Re ( \\tau ) > 1 + l ^ 2 - l \\geq 1 . \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} \\bar { k } = \\textrm { e s s s u p } _ { ( \\omega , t , x ) \\in \\Omega \\times \\partial _ { p } \\mathcal { O } _ t } u ^ + + \\textrm { e s s s u p } _ { ( \\omega , t , x ) \\in \\Omega \\times \\mathcal { O } _ t } \\hat { \\xi } ^ + . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} : a ( z ) b ( z ) : = a ( z ) _ + b ( z ) + b ( w ) a ( z ) _ - , \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\mathcal { F } = \\big \\{ Q ^ k \\subset \\Omega _ T : Q ^ k = B _ { r _ k } ( x _ k ) \\times ( \\tau _ k , T ) , r _ k , \\tau _ k \\in \\Q , x _ k \\in \\Q ^ n \\big \\} . \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} n A _ n = - n - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\left ( 2 i - 1 \\right ) \\Big ( \\log F _ 0 ( e _ i ) + \\log \\left ( 1 - F _ 0 ( e _ i ) \\Big ) \\right ) . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} \\varphi = - \\left ( 1 - \\frac { \\kappa \\gamma ^ { 2 } } { 1 + \\kappa } \\right ) \\operatorname { d i v } \\mathbf { Z } ^ { \\left ( e \\right ) } + \\frac { \\kappa \\gamma ^ { 2 } } { \\left ( 1 + \\kappa \\right ) c } \\mathbf { v } \\cdot \\left ( \\frac { \\partial \\mathbf { Z } ^ { \\left ( e \\right ) } } { c \\partial t } + \\operatorname { c u r l } \\mathbf { Z } ^ { \\left ( m \\right ) } \\right ) \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{gather*} D _ { n - 1 } D _ { n } \\ x P _ n ( x ) \\ ! = \\ ! D _ { n - 1 } ^ { \\ , 2 } \\ , P _ { n + 1 } ( x ) \\ ! + \\ ! \\left ( D _ { n - 1 } D _ { n + 1 } ^ { \\ , \\prime } \\ ! - \\ ! D _ { n } D _ { n } ^ { \\ , \\prime } \\right ) \\ , P _ { n } ( x ) \\ ! + \\ ! D _ { n } ^ { \\ , 2 } \\ , P _ { n - 1 } ( x ) \\ , \\ \\ \\end{gather*}"}
-{"id": "4264.png", "formula": "\\begin{align*} \\mu ( Q ) = \\mu ( T ) - \\mu ( Q _ q ) . \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} \\begin{cases} \\frac { d } { d t } Z _ s ( t ) = \\left ( V _ s ( t ) , 0 \\right ) \\ ; \\ ; \\textnormal { i f } \\ ; \\ ; Z _ s ( t ) \\notin \\partial \\mathcal { D } _ s \\\\ Z _ s ( t ^ + ) = \\left ( Z _ s ( t ^ - ) \\right ) ^ * \\ ; \\ ; \\textnormal { i f } Z _ s ( t ) \\in \\partial \\mathcal { D } _ s \\end{cases} \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} \\sum \\limits _ { t = 1 } ^ { k } \\sum \\limits _ { j = 1 } ^ { n } \\left | \\left [ A ^ { k - t } \\right ] _ { i j } - \\frac { 1 } { n } \\right | & \\leq \\frac { 4 \\log n } { 1 - \\lambda } \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\mathcal { D } _ n = \\{ A : \\exists E > 0 \\mbox { a n d } A ^ { \\prime } \\mbox { s u c h t h a t } E \\times Y \\subset A ^ { \\prime } , \\mu ( A \\triangle A ^ { \\prime } ) < \\frac { 1 } { n } \\mu ( E ) \\} . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } e ^ { i t H _ 0 } J _ a ^ * e ^ { - i t H } v = : \\Omega ^ a v . \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} 0 \\in \\partial \\left ( f ( x ) + r ( x ) + \\iota _ S ( x ) \\right ) = \\nabla f ( x ) + \\partial r ( x ) + \\partial \\iota _ S ( x ) , \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} & ( 1 \\otimes _ \\nabla X _ j ) \\left ( \\delta _ k \\otimes c \\otimes \\delta _ n \\otimes b \\right ) = \\delta _ k \\otimes c \\otimes X _ j \\delta _ n \\otimes b + \\delta _ k \\otimes 1 \\otimes [ X _ j , c ] \\delta _ n \\otimes b \\end{align*}"}
-{"id": "10092.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ p y ^ q ( b y + c z ) ^ r } { z ^ { p + q + r } } \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} \\omega _ { \\xi , \\eta } ( ( \\iota \\otimes m ) \\Delta ( x ) ) & = \\lim _ i \\langle W ^ * ( 1 \\otimes x ) W ( \\xi \\otimes \\xi _ i ) | ( \\eta \\otimes \\xi _ i ) \\rangle \\\\ & = \\lim _ i \\langle ( 1 \\otimes x ) ( \\xi \\otimes \\xi _ i ) | ( \\eta \\otimes \\xi _ i ) \\rangle \\\\ & = \\omega _ { \\xi , \\eta } ( 1 ) \\lim _ { i } \\omega _ { \\xi _ i } ( x ) = \\omega _ { \\xi , \\eta } ( 1 ) m ( x ) . \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} \\frac { \\tilde { \\mu } _ { \\max } } { \\tilde { \\mu } _ 0 } \\sum _ { j = 1 } ^ N \\frac { ( \\alpha ^ { - 1 } - 1 ) } { 2 t _ j } \\cdot \\frac { \\norm { x _ j - y _ j } ^ 2 } { a _ j ^ 2 } & \\le \\frac { \\tilde { \\mu } _ { \\max } } { \\tilde { \\mu } _ 0 } \\left ( \\frac { \\tilde { \\mu } _ 0 } { 2 } \\norm { x ^ * - v _ 0 } ^ 2 + \\rho M ^ 2 \\left ( \\sum _ { j = 1 } ^ N \\frac { 1 } { a _ j } \\right ) + \\frac { N r M ^ 2 } { 2 } \\right ) . \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\mathrm { T r } ^ N _ M \\left ( ( g E _ { \\lambda , N , \\overline { \\chi } } ) ^ \\mu \\right ) = \\sum _ { i = 1 } ^ { d } c '' _ i f _ i , \\ \\ \\textrm { f o r s o m e } \\ c '' _ i \\in \\mathbf C . \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } X _ s ^ { t , x } = \\ x + \\int _ t ^ s b ( r , X _ r ^ { t , x } ) \\mathrm d r + \\int _ t ^ s \\mathrm d W _ r , \\\\ Y _ s ^ { t , x } = \\Phi ( X _ T ^ { t , x } ) - \\int _ s ^ T Z _ r ^ { t , x } \\mathrm d W _ r + \\int _ s ^ T f ( r , X _ r ^ { t , x } , Y _ r ^ { t , x } , Z _ r ^ { t , x } ) \\mathrm d r , \\\\ \\forall s \\in [ t , T ] . \\end{array} \\right . \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} \\rho \\ddot { u } _ i & = \\big ( A _ { i J K j } u _ { j , K } - \\beta _ { J i } \\dot { \\tau } - ( C _ { i J K L I j } u _ { j , I L } + M _ { i J K L } \\tau _ { , L } ) _ { , K } \\big ) _ { , J } - E _ { i j } \\dot { u } _ j , \\\\ a \\ddot { \\tau } & = - \\beta _ { K i } \\dot { u } _ { i , K } + m _ { I J } \\dot { \\tau } _ { , I J } + M _ { j L K I } u _ { j , L K I } + K _ { I J } \\tau _ { , I J } \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} V ^ { * } B ( \\alpha ) V = - B ( - \\alpha ) \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} \\begin{aligned} U _ Y & = J \\Omega _ { [ 1 ] } , \\\\ J \\Omega _ { [ i + 1 ] } & = K \\Omega _ { [ i ] } , \\\\ U _ T & = K \\Omega _ { [ n ] } , \\end{aligned} i = 1 , \\dots , n - 1 , \\end{align*}"}
-{"id": "10117.png", "formula": "\\begin{align*} \\omega = ( b y z ( q + r ) + q c z ^ 2 ) d y - ( ( p + q + r ) b y ^ 2 + c ( p + q ) y z ) d z . \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} z = 1 + s _ { 1 } \\mu _ { \\nu } + s _ { 2 } \\mu _ { \\nu } ^ { 2 } + \\dots + s _ { k } \\mu _ { \\nu } ^ { k } \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} w ( t ) [ x ] = t ^ { - \\frac { d } { 2 } } A ( t , x ) s _ a ( x , \\xi ( x , t ) ) F u ( \\xi ( x , t ) ) + t ^ { - \\frac { d } { 2 } - 1 } r ( t , x ) , \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} \\left | \\frac { \\partial _ { \\xi } M _ 1 ( k , \\xi ) } { M _ 1 ( k , \\xi ) } \\right | & = \\left | \\int _ 0 ^ t \\frac { 2 \\abs { k } ( \\xi - k s ) } { ( k ^ 2 + | \\xi - k s | ^ 2 ) ^ 2 } \\ , d s \\right | \\le \\frac { 2 } { \\abs { k } ^ 2 } \\int _ 0 ^ t \\frac { 1 } { ( 1 + | \\xi / k - s | ^ 2 ) } \\ , d s , \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} ( f _ { u u } \\cos ^ 2 \\theta + 2 f _ { u v } \\cos \\theta \\sin \\theta + f _ { v v } \\sin ^ 2 \\theta ) \\big | _ { u = v = 0 } = 0 \\ ; . \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{gather*} \\big \\langle Q ^ { \\pm 1 } v _ { 0 } , \\psi ^ { \\pm } ( z ) v _ { 0 } \\big \\rangle = \\left \\langle \\psi ^ { \\pm } _ { ( - 1 ) } v _ { 0 } , \\sum _ { k \\in \\mathbb { Z } } z ^ { k } \\psi ^ { \\pm } _ { ( - k - 1 ) } v _ { 0 } \\right \\rangle = 1 , \\end{gather*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\Psi ( A ) = - e ^ { \\frac { 1 } { A } } \\ , \\mathrm { E i } \\left ( - \\frac { 1 } { A } \\right ) \\approx \\begin{cases} A \\ : , & \\\\ \\ln ( 1 + A ) - \\gamma \\ : , & \\end{cases} \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} 1 - \\frac { 1 } { i } = 1 , \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} | \\rho _ i | ^ 2 X _ { 1 1 , i } - \\rho _ i X _ { 1 1 , i } J _ { s _ i } ^ T - \\bar \\rho _ i J _ { s _ i } X _ { 1 1 , i } + J _ { s _ i } X _ { 1 1 , i } J _ { s _ i } ^ T = X _ { 1 1 , i } . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} \\frac { 1 } { \\lambda _ u ^ t \\cdot t ^ { d _ u + 1 } } M ^ t \\vec v _ 0 & = \\frac { 1 } { \\lambda _ u ^ t \\cdot t ^ { d _ u + 1 } } \\left ( \\lambda ^ t \\vec v _ t + \\sum _ { j = 0 } ^ { t - 1 } \\lambda ^ { j } M ^ { t - j - 1 } \\vec u _ { j + 1 } \\right ) \\\\ & = \\frac { 1 } { t ^ { d _ u + 1 } } \\vec v _ t + \\frac { 1 } { \\lambda } \\sum _ { j = 0 } ^ { t - 1 } \\frac { ( { t - j - 1 } ) ^ { d _ u } } { t ^ { d _ u + 1 } } \\frac { 1 } { \\lambda ^ { t - j - 1 } \\cdot ( { t - j - 1 } ) ^ { d _ u } } M ^ { t - j - 1 } \\vec u _ { j + 1 } \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { d + 1 } x _ j \\phi ( { \\bf v } _ j ) = \\sum _ { i = 1 } ^ \\ell \\left ( \\sum _ { j = c _ { i - 1 } + 1 } ^ { c _ i } x _ j ( { \\bf e } _ i + { \\bf w } _ j ) \\right ) = \\sum _ { i = 1 } ^ \\ell \\left ( \\sum _ { j = c _ { i - 1 } + 1 } ^ { c _ i } x _ j \\right ) { \\bf e } _ i + \\sum _ { j = 1 } ^ { d + 1 } x _ j { \\bf w } _ j \\in \\Z ^ { d + 1 } , \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} A ^ { * } = \\bigcap _ { n \\geq 0 } \\mathcal { B } ^ { n } ( A ) = \\bigcap _ { n \\geq 0 } \\ , \\ , \\bigcup _ { \\xi \\in \\Sigma _ { k } ^ { + } } T _ { \\xi _ { 0 } } \\circ \\cdots \\circ T _ { \\xi _ { n - 1 } } ( A ) . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\theta \\circ T ^ n ( a ) = \\omega _ \\xi ( a ) \\ , . \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { 1 } { k } \\| v ^ k ( \\ell ) - [ y ^ { k } ] _ \\ell \\mathbf { 1 } \\| < \\infty \\mbox { a . s . } \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} g ( P ( x , y , z ) , w ) = \\Phi ( x , y , z , w ) . \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} K : = \\biguplus _ { m \\in \\omega } K _ m \\uplus \\{ b \\} . \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} \\mathcal { B } _ { r } ^ c ( \\theta ^ * ) & = \\bigcup _ { l \\ge 1 } ^ { { { L _ r } } - 1 } \\bigcup _ { z _ m \\in \\mathcal { N } _ l ( \\delta _ l ) } \\mathcal { F } _ { l , m } \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\left \\| d _ G ( \\cdot , y _ 0 ) ^ { - 2 } E ^ { y _ 0 , i j } _ 2 \\right \\| _ { L ^ \\infty ( \\mathcal { B } _ { 1 } ^ + ( y _ 0 ) ) } + \\left [ d _ G ( \\cdot , y _ 0 ) ^ { - ( 2 - \\epsilon ) } E ^ { y _ 0 , i j } _ 2 \\right ] _ { \\dot { C } ^ { 0 , \\epsilon } _ \\ast ( \\mathcal { B } _ { 1 } ^ + ( y _ 0 ) ) } \\leq C . \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} \\eta _ { i ; j } = \\left ( - r ^ { - 2 } ( d ^ 2 ) _ { i ; j } + C u _ { i ; j } \\right ) g ' + \\left ( - r ^ { - 2 } ( d ^ 2 ) _ i + C u _ i \\right ) \\left ( - r ^ { - 2 } ( d ^ 2 ) _ j + C u _ j \\right ) g '' . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\mu ^ { ( m ) } & = 2 ^ { - m } \\delta _ { t , t } + \\sum _ { k = 1 } ^ m \\big ( \\tfrac { 1 } { 3 } 2 ^ { k - m } \\delta _ { 2 ^ { - k } t , - 2 ^ { - k } t } + \\tfrac { 1 } { 6 } 2 ^ { k - m } \\delta _ { - 2 ^ { 1 - k } t , 2 ^ { 1 - k } t } \\big ) \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} f = \\sum _ { i \\in I } f _ i , \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} R _ n \\stackrel { d } { = } \\sum _ { i = 1 } ^ { n - 1 } { \\bf 1 } _ { A _ i } Q _ n \\stackrel { d } { = } \\sum _ { i = 1 } ^ { n - 2 } { \\bf 1 } _ { B _ i } , \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} \\Gamma _ 1 ( w \\ , | \\ , a ) & = \\frac { a ^ { w / a - 1 / 2 } } { \\sqrt { 2 \\pi } } \\ , \\Gamma ( w / a ) , \\\\ \\Gamma _ 0 ( w ) & = 1 / w . \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} f ( 1 - m ) + f ( m ) + 1 = 0 . \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} k _ L = \\log _ { 1 / q } n + ( 1 + \\epsilon ) \\psi ( n ) , & & k _ U = \\log _ { 1 / q } n + ( 1 - \\epsilon ) \\psi ( n ) . \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} \\left ( E _ { \\alpha , \\beta } ^ \\gamma ( t ) \\right ) '' = \\gamma ( \\gamma + 1 ) \\ , E _ { \\alpha , 2 \\alpha + \\beta } ^ { \\gamma + 2 } ( t ) \\ , , \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} \\mathbf { n } ^ { T } ( \\mathbf { H } \\nabla \\mathbf { u } ) = \\mathbf { 0 } ( 0 , \\infty ) \\times \\partial G \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} \\gamma _ { 2 n } = \\sum \\limits _ { i = 0 } ^ n \\frac { 2 } { 2 i + r } { - 2 i - r \\choose - 2 n - r } B _ { 2 n - 2 i } \\gamma _ { 2 i + 1 } \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} E _ { k , l } \\left ( \\{ \\alpha ^ { - 1 } q ^ { m } + \\alpha q ^ { - m } \\} \\right ) = \\frac { v _ { m , k } v _ { m , l } } { \\| \\textbf { \\textit { v } } _ { m } \\| ^ { 2 } } = \\left ( 1 - \\alpha ^ { - 2 } q ^ { 2 m } \\right ) \\frac { \\alpha ^ { 2 m } q ^ { - m ( m + 1 ) } } { ( q ; q ) _ { \\infty } ^ { 2 } } f _ { k } \\left ( \\alpha ^ { - 1 } q ^ { m } \\right ) f _ { l } \\left ( \\alpha ^ { - 1 } q ^ { m } \\right ) \\ ! , \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} v = \\nabla \\varphi + \\sigma \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} r ^ { * , \\pi } _ t ( x _ t | y ^ { t - 1 } _ { t - M } , y _ t ) = \\Big ( \\frac { q _ t ( y _ t | y ^ { t - 1 } _ { t - M } , x _ t ) } { \\nu ^ { \\pi } _ { t } ( y _ t | y ^ { t - 1 } _ { t - J } ) } \\Big ) { \\pi } _ { t } ( x _ t | y ^ { t - 1 } _ { t - J } ) , ~ t \\in \\mathbb { N } _ 0 ^ { n - 1 } \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} T ( M ' ) ( a , b ) = \\sum _ { i , j \\leq k } M ' ( ( i , a ) , ( j , b ) ) . \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} \\Pi f ( \\xi ) : = \\sum _ { m \\in \\mathbb { Z } ^ d } f ( \\xi + 2 \\pi m ) . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} \\lambda _ 2 ( B _ 1 ^ { \\pi / 4 } ; { p _ n } ) = \\lambda _ 1 ( B _ { \\rho _ n } ^ { \\pi / 4 } ; { p _ n } ) = \\lambda _ 1 ( B _ 1 ^ { \\pi / 4 } \\setminus \\overline { B _ { \\rho _ n } ^ { \\pi / 4 } } ; { p _ n } ) . \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{gather*} u = 2 \\ln ( \\tau ) _ { x x } \\end{gather*}"}
