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-{"id": "350.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { H ^ { s , p } ( G ) } : = \\Vert \\Lambda _ s f \\Vert _ { L ^ p ( G ) } < \\infty . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} \\mathcal { L } _ { m } ^ { \\Gamma , - } = & \\left ( [ 1 - C _ { - } m ^ { - \\frac { 1 } { 2 } } ] \\mathcal { L } ^ { \\Gamma } - \\frac { \\kappa ^ 2 } { 4 } + K - C _ { - } m ^ { - 1 } \\right ) _ { + } ^ { \\frac { 1 } { 2 } } \\ , , \\\\ \\mathcal { L } _ { m } ^ { \\Gamma , + } = & \\left ( [ 1 + C _ { + } m ^ { - \\frac { 1 } { 2 } } ] \\mathcal { L } ^ { \\Gamma } - \\frac { \\kappa ^ 2 } { 4 } + K + C _ { + } m ^ { - 1 } \\right ) ^ { \\frac { 1 } { 2 } } \\ , . \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} B ^ { - 1 } \\omega \\cdot \\partial _ { \\varphi } B = \\rho ( \\vartheta ) \\omega \\cdot \\partial _ { \\vartheta } , B ^ { - 1 } \\partial _ y B = \\partial _ y , \\rho : = B ^ { - 1 } ( 1 + \\omega \\cdot \\partial _ { \\varphi } \\alpha ) . \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 | q ( \\tau ) | \\ ; d \\tau \\le \\left ( \\int _ { - 1 } ^ 1 d \\tau \\right ) ^ { 1 / 2 } \\left ( \\int _ { - 1 } ^ 1 q ^ 2 ( \\tau ) \\ ; d \\tau \\right ) ^ { 1 / 2 } = 2 . \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} ( \\sigma ^ { \\le q } M ) ^ i : = \\begin{cases} M ^ i & i \\ge q , \\\\ 0 & i < q . \\end{cases} \\end{align*}"}
-{"id": "6898.png", "formula": "\\begin{align*} u ( 0 , t ) = 0 , 0 < t \\leq T \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} ( k - 1 ) \\big ( ( r + 1 ) + ( l + r - 1 ) \\big ) = ( r + 1 ) ( k - l - r ) + ( l + r - 1 ) ( k + r ) . \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} L _ { y } \\left \\lbrace g \\right \\rbrace : = \\sum _ { \\alpha \\in \\mathbb { N } ^ { 2 n } } g _ { \\alpha } y _ { \\alpha } . \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\begin{aligned} K _ 0 ( n , m ) & = \\left ( \\frac N 2 \\right ) ^ { \\frac { n - m } 2 } 2 ^ { - 1 / 2 } N ^ { - 1 / 6 } e ^ { T ( \\xi - \\zeta ) + o ( 1 ) } \\\\ & \\qquad \\times \\int _ \\R \\d U \\ , e ^ { 2 T U } \\varphi _ { N - \\xi N ^ { 1 / 3 } } \\Big ( \\sqrt { 2 N } + \\frac { ( T ^ 2 + R + U ) N ^ { - 1 / 6 } } { \\sqrt 2 } \\Big ) \\\\ & \\qquad \\qquad \\times \\varphi _ { N - \\zeta N ^ { 1 / 3 } } \\Big ( \\sqrt { 2 N } + \\frac { ( T ^ 2 + R - U ) N ^ { - 1 / 6 } } { \\sqrt 2 } \\Big ) \\end{aligned} \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} \\langle U ( x _ 0 ) \\ , | \\ , V ( x _ 0 ) \\rangle _ g = \\langle d e ^ { i ( \\delta _ 1 + \\varepsilon ) } U ( x _ 0 ) \\ , | \\ , d e ^ { i ( \\delta _ 1 + \\varepsilon ) } V ( x _ 0 ) \\rangle , ~ \\forall ~ \\varepsilon \\in ( 0 , \\sigma ) . \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} f ( t ) & = f ( 0 ) + \\int _ 0 ^ t \\left ( \\int _ 0 ^ s f '' ( l ) d l \\right ) d s \\\\ & = f ( 0 ) + \\int _ 0 ^ t \\left ( \\int _ 0 ^ s \\langle x ' _ l , \\nabla ^ 2 u | _ { x _ l } x ' _ l \\rangle d l \\right ) d s \\\\ & \\geq f ( 0 ) - \\int _ 0 ^ t \\left ( \\int _ 0 ^ s \\langle x ' _ l , \\nabla ^ 2 \\phi _ 0 | _ { x _ l } x ' _ l \\rangle d l \\right ) d s \\\\ & \\geq f ( 0 ) - C t ^ 2 , \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} x _ \\gamma = & \\left [ \\begin{array} { c c c c c c c } 1 & V _ { d 1 } & \\ldots & V _ { q n } & V _ { d 1 } ^ 2 & V _ { d 1 } V _ { d 2 } & \\ldots \\end{array} \\right . \\\\ & \\left . \\begin{array} { c c c c c c } \\ldots & V _ { q n } ^ 2 & V _ { d 1 } ^ 3 & V _ { d 1 } ^ 2 V _ { d 2 } & \\ldots & V _ { q n } ^ \\gamma \\end{array} \\right ] ^ \\intercal \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} \\mu = \\int \\mu _ \\omega \\ , d \\mathbb { P } ( \\omega ) \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} m _ P ( p ) = \\sigma _ { F _ p } \\Delta _ l ( p ) = \\| \\delta _ P ( p ) \\| = q ^ { \\langle m _ P ( p ) , \\delta _ P \\rangle } . \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} f _ 0 = S _ 0 f , f _ j = P _ j f , j \\geq 1 . \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} I ^ { \\gamma \\gamma ' } = \\kappa _ { \\rho ^ \\circ } ( \\gamma , \\gamma ' ) I ^ \\gamma \\circ I ^ { \\gamma ' } \\gamma , \\gamma ' \\in \\pi _ 0 ( Z _ G ( \\sigma , y ) ) _ { \\rho ^ \\circ } . \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { r + 1 } \\varepsilon ^ k \\varphi ( x _ 1 , x _ 2 , . . . , \\hat { x _ k } , . . . , x _ { r + 1 } ) \\times \\varphi ( x _ k , y _ 1 , . . . , y _ { r - 1 } ) \\in K _ 0 . \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} s ( x , y ) = \\frac { | [ x , y ] | } { | | y | | _ a | | x | | } , \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} \\sum _ { j ' = 1 } ^ J & \\frac { n _ 0 ^ { ( j ' ) } } { 2 } \\log ( k _ n ^ { ( j ) } ) \\leq _ n \\\\ & \\frac { \\epsilon ' } { 2 } \\frac { n } { 2 } \\log k _ n ^ { ( j ) } + \\sum _ { j ' = 1 } ^ J ( 1 + \\epsilon ' ) \\beta ^ { ( j ' ) } \\ell _ n H _ 2 \\left ( \\alpha _ n ^ { ( j ' ) } \\right ) . \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} \\mu _ 3 ( x ) = & \\frac { - 1 } { 2 \\sqrt { - 1 } } ( ( [ a , b ] - [ a ^ { \\dagger } , b ^ { \\dagger } + i j + j ^ { \\dagger } i ^ { \\dagger } + f g + g ^ { \\dagger } f ^ { \\dagger } ] ) , \\\\ & ( [ a ' , b ' ] - [ a ^ { \\dagger } , b ^ { \\dagger } ] - g f - f ^ { \\dagger } g ^ { \\dagger } ) ) . \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} t \\frac { d ^ 2 w } { d t ^ 2 } + ( b - t ) \\frac { d w } { d t } - a \\cdot w = 0 , \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} \\mu _ { G \\times B } ( g , \\theta , r , s , i , j ) = ( \\mu _ G ( g , \\theta , r , s , i , j ) , \\mu _ B ( g , \\theta , r , s , i , j ) ) . \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} \\Gamma \\left ( b , z \\right ) = \\Gamma \\left ( b \\right ) + z ^ { b } \\left ( - \\frac { 1 } { b } + \\frac { z } { 1 + b } + \\dots \\right ) \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ N \\left ( \\max _ { i \\in \\mathcal { A } ^ { \\star } _ j } \\ell _ { i , s } ^ { \\prime } + \\max _ { i \\in ( [ 1 : N ] \\backslash \\{ j \\} ) \\backslash \\mathcal { A } ^ { \\star } _ j } r _ { i , s } ^ { \\prime } \\right ) \\geq \\sum _ { j = 1 } ^ { N - 1 } \\left ( \\max _ { i \\in \\mathcal { A } _ { { \\rm { F } } j } } \\ell _ { i , s } ^ { \\prime } + \\max _ { i \\in [ 1 : N ] \\backslash \\mathcal { A } _ { { \\rm { F } } j } } r _ { i , s } ^ { \\prime } \\right ) , \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} - \\dim { \\rm t o t } \\phi _ { ( u , v ) } ( { \\cal F } \\boxtimes { \\cal G } , h ) = ( - \\dim { \\rm t o t } \\phi _ u ( { \\cal F } , f ) ) \\cdot ( - \\dim { \\rm t o t } \\phi _ v ( { \\cal G } , g ) ) . \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} \\min \\limits _ { \\stackrel { E _ i \\in Q _ k ^ { ( i ) } } { i = 1 , \\dots , m } } \\max \\limits _ { \\stackrel { x _ j \\in \\mathbb { R } ^ N , } { j = 1 , \\dots , L } } F ( E , x ) . \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} P _ { X | V , S } ( 1 | v , s ) = 0 \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} ( \\mathcal { L } _ { \\omega } - \\mathcal { D } _ { \\omega } ) \\mathrm { R } _ { \\Phi } = \\Pi _ S ^ { \\perp } [ \\partial _ { x x } ( a _ 1 \\partial _ x ) + \\partial _ x ( a _ 0 \\cdot ) ] \\Pi _ S ^ { \\perp } \\mathrm { R } _ { \\Phi } = 0 . \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} h _ { X _ N , \\Omega } = \\sup _ { \\boldsymbol { x } \\in \\Omega } \\min _ { \\boldsymbol { x } _ i \\in X _ N } | | \\boldsymbol { x } - \\boldsymbol { x } _ i | | _ 2 , \\end{align*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\sum _ { t \\in [ 0 , T ] } \\bar { q } ^ { n , Q } ( t ) \\ , \\tfrac { ( T - t ) ^ k } { k ! } = \\tfrac { T ^ { n + k } } { ( n + k ) ! } . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} | | \\partial / \\partial t ^ j | | _ H ^ 2 = \\sum a _ p . \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} \\mathbf { Y } = \\sqrt { \\beta _ K } \\mathbf { G } \\boldsymbol { \\Psi } + \\mathbf { Z } \\ , , \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & * \\\\ 0 & \\chi _ 3 \\end{pmatrix} \\begin{pmatrix} \\chi _ 3 & * \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} ( \\mathfrak { B } _ 1 ) _ j ^ { j ' } ( l ) : = \\begin{cases} - 2 i j \\ , c _ 2 \\ , ( j - j ' ) ^ 2 \\ , \\sqrt { \\lvert j - j ' \\rvert \\xi _ { j - j ' } } - 6 i j \\ , c _ 3 \\ , \\sqrt { \\lvert j - j ' \\rvert \\xi _ { j - j ' } } \\mbox { i f } \\ , \\ , \\ , j - j ' \\in S , l = \\mathtt { l } ( j - j ' ) \\\\ 0 \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n 2 q ^ n } { 1 + q ^ n } \\frac { ( - q ; q ) _ { n - 1 } } { ( q ; q ) _ { n - 1 } } = \\varphi ( - q ) ^ 2 - \\varphi ( - q ) \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} \\lbrack \\tau _ { 2 } ( \\lambda ) , \\Theta ] = 0 , \\Theta \\equiv \\prod _ { n = 1 } ^ { \\mathsf { N } } v _ { n } . \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} ( 1 - 2 x z ) \\frac { \\partial } { \\partial z } Q ( x , q ; z ) = q Q ( x , q ; z ) + 2 x ( 1 - x ) \\frac { \\partial } { \\partial x } Q ( x , q ; z ) . \\end{align*}"}
-{"id": "5815.png", "formula": "\\begin{align*} M _ n ( z ) = \\begin{bmatrix} z + a _ { n + 1 } - a _ n & - b _ n \\\\ c _ { n + 1 } & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { E _ { \\Sigma _ 1 } ^ s } : = \\int _ \\Omega u v \\ , d x + \\iint _ Q \\frac { ( u ( x ) - u ( y ) ) ( v ( x ) - v ( y ) ) } { | x - y | ^ { N + 2 s } } \\ , d x d y , \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} \\{ P _ 0 , g _ i \\} & = 0 , \\{ P _ 0 , E \\} = - 1 . \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} v _ { 0 , 0 } = ( - 1 ) ^ { m + n } ( u _ { 1 , 1 } - u _ { 0 , 0 } ) . \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} z _ { Q _ { 1 } } = \\frac { 1 0 + i \\sqrt { 2 0 } } { 1 2 } , z _ { Q _ { 2 } } = \\frac { 3 4 + i \\sqrt { 2 0 } } { 8 4 } . \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} A ( x ) \\cdot \\left ( \\begin{matrix} \\cos { 2 \\pi y } \\\\ \\sin { 2 \\pi y } \\end{matrix} \\right ) = u ( x , y ) \\left ( \\begin{matrix} \\cos { 2 \\pi ( y + \\psi ( x , y ) ) } \\\\ \\sin { 2 \\pi ( y + \\psi ( x , y ) ) } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} \\frac { 1 } { b - a } \\int _ { a } ^ { b } \\eta '' = \\frac { \\xi ' ( b ) - \\xi ' ( a ) } { b - a } . \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} u ( \\alpha , \\beta ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int p ( \\xi , \\beta ) \\hat f ( \\xi ) e ^ { i \\alpha \\xi } \\ , d \\xi , \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} g _ 1 ( A _ n ) = & 1 - \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( \\sqrt { A _ n / 2 } + \\sqrt { P / 2 } \\big ) \\right ) ^ 2 . \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} \\lim _ { j \\to - \\infty } u ( j + J ( t ) , t ; 0 , u ^ { 0 , * } ) = u ^ + ( t ) , \\lim _ { j \\to \\infty } u ( j + J ( t ) , t ; 0 , u ^ { 0 , * } ) = 0 \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} X _ H ^ { \\gamma } = \\left ( \\frac { p ^ 2 } { 2 m } - V ( q ) - \\alpha S \\right ) \\frac { \\partial } { \\partial S } + \\frac { p } { m } \\frac { \\partial } { \\partial q } \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} p ' ( z ) = - p ( z ) \\sum _ { j = 1 } ^ \\infty p _ j z ^ { j - 1 } \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} & r _ 0 ( T _ { \\delta } ) : = \\sigma _ 0 ( \\Phi _ B ( T _ { \\delta } ) ) , \\sigma _ 0 ( u ) : = ( \\partial _ { u u } f ) ( x , u , u _ x ) - \\partial _ x \\{ ( \\partial _ { u u _ x } f ) ( x , u , u _ x ) \\} , \\\\ & r _ 1 ( T _ { \\delta } ) : = \\sigma _ 1 ( \\Phi _ B ( T _ { \\delta } ) ) , \\sigma _ 1 ( u ) : = - ( \\partial _ { u _ x u _ x } f ) ( x , u , u _ x ) , \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} v ( x ) & = k ( x - \\lambda _ 0 ) \\varphi ( x ) + p g ( x ) , \\\\ u ( x ) & = k ( 2 \\varphi ( x ) + ( x - \\lambda _ 0 ) \\varphi ' ( x ) ) + p f ( x ) \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} c _ 4 ( W ' ) = \\ell ^ 2 c _ 4 , c _ 6 ( W ' ) = \\ell ^ 3 c _ 6 \\Delta ( W ' ) = \\ell ^ 6 \\Delta _ m . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} v = d _ 1 + 4 \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} T _ a = \\{ n \\geq 1 : \\widetilde { S } _ n ^ \\rho + \\eta _ { n } \\ge a \\} , \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} n ^ { - q / 2 } \\sum _ { r _ 1 + \\cdots + r _ q = n } \\exp \\left ( - n \\epsilon \\left | \\frac r n - p \\right | ^ 2 \\right ) & \\le n ^ { - q / 2 } \\sum _ { r _ 1 , \\dots , r _ q = 1 } ^ n \\exp \\left ( - n \\epsilon \\left | \\frac r n - p \\right | ^ 2 \\right ) \\\\ & = \\prod _ { i = 1 } ^ q \\left [ n ^ { - 1 / 2 } \\sum _ { r = 1 } ^ n \\exp \\left ( - n \\epsilon \\left ( \\frac r n - p _ i \\right ) ^ 2 \\right ) \\right ] . \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} { \\bf G } _ { i j } = \\begin{pmatrix} h _ { i j } ^ { R e } & - h _ { i j } ^ { I m } \\\\ h _ { i j } ^ { I m } & h _ { i j } ^ { R e } \\end{pmatrix} . \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} \\bold { E } _ { \\alpha } : = \\sum _ J ( - D ) _ J \\frac { \\partial } { \\partial u _ J ^ { \\alpha } } . \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} e ( p e r m _ m ( C ) ) = { \\binom n m } ^ 2 c ( m ) \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} J ( u , v ) = \\int _ 0 ^ \\infty k \\left ( \\frac { T } { x } \\right ) h \\left ( \\frac { x } { u } \\right ) h \\left ( \\frac { x } { v } \\right ) \\frac { d x } { x } . \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} \\sigma ^ { ( t ) } _ { K , N } ( \\theta ) : = \\left \\{ \\begin{array} { l l } + \\infty , & \\qquad \\textrm { i f } K \\theta ^ 2 \\geq N \\pi ^ 2 , \\\\ \\frac { \\sin ( t \\theta \\sqrt { K / N } ) } { \\sin ( \\theta \\sqrt { K / N } ) } & \\qquad \\textrm { i f } 0 < K \\theta ^ 2 < N \\pi ^ 2 , \\\\ t & \\qquad \\textrm { i f } K \\theta ^ 2 = 0 , \\\\ \\frac { \\sinh ( t \\theta \\sqrt { K / N } ) } { \\sinh ( \\theta \\sqrt { K / N } ) } & \\qquad \\textrm { i f } K \\theta ^ 2 < 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} d \\mu _ a = - \\frac { \\vartheta _ a d r } { 2 r ^ 2 } + O ( \\frac { 1 } { r ^ 3 } ) . \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} \\mathrm { c n } ( x , z ) ^ 2 + \\mathrm { c n } ( x , b ( z ) ) ^ 2 = 1 . \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} g ' = \\frac { 1 } { r ^ 2 } d r ^ 2 + g _ { S ^ 2 } + \\frac { 1 } { V ^ 2 r ^ 2 } ( d \\tau + \\omega ) ^ 2 = d y ^ 2 + g _ { S ^ 2 } + e ^ { - 2 y } V ^ { - 2 } ( d \\tau + \\omega ) ^ 2 , \\end{align*}"}
-{"id": "8532.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { S T B C } } = \\frac { \\gamma _ 0 } { M R _ c } \\frac { \\mathbf { h } ^ { \\dagger } \\times \\mathbf { h } } { 1 + \\sum _ { k = 1 } ^ { K } | g _ k | ^ 2 \\mathcal { P } _ k / \\mathcal { P } _ w } = \\frac { \\gamma _ 0 } { M R _ c } \\frac { \\sum _ { m = 1 } ^ { M } | h _ m | ^ 2 } { 1 + \\sum _ { k = 1 } ^ { K } | g _ k | ^ 2 \\zeta _ k } . \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} \\phi ^ \\diamond ( F ) = \\sum _ { n = 0 } ^ \\infty a _ n F ^ { \\diamond n } , \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} \\frac { [ m ] _ q } { [ n ] _ q } q ^ { m - n } - 1 = \\frac { q ^ { 2 m } - 1 } { q ^ { 2 n } - 1 } - \\frac { q ^ { 2 n } - 1 } { q ^ { 2 n } - 1 } = \\frac { q ^ { 2 m } - q ^ { 2 n } } { q ^ { 2 n } - 1 } = O ( q ^ { 2 m } ) + O ( q ^ { 2 n } ) , \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} \\partial _ t y _ 1 & = \\partial _ { x x } y _ 1 + y _ 2 f _ 2 ( y _ 1 ) - \\sigma y _ 3 f _ 3 ( y _ 1 ) , \\\\ \\partial _ t y _ 2 & = d _ 2 \\partial _ { x x } y _ 2 - y _ 2 f _ 2 ( y _ 1 ) , \\\\ \\partial _ t y _ 3 & = d _ 3 \\partial _ { x x } y _ 3 - \\tau y _ 3 f _ 3 ( y _ 1 ) . \\end{align*}"}
-{"id": "1408.png", "formula": "\\begin{align*} w ( t ) \\ge f _ k ( t ) , k = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{gather*} \\operatorname { T r } B = - x + \\frac { t ^ 2 } { 6 } + \\frac { U } { 3 } . \\end{gather*}"}
-{"id": "1487.png", "formula": "\\begin{align*} \\Delta _ { 4 5 6 } \\ = \\ \\Delta _ { 5 6 7 } \\ , \\Delta _ 1 \\ , \\Delta _ 3 ^ { - 1 } , \\Delta _ { 4 5 7 } \\ = \\ \\Delta _ { 5 6 7 } \\ , \\Delta _ 2 \\ , \\Delta _ 3 ^ { - 1 } , \\Delta _ { 4 6 7 } \\ = \\ \\Delta _ { 5 6 7 } \\ , \\Delta _ 1 \\ , \\Delta _ 3 ^ { - 1 } , \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} r ( S ( T , t ) ) = \\left \\{ \\begin{array} { l l } \\dfrac { 1 } { 2 } ( 3 \\cdot 2 ^ t - 2 t - 3 ) D ( T ) - 2 ( 2 ^ t - t - 1 ) , & D ( T ) \\\\ & \\\\ \\dfrac { 1 } { 2 } \\left ( ( 3 \\cdot 2 ^ t - 2 t - 3 ) D ( T ) - 2 ^ { t + 2 } + 4 t + 5 \\right ) , & \\end{array} \\right . \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} \\log Y \\geq ( 1 - u ) \\log ( 1 ) + u \\log \\{ f ^ * ( X ) / f ( X ) \\} = u \\log \\{ f ^ * ( X ) / f ( X ) \\} . \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} H ( s ; \\mathbf { a } , \\mathbf { z } ) & = \\prod _ { p | q } \\ ( 1 - \\frac { 1 } { p ^ s } \\ ) ^ { \\frac { z _ 1 + \\cdots + z _ l } { \\phi ( q ) } } \\prod _ { \\chi \\neq \\chi _ 0 } ( L ( s , \\chi ) ) ^ { \\frac { \\bar { \\chi } ( b _ 1 ) z _ 1 + \\cdots + \\bar { \\chi } ( b _ l ) z _ l } { \\phi ( q ) } } \\prod _ { j = 1 } ^ l G _ 1 ( s ; b _ j , z _ j ) \\\\ & = \\prod _ p \\ ( 1 - \\frac { 1 } { p ^ s } \\ ) ^ { \\frac { z _ 1 + \\cdots + z _ l } { \\phi ( q ) } } \\prod _ { j = 1 } ^ l \\ ( 1 + \\frac { z _ j \\lambda _ { b _ j } ( p ) } { p ^ s } \\ ) \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} \\left ( L _ { a , n } ( \\lambda ) \\right ) _ { 1 1 } \\left \\vert j _ { n } , n \\right \\rangle = \\left \\vert j _ { n } , n \\right \\rangle _ { n } ( \\lambda q ^ { j _ { n } } ) \\left ( L _ { a , n } ( \\lambda ) \\right ) _ { 2 2 } \\left \\vert j _ { n } , n \\right \\rangle = \\left \\vert j _ { n } , n \\right \\rangle _ { n } ( \\lambda q ^ { - j _ { n } } ) \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} E _ 2 ( - 1 / z ) = z ^ 2 E _ 2 ( z ) - \\frac { 6 i z } { \\pi } . \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} \\mathbb { E } ( \\Phi _ * ) _ { [ \\lambda _ - , \\lambda _ + ] } ( f ) ( y ) = \\frac { 1 } { \\lambda _ + - \\lambda _ - } \\int _ { \\lambda _ - } ^ { \\lambda _ + } f ( \\Phi _ { - t } ( y ) ) \\ : d t . \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} g ( m ) = \\left ( 1 - \\frac { \\log m } { \\log M } \\right ) ^ r \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} D ( m , p _ 0 + 1 ) \\cong _ { \\mathfrak { g } } \\begin{cases} V ( p _ 0 + 1 ) \\oplus V ( p _ 0 ) \\oplus \\cdots \\oplus V ( m - p _ 0 - 1 ) , & 2 p _ 0 + 2 > m , \\\\ V ( p _ 0 + 1 ) , & 2 p _ 0 + 2 \\leq m . \\end{cases} \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} I _ { P ^ { ( \\mathcal { C } _ n ) } } ( M ; \\mathbf { Z } ) \\leq \\frac { 1 } { | \\mathcal { M } _ n | } \\sum _ { m \\in \\mathcal { M } _ n } \\mathsf { D } \\Big ( P ^ { ( \\mathcal { C } _ n ) } _ { \\mathbf { Z } | M = m } \\Big | \\Big | p ^ n _ Z \\Big ) . \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} h ' _ { \\ell } ( w ) & = \\frac { A _ { \\ell } B } { 1 + B w } + \\frac { 1 } { k _ { \\ell } } \\log \\frac { w } { a _ { \\ell } - w } \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\left ( a \\ , x ^ { ( l ) } \\ , \\mu _ { ( l + 1 ) p } ( \\widetilde { g } ) \\right ) ( s ) = a \\ , \\mu _ { ( l + 1 ) p } ( \\widetilde { g } ) ( s ) \\cdot x ^ { ( l ) } = e ^ { 2 \\pi i ( l + 1 ) p s } \\widetilde { g } ( s ) a x ^ { ( l ) } \\ ; , \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} \\rho - \\rho _ { c } ( \\beta ) = \\lim _ { \\eta \\rightarrow + 0 } \\lim _ { \\Lambda } \\frac { 1 } { V } \\sum _ { \\left \\{ k \\in \\Lambda ^ { \\ast } , \\left \\| k \\right \\| \\leq \\eta \\right \\} } \\left \\{ e ^ { \\beta ( \\varepsilon _ { k } - { \\mu _ { \\Lambda } ( \\beta , \\rho ) } ) } - 1 \\right \\} ^ { - 1 } , \\ { \\rm { f o r } } \\ \\ \\rho > \\rho _ { c } ( \\beta ) \\ . \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{align*} \\chi _ \\Delta \\left ( Q \\right ) = \\chi _ \\Delta \\left ( [ \\alpha , \\beta , \\gamma ] \\right ) : = \\begin{cases} \\left ( \\frac { \\Delta } { R } \\right ) & \\operatorname { g c d } ( \\Delta , \\alpha , \\beta , \\gamma ) = 1 Q R \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} h _ { \\ell } ( w ) = A _ { \\ell } \\log ( 1 + B w ) - \\frac { a _ { \\ell } } { k _ { \\ell } } H _ 2 \\left ( \\frac { w } { a _ { \\ell } } \\right ) . \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} V _ t ^ i : = e ^ { \\int _ 0 ^ t \\Lambda \\big ( s , \\tilde \\xi ^ { i , N } _ { s } , u ^ { S ^ N ( \\tilde { \\xi } ) } _ { s } ( \\tilde \\xi ^ { i , N } _ { s } ) \\big ) d s } \\textrm { a n d } \\tilde V _ t ^ i : = e ^ { \\int _ 0 ^ t \\Lambda \\big ( r ( s ) , \\tilde \\xi ^ { i , N } _ { r ( s ) } , \\tilde { u } _ { r ( s ) } ( \\tilde \\xi ^ { i , N } _ { r ( s ) } ) \\big ) d s } \\ , \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{align*} \\C x ( t ) = C z ( t ) \\mbox { o r } \\C x ( t ) = C z ( t - 1 ) , \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} V _ { n , k } ^ { \\mu } = \\int _ { \\mathcal { V } _ { \\alpha , \\theta } } \\ ! \\ ! V _ { n , k } \\ , \\mu ( \\mathrm { d } V ) . \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} | { \\bf H } | = \\beta \\frac { \\sin \\alpha } { \\cos \\alpha } f ^ { - 1 } | \\nabla f | . \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\varphi ( \\psi _ i ( \\chi _ j ) ) = ( \\varphi ( \\psi _ i ) ) ( \\chi _ j ) \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} \\operatorname { D i v } ^ { \\vartriangle } P = Q ^ { \\alpha } F _ { \\alpha } , \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} R _ { 1 2 } ( \\lambda / \\mu ) L _ { 1 , n } ( \\lambda ) L _ { 2 , n } ( \\mu ) = L _ { 2 , n } ( \\mu ) L _ { 1 , n } ( \\lambda ) R _ { 1 2 } ( \\lambda / \\mu ) , \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} E ^ n _ { N F } = E ^ { n } _ { N F , h i g h } + E ^ { n } _ { N F , l o w } , \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\mathsf { D } _ { - } ( \\lambda ) = k ( \\lambda ) \\mathsf { A } _ { - } ( q / \\lambda ) , _ { a } ^ { \\pm } | h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } \\rangle = | h _ { 1 } , . . . , h _ { a } \\pm 1 , . . . , h _ { \\mathsf { N } } \\rangle . \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{gather*} q _ { 2 t } = q _ 2 \\left ( \\frac { 2 } { 3 } \\alpha + \\frac { u _ t } { u } \\frac { 2 - q _ 2 } { 3 } \\right ) + \\frac { u _ t } { u } \\frac { 2 - q _ 2 } { 3 } , \\end{gather*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\tilde { K } _ { 3 } ( t ) g \\right \\| _ { 2 } & \\le C t ^ { 1 - \\ell } ( 1 + t ) ^ { - \\frac { n } { 2 } - k } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla ^ { ( k + \\ell - 1 ) _ { + } } _ { x } g \\| _ { 2 } , \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} \\bold { p r } X = \\xi ^ i \\partial _ { x ^ i } + \\sum _ { i , I } ( S _ I \\xi ^ i ) \\left ( D _ { i ; I } - \\partial _ { x ^ i } \\right ) + \\sum _ { \\alpha , J _ 1 , J _ 2 } ( D _ { J _ 1 } S _ { J _ 2 } Q ^ { \\alpha } ) \\partial _ { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } . \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} N _ 0 ^ { C _ 1 } \\varepsilon ^ { b _ * + 1 } \\gamma ^ { - 2 } < \\delta _ 0 , \\gamma : = \\varepsilon ^ { 2 + a } = \\varepsilon ^ { 2 b } , N _ 0 : = ( \\varepsilon \\gamma ^ { - 1 } ) ^ { \\rho } , b _ * = 6 - 2 b , \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} \\kappa : = \\sum _ { i = 2 } ^ { m + 1 } ( i - 1 ) r _ i + ( m + 2 ) \\sum _ { i = m + 2 } ^ D r _ i , \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} \\lambda = R \\ , \\rfloor \\ , d h + R \\ , \\rfloor \\ , ( V \\rfloor d \\eta ) = R ( h ) - V \\ , \\rfloor \\ , ( R \\ , \\rfloor \\ , d \\eta ) = R ( h ) . \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { W } _ 1 & = X _ 1 M _ 1 ^ 2 X _ 1 ^ t - Y _ 1 M _ 1 ^ 2 Y _ 1 ^ t + i 2 X _ 1 M _ 1 ^ 2 Y _ 1 ^ t \\\\ & = ( X _ 1 + i Y _ 1 ) M _ 1 ^ 2 ( X _ 1 ^ t + i Y _ 1 ^ t ) , \\end{aligned} \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} \\left \\| { A } \\xi - \\hat { A } \\xi \\right \\| _ 2 ^ 2 = \\langle \\bold { B } ^ { - 1 } \\vec { \\bold { a } } , \\vec { \\bold { a } } \\rangle = \\langle \\overline { \\bold { \\Phi } } \\bold { \\Phi } ^ { ' } \\vec { \\bold { a } } , \\vec { \\bold { a } } \\rangle = \\langle \\bold { \\Phi } ^ { ' } \\vec { \\bold { a } } , \\bold { \\Phi } ^ { ' } \\vec { \\bold { a } } \\rangle = \\langle \\bold { A } \\vec { \\varphi } , \\bold { A } \\vec { \\varphi } \\rangle = \\| \\bold { A } \\vec { \\varphi } \\| ^ 2 , \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} \\varphi _ { \\alpha } ( t , \\Phi ( t ) u ) = \\varphi _ { \\alpha } ( t , u ) . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} \\frac { \\partial \\varphi } { \\partial t } ( x , 0 ) = \\log \\frac { ( \\omega + \\sqrt { - 1 } \\partial \\overline { \\partial } \\varphi _ { 0 } ) ^ { n } } { \\omega ^ { n } } - F ( x ) . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} & \\ell _ { i ^ { \\star } } = \\ell _ { j ^ { \\star } } = r _ { i ^ { \\star } } = r _ { j ^ { \\star } } = \\frac { 2 \\frac { N + 2 } { 4 } } { N } , \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} \\alpha _ { 1 } = \\xi , \\ ; \\ ; \\ ; \\alpha _ { 2 } = 2 \\xi - 2 c \\xi ^ { 2 } + \\xi ^ { 3 } , \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} v _ t ( x , t ) = \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ k t ^ { \\delta k - 1 } } { \\Gamma ( \\delta k ) } \\sin x v _ { t t } ( x , t ) = \\sum _ { k = 1 } ^ \\infty \\frac { ( - 1 ) ^ k t ^ { \\delta k - 2 } } { \\Gamma ( \\delta k - 1 ) } \\sin x \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} M ^ 1 ( r , f ) = M ( r , f ) \\quad M ^ { n + 1 } ( r , f ) = M ( M ^ n ( r , f ) , f ) . \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} | e _ N ( 1 ) | \\le \\sqrt { 2 N } \\left ( \\sum _ { i = 1 } ^ N \\frac { \\omega _ i E _ N ^ 2 ( \\tau _ i ) } { 1 - \\tau _ i ^ 2 } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "103.png", "formula": "\\begin{align*} g : = ( \\phi ( \\alpha ) ) : M ^ { j } \\to \\bigoplus _ { \\alpha \\in Q _ { 1 } , \\ , s ( \\alpha ) = j } \\mathbb { S } _ { 2 } ^ { - \\deg ( \\alpha ) } ( M ^ { t ( \\alpha ) } ) \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} \\hat { \\theta } _ { 1 } = - \\theta + { 5 \\over 6 } , \\ \\hat { \\theta } _ { 2 } = - 2 \\theta + { 1 \\over 3 } , \\ \\hat { \\theta } _ { 3 } = 3 \\theta + { 1 \\over 2 } . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} | P _ q ^ c Y | _ \\alpha & = | \\zeta _ q ( Y ) | \\ , | Y _ q ' | _ \\alpha \\le C | Y | _ \\alpha \\ , | Y _ q ' | _ \\alpha \\le C | Y | _ \\beta \\ , | Y _ q ' | _ \\alpha , \\\\ | P _ q ^ c Y | _ 0 & = | \\zeta _ q ( Y ) | \\ , | Y _ q ' | _ 0 \\le C | Y | _ \\alpha \\ , | Y _ q ' | _ 0 \\le C | Y | _ \\beta \\ , | Y _ q ' | _ 0 . \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} { \\cal E } ( z ) : = Z ^ { \\infty } ( z ) \\ , Z ^ { - 1 } ( z ) . \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} { x _ 1 } ( 0 ) = { x _ 1 } ( 1 ) = 0 , \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} \\left \\langle \\Omega \\right \\vert = \\otimes _ { n = 1 } ^ { \\mathsf { N } } \\left \\langle j _ { n } - 1 , n \\right \\vert , \\left \\vert \\bar { \\Omega } \\right \\rangle = \\otimes _ { n = 1 } ^ { \\mathsf { N } } \\left \\vert j _ { n } , n \\right \\rangle . \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} L ^ 2 ( \\mu ) = \\bigoplus _ { n = 0 } ^ \\infty \\mathcal { O P } _ n . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} v ( t ) : = \\pi ^ { ( t ) } ( V _ t ) , \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} E _ { 0 0 } = \\sum _ e \\mathop { \\sum \\sum } _ { ( u , v ) = 1 } \\frac { \\rho ( e u ) \\rho ( e v ) } { e u v } g ( e u ) g ( e v ) \\lambda ( u v ) \\xi ( u v ) . \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { k + 1 } f _ i & = f _ 1 \\cdots f _ k f _ { k + 1 } \\\\ & = ( h _ 1 x ^ { d _ 2 + \\cdots + d _ k } ) f _ { k + 1 } \\\\ & = x ^ { d _ 2 + \\cdots + d _ k } ( h _ 1 f _ { k + 1 } ) \\\\ & = x ^ { d _ 2 + \\cdots + d _ k } ( h x ^ { d _ { k + 1 } } ) \\\\ \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} R ( h ) = ( s - 2 h ) ( 2 h - v ) L _ { 5 } ( h ) \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} a ^ { \\lambda , m , n } = \\frac { 2 - n } { 1 + m - n } + \\frac { 2 \\lambda } { 1 + m - n } , b ^ { \\lambda , m , n } = \\frac { 1 - m } { 1 + m - n } + \\frac { 1 - m + n } { 1 + m - n } \\lambda \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{gather*} q _ 2 = \\frac { \\mu } { \\chi } \\qquad q _ 1 = \\frac { \\nu } { \\chi } , \\end{gather*}"}
-{"id": "6746.png", "formula": "\\begin{align*} P = a \\sqrt { c } = \\pm d \\sqrt { c } = \\pm Q , \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} n _ { j _ 0 , j } \\ = \\ \\begin{cases} \\delta _ { j _ 0 , j } & j _ 0 \\in J _ 0 , \\\\ ( - 1 ) ^ { \\epsilon ( j _ 0 , J ) + \\epsilon ( j , J ) + 1 } \\ , \\frac { \\Delta _ { J - \\{ j \\} } } { \\Delta _ { J _ 0 } } & j _ 0 \\notin J _ 0 \\end{cases} \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} E \\cap B _ { r } ( x ) \\cap ( x + C ( V ^ { \\bot } , \\theta ) ) = \\{ x \\} , \\forall x \\in E . \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} u = \\mu _ 1 - \\mu , v = \\nu - \\nu _ { - 1 } ; \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} x ^ { ( l ) } x ^ { ( l ) * } = \\sum _ { i , j \\in I ^ { ( l ) } } \\psi _ i \\psi _ j ^ * = \\sum _ { i \\in I ^ { ( l ) } } | \\psi _ i | ^ 2 , \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} i _ { p _ { U , W } } ( u , w ) & \\triangleq \\log \\left ( \\frac { d p _ { W | U = u } } { d p _ W } ( w ) \\right ) \\\\ i _ { p _ { U , V , W } } ( u , v , w ) & \\triangleq \\log \\left ( \\frac { d p _ { W | U = u , V = v } } { d p _ W } ( w ) \\right ) . \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} e _ { d C , p } ( T _ C ( P ) , T _ C ( Q ) ) = e _ { C , p } ( P , Q ) . \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} X _ H ^ { \\gamma } = T _ { \\pi } \\circ X _ { H } \\circ \\gamma \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{align*} \\| F _ { \\rho } f \\| _ { L ^ { 2 } ( 0 , T ; H ^ { k } ( \\Omega ) ) } \\leq \\frac { C } { \\sigma ^ { 2 - k } } \\| f \\| _ { L ^ { 2 } ( 0 , T ; H ^ { 1 } ( \\Omega ) ) } , \\ , \\ , \\ , \\ , \\ , k = 1 , \\ , 2 . \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} L ( m , n , [ u ] ) = ( u _ { 1 , 0 } - u _ { 0 , 1 } ) u _ { 0 , 0 } - ( a ( m ) - b ( n ) ) \\ln ( u _ { 1 , 0 } - u _ { 0 , 1 } ) \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} \\Diamond ( p , q ) \\ = \\ \\Diamond ( n - q , n - p ) \\ , . \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} \\widetilde { J } ( ( \\rho , \\mu ) , u ) \\ , : = \\ , { J } ( ( \\rho , \\mu ) , u ) \\ , + \\ , \\frac 1 2 \\| u - \\overline { u } \\| ^ 2 _ { L ^ 2 ( Q ) } \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} \\widehat { E } & = ( I + H H ^ { \\ast } E _ { S _ { A } } ) ( I + E _ { S _ { A } } + H H ^ { \\ast } E _ { S _ { A } } ) ^ { - 1 } \\\\ & = I - E _ { S _ { A } } ( I + E _ { S _ { A } } + H H ^ { \\ast } E _ { S _ { A } } ) ^ { - 1 } \\\\ & = I - ( I + E _ { S _ { A } } + E _ { S _ { A } } H H ^ { \\ast } ) ^ { - 1 } E _ { S _ { A } } . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} \\sigma ( B ) = A B A ^ { - 1 } + \\delta ( A ) A ^ { - 1 } . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\beta ( n ) = \\frac { \\pi ^ 2 } { 1 2 0 } x ^ 2 + O \\left ( x ( \\log x ) ^ { 2 / 3 } ( \\log \\log x ) ^ { 4 / 3 } \\right ) . \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\hat { \\theta } _ { j } = p _ { j } \\theta + \\xi _ { j } , & \\forall 1 \\leq j \\leq k - 1 , \\\\ \\hat { \\theta } _ { k } = p _ { k } \\theta , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} \\frac { z D _ { q } \\mathcal { F } ( z ) } { \\mathcal { F } ( z ) } = 1 + \\frac { 1 - [ 2 _ { q } ] } { 3 } a _ { 2 } z + \\left ( \\frac { [ 3 ] _ { q } - 1 } { 1 0 } a _ { 3 } + \\frac { 1 - [ 2 _ { q } ] } { 9 } a _ { 2 } ^ { 2 } \\right ) z ^ 2 + \\dots \\end{align*}"}
-{"id": "9101.png", "formula": "\\begin{align*} D \\left ( F _ a ( z ) \\right ) = \\frac { \\partial } { \\partial z } F _ a ( z , p ) + \\beta \\sum _ b \\oint \\frac { d \\xi } { \\xi ^ 2 } \\frac { \\phi _ b ^ - ( \\xi ) } { 1 - \\frac { z } { \\xi } } V _ b ( z ) V _ b ^ { - 1 } ( \\xi ) F _ a ( \\xi ) , \\end{align*}"}
-{"id": "2868.png", "formula": "\\begin{align*} \\exists \\ \\{ \\nu _ { j } \\} _ { j \\in I ( \\bar { x } ) } \\geq 0 \\ \\forall y \\in \\ell _ p \\sum _ { k = 1 } ^ { + \\infty } | \\bar { x } _ k - x _ k | ^ { p - 2 } { ( \\bar { x } _ k - x _ k ) } y _ k = - \\sum _ { i \\in I ( \\bar { x } ) } \\nu _ i \\langle f _ i \\ | \\ y \\rangle , \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} { \\Theta } + \\sum _ { j = k } ^ n { \\Theta } ^ { ( j ) } \\lesssim 1 . \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} \\eta ^ j ( v _ 1 \\otimes \\cdots \\otimes v _ j ) : = \\theta _ { v _ 1 } \\circ \\cdots \\circ \\theta _ { v _ j } . \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} \\iota _ m \\circ \\phi ( y _ { j i } ) = \\frac { x _ { j - 1 , i + 1 } \\ , x _ { j , i - 1 } \\ , x _ { j + 1 , i } } { x _ { j - 1 , i } \\ , x _ { j , i + 1 } \\ , x _ { j + 1 , i - 1 } } \\cdot \\frac { X _ { v _ { ( j , i ) , ( j + 1 , i - 1 ) } } } { X _ { v _ { ( j - 1 , i + 1 ) , ( j , i ) } } } . \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} x = \\begin{pmatrix} 0 & 0 & \\dots & 0 & 1 \\\\ 0 & 0 & \\dots & - 1 & 0 \\\\ \\vdots & \\vdots & \\dots & \\vdots & \\vdots \\\\ ( - 1 ) ^ { n + 1 } & 0 & \\dots & 0 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} \\| f _ s \\| _ { K } ^ 2 = \\prod _ { j = s + 1 } ^ \\infty ( 1 + k _ j ( y _ j , y _ j ) ) , \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} \\pi _ { 1 4 } ( d * J d \\beta _ { 1 4 } ) = \\Delta \\beta _ { 1 4 } - \\frac { 3 } { 2 } d _ { 1 4 } ^ 7 d _ 7 ^ { 1 4 } \\beta _ { 1 4 } = 0 \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} ( | b | ^ { 2 } K \\alpha ^ { 2 } - | e | ^ { 2 } L \\beta ^ { 2 } ) 1 + ( \\overline { a } b + a \\overline { b } K ) \\alpha - ( \\overline { d } e + d \\overline { e } L ) \\beta = 0 , \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} \\frac { ( a - a ' ) p ^ m } { q } = \\sum _ { n \\ge 1 } \\frac { c _ n } { p ^ n } . \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} d = R _ \\alpha ( 1 - \\bar Y ) . \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ \\infty ( 1 - | \\alpha _ j | ^ 2 ) = \\exp \\biggl ( \\int _ 0 ^ { 2 \\pi } \\log ( 1 - | f ( e ^ { i \\theta } ) | ^ 2 ) \\ , \\frac { d \\theta } { 2 \\pi } \\biggr ) . \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} F ( \\beta + 1 ) ^ { p - q } - A _ 0 ( \\beta ) F ( \\beta + 1 ) ^ { p } = \\frac { q } { p } \\frac { 1 } { ( \\beta + 1 ) ^ { q - 1 } } f ^ p , \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} \\mathcal E ^ h ( y ^ h ) = \\int _ \\Omega W ^ h ( x , \\nabla _ h y ^ h ) \\dd x - \\int _ \\Omega f ^ h \\cdot y ^ h \\dd x \\ , . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} F _ { \\eta } ( x ) - F _ \\theta ( x ) = \\int _ 0 ^ 1 \\sum _ { k = 1 } ^ K \\alpha ( \\theta _ { i _ k } - \\theta _ { j _ k } + 2 t \\delta _ k ) \\delta _ k f ' _ k ( x | t \\delta ) \\ , { \\rm d } t . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} ( ( \\xi + \\eta ) \\tanh ( \\xi + \\eta ) - \\xi \\tanh \\xi - \\eta \\tanh \\eta ) C ^ h = i \\xi \\eta ( \\xi + \\eta ) + 2 \\xi \\eta B ^ h , \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} \\wedge _ { i } b _ 2 ^ i = \\pm \\nu ( \\mathcal { E } _ B ) ^ { - 1 } ( \\wedge _ { i } \\phi _ B ( b _ 1 ^ i ) ) \\wedge ( \\wedge _ { i } \\psi _ B ^ { - 1 } ( b _ 3 ^ i ) ) . \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} \\tilde \\lambda _ j > \\lambda _ j = m & j \\in B \\\\ \\tilde \\lambda _ j = \\lambda _ j & j \\notin B \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} s ( t ) = 3 ( \\xi ''' ) ^ { 2 } - 2 \\xi '' \\cdot \\xi '''' \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} { \\bf R } _ i = \\sum \\limits _ { l = 1 } ^ { 3 } \\sum \\limits _ { m = 1 } ^ { 2 } p _ { l m } { \\bf G } _ { i l } { \\bf v } _ { l m } { \\bf v } ^ { T } _ { l m } { \\bf G } ^ { T } _ { i l } + \\sigma ^ { 2 } { \\bf I } _ 2 , \\quad \\forall i \\in \\mathcal { I } , \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} \\mathrm { R e s } ( \\tilde { x } \\cdot \\mathrm { d l o g } \\tilde { y } ) = c e \\beta - \\sum _ { j = 0 } ^ { \\infty } p ^ j \\sum _ { ( i , p ) = 1 } c _ i \\sum _ { l | i } l [ a _ { l j } ] ^ { i / l } . \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} - \\Delta _ { p ( x ) } u _ { \\varepsilon } = \\left \\{ \\begin{array} { l l } h ( x ) & d ( x ) > \\varepsilon \\\\ \\tilde { h } ( x ) & d ( x ) < \\varepsilon \\end{array} \\right . , u _ { \\varepsilon } = 0 \\partial \\Omega . \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} \\widehat f _ a ( u ) = \\Big ( 1 + \\sum _ { k = 1 } ^ d a _ k \\| u \\| ^ { 2 k } \\Big ) e ^ { - \\pi \\| u \\| ^ 2 } , \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} h = 2 ^ { 1 / 2 } \\ , h _ { 2 ^ { - 1 / 2 } } \\ast g _ { 2 ^ { - 1 / 2 } } . \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} \\tilde { x } = c \\beta + \\sum _ { ( i , p ) = 1 } c _ i T ^ { - i } \\in W ( k ) [ [ T ^ { - 1 } ] ] . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v = \\frac { 1 } { 2 } \\displaystyle { \\sum _ { i , j = 1 } ^ d } \\partial _ { i j } ^ 2 \\left ( ( \\Phi \\Phi ^ t ) _ { i , j } ( t , x , v ) v \\right ) - d i v \\left ( g ( t , x , v ) v \\right ) + \\Lambda ( t , x , v ) v \\\\ v ( 0 , x ) = v _ 0 \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} \\omega _ 1 = & d \\theta _ 1 \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} \\varPsi ( \\lambda ) = 1 - \\varPhi ( \\lambda ) = 1 - \\frac { 1 } { \\sqrt { 2 \\pi } } \\int \\limits _ { - \\infty } ^ { \\lambda } \\mathrm { e } ^ { - \\frac { 1 } { 2 } z ^ 2 } \\mathrm { d } z , \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} L p _ 0 ( \\lambda _ j ) + \\sum _ { k = 1 } ^ N \\theta ( \\lambda _ j - \\lambda _ k ) = 2 \\pi I _ j . \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{align*} \\mathfrak { h } ' _ n & = \\underset { \\gamma ^ * \\in \\Gamma _ * } { \\sum } \\tau _ { \\gamma ^ * } \\left ( g _ { 1 / \\delta } \\ , \\sharp ^ { \\epsilon , \\kappa } \\ , \\widetilde { \\mathfrak { g } } _ n \\ , \\sharp ^ { \\epsilon , \\kappa } \\ , \\ , \\big [ r ^ { \\epsilon , \\kappa } _ 1 \\big ] ^ { \\ , \\sharp ^ { \\epsilon , \\kappa } \\ , N } \\right ) \\ , , \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} \\sigma _ { g ^ { - 1 } } ( y ^ * ) _ { ( \\vec { 0 } , s ^ { - 1 } ) } & = ( y ^ * ) _ { ( ( i , j ) , h ) ( \\vec { 0 } , s ^ { - 1 } ) } \\\\ & = ( y ^ * ) _ { ( ( i , j ) , h s ^ { - 1 } ) } \\\\ & = ( y ^ { s h ^ { - 1 } } ) _ { \\varphi _ { ( s h ^ { - 1 } ) } ( i , j ) } \\\\ & = ( \\sigma _ { - ( \\varphi _ { ( s h ^ { - 1 } ) } ( i , j ) ) } ( y ^ { s h ^ { - 1 } } ) ) _ { \\vec { 0 } } . \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} J ( k , \\bar { \\theta } ) = \\sup _ { y \\in \\Gamma ( k ) } \\Big \\{ U ( f ( k ) - y , \\bar { \\theta } ) + \\bar { \\delta } J ( y , \\bar { \\theta } ) \\Big \\} , \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\vert U ( x ) \\vert = \\mathcal { O } ( \\langle x \\rangle ^ { - m } ) , \\langle x \\rangle : = \\sqrt { 1 + \\vert x \\vert ^ { 2 } } , \\quad \\mathrm { f o r \\ s o m e } m > 0 . \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} l _ \\sigma ( u ; 0 , 0 ) & = ( \\gamma \\circ c _ \\sigma ) ( b _ 1 b _ 2 ) = \\gamma ( B a _ 1 a _ 2 B ^ { - 1 } ) = ( B ; 0 , 0 ) ( v ^ { - 1 } u ^ { - 1 } ; 1 , 1 ) ( v ; 0 , - 1 ) ( B ^ { - 1 } , 0 , 0 ) \\\\ & = ( B v ^ { - 1 } u ^ { - 1 } \\theta ( 1 , 1 ) ( v ) \\theta ( 1 , 0 ) ( B ) ^ { - 1 } ; 1 , 0 ) = ( B v ^ { - 1 } u ^ { - 1 } B v u ^ { - 2 } B ^ { - 1 } ; 1 , 0 ) = ( B u ^ { - 1 } B ^ { - 1 } ; 1 , 0 ) . \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} p _ k = - k e _ k - \\sum _ { i = 1 } ^ { k - 1 } e _ i p _ { k - i } , \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} F _ n ( \\tau ) = \\left \\{ \\begin{array} { l l } 0 & \\ ; \\mbox { i f $ \\tau \\geq 0 $ } \\\\ e ^ n e ^ { e ^ n \\tau } & \\ ; \\mbox { i f $ \\tau < 0 $ } \\\\ \\end{array} \\right . . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} V _ a ( z ) = \\exp \\left ( \\sum _ { n \\geqslant 0 } z ^ n \\frac { \\partial } { \\partial p _ { n , a } } \\right ) . \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} { { \\bar G } } = { \\mathbb { E } } \\left \\{ { { G _ i } } \\right \\} = \\sum \\nolimits _ { \\ell , k \\in \\left \\{ { { } , { } } \\right \\} } { { G _ { \\ell k } } _ { \\ell k } } . \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} ( 1 / n ) \\sum _ { i = 1 } ^ n ( Y _ i - f ( X _ i ) ) ^ 2 + _ n ( f ) / n . \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} [ F _ { 0 } ] _ { I } = \\begin{cases} s t , & I = \\{ 0 , 1 \\} \\\\ \\lambda b _ { i } s ^ { 2 } , & I = \\{ 0 , i \\} , i \\ge 2 \\\\ - \\lambda a _ { i } t ^ { 2 } , & I = \\{ 1 , i \\} , i \\ge 2 \\\\ \\lambda ^ { 2 } ( a _ { i } b _ { j } - a _ { j } b _ { i } ) , & I = \\{ i , j \\} , i , j \\ge 2 . \\end{cases} \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} \\tilde \\lambda ( p ) = ( \\lambda ( p ) \\xi ( p ) ) ^ 2 = ( 1 + \\chi ( p ) ) ^ 2 ( 1 + \\chi ( p ) / p ) ^ { - 2 } . \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} \\frac { \\xi \\wedge \\eta } { 1 \\wedge \\tau } = \\frac { \\bar \\xi \\eta - \\xi \\bar \\eta } { \\tau - \\bar \\tau } , \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} ( b f ) ( a _ 1 , \\ldots , a _ n ) = & a _ 1 f ( a _ 2 , \\ldots , a _ n ) \\\\ & + \\sum _ { i = 1 } ^ { n - 1 } ( - 1 ) ^ i f ( a _ 1 , \\ldots , a _ { i - 1 } , a _ i a _ { i + 1 } , a _ { i + 2 } , \\ldots , a _ n ) \\\\ & + ( - 1 ) ^ n f ( a _ 1 , \\ldots , a _ { n - 1 } ) a _ n . \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} I \\left ( \\frac { u } { v } \\right ) = 2 \\sum _ { h = 1 } ^ \\infty I _ h \\left ( \\frac { u } { v } \\right ) \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} \\dot { \\rho } = \\frac { 1 } { r } \\sqrt { \\rho ^ 2 - \\frac { \\omega ^ 2 } { \\rho ^ 2 } - a ^ 2 + \\frac { \\omega ^ 2 } { a ^ 2 } } \\\\ \\rho ( a ) = a , \\rho ( b ) = b . \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} ( n _ 2 - n _ 1 ) r _ 1 + ( n _ 4 - n _ 3 ) r _ 3 = 0 \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\langle L _ { i } ( t ) , L _ j ( t ) \\rangle = \\delta _ { i j } \\cdot \\dfrac { 2 } { 2 j + 1 } \\enspace . \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} Q _ 0 = J _ { l , b c } Q + ( I + \\Gamma _ { l , b c } Q ) ^ { - 1 } ( Q \\Gamma _ { l , b c } Q - ( \\Gamma _ { l , b c } Q ) J _ { l , b c } Q ) = : J _ { l , b c } Q + B . \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} \\delta _ { \\pm } [ s ] = e ^ { - s \\mathbb { H } _ m } \\delta _ { \\pm } e ^ { s \\mathbb { H } _ m } . \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} & a = m - 1 + \\delta , \\\\ & \\kappa = 1 + \\frac { 2 \\delta } { n ( m - 1 + \\delta ) } \\\\ & b = 2 \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} f ( \\tau ) = f ^ + ( \\tau ) + f ^ - ( \\tau ) , \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} \\operatorname { R e p } _ A ( \\operatorname { A u t } _ { \\omega } ^ { \\otimes } ) & \\simeq \\varprojlim Q C ( \\operatorname { \\check { C } e c h } ( \\mathsf { S p e c } \\ , A \\to \\mathsf { M } _ { \\mathcal { C } ^ { \\otimes } } ) ) \\\\ & \\simeq \\varprojlim \\operatorname { C o b a r } ( \\omega ) \\\\ \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\| u ^ h _ t \\| ^ 2 _ { H _ l } & = \\| \\psi \\| ^ 2 _ { H ^ l } + \\int _ 0 ^ t \\left [ \\left ( ( L ^ h _ s + I ^ h ) u ^ h _ s , u ^ h _ s \\right ) _ { H ^ l } + ( f _ s , u ^ h _ s ) _ { H ^ l } \\right ] \\ d s \\\\ & \\leq \\| \\psi \\| ^ 2 _ { H ^ l } + N \\int _ 0 ^ t \\| u ^ h _ s \\| ^ 2 _ { H ^ l } \\ d s + \\int _ 0 ^ T \\| f _ s \\| ^ 2 _ { H ^ l } \\ d s < \\infty , \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} & - \\Delta \\psi _ { 1 , \\infty } = \\lambda _ 1 ( \\mathcal { Q } _ { \\infty } ) \\psi _ { 1 , \\infty } \\ , , \\mbox { o n } \\Omega , \\\\ & \\psi _ { 1 , \\infty } = 0 \\ , , \\mbox { o n } \\Gamma \\ , . \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} { R } _ * : = \\tilde { \\mathcal { R } } _ { > 2 } + R _ { H _ { \\geq 5 } } ( T _ { \\delta } ) + R ( \\psi ) , \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} & \\epsilon u _ 1 ( m + v _ 1 ) + \\epsilon u _ 2 ( n + v _ 2 ) - u _ 1 ( m + v _ 1 ) + u _ 2 ( n + v _ 2 ) \\\\ [ 2 m m ] & = ( \\epsilon - 1 ) u _ 1 v _ 1 + ( \\epsilon - 1 ) u _ 2 v _ 2 + ( \\epsilon - 1 ) u _ 1 m + ( \\epsilon - 1 ) u _ 2 n , \\end{align*}"}
-{"id": "3836.png", "formula": "\\begin{align*} R _ 2 ( p _ 2 , \\cdots , p _ s ) = & \\sum _ { j = 3 } ^ s 2 ^ { j - p _ j + p _ 1 } ( 2 ^ { p _ j - p _ { j - 1 } } - 2 ) \\left [ ( 3 \\ 2 ^ { j - 1 } - 1 ) \\Delta _ { 2 ^ { p _ j } + \\cdots + 2 ^ { p _ s } } \\right . \\\\ & \\left . + ( 2 ^ { j - 2 } ( 2 ^ { p _ j - p _ { j - 1 } } - 2 ^ 2 ) + 1 ) ( 1 + 2 ^ { p _ { j + 1 } - p _ j } + \\cdots + 2 ^ { p _ s - p _ j } ) \\right ] . \\\\ \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\ell = U ( J ( \\ell _ 0 ) ) , \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} \\eta ( 1 ; 1 ) & = \\zeta ( 2 ) , \\\\ \\eta ( 2 ; 1 ) & = \\zeta ( 1 , 2 ) + \\zeta ( 3 ) , \\\\ \\eta ( 3 ; 1 ) & = \\zeta ( 1 , 1 , 2 ) + \\zeta ( 2 , 2 ) + \\zeta ( 1 , 3 ) + \\zeta ( 4 ) , \\\\ \\eta ( 2 ; 2 ) & = 2 \\zeta ( 1 , 1 , 2 ) + \\zeta ( 2 , 2 ) + 2 \\zeta ( 1 , 3 ) + \\zeta ( 4 ) . \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} P ( \\mathcal { A } ) & = P \\left ( \\frac { \\tau _ n - \\sigma _ { \\rm w } ^ 2 - \\sigma _ { \\rm a } ^ 2 - \\delta } { \\zeta } \\leq U \\leq \\frac { \\tau _ n - \\sigma _ { \\rm w } ^ 2 + \\delta } { \\zeta } \\right ) \\\\ & \\leq \\frac { \\sigma _ { \\rm a } ^ 2 + 2 \\delta } { \\zeta } . \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{gather*} \\frac { d \\Psi _ 0 } { d x } = \\hat { L } _ 0 \\Psi _ 0 \\end{gather*}"}
-{"id": "8430.png", "formula": "\\begin{align*} A ( r , s ) & = \\bigcup _ { n = 1 } ^ N [ r , r + s ] ^ { \\leftthreetimes 2 ( n - 1 ) } \\times [ 0 , r ] ^ { \\leftthreetimes 2 ( N - n ) } \\subset A ( r , s + t ) , \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} | C ( m , n ; k ) | = \\Big ( \\frac { 1 } { | G _ { m } | } + o ( 1 ) \\Bigr ) | G _ { n } | \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} M _ k ( x ; \\mathbf { a } ) = \\frac { x } { \\log x } \\left \\{ \\frac { 1 } { \\phi ( q ) } Q _ { \\mathbf { k } } \\ ( \\frac { \\log \\log x } { \\phi ( q ) } \\ ) + \\widetilde { R } ( x ) \\right \\} , \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} x ^ { \\ell _ 1 } _ \\gamma = x ^ { \\ell _ 2 } _ \\gamma , \\forall \\ell _ 1 , \\ell _ 2 : \\gamma \\in \\mathcal { I } _ { \\ell _ 1 } ^ 1 \\cap \\mathcal { I } _ { \\ell _ 2 } ^ 1 . \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} U ( \\eta ) = e ^ { i \\frac { 1 } { 2 } \\eta \\sigma _ { y } } \\ ; , \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} \\frac { n ! } { \\pi ^ n } \\frac { \\prod _ { i < j } | z _ i - z _ j | ^ 2 } { \\left ( \\displaystyle \\int \\prod _ { k = 1 } ^ { n } | z - z _ k | ^ 2 d \\nu _ { S ^ 1 } ( z ) \\right ) ^ { n + 1 } } = \\frac { n ! } { \\pi ^ n } \\frac { \\prod _ { i < j } | z _ i - z _ j | ^ 2 } { \\| \\tilde { a } \\| _ 2 ^ { 2 n + 2 } } \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} P _ v ( s ) = \\prod _ { p | v } \\bigl ( 1 - p ^ { \\varepsilon _ 1 - s - 1 } \\bigr ) \\ldots \\bigl ( 1 - p ^ { \\varepsilon _ r - s - 1 } \\bigr ) \\ . \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} 0 & = L ( e _ { i i } ) e _ { k l } - e _ { k l } L ( e _ { i i } ) + e _ { i i } L ( e _ { k l } ) - L ( e _ { k l } ) e _ { i i } + L ( e _ { k i } ) \\\\ & = \\left ( C _ { k k } ^ { i i } - C _ { i i } ^ { i i } \\right ) e _ { k i } + \\left ( C _ { i k } ^ { i i } + C _ { i k } ^ { k k } \\right ) e _ { i i } . \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} D W ^ h ( x , \\nabla _ h \\hat { y } ^ h ) = R ^ h D W ^ h ( x , I + h ^ 2 G ^ h ) = h ^ 2 R ^ h E ^ h \\ , . \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} Q ^ N _ \\ell ( \\cdot , \\cdot ) \\ : = \\ Q ( \\cdot , N ^ \\ell \\cdot ) \\ , . \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} _ { \\tau } ( \\zeta _ { a } ^ { ( 0 ) } ) \\left ( \\begin{array} { c } Q _ { \\tau , a } ^ { ( 0 ) } \\\\ \\vdots \\\\ Q _ { \\tau , a } ^ { ( p - 1 ) } \\end{array} \\right ) = \\left ( \\begin{array} { c } 0 \\\\ \\vdots \\\\ 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} \\tilde T _ 6 ( z ) = \\frac { i } { ( 2 \\pi ) ^ { \\frac 1 2 } } \\int _ { \\xi _ 1 + \\xi _ 2 + \\xi _ 3 = 0 } \\frac { 2 z ^ 2 - \\frac 1 3 ( \\xi _ 1 ^ 2 + \\xi _ 2 ^ 2 + \\xi _ 3 ^ 2 ) } { z ( z ^ 2 - \\xi _ 1 ^ 2 ) ( z ^ 2 - \\xi _ 2 ^ 2 ) ( z ^ 2 - \\xi _ 3 ^ 2 ) } \\hat u ( \\xi _ 1 ) \\hat u ( \\xi _ 2 ) \\hat u ( \\xi _ 3 ) d \\xi _ 1 d \\xi _ 2 \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} a d ( \\mathcal { E } ) = Z \\oplus B , \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} \\xi ' ( 1 - \\epsilon ) + h ^ { 2 } - \\int _ { 0 } ^ { 1 - \\epsilon } \\frac { 1 } { \\phi _ { \\epsilon } \\phi } \\leq \\xi ' ( 1 ) + h ^ { 2 } - \\int _ { 0 } ^ { 1 - \\epsilon } \\frac { 1 } { ( 1 - t ) ^ { 2 } } = \\xi ' ( 1 ) + h ^ { 2 } - \\frac { 1 - \\epsilon } { \\epsilon } . \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} \\mathcal { R } ^ { \\gamma } = \\frac { \\partial } { \\partial t } + p \\frac { \\partial } { \\partial q } . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { s } \\Bigl ( \\int _ { - \\infty } ^ { \\infty } P ( x u ) d F ( x ) \\Bigr ) d u = \\\\ = \\sum _ { j = 0 } ^ k a _ j \\int _ { 0 } ^ { s } \\tt { R e } \\ , f ( j u ) d u + \\sum _ { j = 1 } ^ k b _ j \\int _ { 0 } ^ { s } \\tt { I m } \\ , f ( j u ) d u , \\end{aligned} \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{gather*} \\widetilde { \\Lambda } ^ { 0 , n } ( J ^ \\infty E ) \\ni [ a ] \\mapsto A [ \\phi ] = \\int _ M ( j ^ { \\infty } \\phi ) ^ \\ast ( a ) , \\end{gather*}"}
-{"id": "7202.png", "formula": "\\begin{align*} \\sigma ( Z ) = \\begin{pmatrix} x ^ { 1 / 2 } & 0 \\\\ 0 & - x ^ { 1 / 2 } \\end{pmatrix} Z , \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\int _ \\R \\d u \\ , f _ W ( u ) \\phi _ { 2 ( \\tau _ 2 - \\tau _ 1 ) } ( v - u ) & = e ^ { ( \\tau _ 2 - \\tau _ 1 ) ( \\sqrt 2 r - 2 W ) ^ 2 } f _ W ( v ) , \\\\ \\int _ \\R \\d v \\ , \\phi _ { 2 ( \\tau _ 2 - \\tau _ 1 ) } ( v - u ) g _ Z ( v ) & = e ^ { ( \\tau _ 2 - \\tau _ 1 ) ( \\sqrt 2 r - 2 Z ) ^ 2 } g _ Z ( u ) . \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} u '' + a u - ( b + c n ) ( u _ 1 - u ) + ( b + c ( n - 1 ) ) ( u - u _ { - 1 } ) = 0 , \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} { \\overline { R } } _ { e ^ { * } } = \\frac { 1 } { { \\ln 2 } } \\int _ 0 ^ \\infty { \\frac { { \\left ( { 1 - \\mathcal { P } _ 1 \\left ( { x } \\right ) \\mathcal { P } _ 2 \\left ( { x } \\right ) } \\right ) } } { { 1 + x } } d x } , \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} \\biggl ( \\int _ { \\Omega _ 0 } [ x ( t ) ] ^ p \\textup d t \\biggr ) ^ { 1 / p } = m . \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} [ D ( m , s - m - 1 ) : D ( m + 1 , n - m - 1 ) ] _ q & = [ D ( m + \\delta _ { s _ 1 , 2 } \\delta _ { s _ 0 , 1 } , s - m - 1 ) : D ( m + 1 , n - m - 1 ) ] _ q & \\\\ & + \\delta _ { n , m + 1 } \\delta _ { s _ 0 , 1 } \\delta _ { s _ 1 , 2 } q ^ { m } . \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} v _ 2 \\in W ^ { m - 2 , p } ( B ) \\cap W ^ { m - 1 , q } ( B ) \\subset [ W ^ { m - 2 , p } ( B ) \\ , , \\ , W ^ { m - 1 , q } ( B ) ] _ { \\sigma } = H ^ { s - 2 } ( B ) . \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} d ( h , g ) : = \\sup _ { f } ( f ( h ) - f ( g ) ) \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} R ^ { s h o g u n } & \\le \\frac { \\log _ 9 S _ U ^ { s h o g u n } } { 7 8 3 } = : R _ U ^ { s h o g u n } \\\\ & = \\frac { \\log _ 9 1 . 5 9 9 3 \\times 1 0 ^ { 2 0 8 } } { 7 8 3 } \\approx 0 . 2 7 8 6 . \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} & v ( x , t ) - \\int _ \\tau ^ t \\int _ { { \\bf R } ^ N } G ( x - y , t - s ) v ( y , s ) ^ p \\ , d y \\ , d s \\\\ & = \\int _ { { \\bf R } ^ N } G ( x - y , t - \\tau ) u ( y + z , \\tau + ( 2 \\rho ) ^ \\theta ) \\ , d y \\\\ & \\ge c _ * \\int _ { B ( z , \\rho ) } u ( y , \\tau ) \\ , d y \\int _ { { \\bf R } ^ N } G ( x - y , t - \\tau ) G ( y , \\rho ^ \\theta ) \\ , d y \\\\ & = c _ * \\int _ { B ( z , \\rho ) } u ( y , \\tau ) \\ , d y \\ , G ( x , t - \\tau + \\rho ^ \\theta ) \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} V _ t \\ , \\rfloor \\ , \\eta _ t = h _ t , V _ t \\ , \\rfloor \\ , d \\eta _ t = Y _ t \\ , \\rfloor \\ , d \\eta _ t . \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} \\begin{array} { l l l l } L ( \\pi ) = & \\max \\ & c ' v - \\pi ' ( H v - h ) & \\\\ & \\mbox { s . t . } \\ & A v \\leq b , & \\end{array} \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} r \\phi _ d ^ { - 1 } * a + r \\phi _ d ^ { - 1 } * b & = r * ( r \\phi _ d ^ { - 1 } * a + r \\phi _ d ^ { - 1 } * b ) \\\\ & = r \\phi _ d ^ { - 1 } * r \\phi _ d * ( r \\phi _ d ^ { - 1 } * a + r \\phi _ d ^ { - 1 } * b ) \\\\ & = r \\phi _ d ^ { - 1 } * ( r \\phi _ d * r \\phi _ d ^ { - 1 } * a + r \\phi _ d * r \\phi _ d ^ { - 1 } * b ) \\\\ & = r \\phi _ d ^ { - 1 } * ( r * a + r * b ) \\\\ & = r \\phi _ d ^ { - 1 } * ( a + b ) \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} \\hat { w } = \\mathcal { L } _ { \\omega } ^ { - 1 } [ g _ 3 + \\partial _ x K _ { 1 1 } ( \\varphi ) \\hat { \\eta } ] . \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} X _ H = \\sum _ { i = 1 } ^ n \\left ( p _ i \\frac { \\partial H } { \\partial p _ i } - \\sum _ { i = 1 } ^ n H \\right ) \\frac { \\partial } { \\partial t } - \\sum _ { i = 1 } ^ n \\left ( p _ i \\frac { \\partial H } { \\partial t } + \\frac { \\partial H } { \\partial q ^ i } \\right ) \\frac { \\partial } { \\partial p _ i } + \\sum _ { i = 1 } ^ n \\frac { \\partial H } { \\partial p _ i } \\frac { \\partial } { \\partial q ^ i } \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{align*} F ( z ) : = \\frac { 1 + z f ( z ) } { 1 - z f ( z ) } , \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} \\frac { 1 } { a ( x ) } \\partial ^ 2 _ { t t } p ( t , x ) - \\partial _ { x x } ^ 2 p ( t , x ) = 0 , ( t , x ) \\in [ 0 , T ] \\times D , \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} V ( x ) = x ^ 6 - u | x | ^ 5 + t x ^ 4 - s | x | ^ 3 + r x ^ 2 - q | x | , \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\theta _ 0 ( \\theta _ 0 ( x ) ) = \\theta _ 0 ( x ) \\textrm { a n d } \\theta _ 0 ( x y ) = \\theta _ 0 ( x ) \\cdot \\theta _ 0 ( y ) , \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} \\gamma = \\frac { d } { d t } \\left ( \\frac { \\sqrt { 2 } } { | W | } \\sqrt { C _ 1 y _ 1 ^ 2 + C _ 2 y _ 2 ^ 2 \\pm \\sqrt { 4 C _ 1 C _ 2 - k W ^ 2 y _ 1 y _ 2 } } \\right ) \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} \\lim _ { \\overline \\Omega \\ni y \\to x } u ^ \\varepsilon ( y ) = g ^ \\varepsilon ( x ) \\ \\ \\ x \\in \\partial \\Omega , \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} \\Upsilon ^ { [ 0 ] } = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\tilde k ( \\tau ) d \\tau . \\end{align*}"}
-{"id": "6273.png", "formula": "\\begin{align*} a _ { \\xi } ^ { \\eta ' } ( y ) = 0 \\Longleftrightarrow a _ { \\frac { d \\xi } { a } } ^ { \\eta } \\left ( \\frac { a y } { d } \\right ) = 0 . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} S _ = 2 ^ { N + 1 } S _ { } , \\end{align*}"}
-{"id": "9475.png", "formula": "\\begin{align*} \\psi ( \\gamma ^ { \\dagger } - \\gamma - s \\gamma ^ { \\dagger } ) \\ = \\ \\gamma ^ { \\dagger } . \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} \\beta _ { 0 , q } = 1 , q ( q \\beta _ q + 1 ) ^ n - \\beta _ { n , q } = \\begin{cases} 1 , \\ , \\ , \\ , n = 1 \\\\ 0 , \\ , \\ , \\ , n > 1 , \\end{cases} \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} \\alpha _ { j } = \\sum _ { l = 1 } ^ { r } c _ { j l } \\alpha _ { m l } + \\xi _ { j } , \\ \\forall 1 \\leq j \\leq m . \\end{align*}"}
-{"id": "8909.png", "formula": "\\begin{align*} \\textrm { s i g n } \\left ( ( - 1 ) ^ k \\sin ( \\phi ) \\right ) = \\textrm { s i g n } ( \\cos ( A ) ) . \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} \\sum _ { k , \\ell = 0 } ^ p \\left | \\psi _ n ( k , \\ell ) \\mathcal { E } _ n ( k , \\ell ) - \\psi ( k , \\ell ) \\mathcal { E } ( k , \\ell ) \\right | . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} f _ j = A ( E ) u j = 1 , \\dots L . \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} f ( x ; B _ \\epsilon ( 1 ) ) \\geq \\epsilon f ( x ; x ) = \\frac { \\epsilon x ^ x } { x ! } \\exp ( - x ) \\approx \\frac { \\epsilon } { \\sqrt { 2 \\pi x } } \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} - 4 i \\Omega L _ \\xi & = \\xi \\eta ( \\eta ^ 2 + \\zeta ^ 2 - 2 \\xi ^ 2 ) J ( \\zeta ) ( J ( \\zeta ) - J ( \\xi ) - J ( \\eta ) ) \\\\ & + \\xi ^ 2 \\eta ( \\zeta - \\eta ) J ( \\xi ) ( J ( \\xi ) - J ( \\eta ) - J ( \\zeta ) ) \\\\ & + \\xi \\eta ^ 2 ( \\zeta - \\eta ) J ( \\xi ) ( J ( \\xi ) - J ( \\eta ) + J ( \\zeta ) ) , \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{align*} \\hat { M } _ { a } ^ { s G } ( \\lambda ) \\equiv ( - 1 ) ^ { \\mathsf { N } } \\ , \\sigma _ { a } ^ { y } \\ , \\left ( M _ { a } ^ { s G } ( 1 / \\lambda ) \\right ) ^ { t _ { 0 } } \\ , \\sigma _ { a } ^ { y } = \\left ( - \\sigma _ { a } ^ { x } \\right ) ^ { \\mathsf { x } } \\hat { M } _ { a } ( \\lambda ) , \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = A ( t ) x + c ( t ) , \\\\ x ( 0 ) = x _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\Big \\| \\sup _ { N \\ge 1 } \\big | \\sum _ { n = 1 } ^ N \\alpha _ n D ( \\lambda _ n t ) \\big | \\ , \\Big \\| _ { \\S ^ 2 } \\le C \\Big ( \\sum _ { n \\ge 1 } \\big ( \\sum _ { k \\ , : \\ , n \\le \\lambda _ k < n + 1 } | \\alpha _ k | \\big ) ^ p \\Big ) ^ { 1 / p } \\Big ( \\sum _ { n \\ge 1 } \\big ( \\sum _ { k \\ , : \\ , n \\le \\mu _ k < n + 1 } | \\beta _ k | \\big ) ^ q \\Big ) ^ { 1 / q } \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} F ( \\xi , h ) = \\lim _ { N \\to \\infty } \\frac { 1 } { N } \\log \\int _ { S ^ { N - 1 } ( \\sqrt { N } ) } e ^ { H _ { N } ( \\sigma ) } d v o l _ { N - 1 } a . s . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\vdash n } \\sum _ { \\lambda _ i \\in \\lambda } a ( \\lambda _ i ) = \\sum _ { k = 1 } ^ { n } p ( n - k ) A ( k ) , \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} - c P ' = D ( P ) P '' - M ( P ) P ' + f ( P ) \\ , , \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} H ^ r = \\{ \\chi \\in H ^ 0 , v ( \\chi ( [ T ] ^ { - i } ) ) \\geq \\lceil \\log _ p ( r / i ) \\rceil , ( i , p ) = 1 \\} . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} \\sum _ { \\substack { i , j = 1 \\\\ [ 1 p t ] i < j } } ^ n \\left [ \\theta _ 0 ^ i ( \\delta ^ i ) ^ { t + \\tau } \\right ] \\left [ \\theta _ 0 ^ j ( \\delta ^ j ) ^ { t + \\tau } \\right ] \\left [ ( \\delta ^ i ) ^ { \\Delta { t } } - ( \\delta ^ j ) ^ { \\Delta { t } } \\right ] \\left [ ( \\delta ^ i ) ^ { \\Delta \\tau } - ( \\delta ^ j ) ^ { \\Delta \\tau } \\right ] . \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} [ D _ { \\overline { \\omega } } , \\mathcal { R } _ { \\Phi } ^ { \\varepsilon } ] h = \\varepsilon D _ { \\overline { \\omega } } \\Pi _ S [ ( \\beta _ 1 ) _ x h ] - \\varepsilon \\Pi _ S [ ( \\beta _ 1 ) _ x D _ { \\overline { \\omega } } h ] = \\varepsilon \\Pi _ S [ ( D _ { \\overline { \\omega } } ( \\beta _ 1 ) _ x ) h ] \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} B | _ { [ x , y + m ' ) } = A ' | _ { [ x , y + m ' ) } . \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} R ^ { s u m o } & \\le \\frac { \\log _ 9 S _ { U , s u m o } } { 9 0 9 } = : R _ U ^ { s u m o } \\\\ & = \\frac { \\log _ 9 1 . 7 0 4 5 \\times 1 0 ^ { 2 4 1 } } { 9 0 9 } \\approx 0 . 2 7 8 1 . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} \\langle d e ^ { i \\theta } U ( x _ 0 ) \\ , | \\ , d e ^ { i \\theta } V ( x _ 0 ) \\rangle _ g = \\sum _ { j , k = 1 } ^ { n - 1 } a _ j ( 0 , 0 ) \\overline b _ k ( 0 , 0 ) \\langle d e ^ { i \\theta } Z _ j ( x _ 0 ) \\ , | \\ , d e ^ { i \\theta } Z _ k ( x _ 0 ) \\rangle _ g . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} K \\cap B _ { r } ( x ) \\backslash ( x + C ( V _ { x } , \\theta ) ) = \\emptyset . \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\gamma _ 2 = \\frac { 1 } { 4 8 } \\left ( 8 b _ 3 + 2 b _ 2 c _ 1 + 8 c _ 2 - 3 b ^ 2 _ 2 - 3 c _ 1 ^ 2 \\right ) . \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} z \\in \\bigcap _ { l = 1 } ^ \\infty X _ l = \\bigcap _ { l = 1 } ^ \\infty \\overline { X _ l } . \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} F ( t , x , v ) | _ { t = 0 } = F _ 0 ( x , v ) . \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} \\alpha _ h ^ g \\cdot \\alpha ^ k _ l = \\alpha ^ a _ b , \\hbox { w h e r e } a = ( h \\vee k ) \\cdot h ^ { - 1 } \\cdot g \\hbox { a n d } b = ( h \\vee k ) \\cdot k ^ { - 1 } \\cdot l , \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} \\lambda _ n ^ { ( 1 ) } = C _ \\beta \\ , n ^ { - \\frac { 2 } { \\beta + 2 } } \\int _ \\R V ( x ) \\ ; \\dd x + o \\left ( n ^ { - \\frac { 2 } { \\beta + 2 } } \\right ) , n \\to \\infty . \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} s = - \\frac { a } { 2 } - \\frac { n } { a H ^ 2 } . \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} Y ^ { ( m ) } _ k ( t ) = Y ^ { ( m ) } _ 1 ( t ) + Z _ { k - 1 } ^ { ( m ) } ( t ) + \\ldots + Z _ 1 ^ { ( m ) } ( t ) \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} \\sum _ { \\{ \\Delta : \\Omega \\} = ( 1 , p ) } c ( \\Omega ^ { p ^ { r - 2 } m } ) & = ( \\widetilde \\lambda _ 1 ( p ^ 2 ) \\kappa ( p ^ { r - 2 } ) - \\chi ( p ^ 2 ) p ^ { 2 k - 3 } \\eta ( p ^ { r - 2 } ) ) c ( \\Delta ^ m ) \\\\ & \\quad - \\chi ( p ) p ^ { k - 2 } \\alpha ( \\Delta ^ { p ^ { r - 2 } } ; p ) \\kappa ( p ^ { r - 2 } ) c ( \\Delta ^ m ) \\\\ & = \\eta ( p ^ r ) c ( \\Delta ^ m ) . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { \\infty } \\sum _ { e _ { 1 } \\cdots e _ j } ( \\alpha _ 1 ( e _ { 1 } \\cdots e _ j ) ) ^ { - ( q - 1 ) s } ( \\mu [ e _ { 1 } \\cdots e _ j ] ) ^ q & = \\sum _ { j = 1 } ^ { \\infty } \\sum _ { e _ { 1 } \\cdots e _ j } \\phi ^ s ( e _ 1 \\cdots e _ j ) ^ { 1 - q } ( \\mu [ e _ { 1 } \\cdots e _ j ] ) ^ q \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { 1 - F _ \\rho ( k ) } { 1 - F ( k ) } = \\frac { \\mu _ \\rho } { \\mu } . \\end{align*}"}
-{"id": "4499.png", "formula": "\\begin{align*} f _ 0 = \\langle \\partial _ x ^ { - 1 } u _ 0 , u _ 0 \\rangle _ { L ^ 2 } = \\frac { 1 } { 2 } \\| u _ 0 \\| _ { L ^ 1 } ^ 2 = \\sqrt { 2 \\pi } . \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\left ( \\left [ - \\left ( \\frac { 1 } { 2 } - 8 \\varepsilon \\right ) , \\frac { 1 } { 2 } - 8 \\varepsilon \\right ] \\times \\left [ 0 , 8 \\varepsilon \\right ] \\right ) / \\sim \\ ; . \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} ( u _ { 0 , 0 } - u _ { 1 , 1 } ) ( u _ { 1 , 0 } - u _ { 0 , 1 } ) - a ( m ) + b ( n ) = 0 . \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} h ^ { 0 } ( L _ { 2 } ^ { \\vee } \\otimes L _ { 1 } ) - h ^ { 1 } ( L _ { 2 } ^ { \\vee } \\otimes L _ { 1 } ) + h ^ { 2 } ( L _ { 2 } ^ { \\vee } \\otimes L _ { 1 } ) - h ^ { 3 } ( L _ { 2 } ^ { \\vee } \\otimes L _ { 1 } ) = - 8 \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} H ( P ) = \\prod _ v \\max \\{ \\| x _ 0 \\| _ v , \\dotsc , \\| x _ n \\| _ v \\} , \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} \\partial _ { t } u ( t , x ) = a ^ { i j } ( t ) D _ { i j } u ( t , x ) + f ( t , x ) \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} \\begin{array} { l l l l } L ( \\pi ) = & \\max \\ & c ' v - \\pi ( H v - h ) & \\\\ & \\mbox { s . t . } \\ & A v \\leq b , & \\\\ & & v \\in \\{ 0 , 1 \\} ^ n & \\end{array} \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} \\bold { D } = D _ t - D _ x ^ 3 , \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} \\bar { R } ^ { m _ j } _ j ( \\boldsymbol { x } ) = \\sum _ { k = 1 } ^ { m _ j } ( \\sigma ^ { ( j ) } _ k ) ^ { - 1 } ( f _ { | \\Omega _ j } , \\bar { u } ^ { ( j ) } _ k ) _ { \\ell _ 2 ( X _ { N _ j } ) } \\bar { u } ^ { ( j ) } _ k ( \\boldsymbol { x } ) , \\boldsymbol { x } \\in \\Omega _ j . \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} v ( s - ( \\epsilon - u b ) ' ) & = v ( s - \\epsilon - u b ' - u ' b ) \\\\ & = v ( u ' b ) \\\\ & = v ( u ' b ' ( b ^ { \\dagger } ) ^ { - 1 } ) \\\\ & = v ( u ' ) - \\psi \\textstyle \\int \\gamma + \\gamma \\\\ & > s ^ 2 \\gamma - s \\gamma + \\gamma \\\\ & = - \\textstyle \\int s \\gamma + \\gamma . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} T = \\begin{pmatrix} a & \\alpha \\\\ \\beta & b \\end{pmatrix} \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\{ f g , h \\} = & \\mathrm { d } \\eta ( f a _ g + g a _ f , a _ h ) \\\\ = & f \\mathrm { d } \\eta ( a _ g , a _ h ) + g \\mathrm { d } \\eta ( a _ f , a _ h ) \\\\ = & f \\{ g , h \\} + g \\{ f , h \\} , \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} x ( t ) = \\lambda \\int _ 0 ^ 1 k ( t , s ) f ( s , x ( s ) ) \\textup d s , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} \\lambda _ j ( u ) = \\lambda ( u ) \\sum _ { q \\mid u } \\Lambda _ j ^ * ( q ) \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} F ( 0 , r , p ) = 2 4 \\left ( 1 - r ^ 2 \\right ) . \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} u ^ { n + 1 } ( t , x ) - u ^ n ( t , x ) = \\int _ 0 ^ t \\int _ { \\R } G _ { t - \\theta } ( x - \\eta ) \\left ( b ( u ^ n ( \\theta , \\eta ) ) - b ( u ^ { n - 1 } ( \\theta , \\eta ) ) \\right ) d \\eta d \\theta . \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} L _ { 2 4 } = \\left \\{ \\frac { 1 } { \\sqrt { 2 } } x : x \\in \\mathbb { Z } ^ { 2 4 } , \\ , x 2 \\in \\mathcal { G } _ { 2 4 } \\right \\} . \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} - \\frac { v ' } { u } + v _ 1 - v _ { - 1 } = 0 . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} R _ { j , - p _ j } \\left ( f _ s \\big | _ { \\{ - p ; s \\} , \\gamma } ( z ) \\right ) = ( R _ { j , - p _ j } f _ s ) \\Big | _ { \\{ - p + 2 e _ j ; s \\} , \\gamma } ( z ) = \\left ( s - \\frac { p _ j } { 2 } \\right ) \\cdot f _ s \\big | _ { \\{ - p + 2 e _ j ; s \\} , \\gamma } ( z ) , \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } _ b = 2 ^ { s _ b ( n ) } . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} ( u , v ) = 1 , \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ r _ { p , \\infty } } : = \\sup _ { m \\in \\mathbb { N } } 2 ^ { m r } \\Vert \\sum _ { 2 ^ m \\leq \\langle \\xi \\rangle < 2 ^ { m + 1 } } d _ { \\xi } [ \\xi ( x ) \\widehat { f } ( \\xi ) ] \\Vert _ { L ^ p ( G ) } < \\infty . \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} j _ 1 + \\dots + j _ 5 = 0 , j _ 1 ^ 3 + \\dots + j _ 5 ^ 3 = 0 , \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} \\chi _ k = e ^ { \\mathrm { i } k x } + O ( \\varepsilon ) . \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { n ( n + 1 ) / 2 } } { ( 1 + q ^ n ) ( q ; q ) _ n } = \\sum _ { n \\geq 1 } ( - 1 ) ^ n q ^ { n ^ 2 } . \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} & \\tilde { E } ^ { \\nu } _ i ( x ) \\\\ & = ( - 1 ) ^ \\nu \\int _ { S } \\left \\{ e _ { i j k } \\frac { \\partial \\Gamma ^ { \\nu } ( x , y ) } { \\partial x _ j } m _ k ( y ) - i \\omega \\mu _ { \\nu } \\left ( \\delta _ { i p } + \\frac { 1 } { k ^ 2 _ { \\nu } } \\frac { \\partial } { \\partial y _ i } \\frac { \\partial } { \\partial y _ p } \\right ) \\Gamma ^ { \\nu } ( x , y ) j _ p ( y ) \\right \\} \\ , d S _ y \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} s - r ( 2 \\ell - 5 ) = ( \\tilde b _ { \\ell - \\frac { 3 } { 2 } } + 3 \\tilde b _ { \\ell - 1 } + \\tilde b _ { \\ell - 2 } ) = \\xi _ { \\ell - 1 } + \\xi _ \\ell + z \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} \\| f \\| _ { 1 + \\gamma \\cdot k } ^ 2 = \\left | \\xi ( f ) \\right | ^ 2 + \\frac { 1 } { \\gamma } \\| f - \\xi ( f ) \\| _ { 1 + k _ 1 } ^ 2 = \\| f \\| ^ 2 _ 1 + \\frac { 1 } { \\gamma } \\| f \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} R _ \\mathsf { C H V } \\left ( p _ { V , X | S } \\right ) = I ( V ; Y ) - \\max \\Big \\{ I ( V ; Z ) , I ( V ; S ) \\Big \\} . \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} \\nabla _ { S _ 1 } S _ 2 = \\frac { 1 } { 2 } [ S _ 1 , S _ 2 ] \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} \\bold { E } _ { \\alpha } : = \\sum _ { J _ 1 , J _ 2 } ( - D ) _ { J _ 1 } S _ { - J _ 2 } \\frac { \\partial } { \\partial u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } . \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} a ^ { 2 } - b ^ { 2 } c \\left ( c + 2 \\right ) = \\frac { 2 } { c + 2 } , a ^ { 2 } - e ^ { 2 } c \\left ( c - 2 \\right ) = - \\frac { 2 } { c - 2 } \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} g ( t ) = 1 + \\abs { t } ^ { q - 1 } . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{align*} Z ^ * _ \\psi ( m , n ; 3 / 4 ) = \\lim _ { X \\to \\infty } & \\sum _ { [ Q ] \\in \\mathcal Q _ { N ; \\hat h , \\hat v } ( m n ) / \\Gamma _ 0 ( N ; \\hat h , m \\hat v ) } \\frac { \\chi _ m ( \\varphi _ { \\hat h , m \\hat v } Q ) } { \\omega _ Q } \\\\ & \\sum _ { \\substack { r , N s \\in Z , \\\\ ( r , N s / h ) = 1 \\\\ 0 < c ( Q , r , N s ) < X } } \\frac { 1 } { 4 N c ( Q , r , N s ) } \\exp \\left ( 2 \\pi i \\frac { \\beta } { 2 N c ( Q , r , N s ) } \\right ) . \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} 4 T ^ 0 ( \\xi _ s , I _ s X , Y ) = T ^ 0 ( X , Y ) - T ^ 0 ( I _ s X , I _ s Y ) , \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} \\textstyle { \\sum _ p } \\ , \\Diamond ( p , q ) \\ = \\ h ^ { n - q , q } \\ , . \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} \\nu _ { i } \\left ( a _ { i } , a _ { - i } \\right ) = \\limsup _ { T \\rightarrow \\infty } \\frac { 1 } { T } \\sum _ { t = 1 } ^ T \\omega _ { i } ^ { ( t ) } \\left ( a _ { i } , H _ { i } ( t ) \\right ) . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} | b ^ * ( \\psi _ m , \\psi _ n ) | = | \\overline { b ( \\psi _ n , \\psi _ m ) } | \\leq \\frac { M _ b } { m ^ \\alpha n ^ \\alpha } . \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} H _ n ^ { ( 1 ) } = H _ { n } ^ { ( 2 ) } - W _ n , \\ \\ \\ W _ n = V _ { n } ^ { ( 1 ) } - V _ { n } ^ { ( 2 ) } . \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} ( p , q ) & \\in \\{ ( 0 , \\pm 1 ) , ( \\pm 1 , 0 ) \\} \\mu = 1 , \\\\ ( p , q ) & \\in \\{ ( 0 , \\pm \\omega ) , ( \\pm \\omega , 0 ) \\} \\mu = \\omega , \\\\ ( p , q ) & \\in \\{ ( 0 , \\pm \\omega ^ { 2 } ) , ( \\pm \\omega ^ { 2 } , 0 ) \\} \\mu = \\omega ^ { 2 } . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { d } { d s } H ( \\gamma ( s ) | e ^ { - \\Psi } ) = \\int _ { \\mathbb { S } ^ { 1 } } ( 1 + \\log ( 1 + s f ) ) f e ^ { - \\Psi } d \\omega , \\\\ & \\frac { d ^ 2 } { d s ^ 2 } H ( \\gamma ( s ) | e ^ { - \\Psi } ) = \\int _ { \\mathbb { S } ^ { 1 } } \\frac { f ^ 2 } { 1 + s f } e ^ { - \\Psi } d \\omega , \\\\ & \\frac { d ^ 3 } { d s ^ 3 } H ( \\gamma ( s ) | e ^ { - \\Psi } ) = - \\int _ { \\mathbb { S } ^ { 1 } } \\frac { f ^ 3 } { ( 1 + s f ) ^ 2 } e ^ { - \\Psi } d \\omega . \\end{aligned} \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} K _ 2 ( \\xi , \\eta ) = \\frac 1 2 \\tanh \\xi \\tanh \\eta \\left ( I ( \\xi , \\eta ) + I ( - \\xi , - \\eta ) \\right ) + \\tanh \\xi \\tanh \\eta \\ , K _ 3 ( \\xi , \\eta ) , \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} \\theta = \\underline { \\theta } , \\ ; \\phi = \\underline { \\phi } , \\ ; \\hbox { o n $ \\mathcal { S } $ } , \\ ; \\textbf { f } = \\textbf { \\underline { f } } \\hbox { o n $ \\mathcal { S } \\times \\mathbb { R } ^ n $ } . \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} Y ( M ) = \\sup _ { [ g ] } Y ( M , [ g ] ) . \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} \\tilde { A } _ { | c | , E } ( \\theta , t ) = \\frac { 1 } { \\sqrt { | c | ( \\theta ) | c | ( \\theta - t \\alpha ) } } \\left ( \\begin{matrix} t ( E - v ( \\theta ) ) \\ \\ & - | c | ( \\theta - t \\alpha ) \\\\ | c | ( \\theta ) \\ \\ & 0 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} \\int _ { x _ \\frac \\mu 2 } ^ { x _ \\mu - \\delta } | w u | ^ q \\ , \\dd x \\leq C w ( x _ \\mu ) ^ q a _ \\mu ^ { - 1 } \\begin{cases} \\mu ^ { 1 - \\frac q 4 } , & 1 \\leq q < 4 , \\\\ [ 1 m m ] \\log ( \\mu a _ \\mu ^ { - \\frac 2 3 } ) , & q = 4 , \\\\ [ 1 m m ] a _ \\mu ^ { \\frac 2 3 - \\frac { q } { 6 } } , & q > 4 . \\end{cases} \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} F _ N ( u ) = ( X _ N ( u ) , Y _ N ( u ) , Z _ N ( u ) ) : = F ( u , u ^ N ) = \\bigl ( X ( u , u ^ N ) , Y ( u , u ^ N ) , Z ( u , u ^ N ) \\bigr ) \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} Q ( x , q ; z ) = \\left ( \\sqrt { \\frac { x - 1 } { x - e ^ { 2 z ( x - 1 ) } } } \\right ) ^ q . \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} { \\overline { R } } _ { s , \\mathrm { U L A } } ^ L = \\left [ { \\overline { R } } ^ { \\mathrm { L } } _ \\mathrm { U L A } - { \\overline { R } } _ { e ^ { * } } ^ { \\mathrm { U L A } } \\right ] ^ + . \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} v ( n ) = ( 1 - \\epsilon ) n C , \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } < P _ { k } x ^ { * } _ { n } , \\pi _ { k } x _ { n } > = 0 , k = 1 , 2 , . . . \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} h e _ i = y _ i ^ { ( 1 ) } , \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} 0 = & ( \\nabla _ S \\omega ) ( T , Z ) \\\\ = & \\omega ( \\nabla _ S T , Z ) + \\omega ( T , \\nabla _ S Z ) \\\\ = & \\omega ( \\frac { 1 } { 2 } [ S , T ] , Z ) + \\omega ( T , \\frac { 1 } { 2 } [ S , Z ] ) \\\\ = & \\omega ( T , \\frac { 1 } { 2 } [ S , Z ] ) . \\\\ \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{gather*} \\delta _ Q C ^ \\ast = d A ^ \\ast , \\delta _ Q A ^ \\ast = d \\tilde { F } , \\delta _ Q A = d C , \\delta _ Q C = 0 . \\end{gather*}"}
-{"id": "326.png", "formula": "\\begin{align*} u _ { x , n } = \\left \\{ \\begin{array} { c c } 1 + \\max \\{ i p ^ { n - v ( c _ { x , i } ) - 1 } : ( i , p ) = 1 \\textrm { s . t . } v ( c _ { x , i } ) < n \\} & \\mathrm { i f \\ } \\exists i : v ( c _ { x , i } ) < n \\\\ 0 & \\mathrm { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} X _ N ^ { R , { \\rm r e s c } } ( T ) = \\frac { X _ N ^ R ( 2 ^ { - 1 / 6 } N ^ { 2 / 3 } T ) - N / \\sqrt { 2 } } { 2 ^ { - 5 / 6 } N ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} Q ' ( t ) = \\big ( Q ( t ) \\big ) ^ 2 + ( x - 1 ) \\qquad Q ( 0 ) = 1 . \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\gamma _ j < \\infty , \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} \\leq b \\left ( \\frac { n - 1 } { n - k } \\right ) \\cdot \\ell . \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} F _ { 0 } ^ { \\left ( 3 \\right ) } = F _ { 1 } ^ { \\left ( 3 \\right ) } = 1 , \\thinspace \\thinspace F _ { 2 } ^ { \\left ( 3 \\right ) } = 2 \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} U _ { 2 ^ { p _ 0 } + \\cdots + 2 ^ { p _ s } } = 2 ^ { p _ 0 - p _ 1 } U _ { 2 + 2 ^ { p _ 2 - p _ 1 } + \\cdots + 2 ^ { p _ s - p _ 1 } } + 4 ( 1 - { 2 ^ { p _ 0 - p _ 1 } } ) U _ { 2 ^ { p _ 1 } + \\cdots + 2 ^ { p _ s } } + 1 - 2 ^ { p _ 0 - p _ 1 } . \\\\ \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} N _ i = n ^ { \\rm i m p } _ i + L \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\lambda \\ , \\rho _ { \\infty } ( \\lambda ) , D _ i = d ^ { \\rm i m p } _ i + L \\left \\{ \\int _ { - \\infty } ^ { \\lambda _ { i L } } - \\int _ { \\lambda _ { i R } } ^ { \\infty } \\right \\} d \\lambda \\ , \\rho _ { \\infty } ( \\lambda ) \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} \\sum _ f ( n ^ K ( \\partial f ) + 1 ) = n ^ K ( C ' ) + n ^ K ( - C '' ) + \\ell + E _ + F \\ , . \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} c ( \\xi ) : = M _ { \\varphi , x } [ \\alpha _ { 1 , 2 } + \\alpha _ { 1 , 1 } \\ , ( \\beta _ 1 ) _ x ] . \\end{align*}"}
-{"id": "5947.png", "formula": "\\begin{align*} b _ { n } = - q ^ { 2 j _ { n } - 1 } a _ { n } , \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} \\dot { v } _ j + \\mu _ j ^ { \\infty } v _ j = f _ j ( \\omega t ) , j \\in S ^ c \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} z _ 1 w _ 2 w _ 1 z _ 1 ^ { - 1 } = w _ 2 ( x ^ { - 1 } , y ^ { - 1 } ) ^ { - 1 } w _ 1 ( x ^ { - 1 } , y ^ { - 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} [ j ^ k n ^ { - h } ] \\ , \\ln \\biggl ( 1 + H _ j \\biggr ) = [ j ^ k n ^ { - h } ] \\ , \\ln \\biggl ( 1 + \\sum _ { s = 1 } ^ { j - 1 } \\frac { a _ s ( r , j ) } { n ^ s } \\biggr ) & = 0 , k \\ge h + 2 \\\\ [ j ^ { h + 1 } n ^ { - h } ] \\ , \\ln \\biggl ( 1 + H _ j \\biggr ) = [ j ^ { h + 1 } n ^ { - h } ] \\ , \\ln \\biggl ( 1 + \\sum _ { s = 1 } ^ { j - 1 } \\frac { a _ s ( r , j ) } { n ^ s } \\biggr ) & = \\frac { 1 } { ( h + 1 ) h } \\biggl ( \\frac { 1 } { r ^ h } - 2 \\biggr ) \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} y _ { \\delta } : = y _ 0 + [ \\partial _ { \\varphi } \\theta _ 0 ( \\varphi ) ] ^ { - T } \\rho ( \\varphi ) , \\rho _ j ( \\varphi ) : = \\Delta ^ { - 1 } _ { \\varphi } \\sum _ { k = 1 } ^ { \\nu } \\partial _ { \\varphi _ j } A _ { k \\ , j } ( \\varphi ) , \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} 0 = - \\frac { \\phi } { u } ( u _ 1 - u _ { - 1 } ) + \\frac { \\phi _ t } { u } + \\phi _ u ( u _ 1 - u _ { - 1 } ) - \\xi _ t ( u _ 1 - u _ { - 1 } ) - S \\phi + S _ { - 1 } \\phi . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} D \\alpha ( h ) = \\int \\gamma ' ( h , x ) ^ 2 e ^ { \\Gamma ( h , x ) ( z ) } F ' ( \\dd x ) - \\int \\left ( D \\gamma ' ( h , x ) \\right ) \\left ( e ^ { \\Gamma ( h , x ) ( z ) } - 1 \\right ) F ' ( \\dd x ) . \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} \\bar \\nabla ^ { H } _ { \\frac { \\partial } { \\partial t } } D ( \\pi \\circ \\Phi ) \\ ; \\big | _ 0 \\ w = \\pi ^ * \\bar \\nabla ^ { \\pi ^ { - 1 } T Y } _ w ( D \\pi v ) \\ ; . \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} M _ \\theta ( x ) = \\left ( \\begin{array} { c c } u ( x ) & u ( - x ) \\\\ \\mathrm { e } ^ { - 2 \\pi i \\theta } u ( x - \\alpha ) & \\mathrm { e } ^ { 2 \\pi i \\theta } u ( - ( x - \\alpha ) ) \\end{array} \\right ) ~ \\mbox { , } \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} \\sqrt { n } \\tilde { \\phi } = \\frac { 1 } { \\sqrt { n } } \\sum _ { \\ell = 1 } ^ { L } \\sum _ { i \\in I _ { \\ell } } \\alpha _ { 0 } \\left ( X _ { i } \\right ) [ Y _ { i } - \\hat { \\gamma } _ { \\ell } ( X _ { i } ) ] \\overset { p } { \\longrightarrow } 0 . \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} Q _ n ^ { \\alpha } ( n , u _ n ) = \\frac { \\operatorname { d } \\ ! \\widetilde { u } _ n ^ { \\alpha } } { \\operatorname { d } \\ ! \\varepsilon } \\Big | _ { \\varepsilon = e } . \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} a _ k = \\mathcal { F } _ n ^ \\ast ( D _ { \\phi ( i ) } ) \\sum _ { t = 1 } ^ { i - 1 } D _ { \\phi ( t ) } < k \\le \\sum _ { t = 1 } ^ i D _ { \\phi ( t ) } , \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} P _ { X , Y | V , S } ( x , y | v , s ) = P _ { X | V , S } ( x | v , s ) p _ { Y | X } ( y | x ) , \\quad \\forall ( v , s , x , y ) \\in \\mathcal { V } \\times \\mathcal { S } \\times \\mathcal { X } \\times \\mathcal { Y } . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} F _ { n + 2 } = F _ { n + 1 } + F _ { n } . \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} \\Delta _ { \\Phi _ \\ell ^ * g } ( \\Phi _ \\ell ^ * u ) = \\Phi _ \\ell ^ * \\bar { f } + f _ \\ell = \\Phi _ \\ell ^ * \\bar { f } + \\Delta _ { g _ { C _ \\ell } } \\left ( \\frac { f _ \\ell } { 2 m } r ^ 2 \\right ) . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} \\big ( \\rho ( a _ f ) + \\pi ^ * ( d f ) \\big ) . g = & d g ( a _ f ) + d g ( \\pi ^ * ( d f ) ) \\\\ = & \\omega ( a _ g , a _ f ) + \\pi ( d f , d g ) \\\\ = & \\{ g , f \\} - \\{ g , f \\} \\\\ = & 0 , \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} d d ^ * ( \\kappa d r ) & = - \\kappa '' d r + \\left ( \\frac { m - 1 } { r ^ 2 } \\kappa - \\frac { m - 1 } { r } \\kappa ' \\right ) d r - \\tilde { d } \\kappa ' - \\frac { m - 1 } { r } \\tilde { d } \\kappa , \\\\ d ^ * d ( \\kappa d r ) & = \\tilde { d } \\kappa ' + \\frac { 1 } { r ^ 2 } ( \\tilde { d } ^ * \\tilde { d } \\kappa ) d r + \\frac { m - 3 } { r } \\tilde { d } \\kappa . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} a * b = \\begin{cases} 0 & r _ b \\in M \\\\ r _ a ( \\phi _ a \\phi _ b ) = a \\phi _ b \\ , & r _ b \\not \\in M \\end{cases} \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } w _ 0 ^ \\varepsilon ( x ) & = \\gamma - a \\ \\ \\ x \\in \\overline { \\Omega ( \\delta ) } , \\\\ \\lim _ { \\varepsilon \\to 0 } w _ i ^ \\varepsilon ( x ) & = v _ i \\circ H ( x ) - a \\ \\ \\ x \\in \\overline { \\Omega } _ i \\setminus \\Omega _ i ( \\delta _ i / 2 ) , \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} \\P ( | \\xi | \\ge t ) = O ( \\exp ( - t ^ 2 / K ) ) . \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} \\psi ( - \\gamma ) \\ = \\ \\gamma ^ { \\dagger } < s \\gamma ^ { \\dagger } \\ = \\ \\psi ( \\gamma ^ { \\dagger } - s \\gamma ^ { \\dagger } ) , \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} ( k , d _ 2 ) = ( 3 , 3 ) , ( 3 , 4 ) , ( 3 , 5 ) , ( 4 , 3 ) , ( 5 , 3 ) . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} \\begin{array} { l l l l } Z ( V ) = & \\max \\ & c ' v ^ o + c ' V \\lambda + d ' u & \\\\ & \\mbox { s . t . } \\ & H V \\lambda + G u \\leq h - H v ^ o & ( \\pi ) \\end{array} \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} S ^ { s h o g u n } & \\le S ( 3 ; 0 , 0 ) ^ 8 S ( 3 ; 2 , 2 ) ^ 3 \\\\ & \\le S ( 3 ; 0 , 0 ) ^ 8 S _ U ( 3 ; 2 , 2 ) ^ 3 = : S _ U ^ { s h o g u n } \\\\ & \\approx ( 6 . 6 7 0 9 \\times 1 0 ^ { 2 1 } ) ^ 8 \\times ( 1 . 5 9 7 6 \\times 1 0 ^ { 1 1 } ) ^ 3 \\\\ & \\approx 1 . 5 9 9 3 \\times 1 0 ^ { 2 0 8 } . \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} & \\frac { \\partial a ^ k _ j } { \\partial \\eta ^ \\sigma _ k } = - \\gamma ^ k _ j \\beta ^ k _ j ( y , 0 , \\widehat { \\eta } ) = 0 ; \\\\ & \\frac { \\partial \\beta ^ k _ \\rho } { \\partial \\eta ^ \\sigma _ k } = - g ^ k _ \\rho a ^ k _ \\rho ( y , 0 , \\widehat { \\eta } ) = 0 , \\\\ \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} q _ { s j } = \\frac { 1 } { \\sqrt { 2 } } ( \\widetilde { q } _ { | s - j | } - \\widetilde { q } _ { s + j } ) , s , j \\in \\mathbb { N } . \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} \\Phi _ { m + n } = \\tilde { \\Phi } _ { m + n } = \\sigma _ { m + n } , \\Phi ' _ { m + n } = \\tilde { \\Phi } ' _ { m + n } = \\sigma ' _ { m + n } . \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} \\sigma \\wp \\alpha = \\sigma ( F \\alpha - \\alpha ) = F ( \\sigma \\alpha ) - \\sigma \\alpha = F ( \\alpha + 1 ) - \\alpha + 1 = F \\alpha - \\alpha = \\wp \\alpha . \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} & \\alpha ( q ' _ { j , i } , q ' _ { j + 1 , i } ) - \\alpha ( q ' _ { j , i } , q ' _ { j + 2 , i - 1 } ) - \\alpha ( q ' _ { j + 1 , i - 1 } , q ' _ { j + 1 , i } ) + \\alpha ( q ' _ { j + 1 , i - 1 } , q ' _ { j + 2 , i - 1 } ) \\\\ & = \\alpha ( q _ { j , i } , q _ { j + 1 , i } ) - \\alpha ( q _ { j , i } , q _ { j + 2 , i - 1 } ) - \\alpha ( q _ { j + 1 , i - 1 } , q _ { j + 1 , i } ) + \\alpha ( q _ { j + 1 , i - 1 } , q _ { j + 2 , i - 1 } ) \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} a _ { \\hbar , \\kappa , K } ( \\tau ) = 1 - \\hbar ^ 2 \\kappa \\tau + \\hbar ^ 4 K \\tau ^ 2 \\in ( - 1 / 2 , 1 / 2 ) \\ , . \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} \\| f \\| _ { 1 + k _ { \\gamma , } } ^ 2 = \\int ^ 1 _ 0 \\left | f ( y ) \\right | ^ 2 \\ , { \\rm d } y + \\frac { 1 } { \\gamma } \\left ( \\sum _ { \\nu = 1 } ^ r \\int ^ 1 _ 0 | f ^ { ( \\nu ) } ( y ) | ^ 2 \\ , { \\rm d } y \\right ) . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} a = - \\frac { \\tilde { c } _ 4 } { 3 } , b = - \\frac { \\tilde { c } _ 6 } { 2 7 } \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} \\Psi _ { t , \\Theta _ + } ( \\xi ) = ( 1 + \\| \\xi \\| ^ 2 ) ^ { t / 2 } \\varphi _ + \\left ( \\frac { \\xi } { \\| \\xi \\| } \\right ) \\quad \\mbox { a n d } \\Psi _ { v , \\Theta _ - } ( \\xi ) = ( 1 + \\| \\xi \\| ^ 2 ) ^ { v / 2 } \\varphi _ - \\biggl ( \\frac { \\xi } { \\| \\xi \\| } \\biggr ) \\ , . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} x ( n ) = \\sum \\limits ^ { + \\infty } _ { i = - \\infty } { \\sum \\limits ^ { K - 1 } _ { k = 0 } { \\sum \\limits ^ { M - 1 } _ { m = 0 } { d _ { i , k , m } g _ { k , m } ( n - i ( N + N _ { \\rm c p } ) ) } } } . \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} \\begin{aligned} \\log \\log ( f ^ n ) ^ \\# ( z ) & \\geq \\log x _ n - \\log \\log x _ n - \\log 2 \\\\ & = n \\log ( 1 + \\lambda + \\eta _ n ) - \\log n - \\log \\log ( 1 + \\lambda + \\eta _ n ) - \\log 2 , \\end{aligned} \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} F _ { \\boldsymbol { n _ m } } ( s ; \\chi _ 0 ) = \\sum _ { \\boldsymbol { n } \\in S ^ { ( k ) } _ m } a ^ { ( k ) } _ m ( \\boldsymbol { n } ) F ( \\boldsymbol { n } s ; \\chi _ 0 ) \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} t _ { \\max } \\geq \\frac { 1 } { \\| w \\| } \\left ( \\frac { a ( 2 - q ) ( \\sqrt { \\kappa _ \\alpha } S ( \\alpha , n ) ) ^ { 2 ^ * _ \\alpha } } { ( { 2 ^ * _ \\alpha } - q ) } \\right ) ^ { \\frac { 1 } { 2 ^ * _ \\alpha - 2 } } : = T _ 1 . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} ( k = 1 , \\ , \\ell > p ) , ( k , \\ , \\ell \\le p ) ( k > p , \\ , \\ell = 1 ) . \\end{align*}"}
-{"id": "3771.png", "formula": "\\begin{align*} d _ p = \\begin{cases} 0 & i f ~ i = 2 ^ e , \\\\ 2 & i f ~ i = 0 , \\\\ 3 & i f ~ i = 1 , \\\\ 4 & i f ~ 2 \\leq i \\leq 2 ^ { e - 1 } , \\\\ 3 \\cdot 2 ^ k & i f ~ i = 2 ^ e - 2 ^ { e - k } + 1 , ~ w h e r e ~ 1 \\leq k \\leq e - 2 , \\\\ 2 ^ { k + 2 } & i f ~ 2 ^ e - 2 ^ { e - k } + 2 \\leq i \\leq 2 ^ e - 2 ^ { e - k } + 2 ^ { e - k - 1 } , ~ w h e r e ~ 1 \\leq k \\leq e - 2 , \\\\ 2 ^ e & i f ~ i = 2 ^ e - 1 . \\end{cases} \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} \\gamma ( x , y ) = \\left | \\frac { \\mathrm { c m } ( x , x + y ) } { \\mathrm { c m } ( y , x + y ) } \\right | . \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} Q = \\frac { 1 } { \\sqrt { c - 2 } } ( c + \\sqrt { c ( c - 2 ) } ) ( c - 1 + \\sqrt { c ( c - 2 ) } ) ^ n , \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} \\infty > w ( t ) & \\ge c _ * M _ \\tau \\int _ { { \\bf R } ^ N } G ( x , t - \\tau + \\rho ^ \\theta ) G ( x , t ) \\ , d x \\\\ & \\qquad + \\int _ { { \\bf R } ^ N } \\int _ \\tau ^ t \\int _ { { \\bf R } ^ N } G ( x - y , t - s ) G ( x , t ) v ( y , s ) ^ p \\ , d y \\ , d s \\ , d x \\\\ & \\ge c _ * M _ \\tau G ( 0 , 2 t - \\tau + \\rho ^ \\theta ) + \\int _ { \\rho ^ \\theta } ^ t \\int _ { { \\bf R } ^ N } G ( y , 2 t - s ) v ( y , s ) ^ p \\ , d y \\ , d s \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} ( \\tau _ { x ^ 0 } \\cdot f ) ( x ) = f ( x - x ^ 0 ) , \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ s c _ k q _ k + l _ 1 \\xi _ 1 = 0 . \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} h = y + z \\quad \\mbox { a n d } y \\perp z . \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} - \\| \\omega \\| ^ 2 _ 1 \\int _ { 0 } ^ { 1 } [ ( \\alpha \\cdot x - t ) _ { + } e ^ { i \\| \\omega \\| _ 1 t } + ( - \\alpha \\cdot x - t ) _ { + } e ^ { - i \\| \\omega \\| _ 1 t } ] d t = & \\\\ e ^ { i \\omega \\cdot x } - i \\omega \\cdot x - 1 . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} D _ { i j } = \\dot { L } _ j ( \\tau _ i ) , \\ ; \\mbox { w h e r e } L _ j ( \\tau ) : = \\prod ^ { N } _ { \\substack { l = 0 \\\\ l \\neq j } } \\frac { \\tau - \\tau _ l } { \\tau _ j - \\tau _ l } , \\ ; 1 \\le i \\le N \\mbox { a n d } 0 \\le j \\le N . \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} H _ { \\Lambda , { B } } : = H _ { \\Lambda } + H _ { \\Lambda } ^ { { B } } \\ , \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} \\bold { E } _ { \\alpha } ( A \\cdot B ) = \\sum _ { \\beta , J _ 1 , J _ 2 } ( - D ) _ { J _ 1 } S _ { - J _ 2 } \\left ( \\frac { \\partial A ^ { \\beta } } { \\partial u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } B ^ { \\beta } + \\frac { \\partial B ^ { \\beta } } { \\partial u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } A ^ { \\beta } \\right ) . \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} \\tilde { c } _ k : = c _ k - b _ k = ( B ^ { - 1 } - \\mathrm { I } ) b _ k + ( \\rho ^ { - 1 } - 1 ) B ^ { - 1 } b _ k \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\delta _ L ( X _ i ) = \\sum _ { k = 1 } ^ 3 t _ { i k } \\otimes X _ k , \\delta _ R ( X _ i ) = \\sum _ { k = 1 } ^ 3 X _ k \\otimes t _ { k i } . \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} n = B k + k ^ \\prime , \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ 2 } { d x ^ 2 } - 2 ( x ^ 2 + 2 a x + b ) \\frac { d } { d x } + ( q - 4 a b - 2 ) x - b ^ 2 - 2 a \\right ) \\phi _ n ( x ) = E _ n \\phi _ n ( x ) \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} \\lambda _ 2 ( w ) = \\lambda ( w ) \\left ( \\sum _ { q \\mid u } \\Lambda _ 2 ^ * ( q ) + \\sum _ { r \\mid v } \\Lambda _ 2 ^ * ( r ) \\right ) \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} \\overline { \\chi } ( f , z ) = \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log ( f ^ n ) ^ \\# ( z ) \\quad \\underline { \\chi } ( f , z ) = \\liminf _ { n \\to \\infty } \\frac { 1 } { n } \\log ( f ^ n ) ^ \\# ( z ) \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ n ^ { ( 1 ) } & = \\frac 1 l \\int _ { - l } ^ l V ( x ) \\cos ^ 2 ( \\sqrt { \\mu _ n } ( x + l ) ) \\ , \\dd x \\\\ & = \\frac 1 { 2 l } \\int _ { - l } ^ l V ( x ) \\ , \\dd x + \\frac { 1 } { 2 l } \\int _ { - l } ^ l V ( x ) \\cos ( 2 \\sqrt { \\mu _ n } ( x + l ) ) \\ , \\dd x \\\\ & = \\frac 1 { 2 l } \\int _ { - l } ^ l V ( x ) \\ , \\dd x + o ( 1 ) , n \\to \\infty . \\end{aligned} \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} \\mathcal { V } _ { i } ^ { 1 } = \\left \\{ \\mathbf { z } \\in \\mathbb { R } ^ 2 | \\Vert \\mathbf { z } - \\mathbf { z } _ { i } \\Vert \\leq \\Vert \\mathbf { z } - \\mathbf { z } _ k \\Vert , ~ \\forall ~ \\mathbf { z } _ k \\in \\Phi _ { \\mathrm b } \\ ! \\setminus \\ ! \\mathbf { z } _ { i } \\right \\} . \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{gather*} I _ 1 = I _ 2 = 0 . \\end{gather*}"}
-{"id": "9253.png", "formula": "\\begin{align*} | \\Phi ( x _ n ) - \\Phi ( y _ n ) | = | \\Phi ( x _ n ) - \\Phi _ { \\varepsilon _ n } ( x _ n ) + \\Phi _ { \\varepsilon _ n } ( y _ n ) - \\Phi ( y _ n ) | = | ( \\Phi - \\Phi _ { \\varepsilon _ n } ) ( x _ n ) - ( \\Phi - \\Phi _ { \\varepsilon _ n } ) ( y _ n ) | . \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} \\sum _ { ( u , v ) \\in [ r ] \\times [ s ] } c \\big ( ( x _ 1 , x _ 2 , \\ldots , x _ t ) ; ( u , v ) \\big ) = 0 . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} L _ 1 ( \\Psi ) : = [ \\partial _ { \\psi } \\theta _ 0 ( \\psi ) ] ^ { - T } , L _ 2 ( \\psi ) : = [ ( \\partial _ { \\theta } \\tilde { z } _ 0 ) ( \\theta _ 0 ( \\psi ) ) ] ^ T \\partial _ x ^ { - 1 } . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} \\hat f ( - \\xi ) = \\bar { \\hat { f } } ( \\xi ) , \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} \\rho ( x , y ) : = \\frac { ( 1 - | x | ^ 2 ) _ + ( 1 - | y | ^ 2 ) _ + } { | x - y | ^ 2 } , k _ { N , s } : = \\frac { \\Gamma ( \\frac { N } { 2 } ) } { \\pi ^ \\frac { N } { 2 } 4 ^ s \\Gamma ( s ) ^ 2 } . \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} I \\mathcal { P } _ n = \\bigoplus _ { k = 0 } ^ n \\mathcal { H } ^ { \\otimes _ { \\mathbf a } k } . \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } \\left | a _ j \\right | < \\infty , \\sum _ { j = 0 } ^ { \\infty } ( j + 1 ) \\left | { a } _ j \\right | ^ { 2 } < \\infty . \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} C C { \\cal F } = ( - 1 ) ^ n { \\rm r a n k } \\ { \\cal F } \\cdot [ T ^ * _ X X ] \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} 2 \\int V \\mbox { R e } \\Big ( \\bar { u } ( r \\partial _ r - n / 2 ) ( \\delta A ^ 2 + 1 ) ^ { - 1 } u \\Big ) d x \\stackrel { \\delta \\downarrow 0 } { \\longrightarrow } 2 \\int V \\mbox { R e } \\Big ( \\bar { u } ( r \\partial _ r - n / 2 ) u \\Big ) d x = \\int V \\nabla \\cdot x ( | u | ^ 2 ) d x . \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} v ^ 2 & = \\mathfrak { P f } ( A , \\sigma ) ( v , v ) = \\mathfrak { b } _ K ( h ( v ) , h ( v ) ) = \\mathfrak { b } _ K ( u \\otimes 1 + w \\otimes \\sqrt \\alpha , u \\otimes 1 + w \\otimes \\sqrt \\alpha ) \\\\ & = \\mathfrak { b } ( u , u ) + \\sqrt { \\alpha } \\mathfrak { b } ( u , w ) + \\sqrt { \\alpha } \\mathfrak { b } ( w , u ) + \\alpha \\mathfrak { b } ( w , w ) = \\mathfrak { b } ( u , u ) + \\alpha \\mathfrak { b } ( w , w ) \\in F , \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} m _ M ( I _ o ^ { - 1 } J _ o ) = m _ M ( I _ o ) + m _ M ( J _ o ) - m _ M ( \\{ e _ M \\} ) . \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\mu _ p ( C _ 1 \\cap C _ 2 ) - \\mu _ p ( C _ 1 ) \\mu _ p ( C _ 2 ) & = \\frac { q ^ { 1 0 } } { Z _ p } - \\left ( \\frac { q ^ 4 + q ^ { 1 0 } } { Z _ p } \\right ) ^ 2 = \\frac { - q ^ 8 + q ^ { 1 0 } + q ^ { 1 4 } + 3 q ^ { 1 6 } } { Z _ p ^ 2 } < 0 \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{gather*} \\Psi _ 0 ( x , t ) = i \\Psi ^ { ( 6 ) } _ 0 ( x , t ) \\sigma _ 1 , \\sigma _ 1 = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "4135.png", "formula": "\\begin{align*} ( - 1 ) ^ i h _ i - ( - 1 ) ^ i H ( x _ n ) & = ( - 1 ) ^ i H ( Y ^ \\varepsilon ( \\tau ^ \\varepsilon ( x _ n ) , x _ n ) ) - ( - 1 ) ^ i H ( x _ n ) \\\\ & = \\nu \\int _ 0 ^ { \\tau ^ \\varepsilon ( x _ n ) } | D H ( Y ^ \\varepsilon ( t , x _ n ) ) | \\ , d t \\geq \\nu c _ 0 \\tau ^ \\varepsilon ( x _ n ) , \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} \\Delta _ { 2 ^ { p _ 0 } + 2 ^ { p _ 1 } } = 2 ^ { p _ 0 - p _ 1 } ( 2 ^ { p _ 1 - p _ 0 } - 2 ) ^ 2 \\ge 0 . \\\\ \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} \\underset { m \\to + \\infty } { \\lim \\inf } \\ ; m \\left ( \\lambda _ { 1 , m } - \\lambda _ { 1 , \\infty } \\right ) = \\lim _ { k \\to + \\infty } m _ k \\left ( \\lambda _ { 1 , m _ k } - \\lambda _ { 1 , \\infty } \\right ) \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\alpha ^ 2 + \\left ( \\beta + \\frac { 1 } { 2 } + \\frac { n } { H ^ 2 } \\right ) ^ 2 = \\left ( \\frac { n } { H ^ 2 } - \\frac { 1 } { 2 } \\right ) ^ 2 . \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} ( \\Theta _ { j \\bar k } ^ { T _ B } v , w ) _ H = ( \\Theta _ { j \\bar k } ^ { { \\rm E n d } ( H ) } \\theta _ v , \\theta _ w ) - \\left ( P ^ { \\bot } ( D ^ { { \\rm E n d } ( H ) } _ j \\theta _ v ) , P ^ { \\bot } ( D ^ { { \\rm E n d } ( H ) } _ k \\theta _ w ) \\right ) , \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} \\partial _ u [ \\nabla ( H \\circ \\Phi ) ] ( u ) [ h ] = ( \\partial _ u \\nabla H ) ( \\Phi ( u ) ) [ h ] + \\mathcal { R } ( u ) [ h ] , \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} ( \\bold { D } _ F ) _ { \\alpha \\beta } = \\sum _ J \\frac { \\partial F _ { \\alpha } } { \\partial u _ J ^ { \\beta } } D _ J . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} \\left \\| x + t \\frac { y _ k - x } { \\| y _ k - x \\| } - y _ k \\right \\| - \\| b - y _ k \\| & = \\| x - y _ k \\| \\left ( 1 - \\frac { t } { \\| x - y _ k \\| } \\right ) - \\| b - y _ k \\| \\\\ & = \\| x - y _ k \\| - \\| b - y _ k \\| - t \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} \\nabla ^ \\lambda : \\mathcal { X } ( M ) \\times \\Gamma ( L ) \\rightarrow \\Gamma ( L ) \\\\ \\nabla ^ \\lambda _ X T : = [ \\lambda ( X ) , T ] \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} \\mathcal { F } _ X ( k ) = F _ X ( k ) + F _ X ( k - 1 ) , k \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { \\varphi ( n ) } { n ^ s } = \\frac { 2 ^ s - 3 } { 2 ^ s - 1 } \\cdot \\frac { \\zeta ( s - 1 ) } { \\zeta ( s ) } ( \\Re s > 2 ) . \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} u ( t ) = \\int \\limits _ { 0 } ^ { t } Q ( t , \\tau ) u ( \\tau ) d \\tau + f ( t ) , 0 \\leq t \\leq 1 . \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 - \\cot ^ 2 ( \\alpha ) z ^ 2 = 0 . \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} E _ i ^ { ( k ) } ( \\mu ) & = \\sum _ { \\lambda } q ^ { N _ E ( \\lambda , \\mu ) } \\lambda , \\\\ F _ i ^ { ( k ) } ( \\lambda ) & = \\sum _ { \\mu } q ^ { N _ F ( \\lambda , \\mu ) } \\mu , \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\partial _ t \\Gamma + \\partial _ x ( u _ 2 \\Gamma - D \\partial _ x \\Gamma ) = 0 , \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} ( a _ 1 , a _ 2 , a _ 3 , a _ 4 ) = ( 1 , 1 , 1 , 2 ) , ( 1 , 1 , 2 , 2 ) , ( 1 , 2 , 2 , 2 ) , ( 1 , 1 , 1 , 5 ) , ( 1 , 1 , 5 , 5 ) , ( 1 , 5 , 5 , 5 ) \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} G ( 1 , \\Lambda _ N ) + G ( - 3 , \\Lambda _ N ) = \\left \\{ \\begin{matrix} 0 & \\textrm { i f } & N \\equiv 1 \\pmod 2 \\\\ 4 i e ^ { \\frac { 3 N } { 4 } \\pi i } & \\textrm { i f } & N \\equiv 0 \\pmod 2 . \\end{matrix} \\right . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} W = - ( 2 \\mathcal { P } _ 1 - I ) ( 2 \\mathcal { P } _ 2 - I ) , \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} I _ { 3 , r } = g _ 0 ^ { r + 1 } \\underbrace { \\int _ { 0 } ^ { \\infty } \\ln ( 1 + g _ 0 z ) z ^ r e ^ { - z } d z } _ { I _ { 4 , r } } \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} H ^ { \\alpha _ 0 } ( \\Omega ) \\cap W ^ \\gamma _ \\alpha ( L _ 2 ( \\Omega ) ) \\hookrightarrow B ^ s _ { \\tau } ( L _ \\tau ( \\Omega ) ) , \\frac { 1 } { \\tau } = \\frac { s } { d } + \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "290.png", "formula": "\\begin{align*} H / p H \\cong W ( K ) / ( \\wp W ( K ) + p W ( K ) ) = W ( K ) / ( V ^ 1 W ( K ) + \\wp W ( K ) ) \\cong K / \\wp K . \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} \\nu _ 1 ^ { ( a ) } = \\begin{cases} k + x _ 1 ^ { ( i + 1 ) } + \\cdots + x _ 1 ^ { ( n ) } & a < i , \\\\ x _ 1 ^ { ( a ) } + \\cdots + x _ 1 ^ { ( n ) } & a > i . \\end{cases} \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{gather*} P ( x ) = - \\frac { ( q _ 2 ^ 2 - 1 ) ( x + e _ 1 ) - 2 q _ 2 q _ 1 } { 4 } , Q ( x ) = \\frac { q _ 2 x ^ 2 - q _ 1 x + q _ 0 } { 2 } . \\end{gather*}"}
-{"id": "537.png", "formula": "\\begin{align*} ( \\bold { D } _ A ) _ { \\alpha \\beta } = \\sum _ { J _ 1 , J _ 2 } \\frac { \\partial A ^ { \\alpha } } { \\partial u _ { J _ 1 ; J _ 2 } ^ { \\beta } } D _ { J _ 1 } S _ { J _ 2 } . \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} r _ j = \\rho - ( 1 - \\alpha ) \\rho \\frac { 1 - 2 ^ { - j } } { 1 - 2 ^ { - ( N + 1 ) } } . \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\deg ( I d - H _ 0 , \\tilde \\Gamma , 0 ) = 1 . \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} ( \\beta _ 2 ^ t + i \\beta _ 1 ^ t ) ( \\beta _ 2 ^ t - i \\beta _ 1 ^ t ) ^ { - 1 } = \\beta _ 2 ^ t \\beta _ 2 - \\beta _ 1 ^ t \\beta _ 1 + 2 i ( \\beta _ 2 ^ t \\beta _ 1 ) , \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} z _ \\lambda - y _ \\lambda = \\frac { \\lambda } { 2 } [ ( q + s _ 2 + t _ 1 ) - ( q + s _ 1 + t _ 2 ) ] \\quad \\mbox { i s p a r a l l e l t o } S . \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\int _ { E _ { p + q } } \\prod _ { j = 1 } ^ { q } \\frac { d u _ j } { 1 - u _ j } \\prod _ { i = q + 1 } ^ { p + q } \\frac { d u _ i } { u _ i } \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} \\frac { u _ 1 - u _ { - 1 } } { 2 } + u u ' + u ''' = 0 . \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} \\frac { d } { d \\xi } F _ z ( 1 , ( 0 , 0 ) ) & = p \\| ( u , v ) \\| ^ p - q \\int _ { \\Omega } \\left ( \\lambda | u | ^ { q } + \\mu | v | ^ { q } \\right ) d x - 2 ( \\alpha + \\beta ) \\int _ { \\Omega } | u | ^ { \\alpha } | v | ^ { \\beta } d x \\\\ & = ( p - q ) \\| ( u , v ) \\| ^ p - 2 ( \\alpha + \\beta - q ) \\int _ { \\Omega } | u | ^ { \\alpha } | u | ^ { \\beta } d x \\not = 0 . \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} & \\sum _ { j = 1 , 2 } ( \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } K _ { j M } ( t ) g \\| _ { 2 } + \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } K _ { j H } ( t ) g \\| _ { 2 } ) \\\\ & \\le C e ^ { - c t } t ^ { - \\frac { n } { 2 ( 1 - \\sigma ) } ( \\frac { 1 } { r } - \\frac { 1 } { 2 } ) - \\frac { k - \\tilde { k } } { 2 ( 1 - \\sigma ) } } \\| \\nabla ^ { \\tilde { k } + 2 ( 1 - \\sigma ) \\ell } _ { x } g \\| _ { r } , \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} R _ { H _ 0 } ( z ) = ( H _ 0 - z ) ^ { - 1 } : \\mathcal H \\to D ( H _ 0 ) , R _ H ( z ) = ( H - z ) ^ { - 1 } : \\mathcal H \\to D ( H ) \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\lambda ^ r ( \\lambda ^ s ( x ) ) = P _ { r , s } ( \\lambda ^ 1 ( x ) , \\ldots , \\lambda ^ { r s } ( x ) ) \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} u _ t = \\delta u _ { x x } + g ( u ) \\sqrt { 1 + u _ x ^ 2 } , \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} \\widetilde { g } _ { 0 1 } { _ { | \\mathcal { I } ^ 0 } } = \\theta , \\ ; \\widetilde { g } _ { a b } { _ { | \\mathcal { I } ^ 0 } } = \\Theta _ { a b } , \\ ; \\Phi _ { | \\mathcal { I } ^ 0 } = \\phi , \\ ; \\rho _ { | \\mathcal { I } ^ 0 } = \\textbf { f } , \\ ; \\psi _ { \\mu \\nu } = \\frac { \\partial \\widetilde { g } _ { \\mu \\nu } } { \\partial y ^ 0 } , \\ ; \\partial _ \\mu = \\frac { \\partial } { \\partial y ^ \\mu } , \\ ; d \\widetilde { p } \\ ; ' = d \\widetilde { p } ^ 1 . . . d \\widetilde { p } ^ n , \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} \\sum _ { n } \\lambda ( n ) e \\left ( \\frac { b n } { c } \\right ) V \\left ( \\frac { n } { N } \\right ) = \\frac { N } { c } \\sum _ { n } \\lambda ( n ) e \\left ( - \\frac { \\bar { b } n } { c } \\right ) \\cdot 2 \\pi i ^ \\kappa \\int _ { 0 } ^ { \\infty } V ( x ) J _ { k - 1 } \\left ( \\frac { 4 \\pi \\sqrt { n N x } } { c } \\right ) \\mathrm { d } x \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} F _ B ( \\lambda | \\lambda ' ) = \\frac { \\theta ( \\lambda - \\lambda ' ) - \\pi } { 2 \\pi } + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) F _ B ( \\nu | \\lambda ' ) . \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} u _ t = u _ 0 + \\int _ { 0 } ^ { t } \\eta _ { s } d s + \\int _ { 0 } ^ { t } \\zeta _ { s } d B _ { s } + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } w _ { s } d \\langle B \\rangle _ { s } - \\int _ { 0 } ^ { t } G ( w _ { s } ) d s , \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { 2 q ^ n } { 1 + q ^ n } \\frac { ( - q ; q ) _ { n - 1 } } { ( q ; q ) _ { n - 1 } } = \\sum _ { \\pi \\in \\mathcal { O } } ( - 1 ) ^ { s ( \\pi ) + 1 } q ^ { | \\pi | } . \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} \\mathrm { p e r } A \\le \\prod _ { i = 1 } ^ n r _ i ! ^ { 1 / r _ i } . \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} \\tau _ { \\infty } = \\frac { ( - 1 ) ^ { \\mathsf { N } } \\kappa _ { + } \\kappa _ { - } e ^ { \\epsilon ( \\tau _ { - } - \\tau _ { + } ) } \\prod _ { b = 1 } ^ { \\mathsf { N } } \\alpha _ { b } \\beta _ { b } } { \\left ( \\zeta _ { + } - 1 / \\zeta _ { + } \\right ) \\left ( \\zeta _ { - } - 1 / \\zeta _ { - } \\right ) } \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} & \\mathcal { L } _ { \\omega } \\Phi = \\Phi ( \\mathcal { D } _ { \\omega } + b _ 3 \\partial _ { y y y } + b _ 2 \\partial _ { y y } + b _ 1 \\partial _ y + b _ 0 ) \\Pi _ S ^ { \\perp } + \\mathcal { R } _ { \\mathit { I I } } , \\\\ & \\mathcal { R } _ { \\mathit { I I } } : = \\{ \\Pi _ S ^ { \\perp } ( \\mathcal { A } - \\mathrm { I } ) \\Pi _ S - \\mathcal { R } _ { \\Phi } \\} ( b _ 3 \\partial _ { y y y } + b _ 2 \\partial _ { y y } + b _ 1 \\partial _ y + b _ 0 ) \\Pi _ S ^ { \\perp } + \\mathcal { R } _ { \\mathit { I } } . \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} d ( X ( s ) - \\bar X ( s ) ) = - e ^ { \\mu ( s - t ) } p ( s ) ( X - \\bar X ( s ) ) d s + \\nu X ( s ) d W ( s ) + \\nu _ 0 ( X - \\bar X ( s ) ) d W _ 0 ( s ) . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} ( \\Box ^ { ( q ) } _ { t , b , m } u | u ) _ t = \\| u \\| _ { \\overline S _ t } ^ 2 + q m \\| u \\| _ t ^ 2 + ( R ^ t _ \\ast u | u ) _ t \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{gather*} F _ { \\beta } ( t ) \\equiv E _ { \\beta } ^ { { \\rm s o f t } } \\bigl ( 0 ; ( t , \\infty ) \\bigr ) = \\lim _ { N \\rightarrow \\infty } E _ { \\beta N } ^ { { \\rm s o f t } } \\left ( 0 ; \\left ( \\sqrt { 2 N } + \\frac { t } { \\sqrt { 2 } N ^ { 1 / 6 } } , \\infty \\right ) \\right ) , \\end{gather*}"}
-{"id": "3330.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = A ( t ) x + c ( t ) + \\lambda f _ 1 ^ i ( t , x , y , \\lambda ) \\\\ \\dot y = \\lambda f _ 2 ^ i ( t , x , y , \\lambda ) , \\end{array} \\right . \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\limsup _ { B \\to \\infty } \\frac { \\# \\{ P \\in ( T \\setminus T ' ) \\mid H ( P ) \\le B \\} } { \\# \\{ P \\in T \\mid H ( P ) \\le B \\} } = 0 . \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\Lambda _ \\infty \\rightarrow \\Lambda _ N : \\ \\ p _ k \\rightarrow p _ k ^ { ( N ) } = \\sum _ { i = 1 } ^ N x _ i ^ k , \\ \\ p _ 0 \\rightarrow N . \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} \\omega ( \\xi ) = \\overline { \\omega } + \\mathbb { A } \\ , \\xi , \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} 2 \\ell \\le \\ell ( R ' ) \\le \\ell ( R '' ) \\le \\ell ( R ) = 5 \\ell . \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} \\{ \\varphi _ a ^ - ( x ) , \\mathcal { V } _ b ( y ) \\} = - \\delta _ { a , b } \\frac { \\mathcal { V } _ b ( y ) } { 1 - \\frac { y } { x } } , \\{ \\varphi _ a ^ + ( x ) , \\mathcal { V } _ b ( y ) \\} = 0 . \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} Q : = \\{ v \\in V \\colon \\langle v , w _ i - f ( w _ i ) \\rangle \\leq C i = 1 , \\ldots , m \\} , \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} J _ { \\lambda , \\mu } ( u , v ) & : = \\frac { 1 } { p } \\int _ Q \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { n + p s } } \\ , d x \\ , d y + \\frac { 1 } { p } \\int _ Q \\frac { | v ( x ) - v ( y ) | ^ p } { | x - y | ^ { n + p s } } \\ , d x \\ , d y \\\\ & - \\frac { 1 } { q } \\int _ \\Omega \\left ( \\lambda | u | ^ { q } + \\mu | v | ^ { q } \\right ) d x - \\frac { 2 } { \\alpha + \\beta } \\int _ \\Omega | u | ^ { \\alpha } | v | ^ \\beta d x . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{align*} M _ { a } ^ { s G } ( \\lambda ) = [ L _ { a , 2 \\mathsf { M } } ( \\frac { \\lambda q ^ { - 1 / 2 } } { \\xi _ { 2 \\mathsf { M } } } ) \\sigma _ { a } ^ { x } ] [ \\tilde { L } _ { a , 2 \\mathsf { M } - 1 } ( \\frac { \\lambda q ^ { - 1 / 2 } } { \\xi _ { 2 \\mathsf { M } - 1 } } ) \\sigma _ { a } ^ { x } ] \\cdots \\lbrack L _ { a , 2 } ( \\frac { \\lambda q ^ { - 1 / 2 } } { \\xi _ { 2 } } ) \\sigma _ { a } ^ { x } ] [ \\tilde { L } _ { a , 1 } ( \\frac { \\lambda q ^ { - 1 / 2 } } { \\xi _ { 1 } } ) \\sigma _ { a } ^ { x } ] \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\sup _ { \\mathbf { \\hat { x } } \\in \\Psi ( k _ 0 ) , \\ , \\mathbf { k } \\in \\Pi ( k _ 0 ) } \\ & \\ \\liminf _ { T \\to \\infty } \\ \\sum _ { t = 0 } ^ T \\beta _ t \\ , U _ t \\left ( \\hat { x } _ t \\right ) \\\\ [ 5 p t ] & \\hat { x } _ t + k _ { t + 1 } \\leq f ( k _ t ) , & t = 0 , 1 , \\ldots , \\\\ [ 5 p t ] & k _ 0 > 0 \\ . \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} E _ C ( \\theta ) = - \\frac { 1 } { \\theta } \\ln \\left ( \\textsf { E } \\left \\{ \\left ( 1 + \\frac { \\xi _ 0 } { M R _ c } \\Omega ^ { ( N ) } \\right ) ^ { - \\breve { \\theta } } \\right \\} \\right ) \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} A _ x = \\big ( A \\times Y ) _ { ( x , y ) } \\subset \\pi ^ { - 1 } ( I ) _ { ( x , y ) } = \\tau ^ { - 1 } ( I \\pi ( x , y ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} a x ^ 3 + b y ^ 3 + c z ^ 3 + d t ^ 3 = 0 , \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} \\mathcal { R } _ { \\mathsf { W P C } } ( \\gamma _ { 1 } ) = - \\int _ { 0 } ^ { \\infty } f ( u , \\gamma _ { 1 } ) \\ln \\left ( \\frac { f ( u , \\gamma _ { 1 } ) } { u } \\right ) u + \\ln \\left ( \\frac { 2 \\gamma _ { 1 } } { e } \\right ) , \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} \\bold { E } _ { \\alpha } ^ { \\vartriangle } ( L _ n ) = 0 , \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} \\Big | \\sum _ { \\widetilde \\alpha } ( - 1 ) ^ { \\widetilde { q } ( \\widetilde \\alpha ) } Z _ { \\widetilde \\alpha , \\widetilde { D } _ 0 } ( \\widetilde { G } ) \\Big | = \\Big ( \\sum _ { w _ 1 ( \\alpha ) = 0 } ( - 1 ) ^ { \\frac { q ( \\alpha ) } { 2 } } Z _ { \\alpha , D _ 0 } ( G ) \\Big ) ^ 2 + \\Big ( \\sum _ { w _ 1 ( \\alpha ) = 1 } ( - 1 ) ^ { \\frac { q ( \\alpha ) - 1 } { 2 } } Z _ { \\alpha , D _ 0 } ( G ) \\Big ) ^ 2 \\ , , \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} A = f _ 1 + _ F f _ 2 + _ F \\cdots + _ F f _ n = g _ 1 + _ F g _ 2 + _ F \\cdots + _ F g _ m \\textrm { f o r s o m e } f _ i , g _ i \\in F , n , m \\in \\mathbb { N } \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{gather*} \\partial _ I Q ^ a = 0 , | I | = k , \\end{gather*}"}
-{"id": "4218.png", "formula": "\\begin{align*} & \\log \\left [ \\left ( \\frac { f ( m ) - p } { f ( m ) - q } \\right ) ^ { h ( m ) } \\left ( \\frac { g ( m ) - q } { g ( m ) - p } \\right ) ^ { c \\ , g ( m ) } \\right ] \\\\ & = c \\ , g ( m ) \\ , \\log \\frac { ( f ( m ) - p ) \\ , ( g ( m ) - q ) } { ( f ( m ) - q ) \\ , ( g ( m ) - p ) } \\\\ & + ( h ( m ) - c \\ , g ( m ) ) \\ , \\log \\frac { f ( m ) - p } { f ( m ) - q } . \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\begin{aligned} \\Big ( \\tilde { \\Omega } ^ { ( 1 ) } _ { \\mathcal { P } } ( t _ * ) \\tilde { v } , \\tilde { v } \\Big ) _ { \\mathbb { C } ^ n } & = 2 \\Big ( ( X _ 1 ( t _ * ) ^ t Y _ 1 ' ( t _ * ) - Y _ 1 ^ t ( t _ * ) X _ 1 ' ( t _ * ) ) M _ 1 ( t _ * ) ^ 2 \\Big { \\{ } X _ 1 ( t _ * ) ^ t v _ 1 + Y _ 1 ( t _ * ) ^ t v _ 2 \\Big { \\} } , \\\\ & M _ 1 ( t _ * ) ^ 2 \\Big { \\{ } X _ 1 ( t _ * ) ^ t v _ 1 + Y _ 1 ( t _ * ) ^ t v _ 2 \\Big { \\} } \\Big ) _ { \\mathbb { C } ^ n } , \\end{aligned} \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{gather*} F \\big ( 3 ^ { - 1 / 3 } x , 3 ^ { - 2 / 3 } t ; \\beta = 6 \\big ) = ( 1 - a ^ 2 ) \\kappa u ^ { \\frac { 1 } { 2 } } \\left [ u ^ { - 1 } \\left ( \\frac { 1 + q _ 2 } { 2 } x - \\alpha \\right ) Y _ { 1 2 } ^ { ( 3 ) } ( x , t ) + Y ^ { ( 3 ) } _ { 2 2 } ( x , t ) \\right ] \\\\ \\qquad { } - a \\kappa u ^ { \\frac { 1 } { 2 } } e ^ { \\frac { x ^ 3 } { 3 } - x t } \\left [ u ^ { - 1 } \\left ( \\frac { 1 + q _ 2 } { 2 } x - \\alpha \\right ) Y _ { 1 1 } ^ { ( 3 ) } ( x , t ) + Y ^ { ( 3 ) } _ { 2 1 } ( x , t ) \\right ] . \\end{gather*}"}
-{"id": "8325.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ { j - 1 } Q ( B _ i B _ j ) - Q ( B _ i ) Q ( B _ j ) \\le M Q ( B _ j ) . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} \\tilde { V } ' ( z ) \\tilde { V } ( z ) F = F \\qquad \\forall \\ F \\in \\Lambda _ N . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} \\alpha _ \\beta ( \\gamma , \\nu ) = \\beta \\langle \\gamma , \\nu \\rangle - R ( ( \\gamma , \\nu ) | \\rho ) \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} D \\mid d , \\ ( D , w ) = 1 . \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} M _ { \\ell } ^ { - 1 } \\Big ( \\frac { 1 } { N } \\Big ) = \\left ( \\frac { 2 q } { q + 2 } \\right ) ^ { q / 2 } \\frac { 1 } { q + 2 } \\left ( \\frac { 2 N + e q \\ell e ^ { q / 2 } } { e N \\ell } \\right ) \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} \\mathcal { P } = \\bigotimes _ { i = 1 } ^ { K } \\mathcal { P } _ { i } , \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} \\dim _ F ( \\lambda + u ) A = \\frac { 1 } { 2 } \\dim _ F A . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} C _ b \\left ( P _ { \\pi ( i ) } , P _ { - \\pi ( i ) } , H _ { \\pi ( i ) } \\right ) \\triangleq \\frac { 1 } { 2 } \\log \\left ( 1 + \\frac { H _ { \\pi ( i ) } P _ { \\pi ( i ) } ( H _ { \\pi ( i ) } ) } { 1 + \\sum _ { j = i + 1 } ^ K H _ { \\pi ( j ) } P _ { \\pi ( j ) } ( H _ { \\pi ( j ) } ) } \\right ) . \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} ( k - 1 ) k ( k - l ) P _ { k - 1 } ^ { ( l ) } = ( k - 2 ) P _ k ^ { ( l + 2 ) } Q & + ( 2 k - l - 2 ) P _ k ^ { ( l + 1 ) } R \\\\ & + ( k + 1 ) ( 2 k - l - 2 ) ( 2 k - l - 1 ) P _ k ^ { ( l ) } . \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} { \\mathcal A } = \\left \\{ { \\mathbf u } \\in R ^ { 2 \\delta } _ { \\rm B } ( n ) \\right \\} \\cup \\left \\{ \\max _ { i = 1 , 2 , \\ldots , M } ~ \\sigma _ { \\rm j , i } ^ 2 > c \\right \\} \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} | r _ n ( i , j ) | \\leq n ^ { - 2 } \\sum _ { l , l ' = 1 } ^ 2 | \\partial _ { l l ' } R _ h ( \\psi _ { i j } ^ n ) | | i - j | > 2 , \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} ( D ^ 2 , u v ) = ( D , u v ) . \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} D _ { ( \\gamma , \\widetilde { \\gamma } ) } ^ { g } ( ( x ' , y ' ) , ( x '' , y '' ) ) & = \\sum \\limits _ { k = 1 } ^ { q } \\int _ \\gamma \\left | \\langle \\nabla g _ { k } ^ { H , \\beta } ( x ) , d x \\rangle \\right | ~ + ~ \\sum \\limits _ { k = 1 } ^ { q } \\int _ { \\widetilde { \\gamma } } \\left | \\langle \\nabla g _ { k } ^ { H , \\beta } ( y ) , d y \\rangle \\right | \\\\ & = D _ \\gamma ^ { g } ( x ' , x '' ) ~ + ~ D _ { \\widetilde { \\gamma } } ^ { g } ( y ' , y '' ) , \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} H ( X _ n ) & = \\sum _ { k = 1 } ^ n H ( X _ k ) - H ( X _ { k - 1 } ) \\geq n \\cdot \\big ( H ( X _ n ) - H ( X _ { n - 1 } ) \\big ) . \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} \\textrm { R R M S E } = \\sqrt { \\dfrac { 1 } { s } \\sum _ { i = 1 } ^ { s } \\dfrac { \\left | f ( \\tilde { \\boldsymbol { x } } _ i ) - { \\bar { \\cal I } } ( \\tilde { \\boldsymbol { x } } _ i ) \\right | ^ 2 } { \\left | f ( \\tilde { \\boldsymbol { x } } _ i ) \\right | ^ 2 } } . \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} d ^ * ( A B ) = d ^ * ( A ) + d ^ * ( B ) < 1 , \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} a b l _ \\sigma ( a ) & = ( w , \\beta _ { 1 , 1 } , \\beta _ { 2 , 1 } ) ( z , r _ b , s _ b ) ( x y ^ { - 1 } w ( x ^ { - 1 } , y ^ { - 1 } ) y x ^ { - 1 } , | w | _ x + \\beta _ { 1 , 1 } , | w | _ y + \\beta _ { 2 , 1 } ) \\\\ & = ( w z x y ^ { - 1 } w ( x ^ { - 1 } , y ^ { - 1 } ) y x ^ { - 1 } , 2 \\beta _ { 1 , 1 } + | w | _ x + r _ b , 2 \\beta _ { 2 , 1 } + | w | _ y + s _ b ) . \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} S _ \\varsigma : = S \\cup \\big ( \\bigcup _ { ( l , a ) \\in I _ \\sigma } \\overline \\Omega _ { l , a } \\big ) \\cup \\big ( \\bigcup _ { ( l , a ) \\in I _ \\varsigma } \\delta _ { l , a } \\cup \\Upsilon _ { l , a } \\big ) \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} \\Omega = \\left [ \\frac { \\mu ^ { \\frac { 1 } { \\eta } } \\Gamma \\left ( \\mu \\right ) \\mathbb { E } [ R _ k ] } { \\Gamma \\left ( \\mu + \\frac { 1 } { \\eta } \\right ) } \\right ] ^ \\eta \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} R _ { 1 2 } ( \\lambda / \\mu ) \\ , \\mathcal { U } _ { 1 } ( \\lambda ) \\ , R _ { 1 2 } ( \\lambda \\mu / q ) \\ , \\mathcal { U } _ { 2 } ( \\mu ) = \\mathcal { U } _ { 2 } ( \\mu ) \\ , R _ { 1 2 } ( \\lambda \\mu / q ) \\ , \\mathcal { U } _ { 1 } ( \\lambda ) \\ , R _ { 1 2 } ( \\lambda / \\mu ) . \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} S ( t ) = \\begin{pmatrix} S _ 1 ( t ) & Q ( t ) \\\\ 0 & S _ 2 ( t ) \\end{pmatrix} \\quad Q ( t ) = \\int _ 0 ^ t S _ 1 ( t - s ) \\partial _ V R _ 1 ( 0 , 0 ) S _ 2 ( s ) \\dd s . \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} X ^ { 2 } + 3 Y ^ { 2 } = \\mu \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} \\mu _ i ( f ) ( z ) = \\begin{cases} f ( \\kappa _ i ( z ) ) & \\mid z \\in Z _ i \\\\ 0 & \\mid z \\notin Z _ i . \\end{cases} \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} \\begin{aligned} \\tau & < \\frac 2 3 ( \\beta - 1 ) \\left ( 1 - \\frac 1 { 2 p } \\right ) & & 1 \\leq p < 2 , \\\\ \\tau & < \\frac { \\beta - 2 } { 2 } + \\frac 1 p & & 2 \\leq p \\leq \\infty . \\end{aligned} \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta u _ 1 ( x ) = \\lambda _ 1 f _ 1 ( x , u _ 1 ( x ) , u _ 2 ( x ) ) , & x \\in \\Omega , \\\\ - \\Delta u _ 2 ( x ) = \\lambda _ 2 f _ 2 ( x , u _ 1 ( x ) , u _ 2 ( x ) ) , & x \\in \\Omega , \\\\ u _ 1 ( x ) = u _ 2 ( x ) = 0 , & x \\in \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} { } { \\cal R } ( z ) = I + \\frac { \\sqrt { 2 \\pi } \\left ( a ^ 2 - 1 \\right ) ^ c } { N ^ { 1 / 2 - c } a \\ , \\Gamma ( c ) } \\frac { 1 } { z - \\beta } \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} X _ r = \\sum _ { i = 1 } ^ p f _ i \\frac { \\partial } { \\partial x ^ i } + \\sum _ { \\rho = 1 } ^ q \\varphi _ \\rho \\frac { \\partial } { \\partial \\xi ^ \\rho } \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} \\beta b ( \\phi _ \\beta ( x ) ) ^ { b - 1 } = x - \\phi _ \\beta ( x ) \\ , , \\quad \\forall x > 0 \\ , . \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} A = c _ 4 - 1 2 \\Delta ^ { 1 / 3 } B = c _ 4 ^ 2 + 1 2 c _ 4 \\Delta ^ { 1 / 3 } + ( 1 2 \\Delta ^ { 1 / 3 } ) ^ 2 \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} \\P ( [ X _ g ] _ b = z | J ^ g = d ^ g , X _ 0 = x ) = \\P ( [ X _ g ] _ b = z ) = \\P ( [ X _ 0 ] _ b = z ) , \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} L ( x ^ { 1 0 } ) & = \\{ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 1 0 \\} \\quad \\mbox { i f $ p > 2 $ , } \\\\ L ( x ^ { 1 0 } ) & = \\{ 2 , 3 , 4 , 5 , 6 , 8 , 1 0 \\} \\quad \\quad \\mbox { i f $ p = 2 $ . } \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} H \\left ( { q } ^ i , \\frac { \\partial W } { \\partial { q } ^ i } \\right ) = E . \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} \\frac { \\prod _ { b \\in B } \\xi ( \\frac { 1 } { 2 } + | b - [ f ( b ) ] - 1 / 2 | \\pm z ) } { \\xi ( \\frac { n _ B + n _ C } { 2 } + z ) \\dots \\xi ( \\frac { n _ C - n _ B } { 2 } + 1 + z ) } = \\frac { \\prod _ { b \\in B } \\xi ( \\frac { 1 } { 2 } + | b - f ( b ) | \\pm z ) } { \\xi ( \\frac { n _ B + n _ C } { 2 } + z ) \\dots \\xi ( \\frac { n _ C - n _ B } { 2 } + 1 + z ) } \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} - \\Delta _ { q ( x ) } v _ { 1 } = \\lambda ^ { \\sigma } \\left \\{ \\begin{array} { l l } w _ { 2 } ^ { \\beta _ { 2 } ( x ) } & \\Omega \\backslash \\overline { \\Omega } _ { \\delta } \\\\ d ( x ) ^ { \\alpha _ { 2 } ( x ) + \\beta _ { 2 } ( x ) } & \\Omega _ { \\delta } \\end{array} \\right . , v _ { 1 } = 0 \\partial \\Omega \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} \\tilde { \\theta } & = \\theta _ { 0 } + \\frac { 1 } { n } \\sum _ { \\ell = 1 } ^ { L } \\sum _ { i \\in I _ { \\ell } } Z _ { i } [ \\hat { \\gamma } _ { \\ell } ( X _ { i } ) - \\gamma _ { 0 } ( X _ { i } ) ] + \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } Z _ { i } \\gamma _ { 0 } ( X _ { i } ) - \\theta _ { 0 } \\\\ & = \\theta _ { 0 } + \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\zeta ( W _ { i } ) + o _ { p } ( n ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} \\phi _ 0 ( x , y ) = a x ^ 3 + b x ^ 2 y + c x y ^ 2 + d y ^ 3 \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} \\left ( \\sum _ { k = 1 } ^ n \\Pi _ { n , L } \\right ) W _ r ( L ) \\left ( \\sum _ { k = 1 } ^ n \\Pi _ { n , L } \\right ) \\geq C \\left ( \\sum _ { k = 1 } ^ n \\Pi _ { n , L } \\right ) , \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} \\Psi ( T ) = \\Phi ( T ) T ^ s , \\Phi ( 0 ) \\neq 0 \\mbox { a n d } \\Phi ( 0 ) \\mid J \\eta , \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} B ^ { h } _ t = \\frac { 1 } { c _ { h ( t ) } } \\int _ { \\R } \\frac { \\exp ( i t x ) - 1 } { | x | ^ { h ( t ) + 1 / 2 } } ~ W ( d x ) , \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\Phi ' = \\operatornamewithlimits { i n d \\ , l i m } _ { \\tau \\in T } \\mathcal { H } _ { - \\tau } , \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} \\rho _ i ( \\partial _ t v _ i + ( v _ i \\cdot \\nabla ) v _ i ) = \\mu _ i \\Delta v _ i - \\nabla p _ i - \\rho _ i \\gamma . \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} A _ x = ( A \\times Y ) _ { ( x , y ) } \\subset \\tau ^ { - 1 } ( p ^ { - 1 } ( I _ o ) \\pi ( x , y ) ^ { - 1 } ) = \\tau _ o ^ { - 1 } ( I _ o \\pi _ p ( x , y ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } \\frac { \\partial \\varphi } { \\partial \\nu } g ( x ) d S ( x ) = 0 \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} h ( t ) \\le r \\quad \\mbox { f o r } t \\in [ 0 , 1 ] \\qquad \\mbox { a n d } h ( t ) = r \\quad \\mbox { f o r } t \\in [ 0 , 1 ] \\setminus ( t _ 1 , t _ 2 ) . \\end{align*}"}
-{"id": "2180.png", "formula": "\\begin{align*} c _ 4 ( E _ 3 ) = 2 ^ 4 ( 4 w ^ 2 - 3 u ^ { p } ) , c _ 6 ( E _ 3 ) = 2 ^ 6 w ( 9 u ^ { p } - 8 w ^ 2 ) , \\Delta ( E _ 3 ) = - 2 ^ 7 \\ell ( u v ) ^ { 2 p } . \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} \\Delta f \\left ( x \\right ) = f \\left ( x + 1 \\right ) - f \\left ( x \\right ) . \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} f _ 2 ' ( B _ n ) & = - \\frac { 1 } { 2 \\sqrt { \\pi } } e ^ { - \\left ( \\sqrt { B _ n } - \\sqrt { P } \\right ) ^ 2 } B _ n ^ { - \\frac { 1 } { 2 } } \\le 0 . \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} { K ^ g ( x , z ) \\over \\pi ( z ) } & = { \\P ( [ X _ g ] _ b = z | X _ 0 = x ) \\over \\P ( [ X _ 0 ] _ b = z ) } . \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} v _ t - \\big ( b ( S ) - \\sigma ( S ) \\ , \\sigma ' ( S ) \\ , S _ x \\big ) \\ , v _ x - \\bigg ( \\frac { \\sigma ( S ) ^ 2 } { 2 } \\ , v _ x \\bigg ) _ x = g , v ( 0 , \\cdot ) = 0 \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} d \\mu _ a = - \\frac { \\vartheta _ a d r } { 2 r ^ 2 } + O ( \\frac { \\ln ( r ) } { r ^ 3 } ) . \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} \\lambda \\left ( A _ { \\alpha } \\right ) & = \\max \\left \\{ \\lambda \\left ( A _ { \\alpha } \\left ( H \\right ) \\right ) : H G \\right \\} , \\\\ \\lambda _ { \\min } \\left ( A _ { \\alpha } \\right ) & = \\min \\left \\{ \\lambda _ { \\min } \\left ( A _ { \\alpha } \\left ( H \\right ) \\right ) : H G \\right \\} . \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} X ( n , k ) = \\left \\{ \\begin{array} { r c l } A ( n - 1 ) \\ldots A ( k ) & \\textnormal { i f } & n > k \\\\ \\\\ I & \\textnormal { i f } & n = k \\\\ \\\\ A ^ { - 1 } ( n ) \\ldots A ^ { - 1 } ( k - 1 ) & \\textnormal { i f } & n < k . \\end{array} \\right . \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} a = 2 \\ell , b = 1 , c = 1 , x = - v ^ 2 , y = u , z = w . \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} I _ Q ( V ; Y , S | U ) - I _ Q ( V ; Z | U ) & = I _ P ( S ; Y , S | T ) - I _ P ( S ; Z | T ) = H _ P ( S | T , Z ) , \\\\ I _ Q ( U , V ; Y , S ) - I _ Q ( U , V ; S ) & = I _ P ( S , T ; Y , S ) - I _ P ( S , T ; S ) = I _ P ( T ; Y | S ) , \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\psi ' _ k \\| _ { L ^ q ( \\R ) } & \\leq C \\| \\psi _ k '' \\| _ { L ^ q ( \\R ) } ^ \\frac 1 2 \\| \\psi _ k \\| _ { L ^ q ( \\R ) } ^ \\frac 1 2 \\\\ & \\leq C \\left ( \\mu _ k ^ \\frac 1 2 \\| \\psi _ k \\| _ { L ^ q ( \\R ) } ^ \\frac 1 2 + \\| Q ^ \\frac 1 2 \\psi _ k \\| _ { L ^ q ( \\R ) } ^ \\frac 1 2 \\right ) \\| \\psi _ k \\| _ { L ^ q ( \\R ) } ^ \\frac 1 2 . \\end{aligned} \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} \\Psi ( r , \\theta ) \\mapsto \\Psi _ j ( r , \\theta ) : J _ z \\Psi _ j \\left ( r , \\theta \\right ) = j \\hbar \\Psi _ j \\left ( r , \\theta \\right ) \\ ; , \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} L _ { 1 } & = I - F _ { 1 } ^ { \\ast } V \\widehat { F } V ^ { \\ast } , L _ { 2 } = - ( I + F _ { 3 } ^ { \\ast } V ) \\widehat { F } V ^ { \\ast } , L _ { 3 } = F _ { 1 } ^ { \\ast } V \\widehat { F } , L _ { 4 } = ( I + F _ { 3 } ^ { \\ast } V ) \\widehat { F } , \\\\ R _ { 1 } & = I - U \\widehat { E } U ^ { \\ast } E _ { 1 } ^ { \\ast } , R _ { 2 } = - \\widehat { E } U ^ { \\ast } E _ { 1 } ^ { \\ast } , R _ { 3 } = U \\widehat { E } ( I - U ^ { \\ast } E _ { 2 } ^ { \\ast } ) , R _ { 4 } = \\widehat { E } ( I - U ^ { \\ast } E _ { 2 } ^ { \\ast } ) . \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} \\lbrack _ { q } \\mathcal { U } _ { a , - } ( \\lambda ) , \\mathcal { U } _ { a , - } ( \\mu ) ] = 0 , \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} \\lambda _ 2 ( u v ) = \\lambda ( v ) \\lambda _ 2 ( u ) + \\lambda ( u ) \\lambda _ 2 ( v ) \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} \\bigg | \\phi ( 0 ) - \\dfrac { 1 } { N _ j } \\sum _ { k = 1 } ^ { m _ j } \\alpha ^ { ( j ) } _ k \\bigg | < \\tau , \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} \\kappa ( z , y ) = \\sum _ { [ \\xi ] \\in \\widehat { G } _ 0 } d _ { \\xi } [ \\xi ( y ) \\sigma ( z , \\xi ) ] , \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\psi _ i = \\varphi _ i ^ { \\frac { 1 } { 2 } } \\cdot \\mathrm { e x p } \\left ( \\frac { 2 \\pi i } { M } \\cdot a _ { B _ i , + } \\right ) : B _ i \\to { \\mathbb { C } ^ { } } , \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} x _ \\alpha = y _ \\beta , x _ \\beta = - y _ \\alpha . \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} \\varphi _ { \\alpha } ( t , \\Phi ( s ) u ) & = \\sup \\limits _ { \\xi \\geq t } e ^ { - \\alpha ( \\xi - t ) } \\parallel U ( \\xi , t ) ( \\Phi ( s ) u ) ( t ) \\parallel \\\\ & = \\sup \\limits _ { \\xi \\geq t } e ^ { - \\alpha ( \\xi - t ) } \\parallel U ( \\xi , t ) U ( t , s ) u ( s ) \\parallel \\\\ & = \\sup \\limits _ { \\xi \\geq t } e ^ { - \\alpha ( \\xi - t ) } \\parallel U ( \\xi , s ) x \\parallel . \\end{align*}"}
-{"id": "719.png", "formula": "\\begin{align*} I ( p , q ; a , r ) = \\int _ { E _ { p + q } } ( t _ { p + 1 } ) ^ r \\left ( \\frac { t _ 1 } { t _ { p + 1 } } \\right ) ^ a \\prod _ { j = 1 } ^ p \\frac { d t _ j } { 1 - t _ j } \\prod _ { i = p + 1 } ^ { p + q } \\frac { d t _ i } { t _ i } . \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\Delta _ k ( x ; \\boldsymbol { a } ) & = \\frac { 1 } { k ! \\phi ^ k ( q ) } \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { H } ( 1 , \\delta ) } \\ ( \\ ( F ( s , \\chi _ 0 ) + F ( s , \\chi _ a ) \\ ) ^ k - F ^ k ( s , \\chi _ 0 ) \\ ) \\frac { x ^ s } { s } d s \\\\ & + O \\ ( \\frac { x } { \\log x } ( \\log \\log x ) ^ { k - 3 } \\ ) . \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{align*} F _ { \\alpha } ( x , n , [ u ] ) = F _ { \\alpha } \\left ( x , n , u ^ { \\beta } _ { \\bold { 1 } _ i ; \\bold { 0 } } , \\ldots , u ^ { \\beta } _ { J _ 1 ; \\bold { 0 } } , u ^ { \\beta } _ { \\bold { 0 } , \\bold { 1 } _ i } , u ^ { \\beta } _ { \\bold { 0 } ; - \\bold { 1 } _ i } , \\ldots , u ^ { \\beta } _ { \\bold { 0 } ; J _ 2 } \\right ) \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } | A ( | u _ n | ) - A ( | u _ n - u | ) - A ( | u _ { } | ) | d x = 0 . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} d ^ 2 f = & \\sum _ { i , k } d x ^ i \\wedge d x ^ k \\frac { \\partial ^ 2 f } { \\partial x ^ k \\partial x ^ i } + \\sum _ { i , l } d x ^ i \\wedge d \\xi ^ l \\frac { \\partial ^ 2 f } { \\partial \\xi ^ l \\partial x ^ i } \\\\ & + \\sum _ { j , m } d \\xi ^ j \\wedge d x ^ m \\frac { \\partial ^ 2 f } { \\partial x ^ m \\partial \\xi ^ j } + \\sum _ { j , n } d \\xi ^ j \\wedge d \\xi ^ n \\frac { \\partial ^ 2 f } { \\partial \\xi ^ n \\partial \\xi ^ j } ; \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} \\Theta ( z ) = e ^ { \\pm i z ( \\sqrt { \\frac { m _ 1 } { 2 b } } \\pm \\sqrt { \\frac { m _ 2 } { 2 b } } ) } w _ 2 \\left ( \\frac { z } { \\mathcal { Z } } \\right ) \\left ( \\frac { z } { \\mathcal { Z } } \\right ) ^ { s _ 1 - 2 s _ 2 - 2 s _ 3 - 2 s _ 4 - 2 s _ 5 } . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} \\sum _ K 2 ^ { N - 2 K } { N \\choose K } { N - K \\choose N - 2 K } = { 2 N \\choose N } . \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} D _ { - c } ( \\zeta ) = e ^ { - \\frac { \\zeta ^ 2 } { 4 } } \\zeta ^ { - c } \\left ( \\sum _ { s = 0 } ^ { n - 1 } ( - 1 ) ^ s \\frac { ( c ) _ { 2 s } } { s ! ( 2 \\zeta ^ 2 ) ^ s } + \\varepsilon _ n ( \\zeta ) \\right ) , \\quad | \\rm a r g \\ , \\zeta | < \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\vec { X } _ { 1 1 } & = \\Bigg ( 0 , 1 , - \\frac { \\lambda } { m - n } \\bigg ( \\frac { 1 } { 1 + B ^ { - 1 } } \\bigg ) \\Bigg ) , \\vec { X } _ { 1 2 } = \\bigg ( 1 \\ ; , \\ ; 0 \\ ; , \\ ; - \\frac { \\lambda } { m - n } \\frac { \\lambda c } { 1 + C ^ { - 1 } } \\bigg ) . \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} ^ c D ^ q \\mathbf { x } = A \\mathbf { x } \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} c ( \\Delta ^ { p ^ r m } ) & = ( \\lambda ( p ) \\kappa ( p ^ { r - 1 } ) - \\chi ( p ^ 2 ) p ^ { 2 k - 3 } \\kappa ( p ^ { r - 2 } ) - \\chi ( p ) p ^ { k - 2 } \\eta ( p ^ r ) ) c ( \\Delta ^ m ) \\\\ & = \\kappa ( p ^ r ) c ( \\Delta ^ m ) . \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} \\left \\vert \\theta _ { 1 } ^ { ( 1 ) } - \\frac { V } { U } \\right \\vert = \\left \\vert \\frac { c + 2 } { c } - \\frac { V ^ { 2 } } { U ^ { 2 } } \\right \\vert \\left \\vert \\theta _ { 1 } ^ { ( 2 ) } - \\frac { V } { U } \\right \\vert ^ { - 1 } \\leq \\frac { 2 } { | c | | U | ^ { 2 } } \\sqrt { \\frac { | c | } { | c | - 2 } } = \\frac { 2 } { \\sqrt { | c | ( | c | - 2 ) } } | U | ^ { - 2 } . \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} \\delta E = \\tilde { \\epsilon } ( \\lambda _ p ) + \\sum _ { i a } \\tilde { \\epsilon } ( \\lambda _ { i a } ) [ N _ { i a } - n ^ { \\rm i m p } _ { i a } ] + \\frac { 2 \\pi } { L } \\sum _ { i a } s _ a \\tilde { v } _ { i a } \\left [ n _ { i a } + \\frac { 1 } { 2 } \\left ( \\sum _ { j b } U _ { j b , i a } [ N _ { j b } - n ^ { \\rm i m p } _ { j b } ] \\right ) ^ 2 \\right ] \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} ( p , W , \\ell ) = ( 1 9 , 8 6 4 a 1 , 5 ) , \\ ; ( 1 9 , 8 6 4 b 1 , 7 ) , \\ ; ( 4 3 , 8 6 4 a 1 , 3 1 ) , \\ ; ( 4 3 , 8 6 4 b 1 , 1 3 ) , \\ ; ( 6 7 , 8 6 4 b 1 , 1 9 ) . \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} \\theta ( s ) = \\tilde \\Psi ( s ) \\zeta ( s ) + \\Psi ( 0 ) / 2 s ( s + 1 ) . \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\Pr [ \\mathrm { r a n k } \\ , \\mathbf { X } = k ] = \\sum _ { i = \\max ( 0 , k - b ) } ^ { \\min ( a , k ) } \\mathbb { P } _ { i } ( a , k ) \\mathbb { P } ( b , k - i ) . \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial \\lambda _ { i a } } = L s _ a \\rho ( \\lambda _ { i a } ) \\epsilon ( \\lambda _ { i a } ) = \\sum _ { j b } L s _ a \\rho ( \\lambda _ { i a } ) [ \\delta _ { i a , j b } - s _ b F ( \\lambda _ { i a } | \\lambda _ { j b } ) ] \\tilde { \\epsilon } ( \\lambda _ { j b } ) \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} f ( x ) = \\prod _ { i = 1 } ^ { k } ( x - n _ i ) ^ { r _ i } \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} { \\mathbf { A } } = \\frac { 1 } { N } \\mathbf { F } ^ { \\rm H } \\begin{bmatrix} \\mathbf { \\Phi } _ 0 \\mathbf { C } & \\mathbf { \\Phi } _ 1 \\mathbf { C } & \\cdots & \\mathbf { \\Phi } _ { M - 1 } \\mathbf { C } \\end{bmatrix} , \\end{align*}"}
-{"id": "8583.png", "formula": "\\begin{align*} \\max \\limits _ { m \\in \\mathcal { M } _ n } \\Big | \\Big | Q ^ { ( \\mathcal { B } _ n ) } _ { I , \\mathbf { U } } Q ^ { ( \\mathcal { B } _ n ) } _ { \\mathbf { Z } | M = m , I , \\mathbf { U } } - Q ^ { ( \\mathcal { B } _ n ) } _ { I , \\mathbf { U } } Q ^ n _ { Z | U } \\Big | \\Big | _ { \\mathsf { T V } } \\leq e ^ { - n \\gamma _ 1 } , \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} \\int \\hat { \\Delta } _ { \\ell } ( w ) ^ { 2 } F _ { 0 } ( d w ) & = \\int [ \\hat { \\alpha } _ { \\ell } ( x ) - \\alpha _ { 0 } ( x ) ] ^ { 2 } [ \\lambda ( w , \\hat { \\gamma } _ { \\ell } ) - \\lambda ( w , \\gamma _ { 0 } ) ] ^ { 2 } F _ { 0 } ( d w ) \\\\ & \\leq O _ { p } ( 1 ) \\int [ \\lambda ( w , \\hat { \\gamma } _ { \\ell } ) - \\lambda ( w , \\gamma _ { 0 } ) ] ^ { 2 } F _ { 0 } ( d w ) \\overset { p } { \\longrightarrow } 0 . \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} ( T u ) ( x ) = \\frac { \\partial } { \\partial \\theta } \\left ( u ( e ^ { i \\theta } \\circ x ) \\right ) \\Big | _ { \\theta = 0 } \\ , , \\ : \\ : u \\in C ^ \\infty ( X ) . \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} \\{ 1 , 2 , \\dots , m + n \\} \\setminus A = \\{ j _ 1 , j _ 2 , \\dots , j _ n \\} \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} \\sum \\limits _ { k = 1 } ^ d { | x _ k | ^ 3 } \\prod \\limits _ { l \\neq k } | x _ l | \\leq \\sum _ { \\substack { ( a _ 1 , \\dots , a _ m ) \\\\ \\in \\{ | p _ 1 | , | q _ 1 | , \\dots , | p _ d | , | q _ d | \\} { d + 2 } } } \\prod _ { m = 1 } ^ { d + 2 } a _ m \\leq ( | p _ 1 | + | q _ 1 | + \\dots + | p _ d | + | q _ d ) ^ { d + 2 } \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} ( W _ { ( \\xi , \\xi ' ) } ) _ W = & \\frac { d } { d t } ( \\Psi _ W \\circ \\gamma ) | _ { t = 0 } \\\\ & = ( [ \\xi , a ] , [ \\xi , b ] , \\xi i , - j \\xi , [ \\xi , a ' ] , [ \\xi , b ' ] , \\xi f - f \\xi ' , \\xi ' g - g \\xi ) . \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} f _ { Z | H _ 1 } ( z | H _ 1 ) & = \\frac { e ^ { \\frac { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } { \\zeta } } } { \\pi ^ { n } } \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { z } { v } } e ^ { - \\frac { v } { \\zeta } } d v . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} \\omega ( x ) = \\exp \\left ( c _ 2 ( \\log x ) ( \\log \\log x ) ^ { - 1 } \\right ) , \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\tilde { \\mathcal { E } } _ x = \\tilde { \\mathcal { E } } _ x ( n , L , B , r , R , \\eta ) \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} Z ( r , d ) = - d + \\sqrt { - 1 } r . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} P _ { S _ 1 | Y } ( 1 | y ' ) = \\delta ' ( 1 - \\sigma ) . \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} \\begin{aligned} I _ 5 ( y ) - I _ 5 ( \\bar { y } ) & = \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\left ( \\partial _ y r ( Y _ q + s t y ) - \\partial _ y r ( Y _ q + s t \\bar { y } ) \\right ) t y v \\dd s \\dd t \\\\ & \\qquad + \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\partial _ y r ( Y _ q + t s \\bar { y } ) t v ( y - \\bar { y } ) \\dd s \\dd t \\\\ & \\qquad + \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\partial _ y r ( Y _ q + s t \\bar { y } ) t \\bar { y } ( v - \\bar { v } ) \\dd s \\dd t \\end{aligned} \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} \\lim _ { R \\to 0 } \\frac { h _ { \\rm i n } ( R ) } { g _ { \\rm i n } ( R ) } \\equiv \\frac { J _ { j + \\frac { 1 } { 2 } } ( Z \\alpha _ g ) } { J _ { j - \\frac { 1 } { 2 } } ( Z \\alpha _ g ) } = C _ { \\rm i n } \\ ; , \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} E _ { 1 } - \\widetilde { E } _ { 1 } & = U E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\Phi , E _ { 3 } - \\widetilde { E } _ { 3 } = U E _ { S _ { A } } ( I + H H ^ { \\ast } E _ { S _ { A } } ) ^ { - 1 } , \\\\ F _ { 1 } - \\widetilde { F } _ { 1 } & = \\Psi A ^ { \\dagger } U F _ { S _ { A } } V ^ { \\ast } , \\ F _ { 2 } - \\widetilde { F } _ { 2 } = - ( I + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } F _ { S _ { A } } V ^ { \\ast } . \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\mathcal { H } ( u + h ) = \\mathcal { H } ( u ) + ( ( \\nabla \\mathcal { H } ) ( u ) , h ) _ { L ^ 2 ( \\mathbb { T } ) } + \\frac { 1 } { 2 } ( ( \\partial _ u \\nabla \\mathcal { H } ) ( u ) [ h ] , h ) _ { L ^ 2 ( \\mathbb { T } ) } + O ( h ^ 3 ) . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} \\hat { F } _ e \\left ( { x } \\right ) = \\sum \\limits _ { \\ell , n \\in \\left \\{ { { } , { } } \\right \\} } \\left ( { \\rm { \\mathbf { 1 } } } \\left ( d < \\eta \\left ( G _ \\ell , G _ n ^ e , x \\right ) \\right ) \\frac { d ^ 2 } { 2 } + \\frac { \\varrho ^ 2 - d ^ 2 } { 2 } \\right ) { { { \\Pr } _ { \\ell n } } } \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( - \\Delta ) ^ s \\chi _ 1 & = \\lambda _ 1 \\chi _ 1 & & \\mbox { i n } \\Omega , \\\\ \\chi _ 1 & = 0 & & \\mbox { i n } \\mathbb { R } ^ N \\setminus \\Omega , \\end{aligned} \\right . \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} x = F _ 1 ( x ) + F _ 2 ( x ) . \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} \\widehat { a d ( E ) } ^ { * } _ x = \\mathfrak { p } ^ { \\vee } \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} I _ { \\mathcal { O } } ( \\alpha ) = \\left [ \\mathcal { O } ^ { + } : \\mathbb { Z } [ \\alpha ] ^ { + } \\right ] = \\left [ \\mathcal { O } ^ { + } : \\mathbb { Z } _ { M } [ \\alpha ] ^ { + } \\right ] \\cdot \\left [ \\mathbb { Z } _ { M } [ \\alpha ] ^ { + } : \\mathbb { Z } [ \\alpha ] ^ { + } \\right ] . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\int _ { S } \\frac { d \\mu _ r } { d \\mu _ s } ( n ) \\ , d \\mu ( n ) = \\frac { r } { r - s } \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} C = \\prod _ { p \\in \\P } \\left ( 1 - \\frac 1 { p ( p + 1 ) } \\right ) , D = \\sum _ { p \\in \\P } \\frac { \\log p } { p ^ 2 + p - 1 } . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} C _ n = \\{ c _ 1 e _ 1 + c _ 2 e _ 2 + \\cdots c _ n e _ n : c _ i = 0 1 \\} . \\end{align*}"}
-{"id": "8916.png", "formula": "\\begin{align*} \\overline { E _ f } = T ^ { \\frac { 1 } { 6 } ( n ^ 3 - n ) } \\cdot \\frac { 1 } { n } \\prod _ { i = 2 } ^ n \\frac { 1 } { i ( n - i ) } \\cdot \\xi ( 2 ) ^ { - 1 } \\dots \\xi ( n ) ^ { - 1 } \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} f _ { \\Theta _ \\rho } ( \\theta ) = \\begin{cases} 1 / \\zeta , & 0 < \\theta \\le P _ { \\rm { m a x } } / d ^ { \\alpha } _ { \\rm j , w } , \\rho = 0 \\\\ 1 / \\zeta , & \\sigma _ { \\rm a } ^ 2 < \\theta \\le \\sigma ^ 2 _ { \\rm a } + P _ { \\rm { m a x } } / d ^ { \\alpha } _ { \\rm j , w } , \\rho = 1 , \\\\ 0 , & , \\end{cases} \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} D _ { \\ell _ i - 1 } \\mathcal { G } _ { \\{ \\ell _ 1 , \\dots , \\ell _ i , \\dots , \\ell _ k \\} } = \\mathcal { G } _ { \\{ \\ell _ 1 , \\dots , \\ell _ i - 1 , \\dots , \\ell _ k \\} } , \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} \\mathbb { E } ( \\vert \\langle S ^ N ( \\tilde \\xi _ t ) - m ^ 0 _ t , \\varphi \\rangle \\vert ) \\le C _ \\varphi ( \\frac { 1 } { \\sqrt { N } } + \\delta t ) , { \\rm w h e r e \\ a g a i n } S ^ N ( \\mathbf { \\tilde { \\xi _ t } } ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\delta _ { \\tilde { \\xi } _ t ^ { i , N } } \\ . \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} P _ n ( T \\widetilde { f } | | f ^ { \\star } ) - P _ n ( f | | f ^ { \\star } ) + D ^ { \\prime } _ n ( T f , T \\widetilde { f } ) & \\leq \\frac { 2 v v _ f } { m _ 1 } + \\frac { L _ 2 v ^ 2 _ f v ^ 2 _ 0 } { m _ 0 } + \\frac { L ^ 2 _ 2 v ^ 2 _ f v ^ 4 _ 0 } { 4 m _ 0 } \\\\ & + \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n | Y _ i | \\right ) \\frac { v _ f L _ 2 v ^ 2 _ 0 } { m _ 0 } + \\frac { T _ n } { n } . \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} G _ { \\lambda } ( \\sigma ) : = \\eta _ { 1 } \\left ( \\frac { 2 \\sigma } { \\left \\vert \\lambda \\right \\vert } \\right ) \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} R ( s ) \\ll Q ^ { 1 / 2 } | s | ^ { - \\frac { 1 } { 1 2 } } . \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { k = n + 2 } ^ { \\infty } \\frac { 1 } { k ^ { \\omega } ( k ^ \\gamma - n ^ \\gamma ) } & = \\left ( \\sum _ { k = n + 2 } ^ { 2 n } + \\sum _ { k = 2 n + 1 } ^ { \\infty } \\right ) \\frac { 1 } { k ^ { \\omega } ( k ^ \\gamma - n ^ \\gamma ) } \\\\ & \\leq C \\left ( \\frac { 1 } { n ^ \\omega } \\sum _ { k = n + 2 } ^ { 2 n } \\frac { 1 } { k ^ \\gamma - n ^ \\gamma } + \\sum _ { k = 2 n + 1 } ^ { \\infty } \\frac { 1 } { k ^ { \\omega + \\gamma } } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} \\begin{aligned} & E _ s E _ { s - 1 ; s ; s + 1 } + E _ { s - 1 ; s ; s + 1 } E _ s = - \\sigma _ s \\sigma _ { s + 1 } ( e _ s e _ { s - 1 ; s ; s + 1 } + e _ { s - 1 ; s ; s + 1 } e _ s ) = 0 , \\\\ & F _ s F _ { s - 1 ; s ; s + 1 } + F _ { s - 1 ; s ; s + 1 } F _ s = - \\sigma _ { s - 1 } \\sigma _ { s } ( f _ s f _ { s - 1 ; s ; s + 1 } + f _ { s - 1 ; s ; s + 1 } f _ s ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} F ^ { ( 3 ) } _ { j _ 1 j _ 2 j _ 3 } : = \\begin{cases} \\dfrac { - \\mathrm { i } \\ , c _ 1 \\ , j _ 1 j _ 2 j _ 3 - c _ 2 \\ , j _ 1 j _ 2 + c _ 3 } { \\mathrm { i } ( j _ 1 ^ 3 + j _ 2 ^ 3 + j _ 3 ^ 3 ) } \\mbox { i f } \\ , \\ , ( j _ 1 , j _ 2 , j _ 3 ) \\in \\mathcal { A } _ 3 , \\\\ 0 \\qquad \\ , \\ , \\ , \\ , \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} \\eta ( { \\sigma } ) = \\lambda \\ , { n } \\ , \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} - b _ { j } + T _ { j j } g _ { j } ^ { \\prime } ( z _ { j } ^ { \\ast } ) = \\beta , \\forall j = \\overline { 2 , n } . \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} F ( s , \\chi _ 0 ) = \\log \\zeta ( s ) + G _ 0 ( s ) , \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} \\sum _ { v \\in L } \\sum _ { i , j = 1 } ^ m f ( v + x _ j - x _ i ) \\leq m f ( 0 ) . \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } - \\kappa u _ { y y } - u _ { x x } + u _ y & = & f ( y , x , u , v ) \\Omega , \\\\ - \\kappa v _ { y y } - v _ { x x } + v _ y & = & g ( y , x , u , v ) \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} \\Lambda _ { n , r , R } : = \\int _ { r \\leq | x _ 0 + \\dots + x _ n | \\leq R } \\prod _ { i = 0 } ^ { n } F _ i ( x _ 0 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ n ) \\frac { 1 } { x _ 0 + \\dots + x _ n } d x _ 0 \\dots d x _ n \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} \\tilde { H } _ c = U _ { R } ^ { - 1 } H _ c U _ { R } . \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} H ^ { ( 4 ) } _ 5 ( u ) = \\sum _ { j _ 1 + \\dots + j _ 5 = 0 } H ^ { ( 4 ) } _ { 5 , \\ , j _ 1 , \\dots , j _ 5 } \\ , u _ { j _ 1 } u _ { j _ 2 } u _ { j _ 3 } u _ { j _ 4 } u _ { j _ 5 } , \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} f : = \\sum _ { i = 2 } ^ n \\left ( 1 0 0 ( x _ i - x _ { i - 1 } ^ 2 ) ^ 2 + ( 1 - x _ i ) ^ 2 \\right ) . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} c _ i = \\sum \\limits _ { H \\in { \\cal H } _ i } ( - 1 ) ^ { p ( H ) } 2 ^ { c ( H ) } , \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} \\sup _ { M \\times [ 0 , \\infty ) } | \\nabla ^ { k } \\tilde { \\varphi } ( x , t ) | = \\sup _ { M \\times [ 0 , \\infty ) } | \\nabla ^ { k } \\varphi ( x , t ) | \\leq C _ { k } , \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} A = \\mathop { \\oplus } \\limits _ { a = 1 } ^ n \\left ( - i \\left ( \\lambda _ a + \\frac { m _ a } { 2 r } \\right ) \\frac { d \\tau + \\omega } { V } + \\pi _ k ^ * \\eta _ a \\right ) + O \\left ( \\frac { 1 } { r ^ 2 } \\right ) , \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} E _ s = & \\ - \\frac 1 \\pi \\int _ \\R \\xi ^ 2 ( 1 + \\xi ^ 2 ) ^ { s } \\real \\ln T ( \\xi / 2 ) d \\xi + \\sum _ j \\Xi _ s ( 2 \\kappa _ j ) \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} p ( h ) = 2 J ^ { \\prime } ( h ) M ( h ) - J ( h ) M ^ { \\prime } ( h ) \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} C _ n = & B _ n + V ^ { - 1 } ( B _ { n + 1 } \\cap p \\left ( W _ { n + 1 } ( K ) / \\wp W _ { n + 1 } ( K ) \\right ) \\\\ & + V ^ { - 2 } ( B _ { n + 2 } \\cap p ^ 2 \\left ( W _ { n + 2 } ( K ) / \\wp W _ { n + 2 } ( K ) \\right ) + \\ldots . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} \\gamma ( s , \\xi , \\psi ) \\gamma ( 1 - s , \\xi ^ { - 1 } , \\psi ^ { - 1 } ) = 1 = \\epsilon ( s , \\xi , \\psi ) \\epsilon ( 1 - s , \\xi ^ { - 1 } , \\psi ^ { - 1 } ) . \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} \\mathcal { F } ( \\varphi ) = \\int _ M | \\nabla \\varphi | ^ 2 \\mathrm { e } ^ { - \\varphi } \\ , V o l _ { \\eta } . \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} ( A ) = \\sum _ { x \\in U } \\left ( x ; A \\right ) \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} \\langle f ( b ) , b ' \\rangle _ B = \\langle b , f ^ * ( b ' ) \\rangle _ { B ' } \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} \\bar { x } = x - \\sum \\limits _ { i \\in J } \\tilde { \\nu } _ i u _ i \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} D _ { \\xi , h } ( \\eta ) = \\frac { 1 } { 2 } \\left ( \\int _ { 0 } ^ { 1 } 2 \\sqrt { \\eta '' ( s ) } - \\eta '' ( s ) \\cdot ( 1 - s ) - \\frac { 1 } { 1 - s } \\ , d s - \\eta ( 0 ) + h ^ { 2 } + \\xi ( 1 ) \\right ) . \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} \\sum _ n L ( n , [ \\widetilde { u } ] ) = \\sum _ n L ( n , [ u ] ) . \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ d \\eta ^ { ( j ) } \\circ ( 1 \\otimes a ) = \\Bigg ( \\sum _ { j = 0 } ^ d \\mu ^ { ( j ) } ( 1 ) \\Bigg ) a = a \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{align*} M ( v _ 1 , \\ldots , v _ i ) = M ( v _ { i + 1 } , \\ldots , v _ { 2 i } ) . \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} 2 ( 1 2 + ( 2 6 - N ) ( 2 5 - N ) ) \\lambda ( N ) = H _ n ( N ) + 2 ( 2 6 - N ) H _ h ( N ) + \\tau ( N ) \\mu ( N ) H _ u ( N ) , \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} d e ^ { i \\theta } ( V ( x ) ) = V ( e ^ { i \\theta } \\circ x ) \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} x _ 1 = \\log \\frac { 1 - t _ 1 } { 1 - t _ 2 } , \\ ; x _ 2 = \\log \\frac { 1 } { t _ 2 } , \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } < x ^ { * } _ { n } - x ^ { * } _ { 0 } , e _ { k } > = 0 , k = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} \\frac { \\partial E } { \\partial k _ { i a } } = \\frac { s _ a L } { 2 \\pi } \\tilde { \\epsilon } ( \\lambda _ { i a } ) \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} c ( x ) = \\sum _ { j = 0 } ^ { p ^ k - 1 } x ^ { j p ^ { e - k } } b ( x ) . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} k ^ * ( y ) = r \\theta ( 0 ) - \\frac { \\Psi ( 0 ) \\xi ( w ) } { 2 \\zeta ( 2 ) } \\sum _ { c \\mid w } \\chi ( c ) \\sum _ { \\substack { c d y < 1 \\\\ ( d , D ) = 1 } } ( 1 - c d y ) d ^ { - 1 } + O ( y w \\tau ( D ) ) . \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} \\mathbf { y } = \\sum _ { k = 1 } ^ { K _ g } \\left ( \\mathbf { X } _ k ^ { ( g ) } \\otimes \\mathbf { I } _ N \\right ) \\mathbf { f } _ k ^ { ( g ) } + \\boldsymbol { \\xi } ^ { ( g ) } \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} [ X + r + \\xi , Y + t + \\eta ] = { } & [ X , Y ] + L _ { X } \\eta - i _ { Y } d \\xi + i _ { Y } i _ { X } H _ 0 \\\\ & - [ r , t ] - F _ 0 ( X , Y ) + d ^ { \\theta _ 0 } _ X t - d ^ { \\theta _ 0 } _ Y r \\\\ & + 2 c ( d ^ { \\theta _ 0 } r , t ) + 2 c ( F _ 0 ( X , \\cdot ) , t ) - 2 c ( F _ 0 ( Y , \\cdot ) , r ) . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\widetilde \\lambda _ 1 ( p ^ 2 ) c ( \\Delta ^ m ) = \\chi ( p ) p ^ { k - 2 } \\alpha ( \\Delta ; p ) c ( \\Delta ^ m ) + \\sum _ { \\{ \\Delta : \\Omega \\} = ( 1 , p ) } c ( \\Omega ) \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} V : = H ^ \\bullet _ z ( B _ x B _ y ) U : = H _ z ^ \\bullet ( E ) . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} c _ n = - \\frac { 1 } { n } \\sum _ { j + k = n } k a _ j b _ k , n = 1 , 2 , \\ldots . \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} \\dd Y _ t = \\left ( A Y _ t + \\xi ( Y _ t ) \\right ) \\dd t + a ( Y _ t ) \\dd W _ t + \\int _ E \\gamma ' ( Y _ { t - } , y ) \\left ( \\mu ( \\dd t , \\dd y ) - \\dd t \\otimes F ' ( \\dd y ) \\right ) \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} \\int _ { \\Omega } f w _ 1 d x = 0 , \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} & \\underset { t \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\int _ { { \\bf R } ^ N } \\int _ { { \\bf R } ^ N } G ( x - y , t ) \\eta ( x ) \\ , d x \\ , d \\mu ( y ) \\\\ & = \\int _ { { \\bf R } ^ N } \\eta ( y ) \\ , d \\mu ( y ) + \\lim _ { t \\to + 0 } \\int _ { { \\bf R } ^ N } ( \\eta ( y , t ) - \\eta ( y ) ) \\ , d \\mu ( y ) = \\int _ { { \\bf R } ^ N } \\eta ( y ) \\ , d \\mu ( y ) . \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} \\dot x ( t ) = - \\left | { x ( t ) - 0 . 5 } \\right | + \\frac { { 2 \\ , \\left ( { u ( t ) + 1 } \\right ) } } { { t + 2 } } - 0 . 5 , \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} \\tau _ i \\equiv \\begin{cases} \\theta _ i + t _ 1 \\delta _ i & i < j , \\\\ \\theta _ i - t _ 1 \\sum _ { l = 1 } ^ { j - 1 } \\delta _ l & i = j , \\\\ \\theta _ i & j < i < k ; \\\\ \\theta _ i - t _ 2 \\sum _ { l = k + 1 } ^ n \\delta _ l & i = k , \\\\ \\theta _ i + t _ 2 \\delta _ i & i > k . \\end{cases} \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} M _ j & = \\frac { n ^ j r ^ j } { j ! } ( 1 + H _ j ) \\\\ H _ j & = \\sum _ { h = 1 } ^ { j - 1 } \\frac { a _ h ( r , j ) } { n ^ h } \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} d _ { 2 r } ( p \\cdot x ) = p ^ { r + 1 } \\cdot d _ { 2 r } ( x ) . \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\Phi ^ + = \\Phi _ 1 ^ + \\sqcup \\Phi _ 2 ^ + \\sqcup \\cdots \\sqcup \\Phi _ n ^ + . \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} \\dot r ( t ) = - 2 \\lambda \\Big ( \\big ( r x + y \\phi ( t , x , y ) \\big ) u + \\big ( r y - x \\phi ( t , x , y ) \\big ) v \\Big ) . \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} \\varphi _ { u , v } ' ( t ) = t ^ { p - 1 } \\| ( u , v ) \\| ^ p - t ^ { q - 1 } \\int _ \\Omega ( \\lambda | u | ^ { q } + \\mu | v | ^ { q } ) d x - 2 t ^ { \\alpha + \\beta - 1 } \\int _ \\Omega | u | ^ { \\alpha } | v | ^ \\beta d x , \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} \\Big \\| \\sup _ { n \\ge 1 } | \\sum _ { k = 1 } ^ n a _ k k ^ { i \\cdot } | \\ , \\Big \\| _ { \\S ^ 2 } \\le C \\ , \\Big ( \\sum _ { n \\ge 1 } n | a _ n | ^ 2 \\Big ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} b ( x ) = b _ 0 + \\tilde b ( x ) , \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { g _ n } { p ^ { 2 n } } = \\frac { 1 } { 2 ( p + 1 ) } . \\end{align*}"}
-{"id": "6865.png", "formula": "\\begin{align*} ( f ( z ) \\mid U _ { \\ell ^ m } ) \\mid _ { - 1 } \\gamma = \\sum _ { \\substack { n = n _ 0 \\\\ n \\equiv 0 \\pmod { \\ell ^ m } } } ^ { \\infty } a _ 0 ( n ) q _ { 2 4 \\ell ^ m } ^ { n } , \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} \\phi _ 2 ( t , q ) & : = ( 1 + t ^ { a ( 1 ) } q + t ^ { a ( 1 ) } q ^ 2 + \\ldots ) ( 1 + t ^ { a ( 2 ) } q ^ 2 + t ^ { a ( 2 ) } q ^ 4 + \\ldots ) \\times \\cdots \\\\ & \\times ( 1 + t ^ { a ( n ) } q ^ n + t ^ { a ( n ) } q ^ { 2 n } + \\ldots ) \\times \\cdots . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} \\int _ { \\R _ - } \\d u \\ , f _ W ( u ) g _ Z ( u ) = \\int _ { \\R _ - } \\d u \\ , e ^ { 2 ( Z - W ) u } = \\frac 1 { 2 ( Z - W ) } \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} f _ \\rho ( 0 ) = \\frac { \\mu _ \\rho f ^ \\ast _ \\delta ( 1 ) } { 2 } = f _ \\rho ( 1 ) & = \\frac { \\mu _ \\rho f ^ \\ast _ \\delta ( 1 ) } { 2 } f _ \\rho ( t ) = \\frac { \\mu _ \\rho f ^ \\ast _ \\delta ( t ) } { t } k \\ge 2 . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} d _ { \\lambda \\mu } ( q ) = \\begin{cases} q ^ { \\sum _ { j = 0 } ^ { e - 1 } ( x _ j - c _ j ) } & 0 \\leq c _ j \\leq \\min ( 1 , x _ j ) j \\in [ 0 , \\ , e ) ; \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\bold { E } ^ { \\vartriangle } _ { u } ( L ) \\\\ & = v _ { - 1 , 0 } - v _ { 0 , - 1 } + \\frac { a ( m ) - b ( n ) } { ( u _ { 0 , 0 } - u _ { 1 , 1 } ) ^ 2 } v _ { 0 , 0 } - \\frac { a ( m - 1 ) - b ( n - 1 ) } { ( u _ { - 1 , - 1 } - u _ { 0 , 0 } ) ^ 2 } v _ { - 1 , - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} A _ { x _ o } ^ { - 1 } C = B ^ c \\subset \\tau _ o ^ { - 1 } ( t _ o I ^ { - 1 } ) \\beta ^ { - 1 } ( H _ o ) , \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } ( \\Phi ^ { \\prime } ( u _ { n } , v _ { n } ) - \\Phi ^ { \\prime } ( u , v ) , ( u _ { n } - u , v _ { n } - v ) ) = 0 . \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} \\begin{cases} \\displaystyle { - \\frac { d } { d x } \\left ( \\ , \\ , a ^ \\ast \\frac { d } { d x } \\bar { u } ( x ) \\ , \\right ) } = f ( x ) \\ , \\ , , x \\in ( 0 , 1 ) \\\\ \\bar { u } ( 0 ) = 0 \\ , \\ , , \\bar { u } ( 1 ) = b . \\end{cases} \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\gamma \\left ( a , z \\right ) + \\Gamma \\left ( a , z \\right ) = \\Gamma \\left ( a \\right ) a , z > 0 . \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} { \\cal I } ( \\boldsymbol { x } ) = \\sum _ { j = 1 } ^ { d } \\bar { R } ^ { N _ j } _ j ( \\boldsymbol { x } ) W _ j ( \\boldsymbol { x } ) = \\sum _ { j = 1 } ^ { d } \\left ( \\sum _ { k = 1 } ^ { N _ j } ( \\sigma ^ { ( j ) } _ k ) ^ { - 1 } ( f _ { | \\Omega _ j } , { \\bar u } ^ { ( j ) } _ k ) _ { \\ell _ 2 ( X _ { N _ j } ) } { \\bar u } ^ { ( j ) } _ k ( \\boldsymbol { x } ) \\right ) W _ j ( \\boldsymbol { x } ) , \\boldsymbol { x } \\in \\Omega . \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\begin{aligned} ( S - \\operatorname { i d } ) \\left ( \\frac { u _ 1 + u } { 2 } \\right ) + D _ x \\left ( \\frac { 1 } { 2 } u ^ 2 + u '' \\right ) & = F _ 2 , \\\\ ( S - \\operatorname { i d } ) \\left ( \\frac { u u _ { - 1 } } { 2 } \\right ) + D _ x \\left ( \\frac { 1 } { 3 } u ^ 3 + u u '' - \\frac { 1 } { 2 } ( u ' ) ^ 2 \\right ) & = u F _ 2 , \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} \\limsup _ { | x | \\rightarrow \\infty } \\left \\langle \\frac { x } { | x | } , b ( x ) \\right \\rangle = \\limsup _ { | x | \\rightarrow \\infty } | x | ( 1 - | x | ^ 2 ) = - \\infty . \\end{align*}"}
-{"id": "8577.png", "formula": "\\begin{align*} \\Big | \\Big | Q ^ { ( \\mathcal { B } _ n ) } _ { I , \\mathbf { U } } Q ^ { ( \\mathcal { B } _ n ) } _ { \\mathbf { Z } | M = m , I , \\mathbf { U } } - Q ^ { ( \\mathcal { B } _ n ) } _ { I , \\mathbf { U } } Q ^ n _ { Z | U } \\Big | \\Big | _ { \\mathsf { T V } } \\xrightarrow [ n \\to \\infty ] { } 0 \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} \\chi = 2 \\lim _ { n \\to \\infty } \\mathbb { E } [ \\mathcal { D } ^ { i j } ( 0 ) \\mathcal { D } ^ { l m } ( - n , - 1 ) ] = \\begin{cases} - \\frac p { k - 1 } & i = l , j = m , \\\\ - \\frac p { 2 ( k - 1 ) } & i = l , j \\neq m , \\\\ 0 & i , j , l , m . \\end{cases} \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} \\mathbb { H } ^ 0 ( \\Gamma , k ) = \\mathcal { U } _ { - 2 } ^ + \\oplus \\mathbb { F } _ { ( - 2 ) } \\oplus \\mathbb { F } _ { ( 0 ) } \\oplus \\mathbb { F } _ { ( 0 ) } . \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} b ^ * ( x ) = x - \\delta , \\ x \\le 1 - \\alpha + \\delta , \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\Lambda _ i = \\frac { 1 } { \\rho + \\frac { 1 } { N } \\sum _ { l = 1 } ^ K \\frac { r _ { i i l } ^ { - \\alpha } } { 1 + \\Lambda _ i r _ { i i l } ^ { - \\alpha } } + \\frac { 1 } { N } \\sum _ { l = 1 } ^ { K ' } \\frac { r _ { i \\bar { i } l } ^ { - \\alpha } } { 1 + \\Lambda _ i r _ { i \\bar { i } l } ^ { - \\alpha } } } . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} \\left | \\frac { P } { \\sqrt { c } } \\right | ^ { 2 } - \\left | \\frac { Q } { \\sqrt { c } } \\right | ^ { 2 } \\ \\in \\ V = s p a n ( 1 , \\alpha , \\overline { \\alpha } , | \\alpha | ^ { 2 } , \\beta , \\overline { \\beta } , | \\beta | ^ { 2 } ) . \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} \\| f \\| _ { \\mathcal B ^ 2 } = \\limsup _ { T \\to \\infty } \\Big ( \\frac { 1 } { 2 T } \\int _ { - T } ^ T | f ( t ) | ^ 2 \\dd t \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} f ( P ) = k P ( 1 - P ) \\ , , k > 0 \\ , , \\end{align*}"}
-{"id": "1377.png", "formula": "\\begin{align*} \\partial _ t u + ( - \\Delta ) ^ { \\frac { \\theta } { 2 } } u = u ^ p , x \\in { \\bf R } ^ N , \\ , \\ , t > 0 , u ( 0 ) = \\mu \\ge 0 \\quad \\mbox { i n } \\quad { \\bf R } ^ N , \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} \\delta : = \\min _ { z \\in \\partial B _ 1 } \\max _ { j = 1 , \\ldots , m } d ( z , U ^ c _ j ) > 0 . \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} L ^ * = \\{ y \\in \\mathbb { R } ^ n : x \\cdot y \\in \\mathbb { Z } x \\in L \\} . \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} Q ( z ) d z ^ 2 = \\left ( \\frac { \\partial } { \\partial z } x \\circ \\pi ^ { - 1 } ( z ) \\right ) ^ 2 d z ^ 2 . \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} \\lambda = \\lambda _ 1 = \\lambda _ { - 1 } . \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} \\sum \\limits _ { \\theta \\in { \\rm I r r } ( N _ G ( P ) ) } \\theta ( J T ) \\theta ( J ) = q - 1 + b ( q - 1 ) + \\overline { b } ( q - 1 ) = 0 \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} G ^ { \\prime } ( h ) = \\dfrac { p ( h ) } { \\sqrt { M ( h ) } { \\large ( } J ( h ) + \\sqrt { M ( h ) } { \\large ) } ^ { 2 } } \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} 2 ^ { 1 2 } \\tilde { c } _ 4 ^ 3 - 2 ^ { 2 n } \\tilde { c } _ 6 ^ 2 = 2 ^ { 1 2 } 3 ^ 3 \\tilde { \\Delta } \\Leftrightarrow \\tilde { c } _ 4 ^ 3 - ( 3 \\tilde { \\Delta } ^ { 1 / 3 } ) ^ 3 = ( 2 ^ { n - 6 } \\tilde { c } _ 6 ) ^ 2 \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} h _ { 2 , 0 } ^ 0 = - \\frac { - 2 L _ 1 L _ 2 \\gamma \\hat { \\tilde { V } } _ { 1 , 0 } } { 1 + \\gamma \\hat { \\tilde { V } } _ { 2 , 0 } } . \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} i _ { t _ 0 } ^ * \\mathbf u = u ^ { t _ 0 } , \\ i _ t ^ * \\mathbf u \\in \\mathcal H _ { E _ t } ^ { p , q } , \\ \\forall \\ t \\in B , \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} \\lambda _ { t o p } \\leq \\tilde { c } \\int _ 0 ^ \\infty \\left ( \\frac { d } { d r } \\ , e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( r ^ 2 - 1 \\right ) ^ 2 } \\right ) \\ , d r = - \\tilde { c } \\ , e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } } < 0 . \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\C } 2 ^ { \\nu _ d ( \\pi ) } q ^ { | \\pi | } = \\sum _ { \\pi \\in \\R } \\mu ( \\pi ) q ^ { | \\pi | } . \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} X _ { H _ { \\varepsilon } } = ( \\partial _ y H _ { \\varepsilon } , - \\partial _ { \\theta } H _ { \\varepsilon } , \\partial _ x \\nabla _ z H _ { \\varepsilon } ) \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } u = L u , \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} R _ N ( x ) = e ^ { - c _ 1 \\sqrt { \\log x } } + \\ ( \\frac { c _ 2 N + 1 } { \\log x } \\ ) ^ { N + 1 } , \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} h _ T = \\frac { h _ L [ H ^ 1 ( K , \\hat { T } ) ] [ H ^ 0 _ T ( G , O _ L ^ { * } ) ] } { h _ K [ \\mathbb { I I I } ^ 1 ( T ) ] \\prod _ { p } [ H ^ 0 ( \\hat { \\mathbb { Z } } , H ^ 1 ( I _ p , \\hat { T } ) ) ] } . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} \\prod _ { B } \\frac { \\xi ( \\varepsilon _ b z + m _ b ) } { \\xi ( z + m _ b + j _ b ) } \\leq C ^ { | B | } \\prod _ { b : m _ b = 1 } q ( z ) ^ { - 1 } \\cdot \\log ( 5 + | t | ) \\cdot \\begin{cases} ( 1 + | t | ) ^ { - 0 . 0 2 } , \\mbox { s o m e $ j _ b \\neq 0 $ , } \\\\ ( 1 + | t | ) ^ { 0 . 0 1 / n ^ 2 } , \\mbox { a l l $ j _ b = 0 $ . } \\end{cases} \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} \\frac { 4 } { | x - y | ^ 2 } [ ( s - 1 ) ( 1 - 2 x \\cdot y + | x | ^ 2 | y | ^ 2 ) - ( 1 - | y | ^ 2 ) ( \\frac { N } { 2 } ( | y | ^ 2 - 2 x \\cdot y + 1 ) + ( 2 s - 2 - N ) ( 1 - x \\cdot y ) ) ] \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} ( u ^ { k } _ { t } , \\zeta ) = ( u _ { 0 } , \\zeta ) + \\int _ { 0 } ^ { t } ( u ^ { k } _ { s } , L ^ { k } _ { s } \\zeta ) \\ , d s + \\int _ { 0 } ^ { t } ( h _ { s } f _ { s } , \\zeta ) \\ , d s . \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} \\left ( \\bold { D } _ F \\right ) _ { \\alpha \\beta } ( Q ^ { \\beta } ) = \\sum _ { \\beta , J _ 1 , J _ 2 } \\frac { \\partial F _ { \\alpha } } { \\partial u _ { J _ 1 ; J _ 2 } ^ { \\beta } } \\left ( D _ { J _ 1 } S _ { J _ 2 } Q ^ { \\beta } \\right ) = \\bold { p r } X ( F _ { \\alpha } ) . \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} { \\mathcal A } _ s u = g , u ( t _ \\ell , \\cdot ) = 0 \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} ( \\bar { u } _ i - \\bar { u } _ { i + 1 } ) ^ 2 \\leq \\frac { 1 } { | A _ i | | A _ { i + 1 } | } \\int _ { A _ i } \\int _ { A _ { i + 1 } } | u ( x ) - u ( y ) | ^ 2 \\ , d x \\ , d y \\leq c r _ i ^ 2 \\frac { | A _ i \\cup A _ { i + 1 } | } { | A _ i | | A _ { i + 1 } | } \\int _ { A _ i \\cup A _ { i + 1 } } | \\nabla u | ^ 2 . \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} \\partial Q _ 1 & = \\{ x \\in \\partial Q : \\ x = ( \\pm 1 , x _ 2 ) , \\ | x _ 2 | \\leq 1 \\} \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} ( v - k _ j ) _ - > k _ j - k _ { j + 1 } = \\frac { 1 - a } { 2 ^ { j + 1 } } \\xi \\mu . \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} C _ { k k } ^ { i i } = C _ { l l } ^ { i i } \\mbox { f o r } k < l \\mbox { a n d } k \\neq i \\neq l , \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} \\lambda & W _ i ^ \\varepsilon ( x ) - \\frac { b ( x ) \\cdot D W _ i ^ \\varepsilon ( x ) } { \\varepsilon } + G ( x , D W _ i ^ \\varepsilon ( x ) ) \\\\ & = \\lambda v _ i \\circ H ( x ) - \\lambda a + \\varepsilon \\lambda \\psi _ i ( x ) + ( - 1 ) ^ i \\lambda C ( h _ i - H ( x ) ) - b ( x ) \\cdot D \\psi _ i ( x ) \\\\ & + G \\big ( x , v _ i ' \\circ H ( x ) D H ( x ) + \\varepsilon D \\psi _ i ( x ) - ( - 1 ) ^ i C D H ( x ) \\big ) , \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} | C ( m , n ; k ) | = \\Bigl ( \\frac { 1 } { | G _ { m } | } + o ( 1 ) \\Bigr ) | G _ { n } | \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} 6 | \\gamma _ 2 | \\leq 2 - \\frac { | c _ 1 | ^ 2 } { 2 } + \\frac { 1 } { 8 } \\sqrt { ( d ^ 2 + 1 - 2 d t ) ( d ^ 2 + 9 + 6 d t ) } = : k ( d , q ) . \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { H } ( 0 , X ) } w ^ { - z } ( \\log w ) ^ j e ^ w d w = ( - 1 ) ^ j \\frac { d ^ j } { d z ^ j } \\ ( \\frac { 1 } { \\Gamma ( z ) } \\ ) + E _ { j , z } ( X ) , \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} J _ \\kappa ( g , z ) = \\det ( g ) ^ { \\kappa _ 1 - I } j ( g , z ) ^ { \\kappa _ 2 - \\kappa _ 1 + I } , \\mbox { f o r } g \\in G ( \\R ) \\mbox { a n d } z \\in \\mathfrak { H } ^ I , \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} \\begin{aligned} v _ s ( x ) & = x , \\\\ u _ s ( t ) & = \\partial _ x v _ s ( x , t ) = 1 , \\\\ \\gamma _ s ( t ) & = t + \\gamma _ 0 , \\\\ \\sigma _ s ( t ) & = \\varphi ( t + \\gamma _ 0 ) \\ , , \\end{aligned} \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} p = \\frac { 2 c ^ 4 + 2 c ^ 3 - 7 c ^ 2 - 1 2 c - 5 } { 3 c \\left ( c ^ 3 - 2 c - 1 0 \\right ) } . \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\rm w } \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f = \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f . \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} \\pi ( w _ 2 ^ { l + 1 } | x ) = { \\P ( [ X _ 2 ^ { l + 1 } ] _ b = w ^ { l + 1 } _ 2 | X _ 1 = x ) } , \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} r _ k ' = i ( p _ k \\bar u - \\bar p _ k u ) . \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} \\begin{array} { r l } Z ^ { D W } = \\max \\ & c ' V \\lambda + d ' u \\\\ \\mbox { s . t . } \\ & \\lambda \\cdot 1 = 1 , \\\\ & H V \\lambda + G u \\leq h , \\\\ & \\lambda \\geq 0 . \\end{array} \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} X _ r = \\frac { \\partial } { \\partial x '^ r } + \\sum _ { i = 1 } ^ { r - 1 } f _ i \\frac { \\partial } { \\partial x ^ i } , \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} G _ i = \\sum _ { \\substack { 1 \\leq j \\leq i \\\\ d _ i = d _ j } } \\lambda _ { i j } F _ j + \\sum _ { \\substack { 1 \\leq j \\leq i \\\\ d _ i > d _ j } } \\sum _ { 1 \\leq k \\leq n } \\lambda _ { i j k } x _ k ^ { d _ i - d _ j } F _ j \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} ( D \\varphi _ t ( \\omega , x ) v ) ^ { ( 1 ) } = v ^ { ( 1 ) } \\ , e ^ { \\int _ 0 ^ t ( 1 - 3 | \\varphi _ s ( \\omega , x ) | ^ 2 ) \\ , d s } \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq x \\\\ p \\equiv a \\bmod q } } \\frac { 1 } { p } - \\frac { \\log \\log x } { \\phi ( q ) } = \\frac { 1 } { p ( q , a ) } + O \\ ( \\frac { \\log 2 q } { \\phi ( q ) } \\ ) , \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} t ^ j = \\iota _ j ^ * t + \\sum \\limits _ { \\begin{subarray} { c } \\vec { \\Gamma } ~ ~ ~ d _ { s _ 1 } = 0 , \\\\ p _ { s _ 1 } = j , n _ { s _ 1 } = 0 , ( s _ 1 ) = 1 \\end{subarray} } Q ^ { d _ \\Gamma } _ { \\vec \\Gamma } ( z ) , \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\| R _ { \\hbar } \\| _ { L ^ \\infty ( \\Gamma ) } = \\mathcal { O } ( \\hbar ^ 4 ) \\ , , \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} p ( t ) = \\begin{cases} { g } \\cdot \\alpha _ t ( g ) & \\mid t \\in [ - 2 r , 2 r ] \\\\ 0 & \\mid t \\not \\in [ - 2 r , 2 r ] . \\end{cases} \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } J _ { r } ( \\varphi _ { n } , u _ { n } , \\tau _ { n } ) = \\inf _ { ( \\phi , w , s ) } J _ { r } ( \\phi , w , s ) . \\end{align*}"}
-{"id": "5070.png", "formula": "\\begin{align*} F _ { q } ^ { 2 } - F _ { q + 1 } F _ { q - 1 } = \\left ( - 1 \\right ) ^ { q } , \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} N = N ' A ^ s = A ' , \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} \\sum _ { m _ 1 = 0 } ^ { r p - 1 } \\sum _ { m _ 2 = 0 } ^ { s p - 1 } \\sum _ { m _ 3 = 0 } ^ { t p - 1 } \\binom { m _ 1 + m _ 2 + m _ 3 } { m _ 1 , m _ 2 , m _ 3 } \\equiv _ { p ^ 3 } \\sum _ { m _ 1 = 0 } ^ { r - 1 } \\sum _ { m _ 2 = 0 } ^ { s - 1 } \\sum _ { m _ 3 = 0 } ^ { t - 1 } \\binom { m _ 1 + m _ 2 + m _ 3 } { m _ 1 , m _ 2 , m _ 3 } . \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} Y ( X Y + 2 q _ 4 Z ^ 2 ) = X ^ 3 + p _ 2 X ^ 2 Z + p _ 4 X Z ^ 2 + p _ 6 Z ^ 3 . \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} \\Phi & = \\Phi _ { \\beta } ^ { \\alpha } \\frac { \\partial } { \\partial x ^ { \\alpha } } \\otimes d x ^ { \\beta } \\\\ & = \\left ( g ^ { \\alpha \\gamma } \\varphi _ { \\gamma \\beta } - g ^ { \\alpha \\gamma } B _ { \\gamma \\beta } \\right ) \\frac { \\partial } { \\partial x ^ { \\alpha } } \\otimes d x ^ { \\beta } , \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} 2 \\pi b _ n ^ 2 \\exp ( b _ n ^ 2 ) = n ^ 2 . \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\sum _ { i \\in J - I } ( - 1 ) ^ { \\epsilon ( i , I ) + \\epsilon ( i , J ) } \\ , \\Delta _ { I \\cup \\{ i \\} } \\Delta _ { J - \\{ i \\} } \\ = \\ 0 . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} f \\left ( \\frac { a z + b } { c z + d } \\right ) = ( c z + d ) ^ k f ( z ) \\ ; \\ ; \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in \\mathrm { S L } _ 2 ( \\mathbb { Z } ) , \\ ; z \\in \\mathbb { H } , \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} \\dot Y _ n ( t ) = \\begin{cases} \\cfrac { b ( Y _ n ( t ) ) } { \\varepsilon } - ( - 1 ) ^ i \\nu \\ , \\cfrac { D H ( Y _ n ( t ) ) } { | D H ( Y _ n ( t ) ) | } \\ \\ \\ & t \\in ( 0 , s _ n ) , \\\\ \\cfrac { b ( Y _ n ( t ) ) } { \\varepsilon } + \\alpha _ n ( t - s _ n ) \\ \\ \\ & t \\in ( s _ n , t _ n ) , \\end{cases} \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\left | F _ n ^ \\ast ( k ) + F _ n ^ \\ast ( \\ell ) - 1 - f ^ \\ast ( k ) \\right | & = \\left | F _ n ^ \\ast ( k ) - F ^ \\ast ( k ) + F _ n ^ \\ast ( \\ell ) - 1 + F ^ \\ast ( k - 1 ) \\right | \\\\ & \\le \\left | F _ n ^ \\ast ( k ) - F ^ \\ast ( k ) \\right | + \\left | F _ n ^ \\ast ( \\ell ) - F ^ \\ast ( \\ell ) \\right | , \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} F ( y ) & = \\int _ { 0 } ^ { 1 } \\nabla F ( ( 1 - t ) y + t y _ 1 ( y ) ) \\cdot ( y _ 1 ( y ) - y ) \\ , d t \\\\ & \\geq - d ( y ) \\int _ { 0 } ^ { 1 } | \\nabla F ( ( 1 - t ) y + t y _ 1 ( y ) ) | \\ , d t . \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P _ o } ( - 1 ) ^ { \\nu _ e ( \\pi ) } z ^ { \\nu _ o ( \\pi ) } q ^ { | \\pi | } = \\frac { 1 } { 1 - z q } - 1 , \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} \\eta _ 0 = d z + \\sum _ { j = 1 } ^ n x _ j \\ , d y _ j . \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} \\mathcal { P } _ { - i } = \\bigotimes _ { j = 1 , j \\neq i } ^ { K } \\mathcal { P } _ { j } , \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} Y = X + Q _ { \\ast } ^ { \\alpha } ( x , n , [ u ] , [ v ] ) \\partial _ { v ^ { \\alpha } } \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} { h } ( \\theta ) = \\left ( \\sum _ { j = 0 } ^ { \\infty } a _ j e ^ { i j \\theta } \\right ) - \\left ( \\sum \\limits _ { j = 0 } ^ { \\infty } ( \\bold { B } ^ { - 1 } \\bold { a } ) _ j e ^ { i j \\theta } \\right ) \\left ( b _ { - 1 } e ^ { - i \\theta } + b _ 0 + b _ 1 e ^ { i \\theta } \\right ) , \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} \\widetilde { M } _ k ( x ; \\boldsymbol { a } ) = \\sum _ { \\boldsymbol { a } \\in \\mathcal { A } ^ { \\boldsymbol { k } } _ { \\boldsymbol { b } } } M _ k ( x ; \\boldsymbol { a } ) . \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} & \\rho \\left ( \\mathbb { R } ^ { n } \\times \\left ( \\mathbb { R } ^ { d - n } \\setminus \\left \\{ 0 \\right \\} \\right ) \\right ) = 0 \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} T = A ^ { \\frac 1 2 } ( I - B ( 0 ) ) A ^ { \\frac 1 2 } , \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} \\langle u , \\ G _ { \\epsilon } ( z ) v \\rangle _ { { \\mathcal G } ^ { - 1 } , { \\mathcal G } ^ 1 } = \\overline { \\langle v , \\ G _ { - \\epsilon } ( \\bar { z } ) u \\rangle _ { { \\mathcal G } ^ { - 1 } , { \\mathcal G } ^ 1 } } . \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} \\Delta L ^ * _ { i } = \\sqrt { ( y ^ * _ { i + 1 } - y ^ * _ i ) ^ 2 + ( \\Delta { x ^ * } ) ^ 2 } \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\partial _ t v & = & { \\displaystyle \\frac { 1 } { 2 } \\sum _ { i , j = 1 } ^ d \\partial _ { i , j } ^ 2 \\big ( v ( \\Phi \\Phi ^ t ) _ { i , j } ( t , x , v ) \\big ) } - d i v \\big ( v g ( t , x , v ) \\big ) + v \\Lambda ( t , x , v ) \\ , \\\\ v ( 0 , x ) & = & B _ m ( 2 , x ) f _ { \\mu , A } ( x ) \\quad \\textrm { f o r a l l } \\ x \\in \\R ^ d \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} _ \\delta ( f ) : = \\sup \\big \\{ \\Vert f ( t ) - f ( s ) \\Vert _ 2 \\ \\big | \\ s , t \\in [ 0 , T ] , | t - s | \\leq \\delta \\big \\} \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} I _ { j , l } \\ll _ m \\Gamma ( l + 1 ) = l ! . \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} \\| u - \\pi _ N u \\| _ 0 \\le N ^ { - 1 } | u - \\pi _ N u | _ { \\C { H } ^ 1 ( \\Omega ) } , \\mbox { w h e r e } \\| v \\| _ 0 = \\left ( \\int _ \\Omega \\frac { v ^ 2 ( \\tau ) } { 1 - \\tau ^ 2 } \\ ; d \\tau \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} \\mathcal { H } _ { s } = \\{ x \\mapsto \\pm ( \\alpha \\cdot x - t ) ^ { s - 1 } _ { + } : \\| \\alpha \\| _ 1 = 1 , \\ ; | t | \\leq 1 \\} . \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} n _ c = \\min \\{ v ( c _ i ) : i \\} , \\ n _ 0 = \\min \\{ n _ c , v ( c ) \\} . \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} e ^ { t \\lambda _ { + } + \\frac { t } { \\nu } | \\xi | ^ { 2 ( 1 - \\sigma ) } } - 1 = \\left ( t \\lambda _ { + } + \\frac { t } { \\nu } | \\xi | ^ { 2 ( 1 - \\sigma ) } \\right ) e ^ { \\theta ( t \\lambda _ { + } + \\frac { t } { \\nu } | \\xi | ^ { 2 ( 1 - \\sigma ) } ) } \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{gather*} \\omega = \\delta A \\wedge \\delta A ^ \\ast + \\delta C \\wedge \\delta C ^ \\ast , \\operatorname { g h } ( \\omega ) = - 1 , \\end{gather*}"}
-{"id": "8613.png", "formula": "\\begin{align*} R _ 1 - \\delta _ 1 & > d _ { \\alpha _ 1 } ( Q _ { U , W } , Q _ U , Q _ W ) \\geq d _ 1 ( Q _ { U , W } , Q _ U , Q _ W ) = I ( U ; W ) , \\\\ R _ 1 + R _ 2 - \\delta _ 2 & > d _ { \\alpha _ 2 } ( Q _ { U , V , W } , Q _ { U , V } , Q _ W ) \\geq d _ 1 ( Q _ { U , V , W } , Q _ { U , V } , Q _ W ) = I ( U , V ; W ) . \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} \\mathcal { G } _ n \\setminus \\mathcal { G } _ { n + 1 } = \\bigcup _ { l \\in \\mathbb { Z } ^ { \\nu } , j , k \\in S ^ c \\cup \\{ 0 \\} } R _ { l j k } ( i _ n ) \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 } \\frac { { \\bar \\nu _ \\alpha } } { | \\log \\alpha | } = 0 . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} \\mathbb { P } \\Bigg ( \\max _ { m \\in \\mathcal { M } _ n } \\Big | \\Big | Q ^ { ( \\mathsf { B } _ n ) } _ { I , \\mathbf { U } } Q ^ { ( \\mathsf { B } _ n ) } _ { \\mathbf { Z } | M = m , I , \\mathbf { U } } - Q ^ { ( \\mathsf { B } _ n ) } _ { I , \\mathbf { U } } Q ^ n _ { Z | U } \\Big | \\Big | _ { \\mathsf { T V } } > e ^ { - n \\gamma _ 1 } \\Bigg ) \\leq e ^ { - e ^ { n \\gamma _ 2 } } . \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\sum _ { t \\in [ s , T ] } q ^ { Q , [ s , T ] } ( t ) \\ , \\tfrac { ( T - t ) ^ k } { k ! } = \\int _ s ^ T \\tfrac { ( T - t ) ^ k } { k ! } \\ , d t = \\tfrac { ( T - s ) ^ { k + 1 } } { ( k + 1 ) ! } . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} \\rho _ { 1 } & = \\left \\vert \\alpha + \\beta \\right \\vert & \\theta _ { 1 } & = A r g ( \\alpha + \\beta ) \\in ( - \\pi , \\pi ] \\\\ \\rho _ { 2 } & = \\left \\vert \\alpha \\beta - \\gamma \\right \\vert > 0 & \\theta _ { 2 } & = A r g ( \\alpha \\beta - \\gamma ) \\in ( - \\pi , \\pi ] \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} \\frac { \\| \\prod _ { k = i + 1 } ^ { q _ { n _ l } - 1 } D ^ k \\| } { | \\prod _ { k = i } ^ { q _ { n _ l } - 1 } f _ k | } = \\frac { \\| \\prod _ { k = i + 1 } ^ { q _ { n _ l } - 1 } D ^ k \\| \\cdot | \\prod _ { k = 0 } ^ { i - 1 } f _ k | } { | \\prod _ { k = 0 } ^ { q _ { n _ l } - 1 } f _ k | } \\leq q _ { n _ l + 1 } e ^ { ( L ( E ) - \\delta _ c + 3 \\epsilon ) q _ { n _ l } } . \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} \\mathbf m = \\mathbf X \\mathbf C = \\mathbf X \\mathbf N \\mathbf N ^ * \\mathbf C ^ + . \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} I \\left ( x _ { 2 } , . . . , x _ { n } \\right ) = \\pm 1 \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} u _ n ( t , x ) = \\int _ 0 ^ t a ^ { i j } _ n ( s ) D _ { i j } u _ n ( s , x ) d s + \\int _ 0 ^ t f ( s , x ) d s , \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} ( r _ i ) _ i \\to \\left ( \\sum _ { j = 0 } ^ i p ^ j r _ j ^ { p ^ { i - j } } \\right ) _ i . \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} \\lambda _ c = \\frac { ( \\Delta - 1 ) ^ { \\Delta - 1 } } { ( \\Delta - 2 ) ^ \\Delta } , \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} \\| f \\| _ { 1 } = | \\hat { f } ( 0 ) | \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} \\gamma ^ 1 _ { 4 4 } & = 1 - \\frac { k } { 4 \\ell } e ^ { - y } + O ( e ^ { - 2 y } ) , & \\gamma ^ 2 _ { 3 3 } & = - \\cot ( \\phi ) + O ( e ^ { - 2 y } ) = O ( e ^ { - \\frac { 3 y } { 4 } } ) , \\\\ \\gamma _ { 2 3 } ^ 4 & = \\frac { k } { 4 \\ell } e ^ { - y } + O ( e ^ { - 2 y } ) , & \\gamma _ { 3 2 } ^ 4 & = - \\frac { k } { 4 \\ell } e ^ { - y } + O ( e ^ { - 2 y } ) , \\\\ \\gamma _ { 4 2 } ^ 3 & = - \\frac { k } { 4 \\ell } e ^ { - y } + O ( e ^ { - 2 y } ) . \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} E [ S ^ * ( 0 ) ] = \\frac { 1 2 \\mu - \\lambda } { 8 \\mu ( \\mu - \\lambda ) } E [ D ^ * ( 0 ) ] = \\frac { \\lambda ( 4 \\mu - \\lambda ) } { 4 \\mu ( \\mu - \\lambda ) } . \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} \\mathbf { P } _ f \\mathbf { b } _ i = \\Delta \\mathbf { x } _ i , \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} \\sharp ( T N ^ { \\circ } ) = T N \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} g ( x ) = \\sum _ { y \\in \\Lambda } f ( x - y ) \\ ; , \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{align*} J ( h _ { + } ) = \\dfrac { 2 p _ { 1 } \\left ( s - v \\right ) } { s ^ { 2 } } ( s - 2 h _ { + } ) \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} L ( \\lambda | \\lambda ' ) = \\frac { K ( \\lambda - \\lambda ' ) } { 2 \\pi } + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) L ( \\nu | \\lambda ' ) . \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} v ( t ^ { 2 } - s ^ { 2 } ) - 2 w s t = 0 \\end{align*}"}
-{"id": "2796.png", "formula": "\\begin{align*} j ' _ * \\circ \\pi '^ * \\circ \\rho ^ { \\rm s e m i } _ { Y ' } ( \\alpha ' ) = \\pi '^ * \\circ \\rho ^ { \\rm s e m i } _ { X ' } \\circ j ' _ * ( \\alpha ' ) = 0 . \\end{align*}"}
-{"id": "3148.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { K } \\sum _ { \\ell = 0 } ^ { L } \\mathcal { F } ^ \\ast ( k ) \\mathcal { F } ^ \\ast ( \\ell ) \\psi ( k , \\ell ) \\mathcal { E } ( k , \\ell ) , \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} \\mathrm { e x p } { \\left ( \\frac { 2 \\pi i t } { 8 L } \\right ) } \\cdot \\xi ^ { ( l , \\pm ) } ( \\Phi _ t ( y ) ) = \\mathrm { e x p } { \\left ( \\frac { 2 \\pi i 0 } { 8 L } \\right ) } \\cdot \\xi ^ { ( l , \\pm ) } ( \\Phi _ 0 ( y ) ) = \\xi ^ { ( l , \\pm ) } ( y ) \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{align*} \\tau _ { \\epsilon } = \\epsilon ^ { \\mathsf { x } } \\tau _ { \\epsilon } ^ { s G } , \\kappa _ { \\epsilon } = \\epsilon ^ { \\mathsf { x } } \\kappa _ { \\epsilon } ^ { s G } , \\zeta _ { \\epsilon } = \\left ( \\zeta _ { \\epsilon } ^ { s G } \\right ) ^ { \\epsilon ^ { \\mathsf { x } } } \\epsilon = \\pm , \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} K _ { G \\left ( x , b ; c \\right ) } ^ { - } \\left ( u \\right ) & = \\frac { \\Gamma \\left ( x / b + 1 \\right ) } { \\gamma \\left ( x / b + 1 , c / b \\right ) } K _ { G \\left ( x , b ; c \\right ) } ^ { L } \\left ( u \\right ) \\\\ K _ { G \\left ( x , b ; c \\right ) } ^ { + } \\left ( u \\right ) & = \\frac { \\Gamma \\left ( x / b + 1 \\right ) } { \\Gamma \\left ( x / b + 1 , c / b \\right ) } K _ { G \\left ( x , b ; c \\right ) } ^ { U } \\left ( u \\right ) . \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} \\partial _ i v _ { \\delta } [ \\hat { \\imath } ] = \\sum _ { j \\in S } \\sqrt { \\lvert j \\rvert } \\sqrt { \\xi _ j + \\varepsilon ^ { 2 ( b - 1 ) } ( y _ { \\delta } ) _ j } e ^ { \\mathrm { i } ( \\theta _ 0 ) _ j } \\left ( \\mathrm { i } \\hat { \\Theta } _ j + \\varepsilon ^ { 2 ( b - 1 ) } \\frac { \\hat { y } _ j + ( \\partial _ i G ( i _ 0 ( \\varphi ) ) [ \\hat { \\imath } ] ) _ j } { 2 \\ , \\lvert j \\rvert \\ , ( \\xi _ j + \\varepsilon ^ { 2 ( b - 1 ) } ( y _ { \\delta } ) _ j ) } \\right ) \\ , e ^ { \\mathrm { i } j x } \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} p = \\frac { 3 c ^ 2 r ^ 2 + c ^ 2 - 1 2 r ^ 2 + 4 } { 6 \\left ( c ^ 2 + 4 \\right ) r } . \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} Q _ { 1 } \\left ( x , y , z \\right ) & = x ^ { 2 } + 2 c x y + 2 y ^ { 2 } + 4 \\left ( c ^ { 2 } - 1 \\right ) x z + 6 c y z + z ^ { 2 } \\left ( 4 c ^ { 2 } + 5 \\right ) \\\\ Q _ { 2 } \\left ( x , y , z \\right ) & = y ^ { 2 } - x z + 2 c y z + 2 z ^ { 2 } . \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} d _ { n , k } ^ { m , l } & = \\binom { l } { l - k } ( ( \\theta + m - 1 ) _ { m - n \\downarrow } ) ^ { k } ( \\sum _ { j = 1 } ^ { m - n } ( \\alpha + \\theta ) _ { m - j \\uparrow } ( \\theta + m - 1 ) _ { j - 1 \\downarrow } ) ^ { l - k } \\\\ & = \\binom { l } { k } ( ( \\theta + n ) _ { m - n \\uparrow } ) ^ { k } ( \\sum _ { j = 1 } ^ { m - n } ( \\alpha + \\theta ) _ { m - j \\uparrow } ( \\theta + m - 1 ) _ { j - 1 \\downarrow } ) ^ { l - k } . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} \\frac { t } { ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - 1 } ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } = \\sum _ { n = 0 } ^ \\infty B _ { n , \\lambda } ^ * ( x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} \\tilde \\theta = \\left \\{ \\begin{array} { c c } - M & ~ ~ ~ \\theta < - M ; \\\\ \\theta & ~ ~ ~ | \\theta | < M ; \\\\ M & ~ ~ ~ \\theta > M . \\end{array} \\right . \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} \\frac { 1 } { N } = M _ \\ell \\left ( \\frac { 1 } { \\ell s _ \\theta } \\right ) = \\frac { 1 } { \\ell } \\ , M _ 1 \\left ( \\frac { 1 } { s _ \\theta } \\right ) . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{gather*} s _ 1 = \\overline { s _ 3 } , s _ 2 = \\overline { s _ 2 } . \\end{gather*}"}
-{"id": "721.png", "formula": "\\begin{align*} D _ 2 = \\{ ( x _ 1 , x _ 2 ) \\ ; | \\ ; x _ 1 > 0 , \\ ; x _ 2 > 0 , \\ ; e ^ { x _ 1 } + e ^ { x _ 2 } - e ^ { x _ 1 + x _ 2 } > 0 \\} . \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} \\mathcal I _ s ^ { \\beta , q } ( \\mu ) \\leq \\iint _ { \\underline a , \\underline b : \\underline a \\wedge \\underline b = \\phi } \\sum _ { n = 0 } ^ { \\infty } \\sum _ { c _ 1 \\cdots c _ n } \\frac { \\mu [ c _ 1 \\cdots c _ n ] ^ q } { d ( c _ 1 \\cdots c _ n \\underline a , c _ 1 \\cdots c _ n \\underline b ) ^ { s ( q - 1 ) } } d \\mu ( \\underline a ) d \\mu ( \\underline b ) . \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} d _ { \\lambda \\mu } ( q ) = q ^ { \\delta ( \\lambda , \\mu ) } \\sum _ { \\substack { \\alpha ^ 0 , \\dotsc , \\alpha ^ e \\\\ \\beta ^ 0 , \\dotsc , \\beta ^ { e - 1 } } } \\prod _ { 0 \\leq j \\leq e - 1 } c ^ { \\mu ^ { ( j ) } } _ { \\alpha ^ j \\beta ^ j } c ^ { \\lambda ^ { ( j ) } } _ { \\beta ^ j ( \\alpha ^ { j + 1 } ) ' } , \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} | \\partial \\varphi | _ { \\tilde { g } } ^ { 2 } + \\sum _ { i } \\tilde { g } ^ { i \\overline { i } } & \\geq C ^ { - 1 } \\sum _ { i } \\left ( \\frac { | \\varphi _ { i } | ^ { 2 } } { \\tilde { g } _ { i \\overline { i } } } + \\tilde { g } _ { i \\overline { i } } ^ { \\frac { 1 } { n - 1 } } \\right ) \\\\ & \\geq C ^ { - 1 } \\sum _ { i } | \\varphi _ { i } | ^ { \\frac { 2 } { n } } \\\\ & \\geq C ^ { - 1 } | \\partial \\varphi | _ { g } ^ { \\frac { 2 } { n } } , \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} \\mathcal { U } = \\left \\{ u : X ^ { r e g } \\to \\R : u { \\rm \\ , \\ , i s \\ , \\ , o f \\ , \\ , t y p e \\ , \\ , U } , \\ ; \\int _ { X ^ { r e g } } u \\omega ^ n = 0 , \\ ; \\omega + i \\partial \\bar \\partial u > 0 \\right \\} , \\\\ \\mathcal { F } = \\left \\{ f : X ^ { r e g } \\to \\R : f { \\rm \\ , \\ , i s \\ , \\ , o f \\ , \\ , t y p e \\ , \\ , F } , \\ ; \\int _ { X ^ { r e g } } ( e ^ f - 1 ) \\omega ^ n = 0 \\right \\} . \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{gather*} \\xi _ t = \\frac { 1 } { 3 } \\lambda . \\end{gather*}"}
-{"id": "556.png", "formula": "\\begin{align*} \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 = Q ^ { \\alpha } F _ { \\alpha } . \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} - g ( [ S _ 1 , S _ 2 ] , J S _ 3 ) - g ( [ S _ 1 , J S _ 2 ] , S _ 3 ) - g ( [ J S _ 1 , S _ 2 ] , S _ 3 ) = 0 . \\end{align*}"}