-{"id": "4912.png", "formula": "\\begin{align*} K ^ { ( \\delta ) } = \\biguplus _ { m \\in \\omega } K _ m ^ { ( \\delta ) } \\uplus \\{ b \\} , \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} \\Vert 1 _ A - P _ t 1 _ A \\Vert _ 1 = & \\int _ A ( 1 - P _ t 1 _ A ) d \\mu + \\int _ { A ^ c } P _ t ( 1 _ A ) d \\mu \\\\ = & \\int _ A ( 1 - P _ t 1 _ A ) d \\mu + \\int _ A ( P _ t 1 _ { A ^ c } ) d \\mu \\\\ = & 2 \\left ( \\mu ( A ) - \\int _ A P _ t ( 1 _ A ) d \\mu \\right ) \\\\ = & 2 \\left ( \\mu ( A ) - \\Vert P _ \\frac { t } { 2 } ( 1 _ A ) \\Vert _ 2 ^ 2 \\right ) . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} B v _ z ( t ) = f ( t ) - A \\left ( \\int _ { 0 } ^ { t } z ( s ) d s + u _ 0 \\right ) \\ae . \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} \\gamma _ { 2 n } = \\frac { 2 } { 2 n + r } \\sum \\limits _ { i = - k } ^ n { 2 n + r \\choose 2 i + r } B _ { 2 n - 2 i } \\gamma _ { 2 i + 1 } , \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} I _ \\Delta ( \\lambda ) = \\left \\{ \\begin{array} { r l } 1 & \\lambda \\in \\Delta , \\\\ 0 & \\lambda \\notin \\Delta . \\end{array} \\right . \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} J = \\prod _ { p | \\ell _ 2 , \\P | p } \\P ^ { \\frac { r _ p - 1 } { 2 } } = \\prod _ { p \\in _ f ( A ) , \\P | p } \\P ^ { \\frac { r _ p - 1 } { 2 } } . \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} [ v _ 0 v _ 1 \\dots v _ i ] = [ \\{ v _ 0 , \\dots , v _ i \\} \\ , , \\ , v _ 0 < v _ 1 < \\dots < v _ i ] \\ . \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} A ( z ) : = \\int _ { 0 } ^ { z } e ^ { - \\frac { \\lambda } { \\xi } s } ( 1 - s ) ^ { \\frac { \\gamma } { \\xi } - 1 } d s , \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} f _ g f _ h = c _ { g , h } f _ { g h } \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} \\mathcal { D } ( X ) = 2 | \\nabla H - \\Pi ( X , \\cdot ) | ^ 2 + 2 \\widetilde { R c } ( H \\nu - X , H \\nu - X ) + 2 \\langle \\widetilde { R m } ( X , \\nu ) \\nu , X \\rangle , \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} 0 \\le { \\bf 1 } ^ \\top S _ { 1 2 } S _ { 1 2 } ^ \\ast { \\bf 1 } = \\chi ^ \\top S _ G S _ G ^ \\ast \\chi - { \\bf 1 } ^ \\top S _ \\Gamma S _ \\Gamma ^ \\ast { \\bf 1 } = \\chi ^ \\top S _ G S _ G ^ \\ast \\chi - s ( s ^ 2 - 1 ) / 3 . \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} f = f _ \\xi h \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} g _ { k ; + } ( x ) = 4 - g _ { k ; - } ( 4 - x ) h _ { k ; + } ( x ) = - h _ { k ; - } ( 4 - x ) , \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\frac { 1 - e ( - z ) } { 1 - e ( z ) } = - e ( - z ) . \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi _ i \\colon \\{ x _ 1 , \\dots , x _ n \\} & \\to \\{ \\ 0 , \\ 1 \\} , & \\varphi _ i ( x _ j ) & : = v _ i ( j ) , \\\\ \\theta _ i \\colon \\{ y _ 1 , \\dots , y _ n \\} & \\to \\{ \\ 0 , \\ 1 \\} , & \\theta _ i ( y _ j ) & : = v _ i ( n + j ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} ( w ^ { z , - } _ - , w ^ { z , - } _ + ) = ( w _ - , w _ + ) \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{gather*} D _ { r - 1 } s _ { r + m } ^ { ( r ) } + \\sum \\limits _ { k = 0 } ^ { r - 1 } \\ p _ { r , k } s _ { k + m } ^ { ( r ) } = 0 \\ , , \\ \\ \\ \\ m \\geq 0 \\ , . \\end{gather*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } \\circ I \\circ \\varphi ( z ) = a ( \\bar z + \\bar \\delta ) + b - \\delta = a \\bar z + b + a \\bar \\delta - \\delta . \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\sum _ { p = 1 } ^ { n } X _ { \\left ( A _ { 0 } , \\alpha _ { 0 } \\right ) \\cdots \\left ( A _ { n } , \\alpha _ { n } \\right ) } ^ { \\left ( p \\right ) } = 0 . \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} P ( x , y ) = \\binom { x + y } { y } . \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} { \\bf P } \\bigl [ \\beta _ { M , N } ( a , b ) = 1 \\bigr ] & = \\exp \\Bigl ( - \\int \\limits _ 0 ^ \\infty e ^ { - b _ 0 t } \\frac { \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) } { \\prod \\limits _ { i = 1 } ^ M ( 1 - e ^ { - a _ i t } ) } \\frac { d t } { t } \\Bigr ) , \\\\ & = \\exp \\bigl ( - ( \\mathcal { S } _ N \\log \\Gamma _ M ) ( 0 \\ , | a , \\ , b ) \\bigr ) . \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} K = \\frac { f '' } { f } ; \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\circlearrowleft _ { x , y , z } ( - 1 ) ^ { | x | | z | } \\sum _ { k = 0 } ^ { s } \\sum _ { i = 0 } ^ { s - k } [ \\alpha _ { i } ( x ) , [ y , z ] ' _ { k } ] ' _ { s - i - k } = 0 , \\ s = 0 , 1 , \\dots \\end{align*}"}
-{"id": "10096.png", "formula": "\\begin{align*} d \\alpha \\cdot X _ 1 = X \\circ \\alpha . \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} E _ \\sigma = \\lim _ { n \\to \\infty } E \\left ( w _ 1 ^ n , w _ 2 ^ n \\right ) \\geq E ( u _ 1 , u _ 2 ) , \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} D _ t ^ \\sigma ( - \\Delta ) ^ { \\gamma / 2 } q _ { \\alpha , \\beta } ( t , x ) = t ^ { - \\sigma - \\frac { \\alpha ( d + \\gamma ) } { 2 } + \\alpha - \\beta } D _ t ^ \\sigma ( - \\Delta ) ^ { \\gamma / 2 } q _ { \\alpha , \\beta } ( 1 , x t ^ { - \\frac { \\alpha } { 2 } } ) . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} P ^ - \\sum _ { i = 1 } ^ { \\chi - 1 } U _ i B U _ i ^ * P ^ - \\succeq C . \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} v ( \\theta , t ) = H \\int _ 0 ^ t s ^ { 2 H - 1 } \\left ( e ^ { \\theta s } + e ^ { \\theta ( 2 t - s ) } \\right ) d s . \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} \\varDelta _ { \\mathcal { D } } ( f , g ) = \\widetilde { \\varDelta } ( f , g ) + \\varDelta ( f , g | \\mathcal { D } _ f \\times \\mathcal { D } _ g ) \\to \\inf , \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ r \\frac { | K | } { | C _ K ( H _ { [ i ] } ) | } < \\frac { | K | } { | C _ K ( H ) | } \\Leftrightarrow \\sum _ { i = 1 } ^ r \\frac { 1 } { | N ( H _ { [ i ] } ) | } < 1 , \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} L ( u , \\chi ) = \\prod _ { P \\nmid Q } ( 1 - \\chi ( P ) u ^ { \\deg P } ) ^ { - 1 } \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} \\varphi _ 2 ( x ) = \\sum _ { i = 1 - r } ^ \\infty \\gamma _ i x ^ i = \\sum _ { i = \\lfloor - r / 2 \\rfloor + 1 } ^ \\infty \\delta _ i \\ , x ^ { 2 i } \\ , ( x - 2 ) ^ { - r - 2 i } \\in \\mathcal { F } _ r . \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} \\Re \\Big \\{ \\sum _ { n = 1 } ^ { \\infty } z _ n ^ { m _ 0 } \\Big \\} \\geq \\tfrac { \\epsilon } { 5 0 } | z _ 1 | ^ { m _ 0 } \\geq \\tfrac { \\epsilon } { 5 0 } ( \\alpha + 1 - \\beta ' ) ^ { - 2 m _ 0 } \\geq \\tfrac { \\epsilon } { 5 0 } \\alpha ^ { - 2 m _ 0 } \\exp ( - \\tfrac { 2 m _ 0 } { \\alpha } ( 1 - \\beta ' ) ) , \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} \\partial ( ^ h g ) = g \\partial ( h ) g ^ { - 1 } \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\langle \\mathcal { A } ( \\mathbf { H } ) \\mathbf { u } , \\tilde { \\mathbf { u } } \\rangle _ { \\mathcal { V } ' ; \\mathcal { V } } : = a ( \\mathbf { u } , \\tilde { \\mathbf { u } } ; \\mathbf { H } ) \\mathbf { u } , \\tilde { \\mathbf { u } } \\in \\mathcal { V } . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} d ' _ { r , t } \\triangleq \\max \\limits _ { 1 \\le t ' \\le t } \\left \\{ \\frac { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t ' } \\binom { N _ R - r - 1 } { t ' - 1 } t ' } { \\binom { N _ R - 1 } { r } \\binom { N _ T } { t ' } \\binom { N _ R - r - 1 } { t ' - 1 } t ' + \\binom { N _ R - 1 } { r + 1 } \\binom { N _ R - r - 2 } { t ' - 1 } \\binom { N _ T } { t ' - 1 } } \\right \\} \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} g ( t ) \\triangleq f ( t ) e ^ { - q t } e ^ { - b _ 0 t } \\prod \\limits _ { j = 1 } ^ N ( 1 - e ^ { - b _ j t } ) . \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} { \\displaystyle A = \\begin{pmatrix} \\frac { 1 - a } { a _ { 1 } - a _ { 3 } } & & 0 & & \\frac { a } { a _ { 3 } } - \\frac { 1 - a } { a _ { 1 } - a _ { 3 } } \\\\ \\\\ 0 & & 1 & & 0 \\\\ \\\\ 0 & & 0 & & \\frac { a } { a _ { 3 } } \\end{pmatrix} } \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} \\prod _ { 1 < a < q _ 1 / 2 \\atop ( a , q _ 1 ) = 1 } { \\xi } _ a ^ { 2 d _ a } ~ = ~ \\prod _ { 1 < b < q _ 2 / 2 \\atop ( b , q _ 2 ) = 1 } { \\xi } _ b ^ { - 2 e _ b } \\prod _ { 1 < c < q _ 3 / 2 \\atop ( c , q _ 3 ) = 1 } { \\xi } _ c ^ { - 2 f _ c } ~ = ~ 1 . \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} \\widetilde { f _ j } ( u , w ) = c _ j u ^ { A _ j } w ^ { B _ j } \\prod _ { i = 1 } ^ { l _ j } ( w - \\alpha _ { i , j } ) ^ { e _ { i , j } } . \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} v ( \\xi ) & = \\nabla _ \\xi h _ 0 ( \\xi ) , \\\\ A ( \\xi ) & = { } ^ t \\nabla _ \\xi \\nabla _ \\xi h _ 0 ( \\xi ) = ( \\partial _ { \\xi _ j } \\partial _ { \\xi _ k } h _ 0 ( \\xi ) ) _ { 1 \\leq j , k \\leq d } . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\nabla \\mathfrak { L } ( \\boldsymbol { \\theta } ) \\right ] = \\frac { 2 } { n } \\mathbb { E } \\left [ \\mathbf { X } ^ T ( \\mathbf { x } _ 1 ^ n - \\mathbf { X } \\boldsymbol { \\theta } ) \\right ] = \\frac { 2 } { n } \\mathbb { E } \\left [ \\mathbf { X } ^ T \\mathbf { w } _ 1 ^ n \\right ] = \\mathbf { 0 } . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} \\frac { \\partial F _ 1 } { \\partial r } = \\frac { \\partial F _ 2 } { \\partial s } , \\frac { \\partial F _ 1 } { \\partial s } = - \\frac { \\partial F _ 2 } { \\partial r } . \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} \\left \\langle \\tilde { \\eta } ^ { \\varepsilon } , \\varphi \\right \\rangle = \\varepsilon ^ { \\left ( d / 2 \\right ) - 1 } \\sum _ { b \\in \\left ( D ^ { \\varepsilon } \\right ) ^ { \\ast } } \\nabla \\varphi \\left ( b \\right ) \\tilde { \\eta } ^ { \\varepsilon } \\left ( b \\right ) \\varphi \\in \\mathcal { C } _ { 0 } ^ { \\infty } \\left ( D \\right ) \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} \\| g ( | \\lambda _ k \\cdot | ) \\| _ { L ^ q ( B _ { 2 \\rho } ( 0 ) ) } = ( 4 \\pi \\rho ^ 2 ) ^ { \\frac 1 q } \\bigg ( \\dfrac 1 { | B _ { 2 \\rho \\lambda _ k } ( 0 ) | } \\int _ { B _ { 2 \\rho \\lambda _ k } ( 0 ) } | g ( y ) | ^ q d y \\bigg ) ^ { \\frac 1 q } \\leq ( 4 \\pi \\rho ^ 2 ) ^ { \\frac 1 q } g ( 0 ) + \\varepsilon \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} m \\cdot \\sum _ { \\pi ( S ) \\in \\N ( { } ^ { p ( k ) } \\pi ( H _ { [ i ] } ) ) } { \\bar a } _ { \\pi ( S ) } = 0 , \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} | x _ 0 : \\ldots : 1 : \\ldots : x _ n | = | \\omega _ i ^ { a _ 0 } x _ 0 : \\ldots : \\omega _ i ^ { a _ i } : \\ldots : \\omega _ i ^ { a _ n } x _ n | = | \\omega _ i ^ { a _ 0 } x _ 0 : \\ldots : 1 : \\ldots : \\omega _ i ^ { a _ n } x _ n | . \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} g _ { y } ( v , w ) : = ( y _ n ^ 2 + y _ { n + 1 } ^ 2 ) ^ { - 1 } \\left ( \\sum \\limits _ { j = 1 } ^ { n - 1 } v _ j w _ j \\right ) + v _ n w _ n + v _ { n + 1 } w _ { n + 1 } , \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} \\zeta _ n = \\frac { 1 } { n ^ { m - 1 } } \\int _ 0 ^ { n ^ { m - 1 } } Z _ n ( t ) \\ , d t . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\tilde { F } ( v , y ) = g ( y ) . \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} F _ n = \\begin{cases} \\log _ { 1 / q } n - \\log _ { 1 / q } \\log \\log n + o ( \\log \\log \\log n ) & p > q \\\\ \\log _ { 2 } n - \\log _ 2 \\log n + o ( \\log \\log n ) & p = q = 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} O _ P ( f ) \\{ \\phi \\} = \\frac { 1 } { | \\det ( P ) | } f \\{ O _ { P ^ { - 1 } } ( \\phi ) \\} , \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} ( - z ; q ) _ { \\infty } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { 2 n ^ { 2 } - n } z ^ { 2 n } } { ( q ^ { 2 } ; q ^ { 2 } ) _ { n } } A _ { q } \\left ( - q ^ { 2 n - 1 } z \\right ) , \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{gather*} g _ { a b } ^ { [ k ] ( \\alpha ) } ( z ) = \\epsilon _ { a b } ^ { [ k ] ( \\alpha ) } ( z ) \\prod _ { i = 1 } ^ { \\ell } c ^ { ( \\alpha ) } _ { i } \\left \\langle Q _ { 1 } ^ { k - b } Q _ { 0 } ^ { - k - 1 + b } v _ { 0 } , \\psi _ { a } ^ { - } ( z ) \\prod _ { s = 1 } ^ { \\ell } \\psi _ { 1 } ^ { + } ( z _ { s } ) \\prod _ { t = 1 } ^ { \\ell } \\psi _ { 0 } ^ { - } ( z _ { t } ) v _ { 0 } \\right \\rangle / \\tau _ { k } ^ { ( \\alpha ) } , \\end{gather*}"}
-{"id": "6851.png", "formula": "\\begin{align*} \\tilde { \\mathbf { n } } ^ { T _ E } _ { [ \\ell + 1 : K ] } = \\left ( \\mathbf { H } _ 2 \\cdot { \\mathbf { H } _ 1 } ^ { \\dagger } \\right ) \\mathbf { n } ^ { T _ E } _ { [ 1 : \\ell ] } , \\end{align*}"}