-{"id": "988.png", "formula": "\\begin{align*} - \\Delta F _ { N , s } ( x ) & = - \\kappa _ { N , s } ( \\Delta | x | ^ { 2 s - N } \\ln | x | + 2 \\nabla | x | ^ { 2 s - N } \\nabla \\ln | x | + | x | ^ { 2 s - N } \\Delta \\ln | x | ) \\\\ & = \\kappa _ { N , s - 1 } | x | ^ { 2 s - N - 2 } \\ln | x | + C _ 2 | x | ^ { 2 s - N - 2 } = F _ { N , s - 1 } + C _ 2 | x | ^ { 2 s - N - 2 } , \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} u _ t = u _ { x x } + u g ( v ) , v _ t = \\epsilon v _ { x x } + \\kappa u g ( v ) , \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 = Q ^ { \\alpha } \\bold { E } _ { \\alpha } ( L ) . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\int _ { \\gamma } \\ast d h = \\int _ { \\gamma } \\frac { \\partial h } { \\partial n } d s . \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} F ^ { ( 4 ) } ( u ) : = \\sum _ { \\substack { j _ 1 + j _ 2 + j _ 3 + j _ 4 = 0 , \\\\ a t \\ , \\ , l e a s t \\ , \\ , 3 \\ , \\ , i n d i c e s \\ , \\ , b e l o n g \\ , \\ , t o \\ , \\ , S } } F ^ { ( 4 ) } _ { j _ 1 j _ 2 j _ 3 j _ 4 } \\ , u _ { j _ 1 } u _ { j _ 2 } u _ { j _ 3 } u _ { j _ 4 } , \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} \\mathcal H ^ { p , q } _ E : = \\{ H ^ { p , q } ( E _ t ) \\} _ { t \\in B } , \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} h ( t ) = \\left [ \\begin{array} { c c } 1 _ { \\widetilde { V } } & 0 \\\\ 0 & t ^ { - 1 } 1 _ { \\widehat { V } } \\end{array} \\right ] , h ' ( t ) = \\left [ \\begin{array} { c c } t 1 _ { \\widetilde { V } } & 0 \\\\ 0 & t ^ { - 1 } 1 _ { \\widehat { V ' } } \\end{array} \\right ] . \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} \\lim _ n \\frac { | F _ n \\bigtriangleup g F _ n | } { | F _ n | } = 0 , \\textrm { f o r a l l $ g \\in G $ } . \\end{align*}"}
-{"id": "3813.png", "formula": "\\begin{align*} e _ { 2 j + 1 } = - e _ { 2 j } , \\ e _ { 2 j } ^ 2 = e _ { 2 j + 1 } ^ 2 = e _ j , \\ \\ \\ j \\ge 0 , \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{gather*} z _ 4 ( w ) : = b _ 1 w b _ 2 c ^ { n _ w + 1 } b _ 2 \\ , b _ 1 w b _ 2 c ^ { n _ w + 2 } b _ 2 \\ , b _ 1 w b _ 2 c ^ { n _ w + 3 } b _ 2 \\ , b _ 1 w b _ 2 c ^ { n _ w + 4 } b _ 2 \\end{gather*}"}
-{"id": "7149.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { \\int _ { r T } ^ { T } \\chi _ { Q } ( ( y , [ g ] ) \\Delta ( u _ t ) ) d t } { \\int _ { r T } ^ { T } \\chi _ { Q _ { + } } ( ( y , [ g ] ) \\Delta ( u _ t ) ) d t } = \\frac { \\mu ( Q ) } { \\mu ( Q _ { + } ) } . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} c \\beta + \\sum _ { ( i , p ) = 1 } c _ i [ T '^ { - i } ] \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty V u ^ 2 \\ , \\dd x = \\frac { \\pi } { \\mu ^ \\frac 1 2 } \\int _ 0 ^ \\infty V \\ ; \\dd x \\ ; ( 1 + o ( 1 ) ) , \\mu \\to \\infty . \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { N - 1 } A _ { - 1 } ^ { k } \\left ( B u ( N - k + 0 ) - B u ( N - k - 0 ) \\right ) . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} R _ { a , b } ( z _ a / z _ b ) R _ { a , c } ( z _ a / z _ c ) R _ { b , c } ( z _ b / z _ c ) = R _ { b , c } ( z _ b / z _ c ) R _ { a , c } ( z _ a / z _ c ) R _ { a , b } ( z _ a / z _ b ) \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} Q _ { 2 0 } = & \\frac { 1 / 2 0 } { r ^ { 1 9 } } * ( - 1 6 7 9 6 r ^ 9 + 2 6 4 5 3 7 0 r ^ 8 - 6 8 6 4 3 9 6 0 r ^ 7 \\\\ & + 6 8 6 4 3 9 6 0 0 r ^ 6 - 3 4 4 2 0 0 4 2 8 0 r ^ 5 + 9 7 0 4 5 3 9 8 4 5 r ^ 4 - 1 6 0 8 7 7 3 9 8 5 0 r ^ 3 \\\\ & + 1 5 5 5 7 3 7 4 8 0 0 r ^ 2 - 8 1 1 9 8 5 7 9 0 0 r + 1 7 6 7 2 6 3 1 9 0 ) \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} f | _ x ^ y = \\sum _ { x \\le u \\le v \\le y } f ( u , v ) e _ { u v } . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} G ( y , 2 t - s ) & = ( 2 t - s ) ^ { - \\frac { N } { \\theta } } G \\left ( \\frac { y } { ( 2 t - s ) ^ { \\frac { 1 } { \\theta } } } , 1 \\right ) \\\\ & \\ge \\left ( \\frac { s } { 2 t } \\right ) ^ { \\frac { N } { \\theta } } s ^ { - \\frac { N } { \\theta } } G \\left ( \\frac { y } { s ^ { \\frac { 1 } { \\theta } } } , 1 \\right ) = \\left ( \\frac { s } { 2 t } \\right ) ^ { \\frac { N } { \\theta } } G ( y , s ) \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} C _ j ^ { ( k ) } ( 2 x ^ 2 - 1 ) = \\int _ { - 1 } ^ 1 C _ { 2 j } ^ { ( 2 k ) } ( u x ) \\mu ^ { k } ( d u ) , \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} m + o ( 1 ) & = I ( u _ k ) \\\\ & \\geq \\frac { 1 } { b p ^ 2 } M _ { u _ k } ^ p - \\frac { 1 } { \\mu } \\int _ 0 ^ T f ( t , u _ k ( t ) ) u _ k ( t ) d t - c T - \\frac { a ^ p } { b p ^ 2 } \\\\ & = \\frac { 1 } { b p ^ 2 } M _ { u _ k } ^ p - \\frac { 1 } { \\mu } M _ { u _ k } ^ { p - 1 } \\| u _ k \\| _ { E ^ { \\alpha , p } } ^ p - c T - \\frac { a ^ p } { b p ^ 2 } \\\\ & = M _ { u _ k } ^ { p - 1 } \\left ( \\left ( \\frac { 1 } { p ^ 2 } - \\frac { 1 } { \\mu } \\right ) \\| u _ k \\| _ { E ^ { \\alpha , p } } ^ p + \\frac { a } { b p ^ 2 } \\right ) - c T - \\frac { a ^ p } { b p ^ 2 } . \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} \\sigma = \\star X & & \\\\ \\pi = \\tfrac { 1 } { 2 } \\delta ( X \\wedge \\star X ) - \\delta ( X ) ( \\star X ) & & \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} \\begin{bmatrix} G L _ \\ell & * & * \\\\ & G L _ { m - \\ell } & * \\\\ & & G L _ { n - m } \\end{bmatrix} \\leq \\begin{bmatrix} G L _ { \\ell } & * \\\\ & G L _ { n - \\ell } \\end{bmatrix} \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} \\left | t - t ' + 2 \\sum _ { j = 1 } ^ n ( x _ j ' y _ j - x _ j y _ j ' ) \\right | ^ { 1 / 2 } \\leq C d ( p , q ) \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} u _ 1 ^ { ( 8 ) } + \\lambda [ a ^ { 1 1 } + a ^ { 2 2 } ] u _ 1 ^ { ( 4 ) } + \\lambda ^ 2 \\left [ a ^ { 1 1 } a ^ { 2 2 } - a ^ { 1 2 } a ^ { 2 1 } \\right ] u _ 1 = 0 , t > 0 , \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{gather*} I _ 0 = 2 r _ 0 + U - e _ 1 q _ 2 + 2 q _ 1 , I _ 1 = 2 r _ 1 - 1 - q _ 2 , \\end{gather*}"}
-{"id": "5353.png", "formula": "\\begin{align*} M _ x [ v _ { \\delta } ^ 3 ] - M _ { \\varphi , x } [ v _ { \\delta } ^ 3 ] = ( M _ x [ \\overline { v } ^ 3 ] - M _ { \\varphi , x } [ \\overline { v } ^ 3 ] ) + M _ x [ v _ { \\delta } ^ 3 - \\overline { v } ^ 3 ] - M _ { \\varphi , x } [ v _ { \\delta } ^ 3 - \\overline { v } ^ 3 ] , \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} E : y ^ 2 = x ^ 3 + x ^ 2 - 6 x + 4 \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} M ^ q _ \\delta ( \\mu ) \\ = \\ \\sum _ { Q \\in \\mathcal { M } _ \\delta } \\mu ( Q ) ^ q , \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} ( S - \\operatorname { i d } ) P _ 1 = 1 ( v _ 1 - 2 v + v _ { - 1 } ) , ( S - \\operatorname { i d } ) P _ 2 = n ( v _ 1 - 2 v + v _ { - 1 } ) . \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} Q ( z ) = | z | ^ 2 + \\frac { 2 c } { N } \\log \\frac { 1 } { | z - a | } , c > - 1 , a > 0 . \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} d _ { m - 3 , k } ^ { m , l } = & \\left ( \\left ( \\theta + m - 1 \\right ) \\left ( \\theta + m - 2 \\right ) \\left ( \\theta + m - 3 \\right ) \\right ) ^ { k } \\binom { l } { l - k } \\\\ & \\cdot \\left ( \\left ( \\alpha + \\theta \\right ) _ { m - 1 \\uparrow } + \\left ( \\theta + m - 1 \\right ) \\left ( \\alpha + \\theta \\right ) _ { m - 2 \\uparrow } + \\left ( \\theta + m - 1 \\right ) \\left ( \\theta + m - 2 \\right ) \\left ( \\alpha + \\theta \\right ) _ { m - 3 \\uparrow } \\right ) ^ { l - k } . \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} \\frac { \\mathsf { C } _ { \\mathcal { N } _ { \\mathcal { K } } } } { { \\mathsf { C } } _ { \\mathcal { N } _ { [ 1 : N ] } } } = \\frac { N + 2 } { 2 N } , \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} W _ n ( f , g ) : = \\sqrt { n ( n + 1 ) ( n + 2 ) } ( f _ n g _ { n + 1 } - f _ { n + 1 } g _ n ) , n \\in \\mathbb { N } \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { \\alpha , J } ( D _ J Q ^ { \\alpha } ) \\frac { \\partial L } { \\partial u _ J ^ { \\alpha } } & = \\operatorname { D i v } P - L \\operatorname { D i v } \\xi - \\xi ^ i D _ i L \\\\ & = D _ i ( P ^ i - \\xi ^ i L ) . \\end{aligned} \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} \\tilde { P } ( \\mu ) = \\sup _ { \\substack { \\eta \\in X \\\\ \\eta ' ( 0 ) = - h ^ { 2 } } } \\ , K ( \\mu , \\eta ) . \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} \\tilde { \\mathbf { P } } ^ 2 = \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 2 \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 2 = \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ f \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 2 = \\tilde { \\mathbf { P } } , \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} - \\theta ( \\partial _ c \\ln | \\theta | + \\widetilde { \\Gamma } ^ d _ { d c } ) \\Theta ^ { a b } \\widetilde { \\Gamma } ^ c _ { a b } = \\left ( \\Theta ^ { c d } \\partial _ c \\phi \\partial _ d \\phi + V ( \\phi ) \\right ) + 2 \\int _ { \\mathbb { R } ^ n } \\textbf { f } \\ ; \\frac { \\theta \\sqrt { | \\Theta | } ( \\textbf { m } ^ 2 + \\Theta _ { a b } \\widetilde { p } ^ a \\widetilde { p } ^ b ) } { \\widetilde { p } ^ 1 } \\ ; d \\widetilde { p } \\ ; ' ; \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} C _ 0 : = \\sum _ { n = 2 } ^ { \\infty } \\frac { | \\langle \\partial _ x ^ { - 1 } u _ 0 , u _ n \\rangle _ { L ^ 2 } | ^ 2 } { n - 1 } < \\infty , \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} \\dot { p } ( \\tau ) = \\sum _ { i = 1 } ^ N \\dot { p } ( \\tau _ i ) l _ i ( \\tau ) . \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} f _ { \\geq 5 } ( x , u , u _ x ) : = f _ 5 ( u , u _ x ) + f _ { \\geq 6 } ( x , u , u _ x ) \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} { \\left | A ( e ^ { i \\theta } ) f _ 0 ( \\theta ) - \\sum \\limits _ { j = 0 } ^ { \\infty } ( ( \\bold { B } ^ 0 ) ^ { - 1 } \\bold { R } ^ 0 \\bold { a } ) _ j e ^ { i j \\theta } \\right | ^ 2 } = { ( f _ 0 ( \\theta ) + g _ 0 ( \\theta ) ) ^ 2 } \\left ( \\varphi _ 1 ( \\theta ) + \\gamma _ 2 \\right ) , \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} f _ W ( u ) = e ^ { ( \\sqrt 2 r - 2 W ) u } \\quad \\mbox { a n d } g _ Z ( v ) = e ^ { - ( \\sqrt 2 r - 2 Z ) v } \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} E [ \\lambda ( W , \\gamma _ { 0 } ) | X ] = 0 , \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} \\mathtt { l } \\colon S \\to \\mathbb { Z } ^ { \\nu } , \\mathtt { l } ( \\overline { \\jmath } _ i ) : = \\mathtt { e } _ i , \\mathtt { l } ( - \\overline { \\jmath } _ i ) = - \\mathtt { l } ( \\overline { \\jmath } _ i ) = - \\mathtt { e } _ i , i = 1 , \\dots , \\nu , \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} \\pi _ { \\omega } ( \\tau _ { t } ( A ) ) = U _ { \\omega } ( t ) \\pi _ { \\omega } ( A ) U _ { \\omega } ^ { * } ( t ) \\ , \\ U _ { \\omega } ( t ) \\Omega _ { \\omega } = \\Omega _ { \\omega } \\ , \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} M _ k : = \\mu _ 0 \\mu _ 1 \\mu _ 2 \\cdots \\mu _ k , \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} \\hat { u } = \\frac { \\sigma ( u ) } { u } , \\hat { r } = \\frac { \\sigma ( r ) - r } { u ^ 2 } , \\hat { s } = \\frac { \\sigma ( s ) - s } { u } , \\hat { T } = \\frac { \\sigma ( T ) - T - s ( \\sigma ( r ) - r ) } { u ^ 3 } . \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} \\hat { \\beta } _ { 2 2 } = - \\beta + { 1 \\over 2 } , \\ \\ \\hat { \\beta } _ { 3 2 } = \\beta + { 2 \\over 3 } . \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{gather*} i _ { X } \\delta \\phi ^ a _ I = X ^ a _ I , i _ { X } d x ^ j = X ^ j . \\end{gather*}"}
-{"id": "8717.png", "formula": "\\begin{align*} B _ n ( k / n ) = \\frac { 1 } { n } \\ , \\sum _ { j = 1 } ^ k b ( j / n ) , \\ ; \\ ; \\Sigma _ n ( k / n ) = \\frac { 1 } { n } \\ , \\sum _ { j = 1 } ^ k \\frac { \\sigma ( j / n ) ^ 2 } { 2 } , k = 0 , \\ , 1 , \\ , \\ldots , \\ , n . \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} \\Sigma ( \\lambda ^ { - 1 } A ) = \\Big [ \\frac { a _ { 1 } } { \\lambda } , \\frac { b _ { 1 } } { \\lambda } \\Big ] \\cup \\Big [ \\frac { a _ { 2 } } { \\lambda } , \\frac { b _ { 2 } } { \\lambda } \\Big ] \\cup \\cdots \\cup \\Big [ \\frac { a _ { \\ell - 1 } } { \\lambda } , \\frac { b _ { \\ell - 1 } } { \\lambda } \\Big ] \\cup \\Big [ \\frac { a _ { \\ell } } { \\lambda } , \\frac { b _ { \\ell } } { \\lambda } \\Big ] \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} \\phi ^ * ( y ) = \\langle y , x \\rangle - \\phi ( x ) \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} f _ 1 + _ S f _ 2 + _ S \\cdots + _ S f _ n = g _ 1 + _ S g _ 2 + _ S \\cdots + _ S g _ m . \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} 0 = F _ { \\alpha } ^ { \\ast } ( x , n , [ u ] , [ v ] ) : = \\bold { E } _ { u ^ { \\alpha } } ( v _ n ^ { \\beta } F _ { \\beta } ) . \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} \\mathbf { S } _ { D , o p t } ^ { ( g ) } = \\underset { \\mathbf { S } _ D ^ { ( g ) } } { \\operatorname { a r g m a x } } \\operatorname { d e t } \\left ( \\boldsymbol { \\mathcal { F } } ^ { ( g ) } + \\mathbf { I } _ { T D } \\right ) \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} \\langle T , \\omega \\rangle = \\int _ { \\R ^ d } \\langle \\omega ( x ) , \\vec { T } ( x ) \\rangle \\dd \\| T \\| ( x ) \\ ; . \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} \\limsup _ { | i | \\le c ^ { '' } t , t \\to \\infty } | u _ i ( t + s ; s , u ^ s ) - u ^ + ( t + s ) | = 0 \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} c ( \\ell ) = \\sum _ { m n = \\ell } \\rho ( m ) \\lambda ( n ) g ( m ) h ( n ) \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} F ( x + x _ 1 \\varepsilon _ 1 + x _ 2 \\varepsilon _ 2 + \\cdots + x _ n \\varepsilon _ n ) & = \\sum _ { S \\subseteq \\{ 1 , 2 , \\ldots , n \\} } F ^ { ( | S | ) } ( x ) \\prod _ { i \\in S } x _ i \\varepsilon _ i \\\\ & = \\sum _ { S \\subseteq \\{ 1 , 2 , \\ldots , n \\} } \\left ( \\sum _ { T \\subseteq S } f _ { S - T } ^ { ( | T | ) } ( x ) \\prod _ { i \\in T } x _ i \\right ) \\varepsilon _ S \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} \\begin{aligned} \\int \\Big ( | \\nabla v _ \\lambda | ^ 2 - ( \\partial _ r ( r V ) ) | u | ^ 2 - 2 \\varepsilon \\mbox { I m } [ ( r \\partial _ r u ) \\overline { u } ] + 2 \\varepsilon \\lambda ^ { \\frac 1 2 } r | u | ^ 2 \\Big ) d x \\\\ = \\int \\Big ( \\mbox { R e } [ f ( 2 r \\partial _ r \\overline u + ( n - 1 ) \\overline u ) ] - 2 \\lambda ^ { \\frac 1 2 } \\mbox { I m } ( r f \\overline u ) \\Big ) d x . \\end{aligned} \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} C _ { j i } ^ { i i } + C _ { j i } ^ { j j } = 0 , \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { 1 } { \\ell ^ { p - 1 } ( \\ell + 1 ) ^ { m + 1 } } - \\zeta ( m + p ) + \\zeta ( m + p + 1 ) . \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} J = \\det \\left ( \\left . \\frac { \\partial F _ i } { \\partial x _ j } \\right | _ { j = 1 \\ldots n } \\left . \\frac { \\partial F _ i } { \\partial y _ j } \\right | _ { j = 1 \\ldots m } \\right ) _ { \\ ! i = 1 \\ldots n + m } \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} C _ { i i } ^ { i i } = C _ { l l } ^ { i i } \\mbox { f o r } i < l , \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} \\partial _ { t } u ( t , x , y , \\omega ) = D ^ { 2 } _ { x } u ( t , x , y , \\omega ) + f ( t , x , y - h \\pi _ { t } ( \\omega ) ) \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} y ( t , x ) = \\int _ { x } ^ { \\infty } u _ 0 ( x ' ) w ( t , x - x ' ) d x = \\int _ { - \\infty } ^ 0 u _ 0 ( x - z ) w ( t , z ) d z , \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} \\| g _ { 0 } \\| _ { { \\rm t b } } ^ 2 & : = \\sup _ { t \\in \\R } \\int _ { t } ^ { t + 1 } \\| g _ { 0 } ( y ) \\| ^ 2 \\d y = M _ { 0 } , \\\\ \\| g _ { 1 } \\| _ { { \\rm t b } } ^ 2 & : = \\sup _ { t \\in \\R } \\int _ { t } ^ { t + 1 } \\| g _ { 1 } ( y ) \\| ^ 2 \\d y = M _ { 1 } , \\end{align*}"}
-{"id": "5817.png", "formula": "\\begin{align*} \\tilde { c } & = a ^ 2 N - c - a ( 1 + 2 ( c + n ) ) \\alpha _ n \\beta _ n + \\left ( a ^ 2 N - c - a ( c + n ) \\alpha _ n \\beta _ n \\right ) \\beta _ n \\gamma _ n \\\\ & + ( c + n ) ( c + n + 1 ) \\alpha ^ 3 _ n \\beta ^ 3 _ n + a N ^ 2 b ^ 3 _ n \\left ( 1 + \\beta _ n \\gamma _ n \\right ) ^ 2 , \\\\ & a _ 0 = 0 , b _ 0 = a , \\quad \\alpha _ 0 = 1 , \\quad \\beta _ 0 = 1 + a ^ 2 N , \\quad \\gamma _ 0 = 0 . \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} \\mathcal { T } ^ { - 1 } \\omega \\cdot \\partial _ { \\vartheta } \\mathcal { T } = \\omega \\cdot \\partial _ { \\vartheta } + \\{ \\omega \\cdot \\partial _ { \\vartheta } p ( \\vartheta ) \\} \\partial _ z , \\mathcal { T } ^ { - 1 } \\partial _ y \\mathcal { T } = \\partial _ z . \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} 1 - F _ \\rho ( k ) = \\sum _ { t = k + 1 } ^ \\infty f _ \\rho ( t ) = \\mu _ \\rho \\sum _ { t = k + 1 } ^ \\infty \\frac { f _ \\delta ^ \\ast ( t ) } { t } = \\mu _ \\rho \\sum _ { t = k + 1 } ^ \\infty \\frac { f ^ \\ast ( t ) } { t } = \\frac { \\mu _ \\rho } { \\mu } \\sum _ { t = k + 1 } ^ \\infty f ( t ) , \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} | F _ i ( x ) - F _ \\infty ( g _ i ( \\Phi _ i ( x ) ) ) | & = | g _ i ( F _ 1 ( x ) - F _ \\infty ( \\Phi ( x ) ) ) | \\\\ & \\leq C _ \\delta \\lambda ^ { - ( 1 + \\delta ) d _ 1 i } | F _ 1 ( x ) | ^ 2 \\leq C _ \\delta ^ 2 \\lambda ^ { - ( 1 + \\delta ) d _ 1 i + 2 ( 1 - \\delta ) d _ 1 i } | F _ i ( x ) | ^ 2 . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} T [ f ] ( \\boldsymbol { x } ) = \\int _ { \\Omega } \\Phi ( \\boldsymbol { x } , \\boldsymbol { y } ) f ( \\boldsymbol { y } ) d \\boldsymbol { y } , \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} H _ 1 = A ^ \\dagger ( x ) A ( x ) \\ ; , \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\partial _ i X ^ j = \\delta _ { i j } + \\sum _ { k , l } \\hat { B } ^ { j k } _ { i l } X ^ l \\partial _ k , \\sum _ { k , l } ( { \\sf P } _ { h - } ) ^ { l k } _ { i j } \\ , \\partial _ k \\partial _ l = 0 . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} ( z - T ) ^ { - 1 } - ( z - A ) ^ { - 1 } = K ( z ) \\left ( \\sum _ { m = 1 } ^ \\infty B ( z ) ^ m \\right ) K ( z ) . \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} \\alpha ( \\xi ) = \\overline { \\omega } + \\varepsilon ^ 2 \\ , \\mathbb { M } \\ , \\xi \\ , , \\mathbb { M } : = \\mathbb { A } \\ , D _ S \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{align*} L _ { a , n } ^ { X X Z } ( \\lambda ) = L _ { a , n } ( \\lambda / q ) \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\sum _ { i j } N _ { i j } t _ { i j i j } = \\sqrt n \\sum _ { i j } X _ { i j } t _ { i j i j } + n \\sum _ { i j } \\pi _ i \\pi _ j t _ { i j i j } = O ( n ^ { - 1 / 2 } ) \\sum _ { i j } | X _ { i j } | + \\frac { 1 } { d } \\sum _ { i j } A _ { i i } A _ { j j } \\pi _ i \\pi _ j + O ( n ^ { - 1 } ) ; \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} ( \\Box ^ { ( q ) } _ { b , m } u | u ) = \\| u \\| _ { \\overline S } ^ 2 + q m \\| u \\| ^ 2 + ( R _ \\ast u | u ) . \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} \\alpha _ { s t } = \\frac q n | \\sigma ^ { - 1 } ( s ) \\cap \\tau ^ { - 1 } ( t ) | ; \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} x ( n + 1 ) = A ( n ) x ( n ) , \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} \\pi ^ { ( t ) } ( x ) : = \\sum _ { y \\in V _ t } \\pi ^ { ( t ) } ( x , y ) , \\pi ^ { ( t ) } ( A , B ) : = \\sum _ { x \\in A , \\ , y \\in B } \\pi ^ { ( t ) } ( x , y ) , \\pi ^ { ( t ) } ( A ) : = \\sum _ { x \\in A } \\pi ^ { ( t ) } ( x ) . \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} C _ { * } : = \\frac { 2 \\pi ^ { 2 } e ^ { 2 ( 1 + | \\log C _ { M } | ) } ( 1 + \\frac { 1 } { 1 5 } \\varepsilon _ { * } ) } { 1 - \\frac { 7 } { 6 } \\varepsilon _ { * } } , \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} 0 = \\frac { b ' - b ( u _ 1 - u _ { - 1 } ) } { u } - \\left [ \\xi _ t + a ( t , n + 1 ) \\right ] u _ 1 + \\left [ \\xi _ t + a ( t , n - 1 ) \\right ] u _ { - 1 } + a ' - b ( t , n + 1 ) + b ( t , n - 1 ) . \\end{align*}"}
-{"id": "2355.png", "formula": "\\begin{gather*} \\omega : = u ^ 4 + t u ^ 2 - u _ t ^ { 2 } . \\end{gather*}"}
-{"id": "7336.png", "formula": "\\begin{align*} \\lim _ { T \\rightarrow \\infty } \\frac 1 T \\sum _ { n = n _ 1 } ^ { n _ 1 + T - 1 } \\frac { \\xi ^ n } { n ^ { \\alpha } } = 0 , \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} \\sup _ { x \\in { \\bf R } ^ N } \\| \\mu \\| _ { L ^ { r , \\infty } ( B ( x , \\rho ) ) } \\le c _ 2 \\quad \\mbox { w i t h } r = \\frac { N ( p - 1 ) } { 2 } \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} \\mathcal { A } = \\{ F _ { \\alpha } ( x , [ u ] ) = 0 \\} , \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { n / 2 } \\frac { 2 i } { n } = \\frac { ( n / 2 ) ! 2 ^ { n / 2 } } { n ^ { n / 2 } } = ( 1 + o ( 1 ) ) \\sqrt { \\pi n } e ^ { - n / 2 } \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} \\left ( \\frac { r + \\ell } { n } - p \\right ) ^ 2 = \\left ( \\frac r n - p \\right ) ^ 2 + \\frac { \\ell ^ 2 } { n ^ 2 } + \\frac { 2 \\ell } { n } \\left ( \\frac r n - p \\right ) \\ge \\left ( \\frac r n - p \\right ) ^ 2 + \\frac 1 n . \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} F ( \\lambda | \\lambda ' ) + F ( \\lambda ' | \\lambda ) = \\sum _ { i a } s _ a F ( \\lambda _ { i a } | \\lambda ) F ( \\lambda _ { i a } | \\lambda ' ) . \\end{align*}"}
-{"id": "9123.png", "formula": "\\begin{align*} \\| f \\| _ 1 = \\left | \\xi ( f ) \\right | \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} y = \\mathrm { c m } ( x , y ) x + \\mathrm { s n } ( x , y ) b ( x ) . \\end{align*}"}
-{"id": "6770.png", "formula": "\\begin{align*} v _ { 0 } = & \\epsilon , & v _ { 1 } = & 5 \\epsilon , & v _ { m + 2 } & = 4 v _ { m + 1 } - v _ { m } , \\ ; \\ ; m \\geq 0 , \\\\ u _ { 0 } = & \\epsilon , & u _ { 1 } = & 3 \\epsilon , & u _ { m + 2 } & = 4 u _ { m + 1 } - u _ { m } , \\ ; \\ ; m \\geq 0 , \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} a \\left ( 3 - \\frac 2 { k - 1 } \\right ) + 2 ( 1 - a ) + a - [ a - ( 1 - a ) ] = 3 - \\frac { 2 a } { k - 1 } . \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} U ^ T : = \\tau _ 0 U ^ t \\tau _ 0 , \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} y _ i ' & = ( \\alpha _ { i + 2 } ) ^ { - 1 } \\cdot y _ { i + 1 } ^ { - 1 } \\cdot \\alpha _ i , \\\\ y _ { i ^ - } ' & = ( \\alpha _ { i } ) ^ { - 1 } \\cdot y _ i \\ , y _ i ^ - \\cdot \\alpha _ { i + 1 } , \\\\ y _ { i ^ + } ' & = ( \\alpha _ { i + 1 } ) ^ { - 1 } \\cdot y _ { i + 1 } \\ , y _ i ^ + \\cdot \\alpha _ { i + 2 } , \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\begin{cases} 1 - \\alpha \\ , \\lambda _ 1 \\neq 0 , \\\\ 1 - \\alpha \\ , \\lambda _ 2 \\neq 0 . \\end{cases} \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{align*} u _ 1 : = \\{ \\{ 1 , 2 , \\ldots , n , 1 ' \\} , \\{ 2 ' , 3 ' , \\ldots , n ' \\} \\} , u _ 2 : = \\{ \\{ 1 , 2 , \\ldots , n , 2 ' , 3 ' , \\ldots , n ' \\} , \\{ 1 ' \\} \\} . \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} w ( z ) = w _ 1 z + w _ 2 z ^ 2 + w _ 3 z ^ 3 + \\cdots , \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} \\sum _ { k c } [ \\delta _ { i a , k c } - s _ c F ( \\lambda _ { k c } | \\lambda _ { i a } ) ] [ \\delta _ { j b , k c } - s _ b F ( \\lambda _ { k c } | \\lambda _ { j b } ) ] = \\delta _ { i a , j b } \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} R ( e _ 1 , e _ 2 ) e _ 1 & = \\dfrac { - 2 a ( a - b - c ) - ( a - b + c ) ( a + b - c ) } { 4 } \\ , e _ 2 ; \\\\ R ( e _ 1 , e _ 2 ) e _ 2 & = \\dfrac { 2 a ( a - b - c ) + ( a - b + c ) ( a + b - c ) } { 4 } \\ , e _ 1 ; \\\\ R ( e _ 1 , e _ 3 ) e _ 1 & = \\dfrac { 2 b ( a - b + c ) + ( a - b - c ) ( a + b - c ) } { 4 } \\ , e _ 3 ; \\\\ R ( e _ 1 , e _ 3 ) e _ 3 & = \\dfrac { - 2 b ( a - b + c ) - ( a - b - c ) ( a + b - c ) } { 4 } \\ , e _ 1 ; \\\\ R ( e _ 2 , e _ 3 ) e _ 2 & = \\dfrac { 2 c ( a + b - c ) + ( a - b + c ) ( a - b - c ) } { 4 } \\ , e _ 3 ; \\\\ R ( e _ 2 , e _ 3 ) e _ 3 & = \\dfrac { - 2 c ( a + b - c ) - ( a - b + c ) ( a - b - c ) } { 4 } \\ , e _ 2 . \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} S ( U , f ) = \\frac { 1 } { \\pi } \\int _ U f ^ \\# ( z ) ^ 2 d x \\ : d y . \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} | N | : = \\{ n \\in | M | : J \\subseteq J _ n \\} . \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} \\abs { \\sum _ { i , l = 1 , 2 } \\int _ { - \\infty } ^ { \\infty } \\pi _ \\infty ( i ) p _ \\infty ( i , l ) y ^ { r + 1 } g _ { l } ( y ) \\ , \\d Q ( y ) } \\le \\sum _ { i , l = 1 , 2 } \\pi _ \\infty ( i ) p _ \\infty ( i , l ) \\int _ { - \\infty } ^ { \\infty } | y | ^ { r + 1 } g _ l ( y ) \\ , \\d Q ( y ) . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} \\mathcal { P } _ 2 \\left ( { x } \\right ) = \\exp \\left \\{ - 2 \\pi { \\lambda _ e } \\int _ 0 ^ \\infty ( 1 - { { f _ { \\Pr } } ( { r _ e } ) } ) { r _ e } \\sum \\limits _ { \\ell , n \\in \\left \\{ { { } , { } } \\right \\} } { \\rm { \\mathbf { 1 } } } \\left ( { \\max \\{ { r _ e } , d \\} } < \\big ( \\frac { { P _ t } { G _ \\ell } { G _ n ^ e } \\beta } { x \\sigma _ e ^ 2 } \\big ) ^ { \\frac { 1 } { { \\alpha _ { \\mathrm { N L o S } } } } } \\right ) { { { \\Pr } _ { \\ell n } } } d { r _ e } \\right \\} \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} d ( x ) = \\sum _ { i = 1 } ^ n a _ i ( x ) f _ i ( x ) \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} \\sigma ( B ) = A B A ^ { - 1 } + \\delta ( A ) A ^ { - 1 } - \\tfrac { 1 } { n } \\delta ( \\mathrm { d e t } ( A ) ) \\mathrm { d e t } ( A ) ^ { - 1 } \\cdot I _ n . \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} g ' ( \\mu ) = \\frac { x _ \\mu } { 2 \\mu ^ \\frac 1 2 } \\int _ 0 ^ 1 \\left ( 1 - \\frac { Q ( x t ) } { Q ( x ) } \\right ) ^ { - \\frac 1 2 } \\ ; \\dd t . \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} G _ { 2 n } ^ { ( k , \\ell ) } ( x , y ) = \\prod _ { i = 1 } ^ { 2 n } \\frac { 1 } { 1 - x ^ { a ^ { ( k , \\ell ) } _ i } y ^ { a _ { i - 1 } ^ { ( \\ell , k ) } } } \\\\ \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} \\tau ^ f = 1 , \\sigma ^ e = 1 , \\tau \\sigma \\tau ^ { - 1 } = \\sigma ^ k \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} \\varphi ( x ) & = \\eta ^ 2 ( x ) \\min \\{ \\Delta ^ { 2 , h } \\Sigma ( x ) + 1 , 0 \\} \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} p = \\exp _ { \\O } ( \\rho V ) . \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} \\Delta \\mathbf { x } _ i = \\mathbf { P } _ 1 { \\mathbf { d } } _ { i - 1 } + \\mathbf { P } _ 1 \\mathbf { A } ^ { - 1 } \\mathbf { w } _ { i - 1 } - \\mathbf { P } _ 2 \\mathbf { d } _ i \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} M _ x [ v _ { \\delta } ^ 2 ] - M _ { \\varphi , x } [ v _ { \\delta } ^ 2 ] = \\varepsilon ^ { 2 ( b - 1 ) } \\sum _ { j \\in S } \\lvert j \\rvert ( ( y _ { \\delta } ) _ j - M _ { \\varphi } [ ( y _ { \\delta } ) _ j ] ) . \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} _ { q } \\mathcal { U } _ { a , - } ( \\lambda ) = ( \\lambda ^ { 2 } / q ^ { 2 } - q ^ { 2 } / \\lambda ^ { 2 } ) \\mathsf { A } _ { - } ( \\lambda q ^ { 1 / 2 } ) \\mathsf { A } _ { - } ( q ^ { 1 / 2 } / \\lambda ) . \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} a + c & = ( \\beta + \\lambda \\eta ) + ( \\beta + d ^ 2 \\delta + e ^ 2 \\delta b ) = ( d ^ 2 + d + e ^ 2 b ) \\eta + d ^ 2 \\delta + e ^ 2 \\delta b \\\\ & = d ^ 2 \\eta + d ^ 2 \\delta + d \\eta + e ^ 2 \\eta b + e ^ 2 \\delta b = ( d \\eta ) ^ 2 + d \\eta + ( e \\eta ) ^ 2 b . \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} \\Lambda _ S ( - \\theta , \\mathbf { F } ) = \\log \\left ( \\pi \\sum _ { s = 1 } ^ S w _ s e ^ { - \\theta r _ s } + \\sum _ { \\ell = 1 } ^ L \\pi _ \\ell e ^ { - \\theta r _ \\ell } \\right ) . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} \\Re ( \\lambda _ p ) & = \\Re ( \\alpha ) + \\Re ( \\beta \\omega ^ p + \\gamma \\overline { \\omega } ^ p ) = \\\\ & = \\Re ( \\alpha ) + \\Re ( ( \\beta + \\overline { \\gamma } ) \\omega ^ p ) \\leq \\\\ & = \\Re ( \\alpha ) + | ( \\beta + \\overline { \\gamma } ) \\omega ^ p | = \\\\ & = \\Re ( \\alpha ) + | \\beta + \\overline { \\gamma } | < 0 , \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} \\mathbf { h } ^ { ( g ) } & = \\underbrace { \\left ( \\mathbf { I } _ { K _ g } \\otimes \\mathbf { V } \\right ) } _ { \\triangleq \\boldsymbol { \\Upsilon } _ { U } ^ { ( g ) } } \\mathbf { c } ^ { ( g ) } \\textrm { w h e r e } \\\\ \\mathbf { V } & \\triangleq \\operatorname { b d i a g } \\left [ \\left \\{ \\left ( \\rho _ l ^ { ( g ) } \\right ) ^ { 1 / 2 } \\mathbf { U } _ l ^ { ( g ) } \\left ( \\boldsymbol { \\Lambda } _ l ^ { ( g ) } \\right ) ^ { 1 / 2 } \\right \\} _ { l = 0 } ^ { L _ g - 1 } \\right ] . \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} ( \\rho _ g ( \\bar a b ) , \\rho _ g ( \\bar a c ) ) & = \\rho _ g ( \\bar a b ) ^ * \\rho _ g ( \\bar a c ) \\\\ & = \\alpha _ g ( b ) ^ * \\bar a ^ 2 \\alpha _ g ( c ) \\\\ & = \\alpha _ g ( b ^ * \\bar a ^ 2 c ) , \\qquad \\forall g \\in G \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} \\delta E = \\sum _ { i a } \\tilde { \\epsilon } ( \\lambda _ { i a } ) N _ { i a } + \\frac { 2 \\pi } { L } \\sum _ { i a } s _ a \\tilde { v } _ { i a } \\left [ n _ { i a } + \\frac { 1 } { 2 } \\left ( \\sum _ { j b } U _ { j b , i a } N _ { j b } . \\right ) ^ 2 \\right ] \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} \\ker ( N _ l \\stackrel { \\varphi _ N } { \\longrightarrow } S _ l ) = \\varphi _ M ( M ) \\cap N _ l . \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} \\| P _ k ^ \\pm \\| = \\| \\langle \\cdot , ( g _ k ^ \\pm ) ^ * \\rangle g _ k ^ \\pm \\| = \\| ( g _ k ^ \\pm ) ^ * \\| \\| g _ k ^ \\pm \\| = \\frac { 1 } { \\tau _ k ^ 2 } = \\frac { 1 } { 1 - t _ k ^ 2 } . \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{gather*} u _ t = \\frac { \\partial H } { \\partial p } , p _ t = - \\frac { \\partial H } { \\partial u } , \\end{gather*}"}
-{"id": "1094.png", "formula": "\\begin{align*} [ j ^ k n ^ { - h } ] \\ , \\ln \\biggl ( 1 + H _ j \\biggr ) = [ j ^ k n ^ { - h } ] \\ , \\ln \\biggl ( 1 + \\sum _ { s = 1 } ^ { j - 1 } \\frac { a _ s ( r , j ) } { n ^ s } \\biggr ) = 0 , k \\le h \\\\ \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\epsilon _ { n , m } = \\pm \\lambda \\sqrt { 1 - \\frac { ( 2 m + 1 ) ^ 2 } { ( 2 n - 1 ) ^ 2 } } \\ ; , \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} e A e \\ ; \\stackrel { M } { = } \\ ; \\tilde { e } A \\tilde { e } . \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} \\mu _ b ( u ^ { \\ell _ 1 } ) = \\P ( Z ^ { \\ell _ 1 } = u ^ { \\ell _ 1 } ) , \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\Delta \\mathbf { x } _ i = \\mathbf { P } _ 1 \\bar { \\mathbf { d } } _ { i - 1 } - \\mathbf { P } _ 2 \\mathbf { d } _ i , \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} x _ 1 = \\log \\frac { 1 } { 1 - t _ 1 } , x _ 2 = \\log \\frac { t _ 2 } { t _ 1 } \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} E E _ { k } = \\frac { b _ { k , k ' } \\log _ 2 \\left ( 1 + \\frac { p _ { k , k ' } g _ { k , k ' } } { b _ { k , k ' } N _ 0 } \\right ) } { \\frac { p _ { k , k ' } } { \\xi } + \\frac { q _ k } { \\xi } } , \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{gather*} c _ 0 = \\frac { \\beta } { 2 } \\left ( \\frac { 1 } { 1 2 } - \\zeta ' ( - 1 ) \\right ) + \\frac { \\gamma } { 6 \\beta } - \\frac { \\log 2 \\pi } { 4 } - \\frac { ( \\beta / 2 ) } { 2 } \\\\ \\hphantom { c _ 0 = } { } + \\left ( \\frac { 1 7 } { 8 } - \\frac { 2 5 } { 2 4 } ( \\beta / 2 + 2 / \\beta ) \\right ) \\log 2 + \\int _ { 0 } ^ { \\infty } \\frac { 1 } { e ^ { \\beta t / 2 } - 1 } \\left ( \\frac { t } { e ^ t - 1 } - 1 + \\frac { t } { 2 } - \\frac { t ^ 2 } { 1 2 } \\right ) d t , \\end{gather*}"}
-{"id": "3266.png", "formula": "\\begin{align*} \\tau _ s \\left ( \\varphi \\right ) = \\varphi + s \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} \\phi ( z ) = a z + \\log z - 2 g ( z ) + \\ell , \\ell = \\log \\beta - a \\beta \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} R _ 1 - \\beta ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } = \\frac { \\alpha R _ 1 + ( \\alpha - 1 ) \\big ( \\delta _ 1 + d _ \\alpha ( p _ { U , W } , p _ U p _ W ) \\big ) } { 2 \\alpha - 1 } > 0 , \\quad \\forall \\alpha > 1 . \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{align*} \\begin{array} { l } Y _ t = Y _ 0 + \\int _ 0 ^ t \\Phi ( r ( s ) , Y _ { r ( s ) } , v ( r ( s ) , Y _ { \\cdot \\wedge r ( s ) } ) ) d W _ s + \\int _ 0 ^ t g ( r ( s ) , Y _ { r ( s ) } , v ( r ( s ) , Y _ { \\cdot \\wedge r ( s ) } ) ) d s \\ , \\textrm { f o r a n y } \\ t \\in [ 0 , T ] \\ , \\end{array} \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\hat \\zeta ^ + ( s ) = \\zeta ^ - ( s ) \\hat \\zeta ^ - ( s ) = 3 \\zeta ^ + ( s ) . \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{gather*} \\frac { 2 } { \\beta } \\partial _ t \\Phi = \\left ( H \\left ( t , u , \\frac { 2 } { \\beta } \\partial _ u \\right ) + \\frac { u } { 2 } \\right ) \\Phi . \\end{gather*}"}
-{"id": "4705.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } \\frac { 2 ^ k } { k ! ( n - 2 k ) ! } | a _ n | ^ n & \\prod _ { i < j } | z _ i - z _ j | \\prod _ { k = 0 } ^ { n } g ( a _ i ) d \\ell _ E ( a _ n ) d \\ell _ { n , k } ( z _ 1 , \\dots , z _ n ) \\\\ & = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } \\frac { 2 ^ k } { k ! ( n - 2 k ) ! } p _ { n , k } ( z _ 1 , \\dots , z _ n , a _ n ) d \\ell _ E ( a _ n ) d \\ell _ { n , k } ( z _ 1 , \\dots , z _ n ) . \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} \\mathcal { S } : = \\exp ( \\Pi _ S ^ { \\perp } ( w \\partial _ x ^ { - 1 } ) ) \\Pi _ S ^ { \\perp } = \\Pi _ S ^ { \\perp } ( \\mathrm { I } + w \\partial _ x ^ { - 1 } ) \\Pi _ S ^ { \\perp } + \\hat { \\mathcal { S } } , \\hat { \\mathcal { S } } : = \\sum _ { k \\geq 2 } \\frac { 1 } { k ! } [ \\Pi _ S ^ { \\perp } ( w \\partial _ x ^ { - 1 } ) ] ^ k \\Pi _ S ^ { \\perp } , \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} \\ln T ( z ) \\approx i \\sum _ { j = 0 } ^ { \\infty } H _ { j } ( 2 z ) ^ { - j - 1 } \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} l ( P _ 2 ) = & d _ { S ( G , t - j ) } ( x _ { j + 1 } \\cdots x _ t , ( v _ 1 ) ^ { t - j } ) + 2 ( 2 ^ { t - j } - 1 ) ( s - 1 ) + s \\\\ & + d _ { S ( G , t - j ) } ( ( u _ { s - 1 } ) ^ { t - j } , y _ { j + 1 } \\cdots y _ t ) . \\end{align*}"}
-{"id": "6755.png", "formula": "\\begin{align*} | e | ^ { 2 } | K | ^ { 2 } | \\alpha | ^ { 4 } = 0 , \\ \\overline { a } b = 0 , \\ | b | ^ { 2 } - \\overline { d } e K - d \\overline { e } \\overline { K } = 0 . \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} f ( t ) : = u ( x + t v ) . \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathcal { I } _ { \\mathrm { u } } \\right ] & = \\mathbb { E } \\left [ \\sum _ { x \\in \\Phi _ { \\mathrm { b } } \\setminus i } \\frac { g _ { x i k } } { \\| x \\| ^ { \\alpha } } \\right ] \\\\ & \\stackrel { ( a ) } { = } \\int _ { t } ^ \\infty r ^ { - \\alpha } r d r = \\frac { 2 \\pi \\lambda t ^ { - ( \\alpha - 2 ) } } { \\alpha - 2 } \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} \\omega ( \\tau _ { g } ( A ) ) = \\omega ( A ) , \\ \\forall A \\in { \\cal A } , \\ \\forall g \\in G \\ . \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} \\upsilon _ F ( \\Delta ' ) = \\upsilon _ F ( \\Delta _ m ) - 1 2 \\upsilon _ F ( u ) = 4 \\upsilon ( \\Delta _ m ) - 1 2 \\alpha = 0 . \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\lim _ { \\eta \\rightarrow 0 } \\lim _ { \\Lambda } \\omega _ { \\beta , { \\mu } _ { \\Lambda } ( \\beta , { \\rho } , \\eta ) , \\Lambda , \\eta } ^ { 0 } ( { b _ { q } } / \\sqrt { V } ) = \\lim _ { \\eta \\rightarrow 0 } \\frac { \\eta } { \\mu ( \\beta , { \\rho } , \\eta ) } = e ^ { i \\ , { \\rm { a r g } } ( \\eta ) } \\ , \\sqrt { { \\rho } - \\rho _ { c } ( \\beta ) } \\ , \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} \\Delta _ n ( f , \\widetilde { f } ) & = n [ P _ n ( T \\widetilde { f } | | f ^ { \\star } ) - P _ n ( f | | f ^ { \\star } ) ] + \\\\ & n \\tau ^ { - 1 } [ \\tau _ 1 ^ { - 1 } D ^ { \\prime } _ n ( T f , f ^ { \\star } ) - D ^ { \\prime } _ n ( T \\widetilde { f } , f ^ { \\star } ) ] . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} \\operatorname { D i v } ^ { \\vartriangle } P ( n , [ u ] ) : = \\sum _ { i = 1 } ^ p ( S _ i - \\operatorname { i d } ) P ^ i ( n , [ u ] ) , \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} \\mu _ k & = \\left ( \\frac { \\pi } { \\Omega _ \\beta } k \\right ) ^ \\gamma ( 1 + o ( 1 ) ) , & k \\to \\infty , \\\\ \\mu _ { k + 1 } - \\mu _ k & = \\frac { 2 \\pi } { \\Omega _ \\beta ' } \\left ( \\frac { \\pi } { \\Omega _ \\beta } k \\right ) ^ { \\gamma - 1 } ( 1 + o ( 1 ) ) , & k \\to \\infty , \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} a _ { \\omega } ( 1 ) \\sim \\frac { 2 } { 8 \\sqrt { 2 0 } } \\cdot 2 \\cdot 1 7 . 8 8 8 5 4 3 8 2 0 0 0 0 = 2 . 0 0 0 0 0 0 0 0 0 0 0 0 . \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} \\bold { p r } X = \\xi ^ i ( x ) D _ i + \\sum _ { \\alpha , J _ 1 , J _ 2 } ( D _ { J _ 1 } S _ { J _ 2 } Q ^ { \\alpha } ( x , n , [ u ] ) ) \\partial _ { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } . \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} h ^ { 0 } ( \\mathcal { E } \\otimes \\mathcal { E ^ { \\vee } } ) - h ^ { 1 } ( \\mathcal { E } \\otimes \\mathcal { E ^ { \\vee } } ) + h ^ { 2 } ( \\mathcal { E } \\otimes \\mathcal { E ^ { \\vee } } ) - h ^ { 3 } ( \\mathcal { E } \\otimes \\mathcal { E ^ { \\vee } } ) = - 1 4 \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} u _ t ( t , x ) - v _ x ( t , x ) & = 0 , \\\\ \\frac { 1 } { a ( x ) } v _ t ( t , x ) - u _ x ( t , x ) & = 0 , ( t , { x } ) \\in D _ T , \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} w _ { c , t } = z _ { a , M _ a , t } . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} \\bar g ^ { \\rm R } _ 3 ( C _ n ^ { \\rm R } ) = \\left \\{ \\begin{array} { l l } g _ 3 ^ { \\rm R } ( C _ n ^ { \\rm R } ) , & { \\rm i f } ~ C _ n ^ { \\rm R } \\in [ 0 , \\hat \\tau _ 1 ] \\cup [ \\hat \\tau _ 2 , + \\infty ) \\\\ \\hat c C _ n ^ { \\rm R } + \\hat d , & { \\rm i f } ~ C _ n ^ { \\rm R } \\in ( \\hat \\tau _ 1 , \\hat \\tau _ 2 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} ( A ( s , t ) + r ) = \\bigcup _ { n = 1 } ^ N [ r + s , r + s + t ] ^ { \\leftthreetimes 2 ( n - 1 ) } \\times [ r , r + s ] ^ { \\leftthreetimes 2 ( N - n ) } \\subset A ( r + s , t ) , \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} G = \\left ( { \\bf e } _ 1 , { \\bf e } _ 2 , \\ldots , { \\bf e } _ { k } \\ | \\ { \\bf p } _ 1 , { \\bf p } _ 2 , \\cdots , { \\bf p } _ { n _ 1 } \\right ) , \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} A \\left ( \\frac { \\partial } { \\partial \\tau } \\right ) = - i \\ , \\mathrm { d i a g } ( \\mu _ a ) . \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} \\alpha _ 0 : = \\begin{pmatrix} a & 0 \\\\ c \\ell ^ { m + 2 } & d - w _ 0 c \\ell ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\left \\lfloor { \\sqrt { n } } \\right \\rfloor } p ( n - k ^ 2 ) . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} \\gamma ( x , y _ 1 ) - \\gamma ( x , y _ 2 ) = ( \\gamma x , y _ 1 ) - ( \\gamma x , y _ 2 ) = y _ 1 - y _ 2 = ( x , y _ 1 ) - ( x , y _ 2 ) . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} \\phi ( x ; \\mu , \\sigma ) = \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma } \\exp \\left \\{ - \\frac { ( x - \\mu ) ^ 2 } { 2 \\sigma ^ 2 } \\right \\} . \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} u _ s ( r ) = [ x ^ s ] T _ r . \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} U _ { 2 ^ { p _ 1 } + 2 ^ { p _ 2 } + \\cdots + 2 ^ { p _ s } } = \\frac { 1 } { 2 } U _ { 2 + 2 ^ { p _ 2 - p _ 1 } + \\cdots + 2 ^ { p _ s - p _ 1 } } + 2 U _ { 2 ^ { p _ 2 } + \\cdots + 2 ^ { p _ s } } + \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{gather*} B _ x B _ s = B ^ s _ x \\otimes _ { R ^ s } R \\otimes _ { R ^ s } R ( 1 ) = B _ x ( 1 ) \\oplus B _ x ( - 1 ) . \\end{gather*}"}
-{"id": "1018.png", "formula": "\\begin{align*} P = \\frac { V _ s ( \\rho ) } { ( 1 - | x | ^ 2 ) } \\frac { 1 - | x | ^ 2 | y | ^ 2 } { | x - y | ^ { 2 + N - 2 s } } = \\frac { ( 1 - | x | ^ 2 ) ^ { s - 2 } ( 1 - | y | ^ 2 ) ^ { s - 1 } ( 1 - | x | ^ 2 | y | ^ 2 ) } { \\Big | x | y | - \\frac { y } { | y | } \\Big | ^ { N } } = P _ { s - 1 } ( x , y ) , \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} \\lim _ { j \\to - \\infty } u ( j , t ; 0 , u ^ { 0 , * } ) = u ^ + ( t ) , \\lim _ { j \\to \\infty } u ( j , t ; 0 , u ^ { 0 , * } ) = 0 \\forall \\ , \\ , t \\in \\R . \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} \\nu _ z ( F _ { 2 j } ( \\gamma ) ) \\geq - C , \\ , \\ \\forall \\gamma \\in \\widehat { \\mathfrak { p } } _ { \\theta } , \\ , \\ j = 1 , 2 , \\cdots , n . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} K _ { G \\left ( x , b ; c \\right ) } ^ { L } \\left ( u \\right ) & = \\frac { u ^ { x / b } \\exp \\left ( - u / b \\right ) } { b ^ { x / b + 1 } \\Gamma \\left ( x / b + 1 \\right ) } \\mathbf { 1 } \\left ( 0 \\leq u < c \\right ) \\\\ K _ { G \\left ( x , b ; c \\right ) } ^ { U } \\left ( u \\right ) & = \\frac { u ^ { x / b } \\exp \\left ( - u / b \\right ) } { b ^ { x / b + 1 } \\Gamma \\left ( x / b + 1 \\right ) } \\mathbf { 1 } \\left ( u \\geq c \\right ) . \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} \\scriptstyle f _ N ^ { * } H _ h ( N ) = - 2 \\lambda ( N - 1 ) + f _ N ^ { * } ( B ( N ) - H _ h ( N ) ) - B ( N - 1 ) . \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} P ( T _ 1 < x ) = P ( T _ 2 < x ) = x - x ^ 2 / 2 , \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} \\operatorname { D i v } P = Q ^ { \\alpha } F _ { \\alpha } . \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} x = \\sqrt { 2 r } \\cos \\theta , y = \\sqrt { 2 r } \\sin \\theta , z = z . \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} C _ k ( x ) = \\frac { x } { \\log x } \\frac { ( \\log \\log x ) ^ { k - 1 } } { ( k - 1 ) ! } & \\bigg \\{ h ( 0 ) + \\frac { k - 1 } { \\log \\log x } h ' ( 0 ) + \\frac { ( k - 1 ) ( k - 2 ) } { ( \\log \\log x ) ^ 2 } g \\ ( \\frac { k - 3 } { \\log \\log x } \\ ) \\\\ & + O _ A \\ ( \\frac { B _ 2 ( k - 1 ) ( k - 2 ) ( k - 3 ) } { ( \\log \\log x ) ^ 4 } + \\frac { \\log \\log x } { k } R ( x ) \\ ) \\bigg \\} , \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} \\tilde { \\epsilon } ( \\lambda ) = \\epsilon ( \\lambda ) - \\sum _ { i a } s _ a \\epsilon ( \\lambda _ { i a } ) F ( \\lambda _ { i a } | \\lambda ) . \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} \\varphi ( \\eta ) - M _ 0 = \\kappa _ 1 e ^ \\eta \\vec { X } _ { 0 1 } + \\kappa _ 2 e ^ { 2 \\eta } \\vec { X } _ { 0 2 } + \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} \\min _ { G _ n \\in \\mathcal { G } _ { \\eta , \\ , \\varepsilon } ( f , f ^ \\ast ) } \\rho ( G _ n ) = \\rho ( D _ \\ast , D ^ \\ast ) , \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} \\norm { ( \\sigma - 1 ) ^ { + } ( t ) } _ { L ^ { 2 } } ^ { 2 } \\leq \\norm { ( \\sigma - 1 ) ^ { + } ( 0 ) } _ { L ^ { 2 } } ^ { 2 } = 0 \\forall t \\in ( 0 , T ] , \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} \\dot { v } _ j + \\mu _ j ^ { \\infty } \\ , v _ j = 0 , j \\in S ^ c , \\mu _ j ^ { \\infty } \\in \\mathrm { i } \\mathbb { R } , \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} ( i + 1 \\to X _ { i , i + 1 } \\to i ) \\stackrel { \\mu _ i } { \\mapsto } \\begin{cases} ( 1 \\to X _ { 1 , 2 } ) , ( X _ { 1 , 2 } \\to 1 ^ - ) , ( X _ { 1 , 2 } , n ^ + ) & i = 1 , \\\\ ( i \\to X _ { i , i + 1 } \\to i - 1 ) & i = 2 , \\ldots , n - 2 , \\end{cases} \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} \\log _ 2 \\det ( \\mathbf { I } _ { n } + \\mathbf { A } \\mathbf { A } ^ { \\dagger } ) = \\log _ 2 \\det ( \\mathbf { I } _ { m } + \\mathbf { A } ^ { \\dagger } \\mathbf { A } ) . \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\Xi _ n = \\left \\{ \\sum _ { k , \\ell = 0 } ^ \\infty \\left | h _ n ( k , \\ell ) - h ( k , \\ell ) \\right | \\le K n ^ { - \\delta } \\right \\} . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} B ( s ) = \\sum _ n b \\left ( \\frac { \\log Q ^ 2 | s | ^ 2 / n } { \\log { N } } \\right ) \\overline \\lambda ( n ) n ^ { s - 1 } . \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} V ^ { \\ast } \\Psi K S _ { A } ^ { \\dagger } H + ( I + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } S _ { A } ^ { \\dagger } H = ( S _ { A } - I ) S _ { A } ^ { \\dagger } H + S _ { A } ^ { \\dagger } H = S _ { A } S _ { A } ^ { \\dagger } H . \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} f ( \\tau _ { \\le i } B ) \\subset \\tau _ { \\le i } ( B ' ( m ) ) = ( \\tau _ { \\le i + m } B ' ) ( m ) . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} E [ \\psi ( W , \\gamma , \\alpha _ { 0 } , \\theta _ { 0 } ) ] & = E [ m ( W , \\gamma ) ] - \\theta _ { 0 } + E [ \\alpha _ { 0 } ( X ) \\{ Y - \\gamma ( X ) \\} ] \\\\ & = E [ \\alpha _ { 0 } ( X ) \\{ \\gamma ( X ) - \\gamma _ { 0 } ( X ) \\} ] + E [ \\alpha _ { 0 } ( X ) \\{ \\gamma _ { 0 } ( X ) - \\gamma ( X ) \\} ] = 0 . Q . E . D . \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} \\Lambda M = \\left \\{ \\gamma : S ^ { 1 } \\to M \\mid \\gamma \\ { \\rm i s \\ a b s o l u t e l y \\ c o n t i n u o u s \\ a n d } \\ \\int _ { 0 } ^ { 1 } F ( \\gamma , \\dot { \\gamma } ) ^ { 2 } d t < + \\infty \\right \\} , \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} \\scriptstyle m _ N ^ { - 1 } H _ n ( N ) = H _ n ( I I _ { 2 , 2 + 8 k } \\oplus A _ 1 ) , m _ N ^ { - 1 } H _ h ( N ) = H _ u ( I I _ { 2 , 2 + 8 k } \\oplus A _ 1 ) . \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} f ( p ) = p ( 1 - p ) \\left [ ( r _ 1 - r _ 2 ) - \\left ( \\frac { r _ 1 } { \\alpha _ 1 } \\ , p - \\frac { r _ 2 } { \\alpha _ 2 } ( 1 - p ) \\right ) \\right ] \\ , , \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} { h } ( \\theta ) = A ( e ^ { i \\theta } ) - \\left ( \\sum \\limits _ { j = 0 } ^ { \\infty } ( \\bold { B } ^ { - 1 } \\bold { a } ) _ j e ^ { i j \\theta } \\right ) \\left ( f ( \\theta ) \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} | < x ^ { * } _ { n } , x _ { n } > | = | \\sum _ { k = 1 } ^ { \\infty } < P _ { k } x ^ { * } _ { n } , \\pi _ { k } x _ { n } > | > \\epsilon _ { 0 } , n = 1 , 2 , . . . \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} u _ { t } = u _ { x x } + u f ( t , u ) \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} m _ 1 = m _ 2 = \\ldots = m _ { i _ j - 1 } = 0 . \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} \\tau _ 0 = \\epsilon / ( 1 5 M _ 4 ) . \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} 0 \\leq \\psi \\leq \\frac { 1 } { 2 a ^ * \\sum _ { l = 0 } ^ m | b _ l | } . \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} \\lVert \\mathtt { R } _ k \\rVert _ s ^ { L i p ( \\gamma ) } \\le _ s \\varepsilon ^ 3 + \\varepsilon \\lVert \\mathfrak { I } _ { \\delta } \\rVert _ { s + \\sigma } ^ { L i p ( \\gamma ) } , \\lVert \\partial _ i \\mathtt { R } _ k [ \\hat { \\imath } ] \\rVert _ s \\le _ s \\varepsilon ( \\lVert \\hat { \\imath } \\rVert _ { s + \\sigma } + \\lVert \\mathfrak { I } _ { \\delta } \\rVert _ { s + \\sigma } \\lVert \\hat { \\imath } \\rVert _ { s _ 0 + \\sigma } ) , k = 0 , 1 . \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} \\varepsilon M _ x [ v _ { \\delta } \\tilde { q } ] = \\varepsilon ^ { 2 + b } M _ x [ \\Psi ' _ 2 ( v _ { \\delta } ) v _ { \\delta } z _ 0 ] + \\varepsilon M _ x [ v _ { \\delta } \\ , \\Psi _ 3 ( T _ { \\delta } ) ] , \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} I _ { l o w e r } ( f , w ) = \\log { \\left [ \\frac { | w | | f ' ( 0 ) | } { ( 1 + | w | ) ^ 2 } \\frac { ( 1 + | f ( w ) | ) ^ 2 } { | f ( w ) | } \\right ] } \\leq 0 \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} \\iota _ { \\xi _ L } \\omega _ L = d E _ L \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} \\Delta _ n ( f , \\widetilde { f } ) = n [ P _ n ( \\widetilde { f } | | f ^ { \\star } ) - P _ n ( f | | f ^ { \\star } ) ] - ( n / \\tau ) [ P ^ { \\prime } _ n ( \\widetilde { f } | | f ^ { \\star } ) - P ^ { \\prime } _ n ( f | | f ^ { \\star } ) ] . \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} ( f + g ) ( x , y ) & = f ( x , y ) + g ( x , y ) , \\\\ ( r f ) ( x , y ) & = r f ( x , y ) , \\\\ ( f g ) ( x , y ) & = \\sum _ { x \\le z \\le y } f ( x , z ) g ( z , y ) \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} \\frac { P } { \\sqrt { c } } = \\frac { c + \\sqrt { c ( c + 2 ) } } { \\sqrt { c ( c + 2 ) } } ( c + 1 + \\sqrt { c ( c + 2 ) } ) ^ m = a + b \\alpha = a + \\frac { b _ 1 } { c + 2 } \\alpha , \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} \\sum _ { q = m } ^ { m - p + 1 } & { m \\choose q } \\frac { \\Gamma ( 2 + q ) } { ( m + 1 ) ^ { 1 + q } } = \\\\ & \\frac { \\Gamma ( m - p + 3 ) } { ( m + 1 ) ^ { m - p + 1 } } \\frac { m ( m - 1 ) ( . . . ) ( m - p + 3 ) } { ( p - 1 ) ! } , \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} \\log M & \\leq ( 1 + \\epsilon ) B _ 1 ( n ) - \\frac { H _ 2 ( \\alpha _ n ) } { \\alpha _ n } \\\\ & = ( 1 + \\epsilon - \\theta _ n ) B _ 1 ( n ) , \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} T \\gamma \\left ( \\frac { \\partial } { \\partial t } \\right ) = \\frac { \\partial } { \\partial t } + \\frac { \\partial \\gamma } { \\partial t } \\frac { \\partial } { \\partial p } , T \\gamma \\left ( \\frac { \\partial } { \\partial q } \\right ) = \\frac { \\partial } { \\partial q } + \\frac { \\partial \\gamma } { \\partial q } \\frac { \\partial } { \\partial p } , \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} \\mathcal { T } ( q ^ { \\pm 1 / 2 } ) = \\left ( - 1 \\right ) ^ { \\mathsf { N } } X _ { q } M ( 1 ) , \\mathcal { T } ( i q ^ { \\pm 1 / 2 } ) = \\left ( - 1 \\right ) ^ { \\mathsf { N } } X \\frac { ( \\zeta _ { + } + 1 / \\zeta _ { + } ) } { ( \\zeta _ { + } - 1 / \\zeta _ { + } ) } \\frac { ( \\zeta _ { - } + 1 / \\zeta _ { - } ) } { ( \\zeta _ { - } - 1 / \\zeta _ { - } ) } _ { q } M ( i ) . \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} W \\ ; : \\ ; y ^ 2 = x ^ 3 - 2 a x ^ 2 + ( a ^ 2 - 4 b ) x , \\Delta _ W = 2 ^ 8 b ( a ^ 2 - 4 b ) ^ 2 . \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} L ( s , \\delta _ 1 , \\delta _ 2 ^ \\vee ) = \\prod _ { i = 1 } ^ { \\min ( k _ 1 , k _ 2 ) } L ( s + \\frac { k _ 1 + k _ 2 } 2 - i , \\rho _ 1 , \\rho _ 2 ^ \\vee ) . \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} \\mathbb { E } ( \\kappa ( G , B ) ) = \\sum _ { e \\in E ( G ) } \\mathbb { E } \\bigg ( \\frac { 1 } { d ' ( e ) + 1 } \\bigg ) = \\sum _ { e \\in E ( G ) } \\frac { { n \\choose 2 } } { ( 2 n - 3 ) m } \\bigg ( 1 - \\frac { { { n \\choose 2 } - m \\choose 2 n - 3 } } { { { n \\choose 2 } \\choose 2 n - 3 } } \\bigg ) \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} \\Diamond ( p , q ) \\ = \\ \\Diamond ( q , p ) \\ , . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} j _ 1 & = \\chi _ 1 ( 1 ) + \\chi _ 2 ( 1 ) \\\\ j _ 2 & = \\chi _ 1 ( 1 ) + \\chi _ 2 ( 1 ) + \\chi _ 3 ( 1 ) \\\\ j _ 3 & = 2 \\chi _ 1 ( 1 ) + 2 \\chi _ 2 ( 1 ) + \\chi _ 3 ( 1 ) + \\chi _ 4 ( 1 ) . \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} 0 = n ^ K ( C ' ) + n ^ K ( - C '' ) + E _ + F = n ^ K ( C ) + | C '' | + E _ + F = n ^ K ( C ) + | C ' | + V \\ , , \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} v ( x , t ) = \\int _ { { \\bf R } ^ N } G ( x - y , t - \\tau ) v ( y , \\tau ) \\ , d y + \\int _ \\tau ^ t \\int _ { { \\bf R } ^ N } G ( x - y , t - s ) v ( y , s ) ^ p \\ , d y \\ , d s < \\infty \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} T \\phi _ n = \\lambda _ n \\phi _ n , \\phi _ n \\neq 0 , n > N _ 1 \\geq N . \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} 2 ( 1 3 2 + ( 1 8 - N ) ( 1 7 - N ) \\lambda ( N ) = H _ n ( N ) + 2 ( 1 8 - N ) H _ h ( N ) + \\tau ( N ) \\mu ( N + 8 ) H _ u ( N ) , \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} \\rho _ { \\mathrm { i m p } } ( \\lambda | \\lambda _ p ) = L ( \\lambda | \\lambda _ p ) . \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} A ^ 2 - \\mathrm { T r } A \\cdot A + \\det { A } \\cdot \\mathrm { I d } = 0 , \\\\ A - \\mathrm { T r } A \\cdot \\mathrm { I d } + \\det { A } \\cdot A ^ { - 1 } = 0 . \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} V ^ { 2 } - 3 U ^ { 2 } = - 2 \\mu , U ^ { 2 } + Z ^ { 2 } = 2 \\mu . \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} { \\mathcal { D } } \\left ( \\l \\right ) = H _ 0 ^ 1 ( \\Omega ) . \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} \\begin{aligned} \\lim _ { x \\to + \\infty } 2 \\int _ 0 ^ 1 \\left ( 1 - \\frac { Q ( x t ) } { Q ( x ) } \\right ) ^ \\frac 1 2 \\ , \\dd t & = \\Omega _ \\beta , \\\\ \\lim _ { x \\to + \\infty } 2 \\int _ 0 ^ 1 \\left ( 1 - \\frac { Q ( x t ) } { Q ( x ) } \\right ) ^ { - \\frac 1 2 } \\ , \\dd t & = \\Omega _ \\beta ' . \\end{aligned} \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} H = \\int d x \\left [ \\partial _ x \\Psi ^ { \\dag } ( x ) \\partial _ x \\Psi ( x ) + c \\Psi ^ { \\dag } ( x ) \\Psi ^ { \\dag } ( x ) \\Psi ( x ) \\Psi ( x ) \\right ] , c > 0 . \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} \\delta ( \\rho X \\sigma ) & = \\delta ( \\rho X ) \\sigma + \\rho \\delta ( X \\sigma ) - X \\delta ( \\rho \\sigma ) - \\rho \\delta ( X ) \\sigma + \\rho X \\delta ( \\sigma ) \\\\ & = \\delta ( \\rho X ) \\star X + \\rho \\delta ( X \\wedge \\star X ) - X \\rho V - \\rho \\delta ( X ) \\star X - \\rho X \\wedge \\star d X \\\\ & = \\rho ( \\pi - X V ) + \\delta ( \\rho X ) \\star X + \\frac { \\rho } { 2 } \\delta ( X \\wedge \\star X ) - \\rho X \\wedge \\star d X . \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} - \\Delta _ { p ( x ) } \\underline { u } = \\left \\{ \\begin{array} { l l } \\lambda ^ { \\sigma } w _ { 1 } ^ { \\alpha _ { 1 } ( x ) } & \\Omega \\backslash \\overline { \\Omega } _ { \\delta } \\\\ - w _ { 1 } ^ { \\alpha _ { 1 } ( x ) } & \\Omega _ { \\delta } \\end{array} \\right . , \\underline { u } = 0 \\partial \\Omega \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} H ^ { \\lambda _ n } ( \\mu _ n , p _ n , q _ n ) = ( p _ n , q _ n ) \\in \\textbf { F } _ { \\lambda _ n } . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} \\int _ { B } u \\ , ( - \\Delta ) ^ s \\varphi \\ d x = \\int _ { B } - \\Delta u \\ , ( - \\Delta ) ^ { s - 1 } \\varphi \\ d x . \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} v _ f ^ { 4 / 3 } \\left ( \\frac { \\gamma _ n { v ^ 2 _ 0 } \\log ( d + 1 ) } { n } \\right ) ^ { 1 / 3 } + v _ f \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n | Y _ i | \\right ) \\left ( \\frac { \\gamma _ n { v ^ 2 _ 0 } \\log ( d + 1 ) } { n } \\right ) ^ { 1 / 3 } + \\frac { T _ n } { n } . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} \\Pr ( Y _ { 2 t } = 1 | X _ { t } ) = \\Lambda ( a ( X _ { t } , \\theta _ { 0 } , \\gamma _ { 2 0 } , \\gamma _ { 3 0 } ) ) , a ( x , \\theta , \\gamma _ { 2 } , \\gamma _ { 3 } ) = D ( x ) ^ { \\prime } \\theta + \\delta \\{ \\gamma _ { 2 } ( x ) - \\gamma _ { 3 } \\} . \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} \\delta ^ { - 1 } \\varrho = 0 . \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} h _ g ( f _ 0 , g _ 0 ) = \\frac { \\left | A ( e ^ { i \\theta } ) f _ 0 ( \\theta ) - \\sum \\limits _ { j = 0 } ^ { \\infty } ( ( \\bold { B } ^ 0 ) ^ { - 1 } \\bold { R } ^ 0 \\bold { a } ) _ j e ^ { i j \\theta } \\right | ^ 2 } { ( f _ 0 ( \\theta ) + g _ 0 ( \\theta ) ) ^ 2 } \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} D ^ { ( s ) } = D _ 1 \\times \\dots \\times D _ s \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} & \\left | f \\right | _ { C ^ { \\alpha } ( E ) } : = \\sup _ { \\substack { x _ 1 , x _ 2 \\in E \\\\ x _ 1 \\neq x _ 2 } } \\frac { \\left | f ( x _ 2 ) - f ( x _ 1 ) \\right | } { \\left | x _ 2 - x _ 1 \\right | ^ { \\alpha } } , \\left \\| f \\right \\| _ { L ^ { \\infty } ( E ) } = \\sup _ { x \\in E } \\left | f ( x ) \\right | , \\\\ & \\left \\| f \\right \\| _ { C ^ { \\alpha } ( E ) } : = \\left | f \\right | _ { C ^ { \\alpha } ( E ) } + \\left \\| f \\right \\| _ { L ^ { \\infty } ( E ) } . \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\ddot { u } = 0 , \\ddot { v } = 0 . \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} k ^ * ( y ) = \\sum _ d \\gamma ^ * ( d ) \\phi ( d y ) . \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} | | f _ x | | = \\sup _ { t \\in \\mathbb { R } } \\frac { [ y + t b ( x ) , b ( x ) ] } { | | y + t b ( x ) | | } = \\frac { [ y , b ( x ) ] } { \\inf _ { t \\in \\mathbb { R } } | | y + t b ( x ) | | } , \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} \\nu = g _ { \\| T \\| } \\| T \\| + \\nu _ { \\| T \\| } ^ s \\qquad \\nu _ { \\| T \\| } ^ s \\perp \\| T \\| \\ ; , \\ ; \\int g _ { \\| T \\| } \\dd \\| T \\| > 0 \\ ; . \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} \\left . \\frac { d \\gamma _ { \\partial B } } { d a } \\right | _ y = \\left . \\frac { d \\gamma _ { \\partial B } } { d s } . \\frac { d s } { d a } \\right | _ y = \\frac { b ( y ) } { | | b ( y ) | | } . | | b ( y ) | | = b ( y ) . \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} ( H _ 1 / H _ 2 ) ^ r = H _ 1 ^ r H _ 2 / H _ 2 . \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} | E ( G ) | ! p _ G ( x ) & = \\sum _ k F ( G , k ) x ^ k \\\\ & = \\sum _ { e \\in E ( G ) } \\bigg ( \\sum _ k F ( G - e , k ) x ^ k \\bigg ) \\\\ & = \\sum _ { e \\in E ( G ) } \\bigg ( ( | E ( G ) | - 1 ) ! \\sum _ k P ( G - e , k ) x ^ k \\bigg ) \\\\ & = ( | E ( G ) | - 1 ) ! \\sum _ { e \\in E ( G ) } p _ { G - e } ( x ) . \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} \\nabla ^ 2 _ g \\varphi = - \\frac { R _ g } { n ( n - 1 ) } \\varphi g \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} f ( \\theta ) = \\left | \\sum \\limits _ { j = 0 } ^ { \\infty } \\varphi _ { j } e ^ { - i j \\theta } \\right | ^ { 2 } , \\sum \\limits _ { j = 0 } ^ { \\infty } | \\varphi _ { j } | ^ { 2 } = P , \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} \\sum ( \\Theta _ { j \\bar k } ^ { T _ B } \\partial / \\partial t ^ k , \\partial / \\partial t ^ j ) _ H = - | | \\sum [ \\theta _ k ^ * , \\theta _ k ] | | ^ 2 - \\sum | | P ^ { \\bot } [ \\partial ^ h _ j , \\theta _ k ] | | ^ 2 , \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} H ^ { ( 3 ) } _ 4 ( u ) = \\sum _ { j _ 1 + j _ 2 + j _ 3 + j _ 4 = 0 } H ^ { ( 3 ) } _ { 4 , \\ , j _ 1 j _ 2 j _ 3 j _ 4 } u _ { j _ 1 j _ 2 j _ 3 j _ 4 } . \\end{align*}"}
-{"id": "1349.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": "1284.png", "formula": "\\begin{align*} \\max \\{ g _ i ^ T f _ c ( x ) \\ , | \\ , g _ i ^ T x = b _ i x \\in \\mathcal { P } \\} , i \\in \\mathcal { I } ( m ) . \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} \\sqrt { \\beta _ K } \\hat { \\mathbf { G } } = \\mathbf { Y } _ p \\boldsymbol { \\Psi } _ p ^ { \\dagger } ( \\boldsymbol { \\Psi } _ { p } \\boldsymbol { \\Psi } _ p ^ { \\dagger } ) ^ { - 1 } = \\mathbf { Y } _ p \\boldsymbol { \\Psi } _ p ^ { \\dagger } / \\left ( m _ 0 m _ 1 \\rho _ p \\right ) . \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} Z _ { i j } = \\delta _ { i j } + \\int _ { \\lambda _ { n + 1 - i L } } ^ { \\lambda _ { i R } } d \\nu \\ , F ( \\lambda _ { j R } | \\nu ) . \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} \\| f \\| _ H = \\left ( \\| f \\| _ 1 ^ 2 + \\| f \\| _ 2 ^ 2 \\right ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} N = \\dbinom { K } { 2 M - K } \\prod \\limits _ { m = 1 } ^ { K - M } { \\left ( { 2 m - 1 } \\right ) } . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} \\lambda ( U _ { x _ o } \\cap V _ { x _ o } ) = \\lambda ( ( U \\cap V ) _ { x _ o } ) = \\mu ( U \\cap V ) = \\mu ( U ) . \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} \\mathbb { M } + B ( j , k ) & = ( 2 4 c _ 1 ^ 2 - 1 2 c _ 4 ) D _ S ^ 6 + ( \\frac { 1 4 } { 3 } c _ 2 ^ 2 - 4 c _ 6 ) D _ S ^ 4 + ( 4 c _ 6 - \\frac { 1 6 } { 3 } c _ 2 ^ 2 ) D _ S ^ 3 U D _ S ( \\mathrm { I } - \\frac { 1 } { j ^ 2 + k ^ 2 + j k } D _ S ^ 2 ) \\\\ & + 1 2 ( c _ 2 c _ 3 - c _ 7 ) D _ S ^ 2 - 6 c _ 3 ^ 2 \\mathrm { I } + ( 1 6 c _ 2 c _ 3 - 2 4 c _ 7 ) D _ S U D _ S ( \\mathrm { I } - \\frac { 1 } { j ^ 2 + k ^ 2 + j k } D _ S ^ 2 ) . \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{align*} \\int \\limits _ { 0 } ^ { 1 } u ( x , t ) d x = E ( t ) , 0 \\leq t \\leq T , \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} \\phi ( z ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\varepsilon ) } \\tilde \\Psi ( s ) \\zeta ( s ) z ^ { - s } d s \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} \\sum _ { m _ 1 + m _ 2 + m _ 3 < p } \\binom { m _ 1 + m _ 2 + m _ 3 } { m _ 1 , m _ 2 , m _ 3 } ^ 2 \\left ( H _ { m _ 1 + m _ 2 + m _ 3 } - H _ { m _ 1 } \\right ) \\equiv _ { p } 0 . \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} \\| f \\| _ H ^ 2 = \\langle f , 1 \\rangle _ 1 ^ 2 + \\| f \\| _ 2 ^ 2 , \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} \\int _ { D _ j } \\left | f \\right | \\ , \\mathrm d \\rho _ j = \\sqrt { 2 / \\pi } \\cdot \\| f \\| _ { k _ j } \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} D _ { i j } ^ + : = \\begin{cases} ~ ( j - p ) / p & i \\leq j < p \\\\ ~ j / p & i > j , ~ j < p \\\\ ~ 0 & j = p \\end{cases} \\ , . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} \\| \\sqrt { \\eta } \\epsilon _ { z , 1 } ^ N \\| _ { L ^ 2 ( M \\times M ) } = O ( t ^ a e ^ { - ( N - \\frac { 3 } { 2 } - 2 a ) T } ) . \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} J _ M ^ 2 ( X ) = \\rho \\Big ( J _ \\mathcal { A } \\big ( \\lambda ( \\rho \\big ( J _ \\mathcal { A } ( \\lambda ( X ) ) \\big ) ) \\big ) \\Big ) = \\rho \\Big ( J _ \\mathcal { A } \\big ( \\big ( J _ \\mathcal { A } ( \\lambda ( X ) ) \\big ) ^ E \\big ) \\Big ) \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} Z ( G ) ( \\lambda ) : = \\sum _ { \\substack { I \\subseteq V \\\\ I } } \\lambda ^ { | I | } . \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{align*} \\lVert u \\rVert _ s : = \\lVert u \\rVert _ { H ^ { s } ( \\mathbb { T } ^ { \\nu + 1 } ) } : = \\lVert u \\rVert _ { H ^ s _ { \\varphi , x } } \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} \\tilde { x } = c \\beta + \\sum _ { ( i , p ) = 1 } c _ i T ^ { - i } . \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} \\hat { \\mathbf { P } } = \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ { 1 } . \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{gather*} \\{ A ^ \\ast , A ^ \\ast \\} _ 1 = 0 , \\{ A , A \\} _ 2 = 0 , \\{ C , C \\} _ 3 = - 1 . \\end{gather*}"}
-{"id": "3956.png", "formula": "\\begin{align*} \\gamma ( 1 / 2 , \\pi , \\tau , \\psi ) = 1 . \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} \\tau _ { \\le i } ( B ( m ) ) = ( \\tau _ { \\le i + m } B ) ( m ) . \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} \\# \\big \\lbrace z \\in \\sigma _ { \\textup { d i s c } } \\big ( H ( b _ 0 , e V ) \\big ) : - r _ 0 e ^ 2 \\leq z < - r e ^ 2 \\big \\rbrace = C _ { m } \\left ( \\frac { 1 } { 2 b _ 0 } \\right ) ^ { 1 / m } r ^ { - 1 / m } \\big ( 1 + o ( 1 ) \\big ) , \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{align*} x _ { 1 } ^ { 2 } = \\dfrac { 1 - b ^ { 2 } } { a ^ { 2 } - b ^ { 2 } } \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} \\delta E = \\sum _ { i } \\tilde { \\epsilon } _ i \\tilde { N } _ i + \\frac { 2 \\pi } { L } \\sum _ i \\frac { \\tilde { v } _ i } { 2 } \\left [ \\left ( \\sum _ j [ Z ^ { - 1 } ] _ { i j } \\tilde { N } _ j \\right ) ^ 2 + \\left ( \\sum _ j Z _ { j i } \\tilde { D } _ j \\right ) ^ 2 \\right ] \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} \\| u _ k \\| _ { L ^ 2 ( \\Omega ) } = 1 , [ u _ k ] _ { X ^ s ( \\Omega ) } \\rightarrow 0 \\mbox { a s } k \\rightarrow + \\infty . \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} [ S ( t ) \\mu ] ( x ) & \\le \\sum _ { k = 1 } ^ m \\sum _ { i = 1 } ^ \\infty \\int _ { B _ { k , i } } G ( x - y , t ) \\ , d \\mu ( y ) \\\\ & \\le C t ^ { - \\frac { N } { \\theta } } \\sup _ { z \\in { \\bf R } ^ N } \\mu ( B ( z , t ^ { \\frac { 1 } { \\theta } } ) ) \\sum _ { k = 1 } ^ m \\sum _ { i = 1 } ^ \\infty \\sup _ { y \\in B _ { k , i } } \\left ( 1 + t ^ { - \\frac { 1 } { \\theta } } | x - y | \\right ) ^ { - N - \\theta } . \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} f = a _ { 2 2 } ( 2 | 2 ) _ { A _ 0 } = ( 2 | 2 ) _ { A _ 1 } ( 2 | 2 ) _ { A _ 0 } = \\begin{matrix} ( 2 | 2 ) _ { A _ 0 } \\\\ ( 2 | 2 ) _ { A _ 1 } \\end{matrix} , \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} | \\alpha _ n | < 1 , n = 0 , 1 , 2 , \\dots . \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} c _ k = b _ k + ( B ^ { - 1 } - \\mathrm { I } ) b _ k + ( \\rho ^ { - 1 } - 1 ) \\ , B ^ { - 1 } \\ , b _ k \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} \\bold { D } _ A ( B ) : = \\frac { \\operatorname { d } } { \\operatorname { d } \\ ! \\varepsilon } \\Big | _ { \\varepsilon = 0 } A ( x , n , [ u + \\varepsilon B ( x , n , [ u ] ) ] ) . \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} D _ t P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 = 0 , \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} d _ { S ( T , t ) } ( u ^ t , u ' u ^ { i - 2 } x x _ { i + 1 } \\cdots x _ t ) & > d _ { S ( T , t ) } ( u ^ { t } , u ( u ' ) ^ { t - 1 } ) + d _ { S ( T , t ) } ( u ' u ^ { t - 1 } , u ' u ^ { i - 2 } x x _ { i + 1 } \\cdots x _ t ) \\\\ & > d _ { S ( T , t ) } ( u ' u ^ { t - 1 } , u ' u ^ { i - 2 } x x _ { i + 1 } \\cdots x _ t ) \\\\ & = d _ { S ( T , t ) } ( u ^ t , w ) , \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} J x = - x - k . \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} \\zeta _ 1 ( x ) ( t ) ( s ) = x ( t ) ( s ) h _ 1 ^ { 1 / 2 } ( s - t ) h _ 1 ^ { 1 / 2 } ( s ) \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} q ( \\xi , \\beta ) = \\frac { i \\sinh ( ( \\beta + h ) \\xi ) } { \\cosh ( h \\xi ) } . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} \\sup _ { k \\ge 0 } \\left | F _ n ^ \\ast ( k ) - F ^ \\ast ( k ) \\right | \\le \\sum _ { k = 0 } ^ \\infty | f _ n ^ \\ast ( k ) - f ^ \\ast ( k ) | \\le n ^ { - \\varepsilon } . \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} c _ 0 = ( 1 + \\rho ) ^ { - n _ 0 + 1 } \\sum _ { j = 0 } ^ { n _ 0 - 1 } t _ { j } . \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} \\bar { \\psi } ( \\gamma , \\alpha _ { 0 } , \\theta _ { 0 } ) & = \\frac { d [ \\tau \\bar { \\psi } ( \\gamma , \\alpha _ { 0 } , \\theta _ { 0 } ) ] } { d \\tau } = \\frac { d [ ( 1 - \\tau ) \\bar { \\psi } ( \\gamma _ { 0 } , \\alpha _ { 0 } , \\theta _ { 0 } ) + \\tau \\bar { \\psi } ( \\gamma , \\alpha _ { 0 } , \\theta _ { 0 } ) ] } { d \\tau } \\\\ & = \\frac { \\bar { \\psi } ( ( 1 - \\tau ) \\gamma _ { 0 } + \\tau \\gamma , \\alpha _ { 0 } ) } { d \\tau } = 0 . Q . E . D . \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ 1 \\cap L ^ 2 _ 1 } ^ 2 : = \\int _ { \\mathbb { R } } \\left [ u _ x ^ 2 + \\frac { 1 } { 4 } x ^ 2 u ^ 2 + \\frac { 1 } { 2 } u ^ 2 \\right ] d x \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} | | P _ j ^ { ( a , b ) } | | _ 2 = \\frac { 2 ^ { a + b + 1 } } { 2 j + a + b + 1 } \\frac { \\Gamma ( j + a + 1 ) \\Gamma ( j + b + 1 ) } { \\Gamma ( j + a + b + 1 ) j ! } , \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} - \\left ( \\begin{matrix} \\varphi \\\\ \\psi \\end{matrix} \\right ) '' - A ( x ) \\left ( \\begin{matrix} \\varphi \\\\ \\psi \\end{matrix} \\right ) = \\l \\left ( \\begin{matrix} \\varphi \\\\ \\psi \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} \\Phi _ \\tau ^ n ( u ) = \\frac 1 { 2 ^ { 2 n + 1 } \\tau ^ { \\frac { n + 1 } 2 } \\sqrt \\pi } e ^ { - \\left ( \\frac { ( 1 + 4 \\tau ) r } { 2 \\sqrt { 2 \\tau } } + \\frac u { 2 \\sqrt \\tau } \\right ) ^ 2 } H _ n \\Big ( \\frac { ( 1 + 4 \\tau ) r } { 2 \\sqrt { 2 \\tau } } + \\frac u { 2 \\sqrt \\tau } \\Big ) e ^ { 2 \\tau r ^ 2 + \\sqrt 2 r u } - ( u \\leftrightarrow - u ) \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} _ { a } \\mathcal { W } _ { p , b , c , \\xi } ^ { \\alpha , \\mu } \\left ( z \\right ) : = \\sum _ { k = 0 } ^ { \\infty } \\frac { \\left ( - c \\right ) ^ { k } } { \\Gamma \\left ( \\alpha k + \\mu \\right ) \\Gamma \\left ( a k + \\frac { p } { \\xi } + \\frac { b + 2 } { 2 } \\right ) } \\left ( \\tfrac { z } { 2 } \\right ) ^ { 2 k + p + 1 } ~ ~ ~ ~ ( a \\in \\mathbb { N } , p , b , c \\in \\mathbb { C } ) , \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} \\eta _ t = \\eta _ 0 + t ( \\eta - \\eta _ 0 ) , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} \\lim _ { s \\to 1 / 2 } \\frac { L ( 1 - s , \\pi , \\tau ) } { L ( s , \\pi , \\tau ) } = - 1 . \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} \\partial _ x L \\partial _ x V = 0 , \\partial _ x L V = - \\partial _ x V , \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} P = \\frac { 1 } { \\sqrt { c + 2 } } ( c + \\sqrt { c ( c + 2 ) } ) ( c + 1 + \\sqrt { c ( c + 2 ) } ) ^ m . \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} A ^ h ( \\xi , \\eta ) & = \\frac { 2 i \\eta J ( \\xi ) \\left ( J ( \\zeta ) - J ( \\xi ) + J ( \\eta ) \\right ) } { \\Omega } , \\\\ B ^ h ( \\xi , \\eta ) & = - \\frac { 2 i \\zeta J ( \\xi ) J ( \\eta ) } { \\Omega } , \\\\ C ^ h ( \\xi , \\eta ) & = - \\frac { i \\xi \\eta \\zeta \\left ( J ( \\zeta ) - J ( \\xi ) - J ( \\eta ) \\right ) } { \\Omega } . \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { q _ { n _ l } - 1 } | f ( \\theta + j \\alpha ) | \\geq \\frac { e ^ { ( \\delta _ c - \\epsilon ) q _ { n _ l } } } { q _ { n _ l + 1 } } . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} P _ 1 = \\widehat { P } _ 1 - R _ 1 - L \\xi , P _ 2 = \\widehat { P } _ 2 - R _ 2 . \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} S ( 2 ; 0 , 0 ) = 2 8 8 \\le S _ U ( 2 ; 0 , 0 ) = 3 8 4 . \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} ( A + U V ^ { \\ast } ) ^ { \\dagger } = ( L _ { 1 } N _ { 1 } + L _ { 3 } N _ { 2 } ) R _ { 1 } + ( L _ { 1 } N _ { 3 } + L _ { 3 } N _ { 4 } ) R _ { 2 } . \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} w _ 0 ( - w + v ) = - ( ( - w + v , - w + v ) + ( - w + v , w ) ) / 2 = ( ( w , v ) - ( v , v ) ) / 2 , \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{align*} W = \\bigoplus _ i H ^ { 0 } ( X , K _ X ^ { d _ i } ( d _ i - 1 ) ) \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} S _ { n ^ 1 } ^ { k } u ^ { \\alpha } = h ^ { \\alpha } \\left ( x , n , [ u ] _ { n ^ 1 } , \\left [ S _ { n ^ 1 } ^ { 1 } u \\right ] _ { n ^ 1 } , \\left [ S _ { n ^ 1 } ^ { 2 } u \\right ] _ { n ^ 1 } , \\ldots , \\left [ S _ { n ^ 1 } ^ { k - 1 } u \\right ] _ { n ^ 1 } \\right ) . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} \\frac { p ( 1 - \\beta _ j , 1 - \\alpha _ j ; 1 , y _ j ) } { p ( \\alpha _ j , \\beta _ j ; 1 , y _ j ) } = \\left \\{ \\begin{array} { c c } \\frac { \\Gamma ( 1 - \\beta _ j ) } { \\Gamma ( \\alpha _ j ) } \\cdot \\left | 4 \\pi y _ j \\right | ^ { 1 - \\alpha _ j - \\beta _ j } & \\mbox { i f $ y _ j > 0 $ } \\\\ \\frac { \\Gamma ( 1 - \\alpha _ j ) } { \\Gamma ( \\beta _ j ) } \\cdot \\left | 4 \\pi y _ j \\right | ^ { 1 - \\alpha _ j - \\beta _ j } & \\mbox { i f $ y _ j < 0 $ } \\end{array} \\right . \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} \\left \\vert E [ K _ { h } ( Y - \\hat { \\gamma } ( X ) ) | X ] - E [ K _ { h } ( U ) | X ] \\right \\vert & = \\left \\vert \\int K _ { h } ( u - \\hat { \\Delta } ( X ) ) f ( u | X ) d u - \\int K _ { h } ( u ) f ( u | X ) d u \\right \\vert \\\\ & = \\left \\vert \\int K ( v ) [ f ( h v + \\hat { \\Delta } ( X ) | X ) - f ( h v | X ) ] d v \\right \\vert \\\\ & \\leq \\int \\left \\vert K ( v ) \\right \\vert \\left \\vert f ( h v + \\hat { \\Delta } ( X ) | X ) - f ( h v | X ) \\right \\vert d v \\leq C \\hat { \\Delta } ( X ) . \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} \\sup _ \\ell \\sup _ { \\ell _ s } 2 ^ { \\ell t } 2 ^ { \\ell _ s s } \\sup _ { \\Gamma \\in \\Sigma } \\| \\psi ^ { O p ( \\Gamma ) } _ { \\ell _ s } \\psi ^ { O p } _ \\ell \\varphi \\| _ { L _ p ( \\Gamma ) } = \\sup _ \\ell 2 ^ { \\ell t } \\sup _ { \\Gamma \\in \\Sigma } \\| \\psi ^ { O p } _ \\ell \\varphi \\| _ { B ^ s _ { p , \\infty } ( \\Gamma ) } \\ , . \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} U ^ { 1 - } = U ^ { 2 + } = \\frac { \\partial U ^ { 1 - } } { \\partial n } = \\frac { \\partial U ^ { 2 + } } { \\partial n } = 0 \\ \\mbox { o n } \\partial \\Omega . \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{align*} \\Lambda _ C ( \\theta ) = - \\theta + \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mathbf { E } ( C _ n ^ \\theta ) = \\begin{cases} - \\theta - \\log ( 1 - \\theta ) , & ; \\\\ + \\infty , & . \\end{cases} \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{gather*} V ^ s ( a ) : = \\{ X \\in V \\colon s ( X ) = a X \\} \\end{gather*}"}
-{"id": "1397.png", "formula": "\\begin{align*} & u ( x + z , \\tau + ( 2 \\rho ) ^ \\theta ) \\ge \\int _ { B ( z , \\rho ) } G ( x + z - y , ( 2 \\rho ) ^ \\theta ) u ( y , \\tau ) \\ , d y \\\\ & \\qquad \\ge \\int _ { B ( z , \\rho ) } G ( 2 x , ( 2 \\rho ) ^ \\theta ) u ( y , \\tau ) \\ , d y = 2 ^ { - N } G ( x , \\rho ^ \\theta ) \\int _ { B ( z , \\rho ) } u ( y , \\tau ) \\ , d y \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} Y _ t = D Y _ { x x } + R ( Y ) , x \\in \\mathbb { R } , \\ t \\geq 0 , \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{align*} \\Psi _ N ( z ) : = z B _ { N - 1 } ^ * ( z ) + A _ { N - 1 } ^ * ( z ) , \\Psi _ N ^ * ( z ) : = B _ { N - 1 } ( z ) + z A _ { N - 1 } ( z ) . \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} R ( s ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( - \\varepsilon ) } \\overline L ( 1 - s - z ) ( Q | s | ) ^ { - 2 z } f ( z ) \\eta ( s , z ) d z . \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} 2 \\xi _ u u = 0 . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{align*} \\mathcal { I } ^ { j , k } _ e : = \\{ \\gamma \\in e + z \\mathfrak { g } [ [ z ] ] \\mid \\nu _ t ( F _ j ( \\gamma ) ) \\leq k \\} \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} \\frac { \\mathsf { C } _ { \\mathcal { N } _ { \\mathcal { K } } } } { { \\mathsf { C } } _ { \\mathcal { N } _ { [ 1 : N ] } } } = \\begin{cases} \\frac { 1 } { 4 } , & | \\mathcal { K } | = 1 \\\\ \\frac { 1 } { 2 } , & | \\mathcal { K } | = 2 \\\\ \\end{cases} . \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { b - 1 } f ( k , i ) x ^ { b ^ i k } = \\sum _ { k = 0 } ^ { \\infty } x ^ k \\prod _ { i = 0 } ^ { \\infty } f ( k _ i , i ) . \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\bar c ( | \\nabla u | ) = 0 , \\\\ u ( x , 0 ) = u _ 0 ( x ) . \\end{cases} \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} s - \\epsilon ' \\ = \\ \\frac { \\delta ' } { 1 + \\delta } . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} | X ( - J , - 1 ) | + k - 2 = & \\sum _ { j \\neq i } \\mathcal { D } ^ { i j } ( - J , - 1 ) \\\\ = & - \\sum _ { j \\neq i } \\mathcal { D } ^ { i j } ( 0 ) \\mathcal { D } ^ { i j } ( - J , - 1 ) = - \\frac 1 2 \\sum _ { j \\neq l } \\mathcal { D } ^ { j l } ( 0 ) \\mathcal { D } ^ { j l } ( - J , - 1 ) . \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\rho , \\phi } \\left [ W _ p ( \\Theta _ t ) \\right ] & = \\mathbb { E } _ { \\rho , \\phi } \\left [ W _ p ( \\Theta _ t ) { \\bf 1 } _ { \\{ T _ 0 > t \\} } \\right ] + \\mathbb { E } _ { \\rho , \\phi } \\left [ W _ p ( \\Theta _ t ) { \\bf 1 } _ { \\{ T _ 0 < t \\} } \\right ] \\\\ & = \\mathbb { P } _ { \\rho , \\phi } ( T _ 0 > t ) + \\mathbb { E } _ { \\rho , \\phi } \\left [ W _ p ( \\Theta _ t ) { \\bf 1 } _ { \\{ T _ 0 < t \\} } \\right ] \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{gather*} \\Psi ^ { ( k ) } _ 0 ( x ) \\sim \\left ( I + \\frac { m _ 1 } { x } + \\cdots \\right ) e ^ { \\left ( \\frac { x ^ 3 } { 6 } - \\frac { x t } { 2 } \\right ) \\sigma _ 3 } , x \\to \\infty , \\\\ \\frac { \\pi } { 2 } + \\frac { k - 2 } { 3 } \\pi < \\arg x < \\frac { \\pi } { 2 } + \\frac { k } { 3 } \\pi , k = 1 , 2 , \\dots , 7 . \\end{gather*}"}
-{"id": "3484.png", "formula": "\\begin{align*} h ( z ) = 1 + \\sum _ { n = 1 } ^ \\infty c _ n z ^ n \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} \\mu _ a = \\sum \\mu _ a ^ k ( r ) Y _ k \\left ( \\frac { x } { r } \\right ) , \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} W ' : y '^ 2 + a _ 1 ' x ' y ' + a _ 3 ' y ' = x '^ 3 + a _ 2 ' x '^ 2 + a _ 4 ' x ' + a _ 6 ' . \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} \\kappa ( r , y ) & = \\sum f _ \\lambda ( r ) \\kappa _ \\lambda ( y ) , \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\ ; \\tilde { d } ^ * \\tilde { d } \\kappa _ \\lambda = \\lambda \\kappa _ \\lambda , \\\\ \\eta ( r , y ) & = \\sum g _ \\lambda ( r ) \\tilde { d } \\kappa _ \\lambda ( y ) + h _ \\lambda ( r ) \\eta _ \\lambda ( y ) , \\tilde { d } ^ * \\tilde { d } \\eta _ \\lambda = \\lambda \\eta _ \\lambda , \\ ; \\tilde { d } ^ * { \\eta } _ \\lambda = 0 . \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} \\overline { a } b = a \\overline { b } = | b | ^ { 2 } = \\overline { d } e = d \\overline { e } = | e | ^ { 2 } = 0 . \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\chi _ { \\mathcal { A } } ( e _ { l , j } ( x ) ) = 0 \\ \\mathrm { f o r } \\ \\mathrm { a . e . } \\ x \\ \\mathrm { a n d } \\ \\mathrm { a n y } \\ l , j . \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} ( C C R f _ * { \\cal F } , T ^ * _ Y Y ) _ { T ^ * Y } & \\ = \\chi ( Y , R f _ * { \\cal F } ) \\\\ & \\ = \\chi ( P , { \\cal G } ) = ( C C { \\cal G } , T ^ * _ P P ) _ { T ^ * P } = ( g _ * C C { \\cal G } , T ^ * _ Y Y ) _ { T ^ * Y } . \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} B _ E ^ { - 1 } ( \\theta + \\alpha ) \\tilde { A } _ { | c | , E } ( \\theta ) B _ E ( \\theta ) = R _ { \\rho _ { | c | } ( E ) } . \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} \\| f \\| _ k = \\| f \\| _ { 1 + k _ 1 } = \\| f \\| _ H \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} \\eta = d z + \\sum _ { j = 1 } ^ n x _ j \\ , d y _ j . \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} \\cos \\theta = 0 \\textrm { o r } \\tan \\theta \\in \\mathbb { Q } . \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} d _ 1 & : = \\frac { S _ 1 \\mu } { R _ 1 } \\left ( S _ 1 \\frac { g _ * ^ 3 } { 3 } + S _ 1 \\mu \\left ( \\frac { f _ * ^ 3 } { 3 } + f _ * ^ 2 g _ * + f _ * g _ * ^ 2 \\right ) \\right ) , \\\\ [ 5 p t ] d _ 3 & : = - z ( \\mu f _ * + g _ * ) \\Gamma _ * \\sigma ^ \\prime ( \\Gamma _ * ) + z D . \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} \\Phi ( s ) : = \\int _ t ^ s e ^ { \\mu ( \\tau - t ) } p ( \\tau ) d \\tau + \\frac { 1 } { 2 } ( \\nu ^ 2 + \\nu _ 0 ^ 2 ) ( s - t ) + \\nu ( W ( s ) - W ( t ) ) + \\nu _ 0 ( W _ 0 ( s ) - W _ 0 ( t ) ) . \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} x \\circ y = [ [ x , y ] ] _ c + \\frac { 1 } { 2 } D ( x , y ) ( D o r f m a n \\ B r a c k e t ) \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} T _ { \\underline a \\vert _ k } = T _ { a _ 1 } \\circ \\cdots \\circ T _ { a _ k } . \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} \\frac { \\partial \\Pi _ i } { \\partial z ^ j } ( x , y , z ) & = \\frac { ( y \\times \\nu _ i ( x ) ) \\otimes ( y \\times \\nu _ i ( x ) ) \\Lambda _ j } { \\abs { y \\times \\nu _ i ( x ) } ^ 2 } \\\\ & = A ( y , x ) \\Lambda _ j , \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} \\omega ^ n = i ^ { n ^ 2 } \\Omega \\wedge \\bar \\Omega \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} \\Lambda _ n ^ \\rho = \\frac { e ^ { g ( W _ { n - 1 } , W _ n ) } } { 1 - \\rho } = e ^ { g _ \\rho ( W _ { n - 1 } , W _ n ) } , g _ \\rho ( W _ { i - 1 } , W _ { i } ) = g ( W _ { i - 1 } , W _ { i } ) + | \\log ( 1 - \\rho ) | . \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} \\mathcal { E } ^ { \\epsilon , \\kappa } ( \\alpha , \\beta ) : = \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( \\alpha , \\beta ) \\Lambda ^ \\epsilon ( \\alpha , \\beta ) \\ , \\mathfrak { h } ^ \\epsilon ( \\alpha - \\beta ) = \\Lambda ^ { { \\epsilon , \\kappa } } ( \\alpha , \\beta ) \\ , \\mathfrak { h } ^ \\epsilon ( \\alpha - \\beta ) \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} n = 6 ( g - 1 ) + 3 b + 3 p + h . \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} F = \\sum \\limits _ { j = 1 } ^ l ( \\iota _ j ) _ ! \\displaystyle \\frac { F ^ j } { e _ T ( N _ j ) } . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} T _ { 1 / f } ( x ) = \\begin{cases} x ^ { - 1 } R _ { 1 / f } ( x ) , & \\ 0 < M < \\frac 1 { 2 } ; \\\\ x ^ { - 1 } R _ { 1 / f } ( x ) \\log x , & \\ M = \\frac 1 { 2 } ; \\\\ x ^ { \\log M / \\log 2 } \\max ( \\log x , R _ { 1 / f } ( x ) ) , & \\ \\frac 1 { 2 } < M < 1 . \\end{cases} \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { \\ell } { p } \\right ) ^ r \\left ( \\frac { 3 } { p } \\right ) ^ t = 1 , \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} n ^ { - 1 } \\sum _ { i = 1 } ^ n \\log \\{ 1 + u [ f ( x _ i ; G ^ * ) / f ( x _ i ; B _ k ) - 1 ] \\} > 0 \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} A _ n : = \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ { n } x _ i \\epsilon _ { i j } y _ j \\quad \\epsilon _ { i , j } : = \\mathbf { 1 } _ { j > i } - \\mathbf { 1 } _ { j < i } . \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} \\breve { T } _ { 2 } + T _ { 1 } & = \\alpha _ { 3 0 } \\int \\zeta _ { 2 0 } ( x _ { t } ) H _ { p } ( \\gamma _ { 1 0 } ( x _ { t } ) ) [ y _ { 2 t } - \\hat { \\gamma } _ { 1 } ( x _ { t } ) ] F _ { 0 } ( d w ) \\\\ & = \\alpha _ { 3 0 } \\int \\zeta _ { 2 0 } ( x _ { t } ) H _ { p } ( \\gamma _ { 1 0 } ( x _ { t } ) ) [ \\gamma _ { 1 0 } ( x _ { t } ) - \\hat { \\gamma } _ { 1 } ( x _ { t } ) ] F _ { 0 } ( d w ) . \\end{align*}"}
-{"id": "82.png", "formula": "\\begin{align*} 2 \\imath \\psi _ { t } + \\psi _ { r r } + \\frac { 1 } { r } \\psi _ { r } + \\frac { 1 } { 4 k ^ 2 r ^ 2 } \\psi _ { \\phi \\phi } - \\left ( \\frac { 1 } { 4 r ^ 2 } + \\omega ^ 2 r ^ 2 \\right ) \\psi = 0 . \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\left \\lfloor { \\frac { y ^ { 1 / 2 } } { a } } \\right \\rfloor } ( y - a ^ { 2 } i ^ { 2 } ) ^ { n / 2 } = a ^ { n } \\sum _ { i = 1 } ^ { \\left \\lfloor { \\frac { y ^ { 1 / 2 } } { a } } \\right \\rfloor } \\bigg ( \\left ( \\frac { y ^ { 1 / 2 } } { a } \\right ) ^ 2 - i ^ { 2 } \\bigg ) ^ { n / 2 } . \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} e _ { T \\times \\C ^ * } ( E _ { 0 , n , d } ) ^ { - 1 } = \\displaystyle \\frac { e _ { T \\times \\C ^ * } ( E _ { 0 , n , d } ^ 1 ) } { e _ { T \\times \\C ^ * } ( E _ { 0 , n , d } ^ 0 ) } \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\kappa ^ * ( n ) = \\frac { 5 } { 3 8 } \\widetilde { C } x ^ 2 + O ( R _ { \\kappa ^ * } ( x ) ) . \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} e _ { ( m + 1 ) p _ n + \\frac { m + 1 } { 2 } } ( x ) = e _ { ( m + 1 ) ( p _ n - 1 ) + \\frac { m + 1 } { 2 } + 1 } ( x ) + x ^ { m - 1 } e _ { ( m + 1 ) ( p _ n - 1 ) + \\frac { m + 1 } { 2 } } ( x ) . \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} N _ E = 3 ^ 3 \\cdot 6 1 ^ 2 N _ { E ' } = 3 ^ 3 \\cdot 5 \\cdot 6 1 ^ 2 \\cdot 4 4 9 . \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} 1 \\leq \\sum _ { x \\in L } \\widehat f ( x ) = \\sum _ { x \\in L } f ( x ) \\leq 1 . \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } u \\ , ( - \\Delta ) ^ s \\varphi \\ d x = \\lim _ { n \\to \\infty } \\int _ { \\R ^ N } u \\ , ( - \\Delta ) \\psi _ n \\ d x = \\lim _ { n \\to \\infty } \\int _ { \\R ^ N } - \\Delta u \\ , \\psi _ n \\ d x = \\int _ { \\R ^ N } - \\Delta u \\ , ( - \\Delta ) ^ { s - 1 } \\varphi \\ d x , \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} \\nabla d f = \\frac { d ^ 2 f } { d x _ i d x _ j } d x _ i \\otimes d x _ j \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} K _ { k , j } ^ { ( \\alpha ) } = \\frac { { { 2 ^ { 2 j - 1 } } } } { { { { \\left ( { { \\tau _ k } - { \\tau _ { k - 1 } } } \\right ) } ^ j } } } \\frac { { \\Gamma \\left ( { 2 \\alpha + 1 } \\right ) \\Gamma \\left ( { j + \\alpha } \\right ) } } { { \\Gamma \\left ( { \\alpha + 1 } \\right ) \\Gamma \\left ( { j + 2 \\alpha } \\right ) } } \\forall j \\in \\mathbb { Z } _ 0 ^ + , \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} d \\nu _ 1 \\wedge \\dots \\wedge d \\nu _ { n - 1 } = \\pm d \\mu _ 1 \\wedge \\dots \\wedge d \\mu _ { n - 1 } . \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} Q ^ N _ \\ell ( \\cdot , \\cdot ) \\ : = \\ Q ( \\cdot , N ^ \\ell \\cdot ) \\ , , \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} m = m _ 1 m _ 2 \\dots m _ k \\equiv ( q _ 1 r _ 1 ) ( q _ 2 r _ 2 ) \\dots ( q _ k r _ k ) \\pmod p , \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\Gamma ^ { \\mathrm { h o r i } } ( f , g ) & : = \\frac { 1 } { 2 } ( L ( f g ) - f L g - g L f ) = X ( f ) X ( g ) + Y ( f ) Y ( g ) , \\\\ \\Gamma ^ { \\mathrm { v e r t } } ( f , g ) & : = Z ( f ) Z ( g ) , \\\\ \\Gamma ^ { \\mathrm { e l l i } } ( f , g ) & : = \\Gamma ^ { \\mathrm { h o r i } } ( f , g ) + \\nu \\Gamma ^ { \\mathrm { v e r t } } ( f , g ) . \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} \\| B ( z ) \\| \\leq M _ b \\left ( \\frac { \\max \\{ N _ 0 , N _ 0 ^ { 1 - 2 \\alpha } \\} } { h _ 1 } + \\sum _ { k = N _ 0 + 1 } ^ { \\infty } \\frac { 1 } { k ^ { 2 \\alpha } ( \\mu _ k + h _ 1 ) } \\right ) . \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} & ( \\tau ^ * \\circ \\phi ^ * ) ( y ^ j ) = \\widetilde { x } ^ j + \\sum _ { | P | \\geq 2 } \\widetilde { f } ^ j _ P ( \\widetilde { x } ) \\ , \\widetilde { \\xi } ^ P \\\\ & ( \\tau ^ * \\circ \\phi ^ * ) ( \\eta ^ s _ \\mu ) = \\widetilde { \\xi } ^ s _ \\mu + \\sum _ { | Q | \\geq 2 } \\widetilde { g } ^ j _ { Q } ( \\widetilde { x } ) \\ , \\widetilde { \\xi } ^ Q \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} \\eta ( \\underbrace { 1 , \\ldots , 1 } _ r ; \\underbrace { 1 , \\ldots , 1 } _ r ) = \\zeta ( \\underbrace { 2 , \\ldots , 2 } _ r ) . \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} \\left \\Vert \\bar { \\lambda } ( \\hat { \\gamma } _ { \\ell } ) - \\bar { \\lambda } ( \\gamma _ { 0 } ) \\right \\Vert ^ { 2 } & = \\int [ \\int _ { - \\infty } ^ { \\hat { \\gamma } ( x ) - \\gamma _ { 0 } ( x ) } f ( u | x ) d u - \\int _ { - \\infty } ^ { 0 } f ( u | x ) d u ] ^ { 2 } F _ { 0 } ( d x ) \\\\ & \\leq \\int C | \\hat { \\gamma } ( x ) - \\gamma _ { 0 } ( x ) | ^ { 2 } F _ { 0 } ( d x ) = C \\left \\Vert \\hat { \\gamma } - \\gamma _ { 0 } \\right \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{gather*} U ( t ) = - 2 \\omega - 4 \\alpha - \\frac { u _ t } { u } ( 1 - 2 q _ 2 ) - 6 \\frac { q _ 2 q _ { 2 t } } { 1 - q ^ 2 _ 2 } - \\frac { t ^ 2 } { 2 } . \\end{gather*}"}
-{"id": "1703.png", "formula": "\\begin{align*} q ( G ) ( z ) = k ^ { - | V | } p ( G ) ( \\mathcal { A } ' ( z ) ) . \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{gather*} [ Q , Y ] = 0 , i _ Y \\omega = \\delta C ^ \\ast , \\mathcal { L } _ Y L \\simeq 0 . \\end{gather*}"}
-{"id": "3808.png", "formula": "\\begin{align*} k = 2 ^ { p _ 0 } + \\cdots + 2 ^ { p _ { s } } , \\ \\ 0 \\le p _ 0 < p _ 1 < \\cdots < p _ s . \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} k _ n = \\alpha _ n \\ell _ n . \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} \\tilde { \\mathbf { P } } = \\mathbf { P } ^ { \\dagger } _ 2 \\mathbf { P } _ 2 = \\mathbf { P } ^ { \\rm H } _ 2 \\left ( \\mathbf { P } _ 2 \\mathbf { P } ^ { \\rm H } _ 2 \\right ) ^ { - 1 } \\mathbf { P } _ 2 , \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} & \\tilde { d } _ k : = d _ k - \\varepsilon \\ , \\alpha _ { k , 1 } - \\varepsilon ^ 2 ( \\alpha _ { k , 2 } - \\alpha _ { k , 1 } \\ , ( \\beta _ 1 ) _ x ) , k = 0 , 1 \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} B ( M , N , p ) = \\binom { N } { M } ( 1 - p ) ^ { M } p ^ { N - M } . \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{align*} \\begin{aligned} & H ^ s _ { S ^ { \\perp } } ( \\mathbb { T } ^ { \\nu + 1 } ) : = \\left \\{ u \\in H ^ s ( \\mathbb { T } ^ { \\nu + 1 } ) : u ( \\varphi , \\cdot ) \\in H _ S ^ { \\perp } , \\ , \\ , \\ , \\forall \\varphi \\in \\mathbb { T } ^ { \\nu } \\right \\} , \\\\ & H ^ s _ { S } ( \\mathbb { T } ^ { \\nu + 1 } ) : = \\left \\{ u \\in H ^ s ( \\mathbb { T } ^ { \\nu + 1 } ) : u ( \\varphi , \\cdot ) \\in H _ S \\ , \\ , \\ , \\forall \\varphi \\in \\mathbb { T } ^ { \\nu } \\right \\} . \\end{aligned} \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\bar Z _ 0 = \\Pi \\left \\{ ( F + \\bar { F } ) \\bar X + ( G + \\bar { G } ) \\bar v \\right \\} , \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\frac { 1 } { t } \\big ( G ^ t _ { \\epsilon } ( z ) - G _ { \\epsilon } ( z ) \\big ) = G _ z ( \\epsilon ) \\frac { 1 } { t } \\big ( L _ 0 - L _ t \\big ) G _ { \\epsilon } ( z ) + \\big ( G _ z ^ t ( \\epsilon ) - G _ z ( \\epsilon ) \\big ) \\frac { 1 } { t } \\big ( L _ 0 - L _ t \\big ) G _ { \\epsilon } ( z ) \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} \\frac { 1 - F _ { \\rho } ( t k ) } { t ^ { - \\gamma } ( 1 - F _ { \\rho } ( k ) ) } & = \\left ( \\frac { 1 - F _ { \\rho } ( t k ) } { 1 - F ( t k ) } \\right ) \\left ( \\frac { 1 - F ( k ) } { 1 - F _ { \\rho } ( k ) } \\right ) \\left ( \\frac { 1 - F ( t k ) } { t ^ { - \\gamma } ( 1 - F ( k ) ) } \\right ) \\\\ & = \\left ( \\frac { 1 - F _ { \\rho } ( t k ) } { 1 - F ( t k ) } \\right ) \\left ( \\frac { 1 - F ( k ) } { 1 - F _ { \\rho } ( k ) } \\right ) \\frac { \\mathcal { L } ( t k ) } { \\mathcal { L } ( k ) } . \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} | F _ { \\boldsymbol { n _ m } } ( s ; a ) | \\leq \\sum _ { \\boldsymbol { n } \\in S _ m ^ { ( k ) } } \\frac { k ! } { m ! } 3 ^ { \\frac { k - m } { 2 } } ( \\phi ( q ) ) ^ { k - m } \\leq 2 ^ { k - m } k ^ { k - m } ( 2 \\phi ( q ) ) ^ { k - m } = ( 4 k \\phi ( q ) ) ^ { k - m } . \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} \\delta _ \\ell ( g ) = \\begin{cases} \\ell & \\textrm { i f } g = 1 _ G \\\\ 1 _ L & \\textrm { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} p _ { Y , Z | X , S } = p _ { ( \\tilde { Y } , S _ 2 ) , ( \\tilde { Z } , S _ 3 ) | X , S _ 1 } = p _ { S _ 2 , S _ 3 | S _ 1 } p _ { \\tilde { Y } , \\tilde { Z } | X , S _ 1 , S _ 2 , S _ 3 } , \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} V ( x ) & = x ^ 6 + 6 a | x | ^ 5 + ( 9 a ^ 2 - 4 b ) x ^ 4 - ( 1 2 a b - 2 c ) | x | ^ 3 + ( 4 b ^ 2 + 6 a c - 5 ) x ^ 2 \\\\ & - \\left ( 1 2 a + 4 b c - \\frac { 2 } { c } \\right ) | x | \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} \\Pi _ { n } f ( \\omega ) = \\int f ( \\omega ' ) Q _ { n } ( \\omega , \\mathrm { d } \\omega ' ) , \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { 1 } { p _ 1 } + \\dots + \\frac { 1 } { p _ n } , \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} \\| g + y \\mathbf 1 \\| _ { x } = \\| g \\| _ { x - y } g \\in G , x , y \\in \\mathbb F _ { p } . \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\digamma : = 4 \\Phi ^ + \\ast \\Phi ^ - . \\end{align*}"}
-{"id": "3819.png", "formula": "\\begin{align*} \\vert l _ { 2 j , 2 N + 1 } ( z ) \\vert ^ 2 + \\vert l _ { 2 j + 1 , 2 N + 1 } ( z ) \\vert ^ 2 & \\le \\frac { \\vert z ^ 2 - e _ N \\vert ^ 2 + 2 \\vert z ^ 2 - e _ j \\vert ^ 2 + 2 \\vert e _ j - e _ N \\vert ^ 2 } { 2 \\vert e _ j - e _ N \\vert ^ 2 } \\vert l _ { j , N } ( z ^ 2 ) \\vert ^ 2 \\\\ & = \\frac { 1 } { 2 } \\vert l _ { j , N + 1 } ( z ^ 2 ) \\vert ^ 2 + \\vert l _ { N , N + 1 } ( z ^ 2 ) \\vert ^ 2 \\vert l _ { j , N } ( e _ N ) \\vert ^ 2 + \\vert l _ { j , N } ( z ^ 2 ) \\vert ^ 2 \\\\ \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} \\tau _ I \\mu ( I ) \\ - \\ ( \\beta + 1 ) \\sum _ { J \\in S _ \\phi , J ^ \\star = I } \\mu ( J ) = \\mu ( I ) \\ - \\sum _ { J \\in S _ \\phi , J ^ \\star = I } \\mu ( J ) . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} K _ { 1 2 } & = \\frac { - 2 a ( a - b - c ) - ( a - b + c ) ( a + b - c ) } { 4 } \\\\ K _ { 1 3 } & = \\frac { 2 b ( a - b + c ) + ( a - b - c ) ( a + b - c ) } { 4 } \\\\ K _ { 2 3 } & = \\frac { 2 c ( a + b - c ) + ( a - b + c ) ( a - b - c ) } { 4 } \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} s - ( \\epsilon + \\delta ) ' \\ = \\ s - \\epsilon ' - \\delta ' \\ = \\ \\frac { \\delta ' } { 1 + \\delta } - \\delta ' \\ = \\ \\frac { - \\delta ' \\delta } { 1 + \\delta } \\ \\asymp \\ \\delta ' \\delta \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} i ( c ^ { m _ { l _ 2 } } ) = 2 n \\bar { q } l _ { 2 } + 2 [ Q _ { 0 } ] - 2 i '' . \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} \\textstyle \\int \\gamma \\ > \\ - \\textstyle \\int s \\gamma \\ = \\ - \\chi \\textstyle \\int \\gamma > 0 . \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} \\Phi _ q ( y , z _ 0 ) ( t ) = T _ q ( t ) P _ q ^ { \\mathrm { s } } z _ 0 + \\int _ 0 ^ t T _ q ( t - \\tau ) P _ q ^ { \\mathrm { s } } F _ q ( y ( \\tau ) ) \\dd \\tau - \\int _ t ^ { \\infty } \\ ! \\ ! P _ q ^ { \\mathrm { c } } F _ q ( y ( \\tau ) ) \\dd \\tau , \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} \\Delta ( K _ 1 ) & = q ^ 4 X _ 1 X _ 2 X _ 3 X _ 4 X _ 5 , \\\\ \\Delta ( K _ 2 ) & = q ^ 8 X _ 6 X _ 7 X _ 8 X _ 9 X _ { 1 0 } X _ { 1 1 } X _ { 1 2 } X _ { 1 3 } X _ { 1 4 . } \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathcal { I } _ { \\mathrm { p } } \\right ] & = \\mathbb { E } \\left [ \\sum _ { x \\in \\Phi _ { \\mathrm { P } } \\cap B ^ c ( 0 , R _ { \\mathrm { e } } ) } \\| x \\| ^ { - \\alpha } \\right ] \\\\ & = \\frac { 2 \\left ( \\lambda \\pi / F \\right ) ^ { \\frac { \\alpha } { 2 } } } { \\alpha - 2 } , \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} ( \\varphi \\rtimes G ) ( \\iota ^ \\alpha ( a ) \\lambda ^ \\alpha ( f ) ) = \\iota ^ \\beta ( \\varphi ( a ) ) \\lambda ^ \\beta ( f ) \\quad ~ a \\in A , f \\in C _ c ( G ) . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} & \\left \\{ \\omega \\in \\Omega : \\varphi _ { t _ 1 + t _ 0 } ( \\cdot , B ( x _ 0 , r _ 0 ) ) \\subset \\bar { B } \\left ( \\varphi _ { t _ 1 + t _ 0 } ( \\cdot , x _ 0 ) , \\frac { r _ 0 } { 2 } \\right ) \\right \\} \\\\ & \\quad \\subset \\left \\{ \\omega \\in \\Omega : \\varphi _ { t _ 1 + t _ 0 } ( \\cdot , B ( x _ 0 , r _ 0 ) ) \\subset B \\left ( z _ m , \\frac { 2 } { 3 } r _ 0 \\right ) \\textrm { f o r s o m e } m \\in \\mathbb { N } \\right \\} . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} \\binom { k + ( i + j + 1 ) p } { m + i p } & = \\frac { ( k + p + ( i + j ) p ) \\cdots ( p + ( i + j ) p ) } { ( m + i p ) \\cdots ( 1 + i p ) } \\binom { k + p - m + ( i + j ) p } { i p } \\\\ & \\equiv _ { p ^ 2 } ( i + j + 1 ) \\binom { k + p } { m } \\binom { k - m + p + ( i + j ) p } { i p } \\\\ & \\equiv _ { p ^ 2 } ( i + j + 1 ) \\frac { p ( - 1 ) ^ { m - ( k + 1 ) } } { ( k + 1 ) \\binom { m } { k + 1 } } \\binom { i + j } { i } . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( 1 - x ) ^ { - X } x ^ { m - 1 } d x & = B ( 1 - X , m ) = \\frac { \\Gamma ( 1 - X ) \\Gamma ( m ) } { \\Gamma ( 1 - X + m ) } \\\\ & = \\frac { ( m - 1 ) ! } { ( 1 - X ) ( 2 - X ) \\cdots ( m - X ) } \\\\ & = \\frac { 1 } { m } \\prod _ { a = 1 } ^ m \\biggl ( 1 - \\frac { X } { a } \\biggr ) ^ { - 1 } = \\frac { 1 } { m } \\prod _ { a = 1 } ^ m \\sum _ { n = 0 } ^ \\infty \\frac { X ^ n } { a ^ n } \\\\ & = \\frac { 1 } { m } \\sum _ { k = 0 } ^ \\infty X ^ k \\sum _ { 0 < a _ 1 \\leq \\cdots \\leq a _ k \\leq m } \\frac { 1 } { a _ 1 \\cdots a _ k } \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} \\sum _ { \\iota ' \\in I ' } b ^ 2 _ { \\iota ' } = 1 _ { M \\left ( B \\right ) } . \\end{align*}"}
-{"id": "3267.png", "formula": "\\begin{align*} \\widetilde { e } _ j ( \\widetilde { x } ) = \\left \\{ \\begin{array} { c l } e _ j ( \\widetilde { \\pi } ( \\widetilde { x } ) ) & \\widetilde { x } \\in \\widetilde { \\mathcal { U } } _ i \\\\ 0 & \\widetilde { x } \\notin \\widetilde { \\mathcal { U } } _ j \\end{array} \\right . ; \\ j \\in \\{ 1 , 2 \\} . \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{align*} \\frac { \\kappa _ { + } e ^ { \\epsilon ( \\tau _ { - } - \\tau _ { + } ) } } { i z _ { + } \\alpha _ { - } \\beta _ { - } } = q ^ { 1 + \\mathsf { N } } \\prod _ { n = 1 } ^ { \\mathsf { N } } \\frac { b _ { n } c _ { n } } { \\alpha _ { n } \\beta _ { n } } , \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} \\mathcal { N } i l p _ { X , x , \\theta } ( S ) = \\{ ( E , s ) \\in T ^ * B u n _ { \\mathcal { G } _ { X , x , \\theta } } \\mid s \\mid _ { X - x \\times S } \\} . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} \\begin{aligned} D _ t ( u ' ) + ( S - \\operatorname { i d } ) \\exp ( u _ { - 1 } - u ) & = \\bold { E } ( L ) , \\\\ D _ t ( t u ' - u ) + ( S - \\operatorname { i d } ) ( t \\exp ( u _ { - 1 } - u ) ) & = t \\bold { E } ( L ) , \\\\ D _ t \\left ( \\frac { ( u ' ) ^ 2 } { 2 } + \\exp ( u - u _ 1 ) \\right ) + ( S - \\operatorname { i d } ) ( u ' \\exp ( u _ { - 1 } - u ) ) & = u ' \\bold { E } ( L ) . \\end{aligned} \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} d ( h ) = f d ( k ) = g h ^ 2 = k ^ 2 = 0 . \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} u _ n ( A , X ) = \\inf \\{ t > 0 : t ( 0 , I _ n ) - ( A , X ) \\in \\tilde { Z } _ { a c } ^ n \\} . \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} \\tau \\ltimes M ^ \\circ _ { y , \\sigma , r , \\rho ^ \\circ } = ( \\tau \\ltimes E ^ \\circ _ { y , \\sigma , r , \\rho ^ \\circ } ) / ( \\tau \\ltimes N ^ \\circ ) , \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ 3 R ( \\mathcal { F } _ i ) ^ { ( - 1 ) ^ { i + 1 } } \\right ) = { \\nu ( \\mathcal { H } _ B / \\mathcal { H } ^ 0 ) \\nu ( \\mathcal { H } o m ) _ { \\mathbb { R } } } . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} { \\bf F } _ { i j k } = & { \\bf R } _ i - { \\bf T } _ { i j k } , \\ \\forall \\{ i , j \\} \\in \\mathcal { L } , \\ \\forall k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "9645.png", "formula": "\\begin{align*} \\theta _ 1 = 0 + \\mathcal { O } \\left ( \\delta ^ { p + 3 } \\right ) \\theta _ 2 = 2 \\pi + \\mathcal { O } \\left ( \\delta ^ { p + 3 } \\right ) . \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} [ T ] = [ T _ { a n , b n } ] + \\sum _ { k = 0 } ^ { a n - 1 } [ T _ { k , b n } ] + \\sum _ { l = 0 } ^ { b n - 1 } [ T _ { a n , l } ] \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} F _ j ( b _ j ) - F _ j ( a _ j ) & = \\int _ { 0 } ^ 1 \\frac { d F _ j ( \\gamma _ j ( t ) ) } { d t } d t \\\\ & = ( \\int _ { 0 } ^ 1 d F _ j ( \\gamma _ j ( t ) ) d t ) ( b _ j - a _ j ) , \\end{align*}"}
-{"id": "8926.png", "formula": "\\begin{align*} ( E _ B ) _ N = \\sum _ { \\sigma \\in S _ n } a ^ { \\sigma \\nu + \\rho } \\prod _ { i < j , \\sigma ( i ) > \\sigma ( j ) } \\frac { \\xi ( \\nu _ { i j } ) } { \\xi ( \\nu _ { i j } + 1 ) } \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} C _ E = \\begin{pmatrix} 0 & 5 \\\\ 1 & 0 \\end{pmatrix} C _ { E ' } = \\begin{pmatrix} 0 & 2 \\\\ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} F \\left ( u , v \\right ) = u ^ { 3 } - a _ { 2 } u ^ { 2 } v + \\left ( a _ { 1 } a _ { 3 } - 4 a _ { 4 } \\right ) u v ^ { 2 } + \\left ( 4 a _ { 2 } a _ { 4 } - a _ { 3 } ^ { 2 } - a _ { 1 } ^ { 2 } a _ { 4 } \\right ) v ^ { 3 } , \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} \\phi _ d \\ ; : \\ ; W [ p ] \\to W ' [ p ] \\phi _ d : = T _ { E ' } \\circ \\phi \\circ T _ W , \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} \\Gamma ( k _ t ) : = \\{ k _ { t + 1 } \\in K : 0 \\leq k _ { t + 1 } \\leq f ( k _ t ) \\} , \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} \\left | \\left ( C ^ 0 ( e ^ { i \\theta } ) \\right ) ^ { < \\frac { 1 } { \\alpha - 1 } > } ( f _ 0 ( \\theta ) ) ^ { \\frac { - 1 } { \\alpha - 1 } } \\right | ^ { \\alpha } = \\gamma _ 1 \\left ( f _ 0 ( \\theta ) \\right ) ^ { \\beta - 1 } , \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} \\chi ( D ) = \\frac { D ^ 4 } { 2 4 } + \\frac { 1 } { 2 4 } D ^ 2 . c _ 2 ( X ) + \\chi ( \\mathcal { O } _ X ) \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} k ( \\lambda ) = p _ 0 ( \\lambda ) - \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\nu \\ , p _ 0 ' ( \\nu ) F ( \\nu | \\lambda ) \\end{align*}"}
-{"id": "8822.png", "formula": "\\begin{align*} & \\bar { G } = \\mathbb { E } \\left [ G _ i \\right ] = \\frac { 1 } { N ^ 2 } \\mathbb { E } \\left [ \\frac { { 1 - \\cos ( N { \\mathcal K } _ 1 ( \\xi _ { r _ { i , o } } ) ) } } { { 1 - \\cos ( { \\mathcal K } _ 1 ( \\xi _ { r _ { i , o } } ) ) } } \\right ] \\times \\\\ & \\mathbb { E } \\left [ \\frac { 1 - \\cos ( { N } { { \\mathcal K } _ 2 } ( \\varphi _ { t _ { i , o } } , { \\varphi _ { { t _ i } } } ) ) } { 1 - \\cos ( { { \\mathcal K } _ 2 } ( \\varphi _ { t _ { i , o } } , { \\varphi _ { { t _ i } } } ) ) } \\right ] . \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} ( \\Phi \\circ \\mathcal { J } ) ( t ) & = \\Phi ( r ( t ) , \\sigma _ { } ( t ) , u _ { } ( t ) ) \\\\ & = ( r ( t ) \\theta _ { 1 } ( \\sigma _ { } ( t ) ) , \\dots , r ( t ) \\theta _ { n } ( \\sigma _ { } ( t ) ) , \\tau _ { 1 } ( u _ { } ( t ) ) , \\dots , \\tau _ { n } ( u _ { } ( t ) ) ) , \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { W _ { G } ^ { 1 , 2 ; p } } ^ { p } = \\mathbb { E } [ \\sup _ { s \\in \\lbrack 0 , T ] } | u _ { s } | ^ { p } + \\int _ { 0 } ^ { T } ( | \\mathcal { D } _ { s } u _ { s } | ^ { p } + | \\mathcal { D } _ { x } ^ { 2 } u _ { s } | ^ { p } ) d s + ( \\int _ { 0 } ^ { T } | \\mathcal { D } _ { x } u _ { s } | ^ { 2 } d s ) ^ { p / 2 } ] . \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\lim _ { n \\rightarrow \\infty } \\textrm { d i a m } ( \\varphi _ { t _ n } ( \\cdot , A ) ) = 0 \\right ) \\geq \\mathbb { P } \\left ( A \\subset U \\right ) \\cdot \\mathbb { P } \\left ( \\lim _ { n \\rightarrow \\infty } \\textrm { d i a m } ( \\varphi _ { t _ n } ( \\cdot , U ) ) = 0 \\right ) > 0 . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} 2 \\phi d ^ c r + r d ^ c \\phi = 2 \\psi d r + r d \\psi + r \\eta + 2 \\alpha d r . \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} \\mathcal { H } ^ \\mathsf { R o b } _ { \\hbar , \\kappa , K } = - a _ { \\hbar , \\kappa , K } ^ { - 1 } ( \\tau ) \\partial _ { \\tau } \\left ( a _ { \\hbar , \\kappa , K } ( \\tau ) \\partial _ { \\tau } \\right ) = - \\partial ^ 2 _ { \\tau } + \\frac { \\hbar ^ 2 \\kappa - 2 \\hbar ^ 4 K \\tau } { a _ { \\hbar , \\kappa , K } ( \\tau ) } \\partial _ { \\tau } \\ , , \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} ( \\Psi _ 1 , \\Psi _ { - 1 } ) ( x ) = ( w _ 1 , w _ { - 1 } ) . \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde \\psi ( x ) = \\left ( 1 - \\frac { \\tau _ + ^ r ( x ) } { T ^ r ( x ) } \\right ) \\int _ { \\tau _ - ^ \\delta ( y ) } ^ { \\tau _ + ^ \\delta ( y ) } f ( X ( t , y ) ) \\ , d t . \\end{aligned} \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} 1 _ A \\ ; = \\ ; e ^ \\ast _ 0 + e ^ \\ast _ 1 + \\cdots + e ^ \\ast _ n , \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} { { \\mathsf { R } } ^ { \\lambda } _ { \\mathcal { N } _ { [ 1 : N ] } } \\leq { \\mathsf { R } } ^ { \\lambda } _ { \\mathcal { N } _ { \\mathcal { M } _ { 1 } } } + { \\mathsf { R } } ^ { \\lambda } _ { \\mathcal { N } _ { \\mathcal { M } _ { 2 } } } = { \\mathsf { R } } ^ { \\lambda } _ { \\mathcal { N } _ { \\mathcal { K } } } + { \\mathsf { R } } ^ { \\lambda } _ { \\mathcal { N } _ { [ 1 : N ] \\backslash \\mathcal { K } } } } . \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} w _ { 1 , k } = t _ { - 1 , - k + 1 - R _ { 1 , k } ( 0 ) } . \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } t _ { j } = t _ n + \\sum _ { j = 0 } ^ { n - 1 } t _ { j } \\leq ( 1 + \\rho ) \\sum _ { j = 0 } ^ { n - 1 } t _ { j } . \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} J = \\bigoplus _ { k \\neq 0 } E _ k . \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} \\frac { ( 1 - | x | ^ 2 | y | ^ 2 ) } { ( 1 - | y | ^ 2 ) [ x , y ] ^ { N } } = v ( x ) = 2 k _ { N , 1 } \\int _ { \\partial B } \\frac { 1 - | x | ^ 2 } { | x - \\theta | ^ N [ \\theta , y ] ^ { N } } \\ d \\theta . \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} \\lambda _ 0 : = \\frac { 1 } { c ( \\psi ( 1 ) + 1 ) } \\biggl ( 1 + \\int _ 0 ^ 1 \\bigl ( m ( s ) + \\abs { k ( 0 , s ) } \\bigr ) \\phi ( s ) \\textup d s \\biggr ) ^ { - 1 } , \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\boldsymbol { \\Psi } _ D ^ { ( g ) } \\left [ \\boldsymbol { \\Psi } _ D ^ { ( g ) } \\right ] ^ H = \\left ( \\mathbf { X } ^ { ( g ) } \\otimes \\left [ \\mathbf { S } _ D ^ { ( g ) } \\right ] ^ H \\right ) \\left ( \\mathbf { I } _ { K _ g } \\otimes \\mathbf { V } \\mathbf { V } ^ H \\right ) \\left ( \\left [ \\mathbf { X } ^ { ( g ) } \\right ] ^ H \\otimes \\mathbf { S } _ D ^ { ( g ) } \\right ) \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\lim _ { n \\rightarrow \\infty } \\textrm { d i a m } \\left ( \\varphi _ { t _ n } \\left ( \\cdot , U \\right ) \\right ) = 0 \\right ) > 0 . \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} \\{ ( \\alpha , \\tau ) \\in \\mathfrak A \\times \\R \\mid \\rho ( G _ { \\alpha , \\tau } ) = \\rho \\} \\ = \\ \\{ ( \\alpha , \\tau ) \\in \\mathfrak A \\times \\R \\mid \\tau ^ - _ \\rho ( \\alpha ) \\leq \\tau \\leq \\tau ^ + _ \\rho ( \\alpha ) \\} \\ . \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} \\chi ( \\mathfrak { L } _ { \\mathfrak { m } } ( L / F , \\pi _ B ) ^ { \\epsilon } ) = 0 \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} f _ e ( x | \\eta ) = \\eta e ^ { - x \\eta } . \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} \\mathbf { P } _ f = \\ ! \\ ! \\begin{bmatrix} f ( - N _ { \\rm c p } ) & f _ 1 ( - N _ { \\rm c p } ) & \\ ! \\ ! \\cdots \\ ! \\ ! & f _ V ( - N _ { \\rm c p } ) \\\\ f _ 1 ( - N _ { \\rm c p } ) & f _ 2 ( - N _ { \\rm c p } ) & \\ ! \\ ! \\cdots \\ ! \\ ! & f _ { V + 1 } ( - N _ { \\rm c p } ) \\\\ \\vdots & \\vdots & \\ ! \\ ! \\ ! \\ ! & \\vdots \\\\ f _ V ( - N _ { \\rm c p } ) & f _ { V + 1 } ( - N _ { \\rm c p } ) & \\ ! \\ ! \\cdots \\ ! \\ ! & f _ { 2 V } ( - N _ { \\rm c p } ) \\end{bmatrix} , \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} \\Lambda ^ + _ { \\iota + 1 } \\ = \\ \\bar \\Lambda ^ + _ { \\iota | [ \\hat \\theta _ \\iota , \\hat \\theta _ \\iota ^ * ] } \\cup \\lambda \\cup \\bar \\Lambda ^ + _ { \\iota + 1 | [ \\hat \\theta _ \\iota ^ * + \\delta , \\hat \\theta _ { \\iota + 1 } ] } \\ . \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} \\omega _ 1 = \\frac { 1 } { h ^ 2 } d h \\wedge d \\theta , \\omega _ 2 = \\left ( \\frac { 1 } { h } + \\frac { 1 } { h ^ 2 } \\right ) d h \\wedge d \\theta = \\frac { 1 } { h ^ 2 } d h \\wedge ( h d \\theta + d \\theta ) . \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} w ( q ) = \\frac { 1 } { T } \\int _ 0 ^ T f _ 2 \\big ( t , \\hat \\xi ( t ) , q , 0 \\big ) \\ , d t = \\begin{pmatrix} q _ 2 \\\\ \\frac { 1 } { T } \\int _ 0 ^ T \\phi ( \\hat { x } _ 1 ( t ) - q _ 1 \\big ) \\ , d t \\end{pmatrix} , \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\left \\vert \\mathcal { C } _ i ^ { \\mathrm { E } } \\right \\vert \\right ] = 2 \\mathbb { E } \\left [ \\left \\vert \\mathcal { C } _ i \\right \\vert \\right ] . \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} b _ 3 \\partial _ { y y y } + b _ 1 \\partial _ y + b _ 0 = \\partial _ { y y y } + \\varepsilon \\partial _ y ( \\alpha _ { 1 , 1 } \\ , \\cdot ) + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
-{"id": "1510.png", "formula": "\\begin{align*} \\langle \\overline { Z | b | ^ 2 ( H ) \\Gamma _ H } F , \\widetilde { G } \\rangle _ T & = \\langle \\Gamma _ H F , | b | ^ 2 ( H ) Z ^ * \\widetilde { G } \\rangle _ T \\\\ & = \\langle \\Gamma _ { H _ 0 } F , | b | ^ 2 ( H ) Z ^ * \\widetilde { G } \\rangle _ T - i \\langle \\overline { Y \\Gamma _ { H _ 0 } } F , \\overline { Z b ( H ) \\Gamma _ H ^ * } b ( H ) ^ * Z ^ * \\widetilde { G } \\rangle _ T . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} L ( e _ { i i } ) - L ( e _ { j j } ) & = \\sum _ { x < i } C _ { x i } ^ { i i } e _ { x i } + \\sum _ { x \\in X } C _ { x x } ^ { i i } e _ { x x } + \\sum _ { y > i } C _ { i y } ^ { i i } e _ { i y } \\\\ & - \\sum _ { x < j } C _ { x j } ^ { j j } e _ { x j } - \\sum _ { y \\in X } C _ { y y } ^ { j j } e _ { y y } - \\sum _ { y > j } C _ { j y } ^ { j j } e _ { j y } . \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} d y _ 1 \\wedge \\ldots \\wedge d y _ { g - 1 } \\wedge d y _ g = \\frac { u _ g } { g } d u _ 1 \\wedge \\ldots \\wedge d u _ { g - 1 } \\wedge d t . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} 0 & = \\frac { d } { d t } \\left [ x ( \\gamma ( t ) ) \\right ] = \\frac { d } { d t } \\left [ x \\circ \\pi ^ { - 1 } \\circ \\Gamma ( t ) \\right ] \\\\ & = \\left [ \\frac { \\partial x \\circ \\pi ^ { - 1 } } { \\partial z } \\circ \\Gamma \\cdot \\frac { d \\Gamma } { d t } \\right ] \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} \\sum _ { j = 2 } ^ { \\left \\lfloor x / 2 \\right \\rfloor } ( j ; j | x ) = \\sum _ { j = 2 } ^ { \\left \\lfloor x / 2 \\right \\rfloor } \\mathcal { B } \\left ( \\mathbf { d } _ j ( 0 , x ) , \\frac { 1 } { 2 } \\right ) \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 = Q ^ { \\alpha } \\bold { E } _ { \\alpha } ( L ) , \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} { F _ { { { \\left | { { h _ i ^ m } } \\right | } ^ 2 } } } \\left ( x \\right ) = 1 - { e ^ { - \\left ( { 1 + d _ i ^ \\alpha } \\right ) x } } , \\ ; x \\ge 0 . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} F ( \\boldsymbol { n } s ; a ) : = \\prod _ { j = 1 } ^ l F _ { n _ j } ( n _ j s ; a ) . \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} \\alpha _ P ( x ) : = \\begin{cases} x ^ p & \\mbox { i f } P = \\{ 1 \\} \\\\ x ^ p ( \\prod _ { k = 0 } ^ { p - 1 } \\omega _ P ^ k ( x ) ) ^ { - 1 } & \\mbox { i f } P \\neq \\{ 1 \\} \\mbox { a n d c y c l i c } \\\\ x ^ p & \\mbox { i f } P \\mbox { i s n o t c y c l i c } . \\end{cases} \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} F ( c , 1 , p ) = 2 c \\sqrt { 2 4 c ^ 2 ( p - 1 ) - 1 6 ( p - 1 ) ( 5 + 3 p ) + c ^ 4 ( 2 - 4 p + 3 p ^ 2 ) } . \\end{align*}"}
-{"id": "5454.png", "formula": "\\begin{align*} T : = \\sum _ { k } 2 ^ { - k } T _ k \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} E = L \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\lambda \\ , \\epsilon _ 0 ( \\lambda ) \\rho ( \\lambda ) - \\frac { 1 } { 2 4 L } \\sum _ { i a } \\frac { s _ a \\epsilon _ 0 ' ( \\lambda ) } { \\rho ( \\lambda _ { i a } ) } . \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} W _ t = U ( \\hat { x } _ t , \\theta _ t ) + \\mu ( \\theta _ t ) W _ { t + 1 } , & & t = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} 0 = X _ 1 ( 0 ) \\leq X _ 2 ( 0 ) \\leq X _ 3 ( 0 ) \\leq \\ldots \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\ast d h = \\mbox { R e } \\left ( \\frac { 2 } { i } \\frac { \\partial h } { \\partial z } d z \\right ) \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} N = N ' A ^ s = A ' , \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\epsilon ' = \\min _ { 1 \\leq w _ 2 \\leq \\bar { w } } \\frac { \\phi } { 8 } \\left [ \\log \\left ( 1 + \\frac { w _ 2 P ' } { 3 } \\right ) - \\frac { 1 } { 2 } \\log \\left ( 1 + \\frac { 2 w _ 2 P ' } { 3 } \\right ) \\right ] . \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} 0 = 2 \\lambda Q ( w _ 1 , w _ 2 ) \\geq Q ( w _ 1 ) + \\lambda ^ 2 Q ( w _ 2 ) = Q ( w _ 1 ) \\geq 0 . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} W : y ^ 2 + a _ 1 x y + a _ 3 y = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x + a _ 6 . \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} \\ell _ { i , s } ^ { \\prime } = \\left \\{ \\begin{array} { l l } \\ell _ i & \\ i \\in \\mathcal { L } _ s \\\\ 0 & \\end{array} \\right . , r _ { i , s } ^ { \\prime } = \\left \\{ \\begin{array} { l l } r _ i & \\ i \\in \\mathcal { T } _ s \\\\ 0 & \\end{array} \\right . . \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} n _ + \\big ( r , K _ V ( 0 ) \\big ) = C _ { m } ( 2 b _ 0 ) ^ { - 1 / m } r ^ { - 1 / m } \\big ( 1 + o ( 1 ) \\big ) , r \\searrow 0 . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} R ^ N ( \\boldsymbol { x } ) = \\sum _ { k = 1 } ^ { N } \\sigma ^ { - 1 } _ k ( f , u _ k ) _ { \\ell _ 2 ( X _ N ) } u _ k ( \\boldsymbol { x } ) . \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} \\big ( \\sum _ { j = 1 } ^ d | x _ j + y _ j | ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } \\le \\big ( \\sum _ { j = 1 } ^ d | x _ j | ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } + \\big ( \\sum _ { j = 1 } ^ d | y _ j | ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} [ \\xi _ k + \\eta _ k , A \\xi - \\hat { A } \\xi ] _ { \\alpha } = 0 , \\forall k = - 1 , - 2 , \\dots . \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} \\iota _ Z \\Omega = \\eta , \\iota _ { \\sharp ( Z ) } \\Lambda = Z , \\end{align*}"}
-{"id": "3833.png", "formula": "\\begin{align*} \\Delta _ { 2 ^ { p _ 0 } + \\cdots + 2 ^ { p _ s } } = & 2 ^ { p _ 0 - p _ 1 } ( 2 ^ { p _ 1 - p _ 0 } - 2 ) \\left [ 4 \\Delta _ { 2 ^ { p _ 1 } + \\cdots + 2 ^ { p _ s } } + ( 2 ^ { p _ 1 - p _ 0 } - 2 ) ( 1 + 2 ^ { p _ 2 - p _ 1 } + \\cdots + 2 ^ { p _ s - p _ 1 } ) \\right ] \\\\ & + 2 ^ { p _ 0 - p _ 1 } R _ 1 ( p _ 1 , \\cdots , p _ s ) , \\\\ \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} \\rho = \\begin{cases} 1 / 9 r ( \\Delta - 1 ) & \\theta \\in [ - \\frac { \\pi } { 2 } , \\frac { \\pi } { 2 } ] ; \\\\ | \\sin \\theta | / 6 r ( \\Delta - 1 ) & , \\end{cases} \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} \\forall i \\in I ( \\bar { x } ) \\ \\exists \\tilde { \\nu } _ k > 0 , \\ k \\in I \\sum _ { k \\in I ( \\bar { x } ) } \\lambda _ k ^ i \\nu _ k = \\sum _ { k \\in I } \\lambda _ k ^ i \\tilde { \\nu } _ k . \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} T _ q ( t ) P _ q ^ s = S ( t ) P _ q ^ s + \\int _ { 0 } ^ { t } S ( t - s ) B _ q T _ q ( s ) P _ q ^ s \\dd s . \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{gather*} \\hat { B } _ 0 = - \\frac { x } { 2 } \\sigma _ 3 - \\begin{pmatrix} 0 & u \\\\ u & 0 \\end{pmatrix} , \\end{gather*}"}
-{"id": "4029.png", "formula": "\\begin{align*} e r f ( z ) = \\frac { 2 } { \\sqrt { \\pi } } \\int _ 0 ^ { z } e x p ( - t ^ { 2 } ) d t = \\frac { 2 } { \\sqrt { \\pi } } \\sum _ { n = 0 } ^ \\infty \\frac { ( - 1 ) ^ { n } z ^ { 2 n + 1 } } { ( 2 n + 1 ) n ! } , \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} \\chi ( w _ { \\sigma } \\cdot w ' ) = \\Im \\left ( - \\frac { Z ( w ' ) } { Z ( v ) } \\right ) \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} u ^ { S ^ N ( \\mathbf { \\xi } ) } _ t ( y ) = \\displaystyle \\frac { 1 } { N } \\sum _ { j = 1 } ^ N K ( y - \\xi ^ { j , N } _ t ) \\exp \\left \\{ \\int _ 0 ^ t \\Lambda \\big ( s , \\xi ^ { j , N } _ s , u ^ { S ^ N ( \\mathbf { \\xi } ) } ( s , \\xi ^ { j , N } _ s ) \\big ) d s \\right \\} , \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} \\begin{aligned} ( X _ j - i Y _ j ) ^ { - 1 } & = M _ j ^ 2 ( X _ j ^ t + i Y _ j ^ t ) \\\\ ( X _ j + i Y _ j ) ^ { - 1 } & = M _ j ^ 2 ( X _ j ^ t - i Y _ j ^ t ) , \\end{aligned} \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} ( X _ { \\geq j } ) _ i = \\begin{cases} X _ i & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} a _ 0 ( u ) = x ( u ) , \\ \\ b _ 0 ( u ) = y ( u ) , \\ \\ c _ 0 ( u ) = z ( u ) . \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } \\ast & 0 \\\\ 0 & \\ast \\end{array} \\right ] , \\mbox { f o r $ A ( t ) $ , $ B ( t ) $ , $ A ' ( t ) $ , $ B ' ( t ) $ , $ F ( t ) $ a n d $ G ( t ) $ } , \\\\ I ( t ) = \\left [ \\begin{array} { c } \\ast \\\\ 0 \\end{array} \\right ] , J ( t ) = \\left [ \\begin{array} { c c } \\ast & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\varphi ( n ) } { n ^ s } = \\frac { \\zeta ( s - 1 ) } { \\zeta ( s ) } = \\prod _ { p \\in \\P } \\left ( 1 - \\frac 1 { p ^ s } \\right ) \\left ( 1 - \\frac 1 { p ^ { s - 1 } } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} h ( \\rho ) = \\rho \\ , \\ln ( \\rho ) \\ , + \\ , ( 1 - \\rho ) \\ , \\ln ( 1 - \\rho ) , 0 < \\rho < 1 , h ( 0 ) = h ( 1 ) = 0 , \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} G ( 0 , r , p ) = \\sqrt { 2 4 p r + 1 6 r ^ 2 + 9 } + 1 2 \\left ( 1 - r ^ 2 \\right ) . \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} f _ k ( t ) : = a _ k M _ \\tau ^ { p ^ { k - 1 } } t ^ { - \\frac { N } { \\theta } } \\left ( \\log \\frac { t } { \\rho ^ \\theta } \\right ) ^ { \\frac { p ^ { k - 1 } - 1 } { p - 1 } } , k = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} 0 = & d \\omega ( S , T , Z ) \\\\ = & - \\omega ( [ S , T ] , Z ) + \\omega ( [ S , Z ] , T ) - \\omega ( [ T , Z ] , S ) \\\\ = & - \\omega ( [ S , T ] , Z ) - 2 ( \\nabla _ Z \\omega ) ( S , T ) \\\\ = & - \\omega ( [ S , T ] , Z ) , \\\\ \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} D ( \\omega \\otimes \\eta ) : = ( \\pi ^ * _ + d _ + ) \\omega \\otimes \\eta + ( - 1 ) ^ k \\omega \\otimes ( \\pi ^ * _ - d _ - ) \\eta , \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{align*} u _ { n } | k , n \\rangle = | k + 1 , n \\rangle \\forall k \\in \\{ - l , . . . , l \\} \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{gather*} \\delta _ Q \\Xi = d \\Theta , \\Theta \\approx \\frac 1 2 \\big ( F \\wedge i _ \\xi \\tilde { F } - \\tilde { F } \\wedge i _ \\xi F \\big ) , \\operatorname { g h } ( \\Theta ) = 0 . \\end{gather*}"}
-{"id": "3631.png", "formula": "\\begin{align*} e _ i = \\left | \\{ j \\ | \\ j < i \\ { \\rm a n d } \\ \\pi _ j > \\pi _ i \\} \\right | . \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { A } : & = ( 2 4 c _ 1 ^ 2 - 1 2 c _ 4 ) D _ S ^ 5 \\{ \\mathrm { I } - 2 D _ { - 2 } U D _ S ^ 2 \\} + ( \\frac { 1 4 } { 3 } c _ 2 ^ 2 - 4 c _ 6 ) D _ S ^ 3 + ( 4 c _ 6 - \\frac { 1 6 } { 3 } c _ 2 ^ 2 ) \\{ D _ S ^ 3 U + D _ S U D _ S ^ 2 \\} \\\\ & + 1 2 ( c _ 2 c _ 3 - c _ 7 ) D _ S + ( 2 4 c _ 7 - 1 6 c _ 2 c _ 3 ) D _ S U - 6 c _ 3 ^ 2 D _ S ^ { - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "3839.png", "formula": "\\begin{align*} \\langle \\theta _ 1 , \\theta _ 2 \\rangle : = \\frac { 1 } { N _ G ( P ) } \\sum \\limits _ { g \\in N _ G ( P ) } \\theta _ 1 ( g ) \\overline { \\theta _ 2 ( g ) } . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} h ( ( u _ { k l } ^ t ) ^ * u _ { i j } ^ s ) = \\delta _ { s t } \\delta _ { j l } \\frac { ( \\rho _ s ^ { - 1 } ) _ { i k } } { d _ s } , & & h ( u _ { k l } ^ t ( u _ { i j } ^ s ) ^ * ) = \\delta _ { s t } \\delta _ { k i } \\frac { ( \\rho _ s ) _ { j l } } { d _ s } . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\int \\phi ( w , F _ { \\tau } ) F _ { 0 } ( d w ) \\longrightarrow - \\int \\phi ( w , F _ { 0 } ) H ( d w ) + 0 = - \\int \\phi ( w , F _ { 0 } ) H ( d w ) . \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} q = 4 a b + 2 n + 2 , \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} \\partial _ { t } u ( t , x ) = D ^ { 2 } u ( t , x ) + f ( t , x ) \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} \\mathcal { L } ^ { \\Gamma } - \\frac { \\kappa ^ 2 } { 4 } + K = R ^ { - 2 } A ^ 2 \\ , , \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} V ^ \\bullet : = H _ z ^ \\bullet ( B _ x B _ y ) . \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} & \\left \\| \\nabla ^ { k } _ { x } \\left ( K _ { 3 } ( t ) g - \\mathcal { F } ^ { - 1 } \\left [ e ^ { - \\frac { \\nu t | \\xi | ^ { 2 \\sigma } } { 2 } } \\frac { \\sin ( t | \\xi | ) } { | \\xi | } \\right ] \\ast g \\right ) \\right \\| _ { 2 } \\\\ & \\le C ( 1 + t ) ^ { - \\frac { n } { 4 \\sigma } - 1 - \\frac { k } { 2 \\sigma } + \\frac { 1 } { \\sigma } } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla ^ { ( k - 2 \\sigma ) _ { + } } _ { x } g \\| _ { 2 } , \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} \\tilde { z } _ { 1 } = \\min \\left \\{ \\max \\left \\{ z _ { 1 } , u _ { 0 } \\right \\} , u _ { 1 } \\right \\} \\tilde { z } _ { 2 } = \\min \\left \\{ \\max \\left \\{ z _ { 2 } , v _ { 0 } \\right \\} , v _ { 1 } \\right \\} . \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { \\sigma ^ * ( n ) } = E ^ * \\log x + F ^ * + O \\left ( x ^ { - u } ( \\log x ) ^ { 5 / 3 } ( \\log \\log x ) ^ { 4 / 3 } \\right ) , \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} 2 ( 1 2 + ( 2 6 - N ) ( 2 5 - N ) ) \\lambda ( N ) = H _ n ( N ) + 2 ( 2 6 - N ) H _ h ( N ) + \\tau ( N ) \\mu ( N ) H _ u ( N ) , \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} \\sigma ( L _ { b c } ) = \\sigma _ { ( m ) } \\cup \\{ \\widetilde { \\lambda } _ n ^ \\mp , n \\geq m + 1 \\} , \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} B ^ { - 1 } ( \\theta + \\alpha ) A ( \\theta ) B ( \\theta ) = R _ { \\psi ( \\theta ) } . \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} I ( \\xi u ) & = \\frac { 1 } { b p ^ 2 } M _ { \\xi u } ^ p - \\int _ 0 ^ T F ( t , \\xi u ( t ) ) d t - \\frac { a ^ p } { b p ^ 2 } \\\\ & \\leq \\frac { 1 } { b p ^ 2 } M _ { \\xi u } ^ p - c _ 1 \\int _ 0 ^ T | \\xi u ( t ) | ^ \\mu d t + c _ 2 T - \\frac { a ^ p } { b p ^ 2 } \\\\ & = \\frac { 1 } { b p ^ 2 } ( a + b \\xi ^ p \\| u \\| _ { E ^ { \\alpha , p } } ^ p ) ^ p - c _ 1 \\xi ^ \\mu \\| u \\| _ { L ^ \\mu } ^ \\mu + c _ 2 T - \\frac { a ^ p } { b p ^ 2 } \\\\ & \\rightarrow - \\infty \\ \\ \\mbox { a s } \\ \\xi \\rightarrow \\infty . \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} 0 < h _ { \\min } : = \\min _ { t \\in [ 0 , 1 ] } h ( t ) \\leq h _ { \\max } : = \\max _ { t \\in [ 0 , 1 ] } h ( t ) < \\min \\{ 1 , \\eta \\} , \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} & \\frac { 1 } { P ( \\alpha , \\beta ; ( m + v _ 1 ) z + ( h n + \\mu + v _ 2 ) ) } = \\left ( \\prod _ { j = 1 } ^ g ( h ^ { ( j ) } ) ^ { - \\alpha _ j - \\beta _ j } \\right ) \\cdot P \\left ( - \\alpha , - \\beta ; \\left ( \\frac { m + v _ 1 } { h } \\right ) z + n + \\frac { ( \\mu + v _ 2 ) } { h } \\right ) . \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} \\int _ 0 ^ { x _ { \\sqrt \\mu } } V u ^ 2 \\ , \\dd x = \\frac { \\pi } { \\mu ^ \\frac 1 2 } \\int _ 0 ^ { x _ { \\sqrt \\mu } } V \\ , \\dd x + o ( \\mu ^ { - \\frac 1 2 } ) , \\mu \\to \\infty . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} ( A - \\lambda \\vec u \\vec u ^ \\top ) ^ { - 1 } = A ^ { - 1 } + \\lambda \\dfrac { A ^ { - 1 } \\vec u \\vec u ^ { \\top } A ^ { - 1 } } { 1 - \\lambda \\vec u ^ \\top A ^ { - 1 } \\vec u } \\enspace . \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} s = \\infty . \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\rho _ { 0 } ( \\beta ) = { \\rho } - \\rho _ { c } ( \\beta ) = \\lim _ { \\eta \\rightarrow 0 } \\frac { \\vert \\eta \\vert \\ , ^ { 2 } } { \\mu ( \\beta , { \\rho } , \\eta ) \\ , ^ { 2 } } \\ . \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} ( U ^ * ) ^ { D } = \\left ( U ^ { ( D ^ { - 1 } ) ^ s } \\right ) ^ * . \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\chi : T ^ * B u n _ { \\mathcal { G } _ { X , x , \\theta } } \\rightarrow \\bigoplus \\bigoplus _ { j = 1 } ^ n H ^ { 0 } ( X , K _ X ^ { 2 j } ( C x ) ) \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} T ^ 0 ( X , Y ) = g ( ( T _ { \\xi _ 1 } ^ { 0 } I _ 1 + T _ { \\xi _ 2 } ^ { 0 } I _ 2 + T _ { \\xi _ 3 } ^ { 0 } I _ 3 ) X , Y ) \\ \\ U ( X , Y ) = g ( u X , Y ) \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} \\mathfrak { n } _ { T } & : = \\left \\{ x \\in M \\mid \\| T ( x ^ * x ) \\| _ \\infty < + \\infty \\right \\} , \\\\ \\mathfrak { m } _ { T } & : = ( \\mathfrak { n } _ { T } ) ^ * \\mathfrak { n } _ { T } = \\left \\{ \\sum _ { i = 1 } ^ n x _ i ^ * y _ i \\mid n \\geq 1 , x _ i , y _ i \\in \\mathfrak { n } _ { T } 1 \\leq i \\leq n \\right \\} . \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{gather*} Q = \\partial _ I Q ^ a \\frac { \\partial } { \\partial \\phi _ I ^ a } . \\end{gather*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\lVert \\mathfrak { I } _ { n + 1 } \\rVert _ { s _ 0 + \\mu } \\le \\sum _ { k = 1 } ^ { n + 1 } \\lVert \\hat { \\mathfrak { I } } _ k \\rVert _ { s _ 0 + \\mu } \\le C _ * \\varepsilon ^ { b _ * } \\gamma ^ { - 1 } \\sum _ { k \\geq 1 } N _ { k - 1 } ^ { - \\alpha _ 1 } \\le C _ * \\varepsilon ^ { b _ * } \\gamma ^ { - 1 } \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} \\hat { \\zeta } = M _ { \\varphi } [ g _ 2 ] . \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} u ( x , t ) _ { | x \\in \\partial \\Omega } = 0 . \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} S _ { n } ^ { z } = \\left ( \\begin{array} { c c c c } 2 s & 0 & & \\\\ 0 & \\ddots & \\ddots & \\\\ & \\ddots & \\ddots & 0 \\\\ & & 0 & - 2 s \\end{array} \\right ) . \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} ( \\gamma - \\chi ( \\gamma ) ) - \\gamma \\ = \\ - \\chi ( \\gamma ) \\ \\geq \\ - \\chi ( \\beta ) \\ > \\ \\Delta . \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} & K ( \\pi ^ * ( \\sigma ) \\cdot f ( x , \\psi , \\eta ) ) : = 0 \\\\ & K ( \\pi ^ * ( \\sigma ) \\wedge d \\eta \\ , f ( x , \\psi , \\eta ) ) : = \\pi ^ * ( \\sigma ) \\cdot F ( x , \\psi , \\eta ) , \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} A _ \\beta ( \\infty ) = \\beta e ^ { - ( 2 \\beta - 1 ) T _ \\theta } ( A _ \\beta ^ + ( \\infty ) + A _ \\beta ^ - ( \\infty ) ) . \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} a _ k = b _ m \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} 2 \\xi ( \\phi ) r \\partial _ r + 2 \\phi \\nabla _ \\xi ( r \\partial _ r ) + r ^ 2 \\nabla _ \\xi \\nabla \\phi & = 2 \\phi \\xi + 2 r ^ 2 J \\nabla \\phi + r ^ 2 J \\nabla _ { r \\partial _ r } \\nabla \\phi . \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} L _ { n + 1 } = \\min \\left \\{ j : X _ j > X _ { L _ n } , j > L _ n \\right \\} \\ \\ n \\geq 0 . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} P ( \\widetilde { \\Delta } , L _ { b c } ) = ( I + \\mathcal { U } ) ( I + \\mathcal { V } ) P ( \\Delta , L _ { b c } ^ 0 ) ( I + \\mathcal { V } ) ^ { - 1 } ( I + \\mathcal { U } ) ^ { - 1 } . \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} \\{ \\mathbf { r ' , p ' } \\} = { \\rm a r g } \\max _ { \\mathbf { r , p } } \\ & R _ { \\Sigma } ( \\boldsymbol { \\alpha } ) \\quad { \\rm s . t . } ( \\ref { p o w } ) - ( \\ref { p s d } ) \\\\ \\{ \\mathbf { Q } ^ { * } _ j \\} = { \\rm a r g } \\min _ { \\mathbf { Q } _ j , \\forall j } \\ & \\sum _ { j } { \\rm T r } ( \\mathbf { Q } _ j ) \\quad { \\rm s . t . } \\mathbf { r } \\geq \\mathbf { r ' } , \\ ( \\ref { b } ) , ( \\ref { c } ) , \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} z = \\frac { a z + b } { c z + d } . \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} - 4 i \\Omega L _ \\zeta & = \\xi \\eta ( \\xi ^ 2 + \\eta ^ 2 - 2 \\zeta ^ 2 ) J ( \\zeta ) ( J ( \\zeta ) - J ( \\xi ) - J ( \\eta ) ) \\\\ & - \\xi ^ 2 \\eta ( \\xi - \\eta ) J ( \\zeta ) ( J ( \\xi ) - J ( \\eta ) - J ( \\zeta ) ) \\\\ & - \\xi \\eta ^ 2 ( \\xi - \\eta ) J ( \\zeta ) ( J ( \\xi ) - J ( \\eta ) + J ( \\zeta ) ) . \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n f \\left ( \\mathcal { A } _ i \\right ) \\geq \\sum _ { j = 1 } ^ n f \\left ( \\mathcal { E } _ j ^ { ( n ) } \\right ) . \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} ( - 1 ) ^ n \\omega _ n ( g ) = \\sum _ { j = 1 } ^ { 4 8 } m _ j ( n ) \\chi _ j ( g ) , \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } - \\mathrm { d i v } ( y ^ { 1 - 2 \\alpha } \\nabla w ) & = 0 , \\quad \\textrm { i n } \\ ; \\mathcal { C } , \\\\ w & = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\partial _ L , \\\\ M ( \\| w \\| ^ 2 ) \\frac { \\partial w } { \\partial \\nu } & = \\lambda f ( z ) | w | ^ { q - 2 } w + | w | ^ { 2 ^ * _ \\alpha - 2 } w \\ ; \\ ; \\textrm { o n } \\ ; \\ ; \\Omega \\times \\{ 0 \\} , \\end{array} \\right . \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} A u ^ n = \\left [ \\begin{array} { c } \\langle A _ 1 ^ n , u ^ n \\rangle \\\\ \\langle A _ 2 ^ n , u ^ n \\rangle \\\\ \\vdots \\\\ \\langle A _ m ^ n , u ^ n \\rangle \\\\ \\end{array} \\right ] , \\ ; \\ ; \\ ; \\ ; A v ^ n = \\left [ \\begin{array} { c } \\langle A _ 1 ^ n , v ^ n \\rangle \\\\ \\langle A _ 2 ^ n , v ^ n \\rangle \\\\ \\vdots \\\\ \\langle A _ m ^ n , v ^ n \\rangle \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} \\phi ^ { * } ( \\phi ' ( x ) ) = \\phi ' ( x ) x - \\phi ( x ) . \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} f ( \\mathbf { x } ; n , \\mathbf { \\theta } ) = \\alpha ( \\mathbf { x } ) \\theta _ { 1 } ^ { x _ { 1 } } \\cdots \\theta _ { k } ^ { x _ { k } } , \\quad \\mathrm { f o r } \\ , \\sum _ { i = 1 } ^ { k } x _ { i } = n , \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} f : = \\sum _ { j \\in H } \\left ( 1 0 0 ( x _ { j + 1 } - x _ { j } ^ 2 ) ^ 2 + ( 1 - x _ { j } ) ^ 2 + 9 0 ( x _ { j + 3 } - x _ { j + 2 } ^ 2 ) ^ 2 \\right . \\\\ \\left . + ( 1 - x _ { j + 2 } ) ^ 2 + 1 0 ( x _ { j + 1 } + x _ { j + 3 } - 2 ) ^ 2 + 0 . 1 ( x _ { j + 1 } - x _ { j + 3 } ) ^ 2 \\right ) \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{gather*} q _ { 2 t } = \\frac { \\mu _ t } { \\chi } - \\frac { \\mu } { \\chi ^ 2 } \\chi _ t = \\frac { 2 } { 3 } \\frac { u _ t } { u } - \\frac { \\mu } { \\chi } \\left ( \\frac { 2 } { 3 } \\frac { u _ t } { u } \\frac { \\mu } { \\chi } - \\frac { 1 } { 3 } \\frac { \\nu } { \\chi } \\right ) = \\frac { 2 } { 3 } \\frac { u _ t } { u } - q _ 2 \\left ( \\frac { 2 } { 3 } \\frac { u _ t } { u } q _ 2 - \\frac { 1 } { 3 } q _ 1 \\right ) , \\end{gather*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\tau _ a = \\bigg \\{ \\begin{array} { l l } 1 / 2 & a > 1 , \\\\ 1 & a < 1 . \\end{array} \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} p ( z ) = [ p ( z ) ] _ + + [ p ( z ) ] _ \\bullet + [ p ( z ) ] _ - , \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} \\sqrt { M ( h _ { + } ) } = \\dfrac { 2 \\sqrt { p _ { 1 } } \\left ( s - v \\right ) \\left \\vert 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v \\right \\vert ( s - 2 h _ { + } ) } { s ^ { 2 } \\sqrt { s ^ { 2 } + t ^ { 2 } } } \\allowbreak \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} \\widetilde { \\phi } _ i ^ W : = \\sum _ { \\nu = ( i \\rightarrow j ) \\in Q _ 1 } \\rho _ { \\nu } : W _ i \\rightarrow \\bigoplus _ j W _ j , \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} B \\otimes B ^ { r , k } & \\longrightarrow B ^ { r , k } \\otimes B , \\\\ b \\otimes u _ { k \\varpi _ r } & \\longmapsto \\widetilde { u } \\otimes \\widetilde { b } , \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} \\mathrm { c m } ( x , y ) = \\frac { g ( x , y ) } { | | x | | . | | y | | } , \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} q ^ { 2 } = r ( p - 2 c q ) . \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} ( - \\Delta ) ^ \\sigma u ( x ) = c _ { N , \\sigma } P . V . \\int _ { \\R ^ N } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { N + 2 \\sigma } } \\ d x d y = c _ { N , \\sigma } \\int _ { \\R ^ N } \\frac { 2 u ( x ) - u ( x - y ) - u ( x + y ) } { | y | ^ { N + 2 \\sigma } } \\ d x d y , \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} \\P \\Big ( \\sum _ { i = 1 } ^ m U _ i ^ 2 < m ( 1 - \\tau ) \\Big ) \\leq { \\rm e } ^ { \\frac { m } { 2 } ( \\tau + \\ln ( 1 - \\tau ) ) } \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} E \\int _ { ( 0 , t ] } \\xi _ { s } \\ , d \\pi _ { s } = \\lambda E \\int _ { 0 } ^ { t } \\xi _ { s } \\ , d s . \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} p = \\frac { 2 \\left ( c ^ 2 - 2 \\right ) } { 3 \\left ( c ^ 2 + 4 \\right ) } . \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow 0 } z ^ { \\alpha + \\beta - 1 } \\cdot U ( \\beta , \\alpha + \\beta ; z ) = \\frac { \\Gamma ( \\alpha + \\beta - 1 ) } { \\Gamma ( \\beta ) } . \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} \\frac { \\gamma \\left ( a + 1 , z \\right ) } { \\Gamma \\left ( a + 1 \\right ) } = 1 + \\left \\{ \\begin{array} [ c ] { l l } O \\left [ a ^ { 1 / 2 } \\exp \\left \\{ a \\ln \\left ( e / \\left ( \\rho e ^ { 1 / \\rho } \\right ) \\right ) \\right \\} \\right ] & x / b \\rightarrow \\infty \\\\ O \\left \\{ z ^ { \\kappa } \\exp \\left ( - z \\right ) \\right \\} & x / b \\rightarrow \\kappa \\end{array} \\right . . \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} \\mathcal L _ v w : = \\frac { d } { d t } \\Big | _ { t = 0 } D \\Phi _ { - t } w \\circ \\Phi _ { t } \\ ; . \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} D \\pi _ { \\Phi _ s ( x ) } v ( \\Phi _ s ( x ) ) = \\tilde v \\circ \\pi \\circ \\Phi _ s ( x ) \\ ; . \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} m n = \\frac { \\langle m , n \\rangle } { \\mu } ( W ) + \\deg ( ( W ) ) , \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} S ( F ) : = \\left \\{ A \\subseteq F \\mid A = \\sum _ { i = 1 } ^ n a _ i , \\ : \\ : a _ i \\in F , \\ : n \\in \\mathbb { N } \\right \\} . \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} w _ i ( A ) = \\left \\{ \\begin{array} { l l } 1 , & 1 \\le i \\le c _ 2 n \\\\ n ^ 2 - n , & c _ 2 n < i \\le n ^ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} H = \\alpha \\cdot D + m \\beta \\ , , D = - i \\nabla \\ , , \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} Z _ { \\omega } = \\pi _ { \\omega } ( { \\cal A } ) ^ { ' } \\cap \\pi _ { \\omega } ( { \\cal A } ) ^ { '' } \\ , \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} \\Gamma _ { p _ 1 } : = \\{ \\varphi _ t ( p _ 1 ) , \\ , t \\in \\mathbb { R } \\} , \\end{align*}"}
-{"id": "2687.png", "formula": "\\begin{align*} d ( x , z ) = \\sum _ { n \\geq 0 } d _ n ( x ) \\frac { z ^ n } { n ! } = \\frac { 1 - x } { e ^ { x z } - x e ^ z } . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to + 0 } \\frac { \\partial p ( \\mu , \\lambda ) } { \\partial \\lambda } = \\lim _ { \\lambda \\to + 0 } \\zeta _ { m a x } ( \\lambda ) = \\sqrt { \\rho _ { 0 } } \\ , \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} W H ^ r ( P ) = \\bigoplus _ { \\substack { x \\in P \\\\ \\rho ( x ) = r } } \\widetilde H ^ { r - 2 } ( ( \\hat 0 , x ) ; { \\bf k } ) . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} \\begin{aligned} 1 - F ( 2 \\pi / s ) + F ( - 2 \\pi / s ) \\le \\\\ \\le \\frac { 2 } { s a _ 0 } \\Bigl ( \\sum _ { j = 0 } ^ k a _ j \\int _ { 0 } ^ { s } \\tt { R e } \\ , f ( j u ) d u + \\sum _ { j = 1 } ^ k b _ j \\int _ { 0 } ^ { s } \\tt { I m } \\ , f ( j u ) d u \\Bigr ) . \\end{aligned} \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} t _ \\alpha s _ \\alpha ( t _ \\beta ) = t _ \\beta s _ \\beta ( t _ \\alpha ) . \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} ( A + c d ^ T ) ^ { - 1 } = A ^ { - 1 } - \\frac { A ^ { - 1 } c d ^ T A ^ { - 1 } } { 1 + d ^ T A ^ { - 1 } c } . \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\bar { \\mu } ( Y , s ) = \\dfrac { 1 } { 8 } \\left ( ( \\Gamma ) - w ^ 2 \\right ) . \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} A B = \\bigcup _ { a \\in A } a B = \\big \\{ a b \\ , : \\ , a \\in A , \\enskip b \\in B \\big \\} \\subset Y . \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} \\mathbf { P } _ 1 = \\mathbf { B } \\mathbf { G } , \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{align*} \\Lambda _ C ( \\theta ) = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mathbf { E } \\left ( \\exp { \\left ( \\frac { \\log C _ n - n } { n } \\cdot n \\theta \\right ) } \\right ) \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{align*} \\frac { M _ k ( x ; \\mathbf { a } ) } { \\frac { 1 } { \\phi ^ k ( q ) } \\frac { k ! } { k _ { 1 } ! k _ 2 ! \\cdots k _ l ! } S _ k ( x ) } = 1 + \\frac { k - 1 } { \\log \\log x } \\frac { 1 } { k } \\sum _ { j = 1 } ^ k C ( q , a _ j ) + O _ { q , k , l } \\ ( \\frac { 1 } { ( \\log \\log x ) ^ 2 } \\ ) . \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\dfrac { 1 - t } { 1 - t e ^ { ( 1 - t ) y } } = \\sum _ { n \\ge 0 } A _ n ( t ) \\dfrac { y ^ n } { n ! } . \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} C _ h = \\lim _ { q \\to 1 } \\left ( g ^ { s t } C _ q g \\right ) = \\begin{pmatrix} 0 & 0 & - 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & h \\end{pmatrix} , \\hat { B _ h } = \\lim _ { q \\to 1 } \\left [ ( g \\otimes g ) ^ { - 1 } \\hat { B _ q } ( g \\otimes g ) \\right ] , \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} \\nu _ i ( \\langle x , u _ i \\rangle - \\eta _ i - \\sum _ { j = 1 } ^ n \\nu _ i \\langle u _ j \\ | \\ u _ i \\rangle ) = 0 \\ i = 1 , \\dots , n . \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{gather*} D ( p ) = ( 1 - p ) D _ 1 + p D _ 2 , M ( p ) = ( 1 - p ) M _ 1 + p M _ 2 , \\\\ f ( p , N ) = ( 1 - p ) \\frac { f _ 1 } N - p \\ , \\frac { f _ 2 } { N } \\ , . \\end{gather*}"}
-{"id": "4069.png", "formula": "\\begin{align*} \\varphi _ \\beta ' ( r ) = \\int \\limits _ { S _ \\beta } \\frac { d } { d r } \\varphi _ \\beta ^ 2 ( r s ) \\nu _ \\beta ( d s ) = \\frac { 1 } { r } \\int \\limits _ { S _ \\beta } \\frac { d } { d s } \\varphi _ \\beta ^ 2 ( r s ) \\ , s \\ , \\nu _ \\beta ( d s ) . \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} \\bar y : = x _ { j _ 1 } ^ 2 x _ { j _ 2 } \\ldots x _ { j _ { k - 3 } } x _ j = \\sum _ { s = 2 } ^ { k - 3 } & x _ { j _ s } ^ 2 x _ { j _ 1 } x _ { j _ 2 } \\ldots \\hat x _ { j _ s } \\ldots x _ { j _ { k - 3 } } x _ j \\\\ & + \\sum _ { t = 1 } ^ m y _ t + S q ^ 1 ( g _ 0 + \\tilde y ) + \\sum _ { u \\geqslant 1 } S q ^ { 2 ^ u } ( g _ u ) . \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} \\rho ( u ( \\cdot , t _ 2 ; s , u _ 0 ) , u ( \\cdot , t _ 2 ; s , v _ 0 ) ) & = \\rho ( u ( \\cdot , t _ 2 ; t _ 1 , u ( \\cdot , t _ 1 ; s , u _ 0 ) ) , u ( \\cdot , t _ 2 ; t _ 1 , u ( \\cdot , t _ 1 ; s , v _ 0 ) ) ) \\\\ & < \\rho ( u ( \\cdot , t _ 1 ; s , u _ 0 ) , u ( \\cdot , t _ 1 ; s , v _ 0 ) ) . \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{align*} \\overline { \\omega } : = ( \\overline { \\jmath } _ 1 ^ 3 , \\dots , \\overline { \\jmath } _ { \\nu } ^ 3 ) \\in \\mathbb { N } ^ { \\nu } . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} p ( \\beta , \\mu , \\lambda ) = \\lim _ { V \\to \\infty } p _ { \\beta , \\mu , \\Lambda , \\lambda } \\ , \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { \\infty } \\sum _ { c _ 1 \\cdots c _ m \\theta } ( \\alpha _ 2 ( c _ m \\cdots c _ 1 ) ) ^ { - ( q - 1 ) s } ( \\mu [ c _ m \\cdots c _ 1 ] ) ^ q . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} \\left ( \\sum _ { l = 0 } ^ { | k | - 1 } \\theta _ k ^ l \\delta _ { - h } \\delta _ h \\phi ( x + s _ k l h ) , \\phi ( x ) \\right ) \\leq 0 . \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} G _ i ( t ) = \\sum _ { k = - Q _ i ( 0 ) + 1 } ^ { N _ i ( t ) } 1 _ { \\{ d _ { i , k } \\le w _ { i , k } , \\ t _ { i , k } + d _ { i , k } \\le t \\} } . \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} \\bar { \\mathbf { x } } _ i = { \\mathbf { x } } _ i + \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\left ( \\mathbf { P } _ 1 { \\mathbf { d } } _ { i - 1 } + \\mathbf { P } _ 1 \\mathbf { A } ^ { - 1 } \\mathbf { w } _ { i - 1 } - \\mathbf { P } _ 2 \\mathbf { d } _ i \\right ) . \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\sqrt { n ! } = N ^ { \\frac n 2 } ( 1 - \\xi N ^ { - 2 / 3 } ) ^ { \\frac { N - \\xi N ^ { 1 / 3 } } 2 } e ^ { - \\frac N 2 - \\frac { \\xi N ^ { 1 / 3 } } 2 + o ( 1 ) } ( 2 \\pi N ) ^ { \\frac 1 4 } = ( 2 \\pi ) ^ { \\frac 1 4 } N ^ { \\frac n 2 + \\frac 1 4 } e ^ { - \\frac N 2 + o ( 1 ) } . \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} A ^ h ( \\xi , \\eta ) = - i \\left ( \\eta + \\frac { J ( \\xi ) ^ 2 - \\xi ^ 2 } { 4 J ( \\xi ) } \\right ) + S ( d ^ 2 \\rho ^ { - 1 } ) . \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{gather*} q _ { 2 t } = \\frac { 2 } { 3 } \\alpha q _ 2 + \\frac { u _ t } { u } \\frac { ( 1 + q _ 2 ) ( 2 - q _ 2 ) } { 3 } , \\end{gather*}"}
-{"id": "56.png", "formula": "\\begin{align*} \\eta ( u , v , a ) : = \\frac { u ^ 2 } { 2 } + \\frac { v ^ 2 } { 2 a } , q ( u , v ) : = - u v , \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { 2 n } } { 1 - q ^ { 2 n } } \\frac { ( - q / z ; q ^ 2 ) _ { n - 1 } } { ( q ^ 2 , q ^ 2 ) _ { n - 1 } } ( z q ) ^ { n - 1 } = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( - q z ; q ^ 2 ) _ \\infty } - \\frac { \\tiny 1 } { 1 + z q } \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} E _ f ( g ) = \\sum _ { \\gamma \\in \\Gamma _ B \\backslash \\Gamma } f ( H ( \\gamma g ) ) \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} \\Xi _ { \\Lambda } ( \\mu , \\beta ) : = { \\rm { T r } } _ { { \\cal H } _ { \\Lambda } } \\exp ( - \\beta ( H _ { \\Lambda } - \\mu N _ { \\Lambda } ) ) \\ , \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} | \\sum _ { k = 1 } ^ { \\infty } < x ^ { * } _ { n } - x ^ { * } _ { 0 } , e _ { k } > | = | < x ^ { * * } _ { 0 } , \\sum _ { k = 1 } ^ { \\infty } < x ^ { * } _ { n } - x ^ { * } _ { 0 } , e _ { k } > e ^ { * } _ { k } > | > \\epsilon _ { 0 } , n = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} Z _ v ( u , v ) + X ( u , v ) Y _ v ( v , u ) = 0 , u \\in \\C \\setminus \\{ 0 \\} , \\ v \\in \\C . \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} \\sum _ { i = a } ^ b f ( i ) \\leq f ( a ) + f ( b ) + \\int _ a ^ b f ( x ) \\ , \\dd x ; \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} [ D ( m , m j + k ) : D ( m + 1 , n ) ] _ q & = q ^ { 4 k j } [ D ( m , m j - k ) : D ( m + 1 , n ) ] _ q & \\\\ & = \\begin{cases} q ^ { j ( m j + 2 k ) } & n = k , \\\\ q ^ { ( j + 1 ) ( m j + 2 k - m ) } & n = m - k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} \\operatorname { R N } _ { N , N , N } + \\operatorname { F E } _ { N , N , N } \\le C d \\left [ \\sup _ { ( t , x ) \\in [ 0 , T ] \\times \\R ^ d } \\left \\| U _ { N , N , N } ^ { 0 } ( t , x ) - u ^ { \\infty } ( t , x ) \\right \\| _ { L ^ 2 ( \\P ; \\R ) } \\right ] ^ { - \\left ( 4 + \\delta \\right ) } . \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} ( E _ B ) _ N = \\sum _ { w \\in W } a ^ { w \\nu + \\rho } \\prod _ { w \\alpha < 0 , \\alpha > 0 } \\frac { \\xi ( s _ { \\alpha } ) } { \\xi ( s _ { \\alpha } + 1 ) } \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{gather*} \\| \\sum _ { \\ell = 2 ^ n } ^ k \\frac { \\varepsilon _ \\ell ( \\omega ) \\ell ^ { i t } } { \\ell \\sqrt { \\log ( \\ell + 1 ) } } \\| _ { \\S ^ 2 } \\le \\sum _ { \\ell = 2 ^ n } ^ k \\frac { 1 } { \\ell \\sqrt { \\log ( \\ell + 1 ) } } \\le \\frac { 2 } { \\sqrt n } \\underset { n \\to + \\infty } \\longrightarrow 0 \\ , . \\end{gather*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } h = - A _ { 1 1 } ^ { - 1 } A _ { 1 2 } v , \\\\ A _ { 2 1 } h ' + A _ { 2 2 } v ' = Q ' _ { 2 2 } ( u ^ \\pm ) v . \\end{array} \\right . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} I ( u ) & = \\frac { 1 } { b p ^ 2 } \\left ( a + b \\int _ 0 ^ T | { _ 0 D _ t ^ \\alpha } u ( t ) | ^ p d t \\right ) ^ p - \\int _ 0 ^ T F ( t , u ( t ) ) d t - \\frac { a ^ p } { b p ^ 2 } \\\\ & = \\frac { 1 } { b p ^ 2 } ( a + b \\| u \\| _ { E ^ { \\alpha , p } } ^ p ) ^ p - \\int _ 0 ^ T F ( t , u ( t ) ) d t - \\frac { a ^ p } { b p ^ 2 } . \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{gather*} u ( t ) = \\frac { a } { 2 \\sqrt { \\pi } } t ^ { - \\frac { 1 } { 4 } } e ^ { - \\frac { 2 } { 3 } t ^ { \\frac { 3 } { 2 } } } ( 1 + o ( 1 ) ) , t \\to + \\infty . \\end{gather*}"}
-{"id": "511.png", "formula": "\\begin{align*} u ' - u ( u _ 1 - u _ { - 1 } ) = 0 , \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\begin{bmatrix} \\operatorname { D P [ 0 ] } \\\\ \\mathcal { T } \\end{bmatrix} ( \\Xi _ 1 ) = 0 \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} ( u _ 1 ^ + - \\phi ) ( \\hat h ) & = ( v ^ + - \\phi \\circ H ) ( x _ 0 ) \\leq a - \\phi \\circ H ( \\hat x ) \\\\ & \\leq ( v ^ + - \\phi \\circ H ) ( \\hat x ) = ( u _ 1 ^ + - \\phi ) \\circ H ( \\hat x ) . \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} \\tilde { \\psi } ( \\xi ) = \\frac { \\psi ( \\xi ) } { \\psi ( 1 ) } \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + a x + b , a = - \\frac { c _ 4 } { 4 8 } , b = - \\frac { c _ 6 } { 8 6 4 } \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} \\begin{aligned} \\nu _ i \\geq 0 , F _ i ( \\bar { x } ) \\leq 0 , \\nu _ i F _ i ( \\bar { x } ) = 0 , i \\in N , \\\\ F _ 0 ^ \\prime ( \\bar { x } ) + \\nu _ 1 F _ 1 ^ \\prime ( \\bar { x } ) + \\dots + \\nu _ n F _ n ^ \\prime ( \\bar { x } ) = 0 . \\end{aligned} \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} \\langle p - 1 , . . . , p - 1 | p - 1 , . . . , p - 1 \\rangle = \\prod _ { 1 \\leq b < a \\leq \\mathsf { N } } \\frac { 1 } { X _ { a } ^ { \\left ( p - 1 \\right ) } - X _ { b } ^ { \\left ( p - 1 \\right ) } } , \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} s _ i = s \\textnormal { a n d } s _ i ^ \\prime = s ^ \\prime , \\ , \\ , \\forall \\ , i \\in [ 1 : n ] . \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} \\iota _ q ( \\mu ) : = \\begin{cases} 1 , & q \\neq 4 , \\\\ \\log ( \\mu a _ \\mu ^ { - \\frac 2 3 } ) , & q = 4 . \\end{cases} \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} \\mathcal { P } _ 2 ^ { \\mathrm { U L A } } \\left ( { x } \\right ) = \\exp \\left \\{ - 2 \\pi { \\lambda _ e } \\int _ 0 ^ \\infty \\int _ 0 ^ { 2 \\pi } { \\rm { \\mathbf { 1 } } } \\left ( { \\max \\{ { r _ e } , d \\} } < \\big ( \\frac { { P _ t } { G _ e ( \\varphi _ { t _ { e , o } } ) } \\beta } { x \\sigma _ e ^ 2 } \\big ) ^ { \\frac { 1 } { { \\alpha _ { \\mathrm { N L o S } } } } } \\right ) \\frac { 1 - { { f _ { \\Pr } } ( { r _ e } ) } } { 2 \\pi } { r _ e } d { \\varphi _ { t _ { e , o } } } d { r _ e } \\right \\} \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} \\sigma : = \\sqrt [ d ] { | \\det \\operatorname { C o v } ( N ) | } . \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} \\sum \\limits _ { \\theta \\in { \\rm I r r } ( N _ G ( P ) ) } \\theta ( J ) ^ 2 = q - 1 + 4 c ^ 2 = q ( q - 1 ) , \\end{align*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\bold { E } ( Q \\cdot F ) = \\bold { D } _ Q ^ { \\ast } ( F ) + \\bold { D } _ F ^ { \\ast } ( Q ) , \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} \\begin{aligned} E _ i E _ j E _ k = ( - 1 ) ^ { \\delta _ { i , k + 1 } + \\delta _ { j , k + 1 } + \\delta _ { i , j + 1 } } \\Phi _ { i + 1 } \\Phi _ { j + 1 } \\Phi _ { k + 1 } e _ i e _ j e _ k . \\end{aligned} \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} f ( \\boldsymbol { k } _ { m , n } ( t ) ) = \\lambda \\frac { k _ { m , n } ( t ) x _ { m , n } } { k } + ( 1 - \\lambda ) f ( \\boldsymbol { k } _ { m , n } ( t - 1 ) ) , \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} \\gamma _ { \\sigma , k } : = \\begin{cases} & \\frac { n } { 4 ( 1 - \\sigma ) } - \\frac { \\sigma } { 1 - \\sigma } + \\frac { k } { 2 ( 1 - \\sigma ) } \\ \\ 0 \\le \\sigma < \\frac { 1 } { 2 } , \\\\ & \\frac { n } { 2 } + 1 - k \\ \\ \\sigma = \\frac { 1 } { 2 } , \\\\ & \\frac { n } { 4 \\sigma } - \\frac { 1 } { 2 \\sigma } + \\frac { k } { 2 \\sigma } \\ \\ \\frac { 1 } { 2 } < \\sigma \\le 1 , \\end{cases} \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} a _ x { \\cal F } = { \\rm r a n k } \\ { \\cal F } | _ U - { \\rm r a n k } \\ { \\cal F } _ x + { \\rm S w } _ x { \\cal F } \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} d x ( t ) = a ( x ( t ) ) d W ( t ) + b ( x ( t ) ) d t , \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} L _ { - \\tau } \\psi = \\psi , \\ , \\ , L ^ { * } _ { - \\tau } \\nu ' = \\nu ' , \\ , \\ , \\ , \\ , \\int \\psi d \\nu ' = 1 . \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} \\frac { p ^ 2 + 1 } { p } + \\frac { 1 } { 2 ^ n } = \\bigg ( \\frac { p ^ 2 + 1 } { p } + \\frac { 1 } { 2 ^ { n + 1 } } \\bigg ) + \\frac { 1 } { 2 ^ { n + 1 } } , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} c _ 4 ( F ) = 2 ^ 4 ( 4 ( \\ell - 1 ) + 3 ) , c _ 6 ( F ) = - 2 ^ 6 \\sqrt { \\ell - 1 } ( \\ell + 8 ) , \\Delta ( F ) = 2 ^ 6 \\ell . \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} \\chi = \\mathbb { E } [ | X ( - J , - 1 ) | ] \\in [ 1 , 2 ] . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} V _ 1 ( x ) = a ( \\theta ) r ^ { - \\nu } \\langle r \\rangle ^ { \\nu - \\mu } , V _ 2 ( x ) = a ( \\theta ) r ^ { - 2 } ( 1 + ( \\log r ) ^ 2 ) ^ { - \\delta } , r = | x | , \\ \\theta = x / r , \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} R _ t = \\frac { Q _ { \\alpha t } } { 1 + W _ \\alpha } - \\dfrac { Q _ \\alpha W _ { \\alpha t } } { ( 1 + W _ \\alpha ) ^ 2 } . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} c _ i ( t u , v ) & = c _ i ( t u , v ^ { 2 k + 1 } ) c _ i ( t u , v ^ { 2 k } ) ^ { - 1 } \\ : \\ : \\mbox { ( B y ( i i i ) ) } \\\\ & = c _ i ( t , v ^ { 2 k + 1 } ) c _ i ( u , v ^ { 2 k + 1 } ) c _ i ( t , v ^ { 2 k } ) ^ { - 1 } c _ i ( u , v ^ { 2 k } ) ^ { - 1 } \\ : \\ : \\mbox { ( B y ( L 6 ) a n d ( v ) ) } \\\\ & = c _ i ( t , v ) c _ i ( t , v ^ { 2 k } ) c _ i ( u , v ) c _ i ( u , v ^ { 2 k } ) c _ i ( t , v ^ { 2 k } ) ^ { - 1 } c _ i ( u , v ^ { 2 k } ) ^ { - 1 } \\ : \\ : \\mbox { ( B y ( i i i ) ) } \\\\ & = c _ i ( t , v ) c _ i ( u , v ) \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & - a _ 3 & a _ 2 \\\\ a _ 3 & 0 & - a _ 1 \\\\ - a _ 2 & a _ 1 & 0 \\end{pmatrix} \\mapsto a = ( a _ 1 , a _ 2 , a _ 3 ) . \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} B _ x B _ s B _ y = B _ x B _ y ( 1 ) \\oplus B _ x B _ y ( - 1 ) \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} n ^ { - 1 / 2 } \\sum _ { r = 1 } ^ n \\exp ( - n \\epsilon ( r / n - p ) ^ 2 ) < C ( p , \\epsilon ) . \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} \\Pi _ i ( x , y , z ) = \\frac { \\left [ ( y \\times \\nu _ i ( x ) ) \\otimes ( y \\times \\nu _ i ( x ) ) \\right ] z } { \\abs { y \\times \\nu _ i ( x ) } ^ 2 } + y . \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\frac { 1 } { k 2 ^ k } \\sum _ { j = 1 } ^ { k - 1 } \\frac { 2 ^ j } { j } \\equiv _ { p } 0 , \\end{align*}"}
-{"id": "7110.png", "formula": "\\begin{align*} d _ x ^ * \\circ d _ x = B _ x \\rho - ( x \\rho ) B _ x . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} \\Lambda = \\lambda ' * \\nu \\ . \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\begin{pmatrix} { P } ^ \\prime \\\\ { Q } ^ \\prime \\\\ { R } ^ \\prime \\end{pmatrix} = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\lambda ( h ^ { \\lambda , m , 0 } ) ^ 2 & h ^ { \\lambda , m , 0 } & ( \\frac { m } { \\lambda } + \\lambda p ) h ^ { \\lambda , m , 0 } \\end{pmatrix} \\begin{pmatrix} { P } \\\\ { Q } \\\\ { R } \\end{pmatrix} . \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} A ^ { \\epsilon , \\kappa } ( x ) : = \\epsilon A ^ 0 ( x ) + \\kappa A ( \\epsilon x ) \\ , , \\ , B _ { \\epsilon , \\kappa } = \\partial _ 1 A ^ { \\epsilon , \\kappa } _ 2 - \\partial _ 2 A ^ { \\epsilon , \\kappa } _ 1 \\ , . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} c ' ( x , y ) = c ( x , y ) + f ( x ) + g ( y ) \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c } \\int _ { \\Omega } | \\nabla u | ^ { p ( x ) - 2 } \\nabla u \\nabla \\varphi \\ d x & = \\int _ { \\Omega } u ^ { \\alpha _ { 1 } ( x ) } v ^ { \\beta _ { 1 } ( x ) } \\varphi \\ d x \\\\ \\int _ { \\Omega } | \\nabla v | ^ { q ( x ) - 2 } \\nabla v \\nabla \\psi \\ d x & = \\int _ { \\Omega } u ^ { \\alpha _ { 2 } ( x ) } v ^ { \\beta _ { 2 } ( x ) } \\psi \\ d x \\end{array} \\right . \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} \\alpha ' \\ = \\ ( v ( y - \\epsilon ) ) ' \\ = \\ v ( y ' - \\epsilon ' ) \\ = \\ v ( s - \\epsilon ' ) \\in S . \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} \\log T ( \\lambda \\phi ) \\le \\left \\{ \\begin{array} { l l } C _ 1 \\lambda ^ { - ( p - 1 ) } & \\mbox { i f } A > N , \\\\ C _ 1 \\lambda ^ { - \\frac { p - 1 } { p } } & \\mbox { i f } A = N , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} f _ m ( n ) = \\begin{cases} 3 - \\frac 2 { k } & , \\\\ 3 - \\frac 1 { k } & . \\end{cases} \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} c ( \\ell ) = \\sum _ { m n = \\ell } \\rho ( m ) \\lambda ( n ) \\left ( 1 - \\frac { \\log { m } } { \\log { M } } \\right ) ^ { r } \\left ( 1 - \\frac { \\log { n } } { \\log { N } } \\right ) \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} A = \\ , \\mathop { \\oplus } _ a \\left ( - i ( \\lambda _ a + \\frac { m _ a } { 2 r } ) \\frac { d \\tau + \\omega } { V } + \\pi _ k ^ * \\eta _ a \\right ) + O ( r ^ { - 2 } ) , \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} \\hat { f } _ { - } \\left ( x \\right ) & = E \\left \\{ \\hat { f } _ { - } \\left ( x \\right ) \\right \\} + o _ { p } \\left ( 1 \\right ) . \\\\ E \\left \\{ \\hat { f } _ { - } \\left ( x \\right ) \\right \\} & = E \\left \\{ \\hat { f } _ { - } \\left ( 0 \\right ) \\right \\} + o \\left ( 1 \\right ) . \\\\ E \\left \\{ \\hat { f } _ { - } \\left ( 0 \\right ) \\right \\} & \\rightarrow \\infty . \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} \\gamma = ( P _ { t } / n _ { t } ) \\mathbb { E } \\left \\{ | | \\mathbf { G } | | ^ { 2 } \\right \\} \\beta _ K / \\sigma ^ { 2 } = P _ { t } / \\sigma ^ { 2 } \\ , , \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} \\phi ( r , z ) = \\left ( \\frac { t _ n } t \\right ) ^ { \\frac \\ell 2 } c ^ { 1 - \\frac \\alpha 2 } \\widehat { \\phi } \\left ( \\frac { t - r } t , \\sqrt { \\frac { t _ n } t } ( z - x ) \\right ) \\mathbf { 1 } _ { [ 0 , t ] } ( r ) , \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} { \\bf \\Gamma } ^ { ( t + 1 ) } _ 2 = \\sigma ^ { 2 } { \\bf I } _ 2 + { \\bf G } _ { 1 3 } { \\bf Q } ^ { ( t ) } _ 3 { \\bf G } _ { 1 3 } ^ { T } , \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{gather*} r _ 2 = - \\frac { t } { 2 } \\qquad r _ 1 = \\frac { q _ 2 } { 2 } + \\frac { 1 } { 2 } \\forall \\ , q _ 2 , \\ , \\alpha , \\ , u . \\end{gather*}"}
-{"id": "7689.png", "formula": "\\begin{align*} 2 \\mu _ k ^ \\frac 1 2 x _ { \\mu _ k } \\int _ 0 ^ 1 \\left ( 1 - \\frac { Q ( x _ { \\mu _ k } t ) } { Q ( x _ { \\mu _ k } ) } \\right ) ^ \\frac 1 2 \\ , \\dd t = \\pi k ( 1 + o ( 1 ) ) , k \\to \\infty . \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} p _ s ^ \\star \\coloneqq \\begin{cases} \\dfrac { N p } { N - s p } & s p < N , \\\\ + \\infty & s p \\ge N . \\\\ \\end{cases} \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} \\alpha = - d h + R ( h ) \\eta \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} ( \\mathbb { M } ( \\lambda \\vec { 1 } ) ) ^ { - 1 } = \\frac { \\mathrm { I } } { a ( \\lambda ) } - \\frac { b ( \\lambda ) } { a ( \\lambda ) \\ , ( a ( \\lambda ) + b ( \\lambda ) \\ , \\nu ) } \\ , U \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} A ' = \\left ( \\bigoplus _ { j } \\left ( ( \\mathbb { F } _ { ( n _ j ) } \\oplus \\mathbb { F } _ { ( n _ j + 1 ) } \\right ) \\right ) \\oplus \\left ( \\bigoplus _ { k } \\left ( ( \\mathbb { F } _ { ( n _ k ) } \\oplus \\mathbb { F } _ { ( n _ k + 2 ) } \\right ) \\right ) \\end{align*}"}
-{"id": "633.png", "formula": "\\begin{align*} \\left [ \\eta , \\xi _ n - \\hat { \\xi } _ n \\right ] _ { \\alpha } = 0 , \\eta \\in H ^ - ( \\xi ) . \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} \\left ( \\zeta ( z ) + x \\right ) ^ 2 = N \\phi _ { A } ( z ) - N \\phi _ { A } \\left ( 1 / a \\right ) \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} \\nu _ z ( F _ { 2 j } ( \\gamma ) ) \\geq - 2 j + n ( \\mu , 2 j ) = - ( 2 j - n ( \\mu , 2 j ) ) \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} y _ i ^ m = h _ i ^ m { x ^ m } + { z _ i ^ m } , \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} E _ { - 1 } = & \\ - \\frac 1 \\pi \\int _ \\R \\frac { \\xi ^ 2 } { 1 + \\xi ^ 2 } \\real \\ln T ( \\xi / 2 ) d \\xi + \\sum \\Big [ ( \\ln ( 1 + 2 \\kappa _ j ) - \\ln | 1 - 2 \\kappa _ j | ) - 4 \\kappa _ j \\Big ] \\\\ = & \\ , \\real \\ln T ( i / 2 ) - \\int u \\ , d x . \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ \\infty \\frac { 1 } { v ( t ) } = \\sum _ { l = 0 } ^ \\infty \\frac { t _ { l + 1 } - t _ l } { v ( t _ l ) } = \\infty \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} & 1 + 4 C ( ( i + \\ell - 1 ) ( j + 1 ) ) ^ { - 2 } + \\left ( 1 + 4 C ( ( i + \\ell - 1 ) ( j + 1 ) ) ^ { - 2 } \\right ) j ^ { - 2 } 8 C \\sum _ { k = 1 } ^ { \\ell - 1 } ( i + k - 1 ) ^ { - 2 } \\\\ & \\leq 1 + j ^ { - 2 } 8 C \\sum _ { k = 1 } ^ { \\ell } ( i + k - 1 ) ^ { - 2 } . \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} \\begin{aligned} \\mathrm { R } _ { \\Phi } ( h ) : & = - \\sum _ { k \\in S } ( h , \\beta _ 1 \\partial _ x e ^ { \\mathrm { i } k x } ) _ { L ^ 2 ( \\mathbb { T } ) } e ^ { \\mathrm { i } k x } + \\sum _ { k \\in S } ( h , ( \\beta _ 1 ) _ x e ^ { \\mathrm { i } k x } ) _ { L ^ 2 ( \\mathbb { T } ) } e ^ { \\mathrm { i } k x } + \\sum _ { k \\in S } ( h , \\beta _ 1 \\partial _ x e ^ { \\mathrm { i } k x } ) _ { L ^ 2 ( \\mathbb { T } ) } e ^ { \\mathrm { i } k x } \\\\ [ 2 m m ] & = \\Pi _ S [ ( \\beta _ 1 ) _ x \\ , h ] . \\end{aligned} \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} & \\sigma ( H ^ { \\epsilon , 0 } ) \\cap \\left [ a _ 0 , b _ N \\right ] \\subset \\bigcup _ { k = 0 } ^ N [ a _ k , b _ k ] \\ , , \\\\ & b _ k - a _ k = \\mathcal O ( \\epsilon ^ \\infty ) , \\ ; 0 \\leq k \\leq N , { \\rm a n d } a _ { k + 1 } - b _ k \\geq \\epsilon / C , \\ ; 0 \\leq k \\leq N - 1 \\ , . \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} \\psi ( \\rho , \\nabla \\rho , \\mu ) = - \\mu \\ , g ( \\rho ) + W ( \\rho ) + \\frac 1 2 \\ , | \\nabla \\rho | ^ 2 , \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} & h _ { 2 / 3 , 3 / 4 } ( w _ 1 , w _ 2 ) \\\\ & \\geq \\min \\left \\{ \\frac { \\epsilon } { 4 8 } \\log \\left ( 1 + \\frac { 3 P ' } { 5 } \\right ) , \\right . \\left . \\frac { \\epsilon } { 8 } \\log \\left ( 1 + \\frac { P ' } { 4 } \\right ) \\right \\} \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} \\int _ { \\Omega ^ { 2 } } \\dfrac { L ( w _ 1 , u ) ( x , y ) } { | x - y | ^ { N + s p } } \\ , d x d y = 0 . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} \\lambda ^ { - l } c \\theta ^ 2 ( 1 - 1 6 \\lambda ) ^ { - 1 } & = \\theta ^ { - \\frac { 3 l } { k } } c \\theta ^ 2 ( 1 - 1 6 \\lambda ) ^ { - 1 } \\\\ & \\leq \\frac { c \\theta ^ 2 } { 1 6 c \\theta } ( 1 - 1 6 \\lambda ) ^ { - 1 } \\\\ & \\leq \\frac { \\theta } { 8 } . \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{gather*} \\det ( \\lambda - L ) \\equiv \\left ( \\lambda - \\frac { x ^ 2 - t } { 2 } \\right ) ^ 2 - \\left ( \\frac { x ^ 4 } { 4 } + r _ 2 x ^ 2 + r _ 1 x + r _ 0 \\right ) = 0 , \\end{gather*}"}
-{"id": "5492.png", "formula": "\\begin{align*} A : = \\left \\{ l \\in N : \\left ( \\tilde { \\theta } _ t ^ l - \\frac { 1 } { n } \\right ) \\geq 0 \\right \\} . \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta z = \\lambda ( \\sqrt { z } + \\tan ( z ) ) , & x \\in \\Omega = \\{ ( x _ 1 , x _ 2 ) \\in \\R ^ 2 : x _ 1 ^ 2 + x _ 2 ^ 2 < 1 \\} , \\\\ z ( x ) = 0 , & x \\in \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} \\det ( 1 - \\mathrm { F r } _ \\lambda X \\mid \\mathcal { V } ) = 1 - T _ \\lambda X + S _ \\lambda X ^ 2 . \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} u _ 1 ( t ) = \\displaystyle { \\sum _ { k = 1 } ^ 8 } c _ k e ^ { \\Lambda _ k t } , t \\geq 0 , \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} \\Phi ( z ) = \\sum _ { \\pm } \\sum ^ { \\pm } w _ 0 \\left ( \\frac { 4 \\pi \\sqrt { 2 n b } } { z C } \\right ) W _ { \\pm } \\left ( \\frac { m _ 1 z ^ 2 } { 3 2 \\pi ^ 2 b } , \\frac { m _ 1 H z ^ 2 } { 3 2 \\pi ^ 2 n b } \\right ) V ^ { \\pm } \\left ( \\frac { m _ 2 z ^ 2 } { 3 2 \\pi ^ 2 b } , \\frac { m _ 2 H ' z ^ 2 } { 3 2 \\pi ^ 2 n b } \\right ) e ^ { \\pm i \\sqrt { \\frac { m _ 1 } { 2 b } } z } e ^ { \\pm i \\sqrt { \\frac { m _ 2 } { 2 b } } z } . \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} \\frac { \\Gamma ( 1 - \\alpha _ j ) } { \\Gamma ( \\beta _ j ) } = ( - 1 ) ^ { \\alpha _ j - \\beta _ j } \\cdot \\frac { \\Gamma ( 1 - \\beta _ j ) } { \\Gamma ( \\alpha _ j ) } . \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} i ( c ^ { m _ { l _ { 2 } } + 2 L _ { i } } ) = 2 n \\bar { q } l _ { 2 } + 2 [ Q _ { 0 } ] - 2 i '' , \\ \\forall 1 \\leq i \\leq \\beta . \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = s } \\Phi : & = \\ ; \\big ( D \\Phi \\big ) \\big | _ { s } \\ , \\frac { \\partial } { \\partial t } \\ ; , \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} \\limsup _ { l \\to \\infty } \\limsup _ { k \\to \\infty } \\| \\nabla ( \\zeta _ l \\tilde u _ k ) \\| _ { L ^ p } & = \\limsup _ { l \\to \\infty } \\limsup _ { k \\to \\infty } \\| \\nabla \\zeta _ l \\otimes \\tilde u _ k + \\zeta _ l \\nabla \\tilde u _ k \\| _ { L ^ p } \\\\ & \\leq \\limsup _ { k \\to \\infty } \\| \\nabla \\tilde u _ k \\| _ { L ^ p } < \\infty \\ , . \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\left ( U , V , Z \\right ) & = \\left ( \\pm 1 , \\pm 1 , \\pm 1 \\right ) , \\mu = 1 \\\\ \\left ( U , V , Z \\right ) & = \\left ( \\pm i , \\pm i , \\pm i \\right ) , \\mu = - 1 . \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} X _ x ( \\sqrt { 2 r } \\cos \\theta _ 0 , \\sqrt { 2 r } \\sin \\theta _ 0 , z ) \\sin \\theta _ 0 & - X _ y ( \\sqrt { 2 r } \\cos \\theta _ 0 , \\sqrt { 2 r } \\sin \\theta _ 0 , z ) \\cos \\theta _ 0 \\\\ & = \\sqrt { 2 r } \\left ( \\delta ^ { - 1 } \\alpha + c z - \\delta ^ p \\tilde G ( r , z ) \\right ) . \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} c ( \\lambda ) : = \\int 1 _ { | x | \\leq \\delta } \\frac { 1 } { g ( x ) ^ { \\lambda } } d \\ell _ E ( x ) < \\infty . \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} | \\widehat { \\Phi } ( t ) | ^ 2 \\leq 4 \\big | e ^ { - i \\frac { L t } { 2 } } \\widehat { \\Phi ^ + } ( t ) \\big | ^ 2 = 4 \\big | \\widehat { \\Phi ^ + } ( t ) \\big | ^ 2 . \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} \\varphi _ { \\alpha } ^ { p } ( t , \\Phi ( s ) u ) \\frac { ( t - s ) ^ { n + 1 } } { ( n + 1 ) ! N ^ { n + 1 } } & = \\frac { 1 } { N } \\int _ { s } ^ { t } \\varphi _ { \\alpha } ^ { p } ( t , \\Phi ( s ) u ) \\frac { ( t - \\xi ) ^ { n } } { n ! N ^ { n } } d \\xi \\leq \\int _ { s } ^ { t } \\varphi _ { \\alpha } ^ { p } ( \\xi , \\Phi ( s ) u ) d \\xi \\\\ & \\leq \\int _ { s } ^ { \\infty } \\varphi _ { \\alpha } ^ { p } ( \\xi , \\Phi ( s ) u ) d \\xi \\leq K \\varphi _ { \\alpha } ^ { p } ( s , u ) \\leq N \\varphi _ { \\alpha } ^ { p } ( s , u ) , \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} \\mathsf { D } ( P | | Q ) = \\int _ \\mathcal { X } \\mathsf { d } P \\log \\left ( \\frac { \\mathsf { d } P } { \\mathsf { d } Q } \\right ) , \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\pi \\in \\mathcal { D } _ o , \\\\ | \\pi | = N , \\\\ \\nu _ o ( \\pi ) = k } } ( - 1 ) ^ { \\nu _ e ( \\pi ) } = \\chi ( N = k ^ 2 ) . \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} Q _ 2 = \\frac { 1 / 2 } { r } \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\mathrm { S h } _ { \\ell } ( X ) : = \\mathrm { I n d } ( \\mathrm { S h } _ { \\ell } ^ c ( X ) ) \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{gather*} Y ^ { ( 3 ) } ( x , t ) : = \\Psi _ { 0 } ^ { ( 3 ) } ( x , t ) e ^ { - \\left ( \\frac { x ^ 3 } { 6 } - \\frac { x t } { 2 } \\right ) \\sigma _ 3 } . \\end{gather*}"}
-{"id": "6991.png", "formula": "\\begin{align*} I _ h \\left ( \\frac { u } { v } \\right ) = \\mathop { \\sum \\sum } _ { u m - v n = h } \\Psi \\left ( T \\log { \\frac { u m } { v n } } \\right ) \\frac { \\lambda ( m ) \\lambda ( n ) } { \\sqrt { m n } } h ( m ) h ( n ) \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} W _ i ( t ) = \\left [ t _ { - i , N _ i ( t ) + 1 + Q _ i ( 0 ) - R _ i ( t ) - Q _ { - i } ( 0 ) + R _ { - i } ( t ) } - t \\right ] ^ + . \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} \\mathcal { I } ( \\beta , \\mu ) = \\lim _ { \\Lambda } \\frac { 1 } { V } \\sum _ { k \\in \\Lambda ^ { * } } \\omega _ { \\beta , \\mu , \\Lambda } ^ { 0 } ( N _ k ) = \\frac { 1 } { ( 2 \\pi ) ^ 3 } \\int _ { \\mathbb { R } ^ 3 } d ^ 3 k \\ \\frac { 1 } { e ^ { \\beta \\left ( \\varepsilon _ { k } - \\mu \\right ) } - 1 } \\ , \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} \\left ( { p _ 1 ^ { \\mathrm { O M A } } , p _ 2 ^ { \\mathrm { O M A } } } \\right ) = \\left ( \\frac { { { 2 ^ { 2 { { \\widetilde R } _ 1 } } } - 1 } } { { { 2 \\beta _ 1 } } } , \\frac { { { 2 ^ { 2 { { \\widetilde R } _ 2 } } } - 1 } } { { { 2 \\beta _ 2 } } } \\right ) , \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} \\frac { r ^ { 2 p _ k } ( x _ { p _ k } ) } { r ^ { 2 p _ { k - 1 } } ( x _ { p _ k } ) } = r ( x _ { p _ k } ) ^ { 2 e ^ { 1 - k } - 2 e ^ { - k } } < ( 2 C ) ^ { 2 e ^ { 1 - k } } e ^ { 2 k e ^ { 1 - k } } = : A _ k . \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} g _ { \\ell , j } ( z ) \\mid T _ { Q , \\kappa , \\chi } = \\sum _ { n = 1 } ^ { \\infty } \\left ( b ( Q n ) + \\chi ( Q ) Q ^ { \\kappa - 1 } b ( n / Q ) \\right ) q ^ n \\equiv 0 \\pmod { \\ell ^ j } . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} \\| \\psi _ s - \\underline { \\psi _ s } \\| _ { L ^ { \\frac { 4 } { 3 } } ( M ) } \\geq \\left ( \\int _ { \\{ \\psi _ s = 0 \\} } | \\psi _ s - \\underline { \\psi _ s } | ^ \\frac { 4 } { 3 } \\omega ^ 2 \\right ) ^ { \\frac { 3 } { 4 } } = \\gamma ( s ) ^ { \\frac { 3 } { 4 } } \\underline { \\psi _ s } . \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} \\pi ( z ) = \\P ( [ X _ 1 ] _ b = z ) , \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} a ( x , T , z ) = \\left \\{ \\begin{array} { l l } 0 & x , z \\in \\C ( J ) \\\\ 0 & x \\in \\C ( J ) , z \\in \\C ( J T ) \\\\ q & x \\in \\C ( J ) , z \\in \\C ( J T ^ { - 1 } ) \\\\ \\frac { q ( q - 3 ) } { 4 } & x \\in \\C ( J T ) , z \\in \\C ( J T ) \\\\ \\frac { q ( q - 3 ) } { 4 } & x \\in \\C ( J T ) , z \\in \\C ( J T ^ { - 1 } ) \\\\ \\frac { q ( q + 1 ) } { 4 } & x \\in \\C ( J T ^ { - 1 } ) , z \\in \\C ( J T ) \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} H ( u ) = \\sum _ { j _ 1 , \\dots , j _ n \\in \\mathbb { Z } \\setminus \\{ 0 \\} } H _ { j _ 1 , \\dots , j _ n } u _ { j _ 1 } \\dots u _ { j _ n } \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} p ^ { - 1 } ( \\Gamma g a _ s ) \\cap \\Omega & = \\{ \\Gamma c _ 1 g a _ s u _ { p _ 1 } , \\ldots , \\Gamma c _ n g a _ { s } u _ { p _ n } \\} \\\\ & = \\{ \\Gamma c _ 1 g u _ { p _ 1 } a _ s u _ { b _ 1 } , \\ldots , \\Gamma c _ n g u _ { p _ n } a _ s u _ { b _ n } \\} , \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } - \\kappa u _ { t t } - u _ { x x } + u _ t & = & u ( r _ u - \\gamma _ u ( u + v ) ) + \\mu v - \\mu u \\\\ - \\kappa v _ { t t } - v _ { x x } + v _ t & = & v ( r _ v - \\gamma _ v ( u + v ) ) + \\mu u - \\mu v , \\end{array} \\right . \\end{align*}"}
-{"id": "1794.png", "formula": "\\begin{align*} \\int _ { t = 2 0 } ^ \\infty \\frac { ( \\log t ) ^ { x - 1 } } { ( t \\log \\log t ) ^ 2 } d t \\leq \\int _ { u = 0 } ^ \\infty u ^ { x - 1 } \\exp ( - u ) d u = ( x - 1 ) ! \\end{align*}"}
-{"id": "2295.png", "formula": "\\begin{gather*} \\frac { d B } { d x } - \\frac { d L } { d t } = [ L , B ] , \\end{gather*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\theta } _ j = j \\ , \\ , \\dfrac { \\partial } { \\partial I _ j } \\mathcal { H } _ { \\leq 5 } ( \\theta , I , 0 ) , j \\in S ^ + , \\\\ [ 3 m m ] \\dot { I } _ j = - \\dfrac { \\partial } { \\partial \\theta _ j } \\mathcal { H } _ { \\leq 5 } ( \\theta , I , 0 ) , j \\in S ^ + . \\end{cases} \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} \\Lambda _ 0 & = B ^ T P + D ^ T P C + G ^ T P F + S , \\\\ \\Lambda _ 1 & = ( B ^ T + \\bar { B } ^ T ) \\Pi + ( D ^ T + \\bar { D } ^ T ) P ( C + \\bar { C } ) + ( G ^ T + \\bar { G } ^ T ) \\Pi ( F + \\bar { F } ) + S + \\bar S , \\\\ \\Sigma _ 0 & = D ^ T P D + R , \\\\ \\Sigma _ 1 & = ( D ^ T + \\bar { D } ^ T ) P ( D + \\bar { D } ) + ( G ^ T + \\bar { G } ^ T ) \\Pi ( G + \\bar { G } ) + ( R + \\bar R ) . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} \\theta \\widetilde { c } + \\theta ' \\widetilde { c ' } + \\theta _ { \\infty } \\widetilde { r } & = \\theta c + \\theta ' \\widetilde { c ' } + \\theta _ { \\infty } r - ( \\theta c + \\theta ' c ' + \\theta _ { \\infty } r ) \\\\ & = \\theta ' ( \\widetilde { c ' } - c ' ) < 0 \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} \\begin{aligned} X _ 0 M _ 0 ^ 2 X _ 0 ^ t + Y _ 0 M _ 0 ^ 2 Y _ 0 ^ t & = I ; \\\\ X _ 0 M _ 0 ^ 2 Y _ 0 ^ t - Y _ 0 M _ 0 ^ 2 X _ 0 ^ t & = 0 . \\end{aligned} \\end{align*}"}
-{"id": "7617.png", "formula": "\\begin{align*} \\mu ( i ) = \\frac { \\rho ^ i } { 1 + \\rho + \\ldots + \\rho ^ n } \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} \\limsup _ { | i | \\ge \\gamma t , t \\to \\infty } u _ i ( t ; 0 , u ^ 0 ) = 0 . \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} - \\frac { 1 } { Z _ { \\alpha , \\beta } ( v ) } = x + \\sqrt { - 1 } y . \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} F = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} , w = \\begin{bmatrix} 0 \\\\ 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} y ( t ) = \\Phi _ q ( y , z _ 0 ) ( t ) + z _ c , t \\geq 0 . \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\rho ( \\alpha , A ^ { ( 1 ) } ) = \\rho ( \\alpha , A ^ { ( 2 ) } ) - k \\alpha / 2 . \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} \\bold { D } _ Q ^ { \\ast } ( F ) = \\sum _ { \\beta , J _ 1 , J _ 2 } ( - D ) _ { J _ 1 } S _ { - J _ 2 } \\left ( \\frac { \\partial Q ^ { \\beta } } { \\partial u _ { J _ 1 ; J _ 2 } } F _ { \\beta } \\right ) . \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} \\Phi _ 2 : = \\exp ( \\varepsilon ^ 2 A _ 2 ) = \\mathrm { I } _ { H _ S ^ { \\perp } } + \\varepsilon ^ 2 A _ 2 + \\varepsilon ^ 4 \\hat { A } _ 2 , \\hat { A } _ 2 : = \\sum _ { k \\geq 2 } \\frac { \\varepsilon ^ { 2 ( k - 2 ) } } { k ! } \\ , A _ 2 ^ k \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} \\| f \\| _ { 1 + k _ { \\gamma , } } ^ 2 = \\left | \\int ^ 1 _ 0 f ( y ) \\ , { \\rm d } y \\right | ^ 2 + \\frac { 1 } { \\gamma } \\left ( \\sum _ { \\nu = 1 } ^ { r - 1 } \\left | \\int _ 0 ^ 1 f ^ { ( \\nu ) } ( y ) \\ , { \\rm d } y \\right | ^ 2 + \\int ^ 1 _ 0 | f ^ { ( r ) } ( y ) | ^ 2 \\ , { \\rm d } y \\right ) , \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} X = \\frac { \\partial } { \\partial y ^ 1 _ k } + \\sum _ j g ^ k _ j \\frac { \\partial } { \\partial y ^ j _ k } + \\sum _ \\rho \\gamma ^ k _ \\rho \\frac { \\partial } { \\partial \\eta ^ j _ k } , \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} a ( r , s + t ) | _ { ( [ r , r + s ] \\times [ 0 , r ] ) \\cup ( [ 0 , r ] \\times [ r , r + s ] ) } & = a ( r , s ) | _ { ( [ r , r + s ] \\times [ 0 , r ] ) \\cup ( [ 0 , r ] \\times [ r , r + s ] ) } \\\\ a ( r + s , t ) | _ { [ r + s , r + s + t ] \\times [ 0 , r ] } & = a ( r , s + t ) | _ { [ r + s , r + s + t ] \\times [ 0 , r ] } , \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n f \\left ( \\mathcal { A } _ i \\right ) \\geq \\sum _ { j = 1 } ^ n f \\left ( \\mathcal { E } ^ { ( n ) } _ j \\right ) , \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} \\Lambda = x _ 1 ^ m \\frac { \\partial } { \\partial x _ 1 } \\wedge \\ldots \\wedge \\frac { \\partial } { \\partial x _ n } \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} 0 = A _ 0 \\subsetneq A _ 1 \\subsetneq \\ldots \\subsetneq A _ l \\subsetneq \\ldots \\subset M \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} R _ 1 \\leq \\frac { 1 } { 2 } \\log \\frac { { C ^ { 2 } _ { y _ { 1 2 } } } - { | \\tilde { C } _ { y _ { 1 2 } } | ^ { 2 } } } { { C ^ { 2 } _ { s _ 1 } } - { | \\tilde { C } _ { s _ 1 } | ^ { 2 } } } = L _ 1 , \\\\ R _ 2 \\leq \\frac { 1 } { 2 } \\log \\frac { { C ^ { 2 } _ { y _ { 1 1 } } } - { | \\tilde { C } _ { y _ { 1 1 } } | ^ { 2 } } } { { C ^ { 2 } _ { s _ 1 } } - { | \\tilde { C } _ { s _ 1 } | ^ { 2 } } } = L _ 2 , \\\\ R _ 1 + R _ 2 \\leq \\frac { 1 } { 2 } \\log \\frac { { C ^ { 2 } _ { y _ 1 } } - { | \\tilde { C } _ { y _ 1 } | ^ { 2 } } } { { C ^ { 2 } _ { s _ 1 } } - { | \\tilde { C } _ { s _ 1 } | ^ { 2 } } } = L _ 4 , \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} F _ i ( \\phi _ i ) & = \\int _ M ( \\phi _ i - \\phi _ 0 ) d \\mu _ i \\\\ & = \\int _ M ( \\bar { \\phi } - \\phi _ 0 ) d \\mu + \\int _ M ( \\phi _ i - \\bar { \\phi } ) d \\mu _ i + \\int _ M ( \\bar { \\phi } - \\phi _ 0 ) ( d \\mu _ i - d \\mu ) \\\\ & = F ( \\bar { \\phi } ) + \\int _ M ( \\phi _ i - \\bar { \\phi } ) d \\mu _ i + \\int _ M ( \\bar { \\phi } - \\phi _ 0 ) ( d \\mu _ i - d \\mu ) . \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} u ^ n ( t , \\omega ) & = u ^ n ( 0 , \\omega ) + \\int _ { 0 } ^ { t } \\mathcal { D } _ { s } u ^ n ( s , \\omega ) d s + \\int _ { 0 } ^ { t } \\mathcal { D } _ { x } u ^ n ( s , \\omega ) d B _ { s } + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\mathcal { D } _ { x } ^ { 2 } u ^ n ( s , \\omega ) d \\langle B \\rangle _ { s } \\\\ & = u ^ n ( 0 , \\omega ) + \\int _ { 0 } ^ { t } \\mathcal { A } _ { G } u ^ n ( s , \\omega ) d s + \\int _ { 0 } ^ { t } \\mathcal { D } _ { x } u ^ n ( s , \\omega ) d B _ { s } + K ^ n _ { t } , \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} \\eta = ( u _ 0 , u _ 1 , \\cdots , u _ { n - 2 } , - v _ 0 , - v _ 1 , \\cdots , - v _ { n - 1 } ) ^ T , \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} M _ k ( x ; \\mathbf { a } ) = \\sum _ { P ( \\mathbf { n } ) \\leq x } c ( \\mathbf { k } ( \\mathbf { a } ) , \\mathbf { n } ) = \\ ( \\frac { 1 } { 2 \\pi i } \\ ) ^ l \\oint _ { | z _ l | = r _ l } \\cdots \\oint _ { | z _ 1 | = r _ 1 } \\ ( \\sum _ { P ( \\mathbf { n } ) \\leq x } a ( \\mathbf { n } ; \\mathbf { z } ) \\ ) \\frac { d z _ 1 } { z _ 1 ^ { k _ 1 + 1 } } \\cdots \\frac { d z _ l } { z _ l ^ { k _ l + 1 } } . \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} \\phi _ 2 ( c , r , p ) & = \\left ( c ^ 2 + 4 \\right ) ^ 2 + 2 r ( 4 - c ^ 2 ) ( 4 + c ^ 2 ) ( 2 p + 3 r - 6 p ^ 2 r ) \\\\ & + r ^ 2 ( 4 - c ^ 2 ) ^ 2 ( 4 + 9 r ^ 2 - 1 2 r p ) . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\left \\lfloor { R } \\right \\rfloor } g ( i ) \\leq \\int _ { 0 } ^ { R } g ( i ) \\ , d i - \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ { \\left \\lfloor { R } \\right \\rfloor } ( g ( i - 1 ) - g ( i ) ) - \\frac { 1 } { 2 } \\left ( R - \\left \\lfloor { R } \\right \\rfloor \\right ) g \\left ( \\left \\lfloor { R } \\right \\rfloor \\right ) . \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} \\tilde { F } ^ { \\left ( 2 \\right ) } _ { n } = \\sum _ { k = 0 } ^ { n } \\binom { n - k } { k } _ b \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\mu ^ p _ { \\Delta } \\circ \\mu ^ p _ { \\Lambda } = \\mu ^ p _ { \\Delta } \\Lambda \\subset \\Delta . \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} r : = \\phi _ x - i \\phi _ y = \\frac { q _ \\alpha } { 1 + w _ \\alpha } . \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} S _ { \\omega } ( v _ n ) \\to S _ { \\omega } ( \\phi _ { \\omega } ) = d ( \\omega ) . \\end{align*}"}
-{"id": "1290.png", "formula": "\\begin{align*} \\begin{array} { r l } x \\in \\partial \\mathcal { E } _ 1 \\cap \\partial \\mathcal { E } _ 2 & \\leftrightarrow ~ x ^ T Q _ 1 x = 1 , x ^ T Q _ 2 x = 1 \\\\ x \\in \\partial \\mathcal { E } _ 1 \\cap \\mathcal { E } _ 2 ^ o & \\leftrightarrow ~ x ^ T Q _ 1 x = 1 , x ^ T Q _ 2 x < 1 \\\\ x \\in \\mathcal { E } _ 1 ^ o \\cap \\partial \\mathcal { E } _ 2 & \\leftrightarrow ~ x ^ T Q _ 1 x < 1 , x ^ T Q _ 2 x = 1 . \\end{array} \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} \\mathcal { L } = \\mathcal { U } \\Sigma \\mathcal { V } ^ T , \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} \\dd _ { _ a T _ r } ( s ) = \\zeta _ { \\Z ^ r } ( s - a ) = \\prod _ { i = 0 } ^ { r - 1 } \\zeta ( s - a - i ) . \\end{align*}"}
-{"id": "4068.png", "formula": "\\begin{align*} \\varphi _ { \\frac { 1 } { 2 } } ( r ) = \\varphi _ \\frac { 1 } { 2 } ^ 2 ( r / 2 ) , \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\varphi _ { \\alpha } ^ { p } ( t , \\Phi ( s ) u ) & = \\int _ { t - 1 } ^ { t } \\varphi _ { \\alpha } ^ { p } ( t , \\Phi ( s ) u ) d \\xi \\leq \\int _ { t - 1 } ^ { t } e ^ { p \\alpha ( t - \\xi ) } \\varphi _ { \\alpha } ^ { p } ( \\xi , \\Phi ( s ) u ) d \\xi \\\\ & \\leq e ^ { p \\alpha } \\int _ { s } ^ { \\infty } \\varphi _ { \\alpha } ^ { p } ( \\xi , \\Phi ( s ) u ) d \\xi \\leq e ^ { p \\alpha } K \\varphi _ { \\alpha } ^ { p } ( s , u ) . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\frac { f ( m ) - p } { f ( m ) - q } = 1 + \\frac { q - p } { f ( m ) - q } . \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} \\beta _ x = g ^ { - 1 } ( b _ 3 ^ { - 1 } \\ , a _ 1 - 1 ) - g ^ { - 1 } ( 0 ) \\mbox { a n d } b _ 3 ^ { - 1 } \\ , a _ 1 - 1 = a _ 1 \\ , ( \\phi ( b _ 3 - 1 ) - \\phi ( 0 ) ) + ( a _ 1 - 1 ) . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} A = B + ( A - B ) \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{align*} c _ j = \\int e ^ { - i j \\theta } d \\mu ( \\theta ) . \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\Gamma ( \\ell _ 1 , \\ell _ 2 ; t _ * ) ( v ) = \\frac { 1 } { 2 } \\Big ( \\tilde { \\Omega } ^ { ( 1 ) } _ { \\mathcal { P } } ( t _ * ) \\tilde { v } , \\tilde { v } \\Big ) _ { \\mathbb { C } ^ n } . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} \\hat \\tau = x ^ 4 + y ^ 2 + O _ { x , y } ( 1 ) \\varepsilon - b _ 2 x ^ 2 + O ( 1 ) x + O ( 1 ) y + O ( 1 ) \\varepsilon ^ { - 2 } . \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} X ^ { ( m ) } _ 1 ( 0 ) : = 0 \\leq X ^ { ( m ) } _ 2 ( 0 ) \\leq X ^ { ( m ) } _ 3 ( 0 ) \\leq \\ldots ; \\ \\ \\mbox { a n d } \\ \\ ( Z ^ { ( m ) } _ k ( 0 ) ) _ { k = 1 } ^ { m ^ 2 - 1 } \\sim \\pi ^ { ( m ) } _ a . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} W _ n ( R ) = W ( R ) / V ^ n W ( R ) , \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} c _ h = \\frac { 1 1 } { 8 } f ( h ) \\prod _ { p } \\left ( 1 - \\frac { 1 } { p } \\right ) ^ 2 \\left ( 1 + \\frac { 2 } { p } \\right ) , \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} b ( x , z ) = b ( y , z ) - b ( y , x ) . \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} x _ { n + 1 } = a _ { n } x _ { n } , \\textnormal { w i t h } a _ { n } = \\left \\{ \\begin{array} { r c l } a & \\textnormal { i f } & n \\leq - 1 \\\\ b & \\textnormal { i f } & n \\geq 0 \\end{array} \\right . \\textnormal { a n d } 0 < a \\leq b . \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} \\theta _ { 0 } \\left ( 2 n + 1 \\right ) = \\theta _ { 0 } \\left ( n \\right ) , \\thinspace \\thinspace \\thinspace \\theta _ { 0 } \\left ( 2 n \\right ) = \\theta _ { 0 } \\left ( n \\right ) + \\theta _ { 0 } \\left ( n + 1 \\right ) \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} U _ 1 & = E ( s ^ 2 ) & U _ 2 & = \\min \\{ t > U _ 1 : g _ t = 1 _ G \\} \\\\ U _ 3 & = \\min \\{ t > U _ 2 : | g _ t | > r \\} & U & = U _ 3 - U _ 2 + U _ 1 . \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} ( M ^ { - n } \\cdot \\operatorname { R N } _ { n , M , Q } ) \\leq d \\left ( 1 + \\tfrac { M Q } { M - 1 } \\right ) ( 1 + ( 1 + \\tfrac { 1 } { M } ) Q ) ^ { n - 1 } \\leq \\tfrac { d ( M + ( M + 1 ) Q ) ^ { n } } { M ^ { n - 1 } ( M - 1 ) } . \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} \\rho = \\frac { 1 } { V } \\sum _ { k \\in \\Lambda ^ { * } } \\omega _ { \\beta , \\mu , \\Lambda } ^ { 0 } ( N _ k ) = \\frac { 1 } { V } \\sum _ { k \\in \\Lambda ^ { \\ast } } \\frac { 1 } { e ^ { \\beta \\left ( \\varepsilon _ { k } - \\mu \\right ) } - 1 } \\ , \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} Z ( z ) = e ^ { \\frac { - N \\ell } { 2 } \\sigma _ 3 } Y ( z ) \\ , e ^ { - N g ( z ) \\sigma _ 3 } e ^ { \\frac { N \\ell } { 2 } \\sigma _ 3 } \\begin{bmatrix} 1 & 0 \\\\ \\displaystyle \\star \\ , \\Big ( \\frac { z } { z - a } \\Big ) ^ { c } e ^ { N \\phi ( z ) } & 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} \\min _ { \\mu \\in \\Pr } P ( \\mu ) & = \\max _ { \\substack { \\eta \\in X \\\\ \\eta ' ( 0 ) = - h ^ { 2 } } } \\frac { 1 } { 2 } \\tilde { D } ( \\eta ) . \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} H _ 0 = & \\int | u | ^ 2 d x , \\\\ H _ 1 = & \\frac 1 i \\int u \\partial _ x \\bar u d x , \\\\ H _ 2 = & \\int | u _ x | ^ 2 + | u | ^ 4 d x , \\\\ H _ 3 = & i \\int \\ u _ x \\partial _ x \\overline { u } _ x + 3 | u | ^ 2 u \\overline { u } _ x d x , \\\\ H _ 4 = & \\int | u _ { x x } | ^ 2 + \\frac 3 2 | ( u ^ 2 ) _ x | ^ 2 + | | u | ^ 2 _ x | ^ 2 + 2 | u | ^ 6 d x . \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} \\Omega _ { \\varepsilon } : = \\{ \\alpha ( \\xi ) : \\xi \\in [ 1 , 2 ] ^ { \\nu } \\} , \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} f _ 2 ( B _ n ) = & \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\mathrm { e r f } \\left ( \\sqrt { B _ n } - \\sqrt { P } \\right ) . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} [ A , B ] + I J = 0 \\Leftrightarrow \\left \\{ \\begin{array} { l } [ A ' , B ' ] = 0 \\\\ A ' \\widetilde { B } + \\widetilde { A } B '' - B ' \\widetilde { A } - \\widetilde { B } A '' + \\widetilde { I } J '' = 0 \\end{array} \\right . . \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} \\frac { \\min g } { a } \\le \\ , c ( a ) \\le \\frac { \\max g } { a } h _ { a , c ( a ) } ' \\left ( \\bar z ( a ) \\right ) = a . \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} \\frac { D ^ + _ t \\left ( w ^ n _ j - k \\right ) } { a _ j } = - D ^ - _ x \\left ( w ^ n _ j - k \\right ) \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} \\mu _ 2 ( x ) = & \\frac { - 1 } { 2 } ( ( [ a , b ] + [ a ^ { \\dagger } , b ^ { \\dagger } + i j - j ^ { \\dagger } i ^ { \\dagger } + f g - g ^ { \\dagger } f ^ { \\dagger } ] ) , \\\\ & ( [ a ' , b ' ] + [ a ^ { \\dagger } , b ^ { \\dagger } ] - g f + f ^ { \\dagger } g ^ { \\dagger } ) ) \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\tilde T _ 6 ( z / 2 ) = \\frac { i } { ( 2 \\pi ) ^ { \\frac 1 2 } } \\int _ { \\xi _ 1 + \\xi _ 2 + \\xi _ 3 = 0 } \\frac { 1 } { z ^ 2 ( z + \\xi _ 1 ) ( z + \\xi _ 2 ) } \\left ( \\frac { 1 } { z } + \\frac { 1 } { z - \\xi _ 3 } \\right ) \\hat u ( \\xi _ 1 ) \\hat u ( \\xi _ 2 ) \\hat u ( \\xi _ 3 ) d \\xi _ 1 d \\xi _ 2 , \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{align*} \\int \\ 1 _ G ( \\gamma ( x , y ) ) Y ( x , \\gamma ( x , y ) ) F ( \\dd y ) = \\int \\ 1 _ G ( \\gamma ' ( x , y ) ) F ' ( \\dd y ) . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} L _ \\zeta e ^ { 2 \\zeta } + L _ { - \\xi } e ^ { - 2 \\xi } = - \\frac 1 4 i \\xi \\eta \\zeta ( \\xi - \\eta ) e ^ { 2 \\zeta + 2 \\eta } + \\xi \\eta \\zeta E S ( 1 ) , \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\mathcal { P } ( \\mathcal { X } ) = \\left \\{ P : \\mathcal { X } \\to [ 0 , 1 ] \\Bigg | \\sum _ { x \\in \\mathcal { X } } P ( x ) = 1 ] \\right \\} . \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} { \\cal F } ( \\zeta ) F _ 1 ( \\zeta ) ^ { - 1 } \\cdots F _ k ( \\zeta ) ^ { - 1 } = I + { \\cal O } \\left ( \\frac { 1 } { | \\zeta ^ { k + 1 } | } \\right ) \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} K = \\mathcal { P } ^ * ( R ) , K _ 0 = \\{ T \\in K \\mid 0 _ R \\in T \\} , \\varepsilon = \\{ - 1 _ R \\} \\in K , 0 = \\{ 0 _ R \\} \\in K _ 0 , 1 = \\{ 1 _ R \\} \\in K . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\mathrm { i n d } _ { L ^ 2 } D ^ - = \\sum _ { j = 1 } ^ n \\sum _ { \\sigma = 1 } ^ k \\frac { \\left ( \\lfloor \\lambda _ j / \\ell \\rfloor - v _ { j \\sigma } \\right ) \\left ( \\lfloor \\lambda _ j / \\ell \\rfloor - v _ { j \\sigma } + 1 \\right ) } { 2 } . \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\norm { \\sigma ^ { - } ( t ) } _ { L ^ { 2 } } ^ { 2 } \\leq \\norm { \\sigma ^ { - } ( 0 ) } _ { L ^ { 2 } } ^ { 2 } = 0 \\forall t \\in ( 0 , T ] , \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} \\bar { q } ^ { n , Q } ( t ) = \\sum _ { s \\in [ 0 , t ] } \\bar { q } ^ { n - 1 , Q } ( s ) \\ , q ^ { Q , [ s , T ] } ( t ) , \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} d H _ 0 = c ( F _ 0 \\wedge F _ 0 ) . \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} D _ { i } = x _ i \\frac { \\partial } { \\partial x _ i } + \\beta \\sum _ { j \\neq i , \\sigma ( i ) = \\sigma ( j ) } \\frac { x _ i } { x _ i - x _ j } \\left ( 1 - K _ { i j } \\right ) + \\beta \\sum _ { j , \\sigma ( i ) \\neq \\sigma ( j ) } \\frac { x _ i } { x _ i - x _ j } \\left ( 1 - K _ { i j } \\right ) . \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} N _ 0 ^ { C _ 0 } \\varepsilon ^ { 7 - 2 b } \\gamma ^ { - 2 } = N _ 0 ^ { C _ 0 } \\varepsilon ^ { 1 - 3 a } \\le \\delta _ 0 , \\gamma : = \\varepsilon ^ { 2 + a } , a \\in ( 0 , 1 / 6 ) , \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} D _ j = & \\left ( [ - j \\varepsilon , j \\varepsilon ] \\times \\left [ \\frac { 3 - j } { 3 \\sqrt { d + 2 } } , 1 \\right ] \\right ) / \\sim \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} t ( z ) \\in H ^ * ( Z _ j ; S _ T ) [ \\ ! [ t _ i ^ 1 , \\dotsb , t _ i ^ N ] \\ ! ] [ z , ( z + \\chi ) ^ { - 1 } ] [ \\ ! [ Q ] \\ ! ] , ~ i \\geq 0 , ~ \\chi \\in C ( T ) _ \\Q , ~ N = ( H ^ * ( Z _ j ; \\C ) ) . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} e ( p e r m _ m ( A ) ) = { \\binom n m } ^ 2 a ( m ) \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} \\langle R _ H ( z ) u , v \\rangle & = \\langle R _ { H _ 0 } ( z ) u , v \\rangle - \\langle Z R _ H ( z ) u , Y R _ { H _ 0 } ( \\bar { z } ) v \\rangle \\\\ & = \\langle R _ { H _ 0 } ( z ) u , v \\rangle - \\langle Y R _ { H _ 0 } ( z ) u , Z R _ H ( \\bar { z } ) v \\rangle . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} \\| \\bar { u } _ \\ell \\| _ { C ^ { k + 2 , \\alpha } _ { \\nu _ { \\ell , j } + 2 } ( U _ \\ell ) } + \\sum _ { i = 0 } ^ { I _ { \\ell , j } } | A _ { \\ell , i } | \\leq c c _ j \\left ( \\| \\bar { f } \\| _ { C ^ { k , \\alpha } _ \\nu ( M ) } + | f _ \\ell | \\right ) . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} \\mathcal I _ { M , \\lambda } ( w ) = \\frac { 1 } { 2 } \\widehat M ( \\| w \\| ^ 2 ) - \\frac { \\lambda } { q } \\int _ { \\Omega \\times \\{ 0 \\} } f ( z ) | w ( z , 0 ) | ^ q d z - \\frac { 1 } { 2 ^ * _ \\alpha } \\int _ { \\Omega \\times \\{ 0 \\} } | w ( z , 0 ) | ^ { 2 ^ * _ \\alpha } d z . \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} \\iota _ { X _ t } \\Theta _ t = - \\beta . \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} p _ n ^ + : = - S _ 0 ^ * ( C _ { \\omega _ R } ^ * \\mu _ n ^ + ) \\to - S _ 0 ^ * ( C _ { \\omega _ R } ^ * \\bar \\mu ^ + ) = : \\bar p ^ + \\quad p _ n ^ - : = - S _ 0 ^ * ( C _ { \\omega _ T } ^ * \\mu _ n ^ + ) \\to - S _ 0 ^ * ( C _ { \\omega _ T } ^ * \\bar \\mu ^ + ) = : \\bar p ^ - , \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} \\norm { \\dot z ( t ) } _ { H _ 0 ^ 1 ( \\Omega ) } = \\norm { \\Delta \\dot z ( t ) } _ { H ^ { - 1 } ( \\Omega ) } \\le \\norm { \\dot g ( t ) } _ { H ^ { - 1 } ( \\Omega ) } \\qquad t \\in I \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} \\Lambda _ { E _ { 2 ^ n - 1 } , 2 } = \\sqrt { 2 ^ n - 1 } \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} \\begin{cases} \\zeta _ r ( x ) = 0 \\ \\ \\ & | x | \\leq r , \\\\ 0 \\leq \\zeta _ r ( x ) \\leq 1 \\ \\ \\ & r \\leq | x | \\leq 2 r , \\\\ \\zeta _ r ( x ) = 1 \\ \\ \\ & | x | \\geq 2 r , \\end{cases} \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} X _ n = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n x _ i , Y _ n = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ n y _ i , Z _ n = A _ n + \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } z _ { i } \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} u ( x ) = \\int _ { \\partial B } M _ s ( x , z ) \\ ; d \\mu ( z ) x \\in \\R ^ N \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} l _ n ( G ) = \\sum _ { i = 1 } ^ n \\log \\{ g ( x _ i ; G ) \\} \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} b = \\left \\{ \\begin{array} { c c } \\min \\{ v ( c _ i ) + \\lceil \\log _ p \\left ( \\frac { r } { i } \\right ) \\rceil : ( i , p ) = 1 \\} & \\textrm { i f } r \\geq 1 \\\\ \\min \\{ v ( c _ i ) : ( i , p ) = 1 \\} & \\textrm { i f } r = 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} L ( s , \\pi , \\tau ) = \\epsilon ( s , \\pi , \\tau ) L ( 1 - s , \\pi ^ \\vee , \\tau ^ \\vee ) . \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} & \\| w ^ n _ 1 - w ^ { n - 1 } _ 1 \\| _ { y _ { k + 1 } } + \\| w ^ n _ { - 1 } - w ^ { n - 1 } _ { - 1 } \\| _ { y _ { k + 1 } } \\\\ & \\le c ( 2 \\kappa \\delta ) ^ { n - 1 } \\left ( k \\sum _ { j = 1 } ^ { n - 1 } j ^ k + 1 \\right ) \\le c ( 2 \\kappa \\delta ) ^ { n - 1 } ( k ( n - 1 ) ^ { k + 1 } + 1 ) \\\\ & \\le c ( 2 \\kappa \\delta ) ^ { n - 1 } ( k + 1 ) ( n - 1 ) ^ { k + 1 } . \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 2 ^ { \\ell } - 1 } \\lambda _ { \\ell , j } \\delta _ { B _ { \\ell , j } } \\in \\mathcal { L } ( \\R ^ { n \\times n } ) , \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} x _ { i } ( n + 1 ) = B _ { i } ( n ) x _ { i } ( n ) \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} T ' \\models \\varphi _ i ( x , c ) = \\varphi _ i ( x , c ' ) \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} u ( t ) = b ( t ) \\partial _ x V + y ( t ) , \\langle V , y ( t ) \\rangle _ { L ^ 2 } = \\langle \\partial _ x ^ { - 1 } V , y ( t ) \\rangle _ { L ^ 2 } = 0 , \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} 2 \\ell ( - v ^ 2 ) ^ p + u ^ p = w ^ 2 . \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to 0 } \\frac { { \\sum _ { k = 0 } ^ \\infty | \\log \\omega _ k ^ \\alpha | ^ r \\omega _ k ^ \\alpha } } { | \\log \\alpha | ^ r } = 0 . \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{gather*} \\begin{aligned} L _ { \\mu \\alpha } ^ { \\iota } = 0 & \\mbox { i f } \\mu > \\iota \\mbox { o r } \\alpha < \\iota , \\mbox { a n d } \\\\ L _ { \\alpha \\nu } ^ { \\iota } = 0 & \\mbox { i f } \\alpha > \\iota \\mbox { o r } \\nu < \\iota . \\\\ \\end{aligned} \\end{gather*}"}
-{"id": "8081.png", "formula": "\\begin{align*} U _ { i a , j b } = \\delta _ { i a , j b } - s _ b F ( \\lambda _ { i a } | \\lambda _ { j b } ) , [ U ^ { - 1 } ] _ { i a , j b } = s _ a s _ b U _ { j b , i a } . \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} l _ { 2 j + 1 , 2 N + 1 } ( z ) = \\frac { ( z - e _ { 2 N } ) ( z - e _ { 2 j } ) ( e _ { 2 j } - e _ { 2 N } ) } { 2 e _ { 2 j } ( e _ j - e _ N ) } l _ { j , N } ( z ^ 2 ) . \\\\ \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} H _ { n } ^ { ( 1 ) } = - \\Delta _ x - V _ { n } ^ { ( 1 ) } , \\ \\ \\ H _ n ^ { ( 2 ) } = - \\Delta _ x - V _ { n } ^ { ( 2 ) } . \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} L ( x , n , [ u ] ) - L ( x , n , [ 0 ] ) = \\operatorname { D i v } \\widehat { R } _ 1 + \\operatorname { D i v } ^ { \\vartriangle } \\widehat { R } _ 2 , \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} \\mathbf { b } _ i = \\mathbf { P } ^ { - 1 } _ f \\left ( \\mathbf { P } _ 1 { \\mathbf { d } } _ { i - 1 } + \\mathbf { P } _ 1 \\mathbf { A } ^ { - 1 } \\mathbf { w } _ { i - 1 } - \\mathbf { P } _ 2 \\mathbf { d } _ i \\right ) . \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} F _ { b n + p } ^ { \\left ( b \\right ) } = F _ { p } F _ { n } ^ { \\left ( b \\right ) } \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} R _ - ( z ) = ( H _ - - z ) ^ { - 1 } p + ( H _ - - z ) ^ { - 1 } p ^ \\bot = - z ^ { - 1 } p + ( H _ - - z ) ^ { - 1 } p ^ \\bot . \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} P _ { V } \\left ( K _ { 1 } = j \\right ) = V _ { 1 , j } , \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} e _ a ( c _ n ) = 2 ^ { - n R } \\sum _ { m \\in \\mathcal { M } _ n } e _ m ( c _ n ) \\leq \\frac { \\epsilon } { 2 } , \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} I ( U ' ; Y ) - I ( U ' ; S ) = I ( U , V ; Y ) - I ( U , V ; S ) > 0 \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} \\begin{aligned} \\frac 1 2 S ( \\sqrt { r } , f ) \\log r & = \\int _ { \\sqrt { r } } ^ r \\frac { S ( \\sqrt { r } , f ) } { t } d t \\leq \\int _ { 0 } ^ r \\frac { S ( t , f ) } { t } d t \\\\ & = T _ 0 ( r , f ) \\leq S ( r , f ) \\log r + T _ 0 ( 1 , f ) \\end{aligned} \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} | | g | | _ { L ^ { q ^ { \\prime } , \\sigma ^ { \\prime } } } \\approx \\sup _ { 1 = | | f | | _ { L ^ { q , \\sigma } } } \\left | \\int _ X f g d \\mu \\right | , \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} P _ { R } = P _ { R , 1 } + P _ { R , 2 } + P _ { R , 3 } . \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} \\widetilde { \\Xi } _ 2 ( z ) = \\exp \\Big ( - 2 \\pi \\lambda \\int _ 0 ^ \\infty { f _ { { \\rm { P r } } } } \\left ( u \\right ) ( 1 - \\widetilde { \\Omega } _ 1 ( z , u ) ) u d u - 2 \\pi \\lambda \\int _ 0 ^ \\infty ( 1 - { f _ { { \\rm { P r } } } } \\left ( u \\right ) ) ( 1 - \\widetilde { \\Omega } _ 2 ( z , u ) ) u d u \\Big ) \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} \\frac { 1 } { p } \\| ( u _ k , v _ k ) \\| ^ p - \\frac { 1 } { q } \\int _ \\Omega ( \\lambda | u _ k | ^ q + \\mu | v _ k | ^ q ) d x - \\frac { 2 } { \\alpha + \\beta } \\int _ \\Omega | u _ k | ^ \\alpha | v _ k | ^ \\beta d x & = c + o _ k ( 1 ) , \\\\ \\| ( u _ k , v _ k ) \\| ^ p - \\int _ \\Omega ( \\lambda | u _ k | ^ q + \\mu | v _ k | ^ q ) d x - 2 \\int _ \\Omega | u _ k | ^ \\alpha | v _ k | ^ \\beta d x & = o _ k ( \\| ( u _ k , v _ k ) \\| ) , \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} \\iota _ C : \\mathcal { C } \\stackrel { \\sim } { \\longrightarrow } C : = \\left \\{ \\frac { n _ C - 1 } { 2 } , \\frac { n _ C - 3 } { 2 } , \\dots , - \\frac { n _ C - 1 } { 2 } \\right \\} \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} f ( A ) & = \\phi ( A , \\det A ) \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} \\langle d e ^ { i \\delta _ 1 } U ( x _ 0 ) \\ , | \\ , d e ^ { i \\delta _ 1 } V ( x _ 0 ) \\rangle _ g = \\langle d e ^ { i ( \\delta _ 1 - \\frac { \\sigma } { 2 } ) } U ( x _ 0 ) \\ , | \\ , d e ^ { i ( \\delta _ 1 - \\frac { \\sigma } { 2 } ) } V ( x _ 0 ) \\rangle _ g = \\langle U ( x _ 0 ) \\ , | \\ , V ( x _ 0 ) \\rangle _ g . \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} g ( i ) = ( R ^ { 2 } - i ^ { 2 } ) ^ { n / 2 } . \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} M _ { \\varphi } [ \\hat { \\eta } ] : = - ( M _ { \\varphi } [ M _ 1 ] ) ^ { - 1 } \\{ M _ { \\varphi } [ g _ 1 ] + M _ { \\varphi } [ M _ 2 g _ 2 ] + M _ { \\varphi } [ M _ 3 g _ 3 ] - M _ { \\varphi } [ M _ 2 ( \\partial _ { \\psi } \\theta _ 0 ) ^ T ] \\ , M _ { \\varphi } [ g _ 2 ] \\} . \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} h ^ + ( \\omega ) = \\left \\{ y \\in \\Phi _ { \\mathrm { b } } : 0 \\in \\mathcal { V } _ { y , z } ^ 2 , z \\in \\Phi _ { \\mathrm { b } } \\setminus \\{ y \\} \\right \\} . \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\mathbf { X } ^ { ( g ) } = \\sum _ { \\left \\{ m \\ ; \\vert \\beta _ m > 0 \\right \\} } \\left ( \\beta _ m \\right ) ^ { 1 / 2 } \\boldsymbol { \\phi } _ m \\boldsymbol { \\psi } _ m ^ H \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\frac { 1 } { M } \\sum _ { m = 1 } ^ M X _ m \\geq c \\right ) \\leq e ^ { - \\frac { M \\mu } { B } \\Big ( \\frac { c } { \\mu } \\left ( \\ln \\frac { c } { \\mu } - 1 \\right ) + 1 \\Big ) } . \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} \\mu ( B ) = \\int \\mu _ \\gamma ( B ) \\dd \\pi ( \\gamma ) \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} e _ b ( x , w ) \\star e _ b ( y , w ) = e _ b ( x + y , w ) , \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} H = \\frac { p ^ 2 } { 2 } + \\frac { q ^ 2 } { 2 } + \\alpha \\sin { ( w t ) } \\frac { q ^ 2 p ^ 2 } { 2 } . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} t ( 1 - t ) \\frac { d ^ 2 w } { d t ^ 2 } + \\left [ b - ( a + c + 1 ) t \\right ] \\frac { d w } { d t } - a c \\cdot w = 0 . \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} G ^ j ( z ) = [ G ^ j ( z ) ] _ + + \\sum \\limits _ { \\chi } \\left ( \\sum \\limits _ { i = 0 } ^ { \\infty } \\displaystyle \\frac { \\bar P _ { i , \\chi } ^ j } { ( z + \\chi ) ^ i } \\sum \\limits _ { l = 0 } ^ { i - 1 } \\displaystyle \\frac { \\partial ^ i } { \\partial z ^ l } S _ { u ^ j } ( \\chi ) ( - z - \\chi ) ^ l \\right ) . \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} u _ j : = e ^ { \\imath 2 \\pi \\frac { j } { m } } ; \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} X Y _ v = \\sum _ { j \\ge 0 } a _ j v ^ j \\ , \\ , \\cdotp \\sum _ { k \\ge 1 } b _ k k v ^ { k - 1 } = \\sum _ { n \\ge 1 } \\biggl ( \\ , \\sum _ { j + k = n } k a _ j b _ k \\biggr ) v ^ { n - 1 } . \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} F _ x = ( \\dots \\to { } ^ { - 1 } F _ x = 0 \\to { } ^ 0 F _ x = B _ x \\to { } ^ 1 F _ x \\to \\dots ) \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} \\alpha = \\frac { - 2 } { 1 + m - n } , \\beta = - \\frac { 1 - m + n } { 1 + m - n } . \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} \\tilde { N } _ i = N _ { i R } + N _ { n + 1 - i L } , \\tilde { D } _ i = N _ { i R } - N _ { n + 1 - i L } . \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} c _ { 3 } \\Pr _ x [ T _ { V ( H ) } > \\frac { t } { 2 } ] \\le \\Pr _ x [ T _ { R _ { i - 1 } } < T _ { V ( H ) } ] \\le \\frac { | \\mathrm { B M P _ i } | } { | \\mathrm { G M P _ i } \\cup \\mathrm { B M P _ i } | } = \\frac { | \\mathrm { B M P _ i } | } { 2 ^ { 2 ^ { 2 n + i - 2 } } } \\le 2 ^ { - c _ 2 2 ^ { 2 n + i } } . \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{gather*} 3 ( \\Psi _ { 1 1 } ) _ t + ( \\Psi _ { 1 1 } ) _ { x x } + \\big ( t - x ^ 2 \\big ) ( \\Psi _ { 1 1 } ) _ x \\\\ { } = x ^ 2 \\left ( r _ 2 + \\frac { t } { 2 } \\right ) \\Psi _ { 1 1 } + x ( 3 b - 2 e _ 1 ) \\Psi _ { 2 1 } + x \\left ( r _ 1 - \\frac { 1 } { 2 } - \\frac { q _ 2 } { 2 } \\right ) \\Psi _ { 1 1 } \\\\ { } + ( e _ 2 + 3 c ) \\Psi _ { 2 1 } + \\left ( \\frac { U } { 2 } + \\frac { 3 } { 2 } q _ 2 ( b - e _ 1 ) + q _ 1 + r _ 0 \\right ) \\Psi _ { 1 1 } . \\end{gather*}"}
-{"id": "7999.png", "formula": "\\begin{align*} \\big ( F \\ , \\ , \\sharp ^ B \\ , \\ , G \\big ) ( X ) & : = \\pi ^ { - 4 } \\int _ \\Xi d Y \\int _ \\Xi d Z \\ , e ^ { - 2 i \\sigma ( Y , Z ) } e ^ { - i \\int _ { T ( x , y , z ) } B } F ( X - Y ) \\ , G ( X - Z ) \\\\ & = \\pi ^ { - 4 } \\int _ \\Xi d Y \\int _ \\Xi d Z \\ , e ^ { - 2 i \\sigma ( X - Y , X - Z ) } e ^ { - i \\int _ { \\widetilde { T } ( x , y , z ) } B } F ( Y ) \\ , G ( Z ) \\ , , \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} \\mathbb { E } [ \\xi ] = \\sup _ { Q \\in \\mathcal { P } } E _ { Q } [ \\xi ] \\ \\ \\xi \\in L _ { i p } ( \\Omega _ T ) . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} [ D ( m , m j + k ) : D ( m + 1 , n ) ] _ q & = q ^ { ( 2 j + 1 ) ( 2 k - m ) } [ D ( m , m j + m - k ) : D ( m + 1 , n ) ] _ q & \\\\ & = q ^ { ( 2 j + 1 ) ( 2 k - m ) } q ^ { 4 ( m - k ) j } [ D ( m , m ( j - 1 ) + k ) : D ( m + 1 , n ) ] _ q & \\\\ & = \\begin{cases} q ^ { j ( m j + 2 k ) } & n = k , \\\\ q ^ { ( j + 1 ) ( m j + 2 k - m ) } & n = m - k , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} C _ y = ( X \\times C ) _ { ( x , y ) } \\subset \\tau ^ { - 1 } ( J \\pi ( x , y ) ^ { - 1 } ) , \\textrm { f o r $ \\eta $ - a l m o s t e v e r y $ ( x , y ) $ } . \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} \\langle x \\ | \\ u _ i \\rangle - \\eta _ i = \\sum _ { k \\in I _ m } { \\nu } _ k \\langle u _ k \\ | \\ u _ i \\rangle , i \\in I _ m \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} \\widetilde { r } ( t ) = \\widetilde { P } ( t ) + \\int \\limits _ { 0 } ^ { t } \\widetilde { Q } ( t , \\tau ) \\widetilde { r } ( \\tau ) d \\tau . \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} X = Q ^ { \\alpha } ( x , n , [ u ] ) \\partial _ { u ^ { \\alpha } } . \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} e ^ \\ast _ i A e ^ \\ast _ j \\ ; = \\ ; e ^ \\ast _ i B e ^ \\ast _ j \\oplus \\sum _ { k < i } e ^ \\ast _ i A e ^ \\ast _ k A e ^ \\ast _ j . \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} \\limsup _ { | i | \\le \\gamma t , t \\to \\infty } | u _ i ( t + s ; s , u ^ 0 ) - u ^ + ( t + s ) | = 0 \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left | D \\varphi _ t ( \\omega , x ) v \\right | ^ 2 & = 2 \\left \\langle D b ( \\varphi _ t ( \\omega , x ) ) D \\varphi _ t ( \\omega , x ) v , D \\varphi _ t ( \\omega , x ) v \\right \\rangle \\\\ & \\leq 2 \\left | D \\varphi _ t ( \\omega , x ) v \\right | ^ 2 . \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{gather*} \\Phi ( u , t ; \\beta ) : = F \\bigl ( { - } 2 ^ { 1 / 3 } e ^ { - i \\frac { \\pi } { 3 } } u , 2 ^ { - 1 / 3 } e ^ { i \\frac { \\pi } { 3 } } t ; \\beta \\bigr ) . \\end{gather*}"}
-{"id": "180.png", "formula": "\\begin{align*} \\mathcal { A } ( u , \\phi ) + \\mathcal { A } ( v , \\psi ) = \\int _ \\Omega \\left ( \\lambda | u | ^ { q - 2 } u \\phi + \\mu | v | ^ { q - 2 } v \\psi \\right ) d x + \\frac { 2 \\alpha } { \\alpha + \\beta } \\int _ \\Omega | u | ^ { \\alpha - 2 } u | v | ^ \\beta \\phi d x + \\frac { 2 \\beta } { \\alpha + \\beta } \\int _ \\Omega | u | ^ { \\alpha } | v | ^ { \\beta - 2 } v \\psi d x \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} \\tau _ { i j k } ^ 2 & = 1 - \\frac { r _ { i j k } ^ { - \\alpha } } { \\sum _ { l \\in \\Phi _ { \\mathrm { P } } } r _ { i l k } ^ { - \\alpha } } \\\\ & = \\frac { 1 } { 1 + \\frac { r _ { i j k } ^ { - \\alpha } } { \\sum _ { l \\in \\Phi _ { \\mathrm { P } } \\setminus \\{ j \\} } r _ { i l k } ^ { - \\alpha } } } \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} \\mbox { f o r a l l } \\ \\ m \\ge 2 , \\ \\ k = 1 , \\ldots , m ^ 2 , \\ \\ | g _ k ^ { ( m ) } | \\le g _ { * * } < \\infty . \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} n _ i ^ { \\rm i m p } & = \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\lambda \\ , L ( \\lambda | \\lambda _ p ) = - F ( \\lambda _ { i R } | \\lambda _ p ) + F ( \\lambda _ { i L } | \\lambda _ p ) , \\\\ d _ i ^ { \\rm i m p } & = \\left \\{ \\int _ { - \\infty } ^ { \\lambda _ { i L } } - \\int _ { \\lambda _ { i R } } ^ { \\infty } \\right \\} d \\lambda \\ , L ( \\lambda | \\lambda _ p ) = - F ( \\lambda _ { i R } | \\lambda _ p ) - F ( \\lambda _ { i R } | \\lambda _ p ) . \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} ( p _ 1 ) _ \\# ( a ) & = ( \\beta _ { 1 , 1 } , \\beta _ { 1 , 2 } ) \\ ; \\\\ ( p _ 1 ) _ \\# ( b l _ \\sigma ( b ) ) & = ( r + 1 , s - \\beta _ { 2 , 1 } ) + ( - 1 , 1 + 2 \\beta _ { 2 , 1 } ) + ( r + 1 , s - \\beta _ { 2 , 1 } ) = ( \\beta _ { 1 , 2 } , \\beta _ { 2 , 2 } ) , \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\forall i \\in N \\left \\{ \\begin{array} { l } \\langle x - \\sum _ { k \\in N } \\tilde { \\nu } _ k u _ k \\ | \\ u _ i \\rangle - \\eta _ i \\leq 0 , \\\\ \\tilde { \\nu } _ i ( \\langle x - \\sum _ { k \\in N } \\tilde { \\nu } _ k u _ k \\ | \\ u _ i \\rangle - \\eta _ i ) = 0 , \\\\ \\tilde { \\nu } _ i \\geq 0 \\end{array} \\right . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\mathsf { B } _ n } \\mathbb { P } _ { Q ^ { ( \\mathsf { B } _ n ) } } \\Big ( \\mathbf { U } ( I , \\mathsf { B } _ n ) \\notin \\mathcal { T } _ \\epsilon ^ n ( Q _ U ) \\Big ) = \\mathbb { P } _ { Q _ U ^ n } \\Big ( \\mathbf { U } \\notin \\mathcal { T } _ \\epsilon ^ n ( Q _ U ) \\Big ) \\leq e ^ { - n \\eta ( \\epsilon ) } , \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} \\Sigma ( x _ 1 , y _ 1 ) \\cap \\Sigma ( x _ 2 , y _ 2 ) = \\{ e \\} . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} \\lambda x ( t ) = \\int _ { \\overline { \\Omega } } k ( t , s ) f ( s , x ( s ) ) \\textup d s , t \\in \\overline { \\Omega } , \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} & ( a ^ + ( \\varphi ) + a ^ - ( \\varphi ) ) \\diamond ( a ^ + ( \\psi ) + a ^ - ( \\psi ) ) \\\\ & \\quad = a ^ + ( \\varphi ) a ^ + ( \\psi ) + a ^ + ( \\varphi ) a ^ - ( \\psi ) + q a ^ + ( \\psi ) a ^ - ( \\varphi ) + a ^ - ( \\varphi ) a ^ - ( \\psi ) , \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} ( s - 1 ) - ( n - 1 ) & = ( s - m - 1 ) + ( m + 1 - n ) \\\\ ( s - 1 ) + ( n - 1 ) & \\equiv ( s - m - 1 ) - ( m + 1 - n ) \\pmod { m } , \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} \\tau _ { \\infty } = \\frac { ( - 1 ) ^ { \\mathsf { N } } \\kappa _ { + } ^ { \\prime } \\kappa _ { - } e ^ { \\tau _ { - } - \\tau _ { + } ^ { \\prime } } \\prod _ { b = 1 } ^ { \\mathsf { N } } \\alpha _ { b } \\beta _ { b } } { \\left ( \\zeta _ { + } - 1 / \\zeta _ { + } \\right ) \\left ( \\zeta _ { - } - 1 / \\zeta _ { - } \\right ) } . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} \\forall i \\in J \\left \\{ \\begin{array} { l } \\langle x - \\sum _ { k \\in J } \\tilde { \\nu } _ k u _ k \\ | \\ u _ i \\rangle - \\eta _ i = 0 , \\\\ \\tilde { \\nu } _ i > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} a ^ - ( \\varphi ) a ^ + ( \\psi ) = q a ^ + ( \\varphi ) a ^ - ( \\psi ) + ( \\varphi , \\psi ) _ \\mathcal { H } . \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\frac { 1 } { 3 } - 2 c r ^ { \\frac { 1 } { 2 } } \\leq c \\sum _ { j = 0 } ^ { \\infty } ( 2 ^ { - j } r ) ^ { 1 - \\frac { n + \\alpha } { p } } \\log ^ { - \\frac { \\lambda } { p } } \\left ( e + ( 2 ^ { - j } r ) ^ { - \\frac { 1 } { 2 } } \\right ) \\left ( \\int _ { B ^ n ( x , 2 ^ { - j } r ) ^ b } \\Psi ( \\vert \\nabla u \\vert ) \\ , d \\mu _ { \\alpha } \\right ) ^ { \\frac { 1 } { p } } , \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} I \\ , \\mathcal { O P } _ n = \\mathcal { F } ^ { ( n ) } _ q ( \\mathcal { H } ) . \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} \\sigma ( A _ { b c } - B ) = \\sigma ( A _ { ( m ) } ) \\cup \\Big ( \\bigcup _ { n \\geq m + 1 } \\sigma ( A _ n ) \\Big ) = \\sigma _ { ( m ) } \\cup \\Big ( \\bigcup _ { n \\geq m + 1 } \\sigma _ n \\Big ) , \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} P ^ { \\mu ' } ( A ) = \\int _ { \\mathcal { P } _ { \\mathcal { V } _ { \\alpha , \\theta } } } P ( A ) \\mu ' ( \\mathrm { d } P ) , \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} [ B _ { m _ k } , B _ { m _ { k - 1 } } , \\cdots , B _ 1 ] = - [ A _ { m _ k } , A _ { m _ { k - 1 } } , \\cdots , A _ 1 ] \\left ( I + \\widetilde { M } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} & F ( \\eta _ { E _ 0 , F _ 0 } , \\eta _ { E _ 0 , E _ 1 } , \\eta _ { E _ 1 , F _ 1 } , \\eta _ { F _ 1 , F _ 0 } , \\eta _ { E _ 0 , E _ 2 } , \\eta _ { E _ 2 , F _ 2 } , \\eta _ { F _ 2 , F _ 0 } , \\eta _ { E _ 0 , E _ 3 } , \\eta _ { E _ 3 , F _ 3 } , \\eta _ { E _ 3 , F _ 0 } ) \\\\ = & F _ 1 ( X _ 0 , X _ 1 , X _ 2 , X _ 3 ) , F _ 1 ( x , y , z , t ) = F ( x , y , y , y , z , z , z , t , t , t ) . \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\widetilde { F } _ k ( s ) : = \\sum _ { p _ 1 , \\cdots , p _ k } \\frac { \\chi _ 0 ( p _ 1 ) \\cdots \\chi _ 0 ( p _ k ) } { p ^ s } . \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} \\begin{aligned} \\mu _ k - \\mu _ j & = \\sum _ { i = j + 1 } ^ { k } ( \\mu _ i - \\mu _ { i - 1 } ) \\geq \\kappa \\sum _ { i = j + 1 } ^ { k } ( i - 1 ) ^ { \\gamma - 1 } \\\\ & \\geq \\kappa \\begin{cases} \\int _ j ^ k ( x - 1 ) ^ { \\gamma - 1 } \\dd x , & \\gamma \\geq 1 , \\\\ [ 2 m m ] \\int _ j ^ k x ^ { \\gamma - 1 } \\dd x , & 0 < \\gamma < 1 . \\end{cases} \\end{aligned} \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{gather*} f ( P _ 1 ) = f ( P _ 2 ) = f ( P _ 3 ) = 0 , \\\\ f ( P ) < 0 \\quad \\mbox { f o r a l l $ P \\in ( P _ 1 , P _ 2 ) $ , } f ( P ) > 0 \\quad \\mbox { f o r a l l $ P \\in ( P _ 2 , P _ 3 ) $ . } \\end{gather*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} u _ { n + 1 } \\\\ u _ n \\end{matrix} \\right ) = A _ { c , E } ( \\theta + n \\alpha ) \\left ( \\begin{matrix} u _ n \\\\ u _ { n - 1 } \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} \\tilde { P } ( \\mu ) & \\geq \\int _ { 0 } ^ { 1 } \\xi '' ( s ) \\phi _ { \\mu } ( s ) + 2 \\sqrt { \\eta '' } - \\phi _ { \\mu } \\eta '' - \\frac { 1 } { 1 - s } d s + h ^ { 2 } \\phi _ { \\mu } ( 0 ) \\\\ & = \\int _ { 0 } ^ { 1 } ( \\eta '' ( s ) - \\xi '' ( s ) ) ( 1 - s - \\phi _ { \\mu } ( s ) ) + \\int _ { 0 } ^ { 1 } 2 \\sqrt { \\eta '' } - \\eta '' ( s ) \\cdot ( 1 - s ) - \\frac { 1 } { 1 - s } \\\\ & + \\int _ { 0 } ^ { 1 } \\xi '' ( s ) \\cdot ( 1 - s ) d s + h ^ { 2 } \\phi _ { \\mu } ( 0 ) . \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} B \\cap p \\left ( W ( K ) / \\wp W ( K ) \\right ) = p B \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} Q _ 1 = 1 , ~ ~ Q _ 2 = v ' , ~ ~ Q _ 4 = n . \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} f ( r ) = \\widehat f ( r ) = 0 \\ ; \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow 0 } U ( \\beta , \\alpha + \\beta ; z ) = \\frac { \\Gamma ( 1 - ( \\alpha + \\beta ) ) } { \\Gamma ( 1 - \\alpha ) } . \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} d \\widetilde { \\omega } = \\sum _ { I , J } d ( d \\widetilde { x } ^ I \\wedge d \\widetilde { \\xi } ^ J ) \\widetilde { \\omega } _ { I J } + ( - 1 ) ^ { | I | + | J | } \\ , d \\widetilde { x } ^ I \\wedge d \\widetilde { \\xi } ^ J \\wedge d \\widetilde { \\omega } _ { I J } . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\bar g _ 2 ( B _ n ) = \\left \\{ \\begin{array} { l l } g _ 2 ( B _ n ) , & { \\rm i f } ~ B _ n \\in [ 0 , \\bar \\tau _ 1 ] \\cup [ \\bar \\tau _ 2 , + \\infty ) \\\\ \\bar c B _ n + \\bar d , & { \\rm i f } ~ B _ n \\in ( \\bar \\tau _ 1 , \\bar \\tau _ 2 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} b \\partial _ \\alpha \\P [ R \\bar R _ \\alpha ] - \\P \\left [ b R _ \\alpha \\bar R _ \\alpha \\right ] - \\P \\left [ R \\partial _ \\alpha \\bar \\P ( b \\bar R _ \\alpha ) \\right ] = \\sum \\limits _ { j \\geq 0 } f _ j , \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} \\C ( ( q ) ) \\times \\C [ [ q ] ] \\to \\C , ( f , g ) \\mapsto \\{ f , g \\} : = . \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { b - 1 } f ( k , i ) = \\prod _ { i = 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { b - 1 } \\frac { ( x w ^ { b ^ i } ) ^ { k } } { k ! } = \\prod _ { i = 0 } ^ { \\infty } \\exp \\left ( x w ^ { b ^ i } \\right ) \\frac { \\Gamma ( b , x w ^ { b ^ i } ) } { ( b - 1 ) ! } = \\exp \\left ( \\sum _ { i = 0 } ^ { \\infty } x w ^ { b ^ i } \\right ) \\prod _ { i = 0 } ^ { \\infty } \\left ( \\frac { \\Gamma ( b , x w ^ { b ^ i } ) } { ( b - 1 ) ! } \\right ) . \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} x = \\frac { u } { v ^ 2 } y = \\frac { w } { v ^ p } \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} k ^ { - 1 } = \\mu ( k + 1 ) ^ { - 1 } + ( 1 - \\mu ) i , \\end{align*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\lambda ^ 2 _ { 2 N + 1 , 2 } ( z ) & \\le \\frac { 1 } { 2 } \\lambda ^ 2 _ { N + 1 , 2 } ( z ^ 2 ) + \\vert l _ { N , N + 1 } ( z ^ 2 ) \\vert ^ 2 \\lambda ^ 2 _ { N , 2 } ( e _ N ) + \\lambda ^ 2 _ { N , 2 } ( z ^ 2 ) + \\frac { 1 } { 2 } \\vert l _ { N , N + 1 } ( z ^ 2 ) \\vert ^ 2 . \\\\ \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} f = 0 & & \\\\ V = \\star X & & \\\\ \\pi = \\tfrac { 1 } { 2 } \\delta ( X \\wedge \\star X ) & & \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} \\mathsf { E } _ 8 = \\left \\{ \\frac { 1 } { \\sqrt { 2 } } x : x \\in \\mathbb { Z } ^ 8 , \\ ; x 2 \\in \\mathcal { H } _ 8 \\right \\} . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\mathcal { D } _ i ( c ' ) = \\left \\{ 2 ^ { - n R _ 2 } \\sum _ { j \\in \\mathcal { J } _ n } g _ \\mathbf { w } \\big ( \\mathbf { U } ( i ) , \\mathbf { V } ( i , j ) \\big ) \\geq c ' \\cdot 2 ^ { n \\left ( I ( U ; W ) + \\epsilon ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } \\right ) } \\right \\} , \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} c _ 4 ^ 3 - c _ 6 ^ 2 = 1 2 ^ 3 \\Delta \\Leftrightarrow c _ 4 ^ 3 - ( 1 2 \\Delta ^ { 1 / 3 } ) ^ 3 = c _ 6 ^ 2 \\Leftrightarrow A B = c _ 6 ^ 2 . \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} \\dot { A } = \\lambda A + O ( ( A , B ) ^ 3 ) , \\dot { B } = \\lambda B + O ( ( A , B ) ^ 3 ) . \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} \\Delta \\rho & = \\frac { 2 ( 1 - | y | ^ 2 ) } { | x - y | ^ 4 } \\left ( - N ( | y | ^ 2 - 2 x \\cdot y + 1 ) + 4 ( 1 - x \\cdot y ) \\right ) \\\\ ( x - y ) \\cdot \\nabla \\rho & = - 2 \\frac { 1 - | y | ^ 2 } { | x - y | ^ 2 } ( | x | ^ 2 - x \\cdot y + 1 - | x | ^ 2 ) = - 2 \\frac { ( 1 - | y | ^ 2 ) ( 1 - x \\cdot y ) } { | x - y | ^ 2 } \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} ( \\Lambda - x ) \\cap B _ r ( 0 ) = ( \\Lambda - y ) \\cap B _ r ( 0 ) \\ ; , \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\sum _ { \\lambda \\vdash n } \\sum _ { \\lambda _ i \\in \\lambda } a ( \\lambda _ i ) q ^ n = \\frac { 1 } { ( q ; q ) _ { \\infty } } \\langle f \\rangle _ q = \\sum _ { n = 0 } ^ { \\infty } p ( n ) q ^ n \\sum _ { n = 1 } ^ { \\infty } A ( n ) q ^ n , \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} w ( x , \\tau ) = \\frac { e ^ { \\frac { \\tau } { m - 1 } } } k ( \\delta \\rho ^ 2 ) ^ { \\frac 1 { m - 1 } } u ( x , \\Lambda ^ { - 1 } ( \\tau ) ) \\le \\frac { b _ 1 ^ \\theta } { k } ( \\delta \\rho ^ 2 ) ^ { \\frac 1 { m - 1 } } u ( x , \\Lambda ^ { - 1 } ( \\tau ) ) , \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} E ^ + & = \\bigoplus _ k E ^ { 2 k } , E ^ - = \\bigoplus _ k E ^ { 2 k + 1 } , \\\\ \\nabla ^ + & = \\bigoplus _ k \\nabla ^ { 2 k } , \\nabla ^ - = \\bigoplus _ k \\nabla ^ { 2 k + 1 } . \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} e ^ { i \\theta } ( z _ 1 , z _ 2 ) = ( e ^ { i \\theta } z _ 1 , e ^ { i n \\theta } z _ 2 ) , n \\in \\mathbb Z , n > 0 . \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} u ( t ) = e ^ { - i t H } \\psi - i \\Gamma _ H F ( t ) . \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} a = \\min \\left \\{ \\frac { \\phi } { 1 6 } \\log \\left ( 1 + \\frac { P ' } { 4 } \\right ) , 1 \\right \\} . \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} D _ { q } f ( z ) = \\left \\{ \\begin{array} { l c l } \\dfrac { f ( z ) - f ( q z ) } { ( 1 - q ) z } & f o r & z \\neq 0 , \\\\ f ' ( 0 ) & f o r & z = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} \\hat { \\alpha } _ { 1 } = - \\alpha , \\ \\hat { \\alpha } _ { 2 } = - 2 \\alpha + { 2 \\over 3 } , \\ \\hat { \\alpha } _ { 3 } = 3 \\alpha . \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} \\left ( \\frac { \\delta ^ i } { \\delta ^ j } \\right ) ^ t = \\frac { \\bar { \\theta } ^ j u _ j ' ( \\tilde { x } _ t ^ j ) } { \\bar { \\theta } ^ i u _ i ' ( \\tilde { x } _ t ^ i ) } , & & 1 \\leq j < i ; t = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\int _ { B _ s ( x _ 0 ) \\cap \\Sigma } f d \\mu = \\int _ 0 ^ s \\left ( \\int _ { \\partial B _ { \\tau } ( x _ 0 ) \\cap \\Sigma } \\frac { f } { | \\nabla r | } d \\sigma \\right ) d \\tau , \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} f ( z ) : = \\frac { 1 } { \\eta ( 1 2 z ) ^ 2 } = \\sum _ { n = - 1 } ^ { \\infty } a ( n ) q ^ n . \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} \\overline \\nabla _ { \\nu } = \\overline z ^ 2 \\frac { \\partial ^ 2 } { \\partial \\overline z ^ 2 } + \\overline z \\frac { \\partial } { \\partial \\overline z } + \\overline z ^ 2 - \\nu ^ 2 . \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} \\mathcal { E } ( \\tilde { i } , \\tilde { j } , \\tilde { m } , \\mathcal { C } _ n ) = \\Big \\{ \\big ( \\mathbf { u } ( \\tilde { i } ) , \\mathbf { v } ( \\tilde { i } , \\tilde { j } , \\tilde { m } ) , \\mathbf { Y } \\big ) \\in \\mathcal { T } _ \\epsilon ^ { n } ( p _ { U , V , Y } ) \\Big \\} , \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} \\frac { \\Gamma \\left ( b , x w ^ { b ^ { i } } \\right ) } { \\Gamma \\left ( b \\right ) } = 1 - \\frac { x ^ b w ^ { b ^ { i + 1 } } } { \\Gamma \\left ( b + 1 \\right ) } + \\frac { x ^ { b + 1 } w ^ { b ^ { i + 1 } + b ^ { i } } } { \\left ( b + 1 \\right ) \\Gamma \\left ( b \\right ) } + \\dots \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} Y = \\bold { p r } X + c ^ i \\partial _ { x ^ i } , \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} ( x , y , 0 ) ( x ' , y ' , 0 ) = ( x + x ' , y + y ' , \\frac { 1 } { 2 } ( x y ' - y x ' ) ) \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} W _ { ( \\alpha , \\beta ) } ( n ) = \\sum _ { \\substack { { ( l , k ) \\in \\mathbb { N } _ { 0 } ^ { 2 } } \\\\ { \\alpha \\ , l + \\beta \\ , k = n } } } \\sigma ( l ) \\sigma ( k ) = \\sum _ { \\substack { { ( l , k ) \\in \\mathbb { N } _ { 0 } ^ { 2 } } \\\\ { \\delta \\ , \\alpha _ { 1 } \\ , l + \\delta \\ , \\beta _ { 1 } \\ , k = n } } } \\sigma ( l ) \\sigma ( k ) = W _ { ( \\alpha _ { 1 } , \\beta _ { 1 } ) } ( \\frac { n } { \\delta } ) . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} g ^ 2 = 1 = u ^ 2 , g u = u g , . \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{gather*} F ( x , t ; \\beta = 2 ) = ( \\Psi _ 0 ( x , t ) ) _ { 2 2 } e ^ { \\frac { x ^ 3 } { 6 } - \\frac { 1 } { 2 } t x + \\int ^ { \\infty } _ { t } \\omega ( \\tau ) d \\tau } , \\omega = u ^ 4 + t u ^ 2 - u ^ 2 _ t . \\end{gather*}"}
-{"id": "5307.png", "formula": "\\begin{align*} a _ 1 ( \\varphi , x ) ( 1 + \\beta _ x ( \\varphi , x ) ) ^ 3 = b _ 3 ( \\varphi ) , \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} W : = \\vec ( X ) \\in \\R ^ { q ^ 2 } , \\end{align*}"}
-{"id": "6886.png", "formula": "\\begin{align*} p _ { i j } = \\frac { i - 1 } { j ( j - 1 ) } \\ \\ j \\geq i \\geq 2 ; \\ \\ \\ \\ \\ p _ { i j } = 0 \\ \\ j < i . \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} \\xi : = [ ( 0 _ { 4 + 8 k } , ( \\underbrace { 1 , 0 , \\ldots , 0 } _ { a } ) , 0 _ { 2 6 - N } ) ] \\in A _ { \\Lambda _ N } \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\zeta _ t \\leq \\max \\left \\{ 2 \\sqrt { \\log ( 4 \\log T / \\delta ) } \\sqrt { \\sum _ { t = 1 } ^ T \\textrm { V a r } _ t \\zeta _ t } , 6 G D \\log ( 4 \\log T / \\delta ) \\right \\} . \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{align*} y = m x + B , m \\neq 0 \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rm F A } ( u ) = P \\left ( \\frac { Z } { n } \\geq \\tau _ n | U = u , H _ 0 \\right ) . \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} V \\ = \\ \\bigoplus _ { \\ell } V _ \\ell \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\sum _ { \\ell = j - k } ^ k \\lambda _ { \\ell , j } ^ k = 1 \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{gather*} \\sum _ { n = 1 } ^ { 2 ^ N } \\frac { \\varepsilon _ n ( \\omega ) n ^ { i t } } { n \\sqrt { \\log ( n + 1 ) } } = \\sum _ { n = 1 } ^ { 2 ^ N } ( S _ n ( t ) - S _ { n - 1 } ( t ) ) u _ n = \\sum _ { n = 1 } ^ { 2 ^ N } S _ n ( t ) ( u _ n - u _ { n + 1 } ) + S _ { 2 ^ N } ( t ) u _ { 2 ^ N + 1 } \\ , . \\end{gather*}"}
-{"id": "4420.png", "formula": "\\begin{align*} D ( \\eta _ { \\tau } ) - D ( \\eta ) = \\int _ { a } ^ { b } \\sqrt { \\eta _ { \\tau } '' } - \\sqrt { \\eta '' } \\ , d s \\quad \\forall \\ , \\tau \\end{align*}"}
-{"id": "6474.png", "formula": "\\begin{align*} \\omega _ { \\beta , { n } } ( \\tau _ { g } ( A ) ) = \\omega _ { \\beta , { n } } ( \\tau _ { g } ( \\eta ( A ) ) ) = \\omega _ { \\beta , R { n } } ( \\eta ( A ) ) \\ . \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} \\sum _ { k \\in \\left [ r \\right ] } \\frac { n _ { k } } { \\lambda - \\alpha n + n _ { k } } = \\frac { 1 } { 1 - \\alpha } . \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} X = \\xi ^ { i } ( x ) \\partial _ { x ^ i } + \\phi ^ { \\alpha } ( x , n , [ u ] ) \\partial _ { u ^ { \\alpha } } \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} w = e _ 1 + e _ 2 + \\cdots + e _ n \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} & \\wedge ( \\alpha x ^ i y ^ i \\beta ) + \\wedge ( \\alpha y ^ i x ^ i \\beta ) = 0 \\\\ & \\wedge ( \\alpha x ^ i y ^ j \\beta ) + \\wedge ( \\alpha y ^ j x ^ i \\beta ) + \\wedge ( \\alpha y ^ i x ^ j \\beta ) + \\wedge ( \\alpha x ^ j y ^ i \\beta ) = 0 \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} | \\nabla ^ 2 F | ^ 2 = \\frac { 1 } { 4 n } ( \\Delta F ) ^ 2 + \\frac { 1 } { 4 n } \\sum _ { s = 1 } ^ 3 [ g ( \\nabla ^ 2 F , \\omega _ s ) ] ^ 2 + p ( F ) , \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} \\omega \\cdot \\nabla \\phi ( x ) = - \\omega \\cdot A ( x ) , \\ , \\ , \\ , \\ , \\ , x \\in \\R ^ { n } . \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} | X | & = | X ' | + \\sum _ { i = 1 } ^ s | X _ i | \\leq k - 1 + \\sum _ { i = 1 } ^ s ( p _ i - 1 ) ( k - 1 ) \\\\ & \\leq k - 1 + ( p - s ) ( k - 1 ) \\leq k - 1 + ( p - 2 ) ( k - 1 ) = ( p - 1 ) ( k - 1 ) , \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} ( F _ \\Omega ) _ { k + 1 } = - \\frac { 1 } { 2 } ( \\Omega _ k ) ^ \\sharp + \\textrm { t e r m s n o t c o n t a i n i n g } \\Omega _ k . \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} 0 \\to H ^ 0 ( C , M _ { K _ C } \\otimes L ) \\to H ^ 0 ( C , K _ C ) \\otimes H ^ 0 ( C , L ) & \\stackrel { \\alpha } { \\to } H ^ 0 ( C , K _ C \\otimes L ) \\\\ & \\to H ^ 1 ( C , M _ { K _ C } \\otimes L ) \\to 0 . \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ r _ { p , q } } : = \\Vert \\{ 2 ^ { j r } \\Vert g _ j \\Vert _ { L ^ p } \\} _ { j = 0 } ^ { \\infty } \\Vert _ { l ^ q ( \\mathbb { N } ) } < \\infty , \\ , \\ , \\ , g _ 0 = f _ 0 , \\ , 0 < p , q \\leq \\infty , \\ , r \\in \\mathbb { R } , \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} T N ^ { \\bot } = \\{ \\xi \\in T N | \\Omega ( \\xi , \\chi ) = 0 , \\forall \\chi \\in T N \\} \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} \\int _ { a } ^ { b } ( \\eta _ { \\tau } '' - \\eta '' ) ( 1 - s ) \\ , d s = ( \\eta _ { \\tau } ' - \\eta ' ) ( 1 - s ) | _ { a } ^ { b } + ( \\eta _ { \\tau } - \\eta ) | _ { a } ^ { b } = 0 \\quad \\forall \\ , \\tau \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} | \\{ x \\in \\Omega : f ( x ) > C \\} | = | \\Omega _ C | \\ge \\lambda ^ { \\frac { - q } { q - 1 } } | \\Omega | , \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} J _ { \\lambda , \\mu } ( u _ k , v _ k ) & = \\frac { \\alpha + \\beta - p } { p ( \\alpha + \\beta ) } \\| ( u _ k , v _ k ) \\| ^ p - \\frac { \\alpha + \\beta - q } { q ( \\alpha + \\beta ) } \\int _ \\Omega ( \\lambda | u _ k | ^ { q } + \\mu | v _ k | ^ { q } ) d x \\\\ & \\geq - \\frac { \\alpha + \\beta - q } { q ( \\alpha + \\beta ) } \\int _ \\Omega ( \\lambda | u _ k | ^ { q } + \\mu | v _ k | ^ { q } ) d x . \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} f \\left ( x \\right ) = x ^ { 4 } - 2 c x ^ { 3 } + 2 x ^ { 2 } + 2 c x + 1 , \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 } \\frac { W _ { \\frac { Z \\alpha _ g \\epsilon } { \\omega } , \\nu - \\frac { 1 } { 2 } } \\left ( x \\right ) } { W _ { \\frac { Z \\alpha _ g \\epsilon } { \\omega } , \\nu + \\frac { 1 } { 2 } } \\left ( x \\right ) } = C \\ ; , \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{align*} J ^ { 2 } ( h ) - M ( h ) = 1 6 u \\left ( s - v \\right ) ^ { 2 } R ( h ) \\end{align*}"}
-{"id": "9444.png", "formula": "\\begin{align*} { u ^ * } ( t ) = \\left \\{ \\begin{array} { l } 2 0 0 \\ , t / 9 - 2 0 / 3 , t \\in [ 0 , 0 . 3 ] , \\\\ 0 , t \\in [ 0 . 3 , 0 . 7 ] , \\\\ - 2 0 0 \\ , t / 9 + 1 4 0 / 9 , t \\in [ 0 . 7 , 1 ] , \\end{array} \\right . \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\ell _ U : = u _ 1 v _ 1 + u _ 2 v _ 2 . \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} { \\bf u } ^ { \\infty } = \\Phi ( { \\bf u } ^ { \\infty } ) . \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} \\mathrm { c o s } ( x , y ) = \\frac { ( x , y ) } { | | x | | _ E | | y | | _ E } = \\frac { g _ E ( x , y ) } { | | x | | _ E | | y | | _ E } . \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} & r = 4 b ^ 2 + 6 a c - 2 n - 3 , q = - 2 \\sum _ { i = 1 } ^ n x _ i + 6 a ( n + 1 ) + 4 b c , \\\\ & E = 2 \\sum _ { i = 1 } ^ n x _ i ^ 2 - 6 a \\sum _ { i = 1 } ^ n x _ i - 2 b ( 2 n + 1 ) - c ^ 2 , \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} \\Gamma _ { i j } ^ k = e ^ k ( \\nabla _ { e _ i } e _ j ) \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} \\gamma ( x , y ) = \\frac { \\mathrm { c m } ( x , x + y ) } { \\mathrm { c m } ( y , x + y ) } . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} M ( t , i ) : = \\sum _ { j = 1 } ^ i \\frac { X ( t , j ) - \\pi ( t ) } { 1 - \\pi ( t ) } + \\sum _ { j = i + 1 } ^ { n - a } \\frac { X ( t - 1 , j ) - \\pi ( t - 1 ) } { 1 - \\pi ( t - 1 ) } . \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { \\infty } \\sum _ { k = 0 } ^ { b - 1 } f ( k , i ) = \\sum _ { k = 0 } ^ { \\infty } \\prod _ { i = 0 } ^ { \\infty } f ( k _ i , i ) . \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\lambda _ { \\phi } = \\lambda \\exp ( i \\phi ) \\ \\mbox { w i t h } \\lambda \\geq 0 \\ , , \\ , \\mbox { w h e r e } \\ , { \\rm { a r g } } ( \\lambda ) = \\phi \\in [ 0 , 2 \\pi ) \\ . \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} H ^ { 1 } ( 0 ) = \\left [ \\begin{array} { c c c c c } s & 0 & a _ { 2 } \\lambda & \\cdots & a _ { n } \\lambda \\\\ 0 & 1 & 0 & \\cdots & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "821.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\langle x \\wedge y , v \\wedge w \\rangle _ { \\wedge ^ { 2 } \\mathfrak { g } } & = & \\langle x , v \\rangle \\langle y , w \\rangle - \\langle x , w \\rangle \\langle y , v \\rangle . \\end{array} \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{align*} \\mathsf { T } _ { \\lambda } ^ { { \\small } } ( , ^ { \\prime } ) \\equiv \\langle | \\mathsf { T } _ { \\lambda } ^ { { \\small } } | ^ { \\prime } \\rangle = \\prod _ { n = 1 } ^ { \\mathsf { N } } W _ { _ { n } } ( z _ { n } / z _ { n } ^ { \\prime } ) \\bar { W } _ { _ { n } } ( z _ { n } / z _ { n + 1 } ^ { \\prime } ) , \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { 2 q ^ n } { 1 + q ^ n } \\frac { ( - q ; q ) _ { n - 1 } } { ( q ; q ) _ { n - 1 } } = \\sum _ { n \\geq 0 } \\frac { q ^ { n ( n + 1 ) / 2 } } { ( q ; q ) _ n } \\frac { 2 q ^ { n + 1 } } { 1 - q ^ { 2 ( n + 1 ) } } \\frac { ( - q ^ { n + 2 } ; q ) _ \\infty } { ( q ^ { n + 2 } ; q ) _ \\infty } \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} \\mbox { R e } [ f ( 2 r \\partial _ r \\overline u + ( n - 1 ) \\overline u ) ] - 2 \\lambda ^ { \\frac 1 2 } \\mbox { I m } ( r f \\overline u ) = \\mbox { R e } \\Big ( 2 r f \\overline { e ^ { i \\lambda ^ { \\frac 1 2 } r } \\partial _ r v _ \\lambda } + ( n - 1 ) f \\overline u \\Big ) . \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} F = \\widehat { f } _ { \\mathcal { U } _ 0 ^ + } = \\left ( \\bigoplus _ { U \\in \\mathcal { U } _ 0 } \\widehat { f } _ U \\right ) \\oplus \\left ( \\mathrm { 1 } _ X - \\sum _ { V \\in \\mathcal { U } _ 0 } \\widehat { f } _ { V } \\right ) : Y \\to | \\mathcal { N } ( \\mathcal { U } _ 0 ^ + ) | \\subset | C Z | , \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} \\pi _ i ( x ) = ( x _ 0 ^ 2 - x _ j x _ j ) \\delta _ { i 3 } + 2 x _ 0 \\epsilon _ { i j 3 } x _ j + 2 x _ 3 x _ i \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} \\xi = ( \\xi _ 1 - \\xi _ 2 ) + i ( \\xi _ 3 - \\xi _ 4 ) , \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} c ( m ) = \\sum _ { \\substack { d | m \\\\ d < x ^ { 1 / 3 } } } \\gamma ( d ) \\ . \\end{align*}"}
-{"id": "1289.png", "formula": "\\begin{align*} x ' ( t ) = A x ( t ) . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} h ( t ) = \\sqrt N + \\frac { H ( T ) N ^ { - 1 / 6 } } 2 , t = \\frac { 1 + T N ^ { - 1 / 3 } } 2 . \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} h A h ^ { - 1 } & = A , h A = A h ; \\\\ h B h ^ { - 1 } & = B , h B = B h ; \\\\ h I & = I , ( h - 1 _ { V } ) I = 0 ; \\\\ h A ' h ^ { - 1 } & = A ' , h A ' = A ' h ; \\\\ h B ' h ^ { - 1 } & = B ' , h B ' = B ' h ; \\\\ h F h '^ { - 1 } & = F \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} ( x , z ) \\cdot ( x ' , z ' ) = ( x + x ' , z + z ' + \\frac { 1 } { 2 } \\langle B x , x ' \\rangle ) \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{gather*} \\frac { d \\Psi } { d x } = L \\Psi , \\frac { d \\Psi } { d t } = B \\Psi , \\end{gather*}"}
-{"id": "8953.png", "formula": "\\begin{align*} \\theta = \\sum d t ^ j \\otimes \\theta _ j , \\ \\theta _ j : = \\partial / \\partial t ^ j \\lrcorner ~ \\theta . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} y _ 1 = \\hat { u } ^ 3 \\sigma ( y ' ) + \\hat { u } ^ 2 \\hat { s } \\sigma ( x ' ) + \\hat { t } . \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} \\langle u _ 0 , y \\rangle _ { L ^ 2 } = \\langle \\partial _ x ^ { - 1 } u _ 0 , y \\rangle _ { L ^ 2 } = 0 , \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} r _ j = \\sum _ { i = 1 } ^ m c _ i r _ i = \\sum _ { i = 1 } ^ m c _ i \\frac { p _ 1 \\dots p _ i } { 2 ^ { k _ i } } , \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} 2 \\imath \\psi _ { t } + \\psi _ { r r } + \\frac { 1 } { r } \\psi _ { r } + \\frac { 1 } { 4 k ^ 2 r ^ 2 } \\psi _ { \\phi \\phi } - \\frac { 1 } { 4 r ^ 2 } \\psi = 0 , \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} h = - J ^ - _ n J ^ - _ n - 2 J ^ + _ n - 4 a J ^ 0 _ n - 2 b J ^ - _ n - 2 ( n + 1 ) a - b ^ 2 \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\beta _ { a 0 ( n - a ) } = \\sum _ { i = 1 } ^ { a - 1 } ( - \\rho ^ { i } + \\rho ^ { n - i } ) + \\frac { a - 1 } { 2 } \\rho ^ a + \\frac { n - a + 1 } { 2 } \\rho ^ { n - a } \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{align*} \\hat x ( t ) : = \\Phi ( t ) \\left ( \\big ( I - \\Phi ( T ) \\big ) ^ { - 1 } \\Phi ( T ) \\int _ 0 ^ T \\Phi ( s ) ^ { - 1 } c ( s ) d s + \\int _ 0 ^ t \\Phi ( s ) ^ { - 1 } c ( s ) d s \\right ) \\ ; . \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ \\infty Q ( C \\rho ^ \\alpha g ( l _ i \\rho ) > 1 - \\delta ) \\le \\sum \\limits _ { i = 1 } ^ \\infty Q ( l _ i > t _ 0 ) \\le \\sum \\limits _ { i = 1 } ^ \\infty E _ Q [ T _ 1 ] ^ i / t _ 0 < \\infty . \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} Q ( \\lambda , \\hat { x } , \\theta ) : = \\textstyle \\sum _ i \\theta ^ i u _ i \\left ( \\sigma ^ i ( \\theta _ i , \\lambda ) \\right ) + \\lambda \\left [ \\hat { x } - \\textstyle \\sum _ i \\sigma ^ i ( \\theta ^ i , \\lambda ) \\right ] . \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} \\widetilde { u } _ n = \\widetilde { u } ( \\varepsilon ; n , u _ n ) \\widetilde { u } _ n | _ { \\varepsilon = e } = u _ n . \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} \\widetilde { J } _ i ^ { ( a ) } = \\begin{cases} J _ i ^ { ( a ) } + \\min ( s , \\nu _ i ^ { ( a ) } ) & a = r , \\\\ J _ i ^ { ( a ) } & \\end{cases} \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} \\left ( X \\otimes _ { B } Y \\right ) _ { i } = \\left \\{ \\ , \\sum x \\otimes y \\mid x \\in X _ { j } , \\ , y \\in Y _ { k } , \\ , j + k = i \\ , \\right \\} , \\end{align*}"}
-{"id": "9611.png", "formula": "\\begin{align*} g ^ { 1 1 } \\partial _ 1 [ \\mathfrak { C } _ { 0 i } ] + L _ i ( [ \\mathfrak { C } _ { 0 j } ] ) = 0 ; \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} \\Phi _ { [ - L , L ] } ( \\overline { U } ) \\cap \\Phi _ { [ - L , L ] } ( \\overline { U ' } ) = \\varnothing . \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} \\mu _ { c , \\theta } ( U ) = ( \\delta _ 0 , \\chi _ U ( H _ { c } ( \\theta ) ) \\delta _ 0 ) . \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} E _ { s , 3 } = \\frac { 2 } { 3 ( 2 \\pi ) ^ { \\frac 1 2 } } \\int _ { \\xi _ 1 + \\xi _ 2 + \\xi _ 3 = 0 } \\frac { ( 1 + \\xi _ 1 ^ 2 ) ^ s \\xi _ 1 + ( 1 + \\xi _ 2 ^ 2 ) ^ s \\xi _ 2 + ( 1 + \\xi _ 3 ) ^ s \\xi _ 3 } { \\xi _ 1 \\xi _ 2 \\xi _ 3 } \\hat u ( \\xi _ 1 ) \\hat u ( \\xi _ 2 ) \\hat u ( \\xi _ 3 ) d \\xi _ 1 d \\xi _ 2 \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} \\omega _ { \\alpha } ^ \\beta = g ^ { \\overline \\sigma \\beta } \\partial g _ { \\alpha \\overline \\sigma } \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; J _ 2 } = D _ i \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } - \\left ( S _ { J _ 2 } D _ i \\xi ^ j \\right ) u _ { J _ 1 + \\bold { 1 } _ j ; J _ 2 } ^ { \\alpha } \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} \\| f \\| _ { 1 , ' } ^ 2 & = \\sum _ { \\nu = 0 } ^ { r - 1 } \\int ^ 1 _ 0 | f ^ { ( \\nu ) } ( y ) | ^ 2 \\ , { \\rm d } y , \\\\ \\| f \\| _ { 1 , \\pitchfork ' } ^ 2 & = \\sum _ { \\nu = 0 } ^ { r - 1 } | f ^ { ( \\nu ) } ( a ) | ^ 2 , \\\\ \\| f \\| _ { 1 , ' } ^ 2 & = \\sum _ { \\nu = 0 } ^ { r - 1 } \\left | \\int _ 0 ^ 1 f ^ { ( \\nu ) } ( y ) \\ , { \\rm d } y \\right | ^ 2 . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} | < x ^ { * } , f _ { n } > | & = | \\sum _ { k = p _ { n } } ^ { q _ { n } } < P _ { k } x ^ { * } , x ^ { ( n ) } _ { k } > | \\\\ & \\leq ( \\sum _ { k = p _ { n } } ^ { q _ { n } } \\| x ^ { ( n ) } _ { k } \\| ^ { p } ) ^ { \\frac { 1 } { p } } \\cdot ( \\sum _ { k = p _ { n } } ^ { q _ { n } } \\| P _ { k } x ^ { * } \\| ^ { p ^ { * } } ) ^ { \\frac { 1 } { p ^ { * } } } \\\\ & = ( \\sum _ { k = p _ { n } } ^ { q _ { n } } \\| P _ { k } x ^ { * } \\| ^ { p ^ { * } } ) ^ { \\frac { 1 } { p ^ { * } } } , \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} \\alpha _ { 1 , i } ( s _ { \\iota ( 1 ) , \\sigma _ 1 ^ { - 1 } ( i ) } ) ^ { \\epsilon _ 1 } \\cdots \\alpha _ { l , i } ( s _ { \\iota ( l ) , \\sigma _ l ^ { - 1 } ( i ) } ) ^ { \\epsilon _ l } = g _ i . \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{gather*} \\delta _ Y c ^ a = \\xi ^ a , \\delta _ Y c ^ { a b } = \\xi ^ { a b } , \\delta _ Y ( ) = 0 . \\end{gather*}"}
-{"id": "2094.png", "formula": "\\begin{align*} x = u ^ 2 x ' + r , y = u ^ 3 y ' u = t , r = y _ 0 + y _ 0 t + y _ 2 t ^ 2 , \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} \\left \\langle a , b \\right \\rangle _ { \\widetilde { A } } = \\sum _ { g \\in G } g ( a ^ * b ) . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{gather*} \\frac { \\kappa _ t } { \\kappa } = - \\frac { 1 } { 3 } \\omega - \\frac { 2 } { 3 } \\alpha - \\frac { u _ t } { u } \\frac { 1 - 2 q _ 2 } { 6 } , \\end{gather*}"}
-{"id": "5001.png", "formula": "\\begin{align*} \\mathcal { V } = \\{ v \\in L ^ 2 ( U ) , \\ ; \\tilde \\alpha \\cdot D v \\in L ^ 2 ( U ) , \\ , \\mathsf { s u p p } \\ , v \\subset \\subset U \\} \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{gather*} \\delta _ Q \\alpha _ 0 = d \\alpha _ 1 \\end{gather*}"}
-{"id": "8545.png", "formula": "\\begin{align*} | | p - q | | _ { \\mathsf { T V } } = \\frac { 1 } { 2 } \\sum _ { x \\in \\Omega } \\big | p ( x ) - q ( x ) \\big | . \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} \\frac { \\operatorname { d } } { \\operatorname { d } \\ ! \\varepsilon } L ( x , n , [ \\varepsilon u ] ) = \\sum _ { \\alpha , J _ 1 , J _ 2 } u _ { J _ 1 ; J _ 2 } ^ { \\alpha } \\frac { \\partial } { \\partial u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } L ( x , n , [ \\varepsilon u ] ) . \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} \\widetilde { f } ^ \\gamma ( x ) = \\sum _ { n = 1 } ^ { 4 0 0 } 2 ^ { - \\gamma n } \\cos ( 2 ^ n \\pi x ) + f _ 0 , \\alpha \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x _ 1 = x _ 2 , \\\\ \\dot x _ 2 = - x _ 1 - \\delta ( t ) - \\alpha x _ 2 + \\lambda ^ 2 \\phi ( y _ 1 - x _ 1 ) , \\\\ \\dot y _ 1 = \\lambda y _ 2 , \\\\ \\dot y _ 2 = \\lambda \\phi ( x _ 1 - y _ 1 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} g _ l ( Y _ n ) = ( P _ d ^ l ) ^ { Y _ n } ( 1 - P _ d ^ l ) ^ { 1 - Y _ n } , ~ ~ l = 1 , 2 ; f ( Y _ n ) = ( P _ { f a } ) ^ { Y _ n } ( 1 - P _ { f a } ) ^ { 1 - Y _ n } , \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} e _ i & = e _ { i - 1 } + \\tau \\mathbb { R } _ i e _ i + \\mathbf { F } _ i , \\ i = 1 , , , . n , \\\\ e _ 0 & = 0 , \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} u ( t ) ^ 2 + v ( t ) ^ 2 - r ( t ) \\big ( x ( t ) ^ 2 + y ( t ) ^ 2 \\big ) = 0 . \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\omega } \\hat { \\eta } = g _ 2 - [ \\partial _ { \\psi } \\theta _ 0 ( \\varphi ) ] \\hat { \\zeta } . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} E e ^ { - \\alpha Y _ e } = \\frac { \\eta _ 0 ( \\alpha ) } { \\rho _ 0 \\alpha } \\ . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} D _ t \\widehat { P } _ 1 + ( S - \\operatorname { i d } ) \\widehat { P } _ 2 = Q \\left ( - \\frac { v ' } { u } + v _ 1 - v _ { - 1 } \\right ) + Q _ { \\ast } \\left ( \\frac { u ' } { u } - u _ 1 + u _ { - 1 } \\right ) . \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} T _ i ( x , y ) & = { \\rm T a n } ( \\Sigma ^ i _ x , y ) x \\in G _ { \\varphi _ i } y \\in \\Sigma _ x ^ i . \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} \\widehat { E } _ { 3 } & = I + H H ^ { \\ast } E _ { S _ { A } } - ( I + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } S _ { A } ^ { \\dagger } - H H ^ { \\ast } E _ { S _ { A } } - S _ { A } S _ { A } ^ { \\dagger } + ( I + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } S _ { A } ^ { \\dagger } \\\\ & = E _ { S _ { A } } . \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} i u _ t + u _ { x x } \\pm 2 u | u | ^ 2 = 0 , u ( 0 ) = u _ 0 , \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to - 0 } \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\eta ( b _ { { 0 } } ^ { * } ) \\eta ( b _ { { 0 } } ) ) = \\lim _ { \\lambda \\to + 0 } \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\eta ( b _ { { 0 } } ^ { * } ) \\eta ( b _ { { 0 } } ) ) \\ . \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} u _ { n } ^ { p } = v _ { n } ^ { p } = 1 q = e ^ { - i \\pi \\beta ^ { 2 } } \\beta ^ { 2 } = p ^ { \\prime } / p p ^ { \\prime } p = 2 l + 1 \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} { D } _ { i j } = - \\left ( \\frac { \\omega _ j } { \\omega _ i } \\right ) { D } _ { j i } ^ \\dag , { D } _ { i , N + 1 } ^ \\dag = - \\sum _ { j = 1 } ^ N { D } _ { i j } ^ \\dag , 1 \\leq i \\leq N . \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} \\tau _ { p } ( \\epsilon ) : = \\min \\{ t : \\max _ x \\| \\Pr _ x ^ t - \\pi \\| _ { p , \\pi } \\le \\epsilon \\} . \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathcal { D } ^ { i j } ( 0 ) \\mathcal { D } ^ { l m } ( 0 ) ] = \\begin{cases} \\frac 2 k & i = l , j = m , \\\\ \\frac 1 k & i = l , j \\neq m , \\\\ 0 & i , j , l , m . \\end{cases} \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} ( u , v ) _ { \\mathbb { R } ^ { 2 n } _ J } : = ( u , v ) _ { \\mathbb { R } ^ { 2 n } } - i \\omega ( u , v ) , u , v \\in \\mathbb { R } ^ { 2 n } \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} \\Sigma ( s , t ) & \\subseteq \\bigcup _ { n = 1 } ^ N { [ s , s + t ] ^ { \\leftthreetimes { 2 n - 1 } } \\times [ 0 , s ] ^ { \\leftthreetimes ( 2 ( N - n ) + 1 ) } } \\\\ & = [ s , s + t ] \\leftthreetimes \\left ( \\bigcup _ { n = 1 } ^ N [ s , s + t ] ^ { \\leftthreetimes 2 ( n - 1 ) } \\times [ 0 , s ] ^ { \\leftthreetimes 2 ( N - n ) } \\right ) \\leftthreetimes [ 0 , s ] . \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} f : = \\sum _ { j \\in H } \\left ( ( x _ j + 1 0 x _ { j + 1 } ) ^ 2 + 5 ( x _ { j + 2 } - x _ { j + 3 } ) ^ 2 + ( x _ { j + 1 } - 2 x _ { j + 2 } ) ^ 4 + 1 0 ( x _ { j } - x _ { j + 3 } ) ^ 4 \\right ) \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\lambda _ j ( w ) = \\sum _ { c \\mid w } \\chi ( c ) ( \\log { c / \\sqrt { w } } ) ^ j \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} f ( x ; \\theta ) = \\frac { \\theta ^ x } { x ! } \\exp ( - \\theta ) \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 n T _ i ( n ) \\equiv Q _ i = \\sum _ k \\frac { a _ k ( i ) } { r ^ { k } } \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} \\vartheta _ { \\Lambda _ { 2 4 } } ( z ) = E ^ 3 _ 4 ( z ) - 7 2 0 \\Delta ( z ) = 1 + 1 9 6 5 6 0 \\cdot q ^ 2 + 1 6 7 7 3 1 2 0 \\cdot q ^ 3 + \\cdots \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} c V ^ { 2 } - ( c + 2 ) U ^ { 2 } & = - 2 \\mu , \\\\ \\ ( c - 2 ) U ^ { 2 } - c Z ^ { 2 } & = - 2 \\mu , \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} \\theta h ' ( x ) + h ( x ) = g ( x ) , x > 0 , \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} \\zeta ( x _ \\mu - \\delta ) = - i \\zeta ( x _ \\mu + \\delta _ 1 ) = 1 . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} J _ { \\lambda , \\mu } ( u , v ) & = \\Big ( \\frac { 1 } { p } - \\frac { 1 } { q } \\Big ) \\| ( u , v ) \\| ^ p + 2 \\Big ( \\frac { 1 } { q } - \\frac { 1 } { \\alpha + \\beta } \\Big ) \\int _ \\Omega | u | ^ { \\alpha } | v | ^ \\beta d x \\\\ & < \\Big [ \\Big ( \\frac { 1 } { p } - \\frac { 1 } { q } \\Big ) + \\Big ( \\frac { 1 } { q } - \\frac { 1 } { \\alpha + \\beta } \\Big ) \\frac { p - q } { \\alpha + \\beta - q } \\Big ] \\| ( u , v ) \\| ^ p = - \\frac { ( p - q ) ( \\alpha + \\beta - p ) } { p q ( \\alpha + \\beta ) } \\| ( u , v ) \\| ^ p < 0 . \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} K \\begin{bmatrix} e _ { ( m + 1 ) ( p - 1 ) } ( x ) & e _ { ( m + 1 ) ( p - 1 ) + 1 } ( x ) & \\cdots & e _ { ( m + 1 ) ( p - 1 ) + m } ( x ) \\end{bmatrix} ^ { T } \\\\ = \\begin{bmatrix} e _ { ( m + 1 ) p } ( x ) & e _ { ( m + 1 ) p + 1 } ( x ) & \\cdots & e _ { ( m + 1 ) p + m } ( x ) \\end{bmatrix} ^ { T } . \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} \\Xi _ 2 ( z ) = \\exp \\Big ( - 2 \\pi \\lambda \\int _ 0 ^ \\infty { f _ { { \\rm { P r } } } } \\left ( u \\right ) ( 1 - \\Omega _ 1 ( z , u ) ) u d u - 2 \\pi \\lambda \\int _ 0 ^ \\infty ( 1 - { f _ { { \\rm { P r } } } } \\left ( u \\right ) ) ( 1 - \\Omega _ 2 ( z , u ) ) u d u \\Big ) \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{align*} 2 ^ { k _ m - k _ j } p _ 1 \\dots p _ j = \\sum _ { i = j + 1 } ^ m 2 ^ { k _ m - k _ i } c _ i p _ 1 \\dots p _ i . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} R _ \\mathsf { A } ^ \\mathsf { A l t } \\left ( Q _ { U , V , X | S } \\right ) \\triangleq \\min \\left \\{ \\begin{aligned} I ( V ; Y | U ) & - I ( V ; Z | U ) , \\\\ I ( U , V ; Y ) & - I ( U , V ; S ) , \\\\ I ( U , V ; Y ) & - I ( U ; S ) - I ( V ; Z | U ) \\end{aligned} \\right \\} , \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} \\mathcal { E } _ n ( k , \\ell ) = \\begin{cases} f _ n ^ \\ast ( k ) & \\mbox { o n t h e e v e n t } I _ n \\cap J _ n \\\\ 1 - F ^ \\ast ( k - 1 ) - F ^ \\ast ( \\ell - 1 ) & \\mbox { o n t h e e v e n t } I _ n \\cap J _ n ^ c \\\\ F ^ \\ast ( k ) + F ^ \\ast ( \\ell ) - 1 & \\mbox { o n t h e e v e n t } I _ n ^ c \\cap J _ n \\\\ f _ n ^ \\ast ( \\ell ) & \\mbox { o n t h e e v e n t } I _ n ^ c \\cap J _ n ^ c . \\end{cases} \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} \\| a \\| _ { \\ell ^ 2 } ^ 2 = \\sum _ { n \\in \\mathbb { N } } n | c _ { n + 1 } | ^ 2 = E _ c ( u ) , u \\in X _ c , \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} \\overline { ( x _ 1 , x _ 2 , \\ldots , x _ t ) } ^ { v , p } & = ( x _ 1 , x _ 2 , \\ldots , x _ { a - 1 } , \\overline { x _ { \\overline { v } ^ { \\{ t \\} } } + p } ^ { \\{ r \\} } , x _ { a + 1 } , \\ldots , x _ t ) \\\\ & = ( x _ 1 , x _ 2 , \\ldots , x _ { a - 1 } , \\overline { x _ { a } + p } ^ { \\{ r \\} } , x _ { a + 1 } , \\ldots , x _ t ) . \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} \\dim ( \\mathcal { V } _ \\mathcal { L } ) = m \\dim ( \\mathcal { L } ) . \\end{align*}"}
-{"id": "3365.png", "formula": "\\begin{align*} c ^ { \\mu } _ { m , n } = \\log _ 2 \\left ( 1 + \\frac { p _ n g ( \\boldsymbol { y } _ m , \\boldsymbol { y } _ n ) } { \\sum _ { n ' \\neq n } p _ { n ' } g ( \\boldsymbol { y } _ m , \\boldsymbol { y } _ { n ' } ) + w _ 2 N _ 0 } \\right ) , \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} \\rho ( \\lambda _ j ) = \\frac { 1 } { L ( \\lambda _ { j + 1 } - \\lambda _ j ) } . \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{align*} _ { n } = \\Delta ( _ { n } ) , \\forall n \\in \\{ 1 , . . . , \\mathsf { N } \\} , \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} 2 u _ n ' ( x ) = - \\sqrt { n + 1 } u _ { n + 1 } ( x ) + \\sqrt { n } u _ { n - 1 } ( x ) , n \\in \\mathbb { N } _ 0 . \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} H _ 3 + \\{ H _ 2 , F ^ { ( 3 ) } \\} = H _ { 3 , \\geq 2 } \\end{align*}"}
-{"id": "2553.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } f \\left ( \\mathcal { A } _ i \\right ) \\geq \\sum _ { j = 1 } ^ { n + 1 } f \\left ( \\mathcal { E } _ j ^ { ( n + 1 ) } \\right ) . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} X _ k ^ 2 \\leq X _ 0 ^ 2 + C _ 1 + C _ 2 \\sum _ { i = 0 } ^ k X _ i , \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} \\mathbb C T ^ { \\ast } X = T ^ { \\ast 1 , 0 } X \\oplus T ^ { \\ast 0 , 1 } X \\oplus \\{ \\lambda \\omega _ 0 : \\lambda \\in \\mathbb C \\} . \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} [ \\psi _ { \\lambda , - \\partial - \\lambda } ( a , b ) _ { \\lambda + \\mu } \\alpha ( c ) ] = - [ \\alpha ( c ) _ { - \\partial - \\lambda - \\mu } \\psi _ { \\lambda , - \\partial - \\lambda } ( a , b ) ] = - [ \\alpha ( c ) _ { - \\partial - \\lambda - \\mu } \\psi _ { \\lambda , \\mu } ( a , b ) ] . \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} \\int _ { a \\le | z | \\le b } z ^ \\gamma e ^ { - | z | ^ 2 } \\ , d z = \\bigg ( \\int _ a ^ b t ^ { \\langle \\gamma \\rangle + d - 1 } e ^ { - t ^ 2 } \\ , d t \\bigg ) \\bigg ( \\int _ { S _ + ^ { d - 1 } } w ^ \\gamma \\ , d \\sigma ( w ) \\bigg ) , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} g _ M ( R ( X , J _ M ( Y ) ) U , V ) = & g _ M ( R ( U , V ) X , J _ M ( Y ) \\\\ = & - g _ M ( R ( U , V ) J _ M ( X ) , Y ) \\\\ = & - g _ M ( R ( J _ M ( X ) , Y ) U , V ) . \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } | A _ n | ^ { 1 / n } = 1 \\quad \\mbox { a . s . } , \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} L = v \\left ( \\frac { u ' } { u } - u _ { 1 } + u _ { - 1 } \\right ) , \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} \\sum _ { v = 0 } ^ { \\ell ^ m - 1 } f ( z ) \\mid _ { - 1 } \\alpha _ 0 \\sigma _ { w _ v , \\ell ^ m } & = \\sum _ { v = 0 } ^ { \\ell ^ m - 1 } \\ell ^ { \\frac { m } { 2 } } \\sum _ { n = n _ 0 } ^ { \\infty } a _ 0 ( n ) e ^ { \\frac { 2 \\pi i n ( z + w _ v ) } { 2 4 \\ell ^ m } } \\\\ & = \\ell ^ { \\frac { m } { 2 } } \\sum _ { n = n _ 0 } ^ { \\infty } a _ 0 ( n ) q _ { 2 4 \\ell ^ m } ^ n \\sum _ { v = 0 } ^ { \\ell ^ m - 1 } e ^ { \\frac { 2 \\pi i n w _ v } { 2 4 \\ell ^ m } } . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} \\chi < 2 \\iff p > 1 - 1 / k \\chi = 2 \\iff p \\leq 1 - 1 / k . \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} \\gamma _ { e } ( \\tilde y _ e ) = \\left \\{ \\begin{array} { l l } 2 \\cosh { J _ e } , & \\\\ 2 \\sinh { J _ e } , & \\end{array} \\right . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} M ( t ) = \\int _ 0 ^ t \\int _ { ( 0 , \\infty ) } F ( W ( s - ) + x ) - F ( W ( s - ) ) N _ 0 ( d x , d s ) \\\\ - \\int _ 0 ^ t \\int _ { ( 0 , \\infty ) } F ( W ( s ) + x ) - F ( W ( s ) ) ) \\nu _ 0 ( d x ) d s \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} \\| u _ i \\| _ { 2 \\alpha } ^ 2 \\leq 2 ( \\| u _ i - \\bar { u } _ i \\| _ { 2 \\alpha } ^ 2 + \\| \\bar { u } _ i \\| _ { 2 \\alpha } ^ 2 ) \\leq c r _ i ^ { 2 + m ( \\frac { 1 } { \\alpha } - 1 ) } \\| \\nabla u _ i \\| _ 2 ^ 2 + c r _ i ^ { \\frac { m } { \\alpha } } \\bar { u } _ i ^ 2 \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} \\begin{bmatrix} \\alpha _ 1 & \\alpha _ 2 \\end{bmatrix} = n ; \\begin{bmatrix} \\beta _ 1 & \\beta _ 2 \\end{bmatrix} = n , \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} \\sum _ { n \\ge 0 } \\Big ( \\sum _ { k = 2 ^ n } ^ { 2 ^ { n + 1 } - 1 } | a _ k | \\Big ) ^ 2 < \\infty \\ , . \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} \\nu ( A _ { x _ o } ^ { - 1 } C ) = \\mu ( A ) + \\nu ( C ) < 1 . \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{gather*} ( A , B ) = \\int _ M \\left ( \\frac { \\delta _ r A } { \\delta \\Phi ^ A } \\frac { \\delta _ l B } { \\delta \\Phi ^ \\ast _ A } - \\frac { \\delta _ r A } { \\delta \\Phi ^ \\ast _ A } \\frac { \\delta _ l B } { \\delta \\Phi ^ A } \\right ) d ^ n x . \\end{gather*}"}
-{"id": "3516.png", "formula": "\\begin{align*} \\phi _ 3 ( c , r , p ) & = ( c ^ 3 - 2 c - 1 0 ) ^ 2 + 2 r ( 4 - c ^ 2 ) ( c ^ 3 - 2 c - 1 0 ) ( 2 p + 2 c p - 6 c r p ^ 2 + 3 r c ) \\\\ & + r ^ 2 ( 4 - c ^ 2 ) ^ 2 ( 4 c ^ 2 + 4 + 9 c ^ 2 r ^ 2 + 8 c - 1 2 c ^ 2 r p - 1 2 c r p ) . \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} \\tilde { \\phi } = \\frac { 1 } { n } \\sum _ { \\ell = 1 } ^ { L } \\sum _ { i \\in I _ { \\ell } } \\phi ( W _ { i } , \\hat { \\gamma } _ { \\ell } , \\alpha _ { 0 } , \\theta _ { 0 } ) . \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} d \\psi _ m ( v ) ( [ f ] _ n ) = v ( \\psi ^ * _ m ( [ f ] _ n ) ) \\ ; , \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} F _ \\ell = \\begin{pmatrix} \\frac { a _ \\ell \\beta _ \\ell - \\Delta _ \\ell } { 2 \\beta _ \\ell } & \\frac { \\Delta _ \\ell ( \\beta _ \\ell ^ 2 - \\Delta _ \\ell ) } { 4 \\beta _ \\ell ^ 3 } \\\\ \\beta _ \\ell & \\frac { a _ \\ell \\beta _ \\ell + \\Delta _ \\ell } { 2 \\beta _ \\ell } \\end{pmatrix} . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} \\deg ( I d - G _ 0 , \\tilde \\Gamma , 0 ) = \\deg ( I d - H _ 1 , \\tilde \\Gamma , 0 ) = \\deg ( I d - H _ 0 , \\tilde \\Gamma , 0 ) . \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} \\sum _ { c \\mid w } \\chi ( c ) \\log { \\sqrt { w } / c } = 0 . \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} C C h ^ * { \\cal F } = h ^ ! C C { \\cal F } . \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} a \\in \\mathcal { A } ( \\langle S _ i \\rangle ) \\subseteq \\bigcup _ { i = 1 } ^ n \\mathcal { A } ( \\langle S _ i \\rangle ) . \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} & L _ 2 : = L _ 2 ( 0 , L ; \\R ^ 3 ) , \\\\ [ 5 p t ] & H _ N ^ 2 : = H _ N ^ 2 ( 0 , L ; \\R ^ 3 ) : = \\{ u \\in H ^ 2 ( 0 , L ; \\R ^ 3 ) \\mid \\partial _ x u ( 0 ) = \\partial _ x u ( L ) = 0 \\} . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} \\langle f _ { s _ 1 } , f _ { s _ 2 } \\rangle _ { K } = \\prod _ { j = s _ 2 + 1 } ^ \\infty ( 1 + k _ j ( y _ j , y _ j ) ) = \\| f _ { s _ 2 } \\| _ { K } ^ 2 \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} P _ { Y | S } ( 0 | s ) = P _ { Y | S } ( 0 | s ' ) , \\quad \\forall ( s , s ' ) \\in \\mathcal { S } ^ 2 , \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} S ^ \\eta _ \\pm : = A ^ { - 1 } \\big ( Q ' ( u ^ \\pm ) \\pm \\eta A \\big ) , S ^ { \\mathrm { r } } _ \\pm = \\tilde { A } ^ { - 1 } Q _ { 2 2 } ' ( u ^ \\pm ) . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } a _ k x ^ { k } \\star \\sum _ { l = 0 } ^ { \\infty } b _ l y ^ { l } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { k \\leq n } a _ k b _ { n - k } x ^ { k } y ^ { n - k } . \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} \\hat { A } \\xi = \\int _ { - \\pi } ^ { \\pi } { h } ( \\theta ) \\left ( d Z ^ { \\xi } ( \\theta ) + d Z ^ { \\eta } ( \\theta ) \\right ) . \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} \\begin{aligned} & \\norm { \\varphi } _ { L ^ { \\infty } ( 0 , T ; H ^ { 2 } ) \\cap L ^ { 2 } ( 0 , T ; H ^ { 3 } ) \\cap H ^ { 1 } ( 0 , T ; L ^ { 2 } ) } + \\norm { \\mu } _ { L ^ { 2 } ( 0 , T ; H ^ { 2 } ) \\cap L ^ { \\infty } ( 0 , T ; L ^ { 2 } ) } \\leq C _ { \\mathrm { A P 2 } } . \\end{aligned} \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} \\mathrm { r a n k } \\big ( \\mathbf { M } _ { \\gamma } \\{ y \\} \\big ) = 1 . \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\mathcal { G } _ q ( \\mathcal { H } _ + , \\alpha ) : = \\projlim _ { r \\ge 1 } \\mathcal { G } _ q ( \\mathcal { H } _ + , r , \\alpha ) . \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} y ( n + 1 ) = U ( n ) y ( n ) , \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} ( T _ { \\theta } u ) _ n = \\left \\lbrace \\begin{matrix} \\sqrt { \\frac { \\prod _ { j = 0 } ^ { n - 1 } \\tilde { c } ( \\theta + j \\alpha ) } { \\prod _ { j = 0 } ^ { n - 1 } c ( \\theta + j \\alpha ) } } u _ n \\ \\ & n \\geq 1 , \\\\ u _ n \\ \\ & n = 0 , \\\\ \\sqrt { \\frac { \\prod _ { j = n } ^ { - 1 } c ( \\theta + j \\alpha ) } { \\prod _ { j = n } ^ { - 1 } \\tilde { c } ( \\theta + j \\alpha ) } } u _ n \\ \\ & n \\leq - 1 , \\end{matrix} \\right . \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} S ^ * _ h ( x ) = S ^ * _ { h k } ( x ) + x E _ { h , k } ( x ) \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} \\widetilde u ( s , t ) : = u ( \\Phi ( s , t ) ) . \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} \\rho ( e u ) = \\mu ( e u ) \\lambda ( e u ) , \\ \\rho ( e v ) = \\mu ( e v ) \\lambda ( e v ) . \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} z \\longmapsto M ( z ) : = z \\sum _ { k \\geq 0 } z ^ { k } H _ + ^ { - k - 1 } \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} \\binom { | A ^ { \\ast } | } { w _ 1 } \\binom { \\ell } { w _ 2 } ^ { \\rho } \\ ! \\ ! \\leq \\exp \\left [ | A ^ { \\ast } | H _ 2 \\Big ( \\frac { w _ 1 } { | A ^ { \\ast } | } \\Big ) + \\rho \\ell H _ 2 \\left ( \\frac { w _ 2 } { \\ell } \\right ) \\right ] . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\vdash n } \\sum _ { \\substack { \\lambda _ i \\in \\lambda \\\\ \\lambda _ i } } a ( \\lambda _ i ) = \\sum _ { k = 1 } ^ { n } p ( n - k ) a ( n ) . \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} P _ { n , n , \\beta } ( \\sigma , \\tau ) = \\frac { 1 } { Z _ { n , n } ( \\beta ) } \\exp ( - \\beta H _ n ( \\sigma , \\tau ) ) P _ n \\ ! \\times \\ ! P _ n ( \\sigma , \\tau ) \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{gather*} \\eta = \\frac { 2 \\alpha } { q _ 2 - 1 } - \\frac { u ^ 2 } { \\omega } - \\frac { u _ t } { u } \\frac { 1 + q _ 2 } { 1 - q _ 2 } , \\omega = u ^ 4 + t u ^ 2 - u ^ 2 _ t . \\end{gather*}"}
-{"id": "6773.png", "formula": "\\begin{align*} \\left ( U , Z \\right ) = \\left ( \\pm 1 , \\pm 1 \\right ) , \\mu = 1 \\left ( U , Z \\right ) = \\left ( \\pm i , \\pm i \\right ) , \\mu = - 1 \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n s _ { k i } ^ 2 ( w ) \\leq P . \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} s _ { n + 3 } = s _ { n + 2 } - p \\big ( s _ { n } - s _ { n - 1 } \\big ) + r _ { n + 3 } - r _ { n + 2 } . \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} A _ { 1 1 } h + A _ { 1 2 } v = 0 , s A _ { 2 1 } h + s A _ { 2 2 } v + T _ { 2 2 } v = 0 . \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} f _ i = \\frac { 1 } { \\binom n i ^ 2 } . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } ( \\eta '' - \\xi '' ) ( \\phi _ { \\nu } - \\phi _ { \\mu } ) \\ , d s & = \\int _ { 0 } ^ { 1 } - ( \\eta - \\xi ) ' ( \\phi _ { \\nu } ' - \\phi _ { \\mu } ' ) \\ , d s - \\eta ' ( 0 ) ( \\phi _ { \\nu } ( 0 ) - \\phi _ { \\mu } ( 0 ) ) \\\\ & = \\int _ { 0 } ^ { 1 } ( \\eta - \\xi ) ( \\phi _ { \\nu } '' - \\phi _ { \\mu } '' ) \\ , d s - \\eta ' ( 0 ) ( \\phi _ { \\nu } ( 0 ) - \\phi _ { \\mu } ( 0 ) ) + \\eta ( 0 ) ( \\phi _ { \\nu } ' ( 0 ) - \\phi _ { \\mu } ' ( 0 ) ) . \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} ( L _ q - L ^ - ) Y = B _ q Y B _ q ( x ) = \\partial _ { Y } R ( Y _ q ( x ) ) - \\partial _ Y R ( 0 ) . \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} a _ { 0 } = 0 , \\thinspace \\thinspace a _ { 1 } = 1 \\thinspace \\thinspace \\thinspace \\thinspace a _ { 2 n } = a _ { n } , \\thinspace \\thinspace \\thinspace a _ { 2 n + 1 } = a _ { n } + a _ { n + 1 } , \\thinspace \\thinspace n \\ge 1 . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} \\sum _ { r \\in [ s , T ] } q ^ { n , [ s , T ] } ( r ) \\ , \\psi ( r ) \\leq \\sum _ { r \\in [ t , T ] } q ^ { n , [ t , T ] } ( r ) \\ , \\psi ( r ) . \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} d ^ n f ( s _ 1 , \\ldots , s _ { n + 1 } ) : = \\kappa _ { s _ 1 } f ( s _ 2 , \\ldots , s _ { n + 1 } ) & \\\\ + \\sum _ { i = 1 } ^ n { ( - 1 ) ^ n f ( s _ 1 , \\ldots , s _ i + s _ { i + 1 } , \\ldots , s _ { n + 1 } ) } & + ( - 1 ) ^ { n + 1 } f ( s _ 1 , \\ldots , s _ n ) \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} \\underline { D } ^ q ( \\mu ) = \\liminf _ { \\delta \\to 0 } \\frac { \\log M ^ q _ \\delta ( \\mu ) } { ( q - 1 ) \\log \\delta } \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} G _ t ( x ) = \\begin{cases} \\frac 1 { \\sqrt { 4 \\pi t } } \\exp ( - \\frac { | x | ^ 2 } { 4 t } ) , & ; \\\\ \\frac 1 2 \\mathbf 1 _ { \\{ | x | < t \\} } , & , \\end{cases} \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} & \\mathbb { E } \\left [ \\frac { 1 } { t } \\int _ 0 ^ t \\left ( | \\varphi _ s ( \\cdot , x ) | ^ 2 - | a ( \\theta _ s \\cdot ) | ^ 2 \\right ) \\ , d s \\right ] \\\\ & \\leq \\frac { 1 } { t } \\ , \\mathbb { E } \\left [ \\int _ 0 ^ { t _ 0 } \\left ( | \\varphi _ s ( \\cdot , x ) | ^ 2 - | a ( \\theta _ s \\cdot ) | ^ 2 \\right ) \\ , d s \\right ] + \\frac { t - t _ 0 } { t } \\ , \\frac { \\lambda _ { t o p } } { 2 } \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} \\tau _ { 1 } ( G _ n ' ) \\ge ( n - 1 ) 2 ^ { 1 8 n - 2 } = : \\tau \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { \\beta ( n ) } = K _ 3 \\log x + K _ 4 + O ( x ^ { - u } ) \\end{align*}"}
-{"id": "4427.png", "formula": "\\begin{align*} \\phi _ { \\epsilon } - \\phi = \\begin{cases} \\epsilon - \\phi _ { \\mu } ( 1 - \\epsilon ) & 0 \\leq t < 1 - \\epsilon \\\\ 1 - t - \\phi _ { \\mu } ( t ) & 1 - \\epsilon \\leq t \\leq 1 \\end{cases} . \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} \\left . w ^ { ( v ) } _ i ( n ) \\right | _ { n = - N _ { \\rm c p } } = \\left . \\bar { x } ^ { ( v ) } _ { i - 1 } ( n ) \\right | _ { n = N } - \\left . x ^ { ( v ) } _ { i } ( n ) \\right | _ { n = - N _ { \\rm c p } } \\end{align*}"}
-{"id": "9522.png", "formula": "\\begin{align*} T _ t \\alpha = \\mathbb { E } [ f ( e ^ { - t } ( g , \\xi ) + \\sqrt { 1 - e ^ { - 2 t } } ( g , \\xi ' ) ) / \\xi ] , \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} \\frac { u ' } { u } - u _ 1 + u _ { - 1 } = 0 . \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} \\omega ^ { 0 } _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\cdot ) : = { \\frac { 1 } { \\Xi _ { \\Lambda } } } \\ { \\rm { T r } } [ e ^ { - \\beta H _ { 0 , \\Lambda , \\mu , \\lambda } } \\ ( \\cdot ) ] \\ , \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} \\begin{aligned} \\partial _ u \\nabla ( H _ { \\geq 5 } \\circ \\Phi _ B ) ( T _ { \\delta } ) [ h ] & = ( \\partial _ u \\nabla H _ { \\geq 5 } ) ( \\Phi _ B ( T _ { \\delta } ) ) [ h ] + \\mathcal { R } _ { H _ { \\geq 5 } } ( T _ { \\delta } ) [ h ] = \\\\ & = \\partial _ x ( r _ 1 ( T _ { \\delta } ) \\ , \\partial _ x h ) + r _ 0 ( T _ { \\delta } ) h + \\mathcal { R } _ { H _ { \\geq 5 } } ( T _ { \\delta } ) [ h ] \\end{aligned} \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} \\| { \\tt Q } _ { n } f \\| = q ^ { - ( n + 1 ) d } , \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int ^ \\infty _ 0 \\frac { e ^ { - t } } { ( 1 - e ^ { - 2 t } ) ^ { 3 / 2 } } ( y - e ^ { - t } x ) e ^ { - \\frac { ( y - e ^ { - t } x ) ^ 2 } { 2 ( 1 - e ^ { - 2 t } ) } } d t . \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ \\infty ( 1 - | \\alpha _ j | ^ 2 ) = \\exp \\biggl ( \\int _ 0 ^ { 2 \\pi } \\log ( w ( \\theta ) ) \\ , \\frac { d \\theta } { 2 \\pi } \\biggr ) . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} g _ i = \\sum _ { \\substack { 1 \\leq j \\leq i \\\\ d _ i = d _ j } } \\lambda _ { i j } f _ j + \\sum _ { \\substack { 1 \\leq j \\leq i \\\\ d _ i > d _ j } } \\sum _ { 1 \\leq k \\leq n } \\lambda _ { i j k } x _ k ^ { d _ i - d _ j } f _ j \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} \\frac { d \\mathbf { p } } { d x } = \\mathbb { A } ( x ; \\lambda ) \\mathbf { p } ; \\mathbb { A } ( x ; \\lambda ) = \\begin{pmatrix} 0 & I \\\\ V ( x ) - \\lambda I & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} \\sup _ { x } | \\hat { \\gamma } _ { 2 \\ell } ( x ) - \\beta _ { L } ^ { \\prime } b ( x ) | & \\leq \\max _ { j } \\sup _ { x } \\left \\vert b _ { j } ( x ) \\right \\vert \\sum _ { j = 1 } ^ { p } \\left \\vert \\hat { \\beta } _ { j } - \\beta _ { L j } \\right \\vert = O _ { p } ( ( \\varepsilon _ { n } ^ { 2 } ) ^ { - 1 / ( 2 \\xi _ { 1 } + 1 ) } r _ { 2 } ) \\\\ & = O _ { p } ( n ^ { - d _ { 1 } ( 2 \\xi _ { 1 } - 1 ) / ( 2 \\xi _ { 1 } + 1 ) } \\ln ( n ) ) . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} \\sum _ { a = 1 } ^ { p ' } ( - 1 ) ^ { p ' - a } \\binom { p ' } { a } \\binom { p ' + a } { a } ( 2 H _ { 2 a } - H _ a - H _ { p ' } ) \\equiv _ { p } 0 . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{align*} \\varphi _ { \\hat h , m \\hat v } [ N \\alpha , \\beta , \\gamma / \\hat h ] = [ N \\alpha , \\beta , ( \\gamma - m \\hat v \\alpha ) / \\hat h ] . \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} d = 1 + \\min \\{ | \\xi | , | \\eta | , | \\zeta | \\} , d _ 1 = 1 + \\min \\{ | \\eta | , | \\zeta | \\} . \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\operatorname { R e } \\left ( \\frac { c ^ { 2 } - 4 } { c \\overline { a } ^ { 2 } } \\right ) = 0 . \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} C ( m , n ; k ) & : = \\{ g \\in G _ { n } : g _ { \\tau } \\in C ( 0 , n - m ; k ) \\tau \\in \\Omega _ { m } \\} . \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} C _ \\mathsf { R L N } ( \\alpha , \\sigma ) = \\max _ { P _ X , P _ { A | S } } \\min \\Big \\{ I ( A ; S _ 1 ) , I ( X ; Y ) - I ( A ; S | S _ 1 ) \\Big \\} . \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{gather*} x '' ( t ) + \\omega ^ 2 x ( t ) = \\lambda f ( t , x ( t ) ) , t \\in [ 0 , 1 ] , \\\\ x ( 0 ) = x ( 1 ) , x ' ( 0 ) = x ' ( 1 ) , \\end{gather*}"}
-{"id": "7300.png", "formula": "\\begin{align*} \\sqrt { n } \\hat { \\psi } ( \\theta _ { 0 } ) = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } \\psi ( W _ { i } , \\theta _ { 0 } , \\gamma _ { 0 } , \\alpha _ { 0 } ) + o _ { p } ( 1 ) . \\end{align*}"}
-{"id": "7305.png", "formula": "\\begin{align*} E _ { F _ { n } } [ Y | X ] & = \\gamma _ { 0 } ( X ) + \\left ( \\frac { \\mu } { \\sqrt { n } } \\right ) [ \\bar { \\alpha } ( x ) - \\alpha _ { 0 } ( x ) ] , \\\\ \\theta _ { n } & = E [ Z E _ { F _ { n } } [ Y | X ] ] = E [ \\alpha _ { 0 } ( X ) E _ { F _ { n } } [ Y | X ] ] = \\theta _ { 0 } - \\left ( \\frac { \\mu } { \\sqrt { n } } \\right ) \\bar { \\sigma } ^ { 2 } . \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} f _ j = \\int _ { \\Omega } f ( y ) w _ j ( y ) d y \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} C _ 3 = \\frac { \\operatorname { d i s t } ( I , J ^ c ) ^ 2 } { C _ 2 \\operatorname { d i s t } ( I , J ^ c ) ^ 2 - \\lVert V \\rVert ^ 2 ( C _ 2 + 1 + \\lVert W \\rVert ) } . \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} \\widetilde P \\psi ( x , y , z ) = \\left \\{ \\begin{array} { l l } \\psi ( x , y , z ) , \\ ; & \\mbox { i f } z > 0 \\\\ ( \\psi ^ 1 , - \\psi ^ 2 , - \\psi ^ 3 , \\psi ^ 4 ) ^ T ( x , y , - z ) = \\mathcal { B } ^ 0 \\left ( { \\psi } \\circ S \\right ) ( x , y , z ) , \\ ; & \\mbox { i f } z < 0 \\end{array} \\right . \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} \\partial _ { t } v ( t , x , y ) = D ^ { 2 } _ { x } v ( t , x , y ) + \\lambda [ v ( t , x , y + h ) - v ( t , x , y ) ] + f ( t , x , y ) \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} S _ f - \\bar { S } _ f = k \\xi . \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} F ^ { \\left ( b \\right ) } _ { n } = \\prod _ { i = 0 } ^ { N - 1 } F _ { n _ i } . \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} d \\hat { \\rho } ( x ) = \\frac { 1 } { Z _ \\sigma } e ^ { \\frac { 2 } { \\sigma ^ 2 } ( \\frac { 1 } { 2 } | x | ^ 2 - \\frac { 1 } { 4 } | x | ^ 4 ) } \\ , d x , \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} S _ h ( x ) = B C ( h ) x + O \\bigl ( L ( 1 , \\chi ) x \\log x + x / \\log x \\bigr ) \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} L ( e _ { i i } ) - L ( e _ { j j } ) & = [ L ( e _ { i j } ) , e _ { j i } ] + [ e _ { i j } , L ( e _ { j i } ) ] \\\\ & = \\sum _ { x < i } C _ { x i } ^ { i i } e _ { x i } + C _ { i j } ^ { i j } e _ { i i } - C _ { i j } ^ { i j } e _ { j j } - \\sum _ { y > j } C _ { j y } ^ { j j } e _ { j y } \\\\ & + C _ { j i } ^ { j i } e _ { i i } + \\sum _ { y > i } C _ { i y } ^ { i i } e _ { i y } - \\sum _ { x < j } C _ { x j } ^ { j j } e _ { x j } - C _ { j i } ^ { j i } e _ { j j } . \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} \\beta _ t : = \\prod _ { s = 0 } ^ { t - 1 } \\mu ( \\theta _ s ) , & & t = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} e ^ { - \\beta H _ n } = e ^ { - \\beta ( H _ m - \\delta ) } = \\sum _ { j \\ge 0 } \\int _ { \\mathcal { T } _ j ( \\beta ) } \\delta ( s _ 1 ) \\cdots \\delta ( s _ n ) e ^ { - \\beta H _ m } d s _ 1 \\cdots d s _ n , \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} \\begin{aligned} \\big | \\big | \\nabla ( e ^ { i t A } f - f ) \\big | \\big | _ { L ^ 2 } ^ 2 \\le | t | ^ 2 \\sup _ { | t | \\le 1 } \\big | \\big | \\nabla A e ^ { i t A } f \\big | \\big | _ { L ^ 2 } ^ 2 \\le C | t | ^ 2 | | \\nabla A f | | _ { L ^ 2 } ^ 2 , \\end{aligned} \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} \\mathcal { D } _ t u + G ( \\mathcal { D } _ x ^ 2 u ) = 0 , \\ , \\ , \\ , \\ , u _ T = \\xi . \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\widetilde { E } _ \\eta ( n ) = & \\frac { H H ' } { \\Lambda } \\sum _ { c } \\frac { w _ 0 ( c / C ) } { c ^ 2 } \\sum _ { m _ 1 } \\sum _ { m _ 2 } \\lambda _ { 1 } ( m _ 1 ) \\lambda _ { 2 } ( m _ 2 ) S ( m _ 1 + m _ 2 , 2 n ; c ) \\\\ & \\cdot W ^ { \\star } _ { \\eta H } \\left ( \\frac { m _ 1 n } { c ^ 2 } , \\frac { m _ 1 H } { c ^ 2 } \\right ) V ^ { \\star } _ { - \\eta H ' } \\left ( \\frac { m _ 2 n } { c ^ 2 } , \\frac { m _ 2 H ' } { c ^ 2 } \\right ) . \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} g ( t _ \\epsilon w _ { \\epsilon , \\eta } ) = \\displaystyle \\sup _ { t \\geq 0 } g ( t w _ { \\epsilon , \\eta } ) \\ ; \\textrm { a n d } \\ ; \\frac { d } { d t } g ( t w _ { \\epsilon , \\eta } ) \\mid _ { t = t _ \\epsilon } = 0 . \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } \\beta _ i ( T - \\alpha _ i I ) ^ { - 1 } . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} \\frac { H _ { 1 } } { \\beta } = - \\sum _ a T ^ { a a } _ { 1 } + \\frac { 1 } { 2 } \\sum _ { a , b } T ^ { a b } _ { 0 } T ^ { b a } _ { 0 } - \\frac { s } { 2 } \\sum _ a T ^ { a a } _ { 0 } , \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} T _ 1 \\times ( T _ 2 + T _ 3 ) = \\{ z \\} \\times ( T _ 2 + T _ 3 ) = ( \\{ z \\} \\times T _ 2 ) + ( \\{ z \\} \\times T _ 3 ) = T _ 1 \\times T _ 2 + T _ 1 \\times T _ 3 . \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} & v _ { 0 } = \\epsilon , v _ { 1 } = \\epsilon ( 2 c + 3 ) , \\ v _ { m + 2 } = ( 2 c + 2 ) v _ { m + 1 } - v _ { m } , \\ m \\geq 0 , \\\\ & u _ { 0 } = \\epsilon , u _ { 1 } = \\epsilon ( 2 c + 1 ) , \\ u _ { m + 2 } = ( 2 c + 2 ) u _ { m + 1 } - u _ { m } , \\ m \\geq 0 . \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} \\psi _ 1 ^ h & = u ^ h - u - x _ 2 \\Big ( \\frac { R _ { 2 1 } ^ h } { h } - v _ 2 ' \\Big ) - x _ 3 \\Big ( \\frac { R _ { 3 1 } ^ h } { h } - v _ 3 ' \\Big ) + \\beta _ 1 ^ h \\ , , \\\\ \\psi _ i ^ h & = \\frac { 1 } { h } ( v _ i ^ h - v _ i ) + ( w ^ h - w ) x _ i ^ \\bot + \\beta _ i ^ h \\ , , i = 2 , 3 \\ , . \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} \\lambda \\sigma ^ { - A } = \\sigma ^ { - \\frac { \\theta } { p - 1 } } \\cdot \\lambda \\sigma ^ { \\frac { \\theta } { p - 1 } - A } \\le \\sigma ^ { - \\frac { \\theta } { p - 1 } } \\cdot \\lambda \\tilde { T } _ \\lambda ^ { \\frac { 1 } { p - 1 } - \\frac { A } { \\theta } } = \\delta ^ { \\frac { 1 } { p - 1 } - \\frac { A } { \\theta } } \\sigma ^ { - \\frac { \\theta } { p - 1 } } \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} \\inf \\Psi _ { t , p } = \\inf \\left [ ( 1 - t ) ( q ^ * \\psi _ 0 - p ) + t ( q ^ * \\psi _ 1 - p ) \\right ] \\geq ( 1 - t ) \\inf \\Psi _ { 0 , p } + t \\inf \\Psi _ { 1 , p } . \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\rho ( [ n ] _ v ) = \\rho \\left ( \\frac { v ^ n - v ^ { - n } } { v - v ^ { - 1 } } \\right ) = q ^ { ( 1 - n ) p } ( 1 + q ^ { 2 p } + \\cdots + q ^ { 2 p ( n - 1 ) } ) = n q ^ { ( 1 - n ) p } , \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{gather*} ( 3 5 - 3 \\sqrt { 6 1 } ) { \\large ( } 2 9 x ^ { 2 } - 4 x y + 3 6 y ^ { 2 } ) \\\\ + 4 8 ( \\allowbreak 7 2 - 1 1 \\sqrt { 6 1 } ) ( x + 2 y ) \\\\ + 1 6 ( 8 8 7 - 1 0 5 \\sqrt { 6 1 } ) = 0 \\end{gather*}"}
-{"id": "3269.png", "formula": "\\begin{align*} e _ { \\iota } = g e _ j ^ n ; \\iota = ( g , j ) \\in I _ n \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\omega _ { \\mathcal { O } _ x } : \\ \\ \\mathcal { X } ( \\mathcal { O } _ x ) \\times \\mathcal { X } ( \\mathcal { O } _ x ) & \\longrightarrow C ^ \\infty ( \\mathcal { O } _ x ) \\\\ \\omega _ { \\mathcal { O } _ x } ( X , Y ) ( p ) & : = \\omega ( a , b ) ( p ) , \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } x = - b ( \\varphi , \\tau , x ) . \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} u = \\exp ( v _ 1 - v _ { - 1 } ) , \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} \\frac { \\partial { U ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ i } } & = u _ i ( s ^ i ( \\hat { x } , \\theta ) ) + \\sum _ { j = 1 } ^ n \\theta ^ j u _ j ' ( s ^ j ( \\hat { x } , \\theta ) ) \\frac { \\partial { s ^ j ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ i } } \\\\ & = u _ i ( s ^ i ( \\hat { x } , \\theta ) ) + \\lambda ( \\hat { x } , \\theta ) \\sum _ { j = 1 } ^ n \\frac { \\partial { s ^ j ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ i } } , \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{align*} x _ { _ { n } } = y _ { _ { n } } , y _ { _ { n } } = x _ { _ { n } } , s _ { _ { n } } = s _ { _ { n } } ^ { - 1 } . \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} B _ 1 ( n ) = \\frac { n } { 2 k _ n } \\log ( 1 + k _ n P ) . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} u _ { n } v _ { m } = q ^ { \\delta _ { n , m } } v _ { m } u _ { n } \\forall n , m \\in \\{ 1 , . . . , \\mathsf { N } \\} , \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} P = a \\sqrt { c } = \\pm d \\sqrt { c } = \\pm Q , \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} F = \\kappa ^ { - 1 } _ \\omega \\circ T ^ { - 1 } \\circ \\kappa _ { \\omega ' } : \\kappa _ { \\omega ' } ^ { - 1 } ( V _ { \\omega ' \\omega } ) \\to U _ \\omega \\ , , \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ^ c D ^ { q } z _ 1 ( t ) = - z _ 1 ( t ) + ( 2 - 5 i ) g ( z _ 1 ( t ) ) - ( 2 + i ) g ( z _ 2 ( t ) ) + ( 2 + i ) g ( z _ 3 ( t ) ) \\\\ ^ c D ^ { q } z _ 2 ( t ) = - 2 z _ 2 ( t ) + 3 g ( z _ 1 ( t ) ) + ( 1 + i ) g ( z _ 2 ( t ) ) \\\\ ^ c D ^ { q } z _ 3 ( t ) = - 2 z _ 3 ( t ) + ( 1 - i ) g ( x _ 1 ( t ) ) + ( 1 + i ) g ( z _ 3 ( t ) ) \\end{array} \\right . \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} f \\left ( t , x , 1 \\right ) & = - \\frac { 1 } { 2 } x - \\left \\vert \\sin t \\right \\vert \\sqrt [ 4 ] { \\left \\vert x \\right \\vert } \\sqrt [ 3 ] { x } , \\ ; f \\left ( t , x , 2 \\right ) = - \\frac { 1 } { 2 } x , \\\\ g \\left ( t , y , 1 \\right ) & = - \\sqrt { \\left \\vert b \\left ( t \\right ) \\right \\vert } y \\cos t , \\ ; g \\left ( t , y , 2 \\right ) = \\sqrt { \\left \\vert b \\left ( t \\right ) \\right \\vert } y \\sin t , \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} | | F | | _ { L ^ 2 _ T L ^ { q , \\sigma } } \\approx \\sup _ { 1 = | | G | | _ { L ^ 2 _ T L ^ { q ^ { \\prime } , \\sigma ^ { \\prime } } } } | \\langle F , G \\rangle _ T | . \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} T _ t ^ { 1 , 0 } X = { \\rm S p a n } _ { \\mathbb C } \\left \\{ Z _ j + \\sum _ { k = 1 } ^ { n - 1 } \\Phi _ { j \\overline k } ( \\cdot , t ) \\overline Z _ k , j = 1 , \\ldots , n - 1 \\right \\} \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} 2 \\frac { d b _ n } { d t } = ( n + 1 ) \\sqrt { n + 2 } b _ { n + 1 } - ( n - 1 ) \\sqrt { n + 1 } b _ { n - 1 } , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{align*} Q _ { i j } ( D ) = d _ { i j } \\in z \\mathbb { C } [ [ z ] ] . \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} S ( X , Y ) \\le \\sum _ { d \\le U } \\frac { \\tau ( d ) ^ 6 } { d } \\sum _ { u v = d } S ^ \\flat _ { u v } \\left ( \\frac { X } { d } , \\frac { Y } { d } \\right ) + O ( ( \\log { Y } ) ^ { - 2 } ) . \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} \\lim _ { h _ 1 \\to \\infty } \\sum _ { k = N _ 0 + 1 } ^ { \\infty } \\frac { 1 } { k ^ { 2 \\alpha } ( \\mu _ k + h _ 1 ) } = 0 . \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} x _ 1 = \\log \\frac { 1 - u _ 1 } { 1 - u _ 2 } , \\ ; x _ 2 = \\log \\frac { 1 } { u _ 2 } , \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ N \\left ( \\max _ { i \\in \\mathcal { A } _ j } \\ell _ i + \\max _ { i \\in ( [ 1 : N ] \\backslash \\{ j \\} ) \\backslash \\mathcal { A } _ j } r _ i \\right ) \\geq \\sum _ { j = 1 } ^ { N - 1 } \\left ( \\max _ { i \\in \\mathcal { A } _ { { \\rm { F } } j } } \\ell _ i + \\max _ { i \\in [ 1 : N ] \\backslash \\mathcal { A } _ { { \\rm { F } } j } } r _ i \\right ) . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{gather*} L = \\frac { 1 } { 2 } \\begin{pmatrix} x ^ 2 - t + x ^ 2 q _ 2 - x q _ 1 + q _ 0 & 2 ( x ^ 3 - x ^ 2 e _ 1 + x e _ 2 - e _ 3 ) \\\\ ( x + e _ 1 ) \\dfrac { 1 - q _ 2 ^ 2 } { 2 } + q _ 1 q _ 2 & x ^ 2 - t - x ^ 2 q _ 2 + x q _ 1 - q _ 0 \\end{pmatrix} , \\end{gather*}"}
-{"id": "3332.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = x - \\lambda y , \\\\ \\dot y = \\lambda \\big ( h ( t , y ) + \\phi ( t , y ) x \\big ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} \\iota _ { X _ t } \\Theta _ t = - \\beta . \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} \\langle \\alpha | \\beta \\rangle = \\sum _ { h _ { 1 } , . . . , h _ { \\mathsf { N } } = 0 } ^ { p - 1 } V ( X _ { 1 } ^ { ( h _ { 1 } ) } , . . . , X _ { \\mathsf { N } } ^ { ( h _ { \\mathsf { N } } ) } ) \\prod _ { a = 1 } ^ { \\mathsf { N } } \\alpha _ { a } ^ { ( h _ { a } ) } \\beta _ { a } ^ { ( h _ { a } ) } , \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} \\hat { \\theta } _ { j } = p _ { j } \\theta + \\xi _ { j } , \\ \\forall j \\in \\{ 1 , 2 , \\dots , k \\} \\backslash \\{ j ' , j '' \\} . \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} \\widetilde { J } _ i ^ { ( a ) } = \\begin{cases} J _ i ^ { ( a ) } & i \\neq i ^ { ( a ) } , \\\\ p _ { \\ell ^ { ( a ) } } ^ { ( a ) } ( \\widetilde { \\nu } _ { i ^ { ( a ) } } ^ { ( a ) } , B ^ { 1 , s _ { m - 1 } } \\otimes \\cdots \\otimes B ^ { 1 , s _ 1 } ) & i = i ^ { ( a ) } , \\end{cases} \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} \\pi _ 0 [ u ] = u ( x ) - \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { T } } u ( x ) \\ , d x . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} \\ell ( x - x _ { 0 } ) = o ( \\| x - x _ { 0 } \\| ) . \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} S _ 1 = \\begin{cases} S , E _ \\sigma = 0 \\\\ ? , E _ \\sigma = 1 \\end{cases} . \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} \\kappa _ { a } ^ { \\left ( h + 1 \\right ) } \\kappa _ { a + \\mathsf { N } } ^ { \\left ( h \\right ) } = 1 \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} p _ { j 1 } + \\sum _ { l = 2 } ^ { r } s _ { l } p _ { j l } \\in \\Z \\backslash \\{ 0 \\} , \\ \\forall 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} \\mathrm { c n } ( x , z ) ^ 2 + \\mathrm { c n } ( x , b ( z ) ) ^ 2 = \\left [ x [ z , b ( x ) ] + b ( x ) [ x , z ] , b ( z ) \\right ] , \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} p _ j = \\min \\{ p : ~ h ^ { ( p ) } _ l ( x _ j ) \\not = 0 \\} = \\min \\{ p : ~ h ^ { ( p ) } _ r ( x _ j ) \\not = 0 \\} . \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { f ( n ) } = D _ f \\left ( \\left ( \\frac { 2 } { S _ { 1 / f } ( 1 ) } - 1 \\right ) ( \\log x + E _ f ) + 2 ( \\log 2 ) \\frac { S ' _ { 1 / f } ( 1 ) } { S _ { 1 / f } ( 1 ) ^ 2 } \\right ) + O \\left ( T _ { 1 / f } ( x ) \\right ) , \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} \\Sigma _ { \\lambda , \\alpha } = \\lambda _ 2 \\Sigma _ { \\hat { \\lambda } , \\alpha } . \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} u _ t = \\partial _ x L u , \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{gather*} q _ 2 = q _ 2 , q _ 1 = 2 \\alpha + \\frac { u _ t } { u } ( 1 + q _ 2 ) , q _ 0 = 2 \\alpha \\frac { u _ t } { u } + t + 2 u ^ 2 , \\\\ U ( t ) = 6 \\frac { d } { d t } \\left ( \\log \\frac { \\kappa } { \\sqrt { 1 - q ^ 2 _ 2 } } \\right ) - \\frac { t ^ 2 } { 2 } . \\end{gather*}"}
-{"id": "3877.png", "formula": "\\begin{align*} { \\bf x } _ j = \\sum _ { k = 1 } ^ { 2 } { { \\bf v } _ { j _ k } d _ { j _ k } } = { \\bf V } _ j { \\bf d } _ j , \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} f = X ^ 4 + 2 a X ^ 2 + 4 b X - \\frac { a ^ 2 } { 3 } . \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} \\pi ( w ^ l | w _ 0 ) = { \\P ( [ X _ 2 ^ { l + 1 } ] _ b = w _ 2 ^ { l + 1 } | [ X _ 1 ] _ b = w _ 1 ) } . \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} \\mathcal { T } _ { 1 i } ^ * = \\dot { \\bf x } ^ I ( \\tau _ i ) - \\dot { \\bf x } ^ * ( \\tau _ i ) , 1 \\le i \\le N . \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} C _ { \\ell , p } \\ ; : \\ ; y ^ 2 = x ^ p - \\ell C ' _ { \\ell , p } \\ ; : \\ ; y ^ 2 = x ^ p - 2 \\ell \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} \\frac { 1 } { T } \\sum _ { t = 1 } ^ T \\mathsf { E } _ { a _ { - i \\sim \\psi _ { - i } } } \\left [ C _ { i } ^ { ( t ) } \\left ( a _ { i } ^ { ( t ) } , a _ { - i } \\right ) - C _ { i } ^ { ( t ) } \\left ( a _ { i } , a _ { - i } \\right ) \\right ] \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{gather*} X = X ^ i \\frac { \\partial } { \\partial x ^ i } + X _ { I } ^ a \\frac { \\partial _ l } { \\partial \\phi ^ a _ I } , \\end{gather*}"}
-{"id": "3296.png", "formula": "\\begin{align*} G = \\begin{bmatrix} t _ 0 & 1 \\\\ t _ 1 & 0 \\\\ \\vdots & \\vdots \\\\ t _ { n - 1 } & 0 \\end{bmatrix} , B = \\begin{bmatrix} 1 & 0 \\\\ 0 & \\bar t _ { - 1 } \\\\ \\vdots & \\vdots \\\\ 0 & \\bar t _ { - n + 1 } \\end{bmatrix} . \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} \\mathrm { d } x \\left ( t \\right ) = f \\left ( t , x _ { t } , r \\left ( t \\right ) , u \\left ( t \\right ) \\right ) \\mathrm { d } t + g \\left ( t , x _ { t } , r \\left ( t \\right ) , u \\left ( t \\right ) \\right ) \\mathrm { d } w \\left ( t \\right ) , t \\in J , \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} \\Delta _ { 1 3 } \\ = \\ \\Delta _ 7 \\ ; \\Delta _ { 1 2 } \\ ; \\Delta _ 6 ^ { - 1 } \\Delta _ { 2 3 } \\ = \\ \\Delta _ 9 \\ ; \\Delta _ { 1 2 } \\ ; \\Delta _ 8 ^ { - 1 } \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{gather*} F ( x , t ; \\beta = 6 ) : = \\Psi _ { 1 1 } \\bigl ( 3 ^ { 1 / 3 } x , 3 ^ { 2 / 3 } t \\bigr ) \\end{gather*}"}
-{"id": "8774.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\sigma ( n ) = - \\frac { \\pi ^ 2 } { 4 8 } x ^ 2 + O \\left ( x ( \\log x ) ^ { 2 / 3 } \\right ) . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} L ^ { 0 } _ { t } = L _ { t } + J _ { \\nu _ { t } } , \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} [ X , Y ] ( f ) : = X ( Y f ) - ( - 1 ) ^ { \\langle d e g X , d e g Y \\rangle } Y ( X f ) . \\\\ \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} \\sum \\limits _ r \\eta _ r ( x ) \\eta _ r ( R ^ a ) \\eta _ r ( z ^ { - 1 } ) = n \\sum \\limits _ r \\eta _ r ( R ^ a ) , \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} \\Delta _ { 2 ^ { p _ 0 } + 2 ^ { p _ 1 } + \\cdots + 2 ^ { p _ s } } = \\Delta _ { 1 + 2 ^ { p _ 1 - p _ 0 } + \\cdots + 2 ^ { p _ s - p _ 0 } } . \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} A = \\begin{pmatrix} \\lambda _ 1 & 0 \\\\ 0 & \\lambda _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} \\frac { \\partial p } { \\partial t } = D ( p ) \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } - M ( p ) \\frac { \\partial p } { \\partial x } + f ( p ) \\ , , \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} c _ 1 f _ { x _ 1 } + \\cdots + c _ r f _ { x _ r } = : c \\in K ^ \\circ \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} m _ { 0 } : = \\int _ { \\R ^ { n } } u _ { 0 } ( y ) d y . \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} F _ { 3 n } ^ { \\left ( 3 \\right ) } = F _ { n } ^ { \\left ( 3 \\right ) } F _ { 0 } = F _ { n } ^ { \\left ( 3 \\right ) } \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} \\eta : B _ x \\otimes _ R B _ y & \\to B _ x \\otimes _ R B _ y [ 2 ] \\\\ b \\otimes b ' & \\mapsto b \\otimes \\rho b ' = b \\rho \\otimes b ' \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\sum _ { \\alpha \\in ( \\alpha _ { i - 1 } , \\alpha _ i ) } ( \\alpha _ 0 < \\cdots < \\alpha _ { i - 1 } < \\alpha < \\alpha _ i < \\cdots < \\alpha _ d ) = 0 . \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} B = \\Psi ( 0 ) \\varphi ( D ) / 2 D . \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\tilde { c } _ 6 = 1 + \\alpha _ 1 2 + \\alpha _ 2 2 ^ 2 + \\alpha _ 3 2 ^ 3 + \\alpha _ 4 2 ^ 4 + s ' 2 ^ 5 , \\alpha _ i \\in \\{ 0 , 1 \\} , s ' \\in \\Z _ 2 \\end{align*}"}
-{"id": "9137.png", "formula": "\\begin{align*} I _ s ( f ) = \\int _ { D ^ s } f \\ , { \\rm d } \\rho ^ { s } \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\{ n - k + 1 , \\dots , n \\} } = \\prod _ { i = 1 } ^ { k } \\prod _ { j = 1 } ^ { n - k } \\left ( 1 - \\frac { x _ i } { y _ j } \\right ) , \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} T \\overline \\partial _ b = \\overline \\partial _ b T ~ ~ \\Omega ^ { 0 , q } ( X ) . \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} f ( x ; \\alpha , \\theta _ 1 , \\theta _ 2 ) = ( 1 - \\alpha ) \\phi ( x - \\theta _ 1 ) + \\alpha \\phi ( x - \\theta _ 2 ) . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} Z _ { i , j } | q _ { j } & \\sim ( p _ { j } ) \\ ; & i = 1 , & \\ldots , n ; \\ ; j = 1 , \\dots , N ; \\\\ q _ { j } | \\eta _ { 1 } , \\eta _ { 2 } & \\sim ( \\eta _ { 1 } , \\eta _ { 2 } ) & & j = 1 , \\dots , N . \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } a _ k x ^ { s _ b ( k ) } \\star \\sum _ { l = 0 } ^ { \\infty } b _ l y ^ { s _ b ( l ) } = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { k \\leq _ b n } a _ k b _ { n - k } x ^ { s _ b ( k ) } y ^ { s _ b ( n - k ) } . \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} u ( x ) = \\exp \\left ( - \\frac { 1 } { 1 - \\lvert x \\rvert ^ 2 } \\right ) \\chi _ { \\lvert x \\rvert < 1 } , \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} ( u _ 1 : u _ 2 : u _ 3 : u _ 4 : u _ 5 : u _ 6 ) = & \\left ( \\frac { x _ 1 } { \\lambda _ 3 } : \\frac { x _ 0 } { \\lambda _ 3 } : \\frac { y _ 1 } { \\lambda _ 2 } : \\frac { y _ 0 } { \\lambda _ 2 } : \\frac { z _ 1 } { \\lambda _ 1 } : \\frac { z _ 0 } { \\lambda _ 1 } \\right ) , \\\\ \\left ( \\frac { x _ 1 } { x _ 0 } , \\frac { y _ 1 } { y _ 0 } , \\frac { z _ 1 } { z _ 0 } \\right ) = & \\left ( \\frac { u _ 1 } { u _ 2 } , \\frac { u _ 3 } { u _ 4 } , \\frac { u _ 5 } { u _ 6 } \\right ) , \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} { \\overline { R } } _ { e ^ { * } } = \\frac { 1 } { { \\ln 2 } } \\int _ 0 ^ \\infty { \\frac { 1 - \\exp \\left ( - 2 \\pi { \\lambda _ e } \\hat { F } _ e \\left ( { x } \\right ) \\right ) } { { 1 + x } } d x } , \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} \\begin{aligned} H ( \\rho _ { T _ 0 } | M _ { \\Omega _ { \\rho _ { T _ 0 } } } ) & \\ge H ( \\rho _ { T _ 0 + L / 2 } | M _ { \\Omega _ { \\rho _ { T _ 0 + L / 2 } } } ) \\ge \\int _ { T _ 0 + L / 2 } ^ { T _ 0 + L } I ( \\rho _ s | M _ { \\Omega _ { \\rho _ s } } ) d s \\\\ & \\ge \\frac { L } { 2 } \\min _ { s \\in [ T _ 0 + L / 2 , T _ 0 + L ] } I ( \\rho _ s | M _ { \\Omega _ { \\rho _ s } } ) . \\end{aligned} \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} E ' \\ ; : \\ ; y ^ 2 = x ^ 3 + a _ 2 ' x ^ 2 + a _ 4 ' x , a _ 2 ' = \\frac { a } { u ^ 2 } , a _ 4 ' = \\frac { b } { u ^ 4 } , \\Delta ' = \\Delta _ m u ^ { - 1 2 } \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} S \\left ( \\frac { a } { c } \\right ) = \\psi ( a , c ) \\int g ( x ) d x + T ( a , c ) , \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{align*} & \\sum _ { k : M _ k \\in B _ { j + 1 } ^ { j + 1 } } \\lambda _ { k } | A - M _ { k } | \\lesssim \\sum _ { k : M _ k \\in B _ { j + 1 } ^ { j + 1 } } \\lambda _ { k } j = \\nu ( B _ { j + 1 } ^ { j + 1 } ) j \\lesssim j ^ { 2 m _ 1 ' + 3 m _ 2 ' - 3 n ' + 1 } i ^ { m _ 1 ' } \\leq j ^ { - 2 } . \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} R _ { 1 2 } ( u - v ) t _ 1 ( u ) t _ 2 ( v ) = t _ 2 ( v ) t _ 1 ( u ) R _ { 1 2 } ( u - v ) , \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} \\limsup _ { x \\to x _ 0 } \\frac { \\abs { f ( x ) - f ( x _ 0 ) - d f ( x _ 0 ) \\cdot ( \\phi ( x ) - \\phi ( x _ 0 ) ) } } { \\rho ( x , x _ 0 ) } = 0 . \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} Q _ n ( \\omega , A ' ) = P ( ( K _ { m } ) _ { m \\in \\mathbb { N } } \\in A ' | \\mathcal { F } _ { n } ) ( \\omega ) = P ( ( K _ { m } ) _ { m \\in \\mathbb { N } } \\in A ' | K _ n ) ( \\omega ) \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} M _ { 1 2 } \\geq \\begin{cases} 0 , & N \\geqslant M _ { 1 } + M _ { 2 } \\\\ M _ { 1 } + M _ { 2 } - N , & N < M _ { 1 } + M _ { 2 } \\end{cases} . \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} p _ \\mu ( G ) ( \\mathcal { A } ) = \\sum _ { \\substack { \\phi : V \\to [ k ] \\\\ | \\phi ^ { - 1 } ( i ) | = \\mu _ i } } \\prod _ { e = \\{ u , v \\} \\in E } A ^ { e } _ { \\phi ( u ) , \\phi ( v ) } . \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} F ( \\xi , h ) = \\min _ { \\mu \\in \\Pr ( [ 0 , 1 ] ) } P _ { \\xi , h } ( \\mu ) . \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\nabla \\Psi = 0 , ( d \\phi - \\frac { 1 } { 4 } H ) \\cdot \\Psi = 0 \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} \\langle \\overline { R } ( x \\wedge y ) , v \\wedge w \\rangle _ { \\wedge ^ { 2 } \\mathfrak { g } } = R ( x \\wedge y , v \\wedge w ) = \\langle R ( x , y ) v , w \\rangle = R ( x , y , v , w ) \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} n ^ { \\widetilde { K } } ( \\widetilde { C } ) = n ^ K ( C ^ 0 ) + n ^ K ( - C ^ 0 ) + 2 n ^ K ( C ^ 1 ) = | C ^ 0 | = | C | + w _ 1 ( C ) = | C | + 1 \\ , . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} f _ 1 ( t , \\xi , \\eta , \\lambda ) : = \\begin{pmatrix} 0 \\\\ \\lambda \\phi ( y _ 1 - x _ 1 ) \\end{pmatrix} , f _ 2 ( t , \\xi , \\eta , \\lambda ) : = \\begin{pmatrix} y _ 2 \\\\ \\phi ( x _ 1 - y _ 1 ) \\end{pmatrix} . \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} \\mathbb { X } = & E n d ( V ) ^ { \\oplus ^ 2 } \\oplus H o m ( W , V ) \\oplus H o m ( V , W ) \\oplus \\\\ & \\oplus E n d ( V ' ) ^ { \\oplus ^ 2 } \\oplus H o m ( V ' , V ) \\oplus H o m ( V , V ' ) . \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} r _ { k , n } = b _ { k , n } \\log _ 2 \\left ( 1 + \\frac { p _ { k , n } g _ { k , n } } { b _ { k , n } N _ 0 } \\right ) . \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} r _ h ( c , \\chi ) = \\sideset { } { ^ * } \\sum _ { a \\pmod { c } } \\chi ( a ) e \\left ( \\frac { a h } { c } \\right ) . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} J _ { 2 0 } ( u , v ) = \\lambda ( w ) K ( u / v ) \\log { N } . \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} c ^ { \\ast } = \\begin{cases} M _ 1 + 2 \\sqrt { k D } \\ , , & \\mbox { i f $ M _ 2 \\le M _ 1 + 2 \\sqrt { k D } $ , } \\\\ \\displaystyle \\frac { M _ 1 + M _ 2 } { 2 } + \\displaystyle \\frac { 2 k D } { M _ 2 - M _ 1 } \\ , , & \\mbox { i f $ M _ 2 \\ge M _ 1 + 2 \\sqrt { k D } $ . } \\end{cases} \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} \\mu _ b ( w _ 2 ^ { \\ell _ 2 } | w _ 1 , u ^ { \\ell _ 1 } ) = \\P ( Z _ { \\ell _ 1 + g + 2 } ^ { \\ell _ 1 + g + \\ell _ 2 } = w _ 2 ^ { \\ell _ 2 } | Z _ { \\ell _ 1 + g + 1 } = w _ 1 , Z ^ { \\ell _ 1 } = u ^ { \\ell _ 1 } ) , \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} F _ { b n + p } ^ { \\left ( b \\right ) } = F _ { n } ^ { \\left ( b \\right ) } F _ { p } \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\left \\| { A } \\xi - \\hat { A } \\xi \\right \\| _ \\alpha ^ \\alpha = \\int _ { - \\pi } ^ { \\pi } \\left | A ( e ^ { i \\theta } ) - { h } ( \\theta ) \\right | ^ { \\alpha } f ( \\theta ) d \\theta + \\int _ { - \\pi } ^ { \\pi } \\left | { h } ( \\theta ) \\right | ^ { \\alpha } g ( \\theta ) d \\theta . \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} m _ M ( I _ o ^ { - 1 } J _ o ) = m _ M ( I _ o ) + m _ M ( J _ o ) - m _ M ( \\{ e _ M \\} ) . \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} \\phi _ { \\sigma } ( \\xi ) - 1 = - \\frac { \\nu ^ { 2 } | \\xi | ^ { 4 \\sigma - 2 } } { 4 \\left ( 1 + \\phi _ { \\sigma } ( \\xi ) \\right ) } , \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} ( v , w ) _ H : = ( \\theta _ v , \\theta _ w ) _ { h \\otimes h ^ * } , \\ \\ \\forall \\ v , w \\in T _ B . \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} { Q } _ { \\Lambda } = { M } _ { \\Lambda } = \\sum _ { { x } \\in \\Lambda } { \\sigma } _ { { x } } \\ . \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} \\binom { n + m } { r } _ b = \\sum _ { 0 \\leq k \\leq _ b r } \\binom { n } { k } _ b \\binom { m } { r - k } _ b . \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } D _ { 0 + } ^ { q } [ u ( t ) - u ( 0 ) ] + \\lambda u ( t ) = h ( t ) , t > 0 , \\\\ u ( 0 ) = u _ { 0 } , \\end{array} \\right . \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{align*} \\exp ( | C | p ) = \\omega ( ( | C | p ) ^ { 5 ( r - 1 ) } ) . \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} X = B \\Gamma _ { m , b c } X - ( \\Gamma _ { m , b c } X ) ( J _ { m , b c } B ) - ( \\Gamma _ { m , b c } X ) J _ { m , b c } ( B \\Gamma _ { m , b c } X ) + B . \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} & \\underbrace { \\sum _ { ( u , v ) \\in [ r ] \\times [ s ] } \\lambda ^ p _ { ( u - 1 ) s + v } \\cdot c \\big ( ( x _ 1 , x _ 2 , \\ldots , x _ t ) ; ( u , v ) \\big ) } _ { } \\\\ & + \\underbrace { \\sum _ { v \\in [ s ] } \\rho \\cdot c \\big ( \\overline { ( x _ 1 , x _ 2 , \\ldots , x _ t ) } ^ { v , p } ; ( x _ { \\overline { v } ^ { \\{ t \\} } } , v ) \\big ) } _ { } = 0 . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\sum _ { l = 0 } ^ { d } x ^ { ( l ) } x ^ { ( l ) * } & = \\sum _ { i \\in I } | \\psi _ i | ^ 2 = \\sum _ { i \\in I } \\varphi _ i , \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { p } } _ { \\theta } ^ { \\vee } = \\frac { 1 } { z } ( \\mathfrak { n } _ { \\theta } + z \\mathfrak { g } [ [ z ] ] ) \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} \\epsilon = \\delta ^ \\mu \\ , , \\mbox { f o r s o m e } \\mu \\in ( 2 , 4 ) \\ , . \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} T _ { \\alpha \\beta } = \\partial _ \\alpha \\Phi \\partial _ \\beta \\Phi - \\frac { 1 } { 2 } g _ { \\alpha \\beta } ( g ^ { \\mu \\nu } \\partial _ \\mu \\Phi \\partial _ \\nu \\Phi + V ( \\Phi ) ) - \\int _ { \\{ g ( p , p ) = - \\textbf { m } ^ 2 \\} } \\frac { \\rho ( x ^ \\nu , p ^ \\mu ) p _ \\alpha p _ \\beta \\sqrt { | g | } } { p ^ 0 } \\ ; d p ^ 1 . . . d p ^ n . \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} \\mathfrak { T } _ { \\epsilon , \\kappa , \\delta } ( X , s \\eta , \\zeta ) & : = \\big ( \\partial ^ \\alpha _ \\xi \\lambda ^ \\epsilon \\big ) ( \\xi - s \\epsilon ^ { 1 / 2 } b _ \\kappa ( x ) \\eta ) \\big ( \\partial ^ \\alpha _ { \\xi ^ \\bot } \\widetilde { g } _ { 1 / \\delta } \\big ) ( \\xi - \\epsilon ^ { 1 / 2 } b _ \\kappa ( x ) \\zeta ) \\ , , \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} c = \\sqrt { { \\pi } / { 2 } } \\cdot \\| i _ 1 \\| \\| i _ 2 \\| \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} \\left | m _ { N } ^ { N ^ \\delta } ( f ) - T _ { p } . m _ { N } ^ { N ^ \\delta } ( f ) \\right | & \\leq \\frac { 1 } { N ^ \\delta } \\left | \\sum _ { i = 0 } ^ { N ^ \\delta - 1 } ( T _ { p } ^ { i } ) . m _ { N } ( f ) - ( T _ { p } ^ { i + 1 } ) . m _ { N } ( f ) \\right | \\\\ & = \\frac { 1 } { N ^ \\delta } \\left | m _ { N } ( f ) - T _ { p } ^ { N } . m _ { N } ( f ) \\right | \\\\ & \\leq \\frac { 2 \\| f \\| _ { \\infty } } { N ^ \\delta } \\xrightarrow { N \\to \\infty } 0 . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} \\Lambda _ n : = \\textup { p . v . } \\int _ { \\mathbb { R } ^ { n + 1 } } \\prod _ { i = 0 } ^ { n } F _ i ( x _ 0 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ n ) \\frac { 1 } { x _ 0 + \\dots + x _ n } d x _ 0 \\dots d x _ n . \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} f _ q ( x ) & = f _ { q + r } ( x ) + r ( x ) = f _ { q + r } ( x ) + \\delta \\| x \\| \\\\ & \\geq \\delta \\| x \\| + \\inf f _ { q + r } ( y ) = \\delta \\| x \\| - f ^ * ( q + r ) \\geq \\delta \\| x \\| - C _ \\delta . \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} \\frac { \\tilde { \\beta } _ { t + \\tau } } { \\tilde { \\beta } _ { t ' + \\tau } } - \\frac { \\tilde { \\beta } _ { t + \\tau ' } } { \\tilde { \\beta } _ { t ' + \\tau ' } } = \\frac { \\tilde { \\beta } _ { t + \\tau ' } } { \\tilde { \\beta } _ { t ' + \\tau } } \\left ( \\frac { \\tilde { \\beta } _ { t + \\tau } } { \\tilde { \\beta } _ { t + \\tau ' } } - \\frac { \\tilde { \\beta } _ { t ' + \\tau } } { \\tilde { \\beta } _ { t ' + \\tau ' } } \\right ) = 0 , \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{align*} \\eta ( s ) : = \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac 1 { n ^ s } = \\left ( 1 - \\frac 1 { 2 ^ { s - 1 } } \\right ) \\zeta ( s ) , \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} y ( x y + 2 q _ 4 ) = x ^ 3 + p _ 2 x ^ 2 + p _ 4 x + p _ 6 \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} & f _ { \\Omega ^ { ( N ) } } ( x ) = \\\\ & \\frac { N } { \\mathcal { B } ( M , K ) } \\left [ \\frac { ( - 1 ) ^ { M } ( M - 1 ) ! } { \\mathcal { B } ( M , K ) \\prod _ { m = 1 } ^ { M } ( m - M - K ) } \\right ] ^ { N - 1 } \\sum _ { k = N - 1 } ^ { ( N - 1 ) ( M + K - 1 ) } q _ { k - ( N - 1 ) } \\frac { x ^ { M + k - 1 } } { ( 1 + x ) ^ { N ( M + K - 1 ) + 1 } } \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} \\begin{aligned} - y '' + V ( x ) y & = \\lambda y \\\\ \\alpha _ 1 y ( 0 ) + \\alpha _ 2 y ' ( 0 ) & = 0 \\\\ \\beta _ 1 y ( 1 ) + \\beta _ 2 y ' ( 1 ) & = 0 , \\end{aligned} \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} f ^ { \\Delta _ i } _ { t _ i } ( t ) = \\lim \\limits _ { \\substack { s _ i \\to t _ i \\\\ s _ i \\neq \\sigma _ i ( t _ i ) } } \\frac { f ( t _ 1 , \\ldots , t _ { i - 1 } , \\sigma _ i ( t _ i ) , t _ { t + 1 } , \\ldots , t _ n ) - f ( t _ 1 , \\ldots , t _ { i - 1 } , s _ i , t _ { t + 1 } , \\ldots , t _ n ) } { \\sigma _ i ( t _ i ) - s _ i } , \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} L = - \\frac { ( u ' ) ^ 2 } { 2 } + \\frac { a u ^ 2 } { 2 } + \\frac { b + c n } { 2 } \\left ( u _ 1 - u \\right ) ^ 2 , \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\left ( x ; \\right ) = \\mathcal { B } \\left ( \\sum _ { j = 2 } ^ { \\left \\lfloor x / 2 \\right \\rfloor } \\mathcal { B } \\left ( \\mathbf { d } _ j ( 0 , x ) , \\frac { 1 } { 2 } \\right ) , \\frac { 1 } { 2 } \\right ) \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} \\eta ( \\pi ) : = 2 ^ { \\nu _ { d , e } ( \\pi ) } ( \\chi ( n ) \\cdot ( 1 + \\chi ( f _ { n } = 0 ) \\ , ) + 2 \\cdot f _ { n } \\cdot \\chi ( n ) \\ , ) \\prod _ { 2 j + 1 < n } ( 2 f _ { 2 j + 1 } - 1 ) \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} u = \\frac { 1 } { v _ 1 - v _ { - 1 } } ; \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} \\eta _ { R } = \\frac { 1 } { 2 } ( \\eta + J \\eta ) , \\ \\ \\ \\eta _ { I } = \\frac { 1 } { 2 i } ( \\eta - J \\eta ) . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} \\frac { d \\hat \\P _ n ( G , \\sigma ) } { d \\Q _ n } = \\frac { Y _ n } { \\P _ n ( \\Omega _ n ) } = ( 1 + o ( 1 ) ) Y _ n . \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\theta } _ j = \\omega _ j ( I ) , j \\in S ^ + , \\\\ [ 3 m m ] \\dot { I } _ j = 0 , \\quad \\ , \\ , \\ , \\ , j \\in S ^ + . \\end{cases} \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} f _ 1 ( x ) - 2 f _ 2 ( x ) & = - x _ 1 ^ 2 - 2 x _ 2 ^ 2 + 3 x _ 1 + 4 x _ 2 = - x _ 1 ^ 2 - 2 ( x _ 2 - 2 . 5 ) ^ 2 + 3 x _ 1 - 6 x _ 2 + 1 3 . 5 \\\\ & \\leq 3 ( x _ 1 - 2 x _ 2 ) + 1 3 . 5 \\leq - 6 0 + 1 3 . 5 \\leq - 2 0 . \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} c _ 4 ( E ) = 2 ^ 4 \\cdot 1 9 , c _ 6 ( E ) = - 2 ^ 7 \\cdot 4 1 , \\Delta ( E ) = 2 ^ 6 \\cdot 5 \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\varphi ( x _ { \\sigma ( 1 ) } , . . . , x _ { \\sigma ( r ) } ) = \\varepsilon \\varphi ( x _ 1 , . . . , x _ r ) , \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} \\begin{aligned} & \\norm { \\xi ^ { w } ( s ) } _ { L ^ { 2 } } ^ { 2 } + \\norm { \\theta ^ { w } ( s ) } _ { L ^ { 2 } } ^ { 2 } \\\\ & + \\norm { \\nabla \\theta ^ { w } } _ { L ^ { 2 } ( 0 , s ; L ^ { 2 } ) } ^ { 2 } + \\norm { \\nabla \\xi ^ { w } } _ { L ^ { 2 } ( 0 , s ; L ^ { 2 } ) } ^ { 2 } + \\norm { \\rho ^ { w } } _ { L ^ { 2 } ( 0 , s ; L ^ { 2 } ) } ^ { 2 } \\leq C _ { 1 5 } \\norm { w } _ { L ^ { 2 } ( 0 , s ; L ^ { 2 } ) } ^ { 4 } , \\end{aligned} \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} \\mu _ 1 ( x ) : = & - \\theta _ 1 ( W _ { ( \\xi , \\xi ' ) } ) _ w \\\\ = & ( ( \\frac { 1 } { 2 \\sqrt { - 1 } } [ a , a ^ { \\dagger } ] + [ b , b ^ { \\dagger } ] + i i ^ { \\dagger } - j ^ { \\dagger } j + f f ^ { \\dagger } - g ^ { \\dagger } g ) ) , \\\\ ( & \\frac { 1 } { 2 \\sqrt { - 1 } } ) ( [ a ' , a ^ { \\prime \\dagger } ] + [ b ' , b ^ { \\prime \\dagger } ] - f ^ { \\dagger } f + g g ^ { \\dagger } ) ) . \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\frac x { \\sqrt { 2 t ( 1 - t ) } } = \\frac { ( 1 + 4 \\tau ) r } { 2 \\sqrt { 2 \\tau } } + \\frac u { 2 \\sqrt \\tau } , \\qquad \\frac { 2 r - x } { \\sqrt { 2 t ( 1 - t ) } } = \\frac { ( 1 + 4 \\tau ) r } { 2 \\sqrt { 2 \\tau } } - \\frac u { 2 \\sqrt \\tau } \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} \\phi _ { L _ n ( x ) } ( t ) & \\rightarrow \\exp \\Big \\{ { \\alpha x ^ 2 } \\frac { ( e ^ { i t } - 1 ) } { 2 x - ( 2 x - 1 ) e ^ { i t } } \\Big \\} \\\\ & = \\phi _ { \\ell ( x ) } ( t ) . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} v _ i & = v _ { i - 1 } + \\tau [ ( L ^ h _ { i \\tau } + J ^ h ) v _ i + f _ { i \\tau } ] , \\ i = 1 , . . . , n \\\\ v _ 0 & = \\psi . \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla H _ { 3 , \\le 1 } ( u ) = & - 3 \\ , c _ 1 \\partial _ x ( v _ x ^ 2 ) - 6 \\ , c _ 1 \\partial _ x \\Pi _ S [ v _ x \\ , z _ x ] - c _ 2 \\partial _ { x x } ( v ^ 2 ) - 2 \\ , c _ 2 \\partial _ { x x } \\Pi _ S [ v \\ , z ] + \\\\ & + c _ 2 \\pi _ 0 [ v _ x ^ 2 ] + 2 \\ , c _ 2 \\Pi _ S [ v _ x \\ , z _ x ] + 3 \\ , c _ 3 \\pi _ 0 [ v ^ 2 ] + 6 \\ , c _ 3 \\Pi _ S [ v \\ , z ] . \\end{aligned} \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} \\sum _ { m _ 1 = 0 } ^ { r p - 1 } \\sum _ { m _ 2 = 0 } ^ { s p - 1 } \\sum _ { m _ 3 = 0 } ^ { t p - 1 } \\binom { m _ 1 + m _ 2 + m _ 3 } { m _ 1 , m _ 2 , m _ 3 } ^ 2 \\equiv _ { p ^ 2 } \\sum _ { m _ 1 = 0 } ^ { r - 1 } \\sum _ { m _ 2 = 0 } ^ { s - 1 } \\sum _ { m _ 3 = 0 } ^ { t - 1 } \\binom { m _ 1 + m _ 2 + m _ 3 } { m _ 1 , m _ 2 , m _ 3 } ^ 2 . \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} 1 \\geq \\int _ { F ^ c } \\langle \\delta _ l , P ( x ) \\delta _ l \\rangle = \\int _ { F ^ c } \\sum _ { n \\in \\Z } \\langle \\delta _ l , P _ n ( x - n \\alpha ) \\delta _ l \\rangle \\geq 1 . \\end{align*}"}
-{"id": "2636.png", "formula": "\\begin{align*} \\| f ^ { \\star } - ( 1 - \\alpha _ m ) f _ { m - 1 } - \\alpha _ m c v _ f h _ m \\| ^ 2 & = ( 1 - \\alpha _ m ) ^ 2 \\| f ^ { \\star } - f _ { m - 1 } \\| ^ 2 \\\\ & - 2 \\alpha _ m ( 1 - \\alpha _ m ) \\langle f ^ { \\star } - f _ { m - 1 } , c h _ m v _ f - f ^ { \\star } \\rangle \\\\ & + \\alpha ^ 2 _ m \\| f ^ { \\star } - c h _ m v _ f \\| ^ 2 . \\end{align*}"}
-{"id": "3886.png", "formula": "\\begin{align*} { \\bf T } _ { i j k } = & { \\bf G } _ { i j } { \\bf Q } _ { j k } { \\bf G } ^ { T } _ { i j } , \\ \\forall \\{ i , j \\} \\in \\mathcal { L } , \\ \\forall k \\in \\mathcal { K } , \\\\ { \\bf R } _ i = & \\sum \\limits _ { l = 1 } ^ { 3 } \\sum \\limits _ { m = 1 } ^ { 2 } { \\bf G } _ { i l } { \\bf Q } _ { l m } { \\bf G } ^ { T } _ { i l } + \\sigma ^ { 2 } { \\bf I } _ 2 , \\quad \\forall i \\in \\mathcal { I } , \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} T = - \\frac { \\dd ^ 2 } { \\dd x ^ 2 } + Q ( x ) + V ( x ) \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} L ( f ) & = \\tilde { g } ^ { i \\overline { j } } \\partial \\overline { \\partial } f ( e _ { i } , \\overline { e } _ { j } ) \\\\ & = \\tilde { g } ^ { i \\overline { j } } \\left ( e _ { i } \\overline { e } _ { j } ( f ) - [ e _ { i } , \\overline { e } _ { j } ] ^ { ( 0 , 1 ) } ( f ) \\right ) . \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} J = \\frac { \\gamma I ^ 3 } { g } \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} \\rho _ { \\infty } ( \\lambda ) & = \\frac { p _ 0 ' ( \\lambda ) } { 2 \\pi } + \\sum _ { i } \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) \\rho _ { \\infty } ( \\nu ) , \\\\ \\rho _ { i a } ( \\lambda ) & = \\frac { K ' ( \\lambda - \\lambda _ { i a } ) } { 2 \\pi } + \\sum _ { i } \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) \\rho _ { i a } ( \\nu ) . \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} \\phi _ t ^ * \\eta _ t = \\eta _ 0 , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} | \\log ( 1 + y ) - y | = \\Big | \\int _ 0 ^ y \\frac { z } { 1 + z } d z \\Big | \\leq k _ * \\Big | \\int _ 0 ^ y z d z \\Big | \\leq c y ^ 2 , | y | \\leq \\frac { k _ * } { 1 + k _ * } . \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} \\Delta \\left ( h ( f _ 0 ) ; f \\right ) = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } \\left | \\sum \\limits _ { j = 0 } ^ { \\infty } ( ( \\bold { B } ^ 0 ) ^ { - 1 } \\bold { a } ) _ j e ^ { i j \\theta } \\right | ^ 2 f _ 0 ^ { - 2 } ( \\theta ) f ( \\theta ) d \\theta . \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} C _ q = \\frac { 2 L } { ( 1 - 2 \\epsilon \\kappa ) \\mu } , ~ ~ C _ l = \\frac { 4 \\epsilon \\kappa } { 1 - 2 \\epsilon \\kappa } . \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} E [ Y _ { 2 t } - \\gamma _ { 1 0 } ( X _ { t } ) | X _ { t } ] & = 0 , E [ Y _ { 2 t } \\{ H ( \\gamma _ { 1 0 } ( X _ { t + 1 } ) ) - \\gamma _ { 2 0 } ( X _ { t } ) \\} | X _ { t } ] = 0 , \\\\ E [ Y _ { 1 t } \\{ H ( \\gamma _ { 1 0 } ( X _ { t + 1 } ) ) - \\gamma _ { 3 0 } \\} ] & = 0 . \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{gather*} - \\dfrac { x _ { 1 } } { y _ { 1 } } = - \\dfrac { b ^ { 2 } } { a ^ { 2 } } \\dfrac { x _ { 2 } } { y _ { 2 } } = m \\\\ \\dfrac { 1 } { y _ { 1 } } = \\dfrac { b ^ { 2 } } { y _ { 2 } } = B \\end{gather*}"}
-{"id": "9639.png", "formula": "\\begin{align*} \\Upsilon _ { 0 , f } ^ { [ 1 ] } = \\Upsilon _ { 0 , 0 } ^ { [ 1 ] } + \\Upsilon _ { 0 , 1 } ^ { [ 1 ] } \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} \\varrho ^ { ( 2 k + 1 ) } = t ^ k \\delta ^ { - 1 } \\Omega _ k + \\textrm { t e r m s n o t c o n t a i n i n g } \\Omega _ k . \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} y _ i ( n ) = h _ i ( n ) * x _ i ( n ) + n _ i ( n ) , \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} - \\Delta _ { q ( x ) } v _ { 0 } = \\left \\{ \\begin{array} { l l } 1 & \\Omega \\backslash \\overline { \\Omega } _ { \\delta } \\\\ - 1 & \\Omega _ { \\delta } \\end{array} \\right . , v _ { 0 } = 0 \\partial \\Omega . \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} q ^ { ( S _ { n } ^ { z } + p + 1 ) / 2 } \\overline { \\left \\vert a + 1 , n \\right \\rangle } = \\overline { \\left \\vert a + 1 , n \\right \\rangle } q ^ { ( 2 ( s - a ) + p + 1 ) / 2 } = \\left \\vert p - a , n \\right \\rangle q ^ { p - a } = v _ { n } \\left \\vert p - a , n \\right \\rangle . \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\lim _ { m \\to \\infty } \\sum _ { j = 0 } ^ d x ^ { ( j ) } ( m ) ^ * x ^ { ( j ) } ( m ) a = a \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} { \\overline { R } } ^ { \\mathrm { L } } _ 1 = { \\log _ 2 } \\left ( 1 + \\frac { { G _ { } ^ 2 \\beta } r ^ { - { \\overline { \\alpha } } } } { { { \\lambda \\bar { G } \\Lambda + \\frac { N _ o } { P _ t } } } } \\right ) , \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} \\Theta _ { 0 1 } = \\sum _ { n \\in \\Z } ( - 1 ) ^ n e ^ { \\pi i n ^ 2 z } \\Theta _ { 1 0 } = \\sum _ { n \\in \\Z } e ^ { \\pi i ( n + 1 / 2 ) ^ 2 z } . \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} P ' = P - \\epsilon . \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} L \\left ( [ e _ { x y } , f ] \\right ) = L ( e _ { x y } f - f e _ { x y } ) = L ( e _ { x y } ) f - f L ( e _ { x y } ) + e _ { x y } L ( f ) - L ( f ) e _ { x y } , \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} Q ^ { \\alpha } = \\sum _ J ( - D ) _ J B ^ { \\alpha } _ J , \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} \\sum _ { h } W \\left ( \\frac { h } { H } \\right ) \\sum _ { X \\leq n \\leq 2 X } \\tau _ k ( n ) \\tau ( n + h ) \\tau ( n - h ) & = \\widehat { W } ( 1 ) X H Q _ { k + 1 } ( \\log X ) \\\\ & + O \\left ( X ^ { \\varepsilon } ( H ^ 2 + H X ^ { 1 - \\frac { 1 } { k + 2 } } + X H ^ { 1 / 2 } + X ^ { 3 / 2 } H ^ { - 1 / 2 } ) \\right ) , \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} d = 2 , \\ m = 1 , \\ B = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} \\bar k ( s ) = 2 \\alpha - ( 1 + 2 \\beta ) \\bar v ( s ) = \\frac { \\alpha } { 1 + \\beta } - e ^ { \\mu ( s - t ) } \\left ( \\frac { 1 + 2 \\beta } { 1 + \\beta } \\pi ( s ) \\bar X ( s ) + \\frac { \\alpha ( 1 + 2 \\beta ) } { ( 1 + \\beta ) ^ 2 } \\int _ t ^ s \\pi ( \\tau ) d \\tau \\right ) \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\big ( u _ i ( s _ n + t _ n ; s _ n + t _ n - \\tilde T , \\tilde u ^ n ) - u _ i ( s _ n + t _ n ; s _ n + t _ n - \\tilde T , \\tilde u ^ 0 ) \\big ) = 0 \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} ( A _ 2 ) _ j ^ { j ' } ( l ) : = \\begin{cases} - \\dfrac { T _ j ^ { j ' } ( l ) } { \\mathrm { i } ( \\omega \\cdot l + m _ 3 ( j '^ 3 - j ^ 3 ) ) } \\mbox { i f } \\ , \\ , \\overline { \\omega } \\cdot l + j '^ 3 - j ^ 3 \\neq 0 , \\\\ [ 3 m m ] 0 \\qquad \\qquad \\qquad \\qquad \\qquad \\quad \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} P ( \\lambda \\vec { 1 } ) = \\{ \\lambda ^ 5 \\ , ( 2 4 c _ 4 - 4 8 \\ , c _ 1 ^ 2 ) + \\lambda ^ 3 \\ , ( 4 c _ 6 - \\frac { 1 6 } { 3 } c _ 2 ^ 2 ) \\} \\ , ( \\vec { 1 } \\cdot \\mathbb { M } ( \\lambda \\vec { 1 } ) ^ { - 1 } \\vec { 1 } ) - 1 \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} [ [ \\alpha , \\beta ] ] = \\mathcal { L } _ { \\Pi ( \\alpha ) } \\beta + ( - 1 ) ^ p ( \\Pi ( d \\alpha ) ) \\beta \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} E _ i E _ { i + 1 } ^ 2 & = q \\sum _ { t \\lhd r , t \\rhd s } w _ { i + 1 } ^ r w _ i ^ t w _ { i + 1 } ^ s + q ^ { - 1 } \\sum _ { t \\rhd r , t \\rhd s } w _ { i + 1 } ^ r w _ i ^ t w _ { i + 1 } ^ s \\\\ & + q \\sum _ { t \\lhd r , t \\lhd s } w _ { i + 1 } ^ r w _ i ^ t w _ { i + 1 } ^ s + q ^ { - 1 } \\sum _ { t \\rhd r , t \\lhd s } w _ { i + 1 } ^ r w _ i ^ t w _ { i + 1 } ^ s . \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} \\Phi ^ + _ { P } = \\{ \\beta \\in \\Phi ^ + ; \\ \\mathfrak g _ { - \\beta } \\subset U ^ - _ { P } \\} . \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} ( \\vartheta \\phi ) ( X ) = \\overline { \\phi ( X ) } \\ \\ \\mbox { a . e . $ X $ } \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} ( \\operatorname { D P [ 0 ] } + \\operatorname { Q } ) ( \\Xi _ 2 ) = \\xi _ 1 . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} \\sigma ( j _ E ^ { 1 / 3 } ) = \\sigma ( \\pi ) ^ k c _ 0 = \\zeta _ 3 ^ k \\pi ^ k c _ 0 = \\zeta _ 3 ^ k \\cdot j _ E ^ { 1 / 3 } , \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} - \\Delta _ { p ( x ) } u _ { 1 } = \\lambda ^ { \\sigma } \\left \\{ \\begin{array} { l l } w _ { 1 } ^ { \\alpha _ { 1 } ( x ) } & \\Omega \\backslash \\overline { \\Omega } _ { \\delta } \\\\ d ( x ) ^ { \\alpha _ { 1 } ( x ) + \\beta _ { 1 } ( x ) } & \\Omega _ { \\delta } \\end{array} \\right . , u _ { 1 } = 0 \\partial \\Omega \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\lambda \\int _ { \\mathbb { Z } _ p } \\left ( \\frac { x + y } { \\lambda } \\right ) _ n d \\mu _ { 1 } ( x ) = B _ { n , \\lambda ( x ) } ( n \\geq 0 ) , \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} 2 \\imath \\psi _ { t } + \\psi _ { u u } + \\psi _ { v v } = 0 , \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} P ^ { ( \\mathbf { s } , \\mathcal { B } _ n ) } ( \\mathbf { w } ) = 2 ^ { - n R } \\sum _ { j \\in \\mathcal { J } _ n } p _ { W | V , U } ^ n \\big ( \\mathbf { w } \\big | \\mathbf { v } ( j ) , \\mathbf { u } \\big ) . \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} \\omega ( q ) & = 1 + \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { 2 n ^ { 2 } + 2 n } } { ( 1 - q ) ^ { 2 } ( 1 - q ^ { 3 } ) ^ { 2 } \\cdots ( 1 - q ^ { 2 n + 1 } ) ^ { 2 } } \\\\ & = 1 + 2 q + 3 q ^ { 2 } + 4 q ^ { 3 } + 6 q ^ { 4 } + 8 q ^ { 5 } + 1 0 q ^ { 6 } + 1 4 q ^ { 7 } + \\dots \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} p ^ n ( x , y ) : = \\begin{cases} \\delta _ { x , y } & \\textrm { i f } n = 0 ; \\\\ p ( x , y ) , & \\textrm { i f } n = 1 ; \\\\ \\sum _ { z \\in X } p ^ { n - 1 } ( x , z ) p ( z , y ) , & \\textrm { i f } n > 1 . \\end{cases} \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} v _ s ( x ) = x , \\gamma _ s ( t ) = t + \\gamma _ 0 , \\sigma _ s ( t ) = ( t + \\gamma _ 0 ) ^ { - m } . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} \\mathbf { P } _ 2 = \\mathbf { B } \\mathbf { \\Phi } \\mathbf { G } , \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} \\min \\limits _ { \\stackrel { E _ i \\in Q _ k ^ { ( i ) } } { i = 1 , \\dots , m } } \\max \\limits _ { \\stackrel { \\| x _ j \\| \\leq \\eta , } { j = 1 , \\dots , L } } F ( E , x ) . \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} \\frac { P } { \\sqrt { c } } - \\frac { Q } { \\sqrt { c } } & = - \\frac { 2 } { \\left ( c - 2 \\right ) } \\cdot \\frac { \\sqrt { c } } { Q } + \\frac { 2 } { \\left ( c + 2 \\right ) } \\cdot \\frac { \\sqrt { c } } { P } , \\ \\ a = d , \\\\ \\frac { P } { \\sqrt { c } } + \\frac { Q } { \\sqrt { c } } & = \\frac { 2 } { \\left ( c - 2 \\right ) } \\cdot \\frac { \\sqrt { c } } { Q } + \\frac { 2 } { \\left ( c + 2 \\right ) } \\cdot \\frac { \\sqrt { c } } { P } , \\ \\ a = - d . \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} s A _ { 2 1 } y + s A _ { 1 2 } z + s A _ { 2 2 } v + T _ { 2 2 } v = 0 . \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ k j ( n - 2 ) ^ j \\le k ( n - 1 ) ^ k . \\end{align*}"}
-{"id": "0.png", "formula": "\\begin{align*} C _ q = \\frac { 2 L } { ( 1 - \\epsilon ) \\mu } , C _ l = \\frac { 3 \\epsilon } { 1 - \\epsilon } \\sqrt { \\kappa } . \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} I ( \\mathbf { x } ; C B ^ T ) = \\dim ( \\mathcal { C } _ 2 ^ \\perp \\cap \\mathcal { V } _ \\mathcal { L } ) - \\dim ( \\mathcal { C } _ 1 ^ \\perp \\cap \\mathcal { V } _ \\mathcal { L } ) , \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} \\max \\{ p ( x ) \\mid x \\in \\overline { B ( x _ { 0 } , \\varepsilon ) } \\} = \\max \\{ p ( x ) \\mid x \\in B ( x _ { 0 } , \\varepsilon , \\varepsilon , \\theta ) \\} . \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} 2 \\tau = - ( m - 4 ) \\pm \\sqrt { ( m - 4 ) ^ 2 + 4 \\lambda } . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} M _ { 2 } \\leq \\prod \\limits _ { j = 0 } ^ { n - 1 } \\Big | \\frac { c _ { i _ { r r } } ( j ) } { h _ { i } ( j ) + \\Delta _ { i } ( j ) } \\Big | \\leq M _ { 1 } , \\textnormal { i f $ n \\geq 1 $ } \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} ( x , y ; \\ x ^ { 2 } + y ^ { 2 } < R ^ { 2 } ) = \\mathcal { B } _ { - - } \\left ( x ^ { 2 } + y ^ { 2 } , R ^ { 2 } \\right ) \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} x _ p ^ { \\infty } & = ( x ' ) _ { q } ^ { \\infty } \\ , ( , \\ , x _ { p + i } = x ' _ { q + i } \\ , \\ , \\ , \\ , \\ , i \\in \\mathbb { N } ) ; \\\\ t - t ' & = h ( x ) - h ( x ' ) + \\tau _ p ( x ) - \\tau _ q ( x ' ) ; \\\\ \\xi - \\xi ' & = f _ q ( x ' ) - f _ p ( x ) , \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} \\nu ( A B ) = d ^ * ( A ) + \\nu ( B ) < 1 . \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ r _ { p , q } } = \\Vert \\{ 2 ^ { m r } \\Vert \\textnormal { O p } ( \\chi _ { m } ) f \\Vert _ { L ^ p ( G ) } \\} _ { m = 0 } ^ { \\infty } \\Vert _ { l ^ q ( \\mathbb { N } ) } , \\ , \\ , 0 < p , q \\leq \\infty , \\ , r \\in \\mathbb { R } . \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} z _ { i } ( n + 1 ) = \\Lambda _ { i } ( n ) z _ { i } ( n ) , \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\psi ( k , \\ell ) = \\begin{cases} 1 & \\mbox { f o r a l l } k > p \\ell = 1 \\\\ 1 & \\mbox { f o r a l l } k = 1 \\ell > p \\\\ \\psi ( k , \\ell ) & \\mbox { f o r a l l } k \\le p \\ell \\le p \\\\ 0 & \\mbox { e l s e } \\end{cases} , \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} ( a \\star b ) _ 0 = & a _ 0 b _ 0 - a _ j b _ j \\\\ ( a \\star b ) _ i = & a _ 0 b _ i + a _ i b _ 0 + \\epsilon _ { i j k } a _ j b _ k , \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} & ( - 3 2 \\varepsilon L , 3 2 \\varepsilon L ) = \\\\ & [ - 4 L + t , 4 L + t ] \\setminus \\left ( [ - ( 1 - 4 \\varepsilon ) 8 L , - 4 \\varepsilon \\cdot 8 L ] \\cup [ 4 \\varepsilon \\cdot 8 L , ( 1 - 4 \\varepsilon ) 8 L ] \\right ) \\ ; . \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} \\left \\langle \\tau \\right \\vert = \\left \\langle \\omega \\right \\vert \\prod _ { b = 1 } ^ { \\mathsf { N } _ { Q } } \\mathcal { \\hat { B } } _ { - } ( \\lambda _ { b } ) , | \\tau \\rangle = \\prod _ { b = 1 } ^ { \\mathsf { N } _ { Q } } \\mathcal { \\hat { B } } _ { - } ( \\lambda _ { b } ) | \\bar { \\omega } \\rangle , \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} \\begin{aligned} b = 0 , \\xi _ t + a ( t , n + 1 ) = 0 , a ( t , n + 1 ) = a ( t , n - 1 ) , a ' = 0 , \\end{aligned} \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} e _ a ( \\mathcal { C } _ n ) = \\frac { 1 } { | \\mathcal { M } _ n | } \\sum _ { m \\in \\mathcal { M } _ n } e _ m ( c _ n ) = \\mathbb { P } _ { P ^ { ( \\mathcal { C } _ n ) } } \\big ( \\hat { M } \\neq M \\big ) , \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} \\hat { u } = \\frac { \\sigma ( u ) } { u } , \\hat { r } = \\frac { \\sigma ( r ) - r } { u ^ 2 } , \\hat { s } = \\frac { \\sigma ( s ) - s } { u } , \\hat { t } = \\frac { \\sigma ( t ) - t - s ( \\sigma ( r ) - r ) } { u ^ 3 } . \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} X _ j = \\frac { \\partial } { \\partial x ^ j } + \\sum _ { i = 1 } ^ { j - 1 } a _ { i j } \\frac { \\partial } { \\partial x ^ i } \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\frac { F _ { \\eta ( l ) } ( x ) - F _ \\theta ( x ) } { \\alpha L ( \\delta ( l ) ) } = \\sum _ { k = 1 } ^ K \\lambda _ k f _ k ' ( x | 0 ) , \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} A _ n = \\frac { 1 } { 2 \\pi i } \\int _ { L _ R } \\frac { P _ n ( z ) \\ , d z } { z B _ n ( z ) } , \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} \\partial _ { \\tau } u = \\partial _ x ( b ( \\varphi , \\tau , x ) u ) , b ( \\varphi , \\tau , x ) : = \\frac { \\beta ( \\varphi , x ) } { 1 + \\tau \\beta _ x ( \\varphi , x ) } . \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} [ X , Y ] : = X Y - Y X = Z \\quad [ X , Z ] = [ Y , Z ] = 0 . \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} u ' = f ( u _ { - 1 } , u , u _ 1 ) . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} 2 e = d _ 2 ( f - 3 ) + 3 d _ 1 \\geq 6 ( f - 3 ) + 9 = 6 f - 9 = 2 ( 3 f - 6 ) + 3 \\geq 2 e + 3 \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} A ( \\tau , s ) = A ( \\tau , 0 ) + \\int _ 0 ^ s F ( c ' ( u ) , \\cdot ) ( \\tau , u ) d u . \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} u ( t , \\omega ) = u _ { k } ( t , \\omega ( t ) ; \\omega ( t _ { 1 } ) , \\cdots , \\omega ( t _ { k } ) ) . \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} R _ \\mathsf { A } \\geq R _ \\mathsf { A } ( p _ { U ' , V ' , X | S } ) = \\min \\Big \\{ I ( V ' ; Y | U ' ) - I ( V ' ; Z | U ' ) , I ( U ' , V ' ; Y ) - I ( U ' , V ' ; S ) \\Big \\} \\geq R _ \\mathsf { A } ^ \\mathsf { A l t } , \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} \\begin{aligned} \\overline G _ i ( h , q ) & \\geq \\frac { 1 } { T _ i ( h ) } \\int _ 0 ^ { T _ i ( h ) } \\Big ( \\nu | q | | D H ( X ( t , x ) ) | - M \\Big ) \\ , d t \\\\ & \\geq \\nu c _ 0 | q | - M . \\end{aligned} \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} \\gamma ( 1 / 2 , \\pi , \\pi ' , \\psi ) = 1 . \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} T ( \\Z / p ^ r \\Z ) ^ 2 & \\to \\widehat \\Gamma _ 0 ( p ^ r ) / \\widehat \\Gamma _ 1 ( p ^ r ) \\\\ ( a , d ) & \\mapsto \\begin{pmatrix} a & 0 \\\\ 0 & d \\end{pmatrix} . \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} | \\alpha ^ i | = \\sum _ { j = 0 } ^ { i - 1 } ( | \\lambda ^ { ( j ) } | - | \\mu ^ { ( j ) } | ) , | \\beta ^ i | = | \\mu ^ { ( i ) } | + \\sum _ { j = 0 } ^ { i - 1 } ( | \\mu ^ { ( j ) } | - | \\lambda ^ { ( j ) } | ) \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} W _ n = \\left \\{ \\begin{array} { l } \\sqrt { ( 2 m - 1 ) ( 2 m ) ( 2 m + 1 ) } B _ m V _ m , \\ ; \\ ; n = 2 m - 1 , \\\\ - \\sqrt { ( 2 m ) ( 2 m + 1 ) ( 2 m + 2 ) } B _ m V _ { m + 1 } , n = 2 m \\end{array} \\right . n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} \\tilde { V } ( z ) = \\exp \\left ( \\sum _ { n > 0 } z ^ n \\frac { \\partial } { \\partial p _ n } \\right ) . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\begin{cases} u _ t - F ( \\nabla u , \\nabla ^ 2 u ) = 0 & \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ( x , 0 ) = u _ 0 ( x ) & \\R ^ n , \\end{cases} \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} & \\frac { 1 } { J _ 0 ( t ) } \\int _ 0 ^ t \\frac { 1 } { \\Delta J _ 0 ( s ) } \\int _ 0 ^ { \\Delta J _ 0 ( s ) } f ( W ( s - ) + x ) d x d J _ 0 ( s ) \\\\ & - \\frac { 1 } { t } \\int _ 0 ^ t g ( W ( s ) ) d s \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} \\underbrace { \\sum _ { ( { u } , { v } ) \\in [ r ] \\times [ s ] } \\lambda ^ p _ { ( { u } - 1 ) s + { v } } \\cdot c ( u ^ { \\ast } ; ( { u } , { v } ) ) } _ { } + \\underbrace { \\sum _ { { v } \\in [ s ] } \\rho \\cdot c ( \\overline { u ^ { \\ast } + p } ; ( u ^ { \\ast } , { v } ) ) } _ { } = 0 ~ ~ ~ p \\in \\{ 1 , \\ldots , r - 1 \\} . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} \\Delta ^ { ( k ) } ( N ) = \\Delta ^ { ( k ) } ( \\Lambda _ N , \\xi _ N ) , H ^ { ( k ) } _ h ( N ) = H ^ { ( k ) } _ h ( \\Lambda _ N , \\xi _ N ) H ^ { ( k ) } _ u ( N ) = H ^ { ( k ) } _ u ( \\Lambda _ N , \\xi _ N ) , \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} E _ n : = \\{ \\mathfrak { I } ( \\varphi ) = ( \\Theta , y , z ) ( \\varphi ) : \\Theta = \\Pi _ n \\Theta , y = \\Pi _ n y , z = \\Pi _ n z \\} \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} x = \\sum _ { i = 1 } ^ { n } \\lambda _ { i } \\cdot s _ { i } . \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} p _ { k , \\alpha } ( w ( z ) ) = 1 + p _ 1 w _ 1 + ( p _ 1 w _ 2 + p _ 2 w _ 1 ^ 2 ) z ^ 2 + ( p _ 1 w _ 3 + 2 p _ 2 w _ 1 w _ 2 + p _ 3 w _ 1 ^ 3 ) z ^ 3 + \\cdots . \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ 2 = \\sum _ { k = 1 } ^ n f ( k ) ^ 2 \\ge \\int _ 0 ^ n f ( k ) ^ 2 \\ , \\mathrm { d } k \\ge \\frac { 2 } { 1 5 } n ^ 5 . \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} T ( X , Y ) = \\sum _ { d < M \\Delta ^ { - 3 } } \\mu ( d q ) \\frac { \\tau _ r ( d ) } { d } \\int _ { X / d } ^ { Y / d } T _ d ( y ) \\frac { d y } { y } + O \\left ( \\left ( \\frac { \\log { Y } } { \\log { M } } \\right ) ^ r \\right ) \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{gather*} ( 1 - x ^ m ) A _ { ( m + 1 ) p _ n + r _ n } = A _ { ( m + 1 ) p _ n - ( m - r _ n ) } + x ^ { m - 2 r _ n } A _ { ( m + 1 ) p _ n - r _ n } + x ^ { m - 2 r _ n - 1 } A _ { ( m + 1 ) p _ n - r _ n - 1 } . \\end{gather*}"}
-{"id": "7007.png", "formula": "\\begin{align*} ( d , D ) = 1 . \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} u = \\bar { u } + \\hat \\chi \\left ( p + \\frac { 1 } { 2 } ( r ^ 2 \\circ \\Phi - r ^ 2 ) \\right ) \\circ P ^ { - 1 } & \\ ; \\ , { \\rm w i t h } \\ , \\ , \\bar { u } \\in C ^ { k + 2 , \\alpha } _ { \\nu + 2 } ( X ^ { r e g } ) , \\ , p \\in \\mathcal { P } , \\ , \\Phi \\in G , \\\\ f = \\bar { f } + f _ x & \\ ; \\ , { \\rm w i t h } \\ , \\ , \\bar { f } \\in C ^ { k , \\alpha } _ \\nu ( X ^ { r e g } ) , \\ , f _ x \\in \\R , \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\int \\eta - \\xi \\ , d \\mu = \\inf \\ , \\{ \\eta - \\xi \\} \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} \\mathfrak { g } = \\mathfrak { t } \\oplus \\bigoplus _ { \\beta \\in R } \\mathfrak { g } _ { \\beta } \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} \\Gamma _ { m , d i r } X = \\Gamma _ { d i r } ( X - P _ { ( m ) } X P _ { ( m ) } ) , X \\in \\mathfrak { S } _ 2 ( \\mathcal { H } ) , \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} \\begin{aligned} | b _ V ( \\psi _ m , \\psi _ n ) | & = | ( V , \\psi _ m \\psi _ n ) | \\leq C ( \\| ( \\psi _ m \\psi _ n ) ' \\| ^ 2 + \\| \\psi _ m \\psi _ n \\| ^ 2 ) ^ \\frac s 2 \\| \\psi _ m \\psi _ n \\| ^ { 1 - s } \\\\ & \\leq C _ 1 ( m + n ) ^ s \\leq C _ 2 ( m n ) ^ s . \\end{aligned} \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} \\frac { \\partial \\gamma } { \\partial q } + \\frac { 1 } { 2 } \\gamma + \\alpha m + \\frac { m } { \\gamma } \\left ( V ' ( q ) - V ( q ) - \\alpha S \\right ) = 0 \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} Y ( X Y + 2 q _ 4 Z ^ 2 ) = X ^ 3 + p _ 2 X ^ 2 Z + p _ 4 X Z ^ 2 + p _ 6 Z ^ 3 , \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{gather*} W ( f _ { \\ell + 1 } , z _ 4 ) = \\mu ( W [ f _ 1 , \\dots , f _ { \\ell + 1 } ] ) = \\mu ( W ^ \\tau ) , \\\\ X ( f _ { \\ell + 1 } , z _ 4 ) = \\mu ( X [ f _ 1 , \\dots , f _ { \\ell + 1 } ] ) = \\mu ( X ^ \\tau ) , \\\\ Y ( f _ { \\ell + 1 } , z _ 4 ) = \\mu ( Y [ f _ 1 , \\dots , f _ { \\ell + 1 } ] ) = \\mu ( Y ^ \\tau ) . \\end{gather*}"}
-{"id": "6389.png", "formula": "\\begin{align*} A \\geq \\sum _ { I \\in S _ \\phi } \\Big [ \\mu ( I ) \\frac { y _ I ^ q } { ( \\tau _ I ) ^ { q - 1 } } - \\sum _ { \\substack { J \\in S _ \\phi \\\\ J ^ \\star = I } } \\mu ( J ) \\frac { y _ J ^ q } { ( \\beta + 1 ) ^ { q - 1 } } \\Big ] , \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} \\partial _ t N = \\sum _ { j = 1 } ^ { m } D _ j \\partial _ { x x } n _ j - \\sum _ { j = 1 } ^ m M _ j \\partial _ x n _ j + \\sum _ { j = 1 } ^ m f _ j ( \\vec { n } ) \\ , . \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} \\mathrm { c n } ( x , y ) = \\sqrt { \\mathrm { c m } ( x , y ) \\cdot \\mathrm { c m } ( y , x ) } . \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} 2 ( 1 3 2 + ( 1 8 - N ) ( 1 7 - N ) ) \\lambda ( N ) = H _ n ( N ) + 2 ( 1 8 - N ) H _ h ( N ) + \\tau ( N ) \\mu ( N + 8 ) H _ u ( N ) , \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} Z = \\operatorname { S p e c } \\Big ( C ^ * \\big ( \\mu _ 1 ( C _ 0 ( Y ) ) , \\mu _ 2 ( C _ 0 ( Y ) ) \\big ) \\Big ) . \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} \\nabla F _ s = ( 2 s - N ) F _ { s - 1 } ( x - y ) = ( 2 s - N ) F _ s \\frac { x - y } { | x - y | ^ 2 } - \\Delta \\ F _ s = ( N - 2 s ) 2 ( s - 1 ) F _ { s - 1 } , \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} \\mathbb { P } [ Y _ v = 1 ] & = \\sum _ { j = r } ^ { | B | } \\binom { | B | } { j } p ^ { j } ( 1 - p ) ^ { | B | - j } = ( 1 + o ( 1 ) ) \\binom { | B | } { r } p ^ r ( 1 - p ) ^ { | B | - r } \\\\ & = ( 1 + o ( 1 ) ) \\frac { | B | ^ r } { r ! } p ^ r = ( 1 + o ( 1 ) ) \\frac { ( n p ) ^ { - r / ( 4 ( r - 1 ) ) } } { 4 ^ r r ! } . \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} M = \\lceil \\exp \\left [ ( 1 - \\epsilon ) B ( n ) \\right ] \\rceil \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} x = \\sigma ( u ) ^ 2 x '' + \\sigma ( r ) , y = \\sigma ( u ) ^ 3 y '' + \\sigma ( u ) ^ 2 \\sigma ( s ) x '' + \\sigma ( t ) . \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} & ( a h ) ( t ) = a h ( t ) \\ ; , & & ( h a ) ( t ) = h ( t ) \\alpha _ t ( a ) \\ ; , \\\\ & ( h g ) ( t ) = \\int _ { \\R } h ( s ) \\alpha _ s ( g ( t - s ) ) \\ , d s \\ ; , & & ( f h ) ( t ) = \\int _ { \\R } f ( s ) h ( t - s ) \\ , d s \\ ; , \\\\ & ( a f ) ( t ) = a f ( t ) \\ ; , & & ( f a ) ( t ) = f ( t ) \\alpha _ t ( a ) \\ ; . \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} C \\rho ^ { N - \\frac { \\theta } { p - 1 } } \\ge \\underset { \\tau \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\int _ { B ( z , \\rho ) } u ( y , \\tau ) \\zeta ( y - z ) \\ , d y = \\int _ { { \\bf R } ^ N } \\zeta ( y - z ) \\ , d \\mu ( y ) \\ge \\mu ( B ( z , \\rho / 2 ) ) \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} \\bold { p r } X = \\xi ^ i \\partial _ { x ^ i } + \\phi ^ { \\alpha } \\partial _ { u ^ { \\alpha } } + \\cdots + \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } \\partial _ { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } + \\cdots , \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\begin{aligned} ( \\varepsilon _ i , \\varepsilon _ j ) = ( - 1 ) ^ { \\theta } \\delta _ { i j } , ( \\delta _ \\mu , \\delta _ \\nu ) = - ( - 1 ) ^ { \\theta } \\delta _ { \\mu \\nu } , ( \\varepsilon _ i , \\delta _ \\mu ) = ( \\delta _ \\mu , \\varepsilon _ i ) = 0 , \\end{aligned} \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} \\langle ( - \\Delta ) ^ s u , \\phi \\rangle = \\langle f , \\phi \\rangle \\qquad \\phi \\in C ^ { \\infty } _ c ( \\Omega ) . \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + 2 \\sqrt { \\ell - 1 } x ^ 2 - x , \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{gather*} S _ { \\omega } ( u ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 d x + \\frac { \\omega + 1 } { 2 } \\int _ { \\mathbb { R } ^ N } | u | ^ 2 d x - \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ N } | u | ^ 2 \\log | u | ^ 2 d x , \\\\ I _ { \\omega } ( u ) = \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 d x + \\omega \\int _ { \\mathbb { R } ^ N } | u | ^ 2 d x - \\int _ { \\mathbb { R } ^ N } | u | ^ 2 \\log | u | ^ 2 d x . \\end{gather*}"}
-{"id": "370.png", "formula": "\\begin{align*} Q _ t ( \\eta _ 0 , \\eta ) : = \\sum _ { 0 \\le j < k \\le n } \\int _ { [ 0 , t ] ^ 2 } \\eta ( B _ { 0 , t } ^ j ( s ) - B _ { 0 , t } ^ k ( r ) ) \\eta _ 0 ( \\frac { s - r } t ) d r d s \\ , . \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} \\lambda _ j ( w ) = \\lambda ( w ) \\sum _ { a + b = j } \\binom { j } { a } \\sum _ { q \\mid u } \\sum _ { r \\mid v } \\Lambda _ a ^ * ( q ) \\Lambda _ b ^ * ( r ) . \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} v ( t , x ) = e ^ { c } V ( x - c t - a ) , a , c \\in \\mathbb { R } , \\end{align*}"}
-{"id": "9275.png", "formula": "\\begin{align*} \\varepsilon ^ { i l } + \\varepsilon ^ { - i l } + \\varepsilon ^ { i l k } + \\varepsilon ^ { - i l k } = \\varepsilon ^ { j l } + \\varepsilon ^ { - j l } + \\varepsilon ^ { j l k } + \\varepsilon ^ { - j l k } , \\ , \\ , \\ , l = 1 , k - 1 , \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} x ( t ) = \\alpha [ x ] v ( t ) + \\beta [ x ] w ( t ) + \\lambda \\int _ 0 ^ 1 k ( t , s ) f ( s , x ( s ) ) \\textup d s , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} \\rho ( \\alpha , A ) = \\int \\psi d \\mu \\mod \\Z \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{align*} Q _ { i j } ( e + \\Sigma a _ { p q } v _ { p q } ) = a _ { i j } . \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} { \\frac { 1 } { n ! } B _ n ( x ) = - \\sum _ { p \\neq 0 } \\frac { e ^ { 2 \\pi i p x } } { ( 2 \\pi i p ) ^ n } , } \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} F ( c ' ( s ) , v ) = d A ( c ' ( s ) , v ) , \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} N = N ' A ^ s = A ' , \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} \\sigma _ { v , t } : = \\begin{pmatrix} 1 & v \\\\ 0 & t \\end{pmatrix} . \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} a ( t ) = ( \\delta ^ { i j } ) + \\sum _ { k = 1 } ^ { n } \\nu ^ { 2 } _ { k } ( t ) l _ { k } l _ { k } ^ { * } , \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} P _ { S _ 1 | Y } ( 1 | y ) = P _ { S _ 1 | Y } ( 1 | y ' ) . \\quad \\forall ( y , y ' ) \\in \\mathcal { Y } ^ 2 , \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} S ( - 1 , z ) = \\sqrt { \\frac { 2 } { e ^ { 2 z } + e ^ { - 2 z } } } = \\sqrt { \\sec ( 2 \\imath z ) } . \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} \\mu _ { \\mathbb { C } } ( x ) = & ( \\mu _ 2 + \\sqrt { - 1 } \\mu _ 3 ) ( x ) \\\\ = & - ( ( [ a , b ] + i j + f g ) , [ a ' , b ' ] - g f ) . \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\alpha - \\chi ( \\alpha ) = ( \\underbrace { 0 , \\ldots , 0 } _ n , r _ n , r _ { n + 1 } ) - ( \\underbrace { 0 , \\ldots , 0 } _ { n + 1 } , - 1 , 0 , 0 , \\ldots ) = ( \\underbrace { 0 , \\ldots , 0 } _ n , r _ n , r _ { n + 1 } + 1 , \\ldots ) \\in S \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} \\epsilon \\le u ( t + s _ n ; s _ n , u ^ 0 ) \\le M , \\epsilon \\le u ^ + ( t + s _ n ) \\le M \\forall \\ , \\ , t \\ge s _ n , \\ , \\ , n = 1 , 2 , \\cdots . \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} \\nu _ i - \\nu _ { i + 1 } = 1 , \\ \\ \\mbox { f o r a l l } i \\notin \\{ N _ 1 , N _ 2 , \\dots \\} \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} m _ \\sigma ( F ) \\leq \\frac { \\max \\{ 0 , \\Re W ( E ) \\} - \\Gamma _ E } { 1 - \\epsilon } = : \\Gamma _ E ' . \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} \\nu _ i ^ d ( h ) = \\sup \\{ u ( h ) \\ | \\ u \\in \\mathcal S _ i ^ - \\cap C ( \\bar J _ i ) , \\ u ( 0 ) = d \\} \\ \\ \\ h \\in \\bar J _ i i \\in \\{ 1 , 2 , 3 \\} . \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} ( J f ) _ n : = \\sqrt { n ( n + 1 ) ( n + 2 ) } f _ { n + 1 } + \\sqrt { ( n - 1 ) n ( n + 1 ) } f _ { n - 1 } , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} - \\left ( a + \\varepsilon \\displaystyle \\int _ \\Omega | \\nabla u | ^ 2 d x \\right ) \\Delta u ( x ) = u ^ 5 + \\lambda u ^ { q - 1 } , \\ ; u > 0 \\ ; \\textrm { i n } \\ ; \\Omega , \\ ; \\ ; u = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\textrm { o n } \\ ; \\partial \\Omega , \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} g ( t _ \\epsilon w _ { \\epsilon , \\eta } ) = \\displaystyle \\sup _ { t \\geq 0 } g ( t w _ { \\epsilon , \\eta } ) \\ ; \\textrm { a n d } \\ ; \\frac { d } { d t } g ( t w _ { \\epsilon , \\eta } ) \\mid _ { t = t _ \\epsilon } = 0 . \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} g _ 2 ( B _ n ) = 1 - \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( \\sqrt { B _ n / 2 } - \\sqrt { P / 2 } \\big ) \\right ) ^ 2 . \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} P _ { Y | V , S } ( 1 | v , s ) = p _ { Y | X } ( 1 | 1 ) = \\bar { \\alpha } . \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} u _ { i _ n } ( s _ n + t _ n ; s _ n , u ^ 0 ) & = u _ { i _ n } ( s _ n + t _ n ; s _ n + t _ n - \\tilde T , u ( s _ n + t _ n - \\tilde T ; s _ n , u ^ 0 ) ) \\\\ & = u _ 0 ( s _ n + t _ n ; s _ n + t _ n - \\tilde T , u _ { \\cdot + i _ n } ( s _ n + t _ n - \\tilde T ; s _ n , u ^ 0 ) ) \\\\ & \\ge u _ 0 ( s _ n + t _ n ; s _ n + t _ n - \\tilde T , \\tilde u ^ n ) { \\rm f o r } \\ , \\ , n \\gg 1 . \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} \\widetilde { \\Gamma } _ { a c } ^ d \\widetilde { \\Gamma } ^ c _ { d b } + \\frac { \\theta } { 2 } \\partial _ 1 \\Theta _ { a b } \\chi - \\theta ( \\partial _ c \\ln | \\theta | + \\widetilde { \\Gamma } ^ d _ { d c } ) \\widetilde { \\Gamma } ^ c _ { a b } = \\frac { \\widetilde { R } ^ { ( n - 1 ) } - \\Theta ^ { c d } \\widetilde { T } _ { c d } } { n - 1 } \\Theta _ { a b } + \\widetilde { T } _ { a b } ; \\ ; a , b , c , d , e , f = 2 , . . . , n ; \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} L ^ 0 ( \\lambda ) : = L ( \\lambda ) ^ { B _ + } \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} \\gamma _ { 1 } = \\frac { \\mu _ { h _ { i } \\cap h _ { j } } ( W _ { 1 } | _ { ( h _ { i } + h _ { j } ) } ) } { \\mu _ { h _ { i } + h _ { j } } ( W _ { 1 } | _ { ( h _ { i } ) \\cap ( h _ { j } ) } ) } . \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} t = \\frac { \\lambda } { \\sum _ { i = 1 } ^ k \\sigma _ i ^ 2 + m \\lambda / 3 } . \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} Q _ M : = \\prod Q _ { n _ i } \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} S _ 1 \\setminus \\big ( S \\cup \\big ( \\bigcup _ { ( l , a ) \\in I _ \\varsigma } \\overline \\Omega _ { l , a } \\big ) \\big ) = \\bigcup _ { ( l , a ) \\in I _ 1 } \\delta _ { l , a } \\cup \\Upsilon _ { l , a } \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} W ^ { \\star } _ { \\eta H } ( z , w ) = 2 \\pi i ^ { \\kappa _ 1 } \\int _ { 0 } ^ { \\infty } W _ { \\eta H } ( y ) J _ { \\kappa _ 1 - 1 } ( 4 \\pi \\sqrt { y w + z } ) \\mathrm { d } y . \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} \\Lambda _ S ( - \\theta , \\mathbf { D } ) = \\log \\left ( p e ^ { - \\theta i ( K - k ) c } + \\sum _ { n = 1 } ^ N p _ n e ^ { - \\theta r _ n } \\right ) \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} | \\Lambda _ { n , r , R } | \\leq C \\Big ( \\log \\frac { R } { r } \\Big ) ^ { 1 - 2 ^ { - n + 1 } } \\| F _ 0 \\| _ { { 2 ^ n } } \\prod _ { i = 1 } ^ n \\| F _ i \\| _ { { 2 ^ { n - i + 1 } } } . \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} \\lim _ { J _ i \\times \\mathbb R \\ni ( h , q ) \\to ( 0 , 0 ) } \\overline G _ i ( h , q ) = G ( 0 , 0 ) \\ \\ \\ i \\in \\{ 1 , 2 , 3 \\} . \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} \\tilde { F } _ { n } ^ { \\left ( 2 \\right ) } = a _ { n + 1 } \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} ( \\Theta _ { j \\bar k } ^ { T _ B } v , w ) _ H = ( [ \\Theta ^ h _ { j \\bar k } , \\theta _ v ] , \\theta _ w ) - \\left ( P ^ { \\bot } ( D ^ { { \\rm E n d } ( H ) } _ j \\theta _ v ) , P ^ { \\bot } ( D ^ { { \\rm E n d } ( H ) } _ k \\theta _ w ) \\right ) . \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} D = - \\frac 1 2 ( \\Delta - x ^ 2 + 1 ) \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} D ( f , s ) = \\prod _ { p \\in \\P } \\sum _ { \\nu = 0 } ^ { \\infty } \\frac { f ( p ^ { \\nu } ) } { p ^ { \\nu s } } . \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} { \\mathbf { H } _ i } = \\mathbf { R } _ i ^ { \\frac { 1 } { 2 } } \\mathbf { X } _ i \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} \\beta _ t = \\prod _ { s = 0 } ^ { t - 1 } \\mu _ s = \\sum _ { i = 1 } ^ n \\theta _ 0 ^ i ( \\delta ^ i ) ^ t , t \\geq 1 . \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\theta _ n ^ { ( j ) } = \\frac { 2 \\beta ^ { ( j ) } \\ell _ n H _ 2 \\left ( \\alpha _ n ^ { ( j ) } \\right ) } { n \\log k _ n ^ { ( j ) } } \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} \\sigma ( w ) ^ { o ( w ) } = ( \\sum _ { i = 1 } ^ l d _ i \\rho ^ \\vee ( S _ i ) ) ( - 1 ) = \\prod _ { i = 1 } ^ \\ell z _ { L ( S _ i ) } { } ^ { d _ i / 2 } . \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} S _ h ( x ) = \\sum _ { n \\le x } \\Lambda ( n ) \\Lambda ( n + h ) \\sim B C ( h ) x \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\Lambda _ j = \\max \\{ | g _ t | : t \\in [ T _ j , T _ { j + 1 } ] \\} , \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} & a ( p m I ) a ( p ^ r I ) - \\chi ( p ^ 2 ) p ^ { 2 k - 3 } a ( m I ) a ( p ^ { r - 1 } I ) \\\\ & \\quad + \\epsilon \\chi ( p ) p ^ { k - 2 } a ( m I ) a \\begin{pmatrix} p ^ { r - 1 } m \\\\ & p ^ { r + 1 } m \\end{pmatrix} \\\\ & + \\epsilon \\chi ( p ) p ^ { k - 2 } a ( m I ) \\sum _ { \\substack { 1 \\le u < p / 2 \\\\ u ^ 2 \\not \\equiv - 1 \\ , ( p ) } } a \\left ( p ^ r m \\begin{pmatrix} ( 1 + u ^ 2 ) / p & u \\\\ u & p \\end{pmatrix} \\right ) . \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} \\mathcal { H } ( w ^ n _ { j - 1 } \\vee k , w ^ n _ j \\vee k ) & \\ge \\mathcal { H } ( w ^ n _ { j - 1 } , w ^ n _ j ) = \\frac { w ^ { n + 1 } _ j } { a _ j } , \\\\ \\mathcal { H } ( w ^ n _ { j - 1 } \\vee k , w ^ n _ j \\vee k ) & \\ge \\mathcal { H } ( k , k ) = \\frac { k } { a _ j } , \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( 1 - \\frac { Q ( x t ) } { Q ( x ) } \\right ) ^ { - \\frac 1 2 } & \\leq \\left ( \\frac { Q ( x ) } { Q ' ( x t ) x ( 1 - t ) } \\right ) ^ { \\frac 1 2 } \\leq \\left ( \\frac { Q ( x ) } { Q ' ( \\frac x 2 ) x ( 1 - t ) } \\right ) ^ { \\frac 1 2 } \\\\ & \\leq \\left ( \\frac { Q ( x ) } { 2 Q ( \\frac x 2 ) ( 1 - t ) } \\right ) ^ { \\frac 1 2 } \\to \\frac { 2 ^ \\frac { \\beta - 1 } { 2 } } { ( 1 - t ) ^ \\frac { 1 } { 2 } } . \\end{aligned} \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} \\frac { \\partial \\phi ( t , q ) } { \\partial t } \\Bigr | _ { t = 1 } = \\sum _ { n = 0 } ^ { \\infty } q ^ n \\sum _ { \\lambda \\vdash n } f ( \\lambda ) = \\frac { 1 } { ( q ; q ) _ { \\infty } } \\langle f \\rangle _ q . \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{align*} \\langle | u _ { n } = z _ { n } \\langle | , u _ { n } | \\rangle = z _ { n } | \\rangle z _ { n } \\in \\mathbb { S } _ { p } \\equiv \\{ q ^ { 2 r } , r = 1 , . . , p \\} , \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} D _ { - ( s - 1 ) } = ( I - \\sum _ { i = 1 } ^ { s - 1 } u _ i u _ i ^ \\top ) ( \\lambda I - A ) ^ { - 1 } ( I - \\sum _ { i = 1 } ^ { s - 1 } u _ i u _ i ^ \\top ) ~ . \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } \\circ \\rho = D \\pi \\end{align*}"}
-{"id": "5923.png", "formula": "\\begin{align*} C _ { j + 1 , i } ^ { 1 } = C _ { i , i } ^ { 1 } + \\sum _ { \\ell = i } ^ { j } \\tilde { C } \\left ( M _ { \\ell } \\ell ^ { - 2 } + ( \\ell + 2 - i ) ^ { - 2 } C _ { \\ell , i - 1 } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{align*} \\langle h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } | \\mathcal { D } _ { - } ( 1 / \\xi _ { a } ^ { \\left ( h _ { a } + 1 \\right ) } ) | h _ { 1 } , . . . , h _ { a } + 1 , . . . , h _ { \\mathsf { N } } \\rangle = 0 \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} & \\{ L ( q ) - t L ( q ^ t ) \\mid t = 2 , 4 , 5 , 8 , 1 0 , 2 0 , 4 0 \\} , \\\\ & \\{ E _ { \\chi _ 0 , \\chi _ 1 } ( q ^ t ) , E _ { \\chi _ 1 , \\chi _ 0 } ( q ^ t ) \\mid t = 1 , 2 , 4 , 8 \\} , \\\\ & \\{ E _ { \\chi _ 0 , \\chi _ 2 } ( q ^ t ) , E _ { \\chi _ 2 , \\chi _ 0 } ( q ^ t ) \\mid t = 1 , 5 \\} , \\\\ & \\{ E _ { \\chi _ 0 , \\chi _ 3 } ( q ) , E _ { \\chi _ 1 , \\chi _ 2 } ( q ) , E _ { \\chi _ 2 , \\chi _ 1 } ( q ) , E _ { \\chi _ 3 , \\chi _ 0 } ( q ) \\} \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} \\Gamma \\left ( a + 1 \\right ) = \\sqrt { 2 \\pi } a ^ { a + 1 / 2 } \\exp \\left ( - a \\right ) \\left \\{ 1 + \\frac { 1 } { 1 2 a } + O \\left ( a ^ { - 2 } \\right ) \\right \\} a \\rightarrow \\infty . \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} f _ 2 ( p ) \\Big ( e ^ { i ( p _ 1 + \\cdots + p _ { \\ell } ) \\cdot ( - i \\nabla _ p ) } f _ 1 \\Big ) ( p ) = f _ 2 ( p ) f _ 1 ( p + p _ 1 + \\cdots + p _ { \\ell } ) > 0 \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { r } \\binom { n } { k } \\binom { m } { r - k } = \\binom { n + m } { r } . \\end{align*}"}
-{"id": "1343.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": "5603.png", "formula": "\\begin{align*} \\tilde T _ 4 ( z ) \\sim i \\sum _ { j = 2 } ^ \\infty H _ { j , 4 } ( 2 z ) ^ { - j - 1 } \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } D ( F _ n ) = \\infty . \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} \\sigma ( y _ 1 ) = x ^ { 1 / 2 } y _ 2 \\quad \\sigma ( y _ 2 ) = x ^ { 1 / 2 } y _ 1 . \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} \\eta _ { \\Lambda } ( A ) = \\frac { 1 } { V } \\int _ { \\Lambda } d { x } \\pi _ { \\omega } ( \\tau _ { { x } } ( A ) ) \\ , \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} M & = - A + T ^ { R } D _ { 1 } - T ^ { I } D _ { 2 } + i \\left ( T ^ { R } D _ { 2 } + T ^ { I } D _ { 1 } \\right ) = \\\\ & = - A + \\left ( T ^ { R } + i T ^ { I } \\right ) \\left ( D _ { 1 } + i D _ { 2 } \\right ) = \\\\ & = - A + T \\mathbf { g } ^ { \\prime } ( \\mathbf { z } ^ { \\ast } ) , \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} a _ n = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\left ( - 1 \\right ) ^ k b _ k \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} f ( z ) = z + \\sum _ { n = 2 } ^ { \\infty } a _ n z ^ n . \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} C _ \\infty ^ { - p } ( a + b C _ \\infty ^ { - p } \\| u _ k \\| _ \\infty ^ p ) ^ { p - 1 } \\| u _ k \\| _ \\infty ^ p & \\leq ( a + b \\| u _ k \\| _ { E ^ { \\alpha , p } } ^ p ) ^ { p - 1 } \\| u _ k \\| _ { E ^ { \\alpha , p } } ^ p \\\\ & = \\int _ 0 ^ T f ( t , u _ k ( t ) ) u _ k ( t ) d t \\\\ & \\leq \\varepsilon \\int _ 0 ^ T | u _ k ( t ) | ^ p d t \\\\ & \\leq \\varepsilon T \\| u _ k \\| _ \\infty ^ p , \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\tilde { L } ( x ^ n , y ^ n ) = \\tilde { L } ( x ^ n ) + \\tilde { L } ( y ^ n | x ^ n ) \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} ( A + U V ^ { \\ast } ) ^ { \\dagger } = \\big ( I + A ^ { \\dagger } U F _ { S _ { A } } U ^ { \\ast } ( A ^ { \\dagger } ) ^ { \\ast } \\big ) ^ { - 1 } \\big ( A ^ { \\dagger } - A ^ { \\dagger } U S _ { A } ^ { \\dagger } V ^ { \\ast } A ^ { \\dagger } \\big ) \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ^ { - 1 } , \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} \\frac { \\delta \\mathcal E ^ h ( y ^ h ) } { \\delta y ^ h } [ \\phi ] = \\int _ \\Omega \\left ( D W ^ h ( x , \\nabla _ h y ^ h ) : \\nabla _ h \\phi - h ^ 3 ( f _ 2 \\phi _ 2 + f _ 3 \\phi _ 3 ) \\right ) \\dd x = 0 \\ , , \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} \\mathcal { D } _ { t } u _ t = \\zeta _ t , \\ , \\ , \\ \\mathcal { D } _ { x } u _ t = v _ t , \\ , \\ , \\ \\mathcal { D } _ { x } ^ { 2 } u _ t = w _ t . \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} S _ k \\phi _ { J _ 1 ; J _ 2 } ^ { \\alpha } = \\phi _ { J _ 1 ; J _ 2 + \\bold { 1 } _ k } ^ { \\alpha } \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} X = \\nabla ( r ^ 2 \\phi ) + Z + \\nabla ( \\alpha r ^ 2 ) , \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} p _ i q _ i ( \\kappa _ { i + 1 } ^ { \\epsilon } ) = \\epsilon ^ { - 1 } ( \\kappa _ { i + 1 } ^ { \\epsilon } ) p _ i q _ i . \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} E [ \\phi ( W , \\gamma _ { 0 } , \\alpha , \\theta ) ] & = \\int \\phi ( w , \\gamma _ { 0 } , \\alpha , \\theta ) F _ { 0 } ( d w ) = d \\int g ( w , \\gamma ( F _ { \\tau } ^ { \\alpha } ) , \\theta ) F _ { \\alpha } ( d w ) / d \\tau \\\\ & = d \\int g ( w , \\gamma _ { 0 } , \\theta ) F _ { \\alpha } ( d w ) / d \\tau = 0 . Q . E . D . \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} \\forall _ { x \\in S } \\bigvee _ { j = 1 } ^ n \\exists _ y \\dd ( x , i _ j ( y ) ) \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} \\lambda ( q ) = \\tau ( q / ( q , D ) ) , q \\mid u . \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} - h _ { 1 2 ; 1 } + h _ { 1 3 ; 4 } + h _ { 4 1 ; 3 } & = 0 , & h _ { 1 2 ; 2 } + h _ { 1 3 ; 3 } + h _ { 1 4 ; 4 } & = 0 , \\\\ h _ { 1 2 ; 3 } + h _ { 3 1 ; 2 } - h _ { 1 4 ; 1 } & = 0 , & h _ { 1 2 ; 4 } - h _ { 3 1 ; 1 } + h _ { 4 1 ; 2 } & = 0 . \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} \\bold { E } _ { \\alpha } ( L ) = 0 , \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{gather*} \\Xi = - A ^ \\ast \\wedge L _ \\xi A - C ^ \\ast \\wedge L _ \\xi C , \\operatorname { g h } ( \\Xi ) = - 1 . \\end{gather*}"}
-{"id": "4940.png", "formula": "\\begin{align*} T _ q ( t - \\tau ) = S ( t - \\tau ) + \\int _ { \\tau } ^ { t } S ( t - s ) B _ q T _ q ( s - \\tau ) \\dd s , t \\geq \\tau \\geq 0 , \\ q \\in \\mathbb { R } . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} B _ { k , i } \\cap B _ { k , j } = \\emptyset \\quad \\mbox { i f $ i \\not = j $ } \\qquad \\mbox { a n d } { \\bf R } ^ N = \\bigcup _ { k = 1 } ^ m \\bigcup _ { i = 1 } ^ \\infty B _ { k , i } , \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} g ( v _ i ) ^ 2 = ( u _ i \\otimes 1 + w _ i \\otimes \\eta ) ^ 2 = ( u _ i ^ 2 + \\delta w _ i ^ 2 ) \\otimes 1 + ( u _ i w _ i + w _ i u _ i + w _ i ^ 2 ) \\otimes \\eta . \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{gather*} i _ Y \\omega = \\delta C ^ \\ast , \\end{gather*}"}
-{"id": "5998.png", "formula": "\\begin{align*} Z = \\lambda ^ { 2 p } + \\frac { 1 } { \\lambda ^ { 2 p } } \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} \\begin{aligned} - 2 \\varepsilon \\mbox { I m } [ ( r \\partial _ r u ) \\overline { u } ] + 2 \\varepsilon \\lambda ^ { \\frac 1 2 } r | u | ^ 2 + \\varepsilon \\lambda ^ { - \\frac 1 2 } ( r | \\nabla u | ^ 2 - \\lambda r | u | ^ 2 ) = \\varepsilon \\lambda ^ { - \\frac 1 2 } r | \\nabla v _ \\lambda | ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} D ( A ) _ n = \\sum _ { i = 0 } ^ n \\operatorname { I m } ( \\sigma _ i \\colon A _ { n - 1 } \\rightarrow A _ n ) \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} \\Delta a = \\sum _ { a _ 1 a _ 2 = a } a _ 1 \\otimes a _ 2 . \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} \\Phi _ \\ell ^ * u - \\frac { f _ \\ell } { 2 m } r ^ 2 = \\bar { u } _ \\ell + \\sum _ { i = 0 } ^ { I _ \\ell } A _ { \\ell , i } r ^ { \\mu _ { \\ell , i } ^ + } \\phi _ { \\ell , i } , \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} { \\overline { R } } = \\frac { 1 } { \\ln 2 } \\int _ 0 ^ \\infty { \\frac { 1 } { { z } } ( 1 - \\Xi _ 1 ( z ) ) \\Xi _ 2 ( z ) { e ^ { - z \\sigma _ o ^ 2 } } d z } , \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} L _ { s , \\mathsf { k } } ( \\theta _ \\mathsf { k } ) = \\begin{bmatrix} l _ { s , \\mathsf { k } } & 0 \\\\ 0 & l _ { s , \\mathsf { k } } \\end{bmatrix} + \\mathrm { R } ( 2 \\theta _ \\mathsf { k } ) \\begin{bmatrix} l _ { s a , \\mathsf { k } } & 0 \\\\ 0 & - l _ { s a , \\mathsf { k } } \\end{bmatrix} . \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} 2 r + ( x - \\lambda _ 0 ) ^ 2 f - ( x - \\lambda _ 0 ) r ' - 2 ( x - \\lambda _ 0 ) g & = \\Phi \\varphi ' , \\\\ ( x - \\lambda _ 0 ) ( ( x - \\lambda _ 0 ) g - r ) & = \\Phi \\varphi . \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} P ( C ) = \\left [ ( p ^ r - 1 ) \\mid \\sum _ { k = 0 } ^ { r - 1 } p ^ k c _ k \\right ] . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} ( h ^ { ( j ) } ) ^ { 1 - \\alpha _ j - \\beta _ j } \\cdot \\tau \\left ( \\alpha _ j , \\beta _ j , \\xi ^ { ( j ) } , ( m ^ { ( j ) } + v _ 1 ^ { ( j ) } ) \\frac { y _ j } { h ^ { ( j ) } } \\right ) = \\tau \\left ( \\alpha _ j , \\beta _ j , \\frac { \\xi ^ { ( j ) } } { h ^ { ( j ) } } , ( m ^ { ( j ) } + v _ 1 ^ { ( j ) } ) y _ j \\right ) . \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} | \\Lambda ( C ) | = | S _ { 1 } | + | S _ { 2 } | \\leq ( { \\rm r a n k } ( B ) + 1 ) | \\Lambda ( A ) | + d ( A ) - d ( C ) . \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\left | \\sum \\limits _ { j = 0 } ^ { \\infty } ( ( \\bold { B } ^ 0 ) ^ { - 1 } \\bold { a } ) _ j e ^ { i j \\theta } \\right | ^ 2 f _ 0 ^ { - 2 } ( \\theta ) = \\gamma _ 1 \\left ( f _ 0 ( \\theta ) \\right ) ^ { \\beta - 1 } , \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\frac { \\partial { s ^ j ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ i } } = 0 . \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\underline { \\varepsilon } | \\underline { X } } \\exp \\left \\{ \\frac { 1 } { \\gamma } \\sum _ { i = 1 } ^ n \\varepsilon _ i g ( X _ i ) \\right \\} & = \\prod _ { i = 1 } ^ n \\mathbb { E } _ { \\varepsilon _ i | X _ i } \\exp \\left \\{ \\frac { 1 } { \\gamma } \\varepsilon _ i g ( X _ i ) \\right \\} . \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} h '' - 2 ( h ' ) ^ { 2 } = 0 C ^ { - 1 } \\leq h ' \\leq C , \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} | w _ N | _ { \\C { H } ^ 1 ( \\Omega ) } = \\| w _ N ' \\| _ { \\C { L } ^ 2 ( \\Omega ) } \\le \\frac { \\sqrt { 2 } ( N + 1 ) ^ 2 } { N } \\le c N . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} \\abs { \\sum _ { i , l = 1 , 2 } \\int _ { - \\infty } ^ { \\infty } \\pi _ 0 ( i ) p _ 0 ( i , l ) y ^ { r + 1 } f ( y ) \\ , \\d Q ( y ) } \\le \\int _ { - \\infty } ^ { \\infty } | y | ^ { r + 1 } f ( y ) \\ , \\d Q ( y ) \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} \\partial _ t u + f ( u , \\sigma ^ { * } \\nabla u ) + \\langle \\mu , \\nabla u \\rangle + \\tfrac { 1 } { 2 } \\operatorname { T r a c e } ( \\sigma \\sigma ^ * \\operatorname { H e s s } u ) = 0 , \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\overline \\eta ^ N \\tilde k = \\eta _ 0 \\gamma ^ d k , \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} U : = \\mathcal { C } \\setminus \\overline { \\Omega } _ \\varphi . \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\phi + t v ) ^ * ( p ) = - v ( T _ \\phi ( p ) ) . \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} g _ { \\varepsilon } ( q ) = \\frac { P _ q ( \\varepsilon ) } { \\varphi ( q ) } \\sum _ { c | q ^ { \\infty } } \\gamma ( c ) ( c , q ) c ^ { - \\varepsilon - 1 } \\end{align*}"}
-{"id": "1001.png", "formula": "\\begin{align*} M _ s ( x , \\theta ) = \\frac { k _ { N , s } } { s } \\frac { ( 1 - | x | ^ 2 ) ^ s _ { + } } { | \\theta - x | ^ N } x \\in B , \\ \\theta \\in \\partial B , \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} \\begin{pmatrix} X _ 1 M _ 1 \\\\ Y _ 1 M _ 1 \\end{pmatrix} = \\begin{pmatrix} U _ { 1 1 } & - U _ { 2 1 } \\\\ U _ { 2 1 } & U _ { 1 1 } \\end{pmatrix} \\begin{pmatrix} - Y _ 2 M _ 2 \\\\ X _ 2 M _ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} & H = L _ \\ell ^ 2 ( N ) \\otimes L _ r ^ 2 ( N ) \\otimes L _ \\ell ^ 2 ( \\R ) \\otimes L ^ 2 ( X ) \\otimes L _ r ^ 2 ( \\R ) , \\\\ & K = L ^ 2 ( N ) \\otimes L _ \\ell ^ 2 ( \\R ) \\otimes L ^ 2 ( X ) \\otimes L _ r ^ 2 ( \\R ) . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} \\pi _ { n } \\left ( m _ { 1 } , \\ldots , m _ { k } \\right ) = \\sum _ { j = 0 } ^ { \\infty } \\binom { k + j } { j } \\sum _ { \\underline { z } \\in \\left \\{ 0 , 1 \\right \\} ^ { k } } \\pi _ { n + 1 } ( m _ { 1 } + z _ { 1 } , \\ldots , m _ { k } + z _ { k } , \\underbrace { 1 , \\ldots , 1 } _ { j } ) . \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} \\pi ( w ^ l ) \\mu _ b ( w _ { l + 1 } ^ { \\ell _ 2 } | w ^ l , u ^ { \\ell _ 1 } ) & = \\pi ( w ^ { \\ell _ 2 } ) { \\mu _ b ( w _ { l + 1 } ^ { \\ell _ 2 } | w ^ l , u ^ { \\ell _ 1 } ) \\over \\pi ( w _ { l + 1 } ^ { \\ell _ 2 } | w ^ l ) } . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} g ( x ) \\in T _ x ( \\partial N ) \\abs { g ( x ) } = 1 . \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} \\begin{array} { l l l l } Z ^ L _ j = & \\max \\ & c ' V ^ j \\lambda + d ' u & \\\\ & \\mbox { s . t . } \\ & A V ^ j \\lambda \\leq b & \\\\ & & H V ^ j \\lambda + G u \\leq h & \\end{array} \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\gamma ( 1 / 2 , \\pi , \\tau , \\psi ) = \\lim _ { s \\to 1 / 2 } \\frac { L ( 1 - s , \\pi , \\tau ) } { L ( s , \\pi , \\tau ) } = - 1 . \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\mu ( U , f ^ n ) = \\infty \\quad \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log S ( U , f ^ n ) = \\infty . \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} G ( s ) = A ( s ) + X ( s ) B ( s ) + O ( T ^ { - 1 / 4 8 } ) \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { \\sigma ( n ) } = E \\left ( \\left ( \\frac 2 { K } - 1 \\right ) \\left ( \\log x + \\gamma + F \\right ) + 2 ( \\log 2 ) \\frac { K ' } { K ^ 2 } \\right ) \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} \\mu ( i , j ; c _ 2 ) & : = [ n ^ 2 - ( i - 1 ) n - ( j - 1 ) ] ! ^ { \\frac { n ^ 2 - c _ 2 n } { n ^ 2 - ( i - 1 ) n - ( j - 1 ) } } , \\\\ \\nu ( j ; c _ 2 ) & : = [ n ^ 2 - ( j - 1 ) n ] ! ^ { \\frac { n ^ 2 - c _ 2 n } { n ^ 2 - ( j - 1 ) n } } . \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} ( \\gamma _ { t , x ; s , y } ^ 2 - ( \\sigma _ { t , x } - \\sigma _ { s , y } ) ^ 2 ) ( ( \\sigma _ { t , x } + \\sigma _ { s , y } ) ^ 2 - \\gamma _ { t , x ; s , y } ^ 2 ) = 4 ( \\sigma _ { t , x } ^ 2 \\sigma _ { s , y } ^ 2 - \\sigma _ { t , x ; s , y } ^ 2 ) , \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} X ^ { 2 } + 3 Y ^ { 2 } = \\left ( X - \\sqrt { - 3 } Y \\right ) \\left ( X + \\sqrt { - 3 } Y \\right ) = \\mu , \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} v _ { t x } + v _ x v _ { x x } + v _ { x x x x } = 0 , \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} \\omega : = \\Gamma _ { i j } { } ^ j d x ^ i , \\rho _ { 1 , \\Gamma } = \\Gamma _ { i n } { } ^ i \\Gamma _ { j k } { } ^ n d x ^ j \\otimes d x ^ k , \\rho _ { 2 , \\Gamma } = \\Gamma _ { j n } { } ^ i \\Gamma _ { i k } { } ^ n d x ^ j \\otimes d x ^ k \\ , . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} \\# \\big \\lbrace z \\in & \\sigma _ \\mathrm { d i s c } ( H ( b _ 0 , e V ) ) : - r _ 0 e ^ 2 \\leq z < - r e ^ 2 \\big \\rbrace \\\\ & = \\# \\big \\lbrace z \\in \\sigma \\left ( \\big ( p _ { 0 } U p _ { 0 } \\big ) ^ { \\ast } p _ { 0 } U p _ { 0 } \\right ) : z \\geq 2 r b _ 0 \\big \\rbrace \\big ( 1 + o ( 1 ) \\big ) . \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ q j ^ k a _ j = 0 k = 0 , \\ldots , m - 1 \\sum _ { j = 0 } ^ q j ^ m a _ j \\not = 0 . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} Q _ { 1 } \\left ( x , y , z \\right ) & = x ^ { 2 } - x y a _ { 1 } + y ^ { 2 } a _ { 2 } + x z \\left ( a _ { 1 } ^ { 2 } - 2 a _ { 2 } \\right ) \\\\ [ 0 . 0 7 i n ] & + y z \\left ( a _ { 3 } - a _ { 1 } a _ { 2 } \\right ) + z ^ { 2 } \\left ( - a _ { 1 } a _ { 3 } + a _ { 2 } ^ { 2 } + a _ { 4 } \\right ) , \\\\ Q _ { 2 } \\left ( x , y , z \\right ) & = y ^ { 2 } - x z - y z a _ { 1 } + a _ { 2 } z ^ { 2 } , \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\phi _ n ^ * = \\psi _ n + \\sum _ { k = 1 } ^ j { { \\phi _ n ^ * } ^ { ( k ) } } + { \\rho _ n ^ * } ^ { ( j ) } , j \\in \\N , \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} ( - 1 ) ^ j C _ { j } ^ { ( k ) } ( \\cos ( 2 p \\phi ) ) = C _ j ^ { ( k ) } ( - \\cos ( 2 p \\phi ) ) = C _ j ^ { ( k ) } \\left ( \\cos \\left ( 2 p \\left ( \\frac { \\pi } { 2 p } - \\phi \\right ) \\right ) \\right ) \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} | | B e ^ { - i t H } \\psi | | _ { L ^ 2 _ t \\mathcal H } : = \\Big ( \\int _ { \\R } | | B e ^ { - i t H } \\psi | | ^ 2 d t \\Big ) ^ { 1 / 2 } \\le a | | \\psi | | , \\psi \\in \\mathcal H . \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} d \\widetilde { \\omega } : = \\sum _ { I , J } ( - 1 ) ^ { | I | + | J | } \\ , d \\widetilde { x } ^ I \\wedge d \\widetilde { \\xi } ^ J \\wedge d \\widetilde { \\omega } _ { I J } . \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} \\nu _ { i ^ \\prime } : = \\sum _ { j \\in I \\cup \\{ i ^ \\prime \\} } w _ j B _ I ^ { j } B _ I ^ { i ^ \\prime } \\det G _ { I , ( I \\cup i ^ \\prime ) \\backslash j } i ^ \\prime \\in I ^ \\prime . \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} w _ { 1 , k } = 0 . \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\lim _ { m \\rightarrow \\infty } \\textrm { d i a m } \\left ( \\varphi _ { t _ m } ( \\cdot , B ( x , r ) ) \\right ) = 0 \\right ) > 0 . \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} c = \\frac { 2 - n } { 1 + m - n } - \\frac { 1 - m + n } { ( 1 + m - n ) ( m - n ) } \\lambda \\ , . \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\Theta ^ n : = \\left \\{ \\theta \\in \\mathbb { R } _ + ^ n : \\theta ^ i \\geq 0 , \\ i = 1 , \\ldots , n ; \\ \\ \\textstyle \\sum _ i \\theta ^ i = 1 \\right \\} . \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { \\epsilon \\rightarrow 0 } \\lim _ { k \\rightarrow \\infty } \\int _ { \\Omega \\times \\{ 0 \\} } f ( z ) | w _ k ( z , 0 ) | ^ { q - 1 } \\psi _ { \\epsilon , j } w _ k ( z , 0 ) d z = 0 . \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} Q _ { [ H , i A ] } ( f , g ) & : = Q _ H ( f , i A g ) - Q _ H ( A f , i g ) \\\\ & = Q _ H ( f , g ) - \\langle ( 2 V + x \\cdot \\nabla V ) f , g \\rangle , \\\\ Q _ { [ [ H , i A ] , i A ] } ( f , g ) & : = Q _ H ( f , i A g ) - Q _ H ( A f , i g ) \\\\ & = 2 Q _ { [ H , i A ] } ( f , g ) + \\langle ( 2 x \\cdot \\nabla V + ( x \\cdot \\nabla V ) ^ 2 V ) f , g \\rangle . \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} { \\cal O } \\bigg ( N \\log { N } + \\sum _ { s = 1 } ^ { q } { \\bar n } _ s \\log { { \\bar n } _ s } \\bigg ) \\approx { \\cal O } \\bigg ( \\frac { \\displaystyle 3 } { \\displaystyle 2 } N \\log { N } \\bigg ) . \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\int _ { { \\bf R } ^ N } u ( y , t _ j ) \\eta ( y ) \\ , d y = \\int _ { { \\bf R } ^ N } \\eta ( y ) \\ , d \\mu ( y ) \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} \\sqrt { 2 ^ { s + 1 } - 1 } = \\lambda _ { { E _ k } , 2 } ( e _ k ) \\le \\Lambda _ { { E _ k } , 2 } \\le \\sqrt { 3 ( 2 ^ { s + 1 } - 1 ) } \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} \\| y \\| _ { H ^ 1 _ 0 ( \\Omega ) } = \\| T _ \\rho ' ( v ) \\ , h \\| _ { H ^ 1 _ 0 ( \\Omega ) } \\le \\rho ^ { - 1 / 2 } \\ , \\| h \\| _ { H ^ { - 1 } ( \\Omega ) } . \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\widetilde { \\Xi } _ 1 ( z ) = f _ \\mathrm { P r } \\left ( r \\right ) e ^ { - z { { P _ S } G _ \\mathrm { M } ^ { S } G _ \\mathrm { M } \\beta \\left ( { \\max { { \\{ r , d \\} } } } \\right ) ^ { - { \\alpha _ \\mathrm { L o S } } } } } + ( 1 - f _ \\mathrm { P r } \\left ( r \\right ) ) e ^ { - z { { P _ S } G _ \\mathrm { M } ^ { S } G _ \\mathrm { M } \\beta \\left ( { \\max { { \\{ r , d \\} } } } \\right ) ^ { - { \\alpha _ \\mathrm { N L o S } } } } } \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} \\lambda _ 1 : = \\nu + \\sqrt { \\left ( \\sum _ { i = 1 } ^ { \\nu } \\overline { \\jmath } _ i ^ 2 \\right ) \\left ( \\sum _ { i = 1 } ^ { \\nu } \\overline { \\jmath } _ i ^ { - 2 } \\right ) } , \\lambda _ 2 : = \\nu - \\sqrt { \\left ( \\sum _ { i = 1 } ^ { \\nu } \\overline { \\jmath } _ i ^ 2 \\right ) \\left ( \\sum _ { i = 1 } ^ { \\nu } \\overline { \\jmath } _ i ^ { - 2 } \\right ) } . \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} q ( \\alpha , \\beta ; t _ 1 , d y _ 1 ; t _ 2 , y _ 2 ) = q ( \\alpha , \\beta ; t _ 1 , y _ 1 ; t _ 2 , d y _ 2 ) \\cdot | d | ^ { \\alpha + \\beta } . \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{align*} n ( \\mu , 2 j ) = a , \\ , \\ m _ 1 + \\cdots + m _ { a - 1 } < 2 j \\leq m _ 1 + \\cdots + m _ a . \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} N _ n ( E ) = \\max \\{ \\tilde N _ n ( A ) : A E \\} . \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} n _ 2 = d + \\left \\lceil \\frac { k ( \\delta - 1 ) } { r } \\right \\rceil - \\delta - n _ 1 \\end{align*}"}
-{"id": "7539.png", "formula": "\\begin{align*} a ( { p ^ { r + 1 } } I ) & = \\lambda ( p ) a ( { p ^ r } I ) - \\chi ( p ^ 2 ) p ^ { 2 k - 3 } a ( { p ^ { r - 1 } } I ) \\\\ & - \\chi ( p ) p ^ { k - 2 } \\eta ( p ^ { r + 1 } ) a ( I ) . \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} f \\left ( t \\right ) = t ^ { 4 } - 2 c t ^ { 3 } + 2 t ^ { 2 } + 2 c t + 1 , \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{gather*} \\log F _ { 6 } ( t ) = - \\frac { 1 } { 4 } | t | ^ 3 + \\frac { 2 \\sqrt { 2 } } { 3 } | t | ^ { 3 / 2 } + \\frac { 1 } { 2 4 } \\log | t | + c _ 0 + O \\big ( | t | ^ { - \\frac { 3 } { 2 } } \\big ) \\mbox { a s \\ \\ $ t \\to - \\infty $ } , \\end{gather*}"}
-{"id": "3535.png", "formula": "\\begin{align*} r _ 1 + r _ 2 + r _ 3 = 0 \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} T _ 0 \\mathcal { U } = \\left \\{ u \\in C ^ { k + 2 , \\alpha } _ { \\nu + 2 } ( X ^ { r e g } ) \\oplus \\hat { \\chi } ( \\mathcal { P } \\oplus \\R r ^ 2 \\oplus \\mathcal { H } ) \\circ P ^ { - 1 } : \\int _ { X ^ { r e g } } u \\omega ^ n = 0 \\right \\} , \\\\ T _ 0 \\mathcal { F } = \\left \\{ f \\in C ^ { k , \\alpha } _ \\nu ( X ^ { r e g } ) \\oplus \\mathbb { R } : \\int _ { X ^ { r e g } } f \\omega ^ n = 0 \\right \\} . \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} \\Omega _ m ^ { 0 , q } ( X ) = \\{ u \\in \\Omega ^ { 0 , q } ( X ) : T u = i m u \\} \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathcal { D } ^ { i j } ( 0 ) \\mathcal { D } ^ { l m } ( - J , - 1 ) ] = 0 . \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} c _ { b } = \\frac { 2 } { n + 1 - 2 b } \\left ( z _ b - \\frac { 1 } { n - 1 - 2 b } \\sum _ { i = b + 1 } ^ { n - b - 1 } z _ i \\right ) \\\\ = \\frac { \\sum _ { i = b } ^ { n - b } z _ i } { n + 1 - 2 b } - \\frac { \\sum _ { i = b + 1 } ^ { n - b - 1 } z _ i } { n - 1 - 2 b } . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} \\mathrm { c a } ( x , y ) + \\mathrm { c a } ( x , z ) + \\mathrm { c a } ( y , - z ) = \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} \\sigma ( B ' ) & = \\sigma ( B ) + \\tfrac { \\sigma ( b ) } { n } \\cdot I _ n \\\\ & = A B A ^ { - 1 } + \\delta ( A ) A ^ { - 1 } - \\tfrac { 1 } { n } \\delta ( \\mathrm { d e t } ( A ) ) \\mathrm { d e t } ( A ) ^ { - 1 } \\cdot I _ n + \\tfrac { \\sigma ( b ) } { n } \\cdot I _ n \\\\ & = A ( B + \\tfrac { b } { n } \\cdot I _ n ) A ^ { - 1 } + \\delta ( A ) A ^ { - 1 } - \\tfrac { 1 } { n } \\delta ( \\mathrm { d e t } ( A ) ) \\mathrm { d e t } ( A ) ^ { - 1 } \\cdot I _ n + \\tfrac { \\sigma ( b ) - b } { n } \\cdot I _ n \\\\ & = A B ' A ^ { - 1 } + \\delta ( A ) A ^ { - 1 } , \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } b _ { 1 1 } & = & B _ { 1 2 } a _ { 1 1 } \\\\ f _ 2 & = & 0 \\end{array} \\right . , \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} | | P - Q | | _ { \\mathsf { T V } } = \\frac { 1 } { 2 } \\sum _ { x \\in \\mathcal { X } } \\big | P ( x ) - Q ( x ) \\big | . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} \\epsilon ' ( \\lambda ) & = \\epsilon _ 0 ' ( \\lambda ) + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\nu \\ , \\partial _ { \\lambda } L ( \\lambda | \\nu ) \\epsilon _ 0 ( \\nu ) , \\\\ \\tilde { \\epsilon } ' ( \\lambda ) & = \\epsilon ' ( \\lambda ) + \\sum _ { i a } s _ a \\epsilon ( \\lambda _ { i a } ) L ( \\lambda _ { i a } | \\lambda ) . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} { } _ 1 F _ 1 ( a , c ; y z ) = \\sum _ { j \\geq 0 } \\frac { ( - z ) ^ j \\Gamma ( b + j ) ( a ) _ j } { \\Gamma ( b + 2 j ) j ! } { } _ 2 F _ 1 ( - j , j + b , c ; y ) { } _ 1 F _ 1 ( a + j , b + 1 + 2 j ; z ) , \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} b _ 1 ( \\vartheta + \\omega \\tilde { \\alpha } ( \\vartheta ) , x ) = b _ 1 ( \\vartheta , x ) - \\omega \\cdot \\partial _ { \\vartheta } b _ 1 ( \\vartheta , x ) \\ , \\alpha ( \\vartheta ) + \\mathtt { R } _ { \\tilde { \\alpha } } ( \\vartheta , x ) \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} N ( S _ 1 , S _ 2 ) = - J [ S _ 1 , J S _ 2 ] . \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} v ( x ) = \\left \\{ \\begin{array} { c } - \\frac { 1 } { S _ 1 } U ^ 2 ( x ) x \\in \\mathbb { R } ^ 2 \\setminus \\overline { \\Omega _ 2 } \\\\ \\frac { 1 } { S _ 2 } U ^ 1 ( x ) x \\in \\Omega _ 2 \\end{array} \\right . . \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} - b ( x ) \\cdot D \\varphi ( x ) + f ( x ) = - b ( x ) \\cdot D \\tilde \\psi ( x ) \\zeta _ { 2 r } ( x ) - \\tilde \\psi ( x ) b ( x ) \\cdot D \\zeta _ { 2 r } ( x ) . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} [ d ( H \\circ \\gamma _ t ) ] ^ { V } = \\dot { \\gamma } _ q \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} | V ( S _ \\ell ) | - | U | = 2 p - \\frac 1 5 \\ell \\ge \\frac 4 3 ( \\ell - 3 ) - \\frac 1 5 \\ell = \\ell + \\frac { 2 \\ell - 6 0 } { 1 5 } \\ge \\ell \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} B ( v _ 0 , u ) \\ge \\frac 3 4 \\sup \\{ B ( w , u ) : w \\in V ^ q _ { l c } : \\Vert w \\Vert _ { V ^ p } = 1 \\} . \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} \\max _ { P _ M \\in \\mathcal { P } ( \\mathcal { M } _ n ) } I _ Q ( M ; \\mathbf { Z } ) \\leq \\max _ { m \\in \\mathcal { M } } \\mathsf { D } \\Big ( Q ^ { ( \\mathcal { B } _ n ) } _ { \\mathbf { Z } | M = m , I , \\mathbf { U } } \\Big | \\Big | Q ^ n _ { Z | U } \\Big | Q ^ { ( \\mathcal { B } _ n ) } _ { I , \\mathbf { U } } \\Big ) \\leq e ^ { - n \\lambda } , \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} \\xi ( n ) = \\prod _ { p \\mid n } \\left ( 1 + \\frac { \\chi ( p ) } { p } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in S _ { 2 } } m _ { g } ( C , \\lambda ) + \\sum _ { \\lambda \\in S _ { 2 } } d ( C , \\lambda ) & = \\sum _ { \\lambda \\in S _ { 2 } } m _ { a } ( C , \\lambda ) \\\\ & \\leq n - \\left ( n - | \\Lambda ( A ) | \\cdot { \\rm r a n k } ( B ) - d ( A ) + \\sum _ { \\lambda \\in S _ { 1 } } d ( C , \\lambda ) \\right ) \\\\ & = | \\Lambda ( A ) | \\cdot { \\rm r a n k } ( B ) + d ( A ) - \\sum _ { \\lambda \\in S _ { 1 } } d ( C , \\lambda ) . \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} [ a ] \\oplus [ b ] : = \\{ [ c ] \\in R / U \\mid c \\in a U + b U \\} . \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} \\mathbb { E } \\left ( \\kappa ( G , B ) \\right ) = \\frac { { n \\choose 2 } } { ( 2 n - 3 ) } \\bigg ( 1 - \\frac { { { n \\choose 2 } - m \\choose 2 n - 3 } } { { { n \\choose 2 } \\choose 2 n - 3 } } \\bigg ) . \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\tilde { H } = ( 0 , h _ { 0 } , h _ { 2 } - h _ { 1 } , 0 , - h _ { 1 } - h _ { 2 } , - h _ { 0 } , 0 , h _ { 0 } , h _ { 1 } + h _ { 2 } , 0 , h _ { 1 } - h _ { 2 } , - h _ { 0 } ) ^ { T } \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} \\Delta U = H ( U ) U _ { x } \\wedge U _ { y } \\quad \\quad \\R ^ { 2 } \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} & \\eta ( p ) a ( p ^ r I ) - \\eta ( p ^ { r + 1 } ) a ( I ) \\\\ & = - \\epsilon a \\begin{pmatrix} p ^ { r - 1 } m \\\\ & p ^ { r + 1 } m \\end{pmatrix} - \\epsilon \\sum _ { \\substack { 1 \\le u < p / 2 \\\\ u ^ 2 \\not \\equiv - 1 \\ , ( p ) } } a \\left ( p ^ r m \\begin{pmatrix} ( 1 + u ^ 2 ) / p & u \\\\ u & p \\end{pmatrix} \\right ) . \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } \\frac 1 { 2 j + 1 } \\sum _ { k = 0 } ^ j T ( n , k ) = H _ { 2 n } - \\frac { 5 } { 2 } H _ n + H _ { \\lfloor n / 2 \\rfloor } . \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} \\wedge _ { i } \\psi _ B ^ { - 1 } ( b _ 3 ^ i ) = \\pm \\nu ( \\mathcal { E } _ 3 ) ^ { - 1 } ( \\wedge _ { i } \\psi _ B ^ { - 1 } \\theta _ 3 ( a _ 3 ^ i ) ) \\wedge ( \\wedge _ { i } \\psi _ B ^ { - 1 } \\tau _ 3 ^ { - 1 } ( c _ 3 ^ i ) ) = \\pm \\nu ( \\mathcal { E } _ 3 ) ^ { - 1 } P \\wedge Q . \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\log \\left ( \\frac { 1 } { c _ { \\phi _ { n } } } \\right ) < \\infty . \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} \\psi & = \\frac { 1 } { N } \\mathrm { t r } \\left ( \\mathbf { I } _ N + \\frac { K } { N } \\frac { 1 } { \\phi } \\mathbf { I } _ N \\right ) ^ { - 2 } = \\frac { 1 } { \\left ( 1 + \\frac { \\beta } { \\phi } \\right ) ^ 2 } , \\\\ \\phi & = \\frac { 1 } { N } \\mathrm { t r } \\left [ \\left ( 1 + \\frac { \\beta } { \\phi } \\right ) \\mathbf { I } _ N \\right ] ^ { - 1 } = \\frac { 1 } { 1 + \\frac { \\beta } { \\phi } } . \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} J ( \\alpha ) = \\left [ \\mathbb { Z } _ { M } [ \\alpha ] ^ { + } : \\mathbb { Z } [ \\alpha ] ^ { + } \\right ] = \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} \\frac { ( f ( m ) - p ) \\ , ( g ( m ) - q ) } { ( f ( m ) - q ) \\ , ( g ( m ) - p ) } & = 1 + \\frac { ( f ( m ) - g ( m ) ) \\ , ( p - q ) } { ( f ( m ) - q ) ( g ( m ) - p ) } \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} - z + \\iota _ j ^ * t = \\left [ S ^ { - 1 } _ { u ^ j } ( - z ) [ G ^ j ( z ) ] _ + \\right ] _ + . \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} F _ k ( s ) : = \\sum _ { p _ 1 , \\cdots , p _ k ~ } \\frac { ( \\chi _ 0 ( p _ 1 ) + \\chi _ a ( p _ 1 ) ) \\cdots ( \\chi _ 0 ( p _ k ) + \\chi _ a ( p _ k ) ) } { p ^ s } . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} \\lambda _ n ^ { ( 1 ) } = \\frac { 1 } { \\Omega _ \\beta ' } \\left ( \\frac { \\pi } { \\Omega _ \\beta } n \\right ) ^ { - \\frac { 2 } { \\beta + 2 } } \\int _ \\R V ( x ) \\ ; \\dd x + o \\left ( n ^ { - \\frac { 2 } { \\beta + 2 } } \\right ) , n \\to \\infty , \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} s ( s + 1 ) \\tilde \\Psi ( s ) = \\int _ 0 ^ \\infty \\Psi '' ( z ) z ^ { s + 1 } d z . \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} \\{ d \\lambda _ i \\lambda _ j : i , j = 2 , \\dots , q \\} \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} [ T , Z ] = - i ( n + 1 ) Z . \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} \\sigma ^ i ( \\theta _ t ^ i , \\lambda _ t ) = \\frac { \\gamma } { \\eta } \\left [ \\left ( \\frac { \\eta \\theta _ t ^ i } { \\lambda _ t } \\right ) ^ { \\frac { 1 } { \\gamma } } - \\phi ^ i \\right ] . \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} x = 2 r + p , \\ y = q , \\ z = r , \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} g _ 1 ^ * ( b ) = \\begin{cases} f ^ * ( b ) , b \\neq b _ { k + 1 } \\\\ f ^ * ( b _ k ) , b = b _ { k + 1 } \\end{cases} \\mbox { o r } g _ 2 ^ * ( b ) = \\begin{cases} f ^ * ( b _ { k + 1 } ) , b \\leq b _ { k } \\\\ f ^ * ( b ) , b \\geq b _ { k + 1 } . \\end{cases} . \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} 3 r - s ^ 2 = 3 + 3 \\pi ^ 4 - ( 1 + 2 \\pi ^ 2 + \\pi ^ 4 ) = 2 - 2 \\pi ^ 2 + 2 \\pi ^ 4 \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { s } \\Bigl ( \\int _ { - \\infty } ^ { \\infty } P ( x u ) d F ( x ) \\Bigr ) d u = \\int _ { 0 } ^ { s } \\Bigl ( \\int _ { 0 } ^ { \\infty } P ( x u ) d F _ 1 ( x ) \\Bigr ) d u \\\\ = \\int _ 0 ^ { \\infty } \\Bigl ( \\int _ 0 ^ s P ( x u ) d u \\Bigr ) d F _ 1 ( x ) . \\end{aligned} \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} \\kappa _ 1 = \\frac { \\partial _ x ^ 2 f } { ( 1 + | \\partial _ x f | ^ 2 ) ^ { \\frac { 3 } { 2 } } } , \\kappa _ 2 = \\frac { \\partial _ x ^ 2 ( f + g ) } { ( 1 + | \\partial _ x ( f + g ) | ^ 2 ) ^ { \\frac { 3 } { 2 } } } . \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\inf _ { \\mu \\in Q } \\ , \\tilde { P } ( \\mu ) = \\inf _ { \\mu \\in Q } \\sup _ { \\substack { \\eta \\in X \\\\ \\eta ' ( 0 ) = - h ^ { 2 } } } \\ , K ( \\mu , \\eta ) \\geq \\sup _ { \\substack { \\eta \\in X \\\\ \\eta ' ( 0 ) = - h ^ { 2 } } } \\inf _ { \\mu \\in Q } \\ , K ( \\mu , \\eta ) = \\sup _ { \\substack { \\eta \\in X \\\\ \\eta ' ( 0 ) = - h ^ { 2 } } } \\ , \\tilde { D } ( \\eta ) . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} g _ { | \\mathcal { I } ^ m } = \\widetilde { g } _ { 0 1 } ( d y ^ 0 d y ^ 1 + d y ^ 1 d y ^ 0 ) + \\widetilde { g } _ { a b } d y ^ a d y ^ b ; a , b = 2 , . . . , n . \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} \\mathbf { A } = \\left [ \\mathbf { g } _ { 0 , 0 } \\ ; \\cdots \\ ; \\mathbf { g } _ { K - 1 , 0 } \\ ; \\ ; \\mathbf { g } _ { 0 , 1 } \\ ; \\cdots \\ ; \\mathbf { g } _ { K - 1 , M - 1 } \\right ] \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} \\int _ M f \\cdot C f \\ , V o l _ { \\eta } = - \\int _ M P _ f ( \\nabla f ) \\ , V o l _ { \\eta } \\geq 0 . \\end{align*}"}