-{"id": "10177.png", "formula": "\\begin{align*} \\tilde L _ q ( s , y ) = L ( s _ { 0 } , s _ { 1 } ) + L ( s _ 1 , s _ 2 ) + \\cdots + L ( s _ { q - 1 } , s _ 0 ) , \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} \\dim \\delta H _ 2 ^ g = 3 ( k + 2 ) + k ( k + 1 ) / 2 - ( k + 3 ) ( k + 4 ) / 2 = 0 , \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} \\varphi P _ { \\omega } ( u ) = P _ { \\omega _ { 0 } } ( \\varphi u ) + \\left ( \\mathrm { I d } - P _ { \\omega _ { 0 } } \\right ) \\circ A \\left ( P _ { \\omega } ( u ) \\wedge \\overline { \\partial } \\varphi \\right ) , \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} C _ 2 f _ N ^ { ( 2 ) } ( t , x _ 1 , v _ 1 ) = \\int _ { \\mathbb { R } ^ d \\times \\mathbb { S } ^ { d - 1 } } \\omega \\cdot ( v _ 2 - v _ 1 ) f _ N ^ { ( 2 ) } ( t , x _ 1 , v _ 1 , x _ 1 + \\varepsilon \\omega , v _ 2 ) d \\omega d v _ 2 \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} \\bar { f } ( t , x , \\bar { u } ( t , x ) , \\nabla \\bar { u } ( t , x ) , \\bar { v } ( t , x ) ) & = e ^ { L t } f ( t , x , e ^ { - L t } \\bar { u } ( t , x ) , e ^ { - L t } \\nabla \\bar { u } ( t , x ) , e ^ { - L t } \\bar { v } ( t , x ) ) - L \\bar { u } ( t , x ) \\\\ \\bar { g } ( t , x , \\bar { u } ( t , x ) , \\nabla \\bar { u } ( t , x ) , \\bar { v } ( t , x ) ) & = e ^ { L t } g ( t , x , e ^ { - L t } \\bar { u } ( t , x ) , e ^ { - L t } \\nabla \\bar { u } ( t , x ) , e ^ { - L t } \\bar { v } ( t , x ) ) . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} \\mathbf { \\tilde { U } _ n ^ { * } } = \\mathbf { \\tilde { M } _ { 0 n } ^ * S _ n ^ * } + ( \\mathbf { { S ^ * _ n } } ' \\mathbf { \\tilde { M } _ { 1 n } ^ * S _ n ^ * } , \\ldots , \\mathbf { { S ^ * _ n } } ' \\mathbf { \\tilde { M } _ { p n } ^ * S _ n ^ * } ) ' \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} c & = ( u '^ 2 + v '^ 2 ) ( 1 + u ^ 2 + v ^ 2 ) ^ 2 , \\\\ \\frac { c \\kappa } { 1 + u ^ 2 + v ^ 2 } & = ( u '' v ' - v '' u ' ) ( 1 + u ^ 2 + v ^ 2 ) + 2 ( u '^ 2 + v '^ 2 ) ( v u ' - u v ' ) . \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} w _ k ( z _ 0 ) = v _ k ( z _ 0 ) - \\varphi _ k ( z _ 0 ) = \\varphi _ { k + 1 } ( z _ 0 ) - \\psi ( z _ 0 ) > 0 . \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} v _ { i _ { k + 1 } } ^ { \\prime * } = v _ { i _ { k + 1 } } ^ \\prime + \\omega _ { k + 1 } \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime \\right ) \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } \\frac { q ^ { { ( n ^ 2 + n ) } / { 2 } } ( - 1 ; q ) _ n } { ( q ; q ) ^ 2 _ n } & = \\frac { ( - q ; q ) _ \\infty } { ( q ; q ) _ \\infty } , \\\\ \\sum _ { n \\geq 0 } \\frac { q ^ { ( n ^ 2 + n ) / { 2 } } ( - q ; q ) _ n } { ( q ; q ) ^ 2 _ n } & = \\frac { ( - q ; q ) _ \\infty } { ( q ; q ) _ \\infty } \\left ( 1 + \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { 2 n - 1 } } { ( - q ; q ^ 2 ) _ { n } } \\right ) \\\\ & = \\frac { ( - q ; q ) _ \\infty } { ( q ; q ) _ \\infty } \\sum _ { j \\geq 0 } q ^ { { ( 3 j ^ 2 + j ) } / { 2 } } ( 1 - q ^ { 2 j + 1 } ) . \\end{align*}"}
-{"id": "9977.png", "formula": "\\begin{align*} d ( a , b ) + d ( b , c ) \\geq d ( a , c ) , d ( a , a ) = 0 \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} H ^ { n - 1 } ( \\widetilde { X } ; \\ , ^ { \\omega } \\ ! \\Lambda ^ { \\omega } ) = 0 \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} & \\sp M t ^ { n - 1 } e ^ { \\omega t } \\left | \\frac { e ^ { - t ( r + h + s \\j ) } - e ^ { - t ( r + s \\j ) } } { h } - \\partial _ r e ^ { - t ( r + s \\j ) } \\right | \\\\ & = M t ^ { n - 1 } e ^ { \\omega t } | e ^ { - t ( r + s \\j ) } | \\left | \\frac { e ^ { - t h } - 1 } { h } + t \\right | \\le M t ^ { n - 1 } e ^ { t \\omega } e ^ { - t r } 2 t = 2 M t ^ n e ^ { t ( r - \\omega ) } . \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} \\gamma _ { 2 n + 1 } = \\sum \\limits _ { i = 0 } ^ n - { - k - i - 1 \\brace - k - n - 1 } \\gamma _ { 2 i } . \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} D ^ F ( x ) = D ^ F _ 0 ( x ) + m ^ 2 \\bigl [ f _ 1 ( m ^ 2 x ^ 2 ) \\log ( - m ^ 2 ( x ^ 2 - i 0 ) ) + f _ 2 ( m ^ 2 x ^ 2 ) \\bigr ] , \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} \\widehat N _ 1 ( x , y , z ) = N _ 1 ( z , x , y ) + N _ 1 ( z , y , x ) . \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} & \\left ( \\sigma _ i ^ 2 ( A W ) - \\sigma _ k ^ 2 ( A ) \\right ) \\alpha ^ { ( k ) } _ i + \\sigma _ i ( A W ) y ^ { ( i ) \\intercal } \\beta ^ { ( k ) } = 0 , \\\\ \\Rightarrow & \\alpha _ i ^ { ( k ) } = \\frac { - \\sigma _ i ( A W ) } { \\sigma _ i ^ 2 ( A W ) - \\sigma _ k ^ 2 ( A ) } y ^ { ( i ) \\intercal } \\beta ^ { ( k ) } . \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} \\tilde { F } _ a ( w _ a , y ) : = F _ a ( w _ a + v , y ) = 0 \\mbox { i n } \\mathcal { B } _ { 1 } ^ + . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\delta _ { \\mathsf { P , A c h } } ( \\mu , r ) & = \\lim _ { B \\rightarrow \\infty } \\lim _ { P \\rightarrow \\infty } \\limsup _ { L \\rightarrow \\infty } \\frac { ( B + 1 ) } { B } \\frac { \\max \\left ( T _ F , T _ E \\right ) } { L / \\log ( P ) } \\\\ & = \\max \\left ( \\delta _ F , \\delta _ E \\right ) , \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} \\tau = \\{ ( k _ i , \\lambda ^ { ( i ) } ) \\} _ { i = 1 } ^ r ( ( k _ 1 , \\lambda ^ { ( 1 ) } ) \\leq ( k _ 2 , \\lambda ^ { ( 2 ) } ) \\leq \\cdots \\leq ( k _ r , \\lambda ^ { ( r ) } ) , \\ ; r \\in \\Z _ { \\geq 0 } ) . \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} { \\textstyle \\sum \\limits _ { u \\in \\Gamma ( k ) } } Q ( b _ { i } , u ) = 0 , \\forall i = 1 , 2 , . . . , s , \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} ( - q ; q ^ 2 ) _ \\infty = 1 + \\sum _ { n \\geq 1 } q ^ { 2 n - 1 } ( - q ^ { 2 n + 1 } ; q ^ 2 ) _ \\infty . \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} & \\ \\ \\rho ( \\lambda , \\varepsilon ) \\phi \\\\ & = \\sum _ { k \\neq 0 , \\ \\pm } \\left ( \\int _ { v > 0 } e ^ { i k \\theta _ { \\pm } } \\frac { \\omega _ { \\pm } \\mu _ { \\pm , + } ^ { \\prime } ( e _ { \\pm } ) } { \\omega _ { \\pm } + \\frac { \\lambda } { i k } } \\phi _ { k } ^ { \\pm } ( I _ { \\pm } ) d v + \\int _ { v < 0 } e ^ { i k \\theta _ { \\pm } } \\frac { \\omega _ { \\pm } \\mu _ { \\pm , - } ^ { \\prime } ( e _ { \\pm } ) } { \\omega _ { \\pm } - \\frac { \\lambda } { i k } } \\phi _ { k } ^ { \\pm } ( I _ { \\pm } ) d v \\right ) . \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } \\frac { q ^ { 2 k ^ { 2 } - \\left ( 2 n - 1 \\right ) k } } { \\left ( q ^ { 2 } , q ^ { 3 } ; q ^ { 2 } \\right ) _ { k } } = \\frac { \\left ( - 1 \\right ) ^ { n } \\left ( - q ; q \\right ) _ { n } } { \\left ( q ^ { 3 } ; q ^ { 2 } \\right ) _ { \\infty } q ^ { \\binom { n } { 2 } } } . \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} \\pi ( x ) = \\frac { x } { \\log x - 1 - \\frac { k _ { 1 } } { \\log x } - \\frac { k _ { 2 } } { ( \\log x ) ^ { 2 } } - . . . - \\frac { k _ { n } ( 1 + \\alpha _ { n } ( x ) ) } { ( \\log x ) ^ { n } } } \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} e ^ { 4 i R \\lambda } C ( \\lambda ) v = v , \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} s = \\sigma + i t , \\sigma \\geq 1 - \\frac { 0 . 1 0 3 6 7 } { \\log ( q ( 2 + | t | ) ) } . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} F ^ { 0 } _ { k _ 1 - n + 1 } = c _ 1 F ^ { 0 } _ { k _ 1 } + \\cdots + c _ { n - 2 } F ^ { 0 } _ { k _ 1 - n + 2 } , \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{gather*} M = M _ { n - 1 } \\cdots M _ { 1 } M _ { 0 } , M _ { a } = M _ { a } ^ { + } M _ { a } ^ { - } , a = n - 1 , n - 2 , \\dots , 2 , 1 , \\end{gather*}"}
-{"id": "5320.png", "formula": "\\begin{align*} u _ { 0 . 6 7 } ^ 1 ( f _ 2 ) = [ I - 0 . 6 7 P ^ 1 ( f _ 2 ) ] ^ { - 1 } \\bar { r } ^ 1 ( f _ 2 ) = ( 8 , 1 0 ) . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} f ( 0 , 0 ) = 0 , f _ u ( 0 , 0 ) = 1 , f _ v ( 0 , 0 ) = 0 , \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{gather*} ( - 1 ) ^ { \\frac { k ( k - 1 ) + \\ell ( \\ell - 1 ) } { 2 } + k \\ell + \\frac { n _ c ( n _ c - 1 ) + n _ d ( n _ d - 1 ) + n _ e ( n _ e - 1 ) } { 2 } + n _ c n _ e } = ( - 1 ) ^ { \\frac { n _ d ( n _ d + 1 ) } { 2 } } , \\end{gather*}"}
-{"id": "9287.png", "formula": "\\begin{align*} 3 z + 1 = y 2 ^ { 2 + 2 k } = ( 3 \\xi ( T x ) + 1 ) 4 ^ k . \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} \\ell : = \\zeta _ d - \\frac { \\alpha } { 4 } \\kappa + \\frac { \\alpha } { 2 } \\kappa _ d + \\frac { \\alpha } { 2 } d \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} E _ s ^ { i * } = \\alpha _ i ^ * \\left [ \\frac { 1 } { \\ln 2 ( \\sum _ { j = i } ^ M \\lambda _ j ^ * + \\gamma _ i ^ * h _ { s p } ^ i ) } - \\frac { 1 } { \\theta _ i } \\right ] ^ + \\alpha _ i ^ * = \\theta _ i E _ s ^ { i * } \\left [ \\frac { 1 } { \\ln 2 \\left ( \\log _ 2 ( 1 + \\theta _ i \\beta _ i ^ * ) - \\sum _ { j = i } ^ M \\eta P _ p \\lambda _ j ^ * + \\gamma _ i ^ * P _ { i n t } - \\mu _ i ^ * \\right ) } - 1 \\right ] ^ + \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } - \\Delta _ { g } u \\triangleq \\frac { \\partial u } { \\partial t } - \\frac { 1 } { \\sqrt { d e t g _ { i j } } } \\frac { \\partial } { \\partial x _ { i } } ( g ^ { i j } \\sqrt { d e t g _ { i j } } \\frac { \\partial u } { \\partial x _ { j } } ) = f , \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sup _ { x \\in X } \\int _ X | \\rho _ n ( x , y ) | d \\mu ( y ) = 0 \\end{align*}"}
-{"id": "10090.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { x y ( a x + b y + c z ) } { z ^ 3 } , f ( x , y , z ) = \\dfrac { x y ( a x + b y + c z ) ^ 2 } { z ^ 4 } , \\\\ f ( x , y , z ) = \\dfrac { x y ^ 2 ( a x + b y + c z ) ^ 3 } { z ^ 6 } , \\end{gather*}"}
-{"id": "4185.png", "formula": "\\begin{align*} \\mathfrak { C } _ { d } = \\left ( Z _ { 4 } \\times \\mathfrak { a d s } _ { d } \\right ) _ { H } , \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{gather*} \\rho _ t ^ { Z , 0 \\oplus 1 } ( S _ c ) = S _ c , \\rho _ t ^ { Z , 0 \\oplus 1 } ( S _ d ) = e ^ { 2 \\pi \\sqrt { - 1 } t } S _ d , \\\\ \\rho _ t ^ { Z , 1 \\oplus 0 } ( S _ c ) = e ^ { 2 \\pi \\sqrt { - 1 } t } S _ c , \\rho _ t ^ { Z , 1 \\oplus 0 } ( S _ d ) = S _ d \\end{gather*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\sum _ { \\substack { n \\leq x \\\\ n = a \\mod q } } \\alpha _ { 2 } ( n ) \\sim \\frac { \\zeta ( 3 / 2 ) } { \\zeta ( 3 ) } \\frac { A _ { a , q } } { q } x ^ { 1 / 2 } + \\frac { \\zeta ( 2 / 3 ) } { \\zeta ( 2 ) } \\frac { B _ { a , q } } { q } x ^ { 1 / 3 } \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} H ( \\textbf { q } _ 1 , \\textbf { p } _ 1 , \\cdots , \\textbf { q } _ N , \\textbf { p } _ N , t ) = \\sum _ { j = 1 } ^ N \\left ( \\frac { 1 } { 2 m _ j } \\| \\textbf { p } _ j - e _ j \\textbf { A } _ j ( \\textbf { q } _ j , t ) \\| ^ 2 + e _ j \\phi _ j ( \\textbf { q } _ j , t ) \\right ) + V ( \\textbf { q } _ 1 , \\cdots , \\textbf { q } _ N , t ) . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} \\Im \\alpha _ { k } ^ { k ' } = - ( - 1 ) ^ k \\big ( \\tau \\gamma _ k + ( - 1 ) ^ { k ' } A _ k \\big ) . \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} Z _ { H , a } = \\sum _ { [ i _ 1 , \\ldots , i _ { r - 1 } ] \\in S ^ r _ { r - 1 } } { \\epsilon _ { [ i _ 1 , \\ldots , i _ { r - 1 } ] } } \\cdot a \\cdot \\sigma _ { [ i _ 1 , \\ldots , i _ { r - 1 } ] } . \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} F ( v , y ) = L _ { y _ 0 } v + P _ { y _ 0 } ( y ) + E _ { y _ 0 } ( y ) . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\hat { \\mathbf { u } } ^ { 0 } = S ( T ) \\tilde { \\mathbf { u } } ^ { 0 } T > 0 . \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} { \\bf E } [ e ^ { q \\ , V _ N } ] \\approx e ^ { q ( 2 \\log N - ( 3 / 2 ) \\log \\log N + \\ , { \\rm c o n s t } ) } \\ , { \\bf E } \\Bigl [ M ^ q _ { ( \\tau = 1 , \\lambda _ 1 , \\lambda _ 2 ) } \\Bigr ] \\Gamma ( 1 - q ) , \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } u ^ { - } \\partial _ { t } & \\left [ g _ { 1 - \\alpha , m } * ( u - u _ { 0 } ) \\right ] d x d t \\\\ & = \\int _ { 0 } ^ { T } \\int _ { \\Omega } u ^ { - } \\partial _ { t } \\left [ g _ { 1 - \\alpha , m } * u \\right ] d x d t - \\int _ { 0 } ^ { T } \\int _ { \\Omega } u ^ { - } g _ { 1 - \\alpha , m } u _ { 0 } d x d t . \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} \\int _ { \\Omega } w \\partial _ z \\zeta \\cdot \\zeta = \\frac { 1 } { 2 } \\int _ { \\Omega } \\left ( \\int _ 0 ^ z - \\div _ H v \\right ) \\partial _ z \\zeta ^ 2 = \\frac { 1 } { 2 } \\int _ { \\Omega } ( \\div _ H v ) \\zeta ^ 2 . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} \\rho \\ddot { v } _ i & = \\big ( A _ { i J K j } v _ { j , K } - \\beta _ { J i } \\dot { \\omega } - ( C _ { i J K L I j } v _ { j , I L } + M _ { i J K L } \\omega _ { , L } ) _ { , K } \\big ) _ { , J } , \\\\ a \\ddot { \\omega } & = - \\beta _ { K i } \\dot { v } _ { i , K } + m _ { I J } r _ { I , J } + M _ { j L K I } v _ { j , L K I } + K _ { I J } \\omega _ { , I J } , \\\\ \\kappa \\dot { r } _ i & = \\dot { \\omega } _ { , i } - r _ { i } \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\mathbf { m } ' \\cdot e _ 1 \\wedge \\ldots \\wedge e _ k = e _ { i _ 1 } \\wedge \\ldots \\wedge e _ { i _ k } + , \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} A \\geq \\int _ M u d \\mu \\geq \\int _ { \\cup _ { i = 1 } ^ N B ( z _ i , 0 . 0 1 A ^ { - 2 } ) } u d \\mu \\geq 0 . 1 A ^ { - 1 } \\sum _ { i = 1 } ^ { N } | B ( z _ i , 0 . 0 1 A ^ { - 2 } ) | \\geq 0 . 1 A ^ { - 1 } \\cdot A ^ { - 1 } \\cdot \\left ( 0 . 0 1 A ^ { - 2 } \\right ) ^ m N , \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} g ^ { k } _ { \\beta , \\epsilon } = \\Sigma _ { j = 1 } ^ { k } ( d s _ { j } ^ 2 + a _ { j , \\epsilon } s _ { j } ^ 2 d \\theta _ { j } ^ 2 ) + \\Sigma _ { j = k + 1 } ^ { n } d z _ { j } \\otimes d \\bar { z } _ { j } , \\ \\beta _ { j } ^ { 2 } < a _ { j , \\epsilon } \\leq 1 . \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} p ^ { \\prime } : = \\max \\{ p , 2 \\} \\hbox { a n d } q ^ { \\prime } : = \\max \\{ q , 2 \\} . \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} B _ \\iota ( k , d , r , c ) = \\{ a \\in \\R ^ r : \\| a \\| \\le c | k | ^ { - \\frac { d } { r } } \\} , \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} p _ { 2 i } - p _ i = \\frac 1 2 ( p _ i - p _ { \\lfloor i / 2 \\rfloor } ) \\omega ^ { - 1 } z . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i { m \\choose i } ( m + i ) \\gamma _ { m + i - 1 } = \\sum \\limits _ { i = 0 } ^ m ( - 1 ) ^ i { m \\choose i } \\frac { ( m + i ) ! } { ( m + i - 1 ) ! } \\gamma _ { m + i - 1 } = 0 . \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} F _ a ( u , y ) = F ( u ( \\Phi ^ { - 1 } _ a ( z ) ) , z ) \\big | _ { z = \\Phi _ a ( y ) } . \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} \\Z / m = H _ { 2 n - 2 i - 1 } ( L _ { 2 n - 1 } ( m ) ) \\to H _ { 2 n - 2 i - 1 } ( X ^ \\circ ) \\simeq H ^ { 2 i + 1 } _ c ( X ^ \\circ ) = H ^ { 2 i + 1 } _ c ( X ) \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} \\left | K _ { N } ^ { 1 } ( z , \\zeta ) \\right | \\lesssim \\frac { \\left | \\rho ( \\zeta ) \\right | ^ { k } } { \\varepsilon ^ { k } } \\frac { 1 } { \\prod _ { i = 1 } ^ { n - 1 } \\tau _ { i } } \\frac { 1 } { \\left | z - \\zeta \\right | } , \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} \\pi ( e ^ n ) & = x ^ n , & \\pi ( f ^ n ) & = y ^ n , & \\pi ( h ^ n ) & = \\sum _ { j = 1 } ^ n \\tau _ n ( j ) \\alpha ^ { n - j } z ^ j , & n & \\in \\N , \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} S ( \\delta ; \\phi ) = - \\int _ \\Sigma d v \\ , \\phi ^ i ( x ) \\Delta _ \\gamma \\phi _ i ( x ) \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} \\int _ 0 ^ T \\left \\| \\nabla \\psi \\left ( t \\right ) \\right \\| _ { \\infty } \\ , d t = \\infty , \\end{align*}"}