-{"id": "7573.png", "formula": "\\begin{align*} C ^ { i i } _ { i i } & = C ^ { i i } _ { j j } , \\\\ C ^ { i i } _ { j y } & = 0 , \\ \\mbox { i f } \\ y \\ne i , j . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} \\P ( \\{ M _ n \\le \\lambda ^ n \\mbox { i . o . } \\} ) = 0 \\end{align*}"}
-{"id": "2849.png", "formula": "\\begin{align*} \\forall i \\in I \\left \\{ \\begin{array} { l } \\langle x \\ | \\ u _ i \\rangle - \\eta _ i = \\sum _ { k \\in I } \\nu _ k \\langle u _ k \\ | \\ u _ i \\rangle , \\\\ \\nu _ i > 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} \\epsilon = \\begin{cases} 1 - \\lfloor { \\log ( 4 8 ) / \\log ( \\ell ) } \\rfloor = - 1 & \\ell = 5 \\\\ - \\lfloor { \\log ( 4 8 ) / \\log ( \\ell ) } \\rfloor = - 1 & \\ell = 7 . \\end{cases} \\end{align*}"}
-{"id": "2725.png", "formula": "\\begin{align*} \\omega _ 0 l _ { s f , \\mathsf { k } } i _ { f , \\mathsf { k } } j \\mathrm { r } ( \\theta _ \\mathsf { k } ) = v _ \\mathsf { k } - Z _ { s , \\mathsf { k } } ( \\theta _ \\mathsf { k } ) i _ { s , \\mathsf { k } } = \\nu _ \\mathsf { k } . \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} T _ C ( \\sigma ( P ) ) = \\epsilon ( \\sigma ) \\cdot \\sigma ( T _ C ( P ) ) \\sigma \\in G _ K P \\in C [ P ] . \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{gather*} \\delta _ Y C ^ \\ast = 0 , \\delta _ Y A ^ \\ast = 0 , \\delta _ Y A = 0 , \\delta _ Y C = 1 . \\end{gather*}"}
-{"id": "8134.png", "formula": "\\begin{align*} \\left ( \\varphi f \\right ) ( x ) = \\sum _ { n = 0 } ^ { N - 1 } \\varphi _ n ( x ) f ( n ) , \\qquad \\left ( \\varphi ^ * g \\right ) ( x ) = \\int _ \\R \\d x \\ , \\varphi _ n ( x ) g ( x ) . \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} \\mathbb { P } ( Y ^ 2 - B _ n ^ 2 > t ) & = \\mathbb { P } ( | Y | ^ 2 > t + B _ n ^ 2 ) \\\\ & \\leq \\mathbb { P } ( | \\varepsilon | ^ 2 > ( 1 / 2 ) ( t + B _ n ^ 2 ) - B ^ 2 ) \\\\ & \\leq e ^ { - \\frac { t } { 2 \\nu } } e ^ { - \\frac { 1 } { v } ( \\frac { B _ n ^ 2 } { 2 } - B ^ 2 ) } \\mathbb { E } e ^ { | \\varepsilon | ^ 2 / \\nu } . \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} \\begin{aligned} a [ f ] = \\sum _ { j = 1 } ^ { \\infty } \\mu _ j | f _ j | ^ 2 \\geq C _ 1 \\sum _ { j = 1 } ^ { \\infty } j ^ \\gamma | f _ j | ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} v _ { t } ( x , t ) & = H v ( x , t ) + v ( x , t ) f ( t , u ( x , t ; s , u _ 0 ) ) \\\\ & = H v ( x , t ) + v ( x , t ) f ( t , v ( x , t ) ) + v ( x , t ) f ( t , u ( x , t ; s , u _ 0 ) ) - v ( x , t ) f ( t , v ( x , t ) ) \\\\ & \\ge H v ( x , t ) + v ( x , t ) f ( t , v ( x , t ) ) + \\delta _ 1 \\forall \\ , s < t \\le s + \\tau _ 1 , \\ , \\ , s \\in \\R . \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} A _ { ( m + 1 ) p _ n - ( m - r _ n ) } = A _ { ( m + 1 ) p _ n + r _ n } - x ^ { m - 2 r _ n } A _ { ( m + 1 ) p _ n + ( m - r _ n ) } - x ^ { m - 2 r _ n - 1 } A _ { ( m + 1 ) p _ n - r _ n - 1 } . \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} B _ h ( ( { \\bf u } _ h , { \\bf p } _ h ) , ( { \\bf v } _ h , { \\bf q } _ h ) ) : = a _ h ( { \\bf u } _ h , { \\bf v } _ h ) + b _ h ( { \\bf p } _ h , { \\bf v } _ h ) - b _ h ( { \\bf q } _ h , { \\bf u } _ h ) + c _ h ( { \\bf p } _ h , { \\bf q } _ h ) . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} u ^ { \\alpha } _ J = \\frac { \\partial ^ { | J | } u ^ { \\alpha } } { \\partial ( x ^ 1 ) ^ { j _ 1 } \\partial ( x ^ 2 ) ^ { j _ 2 } \\ldots \\partial ( x ^ p ) ^ { j _ p } } , \\end{align*}"}
-{"id": "6536.png", "formula": "\\begin{align*} \\lambda _ { k } \\left ( A _ { \\alpha } \\right ) - \\lambda _ { k } \\left ( A _ { \\beta } \\right ) \\geq \\left ( \\alpha - \\beta \\right ) \\lambda _ { \\min } \\left ( L \\left ( G \\right ) \\right ) = 0 . \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} A _ { c , E } ( \\theta ) = \\frac { 1 } { f ( \\theta ) } \\left ( \\begin{matrix} \\frac { E - v ( \\theta ) } { g ( \\theta ) } \\ \\ & - \\frac { \\tilde { c } ( \\theta - \\alpha ) } { g ( \\theta ) } \\\\ f ( \\theta ) \\ \\ & 0 \\end{matrix} \\right ) \\triangleq \\frac { 1 } { f ( \\theta ) } D _ { c , E } ( \\theta ) . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\theta _ t ^ i = \\frac { \\theta _ 0 ^ i ( \\delta ^ i ) ^ t } { \\sum _ j \\theta _ 0 ^ j ( \\delta ^ j ) ^ t } , & & t = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\langle \\psi , \\alpha _ 0 \\rangle & = \\frac { 1 } { | N _ G ( P ) | } ( q ( q - 1 ) + \\psi ( X ) ( q - 1 ) ) = 0 \\end{array} \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} \\alpha _ 1 = \\frac { d } { 4 } + \\frac { 1 } { 8 } \\mathrm { R e } ( G ( 2 , \\Lambda _ N ) ) = \\left \\{ \\begin{matrix} 1 & \\textrm { i f } & N \\equiv 1 \\pmod 2 \\\\ 1 & \\textrm { i f } & N \\equiv 0 \\pmod 4 \\\\ \\frac { 3 } { 2 } & \\textrm { i f } & N \\equiv 2 \\pmod 4 . \\end{matrix} \\right . \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} x = \\sum _ { i \\in I ^ { \\prime } } \\alpha _ i b _ i + \\sum _ { j \\in I } \\beta _ j u _ j \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} M ^ * ( x ) = \\sup _ { t \\in [ 0 , \\infty ) } \\big [ x t - M ( t ) \\big ] . \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} B ' = \\{ - \\frac { n _ B - 3 } { 2 } , \\dots , \\frac { n _ B - 3 } { 2 } \\} \\subset B , \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} \\chi _ \\alpha ( \\rho \\oplus \\eta ) = 0 \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\varphi ^ 3 ( - q ) \\varphi ( - q ^ 5 ) = & E _ { \\chi _ 0 , \\chi _ 1 } ( - q ) - 2 E _ { \\chi _ 0 , \\chi _ 1 } ( q ^ 2 ) - 4 E _ { \\chi _ 0 , \\chi _ 1 } ( q ^ 4 ) \\\\ & + 5 E _ { \\chi _ 1 , \\chi _ 0 } ( - q ) + 1 0 E _ { \\chi _ 1 , \\chi _ 0 } ( q ^ 2 ) - 2 0 E _ { \\chi _ 1 , \\chi _ 0 } ( q ^ 4 ) . \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} \\scriptstyle q _ N ^ { - 1 } H _ n ( N ) = H _ n ( I I _ { 2 , 2 + 8 k } \\oplus A _ 2 ) \\cup H _ u ( I I _ { 2 , 2 + 8 k } \\oplus A _ 2 ) , q _ N ^ { - 1 } H _ h ( N ) = H _ u ( I I _ { 2 , 2 + 8 k } \\oplus A _ 2 ) . \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} f ( r ) \\kappa ( y ) d r + g ( r ) \\tilde { d } \\kappa ( y ) , & \\tilde { d } ^ * \\tilde { d } \\kappa = \\lambda \\kappa , \\\\ h ( r ) \\eta ( y ) , & \\tilde { d } ^ * \\tilde { d } \\eta = \\lambda \\eta , \\ ; \\tilde { d } ^ * \\eta = 0 . \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} \\int _ 0 ^ s \\Bigl ( \\int _ { 0 } ^ { \\infty } ( \\cosh ( x u ) - 1 ) d F _ 1 ( x ) \\Bigr ) d u = \\int _ 0 ^ s \\Bigl ( \\frac { f ( i u ) + f ( - i u ) } { 2 } - 1 \\Bigr ) d u , \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\rho , 0 } ( e ^ { i \\lambda \\Theta _ t } ) = \\frac { \\sqrt { \\pi } } { 2 } \\sqrt { \\frac { \\rho ^ 2 } { 2 t } } e ^ { - \\rho ^ 2 / ( 4 t ) } \\left [ { \\it I } _ { ( | \\lambda | - 1 ) / 2 } \\left ( \\frac { \\rho ^ 2 } { 4 t } \\right ) + { \\it I } _ { ( | \\lambda | + 1 ) / 2 } \\left ( \\frac { \\rho ^ 2 } { 4 t } \\right ) \\right ] , \\lambda \\in \\mathbb { R } . \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} & M \\ni x \\mapsto \\pi _ { \\alpha } ( x ) ; \\ ( \\pi _ { \\alpha } ( x ) \\xi ) ( s ) : = \\alpha _ { - s } ( x ) \\xi ( s ) ; \\\\ & L \\R \\ni \\lambda _ t \\mapsto 1 _ M \\otimes \\lambda _ t ; \\ ( ( 1 \\otimes \\lambda _ t ) \\xi ) ( s ) : = \\xi ( s - t ) . \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} \\chi _ \\alpha ( \\rho ) = ( \\pi _ ! \\circ \\tau ^ \\ast ) ( \\alpha ) \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} ( \\mathrm { r e s } \\mid _ { \\mathbb { H } _ F } \\mathcal { E ' } ) ( \\tau ) = ( \\mathcal { E ' } ) ( \\tau ^ * ) . \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} \\Phi ( x ) = \\sup _ { p ( x _ i ) \\leq \\Phi ( x _ i ) , p \\in \\partial \\Phi ( K ) } p ( x _ i ) . \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\ldots \\subset E _ { n - 1 } \\subset E _ n = E \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} \\delta ( T ) = \\frac { \\delta ( \\mathrm { d e t } ( T ) ) } { n \\cdot \\mathrm { d e t } ( T ) } \\cdot T \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} \\sum _ { n = N _ 1 + 1 } ^ \\infty | \\langle f , \\rho _ n ^ { ( j ) } \\rangle | ^ 2 \\leq \\| f \\| ^ 2 \\sum _ { n = N _ 1 + 1 } ^ \\infty \\| \\rho _ n ^ { ( j ) } \\| ^ 2 < \\infty . \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} E _ 0 ( \\gamma , \\rho ) & = \\frac { \\rho } { 2 } \\log \\left ( 1 + \\frac { \\gamma k _ n P ' } { \\rho + 1 } \\right ) . \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} \\frac { 1 } { f ( \\theta ) } = \\sum \\limits _ { p = - \\infty } ^ { \\infty } b _ { p } e ^ { i p \\theta } = \\left | \\sum \\limits _ { j = 0 } ^ { \\infty } \\psi _ { j } e ^ { - i j \\theta } \\right | ^ 2 = \\left | \\sum \\limits _ { j = 0 } ^ { \\infty } \\varphi _ { j } e ^ { - i j \\theta } \\right | ^ { - 2 } . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} \\# \\big \\lbrace z \\in & \\sigma _ { \\mathrm { d i s c } } ( H ( b _ 0 , e V ) ) : - r _ 0 e ^ 2 \\leq z < - r e ^ 2 \\big \\rbrace \\\\ & = \\# \\big \\lbrace z \\in \\sigma ( p _ { 0 } U p _ { 0 } ) : z \\geq \\big ( 2 r b _ 0 \\big ) ^ \\frac { 1 } { 2 } \\big \\rbrace \\big ( 1 + o ( 1 ) \\big ) . \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} \\varphi : K _ { \\Gamma } ^ M \\longrightarrow \\widetilde { H _ { \\Gamma } } ^ M , f \\mapsto i \\circ f : = \\tilde { f } . \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} d ^ { m , l } = & \\sum _ { i = 0 } ^ { l } d _ { 1 , i } ^ { m , l } = \\sum _ { i = 0 } ^ { l } \\binom { l } { i } ( ( \\theta + 1 ) _ { m - 1 \\uparrow } ) ^ { i } ( \\sum _ { j = 1 } ^ { m - 1 } ( \\alpha + \\theta ) _ { m - j \\uparrow } ( \\theta + m - 1 ) _ { j - 1 \\downarrow } ) ^ { l - i } \\\\ & = ( ( \\theta + 1 ) _ { m - 1 \\uparrow } + \\sum _ { j = 1 } ^ { m - 1 } ( \\alpha + \\theta ) _ { m - j \\uparrow } ( \\theta + 1 + m - j ) _ { j - 1 \\uparrow } ) ^ { l } . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{gather*} \\Psi _ { \\rm W K B } ( x ) = \\begin{pmatrix} \\dfrac { 1 } { 2 } ( 1 + q _ 2 ) x - \\alpha & - 1 \\\\ \\dfrac { 1 } { 4 } ( 1 - q _ 2 ^ 2 ) & 0 \\end{pmatrix} \\left [ I + \\frac { M _ 0 } { x } + \\cdots \\right ] e ^ { \\frac { x ^ 3 } { 6 } - \\frac { t x } { 2 } } e ^ { \\left ( \\frac { x ^ 3 } { 6 } - \\frac { t x } { 2 } \\right ) \\sigma _ 3 } , \\end{gather*}"}
-{"id": "8745.png", "formula": "\\begin{align*} \\sum _ { \\nu = 0 } ^ { \\infty } \\frac { f ( p ^ { \\nu } ) } { p ^ { \\nu s } } = \\left ( 1 - \\frac { f ( p ) } { p ^ s } \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} \\Omega ^ { 2 \\gamma } _ { \\infty } : = \\Omega ^ { 2 \\gamma } _ { \\infty } ( i _ { \\delta } ) : = \\left \\{ \\omega \\in \\Omega _ 0 : \\lvert \\mathrm { i } \\omega \\cdot l + \\mu _ j ^ { \\infty } ( \\omega ) - \\mu _ k ^ { \\infty } ( \\omega ) \\rvert \\geq \\frac { 2 \\gamma \\ , \\lvert j ^ 3 - k ^ 3 \\rvert } { \\langle l \\rangle ^ { \\tau } } , \\ , \\ , \\forall l \\in \\mathbb { Z } ^ { \\nu } , \\ , \\ , \\forall j , k \\in S ^ c \\cup \\{ 0 \\} \\right \\} \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} \\Upsilon ^ { [ 0 ] } = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } M ( 0 , \\theta ) d \\theta + \\mathcal { O } ( \\delta ^ { p + 4 } ) = M ^ { [ 0 ] } ( 0 ) + \\mathcal { O } ( \\delta ^ { p + 4 } ) , \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} ( D ^ 2 v ) ^ J = \\frac { 1 } { 2 } ( D ^ 2 v + J ^ T ( D ^ 2 v ) J ) , \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\eta ( X _ H ) = - H . \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} \\mathcal { B } ( x , y + z ) = \\mathcal { B } ( x + y , z ) + \\mathcal { B } ( x - z , y ) \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} \\big | \\langle w _ 2 ^ { { s + i t } } ( - \\Delta - z ) ^ { - s - i t } w _ 1 ^ { { s + i t } } \\varphi , \\psi \\rangle \\big | & \\le C e ^ { C t ^ 2 } | | w _ 1 ^ s | | _ { M ^ { \\frac { 2 n } { \\beta _ 1 } , 2 p } } | | w _ 2 ^ s | | _ { M ^ { \\frac { 2 n } { \\beta _ 2 } , 2 p } } \\\\ & = C e ^ { C t ^ 2 } | | w _ 1 | | _ { M ^ { \\frac { 2 s n } { \\beta _ 1 } , 2 s p } } ^ s | | w _ 2 | | _ { M ^ { \\frac { 2 s n } { \\beta _ 1 } , 2 s p } } ^ s . \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} D ( m , p _ 0 - 1 ) \\cong _ { \\mathfrak { g } } \\begin{cases} V ( p _ 0 - 1 ) \\oplus V ( p _ 0 ) \\oplus \\cdots \\oplus V ( m - p _ 0 + 1 ) , & 2 p _ 0 + 2 > m , \\\\ V ( p _ 0 - 1 ) , & 2 p _ 0 + 2 \\leq m . \\end{cases} \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} M ( t , i ) - M ( ( t , i ) - 1 ) \\stackrel { \\eqref { m a r t i n g a l e d e f } } { = } \\frac { X ( t , i ) - \\pi ( t ) } { 1 - \\pi ( t ) } - \\frac { X ( t - 1 , i ) - \\pi ( t - 1 ) } { 1 - \\pi ( t - 1 ) } . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} \\mathbb { P } ( Y ^ 2 - B _ n ^ 2 > t ) & = \\mathbb { P } ( | Y | > \\sqrt { t + B _ n ^ 2 } ) \\\\ & \\leq \\mathbb { P } ( | \\varepsilon | > \\sqrt { t + B _ n ^ 2 } - B ) \\\\ & \\leq \\mathbb { P } ( | \\varepsilon | > ( 1 / \\sqrt { 2 } ) ( \\sqrt { t } + B _ n ) - B ) \\\\ & \\leq e ^ { - \\frac { 1 } { \\nu } \\sqrt { \\frac { t } { 2 } } } e ^ { - \\frac { 1 } { \\nu } ( \\frac { B _ n } { \\sqrt { 2 } } - B ) } \\mathbb { E } e ^ { | \\varepsilon | / \\nu } . \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n f \\left ( \\mathcal { A } _ i \\right ) + f \\left ( \\mathcal { A } _ { n + 1 } \\right ) \\geq \\sum _ { j = 1 } ^ n f \\left ( \\mathcal { E } _ j ^ { ( n ) } \\right ) + f \\left ( \\mathcal { A } _ { n + 1 } \\right ) , \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} 0 < \\sup _ { G \\not \\in { B _ \\delta ( G ^ * ) } } \\sum _ { i = 1 } ^ n \\log \\{ 1 + u [ f ( x _ i ; G ^ * ) / f ( x _ i ; G ) - 1 ] \\} \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} \\mathcal { H } = H \\circ \\Phi _ B = H _ 2 + \\mathcal { H } _ 3 + \\mathcal { H } _ 4 + \\mathcal { H } _ 5 + \\mathcal { H } _ { \\geq 6 } , \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} G _ o ( D \\cap T ) = \\bigsqcup _ { s \\in D _ { + } } G _ o ( ( D \\cap G _ o s ) \\cap T ) . \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} h ( t , \\mathbf { \\Phi } , \\mathbf { \\Theta } ) = \\sum \\limits _ { n = 1 } ^ N \\sum \\limits _ { k = 1 } ^ { M _ n } a _ { n k } \\delta ( t - \\tau _ { n k } ) \\delta ( \\mathbf { \\Phi } - \\mathbf { \\Phi } _ { n k } ) \\delta ( \\mathbf { \\Theta } - \\mathbf { \\Theta } _ { n k } ) . \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { J + \\bold { 1 } _ i } = D _ i \\phi _ J ^ { \\alpha } - ( D _ i \\xi ^ j ) u _ { J + \\bold { 1 } _ j } ^ { \\alpha } . \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{align*} C _ { j + 1 , j + 1 } ^ { 2 } \\lesssim \\sum _ { \\ell = 1 } ^ { j } M _ { \\ell } \\ell ^ { - 4 } \\lesssim M _ { j } . \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} \\alpha ( q _ i , q _ j ) = \\chi ( i + 1 - j ) - 2 \\chi ( i - j ) + \\chi ( i - 1 - j ) , \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} b K _ { G \\left ( x , b ; c \\right ) } ^ { - } \\left ( x \\right ) = \\left ( \\frac { x } { b } \\right ) ^ { x / b } \\exp \\left ( - \\frac { x } { b } \\right ) \\left \\{ \\frac { \\Gamma \\left ( x / b + 1 \\right ) } { \\gamma \\left ( x / b + 1 , c / b \\right ) } \\right \\} \\left \\{ \\frac { 1 } { \\Gamma \\left ( x / b + 1 \\right ) } \\right \\} . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} g ( { \\mathbf { x } _ { \\sim 1 } } ) = \\left \\{ \\begin{array} { l l } \\mbox { u n d e f i n e d } , ~ & \\mbox { n o } ~ x _ 1 ~ \\mbox { s . t . } ~ \\mathbf { x } \\in { \\rm C } ( n ) \\\\ x _ 1 , ~ & \\mbox { a s i n g l e } ~ x _ 1 ~ \\mbox { s . t . } ~ \\mathbf { x } \\in { \\rm C } ( n ) . \\end{array} \\right . \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} R ^ { 2 s - N } \\int _ 0 ^ { \\frac { r } { R ^ 2 } } \\frac { t ^ { s - 1 } } { ( t + 1 ) ^ { \\frac { N } { 2 } } } \\ d t = R ^ { - N } \\int _ 0 ^ { r } \\frac { t ^ { s - 1 } } { ( t R ^ { - 2 } + 1 ) ^ { \\frac { N } { 2 } } } \\frac { R ^ { \\varepsilon } } { R ^ { \\varepsilon } } \\ d t = R ^ { \\varepsilon - N } \\int _ 0 ^ { r } \\frac { t ^ { s - 1 } } { ( t R ^ { \\delta - 2 } + R ^ \\delta ) ^ { \\frac { N } { 2 } } } \\ d t . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} x _ k = \\left \\{ \\begin{array} { l c l } 1 , & , \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} m _ 1 = \\varepsilon ^ 2 c ( \\xi ) + \\mathtt { r } _ { m _ 1 } , \\mbox { w i t h } \\lvert \\mathtt { r } _ { m _ 1 } \\rvert ^ { L i p ( \\gamma ) } \\le \\varepsilon ^ { 3 - 2 a } . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} \\mathcal { G } _ q ( \\Phi , \\alpha ) : = \\projlim _ { ( \\tau , r ) \\in T \\times [ 1 , \\infty ) } \\mathcal { G } _ q ( \\mathcal { H } _ \\tau , r , \\alpha ) . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} \\begin{aligned} K _ { 0 2 } ( \\psi ) w = & ( \\partial _ u \\nabla \\mathcal { H } ) ( T _ { \\delta } ) [ w ] + 2 \\varepsilon ^ { b - 1 } ( \\partial _ u \\nabla \\mathcal { H } ) ( T _ { \\delta } ) [ U _ 1 w ] + \\\\ & + \\varepsilon ^ { 2 ( b - 1 ) } U _ 1 ^ T ( \\partial _ u \\nabla \\mathcal { H } ) ( T _ { \\delta } ) [ U _ 1 w ] + 2 \\ , \\varepsilon ^ { 2 b - 3 } U _ 2 [ w , \\cdot ] ^ T ( \\nabla \\mathcal { H } ) ( T _ { \\delta } ) . \\end{aligned} \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} J _ T U = 0 ~ ~ \\nabla _ T U = L _ T U ~ ~ U \\in \\Gamma ( T ^ { 1 , 0 } X ) , \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} A \\bold { D } _ F ( B ) = B \\bold { D } _ F ^ { \\ast } ( A ) + \\operatorname { D i v } P \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} \\gamma = \\frac { - u + \\sqrt { u ^ 2 + 4 t ^ 3 } } { 2 } \\bar \\gamma = \\frac { - u - \\sqrt { u ^ 2 + 4 t ^ 3 } } { 2 } . \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} \\psi _ n = \\mathrm { s t r o n g } \\lim _ { \\beta \\to \\infty } \\phi _ { \\beta , n } . \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} \\| r ^ { - 1 - \\frac { m } { 2 } ( \\frac { 1 } { \\alpha } - 1 ) } u \\| _ { 2 \\alpha } ^ 2 \\leq c \\sum _ { i = 0 } ^ \\infty r _ i ^ { - 2 - m ( \\frac { 1 } { \\alpha } - 1 ) } \\| u _ i \\| _ { 2 \\alpha } ^ 2 \\leq c \\| \\nabla u \\| _ 2 ^ 2 + c \\int _ V \\frac { u ^ 2 } { r ^ 2 } . \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} A F - F A ' = B F - F B ' = J F = 0 \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} [ D _ N H ^ + _ \\gamma ( y ) ] ( x ) = \\frac 1 \\gamma [ \\chi ^ + ( y ) ] ( x ) \\coloneqq \\begin{cases} \\frac 1 \\gamma & y ( x ) \\in [ L , L + \\gamma ] , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} \\left | \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) } { | C _ G ( x ) | | C _ G ( y ) | } \\right | _ 3 = q . \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} \\omega | _ { 0 } = \\omega _ 0 : = d x _ 1 \\wedge d y _ 1 + \\ldots + d x _ n \\wedge d y _ n . \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} & \\left \\{ ( z _ v ) _ v \\in | Z | \\ \\bigg | \\ z _ v > z _ { v ' } , \\ v \\in \\sigma \\ \\ v ' \\in Z _ 0 \\setminus \\sigma \\right \\} \\\\ = & \\left \\{ z \\in | Z | \\ \\bigg | \\ d ^ 1 ( z , | \\sigma | ) < d ^ 1 ( z , | \\sigma ' | ) , \\sigma ' \\in Z \\setminus \\{ \\sigma \\} \\ \\mathrm { d i m } ( \\sigma ' ) = l \\right \\} . \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} H ^ { ( 3 ) } _ 4 = \\frac { 1 } { 2 } \\{ \\{ H _ 2 , F ^ { ( 3 ) } \\} , F ^ { ( 3 ) } \\} + \\{ H _ 3 , F ^ { ( 3 ) } \\} + H _ 4 = \\frac { 1 } { 2 } \\{ H _ { 3 , \\le 1 } , F ^ { ( 3 ) } \\} + \\{ H _ { 3 } ^ { ( 3 ) } , F ^ { ( 3 ) } \\} + H _ 4 \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathfrak { B } _ 1 [ h ] : = \\alpha _ { 1 , 1 } \\ , \\partial _ x h + \\alpha _ { 0 , 1 } \\ , h = \\partial _ x \\{ ( 2 c _ 2 v _ { x x } - 6 c _ 3 v ) \\ , h \\} , \\\\ & \\mathfrak { B } _ 2 [ h ] : = \\{ \\alpha _ { 1 , 2 } - ( \\alpha _ { 1 , 1 } ) _ x \\ , \\beta _ 1 \\} \\ , \\partial _ x h + \\{ \\alpha _ { 0 , 2 } - ( \\alpha _ { 0 , 1 } ) _ x \\ , \\beta _ 1 \\} \\ , h - \\partial _ x \\overline { \\mathcal { R } } _ 2 [ h ] . \\end{aligned} \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} C _ k ( x ) = \\frac { x } { \\log x } \\frac { ( \\log \\log x ) ^ { k - 1 } } { ( k - 1 ) ! } \\left \\{ h \\ ( \\frac { k - 1 } { \\log x } \\ ) + O _ A \\ ( \\frac { B _ 1 ( k - 1 ) } { ( \\log \\log x ) ^ 2 } + \\frac { \\log \\log x } { k } R ( x ) \\ ) \\right \\} . \\end{align*}"}
-{"id": "5450.png", "formula": "\\begin{align*} \\| T \\| = \\int _ \\Gamma \\mu _ \\gamma \\dd \\pi ( \\gamma ) \\ ; , \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} ( c + 1 ) ^ c \\leq ( 1 + \\frac { 1 } { c } ) ^ h = ( \\frac { c + 1 } { c } ) ^ h . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} \\mathcal { L } _ \\alpha ( \\varphi ) ( x , t ) = \\Gamma ( \\alpha + 1 ) \\emph { P . V . } \\int _ { \\{ w \\in \\R ^ d : [ x , x + w ] \\subset \\Omega \\} } \\frac { \\varphi ( x + w , t ) - \\varphi ( x , t ) } { | w | ^ { d + \\alpha } } \\d w , \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\tau ^ { 2 s } \\Big | \\ln T ( i \\tau / 2 ) + \\sum _ { j = 1 } ^ { [ s ] } \\tilde T _ { 2 j } ( i \\tau / 2 ) \\Big | d \\tau \\lesssim \\left ( \\frac 1 { ( s + \\frac 1 2 ) ^ 2 } + \\frac { 1 } { [ s ] + 1 - s } \\right ) \\| ( u , v ) \\| _ { \\dot H ^ s } ^ 2 \\| ( u , v ) \\| _ { l ^ 2 D U ^ 2 } ^ { 2 [ s ] } . \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} H _ { N } ( \\sigma ) = \\beta \\sum _ { p = 2 } ^ { \\infty } \\frac { \\beta _ { p } } { N ^ { \\frac { p - 1 } { 2 } } } \\sum _ { i _ { 1 } , \\ldots , i _ { p } = 1 } ^ { N } g _ { i _ { 1 } , \\ldots , i _ { p } } \\sigma _ { i _ { 1 } } \\cdots \\sigma _ { i _ { p } } + h \\sum _ { i = 1 } ^ { N } \\sigma _ { i } , \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} \\zeta _ { \\mathrm { a } , i } = \\mathrm { t r } \\left [ \\left ( \\left \\{ \\hat { \\mathbf { H } } _ { i , \\mathrm { E } } ^ { \\mathrm { H } } \\left ( \\hat { \\mathbf { H } } _ { i , \\mathrm { E } } \\hat { \\mathbf { H } } _ { i , \\mathrm { E } } ^ { \\mathrm { H } } \\right ) ^ { - 1 } \\right \\} _ { [ 1 : K ] } \\right ) ^ 2 \\right ] . \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} Q _ 1 = u _ 0 , Q _ 2 = n u _ 0 , Q _ 3 = u _ n \\ln | u | . \\end{align*}"}
-{"id": "8550.png", "formula": "\\begin{align*} e ( c _ n ) = \\max _ { m \\in \\mathcal { M } _ n } e _ m ( c _ n ) , \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} g _ n = a p ^ { 2 n } + b p ^ n + c . \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} a _ 1 + _ F a _ 2 + _ F \\cdots + _ F a _ n = a _ 1 + _ S a _ 2 + _ S \\cdots + _ S a _ n , \\forall a _ i \\in F , \\forall n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} r _ h ( c ) = \\sideset { } { ^ * } \\sum _ { a \\pmod { c } } e \\left ( \\frac { a h } { c } \\right ) = \\sum _ { d \\mid ( c , h ) } d \\mu ( c / d ) \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} \\mathbb { H } = \\left ( \\begin{array} { c c } H _ 1 & 0 \\\\ 0 & H _ 2 \\end{array} \\right ) \\ ; , \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} V _ { x _ { 0 } } ^ { \\bot } \\cap C ( V _ { x } , \\theta ) = \\{ \\mathbf { 0 } \\} . \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} K _ { n , q } ( w _ 1 , \\cdots w _ { n - 1 } | i , t ) = \\prod _ { l = 1 } ^ { n - 1 } \\sum _ { k _ l = 0 } ^ { w _ l - 1 } q ^ { ( t + 1 ) \\sum _ { j = 1 } ^ { n - 1 } \\big ( \\prod _ { \\substack { i = 1 \\\\ i \\neq j } } ^ { n - 1 } w _ i \\big ) k _ j } \\bigg [ \\sum _ { j = 1 } ^ { n - 1 } \\big ( \\prod _ { \\substack { i = 1 \\\\ i \\neq j } } ^ { n - 1 } w _ i \\big ) k _ j \\bigg ] _ q ^ i . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} \\lim _ { m \\rightarrow \\infty } \\frac { 1 } { m } \\log | D \\varphi _ m ( \\omega , x ) v | \\in \\left \\{ \\lambda _ i \\right \\} _ { i = 1 } ^ N \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{gather*} G ^ { [ 2 ] } = \\P [ Q _ \\alpha \\bar W _ \\alpha - \\bar Q _ \\alpha W _ \\alpha ] , K ^ { [ 2 ] } = - Q _ \\alpha ^ 2 - \\P \\left [ Q _ \\alpha \\bar Q _ \\alpha \\right ] , \\end{gather*}"}
-{"id": "8713.png", "formula": "\\begin{align*} { \\mathcal L } ( \\bar { X } ( t ) ) = \\rho ( t ) , t \\ge 0 . \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} \\mathcal { R } ( u ) = \\mathcal { R } _ 0 ( u ) + \\mathcal { R } _ 1 ( u ) + \\mathcal { R } _ 2 ( u ) \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} & a ( \\lambda ) : = ( 2 4 c _ 1 ^ 2 - 1 2 c _ 4 ) \\lambda ^ 6 + ( \\frac { 1 4 } { 3 } c _ 2 ^ 2 - 4 c _ 6 ) \\lambda ^ 4 + ( 1 2 c _ 2 c _ 3 - 1 2 c _ 7 ) \\ , \\lambda ^ 2 - 6 \\ , c _ 3 ^ 2 , \\\\ & b ( \\lambda ) : = ( - 4 8 c _ 1 ^ 2 + 2 4 c _ 4 ) \\lambda ^ 6 + ( - \\frac { 3 2 } { 3 } c _ 2 ^ 2 + 8 c _ 6 ) \\lambda ^ 4 + ( - 1 6 c _ 2 c _ 3 + 2 4 c _ 7 ) \\ , \\lambda ^ 2 . \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{gather*} L = T \\Lambda T ^ { - 1 } , \\Lambda = \\begin{pmatrix} \\lambda _ + & 0 \\\\ 0 & \\lambda _ - \\end{pmatrix} . \\end{gather*}"}
-{"id": "5079.png", "formula": "\\begin{align*} \\tilde { F } _ { n } ^ { \\left ( 2 \\right ) } = \\sum _ { k \\ge 0 } \\binom { n - k } { k } _ { 2 } \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} \\partial _ x u _ i + \\partial _ z w _ i = 0 \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} \\mathrm { v o l } ( B ^ d _ { \\bf R } ( t ) ) = t ^ { 1 / R _ 1 + . . . + 1 / R _ d } \\ , \\mathrm { v o l } ( B ^ d _ { \\bf R } ) . \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} \\Xi _ n = \\left \\{ \\sum _ { k , \\ell = 0 } ^ \\infty \\left | h _ n ( k , \\ell ) - h ( k , \\ell ) \\right | \\le K n ^ { - \\delta } \\right \\} . \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} t ^ { a b } ( u ) = \\delta _ { a b } + \\beta \\sum _ i \\frac { E ^ { a b } _ { i } } { \\beta u + D _ i ^ { ( N ) } } \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} n _ { + } \\big ( r , \\textup { \\textbf { w } } p \\textup { \\textbf { w } } ^ { \\ast } \\big ) \\leq n _ { + } \\left ( r , \\zeta ^ { - 1 } p \\textbf { \\textup { W } } ( I ) p \\right ) , r > 0 . \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} T \\beta T ^ \\star = \\left ( \\begin{array} { c c } 0 & - i 1 _ 2 \\\\ i 1 _ 2 & 0 \\end{array} \\right ) \\ , , T \\alpha _ k T ^ \\star = \\alpha _ k \\ , , T ( i \\beta \\alpha _ { 3 } ) T ^ \\star = \\left ( \\begin{array} { c c } \\sigma _ 3 & 0 \\\\ 0 & - \\sigma _ 3 \\end{array} \\right ) = : \\mathcal { B } ^ 0 \\ , . \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} \\int 1 _ { \\{ \\prod _ { k = 0 } ^ { n } g ( a _ i ) \\leq e ^ { - \\varepsilon n ^ 2 } \\} } 1 _ { \\| a \\| _ { \\infty } < \\delta } d \\ell _ { E ^ n } ( a _ 0 , \\dots , a _ n ) \\leq e ^ { - K n ^ 2 } . \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} | \\Lambda _ { n , r , R } | \\leq 2 \\Big ( \\log \\frac { R } { r } \\Big ) \\prod _ { i = 0 } ^ n \\| F _ i \\| _ { p _ i } \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\Lambda _ S ( \\theta ) = \\log \\left ( \\sum _ i p _ i e ^ { - \\theta r _ i } \\right ) . \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} p ( s ) = \\frac { e ^ { - \\mu ( s - t ) } } { 1 + T - s } . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} A ^ * = \\frac { A ( 9 - 1 2 \\sqrt { 2 } ) } { 2 3 } , B ^ * = \\frac { B ( 2 ^ { 5 / 3 } - 3 \\cdot 2 ^ { 1 / 3 } - 1 ) } { 2 ^ { 5 / 3 } - 2 ^ { 1 / 3 } + 1 } . \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} i Q _ { H _ 0 } \\big ( A ( \\delta A ^ 2 + 1 ) ^ { - 1 } u , u \\big ) = i Q _ { H _ 0 } \\big ( A u _ { \\delta } , u _ { \\delta } \\big ) + O \\Big ( \\big | \\big | \\delta ^ { \\frac { 1 } { 2 } } A u _ { \\delta } \\big | \\big | _ { { \\mathcal H } ^ 1 } | | u | | _ { { \\mathcal H } ^ 1 } \\Big ) . \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} s = \\frac { 1 } { 2 } + i t , \\ T \\le t \\le 2 T . \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} u a _ 1 ' = 2 s , u ^ 2 a _ 2 ' = 3 r - s ^ 2 , u ^ 3 a _ 3 ' = 2 T , u ^ 4 a _ 4 ' = a _ 4 + 3 r ^ 2 - 2 s T , u ^ 6 a _ 6 ' = a _ 6 + r a _ 4 + r ^ 3 - T ^ 2 . \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} \\Gamma \\left ( s - \\frac { 1 } { 2 } \\right ) \\Gamma ( s ) = 2 ^ { 2 - 2 s } \\sqrt { \\pi } \\cdot \\Gamma \\left ( 2 s - 1 \\right ) . \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} \\mathcal { L } _ { s u b } u = f , \\ , \\ , \\ , f \\in B ^ r _ { p , q } ( \\textnormal { S U } ( 2 ) ) \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} S ^ { > v ( s ) } \\ = \\ \\{ v ( ( g ( 1 + \\epsilon ) ) ' - s ) : \\epsilon \\prec 1 \\} . \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} F ( \\lambda | \\lambda ' ) = \\frac { \\theta ( \\lambda - \\lambda ' ) } { 2 \\pi } + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) F ( \\nu | \\lambda ' ) \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} \\psi ( \\rho , \\nabla \\rho , \\mu ) = - \\mu \\ , \\rho + W ( \\rho ) + \\frac 1 2 \\ , | \\nabla \\rho | ^ 2 , \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} \\left \\| [ 0 , T ] \\times \\R ^ d \\ni ( s , z ) \\mapsto \\sum _ { t \\in [ s , T ] } q ^ { Q , [ s , T ] } ( t ) U ( t , z + W _ t ^ 0 - W _ s ^ 0 ) \\in \\R \\right \\| _ { k , Q } \\leq \\| U \\| _ { k + 1 , Q } \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{gather*} \\ast \\Theta = \\xi ^ \\mu T _ { \\mu \\nu } d x ^ \\nu , \\end{gather*}"}
-{"id": "8539.png", "formula": "\\begin{align*} R = B T _ f R _ c \\log _ 2 \\left ( 1 + \\gamma _ { \\mathrm { S T B C } } ^ { ( N ) } \\right ) = B T _ f R _ c \\log _ 2 \\left ( 1 + \\frac { \\xi _ 0 } { M R _ c } \\Omega ^ { ( N ) } \\right ) \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} \\int _ { Q } L ( \\nabla \\sigma ) \\cdot \\nabla \\eta \\ , d x = 0 , \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} ( E - v ( \\theta ) ) D _ { E , 1 1 } ( \\theta ) & = e ^ { 2 \\pi i \\rho _ { | s | } ( E ) } s ( \\theta ) D _ { E , 1 1 } ( \\theta + \\alpha ) + e ^ { - 2 \\pi i \\rho _ { | s | } ( E ) } \\tilde { s } ( \\theta - \\alpha ) D _ { E , 1 1 } ( \\theta - \\alpha ) , \\\\ ( E - v ( \\theta ) ) D _ { E , 2 1 } ( \\theta ) & = e ^ { - 2 \\pi i \\rho _ { | s | } ( E ) } s ( \\theta ) D _ { E , 2 1 } ( \\theta + \\alpha ) + e ^ { 2 \\pi i \\rho _ { | s | } ( E ) } \\tilde { s } ( \\theta - \\alpha ) D _ { E , 2 1 } ( \\theta - \\alpha ) . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} \\begin{aligned} \\| B ( z ) \\| & \\leq M _ b \\left ( \\frac { C } { h _ 2 } + \\sum _ { k = 2 N } ^ { \\infty } \\frac { 1 } { k ^ { 2 \\alpha } \\sqrt { ( \\mu _ k - \\mu _ { N + 1 } ) ^ 2 + h _ 2 ^ 2 } } \\right ) \\end{aligned} \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} Q _ 7 = \\frac { 6 } { r ^ 4 } - \\frac { 2 4 } { r ^ 5 } + \\frac { 1 3 2 / 7 } { r ^ 6 } \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} C \\ ; : \\ ; y ^ 2 = x ^ 3 + a x + b d C \\ ; : \\ ; y ^ 2 = x ^ 3 + a d ^ 2 x + d ^ 3 b . \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} \\frac 1 y \\int _ 0 ^ { 2 \\pi } \\int _ { 0 } ^ { 1 } x ^ { - 2 \\rho + 1 } d x d \\phi + \\frac 1 y \\int _ 0 ^ { 2 \\pi } \\int _ { 1 } ^ { 4 y ^ { 2 \\delta } } \\sqrt x d x d \\phi = O \\big ( y ^ { 3 \\delta - 1 } \\big ) . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} e ^ { b } - e ^ { a } = ( b - a ) \\int _ 0 ^ 1 e ^ { \\alpha b + ( 1 - \\alpha ) a } d \\alpha \\leq e ^ { \\sup ( a , b ) } \\vert b - a \\vert \\ . \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} \\tilde { d } _ 0 : = \\varepsilon ^ 2 ( \\mathcal { A } ^ T - I ) \\alpha _ { 0 , 2 } + \\mathcal { A } ^ T \\mathtt { R } _ 0 + \\mathcal { R } _ { \\tilde { \\beta } } + ( \\mathcal { T } ^ { - 1 } - \\mathrm { I } ) b _ 0 + \\mathcal { T } ^ { - 1 } \\tilde { c } _ 0 . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} \\mathbb { B } & = D _ S ^ 4 \\left \\{ ( \\frac { 1 4 } { 3 } c _ 2 ^ 2 - 4 c _ 6 ) \\mathrm { I } + ( 4 c _ 6 - \\frac { 1 6 } { 3 } c _ 2 ^ 2 ) \\{ U + D _ { - 2 } U D _ S ^ 2 \\} \\right \\} \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} f ^ * \\circ f = B _ x ( 2 \\rho ) B _ y - x ( \\rho ) B _ x B _ y - B _ x B _ y ( y ^ { - 1 } \\rho ) . \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} \\left ( \\overline { g } \\cdot \\alpha _ { 2 r } ( g ) \\right ) ( x ) = \\sum _ { y \\in \\Lambda } \\sum _ { y ' \\in \\Lambda } \\overline { f ( x - y ) } f ( x - 2 r - y ' ) = 0 \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} & \\mathrm { P r } ( \\hat x _ n \\neq \\bar x _ n | x _ n = \\bar x _ n ) = \\mathrm { P r } ( \\hat x _ n \\neq x _ n ) \\\\ = & \\mathrm { P r } ( | y _ n - \\sqrt { P } x _ n | ^ 2 > | y _ n + \\sqrt { P } x _ n | ^ 2 ) \\\\ = & \\mathrm { P r } ( | z _ n + v _ n | ^ 2 > | 2 \\sqrt { P } x _ n + z _ n + v _ n | ^ 2 ) , \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} H _ { | c | } ( \\theta ) = T _ { \\theta } ^ { - 1 } H _ c ( \\theta ) T _ { \\theta } . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} - a _ y R f _ * { \\cal F } = ( C C { \\cal F } , d f ) _ { T ^ * X , y } \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} P _ 1 = ( 0 , 0 ) , P _ 2 = ( - 1 , \\zeta _ 3 ^ 2 ) , Q = ( - \\zeta _ 3 ^ { 2 \\alpha } , \\zeta _ 3 ^ 2 ) . \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} B ' ( n ) = \\frac { n } { 2 k _ n } \\log k _ n - \\frac { H _ 2 ( \\alpha _ n ) } { \\alpha _ n } . \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} s - \\frac { \\epsilon ' } { 1 + \\epsilon } \\ = \\ \\frac { y ' } { 1 + y } - \\frac { \\epsilon ' } { 1 + \\epsilon } \\ = \\ \\frac { y ' ( 1 + \\epsilon ) - \\epsilon ' ( 1 + y ) } { ( 1 + y ) ( 1 + \\epsilon ) } \\ = \\ \\frac { ( 1 + \\epsilon ) ( y - \\epsilon ) ' - \\epsilon ' ( y - \\epsilon ) } { ( 1 + y ) ( 1 + \\epsilon ) } \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\sum _ { \\substack { p \\leq x \\\\ p \\equiv a \\bmod q } } \\frac { 1 } { p } = \\frac { \\log \\log x } { \\phi ( q ) } + M ( q , a ) + O \\ ( \\frac { 1 } { \\log x } \\ ) , \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} \\mu ( \\theta ) : = \\sum _ { i = 1 } ^ n \\theta ^ i \\delta ^ i , \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} 2 ^ { ( 2 - 2 s ) g } \\pi ^ { g / 2 } \\prod _ { j = 1 } ^ g \\Gamma ( \\alpha _ j ( s ) + \\beta _ j ( s ) - 1 ) = \\prod _ { j = 1 } ^ g \\Gamma ( \\alpha _ j ( s ) ) \\Gamma \\left ( \\beta _ j ( s ) - \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} \\varphi _ s ( a b ) = \\overline { \\varphi _ s ( b ^ * a ^ * ) } . \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\mathrm { d l o g } \\tilde { y } = e / T - \\sum _ { j = 0 } ^ { \\infty } p ^ j \\sum _ { ( i , p ) = 1 } ^ { \\infty } i \\sum _ { k \\geq 1 } [ a _ { i j } ] ^ k T ^ { i k - 1 } . \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} p _ { u } = \\max \\{ 1 \\leq j \\leq l \\mid u _ { j } = u \\} , m _ { u } = \\min \\{ 1 \\leq j \\leq l \\mid u _ { j } = u \\} . \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} \\textrm { d i a m } ( A ) : = \\sup _ { x , y \\in A } | x - y | . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} x ( 0 ) = \\int _ 0 ^ 1 \\widehat { A } ( s ) \\textup d x ( s ) x ( 1 ) = \\int _ 0 ^ 1 \\widehat { B } ( s ) \\textup d x ( s ) \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} F _ i ( w ^ { ( i ) } ) = \\left \\{ \\begin{array} { r l } w ^ { ( i ) } - \\frac { 1 0 ^ { - 6 } } { 2 } , & ~ w ^ { ( i ) } \\geq \\frac { 1 0 ^ { - 6 } } { 2 } \\\\ - w ^ { ( i ) } - \\frac { 1 0 ^ { - 6 } } { 2 } , & ~ w ^ { ( i ) } \\leq - \\frac { 1 0 ^ { - 6 } } { 2 } \\\\ 0 , & . \\end{array} \\right . \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} \\widetilde { \\rho } ( G _ n ) = \\frac { 3 } { L _ n } \\sum _ { i \\to j } \\mathcal { F } _ n ^ \\ast ( D _ i ) \\mathcal { F } _ n ^ \\ast ( D _ j ) - 3 . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} w ( z ) = - z \\frac { \\lambda + z } { 1 + \\lambda z } ( 0 \\le \\lambda \\leq 1 ) \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} \\frac { d } { d u } P _ { j + 1 } ^ { ( a - 1 , b - 1 ) } ( u ) = \\frac { j + a + b } { 2 } P _ { j } ^ { ( a , b ) } ( u ) . \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} | u _ { 0 } | = ( 2 | c | + 3 ) ^ { 0 } , \\ | u _ { 1 } | = | 2 c + 1 | \\leq ( 2 | c | + 1 ) \\leq ( 2 | c | + 3 ) ^ { 1 } . \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} A \\boxplus M : = v _ \\ast ( p r ^ \\ast ( M ) \\otimes p r _ { ( S , 0 ) } ^ * ( A ) ) ) \\simeq M \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} \\liminf \\limits _ { n \\to \\infty } \\frac { t } { n ^ 2 \\ln n } = \\infty . \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} \\mathcal L _ x = - i d \\omega _ 0 = 2 \\sum _ { k , j = 1 } ^ { n - 1 } \\frac { \\partial ^ 2 \\varphi ( z ) } { \\partial z _ k \\partial \\overline z _ j } d z _ k \\wedge d \\overline z _ j . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{gather*} \\{ \\alpha , \\beta \\} = ( - 1 ) ^ { \\epsilon ( X ) } i _ X i _ Y \\omega , \\end{gather*}"}
-{"id": "4261.png", "formula": "\\begin{align*} \\sum _ { i \\in I } \\mathbb { E } ( \\Phi _ * ) _ { \\left [ - \\lambda , \\lambda \\right ] } ( f _ i ) ( x ) = \\ & \\frac { 1 } { 2 \\lambda } \\int _ { - \\lambda } ^ { \\lambda } \\left ( \\sum _ { i \\in I } f _ i ( \\Phi _ { - t } ( x ) ) \\right ) \\ d t \\\\ = \\ & \\frac { 1 } { 2 \\lambda } \\int _ { - \\lambda } ^ { \\lambda } 1 \\ : d t \\ = 1 . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} \\varepsilon ^ * ( x ) = \\sup _ { t \\geq x } \\varepsilon ( t ) . \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} \\nu ( A _ { x _ o } ^ { - 1 } C ) = \\mu ( A ) + \\nu ( C ) < 1 , \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\cdot \\prod ^ { 2 p } _ { j = l + 1 } p _ { \\varepsilon _ 2 } ( x _ { u _ k } ( v _ j ) - ( x ( U _ 1 ) - x ( u _ k ) + z ) ) d \\vec { v } . \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} \\Phi _ 1 ^ { - 1 } = \\exp ( - \\varepsilon A _ 1 ) = \\mathrm { I } _ { H _ S ^ { \\perp } } + \\varepsilon \\check { A } _ 1 , \\ , \\ , \\check { A } _ 1 : = \\sum _ { n \\geq 1 } \\frac { \\varepsilon ^ { n - 1 } } { n ! } ( - A _ 1 ) ^ n , \\ , \\ , \\lvert \\check { A } _ 1 \\partial _ x \\rvert _ s ^ { L i p ( \\gamma ) } + \\lvert \\partial _ x \\check { A } _ 1 \\rvert _ s ^ { L i p ( \\gamma ) } \\le C ( s ) . \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\sigma } u ( x ) & : = c _ { N , \\sigma } P . V . \\int _ { \\R ^ { N } } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { N + 2 \\sigma } } \\ d y : = c _ { N , \\sigma } \\lim _ { \\epsilon \\to 0 ^ + } \\int _ { | x - y | > \\epsilon } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { N + 2 \\sigma } } \\ d y \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} v \\lrcorner ~ \\theta : = \\theta _ v = \\eta ( v ) . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} \\tilde { \\beta } _ t = \\left ( \\sum _ { i = 1 } ^ n \\tilde { \\theta } _ 0 ^ i ( \\tilde { \\delta } ^ i ) ^ t \\right ) ^ \\gamma , & & t = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} F ^ { \\mathbf { \\underline m } _ \\beta } v _ { \\varpi _ j } = \\sum _ { k = 1 } ^ r c _ k F ^ { \\mathbf { \\underline a } ^ k } v _ { \\varpi _ j } . \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} \\big \\{ c \\big ( ( x _ 1 , \\ldots , x _ { a - 1 } , x _ { a } = u ^ { \\ast } , x _ { a + 1 } , \\ldots , x _ t ) ; ( u ^ { \\ast } , v ^ { \\ast } ) \\big ) \\big \\} _ { ( x _ 1 , \\ldots , x _ { a - 1 } , x _ { a + 1 } , \\ldots , x _ t ) \\in [ r ] ^ { t - 1 } } , \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} \\lambda _ { \\underbrace { 0 0 \\ldots 0 } _ { \\frac { N } { 2 } } \\underbrace { 1 1 \\ldots 1 } _ { \\frac { N } { 2 } } } = \\lambda _ { \\underbrace { 1 1 \\ldots 1 } _ { \\frac { N } { 2 } } \\underbrace { 0 0 \\ldots 0 } _ { \\frac { N } { 2 } } } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda } ( \\frac { b _ { { 0 } } ^ { * } b _ { { 0 } } } { V } ) = \\\\ \\lim _ { \\lambda \\to + 0 } \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\eta ( b _ { { 0 } } ^ { * } ) \\eta ( b _ { { 0 } } ) ) \\ \\ \\forall \\phi \\in [ 0 , 2 \\pi ) \\ . \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} d _ { K E } ( f ) = \\lim _ { r \\rightarrow 0 } \\frac { \\log \\sup _ { B _ { \\omega } ( x , r ) } | f | } { \\log r } . \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} \\chi _ \\alpha ( \\rho ) = - ( \\pi _ ! \\circ \\tau ^ \\ast \\circ i ^ * ) ( \\alpha ) \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} ( - 1 ) ^ { m + n } \\left [ u _ { 1 , 0 } - u _ { 0 , 1 } - \\frac { a ( m ) - b ( n ) } { u _ { 0 , 0 } - u _ { 1 , 1 } } - \\left ( u _ { 0 , - 1 } - u _ { - 1 , 0 } - \\frac { a ( m - 1 ) - b ( n - 1 ) } { u _ { - 1 , - 1 } - u _ { 0 , 0 } } \\right ) \\right ] = 0 , \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} ( H _ { c } ( \\theta ) u ) _ n = c ( \\theta + n \\alpha ) u _ { n + 1 } + \\tilde { c } ( \\theta + ( n - 1 ) \\alpha ) u _ { n - 1 } + v ( \\theta + n \\alpha ) u _ { n } , \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} G ^ { \\prime } ( h ) = \\dfrac { 2 5 6 h \\left ( \\dfrac { s - v } { s } \\right ) ^ { 4 } \\left ( v t - w s \\right ) ^ { 2 } \\left ( s - h \\right ) o ( h ) } { \\sqrt { M ( h ) } { \\large ( } J ( h ) + \\sqrt { M ( h ) } { \\large ) } ^ { 2 } } \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} q _ k = P _ { \\mathcal { R } } \\left ( 1 - \\frac { k } { w } \\right ) ^ { d _ v } , k \\in \\{ 0 , 1 , \\ldots , w - 1 \\} , \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} u ( t , x ) = a ( t ) u _ 0 ( x ) + b ( t ) u _ 1 ( x ) + y ( t , x ) , \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} d _ { \\lambda ' \\mu ^ * } ( q ) = q ^ { w } d _ { \\lambda \\mu } ( q ^ { - 1 } ) . \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{align*} u \\cdot ( \\nabla v ) : = \\sum _ { i = 1 } ^ d u _ i \\frac { \\partial v } { \\partial x _ i } , \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} Q ( \\nabla _ { \\hat e _ 1 } h , h ) ( x _ p ) & = - \\frac { p } { 2 r \\sqrt { V } } \\Phi ( h ) ( x _ p ) , \\\\ \\Phi ( \\nabla _ { \\hat e _ 1 } h ) ( x _ p ) & \\geq \\frac { p ^ 2 } { 4 r ^ 2 V } \\Phi ( h ) ( x _ p ) . \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ { p ( x ) } u = \\widetilde { f } ( x , z _ { 1 } , z _ { 2 } ) & \\Omega , \\\\ - \\Delta _ { q ( x ) } v = \\widetilde { g } ( x , z _ { 1 } , z _ { 2 } ) & \\Omega , \\\\ u , v > 0 & \\Omega , \\\\ u , v = 0 & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} \\eta ( z ) = q ^ { 1 / 2 4 } \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { n } ) . \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathcal { D } ^ { i j } ( 0 ) \\mathcal { D } ^ { i l } ( - J , - 1 ) ] = \\frac 1 2 \\mathbb { E } [ \\mathcal { D } ^ { i j } ( 0 ) \\mathcal { D } ^ { i j } ( - J , - 1 ) ] = - \\frac { p ( \\chi + k - 2 ) } { 2 k ( k - 1 ) } . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} & \\partial _ x f = \\partial _ x g = \\partial _ x \\Gamma = 0 , \\\\ [ 5 p t ] & \\partial _ x ^ k f = \\partial _ x ^ k g = 0 . \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} U _ { 2 ^ n m } = U _ m ; \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} A _ \\theta ( r ) = \\lambda \\ ; . \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} X + Y = \\bigcup _ { x \\in X , y \\in Y } ( x + y ) \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} h ^ * ( f ) ( [ a _ 1 | \\dots | a _ n ] ) = ( - 1 ) ^ { ( n - 1 ) w ( a _ 1 ) + ( n - 2 ) w ( a _ 2 ) + \\dots + w ( a _ { n - 1 } ) } f ( [ a _ 1 | \\dots | a _ n ] ) . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} ( f - T f ) ( 2 y - f - T f ) & = 2 y f ( | f | - B _ n ) - ( | f | - B _ n ) ( | f | + B _ n ) \\\\ & \\leq 2 | y | ( | f | - B _ n ) - ( | f | - B _ n ) ( | f | + B _ n ) . \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} \\begin{cases} ( d _ 0 , d _ 1 , d _ 2 , d _ 3 ) \\in I _ 0 \\times I _ 1 \\times I _ 2 \\times I _ 3 , \\\\ \\min _ { i \\in \\{ 1 , 2 , 3 \\} } \\rho _ i ^ { d _ i } ( 0 ) \\geq d _ 0 , \\\\ \\nu _ i ^ { d _ 0 } ( h _ i ) \\geq d _ i \\ \\ \\ i \\in \\{ 1 , 2 , 3 \\} . \\end{cases} \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} ( x , y , z ) \\cdot ( x ' , y ' , z ' ) = ( x + x ' , y + y ' , z + z ' + \\frac { 1 } { 2 } ( x y ' - y x ' ) ) \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} C _ { * } : = \\frac { 2 \\pi ^ { 2 } e ^ { 2 ( 1 + | \\log C _ { M } | ) } ( 1 + \\frac { 1 } { 1 5 } \\varepsilon _ { * } ) } { 1 - \\frac { 7 } { 6 } \\varepsilon _ { * } } , \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} \\left ( I _ { \\ell + 1 } \\cap \\bigcup _ { r = 1 } ^ \\ell I _ r \\right ) \\subseteq I _ \\ell . \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} N = N ' A ^ s = A ' , \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} \\int _ { E _ { r + 1 } } \\left ( \\frac { t _ 1 } { t _ { r + 1 } } \\right ) ^ a \\left ( \\prod _ { j = 1 } ^ r \\frac { d t _ j } { 1 - t _ j } \\right ) \\frac { d t _ { r + 1 } } { t _ { r + 1 } } \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} a ( \\boldsymbol { n } ; \\boldsymbol { z } ) = \\sum _ { \\boldsymbol { k } } c ( \\boldsymbol { k } , \\boldsymbol { n } ) z _ 1 ^ { k _ 1 } \\cdots z _ l ^ { k _ l } . \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} u = - 6 a , t = 9 a ^ 2 - 4 b , s = 1 2 a b - 2 c , \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} | \\Omega _ { \\Lambda } = \\otimes _ { { x } \\in \\Lambda } | { n } ) _ { { x } } \\ . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} & u ( j + x ^ * _ { t , n _ k } + J ( t ) + x ( \\mu ^ * , n _ k ) , t ; - n _ k , \\bar \\phi _ { \\mu ^ * } ( \\cdot , - n _ k ) ) \\\\ & \\begin{cases} \\ge u ( j + x ( \\mu , t , n _ k ) , t ; - n _ k , \\bar \\phi _ \\mu ( \\cdot , - n _ k ) ) \\quad \\forall \\ , \\ , j \\le 0 \\cr \\le u ( j + x ( \\mu , t , n _ k ) , t ; - n _ k , \\bar \\phi _ \\mu ( \\cdot , - n _ k ) ) \\forall \\ , \\ , j > 0 . \\end{cases} \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} X = \\bigcup _ { i \\in \\{ 1 , \\ldots , k \\} } \\{ x \\in X \\ | \\ f _ { x _ i } ( x ) \\ne 0 \\} . \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} U ( \\eta ) \\left ( \\begin{array} { c } g _ j ( r ) \\\\ h _ j ( r ) \\end{array} \\right ) = \\left ( \\begin{array} { c } G _ j ( r ) \\\\ F _ j ( r ) \\end{array} \\right ) \\ ; , \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} \\left \\vert \\widehat { F _ { \\alpha _ { 1 } } } ( \\xi ) \\right \\vert = \\frac { 1 } { \\left \\vert \\frac { 1 } { 2 } + i \\xi \\right \\vert } \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} \\frac { d } { d s _ 1 } \\mathrm { s n } ( x , y ) = \\mathrm { s n } ( b ( x ) , y ) \\ \\mathrm { a n d } \\\\ \\frac { d } { d s _ 2 } \\mathrm { c m } ( x , y ) = \\mathrm { c m } ( x , b ( y ) ) . \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\lim _ { x \\to \\pm \\infty } \\mathcal { E } _ j ^ { \\pm } ( x ; \\lambda ) = 0 ; \\lim _ { x \\to \\pm \\infty } \\tilde { \\mathcal { E } } _ j ^ { \\pm } ( x ; \\lambda ) = 0 . \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} V = X _ { T _ i } ^ { - 1 } X _ { T _ { i + 1 } } ^ { \\vphantom { - 1 } } W = X _ { T _ i } ^ { \\vphantom { - 1 } } X _ { T _ { i + 1 } } ^ { - 1 } X _ { T _ k } ^ { \\vphantom { - 1 } } . \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} T = \\frac { 1 } { \\sqrt { 2 } } \\left ( \\begin{array} { c c } 1 _ 2 & i 1 _ 2 \\\\ i 1 _ 2 & 1 _ 2 \\end{array} \\right ) \\ , . \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} L \\sigma _ z ^ { - 1 } = \\sigma _ z ^ { - 1 } L - \\sigma _ z ^ { - 2 } [ L , c ( \\delta u ) ^ 2 ] + \\sigma _ z ^ { - 3 } [ [ L , c ( \\delta u ) ^ 2 ] , c ( \\delta u ) ^ 2 ] . \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} a ( x ) = \\left ( q ( x ) + \\frac { 1 } { a ^ \\ast } \\right ) ^ { - 1 } = \\left ( \\Phi ( g ( x ) ) + \\frac { 1 } { a ^ \\ast } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } \\mu \\left ( t \\right ) \\mathrm { d } t = - 1 + q \\left ( 1 - c \\right ) e < 0 . \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} \\widetilde { M } _ k ( x ; \\boldsymbol { a } ) : = \\# \\{ p _ 1 p _ 2 \\cdots p _ k \\leq x : p _ 1 < p _ 2 < \\cdots < p _ l , p _ j \\equiv a _ j \\bmod q , 1 \\leq j \\leq k \\} . \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} \\mathbb { E } [ e ^ { t M _ { i - 1 } } ] \\leq \\exp \\left ( t m _ 0 + \\frac { t ^ { 2 } } { 2 } g ( t m ) \\sum _ { j = 1 } ^ { i - 1 } \\sigma _ j ^ 2 \\right ) . \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\mu ( \\mathcal { C } _ n ) = \\prod _ { i \\in \\mathcal { I } _ b } p ^ n _ U \\big ( \\mathbf { u } ( i ) \\big ) \\prod _ { \\substack { \\big ( \\hat { i } , j , m \\big ) \\\\ \\in \\mathcal { I } _ n \\times \\mathcal { J } _ n \\times \\mathcal { M } _ m } } p ^ n _ { V | U } \\Big ( \\mathbf { v } \\big ( \\hat { i } , j , m \\big ) \\Big | \\mathbf { u } ( \\hat { i } ) \\Big ) . \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} \\bold { p r } X = \\sum _ { \\alpha , J } ( D _ J Q ^ { \\alpha } ) \\frac { \\partial } { \\partial u _ J ^ { \\alpha } } . \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { ( 1 + 4 \\tau ) r } { 2 \\sqrt { 2 \\tau } } \\pm \\frac u { 2 \\sqrt \\tau } & = \\sqrt { 2 N } + \\frac { ( T ^ 2 + R \\pm U ) N ^ { - 1 / 6 } } { \\sqrt 2 } \\mp \\frac { T U N ^ { - 1 / 2 } } { \\sqrt 2 } + o ( N ^ { - 1 / 2 } ) , \\\\ 2 \\tau r ^ 2 \\pm \\sqrt 2 r u & = \\frac N 2 + T N ^ { 2 / 3 } + \\left ( T ^ 2 + \\frac R 2 \\pm U \\right ) N ^ { 1 / 3 } + R T + T ^ 3 + o ( 1 ) . \\end{aligned} \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} \\left ( \\le N \\right ) = \\sum _ { x = 2 } ^ { N } \\left ( x ; \\right ) = \\sum _ { x = 2 } ^ { N } \\mathcal { B } \\left ( \\sum _ { j = 2 } ^ { \\left \\lfloor x / 2 \\right \\rfloor } \\mathcal { B } \\left ( \\mathbf { d } _ j ( 0 , x ) , \\frac { 1 } { 2 } \\right ) , \\frac { 1 } { 2 } \\right ) \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} M _ k ( x ; \\mathbf { a } ) = \\frac { x } { \\log x } \\left \\{ \\frac { 1 } { \\phi ( q ) } Q _ { \\mathbf { k } } \\ ( \\frac { \\log \\log x } { \\phi ( q ) } \\ ) + O _ { A , q , l } \\ ( \\frac { 1 } { \\phi ^ k ( q ) } \\frac { ( \\log \\log x ) ^ k } { k _ 1 ! \\cdots k _ l ! \\log x } \\ ) \\right \\} , \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} I _ C ( x ) = \\begin{cases} x - \\log ( x + 1 ) , & ; \\\\ + \\infty , & . \\end{cases} \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} l ^ 2 _ \\sigma D U ^ 2 = D U ^ 2 + \\sigma ^ { \\frac 1 2 } L ^ 2 . \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} f ( z ) = \\prod _ { \\delta } \\eta ^ { r _ { \\delta } } ( \\delta z ) , \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { \\ell } { p } \\right ) ^ r = 1 \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} \\left \\langle h _ { 1 } , . . . , h _ { \\mathsf { N } } \\right \\vert \\mathcal { T } ( \\lambda ) | \\tau \\rangle = \\tau ( \\lambda ) \\langle h _ { 1 } , . . . , h _ { \\mathsf { N } } | \\tau \\rangle \\forall \\lambda \\in \\mathbb { C } , \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} [ G ^ j ( z ) ] _ + = \\left [ e ^ { - u ^ j / z } ( z + \\iota _ j ^ * t ( - z ) ) \\right ] _ + \\in z \\mathcal H ^ j _ + . \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ T \\zeta _ t \\leq & 4 G \\sqrt { \\log ( 4 \\log T / \\delta ) } \\sqrt { D _ T } + 6 G D \\log ( 4 \\log T / \\delta ) \\\\ \\leq & 1 6 \\beta G ^ 2 \\log ( 4 \\log T / \\delta ) + \\frac { 1 } { 4 \\beta } D _ T + 6 G D \\log ( 4 \\log T / \\delta ) . \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} \\begin{aligned} \\ge \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { A _ { m + 1 } } \\Bigl ( \\sum _ { j = 0 } ^ { m - 1 } \\int _ { A _ j s } ^ { A _ { j + 1 } s } P ( v ) d v \\Bigr ) \\Bigl ( F _ 1 ( A _ { m + 1 } - F _ 1 ( A _ m ) ) \\Bigr ) \\ge \\\\ \\ge \\sum _ { m = 1 } ^ { \\infty } \\frac { s m } { 2 \\pi ( m + 1 ) } 2 \\pi a _ 0 \\Bigl ( F _ 1 ( A _ { m + 1 } - F _ 1 ( A _ m ) ) \\Bigr ) \\ge \\\\ \\ge \\frac { s a _ 0 } { 2 } \\Bigl ( 1 - F _ 1 ( A _ 1 ) \\Bigr ) = \\frac { s a _ 0 } { 2 } \\Bigl ( 1 - F ( 2 \\pi / s ) + F ( - 2 \\pi / s ) \\Bigr ) . \\end{aligned} \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} h _ f ( f _ 0 ) = \\left | \\sum \\limits _ { j = 0 } ^ { \\infty } ( ( \\bold { B } ^ 0 ) ^ { - 1 } \\bold { a } ) _ j e ^ { i j \\theta } \\right | ^ 2 f _ 0 ^ { - 2 } ( \\theta ) \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} Z = P \\biggl \\{ C ( X - \\bar X ) + ( C + \\bar { C } ) \\bar X + D ( v - \\bar v ) + ( D + \\bar { D } ) \\bar v \\biggr \\} \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} \\begin{aligned} u ^ \\varepsilon ( x ) = \\inf \\Big \\{ \\int _ 0 ^ { \\tau ^ \\varepsilon } L ( X ^ \\varepsilon ( t , x , \\alpha ) , \\alpha & ( t ) ) e ^ { - \\lambda t } \\ , d t \\\\ & + g ^ \\varepsilon ( X ^ \\varepsilon ( \\tau ^ \\varepsilon , x , \\alpha ) ) e ^ { - \\lambda \\tau ^ \\varepsilon } \\ | \\ \\alpha \\in L ^ \\infty ( \\mathbb R ; \\mathbb R ^ 2 ) \\Big \\} , \\end{aligned} \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty A _ i r ^ { \\mu _ i ^ + } \\phi _ i ( y ) , \\ ; \\ , \\sum _ { i = 0 } ^ \\infty \\bar { u } _ i ( r ) \\phi _ i ( y ) , \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} \\lim _ { \\ell \\to \\infty } \\ell e ^ { - \\delta k _ { \\ell } } = 0 \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} \\langle B x , x ' \\rangle = \\left ( \\langle B ^ { ( 1 ) } x , x ' \\rangle , \\cdots , \\langle B ^ { ( m ) } x , x ' \\rangle \\right ) \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{gather*} F _ { \\beta } ( t ) = \\exp \\left ( - \\beta \\frac { | t | ^ 3 } { 2 4 } + \\frac { \\sqrt { 2 } } { 3 } ( \\beta / 2 - 1 ) | t | ^ { 3 / 2 } \\right . \\\\ \\left . \\hphantom { F _ { \\beta } ( t ) = } { } + \\frac { 1 } { 8 } ( \\beta / 2 + 2 / \\beta - 3 ) \\log | t | + c _ 0 + O \\left ( \\frac { 1 } { | t | ^ { 3 / 2 } } \\right ) \\right ) , t \\to - \\infty . \\end{gather*}"}
-{"id": "2774.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} u _ t + ( - \\Delta ) ^ s u & = { 0 } & & \\mbox { i n } \\Omega \\times ( 0 , + \\infty ) , \\\\ { u } & \\geq 0 & & \\mbox { i n } \\mathbb { R } ^ { N } , \\\\ u & = 0 & & \\mbox { i n } \\Sigma _ 1 \\times ( 0 , + \\infty ) , \\\\ \\mathcal { N } _ s u & = 0 & & \\mbox { i n } \\Sigma _ 2 \\times ( 0 , + \\infty ) , \\\\ u ( x , 0 ) & = u _ 0 ( x ) & & \\mbox { i n } \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} \\left ( A + U V ^ { \\ast } \\right ) ^ { - 1 } = A ^ { - 1 } - A ^ { - 1 } U \\left ( I + V ^ { \\ast } A ^ { - 1 } U \\right ) ^ { - 1 } V ^ { \\ast } A ^ { - 1 } , \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} ( D \\varphi _ t ( \\omega , x ) v ) ^ { ( i ) } = v ^ { ( i ) } \\ , e ^ { \\int _ 0 ^ t ( 1 - | \\varphi _ s ( \\omega , x ) | ^ 2 ) \\ , d s } \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} ( 2 \\cdot 2 ) \\triangledown ( 2 \\cdot 3 ) \\triangledown ( 3 \\cdot 2 ) \\triangledown ( 3 \\cdot 3 ) = 4 \\triangledown 6 \\triangledown 6 \\triangledown 9 = [ 0 , 2 5 ] . \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} S ( \\alpha ) = \\sum _ m a _ m g ( m ) e ( \\alpha m ) \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} \\mathcal { X } _ { k , \\rho } ^ { \\theta } ( s , x , t ) & \\approx X ^ { s , x } _ t , \\\\ \\mathcal { I } _ { k , \\rho } ^ { \\theta } ( s , x , t ) & \\approx \\big ( 1 , \\tfrac { [ \\sigma ( s , x ) ] ^ { * } } { t - s } \\smallint \\nolimits _ s ^ t \\big [ \\sigma ( r , X _ r ^ { s , x } ) ^ { - 1 } D _ r ^ { s , x } \\big ] ^ { * } \\ , d W _ r \\big ) . \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} R _ \\eta \\ , \\rfloor \\ , \\eta = \\eta ( R _ \\eta ) = 1 , R _ \\eta \\ , \\rfloor \\ , d \\eta = 0 . \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} \\left ( L - \\frac { \\partial } { \\partial t } \\right ) f = - | \\partial f | _ { \\tilde { g } } ^ { 2 } . \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} L _ { b c } ( I + \\Gamma _ { l , b c } Q ) = ( I + \\Gamma _ { l , b c } Q ) ( \\widetilde { L } _ { b c } ^ 0 - B ) , \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} \\Theta _ { \\mathfrak { m } \\mathfrak { p } } ( L / F , \\pi _ B ) ^ { \\epsilon } = ( \\lambda _ { \\mathfrak { p } } - ( \\ast ) ) \\Theta _ { \\mathfrak { m } } ( L / F , \\pi _ B ) ^ { \\epsilon } \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ n i \\rho ^ i = \\frac { n } { 3 } \\sum _ { i = 0 } ^ n \\rho ^ i . \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} \\Lambda ^ * ( n ) = \\sum _ { \\substack { a b = n \\\\ b < y } } \\lambda ' ( a ) \\nu ( b ) \\ , \\quad \\Lambda _ * ( n ) = \\sum _ { \\substack { a b = n \\\\ b \\ge y } } \\lambda ' ( a ) \\nu ( b ) \\ . \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} \\big ( \\sum _ { j = 1 } ^ d | x _ j + y _ j | ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } & \\le \\big ( \\sum _ { j = 1 } ^ d ( | x _ j | ^ { R _ j / p } + | y _ j | ^ { R _ j / p } ) ^ { 2 p } \\big ) ^ { 1 / ( 2 p ) } \\\\ & \\le \\big ( \\sum _ { j = 1 } ^ d | x _ j | ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } + \\big ( \\sum _ { j = 1 } ^ d | y _ j | ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } . \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} \\P _ n ( G , \\sigma ) = \\prod _ { v \\in V ( G ) } \\pi _ { \\sigma _ v } \\prod _ { ( u , v ) \\in E ( G ) } \\frac { M _ { \\sigma _ u , \\sigma _ v } } { n } \\prod _ { ( u , v ) \\not \\in E ( G ) } \\left ( 1 - \\frac { M _ { \\sigma _ u , \\sigma _ v } } { n } \\right ) . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} \\tau ^ S = \\bigcap _ { { \\rm p a r a m e t e r \\ i d e a l s \\ } I } ( I : I ^ * ) \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 = Q ^ { \\alpha } ( x , n , [ u ] ) F _ { \\alpha } , \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} X : = c y - c - 2 x , \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { + \\infty } F ( x ) \\cosh \\Big ( \\frac { x } { 2 } \\Big ) \\d x & = 2 \\Big ( \\int _ { 0 } ^ { + \\infty } \\Phi ( x ) \\cosh \\Big ( \\frac { x } { 2 } \\Big ) \\d x \\Big ) ^ 2 , \\\\ \\int _ { 0 } ^ { + \\infty } F ( x ) \\cos ( x t ) \\d x & = 2 \\Big ( \\int _ { 0 } ^ { + \\infty } \\Phi ( x ) \\cos ( x t ) \\d x \\Big ) ^ 2 , \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} N _ \\epsilon ' = d N _ \\epsilon + ( \\frac 1 2 - \\epsilon ) ( d - 1 ) , \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { n } } x ^ { \\beta } I _ { \\alpha } a ( x ) d x = 0 , \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} W = \\bigoplus _ { j = 1 } ^ { n } H ^ { 0 } ( X , K _ X ^ { 2 j } ( 2 j - n ( \\Lambda , 2 j ) ) ) \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} \\frac { n _ k } { n } \\prod _ { i = 1 } ^ { k - 1 } \\frac { n _ i d ( \\nu _ i - \\nu _ k ) } { 2 \\pi i } = \\prod _ { i = 1 } ^ { k - 1 } \\frac { n _ i d \\nu _ i } { 2 \\pi i } . \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} F _ { 2 } ^ { \\mu } = \\overline { \\Pi } _ { J } ^ { \\mu } \\phi _ { J } + \\overline { \\phi } _ { J } \\Pi _ { J } ^ { \\mu } + P _ { J K } ^ { \\alpha \\mu } \\ , a _ { K J \\alpha } + \\frac { \\epsilon } { q } j ^ { \\mu } , \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} \\frac { \\partial \\gamma ^ j } { \\partial t } + \\sum _ { i = 1 } ^ n \\frac { \\partial H } { \\partial p _ i } \\frac { \\partial \\gamma ^ j } { \\partial q ^ i } + \\frac { \\partial H } { \\partial q ^ j } = 0 . \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} q < \\frac { 2 } { \\pi } \\min \\limits _ { p = \\overline { 0 , n - 1 } } \\left | \\arg ( \\alpha + \\beta \\omega ^ p + \\gamma \\overline { \\omega } ^ p ) \\right | , \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} \\begin{aligned} m _ 1 - M _ x [ c _ 1 ] & = ( M _ { \\varphi , x } [ b _ 1 ] - M _ x [ b _ 1 ] ) + ( M _ { \\varphi , x } [ ( \\rho ^ { - 1 } - 1 ) b _ 1 ] - M _ x [ ( \\rho ^ { - 1 } - 1 ) b _ 1 ] ) \\\\ & + ( M _ { \\varphi , x } [ ( B ^ { - 1 } - \\mathrm { I } ) b _ 1 ] - M _ x [ ( B ^ { - 1 } - \\mathrm { I } ) b _ 1 ] ) \\\\ & + ( M _ { \\varphi , x } [ ( \\rho ^ { - 1 } - 1 ) ( B ^ { - 1 } - \\mathrm { I } ) b _ 1 ] - M _ x [ ( \\rho ^ { - 1 } - 1 ) ( B ^ { - 1 } - \\mathrm { I } ) b _ 1 ] ) . \\end{aligned} \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{align*} d i m ( Z _ { S P _ { 2 n } } ( e ) ) = \\frac { 1 } { 2 } ( \\Sigma _ i \\tilde { m } _ i ^ 2 + \\# \\{ i \\mid m _ i \\ , \\ \\} ) \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} \\hat { \\mu } ^ t : = ( d - \\mu _ c , d - \\mu _ { c - 1 } , \\dots , d - \\mu _ 1 ) , \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } | < f _ { n } , x > | & = \\sum _ { n = 1 } ^ { \\infty } | \\sum _ { k = p _ { n } } ^ { q _ { n } } x ( k ) \\alpha _ { n } ( k ) | \\\\ & \\leq \\sum _ { n = 1 } ^ { \\infty } \\sum _ { k = p _ { n } } ^ { q _ { n } } | x ( k ) | | \\alpha _ { n } ( k ) | \\\\ & \\leq \\sum _ { n = 1 } ^ { \\infty } \\sum _ { k = p _ { n } } ^ { q _ { n } } | x ( k ) | \\leq \\| x \\| . \\\\ \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} - | \\inf f - \\inf g | = \\inf f - \\inf g \\geq f ( x _ \\epsilon ) - \\epsilon - g ( x _ \\epsilon ) \\geq - \\epsilon - \\sup | f - g | . \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} r = \\sum _ { i = 1 } ^ k b _ i = \\sum _ { i = 1 } ^ k \\sum _ { j = 1 } ^ { n _ i } r _ { i j } . \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} \\underbrace { e ^ { - s _ 1 A } } _ { \\unrhd 0 } \\underbrace { C } _ { \\unrhd 0 } \\underbrace { e ^ { - s _ 2 A } } _ { \\unrhd 0 } \\cdots \\underbrace { e ^ { - s _ n A } } _ { \\unrhd 0 } \\underbrace { C } _ { \\unrhd 0 } \\underbrace { e ^ { - ( \\beta - \\sum _ { j = 1 } ^ n s _ j ) A } } _ { \\unrhd 0 } \\unrhd 0 \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} h ( x ) = \\left \\{ \\begin{array} { l l } r & \\mbox { i f } x \\neq t _ i \\mbox { f o r } i = 1 , \\dots , k \\\\ h _ i & \\mbox { i f } x = t _ i \\end{array} \\right . \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( - \\Delta ) ^ s u & = f & & \\mbox { i n } \\Omega , \\\\ { u } & \\geq 0 & & \\mbox { i n } \\mathbb { R } ^ { N } , \\\\ u & = 0 & & \\mbox { i n } \\Sigma _ 1 , \\\\ \\mathcal { N } _ s u & = 0 & & \\mbox { i n } \\Sigma _ 2 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} ( X , Y ) \\mapsto ( \\hat { u } ^ 2 X + \\hat { r } \\pmod { t } , \\ ; \\hat { u } ^ 3 Y + \\hat { u } ^ 3 \\hat { s } X + \\hat { T } \\pmod { t } ) = ( X + 1 , Y + X + \\alpha _ 1 + \\omega _ 3 ) , \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} \\alpha + n S \\ : = \\ \\{ \\alpha + n \\gamma : \\gamma \\in S \\} . \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} \\mathcal { E } = ( 1 + O ( A ) ) E _ { 0 } ( W , Q ) . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} [ D ( 2 , 2 n ) : D ( 3 , s ) ] _ q & = q ^ { 2 ( 2 n - s ) } [ D ( 2 , 2 n - 3 ) : D ( 3 , s - 3 ) ] _ q + q ^ { 2 ( 2 n - 1 ) } [ D ( 2 , 2 n - 2 ) : D ( 3 , s ) ] _ q & \\\\ & + q ^ { 6 n - 2 s - 3 } [ D ( 2 , 2 n - 4 ) : D ( 3 , s - 3 ) ] _ q , \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { 2 } { p } \\right ) ^ r = 1 . \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} \\Lambda ( z ^ { ( 0 ) } + \\Delta ) & = \\frac { e ^ { \\frac { \\sigma _ { \\mathrm { a } } ^ 2 } { \\zeta } } \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { ( z ^ { ( 0 ) } + \\Delta ) } { v } } e ^ { - \\frac { v } { \\zeta } } d v } { \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { ( z ^ { ( 0 ) } + \\Delta ) } { v } } e ^ { - \\frac { v } { \\zeta } } d v } . \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y ^ j _ k } = \\frac { \\partial } { \\partial y ^ j _ { k + 1 } } + \\sum _ l \\bigg ( \\frac { \\partial a ^ k _ l } { \\partial y ^ j _ k } \\bigg ) \\frac { \\partial } { \\partial y ^ l _ { k + 1 } } + \\sum _ \\tau \\bigg ( \\frac { \\partial \\beta ^ k _ \\tau } { \\partial y ^ j _ k } \\bigg ) \\frac { \\partial } { \\partial \\eta ^ \\tau _ { k + 1 } } \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} \\tilde P _ { S _ i } = \\tilde P _ S , i \\in [ 1 : n ] , \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} \\beta k - i = 1 , \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} 4 9 9 Y ^ 2 - 4 9 5 X ^ 2 = - 9 8 8 0 0 4 . \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} & \\ell _ i = \\frac { 1 } { 2 } , r _ i = L , \\forall i \\in [ 1 : N - 1 ] , \\\\ & \\ell _ N = L , r _ N = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} L _ n ( G ) = \\prod _ { i = 1 } ^ n f ( x _ i ; G ) . \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} f \\left ( w \\right ) = \\begin{cases} 1 , & 0 \\le w \\le 1 \\\\ 0 , & w > 1 \\end{cases} \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} U _ 1 = 1 , U _ { 2 N } = U _ N , \\ U _ { 2 N + 1 } = \\frac { 1 } { 2 } U _ { N + 1 } + 2 U _ N + \\frac { 1 } { 2 } , \\ \\ \\ N \\ge 1 . \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} \\lambda = \\frac { \\abs { x - \\mu } } { \\sigma } \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} u & = u ( x , \\mu ) = \\left ( \\frac { \\zeta } { \\zeta ' } \\right ) ^ { \\frac 1 2 } K _ { \\frac 1 3 } ( - i \\zeta ) , \\\\ \\zeta & = \\zeta ( x , \\mu ) = \\begin{cases} \\displaystyle \\int _ x ^ { x _ \\mu } ( \\mu - Q ( s ) ) ^ \\frac 1 2 \\ , \\dd s , & 0 < x < x _ \\mu , \\\\ [ 4 m m ] \\displaystyle i \\int _ { x _ \\mu } ^ x ( Q ( s ) - \\mu ) ^ \\frac 1 2 \\ , \\dd s , & x > x _ \\mu ; \\end{cases} \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} N _ k & = \\binom { \\Delta } { \\frac { \\Delta + k - x } { 2 } } - \\binom { \\Delta } { \\frac { \\Delta - x - k } { 2 } } \\\\ & = \\binom { \\Delta } { \\frac { \\Delta + k - x } { 2 } } \\left ( 1 - \\frac { \\Big ( \\frac { \\Delta + k - x } { 2 } \\Big ) ! \\Big ( \\frac { \\Delta - k + x } { 2 } \\Big ) ! } { \\Big ( \\frac { \\Delta + k + x } { 2 } \\Big ) ! \\Big ( \\frac { \\Delta - k - x } { 2 } \\Big ) ! } \\right ) . \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} \\aligned \\langle & G ' _ j , ( 0 , . . . , u _ k , . . . , 0 ) \\rangle = - \\beta _ { j k } \\alpha _ { j k } \\alpha _ { k j } \\int _ { \\R ^ N } | u _ j | ^ { \\alpha _ { j k } } | u _ k | ^ { \\alpha _ { k j } } . \\endaligned \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} a & = 2 \\alpha ( - Y , s ) = \\rho , \\\\ b & = 2 \\beta ( - Y , s ) + 1 = \\rho + 1 , \\\\ c & = 2 \\gamma ( - Y , s ) + 2 = \\begin{cases} 2 \\delta ( - Y , s ) + 2 & \\quad 2 \\delta ( - Y , s ) = \\rho \\mod 4 \\\\ 2 \\delta ( - Y , s ) & \\quad 2 \\delta ( - Y , s ) = \\rho + 2 \\mod 4 . \\ \\end{cases} \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} \\| ( x _ 1 , \\dots , x _ n ) \\| _ p : = \\begin{cases} \\big ( \\sum _ { i = 1 } ^ n | x _ i | ^ p \\big ) ^ { 1 / p } \\ , , & 1 \\leq p < \\infty , \\\\ \\max _ { 1 \\leq i \\leq n } | x _ i | \\ , , & p = \\infty . \\end{cases} \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} ( X , f ) , ( Y , g ) \\mapsto ( X , f ) \\boxplus ( Y , g ) : = ( X \\times _ { S } Y , f \\boxplus g ) \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} \\partial _ t w + A _ G ^ * w = 0 , t > 0 , w ( 0 ) = w ^ 0 . \\end{align*}"}
-{"id": "9144.png", "formula": "\\begin{align*} \\| f _ { s _ 1 } - f _ { s _ 2 } \\| _ { K } ^ 2 = \\| f _ { s _ 1 } \\| _ { K } ^ 2 + \\| f _ { s _ 2 } \\| _ { K } ^ 2 - 2 \\langle f _ { s _ 1 } , f _ { s _ 2 } \\rangle _ { K } = \\| f _ { s _ 1 } \\| _ { K } ^ 2 - \\| f _ { s _ 2 } \\| _ { K } ^ 2 . \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} \\psi _ n ^ { ( \\epsilon ) } ( x ) = e ^ { W ( x ) } \\phi _ n ^ { ( \\epsilon ) } ( x ) , W ( x ) = - \\tfrac { 1 } { 3 } | x | ^ 3 + a x ^ 2 - b | x | , \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} f & = \\frac { k v - f _ 1 ( d _ 1 - d _ 2 ) } { d _ 2 } \\\\ v ( 2 d _ 2 - k d _ 2 + 2 k ) & = 2 f _ 1 d _ 1 + ( 4 - 2 f _ 1 ) d _ 2 \\\\ \\frac { e } { k d _ 2 } \\biggl ( 4 - ( k - 2 ) ( d _ 2 - 2 ) \\biggr ) & = \\Phi ( f _ 1 , d _ 1 , d _ 2 ) , \\end{align*}"}
-{"id": "4914.png", "formula": "\\begin{align*} \\begin{array} { l l } k = \\left ( r - a \\right ) / 2 , & g = \\left ( 2 2 - r - a \\right ) / 2 , \\\\ N = \\left ( 2 + r - a \\right ) / 2 , & N ' = \\left ( 2 2 - r - a \\right ) / 2 , \\\\ r = 1 1 + k - g = 1 0 + N - N ' , & a = 1 1 - k - g = 1 2 - N - N ' . \\end{array} \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\eta ^ \\sigma } = & \\sum _ j \\alpha _ j ( y , 0 , \\widehat { \\eta } ) \\frac { \\partial } { \\partial z ^ j } + \\sum _ \\rho a _ \\rho ( y , 0 , \\widehat { \\eta } ) \\frac { \\partial } { \\partial \\zeta ^ \\rho } . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} \\tau \\sigma \\tau ^ { - 1 } ( \\zeta ) = \\zeta = \\sigma ^ \\ell ( \\zeta ) \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} 9 \\xi ^ 3 + ( 3 t - 3 9 ) \\xi ^ 2 + ( 3 t ^ 2 - 1 8 t + 5 1 ) \\xi + t ^ 3 - 3 t ^ 2 + 1 5 t - 2 1 \\ = \\ 0 \\ , . \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} ( x ^ \\ell + m ) ( x ^ { \\ell + 3 } - m ) ( x ^ { \\ell + 3 } - m x ^ 3 + m ) = x ^ { 3 \\ell + 6 } \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} \\sum \\limits _ r \\eta _ r ( x ) \\eta _ r ( R ^ a ) \\eta _ r ( z ^ { - 1 } ) = n \\sum \\limits _ r \\big { ( } \\eta ( R ^ { a + a _ 1 + a _ 2 } ) + \\eta _ r ( R ^ { a + a _ 1 - a _ 2 } ) + \\eta ( R ^ { a - a _ 1 + a _ 2 } ) + \\eta ( R ^ { a - a _ 1 - a _ 2 } ) \\big { ) } , \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} \\hat { p } ^ { ( 1 ) } ( a ^ k | S ^ { ( i ) , t } ) = { | \\{ j : S ^ { ( i ) } _ j = a ^ k , 1 \\leq j \\leq t \\} | \\over t } . \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} & \\Box ^ { ( 0 ) } _ { t , b , m } N _ { t , m } + S _ { t , m } = I \\ \\ \\mbox { o n $ L ^ 2 _ m ( X ) $ } , \\\\ & N _ { t , m } \\Box ^ { ( 0 ) } _ { t , b , m } + S _ { t , m } = I \\ \\ \\mbox { o n $ { \\rm D o m \\ , } ( \\Box ^ { ( 0 ) } _ { t , b , m } ) $ } . \\\\ \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} d \\eta = 0 , d \\Omega + \\eta \\wedge \\Omega = 0 \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} c ( S , M ) : = \\sum _ { i = 1 } ^ b \\left \\lfloor { \\frac { h _ i } { 2 } } \\right \\rfloor . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} e _ { ( m + 1 ) p _ n + \\frac { m } { 2 } } ( x ) = e _ { ( m + 1 ) ( p _ n - 1 ) + \\frac { m } { 2 } + 1 } ( x ) . \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} \\abs { F _ { n + i + j } ^ { \\langle r \\rangle } } _ { 0 \\leq i , j \\leq r } = ( - 1 ) ^ { n { r + 1 \\choose 2 } + \\binom { r + 2 } { 3 } } ( F _ 1 F _ 2 \\cdots F _ r ) ^ { r + 1 } . \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} \\mu _ i ^ \\pm = - \\frac { m - 2 } { 2 } \\pm \\sqrt { \\frac { ( m - 2 ) ^ 2 } { 4 } + \\lambda _ i } \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} a _ 0 ( y , s ) = \\sum _ { \\mu \\in \\{ s , 1 - s \\} ^ { g } } b _ { \\mu } ( s ) y ^ { \\mu } , \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} q ^ { n , [ a , b ] } ( t ) = \\begin{cases} \\int _ a ^ b \\left [ \\prod _ { \\substack { i \\in \\{ 1 , \\ldots , n \\} , \\\\ c _ i ^ n \\neq \\frac { 2 t - ( a + b ) } { b - a } } } \\tfrac { 2 x - ( b - a ) c _ i ^ n - ( a + b ) } { 2 t - ( b - a ) c _ i ^ n - ( a + b ) } \\right ] \\ , d x & \\colon ( a < b ) \\big ( \\frac { 2 t - ( a + b ) } { b - a } \\in \\{ c _ 1 ^ n , \\ldots , c _ n ^ n \\} \\big ) \\\\ 0 & \\colon \\end{cases} \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} g _ { a , b } ^ * \\circ g _ { a , b } = a B _ x ( \\rho ) B _ s B _ y + b B _ x B _ s ( \\rho ) B _ y - a ( x \\rho ) B _ x B _ x B _ y - b B _ x B _ y ( y ^ { - 1 } \\rho ) . \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} \\int _ { \\Omega } f _ { n } \\cdot x _ { n } d \\mu > \\epsilon , n = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} b ( m , a _ 1 , \\ldots , a _ r , n ) = & ( m a _ 1 , a _ 2 , \\ldots , a _ r , n ) \\\\ & + \\sum _ { i = 1 } ^ { r - 1 } ( - 1 ) ^ i ( m , a _ 1 , \\ldots , a _ { i - 1 } , a _ i a _ { i + 1 } , a _ { i + 2 } , \\ldots , a _ r , n ) \\\\ & + ( - 1 ) ^ r ( m , a _ 1 , \\ldots , a _ { r - 1 } , a _ r n ) . \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} z w z ^ { - 1 } = w ( x ^ { - 1 } , y ^ { - 1 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} \\P ( [ X _ g ] _ b = z | X _ 0 = x ) & = \\sum _ { d ^ g \\in \\{ 0 , 1 \\} ^ g } \\P ( [ X _ g ] _ b = z , J ^ g = d ^ g | X _ 0 = x ) \\\\ & \\stackrel { ( a ) } { = } \\sum _ { d ^ g \\in \\{ 0 , 1 \\} ^ g } \\P ( [ X _ g ] _ b = z | J ^ g = d ^ g , X _ 0 = x ) \\P ( J ^ g = d ^ g ) . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} \\mathfrak { b } _ K ( x , x ) = x ^ 2 \\quad { \\rm f o r } x \\in \\Phi ( A , \\sigma ) . \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} \\limsup _ { | i | \\le \\gamma t , t \\to \\infty } | u _ i ( t ; 0 , u ^ 0 ) - u ^ + ( t ) | = 0 . \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} \\sum _ { v = 0 } ^ { \\ell ^ m - 1 } f ( z ) \\mid _ { - 1 } \\alpha _ 0 \\sigma _ { w _ v , \\ell ^ m } = \\ell ^ { \\frac { 3 } { 2 } } \\sum _ { \\substack { n = n _ 0 \\\\ n \\equiv 0 \\pmod { \\ell ^ m } } } ^ { \\infty } a _ 0 ( n ) q _ { 2 4 \\ell ^ m } ^ { n } . \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} \\Pr ( N = \\alpha n ) \\exp ( n ( 1 + \\epsilon ) ( \\alpha - p ) ^ T A ( \\alpha - p ) ) \\le C \\exp ( - n \\epsilon | \\alpha - p | ^ 2 ) . \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} \\mu ( E ) = \\frac { d ( E ) } { r ( E ) } , \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} & \\zeta = 1 B \\left ( x _ 0 , \\frac 1 2 \\rho \\right ) \\\\ & | \\nabla \\zeta | \\le \\frac { \\widetilde { C } } { \\rho } . \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} u ( t ) \\in [ 0 , 1 ] , \\ ; x ( 0 ) = 1 , \\ ; x ( 1 ) = 0 . \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} g ( h , h ' ) : = & \\ , \\left ( \\partial _ x f ( h ) - \\frac { y - 2 y ' } { 2 } \\partial _ z f ( h ) \\right ) ^ 2 + \\left ( \\partial _ y f ( h ) + \\frac { x - 2 x ' } { 2 } \\partial _ z f ( h ) \\right ) ^ 2 + \\beta ^ 2 \\left ( \\partial _ z f ( h ) \\right ) ^ 2 \\\\ = & \\ , ( ( X + y ' Z ) f ( h ) ) ^ { 2 } + ( ( Y - x ' Z ) f ( h ) ) ^ { 2 } + \\beta ^ { 2 } ( Z f ( h ) ) ^ { 2 } . \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} \\psi _ n ( \\xi ) = \\chi ( 2 ^ { - n } \\| \\xi \\| ) - \\chi ( 2 ^ { - n + 1 } \\| \\xi \\| ) \\ , , n \\ge 1 \\ , . \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} g ( \\theta + \\alpha ) - g ( \\theta ) = \\ln { s ( \\theta ) } - \\ln { \\tilde { s } ( \\theta ) } . \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} M _ c ( \\theta ) = \\left ( \\begin{matrix} 1 \\ \\ & 0 \\\\ 0 \\ \\ & \\sqrt { \\frac { c ( \\theta - \\alpha ) } { \\tilde { c } ( \\theta - \\alpha ) } } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} A ^ { ( i ) } = f _ { i + 1 } ( A / J _ i ) f _ { i + 1 } = e _ i ^ \\ast ( A / J _ i ) e _ i ^ \\ast \\cong e _ i ^ \\ast H e _ i ^ \\ast = H e _ i ^ \\ast = e _ i ^ \\ast H . \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} c V ^ { 2 } - \\left ( c + 2 \\right ) U ^ { 2 } = - 2 \\mu c Z ^ { 2 } - \\left ( c - 2 \\right ) U ^ { 2 } = 2 \\mu \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} \\ell = - a c , m = - a d , n = - b d . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} & f ( \\gamma , 1 ) = \\\\ & \\frac { 1 } { 2 } \\log \\left ( 1 + \\frac { \\gamma k _ n P ' } { 2 } \\right ) - \\frac { ( 1 - \\epsilon ) \\gamma } { 2 } \\log ( 1 + k _ n P ' ) - \\frac { k _ n } { n } H _ 2 ( \\gamma ) . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} \\epsilon - \\delta & < \\sum _ { i = 1 } ^ { \\infty } | x _ { N _ { k } } ( i ) | \\chi _ { F _ { k } } ( i ) \\\\ & = \\sum _ { i = 1 } ^ { n } | x _ { N _ { k } } ( i ) | \\chi _ { F _ { k } } ( i ) + \\sum _ { i = n + 1 } ^ { \\infty } | x _ { N _ { k } } ( i ) | \\chi _ { F _ { k } } ( i ) \\\\ & \\leq \\sum _ { i = 1 } ^ { n } \\chi _ { F _ { k } } ( i ) + \\sup _ { x \\in A } \\sum _ { i = n + 1 } ^ { \\infty } | x ( i ) | . \\\\ \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} \\operatorname { T r } \\left \\{ \\mathbf { n M S E } ^ { ( g ) } \\right \\} \\triangleq \\operatorname { T r } \\left \\{ \\left ( \\sum _ { l = 0 } ^ { L _ g - 1 } \\mathbf { R } ^ { ( g ) } _ { c o d e } ( l ) \\otimes \\mathbf { S N R } ^ { ( g ) } _ { m i m o } ( l ) + \\mathbf { I } _ { T D } \\right ) ^ { - 1 } \\right \\} + \\left ( K _ g \\left ( \\sum _ { l = 0 } ^ { L _ g - 1 } r _ { g , l } \\right ) - T D \\right ) \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} \\frac { r _ T ^ { 2 p } V ^ { - \\frac { 1 } { 2 } } } { \\sqrt { 2 + r ^ { - 1 } } } \\hat e _ 1 | h | ^ 2 = & \\mathrm { d i v } \\left ( \\frac { r _ T ^ { 2 p } | h | ^ 2 \\hat { e } _ 1 } { \\sqrt { 2 + r ^ { - 1 } } } \\right ) - \\frac { 2 | h | ^ 2 } { \\sqrt { 2 } } ( p \\chi _ T + 1 ) \\frac { r _ T ^ { 2 p } } { r V } + O \\left ( | h | ^ 2 r _ T ^ { 2 p } r ^ { - 2 } \\right ) . \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\varphi _ { - p } ( s ) = \\varphi _ { p } ( s ) . \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} R _ { n + 1 } ( x , q ) = 2 n x R _ n ( x , q ) + 2 x ( 1 - x ) \\frac { \\partial } { \\partial x } R _ n ( x , q ) + 2 n x q R _ { n - 1 } ( x , q ) , \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} S ( \\nabla ) \\ , = \\ , \\omega \\circ \\nabla \\ , : \\ , \\mathrm { T } _ X ( - D _ 0 ) \\ , \\longrightarrow \\ , \\mathrm { A t } _ { D _ 0 } ( D - D _ 0 ) \\ , \\longrightarrow \\ , E _ P ( { \\mathfrak g } / { \\mathfrak p } ) ( D - D _ 0 ) \\ , . \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} A _ * = R _ { \\rho ( \\alpha , A ) } . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\int _ \\R H _ n ( x ) H _ m ( x ) e ^ { - x ^ 2 } \\d x = \\sqrt \\pi 2 ^ n n ! \\delta _ { n , m } . \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} E _ \\mu ( u ) = & \\ , - \\frac 1 \\pi \\int \\mu ( \\xi ) \\ln | T ( - \\xi / 2 ) | d \\xi \\\\ = & \\ , \\frac 1 \\pi \\real \\int _ 1 ^ \\infty \\overline { [ \\mu ] ( \\tau ) } \\Big ( \\ln T ( i \\tau / 2 ) + \\sum _ { j = 0 } ^ { N - 1 } i ^ j H _ { j } \\tau ^ { - j - 1 } \\Big ) d t + \\sum _ { j = 0 } ^ { N - 1 } \\frac 1 { j ! } \\overline { \\mu ^ { ( j ) } ( 0 ) } H _ { j } \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} N _ { 0 0 } ( T ) = \\{ 1 + O ( \\delta ) \\} N ( T ) \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} \\wedge _ { i } \\phi _ B ( b _ 1 ^ i ) = \\pm \\nu ( \\mathcal { E } _ 1 ) ^ { - 1 } ( \\wedge _ { i } \\phi _ B \\theta _ 1 ( a _ 1 ^ i ) ) \\wedge ( \\wedge _ { i } \\phi _ B \\tau _ 1 ^ { - 1 } ( c _ 1 ^ i ) ) = \\pm \\nu ( \\mathcal { E } _ 1 ) ^ { - 1 } M \\wedge N . \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} e _ k = e _ k ( x _ 1 , \\dots , x _ n ) : = \\sum _ { 1 \\leq i _ 1 < \\dots < i _ k \\leq n } x _ { i _ 1 } \\cdots x _ { i _ k } . \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{align*} | \\mathcal { U } | ( k - 1 ) \\geq \\sum _ { M \\in \\mathcal { U } } f ( M ) = \\sum _ { i \\in [ k ] } | D _ i | \\geq k \\left ( \\frac { k } { k - 1 } \\right ) ^ t \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} \\Bigg \\| a - a \\cdot \\sum _ { l = 0 } ^ d x ^ { ( l ) } x ^ { ( l ) * } \\Bigg \\| \\leq \\delta \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} w ^ n _ { i , k } = \\left [ t ^ n _ { - i , k + Q ^ n _ i ( 0 ) - R ^ n _ i ( t ^ n _ { i , k } - ) - Q ^ n _ { - i } ( 0 ) + R ^ n _ { - i } ( t ^ n _ { i , k } - ) } - t ^ n _ { i , k } \\right ] ^ + . \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} b \\left ( \\frac { \\log { y } } { \\log { N } } \\right ) = \\frac { - 1 } { 2 \\pi i } \\int _ { ( - 1 ) } f ( z ) y ^ { - z } z ^ { - 2 } d z , \\ y > 0 . \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} \\omega _ { \\beta , \\mu } = \\frac { 1 } { 2 \\pi } \\int _ { 0 } ^ { 2 \\pi } d \\phi \\ \\omega _ { \\beta , \\mu , \\phi } \\ . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} | < z ^ { * } _ { j } , x _ { n _ { j } } > | & = | < e ^ { * } _ { p _ { j } } , x _ { n _ { j } } > - < e ^ { * } _ { q _ { j } } , x _ { n _ { j } } > | \\\\ & \\geq | < e ^ { * } _ { p _ { j } } , x _ { n _ { j } } > | - | < e ^ { * } _ { q _ { j } } , x _ { n _ { j } } > | \\\\ & > \\delta - \\frac { \\delta } { 2 } = \\frac { \\delta } { 2 } . \\\\ \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} ( \\bold { D } _ F ) _ { \\alpha \\beta } Q ^ { \\beta } = 0 , \\{ F _ { \\alpha } ( x , [ u ] ) = 0 \\} , \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} A ( s ) = \\sum _ n a \\left ( \\frac { \\log { n } } { \\log { N } } \\right ) \\lambda ( n ) n ^ { - s } \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{align*} \\frac { \\partial x ( u , t ) } { \\partial t } = \\nu \\frac { \\partial ^ 2 } { \\partial u ^ 2 } x ( u , t ) + F ( x ( u , t ) ) + W . \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} \\{ b _ 1 \\leq b _ 2 \\leq \\cdots \\leq b _ t \\} & = \\{ a _ 1 , a _ 2 , \\dotsc , a _ t \\} , \\\\ \\{ b ' _ 1 \\leq b ' _ 2 \\leq \\cdots \\leq b ' _ t \\} & = \\{ a ' _ 1 , a ' _ 2 , \\dotsc , a ' _ t \\} . \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} \\mathbf m = \\left ( \\mathbf X \\mathbf N \\mathbf M _ p ^ { - 1 } \\right ) \\cdot \\left ( \\mathbf M _ p \\mathbf N ^ * \\mathbf C ^ + \\right ) . \\end{align*}"}
-{"id": "6387.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": "8920.png", "formula": "\\begin{align*} \\int E _ f = \\int _ { \\Gamma \\backslash G } d g \\left ( \\sum _ { \\Gamma _ B \\backslash \\Gamma } f ( H ( \\gamma g ) \\right ) = \\int _ { \\Gamma _ B \\backslash G } f ( H ( g ) ) d g \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} S ( n ) \\le \\prod _ { i = 1 } ^ n \\biggl ( \\prod _ { j = 1 } ^ i \\mu ( i , j ) \\biggr ) \\biggl ( \\prod _ { j = i + 1 } ^ n \\nu ( j ) \\biggr ) = : S _ U ( n ) , \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{align*} b ( x ) = \\sum _ { j \\in \\mathbb { N } } \\left ( a _ j ( x ) \\int _ 0 ^ \\cdot a _ j ( x ) ( z ) \\dd z \\right ) - a ( x ) a ^ * ( x ) \\zeta ( x ) \\end{align*}"}
-{"id": "5483.png", "formula": "\\begin{align*} s ^ i ( \\hat { x } _ t , \\theta _ t ) = a ^ i ( \\theta _ t ) \\ , \\hat { x } _ t + b ^ i ( \\theta _ t ) , \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} \\epsilon ( u ^ { t + 1 } ) = \\epsilon ( ( x ' ) ^ t ) + ( 2 ^ { t + 1 } - 1 ) \\epsilon ( u ) - ( 2 ^ { t } - 1 ) \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} 2 c _ 1 = - \\langle \\partial _ x ^ { - 1 } u _ 0 , y - c _ 1 u _ 1 \\rangle _ { L ^ 2 } = - \\sum _ { n = 2 } ^ { \\infty } c _ n \\langle \\partial _ x ^ { - 1 } u _ 0 , u _ n \\rangle _ { L ^ 2 } . \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} ~ \\mu [ a , b ) - ( b - a ) = \\frac { 1 } { \\pi } \\sum _ { n = 1 } ^ \\infty \\Im ( \\frac { \\hat { \\mu } ( n ) } { n } [ e ( n b ) - e ( n a ) ] ) + \\lim _ { n \\to \\infty } \\frac { 1 } { 2 n } \\sum _ { j = 1 } ^ n \\hat { \\mu } ( j ) [ e ( j a ) - e ( j b ) ] . \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} | ( 1 + s ) x - ( 1 - s ) y | ^ 2 & = | ( 1 + s ) ( x - y ) + 2 s y | ^ 2 = ( 1 + s ) ^ 2 \\Big | x - y + 2 \\frac { s } { 1 + s } y \\Big | ^ 2 \\\\ & \\ge | x - y | ^ 2 - 4 s | \\langle x - y , y \\rangle | \\ge | x - y | ^ 2 - 4 s | y | m ( x ) \\\\ & \\ge | x - y | ^ 2 - 8 s , \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} I _ { 3 , 2 } ( { t , x } ) = \\frac { n p } { t ^ a } \\log \\int _ { | y | \\le K \\sqrt { t t _ n } } u _ 0 ( x + y ) p _ t ( y ) d y \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\mapsto ( - 1 ) ^ k \\begin{bmatrix} d & c \\\\ b & a \\end{bmatrix} . \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} u _ { j } ( x , t ) = e ^ { - i \\big ( ( \\rho _ { j } \\cdot \\rho _ { j } ) t + x \\cdot \\rho _ { j } \\big ) } \\Big ( e ^ { i \\phi _ { j } ( x ) } + w _ { j } ( x , t ) \\Big ) , j = 1 , 2 , \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} Z ( \\lambda , \\nu ) & = \\frac 1 { ( 2 \\pi ) ^ { \\nu - 1 } } \\int _ { - \\pi } ^ \\pi \\cdots \\int _ { - \\pi } ^ \\pi \\exp \\Bigg \\{ \\sum _ { j = 1 } ^ { \\nu - 1 } \\exp \\big \\{ { \\rm i } x _ j \\big \\} \\\\ & + \\lambda \\exp \\Big \\{ - { \\rm i } \\sum _ { j = 1 } ^ { \\nu - 1 } x _ j \\Big \\} \\Bigg \\} \\ , { \\rm d } x _ 1 \\cdots { \\rm d } x _ { \\nu - 1 } \\ , . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} R _ M ( y _ 1 ) & = ( 1 + y _ 3 + y _ 3 y _ 1 ) ^ { - 1 } y _ 2 ^ { - 1 } ( 1 + y _ 1 + y _ 1 y _ 2 ) , \\\\ R _ M ( y _ 1 ^ - ) & = ( 1 + y _ 1 + y _ 1 y _ 2 ) ^ { - 1 } y _ 1 y _ 1 ^ - ( 1 + y _ 2 + y _ 2 y _ 3 ) , \\\\ R _ M ( y _ 1 ^ + ) & = ( 1 + y _ 2 + y _ 2 y _ 3 ) ^ { - 1 } y _ 2 y _ 1 ^ + ( 1 + y _ 3 + y _ 3 y _ 1 ) . \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} | f ( b ) - f ( a ) | = | k _ { 0 } ^ { + } - k _ { 0 } ^ { - } + \\sum _ { j = 2 } ^ { k } p _ { j } ( b - 1 - a ) | = | k _ { 0 } ^ { + } - k _ { 0 } ^ { - } + p _ { 1 } ( - b + 1 + a ) | , \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} \\left ( \\begin{matrix} u _ { n + 1 } \\\\ u _ { n } \\end{matrix} \\right ) = A _ { c , E } ( \\theta + n \\alpha ) \\left ( \\begin{matrix} u _ { n } \\\\ u _ { n - 1 } \\end{matrix} \\right ) \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} ( w _ 1 , w _ 2 ) = ( 1 , 0 ) + L ( w _ 1 , w _ 2 ) \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} u ^ 2 ( x ) = \\frac { \\pi } { ( \\mu - Q ( x ) ) ^ \\frac 1 2 } ( 1 + \\sin 2 \\zeta + R _ 1 ( \\zeta ) ) , \\zeta > 1 , \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\widetilde { f } _ { n } \\cdot x _ { n } d \\mu = \\int _ { \\Omega } f _ { n } \\cdot x _ { n } d \\mu - \\int _ { E _ { 1 } } f _ { n } \\cdot x _ { n } d \\mu > \\epsilon - \\frac { \\delta } { 2 } . \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} e _ { k } ( x ) = e _ { k } ( x _ { 0 } ) + d e _ { k } ( x _ { 0 } ) \\cdot ( \\phi ( x ) - \\phi ( x _ { 0 } ) ) + o ( \\| x - x _ { 0 } \\| ) \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} \\left ( L ^ { \\iota } L ^ { \\gamma } \\right ) _ { \\mu \\nu } = \\sum _ { \\alpha = 1 } ^ n L _ { \\mu \\alpha } ^ { \\iota } L _ { \\alpha \\nu } ^ { \\gamma } = 0 \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} \\widetilde { f } ( x , z _ { 1 } , z _ { 2 } ) = f ( x , \\widetilde { z } _ { 1 } , \\widetilde { z } _ { 2 } ) \\widetilde { g } ( x , z _ { 1 } , z _ { 2 } ) = g ( x , \\widetilde { z } _ { 1 } , \\widetilde { z } _ { 2 } ) \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} a _ n ( x ) = \\begin{cases} a _ { n - 1 } ( x ) - x a _ { n - 2 } ( x ) & n , \\\\ ( 1 + x ) a _ { n - 1 } ( x ) - x a _ { n - 2 } ( x ) & n . \\\\ \\end{cases} \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} \\tilde { \\epsilon } ( \\lambda _ { i a } ) & = \\sum _ { j b } [ \\delta _ { i a , j b } - s _ b F ( \\lambda _ { j b } | \\lambda _ { i a } ) ] \\epsilon ( \\lambda _ { j b } ) , \\\\ \\epsilon ( \\lambda _ { i a } ) & = \\sum _ { j b } [ \\delta _ { i a , j b } - s _ b F ( \\lambda _ { i a } | \\lambda _ { j b } ) ] \\tilde { \\epsilon } ( \\lambda _ { j b } ) . \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} r _ { i + 1 } = \\left ( \\prod _ { i = n } ^ 1 p _ i - \\prod _ { i = n } ^ 1 q _ i \\right ) ( \\kappa _ { i + 1 } ^ { \\epsilon } ) ^ { - 1 } = ( \\kappa _ { i + 1 } ^ { \\epsilon } ) ^ { - 1 } \\left ( \\prod _ { i = n } ^ 1 p _ i - \\prod _ { i = n } ^ 1 q _ i \\right ) . \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} p = \\frac { 3 c ^ 4 r ^ 2 + c ^ 4 + 3 c ^ 3 r ^ 2 + c ^ 3 - 1 2 c ^ 2 r ^ 2 - 2 c ^ 2 - 1 2 c r ^ 2 - 1 2 c - 1 0 } { 6 c \\left ( c ^ 3 - 2 c - 1 0 \\right ) r } . \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} I _ 1 ( y ) & = \\int _ 0 ^ 1 \\left ( \\partial _ u r ( Y _ q + t y ) - \\partial _ u r ( Y _ q ) \\right ) u V _ q \\dd t , \\\\ I _ 2 ( y ) & = \\int _ { 0 } ^ { 1 } \\left ( \\partial _ u r ( Y _ q + t y ) u \\right ) t v \\dd t , \\\\ I _ 3 ( y ) & = \\int _ 0 ^ 1 \\left ( \\partial _ v r ( Y _ q + t y ) - \\partial _ v r ( Y _ q ) \\right ) v V _ q \\dd t , \\\\ I _ 4 ( y ) & = \\int _ 0 ^ 1 \\left ( \\partial _ v r ( Y _ q + t y ) v \\right ) t v \\dd t , \\\\ I _ 5 ( y ) & = \\int _ 0 ^ 1 \\left ( r ( Y _ q + t y ) - r ( Y _ q ) \\right ) v \\dd t , \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\left \\{ \\textrm { d i a m } \\left ( A ( \\theta _ { - t } \\omega ) \\right ) = 0 \\right \\} \\subset \\left \\{ \\textrm { d i a m } ( A ( \\omega ) ) = 0 \\right \\} \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} \\Omega _ t : = \\{ ( x ^ \\alpha ) \\in \\mathbb { R } ^ { n + 1 } / 0 < x ^ 0 < t ; | x ^ i | < K ( 2 t - x ^ 0 ) ; \\ ; i = 1 , 2 , . . . , n \\} , \\ ; \\Lambda _ \\tau : = \\Omega _ t \\cap \\{ x ^ 0 = \\tau \\} , \\ ; 0 \\leq \\tau \\leq t , \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } { \\eta _ { k } } ( \\hat { \\theta } _ { j } ) = p _ { j } { \\eta _ { k } } ( \\theta ) + { \\eta _ { k } } ( \\xi _ { j } ) , & \\forall \\ 1 \\leq j \\leq k - 1 , \\\\ \\hat { \\theta } _ { k , l ' } = ( p _ { k } ) { \\eta _ { k } } ( \\theta ) + { l ' \\over | p _ { k } | } , & \\forall \\ 0 \\leq l ' \\leq | p _ { k } | - 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} g ( \\mu ) : = \\int _ 0 ^ { x _ \\mu } ( \\mu - Q ( x ) ) ^ \\frac 1 2 \\ , \\dd x \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} U ^ { \\mu _ \\gamma } ( z ) = U ^ { \\mu _ \\gamma ^ { ( 2 ) } } ( z ) , z \\notin . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} \\sum _ { k _ j = 1 } ^ \\infty \\frac { 1 } { \\Gamma ( k _ j ) } \\biggl ( \\log \\frac { 1 - x _ { j + 1 } } { 1 - x _ j } \\biggr ) ^ { k _ j - 1 } X _ j ^ { k _ j - 1 } & = \\exp \\biggl ( X _ j \\log \\frac { 1 - x _ { j + 1 } } { 1 - x _ j } \\biggr ) \\\\ & = \\biggl ( \\frac { 1 - x _ { j + 1 } } { 1 - x _ j } \\biggr ) ^ { X _ j } \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} \\P ( S ^ { ( i ) , t } = s ^ t ) & = \\P \\Big ( Z _ { ( k + g ) ( j - 1 ) + i } ^ { ( k + g ) ( j - 1 ) + i + k - 1 } = s _ j , \\ ; j = 1 , \\ldots , t \\Big ) \\\\ & \\stackrel { ( a ) } { \\leq } \\Psi ^ t ( b , g ) \\prod _ { j = 1 } ^ t \\P \\Big ( Z _ { ( k + g ) ( j - 1 ) + i } ^ { ( k + g ) ( j - 1 ) + i + k - 1 } = s _ j \\Big ) , \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} & d f \\big ( \\sum _ { \\sigma \\in S _ n } s g n ( \\sigma ) ^ { m - 1 } [ x _ { \\sigma ( 1 ) } | \\dots | x _ { \\sigma ( n ) } ] \\big ) \\\\ & = \\sum _ { \\sigma \\in S _ n } s g n ( \\sigma ) ^ { m - 1 } ( ( - 1 ) ^ { \\delta _ n ' } + 1 ) f ( [ x _ { \\sigma ( 1 ) } | \\dots | x _ { \\sigma ( n - 1 ) } ] ) x _ { \\sigma ( n ) } \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} a _ k = a _ m \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} d _ i = x _ i \\frac { \\partial } { \\partial x _ i } + \\beta \\sum _ { j < i } \\frac { x _ j } { x _ i - x _ j } \\left ( 1 - K _ { i j } \\right ) + \\beta \\sum _ { i < j } \\frac { x _ i } { x _ i - x _ j } \\left ( 1 - K _ { i j } \\right ) + \\beta \\left ( i - 1 \\right ) \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} b _ \\alpha = \\bar F _ \\alpha + \\frac 1 J \\left ( Q _ { \\alpha \\alpha } - \\frac { Q _ \\alpha W _ { \\alpha \\alpha } } { 1 + W _ \\alpha } - \\frac { Q _ \\alpha \\bar W _ { \\alpha \\alpha } } { 1 + \\bar W _ \\alpha } \\right ) . \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{align*} u ( x ) = \\sum _ { j \\in \\mathbb { Z } \\setminus \\{ 0 \\} } u _ j \\ , e ^ { \\mathrm { i } \\ , j \\ , x } , \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\| \\mu - \\nu \\| _ { p , \\pi } : = \\begin{cases} \\left ( \\sum _ { x } \\pi ( x ) a _ { \\mu , \\nu , \\pi } ^ p \\right ) ^ { 1 / p } , & 1 \\le p < \\infty , \\\\ \\max _ { x \\in \\Omega } a _ { \\mu , \\nu , \\pi } , & p = \\infty \\\\ \\end{cases} \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} d _ { S ( G , t ) } ( x w , x w ' ) > & d _ { S ( G , t - 1 ) } ( w , ( v _ 1 ) ^ { t - 1 } ) + \\sum _ { i = 0 } ^ { \\frac { r - 5 } { 2 } } d _ { S ( G , t - 1 ) } ( ( v _ { 2 i + 1 } ) ^ { t - 1 } , ( v _ { 2 i + 3 } ) ^ { t - 1 } ) \\\\ & + d _ { S ( G , t - 1 ) } ( ( v _ { r - 2 } ) ^ { t - 1 } , ( v _ 0 ) ^ { t - 1 } ) \\sum _ { i = 0 } ^ { \\frac { r - 3 } { 2 } } d _ { S ( G , t - 1 ) } ( ( v _ { 2 i } ) ^ { t - 1 } , ( v _ { 2 i + 2 } ) ^ { t - 1 } ) \\\\ & + d _ { S ( G , t - 1 ) } ( ( v _ { r - 1 } ) ^ { t - 1 } , w ' ) \\\\ \\ge & d _ { S ( G , t - 1 ) } ( w , w ' ) , \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} \\hat { f } ( \\zeta ) = \\frac { - 1 } { 2 \\mathrm { i } \\pi } \\int _ { \\cal L } \\frac { e ^ { s } } { s ^ { c } ( s - \\zeta ) } d s , \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} s - \\frac { \\delta ' } { 1 + \\delta } \\ = \\ \\frac { b b ' } { 1 + b } \\ \\asymp \\ b b ' \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} K ( z ) B ( z ) K ( z ) \\psi _ j = \\sum _ { m = 1 } ^ \\infty \\frac { b ( \\psi _ j , \\psi _ m ) } { ( z - \\mu _ j ) ( z - \\mu _ m ) } \\psi _ m , z \\in \\Gamma _ n . \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} \\mathrm { T r } _ 2 ( g _ 1 ) \\mid x - \\mathrm { T r } _ 2 ( g _ 2 ) \\mid x = p \\mathrm { T r } _ 2 ( g _ 3 ) \\mid x . \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} | N ^ n _ 1 ( t ) + 1 + Q ^ n _ 1 ( 0 ) - G ^ n _ 1 ( t ) - Q ^ n _ { - 1 } ( 0 ) + G ^ n _ { - 1 } ( t ) | = | Q ^ n ( t ) + 1 + N ^ n _ { - 1 } ( t ) | \\le | Q ^ n ( t ) | + 1 + N ^ n _ { - 1 } ( t ) . \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} R _ 3 \\leq { I ( { X _ 3 ; Y _ 2 } ) } & = h ( Y _ 2 ) - h ( Y _ 2 | X _ 3 ) \\\\ & = \\frac { 1 } { 2 } \\log \\frac { | { \\hat { \\bf { C } } } _ { y _ 2 } | } { | { \\hat { \\bf { C } } } _ { s _ 2 } | } \\\\ & = \\frac { 1 } { 2 } \\log \\frac { { C ^ { 2 } _ { y _ 2 } } - { | \\tilde { C } _ { y _ 2 } | ^ { 2 } } } { { C ^ { 2 } _ { s _ 2 } } - { | \\tilde { C } _ { s _ 2 } | ^ { 2 } } } = L _ 3 , \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} & \\left \\vert \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } b ( X _ { i } ) b ( X _ { i } ) ^ { \\prime } \\{ E [ K _ { h } ( Y _ { i } - \\hat { \\gamma } ( X _ { i } ) ) | X _ { i } ] - f ( 0 | X _ { i } ) \\} \\right \\vert _ { \\infty } \\\\ & \\leq C \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\left \\vert \\hat { \\Delta } ( X _ { i } ) \\right \\vert + C h ^ { 2 } = O _ { p } ( n ^ { - d _ { \\gamma } } + h ^ { 2 } ) , \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} m _ 3 = 1 + \\varepsilon ^ 2 d ( \\xi ) + \\mathtt { r } _ { m _ 3 } ( \\omega ) , m _ 1 = \\varepsilon ^ 2 c ( \\xi ) + \\mathtt { r } _ { m _ 1 } ( \\omega ) \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} u _ i ( x _ t ^ i ) : = \\frac { \\gamma } { 1 - \\gamma } \\left [ \\left ( \\phi ^ i + \\frac { \\eta } { \\gamma } \\ , x _ t ^ i \\right ) ^ { 1 - \\gamma } - 1 \\right ] , \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} d _ K ( p , q ) & = \\left ( \\left [ \\sum _ { j = 1 } ^ n ( x _ j - x _ j ' ) ^ 2 + ( y _ j - y _ j ' ) ^ 2 \\right ] ^ 2 + \\left [ t - t ' + 2 \\sum _ { j = 1 } ^ n ( x _ j ' y _ j - x _ j y _ j ' ) \\right ] ^ 2 \\right ) ^ { 1 / 4 } \\\\ & \\approx \\left [ \\sum _ { j = 1 } ^ n ( x _ j - x _ j ' ) ^ 2 + ( y _ j - y _ j ' ) ^ 2 \\right ] ^ { 1 / 2 } + \\left | t - t ' + 2 \\sum _ { j = 1 } ^ n ( x _ j ' y _ j - x _ j y _ j ' ) \\right | ^ { 1 / 2 } \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} E = \\big \\{ f \\in C _ 0 ( [ 0 , 1 ] ^ n \\setminus \\{ \\vec { 0 } \\} , D _ 1 \\otimes _ { \\max } D _ 2 \\otimes _ { \\max } \\cdots \\otimes _ { \\max } D _ n \\mid f ( \\vec { t } ) \\in D ^ { ( \\vec { t } ) } \\big \\} \\ , . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} m _ K \\otimes m _ K ( G ( I ' \\times J ' ) ) = m _ K \\otimes m _ K ( K ( I ' \\times J ' ) ) . \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{gather*} \\delta _ Y C ^ \\ast = 0 , \\delta _ Y A ^ \\ast = 0 , \\delta _ Y A = 0 , \\delta _ Y C = 1 . \\end{gather*}"}
-{"id": "2048.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & * \\\\ 0 & \\chi _ 3 \\end{pmatrix} \\begin{pmatrix} \\chi _ 3 & * \\\\ 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} d \\mu ^ X = \\iota _ X \\omega = \\sum _ i d \\theta _ i = \\omega ( X ) . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} \\tau ^ { 2 f } = 1 , \\sigma ^ 4 = 1 , \\tau \\sigma \\tau ^ { - 1 } = \\sigma ^ { - 1 } . \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} \\Phi _ n ^ * ( z ) = z ^ n \\overline { \\Phi _ n ( 1 / \\overline { z } ) } . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} I = \\int _ { - 1 } ^ { 1 } d \\tau \\int _ { - 1 } ^ { 1 } d t \\left [ \\vec \\psi ( \\tau ) \\delta ( t ) - \\vec \\psi ( t ) \\delta ( \\tau ) \\right ] \\left [ \\vec \\psi ( \\tau ) \\delta ( t ) - \\vec \\psi ( t ) \\delta ( \\tau ) \\right ] ^ { \\top } \\geq 0 \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} z _ { a , m , t } = 1 . \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} \\liminf _ { t - s \\to \\infty } \\frac { J ( t ) - J ( s ) } { t - s } = \\tilde c _ 0 ^ - . \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} { \\widetilde { \\gamma } _ { { e ^ * } } } = \\mathop { \\max } \\limits _ { e \\in { \\Phi _ e } } \\left \\{ { \\frac { { { P _ S } G _ e ^ { S } L \\left ( \\left | X _ e \\right | \\right ) } } { { { { { P _ A } G _ e ^ { A } L \\left ( \\left | X _ e \\right | \\right ) } + \\sigma _ e ^ 2 } } } } \\right \\} . \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} f ( z ) = z + \\sum _ { n = 2 } ^ { \\infty } a _ n z ^ n . \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} K _ S \\simeq \\phi ^ * G \\otimes \\mathcal O _ S \\biggl ( \\sum _ { i = 1 } ^ k ( m _ i - 1 ) \\ , F _ i \\biggr ) , \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} \\tilde { \\mathbf { P } } = \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 2 = \\mathbf { A } ^ { - 1 } \\mathbf { P } _ { \\rm w } , \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} & \\mathcal { W } ^ { ( \\ell ) } = \\left \\{ ( w _ 1 , w _ 2 ) : w _ 1 \\in \\{ 0 , 1 , \\cdots , | A ^ { \\ast } | \\} , \\right . \\\\ & \\left . w _ 2 \\in \\{ 0 , 1 , \\cdots , ( 1 + \\delta _ { \\ell } ) k _ { \\ell } \\} , \\right . \\\\ & \\left . w _ 1 + w _ 2 > 0 , | A ^ { \\ast } | + w _ 2 \\leq ( 1 + \\delta _ { \\ell } ) k _ { \\ell } + w _ 1 \\right \\} . \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} [ a , b ] \\ : = \\ \\{ x \\in S : a \\leq x \\leq b \\} . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} [ x , y ] = \\sqrt { | x | ^ 2 | y | ^ 2 - 2 x \\cdot y + 1 } \\geq 1 - | x | | y | \\geq 1 - | y | \\geq \\frac { 1 } { 2 } ( 1 - | y | ^ 2 ) , \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\frac { Z \\alpha _ g \\epsilon } { \\omega } + \\nu = - n n \\in \\mathbb { N } \\ ; . \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{align*} M _ { a } ^ { s G } ( \\lambda ) = M _ { a } ( \\lambda ) \\left ( \\sigma _ { a } ^ { x } \\right ) ^ { \\mathsf { x } } , \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} & M ^ { i } = ( \\Pi / I ( s _ { u _ { 1 } } s _ { u _ { 2 } } \\cdots s _ { u _ { i } } ) ) e _ { u _ { i } } , M = \\bigoplus _ { i = 1 } ^ { l } M ^ { i } . \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} d ( \\theta ) = \\frac { \\lambda _ 1 } { \\lambda _ 2 } e ^ { - 2 \\pi i ( \\theta + \\frac { \\alpha } { 2 } ) } \\left ( e ^ { 2 \\pi i ( \\theta + \\frac { \\alpha } { 2 } ) } - \\frac { - 1 + \\sqrt { 1 - 4 \\lambda _ 1 \\lambda _ 3 } } { 2 \\lambda _ 1 } \\right ) \\left ( e ^ { 2 \\pi i ( \\theta + \\frac { \\alpha } { 2 } ) } - \\frac { - 1 - \\sqrt { 1 - 4 \\lambda _ 1 \\lambda _ 3 } } { 2 \\lambda _ 1 } \\right ) . \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} \\varphi ( q ) = \\sum _ { n = - \\infty } ^ { \\infty } q ^ { n ^ 2 } . \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{gather*} A ( h _ { + } ) + C ( h _ { + } ) = 4 \\left ( s - v \\right ) \\times \\\\ \\left ( \\dfrac { \\left ( s - v \\right ) ( t ^ { 2 } + s ^ { 2 } ) } { s ^ { 2 } } h _ { + } ^ { 2 } + \\dfrac { 2 w \\left ( v t - w s \\right ) } { s } h _ { + } - \\allowbreak w \\left ( v t - w s \\right ) \\right ) \\end{gather*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\lambda _ \\delta ( t ) : = \\begin{cases} 1 & 0 \\leq t \\leq \\frac { \\delta } { 2 } , \\\\ 2 - 2 \\delta ^ { - 1 } t & \\frac { \\delta } { 2 } < t \\leq \\delta , \\\\ 0 & \\delta < t . \\end{cases} \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{gather*} \\delta \\omega _ 0 = 0 , \\omega _ 0 \\simeq \\omega . \\end{gather*}"}
-{"id": "3134.png", "formula": "\\begin{align*} \\Phi ( t ) = \\int _ 0 ^ t \\frac { d u } { H ( u ) } . \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} B \\cap p \\left ( W ( K ) / \\wp W ( K ) \\right ) = p B . \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} g _ { a , b } : = \\left ( \\begin{matrix} \\sqrt { a } \\cdot d _ x B _ s B _ y \\\\ \\sqrt { b } \\cdot B _ x B _ s d _ y \\end{matrix} \\right ) : B _ x B _ s B _ y \\stackrel { } { \\to } E ( 1 ) : = F B _ s B _ y ( 1 ) \\oplus B _ x B _ s G ( 1 ) . \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} & \\left \\| \\nabla ^ { k } _ { x } \\left ( K _ { 1 L } ( t ) g - \\mathcal { F } ^ { - 1 } \\left [ e ^ { - \\frac { \\nu t | \\xi | ^ { 2 \\sigma } } { 2 } } \\cos ( t | \\xi | ) \\chi _ { L } \\right ] \\ast g \\right ) \\right \\| _ { 2 } \\\\ & \\le C ( 1 + t ) ^ { - \\frac { n } { 2 \\sigma } ( \\frac { 1 } { r } - \\frac { 1 } { 2 } ) - \\frac { k - \\tilde { k } } { 2 \\sigma } - 1 + \\frac { 1 } { 2 \\sigma } } \\| \\nabla ^ { \\tilde { k } } _ { x } g \\| _ { r } , \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} j _ 1 + j _ 2 + j _ 3 = 0 \\Rightarrow j _ 1 ^ 3 + j _ 2 ^ 3 + j _ 3 ^ 3 = 3 \\ , j _ 1 \\ , j _ 2 \\ , j _ 3 \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} \\Omega ( x _ 1 , x _ 2 ) = & ( \\iota ^ * \\omega _ 2 + \\sqrt { - 1 } \\iota ^ * \\omega _ 3 ) ( x _ 1 , x _ 2 ) \\\\ = & ( \\omega _ 2 + \\sqrt { - 1 } \\omega _ 3 ) ( \\iota _ * x _ 1 , \\iota _ * x _ 2 ) \\\\ & = t r ( - a _ 2 b _ 1 + b _ 2 a _ 1 - i _ 2 j _ 1 + i _ 1 j _ 2 - a ' _ 2 b ' _ 1 + b ' _ 2 a ' _ 1 ) . \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} \\psi _ - \\left ( \\frac { a z + b } { c z + d } \\right ) = ( c z + d ) ^ { 2 - n / 2 } \\psi _ - ( z ) \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} \\in \\mathrm { S L } _ 2 ( \\mathbb { Z } ) , a , d b , c \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} F ( x ) ( t ) = \\int _ { \\overline { \\Omega } } k ( t , s ) f ( s , x ( s ) ) \\textup d s , t \\in \\overline { \\Omega } . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{gather*} r _ 0 = \\frac { q _ 0 ^ 2 } { 4 } + e _ 1 e _ 3 \\frac { q _ 2 ^ 2 - 1 } { 4 } - \\frac { 1 } { 2 } e _ 3 q _ 1 q _ 2 . \\end{gather*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\bar v ( s ) = - \\Sigma _ 1 ( s ) ^ { - 1 } ( \\Lambda _ 1 ( s ) \\bar X ( s ) + r ( s ) + \\bar r ( s ) + ( B ^ T ( s ) + \\bar { B } ^ T ( s ) ) \\phi ( s ) ) . \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} \\begin{aligned} \\gamma _ t & = v _ x , \\\\ v _ t & = \\big ( \\gamma ^ { - m } v _ x ^ n \\big ) _ x , \\end{aligned} \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} \\mathcal { A } ( M ) \\subseteq \\bigcup _ { i = 1 } ^ n \\mathcal { A } ( \\langle S _ i \\rangle ) . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} \\mathbb { P } ( m , k ) = \\prod _ { i = 0 } ^ { k - 1 } ( 1 - q ^ { i - m } ) . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} \\mathbf { y } = \\sqrt { \\beta _ K } { \\hat { \\mathbf { G } } } \\boldsymbol { \\psi } + \\sqrt { \\beta _ K } \\tilde { \\mathbf { G } } \\boldsymbol { \\psi } + \\mathbf { z } . \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{align*} \\Psi \\ ; : = \\ ; \\left ( \\begin{array} { c c c c } f _ 1 ^ { } & f _ 2 & \\cdots & f _ r \\\\ \\sigma ^ * f _ 1 & \\sigma ^ * f _ 2 & \\cdots & \\sigma ^ * f _ r \\\\ \\vdots & \\vdots & & \\vdots \\\\ \\sigma ^ { ( r - 1 ) * } f _ 1 & \\sigma ^ { ( r - 1 ) * } f _ 2 & \\cdots & \\sigma ^ { ( r - 1 ) * } f _ r \\end{array} \\right ) \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} \\begin{cases} \\mu \\left ( \\{ \\eta = \\xi \\} \\right ) = 1 \\\\ \\eta '' \\phi _ { \\mu } ^ { 2 } = 1 & L e b - a . e . \\end{cases} \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} s ( z ) = \\int \\frac 1 { x - z } \\rho ( \\dd x ) = \\frac { - z + \\sqrt { z ^ 2 - 4 } } { 2 } \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\epsilon ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } & = \\frac { \\frac { 1 } { 2 } ( R _ 1 - \\delta _ 1 ) + ( \\alpha - 1 ) d _ { \\alpha } ( p _ { U , W } , p _ U p _ W ) } { \\frac { 1 } { 2 } + ( \\alpha - 1 ) } - I ( U ; W ) , \\\\ \\epsilon ^ { ( 2 ) } _ { \\alpha , \\delta _ 2 } & = \\frac { \\frac { 1 } { 2 } ( R _ 1 + R _ 2 - \\delta _ 2 ) + ( \\alpha - 1 ) d _ { \\alpha } ( p _ { U , V , W } , p _ { U , V } p _ W ) } { \\frac { 1 } { 2 } + ( \\alpha - 1 ) } - I ( U , V ; W ) . \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} \\| v \\| _ { \\mathcal { V } } ^ 2 = \\| v \\| _ { L ^ 2 } ^ 2 + \\| \\tilde \\alpha \\cdot D v \\| _ { L ^ 2 } ^ 2 \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} | \\nabla _ { \\omega _ { C _ x } } ^ j ( P ^ * \\omega - \\omega _ { C _ x } ) | _ { \\omega _ { C _ x } } = O ( r ^ { \\lambda - j } ) \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} \\max _ { z _ n ^ { \\rm R } \\le 0 , z _ n ^ { \\rm I } \\le 0 } & \\ln \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( z _ n ^ { \\rm R } + \\sqrt { P / 2 } \\big ) \\right ) + \\ln \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( z _ n ^ { \\rm I } + \\sqrt { P / 2 } \\big ) \\right ) \\\\ \\mathrm { s . t . } ~ & { z _ n ^ { \\rm R } } ^ 2 + { z _ n ^ { \\rm I } } ^ 2 = B _ n . \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} P ( \\overline { \\jmath } _ 1 , \\dots , \\overline { \\jmath } _ { \\nu } ) : = \\{ ( 2 4 c _ 4 - 4 8 \\ , c _ 1 ^ 2 ) v _ 3 + ( 4 c _ 6 - \\frac { 1 6 } { 3 } c _ 2 ^ 2 ) v _ 1 \\} \\cdot \\mathbb { M } ^ { - 1 } \\overline { \\omega } - 1 = 0 . \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} \\lambda _ { \\ell , 0 } = \\prod _ { j = 1 } ^ { \\min \\{ \\ell , n - m _ 1 \\} } \\frac { i + 1 - \\sigma _ j } { i + 1 - ( k + 2 ) ^ { - 1 } } . \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} \\left ( \\frac { p ^ 2 } { 2 m } - V ( q ) - \\alpha S \\right ) \\frac { \\partial \\gamma } { \\partial S } + \\frac { p } { m } \\frac { \\partial \\gamma } { \\partial q } + ( p \\alpha + V ' ( q ) ) = 0 \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\Phi ^ b _ { x x } - A ( x ) \\Phi ^ b = \\lambda _ 1 ^ b \\Phi ^ b \\\\ \\varphi ^ b ( x ) > 0 , \\ ; \\psi ^ b ( x ) > 0 , x \\in ( - b , b ) \\\\ \\varphi ^ b ( \\pm b ) = \\psi ^ b ( \\pm b ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} & d ( \\sum _ { \\sigma \\in S _ n } s g n ( \\sigma ) ^ { m - 1 } a [ x _ { \\sigma ( 1 ) } | x _ { \\sigma ( 2 ) } | \\dots | x _ { \\sigma ( n ) } ] ) \\\\ & = \\sum _ { \\sigma \\in S _ n } s g n ( \\sigma ) ^ { m - 1 } ( 1 + ( - 1 ) ^ { \\epsilon _ n ' } ) a x _ { \\sigma ( 1 ) } [ x _ { \\sigma ( 2 ) } | x _ { \\sigma ( 1 ) } | \\dots | x _ { \\sigma ( n ) } ] \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} g ( t ) = 1 + \\int _ { 0 } ^ { t } \\nu g ( s ) \\ , d s . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} \\mathbf { b } _ i = \\mathbf { P } ^ { - 1 } _ f \\left ( \\mathbf { P } _ 1 { \\mathbf { d } } _ { i - 1 } - \\mathbf { P } _ 2 \\mathbf { d } _ i \\right ) . \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} \\varphi \\colon A \\rtimes _ { \\alpha } \\R \\to B _ { \\sigma } \\subseteq A \\rtimes _ { \\alpha } \\R \\rtimes _ { \\widehat { \\alpha } } \\widehat { \\R } \\quad \\varphi ( a ) = h ^ { 1 / 2 } a h ^ { 1 / 2 } . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} \\langle \\chi , \\alpha _ 0 \\rangle = \\frac { 1 } { | N _ G ( P ) | } \\left ( 3 ^ k ( q - 1 ) ( 1 + q - 1 ) + \\frac { q ( q - 1 ) } { 2 } ( \\chi ( T ) + \\overline { \\chi ( T ) } ) + \\frac { q ^ 2 ( q - 1 ) } { 3 } \\cdot \\varepsilon \\cdot 3 ^ k ( 1 + \\omega + \\overline { \\omega } ) \\right ) = 0 \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} f _ 1 ( A _ n ) = & \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\mathrm { e r f } \\left ( \\sqrt { A _ n } + \\sqrt { P } \\right ) , \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} \\begin{bmatrix} R _ { s , \\mathsf { k } } + \\omega _ 0 j L _ { s , \\mathsf { k } } ( \\theta _ \\mathsf { k } ) & \\omega _ 0 j \\mathrm { r } ( \\theta _ \\mathsf { k } ) l _ { s f , \\mathsf { k } } \\\\ 0 & R _ { f , \\mathsf { k } } \\end{bmatrix} \\begin{bmatrix} i _ { s , \\mathsf { k } } \\\\ i _ { f , \\mathsf { k } } \\end{bmatrix} = \\begin{bmatrix} v _ \\mathsf { k } \\\\ v _ { f , \\mathsf { k } } \\end{bmatrix} . \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} F _ q = \\mathrm { s p a n } _ { \\Z [ G ] } \\{ \\hat { a } _ i , \\hat { b } _ i , i < g \\} \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { p - 1 } \\frac { 2 ^ k } { k } \\sum _ { j = 1 } ^ { k - 1 } \\frac { 1 } { j 2 ^ j } \\equiv _ { p } - 2 q ^ 2 _ p ( 2 ) . \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} \\aligned t _ j & = \\Big ( \\frac { A _ j } { B _ j } + \\frac { A _ j } { B _ j } f _ j ( t _ 1 , . . . , t _ r ) \\Big ) ^ { \\frac { 1 } { \\alpha } } \\\\ & = \\Big ( \\frac { A _ j } { B _ j } \\Big ) ^ { \\frac { 1 } { \\alpha } } \\Big ( 1 + f _ j ( t _ 1 , . . . , t _ r ) \\Big ) ^ { \\frac { 1 } { \\alpha } } , \\ ; \\ ; j = 1 , . . . , r . \\endaligned \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} H ( m _ { N } , P _ { p ^ { N } } ) = - \\sum _ { P _ { p ^ { N } } } m _ { N } ( s ) \\log ( m _ { N } ( s ) ) = \\log ( \\lvert M _ { N } \\rvert ) \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} \\tau _ { \\infty } = \\lim _ { \\log \\lambda \\rightarrow \\pm \\infty } \\lambda ^ { \\mp 2 ( \\mathsf { N } + 2 ) } \\mathcal { T } ( \\lambda ) . \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} D _ h = \\Phi \\approx D = 1 + \\underset { N ' \\rightarrow \\infty } \\lim \\left [ \\dfrac { \\ln ( L / 2 ) } { \\ln ( 2 \\cdot N ' ) } \\right ] . \\end{align*}"}
-{"id": "9200.png", "formula": "\\begin{align*} \\mathcal L _ x ( U , V ) = - d \\omega _ 0 ( J U , V ) , \\forall ~ U , V \\in H _ x X . \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} \\int _ { \\Omega } { \\widetilde { \\widetilde { f } } } _ { n } \\cdot x _ { n } d \\mu & = \\int _ { E ^ { c } _ { 2 } } \\widetilde { f } _ { n } \\cdot x _ { n } d \\mu \\\\ & = \\int _ { \\Omega } \\widetilde { f } _ { n } \\cdot x _ { n } d \\mu - \\int _ { E _ { 2 } } \\widetilde { f } _ { n } \\cdot x _ { n } d \\mu \\\\ & > \\epsilon - \\frac { \\delta } { 2 } - \\frac { \\delta } { 2 ^ { 2 } } . \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n T ( n , k ) H _ { 2 k } = 3 H _ n - H _ { \\lfloor n / 2 \\rfloor } . \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} \\Phi _ B ( T _ { \\delta } ) = T _ { \\delta } + \\Psi _ 2 ( T _ { \\delta } ) + \\Psi _ { \\geq 3 } ( T _ { \\delta } ) = \\varepsilon v _ { \\delta } + \\varepsilon ^ 2 \\Psi _ 2 ( v _ { \\delta } ) + \\tilde { q } , \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} \\lfloor m ; g \\rceil \\star \\lfloor n ; h \\rceil = \\left \\{ \\begin{array} { l l } \\lfloor \\gamma n m ; g + h \\rceil & ( g ) \\cup ( h ) \\cup ( g + h ) = \\C ^ { n } \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} \\bar { r } _ { k - 1 , k } & = r _ { k - 1 , k } - \\lfloor r _ { k - 1 , k } / r _ { k - 1 , k - 1 } \\rceil r _ { k - 1 , k - 1 } , \\\\ \\bar { r } _ { k - 1 , k - 1 } & = \\sqrt { \\bar { r } ^ 2 _ { k - 1 , k } + r _ { k k } ^ 2 } , \\\\ | \\bar { r } _ { k k } | & = | r _ { k - 1 , k - 1 } r _ { k k } / \\bar { r } _ { k - 1 , k - 1 } | . \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\tau ^ { 2 s } \\left | T _ \\Sigma ( i \\tau / 2 ) - \\sum _ { l = 0 } ^ k T _ \\Sigma ^ l ( i \\tau ) ^ { - ( 2 j - 1 + l ) } \\right | d \\tau \\lesssim \\frac { 1 } { | \\sin ( 2 \\pi s ) | } \\| ( u , v ) \\| _ { \\dot H ^ s } ^ 2 \\| ( u , v ) \\| _ { l ^ 2 _ 1 D U ^ 2 } ^ { 2 j - 2 } . \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} \\textrm { d i a m } ( \\varphi _ t ( \\omega , B ( 0 , r ) ) & = \\sup _ { x , y \\in B ( 0 , r ) } \\left | \\varphi _ t ( \\omega , x ) - \\varphi _ t ( \\omega , y ) \\right | \\\\ & \\leq \\sup _ { x , y \\in B ( 0 , r ) } \\left | \\varphi _ t ( \\omega , x ) - a ( \\theta _ t \\omega ) \\right | + \\left | a ( \\theta _ t \\omega ) - \\varphi _ t ( \\omega , y ) \\right | \\\\ & \\rightarrow 0 \\textrm { a s } t \\rightarrow \\infty \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} G ( z ) = \\real \\left ( - z ^ 2 \\ln T ( z / 2 ) + i z \\int u \\ d x \\right ) . \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} r = h ^ { \\lambda , m , 0 } ( p , q ) = \\frac { \\frac { m } { \\lambda } \\frac { 2 } { 1 + m } - \\frac { 1 - m } { 1 + m } + 1 - q } { \\frac { m } { \\lambda } + \\lambda p } , \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} d \\gamma _ t = 0 \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} \\mathbb { E } \\Big ( \\sum ^ { n - 1 } _ { k = 0 } a _ k ( x ( t _ { k + 1 } ) - x ( t _ k ) ) \\Big ) ^ 2 \\leq c \\sum ^ { n - 1 } _ { k = 0 } a ^ 2 _ k \\Delta t _ k . \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{gather*} \\Psi _ { \\rm W K B } ( x ) = T ( x ) \\exp \\left [ \\int \\Lambda d x - \\int \\operatorname { d i a g } \\left ( T ^ { - 1 } \\frac { d T } { d x } \\right ) d x \\right ] , \\end{gather*}"}
-{"id": "7585.png", "formula": "\\begin{align*} 0 & = [ L ( e _ { i j } ) , e _ { k l } ] + [ e _ { i j } , L ( e _ { k l } ) ] \\\\ & = \\left ( C _ { j k } ^ { j j } + C _ { j k } ^ { k k } \\right ) e _ { i l } - \\left ( C _ { l i } ^ { i i } + C _ { l i } ^ { l l } \\right ) e _ { k j } . \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} P \\left ( \\bigcap _ { i = 1 } ^ M \\left \\{ X _ i \\in ( \\sigma _ { \\rm w } ^ 2 + \\sigma _ { \\rm j , i } ^ 2 - \\delta , \\sigma _ { \\rm w } ^ 2 + \\sigma _ { \\rm j , i } ^ 2 + \\delta ) \\right \\} \\right ) > 1 - \\frac { \\epsilon } { 2 } . \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} \\left [ \\sum _ { i = 1 } ^ n \\theta _ 0 ^ i ( \\delta ^ i ) ^ { t + \\tau } \\right ] \\left [ \\sum _ { j = 1 } ^ n \\theta _ 0 ^ j ( \\delta ^ j ) ^ { t + \\Delta { t } + \\tau + \\Delta { \\tau } } \\right ] - \\left [ \\sum _ { i = 1 } ^ n \\theta _ 0 ^ i ( \\delta ^ i ) ^ { t + \\tau + \\Delta { \\tau } } \\right ] \\left [ \\sum _ { j = 1 } ^ n \\theta _ 0 ^ j ( \\delta ^ j ) ^ { t + \\Delta { t } + \\tau } \\right ] . \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} R _ 1 = \\frac { 1 } { 2 } \\log \\frac { | \\sigma ^ { 2 } { \\bf I } _ 2 + \\sum _ { j = 1 } ^ { 3 } { \\bf G } _ { 1 j } { \\bf Q } _ j { \\bf G } _ { 1 j } ^ { T } | } { | \\sigma ^ { 2 } { \\bf I } _ 2 + \\sum _ { j = 2 } ^ { 3 } { \\bf G } _ { 1 j } { \\bf Q } _ j { \\bf G } _ { 1 j } ^ { T } | } , \\\\ R _ 2 = \\frac { 1 } { 2 } \\log \\frac { | \\sigma ^ { 2 } { \\bf I } _ 2 + \\sum _ { j = 2 } ^ { 3 } { \\bf G } _ { 1 j } { \\bf Q } _ j { \\bf G } _ { 1 j } ^ { T } | } { | \\sigma ^ { 2 } { \\bf I } _ 2 + { \\bf G } _ { 1 3 } { \\bf Q } _ 3 { \\bf G } _ { 1 3 } ^ { T } | } , \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} \\mathcal { B } _ 1 Y = ( 1 , 0 ) \\partial _ x Y , \\mathcal { B } _ 2 Y = ( 0 , 1 ) \\partial _ x Y , \\mathcal { B } _ 3 Y = ( 1 , 0 ) \\partial _ x ^ 3 Y , \\mathcal { B } _ 4 Y = ( 0 , 1 ) \\partial _ x ^ 3 Y \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} L _ { m , \\mathsf { k } } ( \\theta _ \\mathsf { k } ) = \\mathrm { R } ( \\theta _ \\mathsf { k } ) \\begin{bmatrix} l _ { s f , \\mathsf { k } } & l _ { s d , \\mathsf { k } } & 0 \\\\ 0 & 0 & - l _ { s q , \\mathsf { k } } \\end{bmatrix} . \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} d e ^ { i \\theta _ 0 } : T _ x ^ { 1 , 0 } X \\rightarrow T ^ { 1 , 0 } _ { e ^ { i \\theta _ 0 } \\circ x } X , \\\\ d e ^ { i \\theta _ 0 } : T _ x ^ { 0 , 1 } X \\rightarrow T ^ { 0 , 1 } _ { e ^ { i \\theta _ 0 } \\circ x } X , \\\\ d e ^ { i \\theta _ 0 } ( T ( x ) ) = T ( e ^ { i \\theta _ 0 } \\circ x ) . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{align*} \\overline { l } = \\begin{cases} r & \\mbox { i f } ~ l ~ ( { \\rm m o d } ~ r ) = 0 , \\\\ l ~ ( { \\rm m o d } ~ r ) & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "8935.png", "formula": "\\begin{align*} \\mathrm { R e } \\left ( \\sigma \\upsilon \\right ) = - \\rho _ { \\mathcal { J } } \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} \\omega _ Q = d q ^ i \\wedge d p _ i \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} \\gamma ' ( d ) = \\chi ( w ) \\gamma ^ * ( d / | D | ) . \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} h _ { \\sigma _ i } ( x _ j ) = \\begin{cases} x _ i x _ { i + 1 } x _ i ^ { - 1 } & \\mbox { i f } j = i , \\\\ x _ i & \\mbox { i f } j = i + 1 , \\\\ x _ j & \\mbox { o t h e r w i s e . } \\\\ \\end{cases} \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} \\mathbf { y } = \\sqrt { \\beta _ 1 } \\mathbf { G } \\left ( \\left [ \\begin{matrix} 1 & x _ 1 \\end{matrix} \\right ] ^ T \\otimes \\mathbf { x } _ 0 \\right ) + \\mathbf { z } . \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} \\tilde { R } ^ \\mathrm { E n c - D e c - C S I } _ \\mathsf { R L N } = \\max _ { p _ X } \\min \\Big \\{ H ( S ) , I ( X ; Y ) \\Big \\} . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} M _ h ( x ) = \\sum _ { \\substack { q < y \\\\ ( q , h ) = 1 } } \\xi _ q \\sum _ { \\substack { d < x ^ { 1 / 3 } \\\\ ( d , h ) = 1 } } \\frac { \\gamma ( d ) } { \\varphi ( [ d , q ] ) } \\sum _ { \\substack { 0 < n \\le x \\\\ ( n , h d q ) = 1 } } \\theta ( n ) a ( n ) \\ , \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} ( R + \\omega _ 0 \\mathcal { J } _ g L ( \\theta ) ) i = \\mathcal { I } _ v ^ { \\top } v + \\mathcal { I } _ f v _ f . \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} J ( k , \\bar { \\theta } ) = \\sup _ { y \\in \\Gamma ( k ) } \\Big \\{ U ( f ( k ) - y , \\bar { \\theta } ) + \\delta ^ 1 J ( y , \\bar { \\theta } ) \\Big \\} . \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} \\pi _ { 0 a ( n - a ) } = \\frac { n } { n - 2 } \\left ( 1 - \\rho ^ n - \\rho ^ { n - 1 } / 2 \\right ) \\geq 0 \\end{align*}"}
-{"id": "285.png", "formula": "\\begin{align*} \\widehat { H } = \\underset { \\underset { i } { \\leftarrow } } \\lim \\ H / p ^ i H . \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} \\phi ( - 3 ; - 7 ) = ( - 1 ; - 7 ) , \\phi ( - 3 ; - 5 ) & = ( 5 ; - 5 ) , \\phi ( - 3 ; - 3 ) = ( 7 ; - 3 ) , \\\\ \\phi ( 0 ; - 4 ) & = ( - 4 ; - 4 ) , \\\\ \\phi ( 4 ; - 2 ) & = ( 9 ; - 1 ) . \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { d } \\left ( a + \\eta \\left ( x \\xi + x \\xi ^ { 2 } + x \\xi ^ { 3 } \\right ) \\right ) , \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} f ( z ) = 1 + \\sum _ { n = 1 } ^ { \\infty } c _ 1 z ^ n \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} B ^ { - 1 } \\mathcal { L } _ 1 B = \\Pi _ S ^ { \\perp } [ \\rho \\ , \\omega \\cdot \\partial _ { \\vartheta } + ( B ^ { - 1 } b _ 3 ) \\partial _ { y y y } + ( B ^ { - 1 } b _ 1 ) \\partial _ y + ( B ^ { - 1 } b _ 0 ) ] \\Pi _ S ^ { \\perp } + B ^ { - 1 } \\mathfrak { R } _ 1 B . \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} & \\left \\| \\nabla ^ { k } _ { x } \\left ( K _ { 1 } ( t ) g - \\mathcal { F } ^ { - 1 } \\left [ e ^ { - \\frac { \\nu t | \\xi | ^ { 2 \\sigma } } { 2 } } \\cos ( t | \\xi | ) \\right ] \\ast g \\right ) \\right \\| _ { 2 } \\\\ & \\le C ( 1 + t ) ^ { - \\frac { n } { 4 \\sigma } - \\frac { k } { 2 \\sigma } - 2 + \\frac { 1 } { 2 \\sigma } } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla ^ { k } _ { x } g \\| _ { 2 } , \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} 2 g ( ( \\nabla _ a J ) b , c ) = g ( N ( b , c ) , J ( a ) ) , \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} D = \\frac { 2 } { r } E , \\ ; \\ , D ' = E '' + \\frac { m - 1 } { r } E ' - \\frac { \\lambda + m - 1 } { r ^ 2 } E . \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} N = M _ 0 \\supset M _ 1 \\supset M _ 2 \\supset \\cdots \\supset M _ { d - 1 } \\supset M _ d = 0 , \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} c _ 4 = - 1 2 ^ 2 b , c _ 6 = - 1 2 ^ 3 a , \\Delta = - 1 2 ^ 3 c ^ p . \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{gather*} \\delta _ Q d + d \\delta _ Q = 0 , \\delta _ Q \\delta + \\delta \\delta _ Q = 0 . \\end{gather*}"}
-{"id": "5020.png", "formula": "\\begin{align*} \\Pi _ \\hbar ^ \\perp = \\mathsf { I d } - \\Pi _ { \\hbar } \\ , . \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} & \\frac { 1 } { \\epsilon } ( a \\ , \\sharp ^ { \\epsilon } \\ , b \\ , - \\ , a \\ , \\sharp ^ { 0 } \\ , b ) ( X ) \\\\ & = \\ , - \\frac { 4 i } { ( 2 \\pi ) ^ { 4 } \\epsilon } \\int _ { \\Xi \\times \\Xi } e ^ { - 2 i \\sigma ( Y , Z ) } \\left ( \\int _ 0 ^ 1 e ^ { - 4 i t F _ { \\epsilon } ( x , y , z ) } d t \\right ) F _ \\epsilon ( x , y , z ) \\ , a ( X - Y ) \\ , b ( X - Z ) \\ , d Y d Z \\ , . \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} \\lim _ { \\substack { m \\to \\infty \\\\ \\frac { \\omega _ { m } } { m ^ { \\alpha } } \\to c } } \\frac { d _ { n , k } ^ { m , \\omega _ { m } } } { d ^ { m , \\omega _ { m } } } = V ^ { 3 I B P , \\alpha , \\theta } _ { n , k } \\Big ( \\frac { c \\alpha \\Gamma ( \\alpha + \\theta ) } { \\Gamma ( \\theta + 1 ) } \\Big ) . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} \\omega _ \\delta ^ n = c _ \\delta e ^ { ( 1 + \\delta ) F } i ^ { n ^ 2 } \\Omega \\wedge \\bar \\Omega . \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} \\mathcal { R } _ H = \\frac { \\partial } { \\partial t } + \\sum _ { i = 1 } ^ n \\frac { \\partial H } { \\partial p _ i } \\frac { \\partial } { \\partial q ^ i } - \\sum _ { i = 1 } ^ n \\frac { \\partial H } { \\partial q ^ i } \\frac { \\partial } { \\partial p _ i } . \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} e _ { r p } ^ p ( \\underline { x } , \\underline { x } ^ \\prime ) = \\sum _ { j = 0 } ^ { \\mathrm { m i n } ( r , a , b ) } e _ { j } ^ p ( \\underline { x } ) \\cdot e ^ p _ { r p - j } ( \\underline { x } ^ \\prime ) , \\end{align*}"}
-{"id": "3529.png", "formula": "\\begin{align*} F ( z ) = \\sum _ { k = 1 } ^ { N } \\mu _ k \\frac { 1 + \\eta ^ k z } { 1 - \\eta ^ k z } \\equiv 1 + \\sum _ { n = 1 } ^ { \\infty } C _ n z ^ n , \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} \\widetilde { R } _ e ^ { * } = \\frac { 1 } { { \\ln 2 } } \\int _ 0 ^ \\infty { \\frac { { \\left ( { 1 - \\widetilde { \\mathcal { P } } _ 1 \\left ( { x } \\right ) \\widetilde { \\mathcal { P } } _ 2 \\left ( { x } \\right ) } \\right ) } } { { 1 + x } } d x } , \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} Q ( t ) = Q ( 0 ) + N _ 1 ( t ) - N _ { - 1 } ( t ) - G _ 1 ( t ) + G _ { - 1 } ( t ) . \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} \\begin{aligned} \\overline G _ i ( h , q ) - G ( 0 , 0 ) + \\gamma & \\geq - \\ , \\frac { C _ 2 } { T _ i ( h ) } \\int _ 0 ^ { T _ i ( h ) } ( 1 + | q | ) | D H ( X ( t , x ) ) | \\ , d t \\\\ & = - ( 1 + | q | ) \\frac { C _ 2 L _ i ( h ) } { T _ i ( h ) } \\geq - \\ , \\frac { C _ 2 L _ i ( h ) } { T _ i ( h ) } - \\gamma C _ 2 L _ i ( h ) . \\end{aligned} \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} V _ h ( x ) = M _ h ( x ) + O \\bigl ( | h | ( \\log x ) ^ { 2 0 } + x ^ { 1 - \\delta } \\bigr ) \\ , \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} | b _ i [ \\psi ] | \\leq C a [ \\psi ] ^ p \\| \\psi \\| ^ { 2 ( 1 - p ) } , i = 1 , 2 , p = \\frac 1 2 , \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} R : = ( \\langle x _ i , L ^ d x _ j \\rangle ) _ { 1 \\le i , j \\le m } \\ ; Q : = ( \\langle p _ i , L ^ d p _ j \\rangle ) _ { 1 \\le i , j \\le n } \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} \\frac { q ^ 2 + 1 } { q } + \\frac { 1 } { 2 } = \\frac { 2 q ^ 2 + q + 2 } { 2 q } \\in S _ q \\setminus \\langle Y \\rangle , \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} \\big ( ( A + U V ^ { \\ast } ) \\widehat { A } \\big ) ^ { \\ast } & = \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ^ { - 1 } \\big ( A A ^ { \\dagger } + ( U E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } ) ^ { \\ast } \\big ) , \\\\ \\big ( \\widehat { A } ( A + U V ^ { \\ast } ) \\big ) ^ { \\ast } & = \\big ( A ^ { \\dagger } A + ( A ^ { \\dagger } U F _ { S _ { A } } V ^ { \\ast } ) ^ { \\ast } \\big ) \\big ( I + A ^ { \\dagger } U F _ { S _ { A } } U ^ { \\ast } ( A ^ { \\dagger } ) ^ { \\ast } \\big ) ^ { - 1 } . \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} \\lbrace r _ { i + 1 } \\looparrowleft r _ i + N ( 0 , 1 ) \\rbrace _ { i = 2 , \\ldots , N - 1 } \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} \\tilde { y } = [ a ] \\cdot T ^ e \\cdot \\prod _ { ( i , p ) = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\infty } ( 1 - [ a _ { i j } ] T ^ i ) ^ { p ^ j } \\in W ( k ) [ [ T ] ] . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} \\small \\ ! F \\ ! = \\ ! \\sum \\limits _ { i \\in \\mathcal { U } } \\ ! \\sum \\limits _ { n _ { i k } \\in \\mathcal { N } ' } \\ ! \\ ! { { \\ ! \\ ! p _ { k i n _ { i k } } } } E _ { k , i } ( \\theta _ { i , n _ { i k } , k } ^ S ) + \\ ! \\ ! \\sum \\limits _ { i \\in \\mathcal { U } } \\sum \\limits _ { n _ { i k } \\in { { \\mathcal { C } ' _ i } } } \\ ! \\ ! { { \\ ! p _ { k i n _ { i k } } } } E _ { k , i } ( \\theta _ { i , n _ { i k } , k } ^ G ) . \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow 0 } \\frac { { } _ { 1 } F _ 1 ( \\alpha , \\beta ; z ) } { \\Gamma ( \\beta ) } = 1 . \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} { n - \\ell \\choose m - \\ell } _ q = \\prod _ { i = 1 } ^ { m - \\ell } \\frac { q ^ { n - \\ell - i + 1 } - 1 } { q ^ i - 1 } \\equiv 1 \\mod q . \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\pi ^ { ( t ) } ( x , y ) : = \\left | \\{ e \\in E _ t : \\ , e = ( x , y ) \\} \\right | \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} u = \\pi ^ 4 , r = 1 + \\pi ^ 4 , s = 1 + \\pi ^ 2 , t = \\pi ^ 4 \\mu . \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} a ( x , y , z ) = \\frac { q ^ 3 ( q - 1 ) } { 3 ^ 2 q ^ 2 } \\left ( \\sum \\limits _ { \\theta \\in \\mathfrak { A } } \\frac { \\theta ( x ) \\theta ( y ) \\theta ( z ^ { - 1 } ) } { \\theta ( 1 ) } + q - 1 + \\frac { \\lambda ( z ^ { - 1 } ) } { q - 1 } \\right ) . \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\lfloor N / \\ell \\rfloor } M _ 1 \\left ( \\frac { 1 } { s _ \\theta } \\right ) = \\lfloor N / \\ell \\rfloor \\frac { \\ell } { N } \\leq 1 . \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} q = \\frac { 2 } { \\sqrt { 2 c _ 2 - c _ 1 ^ 2 } } \\tan ^ { - 1 } { \\left ( \\frac { \\gamma + c _ 1 } { \\sqrt { 2 c _ 2 - c _ 1 ^ 2 } } \\right ) } - \\ln { \\left ( \\frac { 1 } { 2 } \\gamma ^ 2 + c _ 1 \\gamma + c _ 2 \\right ) } \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} D ^ { n + 1 } ( a ) = & q D ^ n ( a b ^ 2 ) \\\\ = & q \\sum _ { k = 0 } ^ n \\binom { n } { k } D ^ k ( a ) D ^ { n - k } ( b ^ 2 ) \\\\ = & q b ^ 2 D ^ n ( a ) + q \\sum _ { k = 0 } ^ { n - 1 } \\binom { n } { k } D ^ k ( a ) 2 ^ { n - k } d ^ { 2 n - 2 k } c ^ 2 A _ { n - k } \\left ( \\frac { c ^ 2 } { d ^ 2 } \\right ) . \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} ( a \\star b ) ^ * = b ^ * \\star a ^ * \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} { ( a + b ) p \\brack a p } _ { \\mathbb { O } _ p } = q ^ { p a b } { a + b \\choose a } = \\rho ( { a + b \\brack a } _ v ) \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} U & = \\{ x \\in \\Omega \\mid p ( x ) \\geq 2 \\} , V = \\{ x \\in \\Omega \\mid 1 < p ( x ) < 2 \\} , \\\\ V _ { n } ^ { - } & = \\{ x \\in V \\mid \\left \\vert \\nabla u _ { n } \\right \\vert + \\left \\vert \\nabla u \\right \\vert < 1 \\} , V _ { n } ^ { + } = \\{ x \\in V \\mid \\left \\vert \\nabla u _ { n } \\right \\vert + \\left \\vert \\nabla u \\right \\vert \\geq 1 \\} , \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} \\frac { f ( y _ { j } ) - f ( x _ { j } ) } { y _ { j } - x _ { j } } = f ' ( t _ { j } ) = g ( t _ { j } ) \\frac { f ( y _ { j + 1 } ) - f ( x _ { j + 1 } ) } { y _ { j + 1 } - x _ { j + 1 } } = f ' ( t _ { j + 1 } ) = g ( t _ { j + 1 } ) . \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} I = \\left [ \\begin{array} { c } \\mu \\\\ 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} \\delta : = h + d : \\psi ( E ) \\longrightarrow \\psi ( E ) . \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} \\{ H , M \\} = 0 , M ( u ) = \\int _ { \\mathbb { T } } u ^ 2 \\ , d x . \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} B ( z ) ^ k \\psi _ { j _ 0 } = \\sum _ { j _ 1 , \\dots , j _ k } \\left ( \\prod _ { l = 1 } ^ k \\langle B ( z ) \\psi _ { j _ { l - 1 } } , \\psi _ { j _ l } \\rangle \\right ) \\psi _ { j _ k } \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} F ( \\phi ) = \\int _ { M } ( \\phi - \\phi _ 0 ) d \\mu + \\int _ { M ^ * } ( \\phi ^ * - \\phi _ 0 ^ * ) d \\nu \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} \\left | V _ i \\right | ^ 2 = f _ { V i } \\left ( V _ d , V _ q \\right ) : = V _ { d i } ^ 2 + V _ { q i } ^ 2 . \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\dot z = T _ \\rho ( g + \\Delta z ) H _ 0 ^ 1 ( \\Omega ) , I , \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} D _ { k } = \\{ z \\ ; : \\ ; \\xi \\leq | z | \\leq \\xi \\lambda ^ { k } \\} , \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} g _ \\beta ( \\lambda ) = \\frac 1 2 \\beta ^ 2 b ^ 2 \\phi _ \\beta ( \\lambda ) ^ { 2 b - 2 } + \\beta \\phi _ \\beta ( \\lambda ) ^ b , \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ n } = \\sum _ { i = 1 } ^ k c _ i \\frac { 1 } { p _ i } . \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { v _ \\ell ( \\Delta _ m ) / v _ \\ell ( \\Delta _ m ' ) } { p } \\right ) = 1 . \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} \\dim W _ i ' = \\sum _ { j \\rightarrow i } \\dim W _ j - \\dim W _ i , \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} \\sum _ { i a } s _ a F ( \\lambda | \\lambda _ { i a } ) F ( \\lambda ' | \\lambda _ { i a } ) = \\sum _ { i a } s _ a F ( \\lambda _ { i a } | \\lambda ) F ( \\lambda _ { i a } | \\lambda ' ) . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} \\nu _ 1 ^ { ( a ) } & = x _ 1 ^ { ( a + 1 ) } + \\cdots + x _ 1 ^ { ( n ) } , \\\\ J _ 1 ^ { ( a ) } & = - x _ 1 ^ { ( a + 1 ) } . \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} \\rho ( \\lambda ) = \\rho _ { \\infty } ( \\lambda ) + \\frac { \\rho _ { \\mathrm { i m p } } ( \\lambda | \\lambda _ p ) } { L } + \\sum _ { i a } \\frac { \\rho _ { i a } ( \\lambda ) } { 2 4 L ^ 2 \\rho _ { \\infty } ( \\lambda _ { i a } ) } \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} z + \\sum _ { n = 2 } ^ { \\infty } n a _ n z ^ n = \\left ( z + \\sum _ { n = 2 } ^ { \\infty } b _ n z ^ n \\right ) \\left ( 1 + \\sum _ { n = 1 } ^ { \\infty } c _ n z ^ n \\right ) . \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} T _ r = \\frac { 2 ( r - 1 ) } { 2 ( r - 1 ) - r + r \\sqrt { 1 - 4 x ( r - 1 ) } } . \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\eta _ { \\Lambda } ( b _ { { 0 } } ^ { * } ) \\eta _ { \\Lambda } ( b _ { { 0 } } ) ) = | \\zeta _ { m a x } ( \\lambda ) | ^ { 2 } \\ . \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} K ( z ) ^ 2 = ( z - A ) ^ { - 1 } , z \\in \\rho ( A ) . \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} F \\big | _ { \\{ [ \\alpha , \\beta ] ; \\mu \\} , \\gamma } ( z ) = \\zeta _ { \\gamma , z } \\cdot F ( z ) , \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\pm \\frac { r x } { 1 - t } + \\frac { ( t \\mp 2 ) r ^ 2 } { 2 ( 1 - t ) } = 2 \\tau r ^ 2 \\pm \\sqrt 2 r u . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} 0 & = \\theta _ t ^ i u _ i ' ( x _ t ^ i ) - \\lambda _ t , & t = 0 , 1 , \\ldots , \\\\ 0 & = \\theta _ t ^ i \\delta ^ i - \\mu _ t \\theta _ { t + 1 } ^ i , & t = 0 , 1 , \\ldots , \\\\ 0 & = z _ t ^ i - V _ { \\theta ^ i } ( k _ { t + 1 } , \\theta _ { t + 1 } ) , & t = 0 , 1 , \\ldots , \\\\ 0 & = - \\lambda _ t + \\mu _ t V _ k ( k _ { t + 1 } , \\theta _ { t + 1 } ) , & t = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} q _ 0 : = \\sup \\{ q \\geq 2 : \\} \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} \\hat { H } = \\frac { 1 } { 4 } k \\partial _ k ^ 2 - k ^ 2 \\partial _ k + k . \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} r _ 1 e ^ { i \\phi / 2 } + r _ 2 e ^ { - i \\phi / 2 } = ( - 1 ) ^ k \\rho _ 1 e ^ { i A } . \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} \\lambda _ n ( \\epsilon ) & \\to \\left ( h ^ { - 1 } ( [ h _ { \\min } , h _ { \\min } + \\epsilon ] ) \\right ) n \\to \\infty , \\\\ [ 1 . 5 e x ] \\left ( h ^ { - 1 } ( [ h _ { \\min } , h _ { \\min } + \\epsilon ] ) \\right ) & \\to \\left ( h ^ { - 1 } ( \\{ h _ { \\min } \\} ) \\right ) = \\sum _ { l = 1 } ^ q ( b _ l - a _ l ) > 0 \\epsilon \\to 0 . \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{gather*} \\alpha _ { t } = \\alpha \\left ( \\frac { 2 } { 3 } \\alpha + \\frac { u _ t } { u } \\frac { 2 - q _ 2 } { 3 } \\right ) - \\frac { t } { 6 } ( 1 + q _ 2 ) - \\frac { u ^ 2 } { 3 } ( 3 + q _ 2 ) . \\end{gather*}"}
-{"id": "9481.png", "formula": "\\begin{align*} v \\left ( \\frac { P ( a _ { \\rho } ) } { Q ( a _ { \\rho } ) } - \\frac { P ( a ) } { Q ( a ) } \\right ) \\ & = \\ v \\left ( \\frac { P ( a _ { \\rho } ) Q ( a ) - P ( a ) Q ( a _ { \\rho } ) } { Q ( a _ { \\rho } ) Q ( a ) } \\right ) \\\\ & = \\ v ( P ( a _ { \\rho } ) Q ( a ) - P ( a ) Q ( a _ { \\rho } ) ) - v ( Q ( a _ { \\rho } ) ) - v ( Q ( a ) ) . \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} \\begin{aligned} H ( \\rho _ 0 | M _ { \\Omega _ { \\rho _ 0 } } ) & \\ge H ( \\rho _ 0 | M _ { \\Omega _ { \\rho _ 0 } } ) - H ( \\rho _ t | M _ { \\Omega _ { \\rho _ t } } ) = - \\int _ { 0 } ^ { t } \\frac { d } { d s } H ( \\rho _ s | M _ { \\Omega _ { \\rho _ s } } ) d s \\\\ & = \\int _ { 0 } ^ { t } I ( \\rho _ s | M _ { \\Omega _ { \\rho _ s } } ) d s \\ge t \\min _ { s \\in [ 0 , t ] } I ( \\rho _ s | M _ { \\Omega _ { \\rho _ s } } ) . \\end{aligned} \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} \\begin{aligned} I _ 1 ( y ) - I _ 1 ( \\bar { y } ) & = \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\partial _ y \\partial _ u r ( Y _ q + s t ( y - \\bar { y } ) + t \\bar { y } ) u V _ q t ( y - \\bar { y } ) \\dd s \\dd t \\\\ & + \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\partial _ y \\partial _ u r ( Y _ q + s t \\bar { y } ) ( u - \\bar { u } ) V _ q t \\bar { y } \\dd s \\dd t . \\end{aligned} \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| u ( t ) - b _ 0 u _ 1 \\| _ { L ^ 2 \\cap L ^ { \\infty } } = 0 . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} h ( k , \\ell ) = \\begin{cases} f ^ \\ast ( k ) & \\mbox { i f } k > 1 \\ell = 1 \\\\ 2 f ^ \\ast ( 1 ) - 1 & \\mbox { i f } k = 1 \\ell = 1 \\\\ f ^ \\ast ( \\ell ) & \\mbox { i f } k = 1 \\ell > 1 \\\\ 0 & \\mbox { e l s e . } \\end{cases} \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{align*} \\chi _ \\alpha ( \\rho ) = ( \\pi ^ { } _ ! \\circ \\tau ^ \\ast ) ( \\alpha ) \\in H ^ * ( G ) \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} P ( T _ 1 < x ) = P ( T _ 2 < x ) = \\frac 2 \\pi \\arcsin x \\asymp \\sqrt { x } , \\ x \\to 0 \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} P _ S ( s ) = \\begin{cases} 1 & \\textnormal { i f } s = s ^ \\star , \\\\ 0 & \\textnormal { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} \\Big ( \\Gamma \\big ( 1 + \\frac { 1 } { 2 g ( { \\bf R } ) } \\big ) \\Big ) ^ { - g ( { \\bf R } ) } \\geq \\Big ( 1 + \\frac { 1 } { 2 g ( { \\bf R } ) } \\Big ) ^ { - \\frac { 1 } { 2 } } = \\Big ( \\frac { 2 d g ( { \\bf R } ) } { d + 2 d g ( { \\bf R } ) } \\Big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} e ( p e r m ( C _ \\alpha ) ) = c ( s ( \\alpha ) ) \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} P _ q ^ { ( n ) } f _ 1 \\otimes \\dots \\otimes f _ n : = \\sum _ { \\pi \\in S _ n } q ^ { \\operatorname { i n v } ( \\pi ) } f _ { \\pi ( 1 ) } \\otimes \\dots \\otimes f _ { \\pi ( n ) } \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} I _ 2 & \\ge t _ 0 ( t - s ) ^ { 2 H } \\int _ { | y | \\ge 1 } \\frac { 1 - \\cos ( | y | ) } { | y | ^ { 2 H + 1 } } d y - \\frac { ( t - s ) ^ { 2 H + 1 } } 2 \\int _ { | y | \\ge 1 } \\frac 1 { | y | ^ { 2 H + 1 } } d y \\\\ & - \\frac { 3 ( t - s ) ^ { 2 H + 1 } } 4 \\int _ { | y | \\ge 1 } \\frac 1 { | y | ^ { 2 + 2 H } } d y \\\\ & = C _ 1 ( t - s ) ^ { 2 H } - \\frac { 5 H + 1 } { 2 H ( 2 H + 1 ) } ( t - s ) ^ { 2 H + 1 } , \\end{align*}"}
-{"id": "2533.png", "formula": "\\begin{align*} \\mathbf { X } ^ { ( g ) } \\left [ \\mathbf { I } _ { K _ g } \\otimes \\mathbf { E } _ { L _ g , l } \\right ] = \\sum _ { \\left \\{ m \\ ; \\vert \\beta _ m ^ l > 0 \\right \\} } \\left ( \\beta _ m ^ l \\right ) ^ { 1 / 2 } \\boldsymbol { \\psi } _ m ^ l \\left [ \\boldsymbol { \\phi } _ m ^ l \\right ] ^ H \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} T ^ { - 1 } = \\begin{pmatrix} d & q ^ { - 1 / 2 } \\beta & - q ^ { - 1 } b \\\\ - q ^ { 1 / 2 } \\delta & e & q ^ { - 1 / 2 } \\alpha \\\\ - q c & - q ^ { 1 / 2 } \\gamma & a \\end{pmatrix} . \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} h _ { T ' } = \\frac { h _ L [ H ^ 1 _ T ( G , O _ L ^ { * } ) ] } { h _ K \\prod _ { v } e _ v ( L / K ) } . \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{align*} \\| G _ { \\rho } f \\| _ { L ^ { 2 } ( 0 , T ; H ^ { k } ( \\Omega ) ) } \\leq \\frac { C } { \\sigma ^ { 2 - k } } \\| f \\| _ { L ^ { 2 } ( 0 , T ; H ^ { 1 } ( \\Omega ) ) } , \\ , \\ , \\ , \\ , \\ , \\ , k = 1 , \\ , 2 . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} \\frac { a ^ 2 - 4 b } { b } = \\frac { a ^ 2 } { b } - 4 \\equiv - 4 \\pmod { \\pi } . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } X ^ { \\prime \\prime } ( x ) + \\mu X ( x ) = 0 , 0 \\leq x \\leq 1 , \\\\ X ( 0 ) = 0 \\\\ ( a \\mu - b ) X ( 1 ) = d X ^ { ^ { \\prime } } ( 1 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{align*} S _ { n } ^ { z } = \\frac { 2 p } { i \\pi p ^ { \\prime } } \\log v _ { n } - ( p + 1 ) \\in \\{ - 2 s , - 2 ( s - 1 ) , . . . , 2 s \\} 2 p \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} N _ { ( 2 , 7 ) } ( n ) & = ~ 8 \\sigma ( \\frac { n } { 2 } ) - 3 2 \\sigma ( \\frac { n } { 8 } ) + 8 \\sigma ( \\frac { n } { 7 } ) - 3 2 \\sigma ( \\frac { n } { 2 8 } ) + 6 4 \\ , W _ { ( 2 , 7 ) } ( n ) + 1 0 2 4 \\ , W _ { ( 2 , 7 ) } ( \\frac { n } { 4 } ) \\\\ & - 2 5 6 \\ , \\biggl ( W _ { ( 7 , 8 ) } ( n ) + W _ { ( 1 , 1 4 ) } ( \\frac { n } { 2 } ) \\biggr ) . \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} A = \\epsilon \\left ( \\frac { m ( m + 1 ) } { 2 } + \\frac { n ( n + 1 ) } { 2 } \\right ) . \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} \\| \\hat { p } ^ { ( k ) } ( \\cdot | Z ^ n ) - \\mu _ k ^ { ( b ) } \\| _ 1 & \\leq { t \\over n - k } \\sum _ { i = 1 } ^ { k + g } \\| \\hat { p } ^ { ( 1 ) } ( \\cdot | S ^ { ( i ) , t } ) - \\mu _ k ^ { ( b ) } ( \\cdot ) \\| _ 1 + | { t ( k + g ) \\over n - k } - 1 | + { k + g \\over n - k } . \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} h _ T = \\frac { h _ L [ L _ 0 : K ] [ H ^ 0 _ T ( G , O _ L ^ { * } ) ] } { h _ K [ \\ker ( H ^ 0 _ T ( G , L ^ { * } ) \\to H ^ 0 _ T ( G , I _ L ) ) ] \\prod _ { v } [ H ^ 0 _ T ( D _ w , O _ w ^ { * } ) ] } . \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} \\iota _ B : \\mathcal { B } \\stackrel { \\sim } { \\longrightarrow } B : = \\left \\{ \\frac { n _ B - 1 } { 2 } , \\frac { n _ B - 3 } { 2 } , \\dots , - \\frac { n _ B - 1 } { 2 } \\right \\} \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} ( X + Y ) ^ { s _ b ( n ) } = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } _ b X ^ { s _ b ( k ) } Y ^ { s _ b ( n - k ) } \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} & \\frac { 1 } { k ! } \\int _ 0 ^ T \\ ( \\ ( \\log q ( | t | + 2 ) \\ ) ^ k + \\sum _ { m = 0 } ^ { k - 2 } \\frac { k ! } { m ! } \\ ( \\log q ( | t | + 2 ) \\ ) ^ { m } C ^ { k - m } \\ ) \\frac { x ^ { 1 - \\frac { c _ 1 } { \\log q T } } } { | t | + 2 } d t \\\\ & \\ll \\frac { 1 } { k ! } ( C k \\log q T ) ^ k x ^ { 1 - \\frac { c _ 1 } { \\log T } } . \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} C C ( { \\cal F } \\boxtimes ^ L _ \\Lambda { \\cal G } ) = C C ( { \\cal F } ) \\boxtimes C C ( { \\cal G } ) . \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} y _ i ^ * = \\frac { y _ i - y _ { m i n } } { y _ { m a x } - y _ { m i n } } \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} g . k = \\tau ( g ) k \\textrm { f o r $ g \\in G $ a n d $ k \\in K $ } . \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} D _ { 1 } ( \\sum _ { k = 1 } ^ { n } | \\alpha _ { k } | ^ { p } ) ^ { \\frac { 1 } { p } } & \\leq \\| \\sum _ { k = 1 } ^ { n } \\alpha _ { k } ( x _ { 2 k - 1 } - x _ { 2 k } ) \\| \\\\ & \\leq \\| S \\| \\cdot \\| \\sum _ { k = 1 } ^ { n } \\alpha _ { k } ( z _ { 2 k - 1 } - z _ { 2 k } ) \\| \\\\ & \\leq \\| S \\| \\cdot C _ { 2 } \\cdot 2 ^ { \\frac { 1 } { q ^ { * } } } ( \\sum _ { k = 1 } ^ { n } | \\alpha _ { k } | ^ { q ^ { * } } ) ^ { \\frac { 1 } { q ^ { * } } } , \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} \\eta '' \\phi _ { \\mu } ^ { 2 } & = 1 L e b - a . e . \\\\ \\mu \\left ( \\left \\{ \\eta ( s ) - \\xi ( s ) = \\inf _ { t } \\left ( \\eta ( t ) - \\xi ( t ) \\right ) \\right \\} \\right ) & = 1 . \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} & \\| ( u _ k , v _ k ) \\| ^ p = \\frac { 2 ( \\alpha + \\beta - q ) } { p - q } \\int _ { \\Omega } | u _ { k } | ^ { \\alpha } | v _ { k } | ^ { \\beta } d x + o _ k ( 1 ) , \\\\ & \\| ( u _ k , v _ k ) \\| ^ p = \\frac { \\alpha + \\beta - q } { \\alpha + \\beta - p } \\int _ \\Omega ( \\lambda | u _ k | ^ q + \\mu | v _ k | ^ q ) d x + o _ k ( 1 ) . \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} \\nu _ { \\bar \\gamma } ( E ) = \\int _ { \\Phi ^ { - 1 } ( \\bar \\gamma ) } \\mu _ \\gamma ( \\varphi ^ { - 1 } ( E ) ) \\dd \\eta _ { \\bar \\gamma } ( \\gamma ) = 0 \\ ; . \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{gather*} \\delta _ X \\Phi = L _ \\xi \\Phi , \\Phi = ( A , C , A ^ \\ast , C ^ \\ast ) . \\end{gather*}"}
-{"id": "19.png", "formula": "\\begin{align*} A _ u = \\Big \\{ \\sup _ { 0 \\le i \\le u } \\pi ^ { ( i ) } ( S _ i ) \\le v ( s ) / 2 \\Big \\} . \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} \\ln T ( z ) \\approx - i \\sum _ { j = 0 } ^ { \\infty } H _ { j } ( 2 z ) ^ { - j - 1 } \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} \\hat D = c ^ m ( e _ m + A _ m ^ { a b } - \\gamma _ { i i } ^ m ) + i c ^ 4 \\Lambda + \\frac { k e ^ { - y } } { 8 \\ell } c ( e ^ 4 \\wedge d V o l _ { S ^ 2 } ) + O ( e ^ { - 2 y } ) , \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\mathbb { P } _ \\mu \\bigg ( D \\Big ( P ^ { ( \\mathbf { u , \\mathsf { B } _ n } ) } _ { \\mathbf { W } } \\Big | \\Big | p _ { W | U = \\mathbf { u } } ^ n \\Big ) > e ^ { - n \\delta _ 1 } \\bigg ) \\leq e ^ { - e ^ { n \\delta _ 2 } } . \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} D \\left ( F _ a ( z ) \\right ) = z \\frac { \\partial } { \\partial z } F _ a ( z ) + \\beta \\sum _ b z \\oint \\frac { d \\xi } { \\xi ^ 2 } \\frac { \\phi _ b ^ - ( \\xi ) } { 1 - \\frac { z } { \\xi } } V _ b ^ { - 1 } ( \\xi ) V _ b ( z ) F _ a ( \\xi ) . \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} \\langle A u , u \\rangle = \\frac { 1 } { 2 \\pi \\omega } \\ , \\mathrm { I m } \\langle w , u \\rangle \\ ; \\mbox { a n d } \\ ; \\langle E ^ \\eta _ \\pm u , u \\rangle = - \\mathrm { R e } \\langle w , u \\rangle . \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} \\mathbf { P } _ 2 \\tilde { \\mathbf { P } } = \\mathbf { P } _ 2 . \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} \\frac { ( c ^ 3 - 5 c + 5 ) \\zeta _ 2 ( c ) } { 3 c ^ 2 \\left ( c ^ 3 - 2 c - 1 0 \\right ) ^ 2 } = 0 \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} \\bar Z = P \\left \\{ ( C + \\bar { C } ) \\bar X + ( D + \\bar { D } ) \\bar v \\right \\} , \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} u _ n = \\left \\{ \\begin{array} { c c } 1 + \\max \\{ i p ^ { n - v ( c _ i ) - 1 } : ( i , p ) = 1 : v ( c _ i ) < n \\} & \\mathrm { i f \\ } \\exists i : v ( c _ { i } ) < n \\\\ 0 & \\mathrm { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} d = d ( \\{ A _ i \\} _ { i \\in \\Lambda } ) = \\inf \\bigg \\{ s : \\sum _ { k = 1 } ^ \\infty \\sum _ { \\underline { a } \\in \\Lambda ^ k } \\phi ^ s ( \\underline { a } ) < \\infty \\bigg \\} . \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} S _ k \\widetilde { u ^ { \\alpha } _ { \\bold { 0 } ; J _ 2 } } = \\widetilde { u ^ { \\alpha } _ { \\bold { 0 } ; J _ 2 + \\bold { 1 } _ k } } . \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} \\overline { \\omega } \\cdot l + j '^ 3 - j ^ 3 = \\overline { \\omega } \\cdot \\mathtt { l } ( j _ 1 ) + \\overline { \\omega } \\cdot \\mathtt { l } ( j _ 2 ) + j '^ 3 - j ^ 3 = j _ 1 ^ 3 + j _ 2 ^ 3 + j '^ 3 - j ^ 3 . \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{gather*} m _ 1 = u v _ 2 v _ 3 , m _ 2 = u v _ 1 v _ 3 , m _ 3 = u v _ 1 v _ 2 , \\\\ [ m _ 1 , m _ 2 ] = [ m _ 1 , m _ 3 ] = [ m _ 2 , m _ 3 ] = [ m _ 1 , m _ 2 , m _ 3 ] = u v _ 1 v _ 2 v _ 3 . \\end{gather*}"}
-{"id": "477.png", "formula": "\\begin{align*} u _ { J _ 1 ; J _ 2 } = D _ { J _ 1 } S _ { J _ 2 } u = S _ { J _ 2 } D _ { J _ 1 } u . \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} 0 \\rightarrow P _ j [ j - 1 ] \\rightarrow ( P _ j = P _ j ) \\rightarrow P _ j [ j ] \\rightarrow 0 , \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} F _ \\infty ( A ) = ( A _ \\infty \\cap A ' ) / \\operatorname { A n n } ( A , A _ \\infty ) . \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\abs { F _ 2 ( x ) ( t _ i ) - F _ 2 ( x ) ( t _ { i - 1 } ) } & \\leq \\int _ 0 ^ 1 \\sum _ { i = 1 } ^ n \\abs { k ( t _ i , s ) - k ( t _ { i - 1 } , s ) } \\phi ( s ) \\psi ( \\norm { x } _ { B V } ) \\textup d s \\\\ & \\leq \\psi ( \\norm { x } _ { B V } ) \\int _ 0 ^ 1 m ( s ) \\phi ( s ) \\textup d s . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} \\alpha [ x ] = \\int _ 0 ^ 1 x ( s ) \\textup d A ( s ) \\beta [ x ] = \\int _ 0 ^ 1 x ( s ) \\textup d B ( s ) . \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ 2 ( \\Omega ) } = 1 . \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} \\sum _ { ( u , v ) \\in [ r ] \\times [ s ] } c \\big ( ( x _ 1 , x _ 2 , \\ldots , x _ t ) ) ; ( u , v ) \\big ) = 0 . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} \\lambda = \\frac { 3 } { 2 } ( n _ 1 ^ 2 + n _ 2 ^ 2 ) + 3 ( n _ 1 + n _ 2 ) - \\frac { 3 } { 4 } k ^ 2 , \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{gather*} Y ( t ) = \\left ( I + \\sum _ { k = 1 } ^ { \\infty } Y _ k ( - t ) ^ { - 3 k / 2 } \\right ) \\begin{pmatrix} ( - t ) ^ { 1 / 4 } & 0 & 0 \\\\ 0 & e ^ { \\frac { 2 \\sqrt 2 } { 9 } ( - t ) ^ { 3 / 2 } } & 0 \\\\ 0 & 0 & e ^ { - \\frac { 2 \\sqrt 2 } { 9 } ( - t ) ^ { 3 / 2 } } \\end{pmatrix} , \\end{gather*}"}
-{"id": "6628.png", "formula": "\\begin{align*} a _ y R f _ * { \\cal F } = \\chi ( X _ { \\bar \\eta } , { \\cal F } ) - \\chi ( X _ { \\bar y } , { \\cal F } ) + { \\rm S w } _ y H ^ * ( X _ { \\bar \\eta } , { \\cal F } ) . \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} [ \\Lambda , \\Lambda ] = 2 Z \\wedge \\Lambda , \\mathcal { L } _ Z \\Lambda = 0 . \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} \\Upsilon = \\Psi = \\frac { 1 } { N } \\frac { 1 } { 1 - \\beta } \\sum _ { l = 1 } ^ K r _ { i i l } ^ \\alpha . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} { \\mathbb E } \\frac { 1 } { N } \\sum _ { j = 1 } ^ n \\delta ( z - z _ j ) \\to \\frac { \\Delta Q } { 4 \\pi } \\chi _ K \\end{align*}"}
-{"id": "5905.png", "formula": "\\begin{align*} \\Psi _ { \\Delta , r } ( z , f ) = \\prod _ { n = 1 } ^ { \\infty } \\prod _ { b ( \\Delta ) } [ 1 - e ( b / \\Delta ) e ( n z ) ] ^ { \\left ( \\frac { \\Delta } { b } \\right ) c _ { f } ^ { + } ( | \\Delta | n ^ { 2 } , r n ) } , \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} \\Gamma _ b & = \\{ ( c _ j ; b + ( j - w ) e ) \\mid j \\in [ 1 , \\ , r ] \\} \\cup \\{ ( b - 1 ; b - 1 - i e ) \\mid i \\in [ 0 , \\ , \\tilde { w } ) \\} ; \\\\ \\Delta _ b & = \\{ ( b ; b - i e ) \\mid i \\in [ 0 , \\ , w ) \\} \\cup \\{ ( c ' _ j ; b - 1 + ( j - \\tilde { w } ) e ) \\mid j \\in [ 1 , \\ , r - l ] \\} \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} \\mathbb { E } _ { P ^ { ( \\mathcal { C } _ n ) } } g _ { \\mathcal { C } _ n } ( M , \\hat { M } ) & = \\mathbb { P } _ { P ^ { ( \\mathcal { C } _ n ) } } \\big ( \\hat { M } \\neq M \\big ) \\\\ \\mathbb { E } _ { Q ^ { ( \\mathcal { C } _ n ) } } g _ { \\mathcal { C } _ n } ( M , \\hat { M } ) & = \\mathbb { P } _ { Q ^ { ( \\mathcal { C } _ n ) } } \\big ( \\hat { M } \\neq M \\big ) . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} \\widehat { c } = - c , \\widehat { \\xi } = - \\hat { \\xi } , \\widehat { M _ 1 } = - M _ 1 , \\widehat { M _ 2 } = - M _ 2 . \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} { \\bf T } _ { i j k } = & p _ { j k } { \\bf G } _ { i j } { \\bf v } _ { j k } { \\bf v } ^ { T } _ { j k } { \\bf G } ^ { T } _ { i j } , \\ \\forall \\{ i , j \\} \\in \\mathcal { L } , \\ \\forall k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} 1 - \\frac { r } { 4 ( r - 1 ) } = \\frac { 3 r - 4 } { 4 ( r - 1 ) } \\stackrel { r \\geq 2 } { \\geq } \\frac { 1 } { 2 ( r - 1 ) } . \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} \\operatorname { D i v } ^ { \\vartriangle } P = Q ^ { \\alpha } \\bold { E } _ { \\alpha } ^ { \\vartriangle } ( L _ n ) . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} \\bigcup _ { \\sigma \\in Z } \\left \\{ ( z _ v ) _ { v } \\in [ 0 , 1 ] ^ { Z _ 0 } ~ \\Bigl | ~ \\sum _ { v \\in \\sigma } z _ v = 1 \\ , , ~ ~ z _ v = 0 ~ ~ v \\in Z _ 0 \\setminus \\sigma \\right \\} . \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} p ( s ) = \\frac { \\lambda e ^ { - \\mu ( s - t ) } } { ( \\lambda + 1 ) e ^ { \\lambda ( T - s ) } - 1 } \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } F ( n , k ) = - \\delta _ { o d d } ( n ) , \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} { { \\overline { R } } _ \\mathrm { s } } = \\left [ { \\overline { R } } - { \\overline { R } } _ { e ^ { * } } \\right ] ^ + , \\end{align*}"}
-{"id": "8551.png", "formula": "\\begin{align*} e _ m ( c _ n ) = \\sum _ { ( \\mathbf { s } , \\mathbf { x } ) \\in \\mathcal { S } ^ n \\times \\mathcal { X } ^ n } p ^ n _ S ( \\mathbf { s } ) f _ n ( \\mathbf { x } | m , \\mathbf { s } ) \\sum _ { \\substack { ( \\mathbf { y } , \\mathbf { z } ) \\in \\mathcal { Y } ^ n \\times \\mathcal { Z } ^ n : \\\\ \\phi _ n ( \\mathbf { y } ) \\neq m } } p _ { Y , Z | X , S } ^ n ( \\mathbf { y } , \\mathbf { z } | \\mathbf { x } , \\mathbf { s } ) . \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} \\sup _ { M \\times [ 0 , T ) } | \\varphi ( x , t ) | & \\leq \\sup _ { M } | \\varphi _ { 0 } ( x ) | + T \\sup _ { M \\times [ 0 , T ) } \\left | \\frac { \\partial \\varphi } { \\partial t } ( x , t ) \\right | \\\\ & \\leq C ( T + 1 ) , \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} \\mathbb { E } [ Y ] = ( 1 + o ( 1 ) ) n \\frac { ( n p ) ^ { - r / ( 4 ( r - 1 ) ) } } { 4 ^ r r ! } \\stackrel { \\eqref { g a i n } } { \\geq } ( 1 + o ( 1 ) ) \\frac { ( n p ) ^ { 1 / ( 2 ( r - 1 ) ) } } { 4 ^ r r ! } p ^ { - 1 } . \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} G & = \\begin{bmatrix} ( Z - I ) A _ 1 ( Z - I ) ^ { - 1 } G _ 2 & G _ 1 & - ( Z - I ) A _ 1 ( Z - I ) ^ { - 1 } e _ 1 \\end{bmatrix} , \\\\ B & = \\begin{bmatrix} B _ 2 & ( Z - I ) A _ 2 ^ * ( Z - I ) ^ { - 1 } B _ 1 & ( Z - I ) A _ 2 ^ * ( Z - I ) ^ { - 1 } e _ 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} A ^ h ( \\xi , \\eta ) = - \\frac { i \\eta ( \\eta + J ( \\eta ) ) } { 2 J ( \\eta ) } \\left ( 1 + \\frac { ( \\eta - J ( \\eta ) ) ^ 2 } { 4 \\xi J ( \\eta ) } \\right ) + S ( d ^ 3 \\rho ^ { - 1 } ) , \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{align*} R _ n ( z ) : = \\Phi _ N ^ * ( z ) - z \\Phi _ N ( z ) A _ n ^ { ( N ) } ( z ) / B _ n ^ { ( N ) } ( z ) \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} \\left ( \\bold { D } _ A \\right ) ^ { \\ast } _ { \\alpha \\beta } = \\sum _ { J _ 1 , J _ 2 } \\left ( ( - D ) _ { J _ 1 } S _ { - J _ 2 } \\right ) \\cdot \\frac { \\partial A ^ { \\beta } } { \\partial u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } . \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} \\begin{aligned} | - b ( x ) \\cdot D \\varphi ( x ) + f ( x ) | & = | \\bar f ( x ) | \\\\ & \\leq \\left ( \\log \\frac { \\kappa } { r } - \\frac { \\log 2 } { 2 } \\right ) ^ { - 1 } \\int _ { \\tau _ - ^ \\delta ( y ) } ^ { \\tau _ + ^ \\delta ( y ) } | f ( X ( t , y ) ) | \\ , d t , \\end{aligned} \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} S ^ * _ h ( x ) = S ^ * _ { h k } ( x ) + O \\bigl ( ( \\psi ( k ) - 1 ) x ( \\log x ) ^ 6 + x ( \\log x ) ^ { - 2 } \\bigr ) \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} S _ h ( \\xi \\star \\eta ) = S _ h ( \\xi ) \\star ' S _ h ( \\eta ) \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} ( 1 + \\mu ) ^ { p ^ l } = 1 + p ^ l \\mu + \\sum _ { k = 2 } ^ { p ^ l } \\binom { p ^ l } { k } \\mu ^ k . \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} L _ 0 w ( x ) = f ( x , 0 ) \\ x \\in \\Omega , w ( x , 0 ) = \\psi ( x , 0 ) x \\in \\partial \\Omega \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} M _ x [ a _ 1 - 1 ] = \\varepsilon ^ 2 M _ x [ a _ { 1 , 2 } ] + M _ x [ \\mathtt { R } _ { a _ 1 } ] . \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} J _ { 2 2 } ( u , v ) = ( \\log { N } ) ^ 2 \\sum _ { c \\mid w } \\chi ( c ) P ( \\delta ) \\end{align*}"}
-{"id": "7472.png", "formula": "\\begin{align*} H ( \\gamma ( 0 ) | e ^ { - \\Psi } ) = 0 = I ( \\gamma ( 0 ) | e ^ { - \\Psi } ) . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} P _ { X , Y | V , S } ( x , y | v , s ) = P _ { X | V , S } ( x | v , s ) P _ { Y | V , S } ( y | v , s ) , \\quad \\forall ( v , s , x , y ) \\in \\mathcal { V } \\times \\mathcal { S } \\times \\mathcal { X } \\times \\mathcal { Y } . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} R _ { l j k } ( i _ n ) : = \\{ \\omega \\in \\mathcal { G } _ n : \\lvert \\mathrm { i } \\omega \\cdot l + \\mu _ j ^ { \\infty } ( i _ n ) - \\mu _ k ^ { \\infty } ( i _ n ) \\rvert < 2 \\ , \\gamma _ n \\ , \\lvert j ^ 3 - k ^ 3 \\rvert \\langle l \\rangle ^ { - \\tau } \\} . \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} \\Big \\| \\sup _ { N \\ge 1 } \\big | \\sum _ { n = 1 } ^ N a _ n { \\rm e } ^ { i \\lambda _ n t } \\big | \\ , \\Big \\| _ { \\S ^ 2 } \\le C \\Big ( \\sum _ { n \\ge 1 } \\Big ( \\sum _ { k \\ , : \\ , n \\le \\lambda _ k \\le n + 1 } | a _ k | \\Big ) ^ 2 \\Big ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} & e ^ { - \\theta j ( K - k ) c } < \\sum _ { s = 1 } ^ S w _ s e ^ { - \\theta r _ s } \\Rightarrow \\\\ & \\Lambda _ S ( - \\theta , \\mathbf { H } ) > \\Lambda _ S ( - \\theta , \\mathbf { F } ) \\forall \\theta \\in [ \\theta ^ * ( \\mathbf { \\Phi } ) , \\theta ^ * ( \\mathbf { D } ) ] . \\end{align*}"}
-{"id": "1135.png", "formula": "\\begin{align*} v ^ { ( j ) } ( n ) = ( 1 - \\epsilon ' ) \\phi ^ { ( j ) } \\frac { n - \\sum _ { j ' = 1 } ^ { J } n _ 0 ^ { ( j ' ) } } { 2 k _ n ^ { ( j ) } } \\log k _ n ^ { ( j ) } , \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\chi ( D ) = \\frac { 1 } { 8 } ( q ( D ) + 4 ) ( q ( D ) + 6 ) . \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} f _ 1 ( A _ n ) = & \\mathrm { P r } ( | \\sqrt { A _ n } x _ n + \\bar v _ n | ^ 2 > | 2 \\sqrt { P } x _ n + \\sqrt { A _ n } x _ n + \\bar v _ n | ^ 2 ) \\\\ = & \\mathrm { P r } \\left ( \\bar v _ n / x _ n > \\sqrt { A _ n } + \\sqrt { P } \\right ) \\\\ = & \\frac { 1 } { 2 } - \\frac { 1 } { 2 } \\mathrm { e r f } \\left ( \\sqrt { A _ n } + \\sqrt { P } \\right ) , \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{align*} \\phi _ a ( x _ { ( n ) } ) = x _ { ( n - 1 ) } n > 0 \\qquad \\phi _ a ( x _ { ( 0 ) } ) = x \\ , . \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} [ D ( m ' , s ) : D ( m , n ) ] _ q = \\sum _ { p \\geq 0 } [ D ( m ' , s ) : D ( \\ell , p ) ] _ q \\ , [ D ( \\ell , p ) : D ( m , n ) ] _ q . \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} \\| f \\| _ 2 = \\| f - \\xi ( f ) \\| \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} & \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( y _ i - T \\widetilde { f } _ m ( x _ i ) ) ^ 2 - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( y _ i - f ( x _ i ) ) ^ 2 + \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( T \\widetilde { f } _ m ( x ^ { \\prime } _ i ) ) ^ 2 - T f ( x ^ { \\prime } _ i ) ) ^ 2 \\\\ & \\leq \\frac { 2 v ^ 2 \\epsilon ^ 2 _ 1 ( 1 + M _ 1 / m _ 0 ) } { m _ 0 } + \\frac { v ^ 2 M _ 1 } { 2 m ^ 2 _ 0 } + \\\\ & 8 B _ n v \\left ( 1 + M _ 1 / m _ 0 \\right ) \\epsilon _ 2 + \\frac { 2 } { n } \\sum _ { i = 1 } ^ n ( | y _ i | ^ 2 - B ^ 2 _ n ) \\mathbb { I } \\{ | y _ i | > B _ n \\} . \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\langle A _ { 2 1 } y , v \\rangle = \\langle y , A _ { 2 1 } ^ * v \\rangle = \\langle y , A _ { 1 2 } v \\rangle = - \\langle y , A _ { 1 1 } z \\rangle = - \\langle A _ { 1 1 } y , z \\rangle = 0 , \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{gather*} Y ^ { ( 6 ) } ( x , t ) : = \\Psi _ { 0 } ^ { ( 6 ) } ( x , t ) e ^ { - \\left ( \\frac { x ^ 3 } { 6 } - \\frac { x t } { 2 } \\right ) \\sigma _ 3 } . \\end{gather*}"}
-{"id": "5968.png", "formula": "\\begin{align*} \\frac { \\langle h _ { 1 } , . . . , h _ { \\mathsf { N } } | h _ { 1 } , . . . , h _ { \\mathsf { N } } \\rangle } { \\langle p - 1 , . . . , p - 1 | p - 1 , . . . , p - 1 \\rangle } = \\prod _ { 1 \\leq b < a \\leq \\mathsf { N } } \\frac { X _ { a } ^ { \\left ( p - 1 \\right ) } - X _ { b } ^ { \\left ( p - 1 \\right ) } } { X _ { a } ^ { \\left ( h _ { a } \\right ) } - X _ { b } ^ { \\left ( h _ { b } \\right ) } } . \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\tilde { \\alpha } _ { \\infty , f } ( x ^ { ( j ) } ) = \\int _ { - \\infty } ^ { \\infty } f ( t ) \\tilde { \\alpha } _ { \\infty , t } ( x ^ { ( j ) } ) ~ d t = x ^ { ( j ) } \\int _ { - \\infty } ^ { \\infty } f ( t ) e ^ { i p t } ~ d t = \\widehat { f } ( - p ) x ^ { ( j ) } \\ , . \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\sqrt { | D _ { M } | } ^ { [ K : M ] } = 1 6 D ^ { 2 } . \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} \\nabla u & = r ^ { \\mu - 1 } ( \\mu \\phi \\partial _ r + \\nabla _ Y \\phi ) , \\\\ \\langle \\nabla _ { \\partial _ r } \\nabla u , \\nabla u \\rangle = \\frac { 1 } { 2 } \\partial _ r | \\nabla u | ^ 2 & = ( \\mu - 1 ) r ^ { 2 \\mu - 3 } ( \\mu ^ 2 \\phi ^ 2 + | \\nabla _ Y \\phi | ^ 2 ) . \\end{align*}"}
-{"id": "7129.png", "formula": "\\begin{gather*} \\langle ( 0 , x _ i ) , L _ { a , b } ^ d ( p _ i , 0 ) \\rangle = d a b ^ { d - 1 } \\langle L x _ i , L ^ { d - 1 } p _ i \\rangle + O ( b ^ { d - 2 } ) = O ( b ^ { d - 2 } ) \\\\ \\langle ( L x _ i , 0 ) , L _ { a , b } ^ d ( p _ i , 0 ) \\rangle = d a b ^ { d - 1 } \\langle L x _ i , L ^ { d - 1 } p _ i \\rangle + O ( b ^ { d - 2 } ) = O ( b ^ { d - 2 } ) \\\\ \\langle ( p _ i , 0 ) , L _ { a , b } ^ d ( p _ i , 0 ) \\rangle = d a b ^ { d - 1 } \\langle p _ i , L ^ { d - 1 } p _ i \\rangle + O ( b ^ { d - 2 } ) . \\end{gather*}"}
-{"id": "5921.png", "formula": "\\begin{align*} & \\sum _ { k : M _ k \\in B _ { j + 1 } ^ { j + 1 } } \\lambda _ { k } \\frac { \\det ( M _ k ) } { \\det ( A ) } | A ^ { - 1 } - M _ { k } ^ { - 1 } | \\lesssim \\sum _ { k : M _ k \\in B _ { j + 1 } ^ { j + 1 } } \\lambda _ { k } j ^ { 2 n ' - 2 m _ 2 ' - m _ 1 ' + 1 } i ^ { - m _ 1 ' } \\\\ & = \\nu ( B _ { j + 1 } ^ { j + 1 } ) j ^ { 2 n ' - 2 m _ 2 ' - m _ 1 ' + 1 } i ^ { - m _ 1 ' } \\lesssim j ^ { 2 m _ 1 ' + 3 m _ 2 ' - 3 n ' } i ^ { m _ 1 ' } j ^ { 2 n ' - 2 m _ 2 ' - m _ 1 ' + 1 } i ^ { - m _ 1 ' } = j ^ { m _ 1 ' + m _ 2 ' - n ' + 1 } \\leq 1 . \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} | i X + Y | ^ 2 + | X + i Y | ^ 2 = 2 | X | ^ 2 + 2 | Y | ^ 2 . \\end{align*}"}
-{"id": "6545.png", "formula": "\\begin{align*} \\alpha : = h \\otimes 1 + 1 \\otimes k \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} \\dot { A } = \\lambda A + 8 \\gamma ^ 2 \\pi ^ 2 L _ 2 ^ 2 \\frac { \\hat { \\tilde { V } } _ { 1 , 0 } } { 1 + \\gamma \\hat { \\tilde { V } } _ { 2 , 0 } } \\left ( 2 \\hat { \\tilde { V } } _ { 2 , 0 } - \\hat { \\tilde { V } } _ { - 1 , 0 } \\right ) | A | ^ 2 A \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} 9 6 \\gamma _ 3 & = 6 t ( 1 - | x | ^ 2 ) ( 4 - c _ 1 ^ 2 ) + c _ 1 ^ 3 + ( 4 b _ 3 - 2 b _ 1 ^ 2 ) c _ 1 + ( 2 b _ 2 ^ 3 - 8 b _ 2 b _ 3 + 2 b _ 4 ) \\\\ & + x ( 4 - c _ 1 ^ 2 ) ( 2 b _ 2 + 2 c _ 1 - 3 c _ 1 x ) . \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} \\eta _ M ( \\varphi ) = \\varphi ( \\eta _ { L _ i } ) = \\varphi . \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} u _ { \\ast } ^ { N } ( r , \\theta ) = \\rho ( r ) e _ r ( \\theta + \\psi ( r ) ) \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} A ( t ) = \\sum _ { n \\geq 0 } \\left ( \\frac { t ^ n } { n ! } \\prod _ { k = 1 } ^ n A ( k t ) \\right ) , \\end{align*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\det ( \\xi ( A _ 1 + B _ 1 ) ) = \\alpha \\cdot \\det ( A _ 1 + B _ 1 ) . \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} g ( y ) = \\begin{cases} \\left ( \\frac { f ( y ) } { \\int _ { - 2 r } ^ { 2 r } f ( \\Phi _ t ( y ) ) \\ , d t } \\right ) ^ { \\frac { 1 } { 2 } } & \\mid y \\in \\Phi _ { [ - r , r ] } ( X ) \\\\ 0 \\ , , & \\mid y \\not \\in \\Phi _ { [ - r , r ] } ( X ) . \\end{cases} \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} \\begin{cases} & x _ 1 ^ { \\frac { 2 ^ \\ast } { 2 } - 1 } + \\nu \\alpha x _ 1 ^ { \\frac { \\alpha } { 2 } - 1 } x _ 2 ^ { \\frac { \\beta } { 2 } } = 1 , \\\\ & x _ 2 ^ { \\frac { 2 ^ \\ast } { 2 } - 1 } + \\nu \\beta x _ 1 ^ { \\frac { \\alpha } { 2 } } x _ 2 ^ { \\frac { \\beta } { 2 } - 1 } = 1 , \\end{cases} \\end{align*}"}
-{"id": "3226.png", "formula": "\\begin{align*} \\gamma _ s ^ * \\omega _ s = \\omega _ 0 . \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p _ { 2 } ( n ) q ^ n & = \\prod _ { n = 1 } ^ { \\infty } \\frac { 1 } { ( 1 - q ^ n ) ^ 2 } = \\frac { q ^ { \\frac { 1 } { 1 2 } } } { \\eta ^ 2 ( z ) } . \\end{align*}"}
-{"id": "4911.png", "formula": "\\begin{align*} F _ U ( x ) : = F \\Big ( \\frac { x } { U } \\Big ) . \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} b _ { i j } = \\# \\left \\{ \\right \\} - \\# \\left \\{ \\right \\} , \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} N _ l = \\sum _ { k = 1 } ^ l ( 2 n _ j + m _ j ) \\quad M _ l = N _ l - 2 n _ l = N _ { l - 1 } + m _ l \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} \\hat { 0 } \\lessdot \\{ 1 \\} ^ { u _ 1 } \\lessdot \\{ 1 , 2 \\} ^ { u _ 1 + u _ 2 } \\lessdot \\cdots \\lessdot [ n ] ^ { \\sum _ { i = 1 } ^ n u _ i } . \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} a _ { f } ( n ) = - \\frac { 1 } { \\sqrt { 2 4 n - 1 } } \\Im \\bigg ( \\sum _ { Q \\in \\mathcal { Q } _ { n } } \\frac { F ( z _ { Q } ) } { \\omega _ { Q } } \\bigg ) , \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} \\tilde { \\mu } ( \\theta _ t ) : = \\left [ \\frac { \\sum _ i ( \\theta _ t ^ i \\delta ^ i ) ^ { \\frac { 1 } { \\gamma } } } { \\sum _ i ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } } \\right ] ^ \\gamma = \\left [ \\textstyle \\sum _ i a ^ i ( \\theta _ t ^ i ) \\ , ( \\delta ^ i ) ^ { \\frac { 1 } { \\gamma } } \\right ] ^ \\gamma . \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} u _ 0 ( x ) = \\begin{cases} u _ 1 \\circ H ( x ) \\ \\ \\ x \\in \\overline \\Omega _ 1 , \\\\ u _ 2 \\circ H ( x ) \\ \\ \\ x \\in \\overline \\Omega _ 2 , \\\\ u _ 3 \\circ H ( x ) \\ \\ \\ x \\in \\overline \\Omega _ 3 , \\end{cases} \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} y ' _ { v } = \\begin{cases} ( y _ { i + 1 } ) ^ { - 1 } & \\mbox { i f $ { v } = i \\in M $ , } \\\\ y _ i \\ , y _ i ^ - & \\mbox { i f $ v = i ^ - \\in M ^ - $ , } \\\\ y _ { i + 1 } \\ , y _ i ^ + & \\mbox { i f $ v = i ^ + \\in M ^ + $ , } \\\\ y _ { v } & \\mbox { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\sum _ { x , y \\in \\mathbb { Z } } ( - 1 ) ^ { x + y } q ^ { x ^ 2 + y ^ 2 } - \\sum _ { n = - \\infty } ^ \\infty ( - 1 ) ^ n q ^ { n ^ 2 } = \\sum _ { N \\geq 1 } ( - 1 ) ^ N r _ 2 ( N ) q ^ N - 2 \\sum _ { n \\geq 1 } ( - 1 ) ^ n q ^ { n ^ 2 } . \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} \\chi ( w _ { \\sigma } \\cdot w ' ) = \\Im \\left ( - \\frac { Z _ { \\alpha , \\beta } ( w ' ) } { Z _ { \\alpha , \\beta } ( v ) } \\right ) \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N } \\mu _ k = 1 \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} e ^ { - \\frac { 1 } { 3 } 2 ^ { n \\left ( R _ 1 - I ( U ; W ) - \\epsilon ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } \\right ) } \\frac { ( c - 1 ) ^ 2 } { c ' } } \\leq e ^ { - \\frac { 1 } { 3 } 2 ^ { n \\left ( R _ 1 - I ( U ; W ) - \\epsilon ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } \\right ) } \\frac { ( c - 1 ) ^ 2 } { ( c ' + 1 ) } } = e ^ { - \\frac { 1 } { 3 } 2 ^ { n \\frac { \\delta _ 2 - \\delta _ 1 } { 2 } } } , \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} x _ \\mu - x _ { \\sqrt \\mu } = Q ^ { - 1 } ( \\mu ) - Q ^ { - 1 } ( \\mu ^ \\frac 1 2 ) \\geq \\frac { \\mu - \\mu ^ \\frac 1 2 } { a _ \\mu } , \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} \\left ( - \\frac { d ^ 2 } { d x ^ 2 } + V ( x ) \\right ) \\psi ( x ) = E \\psi ( x ) \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} \\hat { H } _ n & = p ^ 2 - V _ n ( - i \\nabla _ p ) , \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} \\Delta \\left ( h ( f _ 0 , g _ 0 ) ; f , g \\right ) & = \\left \\| { A } \\xi - \\hat { A } \\xi \\right \\| _ \\alpha ^ \\alpha \\\\ & = \\int _ { - \\pi } ^ { \\pi } \\left | A ( e ^ { i \\theta } ) - { h } ^ 0 ( \\theta ) \\right | ^ { \\alpha } f ( \\theta ) d \\theta + \\int _ { - \\pi } ^ { \\pi } \\left | { h } ^ 0 ( \\theta ) \\right | ^ { \\alpha } g ( \\theta ) d \\theta . \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} | { n } ) _ { { x } } \\mbox { s u c h t h a t } { \\sigma } _ { { x } } | { n } ) _ { { x } } = | { n } ) _ { { x } } \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} \\begin{array} { l l l l } L ( \\pi ) = & \\max \\ & c ' v - \\pi ( H v - h ) & \\\\ & \\mbox { s . t . } \\ & A v = b , & \\\\ & & v \\in \\{ 0 , 1 \\} ^ n . & \\end{array} \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} p = \\frac { 3 c ^ 4 r ^ 2 + c ^ 4 + 3 c ^ 3 r ^ 2 + c ^ 3 - 1 2 c ^ 2 r ^ 2 + 2 c ^ 2 - 1 2 c r ^ 2 + 8 c + 6 } { 6 c \\left ( c ^ 3 + 2 c + 6 \\right ) r } . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\eta ^ \\sigma _ k } = \\frac { \\partial } { \\partial \\xi ^ \\sigma } + \\sum _ l \\bigg ( \\frac { \\partial h ^ k _ l } { \\partial \\eta ^ \\sigma _ k } \\bigg ) \\frac { \\partial } { \\partial x ^ l } + \\sum _ \\tau \\bigg ( \\frac { \\partial \\omega ^ k _ l } { \\partial \\eta ^ \\sigma _ k } \\bigg ) \\frac { \\partial } { \\partial \\xi ^ \\tau } , \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} \\{ a _ { 0 } = g , a _ { 1 } , \\dots \\} , \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { N - 1 } \\binom { X + Y + n _ i - 1 } { n _ i } = \\sum _ { 0 \\leq k \\leq _ b n } \\prod _ { i = 0 } ^ { N - 1 } \\binom { X + k _ i - 1 } { k _ i } \\binom { Y + n _ i - k _ i - 1 } { k _ i } . \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{align*} Q _ { i j } ( D ) = Q _ { i j } ( e + \\Sigma d _ { p q } v _ { p q } ) . \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} h '' ( x ) = - g '' ( f _ d ( x ) ) ( f _ d ( x ) ) ^ 2 - g ' ( f _ d ( x ) ) f _ d '' ( x ) . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} A = B \\oplus A _ { - } B . \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} B ( n ) = \\begin{cases} \\frac { n } { 2 } \\log ( 1 + P ) & \\alpha = 0 , \\\\ \\frac { n } { 2 \\ell } \\log ( 1 + \\ell P ) & \\alpha > 0 . \\end{cases} \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} \\bold { D } ^ { \\ast } _ F = B ( x , [ u ] ) \\bold { D } _ F \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} f ( x ) = \\sum _ { i = 0 } ^ { k - 1 } x ^ i g _ i ( x ^ k ) , \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} \\tau ^ 2 = g \\xi \\tanh \\xi . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} V _ { n , k } ^ { B B , \\alpha , \\theta } ( N ) = \\frac { \\binom { N } { k } \\big ( \\frac { - \\alpha \\Gamma ( \\theta + \\alpha ) } { \\Gamma ( \\theta + \\alpha + n ) } \\big ) ^ k } { \\big ( \\frac { \\Gamma ( \\theta + \\alpha ) \\Gamma ( \\theta + n ) } { \\Gamma ( \\theta + \\alpha + n ) \\Gamma ( \\theta ) } \\big ) ^ { N } } . \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} \\underleftarrow { \\bf u } ^ { * } _ { j k } = { \\bf v } ^ { * } _ { j k } = \\frac { ( \\underleftarrow { \\bf F } _ { i j k } ) ^ { - 1 } { \\bf G } _ { i j } { \\bf u } ^ { * } _ { i j k } } { \\norm { ( \\underleftarrow { \\bf F } _ { i j k } ) ^ { - 1 } { \\bf G } _ { i j } { \\bf u } ^ { * } _ { i j k } } } , \\forall \\{ i , j \\} , k \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} [ D ( m , m j ) : D ( m + 1 , n ) ] _ q & = q ^ { m ( 2 j - 1 ) } [ D ( m , ( j - 1 ) m ) : D ( m + 1 , n ) ] _ q & \\\\ & = \\begin{cases} q ^ { j ( m j ) } & n = 0 , \\\\ q ^ { ( j + 1 ) ( m j - m ) } & n = m , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} u ( t , x + h b _ { t } ) = \\int _ { 0 } ^ { t } [ a ^ { i j } ( s ) D _ { i j } u ( s , x + h b _ { s } ) + f ( s , x ) ] \\ , d s + \\int _ { ( 0 , t ] } g ( s , x ) \\ , d \\pi _ { s } , \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} a _ 1 , \\ldots , a _ n \\in K ^ \\times , \\sum _ { i = 1 } ^ n a _ i \\in K _ 0 \\sum _ { i = 1 } ^ n f ( a _ i ) \\in L _ 0 . \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} & ( \\mathtt { H } 1 ) d ( \\xi ) - 1 \\neq 0 \\quad \\mbox { a t } \\xi = \\mathbb { M } ^ { - 1 } \\overline { \\omega } , \\\\ [ 2 m m ] & ( \\mathtt { H } 2 ) _ { j , k } \\mbox { F i x e d } \\ , \\ , j , k \\in S ^ c , \\ , j \\neq k , \\quad \\det ( \\mathbb { M } + B ( j , k ) ) \\neq 0 , \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} \\iota \\circ \\widetilde { I } = I \\circ 1 _ { W } . \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} \\langle f _ i \\ | \\ \\bar { x } \\rangle = \\eta _ i \\iff \\sum _ { j \\in \\mathbb { N } } \\lambda _ j ^ i \\bar { x } _ j = \\eta _ i . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\sup _ { t \\in [ 0 , e ^ s ] } d ( \\mathcal { Y } ( x u ^ { + } _ r a _ { - s } ) u ^ { + } _ { - e ^ { - s } r } u _ t , \\mathcal { Y } ( x a _ { - s } ) u _ t ) = O ( 1 ) . \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} V ( z ) = \\exp \\left ( \\sum _ { n \\geqslant 0 } z ^ n \\frac { \\partial } { \\partial p _ n } \\right ) . \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} Z ( \\widetilde { G } ) = Z _ { 0 } ( G ) ^ 2 + Z _ { 1 } ( G ) ^ 2 \\ , , \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} | X | & \\leq 2 1 0 ( k ' + 1 ) ^ 2 \\log k + 1 0 \\ell k ' + 2 1 0 ( k - k ' ) ^ 2 \\log k + 1 0 \\ell ( k - k ' - 1 ) \\\\ & \\le 2 1 0 \\log k \\left ( k '^ 2 + 2 k ' + 1 + k ^ 2 - 2 k k ' + k '^ 2 \\right ) + 1 0 \\ell ( k - 1 ) \\\\ & = 2 1 0 \\log k \\left ( 2 k ' ( k ' + 1 - k ) + 1 + k ^ 2 \\right ) + 1 0 \\ell ( k - 1 ) \\\\ & \\le 2 1 0 k ^ 2 \\log k + 1 0 \\ell ( k - 1 ) = f ( k , \\ell ) \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} M ^ j _ k = \\left [ \\begin{array} { c c c } p _ j ( 1 , 1 ) f _ j ( Y _ k | X _ { k } = 1 , Y _ { k - 1 } ) & \\cdots & p _ j ( d , 1 ) f _ j ( Y _ k | X _ { k } = 1 , Y _ { k - 1 } ) \\\\ \\vdots & \\ddots & \\vdots \\\\ p _ j ( 1 , d ) f _ j ( Y _ k | X _ { k } = d , Y _ { k - 1 } ) & \\cdots & p _ j ( d , d ) f _ j ( Y _ k | X _ { k } = d , Y _ { k - 1 } ) \\end{array} \\right ] \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} \\big \\vert R _ q ( x ) \\big \\vert = \\big ( o ( 1 ) + V _ m ^ 2 / m ! \\big ) L ( \\vert x \\vert ) ^ { 2 m } \\vert x \\vert ^ { - 2 ( 1 - H ) } \\ , , \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} { { \\bar G } _ S } = \\sum \\nolimits _ { \\ell , k \\in \\left \\{ { { } , { } } \\right \\} } { { G _ { \\ell } ^ S G _ k } _ { \\ell k } ^ S } , { { \\bar G } _ A } = \\sum \\nolimits _ { \\nu , k \\in \\left \\{ { { } , { } } \\right \\} } { { G _ { \\nu } ^ A G _ k } _ { \\nu k } ^ A } . \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} & \\Psi ^ q ( u ) \\Psi ^ q ( v ) = \\Psi ^ q ( u + v ) \\\\ & \\Psi ^ q ( v ) \\Psi ^ q ( u ) = \\Psi ^ q ( u ) \\Psi ^ q ( q v u ) \\Psi ^ q ( v ) \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} & g _ { \\lambda , \\rho } ^ 2 ( w _ 2 ) = \\frac { n _ 0 } { 4 k _ { \\ell } } \\log \\left ( 1 + \\lambda ( 1 - \\lambda \\rho ) w _ 2 P ' \\right ) \\\\ & - \\frac { ( 1 - \\rho ) n _ 0 } { 2 k _ { \\ell } } \\log \\left ( 1 + \\lambda w _ 2 P ' \\right ) - \\frac { \\rho \\ell } { k _ { \\ell } } H _ 2 \\left ( \\frac { w _ 2 } { \\ell } \\right ) . \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} \\bold { p r } X ( L ) + L \\operatorname { D i v } \\xi = \\operatorname { D i v } P . \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} \\nabla _ w ( R ( x , y ) z ) = R ( \\nabla _ w x , y ) z + R ( x , \\nabla _ w y ) z + R ( x , y ) \\nabla _ w z , \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{gather*} F ( x , t ; \\beta = 6 ) = \\Psi _ { 1 1 } \\bigl ( 3 ^ { 1 / 3 } x , 3 ^ { 2 / 3 } t \\bigr ) , \\end{gather*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\phi = \\psi _ { R _ 2 } \\psi _ g \\psi _ { R _ 1 } = \\psi _ { R _ 2 } \\psi _ { g ( R _ 1 ) } \\psi _ g = \\psi _ { R _ 2 \\ominus g ( R _ 1 ) } \\psi _ g = \\psi _ { g , R _ 2 \\ominus g ( R _ 1 ) } . \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} u _ { 1 , 0 } - u _ { 0 , 1 } - \\frac { a ( m ) - b ( n ) } { u _ { 0 , 0 } - u _ { 1 , 1 } } = 0 . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} R _ { i \\to j } ^ m = { \\log _ 2 } \\left ( { 1 + \\frac { { p _ j ^ m { { \\left | { h _ i ^ m } \\right | } ^ 2 } } } { { p _ i ^ m { { \\left | { h _ i ^ m } \\right | } ^ 2 } + \\sigma ^ 2 } } } \\right ) . \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} \\big | L ( u _ { n } , z ) \\big | \\leq \\sum _ { j = 1 } ^ { m } \\big | L ( u _ { n } , p _ { j } z ) \\big | + C | z | ^ { \\tau } e ^ { - | z | ^ { 1 - \\alpha } } n ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} W ' \\ ; : \\ ; y '^ 2 = x '^ 3 + \\frac { - 2 a } { \\pi ^ { 2 \\alpha } } x '^ 2 + \\frac { a ^ 2 - 4 b } { \\pi ^ { 4 \\alpha } } x ' , \\upsilon _ F ( \\frac { - 2 a } { \\pi ^ { 2 \\alpha } } ) > 0 , \\upsilon _ F ( \\frac { a ^ 2 - 4 b } { \\pi ^ { 4 \\alpha } } ) = 0 . \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} \\log f ( x ; G ^ * ) \\leq \\log \\{ ( x - 1 ) ! / x ! \\} = - \\log x . \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} \\exists \\ \\{ \\nu _ { i } \\} _ { i \\in I ( \\bar { x } ) } \\geq 0 \\ \\forall _ { k \\in \\mathbb { N } \\backslash W } & | \\bar { x } _ k - x _ k | ^ { p - 2 } { ( \\bar { x } _ k - x _ k ) } = - \\sum _ { i \\in I ( \\bar { x } ) } \\lambda _ k ^ i \\nu _ i , \\\\ \\forall _ { k \\in W } & | \\bar { x } _ k - x _ k | ^ { p - 2 } { ( \\bar { x } _ k - x _ k ) } = 0 . \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} h ( x + t z ) = h ( x ) - t \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} 0 \\geq Q ( w _ 1 - \\lambda w _ 2 ) = Q ( w _ 1 ) + \\lambda ^ 2 Q ( w _ 2 ) - 2 \\lambda Q ( w _ 1 , w _ 2 ) . \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{align*} \\pi \\in B ^ { s _ 2 } _ { \\tau _ 2 } ( L _ { \\tau _ 2 } ( \\Omega ) ) , \\frac { 1 } { \\tau _ 2 } = \\frac { s _ 2 } { d } + \\frac { 1 } { 2 } , 0 < s _ 2 < \\frac { 1 } { 2 } \\cdot \\frac { d } { d - 1 } , \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} f : = \\sum _ { i = 1 } ^ n \\left ( x _ { i } ( 2 + 1 0 x _ { i } ^ 2 ) + 1 - \\sum _ { j \\in H _ i } ( 1 + x _ { j } ) x _ { j } \\right ) ^ 2 , \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} c = \\widehat { c } = \\frac { M _ 1 + M _ 2 } { 2 } + \\frac { 2 f ' ( 0 ) D } { M _ 2 - M _ 1 } = \\frac { M _ 1 + M _ 2 } { 2 } - \\frac { 2 f ' ( 0 ) D } { M _ 1 - M _ 2 } \\ , , \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} ( a , b ) ( c , d ) = \\left \\{ \\begin{array} { l l } ( c \\cdot b ^ { - 1 } \\cdot a , d ) , & \\hbox { i f } \\ ; b < c ; \\\\ ( a , d ) , & \\hbox { i f } \\ ; b = c ; \\\\ ( a , b \\cdot c ^ { - 1 } \\cdot d ) , & \\hbox { i f } \\ ; b > c , \\end{array} \\right . \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} \\hat y ^ h & \\to x _ 1 e _ 1 \\quad H ^ 1 ( \\Omega , \\R ^ 3 ) \\ , , \\\\ u ^ h & \\rightharpoonup u H ^ 1 ( 0 , L ) \\ , , \\\\ v _ i ^ h & \\to v _ i H ^ 1 ( 0 , L ) \\ , , v _ i \\in H ^ 2 ( 0 , L ) i = 2 , 3 \\ , , \\\\ w ^ h & \\rightharpoonup w H ^ 1 ( 0 , L ) \\ , . \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} \\bar g _ 3 ( C _ n ^ { \\rm R } , C _ n ^ { \\rm I } ) = 1 - \\bar g _ 3 ^ { \\rm R } ( C _ n ^ { \\rm R } ) g _ 3 ^ { \\rm I } ( C _ n ^ { \\rm I } ) , \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} \\rho ( \\lambda ) = \\frac { p _ 0 ' ( \\lambda ) } { 2 \\pi } + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) \\rho ( \\nu ) + \\frac { 1 } { 2 4 L ^ 2 } \\sum _ { i a } \\frac { s _ a K ' ( \\lambda - \\lambda _ { i a } ) } { 2 \\pi \\rho ( \\lambda _ { i a } ) } \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} 9 f _ \\rho ^ \\ast ( 1 ) ^ 2 - 6 f _ \\rho ^ \\ast ( 1 ) ^ 3 - 3 = \\rho + \\epsilon . \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\omega _ { \\beta , { n } } = \\lim _ { B \\to + 0 } \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\Lambda , { n } } \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} \\tau _ b ( x ) ( y ) : = d ( y , x ) - d ( b , x ) \\quad y \\in X . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{align*} d e t ( Q - T I d _ { m \\times m } ) = T ^ m + F _ 1 ( Q ) T ^ { m - 1 } + \\ldots + F _ m ( Q ) . \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} \\mathcal { D } _ { - } ( \\lambda ) = \\frac { ( \\lambda ^ { 2 } / q - q / \\lambda ^ { 2 } ) } { ( \\lambda ^ { 2 } - 1 / \\lambda ^ { 2 } ) } \\mathcal { A } _ { - } ( \\lambda ^ { - 1 } ) + \\frac { ( q - 1 / q ) } { ( \\lambda ^ { 2 } - 1 / \\lambda ^ { 2 } ) } \\mathcal { A } _ { - } ( \\lambda ) , \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} [ D ( m ' , s ) : D ( m , n ) ] _ q = \\delta _ { n , s } , \\mbox { i f $ n \\geq s $ } , [ D ( m , s ) : D ( m , n ) ] _ q = \\delta _ { n , s } \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{gather*} r _ 0 = \\omega + \\frac { t ^ 2 } { 4 } - \\frac { u _ t } { u } \\frac { 1 + q _ 2 } { 2 } , \\end{gather*}"}
-{"id": "601.png", "formula": "\\begin{align*} L = v _ n ^ { \\beta } F _ { \\beta } \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; J _ 2 } = D _ i \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } - ( D _ i \\xi ^ j ) u ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ j ; J _ 2 } \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{gather*} L = \\frac 1 2 A \\wedge d A + d C \\wedge A ^ \\ast , i _ Q \\omega \\simeq \\delta L . \\end{gather*}"}
-{"id": "6817.png", "formula": "\\begin{align*} H _ \\xi ( a ) = \\left \\{ \\begin{array} { c l } \\frac { 1 } { 2 } a ^ 2 , & , \\\\ \\xi ( | a | - \\frac { 1 } { 2 } \\xi ) , & . \\end{array} \\right . \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\varphi ( 1 ) & = 1 , & \\varphi ( 3 ) & = 2 , & \\varphi ( 5 ) & = 4 , & \\varphi ( 7 ) & = 6 , \\\\ \\varphi ( 1 6 ) & = 8 , & \\varphi ( 3 1 ) & = 3 0 , & \\varphi ( 7 1 ) & = 7 0 , & \\varphi ( 6 2 5 ) & = 5 0 0 , \\\\ \\varphi ( 1 0 5 7 ) & = 9 0 0 , & \\varphi ( 9 9 1 ) & = 9 9 0 , & \\varphi ( 5 5 9 1 ) & = 5 5 9 0 , & \\varphi ( 9 5 5 1 ) & = 9 5 5 0 , \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} C _ q \\Big ( \\frac { 1 } { 2 } , s , s + \\frac { 1 } { 2 } ; \\pm \\frac { 1 } { 2 } , j , j \\pm \\frac { 1 } { 2 } \\Big ) & = q ^ { \\frac { 1 } { 2 } ( \\mp s + j ) } \\Big ( \\frac { [ s \\pm j + 1 ] _ q } { [ 2 s + 1 ] _ q } \\Big ) ^ { \\frac { 1 } { 2 } } \\\\ C _ q \\Big ( \\frac { 1 } { 2 } , s , s - \\frac { 1 } { 2 } ; \\pm \\frac { 1 } { 2 } , j , j \\pm \\frac { 1 } { 2 } \\Big ) & = \\mp q ^ { \\frac { 1 } { 2 } ( \\pm s + j \\pm 1 ) } \\Big ( \\frac { [ s \\mp j ] _ q } { [ 2 s + 1 ] _ q } \\Big ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} H = \\lim _ { \\rightarrow \\atop { t } } S / ( x _ 1 ^ t , \\dots , x _ d ^ t ) \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} \\hat { \\pi } ( \\tau ) - \\hat { \\pi } ( \\tau - 1 ) = \\binom { \\tau - 1 } { r - 1 } p ^ r ( 1 - p ) ^ { \\tau - r } \\leq \\binom { \\tau } { r - 1 } p ^ r ( 1 - p ) ^ { \\tau - r } \\\\ \\stackrel { p = o ( 1 ) } { \\leq } \\binom { \\tau } { r - 1 } p ^ { r - 1 } ( 1 - p ) ^ { \\tau - r + 1 } \\leq \\sum _ { i = 0 } ^ { r - 1 } \\binom { \\tau } { i } p ^ i ( 1 - p ) ^ { \\tau - i } = 1 - \\hat { \\pi } ( \\tau ) \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} e ^ 2 = e , f ^ 2 = f , e f = 0 , a h = a , b + b = \\{ 0 , b \\} , b + c = H \\cup \\{ e , f \\} \\smallsetminus \\{ b , c \\} \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} f ( t ) = \\frac { e ^ { - | t | ^ p } } { 2 \\Gamma \\big ( 1 + { 1 / p } \\big ) } , t \\in \\R \\ , . \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} s _ n = \\ell s _ { n - 1 } + m s _ { n - 2 } , \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} \\hat L _ { t } ^ { * } = L ^ { * } _ { t } + \\sum _ { i = 1 } ^ { n } \\lambda _ { i } ( g _ { t } ^ { ( i ) } - 1 ) , \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} \\left ( x ; S ^ { ( 1 ) } \\right ) = \\mathcal { B } \\left ( 1 + 1 - 2 \\left ( x ; S ^ { ( 1 ) } \\right ) , \\ 0 + 1 \\right ) = \\mathcal { B } \\left ( 2 - 2 \\left ( x ; S ^ { ( 1 ) } \\right ) , \\ 1 \\right ) \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} \\frac { ( e ^ { i t } - 1 ) } { 2 x - ( 2 x - 1 ) e ^ { i t } } - i t & = \\frac { ( e ^ { i t } - 1 ) - i t ( 2 x - ( 2 x - 1 ) e ^ { i t } ) } { 2 x - ( 2 x - 1 ) e ^ { i t } } , \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} I _ d : \\ , W _ 2 ^ { \\bf R } ( \\Bbb T ^ d ) \\longrightarrow L _ 2 ( \\Bbb T ^ d ) , \\ \\ { \\bf R } = ( R _ 1 , R _ 2 , \\dots , R _ d ) \\in \\Bbb R _ + ^ d , \\ \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} j ( \\gamma _ 1 \\gamma _ 2 , z ) = j ( \\gamma _ 1 , \\gamma _ 2 z ) \\cdot j ( \\gamma _ 2 , z ) , \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} p \\begin{cases} \\geq 1 , n = 1 , \\\\ \\geq 2 , n \\geq 2 , \\end{cases} \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} a ( d _ { \\tilde { w } } ) = \\sup \\left \\{ a > 0 : \\int _ { V } r _ { \\tilde { w } } ^ { - 2 a } i ^ { n ^ 2 } \\Omega \\wedge \\bar \\Omega < \\infty \\right \\} \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} \\frac { x _ { \\frac 3 2 \\mu } } { x _ \\mu } = \\frac { x _ \\mu + Q ^ { - 1 } ( \\frac 3 2 \\mu ) - Q ^ { - 1 } ( \\mu ) } { x _ \\mu } \\leq \\frac { x _ \\mu + \\frac { \\mu } { 2 a _ \\mu } } { x _ \\mu } \\leq \\frac 3 2 . \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} \\mathcal { G } _ { \\{ n - k + 1 , \\dots , n \\} } = \\frac { \\det \\left ( x _ j ^ { k - i } \\prod _ { m = 1 } ^ { n - k + i - 1 } ( 1 - x _ j / y _ m ) \\right ) _ { 1 \\leq i , j \\leq k } } { \\prod _ { 1 \\leq i < j \\leq k } ( x _ i - x _ j ) } . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} f _ { \\beta } ( g ) = \\begin{cases} 0 & g \\in \\mathfrak { t } \\\\ 0 & g \\in \\mathfrak { g } _ { \\alpha } , \\alpha \\neq \\beta . \\\\ c & g = c x _ { \\beta } . \\end{cases} \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} \\sigma _ 1 = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ) , \\sigma _ 2 = \\left ( \\begin{array} { c c } 0 & - i \\\\ i & 0 \\end{array} \\right ) , \\sigma _ 3 = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) \\ , , \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\sigma ( y ) : = ( \\sigma _ { J _ o } ^ { - 1 } \\circ \\sigma _ C ) ( y ) = \\pi _ p ( x , y ) , \\textrm { f o r $ \\eta $ - a l m o s t e v e r y $ ( x , y ) $ } , \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} \\| f \\| _ { 1 + k _ { \\gamma , \\pitchfork } } ^ 2 = | f ( a ) | ^ 2 + \\frac { 1 } { \\gamma } \\left ( \\sum _ { \\nu = 1 } ^ { r - 1 } | f ^ { ( \\nu ) } ( a ) | ^ 2 + \\int ^ 1 _ 0 | f ^ { ( r ) } ( y ) | ^ 2 \\ , { \\rm d } y \\right ) , \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} \\sigma _ A ( x , \\xi ) = \\xi ( x ) ^ { * } ( A \\xi ) ( x ) : = \\xi ( x ) ^ { * } [ A \\xi _ { i j } ( x ) ] _ { i , j = 1 } ^ { d _ \\xi } . \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\ell \\geq \\frac { n - k } { b } ~ ~ ~ ~ \\ell = \\Omega ( n - k ) . \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\begin{aligned} ( u , v ) . f ( A , C , S ) & = f ( u ^ { - 1 } A v , u ^ { - 1 } C v , u ^ { - 1 } S v ) \\\\ & = ( a _ { 1 1 } c _ { 2 2 } - a _ { 2 2 } c _ { 1 1 } ) ( s _ { 1 2 } - s _ { 2 2 } u _ { 1 2 } + s _ { 1 1 } v _ { 1 2 } ) + ( s _ { 1 1 } a _ { 2 2 } - s _ { 2 2 } a _ { 1 1 } ) ( c _ { 1 2 } - c _ { 2 2 } u _ { 1 2 } + c _ { 1 1 } v _ { 1 2 } ) \\\\ & + ( c _ { 1 1 } s _ { 2 2 } - c _ { 2 2 } s _ { 1 1 } ) ( a _ { 1 2 } - a _ { 2 2 } u _ { 1 2 } + a _ { 1 1 } v _ { 1 2 } ) = f ( A , C , S ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} \\widetilde { u _ { J _ 1 ; J _ 2 } ^ { \\alpha } } : = S _ { J _ 2 } \\widetilde { D _ { J _ 1 } } \\widetilde { u } ^ { \\alpha } . \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} \\| f ^ { ( n ) } \\| _ { \\mathbb { F } _ q ^ { ( n ) } ( \\mathcal { H } ) } = \\| P _ q ^ { ( n ) } f ^ { ( n ) } \\| _ { \\mathcal { H } ^ { \\otimes n } } . \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} T _ { \\delta } ( \\psi ) : = A _ { \\varepsilon } ( G _ { \\delta } ( \\psi , 0 , 0 ) ) = \\varepsilon v _ { \\delta } ( \\psi ) + \\varepsilon ^ b z _ 0 ( \\psi ) , v _ { \\delta } ( \\psi ) : = v _ { \\varepsilon } ( \\theta _ 0 ( \\psi ) , y _ { \\delta } ( \\psi ) ) \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} & \\| F ( U ) - F ( V ) \\| _ { k , Q } \\leq L \\| U - V \\| _ { k , Q } . \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\rm w } \\mathbf { A } ^ { - 1 } \\tilde { \\mathbf { y } } _ i = \\mathbf { P } _ { \\rm w } \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 1 \\bar { \\mathbf { d } } _ { i - 1 } + \\mathbf { P } _ { \\rm w } \\mathbf { A } ^ { - 1 } \\mathbf { H } ^ { - 1 } _ i \\mathbf { n } _ i . \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\ddot x = - x - \\delta ( t ) - \\alpha \\dot x + \\mu \\phi ( y - x ) , \\\\ \\ddot y = \\mu \\phi ( x - y ) , \\end{array} \\right . \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} \\nabla h ( x ) = 2 \\sum _ { j = 1 } ^ n \\left ( g _ j ( x ) \\nabla f _ j ( x ) - f _ j ( x ) \\nabla g _ j ( x ) \\right ) x \\in \\Omega . \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{gather*} F ( x , t ; \\beta = 6 ) = \\Psi _ { 1 1 } \\bigl ( 3 ^ { 2 / 3 } t , 3 ^ { 1 / 3 } x \\bigr ) \\end{gather*}"}
-{"id": "6645.png", "formula": "\\begin{align*} I I + \\nabla _ g ^ { 2 } f = \\mu g , \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} e ^ { - t D _ { A _ R } ^ 2 } - K _ t ^ N & = \\frac { - i } { 2 \\pi } \\int _ { C _ t } e ^ { - t z } ( ( D _ { A _ R } ^ 2 - z ) ^ { - 1 } - Q ^ N _ z ) d z \\\\ & = \\frac { - i } { 2 \\pi } \\int _ { C _ t } e ^ { - t z } ( D _ { A _ R } ^ 2 - z ) ^ { - 1 } \\epsilon _ z ^ N d z \\\\ & = \\frac { - i } { 2 \\pi } \\int _ { C _ t } e ^ { - t z } Q _ z ^ N \\epsilon _ z ^ N d z + \\frac { - i } { 2 \\pi } \\int _ { C _ t } e ^ { - t z } ( D _ { A _ R } ^ 2 - z ) ^ { - 1 } ( \\epsilon _ z ^ N ) ^ { 2 } d z . \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} I ( A ; S | S _ 1 ) = I ( S ; S | S _ 1 , E _ \\sigma ) = \\bar { \\sigma } H ( S | S , E _ \\sigma = 0 ) + \\sigma H ( S | ? , E _ \\sigma = 1 ) = \\sigma \\big [ 1 - h ( \\alpha ) \\big ] . \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} \\frac 1 j + \\frac 1 { p - j } = \\frac { p } { j ( p - j ) } , 1 \\le j \\le \\frac { p - 1 } 2 , \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} L ( e _ { x y } f | _ x ^ y - f | _ x ^ y e _ { x y } ) ( x , y ) & = \\left ( L ( e _ { x y } ) f | _ x ^ y ) ( x , y ) - ( f | _ x ^ y L ( e _ { x y } ) \\right ) ( x , y ) \\\\ & + L ( f | _ x ^ y ) ( y , y ) - L ( f | _ x ^ y ) ( x , x ) . \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} \\iint _ { \\underline a , \\underline b : \\underline a \\wedge \\underline b = \\phi } \\sum _ { n = 0 } ^ { \\infty } \\sum _ { c _ 1 \\cdots c _ n } \\frac { \\mu [ c _ 1 \\cdots c _ n ] ^ q } { d ( c _ 1 \\cdots c _ n \\underline a , c _ 1 \\cdots c _ n \\underline b ) ^ { s ( q - 1 ) } } d \\mu ( \\underline a ) d \\mu ( \\underline b ) < \\infty \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} u _ t - u _ { x x x } = 0 . \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} { Z } _ { t n + i } = \\begin{cases} \\frac { f _ i } { d _ { \\rm a , w } ^ { \\alpha / 2 } } + \\frac { g _ { t n + i } } { d _ { \\rm j , w } ^ { \\alpha / 2 } } + { N } _ { t n + i } ^ { \\rm ( w ) } , & \\ ! \\ ! \\mbox { A l i c e t r a n s m i t s a n d } ~ t = 0 \\\\ \\frac { g _ { t n + i } } { d _ { \\rm j , w } ^ { \\alpha / 2 } } + { N } _ { t n + i } ^ { \\rm ( w ) } , & \\ ! \\ ! \\mbox { e l s e , } \\end{cases} \\end{align*}"}
-{"id": "8024.png", "formula": "\\begin{align*} \\hat { \\rho } _ { \\rm G G E } = Z _ { \\rm G G E } ^ { - 1 } \\exp \\{ - \\sum _ n \\beta _ n \\hat { Q } _ n \\} . \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{gather*} d \\alpha = i _ Q \\chi . \\end{gather*}"}
-{"id": "2821.png", "formula": "\\begin{align*} \\alpha _ 2 = 1 + \\left \\{ \\frac { N + 2 } { 6 } \\right \\} . \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} n f _ n ( x ) = f _ 0 ( x ) f _ { n - 1 } ( x ) + f _ 1 ( x ) f _ { n - 2 } ( x ) + \\cdots + f _ { n - 1 } ( x ) f _ 0 ( x ) . \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} \\mathfrak { R } _ 2 : = - \\rho ^ { - 1 } B ^ { - 1 } \\mathfrak { R } _ 1 B = - \\varepsilon ^ { 2 } \\Pi _ S ^ { \\perp } \\partial _ x \\mathcal { R } _ 2 + \\mathcal { R } _ * \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} K ^ { g + 1 } ( x _ { \\ell _ 1 } , w _ 1 ) = \\P ( [ X _ { g + 1 } ] _ b = w _ 1 | X _ 0 = x _ { \\ell _ 1 } ) . \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} X = \\xi ^ i ( x , [ u ] ) \\frac { \\partial } { \\partial x ^ i } + \\phi ^ { \\alpha } ( x , [ u ] ) \\frac { \\partial } { \\partial u ^ { \\alpha } } \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\nabla _ { F _ 1 } ( M _ 1 - L D ^ { - 1 } U ) = G _ 1 B _ 1 ^ * , \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} \\left ( \\boldsymbol { \\sigma } \\cdot \\boldsymbol { a } \\right ) \\left ( \\boldsymbol { \\sigma } \\cdot \\boldsymbol { b } \\right ) = \\left ( \\boldsymbol { a } \\cdot \\boldsymbol { b } \\right ) I _ { 2 \\times 2 } + i \\ , \\boldsymbol { \\sigma } \\cdot \\left ( \\boldsymbol { a } \\times \\boldsymbol { b } \\right ) , \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} c _ 4 ( E _ 1 ) = 2 ^ 4 ( 4 w ^ 2 - 3 u ^ p ) , c _ 6 ( E _ 1 ) = 2 ^ 6 w ( 9 u ^ p - 8 w ^ 2 ) , \\Delta ( E _ 1 ) = - 2 ^ 6 \\ell ( u v ) ^ { 2 p } \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ 0 = \\int _ { \\Omega } \\frac { u ( \\tau ) v ( \\tau ) } { 1 - \\tau ^ 2 } \\ ; d \\tau . \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} \\hat { g } ( \\theta _ { 0 } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\psi ( W _ { i } , \\gamma _ { 0 } , \\alpha _ { 0 } , \\theta _ { 0 } ) + o _ { p } ( n ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{gather*} u = - m _ { 1 , 1 2 } = m _ { 1 , 2 1 } . \\end{gather*}"}
-{"id": "3425.png", "formula": "\\begin{align*} \\lambda _ a ( n ) = \\begin{cases} 1 & n ~ , ~ p | n \\Rightarrow p \\equiv a \\bmod q , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} \\psi _ k = ( \\mathcal { A } ^ T - \\mathrm { I } ) e ^ { \\mathrm { i } k x } = \\varepsilon \\partial _ x ( \\beta _ 1 e ^ { \\mathrm { i } k x } ) + O ( \\varepsilon ^ 2 ) = \\varepsilon ( \\beta _ 1 ) _ x e ^ { \\mathrm { i } k x } + \\varepsilon \\beta _ 1 \\ , \\partial _ x e ^ { \\mathrm { i } k x } + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} { h } ( \\theta ) = A ( e ^ { i \\theta } ) - \\left ( \\overline { C ( e ^ { i \\theta } ) } ) \\right ) ^ { < \\frac { 1 } { \\alpha - 1 } > } \\left ( f ( \\theta ) \\right ) ^ { \\frac { - 1 } { \\alpha - 1 } } , \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} & 2 \\log \\left ( 1 + \\lambda ( 1 - \\lambda \\rho ) w _ 2 P ' + \\lambda \\rho ( 1 - \\lambda \\rho ) w _ 1 P ' \\right ) \\geq \\\\ & \\log \\left ( 1 + \\lambda ( 1 - \\lambda \\rho ) w _ 2 P ' \\right ) + \\log \\left ( 1 + \\lambda \\rho ( 1 - \\lambda \\rho ) w _ 1 P ' \\right ) . \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} e _ b ( x , w ) = \\exp \\left ( x \\sum _ { i = 0 } ^ { \\infty } w ^ { b ^ i } \\right ) \\prod _ { i = 0 } ^ { \\infty } \\left ( \\frac { \\Gamma ( b , x w ^ { b ^ i } ) } { ( b - 1 ) ! } \\right ) \\simeq \\exp \\left ( x \\sum _ { i = 0 } ^ { \\infty } w ^ { b ^ i } \\right ) \\simeq e ^ { x w + x w ^ { b } } \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} [ Q ] = \\frac { [ \\mathbb { I I I } ^ 1 ( T ) ] \\prod _ p \\frac { [ H ^ 0 ( \\hat { \\mathbb { Z } } , H ^ 1 ( I _ p , \\hat { T } ) ) ^ { D } ] } { [ S _ p ] } [ R ] } { [ \\mathrm { c o k } \\Delta ] } = \\frac { [ \\mathbb { I I I } ^ 1 ( T ) ] [ \\mathbb { I I I } ^ 2 ( T ) ] [ R ] \\prod _ p [ H ^ 0 ( \\hat { \\mathbb { Z } } , H ^ 1 ( I _ p , \\hat { T } ) ) ^ { D } ] } { [ H ^ 1 ( K , \\hat { T } ) ] \\prod _ p [ S _ p ] } . \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} \\chi _ \\alpha ( \\rho \\vee \\eta ) & = - \\sum \\chi _ { \\alpha _ { ( 1 ) } } ( \\rho ) \\times \\chi _ { \\alpha _ { ( 2 ) } } ( \\eta ) \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} \\mathcal { L } _ X ( f ) : = \\dfrac { d } { d t } ( \\phi _ t ^ * f ) | _ { t = 0 } = \\lim _ { t \\rightarrow 0 } \\dfrac { \\phi _ t ^ * f - f } { t } , \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} ( \\gamma ^ { - 1 } \\eta ^ { - 1 } \\Gamma \\eta \\gamma ) \\cap \\Gamma \\cap \\mathcal { U } ( K ) = ( \\eta ^ { - 1 } \\Gamma \\eta ) \\cap \\Gamma \\cap \\mathcal { U } ( K ) , \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{align*} a u _ { x x } ( 1 , t ) + d u _ { x } ( 1 , t ) + b u ( 1 , t ) = 0 0 < t \\leq T \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\sum _ { \\ell \\in I ( k , k ' ) } F ( k \\ell ) \\overline { F ( k ' \\ell ) } = \\sum _ { Q \\leq q , q ' < 2 Q } \\sum _ { \\substack { \\ell \\in I ( k , k ' , q , q ' ) \\\\ k \\ell \\equiv a _ q \\pmod { q } \\\\ k ' \\ell \\equiv a _ { q ' } \\pmod { q ' } } } \\psi _ q ( ( k \\ell - a _ q ) / q ) \\overline { \\psi _ { q ' } ( ( k ' \\ell - a _ { q ' } ) / q ' ) } , \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\{ \\mathbf { h } _ l ^ { ( g _ k ) } \\left ( \\mathbf { h } _ { l ' } ^ { ( g ' _ { k ' } ) } \\right ) ^ H \\right \\} = \\rho _ l ^ { ( g ) } \\mathbf { R } _ l ^ { ( g ) } \\delta _ { g g ' } \\delta _ { k k ' } \\delta _ { l l ' } , \\textrm { w h e r e } \\sum _ { l = 0 } ^ { L _ g - 1 } \\rho _ l ^ { ( g ) } = 1 , \\ ; \\operatorname { T r } \\left \\{ \\mathbf { R } _ l ^ { ( g ) } \\right \\} = 1 \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ p ) ^ s u = \\lambda _ n | u | ^ { p - 2 } u + \\dfrac { f ( x ) } { \\| u _ n \\| _ { \\scriptstyle L ^ { \\infty } ( \\Omega ) } } & \\Omega , \\\\ u = 0 & \\Omega ^ c . \\end{cases} \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} P _ { \\xi , h } ( \\mu ) = \\frac { 1 } { 2 } \\left ( \\int _ { 0 } ^ { 1 } \\xi '' \\phi _ { \\mu } \\ , d s + \\int _ { 0 } ^ { 1 } \\left ( \\frac { 1 } { \\phi _ { \\mu } } - \\frac { 1 } { 1 - s } \\right ) \\ , d s + h ^ { 2 } \\phi _ { \\mu } ( 0 ) \\right ) . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} P ( K _ { n } = k | K _ { m } = l ) = \\frac { d _ { n , k } ^ { m , l } } { d ^ { m , l } } d ^ { n , k } , \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} h _ { K _ { N , \\ell , q } } ( \\theta ) & \\geq N ^ { - 1 / q } \\bigg ( \\sum _ { j = 1 } ^ k \\sum _ { i \\in \\sigma _ j } | \\langle X _ i , \\theta \\rangle | ^ q \\bigg ) ^ { 1 / q } \\cr & \\geq N ^ { - 1 / q } \\bigg ( \\sum _ { j = 1 } ^ k \\Big ( \\max _ { i \\in \\sigma _ j } | \\langle X _ i , \\theta \\rangle | \\Big ) ^ q \\ , \\bigg ) ^ { 1 / q } . \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} R _ { \\ast } ^ t \\overline e _ { t , k _ j } = - \\sum _ { i = 1 } ^ { n - 1 } R ^ t ( e _ { t , i } , \\overline e _ { t , i } ) \\overline e _ { t , k _ j } . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} D ( f , s ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { f ( n ) } { n ^ s } \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} ( \\nabla _ X \\alpha ) ( Y ) = - \\alpha ( \\nabla _ X Y ) \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} \\mathcal { B } \\left ( m - \\sum _ { j = 2 } ^ { \\left \\lfloor x / 2 \\right \\rfloor } \\mathcal { B } \\left ( \\mathbf { d } _ j ( 0 , x ) , \\frac { 1 } { 2 } \\right ) , \\frac { 1 } { 2 } \\right ) \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} \\sharp ( T N ^ { \\circ } ) = T N \\cap \\mathcal { C } , \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} Z _ T ( G ) ( q , w ) : = \\sum _ { A \\subseteq E } q ^ { k ( A ) } w ^ { | A | } , \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} \\lim _ { z \\to \\pm \\infty } \\frac { \\chi ( z ) } { z } = 1 . \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} f \\left ( x \\right ) = x ^ { 4 } - 2 c x ^ { 3 } + 2 x ^ { 2 } + 2 c x + 1 , \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} \\Big \\| \\sup _ { N \\ge 1 } \\big | \\sum _ { n = 1 } ^ N \\alpha _ n D ( \\lambda _ n t ) \\big | \\ , \\Big \\| _ { \\S ^ 2 } \\le C \\Big ( \\sum _ { n \\ge 1 } | \\beta _ n | \\Big ) \\Big ( \\sum _ { n \\ge 1 } \\big ( \\sum _ { k \\ , : \\ , n \\le \\lambda _ k < n + 1 } | \\alpha _ k | \\big ) ^ 2 \\Big ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} ( f ^ n ) ^ \\# ( \\xi ) > \\frac { L } { 1 + K ^ 2 } = \\frac { K } { r ( 1 + K ^ 2 ) } \\log \\frac { M ^ { n - m } ( R , f ) } { K } . \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} F = S _ { p _ 1 } \\times \\cdots \\times S _ { p _ s } . \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} \\dot { u } _ { i } ( t ) = u _ { i + 1 } ( t ) - 2 u _ { i } ( t ) + u _ { i - 1 } ( t ) + u _ i ( t ) f ( t , u _ { i } ( t ) ) , i \\in \\Z , \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{align*} V _ t ^ i : = e ^ { \\int _ 0 ^ t \\Lambda \\big ( s , \\tilde \\xi ^ { i , N } _ { s } , u ^ { S ^ N ( \\tilde { \\xi } ) } _ { s } ( \\tilde \\xi ^ { i , N } _ { s } ) \\big ) d s } \\textrm { a n d } \\tilde V _ t ^ i : = e ^ { \\int _ 0 ^ t \\Lambda \\big ( r ( s ) , \\tilde \\xi ^ { i , N } _ { r ( s ) } , \\tilde { u } _ { r ( s ) } ( \\tilde \\xi ^ { i , N } _ { r ( s ) } ) \\big ) d s } \\ , \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} w = \\frac { ( u ^ { \\frac { m + 1 } 2 } - k ^ { \\frac { m + 1 } 2 } ) _ - } { k ^ { \\frac { m + 1 } 2 } } . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} d = \\frac { 1 } { \\lambda ^ 2 } \\ , . \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} \\psi _ \\beta ( w ) = \\frac 1 2 \\beta ^ 2 b ^ 2 w ^ { 2 b - 2 } + \\beta w ^ { b } \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} \\Phi _ B ( u ) = u + \\Psi ( u ) , \\Psi ( u ) : = \\Pi _ E \\Psi ( \\Pi _ E u ) , \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} \\begin{aligned} & \\gamma _ \\rho ( t , x ) = \\rho ^ a \\gamma ( \\rho ^ { - 1 } t , \\rho ^ \\lambda x ) , v _ \\rho ( t , x ) = \\rho ^ b v ( \\rho ^ { - 1 } t , \\rho ^ \\lambda x ) , \\\\ & s = \\rho ^ { - 1 } t , y = \\rho ^ \\lambda x . \\end{aligned} \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\frac 1 { \\beta ( n ) } = K _ 1 \\log x + K _ 2 + O ( x ^ { - 1 + \\varepsilon } ) , \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} K _ \\sigma : = \\overline { \\mathring S _ \\sigma } , \\Gamma _ \\sigma : = \\overline { S _ \\sigma \\setminus K _ \\sigma } = \\bigcup _ { ( l , a ) \\in I _ \\sigma } \\delta _ { l , a } \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} \\frac { d } { d t } D \\varphi _ t ( \\omega , x ) = D b ( \\varphi _ t ( \\omega , x ) ) D \\varphi _ t ( \\omega , x ) , D \\varphi _ 0 ( \\omega , x ) = I d . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} w ^ { p ^ 2 } = [ a , d ] . \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} \\eta ( t , x ) = A ( t ) e _ { 1 , 0 } + A ^ \\ast ( t ) e _ { - 1 , 0 } + H [ A , A ^ \\ast ] ( x ) , \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} i \\hat { u } _ t = k \\hat { L } \\hat { u } , \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} \\sum _ { T \\subseteq U } ( - 1 ) ^ { | T | } v _ { i , T } = 0 \\forall v = ( v _ T ) _ T \\in V , i = 0 , 1 . \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} \\partial g ^ + ( v ) = \\begin{cases} \\{ 0 \\} & v < L , \\\\ \\{ 1 \\} & v > L , \\\\ [ 0 , 1 ] & v = L , \\end{cases} \\qquad \\partial g ^ - ( v ) = \\begin{cases} \\{ - 1 \\} & v < U , \\\\ \\{ 0 \\} & v > U , \\\\ [ - 1 , 0 ] & v = U . \\end{cases} \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} H ( q , q _ k ^ { 0 } , q _ k ^ { 1 } , \\dots , q _ n ^ { 0 } , q _ n ^ { 1 } ) : = & g _ { t \\alpha } ( q + x _ 0 + \\dots + x _ { k - 1 } ) \\\\ & h _ { t \\alpha _ j } ( q _ j ^ { 0 } - x _ j ^ { 0 } ) { h } _ { t \\alpha _ j } ( q _ j ^ { 1 } - x _ j ^ { 1 } ) \\Big ( \\prod _ { \\substack { i = k \\\\ i \\neq j } } ^ { n } g _ { t \\alpha _ i } ( q _ i ^ { 0 } - x _ i ^ { 0 } ) g _ { t \\alpha _ i } ( q _ i ^ { 1 } - x _ i ^ { 1 } ) \\Big ) \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\| i \\| \\leq ( \\pi / 2 ) ^ { ( s - 1 ) / 2 } \\prod _ { j = 1 } ^ s \\| i _ j \\| , \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } \\int _ { \\Omega } \\frac { 1 } { p ( x ) } \\left \\vert \\nabla u _ { n } \\right \\vert ^ { p ( x ) } d x = \\int _ { \\Omega } \\frac { 1 } { p ( x ) } \\left \\vert \\nabla u \\right \\vert ^ { p ( x ) } d x . \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} { \\mathbb E } A _ \\beta ^ p ( t ) = e ^ { - [ ( 2 \\beta - 1 ) p + 1 ] t } + \\beta ^ p \\int _ 0 ^ t e ^ { - [ ( 2 \\beta - 1 ) p + 1 ] s } { \\mathbb E } ( A _ \\beta ^ + ( t - s ) + A _ \\beta ^ - ( t - s ) ) ^ p d s . \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\kappa ^ * ( n ) = \\frac 1 { 2 } \\widetilde { C } x ^ 2 + O ( R _ { \\kappa ^ * } ( x ) ) , \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} S ^ { b e l t } ( \\ell ) & \\le S ( 3 ; 0 , 0 ) S ( 3 ; 1 , 3 ) ^ { \\ell - 1 } \\\\ & \\le S ( 3 ; 0 , 0 ) S _ U ( 3 ; 1 , 3 ) ^ { \\ell - 1 } = : S _ U ^ { b e l t } ( \\ell ) \\\\ & \\approx 1 0 ^ { 2 1 . 8 2 4 1 + 1 4 . 0 5 2 0 ( \\ell - 1 ) } . \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} e _ { r p } ^ p ( \\underline { x } ^ { ( 1 ) } , \\dots , \\underline { x } ^ { ( k ) } ) = \\sum _ { j _ 1 + \\dots + j _ k = r p } e _ { j _ 1 } ^ p ( \\underline { x } ^ { ( 1 ) } ) \\cdots e _ { j _ k } ^ p ( \\underline { x } ^ { ( k ) } ) . \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} \\max \\limits _ { f \\in D _ f } \\Delta \\left ( h ( f _ 0 ) ; f \\right ) = \\Delta \\left ( h ( f _ 0 ) ; f _ 0 \\right ) , \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} F = \\bigcup _ { i \\in \\Lambda } T _ i ( F ) . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} \\rho ^ 2 \\leq \\frac { 2 H ^ 2 n } { 8 ( H ^ 2 ) ^ 2 } = \\frac { n } { 4 H ^ 2 } < \\left ( \\frac { n } { H ^ 2 } - \\frac { 1 } { 2 } \\right ) ^ 2 . \\end{align*}"}
-{"id": "480.png", "formula": "\\begin{align*} \\frac { \\partial ^ k u ^ { \\alpha } } { \\partial ( x ^ 1 ) ^ k } = f ^ { \\alpha } \\left ( x , n , [ u ] _ { x ^ 1 } , \\left [ \\frac { \\partial ^ 1 u } { \\partial ( x ^ 1 ) ^ 1 } \\right ] _ { x ^ 1 } , \\left [ \\frac { \\partial ^ 2 u } { \\partial ( x ^ 1 ) ^ 2 } \\right ] _ { x ^ 1 } , \\ldots , \\left [ \\frac { \\partial ^ { k - 1 } u } { \\partial ( x ^ 1 ) ^ { k - 1 } } \\right ] _ { x ^ 1 } \\right ) , \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} A _ { 1 1 } z + A _ { 1 2 } v = 0 . \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} G ^ * ( \\{ \\log n \\} ) = c \\{ n ( \\log n ) ( \\log \\log n ) ^ 2 \\} ^ { - 1 } \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\zeta ( x , s ) & = 2 ( 2 \\pi ) ^ { s - 1 } \\Gamma ( 1 - s ) \\sin \\left ( \\pi s / 2 \\right ) \\Psi ( x , 1 - s ) . \\end{align*}"}
-{"id": "740.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { p } } ^ { \\vee } = \\{ u \\in \\mathfrak { g } ( ( z ) ) \\mid \\kappa ( u , v ) \\in \\mathbb { C } [ [ t ] ] , \\ , \\ \\forall v \\in \\mathfrak { p } \\} \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} | A _ n | \\le \\frac { 1 } { 2 \\pi } \\int _ { | z | = R } \\frac { | P _ { n _ m } ( z ) | \\ , | d z | } { R ^ { n + 1 } } \\le q ^ n , c n _ m \\le n \\le n _ m . \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\deg ( f , G , a ) = \\deg ( f , G _ 1 , a ) + \\deg ( f , G _ 2 , a ) . \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} R _ \\mathsf { A l t } ^ \\mathrm { E n c - D e c - C S I } = \\max _ { Q _ { U , V , X | S } } R _ \\mathsf { A l t } ^ \\mathrm { E n c - D e c - C S I } \\left ( Q _ { U , V , X | S } \\right ) , \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\int _ \\Omega \\Big ( E _ { 1 1 } \\phi _ { 1 1 } ' & + \\sum _ { i = 1 , j = 2 } ^ 3 E _ { i j } \\phi _ { i j } \\ , + x _ 2 E _ { 1 1 } \\Phi _ { 1 2 } ' ( x _ 1 ) + x _ 3 E _ { 1 1 } \\Phi _ { 1 3 } ' ( x _ 1 ) \\\\ & + x _ 3 E _ { 1 2 } \\Phi ' _ { 2 3 } ( x _ 1 ) - x _ 2 E _ { 1 3 } \\Phi ' _ { 2 3 } ( x _ 1 ) \\Big ) \\dd x = - \\int _ 0 ^ L ( f _ 2 \\tilde \\Phi _ { 1 2 } + f _ 3 \\tilde \\Phi _ { 1 3 } ) \\dd x _ 1 \\ , . \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} P ( b ) & = \\prod _ { i = 1 } ^ N \\left [ \\Sigma \\left ( \\frac { 2 \\pi m _ i } { b } \\right ) ^ { 3 / 2 } \\frac { 1 - \\exp ( - m _ i g b h ) } { m _ i g b } \\right ] \\ , , \\\\ \\rho _ b & = \\frac { 1 } { P ( b ) } \\exp \\left [ - b \\sum _ { i = 1 } ^ N \\left ( \\frac { \\Vert \\vect { p _ i } \\Vert ^ 2 } { 2 m _ i } + m _ i g z _ i \\right ) \\right ] \\ , . \\\\ \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} \\hat { u } ( t ) = ( e ^ { - t | \\xi | } + t e ^ { - t | \\xi | } | \\xi | ) \\hat { u } _ { 0 } + t e ^ { - t | \\xi | } \\hat { u } _ { 1 } \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} \\varrho = n ^ { - 4 } \\hat \\varrho \\left ( 1 + \\frac { 3 } { 2 } \\left ( n ^ { - 2 } \\varpi ^ 2 - 1 \\right ) \\right ) . \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} - \\tilde { W } _ 1 \\tilde { W } _ D ( \\tilde { W } _ 2 \\tilde { W } _ D ) ^ { - 1 } = - \\tilde { W } _ 1 ( \\tilde { W } _ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\frac { d } { d \\epsilon } G _ { \\epsilon } ( z ) = G _ { \\epsilon } ( z ) i \\tilde { S } G _ { \\epsilon } ( z ) \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} \\begin{aligned} & \\norm { \\varphi } _ { L ^ { \\infty } ( 0 , T ; H ^ { 2 } ) \\cap L ^ { 2 } ( 0 , T ; H ^ { 3 } ) \\cap H ^ { 1 } ( 0 , T ; L ^ { 2 } ) } \\\\ & + \\norm { \\mu } _ { L ^ { 2 } ( 0 , T ; H ^ { 2 } ) \\cap L ^ { \\infty } ( 0 , T ; L ^ { 2 } ) } + \\norm { \\sigma } _ { L ^ { 2 } ( 0 , T ; H ^ { 2 } ) \\cap L ^ { \\infty } ( 0 , T ; H ^ { 1 } ) \\cap H ^ { 1 } ( 0 , T ; L ^ { 2 } ) } \\leq \\overline { C } , \\end{aligned} \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} \\begin{aligned} E ^ { ( 3 ) } _ { h i g h } ( w , r ) : = & E _ { l i n } ^ { ( 2 ) } ( w , r ) - \\frac { 1 } { 4 } E ^ { ( 2 ) } _ { \\omega , l i n } ( w , r ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{gather*} F _ { 6 } \\big ( 3 ^ { - 2 / 3 } t \\big ) = \\frac { ( q _ 2 - 1 ) } { 2 q _ 2 } \\exp \\left ( \\frac { 1 } { 3 } \\int _ { t } ^ { \\infty } \\omega ( s ) d s - \\frac { 2 } { 3 } \\int _ { t } ^ { \\infty } \\frac { u _ s ( s ) } { u ( s ) } \\frac { 1 + q _ 2 ( s ) } { q _ 2 ( s ) } d s \\right ) . \\end{gather*}"}
-{"id": "1521.png", "formula": "\\begin{align*} \\big | \\big | S ^ { 1 / 2 } G _ { \\epsilon } ( z ) B ^ * f \\big | \\big | _ { L ^ 2 } ^ 2 & = Q _ { [ H , i A ] } ( G _ { \\epsilon } ( z ) B ^ * f , G _ { \\epsilon } ( z ) B ^ * f ) \\\\ & \\le \\frac { 1 } { \\epsilon } \\Big ( \\epsilon Q _ { [ H , i A ] } ( G _ { \\epsilon } ( z ) B ^ * f , G _ { \\epsilon } ( z ) B ^ * f ) + \\mbox { I m } ( z ) | | G _ { \\epsilon } ( z ) B ^ * f | | ^ 2 \\Big ) , \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} \\sigma ^ 2 ( Y ) = \\begin{pmatrix} q ^ { 1 / 2 } x & 0 \\\\ 0 & q ^ { 3 / 2 } x \\end{pmatrix} Y , \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} | \\zeta | \\geq \\lambda ^ * ( \\Delta ) = \\frac { ( \\Delta - 1 ) ^ { \\Delta - 1 } } { \\Delta ^ { \\Delta } } > \\frac { 1 } { e ( \\Delta - 1 ) } , \\end{align*}"}
-{"id": "9413.png", "formula": "\\begin{align*} d f = \\sum _ i d x ^ i \\cdot \\frac { \\partial f } { \\partial x ^ i } + \\sum _ j d \\xi ^ j \\cdot \\frac { \\partial f } { \\partial \\xi ^ j } \\ ; . \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} x _ i ^ t = x _ i ^ { - 1 } \\mbox { a n d } y _ i ^ t = y _ i ^ { - 1 } \\mbox { f o r } i = 1 , 2 . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\beta = \\frac { \\Lambda - \\lambda _ { \\textnormal { m i n } } } { \\lambda _ { \\textnormal { m a x } } - \\lambda _ { \\textnormal { m i n } } } , \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} E ^ { ( a ) } 1 _ n : = { \\theta } ^ { ( a ) } 1 _ n \\otimes 1 , F ^ { ( a ) } 1 _ { n } : = \\vartheta ^ { ( a ) } 1 _ n \\otimes 1 . \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} A B = A ^ 2 B _ o \\subset F A B _ o = F B . \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} \\begin{array} { r l } Z ^ { I P } = \\max \\ \\ ; & c ' z + d ' u \\\\ \\mbox { s . t . } \\ \\ ; & A z \\leq b , \\\\ & H z + G u \\leq h , \\\\ & z \\in \\{ 0 , 1 \\} ^ n . \\end{array} \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rm F A } + \\mathbb { P } _ { \\rm M D } & = E _ U \\left [ \\mathbb { P } _ { \\rm F A } ( U ) + \\mathbb { P } _ { \\rm M D } ( U ) \\right ] \\\\ & \\geq E _ U \\left [ \\mathbb { P } _ { \\rm F A } ( U ) + \\mathbb { P } _ { \\rm M D } ( U ) | { \\mathcal { A } ^ c } \\right ] P ( { \\mathcal { A } ^ c } ) \\\\ & > 1 - \\epsilon . \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} \\hat { h } ( \\Delta ) = \\begin{cases} 3 h ( \\Delta ) & \\Delta > 0 \\\\ h ( \\Delta ) & \\Delta < 0 . \\end{cases} \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} a _ 0 = 1 , \\ \\ \\ \\ \\ \\ \\ a _ n = [ a _ { n - 1 } \\exp ( W _ n ) ] , \\ \\ \\ \\ \\ \\ \\ A _ n = a _ n + 1 , \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} u = Q _ { 1 } \\left ( x , y , z \\right ) , \\ \\ v = Q _ { 2 } \\left ( x , y , z \\right ) , \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} B ^ T \\bar Y + D ^ T \\bar Z + G ^ T \\bar Z _ 0 + ( R + \\bar R ) \\bar v + ( S + \\bar S _ 1 ) \\bar X + r = 0 . \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} \\alpha = ( \\underbrace { 0 , \\ldots , 0 } _ n , r _ n , r _ { n + 1 } , \\ldots ) \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} \\widetilde { c } \\theta + \\widetilde { c ' } \\theta ' = \\theta ' \\cdot \\dim ( \\ker ( F ) ) > 0 \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} D _ { { - c } } ( \\zeta ) : = \\frac { e ^ { \\frac { \\zeta ^ 2 } { 4 } } } { \\mathrm { i } \\sqrt { 2 \\pi } } \\int _ { \\epsilon - \\mathrm { i } \\infty } ^ { \\epsilon + \\mathrm { i } \\infty } e ^ { - \\zeta s + \\frac { s ^ 2 } { 2 } } s ^ { - c } d s , \\epsilon > 0 . \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} \\binom { k + ( i + j ) p } { m + i p } & = \\sum _ { l = 0 } ^ { k } \\binom { ( i + j ) p } { m - l + i p } \\binom { k } { l } \\\\ & = \\binom { i + j } { j } \\binom { k } { m } + \\sum _ { l = 1 } ^ { m } \\binom { ( i + j ) p } { l + i p } \\binom { k } { m - l } + \\sum _ { l = 1 } ^ { k - m } \\binom { ( i + j ) p } { l + j p } \\binom { k } { m + l } . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} S _ j e _ i = \\left \\{ \\begin{array} { r c } e _ i & i = j \\\\ - e _ i & i \\ne j \\end{array} \\right \\} \\ , . \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} \\begin{cases} M _ { R } = 1 , \\ldots , N _ { S } ; \\ , M _ { D } = 0 , \\ldots , N _ { S } ; \\\\ M _ { R D } = \\max ( 0 , M _ { R } + M _ { D } - N _ { S } ) , \\ldots , \\min ( M _ { R } , M _ { D } ) . \\end{cases} \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\frac { d ^ 2 } { d s ^ 2 } \\mathrm { s n } \\left ( \\gamma _ { \\partial B } ( s ) , x _ 0 \\right ) = \\frac { d } { d s } \\mathrm { s n } \\left ( b \\left ( \\gamma _ { \\partial B } ( s ) \\right ) , x _ 0 \\right ) = \\frac { d } { d s } \\left [ b \\left ( \\gamma _ { \\partial B } ( s ) \\right ) , x _ 0 \\right ] = - \\rho ( s ) . \\left [ \\gamma _ { \\partial B } ( s ) , x _ 0 \\right ] = \\\\ = - \\rho ( s ) . \\mathrm { s n } \\left ( \\gamma _ { \\partial B } ( s ) , x _ 0 \\right ) , \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} N _ b = \\inf \\{ n \\ge 1 : \\widetilde { S } _ { n } ^ { \\rho } \\ge b \\} , \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} T _ { j , j + 1 } g _ { j + 1 } ^ { \\prime } ( z _ { j + 1 } ^ \\ast ) = \\beta , \\quad \\forall ~ j = \\overline { 1 , n } \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} p ( G ) ( z ) : = \\sum _ { i = 0 } ^ { d ( G ) } e _ { i } ( G ) z ^ { i } \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} \\mathcal { C } _ i ^ { \\mathrm { N } } = \\mathcal { C } _ i ^ { \\mathrm { E } } \\setminus \\mathcal { C } _ { i } . \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} \\mathbb { E } [ e ^ { t ( M _ k - m _ 0 ) } ] \\leq \\exp \\left ( \\frac { t ^ { 2 } } { 2 } g ( t m ) \\sum _ { i = 1 } ^ k \\sigma _ i ^ 2 \\right ) . \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} \\mathrm { c m } ( x , y ) = \\frac { [ y , b ( x ) ] } { | | y | | } . \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} \\hat { \\theta } _ { j } = \\sum _ { l = 1 } ^ { r } p _ { j l } \\theta _ { k _ l } + \\xi _ { j } , \\ \\forall 1 \\leq j \\leq k . \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} ( \\widetilde { L } _ { b c } ^ 0 - B ) ( I + \\Gamma _ { m , b c } X _ * ) = ( I + \\Gamma _ { m , b c } X _ * ) ( \\widetilde { L } _ { b c } ^ 0 - J _ { m , b c } X _ * ) . \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} \\mathcal { N } _ 2 ( x , u , u _ x , u _ { x x } , u _ { x x x } ) : = - \\partial _ x [ ( \\partial _ u f ) ( x , u , u _ x ) - \\partial _ x ( ( \\partial _ { u _ x } f ) ( x , u , u _ x ) ) ] \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { f ( n ) } { n ^ s } = D ( f , s ) \\left ( 2 \\left ( \\sum _ { \\nu = 0 } ^ { \\infty } \\frac { f ( 2 ^ \\nu ) } { 2 ^ { \\nu s } } \\right ) ^ { - 1 } - 1 \\right ) , \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} R _ \\nabla \\cdot \\Psi = 0 , F _ A \\cdot \\Psi = 0 . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} N _ { ( 2 , 3 ) } ( n ) & = 8 \\ , \\sigma ( \\frac { n } { 2 } ) - 3 2 \\ , \\sigma ( \\frac { n } { 8 } ) + 8 \\ , \\sigma ( \\frac { n } { 3 } ) - 3 2 \\ , \\sigma ( \\frac { n } { 1 2 } ) + 6 4 \\ , W _ { ( 1 , 3 ) } ( n ) + 1 0 2 4 \\ , W _ { ( 1 , 3 ) } ( \\frac { n } { 4 } ) \\\\ & - 2 5 6 \\ , \\biggl ( W _ { ( 3 , 8 ) } ( n ) + W _ { ( 1 , 1 2 ) } ( n ) \\biggr ) , \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\displaystyle \\nu = | w | ^ { 2 ^ { * } _ { \\alpha } } + \\sum _ { j \\in J } \\nu _ { j } \\delta _ { x _ { j } } , \\ ; \\ ; \\ ; \\displaystyle \\mu \\geq y ^ { 1 - 2 \\alpha } | \\nabla w _ 0 | ^ { 2 } + \\sum _ { j \\in J } \\mu _ { j } \\delta _ { x _ { j } } , \\ ; \\ ; \\ ; \\displaystyle \\mu _ { j } \\geq S ( \\alpha , n ) \\nu _ { j } ^ { \\frac { 2 } { 2 ^ * _ { \\alpha } } } . \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} \\lambda _ { c _ n \\cdots c _ 1 } ( \\theta ) & = \\frac { d ( c _ n \\cdots c _ 1 \\underline a , c _ n \\cdots c _ 1 \\underline b ) } { d ( c _ m \\cdots c _ 1 \\underline a , c _ m \\cdots c _ 1 \\underline b ) } \\frac { d ( c _ m \\cdots c _ 1 \\underline a , c _ m \\cdots c _ 1 \\underline b ) } { d ( \\underline a , \\underline b ) } \\\\ & = \\lambda _ { c _ n \\cdots c _ { m + 1 } } ( \\phi _ { c _ 1 \\cdots c _ m } ^ { - 1 } ( \\theta ) ) \\lambda _ { c _ m \\cdots c _ 1 } ( \\theta ) . \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} \\partial \\omega = i \\{ \\partial ^ { { \\rm E n d } ( H ) } \\theta , \\theta \\} , \\end{align*}"}
-{"id": "4332.png", "formula": "\\begin{align*} f = \\sum _ { | K | \\leq d } a _ { i j } ^ K f _ { i j , K } , \\mbox { w h e r e } f _ { i j , K } \\in \\mathbb { C } [ \\{ { } _ { ( \\alpha ) } a _ { s t } : ( s , t ) \\not = ( i , j ) \\} ] , a _ { i j } ^ K : = \\prod _ { \\alpha \\in Q _ 1 } { } _ { ( \\alpha ) } a _ { i j } ^ { k _ { \\alpha } } , \\mbox { a n d } | K | = \\sum _ { \\alpha = 1 } ^ { Q _ 1 } k _ { \\alpha } . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} \\frac { 4 \\left ( 3 c ^ 8 - 1 6 0 c ^ 4 - 5 1 2 c ^ 2 + 2 0 4 8 \\right ) } { 3 \\left ( c ^ 2 + 4 \\right ) ^ 2 } = 0 . \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} r \\phi * ( a + b ) & = r \\phi * r \\phi _ d ^ { - 1 } * r \\phi _ d * ( a + b ) \\\\ & = r \\phi * r \\phi _ d ^ { - 1 } * ( r \\phi _ d * a + r \\phi _ d * b ) \\\\ & = r \\phi * r \\phi _ d ^ { - 1 } * ( r \\phi _ d * r * a + r \\phi _ d * r * b ) \\\\ & = r \\phi * r \\phi _ d ^ { - 1 } * r \\phi _ d * ( r * a + r * b ) \\\\ & = r \\phi * ( r * a + r * b ) \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 + \\lambda t ) ^ { \\frac { x + y } { \\lambda } } d \\mu _ { q } ( x ) = \\frac { \\log ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } } { ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - 1 } ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } ( \\textnormal { s e e } \\ , \\ , [ 7 ] ) , \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} d d ^ * \\eta & = \\left ( - \\frac { 2 } { r ^ 3 } \\tilde { d } ^ * \\eta + \\frac { 1 } { r ^ 2 } \\tilde { d } ^ * \\eta ' \\right ) d r + \\frac { 1 } { r ^ 2 } \\tilde { d } \\tilde { d } ^ * \\eta , \\\\ d ^ * d \\eta & = - \\eta '' - \\frac { m - 3 } { r } \\eta ' - \\frac { 1 } { r ^ 2 } ( \\tilde { d } ^ * \\eta ' ) d r + \\frac { 1 } { r ^ 2 } \\tilde { d } ^ * \\tilde { d } \\eta , \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} & e ^ { - \\theta i ( K - k ) c } < \\sum _ { m = 1 } ^ M q _ m e ^ { - \\theta r _ m } \\\\ \\Rightarrow & e ^ { - \\theta i ( K - k ) c } e ^ { - \\theta ( j - i ) ( K - k ) c } \\\\ < & \\sum _ { m = 1 } ^ M q _ m e ^ { - \\theta \\left ( r _ m + ( j - i ) ( K - k ) c \\right ) } \\\\ \\Rightarrow & e ^ { - \\theta j ( K - k ) c } < \\sum _ { s = 1 } ^ S w _ s e ^ { - \\theta \\left ( r _ m + ( j - i ) ( K - k ) c \\right ) } . \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} \\frac { \\partial S } { \\partial t } + H \\left ( q ^ i , \\frac { \\partial S } { \\partial q ^ i } \\right ) = 0 \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} d ^ c u = ( \\partial _ r u ) d ^ c r - ( J \\partial _ r u ) d r + d ^ c _ b u , \\\\ d d ^ c v = ( \\partial _ r ^ 2 v ) d r \\wedge d ^ c r + ( \\partial _ r v ) d d ^ c r - d ( J \\partial _ r v ) \\wedge d r + d d ^ c _ b v , \\\\ \\partial _ r \\ , \\lrcorner \\ , d d ^ c v = ( \\partial _ r ^ 2 v + r ^ { - 1 } \\partial _ r v + ( J \\partial _ r ) ^ 2 v ) d ^ c r + d _ b ( J \\partial _ r v ) + \\mathcal { L } _ { \\partial _ r } ( d ^ c _ b v ) , \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} u ( x , t ) \\le u _ 0 ( x ) + t \\int _ 0 ^ 1 g ( s ) d s = u _ { 0 , \\bar x } ( x ) + t \\int _ 0 ^ 1 g ( s ) d s \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{gather*} U ( t ) = 6 \\frac { \\kappa _ t } { \\kappa } - 6 \\frac { q _ 2 q _ { 2 t } } { 1 - q ^ 2 _ 2 } - \\frac { t ^ 2 } { 2 } , \\end{gather*}"}
-{"id": "3327.png", "formula": "\\begin{align*} \\big \\{ ( 0 , \\hat x ( 0 ) , q ) \\in \\mathcal { U } _ \\Omega : w ( q ) = 0 \\big \\} \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} ( T _ f \\phi ) ( z ) = \\int _ { Z } f ( z , z ' ) \\phi ( z ' ) \\ , d \\eta ( z ' ) . \\end{align*}"}
-{"id": "8710.png", "formula": "\\begin{align*} G _ t = \\big ( b ( R ) \\ , G \\big ) _ x + \\bigg ( \\frac { \\sigma ( R ) ^ 2 } { 2 } \\ , G \\bigg ) _ { x x } + \\sigma ( R ) \\ , R _ x ^ { 1 / 2 } \\ , \\dot { W } , G ( 0 , \\cdot ) = \\beta ( F _ \\lambda ( \\cdot ) ) , \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} f ( z ) = \\eta E _ \\alpha ( z ) \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} \\lim _ { V \\to \\infty } \\omega ^ { ' } \\left \\{ \\left ( \\frac { 1 } { V } \\ , \\int _ { \\Lambda } d { x } \\ \\tau _ { { x } } ( A ) \\right ) ^ { 2 } - \\left ( \\frac { 1 } { V } \\ , \\int _ { \\Lambda } d { x } \\ \\omega ^ { ' } ( \\tau _ { { x } } ( A ) ) \\right ) ^ { 2 } \\right \\} = 0 \\ . \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\omega ^ { \\kappa \\epsilon B _ { \\epsilon } } ( x , y , z ) = \\exp \\left \\{ - 2 i \\kappa \\epsilon B ( \\epsilon x ) ( y \\wedge z ) \\right \\} \\left ( 1 \\ , + \\ , i \\ , \\kappa \\epsilon ^ 2 \\Psi ^ \\epsilon ( \\epsilon x , y , z ) \\int _ 0 ^ 1 \\exp \\big \\{ i \\ , \\tau \\kappa \\epsilon ^ 2 \\Psi ^ \\epsilon ( \\epsilon x , y , z ) \\big \\} d \\tau \\right ) \\ , . \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} | b | ^ { 2 } \\underbrace { K \\alpha ^ { 2 } } _ { \\not = 0 } = \\overline { a } b + a \\overline { b } K = \\overline { d } e = d \\overline { e } = | e | ^ { 2 } = 0 . \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} K = \\frac { a \\overline { b } } { \\overline { d } e } = \\pm \\frac { a \\overline { b } } { \\overline { a } e } . \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} f ( k ) = F ( k ) - F ( k - 1 ) = k ^ { - \\gamma } - ( k + 1 ) ^ { - \\gamma } . \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} d X _ i = ( A X _ i + B v _ i ) d s + ( C X _ i + D v _ i ) d W _ i + ( F X _ i + G v _ i ) d W _ 0 , \\ \\ X _ i ( t ) = x _ i \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} | | x | | _ { E } ^ { \\mu } = 1 - 2 \\int 1 _ { E } ( t ) 1 _ { E ^ { - 1 } } ( t ^ { - 1 } x ) \\frac { d \\mu } { d \\eta } d \\eta ( t ) . \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} S \\ \\ni \\ v ( s - ( \\epsilon + \\delta ) ' ) \\ = \\ v ( \\delta ' \\delta ) \\ = \\ \\gamma + \\textstyle \\int \\gamma . \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P _ e } ( - 1 ) ^ { \\nu ( \\pi ) } z ^ { \\nu _ o ( \\pi ) } q ^ { | \\pi | } = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( - q z ; q ^ 2 ) _ \\infty } - \\frac { \\tiny 1 } { 1 + z q } , \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} \\int _ \\Omega ( \\lambda + V _ 0 ) u ^ 0 \\varphi \\ , d x = \\int _ \\Omega P g \\varphi \\ , d x , \\quad \\forall \\ , \\varphi \\in X _ 0 ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} ( \\omega + d d ^ c \\varphi ) ^ n = e ^ { f - \\lambda \\varphi } \\omega ^ n . \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} F _ { \\mathcal { B } } = \\mathfrak { R } _ { Y / X } ^ { \\Gamma } ( N _ B \\wedge E _ B ) \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\pi \\in \\P _ o , \\\\ | \\pi | = N , \\\\ \\nu _ o ( \\pi ) = k } } ( - 1 ) ^ { \\nu _ e ( \\pi ) } = \\chi ( N = k ) . \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} f * g ( t ) ( x ) & = \\int _ { \\R } f ( s ) ( x ) ( \\sigma _ s \\otimes \\alpha _ s ) ( g ( t - s ) ) ( x ) ~ d s \\\\ & = \\int _ { \\R } f ( s ) ( x ) \\alpha _ s ( g ( t - s ) ( x - s ) ) ~ d s . \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} X \\times C = \\pi _ p ^ { - 1 } ( J _ o ) , \\textrm { m o d u l o $ \\eta $ - n u l l s e t s } , \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} \\int \\left ( \\eta '' - \\xi '' \\right ) \\left ( \\mbox { \\ensuremath { \\phi _ { \\nu } } - \\ensuremath { \\phi _ { \\mu } } } \\right ) \\ , d t = - \\eta ' ( 0 ) \\cdot ( \\phi _ { \\nu } ( 0 ) - \\phi _ { \\mu } ( 0 ) ) - \\int \\left ( \\eta - \\xi \\right ) d ( \\nu - \\mu ) . \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\nabla _ X Y - \\nabla _ Y X - [ X , Y ] = 0 \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} F _ { w } ( U \\ , ; z ) = T _ 1 + T _ 2 + T _ 3 \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} \\cosh ( r ) = \\cot ( \\theta ) \\cot \\left ( \\frac { n \\pi } { 6 n + 1 } \\right ) . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} v ( x , t ) : = \\min \\left [ u _ 0 ( x ) + t \\int _ 0 ^ 1 g ( s ) d s , \\ t \\left ( \\int _ 0 ^ 1 \\frac { 1 } { g ( s ) } d s \\right ) ^ { - 1 } \\right ] . \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} Q _ n = \\sqrt { \\frac { \\sum \\limits _ { k = 1 } ^ { M _ n } \\left ( \\theta ^ { \\mathrm { A } } _ { n k } - \\bar { \\theta } ^ { \\mathrm { A } } _ n \\right ) ^ 2 P _ { n k } } { \\sum \\limits _ { k = 1 } ^ { M _ n } P _ { n k } } } . \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} ( E , d _ E ) \\mapsto \\mathcal { E } ( E , d _ E ) : = ( E \\otimes _ A A [ u ] , d _ E + \\epsilon ) \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} \\psi _ n ^ { ( \\epsilon ) } ( x ) = e ^ { W ( x ) } \\phi _ n ^ { ( \\epsilon ) } ( x ) , W ( x ) = - \\frac { 1 } { 4 } x ^ 4 - a | x | ^ 3 + b x ^ 2 - c | x | , \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} d _ { \\eta } : = d - \\eta \\wedge \\cdot \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} \\tilde { W } ( x ; \\lambda ) = - ( X ^ - ( x ; \\lambda ) + i Y ^ - ( x ; \\lambda ) ) ( X ^ - ( x ; \\lambda ) - i Y ^ - ( x ; \\lambda ) ) ^ { - 1 } ( R ^ + - i S ^ + ( \\lambda ) ) ( R ^ + + i S ^ + ( \\lambda ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\mu ( P ) : = ( P \\Omega , \\Omega ) _ { \\mathcal { F } _ q ( \\mathcal { H } ) } , P \\in \\mathcal { P } , \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} f ( i , k ) = \\frac { \\binom n i ^ 2 \\binom i k ^ 2 } { \\binom n k ^ 2 \\binom n { i - k } ^ 2 } . \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} g ( z ) = & g ( r _ 3 ) + ( z - r _ 3 ) g ' ( r _ 3 ) + ( z - r _ 3 ) ^ 2 \\int _ 0 ^ 1 ( 1 - t ) g '' ( r _ 3 + t ( z - r _ 3 ) ) d t \\\\ = & g ( r _ 3 ) + ( z - r _ 3 ) g ' ( r _ 3 ) + O \\ ( B _ 2 | z - r _ 3 | ^ 2 \\ ) . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} \\Psi ( s ) : = \\frac { 1 } { 1 + \\beta } \\int _ t ^ s e ^ { \\mu ( \\tau - t ) } \\pi ( \\tau ) d \\tau + \\frac { \\nu _ 0 ^ 2 } { 2 } ( s - t ) + \\nu _ 0 ( W _ 0 ( s ) - W _ 0 ( t ) ) , \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} A ( \\tau , s ) ( c ' ( s ) ) = 0 , A \\wedge A = 0 . \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} \\sum _ { n _ 1 \\cdots n _ l \\leq x } a ( \\boldsymbol { n } ; \\boldsymbol { z } ) = x ( \\log x ) ^ { z _ 1 + \\cdots + z _ l - 1 } \\left \\{ \\sum _ { 0 \\leq j \\leq N } \\frac { ( z _ 1 + \\cdots + z _ l ) h _ j ( z _ 1 , \\cdots , z _ l ) } { ( \\log x ) ^ j } + O _ A \\ ( R _ N ( x ) \\ ) \\right \\} . \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} M ( a ; b ; z ) = { _ 1 F _ 1 } ( a ; b ; z ) = \\frac { \\Gamma ( b ) } { \\Gamma ( a ) } \\sum _ { n = 0 } ^ \\infty \\frac { \\Gamma ( a + n ) } { \\Gamma ( b + n ) n ! } z ^ n . \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} & \\frac { ( n + 1 - 2 b ) ( n - 1 - 2 b ) c _ b } { 2 } \\\\ & = ( n - 1 - 2 b ) z _ { b } - - \\sum _ { i = b + 1 } ^ { n - b - 1 } z _ i \\\\ & = ( n - 1 - 2 b ) \\Delta _ b + ( n - 1 - 2 b ) z _ { b + 1 } - \\sum _ { i = b + 1 } ^ { n - b - 1 } z _ i \\\\ & = ( n - 1 - 2 b ) \\Delta _ b + ( n - 1 - 2 ( b + 1 ) ) z _ { b + 1 } - \\sum _ { i = b + 2 } ^ { n - b - 2 } z _ i \\\\ & = ( n - 1 - 2 b ) \\Delta _ b + \\frac { ( n - 1 - 2 b ) ( n - 3 - 2 b ) c _ { b + 1 } } { 2 } \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} \\frac { M _ k ( x ; \\mathbf { a } ) } { \\frac { 1 } { \\phi ^ k ( q ) } \\frac { k ! } { k _ 1 ! k _ 2 ! \\cdots k _ l ! } S _ k ( x ) } & = 1 + \\frac { M _ k ( x ; \\mathbf { a } ) - \\frac { 1 } { \\phi ^ k ( q ) } \\frac { k ! } { k _ 1 ! k _ 2 ! \\cdots k _ l ! } S _ k ( x ) } { \\frac { 1 } { \\phi ^ k ( q ) } \\frac { k ! } { k _ 1 ! k _ 2 ! \\cdots k _ l ! } S _ k ( x ) } \\\\ & = 1 + \\frac { k - 1 } { \\log \\log x } \\frac { 1 } { k } \\sum _ { j = 1 } ^ k C ( q , a _ j ) + O _ { q , k , l } \\ ( \\frac { 1 } { ( \\log \\log x ) ^ 2 } \\ ) . \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} g . ( g ' , \\theta , r , s , i , j ) = ( g ' g ^ { - 1 } , g \\theta g ^ { - 1 } , r , s , g i , j g ^ { - 1 } ) \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ 3 [ \\mathrm { c o k } ( \\delta _ { i , t o r } ) ] ^ { ( - 1 ) ^ { i + 1 } } \\right ) = \\frac { \\nu ( \\mathcal { H } ^ 0 ) _ { \\mathbb { R } } \\nu ( \\mathcal { H } _ B / \\mathcal { H } ^ 0 ) } { \\nu ( \\mathcal { H } _ B ) _ { \\mathbb { R } } } . \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{gather*} q _ 2 = \\frac { \\mu _ + + \\mu _ - } { \\mu _ + - \\mu _ - } \\qquad q _ 1 = \\frac { 2 \\nu } { \\mu _ + - \\mu _ - } \\end{gather*}"}
-{"id": "367.png", "formula": "\\begin{align*} B ^ j ( s ) = B _ { 0 , t } ^ j ( s ) + \\frac s t B ^ j ( t ) , \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} \\Xi _ 1 ( z ) = f _ \\mathrm { P r } \\left ( r \\right ) e ^ { - z { { P _ t } G _ { } ^ 2 \\beta \\left ( { \\max { { \\{ r , d \\} } } } \\right ) ^ { - { \\alpha _ \\mathrm { L o S } } } } } + ( 1 - f _ \\mathrm { P r } \\left ( r \\right ) ) e ^ { - z { { P _ t } G _ { } ^ 2 \\beta \\left ( { \\max { { \\{ r , d \\} } } } \\right ) ^ { - { \\alpha _ \\mathrm { N L o S } } } } } \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{align*} K _ { a , + } ( \\lambda | \\tau _ { \\epsilon } , \\kappa _ { \\epsilon } , \\zeta _ { \\epsilon } ) = \\left ( \\sigma _ { a } ^ { x } \\right ) ^ { \\mathsf { x } } K _ { a , + } ^ { s G } ( \\lambda | \\tau _ { - } ^ { s G } , \\kappa _ { - } ^ { s G } , \\zeta _ { - } ^ { s G } ) \\left ( - \\sigma _ { a } ^ { x } \\right ) ^ { \\mathsf { x } } , \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} { \\bf H } & = \\begin{bmatrix} h _ { 1 1 } & h _ { 1 2 } & h _ { 1 3 } \\\\ h _ { 2 1 } & h _ { 2 2 } & h _ { 2 3 } \\end{bmatrix} , \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} P ( f _ { n , 1 } , \\ldots , f _ { n , k } ) = \\pi _ { n } ( m _ { 1 } , \\ldots , m _ { k } ) , \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} l _ \\sigma ( v ^ { 2 ( k + 1 ) } ; 0 , 0 ) & = l _ \\sigma ( v ^ { 2 k } ; 0 , 0 ) l _ \\sigma ( v ^ { 2 } ; 0 , 0 ) = ( ( u v ) ^ { - 2 k } ; 0 , 2 k ) ( ( u v ) ^ { - 2 } ; 0 , 2 ) \\\\ & = ( ( u v ) ^ { - 2 k } ( \\theta ( 0 , 2 k ) ( u ) \\theta ( 0 , 2 k ) ( v ) ) ^ { - 2 } ; 0 , 2 k + 2 ) \\\\ & = ( ( u v ) ^ { - 2 k } ( u v ) ^ { - 2 } ; 0 , 2 ( k + 1 ) ) = ( ( u v ) ^ { - 2 ( k + 1 ) } ; 0 , 2 ( k + 1 ) ) . \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} \\bar { x } = x - \\sum \\limits _ { i \\in I } \\nu _ i u _ i \\in C , \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} L _ 1 = - \\frac { v ' } { 2 } ( v _ 1 - v ) - \\frac { ( v _ 1 - v ) ^ 3 } { 6 } + \\frac { ( v _ 1 - 2 v + v _ { - 1 } ) ^ 2 } { 2 } . \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} z ^ 2 \\in A z ^ n = - 1 . \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} \\mathbb { E } [ \\mathcal { D } ^ { i j } ( 0 ) \\mathcal { D } ^ { i j } ( - J , - 1 ) ] = - \\frac { p ( \\chi + k - 2 ) } { k ( k - 1 ) } , \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} | X | \\leq l ^ L q ^ { l ^ 2 L ^ 2 + 2 ( b _ 1 + \\cdots + b _ L ) ^ 2 } | G | ^ d \\leq l ^ L q ^ { l ^ 2 L ^ 2 + 2 l ^ 2 L ^ 2 } | G | ^ d = l ^ L q ^ { 3 l ^ 2 L ^ 2 } | G | ^ d . \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} { \\Theta } ^ { ( k ) } \\leq \\sum _ { j = k } ^ n { \\Theta } ^ { ( j ) } \\lesssim 1 + | \\Theta | \\leq 1 + \\Big ( 1 + \\alpha ^ { - 2 } \\sum _ { j = k } ^ { n } \\alpha _ j ^ 2 \\Big ) \\ , \\widetilde { \\Lambda } ^ { k - 1 } _ { \\alpha 2 ^ { - 1 / 2 } , \\alpha 2 ^ { - 1 / 2 } , \\alpha _ k , \\dots , \\alpha _ n } . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\omega } h : = \\omega \\cdot \\partial _ { \\varphi } h - \\partial _ x K _ { 0 2 } h = \\Pi _ S ^ { \\perp } ( \\omega \\cdot \\partial _ { \\varphi } h + \\partial _ { x x } ( a _ 1 \\ , h _ x ) + \\partial _ x ( a _ 0 h ) - \\varepsilon ^ 2 \\partial _ x \\mathcal { R } _ 2 h - \\partial _ x \\mathcal { R } _ * h ) \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} - ( \\Delta _ g \\lambda ) g + \\nabla ^ 2 _ g \\lambda - \\lambda { \\rm R i c } _ g = g { \\rm o n } { \\rm i n t } \\ , M , \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} w _ n : = \\varepsilon \\gamma ^ { - 2 } \\lVert \\mathcal { F } ( U _ n ) \\rVert _ { s _ 0 } , B _ n : = \\varepsilon \\gamma ^ { - 1 } \\lVert \\mathfrak { I } _ n \\rVert _ { s _ 0 + \\beta _ 1 } + \\varepsilon \\gamma ^ { - 2 } \\lVert \\mathcal { F } ( U _ n ) \\rVert _ { s _ 0 + \\beta _ 1 } . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} \\omega _ t ^ n = c _ t e ^ { t F } i ^ { n ^ 2 } \\Omega \\wedge \\bar \\Omega , \\ ; \\ , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} \\left | \\left ( \\frac { - \\lambda _ { - } } { \\lambda _ { + } - \\lambda _ { - } } - 1 \\right ) \\chi _ { L } \\right | & \\le C | \\xi | ^ { 2 ( 1 - 2 \\sigma ) } \\chi _ { L } \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} \\Delta _ { 2 ^ { p _ 0 } } = \\Delta _ 1 = 0 . \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} \\int _ { Q } \\nabla \\Phi \\cdot \\nabla \\sigma \\ , d x & = \\int _ { K } \\Phi \\left ( 1 / 2 , x _ 2 \\right ) \\sigma _ { , _ { 1 } } \\left ( 1 / 2 , x _ 2 \\right ) \\ , d x _ 2 - \\int _ { P } \\Phi _ { , _ { 2 } } \\ , d x . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } u ( t ) \\| _ { 2 } & \\ge | m _ { 1 } | \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\mathcal { F } ^ { - 1 } [ \\mathcal { G } _ { \\sigma , \\nu } ( t ) ] \\| _ { 2 } - \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } ( u ( t ) - m _ { 1 } \\mathcal { F } ^ { - 1 } [ \\mathcal { G } _ { \\sigma , \\nu } ( t ) ] ) \\| _ { 2 } \\\\ & = C t ^ { - \\gamma _ { \\sigma , k } - \\ell } - o ( t ^ { - \\gamma _ { \\sigma , k } - \\ell } ) \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} \\left ( I _ { 0 { + } } ^ { \\lambda , \\lambda ^ { \\prime } , \\xi , \\xi ^ { \\prime } , \\gamma } f \\right ) ( x ) = \\frac { x ^ { - \\lambda } } { \\Gamma ( \\gamma ) } \\int _ { 0 } ^ { x } ( x - t ) ^ { \\gamma - 1 } t ^ { - \\lambda ^ { \\prime } } F _ { 3 } \\left ( \\lambda , \\lambda ^ { \\prime } , \\xi , \\xi ^ { \\prime } ; \\gamma ; 1 - \\frac { t } { x } , 1 - \\frac { x } { t } \\right ) f ( t ) \\ , \\mathrm { d } t \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} \\hat { d } _ { i , k , m , ( r ) } = \\arg \\min \\limits _ { d \\in \\mathcal { C } } \\left \\{ \\left | \\hat { { y } } _ { i , k , m , ( r ) } - d \\right | ^ 2 \\right \\} , \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot { a } ( t ) = 2 w ( t , 0 ) , \\\\ \\dot { b } ( t ) = \\frac { 1 } { 2 } a ( t ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} \\langle ( b _ 1 , b _ 2 ) , ( b _ 1 ' , b _ 2 ' ) \\rangle = \\langle b _ 1 , b _ 2 ' \\rangle + \\langle b _ 2 , b _ 1 ' \\rangle + \\langle \\rho b _ 2 , b _ 2 ' \\rangle . \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} g _ 3 ( C ^ { \\rm R } _ n , C ^ { \\rm I } _ n ) = & 1 - \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( \\sqrt { C ^ { \\rm R } _ n } - \\sqrt { P / 2 } \\big ) \\right ) \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( \\sqrt { C ^ { \\rm I } _ n } + \\sqrt { P / 2 } \\big ) \\right ) . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} \\phi _ 1 ( c , r , p ) & = \\left ( \\frac { c ^ 3 } { 2 } + c + 3 \\right ) ^ 2 + \\left ( 4 - c ^ 2 \\right ) ^ 2 r ^ 2 \\left ( - 3 c ^ 2 p r + \\frac { 9 } { 4 } c ^ 2 r ^ 2 + c ^ 2 - 3 c p r + 2 c + 1 \\right ) \\\\ & + 2 \\left ( \\frac { c ^ 3 } { 2 } + c + 3 \\right ) \\left ( 4 - c ^ 2 \\right ) r \\left ( \\frac { 3 } { 2 } c r - 3 c p ^ 2 r - 1 + c p + p \\right ) . \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} L ( e _ { x y } f - f e _ { x y } ) ( x , y ) = \\left ( L ( e _ { x y } ) f \\right ) ( x , y ) - \\left ( f L ( e _ { x y } ) \\right ) ( x , y ) + L ( f ) ( y , y ) - L ( f ) ( x , x ) . \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} K ( \\mu , \\eta ) = \\int _ { 0 } ^ { 1 } ( \\eta - \\xi ) d \\mu + \\int _ { 0 } ^ { 1 } 2 \\sqrt { \\eta '' } - \\eta '' ( s ) \\cdot ( 1 - s ) - \\frac { 1 } { 1 - s } d s - \\eta ( 0 ) + h ^ { 2 } + \\xi ( 1 ) . \\end{align*}"}
-{"id": "1365.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": "6091.png", "formula": "\\begin{align*} V ( x ) = x ^ 6 + 6 a | x | ^ 5 + ( 9 a ^ 2 - 4 b ) x ^ 4 - 1 2 a b | x | ^ 3 + ( 4 b ^ 2 - 5 ) x ^ 2 - 1 2 a | x | \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} e _ N ( \\tau ) = E _ N ^ I ( \\tau ) = \\sum _ { i = 1 } ^ N E _ N ( \\tau _ i ) \\left ( \\frac { ( \\tau + 1 ) P _ N ( \\tau ) } { ( \\tau _ i + 1 ) P _ N ' ( \\tau _ i ) ( \\tau - \\tau _ i ) } \\right ) , \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} \\gamma ( s , \\pi , \\tau , \\psi ) = \\prod _ { i = 1 } ^ k \\gamma ( s , \\pi , \\tau _ i , \\psi ) . \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} \\delta = - \\ln \\big ( 1 - \\frac { \\delta _ 2 \\epsilon } { M } \\big ) . \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} \\sigma ( \\sum B _ i \\mathrm { l o g } ^ i ( x ) ) A & = A \\sum B _ i \\mathrm { l o g } ^ i ( x ) + \\delta ( A ) \\\\ \\sum q ^ i \\mathrm { l o g } ^ i ( x ) \\sigma ( B _ i ) A & = \\sum \\mathrm { l o g } ^ i ( x ) A B _ i + \\mathrm { l o g } ( x ) \\partial ( A ) , \\intertext { w h e n c e t h e e q u a l i t y o f $ \\mathrm { l o g } ^ i ( x ) $ t e r m s f o r $ i = 1 $ : } q \\mathrm { l o g } ( x ) \\sigma ( B _ 1 ) A & = \\mathrm { l o g } ( x ) ( A B _ 1 + \\partial ( A ) ) , \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{gather*} r _ 2 = - \\frac { t } { 2 } \\qquad \\nu = \\frac { 1 } { 2 } , \\end{gather*}"}
-{"id": "6257.png", "formula": "\\begin{align*} U ( s , 2 s ; 2 z ) = \\frac { ( 2 z ) ^ { \\frac { 1 } { 2 } - s } } { \\sqrt { \\pi } } e ^ { z } K _ { s - \\frac { 1 } { 2 } } ( z ) . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} \\Big ( \\prod \\limits _ { j = 0 } ^ { 2 m - 2 } | U ( N _ { j + 1 } ) U ^ { - 1 } ( N _ { j } ) | | S ( N _ { j + 2 } ) S ^ { - 1 } ( N _ { j + 1 } ) | \\Big ) | U ( n ) U ^ { - 1 } ( N _ { 2 m } ) | \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} E ( a , \\emptyset , \\{ 1 , 2 , 3 \\} ) \\ = \\ \\Delta _ 5 \\ ; \\Delta _ { 1 3 } \\ - \\ \\Delta _ 4 \\ ; \\Delta _ { 2 3 } \\ = \\ 0 , \\\\ E ( b , \\emptyset , \\{ 1 , 2 , 3 \\} ) \\ = \\ \\Delta _ 6 \\ ; \\Delta _ { 1 3 } \\ - \\ \\Delta _ 7 \\ ; \\Delta _ { 1 2 } \\ = \\ 0 , \\\\ E ( c , \\emptyset , \\{ 1 , 2 , 3 \\} ) \\ = \\ \\Delta _ 9 \\ ; \\Delta _ { 1 2 } \\ - \\ \\Delta _ 8 \\ ; \\Delta _ { 2 3 } \\ = \\ 0 . \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} A _ { l } & : = C ( 0 , n _ { 1 } ; 3 ) + C ( n _ { 1 } , n _ { 2 } ; 6 ) + \\cdots + C ( n _ { l - 1 } , n _ { l } ; 3 l ) , \\\\ A _ { l } ' & : = C ( 0 , n _ { 1 } ; 2 ) + C ( n _ { 1 } , n _ { 2 } ; 4 ) + \\cdots + C ( n _ { l - 1 } , n _ { l } ; 2 l ) . \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} \\psi _ 1 ' = & \\psi _ 2 \\\\ \\psi _ 2 ' = & - 2 \\tau \\psi _ 2 + u \\psi _ 1 . \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} { { \\mathsf { R } } ^ { \\lambda } _ { \\mathcal { N } _ { [ 1 : N ] } } } & = \\min _ { \\mathcal { A } _ { \\rm { F } } \\subseteq { [ 1 : N ] } } \\sum _ { s \\in [ 0 : 1 ] ^ N } \\lambda _ s \\left ( \\max _ { i \\in \\mathcal { A } _ { \\rm { F } } } \\ell _ { i , s } ^ { \\prime } + \\max _ { i \\in { [ 1 : N ] } \\backslash \\mathcal { A } _ { \\rm { F } } } r _ { i , s } ^ { \\prime } \\right ) , \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\Lambda _ S ( - \\theta , \\mathbf { G } ) = \\log \\left ( p \\sum _ { m = 1 } ^ M q _ m e ^ { - \\theta r _ m } + \\sum _ { n = 1 } ^ N p _ n e ^ { - \\theta r _ n } \\right ) , \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} \\begin{aligned} \\left ( A \\norm { \\Psi _ { n } ( \\varphi ( s ) ) } _ { L ^ { 1 } } + \\frac { B } { 2 } \\norm { \\nabla \\varphi _ { n } ( s ) } _ { L ^ { 2 } } ^ { 2 } \\right ) + \\frac { 1 } { 2 } \\norm { \\nabla \\mu _ { n } } _ { L ^ { 2 } ( 0 , s ; L ^ { 2 } ) } ^ { 2 } \\leq ( c _ { 0 } + d _ { 0 } T ) e ^ { k _ { 0 } C _ { u } s } \\end{aligned} \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} X = \\frac { \\partial } { \\partial z ^ 1 } + \\sum _ j a _ j \\ , \\frac { \\partial } { \\partial z ^ j } + \\sum _ \\rho \\beta _ \\rho \\ , \\frac { \\partial } { \\partial \\zeta ^ \\rho } . \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} \\lim _ { z \\to \\tilde \\theta , z \\in B } \\frac { \\int _ { \\partial B } M _ s ( z , \\theta ) g ( \\theta ) \\ d \\theta } { ( 1 - | z | ^ 2 ) ^ { s - 1 } } = { \\frac { k _ { N , s } } { 2 k _ { N , 1 } { s } } } g ( \\tilde \\theta ) \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} \\big ( \\lambda _ \\epsilon - \\epsilon a \\big ) \\ , \\ , \\sharp ^ { \\epsilon , \\kappa } \\ , \\ , \\widetilde { r } _ { \\lambda } ( \\epsilon a ) \\ = \\ 1 \\ , + \\ , \\mathfrak { r } _ { \\delta , a } , \\ ; \\mbox { w i t h } \\| \\mathfrak { O p } ^ { \\epsilon , \\kappa } ( \\mathfrak { r } _ { \\delta , a } ) \\| \\ , \\leq \\ , C \\ ; \\epsilon ^ { 1 / 3 } \\ , . \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} v = \\frac { 2 ( d _ 1 + d _ 2 ) } { 4 - ( k - 2 ) ( d _ 2 - 2 ) } \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} g _ f / f ^ { p ^ j } = ( h _ 1 / f ^ { p ^ { j - 1 } } ) ^ { p } + h _ 2 / f ^ { p ^ { j } - 1 } \\equiv h _ 1 / f ^ { p ^ { j - 1 } } + h _ 2 / f ^ { p ^ { j - 1 } } \\pmod { \\wp K } . \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{align*} M _ k ( x ; \\boldsymbol { a } ) - \\frac { 1 } { \\phi ^ k ( q ) } S _ k ( x ) & = \\frac { 1 } { ( k - 2 ) ! \\phi ^ k ( q ) } \\frac { x } { \\log x } ( \\log \\log x ) ^ { k - 2 } \\bigg \\{ C ( q , a ) + \\frac { k - 2 } { 2 \\log \\log x } f \\ ( \\frac { k - 3 } { 3 \\log \\log x } \\ ) \\\\ & + O _ { A , q } \\ ( \\frac { k } { ( \\log \\log x ) ^ 2 } \\ ) \\bigg \\} , \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } | f _ { n } ( k ) | \\leq 1 , k = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} \\hat { \\alpha } _ { 1 } = - \\alpha , \\ \\hat { \\alpha } _ { 2 } = - 2 \\alpha + { 2 \\over 3 } , \\ \\hat { \\alpha } _ { 3 1 } = \\alpha , \\ \\hat { \\alpha } _ { 3 2 } = \\alpha + { 1 \\over 3 } , \\ \\hat { \\alpha } _ { 3 3 } = \\alpha + { 2 \\over 3 } . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} q _ { j } = - \\left ( \\sigma _ z ^ { j } L \\sigma _ z ^ { - j } q _ { j - 1 } \\psi \\right ) \\psi ^ { - 1 } = ( - 1 ) ^ { j } \\sigma _ z ^ { j } \\left ( ( L \\sigma _ z ^ { - 1 } ) ^ { j } \\psi \\right ) \\psi ^ { - 1 } , \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 + \\bold { 1 } _ k } = S _ k \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } + D _ { J _ 1 } \\left ( \\left ( S _ k \\xi ^ i - \\xi ^ i \\right ) u ^ { \\alpha } _ { \\bold { 1 } _ i ; J _ 2 + \\bold { 1 } _ k } \\right ) - \\left ( S _ k \\xi ^ i - \\xi ^ i \\right ) u ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; J _ 2 + \\bold { 1 } _ k } . \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} \\bold { D } _ F ^ { \\ast } ( Q ) = 0 \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } Y ( z ) \\mathbb { C } \\setminus \\Gamma , \\\\ \\\\ Y _ + ( z ) = Y _ - ( z ) \\begin{bmatrix} 1 & \\omega _ { n , N } ( z ) \\\\ 0 & 1 \\end{bmatrix} , & z \\in \\Gamma , \\\\ \\\\ \\displaystyle Y ( z ) = \\left ( I + \\mathcal { O } \\left ( \\frac { 1 } { z } \\right ) \\right ) \\begin{bmatrix} z ^ n & 0 \\\\ 0 & z ^ { - n } \\end{bmatrix} , & z \\to \\infty , \\end{array} \\right . \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\sum _ { A \\in \\mathcal { F } } \\left ( \\frac { q } { q + 1 } \\right ) ^ { | A | } & \\leqslant \\sum _ { A \\in \\mathcal { F } } 2 ^ { - | A | / q } \\leqslant \\binom { n } { \\lceil n / 2 \\rceil } 2 ^ { - \\binom { \\lceil n / 2 \\rceil } { k } / q } \\\\ & \\leqslant \\left ( \\frac { e n } { \\lceil n / 2 \\rceil } \\right ) ^ { \\lceil n / 2 \\rceil } 2 ^ { - ( 1 + 1 / \\ln 2 ) n } < \\left ( \\frac { e } { 2 ^ { 1 / \\ln 2 } } \\right ) ^ n = 1 , \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} \\binom { B } { B _ 1 ; \\cdots ; B _ n } = \\binom { b _ 0 } { b _ { 0 , 1 } , \\dots , b _ { 0 , n } } \\cdots \\binom { b _ { r - 1 } } { b _ { r - 1 , 1 } , \\ldots , b _ { r - 1 , n } } , \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} H _ n = \\sum _ { i } d _ i ^ n . \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} X _ r = \\sum _ { i = 1 } ^ { r - 1 } f _ i \\frac { \\partial } { \\partial x ^ i } + \\sum _ { k = r } ^ p f _ k \\frac { \\partial } { \\partial x ^ k } + \\sum _ { \\rho = 1 } ^ q \\varphi _ \\rho \\frac { \\partial } { \\partial \\xi ^ \\rho } \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\beta ( v _ i ) = \\sum _ { j = 1 } ^ m \\beta ( w _ j ) . \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} \\left ( b \\alpha \\right ) ^ { 2 } = b ^ { 2 } c \\left ( c + 2 \\right ) = a ^ { 2 } - \\frac { 2 } { v i + 2 } . \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} R _ \\mathsf { A } ^ \\mathsf { A l t } = R _ \\mathsf { A } . \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} \\varepsilon ( \\mathcal { A } ^ T - \\mathrm { I } ) \\alpha _ { k , 1 } ( \\varphi , y ) = \\varepsilon ( \\alpha _ { k , 1 } ( \\varphi , y + \\tilde { \\beta } ( \\varphi , y ) ) - \\alpha _ { k , 1 } ( \\varphi , y ) ) = \\varepsilon \\partial _ y ( \\alpha _ { k , 1 } ) ( \\varphi , y ) \\ , \\tilde { \\beta } ( \\varphi , y ) + \\mathtt { R } _ { \\tilde { \\beta } } , \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} \\| ( j _ n ( x y ) - & j _ \\infty ( x ) j _ \\infty ( y ) ) \\xi \\| = \\| ( j _ n ( x ) j _ n ( y ) - j _ \\infty ( x ) j _ \\infty ( y ) ) \\xi \\| \\\\ & \\leq \\| ( j _ n ( x ) j _ n ( y ) - j _ n ( x ) j _ \\infty ( y ) ) \\xi \\| + \\| ( j _ n ( x ) j _ \\infty ( y ) - j _ \\infty ( x ) j _ \\infty ( y ) ) \\xi \\| \\\\ & \\leq \\sup _ m \\| j _ m ( x ) \\| \\| ( j _ n ( y ) - j _ \\infty ( y ) ) \\xi \\| + \\| ( j _ n ( x ) - j _ \\infty ( x ) ) j _ \\infty ( y ) \\xi \\| , \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} e ^ { - 2 \\eta } \\left [ \\begin{pmatrix} p ( \\eta ) \\\\ q ( \\eta ) \\\\ r ( \\eta ) \\end{pmatrix} - M _ 0 \\right ] \\rightarrow \\kappa \\vec { X } _ { 0 2 } , . \\end{align*}"}
-{"id": "5820.png", "formula": "\\begin{align*} \\tau _ 0 = - 1 \\mbox { a n d } \\tau _ { N + 1 } = + 1 . \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} U = \\{ | g _ 1 | \\le | g _ 0 | , \\ldots , | g _ r | \\le | g _ 0 | \\} \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} g ( x , y ) = \\mathrm { s g n } \\left ( \\lim _ { t \\rightarrow 0 } \\frac { | | x + t y | | - | | x | | } { t } \\right ) \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} \\lambda ^ r ( x y ) = P _ r ( \\lambda ^ 1 ( x ) , \\dots , \\lambda ^ r ( x ) , \\lambda ^ 1 ( y ) , \\dots , \\lambda ^ r ( y ) ) \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{gather*} V ( B [ f _ 1 , f _ 2 ] _ c ^ { \\textsf { c o m } } ) \\underset { \\ne } { \\subset } V ( B [ f _ 1 , f _ 2 ] _ { f _ 2 } ^ { \\textsf { c o m } } ) , \\\\ | V ( B [ f _ 1 , f _ 2 ] _ { f _ 2 } ^ { \\textsf { c o m } } ) | = | V ( B [ f _ 1 ] _ { f _ 2 } ^ { \\textsf { c o m } } ) | < | V ( B [ f _ 1 ] _ { f _ 1 } ^ { \\textsf { c o m } } ) | = | V ( B _ { f _ 1 } ^ { \\textsf { c o m } } ) | . \\end{gather*}"}
-{"id": "9576.png", "formula": "\\begin{align*} \\deg ( f ) & \\le \\max \\{ \\deg ( r ' ) - 1 , \\deg ( \\varphi ) - 1 , \\deg ( \\varphi ' ) \\} = \\deg ( \\phi ) - 3 , \\\\ \\deg ( g ) & \\le \\max \\{ \\deg ( r ) - 1 , \\deg ( \\varphi ) \\} = \\deg ( \\phi ) - 2 , \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{align*} ( u + \\epsilon ) ( x ) & ( u + \\epsilon ) ( y ) ( \\varphi _ \\epsilon ( x ) - \\varphi _ \\epsilon ( y ) ) ^ 2 \\\\ \\leq & \\left [ u ( y ) ( v ( x ) - v ( y ) ) - v ( y ) ( u ( x ) - u ( y ) ) \\right ] ( \\varphi _ \\epsilon ( x ) - \\varphi _ \\epsilon ( y ) ) \\\\ & + \\epsilon ( v ( x ) - v ( y ) ) ( \\varphi _ \\epsilon ( x ) - \\varphi _ \\epsilon ( y ) ) , \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} \\Psi ( u ) : = \\Psi ( u ) ( v ) = \\kappa ( u , v ) , \\ , \\ \\forall v \\in \\widehat { \\mathfrak { p } } , \\ , \\ u \\in \\widehat { \\mathfrak { p } } ^ { \\vee } . \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} \\dd Z _ t & = Z _ t \\left ( \\zeta ( X _ t ) \\dd X ^ c _ t + \\int _ E ( Y ( X _ { t - } , y ) - 1 ) \\left ( \\mu ^ X ( \\dd t , \\dd y ) - \\nu ^ X ( \\dd t , \\dd y ) \\right ) \\right ) , \\\\ Z _ 0 & = 1 , \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} J _ { \\lambda , \\mu } ( u , 0 ) = \\frac { 1 } { p } \\int _ Q \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { n + p s } } \\ , d x \\ , d y - \\frac { \\lambda } { q } \\int _ \\Omega | u | ^ { q } d x = - \\frac { p - q } { p q } \\| u \\| _ { X _ 0 } ^ p < 0 . \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} u _ 1 ( t , x , z ) = & - \\left ( G _ 1 \\partial _ x f ( t , x ) + G _ 2 \\mu \\partial _ x g ( t , x ) - \\sigma _ 2 ^ c \\mu \\partial _ x ^ 3 ( f + g ) ( t , x ) - \\sigma _ 1 ^ c \\partial _ x ^ 3 f ( t , x ) \\right ) \\\\ [ 5 p t ] & \\quad \\times \\left ( f ( t , x ) z - \\frac { 1 } { 2 } z ^ 2 \\right ) + \\displaystyle { \\mu } \\partial _ z u _ 2 ( t , x , f ) z . \\end{align*}"}
-{"id": "7381.png", "formula": "\\begin{align*} D = \\begin{pmatrix} 0 & D ^ + \\\\ D ^ - & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} \\frac { 1 } { \\Gamma ( k _ j ) } \\biggl ( \\log \\frac { 1 - x _ { j + 1 } } { 1 - x _ j } \\biggr ) ^ { k _ j - 1 } & = \\frac { 1 } { ( k _ j - 1 ) ! } \\biggl ( \\int _ { x _ { j + 1 } } ^ { x _ j } \\frac { d x } { 1 - x } \\biggr ) ^ { k _ j - 1 } \\\\ & = \\int _ { x _ j > x _ { j 2 } > \\cdots > x _ { j k _ j } > x _ { j + 1 } } \\frac { d x _ { j 2 } } { 1 - x _ { j 2 } } \\cdots \\frac { d x _ { j k _ j } } { 1 - x _ { j k _ j } } , \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} g ^ + ( v ) : = | ( v - L ) ^ + | = \\begin{cases} 0 & v \\leq L , \\\\ v - L & v \\geq L , \\end{cases} g ^ - ( v ) : = | ( v - U ) ^ - | = \\begin{cases} U - v & v \\leq U , \\\\ 0 & v \\geq U . \\end{cases} \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} \\widetilde { z } _ { 1 } = \\min \\left \\{ \\max \\left \\{ z _ { 1 } , \\underline { u } \\right \\} , \\overline { u } \\right \\} \\widetilde { z } _ { 2 } = \\min \\left \\{ \\max \\left \\{ z _ { 2 } , \\underline { v } \\right \\} , \\overline { v } \\right \\} . \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} P ( \\left | x - \\mu \\right | \\geq \\lambda \\sigma ) \\leq \\frac { 4 } { 9 \\lambda ^ 2 } = \\epsilon . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} a \\overline { b } - \\overline { d } e K = 0 , \\ | b | ^ { 2 } - | e | ^ { 2 } | K | ^ { 2 } = 0 . \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + a x + b , a = - \\frac { c _ 4 } { 4 8 } , b = - \\frac { c _ 6 } { 8 6 4 } . \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} d ^ n _ { \\lambda } ( p , r | x ^ n ) : = - \\frac { 1 } { 1 - \\lambda } \\log E _ { p ( y ^ n | x ^ n ) } \\Bigl ( \\frac { r ( y ^ n | x ^ n ) } { p ( y ^ n | x ^ n ) } \\Bigr ) ^ { 1 - \\lambda } . \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{gather*} \\int _ x ^ { x + 1 } \\sup _ { N \\ge 1 } \\Big | \\sum _ { n = 1 } ^ N \\alpha _ n { \\rm e } ^ { i t \\lambda _ n \\mu _ k } \\Big | ^ 2 d t = \\frac 1 { \\mu _ k } \\int _ { \\mu _ k x } ^ { \\mu _ k ( x + 1 ) } \\sup _ { N \\ge 1 } \\Big | \\sum _ { n = 1 } ^ N \\alpha _ n { \\rm e } ^ { i t \\lambda _ n } \\Big | ^ 2 d t \\\\ \\le \\frac { [ \\mu _ k ] + 1 } { \\mu _ k } \\Big \\| \\sup _ { N \\ge 1 } \\big | \\sum _ { n = 1 } ^ N \\alpha _ n { \\rm e } ^ { i t \\lambda _ n } \\big | \\ , \\Big \\| _ { \\S ^ 2 } ^ 2 \\ , , \\end{gather*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\| f \\| _ { 1 + k _ \\gamma } ^ 2 = \\| f \\| _ 1 ^ 2 + \\frac { 1 } { \\gamma } \\| f \\| _ 2 ^ 2 \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} \\mathcal { N } _ { \\chi } ^ { s s } ( r , c , c ' ) = \\mathbb { X } _ 0 ( r , c , c ' ) / / _ { \\chi } \\mathcal { G } : = \\mbox { P r o j } ( \\oplus _ { n \\geq 0 } A ( \\mathbb { X } _ 0 ( r , c , c ' ) ) ^ { \\mathcal { G } , \\chi ^ { n } } ) , \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} ( D _ { \\bar j } u ) ( t ) = \\mathbb H ^ t ( i _ t ^ * [ d ^ E , \\delta _ { \\overline { V _ j } } ] \\mathbf { u } ) , \\ ( D _ { j } u ) ( t ) = \\mathbb H ^ t ( i _ t ^ * [ d ^ E , \\delta _ { V _ j } ] \\mathbf { u } ) , \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{align*} \\dot z ( t ) = A _ { - 1 } z ( t - 1 ) + A _ 0 z ( t ) + A _ 1 z ( t - 1 ) + B u ( t ) . \\end{align*}"}
-{"id": "7894.png", "formula": "\\begin{align*} G ^ { \\lambda , m , 0 } = \\{ ( p , q , r ) \\in \\bar { D } \\ ; | \\ ; r = h ^ { \\lambda , m , 0 } ( p , q ) \\} . \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} \\psi ( E ) _ 0 = \\oplus _ { n } E _ { 2 n } \\psi ( E ) _ 1 = \\oplus _ { n } E _ { 2 n + 1 } . \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} \\mathbf { W } _ { \\mathrm { u } , i } = \\frac { 1 } { \\sqrt { \\zeta _ { \\mathrm { u } , i } } } \\cdot \\hat { \\mathbf { H } } _ i ^ { \\mathrm { H } } \\left ( \\hat { \\mathbf { H } } _ i \\hat { \\mathbf { H } } _ i ^ { \\mathrm { H } } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} \\Psi _ { u , v } ( t ) = 2 \\int _ \\Omega | u | ^ { \\alpha } | v | ^ \\beta d x . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} P _ { M } ( N , K , \\mathbf { p } ) = \\sum _ { \\mathbf { M } } B ^ { * } ( \\mathbf { M } , N , \\mathbf { p } ) \\mathbb { P } ^ { * } ( \\mathbf { M } , K ) , \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} & G _ i = \\left | \\frac { \\textbf { a } _ { r } ^ H ( \\xi _ { r _ o } , N ) } { \\sqrt { N } } { \\bf { A } } \\left ( \\xi _ { r _ { i , o } } , \\varphi _ { t _ { i , o } } \\right ) \\frac { \\textbf { a } _ { t } ( \\varphi _ { t _ i } , N ) } { \\sqrt { N } } \\right | ^ 2 . \\end{align*}"}
-{"id": "7590.png", "formula": "\\begin{align*} C _ { i i } ^ { i i } = C _ { j j } ^ { j j } , \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} y _ { 1 } ( n + 1 ) = A _ { 1 } ( n ) y _ { 1 } ( n ) \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} \\Delta _ { j , p _ j } G ( z , s ) = s ( 1 - s ) G ( z , s ) , \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} f ( y ) = \\inf _ { x \\in K } \\inf _ { p \\in \\partial \\Phi ( x ) } \\Phi _ p ( y ) \\geq \\inf _ { p \\in \\cup V _ { p _ j } } \\Phi _ p ( y ) \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} \\langle f \\rangle _ q : = \\frac { \\sum _ { \\lambda \\in \\mathcal { P } } f ( \\lambda ) q ^ { | \\lambda | } } { \\sum _ { \\lambda \\in \\mathcal { P } } q ^ { | \\lambda | } } = ( q ; q ) _ { \\infty } \\sum _ { n = 0 } ^ { \\infty } q ^ n \\sum _ { \\lambda \\vdash n } f ( \\lambda ) \\in \\C [ [ q ] ] , \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} k a _ k = - \\sum _ { i = 0 } ^ { k - 1 } a _ i p _ { k - i } . \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} p ( y _ n ^ { \\rm R } , y _ n ^ { \\rm I } ) = \\frac { 1 } { \\sqrt { \\pi } } e ^ { - ( y _ n ^ { \\rm R } - \\sqrt { P / 2 } - z _ n ^ { \\rm R } ) ^ 2 } e ^ { - ( y _ n ^ { \\rm I } - \\sqrt { P / 2 } - z _ n ^ { \\rm I } ) ^ 2 } . \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} m ^ p & = \\int _ { \\Omega _ 0 } \\abs { x ( t ) } ^ p \\textup d t = \\int _ { \\Omega _ 1 } \\abs { x ( t ) } ^ p \\textup d t + \\int _ { \\Omega _ 0 \\setminus \\Omega _ 1 } \\abs { x ( t ) } ^ p \\textup d t \\leq r ^ p \\mu ( \\Omega _ 1 ) + \\dfrac { \\vartheta ^ p m ^ p } { \\mu ( \\Omega _ 0 ) } \\cdot \\mu ( \\Omega _ 0 \\setminus \\Omega _ 1 ) \\\\ [ 2 m m ] & \\leq r ^ p \\mu ( \\Omega _ 1 ) + \\vartheta ^ p m ^ p . \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} \\max _ { x \\in \\mathbb { R } ^ n } \\{ \\alpha g ( x ) - g ( f _ d ( x ) ) \\ , | \\ , \\alpha \\nabla _ x g ( x ) - \\nabla _ x ( g ( f _ d ( x ) ) ) = 0 \\} . \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} D ( s ) = \\sum _ d \\frac { \\gamma ( d ) } { \\varphi ( d ) } d ^ { - s } = \\zeta ( s + 1 - \\varepsilon _ 1 ) \\ldots \\zeta ( s + 1 - \\varepsilon _ r ) \\ . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} [ \\beta , 1 - a T ^ i ) = 0 . \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} X = \\xi ^ i ( x ) \\partial _ { x ^ i } + \\phi ^ { \\alpha } ( x , n , [ u ] ) \\partial _ { u ^ { \\alpha } } . \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} ( - 1 ) ^ { n - 1 } f ( n ) = \\sum _ { d j = n } h _ f ( d ) f ( j ) ( n \\ge 1 ) \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} d ( \\omega ) & = { \\inf } \\{ S _ { \\omega } ( u ) : \\ , u \\in W ( \\mathbb { R } ^ N ) \\setminus \\{ 0 \\} , I _ { \\omega } ( u ) = 0 \\} \\\\ & = \\frac { 1 } { 2 } \\ , { \\inf } \\{ \\| u \\| _ { L ^ 2 } ^ 2 : u \\in W ( \\mathbb { R } ^ N ) \\setminus \\{ 0 \\} , I _ { \\omega } ( u ) = 0 \\} , \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} \\langle \\alpha _ s , \\alpha _ t ^ \\vee \\rangle = - 2 \\cos ( \\pi / m _ { s t } ) , \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{align*} \\Big | \\sum _ { \\widetilde \\alpha } ( - 1 ) ^ { \\widetilde { q } ( \\widetilde \\alpha ) } Z _ { \\widetilde \\alpha , \\widetilde { D } _ 0 } ( \\widetilde { G } ) \\Big | = \\Big ( \\sum _ { w _ 1 ( \\alpha ) = 0 } ( - 1 ) ^ { \\frac { q ( \\alpha ) } { 2 } } Z _ { \\alpha , D _ 0 } ( G ) \\Big ) ^ 2 + \\Big ( \\sum _ { w _ 1 ( \\alpha ) = 1 } ( - 1 ) ^ { \\frac { q ( \\alpha ) - 1 } { 2 } } Z _ { \\alpha , D _ 0 } ( G ) \\Big ) ^ 2 \\ , . \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} & \\left ( \\frac { \\partial \\phi ^ * ( y ^ k ) } { \\partial x ^ i } \\Big | _ { m } \\right ) _ { i , k = 1 , \\ldots , r } \\mbox { o f t h e d i a g o n a l b l o c k $ B _ 0 $ } \\\\ \\mbox { a n d } & \\left ( \\frac { \\partial \\phi ^ * ( \\theta _ { \\mu } ^ a ) } { \\partial \\xi _ { \\mu } ^ b } \\Big | _ { m } \\right ) _ { a , b = 1 , \\ldots , s _ { \\mu } } \\mbox { o f t h e d i a g o n a l b l o c k s $ B _ { \\mu } $ } \\ ; , \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} f ( a + _ R b ) \\subseteq f ( a ) + _ S f ( b ) , f ( a \\times _ R b ) = f ( a ) \\times _ S f ( b ) . \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\sharp ( T N ^ { \\circ } ) = T N \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} U = : U _ 0 \\leftarrow U _ 1 \\leftarrow \\dotsb \\leftarrow U _ k \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} I = \\ < 1 , \\dfrac { s + \\sqrt { \\Delta } } { 2 t } \\ > , \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} ( r _ 0 , \\ldots ) + ( r _ 0 ' , \\ldots ) = ( r _ 0 + r _ 0 ' , \\ldots ) , \\\\ ( r _ 0 , \\ldots ) \\cdot ( r _ 0 ' , \\ldots ) = ( r _ 0 \\cdot r _ 0 ' , \\ldots ) . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{gather*} \\alpha ^ t = h _ 1 ^ { - t } \\alpha ^ t = ( h _ 1 ^ { - 1 } \\alpha ) ^ t = ( h _ 2 h _ 1 ^ { - 1 } h _ 2 ^ { - 1 } ) ^ t = h _ 2 h _ 1 ^ { - t } h _ 2 ^ { - 1 } = 1 \\intertext { a n d } \\alpha ^ p = \\alpha ^ p h _ 2 ^ p = ( \\alpha h _ 2 ) ^ p = h _ 1 h _ 2 ^ p h _ 1 ^ { - 1 } = 1 , \\end{gather*}"}
-{"id": "4514.png", "formula": "\\begin{align*} \\frac { d f } { d t } = \\frac { i } { 2 } J f , \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} & \\sum _ { r \\in [ s , T ] } q ^ { n , [ s , T ] } ( r ) \\ , \\psi ( r ) \\leq \\sum _ { i = 1 } ^ { n } q ^ { n , [ t , T ] } ( \\tfrac { T - t } { 2 } c _ i ^ n + \\tfrac { T + t } { 2 } ) \\ , \\psi ( \\tfrac { T - t } { 2 } c _ i ^ n + \\tfrac { T + t } { 2 } ) = \\sum _ { r \\in [ t , T ] } q ^ { n , [ t , T ] } ( r ) \\ , \\psi ( r ) . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} ( E _ 2 \\blacktriangle S ) ( X ) = \\sum _ { x \\in X } \\{ ( \\{ x , \\ast _ { \\{ x \\} } \\} , f ) \\ , | \\ , , f \\in S ( X \\backslash \\{ x \\} \\cup \\{ \\ast _ { X \\backslash \\{ x \\} } \\} ) \\} . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} u ( n + 1 ) = \\frac { c _ { i _ { r r } } ( n ) } { \\big ( a _ { i } - \\frac { \\delta } { 2 } \\big ) } u ( n ) \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{gather*} g _ { i _ 1 } = f _ 1 , \\ g _ { i _ 2 } = f _ 2 , \\dots , g _ { i _ m } = f _ m , \\\\ g _ { j _ 1 } = f _ { m + 1 } , \\ g _ { j _ 2 } = f _ { m + 2 } , \\dots , g _ { j _ n } = f _ { m + n } \\end{gather*}"}
-{"id": "6711.png", "formula": "\\begin{align*} U = p ^ { 2 } + q ^ { 2 } , \\ V = p ^ { 2 } + 2 p q - q ^ { 2 } , \\ Z = - p ^ { 2 } + 2 p q + q ^ { 2 } . \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} \\lambda ^ r ( ( a _ 0 , a _ 1 , a _ 2 , \\dots ) ) = \\left ( \\lambda ^ r ( a _ 0 ) , \\sum _ { i = 0 } ^ { r - 1 } \\lambda ^ i ( a _ 0 ) \\lambda ^ { r - i } ( a _ 1 ) , \\sum _ { i = 0 } ^ { r - 1 } \\lambda ^ i ( a _ 0 ) \\lambda ^ { r - i } ( a _ 2 ) , \\dots \\right ) . \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} \\lambda ' \\mathbb { E } ^ { 0 ' } \\left [ \\mathrm { c a r d } \\left ( H ^ + ( 0 ) \\right ) \\right ] = \\lambda \\mathbb { E } ^ { 0 } \\left [ \\mathrm { c a r d } \\left ( H ^ - ( 0 ) \\right ) \\right ] \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} M : = \\Big \\{ ( X , Y , U , V , R ) \\in \\R ^ 5 : X ^ 2 + Y ^ 2 = 1 , \\ , X U + Y V = 0 , \\ , U ^ 2 + V ^ 2 - R = 0 \\big \\} , \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} b _ 1 = 2 \\Re R , b _ 2 = - 2 \\Re \\P [ R \\bar Y ] . \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} [ A , B ] + I J = 0 , [ A ' , B ' ] = 0 , G \\equiv 0 . \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} x \\equiv c \\beta + \\sum _ { ( i , p ) = 1 } c _ i [ T ] ^ { - i } \\pmod { \\wp W ( K ) } \\end{align*}"}
-{"id": "7592.png", "formula": "\\begin{align*} \\tilde C _ { i j } ^ { i i } + \\tilde C _ { i j } ^ { j j } & = 0 , \\mbox { i f } i < j , \\\\ \\tilde C ^ { i j } _ { i j } + \\tilde C ^ { j k } _ { j k } & = \\tilde C ^ { i k } _ { i k } , \\mbox { i f } i \\le j \\le k , \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{align*} 0 = y _ i ( T ) - y _ i ( 0 ) = \\lambda _ i \\int _ 0 ^ T f _ 2 \\big ( t , x _ i ( t ) , y _ i ( t ) , \\lambda _ i \\big ) \\ , d t , \\quad . \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} \\left | { h } ^ 0 ( \\theta ) \\right | ^ { \\alpha } = \\left ( \\varphi _ 1 ( \\theta ) + \\gamma _ 2 \\right ) , \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} M ( \\lambda , \\vec \\gamma ) \\leq \\dfrac { \\lambda ^ 2 } { 2 } p + \\lambda ^ 3 \\cdot \\dfrac { 8 C _ { V , f } ^ 3 n } { ( \\sqrt { n h ^ d } ) ^ 3 } = \\dfrac { \\lambda ^ 2 } { 2 } \\left ( p + 2 \\lambda \\dfrac { 8 C _ { V , f } ^ 3 } { \\sqrt { n h ^ { 3 d } } } \\right ) \\leq \\dfrac { \\lambda ^ 2 } { 2 } \\left ( p + \\dfrac { 1 6 \\mathfrak g C _ { V , f } ^ 3 } { \\sqrt { n h ^ { 3 d } } } \\right ) \\enspace . \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { \\mathbb { R } ^ N } | f ( x ) | ^ 2 \\log | f ( x ) | ^ 2 d x \\\\ & \\leq \\frac { \\alpha ^ 2 } { \\pi } \\| \\nabla f \\| ^ 2 _ { L ^ 2 } + ( \\log \\| f \\| ^ 2 _ { L ^ 2 } - N ( 1 + \\log \\ , \\alpha ) ) \\| f \\| ^ 2 _ { L ^ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} X _ \\sigma = \\lim \\limits _ { n \\to \\infty } \\sum \\limits _ { | \\tau | = n } l _ { \\sigma * \\tau } ^ \\alpha . \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} - \\sum _ { k = \\iota + 1 } ^ n s _ { k \\iota } = s _ { \\iota \\iota } - \\sum _ { k = \\iota } ^ n s _ { k , \\iota - 1 } . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} L = \\frac { K } { r } \\log \\frac { M ^ { n - m } ( R , f ) } { K } \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} \\xi ( \\omega ) = \\varphi ( \\omega ( t _ { 1 } ) , \\cdots , \\omega ( t _ { n } ) ) , \\omega \\in \\Omega _ { T } , \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{align*} x \\ ; = \\ ; y _ 1 x - y _ 2 x , \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{gather*} F _ { 6 } \\big ( 3 ^ { - 2 / 3 } t \\big ) = \\frac { ( q _ 2 - 1 ) } { 2 q _ 2 } \\exp \\left ( \\frac { 1 } { 3 } \\int _ { t } ^ { \\infty } \\omega ( s ) d s - \\frac { 2 } { 3 } \\int _ { t } ^ { \\infty } \\frac { u _ s ( s ) } { u ( s ) } \\frac { 1 + q _ 2 ( s ) } { q _ 2 ( s ) } d s \\right ) , \\end{gather*}"}
-{"id": "3710.png", "formula": "\\begin{align*} | X | \\leq ( p - 1 ) ( k - 1 ) + ( | A | - p ) ( k - 1 ) = ( | A | - 1 ) ( k - 1 ) , \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { D U ^ p } = \\sup \\left \\{ \\int f \\phi d t : \\Vert \\phi \\Vert _ { V ^ q } \\le 1 , \\phi \\in C ^ \\infty _ 0 \\right \\} \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} \\left [ 1 , \\tau = \\frac { s + \\sqrt { \\Delta } } { - 2 t } \\right ] \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} \\big ( L _ { b , j } - ( 2 j + 1 ) B _ 0 \\big ) P _ { j , b } = \\mathcal { O } ( \\beta ) \\ , , \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} \\delta _ { \\ell } : = \\frac { Q \\ell ^ { m _ { \\ell } } \\beta _ { \\ell } + 1 } { 2 4 } , \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} ( - 1 ) ^ { \\sum _ j ( n ^ K ( C _ j ) + 1 ) } \\ , i ^ { \\sum _ j \\omega ( C _ j \\setminus D _ 0 ) - \\omega ( C _ j \\cap D _ 0 ) } = 1 \\ , . \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} \\frac 1 4 e ^ { 2 \\eta + 2 \\zeta } ( i \\xi ) ^ { n + 1 } ( i \\zeta ) ^ n \\eta + \\frac { n } { 4 } i e ^ { 2 \\eta + 2 \\zeta } ( i \\xi ) ^ { n + 1 } ( i \\zeta ) ^ { n - 1 } \\eta ^ 2 + . \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} \\hat R ( \\xi ) = ( 1 - \\tanh \\xi ) \\widehat { \\Re R } ( \\xi ) . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} | \\bar { x } _ j - x _ j | ^ { p - 2 } { ( \\bar { x } _ j - x _ j ) } = - \\sum _ { i \\in I } \\lambda _ j ^ i \\frac { 1 } { \\det L _ { I , I } } \\left ( \\sum _ { k \\in I } \\xi _ k B _ { I } ^ k B _ { I } ^ i \\det L _ { I \\backslash \\{ k \\} , I \\backslash \\{ i \\} } \\right ) \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} \\left ( R ^ i \\sigma _ { * } \\o _ { \\tilde { X } } ( ( j - 1 ) E ) \\right ) ^ { \\wedge } _ { Q } = \\left \\{ \\begin{array} { l r } 0 & Q \\neq P , \\\\ \\varprojlim H ^ i ( m E , \\o _ { m E } ( ( j - 1 ) E ) ) & Q = P . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} P _ { V } \\left ( K _ { n + 1 } = j + k | K _ { n } = k \\right ) = \\binom { k + j } { j } \\left ( \\left ( \\alpha + \\theta \\right ) _ { n \\uparrow } \\right ) ^ { j } \\left ( \\theta + n \\right ) ^ { k } \\frac { V _ { n + 1 , k + j } } { V _ { n , k } } , \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} W ( r ) = ( 1 + r ) \\ln ( 1 + r ) + ( 1 - r ) \\ln ( 1 - r ) - c r ^ 2 \\ , , r \\in ( - 1 , 1 ) , \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} U ^ t ( \\ell ) = J ( \\ell _ 0 ) . \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} \\mu _ n ( - m ) \\underset { m \\to + \\infty } { = } \\widetilde { \\mu } ^ { \\frac { 1 } { 2 } } _ n + \\mathcal { O } ( m ^ { - \\frac { 1 } { 2 } } ) , \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} y = \\mathrm { c m } ( z , y ) z + \\mathrm { c m } ( b ( z ) , y ) b ( z ) . \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} \\tilde \\kappa ( j ) : = \\frac { \\lambda _ j } { \\tilde \\lambda _ j } \\kappa ( j ) \\ , . \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow ( - 1 ) ^ s \\left ( \\frac { \\beta _ \\ell ' / \\beta _ \\ell } { p } \\right ) = 1 . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\displaystyle ( | v ' ( t ) | ^ { p - 2 } v ' ( t ) ) ' + q ( t ) f ( v ( t ) ) = 0 & \\textup { i n } & ( 0 , 1 ) , \\\\ \\displaystyle v ( 0 ) = v ( 1 ) = 0 , & & \\end{array} \\right . \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\psi ( t ) : = x + \\frac { t } { t _ 0 } ( y - x ) \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} L ( s ) = X ( s ) \\overline L ( 1 - s ) \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} 2 \\frac { d a _ n } { d t } = \\sqrt { n ( n + 1 ) ( n + 2 ) } a _ { n + 1 } - \\sqrt { ( n - 1 ) n ( n + 1 ) } a _ { n - 1 } , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} \\mathcal H _ E ^ k : = \\oplus _ { p + q = k } \\mathcal H ^ { p , q } _ E , \\ \\mathcal H ^ { p , q } _ E : = R ^ q \\pi _ * \\mathcal O ( E \\otimes \\wedge ^ p T ^ * _ { \\mathcal X / B } ) . \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} u _ n ( x ) = \\frac { 1 } { \\sqrt { 2 ^ n n ! \\sqrt { 2 \\pi } } } H _ n \\left ( \\frac { x } { \\sqrt { 2 } } \\right ) e ^ { - \\frac { x ^ 2 } { 4 } } , n \\in \\mathbb { N } _ 0 . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\dfrac { 1 } { N ^ 2 } \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { j = 0 } ^ { N - 1 } f ( a ^ { p ^ i q ^ j } ) & = \\lim _ { N \\to \\infty } \\dfrac { 1 } { N ^ 2 } \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { j = 0 } ^ { N - 1 } f ( c ^ { p ^ i q ^ j } ) \\\\ & = \\lim _ { N \\to \\infty } \\dfrac { 1 } { N ^ 2 } \\sum _ { i = 0 } ^ { N - 1 } \\sum _ { j = 0 } ^ { N - 1 } f ( b ^ { p ^ i q ^ j } ) \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} h ( y ) = \\sum _ { i = 0 } ^ { k - 1 } g _ i ( y ) \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} r _ w ( y ) ^ 2 = \\sum _ { i = 1 } ^ N | y _ i | ^ { \\frac { 2 } { w _ i } } . \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 \\beta } { \\beta + 2 } , \\alpha = \\frac { 1 - 2 \\beta s } { 2 ( \\beta + 2 ) } . \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} J _ { 2 2 } ( u , v ) = \\lambda _ 2 ( u v ) A _ 2 + \\lambda _ 4 ( u v ) A _ 4 ( \\log { N } ) ^ { - 2 } \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} \\omega ^ n = e ^ F i ^ { n ^ 2 } \\Omega \\wedge \\bar \\Omega . \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} x _ 1 \\wedge x _ 2 \\wedge \\cdots \\wedge x _ n & = \\frac { 1 } { n ! } \\sum _ { \\sigma \\in S _ n } \\epsilon ( \\sigma ) x _ { \\sigma ( 1 ) } \\otimes x _ { \\sigma ( 2 ) } \\otimes \\cdots \\otimes x _ { \\sigma ( n ) } , \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} \\gamma ^ * ( d ) = \\frac { ( d , w ) } { \\zeta ( 2 ) d } \\chi ( ( d , w ) ) \\xi ( w / ( d , w ) ) \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} \\begin{pmatrix} \\theta \\\\ y \\\\ z \\end{pmatrix} : = G _ { \\delta } \\begin{pmatrix} \\psi \\\\ \\eta \\\\ w \\end{pmatrix} : = \\begin{pmatrix} \\theta _ 0 ( \\psi ) \\\\ y _ { \\delta } ( \\psi ) + [ \\partial _ { \\psi } \\theta _ 0 ( \\psi ) ] ^ { - T } \\eta + [ ( \\partial _ { \\theta } \\tilde { z } _ 0 ) ( \\theta _ 0 ( \\psi ) ) ] ^ T \\partial _ x ^ { - 1 } w \\\\ z _ 0 ( \\psi ) + w \\end{pmatrix} \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} = \\frac { \\theta } { 2 } \\left ( \\left [ \\frac { \\partial \\Phi } { \\partial y ^ 0 } \\right ] ^ 2 - 2 \\int _ { \\mathbb { R } ^ n } \\textbf { f } \\ ; | \\theta | ^ 3 \\sqrt { | \\Theta | } \\ ; \\widetilde { p } ^ 1 \\ ; d \\widetilde { p } ' \\right ) - \\frac { \\theta ^ 2 } { 2 } ( \\widetilde { R } ^ { ( n - 1 ) } + 2 \\frac { \\Theta ^ { a b } \\widetilde { T } _ { a b } } { n - 1 } ) . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} p _ T ( G ) ( z ) : = z ^ { | V | } Z _ T ( G ) ( 1 / z , w ) , \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} \\hat { \\theta } _ { j } = p _ { j } \\theta + \\xi _ { j } , \\ \\forall 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\theta y = q ^ { - 1 } y \\theta , \\theta z = q z \\theta , y z = q ^ 2 z y , \\theta ^ 2 = q ^ { 1 / 2 } ( q - 1 ) z y . \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{align*} \\bar { \\psi } ( \\gamma , \\alpha , \\theta ) & = E [ \\psi ( W , \\gamma , \\alpha , \\theta ) ] = E [ Z \\gamma ( X ) ] - \\theta + E [ \\alpha ( X ) \\{ Y - \\gamma ( X ) \\} ] \\\\ & = E [ \\alpha _ { 0 } ( X ) \\gamma ( X ) ] - \\theta + E [ \\alpha ( X ) \\{ \\gamma _ { 0 } ( X ) - \\gamma ( X ) \\} ] . \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} \\hat { p } _ { r } = - \\imath \\left ( \\partial _ { r } + F ( r ) \\right ) , \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{gather*} B = R \\psi ^ { \\sigma _ 3 } \\hat { B } _ 0 \\psi ^ { - \\sigma _ 3 } R ^ { - 1 } + R _ t R ^ { - 1 } - \\frac { x } { 2 } I + \\frac { \\kappa _ t } { \\kappa } I + \\frac { \\psi _ t } { \\psi } R \\sigma _ 3 R ^ { - 1 } \\equiv - x ^ 2 J + x { B } _ 1 + { B } _ 0 , \\end{gather*}"}
-{"id": "713.png", "formula": "\\begin{align*} \\Omega _ { j } = \\left \\{ \\begin{array} { l l } d t _ j / ( 1 - t _ j ) & \\mbox { i f } \\ ; j = 1 , \\alpha _ 1 + 1 , \\alpha _ 1 + \\alpha _ 2 + 1 , \\ldots , \\alpha _ 1 + \\alpha _ 2 + \\ldots + \\alpha _ { r - 1 } + 1 ; \\\\ d t _ j / t _ j & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} n ^ { - q / 2 } \\sum _ { r _ 1 + \\cdots + r _ q = n } \\exp \\left ( - n \\epsilon \\left | \\frac r n - p \\right | ^ 2 \\right ) \\le C . \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} b _ m : \\real ^ d \\to \\real , b _ m ( x ) = 2 ^ { d m } \\cdot b ( 2 ^ m x ) \\ , , \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} \\operatorname { D i v } \\widehat { P } _ 1 + \\operatorname { D i v } ^ { \\vartriangle } \\widehat { P } _ 2 = Q ^ { \\alpha } F _ { \\alpha } ^ { \\ast } - v ^ { \\alpha } \\bold { p r } X ( F _ { \\alpha } ) , \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} B ( 0 ) = \\bigg ( \\frac { | J _ { \\rho _ 0 } | e ^ { - T _ { 0 } } } { 2 | J _ { \\rho _ 0 } | e ^ { - T _ { 0 } } + 2 } \\bigg ( { \\rm \\exp } \\bigg [ \\bigg ( 2 + \\frac { 2 } { | J _ { \\rho _ { 0 } } | e ^ { - T _ { 0 } } } \\bigg ) T _ { 0 } \\bigg ] - 1 \\bigg ) + C _ * { \\rm \\exp } \\bigg [ \\bigg ( 2 + \\frac { 2 } { | J _ { \\rho _ { 0 } } | e ^ { - T _ { 0 } } } \\bigg ) T _ { 0 } \\bigg ] \\bigg ) ^ { - 1 } . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{align*} \\sigma ^ { l \\ast } ( S _ j \\zeta ^ { j } ) \\ ; = \\ ; \\sum _ { i = 0 } ^ j \\bigl ( U _ { j - i } \\zeta ^ { q ^ l ( j - i ) } \\bigr ) ( S _ i \\zeta ^ { i } ) \\ , \\zeta ^ { i ( q ^ l - 1 ) } \\ , . \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} \\int _ Y \\langle d _ b \\phi , d _ b \\psi \\rangle = \\int _ Y ( d _ b ^ * d _ b \\phi ) \\psi = \\int _ Y ( - \\Delta _ Y \\phi + ( J \\partial _ r ) ^ 2 \\phi ) \\psi , \\\\ \\int _ Y \\langle d ^ c _ b \\phi , d _ b \\psi \\rangle = \\int _ Y ( d _ b ^ * d ^ c _ b \\phi ) \\psi = ( 2 - 2 n ) \\int _ Y ( J \\partial _ r \\phi ) \\psi . \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\abs { k ( t _ i , s ) - k ( t _ { i - 1 } , s ) } \\leq m ( s ) , \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} \\begin{cases} 2 \\alpha + \\gamma > \\frac 3 2 & \\alpha \\leq \\frac 1 2 , \\\\ [ 2 m m ] \\gamma > \\frac 1 2 & \\alpha > \\frac 1 2 . \\end{cases} \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} Y : = Y _ { t , p , \\Omega } : = B ^ { t - 2 } _ { p } ( L _ p ( \\Omega ) ) ^ d \\times B ^ { t - 1 / p , 0 } _ { p } ( L _ p ( \\partial \\Omega ) ) ^ d . \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} \\mathfrak { C } _ { 0 0 } = \\mathfrak { C } _ { 0 1 } , \\ ; \\mathfrak { C } _ { 1 1 } = \\mathfrak { C } _ { 0 1 } , \\ ; \\mathfrak { C } _ { 1 a } = \\mathfrak { C } _ { 0 a } , \\ ; \\mathfrak { C } _ { a b } = - \\frac { \\mathfrak { C } _ { 0 1 } } { g _ { 1 1 } } g _ { a b } , \\ ; \\mathfrak { \\widetilde { C } } = - \\frac { n - 1 } { g _ { 1 1 } } \\mathfrak { C } _ { 0 1 } ; \\ ; a , b = 2 , . . . , n ; \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} P _ { S _ 1 | V , Y } ( 1 | v , y ) = p _ { S _ 1 | S } ( 1 | 1 ) = \\bar { \\sigma } . \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} \\lbrace r _ { i + 1 } \\looparrowleft r _ i + ( 2 \\cdot U [ 0 , 1 ] - 1 ) \\rbrace _ { i = 2 , \\ldots , N - 1 } \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\eta ^ \\rho _ k } = \\frac { \\partial } { \\partial \\eta ^ \\rho _ { k + 1 } } + \\sum _ l \\bigg ( \\frac { \\partial a ^ k _ l } { \\partial \\eta ^ \\rho _ k } \\bigg ) \\frac { \\partial } { \\partial y ^ l _ { k + 1 } } + \\sum _ \\tau \\bigg ( \\frac { \\partial \\beta ^ k _ \\tau } { \\partial \\eta ^ \\rho _ k } \\bigg ) \\frac { \\partial } { \\partial \\eta ^ \\tau _ { k + 1 } } . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} \\max _ { [ 0 , a _ k ] } F = \\max _ { [ 0 , b _ k ] } F . \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} \\phi ^ { * * } ( x ) & = \\sup _ { q \\in \\Gamma ( \\Omega , K ) , \\Phi _ q \\geq 0 } q ( x ) \\\\ \\phi ^ { * * } ( x ) & = \\phi ( x ) \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} \\| f \\| _ { 2 } = \\| f - P ( f ) \\| _ k , \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} X _ i ^ { Y _ 1 ^ { C ' ( i , j ) } \\cdots Y _ n ^ { C ' ( i , j ) } } = X _ j ^ { Y _ 1 ^ { C '' ( i , j ) } \\cdots Y _ n ^ { C '' ( i , j ) } } , \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} \\tau _ { \\lambda } ( b ^ { * } ( f ) ) = \\exp ( i \\lambda ) b ^ { * } ( f ) \\\\ \\tau _ { \\lambda } ( b ( f ) ) = \\exp ( - i \\lambda ) b ( f ) \\ , \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} D _ { \\widetilde { x } } \\widetilde { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } = \\left ( D _ x \\widetilde { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } \\right ) ( D _ x \\widetilde { x } ) ^ { - 1 } \\frac { \\operatorname { d } } { \\operatorname { d } \\ ! \\varepsilon } ( M ( \\varepsilon ) ^ { - 1 } ) = - M ( \\varepsilon ) ^ { - 1 } \\frac { \\operatorname { d } \\ ! M ( \\varepsilon ) } { \\operatorname { d } \\ ! \\varepsilon } M ( \\varepsilon ) ^ { - 1 } . \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\boldsymbol { k } } _ { \\boldsymbol { b } } : = \\{ ( a _ 1 , a _ 2 , \\cdots , a _ l ) : a _ i k _ i \\boldsymbol { a } \\} . \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} [ n ] _ { q } = \\frac { 1 - q ^ { n } } { 1 - q } , \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} V ^ { \\ast } \\Psi A ^ { \\dagger } - F _ { S _ { A } } K ^ { \\ast } \\Psi A ^ { \\dagger } = V ^ { \\ast } A ^ { \\dagger } - F _ { S _ { A } } K ^ { \\ast } A ^ { \\dagger } + F _ { S _ { A } } K ^ { \\ast } A ^ { \\dagger } = V ^ { \\ast } A ^ { \\dagger } . \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } n ! \\| f _ n ( \\cdot , t , x ) \\| ^ 2 _ { \\mathcal { H } ^ { \\otimes n } } < \\infty \\ , . \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} ( k - l + 2 ) & P _ { k + 1 } ^ { ( l ) } = ( k + 2 ) P _ k ^ { ( l ) } Q + l P _ k ^ { ( l - 1 ) } R + ( k - 1 ) ( l - 1 ) l P _ k ^ { ( l - 2 ) } . \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} \\eta ( A ) : = s - \\lim _ { V \\to \\infty } \\eta _ { \\Lambda } ( A ) \\ , \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { F _ 1 ( y ^ G _ 1 ) } p _ { Y _ 1 } = \\int _ { - \\infty } ^ { y ^ G _ 1 } p _ { Y ^ G _ 1 } . \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ { p ( x ) } u = \\lambda u ^ { \\alpha _ { 1 } ( x ) } v ^ { \\beta _ { 1 } ( x ) } & \\Omega , \\\\ - \\Delta _ { q ( x ) } v = \\lambda u ^ { \\alpha _ { 2 } ( x ) } v ^ { \\beta _ { 2 } ( x ) } & \\Omega , \\\\ u , v > 0 & \\Omega , \\\\ u , v = 0 & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} E \\ ; : \\ ; y ^ 2 = x ^ 3 + a x ^ 2 + b x , \\Delta = \\Delta _ m = 2 ^ 4 b ^ 2 ( a ^ 2 - 4 b ) \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{gather*} 9 \\zeta _ { t t } + 9 \\zeta \\zeta _ t + \\zeta ^ 3 + P ( t ) \\zeta + Q ( t ) = 0 , \\\\ 9 \\lambda _ { t t } + 9 \\zeta \\lambda _ t + \\bigl ( 9 \\zeta _ t + 3 \\zeta ^ 3 + P ( t ) \\bigr ) \\lambda = 0 , \\end{gather*}"}
-{"id": "6331.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\tilde { f } ( a _ i ) \\times \\tilde { f } ( b _ i ) \\in \\tilde { R } _ 2 ^ 0 \\textrm { w h e n e v e r } \\sum _ { i = 1 } ^ n a _ i \\times b _ i \\in \\tilde { R } ^ 0 _ 1 \\textrm { f o r } a _ i , b _ i \\in \\tilde { R } _ 1 . \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{gather*} \\frac { d \\Psi _ 0 } { d x } = \\hat { L } _ 0 \\Psi _ 0 , \\frac { d \\Psi _ 0 } { d t } = \\hat { B } _ 0 \\Psi _ 0 , \\end{gather*}"}
-{"id": "226.png", "formula": "\\begin{align*} \\tau _ \\varepsilon = \\sup \\{ n \\ge 1 : | \\xi _ n | > \\varepsilon \\} ( \\sup \\{ \\varnothing \\} = 0 ) \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} S _ { m , l } ^ { ( k ) } : = \\{ ( n _ 1 , \\cdots , n _ l ) ~ | ~ n _ 1 + \\cdots + n _ l = k - m , 2 \\leq n _ 1 \\leq n _ 2 \\leq \\cdots \\leq n _ l , n _ j \\in \\mathbb { N } ( 1 \\leq j \\leq l ) \\} \\end{align*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\left | F _ n ^ \\ast ( \\ell ) + F _ n ^ \\ast ( k ) - 1 \\right | & = \\left | F _ n ^ \\ast ( \\ell ) - F ^ \\ast ( \\ell ) + ( 1 - F ^ \\ast ( k ) ) + F _ n ^ \\ast ( k ) - 1 \\right | \\\\ & \\le \\left | F _ n ^ \\ast ( \\ell ) - F ^ \\ast ( \\ell ) \\right | + \\left | F _ n ^ \\ast ( k ) - F ^ \\ast ( k ) \\right | , \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\mathcal { Z } _ { S ' } ( \\mathfrak { q } ' ) \\cap U _ { S ' / R } \\not = \\emptyset , \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\Omega _ i ( h ) = \\{ x \\in \\overline \\Omega _ i \\ | \\ 0 \\leq | H ( x ) | < h \\} \\ i \\in \\{ 1 , 2 , 3 \\} \\ \\ \\ \\ \\Omega ( h ) = \\bigcup _ { i = 1 } ^ 3 \\Omega _ i ( h ) . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} - b ( x ) \\cdot D \\varphi ( x ) + f ( x ) = 0 \\ \\ \\ x \\in \\overline B _ { 2 r } . \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{gather*} J ^ { 2 } ( h _ { + } \\allowbreak ) - M \\left ( h _ { + } \\right ) = \\dfrac { 4 p _ { 1 } \\left ( s - v \\right ) ^ { 2 } ( s - 2 h _ { + } ) ^ { 2 } } { s ^ { 4 } } \\times \\\\ \\left ( p _ { 1 } - \\dfrac { { \\large ( } 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v { \\large ) } ^ { 2 } } { s ^ { 2 } + t ^ { 2 } } \\right ) \\end{gather*}"}
-{"id": "7281.png", "formula": "\\begin{align*} \\hat { V } = ( \\hat { G } ^ { \\prime } \\hat { \\Upsilon } \\hat { G } ) ^ { - 1 } \\hat { G } ^ { \\prime } \\hat { \\Upsilon } \\hat { \\Psi } \\hat { \\Upsilon } \\hat { G } ( \\hat { G } ^ { \\prime } \\hat { \\Upsilon } \\hat { G } ) ^ { - 1 } , \\hat { G } = \\frac { \\partial \\hat { \\psi } ( \\hat { \\theta } ) } { \\partial \\theta } . _ { { } } \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} \\tilde { v } _ { i L } = - \\tilde { v } _ { n + 1 - i R } U _ { i a , j b } = U _ { n + 1 - i \\bar { a } , n + 1 - j \\bar { b } } \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} \\varphi _ { n } ( x , t ) = \\sum _ { i = 1 } ^ { n } a _ { n , i } ( t ) w _ { i } ( x ) , \\mu _ { n } ( x , t ) : = \\sum _ { i = 1 } ^ { n } b _ { n , i } ( t ) w _ { i } ( x ) , \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} f _ { C i } \\left ( V _ d , V _ q \\right ) : = c _ { 2 i } \\left ( f _ { P i } \\left ( V _ d , V _ q \\right ) \\right ) ^ 2 + c _ { 1 i } f _ { P i } \\left ( V _ d , V _ q \\right ) + c _ { 0 i } . \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} \\Gamma _ 2 ^ { \\mathrm { h o r i } } ( f , f ) & = ( X ^ 2 f ) ^ 2 + ( Y ^ 2 f ) ^ 2 + ( X Y f ) ^ 2 + ( Y X f ) ^ 2 - 2 ( X f ) ( Y Z f ) + 2 ( Y f ) ( X Z f ) \\\\ \\Gamma _ 2 ^ { \\mathrm { v e r t } } ( f , f ) & = ( X Z f ) ^ 2 + ( Y Z f ) ^ 2 . \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} \\prod _ { i , j \\in [ q ] } \\exp \\left ( \\frac { n s _ { a _ u b _ u i j } } { d } \\cdot \\pi _ i \\pi _ j \\right ) & = \\prod _ { i , j \\in [ q ] } \\exp \\left ( \\frac { \\pi _ i \\pi _ j A _ { a _ u i } A _ { b _ u j } } { d } \\right ) \\\\ & = \\exp \\left ( \\frac { \\left ( \\sum _ { i \\in [ q ] } \\pi _ i A _ { a _ u i } \\right ) \\left ( \\sum _ { j \\in [ q ] } \\pi _ j A _ { b _ u j } \\right ) } { d } \\right ) = 1 , \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} 0 & = L ( e _ { i i } ) e _ { k l } - e _ { k l } L ( e _ { i i } ) + e _ { i i } L ( e _ { k l } ) - L ( e _ { k l } ) e _ { i i } \\\\ & = \\left ( C _ { k k } ^ { i i } - C _ { l l } ^ { i i } \\right ) e _ { k l } + \\left ( C _ { i k } ^ { i i } + C _ { i k } ^ { k k } \\right ) e _ { i l } - \\left ( C _ { l i } ^ { i i } + C _ { l i } ^ { l l } \\right ) e _ { k i } . \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} y = L ( x ) = \\dfrac { t } { 2 } + \\dfrac { w + u - t } { v - s } \\left ( x - \\dfrac { s } { 2 } \\right ) , x \\in I \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} \\varphi ( P ) = ( X , Y ) = ( x ' + ( \\pi _ L ) , y ' + ( \\pi _ L ) ) , \\varphi ( \\sigma ( P ) ) = ( X _ 1 , Y _ 1 ) = ( x _ 1 + ( \\pi _ L ) , y _ 1 + ( \\pi _ L ) ) . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} f _ { \\mathrm { s } | \\mathrm { c } } ( s , t ) = 2 \\pi \\lambda s e ^ { - \\lambda \\pi \\left ( s ^ 2 - t ^ 2 \\right ) } . \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} \\hat { x } _ t = \\sum _ { j < i } \\hat { x } _ t ^ j + \\hat { x } _ t ^ i = f ( k _ t ) - \\bar { y } _ t , & & t = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} f _ { \\tilde { v } } ( n ) = \\frac { 1 } { N } \\sum \\limits ^ { N - 1 } _ { l = 0 } { \\left ( j 2 \\pi \\frac { l } { N } \\right ) ^ { \\tilde { v } } F _ 0 ( l ) e ^ { j 2 \\pi \\frac { l + N _ { \\rm c p } } { N } n } } , \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} \\big ( \\rho ( a ) - \\pi ^ * ( \\psi ( a ) ) \\big ) . f = & \\omega ( a _ f , a ) - \\pi ( \\psi ( a ) , d f ) \\\\ = & \\omega ( a _ f , a ) + \\psi ( a ) ( \\pi ^ * ( d f ) ) \\\\ = & \\omega ( a _ f , a ) - \\omega ( a , \\lambda ( \\pi ^ * ( d f ) ) ) \\\\ = & \\omega ( a _ f + \\lambda ( \\pi ^ * ( d f ) ) , a ) \\\\ = & 0 . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\P \\left ( \\bigcap _ { \\ell = 1 } ^ k \\{ X _ N ^ { R , { \\rm r e s c } } ( T _ \\ell ) \\leq U _ \\ell \\} \\right ) = \\frac { \\P \\left ( \\bigcap _ { \\ell = 1 } ^ k \\{ { \\mathcal A } _ 2 ( T _ \\ell ) - T _ \\ell ^ 2 \\leq U _ \\ell \\} \\cap \\{ { \\mathcal A } _ 2 ( 0 ) \\leq R ) \\} \\right ) } { \\P ( { \\mathcal A } _ 2 ( 0 ) \\leq R ) } \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} \\infty > u ( x , t + \\tau ) & \\ge \\int _ \\tau ^ { t + \\tau } \\int _ { { \\bf R } ^ N } G ( x - y , t + \\tau - s ) [ S ( s - \\tau ) u ( \\tau ) ] ( y ) ^ p \\ , d y \\ , d s \\\\ & \\ge \\int _ \\tau ^ { t + \\tau } \\biggr ( \\int _ { { \\bf R } ^ N } G ( x - y , t + \\tau - s ) [ S ( s - \\tau ) u ( \\tau ) ] ( y ) \\ , d y \\biggr ) ^ p \\ , d s \\\\ & = \\int _ \\tau ^ { t + \\tau } [ S ( t ) u ( \\tau ) ] ( x ) ^ p \\ , d s = t [ S ( t ) u ( \\tau ) ] ( x ) ^ p \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} T _ t ^ { 1 , 0 } X = { \\rm s p a n } _ { \\mathbb C } \\{ Z + \\Phi ( z , t ) \\overline Z \\} . \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} B ^ h ( \\xi , 0 ) = \\frac { 2 i \\xi J ( \\xi ) } { \\Lambda ( \\xi ) } . \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} \\beta ^ { ( k ) } ( N ) = \\frac { 1 } { k } , k \\in \\{ 1 , \\ldots , N \\} . \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} P _ { S , S _ 1 | V , Y } ( s , s _ 1 | v , y ) = P _ { S | V , Y } ( s | v , y ) p _ { S _ 1 | S } ( s _ 1 | s ) , \\quad \\forall ( v , y , s , s _ 1 ) \\in \\mathcal { V } \\times \\mathcal { Y } \\times \\mathcal { S } \\times \\mathcal { S } _ 1 . \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} V _ s ' ( v ) & = ( s - 1 ) \\frac { v ^ { s - 2 } } { ( v + 1 ) ^ { \\frac { N } { 2 } } } - \\frac { N } { 2 } \\frac { v ^ { s - 1 } } { ( v + 1 ) ^ { \\frac { N } { 2 } + 1 } } = V _ s ( v ) \\frac { ( s - 1 ) ( v + 1 ) - \\frac { N } { 2 } v } { v ( v + 1 ) } \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} L _ b \\ , e ^ { i \\phi _ b ( x , x ' ) } = e ^ { i \\phi _ b ( x , x ' ) } \\big ( L _ 0 + a ^ 2 ( x , x ' ) - 2 a ( x , x ' ) \\cdot ( - i \\nabla _ x - \\widehat A _ 0 ( x ) ) + i \\nabla _ x a ( x , x ' ) \\big ) \\ , , \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} \\dim ( E _ 2 ( \\Gamma _ 0 ( 4 0 ) ) ) = 7 , ~ \\dim ( S _ 2 ( \\Gamma _ 0 ( 4 0 ) ) ) = 3 . \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\int _ { \\R _ + } \\d u \\ , f _ W ( u ) g _ Z ( u ) = \\int _ { \\R _ + } \\d u \\ , e ^ { 2 ( Z - W ) u } = - \\frac 1 { 2 ( Z - W ) } \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} 0 & \\geq \\lambda u _ 1 ^ + ( \\hat h ) - f ( \\hat x ) + g ( \\hat x ) + \\frac { 1 } { T _ 1 ( \\hat h ) } \\int _ 0 ^ { T _ 1 ( \\hat h ) } f ( X ( t , \\hat x ) ) \\ , d t \\\\ & > \\lambda u _ 1 ^ + ( \\hat h ) - \\frac { \\eta } { 2 } + \\frac { 1 } { T _ 1 ( \\hat h ) } \\int _ 0 ^ { T _ 1 ( \\hat h ) } \\Big ( g ( X ( t , \\hat x ) ) - \\frac { \\eta } { 2 } \\Big ) \\ , d t \\\\ & = \\lambda u _ 1 ^ + ( \\hat h ) + \\overline G _ 1 ( \\hat h , \\phi ' ( \\hat h ) ) - \\eta . \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} \\bar \\Gamma ^ \\pm \\ = \\ \\bigcup _ { i = 1 } ^ N \\Lambda _ i ^ \\pm \\cup \\bigcup _ { i = 1 } ^ { N } S ^ \\pm _ i \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} [ F _ { 0 } ] _ { I } = \\begin{cases} \\lambda ( b _ { i } s ^ { 2 } - a _ { i } t ^ { 2 } ) , & I = \\{ 0 , i \\} \\\\ \\lambda ^ { 2 } ( a _ { i } b _ { j } - a _ { j } b _ { i } ) s t , & I = \\{ i , j \\} , 0 \\notin I \\end{cases} . \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} \\bar { N } ^ K _ { k G } ( Q ) & = \\bar { N } _ { k N } ^ { K } ( Q ) \\bigoplus \\left ( \\bigoplus _ { \\substack { z \\in [ N _ G ^ K ( Q ) / C _ N ( Q ) ] \\\\ z \\notin N } } \\bar { N } _ { \\mathcal { O } N z } ^ { \\varphi _ x } ( Q ) \\right ) \\\\ & = k N _ N ( Q ) \\bigoplus \\left ( \\bigoplus _ { \\substack { z \\in [ N _ G ^ K ( Q ) / C _ N ( Q ) ] \\\\ z \\notin N } } \\bar { N } _ { \\mathcal { O } N z } ^ { \\varphi _ x } ( Q ) \\right ) . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} \\mathcal { S } ( \\omega , x ) : = \\left \\{ y \\in \\mathbb { R } ^ d : | \\varphi _ m ( \\omega , y ) - \\varphi _ m ( \\omega , x ) | \\leq \\beta ( \\omega , x ) e ^ { \\varepsilon m } \\textrm { f o r a l l } m \\in \\mathbb { N } \\right \\} \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} & Q = \\bar Q = \\bar H = S = \\bar S = q = \\bar q = \\bar r = 0 , \\\\ & R ( s ) = e ^ { - \\mu ( s - t ) } , \\bar R ( s ) = \\beta e ^ { - \\mu ( s - t ) } , r ( s ) = - \\alpha e ^ { - \\mu ( s - t ) } , H = e ^ { - \\mu ( T - t ) } . \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} { \\hat { R } } = \\frac { 1 } { \\ln 2 } \\int _ 0 ^ \\infty { \\frac { 1 } { { z } } ( 1 - e ^ { - z { { P _ t } G _ { } ^ 2 L \\left ( r \\right ) } } ) \\hat { \\Xi } _ 2 ( z ) { e ^ { - z \\sigma _ o ^ 2 } } d z } , \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} P _ 3 = W ( P _ 2 ) \\colon W ( W ( k ) ( ( T ) ) ) \\to & W ( K ) = W ( k ( ( T ) ) ) \\\\ ( \\sum _ j a _ { i j } T ^ j ) _ i \\mapsto & ( \\sum _ j \\pi _ 1 ( a _ { i j } ) T ^ j ) _ i . \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} ( H _ { | c | } ( \\theta ) u ) _ n = | c | ( \\theta + n \\alpha ) u _ { n + 1 } + | c | ( \\theta + ( n - 1 ) \\alpha ) u _ { n - 1 } + v ( \\theta + n \\alpha ) u _ n . \\end{align*}"}
-{"id": "7684.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } Q ( x ) = + \\infty . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} \\Psi ( z , f ) = e ( \\rho _ { f , \\infty } z ) \\prod _ { n = 1 } ^ { \\infty } ( 1 - e ( n z ) ) ^ { c _ { f } ^ { + } ( n ^ { 2 } , n ) } \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} \\begin{aligned} \\left | ( F ^ n ) ' ( u ) \\right | & \\geq \\frac { 1 } { ( 9 6 \\pi ) ^ n } \\prod _ { j = 0 } ^ { n - 1 } h ( x _ j ) \\\\ & \\geq \\exp \\ ! \\left ( \\sum _ { j = 0 } ^ { n - 1 } ( \\lambda - \\delta ( x _ j ) ) x _ j - n \\log \\ ! \\left ( 9 6 \\pi \\right ) \\right ) \\\\ & \\geq \\exp \\ ! \\left ( ( 1 + \\delta ( x _ { n - 1 } ) ) x _ n - n \\log \\ ! \\left ( 9 6 \\pi \\right ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} \\frac { v _ 1 ' - v _ { - 1 } ' } { 2 } + \\frac { ( v _ 1 - v ) ^ 2 - ( v - v _ { - 1 } ) ^ 2 } { 2 } + v _ 2 - 4 v _ 1 + 6 v - 4 v _ { - 1 } + v _ { - 2 } = 0 . \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} Y ( X Y + 2 q _ 4 Z ^ 2 ) = X ^ 3 + p _ 2 X ^ 2 Z + p _ 4 X Z ^ 2 + p _ 6 Z ^ 3 , \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} \\frac { d } { d t } E _ { \\delta } ( t ^ \\delta ) = \\sum _ { k = 1 } ^ \\infty \\frac { t ^ { k \\delta - 1 } } { \\Gamma ( \\delta k ) } \\ t > 0 . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\ell ( p _ M , c _ n ) = I _ P ( M ; \\mathbf { Z } ) , \\end{align*}"}
-{"id": "8447.png", "formula": "\\begin{align*} Y _ k ^ { ( m ) } ( t ) = Y ^ { ( m ) } _ k ( 0 ) + g _ k ^ { ( m ) } t + B ^ { ( m ) } _ k ( t ) + \\tfrac 1 2 L _ { ( k - 1 , k ) } ^ { ( m ) } ( t ) - \\tfrac 1 2 L _ { ( k , k + 1 ) } ^ { ( m ) } ( t ) , \\ \\ k = 1 , \\ldots , m ^ 2 , \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } \\int _ { \\Omega } \\frac { 1 } { p ( x ) } \\left \\vert \\nabla u _ { n } ( x ) - \\nabla u ( x ) \\right \\vert ^ { p ( x ) } d x = 0 . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} | \\textbf { H } | = \\beta \\frac { \\sin ^ 2 \\alpha } { \\cos ^ 2 \\alpha } | \\nabla \\alpha | . \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} t ^ { 1 2 } a _ 4 ' = a + 3 r ^ 2 \\Leftrightarrow a _ 4 ' = - \\frac { \\mu ^ 4 \\tilde { c } _ 4 } { 1 6 } + \\mu t ( a _ 0 + y _ 1 t ) ^ 2 . \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\Gamma ( s ) \\Gamma ( s + 1 / 2 ) = 2 ^ { 1 - 2 s } \\cdot \\sqrt { \\pi } \\cdot \\Gamma ( 2 s ) . \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} \\tfrac 1 2 L _ { ( 1 , 2 ) } ^ { ( m ) } \\to \\tfrac 1 2 L _ { ( 1 , 2 ) } C [ 0 , T ] , \\ \\tfrac 1 2 L _ { ( 1 , 2 ) } ( t ) : = - Y _ 1 ( t ) + Y _ 1 ( 0 ) + g _ 1 t + B _ 1 ( t ) . \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} \\zeta ( m + p ) - \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { 1 } { \\ell ^ { p - 1 } ( \\ell + 1 ) ^ { m + 1 } } \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} \\chi ( p ) p ^ { k - 2 } \\widetilde \\lambda _ 1 ( p ^ 2 ) a ( m I ) & = \\chi ( p ^ 2 ) p ^ { 2 k - 4 } ( \\alpha ( I ; p ) - p ) a ( m I ) \\\\ & + \\lambda ( p ) a ( p m I ) - a ( p ^ 2 m I ) \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} | | u | | _ { L ^ 2 _ T L ^ { 2 ^ * , 2 } } = \\gamma ( z , T ) | | f | | _ { L ^ { 2 ^ * , 2 } } , | | F | | _ { L ^ 2 _ T L ^ { 2 _ * , 2 } } = \\gamma ( z , T ) | | ( - \\Delta - z ) f | | _ { L ^ { 2 _ * , 2 } } , \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} f _ m ( z ) \\mid _ { - 1 } \\gamma = \\sum _ { \\substack { n = 0 \\\\ n \\equiv 0 \\pmod { \\ell ^ m } } } ^ { \\infty } a _ 0 ( n ) q _ { 2 4 \\ell ^ m } ^ n - \\sum _ { \\substack { n = 0 \\\\ n \\equiv 0 \\pmod { \\ell ^ { m + 1 } } } } ^ { \\infty } a _ 0 ( n ) q _ { 2 4 \\ell ^ m } ^ n , \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\sum _ { m _ 1 + m _ 2 = 2 n } \\lambda _ { 1 } ( m _ 1 ) \\lambda _ { 2 } ( m _ 2 ) W \\left ( \\frac { m _ 1 - n } { H } \\right ) V \\left ( \\frac { n - m _ 2 } { H ' } \\right ) . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} \\Big | T _ \\Sigma ( i \\tau / 2 ) - \\sum _ { l = 0 } ^ k ( i \\tau ) ^ { - ( 2 j - 1 + l ) } T _ { \\Sigma } ^ l \\Big | \\lesssim \\sum _ { \\substack { k + 1 \\leq | \\alpha | + | \\beta | \\leq 2 j - 1 + k \\\\ \\max \\{ \\alpha _ l , \\beta _ l \\} \\leq [ \\frac { k } 2 ] + 1 } } | \\tau | ^ { - | \\alpha | - | \\beta | } \\prod _ k \\| \\partial ^ { \\alpha _ k } u _ k \\| _ { l ^ { 2 j } _ { \\tau } D U ^ 2 } \\| \\partial ^ { \\beta _ k } v _ k \\| _ { l ^ { 2 j } _ { \\tau } D U ^ 2 } . \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} \\varphi _ \\ell ( P _ 1 ) = ( 0 , 0 ) , \\varphi _ \\ell ( P _ 2 ) = ( - 1 , \\omega _ 3 ^ 2 ) , \\varphi _ \\ell ( Q ) = ( - \\omega _ 3 ^ { 2 \\alpha } , \\omega _ 3 ^ 2 ) \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} \\mathcal I _ s ^ q ( \\mu ) = \\int _ { \\underline a \\in \\Sigma } \\left ( \\sum _ { n = 0 } ^ { \\infty } \\int _ { \\underline b ' \\in \\Sigma : \\underline b ' \\wedge \\sigma ^ n \\underline a = \\phi } \\frac { \\mu [ a _ 1 \\cdots a _ n ] } { d ( \\underline a , a _ 1 \\cdots a _ n \\underline b ' ) ^ s } d \\mu ( \\underline b ' ) \\right ) ^ { q - 1 } d \\mu ( \\underline a ) \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} [ \\Theta ^ h _ { j \\bar k } , \\theta _ v ] = [ [ \\theta _ k ^ * , \\theta _ j ] , \\theta _ v ] = [ [ \\theta _ k ^ * , \\theta _ v ] , \\theta _ j ] . \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\vdash n } \\sum _ { \\lambda _ i \\in \\lambda } \\lambda _ i ^ \\alpha = \\sum _ { \\lambda \\vdash n } \\sum _ { \\substack { \\lambda _ i \\in \\lambda \\\\ \\lambda _ i } } \\sigma _ \\alpha ( \\lambda _ i ) = \\sum _ { k = 1 } ^ { n } \\sigma _ { \\alpha } ( k ) p ( n - k ) . \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} \\frac { 1 } { r _ { \\ast } } = \\frac { 1 } { k } + \\frac { a ^ 2 } { a ^ 2 + \\omega ^ 2 } . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} \\max _ { { \\hat z _ n ^ { \\rm R } } \\ge 0 , { \\hat z _ n ^ { \\rm R } } \\ge 0 } & \\ln \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( \\sqrt { \\hat z _ n ^ { \\rm R } } - \\sqrt { P / 2 } \\big ) \\right ) + \\ln \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( \\sqrt { \\hat z _ n ^ { \\rm I } } - \\sqrt { P / 2 } \\big ) \\right ) \\\\ \\mathrm { s . t . } ~ & { \\hat z _ n ^ { \\rm R } } + { \\hat z _ n ^ { \\rm I } } = B _ n . \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} Q _ { \\ast } ^ { \\beta } ( x , n , [ u ] , [ v ] ) = - \\sum _ { \\alpha , J _ 1 , J _ 2 } ( - D ) _ { J _ 1 } S _ { - J _ 2 } \\left ( v ^ { \\alpha } K ^ { \\beta } _ { \\alpha ; J _ 1 , J _ 2 } \\right ) . \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} f _ a ( x ) = \\Big ( 1 + \\sum _ { k = 1 } ^ d a _ k k ! \\pi ^ { - k } L _ k ^ { n / 2 - 1 } ( \\pi \\| x \\| ^ 2 ) \\Big ) e ^ { - \\pi \\| x \\| ^ 2 } \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} y _ i : = e _ { p } ^ p ( \\underline { x } ^ { ( i ) } ) = e _ p ^ p ( x _ { ( i - 1 ) p + 1 } , \\dots , x _ { i p } ) . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} \\begin{cases} M _ { 1 } , M _ { 2 } = K , \\ldots , N ; \\\\ M _ { 1 2 } = \\max ( 0 , M _ { 1 } + M _ { 2 } - N ) , \\ldots , \\min ( M _ { 1 } , M _ { 2 } ) . \\end{cases} \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} \\ker ( T + s A ) = \\{ 0 \\} \\quad \\mbox { f o r a n y } s \\in ( - \\eta _ 1 , \\eta _ 1 ) \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} k ( t , s ) = \\begin{cases} s ( 1 - t ) , & , \\\\ t ( 1 - s ) , & , \\end{cases} \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} ^ { c } D ^ { q } \\mathbf { z } ( t ) = - A \\mathbf { z } ( t ) + T \\mathbf { g } ( \\mathbf { z } ( t ) ) + \\mathbf { I } . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} p _ j \\frac { \\partial H } { \\partial t } + \\frac { \\partial H } { \\partial q ^ j } + \\sum _ { i = 1 } ^ n \\left ( p _ i \\frac { \\partial H } { \\partial p _ i } - H \\right ) \\frac { \\partial \\gamma ^ j } { \\partial t } + \\sum _ { i = 1 } ^ n \\frac { \\partial H } { \\partial p _ i } \\frac { \\partial \\gamma ^ j } { \\partial q ^ i } = 0 \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} { } _ 1 F _ 1 \\left ( a , b , v \\right ) & = \\frac { \\Gamma ( b ) } { \\Gamma ( a ) \\Gamma ( b - a ) } \\int _ 0 ^ 1 e ^ { u v } u ^ { a - 1 } ( 1 - u ) ^ { b - a - 1 } d u , b > a > 0 , \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} 2 \\rho = \\sum _ i { i ( n - i ) } \\alpha _ i . \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\underset { \\tau \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\ , \\left [ \\int _ { { \\bf R } ^ N } G ( y , t ) u ( y , \\tau ) \\eta _ n ( y ) \\ , d y - \\int _ { { \\bf R } ^ N } G ( y , t ) \\eta _ n ( y ) \\ , d \\mu ( y ) \\right ] = 0 . \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} \\phi ( z ) = \\sum _ k \\Psi ( k z ) . \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} U ^ { 2 } + Z ^ { 2 } = 2 . \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} & \\frac { \\partial E [ S ^ * ( \\boldsymbol { \\mu } ; 0 ) ] } { \\partial \\mu _ k } \\\\ = & \\lim _ { \\theta \\to 0 } \\frac { E [ f ( \\mathbf { W } ^ * ( \\boldsymbol { \\mu } + \\theta \\cdot \\mathbf { e } ^ k ; 0 ) , \\mathbf { J } ( 0 ) , \\boldsymbol { \\mu } + \\theta \\cdot \\mathbf { e } ^ k ) ] - E [ f ( \\mathbf { W } ^ * ( \\boldsymbol { \\mu } ; 0 ) , \\mathbf { J } ( 0 ) , \\boldsymbol { \\mu } ) ] } { \\theta } . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} q _ z ^ N ( x , x ' ) : = \\sum _ { j = 0 } ^ N \\zeta ( x , x ' ) \\int e ^ { 2 \\pi i ( x - x ' ) \\cdot u } \\sigma _ z ^ { - j - 1 } q _ j ( x , x ' ) \\psi ( x , x ' ) d u , \\end{align*}"}
-{"id": "7263.png", "formula": "\\begin{align*} \\sigma ^ { 2 } _ { \\wedge } ( \\Delta _ { g _ { _ { M } } } ) = \\sigma ^ { 1 } _ { \\wedge } ( \\operatorname { D P [ 0 ] } ) \\circ \\sigma ^ { 1 } _ { \\wedge } ( \\operatorname { D P [ 0 ] } ) \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{gather*} \\alpha _ { t } = \\alpha \\left ( \\frac { 2 } { 3 } \\alpha + \\frac { u _ t } { u } \\frac { 2 - q _ 2 } { 3 } \\right ) - \\frac { t } { 6 } ( 1 + q _ 2 ) - \\frac { u ^ 2 } { 3 } ( 3 + q _ 2 ) . \\end{gather*}"}
-{"id": "8144.png", "formula": "\\begin{align*} u = \\frac { U N ^ { - 1 / 6 } } { \\sqrt { 2 } } , n = N - \\xi N ^ { 1 / 3 } , m = N - \\zeta N ^ { 1 / 3 } \\end{align*}"}
-{"id": "3424.png", "formula": "\\begin{align*} \\ll _ A \\frac { x R _ N ( x ) } { \\log x } \\prod _ { j = 1 } ^ l \\oint _ { | z _ j | = r _ j } ( \\log x ) ^ { \\Re ( z _ j ) } \\frac { | d z _ j | } { | z _ j | ^ { k _ j + 1 } } \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} q ( \\alpha , \\beta ; t _ 1 , y _ 1 ; t _ 2 , y _ 2 ) = q ( 1 - \\beta , 1 - \\alpha ; t _ 2 , y _ 2 ; t _ 1 , y _ 1 ) \\cdot \\left ( \\frac { \\sin ( \\pi \\beta ) } { \\sin ( \\pi \\alpha ) } \\right ) ^ { \\epsilon } \\cdot \\frac { | t _ 2 y _ 2 | } { | t _ 1 y _ 1 | } , \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} W ( z ) = H ( z ) { \\cal W } ( \\zeta ( z ) ) , z \\in D _ \\beta . \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} \\left ( L ( f | _ x ^ y ) g \\right ) ( x , y ) & = \\sum _ { x \\le z \\le y } L ( f | _ x ^ y ) ( x , z ) g ( z , y ) \\\\ & = \\sum _ { x \\le z \\le y } L \\left ( ( f | _ x ^ y ) | _ { x } ^ { z } \\right ) ( x , z ) g ( z , y ) \\\\ & = \\sum _ { x \\le z \\le y } L ( f | _ { x } ^ { z } ) ( x , z ) g ( z , y ) \\\\ & = \\sum _ { x \\le z \\le y } \\hat { L } ( f ) ( x , z ) g ( z , y ) = ( \\hat { L } ( f ) g ) ( x , y ) . \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} - \\left ( \\begin{matrix} \\varphi ^ R \\\\ \\psi ^ R \\end{matrix} \\right ) '' - A ( x ) \\left ( \\begin{matrix} \\varphi ^ R \\\\ \\psi ^ R \\end{matrix} \\right ) = \\lambda _ 1 ^ { R } \\left ( \\begin{matrix} \\varphi ^ R \\\\ \\psi ^ R \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} y : = u - c _ 1 u _ 1 = \\sum _ { n \\in \\mathbb { N } } b _ n u _ { n + 1 } = \\sum _ { n \\in \\mathbb { N } } \\frac { i ^ n } { \\sqrt { n } } f _ n u _ { n + 1 } = y _ { \\rm o d d } + i y _ { \\rm e v e n } , \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\bar { n } = \\langle \\bar { n } , \\nu \\rangle \\nu + \\langle \\bar { n } , \\tau \\rangle \\tau . \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } _ b s _ b ( k ) = s _ b ( n ) 2 ^ { s _ b ( n ) - 1 } . \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} X _ N ^ { R , { \\rm r e s c } } ( T ) = \\frac { X _ N ^ R ( 2 ^ { - 1 / 6 } N ^ { 2 / 3 } T ) - N / \\sqrt { 2 } } { 2 ^ { - 5 / 6 } N ^ { 1 / 3 } } . \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} L ( x , n , [ u ] ) = \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} P _ \\mathrm { c } ( \\theta ) = \\mathbb { P } \\left ( \\gamma \\geq \\theta \\right ) , \\theta > 0 . \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} \\Psi ( k _ 0 ) : = \\left \\{ \\mathbf { \\hat { x } } \\in \\ell _ + : 0 \\leq \\hat { x } _ t \\leq f ( k _ t ) - k _ { t + 1 } , \\ t = 0 , 1 , \\ldots ; \\ \\ \\mathbf { k } \\in \\Pi ( k _ 0 ) \\right \\} . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} F ( \\xi , h ) = \\max _ { \\eta \\in \\Lambda _ { \\xi , h } } D _ { \\xi , h } ( \\eta ) . \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{align*} v ( 0 , \\cdot ) = v _ 0 ( \\cdot ) \\textrm { a n d } v ( t , x ) = B _ m ( t + 2 , x ) f _ { \\mu , A } ( x ) \\ , \\textrm { f o r a l l } \\ ( t , x ) \\in [ 0 , T ] \\times \\R ^ d \\ ; , \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { m \\geq 1 } m \\psi ( m ) e ( m ^ 2 t z ) \\in S _ { 3 / 2 } ^ { ( 1 ) } ( 4 r ^ 2 t , \\psi _ t ) , \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} f ( x ) = \\prod _ { i = 1 } ^ n ( x + r _ i ) = \\sum _ { j = 0 } ^ n x ^ j e _ { n - j } . \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} 2 \\imath \\Psi _ t + \\sum _ { k , j = 1 } ^ { N } f _ { k j } ( \\mathbf { x } ) \\Psi _ { x _ j x _ k } + \\sum _ { k = 1 } ^ { N } h _ k ( \\mathbf { x } ) \\Psi _ { x _ k } + f _ 0 ( \\mathbf { x } ) \\Psi = 0 \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} ~ c ( x ) = \\sum _ { n < m ( x ) } \\dfrac { \\frac { x _ n } { 2 } } { 2 ^ n } + \\frac { 1 } { 2 ^ { m ( x ) } } \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} S _ m & = ( N _ { t , m } \\Box ^ { ( 0 ) } _ { t , b , m } + S _ { t , m } ) S _ m \\\\ & = N _ { t , m } ( \\Box ^ { ( 0 ) } _ { t , b , m } - \\Box ^ { ( 0 ) } _ { b , m } ) S _ m + S _ { t , m } S _ m . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} \\ell ( C ) & \\leq \\ell ( C ' ) + \\ell ( C _ { r - 1 } ) + \\ell ( C _ r ) \\\\ & < \\ell + \\ell + \\ell = 3 \\ell , \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{gather*} \\sum _ { d > x / 2 } \\frac { | h _ { 1 / \\tau } ( d ) | } { d } ( \\log d ) ^ { \\ell } = \\sum _ { d = 2 ^ { \\nu } > x / 2 } \\frac { | h _ { 1 / \\tau } ( 2 ^ { \\nu } ) | } { 2 ^ \\nu } ( \\log 2 ^ \\nu ) ^ { \\ell } \\ll \\sum _ { \\nu > \\log x / \\log 2 } \\frac { \\nu ^ { \\ell - 1 } } { 2 ^ { \\nu } } \\ll \\frac { ( \\log x ) ^ { \\ell - 1 } } { x } . \\end{gather*}"}
-{"id": "439.png", "formula": "\\begin{align*} ( S f ) g = ( S - \\operatorname { i d } ) ( f S _ { - 1 } g ) + f S _ { - 1 } g = \\operatorname { D i v } ^ { \\vartriangle } ( f S _ { - 1 } g ) + f S _ { - 1 } g \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} f _ 1 = x _ 0 ^ 2 + \\ldots + x _ n ^ 2 , f _ 2 = \\lambda _ 0 x _ 0 ^ 2 + \\ldots + \\lambda _ n x _ n ^ 2 \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} \\| \\Pr _ x ^ { t } - \\pi \\| _ { \\ell , \\pi } = \\Pr _ x [ A ] \\| \\Pr _ x ^ t [ \\cdot \\mid A ] - \\pi ( \\cdot ) \\| _ { \\ell , \\pi } + \\Pr _ x [ A ^ { c } ] \\| \\Pr _ x ^ t [ \\cdot \\mid A ^ { c } ] - \\pi ( \\cdot ) \\| _ { \\ell , \\pi } . \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} \\rho ( D _ \\ast , D ^ \\ast ) = \\sum _ { k , \\ell = 0 } ^ \\infty \\mathcal { F } ^ \\ast ( k ) \\mathcal { F } ^ \\ast ( \\ell ) \\psi ( k , \\ell ) \\mathcal { E } ( k , \\ell ) , \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} C ^ h ( \\xi , \\eta ) = - \\frac { i \\eta ( \\eta + J ( \\eta ) ) } { 4 J ( \\eta ) } \\left ( \\xi - \\frac { ( \\eta - J ( \\eta ) ) ^ 2 } { 4 J ( \\eta ) } \\right ) + S ( d ^ 3 \\rho ^ { - 1 } ) . \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} ( W _ n , Q _ n ) ( 0 ) = P _ { > - n } ( W , Q ) ( 0 ) . \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\triangle ( ( t , x ) ; ( s , y ) ) : = \\begin{cases} | t - s | ^ { \\frac 1 2 } + | x - y | , & ; \\\\ | t - s | + | x - y | , & , \\end{cases} \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} c _ n = t b _ n ^ { t - 2 } - 2 b _ n ^ { - 2 } \\mathbb { I } \\{ t = 2 \\} , d _ n = b _ n ^ t - 2 b _ n ^ { - 2 } \\mathbb { I } \\{ t = 2 \\} \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} a _ k = k ^ \\gamma , | b _ k ^ { ( j ) } | \\leq M k ^ \\omega , j \\in \\{ - 1 , 0 , 1 \\} , \\ k \\in \\N . \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} P ^ { - d + 1 } : = \\ker L ^ { d } : V ^ { - d + 1 } \\to V ^ { d + 1 } . \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} \\frac { \\sum _ { 0 \\not = x \\in \\mathfrak { a } _ { d - 1 - l } } a ^ { 1 - q ^ { 2 } } } { \\left ( \\sum _ { 0 \\not = x \\in \\mathfrak { a } _ { d - 1 - l } } a ^ { 1 - q } \\right ) ^ { 1 + q } } = \\tilde { J } ( \\mathfrak { a } _ { d - 1 - l } ) , \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} \\langle \\omega , \\varphi \\rangle = \\int _ { \\mathbb { R } _ + } \\omega ( t ) \\varphi ( t ) d t = a ^ + ( \\varphi ) + a ^ - ( \\varphi ) \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} J ^ + _ n = x ^ 2 \\frac { d } { d x } - n x , J ^ 0 _ n = x \\frac { d } { d x } - \\frac { n } { 2 } , J ^ - _ n = \\frac { d } { d x } \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{align*} \\begin{aligned} & \\lVert u \\rVert _ E ^ { \\sup } : = \\lVert u \\rVert _ { E , \\Omega _ 0 } ^ { \\sup } : = \\sup _ { \\omega \\in \\Omega _ 0 } \\lVert u ( \\omega ) \\rVert _ E , \\\\ & \\lVert u \\rVert _ { E } ^ { l i p } : = \\lVert u \\rVert _ { E , \\Omega _ 0 } ^ { l i p } : = \\sup _ { \\omega _ 1 \\neq \\omega _ 2 } \\frac { \\lVert u ( \\omega _ 1 ) - u ( \\omega _ 2 ) \\rVert _ E } { \\lvert \\omega _ 1 - \\omega _ 2 \\rvert } , \\end{aligned} \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} w ^ h ( x _ 1 ) & = \\frac { 1 } { \\mu ( \\omega ) } \\int _ \\omega x _ 2 \\left ( \\frac { h ^ { - 1 } \\hat y _ 3 ^ h - x _ 3 } { h } - \\frac { 1 } { h ^ 2 } \\int _ \\omega \\hat y _ 3 ^ h \\dd x ' \\right ) \\dd x ' \\\\ & - \\frac { 1 } { \\mu ( \\omega ) } \\int _ \\omega x _ 3 \\left ( \\frac { h ^ { - 1 } \\hat y _ 2 ^ h - x _ 2 } { h } - \\frac { 1 } { h ^ 2 } \\int _ \\omega \\hat y _ 2 ^ h \\dd x ' \\right ) \\dd x ' \\ , . \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} \\frac { q ^ n } { 1 + q ^ n } = \\sum _ { k \\geq 1 } ( - 1 ) ^ { k + 1 } q ^ { n k } = q ^ n - q ^ { 2 n } + q ^ { 3 n } \\dots \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} ( w , q ) = \\frac { d } { d n } ( W _ n , Q _ n ) \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} N _ { ( 1 , 3 ) } ( n ) & = 8 \\sigma ( n ) - 3 2 \\sigma ( \\frac { n } { 4 } ) + 8 \\sigma ( \\frac { n } { 3 } ) - 3 2 \\sigma ( \\frac { n } { 1 2 } ) + 6 4 \\ , W _ { ( 1 , 3 ) } ( n ) + 1 0 2 4 \\ , W _ { ( 1 , 3 ) } ( \\frac { n } { 4 } ) \\\\ & - 2 5 6 \\ , \\biggl ( W _ { ( 3 , 4 ) } ( n ) + W _ { ( 1 , 1 2 ) } ( n ) \\biggr ) , \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} \\phi _ { \\epsilon } = \\phi _ { \\mu _ { \\epsilon } } = \\begin{cases} \\phi _ { \\mu } ( t ) - \\phi _ { \\mu } ( 1 - \\epsilon ) + \\epsilon & 0 \\leq t < 1 - \\epsilon \\\\ 1 - t & 1 - \\epsilon \\leq t \\leq 1 \\end{cases} , \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} P = \\sum _ j \\frac { 2 \\pi } { L } I _ j = \\sum _ j p _ 0 ( \\lambda _ j ) , E = \\sum _ j \\epsilon _ 0 ( \\lambda _ j ) \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} d \\mu ^ X = \\iota _ X \\omega = \\sum _ i d \\theta _ i = \\omega ( X ) . \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} c _ 4 ( E _ 2 ) = 2 ^ 4 ( 4 w ^ 2 + 3 \\ell v ^ { 2 p } ) , c _ 6 ( E _ 2 ) = - 2 ^ 6 w ( 9 \\ell v ^ { 2 p } + 8 w ^ 2 ) , \\Delta ( E _ 2 ) = 2 ^ 6 \\ell ^ 2 ( u v ^ 4 ) ^ p . \\end{align*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\eta ( z ) = e ^ { \\pi i z / 1 2 } \\prod _ { n = 1 } ^ { \\infty } ( 1 - e ^ { 2 \\pi i n z } ) . \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\int _ { \\partial R } \\left ( v \\ast d u - u \\ast d v \\right ) = \\iint _ R \\left ( v \\cdot d \\ast d u - u \\cdot d \\ast d v \\right ) . \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} \\mathbb { Z } , \\mathbb { Z } \\begin{pmatrix} 2 \\\\ 0 \\end{pmatrix} + \\mathbb { Z } \\begin{pmatrix} - 1 \\\\ \\sqrt { 3 } \\end{pmatrix} , \\mathbb { Z } \\begin{pmatrix} - \\sqrt { 2 } \\\\ - \\sqrt { 2 } \\\\ 0 \\end{pmatrix} + \\mathbb { Z } \\begin{pmatrix} \\sqrt { 2 } \\\\ - \\sqrt { 2 } \\\\ 0 \\end{pmatrix} + \\mathbb { Z } \\begin{pmatrix} 0 \\\\ \\sqrt { 2 } \\\\ - \\sqrt { 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} \\Delta ^ { 2 , h } \\Sigma ( x _ 1 , x _ 2 ) = \\frac { \\Sigma ( x _ 1 , x _ 2 + h ) - \\Sigma ( x _ 1 , x _ 2 ) } { h } \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} \\omega = \\alpha \\wedge \\frac { d f } { f } + \\beta , \\alpha \\in \\Omega ^ { k - 1 } ( M ) \\beta \\in \\Omega ^ k ( M ) . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } F _ { k } = F _ { n + 2 } - 1 \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} ( g ( 1 + \\epsilon ) ) ' - s \\ = \\ g ' + g \\epsilon ' + g ' \\epsilon - s . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} f = \\sum _ K a _ { i j } ^ K f _ { i j , K } , \\mbox { w h e r e } f _ { i j , K } \\in \\mathbb { C } [ \\{ { } _ { ( \\alpha ) } a _ { s t } : ( s , t ) \\not = ( i , j ) \\} ] \\mbox { a n d } a _ { i j } ^ K : = \\prod _ { \\alpha = 1 } ^ { r + 1 } { } _ { ( \\alpha ) } a _ { i j } ^ { k _ { \\alpha } } . \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\left ( \\int _ 0 ^ 1 g ( s ) ^ { - 1 } d s \\right ) ^ { - 1 } \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ( x , 0 ) = u _ 0 ( x ) . \\end{cases} \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} \\begin{pmatrix} ( X _ 1 M _ 1 ) ^ t \\\\ ( Y _ 1 M _ 1 ) ^ t \\end{pmatrix} = V \\begin{pmatrix} U _ { 1 1 } ^ t \\\\ U _ { 2 1 } ^ t \\end{pmatrix} ; V = \\begin{pmatrix} - ( Y _ 2 M _ 2 ) ^ t & - ( X _ 2 M _ 2 ) ^ t \\\\ ( X _ 2 M _ 2 ) ^ t & - ( Y _ 2 M _ 2 ) ^ t \\end{pmatrix} . \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\psi _ { \\Theta , \\ell , \\sigma } ( \\xi ) = \\psi _ \\ell ( \\xi ) \\varphi _ \\sigma \\left ( \\frac { \\xi } { \\| \\xi \\| } \\right ) \\ , , \\ , \\ , \\tilde \\psi _ { \\Theta , \\ell , \\sigma } ( \\xi ) = \\tilde \\psi _ \\ell ( \\xi ) \\varphi _ \\sigma \\left ( \\frac { \\xi } { \\| \\xi \\| } \\right ) \\ , . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} \\operatorname { D i v } \\widehat { P } _ 1 + \\operatorname { D i v } ^ { \\vartriangle } \\widehat { P } _ 2 = \\sum _ { \\alpha , J _ 1 , J _ 2 } K _ { J _ 1 , J _ 2 } ^ { \\alpha } ( D _ { J _ 1 } S _ { J _ 2 } F _ { \\alpha } ) \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} V ( x ) & = x ^ 6 + 6 a | x | ^ 5 + ( 9 a ^ 2 - 4 b ) x ^ 4 - ( 1 2 a b - 2 c ) | x | ^ 3 + ( 4 b ^ 2 + 6 a c - 7 ) x ^ 2 \\\\ & - ( 1 8 a + 4 b c - 2 x _ 1 - 2 x _ 2 ) | x | , \\\\ E _ 2 ^ { ( + ) } & = - 1 0 b - c ^ 2 - 6 a ( x _ 1 + x _ 2 ) + 2 ( x _ 1 ^ 2 + x _ 2 ^ 2 ) , \\\\ \\psi _ 2 ^ { ( - ) } ( x ) & = e ^ { - \\frac { 1 } { 4 } x ^ 4 - a | x | ^ 3 + b x ^ 2 - c | x | } ( | x | + x _ 1 ) ( | x | + x _ 2 ) , \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} y _ { z _ 0 } ^ q ( t ) = T _ q ( t ) z _ 0 + \\int _ 0 ^ t T _ q ( t - \\tau ) P _ q ^ s F _ q ( y _ { z _ 0 } ^ q ( \\tau ) ) \\dd \\tau - \\int _ t ^ { \\infty } P _ q ^ c F _ q ( y _ { z _ 0 } ^ q ( \\tau ) ) \\dd \\tau \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} B ( w , U ) = B ( w v , u ) w \\in V ^ p _ C \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} | b | ^ { 2 } K \\alpha ^ { 2 } = | e | ^ { 2 } L \\beta ^ { 2 } , \\ \\overline { a } b = - a \\overline { b } K , \\ \\overline { d } e = - d \\overline { e } L . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} F _ { \\alpha } ^ { \\ast } ( x , n , [ u ] , [ v ] ) = 0 \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} U _ 0 ( \\hat { x } ) : = \\lim _ { t \\to 0 ^ + } U _ t ( \\hat { x } ) = \\frac { \\gamma } { 1 - \\gamma } \\left [ \\left ( \\textstyle \\sum _ i ( \\theta _ 0 ^ i ) ^ { \\frac { 1 } { \\gamma } } \\right ) ^ \\gamma \\left ( \\hat { \\phi } + \\frac { \\eta } { \\gamma } \\ , \\hat { x } \\right ) ^ { 1 - \\gamma } - 1 \\right ] , \\ \\hat { x } \\in \\hat { X } , \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{align*} \\chi _ \\alpha ( \\rho ) = \\rho ^ \\ast ( \\chi ^ { ( N ) } _ \\alpha ) \\in H ^ * ( G ) . \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} N _ E = 2 \\cdot 1 3 ^ 2 , \\Delta _ m = - 2 ^ { 1 4 } \\cdot 1 3 ^ 8 N _ { E ' } = 2 \\cdot 1 3 ^ 2 \\cdot 3 7 , \\Delta _ m ' = - 2 ^ { 7 } \\cdot 1 3 ^ 4 \\cdot 3 7 ^ 7 . \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} \\psi _ x = & \\left ( \\begin{matrix} - i z & u \\\\ - \\bar u & i z \\end{matrix} \\right ) \\psi \\\\ [ 2 m m ] \\psi _ t = & i \\left ( \\begin{matrix} - [ 2 z ^ 2 - | u | ^ 2 ] & - 2 i z u + u _ x \\\\ 2 i z \\bar u + \\bar u _ x & 2 z ^ 2 - | u | ^ 2 \\end{matrix} \\right ) \\psi \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} \\frac { Q } { \\sqrt { c } } = d + e \\beta = d + \\frac { e _ 1 } { c - 2 } \\beta , \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} \\alpha ( d _ 1 , d _ 2 ) & : = 2 ( 1 - d _ 1 d _ 2 ) , \\\\ \\beta ( d _ 1 , d _ 2 ) & : = - \\frac 5 2 + ( 1 - d _ 1 ) d _ 2 \\ln ( 1 - d _ 1 ) + d _ 1 d _ 2 + \\frac { d _ 1 ^ 2 d _ 2 } 2 . \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} \\eta ^ t ( 0 ) = \\lim _ { s \\to 0 } \\eta ^ t ( s ) = 0 . \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\left ( \\frac { \\mathsf { c } _ { 0 } \\alpha _ { n } } { q ^ { 1 / 2 } a _ { n } } \\right ) ^ { p } + \\left ( \\frac { q ^ { 1 / 2 } \\mathsf { c } _ { 0 } \\alpha _ { n } } { c _ { n } } \\right ) ^ { p } = k \\left ( 1 + \\left ( \\frac { \\mathsf { c } _ { 0 } ^ { 2 } \\alpha _ { n } ^ { 2 } } { c _ { n } a _ { n } } \\right ) ^ { p } \\right ) . \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} \\Phi _ t : = \\phi _ t \\big ( \\frac { v ( t ) } { 2 } \\big ) . \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} \\tau = \\frac { s + \\sqrt { \\Delta } } { q } \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} d _ { i , \\zeta } \\mathcal { F } ( i _ 0 , \\zeta _ 0 ) [ \\hat { \\imath } , \\hat { \\zeta } ] = \\mathcal { D } _ { \\omega } \\hat { \\imath } - d _ i X _ { H _ { \\varepsilon } } ( i _ 0 ( \\varphi ) ) [ \\hat { \\imath } ] + ( 0 , \\hat { \\zeta } , 0 ) \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} \\mathbb { K } ( X ) = E ^ { \\bullet } _ { X } , \\mathbb { K } ( \\xi _ 1 , \\xi _ 2 ) = \\xi ^ { \\bullet } . \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} & \\| r ^ { 2 } F _ A \\| _ { L ^ \\infty ( M ) } \\leq e \\max \\left \\{ \\| r ^ { 2 } F _ A \\| _ { L ^ \\infty ( \\pi _ k ^ { - 1 } ( | x | = e ^ { N _ 0 } ) } , \\sqrt { \\tilde C _ 3 } \\right \\} , \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} \\frac { 1 } { { \\cal Z } _ n } \\prod _ { i < j } | z _ i - z _ j | ^ 2 \\cdot \\exp \\bigg ( - N \\sum _ { j = 1 } ^ n Q ( z _ j ) \\bigg ) \\cdot \\prod _ { j = 1 } ^ n d A ( z _ j ) \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} q ( G ) ( z ) : = k ^ { - | E | } p ( G ) ( \\mathcal { H } ( z ) ) . \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tau _ n ( K ) = 0 , \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} \\begin{aligned} & B ( W , W , W ) = \\frac 2 { \\sqrt { 2 \\pi } } \\Re \\int _ { \\xi + \\eta + \\zeta = 0 } B ( \\xi , \\eta , \\zeta ) \\hat W ( \\xi ) \\hat W ( \\eta ) \\hat W ( \\zeta ) \\ , d \\xi d \\eta , \\\\ & A ( W , Q , Q ) = \\frac 2 { \\sqrt { 2 \\pi } } \\Re \\int _ { \\xi + \\eta + \\zeta = 0 } A ( \\zeta , \\xi , \\eta ) \\hat W ( \\zeta ) \\hat Q ( \\xi ) \\hat Q ( \\eta ) \\ , d \\xi d \\eta . \\end{aligned} \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} P _ K ( x ) = x - \\sum _ { j \\in N } \\beta _ i u _ i . \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} v _ { l , j } ( x + p \\alpha , \\cdot - p ) = v _ { l , j + p } ( x , \\cdot ) . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} \\Lambda _ T ( u , v , k , \\ell , \\phi ) : = \\int _ { D _ T } \\left ( \\frac { ( u - k ) ^ 2 } { 2 } + \\frac { ( v - \\ell ) ^ 2 } { 2 a } \\right ) \\phi _ t - ( u - k ) ( v - \\ell ) \\phi _ x \\ , d x d t \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} \\hat { x } : = \\sum _ { i = 1 } ^ n x ^ i , \\hat { x } _ t : = \\sum _ { i = 1 } ^ n x _ t ^ i , \\mathbf { \\hat { x } } : = \\sum _ { i = 1 } ^ n \\mathbf { x } ^ i , \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} f ( M _ n ) = \\frac { \\Gamma ( M _ n + r ) } { M _ n ! \\Gamma ( r ) } \\cdot p ^ { M _ n } ( 1 - p ) ^ { r } \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} L _ 0 w ( x ) : = - \\sum _ { i , j = 1 } ^ n p _ { i j } ( x , 0 ) \\frac { \\partial ^ 2 w } { \\partial x _ i \\partial x _ j } + \\sum _ { i = 1 } ^ n q _ i ( x , 0 ) \\frac { \\partial w } { \\partial x _ i } + r ( x , 0 ) w \\ x \\in \\Omega . \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{gather*} \\| f _ \\varepsilon ( 1 + i t ) \\| _ { X _ 2 } \\le { \\rm e } ^ \\varepsilon \\sum _ { n \\ge 1 } ( \\sum _ { n \\le \\lambda _ k < n + 1 } | a _ k | ) ^ { 2 / ( 2 - s ) } = { \\rm e } ^ \\varepsilon \\ , . \\end{gather*}"}
-{"id": "5230.png", "formula": "\\begin{align*} \\Lambda _ { ( \\theta , y , z ) } [ \\hat { \\theta } , \\hat { y } , \\hat { z } ] : = - y \\cdot \\hat { \\theta } + \\frac { 1 } { 2 } ( \\partial _ x ^ { - 1 } z , \\hat { z } ) _ { L ^ 2 ( \\mathbb { T } ) } . \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} u _ t + u _ { x x x } + \\mathcal { N } _ 2 ( x , u , u _ x , u _ { x x } , u _ { x x x } ) = 0 , \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} v _ t + v _ { x x x } + ( v \\log | v | ) _ x = 0 , ( x , t ) \\in \\mathbb { R } \\times \\mathbb { R } . \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} A = 2 ^ 4 ( \\tilde { c } _ 4 - 6 \\tilde { \\Delta } ) = 2 ^ 4 \\tilde { A } B = 2 ^ 8 ( \\tilde { c } _ 4 ^ 2 + 6 \\tilde { c } _ 4 \\tilde { \\Delta } ^ { 1 / 3 } + ( 6 \\tilde { \\Delta } ^ { 1 / 3 } ) ^ 2 ) = 2 ^ { 8 } \\tilde { B } \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} M ( \\| w \\| ^ 2 ) \\kappa _ \\alpha \\int _ \\mathcal C y ^ { 1 - 2 \\alpha } \\nabla w . \\nabla \\phi d z d y & = \\lambda \\int _ { \\Omega \\times \\{ 0 \\} } f ( z ) | w ( z , 0 ) | ^ { q - 2 } w ( z , 0 ) \\phi ( z , 0 ) d z \\\\ & + \\int _ { \\Omega \\times \\{ 0 \\} } | w ( z , 0 ) | ^ { 2 ^ * _ \\alpha - 2 } w ( z , 0 ) \\phi ( z , 0 ) d z . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} \\Lambda ( \\alpha , \\beta ) = < \\beta , \\sharp ( \\alpha ) > = - < \\alpha , \\sharp ( \\beta ) > , \\alpha , \\beta \\in T ^ { * } M \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} h ( X ) = \\frac { 1 } { 2 } \\log ( { ( 2 \\pi { e } ) ^ { 2 } | { \\hat { \\bf { C } } } _ X | } ) . \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\left ( \\prod _ { i = 1 } ^ m \\binom { \\lambda _ i + k } { k } _ q \\right ) x ^ k = \\frac { C _ { \\lambda } ( x , q ) } { \\prod _ { i = 0 } ^ { N } ( 1 - x q ^ i ) } . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} U ^ { 2 } + Z ^ { 2 } = U ^ { 2 } - \\left ( - 1 \\right ) Z ^ { 2 } = \\left ( U - i Z \\right ) \\left ( U + i Z \\right ) = 2 \\mu , \\mu = 1 , - 1 . \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} J ^ * ( u , v ) = \\frac { \\xi ( w ) } { \\zeta ( 2 ) } \\left ( J _ 1 ( u , v ) + J _ 2 ( u , v ) \\right ) . \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} [ T , Z + \\Phi ( z , t ) \\overline Z ] = - i ( n + 1 ) ( Z + \\Phi ( z , t ) \\overline Z ) \\in \\Gamma ( T _ t ^ { 1 , 0 } X ) . \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} x = \\sigma ( u ) ^ 2 x '' + \\sigma ( r ) , y = \\sigma ( u ) ^ 3 y '' + \\sigma ( u ) ^ 2 \\sigma ( s ) x '' + \\sigma ( t ) \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} \\overline E & = 0 ( 0 , L ) \\ , . \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} Y _ n & = \\sum _ { \\sigma \\in \\Omega _ n } \\frac { \\P _ n ( G , \\sigma ) } { \\Q _ n ( G ) } \\\\ & = \\sum _ { \\sigma \\in \\Omega _ n } \\P _ n ( \\sigma ) \\prod _ { u , v } W _ { u v } ( G , \\sigma ) . \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} L ( s , \\xi _ { u , m } ) = \\Gamma _ \\C ( s + u + \\abs { m } ) . \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} r = \\chi ( w _ { \\sigma } \\cdot ( 0 , 0 , 1 ) ) = \\Im ( ( x + \\sqrt { - 1 } y ) Z _ { \\alpha , \\beta } ( 0 , 0 , 1 ) ) = - y . \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\nabla _ X \\omega = 0 \\textrm { a n d } \\nabla _ X r = 0 . \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} v ( t , x , y ) : = E u ( t , x , y + h \\pi _ { t } ) . \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{gather*} \\omega = \\delta e ^ a \\wedge \\delta e ^ \\ast _ a + \\delta \\omega ^ { a b } \\wedge \\delta \\omega ^ \\ast _ { a b } + \\delta c ^ a \\wedge \\delta c ^ \\ast _ a + \\delta c ^ { a b } \\wedge \\delta c _ { a b } ^ \\ast . \\end{gather*}"}
-{"id": "6088.png", "formula": "\\begin{align*} \\sum _ { \\substack { j = 1 \\\\ j \\ne i } } ^ n \\frac { 1 } { x _ i - x _ j } - x _ i ^ 3 + 3 a x _ i ^ 2 + 2 b x _ i + c = 0 , i = 1 , 2 , \\ldots , n . \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha _ 1 \\alpha _ 2 ^ t - \\alpha _ 2 \\alpha _ 1 ^ t & = 0 ; \\\\ \\beta _ 1 \\beta _ 2 ^ t - \\beta _ 2 \\beta _ 1 ^ t & = 0 . \\end{aligned} \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} F ( u ^ k ) & : = S _ 0 ^ * \\left ( \\alpha ( S u ^ k - z ) + \\beta _ 1 C ^ * _ { \\omega _ R } H ^ + _ \\gamma ( C _ { \\omega _ R } S u ^ k ) + \\beta _ 2 C ^ * _ { \\omega _ T } H ^ - _ \\gamma ( C _ { \\omega _ T } S u ^ k ) \\right ) , \\\\ D _ N F ( u ^ k ) & = S _ 0 ^ * \\left ( \\alpha + \\tfrac { \\beta _ 1 } \\gamma C ^ * _ { \\omega _ R } \\chi ^ + ( C _ { \\omega _ R } S _ 0 u ^ k ) C _ { \\omega _ R } + \\tfrac { \\beta _ 2 } { \\gamma } C ^ * _ { \\omega _ T } \\chi ^ - ( C _ { \\omega _ T } S _ 0 u ^ k ) C _ { \\omega _ T } \\right ) S _ 0 , \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} \\begin{aligned} & H ^ { ( 3 ) } : = H \\circ \\Phi ^ { ( 3 ) } = H _ 2 + H _ { 3 } ^ { ( 3 ) } + H ^ { ( 3 ) } _ 4 + H ^ { ( 3 ) } _ { \\geq 5 } , \\\\ & H ^ { ( 3 ) } _ 3 = H _ 3 + \\{ H _ 2 , F ^ { ( 3 ) } \\} , H ^ { ( 3 ) } _ 4 = \\frac { 1 } { 2 } \\{ \\{ H _ 2 , F ^ { ( 3 ) } \\} , F ^ { ( 3 ) } \\} + \\{ H _ 3 , F ^ { ( 3 ) } \\} + H _ 4 , \\end{aligned} \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} x \\oplus y = \\left \\{ \\begin{array} { l l } \\max \\{ x , y \\} & \\textrm { i f $ x \\neq y $ } \\\\ \\left [ 0 , x \\right ] & \\textrm { i f $ x = y $ } \\end{array} \\right . \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} ( f ^ n ) ^ \\# ( z ) \\leq \\frac { | ( f ^ n ) ' ( z ) | } { | ( f ^ n ) ( z ) | ^ 2 } \\leq \\frac { C ^ n } { | z _ n | } \\prod _ { j = 0 } ^ { n - 1 } | z _ { j } | ^ { \\rho } . \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} G _ 2 = \\left \\{ \\begin{pmatrix} \\alpha & 0 \\\\ 0 & \\alpha \\end{pmatrix} , \\ \\begin{pmatrix} \\alpha & 0 \\\\ 0 & - \\alpha \\end{pmatrix} \\ \\middle | \\ \\alpha \\in C ^ \\times \\right \\} , \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} X = \\{ \\eta : \\eta \\in C ^ { 1 } ( [ 0 , 1 ) ) , \\eta ' \\in L i p _ { } ( [ 0 , 1 ) ) \\} . \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} s ^ i ( \\hat { x } _ t , \\tilde { \\theta } _ t ) & = \\tilde { \\theta } _ t ^ i \\ , \\hat { x } _ t + \\frac { \\gamma \\hat { \\phi } } { \\eta } \\left ( \\tilde { \\theta } _ t ^ i - \\tilde { \\phi } ^ i \\right ) , & & i = 1 , \\ldots , n ; t = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{align*} \\chi _ \\alpha ( \\rho ) = \\chi _ \\alpha ( J _ 1 ( \\rho ) ) , \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} | r y - \\theta | ^ 2 & = | ( r - 1 ) y + y - \\theta | ^ 2 = ( 1 - r ) ^ 2 | y | ^ 2 - 2 ( 1 - r ) \\langle y , y - \\theta \\rangle + | y - \\theta | ^ 2 \\\\ & \\geq - 2 ( 1 - r ) \\langle \\theta , y - \\theta \\rangle - 2 ( 1 - r ) | y - \\theta | ^ 2 + | y - \\theta | ^ 2 \\\\ & \\geq - 2 ( 1 - r ) | y | + 2 ( 1 - r ) - 2 ( 1 - r ) | y - \\theta | ^ 2 + | y - \\theta | ^ 2 \\\\ & \\geq - 2 ( 1 - r ) | y - \\theta | ^ 2 + | y - \\theta | ^ 2 = \\frac { | y - \\theta | ^ 2 } { 2 } , \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{align*} ( g , A ) \\cdot f = g \\circ f \\circ A ^ { - 1 } , \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} \\Lambda ^ - ( \\hat \\theta ^ * ) \\ = \\ \\lim _ { \\hat \\vartheta \\rightarrow \\hat \\theta ^ { * } } \\Lambda ^ - ( \\hat \\vartheta ) = \\hat x ^ * \\ . \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\omega - \\overline { \\omega } } \\overline { v } = \\sum _ { j \\in S } \\mathrm { i } ( \\omega - \\overline { \\omega } ) \\cdot \\mathtt { l } ( j ) \\ , \\sqrt { \\lvert j \\rvert \\xi _ j } \\ , e ^ { \\mathrm { i } \\mathtt { l } ( j ) \\cdot \\varphi } \\ , e ^ { \\mathrm { i } j x } . \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} \\langle \\lambda , \\alpha _ 0 \\rangle = \\frac { 1 } { | N _ G ( P ) | } \\left ( q - 1 + ( q - 1 ) ^ 2 + 2 \\lambda ( T ) \\frac { q ( q - 1 ) } { 2 } - 3 \\frac { q ^ 2 ( q - 1 ) } { 3 } \\right ) = 0 \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} \\pi _ { \\omega } ( \\tau _ { t } ( A ) ) = e ^ { i t H _ { \\omega } } \\pi _ { \\omega } ( A ) e ^ { - i t H _ { \\omega } } \\ , \\ H _ { \\omega } \\Omega _ { \\omega } = 0 \\ . \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} A \\log { N } = \\frac { \\varphi ( D ) } { D } \\int _ 0 ^ \\infty \\phi _ 0 ( z ) z ^ { - 1 } d z + \\Psi ( 0 ) \\frac { \\varphi ( D ) } { 2 D } ( \\log { T } - \\gamma + 1 - \\alpha ( D ) ) \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ M q _ m e ^ { - \\theta r _ m } > e ^ { - \\theta i ( K - k ) c } \\ , \\ , \\forall \\theta \\in [ \\theta ^ * ( \\mathbf { G } ) , \\theta ^ * ( \\mathbf { D } ) ] . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} { F } ( i , \\zeta ) = 0 \\end{align*}"}
-{"id": "1145.png", "formula": "\\begin{align*} t _ 0 & = \\frac { 1 } { \\sqrt { 2 \\pi } } \\left ( \\sqrt { \\frac { t _ 3 } { v _ 2 } } \\right ) ^ { \\rho } \\sqrt { \\frac { t _ 4 } { v _ 1 } } \\\\ t _ 5 & = \\left ( 1 - \\frac { \\rho \\lambda ^ 2 v _ 2 } { 1 + \\lambda v _ 2 } - \\frac { ( 1 - \\lambda \\rho ) ^ 2 v _ 1 } { 1 + ( 1 - \\lambda \\rho ) v _ 1 } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} r ( x ) = ( x - \\lambda _ 0 ) \\hat { r } ( x ) + p r _ 0 ( x ) . \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} \\tilde { \\mathbf { P } } = \\hat { \\mathbf { P } } \\hat { \\mathbf { P } } ^ { \\rm H } . \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} P _ q ^ c Y = \\pi _ q ( Y ) Y _ q ' \\qquad \\pi _ q ( Y ) = \\int _ { \\mathbb { R } } \\langle Z _ q ( x ) , \\gamma _ \\alpha ( x ) Y ( x ) \\rangle \\dd x \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} \\begin{aligned} | b _ V ( \\psi _ m , \\psi _ n ) | & \\leq \\int _ { \\R } w ^ { - 1 } | V | \\ , w | \\psi _ m | \\ , | \\psi _ n | \\ , \\dd x \\leq \\| w ^ { - 1 } | V | \\| _ { L ^ p ( \\R ) } \\| w \\psi _ m \\psi _ n \\| _ { L ^ q ( \\R ) } \\\\ & \\leq C \\| w ^ \\frac 1 2 \\psi _ m \\| _ { L ^ { 2 q } ( \\R ) } \\| w ^ \\frac 1 2 \\psi _ n \\| _ { L ^ { 2 q } ( \\R ) } , \\end{aligned} \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} \\sqrt { \\beta _ K } \\tilde { \\mathbf { G } } = \\mathbf { Z } _ { p } \\boldsymbol { \\Psi } _ p ^ { \\dagger } ( \\boldsymbol { \\Psi } _ { p } \\boldsymbol { \\Psi } _ p ^ { \\dagger } ) ^ { - 1 } = \\mathbf { Z } _ { p } \\boldsymbol { \\Psi } _ p ^ { \\dagger } / \\left ( m _ 0 m _ 1 \\rho _ p \\right ) . \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} T _ { j , j - 1 } g _ { j - 1 } ^ { \\prime } ( z _ { j - 1 } ^ \\ast ) = \\gamma , \\quad \\forall ~ j = \\overline { 1 , n } \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{align*} \\langle k , n | k ^ { \\prime } , n \\rangle = ( ( \\langle k , n | ) ^ { \\dagger } , | k ^ { \\prime } , n \\rangle ) \\equiv \\delta _ { k , k ^ { \\prime } } , \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} L ( e _ { x y } f | _ x ^ y - f | _ x ^ y e _ { x y } ) ( x , y ) & = \\left ( L ( e _ { x y } ) f | _ x ^ y ) ( x , y ) - ( f | _ x ^ y L ( e _ { x y } ) \\right ) ( x , y ) \\\\ & + L ( f ) ( y , y ) - L ( f ) ( x , x ) . \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} \\widehat { \\delta } ( \\eta _ X ( x ) , \\eta _ X ( A ) ) & = \\sup \\{ \\widehat { \\psi } ( \\delta ( - , \\{ x \\} ) ) \\mid \\psi \\in \\mathcal { R } X , \\forall a \\in A , \\widehat { \\psi } ( \\delta ( - , \\{ a \\} ) ) = 0 \\} \\\\ & = \\sup \\{ \\psi ( x ) \\mid \\psi \\in \\mathcal { R } X , \\forall a \\in A , \\psi ( a ) = 0 \\} \\\\ & = \\delta ( x , A ) . \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{align*} Y = f _ l ^ { - 1 } ( Y _ l ) . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} A = \\{ B ^ c _ \\epsilon ( \\theta ^ * ) \\} \\cap \\{ \\bar { B } _ r ( \\theta ^ * ) \\} \\end{align*}"}
-{"id": "7904.png", "formula": "\\begin{align*} p _ { K _ n } ( x ) = \\sum _ k \\frac { { n - 1 \\choose n - 2 k , k , k - 1 } 2 ^ { n - 2 k } } { { 2 n - 2 \\choose n } } x ^ k . \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{gather*} ( - \\partial ) _ I = ( - 1 ) ^ { | I | } \\partial _ I , \\partial _ I = \\partial _ { i _ 1 } \\cdots \\partial _ { i _ k } , \\partial _ i = \\frac { \\partial } { \\partial x ^ i } + \\phi ^ a _ { I i } \\frac { \\partial _ l } { \\partial \\phi ^ a _ { I } } . \\end{gather*}"}
-{"id": "1906.png", "formula": "\\begin{align*} q = \\frac { c _ 1 } { \\sqrt { c _ 1 ^ 2 - 2 c _ 2 } } \\ln { \\left ( \\frac { \\gamma + c _ 1 - \\sqrt { c _ 1 ^ 2 - 2 c _ 2 } } { \\gamma + c _ 1 + \\sqrt { c _ 1 ^ 2 - 2 c _ 2 } } \\right ) } - \\ln { \\left ( \\frac { 1 } { 2 } \\gamma ^ 2 + c _ 1 \\gamma + c _ 2 \\right ) } \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} g _ n ( s , y , t , z ) = \\frac { 1 } { n ! } p _ { t - s _ { \\sigma ( n ) } } ( x - y _ { \\sigma ( n ) } ) \\cdots p _ { s _ { \\sigma ( 2 ) } - s _ { \\sigma ( 1 ) } } ( y _ { \\sigma ( 2 ) } - y _ { \\sigma ( 1 ) } ) p _ { s _ { \\sigma ( 1 ) } } ( y _ { \\sigma ( 1 ) } - z ) \\ , . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} \\mathcal { I } = \\sum _ { j \\neq i } \\frac { g _ { j i k } } { r _ { j i k } ^ { \\alpha } } , \\end{align*}"}
-{"id": "9557.png", "formula": "\\begin{align*} u ( x ) f ( x ) = v ( x ) g ( x ) . \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} \\langle d e ^ { i \\theta } U ( x _ 0 ) \\ , | \\ , d e ^ { i \\theta } V ( x _ 0 ) \\rangle _ g = \\langle U ( x _ 0 ) \\ , | \\ , V ( x _ 0 ) \\rangle _ g , \\forall ~ 0 \\leq \\theta < \\delta _ 1 . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} ( c + 2 ) Y ^ 2 - ( c - 2 ) X ^ 2 = d ^ 2 ( - 1 ) ^ k ( u ^ 2 t _ { k + 1 } + 2 r u s _ { k + 2 } - r ^ 2 t _ { k + 2 } ) , \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} C _ \\mathsf { R L N } ( \\alpha , \\sigma ) = \\bar { \\sigma } \\big [ 1 - h ( \\alpha ) \\big ] , \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} | \\hat { M } ( \\tau , i ) - \\hat { M } ( ( \\tau , i ) - 1 ) | \\leq 1 + \\frac { \\hat { \\pi } ( \\tau - 1 ) } { 1 - \\hat { \\pi } ( \\tau - 1 ) } = \\frac { 1 } { 1 - \\hat { \\pi } ( \\tau - 1 ) } \\stackrel { \\tau \\leq t } { \\leq } \\frac { 1 } { 1 - \\hat { \\pi } ( t ) } . \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} - i L _ \\xi = - \\frac 1 { 1 2 } 2 n \\xi ^ { 2 n } \\eta + \\frac 1 { 4 8 } \\zeta ^ { 2 n } \\frac { ( \\eta + J ( \\eta ) ) ^ 2 } { J ( \\eta ) } . \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} ( f ( z ) \\mid U _ { \\ell ^ { m _ \\ell } } ) \\mid \\gamma = \\sum _ { \\substack { n = 0 \\\\ n \\equiv 0 \\pmod { \\ell ^ { m _ { \\ell } } } } } ^ { \\infty } a _ 0 ( n ) q _ { 2 4 \\ell ^ { m _ \\ell } } ^ { n } \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} N A N ^ { - 1 } = A ^ { - 1 } , A ^ s = \\Theta A \\Theta ^ { - 1 } , N = \\Theta N \\Theta ^ { - 1 } , A ^ 3 = 1 . \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} \\Lambda _ { 2 ^ n , 2 } = \\Lambda _ { 1 , 2 } = 1 , \\ \\ \\ n \\ge 0 . \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} { \\eta } ( \\hat { \\theta } _ { j } ) = p _ { j } { \\eta } ( \\theta ) + { \\eta } ( \\xi _ { j } ) , \\ \\forall 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} \\zeta _ 1 = \\widetilde { \\lambda } _ { M / 2 } \\circ \\zeta _ 0 . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} x ( n + 1 ) = \\lambda ^ { - 1 } A ( n ) x ( n ) \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} T ( - \\bar z ) = \\bar T ( z ) . \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} R _ 3 : = \\tilde { d } _ 1 \\partial _ x ( \\Phi _ 1 - \\mathrm { I } ) + \\tilde { d } _ 0 \\Phi _ 1 + \\tilde { \\mathcal { R } } _ * \\Phi _ 1 + \\varepsilon ^ 2 \\mathfrak { B } _ 2 ( \\Phi _ 1 - \\mathrm { I } ) + \\varepsilon ^ 3 \\{ \\mathcal { D } _ { \\omega } \\hat { A } _ 1 + m _ 3 [ \\partial _ { x x x } , \\hat { A } _ 1 ] + \\frac { 1 } { 2 } \\mathfrak { B } _ 1 A _ 1 ^ 2 + \\varepsilon \\mathfrak { B } _ 1 \\hat { A } _ 1 \\} . \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} L ( s , \\pi , \\tau ) = L ( s + \\alpha + \\beta , \\rho , \\rho ^ \\vee ) L ( s + \\alpha - \\beta , \\rho , \\rho ^ \\vee ) L ( s - \\alpha + \\beta , \\rho , \\rho ^ \\vee ) L ( s - \\alpha - \\beta , \\rho , \\rho ^ \\vee ) . \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} \\omega _ { k , n } = \\frac { \\omega _ k } { \\sum _ { j = n + 1 } ^ \\infty \\omega _ j } = \\begin{cases} \\frac { \\tilde { \\omega } _ k } { \\sum _ { j = n + 1 } ^ \\infty \\tilde { \\omega } _ j } & ~ k \\ge 1 , \\\\ \\frac { \\omega _ 0 } { ( 1 - \\omega _ 0 ) \\sum _ { j = n + 1 } ^ \\infty \\tilde { \\omega } _ j } & ~ k = 0 . \\end{cases} \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\mathcal { A } = \\{ F _ { \\alpha } ( x , n , [ u ] ) = 0 \\} \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} d _ { S ( G , t ) } ( x w , x w ' ) > & d _ { S ( G , t - 1 ) } ( w , ( v _ 1 ) ^ { t - 1 } ) + \\sum _ { i = 0 } ^ { \\frac { r - 4 } { 2 } } d _ { S ( G , t - 1 ) } ( ( v _ { 2 i + 1 } ) ^ { t - 1 } , ( v _ { 2 i + 3 } ) ^ { t - 1 } ) \\\\ & + d _ { S ( G , t - 1 ) } ( ( v _ { r - 1 } ) ^ { t - 1 } , w ' ) \\\\ \\ge & d _ { S ( G , t - 1 ) } ( w , w ' ) , \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} j & : = - \\frac { 1 } { 2 } \\left ( \\frac { S _ 1 \\mu } { R _ 1 } \\left ( \\mu \\frac { f _ * ^ 2 } { 2 } + \\mu f _ * g _ * + \\frac { g _ * ^ 2 } { 2 } \\right ) \\sigma ^ \\prime ( \\Gamma _ * ) - z \\left ( S _ 1 \\frac { g _ * ^ 2 } { 2 } + S _ 1 \\mu \\left ( \\frac { f _ * ^ 2 } { 2 } + f _ * g _ * \\right ) \\right ) \\Gamma _ * \\right ) , \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} ( a _ t \\delta ^ h \\delta ^ h u _ t , u _ t ) _ { l _ 2 ( \\mathbb { G } _ h ) } = & - ( \\delta ^ h u _ t , \\delta ^ h ( a _ t u _ t ) ) _ { l _ 2 ( \\mathbb { G } _ h ) } \\\\ = & - ( \\delta ^ h u _ t , ( \\delta ^ h a _ t ) T ^ h u _ t ) _ { l _ 2 ( \\mathbb { G } _ h ) } - ( \\delta ^ h u _ t , ( T ^ h a _ t ) \\delta ^ h u _ t ) _ { l _ 2 ( \\mathbb { G } _ h ) } . \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} 0 \\subset \\mathfrak { t } _ 1 \\cap S ' \\subseteq \\mathfrak { t } _ 2 \\cap S ' \\subseteq \\cdots \\subseteq \\mathfrak { t } _ m \\cap S ' = \\mathfrak { q } . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} D _ t P _ 1 + ( S - \\operatorname { i d } ) P _ 2 = u \\left ( \\frac { u ' } { u } - u _ 1 + u _ { - 1 } \\right ) . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} t x _ i t ^ { - 1 } = x _ 1 ^ { k _ { 1 i } } \\cdots x _ n ^ { k _ { n i } } , \\ \\mathrm { f o r \\ a l l } \\ i = 1 , . . . , n . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ \\infty \\exp \\Big \\{ - \\sum _ { u = \\lfloor t / 2 \\rfloor } ^ { t - 1 } \\Phi _ u ^ 2 \\Big \\} < \\infty . \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} 2 \\mbox { R e } \\langle H u , i A ( \\delta A ^ 2 + 1 ) ^ { - 1 } u \\rangle = i Q _ { H } \\big ( A ( \\delta A ^ 2 + 1 ) ^ { - 1 } u , u \\big ) - i Q _ H ( u , A ( \\delta A ^ 2 + 1 ) ^ { - 1 } u ) . \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} \\theta _ { a } = \\frac { k _ { a } ^ { \\left ( h _ { a } + 1 \\right ) } ( q - 1 / q ) \\mathsf { A } _ { - } ( 1 / \\xi _ { a } ^ { \\left ( h _ { a } \\right ) } ) } { ( ( \\xi _ { a } ^ { ( h _ { a } + 1 ) } ) ^ { 2 } - 1 / ( \\xi _ { a } ^ { ( h _ { a } + 1 ) } ) ^ { 2 } ) } \\langle h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } | h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } \\rangle , \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{align*} \\Sigma _ { j = 1 } ^ n n ( \\mu , 2 j ) = \\frac { n } { 2 } + \\frac { 1 } { 2 } ( d i m ( Z _ { S p _ { 2 n } } ( e ) ) ) \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} \\phi ( \\xi ) = \\frac { 1 } { \\omega _ C } 1 \\wedge \\xi \\wedge \\xi ^ 2 . \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} \\eta ( \\left ( U \\cap \\partial \\Omega \\right ) \\times ( 0 , T ) ) = \\Omega \\cap U , \\eta ( \\left ( U \\cap \\partial \\Omega \\right ) \\times \\{ 0 \\} ) = \\partial \\Omega \\cap U . \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} p ^ 2 s < n < \\begin{cases} \\infty & , \\\\ \\frac { p s } { 2 - p } & , \\end{cases} \\frac { n ( p - 1 ) } { n - p s } \\leq q < p , \\alpha + \\beta = \\frac { n p } { n - p s } . \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } u _ t - u _ { x x } & = & u ( r _ u - \\gamma _ u ( u + v ) ) + \\mu v - \\mu u \\\\ v _ t - v _ { x x } & = & v ( r _ v - \\gamma _ v ( u + v ) ) + \\mu u - \\mu v \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} U _ i ( r ) = W _ i ( r ) \\left ( W _ i ^ { - 1 } ( 1 ) U _ i ( 1 ) + \\int _ 1 ^ r W _ i ( s ) ^ { - 1 } \\begin{pmatrix} 0 \\\\ f _ i ( s ) \\end{pmatrix} \\ , d s \\right ) , \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} 0 = \\partial | \\dot z | _ \\rho - \\rho \\Delta \\dot z - \\Delta z - g H ^ { - 1 } ( \\Omega ) , ( 0 , T ) , z ( 0 ) = 0 . \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} W ( z ) = e ^ { - \\frac { \\ell _ x } { 2 } \\sigma _ 3 } \\zeta ( z ) ^ { \\frac { c } { 2 } \\sigma _ 3 } S \\cdot \\mathcal { P } ( z ) \\cdot { T ( z ) ^ { - 1 } \\ , S ^ { - 1 } } , z \\in D _ 1 , \\end{align*}"}
-{"id": "3979.png", "formula": "\\begin{align*} & F _ r ( X , \\ldots , X ; Y , \\ldots , Y ) \\\\ & = \\sum _ { k , l \\geq 0 } X ^ { k } Y ^ { l } \\sum _ { m _ 1 > \\cdots > m _ r > 0 } \\frac { 1 } { m _ 1 ^ 2 \\cdots m _ r ^ 2 } \\sum _ { \\substack { 0 < a _ 1 \\leq \\cdots \\leq a _ k \\leq m _ 1 \\\\ 0 < b _ 1 \\leq \\cdots \\leq b _ l \\leq m _ 1 } } \\frac { 1 } { a _ 1 \\cdots a _ k b _ 1 \\cdots b _ l } . \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} \\begin{aligned} S ( U , f ^ { n + k } ) & \\geq \\frac 1 3 \\frac { \\log M ^ n ( \\frac 1 4 r _ 2 , f ^ n ) } { \\log r _ 2 } - 2 \\frac { \\log ^ + M ^ n ( r _ 1 ^ 2 , f ) } { \\log r _ 1 } - \\frac { 2 C } { \\log r _ 1 } \\\\ & \\geq \\frac 1 3 \\frac { \\log M ^ n ( \\frac 1 4 r _ 2 , f ^ n ) } { \\log r _ 2 } - 3 \\frac { \\log ^ + M ^ n ( r _ 1 ^ 2 , f ) } { \\log r _ 1 } , \\end{aligned} \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} P _ n ( z ) = \\sum _ { k = 0 } ^ { n } a _ k z ^ k = a _ n \\prod _ { i = 1 } ^ n ( z - z _ i ) \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} A _ { i j } = t p \\sum _ { k = 0 } ^ { p - 1 } \\sum _ { m = 0 } ^ { k } \\cdots & \\equiv _ { p ^ 3 } \\frac { t } { i + j + 1 } \\binom { i + j + t } { i , j , t } , \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} w = \\frac { ( u ^ { \\frac { m + 1 } 2 } - k ^ { \\frac { m + 1 } 2 } ) _ - } { k ^ { \\frac { m + 1 } 2 } } < 1 - \\left ( \\frac \\gamma 8 \\right ) ^ { \\frac { m + 1 } 2 } . \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} a ^ p = [ a , c ] , \\ , \\ , \\ , \\ , b ^ p = [ a , b c d ] , \\ , \\ , \\ , \\ , c ^ p = [ b , c d ] , \\ , \\ , \\ , \\ , d ^ p = [ b , d ] . \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} \\left ( A ( e ^ { i \\theta } ) - { h } ^ 0 ( \\theta ) \\right ) ^ { < \\alpha - 1 > } f _ 0 ( \\theta ) - \\left ( { h } ^ 0 ( \\theta ) \\right ) ^ { < \\alpha - 1 > } g _ 0 ( \\theta ) = C ^ 0 ( e ^ { i \\theta } ) , \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} A ( V _ p ) \\subset V _ { p + 1 } , \\ A ( V _ k ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} \\varphi ( - q ) & = \\sum _ { n = - \\infty } ^ \\infty ( - 1 ) ^ n q ^ { n ^ 2 } = \\frac { ( q ; q ) _ \\infty } { ( - q ; q ) _ \\infty } , \\\\ \\psi ( q ) & = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( q ; q ^ 2 ) _ \\infty } . \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} R _ 1 ( p _ 1 , \\cdots , p _ s ) : = & 2 ^ { 2 + p _ 1 - p _ 2 } ( 2 ^ { p _ 2 - p _ 1 } - 2 ) \\left [ 5 \\Delta _ { 2 ^ { p _ 2 } + \\cdots + 2 ^ { p _ s } } + ( 2 ^ { p _ 2 - p _ 1 } - 3 ) ( 1 + 2 ^ { p _ 3 - p _ 2 } + \\cdots + 2 ^ { p _ s - p _ 2 } ) \\right ] \\\\ & + R _ 2 ( p _ 2 , \\cdots , p _ s ) \\\\ \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} Z ( \\widetilde { G } ) = Z _ { 0 , D _ 0 } ( G ) ^ 2 + Z _ { 1 , D _ 0 } ( G ) ^ 2 \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} \\Lambda _ 0 & = B ^ T P + D ^ T P C + G ^ T P F + S , \\\\ \\Lambda _ 1 & = ( B ^ T + \\bar { B } ^ T ) \\Pi + ( D ^ T + \\bar { D } ^ T ) P ( C + \\bar { C } ) + ( G ^ T + \\bar { G } ^ T ) \\Pi ( F + \\bar { F } ) + S + \\bar S , \\\\ \\Sigma _ 0 & = D ^ T P D + R , \\\\ \\Sigma _ 1 & = ( D ^ T + \\bar { D } ^ T ) P ( D + \\bar { D } ) + ( G ^ T + \\bar { G } ^ T ) \\Pi ( G + \\bar { G } ) + ( R + \\bar R ) . \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} R ( U , 0 ) = ( A _ 1 U , 0 ) \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} < \\widetilde { f } _ { n } , x _ { n } > = \\sum _ { k = 1 } ^ { \\infty } f _ { n } ( k ) x _ { n } ( k ) - \\sum _ { k \\in E _ { 1 } } f _ { n } ( k ) x _ { n } ( k ) > \\epsilon - \\frac { \\delta } { 2 } , \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\Big [ \\int _ { \\mathbb { R } ^ N } | \\nabla v _ n | ^ 2 d x + \\int _ { \\mathbb { R } ^ N } A ( | v _ n ( x ) | ) d x \\Big ] = \\int _ { \\mathbb { R } ^ N } | \\nabla \\varphi | ^ 2 d x + \\int _ { \\mathbb { R } ^ N } A ( | \\varphi ( x ) | ) d x . \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} \\deg ( I d - F _ 1 , \\Gamma , 0 ) = \\deg ( I d - F _ 0 , \\Gamma , 0 ) . \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} [ a , \\ , b ] & = \\{ x \\in \\mathbb { Z } \\mid a \\leq x \\leq b \\} ; \\\\ ( a , \\ , b ] & = \\{ x \\in \\mathbb { Z } \\mid a < x \\leq b \\} ; \\\\ [ a , \\ , b ) & = \\{ x \\in \\mathbb { Z } \\mid a \\leq x < b \\} ; \\\\ ( a , \\ , b ) & = \\{ x \\in \\mathbb { Z } \\mid a < x < b \\} . \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} \\begin{aligned} | \\Omega _ { \\rho _ { t } } - \\Omega _ { \\infty } | & \\le \\frac { 2 | J _ { \\rho _ t } - J _ { \\infty } | } { | J _ { \\infty } | } \\le \\frac { 4 } { m } | J _ { \\rho _ t } - J _ { \\infty } | \\\\ & \\le C \\frac { 4 } { m } H ( \\rho _ { 0 } | M _ { \\Omega _ { \\rho _ { 0 } } } ) \\int _ { t } ^ { \\infty } e ^ { - \\int _ { 0 } ^ { r } B ( s ) d s } d r . \\end{aligned} \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} \\cos a + \\cos b = 2 \\cos \\frac { a + b } { 2 } \\cos \\frac { a - b } { 2 } , \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} u ^ 2 x _ 1 + r = \\sigma ( u ) ^ 2 \\sigma ( x ' ) + \\sigma ( r ) \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} Q = \\sqrt { | D | } / 2 \\pi . \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} u ( x , t ) = u _ 0 ( x ) + t \\int _ 0 ^ 1 g ( s ) d s . \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} M - F M F ^ * = \\begin{bmatrix} - G _ q B _ q ^ * & G _ p B _ p ^ * \\\\ e _ 1 e _ 1 ^ * & 0 \\end{bmatrix} = \\begin{bmatrix} - G _ q & G _ p & 0 \\\\ 0 & 0 & e _ 1 \\end{bmatrix} \\begin{bmatrix} B _ q ^ * & 0 \\\\ 0 & B _ p ^ * \\\\ e _ 1 ^ * & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\chi _ { q , q ' } \\circ g _ { k , k ' , q , q ' } ( m ) = \\sum _ { i = 0 } ^ s \\beta _ i ( k , k ' , q , q ' ) m ^ i \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} | X | & = | X ' | + | E ( C ) | \\le 2 1 0 ( k - 1 ) ^ 2 \\log ( k - 1 ) + 1 0 \\ell ( k - 2 ) + 1 0 \\ell \\\\ & \\le 2 1 0 k ^ 2 \\log k + 1 0 \\ell ( k - 1 ) = f ( k , \\ell ) . \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} \\limsup _ { i \\ge ( c ^ { '' } - c ^ { ' } ) t + \\int _ s ^ { t + s } c ( \\tau ) d \\tau , t \\to \\infty } u _ i ( t + s ; s , u ^ s ) = 0 \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\bigotimes _ { n = 1 } ^ \\infty A _ n : = \\lim _ { \\longrightarrow } ( A _ n ' , \\pi _ n ) , \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} Y _ i = \\langle A _ i ^ n , v ^ n \\rangle . \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} f - \\underbrace { ( f ^ * - \\varepsilon ) } _ { t } \\ , = \\ , \\sum _ { \\ell = 1 } ^ p f ^ \\ell \\quad \\mbox { w i t h } f ^ \\ell \\ , = \\ , \\sum _ { ( \\alpha , \\beta ) \\in N ^ { \\ell } } c _ { \\alpha \\beta } ^ \\ell \\ , h _ { \\alpha \\beta } , \\quad \\ell = 1 , \\ldots , p , \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} \\nabla ^ { W ( a ) } _ { X } \\tilde w _ a ( x ) = \\frac { \\langle \\nabla _ { X ^ h } w _ a , w _ a \\rangle _ { L ^ 2 ( \\pi _ k ^ { - 1 } ( x ) ) } } { \\| w _ a \\| ^ 2 _ { L ^ 2 ( \\pi _ k ^ { - 1 } ( x ) ) } } \\tilde w _ a ( x ) , \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} N _ { F / E } ( a \\theta _ 1 ^ 2 + b \\theta _ 1 - 1 ) & = 4 a ^ 3 + 2 b ^ 3 + 6 a b - 1 & = 3 v , \\\\ N _ { F / E } ( a \\theta _ 1 ^ 2 + b \\theta _ 1 ) & = 4 a ^ 3 + 2 b ^ 3 & = 4 v , \\\\ N _ { F / E } ( a \\theta _ 1 ^ 2 + b \\theta _ 1 + 1 ) & = 4 a ^ 3 + 2 b ^ 3 - 6 a b + 1 & = 5 v , \\\\ N _ { F / E } ( a \\theta _ 1 ^ 2 + ( b + 1 ) \\theta _ 1 ) & = 4 a ^ 3 + 2 b ^ 3 + 6 b ^ 2 + 6 b + 2 & = 6 v , \\\\ \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} - \\dim { \\rm t o t } \\phi _ v ( R f _ * { \\cal F } , g ) & = \\sum _ { u \\in Z \\times _ Y v } - \\dim { \\rm t o t } \\phi _ u ( { \\cal F } , g \\circ f ) \\\\ & = \\sum _ { u \\in Z \\times _ Y v } ( C C { \\cal F } , d ( g \\circ f ) ) _ { T ^ * X , u } = ( f _ * C C { \\cal F } , d g ) _ { T ^ * Y , v } \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} \\left \\{ \\frac { \\partial \\widetilde { u } _ n ^ { \\gamma } } { \\partial u _ n ^ { \\beta } } F _ { \\gamma } = 0 \\right \\} \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} T _ { i j } = \\frac 1 d \\pi _ i M _ { i j } , \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} \\phi ^ s ( \\underline a ) = \\left \\{ \\begin{array} { c c } \\alpha _ 1 ^ s ( \\underline a ) & s \\leq 1 \\\\ \\alpha _ 1 ( \\underline a ) \\alpha _ 2 ^ { s - 1 } ( \\underline a ) & s > 1 \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} G ( s ; z ) : = F ( s ; a , z ) \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} \\Lambda _ { E _ k } = O ( k \\ln k ) . \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} 0 = \\langle F _ { 1 2 ; 3 } - F _ { 1 3 ; 2 } - F _ { 1 4 ; 1 } , F _ { 1 4 } \\rangle . \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} i n d ( F , N ) & = i n d ( \\tilde { G } , N ) \\\\ & = \\chi ( N ) - i n d ( P _ { \\partial N } \\tilde { G } , \\partial \\_ N [ P _ { \\partial N } \\tilde { G } ] ) \\\\ & = \\chi ( N ) - i n d ( g , \\partial \\_ N [ g ] ) \\\\ & = 0 . \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\delta _ c ( \\alpha , \\theta ) = \\limsup _ { n \\rightarrow \\infty } \\frac { \\sum _ { j = 1 } ^ m \\ln { \\| q _ n ( \\theta - \\theta _ j ) \\| } + \\ln { q _ { n + 1 } } } { q _ n } . \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} \\| E _ \\alpha ( u ) \\| = \\| u \\| _ { H ^ { \\alpha } _ { 0 } ( \\Omega ) } \\ , , \\forall \\ , u \\in H _ { 0 } ^ { \\alpha } ( \\Omega ) . \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} _ { n } ( \\lambda q ^ { j _ { n } - 1 } ) _ { n } ( \\lambda q ^ { - j _ { n } } ) = - q \\frac { \\beta _ { n } a _ { n } c _ { n } } { \\alpha _ { n } } ( \\frac { 1 } { \\lambda } - q ^ { 2 ( j _ { n } - 1 ) } \\frac { \\alpha _ { n } } { \\beta _ { n } } \\lambda ) ( \\frac { 1 } { \\lambda } + q ^ { - 1 } \\frac { d _ { n } \\alpha _ { n } } { c _ { n } \\beta _ { n } } \\lambda ) . \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{align*} H ( { \\boldsymbol \\sigma } ) = - \\sum _ { x , y \\in \\Lambda } J _ { x y } \\sigma _ x \\sigma _ y , \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} \\partial _ i ( g ( a _ 1 - 1 ) - g ( 0 ) ) [ \\hat { \\imath } ] = g ' ( 0 ) \\partial _ i a _ 1 [ \\hat { \\imath } ] + g '' ( 0 ) ( a _ 1 - 1 ) \\ , \\partial _ i a _ 1 [ \\hat { \\imath } ] + \\frac { g ''' ( 0 ) } { 2 } ( a _ 1 - 1 ) ^ 2 \\ , \\partial _ i a _ 1 [ \\hat { \\imath } ] + \\mathtt { T } ( i _ { \\delta } , \\hat { \\imath } ) \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} z f ' ( z ) = \\frac { z } { 1 - z + z ^ 2 } h ( z ) . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} H _ { \\varepsilon , \\zeta } ( \\theta , y , z ) : = H _ { \\varepsilon } ( \\theta , y , z ) + \\zeta \\cdot \\theta , \\zeta \\in \\mathbb { R } ^ { \\nu } . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} Q ^ { \\alpha } _ { J _ 2 } : = S _ { J _ 2 } \\phi ^ { \\alpha } - \\xi ^ i u ^ { \\alpha } _ { \\bold { 1 } _ i ; J _ 2 } . \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} f _ m ( n ) = \\begin{cases} 3 - \\frac 2 { k } & \\\\ 3 - \\frac 1 { k } & . \\end{cases} \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} \\bold { E } ^ { \\vartriangle } ( A \\cdot B ) = ( \\bold { D } ^ { \\vartriangle } _ A ) ^ { \\ast } ( B ) + ( \\bold { D } ^ { \\vartriangle } _ B ) ^ { \\ast } ( A ) . \\end{align*}"}
-{"id": "1348.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": "7983.png", "formula": "\\begin{align*} & \\sigma ( H ^ { \\epsilon , 0 } ) \\cap ( - \\infty , E ) \\subset \\bigcup _ { k = 0 } ^ N [ a _ k , b _ k ] \\ , , \\\\ & | b _ k - a _ k | \\leq C \\ , \\epsilon ^ L \\mbox { f o r } 0 \\leq k \\leq N ( \\epsilon ) \\ , , { \\rm a n d } a _ { k + 1 } - b _ k \\geq \\epsilon / C \\mbox { f o r } 0 \\leq k \\leq N ( \\epsilon ) - 1 \\ , . \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{align*} \\mathcal { H } _ { s } = c _ { 0 } \\frac { d } { d \\lambda } \\mathcal { T } _ { p } ( \\lambda ) _ { \\ , \\vrule h e i g h t 1 3 p t d e p t h 1 p t \\ > { \\lambda = q } ^ { s } } + \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} S _ { A } = I + V ^ { \\ast } & A ^ { \\dagger } U , H = - V ^ { \\ast } A ^ { \\dagger } , K = A ^ { \\dagger } U , \\\\ \\Phi = ( I + H ^ { \\ast } E _ { S _ { A } } H ) ^ { - 1 } , \\Psi & = ( I + K F _ { S _ { A } } K ^ { \\ast } ) ^ { - 1 } , \\Sigma = \\Psi ( A ^ { \\dagger } + K S _ { A } ^ { \\dagger } H ) \\Phi . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} \\hat { \\mathbf { x } } _ { i j k } = \\sqrt { 1 - \\tau _ { i j k } ^ 2 } \\mathbf { x } _ { i j k } + \\tau _ { i j k } \\mathbf { q } _ { i j k } , \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} - \\dim { \\rm t o t } \\phi _ u ( { \\cal F } , f ) = ( C C { \\cal F } , d f ) _ { T ^ * U , u } \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} z f ' ( z ) = g ( z ) h ( z ) . \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} \\Xi _ { \\Lambda } ( \\beta , \\mu , \\lambda ) = ( 1 - \\exp ( \\beta \\mu ) ) ^ { - 1 } \\exp ( - \\frac { \\beta | \\lambda | ^ { 2 } V } { \\mu } ) \\ \\Xi ^ { \\prime } _ { \\Lambda } \\ , \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} Y _ t = L _ q Y + F _ q ( Y ) , L _ q Y = D Y _ { x x } + c Y _ x + \\partial _ Y R ( Y _ q ) Y . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} [ X _ H ( u ) ] _ j = \\mathrm { i } \\ , j \\ , \\partial _ { u _ { - j } } H ( u ) , \\{ F , G \\} ( u ) = - \\sum _ { j \\neq 0 } \\mathrm { i } \\ , j \\ , ( \\partial _ { u _ { - j } } F ) ( u ) ( \\partial _ { u _ j } G ) ( u ) . \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} \\sigma _ t ^ { \\hat \\psi } ( x ) = \\rho ^ { - i t } x \\rho ^ { i t } ; & & \\sigma _ t ^ { \\hat \\phi } ( x ) = \\rho ^ { i t } x \\rho ^ { - i t } , & & ( x \\in c _ { 0 0 } ( \\hat G ) ) . \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} \\eta ( D ) = L ( 1 , \\chi ) \\log D \\ . \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} C = X + \\partial S , \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} A : = \\lim _ { \\longrightarrow } ( A _ n , \\pi _ n ) , \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} | \\hat { H } _ k ( [ X ^ n ] _ b ) - H ( [ X _ { k + 1 } ] _ b | [ X ^ k ] _ b ) | & = | H ( U ^ { k + 1 } ) - H ( U ^ k ) - H ( [ X ^ { k + 1 } ] _ b ) + H ( [ X ^ k ] _ b ) | \\\\ & \\leq | H ( U ^ { k + 1 } ) - H ( [ X ^ { k + 1 } ] _ b ) | + | H ( U ^ k ) - H ( [ X ^ k ] _ b ) | \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} C = ( x + \\beta y ) H - \\frac { y } { 2 } K _ X = y s _ W H - \\frac { y } { 2 } K _ X . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} \\left ( x ; A \\cup B \\right ) = \\left ( x ; A \\right ) + \\left ( x ; B \\right ) \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} y _ n = \\sqrt { P } x _ n + z _ n + v _ n . \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} f ( x ) = \\chi ( x ) ( \\delta _ h q ( x ) ) = \\chi ( x ) ( q ( x + h ) - q ( x ) ) \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} [ X , [ Y , Z ] ] + & ( - 1 ) ^ { \\langle \\deg ( X ) , \\ , \\deg ( Y ) \\rangle + \\langle \\deg ( X ) , \\ , \\deg ( Z ) \\rangle } [ Y , [ Z , X ] ] \\\\ & + ( - 1 ) ^ { \\langle \\deg ( X ) , \\ , \\deg ( Z ) \\rangle + \\langle \\deg ( Y ) , \\ , \\deg ( Z ) \\rangle } [ Z , [ X , Y ] ] = 0 \\ ; . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} a ( r , s + t ) | _ { [ 0 , r ] \\times [ r , r + s ] } & = a ( r , s ) | _ { [ 0 , r ] \\times [ r , r + s ] } \\\\ a ( r + s , t ) | _ { [ 0 , r ] \\times [ r + s , r + s + t ] } & = a ( r , s + t ) | _ { [ 0 , r ] \\times [ r + s , r + s + t ] } , \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} T _ q ( t ) P _ q ^ s - T _ { \\bar { q } } ( t ) P _ { \\bar { q } } ^ s & = ( T _ q ( t - n ) - T _ { \\bar { q } } ( t - n ) ) T _ q ( n ) P _ q ^ s + T _ { \\bar { q } } ( t - n ) T _ q ( n ) P _ q ^ s ( P _ q ^ s - \\ ! P _ { \\bar { q } } ^ s ) \\\\ & + T _ { \\bar { q } } ( t - n ) \\sum _ { k = 0 } ^ { n - 1 } T _ q ( n - k - 1 ) P _ q ^ s ( T _ q ( 1 ) - T _ { \\bar { q } } ( 1 ) ) T _ { \\bar { q } } ( k ) P _ { \\bar { q } } ^ s \\\\ & + T _ { \\bar { q } } ( t - n ) ( P _ q ^ s - P _ { \\bar { q } } ^ s ) T _ { \\bar { q } } ( n ) P _ { \\bar { q } } ^ s . \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} \\mathrm { c m } ( x , y ) = \\mathrm { s g n } \\left ( [ y , b ( x ) ] \\right ) \\frac { \\inf _ { t \\in \\mathbb { R } } | | y + t b ( x ) | | } { | | y | | } , \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\left ( \\textstyle \\sum _ i ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } \\right ) ^ \\gamma \\left ( \\hat { \\phi } + \\tfrac { \\eta } { \\gamma } \\ , \\hat { x } _ t \\right ) ^ { - \\gamma } \\ ! = \\left ( \\textstyle \\sum _ i \\theta _ t ^ i \\delta ^ i \\right ) \\left ( \\textstyle \\sum _ i ( \\theta _ { t + 1 } ^ i ) ^ { \\frac { 1 } { \\gamma } } \\right ) ^ \\gamma \\left ( \\hat { \\phi } + \\tfrac { \\eta } { \\gamma } \\ , \\hat { x } _ { t + 1 } \\right ) ^ { - \\gamma } f ' ( k _ { t + 1 } ) . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} \\norm b ^ 2 & = \\norm { b ^ * b } \\\\ & = \\norm { \\phi _ 1 ( a _ 1 ) ^ * \\phi _ 2 ( a _ 2 ) } \\\\ & = \\norm { \\phi _ 1 ( a _ 1 ^ * ) \\phi _ 2 ( a _ 2 ) } \\\\ & = 0 \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\left \\| \\frac { w ( s ) ^ p } { \\Psi _ { \\beta , L } ( w ( s ) ) } \\right \\| _ { L ^ \\infty ( { \\bf R } ^ N ) } \\le C \\gamma ^ { \\frac { \\theta } { N } } s ^ { - 1 } \\biggr [ \\log \\biggr ( e + \\frac { 1 } { s } \\biggr ) \\biggr ] ^ { - 1 - \\beta } \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} \\int _ { A ( a , R ) } \\det \\nabla \\varphi \\ , d x = \\frac { 1 } { 2 } \\int _ { S _ R } J \\varphi \\cdot \\varphi _ { , _ { \\tau } } \\ , d S . \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} 2 ^ n x A _ n ( x ) = \\sum _ { k = 0 } ^ n \\binom { n } { k } N _ k ( x ) N _ { n - k } ( x ) , \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\lambda ( u , v ) = \\lambda _ 2 ( w ) - \\lambda ( w ) \\left ( \\frac { 1 } { 2 } \\log { \\frac { u } { v } } \\right ) ^ 2 . \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} m * n - m * ( n + k ) & = m * n - r _ m \\phi _ m \\phi _ { n + k } \\\\ & = m * n - r _ m \\phi _ m \\phi _ n \\\\ & = m * n - m * n = 0 \\in K \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} q ^ 2 ( \\tau ) - 1 = r ( \\tau ) P _ { N - 1 } ^ { ( 1 . 0 ) } ( \\tau ) ( \\tau _ N - \\tau ) , \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} d X _ t & = \\left ( X _ t - \\left | \\left ( \\begin{array} { c } X _ t \\\\ Y _ t \\end{array} \\right ) \\right | ^ { 2 } X _ t \\right ) d t + \\sigma \\ , d W _ t \\textrm { o n } \\ ; \\mathbb { R } ^ n \\\\ d Y _ t & = \\left ( Y _ t \\ ; - \\left | \\left ( \\begin{array} { c } X _ t \\\\ Y _ t \\end{array} \\right ) \\right | ^ { 2 } Y _ t \\ ; \\right ) d t \\textrm { o n } \\ ; \\mathbb { R } ^ { d - n } \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} \\mathcal { R } h = \\sum _ { \\lvert j \\rvert \\le C } \\int _ 0 ^ 1 ( h , g _ j ( \\tau ) ) _ { L ^ 2 ( \\mathbb { T } ) } \\chi _ j ( \\tau ) \\ , d \\tau , \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} \\alpha = d t + 2 \\sum _ { j = 1 } ^ n ( x _ j d y _ j - y _ j d x _ j ) . \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} \\left ( A \\otimes B \\right ) _ { i } = \\left \\{ \\ , \\sum a \\otimes b \\mid a \\in A _ { j } , \\ , b \\in B _ { k } , \\ , j + k = i \\ , \\right \\} , \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} u f ( x ; G ^ * ) + ( 1 - u ) f ( x ; G ) = f ( x ; u G ^ * + ( 1 - u ) G ) , \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} [ \\tilde M ^ n ] _ t & = \\frac { 1 } { n } \\sum _ { k = - \\lfloor n t \\rfloor + 1 } ^ { 0 } [ 1 _ { \\{ d ^ n _ { 1 , k } \\le w ^ n _ { 1 , k } \\} } - F ^ n _ 1 ( w ^ n _ { 1 , k } ) ] ^ 2 , \\ t \\ge 0 . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} x _ { k + 1 } = f _ d ( x _ k ) , \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} [ D ( 2 , 3 s + r + 2 p ) : D ( 3 , 3 s + r ) ] _ q & = q ^ { 4 p } [ D ( 2 , 3 ( s - 1 ) + r + 2 p ) : D ( 3 , 3 ( s - 1 ) + r ) ] _ q & \\\\ & + q ^ { 2 ( 3 s + r + 2 p - 1 ) } [ D ( 2 , 3 s + r + 2 ( p - 1 ) ) : D ( 3 , 3 s + r ) ] _ q & \\\\ & + q ^ { 3 s + r + 6 p - 3 } [ D ( 2 , 3 ( s - 1 ) + r + 2 ( p - 1 ) + 1 ) : D ( 3 , 3 ( s - 1 ) + r ) ] _ q . \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} \\lVert \\mathfrak { I } \\rVert _ { s } : = \\lVert \\Theta \\rVert _ { H _ { \\varphi } ^ s } + \\lVert y \\rVert _ { H ^ s _ { \\varphi } } + \\lVert z \\rVert _ s \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{gather*} \\operatorname { g h } ( C ^ \\ast ) = - 2 , \\qquad \\operatorname { g h } ( A ^ \\ast ) = - 1 , \\operatorname { g h } ( A ) = 0 , \\operatorname { g h } ( C ) = 1 , \\end{gather*}"}
-{"id": "8913.png", "formula": "\\begin{align*} \\textrm { s i g n } \\left ( \\sin \\left ( \\frac { \\phi } { 2 } \\pm A \\right ) \\right ) & = \\textrm { s i g n } \\left ( \\sin \\left ( \\pi - B \\pm A \\right ) \\right ) \\\\ & = \\textrm { s i g n } \\left ( \\sin \\left ( B \\pm A \\right ) \\right ) = \\textrm { s i g n } \\left ( \\cos ( A ) \\right ) \\\\ & = \\textrm { s i g n } \\left ( ( - 1 ) ^ k \\sin ( \\phi ) \\right ) , \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\widetilde { \\Theta } ^ { ( j ) } ( F ) \\lesssim 1 \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} P _ { n + 1 } ( x , y , q ) = q y P _ n ( x , y , q ) + q x \\sum _ { k = 0 } ^ { n - 1 } \\binom { n } { k } P _ k ( x , y , q ) 2 ^ { n - k } A _ { n - k } ( x ) , \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} \\begin{aligned} \\bold { p r } X & = \\xi ^ i D _ i + \\sum _ { \\alpha , J _ 1 , J _ 2 } \\left ( D _ { J _ 1 } Q ^ { \\alpha } _ { J _ 2 } \\right ) \\partial _ { u _ { J _ 1 ; J _ 2 } ^ { \\alpha } } \\\\ & = \\xi ^ i D _ i + \\sum _ { \\alpha , J _ 1 , J _ 2 } \\left ( D _ { J _ 1 } S _ { J _ 2 } Q ^ { \\alpha } \\right ) \\partial _ { u _ { J _ 1 ; J _ 2 } ^ { \\alpha } } , \\end{aligned} \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} q \\det t ( u ) = \\sum _ { \\sigma \\in S _ m } ( - 1 ) ^ { s g n ( \\sigma ) } t ^ { \\sigma ( 1 ) , 1 } ( u ) t ^ { \\sigma ( 2 ) , 2 } ( u - 1 ) . . . t ^ { \\sigma ( m ) , m } ( u - m + 1 ) . \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} K = ( r + 2 M ) ^ { - 2 } . \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} f = x ^ { d } + a _ { d - 1 } x ^ { d - 1 } + \\cdots + a _ { j } x ^ { j } = x ( x ^ { d - 1 } + a _ { d - 1 } x ^ { d - 2 } + \\cdots + a _ { j } x ^ { j - 1 } ) \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} ( - \\Delta ) ^ m _ { y } { | y - x | ^ { - N - 2 \\sigma } } & = - ( N + 2 \\sigma ) ( 2 \\sigma + 2 ) ( - \\Delta ) _ y ^ { m - 1 } { | y - x | ^ { - N - 2 \\sigma - 2 } } \\\\ & = ( - 1 ) ^ { m } \\prod \\limits _ { i = 0 } ^ { m - 1 } ( N + 2 \\sigma + 2 i ) ( 2 \\sigma + 2 ( i + 1 ) ) { | y - x | ^ { - N - 2 \\sigma - 2 m } } . \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} Z ( F _ { i - 1 } + F _ i ) - Z ( F _ { i - 1 } \\cap F _ i ) = ( v _ { i - 1 } - Z ( F _ { i - 1 } \\cap F _ i ) ) + ( v _ i - Z ( F _ { i - 1 } \\cap F _ i ) ) . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} g _ k ( \\tau , x ) : = - ( \\Phi ^ { \\tau } ) ^ T [ b ( \\tau ) \\partial _ x e ^ { \\mathrm { i } k x } ] , \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { L _ n } a _ k a _ { \\sigma ^ \\ast ( k ) } = \\min _ { \\sigma \\in \\mathcal { P } _ { L _ n } } \\sum _ { k = 1 } ^ { L _ n } a _ k a _ { \\sigma ( k ) } , \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} \\widehat F ( t ) & = 3 0 \\big ( \\widehat \\phi ( t ) \\big ) ^ 2 = 1 2 0 \\frac { \\big ( 2 \\sin \\big ( \\frac { t } { 2 } \\big ) - t \\cos \\big ( \\frac { t } { 2 } \\big ) \\big ) ^ 2 } { t ^ 6 } . \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{align*} \\det \\begin{pmatrix} \\Phi _ N ( z ) & \\Psi _ N ( z ) \\\\ \\Phi _ N ^ * ( z ) & - \\Psi _ N ^ * ( z ) \\end{pmatrix} = 2 z ^ N \\omega _ { N - 1 } = 2 z ^ N \\prod _ { j = 0 } ^ { N - 1 } ( 1 - | \\alpha _ j | ^ 2 ) , \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} \\| 1 \\| _ { K } = 1 . \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} P _ { X | V , S } ( 1 | v , s ) = 1 . \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} \\beta _ \\# ( 1 , 0 ) & = ( p _ 1 ) _ \\# ( \\varphi ( 1 , 0 ) ) = ( p _ 1 ) _ \\# ( a l _ \\sigma ( a ) ) + ( p _ 1 ) _ \\# ( B , 0 , 0 ) = ( p _ 1 ) _ \\# ( a l _ \\sigma ( a ) ) , \\ ; \\\\ \\beta _ \\# ( 0 , 1 ) & = ( p _ 1 ) _ \\# ( \\varphi ( 0 , 1 ) ) = ( p _ 1 ) _ \\# ( b ) , \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} & h ^ - ( \\omega ) = \\left \\{ y \\in \\Phi _ { \\mathrm { b } } ' : \\mathcal { V } _ { 0 , y } ^ 2 \\subseteq \\mathcal { C } _ 0 ^ { \\mathrm { E } } , y \\in \\Phi _ { \\mathrm { b } } ' \\right \\} , \\\\ & H ^ - \\left ( Y \\right ) = Y + h ^ - \\left ( T _ Y \\right ) . \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} H = B _ 0 \\cong \\bigoplus _ { i = 1 } ^ n c _ { \\ge i } ^ \\ast S \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} \\sigma _ { k } ( n ) = \\sum _ { 0 < \\delta | n } \\delta ^ { k } . \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} \\bold { p r } X = \\xi \\partial _ t + \\phi \\partial _ u + ( S \\phi ) \\partial _ { u _ 1 } + \\left ( S _ { - 1 } \\phi \\right ) \\partial _ { u _ { - 1 } } + \\left ( D _ t \\phi - \\left ( D _ t \\xi \\right ) u ' \\right ) \\partial _ { u ' } + \\cdots , \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} f _ k ( v ) = \\exp \\Bigl ( \\sum _ { j = k _ * + 1 } ^ { k } \\big ( - \\log \\big ( 1 - \\tfrac { v } { \\tilde { c } _ * j } \\big ) - \\tfrac { v } { \\tilde { c } _ * j } \\big ) \\Bigr ) . \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} I _ { q } ( f ) & = \\int _ { \\mathbb { Z } _ p } f ( x ) d \\mu _ { q } ( x ) = \\lim _ { N \\rightarrow \\infty } \\sum _ { x = 0 } ^ { p ^ N - 1 } f ( x ) \\mu _ { q } ( x + p ^ N \\mathbb { Z } _ p ) \\\\ & = \\lim _ { N \\rightarrow \\infty } \\frac { 1 } { [ p ^ N ] _ q } \\sum _ { x = 0 } ^ { p ^ N - 1 } f ( x ) q ^ x , ( \\ , \\ , [ 1 3 ] ) . \\end{align*}"}