-{"id": "10161.png", "formula": "\\begin{align*} \\rho : = \\liminf _ { n \\rightarrow + \\infty } \\frac { \\mathbb E [ Z ^ * _ { n + \\lfloor \\varepsilon n \\rfloor } ] } { \\mathbb E [ Z _ n ^ * ] } > 1 . \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} 0 = - \\frac { ( c - j + 1 ) \\gamma } { \\xi } k _ { 1 } ^ { j - 1 } ( 1 ) - j \\left ( 1 + \\frac { \\mu } { \\xi } \\right ) \\sum _ { n = 0 } ^ { j } p _ { j , n } k _ { 2 } ( 1 ) + \\frac { \\mu } { \\xi } \\sum _ { n = 1 } ^ { j } n p _ { j , n } k _ { 3 } ( 1 ) - \\frac { \\mu } { \\xi } \\sum _ { n = 1 } ^ { j + 1 } n p _ { j + 1 , n } k _ { 4 } ( 1 ) \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} A _ { q } ( z ) : = { } _ { 0 } \\phi _ { 1 } \\left ( - ; 0 ; q , - q z \\right ) \\ ! , z \\in \\C . \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\begin{aligned} \\dot { x } & = \\omega + \\xi ( y , z , \\omega , \\sigma , \\mu ) + f ( x , y , z , \\omega , \\sigma , \\mu ) , \\\\ \\dot { y } & = \\sigma + \\eta ( y , z , \\omega , \\sigma , \\mu ) + g ( x , y , z , \\omega , \\sigma , \\mu ) , \\\\ \\dot { z } & = Q ( \\omega , \\mu ) z + \\zeta ( y , z , \\omega , \\sigma , \\mu ) + h ( x , y , z , \\omega , \\sigma , \\mu ) . \\end{aligned} \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} A + B = 2 \\rho . \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} ( Z _ 0 ) _ { r e d } = \\ell _ 0 \\sqcup ( \\ell _ 1 \\cup \\ell _ 2 ) , \\ \\ \\ w : = \\ell _ 1 \\cap \\ell _ 2 = \\{ \\mathrm { p t } \\} . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial } { \\partial t } + V _ s \\cdot \\nabla _ { X _ s } \\right ) f _ \\infty ^ { ( s ) } ( t , Z _ s ) = \\ell ^ { - 1 } C _ { s + 1 } ^ 0 f _ \\infty ^ { ( s + 1 ) } ( t , Z _ s ) \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} v ^ { ( 0 , l ) } _ { 0 , 0 } + v ^ { ( 1 , l ) } _ { 0 , 0 } + \\ldots + v ^ { ( n - 1 , l ) } _ { 0 , 0 } = 0 , \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} \\Phi _ A ( \\psi ) = \\sup _ { x \\in \\bigcup _ { i \\in I } A _ i } \\psi ( x ) = \\sup _ { i \\in I } \\Phi _ { A _ i } ( \\psi ) . \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} \\tau = 1 / \\beta ^ 2 , \\ ; 0 < \\beta < 1 . \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} T ^ { - 1 } ( d _ \\lambda ( u ) ) = D _ \\lambda ( T ^ { - 1 } ( u ) ) , . \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} \\phi _ \\mathcal { G } ( \\gamma ) = [ r ( \\gamma ) , \\alpha _ A , s ( \\gamma ) ] \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} \\langle \\omega \\cdot v ^ 0 , v ^ 0 \\rangle _ { L ^ 2 ( X ) } = \\int _ { \\widehat { G } ^ { \\rm s p h } } \\widehat { \\omega } ( \\nu ) \\ , d \\mu _ X ^ ( \\nu ) \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} \\hat { \\mathcal { U } } _ s ^ \\eta = \\left \\{ Z _ s = ( X _ s , V _ s ) \\in \\mathcal { U } _ s ^ \\eta \\left | \\forall \\tau , \\tau ^ \\prime > 0 , ( \\psi _ s ^ { - \\tau } Z _ s , \\psi _ s ^ { - \\tau ^ \\prime } Z _ s ) \\in \\mathcal { V } _ s ^ \\eta \\right . \\right \\} \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} \\overline \\theta _ \\alpha ( \\overline Z ) = \\theta _ \\alpha ( Z ) + \\theta _ \\alpha ^ \\prime ( Z ^ \\prime ) , \\alpha = 1 , 2 , 3 , \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} c _ { 1 } \\cdots c _ { n - k } \\left ( \\sum _ { \\substack { \\left | \\delta \\right | = n \\\\ \\delta _ { 1 } = \\dots = \\delta _ { n - k } = 1 } } \\alpha _ { \\delta } \\right ) + \\dots + c _ { k + n + 1 } \\cdots c _ { 2 n } \\left ( \\sum _ { \\substack { \\left | \\delta \\right | = n \\\\ \\delta _ { k + n + 1 } = \\dots = \\delta _ { 2 n } = 1 } } \\alpha _ { \\delta } \\right ) = 0 . \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\overrightarrow { H } = \\frac { 1 } { 2 } \\left \\{ \\left ( \\frac { \\kappa ^ { 2 } c ^ { 2 } + 1 } { c ^ { 2 } \\kappa } \\right ) N _ { 1 } - \\frac { \\kappa _ { 1 } } { \\kappa \\lambda \\cos \\left ( \\frac { u } { c } \\right ) } N _ { 2 } \\right \\} . \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} P ( t ^ { - 1 } , t u , x ) = H ( t , u , x ) + \\big ( 1 + H ( t , u , 0 ) \\big ) P ( t , u , x ) . \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} \\left ( \\theta _ { \\lambda } p \\right ) \\left ( x \\right ) = \\frac { p \\left ( x \\right ) - p \\left ( \\lambda \\right ) } { x - \\lambda } , \\left ( \\sigma _ s p \\right ) ( x ) : = p ( x ^ s ) , \\ p \\in \\mathcal { P } . \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} E ( t ) \\le E ( m T ) \\le S ( m ) = S \\left ( \\frac { t - \\tau } { T } \\right ) \\le S \\left ( \\frac { t } { T } - 1 \\right ) , \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} ( v ) & = \\sum ^ { k + 1 } _ { i = 1 } ( - 1 ) ^ { i } \\left [ \\sum ^ { k } _ { j = 1 } ( - 1 ) ^ { j } f \\circ ( \\alpha ^ { j - 1 , 1 } _ { k } ) ^ { - 1 } \\circ \\mu ^ { j - 1 } _ { k + 1 } \\circ \\alpha ^ { j - 1 , 2 } _ { k + 1 } \\right ] \\circ ( \\alpha ^ { i - 1 , 1 } _ { k + 1 } ) ^ { - 1 } \\circ \\mu ^ { i - 1 } _ { k + 2 } \\circ \\alpha ^ { i - 1 , 2 } _ { k + 2 } \\\\ & = \\sum ^ { k + 1 } _ { i = 1 } \\left [ \\sum ^ { k } _ { j = 1 } b _ j \\right ] a _ i = \\sum ^ { k + 1 } _ { i = 1 } \\sum ^ { k } _ { j = 1 } b _ j a _ i = 0 , \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} \\mathcal { B } _ { I I I } ^ + = \\left \\{ \\begin{aligned} & ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } ) \\in \\mathcal { A } ^ + \\backslash \\mathcal { B } _ { I I } \\textnormal { s u c h t h a t } \\\\ & \\exists \\ ; i \\in \\left \\{ 1 , 2 , \\dots , s + k \\right \\} , \\ ; t ^ \\prime \\geq 0 \\ ; : \\ ; \\left | v _ { s + k + 1 } ^ * - v _ i ^ \\prime ( \\tau ; t ^ \\prime ) \\right | \\leq \\eta \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} ( \\Delta _ { N , 1 2 } ) = \\left [ \\Gamma ( 1 ) : \\Gamma _ 0 ( N ) \\right ] \\textbf { a } _ \\infty = N \\prod _ { p | N } ( 1 + 1 / p ) \\textbf { a } _ \\infty . \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} v _ \\alpha ^ 2 ( f , g _ 1 ) - v _ \\alpha ^ 2 ( f , g _ 2 ) = \\left [ \\frac { p ( 1 2 + 7 \\alpha ) - ( 4 + \\alpha ) } { \\alpha ( \\alpha + 2 ) } , \\frac { p ( 1 2 + 7 \\alpha ) - ( 4 + \\alpha ) } { \\alpha ( \\alpha + 2 ) ( 1 + \\alpha ) } \\right ] ^ T . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} F \\bigl ( v ( \\omega , \\mu ) , \\ , \\mu + w ( \\omega , \\mu ) \\bigr ) = \\omega + u ( \\omega , \\mu ) \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\emptyset \\neq \\left ( \\bigcap _ { s \\in F : a _ s = 1 } ( W + h - s ^ * ) \\setminus \\bigcup _ { s \\in F : a _ s = 0 } ( W + h - s ^ * ) \\right ) \\cap ( L ^ * + h - \\vartheta ) \\ . \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} \\phi ( t ; y , \\xi ) : = t h _ 0 ( \\xi ) + \\varphi _ a ( y , \\xi ) \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} \\left [ ( \\lambda z - \\mu ) ( 1 - z ) + \\xi ( 1 - z ) \\right ] P _ { 1 } ( z ) - \\xi z ( 1 - z ) P ' _ { 1 } ( z ) = \\gamma z P _ { 0 } ( z ) - ( \\gamma p _ { 0 , 0 } + \\mu p _ { 1 , 1 } ) + \\xi p _ { 1 , 1 } . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N _ 0 - 1 } w _ k ^ 4 L _ k + \\frac { 1 } { d - \\sum _ { k = 1 } ^ { N _ 0 - 1 } L _ k } \\left ( \\sum _ { k = N _ 0 } ^ { K } w _ k ^ 2 L _ k \\right ) ^ 2 . \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} g _ { 1 } ' \\left ( z \\right ) & = \\frac { \\mathrm { d } } { \\mathrm { d } z } \\left ( \\left ( 1 - z \\right ) \\cdot G \\left ( z \\right ) \\cdot \\frac { 1 } { 1 - c _ { o } z ^ { o } } \\right ) \\left ( z \\right ) \\\\ & = - G \\left ( z \\right ) \\cdot \\frac { 1 } { 1 - c _ { o } z ^ { o } } + \\left ( 1 - z \\right ) \\cdot \\underset { \\eqqcolon \\left ( \\star \\right ) } { \\underbrace { \\frac { \\mathrm { d } } { \\mathrm { d } z } \\left ( G \\left ( z \\right ) \\cdot \\frac { 1 } { 1 - c _ { o } z ^ { o } } \\right ) } } . \\end{align*}"}
-{"id": "10150.png", "formula": "\\begin{align*} \\varphi ( x _ k ) - \\varphi ( x _ { k + 1 } ) & = \\varphi ( x _ k ) - \\min \\varphi - ( \\varphi ( x _ { k + 1 } ) - \\min \\varphi ) \\\\ & \\leq ( \\frac { 1 } { \\nu } - \\frac { 1 } { 2 L } ) \\| \\mathcal { R } _ { 1 / L } ( x _ { k - 1 } ) \\| ^ 2 - \\frac { 1 } { 2 L } \\sum _ { i = 1 } ^ \\infty \\| \\mathcal { R } _ { 1 / L } ( x _ { k + i } ) \\| ^ 2 , ~ k \\geq 1 . \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} \\left < \\Phi _ N , F _ N \\right > = \\sum _ { s = 1 } ^ N \\frac { 1 } { s ! } \\int _ { \\mathcal { D } _ s } \\phi _ N ^ { ( s ) } ( Z _ s ) f _ N ^ { ( s ) } ( Z _ s ) d Z _ s \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} E '' ( z ) = \\frac { 1 } { 1 - z } E ' ( z ) + E ( z ) E ' ( z ) , E ( 0 ) = 1 , E ' ( 0 ) = 2 . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} F ( x ) = \\sum _ { i = 1 } ^ { n } f _ { i } ( x _ { i } ) e _ { i } \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} _ { 2 } \\phi _ { 2 } \\left ( \\sqrt { b } , - \\sqrt { b } ; \\sqrt { c } , - \\sqrt { c } ; \\sqrt { q } , - \\frac { c } { b } \\right ) = \\frac { \\left ( c / b ; q \\right ) _ { \\infty } } { \\left ( c ; q \\right ) _ { \\infty } } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } / 2 } } { \\left ( q ; q \\right ) _ { n } } \\left ( - \\frac { c } { b } \\right ) ^ { n } A _ { q } \\left ( - c q ^ { n - 1 / 2 } \\right ) \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\| B _ { j k } u \\| ^ 2 = \\Im \\langle [ P _ j , P _ k ] u , B _ { j k } u \\rangle \\leq | \\langle P _ k u , P _ j B _ { j k } u \\rangle | + | \\langle P _ j u , P _ k B _ { j k } u \\rangle | \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} \\gamma _ n c _ { i , i + 1 } = \\sigma _ { i , i + 1 } \\gamma _ n & & n \\geq 2 , \\ , i \\in \\{ 1 , \\dots , n - 1 \\} . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} \\Pi ^ n ( t ) = \\begin{pmatrix} \\pi ^ 0 ( t ) \\\\ \\vdots \\\\ \\pi ^ n ( t ) \\end{pmatrix} , \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} 1 - \\| \\rho \\| \\ , \\epsilon < \\theta \\circ T ^ n ( q ) = \\| \\rho \\| \\ , \\big ( \\frac { 1 } { \\| \\rho \\| } \\rho \\circ T ^ n ( q ) \\big ) + \\| \\theta - \\rho \\| \\ , \\big ( \\frac { 1 } { \\| \\theta - \\rho \\| } ( \\theta - \\rho ) \\circ T ^ n ( q ) \\big ) . \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\deg ( K _ { C } + \\mathrm { D i f f } _ { C } ( 0 ) ) = ( K _ { X } + C . C ) = ( C ^ { 2 } ) \\ge \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} & \\sum _ { \\alpha \\in \\Q ^ \\times / ( \\Q ^ \\times _ S ) ^ 2 \\cap \\Q ^ \\times } a ^ G ( S , z \\delta _ 1 ( 1 , \\alpha ) ) \\ , J _ G ( z \\delta _ 1 u _ 1 ( 1 , \\alpha ) , f _ \\xi ) \\ , J _ G ( z \\delta _ 1 u _ 1 ( 1 , \\alpha ) , h ) \\\\ & = 2 c ( z \\delta _ 1 u _ 1 ) \\{ ( - 1 ) ^ { l _ 2 } - ( - 1 ) ^ { l _ 1 } \\} \\sum _ \\chi L ^ S ( 1 , \\chi ) ^ 2 \\sum _ { \\alpha \\in \\Q ^ \\times / ( \\Q ^ \\times _ { S _ 0 } ) ^ 2 \\cap \\Q ^ \\times } \\chi _ { S _ 0 } ( \\alpha ) \\ , J _ G ( z \\delta _ 1 u _ 1 ( 1 , \\alpha ) , h ) \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} f _ 1 ( x ) : = e ^ { - \\abs { x } } \\qquad ( x \\in \\R ) \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} \\bar { g } ^ { i j } \\leq \\bar { g } ^ { i j } + v ^ { - 2 } \\check { u } ^ i \\check { u } ^ j = g ^ { i j } , \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} | \\partial _ \\xi ^ \\beta s _ a ( x , \\xi ) | \\leq \\begin{cases} C _ { \\beta , a } \\langle x \\rangle ^ { - 1 - \\varepsilon } , & | \\cos ( x , v ( \\xi ) ) | \\geq \\frac { 1 } { 2 } , \\\\ C _ { \\beta , a } \\langle x \\rangle ^ { - \\varepsilon } , & | \\cos ( x , v ( \\xi ) ) | \\leq \\frac { 1 } { 2 } . \\end{cases} \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} \\min _ { \\| x \\| = 1 , x \\ge 0 } v ^ T x \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} f _ 4 = a _ i e ^ { n + i } , f _ 8 = b _ i e ^ { i } , \\lambda = \\lambda _ I e ^ I \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} \\mu \\circ \\theta ^ { - 1 } \\left ( B \\right ) = \\int \\ 1 _ { B } \\circ \\theta \\ : \\mbox { d } \\mu = \\int \\mathcal { L } _ { \\varphi } \\left ( \\ 1 _ { B } \\circ \\theta \\right ) \\ : \\mbox { d } \\mu = \\int \\ 1 _ { B } \\cdot \\mathcal { L } _ { \\varphi } \\left ( \\ 1 \\right ) \\ : \\mbox { d } \\mu = \\mu \\left ( B \\right ) . \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} & \\int _ { \\Omega } \\psi \\partial _ { t } \\left [ g _ { 1 - \\alpha , m } * ( u - u _ { 0 } ) \\right ] d x + a ( h _ { m } * u , \\psi ) \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\ , \\ , \\ , \\ , \\geq \\int _ { \\Omega } ( h _ { m } * f ) \\psi d x \\ , \\ , \\ , t \\in ( 0 , T ) , \\ , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\omega ^ { \\alpha } = \\omega ^ { \\widetilde { \\alpha } } = \\sup \\{ \\omega ^ { \\beta _ k } : k \\in \\omega \\} \\le \\tau . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} q _ 1 = \\sum _ { j = 0 } ^ k z ^ j . \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} r _ { m , \\mu } ( x ) & = - ( - i ) ^ m x ^ { m / 2 } h _ { m - 2 , \\mu + 2 } ( ( 2 \\sqrt { x } i ) ^ { - 1 } ) \\\\ & = ( - 1 ) ^ m ( \\mu + 2 ) _ { m - 2 } { \\ ; } _ 2 F _ 3 \\left ( { - ( m - 2 ) / 2 , - ( m - 3 ) / 2 \\atop \\mu + 2 , - m + 2 , 1 - m - \\mu } \\Big { | } 4 x \\right ) x \\\\ & = ( - 1 ) ^ m \\sum _ { j = 0 } ^ { \\lfloor ( m - 2 ) / 2 \\rfloor } \\binom { m - j - 2 } { j } ( \\mu + j + 2 ) _ { m - 2 j - 2 } x ^ { j + 1 } , \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} \\| \\sin \\Theta ( U , \\hat { U } ) \\| & \\leq \\frac { \\alpha z _ { 2 1 } + \\beta z _ { 1 2 } } { \\alpha ^ 2 - \\beta ^ 2 - z _ { 2 1 } ^ 2 \\wedge z _ { 1 2 } ^ 2 } \\wedge 1 , \\\\ \\| \\sin \\Theta ( U , \\hat { U } ) \\| _ F & \\leq \\frac { \\alpha \\| Z _ { 2 1 } \\| _ F + \\beta \\| Z _ { 1 2 } \\| _ F } { \\alpha ^ 2 - \\beta ^ 2 - z _ { 2 1 } ^ 2 \\wedge z _ { 1 2 } ^ 2 } \\wedge \\sqrt { r } . \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} ( k + 1 ) \\cdot p ^ { \\beta - 1 } = k \\cdot p ^ { \\beta - 1 } + p ^ { \\beta - 1 } & \\leq k \\cdot p ^ { \\beta - 1 } + k \\cdot ( p - 1 ) \\cdot p ^ { \\beta - 1 } \\\\ & = k \\cdot p ^ { \\beta } \\leq k \\cdot p ^ { \\alpha } \\leq k \\cdot a < n . \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} & \\sum \\limits _ { \\delta | N } \\delta ' r _ \\delta = p _ 1 p _ 2 r + p _ 1 p _ 2 r - p _ 1 p _ 2 r - p _ 1 p _ 2 r = 0 \\\\ & \\sum \\limits _ { \\delta | N } r _ \\delta = r ( 1 + p _ 1 p _ 2 - p _ 1 - p _ 2 ) = \\frac { 2 4 } { ( p _ 1 - 1 ) ( p _ 2 - 1 ) } ( p _ 1 - 1 ) ( p _ 2 - 1 ) = 2 4 . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } e ^ { - \\beta t } & p _ D ( t , x _ 1 , y _ 1 ) p _ D ( t , x _ 2 , y _ 2 ) \\d t \\\\ & \\geq \\int _ { t _ 0 } ^ { \\infty } e ^ { - \\beta t } p _ D ( t , x _ 1 , y _ 1 ) p _ D ( t , x _ 2 , y _ 2 ) \\d t \\\\ & \\geq c _ 1 \\int _ { t _ 0 } ^ { \\infty } e ^ { - \\beta t } e ^ { - 2 \\mu _ 1 t } \\d t \\\\ & = \\frac { c _ 2 e ^ { - ( \\beta + 2 \\mu _ 1 ) t _ 0 } } { \\beta + 2 \\mu _ 1 } . \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} \\tilde U ( X ) & = 4 0 X ^ { 1 2 } + 1 6 X ^ { 1 1 } - 4 3 X ^ { 1 0 } - 2 6 X ^ 9 + 9 X ^ 8 - 3 4 X ^ 7 \\\\ & \\qquad \\qquad + 3 4 X ^ 6 - 9 X ^ 5 - 2 6 X ^ 4 - 4 3 X ^ 3 + 1 6 X ^ 2 - 4 0 X + 6 5 , \\\\ \\tilde V ( X ) & = - 1 8 X ^ { 1 2 } + 4 5 X ^ { 1 1 } - 3 2 X ^ { 1 0 } + 6 3 X ^ 9 - 5 1 X ^ 8 + 5 2 X ^ 7 \\\\ & \\qquad \\qquad + 5 2 X ^ 6 - 5 1 X ^ 5 + 6 3 X ^ 4 - 3 2 X ^ 3 + 4 5 X ^ 2 - 1 8 X + 1 . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} \\tilde { \\mathcal { U } } _ s ^ \\eta = \\left \\{ Z _ s = ( X _ s , V _ s ) \\in \\overline { \\mathcal { D } _ s } \\left | \\inf _ { 1 \\leq i < j \\leq s } \\left ( \\frac { | v _ i - v _ j | } { \\eta } + \\frac { \\iota ( x _ i - x _ j , v _ i - v _ j ) } { \\eta \\log \\frac { 1 } { \\eta } } \\right ) > 1 \\right . \\right \\} \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} \\Delta _ u = \\sum _ { n = 0 } ^ \\infty ( - u ) ^ n \\Delta _ { e _ n } . \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} M ' _ { \\delta } = M _ { \\delta } z ^ { 2 + 2 \\delta } M _ { \\delta } = B _ { \\delta } \\big \\{ n _ K ^ { n _ K / 4 } D _ K ^ { 1 / 2 } Q ^ { 1 / 2 } T ^ { n _ K / 2 + 1 } \\big \\} ^ { 1 + \\delta } . \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\left | A ( z , \\zeta ) \\right | ^ { 1 + \\mu _ { 0 } } \\delta _ { \\Omega } ^ { - \\gamma \\mu _ { 0 } - \\varepsilon } ( \\zeta ) \\lesssim \\delta _ { \\Omega } ( z ) ^ { - \\mu _ { 0 } ( \\gamma + n ) - \\varepsilon } \\prod _ { j = 1 } ^ { n - 1 } \\tau _ { j } ^ { 2 } \\left ( z , \\delta _ { \\Omega } ( z ) \\right ) . \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} C _ { } & = \\max _ { 0 \\leq p \\leq 1 } ( 1 - \\epsilon ) \\frac { p + \\epsilon H _ 2 ( p ) } { \\epsilon + ( 1 - \\epsilon ) p } . \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} \\textstyle q = \\sum _ { j = 1 } ^ n \\sigma _ j > 2 . \\end{align*}"}
-{"id": "10023.png", "formula": "\\begin{align*} n _ t + n _ a = | S T | = q ^ t + 1 , \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} \\lambda _ i = \\beta _ i \\sigma , i = 0 , 1 . , \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} \\lambda _ R ( F ) = \\det H _ \\mathrm { r e l } ^ \\bullet ( Z _ { 1 , R } , F ) \\otimes \\big ( \\det H ^ \\bullet ( Z _ R , F ) \\big ) ^ { - 1 } \\otimes \\det H ^ \\bullet _ \\mathrm { a b s } ( Z _ { 2 , R } , F ) . \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} \\partial ^ * f ( g ) = - \\int f g ' \\ d m = \\int f ' g \\ d m + \\sum _ { x \\in \\partial I } \\pm f ( x ) g ( x ) . \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} g ( a b + a + b ) + 1 = f ( a b + a + b + 1 ) = f ( a + 1 ) f ( b + 1 ) = g ( a ) + g ( b ) + g ( a b ) + 1 \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} { \\mathcal { R } } ( 1 ) ( x ) = \\int _ { M } \\chi \\bigl ( b ( \\rho ( y ) ) d ( x , y ) \\bigr ) d m ( y ) > 0 . \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} P _ t ^ D f ( x ) = E ^ x ( f ( X _ t ) , \\ , \\tau _ D > t ) , x \\in D , \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} C _ { p , g } ^ { ( \\ell ) } & = \\sum _ { j = 1 } ^ { p } ( - 1 ) ^ { j - 1 } \\sum _ { t = 0 } ^ { \\lfloor \\frac { p - j } { 2 } \\rfloor } \\binom { n - p + j + 2 t } { t } \\sum _ { \\beta = 0 } ^ { p - j - 2 t } 2 ^ { p - j - 2 t - \\beta } \\binom { n - \\ell } { \\beta } \\binom { \\ell } { p - j - 2 t - \\beta } \\\\ & \\sum _ { i = 0 } ^ { j - 1 } \\binom { \\beta } { p + i - j - t - g } . \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} H _ { e } ^ { s } ( \\Omega ) : = \\left \\{ u \\in W ^ { s , 2 } ( \\mathbb { R } ^ { N } ) \\ , : \\ , u = 0 \\mathbb { R } ^ { N } \\backslash \\Omega \\right \\} , \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} a ^ x = \\sum _ { r = 0 } ^ { \\infty } \\frac { 1 } { r ! } ( k x ) ^ r \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} \\mathcal { A } \\big ( \\mathbf { H } ( t , \\cdot ) \\big ) = \\mathbf { g } ( t , \\cdot ) t \\in [ 0 , T ] \\mathbf { g } = \\partial _ { t } \\mathbf { u } \\in L ^ { 2 } ( 0 , T ; \\mathcal { H } ) , \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\mu _ k ( p ) = \\min \\limits _ { 0 < r _ 1 < \\dots < r _ { k - 1 } < 1 } \\max \\left \\{ \\mu _ 1 ^ { ( 0 , r _ 1 ) } ( p ) , \\mu _ 1 ^ { ( r _ 1 , r _ 2 ) } ( p ) , \\dots , \\mu _ 1 ^ { ( r _ { k - 1 } , 1 ) } ( p ) \\right \\} , \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} \\nabla _ { \\gamma ^ \\prime } \\gamma ^ \\prime = \\kappa J ^ { 9 0 } _ \\gamma ( \\gamma ' ) \\ ; . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} \\partial ^ { \\alpha } _ t u = \\left ( a ^ { i j } u _ { x ^ i x ^ j } + b ^ i u _ { x ^ i } + c u + f ( u ) \\right ) + \\partial ^ { \\beta } _ t \\int ^ t _ 0 h ( u ) \\ , d B _ t \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} x _ { i j } ^ \\alpha = - \\sigma h _ { i j } \\nu ^ { \\alpha } , \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} y ( x ; \\xi , \\eta ) : = \\int _ 0 ^ 1 \\nabla _ \\xi \\varphi _ a ( x , \\eta + \\theta ( \\xi - \\eta ) ) d \\theta , \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} [ a _ { ( m ) } , b _ { ( n ) } ] = \\sum _ { i \\geq 0 } \\begin{pmatrix} m \\\\ i \\end{pmatrix} ( a _ { ( i ) } b ) _ { ( m + n - i ) } , m , n \\in \\Z _ + . \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} \\prod _ { v \\in \\{ \\} } ( a , b ) _ v \\prod _ { p \\in \\{ \\} } ( a , b ) _ p = 1 . \\end{align*}"}
-{"id": "10171.png", "formula": "\\begin{align*} \\mathbb E [ \\mathcal V _ n ] = n + 2 \\sum _ { 1 \\leq i < j \\leq n } \\mathbb P ( Z _ { j - i } = 0 ) \\sim C ' \\frac { n ^ 2 } { a _ n } , \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\varphi ( t , x ) = \\mu | x + R t ( 1 - t ) e _ 1 | ^ 2 + \\frac { R ^ 2 t ( 1 - t ) ( 1 - 2 t ) } { 6 } - \\frac { R ^ 2 t ( 1 - t ) } { 1 6 \\mu } , \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} ( m - 4 ) \\alpha ^ { \\frac 4 3 } - ( m - 1 ) \\alpha - \\alpha ^ { \\frac 1 3 } + 1 = 0 . \\end{align*}"}
-{"id": "9945.png", "formula": "\\begin{align*} a _ i ( t ) w _ { r - j } = e ^ { ( \\delta _ { i } + 2 j ) t } w _ { r - j } \\delta _ { i } + 2 j \\geq - \\delta _ { i } . \\end{align*}"}
-{"id": "10155.png", "formula": "\\begin{align*} z _ { k + 1 } = \\frac { 1 } { 2 } ( 1 + \\sqrt { \\frac { L } { \\mu } } ) x _ { k + 1 } + \\frac { 1 } { 2 } ( 1 - \\sqrt { \\frac { L } { \\mu } } ) x _ k . \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} \\begin{cases} u _ t + \\alpha u _ { x x x } = f , & \\ , \\ , ( 0 , L ) \\times ( 0 , T ) , \\\\ u ( 0 , t ) = h _ 0 ( t ) , \\ , \\ , u ( L , t ) = h _ 1 ( t ) , \\ , \\ , u _ { x } ( L , t ) = h _ 2 ( t ) , & \\ , \\ , ( 0 , T ) , \\\\ u ( x , 0 ) = u ^ 0 ( x ) , & \\ , \\ , ( 0 , L ) . \\end{cases} \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} 0 = p _ { \\tilde { 0 } } = \\sum _ { \\alpha } ( x _ \\alpha ) ^ 2 - \\sum _ { \\mu } ( y _ \\mu ) ^ 2 . \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} \\int f g d \\mu : = \\sum _ { i = 1 } ^ { + \\infty } a _ i \\int _ 0 ^ 1 f _ i ( x ) g _ i ( x ) d x \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { { c _ 1 } / 2 } q _ { \\alpha , \\beta } ( T ( t - s ) , x ) = T ^ { - \\frac { \\alpha ( d + { c _ 1 } ) } { 2 } + \\alpha - \\beta } ( - \\Delta ) ^ { { c _ 1 } / 2 } q _ { \\alpha , \\beta } ( t - s , T ^ { - \\frac { \\alpha } { 2 } } x ) , \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} \\left ( x q ; q ^ { 2 } \\right ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } z ^ { n } } { \\left ( q ; q \\right ) _ { n } \\left ( x q ; q ^ { 2 } \\right ) _ { n } } = \\sum _ { n = 0 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } \\left ( - x \\right ) ^ { n } } { \\left ( q ^ { 2 } ; q ^ { 2 } \\right ) _ { n } } A _ { q } \\left ( - q ^ { 2 n } z \\right ) , \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} g _ \\infty ^ { \\left ( s \\right ) } \\left ( t \\right ) = g _ \\infty ^ { \\left ( s \\right ) } \\left ( 0 \\right ) + \\ell ^ { - 1 } \\int _ { 0 } ^ { t } V _ { s + 1 } ^ { 0 } \\left ( \\tau \\right ) g _ \\infty ^ { \\left ( s + 1 \\right ) } \\left ( \\tau \\right ) d \\tau \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} \\tilde { Z } _ { s , s + k + 1 } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ { k + 1 } \\right ] = \\left ( X _ { s + k } ^ \\prime , V _ { s + k } ^ \\prime , x _ { i _ { k + 1 } } ^ \\prime + \\varepsilon \\omega _ { k + 1 } , v _ { s + k + 1 } \\right ) \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\frac { \\partial T _ { p q } } { \\partial x _ { q } } = \\mathcal { F } _ { p } + \\frac { \\partial \\mathcal { G } _ { p } } { \\partial t } , \\qquad \\overrightarrow { \\mathcal { G } } = \\frac { 1 } { 4 \\pi c } \\left ( \\mathbf { D } \\times \\mathbf { B } \\right ) , \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} \\mathcal { I } \\cap \\{ g _ k ( x , y ) : x \\in \\mathcal { I } , y \\in \\{ 0 , 1 \\} \\} = \\emptyset . \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} \\| D _ H f ( x ) \\| ^ Q & = \\| g ^ { - 1 } f ^ * h ( x ) \\| ^ { Q / 2 } \\leq K ^ { ( 2 n + 2 ) / 2 n } \\det ( g ^ { - 1 } f ^ * h ( x ) ) ^ { ( 2 n + 2 ) / 4 n } \\\\ & = K ^ { ( n + 1 ) / n } H J ( x , f ) ^ { ( n + 1 ) / n } = K ^ { ( n + 1 ) / n } J ( x , f ) . \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} \\mathrm { T r } ^ N _ M ( ( g E _ { \\lambda , N , \\overline { \\chi } } ) ^ \\mu ) = c _ M \\sum _ { i = 1 } ^ { d } \\frac { D ( k \\mu - 1 , f _ i , g ^ \\mu E ^ { \\mu - 1 } _ { \\lambda , N , \\overline { \\chi } } ) } { \\pi ^ { k \\mu } \\langle f _ i , f _ i \\rangle } f _ i , \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} \\sum ( m ^ 2 - l ^ 2 ) a _ m ^ k a _ l ^ j = 0 , \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} H _ { k } ^ { \\pm , + } \\left ( \\omega _ { \\pm } \\right ) = \\omega _ { \\pm } \\mu _ { \\pm , + } ^ { \\prime } \\phi _ { k } ^ { \\pm } ( I _ { \\pm } ) \\bar { \\Psi } _ { k } ^ { \\pm } ( I _ { \\pm } ) \\frac { 1 } { \\omega _ { \\pm } ^ { \\prime } \\left ( I _ { \\pm } \\right ) } . \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} c _ j - f _ j = \\begin{pmatrix} 0 & v \\\\ - v ^ { t r } / s \\sqrt { 2 } & w \\end{pmatrix} , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} g ( u ) & = u ^ 2 Q ( u ) - u \\psi ( u ) \\\\ & < u \\psi ( u ) - u \\psi ( u ) \\\\ & = 0 \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\mathcal { W } _ { x _ 0 } ( x ) = a ( x _ 0 ) w _ { 3 / 2 } \\left ( \\frac { ( x - x _ 0 ) \\cdot \\nu _ { x _ 0 } } { ( \\nu _ { x _ 0 } \\cdot A ( x _ 0 ) \\nu _ { x _ 0 } ) ^ { 1 / 2 } } , \\frac { x _ { n + 1 } } { ( a ^ { n + 1 , n + 1 } ( x _ 0 ) ) ^ { 1 / 2 } } \\right ) , \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} \\begin{aligned} & b _ { s , s + k } \\left [ Z _ s , t + \\tau ; t _ 1 + \\tau , \\dots , t _ k + \\tau ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] = \\\\ & \\ ; \\ ; = b _ { s , s + k } \\left [ Z _ s , t ; t _ 1 , \\dots , t _ k ; v _ { s + 1 } , \\dots , v _ { s + k } ; \\omega _ 1 , \\dots , \\omega _ k ; i _ 1 , \\dots , i _ k \\right ] \\ ; \\ ; \\left ( = 0 \\right ) \\end{aligned} \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{gather*} P _ { r } ^ { \\ , \\prime } ( \\lambda ) P _ { r - 1 } ( \\lambda ) = D _ { r - 1 } ^ { 2 } \\sum _ { k = 0 } ^ { r - 1 } \\frac { P _ { k } ( \\lambda ) ^ { 2 } } { D _ { k } D _ { k - 1 } } \\ , \\end{gather*}"}