-{"id": "9312.png", "formula": "\\begin{align*} f '' ( z ) = - z ^ { - 2 } ( f _ { \\theta \\theta } + z f ' ( z ) ) \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} \\lambda _ n = \\mu _ n + \\sum _ { k = 1 } ^ j \\lambda _ n ^ { ( k ) } + r _ n ^ { ( j ) } , j \\in \\N , n > N , \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} \\Pi : = K \\overline { Q } / \\langle \\displaystyle \\sum \\limits _ { \\alpha \\in Q _ 1 } \\alpha \\alpha ^ { \\ast } - \\alpha ^ { \\ast } \\alpha \\rangle . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} ( \\tilde { c } _ 4 - 3 \\tilde { \\Delta } ^ { 1 / 3 } ) ( \\tilde { c } _ 4 ^ 2 + 3 \\tilde { c } _ 4 \\tilde { \\Delta } ^ { 1 / 3 } + ( 3 \\tilde { \\Delta } ^ { 1 / 3 } ) ^ 2 ) = ( 2 ^ { n - 9 } \\tilde { c } _ 6 ) ^ 2 \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} A _ { [ n ] } = \\bigoplus _ { q = n 4 } A _ q B _ { [ n ] } = \\bigoplus _ { q = n 4 } B _ q \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} & \\sigma _ t ^ { \\widehat { \\varphi } } | _ M = \\sigma _ t ^ \\varphi \\sigma _ t ^ { \\widehat { \\varphi } } ( \\lambda _ s ) = \\lambda _ s [ D ( \\varphi \\circ \\alpha _ s ) , D \\varphi ] _ t , s , t \\in \\R ; \\\\ & ( J _ { \\widehat { \\varphi } } \\xi ) ( t ) = u ^ * ( t ) J _ \\varphi \\xi ( - t ) , t \\in \\R \\xi \\in L ^ 2 ( \\R , L ^ 2 ( M ) ) , \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} j = 0 , [ a , B ] + [ A , b ] = 0 , f A ' + F a ' - a F - A f = 0 , f B ' + F b ' - b F - B f = 0 . \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} U ^ \\nu ( x ) = ( - 1 ) ^ \\nu \\left ( \\frac { 1 } { S _ \\nu } \\int _ { \\partial \\Omega } G ^ \\nu ( x , y ) q ( y ) d s _ y - \\int _ { \\partial \\Omega } \\frac { \\partial G ^ \\nu ( x , y ) } { \\partial n _ y } u ( y ) d s _ y \\right ) , \\ \\nu = 1 , 2 \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} A ( x ) = \\frac { d } { d x } + W ( x ) \\ ; , A ^ \\dagger ( x ) = - \\frac { d } { d x } + W ( x ) \\ ; . \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} \\sum _ { e \\in C } ( - 1 ) ^ { d ( e ) } \\left ( C _ 1 ( e ) Y _ 1 + \\ldots + C _ n ( e ) Y _ n \\right ) = 0 , \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} \\tilde { g } ^ { i \\overline { i } } ( \\nabla _ { V _ { 1 } } V _ { 1 } ) ( \\tilde { g } _ { i \\overline { i } } ) = ( \\nabla _ { V _ { 1 } } V _ { 1 } ) F + ( \\nabla _ { V _ { 1 } } V _ { 1 } ) \\frac { \\partial \\varphi } { \\partial t } . \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} { \\rm R e } \\ , \\phi _ { A } ( z ) = \\eta . \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} \\frac { \\epsilon _ 1 ' } { 1 + \\epsilon _ 1 } - \\frac { \\epsilon _ 0 ' } { 1 + \\epsilon _ 0 } \\ & = \\ \\frac { \\epsilon _ 1 ' } { 1 + \\epsilon _ 1 } - ( 1 + \\delta + \\epsilon _ 1 + \\delta \\epsilon _ 1 ) ^ { \\dagger } \\\\ & = \\ \\frac { \\epsilon _ 1 ' } { 1 + \\epsilon _ 1 } - ( ( 1 + \\delta ) ( 1 + \\epsilon _ 1 ) ) ^ { \\dagger } \\\\ & = \\ \\frac { \\epsilon _ 1 ' } { 1 + \\epsilon _ 1 } - \\frac { \\delta ' } { 1 + \\delta } - \\frac { \\epsilon _ 1 ' } { 1 + \\epsilon _ 1 } \\\\ & = \\ - \\frac { \\delta ' } { 1 + \\delta } \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} c \\theta + c ' \\theta ' + r \\theta _ { \\infty } = 0 . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} \\mathbb { E } _ \\mu \\mathsf { D } \\Big ( Q ^ { ( \\mathsf { C } _ n ) } _ { \\mathbf { S } | M = m } \\Big | \\Big | p _ S ^ n \\Big ) \\leq e ^ { - n \\tilde { \\alpha } } . \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} \\sum _ { \\iota \\in I _ n } e _ { \\iota } ( g e ^ * _ \\iota ) = \\sum _ { \\iota \\in I _ n } e _ { \\iota } ( g e _ \\iota ) = \\left \\{ \\begin{array} { c l } 1 _ { C \\left ( S ^ 1 _ v \\right ) } & g \\in G \\left ( S ^ 1 _ v ~ | ~ S ^ 1 _ u \\right ) \\\\ 0 & g \\in G \\left ( S ^ 1 _ v ~ | ~ S ^ 1 _ u \\right ) \\end{array} \\right . . \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = - x + 1 + \\frac { 1 } { 1 0 } \\sin t - \\lambda \\big | y - x \\big | , \\\\ \\dot y = - \\lambda \\left ( \\frac { 1 } { 2 } + y + 2 x \\sin t \\right ) . \\end{array} \\right . \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{align*} P _ j ^ { ( a , b ) } ( 1 ) = \\frac { ( a + 1 ) _ j } { j ! } , P _ j ^ { ( a , b ) } ( - 1 ) = ( - 1 ) ^ j \\frac { ( b + 1 ) _ j } { j ! } , \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} \\tilde { \\gamma } _ { \\sigma , k } : = \\begin{cases} & \\frac { n } { 4 ( 1 - \\sigma ) } - \\frac { k } { 2 ( 1 - \\sigma ) } \\ \\ 0 \\le \\sigma \\le \\frac { 1 } { 2 } , \\\\ & \\frac { n } { 2 } - k \\ \\ \\sigma = \\frac { 1 } { 2 } , \\\\ & \\frac { n } { 4 \\sigma } - \\frac { k } { 2 \\sigma } \\ \\ \\frac { 1 } { 2 } \\le \\sigma \\le 1 , \\end{cases} \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{gather*} I _ 0 = \\frac { q ^ 2 _ 0 } { 2 } + e _ 1 e _ 2 \\frac { q ^ 2 _ 2 - 1 } { 2 } - e _ 3 q _ 1 q _ 2 + U - e _ 1 q _ 2 + 2 q _ 1 . \\end{gather*}"}
-{"id": "3107.png", "formula": "\\begin{gather*} p _ 1 \\asymp \\dotsb \\asymp p _ { m } \\asymp \\xi , p _ 0 \\asymp \\xi ^ 2 \\\\ ( p _ 0 p _ 1 \\dotsb p _ m , D ! ) = 1 , \\end{gather*}"}
-{"id": "1569.png", "formula": "\\begin{align*} [ A '' , B '' ] + I '' J '' & = [ A | _ { V '' } , B | _ { V '' } ] + ( \\pi '' \\circ I ) \\circ ( J | _ { V '' } ) \\\\ & = [ A , B ] | _ { V '' } + ( I J ) | _ { V '' } \\\\ & = ( [ A , B ] + I J ) | _ { V '' } \\\\ & = 0 \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} u \\delta ' \\ = \\ g ' + g \\epsilon ' + g ' \\epsilon - s . \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} \\chi ( \\det G ) F ( \\tau ) & = F ( \\tau ) | \\gamma \\\\ & = ( \\det G ) ^ k F ( G ^ { - 1 } \\tau \\ , ^ t G ^ { - 1 } ) \\\\ & = ( \\det G ) ^ k \\sum _ T a ( \\ , ^ t G T G ) \\exp ( 2 \\pi i T r ( T \\tau ) ) . \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} \\Sigma ( A ) = [ a _ { 1 } , b _ { 1 } ] \\cup [ a _ { 2 } , b _ { 2 } ] \\cup \\cdots \\cup [ a _ { \\ell - 1 } , b _ { \\ell - 1 } ] \\cup [ a _ { \\ell } , b _ { \\ell } ] , \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} F _ { 3 n + 1 } ^ { \\left ( 3 \\right ) } = F _ { n } ^ { \\left ( 3 \\right ) } F _ { 1 } = F _ { n } ^ { \\left ( 3 \\right ) } \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 f ( x ) \\ , d x & = \\int _ 0 ^ 1 [ \\sum _ { j = 0 } ^ { n - 1 } [ f ( \\frac { x + j } { n } ) - f ( \\frac { j } { n } ) ] \\ , d x \\\\ & = \\sum _ { j = 0 } ^ { n - 1 } n \\int _ { \\frac { j } { n } } ^ { \\frac { j + 1 } { n } } f ( x ) \\ , d x - \\sum _ { j = 0 } ^ { n - 1 } f ( \\frac { j } { n } ) \\\\ & = n \\int _ 0 ^ 1 f ( x ) - \\sum _ { j = 0 } ^ { n - 1 } f ( \\frac { j } { n } ) . \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} \\textrm { V a r } _ t X _ t = \\textrm { V a r } ( X _ t | X _ 1 , \\ldots , X _ { t - 1 } ) . \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} f _ { \\mu , A } ( x ) = C e ^ { - \\frac { 1 } { 2 } \\langle x - \\mu , A ( x - \\mu ) \\rangle } \\ , \\quad \\textrm { n o r m a l i z e d b y } { \\displaystyle C = \\left [ \\int _ { \\R ^ d } B _ m ( 2 , x ) e ^ { - \\frac { 1 } { 2 } ( x - \\mu ) \\cdot A ( x - \\mu ) } \\right ] ^ { - 1 } } \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} u ( x ( \\mu ^ * , n ) , 0 ; - n , \\bar \\phi _ { \\mu ^ * } ( \\cdot , - n ) ) = \\frac { u ^ + ( 0 ) } { 2 } , u ( x ( \\mu , n ) , 0 ; - n , \\bar \\phi _ \\mu ( \\cdot , - n ) ) = \\frac { u ^ + ( 0 ) } { 2 } . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} s = \\sum _ { i = 1 } ^ n s ' _ i = \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ { n _ i } s _ { i j } . \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\sup _ { M \\times [ 0 , T ) } | \\tilde { F } ( x , t ) | & \\leq \\| F \\| _ { L ^ { \\infty } ( M ) } + \\sup _ { M \\times [ 0 , T ) } \\left | \\frac { \\partial \\varphi } { \\partial t } ( x , t ) \\right | \\\\ & \\leq 2 \\| F \\| _ { L ^ { \\infty } ( M ) } + \\left \\| \\log \\frac { ( \\omega + \\sqrt { - 1 } \\partial \\overline { \\partial } \\varphi _ { 0 } ) ^ { n } } { \\omega ^ { n } } \\right \\| _ { L ^ { \\infty } ( M ) } . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\lim _ { \\eta \\rightarrow 0 } \\lim _ { \\Lambda } \\omega _ { \\beta , { \\mu } _ { \\Lambda } ( \\beta , { \\rho } , \\eta ) , \\Lambda , \\eta } ^ { 0 } ( { b ^ { * } _ { k } b _ { k } } / { V } ) = \\lim _ { \\eta \\rightarrow 0 } \\lim _ { \\Lambda } \\frac { 1 } { V } \\frac { 1 } { e ^ { \\beta ( \\varepsilon _ { { k } } - { \\mu } _ { \\Lambda } ( \\beta , { \\rho } , \\eta ) ) ) } - 1 } = 0 \\ , \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} M _ { 2 } \\leq \\prod \\limits _ { j = n } ^ { - 1 } \\Big | \\Big ( \\frac { c _ { i _ { r r } } ( j ) } { h _ { i } ( j ) + \\Delta _ { i } ( j ) } \\Big ) ^ { - 1 } \\Big | \\leq M _ { 1 } , \\textnormal { i f $ n \\leq - 1 $ } \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} \\Upsilon ( t ) - \\Upsilon ( 0 ) = \\Upsilon ' ( 0 ) \\ , t + \\Upsilon _ { \\geq 2 } [ t ] , \\Upsilon _ { \\geq 2 } [ t ] : = \\sum _ { k \\geq 2 } \\frac { \\Upsilon ^ { ( k ) } ( 0 ) } { k ! } \\ , t ^ k , \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\partial _ t h _ i + u _ 2 \\partial _ x h _ i = w _ 2 \\mbox { o n } z = h _ i , \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} - b \\cdot D \\psi = - f + \\bar f \\ \\ \\ \\Omega ( r ^ 2 ) \\setminus \\overline B _ r . \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} \\alpha ( t ) = \\left \\{ \\begin{array} { l l } ( g ^ E _ 0 , \\nabla ^ E _ 0 , T ^ H _ { 2 t } X , g ^ { T ^ V X } _ { 2 t } ) , & \\textrm { f o r } t \\in [ 0 , \\frac { 1 } { 2 } ] , \\\\ ( g ^ E _ { 2 t - 1 } , \\nabla ^ E _ { 2 t - 1 } , T ^ H _ 1 X , g ^ { T ^ V X } _ 1 ) , & \\textrm { f o r } t \\in [ \\frac { 1 } { 2 } , 1 ] \\end{array} \\right . . \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} g ( x , y ) = | | x | | \\lim _ { t \\rightarrow 0 } \\frac { | | x + t y | | - | | x | | } { t } , \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} X = \\xi ^ i ( x ) \\partial _ { x ^ i } + \\phi ^ { \\alpha } ( x , n , [ u ] ) \\partial _ { u ^ { \\alpha } } \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\begin{array} { r c l } - u ^ n _ { t t } - u ^ n _ { x x } + \\frac { 1 } { \\sqrt { \\kappa _ n } } u ^ n _ { t } & = & u ^ n ( r _ u - \\gamma _ u ( u ^ n + v ^ n ) ) + \\mu v ^ n - \\mu u ^ n \\\\ - v ^ n _ { t t } - v ^ n _ { x x } + \\frac { 1 } { \\sqrt { \\kappa _ n } } v ^ n _ { t } & = & v ^ n ( r _ v - \\gamma _ v ( u ^ n + v ^ n ) ) + \\mu u ^ n - \\mu v ^ n \\\\ \\end{array} \\\\ \\underset { x - { \\sqrt { \\varepsilon _ n } } t \\in ( - a _ 0 , a _ 0 ) } { \\sup } \\ , u ^ n ( t , x ) + v ^ n ( t , x ) = \\nu , \\end{array} \\right . \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty ( 1 - x ^ 2 ) \\ , e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( x ^ 2 - 1 \\right ) ^ 2 } \\ , d x & \\geq \\frac { \\sigma ^ 2 } { 2 } + \\frac { \\sigma ^ 2 } { 2 } \\left ( e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } \\ , \\frac { 9 } { 1 6 } } - 2 e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } } \\right ) - \\frac { \\sigma ^ 2 } { 2 } \\\\ & = \\frac { \\sigma ^ 2 } { 2 } \\ , e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } } \\left ( e ^ { \\frac { 1 } { 2 \\sigma ^ 2 } \\ , \\frac { 7 } { 1 6 } } - 2 \\right ) > 0 \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} \\mu ( \\pi ) = 2 ^ { \\delta ( \\pi ) } \\cdot \\gamma ( \\pi ) , \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} \\lim _ { \\ell \\to \\infty } \\frac { \\ell } { k _ { \\ell } } H _ 2 \\left ( \\frac { \\bar { w } } { \\ell } \\right ) = 0 . \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} \\mathrm { \\cfrac { d } { d t } } \\ , F ( t , x ) \\Big | _ { t = \\tau _ { \\pm } ^ r ( x ) } = D \\varphi \\big ( X ( \\tau _ { \\pm } ^ r ( x ) , x ) \\big ) \\cdot \\dot X ( \\tau _ { \\pm } ^ r ( x ) , x ) = 8 X _ 1 ( \\tau _ { \\pm } ^ r ( x ) , x ) X _ 2 ( \\tau _ { \\pm } ^ r ( x ) , x ) \\not = 0 , \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} \\omega _ { \\beta } ( A ) = \\int d \\mu _ { { n } } \\omega _ { \\beta , { n } } ( A ) \\ , \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} ( x _ 1 , y _ 1 , \\dots , x _ n , y _ n , t ) & * ( x _ 1 ' , y _ 1 ' , \\dots , x _ n ' , y _ n ' , t ' ) \\\\ & = \\Big ( x _ 1 + x _ 1 ' , y _ 1 + y _ 1 ' , \\dots , x _ n + x _ n ' , y _ n + y _ n ' , t + t ' + 2 \\sum _ { j = 1 } ^ n ( x _ j ' y _ j - x _ j y _ j ' ) \\Big ) \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} 2 \\mu = - ( m - 2 ) \\pm \\sqrt { ( m - 2 ) ^ 2 + 4 \\lambda } . \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} I ( V ; W | U ) - R _ 2 - \\epsilon ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } + \\epsilon ^ { ( 2 ) } _ { \\alpha , \\delta _ 2 } + 2 \\beta ^ { ( 2 ) } _ { \\alpha , \\delta _ 2 } + \\frac { \\delta _ 2 } { 2 } & \\stackrel { ( a ) } = R _ 1 - I ( U ; W ) - \\frac { \\delta _ 2 } { 2 } - \\epsilon ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } \\\\ & = \\frac { 2 ( \\alpha - 1 ) \\Big [ R _ 1 - d _ \\alpha ( p _ { U , W } , p _ U p _ W ) - \\delta _ 1 \\Big ] + \\frac { 2 \\alpha - 1 } { 2 } ( 2 \\delta _ 1 - \\delta _ 2 ) } { 2 \\alpha - 1 } \\\\ & > 0 \\end{align*}"}
-{"id": "6850.png", "formula": "\\begin{align*} \\beta _ { \\ell } : = \\frac { 2 3 } { Q \\ell ^ { m _ \\ell } } \\pmod { 2 4 } . \\end{align*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\vec { X } _ { 1 1 } & = \\Bigg ( 0 , 1 , - \\frac { \\lambda } { m - n } \\bigg ( \\frac { 1 } { 1 + B ^ { - 1 } } \\bigg ) \\Bigg ) , \\\\ \\vec { X } _ { 1 2 } & = \\Bigg ( \\frac { - \\frac { 1 - m + n } { m - n } + 1 } { e c } \\ ; , \\ ; 1 \\ ; , \\ ; - \\frac { \\lambda } { m - n } \\frac { \\big ( - \\frac { 1 - m + n } { m - n } + 1 \\big ) \\lambda + e } { e ( 1 + C ^ { - 1 } ) } \\Bigg ) . \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} \\mu ( f _ { - } ) = \\lambda ( S _ { x _ o } f _ { - } ) \\leq \\lambda ( U _ { x _ o } ) = \\lambda ( S _ { x _ o } \\chi _ U ) \\leq \\lambda ( S _ { x _ o } f _ { + } ) = \\mu ( f _ { + } ) , \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} \\widetilde { E } ( n , \\mathcal { M } _ 1 , \\mathcal { M } _ 2 ) = \\frac { H H ' n ^ { - \\frac { 1 } { 2 } } } { \\Lambda } \\sum _ { b \\asymp \\mathcal { M } _ 1 + \\mathcal { M } _ 2 } \\sum _ { \\substack { m _ 1 + m _ 2 = b \\\\ m _ 1 \\asymp \\mathcal { M } _ 1 , \\ m _ 2 \\asymp \\mathcal { M } _ 2 } } \\lambda _ { 1 } ( m _ 1 ) m _ 1 ^ { - \\frac { 1 } { 4 } } \\lambda _ { 2 } ( m _ 2 ) m _ 2 ^ { - \\frac { 1 } { 4 } } \\sum _ { c } \\frac { S ( b , 2 n ; c ) } { c } \\Phi \\left ( \\frac { 4 \\pi \\sqrt { 2 n b } } { c } \\right ) , \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} \\binom { n } { k } _ b : = \\prod _ { i = 0 } ^ { N - 1 } \\binom { n _ i } { k _ i } , \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} L ^ { \\Psi } ( \\Omega , \\mu ) = \\{ u : \\Omega \\rightarrow [ - \\infty , \\infty ] : u ~ , ~ \\int _ { \\Omega } \\Psi ( \\alpha \\vert u \\vert ) \\ , d \\mu < \\infty ~ ~ \\alpha > 0 \\} . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } \\frac { 2 ^ k n ! } { k ! ( n - 2 k ) ! } & \\frac { \\prod _ { i < j } | z _ i - z _ j | } { ( \\prod _ { i = 1 } ^ n | 1 - z _ i | ) ^ { ( n + 1 ) } } d \\ell _ { n , k } ( z _ 1 , \\dots , z _ n ) \\\\ & = \\sum _ { k = 0 } ^ { \\lfloor n / 2 \\rfloor } \\frac { 2 ^ k n ! } { k ! ( n - 2 k ) ! } \\frac { \\prod _ { i < j } | z _ i - z _ j | } { \\| \\tilde { a } \\| _ 1 ^ { n + 1 } } d \\ell _ { n , k } ( z _ 1 , \\dots , z _ n ) . \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} c _ 4 ^ 2 - c _ 6 ^ 3 = 1 2 ^ 3 \\Delta _ m \\Leftrightarrow \\tilde { c } _ 4 ^ 2 - \\tilde { c } _ 6 ^ 3 = 2 ^ 3 3 ^ 3 \\tilde { \\Delta } , \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} 2 e & = k v \\\\ 2 e & = f _ 1 d _ 1 + f _ 2 d _ 2 . \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} \\| P \\psi \\| ^ 2 _ { H ^ 1 ( \\R ^ 3 ) } = \\| P \\psi \\| ^ 2 _ { L ^ 2 ( \\R ^ 3 ) } + \\| \\nabla P \\psi \\| ^ 2 _ { L ^ 2 ( \\R ^ 3 ) } = 2 \\left ( \\| \\psi \\| ^ 2 _ { L ^ 2 ( \\R ^ 3 _ + ) } + \\| \\alpha \\cdot D \\psi \\| ^ 2 _ { L ^ 2 ( \\R ^ 3 _ + ) } \\right ) . \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} \\hat { q } ^ { ( 2 ) } _ { k + 1 } ( a _ { k + 1 } | a ^ k ) = { \\hat { q } _ { k + 1 } ^ { ( 2 ) } ( a ^ { k + 1 } ) \\over \\hat { q } _ { k } ^ { ( 2 ) } ( a ^ { k } ) } = { \\hat { p } ^ { ( k + 1 ) } ( a ^ { k + 1 } | [ X ^ n ] _ b ) \\over \\hat { p } ^ { ( k ) } ( a ^ k | [ X ^ n ] _ b ) } . \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} \\frac { 1 } { | { \\mathcal K } | } \\sum _ { k \\in { \\mathcal K } } S _ { h k } ( x ) = B C ( h ) x \\{ 1 + O ( 1 / \\log x ) \\} \\ . \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{gather*} - ( - 1 ) ^ { \\epsilon { ( X ) } } i _ X \\delta _ Q \\omega = - ( - 1 ) ^ { \\epsilon { ( X ) } } \\delta \\delta _ Q \\alpha - d \\delta _ Q \\alpha ' , \\\\ - ( - 1 ) ^ { \\epsilon { ( X ) } } i _ X d \\omega _ 1 = - ( - 1 ) ^ { \\epsilon { ( X ) } } \\delta d \\alpha _ 1 - d \\delta _ Q \\alpha ' , \\\\ d i _ X \\omega _ 1 = d \\delta \\alpha _ 1 + d \\delta _ Q \\alpha ' . \\end{gather*}"}
-{"id": "1311.png", "formula": "\\begin{align*} \\begin{array} { l l l l } Z ^ * = & \\max \\ & c ' z + d ' u & \\\\ & \\mbox { s . t . } \\ & A z \\leq b & \\\\ & & H z + G u \\leq h & \\end{array} \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} \\omega _ t = \\frac { \\nu ^ 2 } { 2 } \\omega _ { \\xi \\xi } + \\left ( r - \\lambda \\kappa - \\frac { \\nu ^ 2 } { 2 } \\right ) \\omega _ \\xi - ( r + \\lambda ) \\omega + \\lambda \\int _ { - \\infty } ^ \\infty \\omega ( \\xi + \\eta , t ) \\phi ( \\eta ) d \\eta , \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} \\Gamma _ c ^ + : = \\{ \\gamma \\in \\Gamma : \\gamma c = c \\ ; \\ ; \\mbox { a n d t h e e i g e n v a l u e s o f $ \\gamma $ a r e t o t a l l y p o s i t i v e } \\} . \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} B ( n ) = ( 1 - \\theta _ n ) B _ 1 ( n ) . \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} | z _ n | \\leq \\prod _ { j = 0 } ^ { n - 1 } | z _ { j } | ^ { \\rho } \\quad \\ n \\geq n _ 0 . \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} K ( x , z ) = \\P ( [ X _ 2 ] _ b = z | X _ 1 = x ) , \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} m ( D _ 1 , D _ 2 , \\alpha ) = \\int _ { \\partial \\tilde { D } _ 2 } q _ 1 \\ast d q _ 2 \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{align*} \\tilde { v } _ { i a } = \\left . \\frac { \\partial \\varepsilon } { \\partial k } \\right | _ { k = k _ { i a } } = \\frac { \\tilde { \\epsilon } ' ( \\lambda _ { i a } ) } { 2 \\pi \\rho ( \\lambda _ { i a } ) } . \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} \\begin{aligned} \\left | ( f ^ n ) ' ( z ) \\right | & = \\left | ( f ^ { n - M _ { l + 1 } } \\circ f ^ { m _ { l + 1 } } \\circ f ^ { 2 n _ { l } } \\circ \\dots \\circ f ^ { m _ 2 } \\circ f ^ { 2 n _ { 1 } } \\circ f ^ { m _ { 1 } } ) ( z ) \\right | \\\\ & \\geq \\lambda _ { l + 1 } ^ { ( n - M _ { l + 1 } ) / 2 } \\cdot \\prod _ { k = 1 } ^ { l } \\lambda _ k ^ { n _ k } \\cdot \\prod _ { k = 1 } ^ { l + 1 } \\alpha _ k \\geq \\lambda _ { l + 1 } ^ { ( n - M _ { l + 1 } ) / 2 } ( 1 + \\beta _ { l + 1 } ^ 2 ) \\lambda _ l ^ { n _ l / 2 } \\end{aligned} \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\pi ( s ) = \\frac { ( 1 + \\beta ) \\lambda e ^ { - \\mu ( s - t ) } } { ( ( 1 + \\beta ) \\lambda + 1 ) e ^ { \\lambda ( T - s ) } - 1 } \\ \\ \\lambda \\neq 0 , \\ \\frac { ( 1 + \\beta ) e ^ { - \\mu ( s - t ) } } { 1 + \\beta + T - s } \\ \\ \\lambda = 0 . \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} \\Pi _ S ^ { \\perp } ( \\mathcal { D } _ { \\omega } A _ 1 + m _ 3 [ \\partial _ { x x x } , A _ 1 ] + \\mathfrak { B } _ 1 ) \\Pi _ S ^ { \\perp } = 0 . \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\pi \\in \\mathcal { D } _ o , \\\\ | \\pi | = N } } ( - 1 ) ^ { \\nu ( \\pi ) } = ( - 1 ) ^ N \\chi ( N = \\square \\ , ) , \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} \\mathcal { B } : = \\{ ( j _ 1 , j _ 2 , j _ 3 , j _ 4 ) \\in S ^ 4 \\ , \\ , : \\ , \\ , j _ 1 + j _ 2 + j _ 3 + j _ 4 = 0 , \\ , \\ , & j _ 1 ^ 3 + j _ 2 ^ 3 + j _ 3 ^ 3 + j _ 4 ^ 3 = 0 , \\ , \\ , j _ 1 + j _ 2 \\neq 0 \\} \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} f _ a ( r ) = ( 1 - r ) b ( r ) ^ { \\sf T } Q _ 2 b ( r ) - b ( r ) ^ { \\sf T } Q _ 1 b ( r ) . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} \\mathrm { d } x \\left ( t \\right ) = f \\left ( t , x \\left ( t \\right ) , r \\left ( t \\right ) , u \\left ( t \\right ) \\right ) \\mathrm { d } t + g \\left ( t , x \\left ( t - d \\left ( t , r ( t ) \\right ) \\right ) , r \\left ( t \\right ) , u \\left ( t \\right ) \\right ) \\mathrm { d } w \\left ( t \\right ) , t \\in J , \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} \\sum _ { \\substack { d < X \\\\ ( d , D ) = 1 } } ( 1 - d / X ) d ^ { - 1 } = \\frac { \\varphi ( D ) } { D } \\left ( \\log { X } + \\gamma - 1 - \\alpha ( D ) \\right ) + O ( \\tau ( D ) / X ) , \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to + 0 } \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\eta ( b _ { { 0 } } ^ { * } ) \\eta ( b _ { { 0 } } ) ) = \\lim _ { \\lambda \\to + 0 } | \\zeta _ { m a x } ( \\lambda ) | ^ { 2 } = : \\rho _ { { 0 } } > 0 \\ . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} X _ \\alpha Y _ t - Y _ \\alpha X _ t = - \\Theta _ \\alpha . \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\alpha ( n ) = \\bigg ( \\frac { \\| q \\| _ 2 ^ 2 } { n ^ 2 } + \\sum \\limits _ { | p | \\leq n , \\ ; p \\ne 0 } \\frac { | q _ { p - n } | ^ 2 } { p ^ 2 } \\bigg ) ^ \\frac { 1 } { 2 } , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} L ( s , \\pi , \\tau ) = \\prod _ { v \\in V _ E } L ( s , \\pi _ v , \\tau _ v ) \\ \\ \\ \\ \\ \\ \\epsilon ( s , \\pi , \\tau ) = \\prod _ { v \\in V _ E } \\epsilon ( s , \\pi _ v , \\tau _ v , \\psi _ v ) . \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\frac { ( - 1 ) ^ n q ^ { 2 n } } { 1 - q ^ { 2 n } } = \\sum _ { k \\geq 1 } ( - 1 ) ^ n q ^ { 2 k n } \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\| \\psi _ n - \\phi _ n \\| ^ 2 < \\infty . \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} \\nu ( K _ { \\check \\varphi } ( x ) \\theta _ 0 ( a ) ) = \\nu ( K _ { \\check \\varphi } ( x ) a ) \\textrm { f o r } a \\in \\pi _ \\nu ( \\tilde M ( \\hat G , \\varphi ) ) '' \\textrm { a n d } x \\in c _ { 0 0 } ( \\hat G ) . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} \\begin{array} { l | l } h _ 8 = 1 & \\\\ h _ { 1 6 } = 2 & \\\\ h _ { 2 4 } = 2 4 & \\\\ h _ { 3 2 } \\geq 1 1 6 2 1 0 9 0 2 4 & \\end{array} \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} Q = \\frac { 1 } { \\sqrt { c - 2 } } ( c + \\sqrt { c } \\sqrt { c - 2 } ) ( c - 1 + \\sqrt { c } \\sqrt { c - 2 } ) ^ n . \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} a ( s ) = \\sum _ { i = 0 } ^ s \\binom s i ^ 2 b ( i ) c ( s - i ) . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{align*} \\Gamma = \\Gamma _ 0 ( N ) \\cap \\Gamma _ 1 ( M ) \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} ( A _ { \\varepsilon } \\circ G _ { \\delta } ) ( \\psi , 0 , w ) = T _ { \\delta } ( \\psi ) + T _ 1 ( \\psi ) w + T _ 2 ( \\psi ) [ w , w ] + T _ { \\geq 3 } ( \\psi , w ) , \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\omega } : = \\omega \\cdot \\partial _ { \\varphi } - \\partial _ x K _ { 0 2 } ( \\varphi ) . \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} \\widetilde { { F } } _ k ( z ) : = \\left ( N ^ { \\frac { c } { 2 } } \\eta ( z ) \\right ) ^ { \\sigma _ 3 } H _ { k - 1 } ( z ) F _ k ( \\zeta ( z ) ) H _ { k - 1 } ^ { - 1 } ( z ) \\left ( N ^ { \\frac { c } { 2 } } \\eta ( z ) \\right ) ^ { - \\sigma _ 3 } . \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} f ( j ) = \\sqrt { \\sinh j \\eta \\sinh ( p - j ) \\eta } = i ( q ^ { j } - q ^ { - j } ) / 2 , \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ p ) ^ s u = \\lambda | u | ^ { p - 2 } u + f ( x ) & \\Omega , \\\\ u = 0 & \\Omega ^ c \\coloneqq \\mathbb { R } ^ N \\setminus \\Omega . \\end{cases} \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} \\mu _ a = \\frac { \\lambda _ a } { \\ell } + \\frac { \\vartheta _ a } { 2 r } + O \\left ( \\frac { 1 } { r ^ 2 } \\right ) , \\end{align*}"}
-{"id": "9272.png", "formula": "\\begin{align*} R _ { \\mathfrak { t } \\cap R } = R _ { \\mathfrak { n } \\cap R } = S _ { \\mathfrak { n } } . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} [ a ] \\ : = \\ \\{ g \\in G : | a | \\leq n | g | | g | \\leq n | a | n \\geq 1 \\} . \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v = \\Delta ( \\beta ( v ) ) + d i v \\big ( \\alpha ( v ) \\big ) + \\phi ( x ) \\\\ v ( 0 , \\cdot ) = v _ 0 \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "638.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } e ^ { i \\theta k } \\left [ \\left ( A ( e ^ { i \\theta } ) - { h } ( \\theta ) \\right ) ^ { < \\alpha - 1 > } f ( \\theta ) - \\left ( { h } ( \\theta ) \\right ) ^ { < \\alpha - 1 > } g ( \\theta ) \\right ] d \\theta = 0 , \\ , k = - 1 , - 2 , \\dots . \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} \\bar c ( | p | ) = \\begin{cases} \\int _ 0 ^ 1 g ( \\chi _ p ( z ) ) d z = \\frac { \\int _ 0 ^ 1 g ( s ) d s } { \\int _ 0 ^ 1 ( \\chi _ p ' ( z ) ) ^ 2 d z } & p \\neq 0 \\\\ \\left ( \\int _ 0 ^ 1 \\frac { 1 } { g ( s ) } d s \\right ) ^ { - 1 } & p = 0 . \\end{cases} \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} x = \\frac { 1 } { A _ 1 } + \\frac { 1 } { A _ 1 A _ 2 } + \\cdots + \\frac { 1 } { A _ 1 A _ 2 \\cdots A _ n } + \\cdots , \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} Q ( F ^ p , F ^ q ) \\ = \\ 0 \\ , , \\hbox { f o r a l l } p + q = n + 1 \\ , . \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} \\lim _ { t \\to \\tau } \\int _ 0 ^ 1 \\abs { k ( t , s ) - k ( \\tau , s ) } \\phi ( s ) \\textup d s = 0 . \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{align*} \\partial _ x ^ { - 1 } e ^ { \\mathrm { i } j x } = \\frac { 1 } { \\mathrm { i } j } \\ , e ^ { \\mathrm { i } \\ , j \\ , x } \\quad \\mbox { i f } \\ , \\ , j \\neq 0 , \\qquad \\partial _ x ^ { - 1 } 1 = 0 . \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} \\left ( \\int _ \\Omega | u \\varphi | ^ { 2 ^ * _ s } \\ , d x \\right ) ^ { 2 / 2 ^ * _ s } \\leq & \\ , C \\left ( \\int _ \\Omega \\varphi ^ 2 \\ , d x + \\iint _ Q u ( x ) u ( y ) \\frac { ( \\varphi ( x ) - \\varphi ( y ) ) ^ 2 } { | x - y | ^ { N + 2 s } } \\ , d x d y \\right . \\\\ & \\left . + \\int _ \\Omega u ^ 2 \\varphi ^ 2 \\ , d x \\right ) \\\\ \\leq & \\ , C \\left ( \\iint _ Q u ( x ) u ( y ) \\frac { ( \\varphi ( x ) - \\varphi ( y ) ) ^ 2 } { | x - y | ^ { N + 2 s } } \\ , d x d y + \\int _ \\Omega u ^ 2 \\varphi ^ 2 \\ , d x \\right ) , \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} f _ { n + 1 } = \\frac { \\sqrt { n } } { \\sqrt { n + 1 } } f _ { n - 1 } , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} ( a , b ) * ( c , d ) & = \\begin{cases} 0 & d = 0 \\\\ ( a \\phi _ d , b \\phi _ d ) & \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta ( a b ) & = ( m \\otimes m ) \\left ( \\sum a _ { ( 1 ) } \\otimes \\tau R ( a _ { ( 2 ) } \\otimes b _ { ( 1 ) } ) \\otimes b _ { ( 2 ) } \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} & \\hat \\tau = ( 1 + O ( \\varepsilon ) ) ( x - x _ * ) ^ 4 + ( 1 + O ( \\varepsilon ) ) ( y - y _ * ) ^ 2 + \\cr & ( 4 x _ * + O ( \\varepsilon ) ) ( x - x _ * ) ^ 3 + ( 6 x _ * ^ 2 - b _ 2 + O ( \\varepsilon ) ) ( x - x _ * ) ^ 2 + c o n s t + O _ { x , y } ( 1 ) \\varepsilon . \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} q ( C ) = 2 ( n ^ { K } ( C ) + \\ell _ { D _ 0 } ( C ) + 1 ) \\ , , \\end{align*}"}
-{"id": "3288.png", "formula": "\\begin{align*} G & = \\begin{bmatrix} - ( Z - I ) q ( T ) ^ { - 1 } ( Z - I ) ^ { - 1 } G _ q & ( Z - I ) q ( T ) ^ { - 1 } ( Z - I ) ^ { - 1 } G _ p & e _ 1 , \\end{bmatrix} \\\\ B & = \\begin{bmatrix} ( Z - I ) p ( T ) ^ * q ( T ) ^ { - * } ( Z - I ) ^ { - 1 } B _ q & B _ p & ( Z - I ) p ( T ) ^ * q ( T ) ^ { - * } ( Z - I ) ^ { - 1 } e _ 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} C _ { m } : = \\frac { b _ { 0 } } { 4 \\pi } \\int _ { \\mathbb { S } ^ { 1 } } U _ { 0 } ( t ) ^ { 2 / m } d t . \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} A _ { ( m + 1 ) p _ n - \\frac { m } { 2 } } & = A _ { ( m + 1 ) p _ n + \\frac { m } { 2 } } - x ^ m A _ { ( m + 1 ) p _ n + \\frac { m } { 2 } } & \\\\ & = ( 1 - x ^ m ) A _ { ( m + 1 ) p _ n + \\frac { m } { 2 } } . \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} 0 = \\int _ { \\partial \\Omega } \\sum _ { \\beta = 1 } ^ d \\frac { \\partial \\varphi _ { \\beta } } { \\partial \\nu } \\langle \\bar { n } , \\nu \\rangle \\left ( \\nu _ \\beta - \\bar { n } _ \\beta \\langle \\bar { n } , \\nu \\rangle \\right ) d S . \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} F ( z , s ) : = G ^ 0 ( z , 1 - s ) , \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} w ( \\gamma ) = \\displaystyle { \\prod _ { i = 1 } ^ { k } s _ i ^ { n _ i ( \\gamma ) } } , \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} \\Delta _ { g _ { C _ \\ell } } \\left ( u - \\frac { f _ \\ell } { 2 m } r ^ 2 \\right ) = \\bar { f } + E _ \\ell \\left ( \\frac { f _ \\ell } { 2 m } r ^ 2 \\right ) + E _ \\ell \\left ( u - \\frac { f _ \\ell } { 2 m } r ^ 2 \\right ) , \\\\ E _ \\ell = \\Delta _ { g _ { C _ \\ell } } - \\Delta _ g = ( g _ { C _ \\ell } - g ) \\ast \\nabla ^ 2 _ { g _ { C _ \\ell } } + \\nabla _ { g _ { C _ \\ell } } g \\ast \\nabla _ { g _ { C _ \\ell } } . \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} s ( t ) = \\beta _ { k } ^ { 4 } k ^ { 3 } ( k - 1 ) ^ { 2 } ( k - 2 ) t ^ { 2 k - 6 } + O ( t ^ { 2 k - 5 } ) . \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} \\hat { \\psi } _ { \\gamma } ( \\delta , \\alpha ) & = \\frac { d } { d \\tau } \\frac { 1 } { n - n _ { \\ell } } \\sum _ { i \\notin I _ { \\ell } } \\left \\{ Z _ { i } [ \\hat { \\gamma } _ { \\ell } ( X _ { i } ) + \\tau \\delta ( X _ { i } ) ] + \\alpha ( X _ { i } ) [ Y _ { i } - \\hat { \\gamma } _ { \\ell } ( X _ { i } ) - \\tau \\delta ( X _ { i } ) ] - \\tilde { \\theta } _ { \\ell } \\right \\} \\\\ & = \\frac { 1 } { n - n _ { \\ell } } \\sum _ { i \\notin I _ { \\ell } } \\left [ Z _ { i } - \\alpha ( X _ { i } ) \\right ] \\delta ( X _ { i } ) . \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} S _ 1 - S _ 2 + S _ 3 - \\cdots + ( - 1 ) ^ { p - 1 } S _ p = \\zeta ( m + p ) - \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { 1 } { \\ell ^ { p - 1 } ( \\ell + 1 ) ^ { m + 1 } } . \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{align*} \\Phi ^ + _ P = \\{ \\beta _ N = w _ L ( \\alpha _ { i _ N } ) , \\beta _ { N - 1 } = w _ L s _ { i _ N } ( \\alpha _ { i _ { N - 1 } } ) , \\ldots , \\beta _ 1 = w _ L s _ { i _ { N } } \\cdots s _ 2 ( \\alpha _ { i _ 1 } ) \\} . \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} \\int _ { F _ { 1 } } | x _ { N _ { 1 } } | d \\mu \\geq \\int _ { \\Omega } \\widetilde { \\widetilde { f } _ { 2 } } \\cdot x _ { N _ { 1 } } d \\mu = \\int _ { E _ { 2 } } \\widetilde { f } _ { N _ { 1 } } \\cdot x _ { N _ { 1 } } d \\mu > \\epsilon - \\frac { \\delta } { 2 } . \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} \\bold { p r } X ( F _ { \\alpha } ) = \\sum _ { \\beta , k \\in \\mathbb { Z } } A ^ { \\beta } _ { \\alpha , k } ( n , [ T ( n , u _ n ) ] ) ( S ^ k F _ { \\beta } ) , \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} \\widehat { u } . x _ { s t } = \\begin{cases} x _ { i t } - x _ { j t } u & \\mbox { i f } s = i , \\\\ x _ { s t } & \\mbox { o t h e r w i s e } . \\\\ \\end{cases} \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\chi _ { \\rm o u t } ( x ) = U ^ { \\dag } \\left ( \\eta \\right ) \\left ( \\begin{array} { c } G \\left ( x \\right ) \\\\ F \\left ( x \\right ) \\end{array} \\right ) \\equiv \\left ( \\begin{array} { c } h _ { \\rm o u t } \\left ( x \\right ) \\\\ g _ { \\rm o u t } \\left ( x \\right ) \\end{array} \\right ) \\ ; . \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} \\hat { p } _ { r } = - \\imath \\left ( \\partial _ { r } + \\frac { 1 } { 2 r } \\right ) , \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} & k _ N ^ R ( t , x , x ' ) - k ^ R ( t , x , x ' ) = \\int _ 0 ^ t \\int _ M k ^ R ( t - s , x , w ) \\epsilon _ N ( s , w , x ' ) d w d s \\\\ & = \\int _ 0 ^ t \\int _ M k _ N ^ R ( t - s , x , w ) \\epsilon _ N ( s , w , x ' ) d w d s \\\\ & - \\int _ 0 ^ t \\int _ 0 ^ { t - s } \\int _ M \\int _ M k ^ R ( t - s - u , x , z ) \\epsilon _ N ( u , z , w ) \\epsilon _ N ( s , w , x ' ) d z d w d u d s . \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} & D ( H _ 0 ) \\cup D ( H ) \\subset D ( Y ) \\cap D ( Z ) , \\\\ & \\langle H f , g \\rangle = \\langle f , H _ 0 g \\rangle + \\langle Z f , Y g \\rangle \\quad f \\in D ( H ) , \\ g \\in D ( H _ 0 ) . \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\nu _ { \\bar \\gamma } : = \\int _ { \\Phi ^ { - 1 } ( \\bar \\gamma ) } \\phi _ \\# ( \\mu _ \\gamma ) \\dd \\eta _ { \\bar \\gamma } ( \\gamma ) . \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} f ' ( z ) = ( 1 / ( i z ) ) \\int _ 0 ^ z f _ { \\theta \\theta } ( \\zeta ) / ( i \\zeta ) \\ , d \\zeta . \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} \\det ( M ) \\lesssim \\begin{cases} j ^ { m _ 1 ' - n ' } i ^ { m _ 1 ' } & M \\in A _ { j } ^ { i } , \\\\ j ^ { n ' - m _ 2 ' } i ^ { - m _ 2 ' } & M \\in B _ { i } ^ { j } , \\\\ k ^ { - a } i ^ { n ' - a } & M \\in S _ { k , i } ^ { a } . \\end{cases} \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} \\sum _ { n _ 1 \\cdots n _ l \\leq x } a ( \\mathbf { n } ; \\mathbf { z } ) = x ( \\log x ) ^ { \\frac { z _ 1 + \\cdots + z _ l } { \\phi ( q ) } - 1 } \\left \\{ u _ 0 ( \\mathbf { a } ; \\mathbf { z } ) + O _ A \\ ( \\frac { 1 } { \\log x } \\ ) \\right \\} , \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} & \\Lambda _ { 1 / 2 } = \\pm \\frac { 1 } { \\sqrt { 2 } } ( 1 + i ) \\sqrt [ 4 ] { - E _ + } , \\Lambda _ { 3 / 4 } = \\pm \\frac { 1 } { \\sqrt { 2 } } ( 1 - i ) \\sqrt [ 4 ] { - E _ + } \\\\ [ 5 p t ] & \\Lambda _ { 5 / 6 } = \\pm \\frac { 1 } { \\sqrt { 2 } } ( 1 + i ) \\sqrt [ 4 ] { - E _ - } , \\Lambda _ { 7 / 8 } = \\pm \\frac { 1 } { \\sqrt { 2 } } ( 1 - i ) \\sqrt [ 4 ] { - E _ - } . \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} \\phi _ q ( z _ 0 ) = - \\int _ 0 ^ { \\infty } P _ q ^ c F _ q ( y _ { z _ 0 } ^ q ( \\tau ) ) \\dd \\tau . \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} [ X _ { i } , X _ { j } ] = 0 , [ X _ { i } , X _ { 4 } ] = X _ { i } , ( i , j = 1 , 2 , 3 ) \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} \\infty > \\underset { T / 4 < \\tau < T / 2 } { \\mbox { { \\rm e s s s u p } } } \\int _ { B ( 0 , 1 ) } u ( x , \\tau ) \\ , d x & \\ge \\underset { T / 4 < \\tau < T / 2 } { \\mbox { { \\rm e s s s u p } } } \\ , \\int _ { B ( 0 , 1 ) } \\int _ { { \\bf R } ^ N } G ( x - y , \\tau ) \\ , d \\mu ( y ) \\ , d x \\\\ & \\ge C \\int _ { { \\bf R } ^ N } ( 1 + | y | ) ^ { - N - \\theta } \\ , d \\mu ( y ) . \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} A \\ ; = \\ ; B \\ , \\oplus \\ , A _ - A . \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{align*} n ( \\mu , 2 ) + n ( \\mu , 4 ) = 1 + 2 = 3 . \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{gather*} \\eta = 3 \\frac { d \\ln f } { d t } , \\end{gather*}"}
-{"id": "479.png", "formula": "\\begin{align*} \\mathcal { A } = \\{ F _ { \\alpha } ( x , n , [ u ] ) = 0 \\} \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} L _ 2 & = P L _ 2 \\oplus ( 1 - P ) L _ 2 , \\\\ [ 5 p t ] H _ N ^ 2 & = P H _ N ^ 2 \\oplus ( 1 - P ) H _ N ^ 2 \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} \\bar { \\mathbf { x } } _ i = \\left \\{ \\begin{matrix} \\mathbf { x } _ i + \\left [ \\begin{array} { c } \\tilde { \\mathbf { Q } } { \\mathbf { b } } _ i \\\\ \\mathbf { 0 } _ { ( N + N _ { \\rm c p } - L ) \\times 1 } \\end{array} \\right ] , & 0 \\leq i \\leq N _ { \\rm s } - 1 , \\\\ \\tilde { \\mathbf { Q } } { \\mathbf { b } } _ i , & i = N _ { \\rm s } , \\end{matrix} \\right . \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} J _ { x _ i , t } ^ { N , i } ( v _ i ; \\{ v ^ * _ j \\} _ { j \\neq i } ) & = J _ { x _ i , t } ^ { m f g } ( v _ i ; \\bar X _ i ^ * , \\bar v ^ * _ i ) + o ( 1 ) \\\\ & \\geq J _ { x _ i , t } ^ { m f g } ( v ^ * _ i ; \\bar X _ i ^ * , \\bar v ^ * _ i ) + o ( 1 ) \\\\ & = J _ { x _ i , t } ^ { N , i } ( v ^ * _ i ; \\{ v ^ * _ j \\} _ { j \\neq i } ) + o ( 1 ) , \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} M _ k ( N ) = M _ { k , 0 } ( N ) \\oplus \\C E . \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} \\psi ( \\alpha \\star \\beta ) = \\psi ( \\lfloor p _ { i } ; h _ { i } \\rceil \\star \\lfloor p _ { j } ; h _ { j } \\rceil ) = \\psi ( \\lfloor \\gamma _ { 1 } p _ { i } p _ { j } ; h _ { i } + h _ { j } \\rceil ) = \\lfloor \\phi ( \\gamma _ { 1 } p _ { i } p _ { j } ) ; h _ { i } + h _ { j } \\rceil = \\lfloor \\phi ( \\gamma _ { 1 } ) \\phi ( p _ { i } p _ { j } ) ; h _ { i } + h _ { j } \\rceil . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} \\inf _ { \\mu \\in Q } \\ , K ( \\mu , \\eta ) = \\tilde { D } ( \\eta ) . \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} Q ( x _ \\mu ) = \\mu , x _ \\mu > 0 , a _ \\mu : = Q ' ( x _ \\mu ) . \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} S = \\bigcup _ { b \\in B } S _ b , \\ \\ S _ b = \\{ r \\in R \\mid r b M \\} . \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ 3 [ H ^ 0 _ { e t } ( X , \\mathcal { F } _ { i , B } ) _ { t o r } ] ^ { ( - 1 ) ^ { i + 1 } } \\right ) = \\nu ( \\mathcal { H } _ B ) _ { \\mathbb { R } } . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} H ( \\gamma ( s ) | e ^ { - \\Psi } ) = \\int _ { \\mathbb { S } ^ { 1 } } ( 1 + s f ) e ^ { - \\Psi } \\log ( 1 + s f ) d \\omega , \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\frac 1 2 \\Delta u + u \\diamond \\frac { \\partial ^ { \\ell + 1 } W } { \\partial t \\partial x _ 1 \\dots \\partial x _ \\ell } \\ , , \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} \\eta \\otimes \\eta ( G ( A \\times B ) ) = m _ { K / L } \\otimes m _ { K / L } ( G ( I \\times J ) ) . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} \\Lambda _ { E _ k } : = \\sup _ { \\Vert f \\Vert \\le 1 } \\Vert L _ { E _ k } ( f ) \\Vert = \\sup _ { z \\in K } \\vert \\lambda _ { E _ k } ( z ) \\vert , \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} V _ r ( r ) = - \\frac { Z \\alpha _ g } { R } \\ ; , \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{gather*} h = \\int _ \\gamma A , \\end{gather*}"}
-{"id": "4093.png", "formula": "\\begin{align*} B ( h _ { + } ) = - \\dfrac { 4 \\left ( s - v \\right ) w { \\large ( } 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v { \\large ) } } { s ^ { 2 } + t ^ { 2 } } { \\large ( } s - 2 h _ { + } ) \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} C = \\begin{pmatrix} 0 & 0 & - 1 \\\\ 0 & 1 & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\frac { C ( \\alpha ( s ) , \\beta ( s ) ; z ) } { C ( \\alpha ( 1 - s ) , \\beta ( 1 - s ) ; z ) } = \\pi ^ { - 2 g ( s - \\frac { 1 } { 2 } ) } \\prod _ { j = 1 } ^ g \\left ( ( - 1 ) ^ { \\epsilon _ j ( \\alpha _ j - \\beta _ j ) } \\cdot \\frac { \\Gamma ( 1 - \\beta _ j ) } { \\Gamma ( \\alpha _ j ) } \\right ) . \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} d \\bar X = \\left \\{ ( A + \\bar { A } ) \\bar X + ( B + \\bar { B } ) \\bar v \\right \\} d s + \\left \\{ ( F + \\bar { F } ) \\bar X + ( G + \\bar { G } ) \\bar v \\right \\} d W _ 0 , \\ \\ \\bar X ( t ) = \\bar X \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} W = - \\tilde R ( 1 ) ( L ( 1 , \\chi ) \\log { N } ) ^ 2 E _ { 0 0 } + O ( T ^ { - 1 / 4 } ) . \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} ( e ^ { 2 \\eta } - 1 ) \\frac { i \\eta ( J ( \\eta ) + \\eta ) } { 8 J ( \\eta ) } = \\frac 1 4 i \\eta e ^ { 2 \\eta } , \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} { 1 \\over m } \\langle A u ^ n , A v ^ n \\rangle = { 1 \\over m } \\sum _ { i = 1 } ^ m \\langle A _ i ^ n , u ^ n \\rangle \\langle A _ i ^ n , v ^ n \\rangle . \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} \\partial _ t p _ { t o t a l } = ( 1 - p _ { t o t a l } ) \\left ( \\sum _ { i = 1 } ^ { m } D _ i \\partial _ { x x } p _ i - \\sum _ { i = 1 } ^ m M _ i \\partial _ x p _ i + \\frac { 1 } { N } \\sum _ { i = 1 } ^ m f _ i ( \\vec { p } N ) \\right ) \\ , . \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } \\int _ { \\Omega } \\left \\vert \\nabla u _ { n } ( x ) - \\nabla u ( x ) \\right \\vert ^ { p ( x ) } d x = 0 . \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} T _ a = \\inf \\{ n \\ge 1 : \\widetilde { S } _ { n } ^ \\rho + \\eta _ { n } \\ge a \\} , \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\frac { 1 } { \\sqrt { \\eta '' } } = \\phi _ { \\mu } \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} R '' = \\min \\{ R , R ' \\} \\kappa _ j '' \\ll _ n \\sum _ { j _ 1 + j _ 2 = j } \\kappa _ { j _ 1 } \\kappa ' _ { j _ 2 } . \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} p _ { i j } = \\frac { i } { j ( j - 1 ) } \\ \\ j \\geq i + 1 ; \\ \\ \\ \\ \\ p _ { i j } = 0 \\ \\ j \\leq i . \\end{align*}"}
-{"id": "7258.png", "formula": "\\begin{align*} \\operatorname { P } = r ^ { - l } \\sum \\limits _ { i \\leq l } a _ { i , \\alpha } ( r , u ) ( - r \\partial _ { r } ) ^ { i } ( r D _ u ) ^ { \\alpha } \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} ( v , w ) = \\pm 4 , ( v + w ) / 2 \\in \\Lambda . \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} ( x - 1 ) ^ { ( \\beta + 1 ) p ^ { e - 1 } } = \\sum _ { j = 0 } ^ { \\beta + 1 } ( - 1 ) ^ { \\beta + 1 - j } { \\beta + 1 \\choose j } x ^ { j p ^ { e - 1 } } . \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t \\bar v = \\frac { 1 } { 2 } \\displaystyle { \\sum _ { i , j = 1 } ^ d } \\partial _ { i j } ^ 2 \\left ( ( \\Phi \\Phi ^ t ) _ { i , j } ( t , x , K \\ast \\bar v ) \\bar v \\right ) - d i v \\left ( g ( t , x , K \\ast \\bar v ) \\bar v \\right ) + \\Lambda ( t , x , K \\ast \\bar v ) \\bar v , \\\\ \\bar v ( 0 , x ) = v _ 0 \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} \\chi ( x ) = 1 , \\mbox { f o r $ x \\le 1 $ , } \\chi ( x ) = 0 , \\mbox { f o r $ x \\ge 2 $ . } \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} \\Lambda ^ + ( \\hat \\theta ^ * ) \\ = \\ \\lim _ { \\hat \\vartheta \\rightarrow \\hat \\theta ^ { * } } { \\Lambda } ^ + ( \\hat \\vartheta ) \\ = \\ \\hat x ^ * . \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{align*} \\phi ( ) = 1 \\leq \\frac { 0 } { 2 } + \\frac { 1 } { 2 } \\log _ 2 ( 0 + 1 ) + 1 = 1 . \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} P _ 1 ( x , n , [ u ] ) = \\widehat { R } _ 1 + B ( x , n ) , P _ 2 ( x , n , [ u ] ) = \\widehat { R } _ 2 . \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} u ^ 0 ( t , x ) : & = \\omega ( t , x ) ; \\\\ u ^ { n + 1 } ( t , x ) : & = \\omega ( t , x ) + \\int _ 0 ^ t \\int _ { \\R } G _ { t - \\theta } ( x - \\eta ) b ( u ^ n ( \\theta , \\eta ) ) d \\eta d \\theta \\\\ & + \\int _ 0 ^ t \\int _ { \\R } G _ { t - \\theta } ( x - \\eta ) \\sigma W ( d \\theta , d \\eta ) \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} L _ 1 ( \\theta | x ^ n ) : = \\mu _ 1 \\| \\theta \\| _ { w , 1 } + \\mu _ 2 . \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} - \\Delta _ { x _ 1 } - \\Delta _ { x _ 2 } = - \\Delta _ { X _ 1 } - \\Delta _ { X _ 2 } . \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\beta = \\beta ( \\theta , \\eta ) : = u ( A ) \\Delta ( A ) + u ( B ) \\Delta ( B ) < 0 . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} m _ { K } ( p ) : = \\int ^ { 1 } _ { 0 } K ( s p ) s ^ { 2 } ~ \\ ! d s \\quad Q _ { K } ( p ) : = m _ { K } ( p ) p \\quad \\forall p \\in \\R ^ { 3 } \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} \\delta ( F h ) = F \\diamond W ( h ) . \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} X _ 1 = \\frac { x _ 2 - x _ 1 } { \\sqrt { 2 } } , \\ \\ \\ X _ 2 = \\frac { x _ 2 + x _ 1 } { \\sqrt { 2 } } . \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} \\begin{aligned} \\eta _ s ( \\xi _ k ) = \\delta _ { s k } , ( \\xi _ s \\lrcorner d \\eta _ s ) _ { | H } = 0 , ( \\xi _ s \\lrcorner d \\eta _ k ) _ { | H } = - ( \\xi _ k \\lrcorner d \\eta _ s ) _ { | H } , \\end{aligned} \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} D : = \\left \\{ j \\in N : \\left ( \\tilde { \\theta } _ t ^ j - \\frac { 1 } { n } \\right ) < 0 \\right \\} , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} H _ 2 ( A ( x ) \\psi _ n ^ { ( 1 ) } ( x ) ) = A ( x ) A ^ \\dagger ( x ) A ( x ) \\psi _ n ^ { ( 1 ) } ( x ) = E _ n ^ { ( 1 ) } ( A ( x ) \\psi _ n ^ { ( 1 ) } ( x ) ) \\ ; , \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { N - 1 } S \\left ( n _ i , i \\right ) = \\sum _ { 0 \\leq k \\leq _ b n } \\prod _ { i = 0 } ^ { N - 1 } f ( n _ i , k _ i , i ) . \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} \\sum _ { k = N _ 0 + 1 } ^ { \\infty } \\frac { 1 } { k ^ { 2 \\alpha } \\mu _ k } < \\infty , \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} \\Phi ( s ) = \\delta ^ { - 1 } _ 2 \\int _ { \\Omega _ 0 } k ( t , s ) \\textup d t , s \\in \\overline { \\Omega } , \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\bold { E } _ { \\alpha } ( L ( x , n , [ u ] ) ) = 0 , \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} \\mathcal { U } _ { a } ( \\lambda ) = M _ { a } ( \\lambda ) K _ { a } ( \\lambda ) \\hat { M } _ { a } ( \\lambda ) , \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} \\epsilon ^ { \\alpha + \\beta } a _ { \\xi \\epsilon ^ { - 2 } } ( \\epsilon ^ 2 y ) = a _ { \\xi } ( y ) , \\end{align*}"}
-{"id": "1140.png", "formula": "\\begin{align*} & \\geq _ n \\phi ^ { ( j ) } ( 1 - \\epsilon ) \\\\ & \\times \\left [ \\frac { n } { 2 } \\log \\left ( \\sum _ { j ' = 1 } ^ J k _ n ^ { ( j ' ) } \\right ) - \\sum _ { j ' = 1 } ^ J \\beta ^ { ( j ' ) } \\ell _ n H _ 2 \\left ( \\alpha _ n ^ { ( j ' ) } \\right ) \\right ] \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} & h _ { c , \\gamma } \\left \\lbrace f \\right \\rbrace : = \\max _ { y } L _ { y } \\left \\lbrace c f \\right \\rbrace \\\\ & \\eqref { e q : m s o s r _ P m i n } \\eqref { e q : m s o s r _ y 0 } \\gamma \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} C _ { 0 } \\simeq \\bigoplus _ { \\alpha \\in Q _ { 1 } , \\ , t ( \\alpha ) = j } \\mathbb { S } _ { 2 } ^ { \\deg ( \\alpha ) } ( M ^ { s ( \\alpha ) } ) , C _ { 1 } \\simeq \\bigoplus _ { \\alpha \\in Q _ { 1 } , \\ , s ( \\alpha ) = j } \\mathbb { S } _ { 2 } ^ { 1 - \\deg ( \\alpha ) } ( M ^ { t ( \\alpha ) } ) . \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} \\Lambda ( Z ) = \\frac { E _ { \\Theta _ 1 } [ f _ { Z ( \\theta ) } ( Z ) ] } { E _ { \\Theta _ 0 } [ f _ { Z ( \\theta ) } ( Z ) ] } \\mathop { \\gtrless } _ { H _ 0 } ^ { H _ 1 } \\gamma . \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} X = T \\cup \\big ( \\bigcup _ { p \\in P } S _ p \\big ) . \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} n _ { i , i _ 0 } \\ = \\ \\begin{cases} \\delta _ { i , i _ 0 } & i \\in I _ 0 , \\\\ ( - 1 ) ^ { \\epsilon ( i , I ) + \\epsilon ( i _ 0 , I ) } \\ , \\frac { \\Delta _ { I \\cup \\{ i \\} } } { \\Delta _ { I _ 0 } } & i \\notin I _ 0 \\end{cases} \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} \\lambda _ t = \\left ( \\sum _ { i = 1 } ^ n ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } \\right ) ^ \\gamma \\eta \\left ( \\hat { \\phi } + \\frac { \\eta } { \\gamma } \\ , \\hat { x } _ t \\right ) ^ { - \\gamma } , \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} H ( u ) = \\int \\frac { u _ x ^ 2 } { 2 } + f ( x , u , u _ x ) \\ , d x \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} y ' ( t ) = H ( \\| w \\| _ t ) \\le H ( \\| x \\| _ T + y ( t ) ) , \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} a * b = a \\circ b + \\hbar \\{ a , b \\} + \\hbar ^ 2 \\alpha _ 2 ( a , b ) + \\cdots = \\sum _ { n \\ge 0 } \\hbar ^ n \\alpha _ n ( a , b ) \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} \\left \\lvert \\operatorname { T r } \\left ( f ( H _ { B , L } ^ \\omega \\mid _ { \\omega _ j = \\omega _ + } ) - f ( H _ { B , L } ^ \\omega \\mid _ { \\omega _ j = \\omega _ - } ) \\right ) \\right \\rvert \\leq C _ \\theta \\lvert I \\rvert ^ { \\theta - 1 } . \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\log ( ( 1 - p ) ^ g 2 ^ b ) = g \\log ( 1 - p ) + b \\\\ \\leq r ( \\gamma \\log ( 1 - p ) + 1 ) \\log \\log n . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} \\dim ( Q P _ k ) _ n & = \\sum _ { \\deg \\omega = n } \\dim Q P _ k ( \\omega ) \\\\ & \\geqslant \\dim Q P _ k ( \\omega _ { ( k , d ) } ) + \\dim Q P _ k ( \\bar \\omega _ { ( k , d ) } ) + \\dim Q P _ k ( \\widetilde \\omega _ { ( k , d ) } ) \\\\ & > \\dim Q P _ k ( \\omega _ { ( k , d ) } ) + \\dim Q P _ k ( \\bar \\omega _ { ( k , d ) } ) \\geqslant c ( k , d ) . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} S : = M _ 1 \\cup \\big ( \\bigcup _ { ( l , a ) \\in I } \\gamma _ { l , a } \\big ) \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} ( 1 - \\epsilon ) B ( n ) = \\frac { ( 1 - \\epsilon ) ( 1 - \\theta _ n ) n } { 2 k _ n } \\log \\left ( 1 + k _ n P \\right ) . \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} \\Delta _ 5 \\ ; \\Delta _ 7 \\ ; \\Delta _ 8 \\ = \\ \\Delta _ 4 \\ ; \\Delta _ 6 \\ ; \\Delta _ 9 . \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} x ^ m = \\sqrt { { p ^ m _ i } } { s ^ m _ i } + \\sqrt { { p ^ m _ j } } { s ^ m _ j } . \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} \\widehat { \\delta } ( \\phi , A ) = \\sup \\{ \\widehat { \\psi } ( \\phi ) \\mid \\psi \\in \\mathcal { R } X , \\forall \\xi \\in A , \\widehat { \\psi } ( \\xi ) = 0 \\} \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{gather*} \\chi _ L ( \\xi ) = \\begin{cases} 1 , & | \\xi | \\leq \\frac { \\rho } { 2 } , \\\\ 0 , & | \\xi | \\geq \\rho , \\end{cases} \\chi _ H ( \\xi ) = \\begin{cases} 1 , & | \\xi | \\geq 2 , \\\\ 0 , & | \\xi | \\leq 4 , \\end{cases} \\\\ \\chi _ M ( \\xi ) = 1 - \\chi _ L ( \\xi ) - \\chi _ H ( \\xi ) . \\end{gather*}"}
-{"id": "5111.png", "formula": "\\begin{align*} \\nu ( A _ { x _ o } ^ { - 1 } C ) \\geq \\eta \\otimes \\eta ( G ( A ' \\times C ' ) ) = m _ K ( I ^ { - 1 } J ) , \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} q _ { n + t } = \\sum _ { i = 0 } ^ t Q _ i a _ { n + i } , \\ \\ r _ { n + t - 1 } = \\sum _ { i = 0 } ^ { t - 1 } R _ i a _ { n + i } \\ \\ \\mbox { f o r } n \\ge 0 . \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} \\tilde { h } ( I ) : = \\mathcal { H } _ { \\leq 5 } ( \\theta , I , 0 ) : = \\sum _ { j \\in S ^ + } j ^ 2 \\ , I _ j + H _ { 4 , 0 } ^ { ( 4 ) } ( I ) \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} A U = U \\begin{bmatrix} \\tilde { B } & C \\\\ O & D \\end{bmatrix} , \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} X = B \\Gamma _ { m , b c } X - ( \\Gamma _ { m , b c } X ) ( J _ { m , b c } B ) - ( \\Gamma _ { m , b c } X ) J _ { m , b c } ( B \\Gamma _ { m , b c } X ) + B = : \\Phi ( X ) . \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} F _ { i } ^ { \\mathbf { y } } = p _ m \\mathbf { y } \\cdot \\nabla F _ { i } , \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\tilde p _ n ( \\sigma ) = - n ^ { - 1 } \\{ \\sigma ^ { - 2 } + \\log \\sigma ^ 2 \\} . \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} f : = ( \\phi ( \\alpha ) ) : \\bigoplus _ { \\alpha \\in Q _ { 1 } , \\ , t ( \\alpha ) = j } \\mathbb { S } _ { 2 } ^ { \\deg ( \\alpha ) } ( M ^ { s ( \\alpha ) } ) \\to M ^ { j } \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\sum _ { C \\odot x _ 0 } w ( | C | ) = \\sum _ { n = 1 } ^ { \\infty } \\sum _ { \\substack { C \\odot x _ 0 \\\\ | C | = n } } w ( | C | ) = \\sum _ { n = 1 } ^ { \\infty } w ( n ) \\sum _ { \\substack { C \\odot x _ 0 \\\\ | C | = n } } 1 , \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} F ( x ) & = \\int _ 0 ^ x f ( y ) d y , c ^ \\ast = a ^ \\ast b + \\int _ 0 ^ 1 F ( y ) \\ , d y , \\\\ F ( x , y ) & = \\big [ c ^ \\ast - F ( y ) \\big ] \\textbf { 1 } _ { [ 0 , x ] } ( y ) + x \\Big ( F ( y ) - \\int _ 0 ^ 1 F ( z ) d z - a ^ \\ast b \\Big ) \\textbf { 1 } _ { [ 0 , 1 ] } ( y ) , \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} U = ( u _ { i j } ) _ { i , j } \\textrm { i s u n i t a r y } , U = F U ^ c F ^ { - 1 } , \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} \\mathbf { A } _ { \\operatorname { D P [ 0 ] } } = \\begin{bmatrix} \\operatorname { D P [ 0 ] } & \\operatorname { C } \\\\ \\mathcal { T } & \\operatorname { B } \\end{bmatrix} : \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} \\gamma \\leq \\gamma _ \\star : = \\frac { ( 1 - \\rho ) ( p + 1 ) } { 2 ( p - 1 ) } . \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} \\tilde { \\mathcal { Q } } \\triangleq \\left \\{ \\tilde { \\mathbf { q } } _ { \\tilde { v } } \\left | \\tilde { \\mathbf { q } } _ { \\tilde { v } } = \\left [ \\tilde { f } _ { \\tilde { v } } ( - N _ { \\rm c p } ) , \\tilde { f } _ { \\tilde { v } } ( - N _ { \\rm c p } + 1 ) , \\ldots , \\right . \\tilde { f } _ { \\tilde { v } } ( - N _ { \\rm c p } + L - 1 ) \\right ] ^ { \\rm T } , \\tilde { v } \\in \\mathcal { U } _ { 2 V } \\right \\} . \\end{align*}"}
-{"id": "8256.png", "formula": "\\begin{align*} \\norm { \\phi } _ { L ^ { 2 } ( Q ) } \\leq M \\phi \\in L ^ { 2 } ( Q ) \\lambda \\in [ 0 , 1 ] \\phi = \\lambda \\mathcal { M } ( \\phi ) . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} \\phi _ { \\ell ( x ) } ( t ) & = \\exp \\left \\{ \\frac { \\alpha x } { 2 } \\left ( \\frac { \\frac { 1 } { 2 x } e ^ { i t } } { 1 - \\Big ( 1 - \\frac { 1 } { 2 x } \\Big ) e ^ { i t } } - 1 \\right ) \\right \\} \\\\ & = \\exp \\left \\{ { \\alpha x ^ 2 } \\frac { ( e ^ { i t } - 1 ) } { 2 x - ( 2 x - 1 ) e ^ { i t } } \\right \\} . \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta u _ 1 = \\lambda _ 1 ( | ( u _ 1 , u _ 2 ) | ^ \\frac { 1 } { 2 } + \\tan | ( u _ 1 , u _ 2 ) | ) , & \\Omega , \\\\ - \\Delta u _ 2 = \\lambda _ 2 | ( u _ 1 , u _ 2 ) | ^ 2 , & \\Omega , \\\\ u _ 1 , u _ 2 = 0 , & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} \\lim _ j \\frac { m _ H ( D \\cap s B _ j ) } { m _ H ( B _ j ) } = 1 , \\textrm { f o r a l l $ s \\in D $ } . \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{gather*} \\partial _ 1 \\pi _ m = \\partial _ 1 \\end{gather*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\deg \\big ( w _ i , \\mathcal { O } _ 2 ( \\mathcal { W } ) \\big ) = \\deg \\big ( w , \\mathcal { O } _ 2 ( \\mathcal { W } ) \\big ) = \\deg \\big ( w , \\mathcal { O } _ 2 ( \\Omega ) \\big ) . \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} G ( c , 0 , p ) = 1 2 - 3 c ^ 2 + \\frac { 1 } { 2 } \\left ( c ^ 3 + 2 c + 6 \\right ) . \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} \\mathcal { L } _ 1 : = \\Phi ^ { - 1 } \\mathcal { L } _ { \\omega } \\Phi = \\Pi _ S ^ { \\perp } ( \\omega \\cdot \\partial _ { \\varphi } + b _ 3 ( \\varphi ) \\partial _ { y y y } + b _ 1 \\partial _ y + b _ 0 ) \\Pi _ S ^ { \\perp } + \\mathfrak { R } _ 1 \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} \\eta _ k ( x ) = \\eta ( 1 + \\frac { 2 ^ { k - 1 } } { \\lambda } ( \\frac { \\mathrm { d i s t } ( x , p ) } { R } - 1 ) ) . \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{align*} u ( x , t ) = e ^ { - i \\big ( ( \\rho \\cdot \\rho ) t + x \\cdot \\rho \\big ) } \\big ( e ^ { i \\phi ( x ) } + w ( x , t ) \\big ) , \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} Q = \\frac { 1 } { \\sqrt { c - 2 } } ( c - \\sqrt { c ( c - 2 ) } ) ( c - 1 - \\sqrt { c ( c - 2 ) } ) ^ n . \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} f ( \\ell ) = ( \\sqrt { 3 2 ( \\ell + 1 ) } + 1 ) ^ { 8 ( \\ell - 1 ) } . \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} \\omega _ { \\infty , \\mu , \\Lambda } ( A ) : = ( \\Omega _ { \\Lambda } , A \\Omega _ { \\Lambda } ) \\mbox { f o r } A \\in { \\cal A } _ { \\Lambda } \\ . \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} \\mathbf { y '' } ( s ) + 2 A _ 1 ( s ) \\cdot \\mathbf { y } ' ( s ) + A _ 0 ( s ) \\cdot \\mathbf { y } ( s ) + \\mathbf { b } ( s ) = 0 , \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} ( X + Y ) ^ { s _ 2 ( n ) } = \\sum _ { s _ 2 ( k ) + s _ 2 ( n - k ) = s _ 2 ( n ) } X ^ { s _ 2 ( k ) } Y ^ { s _ 2 ( n - k ) . } \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} \\left ( \\mathcal { B } \\circ \\Pi \\right ) \\left ( \\varphi + \\left ( \\mathcal { B } \\circ \\Pi \\right ) { \\varphi } \\circ S \\right ) ( x , y , 0 ) = \\left ( \\varphi + \\left ( \\mathcal { B } \\circ \\Pi \\right ) { \\varphi } \\circ S \\right ) ( x , y , 0 ) \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} \\delta ( \\epsilon , k ) = \\epsilon + \\gamma _ k \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\left | \\xi ^ { ( l , \\sigma ) } \\left ( \\Phi _ s ( y ) \\right ) \\right | \\stackrel { \\eqref { e q : t h m a b o u t r e l a t i o n b e t w e e n b o x d i m e n s i o n a n d R o k h l i n d i m e n s i o n - e q u i v a r i a n t - s } } { = } \\left | \\xi ^ { ( l , \\sigma ) } \\left ( y \\right ) \\right | \\in ( 0 , 8 \\varepsilon ] \\ ; , \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} F ^ n _ i ( x ) = \\begin{cases} F ^ n ( x ) , & x \\in [ 0 , \\delta ) \\\\ 1 , & x \\in [ \\delta , \\infty ) , \\end{cases} \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} w ^ k _ j & = \\left ( 1 - \\lambda a _ j \\right ) w ^ { k - 1 } _ j + \\lambda a _ j w ^ { k - 1 } _ { j - 1 } \\\\ & = \\left ( 1 - \\lambda a _ j \\right ) \\left \\{ \\left ( 1 - \\lambda a _ j \\right ) w ^ { k - 2 } _ j + \\lambda a _ j w ^ { k - 2 } _ { j - 1 } \\right \\} + \\lambda a _ j \\left \\{ \\left ( 1 - \\lambda a _ { j - 1 } \\right ) w ^ { k - 2 } _ { j - 1 } + \\lambda a _ { j - 1 } w ^ { k - 2 } _ { j - 2 } \\right \\} \\\\ & = \\dots = \\sum _ { \\ell = j - k } ^ j \\lambda _ { \\ell , j } ^ k w ^ 0 _ { \\ell } \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} P ( \\pm 1 ) = \\dfrac 1 2 \\sum _ { j = 0 } ^ { p - 1 } ( 2 j + 1 ) = \\dfrac { p ^ 2 } { 2 } \\enspace . \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} M = \\bigg \\langle \\frac { 1 } { d _ 1 } , \\frac { 1 } { d _ 2 } , \\dots \\bigg \\rangle , \\ \\ d _ n = p _ 1 \\dots p _ n \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} n _ { \\textnormal { b i t } } = n _ 1 + n _ 2 . \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} \\rho ( a , c ) = \\begin{cases} \\chi ( c ) / c & D \\nmid c , \\\\ \\chi ( a ) \\tau ( \\chi ) / c & D \\mid c \\end{cases} \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} ( \\operatorname { D P [ 0 ] } + \\operatorname { Q } ) ( \\Xi _ 2 ) = - \\operatorname { Q } ( \\Xi _ 1 ) - \\sum \\limits _ { j \\geq 2 } \\operatorname { Q } _ { i _ { 1 } } ( \\Xi _ { \\bullet } ) \\cdots \\operatorname { Q } _ { i _ { j } } ( \\Xi _ { \\bullet } ) . \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} Y _ t = D Y _ { x x } + c Y _ { x } + R ( Y ) , x \\in \\mathbb { R } , \\ t \\geq 0 . \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\ldots \\subset E _ { n - 1 } \\subset E _ n = E \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} \\xi = \\xi ( t , n ) , \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} a ( h ) = \\sum _ { n = 0 } ^ \\infty ( p _ n ( 1 _ G ) - p _ n ( h ) ) , \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} ( z _ 1 z _ 2 ) ^ { \\alpha } z _ 1 ^ { - \\alpha } z _ 2 ^ { - \\alpha } = e ^ { \\alpha ( \\arg ( w _ 1 w _ 2 ) - \\arg ( w _ 1 ) - \\arg ( w _ 2 ) ) } . \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\rho ( N _ { J _ \\mathcal { A } } ( a , b ) ) = N _ { J _ M } ( \\rho ( a ) , \\rho ( b ) ) \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} \\lambda _ l ^ { n _ l / 2 } \\geq \\frac { 1 + \\beta _ { l + 1 } ^ 2 } { \\prod _ { k = 1 } ^ { l + 2 } \\alpha _ k } \\geq \\frac { 1 + \\beta _ { l + 1 } ^ 2 } { \\prod _ { k = 1 } ^ { l + 1 } \\alpha _ k } \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} e ^ { - i \\varphi } \\Delta _ { A } e ^ { i \\varphi } = \\Delta _ { A + \\nabla \\varphi } , \\ , \\ , \\ , \\ , \\ , e ^ { - i \\varphi } \\Lambda _ { A } e ^ { i \\varphi } = \\Lambda _ { A + \\nabla \\varphi } , \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{gather*} \\Sigma ^ \\infty = \\lim _ { \\longleftarrow } \\Sigma ^ k . \\end{gather*}"}
-{"id": "9526.png", "formula": "\\begin{align*} \\lim _ { t _ 2 - t _ 1 \\to 0 } ( t _ 2 - t _ 1 ) ^ { 2 \\alpha - 1 } = + \\infty . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{gather*} \\omega _ 2 = \\delta C \\wedge \\delta \\tilde { F } , \\delta _ Q \\omega _ 1 = d \\omega _ 2 , \\operatorname { g h } ( \\omega _ 2 ) = 1 . \\end{gather*}"}
-{"id": "1335.png", "formula": "\\begin{align*} ( k - 1 ) \\big ( C _ { l + r - 2 } ^ { l - 2 } + C _ { l + r - 1 } ^ { l - 2 } \\big ) C _ { 2 k - l } ^ { k - l - r } = ( 2 k - l ) \\big ( C _ { l + r - 2 } ^ { l - 2 } C _ { 2 k - l - 1 } ^ { k - l - r - 1 } + C _ { l + r - 1 } ^ { l - 2 } C _ { 2 k - l - 1 } ^ { k - l - r } \\big ) . \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} ( u | v ) _ t = \\int _ X \\langle u | v \\rangle _ t d v _ X , \\end{align*}"}
-{"id": "3429.png", "formula": "\\begin{align*} F ( s ; \\mathbf { a } , \\mathbf { z } ) : = \\prod _ { j = 1 } ^ l F ( s ; b _ j , z _ j ) = \\prod _ { j = 1 } ^ l \\prod _ p \\ ( 1 + \\frac { z _ j \\lambda _ { b _ j } ( p ) } { p ^ s } \\ ) . \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} | C ( m , n ; k ) | & = | C ( 0 , n - m ; k ) | ^ { | \\Omega _ { m } | } \\\\ & = \\Bigl ( ( \\tfrac 1 2 + o ( 1 ) ) | G _ { n - m } | \\Bigr ) ^ { | \\Omega _ { m } | } & & ( \\ref { e q n : B a s e C a r d i n a l i t y } ) \\\\ & = \\Bigl ( \\frac { 1 } { 2 ^ { | \\Omega _ { m } | } } + o ( 1 ) \\Bigr ) | G _ { n - m } | ^ { | \\Omega _ { m } | } \\\\ & = \\Bigl ( \\frac { 1 } { | G _ { m } | } + o ( 1 ) \\Bigr ) | G _ { n } | . \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\beta u _ t - \\kappa u _ { t t } - u _ { x x } & = & u ( r _ u - \\gamma _ u ( u + v ) ) + \\mu v - \\mu u \\R ^ 2 \\\\ \\beta v _ t - \\kappa v _ { t t } - v _ { x x } & = & v ( r _ v - \\gamma _ v ( u + v ) ) + \\mu u - \\mu v \\R ^ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} ( \\delta f ) ^ { - 1 } ( 0 ) = D _ M : = \\{ ( u , u ) : u \\in M \\} . \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} \\mathcal { S } ( \\omega , x ) : = \\left \\{ y \\in \\mathbb { R } ^ d : | \\varphi _ m ( \\omega , y ) - \\varphi _ m ( \\omega , x ) | \\leq \\beta ( \\omega , x ) e ^ { \\varepsilon m } \\textrm { f o r a l l } m \\in \\mathbb { N } \\right \\} \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} f _ i = f ( n , i ) . \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} \\iota ^ * \\Gamma _ 1 ( a , b , i , j , a ' , b ' , f , 0 ) = & \\Gamma _ 1 ( \\iota _ * a , \\iota _ * b , \\iota _ * i , \\iota _ * j , \\iota _ * a ' , \\iota _ * b ' , \\iota _ * f , 0 ) \\\\ = & ( \\sqrt { - 1 } a , \\sqrt { - 1 } b , \\sqrt { - 1 } i , \\sqrt { - 1 } j , \\sqrt { - 1 } a ' , \\sqrt { - 1 } b ' , \\sqrt { - 1 } f , 0 ) \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} \\Delta _ y ( x ) = e ^ { \\frac { x ^ 2 - y ^ 2 } { 2 } } ( \\Phi ( x ) - 1 _ { x > y } ) \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} \\operatorname { i n v } ( \\pi ) = \\operatorname { i n v } ( \\theta ) + \\operatorname { i n v } ( \\nu ) + \\operatorname { i n v } ( A ) . \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N | r _ i | = \\sum _ { i = 1 } ^ N \\sqrt { \\omega _ i } \\left ( | r _ i | / \\sqrt { \\omega _ i } \\right ) \\le \\left ( \\sum _ { i = 1 } ^ N \\omega _ i \\right ) ^ { 1 / 2 } \\left ( \\sum _ { i = 1 } ^ N r _ i ^ 2 / \\omega _ i \\right ) ^ { 1 / 2 } \\le 2 \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} d ^ n _ \\lambda ( p , r ) = - \\frac { 1 } { 1 - \\lambda } \\log E _ { q _ * ( x ^ n ) p ( y ^ n | x ^ n ) } \\left ( \\frac { r ( y ^ n | x ^ n ) } { p ( y ^ n | x ^ n ) } \\right ) ^ { 1 - \\lambda } \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} X _ { 1 , t + 1 } & = 1 + 1 ( Y _ { 2 t } = 1 ) X _ { 1 t } + S _ { t + 1 } ^ { 2 } , S _ { t + 1 } | X _ { t } \\sim N ( . 2 + \\sum _ { k = 1 } ^ { 5 } c _ { k } X _ { t , k + 1 } , 1 ) , \\\\ c & = ( . 1 , . 0 2 5 , . 0 1 1 1 , . 0 0 6 3 , . 0 0 4 ) ; \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\sum _ { v = 0 } ^ { \\ell ^ m - 1 } e ^ { \\frac { 2 \\pi i n w _ v } { 2 4 \\ell ^ m } } = \\sum _ { v = 0 } ^ { \\ell ^ m - 1 } e ^ { \\frac { 2 \\pi i n v } { \\ell ^ m } } = \\begin{cases} \\ell ^ m & n \\equiv 0 \\pmod { \\ell ^ m } \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{gather*} K _ { \\ell } : = \\sum _ { d = 1 } ^ { \\infty } \\frac { h _ { 1 / \\tau } ( d ) } { d } ( \\log d ) ^ { \\ell } = \\sum _ { \\substack { d = 2 ^ { \\nu } \\\\ \\nu \\ge 0 } } \\frac { h _ { 1 / \\tau } ( 2 ^ { \\nu } ) } { 2 ^ \\nu } ( \\log 2 ^ \\nu ) ^ { \\ell } \\end{gather*}"}
-{"id": "3991.png", "formula": "\\begin{align*} B _ h ( ( { \\bf v } , { \\bf q } ) , ( { \\bf w } _ h , { \\bf r } _ h ) ) = a _ h ( { \\bf v } , { \\bf w } _ h ) + b _ h ( { \\bf q } , { \\bf w } _ h ) - b _ h ( { \\bf r } _ h , { \\bf v } ) + c _ h ( { \\bf q } , { \\bf r } _ h ) . \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} \\hat { \\alpha } _ { L \\ell } ( x ) = \\hat { \\rho } _ { L \\ell } ^ { \\prime } b ( x ) , \\hat { \\rho } _ { L \\ell } = \\arg \\min _ { \\rho } \\left \\{ - 2 \\hat { M } _ { \\ell } ^ { \\prime } \\rho + \\rho ^ { \\prime } \\hat { Q } _ { \\ell } \\rho + 2 r \\sum _ { j = 1 } ^ { p } \\left \\vert \\rho _ { j } \\right \\vert \\right \\} . \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} \\sigma ( Y ( x ) ) = A ( x ) Y ( x ) , \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} G _ e \\left ( \\varphi _ { t _ { e , o } } \\right ) & = \\left | \\frac { \\textbf { a } _ { r } ^ H ( \\xi _ { r _ { e , o } } , N _ e ) } { \\sqrt { N } } { \\bf { A } } \\left ( \\xi _ { r _ { e , o } } , \\varphi _ { t _ { e , o } } \\right ) \\frac { \\textbf { a } _ { t } ( \\varphi _ { t _ o } , N ) } { \\sqrt { N } } \\right | ^ 2 \\\\ & = \\left ( \\frac { N _ e } { N } \\right ) ^ 2 \\frac { { 1 - \\cos ( N { { \\mathcal K } _ 3 } ( \\varphi _ { t _ { e , o } } ) ) } } { { 1 - \\cos ( { { \\mathcal K } _ 3 } ( \\varphi _ { t _ { e , o } } ) ) } } , \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } p ( n ) q ^ n = \\frac { 1 } { ( q ; q ) _ \\infty } = ( 1 + q + q ^ 2 + \\ldots ) ( 1 + q ^ 2 + q ^ 4 + \\ldots ) \\cdots . \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} A f ( x ) & = \\sum _ { [ \\xi ] \\in \\widehat { G } _ 0 } d _ { \\xi } [ \\xi ( x ) \\sigma ( x , \\xi ) \\widehat { f } ( \\xi ) ] \\\\ & = \\int _ { M } [ \\sum _ { [ \\xi ] \\in \\widehat { G } _ 0 } d _ { \\xi } [ \\xi ( y ^ { - 1 } x ) \\sigma ( x , \\xi ) ] ] f ( y ) d y \\\\ & = \\int _ { M } [ \\sum _ { [ \\xi ] \\in \\widehat { G } _ 0 } d _ { \\xi } [ \\xi ( y ) \\sigma ( x , \\xi ) ] ] f ( x y ^ { - 1 } ) d y . \\\\ \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} q ( \\alpha ) - w ( \\alpha ) \\ ; \\ ; 2 \\Z _ 2 \\subset \\Z _ 4 \\quad q ( \\alpha + \\beta ) = q ( \\alpha ) + q ( \\beta ) + 2 ( \\alpha \\cdot \\beta ) \\in \\Z _ 4 \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} M _ n : = \\max _ { n - \\alpha _ n < k \\le n } | A _ k | . \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} P ^ { ( \\mathcal { B } _ n ) } ( \\mathbf { w } ) = 2 ^ { - n ( R _ 1 + R _ 2 ) } \\sum _ { ( i , j ) \\in \\mathcal { I } _ n \\times \\mathcal { J } _ n } p ^ n _ { W | U , V } \\big ( \\mathbf { w } \\big | \\mathbf { u } ( i ) , \\mathbf { v } \\big ( i , j ) \\big ) . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} z _ { a , m , t } = 0 . \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} \\widetilde { u _ { J _ 1 ; J _ 2 } ^ { \\alpha } } : = \\widetilde { D _ { J _ 1 } } S _ { J _ 2 } \\widetilde { u } ^ { \\alpha } . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} \\| y \\| _ { H ^ 1 \\cap L ^ 2 _ 1 } ^ 2 \\leq C E _ c ( y ) , \\langle V , y \\rangle _ { L ^ 2 } = \\langle \\partial _ x ^ { - 1 } V , y \\rangle _ { L ^ 2 } = 0 , \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} k ^ * ( y ) = \\lambda ( w ) \\xi ( w ) \\frac { \\varphi ( D ) } { 2 \\zeta ( 2 ) D } \\{ \\Psi ( 0 ) \\log { y \\sqrt { w } } + \\alpha ( D ) + \\alpha _ 0 \\} + O ( y w \\tau ( D ) ) \\end{align*}"}
-{"id": "7062.png", "formula": "\\begin{align*} \\partial _ t u = \\tfrac { 1 } { 2 } \\Delta u + u - u ^ 3 \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} T ^ r ( x ) & \\geq 2 ( \\sigma ^ { \\kappa } ( h ) - \\sigma ^ r ( h ) ) \\geq \\frac { 1 } { 2 } \\log \\frac { \\kappa ^ 2 } { h } - \\frac { 1 } { 2 } \\log \\frac { 2 r ^ 2 } { h } \\\\ & = \\log \\frac { \\kappa } { r } - \\frac { \\log 2 } { 2 } . \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} \\max \\{ \\alpha \\ , | \\ , x ^ T Q _ 1 x = 1 , x ^ T Q _ 2 x = 1 , \\alpha = x ^ T A ^ T Q _ 1 x \\} , \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{gather*} u _ { t t } = t u + 2 u ^ 3 . \\end{gather*}"}
-{"id": "8355.png", "formula": "\\begin{align*} \\nabla _ { x _ 1 } = - \\frac { 1 } { \\sqrt { 2 } } \\nabla _ { X _ 1 } + \\frac { 1 } { \\sqrt { 2 } } \\nabla _ { X _ 2 } , \\ \\ \\ \\nabla _ { x _ 2 } = \\frac { 1 } { \\sqrt { 2 } } \\nabla _ { X _ 1 } + \\frac { 1 } { \\sqrt { 2 } } \\nabla _ { X _ 2 } , \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\tilde { \\theta } _ 0 ^ i = \\frac { ( \\theta _ 0 ^ i ) ^ { \\frac { 1 } { \\gamma } } } { \\sum _ j ( \\theta _ 0 ^ j ) ^ { \\frac { 1 } { \\gamma } } } , & & i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} \\zeta ( z ) = \\left \\{ \\begin{array} { l } \\displaystyle \\sqrt { 2 N \\phi _ { A } ( z ) } = a \\sqrt { N } ( z - \\beta ) ( 1 + \\mathcal { O } ( z - \\beta ) ) \\qquad \\ , a > 1 , \\\\ \\displaystyle - N \\phi _ { A } ( z ) = \\frac { 1 - a ^ 2 } { a } N ( z - \\beta ) \\left ( 1 + \\mathcal { O } ( z - \\beta ) \\right ) \\quad a < 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} \\nabla _ F ( A ) : = A - F A F ^ * . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} { \\cal F } _ b \\left ( z \\right ) = \\sum _ { n \\ge 0 } F _ { n } ^ { \\left ( b \\right ) } z ^ n = \\prod _ { k \\ge 0 } \\left ( \\sum _ { l = 0 } ^ { b - 1 } F _ l z ^ { b ^ l } \\right ) \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\lim _ { m \\rightarrow \\infty } \\textrm { d i a m } \\left ( \\varphi _ { t _ m } ( \\cdot , B ( x , r ) ) \\right ) = 0 \\right ) > 0 . \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} \\delta \\| v \\| ^ 2 \\leq - \\langle T _ { 2 2 } v , v \\rangle = \\Big | s \\big \\langle \\big ( A _ { 2 2 } - A _ { 2 1 } ( \\tilde { A } _ { 1 1 } ) ^ { - 1 } A _ { 1 2 } \\big ) v , v \\big \\rangle \\Big | \\leq | s | \\big \\| A _ { 2 2 } - A _ { 2 1 } ( \\tilde { A } _ { 1 1 } ) ^ { - 1 } A _ { 1 2 } \\big \\| \\| v \\| ^ 2 \\leq \\frac { \\delta } { 2 } \\| v \\| ^ 2 , \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} \\nabla _ F ( M ) = G B ^ * , G , B \\in \\C ^ { n \\times r } , \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{gather*} B = \\begin{pmatrix} - \\dfrac { x } { 2 } ( 1 + q _ 2 ) + { a } & - x ^ 2 + x b + c \\\\ \\dfrac { q _ 2 ^ 2 - 1 } { 4 } & - \\dfrac { x } { 2 } ( 1 - q _ 2 ) + { d } \\end{pmatrix} , \\end{gather*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to - 0 } \\frac { \\partial p ( \\mu , \\lambda ) } { \\partial \\lambda } = - \\lim _ { \\lambda \\to + 0 } \\zeta _ { m a x } ( \\lambda ) = - \\sqrt { \\rho _ { 0 } } \\ . \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} h _ T = \\frac { h _ L [ L : K ] [ H ^ 0 _ T ( G , O _ L ^ { * } ) ] } { h _ K \\prod _ { v } e _ v ( L / K ) } . \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} t _ 0 p = O \\left ( \\left ( \\frac { 1 } { n p ^ r } \\right ) ^ { 1 / ( r - 1 ) } p \\right ) = O \\left ( \\left ( \\frac { 1 } { n p } \\right ) ^ { 1 / ( r - 1 ) } \\right ) \\stackrel { n p = \\omega ( 1 ) } { = } o ( 1 ) . \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\bold { p r } X \\left ( \\frac { u ' } { u } - u _ 1 + u _ { - 1 } \\right ) \\Big | _ { \\frac { u ' } { u } - u _ 1 + u _ { - 1 } = 0 } \\\\ & = - \\frac { \\phi } { u } ( u _ 1 - u _ { - 1 } ) + \\frac { \\phi _ t } { u } + \\phi _ u ( u _ 1 - u _ { - 1 } ) - ( \\xi _ t + \\xi _ u u ( u _ 1 - u _ { - 1 } ) ) ( u _ 1 - u _ { - 1 } ) - S \\phi + S _ { - 1 } \\phi . \\end{aligned} \\end{align*}"}
-{"id": "5010.png", "formula": "\\begin{align*} \\delta = h ^ { \\frac { 1 } { 4 } } \\ , . \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { \\kappa ^ * ( n ) } = \\frac { A ^ * \\zeta ( 3 / 2 ) } { \\zeta ( 3 ) } x ^ { 1 / 2 } + \\frac { B ^ * \\zeta ( 2 / 3 ) } { \\zeta ( 2 ) } x ^ { 1 / 3 } + O ( x ^ { 1 / 5 } ) , \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} ( h - 1 _ { V } ) v & = 0 \\Rightarrow h v = v . \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{align*} A _ { ( l _ 1 , j _ 1 ) } ^ { ( l _ 2 , j _ 2 ) } : = A _ { j _ 1 } ^ { j _ 2 } ( l _ 1 - l _ 2 ) , \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} \\tilde { T } _ { 2 } & = \\int \\alpha _ { 2 0 } ( x _ { t } , y _ { 2 t } ) [ H ( \\hat { \\gamma } _ { 1 } ( x _ { t + 1 } ) ) - H ( \\gamma _ { 1 0 } ( x _ { t + 1 } ) ) ] F _ { 0 } ( d w ) + \\int \\alpha _ { 2 0 } ( x _ { t } , y _ { 2 t } ) [ H ( \\gamma _ { 1 0 } ( x _ { t + 1 } ) ) - \\gamma _ { 2 0 } ( x _ { t } ) ] F _ { 0 } ( d w ) \\\\ & = \\int \\alpha _ { 2 0 } ( x _ { t } , y _ { 2 t } ) [ H ( \\hat { \\gamma } _ { 1 } ( x _ { t + 1 } ) ) - H ( \\gamma _ { 1 0 } ( x _ { t + 1 } ) ) ] F _ { 0 } ( d w ) . \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} & \\left \\| \\nabla ^ { k } _ { x } \\left ( \\partial _ { t } K _ { 1 } ( t ) g + \\nabla _ { x } \\mathcal { F } ^ { - 1 } \\left [ e ^ { - \\frac { \\nu t | \\xi | ^ { 2 \\sigma } } { 2 } } \\sin ( t | \\xi | ) \\right ] \\ast g \\right ) \\right \\| _ { 2 } \\\\ & \\le C ( 1 + t ) ^ { - \\frac { n } { 4 \\sigma } - 1 - \\frac { k } { 2 \\sigma } } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla ^ { ( k + 2 ( 1 - \\sigma ) ) } _ { x } g \\| _ { 2 } , \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} L = \\nu ( \\mu _ 1 ' - \\mu ' ) - a \\ln \\left \\{ 1 - ( \\mu _ 2 - \\mu _ 1 ) ( \\nu _ 1 - \\nu ) \\right \\} + b \\ln \\left \\{ 1 + ( \\mu _ 1 - \\mu ) ( \\nu _ 1 - \\nu ) \\right \\} . \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} b _ { \\nu } = \\sum _ { k = 1 } ^ { \\nu } ( - 1 ) ^ k \\sum _ { \\substack { j _ 1 , \\ldots , j _ k \\ge 1 \\\\ j _ 1 + \\cdots + j _ k = \\nu } } a _ { j _ 1 } \\cdots a _ { j _ k } \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} U _ R \\psi ( x , n ) = \\hat { \\psi } ( n , x + \\alpha n ) , \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} \\widetilde { x } = \\widetilde { x } ( \\varepsilon ; x , u ) , \\widetilde { u } = \\widetilde { u } ( \\varepsilon ; x , u ) , \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} G ^ { \\lambda } G ^ { \\mu } = \\sum _ { \\nu } c ^ { \\lambda , \\mu } _ { \\nu } G ^ { \\nu } , c ^ { \\lambda , \\mu } _ { \\nu } \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} 1 = \\lim _ { i \\rightarrow \\infty } \\int _ { \\Omega _ { \\varphi _ i } } \\abs { v _ i } ^ 2 d x = \\int _ { \\Omega _ \\varphi } \\abs { v } ^ 2 d x \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} g ( h , h ' ) : = { \\left ( \\partial _ x f ( h ) - \\frac { y - 2 y ' } { 2 } \\partial _ z f ( h ) \\right ) ^ 2 } + { \\left ( \\partial _ y f ( h ) + \\frac { x - 2 x ' } { 2 } \\partial _ z f ( h ) \\right ) } ^ 2 . \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} D _ Q ( f , s ) : = \\sum _ { n = 1 } ^ { \\infty } t _ Q ( n ) \\frac { f ( n ) } { n ^ s } . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} \\dot { w } = \\partial _ x K _ { 0 2 } ( \\omega t ) w + \\partial _ x K _ { 1 1 } ( \\omega t ) \\eta ( 0 ) . \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} e ^ { - 2 L D } - R _ { - L , L } = \\Gamma _ 1 \\Gamma _ 2 \\Gamma _ 3 \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} \\Lambda _ S ( - \\theta , \\mathbf { H } ) = \\log \\left ( \\pi e ^ { - \\theta i ( K - \\ell ) c } + \\sum _ { j = 1 } ^ J \\pi _ j e ^ { - \\theta r _ j } \\right ) , \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} \\underline { A } = A / A e _ { F } A . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{align*} L _ { a , n } ^ { X X Z } ( \\lambda ) = \\left ( \\begin{array} { c c } \\left [ \\lambda q ^ { s + S _ { n } ^ { z } / 2 } - 1 / ( \\lambda q ^ { s + S _ { n } ^ { z } / 2 } ) \\right ] / 2 & S _ { n } ^ { - } \\\\ S _ { n } ^ { + } & \\left [ \\lambda q ^ { s - S _ { n } ^ { z } / 2 } - 1 / ( \\lambda q ^ { s - S _ { n } ^ { z } / 2 } ) \\right ] / 2 \\end{array} \\right ) \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} e _ b ( x , w ) : = \\sum _ { k = 0 } ^ { \\infty } \\frac { x ^ { s _ b ( k ) } } { ( k ! ) _ b } w ^ k . \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} 1 \\ , & = \\ , \\frac { 1 } { \\lambda } A ( k , k ) + \\sum _ { i \\neq k } \\sum _ { i ' = 1 } ^ N \\frac { x _ { i ' } } { \\lambda ^ 2 x _ k } A ( i ' , i ) A ( i , k ) \\cr & = \\ , \\frac { 1 } { \\lambda } A ( k , k ) + \\sum _ { i \\neq k } \\frac { 1 } { \\lambda ^ 2 } A ( k , i ) A ( i , k ) \\ , + \\ , \\sum _ { i , i ' \\neq k } \\frac { x _ { i ' } } { \\lambda ^ 2 x _ k } A ( i ' , i ) A ( i , k ) \\ , . \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} f _ X ( t ) = \\begin{cases} \\gamma t ^ { - 1 - \\gamma } & \\mbox { i f } t \\ge 1 \\\\ 0 & \\mbox { e l s e } , \\end{cases} 1 - F _ X ( t ) = \\begin{cases} t ^ { - \\gamma } & \\mbox { i f } t \\ge 1 \\\\ 1 & \\mbox { e l s e } , \\end{cases} \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} ( - \\Delta ) ^ \\alpha u = \\lambda f ( x ) | u | ^ { q - 2 } u + | u | ^ { 2 _ \\alpha ^ * - 2 } u \\ ; \\textrm { i n } \\ ; \\Omega , \\ ; \\ ; u = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\textrm { o n } \\ ; \\mathbb R ^ n \\setminus \\Omega , \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} \\partial _ { \\tau } u = \\Pi _ S ^ { \\perp } \\partial _ x ( b ( \\tau , x ) u ) = \\partial _ x ( b ( \\tau , x ) u ) - \\Pi _ S \\partial _ x ( b ( \\tau , x ) u ) , u \\in H _ S ^ { \\perp } \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} \\alpha ( \\lambda ) = \\frac { s ( \\lambda ) } { s ( q / \\lambda ) } k ( \\lambda ) , s ( \\lambda ) = \\frac { \\lambda ^ { 2 } q - 1 / \\left ( q \\lambda ^ { 2 } \\right ) } { \\lambda ^ { 2 } - 1 / \\lambda ^ { 2 } } . \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} \\tilde { V } ' ( z ) = \\exp \\left ( \\sum _ { n > 0 } - z ^ n \\frac { \\partial } { \\partial p _ n } \\right ) . \\end{align*}"}
-{"id": "5663.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j = 2 } ^ { k } \\left \\{ m _ { l } \\hat { \\theta } _ { j } \\right \\} & = \\sum _ { j = 2 } ^ { k _ 1 } \\left \\{ m _ { l } \\hat { \\theta } _ { j } \\right \\} + \\sum _ { j = k _ { 1 } + 1 } ^ { k } \\left \\{ m _ { l } \\hat { \\theta } _ { j } \\right \\} \\\\ & = \\sum _ { j = 2 } ^ { k _ 1 } \\left \\{ p _ { j } \\{ m _ { l } \\theta \\} + \\xi _ { j } \\right \\} + \\sum _ { j = k _ { 1 } + 1 } ^ { k } \\left \\{ p _ { j } \\{ m _ { l } \\theta \\} \\right \\} . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "2357.png", "formula": "\\begin{gather*} \\frac { \\kappa _ t } { \\kappa } = - \\frac { 1 } { 3 } \\omega - \\frac { 2 } { 3 } \\alpha - \\frac { u _ t } { u } \\frac { 1 - 2 q _ 2 } { 6 } . \\end{gather*}"}
-{"id": "1945.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n - 1 } ( \\lambda - \\delta ( x _ j ) ) x _ j \\geq ( 1 + \\delta ( x _ { n - 1 } ) ) x _ n \\end{align*}"}
-{"id": "4869.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { p - 1 } \\binom { p - 1 + ( i + j ) p } { m + i p } \\equiv _ { p ^ 3 } \\binom { i + j } { i } \\left ( 1 + ( i + j + 1 ) p q _ p ( 2 ) + \\binom { i + j + 1 } { 2 } p ^ 2 q ^ 2 _ p ( 2 ) \\right ) . \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} \\gamma ( x , y ) = \\frac { 1 + [ y , b ( x ) ] } { 1 + [ x , b ( y ) ] } . \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\varphi _ t ( \\omega , D ( \\omega ) ) = D ( \\theta _ t \\omega ) \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} 1 + \\sum _ { v \\geq 1 } b _ z ( p ^ v ) \\xi ^ v = \\ ( 1 + z \\lambda _ a ( p ) \\xi \\ ) \\ ( 1 - \\chi _ 0 ( p ) \\xi \\ ) ^ { \\frac { z } { \\phi ( q ) } } \\prod _ { \\chi \\neq \\chi _ 0 } ( 1 - \\chi ( p ) \\xi ) ^ { \\frac { \\bar { \\chi } ( a ) z } { \\phi ( q ) } } , ( | \\xi | < 1 ) . \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} \\tilde T _ 4 ( z ) = \\frac { i } { 2 \\pi } \\int _ { \\xi _ 1 + \\xi _ 2 = \\eta _ 1 + \\eta _ 2 } \\frac { 1 } { ( 2 z + \\xi _ 1 ) ( 2 z + \\eta _ 1 ) ( 2 z + \\eta _ 2 ) } \\real \\{ \\overline { \\hat u ( \\xi _ 1 ) \\hat u ( \\xi _ 2 ) } \\hat u ( \\eta _ 1 ) \\hat u ( \\eta _ 2 ) \\} \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} f ( z ) = \\frac { A _ { N - 1 } ( z ) + z B _ { N - 1 } ^ * ( z ) f _ N ( z ) } { B _ { N - 1 } ( z ) + z A _ { N - 1 } ^ * ( z ) f _ N ( z ) } , \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} f _ { \\mathrm { c } } ( r ) = 2 \\pi \\lambda r e ^ { - \\lambda \\pi r ^ 2 } . \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} u _ t + b ( R ) \\ , u _ x + \\frac { \\sigma ( R ) ^ 2 } { 2 } \\ , u _ { x x } = f , u ( T , \\cdot ) = 0 , \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} R : = \\frac { 1 } { 2 } \\left ( 1 \\otimes 1 + 1 \\otimes u + u \\otimes 1 - u \\otimes u \\right ) \\in \\C [ \\Z _ 2 ] \\otimes \\C [ \\Z _ 2 ] . \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} - \\Delta F _ { N , s } ( x ) & = \\kappa _ { N , s } ( 2 s - N ) 2 ( s - 1 ) | x | ^ { 2 ( s - 1 ) - N } = F _ { N , s - 1 } ( x ) \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} p ( \\xi , \\beta ) = \\frac { \\cosh ( ( \\beta + h ) \\xi ) } { \\cosh ( h \\xi ) } . \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} B ^ { ( l , \\sigma ) } = \\Phi _ { \\left [ - \\left ( \\frac { 1 } { 2 } - 4 \\varepsilon \\right ) 8 L , \\left ( \\frac { 1 } { 2 } - 4 \\varepsilon \\right ) 8 L \\right ] } ( \\Delta ^ { ( l , \\sigma ) } _ 1 \\cap K ' ) \\cap \\left ( \\xi ^ { ( l , \\sigma ) } \\right ) ^ { - 1 } ( V ) \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} M = M _ { a , s } : = \\left ( \\begin{array} { c c c } a ^ { g s } & a ^ { g ( 1 - s ) } & w _ a \\\\ a ^ { 2 g s } & a ^ { 2 g ( 1 - s ) } & w _ { a ^ 2 } \\\\ a ^ { 3 g s } & a ^ { 3 g ( 1 - s ) } & w _ { a ^ 3 } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} \\bar \\phi _ { \\mu ^ * } ( j + x ( \\mu ^ * , n ) , - n ) \\begin{cases} \\ge \\bar \\phi _ \\mu ( j + x ( \\mu , n ) , - n ) { \\rm f o r } j \\le j _ n \\cr < \\bar \\phi _ \\mu ( j + x ( \\mu , n ) , - n ) { \\rm f o r } j > j _ n . \\end{cases} \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} U ( a , b ; z ) = \\frac { \\Gamma ( 1 - b ) } { \\Gamma ( 1 + a - b ) } { } _ { 1 } F _ { 1 } ( a , b ; z ) + \\frac { \\Gamma ( b - 1 ) } { \\Gamma ( a ) } x ^ { 1 - b } { } _ { 1 } F _ 1 ( 1 + a - b , 2 - b ; z ) . \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} \\lambda _ j ( u v ) = \\sum _ { a + b = j } \\binom { j } { a } \\lambda _ a ( u ) \\lambda _ b ( v ) . \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\exp \\left ( \\{ x = a \\} \\cap D ^ { * } \\right ) = \\left \\{ e ^ { a } e ^ { i y } \\ , \\left | \\ , | y | < \\frac { 1 } { 2 } l ( a ) \\right . \\right \\} . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\left \\vert p - a , n \\right \\rangle = \\overline { \\left \\vert a + 1 , n \\right \\rangle } \\forall a \\in \\{ 0 , . . . , p - 1 \\} , \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} \\limsup _ { | i | \\le c ^ { '' } t , t \\to \\infty } | u _ i ( t + s ; s , u ^ 0 ) - u ^ + ( t + s ) | = 0 \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} n _ 0 ^ { ( j ) } = \\begin{cases} n ( 1 + \\epsilon ' / 2 ) \\theta _ n ^ { ( j ) } , \\ ; \\ ; & \\theta ^ { ( j ) } > 0 \\\\ n \\epsilon ' / ( 2 J ) , & \\theta ^ { ( j ) } = 0 \\end{cases} \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} \\bar { w } = \\max \\left \\{ \\frac { 4 } { P ' } e ^ { ( 8 + 4 \\epsilon ) / \\phi } , 1 \\right \\} . \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\frac { \\mathsf { C _ { \\mathcal { N } _ { \\mathcal { K } } } } } { { \\mathsf { C } } _ { \\mathcal { N } _ { [ 1 : N ] } } } \\geq \\begin{cases} \\max \\left \\{ \\frac { 1 } { N } \\ , \\frac { 1 } { 4 } \\right \\} , & k = 1 \\\\ \\max \\left \\{ \\frac { k } { N } \\ , \\frac { 1 } { 2 } \\right \\} , & N \\geq k \\geq 2 \\\\ \\end{cases} . \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} S _ h ( x ) = S _ h ^ * ( x ) + \\tfrac 1 2 T _ h ( x ) + \\tfrac 1 2 T _ { - h } ( x ) + O \\bigl ( ( h + z ) ( \\log x ) ^ 2 \\bigr ) \\ , \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} K _ 3 ( \\xi - \\eta ) = \\int g ( \\xi - N ) g ( \\eta - N ) \\ , d N , \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} \\gamma ( 1 / 2 , \\pi _ 1 , \\pi _ 2 , \\psi ) = 1 . \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} \\textstyle \\int ( \\gamma ' - \\textstyle \\int s \\gamma ' ) \\ = \\ \\gamma + ( s \\gamma ^ { \\dagger } - \\gamma ^ { \\dagger } ) \\ = \\ \\gamma - \\chi ( \\gamma ) . \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} I ( \\xi , \\eta ) : = & \\ \\int ( 1 + \\tanh N ) ( 1 + \\tanh ( \\xi - N ) ) ( 1 + \\tanh ( \\eta - N ) ) \\ , d N \\\\ = & \\ \\xi \\left ( 1 + \\frac { 1 } { \\tanh \\xi } \\right ) \\left ( 1 + \\frac { 1 } { \\tanh ( \\eta - \\xi ) } \\right ) + \\eta \\left ( 1 + \\frac { 1 } { \\tanh \\eta } \\right ) \\left ( 1 + \\frac { 1 } { \\tanh ( \\xi - \\eta ) } \\right ) , \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\Delta _ { x , y } \\beta = 0 \\ \\ \\Omega \\\\ & \\beta ( x , - 1 ) = - 1 \\\\ & \\beta ( x , y ) = 0 \\ \\ \\Gamma . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} H ^ 0 ( \\mathcal { C } ( \\mathcal { A } ) ) = H ^ 2 ( \\mathcal { C } ( \\mathcal { A } ) ) = 0 . \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} a _ 2 b _ 2 a _ 2 ^ { - 1 } = [ a _ 2 , b _ 2 ] b _ 2 \\stackrel { } { = } a _ 2 B a _ 2 ^ { - 1 } b _ 2 . \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} \\bigoplus _ { i = 0 } ^ { \\ell } ( - 1 ) ^ { \\ell - i } W H ^ i ( P ) \\cong _ { G } 0 . \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} U _ { \\Gamma , E } E ^ { - 1 } U _ { \\Gamma , E } ^ * = - I d _ { L ^ 2 ( \\Lambda , \\mu ) } ; \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} a = a ' + \\sum _ { n \\ge 1 } c _ n \\frac { q } { p ^ { m + n } } , \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} q ^ \\ast = \\left \\{ \\begin{array} { l l } \\displaystyle \\min \\left \\{ \\frac { 2 } { \\pi } | A r g ( \\beta ) | , q ^ { \\ast \\ast } \\right \\} & , ~ \\textrm { i f } n \\geq 3 \\\\ q ^ { \\ast \\ast } & , ~ \\textrm { i f } n = 2 \\end{array} \\right . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta \\log \\frac { \\rho } { e ^ { - \\Psi } } = \\mbox { d i v } \\frac { \\nabla \\frac { \\rho } { e ^ { - \\Psi } } } { \\frac { \\rho } { e ^ { - \\Psi } } } = - \\frac { \\bigg | \\nabla \\frac { \\rho } { e ^ { - \\Psi } } \\bigg | ^ { 2 } } { \\bigg ( \\frac { \\rho } { e ^ { - \\Psi } } \\bigg ) ^ { 2 } } + \\frac { \\Delta \\frac { \\rho } { e ^ { - \\Psi } } } { \\frac { \\rho } { e ^ { - \\Psi } } } = : I _ 1 + I _ 2 . \\end{aligned} \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} a : = 2 \\Im \\P _ h [ R \\bar R _ \\alpha ] , \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} - \\dot q _ \\rho ( t ) = \\Delta T _ \\rho ' ( g _ \\rho ( t ) + \\Delta z _ \\rho ( t ) ) \\ , q _ \\rho ( t ) + j _ 1 ' ( z _ \\rho ( t ) ) H ^ { - 1 } ( \\Omega ) , t \\in I . \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\begin{bmatrix} \\zeta ^ + ( 1 - s ) \\\\ \\zeta ^ - ( 1 - s ) \\end{bmatrix} = \\begin{bmatrix} c _ 1 ( s ) & c _ 2 ( s ) \\\\ c _ 3 ( s ) & c _ 4 ( s ) \\end{bmatrix} \\begin{bmatrix} \\hat \\zeta ^ + ( s ) \\\\ \\hat \\zeta ^ - ( s ) \\end{bmatrix} \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} L ( x ^ n ) & = \\{ 3 , 4 , 5 , \\ldots , n - 2 \\} \\cup \\{ n \\} \\\\ L ( x ^ n ) & = \\{ 2 , 3 , 4 , 5 , \\ldots , n - 2 \\} \\cup \\{ n \\} \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} { \\cal E } _ { + } ( z ) { \\cal E } ^ { - 1 } _ { - } ( z ) = I + \\mathcal { O } \\left ( \\frac { 1 } { N ^ { 2 - c } } \\right ) , z \\in \\partial D _ \\beta , \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\lambda _ { t o p } & \\leq \\tilde { c } \\left ( \\left [ r ^ { n - 2 } e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( r ^ 2 - 1 \\right ) ^ 2 } \\right ] _ 0 ^ \\infty - \\int _ 0 ^ \\infty ( n - 2 ) r ^ { n - 3 } \\ , e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( r ^ 2 - 1 \\right ) ^ 2 } \\ , d r \\right ) \\\\ & = - \\tilde { c } ( n - 2 ) \\int _ 0 ^ \\infty r ^ { n - 3 } \\ , e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( r ^ 2 - 1 \\right ) ^ 2 } \\ , d r < 0 . \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} H _ 2 ( ( 1 + \\delta ) \\alpha _ n ) - H _ 2 ( \\alpha _ n ) = \\delta \\alpha _ n \\log \\frac { 1 - \\gamma ' _ n } { \\gamma ' _ n } , \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} \\mathfrak n ^ - _ 0 = 0 \\subset \\mathfrak n ^ - _ 1 \\subset \\mathfrak n ^ - _ 2 \\subset \\ldots \\subset \\mathfrak n ^ - _ n = \\mathfrak n ^ - . \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} X ^ t Y - Y ^ t X = 0 . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} \\Psi ^ { ( j ) } ( z ) \\ll ( 1 + | z | ) ^ { - A } , \\ j = 0 , 1 , 2 , \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} \\nu _ 1 & = - \\frac { 1 } { 2 d } \\sum _ { i j } A _ { i i } A _ { j j } \\pi _ i \\pi _ j , \\\\ \\nu _ 2 & = - \\frac { 1 } { 4 d ^ 2 } \\sum _ { i j k \\ell } A _ { i k } ^ 2 A _ { j \\ell } ^ 2 \\pi _ i \\pi _ j \\pi _ k \\pi _ \\ell , \\mbox { a n d } \\\\ \\xi _ n & = O ( n ^ { - 1 / 2 } ) \\sum _ { i j } | X _ { i j } | + O ( n ^ { - 1 } ) \\left ( \\sum _ { i j } | X _ { i j } | \\right ) ^ 2 + O ( n ^ { - 1 } ) . \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} 4 \\int _ { 0 } ^ { + \\infty } F _ U ( x ) \\cosh \\Big ( \\frac { x } { 2 } \\Big ) \\d x & = 4 \\int _ { 0 } ^ { + \\infty } F \\Big ( \\frac { x } { U } \\Big ) \\cosh \\Big ( \\frac { x } { 2 } \\Big ) \\d x \\\\ & = 4 U \\int _ { 0 } ^ 1 ( 1 - 5 x ^ 2 + 5 x ^ 3 - x ^ 5 ) \\cosh \\Big ( \\frac { U x } { 2 } \\Big ) \\d x \\\\ & = 9 6 0 \\frac { \\big ( ( U - 4 ) e ^ { \\frac U 4 } + ( U + 4 ) e ^ { - \\frac U 4 } \\big ) ^ 2 } { U ^ 5 } = S ( U ) . \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} \\Omega ^ { k } ( M ) = \\mathcal { H } ^ k _ \\theta ( M ) \\oplus d _ \\theta \\Omega ^ { k - 1 } ( M ) \\oplus \\delta _ \\theta \\Omega ^ { k + 1 } ( M ) \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} z f ' ( z ) = \\frac { z } { 1 - z ^ 2 } h ( z ) . \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} x ^ \\prime & = f ( x , y , \\epsilon ) , \\\\ y ^ \\prime & = \\epsilon g ( x , y , \\epsilon ) \\end{align*}"}
-{"id": "3963.png", "formula": "\\begin{align*} \\lambda ( \\pi ) = \\delta _ 1 \\times \\cdots \\times \\delta _ k . \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} u ( t ) = \\sum _ { n \\in \\mathbb { N } _ 0 } c _ n ( t ) u _ n , c _ n ( t ) = \\langle u _ n , u ( t ) \\rangle _ { L ^ 2 } , \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} G _ v ( \\varepsilon ) = \\sum _ { \\substack { q < y \\\\ ( q , v ) = 1 } } \\xi _ q g _ { \\varepsilon } ( q ) \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} \\int _ \\mathbb { C } P _ { n , N } ( z ) \\overline { P _ { m , N } ( z ) } e ^ { - N Q ( z ) } d A ( z ) = h _ { n , N } \\delta _ { n m } \\quad ( n , m = 0 , 1 , 2 , \\ldots ) , \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} \\eta = d z + x d y , \\eta \\wedge d \\eta = d x \\wedge d y \\wedge d z . \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { R } g ( i ) \\ , d i & = R ^ { n + 1 } \\int _ { 0 } ^ { 1 } ( 1 - t ^ 2 ) ^ { n / 2 } \\ , d t \\\\ & = \\frac { R ^ { n + 1 } } { 2 } \\int _ { 0 } ^ { 1 } ( 1 - s ) ^ { n / 2 } \\frac { 1 } { \\sqrt { s } } \\ , d s = \\frac { \\sqrt { \\pi } } { 2 } \\frac { \\Gamma \\left ( \\frac { n + 2 } { 2 } \\right ) } { \\Gamma \\left ( \\frac { n + 3 } { 2 } \\right ) } R ^ { n + 1 } , \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{gather*} \\omega _ 1 = \\delta C \\wedge \\delta A ^ \\ast + \\delta A \\wedge \\delta \\tilde { F } , \\delta _ Q \\omega = d \\omega _ 1 , \\operatorname { g h } ( \\omega _ 1 ) = 0 . \\end{gather*}"}
-{"id": "4203.png", "formula": "\\begin{align*} d ^ { m , l } = ( ( \\theta + 1 ) _ { m - 1 \\uparrow } + \\sum _ { j = 1 } ^ { m - 1 } ( \\theta + \\alpha ) _ { m - j \\uparrow } ( \\theta + 1 + m - j ) _ { j - 1 \\uparrow } ) ^ { l } , \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} \\delta ^ { - 1 } ( a ) = \\begin{cases} 0 , & \\textrm { i f } k + \\ell = 0 \\\\ \\frac { 1 } { k + \\ell } \\delta ^ * ( a ) & \\textrm { e l s e , } \\end{cases} \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} \\sum _ { \\kappa \\in \\mathbb { Z } } e ^ { - 4 \\pi ^ 2 s ( \\kappa + a ) ^ 2 } & = \\sum _ { p \\in \\mathbb { Z } } ( 4 \\pi s ) ^ { - 1 / 2 } e ^ { - p ^ 2 / 4 s } e ^ { 2 \\pi i p a } , \\\\ \\sum _ { \\kappa \\in \\mathbb { Z } } e ^ { - 4 \\pi ^ 2 s ( \\kappa + a ) ^ 2 } 4 \\pi ^ 2 s ( \\kappa + a ) ^ 2 & = \\sum _ { p \\in \\mathbb { Z } } ( 4 \\pi s ) ^ { - 1 / 2 } e ^ { - p ^ 2 / 4 s } e ^ { 2 \\pi i p a } \\left ( \\frac { 1 } { 2 } - \\frac { p ^ 2 } { 4 s } \\right ) . \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} \\frac { H _ { 2 } } { \\beta ^ 2 } = \\Delta _ 2 - \\Delta _ 0 \\Delta _ 1 + \\Delta _ 1 + \\frac { 1 } { 3 } \\Delta _ 0 ^ 3 - \\frac { 1 } { 2 } \\Delta _ 0 ^ 2 + \\frac { 1 } { 6 } \\Delta _ 0 . \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\mapsto ( - 1 ) ^ t \\begin{bmatrix} 0 & 1 \\\\ - 1 & 0 \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} 0 & - 1 \\\\ 1 & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} \\mathcal { E } v = \\sum _ { k \\in \\mathbb { Z } ^ d } \\langle \\mathcal { E } v , \\widetilde { \\phi } _ k \\rangle \\ , \\phi _ k + \\sum _ { i = 1 } ^ { 2 ^ { d } - 1 } \\sum _ { j \\in \\mathbb { N } _ 0 } \\sum _ { k \\in \\mathbb { Z } ^ d } \\langle \\mathcal { E } v , \\widetilde { \\psi } _ { i , j , k } \\rangle \\ , \\psi _ { i , j , k } \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} M _ { r \\sigma } ( x ) : = \\left ( \\sum _ { i = 1 } ^ n \\sigma _ i x _ i ^ { 1 / r } \\right ) ^ { 1 / r } , \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} R _ 1 ( p _ 1 , \\cdots , p _ s ) : = & - 2 ( 2 ^ { p _ 2 - p _ 1 } - 2 ) \\left [ ( 1 + 2 ^ { p _ 3 - p _ 2 } + \\cdots + 2 ^ { p _ s - p _ 2 } ) + 2 ^ { 1 + p _ 1 - p _ 2 } \\Delta _ { 2 ^ { p _ 2 } + \\cdots + 2 ^ { p _ s } } \\right ] \\\\ & + 6 \\Delta _ { 2 ^ { p _ 1 } + \\cdots + 2 ^ { p _ s } } - 2 ^ { 3 + p _ 1 - p _ 2 } \\Delta _ { 2 ^ { p _ 2 } + \\cdots + 2 ^ { p _ s } } \\\\ \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} \\boldsymbol { \\mathcal { F } } ^ { ( g ) } \\triangleq \\sum _ { l = 0 } ^ { L _ g - 1 } \\mathbf { R } ^ { ( g ) } _ { c o d e } ( l ) \\otimes \\mathbf { S N R } ^ { ( g ) } _ { m i m o } ( l ) . \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} u ( t , x , y + h \\pi _ { t } ) = \\int _ { 0 } ^ { t } [ D ^ { 2 } _ { x } u ( s , x , y + h \\pi _ { s } ) + f ( s , x , y ) ] \\ , d s + \\int _ { ( 0 , t ] } g ( s , x , y ) \\ , d \\pi _ { s } \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} b _ 1 '' ( u ) = b _ 1 ( u _ 0 ) - \\int _ { u _ 0 } ^ u \\frac { d w ' + \\sum _ { j = 2 } ^ n a _ j ' d b _ j ' } { a _ 1 ' } , u \\in S _ 1 ; \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} x ( \\zeta ) = \\beta \\log { \\frac { \\zeta + i \\sqrt { r } } { \\zeta - i \\sqrt { r } } } - \\overline { \\beta } \\log { \\frac { 1 - i \\sqrt { r } \\zeta } { 1 + i \\sqrt { r } \\zeta } } \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} ( \\bold { D } _ F ^ { \\vartriangle } ) ^ { \\ast } _ { \\alpha \\beta } = \\sum _ { J } S _ { - J } \\cdot \\frac { \\partial F _ { \\beta } } { \\partial u _ J ^ { \\alpha } } . \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} d ^ 2 ( - 1 ) ^ k ( u ^ 2 t _ { k + 1 } + 2 r u s _ { k + 2 } - r ^ 2 t _ { k + 2 } ) = - 1 9 9 6 c + 4 0 0 8 . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} \\sum _ { m _ 1 , m _ 2 , m _ 3 = 0 } ^ { p - 1 } \\binom { m _ 1 + m _ 2 + m _ 3 } { m _ 1 , m _ 2 , m _ 3 } & = C T \\frac 1 { ( x y z ) ^ { p - 1 } } \\prod _ { c y c } \\frac { ( x + y + z ) ^ p - x ^ p } { y + z } \\\\ & = C T \\frac 1 { ( x y z ) ^ { p - 1 } } \\prod _ { c y c } \\left ( ( y + z ) ^ { p - 1 } + \\sum _ { i = 1 } ^ { p - 1 } \\binom { p } i x ^ { p - 1 } ( y + z ) ^ { i - 1 } \\right ) . \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} \\hat { r } = \\frac { \\sigma ( r ) - r } { t ^ 2 } = y _ 0 \\frac { ( \\sigma ( t ) - t ) } { t ^ 2 } + y _ 2 \\frac { \\sigma ( t ) - t } { t ^ 2 } ( \\sigma ( t ) + t ) \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\{ | | \\mathbf { g } _ { d k } | | ^ { 2 } \\right \\} / n _ { d } = 1 , \\mathbb { E } \\left \\{ | | \\mathbf { G } | | ^ { 2 } \\right \\} / n _ { t } = n _ { r } ( K | \\alpha | ^ { 2 } + 1 ) . \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} M ^ { \\dagger } = ( I + L ^ { \\ast } ) ( I + L L ^ { \\ast } ) ^ { - 1 } Y ^ { - 1 } N ^ { \\dagger } X ^ { - 1 } ( I + R ^ { \\ast } R ) ^ { - 1 } ( I + R ^ { \\ast } ) , \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{gather*} d \\xi ^ a = \\xi ^ { a b } h _ b , d \\xi ^ { a b } = 0 . \\end{gather*}"}
-{"id": "4888.png", "formula": "\\begin{align*} S _ 2 & = \\binom { p - 1 } { p ' } \\sum _ { a = 0 } ^ { p ' } \\binom { p ' + a } { p ' } ^ 2 \\\\ & \\equiv _ { p ^ 2 } ( - 1 ) ^ { p ' } \\sum _ { a = 0 } ^ { p ' } ( - 1 ) ^ a \\binom { p ' } { a } \\binom { p ' + a } { a } \\left ( 1 + p ( 2 H _ { 2 a } - H _ a - H _ { p ' } ) \\right ) \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} \\mu \\{ \\underline b ' : \\sigma ^ n \\underline a \\wedge \\underline b ' = \\phi \\} \\geq 1 - \\max _ { i \\in \\Lambda } \\mu [ i ] \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} \\pi ^ * \\bar \\nabla ^ { \\pi ^ { - 1 } T Y } _ { v } D \\pi w = ( \\mathcal L ^ { \\bar \\nabla } _ v D \\pi ) w + D \\pi \\mathcal L _ v w , \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} \\gamma \\left ( a + 1 , z \\right ) & = a \\gamma \\left ( a , z \\right ) - z ^ { a } \\exp \\left ( - z \\right ) a , z > 0 . \\\\ \\Gamma \\left ( a + 1 , z \\right ) & = a \\Gamma \\left ( a , z \\right ) + z ^ { a } \\exp \\left ( - z \\right ) a , z > 0 . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\begin{aligned} d ( \\xi ) : & = ( 1 2 c _ 4 - 2 4 \\ , c _ 1 ^ 2 ) \\ , M _ { \\varphi , x } [ \\overline { v } _ x ^ 2 ] + \\varepsilon ^ 2 ( 2 c _ 6 - \\frac { 8 } { 3 } c _ 2 ^ 2 ) M _ { \\varphi , x } [ \\overline { v } ^ 2 ] \\\\ & = ( 2 4 c _ 4 - 4 8 c _ 1 ^ 2 ) v _ 3 \\cdot \\xi + ( 4 c _ 6 - \\frac { 1 6 } { 3 } c _ 2 ^ 2 ) v _ 1 \\cdot \\xi \\end{aligned} \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} W ( F ) = \\frac { 1 } { 2 } | F | ^ 2 + h ( \\det F ) \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} \\| x - f _ t ( x ) \\| & = \\| x - x _ 0 - ( f _ t ( x ) - f _ t ( x _ 0 ) ) \\| \\\\ & \\ge r - \\| f _ t ( x ) - f _ t ( x _ 0 ) \\| \\\\ & \\ge r - ( 1 - t ) \\| x - x _ 0 \\| = t r > 0 , \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} A ( r , s + t ) \\cap A ( r , s ) & = A ( r , s ) \\subset B ( r , s , t ) , \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} E _ n = \\omega \\left ( 2 n + \\frac { 1 } { 2 } \\sqrt { 1 + \\frac { 4 p ^ 2 } { k ^ 2 } } + 1 \\right ) , \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} S ^ { s t a i r } ( \\ell ) & \\le S ( 3 ; 0 , 0 ) S ( 3 ; 2 , 2 ) ^ { \\ell - 1 } \\\\ & \\le S ( 3 ; 0 , 0 ) S _ U ( 3 ; 2 , 2 ) ^ { \\ell - 1 } = : S _ U ^ { s t a i r } ( \\ell ) \\\\ & \\approx 1 0 ^ { 2 1 . 8 2 4 1 + 1 1 . 2 0 3 4 ( \\ell - 1 ) } . \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\mathrm { c m } ( x , y ) = \\frac { f _ x ( y ) } { | | f _ x | | _ * | | y | | } , \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} n _ { \\textnormal { b i t } } = n _ 1 + n _ 2 . \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ s \\Bigl ( \\int _ { 0 } ^ { \\infty } ( \\cosh ( x u ) - 1 ) d F _ 1 ( x ) \\Bigr ) d u = \\int _ { 0 } ^ { \\infty } \\Bigl ( \\int _ 0 ^ s ( \\cosh ( x u ) - 1 ) d u \\Bigr ) d F _ 1 ( x ) = \\\\ = s \\int _ { 0 } ^ { \\infty } \\Bigl ( \\frac { \\sinh ( s x ) } { s x } - 1 \\Bigr ) d F _ 1 ( x ) \\ge s \\int _ { A } ^ { \\infty } \\Bigl ( \\frac { \\sinh ( s x ) } { s x } - 1 \\Bigr ) d F _ 1 ( x ) \\ge \\\\ \\ge s \\ , \\Bigl ( \\frac { \\sinh ( s A ) } { s A } - 1 \\Bigr ) ( 1 - F _ 1 ( A ) ) . \\end{aligned} \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} t = \\frac { 1 + T N ^ { - 1 / 3 } } { 2 } , r = \\sqrt N + \\frac { R N ^ { - 1 / 6 } } { 2 } , h = \\sqrt N + \\frac { ( R + H ) N ^ { - 1 / 6 } } { 2 } \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} [ D ( 2 , n + p ) : D ( 3 , n ) ] _ q = q ^ { p ( 4 s + r - \\tilde r + \\lceil \\frac { p } { 2 } \\rceil ) + _ 2 ( p ) ( r ' + \\tilde r - \\lceil \\frac { p } { 2 } \\rceil ) } [ D ( 2 , n + p ) : D ( 3 , n ) ] _ q ^ w . \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} M ( s ) = \\sum _ { m \\le M } \\rho ( m ) g ( m ) m ^ { - s } \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} \\widehat a _ \\hbar ( s , \\tau ) = \\tilde a ( s , h ^ { \\frac { 1 } { 2 } } \\tau ) \\ , , \\widehat g _ \\hbar ( s , \\tau ) = \\tilde g ( s , h ^ { \\frac { 1 } { 2 } } \\tau ) \\ , . \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{align*} A ( \\varphi ) : = ( A _ { j _ 1 } ^ { j _ 2 } ( \\varphi ) ) _ { j _ 1 , j _ 2 \\in \\mathbb { Z } } , A _ { j _ 1 } ^ { j _ 2 } ( \\varphi ) : = \\sum _ { l \\in \\mathbb { Z } ^ { \\nu } } A _ { j _ 1 } ^ { j _ 2 } ( l ) \\ , e ^ { \\mathrm { i } \\ , l \\cdot \\varphi } \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} \\tau _ { \\varphi _ p } \\Big ( \\sum _ { i = 1 } ^ n \\xi _ i \\Big ) ^ r \\le \\sum _ { i = 1 } ^ n \\tau _ { \\varphi _ p } ( \\xi _ i ) ^ r \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} \\delta ( x , A ) = \\sup \\{ \\phi ( x ) \\mid \\phi \\in \\mathcal { S } , \\forall a \\in A , \\phi ( a ) = 0 \\} . \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} C ( x ; \\boldsymbol { k } ) & : = \\sum _ { n _ 1 \\cdots n _ l \\leq x } c ( \\boldsymbol { k } , \\boldsymbol { n } ) \\\\ & = \\frac { x } { \\log x } \\left \\{ \\sum _ { 0 \\leq j \\leq N } \\frac { Q _ { j , \\boldsymbol { k } } ( \\log \\log x ) } { ( \\log x ) ^ j } + O _ A \\ ( \\frac { ( \\log \\log x ) ^ k } { k _ 1 ! \\cdots k _ l ! } R _ N ( x ) \\ ) \\right \\} , \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} C _ { \\lambda } ( x ^ { - 1 } , q ^ { - 1 } ) = x ^ { - d _ 1 } q ^ { - d _ 2 } C _ { \\lambda } ( x , q ) , \\end{align*}"}
-{"id": "1513.png", "formula": "\\begin{align*} e ^ { i t A } f ( x ) = e ^ { t n / 2 } f ( e ^ { t } x ) , \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} Q _ { [ [ H , i A ] , i A ] } ( f , g ) = \\left . \\frac { d } { d t } Q _ { [ H , i A ] } ( e ^ { i t A } f , e ^ { i t A } g ) \\right | _ { t = 0 } . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} \\left . \\frac { d } { d t } L _ t \\right | _ { t = 0 } = \\tilde { S } - i \\epsilon \\tilde { S } ^ { \\prime } . \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} L ( x ) = \\sum _ { \\substack { d < x ^ { 1 / 3 } \\\\ ( d , h n ) = 1 } } \\frac { \\gamma _ d } { \\varphi \\bigl ( [ d , q ] \\bigr ) } = \\frac { 1 } { \\varphi ( q ) } \\sum _ { c | q ^ { \\infty } } \\gamma _ c \\frac { ( c , q ) } { c } \\sum _ { \\substack { d < x ^ { 1 / 3 } / c \\\\ ( d , h n q ) = 1 } } \\frac { \\gamma ( d ) } { \\varphi ( d ) } \\ . \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} w _ i ( A ) = \\left \\{ \\begin{array} { l l } 1 , & 1 \\le i \\le c _ 2 n \\\\ n ^ 2 - 2 n , & c _ 2 n < i \\le n ^ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} A _ n ( x ) = A ( x + ( n - 1 ) \\alpha ) \\cdots A ( x ) \\ \\ \\mathrm { f o r } \\ n \\geq 0 . \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} 2 \\eta ( E ) - 2 \\eta ( E \\cap x E ) = \\eta ( E \\triangle x E ) = 1 - 2 \\int 1 _ { E } ( t ) 1 _ { E ^ { - 1 } } ( t ^ { - 1 } x ) d \\eta ( t ) \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} E _ { q ^ { \\lambda } _ { \\theta } } \\left [ x _ { j _ 1 } x _ { j _ 2 } x _ { j _ 3 } x _ { j _ 4 } \\right ] = S _ { j _ 1 j _ 2 } S _ { j _ 3 j _ 4 } + S _ { j _ 1 j _ 3 } S _ { j _ 2 j _ 4 } + S _ { j _ 2 j _ 3 } S _ { j _ 1 j _ 4 } . \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} r = \\frac { \\sqrt { c ^ 3 - 2 c - 1 0 } } { \\sqrt { 3 c ^ 3 - 1 2 c } } . \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} \\lim _ { t \\to \\tau } \\int _ { \\overline { \\Omega } } \\abs { k ( t , s ) - k ( \\tau , s ) } g _ r ( s ) \\textup d s = 0 ; \\end{align*}"}
-{"id": "2262.png", "formula": "\\begin{align*} ( W _ { 2 } ) = \\frac { \\mu \\phi ( m _ { 1 } ) \\phi ( m _ { 2 } ) } { \\langle \\phi ( m _ { 1 } ) , \\phi ( m _ { 2 } ) \\rangle _ { B _ { 2 } } } = \\frac { \\mu \\phi ( m _ { 1 } m _ { 2 } ) } { \\langle \\phi ( m _ { 1 } ) , \\phi ( m _ { 2 } ) \\rangle _ { B _ { 2 } } } . \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} f ( q ) & = 1 + \\sum _ { n = 1 } ^ { \\infty } \\frac { q ^ { n ^ { 2 } } } { ( 1 + q ) ^ { 2 } ( 1 + q ^ { 2 } ) ^ { 2 } \\cdots ( 1 + q ^ { n } ) ^ { 2 } } = 1 + \\sum _ { n = 1 } ^ { \\infty } a _ { f } ( n ) q ^ { n } \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} r _ s ^ q ( \\theta ) : = \\sum _ { n = 0 } ^ { \\infty } \\sum _ { c _ 1 \\cdots c _ n } ( \\mu [ c _ 1 \\cdots c _ n ] ) ^ q ( \\lambda _ { c _ 1 \\cdots c _ n } ( \\theta ) ) ^ { - s ( q - 1 ) } . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} \\frac { \\mu _ 3 ^ 2 } { t _ 3 } & = \\frac { \\lambda ^ 2 v _ 2 ( y - Z _ 3 ) ^ 2 } { 1 + \\lambda v _ 2 } \\\\ \\frac { \\mu _ 4 ^ 2 } { t _ 4 } & = \\frac { ( 1 - \\lambda \\rho ) ^ 2 ( y - Z _ 3 ) ^ 2 v _ 1 } { 1 + ( 1 - \\lambda \\rho ) v _ 1 } . \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{align*} \\space M ( W _ i ) = M ( W _ j ) . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} \\int _ \\Omega p ( t ) \\ , d t = \\sum _ { i = 1 } ^ N \\omega _ i p ( \\tau _ i ) . \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} P \\sp { ( \\alpha , \\alpha ) } _ k = \\frac { ( k + 2 \\alpha + 1 ) ( k + 2 \\alpha + 2 ) } { ( 2 k + 2 \\alpha + 1 ) ( 2 k + 2 \\alpha + 2 ) } P \\sp { ( \\alpha + 1 , \\alpha + 1 ) } _ k - \\frac { k + \\alpha } { 2 ( 2 k + 2 \\alpha + 1 ) } P \\sp { ( \\alpha + 1 , \\alpha + 1 ) } _ { k - 2 } , \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} E _ { 1 } ^ { \\ast } = E _ { 1 } , E _ { 3 } ^ { \\ast } & = E _ { 2 } , E _ { 4 } ^ { \\ast } = E _ { 4 } , F _ { 1 } ^ { \\ast } = F _ { 1 } , F _ { 2 } ^ { \\ast } = F _ { 3 } , F _ { 4 } ^ { \\ast } = F _ { 4 } , \\\\ & E _ { 2 } E _ { 3 } + E _ { 4 } ^ { 2 } = E _ { 4 } , F _ { 2 } F _ { 3 } + F _ { 4 } ^ { 2 } = F _ { 4 } . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} D _ t ^ \\delta E _ { \\delta } ( t ^ \\delta ) = E _ { \\delta } ( t ^ \\delta ) \\ t > 0 . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} [ k _ n : k ] ( 2 g _ n - 2 ) = & p ^ { n } ( 2 g _ 0 - 2 ) + \\deg _ k \\mathfrak { D } _ n . \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} V _ { 1 , 2 } ( r ) = W _ m ^ 2 ( r ) \\pm W _ m ' ( r ) \\ ; , \\end{align*}"}
-{"id": "47.png", "formula": "\\begin{align*} \\lambda _ { \\ell , j } ^ k = \\begin{cases} 1 , & \\ell = j , \\ , k = 0 ; \\\\ 0 , & k + \\ell < j \\ , \\mathrm { o r } \\ , \\ell > j , \\\\ s _ { \\ell } \\lambda _ { \\ell , j - 1 } ^ { k - 1 } + v _ { \\ell } \\lambda _ { \\ell , j } ^ { k - 1 } ; & \\mathrm { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} i _ { f , \\mathsf { k } } \\eta _ { f , \\mathsf { k } } = \\mathrm { R } ( \\theta _ \\mathsf { k } ) ^ \\top \\eta _ { c , \\mathsf { k } } + \\mathrm { R } ( \\theta _ \\mathsf { k } ) \\eta _ { s a , \\mathsf { k } } , \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { d } { d t } H ( \\rho _ { t } | M _ { \\Omega _ { \\rho _ { t } } } ) = - I ( \\rho _ { t } | M _ { \\Omega _ { \\rho _ { t } } } ) . \\end{aligned} \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} U _ { 2 ^ n m + 1 } = 2 ^ { - n } U _ { m + 1 } + 4 ( 1 - 2 ^ { - n } ) U _ m + 1 - 2 ^ { - n } . \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\widetilde { W } ( y | x , \\sigma ) = \\begin{cases} 1 & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} S _ { \\rho } : = \\{ z \\in \\mathbb { C } \\mid - \\rho \\leq \\Re ( z ) \\leq 1 + 2 \\rho \\ : \\ : \\ : \\ : \\ : \\ : - 2 \\rho \\leq \\Im ( z ) \\leq 2 \\rho \\} . \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { \\infty } \\sum _ { k _ i = 0 } ^ { b - 1 } f ( k _ i , i ) = \\sum _ { k = 0 } ^ { \\infty } \\prod _ { i = 0 } ^ { \\infty } f ( k _ i , i ) . \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\hat { \\omega } ^ 0 ( t ) : = \\frac { 1 } { \\sigma } \\left ( \\psi ( t ) - x - \\int _ 0 ^ t b ( \\psi ( s ) ) \\ , d s \\right ) \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} V _ h ( x ) = \\sum _ { \\substack { q < y \\\\ ( q , h ) = 1 } } \\xi _ q \\sum _ { \\substack { d < x ^ { 1 / 3 } \\\\ ( d , h ) = 1 } } \\gamma ( d ) \\sum _ { \\substack { n \\equiv - h ( [ d , q ] ) \\\\ ( n , h ) = 1 } } \\theta ( n ) a ( n ) \\ , \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} x ^ { ( t + 1 ) } _ j = - \\min \\left ( \\frac { \\eta } { \\| s _ { t + 1 } ^ { x _ j } \\| _ { 2 , * } } , \\frac { 1 } { \\beta _ { t + 1 } ( 1 - \\tau ) } \\right ) s _ { t + 1 } ^ { x _ j } . \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} L _ { b c } = ( I + \\mathcal { U } ) ( I + \\mathcal { V } ) ( \\widetilde { L } _ { b c } ^ 0 - J _ { m , b c } X _ * ) ( I + \\mathcal { V } ) ^ { - 1 } ( I + \\mathcal { U } ) ^ { - 1 } . \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} s = 4 \\left ( \\sum _ { i = 1 } ^ n \\tilde \\tau _ i \\right ) \\cdot \\log \\left ( \\frac { 4 d } { \\delta } \\right ) \\cdot \\frac { 1 } { \\epsilon ^ 2 } \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\eta _ R ( y ) : = \\eta \\left ( \\frac { | y | } { R } \\right ) . \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} | A | = \\frac { \\sqrt { \\lambda } } { 2 \\sqrt { 2 } \\gamma \\pi L _ 2 } \\sqrt { \\frac { 1 + \\gamma \\hat { \\tilde { V } } _ { 2 , 0 } } { | \\hat { \\tilde { V } } _ { 1 , 0 } | ( 2 \\hat { \\tilde { V } } _ { 2 , 0 } - \\hat { \\tilde { V } } _ { - 1 , 0 } ) } } . \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} d \\Phi _ 0 ( x ) = p \\Leftrightarrow d \\Phi _ 0 ( \\gamma ( x ) ) = \\gamma . p \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } { \\eta _ { k } } ( \\hat { \\theta } _ { j } ) = p _ { j } { \\eta _ { k } } ( \\theta ) + { \\eta _ { k } } ( \\xi _ { j } ) , \\ \\forall 1 \\leq j \\leq k - 1 , \\\\ { \\eta _ { k } } ( \\hat { \\theta } _ { k } ) = p _ { k } { \\eta _ { k } } ( \\theta ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} \\theta ^ { w } ( 0 ) = 0 , \\xi ^ { w } ( 0 ) = 0 . \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} \\phi ^ * ( p ) : = - q + \\sup _ { x \\in M } q ( x ) - \\phi ( x ) = - q + \\sup _ { x \\in M } - \\Phi _ q ( x ) , \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} U ^ { - 1 } \\psi ( A ) ^ 2 U = \\begin{bmatrix} O & C ^ { * } \\\\ O & \\psi ( D ) ^ 2 \\end{bmatrix} , \\end{align*}"}
-{"id": "1652.png", "formula": "\\begin{align*} q ( x ) = \\Phi \\big ( g ( x ) \\big ) \\ , , \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} \\mathcal { H } ^ j : = H ^ * ( Z _ j ; S _ T ) \\left [ z , \\frac { 1 } { z } \\right ] \\left [ \\mathfrak S \\right ] [ \\ ! [ Q ] \\ ! ] , \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} \\mathbb { P } ( \\bigcap _ i \\mathcal { A } _ i ) = \\prod _ { i = 1 } ^ { n - 1 } \\mathbb { P } ( \\mathcal { A } _ { i } | \\bigcap _ { j < i } \\mathcal { A } _ j ) , \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} { \\bf u } ^ { * } _ { i j k } = \\frac { ( { \\bf F } _ { i j k } ) ^ { - 1 } { \\bf G } _ { i j } { \\bf v } _ { j k } } { \\norm { ( { \\bf F } _ { i j k } ) ^ { - 1 } { \\bf G } _ { i j } { \\bf v } _ { j k } } } , \\forall \\{ i , j \\} \\in \\mathcal { L } , \\ \\forall k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} \\rho ( X , Y ) = \\rho _ f ( X , Y ) = ( Y - f ( X ) ) ^ 2 - ( Y - f ^ { \\star } ( X ) ) ^ 2 . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} \\bar c ^ \\varepsilon = \\inf \\left \\{ c \\geq 0 , \\exists \\lambda > 0 , \\mu _ { c , \\varepsilon } ( \\lambda ) = 0 \\right \\} , \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\Pr [ \\mathrm { r a n k } \\ , \\mathbf { X } = k ] = \\sum _ { i = \\max ( 0 , k - b ) } ^ { \\min ( a , k ) } \\mathbb { P } _ { i } ( a , k ) \\mathbb { P } ( b , k - i ) . \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} V = \\oplus _ { 0 \\leq p \\leq k } V _ p , \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\Delta ( F _ 1 ) = - 2 ^ 7 \\ell ^ 2 \\Delta ( F _ 2 ) = 2 ^ 8 \\ell . \\end{align*}"}
-{"id": "7797.png", "formula": "\\begin{align*} a _ i = \\| f ^ { ( i ) } \\| _ { \\mathbb { F } _ q ^ { ( i ) } ( \\mathcal { H } _ - ) } , b _ { n - i } = \\| g ^ { ( n - i ) } \\| _ { \\mathbb { F } _ q ^ { ( n - i ) } ( \\mathcal { H } _ - ) } . \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} \\bar { \\hat W } ( - \\xi ) = e ^ { 2 \\xi } \\hat W ( \\xi ) , \\bar { \\hat Q } ( - \\xi ) = e ^ { 2 \\xi } \\hat Q ( \\xi ) . \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} g = - \\tau ^ 2 ( d x ^ 0 ) ^ 2 + \\overline { g } _ { i j } d x ^ i d x ^ j . \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} q = \\bigg \\lceil \\frac { \\displaystyle 1 } { \\displaystyle \\delta } \\bigg \\rceil . \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} | p ( t ) | = \\left | \\int _ { - 1 } ^ t \\dot { p } ( \\tau ) \\ ; d \\tau \\right | = \\left | \\sum _ { i = 1 } ^ N \\dot { p } ( \\tau _ i ) \\int _ { - 1 } ^ t l _ i ( \\tau ) \\ ; d \\tau \\right | \\le \\sum _ { i = 1 } ^ N \\left | \\int _ { - 1 } ^ t l _ i ( \\tau ) \\ ; d \\tau \\right | . \\end{align*}"}
-{"id": "2056.png", "formula": "\\begin{align*} b = - \\frac { 2 ^ 7 \\tilde { c } _ 6 } { 3 ^ 3 2 ^ 5 } = - \\frac { 4 \\tilde { c } _ 6 } { 2 7 } \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} \\mathrm { R e } \\left ( G ( 2 , \\Lambda _ N ) \\right ) = \\left \\{ \\begin{matrix} 2 & \\textrm { i f } & N \\equiv 1 \\pmod 2 \\\\ 0 & \\textrm { i f } & N \\equiv 0 \\pmod 4 \\\\ 4 & \\textrm { i f } & N \\equiv 2 \\pmod 4 , \\end{matrix} \\right . \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} G = \\langle \\ , x _ 1 , x _ 2 , x _ 3 \\mid x _ 3 ^ { - 1 } \\ , v \\ , x _ 3 = v \\ , \\rangle , \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} d ( x ) = \\sqrt { \\frac { m ! } { H ( 2 H - 1 ) } } x ^ H L ( x ) ^ m . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} | d \\rho ( 0 ) | = O ( \\mu ) . \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} u = h _ 0 + h _ 1 + \\cdots + h _ I + \\bar { u } , \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} M \\cdot V = \\left ( \\begin{array} { c } T ( z _ { a } , s ) \\\\ T ( z _ { a ^ 2 } , s ) \\\\ T ( z _ { a ^ 3 } , s ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} \\frac { s ( \\theta ) } { \\tilde { s } ( \\theta ) } = e ^ { g ( \\theta + \\alpha ) - g ( \\theta ) } . \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} P ( 0 , x _ 0 ; t , y ) \\le \\frac { \\pi ^ { ( t ) } ( y ) } { \\pi ^ { ( 0 ) } ( x _ 0 ) } \\Big [ \\frac { 2 \\pi ^ { ( 0 ) } ( x _ 0 ) } { v ( s ) } + \\frac { \\pi ^ { ( 0 ) } ( x _ 0 ) L _ t } { \\pi ^ { ( t ) } ( y ) ^ \\alpha } \\Big ] = \\frac { 2 \\pi ^ { ( t ) } ( y ) } { v ( s ) } + \\pi ^ { ( t ) } ( y ) ^ { 1 - \\alpha } L _ t . \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} H = \\begin{bmatrix} 1 & h ( z ) \\\\ 0 & 1 \\end{bmatrix} , h ( z ) = \\frac { \\sqrt { 2 \\pi } \\left ( a ^ 2 - 1 \\right ) ^ c } { \\sqrt N \\eta ^ 2 ( z ) a \\ , \\Gamma ( c ) } \\frac { 1 } { z - \\beta } - \\frac { 1 } { \\zeta ( z ) } \\frac { \\sqrt { 2 \\pi } e ^ { { \\mathrm i } \\pi c } } { \\Gamma ( c ) } = { { \\cal O } \\left ( \\frac { c } { \\sqrt N } \\right ) . } \\end{align*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n } \\left ( x ; S ^ { ( j ) } \\right ) = m \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} h ( w ) = \\frac { \\delta n } { 2 s _ n } \\log ( 1 + w P ' / 4 ) - \\frac { \\ell _ n } { s _ n } H _ 2 \\left ( \\frac { w } { \\ell _ n } \\right ) . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} \\tilde { c } _ 4 = 1 \\cdot 2 ^ 0 + 1 \\cdot 2 + \\alpha _ 1 \\cdot 2 ^ 2 + O ( 2 ^ 3 ) \\alpha _ 1 \\in \\{ 0 , 1 \\} ; \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} L ^ + _ 0 \\oplus G _ 0 & \\cong L ^ + \\oplus H _ 0 , \\\\ L ^ - _ 0 \\oplus G _ 0 & \\cong L ^ - \\oplus H _ 0 , \\\\ L ^ + _ 1 \\oplus G _ 1 & \\cong L ^ + \\oplus H _ 1 , \\\\ L ^ - _ 1 \\oplus G _ 1 & \\cong L ^ - \\oplus H _ 1 . \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} c ( x ) = \\sum _ { j = 0 } ^ { p ^ k - 1 } x ^ { j p ^ { e - k } } a ( x ) . \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} M = \\Q _ 2 ( \\sqrt { A } ) = \\Q _ 2 ( \\sqrt { B } ) . \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} ( ( \\xi + \\eta ) \\tanh ( \\xi + \\eta ) - \\xi \\tanh \\xi - \\eta \\tanh \\eta ) B ^ h = 2 \\tanh \\xi \\tanh \\eta C ^ h . \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} F _ { \\ell } ( z ) : = \\frac { \\eta ^ { \\ell ^ 2 } ( z ) } { \\eta ( \\ell ^ 2 z ) } \\in M _ { \\frac { \\ell ^ 2 - 1 } { 2 } } ( \\Gamma _ 0 ( \\ell ^ 2 ) ) . \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} a ^ { \\lambda , m , n } = \\frac { 2 - n } { 1 + m - n } + \\frac { 2 } { 1 + m - n } \\lambda , b ^ { \\lambda , m , n } = \\frac { 1 - m } { 1 + m - n } + \\frac { 1 - m + n } { 1 + m - n } \\lambda . \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} \\sharp ( T N ^ { \\circ } ) = T N \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} | \\sum _ { k = k _ { j - 1 } + 1 } ^ { k _ { j } } < x ^ { * } _ { n _ { j } } - x ^ { * } _ { 0 } , e _ { k } > | > \\frac { \\epsilon _ { 0 } } { 2 } , j = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} \\partial _ s P _ s f ( x ) = \\frac { 1 } { 2 } \\partial ^ 2 _ x P _ s f ( x ) , \\ 0 < s < 1 , x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} G ( \\mathcal { W } , \\mathcal { B } ) = \\left ( { \\bf e } _ 1 , { \\bf e } _ 2 , \\cdots , { \\bf e } _ k \\ \\left | \\right . \\ { \\bf p } _ 1 ^ { B _ { 1 , 1 } } , \\ldots , { \\bf p } _ 1 ^ { B _ { 1 , | \\mathcal { B } _ 1 | } } , { \\bf p } _ 2 ^ { B _ { 2 , 1 } } , \\ldots , { \\bf p } _ u ^ { B _ { u , 1 } } , \\ldots , { \\bf p } _ u ^ { B _ { u , | \\mathcal { B } _ u | } } , { \\bf p } _ { u + 1 } , \\ldots , { \\bf p } _ { n - k } \\right ) \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} \\zeta _ 0 = \\zeta \\colon B _ { \\sigma } \\stackrel { \\cong } { \\longrightarrow } B _ { \\lambda } \\subseteq ( C ( \\R / M \\Z ) \\otimes A ) \\rtimes _ { \\lambda \\otimes \\alpha } \\R \\ , . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } z _ 1 = T _ 0 g ( z _ 1 ) + T _ 1 g ( z _ 2 ) + T _ 2 g ( z _ 3 ) \\\\ z _ 2 = T _ 2 g ( z _ 1 ) + T _ 0 g ( z _ 2 ) + T _ 1 g ( z _ 3 ) \\\\ z _ 3 = T _ 1 g ( z _ 1 ) + T _ 2 g ( z _ 2 ) + T _ 0 g ( z _ 3 ) \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} \\{ \\xi _ \\frac 1 2 Z _ { | g | , \\psi _ { [ g ] } } , Z _ { | g | , \\psi _ { [ g ] } } \\} \\overset { . } = c _ { Z _ { | g | , \\psi _ { [ g ] } } } ^ - ( 3 ) , \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} \\beta ^ { ( 1 ) } _ { \\alpha , \\epsilon _ 1 } & = ( \\alpha - 1 ) \\big ( I ( U ; W ) + \\epsilon _ 1 - d _ { \\alpha } ( p _ { U , W } , p _ U p _ W ) \\big ) , \\\\ \\beta ^ { ( 2 ) } _ { \\alpha , \\epsilon _ 2 } & = ( \\alpha - 1 ) \\big ( I ( U , V ; W ) + \\epsilon _ 2 - d _ { \\alpha } ( p _ { U , V , W } , p _ { U , V } p _ W ) \\big ) , \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} 1 = [ x , b ( x ) ] = [ x , \\lambda ( \\alpha x + y ) ] = \\lambda [ x , y ] \\ \\mathrm { a n d } \\\\ 1 = [ y , b ( y ) ] = [ y , \\sigma ( \\beta y + x ) ] = \\sigma [ y , x ] , \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} S ( 3 ; 0 , 0 ) & = 6 , 6 7 0 , 9 0 3 , 7 5 2 , 0 2 1 , 0 7 2 , 9 3 6 , 9 6 0 \\\\ & \\approx 6 . 6 7 0 9 \\times 1 0 ^ { 2 1 } \\\\ & \\le S _ U ( 3 ; 0 , 0 ) \\approx 1 . 7 0 7 1 \\times 1 0 ^ { 2 6 } . \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} f ^ \\gamma ( x ) = \\sum _ { n = 1 } ^ \\infty 2 ^ { - \\gamma n } \\cos ( 2 ^ n \\pi x ) , \\quad \\gamma \\in ( 0 , 1 ) , \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} ( n - 1 ) r ^ { t - 1 } + \\frac { r ^ { t - 1 } } { t } ( n - 1 ) = \\left ( 1 + \\frac { 1 } { t } \\right ) ( n - 1 ) r ^ { t - 1 } = \\left ( 1 + \\frac { 1 } { t } \\right ) \\left ( \\frac { n - 1 } { n - k } \\right ) \\cdot \\ell , \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\varphi ( n ) = \\frac 1 { \\pi ^ 2 } x ^ 2 + O \\left ( x ( \\log x ) ^ { 2 / 3 } ( \\log \\log x ) ^ { 4 / 3 } \\right ) . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} \\sum _ k k P ( K _ n , k ) = \\sum _ k \\frac { k 2 ^ { n - 2 k } { n - 1 \\choose n - 2 k , k , k - 1 } } { { 2 n - 2 \\choose n } } = \\frac { n ( n - 1 ) } { 4 n - 6 } . \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} E _ r : = ( 3 , \\underbrace { 1 , \\ldots , 1 } _ { r } ) ^ { \\bot } \\subset ( 1 ) \\oplus ( - 1 ) ^ r . \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} b . ( g ' , \\theta , r , s , i , j ) = ( g ' b ^ { - 1 } , b \\theta b ^ { - 1 } , b r b ^ { - 1 } , b s b ^ { - 1 } , i , j ) . \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} W ^ { \\star } ( z , w ) = W _ + ( z , w ) z ^ { - 1 / 4 } e ( 2 \\sqrt { z } ) + W _ - ( z , w ) z ^ { - 1 / 4 } e ( - 2 \\sqrt { z } ) + O _ A ( C ^ { - A } ) \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} ( C _ { i j } ^ { i j } + C _ { j i } ^ { j i } ) e _ { i i } - ( C _ { i j } ^ { i j } + C _ { j i } ^ { j i } ) e _ { j j } & = ( C _ { i i } ^ { i i } - C _ { i i } ^ { j j } ) e _ { i i } + ( C _ { j j } ^ { i i } - C _ { j j } ^ { j j } ) e _ { j j } \\\\ & + \\sum _ { x \\neq i , j } C _ { x x } ^ { i i } e _ { x x } - \\sum _ { y \\neq i , j } C _ { y y } ^ { j j } e _ { y y } . \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} a _ n ( T : X \\rightarrow Y ) : & = \\inf _ { r a n k A < n } \\sup _ { \\| x | X \\| \\leq 1 } \\| T x - A x | Y \\| \\\\ & = \\inf _ { r a n k A < n } \\| T - A : X \\rightarrow Y \\| , \\ \\ n \\in \\Bbb N _ + . \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} \\exists \\ \\{ \\nu _ { i } \\} _ { i \\in I ( \\bar { x } ) } \\geq 0 \\ \\forall _ { k \\in I ( \\bar { x } ) } & | \\bar { x } _ k - x _ k | ^ { p - 2 } { ( \\bar { x } _ k - x _ k ) } = - \\delta _ k \\nu _ k \\\\ \\forall i \\in J \\backslash I ( \\bar { x } ) & \\bar { x } _ i = x _ i . \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} \\lambda ( d , b ) & = \\min \\{ d _ { S ( C _ 4 , 2 ) } ( a a , a b ) + d _ { S ( C _ 4 , 2 ) } ( a a , d c ) , d _ { S ( C _ 4 , 2 ) } ( c c , a b ) + d _ { S ( C _ 4 , 2 ) } ( c c , d c ) \\} \\\\ & = \\min \\{ 1 + 4 , 5 + 2 \\} \\\\ & = 5 . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} \\left | \\frac { \\partial ( \\phi \\circ F ) } { \\partial x _ k } ( x ) \\right | \\leq C \\ , K \\ , \\sqrt { \\sum _ { j = 1 } ^ n \\left ( \\frac { \\partial f _ j } { \\partial x _ k } ( x ) \\right ) ^ 2 + \\left ( \\frac { \\partial g _ j } { \\partial x _ k } ( x ) \\right ) ^ 2 } \\leq C \\ , K \\ , \\left | \\frac { \\partial F } { \\partial x _ k } ( x ) \\right | . \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} a _ { \\mu } ^ { \\frac 1 3 - \\frac { 1 } { 3 q } } \\mu ^ { - \\frac 1 4 } + a _ { \\mu } ^ { \\frac 1 2 - \\frac 1 q } \\mu ^ { - \\frac 1 2 + \\frac 1 q } = a _ { \\mu } ^ { \\frac 1 3 - \\frac { 1 } { 3 q } } \\mu ^ { - \\frac 1 4 } \\left ( 1 + ( a _ \\mu ^ { \\frac 1 6 } \\mu ^ { - \\frac 1 4 } ) ^ \\frac { q - 4 } { q } \\right ) \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} \\mathcal { Y } : = & \\left [ L ^ { 2 } ( 0 , T ; H ^ { 2 } ) \\cap H ^ { 1 } ( 0 , T ; ( H ^ { 2 } ) ^ { * } ) \\cap C ^ { 0 } ( [ 0 , T ] ; L ^ { 2 } ) \\right ] \\\\ & \\times L ^ { 2 } ( Q ) \\times \\left [ L ^ { \\infty } ( 0 , T ; H ^ { 1 } ) \\cap H ^ { 1 } ( 0 , T ; L ^ { 2 } ) \\right ] . \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} d M | _ { \\mu } = \\lambda ( \\phi - \\phi _ 0 ) + \\log \\rho , \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} [ H ^ 0 _ T ( D _ w , O _ w ^ { * } ) ] = [ H ^ 0 ( \\hat { \\mathbb { Z } } , H ^ 1 ( I _ p , \\hat { T } ) ) ] . \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} a ( F , \\ ( \\begin{smallmatrix} n & r / 2 \\\\ r / 2 & m \\end{smallmatrix} \\ ) ) = \\epsilon a ( F , \\ ( \\begin{smallmatrix} m / N & - r / 2 \\\\ - r / 2 & n N \\end{smallmatrix} \\ ) ) . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} \\mathbb { E } ( \\kappa ( G , B ) ) = \\sum _ { e \\in E ( G ) } \\frac { 1 } { d ' ( e ) + 1 } . \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} \\omega ( t ) : = \\frac { d } { d t } \\Big | _ { t = 0 } B ( t ) = a ^ + ( \\delta _ t ) + a ^ - ( \\delta _ t ) , \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} \\rho ( r ) = \\begin{cases} a , & r \\in [ a , k ] \\\\ \\frac { 1 } { 2 } \\left ( \\left ( a ^ 2 + \\frac { \\omega ^ 2 } { a ^ 2 } \\right ) \\frac { r ^ 2 } { k ^ 2 } + \\left ( a ^ 2 + \\frac { \\omega ^ 2 } { a ^ 2 } \\right ) \\frac { k ^ 2 } { r ^ 2 } + 2 \\left ( a ^ 2 - \\frac { \\omega ^ 2 } { a ^ 2 } \\right ) \\right ) ^ { \\frac { 1 } { 2 } } , & r \\in ( k , b ] \\end{cases} \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} \\mathcal { C } _ i ^ { \\mathrm { E } } = \\cup _ j \\mathcal { V } _ { i , j } ^ { 2 } , \\forall ~ \\mathbf { z } _ j \\in \\Phi _ { \\mathrm b } \\setminus \\mathbf { z } _ { i } . \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} B _ 0 = \\partial _ 1 A ^ 0 _ 2 - \\partial _ 2 A ^ 0 _ 1 \\ , , B = \\partial _ 1 A _ 2 - \\partial _ 2 A _ 1 \\ , , \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} \\frac { \\partial \\gamma } { \\partial t } + \\gamma \\frac { \\partial \\gamma } { \\partial q } = \\frac { k } { q ^ 3 } - \\omega ^ 2 ( t ) q . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} [ y , b ( x ) ] = \\lim _ { t \\rightarrow 0 } \\frac { | | x + t y | | - | | x | | } { t } , \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} \\phi ( x , y ) = a x ^ 3 + b x ^ 2 y + c x y ^ 2 + d y ^ 3 , a , b , c , d \\in A \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} \\rho ( \\mathcal { L } , \\mathcal { M } ) = ( d ( \\ell _ 1 , m _ 1 ) ^ 2 + d ( \\ell _ 2 , m _ 2 ) ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ X ( t , i ) = 0 | X ( t - 1 , i ) = 0 , t \\leq T \\right ] = \\frac { 1 - \\hat { \\pi } ( t ) } { 1 - \\hat { \\pi } ( t - 1 ) } = 1 - \\frac { \\hat { \\pi } ( t ) - \\hat { \\pi } ( t - 1 ) } { 1 - \\hat { \\pi } ( t - 1 ) } . \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} F _ { k + 1 } - F ' _ { k + 1 } = \\partial T _ { k + 1 } . \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} e _ k = \\exp ( i \\pi \\sum _ { j = 0 } ^ s 2 ^ { - p _ j } ) \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} I ( t ) = \\left [ \\begin{array} { c } \\ast \\\\ 0 \\end{array} \\right ] , J ( t ) = \\left [ \\begin{array} { c c } \\ast & t \\ast \\end{array} \\right ] . \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} N _ { 1 } E _ { 1 } + N _ { 3 } E _ { 2 } = 0 , F _ { 1 } N _ { 1 } + F _ { 3 } N _ { 2 } = 0 , F _ { 1 } N _ { 3 } + F _ { 3 } N _ { 4 } = 0 . \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} { \\eta } ( \\theta ) = \\theta + \\eta , \\ { \\eta } ( \\hat { \\theta } _ { j } ) = \\hat { \\theta } _ { j } - \\left [ \\xi _ { j } - p _ { j } \\eta \\right ] \\ \\ { \\eta } ( { \\xi } _ { j } ) = \\left \\{ \\xi _ { j } - p _ { j } \\eta \\right \\} , \\ \\forall 1 \\leq j \\leq k , \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} P ^ i ( \\dd \\omega , \\dd \\sigma ) = \\mathcal { P } ^ i ( \\dd \\omega , \\sigma ) P ^ o ( \\dd \\sigma ) , \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} \\partial _ { t } u = \\Delta u + f \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} \\mathcal { M } ( u ) = \\log \\frac { ( \\omega + i \\partial \\bar \\partial u ) ^ n } { \\omega ^ n } \\in C ^ 0 _ { l o c } ( X ^ { r e g } ) \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} f _ n ( z ) = \\frac { \\alpha _ n + z f _ { n + 1 } ( z ) } { 1 + \\bar \\alpha _ n z f _ { n + 1 } ( z ) } , n = 0 , 1 , 2 , \\dots , \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} \\iota _ m ( q _ { j , i } ) = \\frac { x _ { j - 1 , i } \\ , x _ { j , i + 1 } } { x _ { j - 1 , i + 1 } \\ , x _ { j , i } } X _ { v _ { ( j - 1 , i + 1 ) , ( j , i ) } } . \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} h _ k = h _ k ( x _ 1 , \\dots , x _ n ) : = \\sum _ { 1 \\leq i _ 1 \\leq \\dots \\leq i _ k \\leq n } x _ { i _ 1 } \\cdots x _ { i _ k } , \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} \\Delta _ 1 \\ , \\Delta _ { 4 5 6 } \\ = \\ 0 \\Delta _ 2 \\ , \\Delta _ { 4 5 7 } \\ - \\ \\Delta _ 1 \\ , \\Delta _ { 4 6 7 } \\ = \\ 0 . \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\Vert v u \\Vert _ { D U ^ 2 } \\le & \\ 2 \\Vert v \\Vert _ { V ^ 2 } \\Vert u \\Vert _ { D U ^ 2 } \\\\ \\Vert v u \\Vert _ { D V ^ 2 } \\le & \\ \\Vert v \\Vert _ { D V ^ 2 } \\Vert u \\Vert _ { U ^ 2 } \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} f = X ^ 4 + 2 a X ^ 2 + 4 b X - \\frac { a ^ 2 } { 3 } . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} \\phi _ 1 ( U ^ r ) = ( p ^ { \\lceil \\log _ p \\left ( \\frac { r } { i } \\right ) \\rceil } W ( k ) ) _ i \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} \\mathbf { w } _ { i , ( r ) } = \\mathbf { P } _ { \\rm w } \\left ( \\mathbf { A } ^ { - 1 } \\tilde { \\mathbf { y } } _ i \\right ) - \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 2 \\hat { \\mathbf { d } } _ { i , ( r - 1 ) } , \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} \\begin{aligned} \\psi ( X ( - s , x ) ) & = \\int _ 0 ^ { \\tau _ 1 ( X ( - s , x ) ) } \\Big ( f \\big ( X ( t , X ( - s , x ) ) \\big ) - \\bar f ( X ( - s , x ) ) \\Big ) \\ , d t \\\\ & = \\int _ { - s } ^ { \\tau _ 1 ( x ) } \\Big ( f ( X ( t , x ) ) - \\bar f ( x ) \\Big ) \\ , d t . \\end{aligned} \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N { \\mathsf { R } } ^ { \\lambda } _ { \\bar { \\mathcal { N } } _ i } = \\sum _ { s \\in [ 0 : 1 ] ^ N } \\lambda _ s \\left [ \\sum _ { j = 1 } ^ N \\left ( \\max _ { i \\in \\mathcal { A } ^ { \\star } _ j } \\ell _ { i , s } ^ { \\prime } + \\max _ { i \\in ( [ 1 : N ] \\backslash \\{ j \\} ) \\backslash \\mathcal { A } ^ { \\star } _ j } r _ { i , s } ^ { \\prime } \\right ) \\right ] , \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} K ( u / v ) = \\int _ X ^ \\infty \\left ( A - B \\log { x } / \\log { N } \\right ) h ( x \\sqrt { u / v } ) h ( x \\sqrt { v / u } ) x ^ { - 1 } d x \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} m p ^ a \\le n < ( m + 1 ) p ^ a , a = 1 , 2 , \\ldots \\ , . \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} w _ i ( A ) = \\left \\{ \\begin{array} { l l } 1 , & 1 \\le i \\le c _ 2 n \\\\ n ^ 2 , & c _ 2 n < i \\le n ^ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} J _ { m , b c } X = J _ { b c } ( X - \\mathbb { P } _ { ( m ) } X \\mathbb { P } _ { ( m ) } ) + \\mathbb { P } _ { ( m ) } X \\mathbb { P } _ { ( m ) } , X \\in \\mathfrak { S } _ 2 ( \\mathcal { H } ) , \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} \\mathcal { E } _ n ( k , \\ell ) = \\min ( 1 - F _ n ^ \\ast ( k - 1 ) , F _ n ^ \\ast ( \\ell ) ) - \\max ( 1 - F _ n ^ \\ast ( k ) , F _ n ^ \\ast ( \\ell - 1 ) ) . \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} 4 ( 1 - | y | ^ 2 ) \\frac { ( s - 1 - \\frac { N } { 2 } ) ( 1 - | x | ^ 2 ) ( 1 - | y | ^ 2 ) + ( s - 1 ) | x - y | ^ 2 } { ( 1 - | x | ^ 2 ) | x - y | ^ 2 } . \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} \\tilde { U } ' ( \\hat { x } _ t ) = \\tilde { \\mu } ( \\theta _ t ) \\tilde { U } ' ( \\hat { x } _ { t + 1 } ) f ' ( k _ { t + 1 } ) . \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} U _ { g } \\pi _ { \\omega } ( A ) \\Omega _ { \\omega } = \\pi _ { \\omega } ( \\tau _ { g } ( A ) ) \\Omega _ { \\omega } \\ , \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} X : = \\partial _ x - \\frac { y } { 2 } \\partial _ z , Y : = \\partial _ y + \\frac { x } { 2 } \\partial _ z , Z : = \\partial _ z . \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } \\frac { 1 } { k ^ { \\omega } | \\mu _ k - z | } = \\left ( \\sum _ { k = 1 } ^ { N _ 0 } + \\sum _ { k = N _ 0 + 1 } ^ { \\infty } \\right ) \\frac { 1 } { k ^ { \\omega } | \\mu _ k - z | } \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} w _ i ( n ) = \\left \\{ \\begin{matrix} \\sum \\limits ^ { V } _ { v = 0 } { { b } _ { i , v } \\tilde { f } _ v ( n ) } , & n \\in \\mathcal { L } , \\\\ 0 , & n \\in \\mathcal { N } \\backslash \\mathcal { L } , \\end{matrix} \\right . \\end{align*}"}
-{"id": "2607.png", "formula": "\\begin{align*} | \\Sigma N | ^ 2 _ e = ( 1 - n ^ { - 2 } \\varpi ^ 2 ) \\frac { r ^ 2 } { ( r + 2 M ) ^ 2 } , \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ K ' \\right ] = K . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} \\mathbb { E } \\exp \\Big ( \\lambda \\sum _ { i = 1 } ^ n \\xi _ i \\Big ) \\le \\exp \\Big ( \\frac { \\lambda ^ 2 \\sum _ { i = 1 } ^ n d _ i ^ 2 } { 2 } \\Big ) ; \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} i _ { \\kappa - 1 / 2 } ( u ) = \\int e ^ { z u } \\mu ^ { \\kappa } ( d u ) , \\kappa > - 1 / 2 . \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{align*} \\mathbb { P } [ M _ k - m _ 0 \\geq \\lambda ] & = \\mathbb { P } [ e ^ { t ( M _ k - m _ 0 ) } \\geq e ^ { t \\lambda } ] \\\\ & \\leq e ^ { - t \\lambda } \\mathbb { E } [ e ^ { t ( M _ k - m _ 0 ) } ] \\\\ & \\stackrel { \\eqref { e x p _ f i n a l } } { \\leq } \\exp \\left ( - t \\lambda + \\frac { t ^ { 2 } } { 2 } g ( t m ) \\sum _ { i = 1 } ^ k \\sigma _ i ^ 2 \\right ) . \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} \\Phi _ { t , j } - \\Phi _ j = S _ { t , m _ j } \\Phi _ j - S _ { m _ j } \\Phi _ j = ( S _ { t , m _ j } - S _ { m _ j } ) \\Phi _ j , 1 \\leq j \\leq N , \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} D _ i \\xi ^ j = 0 , \\phi ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; J _ 2 } = D _ i \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} \\mathrm { d } x \\left ( t \\right ) = f _ { 0 } \\left ( t , x _ { t } , u ( t ) \\right ) \\mathrm { d } t + g _ { 0 } \\left ( t , x _ { t } , u ( t ) \\right ) \\mathrm { d } w \\left ( t \\right ) , t \\in J , \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} Y _ { t } : = u _ t , \\ , \\ , \\ , Z _ { t } : = \\mathcal { D } _ { x } u _ t , \\ , \\ , \\ , K _ { t } : = \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } \\mathcal { D } _ { x } ^ { 2 } u _ s d \\langle B \\rangle _ { s } - \\int _ { 0 } ^ { t } G ( \\mathcal { D } _ { x } ^ { 2 } u _ s ) d s \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\Delta _ { \\mathcal { B } _ n , j } ( \\mathbf { w } ) \\triangleq \\frac { d P _ { \\mathcal { B } _ n , j } } { d p _ W ^ n } ( \\mathbf { w } ) , j = 1 , 2 . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} \\mathbb { H } _ 0 = - \\Delta _ { X _ 1 } - \\Delta _ { X _ 2 } = \\mathcal { L } \\Big ( - \\Delta _ X \\Big ) + \\mathcal { R } \\Big ( - \\Delta _ X \\Big ) . \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} \\bar { x } = \\sum _ { k \\in N } \\alpha _ k b _ k , \\alpha _ k \\geq 0 \\ \\ k \\in N . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} \\Lambda _ \\pm : = \\{ \\lambda \\in \\Lambda : \\pm H _ { \\Gamma , E } ( \\lambda ) \\geq \\nu \\} \\Lambda = \\Lambda _ + \\cup \\Lambda _ - , \\Lambda _ + \\cap \\Lambda _ - = \\emptyset . \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} \\langle h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } | _ { a } ^ { \\pm } = \\langle h _ { 1 } , . . . , h _ { a } \\pm 1 , . . . , h _ { \\mathsf { N } } | , \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} \\hat { H } _ { k _ 1 , k _ 2 } [ A , A ^ \\ast ] = \\left \\{ \\begin{array} { l l l } A ^ { k _ 1 } h _ { k _ 1 , 0 } ( \\sigma ) & { \\mbox f o r } & k _ 1 \\geq 0 , \\ k _ 2 = 0 \\\\ ( A ^ \\ast ) ^ { - k _ 1 } h _ { k _ 1 , 0 } ( \\sigma ) & { \\mbox f o r } & k _ 1 < 0 , \\ k _ 2 = 0 \\\\ 0 & { \\mbox f o r } & k _ 2 \\neq 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} \\xi _ { - j } = \\xi _ j , \\xi _ j > 0 , y _ { - j } = y _ { j } , \\theta _ { - j } = - \\theta _ j , \\theta _ j \\in \\mathbb { T } , \\ , \\ , y _ j \\in \\mathbb { R } , \\forall j \\in S . \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} \\partial _ t z + A _ G ^ * z = ( A _ G ^ * - A _ G ( z + u _ * ) \\big | _ { P H _ N ^ 2 } ) z = : F ( z ) , \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} & \\log | { \\mathbf { \\Gamma } _ 2 } | + { \\rm T r } ( { \\mathbf { \\Gamma } _ 2 } ^ { - 1 } ( \\sigma ^ { 2 } { \\bf I } _ 2 + { \\bf G } _ { 1 3 } { \\bf Q } _ 3 { \\bf G } _ { 1 3 } ^ { T } ) - { \\rm T r } ( { \\bf I } _ 2 ) \\\\ & = \\log | \\sigma ^ { 2 } { \\bf I } _ 2 + { \\bf G } _ { 1 3 } { \\bf Q } _ 3 { \\bf G } _ { 1 3 } ^ { T } | + \\epsilon . \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} & \\partial _ x f = \\partial _ x g = \\partial _ x \\Gamma = 0 , \\mbox { a t } x = 0 , L . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} \\frac { \\tilde { \\beta } _ { t ' + \\tau } } { \\tilde { \\beta } _ t ' } \\ , \\tilde { U } ( b ) = \\frac { \\tilde { \\beta } _ { t ' + \\tau ' } } { \\tilde { \\beta } _ t ' } \\ , \\tilde { U } ( c ) . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} a \\left ( 3 n \\right ) = a \\left ( n \\right ) , \\thinspace \\thinspace a \\left ( 3 n + 1 \\right ) = a \\left ( n \\right ) , \\thinspace \\thinspace a \\left ( 3 n + 2 \\right ) = 2 a \\left ( n \\right ) \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} V = V _ 1 \\oplus \\ldots \\oplus V _ s \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} \\Lambda \\left ( \\xi ^ { ( l , \\sigma ) } ( y ) \\right ) = \\Lambda \\left ( \\xi ^ { ( l , \\sigma ) } ( z ) \\right ) - \\frac { t } { 8 L } \\ ; . \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} G ^ { \\eta } ( z ) : = G \\big | _ { \\{ [ \\alpha , \\beta ] ; \\mu \\} , \\eta } ( z ) , \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\mathcal { Q } = \\left \\{ \\mathbf { q } _ { \\tilde { v } } \\left | \\mathbf { q } _ { \\tilde { v } } = \\left [ f _ { \\tilde { v } } ( - N _ { \\rm c p } ) , f _ { \\tilde { v } } ( - N _ { \\rm c p } + 1 ) , \\ldots , f _ { \\tilde { v } } ( N - 1 ) \\right ] ^ { \\rm T } , \\tilde { v } \\in \\mathcal { U } _ { 2 V } \\right . \\right \\} . \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} & 2 \\Re \\int i \\bar Q ^ { ( n + 1 ) } W ^ { ( n ) } Q _ \\alpha \\ , d \\alpha + 2 n \\Re \\int i \\bar Q ^ { ( n + 1 ) } W ^ { ( n - 1 ) } Q _ { \\alpha \\alpha } \\ , d \\alpha , \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ( A + U V ^ { \\ast } ) & = A + U V ^ { \\ast } + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } A + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } U V ^ { \\ast } \\\\ & = A + U V ^ { \\ast } - ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } F _ { A } + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } S _ { A } V ^ { \\ast } \\\\ & = A + U V ^ { \\ast } . \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} X = \\xi ^ i ( x , u ) \\partial _ { x ^ i } + \\phi ^ { \\alpha } ( x , u ) \\partial _ { u ^ { \\alpha } } , \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} B = \\cdots \\otimes B ^ { 1 , s _ 3 } \\otimes B ^ { 1 , s _ 2 } \\otimes B ^ { 1 , s _ 1 } \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} \\Gamma ( f ) : = ( X f ) ^ 2 + ( Y f ) ^ 2 + \\beta ^ 2 ( Z f ) ^ 2 \\leq 1 . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} \\tan \\alpha = \\dfrac { \\allowbreak 2 s \\left ( v t - w s \\right ) } { \\left \\vert ( t ^ { 2 } - s ^ { 2 } ) v - 2 w t s \\right \\vert } \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} \\| f | { { { W } } _ { 2 } ^ { { \\bf R } } ( \\Bbb T ^ d ) } \\| & = \\Big ( \\| f | L _ 2 ( \\Bbb T ^ d ) \\| + \\sum _ { j = 1 } ^ { d } \\Big \\| \\frac { \\partial ^ { R _ { j } } } { \\partial x _ { j } ^ { R _ { j } } } f \\ , \\big | L _ { 2 } ( \\Bbb T ^ d ) \\Big \\| ^ { 2 } \\Big ) ^ { 1 / 2 } \\\\ & = \\Big ( \\sum _ { { \\bf k } \\in \\Bbb Z ^ d } \\big ( 1 + \\sum _ { j = 1 } ^ d | k _ j | ^ { 2 R _ j } \\big ) | \\hat f ( { \\bf k } ) | ^ 2 \\Big ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} \\sigma ^ 2 ( y ) = q ^ { 1 / 2 } x y \\sigma ( Y ) = \\begin{pmatrix} 0 & 1 \\\\ q ^ { 1 / 2 } x & 0 \\end{pmatrix} Y \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} G _ { J , J } [ \\tilde { \\nu } _ i ] _ { i \\in J } = [ \\langle x \\ | \\ u _ i \\rangle - \\eta _ i ] _ { i \\in J } \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{gather*} \\{ L , L \\} = - i ^ 2 _ Q \\omega \\simeq 0 . \\end{gather*}"}
-{"id": "3867.png", "formula": "\\begin{align*} a ( x , y , z ) = \\frac { | G | } { | C _ G ( x ) | | C _ G ( y ) | } \\sum _ { \\theta \\in { \\rm I r r } ( G ) } \\frac { \\theta ( x ) \\theta ( y ) \\theta ( z ^ { - 1 } ) } { \\theta ( 1 ) } . \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} \\begin{aligned} & \\alpha _ { 0 , 1 } = 2 \\ , c _ 2 \\overline { v } _ { x x x } - 6 \\ , c _ 3 \\overline { v } _ x , \\\\ & \\alpha _ { 0 , 2 } = \\partial _ x L _ { \\overline { \\omega } } [ \\beta _ 2 ] - 3 \\partial _ x [ ( \\beta _ 1 ) _ x \\ , ( \\beta _ 1 ) _ { x x x } ] - 3 \\partial _ x [ ( \\beta _ 1 ) _ { x x } ^ 2 ] + a _ { 0 , 1 } ( \\beta _ 1 ) _ { x x } + ( a _ { 0 , 2 } ) _ x . \\end{aligned} \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} T = \\left ( \\begin{array} { r r r } \\cos ( \\theta ) & \\sin ( \\theta ) & 0 \\\\ - \\sin ( \\theta ) & \\cos ( \\theta ) & 0 \\\\ 0 & 0 & 1 \\end{array} \\right ) \\ , . \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} \\Pr \\left ( \\left \\vert \\hat { f } _ { - } \\left ( x \\right ) - E \\left \\{ \\hat { f } _ { - } \\left ( x \\right ) \\right \\} \\right \\vert \\geq \\epsilon \\right ) & = \\Pr \\left ( \\left \\vert \\bar { K } - E \\left ( K _ { i } \\right ) \\right \\vert \\geq b \\epsilon \\right ) \\\\ & \\leq 2 \\exp \\left \\{ - 2 \\left ( \\frac { \\epsilon } { C } \\right ) ^ { 2 } n b ^ { 2 } \\right \\} \\rightarrow 0 . \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} N ^ 2 ( x , z ) = A ( x , 2 z ) . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} & \\max \\tau ^ { * } < \\epsilon ^ * , \\ , \\ , \\max | h | < \\epsilon ^ * , \\ , \\ , d _ { \\max } < \\epsilon ^ * , \\ , \\ , \\max \\psi < \\epsilon ^ * , \\\\ & \\operatorname { d i a m } ( \\pi \\{ ( x , \\xi _ 0 , s ) : x _ 0 = a _ 0 , 0 \\leq s < \\tau ^ * ( x ) \\} ) < \\epsilon ^ * \\ , \\ , \\ , \\ , a _ 0 , \\ , \\xi _ 0 . \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} \\theta ( z _ 1 , z _ 2 ) \\ = \\ \\exp ( z _ 1 N _ 1 + z _ 2 N _ 2 ) \\cdot F \\ , . \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} M _ s ( x , \\theta ) & = k _ { N , s } ( 1 - | x | ^ 2 ) ^ s \\lim _ { z \\to \\theta , z \\in B } \\int _ 0 ^ 1 \\frac { t ^ { s - 1 } } { ( ( 1 - | x | ^ 2 ) _ + ( 1 - | z | ^ 2 ) _ + t + | x - z | ^ 2 ) ^ { \\frac { N } { 2 } } } \\ d t \\\\ & = k _ { N , s } \\frac { ( 1 - | x | ^ 2 ) ^ s } { | x - \\theta | ^ { N } } \\int _ 0 ^ 1 t ^ { s - 1 } \\ d t = \\frac { k _ { N , s } } { s } \\frac { ( 1 - | x | ^ 2 ) ^ s } { | x - \\theta | ^ { N } } \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} & S _ { u ^ j } ( - z ) = \\left ( S _ { u ^ j } ( \\chi ) + \\displaystyle \\frac { \\partial } { \\partial z } S _ { u ^ j } ( \\chi ) ( - z - \\chi ) + \\dotsc + \\displaystyle \\frac { 1 } { K ! } \\frac { \\partial ^ K } { \\partial z ^ K } S _ { u ^ j } ( \\chi ) ( - z - \\chi ) ^ K \\right ) + ( - z - \\chi ) ^ { K + 1 } R ( z ^ { - 1 } ) \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} \\Big | \\tilde T _ { 2 j } ( i \\tau / 2 ) - \\sum _ { l = 0 } ^ N ( - 1 ) ^ { j + l } T _ { 2 j , l } \\tau ^ { - 2 j - 2 l - 1 } \\Big | \\lesssim \\tau ^ { - 3 - 2 s } \\Vert u \\Vert _ { H ^ { s - \\frac 1 4 } } ^ 2 \\Vert u \\Vert _ { H ^ { - 1 } } ^ { j - 1 } . \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\P ( \\{ M _ n \\le \\lambda ^ n \\} ) \\le \\sum _ { n = 1 } ^ \\infty p ^ { \\alpha _ n } < \\infty , \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{align*} \\tilde { f } ( u ) ( x ) : = f ( x , u ( x ) , D u ( x ) , \\dots , D ^ p u ( x ) ) \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\sum _ { m = 3 } ^ \\infty \\mu _ m \\delta _ m ^ 2 & = \\frac 1 2 \\sum _ { m = 3 } ^ \\infty \\frac { d ^ m } { m } \\sum _ { i , j = 2 } ^ q \\lambda _ i ^ m \\lambda _ j ^ m \\\\ & = \\frac 1 2 \\sum _ { i , j = 2 } ^ q \\sum _ { m = 3 } ^ \\infty \\frac { ( d \\lambda _ i \\lambda _ j ) ^ m } { m } \\\\ & = \\sum _ { i , j = 2 } ^ q \\log \\psi ( d \\lambda _ i \\lambda _ j ) , \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} ( - 1 ) ^ { r } = \\epsilon ( - 1 ) \\cdot \\varepsilon \\cdot \\varepsilon _ { n _ 2 } . \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} \\left ( \\operatorname { M o d } _ A \\otimes _ { \\mathcal { C } } \\operatorname { M o d } _ A \\right ) ^ { \\otimes } & = Q C ( \\operatorname { S p e c } \\ , A \\times _ { \\mathsf { M } _ { \\mathcal { C } ^ { \\otimes } } } \\operatorname { S p e c } \\ , A ) \\\\ & = Q C ( \\operatorname { A u t } ^ { \\otimes } _ { \\omega } ) ^ { \\otimes } \\\\ & \\simeq \\operatorname { M o d } _ { \\mathcal { H } } ^ { \\otimes } \\\\ \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\displaystyle ( | v ' ( t ) | ^ { p - 2 } v ' ( t ) ) ' + h ( t ) k ( t ) f ( v ( t ) ) = 0 & \\textup { i n } & ( 0 , 1 ) , \\\\ \\displaystyle v ( 0 ) = v ( 1 ) = 0 , & & \\end{array} \\right . \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} & v _ 1 + b v _ 0 = 0 , \\\\ & v _ 0 = 0 . \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\rm w } = \\left ( \\mathbf { P } _ { \\rm w } \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\right ) \\mathbf { P } _ 2 = \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 2 . \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} & \\beta ( t ) : = \\frac { t } { 1 - e ^ { - s } r t } , \\\\ & g _ t : = \\begin{pmatrix} ( 1 - e ^ { - s } r t ) ^ { - 1 } & 0 \\\\ - e ^ { - s } r & 1 - e ^ { - s } r t \\end{pmatrix} . \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} p ( G ) ( \\mathcal { A } ) = \\sum _ { \\phi : V \\to [ k ] } \\prod _ { e = \\{ u , v \\} \\in E } A ^ { e } _ { \\phi ( u ) , \\phi ( v ) } . \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} U ^ + _ v : = \\bigoplus _ { a \\in \\N } \\Z [ v ^ { \\pm 1 } ] \\cdot \\theta ^ { ( a ) } . \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} \\dot r ( t ) = 2 \\big ( \\dot u ( t ) u ( t ) + \\dot v ( t ) v ( t ) \\big ) . \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{gather*} \\{ A , B \\} = ( - 1 ) ^ { \\epsilon ( X _ A ) } i _ { X _ A } i _ { X _ B } \\omega . \\end{gather*}"}
-{"id": "2806.png", "formula": "\\begin{align*} \\Lambda _ N : = U ^ 2 \\oplus D _ { N - 2 } . \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} E \\left \\{ \\left ( \\mathbf { A } ^ { - 1 } \\mathbf { w } _ i \\right ) ^ { \\rm H } \\mathbf { A } ^ { - 1 } \\mathbf { w } _ i \\right \\} = { \\rm T r } \\left \\{ \\hat { \\mathbf { P } } \\hat { \\mathbf { P } } ^ { \\rm H } + \\tilde { \\mathbf { P } } \\tilde { \\mathbf { P } } ^ { \\rm H } \\right \\} = 2 \\ ; { \\rm T r } \\left \\{ \\tilde { \\mathbf { P } } \\right \\} = 2 \\ ; { \\rm r a n k } \\left \\{ \\tilde { \\mathbf { P } } \\right \\} = 2 ( V + 1 ) . \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} g _ { n } ( x ) = g _ { n - 1 } ( a ) + ( g _ { n - 1 } ( b ) - g _ { n - 1 } ( a ) ) \\cdot g _ { 1 } \\Big ( \\frac { x - a } { b - a } \\Big ) = \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\tau _ n ( A ) = 0 , \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} \\Gamma ( a , z ) : = \\int _ { z } ^ { \\infty } t ^ { a - 1 } e ^ { - t } d t . \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = { A } ( t ) x + c ( t ) + \\lambda f _ 1 ( t , x , y , \\lambda ) , \\\\ \\dot y = \\lambda f _ 2 ( t , x , y , \\lambda ) , \\\\ ( x ( 0 ) , y ( 0 ) ) = ( p , q ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} | ( \\Lambda _ { 2 4 } ) | = 2 ^ { 2 2 } \\cdot 3 ^ 9 \\cdot 5 ^ 4 \\cdot 7 ^ 2 \\cdot 1 1 \\cdot 1 3 \\cdot 2 3 = 8 3 1 5 5 5 3 6 1 3 0 8 6 7 2 0 0 0 0 , \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} T : = T _ { t , p , \\Omega } ^ { f , g , \\nu } : X _ { t , p , \\Omega } & \\to X _ { t , p , \\Omega } \\\\ ( u , \\pi ) & \\mapsto L _ { t , p , \\Omega } ( f - \\nu u \\cdot ( \\nabla u ) , g ) , \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{align*} a _ { i } = a _ { i + 1 } + 1 , \\ , \\ i = 1 , 3 , \\ldots , 2 q - 1 . \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P _ { 2 , 4 } } ( - 1 ) ^ { \\nu ( \\pi ) } q ^ { | \\pi | } = \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { 2 n } } { 1 - q ^ { 4 n } } \\frac { ( - q ; q ^ 2 ) _ { n - 1 } } { ( q ^ 2 , q ^ 2 ) _ { n - 1 } } q ^ { n - 1 } . \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} R = \\begin{pmatrix} 1 & - \\tfrac 1 2 & 0 & 0 & \\ldots \\\\ - \\tfrac 1 2 & 1 & - \\tfrac 1 2 & 0 & \\ldots \\\\ 0 & - \\tfrac 1 2 & 1 & - \\tfrac 1 2 & \\ldots \\\\ 0 & 0 & - \\tfrac 1 2 & 1 & \\ldots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\ddots \\end{pmatrix} . \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} \\underset { m \\rightarrow \\infty } { \\lim } P ( K _ { n } = k | K _ { m } = \\omega _ { m } ) = \\underset { m \\rightarrow \\infty } { \\lim } \\frac { d _ { n , k } ^ { m , \\omega _ { m } } } { d ^ { m , \\omega _ { m } } } d ^ { n , k } = \\bar { V } _ { n , k } d ^ { n , k } \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} \\pi _ k ^ * F _ { \\eta _ a } = ( F _ A ) _ a ^ a + i d \\mu _ a \\wedge ( d \\tau + \\omega ) + i \\mu _ a d \\omega + O ( r ^ { - 5 } ) . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} \\mathcal { D } = \\mathfrak { B } \\sqcup \\{ c T ^ { - i } : c \\in \\mathfrak { C } , \\ ( i , p ) = 1 , i \\geq 1 \\} . \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} - \\frac { ( e ^ L y - e ^ { - L } x ) ^ 2 } { e ^ { 2 L } - e ^ { - 2 L } } + \\frac { ( e ^ L y - e ^ { - L } x - ( e ^ { 2 L } - e ^ { - 2 L } ) \\frac r { \\sqrt 2 } ) ^ 2 } { e ^ { 2 L } - e ^ { - 2 L } } = \\frac { ( e ^ { 2 L } - e ^ { - 2 L } ) r ^ 2 } 2 - \\sqrt 2 r ( e ^ L y - e ^ { - L } x ) \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} [ [ \\alpha , \\beta ] ] _ I = \\mathcal { L } _ { \\Pi ( \\alpha ) } \\beta + ( - 1 ) ^ p \\ ( \\Pi ( d \\alpha ) ) \\beta \\forall \\alpha , \\beta \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} d _ { S ( T , t ) } ( u ^ t , w ) = d _ { S ( T , t - 1 ) } ( x ' \\cdots x ' , x _ { 2 } \\cdots x _ t ) + ( 2 ^ { t } - 1 ) d _ T ( u , x ) - ( 2 ^ { t - 1 } - 1 ) \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{align*} \\Phi ( t , x , z ) = f _ { \\mu , A } ^ { \\frac { 1 - m } { 2 } } ( x ) z ^ { \\frac { m - 1 } { 2 } } I _ d \\ , \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\sum _ { \\mathcal { A } \\in \\mathcal { F } } 2 ^ { - | \\mathcal { A } | / ( 2 q - 1 ) } < 2 ^ { n - \\binom { n } { k } / ( 2 q ) } = 1 , \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} I ( \\gamma ( s ) | e ^ { - \\Psi } ) = \\int _ { \\mathbb { S } ^ { 1 } } ( 1 + s f ) e ^ { - \\Psi } | \\nabla \\log ( 1 + s f ) | ^ 2 d \\omega . \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} x ( t ) = \\int _ 0 ^ 1 ( 1 - t ) A ( s ) \\textup d x ( s ) + \\int _ 0 ^ 1 t B ( s ) \\textup d x ( s ) + \\lambda \\int _ 0 ^ 1 k ( t , s ) f ( s , x ( s ) ) \\textup d s , \\ t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\phi ( z ) = \\phi _ { \\rho } ( z ) = \\frac { 1 } { \\sigma } \\sum _ { i = 1 } ^ N \\frac { ( \\alpha z ) ^ i } { i } \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} u & = X ^ \\sharp . \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} \\theta _ n = \\frac { 2 \\ell _ n H _ 2 ( \\alpha _ n ) } { n \\log ( 1 + k _ n P ) } , \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} \\epsilon ( \\lambda ) = \\epsilon _ 0 ( \\lambda ) + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) \\epsilon ( \\lambda ) . \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\Lambda ( Z ) = \\frac { E _ { \\Theta _ 1 } [ f _ { Z ( \\theta ) } ( Z ) ] } { E _ { \\Theta _ 0 } [ f _ { Z ( \\theta ) } ( Z ) ] } \\mathop { \\gtrless } _ { H _ 0 } ^ { H _ 1 } \\gamma . \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 2 & 0 \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} = \\begin{pmatrix} a ' & b ' \\\\ c ' & d ' \\end{pmatrix} \\begin{pmatrix} \\alpha & \\beta \\\\ 0 & \\delta \\end{pmatrix} \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} \\frac { d } { d t } D \\varphi _ t ( \\omega , x ) v = D b ( \\varphi _ t ( \\omega , x ) ) D \\varphi _ t ( \\omega , x ) v , D \\varphi _ 0 ( \\omega , x ) v = v \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} l _ \\sigma ( u ^ { r + 1 } ; 0 , 0 ) & = l _ \\sigma ( u ; 0 , 0 ) l _ \\sigma ( u ^ r ; 0 , 0 ) = ( B u ^ { - 1 } B ^ { - 1 } ; 1 , 0 ) ( ( B u ^ { - 1 } ) ^ r B ^ { - r } ; r , 0 ) \\\\ & = ( B u ^ { - 1 } B ^ { - 1 } \\left ( \\theta ( 1 , 0 ) ( B ) \\theta ( 1 , 0 ) ( u ) ^ { - 1 } \\right ) ^ r \\theta ( 1 , 0 ) ( B ) ^ { - r } ; r + 1 , 0 ) \\\\ & = ( B u ^ { - 1 } B ^ { - 1 } \\left ( B ( B u B ^ { - 1 } ) ^ { - 1 } \\right ) ^ r B ^ { - r } ; r + 1 , 0 ) = ( ( B u ^ { - 1 } ) ^ { r + 1 } B ^ { - ( r + 1 ) } ; r , 0 ) . \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} v = 2 \\left ( \\frac { d _ 1 + 3 } { 3 } \\right ) \\geq d _ 1 + 1 . \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} y ' | _ { ( B _ m ( \\texttt { S u b } _ { ( 1 , 1 ) } ( y ' ) ) , 1 _ H ) } & = \\sigma _ { ( \\vec { 0 } , s ) } ( y ) | _ { ( B _ m ( \\texttt { S u b } _ { ( 1 , 1 ) } ( y ' ) ) , 1 _ H ) } \\\\ & = y | _ { ( \\vec { 0 } , s ^ { - 1 } ) ( B _ m ( \\texttt { S u b } _ { ( 1 , 1 ) } ( y ' ) ) , 1 _ H ) } \\\\ & = y | _ { ( \\varphi _ { s ^ { - 1 } } ( B _ m ( \\texttt { S u b } _ { ( 1 , 1 ) } ( y ' ) ) ) , s ^ { - 1 } ) } . \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} \\begin{aligned} \\widetilde { u ^ { \\alpha } _ { J _ 1 + \\bold { 1 } ; J _ 2 } } & = S _ { J _ 2 } \\left ( D _ { \\widetilde { x } } \\widetilde { u ^ { \\alpha } _ { J _ 1 ; \\bold { 0 } } } \\right ) = ( S _ { J _ 2 } D _ { \\widetilde { x } } ) \\widetilde { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } \\\\ & = \\left ( D _ x \\widetilde { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } \\right ) S _ { J _ 2 } ( D _ x \\widetilde { x } ) ^ { - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} e _ 1 = \\bar e _ 1 & = \\partial _ y , & e _ 2 & = \\bar { e } _ 2 - \\omega ( \\bar { e } _ 2 ) \\partial _ \\tau , & e _ 3 & = \\bar { e _ 3 } - \\omega ( \\bar { e } _ 3 ) \\partial _ \\tau , & e _ 4 & = e ^ y V \\partial _ \\tau , \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} \\partial g ^ - _ \\gamma ( v ) = \\begin{cases} - 1 & v < U - \\gamma , \\\\ \\frac 1 \\gamma ( v - U ) & v \\in [ U - \\gamma , U ] , \\\\ 0 & v > U . \\end{cases} \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { 1 - F _ \\rho ( k ) } { 1 - F ( k ) } = \\frac { \\mu _ \\rho } { \\mu } . \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} F _ k : \\sum _ { i = 0 } ^ n \\sum _ { j = 0 } ^ m a ^ { ( k ) } _ { i , j } x _ i y _ j = 0 , 1 \\le k \\le n + m . \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} \\| \\beta \\| _ { k , \\ell } ^ 2 = \\sum _ { p = k } ^ { \\ell - 1 } \\int _ { A _ { p , p + 1 } } \\lambda ^ { - 2 n p } | \\beta | _ { \\omega _ C } ^ 2 . \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} q ^ * = \\mathop { } \\limits _ { S \\in \\mathcal { F } } \\frac { R _ { \\rm { \\rm { t o t } } } ( S ) } { P _ { \\rm { \\rm { t o t } } } ( S ) } , \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} U _ { i a , j b } = \\delta _ { i a , j b } - s _ b F ( \\lambda _ { j b } | \\lambda _ { i a } ) , [ U ^ { - 1 } ] _ { i a , j b } = \\delta _ { i a , j b } - s _ b F ( \\lambda _ { i a } | \\lambda _ { j b } ) . \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} \\max \\{ ( f _ c ( x ) ) ^ T Q x \\ , | \\ , x ^ T Q x = 1 \\} . \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} ( a + b ) ( c + d ) = a c + a d + b c + b d . \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} Q ^ { \\alpha } _ { \\ast } F _ { \\alpha } = - v ^ { \\alpha } \\bold { p r } X ( F _ { \\alpha } ) + \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} m = \\left ( \\begin{array} { c } u ' + \\frac 1 2 \\left ( ( v _ 2 ' ) ^ 2 + ( v _ 3 ' ) ^ 2 \\right ) - v _ 2 '' x _ 2 - v _ 3 '' x _ 3 \\\\ - w ' x _ 3 \\\\ w ' x _ 2 \\end{array} \\right ) \\ , , \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} \\omega ^ { 0 } _ { \\beta , \\mu , \\phi } : = \\lim _ { \\lambda \\to + 0 } \\lim _ { V \\to \\infty } \\omega ^ { 0 } _ { \\beta , \\mu , \\Lambda , \\lambda _ { \\phi } } \\ , \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\abs { \\sum _ { i , l = 1 , 2 } \\pi _ \\theta ( i ) p _ \\theta ( i , l ) \\int _ { - \\infty } ^ { \\infty } y ^ { r + 1 } \\varphi ( y - \\mu _ \\theta ^ { ( l ) } ) \\ , \\d y } \\le \\sum _ { i , l = 1 , 2 } \\pi _ \\theta ( i ) p _ \\theta ( i , l ) \\int _ { - \\infty } ^ { \\infty } | y | ^ { r + 1 } \\varphi ( y - \\mu _ \\theta ^ { ( l ) } ) \\ , \\d y . \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} \\partial _ t u + A _ G ( u ) u = 0 , t > 0 , u ( 0 ) = u ^ 0 , \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} \\gamma _ { i k } = \\frac { \\vert \\mathbf { h } _ { i i k } ^ { \\mathrm { H } } \\mathbf { w } _ { i k } \\vert ^ 2 } { \\sum _ { l = 1 , l \\neq k } ^ K \\vert \\mathbf { h } _ { i i k } ^ { \\mathrm { H } } \\mathbf { w } _ { i l } \\vert ^ 2 + \\mathcal { I } } , \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} p ^ { n _ c } ( 2 g _ n - 2 ) \\geq & p ^ n ( 2 g _ 0 - 2 ) + \\sum _ { i = n _ u + 1 } ^ n \\varphi ( p ^ i ) \\left ( 1 + p ^ { i - n _ u - 1 } \\right ) \\\\ = & p ^ n ( 2 g _ 0 - 2 ) + p ^ n - p ^ { n _ u } + p ^ { n _ u } \\frac { p ^ { 2 ( n - n _ u ) } - 1 } { p + 1 } . \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} \\int _ { \\epsilon } ^ { b } \\frac { 1 } { \\sqrt { \\xi '' } } ( \\eta '' - \\xi '' ) = \\int _ { \\epsilon } ^ { b } - \\mathfrak { d } ( \\eta - \\xi ) - \\frac { 1 } { \\sqrt { \\xi '' } } ( \\eta ' - \\xi ' ) ( \\epsilon ) + ( \\frac { 1 } { \\sqrt { \\xi '' } } ) ' ( \\eta - \\xi ) ( \\epsilon ) \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} - ( H - E _ j ) ^ { - 1 } \\chi _ { J ^ c } ( H ) V \\chi _ j = \\chi _ { J ^ c } ( H ) \\phi _ j . \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} N ( E , H ) = \\lim _ { L \\to \\infty } \\frac { \\operatorname { T r } \\left ( \\chi _ { ( - \\infty , E ] } ( H _ L ) \\right ) } { L ^ 2 } , \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} & H _ { 3 , \\le 1 } = \\int _ { \\mathbb { T } } \\left \\{ c _ 1 ( v _ x ^ 3 + 3 \\ , v _ x ^ 2 \\ , z _ x ) + c _ 2 ( v _ x ^ 2 \\ , v + 2 \\ , v _ x \\ , v \\ , z _ x + v _ x ^ 2 \\ , z ) + c _ 3 ( v ^ 3 + 3 \\ , v ^ 2 \\ , z ) \\right \\} \\ , d x , \\\\ & H _ { 3 , \\geq 2 } = \\int _ { \\mathbb { T } } \\{ c _ 1 ( z _ x ^ 3 + 3 \\ , z _ x ^ 2 \\ , v _ x ) + c _ 2 ( z _ x ^ 2 \\ , z + z _ x ^ 2 \\ , v + 2 \\ , z _ x \\ , z \\ , v _ x ) + c _ 3 ( z ^ 3 + 3 \\ , v ^ 2 \\ , z ) \\} \\ , d x , \\\\ & H _ { 4 , 0 } = \\int _ { \\mathbb { T } } \\{ c _ 4 \\ , v _ x ^ 4 + c _ 5 \\ , v _ x ^ 3 \\ , v + c _ 6 \\ , v _ x ^ 2 \\ , v ^ 2 + c _ 7 \\ , v ^ 4 \\} \\ , d x . \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} \\left | \\varphi _ s ( \\omega , x ) - \\varphi _ s ( \\omega , y ) \\right | & \\leq \\left | \\varphi _ { \\hat { \\tau } ( \\omega ) } ( \\omega , x ) - \\varphi _ { \\hat { \\tau } ( \\omega ) } ( \\omega , y ) \\right | \\ ; e ^ { - 2 \\left ( s - \\hat { \\tau } ( \\omega ) \\right ) } \\\\ & = 4 \\ ; e ^ { - 2 \\left ( s - \\hat { \\tau } ( \\omega ) \\right ) } \\leq 4 \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} ~ \\lim _ { n \\to \\infty } \\sigma _ n ^ d ( f , x ) = L ( f , x ) : = \\frac { 1 } { 2 ^ d } \\sum _ { k \\in \\{ 0 , 1 \\} ^ d } f ( x , k ) . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} \\Delta ( \\lambda ) \\ ; = \\ ; \\Delta _ { H e ^ 0 _ \\lambda } \\ ; = \\ ; A \\otimes _ B H e ^ 0 _ \\lambda , \\qquad \\mbox { a n d } \\overline { \\Delta } ( \\lambda ) \\ ; = \\ ; \\Delta _ { L ^ 0 ( \\lambda ) } \\ ; = \\ ; A \\otimes _ B L ^ 0 ( \\lambda ) . \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} E _ n ( j _ m ( x ) ) = j _ n ( P _ \\varphi ^ { m - n } ( x ) ) . \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} s ( x , y ) = \\frac { \\inf _ { t \\in \\mathbb { R } } | | x + t y | | } { | | x | | } \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} \\kappa _ n = \\sum _ { m = 1 } ^ { M } x _ { m , n } . \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} \\frac { \\partial N _ { i a } } { \\partial \\lambda _ { j b } } = s _ a L \\left \\{ \\rho ( \\lambda _ { i a } ) \\delta _ { i a , j b } + \\frac { 1 } { 2 } \\int _ { - \\infty } ^ { \\infty } d \\lambda \\ , s _ a \\mathrm { s g n } ( \\lambda _ { i a } - \\lambda ) \\frac { d \\rho } { d \\lambda _ { j b } } ( \\lambda ) \\right \\} . \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} \\delta = \\delta ( s ) = \\frac { 2 \\log { | s | / T } } { \\log { N } } , \\ 0 \\le \\delta \\le \\frac { \\log { 4 } } { \\log { N } } . \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} \\lim _ { \\mu \\to + \\infty } \\frac { \\mu ^ \\frac 1 \\beta } { x _ \\mu } = \\lim _ { \\mu \\to + \\infty } \\left ( \\frac { Q ( x _ \\mu ) } { x _ \\mu ^ \\beta } \\right ) ^ \\frac 1 \\beta = 1 . \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} \\mathcal { H } ^ q _ { b , m } ( X ) = \\left \\{ u \\in \\mathrm { D o m } ( \\Box ^ { ( q ) } _ { b , m } ) : \\Box ^ { ( q ) } _ { b , m } u = 0 \\right \\} . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} U _ { n + 1 } : = U _ n + H _ { n + 1 } , H _ { n + 1 } : = ( \\hat { \\mathfrak { I } } _ { n + 1 } , \\hat { \\zeta } _ { n + 1 } ) : = - \\tilde { \\Pi } _ n \\textbf { T } _ n \\Pi _ n \\mathcal { F } ( U _ n ) \\in E _ n \\times \\mathbb { R } ^ { \\nu } , \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\int _ S | D H ( X ( t , x ) ) | \\ , d t = L ( c _ i ( h ) \\cap B _ { R / ( c _ 0 | q | ) } ) \\leq \\frac { 4 R } { c _ 0 | q | } . \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} c b _ { 1 } ^ { 2 } - \\left ( c + 2 \\right ) a ^ { 2 } = - 2 , \\left ( c - 2 \\right ) a ^ { 2 } - c e _ { 1 } ^ { 2 } = - 2 , \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} { \\cal I } ( \\boldsymbol { x } _ i ) = \\sum _ { j = 1 } ^ { d } \\bar { R } ^ { N _ j } _ j ( \\boldsymbol { x } _ i ) W _ j ( \\boldsymbol { x } _ i ) = \\sum _ { j \\in I ( \\boldsymbol { x } _ i ) } f ( \\boldsymbol { x } _ i ) W _ j ( \\boldsymbol { x } _ i ) = f ( \\boldsymbol { x } _ i ) . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} \\mathcal { A } _ { \\perp } : = \\Pi _ S ^ { \\perp } \\mathcal { A } \\Pi _ S ^ { \\perp } , ( \\mathcal { A } h ) ( \\varphi , x ) : = ( 1 + \\beta _ x ( \\varphi , x ) ) \\ , h ( \\varphi , x + \\beta ( \\varphi , x ) ) , \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } f ( n ) = \\sum _ { d \\le x } h _ f ( d ) \\sum _ { j \\le x / d } f ( j ) , \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} F _ n ( n s ; a ) = \\sum _ p \\frac { ( \\chi _ 0 ( p ) + \\chi _ a ( p ) ) ^ n } { p ^ { n s } } . \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} A & = \\theta _ { 1 } - \\dfrac { \\theta _ { 2 } } { 2 } \\in \\left ( - \\frac { 3 \\pi } { 2 } , \\frac { 3 \\pi } { 2 } \\right ) \\\\ B & = \\dfrac { 1 } { 2 } \\arccos \\left \\{ \\dfrac { 1 } { 2 } \\left [ \\dfrac { \\rho _ { 1 } ^ { 2 } } { 2 \\rho _ { 2 } } - \\sqrt { \\dfrac { \\rho _ { 1 } ^ { 4 } } { 4 \\rho _ { 2 } ^ { 2 } } + 4 - 2 \\dfrac { \\rho _ { 1 } ^ { 2 } } { \\rho _ { 2 } } \\cos ( 2 A ) } \\right ] \\right \\} \\in \\left [ 0 , \\frac { \\pi } { 2 } \\right ] . \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} W _ { u v } ( G , \\sigma ) = \\begin{cases} \\frac { M _ { \\sigma _ u , \\sigma _ v } } { d } & \\\\ \\frac { 1 - \\frac { M _ { \\sigma _ u , \\sigma _ v } } { n } } { 1 - \\frac d n } & \\end{cases} \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} \\mathbf { R } _ l ^ { ( g ) } = \\mathbf { U } _ l ^ { ( g ) } \\boldsymbol { \\Lambda } _ l ^ { ( g ) } \\left ( \\mathbf { U } _ l ^ { ( g ) } \\right ) ^ H , \\ ; l = 0 , \\ldots , L _ g - 1 , \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} \\left ( x ; A \\cup B \\cup C \\right ) & = \\left ( x ; A \\right ) + \\left ( x ; B \\right ) + \\left ( x ; C \\right ) \\\\ & - \\left ( x ; A \\cap B \\right ) - \\left ( x ; A \\cap C \\right ) - \\left ( x ; B \\cap C \\right ) \\\\ & + \\left ( x ; A \\cap B \\cap C \\right ) \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} t ^ { \\alpha + \\beta - 1 } \\Psi _ { u , v } ' ( t ) = \\varphi _ { u , v } '' ( t ) = t ^ { - 2 } \\varphi '' _ { t u , t v } ( 1 ) . \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} k ( U ^ H ) = - \\sum _ { k = 1 } ^ n g ^ { T X } ( S ( e _ k ) e _ k , U ^ H ) . \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} \\min _ { \\mu \\in \\Pr ( [ 0 , 1 ] ) } P _ { \\xi , h } ( \\mu ) = \\max _ { \\eta \\in \\Lambda _ { \\xi , h } } D _ { \\xi , h } ( \\eta ) . \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} L ( p , \\tau ) : = \\left \\{ v : ( 1 + x ^ 2 ) ^ { - \\frac \\tau 2 } | v ( x ) | \\in L ^ p ( \\R ) \\right \\} , 1 \\leq p \\leq \\infty , \\ \\ \\tau \\in \\R , \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} D _ t P _ 1 + ( S - \\operatorname { i d } ) P _ 2 = Q \\bold { E } ( L ) . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} \\norm { r ( 2 ^ { - \\sigma } T ) ^ { 2 ^ k } - \\exp ( T _ k ) } & = \\norm { \\exp ( T _ k + E _ k ) - \\exp ( T _ k ) } \\\\ & \\le u 2 ^ { - k } \\norm { T } \\exp ( u 2 ^ { - k } \\norm { T } ) \\le \\tau . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} \\Phi _ { \\Gamma , E } ( \\tau , y ) = \\left \\{ \\begin{array} { l l } H _ { \\Gamma , E } ( \\lambda ) e ^ { \\tau H _ { \\Gamma , E } ( \\lambda ) } & \\ ; \\mbox { i f $ \\tau > 0 $ a n d $ H _ { \\Gamma , E } ( \\lambda ) < 0 $ } \\\\ - H _ { \\Gamma , E } ( \\lambda ) e ^ { \\tau H _ { \\Gamma , E } ( \\lambda ) } & \\ ; \\mbox { i f $ \\tau < 0 $ a n d $ H _ { \\Gamma , E } ( \\lambda ) > 0 $ } \\\\ 0 & \\ ; \\mbox { o t h e r w i s e } \\end{array} \\right . . \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} \\xi ^ * = \\xi - h \\eta , x ^ * = { \\bf i } \\ , x , \\eta ^ * = \\eta , \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} L : = \\bigg ( 2 + \\frac { 4 } { m } \\bigg ) ^ { - 1 } \\log 2 , \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} \\Omega _ X ( u , v ) = 2 a _ { 1 1 } \\overline { a _ { 1 1 } } + a _ { 2 2 } \\overline { a _ { 2 2 } } , \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} Z ( x ) & \\\\ = K ( m , H _ 0 ) & \\left \\{ \\int _ { - \\infty } ^ \\infty d B _ { \\xi _ 1 } \\int _ { - \\infty } ^ { \\xi _ 1 } d B _ { \\xi _ 2 } \\ldots \\int _ { - \\infty } ^ { \\xi _ { m - 1 } } d B _ { \\xi _ m } \\int _ 0 ^ x \\prod _ { i = 1 } ^ m ( s - \\xi _ i ) ^ { H _ 0 - \\frac { 3 } { 2 } } { \\bf 1 } _ { ( \\xi _ i < s ) } \\ , d s \\right \\} \\ , \\ , , \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} \\mu _ t = \\frac { \\sum _ i \\theta _ 0 ^ i ( \\delta ^ i ) ^ { t + 1 } } { \\sum _ i \\theta _ 0 ^ i ( \\delta ^ i ) ^ t } , & & t = 0 , 1 , \\ldots . \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} ( - i \\nabla - \\epsilon A ^ 0 ( x ) - \\kappa A ( \\epsilon x ) ) \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( x , \\tilde { \\beta } ) = \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( x , \\tilde { \\beta } ) \\left \\{ ( - i \\nabla - \\epsilon A ^ 0 ( x ) ) \\ , + \\ , \\kappa a _ \\epsilon ( x , \\tilde { \\beta } ) \\right \\} \\ , , \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} C ^ { + } = \\sup \\limits _ { n \\in \\mathbb { Z } } \\| C ( n ) \\| . \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} \\langle 0 = \\theta , \\psi \\rangle = \\left \\{ \\begin{array} { l l } \\frac { 1 } { | N _ G ( P ) | } ( q ( q - 1 ) ( q - 1 ) - q \\lambda ( X ) ( q - 1 ) ) & \\theta = \\lambda ; \\\\ \\frac { 1 } { | N _ G ( P ) | } ( q ( q - 1 ) 3 ^ k ( q - 1 ) - q ( q - 1 ) \\chi ( X ) ) & \\theta = \\chi . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} f ^ { \\star } ( x ) - x \\cdot \\nabla f ^ { \\star } ( 0 ) - f ^ { \\star } ( 0 ) = & \\\\ v \\int _ { \\{ - 1 , 1 \\} \\times [ 0 , 1 ] \\times \\mathbb { R } ^ d } h ( z , t , \\alpha ) ( x ) d p ( z \\times t \\times \\omega ) . \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{align*} \\exp \\left ( \\int _ 0 ^ { 2 \\pi } \\log ( | \\Phi _ N ^ * ( e ^ { i \\theta } ) - e ^ { i \\theta } \\Phi _ N ( e ^ { i \\theta } ) f _ N ( e ^ { i \\theta } ) | ^ 2 ) \\frac { d \\theta } { 2 \\pi } \\right ) = \\prod _ { j = 0 } ^ m | \\lambda _ { j } | ^ { - 2 } . \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} W _ \\nu ( r ) = \\frac { \\nu } { r } - \\frac { Z \\alpha _ g \\epsilon } { \\nu } \\ ; , \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} \\tilde { \\Omega } _ S = \\sum _ { j \\in S ^ + } d \\theta _ j \\wedge \\frac { 1 } { j } \\ , d I _ j . \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} A _ n & = \\left \\{ 1 - F _ n ^ \\ast ( k ) < F _ n ^ \\ast ( \\ell ) \\right \\} \\\\ B _ n & = \\left \\{ 1 - F _ n ^ \\ast ( k - 1 ) > F _ n ^ \\ast ( \\ell - 1 ) \\right \\} , \\\\ I _ n & = \\left \\{ 1 - F _ n ^ \\ast ( k - 1 ) \\le F _ n ^ \\ast ( \\ell ) \\right \\} , \\\\ J _ n & = \\left \\{ 1 - F _ n ^ \\ast ( k ) \\ge F _ n ^ \\ast ( \\ell - 1 ) \\right \\} . \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} P _ { p t p } ( N , K , p ) = \\sum _ { M = K } ^ { N } B ( M , N , p ) \\mathbb { P } ( M , K ) , \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} H _ z ^ \\bullet ( B _ x B _ s B _ y ) = V ^ \\bullet ( 1 ) \\oplus V ^ \\bullet ( - 1 ) . \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} \\Theta = \\frac { 1 } { x _ 1 ^ m } d x _ 1 \\wedge \\ldots \\wedge d x _ n \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} \\widehat { b } _ { { 0 } } : = b _ { { 0 } } + \\frac { \\lambda _ { \\phi } \\sqrt { V } } { \\mu } \\ , \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} O ^ - ( z ) = \\bigcup _ { n = 0 } ^ \\infty f ^ { - n } ( z ) \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} d _ { 2 r } ( p \\cdot x ) = p \\cdot d _ { 2 r } ( x ) . \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} T = \\begin{bmatrix} t _ 0 & t _ { - 1 } & \\cdots & t _ { - n + 1 } \\\\ t _ 1 & t _ 0 & \\ddots & \\vdots \\\\ \\vdots & \\ddots & \\ddots & t _ { - 1 } \\\\ t _ { n - 1 } & \\cdots & t _ 1 & t _ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} \\tilde { f } = \\left ( \\begin{array} { c } f _ { i _ { 1 } } \\\\ \\vdots \\\\ f _ { i _ { k } } \\end{array} \\right ) : M \\to \\bigoplus _ { s = 1 } ^ { k } N ( i _ { k } ) , \\tilde { g } = ( g _ { - i _ { 1 } } , \\cdots , g _ { - i _ { k } } ) : \\bigoplus _ { s = 1 } ^ { k } N ( i _ { k } ) \\to M , \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} q _ i ^ T q _ j = 0 ~ \\mbox { o v e r } ~ \\mathbb { F } _ p , ~ i , j = 1 , 2 , \\cdots , s . \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{gather*} H = \\frac { p ^ 2 } { 2 } - \\left ( u ^ 2 + \\frac { t } { 2 } \\right ) p - \\frac { u } { 2 } . \\end{gather*}"}
-{"id": "7396.png", "formula": "\\begin{align*} e ^ { { - { i } \\tau \\kappa } } e ^ { - 2 \\pi i b \\cdot v } D _ { A _ R } e ^ { 2 \\pi i b \\cdot v } e ^ { { { i } \\tau \\kappa } } = D _ { A _ R } + c ( d u ) , \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\underbrace { \\sum _ { ( u , v ) \\in [ r ] \\times [ s ] } \\lambda ^ p _ { ( u - 1 ) s + v } \\cdot c ( x ; ( u , v ) ) } _ { } + \\underbrace { \\sum _ { v \\in [ s ] } \\rho \\cdot c ( \\overline { x + p } ; ( x , v ) ) } _ { } = 0 , \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} c ( x ) = \\sum _ { j = 0 } ^ { p ^ k - 1 } x ^ { j p ^ { e - k } } b ( x ) . \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} | I | \\lesssim \\frac { 1 } { \\tau ^ { 2 j - 1 + l } } \\| v ^ { - } \\| _ { l ^ { \\frac { 2 j } { j ^ - + 1 } } _ \\tau D U ^ 2 } \\| v ^ { + } \\| _ { l ^ { \\frac { 2 j } { j ^ + + 1 } } _ \\tau D U ^ 2 } \\prod _ { i = 2 + j ^ - } ^ { 2 j - j ^ + - 1 } \\| v _ i \\| _ { l ^ { 2 j } _ \\tau D U ^ 2 } \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} H _ { 2 } = \\sum _ i D _ i ^ 2 + \\beta \\sum _ { i , j , a , b } E ^ { a b } _ i E ^ { b a } _ j D _ j - \\beta s \\sum _ i D _ i + \\\\ + \\frac { \\beta ^ 2 } { 3 } \\sum _ { a , b , c } \\mathbb { E } ^ { a b } \\mathbb { E } ^ { b c } \\mathbb { E } ^ { c a } - \\frac { 2 s \\beta ^ 2 } { 3 } \\sum _ { a , b } \\mathbb { E } ^ { a b } \\mathbb { E } ^ { b a } + \\beta ^ 2 \\frac { 2 s ^ 2 + p _ 0 - 1 } { 6 } p _ 0 , \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} A _ n ( z ) = \\begin{bmatrix} - N a & 0 \\\\ 0 & 0 \\end{bmatrix} + \\frac { 1 } { z } \\begin{bmatrix} n & N a b _ n \\\\ - N a c _ n & 0 \\end{bmatrix} + \\mathcal { O } \\left ( z ^ { - 2 } \\right ) . \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} F l ^ { - 1 } = ( F _ L l ^ { - 1 } ) ( l F _ N l ^ { - 1 } ) \\subset L \\Lambda = S , \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} J _ { \\lambda , \\mu } ( u _ k , v _ k ) = \\Big ( \\frac { 1 } { p } - \\frac { 1 } { \\alpha + \\beta } \\Big ) \\| ( u _ k , v _ k ) \\| ^ p - \\Big ( \\frac { 1 } { q } - \\frac { 1 } { \\alpha + \\beta } \\Big ) \\int _ { \\Omega } ( \\lambda | u _ { k } | ^ q + \\mu | v _ { k } | ^ q ) d x < \\frac { c _ { \\lambda , \\mu } } { 2 } . \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} \\textbf { ( M ) } _ m : \\hbox { t h e v e c t o r f i e l d s $ \\frac { \\partial } { \\partial y ^ { 1 - m } } $ i s t a n g e n t t o t h e n u l l g e o d e s i c s g e n e r a t i n g $ \\mathcal { I } ^ m , \\ ; m = 0 , 1 $ ; } \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} M : = \\left \\{ ( x _ 1 , x _ 2 , \\dots , x _ d ) \\in \\mathbb { R } ^ d : x _ i = 0 \\textrm { f o r } i > n \\right \\} \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} \\mathcal { P } _ 1 \\left ( { x } \\right ) = \\exp \\left \\{ - 2 \\pi { \\lambda _ e } \\int _ 0 ^ \\infty { { f _ { \\Pr } } ( { r _ e } ) } { r _ e } \\sum \\limits _ { \\ell , n \\in \\left \\{ { { } , { } } \\right \\} } { \\rm { \\mathbf { 1 } } } \\left ( { \\max \\{ { r _ e } , d \\} } < \\big ( \\frac { { P _ t } { G _ \\ell } { G _ n ^ e } \\beta } { x \\sigma _ e ^ 2 } \\big ) ^ { \\frac { 1 } { { \\alpha _ { \\mathrm { L o S } } } } } \\right ) { { { \\Pr } _ { \\ell n } } } d { r _ e } \\right \\} \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\frac { \\mathsf { C } _ { \\mathcal { N } _ { \\mathcal { K } } } } { { \\mathsf { C } } _ { \\mathcal { N } _ { [ 1 : N ] } } } = \\frac { N + 2 } { 4 N } , \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} 0 = & \\| \\nabla ( \\eta h ) \\| ^ 2 _ { L ^ 2 } - \\| | d \\eta | h \\| ^ 2 _ { L ^ 2 } + \\langle c ( F ) \\eta h , \\eta h \\rangle _ { L ^ 2 } \\\\ \\geq & \\| \\nabla _ { \\hat e _ 1 } ( \\eta h ) \\| ^ 2 _ { L ^ 2 } + \\| \\frac { \\eta \\nabla _ { \\hat e _ 1 } h } { \\sqrt { 2 + r ^ { - 1 } } } \\| ^ 2 _ { L ^ 2 } - \\| | d \\eta | h \\| ^ 2 _ { L ^ 2 } \\\\ & - \\int O ( r ^ { - 3 } \\eta ^ 2 | h | ^ 2 ) d v - \\int O ( \\eta ^ 2 r ^ { 1 - N } ) d v , \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} n _ { + } \\big ( r , \\textup { \\textbf { w } } p \\textup { \\textbf { w } } ^ { \\ast } \\big ) = n _ { + } \\big ( r , p \\textup { \\textbf { w } } ^ { \\ast } \\textup { \\textbf { w } } p \\big ) = n _ { + } \\Big ( r , p \\textbf { \\textup { W } } \\big ( H _ + ^ { - 1 } \\big ) p \\Big ) , r > 0 . \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} A = 2 ^ 6 ( 2 \\tilde { c } _ 4 - 3 \\tilde { \\Delta } ) = 2 ^ 6 \\tilde { A } B = 2 ^ { 1 2 } ( 2 ^ 2 \\tilde { c } _ 4 ^ 2 + 6 \\tilde { c } _ 4 \\tilde { \\Delta } ^ { 1 / 3 } + ( 3 \\tilde { \\Delta } ^ { 1 / 3 } ) ^ 2 ) = 2 ^ { 1 2 } \\tilde { B } \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} \\overline { v } ( \\varphi , x ) : = \\sum _ { j \\in S } \\sqrt { \\lvert j \\rvert \\xi _ j } \\ , e ^ { \\mathrm { i } \\mathtt { l } ( j ) \\cdot \\varphi } \\ , e ^ { \\mathrm { i } j x } \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\ge X \\\\ ( m , k ) = 1 } } \\mu ( m ) \\tau _ r ( m ) ( \\log { m } ) ^ a \\ll \\sigma _ { - 1 } ( k ) \\exp ( - c \\sqrt { \\log { x } } ) . \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} F _ 1 : y ^ 2 = x ^ 3 + 2 \\sqrt { \\ell - 2 } x ^ 2 + \\ell x F _ 2 : y ^ 2 = x ^ 3 + 2 \\sqrt { \\ell - 2 } x ^ 2 - 2 x . \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} C _ \\mathsf { R L N } = \\max _ { p _ X , p _ { A | S } , p _ { B | A } } R _ \\mathsf { R L N } \\left ( p _ X , p _ { A | S } , p _ { B | A } \\right ) . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} L _ 2 = - \\frac { v _ 1 - v } { 2 } \\frac { v _ 1 ' + v ' } { 2 } - \\frac { ( v ' ) ^ 3 } { 6 } + \\frac { ( v '' ) ^ 2 } { 2 } . \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} 1 = \\sum _ { l = 1 } ^ { d + 1 } z ^ { l } = \\sum _ { l = 1 } ^ { d + 1 } l \\cdot ( z ^ l - z ^ { l + 1 } ) \\le \\sum _ { l = 1 } ^ { d + 1 } l \\cdot \\max _ k ( z ^ k - z ^ { k + 1 } ) , \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} \\left \\| k \\ast g \\right \\| _ { L ^ { r } ( U ) } = \\left \\| \\ ; \\int _ { \\R ^ { N } } k ( \\cdot - y ) g ( y ) \\ d y \\right \\| _ { L ^ { r } ( U ) } \\leq C \\| k \\| _ { L ^ { q , w } ( \\R ^ { N } ) } \\| g \\| _ { L ^ { p } ( U ) } . \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} \\Delta ( z ) = \\frac { E _ 4 ( z ) ^ 3 - E _ 6 ( z ) ^ 2 } { 1 7 2 8 } . \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{align*} \\mathcal { B } _ { - } ( \\lambda ) = - a _ { - } ( \\lambda ) A ( \\lambda ) B ( 1 / \\lambda ) + b _ { - } ( \\lambda ) A ( \\lambda ) A ( 1 / \\lambda ) - c _ { - } ( \\lambda ) B ( \\lambda ) B ( 1 / \\lambda ) + d _ { - } ( \\lambda ) B ( \\lambda ) A ( 1 / \\lambda ) , \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} - L ( e _ { k j } ) & = [ L ( e _ { i j } ) , e _ { k i } ] + [ e _ { i j } , L ( e _ { k i } ) ] \\\\ & = C _ { j k } ^ { j j } e _ { i i } - C _ { i j } ^ { i j } e _ { k j } - \\sum _ { y > j } C _ { j y } ^ { j j } e _ { k y } \\\\ & + C _ { j k } ^ { k k } e _ { i i } - \\sum _ { x < k } C _ { x k } ^ { k k } e _ { x j } - C _ { k i } ^ { k i } e _ { k j } \\\\ & = - \\sum _ { x < k } C _ { x k } ^ { k k } e _ { x j } - \\left ( C _ { i j } ^ { i j } + C _ { k i } ^ { k i } \\right ) e _ { k j } - \\sum _ { y > j } C _ { j y } ^ { j j } e _ { k y } + \\left ( C _ { j k } ^ { j j } + C _ { j k } ^ { k k } \\right ) e _ { i i } . \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{gather*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } | \\nabla v _ n | ^ 2 d x = \\int _ { \\mathbb { R } ^ N } | \\nabla \\varphi | ^ 2 d x , \\\\ \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } A ( | v _ n ( x ) | ) d x = \\int _ { \\mathbb { R } ^ N } A ( | \\varphi ( x ) | ) d x . \\end{gather*}"}