-{"id": "5158.png", "formula": "\\begin{align*} - L u + g \\circ u & = \\mu ^ { \\# } \\ , \\ , \\mbox { i n } \\ , \\ , \\Omega \\\\ u & = \\nu ^ { \\# } \\ , \\ , \\mbox { o n } \\ , \\ , \\partial \\Omega . \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} \\begin{aligned} & \\tilde { b } _ { s , s + k + 1 } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ { k + 1 } \\right ] = \\\\ & = \\tilde { b } _ { s , s + k } \\left [ Z _ s , t ; \\left \\{ t _ j , v _ { s + j } , \\omega _ j , i _ j \\right \\} _ { j = 1 } ^ k \\right ] \\times \\omega _ { k + 1 } \\cdot \\left ( v _ { s + k + 1 } - v _ { i _ { k + 1 } } ^ \\prime \\right ) \\end{aligned} \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{gather*} D _ n : = \\det { \\mathcal { H } } _ { n } > 0 \\ , \\ \\ \\ 0 \\leq n \\leq n _ { 0 } - 1 \\ , \\ \\ \\det { \\mathcal { H } } _ { n } = 0 \\ , \\ \\ \\ n \\geq n _ { 0 } \\ . \\end{gather*}"}
-{"id": "6617.png", "formula": "\\begin{align*} \\Omega \\triangleq \\sum _ { i = 1 } ^ M n _ i \\ , a _ i , \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } ' : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\lim _ { b \\to + \\infty } \\sqrt { \\frac { 4 b } { \\pi } } e ^ { 2 b x } \\rho _ { \\nu + j , b } ( x ^ 2 ) = x ^ { \\nu + j - \\frac { 1 } { 2 } } , j = 0 , 1 , \\ldots , n , \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} L _ { y _ 0 } v = L _ { y _ 0 } v _ { y _ 0 } + \\tilde { f } \\mathcal { B } _ { 1 / 2 } ^ + ( y _ 0 ) , \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} n : = [ K _ T ( \\alpha ^ { p ^ i } ) : K _ T ] = [ K ( \\alpha ^ { p ^ i } ) : K ] \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{gather*} \\tau ^ { \\alpha , \\beta } _ { k , \\ell } = \\sum _ { n _ { c } + n _ { d } = k , n _ { d } + n _ { e } = \\ell \\atop n _ { c } , n _ { d } , n _ { e } \\ge 0 } c ^ { ( \\alpha , \\beta ) } _ { n _ { c } , n _ { d } , n _ { e } } , \\end{gather*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\vec { x } ^ * = \\vec { S } ^ { \\dagger } \\Bigl \\lbrace \\mbox { I S L R } \\bigl ( \\vec { S ( y ) } ; \\lambda _ i , a _ i \\bigr ) \\Bigr \\rbrace , i = 0 , 1 . , \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} f _ { n , k } ( s ) = g _ { n , k } ( s ) = P ( X _ { 1 } > T _ { ( k ) } ) , \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} x ( v \\ * u ) = ( x v ) \\ * u + v \\ * x u , ( v \\ * u ) x = v \\ * ( u x ) , \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} B = U + U ^ { * } + \\alpha V \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} I ( A _ 0 , D ) = P _ { \\mathcal { T } ( X ) } ( A _ 0 + D ) , I ( A _ 0 , D ) , A _ 0 \\in \\mathcal { T } ( X ) . \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} & T _ { p q } = \\frac { 1 } { 8 \\pi } \\left [ E _ { p } D _ { q } + E _ { q } D _ { p } + H _ { p } B _ { q } + H _ { q } B _ { p } \\right . \\\\ & \\qquad \\qquad \\left . - \\delta _ { p q } \\left ( \\mathbf { E } \\cdot \\mathbf { D } + \\mathbf { H } \\cdot \\mathbf { B } \\right ) \\right ] \\qquad \\left ( p , q = 1 , 2 , 3 \\right ) , \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} \\langle f , \\star \\partial g \\rangle _ 2 = \\langle \\star f , \\partial g \\rangle _ \\mathcal { H } = \\langle f \\omega , \\partial g \\rangle _ \\mathcal { H } = \\langle \\omega , f \\partial g \\rangle _ \\mathcal { H } = - \\langle \\omega , g \\partial f \\rangle _ \\mathcal { H } = - \\langle \\star \\partial f , g \\rangle _ 2 , \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} \\varrho _ { T } ^ { A } ( E ) = \\inf \\limits _ { E = x _ 1 + x _ 2 } [ \\varrho _ { T _ 1 } ^ { A _ 1 } ( x _ 1 ) + \\varrho _ { T _ 2 } ^ { A _ 2 } ( x _ 2 ) ] , \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} & \\sigma _ j \\sigma _ 1 ^ { - 1 } \\gamma _ { 1 k } = \\sigma _ j \\sigma _ 1 ^ { - 1 } \\gamma _ { * 1 k } - i \\sigma _ j \\Phi \\Phi ^ * \\sigma _ k + i \\sigma _ j \\sigma _ 1 ^ { - 1 } \\sigma _ k \\Phi \\Phi ^ * \\sigma _ 1 \\\\ & \\gamma _ { 1 k } \\sigma _ 1 ^ { - 1 } \\sigma _ j = \\sigma _ j \\sigma _ 1 ^ { - 1 } \\gamma _ { * 1 k } - i \\sigma _ 1 \\Phi \\Phi ^ * \\sigma _ k \\sigma _ 1 ^ { - 1 } \\sigma _ j + i \\sigma _ k \\Phi \\Phi ^ * \\sigma _ j \\\\ \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} \\Delta { C } ^ 2 _ { t } = ( \\beta _ t - \\gamma _ t ) \\Big ( \\Delta { C } ^ 2 _ { t + 1 } + \\log \\big ( 1 + 2 ^ { \\Delta { C } ^ 1 _ { t + 1 } } \\big ) \\Big ) , ~ \\Delta { C } ^ 2 _ { n + 1 } = 0 , ~ t \\in \\{ n , \\ldots , 0 \\} . \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} \\mathcal { D } _ s = \\left \\{ \\left . Z _ s = \\left ( X _ s , V _ s \\right ) \\in \\mathbb { R } ^ { d s } \\times \\mathbb { R } ^ { d s } \\right | | x _ i - x _ j | > \\varepsilon \\ ; \\forall \\ ; 1 \\leq i < j \\leq s \\right \\} \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} \\begin{cases} u ( 0 , t ) = 0 , \\ , \\ , u ( L , t ) = 0 , \\ , \\ , u _ { x } ( L , t ) = h _ 2 ( t ) , \\\\ v ( 0 , t ) = 0 , \\ , \\ , v ( L , t ) = 0 , \\ , \\ , v _ { x } ( L , t ) = g _ 2 ( t ) . \\end{cases} \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} f ( t ) = \\alpha { \\bf 1 } _ { [ 0 , 1 - p ] } ( t ) + a ( t ) + \\beta { \\bf 1 } _ { ( 1 - p , p ) } ( t ) + b ( t ) + \\gamma { \\bf 1 } _ { [ p , 1 ] } ( t ) + c ( t ) , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{gather*} a \\star b = \\mu \\mathcal { F } ( \\blacktriangleright \\otimes \\blacktriangleright ) ( f \\otimes g ) = \\big ( \\mathcal { F } ^ { ( 1 ) } \\blacktriangleright a \\big ) \\big ( \\mathcal { F } ^ { ( 2 ) } \\blacktriangleright b \\big ) \\end{gather*}"}
-{"id": "2004.png", "formula": "\\begin{align*} \\mathcal { Q } _ { k } = ( \\tau _ k , ( \\mathcal { D } _ { k } - d _ 0 ) + \\varepsilon _ { k } \\tau _ k ) , k = 1 , 2 , \\cdots r , \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} - Z _ 0 ( 1 / 2 ) = - \\left ( 1 - \\frac { 1 } { \\sqrt { 2 } } \\right ) \\zeta ( 1 / 2 ) = \\left ( 1 - \\frac { 1 } { \\sqrt { 2 } } \\right ) \\lim _ { n \\to \\infty } a _ n , \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} \\tilde { h } _ { i j } = h _ { i j } , \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} ( h \\pitchfork u ) ( x ) = \\hom ( u , h ( x ) ) , \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} \\begin{aligned} \\sigma _ j ( Z ) & = & \\tilde { B _ j } Z , \\ j = 1 , 2 \\end{aligned} \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} \\frac { 2 } { 1 - q ^ n } = 2 + 2 q ^ n + 2 q ^ { 2 n } + 2 q ^ { 3 n } + \\dots \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\Pi ( x ) = \\prod _ { p \\leq x } p , \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} u ^ { \\prime } + \\xi & = f ( u ) ( 0 , T ) , \\\\ \\xi & \\in \\partial \\phi ( u ) ( 0 , T ) , \\\\ u ( 0 ) & = u _ { 0 } . \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} \\xi \\cdot U \\psi \\le \\xi \\cdot U \\psi ' & & & & \\xi \\cdot U ( \\psi \\otimes u ) = ( \\xi \\cdot U \\psi ) \\otimes u , \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} ( a ) _ k = \\frac { \\Gamma ( a + k ) } { \\Gamma ( a ) } = a ( a + 1 ) \\cdots ( a + k - 1 ) \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} g ^ a = g ( a ^ { - 1 } ) ^ g a = x . \\big ( x ^ { 2 \\alpha } y ^ { 2 \\beta } z ^ { 2 \\gamma } \\cdot ( y ^ { - 2 \\lambda } z ^ { - 2 \\mu } ) ^ x \\cdot y ^ { 2 \\lambda } z ^ { 2 \\mu } \\big ) = x . ( x ^ { 2 \\alpha } y ^ { 2 \\beta + 4 \\lambda } z ^ { 2 \\gamma + 4 \\mu } ) ; \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} \\sigma ( \\delta + \\delta ^ { t r } ) \\sigma - \\sigma \\Delta h + h ^ { t r } \\sigma \\Delta = 0 , \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} y ^ k = y ^ 0 + \\sum _ { s = 1 } ^ k \\sum _ { j = 1 } ^ N \\frac { 1 } { N } \\left ( x ^ s _ j - x ^ { s - 1 } _ j \\right ) = \\sum _ { \\ell = 1 } ^ { N } \\frac { 1 } { N } v _ \\ell ^ 0 + \\sum _ { s = 1 } ^ k \\sum _ { j = 1 } ^ N \\frac { 1 } { N } \\left ( x ^ s _ j - x ^ { s - 1 } _ j \\right ) , \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} P _ { n } = - a - \\frac { b } { z ^ { 2 n + 2 } } + o ( 1 ) , \\mbox { a s } n \\to \\infty . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} r = j k ^ { - 1 } , k = \\partial _ { x x x } - \\partial _ x , j = - \\partial _ x \\ , u \\ , ( \\partial _ x ) ^ { - 1 } \\ , u \\ , \\partial _ x . \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} \\sum _ { \\gamma _ { s } \\in \\Gamma ( k + ) } \\left ( \\frac { 1 } { \\mid \\gamma + t \\mid ^ { 2 } - \\mid \\gamma + \\gamma _ { 1 } + \\gamma _ { 2 } + . . . + \\gamma _ { s } + t \\mid ^ { 2 } } \\right ) ^ { 2 } = O ( s ^ { - 1 } ) . \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} \\mathrm { R e } \\ , h ^ { W ^ \\bullet } _ t \\Big ( w ( t ) , \\frac { \\partial } { \\partial t } w ( t ) \\Big ) = 0 . \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} & \\mathbf { Y } _ { [ 1 : K ] } ^ { T _ E } = \\mathbf { H } _ { [ 1 : K ] } ^ { [ 1 : M ] } ~ \\mathbf { X } _ { [ 1 : M ] } ^ { T _ E } + \\mathbf { n } _ { [ 1 : K ] } ^ { T _ E } , \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} \\det \\left [ \\begin{array} { c c c c c c c c c c } 0 & 1 & \\cdots & k - 1 \\\\ 0 & 1 & \\cdots & k - 1 \\end{array} \\right ] _ N \\in I ^ { \\frac { p ( 0 + 1 + \\cdots + ( k - 1 ) ) - ( 0 + 1 + \\cdots + ( k - 1 ) ) } { d } } = I ^ { \\lambda ' _ k } . \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} \\alpha _ k = \\alpha _ { k , K ^ { * } } \\leq \\frac { 1 } { { \\binom { k - 1 } { l - 1 } } } \\sum _ { \\{ L _ i \\subseteq K ^ { * } , \\ ; | L _ i | = l \\} } \\alpha _ { l , L _ i } \\leq \\frac { 1 } { { \\binom { k - 1 } { l - 1 } } } \\sum _ { i = 1 } ^ { \\binom { k } { l } } \\alpha _ { l , L _ i } . \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} Q _ 0 ( s ) & : = \\nabla _ x q ( s , t ) - I , \\\\ P _ 0 ( s ) & : = \\nabla _ x p ( s , t ) , \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} \\exp \\Phi ^ { - 1 } ( t ) = | \\omega | q ^ { - 2 \\lfloor \\log _ { q ^ { 2 } } | \\omega | \\rfloor } \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} w ( x , y , z ) = 0 , w _ x ^ 2 + w _ y ^ 2 + w _ z ^ 2 \\ne 0 . \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} F ( \\delta _ \\lambda & ( x ) , d _ \\lambda ( y , \\eta ) ) = ( \\delta _ \\lambda ( x ) , 0 ) ^ { - 1 } \\star d _ \\lambda ( y , \\eta ) \\\\ & \\stackrel { \\eqref { s e c . t w o _ 2 : e q _ D s t a r d e l t a X } } { = } ( d _ \\lambda ( x , 0 ) ) ^ { - 1 } \\star d _ \\lambda ( y , \\eta ) = d _ \\lambda ( ( x , 0 ) ^ { - 1 } \\star ( y , \\eta ) ) = d _ \\lambda ( F ( x , ( y , \\eta ) ) ) ; \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} 2 n - 1 - ( n + h + 2 ) = n - h - 3 , \\end{align*}"}
-{"id": "10098.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 2 } { x z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ 2 z ^ { 4 } } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ { 4 } z ^ { 2 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ { 5 } z } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 4 } { x ^ { 5 } z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 4 } { x ^ { 7 } z } , \\end{gather*}"}
-{"id": "753.png", "formula": "\\begin{align*} \\mathfrak { S } = O _ M ( \\Pi _ H \\omega _ S \\omega _ S ^ * ( 1 ) \\beta ( \\xi ) Q ^ { - M } ) + O ( P ^ A N ^ { - \\delta } \\beta ( \\xi ) ( 1 + \\| \\xi \\| ) ^ { - \\delta } ) . \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} & \\bar \\tau = x - i y = n + m ( x + i y ) \\\\ \\Leftrightarrow \\ \\ & x - n - m x - i ( m y + y ) = 0 \\\\ \\Leftrightarrow \\ \\ & m = - 1 \\ \\ \\ \\ x ( 1 - m ) = n . \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} & \\lim _ { n \\rightarrow \\infty } \\int _ t ^ T \\langle \\Phi ' ( s , ( u ^ n ) ^ + ( s ) ) , f ^ n ( s ) 1 _ { \\{ u ^ n > 0 \\} } \\rangle _ { 1 , - 1 } \\ , d s \\\\ = & ~ \\int _ { \\mathcal { O } _ t } \\Phi ' ( s , x , u ^ + ( s , x ) ) \\ , \\mu ( d s d x ) ~ a . s . . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} & \\left \\{ \\alpha _ { { \\mathcal { R } } , { \\mathcal { T } } } ^ { \\bar { \\mathcal { R } } _ i } \\tilde { h } _ { q , \\mathcal { T } } ^ { \\bar { \\mathcal { R } } _ i } m _ { { \\mathcal { R } } , { \\mathcal { T } } , n } ^ { \\bar { \\mathcal { R } } _ i } : \\mathcal { R } \\ni q , | \\mathcal { R } | = r + 1 , | \\mathcal { T } | = t , i \\in [ t ] , n \\in [ N ^ { \\binom { N _ R - 1 } { r + 1 } \\binom { N _ T } { t } } \\right \\} \\\\ & \\cup \\{ m : m \\in \\mathcal { M } _ q [ N + 1 ] \\} . \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} \\begin{cases} \\partial _ t u _ t ( x ) = \\mathcal { L } u _ t ( x ) + \\xi \\sigma ( u _ t ( x ) ) \\dot { F } ( t , x ) , \\ \\ \\ x \\in B _ R ( 0 ) , \\ \\ t > 0 \\\\ u _ t ( x ) = 0 , \\ \\ \\ x \\in { B _ R ( 0 ) ^ c } . \\end{cases} \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} T ( B _ 1 ^ + \\setminus B ' _ 1 ) & \\subset \\{ y _ n > 0 , y _ { n + 1 } < 0 \\} , \\\\ T ( \\Lambda _ w ) & \\subset \\{ y _ n = 0 , y _ { n + 1 } \\leq 0 \\} , \\ T ( B ' _ 1 \\setminus \\Lambda _ w ) \\subset \\{ y _ n > 0 , y _ { n + 1 } = 0 \\} , \\\\ T ( \\Gamma _ w ) & \\subset \\{ y _ n = y _ { n + 1 } = 0 \\} . \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\rho _ t ^ { A , f } ( S _ i ) = U _ t ( f ) S _ i , i = 1 , \\dots , N . \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} & \\underset { z , \\{ K : \\ ; | K | \\leq k \\} } { } \\ ; \\ ; \\| z _ K \\| _ { 1 } \\\\ & \\ ; \\ ; \\ ; \\ ; \\ ; \\| z \\| _ { 1 } \\leq 1 , \\ ; A z = 0 , \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} \\lambda ( x , \\beta ) = e ^ { f ( x ) ^ T \\beta } . \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} a _ { k , n + 1 } = a _ { k - 1 , n } - \\Big ( 2 + \\frac { \\alpha } { n + 1 } \\Big ) \\ , a _ { k , n } - \\Big ( 1 + \\frac { \\alpha } { n } \\Big ) a _ { k , n - 1 } \\ , , k = 0 , \\ldots , n \\ , . \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} & \\left | \\left ( f ( y _ 1 ) - f _ 0 ( y '' _ 1 ) ( y _ 1 ) _ n \\right ) - \\left ( f ( y _ 2 ) - f _ 0 ( y '' _ 1 ) ( y _ 2 ) _ n \\right ) \\right | \\\\ & = \\left | ( f ( y _ 1 ) - P _ { y '' _ 1 } ( y _ 1 ) ) - ( f ( y _ 2 ) - P _ { y '' _ 1 } ( y _ 2 ) ) \\right | \\lesssim r ( y _ 1 ) ^ { 1 + 2 \\alpha - \\epsilon } d _ G ( y _ 1 , y _ 2 ) ^ { \\epsilon } . \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} P ( y ) & = c _ 0 + \\sum _ { i = 1 } ^ { n + 1 } a _ i y _ i + \\sum _ { k \\in \\{ 1 , \\dots , n - 1 \\} , \\ell \\in \\{ n , n + 1 \\} } a _ { k \\ell } y _ k y _ \\ell \\\\ & + \\left ( c _ 1 y _ n ^ 3 + c _ 2 y _ n ^ 2 y _ { n + 1 } + c _ 3 y _ n y _ { n + 1 } ^ 2 + c _ 4 y _ { n + 1 } ^ 3 \\right ) . \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} v _ { e a } ^ 2 ( f _ { a v g } ^ * , g ) & = \\lim _ { \\beta \\uparrow 1 } ( 1 - \\beta ) v _ \\beta ^ 2 ( f _ \\beta ^ * , g ) \\\\ & \\leq \\lim _ { \\beta \\uparrow 1 } ( 1 - \\beta ) v _ \\beta ^ 2 ( f _ \\beta ^ * , g _ \\beta ^ * ) \\\\ & = v _ { e a } ^ 2 ( f ^ * _ { a v g } , g ^ * _ { a v g } ) . \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} \\boldsymbol { \\varepsilon } = \\frac { 1 } { 2 } ( \\nabla \\mathbf { u } ^ { T } + ( \\nabla \\mathbf { u } ) \\big ) , \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} a ( x _ 0 , x ) = \\bigwedge _ { z \\in A } \\hom ( \\psi ( z ) , a ( h ( z ) , x ) ) . \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} y = T ( x ) = ( x ' , \\partial _ { x _ n } w , \\partial _ { x _ { n + 1 } } w ) , x = T ^ { - 1 } ( y ) = ( y ' , - \\partial _ { y _ n } v , - \\partial _ { y _ { n + 1 } } v ) , \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} ^ { - 1 } = \\begin{bmatrix} A ^ { - 1 } + A ^ { - 1 } B K ^ { - 1 } C A ^ { - 1 } & - A ^ { - 1 } B K ^ { - 1 } \\\\ - K ^ { - 1 } C A ^ { - 1 } & K ^ { - 1 } \\end{bmatrix} , \\end{align*}"}
-{"id": "9880.png", "formula": "\\begin{align*} z = ( 2 K + 1 + 2 m + \\sigma ) ( 2 K + \\sigma ) . \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} E _ { 1 } ( k , \\alpha ) = 3 \\left ( \\frac { \\alpha } { 2 | k | ^ { 1 - \\alpha / 2 } } \\right ) - \\frac { \\alpha } { 2 | k | ^ { 2 } } \\left ( \\frac { \\alpha } { 2 } - 1 \\right ) . \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} \\mathcal { M } '' : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} \\hat { W } ^ { [ { \\sf u } ] } _ { k j } = \\chi _ { k j } ( \\mathbf { y } ^ { [ { \\sf b s } ] } _ k [ 1 ] , \\cdots , \\mathbf { y } _ k ^ { [ { \\sf b s } ] } [ n ] , W _ { k 1 } ^ { [ { \\sf d } ] } , \\cdots , W _ { k N } ^ { [ { \\sf d } ] } ) j \\in [ 1 : N ] . \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} a \\partial _ \\theta v = L v + N ( v ) , \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} H ( p , x ) = \\frac { 1 } { 2 } | p | ^ 2 - f ( x ) , \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} T x _ { i ( n ) } = - n x _ { i ( n - 1 ) } . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} \\left ( \\partial _ t + v \\cdot \\nabla _ x \\right ) f _ N ^ { ( 1 ) } ( t ) = C _ 2 f _ N ^ { ( 2 ) } ( t ) \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} ( 2 \\pi i ) ^ { \\sum _ { j } n _ { j } [ K _ { j } : \\Q ] k n } = ( 2 \\pi i ) ^ { [ K : \\Q ] k n } \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} z ( e _ { k + 1 } \\psi _ k e _ k ) & = \\sum _ { j = 0 } ^ { k - 1 } g _ { k + 1 , j } . y _ { k + 1 } ^ j e _ { k + 1 } \\psi _ k e _ k \\\\ & = \\sum _ { j = 0 } ^ { k - 1 } g _ { k + 1 , j } . \\psi _ k y _ k ^ j e _ k \\\\ & = \\sum _ { j = 0 } ^ { k - 2 } ( g _ { k + 1 , j } + ( - 1 ) ^ { k - j } E _ { k - 1 - j } ( x _ 1 , \\ldots , x _ { k - 1 } ) g _ { k + 1 , k - 1 } ) . \\psi _ k y _ k ^ j e _ k \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} ( \\lambda + \\gamma + n \\xi ) p _ { 0 , n } = \\lambda p _ { 0 , n - 1 } + ( n + 1 ) \\xi p _ { 0 , n + 1 } , n \\geq 1 . \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\tau ( 1 ) = \\int _ { \\widehat { G } ^ { \\rm s p h } } \\widehat { \\tau } ( \\nu ) d \\mu _ G ^ ( \\nu ) = 0 . \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} | \\phi _ j ( x ) | & = e ^ { \\lambda _ j t _ 0 } \\left | \\int _ X p _ { t _ 0 } ( x , y ) \\phi _ j ( y ) d \\mu ( y ) \\right | \\\\ & \\le e ^ { \\lambda _ j t _ 0 } \\left ( \\int _ X p _ { t _ 0 } ( x , y ) ^ 2 d \\mu ( y ) \\right ) ^ { 1 / 2 } \\\\ & \\le M e ^ { \\lambda _ j t _ 0 } . \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} \\check { u } ^ i = \\bar { g } ^ { i j } u _ j , \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} \\abs { a } _ \\nu = q _ \\nu ^ { - \\nu ( a ) } , \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} \\biggl < \\sum _ { j = 1 } ^ m \\pi ( \\tau _ j ) \\eta _ j \\biggm | \\sum _ { i = 1 } ^ n \\pi ( \\phi * \\psi _ i ) \\xi _ i \\biggr > = \\biggl < \\sum _ { j = 1 } ^ m \\pi ( \\phi ^ * * \\tau _ i ) \\eta _ i \\biggm | \\sum _ { i = 1 } ^ n \\pi ( \\psi _ i ) \\xi _ i \\biggr > \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} \\mathcal { F } ( \\mathbf { u } ) : = \\mathbf { F } ( \\nabla \\mathbf { u } ) \\mathbf { u } \\in H ^ { 1 } ( G , \\mathbb { R } ^ { d } ) . \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} \\varphi _ { k } = \\max \\{ \\varphi _ { k - 1 } , v _ { k - 1 } \\} \\geq \\varphi _ { k - 1 } \\geq \\ldots \\geq \\varphi _ 0 = \\psi \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} A = \\frac { \\left ( z ^ { - 1 } \\xi ; q \\right ) _ { \\infty } } { ( q ; q ) _ { \\infty } } \\left ( \\frac { z ^ { 2 } } { z ^ { 2 } - 1 } { } _ { 1 } \\phi _ { 1 } \\left ( q ; q z ^ { - 2 } ; q , q z ^ { - 1 } \\xi ^ { - 1 } \\right ) + { } _ { 1 } \\phi _ { 1 } \\left ( q ; z ^ { - 1 } \\xi ; q , z \\xi \\right ) - 1 \\right ) \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} g _ 1 ^ n ( t ) = \\int ^ t _ 0 P _ p ( { t - r } ) \\nabla u ^ n ( r ) b ^ n ( r ) \\d r \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} \\dim ( X _ 1 \\cup \\cdots \\cup X _ n ) = \\max ( \\dim ( X _ i ) : 1 \\leq i \\leq n ) . \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} | \\gamma _ { \\theta } | _ { g ( t _ 2 ) } \\leq e ^ { C t _ 2 \\xi ^ { - 1 } \\sqrt { S } } | \\gamma _ { \\theta } | _ { g ( 0 ) } \\leq e ^ { C T _ 0 \\xi ^ { - 1 } \\sqrt { S } } | \\gamma _ { \\theta } | _ { g ( 0 ) } = e ^ { C T _ 0 \\xi ^ { - 1 } \\sqrt { S } } L . \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} F ( x ) - F \\left ( 0 \\right ) = \\sum _ { i = 1 } ^ { n } f ( \\alpha _ { i } ) w _ { i } . \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} \\Pr [ | H _ n / \\log n - 1 / \\log ( 1 / p ) | > \\epsilon ] = O ( e ^ { - \\Theta ( \\log \\log n ) ^ 2 } ) . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} R _ 1 ^ { - H } \\Psi R _ 1 ^ { - 1 } = \\left [ \\begin{array} { c c } x & \\beta + \\j z \\\\ \\beta - \\j z & y \\end{array} \\right ] \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} \\gamma ^ l ( \\alpha ^ l ) ^ 2 = \\frac { 2 l } { \\delta ^ 1 ( l + 1 ) } \\ge \\frac { 1 } { \\delta ^ 1 } \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} \\Sigma ^ \\partial = \\coprod _ { S \\in \\Pi ^ \\partial } \\Sigma _ S ^ \\partial . \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} \\lim _ { p \\to + \\infty } \\lambda _ 2 ^ { 1 / p } ( B _ 1 ^ { \\pi / 4 } ; p ) = \\frac { 1 } { r _ 2 } , \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} \\tilde { l } _ \\alpha ( | x - y _ c ( B ) | ) \\geq \\tilde { l } _ \\alpha ( | x - y | + | y - y _ c ( B ) | ) \\geq \\tilde { l } _ \\alpha ( | x - y | + \\alpha \\sqrt { d } / 2 ) = l ( | x - y | ) . \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} H _ { n } ( t ) : = \\frac { 1 } { n } \\log \\int \\exp \\left ( t \\ , S _ { n } \\varphi \\right ) \\ , d \\mu , \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} \\| \\theta \\| ( P _ { 1 } + \\dots + P _ { g } - Q ) ^ { g - 1 } = \\exp \\left ( \\tfrac { 1 } { 8 } \\delta ( X ) \\right ) \\frac { \\| J \\| ( P _ { 1 } , \\dots , P _ { g } ) } { \\prod _ { j < k } G ( P _ { j } , P _ { k } ) } \\prod _ { j = 1 } ^ g G ( P _ { j } , Q ) ^ { g - 1 } . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} T _ s T _ { t _ 0 } f = T _ s g + T _ s h _ { \\max } - T _ s e _ { t _ 0 } \\ge ( 1 + \\delta ) h _ { \\max } - T _ s e _ { t _ 0 } - \\tilde e _ s . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} { \\bf y } _ r = & \\sum \\limits _ { i = 1 } ^ 3 \\sum \\limits _ { j \\neq i } { \\bf H } _ { i , r } { \\bf V } _ { i , j } ^ p { \\bf s } _ { i , j } ^ p + \\sum \\limits _ { ( i , j ) \\in { \\cal S } } { \\bf H } _ { i , r } { \\bf V } _ { i , j } ^ c { \\bf s } _ { i , j } ^ c \\\\ & + { \\bf H } _ { 2 , r } { \\bf V } _ { 2 , 3 } ^ r { \\bf s } _ { 2 , 3 } ^ r + { \\bf H } _ { 3 , r } { \\bf V } _ { 3 , 1 } ^ r { \\bf s } _ { 3 , 1 } ^ r + { \\bf n } _ r , \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} \\Lambda ( x , y ) ( \\pi ( t ) f ) = \\Lambda ( x , y + t ) ( f ) \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} \\alpha \\left ( \\beta + \\gamma \\right ) + A \\beta + A \\gamma & = \\left ( a + A \\right ) \\left ( \\beta + \\gamma \\right ) + A \\beta + A \\gamma \\\\ & = a \\left ( \\beta + \\gamma \\right ) + A \\left ( \\beta + \\gamma \\right ) + A \\beta + A \\gamma \\\\ & = a \\beta + a \\gamma + A \\left ( \\beta + \\gamma \\right ) + A \\beta + A \\gamma \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{gather*} \\eta i _ 1 = i _ 1 g f \\\\ = G i _ 2 f = G F i _ 1 \\end{gather*}"}
-{"id": "8548.png", "formula": "\\begin{align*} M _ { 1 } ^ h = \\sum _ { f \\in H _ { 2 k } ^ { * } ( N ) } ^ { h } L _ f ( 1 / 2 ) X ( f ) = \\sum _ { l \\leq L } \\frac { x _ l } { \\sqrt { l } } M _ 1 ( l , 0 , 0 ) \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} \\tilde { p } _ { k , \\varphi } ( \\omega , \\lambda ) & = a _ { n n } ^ { ( k ) } \\prod _ { k ' = 1 , 2 } \\Big ( ( - 1 ) ^ k \\lambda + \\sum _ { j = 1 } ^ { n - 1 } a _ { n j } ^ { ( k ) } / a _ { n n } ^ { ( k ) } ( x _ 0 ) \\xi _ j + i \\tau \\gamma _ k + ( - 1 ) ^ { k ' } ( B _ k + i A _ k ) \\Big ) . \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\tilde { H } _ { \\rm r } : = \\sqrt { - \\Delta + m ^ 2 } - m + \\tilde W ( x ) , \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{gather*} S _ { x } ^ { \\alpha } \\colon \\ B \\to B , S _ { x } ^ { \\alpha } ( 1 ) = 0 , S _ { x } ^ { \\alpha } ( x _ { k } ) = x _ { k + \\alpha } , \\\\ \\hphantom { S _ { x } ^ { \\alpha } \\colon } \\ S _ { x } ^ { \\alpha } ( y _ { k } ) = y _ { k } , y \\ne x , \\alpha \\in \\mathbb { Z } . \\end{gather*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\psi : = \\psi _ p \\circ \\prod \\limits _ { j = 1 } ^ { \\ell } E _ { c _ j f } ( x ) _ { \\pi _ j } : \\widetilde { \\bold { B } } \\longrightarrow \\widetilde { \\bold { B } } , \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} \\int _ M f ^ p \\leq { \\int _ M f ^ p } \\big | _ { t = 0 } \\cdot \\exp ( 5 C p T ^ * ) , \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} \\mathcal { V } _ s ^ \\eta = \\left \\{ ( Z _ s , Z _ s ^ \\prime ) \\in \\overline { \\mathcal { D } _ s } \\times \\overline { \\mathcal { D } _ s } \\left | \\begin{aligned} & \\inf _ { 1 \\leq i \\neq j \\leq s } | v _ i - v _ j ^ \\prime | > \\eta \\\\ & \\textnormal { a n d } \\\\ & \\inf _ { 1 \\leq i \\leq s \\ ; : \\ ; v _ i \\neq v _ i ^ \\prime } | v _ i - v _ i ^ \\prime | > \\eta \\end{aligned} \\right . \\right \\} \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} \\tau ( \\zeta , v _ { i } , \\varepsilon ) = \\tau _ { i } ( \\zeta , \\varepsilon ) . \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} h & > { \\log \\Bigl ( { 2 \\delta _ T \\over T + 1 } \\Bigr ) \\over \\log \\Bigl ( { \\gamma \\over 2 + \\gamma } \\Bigr ) } = { \\log \\Bigl ( ( { 2 + \\gamma \\over \\gamma } ) { T - T _ \\gamma \\over T + 1 } \\Bigr ) \\over \\log \\Bigl ( { \\gamma \\over 2 + \\gamma } \\Bigr ) } \\ , . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\chi - 1 } U _ i B U _ i ^ * - \\sum _ { i = 1 } ^ { \\chi - 1 } U _ i C U _ i ^ * = C - B . \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\rho '' + \\frac { \\rho ' } { r } - \\frac { ( \\rho ' ) ^ 2 } { 2 \\rho } - \\frac { 2 \\rho } { r ^ 2 } + 2 ( 1 - \\rho ) \\rho = 0 , \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} V ^ { 2 , G } ( \\Omega _ T ) : = L ^ \\infty \\left ( 0 , T ; L ^ 2 ( \\Omega ) \\right ) \\cap V ^ G ( \\Omega _ T ) \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} c _ 2 = f _ 2 = \\delta _ 2 = 0 \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} L u = \\sum _ { i , j = 1 } ^ { N } \\frac { \\partial } { \\partial x _ i } \\left ( a _ { i j } ( x ) \\frac { \\partial u } { \\partial x _ j } \\right ) - \\sum _ { j = 1 } ^ { N } b _ j ( x ) \\frac { \\partial u } { \\partial x _ j } + \\sum _ { j = 1 } ^ { N } \\frac { \\partial ( c _ j ( x ) u ) } { \\partial x _ j } - d u , \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { p - 1 } i C _ { \\tau , ( p - i - 1 , b ) } - \\sum _ { i = 0 } ^ { p - 1 } i C _ { \\tau , ( p - i - 1 , 0 ) } + p \\sum _ { l = 1 } ^ { b } C _ { \\sigma , ( 0 , l ) } & = \\sum _ { l = 1 } ^ { b } \\sum _ { k = 0 } ^ { p - 1 } \\left ( C _ { \\sigma , ( 0 , l ) } + \\sum _ { i = 0 } ^ { k } ( C _ { \\tau , ( i , l ) } - C _ { \\tau , ( i , l - 1 ) } ) \\right ) \\\\ & = \\sum _ { l = 1 } ^ { b } \\sum _ { k = 0 } ^ { p - 1 } C _ { \\sigma , ( k + 1 , l ) } = \\frac { b } { q } C _ \\sigma . \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} T _ 0 & = \\min \\{ T _ + , \\mu ( t , T ) \\} - \\min \\{ T _ - , \\mu ( t , T ) \\} , \\\\ S _ 0 & = \\min \\{ S _ + , \\mu ( t , S ) \\} - \\min \\{ S _ - , \\mu ( t , S ) \\} . \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} v \\left ( \\sum _ { ( i , j ) \\in I } a ( i , j ) x ^ i ( f ( x ) ) ^ j \\right ) = v \\left ( \\sum _ { ( i , j ) \\in I } \\tilde { a } ( i , j ) x ^ i ( f ( x ) ) ^ j + \\sum _ { ( i , j ) \\in I } b ( i , j ) x ^ i ( f ( x ) ) ^ j \\right ) . \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} d ( Y _ { 1 } \\sharp Y _ { 2 } , s _ { 1 } \\sharp s _ { 2 } ) = d ( Y _ { 1 } , s _ { 1 } ) + d ( Y _ { 2 } , s _ { 2 } ) , \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} \\forall t \\in [ 0 , T ] , \\int _ { \\mathbb { R } ^ { 2 d } } \\left ( x \\cdot v - | v | ^ 2 t \\right ) f ( t , x , v ) d x d v = \\int _ { \\mathbb { R } ^ { 2 d } } x \\cdot v f ( 0 , x , v ) d x d v \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\lim _ { c \\to \\infty } \\frac { 1 } { c ^ 2 } \\norm { k ^ 4 \\widehat { u } _ c } _ { L ^ 1 } = 0 . \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} \\left ( \\partial _ t + v \\cdot \\nabla _ x \\right ) g _ { \\varepsilon } ( t ) = \\ell ^ { - 1 } \\tilde { C } _ 2 \\left ( g _ { \\varepsilon } ( t ) \\otimes g _ { \\varepsilon } ( t ) \\right ) \\end{align*}"}