-{"id": "2147.png", "formula": "\\begin{align*} \\begin{pmatrix} \\chi _ 3 & * \\\\ 0 & 1 \\end{pmatrix} \\begin{pmatrix} 1 & * \\\\ 0 & \\chi _ 3 \\end{pmatrix} , \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} \\sum _ { c } \\frac { S ( b , 2 n ; c ) } { c } \\Theta \\left ( \\frac { 4 \\pi \\sqrt { 2 n b } } { c } \\right ) = \\mathcal { H } ( n ) + \\mathcal { M } ( n ) + \\mathcal { E } ( n ) + , \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} a _ 6 ' = \\frac { a _ 6 + r a _ 4 + r ^ 3 - t ^ 2 } { u ^ 6 } a _ 6 = - \\frac { 2 ^ { n - 5 } \\tilde { c } _ 6 } { 3 ^ 3 } \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} u ( x , t ) \\ge \\sup _ { a , b : \\ , v _ { a , b } \\le u _ 0 } u _ { a , b } ( x , t ) = \\sup _ { a , b : \\ , v _ { a , b } \\le u _ 0 } ( a \\cdot x + b ) + t \\int _ 0 ^ 1 g ( s ) d s = u _ 0 ( x ) + t \\int _ 0 ^ 1 g ( s ) d s . \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} & \\mathcal { N } _ { \\lambda , \\mu } ^ + : = \\big \\{ ( u , v ) \\in \\mathcal { N } _ { \\lambda , \\mu } : \\varphi _ { u , v } '' ( 1 ) > 0 \\big \\} , \\\\ & \\mathcal { N } _ { \\lambda , \\mu } ^ - : = \\big \\{ ( u , v ) \\in \\mathcal { N } _ { \\lambda , \\mu } : \\varphi _ { u , v } '' ( 1 ) < 0 \\big \\} , \\\\ & \\mathcal { N } _ { \\lambda , \\mu } ^ 0 : = \\big \\{ ( u , v ) \\in \\mathcal { N } _ { \\lambda , \\mu } : \\varphi _ { u , v } '' ( 1 ) = 0 \\big \\} . \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} & \\Pr _ x [ T _ { V ( H ) } > t / 2 , T _ { R _ i \\cup R _ { i - 1 } } > t / 2 ] \\| \\Pr _ x ^ { t / 2 } [ \\cdot \\mid T _ { V ( H ) } > t / 2 , T _ { R _ i \\cup R _ { i - 1 } } > t / 2 ] - \\pi ( \\cdot ) \\| _ { \\ell , \\pi } = o ( 1 ) \\\\ \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} [ a ] \\odot [ b ] : = [ a b ] . \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\psi ( n ) = - \\frac { 3 } { 2 \\pi ^ 2 } x ^ 2 + O \\left ( x ( \\log x ) ^ { 2 / 3 } \\right ) , \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} \\gamma : = \\lim _ { m \\to \\infty } \\left | \\sum _ { k = 1 } ^ { m - 1 } \\frac { 1 } { k } - \\log ( m ) \\right | . \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} v ( s ) = \\gamma + \\delta \\bar k ( s ) - k ( s ) \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} & \\lambda ^ \\mathsf { a p p } _ 1 ( \\mathcal { Q } _ m ) = \\lambda _ 1 ( \\mathcal { Q } _ { \\infty } ) + \\frac { \\lambda } { m } + \\mathcal { O } ( m ^ { - 2 } ) \\ , , \\\\ & \\psi ^ { \\mathsf { a p p } } _ { 1 , m } = \\psi _ { 1 , \\infty } + m ^ { - 1 } \\varphi + \\mathcal { O } ( m ^ { - 2 } ) \\ , , \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} \\phi = \\lim _ { \\ell \\to \\infty } \\phi _ { \\ell } , \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{align*} \\delta E ^ { ( 2 ) } = \\frac { 1 } { 2 } \\sum _ { i a , j b } \\frac { \\partial E } { \\partial k _ { i a } \\partial k _ { j b } } \\delta k _ { i a } \\delta k _ { j b } . \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} \\frac { d } { d t } ( \\phi + t v ) ^ * ( p ) = - v ( T _ \\phi ( p ) ) . \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} F \\left ( u , v \\right ) = \\delta \\varepsilon , \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} x _ 1 = \\log \\frac { 1 - t _ 1 } { 1 - t _ 2 } , x _ 2 = \\log \\frac { 1 } { t _ 2 } \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} \\begin{cases} d X _ t ^ \\varepsilon = \\cfrac { 1 } { \\varepsilon } \\ , b ( X _ t ^ \\varepsilon ) d t + d W _ t \\ \\ \\ t \\in \\mathbb R , \\\\ X ( 0 ) = x \\in \\mathbb R ^ 2 , \\end{cases} \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} \\psi ( t , y ' ) = \\left | f ( \\Phi _ { - t } ( y ) ) - f ( \\Phi _ { - t } ( y ' ) ) \\right | \\ ; . \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} \\widetilde { R } ( x ) \\ll _ A \\frac { 1 } { ( 2 \\pi ) ^ l \\log x } \\prod _ { j = 1 } ^ l \\oint _ { | z _ j | = r _ j } ( \\log x ) ^ { \\frac { \\Re ( z _ j ) } { \\phi ( q ) } } \\frac { | d z _ j | } { | z _ j | ^ { k _ j + 1 } } . \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{gather*} F _ { 6 } \\big ( 3 ^ { - 2 / 3 } t \\big ) = \\frac { 1 - q _ 2 } { 2 } \\exp \\left ( \\frac { 1 } { 3 } \\int _ { t } ^ { \\infty } \\omega ( s ) d s + \\frac { 2 } { 3 } \\int _ { t } ^ { \\infty } \\alpha ( s ) d s - \\frac { 1 } { 3 } \\int _ { t } ^ { \\infty } \\frac { u _ s ( s ) } { u ( s ) } ( 1 + q _ 2 ( s ) ) d s \\right ) , \\end{gather*}"}
-{"id": "1287.png", "formula": "\\begin{align*} \\max \\{ \\alpha \\ , | \\ , \\alpha = ( \\nabla g ( x ) ) ^ T f _ c ( x ) , ~ g ( x ) = 0 \\} . \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} ( \\pi ^ \\Lambda _ * F ) _ b : = \\Gamma ( Z _ b , ( \\Lambda ( T ^ V X ) ^ * \\otimes F ) | _ { Z _ b } ) . \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} J ^ * ( u , v ) = \\frac { \\xi ( w ) } { \\zeta ( 2 ) } \\sum _ { c \\mid w } \\chi ( c ) \\sum _ { ( d , D ) = 1 } d ^ { - 1 } \\int _ 0 ^ \\infty \\phi \\left ( \\frac { d T } { x } \\right ) h \\left ( \\frac { c x } { u } \\right ) h \\left ( \\frac { c x } { v } \\right ) \\frac { d x } { x } . \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} I _ j : = [ A _ { j - 1 } , A _ j ) , \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} | \\tau \\rangle = \\sum _ { h _ { 1 } , . . . , h _ { \\mathsf { N } } = 0 } ^ { p - 1 } \\prod _ { a = 1 } ^ { \\mathsf { N } } Q _ { \\tau , a } ^ { ( h _ { a } ) } \\prod _ { 1 \\leq b < a \\leq \\mathsf { N } } ( X _ { a } ^ { ( h _ { a } ) } - X _ { b } ^ { ( h _ { b } ) } ) | h _ { 1 } , . . . , h _ { \\mathsf { N } } \\rangle , \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} U ( A ) f _ 1 \\otimes f _ 2 \\otimes \\dots \\otimes f _ { m + n } : = g _ 1 \\otimes g _ 2 \\otimes \\dots \\otimes g _ { m + n } , \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} & \\frac { d f ( \\gamma , 1 ) } { d \\gamma } = \\\\ & \\frac { k _ n P ' } { 4 + 2 \\gamma k _ n P ' } - \\frac { 1 - \\epsilon } { 2 } \\log ( 1 + k _ n P ' ) + \\frac { k _ n } { n } \\log \\frac { \\gamma } { 1 - \\gamma } , \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} \\epsilon ( s , \\lambda , \\xi , \\psi ) = \\prod _ { i = 1 } ^ k \\prod _ { j = 1 } ^ l \\epsilon ( s , \\delta _ i , \\delta _ j ' , \\psi ) . \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} a ( x ) + b ( x ) = 1 - x , \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} K \\cap B _ { r } ( x ) \\cap ( x + C ( V _ { x _ { 0 } } ^ { \\bot } , \\tilde { \\theta } ) ) = x \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ { p ( x ) } u = h ( x ) & \\Omega , \\\\ u = 0 & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{align*} F ( z ) = \\frac { \\Psi _ N ^ * ( z ) + z \\Psi _ N ( z ) f _ N ( z ) } { \\Phi _ N ^ * ( z ) - z \\Phi _ N ( z ) f _ N ( z ) } , \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} Q ( \\cdot , [ u ] ) = \\bold { p r } X _ 1 ( Q _ 2 ) - \\bold { p r } X _ 2 ( Q _ 1 ) . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} 0 & \\le \\int _ { \\R ^ { 2 N } } \\dfrac { | u ( x ) - u ( y ) | ^ { p - 2 } ( u ( x ) - u ( y ) ) ( v ( x ) - v ( y ) ) } { | x - y | ^ { N + s p } } d x d y \\\\ & = - 2 \\int _ { U } \\int _ { U ^ c } \\dfrac { | u ( x ) | ^ { p - 2 } u ( x ) v ( y ) } { | x - y | ^ { N + s p } } d x d y \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} S _ k \\widetilde { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } = S _ k \\left ( S _ { J _ 2 } \\left ( \\widetilde { D _ { J _ 1 } } \\widetilde { u } ^ { \\alpha } \\right ) \\right ) = S _ { J _ 2 + \\bold { 1 } _ k } \\left ( \\widetilde { D _ { J _ 1 } } \\widetilde { u } ^ { \\alpha } \\right ) = \\widetilde { u ^ { \\alpha } _ { J _ 1 ; J _ 2 + \\bold { 1 } _ k } } . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} R _ { \\theta } = \\left ( \\begin{matrix} \\cos { 2 \\pi \\theta } \\ \\ & - \\sin { 2 \\pi \\theta } \\\\ \\sin { 2 \\pi \\theta } \\ \\ & \\cos { 2 \\pi \\theta } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} c _ { u v } ( k ) = \\sum _ { m n = k } \\rho ( m ) \\lambda ( n ) g ( u m ) h ( v n ) . \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} & \\mu _ j ^ { \\infty } ( \\omega ) : = \\mu _ j ^ { \\infty } ( \\omega , i _ { \\delta } ( \\omega ) ) : = - \\mathrm { i } \\tilde { m } _ 3 ( \\omega ) \\ , j ^ 3 + \\mathrm { i } \\tilde { m } _ 1 ( \\omega ) j + r _ j ^ { \\infty } ( \\omega ) , j \\in S ^ c , \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} F = ( F _ 1 + \\Z \\alpha ) \\cup ( F _ 2 + \\Z \\alpha ) \\cup \\{ x \\in [ - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } ] \\ | \\ 2 x \\in \\Z \\alpha + \\Z \\} . \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} \\rho ( \\lambda ) = \\rho _ { \\infty } ( \\lambda ) + \\sum _ { i a } \\frac { s _ a \\rho _ { i a } ( \\lambda ) } { 2 4 L ^ 2 \\rho _ { \\infty } ( \\lambda _ { i a } ) } \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} \\Psi _ \\beta ( s ) : = s [ \\log ( e + s ) ] ^ \\beta , \\rho ( s ) : = s ^ { - N } \\biggr [ \\log \\biggr ( e + \\frac { 1 } { s } \\biggr ) \\biggr ] ^ { - \\frac { N } { \\theta } } . \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} ( N - 1 ) \\mathsf { C } _ { \\mathcal { N } _ { [ 1 : N ] } } \\leq \\sum _ { i = 1 } ^ N { \\mathsf { R } } ^ { \\lambda ^ \\star } _ { \\bar { \\mathcal { N } } _ i } & \\leq \\sum _ { i = 1 } ^ N { \\mathsf { C } } _ { \\bar { \\mathcal { N } } _ i } \\leq N \\max _ { i \\in [ 1 : N ] } { \\mathsf { C } } _ { \\bar { \\mathcal { N } } _ i } . \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} C _ 0 = 1 , \\ \\ \\ \\ \\ \\ \\ \\ C _ n = [ C _ { n - 1 } \\exp ( W _ n ) ] + 1 , \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} d _ { S ( G , t + 1 ) } ( x ^ { t + 1 } , y ^ { t + 1 } ) = & d _ { S ( G , t + 1 ) } ( v _ 0 ^ { t + 1 } , v _ 0 v _ 1 ^ t ) + \\sum _ { i = 0 } ^ { r - 2 } d _ { S ( G , t + 1 ) } ( v _ { i + 1 } v _ i ^ t , v _ { i + 1 } v _ { i + 2 } ^ t ) + \\\\ & + d _ { S ( G , t + 1 ) } ( v _ r v _ { r - 1 } ^ t , v _ r ^ { t + 1 } ) + r . \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} F _ N F _ L \\Lambda L \\Lambda \\cap N = N . \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} f = \\sum _ K a _ { i j } ^ K f _ { i j , K } , \\mbox { w h e r e } f _ { i j , K } \\in \\mathbb { C } [ \\{ { } _ { ( \\alpha ) } a _ { s t } : ( s , t ) \\not = ( i , j ) \\} ] , \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ p ) ^ s u = \\lambda _ n | u | ^ { p - 2 } u + \\dfrac { f ( x ) } { \\| u _ n \\| _ { \\scriptstyle L ^ { q } ( \\Omega ) } ^ { p - 1 } } & \\Omega , \\\\ u = 0 & \\Omega ^ c . \\end{cases} \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} R _ { a , b } ( x _ a / x _ b ) R _ { a , c } ( x _ a / y _ c ) R _ { b , c } ( x _ b / y _ c ) = R _ { b , c } ( x _ b / y _ c ) R _ { a , c } ( x _ a / y _ c ) R _ { a , b } ( x _ a / x _ b ) , \\\\ R _ { b , a } ( y _ b / y _ a ) R _ { c , a } ( x _ c / y _ a ) R _ { c , b } ( x _ c / y _ b ) = R _ { c , b } ( x _ c / y _ b ) R _ { c , a } ( x _ c / y _ a ) R _ { b , a } ( y _ b / y _ a ) . \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} \\gamma = \\varepsilon ^ { 2 + a } , \\mbox { f o r \\ , \\ , s o m e } \\ , \\ , a > 0 . \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} \\delta ( x ) = \\exp \\left ( - c _ 1 ( \\log x ) ^ { 3 / 5 } ( \\log \\log x ) ^ { - 1 / 5 } \\right ) , \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\vdash n } \\sum _ { \\substack { \\lambda _ i \\in \\lambda \\\\ \\lambda _ i } } A ( \\lambda _ i ) = \\sum _ { k = 1 } ^ { n } p ( n - k ) \\sum _ { d | k } a ( d ) = \\sum _ { \\lambda \\vdash n } \\sum _ { \\lambda _ i \\in \\lambda } a ( \\lambda _ i ) . \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\int \\left ( \\eta '' - \\xi '' \\right ) \\left ( \\mbox { \\ensuremath { \\phi _ { \\nu } } - \\ensuremath { \\phi _ { \\mu } } } \\right ) d t = h ^ { 2 } ( \\phi _ { \\nu } ( 0 ) - \\phi _ { \\mu } ( 0 ) ) - \\int \\left ( \\eta - \\xi \\right ) d ( \\nu - \\mu ) . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} H ( z ) = \\left ( N ^ { c / 2 } \\eta ( z ) \\right ) ^ { - \\sigma _ 3 } { \\cal R } ( z ) \\left ( N ^ { c / 2 } \\eta ( z ) \\right ) ^ { \\sigma _ 3 } F _ 1 ( \\zeta ( z ) ) ^ { - 1 } , \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} P _ 2 \\colon W ( k ) ( ( T ) ) \\to & K = k ( ( T ) ) \\\\ \\sum _ i a _ i T ^ i \\mapsto & \\sum _ i \\pi _ 1 ( a _ i ) T ^ i , \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} \\operatorname { D i v } P = \\sum _ { \\alpha , J } B ^ { \\alpha } _ J ( D _ J F _ { \\alpha } ) . \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} ( e _ 1 + e _ 2 + \\cdots e _ n , e _ i ) = ( e _ i , e _ i ) \\mod 2 \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} ( q ^ { \\frac { n - k + 2 } { 2 } } + m ) ( q ^ { \\frac { n - k + 2 } { 2 } } - m ) = q ^ { n - k + 2 } \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} C _ { n } ( x ) = n \\phi ( z _ n ) \\frac { d z _ n } { d x } \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} \\bold { D } _ F ( Q ) = 0 \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} b . ( g ' , \\theta , r , s , i , j ) = ( g ' b ^ { - 1 } , b \\theta b ^ { - 1 } , b r b ^ { - 1 } , b s b ^ { - 1 } , i , j ) . \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} \\langle f , g \\rangle = \\int _ { - 1 } ^ { 1 } f ( t ) g ( t ) d t \\enspace . \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} D ( F _ { n _ 1 } + \\ldots + F _ { n _ { k - 1 } } + F _ n ) = D ( F _ { n _ 1 } + \\ldots + F _ { n _ { k - 1 } } ) + D ( F _ n ) - D ( ( F _ { n _ 1 } + \\ldots + F _ { n _ { k - 1 } } ) \\cap F _ n ) . \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} \\int _ 0 ^ { x _ { \\sqrt \\mu } } V \\sin 2 \\zeta \\ , \\dd x = o ( 1 ) , \\mu \\to \\infty \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} \\int _ { \\Omega _ { \\varphi _ i } } \\langle \\nabla u _ i , \\nabla \\zeta \\rangle - \\abs { \\nabla u _ i } ^ 2 u _ i \\cdot \\zeta d x = 0 , \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} A = - \\frac { \\dd ^ 2 } { \\dd x ^ 2 } + Q ( x ) . \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} R _ { n , d } ( \\mathcal { F } ) = \\inf _ { \\hat { f } } \\sup _ { f \\in \\mathcal { F } } \\mathbb { E } \\| f - \\hat { f } \\| ^ 2 , \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ X ( t , i ) | X ( t - 1 , i ) = 0 , t \\leq T \\right ] = \\frac { \\hat { \\pi } ( t ) - \\hat { \\pi } ( t - 1 ) } { 1 - \\hat { \\pi } ( t - 1 ) } , \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} g '' _ 2 ( B _ n ) & = B _ n ^ { - 1 } e ^ { - \\left ( \\sqrt { B _ n / 2 } - \\sqrt { P / 2 } \\right ) ^ 2 } \\cdot \\left ( - \\frac { 1 } { 4 \\pi } e ^ { - \\left ( \\sqrt { B _ n / 2 } - \\sqrt { P / 2 } \\right ) ^ 2 } \\right . \\\\ & \\left . + \\frac { 1 } { 4 \\sqrt { 2 \\pi } } ( 1 + { \\rm { e r f } } ( \\sqrt { B _ n / 2 } - \\sqrt { P / 2 } ) ) \\left ( B _ n ^ { 1 / 2 } - \\sqrt { P } + B _ n ^ { - 1 / 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} Z _ v = \\sum _ { n \\ge 1 } c _ n n v ^ { n - 1 } \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\phi \\otimes \\psi + \\phi ^ \\vdash ( y ) & = \\inf _ { x \\in X } ( \\phi ( x ) + \\psi ( x ) ) + \\phi ^ \\vdash ( y ) \\\\ & = \\inf _ { x \\in X } ( \\phi ( x ) + \\psi ( x ) + \\phi ^ \\vdash ( y ) ) \\\\ & \\geq \\inf _ { x \\in X } ( d ( x , y ) + \\psi ( x ) ) \\\\ & \\geq \\psi ( y ) , \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} f = f _ 0 + \\sum _ { j \\geq 1 } g _ j , \\ , \\ , \\ , g _ j = f _ { j + 1 } - f _ { j } , \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\rm w } = \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 2 , \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} \\frac { d c _ 1 } { d t } = \\frac { 1 } { \\sqrt { 2 } } c _ 2 . \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} \\partial ^ n _ s \\partial ^ m _ x P _ s f ( x ) = \\int _ { \\mathbb { R } } f ( y ) \\partial ^ n _ s \\partial ^ m _ x p _ s ( x - y ) d y . \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} \\frac { d } { d t } | \\varphi _ t ( \\omega ^ 0 , x ) - \\varphi _ t ( \\omega ^ 0 , y ) | ^ 2 & = 2 \\left \\langle b ( \\varphi _ t ( \\omega ^ 0 , x ) ) - b ( \\varphi _ t ( \\omega ^ 0 , y ) ) , \\varphi _ t ( \\omega ^ 0 , x ) - \\varphi _ t ( \\omega ^ 0 , y ) \\right \\rangle \\\\ & \\leq - 4 \\left | \\varphi _ t ( \\omega ^ 0 , x ) - \\varphi _ t ( \\omega ^ 0 , y ) \\right | ^ 2 . \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} Q ( z ) d z ^ 2 = - e ^ { - i \\psi } \\frac { ( e ^ { i \\psi } + z ) ^ 2 } { z ^ 2 ( z - r ) ( z - 1 / r ) } d z ^ 2 . \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} L = \\ln ( ( 1 + u v ) ( 1 - u v _ 1 ) ) \\frac { u ' } { u } - a ( u v _ { - 1 } - u v - u u _ 1 v v _ 1 ) - b ( u _ { - 1 } v _ 1 - u v _ 1 + u _ { - 1 } u v v _ 1 ) . \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} \\int _ { S ^ 1 } | \\nabla \\sigma ( \\tau ) | ^ 2 d \\tau & = \\sum _ a \\sum _ k \\int _ 0 ^ { 2 \\pi } | \\sigma _ { a k } | ^ 2 ( k - \\mu _ a ) ^ 2 d \\tau \\\\ & \\geq \\kappa ^ 2 \\int _ { S ^ 1 } | \\sigma ( \\tau ) | ^ 2 d \\tau . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\det \\mathbb { M } : = \\det ( D _ S ) \\ , \\det \\left ( \\dfrac { \\partial ^ 2 } { \\partial I _ j \\ , I _ k } \\tilde { h } ( I ) \\right ) _ { j , k \\in \\{ 1 , \\dots , \\nu \\} } \\ , \\det ( D _ S ) \\neq 0 . \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} \\vert ( z + e _ { 2 j } ) ( e _ { 2 j } + e _ { 2 N } ) \\vert ^ 2 + \\vert ( z - e _ { 2 j } ) ( e _ { 2 j } - e _ { 2 N } ) \\vert ^ 2 = 2 \\vert z + e _ { 2 N } \\vert ^ 2 + 2 \\vert z e _ { 2 N } + e _ j \\vert ^ 2 \\\\ \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\begin{gathered} X _ { 1 } = \\partial _ { \\tau } , X _ { 2 } = \\partial _ { u } , X _ { 3 } = \\partial _ { v } , \\\\ X _ { 4 } = \\tau \\partial _ { \\tau } + u \\partial _ { u } + v \\partial _ { v } . \\end{gathered} \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} f _ { \\Theta _ \\rho } ( \\theta ) = \\begin{cases} \\frac { 1 } { \\zeta } e ^ { - \\theta / \\zeta } , & 0 < \\theta , \\rho = 0 , \\\\ \\frac { 1 } { \\zeta } e ^ { - ( \\theta - \\sigma ^ 2 _ { \\rm a } ) / \\zeta } , & \\sigma _ { \\rm a } ^ 2 < \\theta , \\rho = 1 , \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} \\lim _ { | \\lambda | \\to 0 } \\lim _ { V \\to \\infty } \\frac { \\alpha ( | \\lambda | , V ) } { | \\lambda | } = 0 \\ . \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} \\{ A , B \\} : = [ A , B ] - ( A , B ) , \\forall A , B \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} \\Phi _ q ( y , z _ 0 ) - \\Phi _ q ( \\bar { y } , z _ 0 ) & = \\int _ 0 ^ t T _ q ( t - \\tau ) P _ q ^ { \\mathrm { s } } \\big ( F _ q ( y ( \\tau ) ) - F _ q ( \\bar { y } ( \\tau ) ) \\big ) \\dd \\tau \\\\ & - \\int _ t ^ { \\infty } P _ q ^ { \\mathrm { c } } \\big ( F _ q ( y ( \\tau ) ) - F _ q ( \\bar { y } ( \\tau ) ) \\big ) \\dd \\tau . \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{gather*} \\alpha = \\beta + d \\gamma . \\end{gather*}"}
-{"id": "4435.png", "formula": "\\begin{align*} \\mu _ k ( { \\bf { x } } , Q ) : = ( { \\bf { x } } ' , \\mu _ k ( Q ) ) , { \\bf { x } } ' : = ( { \\bf { x } } \\backslash \\{ x _ k \\} ) \\cup \\{ x ' _ k \\} \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} \\max \\{ \\beta \\ , | \\ , x ^ T Q _ 1 x = 1 , x ^ T Q _ 2 x = 1 , \\beta = x ^ T A ^ T Q _ 2 x \\} . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} & \\left \\| \\nabla ^ { k } _ { x } \\left ( \\partial _ { t } K _ { 3 } ( t ) g - \\mathcal { F } ^ { - 1 } \\left [ e ^ { - \\frac { \\nu t | \\xi | ^ { 2 \\sigma } } { 2 } } \\cos ( t | \\xi | ) \\right ] \\ast g \\right ) \\right \\| _ { 2 } \\\\ & \\le C ( 1 + t ) ^ { - \\frac { n } { 4 \\sigma } - 1 + \\frac { 1 } { 2 \\sigma } - \\frac { k } { 2 \\sigma } } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla ^ { k } _ { x } g \\| _ { 2 } , \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} a ^ i ( \\theta _ t ) = \\frac { ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } } { \\sum _ j ( \\theta _ t ^ j ) ^ { \\frac { 1 } { \\gamma } } } b ^ i ( \\theta _ t ) = \\frac { \\gamma \\hat { \\phi } } { \\eta } \\left [ \\frac { ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } } { \\sum _ j ( \\theta _ t ^ j ) ^ { \\frac { 1 } { \\gamma } } } - \\frac { \\phi ^ i } { \\hat { \\phi } } \\right ] , \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{gather*} F \\big ( 3 ^ { - 1 / 3 } x , 3 ^ { - 2 / 3 } t ; \\beta = 6 \\big ) = \\kappa u ^ { \\frac { 1 } { 2 } } \\left ( \\frac { 1 - q _ 2 } { 2 } + O \\left ( \\frac { 1 } { x } \\right ) \\right ) , x \\to + \\infty , \\end{gather*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\bigl \\langle \\Xi ( X _ 1 - X ' _ 1 ) , Y \\bigr \\rangle & = \\bigl \\langle \\Xi ( X _ 1 - X ' _ 1 ) , [ Y _ 1 , b ] \\bigr \\rangle \\\\ & = - \\bigl \\langle \\Xi ( Y _ 1 ) , [ b , X _ 1 - X ' _ 1 ] \\bigr \\rangle - \\bigl \\langle \\Xi ( b ) , [ X _ 1 - X ' _ 1 , Y _ 1 ] \\bigr \\rangle \\\\ & = 0 \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} \\phi _ t ( r ) : = \\inf _ { \\substack { A \\subseteq V _ t : \\ , \\\\ \\pi ^ { ( t ) } ( A ) \\le \\frac { v ( t ) } { 2 } \\wedge r } } \\Big \\{ \\frac { \\pi ^ { ( t ) } ( A , A ^ c ) } { \\pi ^ { ( t ) } ( A ) } \\Big \\} , r \\ge 0 \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} G _ { i , j } = \\frac { \\sigma _ i - \\sigma _ { j } } { \\sigma _ i } ~ . \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} 2 \\imath \\psi _ { t } + \\psi _ { r r } + \\frac { 1 } { r } \\psi _ { r } + \\frac { 1 } { k ^ 2 r ^ 2 } \\psi _ { \\phi \\phi } - \\omega ^ 2 r ^ 2 \\psi = 0 . \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} u = \\left ( \\begin{pmatrix} 0 & a _ { 1 2 } \\\\ 0 & 0 \\end{pmatrix} , \\begin{pmatrix} b _ { 1 1 } & b _ { 1 2 } \\\\ 0 & 0 \\end{pmatrix} , \\begin{pmatrix} i _ 1 \\\\ i _ 2 \\end{pmatrix} , 0 , 0 , b _ { 1 1 } + f _ 2 , \\begin{pmatrix} f _ 1 \\\\ f _ 2 \\end{pmatrix} \\right ) \\\\ \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} \\phi _ { \\ell ( x ) - \\alpha x ^ 2 } ( t ) & = \\exp \\left \\{ { \\alpha x ^ 2 } \\frac { ( e ^ { i t } - 1 ) } { 2 x - ( 2 x - 1 ) e ^ { i t } } - \\alpha x ^ 2 i t \\right \\} \\\\ & = \\exp \\left \\{ { \\alpha x ^ 2 } \\left ( \\frac { ( e ^ { i t } - 1 ) } { 2 x - ( 2 x - 1 ) e ^ { i t } } - i t \\right ) \\right \\} . \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} c _ a = - \\frac { \\sinh ( b - x ) } { \\sinh ( b - a ) } , c _ b = - \\frac { \\sinh ( x - a ) } { \\sinh ( b - a ) } , \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} C _ q = \\frac { 2 L } { ( 1 - \\epsilon ) \\mu } , ~ ~ C _ l = \\frac { 3 \\epsilon \\sqrt { \\kappa } } { 1 - \\epsilon } . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} \\Phi = \\projlim _ { \\tau \\in T } \\mathcal { H } _ \\tau \\ , , \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} \\gamma _ u = \\prod _ { j \\in u } \\gamma _ j . \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{align*} \\dot x = A ( t ) x + c ( t ) , \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} \\sum _ { T \\subseteq U } ( - 1 ) ^ { | T | } v _ T = 0 \\forall v = ( v _ T ) _ T \\in V \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} | \\xi _ 4 ( 1 ) | _ 3 = q < q ^ 2 \\leq \\left | \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) } { | C _ G ( x ) | | C _ G ( y ) | } \\right | _ 3 , \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\pi \\in \\P _ e , \\\\ | \\pi | = N } } ( - 1 ) ^ { \\nu ( \\pi ) } = ( - 1 ) ^ { N + 1 } \\chi ( N \\not = \\triangle \\ , ) . \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} { \\cal F } ( \\zeta ) F _ 1 ( \\zeta ) ^ { - 1 } = I + { \\cal O } \\left ( \\frac { 1 } { | \\zeta ^ 2 | } \\right ) \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} S _ h ( x ) = S ^ * _ h ( x ) + x E _ h ( x ) \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} & v ^ + ( x ) = \\lim _ { r \\to 0 + } \\sup \\{ u ^ \\varepsilon ( y ) \\ | \\ y \\in B _ r ( x ) \\cap \\overline \\Omega , \\ \\varepsilon \\in ( 0 , r ) \\} , \\\\ & v ^ - ( x ) = \\lim _ { r \\to 0 + } \\inf \\{ u ^ \\varepsilon ( y ) \\ | \\ y \\in B _ r ( x ) \\cap \\overline \\Omega , \\ \\varepsilon \\in ( 0 , r ) \\} , \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} \\sum _ { \\substack { j _ 1 , \\ldots , j _ k \\ge 1 \\\\ j _ 1 + \\cdots + j _ k = \\nu } } a _ { j _ 1 } \\cdots a _ { j _ k } = \\sum _ { \\substack { t _ 1 , \\ldots , t _ { \\nu } \\ge 0 \\\\ t _ 1 + 2 t _ 2 + \\cdots + \\nu t _ { \\nu } = \\nu \\\\ t _ 1 + \\cdots + t _ { \\nu } = k } } \\binom { t _ 1 + \\cdots + t _ { \\nu } } { t _ 1 , \\ldots , t _ { \\nu } } a _ 1 ^ { t _ 1 } \\cdots a _ { \\nu } ^ { t _ { \\nu } } . \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} R ^ { s t a i r } ( \\ell ) & = \\frac { \\log _ 9 S _ { U , s t a i r } ( \\ell ) } { 8 1 + 5 4 ( \\ell - 1 ) } = : R _ U ^ { s t a i r } ( \\ell ) \\\\ & \\approx \\frac { 2 2 . 8 7 0 6 + 1 1 . 7 4 0 7 ( \\ell - 1 ) } { 8 1 + 4 5 ( \\ell - 1 ) } . \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\psi } = K _ { 2 0 } ( \\omega t ) \\eta + K _ { 1 1 } ^ T ( \\omega t ) w \\\\ \\dot { \\eta } = 0 \\\\ \\dot { w } = \\partial _ x K _ { 0 2 } ( \\omega t ) w + \\partial _ x K _ { 1 1 } ( \\omega t ) \\eta . \\end{cases} \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} V _ { n } = \\sum _ { j = 1 } ^ { n - 1 } ( 1 - \\rho ) ^ { j } e ^ { - \\sum _ { s = 1 } ^ j g ( W _ { s - 1 } , W _ { s } ) } . \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} \\mathcal { Y } ( x ^ 1 ) = \\operatorname { s u p p } \\mu ^ { \\pi _ 1 } _ { x ^ 1 } . \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} D _ { \\tau } ( \\lambda q ) = C _ { p \\rightarrow 1 } R _ { p \\rightarrow 1 } ( D _ { \\tau } ( \\lambda ) ) , \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} \\begin{cases} \\dot { \\theta } = \\partial _ y H _ { \\varepsilon } ( \\theta , y , z ) , \\\\ \\dot { y } = - \\partial _ { \\theta } H _ { \\varepsilon } ( \\theta , y , z ) , \\\\ \\dot { z } = \\partial _ x \\nabla _ z H _ { \\varepsilon } ( \\theta , y , z ) , \\end{cases} H _ { \\varepsilon } : = \\varepsilon ^ { - 2 b } \\ , \\mathcal { H } \\circ A _ { \\varepsilon } , \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} ^ { c } D ^ { q } x _ { k } ( t ) = - a _ { k } x _ { k } ( t ) + \\sum _ { j = 1 } ^ { n } T _ { k j } g _ { j } ( x _ { j } ( t ) ) + I _ { k } , \\forall k = \\overline { 1 , n } , ~ \\forall ~ t > 0 \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\iota _ m \\circ \\phi ( y _ { v } ) = \\prod _ { u \\in \\tilde Q ' _ { n , m } } x _ { u } ^ { b _ { u v } } . \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} h _ { \\mu } ( T _ { p } ) = h _ { \\mu } \\left ( T _ { p } , P _ { p } \\right ) = \\lim _ { N \\to \\infty } \\frac { 1 } { N } H _ { \\mu } \\left ( P _ { p ^ { N } } \\right ) . \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} - \\left . \\frac { \\partial p } { \\partial \\nu } \\right | _ { \\Gamma ( t ) } = \\frac 1 J \\partial _ \\beta \\left . \\left ( \\phi _ t + \\frac 1 2 | \\nabla \\phi | ^ 2 + g y \\right ) \\right | _ { \\{ \\beta = 0 \\} } . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} r = \\frac { c + 1 } { 3 ( 2 - c ) } . \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} k ^ \\prime ( u , v ) = \\sum _ { i = 1 } ^ { + \\infty } | u _ i - x _ i | ^ { p - 2 } ( u _ i - x _ i ) v _ i . \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} Q _ 1 = 1 , ~ Q _ 2 = v _ x , ~ Q _ 3 = v _ x ^ 2 + 2 v _ { x x x } , ~ Q _ 4 = t . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} \\underset { t \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\int _ 0 ^ t \\int _ { { \\bf R } ^ N } \\int _ { { \\bf R } ^ N } G ( x - y , t - s ) u ( y , s ) ^ p \\eta ( x ) \\ , d x \\ , d y \\ , d s = 0 . \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} G _ 1 = G , G _ { i + 1 } & = \\begin{bmatrix} P _ G ^ i G & P _ G ^ { i - 1 } G & \\hdots & G & - P _ G e _ 1 & \\hdots & - P _ G ^ i e _ 1 \\end{bmatrix} \\\\ B _ 1 = B , B _ { i + 1 } & = \\begin{bmatrix} B & P _ B B & \\hdots & P _ B ^ i B & P _ B ^ i e _ 1 & \\hdots & P _ B e _ 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} A _ { n + 1 } ( x ) = ( 1 + n x ) A _ n ( x ) + x ( 1 - x ) A _ n ' ( x ) , \\end{align*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\sigma \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) = \\left ( \\begin{array} { c c } a & c \\beta ^ { - 1 } \\\\ b \\beta & d \\end{array} \\right ) . \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} \\Phi _ B ( T _ { \\delta } ) ^ 2 = \\varepsilon ^ 2 v _ { \\delta } ^ 2 + \\varepsilon v _ { \\delta } \\tilde { q } + \\varepsilon ^ 3 v _ { \\delta } \\Psi _ 2 ( v _ { \\delta } ) + \\tilde { \\mathtt { Q } } _ 2 ( T _ { \\delta } ) , \\Phi _ B ( T _ { \\delta } ) ^ 3 = \\varepsilon ^ 3 v _ { \\delta } ^ 3 + \\tilde { \\mathtt { Q } } _ 3 ( T _ { \\delta } ) , \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} \\dot { U } _ { \\mathbb { O } _ p } : = \\dot { U } _ v \\otimes _ { \\Z [ v ^ { \\pm 1 } ] } \\mathbb { O } _ p . \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} \\lim _ { k _ { \\ell } \\to \\infty } \\log \\left ( 1 + \\frac { c k _ { \\ell } P ' } { 4 } \\right ) - \\log ( 1 + k _ { \\ell } P ' ) = \\log \\frac { c } 4 . \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} \\eta ( s ' _ 1 , \\ldots , s ' _ r ; s _ 1 , \\ldots , s _ r ) = \\eta ( s _ 1 , \\ldots , s _ r ; s ' _ 1 , \\ldots , s ' _ r ) . \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} \\pi _ 0 ( M ( y ) ) _ { \\sigma _ 0 } ^ \\circ \\cong \\pi _ 0 ( Z _ { G ^ \\circ } ( \\sigma _ 0 , y ) ) = \\pi _ 0 ( Z _ { Q ^ \\circ } ( \\sigma _ 0 , y ) ) = \\pi _ 0 ( Z _ { Q ^ \\circ } ( y ) ) . \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} U ( \\hat { x } _ t , \\theta _ t ) = \\frac { \\gamma } { 1 - \\gamma } \\left [ \\left ( \\sum _ i ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } \\right ) ^ \\gamma \\left ( \\hat { \\phi } + \\frac { \\eta } { \\gamma } \\ , \\hat { x } _ t \\right ) ^ { 1 - \\gamma } - 1 \\right ] , \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\tau _ i ( x ) = \\inf \\{ t > 0 \\ | \\ X ( t , x ) \\in l _ i \\} . \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} - \\phi _ 0 '' ( x ) = 0 \\ ( 0 , \\pi ) , \\phi _ 0 ( 0 ) = \\phi _ 0 ( \\pi ) = 0 . \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} \\int _ { E _ 2 } t _ 2 ^ 2 \\frac { d t _ 1 d t _ 2 } { ( 1 - t _ 1 ) t _ 2 } = \\sum _ { k = 1 } ^ { \\infty } \\frac { 1 } { k ( k + 2 ) } = \\frac { 3 } { 4 } \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} \\partial _ { t } v ( t , x , y ) = \\Delta v ( t , x , y ) + f ( t , x , y ) \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} | e _ N ( 1 ) | \\le \\sum _ { i = 1 } ^ N \\frac { 2 | E _ N ( \\tau _ i ) | } { ( 1 - \\tau _ i ^ 2 ) | P _ N ' ( \\tau _ i ) | } \\le 2 \\sqrt { N } \\left ( \\sum _ { i = 1 } ^ N \\frac { E _ N ^ 2 ( \\tau _ i ) } { ( 1 - \\tau _ i ^ 2 ) ^ 2 P _ N ' ( \\tau _ i ) ^ 2 } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} Q _ \\nu ( r ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ A _ \\nu ( r ) & 0 \\end{array} \\right ) \\quad \\Rightarrow Q ^ \\dagger _ \\nu ( r ) = \\left ( \\begin{array} { c c } 0 & A ^ \\dagger _ \\nu ( r ) \\\\ 0 & 0 \\end{array} \\right ) \\ ; , \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j \\leq ( 1 + \\delta ) k _ n } \\binom { \\ell _ n } { j } & \\leq 2 ^ { \\ell _ n } \\\\ & \\leq \\exp ( 2 ( 1 + \\delta ) k _ n \\log 2 ) . \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } x = - b ( \\tau , x ) . \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} ( \\operatorname { D P [ 0 ] } + \\operatorname { Q } ) ( \\Xi _ 1 + \\Xi _ 2 ) = 0 \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} V ^ { \\star } _ { - \\eta H ' } ( z , w ) = 2 \\pi i ^ { \\kappa _ 2 } \\int _ { 0 } ^ { \\infty } V _ { - \\eta H ' } ( y ) J _ { \\kappa _ 2 - 1 } ( 4 \\pi \\sqrt { - y w + z } ) \\mathrm { d } y . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} q ( \\alpha , \\beta ; t _ 1 , y _ 1 ; t _ 2 , y _ 2 ) : = \\frac { p ( \\beta , \\alpha ; t _ 1 , y _ 1 ) } { p ( \\alpha , \\beta ; t _ 2 , y _ 2 ) } . \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} K _ s ( z ) & : = \\int _ { 0 } ^ { \\infty } e ^ { - z \\cosh ( t ) } \\cosh ( s t ) d t \\\\ & = \\frac { 1 } { 2 } \\int _ { 0 } ^ { \\infty } e ^ { - \\frac { 1 } { 2 } z ( u + 1 / u ) } ( u ^ s + u ^ { - s } ) \\frac { d u } { u } , \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} \\lambda _ { 2 } = K - \\frac { \\kappa ^ 2 } { 4 } \\ , . \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{align*} \\Lambda _ A ( \\theta ) = \\begin{cases} - \\theta - \\log 2 , & ; \\\\ - \\theta - \\log ( 1 - \\theta ) , & ; \\\\ + \\infty , & . \\end{cases} \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} \\begin{aligned} ^ 2 ( \\ell _ 1 , \\ell _ 2 ) & : = \\det \\tilde { W } = - \\det \\Big { \\{ } \\Big ( ( X _ 1 + i Y _ 1 ) M _ 1 ^ 2 ( X _ 1 ^ t + i Y _ 1 ^ t ) \\Big ) \\cdot \\Big ( ( X _ 2 - i Y _ 2 ) M _ 2 ^ 2 ( X _ 2 ^ t - i Y _ 2 ^ t ) \\Big ) \\Big { \\} } \\\\ & = - \\frac { \\det ^ 2 ( X _ 1 + i Y _ 1 ) } { \\det ( X _ 1 ^ t X _ 1 + Y _ 1 ^ t Y _ 1 ) } \\cdot \\frac { \\det ^ 2 ( X _ 2 - i Y _ 2 ) } { \\det ( X _ 2 ^ t X _ 2 + Y _ 2 ^ t Y _ 2 ) } . \\end{aligned} \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} i \\partial _ t u = H u + F ( t ) ; u _ { | _ { t = 0 } } = \\psi , \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} \\begin{aligned} T ^ 0 ( X , Y ) + T ^ 0 ( I _ 1 X , I _ 1 Y ) + T ^ 0 ( I _ 2 X , I _ 2 Y ) + T ^ 0 ( I _ 3 X , I _ 3 Y ) = 0 , \\\\ U ( X , Y ) = U ( I _ 1 X , I _ 1 Y ) = U ( I _ 2 X , I _ 2 Y ) = U ( I _ 3 X , I _ 3 Y ) . \\end{aligned} \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } E _ { 1 } ( t ) g \\right \\| _ { 2 } + \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } E _ { 2 } ( t ) g \\right \\| _ { 2 } & \\le C ( 1 + t ) ^ { - \\frac { n } { 2 } - ( \\ell + k ) } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla _ { x } ^ { \\ell + k } g \\| _ { 2 } , \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} \\lambda _ j = \\underset { i \\in \\phi ^ { - 1 } ( j ) } \\Sigma \\ , \\lambda _ i . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} E _ k ( z ) = 1 + \\frac { 2 } { \\zeta ( 1 - k ) } \\sum _ { n = 1 } ^ \\infty \\sigma _ { k - 1 } ( n ) e ^ { 2 \\pi i n z } , \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} ( \\nabla ^ { F , u } ) ^ 2 = - \\frac { 1 } { 4 } \\omega ( F , g ^ F ) ^ 2 . \\end{align*}"}
-{"id": "4170.png", "formula": "\\begin{align*} - b ( x ) \\cdot D \\tilde \\psi ( x ) \\zeta _ { 2 r } ( x ) & \\leq | - b ( x ) \\cdot D \\tilde \\psi ( x ) | \\\\ & = | \\bar f ( x ) | \\leq \\left ( \\log \\frac { \\kappa } { r } - \\frac { \\log 2 } { 2 } \\right ) ^ { - 1 } \\int _ { \\tau _ - ^ \\delta ( y ) } ^ { \\tau _ + ^ \\delta ( y ) } | f ( X ( t , y ) ) | \\ , d t , \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} - \\lim _ { q \\to \\infty } \\frac { \\ln \\left ( \\textrm { P r } \\{ Q ( \\infty ) > q \\} \\right ) } { q } = \\theta , \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\mathbb { E } \\sup _ { g \\in \\mathcal { G } } \\left \\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\varepsilon _ i g ( X _ i ) - \\frac { \\gamma _ 2 } { n } L ( g ) - \\frac { 1 } { A _ 2 n } D _ n ( g ) \\right \\} \\leq 0 , \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} x ( t ) = \\alpha [ x ] v ( t ) + \\beta [ x ] w ( t ) + \\lambda \\int _ 0 ^ 1 k ( t , s ) f ( s , x ( s ) ) \\textup d s , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} \\chi = \\sum _ j \\alpha _ j ( z , \\zeta ) \\frac { \\partial } { \\partial z ^ j } + \\sum _ \\rho a _ \\rho ( z , \\zeta ) \\frac { \\partial } { \\partial \\zeta ^ \\rho } . \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\omega _ { \\beta , \\mu , \\phi } ( \\eta ( b _ { { 0 } } ^ { * } ) \\eta ( b _ { { 0 } } ) = \\omega _ { \\beta , \\mu } ( ( \\eta ( b _ { { 0 } } ^ { * } ) \\eta ( b _ { { 0 } } ) ) = \\rho _ { { 0 } } \\ \\ \\forall \\phi \\in [ 0 , 2 \\pi ) \\ , \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} \\dd _ { T _ r } ( s ) = \\zeta _ { \\Z ^ r } ( s ) = \\sum _ { n = 1 } ^ \\infty a _ n ( \\Z ^ r ) n ^ { - s } = \\prod _ { i = 0 } ^ { r - 1 } \\zeta ( s - i ) ; \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} a _ c = - \\min _ { t \\leq t _ 0 } \\frac { n \\hat { \\pi } ( t ) - t } { 1 - \\hat { \\pi } ( t ) } = \\frac { t _ c - n \\hat { \\pi } ( t _ c ) } { 1 - \\hat { \\pi } ( t _ c ) } . \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { p } } = \\mathfrak { t } [ [ z ] ] \\oplus \\bigoplus _ { \\beta \\in R ^ { + } } \\mathfrak { g } _ { \\beta } \\otimes \\mathbb { C } [ [ z ] ] \\oplus \\bigoplus _ { \\alpha \\in R ^ * \\subset R ^ { + } } \\mathfrak { g } _ { - \\alpha } \\otimes \\mathbb { C } [ [ z ] ] \\oplus \\bigoplus _ { \\delta \\notin R ^ * } \\mathfrak { g } _ { - \\delta } \\otimes z \\mathbb { C } [ [ z ] ] \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} Q ( \\lambda ) = \\sum _ { a = 1 } ^ { ( p - 1 ) \\mathsf { N } + 1 } \\prod _ { b = 1 , b \\neq a } ^ { ( p - 1 ) \\mathsf { N } + 1 } \\frac { \\Lambda - w _ { b } } { w _ { a } - w _ { b } } Q ( \\xi _ { a } ) , \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} y ( t ) & = T _ q ( t ) y _ 0 + \\int _ 0 ^ t T _ q ( t - \\tau ) F _ q ( y ( \\tau ) ) \\dd \\tau \\\\ & = T _ q ( t ) P _ q ^ s y _ 0 + \\int _ 0 ^ t T _ q ( t - \\tau ) P _ q ^ s F ( y ( \\tau ) ) \\dd \\tau - \\int _ t ^ { \\infty } P _ q ^ c F ( y ( \\tau ) ) \\dd \\tau \\\\ & + P _ q ^ c y _ 0 + \\int _ t ^ { \\infty } P _ q ^ c F ( y ( \\tau ) ) \\dd \\tau + \\int _ 0 ^ t P _ q ^ c F ( y ( \\tau ) ) \\dd \\tau , \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} L _ b : = \\sup _ { u _ 1 \\neq u _ 2 } \\frac { | b ( u _ 1 ) - b ( u _ 2 ) | } { | u _ 1 - u _ 2 | } < \\infty . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} \\phi _ b ( b ; b - i e ) & = \\begin{cases} ( c _ { w - i } ; b - i e ) , & \\max \\{ 0 , w - r \\} \\leq i < w , \\\\ ( b - 1 ; b - 1 - i e ) , \\ ; \\ , & 0 \\leq i < w - r ; \\end{cases} \\\\ \\phi _ b ( c ' _ j ; b - 1 + ( j - \\tilde { w } ) e ) & = \\begin{cases} ( b - 1 ; b - 1 + ( j - \\tilde { w } ) e ) , & 1 \\leq j \\leq \\min \\{ r - l , \\tilde { w } \\} , \\\\ ( c _ { j + l } ; b + ( j - \\tilde { w } ) e ) , & \\tilde { w } < j \\leq r - l ; \\end{cases} \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} \\underset { x \\in ( - a _ 0 , a _ 0 ) } { \\sup } ( u ( 0 , x ) + v ( 0 , x ) ) = \\nu . \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{align*} F _ j ( \\tilde { D } ) = F _ j ( D ) = F _ j ( e + \\Sigma d _ { p q } v _ { p q } ) , \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} H = \\bigl ( H _ 1 ( \\cdotp , \\zeta ' ) , H _ 2 ( \\cdotp , \\zeta ' ) , H _ 3 ) : M \\to \\C ^ 3 \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} a = 2 \\alpha ( Y , s ) , b = 2 \\beta ( Y , s ) + 1 , c = 2 \\gamma ( Y , s ) + 2 . \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} a _ 1 : = c _ * 3 ^ { - \\frac { N } { \\theta } } G ( 0 , 1 ) , a _ { k + 1 } : = 2 ^ { - \\frac { N } { \\theta } } a _ k ^ p \\ , \\frac { p - 1 } { p ^ k - 1 } \\quad ( k = 1 , 2 , \\dots ) . \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} { } J = \\int _ 0 ^ 2 { u ( t ) \\left ( { u ( t ) - t } \\right ) \\ , d t } \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} F _ { m } ^ { \\left ( b \\right ) } = \\prod _ { i } F _ { m _ { i } } . \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} ( d K - K d ) ( \\omega ) & = - K \\bigg ( \\pi ^ * ( d \\sigma ) \\cdot f + ( - 1 ) ^ k \\pi ^ * ( \\sigma ) \\bigg [ d x ^ i \\cdot \\frac { \\partial f } { \\partial x ^ i } + d \\psi ^ j \\cdot \\frac { \\partial f } { \\partial \\psi ^ j } + d \\eta \\cdot \\frac { \\partial f } { \\partial \\eta } \\bigg ] \\bigg ) \\\\ & = ( - 1 ) ^ { k - 1 } \\pi ^ * ( \\sigma ) [ f - f ( x , \\psi , 0 ) ] , \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{gather*} \\sum _ { n = 1 } ^ N a _ n n ^ { i t } = \\sum _ { n = 1 } ^ N a _ n \\int _ n ^ { n + 1 } \\big ( { \\rm e } ^ { i t \\log n } - { \\rm e } ^ { i t \\log x } \\big ) d x + \\int _ 1 ^ { N + 1 } h ( x ) { \\rm e } ^ { i t \\log x } d x \\\\ = \\sum _ { n = 1 } ^ N a _ n \\int _ n ^ { n + 1 } \\big ( { \\rm e } ^ { i t \\log n } - { \\rm e } ^ { i t \\log x } \\big ) d x + \\int _ 0 ^ { \\log ( N + 1 ) } { \\rm e } ^ x h ( { \\rm e } ^ x ) { \\rm e } ^ { i t x } d x \\ , . \\end{gather*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\lim _ { | \\eta | \\to 0 } \\eta ^ { - 2 } \\Omega ( \\xi , \\eta , \\zeta ) = J ' ( \\xi ) ^ 2 - 4 J ( \\xi ) = : \\Lambda ( \\xi ) < 0 . \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\int _ 0 ^ { T _ 1 \\circ H ( x ) } \\Big ( f ( X ( t , x ) ) - \\bar f ( x ) \\Big ) \\ , d t = 0 \\ \\ \\ x \\in \\Omega _ 1 . \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} \\mu = \\int _ { y \\in Y } \\tau _ y d \\nu ( y ) , \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} T _ { n + 1 } = v + J _ a ( T _ n ) , n \\geq 1 \\ . \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} \\varphi ( \\alpha ) = \\alpha - \\sum _ { i = 1 } ^ N \\eta _ i ( \\alpha ) \\ , \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} e ^ { - \\beta B } & = \\sum _ { n = 0 } ^ { \\infty } D _ n ( \\beta ) , \\\\ D _ n ( \\beta ) & = \\int _ { S _ n ( \\beta ) } e ^ { - s _ 1 A } C e ^ { - s _ 2 A } C \\cdots e ^ { - s _ n A } C e ^ { - ( \\beta - \\sum _ { j = 1 } ^ n s _ j ) A } , \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} \\upsilon ( \\Delta _ m ) = 9 , \\upsilon ( a _ 2 ) \\ge 2 , \\upsilon ( a _ 4 ) = 3 , \\upsilon ( a _ 6 ) \\ge 5 . \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { 1 - F _ { \\rho } ( t k ) } { t ^ { - \\gamma } ( 1 - F _ { \\rho } ( k ) ) } = 1 , \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} E _ k ( z ) = \\frac { 1 } { 2 \\zeta ( k ) } \\sum _ { ( c , d ) \\in \\mathbb { Z } ^ 2 \\setminus \\{ 0 \\} } \\frac { 1 } { ( c z + d ) ^ k } , \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\big ( A ^ { \\dagger } - A ^ { \\dagger } U S _ { A } ^ { \\dagger } V ^ { \\ast } A ^ { \\dagger } \\big ) ( A + U V ^ { \\ast } ) = A ^ { \\dagger } A . \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} x ( t ) = \\Phi ( t ) x _ 0 + \\Phi ( t ) \\int _ 0 ^ t \\Phi ( s ) ^ { - 1 } c ( s ) d s . \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} \\mathrm { c m } ( x , y ) + \\mathrm { c m } ( x , z ) = 1 . \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} \\rho ( z ) = R e ( t _ \\zeta ( z ) ) + Q _ \\zeta ( z - \\zeta ) ) + \\mathcal L _ \\zeta ( z - \\zeta ) + o ( | | z - \\zeta | | ^ 2 ) , \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} \\hat \\psi ( x h ) = \\varphi _ \\mu ^ \\infty ( K _ { \\bar \\mu } ( x ) h ) . \\end{align*}"}
-{"id": "2245.png", "formula": "\\begin{align*} L ( \\phi ) = \\bigoplus _ { s \\in M - \\{ 0 \\} } \\phi _ s / \\left \\langle \\phi _ { s + t } \\mid t \\in M - \\{ 0 \\} \\right \\rangle . \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{align*} \\lVert u \\rVert _ E ^ { L i p ( \\gamma ) } : = \\lVert u \\rVert _ { E , \\Omega _ 0 } ^ { L i p ( \\gamma ) } : = \\lVert u \\rVert _ E ^ { \\sup } + \\gamma \\lVert u \\rVert _ { E } ^ { l i p } . \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} F _ i ( \\phi _ i ) & \\leq F _ i ( \\phi ) = \\int _ M ( \\phi - \\phi _ 0 ) d \\mu _ i \\\\ & = \\int _ M ( \\phi - \\phi _ 0 ) d \\mu + \\int _ M ( \\phi - \\phi _ 0 ) ( d \\mu _ i - d \\mu ) . \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} \\mathcal { L } _ 6 : = \\mathcal { S } ^ { - 1 } \\mathcal { L } _ 4 \\mathcal { S } = \\Pi _ S ^ { \\perp } ( \\mathcal { D } _ { \\omega } + m _ 3 \\partial _ { x x x } + m _ 1 \\partial _ x ) \\Pi _ S ^ { \\perp } + R _ 6 , R _ 6 : = \\mathcal { S } ^ { - 1 } \\tilde { R } _ 6 . \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} [ \\widetilde { \\lim _ { k \\to \\infty } f _ k } ] ( x ) = \\lim _ { k \\to \\infty } \\widetilde { f _ k } ( x ) . \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} z ^ { ( 1 ) } _ { n } ( b ) & : = b \\ln ( n / k ) \\bigg ( \\frac { [ n x ] - 2 } { n } - x \\bigg ) ^ { p } , z ^ { ( 2 ) } _ { n } ( b ) : = b \\ln ( n / k ) \\bigg ( \\frac { [ n ( x - \\epsilon ) ] + 1 } { n } - x \\bigg ) ^ { p } , \\\\ z ^ { ( 3 ) } _ { n } ( b ) & : = b \\ln ( n / k ) \\bigg ( \\frac { [ n x ] - 1 } { n } - x \\bigg ) ^ { p } , z ^ { ( 4 ) } _ { n } ( b ) : = b \\ln ( n / k ) \\bigg ( \\frac { [ n ( x - \\epsilon ) ] + 2 } { n } - x \\bigg ) ^ { p } . \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\frac { d } { d t } | \\varphi _ t ( \\omega , x ^ \\prime ) - \\varphi _ t ( \\omega , x ) | ^ 2 & = 2 \\left \\langle b ( \\varphi _ t ( \\omega , x ^ \\prime ) ) - b ( \\varphi _ t ( \\omega , x ) ) , \\varphi _ t ( \\omega , x ^ \\prime ) - \\varphi _ t ( \\omega , x ) \\right \\rangle \\\\ & \\leq 2 | \\varphi _ t ( \\omega , x ^ \\prime ) - \\varphi _ t ( \\omega , x ) | ^ 2 \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} Y _ 1 & = X _ 1 + Z \\\\ Y _ 2 & = X _ 2 + \\sqrt { a } X _ 1 + Z . \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} \\sum _ { m _ 1 + m _ 2 + m _ 3 < p } \\binom { m _ 1 + m _ 2 + m _ 3 } { m _ 1 , m _ 2 , m _ 3 } ^ 2 \\equiv _ { p ^ 2 } 1 \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{gather*} i _ X d + ( - 1 ) ^ { \\epsilon ( X ) } d i _ X = 0 . \\end{gather*}"}
-{"id": "451.png", "formula": "\\begin{align*} ( \\bold { D } ^ { \\vartriangle } _ F ) ^ { \\ast } ( Q ) = 0 \\{ F _ { \\alpha } ( n , [ u ] ) = 0 \\} . \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} \\alpha \\cdot D P \\psi = \\chi _ { \\R ^ 3 _ + } \\left ( \\alpha \\cdot D { \\psi } \\right ) + \\chi _ { \\R ^ 3 _ - } \\left ( \\mathcal { B } \\circ \\Pi \\right ) { \\left ( \\alpha \\cdot D { \\psi } \\right ) \\circ S } \\in L ^ 2 ( \\R ^ 3 ) \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} U = \\bigcup _ { k \\in \\N } U _ k , \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\mathsf { D } \\Big ( P ^ { ( \\mathcal { B } _ n ) } _ { \\mathbf { W } } \\Big | \\Big | p _ W ^ n \\Big ) = \\int d P ^ { ( \\mathcal { B } _ n ) } _ { \\mathbf { W } } \\log \\Delta _ { \\mathcal { B } _ n } . \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { x _ \\frac \\mu 2 } ^ { x _ \\mu - \\delta } \\frac { Q ' \\ , \\dd x } { ( \\mu - Q ) ^ \\frac q 4 } \\leq \\frac { q - 4 } { 4 ( \\mu - Q ( x _ \\mu - \\delta ) ) ^ { \\frac q 4 - 1 } } \\leq \\frac { C } { ( a _ \\mu \\delta ) ^ { \\frac q 4 - 1 } } \\leq C a _ \\mu ^ { \\frac 2 3 - \\frac { q } { 6 } } . \\end{aligned} \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\sigma ( T ) & = \\int _ 0 ^ 1 f ( x ) \\ , d x = \\int _ 0 ^ b f ( x ) \\ , d x + \\int _ b ^ 1 f ( x ) \\ , d x \\\\ & \\geq ( k + 1 ) b + 2 m b + 3 ( 1 - b - m b ) = 3 + ( k - 2 - m ) b \\\\ & \\geq 3 - b ; \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{gather*} U = 3 ( a + d ) - \\frac { t ^ 2 } { 2 } . \\end{gather*}"}
-{"id": "1845.png", "formula": "\\begin{align*} \\frac { d } { d t } \\left ( \\frac { \\partial L } { \\partial \\dot { q } ^ i } \\right ) - \\frac { \\partial L } { \\partial q ^ i } = 0 , \\forall i = 1 , \\dots , n . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} ( f ( z ) \\mid U _ { \\ell ^ m } ) \\mid _ { - 1 } \\gamma = ( \\ell ^ m ) ^ { - \\frac { 3 } { 2 } } \\sum _ { v = 0 } ^ { \\ell ^ m - 1 } f ( z ) \\mid _ { - 1 } \\alpha _ 0 \\sigma _ { w _ v , \\ell ^ m } . \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} C _ j ^ { ( \\nu ) } ( z ) = \\frac { ( 2 \\nu ) _ j } { ( \\nu + 1 / 2 ) _ j } P _ j ^ { ( \\nu - 1 / 2 , \\nu - 1 / 2 ) } ( z ) = \\frac { ( 2 \\nu ) _ j } { j ! } { } _ 2 F _ 1 \\left ( - j , j + 2 \\nu , \\nu + \\frac { 1 } { 2 } , \\frac { 1 - z } { 2 } \\right ) . \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 ( w _ 1 ) = \\left \\{ A _ 1 : A _ 1 \\subseteq A ^ { \\ast } , | A _ 1 | = w _ 1 \\right \\} \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} \\pi _ { n } ( m _ { 1 } , \\ldots , m _ { k } ) = V _ { n , k } \\prod _ { l = 1 } ^ { k } W _ { m _ { l } } U _ { n - m _ { l } } , \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{gather*} \\omega _ t = u ^ 2 . \\end{gather*}"}
-{"id": "8840.png", "formula": "\\begin{align*} Z _ 2 ^ { \\rm { L } } = \\ln \\bigg ( \\frac { 1 } { \\underbrace { \\mathbb { E } \\Big [ \\sum \\nolimits _ { i \\in \\Phi / { o } } { { P _ t } { G _ i } \\beta { { \\left | { { X _ { i , o } } } \\right | } ^ { - { \\alpha _ i } } } } \\Big ] } _ { \\overline { \\Lambda } } + N _ o } \\bigg ) . \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} Y \\ , : = \\ , \\{ v \\in H ^ 1 ( 0 , T ; H ) : v ( 0 ) = 0 \\} , \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{gather*} \\delta _ Q C ^ \\ast = d A ^ \\ast , \\delta _ Q A ^ \\ast = d \\tilde { F } , \\delta _ Q \\tilde { F } = 0 . \\end{gather*}"}
-{"id": "3309.png", "formula": "\\begin{align*} w ( y , z ) = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } \\begin{pmatrix} z y + z \\sin ( t ) \\hat x ( t ) \\\\ - y ^ 2 - y \\sin ( t ) \\hat x ( t ) \\end{pmatrix} d t = \\begin{pmatrix} z y \\\\ - y ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} j = x _ \\alpha y _ \\beta - x _ \\beta y _ \\alpha = x _ \\alpha ^ 2 + y _ \\alpha ^ 2 . \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} ( f ( z ) \\mid U _ { \\ell ^ m } ) \\mid _ { - 1 } \\gamma = ( \\ell ^ m ) ^ { - \\frac { 3 } { 2 } } \\sum _ { v = 0 } ^ { \\ell ^ m - 1 } f ( z ) \\mid _ { - 1 } \\sigma _ { v , \\ell ^ m } \\gamma . \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{gather*} q _ { 2 t } = \\frac { 1 } { 3 } q _ 1 q _ 2 + \\frac { 2 } { 3 } \\frac { u _ t } { u } \\big ( 1 - q ^ 2 _ 2 \\big ) \\end{gather*}"}
-{"id": "4937.png", "formula": "\\begin{align*} L _ q - L _ p & = \\partial _ { Y } R ( Y _ 0 ( \\cdot - q ) ) - \\partial _ { Y } R ( Y _ 0 ( \\cdot - p ) ) \\\\ & = \\int _ 0 ^ 1 \\partial _ { Y Y } R ( s Y _ q + ( 1 - s ) Y _ p ) \\dd s \\ ; [ Y _ 0 ( \\cdot - q ) - Y _ 0 ( \\cdot - p ) ] , \\\\ Y _ 0 ( x - q ) - Y _ 0 ( x - p ) & = - \\int _ 0 ^ 1 Y _ 0 ' ( x - p - s ( q - p ) ) ( q - p ) \\dd s . \\end{align*}"}
-{"id": "8801.png", "formula": "\\begin{align*} f ^ { \\Delta } ( t ) = \\frac { f ( \\sigma ( t ) ) - f ( t ) } { \\mu ( t ) } = \\frac { f ( \\sigma ( t ) ) - f ( t ) } { \\sigma ( t ) - t } , \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} \\frac { N ^ { - 1 / 6 } } { \\sqrt 2 } T _ { \\tau _ 1 , \\tau _ 2 } ( u _ 1 , u _ 2 ) = T _ { T _ 1 , T _ 2 } ( U _ 1 , U _ 2 ) . \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} \\begin{cases} \\zeta = 1 B ( x _ 0 , 4 \\rho ) \\times ( T , \\theta T ) , \\\\ | \\nabla \\zeta | \\le \\frac { \\tilde { C } } { 4 \\rho } \\\\ | \\zeta _ t | \\le \\frac { \\tilde { C } } { T } . \\end{cases} \\end{align*}"}
-{"id": "1966.png", "formula": "\\begin{align*} M _ { \\boldsymbol \\alpha } ( \\mathbf { c } ) = ( c _ { i , j } ) _ { 1 \\leq i \\leq m , 1 \\leq j \\leq n } , \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} \\tau _ n = \\frac { 1 } { n } \\sum _ { k = 1 } ^ n \\delta _ { Z _ k } . \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} ( C C R f _ * { \\cal F } , T ^ * _ Y Y ) _ { T ^ * Y } & \\ = \\chi ( Y , R f _ * { \\cal F } ) \\\\ & \\ = \\chi ( X , { \\cal F } ) = ( C C { \\cal F } , T ^ * _ X X ) _ { T ^ * X } = ( f _ * C C { \\cal F } , T ^ * _ Y Y ) _ { T ^ * Y } \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} \\tilde { f } ( b s ^ { - k } p ) = ( - 1 ) ^ { - k | b | } \\tilde { f } ( s ^ { - k } b p ) & = ( - 1 ) ^ { - k | b | + k ( i - k ) } f _ k ( b p ) \\\\ & = ( - 1 ) ^ { - k | b | + k ( i - k ) + ( i - k ) | b | } b f _ k ( p ) = ( - 1 ) ^ { i | b | } b \\tilde { f } ( s ^ { - k } p ) , \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\nu _ { \\frac { 1 } { 2 } } ( d s ) = \\delta _ { \\frac { 1 } { 2 } } ( d s ) , \\nu _ \\beta ( d s ) = \\frac { \\left ( { s } / { \\beta } \\right ) ^ \\frac { 1 } { 2 \\beta - 1 } } { | 2 \\beta - 1 | } \\frac { d s } { s } , \\ \\beta \\neq \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} ( K _ { 0 2 } w , w ) _ { L ^ 2 ( \\mathbb { T } ) } = ( ( \\partial _ u \\nabla \\mathcal { H } ) ( T _ { \\delta } ) [ w ] , w ) _ { L ^ 2 ( \\mathbb { T } ) } + ( R ( \\psi ) w , w ) _ { L ^ 2 ( \\mathbb { T } ) } \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} \\frac { 1 } { \\phi ( q ) } \\sum _ { \\chi \\bmod q } \\bar { \\chi } ( a ) \\chi ( p ) = \\begin{cases} 1 , & ~ p \\equiv a \\bmod q , \\\\ 0 , & . \\end{cases} \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} i \\partial _ { t } u + \\Delta u + u \\log | u | ^ 2 = 0 , \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} A ^ h ( \\xi , 0 ) = \\frac { 2 i J ( \\xi ) J ' ( \\xi ) } { \\Lambda ( \\xi ) } , A ^ h ( 0 , \\eta ) = \\frac { 4 i \\eta J ( \\eta ) } { \\Lambda ( \\eta ) } . \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} \\xi ( k ( \\cdot , x ) ) = 0 \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} M ( \\| w \\| ^ 2 ) \\| w \\| ^ 2 - \\lambda \\int _ { \\Omega \\times \\{ 0 \\} } f ( z ) | w ( z , 0 ) | ^ q d z - \\int _ { \\Omega \\times \\{ 0 \\} } | w ( z , 0 ) | ^ { 2 ^ * _ \\alpha } d z = 0 . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} u ( j + x ( \\mu ^ * , n ) , 0 ; - n , \\bar \\phi _ { \\mu ^ * } ( \\cdot , - n ) ) \\begin{cases} \\ge u ( j + x ( \\mu , n ) , 0 ; - n , \\bar \\phi _ \\mu ( \\cdot , - n ) ) { \\rm f o r } j \\le 0 \\cr \\le u ( j + x ( \\mu , n ) , 0 ; - n , \\bar \\phi _ \\mu ( \\cdot , - n ) ) { \\rm f o r } j > 0 . \\end{cases} \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} \\| g ( \\tau ) \\| ^ 2 _ { \\ell ^ 2 } = \\sum _ { n = 1 } ^ { \\infty } \\chi ^ 2 _ { [ e ^ { - ( n + 1 ) } , e ^ { - n } ) } ( \\tau ) = \\sum _ { n = 1 } ^ { \\infty } \\chi _ { [ e ^ { - ( n + 1 ) } , e ^ { - n } ) } ( \\tau ) = \\chi _ { ( 0 , e ^ { - 1 } ) } ( \\tau ) \\quad \\mbox { f o r a n y } \\tau \\geq 0 . \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} ( \\mathcal { T } w ) ( \\vartheta , y ) = w ( \\vartheta , y + p ( \\vartheta ) ) , ( \\mathcal { T } ^ { - 1 } h ) ( \\vartheta , z ) = h ( \\vartheta , z - p ( \\vartheta ) ) . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} 6 | \\gamma _ 2 | \\leq 2 - \\frac { | c _ 1 | ^ 2 } { 2 } + \\frac { 1 } { 8 } \\sqrt { ( d ^ 2 + 5 + 2 d q ) ^ 2 - 1 6 d ^ 2 ( 1 - q ^ 2 ) } = : g ( d , q ) . \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} A _ \\nu ( r ) = \\frac { d } { d r } + W _ \\nu ( r ) \\ ; , A ^ { \\dag } _ \\nu ( r ) = - \\frac { d } { d r } + W _ \\nu ( r ) \\ ; . \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} H _ { m } = \\begin{pmatrix} \\alpha _ 1 & \\beta _ 2 & \\cdots & 0 \\\\ \\beta _ 2 & \\alpha _ 2 & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\beta _ { m - 1 } \\\\ 0 & \\cdots & \\beta _ { m - 1 } & \\alpha _ m \\\\ 0 & \\cdots & 0 & \\beta _ m \\end{pmatrix} \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ \\infty \\lambda ( m ) e \\left ( \\frac { a } { c } m \\right ) g ( m ) = \\rho ( a , c ) L ( 1 , \\chi ) \\int g ( x ) d x + T ( a , c ) \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} S I N R _ { i j k } = & \\frac { { \\bf u } ^ { T } _ { i j k } { \\bf T } _ { i j k } { \\bf u } _ { i j k } } { { \\bf u } _ { i j k } ^ { T } { \\bf F } ^ { ' } _ { i j k } { \\bf u } _ { i j k } } , \\ \\forall \\{ i , j \\} \\in \\mathcal { L } , \\ \\forall k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} u = ( H - \\lambda - i \\varepsilon ) f . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( 1 - c _ { \\phi _ { n } } ) < \\infty . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} c _ x : = { \\bigg ( \\frac { 2 \\cos ( \\pi x ) \\Gamma ( 2 - 2 x ) } { x ( 1 - 2 x ) } \\bigg ) } ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} \\frac { ( ( 1 + X ) - X ) ^ { 2 k } } { ( 1 + X ) ^ { k + r } } = \\frac { 1 } { ( 1 + X ) ^ { k + r } } . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{align*} Q ( \\lambda ) = \\prod _ { b = 1 } ^ { \\mathsf { N } _ { Q } } ( \\Lambda - \\Lambda _ { b } ) \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} W ^ { r , 2 } [ 0 , 1 ] = \\{ f \\in L ^ 2 [ 0 , 1 ] \\mid f ^ { ( \\nu ) } \\in L ^ 2 [ 0 , 1 ] , 1 \\leq \\nu \\leq r \\} , \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{gather*} s _ 1 - s _ 2 + s _ 3 + s _ 1 s _ 2 s _ 3 = 0 . \\end{gather*}"}
-{"id": "5131.png", "formula": "\\begin{align*} U _ { y _ o } = A _ { x _ o } B _ { y _ o } \\supset Q _ o \\cap T , \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} \\P ( | a _ { i 1 } x _ 1 + \\dots + a _ { i n } x _ n - y _ i | \\le \\frac { r \\kappa } { D \\sqrt { n } } ) = O ( \\frac { r \\kappa } { D \\sqrt { n } } ) . \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} P ( T _ 1 ^ \\alpha + T _ 2 ^ \\alpha < x ) = E _ P [ \\ 1 _ { \\{ T _ 1 ^ \\alpha + T _ 2 ^ \\alpha < x \\} } ] = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 1 / 2 } \\int _ 0 ^ { 1 / 2 } \\ 1 _ { \\sqrt { v } + \\sqrt { t } < x } ( v t ) ^ { - 1 / 2 } ( 1 - v - t ) ^ { - 3 / 2 } d v d t \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\bigg ( \\frac { r _ 1 } s \\bigg ) ^ n \\sum _ { i = 0 } ^ n { n \\choose i } _ q ^ \\alpha = \\sum _ { n = 0 } ^ \\infty \\sum _ { i = 0 } ^ n \\left [ z ^ n { n \\choose i } _ q \\right ] ^ \\alpha \\le \\left [ \\sum _ { n = 0 } ^ \\infty \\sum _ { i = 0 } ^ n z ^ n { n \\choose i } _ q \\right ] ^ \\alpha . \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{align*} \\lVert \\tilde { f } ( u + h ) - \\sum _ { i = 0 } ^ k \\frac { \\tilde { f } ^ { ( i ) } ( u ) } { i ! } [ h ^ i ] \\rVert _ r \\le C \\ , \\lVert f \\rVert _ { C ^ { r + 2 } } \\lVert h \\rVert ^ k _ { L ^ { \\infty } } ( \\lVert h \\rVert _ { r + p } + \\lVert h \\rVert _ { L ^ { \\infty } } \\lVert u \\rVert _ { r + p } ) . \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{gather*} \\sum _ { d = 1 } ^ { \\infty } h _ { 1 / f } ( d ) = \\frac { 2 } { S _ { 1 / f } ( 1 ) } - 1 , \\\\ \\sum _ { d = 1 } ^ { \\infty } h _ { 1 / f } ( d ) \\log d = - 2 ( \\log 2 ) \\frac { S ' _ { 1 / f } ( 1 ) } { S _ { 1 / f } ( 1 ) ^ 2 } . \\end{gather*}"}
-{"id": "1317.png", "formula": "\\begin{align*} \\begin{array} { l l l l } L ( \\pi ^ k ) = & \\max \\ & c ' v + d ' u - \\pi ^ k ( H v + G u - h ) & \\\\ & \\mbox { s . t . } \\ & A v = b , & \\\\ & & v \\in \\{ 0 , 1 \\} ^ n . & \\end{array} \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} n _ l = \\frac 1 2 ( N _ l - M _ { l } ) \\geq \\frac 1 4 ( N _ l + m _ { l + 1 } ) + 1 = \\frac 1 4 M _ { l + 1 } + 1 \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} \\partial _ { x _ i } g ( s _ n ) & = \\frac { 1 } { \\sqrt n } \\Bigr ( \\partial _ x f - \\frac { Y _ { n , i } } { 2 } \\partial _ z f \\Bigr ) ( s _ n ) , \\\\ \\partial _ { y _ i } g ( s _ n ) & = \\frac { 1 } { \\sqrt n } \\Bigr ( \\partial _ y f + \\frac { X _ { n , i } } { 2 } \\partial _ z f \\Bigr ) ( s _ n ) , \\\\ \\partial _ { z _ i } g ( s _ n ) & = \\frac { \\beta } { \\sqrt n } ( \\partial _ z f ) ( s _ n ) . \\end{align*}"}
-{"id": "2179.png", "formula": "\\begin{align*} E _ 3 : Y ^ 2 = X ^ 3 + 2 w X ^ 2 + u ^ { p } X \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\bigg ( \\frac { r _ 1 } s \\bigg ) ^ n \\sum _ { i = 0 } ^ n { n \\choose i } _ q ^ \\alpha \\le \\left [ \\sum _ { n = 0 } ^ \\infty z ^ n \\prod _ { i = 0 } ^ \\infty \\frac 1 { 1 - z | q | ^ i } \\right ] ^ \\alpha . \\end{align*}"}
-{"id": "4663.png", "formula": "\\begin{align*} \\frac 1 2 \\zeta ^ { 2 n } ( e ^ { 2 \\zeta } + e ^ { - 2 \\zeta } ) C ^ h ( \\xi , \\eta ) + \\frac 1 4 \\xi ^ { 2 n + 1 } ( e ^ { 2 \\xi } - e ^ { - 2 \\xi } ) A ^ h ( \\zeta , \\eta ) . \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} \\bold { p r } X ( F _ { \\alpha } ) = 0 , \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} W _ j ( \\boldsymbol { x } ) = \\frac { \\varphi _ j ( \\boldsymbol { x } ) } { \\sum _ { k \\in I ( \\boldsymbol { x } ) } \\varphi _ k ( \\boldsymbol { x } ) } , j = 1 , \\ldots , d , \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} R = R ^ s \\oplus \\rho R ^ s \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{gather*} \\eta ( t ) = \\lambda ( t ) W ( z ) + \\zeta ( t ) , z = \\xi ( t ) , \\end{gather*}"}
-{"id": "4922.png", "formula": "\\begin{align*} \\begin{array} { l l l } h ^ 0 ( H _ { S ^ { [ 2 ] } } ) & = & \\frac { 1 } { 8 } \\left ( ( H \\cdot H ) + 4 \\right ) \\left ( ( H \\cdot H ) + 6 \\right ) , \\\\ h ^ 0 ( H _ { Z } ) & = & \\frac { 1 } { 2 } \\left ( h ^ 0 ( H ) \\right ) ^ 2 + \\frac { 1 } { 1 6 } h _ { \\Sigma _ { S \\times S } } , \\\\ h ^ 0 ( H _ { Y } ) & = & \\frac { 1 } { 2 } h ^ 0 ( H _ Z ) + \\frac { 1 } { 1 6 } h _ { \\Sigma _ { Z } } + 1 = \\frac { 1 } { 4 } \\left ( h ^ 0 ( H ) \\right ) ^ 2 + \\frac { 1 } { 3 2 } h _ { \\Sigma _ { S \\times S } } + \\frac { 1 } { 1 6 } h _ { \\Sigma _ { Z } } + 1 . \\end{array} \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} \\left ( a \\widetilde { g } \\ , x ^ { ( l ) } \\right ) ( s ) = \\widetilde { g } ( s ) a \\cdot \\alpha _ { s } ( x ^ { ( l ) } ) \\in A \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} X = X _ 0 + B \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} A \\times Y = \\pi _ p ^ { - 1 } ( I _ o ) , \\textrm { m o d u l o $ \\eta $ - n u l l s e t s } , \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} \\tan ^ { 2 } 2 \\theta = \\dfrac { 4 G ( h _ { + } ) } { { \\large ( } 1 - G ( h _ { + } ) { \\large ) } ^ { 2 } } \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} \\| \\mathcal { L } _ { q , \\alpha } - \\mathcal { L } _ { 0 , \\alpha } \\| _ { \\mathcal { B } ( \\mathcal { E } _ { \\alpha } ) } = \\sup \\limits _ { x \\in \\mathbb { R } } \\| \\partial _ { Y } R ( Y _ 0 ( x - q ) ) - \\partial _ { Y } R ( Y _ 0 ( x ) ) \\| _ { \\mathbb { R } ^ { n \\times n } } \\leq C | q | . \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\pi \\in \\P _ { 3 , 4 } , \\\\ | \\pi | = N } } ( - 1 ) ^ { \\nu ( \\pi ) } - \\sum _ { \\substack { \\pi \\in \\P _ { 2 , 4 } , \\\\ | \\pi | = N } } ( - 1 ) ^ { \\nu ( \\pi ) } = ( - 1 ) ^ N \\chi ( N \\not = \\square ) . \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} \\hat { G } & = \\begin{bmatrix} e _ 1 & 0 \\end{bmatrix} + ( Z - I ) T ^ { - 1 } ( Z - I ) ^ { - 1 } \\begin{bmatrix} - g & e _ 1 \\end{bmatrix} , \\\\ \\hat { B } & = \\begin{bmatrix} 0 & e _ 1 \\end{bmatrix} + ( Z - I ) T ^ { - * } ( Z - I ) ^ { - 1 } \\begin{bmatrix} e _ 1 & - b \\end{bmatrix} \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{gather*} H ^ { 2 a } ( B ^ \\vee X Z ) = H ^ { a } ( H ^ { a } ( B ^ \\vee X ) Z ) = H ^ { a } ( B ^ \\vee H ^ { a } ( X Z ) ) , \\\\ H ^ { - 2 a } ( B ^ \\vee X Z ) = H ^ { - a } ( H ^ { - a } ( B ^ \\vee X ) Z ) = H ^ { - a } ( B ^ \\vee H ^ { - a } ( X Z ) ) \\end{gather*}"}
-{"id": "2507.png", "formula": "\\begin{align*} \\mathbf { h } _ l ^ { ( g _ k ) } = \\left ( \\rho _ l ^ { ( g ) } \\right ) ^ { 1 / 2 } \\mathbf { U } _ l ^ { ( g ) } \\left ( \\boldsymbol { \\Lambda } _ l ^ { ( g ) } \\right ) ^ { 1 / 2 } \\mathbf { c } _ l ^ { ( g _ k ) } , \\ ; l = 0 , \\ldots , L _ g - 1 , \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} \\tilde { g } ^ { i \\overline { i } } V _ { 1 } V _ { 1 } ( \\tilde { g } _ { i \\overline { i } } ) = \\tilde { g } ^ { p \\overline { p } } \\tilde { g } ^ { q \\overline { q } } | V _ { 1 } ( \\tilde { g } _ { p \\overline { q } } ) | ^ { 2 } + V _ { 1 } V _ { 1 } ( F ) + V _ { 1 } V _ { 1 } \\left ( \\frac { \\partial \\varphi } { \\partial t } \\right ) . \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} \\mathrm { p e r } A : = \\sum _ { \\sigma \\in \\mathcal { S } _ n } \\prod _ { i = 1 } ^ n a _ { i , \\sigma ( i ) } , \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} { \\bf \\hat y } _ { i } = \\sum \\limits _ { j = 1 } ^ { J } { \\bf U } _ { i } ^ { T } { \\bf G } _ { i j } { \\bf V } _ j { \\bf d } _ j + { \\bf U } _ { i } ^ { T } { \\bf n } _ i . \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} d \\beta _ i = 0 , \\ ; \\ , d ^ * \\beta _ i = d ^ c ( H ( \\beta _ i ) - f _ i ) , \\\\ f _ i = \\log ( ( \\omega _ C + \\beta _ i ) ^ n / \\omega _ C ^ n ) , \\ ; \\ , H ( \\beta _ i ) = f _ i - { \\rm t r } _ { \\omega _ C } \\beta _ i . \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} E \\xi _ { n } = \\sum _ { k = 0 } ^ { 2 ^ { n } - 1 } E g ( k 2 ^ { - n } ) ( \\pi _ { ( k + 1 ) 2 ^ { - n } } - \\pi _ { k 2 ^ { - n } } ) . \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\gamma \\left ( a + 2 , a \\right ) & = \\left ( a + 1 \\right ) \\gamma \\left ( a + 1 , a \\right ) - a ^ { a + 1 } \\exp \\left ( - a \\right ) , \\\\ \\gamma \\left ( a + 3 , a \\right ) & = \\left ( a + 2 \\right ) \\left ( a + 1 \\right ) \\gamma \\left ( a + 1 , a \\right ) - 2 \\left ( a + 1 \\right ) a ^ { a + 1 } \\exp \\left ( - a \\right ) . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} I ( u ) & = \\frac { 1 } { b p ^ 2 } M _ u ^ p - \\int _ 0 ^ T F ( t , u ( t ) ) d t - \\frac { a ^ p } { b p ^ 2 } \\\\ & \\geq \\frac { a ^ { p - 1 } } { p } \\| u \\| _ { E ^ { \\alpha , p } } ^ p - \\frac { ( 1 - \\varepsilon ) a ^ { p - 1 } } { p C _ p ^ p } \\int _ 0 ^ T | u ( t ) | ^ p d t \\\\ & \\geq \\frac { a ^ { p - 1 } } { p } \\| u \\| _ { E ^ { \\alpha , p } } ^ p - \\frac { ( 1 - \\varepsilon ) a ^ { p - 1 } } { p } \\| u \\| _ { E ^ { \\alpha , p } } ^ p \\\\ & = \\frac { \\varepsilon a ^ { p - 1 } } { p } \\| u \\| _ { E ^ { \\alpha , p } } ^ p = \\sigma , \\ \\ \\forall u \\in \\partial B _ \\rho ( 0 ) . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L z ( x ) = g ( x ) , & x \\in \\Omega , \\\\ B z ( x ) = 0 , & x \\in \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\partial _ t u ( x , t ) & = - c _ { d , \\alpha } \\ , \\mbox { P . V . } \\int _ { \\R ^ d } \\frac { u ( x , t ) - u ( y , t ) } { | x - y | ^ { d + \\alpha } } \\d y & \\mbox { i n } \\Omega , t > 0 , \\\\ u ( x , t ) & = 0 & \\mbox { i n } \\R ^ d \\setminus \\Omega , \\\\ u ( x , 0 ) & = u ^ { i n } ( x ) & \\mbox { i n } \\Omega , \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} \\phi _ \\ell ( \\sigma , g ) = ( \\tau , g ) \\textrm { w i t h } \\tau ( k ) = \\begin{cases} \\sigma ( k ) & \\textrm { i f } k \\neq 1 _ G \\\\ \\ell & \\textrm { o t h e r w i s e . } \\end{cases} \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} N _ j = \\bigoplus _ { \\chi \\in C ( T ) } N _ { j , \\chi } . \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} \\mathbb C T _ x X = \\mathbb C T ( x ) \\oplus T _ x ^ { 1 , 0 } ( X ) \\oplus T _ x ^ { 0 , 1 } X . \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} T ( q ^ * ) = \\mathop { } \\limits _ { S \\in \\mathcal { F } } \\Big \\{ R _ { \\rm { \\rm { t o t } } } ( S ) - q ^ * P _ { \\rm { \\rm { t o t } } } ( S ) \\Big \\} = 0 . \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} \\mathbf { M } _ \\gamma \\left \\lbrace y \\right \\rbrace : = L _ { y } \\left \\lbrace x _ \\gamma ^ { \\vphantom { \\intercal } } x _ \\gamma ^ \\intercal \\right \\rbrace . \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} m _ { \\ell } : = \\begin{cases} 2 & 5 \\leq \\ell \\leq 2 3 \\\\ 1 & \\ell \\geq 2 9 , \\end{cases} \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} u _ t + u _ { x x x } - 6 u u _ x = 0 \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} C ' _ 2 = \\frac { 1 } { 2 } \\log \\Bigl ( 1 + \\frac { P _ 2 } { a P _ 1 + N } \\Bigr ) . \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} & g _ { 2 / 3 , 3 / 4 } ^ 2 ( w _ 2 ) \\\\ & \\geq \\frac { \\phi _ { \\ell } } { 4 } \\left [ \\log \\left ( 1 + \\frac { w _ 2 P ' } { 3 } \\right ) - \\frac { 1 } { 2 } \\log \\left ( 1 + \\frac { 2 w _ 2 P ' } { 3 } \\right ) \\right ] \\\\ & - \\frac { 3 \\ell } { 4 k _ { \\ell } } H _ 2 ( \\bar { w } / \\ell ) \\\\ & \\geq \\epsilon ' - \\frac { \\epsilon ' } { 2 } \\\\ & = \\frac { \\epsilon ' } { 2 } . \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} \\mathcal { A } _ { - } ( \\lambda ) = \\sum _ { a = 0 } ^ { 2 \\mathsf { N } + 1 } \\lambda ^ { \\left ( 2 a - 2 \\mathsf { N } + 1 \\right ) } \\mathcal { A } _ { - , a } , \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} \\gamma ( s , \\pi , \\tau , \\psi ) = \\epsilon ( s , \\pi , \\tau , \\psi ) \\frac { L ( 1 - s , \\pi ^ \\vee , \\tau ^ \\vee ) } { L ( s , \\pi , \\tau ) } . \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} f \\left ( \\frac { a z + b } { c z + d } \\right ) = \\chi ( d ) ( c z + d ) ^ k f ( z ) \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { n - 1 } f _ { j k } = \\sum _ { k = 1 } ^ { n - 1 } \\min \\{ j , k \\} - \\frac { j k } { n } = \\frac { j ( n - j ) } { 2 } \\le \\frac { n ^ 2 } { 8 } , \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} g _ i \\phi ( x _ i ) g _ i ^ { - 1 } = x _ i ^ { - 1 } \\mbox { a n d } g _ i \\phi ( y _ i ) g _ i ^ { - 1 } = y _ i ^ { - 1 } \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} R _ \\alpha = \\frac { 1 } { 1 + W _ \\alpha } \\left ( Q _ { \\alpha \\alpha } - \\dfrac { Q _ \\alpha W _ { \\alpha \\alpha } } { 1 + W _ \\alpha } \\right ) , \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { \\sigma ^ { * * } ( n ) } = A _ 1 ^ { * * } \\log x + B _ 1 ^ { * * } + O ( x ^ { c } ( \\log x ) ^ { 1 4 / 3 } ( \\log \\log x ) ^ { 4 / 3 } ) , \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} c ( \\Delta ^ { p ^ { r } m } ) = \\kappa ( p ^ { r } ) c ( \\Delta ^ m ) . \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} A _ a \\ = \\ \\begin{psmallmatrix} 1 & 0 \\\\ 0 & 1 \\\\ 0 & 0 \\\\ \\end{psmallmatrix} , A _ b \\ = \\ \\begin{psmallmatrix} 0 & 0 \\\\ 1 & 0 \\\\ 0 & 1 \\\\ \\end{psmallmatrix} , A _ c \\ = \\ \\begin{psmallmatrix} 1 & 0 \\\\ 0 & 0 \\\\ 0 & 1 \\\\ \\end{psmallmatrix} . \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} \\Phi ( 3 , d _ 1 , d _ 2 ) \\geq 2 + 3 \\left ( \\frac { 3 } { 5 } - 1 \\right ) = \\frac { 4 } { 5 } > 0 . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} & M _ k ( x ; \\mathbf { a } ) - \\frac { 1 } { \\phi ^ k ( q ) } \\frac { k ! } { k _ 1 ! k _ 2 ! \\cdots k _ l ! } S _ k ( x ) \\\\ & = \\frac { 1 } { \\phi ^ k ( q ) } \\frac { k ( k - 1 ) } { k _ 1 ! k _ 2 ! \\cdots k _ l ! } \\frac { x } { \\log x } ( \\log \\log x ) ^ { k - 2 } \\left \\{ \\frac { 1 } { k } \\sum _ { j = 1 } ^ k C ( q , a _ j ) + O _ { k , q , l } \\ ( \\frac { 1 } { \\log \\log x } \\ ) \\right \\} . \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} 1 + \\partial _ 2 \\sigma ( x _ 1 , \\pm 1 ) + h _ { 0 } ' ( \\partial _ 2 \\sigma ( x _ 1 , \\pm 1 ) + 1 ) = 0 \\ \\textrm { a . e . } x _ 1 \\in ( - 1 , 1 ) . \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} \\hat { \\sigma } _ h : = - \\nabla u _ h + \\bar { p } _ h I - \\frac { \\alpha _ { v } } { h _ K } ( \\bar { u } _ h - u _ h ) \\otimes n , \\hat { u } _ h : = u _ h - \\alpha _ { p } h _ K ( \\bar { p } _ h - p _ h ) n , \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} \\frac { \\partial k _ { i a } } { \\partial \\lambda _ { j b } } & = [ \\delta _ { i a , j b } - s _ b F ( \\lambda _ { j b } | \\lambda _ { i a } ) ] 2 \\pi \\rho ( \\lambda _ { j b } ) , \\\\ \\frac { \\partial \\lambda _ { i a } } { \\partial k _ { j b } } & = \\frac { 1 } { 2 \\pi \\rho ( \\lambda _ { i a } ) } [ \\delta _ { i a , j b } - s _ b F ( \\lambda _ { i a } | \\lambda _ { j b } ) ] \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} L = \\overset { N ' } { \\underset { i = 1 } { \\sum } } \\sqrt { \\Delta y _ i ^ 2 + N '^ { - 2 } } \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} l _ \\sigma ( v ^ { 2 k } ; 0 , 0 ) & = l _ \\sigma ( v ^ { - 2 k } ; 0 , 0 ) ^ { - 1 } = ( ( u v ) ^ { 2 k } ; 0 , - 2 k ) ^ { - 1 } = ( ( \\theta ( 0 , 2 k ) ( ( u v ) ^ { 2 k } ) ) ^ { - 1 } ; 0 , 2 k ) \\\\ & = ( ( ( \\theta ( 0 , 2 k ) ( u ) \\theta ( 0 , 2 k ) ( v ) ) ^ { 2 k } ) ^ { - 1 } ; 0 , 2 k ) = ( ( u v ) ^ { - 2 k } ; 0 , 2 k ) , \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} P _ { \\mathcal { R } } \\doteq \\frac { \\left ( 1 - \\frac 1 { d _ c } \\right ) ^ { d _ v } } { \\sum \\limits _ { \\ell = 0 } ^ { d _ v } \\binom { d _ v } { \\ell } \\binom { w M \\frac { d _ v } { d _ c } - d _ v } { d _ v - \\ell } \\left ( 1 - \\frac 1 { d _ c } \\right ) ^ \\ell } . \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} \\tilde { \\theta } _ t ^ i : = \\frac { ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } } { \\sum _ j ( \\theta _ t ^ j ) ^ { \\frac { 1 } { \\gamma } } } , & & t = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} \\sum _ { c } c s _ { i } ( c ) = 0 = \\sum _ { c } s _ { i } ( c ) . \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} | { \\rm d i m } H ^ 0 _ { t , b , m } ( X ) - { \\rm d i m } H ^ 0 _ { b , m } ( X ) | = \\left | \\int _ X S _ { t , m } ( x , x ) - S _ m ( x , x ) d v _ X \\right | \\rightarrow 0 \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} b ( x ) = 1 \\quad \\mbox { i f $ \\| x \\| \\le 1 $ } , b ( x ) = \\| x \\| ^ { - d - 1 } \\quad \\mbox { i f $ \\| x \\| > 1 $ , } \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} & \\dim ( E _ 2 ( \\Gamma _ 0 ( 4 0 ) , \\chi _ 1 ) ) = 8 , ~ \\dim ( S _ 2 ( \\Gamma _ 0 ( 4 0 ) , \\chi _ 1 ) ) = 2 , \\\\ & \\dim ( E _ 2 ( \\Gamma _ 0 ( 4 0 ) , \\chi _ 2 ) ) = 4 , ~ \\dim ( S _ 2 ( \\Gamma _ 0 ( 4 0 ) , \\chi _ 2 ) ) = 4 , \\\\ & \\dim ( E _ 2 ( \\Gamma _ 0 ( 4 0 ) , \\chi _ 3 ) ) = 4 , ~ \\dim ( S _ 2 ( \\Gamma _ 0 ( 4 0 ) , \\chi _ 3 ) ) = 4 . \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} \\nabla ^ H ( s _ 1 \\otimes s _ 2 ) = \\nabla ^ { H _ 1 } s _ 1 \\otimes s _ 2 + s _ 1 \\otimes \\nabla ^ { H _ 1 } s _ 2 , \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} U \\left ( \\frac { c } { 2 } , \\frac { 1 } { 2 } , \\frac { \\zeta ^ 2 } { 2 } \\right ) = \\left ( \\frac { \\zeta ^ 2 } { 2 } \\right ) ^ { - \\frac { c } { 2 } } \\sum _ { s = 0 } ^ { n - 1 } \\left ( - \\frac { \\zeta ^ 2 } { 2 } \\right ) ^ { - s } \\frac { \\left ( \\frac { c } { 2 } \\right ) _ { s } \\left ( \\frac { c + 1 } { 2 } \\right ) _ { s } } { s ! ( 2 \\zeta ^ 2 ) ^ s } + \\widehat { \\varepsilon _ n } \\left ( \\frac { \\zeta ^ 2 } { 2 } \\right ) , \\quad | \\rm a r g \\ , \\zeta | < \\frac { \\pi } { 2 } . \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} \\frac { \\hat y _ 1 ^ h - x _ 1 } { h ^ 2 } & = u - x _ 2 v _ 2 ' - x _ 3 v _ 3 ' + \\psi _ 1 ^ h \\ , , \\\\ \\frac { \\hat y _ i ^ h - h x _ i } { h ^ 2 } & = \\frac { v _ i } { h } + w x _ i ^ \\bot + \\psi _ i ^ h \\ , , i = 2 , 3 \\ , , \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} \\overline { l } ^ { \\{ m \\} } = \\begin{cases} m & \\mbox { i f } ~ l ~ ( { \\rm m o d } ~ m ) = 0 , \\\\ l ~ ( { \\rm m o d } ~ m ) & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} q = \\nabla \\cdot v _ q \\mbox { a n d } \\beta _ c \\norm { v _ q } _ { 1 , \\Omega } \\le \\norm { q } _ { 0 , \\Omega } \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} \\widetilde { R } _ j ^ m = \\frac { { { R _ j } } } { L } \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & \\partial _ t \\eta - G ( \\eta ) \\psi = 0 \\\\ & \\partial _ t \\psi + g \\eta + \\frac 1 2 | \\nabla \\psi | ^ 2 - \\frac 1 2 \\frac { ( \\nabla \\eta \\cdot \\nabla \\psi + G ( \\eta ) \\psi ) ^ 2 } { 1 + | \\nabla \\eta | ^ 2 } = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\mathcal { O } _ { n } } q ^ { | \\pi | } = \\sum _ { \\pi \\in \\mathcal { U } _ { n } } 2 ^ { \\nu _ d ( \\pi ) } q ^ { | \\pi | } , \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} [ 0 , t ] ^ n _ < : = \\{ ( t _ 1 , \\dots , t _ n ) : 0 < t _ 1 < \\cdots < t _ n < t \\} . \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{gather*} h _ { a b c } = \\varepsilon _ { a b c d } h ^ d , H _ { a b } = h ^ c \\wedge h _ { a b c } , \\end{gather*}"}
-{"id": "5945.png", "formula": "\\begin{align*} \\mathcal { T } ( \\lambda ) = \\mathcal { T } ( 1 / \\lambda ) , \\mathcal { T } ( - \\lambda ) = \\mathcal { T } ( \\lambda ) , \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} | E ( x ) | \\ll x d ^ U \\sum \\frac { 1 } { m _ 1 } , \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{align*} x _ { 2 } ^ { 2 } = a ^ { 4 } \\left ( \\dfrac { 1 - b ^ { 2 } } { a ^ { 2 } - b ^ { 2 } } \\right ) \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\operatorname { T r } ( \\chi _ { I } ( H + V ) ) = \\operatorname { T r } ( \\chi _ I ( H + V ) \\chi _ { J } ( H ) ) + \\operatorname { T r } ( \\chi _ I ( H + V ) \\chi _ { J ^ c } ( H ) ) . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} b = \\frac { 2 a } { a + 1 } = \\frac { 4 - \\alpha - 2 \\alpha _ 0 } { 3 - \\alpha - \\alpha _ 0 } . \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} B _ { d } ( y ; p ; s ) : = \\prod _ { j = 1 } ^ g \\tau \\left ( \\alpha _ j ( s ) , \\beta _ j ( s ) , 1 , d ^ { ( j ) } y _ j \\right ) , \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} \\Delta _ { Z _ 1 , Z _ { - 1 } } ( A ) Z _ { - 1 } ^ * & = Z _ 1 A Z _ { - 1 } ^ * - A = ( Z + e _ 1 e _ n ^ * ) A ( Z ^ * - e _ n e _ 1 ^ * ) - A \\\\ & = Z A Z ^ * - A + e _ 1 e _ n ^ * A Z ^ * - Z A e _ n e _ 1 ^ * - e _ 1 e _ n ^ * A e _ n e _ 1 ^ * \\\\ & = - \\nabla ( A ) + \\begin{bmatrix} \\alpha & r \\\\ c & 0 _ { n - 1 , n - 1 } \\end{bmatrix} = - \\nabla ( A ) + \\nabla ( T ) . \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} w _ { a ^ 2 } = \\log { 1 \\over \\P ( [ X _ { 2 } ] _ b = a _ { 2 } | [ X _ { 1 } ] _ b = a _ { 1 } ) } . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} H _ 0 ^ 1 ( \\mathbb { T } _ x ) : = \\left \\{ u \\in H ^ 1 ( \\mathbb { T } , \\mathbb { R } ) : \\int _ { \\mathbb { T } } u ( x ) \\ , d x = 0 \\right \\} \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} _ { p } F _ { q } \\left [ \\begin{array} { r } \\left ( a _ { p } \\right ) ; \\\\ \\left ( b _ { q } \\right ) ; \\end{array} z \\right ] = \\sum \\limits _ { n = 0 } ^ { \\infty } \\frac { \\Pi _ { j = 1 } ^ { p } \\left ( a _ { j } \\right ) _ { n } } { \\Pi _ { j = 1 } ^ { q } \\left ( b _ { j } \\right ) _ { n } } \\frac { z ^ { n } } { n ! } \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} a _ 4 = \\frac { - 2 ^ 3 } { 3 } \\tilde { c } _ 4 \\equiv \\frac { - \\mu ^ 3 } { 3 } ( t ^ { 1 2 } + \\beta _ 2 t ^ { 2 0 } ) \\equiv t ^ { 1 2 } + t ^ { 1 8 } + \\beta _ 2 t ^ { 2 0 } \\pmod { t ^ { 2 1 } } \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} A ( n ) = \\sum _ { d | n } a ( d ) . \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} \\xi = \\sum _ { ( k , x ) \\in F } \\overline { \\beta } _ { k , x } \\lambda _ a \\eta ( k , x ) \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} r _ k = w _ k \\log _ 2 \\left ( 1 + \\frac { q _ k h _ k } { w _ k N _ 0 } \\right ) , \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} u _ t ( x , t ) = \\int _ { \\R } \\kappa ( y - x ) u ( y , t ) d y - u ( x , t ) + u f ( t , u ) , \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} & S ( E _ i ) = - K _ i ^ { - 1 } E _ i , & & S ( K _ i ) = K _ i ^ { - 1 } , \\\\ & S ( F _ i ) = - F _ i K _ i , & & S ( K ' _ i ) = ( K ' _ i ) ^ { - 1 } , \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} F _ { 3 n + 2 } ^ { \\left ( 3 \\right ) } = F _ { n } ^ { \\left ( 3 \\right ) } F _ { 2 } = 2 F _ { n } ^ { \\left ( 3 \\right ) } . \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} | A _ { x _ o } \\cap F _ { n _ k } g _ { n _ k } | = | A _ { g _ { n _ k } . x _ o } \\cap F _ { n _ k } | = | A _ x \\cap F _ { n _ k } | , \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} \\phi _ { A } ( z ) = \\frac { a ^ 2 - 1 } { a } ( z - \\beta ) \\left ( 1 + \\mathcal { O } ( z - \\beta ) \\right ) . \\end{align*}"}
-{"id": "3965.png", "formula": "\\begin{align*} \\gamma ( s , \\xi , \\psi ) = \\frac { \\epsilon ( s , \\xi , \\psi ) L ( 1 - s , \\xi ^ { - 1 } ) } { L ( s , \\xi ) } . \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} \\min _ { 1 / k _ n \\leq \\gamma \\leq 1 } f ( \\gamma , 1 ) = \\min \\left \\{ f \\left ( \\frac 1 { k _ n } , 1 \\right ) , \\min _ { \\gamma ' \\leq \\gamma \\leq 1 } f ( \\gamma , 1 ) \\right \\} . \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot \\xi = A \\xi + c ( t ) + \\lambda f _ 1 ( t , \\xi , \\eta , \\lambda ) , \\\\ \\dot \\eta = \\lambda f _ 2 ( t , \\xi , \\eta , \\lambda ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{gather*} i _ X \\omega _ k = \\delta \\alpha _ k + \\delta _ Q \\alpha ' _ { k - 1 } + d \\alpha ' _ k \\end{gather*}"}
-{"id": "2164.png", "formula": "\\begin{align*} x = \\frac { u } { v ^ 2 } y = \\frac { w } { v ^ p } \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} \\frac { \\partial p } { \\partial t } = D \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } + f ( p ) \\ , , \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} \\Lambda ^ r ( X \\oplus Y ) \\cong \\Lambda ^ r ( X ) \\oplus \\left ( \\bigoplus _ { i = 1 } ^ { r - 1 } \\Lambda ^ { r - i } ( X ) \\otimes \\Lambda ^ i ( Y ) \\right ) \\oplus \\Lambda ^ r ( Y ) . \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} \\Lambda _ { n , k } ( \\gamma , \\epsilon ) & : = \\frac { 1 } { d _ { \\gamma } } \\cdot u _ { n , k , p } ( d _ { \\gamma } ) + \\frac { 1 } { | d ' _ { \\gamma } | } \\cdot u ' _ { n , k , p } ( d ' _ { \\gamma } ) , \\\\ \\Lambda ' _ { n , k } ( \\gamma , \\epsilon ) & : = \\frac { 1 } { d _ { \\gamma } } \\cdot v _ { n , k , p } ( d _ { \\gamma } ) + \\frac { 1 } { | d ' _ { \\gamma } | } \\cdot v ' _ { n , k , p } ( d ' _ { \\gamma } ) . \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} \\Big \\| \\sup _ { N \\ge 1 } \\big | \\sum _ { n = 1 } ^ N \\alpha _ n D ( \\lambda _ n t ) \\big | \\ , \\Big \\| _ { \\S ^ 2 } \\le C \\Big ( \\sum _ { n \\ge 1 } | \\alpha _ n | \\Big ) \\Big ( \\sum _ { n \\ge 1 } \\big ( \\sum _ { k \\ , : \\ , n \\le \\mu _ k < n + 1 } | \\beta _ k | \\big ) ^ 2 \\Big ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "7621.png", "formula": "\\begin{align*} \\beta _ { 0 0 n } = n \\rho ^ n , \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\begin{cases} \\bar c ( 0 ) \\chi ' ( z ) - g ( \\chi ( z ) ) = 0 & z \\in ( 0 , 1 ) \\\\ \\chi ( 1 ) = 1 , \\ \\chi ( 0 ) = 0 & \\\\ \\chi ' ( 0 ) = \\chi ' ( 1 ) . \\end{cases} \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} \\rho = \\sigma \\omega + y , \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} ( h ^ { 1 / 2 } \\cdot x \\cdot h ^ { 1 / 2 } ) ( t ) ( s ) = x ( t ) ( s ) h ^ { 1 / 2 } ( s - t ) h ^ { 1 / 2 } ( s ) \\ , . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} \\zeta _ { E H } ' ( 0 ; \\alpha _ 1 , \\ldots , \\alpha _ d ; \\lambda _ 1 , \\ldots , \\lambda _ d ) & = \\int _ 1 ^ \\infty \\theta ^ \\infty ( t ) \\frac { d t } { t } + \\int _ 0 ^ 1 \\Big ( \\theta ^ \\infty ( t ) - \\Big ( \\prod _ { i = 1 } ^ { d } \\alpha _ i \\Big ) ( 4 \\pi t ) ^ { - \\frac { d } { 2 } } \\Big ) \\frac { d t } { t } \\\\ & - \\frac { 2 } { d } \\Big ( \\prod _ { i = 1 } ^ { d } \\alpha _ i \\Big ) ( 4 \\pi ) ^ { - \\frac { d } { 2 } } . \\end{align*}"}
-{"id": "3861.png", "formula": "\\begin{align*} a ( x , y , z ) = 3 ^ a \\cdot \\frac { q - 1 } { s } \\left ( \\sum \\limits _ { i = 0 } ^ { q - 2 } \\alpha _ i ( x ) \\alpha _ i ( y z ^ { - 1 } ) \\right ) , \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} 1 - g + \\mu ( E ) - \\mu ( F ) = 0 . \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} f ( x _ 1 , x _ 2 ) & = \\frac { 3 } { 4 } \\exp \\left [ - \\frac { ( 9 x _ 1 - 2 ) ^ 2 + ( 9 x _ 2 - 2 ) ^ 2 } { 4 } \\right ] + \\frac { 3 } { 4 } \\exp \\left [ - \\frac { ( 9 x _ 1 + 1 ) ^ 2 } { 4 9 } - \\frac { 9 x _ 2 + 1 } { 1 0 } \\right ] \\\\ & + \\frac { 1 } { 2 } \\exp \\left [ - \\frac { ( 9 x _ 1 - 7 ) ^ 2 + ( 9 x _ 2 - 3 ) ^ 2 } { 4 } \\right ] - \\frac { 1 } { 5 } \\exp \\left [ - ( 9 x _ 1 - 4 ) ^ 2 - ( 9 x _ 2 - 7 ) ^ 2 \\right ] . \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} { V } = \\mathcal { A } \\partial _ { r } + \\sum \\limits ^ { m } _ { k = 1 } \\mathcal { B } _ k \\frac { 1 } { r } \\partial _ { k } + \\sum \\limits ^ { q } _ { l = 1 } \\mathcal { C } _ l \\partial _ { u _ l } \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} \\tilde { J } ( \\mathbf { u } ^ { \\ast } ) = - \\tilde { A } + \\tilde { T } D \\tilde { \\mathbf { g } } ( \\mathbf { u } ^ { \\ast } ) , \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\| u ^ { \\infty } \\| _ { i , Q } \\leq \\left [ \\sup _ { ( t , x ) \\in [ 0 , T ] \\times \\R ^ d } | u ^ { \\infty } ( t , x ) | \\right ] \\left [ \\sum _ { s \\in [ 0 , T ] } \\bar { q } ^ { i , Q } ( s ) \\right ] = \\left [ \\sup _ { ( t , x ) \\in [ 0 , T ] \\times \\R ^ d } | u ^ { \\infty } ( t , x ) | \\right ] \\frac { T ^ { i } } { i ! } . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\left ( \\max _ { 0 \\le k \\le n } | A _ k | \\right ) ^ { 1 / n } = 1 \\quad \\mbox { a . s . } \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} & E _ { \\chi _ 0 , \\chi _ 1 } ( - q ) = - E _ { \\chi _ 0 , \\chi _ 1 } ( q ) - 2 E _ { \\chi _ 0 , \\chi _ 1 } ( q ^ 2 ) + 4 E _ { \\chi _ 0 , \\chi _ 1 } ( q ^ 4 ) , \\\\ & E _ { \\chi _ 1 , \\chi _ 0 } ( - q ) = - E _ { \\chi _ 1 , \\chi _ 0 } ( q ) + 2 E _ { \\chi _ 1 , \\chi _ 0 } ( q ^ 2 ) + 4 E _ { \\chi _ 1 , \\chi _ 0 } ( q ^ 4 ) . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} \\int _ { B ^ n ( x , r _ i ) } \\Psi ( \\vert \\nabla u \\vert ) \\ , d \\mu _ { \\alpha } \\geq \\begin{cases} C R _ i \\log ^ { \\lambda - ( n + \\alpha - 1 ) } \\left ( \\frac { 1 } { R _ i } \\right ) & ~ ~ p = n + \\alpha - 1 , ~ \\lambda > n + \\alpha - 1 , \\\\ C R _ i ^ { n + \\alpha - p } \\log ^ { \\lambda } \\left ( \\frac { 1 } { R _ i } \\right ) & ~ . \\end{cases} \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} \\beta _ i ( k , k ' ) = \\left ( \\frac { k [ q , q ' ] } { q } \\right ) ^ i \\sum _ { i \\leq j \\leq s } \\alpha _ j b ^ { j - i } + \\left ( \\frac { k ' [ q , q ' ] } { q ' } \\right ) ^ i \\sum _ { i \\leq j \\leq s } \\alpha _ j ' b '^ { j - i } . \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} \\theta _ { \\overline { \\jmath } _ i } : = \\theta _ i , y _ { \\overline { \\jmath } _ i } : = y _ { i } , \\xi _ { \\overline { \\jmath } _ i } : = \\xi _ i , \\omega _ { \\overline { \\jmath } _ i } = \\omega _ i , i = 1 , \\dots , \\nu . \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} \\widehat { F } & = ( I + V ^ { \\ast } F _ { 1 } F _ { 1 } ^ { \\ast } V + V ^ { \\ast } F _ { 3 } F _ { 3 } ^ { \\ast } V ) ^ { - 1 } = ( I + F _ { 2 } F _ { 2 } ^ { \\ast } + F _ { 4 } F _ { 4 } ^ { \\ast } ) ^ { - 1 } \\\\ & = ( I + F _ { 2 } F _ { 3 } + F _ { 4 } ^ { 2 } ) ^ { - 1 } = ( I + F _ { 4 } ) ^ { - 1 } , \\\\ \\widehat { E } & = ( I + U ^ { \\ast } E _ { 1 } ^ { \\ast } E _ { 1 } U + U ^ { \\ast } E _ { 2 } ^ { \\ast } E _ { 2 } U ) ^ { - 1 } = ( I + E _ { 3 } ^ { \\ast } E _ { 3 } + E _ { 4 } ^ { \\ast } E _ { 4 } ) ^ { - 1 } \\\\ & = ( I + E _ { 2 } E _ { 3 } + E _ { 4 } ^ { 2 } ) ^ { - 1 } = ( I + E _ { 4 } ) ^ { - 1 } . \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} ( - \\Delta ) ^ s u ( x ) = a _ { N , s } \\int _ { \\R ^ N } \\frac { u ( x ) - u ( y ) } { | x - y | ^ { N + 2 s } } d y , \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\mathcal { R } ^ { \\gamma } _ H = \\frac { \\partial } { \\partial t } + \\left ( p + \\alpha \\sin { ( w t ) } q ^ 2 p \\right ) \\frac { \\partial } { \\partial q } . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\int _ { \\gamma } ( 1 + z ^ 2 ) ^ s g ( z ) d z = 2 \\sin ( \\pi s ) \\int _ 1 ^ \\infty ( \\tau ^ 2 - 1 ) ^ s g ( i \\tau ) d \\tau , \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} ( \\omega _ { C _ x } + \\eta + i \\partial \\bar \\partial { u } ) ^ n = e ^ { f } \\omega _ { C _ x } ^ n , \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} \\lambda _ { \\pm } : = - \\frac { \\nu | \\xi | ^ { 2 \\sigma } } { 2 } \\pm \\sqrt { \\frac { \\nu ^ { 2 } } { 4 } | \\xi | ^ { 4 \\sigma } - | \\xi | ^ { 2 } } , \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} \\mathbb { P } _ n X _ * \\mathbb { P } _ n = \\mathbb { P } _ n B \\mathbb { P } _ n + \\mathbb { P } _ n ( B \\Gamma _ { m , b c } X _ * ) \\mathbb { P } _ n , n \\geq m + 1 . \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} \\underset { t \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\int _ { { \\bf R } ^ N } u ( y , t ) \\phi ( y ) \\ , d y = \\int _ { { \\bf R } ^ N } \\phi ( y ) \\ , d \\mu ( y ) \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} a ^ { ( l , \\sigma ) } _ - ( y ) \\stackrel { \\eqref { e q : t h m a b o u t r e l a t i o n b e t w e e n b o x d i m e n s i o n a n d R o k h l i n d i m e n s i o n - a m i n u s } } { = } \\left ( \\Lambda \\left ( \\xi ^ { ( l , \\sigma ) } ( y ) \\right ) - \\left ( \\frac { 1 } { 2 } - 8 \\varepsilon \\right ) \\right ) \\cdot 8 L \\ , \\leq 0 \\ ; . \\\\ \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} | \\Psi | _ { g ' } = V ^ { \\frac { p } { 2 } } e ^ { p y } | \\Psi | _ { g } , \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} a ^ 2 + b ^ 3 = c ^ p \\gcd ( a , b , c ) = 1 \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} k _ { i a } , a = R , L , i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} \\ , \\ , c _ i \\ , \\ , \\ , \\ , \\gamma , c _ i \\gamma \\in \\Gamma c _ j \\ , \\ , \\ , \\ , j ; \\\\ p ^ { - 1 } ( p ( \\Gamma g ) ) \\cap \\Omega = \\{ \\Gamma c _ 1 g u _ { p _ 1 } , \\ldots , \\Gamma c _ n g u _ { p _ n } \\} \\ , \\ , \\ , \\ , \\tilde { m } \\ , g . \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} \\Delta ( \\sigma , \\tau ) = \\{ x \\in V : \\sigma _ 1 ( x ) \\neq \\tau ( x ) \\} \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ \\infty e ^ { - \\sum _ { u = \\lfloor t / 2 \\rfloor } ^ { t - 1 } \\Phi ^ 2 _ u } = \\sum _ { t = 1 } ^ \\infty e ^ { - c ( d ) t ^ { 1 - 2 \\beta / d } } < \\infty . \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} ( V ( B ^ h ; k ) ^ n ) \\leq C n ^ { - 4 h _ { \\min } + 1 } \\sum _ { i = 1 } ^ n i ^ { 4 h _ { \\min } - 4 } \\leq C \\begin{cases} n ^ { - 4 h _ { \\min } + 1 } & h _ { \\min } \\in ( 0 , 3 / 4 ) \\\\ \\ln n \\cdot n ^ { - 2 } & h _ { \\min } = 3 / 4 \\\\ n ^ { - 2 } & h _ { \\min } \\in ( 3 / 4 , 1 ) \\end{cases} \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} a ( n ) = \\sum _ { \\substack { a b = n \\\\ b \\ge y } } \\tau ( a ) \\lambda ( b ) \\ , b ( m ) = \\sum _ { \\substack { a b = m \\\\ b < y } } \\tau ( a ) \\tau ( b ) \\ . \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} \\tilde { c } _ 6 = a _ 0 + a _ 1 3 + a _ 2 3 ^ 2 + O ( 3 ^ 3 ) , a _ i \\in \\{ 0 , 1 , 2 \\} , a _ 0 \\ne 0 . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} \\xi = \\varphi ( \\alpha ) \\ , h ' ( \\rho ) , \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} u = \\P _ h u + \\bar \\P _ h u . \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} \\omega ( \\tau _ { { x } } ( A ) ) = \\omega ( A ) \\ \\ \\forall A \\in { \\cal A } , \\ \\forall { x } \\ , \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} \\operatorname { T r } \\left ( \\chi _ I ( H + V ) \\chi _ { J ^ c } ( H ) \\right ) & = \\sum _ j \\left \\langle \\phi _ j , \\chi _ { J ^ c } ( H ) \\phi _ j \\right \\rangle = \\\\ \\sum _ j \\left \\langle \\phi _ j , \\left ( V \\frac { \\chi _ { J ^ c } ( H ) } { ( H - E _ j ) ^ 2 } V \\right ) \\phi _ j \\right \\rangle & \\leq \\frac { \\lVert V \\rVert ^ 2 } { \\operatorname { d i s t } ( I , J ^ c ) ^ 2 } \\operatorname { T r } \\left ( \\chi _ I ( H + V ) \\right ) . \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} & a d ( - a ^ 2 - b ^ 2 + 5 c ^ 2 - 5 d ^ 2 ) + b c ( 5 a ^ 2 - 8 b ^ 2 - 5 c ^ 2 + 1 0 d ^ 2 ) = 1 2 D _ 1 ( q ) , \\\\ & a d ( 2 a ^ 2 - b ^ 2 - 4 c ^ 2 + d ^ 2 ) + b c ( - a ^ 2 + b ^ 2 - 5 c ^ 2 + 7 d ^ 2 ) = 2 4 D _ 2 ( q ) , \\\\ & a d ( - a ^ 2 + 2 b ^ 2 - 7 c ^ 2 + 1 0 d ^ 2 ) + b c ( - a ^ 2 + 4 b ^ 2 + c ^ 2 - 8 d ^ 2 ) = 4 8 D _ 3 ( q ) , \\\\ & a d ( a ^ 2 - 8 b ^ 2 - 5 c ^ 2 + 2 0 d ^ 2 ) + b c ( 7 a ^ 2 - 1 0 b ^ 2 + 5 c ^ 2 - 1 0 d ^ 2 ) = 4 8 D _ 4 ( q ) . \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} \\partial _ t u - \\Delta u = u ^ p , x \\in { \\bf R } ^ N , \\ , \\ , t > 0 , u ( 0 ) = \\mu \\ge 0 \\quad \\mbox { i n } \\quad { \\bf R } ^ N , \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} \\hat X : = \\partial _ x + \\frac { y } { 2 } \\partial _ z \\quad \\hat Y : = \\partial _ y - \\frac { x } { 2 } \\partial _ z \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} u ( x ) = u _ { \\varepsilon } ( x ) = \\frac { \\varepsilon ^ { ( N - 2 \\alpha ) \\slash 2 } } { ( | x | ^ 2 + \\varepsilon ^ 2 ) ^ { ( N - 2 \\alpha ) \\slash 2 } } \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} \\tilde { G } & = \\begin{bmatrix} - ( Z - I ) A ^ { - 1 } ( Z - I ) ^ { - 1 } G & ( Z - I ) A ^ { - 1 } ( Z - I ) ^ { - 1 } e _ 1 & e _ 1 \\end{bmatrix} , \\\\ \\tilde { B } & = \\begin{bmatrix} ( Z - I ) A ^ { - * } ( Z - I ) B & e _ 1 & ( Z - I ) A ^ { - * } ( Z - I ) e _ 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} & ( r _ 2 ( 4 ) - 2 \\cdot 1 ) = 4 - 2 = 2 , \\\\ & ( r _ 2 ( 5 ) - 2 \\cdot 0 ) = 8 . \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} M ^ { \\dagger } = \\begin{pmatrix} \\Sigma & \\Sigma H ^ { \\ast } E _ { S _ { A } } - \\Psi K S _ { A } ^ { \\dagger } \\\\ F _ { S _ { A } } K ^ { \\ast } \\Sigma - S _ { A } ^ { \\dagger } H \\Phi & S _ { A } ^ { \\dagger } - S _ { A } ^ { \\dagger } H \\Phi H ^ { \\ast } E _ { S _ { A } } - F _ { S _ { A } } K ^ { \\ast } \\Psi K S _ { A } ^ { \\dagger } + F _ { S _ { A } } K ^ { \\ast } \\Sigma H ^ { \\ast } E _ { S _ { A } } \\end{pmatrix} , \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{align*} \\bar { b } _ { - , m } ^ { p } = q ^ { p / 2 } \\mu _ { m , + } ^ { p } , \\forall m \\in \\left \\{ 1 , . . . , N \\right \\} , \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal { L } _ { \\infty } ( \\omega ) : = \\Phi _ { \\infty } ^ { - 1 } ( \\omega ) \\circ \\mathcal { L } _ 5 \\circ \\Phi _ { \\infty } ( \\omega ) = \\omega \\cdot \\partial _ { \\varphi } + \\mathcal { D } _ { \\infty } ( \\omega ) , \\\\ & \\mathcal { D } _ { \\infty } ( \\omega ) : = \\mbox { d i a g } _ { j \\in S ^ c } \\{ \\mu _ j ^ { \\infty } ( \\omega ) \\} . \\end{aligned} \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} h _ { K _ { N , \\ell , q } } ( \\theta ) \\geq N ^ { - 1 / q } \\left ( \\sum _ { i = 1 } ^ N | \\langle X _ i , \\theta \\rangle | ^ q \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} V ( k , \\theta ) = & \\sup _ { ( \\hat { x } , y ) \\gg 0 } \\ \\Big [ U ( \\hat { x } , \\theta ) + \\mu ( \\theta ) V ( y , F ( \\theta ) ) : \\hat { x } + y \\leq f ( k ) \\Big ] , \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} R _ k = \\sum _ { l = 1 , l \\neq k } ^ K r _ { i i l } ^ { \\alpha } . \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} - \\sum _ { k = \\iota + 1 } ^ n s _ { k \\iota } = s _ { \\iota \\iota } - \\sum _ { k = \\iota } ^ n s _ { k , \\iota - 1 } \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} A _ { \\alpha } \\left ( G \\right ) : = \\alpha D \\left ( G \\right ) + \\left ( 1 - \\alpha \\right ) A \\left ( G \\right ) , 0 \\leq \\alpha \\leq 1 . \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} \\epsilon ( \\frac 1 2 , \\pi , \\tau , \\psi ) = 1 . \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} \\Theta _ { j \\bar k } ^ { { \\rm E n d } ( H ) } \\theta _ v = [ \\Theta ^ h _ { j \\bar k } , \\theta _ v ] , \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} L ^ * & = K ^ * / \\Pi \\\\ M ^ * & = \\Omega ^ * / \\Pi . \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} v \\rho _ g ( | x | b ) & = v | x | \\alpha _ g ( b ) \\\\ & = | x ^ * | v \\alpha _ g ( b ) \\\\ & = \\sigma _ g ( | x ^ * | v ) \\alpha _ g ( x b ) \\\\ & = \\sigma _ g ( | x ^ * | v b ) \\\\ & = \\sigma _ g ( v | x | b ) \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ 3 [ H ^ 0 _ { e t } ( X , \\mathcal { F } _ i ) _ { t o r } ] ^ { ( - 1 ) ^ { i + 1 } } \\right ) = \\nu ( \\mathcal { H } ^ 0 ) _ { \\mathbb { R } } [ S ] . \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} H G = G A , ~ ~ H b \\leq b . \\end{align*}"}
-{"id": "4270.png", "formula": "\\begin{align*} a _ { B _ i , \\pm } ( \\Phi _ t ( y ) ) = a _ { B _ i , \\pm } ( y ) - t \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} C _ 3 = \\frac { ( B / 4 ) ^ 2 } { \\tilde C ( B / 4 ) ^ 2 - \\lVert V _ \\omega \\rVert _ \\infty ^ 2 \\lambda ^ 2 ( \\tilde C + 2 ) } . \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} \\bar { \\tau } _ n = \\frac { \\sum \\limits _ { k = 1 } ^ { M _ n } \\tau _ { n k } P _ { n k } } { \\sum \\limits _ { k = 1 } ^ { M _ n } P _ { n k } } - \\tilde { \\tau } _ n \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} f A ' + F a ' - a F - A f = 0 \\Leftrightarrow \\left \\{ \\begin{array} { l c l } a ' & = & a _ { 1 1 } + f _ { 2 } \\\\ a _ { 2 1 } & = & 0 \\end{array} \\right . . \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} \\left | \\sum \\log \\Gamma \\left ( b , x w ^ { b ^ { i } } \\right ) - \\log \\Gamma \\left ( b \\right ) \\right | = \\frac { 1 } { b } \\sum _ { i \\ge 0 } x ^ b w ^ { b ^ { i + 1 } } \\le \\frac { 1 } { b } \\sum _ { i \\ge 0 } w ^ { b ^ { i + 1 } } \\le \\frac { 1 } { b } \\frac { w ^ { b } } { 1 - w } \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} B _ m ( t , x ) = \\frac { 1 } { 2 } ( D - \\kappa t ^ { - 2 \\beta } \\vert x \\vert ) _ + ^ { \\frac { 1 } { m - 1 } } t ^ { - \\alpha } , \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} E _ C ^ { ( \\infty ) } = \\lim _ { \\theta \\to \\infty } E _ C ( \\theta ) = B T \\log _ 2 ( 1 + \\alpha ) \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} - ( \\Delta _ g \\lambda ) g + \\nabla ^ 2 _ g \\lambda - \\lambda { \\rm R i c } _ g = 0 { \\rm i n } { \\rm i n t } \\ , M , \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} \\lim _ { \\xi \\rightarrow - \\infty } ( P ( \\xi ) , Q ( \\xi ) ) = ( P _ 2 , 0 ) , \\lim _ { \\xi \\rightarrow \\infty } ( P ( \\xi ) , Q ( \\xi ) ) = ( P _ 1 , 0 ) , \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} \\sigma ^ { l \\ast } ( S _ 0 ^ { - 1 } S _ j ) - S _ 0 ^ { - 1 } S _ j \\ ; = \\ ; \\sum _ { i = 0 } ^ { j - 1 } S _ 0 ^ { - 1 } U _ 0 ^ { - 1 } U _ { j - i } S _ i \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} \\alpha = ( \\underbrace { 0 , \\ldots , 0 } _ { n } , \\underbrace { r _ n } _ { \\neq 0 } , r _ { n + 1 } , \\ldots ) \\mapsto \\chi ( \\alpha ) = ( \\underbrace { 0 , \\ldots , 0 } _ { n + 1 } , - 1 , 0 , 0 , \\ldots ) : \\Gamma _ { \\log } \\to \\Gamma ^ { \\leq } _ { \\log } \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} & u _ { 0 } ^ { \\prime } = \\epsilon , u _ { 1 } ^ { \\prime } = \\epsilon ( 2 c - 1 ) , \\ u _ { n + 2 } ^ { \\prime } = ( 2 c - 2 ) u _ { n + 1 } ^ { \\prime } - u _ { n } ^ { \\prime } , \\ n \\geq 0 , \\\\ & z _ { 0 } = \\epsilon , z _ { 1 } = \\epsilon ( 2 c - 3 ) , \\ z _ { n + 2 } = ( 2 c - 2 ) z _ { n + 1 } - u _ { n } , \\ n \\geq 0 . \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} I ( f ) = \\{ x \\in \\R ^ m \\colon f ^ n ( x ) \\to \\infty \\} , \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} & \\int \\eta _ n | r ^ { m + 1 } \\nabla ^ { m + 1 } h | ^ 2 d v \\leq \\tilde C _ m \\int \\sum _ { i = 0 } ^ m | r ^ { m - i } \\nabla ^ { m - i } h | | r ^ { m } \\nabla ^ { m } h | d v . \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} \\| F _ 0 \\| _ { { 2 ^ n } } = \\| F _ 1 \\| _ { { 2 ^ n } } = \\| F _ 2 \\| _ { { 2 ^ { n - 1 } } } = \\dots = \\| F _ n \\| _ { { 2 ^ { 1 } } } = 1 . \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{align*} \\dot { v } _ j + \\mu _ j ^ { \\infty } \\ , v _ j = 0 , j \\in S ^ c , \\mu _ j ^ { \\infty } \\in \\mathrm { i } \\mathbb { R } . \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} & B ^ { - 1 } \\mathcal { L } _ 1 B = \\rho \\ , \\mathcal { L } _ 2 , \\mathcal { L } _ 2 : = \\Pi _ S ^ { \\perp } ( \\omega \\cdot \\partial _ { \\vartheta } + m _ 3 \\partial _ { y y y } + c _ 1 \\partial _ y + c _ 0 ) \\Pi _ S ^ { \\perp } + \\mathfrak { R } _ 2 , \\\\ & c _ 1 : = \\rho ^ { - 1 } ( B ^ { - 1 } b _ 1 ) , c _ 0 : = \\rho ^ { - 1 } ( B ^ { - 1 } b _ 0 ) , \\mathfrak { R } _ 2 : = \\rho ^ { - 1 } B ^ { - 1 } \\mathfrak { R } _ 1 B . \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} \\beta = x ( - t , w ^ { 3 \\theta - 1 } + t ^ { 3 \\theta + 1 } , - v - t ^ { 3 \\theta + 1 } - t w ^ { 3 \\theta - 1 } ) h ( - 1 ) . \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} - h _ \\alpha \\varphi + \\mathcal { L } _ \\alpha ( \\varphi ) = \\Gamma ( \\alpha + 1 ) \\ , \\mbox { P . V . } \\int _ { \\R ^ d } \\frac { \\varphi ( y ) { \\rm \\bf 1 } _ { \\mathcal { S } _ \\Omega ( x ) } ( y ) - \\varphi ( x ) } { | x - y | ^ { d + \\alpha } } d y \\ , , \\end{align*}"}
-{"id": "9530.png", "formula": "\\begin{align*} \\mu ( z ) = \\frac { 2 } { 1 + | z | ^ { 2 } } \\quad z = ( x , y ) \\in \\R ^ { 2 } . \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} w ( x , y ) = - \\left [ \\int _ 0 ^ 1 \\int _ 0 ^ 1 H _ u ( x + ( \\tau - t ) y ) \\ d \\tau d t \\right ] y \\cdot y x \\in V , \\ y \\in \\R ^ N , \\ x \\pm y \\in V . \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} ( q - 1 ) \\mid \\sum _ { k = 0 } ^ { r - 1 } p ^ k \\left ( a _ k + b _ k \\right ) , \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} \\frac { 5 m _ 0 m _ 1 \\log ( d + 1 ) } { n } + \\frac { 2 v v _ f } { m _ 1 } + \\frac { L _ 2 v ^ 2 _ f v ^ 2 _ 0 } { m _ 0 } + \\frac { L ^ 2 _ 2 v ^ 2 _ f v ^ 4 _ 0 } { 4 m ^ 2 _ 0 } + \\left ( \\frac { 1 } { n } \\sum _ { i = 1 } ^ n | Y _ i | \\right ) \\frac { v _ f L _ 2 v ^ 2 _ 0 } { m _ 0 } + \\frac { T _ n } { n } . \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} A ^ { \\downarrow } \\ : = \\ \\{ s \\in S : \\} , \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} k _ \\gamma = \\gamma \\cdot k \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{align*} \\Phi _ N ( z ) = z B _ { N - 1 } ^ * ( z ) - A _ { N - 1 } ^ * ( z ) , \\Phi _ N ^ * ( z ) = B _ { N - 1 } ( z ) - z A _ { N - 1 } ( z ) . \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} m _ \\beta ( t ) = e ^ { ( 2 \\beta - 1 ) t } , t \\ge 0 . \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} \\gamma _ j ^ { m _ j } \\ = \\ \\exp ( m _ j \\ , N _ j ) \\ , . \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{align*} \\mathcal { I } _ e ^ { j , k } = f _ { k + 1 } ^ { - 1 } ( ( f ^ { k + 1 } _ 0 ) ^ { - 1 } ( \\{ e \\} ) \\cap ( F ^ k _ j ) ^ { - 1 } ( 0 ) \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} \\gamma _ 1 = \\frac { 1 } { 4 } \\left ( b _ 2 + c _ 1 \\right ) . \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { \\beta + 2 } . \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} | x ^ j g ^ { ( j ) } ( x ) | \\le 1 , \\ j = 0 , 1 , 2 . \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} L ( f e _ { x x } ) ( x , y ) & = \\left ( L ( f e _ { x x } ) e _ { y y } \\right ) ( x , y ) \\\\ & = - L \\left ( [ e _ { y y } , f e _ { x x } ] \\right ) ( x , y ) + \\left ( L ( e _ { y y } ) f e _ { x x } \\right ) ( x , y ) \\\\ & - \\left ( f e _ { x x } L ( e _ { y y } ) \\right ) ( x , y ) + \\left ( e _ { y y } L ( f e _ { x x } ) \\right ) ( x , y ) \\\\ & = - L ( e _ { y y } f e _ { x x } ) ( x , y ) - f ( x , x ) L ( e _ { y y } ) ( x , y ) \\\\ & = - f ( y , x ) L ( e _ { y x } ) ( x , y ) - f ( x , x ) L ( e _ { y y } ) ( x , y ) . \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\frac { \\delta n } { 2 \\sigma ^ 2 } \\sum _ { j = 1 } ^ p \\frac { w _ j ^ 2 } { w _ j ^ * } | \\theta _ j | + \\frac { \\log 4 p } { \\beta \\delta } \\| \\theta \\| _ { w ^ * , 1 } + \\frac { \\log 2 } { \\beta } . \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta | \\nabla ^ { k } F _ A | ^ 2 & \\leq - | \\nabla ^ { k + 1 } F _ A | ^ 2 + C _ k \\sum _ { m = 0 } ^ k | \\nabla ^ { m } { \\cal R } | | \\nabla ^ { k - m } F _ A | | \\nabla ^ { k } F _ A | \\\\ & + C _ k \\sum _ { m = 0 } ^ k | \\nabla ^ { m } F _ A | | ( \\nabla ^ { k - m } F _ A ) ^ B | | \\nabla ^ { k } F _ A | . \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\mu _ { \\epsilon } ( [ 0 , t ] ) = \\begin{cases} \\mu ( [ 0 , t ] ) & 0 \\leq t < 1 - \\epsilon \\\\ 1 & 1 - \\epsilon \\leq t \\leq 1 \\end{cases} . \\end{align*}"}
-{"id": "9391.png", "formula": "\\begin{align*} d \\mu _ { \\delta _ k , x } ( \\mathcal { A } ) = \\langle \\delta _ k , \\chi _ { \\mathcal { A } } ( \\tilde { H } _ s ( x ) ) \\delta _ k \\rangle = \\sum _ { l , j } | v _ { l , j } ( x , k ) | ^ 2 \\chi _ { \\mathcal { A } } ( e _ { l , j } ( x ) ) . \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} \\Pi _ S ^ { \\perp } [ \\partial _ { x x } ( a _ 1 \\partial _ x ) + \\partial _ x ( a _ 0 \\cdot ) ] \\Pi _ S ( \\mathcal { A } - \\mathrm { I } ) \\Pi _ S ^ { \\perp } = \\varepsilon ^ 2 \\Pi _ S ^ { \\perp } [ \\partial _ { x x } ( a _ { 1 , 1 } \\partial _ { x x } \\Pi _ S [ \\beta _ 1 \\ , \\cdot ] ) + \\partial _ x ( a _ { 0 , 1 } \\ , \\partial _ x \\Pi _ S [ \\beta _ 1 \\ , \\cdot ] ) ] + o ( \\varepsilon ^ 2 ) . \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} K ( c , 1 , p ) = \\sqrt { \\psi _ 3 ( c , p ) + 2 \\left ( c ^ 3 - 2 c - 1 0 \\right ) \\left ( c ^ 2 - 4 \\right ) \\left ( 6 c p ^ 2 - 2 c p - 2 p - 3 c \\right ) } \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} A _ i = \\frac { 1 } { \\mu _ i ^ + - \\mu _ i ^ - } \\left ( u _ i ' ( 1 ) - \\mu _ i ^ - u _ i ( 1 ) + \\int _ { \\delta _ i } ^ 0 s ^ { 1 - \\mu _ i ^ + } f _ i ( s ) \\ , d s \\right ) , \\\\ \\bar { u } _ i ( r ) = \\frac { 1 } { \\mu _ i ^ + - \\mu _ i ^ - } \\left ( r ^ { \\mu _ i ^ + } \\int _ { 1 - \\delta _ i } ^ r s ^ { 1 - \\mu _ i ^ + } { f } _ i ( s ) \\ , d s - r ^ { \\mu _ i ^ - } \\int _ 0 ^ r s ^ { 1 - \\mu _ i ^ - } { f } _ i ( s ) \\ , d s \\right ) . \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} a _ \\beta ( V ^ { ( 1 ) } ( t ) ) = 1 [ T _ \\theta > t ] + 1 [ T _ \\theta \\le t ] \\beta \\{ a _ \\beta ( V ^ { ( 1 ) + } ( t - T _ \\theta ) ) + a _ \\beta ( V ^ { ( 1 ) - } ( t - T _ \\theta ) ) \\} , \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} B _ n ( x ) = \\sum _ { k = 0 } ^ n \\binom { n } { k } N _ k ( x ) M _ { n - k } ( x ) . \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} \\langle \\bar { n } , \\nu \\rangle \\left ( \\nu _ \\beta - \\bar { n } _ \\beta \\langle \\bar { n } , \\nu \\rangle \\right ) = 0 \\beta = 1 , \\ldots , d . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P _ o } ( - 1 ) ^ { \\nu ( \\pi ) } q ^ { | \\pi | } = \\frac { 1 } { 1 + q } - 1 , \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} \\left ( 2 \\ , U - \\mathrm { I } \\right ) ^ { - 1 } = \\mathrm { I } - \\frac { 2 } { 2 \\ , \\nu + 1 } \\ , U , \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} F _ { \\epsilon } ( z ) ^ * = F _ { - \\epsilon } ( \\bar { z } ) \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{align*} \\exp \\biggl ( \\int _ 0 ^ { 2 \\pi } \\log \\omega _ { N - 1 } \\ , \\frac { d \\theta } { 2 \\pi } \\biggr ) = \\omega _ { N - 1 } = \\prod _ { j = 0 } ^ { N - 1 } ( 1 - | \\alpha _ j | ^ 2 ) \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} d N _ { c } ( E ) = d N _ { \\tilde { H } _ c } ( E ) . \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to + 0 } \\lim _ { V \\to \\infty } \\omega ^ { 0 } _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\eta _ { \\Lambda } ( b _ { { 0 } } ^ { * } ) ) = \\lim _ { \\lambda \\to + 0 } \\lim _ { V \\to \\infty } \\frac { \\partial } { \\partial \\lambda _ { \\phi } } p _ { \\beta , \\mu , \\Lambda , \\lambda _ { \\phi } } \\ , \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} g ( z ) = \\frac { z } { c _ 1 + c _ 2 | z | } \\qquad c _ 1 , c _ 2 > 0 . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} \\widetilde { \\Phi } _ k : = { \\phi } ^ 0 \\oplus \\left ( \\oplus _ { i = 1 } ^ { p } \\oplus _ { \\alpha \\in P ( i , k p - i ) } \\phi ^ i _ { \\alpha } \\right ) , \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} H = \\sum _ { i = 1 } ^ { N } \\left ( x _ i \\frac { \\partial } { \\partial x _ i } \\right ) ^ 2 + \\beta \\sum _ { i < j } \\frac { x _ i + x _ j } { x _ i - x _ j } \\left ( x _ i \\frac { \\partial } { \\partial x _ i } - x _ j \\frac { \\partial } { \\partial x _ j } \\right ) . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} \\dfrac { 4 G ( h _ { + } ) } { { \\large ( } 1 - G ( h _ { + } ) { \\large ) } ^ { 2 } } = \\left ( \\dfrac { 2 s \\left ( v t - w s \\right ) } { ( t ^ { 2 } - s ^ { 2 } ) v - 2 w t s } \\right ) ^ { 2 } \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} W H ^ r ( P ) : = \\bigoplus _ { x \\in P } \\widetilde H ^ { r - 2 } ( ( \\hat 0 , x ) ; { \\bf k } ) . \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} \\gamma _ { \\rm S I R } = \\frac { N } { 2 ( V + 1 ) } . \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\Lambda = \\frac { 1 } { 2 \\pi } \\mathrm { I m } \\log : \\left ( \\left ( - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } \\right ) \\times \\left ( 0 , 1 \\right ] \\right ) / \\sim \\ \\to \\left ( - \\frac { 1 } { 2 } , \\frac { 1 } { 2 } \\right ) , \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} \\sum \\limits _ { \\theta \\in { \\rm I r r } ( N _ G ( P ) ) } \\theta ( J T ) \\overline { \\theta ( J T ) } = q - 1 + 4 ( b \\overline { b } ) = 2 q \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} \\mathrm { d } f = i _ { S _ f } \\mathrm { d } \\eta + h \\eta . \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} & ( a ^ + ( \\varphi ) + a ^ - ( \\varphi ) ) \\diamond ( a ^ + ( \\psi ) + a ^ - ( \\psi ) ) \\\\ & \\quad = a ^ + ( \\varphi ) a ^ + ( \\psi ) + a ^ + ( \\varphi ) a ^ - ( \\psi ) + a ^ + ( \\psi ) a ^ - ( \\varphi ) + a ^ - ( \\varphi ) a ^ - ( \\psi ) . \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\mathcal { B } \\ast \\mathcal { C } = \\{ \\psi \\ast \\phi : \\psi \\in \\mathcal { B } , \\phi \\in \\mathcal { C } \\} . \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} \\nu _ { \\ell , 0 } = \\min \\{ \\nu _ \\ell , \\lambda _ \\ell - 2 \\} , \\ ; \\ , \\nu _ { \\ell , j + 1 } = \\min ( \\{ \\nu _ \\ell , \\nu _ { \\ell , j } + \\lambda _ \\ell \\} \\cup \\{ \\mu _ { \\ell , i } ^ + - 2 + Q _ { \\ell , i } : i \\in \\N _ 0 \\} ) . \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\bold { p r } X = \\xi ^ i D _ i + \\sum _ { \\alpha , J } ( D _ J Q ^ { \\alpha } ) \\frac { \\partial } { \\partial u _ J ^ { \\alpha } } . \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} E _ { a b } = \\sum _ e \\mathop { \\sum \\sum } _ { ( u , v ) = 1 } \\frac { \\rho ( e u ) \\rho ( e v ) } { e u v } g ( e u ) g ( e v ) \\lambda ( u v ) \\xi ( u v ) \\sum _ { q \\mid u } \\Lambda _ a ^ * ( q ) \\sum _ { r \\mid v } \\Lambda _ b ^ * ( r ) . \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} \\left [ \\int f ( t ) d Z ( t ) , \\int g ( t ) d Z ( t ) \\right ] _ \\alpha = \\int f ( t ) ( g ( t ) ) ^ { < \\alpha - 1 > } d \\mu . \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} \\varepsilon | | v _ \\lambda | | _ { L ^ 2 } ^ 2 = \\varepsilon | | u | | _ { L ^ 2 } ^ 2 \\le \\int | f u | d x \\le C _ 1 ^ { - 1 } \\delta _ 1 ^ 2 | | \\nabla v _ \\lambda | | _ { L ^ 2 } ^ 2 + C \\delta _ 1 ^ { - 2 } | | r f | | _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} I = \\int _ { d ( w , \\tilde { y } ) < \\frac { 1 } { r _ 0 ^ 2 } } \\ldots + \\sum _ { k = 1 } ^ \\infty \\int _ { \\frac { 2 ^ { k - 1 } } { r _ 0 ^ 2 } \\le d ( w , \\tilde { y } ) < \\frac { 2 ^ k } { r _ 0 ^ 2 } } \\ldots , \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} { \\bf H } _ 1 & = \\begin{bmatrix} 2 . 0 3 e ^ { - i 0 . 6 8 } & 2 . 1 e ^ { i 2 . 6 4 } & 3 . 2 e ^ { i 1 . 4 8 } \\\\ 4 . 7 e ^ { i 1 . 9 7 } & 4 . 5 e ^ { - i 0 . 6 6 } & 2 . 8 5 e ^ { i 2 . 4 1 } \\end{bmatrix} , \\\\ { \\bf H } _ 2 & = \\begin{bmatrix} 3 . 2 e ^ { - i 0 . 7 2 } & 2 . 3 e ^ { i 2 . 5 2 } & 1 . 9 e ^ { i 1 . 3 5 } \\\\ 2 . 8 e ^ { i 1 . 6 8 } & 2 . 5 e ^ { - i 0 . 7 6 } & 3 . 4 e ^ { i 2 . 2 3 } \\end{bmatrix} . \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} y _ { i } ( n + 1 ) = C _ { i } ( n ) y _ { i } ( n ) , \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} \\bold { p r } \\widetilde { X } ( \\widetilde { P } ) = 0 . \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} \\begin{aligned} & B _ 1 [ h ] : = \\alpha _ { 1 , 2 } \\ , h _ x , B _ 2 [ h ] : = \\alpha _ { 0 , 2 } \\ , h , B _ 3 [ h ] : = - ( \\alpha _ { 1 , 1 } ) _ x \\ , \\beta _ 1 \\ , h _ x , \\\\ [ 2 m m ] & B _ 4 [ h ] : = - ( \\alpha _ { 0 , 1 } ) _ x \\ , \\beta _ 1 , B _ 5 [ h ] : = - \\partial _ x \\overline { \\mathcal { R } } _ 2 [ h ] . \\end{aligned} \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} ( - D ) _ J = ( - 1 ) ^ { | J | } D _ J . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} \\langle \\psi _ k , \\psi _ k \\rangle _ 1 = \\langle ( 1 - \\tau ^ 2 ) P _ k ' , ( 1 - \\tau ^ 2 ) P _ k ' \\rangle _ 1 = k ^ 2 ( k + 1 ) ^ 2 \\langle P _ k , P _ k \\rangle _ { \\C { L } ^ 2 ( \\Omega ) } = \\frac { 2 k ^ 2 ( k + 1 ) ^ 2 } { 2 k + 1 } . \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} \\{ k : 0 \\leq k \\leq _ b n , 0 \\leq n _ i \\leq b - 1 , 0 \\leq i < \\infty \\} = \\{ k : 0 \\leq k \\leq \\infty \\} . \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} \\frac { H _ { 2 } } { \\beta ^ 2 } = \\sum _ a T ^ { a a } _ { 2 } - \\sum _ { a , b } T ^ { a b } _ { 0 } T ^ { b a } _ { 1 } + s \\sum _ a T ^ { a a } _ { 1 } + \\frac { 1 } { 3 } \\sum _ { a , b , c } T ^ { a b } _ { 0 } T ^ { b c } _ { 0 } T ^ { c a } _ { 0 } \\\\ - \\frac { 2 s } { 3 } \\sum _ { a , b } T ^ { a b } _ { 0 } T ^ { b a } _ { 0 } + \\frac { 1 } { 6 } \\sum _ { a , b } T ^ { a a } _ { 0 } T ^ { b b } _ { 0 } + \\frac { 2 s ^ 2 - 1 } { 6 } \\sum _ a T ^ { a a } _ { 0 } \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\kappa ( y _ e ) = \\left \\{ \\begin{array} { l l } 2 \\cosh ( 2 J ) , & \\\\ 2 , & \\end{array} \\right . \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} t ^ { 1 8 } a _ 6 ' = b + r a + r ^ 3 \\Leftrightarrow 3 2 t ^ 3 a _ 6 ' = - \\mu ^ 5 \\tilde { c } _ 6 - 2 \\mu ^ 4 \\tilde { c } _ 4 t ^ 2 ( a _ 0 + y _ 1 t ) + 3 2 ( a _ 0 + y _ 1 t ) ^ 3 \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{gather*} u _ { t t } = t u + 2 u ^ 3 , \\end{gather*}"}
-{"id": "9323.png", "formula": "\\begin{align*} & \\biggl ( \\prod _ { j = 1 } ^ i [ n ^ 2 - ( i - 1 ) n - ( j - 1 ) ] ! ^ { \\frac { n ^ 2 - c _ 2 n } { n ^ 2 - ( i - 1 ) n - ( j - 1 ) } } \\biggr ) \\\\ & \\times \\biggl ( \\prod _ { j = i + 1 } ^ n [ n ^ 2 - ( j - 1 ) n ] ! ^ { \\frac { n ^ 2 - c _ 2 n } { n ^ 2 - ( j - 1 ) n } } \\biggr ) . \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} C _ { 1 } ( \\sum _ { k = 1 } ^ { n } | \\alpha _ { k } | ^ { q ^ { * } } ) ^ { \\frac { 1 } { q ^ { * } } } \\leq \\| \\sum _ { k = 1 } ^ { n } \\alpha _ { k } z _ { k } \\| \\leq C _ { 2 } ( \\sum _ { k = 1 } ^ { n } | \\alpha _ { k } | ^ { q ^ { * } } ) ^ { \\frac { 1 } { q ^ { * } } } . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log S ( U , f ^ n ) = \\log d . \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} \\hat { a } _ n = \\hat { g } _ 1 ( n ) \\alpha _ 1 ^ n + \\dots + \\hat { g } _ h ( n ) \\alpha _ h ^ n \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} \\frac { \\partial \\rho } { \\partial \\lambda _ { j b } } ( \\lambda ) = s _ b \\rho ( \\lambda _ { j b } ) L ( \\lambda _ { j b } | \\lambda ) = - s _ b \\rho ( \\lambda _ { j b } ) \\partial _ { \\lambda } F ( \\lambda _ { j b } | \\lambda ) \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} V _ { a , b } : = \\bigoplus _ { \\lambda \\in P ( b p , a p ) } \\Bbbk \\pi _ { \\lambda } , \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} X _ 1 = \\partial _ { v } , X _ 2 = n \\partial _ { v } . \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\delta ( X ) & = 0 \\\\ \\delta ( V ) & = 0 \\\\ - \\dot { V } - \\delta ( X V ) & = 0 . \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} & \\mathcal { L } _ 3 : = \\mathcal { T } ^ { - 1 } \\mathcal { L } _ 2 \\mathcal { T } = \\Pi _ S ^ { \\perp } ( \\omega \\cdot \\partial _ { \\vartheta } + m _ 3 \\ , \\partial _ { z z z } + D _ S \\ , \\partial _ z + d _ 0 ) \\Pi _ S ^ { \\perp } + \\mathfrak { R } _ 3 , \\\\ & d _ 1 : = ( \\mathcal { T } ^ { - 1 } c _ 1 ) + \\omega \\cdot \\partial _ { \\vartheta } p , d _ 0 : = \\mathcal { T } ^ { - 1 } c _ 0 , \\mathfrak { R } _ 3 : = \\mathcal { T } ^ { - 1 } \\mathfrak { R } _ 2 \\mathcal { T } \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} ( \\gamma \\cdot \\alpha ^ n \\cdot F + \\gamma \\cdot e _ n + b ) = \\alpha ^ n \\cdot \\gamma \\cdot ( F ) = \\alpha ^ n \\cdot c = \\beta ^ { \\frac { \\log \\alpha } { \\log \\beta } \\cdot n } \\cdot c \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} D = \\frac { 1 } { 2 } K _ X ^ { [ n ] } + \\left ( \\frac { a } { 2 } + \\frac { n } { a H ^ 2 } \\right ) H ^ { [ n ] } - \\frac { 1 } { 2 } E \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} B ( s ) = \\frac { - 1 } { 2 \\pi i } \\int _ { ( - 1 ) } \\overline L ( 1 - s - z ) ( Q | s | ) ^ { - 2 z } f ( z ) z ^ { - 2 } d z \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} a ( x , y , z ) = 3 ^ a \\cdot \\frac { q - 1 } { s } \\left ( \\sum \\limits _ { i = 0 } ^ { q - 2 } \\alpha _ i ( x ) \\alpha _ i ( y ) \\alpha _ i ( z ^ { - 1 } ) \\right ) = 3 ^ a \\cdot \\frac { q - 1 } { s } \\left ( \\sum \\limits _ { i = 0 } ^ { q - 2 } \\alpha _ i ( x ) \\alpha _ i ( y z ^ { - 1 } ) \\right ) \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\adjustlimits \\sup _ { x \\in X , \\ , y \\geq 0 , \\ , z \\in \\mathcal { U } \\ } \\inf _ { \\tau \\in \\Theta ^ n } \\ & \\sum _ { i = 1 } ^ n \\theta ^ i \\left [ u _ i ( x ^ i ) + \\delta ^ i z ^ i \\right ] , \\\\ [ 5 p t ] & \\sum _ { i = 1 } ^ n x ^ i + y \\leq f ( k ) , \\\\ & \\sum _ { i = 1 } ^ n \\tau ^ i z ^ i - V ( y , \\tau ) \\leq 0 . \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} \\int _ { C } \\langle d d ^ c u , d d ^ c v \\rangle f = \\int _ { C } ( \\Delta u ) ( \\Delta v ) f + \\int _ { C } ( \\partial _ r u ) ( \\Delta v ) f ' - \\int _ { C } \\langle d ^ c u , \\partial _ r \\ , \\lrcorner \\ , d d ^ c v \\rangle f ' . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} I _ Q ( V ; Y , S | U ) - I _ Q ( V ; Z | U ) & = I _ P ( S , T ; Y , S ) - I _ P ( S , T ; Z ) \\\\ & = I _ P ( T ; Y | S ) - I _ P ( T ; Z | S ) + H _ P ( S | Z ) , \\\\ I _ Q ( U , V ; Y , S ) - I _ Q ( U , V ; S ) & = I _ P ( S , T ; Y , S ) - I _ P ( S , T ; S ) = I _ P ( T ; Y | S ) , \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} & \\biggl ( - \\frac { d ^ 2 } { d x ^ 2 } + 2 ( x ^ 3 - 3 a x ^ 2 - 2 b x - c ) \\frac { d } { d x } + ( r - 4 b ^ 2 - 6 a c + 3 ) x ^ 2 + ( q - 4 b c - 6 a ) x \\\\ & - c ^ 2 - 2 b \\biggr ) \\phi _ n ( x ) = E _ n \\phi _ n ( x ) , \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} \\| P _ { n - 1 } \\| _ { L _ R } \\le \\| P _ n \\| _ { L _ R } + | A _ n | \\| B _ n \\| _ { L _ R } \\le \\| P _ n \\| _ { L _ R } \\left ( 1 + \\frac { | L _ R | } { 2 \\pi d } \\frac { c _ 1 } { c _ 2 } \\right ) = : C \\ , \\| P _ n \\| _ { L _ R } , n \\in \\N . \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} w '' ( u ) = w ( u _ 0 ) - \\int _ { u _ 0 } ^ u \\sum _ { j = 1 } ^ n a _ j ' d b _ j ' , u \\in S _ 1 . \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} I = \\left [ \\begin{array} { c } \\mu \\\\ 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} \\gamma ( d ) = \\frac { \\varphi ( d ) } { d } \\sum _ { d _ 1 \\ldots d _ r = d } d _ 1 ^ { \\varepsilon _ 1 } \\ldots d _ r ^ { \\varepsilon _ r } \\ , \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} \\mathrm { T r } _ 2 ( g ) = \\mathrm { T r } ^ { G L _ 2 } _ { S L _ 2 } ( g ) = \\mathcal { I } \\circ \\mathrm { T r } ( g ) = \\mathcal { I } \\circ \\left ( \\dfrac { 1 } { 2 ^ { d - 1 } } \\sum _ { \\epsilon _ i \\in R } \\psi ' ( \\epsilon _ i ^ { - 1 } ) g \\mid _ { k , w } \\epsilon _ i \\right ) \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} - \\frac { 2 \\sin ( \\pi s ) } { \\pi } ( - 1 ) ^ j \\int _ 1 ^ \\infty ( t ^ 2 - 1 ) ^ s t ^ { - 2 j - 1 } d t = \\frac { i } \\pi \\int _ { - \\infty + i 0 } ^ { \\infty + i 0 } ( 1 + z ^ 2 ) ^ s z ^ { - 2 j - 1 } d z = \\binom { s } { j } . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} & \\hat { f } ( n ) = \\int _ 0 ^ 1 f ( x ) e ( - n x ) \\ , d x = \\int _ 0 ^ 1 f ( x ) \\ , d [ \\frac { 1 } { - 2 \\pi i n } e ( - n x ) ] \\\\ & = \\frac { 1 } { - 2 \\pi i n } [ D _ \\mu ( x ) - x ] e ( - n x ) | _ 0 ^ 1 + \\frac { 1 } { 2 \\pi i n } \\int _ 0 ^ 1 e ( - n x ) \\ , d [ D _ \\mu ( x ) - x ] \\\\ & = \\frac { 1 } { 2 \\pi i n } \\int _ 0 ^ 1 e ( - n x ) \\ , d \\mu ( x ) = \\frac { \\hat { \\mu } ( n ) } { 2 \\pi i n } , \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\frac { n _ 0 ^ { ( j ' ) } } { 2 } \\log k _ n ^ { ( j ) } & = \\left ( 1 + \\frac { \\epsilon ' } { 2 } \\right ) \\beta ^ { ( j ' ) } \\ell _ n H _ 2 \\left ( \\alpha _ n ^ { ( j ' ) } \\right ) \\frac { \\log k _ n ^ { ( j ) } } { \\log k _ n ^ { ( j ' ) } } \\\\ & \\leq _ n ( 1 + \\epsilon ' ) \\beta ^ { ( j ' ) } \\ell _ n H _ 2 \\left ( \\alpha _ n ^ { ( j ' ) } \\right ) . \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\delta _ k ( \\cdot ) = \\sum _ { l , j } \\langle \\delta _ k ( \\cdot ) , v _ { l , j } ( x , \\cdot ) \\rangle v _ { l , j } ( x , \\cdot ) = \\sum _ { l , j } v _ { l , j } ( x , k ) v _ { l , j } ( x , \\cdot ) . \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} M ^ I = \\{ v \\in M \\ , | \\ , x v = 0 , \\quad \\mbox { f o r a l l $ x \\in I $ } \\} , \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} \\P ( \\sigma _ 0 \\ge N / 2 | X _ 0 = 0 ) \\le c _ 1 ^ { - 1 } e ^ { - c _ 1 N } . \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} \\Delta _ 2 ^ 2 \\ , \\Delta _ 3 \\ - \\ \\Delta _ 1 ^ 3 \\ + \\ \\Delta _ 1 \\ , \\Delta _ 3 ^ 2 \\ = \\ 0 . \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} a _ 4 = - \\frac { \\tilde { c } _ 4 } { 3 } a _ 4 ' = \\frac { a _ 4 + 3 r ^ 2 - 2 s t } { u ^ 4 } , \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} \\omega | _ \\Sigma = \\cos \\alpha d \\mu _ \\Sigma \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} \\bold { E } ( A \\cdot B ) = \\bold { D } _ A ^ { \\ast } ( B ) + \\bold { D } _ B ^ { \\ast } ( A ) . \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} p _ 1 ^ * h ^ { \\prime * } ( { \\cal F } \\boxtimes { \\cal G } ) = p _ 1 ^ * a ^ { \\prime * } { \\cal F } \\otimes p _ 1 ^ * b ^ { \\prime * } { \\cal G } \\to R \\Psi _ { c ' } h ^ { \\prime * } ( { \\cal F } \\boxtimes { \\cal G } ) \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} I ^ \\gamma \\circ \\rho ^ \\circ ( \\gamma ^ { - 1 } z \\gamma ) = \\rho ^ \\circ ( z ) \\circ I ^ \\gamma z \\in \\pi _ 0 ( Z _ { G ^ \\circ } ( \\sigma , y ) ) . \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { \\int _ { 0 } ^ { T } f ( x \\Delta ( u _ t ) ) d t } { \\int _ { 0 } ^ { T } g \\circ \\pi _ 1 ( x \\Delta ( u _ t ) ) d t } = \\frac { \\mu ( f ) } { \\mu ( g \\circ \\pi _ 1 ) } . \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} \\beta \\ = \\ v \\left ( \\left ( g \\left ( 1 + \\epsilon - \\frac { u \\delta } { g } \\right ) \\right ) ' - s \\right ) . \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} M _ j = [ x ^ j ] \\exp { \\left ( n r x - \\sum _ { s \\geq 2 } \\frac { n u _ s ( r ) } { s } ( - x ) ^ s \\right ) } . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} H _ s y : = - y '' + s y ' + V ( x ) y = \\lambda y , \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} n _ 0 = \\begin{cases} \\epsilon n , & \\theta = 0 \\\\ \\left ( 1 + \\epsilon \\right ) \\theta _ n n , & . \\end{cases} \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\ge X \\\\ ( m , k ) = 1 } } \\mu ( m ) \\tau _ r ( m ) m ^ { - 1 } \\ll \\sigma _ { - 1 } ( k ) \\exp ( - c \\sqrt { \\log { x } } ) . \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{align*} H _ 0 ^ 1 ( \\mathbb { T } ) : = H _ S \\oplus H _ S ^ { \\perp } , H _ S : = \\mbox { s p a n } \\{ e ^ { \\mathrm { i } \\ , j \\ , x } : j \\in S \\} , H _ S ^ { \\perp } : = \\{ u = \\sum _ { j \\in S ^ c } u _ j \\ , e ^ { \\mathrm { i } \\ , j \\ , x } \\in H _ 0 ^ 1 ( \\mathbb { T } ) \\} , \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\delta _ L ( X ) = T \\dot { \\otimes } X , \\delta _ L ( \\hat { X } ) = T \\dot { \\otimes } \\hat { X } \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} \\xi _ i ( x ) & = \\lambda _ { \\delta _ i } ( { \\rm d i s t } ( u _ i ( x ) , \\Sigma _ 0 ) ) \\cdot \\Pi _ i ( x , u _ i ( x ) , \\xi ( x ) ) \\\\ & = \\Pi _ i ( x , u _ i ( x ) , \\xi ( x ) ) . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} ( u - \\phi ^ h ) ( x _ h , t _ h ) & \\leq u ( x _ 0 , t _ 0 ) - \\phi ^ h ( x _ 0 , t _ 0 ) = \\phi ( x _ 0 , t _ 0 ) - \\phi ( x _ 0 + h \\eta , t _ 0 ) \\\\ & = - \\frac { 1 } { 2 } h ^ 2 \\eta ^ t \\nabla ^ 2 \\phi ( x _ 0 , t _ 0 ) \\eta + o ( h ^ 2 ) < 0 , \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\ldots \\subset E _ { n - 1 } \\subset E _ n = E \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } - \\bigtriangleup _ { p ( x ) } u = f ( x , u ) \\Omega \\\\ u = 0 \\partial \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} \\phi ( z ) = \\phi _ 0 ( z ) - \\frac { 1 } { 2 } \\Psi ( 0 ) ( 1 - z ) ^ + \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} g _ f ( x ) : = \\frac { f ( x ) } { f ( x ) _ n } \\mbox { f o r a l l } x \\in \\Sigma _ 0 : = \\{ x \\in \\R ^ n _ { > 0 } \\colon x _ n = 1 \\} , \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} q _ { k + 1 } ( a _ { k + 1 } | a ^ k ) = \\P ( [ X _ { k + 1 } ] _ b = a _ { k + 1 } | [ X ^ k ] _ b = a ^ k ) , \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} Z _ { \\alpha , D _ 0 } ( G ) = \\sum _ { [ D \\Delta D _ 0 ] = \\alpha } \\nu ( D ) \\ , , \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} ( z - T ) ^ { - 1 } = K ( z ) ( I - B ( z ) ) ^ { - 1 } K ( z ) , \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} \\mathcal { V } = \\mathcal { A } \\partial _ { r } + \\sum \\limits ^ { m } _ { k = 1 } \\mathcal { B } _ { k } \\frac { 1 } { r } \\partial _ { k } \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} \\begin{array} { c c l } \\frac { d } { d t } X _ i & = & V _ i \\\\ \\frac { d } { d t } V _ i & = & - ( V \\cdot V ) X _ i \\end{array} \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} t ^ 6 a _ 2 ' = 3 r , t ^ { 1 2 } a _ 4 ' = a + 3 r ^ 2 , t ^ { 1 8 } a _ 6 ' = b + r a + r ^ 3 . \\end{align*}"}
-{"id": "841.png", "formula": "\\begin{align*} J _ { m , d i r } X = J _ { d i r } ( X - P _ { ( m ) } X P _ { ( m ) } ) + P _ { ( m ) } X P _ { ( m ) } , X \\in \\mathfrak { S } _ 2 ( \\mathcal { H } ) , \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{align*} S ^ + : = \\{ \\overline { \\jmath } _ 1 , \\dots , \\overline { \\jmath } _ { \\nu } \\} , S : = S ^ + \\cup ( - S ^ + ) = \\{ \\pm j : j \\in S ^ + \\} , \\overline { \\jmath } _ i \\in \\mathbb { N } \\setminus \\{ 0 \\} , \\forall i = 1 , \\dots , \\nu \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} A = \\begin{bmatrix} A ' & 1 \\\\ 0 & A ' \\end{bmatrix} , B = \\begin{bmatrix} B ' & B _ { 1 2 } \\\\ 0 & B ' \\end{bmatrix} , F = \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} \\omega ( \\pi ) : = ( - 1 ) ^ { s ( \\pi ) + \\nu ( \\pi ) + 1 } 2 ^ { \\nu _ d ( \\pi ) } , \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} \\underset { t \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\int _ { { \\bf R } ^ N } u ( y , t ) \\eta ( y ) \\ , d y = \\int _ { { \\bf R } ^ N } \\eta ( y ) \\ , d \\mu ( y ) \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} P _ q ^ c Y = \\zeta _ q ( Y ) \\ , Y _ q ' , \\zeta _ q ( Y _ q ' ) = 1 , \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} E _ { s , 4 } = \\| u \\| _ { H ^ s } ^ 2 \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} e _ i ( x _ 1 , \\dotsc , x _ i , x _ { i + 1 } , \\dotsc , x _ n ) & = \\begin{cases} ( x _ 1 , \\dotsc , x _ i + 1 , x _ { i + 1 } - 1 , \\dotsc , x _ n ) & x _ { i + 1 } > 0 , \\\\ 0 & , \\end{cases} \\\\ f _ i ( x _ 1 , \\dotsc , x _ i , x _ { i + 1 } , \\dotsc , x _ n ) & = \\begin{cases} ( x _ 1 , \\dotsc , x _ i - 1 , x _ { i + 1 } + 1 , \\dotsc , x _ n ) & x _ i > 0 , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} & \\phi _ { A } ( z ) = a ( z - \\beta ) - \\log \\frac { z } { \\beta } , \\\\ & \\phi ( z ) = \\left \\{ \\begin{array} { l } \\displaystyle \\phi _ { A } ( z ) , z \\in { { \\rm E x t } \\ , { \\cal S } } , \\\\ \\displaystyle - \\phi _ { A } ( z ) , \\quad \\ , z \\in { { \\rm I n t } \\ , { \\cal S } } . \\end{array} \\right . \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} K ( \\xi , \\eta ) = 2 ( \\xi - \\eta ) J ( \\xi ) J ( \\eta ) - ( \\eta J ( \\xi ) - \\xi J ( \\eta ) ) ( J ( \\zeta ) - J ( \\xi ) - J ( \\eta ) ) . \\end{align*}"}
-{"id": "8827.png", "formula": "\\begin{align*} \\widetilde { R } _ S = { \\left [ \\widetilde { R } - \\widetilde { R } _ e ^ { * } \\right ] ^ + } , \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} \\begin{aligned} | J _ { \\rho _ t } - J _ { \\infty } | & \\le \\int _ { t } ^ { \\infty } \\bigg | \\frac { d } { d r } J _ { \\rho _ r } \\bigg | d r \\\\ & \\le C H ( \\rho _ { 0 } | M _ { \\Omega _ { \\rho _ { 0 } } } ) \\int _ { t } ^ { \\infty } e ^ { - \\int _ { 0 } ^ { r } B ( s ) d s } d r . \\end{aligned} \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} N _ 1 ( x , y ) & \\le ( x y ) ^ { - 2 \\beta } \\sup _ { 0 < s \\le 1 / 4 } s ^ { - 1 / 2 } \\big ( x + \\sqrt { s } \\big ) ^ { 2 \\beta - 1 } \\exp \\bigg ( - \\frac { ( x \\vee y ) ^ 2 } { 5 1 2 s } \\bigg ) e ^ { x ^ 2 } \\\\ & \\lesssim ( x y ) ^ { - 2 \\beta } ( x \\vee y ) ^ { 2 \\beta - 2 } e ^ { x ^ 2 } = ( x \\wedge y ) ^ { - 2 \\beta } ( x \\vee y ) ^ { - 2 } e ^ { x ^ 2 } . \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} M _ k = : k ! \\ , m _ k . \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} M = \\begin{bmatrix} - I & T _ { k - 1 } \\\\ T & a _ k I \\end{bmatrix} . \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} N ( S _ 1 , S _ 2 ) = [ J S _ 1 , J S _ 2 ] - [ S _ 1 , S _ 2 ] - J [ J S _ 1 , S _ 2 ] - J [ S _ 1 , J S _ 2 ] = [ J S _ 1 , J S _ 2 ] . \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} \\lambda ( s ) = \\frac { \\tilde { \\lambda } _ 0 } { s ^ 2 } + \\frac { \\tilde { \\lambda } _ 1 } { s } + \\tilde { \\lambda } _ 2 + \\dots . \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} \\begin{aligned} \\min _ { x \\in H } F _ 0 ( x ) \\\\ F _ i ( x ) \\leq 0 , i \\in N , \\end{aligned} \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} \\min \\left \\{ \\sum _ { t = k } ^ \\infty t N _ t , \\sum _ { t = 1 } ^ \\ell t N _ t \\right \\} - \\max \\left \\{ \\sum _ { t = k + 1 } ^ \\infty t N _ t , \\sum _ { t = 1 } ^ { \\ell - 1 } t N _ t \\right \\} . \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\tau = \\frac { { 2 \\ , t - { t _ 0 } - { t _ f } } } { { { t _ f } - { t _ 0 } } } , \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} V ( x ) & = x ^ 6 + 6 a | x | ^ 5 + ( 9 a ^ 2 - 4 b ) x ^ 4 - ( 1 2 a b - 2 c ) | x | ^ 3 + ( 4 b ^ 2 + 6 a c - 7 ) x ^ 2 \\\\ & - \\left ( 1 8 a + 4 b c - \\frac { 2 } { c } \\right ) | x | , \\\\ E _ 2 ^ { ( - ) } & = - 1 0 b - c ^ 2 - \\frac { 6 a } { c } + \\frac { 2 } { c ^ 2 } , \\\\ \\psi _ 2 ^ { ( - ) } ( x ) & = e ^ { - \\frac { 1 } { 4 } x ^ 4 - a | x | ^ 3 + b x ^ 2 - c | x | } x \\left ( | x | + \\frac { 1 } { c } \\right ) , \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} P _ { S | V , Y } ( 1 | v , y ) = 0 \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} h '' _ { \\ell } ( w ) & = \\frac { a _ { \\ell } } { k _ { \\ell } w ( a _ { \\ell } - w ) } - \\frac { A _ { \\ell } B ^ 2 } { ( 1 + B w ) ^ 2 } \\\\ & = \\frac { a _ { \\ell } g _ { \\ell } ( w ) } { k _ { \\ell } w ( a _ { \\ell } - w ) ( 1 + B w ) ^ 2 } , \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} H = \\Big { \\langle } x _ 1 , x _ 2 , \\ldots , x _ n \\colon & [ x _ i , x _ j , x _ k ] = 1 , \\\\ & [ x _ i , x _ j ] ^ p = 1 , \\\\ & x _ i ^ p = \\prod _ { j < k } [ x _ j , x _ k ] ^ { c _ { i j k } } , \\\\ & \\prod _ { j < k } [ x _ j , x _ k ] ^ { d _ { l j k } } = 1 , l = 1 , 2 , \\ldots , t \\Big { \\rangle } , \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} \\int _ \\Omega \\rho ^ { i n } \\varphi ( t = 0 ) d x + \\int _ 0 ^ \\infty \\int _ \\Omega \\rho \\ , \\partial _ t \\varphi \\ , d x \\ , d t = \\int _ 0 ^ \\infty \\int _ \\Omega \\rho ( h _ \\alpha \\varphi - \\mathcal { L } _ \\alpha ( \\varphi ) ) d x \\ , d t , \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} E _ f = \\sum _ { M } V _ M ^ { - 1 } \\frac { 1 } { | W _ M | } \\int _ { \\nu \\in \\mathfrak { a } _ { M , 0 } ^ * } d \\nu \\langle E _ f , E _ { M , \\nu } \\rangle E _ { M , \\nu } \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} Y _ { i } & = ( 1 - S _ i ) h _ { i s } X _ 0 + Z _ i , \\forall i \\in [ 1 : N ] , \\\\ Y _ { N + 1 } & = \\sum _ { i = 1 } ^ N S _ i h _ { d i } X _ i + Z _ { N + 1 } , \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} q = \\frac { x x ' } { x '' x ''' } X . \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} B ( z ) \\psi _ j = \\sum _ { m = 1 } ^ { \\infty } \\langle B ( z ) \\psi _ j , \\psi _ m \\rangle \\psi _ m , \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ^ { - 1 } A A ^ { \\dagger } & = A A ^ { \\dagger } \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ^ { - 1 } , \\\\ \\big ( I + A ^ { \\dagger } U F _ { S _ { A } } U ^ { \\ast } ( A ^ { \\dagger } ) ^ { \\ast } \\big ) ^ { - 1 } A ^ { \\dagger } A & = A ^ { \\dagger } A \\big ( I + A ^ { \\dagger } U F _ { S _ { A } } U ^ { \\ast } ( A ^ { \\dagger } ) ^ { \\ast } \\big ) ^ { - 1 } . \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} \\langle x \\cdot \\nabla p , q \\rangle _ { \\alpha + 1 } = \\langle x \\cdot \\nabla s _ p , q \\rangle _ { \\alpha + 1 } = k \\langle s _ p , q \\rangle _ { \\alpha + 1 } = k \\langle p , q \\rangle _ { \\alpha + 1 } . \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} \\mathbf { \\Phi } = { \\rm d i a g } \\left ( e ^ { j \\varphi 0 } \\ ; e ^ { j \\varphi 1 } \\ ; \\cdots \\ ; e ^ { j \\varphi ( N - 1 ) } \\right ) , \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} \\operatorname { R N } _ { N , N , N } \\leq \\tfrac { N d } { N - 1 } ( N + ( N + 1 ) N ) ^ N = \\tfrac { N } { N - 1 } ( 1 + \\tfrac { 2 } { N } ) ^ N d N ^ { 2 N } \\leq 8 d N ^ { 2 N } . \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} r ( t ) = P ( t ) + \\int \\limits _ { 0 } ^ { t } Q ( t , \\tau ) r ( \\tau ) d \\tau \\ \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} { \\mathcal A } ( E ) : = \\int _ { x _ 1 ^ * ( E ) } ^ { x _ 1 ( E ) } \\sqrt { E - V _ 1 ( t ) } \\ , d t , \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} \\mathrm { T r } _ 2 ( g _ j ) = \\mathrm { T r } ^ { G L _ 2 } _ { S L _ 2 } ( g _ j ) = \\mathcal { I } \\circ \\mathrm { T r } ( g _ j ) = \\mathcal { I } \\circ \\left ( \\dfrac { 1 } { 2 ^ { d - 1 } } \\sum _ { \\epsilon _ i \\in R } \\psi ' ( \\epsilon _ i ^ { - 1 } ) g _ j \\mid _ { k , w } \\epsilon _ i \\right ) . \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} D ^ 2 _ X . \\pi _ * ( [ \\tilde { E } ] ^ 2 ) = D ^ 2 _ X . \\pi _ * ( \\frac { 1 } { 2 } \\pi ^ * E . \\tilde { E } ) = \\frac { 1 } { 2 } D ^ 2 _ X . E ^ 2 = \\frac { 1 } { 2 } \\pi _ * ( \\tilde { D } ^ 2 . \\tilde { E } ^ 2 ) = - \\frac { 1 } { 2 } \\pi _ * ( \\beta ^ * ( ( D _ { | \\Sigma } ) ^ 2 ) ) . \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} ( \\partial _ t + b \\partial _ \\alpha ) a & = 2 \\Im [ \\P , \\P \\left [ R _ t + b R _ \\alpha \\right ] ] \\bar R _ \\alpha + 2 \\Im [ \\P , R ] \\partial _ \\alpha \\bar \\P \\left [ \\bar R _ t + b \\bar R _ \\alpha \\right ] \\\\ & + 2 \\Im \\left ( b \\partial _ \\alpha \\P [ R \\bar R _ \\alpha ] - \\P \\left [ b R _ \\alpha \\bar R _ \\alpha \\right ] - \\P \\left [ R \\partial _ \\alpha \\bar \\P ( b \\bar R _ \\alpha ) \\right ] \\right ) . \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} Q _ 2 ( z ) d z ^ 2 = 4 \\left ( \\frac { \\partial q _ 2 \\circ \\pi _ 1 ^ { - 1 } } { \\partial z } ( z ) \\right ) ^ 2 d z ^ 2 . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} m _ { 2 q + 1 } = \\bar { \\beta } _ { 2 q + 1 } = 0 \\ { \\rm a n d } \\ m _ { 2 q } = \\bar { \\beta } _ { 2 q } , \\forall \\ q \\in \\N \\cup \\{ 0 \\} . { } \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} \\Omega _ X ( u , w ) = 2 b _ { 1 1 } \\overline { b _ { 1 1 } } + b _ { 2 2 } \\overline { b _ { 2 2 } } \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} S ( U ) : = 9 6 0 \\frac { \\big ( ( U - 4 ) e ^ { \\frac { U } { 4 } } + ( U + 4 ) e ^ { - \\frac { U } { 4 } } \\big ) ^ 2 } { U ^ 5 } . \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { \\varphi ^ * ( n ) } = T ^ * \\log x + U ^ * + O \\left ( x ^ { - u } ( \\log x ) ^ { 5 / 3 } \\right ) . \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} \\langle \\partial T , f \\rangle = \\langle T , d f \\rangle = \\int f ( \\psi _ { \\nu _ \\gamma } ( 1 ) ) - f ( \\psi _ { \\nu _ \\gamma } ( 0 ) ) \\dd \\pi ( \\gamma ) \\ ; , \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ k \\binom { k + ( i + j ) p } { m + i p } \\equiv _ { p ^ 2 } 2 ^ k \\binom { i + j } { i } \\left ( 1 + p ( i + j ) \\sum _ { m = 1 } ^ { k } \\frac { 1 } { m 2 ^ m } \\right ) . \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\begin{cases} \\dot X ^ \\varepsilon ( t ) = \\cfrac { 1 } { \\varepsilon } \\ , b ( X ^ \\varepsilon ( t ) ) + \\alpha ( t ) \\ \\ \\ t \\in \\mathbb R , \\\\ X ^ \\varepsilon ( 0 ) = x \\in \\mathbb R ^ 2 , \\\\ \\alpha \\in L ^ \\infty ( \\mathbb R ; \\mathbb R ^ 2 ) , \\end{cases} \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\frac { \\gamma \\left ( a + 1 , a \\right ) } { \\Gamma \\left ( a + 1 \\right ) } & = \\frac { \\gamma \\left ( a , a \\right ) } { \\Gamma \\left ( a \\right ) } - \\frac { a ^ { a } \\exp \\left ( - a \\right ) } { \\Gamma \\left ( a + 1 \\right ) } \\\\ & = \\frac { 1 } { 2 } + \\frac { 1 } { \\sqrt { 2 \\pi } } \\left ( - \\frac { 2 } { 3 } a ^ { - 1 / 2 } + \\frac { 2 3 } { 2 7 0 } a ^ { - 3 / 2 } \\right ) + O \\left ( a ^ { - 5 / 2 } \\right ) . \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} u '' + \\exp ( u - u _ 1 ) - \\exp ( u _ { - 1 } - u ) = 0 , \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{gather*} q _ { 2 t t } = \\frac { 2 q _ { 2 t } } { q _ { 2 } } \\left ( q _ { 2 t } - \\frac { u _ t } { u } \\right ) + \\frac { 4 } { 9 } \\left ( q ^ 2 _ 2 - \\frac { 3 } { 2 } \\right ) \\left ( \\frac { u ^ 2 _ t } { u ^ 2 } - t - 2 u ^ 2 \\right ) - \\frac { 2 } { 9 } q _ 2 \\left ( 3 \\frac { u ^ 2 _ t } { u ^ 2 } - t \\right ) + \\frac { 4 } { 9 q _ 2 } \\frac { u ^ 2 _ t } { u ^ 2 } . \\end{gather*}"}
-{"id": "3121.png", "formula": "\\begin{align*} w _ { 1 , k } = t _ { - 1 , k + Q _ 1 ( 0 ) - R _ 1 ( t _ { 1 , k } - ) + R _ { - 1 } ( t _ { 1 , k } - ) } - t _ { 1 , k } . \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} \\Delta _ k ( x ; \\boldsymbol { a } ) = \\frac { 1 } { \\phi ^ k ( q ) } \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { H } ( 1 , \\delta ( x ) ) } \\ ( F _ k ( s ) - \\widetilde { F } _ k ( s ) \\ ) \\frac { x ^ s } { s } d s + O \\ ( \\frac { 1 } { k ! } \\frac { x } { e ^ { c _ 2 \\sqrt { \\log x } } } \\ ) \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} T _ \\theta : = \\left ( \\begin{array} { r r } \\cos ( \\theta ) & \\sin ( \\theta ) \\\\ - \\sin ( \\theta ) & \\cos ( \\theta ) \\end{array} \\right ) \\tilde T _ a : = \\left ( \\begin{array} { l l } a & 0 \\\\ 0 & a ^ { - 1 } \\end{array} \\right ) \\ , . \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} \\mathcal { F } ( i _ { \\infty } , 0 ) = 0 \\mbox { w i t h } \\lVert \\mathfrak { I } _ { \\infty } \\rVert _ { s _ 0 + \\mu , \\mathcal { G } _ { \\infty } } ^ { L i p ( \\gamma ) } \\le C \\ , \\varepsilon ^ { 6 - 2 b } \\gamma ^ { - 1 } \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} | \\sin ( t | \\xi | \\phi _ { \\sigma } ( \\xi ) ) - \\sin ( t | \\xi | ) | \\chi _ { L } \\le C t | \\xi | ^ { 4 \\sigma - 1 } \\chi _ { L } , \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} D ( \\eta _ { 1 } ) - D ( \\eta _ { \\tau } ) = \\int _ { a } ^ { b } \\sqrt { \\eta _ { 1 } '' } - \\sqrt { \\eta _ { \\tau } '' } = \\int _ { a } ^ { b } \\sqrt { \\xi '' } - \\sqrt { \\xi '' + ( 1 - \\tau ) ( \\eta '' - \\xi '' ) } . \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} w _ { i , k } = \\left [ t _ { - i , k + Q _ i ( 0 ) - R _ i ( t _ { i , k } - ) - Q _ { - i } ( 0 ) + R _ { - i } ( t _ { i , k } - ) } - t _ { i , k } \\right ] ^ + . \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} \\frac { 1 } { ( 1 - u _ 1 ) ^ { r + 1 } } = \\sum _ { \\ell _ 1 = 1 } ^ { \\infty } \\binom { \\ell _ 1 + r - 1 } { r } u _ 1 ^ { \\ell _ 1 - 1 } , \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} p _ { i , l } ^ { B _ { i , j } } = \\left \\{ \\begin{array} { l l } p _ { i , l } , \\ \\ & l \\in B _ { i , j } \\\\ 0 , \\ \\ & l \\in [ k ] ~ \\mathrm { a n d } ~ l \\notin B _ { i , j } \\end{array} \\right . \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} \\tilde { W } = - ( X _ 1 + i Y _ 1 ) ( X _ 1 - i Y _ 1 ) ^ { - 1 } ( X _ 2 - i Y _ 2 ) ( X _ 2 + i Y _ 2 ) ^ { - 1 } , \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} u \\cdot \\j = 0 , y = - h . \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} \\frac { \\pi ^ 2 } { 6 } = \\zeta ( 2 ) = \\frac { \\pi } { 2 } + 4 \\pi ^ 2 \\sum _ { n \\geq 1 } \\sigma _ 1 ( n ) e ^ { - 2 \\pi n } . \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} - a + T _ { 1 1 } g _ { 1 } ^ { \\prime } ( z _ { 1 } ^ { \\ast } ) = \\alpha , \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} \\begin{cases} \\sigma ( Y ) = A Y \\\\ \\partial ( Y ) = B Y \\end{cases} \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} | z - \\zeta _ t | \\leq | G ' ( F ( \\zeta _ t ) ) | \\frac { 4 \\pi / t } { ( 1 - 4 \\pi / t ) ^ 2 } \\leq | G ' ( F ( \\zeta _ t ) ) | \\frac { 1 6 \\pi } { t } = \\frac { 1 6 \\pi } { | F ' ( \\zeta _ t ) | t } \\leq \\frac { 1 2 8 \\pi ^ 2 } { h ( x ) t } . \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{gather*} I _ 2 = 2 r _ 2 + t \\end{gather*}"}
-{"id": "1125.png", "formula": "\\begin{align*} E _ r = \\min _ { \\frac { 1 } { k _ n } \\leq \\gamma \\leq 1 } \\max _ { 0 \\leq \\rho \\leq 1 } f ( \\gamma , \\rho ) . \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} \\sqrt { \\pi } \\Gamma ( 2 x + 1 ) = 2 ^ { 2 x - 1 } \\Gamma \\left ( x + \\frac { 1 } { 2 } \\right ) \\Gamma ( x + 1 ) . \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{gather*} \\mu _ { + t } = \\frac { 2 } { 3 } \\frac { u _ t } { u } \\mu _ + - \\frac { 1 } { 3 } \\nu , \\\\ \\mu _ { - t } = - \\frac { 2 } { 3 } \\frac { u _ t } { u } \\mu _ - + \\frac { 1 } { 3 } \\nu , \\\\ \\nu _ { t } = \\frac { 2 } { 3 } u ^ 2 \\mu _ - + \\frac { 2 } { 3 } \\frac { \\omega } { u ^ 2 } \\mu _ + \\end{gather*}"}
-{"id": "3875.png", "formula": "\\begin{align*} { \\bf y } _ 1 = \\sum _ { j = 1 } ^ { J - 1 } { \\bf G } _ { 1 j } { \\bf x } _ j + { \\bf G } _ { 1 J } { \\bf x } _ J + { \\bf z } _ 1 , \\\\ { \\bf y } _ 2 = { \\bf G } _ { 2 J } { \\bf x } _ J + \\sum _ { j = 1 } ^ { J - 1 } { \\bf G } _ { 2 j } { \\bf x } _ j + { \\bf z } _ 2 , \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{gather*} \\begin{aligned} L _ { \\mu \\alpha } ^ { \\iota } = 0 & \\mbox { i f } \\mu > \\iota \\mbox { o r } \\alpha < \\iota , \\mbox { a n d } \\\\ L _ { \\alpha \\nu } ^ { \\iota } = 0 & \\mbox { i f } \\alpha > \\iota \\mbox { o r } \\nu < \\iota . \\\\ \\end{aligned} \\end{gather*}"}
-{"id": "2419.png", "formula": "\\begin{align*} & \\max ~ \\prod _ { i = 1 } ^ { K } ( \\nu _ i - \\delta _ i ) \\\\ & \\mathsf { s u b j e c t ~ t o } ~ \\nu _ i \\geq \\delta _ i , i = 1 , \\ldots , K , ~ ( \\nu _ 1 , \\ldots , \\nu _ { K } ) \\in \\mathcal { V } . \\square \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} 1 > \\mathbb { P } ( \\mathcal { A } _ i | \\bigcap _ { j < i } \\mathcal { A } _ j ) > \\frac { 2 d i \\Phi } { i d + \\Phi i d } = \\frac { 2 \\Phi } { 1 + \\Phi } . \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} ( T ) & = \\int _ { \\mathbb { T } ^ n } \\sum _ { \\xi \\in \\mathbb { Z } } \\sigma _ { e ^ { - t T ^ * T } - e ^ { - T T ^ * } } ( x , e _ { \\xi } ) d x . \\\\ & = \\int _ { \\mathbb { T } ^ n } \\sum _ { \\xi \\in \\mathbb { Z } } \\sigma _ { [ T , S ] } ( x , e _ { \\xi } ) d x , \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} h ^ t z _ j = u _ j , \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} w _ i ( n ) = \\sum \\limits ^ { V } _ { v = 0 } { b _ { i , v } f _ v ( n ) } , \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} \\operatorname { K e r } ( p r o j _ C ) | _ { A } = \\{ 0 \\} \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\mathcal { I } _ { \\mathrm { a } } \\right ] = \\mathbb { E } \\left [ \\sum _ { x \\in \\Phi _ { \\mathrm { b } } \\setminus \\{ i , \\bar { i } \\} } \\frac { g _ { x i k } } { \\Vert x \\Vert ^ { \\alpha } } \\right ] = \\frac { 2 \\pi \\lambda s ^ { - ( \\alpha - 2 ) } } { \\alpha - 2 } . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} \\begin{cases} Y ' ( h ) = \\cfrac { D H ( Y ( h ) ) } { | D H ( Y ( h ) ) | ^ 2 } \\ \\ \\ h \\in \\bar J _ i \\setminus \\{ 0 \\} , \\\\ Y ( 0 ) = p _ i , \\end{cases} \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} u ' ( x ) = \\frac { \\theta e ^ { x / \\theta } g ' ( x ) ( g ''^ 2 ( x ) - g ' ( x ) g ''' ( x ) ) } { ( \\theta g '' ( x ) + g ' ( x ) ) ^ 2 } < 0 , x > \\alpha - 1 . \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} - T ^ { 2 q } + 2 T ^ { q + 1 } + T ^ { q ( q + 1 ) } \\sum _ { c } ( c + T ) ^ { 1 - q } & = \\\\ \\frac { ( - T ^ { 2 q } + 2 T ^ { q + 1 } ) \\cdot \\prod _ { c } ( c + T ^ { q } ) + T ^ { q ( q + 1 ) } \\cdot ( T ^ { q } - T ) } { \\prod _ { c } ( c + T ^ { q } ) } & = \\\\ \\frac { T ^ { q ( q + 1 ) + 1 } + } { \\prod _ { c } ( c + T ^ { q } ) } . \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} b _ { k + 1 } - b _ k = p ^ { - k - 1 } \\log \\left [ 2 ^ { \\frac { N } { \\theta } } \\frac { p ^ k - 1 } { p - 1 } \\right ] \\le p ^ { - k - 1 } ( C k + C ) , k = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} ( \\lambda _ 1 ( s , 2 ) - \\lambda _ n ) \\int _ { \\Omega } w _ 1 v _ n d x = \\dfrac 1 { \\| u \\| _ { \\scriptstyle L ^ { \\infty } ( \\Omega ) } } \\int _ { \\Omega } f ( x ) w _ 1 d x > 0 \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} H = \\bigoplus _ { i = 1 } ^ l H ^ { 0 } ( X , K _ X ^ { d _ i } ) \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{gather*} 9 \\eta _ { t t } + 9 \\eta \\eta _ t + \\eta ^ 3 + P ( t ) \\eta + Q ( t ) = 0 , \\end{gather*}"}
-{"id": "7464.png", "formula": "\\begin{align*} F ( x + m \\varepsilon ) = f ( x ) + ( f ' ( x ) m + h ( x ) ) \\varepsilon \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} f ( \\mathbf { u } , \\mathbf { v } ) \\triangleq \\mathbb { P } _ { p ^ n _ { W | U , V } } \\Big ( ( \\mathbf { u } , \\mathbf { v } , \\mathbf { W } ) \\notin \\mathcal { A } _ { \\epsilon ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } , \\epsilon ^ { ( 2 ) } _ { \\alpha , \\delta _ 2 } } \\Big | \\mathbf { U } = \\mathbf { u } , \\mathbf { V } = \\mathbf { v } \\Big ) , \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { j = 1 } ^ { q } \\beta _ { j } - \\displaystyle \\sum _ { i = 1 } ^ { p } \\alpha _ { i } > - 1 , \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} u _ i '' + \\frac { m - 1 } { r } u _ i ' - \\frac { \\lambda _ i } { r ^ 2 } u _ i = { f } _ i \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} | \\{ u ^ n : \\ell _ { \\rm L Z } ( u ^ n ) \\leq r \\} | \\leq \\sum _ { i = 1 } ^ r 2 ^ i \\leq 2 ^ { r + 1 } . \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} T \\Box _ { b } ^ { ( q ) } = \\Box ^ { ( q ) } _ { b } T ~ ~ \\Omega ^ { 0 , q } ( X ) , \\forall q = 0 , 1 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} & \\mu _ { p , \\Lambda } ^ { \\eta } ( A ) = \\sum _ { \\omega \\in A } \\mu _ { p , \\Lambda } ^ { \\eta } ( \\omega ) = \\sum _ { \\omega \\in A } \\gamma _ { \\beta , \\Lambda } ^ { c _ \\eta } ( c _ { \\eta , \\Lambda } ( \\omega ) ) = \\sum _ { \\sigma \\in B } \\gamma _ { \\beta , \\Lambda } ^ { c _ \\eta } ( \\sigma ) = \\gamma _ { \\beta , \\Lambda } ^ { c _ \\eta } ( B ) . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ p ) ^ s u = f ( x ) & \\mbox { i n } \\Omega , \\\\ u = 0 & \\mbox { i n } \\Omega ^ c , \\end{cases} \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} n - 1 & = H ( V _ { 1 } ^ { n } ) \\\\ & = H ( V _ { m _ { n } + 1 } ^ { n - m _ { n } } ) + H ( V _ { 1 } ^ { m _ { n } } | V _ { m _ { n } } ^ { n - m _ { n } } ) + H ( V _ { n - m _ { n } + 1 } ^ { n } | V _ { 1 } ^ { n - m _ { n } } ) \\\\ & \\geq H ( V _ { m _ { n } + 1 } ^ { n - m _ { n } } ) + H ( V _ { 1 } ^ { m _ { n } } | V _ { m _ { n } } ^ { n } ) + H ( V _ { n - m _ { n } + 1 } ^ { n } | V _ { 1 } ^ { n - m _ { n } } ) . \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} \\xi ^ i - S _ j \\xi ^ i = 0 , \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } = S _ j \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 - \\bold { 1 } _ j } \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} P = \\bigcup _ { v \\in L } \\bigcup _ { i = 1 } ^ m ( v + x _ i + B _ n ( r _ 1 / 2 ) ) . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} & \\psi _ w ( t ) = a t ^ { 2 - q } \\| w \\| ^ 2 + \\epsilon t ^ { 4 - q } \\| w \\| ^ { 4 } - t ^ { 2 ^ * _ \\alpha - q } \\int _ { \\Omega \\times \\{ 0 \\} } | w ( z , 0 ) | ^ { 2 ^ * _ \\alpha } d z . \\ ; \\textrm { T h e n } \\ ; \\\\ & \\psi _ w ^ { \\prime } ( t ) = a ( 2 - q ) t ^ { 1 - q } \\| w \\| ^ 2 + \\epsilon ( 4 - q ) t ^ { 3 - q } \\| w \\| ^ { 4 } - ( 2 ^ * _ \\alpha - q ) t ^ { 2 ^ * _ \\alpha - 1 - q } \\int _ { \\Omega \\times \\{ 0 \\} } | w ( z , 0 ) | ^ { 2 ^ * _ \\alpha } d z . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} \\iota _ { X _ t ^ G } \\Theta _ t = - \\tilde { \\beta } \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} \\big ( Z _ { G \\times \\C ^ \\times } ( \\sigma , r ) \\cap M ( y ) \\big ) / \\big ( Z _ { G \\times \\C ^ \\times } ( \\sigma , r ) \\cap M ( y ) \\big ) ^ \\circ = \\pi _ 0 \\big ( Z _ { G \\times \\C ^ \\times } ( \\sigma , r ) \\cap M ( y ) \\big ) . \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} K = K _ s \\otimes K ^ s , \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} 0 = \\sum _ f ( n ^ K ( \\partial f ) + 1 ) = n ^ K ( C ) + E _ + F = n ^ K ( C ) + V + E + F = n ^ K ( C ) + 1 \\ , , \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\alpha = x _ 0 \\cdots x _ { r - 1 } ( y _ 0 \\cdots y _ { r - 1 } ) ^ { p ^ n - p ^ { n - 1 } - 1 } . \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{align*} \\Phi ^ + _ P = \\{ \\beta _ 1 = w _ L ( \\alpha _ { i _ 1 } ) , \\beta _ 2 = w _ L s _ { i _ 1 } ( \\alpha _ { i _ 2 } ) , \\ldots , \\beta _ N = w _ L s _ { i _ 1 } \\cdots s _ { i _ { N - 1 } } ( \\alpha _ { i _ N } ) \\} ; \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} \\{ u \\in L ^ 2 ( \\R ^ 3 ) ^ 4 : \\ ; \\alpha \\cdot D u \\in L ^ 2 ( \\R ^ 3 ) ^ 4 \\} = H ^ 1 ( \\R ^ 3 ) ^ 4 , \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P } ( - 1 ) ^ { \\nu ( \\pi ) } z ^ { \\nu _ o ( \\pi ) } q ^ { | \\pi | } = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( - q z ; q ^ 2 ) _ \\infty } - 1 . \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} T \\gamma \\left ( \\frac { \\partial } { \\partial t } \\right ) = \\frac { \\partial } { \\partial t } + \\sum _ { j = 1 } ^ n \\frac { \\partial \\gamma ^ j } { \\partial t } \\frac { \\partial } { \\partial p _ j } , T \\gamma \\left ( \\frac { \\partial } { \\partial q ^ i } \\right ) = \\frac { \\partial } { \\partial q ^ i } + \\sum _ { j = 1 } ^ n \\frac { \\partial \\gamma ^ j } { \\partial q ^ i } \\frac { \\partial } { \\partial p _ j } \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} L ^ + _ 0 \\oplus ( G _ 0 \\oplus H _ 1 ) & \\cong L ^ + \\oplus H _ 0 \\oplus H _ 1 \\cong L ^ + \\oplus H _ 1 \\oplus H _ 0 \\cong L ^ + _ 1 \\oplus ( G _ 1 \\oplus H _ 0 ) , \\\\ L ^ - _ 0 \\oplus ( G _ 0 \\oplus H _ 1 ) & \\cong L ^ - \\oplus H _ 0 \\oplus H _ 1 \\cong L ^ - \\oplus H _ 1 \\oplus H _ 0 \\cong L ^ - _ 1 \\oplus ( G _ 1 \\oplus H _ 0 ) . \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} \\rho ( ( \\theta _ 1 , \\theta _ 2 ) , ( \\eta _ 1 , \\eta _ 2 ) ) = | \\theta _ 1 - \\eta _ 1 | . \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} T _ 2 & = C _ 2 \\\\ T _ 3 & = C _ 3 \\\\ T _ 4 & = C _ 4 - \\tfrac 1 2 T _ 2 ^ 2 \\\\ T _ 5 & = C _ 5 - T _ 3 T _ 2 \\\\ T _ 6 & = C _ 6 - T _ 4 T _ 2 - \\tfrac 1 6 T _ 2 ^ 3 - \\tfrac 1 2 T _ 3 ^ 2 . \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} U ^ { 2 } + Z ^ { 2 } = 2 \\mu , \\mu \\in \\left \\{ 1 , - 1 , \\omega , \\omega ^ { 2 } \\right \\} \\cap \\mathbb { Q } \\left ( \\sqrt { - D } \\right ) . \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} g ( \\mu _ { k + 1 } ) - g ( \\mu _ k ) = g ' ( \\eta _ k ) ( \\mu _ { k + 1 } - \\mu _ k ) , \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} z _ n = ( c _ n x + d _ n ) ^ { 1 / t } \\mbox { a n d } \\alpha _ n = b ^ 2 _ n / 4 ( 4 + t ) \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} F _ { 3 n + 2 } ^ { \\left ( 3 \\right ) } = F _ { 3 n } ^ { \\left ( 3 \\right ) } + F _ { 3 n + 1 } ^ { \\left ( 3 \\right ) } \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} \\frac { \\langle h _ { 1 } , . . . , h _ { a } + 1 , . . . , h _ { \\mathsf { N } } | h _ { 1 } , . . . , h _ { a } + 1 , . . . , h _ { \\mathsf { N } } \\rangle } { \\langle h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } | h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } \\rangle } = \\prod _ { \\substack { b = 1 \\\\ b \\neq a } } ^ { \\mathsf { N } } \\frac { X _ { a } ^ { ( h _ { a } ) } - X _ { b } ^ { ( h _ { b } ) } } { X _ { a } ^ { ( h _ { a } + 1 ) } - X _ { b } ^ { ( h _ { b } ) } } , \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} \\P ( \\{ M _ n \\le \\lambda ^ n \\} ) = \\P ( \\{ | A _ 0 | \\le \\lambda ^ n \\} ) ^ { \\alpha _ n } . \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} & \\hat { f } ( 0 ) = \\mu \\{ 0 \\} - \\frac { 1 } { \\pi } \\sum _ { j = 1 } ^ n \\Im ( \\frac { \\hat { \\mu } ( j ) } { j } ) \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\sum _ { j = 1 } ^ n \\hat { \\mu } ( j ) - \\frac { 1 } { \\pi } \\sum _ { j = 1 } ^ n \\Im ( \\frac { \\hat { \\mu } ( j ) } { j } ) . \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial { \\theta ^ j } } \\left [ \\frac { \\partial { U ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ i } } \\right ] = u _ i ' ( s ^ i ( \\hat { x } , \\theta ) ) \\frac { \\partial { s ^ i ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ j } } , \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} n _ 0 = \\begin{cases} \\left ( 1 + \\epsilon \\right ) n ( \\ell ) , & \\lim \\limits _ { k _ { \\ell } \\to \\infty } n ( \\ell ) / k _ { \\ell } > 0 \\\\ \\epsilon k _ { \\ell } , & \\lim \\limits _ { k _ { \\ell } \\to \\infty } n ( \\ell ) / k _ { \\ell } = 0 . \\end{cases} \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} \\operatorname { K e r } \\begin{bmatrix} \\operatorname { D P [ 0 ] } \\\\ \\mathcal { T } \\end{bmatrix} . \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} \\lambda S _ { \\rho } \\cap \\mathbb { R } ^ - \\subseteq \\begin{cases} [ - \\sqrt { 5 } \\rho r , 0 ] & \\theta \\in [ - \\frac { \\pi } { 2 } , \\frac { \\pi } { 2 } ] ; \\\\ [ - 2 \\rho r / | \\sin \\theta | , 0 ] & . \\end{cases} \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} B _ { 1 2 } \\widetilde { a _ { 1 1 } } - B _ { 1 2 } \\widetilde { a _ { 2 2 } } + \\widetilde { b _ { 2 2 } } - \\widetilde { b _ { 1 1 } } = 0 \\Rightarrow \\widetilde { b _ { 2 2 } } - B _ { 1 2 } \\widetilde { a _ { 2 2 } } = \\widetilde { b _ { 1 1 } } - B _ { 1 2 } \\widetilde { a _ { 1 1 } } . \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} u ' = \\frac { 1 } { u _ 1 - u } + \\frac { 1 } { u - u _ { - 1 } } . \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} \\mathcal { L } _ { p , F } = \\dfrac { l _ \\lambda e T r _ 1 \\left ( \\left ( T r _ 2 ( \\mathcal { E } _ c ^ { \\chi \\chi _ { - 1 } } * \\Theta _ \\chi \\mid [ \\mathfrak { l } ^ 2 ] ) \\right ) \\mid \\Psi _ 2 \\right ) } { \\Delta ( X , Y ) \\mathcal { E } _ 2 ( X , Y ) H } , \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} \\gamma _ 3 = \\frac { 1 } { 4 8 } \\left ( 6 c _ 3 - b _ 2 ^ 2 c _ 1 - b _ 2 c _ 1 ^ 2 + 2 b _ 2 c _ 2 + 2 b _ 3 c _ 1 + b _ 2 ^ 3 - 4 b _ 3 b _ 2 + 6 b _ 4 + c _ 1 ^ 3 - 4 c _ 1 c _ 2 \\right ) . \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} F ^ { \\otimes ( - 1 ) } = ( f ^ { ( 0 ) } ) ^ { - 1 } \\sum _ { n = 0 } ^ \\infty \\big ( \\Omega - ( f ^ { ( 0 ) } ) ^ { - 1 } F \\big ) ^ { \\otimes n } . \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} \\langle \\lfloor p _ { i } ; h _ { i } \\rceil , \\lfloor p _ { j } ; h _ { j } \\rceil \\rangle _ { B _ { 1 } } = \\langle p _ { i } , p _ { j } \\rangle _ { \\mathcal { Q } _ { W _ { 1 } } } = \\langle \\phi ( p _ { i } ) , \\phi ( p _ { j } ) \\rangle _ { \\mathcal { Q } _ { W _ { 2 } } } = \\langle \\lfloor \\phi ( p _ { i } ) ; h _ { i } \\rceil , \\lfloor \\phi ( p _ { j } ) ; h _ { j } \\rceil \\rangle _ { B _ { 2 } } . \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} C ^ a ( \\xi , 0 ) & = \\frac { i \\xi } { \\Lambda ( \\xi ) ( e ^ { 2 \\xi } - 1 ) } \\left [ 2 J ( \\xi ) - \\xi J ' ( \\xi ) + J ( \\xi ) J ' ( \\xi ) \\right ] , \\\\ C ^ a ( 0 , - \\eta ) & = - \\frac { i \\eta } { \\Lambda ( \\eta ) ( e ^ { 2 \\eta } - 1 ) } \\left [ 2 J ( \\eta ) - \\eta J ' ( \\eta ) - J ( \\eta ) J ' ( \\eta ) \\right ] . \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} B e _ { 2 k - 1 } : = - d _ k t _ k e _ { 2 k } , B e _ { 2 k } : = d _ k t _ k e _ { 2 k - 1 } , k \\in \\N , \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\begin{cases} h '' ( z ) + c ( | p | ) h ' ( z ) - g ( h ( z ) ) = 0 & z \\in \\left ( 0 , \\frac { 1 } { | p | } \\right ) \\\\ h ( 0 ) = 0 , \\ \\ h \\left ( \\frac { 1 } { | p | } \\right ) = 1 & \\\\ h ' \\left ( \\frac { 1 } { | p | } \\right ) = h ' ( 0 ) . \\end{cases} \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} \\tilde { B } _ E ^ { - 1 } ( \\theta + \\alpha ) \\tilde { A } _ { | s | , E } ( \\theta ) \\tilde { B } _ E ( \\theta ) = \\left ( \\begin{matrix} e ^ { 2 \\pi i \\rho _ { | s | } ( E ) } \\ \\ & 0 \\\\ 0 \\ \\ & e ^ { - 2 \\pi i \\rho _ { | s | } ( E ) } \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} ( T _ n - \\kappa _ n ^ { n + 1 } ) ( T _ n - \\kappa _ n ^ n ) = 0 , \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} E [ v _ { u } ( Y - \\gamma ( X ) ) | X ] & = \\lambda - \\int _ { - \\infty } ^ { \\gamma ( X ) - \\gamma _ { 0 } ( X ) } f ( u | X ) d u \\\\ & = - f ( 0 | X ) [ \\gamma ( X ) - \\gamma _ { 0 } ( X ) ] - [ \\partial f ( \\delta ( X ) | X ) / \\partial u ] [ \\gamma ( X ) - \\gamma _ { 0 } ( X ) ] ^ { 2 } , \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 1 } ( \\eta - \\xi ) ( \\phi _ { \\nu } '' - \\phi _ { \\mu } '' ) \\ , d s = - \\int _ { 0 } ^ { 1 } ( \\eta - \\xi ) \\ , d ( \\nu - \\mu ) \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} h ( n ) = \\arg \\min _ h \\left ( O ( h ^ p ) + O _ p \\big ( ( n h ^ d ) ^ { - 1 / 2 } \\big ) \\right ) \\enspace , \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} d z + \\sum _ { j = 1 } ^ n x _ j d y _ j = 0 . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { g ( { \\bf R } ) - \\frac 1 2 } a _ n ( I _ d \\ : : W _ 2 ^ { \\bf R } ( \\Bbb T ^ d ) \\rightarrow L _ \\infty ( \\Bbb T ^ d ) ) = ( 2 g ( { \\bf R } ) - 1 ) ^ { - \\frac 1 2 } ( \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) ) ^ { g ( { \\bf R } ) } . \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} { { \\dot x } _ 1 } ( t ) = { x _ 2 } ( t ) , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} \\hat { m } ( \\zeta , \\theta ) = \\sum _ { n \\geq 0 } m _ n ( \\theta ) \\frac { \\zeta ^ { n + i k } } { \\Gamma ( n + 1 + i k ) } . \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} \\partial _ t u + A _ c ( u ) u = 0 , t > 0 , u ( 0 ) = u ^ 0 , \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} \\mathcal I _ s ^ { \\beta , q } ( \\mu ) \\leq C ' \\int _ { \\underline a \\in \\Sigma } \\dfrac { \\sum _ { n = 0 } ^ { \\infty } \\beta ^ n \\int _ { \\underline b ' \\in \\Sigma : \\underline b ' \\wedge \\sigma ^ n \\underline a = \\phi } \\left ( \\frac { \\mu [ a _ 1 \\cdots a _ n ] } { d ( \\underline a , a _ 1 \\cdots a _ n \\underline b ' ) ^ { s ' } } \\right ) ^ { q - 1 } d \\mu ( \\underline b ' ) } { \\sum _ { n = 1 } ^ { \\infty } \\beta ^ n } d \\mu ( \\underline a ) \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} k ^ R _ N ( t , x , x ' ) : = \\frac { - 1 } { 2 \\pi i } \\int \\limits _ C e ^ { - t z } q _ z ^ N ( x , x ' ) d z , \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} q _ 0 : = \\sup \\{ q \\geq 2 : \\} \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} Q ( w ) = Q ( w _ 1 ) + Q ( w _ 2 ) + 2 Q ( w _ 1 , w _ 2 ) \\geq 0 . \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{gather*} X _ { \\{ A , B \\} } = [ X _ A , X _ B ] ( \\mathrm { m o d } \\operatorname { k e r } \\omega ) \\forall \\ , A , B \\in \\Lambda _ \\omega ^ { m } ( J ^ \\infty E ) , \\end{gather*}"}
-{"id": "1518.png", "formula": "\\begin{align*} \\mbox { R e } \\langle ( \\tilde { H } - z - i \\epsilon \\tilde { S } ) f , f \\rangle _ { { \\mathcal G } ^ { - 1 } , { \\mathcal G } ^ 1 } & = Q _ H ( f , f ) - \\mbox { R e } ( z ) | | f | | _ { L ^ 2 } ^ 2 \\\\ - \\mbox { I m } \\langle ( \\tilde { H } - z - i \\epsilon \\tilde { S } ) f , f \\rangle _ { { \\mathcal G } ^ { - 1 } , { \\mathcal G } ^ 1 } & = \\mbox { I m } ( z ) | | f | | _ { L ^ 2 } ^ 2 + \\epsilon Q _ { [ H , i A ] } ( f , f ) \\ \\geq \\ \\mbox { I m } ( z ) | | f | | _ { L ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} b = \\boxed { 1 ^ { 1 7 } \\ , 2 ^ 8 \\ , 3 ^ { 4 6 } } \\otimes \\boxed { 1 ^ { 4 8 } \\ , 2 ^ { 4 2 } \\ , 3 ^ { 3 6 } } \\otimes \\boxed { 1 ^ { 2 9 } \\ , 2 ^ { 5 0 } \\ , 3 ^ { 1 1 } } \\ , \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\nabla _ s \\sigma _ 2 \\cdot t _ 2 = \\displaystyle { \\frac { \\partial _ x \\sigma _ 2 ( \\Gamma ) } { \\sqrt { 1 + | \\partial _ x ^ 2 ( f + g ) | } } } . \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} \\{ x z \\} \\{ y _ 1 , \\ldots , y _ n \\} = \\sum _ { i = 0 } ^ n \\pm x \\{ y _ 1 , \\ldots , y _ i \\} z \\{ y _ { i + 1 } , \\ldots , y _ n \\} . \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} g _ 3 = x ^ 8 + 2 0 x ^ 4 + 5 2 g _ 4 = x ^ 8 + 4 x ^ 4 + 8 4 ; \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} \\mathcal L \\psi = z ^ 2 \\psi \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} = 2 c _ 2 ( X ) + \\frac { 9 } { 2 } E ^ 2 . \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\omega } \\hat { \\psi } = g _ 1 + K _ { 2 0 } \\hat { \\eta } - K _ { 1 1 } ^ T ( \\varphi ) \\hat { w } . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} \\iota _ { N , i } ( F ) = \\tilde { V } ( x _ i ) F . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} ( A + U V ^ { \\ast } ) ^ { \\dagger } = \\big ( I + A ^ { \\dagger } U F _ { S _ { A } } U ^ { \\ast } ( A ^ { \\dagger } ) ^ { \\ast } \\big ) ^ { - 1 } \\big ( A ^ { \\dagger } - A ^ { \\dagger } U S _ { A } ^ { \\dagger } V ^ { \\ast } A ^ { \\dagger } \\big ) \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ^ { - 1 } , \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} H ^ * _ k \\doteq 1 ( k = h ( \\mathbf { W } ^ * ( 0 ) , \\mathbf { J } ( 0 ) , \\boldsymbol { \\mu } ) ) \\left ( V ^ * _ k - \\frac { J _ { k } ( 0 ) } { \\mu _ k ^ 2 } \\right ) \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} M ( \\psi ( \\N ) ) = \\lbrace & 1 , 3 , 4 , 6 , 8 , 2 0 , 7 2 , 9 0 , 2 2 2 , 2 5 2 , 5 0 0 , 5 2 2 , 5 5 2 , 5 7 0 , 5 9 2 , 7 5 0 , 7 7 0 , \\\\ & 9 9 2 , 7 0 0 0 , \\underbrace { 5 5 \\ldots 5 } _ { 6 9 } 0 \\rbrace . \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} I _ x : = \\begin{cases} [ x _ { i - 1 } , x _ { i + 1 } ] & \\\\ ( - \\infty , x _ { \\min I } ] & \\\\ [ x _ { \\max I } , + \\infty ) & \\end{cases} \\end{align*}"}
-{"id": "9268.png", "formula": "\\begin{align*} U _ { S / R } = \\left \\{ \\mathfrak { n } \\in \\operatorname { M a x } S \\ | \\ R _ { \\mathfrak { n } \\cap R } \\right \\} \\not = \\emptyset . \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} \\widetilde { C } = \\prod _ { p \\in \\P } \\left ( 1 - \\frac { p ^ 2 + p - 1 } { p ^ 3 ( p + 1 ) } \\right ) , \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} 1 7 2 8 \\Delta _ m = 2 ^ 6 3 ^ 3 2 ^ 8 \\tilde { \\Delta } = c _ 4 ^ 3 - c _ 6 ^ 2 = 2 ^ { 1 2 } ( \\tilde { c } _ 4 ^ 3 - \\tilde { c } _ 6 ^ 2 ) \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} I _ { n _ 0 \\ldots n _ N } ( u _ k ; u _ { k + 1 } ) = \\Delta _ { n _ 0 } ( u _ k ; s _ 1 ) \\times \\Delta _ { n _ 1 } ( s _ 1 ; s _ 2 ) \\times \\ldots \\times \\Delta _ { n _ N } ( s _ N ; u _ { k + 1 } ) . \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} \\dot { x } ( t ) = f _ c ( x ( t ) ) , \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} \\min _ { \\sigma \\in \\mathcal { P } _ m } \\sum _ k a _ k a _ { \\sigma ( k ) } = \\sum _ k a _ k a _ { m - k + 1 } \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ J k _ n ^ { ( j ) } B ^ { ( j ) } ( n ) \\leq \\frac { n } { 2 } \\log \\left ( \\sum _ { j = 1 } ^ J k _ n ^ { ( j ) } \\right ) - \\sum _ { j = 1 } ^ J \\beta ^ { ( j ) } \\ell _ n H _ 2 \\left ( \\alpha _ n ^ { ( j ) } \\right ) . \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { p - 1 } f ( m ) = \\binom { K } 2 = \\frac 1 2 K ( K - 1 ) . \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} \\int _ { \\Omega } u \\partial _ j \\phi \\ , d x = - \\int _ { \\Omega } \\phi \\partial _ j u \\ , d x \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} | \\sum _ { k = k _ { j } + 1 } ^ { \\infty } < P _ { k } x ^ { * } _ { n _ { j } } , \\pi _ { k } x _ { n _ { j } } > | < \\frac { \\epsilon _ { 0 } } { 4 } , j = 1 , 2 , . . . \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} ( H w ) ( z ) = - z w _ { z z } - 3 w _ z + \\frac { 1 } { 4 } ( z ^ 2 w ) _ z , z < 0 . \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} \\dim \\mathcal { F } ^ { 0 } ( r , n , l ) = & \\dim \\mathcal { M } ^ { 0 } ( r , n ) + \\dim S ^ l ( \\mathbb { C } ^ 2 ) ^ 0 + l ( r - 1 ) \\\\ = & 2 r n + 2 l + l ( r - 1 ) = 2 r n + l ( r + 1 ) . \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} \\abs { x ( t ) } = m \\mu ( \\Omega _ 0 ) ^ { - 1 / p } \\leq c r \\mu ( \\Omega _ 0 ) ^ { - 1 } \\leq r c \\norm { x } _ { \\infty } = c m \\mu ( \\Omega _ 0 ) ^ { - 1 / p } \\leq \\int _ { \\Omega _ 0 } x ( t ) \\textup d t . \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} \\sigma ( 1 ) = \\sigma ' ( n ) , \\ ; \\sigma ( 2 ) = \\sigma ' ( 1 ) , \\dots , \\sigma ( n ) = \\sigma ' ( n - 1 ) . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} \\phi ( c ( x ) ) & = \\frac { 1 } { \\omega _ C } 1 \\wedge c ( x ) \\wedge c ( x ) ^ 2 \\\\ & = \\frac { 1 } { \\omega _ C } 1 \\wedge c ( x ) \\wedge c \\left ( \\frac { \\bar x ^ 2 \\gamma } { t } \\right ) \\\\ & = \\frac { 1 } { \\omega _ I } x \\wedge \\frac { \\bar x ^ 2 \\gamma } { t } \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} I ( t ) = \\left [ \\begin{array} { c } t \\ast \\\\ \\ast \\end{array} \\right ] , J ( t ) = \\left [ \\begin{array} { c c } 0 & \\ast \\end{array} \\right ] . \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\chi _ \\alpha ( \\rho ) = \\rho ^ \\ast \\chi ^ { ( N ) } _ \\alpha \\in H ^ \\ast ( G ) \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ { n } [ n ^ 2 - ( j - 1 ) n ] ! ^ { \\frac { n ^ 2 - c _ 2 n } { n ^ 2 - ( j - 1 ) n } } . \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} R _ \\alpha \\bar Y = T _ { R _ \\alpha } \\bar Y + T _ { \\bar Y } R _ \\alpha + \\Pi [ R _ \\alpha , \\bar Y ] . \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} N _ { ( a , b ) } ( n ) = ~ & 8 \\sigma ( \\frac { n } { a } ) - 3 2 \\sigma ( \\frac { n } { 4 a } ) + 8 \\sigma ( \\frac { n } { b } ) - 3 2 \\sigma ( \\frac { n } { 4 b } ) + 6 4 \\ , W _ { ( a , b ) } ( n ) + 1 0 2 4 \\ , W _ { ( a , b ) } ( \\frac { n } { 4 } ) \\\\ & - 2 5 6 \\ , \\biggl ( W _ { ( 4 a , b ) } ( n ) + W _ { ( a , 4 b ) } ( n ) \\biggr ) . \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} A ( f ) = \\left \\{ z \\in \\C \\colon l \\in \\N | f ^ { n } ( z ) | > M ( R , f ^ { n - l } ) n \\geq l \\right \\} , \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} H ^ n _ W ( X , \\mathcal { F } ) = \\left \\{ \\begin{array} { l l } H ^ n _ { e t } ( X , \\mathcal { F } ) & \\mbox { $ n = 0 , 1 $ } \\\\ \\mathrm { H o m } _ X ( \\mathcal { F } , \\mathbb { G } _ m ) _ { t o r } ^ D & \\mbox { $ n = 3 $ } \\\\ 0 & \\mbox { $ n > 3 $ } . \\end{array} \\right . \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} 4 n ^ 2 \\sin ^ 2 \\Big ( \\frac { \\pi ( j + \\lambda _ i ( n ) ) } { a _ i ( n ) } \\Big ) & = 4 n ^ 2 \\sin ^ 2 \\Big ( \\frac { \\pi ( a _ i ( n ) - j - \\lambda _ i ( n ) ) } { a _ i ( n ) } \\Big ) \\\\ & \\geq \\frac { 1 } { \\alpha _ i ^ 2 } \\Big ( \\pi ( a _ i ( n ) - j - \\lambda _ i ( n ) ) ( 1 - \\frac { \\pi ^ 2 } { 2 4 } ) \\Big ) ^ 2 \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} \\hat { \\eta } = \\mathcal { D } _ { \\omega } ^ { - 1 } ( g _ 2 - [ \\partial _ { \\psi } \\theta _ 0 ( \\varphi ) ] ^ T M _ { \\varphi } [ g _ 2 ] ) + M _ { \\varphi } [ \\hat { \\eta } ] , M _ { \\varphi } [ \\hat { \\eta } ] \\in \\mathbb { R } ^ { \\nu } , \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} Q _ { n } ( \\omega , A ) = P _ { V } ( ( K _ { m } ) _ { m \\in \\mathbb { N } } \\in A | \\mathcal { F } _ { n } ) ( \\omega ) , \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} \\int _ { E _ { k } } | x _ { k } | d \\mu > \\epsilon , k = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} \\mu \\sigma ( B ) A = A B + \\partial ( A ) , \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} \\alpha ( 1 / \\lambda ) \\alpha ( q \\lambda ) = 1 , \\prod _ { a = 0 } ^ { p - 1 } \\alpha ( \\lambda q ^ { a } ) = 1 \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} F ^ { \\mathbf { \\underline m } } v _ { \\varpi _ j } = \\sum _ { k = 1 } ^ r c _ k F ^ { \\mathbf { \\underline a } ^ k } v _ { \\varpi _ j } . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\omega _ { \\beta , \\mu , \\phi } ( \\eta ( b _ { { 0 } } ) ) = \\sqrt { \\rho _ { 0 } } \\exp ( - i \\phi ) \\ , \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} i _ 0 ^ * \\mathcal { W } = d \\ , i _ 0 ^ * \\Lambda = \\sum _ { 1 \\le k < j \\le \\nu } A _ { k \\ , j } ( \\varphi ) \\ , d \\varphi _ k \\wedge d \\varphi _ j , A _ { k \\ , j } ( \\varphi ) : = \\partial _ { \\varphi _ k } a _ j ( \\varphi ) - \\partial _ { \\varphi _ j } a _ k ( \\varphi ) . \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} \\omega = \\frac { d x } { x ^ m } \\wedge ( \\sum _ { i = 0 } ^ { m - 1 } \\pi ^ * ( \\alpha _ { i } ) x ^ i ) + \\beta \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} X \\left ( t , \\frac { 1 } { \\sqrt { 2 } } ( r , r ) \\right ) = \\frac { r } { \\sqrt { 2 } } \\ , ( e ^ { 2 t } , e ^ { 2 t } ) \\ \\ \\ t \\in \\left [ 0 , \\ , \\frac { 1 } { 2 } \\log \\frac { s } { r } \\right ] , \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} \\overline { D } ^ q ( \\mu ) = \\limsup _ { \\delta \\to 0 } \\frac { \\log M ^ q _ \\delta ( \\mu ) } { ( q - 1 ) \\log \\delta } \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} h ( x , y , z , x ' , y ' ) : = ( ( \\partial _ x - \\frac { y ' } { 2 } \\partial _ z ) f ( x , y , z ) ) ^ 2 + ( ( \\partial _ y + \\frac { x ' } { 2 } \\partial _ z ) f ( x , y , z ) ) ^ 2 + \\beta ^ 2 ( \\partial _ z f ( x , y , z ) ) ^ 2 \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} \\tau ( \\pi ) = \\sum _ { i = 1 } ^ { m ( \\pi ) } \\chi ( f _ { i } \\equiv 1 ( 2 ) ) 2 ^ { \\nu _ d ( \\pi , i ) } . \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} S \\ : = \\ \\{ v ( s - \\epsilon ' ) : \\epsilon \\in K ^ { \\prec 1 } \\} \\ \\subseteq \\ ( \\Gamma ^ { > } ) ' \\ \\subseteq \\ \\Gamma _ { \\infty } . \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} X _ H = \\left ( \\frac { p ^ 2 } { 2 m } - V ( q ) - \\alpha S \\right ) \\frac { \\partial } { \\partial S } - \\left ( \\alpha p + V ' ( q ) \\right ) \\frac { \\partial } { \\partial p } + \\frac { p } { m } \\frac { \\partial } { \\partial q } \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} & Z _ - = q ^ { 2 \\theta ( i ) } \\cdot X _ 1 X _ 2 \\dots X _ { 2 \\theta ( i ) + 1 } \\cdot Y _ 1 , \\\\ & Z _ 0 = q ^ { - 2 n } \\cdot X _ { 2 \\theta ( i ) } ^ { - 1 } \\dots X _ 2 ^ { - 1 } X _ 1 ^ { - 1 } \\cdot Y _ { 2 i } ^ { - 1 } \\dots Y _ 2 ^ { - 1 } Y _ 1 ^ { - 1 } , \\\\ & Z _ + = q ^ { 2 i } \\cdot X _ 1 \\cdot Y _ 1 Y _ 2 \\dots Y _ { 2 i + 1 } . \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{align*} B ^ t _ p ( L _ p ( \\Omega ) ) = W ^ t ( L _ p ( \\Omega ) ) \\hookrightarrow L _ \\infty ( \\Omega ) , \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} & - \\int _ r ^ R \\pi t \\partial _ t ( G _ t ( \\eta , \\xi _ k , \\dots , \\xi _ n ) ) \\frac { d t } { t } \\\\ & = \\int _ r ^ R 2 \\pi ^ 2 t ^ 2 \\Big ( \\alpha ^ 2 \\eta ^ 2 + \\sum _ { j = k } ^ n \\alpha _ j ^ 2 ( \\xi _ j ^ 2 + ( \\xi _ j + \\eta ) ^ 2 ) \\Big ) \\ , G _ t ( \\eta , \\xi _ k , \\dots , \\xi _ n ) \\frac { d t } { t } . \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} E ( \\lambda ) : = \\left \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in \\R ^ { 3 } : \\frac { x _ 1 ^ 2 } { a _ 1 ^ 2 } + \\frac { x _ 2 ^ 2 } { a _ 2 ^ 2 } + \\frac { x _ 3 ^ 2 } { a _ 3 ^ 2 } \\leq \\frac { \\lambda } { \\pi ^ { 2 } } \\right \\} . \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} \\limsup _ { x \\to x _ 0 } \\frac { \\abs { f ( x ) - f ( x _ 0 ) - d f ( x _ 0 ) \\cdot ( \\phi ( x ) - \\phi ( x _ 0 ) ) } } { \\rho ( x , x _ 0 ) } = 0 . \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} B ( x , r ) : = \\left \\{ y \\in \\mathbb { R } ^ d : | y - x | < r \\right \\} \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} \\begin{aligned} S ( U , f ^ n ) & \\leq S \\ ! \\left ( \\sqrt { R } , f ^ n \\right ) \\leq \\frac { 2 T _ 0 ( R , f ^ n ) } { \\log R } \\leq \\frac 1 2 \\left ( T ( R , f ^ n ) + C \\right ) \\\\ & \\leq \\frac 1 2 \\left ( \\log M ( R , f ^ n ) + C \\right ) \\leq \\log M ^ n ( R , f ) \\end{aligned} \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} L U + \\partial _ s U & = ( P _ { t - s } f ) \\ , \\Gamma ^ { \\mathrm { h o r i } } ( \\log ( P _ { t - s } f ) ) \\leq ( P _ { t - s } f ) \\ , \\Gamma ^ { \\mathrm { e l l i } } ( \\log ( P _ { t - s } f ) ) = V ( s ) \\\\ L V + \\partial _ s V & = 2 ( P _ { t - s } f ) \\ , \\Gamma _ 2 ^ { \\mathrm { m i x } } ( \\log ( P _ { t - s } f ) ) \\geq - \\frac { 2 } { \\nu } V ( s ) . \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} F _ \\infty ^ { ( \\alpha ) } ( A ) = \\{ x \\in F _ \\infty ( A ) \\mid g \\mapsto \\tilde { \\alpha } _ { \\infty , g } ( x ) ~ \\} . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{gather*} L _ X = i _ X D + ( - 1 ) ^ { \\epsilon ( X ) } D i _ X . \\end{gather*}"}
-{"id": "970.png", "formula": "\\begin{align*} m _ z \\ ! & = \\ ! \\left \\{ \\ ! \\ ! \\begin{array} { l l } 0 & z < z _ 0 \\\\ \\min \\{ b M , z _ 0 M - s M + 1 \\} & z = z _ 0 \\\\ \\max \\{ 0 , \\min \\{ M , b M \\ ! + \\ ! s M \\ ! - 1 \\ ! - ( z \\ ! - \\ ! 1 ) M \\} \\} & z > z _ 0 \\end{array} \\right . \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} \\psi _ { p , \\epsilon } = J _ { \\frac { | p | } { k } } \\left ( \\sqrt { 2 \\epsilon } \\ , | r | \\right ) e ^ { - \\imath ( \\epsilon t + p \\phi ) } , \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} f ( 0 , \\cdot ) = f ^ 0 , g ( 0 , \\cdot ) = g ^ 0 , \\Gamma ( 0 , \\cdot ) = \\Gamma ^ 0 . \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} p _ { K _ { s , t } } ( x ) = \\sum _ k \\frac { k ( s + t ) { s \\choose k } { t \\choose k } } { s t { s + t \\choose s } } x ^ k . \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\alpha = x ( - t , - w ^ { 3 \\theta - 1 } + t ^ { 3 \\theta + 1 } , - v - t ^ { 3 \\theta + 1 } + t w ^ { 3 \\theta - 1 } ) h ( - 1 ) \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} \\sum _ { t \\in [ 0 , T ] } \\bar { q } ^ { 0 , Q } ( t ) \\ , \\tfrac { ( T - t ) ^ k } { k ! } = \\sum _ { t \\in [ 0 , T ] } \\ 1 _ { \\{ 0 \\} } ( t ) \\ , \\tfrac { ( T - t ) ^ k } { k ! } = \\tfrac { T ^ { k } } { k ! } . \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\begin{aligned} f ( b ) & = k _ { 0 } ^ { + } + \\sum _ { j = 2 } ^ { k } p _ { j } ( b - 1 ) + \\sum _ { j = 2 } ^ { k _ 1 } \\xi _ { j } . \\end{aligned} \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} { a + b \\brack a } _ v : = \\frac { [ a + b ] _ v ! } { [ a ] _ v ! [ b ] _ v ! } . \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} E _ { k , 3 } = \\sum _ { \\alpha _ 1 + \\alpha _ 2 = 2 k - 2 } \\int u ^ { ( \\alpha _ 1 ) } u ^ { ( \\alpha _ 2 ) } u d x + \\sum _ { \\alpha _ 1 + \\alpha _ 2 + 2 \\alpha _ 3 = k - 2 } ( - 1 ) ^ { k + 1 + \\alpha _ 3 } \\int u ^ { ( \\alpha _ 1 ) } u ^ { ( \\alpha _ 2 ) } u ^ { ( \\alpha _ 3 ) } d x . \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} F _ N F _ L \\Lambda L \\Lambda \\cap N = F _ N \\Big ( \\bigcup _ { l \\in F _ L } l \\Lambda l ^ { - 1 } \\Big ) \\Lambda = N . \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} f ( x ) & = a _ 0 x ^ n + a _ 1 x ^ { n - 1 } + \\cdots + a _ { n - 1 } x + a _ n , \\\\ g ( x ) & = b _ 0 x ^ m + b _ 1 x ^ { m - 1 } + \\cdots + b _ { m - 1 } x ^ x + b _ m , \\\\ u ( x ) & = u _ 0 x ^ { m - 1 } + u _ 1 x ^ { m - 2 } + \\cdots + u _ { m - 2 } x + u _ { m - 1 } , \\\\ v ( x ) & = v _ 0 x ^ { n - 1 } + v _ 1 x ^ { n - 2 } + \\cdots + v _ { n - 2 } x + v _ { n - 1 } . \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} \\underset { t \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\int _ { { \\bf R } ^ N } u ( x , t ) \\eta ( x ) \\ , d x = \\int _ { { \\bf R } ^ N } \\eta ( x ) \\ , d \\mu ( x ) \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} E \\Big [ \\sum \\limits _ { i = 1 } ^ n T _ i ^ \\alpha \\Big ] = 1 . \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} 2 & ( \\Delta _ { i } - \\Delta _ { i + 1 } ) = ( i + 1 ) \\rho ^ i - ( 2 i + 2 ) \\rho ^ { i + 1 } + ( i + 1 ) \\rho ^ { i + 2 } \\\\ & + ( n - i - 1 ) \\rho ^ { n - i - 2 } - 2 ( n - i - 1 ) \\rho ^ { n - i - 1 } + ( n - i - 1 ) \\rho ^ { n - i } \\\\ & = ( 1 - \\rho ) ^ 2 ( ( i + 1 ) \\rho ^ i + ( n - i - 1 ) \\rho ^ { n - i - 2 } ) . \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} \\sum _ { 2 i + j = n } \\left ( \\binom { i + j } { i } \\mod 2 \\right ) = a _ { n + 1 } . \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} H _ { 0 , \\Lambda , \\mu } = \\sum _ { { k } } { k } ^ { 2 } b _ { { k } } ^ { * } b _ { { k } } - \\mu N _ { \\Lambda } \\ , \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} V ( x ) = x ^ 4 - 4 a | x | ^ 3 + 4 a ^ 2 x ^ 2 - 4 | x | \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} g ( x ) \\leq f _ p ( x ) = \\lim f _ { p _ i } ( x _ i ) = \\lim g ( x _ i ) , \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} \\mathcal { L } _ { V } \\left ( t , x _ { t } , i \\right ) & = V _ { t } \\left ( t , x , i \\right ) + V _ { x } \\left ( t , x , i \\right ) f \\left ( t , x _ { t } , i , u \\right ) \\\\ & + \\frac { 1 } { 2 } \\mathrm { t r a c e } \\left ( g ^ { \\mathrm { T } } \\left ( t , x _ { t } , i , u \\right ) V _ { x x } \\left ( t , x , i \\right ) g \\left ( t , x _ { t } , i , u \\right ) \\right ) + \\sum \\limits _ { j = 1 } ^ { N } \\gamma _ { i j } V \\left ( t , x , j \\right ) , \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} \\xi ^ \\star = q ^ { 1 / 2 } \\eta , x ^ \\star = x , \\eta ^ \\star = - q ^ { - 1 / 2 } \\xi \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y ^ j _ k } = \\frac { \\partial } { \\partial y ^ j _ { k + 1 } } + \\sum _ l \\bigg ( \\frac { \\partial a ^ k _ l } { \\partial y ^ j _ k } \\bigg ) \\frac { \\partial } { \\partial y ^ l _ { k + 1 } } + \\sum _ \\tau \\bigg ( \\frac { \\partial \\beta ^ k _ \\tau } { \\partial y ^ j _ k } \\bigg ) \\frac { \\partial } { \\partial \\eta ^ \\tau _ { k + 1 } } \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} K = - \\frac { \\overline { a } b } { a \\overline { b } } , \\ L = - \\frac { \\overline { d } e } { d \\overline { e } } , \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} \\xi : = \\varepsilon ^ { - 2 } \\ , \\mathbb { M } ^ { - 1 } [ \\omega - \\overline { \\omega } ] , \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} R ^ { b e l t } ( \\ell ) & \\le \\frac { \\log _ 9 S _ U ^ { b e l t } ( \\ell ) } { 8 1 + 5 4 ( \\ell - 1 ) } = : R _ U ^ { b e l t } ( \\ell ) \\\\ & \\approx \\frac { 2 2 . 8 7 0 6 + 1 4 . 7 2 5 8 ( \\ell - 1 ) } { 8 1 + 5 4 ( \\ell - 1 ) } . \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} 2 ( \\l - 1 ) P & = - ( \\l - 1 ) ( \\psi ' ) ^ 2 + \\l ^ 2 \\psi ^ 2 + \\l \\psi '' \\psi , \\\\ \\psi ( 0 ) & = \\psi ( 2 \\pi ) . \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} 2 f _ 1 ( x ) - f _ 2 ( x ) & = - 2 x _ 1 ^ 2 - x _ 2 ^ 2 + 5 x _ 2 \\\\ & = - 2 ( x _ 1 + 0 . 2 5 ) ^ 2 - ( x _ 2 - 3 . 5 ) ^ 2 + x _ 1 - 2 x _ 2 + 1 2 . 3 7 5 \\\\ & \\leq x _ 1 - 2 x _ 2 + 1 2 . 3 7 5 \\leq - 2 0 + 1 2 . 3 7 5 \\leq 1 0 . \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} \\frac { 1 } { p } \\| ( u _ k , v _ k ) \\| ^ p - \\frac { 1 } { q } \\int _ \\Omega ( \\lambda | u _ k | ^ q + \\mu | v _ k | ^ q ) d x - \\frac { 2 } { p _ s ^ \\ast } \\int _ \\Omega | u _ k | ^ \\alpha | v _ k | ^ \\beta d x & = c + o _ k ( 1 ) , \\\\ \\| ( u _ k , v _ k ) \\| ^ p - \\int _ \\Omega ( \\lambda | u _ k | ^ q + \\mu | v _ k | ^ q ) d x - 2 \\int _ \\Omega | u _ k | ^ \\alpha | v _ k | ^ \\beta d x & = o _ k ( 1 ) . \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} \\Upsilon ^ { [ 0 ] } = \\frac { 1 } { 2 \\pi } \\int _ { \\theta _ 1 } ^ { \\theta _ 2 } \\frac { \\Delta ( 0 , \\theta ) } { 1 + P _ 1 ( 0 , \\theta ) } ( 1 + \\partial _ \\theta C ( 0 , \\theta ) ) d \\theta , \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\mathbf { W } _ { \\mathrm { a } , i } = \\frac { 1 } { \\sqrt { \\zeta _ { \\mathrm { a } , i } } } \\cdot \\left \\{ \\hat { \\mathbf { H } } _ { i , \\mathrm { E } } ^ { \\mathrm { H } } \\left ( \\hat { \\mathbf { H } } _ { i , \\mathrm { E } } \\hat { \\mathbf { H } } _ { i , \\mathrm { E } } ^ { \\mathrm { H } } \\right ) ^ { - 1 } \\right \\} _ { [ 1 : K ] } , \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} w ( q ) : = \\frac { 1 } { T } \\int _ 0 ^ T h \\big ( t , q ) d t . \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} \\int _ G \\lambda ( g A \\cap B ) \\ , d \\eta ( g ) = \\lambda ( A ) \\ , \\lambda ( B ) , \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} \\pi _ N u = \\sum _ { i = 1 } ^ { N - 1 } u _ k \\psi _ k , u _ k = \\frac { \\langle u , \\psi _ k \\rangle _ 1 } { \\langle \\psi _ k , \\psi _ k \\rangle _ 1 } . \\end{align*}"}