-{"id": "10081.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ p y ^ q ( b y + c z ) ^ r } { z ^ { p + q + r } } \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\tau = \\frac { N _ T } { \\frac { N _ T } { N _ T + N _ R - 1 } } a ^ * _ { 0 , 1 } = \\frac { N _ T + N _ R - 1 } { N _ T } ( 1 - \\mu _ R ) , \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t \\phi ( t ; x , \\xi ) = h _ \\rho ( t , \\nabla _ \\xi \\phi ( t ; x , \\xi ) , \\xi ) , \\\\ \\phi ( 0 ; x , \\xi ) = x \\cdot \\xi , \\end{array} \\right . \\end{align*}"}
-{"id": "10036.png", "formula": "\\begin{align*} \\dim W \\leq n - ( k - 1 ) { d } = { d } + r _ d . \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} h ( x , \\xi ) : = \\sum _ { j = 1 } ^ n | x _ j | ^ { \\displaystyle { 1 } / { \\sigma _ j } } + \\sum _ { k = 1 } ^ p | \\xi _ k | ^ { \\displaystyle { 1 } / { \\sigma ^ * _ k } } . \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} \\psi ^ { - 1 } ( x , y ) = \\phi \\Bigl ( \\frac { x + y } { 2 } + \\frac { x - y } { 2 } S \\Bigr ) = \\frac { x + y } { 2 } + \\frac { x - y } { 2 } \\cdot \\frac { T } { \\varpi ' } = \\frac { x + y } { 2 } + \\frac { x - y } { 2 } \\cdot \\frac { T ^ { 1 / p ^ m } } { \\varpi '^ { 1 / p ^ m } } . \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} y = f ( z ) = \\frac { \\alpha z + \\beta } { \\gamma z + \\delta } \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\partial _ t \\zeta + ( v \\cdot \\nabla _ H \\zeta + w \\partial _ z \\zeta ) - \\Delta \\zeta = g , t \\geq 0 . \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} g _ 0 = d r ^ 2 + f \\left ( \\theta \\right ) ^ 2 d \\theta ^ 2 \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} \\overrightarrow { C } _ { ( 4 x : n ) } = \\bigoplus _ { ( i , \\alpha ) } H _ { ( 2 x ) } ( i , \\alpha ) \\varphi ( i , \\alpha ) . \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} \\Delta _ v ^ { - 1 } \\tau _ u \\Delta _ v = \\nabla ^ { - 1 } \\tau _ { u v } \\nabla \\tau _ u , \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} F _ { p , i } ( x _ { p } , x _ { i } ) = u _ { 0 } ( 0 ) + u _ { \\delta ^ { i } } ( 0 ) f _ { i , 0 } ( x _ { i } ) + u _ { \\delta ^ { p } } ( 0 ) f _ { p , 0 } ( x _ { p } ) + u _ { \\delta ^ { i , p } } ( 0 ) f _ { i , 0 } ( x _ { i } ) f _ { p , 0 } ( x _ { p } ) \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\int _ { \\Omega } d v _ m ^ 2 = \\int _ { \\Omega } d u ^ 2 . \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } q ^ { \\alpha ^ { 2 } } \\left ( - i q ^ { - n / 2 } \\right ) ^ { \\nu - \\alpha } S _ { n } \\left ( - q ^ { \\nu + \\alpha } ; q \\right ) J _ { \\nu - \\alpha } ^ { ( 2 ) } \\left ( 2 i q ^ { n / 2 } ; q \\right ) d \\alpha = \\frac { 1 } { \\log q ^ { - 1 } } \\int _ { - \\infty } ^ { \\infty } \\frac { \\exp \\left ( \\frac { x ^ { 2 } } { \\log q } \\right ) } { \\left ( q ; q \\right ) _ { \\infty } \\left ( q ; q \\right ) _ { n } } d x , \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} \\textbf { a } _ { B S } = \\dfrac { 1 } { \\sqrt { N _ { B S } } } [ 1 , e ^ { j ( 2 \\pi / \\lambda ) d \\sin ( \\theta ) } , . . . , e ^ { j ( N _ { B S } - 1 ) ( 2 \\pi / \\lambda ) d \\sin ( \\theta ) } ] ^ T \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} \\frac { d } { d t } \\int _ M f ^ p = p \\int _ M f ^ { p - 1 } f ' - \\int _ M H F f ^ p , \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\delta ^ * ( 1 , 0 ) \\geq \\frac { K } { \\min \\{ M , K \\} } , ~ ~ ~ ~ r = 0 . \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} \\theta _ q \\circ T ( a ) = \\theta ( q \\ , T ( a ) q ) = \\theta ( q p \\ , T ( a ) p q ) = \\theta ( q p \\ , T ( p a p ) p q ) \\end{align*}"}
-{"id": "9978.png", "formula": "\\begin{align*} d \\bigl ( ( a , b ) , ( a ' , b ' ) \\bigr ) = d _ A ( a , a ' ) + d _ B ( b , b ' ) . \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} I _ m ( B '' ) = \\sum _ { i = 0 } ^ { n } \\varepsilon _ 2 ^ { m - i } { n + 1 - i \\choose n + 1 - m } I _ i ( B ) ( 1 + ( k + 1 ) \\varepsilon _ 2 ) ^ i . \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} d \\eta = \\Delta _ M , \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} \\overrightarrow { H } = \\frac { 1 } { 2 W ^ { 2 } } \\sum _ { k = 1 } ^ { n - 2 } ( L _ { 1 1 } ^ { k } g _ { 2 2 } + L _ { 2 2 } ^ { k } g _ { 1 1 } - 2 L _ { 1 2 } ^ { k } g _ { 1 2 } ) N _ { k } . \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} N ( \\theta ) ( a , a ^ { * } ) = ( [ \\theta ( a ^ { * } ) , a ] + a ^ { * } \\theta ( a ) , \\theta ( a ^ { * } ) a ^ { * } ) \\ , , \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} B = ( 0 , 0 , \\cdots , 0 , \\xi _ r , \\cdots , \\xi _ 1 ) \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} & \\ ! \\ ! \\ ! \\ ! \\Pi _ { \\rm S I C } ( v , n ) = \\Pi _ { \\rm N C } ( v ) \\\\ & \\ ! \\ ! \\ ! \\ ! + \\sum _ { i = 1 } ^ n \\left ( \\prod _ { j = 0 } ^ { i - 1 } \\left ( 1 - \\Pi _ { \\rm C } ( v , j ) \\right ) \\right ) \\left ( \\prod _ { j = 0 } ^ { i } \\Pi _ { \\rm D } ( v , j ) \\right ) \\Pi _ { \\rm C } ( v , i ) . \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} w _ K ( u ) = h _ K ( u ) + h _ K ( - u ) . \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} \\sum _ { j = m + 1 } ^ { \\infty } f _ { j } ^ { 2 } ( x ) = - \\frac { x ^ { 2 } } { x ^ { 2 } - 1 } W _ { m } ( f ' ( x ) , f ( x ) ) , \\mbox { f o r } 0 < | x | < 1 . \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\boldsymbol { f } } \\left ( \\bigcup _ { \\theta \\in B ^ c } \\left \\lbrace \\bar { v } _ { k } ( \\theta ) \\geq \\frac { 1 } { 2 } \\mathbb { E } _ { \\boldsymbol { f } } \\left [ \\bar { v } _ { k } ( \\theta ) \\right ] \\right \\rbrace \\right ) \\leq \\sum \\limits _ { \\theta \\in B ^ c } \\exp \\left ( \\frac { \\left ( \\mathbb { E } _ { \\boldsymbol { f } } \\left [ \\bar { v } _ { k } ( \\theta ) \\right ] \\right ) ^ 2 } { 8 k \\log ^ 2 \\frac { 1 } { \\alpha } } \\right ) \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} \\Phi _ { * y } ( w ) = ( \\lambda _ { y ' } ) _ { * } \\sigma ^ \\Phi _ { * } ( w , w ' ) + ( \\lambda _ { \\sigma ^ \\Phi ( y , y ' ) } ) _ { * } ( w ' ) . \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} y ^ 2 + x y & = x ^ 3 - 3 0 4 2 4 1 1 6 9 8 1 1 5 3 2 7 1 2 9 7 9 3 1 5 9 9 0 x \\\\ & + 2 0 6 5 9 8 6 4 4 6 4 4 8 9 6 5 0 8 9 5 9 4 6 7 9 1 0 5 2 1 5 8 9 0 3 2 8 1 0 0 . \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\langle \\gamma '' , \\gamma ' \\times n \\rangle = \\kappa \\sin ^ 2 \\theta \\ ; . \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} \\frac { \\dd ( \\alpha \\circ \\tilde \\lambda ) } { \\dd ( \\alpha \\circ \\lambda ) } ( g ) \\frac { \\dd ( \\alpha \\circ \\tilde \\nu ) } { \\dd ( \\alpha \\circ \\nu ) } ( l ) = \\frac { \\dd ( \\alpha \\circ \\tilde \\nu ) } { \\dd ( \\alpha \\circ \\nu ) } ( g l ) \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} \\partial ^ { \\alpha } _ t u = D _ { x ^ i } ( a ^ { i j } u _ { x ^ i x ^ j } + f ^ i ) + h \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} P _ 2 u _ 2 = f _ 2 , { u _ 2 } _ { | S } = ( g _ 1 - g _ 2 ) / 2 , \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} \\int _ M | u ( t ) | ^ p d \\mu _ { g ( t ) } = 1 , \\ \\ a n d \\ \\ \\int _ M | u ( t ) | ^ { p - 2 } u ( t ) d \\mu _ { g ( t ) } = 0 . \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} \\{ u _ { i _ { \\chi } + 1 } , \\ldots , u _ { i _ { \\chi ' } } \\} = \\{ b _ j , c _ j ^ { p _ j } , \\ldots , c _ n ^ { p _ n } , d _ { 1 k _ 1 } , \\ldots , d _ { m k _ m } \\} \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} & \\mu _ n \\sigma _ { i , i + 1 } = \\sigma _ { i , i + 1 } \\mu _ n & & n \\geq 2 , \\ , i \\in \\{ 2 , \\dots , n - 1 \\} . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} d \\omega _ R ( z ) & = ( d R ^ { - 1 } ( z ) ) \\wedge d R ( z ) \\\\ & = - \\omega _ R ( z ) \\wedge \\omega _ R ( z ) \\\\ & = - \\left ( R ^ { - 1 } ( z ) a R ^ { - 1 } ( z ) t - R ^ { - 1 } ( z ) t R ^ { - 1 } ( z ) a \\right ) d z _ 1 \\wedge d z _ 2 . \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} { \\lim _ { n \\rightarrow \\infty } s u p \\frac { ( p _ { n + 1 } - p _ { n } ) } { ( \\log p _ { n } ) ^ { 2 } } = 1 } \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ n \\frac { 1 } { i + k } { - 2 i - 2 k \\choose - 2 n - 2 k } B _ { 2 n - 2 i } \\sum \\limits _ { j = 0 } ^ i - { - k - j \\brack - k - i } \\gamma _ { 2 j } . \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} M _ { a , b , c } ( a + 1 ) & = M _ { a , b , c } , & M _ { a , b , c } ( 1 ) & = M _ { a , b - 1 , c } . \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\mathcal { B } ^ + _ { I V } = \\left \\{ \\begin{aligned} & \\left ( \\tau , v _ { s + k + 1 } , \\omega _ { k + 1 } \\right ) \\in \\mathcal { A } ^ + \\backslash \\mathcal { B } ^ + _ { I I } \\ ; \\textnormal { s u c h t h a t } \\\\ & \\inf _ { i \\in \\left \\{ 1 , \\dots , s , s + 1 , \\dots , s + k \\right \\} \\backslash \\left \\{ i _ { k + 1 } \\right \\} } \\left | v _ { i _ { k + 1 } } ^ { \\prime * } - v _ i ^ \\prime \\right | \\leq \\eta \\end{aligned} \\right \\} \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} \\tau _ 4 = a _ i ( e ^ k \\otimes e _ { k i } ) , \\tau _ 8 = b _ i ( e ^ { n + k } \\otimes e ^ { k i } ) , \\tilde \\lambda = \\lambda _ I e ^ I \\otimes ( e ^ k \\otimes e _ k - e ^ { n + k } \\otimes e _ { n + k } ) . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} J _ { \\epsilon } ( P U _ { \\delta , \\xi } + \\phi _ { d , \\xi } ^ { \\epsilon } ) = J _ { \\epsilon } ( P U _ { \\delta , \\xi } ) + o ( \\epsilon ) . \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} \\varphi _ { n } ( x ) = ( - 1 ; q ) _ { \\infty } \\ , x ^ { n } q ^ { n ( n - 1 ) / 2 } \\ , _ { 0 } \\phi _ { 1 } \\left ( - ; 0 ; q ^ { 2 } , - q ^ { - 2 n + 4 } x ^ { - 2 } \\right ) \\ ! , n \\in \\Z , \\ x \\neq 0 . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} I _ { \\ell \\cup \\ell ' } = ( x _ { 0 } x _ { 2 } , x _ { 1 } x _ { 2 } , x _ { 0 } x _ { 3 } , x _ { 1 } x _ { 3 } , x _ { 4 } , x _ { 5 } , \\dots x _ { r } ) . \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} f _ 3 ( x ) : = \\frac { 1 } { 1 + x ^ 6 } \\qquad ( x \\in \\R ) \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} [ \\bar { e } _ { i , k } , \\bar { f } _ { j , l } ] = \\delta _ { i , j } \\bar { h } _ { i , k + l } + k \\delta _ { i , j } \\delta _ { k , - l } \\bar { c } , \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} C _ { \\gamma } \\left ( \\overline { z } ^ { m - j } z ^ { n - j } \\left ( 1 - | { z } | ^ { 2 } \\right ) ^ { j } \\right ) & = \\dfrac { - z ^ { n } \\overline { z } ^ { m + 1 } } { ( m - j + 1 ) } \\left ( 1 - | z | ^ 2 \\right ) ^ { \\gamma + 1 } \\\\ & \\times \\left ( \\dfrac { 1 - | z | ^ 2 } { | z | ^ { 2 } } \\right ) ^ { j } { _ 2 F _ 1 } \\left ( \\begin{array} { c } 1 , \\gamma + m + 2 \\\\ m - j + 2 \\end{array} \\bigg | | z | ^ 2 \\right ) . \\end{align*}"}
-{"id": "9878.png", "formula": "\\begin{align*} w _ { \\rm r e g } ( \\eta ) = u _ { \\rm r e g } ( 1 - \\eta ) \\ , . \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} \\left | x \\wedge y \\right | \\coloneqq \\max \\left \\{ n \\in \\mathbb { N } _ { 0 } \\middle | \\ ; \\forall 1 \\leq i \\leq n : \\ , x _ { i } = y _ { i } \\right \\} . \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} \\tilde { F } ^ i ( x ) = \\lambda ^ { \\frac { 1 } { 2 } - \\alpha } F ^ i ( x _ 0 + \\lambda x ) , \\\\ \\tilde { g } _ 1 ( x ) = \\lambda ^ { \\frac { 3 } { 2 } - \\alpha } g _ 1 ( x _ 0 + \\lambda x ) , \\\\ \\tilde { g } _ 2 ( x ) = \\lambda ^ { \\frac { 3 } { 2 } - \\alpha } g _ 2 ( x _ 0 + \\lambda x ) . \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} C ( \\Delta ( r + 2 ) ) - d - 1 + \\left \\lfloor \\frac { 1 7 s - 4 } { 6 } \\right \\rfloor \\geq \\frac { 2 ^ { r + 2 } - 2 ^ { r - 1 } - 1 } { 3 } - ( 2 ^ { r + 2 } - 1 ) + \\frac { 1 7 \\cdot 2 ^ { r - 1 } - 2 } { 3 } = 0 \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} w _ { r + 1 , \\theta } ( z ) = ( \\mathcal { D } _ { r + 1 } - d _ 0 ) + \\varepsilon _ { r + 1 } z . \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} \\norm { g _ n } = \\norm { T _ { t _ 0 } f - h _ n + e _ n } = \\norm { T _ { t _ 0 } f + e _ n } - \\norm { h _ n } \\ge 1 - \\norm { h _ n } , \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} \\wp : = \\{ x \\in \\mathbb { G } : \\ , | x | = 1 \\} , \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} \\left [ \\partial ^ { \\tau } \\partial _ { \\tau } + \\kappa \\left ( u ^ { \\tau } \\partial _ { \\tau } \\right ) ^ { 2 } \\right ] Z ^ { \\lambda \\nu } = - 4 \\pi p ^ { \\lambda \\nu } . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} M \\vec v _ { t - 1 } = \\lambda \\vec v _ t + \\vec u _ t . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} \\begin{aligned} u ^ n ( s , { X } _ s ) = \\ & \\Phi ( { X } _ T ) - w ^ n ( s , { X } _ s ) \\\\ & + \\int ^ T _ s f ( r , { X } _ r , u ^ n ( r , { X } _ r ) , \\nabla u ^ n ( r , { X } _ r ) ) \\d r \\\\ & - \\int ^ T _ s \\nabla w ^ n ( r , { X } _ r ) \\d { W } _ r - \\int ^ T _ s \\nabla u ^ n ( r , { X } _ r ) \\d { W } _ r . \\end{aligned} \\end{align*}"}