-{"id": "1347.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": "6465.png", "formula": "\\begin{align*} \\lim _ { V \\to \\infty } \\frac { 1 } { V } \\ , \\int _ { \\Lambda } d { x } \\ \\omega ^ { ' } ( \\tau _ { { x } } ( A ) B ) = \\omega ^ { ' } ( A ) \\ , \\omega ^ { ' } ( B ) \\ \\ \\ \\forall A , B \\in { \\cal A } \\ . \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} q ^ T A q - \\lambda _ 0 q ^ T q & = ( \\sum _ { i , j } k _ i k _ j \\bar { q } _ i ^ T A \\bar { q } _ j + 2 p \\bar { \\beta } ^ T A \\sum _ { i } k _ i \\bar { q } _ i + p ^ 2 \\hat { \\beta } ^ T \\hat { \\beta } ) - \\lambda _ 0 q ^ T q \\\\ & \\equiv 2 p \\hat { \\beta } ^ T A \\sum _ { i } k _ i \\bar { q } _ i - 2 p \\lambda _ 0 \\hat { \\beta } ^ T q ~ ( { \\rm m o d } ~ p ^ 2 ) , \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\mu ( \\theta _ t ) : = \\sum _ { i = 1 } ^ n \\theta _ t ^ i \\delta ^ i , & & t = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} \\begin{aligned} u ^ \\varepsilon ( z _ 3 ) & \\leq \\int _ 0 ^ { t _ 3 + t _ 4 } L \\big ( \\xi ^ \\varepsilon ( s , z _ 1 , - \\gamma ) , - \\gamma ( \\xi ^ \\varepsilon ( s , x , - \\gamma ) ) \\big ) e ^ { - \\lambda s } \\ , d s + u ^ \\varepsilon ( z _ 4 ) e ^ { - \\lambda ( t _ 3 + t _ 4 ) } \\\\ & \\leq C ( 1 + \\lambda ) \\cfrac { 4 \\sqrt { h } } { c _ 0 \\mu } + u ^ \\varepsilon ( z _ 4 ) . \\end{aligned} \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} y = \\sqrt { b } a \\left ( \\sum _ { l = 0 } ^ d x ^ { ( l ) } \\cdot \\mu _ { ( l + 1 ) p } ( g ) \\right ) \\ ; . \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} M _ \\ell \\left ( \\frac { 1 } { \\ell } \\left ( \\frac { 2 q } { q + 2 } \\right ) ^ { q / 2 } \\right ) = \\frac { e ^ { - \\frac { q + 2 } { 2 } } } { \\ell } , \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\Psi ' ( \\varphi ^ { w } ) - \\Psi ' ( \\overline { \\varphi } ) - \\Psi '' ( \\overline { \\varphi } ) \\Phi ^ { w } & = \\Psi '' ( \\overline { \\varphi } ) \\theta ^ { w } + ( \\varphi ^ { w } - \\overline { \\varphi } ) ^ { 2 } R _ { 1 } ^ { w } , \\\\ h ( \\varphi ^ { w } ) - h ( \\overline { \\varphi } ) - h ' ( \\overline { \\varphi } ) \\Phi ^ { w } & = h ' ( \\overline { \\varphi } ) \\theta ^ { w } + ( \\varphi ^ { w } - \\overline { \\varphi } ) ^ { 2 } R _ { 2 } ^ { w } , \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} F _ n = \\sum _ { k = 0 } ^ { n } \\binom { n - k } { k } . \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} \\phi _ { 0 } ( z ) = \\phi _ { 0 } ( z _ \\infty ) . \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} Y _ { t } = \\xi + \\int _ { t } ^ { T } f ( s , Y _ { s } , Z _ { s } ) d s - \\int _ { t } ^ { T } Z _ { s } d B _ { s } - ( K _ { T } - K _ { t } ) , \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} \\varphi ( 0 , a , b ) ( x ) = a ( x ) \\varphi ( 1 , a , b ) ( x ) = b ( x ) . \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} \\frac { ( - q ; q ^ 3 ) _ \\infty ( - q ^ 2 ; q ^ 3 ) _ \\infty } { ( q ; q ^ 3 ) _ \\infty ( q ^ 2 ; q ^ 3 ) _ \\infty } - 1 = \\sum _ { n \\geq 1 } \\frac { ( - q ; q ) _ { n - 1 } } { ( q ; q ) _ { n - 1 } } \\frac { 2 q ^ n } { 1 - q ^ n } \\frac { q ^ { n ^ 2 - n } } { ( q ; q ^ 2 ) _ n } . \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { p } } ^ { \\vee } = < f _ i , \\{ f _ { \\beta } \\} _ { \\beta \\in R ^ { + } } , \\{ f _ { - \\alpha } \\} _ { \\alpha \\in R ^ * } , \\{ \\frac { 1 } { z } f _ { - \\delta } \\} _ { \\delta \\notin R ^ { * } } \\} > \\mathbb { C } [ [ z ] ] . \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} ( - 1 ) ^ { n + 1 } \\left ( u ' - \\frac { 1 } { u _ 1 - u } - \\frac { 1 } { u - u _ { - 1 } } \\right ) = 0 , \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} D _ X ^ 2 . \\pi _ * \\beta ^ * j _ * [ \\Sigma ] = \\pi _ * ( \\tilde { D } ^ 2 . \\beta ^ * j _ * [ \\Sigma ] ) = \\pi _ * ( \\beta ^ * ( D ^ 2 . j _ * [ \\Sigma ] ) ) = \\pi _ * ( \\beta ^ * ( ( D _ { | \\Sigma } ) ^ 2 ) ) , \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} H _ { \\Lambda } ^ { \\lambda } = \\sqrt { V } ( \\bar { \\lambda } _ { \\phi } b _ { { 0 } } + \\lambda _ { \\phi } b _ { { 0 } } ^ { * } ) \\ . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} \\gamma ( 1 / 2 , \\pi , \\chi , \\psi _ a ) = c _ \\pi ( a ) \\gamma ( 1 / 2 , \\pi , \\chi , \\psi ) , \\ \\ \\ a \\in F ^ \\times , \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} \\left ( x ; A \\cap B \\right ) = \\left ( x ; A \\right ) \\left ( x ; B \\right ) \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} L ( f ) ( x , y ) & = \\left ( f L ( e _ { x x } ) \\right ) ( x , y ) - \\left ( L ( e _ { x x } ) f \\right ) ( x , y ) \\\\ & + f ( x , y ) L ( e _ { x y } ) ( x , y ) - \\left ( f L ( e _ { y y } ) \\right ) ( x , y ) + L ( e _ { y y } ) ( x , x ) f ( x , y ) \\\\ & f ( x , x ) L ( e _ { y y } ) ( x , y ) + f ( y , x ) L ( e _ { y x } ) ( x , y ) . \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} ( U _ R S _ l \\psi ) ( x , n ) = & e ^ { 2 \\pi i l x } ( U _ R \\psi ) ( x , n ) \\\\ ( U _ { R } ^ { - 1 } S _ l \\psi ) ( x , n ) = & e ^ { - 2 \\pi i l x } ( U _ { R } ^ { - 1 } \\psi ) ( x , n ) \\\\ ( U _ { k } S _ l \\psi ) ( x , n ) = & e ^ { 2 \\pi i l k ( x + \\frac { l \\alpha } { 2 } ) } ( S _ l U _ { k } \\psi ) ( x , n ) . \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\gamma _ j < \\infty , \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\phi _ t ^ * \\eta = \\lambda _ t \\eta \\quad \\ M _ 0 , \\ \\ t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} w \\left ( \\{ x : | T _ { \\alpha , m } f ( x ) | \\geq \\lambda \\} \\right ) \\leq C \\lambda ^ { - \\frac { n } { n - \\alpha } } \\sum _ { i = 1 } ^ { m } \\left ( \\int _ { \\mathbb { R } ^ { n } } | f ( x ) | [ w _ { A _ i ^ { - 1 } } ( x ) ] ^ { \\frac { n - \\alpha } { n } } d x \\right ) ^ { \\frac { n } { n - \\alpha } } . \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} \\mathbb { S } ^ i ( \\mathbb { Z } ^ s , w ) = \\bigoplus _ n H ^ i ( S _ n , \\mathbb { F } ) , \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} \\boldsymbol { \\Psi } _ D ^ { ( g ) } = \\left ( \\mathbf { X } ^ { ( g ) } \\otimes \\left [ \\mathbf { S } _ D ^ { ( g ) } \\right ] ^ H \\right ) \\left ( \\mathbf { I } _ { K _ g } \\otimes \\mathbf { V } \\right ) . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} \\tau ^ 1 _ x ( k ) & = \\inf \\{ t \\in I _ k : X _ t = x \\} , \\\\ \\tau ^ 2 _ x ( k ) & = \\inf \\{ t > \\tau ^ 1 _ x ( k ) + T : X _ t = x \\} . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} a ^ \\epsilon _ { j k } \\ , & = \\ , a _ { j k } \\ , + \\ , \\epsilon \\big ( \\partial _ j \\partial _ k \\rho ^ 0 \\big ) ( 0 ) + \\ , \\mathcal { O } ( \\epsilon ^ 2 ) \\ , . \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} A _ { ( m + 1 ) p _ n - m } & = A _ { ( m + 1 ) p _ n } - x ^ m A _ { ( m + 1 ) p _ n + m } - x ^ { m - 1 } A _ { ( m + 1 ) p _ n - 1 } & \\\\ & = ( 1 - x ^ m ) A _ { ( m + 1 ) p _ n + m } - x ^ { m - 1 } A _ { ( m + 1 ) p _ n - 1 } . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} \\gamma \\| h \\| ^ 2 \\leq \\langle \\big ( \\mathrm { R e } ( K A - T ) \\big ) h , h \\rangle = \\mathrm { R e } \\langle ( K A - T ) h , h \\rangle = \\mathrm { R e } \\langle \\lambda K h , h \\rangle = \\lambda \\mathrm { R e } \\langle K h , h \\rangle . \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} f ' \\ , { \\rm c o k e r } ( A ^ n \\to M ) = 0 . \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} L _ n = v v _ 1 - v ^ 2 . \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} z \\phi ( z ) = \\int _ 0 ^ \\infty \\{ \\xi z \\} \\Phi ' ( \\xi ) d \\xi \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} V _ n ^ 2 ( f ) = \\mathrm { V a r } _ { f } \\ ; ( \\nabla L ) \\enspace . \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} \\forall \\epsilon = ( \\epsilon _ i ) _ { 1 \\le i \\le 3 } \\in \\{ 0 , 1 \\} ^ 3 \\mu _ p ( X _ i = \\epsilon _ i , \\ ; 1 \\le i \\le 3 ) = \\frac { q ^ { N ( \\epsilon ) } } { Z ( q ) } , \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} ( s + m - 2 s _ 0 + j - 1 ) - ( n - 1 ) & = ( s - 2 s _ 0 + j - 1 ) + ( m + 1 - n ) \\\\ ( s + m - 2 s _ 0 + j - 1 ) + ( n - 1 ) & \\equiv ( s - 2 s _ 0 + j - 1 ) - ( m + 1 - n ) \\pmod { m } , \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} \\beta ( n ) = \\bigg ( \\frac { \\| q \\| _ 2 ^ 2 } { n ^ 2 } + \\sum \\limits _ { | p | \\leq n , p \\ne 0 } \\frac { \\widetilde { q } ( p , n ) } { p ^ 2 } \\bigg ) ^ \\frac { 1 } { 2 } , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} \\delta _ 1 = \\epsilon _ 1 ^ 2 e ^ \\sigma ( 1 - e ^ \\sigma ) \\sup _ { t \\in \\R , u \\in [ \\epsilon _ 1 , M _ 1 M / \\epsilon ] } f _ u ( t , u ) . \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} \\left ( \\begin{bmatrix} p & q \\\\ r & s \\end{bmatrix} . \\ \\phi \\right ) ( x , y ) = \\frac { 1 } { p s - q r } \\phi ( p x + r y , q x + s y ) . \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} \\left | T _ \\Sigma ( i \\tau / 2 ) - \\sum _ { l = 0 } ^ k T _ \\Sigma ^ l ( i \\tau ) ^ { - ( 2 j - 1 + l ) } \\right | \\lesssim \\tau ^ { - 2 s - 1 } \\| ( u , v ) \\| _ { \\dot H ^ s } ^ 2 \\| ( u , v ) \\| _ { l ^ 2 _ 1 D U ^ 2 } ^ { 2 j - 2 } , \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} a _ 0 & = 1 \\\\ a _ h & = 0 h \\geq j \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} M _ { \\Omega } ( \\omega ) : = C _ M e ^ { \\omega \\cdot \\Omega } , \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} N ^ { \\beta } _ { 2 } f ( x ) & = \\chi _ { ( 0 , 1 ) } ( x ) \\int _ { 0 } ^ { \\infty } \\bigg [ ( x y ) ^ { - 2 \\beta } \\sup _ { 1 / 4 < s < 1 } ( 1 - s ) ^ { - 2 \\beta } K _ s ^ { - \\beta } ( x , y ) \\bigg ] f ( y ) \\ , d \\nu _ { \\beta } ( y ) , f \\ge 0 , \\\\ N ^ { \\gamma } _ { 3 } f ( x ) & = \\chi _ { ( 0 , 1 ) } ( x ) \\int _ { 0 } ^ { \\infty } \\bigg [ \\sup _ { 1 / 4 < s < 1 } K _ s ^ { \\gamma } ( x , y ) \\bigg ] f ( y ) \\ , d \\nu _ { \\gamma } ( y ) , f \\ge 0 , \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} K ( c , 0 , p ) = 6 ( 4 - c ^ 2 ) + | c ^ 3 - 2 c - 1 0 | . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} \\lim _ { h _ 2 \\to \\infty } \\sum _ { k = 2 N } ^ { \\infty } \\frac { 1 } { k ^ { 2 \\alpha } \\sqrt { ( \\mu _ k - \\mu _ { N + 1 } ) ^ 2 + h _ 2 ^ 2 } } = 0 . \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} F ^ { \\left ( b \\right ) } _ { n } = \\sum _ { k \\le _ { b } n } \\binom { n - k } { k } _ b . \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} S = k _ B \\ln ( W ) \\Leftrightarrow \\iota = - k _ B \\ln ( W ) . \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} \\Sigma = \\sum _ { \\substack { m \\leq X / Y \\\\ \\mu _ { [ Y , Z ) } ^ 2 ( m ) = 1 } } w ( m ) f ( m ) \\sum _ { \\substack { Y \\leq p < Z \\\\ p \\leq X / m \\\\ ( m , p ) = 1 } } f ( p ) F ( p m ) . \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} Y _ t = L ^ - Y + G ( Y ) , L ^ - Y = D Y _ { x x } + c Y _ x + \\partial _ { Y } R ( 0 ) Y \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} 1 = g \\cdot g ^ { - 1 } = t _ 1 t _ 2 \\cdots t _ l \\cdot \\alpha _ l ( g _ { \\iota ( l ) } ) ^ { - \\epsilon _ l } \\cdots \\alpha _ 2 ( g _ { \\iota ( 2 ) } ) ^ { - \\epsilon _ 2 } \\alpha _ 1 ( g _ { \\iota ( 1 ) } ) ^ { - \\epsilon _ 1 } . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} \\dot { x } \\left ( t \\right ) = f \\left ( t , x _ { t } , u \\left ( t \\right ) \\right ) , \\ ; t \\in J , \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} u ' = f ( x , n , u _ { - l _ 1 } , u _ { - l _ 1 + 1 } , \\ldots , u , \\ldots , u _ { l _ 2 } ) , \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} \\| \\Gamma _ { m , b c } \\| _ 2 \\leq \\frac { \\omega ^ { 2 k } } { \\pi ^ { 2 k } ( 2 m + 1 ) } \\cdot \\begin{cases} 1 / 4 ( 2 m + \\theta ) ^ { 2 k - 2 } , \\ , m \\in \\mathbb { Z } _ + , \\ , b c \\in \\{ p e r , a p \\} , \\\\ 1 / m ^ { 2 k - 2 } , m \\in \\mathbb { N } , b c = d i r . \\end{cases} \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} \\frac { \\partial \\gamma } { \\partial t } + \\left ( p + \\alpha \\sin { ( w t ) } q ^ 2 p \\right ) \\frac { \\partial \\gamma } { \\partial q } = q + \\alpha \\sin { ( w t ) } p ^ 2 q . \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} \\gamma _ { 0 } ( X ) = \\arg \\min _ { \\gamma \\in \\Gamma } E [ v ( Y - \\gamma ( X ) ) ] , E [ m ( W , \\gamma ) ] = E [ \\bar { \\alpha } ( X ) \\gamma ( X ) ] , E [ \\gamma ( X ) ^ { 2 } ] < \\infty , \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} \\left \\{ \\ ! \\ ! \\ ! \\begin{array} { l } ^ { c } D ^ { q } z _ { 1 } ( t ) = - a z _ { 1 } ( t ) + \\sum _ { j = 1 } ^ { n } T _ { 1 j } g _ { j } ( z _ { j } ( t ) ) + I _ { 1 } \\\\ ^ { c } D ^ { q } z _ { j } ( t ) = - b _ { j } z _ { j } ( t ) + T _ { j 1 } g _ { 1 } ( z _ { 1 } ( t ) ) + T _ { j j } g _ { j } ( z _ { j } ( t ) ) + I _ { j } , \\qquad \\forall ~ j = \\overline { 2 , n } \\end{array} \\right . \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} \\sigma _ { g ^ { - 1 } } ( y ^ * ) _ { ( ( i ' , j ' ) , 1 _ H ) } & = ( y ^ * ) _ { g ( ( i ' , j ' ) , 1 _ H ) } \\\\ & = ( y ^ * ) _ { ( ( i , j ) + \\varphi _ h ( i ' , j ' ) , h ) } \\\\ & = ( y ^ { h ^ { - 1 } } ) _ { ( i ' , j ' ) + \\varphi _ { h ^ { - 1 } } ( i , j ) } \\\\ & = ( \\sigma _ { - \\varphi _ { h ^ { - 1 } } ( i , j ) } ( y ^ { h ^ { - 1 } } ) ) _ { ( i ' , j ' ) } . \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} \\nu = \\left ( \\frac 1 { C 4 ^ m ( 2 \\theta - 1 ) } \\right ) ^ { ( n + 2 ) / 2 } ( 2 ( \\theta - 1 ) ) ^ { n / 2 } 2 ^ { - ( n + 2 ) ^ 2 } . \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} \\tilde { \\beta } _ t \\ , \\tilde { U } ' ( \\hat { x } _ t ) & = \\tilde { \\beta } _ { t + 1 } \\ , \\tilde { U } ' ( \\hat { x } _ { t + 1 } ) f ' ( k _ { t + 1 } ) , \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} \\underset { j = 1 } { \\overset { \\kappa } { \\prod } } \\eta ( j z ) ^ { e _ { j } } . \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\mathbb { F } _ q ( \\Phi ' , - 2 ) : = \\operatornamewithlimits { l i m \\ , i n d } _ { ( \\tau , r ) \\in T \\times [ 1 , \\infty ) } \\mathbb { F } _ q ( \\mathcal { H } _ { - \\tau } , r ^ { - 1 } , - 2 ) . \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} B ^ { s y m } = L _ \\xi e ^ { 2 \\xi } + , \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} g '' _ 1 ( A _ n ) = & A _ n ^ { - 1 } e ^ { - \\left ( \\sqrt { A _ n / 2 } + \\sqrt { P / 2 } \\right ) ^ 2 } \\cdot \\left ( - \\frac { 1 } { 4 \\pi } e ^ { - \\left ( \\sqrt { A _ n / 2 } + \\sqrt { P / 2 } \\right ) ^ 2 } \\right . \\\\ & \\left . + \\frac { 1 } { 4 \\sqrt { 2 \\pi } } ( 1 + { \\rm { e r f } } ( \\sqrt { A _ n / 2 } + \\sqrt { P / 2 } ) ) \\left ( A _ n ^ { 1 / 2 } + \\sqrt { P } + A _ n ^ { - 1 / 2 } \\right ) \\right ) . \\end{align*}"}
-{"id": "2570.png", "formula": "\\begin{align*} \\partial _ z u _ 1 ( t , x , z ) = & - \\left ( G _ 1 \\partial _ x f ( t , x ) + G _ 2 \\mu \\partial _ x g ( t , x ) - \\sigma _ 2 ^ c \\mu \\partial _ x ^ 3 ( f + g ) ( t , x ) - \\sigma _ 1 ^ c \\partial _ x ^ 3 f ( t , x ) \\right ) ( f ( t , x ) - z ) \\\\ [ 5 p t ] & + \\displaystyle { \\mu } \\partial _ z u _ 2 ( t , x , f ) . \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\frac { 1 } { M } \\sum _ { m = 1 } ^ M X _ m \\geq c \\right ) \\leq e ^ { - \\frac { M \\mu } { 3 B } \\left ( \\frac { c } { \\mu } - 1 \\right ) ^ 2 } . \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} \\phi ( g \\cdot a _ { i } ) & = \\phi ( \\det ( g ) a _ { i } \\circ g ) = \\det ( g ) \\phi ( a _ { i } \\circ g ) = \\det ( g ) c _ { i } ( b _ { i } \\circ g ) . \\\\ g \\cdot \\phi ( a _ { i } ) & = g \\cdot c _ { i } b _ { i } = \\det ( g ) c _ { i } ( b _ { i } \\circ g ) . \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{align*} \\pi \\in B ^ { s _ 2 } _ { \\tau _ 2 } ( L _ { \\tau _ 2 } ( \\Omega ) ) , \\frac { 1 } { \\tau _ 2 } = \\frac { s _ 2 } { d } + \\frac { 1 } { 2 } , 0 < s _ 2 < \\frac { 1 } { 2 } \\cdot \\frac { d } { d - 1 } . \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} P _ f = \\sum _ { n \\leq 0 } a ( n ) q ^ { n } \\in \\C [ q ^ { - 1 } ] \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} x ^ \\prime & = f ( x , y , 0 ) , \\\\ y ^ \\prime & = 0 . \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} c _ j s _ { j - 1 } = c _ { j - 1 } s _ j + \\gcd ( s _ j , s _ { j - 1 } ) \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\hat { \\beta } _ { 1 } = - \\beta + { 2 \\over 3 } , \\ \\hat { \\beta } _ { 2 } = - 2 \\beta , \\ \\hat { \\beta } _ { 3 1 } = \\beta + { 1 \\over 3 } , \\ \\hat { \\beta } _ { 3 2 } = \\beta + { 2 \\over 3 } , \\ \\hat { \\beta } _ { 3 3 } = \\beta . \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} D ( f , s ) = \\prod _ { p \\in \\P } \\left ( 1 - \\frac { f ( p ) } { p ^ s } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} { Y _ i ^ { ( k ) } } \\left ( { { \\tau ^ { ( k ) } } } \\right ) = { R _ { { m _ k , i } } } \\left ( { { \\tau ^ { ( k ) } } } \\right ) - \\frac { { { R _ { { m _ k , i } } } \\left ( { { \\tau ^ { ( k ) } } } \\right ) } } { { \\hat G _ { { m _ k } + 1 , k } ^ { \\left ( \\alpha _ i ^ { ( k ) , * } \\right ) } \\left ( { { \\tau ^ { ( k ) } } } \\right ) } } \\hat G _ { { m _ k } + 1 , k } ^ { \\left ( \\alpha _ i ^ { ( k ) , * } \\right ) } \\left ( { { \\tau ^ { ( k ) } } } \\right ) = 0 . \\end{align*}"}
-{"id": "993.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { q , w } ( \\R ^ { N } ) } & = \\sup _ { A \\subset \\R ^ { N } , 0 < | A | < \\infty } | A | ^ { - \\frac { 1 } { r } } \\int _ { A } | x | ^ { - \\lambda } \\ d x \\\\ & = \\sup _ { R > 0 } \\left ( N | B | R ^ { N } \\right ) ^ { - 1 / r } N | B | \\int _ { 0 } ^ { R } m ^ { - \\lambda + N - 1 } \\ d m = \\frac { \\left ( N | B | \\right ) ^ { 1 - r ^ { - 1 } } } { N - \\lambda } \\sup _ { R > 0 } R ^ { - N / r + N - \\lambda } . \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} \\tau \\wedge ( d \\tau ) ^ n = \\tau \\wedge d \\tau \\ , \\wedge \\stackrel { } { \\cdots } \\wedge \\ , d \\tau \\neq 0 . \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} b _ { + } \\left ( \\lambda \\right ) = 0 b _ { - } \\left ( \\lambda \\right ) \\neq 0 . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} k B \\cap k ^ { \\prime } B = \\emptyset ( k , k ^ { \\prime } \\in H ^ { - 1 } ) . \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} p _ k = \\begin{cases} { M + K - 1 \\choose k } + \\frac { ( - 1 ) ^ { k + 1 } \\prod _ { \\mathfrak { k } = 1 } ^ { k } ( \\mathfrak { k } - M - K ) } { k ! } & , 1 \\le k \\le M - 1 \\\\ { M + K - 1 \\choose k } & , M \\le k \\le M + K - 1 \\end{cases} . \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\tilde { K } _ { 3 } ( t ) g \\right \\| _ { 2 } & \\le C t ^ { 1 - \\ell } ( 1 + t ) ^ { - n ( \\frac { 1 } { r _ { 1 } } - \\frac { 1 } { 2 } ) - ( k - \\tilde { k } _ { 3 } ) } \\| \\nabla ^ { \\tilde { k } _ { 3 } } _ { x } g \\| _ { r _ { 1 } } \\\\ & + C e ^ { - c t } t ^ { ( 1 - \\ell ) - n ( \\frac { 1 } { r _ { 2 } } - \\frac { 1 } { 2 } ) - ( k - \\tilde { k } _ { 4 } ) } \\| \\nabla ^ { \\tilde { k } _ { 4 } } _ { x } g \\| _ { r _ { 2 } } , \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} a = a _ 0 + \\sum _ { i = 1 } ^ m a _ i X ^ i + \\sum _ { f } \\sum _ { i = 1 } ^ { \\infty } \\frac { g _ { f , i } } { f ^ i } \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} ( q - p \\hat { \\beta } ) ^ T ( q - p \\hat { \\beta } ) = \\sum _ { i , j } k _ i k _ j \\bar { q } _ i ^ T \\bar { q } _ j \\equiv 0 ~ ( { \\rm m o d } ~ p ^ 2 ) , \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} d ( F ( x + s _ 2 e _ k ) , F ( x + s _ 1 e _ k ) ) - \\varepsilon & \\leq d ( F ( x + s _ 2 e _ k ) , F ( x + s _ 1 e _ k ) ) - 2 d ( F ( x + s _ 1 e _ k ) , p _ i ) \\\\ & \\leq d ( F ( x + s _ 2 e _ k ) , p _ i ) - d ( F ( x + s _ 1 e _ k ) , p _ i ) \\\\ & = \\phi _ i ( F ( x + s _ 2 e _ k ) ) - \\phi _ i ( F ( x + s _ 1 e _ k ) ) \\\\ & = \\int _ { s _ 1 } ^ { s _ 2 } \\frac { d } { d \\tau } \\ , ( \\phi _ i \\circ F ) ( x + \\tau e _ k ) \\ , d \\tau \\\\ & \\leq \\int _ { s _ 1 } ^ { s _ 2 } g ( x + \\tau e _ k ) \\ , d \\tau . \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} \\begin{pmatrix} \\xi ' \\\\ x ' \\\\ \\eta ' \\end{pmatrix} = \\begin{pmatrix} a & \\alpha & b \\\\ \\gamma & e & \\beta \\\\ c & \\delta & d \\end{pmatrix} \\begin{pmatrix} \\xi \\\\ x \\\\ \\eta \\end{pmatrix} \\ , \\ , \\mbox { a n d } \\ , \\ , ( \\xi '' , x '' , \\eta '' ) = ( \\xi , x , \\eta ) \\begin{pmatrix} a & \\alpha & b \\\\ \\gamma & e & \\beta \\\\ c & \\delta & d \\end{pmatrix} . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} v _ j ( t ) = c _ j e ^ { \\mu _ j ^ { \\infty } t } + \\tilde { v } _ j ( t ) , \\tilde { v } _ j ( t ) : = \\sum _ { l \\in \\mathbb { Z } ^ { \\nu } } \\frac { f _ { j l } } { \\mathrm { i } \\omega \\cdot l + \\mu _ j ^ { \\infty } } e ^ { \\mathrm { i } \\omega \\cdot l t } . \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} F ( x + y \\rho ) & = f ( x ) + y \\Delta _ { y r } f ( x ) \\cdot \\rho + ( g ( x ) + y \\Delta _ { y r } g ( x ) \\cdot \\rho ) \\rho \\\\ & = f ( x ) + ( y \\Delta _ { y r } f ( x ) + g ( x ) + y r \\Delta _ { y r } g ( x ) ) \\rho \\\\ & = f ( x ) + ( y \\Delta _ { y r } f ( x ) + g ( x + y r ) ) \\rho . \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} Z _ i = \\operatorname { S p e c } \\Big ( \\mu _ i ( C _ 0 ( Y ) ) C _ 0 ( Z ) \\Big ) , i = 1 , 2 . \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} \\max \\{ \\alpha \\ , | \\ , x ^ T Q _ 1 x = 1 , x ^ T Q _ 2 x \\leq 1 , \\alpha = x ^ T A ^ T Q _ 1 x \\} . \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} \\left . \\mathcal { G } _ { \\{ n - k + 1 , \\dots , n \\} } \\right | _ { x _ k , \\dots , x _ 1 \\rightarrow 0 } = 1 . \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} - m < m + 1 - 2 \\sum _ { j = 1 } ^ { n } \\left ( x ; S ^ { ( j ) } \\right ) < m \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{align*} h _ K ( y ) : = \\max _ { x \\in K } \\ , \\langle x , y \\rangle \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} | C ( m , n ; k ) | = | C ( 0 , n - m ; k ) | ^ { | \\Omega _ { m } | } . \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} u _ { m } & = ( 2 c + 2 ) u _ { m - 1 } - u _ { m - 2 } \\equiv ( 2 c + 2 ) \\epsilon ( 1 + ( m - 1 ) m c ) - \\epsilon ( 1 + ( m - 2 ) ( m - 1 ) c ) \\\\ & \\equiv \\epsilon ( 1 + 2 m ( m - 1 ) c ^ { 2 } + m ( m + 1 ) c ) \\equiv \\epsilon ( 1 + m ( m + 1 ) c ) \\pmod { 4 c ^ 2 } , \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} & f _ { \\pi ( 1 ) } \\otimes f _ { \\pi ( 2 ) } \\otimes \\dots \\otimes f _ { \\pi ( m + n ) } \\\\ & = U ( A ) f _ { \\theta ( 1 ) } \\otimes f _ { \\theta ( 2 ) } \\otimes \\dots \\otimes f _ { \\theta ( m ) } \\otimes f _ { m + \\nu ( 1 ) } \\otimes f _ { m + \\nu ( 2 ) } \\otimes \\dots \\otimes f _ { m + \\nu ( n ) } . \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} \\lambda _ { p ( \\cdot ) } = \\underset { u \\in W _ { 0 } ^ { 1 , p ( \\cdot ) } ( \\Omega ) \\backslash \\{ 0 \\} } { \\inf } \\frac { \\int _ { \\Omega } \\frac { 1 } { p ( x ) } \\left \\vert \\nabla u \\right \\vert ^ { p ( x ) } d x } { \\int _ { \\Omega } \\frac { 1 } { p ( x ) } \\left \\vert u \\right \\vert ^ { p ( x ) } d x } \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} S S ( { \\cal F } \\boxtimes ^ L _ \\Lambda { \\cal G } ) = S S ( { \\cal F } ) \\boxtimes S S ( { \\cal G } ) . \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} \\lim _ n \\frac { 1 } { | F _ n | } \\sum _ { g \\in F _ n } \\chi _ A ( g x ) = \\lim _ { n } \\frac { | A _ x \\cap F _ n | } { | F _ n | } , \\textrm { f o r $ x \\in X $ } , \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} \\lambda _ { k , n } ^ { ( \\alpha ) } = { \\left ( { \\frac { { { \\tau _ k } - { \\tau _ { k - 1 } } } } { 2 } } \\right ) ^ { 2 \\alpha } } \\lambda _ n ^ { ( \\alpha ) } , \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} ( t , x ) = ( 2 ^ { - 1 / 6 } N ^ { 2 / 3 } T , N / \\sqrt { 2 } + 2 ^ { - 5 / 6 } N ^ { 1 / 3 } U ) . \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 q ^ 2 ( \\tau ) \\ ; d \\tau = 2 . \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} q _ i ^ T [ \\psi ( A ) , q _ 1 , q _ 2 , \\cdots , q _ s ] = O , { \\rm f o r } ~ i = 1 , 2 , \\cdots , s . \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} ( u ^ { k } _ { t } , \\zeta ) = ( u _ { 0 } , \\zeta ) + \\int _ { 0 } ^ { t } ( u ^ { k } _ { s } , L ^ { k } _ { s } \\zeta ) \\ , d s + \\int _ { 0 } ^ { t } ( f _ { s } , \\zeta ) \\ , d s . \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} [ [ \\alpha , \\beta ] ] = \\mathcal { L } _ { \\Pi ( \\alpha ) } \\beta - \\imath _ { \\Pi ( \\beta ) } d \\alpha \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} \\gamma ' ( d ) = \\frac { \\mu ( q ) } { \\zeta _ q ( 2 ) } \\frac { ( d _ 1 , w _ 1 ) ( d _ 1 , D ) ^ 2 } { d D } \\chi \\xi \\left ( \\frac { w _ 1 } { ( d _ 1 , w _ 1 ) } \\right ) \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} \\lim _ { \\xi \\rightarrow - \\infty } ( P ( \\xi ) , Q ( \\xi ) ) = ( P _ 3 , 0 ) , \\lim _ { \\xi \\rightarrow \\infty } ( P ( \\xi ) , Q ( \\xi ) ) = ( P _ 1 , 0 ) , \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} \\frac { d } { d t } | D \\varphi _ t ( \\omega , x ) v | ^ 2 & = 2 \\left \\langle D b ( \\varphi _ t ( \\omega , x ) ) D \\varphi _ t ( \\omega , x ) v , D \\varphi _ t ( \\omega , x ) v \\right \\rangle \\\\ & \\leq 2 \\left ( 1 - \\left | \\varphi _ t ( \\omega , x ) \\right | ^ 2 \\right ) | D \\varphi _ t ( \\omega , x ) v | ^ 2 . \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} [ d ( H \\circ \\gamma ) ] ^ { V } = - H \\dot { \\gamma } _ q \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} \\langle d e ^ { i \\theta } U ( x _ 0 ) \\ , | \\ , d e ^ { i \\theta } V ( x _ 0 ) \\rangle _ g = \\langle U ( x _ 0 ) \\ , | \\ , V ( x _ 0 ) \\rangle _ g , \\forall ~ | \\theta | < \\delta . \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} e ^ { - 2 \\eta } \\left [ \\begin{pmatrix} p ( \\eta ) \\\\ q ( \\eta ) \\\\ r ( \\eta ) \\end{pmatrix} - M _ 0 \\right ] \\rightarrow \\kappa \\vec { X } _ { 0 2 } , \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} Z _ 0 = P \\left \\{ F ( X - \\bar X ) + G ( v - \\bar v ) \\right \\} + \\Pi \\left \\{ ( F + \\bar { F } ) \\bar X + ( G + \\bar { G } ) \\bar v \\right \\} \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} M _ k ( x ; \\boldsymbol { a } ) = \\frac { 1 } { \\phi ^ k ( q ) } \\sum _ { \\substack { p _ 1 , \\cdots , p _ k ~ \\\\ p _ 1 \\cdots p _ k \\leq x } } ( \\chi _ 0 ( p _ 1 ) + \\chi _ a ( p _ 1 ) ) \\cdots ( \\chi _ 0 ( p _ k ) + \\chi _ a ( p _ k ) ) . \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} L R _ n = \\frac { P _ { \\theta _ 1 , n } + \\widetilde { P } _ { \\theta _ 1 , n } } { P _ { \\theta _ 0 , n } + \\widetilde { P } _ { \\theta _ 0 , n } } . \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\leq 2 \\sum _ { j = 1 } ^ k { k \\choose j } \\pi ^ j \\int _ 0 ^ { \\delta } | \\log \\sigma | ^ { k - j } \\sigma ^ m x ^ { - \\sigma } d \\sigma . \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} 4 c _ 4 = c _ { 1 } ^ { 3 } + 2 ( 4 - c _ { 1 } ^ { 2 } ) c _ 1 x - c _ 1 ( 4 - c _ { 1 } ^ { 2 } ) x ^ 2 + 2 ( 4 - c _ { 1 } ^ { 2 } ) ( 1 - | x | ^ 2 ) t . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\hat g ^ 2 = \\frac { 1 } { \\sqrt { 2 \\pi } } \\hat K _ 3 , \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} E _ 1 ^ { ( + ) } = \\frac { 2 b ^ 3 + 1 } { b } , \\psi _ 1 ^ { ( + ) } ( x ) = e ^ { - \\frac { 1 } { 3 } | x | ^ 3 - \\frac { b ^ 3 + 1 } { 2 b } x ^ 2 - b | x | } \\left ( | x | + \\frac { 1 } { b } \\right ) , \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\max \\limits _ { ( f , g ) \\in D _ f \\times D _ g } \\Delta \\left ( h ( f _ 0 , g _ 0 ) ; f , g \\right ) = \\Delta \\left ( h ( f _ 0 , g _ 0 ) ; f _ 0 , g _ 0 \\right ) , \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} h _ g ( f _ 0 , g _ 0 ) = \\left | { h } ^ 0 ( \\theta ) \\right | ^ { \\alpha } , \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} x ( 0 ) = \\int _ 0 ^ 1 x ( s ) \\textup d A ( s ) x ( 1 ) = \\int _ 0 ^ 1 x ( s ) \\textup d B ( s ) ; \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} | C ( 0 , n ; k ) | = \\bigl ( \\tfrac { 1 } { 2 } + o ( 1 ) \\bigr ) | G _ { n } | , \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} [ \\tau ^ { - 1 } ( D ) \\Delta D _ 0 ] = [ D \\Delta \\tau ( D _ 0 ) ] = [ D \\Delta D _ 0 ] + [ D _ 0 \\Delta \\tau ( D _ 0 ) ] = [ D \\Delta D _ 0 ] + 1 \\ , . \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} \\sum _ i \\pi _ i t _ { i j k \\ell } = - \\sum _ i \\pi _ i \\frac { ( A _ { i k } A _ { j \\ell } ) ^ 2 } { 2 d ^ 2 n ^ 2 } . \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} D ( \\phi ) & = \\int _ { M ^ * } \\left ( \\phi ^ * - \\phi _ 0 ^ * \\right ) d \\nu - \\frac { 1 } { \\lambda } \\log \\int _ M e ^ { - \\lambda ( \\phi - \\phi _ 0 ) } \\mu _ 0 \\\\ M ( \\mu ) & = \\lambda \\inf _ { \\phi \\in C ^ 0 ( M , L ) } F _ { \\mu , \\nu } ( \\phi ) + \\int _ M \\log \\frac { \\mu } { \\mu _ 0 } d \\mu , \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} 0 = \\iint _ { D _ 1 } ( X _ x \\sin \\theta _ 0 - X _ y \\cos \\theta _ 0 ) d S - \\iint _ { D _ 2 } ( X _ x \\sin \\theta _ 0 - X _ y \\cos \\theta _ 0 ) d S , \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} ( C _ 2 ) = \\left ( \\int _ { C _ 2 } \\omega \\right ) = \\left ( \\lim _ { \\varepsilon \\rightarrow 0 } \\int _ { - \\pi } ^ { \\pi } \\int _ { - 1 } ^ { - \\varepsilon } \\frac { 1 } { h } d h \\wedge d \\theta \\right ) = ( \\lim _ { \\varepsilon \\rightarrow 0 } 2 \\pi \\log | \\varepsilon | ) \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} \\begin{pmatrix} \\frac { a _ \\ell } { 2 } & 0 \\\\ \\beta _ \\ell & \\frac { a _ \\ell } { 2 } \\end{pmatrix} = \\begin{pmatrix} a & 0 \\\\ c & d \\end{pmatrix} \\begin{pmatrix} \\frac { a _ \\ell } { 2 } & 0 \\\\ \\beta _ \\ell ' & \\frac { a _ \\ell } { 2 } \\end{pmatrix} \\begin{pmatrix} a & 0 \\\\ c & d \\end{pmatrix} ^ { - 1 } \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} C ^ { j j } _ { j j } & = C ^ { j j } _ { i i } , \\\\ C ^ { j j } _ { x i } & = 0 , \\ \\mbox { i f } \\ x \\neq i , j . \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} M = \\Q _ 2 ( \\sqrt { - 5 } ) = \\Q _ 2 ( \\sqrt { 3 } ) \\subset F _ 1 M = \\Q _ 2 ( \\sqrt { - 1 } ) = \\Q _ 2 ( \\sqrt { 7 } ) \\subset F _ 2 , \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} \\mathbf { V } \\mathbf { V } ^ H = \\sum _ { l = 0 } ^ { L _ g - 1 } \\rho _ l \\mathbf { E } _ { L _ g , l } \\otimes \\mathbf { R } _ l ^ { ( g ) } \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} \\small G = \\left ( \\begin{array} { r c c c c c c c c c c c c c c c l } 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 \\\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 \\\\ \\end{array} \\right ) \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} \\begin{pmatrix} y _ 1 & 0 \\\\ 0 & x ^ { 1 / 2 } y _ 1 \\end{pmatrix} = \\begin{pmatrix} w _ 1 & 0 \\\\ 0 & w _ 2 \\end{pmatrix} \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} \\epsilon ( s , \\pi , \\tau , \\psi ) = 1 . \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} \\mathcal { E } ( \\alpha _ 0 , \\theta \\gamma ) = \\theta ^ { \\frac 2 { 2 - \\alpha } } \\mathcal { E } ( \\alpha _ 0 , \\gamma ) \\ , . \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} j _ 1 ^ 3 + j _ 2 ^ 3 + j _ 3 ^ 3 + j _ 4 ^ 3 = - 3 ( j _ 1 + j _ 2 ) ( j _ 1 + j _ 3 ) ( j _ 2 + j _ 3 ) . \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} x _ 1 = \\hat { u } ^ 2 \\sigma ( x ' ) + \\hat { r } . \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} \\mathfrak { I } ( \\varphi ) : = i ( \\varphi ) - ( \\varphi , 0 , 0 ) : = ( \\Theta ( \\varphi ) , y ( \\varphi ) , z ( \\varphi ) ) , \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} F \\in C ^ 1 ( P H _ N ^ 2 , P L _ 2 ) \\mbox { w i t h } F ( 0 ) = F ^ \\prime ( 0 ) = 0 , \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { u } x ^ { a - 1 } e ^ { - b x } d x = b ^ { - a } \\gamma ( a , b u ) , \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} f \\ , = \\ , \\sigma + \\sum _ { \\alpha , \\beta \\in ( \\N _ 0 ) ^ m } c _ { \\alpha \\beta } \\prod _ { j = 1 } ^ m g _ j ^ { \\alpha _ j } ( 1 - g _ j ) ^ { \\beta _ j } , \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} \\lim _ { | \\theta | \\to \\infty } f ( x ; \\theta , \\sigma ^ 2 ) = 0 . \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} K _ { 0 , t } ( x , y ) = \\P ( X _ t = y | X _ 0 = x ) = P ( 0 , x ; t , y ) . \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} x ( 0 ) = \\int _ 0 ^ 1 A ( s ) \\textup d x ( s ) x ( 1 ) = \\int _ 0 ^ 1 B ( s ) \\textup d x ( s ) , \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{align*} \\lvert A \\rvert _ s ^ 2 = \\sum _ { j \\in \\mathbb { Z } , l \\in \\mathbb { Z } ^ { \\nu } } \\left ( \\sup _ { j _ 1 - j _ 2 = j } \\lvert A _ { j _ 1 } ^ { j _ 2 } ( l ) \\rvert \\right ) ^ 2 \\langle l , j \\rangle ^ { 2 \\ , s } . \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} \\mathbf { G } = \\begin{bmatrix} \\mathbf { G } _ { 0 , 0 } & \\cdots & \\mathbf { G } _ { K - 1 , 0 } & \\cdots & \\mathbf { G } _ { 0 , M - 1 } & \\cdots & \\mathbf { G } _ { K - 1 , M - 1 } \\end{bmatrix} = \\mathbf { F } \\mathbf { A } , \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} \\mathbf { A } ^ { \\rm H } \\mathbf { A } = \\mathbf { I } _ N , \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} ( \\Gamma ^ { \\neq } ) ' \\ : = \\ \\{ \\gamma ' : \\gamma \\in \\Gamma ^ { \\neq } \\} , ( \\Gamma ^ { > } ) ' \\ : = \\ \\{ \\gamma ' : \\gamma \\in \\Gamma ^ { > } \\} , \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} C ^ { p ' } ( G ) = \\langle x \\in G | \\ , p \\mbox { d i v i d e s } | C _ G ( x ) | \\rangle , \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} E [ \\psi ( W , \\gamma , \\alpha , \\theta _ { 0 } ) ] & = E [ m ( W , \\gamma ) ] - \\theta _ { 0 } + E [ \\alpha _ { 0 } ( X ) \\{ Y - \\gamma ( Z ) \\} ] \\\\ & = E [ m ( W , \\gamma ) ] - \\theta _ { 0 } + E [ \\alpha _ { 0 } ( X ) \\{ Y - \\gamma _ { 0 } ( Z ) + \\gamma _ { 0 } ( Z ) - \\gamma ( Z ) \\} ] \\\\ & = E [ \\alpha _ { 0 } ( X ) \\{ \\gamma ( Z ) - \\gamma _ { 0 } ( Z ) \\} ] + E [ \\alpha _ { 0 } ( X ) \\{ \\gamma _ { 0 } ( Z ) - \\gamma ( Z ) \\} ] = 0 . Q . E . D . \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} \\left ( E _ { M , \\nu } \\right ) _ N = \\sum _ { \\sigma \\in S [ M ] } \\prod _ { A < B } \\prod _ { i \\in A , j \\in B , \\sigma ( i ) > \\sigma ( j ) } \\frac { \\xi ( \\upsilon _ { i j } ) } { \\xi ( \\upsilon _ { i j } + 1 ) } a ^ { \\sigma \\upsilon + \\rho } \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} \\sum _ { \\{ \\Delta : \\Omega \\} = ( 1 / p , 1 ) } c ( \\Omega ^ { p ^ r m } ) = \\sum _ { \\{ \\Delta : \\Omega \\} = ( 1 , p ) } c ( \\Omega ^ { p ^ { r - 2 } m } ) = \\eta ( p ^ { r } ) c ( \\Delta ^ m ) \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} g _ { 3 / 5 , 5 / 6 } ^ 1 ( w _ 1 ) = \\frac { \\epsilon } { 4 } \\log \\left ( 1 + \\frac { w _ 1 P ' } { 4 } \\right ) - \\frac { | A ^ { \\ast } | } { k _ { \\ell } } H _ 2 \\left ( \\frac { w _ 1 } { | A ^ { \\ast } | } \\right ) . \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} ( 1 - | y | ^ 2 ) & \\Big [ | y | ^ 2 - 2 x \\cdot y + 1 - \\frac { ( 1 - | y | ^ 2 ) } { \\rho } \\Big ] - ( \\rho + 1 ) | x - y | ^ 2 \\\\ & = - | y | ( | y | ^ 2 - 2 x \\cdot y + | x | ^ 2 ) - \\frac { 1 - | y | ^ 2 } { 1 - | x | ^ 2 } | x - y | ^ 2 = - | x - y | ^ 2 \\Big ( | y | ^ 2 + \\frac { 1 - | y | ^ 2 } { 1 - | x | ^ 2 } \\Big ) . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 \\beta } { \\beta + 2 } , \\alpha = \\frac { 1 - \\beta s } { \\beta + 2 } . \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } | B ( | u _ n | ) - B ( | u _ n - u | ) - B ( | u _ { } | ) | d x = 0 . \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} \\Psi ( \\Phi ( \\{ u , v \\} _ 1 ) ) & = \\Psi ( [ u ^ r , v ] _ 1 [ u ^ { \\frac { b } { h } } , v ] _ 2 ) \\\\ & = \\{ u ^ { \\frac { r a } { h } } , v \\} _ 1 \\{ u ^ { - \\frac { b r } { h } } , v \\} _ 2 \\{ u ^ { \\frac { b s } { h } } , v \\} _ 1 \\{ u ^ { \\frac { b r } { h } } , v \\} _ 2 \\\\ & = \\{ u ^ { \\frac { r a + b s } { h } } , v \\} \\ : \\ : \\mbox { ( B y ( A 1 ) a n d ( i i i ) ) } \\\\ & = \\{ u , v \\} _ 1 \\end{align*}"}
-{"id": "9026.png", "formula": "\\begin{align*} \\mathbf { w } _ i = \\mathbf { Q } \\mathbf { b } _ { i } , \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} \\Psi _ \\tau ^ m ( u ) = \\frac { ( 2 \\sqrt \\tau ) ^ m } { m ! } H _ m \\Big ( \\frac { ( 1 + 4 \\tau ) r } { 2 \\sqrt { 2 \\tau } } + \\frac u { 2 \\sqrt \\tau } \\Big ) e ^ { - 2 \\tau r ^ 2 - \\sqrt 2 r u } - ( u \\leftrightarrow - u ) . \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} a = - \\frac { \\mu ^ 4 t ^ { 1 2 } \\tilde { c } _ 4 } { 1 6 } b = - \\frac { \\mu ^ 5 t ^ { 1 5 } \\tilde { c } _ 6 } { 3 2 } . \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} E [ V ( X _ { t + 1 } ) | X _ { t } , Y _ { 2 t } = 1 ] - E [ V ( X _ { t + 1 } ) | Y _ { 1 t } = 1 ] = \\gamma _ { 2 0 } ( X _ { t } ) - \\gamma _ { 3 0 } . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} \\mathcal { U } _ { \\mathrm { a d } } = \\left \\{ u \\in L ^ { \\infty } ( 0 , T ; L ^ { \\infty } ) : 0 \\leq u \\leq 1 Q \\right \\} . \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} & \\big ( a ^ + ( f _ 1 ) \\dotsm a ^ + ( f _ n ) a ^ - ( g _ 1 ) \\dotsm a ^ - ( g _ m ) \\big ) \\diamond \\big ( a ^ + ( \\varphi _ 1 ) \\dotsm a ^ + ( \\varphi _ k ) a ^ - ( \\psi _ 1 ) \\dotsm a ^ - ( \\psi _ l ) \\big ) \\\\ & \\qquad : = q ^ { k m } \\ , a ^ + ( f _ 1 ) \\dotsm a ^ + ( f _ n ) a ^ + ( \\varphi _ 1 ) \\dotsm a ^ + ( \\varphi _ k ) \\\\ & \\qquad \\qquad \\qquad \\qquad \\times a ^ - ( g _ 1 ) \\dotsm a ^ - ( g _ m ) a ^ - ( \\psi _ 1 ) \\dotsm a ^ - ( \\psi _ l ) , \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} \\rho : = 8 \\lambda ^ { - 1 } \\log 2 + \\int _ { \\min _ x \\pi ( x ) } ^ { 1 / 2 } \\frac { 4 d v } { v \\Lambda ( v ) } , \\Lambda ( v ) : = \\inf _ { A \\subset \\Omega : \\pi ( A ) \\le v } \\lambda ( A ) . \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} \\Big | \\theta _ i ^ { G _ n } ( t ) - a _ i ( n ) e ^ { - 2 t } I _ 0 ( 2 t ) \\Big | & = \\Big | a _ i ( n ) e ^ { - 2 t } \\sum _ { k \\neq 0 } I _ { k a _ i ( n ) } ( 2 t ) \\prod _ { j = 0 } ^ { a _ i ( n ) - 1 } w _ { i , j } ( n ) ^ { - k } \\Big | \\\\ & \\leq 2 a _ i ( n ) e ^ { - 2 t } \\sum _ { k \\geq 1 } I _ { k a _ i ( n ) } \\\\ & \\leq 2 a _ i ( n ) e ^ { - 2 t } I _ 0 ( 2 t ) \\frac { t ^ { a _ i ( n ) } } { 1 - t ^ { a _ i ( n ) } } , \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} \\omega _ H ( c ( x ) ) = \\omega _ H ( ( x - 1 ) ^ { \\beta p ^ { e - 1 } } g ( x ) ) = \\omega _ H ( ( x - 1 ) ^ { \\beta p ^ { e - 1 } } g _ 1 ( x ) ) + \\omega _ H ( ( x - 1 ) ^ { \\beta p ^ { e - 1 } } g _ 2 ( x ) ) . \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{gather*} \\nu ( s , n ) = \\nu ( s - m - 1 , n - m - 1 ) + \\nu ( s - 2 s _ 0 , n ) , . \\end{gather*}"}
-{"id": "9215.png", "formula": "\\begin{align*} \\langle d e ^ { i \\delta _ 1 } U ( x _ 0 ) \\ , | \\ , d e ^ { i \\delta _ 1 } V ( x _ 0 ) \\rangle _ g = \\langle U ( x _ 0 ) \\ , | \\ , V ( x _ 0 ) \\rangle _ g . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} e ^ { - 2 \\pi i ( x - x ' ) u } ( D _ { A _ R } ^ 2 - z ) e ^ { 2 \\pi i ( x - x ' ) u } & = L + \\sigma _ z . \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} v \\left ( s - \\frac { \\delta ' } { 1 + \\delta } \\right ) \\ & \\geq \\ \\min ( v ( u ^ 2 b b ' ) , v ( u ' b ) ) \\\\ & \\geq \\ \\min ( \\gamma + \\textstyle \\int \\gamma , - \\textstyle \\int s \\gamma + \\gamma ) \\\\ & = \\ \\gamma - \\textstyle \\int s \\gamma \\ > \\ \\gamma . \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} E x t ^ n _ X ( j _ { * } \\hat { T } , \\mathbb { G } _ m ) \\simeq \\left \\{ \\begin{array} { c l } \\mathbb { Z } / 6 \\mathbb { Z } \\oplus \\mathbb { Z } & \\mbox { $ n = 0 $ } \\\\ 0 & \\mbox { $ n = 1 , 2 $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} \\nu _ { j } = \\sum _ { k = 1 } ^ { N } \\lambda _ { k } \\delta _ { A _ k } \\in \\mathcal { L } \\left ( \\R ^ { n \\times n } \\right ) , \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} L _ \\xi = & \\ \\frac 1 4 ( \\xi ^ { 2 n + 1 } A ^ h ( \\zeta , \\eta ) + \\eta ^ { 2 n + 1 } D ^ a ( - \\xi , \\zeta ) ) \\\\ = & \\ \\frac 1 4 \\left [ - i \\xi ^ { 2 n + 1 } \\left ( \\eta + \\frac { J ( \\zeta ) ^ 2 - \\zeta ^ 2 } { 4 J ( \\zeta ) } \\right ) + i \\eta ^ { 2 n + 1 } \\xi \\right ] \\\\ = & \\ i \\eta \\xi ^ { 2 n } \\left ( \\frac { n } 2 \\zeta + \\frac { J ( \\zeta ) ^ 2 - \\zeta ^ 2 } { 1 6 J ( \\zeta ) } \\right ) , \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} - y \\cdot \\nabla u ( y ) \\geq C _ u t \\chi _ { B ( x _ 0 , r ) } ( t y ) = C _ u t \\chi _ { \\frac { 1 } { t } B ( x _ 0 , r ) } ( y ) y , t \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\widetilde { f } ( x , z _ { 1 } , z _ { 2 } ) = f ( x , \\tilde { z } _ { 1 } , \\tilde { z } _ { 2 } ) + \\rho \\tilde { z } _ { 2 } \\max \\{ d ( x ) ^ { \\gamma _ { 1 } ( x ) } , | \\tilde { z } _ { 1 } | ^ { p ( x ) - 2 } \\tilde { z } _ { 1 } \\} \\\\ \\widetilde { g } ( x , z _ { 1 } , z _ { 2 } ) = g ( x , \\tilde { z } _ { 1 } , \\tilde { z } _ { 2 } ) + \\rho \\tilde { z } _ { 1 } \\max \\{ d ( x ) ^ { \\gamma _ { 2 } ( x ) } , | \\tilde { z } _ { 2 } | ^ { q ( x ) - 2 } \\tilde { z } _ { 2 } \\} , \\end{array} \\right . \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} \\left ( x ; \\bigcap _ { m = 1 } ^ k S ^ { ( j _ { m } ) } \\right ) = \\prod _ { m = 1 } ^ { k } \\left ( x ; S ^ { ( j _ { m } ) } \\right ) \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{gather*} \\Psi ^ { ( 7 ) } _ 0 ( x ) = \\Psi ^ { ( 1 ) } _ 0 ( x ) . \\end{gather*}"}
-{"id": "5104.png", "formula": "\\begin{align*} f = \\sum _ j \\lambda _ j \\psi _ j \\otimes \\overline { \\psi _ j } , \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} \\delta ( \\rho \\sigma ) - \\rho V & = 0 \\\\ \\delta ( \\rho V \\sigma + \\rho \\Psi ) & = 0 \\\\ - \\rho V \\sigma - \\rho \\Psi & = 0 . \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} \\phi ( x , y ) & = 1 \\wedge ( \\alpha x + \\beta y ) \\wedge ( \\alpha x + \\beta y ) ^ 2 \\\\ & = 1 \\wedge ( \\alpha x + \\beta y ) \\wedge [ ( \\ell - b \\alpha + a \\beta ) x ^ 2 + m x y + ( n - d \\alpha + c \\beta ) ] \\\\ & = ( a x ^ 3 + b x ^ 2 y + c x y ^ 2 + d y ^ 3 ) ( 1 \\wedge \\alpha \\wedge \\beta ) . \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} \\phi ( z ) = e ^ { \\pm i z \\alpha } w \\left ( \\frac { z } { \\mathcal { Z } } \\right ) \\left ( \\frac { z } { \\mathcal { Z } } \\right ) ^ { i \\tau } , \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{align*} C _ a = \\{ j \\mid \\mu ( j ) = a \\} \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} \\begin{aligned} & \\mathcal { R } _ 0 ( u ) : = ( \\partial _ u \\nabla H ) ( \\Phi ( u ) ) \\partial _ u \\Psi ( u ) , \\mathcal { R } _ 1 ( u ) : = [ \\partial _ u \\{ \\Psi ' ( u ) ^ T \\} ] [ \\cdot , \\nabla H ( \\Phi ( u ) ) ] , \\\\ & \\mathcal { R } _ 2 ( u ) : = [ \\partial _ u \\Psi ( u ) ] ^ T ( \\partial _ u \\nabla H ) ( \\Phi ( u ) ) \\partial _ u \\Phi ( u ) . \\end{aligned} \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} e _ i ( \\underline { x } , \\underline { x } ^ \\prime ) = \\sum _ { j = 0 } ^ { \\mathrm { m i n } ( i , a p , b p ) } e _ { j } ( \\underline { x } ) \\cdot e _ { i - j } ( \\underline { x } ^ \\prime ) . \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { m } ( r - w _ { j _ t } ) \\leq \\left \\lceil { k \\Delta \\over r } \\right \\rceil r - k \\Delta \\leq r - 1 \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} \\nu _ B = \\nu _ { t - 1 } , \\ \\ \\nu _ C = \\nu _ { r - 1 } . \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} \\P ( X _ t > N / 2 , \\ , \\forall t \\in [ 0 , T _ N ] | X _ 0 = N ) \\ge 1 - c _ 1 ^ { - 1 } e ^ { - c _ 1 N } . \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { B ^ r _ { p , q } } : = \\left ( \\sum _ { m = 0 } ^ { \\infty } 2 ^ { m r q } \\Vert \\sum _ { 2 ^ m \\leq \\langle \\xi \\rangle < 2 ^ { m + 1 } } d _ { \\xi } [ \\xi ( x ) \\widehat { f } ( \\xi ) ] \\Vert ^ q _ { L ^ p ( G ) } \\right ) ^ { \\frac { 1 } { q } } < \\infty . \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} \\chi _ { q , q ' } ^ { ( 1 ) } \\circ g _ q ( n ) = \\sum _ { i = 0 } ^ s \\alpha _ i ( q , q ' ) n ^ i , \\ \\ \\chi _ { q , q ' } ^ { ( 2 ) } \\circ g _ { q ' } ( n ) = \\sum _ { i = 0 } ^ s \\alpha _ i ' ( q , q ' ) n ^ i , \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} \\log L ( s , \\chi ) = \\sum _ { p } \\sum _ { m = 1 } ^ { \\infty } \\frac { \\chi ( p ^ m ) } { m p ^ { m s } } , \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{align*} \\left ( 1 - t \\right ) f \\left ( x \\right ) + \\kappa \\ , t \\ , g \\left ( x \\right ) = 0 \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} 2 ^ 6 3 ^ 3 \\Delta ( W ) = c _ 4 ( W ) ^ 3 - c _ 6 ( W ) ^ 2 = c _ 4 ^ 3 - c _ 6 ^ 2 = 2 ^ 6 3 ^ 3 \\Delta _ m \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} U ( a , b ; t ) : = \\frac { 1 } { \\Gamma ( a ) } \\sigma ( t ; b - a , a ) , \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\Pi ( d f _ 1 \\wedge \\cdots \\wedge d f _ { p - 1 } ) } ( \\Pi ) = 0 \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} \\eta ( { \\sigma } ) = s - \\lim _ { V \\to \\infty } ( \\eta _ { \\Lambda } = \\frac { 1 } { V } \\sum _ { { x } \\in \\Lambda } { \\sigma } _ { { x } } ) \\ , \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} \\gamma ^ * ( d ) = \\frac { 1 } { d } \\sum _ { ( c d , D ) = 1 } \\chi ( ( c d , w ) ) ( c d , w ) \\mu ( c ) c ^ { - 2 } \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} N ( 1 , 1 , 1 , 1 ; n ) = 8 \\sigma ( n ) - 3 2 \\sigma ( n / 4 ) . \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} X ' : = \\{ ( \\vec { g } , \\vec { z } ) \\in G ^ d \\times \\Delta ^ L \\mid & \\forall i \\not = j : d _ { \\vec { g } } ^ { ( \\alpha _ 1 , \\ldots , \\alpha _ l ) } ( z _ i , z _ j ) > 2 l + 2 \\\\ & \\forall i : \\exists j _ 1 , j _ 2 \\in \\{ 0 , \\ldots , l \\} : ( j _ 1 \\not = j _ 2 \\nu _ { j _ 1 } ( z _ i ) \\nu _ { j _ 2 } ( z _ i ) \\\\ & ) \\} . \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} [ D ( 1 , s + 1 ) : D ( m , p ) ] _ { q = 1 } & = [ D ( 1 , s ) : D ( m , p - 1 ) ] _ { q = 1 } + & \\\\ & + ( 1 - \\delta _ { 2 \\widetilde { p _ 0 } , m } - \\delta _ { 2 \\widetilde { p _ 0 } . m + 1 } - \\delta _ { \\widetilde { p _ 0 } , m } ) [ D ( 1 , s ) : D ( m , p + 1 ) ] _ { q = 1 } & \\\\ & + ( 1 - \\delta _ { 2 p _ 0 , m } - \\delta _ { 2 p _ 0 , m - 1 } ) [ D ( 1 , s ) : D ( m , p ) ] _ { q = 1 } \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} 3 a - \\frac { a } { n - 1 } + 2 ( 1 - a ) + a - [ a - ( 1 - a ) ] = 3 - \\frac { a } { k - 1 } . \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} L ( e _ { i j } ) = \\sum _ { x < i } C _ { x i } ^ { i i } e _ { x j } + C _ { i j } ^ { i j } e _ { i j } - \\left ( C _ { j i } ^ { i i } + C _ { j i } ^ { j j } \\right ) e _ { i i } + C _ { j i } ^ { i j } e _ { j i } + \\sum _ { y > j } C _ { j y } ^ { j j } e _ { i y } . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} \\gamma ( s , \\pi , \\chi ' , \\psi ) = \\epsilon ( s , \\pi , \\chi ' , \\psi ) = c _ \\pi ( c ) ^ { - 1 } \\epsilon ( s , \\mathbf { 1 } , \\chi ' , \\psi ) ^ n . \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\frac { \\partial _ 1 u ( x \\pm y ) } { | y | ^ { N + 2 \\sigma } } 1 _ { \\{ | y | \\geq \\varepsilon \\} } \\ d y & = \\lim _ { t \\to 0 } \\int _ { \\R ^ N } \\int _ 0 ^ 1 \\frac { \\partial _ 1 u ( s t e _ 1 + x \\pm y ) } { | y | ^ { N + 2 \\sigma } } 1 _ { \\{ | y | \\geq \\varepsilon \\} } \\ d s d y \\\\ & = \\lim _ { t \\to 0 } \\int _ { \\R ^ N } \\frac { u ( x + t e _ 1 \\pm y ) - u ( x \\pm y ) } { | y | ^ { N + 2 \\sigma } } 1 _ { \\{ | y | \\geq \\varepsilon \\} } \\ d y . \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} \\langle d e ^ { i \\theta } U _ 1 ( y _ 0 ) \\ , | \\ , d e ^ { i \\theta } V _ 1 ( y _ 0 ) \\rangle _ g = \\langle U _ 1 ( y _ 0 ) \\ , | \\ , V _ 1 ( y _ 0 ) \\rangle _ g , ~ \\forall | \\theta | < \\sigma . \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ \\infty f _ \\delta ( t ) & = \\delta + \\sum _ { t = 2 } ^ { K _ \\delta + 1 } f _ \\delta ( t ) + \\sum _ { t = K _ \\delta + 2 } ^ \\infty f ( t ) \\\\ & = \\delta + x + f ( K _ \\delta + 1 ) + 1 - F ( K _ \\delta + 1 ) = 1 . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} \\nu _ i ( U ) = \\int _ { U \\cap \\Sigma _ i } | { \\textbf A } _ i | ^ 2 \\leq C _ 2 , \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} ( \\lambda ( N ) + \\beta \\Delta ( N ) ) | _ { X } = ( 1 - t _ N ( X ) \\cdot \\beta ) \\lambda ( N ) | _ { X } + D _ X , \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\biggl ( \\int _ { \\Omega _ 0 } \\abs { x ( t ) } ^ p \\textup d t \\biggr ) ^ { 1 / p } = m . \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} I ( X ; X + & Z _ { u ' } ) - I ( X ; X + Z _ u ) \\\\ & \\geq \\mu \\cdot \\bigl ( I ( X ; X + Z _ { u '' } ) - I ( X ; X + Z _ { u ' } ) \\bigr ) \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} E : y ^ 2 = x ^ 3 + x ^ 2 - 2 x \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} \\mathbb { E } \\| \\tilde { f } _ { m _ 0 , m _ 1 } - f \\| ^ 2 = \\| \\tilde { f } _ { m _ 0 , m _ 1 } - \\mathbb { E } \\tilde { f } _ { m _ 0 , m _ 1 } \\| ^ 2 + \\| f - \\mathbb { E } \\tilde { f } _ { m _ 0 , m _ 1 } \\| ^ 2 . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} \\int _ { c ^ + ( \\xi , r ) } \\Big ( \\prod _ { i = 1 } ^ d ( w _ i + r ) ^ { - \\gamma _ i } \\Big ) w ^ \\gamma \\ , d \\sigma ( w ) \\simeq \\Big ( \\prod _ { i \\in J } r ^ { - \\gamma _ i } \\Big ) \\int _ { c ^ + ( \\xi , r ) } \\Big ( \\prod _ { i \\in J } w _ i ^ { \\gamma _ i } \\Big ) \\ , d \\sigma ( w ) . \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} g _ { \\ell } ( w ) & = ( B ^ 2 + k _ { \\ell } A _ { \\ell } B ^ 2 / a _ { \\ell } ) w ^ 2 + ( 2 B - k _ { \\ell } A _ { \\ell } B ^ 2 ) w + 1 . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} S ( X , Y ) = \\sum _ { X < \\ell \\le Y } c ( \\ell ) ^ 2 \\ell ^ { - 1 } \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{gather*} - ( - 1 ) ^ { \\epsilon { ( X ) } } i _ X \\delta _ Q \\omega _ 1 = - ( - 1 ) ^ { \\epsilon { ( X ) } } \\delta \\delta _ Q \\alpha _ 1 - d \\delta _ Q \\alpha ' _ 1 . \\end{gather*}"}
-{"id": "9343.png", "formula": "\\begin{align*} ( H _ { \\lambda , \\alpha , \\theta } u ) _ n = c ( \\theta + n \\alpha ) u _ { n + 1 } + \\tilde { c } ( \\theta + ( n - 1 ) \\alpha ) u _ { n - 1 } + 2 \\cos { 2 \\pi ( \\theta + n \\alpha ) } u _ { n } . \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\max _ { z _ n ^ { \\rm R } \\ge 0 , z _ n ^ { \\rm I } \\ge 0 } & \\ln \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( z _ n ^ { \\rm R } + \\sqrt { P / 2 } \\big ) \\right ) + \\ln \\left ( \\frac { 1 } { 2 } + \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( z _ n ^ { \\rm I } + \\sqrt { P / 2 } \\big ) \\right ) \\\\ \\mathrm { s . t . } ~ ~ ~ & { z _ n ^ { \\rm R } } ^ 2 + { z _ n ^ { \\rm I } } ^ 2 = A _ n . \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\mathcal { U } _ { - } ( q ^ { 1 / 2 } ) = \\left ( - 1 \\right ) ^ { \\mathsf { N } } _ { q } M ( 1 ) I _ { 0 } , \\mathcal { U } _ { - } ( i q ^ { 1 / 2 } ) = i ( - 1 ) ^ { \\mathsf { N } + 1 } \\frac { \\zeta _ { - } + 1 / \\zeta _ { - } } { \\zeta _ { - } - 1 / \\zeta _ { - } } _ { q } M ( i ) \\sigma _ { 0 } ^ { z } , \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} \\omega _ { \\beta , \\mu } ( A ) = \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda } ( A ) \\ \\ { \\rm { a n d } } \\ \\ \\omega _ { \\infty , \\mu } ( A ) = \\lim _ { V \\to \\infty } \\omega _ { \\infty , \\mu , \\Lambda } ( A ) \\ , \\ A \\in \\bigcup _ { \\Lambda \\subset ( \\mathbb { R } ^ d \\ , { \\rm { o r } } \\ , \\mathbb { Z } ^ d ) } { \\cal A } _ { \\Lambda } \\ . \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} \\leq C \\sum _ { i = 1 } ^ { m } \\int _ { \\mathbb { R } ^ { n } } \\left \\{ \\sum _ { j = 1 } ^ { k } \\left ( \\frac { \\lambda _ { j } | B _ j | ^ { \\frac { \\alpha } { n } } \\chi _ { B _ j } ( x ) } { \\| \\chi _ { B _ j } \\| _ { p ( . ) } } \\right ) ^ { p _ 0 } \\right \\} ^ { \\frac { q _ 0 } { p _ 0 } } w _ { A _ { i } ^ { - 1 } } ( x ) d x \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} d _ { n , k } ^ { m , l } = \\binom { l } { k } ( ( \\theta + n ) _ { m - n \\uparrow } ) ^ { k } ( \\sum _ { j = 1 } ^ { m - n } ( \\alpha + \\theta ) _ { m - j \\uparrow } ( \\theta + m - 1 ) _ { j - 1 \\downarrow } ) ^ { l - k } . \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} q ( x ) = \\frac { \\Gamma \\left ( - 2 j \\right ) } { \\Gamma \\left ( 2 j \\right ) } \\frac { \\Gamma \\left ( - \\frac { j } { k \\ell _ { \\lambda } } + j \\right ) } { \\Gamma \\left ( - \\frac { j } { k \\ell _ { \\lambda } } - j \\right ) } x ^ { 2 j } \\ ; , \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} \\left \\{ x \\in B ( x _ 0 , \\rho ) : u ( x , t ) \\le \\frac \\gamma 8 k \\right \\} = \\left \\{ x \\in B ( x _ 0 , \\rho ) : ( u ( x , t ) - k ) _ - \\ge \\left ( 1 - \\frac \\gamma 8 \\right ) k \\right \\} \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} B = \\bigg ( \\frac { | J _ { \\rho _ 0 } | e ^ { - T _ { 0 } } } { 2 | J _ { \\rho _ 0 } | e ^ { - T _ { 0 } } + 2 } \\bigg ( { \\rm \\exp } \\bigg [ \\bigg ( 2 + \\frac { 2 } { | J _ { \\rho _ { 0 } } | e ^ { - T _ { 0 } } } \\bigg ) T _ { 0 } \\bigg ] - 1 \\bigg ) + C _ * { \\rm \\exp } \\bigg [ \\bigg ( 2 + \\frac { 2 } { | J _ { \\rho _ { 0 } } | e ^ { - T _ { 0 } } } \\bigg ) T _ { 0 } \\bigg ] \\bigg ) ^ { - 1 } , \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{align*} F ( z ) = \\frac { ( \\Psi _ N ^ * ( z ) - \\Psi _ N ( z ) ) + ( \\Psi _ N ^ * ( z ) + \\Psi _ N ( z ) ) F _ N ( z ) } { ( \\Phi _ N ^ * ( z ) + \\Phi _ N ( z ) ) + ( \\Phi _ N ^ * ( z ) - \\Phi _ N ( z ) ) F _ N ( z ) } , \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} \\mathcal { M } [ ( \\hat { x } _ t , \\hat { x } _ { t + 1 } ) ; ( \\theta _ t , \\theta _ { t + 1 } ) ] : = \\beta _ t \\ , \\frac { \\partial { U ( \\hat { x } _ t , \\theta _ t ) } } { \\partial { \\hat { x } } } \\left [ \\beta _ { t + 1 } \\frac { \\partial { U ( \\hat { x } _ { t + 1 } , \\theta _ { t + 1 } ) } } { \\partial { \\hat { x } } } \\right ] ^ { - 1 } . \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} ( x - 1 ) ^ { p ^ e - p ^ { e - k } + p ^ { e - k - 1 } } = \\sum _ { j = 0 } ^ { p ^ k - 1 } x ^ { j p ^ { e - k } + p ^ { e - k - 1 } } - \\sum _ { j = 0 } ^ { p ^ k - 1 } x ^ { j p ^ { e - k } } . \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{gather*} G \\left ( h _ { + } \\right ) \\allowbreak = \\\\ \\dfrac { \\sqrt { s ^ { 2 } + t ^ { 2 } } \\sqrt { p _ { 1 } } - \\allowbreak \\left \\vert 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v \\right \\vert } { \\sqrt { s ^ { 2 } + t ^ { 2 } } \\sqrt { p _ { 1 } } + \\allowbreak \\left \\vert 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v \\right \\vert } \\end{gather*}"}
-{"id": "2857.png", "formula": "\\begin{align*} \\forall i \\in I \\left \\{ \\begin{array} { l } \\langle x \\ | \\ u _ i \\rangle - \\eta _ i = \\sum _ { k \\in I } \\tilde { \\nu } _ k \\langle u _ k \\ | \\ u _ i \\rangle , \\\\ \\tilde { \\nu } _ i > 0 \\end{array} \\right . \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} x \\sim p _ { \\theta } ( x ) : = \\exp ( \\theta x - \\psi ( \\theta ) ) , \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} \\hat { u } = 1 \\hat { r } = \\hat { s } = \\hat { t } = 0 , \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\mathfrak { g } = U _ 0 \\oplus U _ 1 \\oplus \\cdots \\oplus U _ q \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} 0 _ F \\in \\bigoplus _ { k = 1 } ^ { r + 1 } ( - 1 ) ^ k \\varphi ( x _ 1 , x _ 2 , . . . , \\hat { x _ k } , . . . , x _ { r + 1 } ) \\odot \\varphi ( x _ k , y _ 1 , . . . , y _ { r - 1 } ) . \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} \\begin{aligned} \\nabla H _ { 3 } ^ { ( 3 ) } ( u ) = & - 3 \\ , c _ 1 \\ , \\partial _ x ( z _ x ^ 2 ) - 6 \\ , c _ 1 \\ , \\partial _ x \\Pi _ S ^ { \\perp } [ v _ x \\ , z _ x ] - c _ 2 \\partial _ { x x } ( z ^ 2 ) + c _ 2 \\pi _ 0 [ z _ x ^ 2 ] - 2 \\ , c _ 2 \\partial _ { x x } \\Pi _ S ^ { \\perp } [ v \\ , z ] + \\\\ & + 2 \\ , c _ 2 \\Pi _ S ^ { \\perp } [ v _ x \\ , z _ x ] + 3 \\ , c _ 3 \\pi _ 0 [ z ^ 2 ] + 2 \\ , c _ 3 \\Pi _ S ^ { \\perp } [ v \\ , z ] . \\end{aligned} \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} | \\epsilon _ N ( t , x , x ' ) | \\leq \\left [ C _ 0 e ^ { - N ( 2 + \\delta ) y } + C _ 1 \\left ( e ^ { 2 y } + \\frac { e ^ { ( 2 + \\delta ) y } } { 2 \\ell } \\right ) e ^ { - \\frac { e ^ { \\delta y } } { 6 4 \\ell ^ 2 } } \\right ] \\chi _ { \\mathcal { N } } = O ( e ^ { - 2 N y } ) . \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\int _ 1 ^ R | \\nabla ^ { 0 , 1 } \\nabla u | ^ 2 + { \\rm R i c } ( \\nabla u , \\nabla u ) & = \\frac { 1 } { 2 } ( \\mu - 2 ) ( R ^ { 2 \\mu + 2 n - 4 } - 1 ) \\int _ Y ( \\mu ^ 2 \\phi ^ 2 + | \\nabla _ Y \\phi | ^ 2 ) . \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} \\langle A u , u \\rangle = \\langle Q ' ( u ^ \\pm ) u , u \\rangle = 0 . \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} \\begin{dcases*} \\operatorname { R e } ( e ^ { - \\sqrt { - 1 } \\theta _ { 0 } } \\Omega ) \\big | _ { M } = d V _ { M } \\\\ \\operatorname { I m } ( e ^ { - \\sqrt { - 1 } \\theta _ { 0 } } \\Omega ) \\big | _ { M } = 0 \\end{dcases*} . \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} V _ \\sigma = \\left \\{ ( z _ v ) _ v \\in | Z | \\ \\bigg | \\ z _ v > z _ { v ' } , \\ v \\in \\sigma \\ \\ v ' \\in Z _ 0 \\setminus \\sigma \\right \\} \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} \\widetilde { t } = \\widetilde { t } ( \\varepsilon ; t , n , u ) , \\widetilde { u } = \\widetilde { u } ( \\varepsilon ; t , n , u ) . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} \\frac { \\partial \\tilde { \\epsilon } } { \\partial \\lambda _ { j b } } ( \\lambda ) = - s _ b \\tilde { \\epsilon } ' ( \\lambda _ { j b } ) F ( \\lambda _ { j b } | \\lambda ) . \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} \\begin{aligned} X u - Y v & = 0 \\\\ Y u + X v & = 0 . \\end{aligned} \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} \\mathbb { P } \\bigg ( \\mathbb { P } _ { Q ^ { ( \\mathsf { B } _ n ) } } \\Big ( \\mathbf { U } ( I , \\mathsf { B } _ n ) \\notin \\mathcal { T } _ \\epsilon ^ n ( Q _ U ) \\Big ) > 2 e ^ { - n \\eta ( \\epsilon ) } \\bigg ) \\leq e ^ { \\frac { 2 ^ { n R _ 1 } e ^ { - n \\eta ( \\epsilon ) } } { 3 } } = e ^ { - \\frac { 1 } { 3 } e ^ { n \\big ( R _ 1 \\ln 2 - \\eta ( \\epsilon ) \\big ) } } , \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\searrow 0 } \\ , \\varphi ( \\alpha ) = 0 . \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\omega _ 0 l _ { s f , \\mathsf { k } } i _ { f , \\mathsf { k } } & = \\sigma _ \\mathsf { k } \\norm { \\nu _ \\mathsf { k } ( \\theta _ \\mathsf { k } ) } , \\\\ j \\mathrm { r } ( \\theta _ \\mathsf { k } ) \\norm { \\nu _ \\mathsf { k } ( \\theta _ \\mathsf { k } ) } & = \\sigma _ \\mathsf { k } \\nu _ \\mathsf { k } ( \\theta _ \\mathsf { k } ) , \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ l \\kappa _ i \\int _ { \\Sigma _ i } \\left ( R _ { \\gamma _ i } - ( n - 2 ) ( n - 1 ) \\right ) d \\Sigma _ i \\geq 0 , \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} & = \\sum _ { l = 0 } ^ { | k | - 1 } \\frac { 2 | k | - ( 2 l + 1 ) } { 2 k ^ 2 h ^ 2 } \\left [ \\phi ( x + ( l - 1 ) h ) - 2 \\phi ( x + l h ) + \\phi ( x + ( l + 1 ) h ) \\right ] \\\\ & = \\frac { 1 } { 2 k ^ 2 h ^ 2 } \\left ( \\phi ( x + k h ) + \\phi ( x + ( k - 1 ) h ) + ( 2 k - 1 ) \\phi ( x - h ) - ( 2 k + 1 ) \\phi ( x ) \\right ) , \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} V _ { 0 0 } = c ( 0 , 0 ) \\zeta ( 2 ) ^ { - 1 } ( L ( 1 , \\chi ) \\log { N } ) ^ 2 E _ { 0 0 } . \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} v _ { i a } = \\frac { \\epsilon ' ( \\lambda _ { i a } ) } { 2 \\pi \\rho _ { \\infty } ( \\lambda _ { i a } ) } \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} & \\mathbb { P } \\left ( \\gamma _ { \\mathrm { u } , i k } \\geq \\theta \\right ) \\approx 1 - { 2 \\beta \\theta } = 1 - \\frac { 2 K \\theta } { N } , \\\\ & \\mathbb { P } \\left ( \\gamma _ { \\mathrm { a } , i k } \\geq \\theta \\right ) \\approx 1 - \\left ( { 2 \\beta \\theta } \\right ) ^ 2 = 1 - \\frac { 4 K ^ 2 \\theta ^ 2 } { N ^ 2 } . \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} Q ^ { \\alpha } = \\phi ^ { \\alpha } - \\xi ^ i u ^ { \\alpha } _ { \\bold { 1 } _ i ; \\bold { 0 } } . \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} \\binom { k + ( i + j ) p } { m + i p } \\equiv _ { p ^ 2 } \\binom { i + j } { i } \\binom { k } { m } ( 1 + p ( ( i + j ) H _ k - j H _ { k - m } - i H _ m ) ) , \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} t ^ * = \\max \\{ t : S _ { \\rho _ * , \\hat u _ 0 } \\} , t _ * = \\min \\{ t : S _ { \\rho _ * , \\hat u _ 0 } \\} . \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} P _ 1 = v - v _ { - 1 } , P _ 2 = ( n - 1 ) v - n v _ { - 1 } . \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} \\gamma = \\frac { 2 \\beta } { \\beta + 2 } , \\Omega _ \\beta = 2 \\int _ 0 ^ 1 ( 1 - t ^ \\beta ) ^ \\frac 1 2 \\ , \\dd t , \\Omega _ \\beta ' = 2 \\int _ 0 ^ 1 \\frac { \\dd t } { ( 1 - t ^ \\beta ) ^ \\frac 1 2 } . \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} \\bold { p r } \\widetilde { X } ( \\widetilde { F } _ { \\alpha } ) = \\sum _ { \\beta , k \\in \\mathbb { Z } } A ^ { \\beta } _ { \\alpha , k } ( n , [ \\widetilde { u } ] ) ( S ^ k \\widetilde { F } _ { \\beta } ) , \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} D _ { i ; I } = \\partial _ { x ^ i } + \\frac { \\partial u _ { \\bold { 0 } ; I } ^ { \\alpha } } { \\partial x ^ i } \\partial _ { u _ { \\bold { 0 } ; I } ^ { \\alpha } } + \\cdots + \\sum _ { \\alpha , J _ 1 } u ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; I } \\partial _ { u ^ { \\alpha } _ { J _ 1 ; I } } . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} \\sum \\limits _ r \\eta _ r ( x ) \\eta _ r ( R ^ a ) \\eta _ r ( z ^ { - 1 } ) = n \\sum \\limits _ r \\big { ( } \\eta ( R ^ { a + a _ 1 } ) + \\eta _ r ( R ^ { a - a _ 1 } ) \\big { ) } , \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} { R _ { m _ k , i } } \\left ( \\tau ^ { ( k ) } \\right ) = g \\left ( \\tau ^ { ( k ) } \\right ) - { P _ { m _ k , i } } g \\left ( \\tau ^ { ( k ) } \\right ) \\forall i , \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} X _ 1 = - t \\partial _ t + u \\partial _ u , X _ 2 = ( - 1 ) ^ n t \\partial _ t + ( - 1 ) ^ n u \\partial _ u , X _ 3 = c _ 3 ( n ) \\partial _ t . \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} M _ 1 : = E _ Q [ T _ 1 ^ \\alpha ] < 1 , \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} \\mathbb { P } \\bigg ( \\Big | \\Big | \\Lambda ^ { \\big ( \\mathsf { B } _ V ^ { ( n ) } ( i ) , m \\big ) } _ { \\mathbf { Z } | \\mathbf { U } = \\mathbf { u } } - Q ^ n _ { Z | U = \\mathbf { u } } \\Big | \\Big | _ { \\mathsf { T V } } > e ^ { - n \\delta _ 1 } \\bigg ) \\leq e ^ { - e ^ { n \\delta _ 2 } } , \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} Q ( t ) = \\sqrt { x - 1 } \\tan \\bigg ( t \\sqrt { x - 1 } + \\arctan \\bigg ( \\frac 1 { \\sqrt { x - 1 } } \\bigg ) \\bigg ) . \\end{align*}"}
-{"id": "6835.png", "formula": "\\begin{align*} & \\varphi ^ 3 ( - q ) \\varphi ( - q ^ 5 ) = - E _ { \\chi _ 0 , \\chi _ 1 } ( q ) - 4 E _ { \\chi _ 0 , \\chi _ 1 } ( q ^ 2 ) - 5 E _ { \\chi _ 1 , \\chi _ 0 } ( q ) + 2 0 E _ { \\chi _ 1 , \\chi _ 0 } ( q ^ 2 ) . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{gather*} { a } = { d } + q _ 2 ( b - e _ 1 ) + q _ 1 . \\end{gather*}"}
-{"id": "7188.png", "formula": "\\begin{align*} | { \\mathcal K } | = K \\prod _ { p | h P } \\bigl ( 1 - \\frac { 1 } { p } \\bigr ) \\{ 1 + O ( 1 / \\log x ) \\} \\ , \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} \\sum _ { A \\in \\mathcal { F } } 2 ^ { - | A | / ( 2 q - 1 ) } \\leqslant \\binom { n } { \\lceil n / 2 \\rceil } 2 ^ { - \\binom { \\lceil n / 2 \\rceil } { k } / ( 2 q ) } \\leqslant \\left ( \\frac { e n } { \\lceil n / 2 \\rceil } \\right ) ^ { \\lceil n / 2 \\rceil } 2 ^ { - ( ( 1 + 1 / \\ln 2 ) n + 1 ) } < \\frac { 1 } { 2 } \\left ( \\frac { e } { 2 ^ { 1 / \\ln 2 } } \\right ) ^ n = \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} 0 < g ( x ) < 1 x \\in ( 0 , 1 ) g ( 0 ) = 0 , \\ g ( 1 ) = 1 . \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} \\Delta \\left ( h ( f _ 0 , g _ 0 ) ; f , g \\right ) & = \\left \\| { A } \\xi - \\hat { A } \\xi \\right \\| _ \\alpha ^ \\alpha \\\\ & = \\int _ { - \\pi } ^ { \\pi } \\left | A ( e ^ { i \\theta } ) - { h } ^ 0 ( \\theta ) \\right | ^ { \\alpha } f ( \\theta ) d \\theta + \\int _ { - \\pi } ^ { \\pi } \\left | { h } ^ 0 ( \\theta ) \\right | ^ { \\alpha } g ( \\theta ) d \\theta . \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} ( \\Theta _ { j \\bar k } ^ { T _ B } v , w ) _ H = - ( [ \\theta _ k ^ * , \\theta _ v ] , [ \\theta _ j ^ * , \\theta _ w ] ) - \\left ( P ^ { \\bot } ( D ^ { { \\rm E n d } ( H ) } _ j \\theta _ v ) , P ^ { \\bot } ( D ^ { { \\rm E n d } ( H ) } _ k \\theta _ w ) \\right ) , \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} p ( G ) ( A ) = \\sum _ { \\phi : V \\to [ k ] } \\prod _ { \\{ u , v \\} \\in E } A _ { \\phi ( u ) , \\phi ( v ) } . \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\begin{aligned} Q ( t ) - Q ( 0 ) = Q _ H ( e ^ { i t A } f - f , e ^ { i t A } f - f ) + Q _ H ( f , e ^ { i t A } f - f \\ ) + Q _ H ( e ^ { i t A } f - f , f ) , \\end{aligned} \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} T \\gamma \\left ( \\frac { \\partial } { \\partial t } \\right ) = \\frac { \\partial } { \\partial t } + \\sum _ { j = 1 } ^ n \\frac { \\partial \\gamma ^ j } { \\partial t } \\frac { \\partial } { \\partial p _ j } , T \\gamma \\left ( \\frac { \\partial } { \\partial q ^ i } \\right ) = \\frac { \\partial } { \\partial q ^ i } + \\sum _ { j = 1 } ^ n \\frac { \\partial \\gamma ^ j } { \\partial q ^ i } \\frac { \\partial } { \\partial p _ j } , \\\\ \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} U ^ + _ \\rho : = U ^ + _ v \\otimes _ { \\Z [ v ^ { \\pm 1 } ] , \\rho } \\mathbb { O } _ p \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} \\alpha = \\left [ \\begin{array} { c } z A + x 1 _ { V } \\\\ z B + y 1 _ { V } \\\\ z J \\end{array} \\right ] , \\beta = \\left [ \\begin{array} { c c c } - z B - y 1 _ { V } & z A + x 1 _ { V } & z I \\end{array} \\right ] . \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} \\hat { D } ( f ) ( x , y ) = D ( f | _ x ^ y ) ( x , y ) = L ( f | _ x ^ y ) ( x , y ) - L ( f | _ x ^ y ) _ d ( x , y ) . \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} _ { p } D _ { \\tau } ( q \\lambda ) = _ { p } D _ { \\tau } ( \\lambda ) , \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} d ^ 2 ( d x ^ I \\wedge d \\xi ^ J \\cdot \\widetilde { \\omega } _ { I J } ) & = d ( d x ^ I \\wedge d \\xi ^ J ) \\wedge d \\omega _ { I J } + ( - 1 ) ^ { | I | + | J | } d x ^ I \\wedge d \\xi ^ J \\wedge d ^ 2 \\omega _ { I J } \\\\ & = 0 . \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} C ' _ 1 & = \\frac { 1 } { 2 } \\log \\Bigl ( 1 + \\frac { a P _ 1 + P _ 2 } { N } \\Bigr ) - C _ 2 \\\\ & = \\frac { 1 } { 2 } \\log \\Bigl ( 1 + \\frac { a P _ 1 } { P _ 2 + N } \\Bigr ) \\\\ C ' _ 2 & = \\frac { 1 } { 2 } \\log \\Bigl ( 1 + \\frac { a P _ 1 + P _ 2 } { N } \\Bigr ) - C _ 1 \\\\ & = \\frac { 1 } { 2 } \\log \\Bigl ( 1 + \\frac { ( a - 1 ) P _ 1 + P _ 2 } { P _ 1 + N } \\Bigr ) . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{align*} & \\underset { \\tau \\to + 0 } { \\mbox { { \\rm e s s l i m s u p } } } \\ , \\left | \\int _ { { \\bf R } ^ N } [ G ( y , t - \\tau ) - G ( y , t ) ] u ( y , \\tau ) \\eta _ n ( y ) \\ , d y \\right | \\\\ & \\le \\sup _ { y \\in B ( 0 , 2 n t ^ { \\frac { 1 } { \\theta } } ) , s \\in ( t / 2 , t ) } \\ , | \\partial _ t G ( y , s ) | \\ , \\underset { \\tau \\to + 0 } { \\mbox { { \\rm e s s l i m s u p } } } \\ , \\biggr [ \\tau \\int _ { B ( 0 , 2 n t ^ \\frac { 1 } { \\theta } ) } u ( y , \\tau ) \\ , d y \\biggr ] = 0 . \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} x = 2 ( \\lambda \\epsilon n + 1 - \\lambda \\epsilon ) C ( 1 + Q ) S ^ { - \\frac 1 2 } , \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda } ( \\frac { b _ { { 0 } } ^ { * } b _ { { 0 } } } { V } ) = \\rho _ { { 0 } } > 0 \\ . \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} q _ { i } : = \\lceil 2 ^ { 8 n - \\frac { i } { 2 } } \\rceil . \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} \\varphi ( \\sigma ( P ) ) = \\gamma _ E ( \\sigma ) ( \\varphi ( P ) ) , \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\theta } : = < T ( A ) , z ^ { m _ r ( \\theta ) } U _ r ( A ) > \\subset G ( K ) \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} \\delta \\mu ( a , \\pi ) & = \\phi ( \\delta a ) + \\phi ( d a ) \\\\ \\pi \\mu ( a , \\pi ) & = \\pi \\phi ( a ) \\\\ ( \\delta + \\pi ) \\mu ( a , \\pi ) & = \\mu ( \\delta a + d a , \\pi ) , \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} e ^ { - \\beta \\mathbb { H } _ 0 } = \\mathcal { L } \\big [ e ^ { \\beta \\Delta _ X } \\big ] \\mathcal { R } \\big [ e ^ { \\beta \\Delta _ X } \\big ] \\unrhd 0 \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{gather*} \\operatorname { g h } ( \\Phi ^ \\ast _ A ) = - \\operatorname { g h } \\big ( \\Phi ^ A \\big ) - 1 , \\epsilon ( \\Phi ^ \\ast _ A ) = \\epsilon \\big ( \\Phi ^ A \\big ) + 1 ( \\mbox { m o d } \\ , 2 ) . \\end{gather*}"}
-{"id": "2221.png", "formula": "\\begin{align*} h _ M ( t ) : = \\inf _ { k \\in \\N } M _ k t ^ k , t > 0 , h _ M ( 0 ) : = 0 . \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{gather*} V ( B [ f _ 1 , \\ldots , f _ j ] _ c ^ { \\textsf { c o m } } ) \\underset { \\ne } { \\subset } V ( B [ f _ 1 , \\ldots , f _ j ] _ { f _ j } ^ { \\textsf { c o m } } ) , \\\\ | V ( B [ f _ 1 , \\ldots , f _ j ] _ { f _ j } ^ { \\textsf { c o m } } ) | < | V ( B [ f _ 1 , \\ldots , f _ { j - 1 } ] _ { f _ { j - 1 } } ^ { \\textsf { c o m } } ) | , \\\\ | V ( B [ f _ 1 , \\ldots , f _ j ] _ { f _ j } ^ { \\textsf { c o m } } ) | = | V ( B [ f _ 1 , \\ldots , f _ { j - 1 } ] _ { f _ j } ^ { \\textsf { c o m } } ) | . \\end{gather*}"}
-{"id": "3149.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty k ^ { 1 + \\eta } f ( k ) < \\infty , \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} \\mu _ b ( w _ 2 ^ { \\ell _ 2 } | w _ 1 , u ^ { \\ell _ 1 } ) & = \\int \\mu _ b ( w _ { 2 } ^ { \\ell _ 2 } | x , w _ 1 , u ^ { \\ell _ 1 } ) d \\mu ( x | w _ 1 , u ^ { \\ell _ 1 } ) \\\\ & = \\int \\mu _ b ( w _ 2 ^ { \\ell _ 2 } | x ) d \\mu ( x | w _ 1 , u ^ { \\ell _ 1 } ) \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} t _ i = \\frac { 1 + T _ i N ^ { - 1 / 3 } } 2 , r = \\sqrt N + \\frac { R N ^ { - 1 / 6 } } 2 , x _ i = \\sqrt N + \\frac { ( R + U _ i ) N ^ { - 1 / 6 } } 2 . \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} \\boldsymbol { \\psi } = \\left [ \\begin{matrix} 1 & \\mathbf { x } _ 1 \\end{matrix} \\right ] ^ T \\otimes \\mathbf { x } _ 0 = \\left [ \\begin{matrix} \\mathbf { x } _ 0 & \\mathbf { x } _ 0 x _ { 1 1 } & \\cdots & \\mathbf { x } _ 0 x _ { 1 K } \\end{matrix} \\right ] ^ T \\ , , \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{gather*} \\dfrac { d \\Psi _ 0 } { d x } = \\hat { L } _ 0 \\Psi _ 0 , \\dfrac { d \\Psi _ 0 } { d t } = \\hat { B } _ 0 \\Psi _ 0 , \\end{gather*}"}
-{"id": "7695.png", "formula": "\\begin{align*} \\mu _ { k + 1 } - \\mu _ k = \\frac { \\pi } { 2 g ' ( \\eta _ k ) } ( 1 + o ( 1 ) ) = \\frac { 2 \\pi } { \\Omega _ { \\beta } ' } \\mu _ k ^ \\frac { \\beta - 2 } { 2 \\beta } ( 1 + o ( 1 ) ) . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} D ( 1 , s + p ) : D ( 2 , s ) ] _ q = q ^ { p ( s + p + { _ 2 ( s ) } ) } [ D ( 1 , s + p ) : D ( 2 , s ) ] ^ w _ q . \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} \\widetilde { u _ { \\bold { 1 } _ i } ^ { \\alpha } } = \\frac { \\partial \\widetilde { u } ^ { \\alpha } } { \\partial \\widetilde { x } ^ i } = \\sum _ j \\frac { D _ j \\widetilde { u } ^ { \\alpha } } { D _ j \\widetilde { x } ^ i } . \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} \\lambda = \\chi * 1 \\ , \\lambda ' = \\chi * \\log = \\lambda * \\Lambda \\ , \\nu = \\mu \\chi * \\mu \\ . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} \\sum _ { n = l + r } ^ k ( - ) ^ n C _ { 2 k } ^ { k - n } C _ { n - r } ^ { n - l - r } = ( - ) ^ { l + r } C _ { 2 k - l - 1 } ^ { k - l - r } . \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\tilde { P } ( \\mu ) = K ( \\mu , \\eta ) = \\tilde { D } ( \\eta ) . \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} k ( \\lambda ) = \\left ( \\lambda ^ { 2 } - 1 / \\lambda ^ { 2 } \\right ) / ( \\lambda ^ { 2 } / q ^ { 2 } - q ^ { 2 } / \\lambda ^ { 2 } ) , \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} f _ 0 ( \\theta ) = C \\left | \\sum _ { j = 0 } ^ { \\infty } c _ j e ^ { - i j \\theta } \\right | ^ \\frac { \\alpha } { \\alpha + ( \\alpha - 1 ) ( \\beta - 1 ) } . \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} \\abs { \\sum _ { i , l = 1 , 2 } \\pi _ \\theta ( i ) p _ \\theta ( i , l ) \\int _ { - \\infty } ^ { \\infty } y ^ { r + 1 } \\varphi ( y - \\mu _ \\theta ^ { ( l ) } ) \\ , \\d y } < \\infty ~ \\theta = \\theta _ 0 , \\theta _ 1 , \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{gather*} \\frac { d \\Psi } { d x } = L \\Psi , \\frac { d \\Psi } { d t } = B \\Psi , \\end{gather*}"}
-{"id": "5533.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\theta ^ i \\frac { \\partial { U ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ i } } = \\sum _ { i = 1 } ^ n \\theta ^ i u _ i ( s ^ i ( \\hat { x } , \\theta ) ) = U ( \\hat { x } , \\theta ) , \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} R _ t ( h _ t ) = R _ t \\rfloor d h _ t = R _ t \\rfloor ( \\eta _ 0 - \\eta ) , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} \\eta ''' = - \\frac { \\mu ( [ 0 , s ] ) } { \\phi _ { \\mu } ^ { 2 } } \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} Z ( \\lambda ) = 1 + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) Z ( \\nu ) \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} w _ i ^ 2 = \\alpha _ i \\in F , \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} \\tilde { W } = ( X _ 1 + i Y _ 1 ) ( X _ 1 - i Y _ 1 ) ^ { - 1 } , \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} \\langle \\omega _ 0 , T \\rangle = 1 , \\langle \\omega _ 0 , H X \\rangle = 0 . \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} A _ 0 u _ t + A u _ x = Q ( u ) , \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} N _ { \\Lambda } = \\sum _ { { k } } b _ { { k } } ^ { * } b _ { { k } } \\ , \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} \\tilde { \\epsilon } ( \\lambda ) = \\epsilon _ 0 ( \\lambda ) - \\sum _ { i a } s _ a \\epsilon _ 0 ( \\lambda _ { i a } ) F ( \\lambda _ { i a } | \\lambda ) + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\nu \\ , \\epsilon _ 0 ( \\nu ) \\partial _ { \\nu } F ( \\nu | \\lambda ) \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} M ( s ) = \\sum _ { m \\le M } v ( m ) m ^ { - s } \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} \\Bigg | \\int _ { \\R ^ N \\setminus B } \\frac { \\phi ( x + y ) } { | y | ^ { N + 2 \\sigma } } \\ d y \\Bigg | = \\Bigg | \\int _ { \\R ^ N \\setminus B } \\frac { ( - \\Delta ) ^ m f ( x + y ) } { | y | ^ { N + 2 \\sigma } } \\ d y \\Bigg | \\leq C \\frac { \\| f \\| _ { 2 m + 2 , s } } { 1 + | x | ^ { N + 2 s } } + C \\int _ { \\R ^ N \\setminus B } \\frac { | f ( x + y ) | } { | y | ^ { N + 2 \\sigma + 2 m } } \\ d y . \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} 0 < \\lambda _ j \\leq \\frac { \\beta _ j } { m _ j ( \\beta _ 1 , \\beta _ 2 , \\ldots , \\beta _ n ) \\| K ( 1 ) \\| _ { \\infty } } \\mbox { f o r a l l $ j = 1 , 2 , \\ldots , n $ , $ j \\not = i _ 0 $ , } \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} r = \\left \\{ \\begin{array} { l l } 0 , \\ ; \\hbox { i f } \\ ; ( x ^ \\alpha , p ^ \\delta ) \\in \\mathbb { P } & \\hbox { } \\\\ \\mathcal { L } _ { Z _ 0 } \\varrho ( x ^ \\alpha , 0 , . . . , 0 ) , \\ ; \\hbox { i f } \\ ; ( x ^ \\alpha ) \\in ( \\Omega _ T \\times \\mathbb { R } ^ n ) - \\mathbb { P } , & \\hbox { } \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} \\hat { \\mathbf { h } } _ { e f f } ^ { ( g ) } \\approx \\sum _ { l = 0 } ^ { L _ g - 1 } \\left ( \\underbrace { \\operatorname { p i n v } \\left \\{ \\mathbf { X } ^ { ( g ) } \\left [ \\mathbf { I } _ { K _ g } \\otimes \\mathbf { E } _ { L _ g , l } \\right ] \\right \\} } _ { \\textrm { ( t e m p o r a l ) c o r r e l a t o r } } \\otimes \\underbrace { \\left [ \\mathbf { S } _ D ^ { ( g ) } \\mathbf { E } _ { D , l } \\right ] ^ H } _ { \\textrm { p r e - b e a m f o r m e r } } \\right ) \\mathbf { y } \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{gather*} H _ 0 ( t , x , p ) : = - p ^ 2 - \\big ( t - x ^ 2 \\big ) p \\end{gather*}"}
-{"id": "4240.png", "formula": "\\begin{align*} x ( t ) ( s ) = \\left \\{ \\begin{matrix} x ( t ) & \\mid & s \\in ( - M / 2 , M / 2 ) \\\\ 0 & \\mid & \\mbox { e l s e } \\end{matrix} \\right . \\ , . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} d ^ 2 ( 4 u ^ 2 + 2 ( c - 4 ) r u - r ^ 2 ( 2 c - 5 ) ) = - 1 9 9 6 c + 4 0 0 8 . \\end{align*}"}
-{"id": "8265.png", "formula": "\\begin{align*} u ^ + = u ^ - \\ , ( = u ) , S _ 1 \\frac { \\partial u ^ + } { \\partial n } = S _ 2 \\frac { \\partial u ^ - } { \\partial n } \\ , ( = q ) \\quad \\mbox { o n } \\partial \\Omega , \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} \\tilde { U } ' ( \\hat { x } _ t ) = \\delta ^ i \\ , \\tilde { U } ' ( \\hat { x } _ { t + 1 } ) \\ , f ' ( \\bar { y } _ t ) , & & t = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} \\delta ( \\alpha \\wedge \\beta \\wedge \\gamma ) & = \\delta ( \\alpha \\wedge \\beta ) \\wedge \\gamma + ( - 1 ) ^ { | \\alpha | } \\alpha \\wedge \\delta ( \\beta \\wedge \\gamma ) + ( - 1 ) ^ { | \\beta | | \\gamma | } \\delta ( \\alpha \\wedge \\gamma ) \\wedge \\beta \\\\ & \\quad { } - \\delta ( \\alpha ) \\wedge \\beta \\wedge \\gamma - ( - 1 ) ^ { | \\alpha | } \\alpha \\wedge \\delta ( \\beta ) \\wedge \\gamma - ( - 1 ) ^ { | \\alpha | + | \\beta | } \\alpha \\wedge \\beta \\wedge \\delta ( \\gamma ) \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} S _ n : = ( X _ n , Y _ n , Z _ n ) : = \\Bigr ( \\frac { x _ 1 } { \\sqrt { n } } , \\frac { y _ 1 } { \\sqrt { n } } , \\frac { z _ 1 } { \\sqrt { n } } \\Bigr ) \\cdots \\Bigr ( \\frac { x _ n } { \\sqrt { n } } , \\frac { y _ n } { \\sqrt { n } } , \\frac { z _ n } { \\sqrt { n } } \\Bigr ) . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { g ( { \\bf R } ) } a _ n ( I _ d \\ : : W _ 2 ^ { \\bf R } ( \\Bbb T ^ d ) \\rightarrow L _ 2 ( \\Bbb T ^ d ) ) = ( \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) ) ^ { g ( { \\bf R } ) } , \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} \\phi _ { x } ( k ' ) = \\phi _ { j x } ( - k ' ) \\end{align*}"}
-{"id": "8530.png", "formula": "\\begin{align*} \\mathbf { r } = \\mathbf { S } \\times \\mathbf { h } + \\mathbf { C } \\times \\mathbf { g } + \\mathbf { w } \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} \\rho _ \\zeta ( z ) = 2 \\mathrm { R e } ( z _ 1 + Q _ \\zeta ( z ) ) + \\mathcal L _ \\zeta ( z ) + \\mathrm { h . o . t } \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} ^ { c } D ^ { q } \\mathbf { u } ( t ) = - \\tilde { A } \\mathbf { u } ( t ) + \\tilde { T } \\mathbf { \\tilde { g } } ( \\mathbf { u } ( t ) ) + \\mathbf { \\tilde { I } } . \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} \\Delta _ k : = 2 ^ { - p _ 0 } k - U _ k \\ge 0 . \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} & \\left | \\frac { \\lambda _ { + } } { \\lambda _ { + } - \\lambda _ { - } } + \\frac { 1 } { \\nu ^ { 2 } } | \\xi | ^ { 2 ( 1 - 2 \\sigma ) } \\right | \\\\ & = \\frac { 1 } { \\nu ^ { 2 } } | \\xi | ^ { 2 ( 1 - 2 \\sigma ) } \\left ( \\frac { - 2 } { \\left ( 1 + \\sqrt { 1 - \\frac { 4 } { \\nu ^ { 2 } } | \\xi | ^ { 2 ( 1 - 2 \\sigma ) } } \\right ) \\sqrt { 1 - \\frac { 4 } { \\nu ^ { 2 } } | \\xi | ^ { 2 ( 1 - 2 \\sigma ) } } } + 1 \\right ) \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{gather*} \\pi _ m ^ 2 [ i , j ] = \\pi _ m \\left ( [ i , j ] - \\sum _ { k \\not \\in \\{ i , j \\} } m _ k \\partial _ 2 ( [ i , j , k ] ) \\right ) = \\pi _ m [ i , j ] - \\sum _ { k \\not \\in \\{ i , j \\} } m _ k \\pi _ m \\partial _ 2 [ i , j , k ] = \\pi _ m [ i , j ] . \\end{gather*}"}
-{"id": "4252.png", "formula": "\\begin{align*} \\varphi ^ { ( i ) } : A \\to C ( \\R / M \\Z ) \\otimes A \\cong C ( \\R / M \\Z , A ) , i = 0 , 1 \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{gather*} { \\large ( } 1 - G ( h _ { + } ) { \\large ) } ^ { 2 } = \\\\ \\dfrac { 4 { \\large ( } 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v { \\large ) } ^ { 2 } } { { \\large ( } \\sqrt { s ^ { 2 } + t ^ { 2 } } \\sqrt { p _ { 1 } } + \\left \\vert 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v \\right \\vert { \\large ) } ^ { 2 } } \\end{gather*}"}
-{"id": "4167.png", "formula": "\\begin{align*} \\int _ { \\tau _ - ^ r ( z ) } ^ { \\tau _ + ^ r ( z ) } f ( X ( t , z ) ) \\ , d t = \\int _ { \\tau _ - ^ \\delta ( y ) } ^ { \\tau _ + ^ \\delta ( y ) } f ( X ( t , y ) ) \\ , d t \\ \\ \\ z \\in \\Omega ( r ^ 2 ) \\setminus \\overline B _ r . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} F ( s ; \\boldsymbol { a } , \\boldsymbol { z } ) : = \\prod _ { i = 1 } ^ l \\ ( F ( s ; a _ i , z _ i ) - 1 ) \\ ) . \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\varphi _ { \\xi ^ - ( \\omega ) ( u - \\omega _ 1 ) , \\xi ^ - ( \\omega ) ( v - \\omega _ 2 ) } '' ( 1 ) & = ( p - q ) \\| ( \\xi ^ - ( \\omega ) ( u - \\omega _ 1 ) , \\xi ^ - ( \\omega ) ( v - \\omega _ 2 ) ) \\| ^ p \\\\ & - 2 ( ( \\alpha + \\beta ) - q ) \\int _ \\Omega | \\xi ^ - ( \\omega ) ( u - \\omega _ 1 ) | ^ { \\alpha } | \\xi ^ - ( \\omega ) ( v - \\omega _ 2 ) | ^ \\beta d x < 0 , \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} z - T = K ( z ) ^ { - 1 } ( I - B ( z ) ) K ( z ) ^ { - 1 } \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} \\vec { X } _ { 1 1 } & = \\Bigg ( 0 , 1 , - \\frac { \\lambda } { m - n } \\bigg ( \\frac { 1 } { 1 + B ^ { - 1 } } \\bigg ) \\Bigg ) , \\vec { X } _ { 1 2 } ' = \\Bigg ( 1 \\ ; , \\ ; - \\lambda c - 2 \\lambda n \\frac { e } { 1 + B ^ { - 1 } } \\ ; , \\ ; 0 \\Bigg ) . \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} W ( z ) = \\zeta ( z ) ^ { - \\frac { c } { 2 } \\sigma _ 3 } S \\mathcal { P } ( z ) S ^ { - 1 } \\zeta ( z ) ^ { \\frac { c } { 2 } \\sigma _ 3 } , \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{align*} \\xi ( x ) & = b ( x ) + a ( x ) a ^ * ( x ) \\zeta ( x ) + \\int \\gamma ( x , y ) ( Y ( x , \\gamma ( x , y ) ) - 1 ) F ( \\dd y ) , \\end{align*}"}
-{"id": "7164.png", "formula": "\\begin{align*} \\theta ( m ) = \\sum _ { q \\mid m } \\xi _ q \\ge 0 \\ . \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} e _ n = \\sup \\ ! \\left \\{ \\sqrt { M ^ { - j } } \\left \\| U _ { n , M , Q } ^ { 0 } - u ^ { \\infty } \\right \\| _ { k , Q } \\colon k , j \\in \\N _ 0 , k + j + n = N \\right \\} . \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} E ( \\gamma ) = \\frac { 1 } { 2 } \\int _ { 0 } ^ { 1 } F ( \\gamma , \\dot { \\gamma } ) ^ { 2 } d t . \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} \\begin{aligned} & R i c ( X , Y ) = ( 2 n + 2 ) T ^ 0 ( X , Y ) + ( 4 n + 1 0 ) U ( X , Y ) + 2 ( n + 2 ) S g ( X , Y ) , \\\\ & T ( \\xi _ { i } , \\xi _ { j } ) = - S \\xi _ { k } - [ \\xi _ { i } , \\xi _ { j } ] _ { | H } , S = - g ( T ( \\xi _ 1 , \\xi _ 2 ) , \\xi _ 3 ) . \\end{aligned} \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} \\norm { u - u _ h } ^ 2 _ { 0 , \\Omega } = B _ h ( ( { \\bf u } - { \\bf u } _ h , { \\bf p } - { \\bf p } _ h ) , ( ( \\zeta , \\gamma ( \\zeta ) ) , ( \\xi , \\gamma ( \\xi ) ) ) ) . \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} \\| f \\| = \\Bigl ( \\| f \\| _ 1 ^ 2 + \\frac { 1 } { \\gamma } \\| f \\| _ 2 ^ 2 \\Bigr ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} f _ m ( x ) = f _ m ( x , \\zeta ) = \\sum _ { k = 1 } ^ m c _ k \\phi ( a _ k \\cdot x + b _ k ) , \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} z f ' ( z ) = \\frac { z } { 1 - z } ~ h ( z ) . \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} \\pi _ \\alpha \\cdot e _ 1 = \\sum _ { \\beta = \\alpha + \\square } \\pi _ { \\beta } \\end{align*}"}
-{"id": "7373.png", "formula": "\\begin{align*} F _ A = i \\mathrm { d i a g } \\left ( \\frac { \\vartheta _ a d r \\wedge ( d \\tau + \\omega ) } { 2 r ^ 2 } + \\epsilon \\frac { V \\vartheta _ a } { 2 } d V o l _ { S ^ 2 } \\right ) + O \\left ( \\frac { 1 } { r ^ 3 } \\right ) , \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} G = \\overline P _ 1 \\cup \\overline P _ 2 . \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} \\mathcal { D } ( c ' ) = \\bigcup _ { i \\in \\mathcal { I } _ n } \\mathcal { D } _ i ( c ' ) . \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} \\epsilon ( s , \\pi , \\tau , \\psi ) = \\epsilon ( s , \\pi _ 1 , \\tau _ 1 , \\psi _ 0 ) \\epsilon ( s , \\pi _ 2 , \\tau _ 2 , \\psi _ 0 ^ { - 1 } ) . \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} = 4 \\left ( \\partial _ 1 \\phi \\right ) ^ 2 - 2 \\int _ { \\mathbb { R } ^ n } \\textbf { f } \\ ; \\frac { | \\theta | \\sqrt { | \\Theta | } ( \\textbf { m } ^ 2 + \\Theta _ { a b } \\widetilde { p } ^ a \\widetilde { p } ^ b ) ^ 2 } { ( \\widetilde { p } ^ 1 ) ^ 2 } d \\widetilde { p } \\ ; ' \\textbf { ; } \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { L _ n } a _ k a _ { \\sigma ^ \\ast ( k ) } = \\min _ { \\sigma \\in \\mathcal { P } ( { \\bf D } _ n ) } \\sum _ { k = 1 } ^ { L _ n } a _ k a _ { \\sigma ^ \\ast ( k ) } , \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} \\tau _ i = \\frac 1 4 \\frac { t _ i } { 1 - t _ i } , u _ i = \\frac { x _ i - r } { \\sqrt 2 ( 1 - t _ i ) } \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} \\alpha \\wedge \\beta = ( - 1 ) ^ { \\langle \\deg ( \\alpha ) , \\deg ( \\beta ) \\rangle + | \\alpha | | \\beta | } \\ , \\beta \\wedge \\alpha \\ ; . \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} \\begin{aligned} D _ t u + D _ x \\left ( \\frac { 1 } { 2 } u ^ 2 + u _ { x x } \\right ) & = F , \\\\ D _ t \\left ( \\frac { 1 } { 2 } u ^ 2 \\right ) + D _ x \\left ( \\frac { 1 } { 3 } u ^ 3 + u u _ { x x } - \\frac { 1 } { 2 } u _ x ^ 2 \\right ) & = u F , \\\\ D _ t \\left ( \\frac { 1 } { 3 } u ^ 3 - u _ x ^ 2 \\right ) + D _ x \\left ( \\frac { 1 } { 4 } u ^ 4 + u ^ 2 u _ { x x } + 2 u _ x u _ t + u _ { x x } ^ 2 \\right ) & = \\left ( u ^ 2 + 2 u _ { x x } \\right ) F , \\end{aligned} \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} T _ k : = A _ k + B _ k = \\begin{pmatrix} \\mu _ { 2 k - 1 } + d _ k & 0 \\\\ 0 & \\mu _ { 2 k - 1 } + d _ k \\end{pmatrix} + d _ k \\begin{pmatrix} - 1 & t _ k \\\\ - t _ k & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} r ( t ) - a ( t ) = \\int \\limits _ { 0 } ^ { t } Q ( t , \\tau ) \\left [ r ( \\tau ) - a ( \\tau ) \\right ] d \\tau . \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} A ( v ' , s '' ) & = ( A | _ { V ' } ( v ' ) , A | _ { V '' } ( s '' ) ) \\\\ & \\in V ' \\oplus S '' , \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} \\Phi ( x ) = \\sup _ { p \\ , , p \\leq \\Phi } p ( x ) . \\end{align*}"}
-{"id": "5578.png", "formula": "\\begin{align*} \\psi _ x = & \\left ( \\begin{matrix} - i z & u \\\\ \\bar u & i z \\end{matrix} \\right ) \\psi \\\\ \\psi _ t = & i \\left ( \\begin{matrix} - [ 2 z ^ 2 + | u | ^ 2 ] & - 2 i z u + u _ x \\\\ - 2 i z \\bar u - \\bar u _ x & 2 z ^ 2 + | u | ^ 2 \\end{matrix} \\right ) \\psi \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} ( \\psi \\rtimes G ) ( \\iota ^ { C \\alpha } ( x ) \\lambda ^ { C \\alpha } ( f ) ) = \\iota ^ \\beta ( \\psi ( x ) ) \\lambda ^ \\beta ( f ) \\ ; ~ x \\in C _ 0 ( ( 0 , 1 ] , A ) , f \\in C _ c ( G ) . \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} \\mathbb { P } [ X _ v = 1 , d _ { Z ' } ( v ) < r ] & = \\binom { t _ 1 } { r - 1 } p ^ { r - 1 } ( 1 - p ) ^ { t _ 1 - r + 1 } \\\\ & = ( 1 + O ( t _ 1 ^ { - 1 } ) ) \\frac { t _ 1 ^ { r - 1 } } { ( r - 1 ) ! } p ^ { r - 1 } ( 1 - p ) ^ { t _ 1 - r + 1 } . \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\Lambda _ L \\circ \\phi _ B ( \\psi ( \\alpha ) ) = \\Lambda _ K \\circ N m ( \\phi ) _ B ( \\alpha ) . \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} W \\bigl ( y \\bigl | x ( s , y ) , s \\bigr ) = 0 . \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} \\xi _ { \\Gamma , n } = ( - 1 ) ^ n \\det ( \\rho _ { s , \\Gamma } ) ^ { - n - 1 } \\tilde \\xi _ { \\Gamma , n } \\ , . \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} \\Lambda _ R ( f ( \\rho s , \\cdot ) ) = \\sup _ { g \\in \\mathcal { F } _ \\ell ( B _ R ) } \\left \\{ \\int _ { B _ R } f ( \\rho s , x ) g ^ 2 ( x ) d x - \\frac 1 2 \\int _ { B _ R } | \\nabla g ( x ) | ^ 2 d x \\right \\} , \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} \\Lambda ( \\mathbf { \\tilde Z } ) = \\frac { f _ { \\mathbf { \\tilde Z } | H _ 1 } ( \\mathbf { \\tilde Z } | H _ 1 ) } { f _ { \\mathbf { \\tilde Z } | H _ 0 } ( \\mathbf { \\tilde Z } | H _ 0 ) } \\mathop { \\gtrless } _ { H _ 0 } ^ { H _ 1 } \\gamma , \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} | V - \\theta _ { 1 } ^ { ( 2 ) } U | & \\geq \\frac { 1 } { 2 } ( | V - \\theta _ { 1 } ^ { ( 1 ) } U | + | V - \\theta _ { 1 } ^ { ( 2 ) } U | ) \\\\ & \\geq \\frac { 1 } { 2 } | U | | \\theta _ { 1 } ^ { ( 1 ) } - \\theta _ { 1 } ^ { ( 2 ) } | = | U | \\left \\vert \\sqrt { \\frac { c + 2 } { c } } \\right \\vert \\geq | U | \\sqrt { \\frac { | c | - 2 } { | c | } } , \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} u ( t , x ) = p _ t * u _ 0 ( x ) + \\int _ 0 ^ t \\int _ { \\mathbb { R } ^ \\ell } p _ { t - s } ( x - y ) u ( { s , y } ) \\ , \\delta W _ { s , y } . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} g ( U , V ) = \\mathcal L _ x ( U , V ) , g ( U , T ) = g ( T , U ) = 0 , g ( T , T ) = 1 , \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\sigma = \\sigma ( \\gamma , \\gamma _ t ) = \\varphi ( \\gamma ) \\gamma _ t ^ n \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} F _ { k } ( \\zeta ) = I + \\frac { c _ k } { \\zeta ^ k } \\begin{bmatrix} 0 & 1 \\\\ 0 & 0 \\end{bmatrix} , c _ k = \\frac { 1 } { 2 \\mathrm { i } \\pi } \\int _ { \\cal L } \\frac { s ^ { k - 1 } e ^ { s } } { s ^ { c } } d s = \\frac { \\sin ( c \\pi ) \\ , \\Gamma ( k - c ) } { \\pi ( - 1 ) ^ { k - 1 } } . \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} \\varphi ^ \\star ( \\eta ) - M _ 1 = \\kappa ' _ 1 e ^ { - \\eta } \\vec { X } _ { 1 1 } + \\kappa ' _ 2 \\eta e ^ { - \\eta } \\vec { X } _ { 1 2 } ' + \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} X = \\xi ^ i ( x ) \\partial _ { x ^ i } + \\phi ^ { \\alpha } ( x , n , [ u ] ) \\partial _ { u ^ { \\alpha } } . \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} \\partial ^ 2 _ 1 \\sigma + ( 1 + h _ 0 '' ( 1 + \\partial _ 2 \\sigma ) ) \\partial ^ 2 _ 2 \\sigma = 0 \\textrm { a . e . \\ @ i n } \\ Q . \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} R _ { \\mathrm { C E G } } \\left ( P _ { T , X | S } \\right ) = \\min \\Big \\{ I _ P ( T ; Y | S ) , H _ P ( S | T , Z ) \\Big \\} , \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} \\phi ^ * ( p ) = \\sup _ x p x - \\phi ( x ) \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} \\phi _ 0 ( x , y ) = a x ^ 3 + b x ^ 2 y + c x y ^ 2 + d y ^ 3 \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} L z ( x ) = - \\sum _ { i , j = 1 } ^ m a _ { i j } ( x ) \\frac { \\partial ^ 2 z } { \\partial x _ i \\partial x _ j } ( x ) + \\sum _ { i = 1 } ^ m a _ { i } ( x ) \\frac { \\partial z } { \\partial x _ i } ( x ) + a ( x ) z ( x ) , \\mbox { f o r $ x \\in \\Omega $ , } \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} \\zeta _ n \\geq \\zeta ' _ n , \\ n = 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} \\sum _ { \\tau = 1 } ^ { t } X _ { \\tau } \\leq \\sqrt { 3 \\log ( 1 / \\delta ) \\sum _ { \\tau = 1 } ^ { t } 1 6 G ^ 2 D ^ 2 } = 4 G D \\sqrt { 3 \\log ( 1 / \\delta ) t } . \\end{align*}"}
-{"id": "7530.png", "formula": "\\begin{align*} F ( \\tau ) | \\gamma = \\chi ( \\det D _ \\gamma ) F ( \\tau ) . \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\mathbf { R _ e } = \\mathbf { R _ e } ^ { m m s e } + \\left ( \\mathbf { W } ^ { ( g ) } - \\mathbf { W } ^ { ( g ) } _ { m m s e } \\right ) ^ H \\mathbf { R } _ { \\mathbf { y } } \\left ( \\mathbf { W } ^ { ( g ) } - \\mathbf { W } ^ { ( g ) } _ { m m s e } \\right ) \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{gather*} p _ { t t } = 4 p ^ 2 + 2 p t + \\frac { p ^ 2 _ t } { 2 p } , \\end{gather*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\bar { R } _ { \\alpha } R _ { \\alpha } - i a _ { \\alpha } = 2 \\bar \\P [ R _ \\alpha \\bar R _ \\alpha ] + i \\Im \\P [ R \\bar R _ { \\alpha \\alpha } ] . \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} 2 \\tau _ { 2 } ( \\sqrt { a } ) = \\tau _ { \\infty } ( a ) , \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{gather*} L _ 1 = \\frac 3 2 C \\wedge d A , d L _ 1 = \\delta _ Q L . \\end{gather*}"}
-{"id": "5664.png", "formula": "\\begin{align*} f _ { L } ( x ) = \\sum _ { j = 2 } ^ { k _ 1 } \\left \\{ \\left \\{ p _ { j } x + \\xi _ { j } \\right \\} + 2 L \\hat { \\theta } _ { j } \\right \\} + \\sum _ { j = k _ { 1 } + 1 } ^ { k } \\left \\{ \\left \\{ p _ { j } x \\right \\} + 2 L \\hat { \\theta } _ { j } \\right \\} , \\ \\forall x \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} m _ M ( I _ o ^ { - 1 } J _ o ) = m _ M ( I _ o ) + m _ M ( J _ o ) - m _ M ( \\{ e _ M \\} ) . \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} \\int _ { E _ { p + q } } \\prod _ { j = 1 } ^ p \\frac { d t _ j } { 1 - t _ j } \\prod _ { i = p + 1 } ^ { p + q } \\frac { d t _ i } { t _ i } . \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} ( \\delta + \\pi ) \\mu ( d f , \\pi ) & = - ( \\delta + \\pi ) \\mu ( \\delta f , \\pi ) \\\\ & = - ( \\delta + \\pi ) \\phi ( \\delta f ) \\\\ & = - \\phi ( d \\delta f ) - \\pi \\phi ( \\delta f ) \\\\ & = \\mu ( - d \\delta f , \\pi ) \\\\ & = \\mu ( ( \\delta - d ) d f , \\pi ) , \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} & \\mathbb { P } \\left ( \\varphi _ { t _ 1 + t _ 0 } ( \\cdot , A ) \\subset B \\left ( x _ 1 , \\frac { 2 } { 3 } r _ 0 \\right ) \\right ) \\\\ & \\geq \\mathbb { P } \\left ( A \\subset B ( x _ 0 , r _ 0 ) \\right ) \\cdot \\mathbb { P } \\left ( \\varphi _ { t _ 1 + t _ 0 } ( \\cdot , B ( x _ 0 , r _ 0 ) ) \\subset B \\left ( x _ 1 , \\frac { 2 } { 3 } r _ 0 \\right ) \\right ) > 0 . \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} T = \\int _ \\Gamma T _ \\gamma \\dd \\pi ( \\gamma ) \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial d _ j } & = 2 \\sum _ { i = j } ^ { n - 1 } ( \\theta _ 1 + d _ 1 + \\dots + d _ i ) - 2 \\sum _ { i = 0 } ^ { n - 1 } ( \\theta _ 1 + d _ 1 + \\dots + d _ i ) \\frac { n - j } { n } , \\\\ \\frac { \\partial ^ 2 f } { \\partial d _ j \\partial d _ k } & = 2 \\sum _ { i = \\max \\{ j , k \\} } ^ { n - 1 } 1 - 2 \\sum _ { i = k } ^ { n - 1 } \\frac { n - j } { n } - 2 \\sum _ { i = j } ^ { n - 1 } \\frac { n - k } { n } + 2 \\sum _ { i = 0 } ^ { n - 1 } \\frac { n - k } { n } \\frac { n - j } { n } \\\\ & = 2 ( \\min \\{ j , k \\} - \\frac { j k } { n } ) \\ge 0 . \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{align*} g = V d x ^ 2 + \\frac { \\varpi ^ 2 } { V } . \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} & \\Delta v + k _ 2 ^ 2 v = 0 \\ \\mbox { i n } \\ \\mathbb { R } ^ 2 \\setminus \\overline { \\Omega _ 2 } , \\Delta v + k _ 1 ^ 2 v = 0 \\ \\mbox { i n } \\ { \\Omega _ 2 } , \\\\ & v ^ { - } = v ^ { + } , S _ 1 \\frac { \\partial v ^ - } { \\partial n } = S _ 2 \\frac { \\partial v ^ + } { \\partial n } \\ \\mbox { o n } \\partial \\Omega \\\\ & \\mbox { o u t g o i n g r a d i a t i o n c o n d i t i o n w i t h } k _ 2 \\mbox { i n } \\mathbb { R } ^ 2 \\setminus \\overline { \\Omega _ 2 } . \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\| u ^ h _ t - u ^ h _ s \\| ^ 2 _ { m - 1 } & = 2 \\int _ s ^ t \\left ( u ^ h _ r - u ^ h _ s , ( L ^ h _ r + J ^ h ) u _ r + f _ r \\right ) _ { m - 1 } d r \\\\ & \\leq N \\int _ s ^ t \\| u ^ h _ r - u ^ h _ s \\| ^ 2 _ { H ^ m } + \\| ( L ^ h _ r + J ^ h ) u ^ h _ r \\| ^ 2 _ { H ^ { m - 2 } } + \\| f _ r \\| ^ 2 _ { H ^ { m - 2 } } d r \\\\ & \\leq N \\int _ s ^ t \\sup _ { t ' \\leq T } \\| u ^ h _ { t ' } \\| ^ 2 _ { H ^ m } + \\| f _ r \\| ^ 2 _ { H ^ { m - 2 } } d r \\\\ & \\leq N ( \\mathcal { K } _ m ^ 2 + K ' ) ( t - s ) , \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} f _ p ( x | \\eta ) = \\dfrac { \\eta ^ x } { x ! } e ^ { - \\eta } . \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} f _ 0 ( \\theta ) = C \\left | \\sum _ { j = 0 } ^ { \\infty } c _ j e ^ { - i j \\theta } \\right | . \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} \\beta _ 1 ^ h ( x ) & = \\frac { \\hat y _ 1 ^ h ( x ) - x _ 1 } { h ^ 2 } - u ^ h ( x _ 1 ) + x _ 2 \\frac { R ^ h _ { 2 1 } ( x _ 1 ) } { h } + x _ 3 \\frac { R ^ h _ { 3 1 } ( x _ 1 ) } { h } \\ , ; \\\\ \\beta _ i ^ h ( x ) & = \\frac { 1 } { h } \\left ( \\frac { \\hat y _ i ^ h ( x ) - h x _ i } { h } - v _ i ^ h ( x _ 1 ) - h w ^ h ( x _ 1 ) x _ i ^ \\bot \\right ) \\ , , i = 2 , 3 \\ , . \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} q ( G ) ( z ) & = k ^ { - n } \\sum _ { \\phi : V \\to [ k ] } \\prod _ { e = \\{ u , v \\} \\in E } ( J + ( z ( A ^ { e } - J ) ) ) _ { \\phi ( u ) , \\phi ( v ) } \\\\ & = k ^ { - n } \\sum _ { i = 0 } ^ { | E | } z ^ i \\bigg ( \\sum _ { \\substack { F \\subseteq E \\\\ | F | = i } } \\sum _ { \\phi : V \\to [ k ] } \\prod _ { e = \\{ u , v \\} \\in F } ( A ^ { e } - J ) _ { \\phi ( u ) , \\phi ( v ) } \\bigg ) . \\end{align*}"}
-{"id": "6026.png", "formula": "\\begin{align*} \\Delta : = ( a _ { } , b _ { } , c _ { } , d _ { } ) \\in \\mathcal { C } _ { k } \\rightarrow \\Delta ( ) = ( b _ { } , a _ { } , d _ { } , c _ { } ) \\in \\mathcal { C } _ { k } , \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} E ^ + \\oplus G & \\cong F ^ + \\oplus H , \\\\ E ^ - \\oplus G & \\cong F ^ - \\oplus H . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} \\Omega ( f ( z ) , g ( z ) ) = _ { z = 0 } ( f ( - z ) , g ( z ) ) _ E d z \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} \\mathrm { d } \\bar { X } ^ { ( n ) } _ i ( t ) = b \\big ( R ( t , \\bar { X } ^ { ( n ) } _ i ( t ) ) \\big ) \\ , \\mathrm { d } t + \\sigma \\big ( R ( t , \\bar { X } ^ { ( n ) } _ i ( t ) ) \\big ) \\ , \\mathrm { d } B ^ { ( n ) } _ i ( t ) , \\ ; \\ ; \\ ; \\bar { X } ^ { ( n ) } _ i ( 0 ) = X ^ { ( n ) } _ i ( 0 ) , \\\\ i = 1 , \\ , 2 , \\ , \\ldots , \\ , n , \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} \\tilde { c } _ 4 \\equiv 2 9 a \\pmod { 3 2 } b = - \\frac { 2 \\tilde { c } _ 6 } { 2 7 } , \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} \\theta \\notin \\Theta \\triangleq \\cup _ { j = 1 } ^ m \\theta _ j + \\Z \\alpha + \\Z \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} I ( p ) = \\begin{cases} \\frac { 2 \\pi } { 3 } ( 3 - | p | ^ 2 ) & \\hbox { i f } \\ | p | \\leq 1 , \\\\ \\frac { 4 \\pi } { 3 | p | } & \\hbox { i f } \\ | p | > 1 . \\end{cases} \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{align*} \\# = 2 ^ h > ( h + 1 ) ^ c \\geq | \\mathcal { M } _ { h , c } | . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} V ( t , x ) = \\int _ 0 ^ t \\int _ { \\R } G _ { t - s } ( x , y ) W _ 1 ( d s , d y ) \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} ( \\tilde { H } _ c ( x ) u ) _ m = \\sum _ { m ^ { \\prime } } d _ { m ^ { \\prime } } ( c , v ) ( x ) u _ { m - m ^ { \\prime } } , \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} k _ { \\epsilon } ( z , z ' ) : = \\left \\{ \\begin{array} { c c } 1 & \\mbox { i f $ d ( z , z ' ) < \\epsilon $ } \\\\ 0 & \\mbox { i f $ d ( z , z ' ) \\geq \\epsilon $ } \\end{array} \\right . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} \\Delta E ( \\lambda _ p , \\lambda _ h ) = \\epsilon _ 0 ( \\lambda _ p ) - \\epsilon _ 0 ( \\lambda _ h ) + \\sum _ j [ \\epsilon _ 0 ( \\tilde { \\lambda } _ j ) - \\epsilon _ 0 ( \\lambda _ j ) ] \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} { Y } _ { t n + i } = \\begin{cases} \\frac { f _ i } { d _ { \\rm a , b } ^ { \\alpha / 2 } } + \\frac { g _ { t n + i } } { d _ { \\rm j , b } ^ { \\alpha / 2 } } + { N _ { t n + i } ^ { \\rm ( b ) } } , & \\ ! \\ ! \\mbox { A l i c e t r a n s m i t s a n d } ~ t = 0 \\\\ \\frac { g _ { t n + i } } { d _ { \\rm j , b } ^ { \\alpha / 2 } } + { N _ { t n + i } ^ { \\rm ( b ) } } , & \\ ! \\ ! \\mbox { e l s e , } \\end{cases} \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} \\sigma ( y ; \\alpha , \\beta ) : = \\int _ { 0 } ^ { \\infty } ( u + 1 ) ^ { \\alpha - 1 } u ^ { \\beta - 1 } e ^ { - y u } d u , \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} & \\mathcal { E } _ k ( u ) : = \\displaystyle { \\int _ 0 ^ L } \\left \\{ \\frac { 1 } { 2 } \\left ( R _ k f ^ 2 + S _ k \\mu ( f + g ) ^ 2 \\right ) + \\mu \\Phi ( \\Gamma ) \\right \\} \\ , d x , \\mbox { f o r } k = 1 , \\\\ [ 1 0 p t ] & \\mathcal { E } _ k ( u ) : = \\displaystyle { \\int _ 0 ^ L } \\left \\{ - \\frac { 1 } { 2 } \\left ( R _ k | \\partial _ x f | ^ 2 + S _ k \\mu | \\partial _ x ( f + g ) | ^ 2 \\right ) + \\mu \\Phi ( \\Gamma ) \\right \\} \\ , d x , \\mbox { f o r } k = 3 \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} \\alpha _ { m , n } ( \\pi ) = \\begin{cases} \\hat { \\alpha } & \\ , \\ , n \\in \\mathcal { N } _ 1 , \\\\ \\hat { \\alpha } ^ \\frac { 1 } { \\log \\left ( ( 1 - \\beta ) f ( k _ { m , n } ) \\right ) } , & \\ , \\ , n \\in \\mathcal { N } _ 2 . \\end{cases} \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} E \\ ; : \\ ; y ^ 2 = x ^ 3 + a x ^ 2 + b x , \\Delta = \\Delta _ m = 2 ^ 4 b ^ 2 ( a ^ 2 - 4 b ) . \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} & \\mu _ p ( X _ i = \\epsilon _ i , \\ ; 0 \\le i \\le 3 ) = \\ 1 _ { \\{ \\epsilon _ 0 + \\epsilon _ 1 + \\epsilon _ 2 + \\epsilon _ 3 \\} } \\mu _ p ( X _ i = \\epsilon _ i , \\ ; 1 \\le i \\le 3 ) . \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} \\beta _ { n , q } ( x ) = \\sum _ { l = 0 } ^ n { n \\choose l } q ^ { l x } \\beta _ { l , q } [ x ] _ q ^ { n - l } , ( \\ , \\ , [ 1 , 1 3 ] ) . \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} I ( P _ 1 \\diamond P _ 2 ) = I P _ 1 \\otimes I P _ 2 . \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} F ( s ; \\mathbf { a } , \\mathbf { z } ) & = ( L ( s , \\chi _ 0 ) ^ { \\frac { z _ 1 + \\cdots + z _ l } { \\phi ( q ) } } \\prod _ { \\chi \\neq \\chi _ 0 } ( L ( s , \\chi ) ) ^ { \\frac { \\bar { \\chi } ( b _ 1 ) z _ 1 + \\cdots + \\bar { \\chi } ( b _ l ) z _ l } { \\phi ( q ) } } \\prod _ { j = 1 } ^ l G _ 1 ( s ; b _ j , z _ j ) \\\\ & = ( \\zeta ( s ) ) ^ { \\frac { z _ 1 + \\cdots + z _ l } { \\phi ( q ) } } H ( s ; \\mathbf { a } , \\mathbf { z } ) , \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} \\tilde { \\eta } \\left ( t \\right ) = \\begin{cases} \\eta \\left ( t \\right ) & t \\notin \\left [ a , b \\right ] \\\\ \\xi \\left ( t \\right ) & t \\in \\left [ a , b \\right ] \\end{cases} . \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} \\Omega ( a ) = \\sum _ { i = 1 } ^ K \\gamma _ i \\nu _ i ( a ) , \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} [ D ( 2 , 3 s + r + 2 p ) : D ( 3 , 3 s + r ) ] _ q & = q ^ { 4 p } [ D ( 2 , 3 ( s - 1 ) + r + 2 p ) : D ( 3 , 3 ( s - 1 ) + r ) ] _ q & \\\\ & + q ^ { 6 s + 2 r + 4 p - 2 } [ D ( 2 , 3 s + r + 2 p - 2 ) : D ( 3 , 3 s + r ) ] _ q , \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{gather*} \\left ( J ( h _ { + } \\allowbreak ) + \\sqrt { M \\left ( h _ { + } \\right ) } \\right ) ^ { 2 } = \\dfrac { 4 p _ { 1 } \\left ( s - v \\right ) ^ { 2 } ( s - 2 h _ { + } ) ^ { 2 } } { s ^ { 4 } } \\times \\\\ \\left ( \\sqrt { p _ { 1 } } + \\dfrac { \\allowbreak \\left \\vert 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v \\right \\vert } { \\sqrt { s ^ { 2 } + t ^ { 2 } } } \\right ) ^ { 2 } \\end{gather*}"}
-{"id": "2749.png", "formula": "\\begin{align*} \\mathrm { c m } ( x , y ) + \\mathrm { c m } ( x , z ) = [ y , b ( x ) ] + [ z , b ( x ) ] = [ x , b ( x ) ] = 1 , \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} \\overline B ( s ) = \\sum _ n b ^ * \\left ( \\frac { \\log { n } } { \\log { N } } \\right ) \\lambda ( n ) n ^ { - s } \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} \\int _ 0 ^ { a } V _ s ( v ) \\ d v = \\frac { 2 } { 2 s - N } \\frac { a ^ { s - 1 } } { ( a + 1 ) ^ { \\frac { N } { 2 } - 1 } } - \\frac { 2 ( s - 1 ) } { 2 s - N } \\int _ 0 ^ { a } V _ { s - 1 } ( v ) \\ d v . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} H ( V _ { n - m _ { n } + 1 } ^ { n } | V _ { 1 } ^ { n - m _ { n } } ) & \\geq m _ { n } - \\frac { 4 m _ { n } ^ { 2 } } { \\sqrt { \\pi ( n - m _ { n } ) } } \\\\ & = m _ { n } - o ( 1 ) . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} \\Phi _ j ( x _ 1 , y _ 1 , \\ldots , x _ n , y _ n , z ) = ( x _ 1 ' , y _ 1 ' , \\ldots , x _ n ' , y _ n ' , z ' ) , \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\lim _ { G \\to G _ 0 } f ( x ; G ) = f ( x ; G _ 0 ) . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } ( n + d i m ( Z _ { \\mathfrak { g } } ( e ) ) ) = \\frac { 1 } { 2 } ( 2 + 3 ^ 2 + 2 ^ 2 - 1 ) = 4 . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} v - e + f = 2 . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} \\cos ^ 3 \\alpha { \\bf { H } } - \\beta ( J ( J \\nabla \\cos \\alpha ) ^ \\top ) ^ \\bot = 0 , \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} & T = \\frac { \\partial } { \\partial \\theta } \\\\ & Z _ j = \\frac { \\partial } { \\partial z _ j } + i \\frac { \\partial \\varphi ( z ) } { \\partial z _ j } \\frac { \\partial } { \\partial \\theta } , j = 1 , \\ldots , n - 1 \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\frac { h _ { \\rm i n } ( R ) } { g _ { \\rm i n } ( R ) } = \\frac { h _ { \\rm o u t } ( R ) } { g _ { \\rm o u t } ( R ) } \\ ; . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} W ( z ) : = \\zeta ( z ) ^ { - { c } \\sigma _ 3 } S \\cdot \\mathcal { P } ( z ) \\cdot { T ( \\zeta ( z ) ) ^ { - 1 } \\ , S ^ { - 1 } } , \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} \\phi = \\psi _ { g , R _ 2 \\ominus g ( R _ 1 ) } \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} \\Phi ( x ) = \\sup _ { p ( x _ i ) \\leq \\Phi ( x _ i ) , p \\in \\partial \\Phi ( K ) , x _ i \\in \\{ f \\leq 1 \\} } p ( x ) , \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\chi _ { \\nu } ( z _ 1 , z _ 1 , \\dots , , z _ k , z _ k ) = \\prod z _ i ^ { n _ i \\nu _ i } \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} \\langle \\mu _ i , \\psi \\rangle = \\frac { 1 } { | N _ G ( P ) | } \\left ( q ( q - 1 ) \\frac { 3 ^ k ( q - 1 ) } { 2 } - q ( q - 1 ) \\mu _ i ( X ) \\right ) = 0 \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} ( a , K _ { \\check \\varphi } ( x ) ) _ \\nu = ( K _ { \\check \\varphi } ^ * ( a ) , x ) _ { \\hat \\psi } , ( a \\in \\pi _ \\nu ( \\tilde M ( \\hat G , \\varphi ) ) '' , \\ , x \\in c _ { 0 0 } ( \\hat G ) ) , \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} \\mathcal { W } : = \\sum _ { i = 1 } ^ { \\nu } d \\theta _ i \\wedge d y _ i + \\frac { 1 } { 2 } \\sum _ { j \\in S ^ c } \\frac { 1 } { \\mathrm { i } j } \\ , d z _ j \\wedge d z _ { - j } = \\left ( \\sum _ { i = 1 } ^ { \\nu } d \\theta _ i \\wedge d y _ i \\right ) \\oplus \\Omega _ { S ^ { \\perp } } = d \\Lambda , \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} f _ 0 ( z ) : = \\frac { 1 } { z } \\ , \\frac { F _ * ( z ) - F _ * ( 0 ) } { F _ * ( z ) + F _ * ( 0 ) } \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{align*} u = \\allowbreak \\dfrac { v t - w s } { s } \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} - j \\left [ \\frac { 1 } { k \\ell _ { \\lambda } } - 1 \\right ] = - \\breve { n } \\breve { n } \\in \\mathbb { N } \\ ; . \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} H _ k ( z ) = \\left ( N ^ { \\frac { c } { 2 } } \\eta ( z ) \\right ) ^ { - \\sigma _ 3 } R _ k ( z ) \\widetilde { { F } } ^ { - 1 } _ k ( z ) \\left ( N ^ { \\frac { c } { 2 } } \\eta ( z ) \\right ) ^ { \\sigma _ 3 } H _ { k - 1 } ( z ) . \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} d ^ * ( A _ { x _ o } ) \\geq \\lim _ k \\frac { | A _ { x _ o } \\cap F _ { n _ k } g _ { n _ k } | } { | F _ { n _ k } | } = \\mu ( A ) , \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} T ( z ) & = \\exp \\left ( \\frac { N } { 2 } ( - 1 ) ^ \\nu \\phi _ A ( z ) \\sigma _ 3 \\right ) = \\exp \\left [ \\frac { ( - 1 ) ^ { \\nu } } { 2 } \\left ( \\zeta ( z ) ^ 2 + 2 x \\zeta ( z ) + \\ell _ x \\right ) \\sigma _ 3 \\right ] , \\\\ \\ell _ x & = x ^ 2 + N \\phi _ { A } ( 1 / a ) , S = S ( z ) = \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} \\cdot \\begin{bmatrix} 0 & 1 \\\\ - 1 & 0 \\end{bmatrix} ^ { \\nu } , \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} \\kappa = \\frac { \\varepsilon + \\delta } { 2 } . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} { \\cal F } ( \\zeta ) : = { \\cal W } ( \\zeta ) \\ , \\zeta ^ { { c } \\sigma _ 3 } e ^ { \\frac { \\zeta ^ 2 } { 4 } \\sigma _ 3 } = I + \\frac { C _ 1 } { \\zeta } + \\frac { C _ 2 } { \\zeta ^ { 2 } } + { \\cal O } \\left ( \\frac { 1 } { \\zeta ^ { 3 } } \\right ) \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} S _ n = \\sqrt { \\frac { \\sum \\limits _ { k = 1 } ^ { M _ n } \\left ( \\tau _ { n k } - \\bar { \\tau } _ n - \\tilde { \\tau } _ n \\right ) ^ 2 P _ { n k } } { \\sum \\limits _ { k = 1 } ^ { M _ n } P _ { n k } } } \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} \\sup _ { B _ { \\sigma _ i } ( y _ i ) \\cap \\Sigma _ { \\beta _ i } } | \\textbf { A } _ i | ^ 2 \\leq 4 | \\textbf { A } _ i | ^ 2 ( y _ i ) = 4 \\sigma _ i ^ { - 2 } . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} r ^ 2 \\dot { \\rho } ^ 2 + \\frac { \\omega ^ 2 } { \\rho ^ 2 } - \\rho ^ 2 = - a ^ 2 + \\frac { \\omega ^ 2 } { a ^ 2 } \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} \\varphi _ p ( 1 - s ) \\cdot \\psi _ p ( s ) = \\left ( \\prod _ { j = 1 } ^ g \\frac { \\Gamma ( 2 s - 1 ) ) } { \\Gamma ( s - p _ j / 2 ) \\Gamma ( s + p _ j / 2 ) } \\right ) . \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} \\begin{cases} \\tilde z _ k = z _ k , \\ \\ k = 1 , \\cdots , n - 1 , \\\\ \\tilde z _ n = z _ n + \\sum _ { j = 1 } ^ M \\frac { 1 } { M } Q f ^ { N _ j } \\end{cases} \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} U = \\left \\{ u _ t : = \\begin{pmatrix} 1 & t \\\\ 0 & 1 \\end{pmatrix} : t \\in \\mathbb { R } \\right \\} \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} \\mathcal { L } _ \\alpha ( \\varphi ) ( x , t ) = \\Gamma ( \\alpha + 1 ) \\ , \\emph { P . V . } \\int _ { \\mathcal { S } _ \\Omega ( x ) } \\frac { \\varphi ( y , t ) - \\varphi ( x , t ) } { | y - x | ^ { d + \\alpha } } d y \\ , . \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ K ' \\right ] & \\stackrel { ( a ) } { = } \\lambda _ { \\mathrm { u } } \\mathbb { E } \\left [ \\left | \\mathcal { C } _ { i } ^ { \\mathrm { E } } \\right | \\right ] - \\lambda _ { \\mathrm { u } } \\mathbb { E } \\left [ \\left | \\mathcal { C } _ { i } \\right | \\right ] = K \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} w _ t = H w , \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} ( b \\alpha - e \\beta ) ( b \\alpha + e \\beta ) = 0 \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\Psi ( s ) = \\int _ 0 ^ s \\psi ( t ) \\ , d t , \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( d ' ( e ) = d \\right ) = \\frac { { K \\choose d } { N - K \\choose t - d } } { { N \\choose t } } , \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} C ^ h ( \\xi , 0 ) = \\frac { i \\xi ^ 2 J ' ( \\xi ) } { \\Lambda ( \\xi ) } . \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{align*} A ( \\varphi ) : h ( x ) = \\sum _ { j \\in \\mathbb { Z } } h _ j \\ , e ^ { \\mathrm { i } \\ , j \\ , x } \\mapsto A ( \\varphi ) h ( x ) = \\sum _ { j _ 1 , j _ 2 \\in \\mathbb { Z } } A _ { j _ 1 } ^ { j _ 2 } ( \\varphi ) h _ { j _ 2 } \\ , e ^ { \\mathrm { i } \\ , j _ 1 \\ , x } . \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ d k a _ k z ^ { k - 1 } = - \\sum _ { i = 0 } ^ d a _ i z ^ i \\sum _ { j = 1 } ^ \\infty p _ j z ^ { j - 1 } . \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} \\frac { s ( \\theta ) } { | s | ( \\theta ) } = e ^ { f ( \\theta + \\alpha ) - f ( \\theta ) } , \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\omega } \\Phi = \\Phi \\mathcal { D } _ { \\omega } + \\Pi _ S ^ { \\perp } \\mathcal { A } ( b _ 3 \\partial _ { y y y } + b _ 2 \\partial _ { y y } + b _ 1 \\partial _ y + b _ 0 ) \\Pi _ S ^ { \\perp } + \\mathcal { R } _ { \\mathit { I } } , \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} T ( X , Y ) = \\sum _ { X < \\ell \\le Y } \\phi ( \\ell ) \\theta ( \\ell ) ^ 2 \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { n } \\nu ( \\mathcal { C } _ i ) ^ { ( - 1 ) ^ { i } } = \\prod _ { i = 0 } ^ { m } \\nu ( \\mathcal { R } _ i ) ^ { ( - 1 ) ^ { i } } . \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} F ( \\lambda ) = \\prod _ { b = 1 } ^ { 2 \\mathsf { N } } \\left ( \\frac { \\lambda ^ { p } } { \\left ( \\zeta _ { b } ^ { ( 0 ) } \\right ) ^ { p } } - \\frac { \\left ( \\zeta _ { b } ^ { ( 0 ) } \\right ) ^ { p } } { \\lambda ^ { p } } \\right ) , \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} \\Xi _ 2 ( z ) & = \\mathbb { E } \\left [ e ^ { - z \\mathcal { I } } \\right ] = \\mathbb { E } \\left [ { e ^ { - z ( \\mathcal { I } _ { \\mathrm { L o S } } + \\mathcal { I } _ { \\mathrm { N L o S } } ) } } \\right ] \\\\ & = \\mathbb { E } \\left [ { e ^ { - z \\mathcal { I } _ { \\mathrm { L o S } } } } \\right ] \\mathbb { E } \\left [ { e ^ { - z \\mathcal { I } _ { \\mathrm { N L o S } } } } \\right ] \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} f _ 1 = \\langle \\partial _ x ^ { - 1 } u _ 0 , u _ 1 \\rangle _ { L ^ 2 } = 2 \\| u _ 0 \\| _ { L ^ 2 } ^ 2 = 2 . \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} v = \\left ( \\begin{array} { c c } 0 & 1 \\\\ \\beta & 0 \\end{array} \\right ) { \\rm a n d } w = \\left ( \\begin{array} { c c } \\eta & 1 \\\\ \\beta & \\eta \\end{array} \\right ) . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} \\delta = \\left ( 1 + \\frac 2 n \\right ) ^ { - ( N + 1 ) } ( q - ( m - 1 ) ) , N . \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} q _ d : = \\prod _ { i = 0 } ^ { \\min \\{ m , d - 2 \\} } p _ { m - i } . \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\ge X \\\\ ( m , k ) = 1 } } \\mu ( m ) \\tau _ r ( m ) f ( m ) m ^ { - 1 } \\ll \\sigma _ { - 1 } ( k ) \\exp ( - c \\sqrt { \\log { x } } ) \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} p = \\frac { ( c + 1 ) \\sqrt { c \\left ( c ^ 2 - 4 \\right ) } } { \\sqrt { 3 } c \\sqrt { c ^ 3 - 2 c - 1 0 } } . \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} F ^ { ( N ) } = \\sum _ { j _ 1 + \\dots + j _ N = 0 } F ^ { ( N ) } _ { j _ 1 \\dots j _ N } u _ { j _ 1 } \\dots u _ { j _ N } , \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\eta ' _ 1 = \\eta '' _ 1 \\mbox { a n d } \\eta ' _ 2 = \\eta '' _ 2 . \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} \\begin{aligned} & \\lvert \\tilde { m } _ 3 - 1 \\rvert ^ { L i p ( \\gamma ) } \\le C \\varepsilon ^ 2 , \\lvert \\tilde { m } _ 1 - \\varepsilon ^ 2 c ( \\xi ) \\rvert ^ { L i p ( \\gamma ) } \\le C \\varepsilon ^ { 3 - 2 a } , \\end{aligned} \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\operatornamewithlimits { i n d \\ , l i m } _ { r \\ge 1 } \\mathbb { F } _ q ( \\mathcal { H } _ - , r ^ { - 1 } , - \\alpha ) = : \\mathbb { F } _ q ( \\mathcal { H } _ - , - \\alpha ) . \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} 5 \\ell \\geq \\ell ( R ) = \\ell ( R ' ) + 1 \\geq 5 \\ell - 1 + 1 = 5 \\ell . \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} \\cos \\frac { { \\mathcal A } ( E ) } { h } = h ^ { \\frac 4 3 } \\left ( \\sin \\frac { { \\mathcal A } ( E ) } { h } \\right ) G ( E , h ) . \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ 1 = \\int _ { \\Omega } u ' ( \\tau ) v ' ( \\tau ) \\ ; d \\tau . \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} B _ 0 = & \\{ f \\in L ^ 1 ( \\R , C _ 0 ( ( - M / 2 , M / 2 ) , A ) ) \\mid \\\\ & f ( t ) ( x ) = 0 \\mbox { w h e n e v e r } x - t \\not \\in ( - M / 2 , M / 2 ) \\} \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} \\mu ( d \\xi ) = c _ H | \\xi | ^ { 1 - 2 H } d \\xi , c _ H = \\frac { \\Gamma ( 2 H + 1 ) \\sin ( \\pi H ) } { 2 \\pi } , \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\left | \\int _ { - 1 } ^ t l _ i ( \\tau ) \\ ; d \\tau \\right | = \\sum _ { i = 1 } ^ N a _ i \\int _ { - 1 } ^ t l _ i ( \\tau ) \\ ; d \\tau = \\int _ { - 1 } ^ t q ( \\tau ) \\ ; d \\tau . \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} F ( \\widetilde { V ' } ) \\subseteq \\widetilde { V } , G ( \\widetilde { V } ) \\subseteq \\widetilde { V ' } , A ( \\widetilde { V } ) , \\mbox { } B ( \\widetilde { V } ) \\subseteq \\widetilde { V } , \\\\ A ' ( \\widetilde { V ' } ) , \\mbox { } B ' ( \\widetilde { V ' } ) \\subseteq \\widetilde { V ' } , J ( \\widetilde { V } ) = 0 . \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} g _ k ( \\tau , x ) = - b ( \\tau ) \\partial _ x e ^ { \\mathrm { i } k x } + ( \\mathrm { I } _ { H _ S ^ { \\perp } } - ( \\Phi ^ { \\tau } ) ^ T ) [ b ( \\tau ) \\partial _ x e ^ { \\mathrm { i } k x } ] \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\omega _ { \\beta , \\mu , \\phi } ( \\eta ( b _ { { 0 } } ^ { * } ) ) = \\sqrt { \\rho _ { 0 } } \\exp ( i \\phi ) \\ , \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} \\begin{aligned} H ( s ) u : = - u '' + s ^ 2 V ( s \\xi ) y & = s ^ 2 \\lambda u \\\\ \\alpha _ 1 u ( 0 ) + \\frac { 1 } { s } \\alpha _ 2 u ' ( 0 ) & = 0 \\\\ \\beta _ 1 u ( 1 ) + \\frac { 1 } { s } \\beta _ 2 u ' ( 1 ) & = 0 . \\end{aligned} \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} & \\frac { \\mu [ 0 , x ) + \\mu [ 0 , x ] } { 2 } - x = \\lim _ { h \\to 0 } \\frac { f ( x + h ) + f ( x - h ) } { 2 } \\\\ & = \\frac { 1 } { \\pi } \\sum _ { n = 1 } ^ \\infty \\Im ( \\frac { \\hat { \\mu } ( n ) } { n } [ e ( n x ) - 1 ] ) + \\lim _ { n \\to \\infty } \\frac { 1 } { 2 n } \\sum _ { j = 1 } ^ n \\hat { \\mu } ( j ) \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{align*} n ( \\mu , j ) = a ; \\ , \\ \\ , \\ m _ 1 + \\cdots + m _ { a - 1 } < j \\leq m _ 1 + \\cdots + m _ a . \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} \\left ( \\sum a _ i \\right ) \\times \\left ( \\sum b _ j \\right ) = \\sum a _ i b _ j \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} F = A f _ b + L f _ s + D + o ( 1 ) \\ , , \\end{align*}"}
-{"id": "524.png", "formula": "\\begin{align*} \\phi = \\left ( c _ 1 + c _ 2 ( - 1 ) ^ n \\right ) u . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} A : = \\frac { 1 } { 2 i } ( x \\cdot \\nabla + \\nabla \\cdot x ) . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} ( w _ { t } , \\zeta ) = ( u _ { 0 } , \\zeta ) + \\int _ { 0 } ^ { t } \\big [ ( u _ { s } , L ^ { 0 } _ { s } \\zeta ) + ( h _ { s } f _ { s } , \\zeta ) \\big ] \\ , d s . \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\hat { \\theta } _ { j } = p _ { j } \\theta + \\xi _ { j } , & \\forall \\ 1 \\leq j \\leq k - 1 , \\\\ \\hat { \\theta } _ { k , l } = ( p _ { k } ) \\theta + { l \\over | p _ { k } | } , & \\forall \\ 0 \\leq l \\leq | p _ { k } | - 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} K ( z ) f : = \\sum _ { k \\in \\N } ( z - \\mu _ k ) ^ { - \\frac 1 2 } \\langle f , \\psi _ k \\rangle \\psi _ k , z \\in \\rho ( A ) , \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty r _ i ^ { m - 2 } \\bar { u } _ i ^ 2 \\leq c \\sum _ { k < \\ell } r _ k ^ { m - 2 } r _ \\ell ^ { m - 2 } ( \\bar { u } _ k - \\bar { u } _ \\ell ) ^ 2 \\leq c \\sum _ { k < \\ell } r _ k ^ { m - 2 } r _ \\ell ^ { m - 2 } \\left ( \\sum _ { i = k } ^ { \\ell - 1 } \\alpha _ { k \\ell i } ( \\bar { u } _ { i } - \\bar { u } _ { i + 1 } ) ^ 2 \\right ) \\left ( \\sum _ { j = k } ^ { \\ell - 1 } \\frac { 1 } { \\alpha _ { k \\ell j } } \\right ) . \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} L = \\frac { 1 } { 2 } ( X ^ 2 + Y ^ 2 + \\beta ^ 2 Z ^ 2 ) \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\alpha _ { n } \\beta _ { n } = a _ { n } c _ { n } \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} \\varphi ^ 2 ( q ) \\varphi ^ 2 ( q ^ { 1 0 } ) - \\varphi ^ 2 ( q ^ 2 ) \\varphi ^ 2 ( q ^ 5 ) = 4 A _ 1 ( q ) + 8 A _ 3 ( q ) . \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} I ( T ) = \\int _ T ^ { 2 T } \\left | F \\left ( \\frac { 1 } { 2 } + i t \\right ) \\right | d t . \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} K ( m , n , c ) = K ( m \\overline { c ' } , n \\overline { c ' } , 2 ^ \\ell ) \\cdot S ( m \\overline { 2 ^ \\ell } , n \\overline { 2 ^ \\ell } , c ' ) , \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} \\int _ \\Omega p _ \\varepsilon \\nabla u ^ \\varepsilon \\nabla \\varphi \\ , d x + \\int _ \\Omega ( \\lambda + V _ \\varepsilon ) u ^ \\varepsilon \\varphi \\ , d x = \\int _ \\Omega g \\varphi \\ , d x , \\quad \\forall \\ , \\varphi \\in X _ \\varepsilon ^ { \\frac { 1 } { 2 } } , \\ , \\ , \\varepsilon \\in ( 0 , \\varepsilon _ 0 ] ; \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} 1 _ { M \\left ( A \\right ) } = \\sum _ { \\iota \\in I } \\left . a _ \\iota \\right \\rangle \\left \\langle a _ \\iota \\right . \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} \\hat { \\psi } ( \\theta ) = \\hat { g } ( \\theta ) + \\hat { \\phi } , \\hat { g } ( \\theta ) = \\frac { 1 } { n } \\sum _ { \\ell = 1 } ^ { L } \\sum _ { i \\in I _ { \\ell } } g ( W _ { i } , \\hat { \\gamma } _ { \\ell } , \\theta ) , \\hat { \\phi } = \\frac { 1 } { n } \\sum _ { \\ell = 1 } ^ { L } \\sum _ { i \\in I _ { \\ell } } \\phi ( W _ { i } , \\hat { \\gamma } _ { \\ell } , \\hat { \\alpha } _ { \\ell } , \\tilde { \\theta } _ { \\ell } ) . \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} [ u ( A ) , u ( B ) ] : = \\left [ \\frac { 3 - \\eta } { 6 - 4 \\eta } , \\frac { 3 - 3 \\eta } { 6 - 4 \\eta } \\right ] . \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} \\begin{aligned} & E _ 0 = \\Phi _ 1 \\prod _ { j \\in \\tau } ( \\tilde { \\Phi } _ 1 \\tilde { \\Phi } _ { j + 1 } ) \\cdot e _ 0 , F _ 0 = \\prod _ { j \\in \\tau } ( \\tilde { \\Phi } _ 1 \\tilde { \\Phi } _ { j + 1 } ) \\cdot f _ 0 , K _ 0 = \\Phi _ 1 \\cdot k _ 0 ; \\\\ & E ' _ 0 = \\tilde { \\Phi } ' _ 1 \\prod _ { j \\in \\tau ' } ( \\tilde { \\Phi } ' _ 1 \\tilde { \\Phi } ' _ { j + 1 } ) \\cdot e ' _ 0 , F ' _ 0 = \\prod _ { j \\in \\tau ' } ( \\tilde { \\Phi } ' _ 1 \\tilde { \\Phi } ' _ { j + 1 } ) \\cdot f ' _ 0 , K ' _ 0 = \\Phi ' _ 1 \\cdot k ' _ 0 . \\end{aligned} \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P _ e } ( - 1 ) ^ { \\nu ( \\pi ) } q ^ { | \\pi | } = \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { 2 n } } { 1 - q ^ { 2 n } } \\frac { ( - q ; q ^ 2 ) _ { n - 1 } } { ( q ^ 2 , q ^ 2 ) _ { n - 1 } } q ^ { n - 1 } . \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} \\begin{aligned} S _ k : f ( n ) \\mapsto f ( S _ k n ) . \\end{aligned} \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} \\theta ^ i \\delta ^ i = \\mu \\tau ^ i , \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} \\mathsf { E } ^ { ( a ) } 1 _ n : = \\theta ^ { ( a ) } 1 _ n \\otimes _ { \\rho } 1 , \\quad \\mathsf { F } ^ { ( a ) } 1 _ n : = \\vartheta ^ { ( a ) } 1 _ n \\otimes _ { \\rho } 1 . \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} u _ { k } ( t _ { k } , x ; x _ { 1 } , \\cdots , x _ { k - 1 } , x ) = u _ { k - 1 } ( t _ { k } , x ; x _ { 1 } , \\cdots , x _ { k - 1 } ) \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} { \\bf S } _ { i j } = \\mathbf { I } _ { N \\times N } \\otimes { \\bf G } _ { i j } , \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} c ( K ) : = \\min \\{ | S | \\colon S \\subset V K \\} . \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} B ( \\theta + \\alpha ) A ( \\theta ) B ^ { - 1 } ( \\theta ) = R _ { \\varphi ( \\theta ) } , \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} \\lim _ { \\Lambda } \\omega _ { \\beta , \\mu , \\Lambda } ^ { 0 } ( \\frac { b ^ { * } _ { k } b _ { k } } { V } ) = \\lim _ { \\Lambda } \\frac { 1 } { V } \\left \\{ e ^ { \\beta ( \\varepsilon _ { k } - { \\mu _ { \\Lambda } ( \\beta , \\rho ) } ) } - 1 \\right \\} ^ { - 1 } = 0 \\ , \\ \\forall k \\in \\Lambda ^ { * } \\ , \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} \\tilde { D } ( \\eta ) & = \\int _ { 0 } ^ { 1 } 2 \\sqrt { \\eta '' } - \\eta '' ( s ) \\cdot ( 1 - s ) - \\frac { 1 } { 1 - s } d s - \\eta ( 0 ) + \\inf _ { t \\in [ 0 , 1 ) } \\left \\{ \\eta ( t ) - \\xi ( t ) \\right \\} + h ^ { 2 } + \\xi ( 1 ) \\\\ \\tilde { P } ( \\mu ) & = 2 P ( \\mu ) . \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { p - 1 } \\binom { p - 1 } { l } \\sum _ { r = 1 } ^ { l } \\binom { p } { r } H _ { r - 1 } & \\equiv _ { p ^ 2 } - p \\sum _ { r = 1 } ^ { p - 1 } \\frac { ( - 1 ) ^ r H _ { r - 1 } } { r } \\sum _ { r = l } ^ { p - 1 } ( - 1 ) ^ l \\\\ & = - \\frac { p } { 2 } \\left ( H _ { p - 1 } ( 1 , - 1 ; - 1 ) + H _ { p - 1 } ( 1 , 1 ) \\right ) \\equiv _ { p ^ 2 } - \\frac { p q _ p ^ 2 ( 2 ) } { 2 } . \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} \\rho ^ { n + 1 } \\geq \\rho ^ { n + 2 } = \\frac { 1 } { ( 1 + \\alpha / ( n + 2 ) ) ^ { n + 2 } } \\geq e ^ { - \\alpha } . \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} \\tilde { f } _ { m _ 0 , m _ 1 } ( x ) = \\frac { v } { m _ 1 } \\sum _ { j = 1 } ^ { m _ 1 } ( \\beta _ { H _ j } ) \\phi \\left ( \\frac { v _ 0 } { m _ 0 } \\sum _ { k = 1 } ^ { m _ 0 } \\widetilde { \\theta } _ { k , H _ j } \\cdot x \\right ) . \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( y _ i - f _ m ( x _ i ) ) ^ 2 - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( y _ i - f ( x _ i ) ) ^ 2 + & \\\\ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( f _ m ( x ^ { \\prime } _ i ) - f ( x ^ { \\prime } _ i ) ) ^ 2 \\leq \\frac { 2 v ^ 2 \\epsilon ^ 2 _ 1 ( 1 + M _ 1 / m _ 0 ) } { m _ 0 } + \\frac { v ^ 2 M _ 1 } { 2 m ^ 2 _ 0 } . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} e _ 2 ( x , w ) = \\prod _ { i = 0 } ^ { \\infty } \\left ( 1 + x w ^ { 2 ^ i } \\right ) . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { M ( t ) } { t } = 0 \\quad \\quad \\lim _ { t \\to \\infty } \\frac { M ( t ) } { t } = \\infty . \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} \\Psi _ n ( T ) = \\Phi _ n ( T ) T ^ s , \\Phi _ n ( 0 ) \\neq 0 \\mbox { a n d } \\Phi _ n ( 0 ) \\mid J \\eta , \\end{align*}"}
-{"id": "5203.png", "formula": "\\begin{align*} H _ 3 ^ { ( 3 ) } = c _ 1 \\int _ { \\mathbb { Z } } ( z _ x ^ 3 + 3 \\ , z _ x ^ 2 \\ , v _ x ) \\ , d x + c _ 2 \\int _ { \\mathbb { Z } } ( z _ x ^ 2 \\ , z + z _ x ^ 2 \\ , v + 2 \\ , v _ x \\ , z _ x \\ , z ) \\ , d x + c _ 3 \\int _ { \\mathbb { T } } ( z ^ 3 + v \\ , z ^ 2 ) \\ , d x . \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\bar c ( | \\nabla u | ) = 0 \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} \\det A ^ { \\widetilde { K } } ( \\widetilde { G } ) = \\begin{vmatrix} M _ 1 & i M _ 2 \\cr i M _ 2 & M _ 1 \\end{vmatrix} = \\begin{vmatrix} M _ 1 + i M _ 2 & i M _ 2 \\cr M _ 1 + i M _ 2 & M _ 1 \\end{vmatrix} = \\begin{vmatrix} M _ 1 + i M _ 2 & i M _ 2 \\cr 0 & M _ 1 - i M _ 2 \\end{vmatrix} = \\left | \\det A ^ { K , \\omega } ( G ) \\right | ^ 2 \\ , . \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} P _ 1 = u , P _ 2 = - u u _ { - 1 } . \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} \\int _ { \\Omega } | \\bar { x } ( t ) - x ( t ) | ^ { p - 2 } ( \\bar { x } ( t ) - x ( t ) ) y ( t ) \\ , \\mu ( d t ) = - \\sum _ { i \\in I ( \\bar { x } ) } \\nu _ i \\int _ { \\Omega } f _ i ( t ) y ( t ) \\ , \\mu ( d t ) , \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} \\flat ( X ) = i _ X d \\eta + \\eta ( X ) \\eta \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} \\Phi _ 1 : = \\exp ( \\varepsilon A _ 1 ) = \\mathrm { I } _ { H _ S ^ { \\perp } } + \\varepsilon A _ 1 + \\varepsilon ^ 2 \\frac { A _ 1 ^ 2 } { 2 } + \\varepsilon ^ 3 \\hat { A } _ 1 , \\hat { A } _ 1 : = \\sum _ { k \\geq 3 } \\frac { \\varepsilon ^ { k - 3 } } { k ! } \\ , A _ 1 ^ k , \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} r _ n ( i , j ) = R _ h \\left ( \\frac { i + k } { n } , \\frac { j + k } { n } \\right ) - R _ h \\left ( \\frac { i } { n } , \\frac { j + k } { n } \\right ) - R _ h \\left ( \\frac { i + k } { n } , \\frac { j } { n } \\right ) + R _ h \\left ( \\frac { i } { n } , \\frac { j } { n } \\right ) . \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} \\mu _ k ^ { ( b ) } ( z ^ k ) = \\P ( Z ^ k = z ^ k ) . \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} \\mathbb { V } = \\bigoplus _ { k = 0 } ^ { n } \\mathbb { V } [ k ] , \\end{align*}"}
-{"id": "1612.png", "formula": "\\begin{align*} \\Omega _ X ( u , v ) = 2 ( a _ { 1 1 } \\widetilde { b _ { 1 1 } } - \\widetilde { a _ { 1 1 } } b _ { 1 1 } ) + a _ { 2 2 } \\widetilde { b _ { 2 2 } } - \\widetilde { a _ { 2 2 } } b _ { 2 2 } . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} \\begin{array} { r c c c c } [ A , B ] + I J = 0 , & [ A ' , B ' ] = 0 , & A F - F A ' = 0 , & B F - F B ' = 0 , & J F = 0 , \\\\ G I = 0 , & F G = 0 , & G A - A ' G = 0 , & G B - B ' G = 0 . & \\end{array} \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} \\sum ( \\Theta _ { j \\bar k } ^ { T _ B } \\partial / \\partial t ^ k , \\partial / \\partial t ^ j ) _ H = - | | \\sum [ \\theta _ k ^ * , \\theta _ k ] | | ^ 2 - \\left ( P ^ { \\bot } ( D ^ { { \\rm E n d } ( H ) } _ j \\theta _ k ) , P ^ { \\bot } ( D ^ { { \\rm E n d } ( H ) } _ k \\theta _ j ) \\right ) . \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} A _ n & = \\left \\{ \\sum _ { k = 0 } ^ \\infty \\left | \\sum _ { t = 0 } ^ k f _ n ( t ) - f ( t ) \\right | \\le n ^ { - \\eta / ( 2 + 2 \\eta ) } \\right \\} \\\\ B _ n & = \\left \\{ \\sum _ { k = 0 } ^ \\infty \\left | f _ n ^ \\ast ( k ) - f ^ \\ast ( k ) \\right | \\le n ^ { - \\varepsilon } \\right \\} \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} b \\alpha - \\overline { b } \\overline { \\alpha } = \\frac { 2 } { 2 - v i } \\cdot \\frac { 1 } { \\overline { a } + \\overline { b } \\overline { \\alpha } } + \\frac { 2 } { 2 + v i } \\cdot \\frac { 1 } { a + b \\alpha } . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} p ( ( h , h ' ) \\cdot X _ 0 ) = \\chi ( h , h ' ) ^ l p ( X _ 0 ) , \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\nu ( s ) : = - \\frac { n } { 2 s } , \\quad \\eta ( \\nu ) = - \\frac { n } { 2 \\nu } . \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} h : = \\sum _ { \\alpha \\in I _ X } \\varphi _ \\alpha \\cdot f \\ ; . \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} \\left | \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ m \\log \\left ( 1 - \\frac { a } { 2 k } \\right ) + \\frac { a } { 4 } \\log ( m ) \\right | \\leq C , \\forall m \\in \\mathbb { N } , \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} P _ { V } ( \\{ \\omega \\in \\mathbb { N } _ { 0 } ^ { \\infty } : \\forall f \\in \\mathcal { C } ( \\mathbb { N } _ { 0 } ^ { \\infty } ) , \\ ; \\underset { n \\rightarrow \\infty } { \\lim } \\Pi _ { n } f ( \\omega ) = \\textstyle \\int \\ ! f \\ , \\mathrm { d } P _ { V } \\} ) = 1 . \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} n ( i ) = 2 ( i + 1 ) ^ { \\lceil 2 s _ { m a x } \\rceil } \\cdot i ^ { 2 \\lceil s _ { m a x } \\rceil } . \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} v _ t + ( - \\Delta ) ^ s v & = 0 & & \\mbox { i n } \\Omega \\times ( 0 , T ) , \\\\ v & = 0 & & \\mbox { i n } \\Sigma _ 1 \\times ( 0 , T ) , \\\\ \\mathcal { N } _ s v & = 0 & & \\mbox { i n } \\Sigma _ 2 \\times ( 0 , T ) , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { n } t _ { j } \\leq c _ 0 ( 1 + \\rho ) ^ n \\quad \\ n \\geq n _ 0 - 1 , \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} F _ { p } ^ { \\left ( b \\right ) } = F _ { p } . \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} \\left . w ^ { ( v ) } _ i ( n ) \\right | _ { n = - N _ { \\rm c p } } = \\left . { x } ^ { ( v ) } _ { i - 1 } ( n ) \\right | _ { n = N } - \\left . x ^ { ( v ) } _ { i } ( n ) \\right | _ { n = - N _ { \\rm c p } } . \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} F _ n F _ { n + i + j } - F _ { n + i } F _ { n + j } = ( - 1 ) ^ { n + 1 } F _ i F _ j \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} - S _ 1 U ^ { 1 - } = S _ 2 U ^ { 2 + } , \\ - \\frac { \\partial U ^ { 1 - } } { \\partial n } = \\frac { \\partial U ^ { 2 + } } { \\partial n } \\mbox { o n } \\partial \\Omega . \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} & | \\{ x \\in B ( x _ 0 , 4 \\rho ) : w ( x , t ) > k _ j \\} | \\ge | \\{ x \\in B ( x _ 0 , \\rho ) : w ( x , t ) > k _ 0 \\} | \\\\ & \\ge \\frac \\gamma 8 | B ( x _ 0 , \\rho ) | = \\gamma \\tilde C \\rho ^ n . \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} q ( t ) = \\left ( \\frac { p - 1 } { N - p } \\right ) ^ { p } \\frac { A ^ { \\frac { ( p - 1 ) p } { N - p } } } { ( B - t ) ^ { \\frac { p ( N - 1 ) } { N - p } } } . \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\Delta E ( \\lambda _ p , \\lambda _ h ) = \\tilde { \\epsilon } ( \\lambda _ p ) - \\tilde { \\epsilon } ( \\lambda _ h ) \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} V ( x ) = x ^ 4 - 4 a | x | ^ 3 + 4 a ^ 2 x ^ 2 - 2 | x | \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} e ^ { - ( B - A ) D } ( x , y ) = e ^ { ( y ^ 2 - x ^ 2 ) / 2 + B } \\frac { \\exp \\left ( - \\frac { ( e ^ B y - e ^ A x ) ^ 2 } { 4 ( \\beta - \\alpha ) } \\right ) } { \\sqrt { 4 \\pi ( \\beta - \\alpha ) } } \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} P ^ { \\mu ' } ( K _ { n } = k ) & = \\int _ { \\mathcal { P } _ { \\mathcal { V } _ { \\alpha , \\theta } } } P ( K _ { n } = k ) \\mu ' ( \\mathrm { d } P ) = \\int _ { \\mathcal { P } _ { \\mathcal { V } _ { \\alpha , \\theta } } } P ( K _ { n } = k ) \\mu \\circ T ^ { - 1 } ( \\mathrm { d } P ) \\\\ & = \\int _ { \\mathcal { V } _ { \\alpha , \\theta } } d _ { n , k } V _ { n , k } \\mu ( \\mathrm { d } V ) = d _ { n , k } V ^ { \\mu } _ { n , k } . \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} \\Theta = \\Theta _ 0 \\wedge \\frac { d f } { f ^ m } , \\Theta _ 0 \\in \\Omega ^ { n - 1 } ( M ) . \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} s _ n = p s _ { n - 3 } + 1 + r _ n , \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} A ^ T Z ^ m = \\left [ \\begin{array} { c } \\langle A _ 1 ^ m , Z ^ m \\rangle \\\\ \\langle A _ 2 ^ m , Z ^ m \\rangle \\\\ \\vdots \\\\ \\langle A _ n ^ m , Z ^ m \\rangle \\\\ \\end{array} \\right ] , \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} \\dim ( S _ { k , \\Lambda _ N } ) = d + \\frac { d k } { 1 2 } - \\alpha _ 1 - \\alpha _ 2 - \\alpha _ 3 - \\alpha _ 4 , \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} H u = f , \\ , \\ , \\ , f \\in B ^ r _ { p , q } ( \\textnormal { S U } ( 2 ) ) . \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} \\nabla ^ \\alpha \\mathfrak { C } _ { \\alpha \\beta } { _ { | \\mathcal { I } ^ 0 } } = 0 , \\ ; \\left ( \\partial _ 0 ( R _ { i j } - \\Lambda _ { i j } ) - \\overline { \\nabla } _ { i } \\mathfrak { C } _ { j 0 } - \\overline { \\nabla } _ { j } \\mathfrak { C } _ { i 0 } \\right ) { _ { | \\mathcal { I } ^ 0 } } = 0 . \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{gather*} 1 \\mapsto ( \\partial _ s ( - s \\rho ) , \\rho \\partial _ s ( 1 ) ) = ( 1 , 0 ) \\\\ \\rho \\mapsto ( \\partial _ s ( - \\rho s ( \\rho ) ) , \\rho \\partial _ s ( \\rho ) = ( 0 , \\rho ) . \\end{gather*}"}
-{"id": "4084.png", "formula": "\\begin{align*} 2 ( v s + w t ) - ( s ^ { 2 } + t ^ { 2 } ) = 0 \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} S _ h ( x ) = \\frac { 1 } { | { \\mathcal K } | } \\sum _ { k \\in { \\mathcal K } } S _ { h k } ( x ) + O \\bigl ( L ( 1 , \\chi ) x \\log x + x ( \\log x ) ^ { - 2 } \\bigr ) \\ . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} [ F _ { 1 } ] _ { I } = \\begin{cases} ( b _ { i } s ^ { 2 } - a _ { i } t ^ { 2 } ) , & I = \\{ 0 , i \\} \\\\ \\lambda ( a _ { i } b _ { j } - a _ { j } b _ { i } ) s t , & I = \\{ i , j \\} , 0 \\notin I \\end{cases} . \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} \\Phi ( M _ q ( \\varphi ) x , M _ q ( \\varphi ) y ) = \\Phi ( x , y ) \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} N = Q ^ 2 T ^ 2 . \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} T _ m ( f ) ( t ) : = f ( 0 ) + \\sum _ { j = 1 } ^ m \\frac { t ^ j } { j ! } f ^ { ( j ) } ( 0 ) . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} D ^ q ( \\mu ) = d ( q ) . \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} \\mathfrak { d } = \\left ( \\frac { 1 } { \\sqrt { \\xi '' } } \\right ) '' . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} M _ { N } = \\underbrace { ( \\mathbb { C } ^ s \\otimes \\mathbb { C } [ z ] ) \\otimes \\dots \\otimes ( \\mathbb { C } ^ s \\otimes \\mathbb { C } [ z ] ) \\otimes ( \\mathbb { C } ^ s \\otimes \\mathbb { C } [ z ] ) } _ N . \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{align*} x _ 0 = x ( T ) = \\Phi ( T ) x _ 0 + \\Phi ( T ) \\int _ 0 ^ T \\Phi ( s ) ^ { - 1 } c ( s ) d s , \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} I \\mathcal { P } = \\mathcal { G } _ { \\mathrm { f i n } } ^ { \\mathrm a } ( \\mathcal { H } ) \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} L _ \\xi e ^ { 2 \\xi } = \\frac 1 6 i \\xi \\eta \\zeta e ^ { 2 \\xi } - \\frac 1 { 2 4 } i \\xi \\zeta ( \\eta + J ( \\eta ) ) ^ 2 J ( \\eta ) ^ { - 1 } e ^ { 2 \\xi } + \\xi \\eta \\zeta E S ( \\rho ^ { - 1 } ) . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} 1 + \\frac { 1 } { b } \\left ( \\varrho \\left ( \\frac { z D _ { q } \\mathcal { F } ( z ) } { \\mathcal { F } ( z ) } \\right ) - \\varrho \\right ) = \\varphi ( z ) \\left ( \\phi ( \\omega ( z ) ) - 1 \\right ) . \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} & v _ 1 + c v _ 0 = 0 , \\\\ & v _ 0 = 0 . \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} \\mathcal D _ 0 = \\{ ( d _ 0 , & d _ 1 , d _ 2 , d _ 3 ) \\in \\mathbb R ^ 4 \\mid \\\\ & ( d _ 0 + a , d _ 1 + a , d _ 2 + a , d _ 3 + a ) \\in \\mathcal D \\} . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\prod _ { j _ { 1 } = 1 } ^ { 4 } \\prod _ { j _ { 2 } = 1 } ^ { 4 } \\left \\vert \\alpha ^ { ( 1 , j _ { 1 } ) } - \\alpha ^ { ( 2 , j _ { 2 } ) } \\right \\vert \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} \\rho ( p ) = - \\lambda ( p ) , \\ \\rho ( p ^ 2 ) = \\chi ( p ) , \\ \\rho ( p ^ \\alpha ) = 0 \\alpha > 2 . \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} S \\ : = \\ \\left \\{ v \\left ( s - \\frac { \\epsilon ' } { 1 + \\epsilon } \\right ) : \\epsilon \\in K ^ { \\prec 1 } \\right \\} \\ \\subseteq \\ ( \\Gamma ^ { > } ) ' \\ \\subseteq \\ \\Gamma _ { \\infty } . \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} \\mathrm { w } _ \\varphi ( q ) = \\frac { 1 } { T } \\int _ 0 ^ T \\varphi ( t , q , 0 ) \\ , d t , q \\in N , \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} \\begin{aligned} & \\int _ { \\R _ - } \\d u \\ , ( f _ W ( u ) - f _ W ( - u ) ) T _ { \\tau _ 1 , \\tau _ 2 } ( u , v ) \\\\ & \\qquad = \\int _ \\R \\d u \\ , f _ W ( u ) \\phi _ { 2 ( \\tau _ 2 - \\tau _ 1 ) } ( v - u ) - \\int _ \\R \\d u \\ , f _ W ( u ) \\phi _ { 2 ( \\tau _ 2 - \\tau _ 1 ) } ( v + u ) \\\\ & \\qquad = e ^ { ( \\tau _ 2 - \\tau _ 1 ) ( \\sqrt 2 r - 2 W ) ^ 2 } ( f _ W ( v ) - f _ W ( - v ) ) \\end{aligned} \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} H ^ + \\left ( X \\right ) = X + h ^ + \\left ( T _ { X } \\right ) \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} A P _ m = P _ { m + 1 } \\bar { H } _ m , \\bar { H } _ m = \\begin{pmatrix} H _ m \\\\ \\bar { h } \\boldsymbol { e } ^ { T } _ m \\end{pmatrix} , \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} \\begin{cases} p ' = q r ( p + r ) ^ { - 1 } = ( p + r ) ^ { - 1 } r q , \\\\ q ' = p + r , \\\\ r ' = q p ( p + r ) ^ { - 1 } = ( p + r ) ^ { - 1 } p q . \\end{cases} \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} \\delta _ i = \\frac { 1 } { T _ { \\delta } } \\sum _ { t = 1 } ^ { T _ { \\delta } } \\omega _ i ^ { ( t ) } ( a _ i , H _ i ( t ) ) . \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} \\liminf _ { t \\to \\infty } \\left \\{ \\frac { \\sum _ { u = \\lfloor t / 2 \\rfloor } ^ { t - 1 } \\Phi ^ 2 _ u } { \\log v ( \\lfloor t / 2 \\rfloor ) } \\right \\} = : \\zeta > 0 , \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} T ( \\lambda ) & = \\# \\{ ( x _ { 1 } , x _ { 2 } , x _ { 3 } ) \\in \\Z ^ { 3 } \\cap E ( \\lambda ) \\} , \\\\ T _ { x _ { 1 } } ( \\lambda ) & = \\# \\{ ( 0 , x _ { 2 } , x _ { 3 } ) \\in ( \\{ 0 \\} \\times \\Z ^ { 2 } ) \\cap E ( \\lambda ) \\} , \\\\ T _ { x _ { 1 } } ^ { + } ( \\lambda ) & = \\# \\{ ( 0 , x _ { 2 } , x _ { 3 } ) \\in ( \\{ 0 \\} \\times \\N ^ { 2 } ) \\cap E ( \\lambda ) \\} . \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} \\P \\Big ( \\sum _ { i = 1 } ^ m U _ i ^ 2 > m ( 1 + \\tau ) \\Big ) \\leq { \\rm e } ^ { - \\frac { m } { 2 } ( \\tau - \\ln ( 1 + \\tau ) ) } . \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} a _ n = g _ 1 ( n ) \\alpha _ 1 ^ n + \\dots + g _ h ( n ) \\alpha _ h ^ n + O ( 1 ) \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} r _ d \\phi * ( a + b ) & = r _ d \\phi \\phi _ { a + b } \\\\ & = r _ d \\phi _ { a + b } \\phi \\\\ & = r _ d ( a + b ) \\phi \\\\ & = r _ d \\phi * a + r _ d \\phi * b \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} d X ( s ) = - v ( s ) d s + \\nu X ( s ) d W ( s ) + \\nu _ 0 X ( s ) d W _ 0 ( s ) , \\ X ( t ) = x . \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} \\vec { X } _ { 2 1 } & = ( 1 , 0 , 0 ) , \\mu _ { 2 1 } = - \\frac { g } { \\lambda } , \\\\ \\vec { X } _ { 2 2 } & = ( 0 , 1 , 0 ) , \\mu _ { 2 2 } = - 1 , \\\\ \\vec { X } _ { 2 3 } & = ( 0 , 0 , 1 ) , \\mu _ { 2 3 } = - B . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} f _ 1 ( x ) & = \\frac { x ^ { \\nu - 1 } } { \\Gamma ( \\nu ) } E \\left [ e ^ { - x / S } S ^ { - \\nu } \\right ] ; \\\\ f ' _ 1 ( x ) & = \\frac { \\nu - 1 } { x } f _ 1 ( x ) - \\frac { x ^ { \\nu - 1 } } { \\Gamma ( \\nu ) } E \\left [ e ^ { - x / S } S ^ { - \\nu - 1 } \\right ] . \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} ( - i \\nabla - \\epsilon A ^ 0 ( x ) - \\kappa A ( \\epsilon x ) ) ^ 2 \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( x , \\tilde { \\beta } ) = \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( x , \\tilde { \\beta } ) \\left \\{ ( - i \\nabla - \\epsilon A ^ 0 ( x ) ) \\ , + \\ , \\kappa a _ \\epsilon ( x , \\tilde { \\beta } ) \\right \\} ^ 2 . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} F _ q ( Y ) = \\int _ { 0 } ^ { 1 } \\left ( \\partial _ Y R ( Y _ q + t Y ) - \\partial _ Y R ( Y _ q ) \\right ) Y \\dd t . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{gather*} ( 1 - x ^ m ) A _ { ( m + 1 ) p _ n + r _ n } = A _ { ( m + 1 ) p _ n - ( m - r _ n ) } + x ^ { 2 m - 2 r _ n } A _ { ( m + 1 ) p _ n - r _ n } ^ { m \\rightarrow m + 1 } ( x ) + x ^ { m - 1 } A _ { ( m + 1 ) p _ n + r _ n - m - 1 } . \\end{gather*}"}
-{"id": "8232.png", "formula": "\\begin{align*} W _ m ( r ) = \\frac { j } { r } - \\bigskip \\frac { 1 } { \\ell _ { \\lambda } } \\ ; . \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} \\varphi ^ h _ A ( x ) = \\left ( \\begin{array} { c } \\Phi _ { 2 3 } v _ 3 ' + \\Phi _ { 1 3 } w \\\\ \\Phi _ { 1 2 } v _ 2 ' + \\Phi _ { 2 3 } w \\\\ \\Phi _ { 1 3 } v _ 2 ' + \\Phi _ { 1 2 } v _ 3 ' \\end{array} \\right ) + h x _ 3 \\left ( \\begin{array} { c } - \\Phi _ { 2 3 } v _ 2 ' - \\Phi _ { 1 2 } w \\\\ \\Phi _ { 1 3 } v _ 2 ' + \\Phi _ { 1 2 } v _ 3 ' \\\\ \\Phi _ { 1 3 } v _ 3 ' + \\Phi _ { 2 3 } w \\end{array} \\right ) \\ , , \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} X \\left ( t , ( 0 , \\sqrt { h } ) \\right ) = \\frac { \\sqrt { h } } { 2 } ( e ^ { 2 t } - e ^ { - 2 t } , e ^ { 2 t } + e ^ { - 2 t } ) \\ \\ \\ t \\in [ - \\sigma ^ s ( h ) , \\sigma ^ s ( h ) ] , \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} = c ( \\vec { s } , k ) \\int _ { \\Delta _ k } \\frac { d \\vec { v } } { \\sqrt { G ( ( I - \\widetilde { P } ) { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ 0 ; v _ 1 ] } , \\ldots , ( I - \\widetilde { P } ) { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ 0 ; v _ k ] } , { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ 0 ; 1 ] } ) } } , \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} E ' \\ ; : \\ ; y '^ 2 = x '^ 3 + \\frac { a } { \\pi ^ { 2 \\alpha } } x '^ 2 + \\frac { b } { \\pi ^ { 4 \\alpha } } x ' , \\upsilon _ F ( \\frac { a } { \\pi ^ { 2 \\alpha } } ) > 0 , \\upsilon _ F ( \\frac { b } { \\pi ^ { 4 \\alpha } } ) = 0 \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} P _ { S | V , Y } ( 1 | v , y ) = 1 . \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} M _ x [ a _ 1 - 1 ] & = M _ x [ \\Phi _ B ( T _ { \\delta } ) ^ 2 ] + M _ x [ r _ 1 ( T _ { \\delta } ) ] , M _ x [ ( a _ 1 - 1 ) ^ 2 ] = M _ x [ \\Phi _ B ( T _ { \\delta } ) ^ 2 ] + 2 M _ x [ \\Phi _ B ( T _ { \\delta } ) ^ 3 ] + \\mathtt { Q } _ 2 ( T _ { \\delta } ) , \\\\ [ 1 . 5 m m ] M _ x [ ( a _ 1 - 1 ) ^ 3 ] & = 4 M _ x [ \\Phi _ B ( T _ { \\delta } ) ^ 3 ] + \\mathtt { Q } _ 3 ( T _ { \\delta } ) , \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} e _ 4 ( x ) = e _ 5 ( x ) + e _ 6 ( x ) \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } \\sum _ { \\beta , \\alpha = 1 } ^ d \\varphi _ { \\beta , \\alpha } \\bar { n } _ \\alpha \\left ( \\nu _ \\beta - \\bar { n } _ \\beta \\langle \\bar { n } , \\nu \\rangle \\right ) d S = 0 , \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} \\lambda \\leq \\lambda _ 0 : = B \\sqrt { \\tilde C / ( 3 2 V _ \\infty ^ 2 ( \\tilde C + 2 ) ) } , \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\nu _ x ^ 2 ( w , \\sigma ) : = \\sup _ { \\delta \\leq \\sigma } \\int _ { \\R } | w ( x + \\delta ) - w ( x ) | ^ 2 \\ , d x \\leq C \\ , \\sigma ^ { 2 \\gamma } . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} Z _ 1 = \\ln \\left ( { P _ t } G _ \\mathrm { M } ^ 2 \\beta \\right ) - \\left ( f _ \\mathrm { P r } \\left ( r \\right ) \\alpha _ { \\mathrm { L o S } } + \\left ( 1 - f _ \\mathrm { P r } \\left ( r \\right ) \\right ) \\alpha _ { \\mathrm { N L o S } } \\right ) \\ln r , \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} v ^ n ( \\cdot , t ; s ) = & e ^ { H t } v ^ n ( \\cdot , 0 ; s ) + \\int _ 0 ^ t e ^ { H ( t - \\tau ) } a _ n ( \\tau , \\cdot ; s ) v ^ n ( \\cdot , \\tau ; s ) d \\tau \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} d ^ * ( A _ { x _ o } ) \\geq d ^ * ( A _ o ) \\geq \\lambda ( A _ o ) = \\mu ( A ) = \\xi ( A _ { x _ o } ) = d ^ * ( A _ { x _ o } ) , \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} a _ { 2 } = \\frac { 3 b p _ 1 w _ 1 } { ( 1 + i \\tan \\beta ) ( 1 - [ 2 ] _ { q } ) } , \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} F ( c , 0 , p ) = 2 4 - 6 c ^ 2 + c \\left ( c ^ 2 + 4 \\right ) . \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} d _ p = \\begin{cases} 0 & i f ~ i = p , \\\\ i + 2 & i f ~ 0 \\leq i \\leq p - 2 , \\\\ p & i f ~ i = p - 1 . \\end{cases} \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\bar \\xi ^ 2 \\bar c _ { n - 1 } \\bar c _ n = \\bar c _ { n - 1 } \\bar c _ n \\bar \\xi ^ 2 = 0 , \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} \\left \\Vert \\bar { \\lambda } ( \\hat { \\gamma } _ { \\ell } ) - \\bar { \\lambda } ( \\gamma _ { 0 } ) \\right \\Vert ^ { 2 } & = \\int [ \\int _ { - \\infty } ^ { \\hat { \\gamma } ( x ) - \\gamma _ { 0 } ( x ) } f ( u | x ) d u - \\int _ { - \\infty } ^ { 0 } f ( u | x ) d u ] ^ { 2 } F _ { 0 } ( d x ) \\\\ & \\leq \\int C | \\hat { \\gamma } ( x ) - \\gamma _ { 0 } ( x ) | ^ { 2 } F _ { 0 } ( d x ) = C \\left \\Vert \\hat { \\gamma } - \\gamma _ { 0 } \\right \\Vert ^ { 2 } . \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} A = \\frac { \\zeta ( 2 ) \\zeta ( 3 ) } { \\zeta ( 6 ) } = \\frac { 3 1 5 \\zeta ( 3 ) } { 2 \\pi ^ 4 } , B = \\sum _ { p \\in \\P } \\frac { \\log p } { p ^ 2 - p + 1 } . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\bar c _ { - } ( | \\nabla u | ) = 0 \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} \\ddot { u } + \\omega ^ 2 u = 0 , \\ddot { v } + \\omega ^ 2 v = 0 , \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} \\partial _ { t } u ^ { r } _ { t } ( x ) + B ^ { r k } _ { j } D _ { j } u ^ { k } _ { t } ( x ) = g ^ { r } _ { t } ( x ) \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} I ( U ; Y ) = 0 . \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} I \\left ( \\frac { u } { v } \\right ) = \\mathop { \\sum \\sum } _ { u m \\neq v n } \\Psi \\left ( T \\log { \\frac { u m } { v n } } \\right ) \\frac { \\lambda ( m ) \\lambda ( n ) } { \\sqrt { m n } } h ( m ) h ( n ) \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} V _ t ( y , z ) : = \\exp \\left ( \\int _ 0 ^ t \\Lambda ( s , y _ s , z _ s ) d s \\right ) \\textrm { f o r a n y } \\ t \\in [ 0 , T ] \\ . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Delta | \\nabla ^ { m } h | ^ 2 & \\leq - | \\nabla ^ { m + 1 } h | ^ 2 + C _ m \\sum _ { i = 0 } ^ m ( 1 + r ) ^ { - 2 - i } | \\nabla ^ { m - i } h | | \\nabla ^ { m } h | . \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} A _ w = \\{ b \\in B | \\forall \\ x \\in X , \\ ( b _ x ) ^ { p ^ m } \\in A _ x + R ( B _ x ) \\ \\mbox { f o r \\ s o m e } \\ m \\in \\N \\} . \\end{align*}"}
-{"id": "1809.png", "formula": "\\begin{align*} \\ell _ n ( G ) = \\sum _ { i = 1 } ^ n \\log f ( x _ i ; G ) . \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} n ^ { \\rm i m p } _ { i a } = \\frac { n _ i ^ { \\rm i m p } + s _ a d _ i ^ { \\rm i m p } } { 2 } = - s _ a F ( \\lambda _ { i a } | \\lambda _ p ) \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} \\partial _ i \\left ( \\frac { b _ 3 - m _ 3 } { m _ 3 } \\right ) [ \\hat { \\imath } ] = \\frac { 1 } { m _ 3 ^ 2 } \\left [ m _ 3 \\partial _ i ( b _ 3 - m _ 3 ) [ \\hat { \\imath } ] - ( b _ 3 - m _ 3 ) \\partial _ i m _ 3 [ \\hat { \\imath } ] \\right ] . \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} n _ 1 = \\left \\lceil \\frac { k ( \\delta - 1 + l ) } { r } \\right \\rceil = l + \\left \\lceil \\frac { k ( \\delta - 1 ) } { r } \\right \\rceil \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} \\bold { p r } ^ { ( k ) } X ( F _ { \\alpha } ( x , [ u ] ) ) = 0 , \\{ F _ { \\alpha } ( x , [ u ] ) = 0 \\} , \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} u ' = u ^ 2 ( u _ 1 - u _ { - 1 } ) . \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\tau _ { \\varphi } \\Big ( \\sum _ { i = 1 } ^ n \\xi _ i \\Big ) ^ r \\le \\sum _ { i = 1 } ^ n \\tau _ { \\varphi } ( \\xi _ i ) ^ r ; \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} { \\overline { R } } _ { e ^ { * } } & = \\mathbb { E } \\left [ { { \\log } _ 2 } \\left ( { 1 + { \\gamma _ { e ^ { * } } } } \\right ) \\right ] \\\\ & = \\frac { 1 } { { \\ln 2 } } \\int _ 0 ^ \\infty { \\frac { { \\left ( { 1 - { F _ { { \\gamma _ { { e ^ * } } } } } \\left ( x \\right ) } \\right ) } } { { 1 + x } } d x } , \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} \\frac { \\partial \\Pi _ i } { \\partial z ^ j } ( x , y , z ) & = \\frac { \\left [ ( y \\times \\nu _ i ( x ) ) \\otimes ( y \\times \\nu _ i ( x ) ) \\right ] ( \\delta _ { 1 j } , \\delta _ { 2 j } , \\delta _ { 3 j } ) } { \\abs { y \\times \\nu _ i ( x ) } ^ 2 } \\\\ \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} \\alpha = A + \\varepsilon \\cdot \\alpha _ { 0 } , \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} - a _ 0 ( { \\cal K } * { \\cal L } ) = ( - a _ 0 { \\cal K } ) \\cdot ( - a _ 0 { \\cal L } ) . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} \\begin{cases} \\dot X ^ \\varepsilon ( t ) = \\cfrac { 1 } { \\varepsilon } \\ , b ( X ^ \\varepsilon ( t ) ) + \\beta ( X ^ \\varepsilon ( t ) ) \\ \\ \\ t \\in \\mathbb R , \\\\ X ^ \\varepsilon ( 0 ) = x \\in \\Omega \\setminus \\{ 0 \\} . \\end{cases} \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} C _ q = \\begin{pmatrix} 0 & 0 & - q ^ { - 1 / 2 } \\\\ 0 & 1 & 0 \\\\ q ^ { 1 / 2 } & 0 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} u \\wedge w ( x ) = \\langle u , x \\rangle w - \\langle w , x \\rangle u . \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} Z ( \\varphi ) : = ( Z _ 1 , Z _ 2 , Z _ 3 ) ( \\varphi ) : = \\mathcal { F } ( i _ 0 , \\zeta _ 0 ) ( \\varphi ) = \\omega \\cdot \\partial _ { \\varphi } i _ 0 ( \\varphi ) - X _ { H _ { \\varepsilon , \\zeta _ 0 } } ( i _ 0 ( \\varphi ) ) , \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} \\mathcal { T } _ k ^ * = \\{ S \\cup \\{ v \\} : S \\in \\mathcal { T } _ { k - 1 } v \\in N _ G ( S ) \\} . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} - \\frac { \\sqrt { N } } { 4 \\pi } ( \\theta _ { 1 / 2 } , \\theta _ { 1 / 2 } ) & = 4 \\sum _ { n \\geq 0 } a ( - n ) \\sigma _ { 1 } ( n ) - 4 \\sum _ { \\substack { b > 0 \\\\ b \\equiv 0 ( N ) } } b \\sum _ { n > 0 } a ( - b n ) - 2 \\sum _ { \\substack { b > 0 \\\\ b \\not \\equiv 0 ( N ) } } b \\sum _ { n > 0 } a ( - b n ) \\\\ & = 4 \\sigma _ { 1 } ( 0 ) + 2 \\sum _ { n > 0 } a ( - n ) ( \\sigma _ { 1 } ( n ) - N \\sigma _ { 1 } ( n / N ) ) = - \\frac { 1 + N } { 1 2 } . \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} 2 x _ { u } x _ { v } = 2 \\alpha x _ { u } x _ { v } + 2 \\left ( 1 - \\alpha \\right ) x _ { u } x _ { v } \\leq \\alpha x _ { u } ^ { 2 } + 2 \\left ( 1 - \\alpha \\right ) x _ { u } x _ { v } + \\alpha x _ { v } ^ { 2 } . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} \\mathcal { A } _ n = \\begin{pmatrix} c _ { 1 1 } & 0 \\\\ 0 & c _ { 2 2 } \\end{pmatrix} + \\begin{pmatrix} - \\sqrt { a _ n b _ n } & 0 \\\\ 0 & \\sqrt { a _ n b _ n } \\end{pmatrix} + U _ n \\mathcal { D } _ n U _ n ^ { - 1 } . \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} x = \\sum _ { ( i , p ) = 1 } ^ d [ b _ i X ^ { i } ] = \\sum _ { ( i , p ) = 1 } ^ d [ b _ i ] [ X ^ { i } ] \\in W ( K ) \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} k ! \\widetilde { F } _ k ( s ) = F ^ k ( s , \\chi _ 0 ) + \\sum _ { m = 0 } ^ { k - 2 } F ^ m ( s , \\chi _ 0 ) F _ { \\boldsymbol { n _ m } } ( s ; \\chi _ 0 ) , \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} a ^ { ( l , \\sigma ) } _ + ( y ) - a ^ { ( l , \\sigma ) } _ - ( y ) = ( 1 - 1 6 \\varepsilon ) 8 L = l _ { B ^ { ( l , \\sigma ) } } \\ ; . \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} \\mathcal { I } _ s ^ q ( \\mu ) \\ : = \\ \\int \\left ( \\int \\frac { d \\mu ( x ) } { \\lvert x - y \\rvert ^ s } \\right ) ^ { q - 1 } d \\mu ( y ) \\ < \\ \\infty \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} f ( x ; G ; \\sigma ^ 2 ) = \\sum _ { j = 1 } ^ m \\alpha _ j \\phi ( x ; \\theta _ j , \\sigma ^ 2 ) \\leq \\max _ j \\phi ( x ; \\theta _ j , \\sigma ^ 2 ) . \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\log ^ + ( | x | ) \\ , d \\rho ( x ) = \\int _ { \\mathbb { R } ^ d } \\log ^ + ( | x | ) e ^ { \\frac { 2 } { \\sigma ^ 2 } ( \\frac { 1 } { 2 } | x | ^ 2 - \\frac { 1 } { 4 } | x | ^ 4 ) } \\ , d x < \\infty \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} \\frac { d } { d t } D \\varphi _ t ( \\omega , x ) = D b ( \\varphi _ t ( \\omega , x ) ) D \\varphi _ t ( \\omega , x ) , D \\varphi _ 0 ( \\omega , x ) = \\textrm { I d } . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} & h ( \\theta ) = \\frac { A ( e ^ { i \\theta } ) f ( \\theta ) - C ( e ^ { i \\theta } ) } { f ( \\theta ) + g ( \\theta ) } = \\\\ & = A ( e ^ { i \\theta } ) - \\frac { A ( e ^ { i \\theta } ) g ( \\theta ) + C ( e ^ { i \\theta } ) } { f ( \\theta ) + g ( \\theta ) } , \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} \\{ p _ \\gamma \\} _ { \\gamma > 0 } : = \\left \\{ S _ 0 ^ * \\left ( \\alpha ( S u _ { \\gamma } - z ) + \\beta _ 1 C ^ * _ { \\omega _ R } \\mu _ { \\gamma } ^ + + \\beta _ 2 C ^ * _ { \\omega _ T } \\mu _ { \\gamma } ^ - \\right ) \\right \\} _ { \\gamma > 0 } \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} A f ( x ) = \\sum _ { [ \\xi ] \\in \\widehat { G } } d _ { \\xi } [ \\xi ( x ) \\sigma _ A ( x , \\xi ) \\widehat { f } ( \\xi ) ] . \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} u _ 0 ( \\mathbf { a } ; \\mathbf { z } ) = \\frac { z _ 1 + \\cdots + z _ l } { \\phi ( q ) } u ( \\mathbf { a } ; \\mathbf { z } ) , u ( \\mathbf { a } ; \\mathbf { z } ) : = \\frac { H ( 1 ; \\mathbf { a } , \\mathbf { z } ) } { \\Gamma \\ ( \\frac { z _ 1 + \\cdots + z _ l } { \\phi ( q ) } + 1 \\ ) } . \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} H ( \\tau + 1 ) = \\begin{pmatrix} \\zeta _ { 2 4 } ^ { - 1 } & 0 & 0 \\\\ 0 & 0 & \\zeta _ { 3 } \\\\ 0 & \\zeta _ { 3 } & 0 \\end{pmatrix} H ( \\tau ) , H \\left ( - \\frac { 1 } { \\tau } \\right ) = \\sqrt { - i \\tau } \\begin{pmatrix} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & - 1 \\end{pmatrix} H ( \\tau ) . \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} e ^ \\ast _ j A \\ ; = \\ ; e ^ \\ast _ j B \\ , \\oplus \\ , e ^ \\ast _ j J _ { j } , \\qquad \\mbox { f o r a l l } \\ , 0 \\le j \\le n . \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} Q ^ { + } = \\cup _ { u \\in U } Q u Q ^ { - } = \\cap _ { u \\in U } Q u . \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} \\int _ M \\left ( | \\widetilde \\nabla ^ g \\psi | ^ 2 - | \\widetilde D ^ g \\psi | ^ 2 + \\frac { R _ g - \\varepsilon n ( n - 1 ) } { 4 } | \\psi | ^ 2 \\right ) d M = \\int _ { \\Sigma } \\left \\langle { \\widetilde D } ^ \\gamma \\psi - \\frac { H } { 2 } \\psi , \\psi \\right \\rangle d \\Sigma , \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} \\lim _ { a \\to 0 } \\bar z ( a ) = + \\infty \\lim _ { a \\to + \\infty } \\bar z ( a ) = 0 . \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} \\sum _ { p \\in \\P } \\left | \\frac 1 { f ( p ) } - \\frac 1 { p } \\right | < \\infty , \\sum _ { p \\in \\P } \\sum _ { \\nu = 2 } ^ { \\infty } \\frac 1 { | f ( p ^ \\nu ) | } < \\infty . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} I ( S ; S _ 1 ) = I ( S ; S _ 1 , E _ \\sigma ) = I ( S ; S _ 1 | E _ \\sigma ) = \\bar { \\sigma } I ( S ; S | E _ \\sigma = 0 ) + \\sigma I ( S ; ? | E _ \\sigma = 1 ) = \\bar { \\sigma } H ( S ) = \\bar { \\sigma } \\big [ 1 - h ( \\alpha ) \\big ] . \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} \\frac { 1 } { p } = \\frac { j } { d } + \\left ( \\frac { 1 } { r } - \\frac { m } { d } \\right ) \\alpha + \\frac { 1 - \\alpha } { q } , \\frac { j } { m } \\leq \\alpha \\leq 1 . \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} [ x ] _ b \\triangleq \\lfloor x \\rfloor + \\sum _ { i = 1 } ^ b 2 ^ { - i } a _ i , \\end{align*}"}
-{"id": "2261.png", "formula": "\\begin{align*} \\phi ( ( W _ { 1 } ) ) = \\phi \\left ( \\frac { \\mu ( m _ { 1 } m _ { 2 } ) } { \\langle m _ { 1 } , m _ { 2 } \\rangle _ { B _ { 1 } } } \\right ) = \\frac { \\mu \\phi ( m _ { 1 } m _ { 2 } ) } { \\langle m _ { 1 } , m _ { 2 } \\rangle _ { B _ { 1 } } } = \\frac { \\mu \\phi ( m _ { 1 } m _ { 2 } ) } { \\langle \\phi ( m _ { 1 } ) , \\phi ( m _ { 2 } ) \\rangle _ { B _ { 2 } } } . \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} T _ { \\nu } f \\left ( x , t \\right ) = \\left ( f \\ast \\nu \\right ) \\left ( x , t \\right ) = \\int _ { \\mathbb { R } ^ { 2 n } } f \\left ( \\left ( x , t \\right ) \\cdot \\left ( w , \\varphi \\left ( w \\right ) \\right ) ^ { - 1 } \\right ) \\eta \\left ( w \\right ) d w . \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} { P _ { m _ k , i } } g \\left ( \\tau ^ { ( k ) } \\right ) = \\sum \\limits _ { j = 0 } ^ { { m _ k } } { a _ { i , j } ^ { ( k ) } \\hat G _ { j , k } ^ { \\left ( \\alpha _ i ^ { ( k ) , * } \\right ) } \\left ( { \\tau ^ { ( k ) } } \\right ) } \\forall i , \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} N ( T ) = \\frac { T } { \\pi } \\log { Q T } + O ( T ) \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} \\min _ { \\mu \\in Q } \\tilde { P } ( \\mu ) = \\max _ { \\substack { \\eta \\in X \\\\ \\eta ' ( 0 ) = - h ^ { 2 } } } \\tilde { D } ( \\eta ) . \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\mathcal { I } _ { \\mathrm { L o S } } & = \\sum \\nolimits _ { i \\in \\Phi _ { \\mathrm { L o S } } } { { P _ t } { G _ i } L \\left ( \\left | X _ i \\right | \\right ) } , \\\\ \\mathcal { I } _ { \\mathrm { N L o S } } & = \\sum \\nolimits _ { i \\in \\Phi _ { \\mathrm { N L o S } } } { { P _ t } { G _ i } L \\left ( \\left | X _ i \\right | \\right ) } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\pi \\in \\P _ { 3 , 4 } , \\\\ | \\pi | = N } } ( - 1 ) ^ { \\nu _ e ( \\pi ) } - \\sum _ { \\substack { \\pi \\in \\P _ { 2 , 4 } , \\\\ | \\pi | = N } } ( - 1 ) ^ { \\nu _ e ( \\pi ) } = \\chi ( N \\not = \\square ) . \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\int _ 1 ^ { \\infty } ( 1 + x ) ^ { \\lambda - 1 } e ^ { - \\frac { c } { K ^ 2 } n x } d x \\le \\int _ 1 ^ { \\infty } e ^ { x \\frac { c } { 2 K ^ 2 } n ( t - 2 ) } d x = O ( 1 ) . \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} M _ { - j } : = e ^ \\ast _ { j } M , \\quad \\mbox { f o r } \\ ; j \\in \\{ 0 , 1 , \\cdots , n \\} . \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} \\epsilon ( 1 / 2 , \\rho _ k , \\psi ) = \\gamma ( 1 / 2 , \\rho _ k , \\psi ) = 1 , \\ \\ \\ k \\in \\Z . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} c ^ * ( \\ell ) = \\sum _ { m n = \\ell } \\rho ( m ) \\lambda ( n ) \\left ( 1 - \\frac { \\log { m } } { \\log { M } } \\right ) ^ r \\left ( \\frac { \\log { n } } { \\log { N } } - \\delta \\right ) . \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} ( \\mathcal L ^ { \\bar \\nabla } _ v D \\pi ) _ i ^ a = \\sum _ b \\phi ^ a { } _ b ( D \\pi ) _ i ^ b \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} \\int _ { { \\tau _ { k - 1 } } } ^ { { y _ i ^ { ( k ) } } } { g \\left ( { { \\tau ^ { ( k ) } } } \\right ) \\ , d { \\tau ^ { ( k ) } } } = \\sum \\limits _ { j = 0 } ^ { { m _ k } } { _ k { p } _ { O B , i , j } ^ { ( 1 ) } \\ , g \\left ( { \\hat z _ { { m _ k } , i , j } ^ { ( k ) , \\alpha _ i ^ { ( k ) , * } } } \\right ) } + E _ { { m _ k } } ^ { \\left ( { \\alpha _ i ^ { ( k ) , * } } \\right ) } \\left ( { y _ i ^ { ( k ) } , \\xi _ i ^ { ( k ) } } \\right ) , \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} X = e ^ { - L } x - \\frac { ( 1 + e ^ { - 2 L } ) r } { \\sqrt 2 } , Y = e ^ L y - \\frac { ( 1 + e ^ { 2 L } ) r } { \\sqrt 2 } , W = e ^ L w , Z = e ^ { - L } z , \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} x _ { c , t } & = \\left \\{ \\begin{array} { r l } 1 & c t \\\\ 0 & \\end{array} \\right . \\\\ y _ { a , m , t } & = \\left \\{ \\begin{array} { r l } 1 & a t m \\\\ 0 & \\end{array} \\right . \\\\ \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\mathcal { G } _ q ( \\mathcal { H } ) : = \\bigoplus _ { n = 0 } ^ \\infty \\mathcal { H } ^ { \\otimes n } [ n ] _ q ! \\ , . \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} B ^ a ( \\xi , - \\eta ) = - i \\xi + S ( d ^ 2 \\rho ^ { - 1 } ) . \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} W _ \\ell ( N ) \\ = \\ \\bigoplus _ { m \\le \\ell } V _ m \\ , . \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { k = 1 } ^ { n - 1 } \\frac { 1 } { k ^ { \\omega } ( n ^ \\gamma - k ^ \\gamma ) } & = \\left ( \\sum _ { k = 1 } ^ { [ \\frac n 2 ] } + \\sum _ { k = [ \\frac n 2 ] } ^ { n - 1 } \\right ) \\frac { 1 } { k ^ { \\omega } ( n ^ \\gamma - k ^ \\gamma ) } \\\\ & \\leq C \\left ( \\frac { 1 } { n ^ \\gamma } \\sum _ { k = 1 } ^ { n } \\frac { 1 } { k ^ { \\omega } } + \\frac { 1 } { n ^ \\omega } \\sum _ { k = [ \\frac n 2 ] } ^ { n - 1 } \\frac { 1 } { n ^ \\gamma - k ^ \\gamma } \\right ) \\end{aligned} \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} \\left \\| { A } \\xi - \\hat { A } \\xi \\right \\| _ 2 ^ 2 = \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } \\left | \\sum \\limits _ { j = 0 } ^ { \\infty } ( \\bold { B } ^ { - 1 } \\bold { a } ) _ j e ^ { i j \\theta } \\right | ^ 2 f ^ { - 1 } ( \\theta ) d \\theta = < \\bold { B } ^ { - 1 } \\vec { \\bold { a } } , \\vec { \\bold { a } } > , \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} f ( x , \\phi ; \\theta ) = 4 x \\cos \\phi - 2 x ^ 2 \\cos ( 2 \\phi - \\theta ) . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} \\frac { | E ( X , V \\setminus X ) | } { \\frac { 1 } { 2 } \\big ( ( n - i ) d - | E ( X , V \\setminus X ) | \\big ) + | E ( X , V \\setminus X ) | } = \\frac { 2 | E ( X , V \\setminus X ) | } { ( n - i ) d + | E ( X , V \\setminus X ) | } . \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{gather*} \\sum _ { k = 2 ^ n } ^ { 2 ^ { n + 1 } - 1 } a _ k k ^ { i t } = \\sum _ { k = 2 ^ n } ^ { 2 ^ { n + 1 } - 1 } ( S _ { k , n } - S _ { k - 1 , n } ) k ^ { i t } = \\sum _ { k = 2 ^ n } ^ { 2 ^ { n + 1 } - 1 } S _ { k , n } ( k ^ { i t } - ( k + 1 ) ^ { i t } ) + 2 ^ { ( n + 1 ) i t } S _ { 2 ^ { n + 1 } - 1 , n } \\ , . \\end{gather*}"}
-{"id": "1123.png", "formula": "\\begin{align*} E _ 0 ( \\gamma , \\rho ) & = - \\log m _ { \\lambda , \\rho } ( w _ 1 , w _ 2 ) | _ { w _ 1 = w _ 2 = \\gamma k _ n , \\lambda = \\frac { 1 } { 1 + \\rho } } . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} F ( x ) = A W _ { \\frac { Z \\alpha _ g \\epsilon } { w } , \\nu - \\frac { 1 } { 2 } } \\left ( x \\right ) \\ ; , \\bigskip G ( x ) = B W _ { \\frac { Z \\alpha _ g \\epsilon } { w } , \\nu + \\frac { 1 } { 2 } } \\left ( x \\right ) \\ ; , \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} S _ { n } ^ { + } = \\left ( \\begin{array} { c c c c } 0 & f ( 1 ) & & \\\\ & \\ddots & & \\\\ & & \\ddots & f ( 2 s ) \\\\ & & & 0 \\end{array} \\right ) , S _ { n } ^ { - } = \\left ( \\begin{array} { c c c c } 0 & & & \\\\ f ( 1 ) & \\ddots & & \\\\ & \\ddots & \\ddots & \\\\ & & f ( 2 s ) & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{gather*} A ( h _ { + } ) - C ( h _ { + } ) = 4 \\left ( s - v \\right ) \\times \\\\ \\left ( \\dfrac { K ( t ^ { 2 } - s ^ { 2 } ) ( s - 2 h _ { + } ) } { 2 s ^ { 2 } \\left ( s ^ { 2 } + t ^ { 2 } \\right ) } + \\dfrac { 2 w \\left ( v t - w s \\right ) } { s } h _ { + } - \\allowbreak w \\left ( v t - w s \\right ) \\right ) \\end{gather*}"}
-{"id": "7082.png", "formula": "\\begin{align*} A _ { i j } = \\Phi ( \\boldsymbol { x } _ i , \\boldsymbol { x } _ j ) , i , j = 1 , \\ldots , N . \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} \\mathcal { \\hat { B } } _ { - } ( \\lambda ) = \\frac { ( \\zeta _ { - } - 1 / \\zeta _ { - } ) } { \\kappa _ { - } e ^ { \\tau _ { - } } ( \\lambda ^ { 2 } / q - q / \\lambda ^ { 2 } ) \\prod _ { n = 1 } ^ { \\mathsf { N } } \\alpha _ { n } \\beta _ { n } } \\mathcal { B } _ { - } ( \\lambda ) , \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} { \\rho } = \\omega _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\frac { N _ { \\Lambda } } { V } ) = \\frac { 1 } { V ( \\exp ( - \\beta \\mu ) - 1 ) } + \\frac { | \\lambda | ^ { 2 } } { \\mu ^ { 2 } } + \\\\ \\frac { 1 } { V } \\sum _ { { k } \\ne { 0 } } \\frac { 1 } { \\exp ( \\beta ( \\epsilon _ { { k } } - \\mu ) ) - 1 } \\ . \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} \\omega = i \\{ \\theta , \\theta \\} , \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} \\begin{cases} \\dot X ( t ) = b ( X ( t ) ) \\ \\ \\ t \\in \\mathbb R , \\\\ X ( 0 ) = x \\in \\mathbb R ^ 2 , \\end{cases} \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} \\Gamma ^ k _ { i j } - \\Gamma ^ k _ { j i } = C _ { i j } ^ k \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} - \\lim _ { q \\to \\infty } \\frac { \\ln \\left ( \\textrm { P r } \\{ Q ( \\infty ) > q \\} \\right ) } { q } = \\theta , \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} X _ c : = \\left \\{ u \\in H ^ 1 ( \\mathbb { R } ) \\cap L ^ 2 _ 1 ( \\mathbb { R } ) : \\langle V , u \\rangle _ { L ^ 2 } = 0 \\right \\} , \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ s \\Bigl ( \\| f _ j \\| ^ 2 _ { 1 , } + \\frac { 1 } { \\gamma _ j } \\| f _ j \\| ^ 2 _ { 2 , } \\Bigr ) = \\prod _ { j = 1 } ^ s \\Bigl ( 1 + \\frac { 3 } { \\gamma _ j } \\Bigr ) , \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{gather*} \\frac { d ^ 2 W } { d z ^ 2 } = - 3 W \\frac { d W } { d z } - W ^ 3 + v ( z ) \\left \\{ \\frac { d W } { d z } + W ^ 2 \\right \\} , \\end{gather*}"}
-{"id": "8620.png", "formula": "\\begin{align*} R _ { \\mathrm { C E G } } \\left ( P _ { T , X | S } \\right ) = \\min \\Big \\{ I _ P ( T ; Y | S ) , I _ P ( T ; Y | S ) - I _ P ( T ; Z | S ) + H _ P ( S | Z ) \\Big \\} . \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} \\pi _ 2 T _ q ( t - \\tau ) = S _ 2 ( t - \\tau ) \\pi _ 2 + \\int _ { \\tau } ^ { t } S _ 2 ( t - s ) \\pi _ 2 B _ q T _ q ( s - \\tau ) \\dd s , 0 \\le \\tau \\le t . \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} S I N R _ { i j k } = & \\frac { { \\bf u } ^ { T } _ { i j k } { \\bf T } _ { i j k } { \\bf u } _ { i j k } } { { \\bf u } _ { i j k } ^ { T } { \\bf F } _ { i j k } { \\bf u } _ { i j k } } , \\ \\forall \\{ i , j \\} \\in \\mathcal { L } , \\ \\forall k \\in \\mathcal { K } , \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} D ^ H = D ^ h + \\theta + \\theta ^ * , \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} R _ \\ast u ( \\overline U _ 1 , \\ldots , \\overline U _ { q } ) = \\sum _ { j = 1 } ^ q u ( \\overline U _ 1 , \\ldots , R _ \\ast \\overline U _ j , \\ldots , \\overline U _ q ) \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} C = X + \\partial S \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\gamma _ { 1 } ( \\mathbf { x } _ 0 ) = \\beta _ { 1 } \\mathbf { x } _ 0 ^ \\dagger \\mathbf { G } _ 1 ^ \\dagger \\mathbf { G } _ 1 \\mathbf { x } _ 0 / \\left ( n _ { t } \\sigma ^ 2 \\right ) . \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} V ( x ) = x ^ 6 + 6 a | x | ^ 5 + ( 9 a ^ 2 - 4 b ) x ^ 4 - 1 2 a b | x | ^ 3 + ( 4 b ^ 2 - 3 ) x ^ 2 - 6 a | x | \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} [ A , B ] = 0 , A F - F A ' = 0 , B F - F B ' = 0 . \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} \\rho ( \\alpha , B _ { n _ k } ( \\theta + \\alpha ) \\tilde { A } _ { | d | , E } ( \\theta ) B ^ { - 1 } _ { n _ k } ( \\theta ) ) = \\rho ( \\alpha , \\tilde { A } _ { | d | , E } ) + \\deg { B _ { n _ k } } \\alpha . \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} \\bold { p r } [ X _ 1 , X _ 2 ] = [ \\bold { p r } X _ 1 , \\bold { p r } X _ 2 ] . \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} R ^ \\mathrm { R L N } _ \\mathsf { A } ( Q _ { U , V , X | S } ) = \\min \\Big \\{ I ( V ; Y , S _ 1 | U ) - I ( V ; Z , S _ 2 | U ) , I & ( U , V ; Y , S _ 1 ) - I ( U , V ; S ) \\\\ & , I ( U , V ; Y , S _ 1 ) - I ( U ; S ) - I ( V ; Z , S _ 2 | U ) \\Big \\} \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { 2 } { p } \\right ) ^ r = 1 \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} E _ k = \\big \\{ ( j , i ) : ~ a _ j ^ i ( k ) > 0 \\big \\} , \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{gather*} \\frac { 1 + q _ 2 } { 2 } u \\psi ^ 2 = - \\frac { 1 + q _ 2 } { 2 } , \\end{gather*}"}
-{"id": "5485.png", "formula": "\\begin{align*} \\hat { x } _ t = \\sum _ { i = 1 } ^ n \\sigma ^ i ( \\theta _ t ^ i , \\lambda _ t ) = \\frac { \\gamma } { \\eta } \\left [ \\left ( \\frac { \\eta } { \\lambda _ t } \\right ) ^ { \\frac { 1 } { \\gamma } } \\sum _ { i = 1 } ^ n ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } - \\hat { \\phi } \\right ] , \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} ( n _ 2 - n _ 1 ) ( n _ 3 + n _ 4 ) r _ 1 + ( n _ 4 - n _ 3 ) ( n _ 1 + n _ 2 ) r _ 3 = 0 . \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} D _ i = \\partial _ { x ^ i } + \\sum _ { I } \\left ( D _ { i ; I } - \\partial _ { x ^ i } \\right ) . \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\vdash n } \\sum _ { \\lambda _ i \\in \\lambda } J _ \\alpha ( \\lambda _ i ) = \\sum _ { \\lambda \\vdash n } \\sum _ { \\substack { \\lambda _ i \\in \\lambda \\\\ \\lambda _ i } } \\lambda _ i ^ \\alpha = \\sum _ { k = 1 } ^ { n } J _ { \\alpha } ( k ) p ( n - k ) . \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } [ x + y ] _ q } d \\mu _ { q } ( y ) = \\sum _ { n = 0 } ^ \\infty \\beta _ { n , \\lambda , q } ( x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} n ^ { - 4 } \\hat \\varrho = - \\frac { 4 M } { ( r + 2 M ) ^ 3 } . \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} M _ k ( x , \\mathbf { a } ) - \\frac { 1 } { \\phi ^ k ( q ) } S _ k ( x ) & = \\frac { 1 } { \\phi ^ k ( q ) } \\frac { x } { \\log x } \\frac { ( \\log \\log x ) ^ { k - 2 } } { ( k - 2 ) ! } \\bigg \\{ C ( q , a ) + \\frac { k - 2 } { \\log \\log x } \\phi ^ 2 ( q ) \\widetilde { h } \\ ( a ; \\frac { ( k - 3 ) \\phi ( q ) } { \\log \\log x } \\ ) \\\\ & - \\frac { k - 2 } { \\log \\log x } \\widetilde { g } \\ ( \\frac { k - 3 } { \\log \\log x } \\ ) + O _ { A , q } \\ ( \\frac { k } { ( \\log \\log x ) ^ 2 } \\ ) \\bigg \\} . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} \\Lambda ( Z ) & = \\frac { e ^ { \\frac { \\sigma _ { \\mathrm { a } } ^ 2 } { \\zeta } } \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { Z } { v } } e ^ { - v / \\zeta } d v } { \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { Z } { v } } e ^ { - \\frac { v } { \\zeta } } d v } \\mathop { \\gtrless } _ { H _ 0 } ^ { H _ 1 } \\gamma . \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} \\mathcal H ^ { p , q } _ E : = \\{ \\mathcal H ^ { p , q } _ { E _ t } \\} _ { t \\in B } . \\end{align*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\rho ^ { * } H _ n ( \\Lambda , \\xi ) = H _ 0 ( \\Lambda ) , \\rho ^ { * } H _ h ( \\Lambda , \\xi ) = 2 H _ { \\xi } ( \\Lambda ) , \\rho ^ { * } H _ u ( \\Lambda , \\xi ) = H _ { \\zeta } ( \\Lambda ) + H _ { \\zeta ' } ( \\Lambda ) . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{align*} | D | = & \\left | ( 1 - i ) ( \\delta _ 1 ) \\frac { K _ { \\psi } ( - 3 , n , 4 N ) } { 4 N } \\left ( I _ { \\frac 1 2 } \\left ( \\frac { \\pi \\sqrt { 3 n } } { N } \\right ) - \\sqrt { \\frac { 2 \\sqrt { 3 n } } { N } } \\right ) \\right | \\\\ \\leq & \\sqrt { 2 } \\frac { \\delta _ 1 } { 2 } \\frac { \\sqrt { N } } { \\pi \\sqrt { 2 } ( 3 n ) ^ \\frac 1 4 } e ^ { \\frac { \\pi \\sqrt { 3 n } } { N } } \\\\ \\leq & \\frac { 2 \\sqrt { N } } { \\pi ( 3 n ) ^ \\frac 1 4 } e ^ { \\frac { \\pi \\sqrt { 3 n } } { N } } . \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} { x _ 2 } ( 0 ) = - { x _ 2 } ( 1 ) = 1 , \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} \\mathbf { \\Phi _ u } : = [ \\phi _ u , \\phi _ u - 1 , \\phi _ u - 2 , \\ldots , 0 ] \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} \\gamma _ { 2 } = \\frac { \\mu _ { h _ { i } \\cap h _ { j } } ( W _ { 2 } | _ { ( h _ { i } + h _ { j } ) } ) } { \\mu _ { h _ { i } + h _ { j } } ( W _ { 2 } | _ { ( h _ { i } ) \\cap ( h _ { j } ) } ) } . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\beta + 1 = \\omega _ p \\ ! \\left ( \\frac { f ^ p } { F } \\right ) \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} \\widehat { 2 \\Im \\P [ R \\bar R _ \\alpha ] } ( \\zeta ) = \\int _ { \\xi - \\eta = \\zeta } \\hat R ( \\xi ) \\bar { \\hat R } ( \\eta ) K ( \\xi , \\eta ) \\ , d \\eta , \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} D ( u + \\varphi ) & = D ( u ) + 2 \\langle \\nabla u , \\nabla \\varphi \\rangle + D ( \\varphi ) \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} \\phi ^ + ( \\xi ) = \\sum _ { n = 1 } ^ { \\infty } \\xi ^ n \\frac { n \\partial } { \\partial p _ n } , \\phi ^ - ( \\xi ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { p _ n } { \\xi ^ n } , \\end{align*}"}
-{"id": "9416.png", "formula": "\\begin{align*} d \\widetilde { \\omega } = \\sum _ { I , J } ( - 1 ) ^ { | I | + | J | } \\ , d \\widetilde { x } ^ I \\wedge d \\widetilde { \\xi } ^ J \\wedge d \\widetilde { \\omega } _ { I J } . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} \\mathcal { V } _ { i , j } ^ { 2 } = & \\big \\{ \\mathbf { z } \\in \\mathbb { R } ^ 2 | \\cap _ { l \\in \\{ i , j \\} } \\{ \\Vert \\mathbf { z } - \\mathbf { z } _ { l } \\Vert \\leq \\Vert \\mathbf { z } - \\mathbf { z } _ k \\Vert \\} \\\\ & \\forall ~ \\mathbf { z } _ k \\in \\Phi _ { \\mathrm b } \\setminus \\{ \\mathbf { z } _ { i } , \\mathbf { z } _ { j } \\} \\big \\} . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} T _ \\infty ( \\delta ) : = \\inf \\Big \\{ t \\ge 0 : \\max _ { x , y , z \\in V _ N } \\left | \\frac { K _ { 0 , t } ( x , z ) } { K _ { 0 , t } ( y , z ) } - 1 \\right | < \\delta \\Big \\} \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} \\Lambda _ { 2 4 } = & \\left \\{ x \\in L _ { 2 4 } : \\sqrt { 2 } \\sum _ { i = 1 } ^ { 2 4 } x _ i = 0 \\right \\} \\\\ & \\cup \\left \\{ ( 1 , \\ldots , 1 ) + x : x \\in L _ { 2 4 } , \\ , \\sqrt { 2 } \\sum _ { i = 1 } ^ { 2 4 } x _ i = 2 \\right \\} . \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} \\sum _ { \\substack { m \\ge X \\\\ ( m , k ) = 1 } } \\mu ( m ) m ^ { - 1 } \\ll \\sigma _ { - 1 } ( k ) \\exp ( - c \\sqrt { \\log { x } } ) . \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} w ^ h ( x _ 1 ) = \\frac { 1 } { \\mu ( \\omega ) } \\int _ \\omega \\frac { x _ 2 \\hat y _ 3 ^ h - x _ 3 \\hat y _ 2 ^ h } { h ^ 2 } \\dd x ' \\ , , \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} E _ { 3 } - \\widetilde { E } _ { 3 } = U ( I - N _ { 4 } + V ^ { \\ast } N _ { 3 } ) = U \\widehat { E } _ { 3 } ( I + H H ^ { \\ast } E _ { S _ { A } } ) ^ { - 1 } , \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} b _ { 3 } ( \\varphi , y ) = b _ 3 ( \\varphi ) \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} h _ 1 = x ^ { 1 2 } - 3 x ^ { 1 1 } - 3 x ^ { 1 0 } + 3 x ^ 9 + 3 x ^ 5 - 3 x ^ 4 + 3 x ^ 3 + 3 h _ 2 = x ^ { 1 2 } + 3 x ^ 4 + 3 ; \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} \\ell _ i : = \\log \\left ( 1 + | h _ { i s } | ^ 2 \\right ) , \\forall i \\in [ 1 : N ] , \\\\ r _ i : = \\log \\left ( 1 + | h _ { d i } | ^ 2 \\right ) , \\forall i \\in [ 1 : N ] . \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} B ( m _ 1 , { \\bf R } , d ) & \\le ( 3 + b _ { \\bf R } ) ^ { p / g ( { \\bf R } ) } \\ , \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) \\le ( 4 d ^ { 1 / ( 2 p ) } ) ^ { p / g ( { \\bf R } ) } \\ , \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) \\\\ & \\le \\Big ( 4 ^ { p / v } 2 ^ { u / v } ( 1 + 1 / ( 2 v ) ) ^ { u / ( 2 v ^ 2 ) } ( 2 e p ) ^ { 1 / ( 2 v ) } \\Big ) ^ d = E ^ d . \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} \\theta _ t ^ i z _ t ^ i & = \\theta _ t ^ i u _ i ( x _ t ^ i ) + \\mu _ t \\theta _ { t + 1 } ^ i z _ { t + 1 } ^ i , & & t = 0 , 1 , \\ldots . \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\hat { \\mathbf { y } } _ { i , ( r ) } = \\mathbf { A } ^ { - 1 } \\left ( \\tilde { \\mathbf { y } } _ i - \\mathbf { w } _ { i , ( r ) } \\right ) , \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} \\Omega ( u , v ) : = \\int _ { \\mathbb { T } } ( \\partial _ x ^ { - 1 } u ) \\ , v \\ , d x , \\forall u , v \\in H _ 0 ^ 1 ( \\mathbb { T } _ x ) , \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\lambda _ { j b } } \\left ( \\frac { \\partial E } { \\partial k _ { i a } } \\right ) = \\frac { s _ a L } { 2 \\pi } [ \\delta _ { i a , j b } - s _ b F ( \\lambda _ { j b } | \\lambda _ { i a } ) ] \\tilde { \\epsilon } ' ( \\lambda _ { j b } ) \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} ( n _ 2 + n _ 3 ) r _ 1 + ( n _ 1 + n _ 3 ) r _ 2 + ( n _ 1 + n _ 2 ) r _ 3 = 0 \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} | u _ { m + 1 } | \\geq | 2 c + 2 | | u _ { m } | - | u _ { m - 1 } | \\geq | 2 c + 2 | | u _ { m } | - | u _ { m } | = ( 2 | c | - 3 ) | u _ { m } | \\geq | u _ { m } | . \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} \\left \\Vert f \\right \\Vert _ { \\theta } : = | f | _ { \\theta } + \\| f \\| , \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\mu ( Y \\cap \\pi _ 1 ^ { - 1 } ( C ) ) = \\int _ { C } \\mu _ { x ^ 1 } ^ { \\pi _ 1 } ( Y ) d m _ { \\Gamma _ 1 } ( x ^ 1 ) . \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} R _ { \\rm { t o t } } = \\sum _ { n = 1 } ^ { N } R _ n = \\sum _ { n = 1 } ^ N r ^ n _ { \\mathrm { S C } } + \\sum _ { n = 1 } ^ N \\sum _ { k = 1 } ^ K x _ k r _ { k , n } . \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\lim _ { J _ i \\ni h \\to 0 } \\overline G _ i ( h , 0 ) = G ( 0 , 0 ) \\ \\ \\ i \\in \\{ 1 , 2 , 3 \\} . \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} I ( V ; Y | U ) - I ( V ; Z | U ) = I ( V ; Y ) - I ( V ; Z ) + I ( U ; Z ) - I ( U ; Y ) , \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} & B _ { 2 { \\bf R } } ^ d \\Big ( \\big ( m - \\big ( \\sum _ { j = 1 } ^ { d } \\big ( \\frac { 1 } { 2 } \\big ) ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } \\big ) ^ { 2 p } _ + \\Big ) \\subset \\bigcup _ { \\substack { { \\bf k } \\in \\Bbb Z ^ d \\\\ \\sum _ { j = 1 } ^ { d } | k _ j | ^ { 2 R _ j } \\leq m ^ { 2 p } } } Q _ { \\bf k } \\\\ & \\subset B _ { 2 { \\bf R } } ^ d \\Big ( \\big ( m + \\big ( \\sum _ { j = 1 } ^ { d } \\big ( \\frac { 1 } { 2 } \\big ) ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } \\big ) ^ { 2 p } \\Big ) , \\end{align*}"}
-{"id": "5818.png", "formula": "\\begin{align*} \\{ v _ 1 , v _ 2 \\} = \\{ & ( 2 , 6 ) ( 4 , 1 1 ) ( 7 , 9 ) ( 8 , 1 3 ) ( 1 0 , 1 4 ) ( 1 2 , 1 6 ) , \\\\ & ( 1 , 2 , 7 , 4 ) ( 3 , 8 , 6 , 1 0 ) ( 5 , 9 , 1 3 , 1 2 ) ( 1 1 , 1 5 ) ( 1 4 , 1 7 ) ( 1 6 , 1 8 ) \\} \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} \\lambda _ { k } \\left ( A _ { \\alpha } \\left ( G \\right ) \\right ) = \\alpha d + \\left ( 1 - \\alpha \\right ) \\lambda _ { k } \\left ( A \\left ( G \\right ) \\right ) , 1 \\leq k \\leq n . \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} S = \\nu ( s - 1 , n - 1 ) + \\nu ( s + m - 2 s _ 0 , n + m ) - \\nu ( s - m - 1 , n + 2 r _ n - m - 1 ) \\\\ - \\nu ( s - 2 s _ 0 , n + 2 r _ n ) - \\nu ( s - m - 1 , n - m - 1 ) - \\nu ( s - 2 s _ 0 , n ) . \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} - n + m < n + m + 1 - 2 \\sum _ { j = 1 } ^ { n } \\left ( x ; S ^ { ( j ) } \\right ) < n - m \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} h ( t ) = \\left [ \\begin{array} { c c } t 1 _ { \\widetilde { V } } & 0 \\\\ 0 & 1 _ { \\widehat { V } } \\end{array} \\right ] , h ' ( t ) = \\left [ \\begin{array} { c c } t 1 _ { \\widetilde { V } } & 0 \\\\ 0 & 1 _ { \\widehat { V ' } } \\end{array} \\right ] . \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} \\underline { D } ^ 1 ( \\mu ) = \\liminf _ { \\delta \\to 0 } \\frac { \\sum _ { Q \\in \\mathcal { M } _ \\delta } \\mu ( Q ) \\log \\mu ( Q ) } { \\log \\delta } \\end{align*}"}
-{"id": "2348.png", "formula": "\\begin{gather*} \\mu _ { t } = \\frac { 2 } { 3 } \\frac { u _ t } { u } \\chi , \\\\ \\chi _ { t } = \\frac { 2 } { 3 } \\frac { u _ t } { u } \\mu - \\frac { 1 } { 3 } \\nu , \\\\ \\nu _ { t } = \\frac { 2 } { 3 } \\left ( \\frac { u _ t } { u } \\right ) _ t \\mu + \\frac { 2 } { 3 } \\left ( t - \\frac { u ^ 2 _ t } { u ^ 2 } \\right ) \\chi , \\end{gather*}"}
-{"id": "5954.png", "formula": "\\begin{align*} \\mathcal { B } _ { - } ( \\lambda ) = \\left ( - 1 \\right ) ^ { \\mathsf { N } } ( - a _ { - } ( \\lambda ) A ( \\lambda ) B ( 1 / \\lambda ) + b _ { - } ( \\lambda ) A ( \\lambda ) A ( 1 / \\lambda ) - c _ { - } ( \\lambda ) B ( \\lambda ) B ( 1 / \\lambda ) + d _ { - } ( \\lambda ) B ( \\lambda ) A ( 1 / \\lambda ) ) , \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} \\mathfrak { m } \\supset \\mathfrak { m } ^ 2 \\supset \\dots \\supset \\mathfrak { m } ^ r = \\{ 0 \\} . \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} \\tilde G = \\arg \\max \\tilde \\ell _ n ( G ) . \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} a ( x , y , z ) = \\frac { q - 1 } { 2 } \\left ( \\frac { q } { 3 } + 3 ^ { 2 k - 1 } \\cdot \\frac { s _ 1 } { s _ 2 } \\right ) \\equiv 0 , \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} G = \\langle g _ 1 , . . . , g _ { 1 3 } & \\mid [ g _ { 1 0 } , g _ 6 ] = g _ { 1 1 } , [ g _ { 1 0 } , g _ 7 ] = g _ { 1 2 } , \\\\ & \\ ; \\ ; \\ ; [ g _ 2 , g _ 1 ] = [ g _ 4 , g _ 3 ] = [ g _ 6 , g _ 5 ] = [ g _ 8 , g _ 7 ] = [ g _ { 1 0 } , g _ 9 ] = g _ { 1 3 } , p \\rangle . \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} 5 5 5 0 & = 2 \\cdot 3 \\cdot 5 ^ 2 \\cdot 3 7 , & 5 5 5 5 0 & = 2 \\cdot 5 ^ 2 \\cdot 1 1 \\cdot 1 0 1 , \\\\ 5 5 5 5 5 0 & = 2 \\cdot 5 ^ 2 \\cdot 4 1 \\cdot 2 7 1 , & 5 5 5 5 5 5 0 & = 2 \\cdot 3 \\cdot 5 ^ 2 \\cdot 7 \\cdot 1 1 \\cdot 1 3 \\cdot 3 7 , \\\\ 5 5 5 5 5 5 5 0 & = 2 \\cdot 5 ^ 2 \\cdot 2 3 9 \\cdot 4 6 4 9 , & 5 5 5 5 5 5 5 5 0 & = 2 \\cdot 5 ^ 2 \\cdot 1 1 \\cdot 7 3 \\cdot 1 0 1 \\cdot 1 3 7 , \\\\ 5 5 5 5 5 5 5 5 5 0 & = 2 \\cdot 3 ^ 2 \\cdot 5 ^ 2 \\cdot 3 7 \\cdot 3 3 3 6 6 7 , & \\ ! \\ ! \\ ! \\ ! 5 5 5 5 5 5 5 5 5 5 0 & = 2 \\cdot 5 ^ 2 \\cdot 1 1 \\cdot 4 1 \\cdot 2 7 1 \\cdot 9 0 9 1 . \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} \\bar { \\psi } _ { \\gamma } ( \\delta , \\alpha _ { 0 } , \\theta _ { 0 } ) = 0 , \\delta \\in \\Gamma . \\end{align*}"}
-{"id": "1411.png", "formula": "\\begin{align*} B _ { k , i } \\cap B _ { k , j } = \\emptyset \\quad \\mbox { i f $ i \\not = j $ } \\qquad \\mbox { a n d } { \\bf R } ^ N \\setminus B ( 0 , n t ^ \\frac { 1 } { \\theta } ) \\subset \\bigcup _ { k = 1 } ^ m \\bigcup _ { i = 1 } ^ \\infty B _ { k , i } , \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{gather*} s _ 2 = 0 , s _ 1 = - i a = - s _ 3 , a \\in { \\mathbb R } , | a | \\leq 1 . \\end{gather*}"}
-{"id": "278.png", "formula": "\\begin{align*} y = a \\cdot T ^ e \\cdot \\prod _ { ( i , p ) = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\infty } ( 1 - a _ { i j } T ^ i ) ^ { p ^ j } \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} d _ { M , r } ( \\mathcal { C } ) = d _ { M , r } ( \\mathcal { C } , \\{ 0 \\} ) . \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{gather*} F _ 2 ( t ) = e ^ { \\int ^ { \\infty } _ { t } \\omega ( \\tau ) d \\tau } , \\end{gather*}"}
-{"id": "2693.png", "formula": "\\begin{align*} \\rho _ { \\Gamma , j k } = \\Gamma _ { i n } { } ^ i \\Gamma _ { j k } { } ^ n - \\Gamma _ { j n } { } ^ i \\Gamma _ { i k } { } ^ n \\ , . \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} U ( \\hat { x } , \\theta ) : = \\sum _ { i = 1 } ^ n \\theta ^ i u _ i \\left ( s ^ i ( \\hat { x } , \\theta ) \\right ) , \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} \\tilde { U } ( \\hat { x } _ t ) = \\frac { \\gamma } { 1 - \\gamma } \\left [ \\left ( \\hat { \\phi } + \\frac { \\eta } { \\gamma } \\ , \\hat { x } _ t \\right ) ^ { 1 - \\gamma } - 1 \\right ] . \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} \\begin{aligned} & G ^ { [ 3 ] } = W _ \\alpha \\P \\left [ Q _ \\alpha \\bar W _ \\alpha - \\bar Q _ \\alpha W _ \\alpha \\right ] - \\P \\left [ ( Q _ \\alpha \\bar W _ \\alpha - \\bar Q _ \\alpha W _ \\alpha ) ( W _ \\alpha + \\bar W _ \\alpha ) \\right ] , \\\\ & K ^ { [ 3 ] } = Q _ \\alpha ^ 2 W _ \\alpha + Q _ \\alpha \\P \\left [ Q _ \\alpha \\bar W _ \\alpha - \\bar Q _ \\alpha W _ \\alpha \\right ] + \\P \\left [ Q _ \\alpha \\bar Q _ \\alpha ( W _ \\alpha + \\bar W _ \\alpha ) \\right ] , \\end{aligned} \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} \\epsilon _ z ^ N ( x , x ' ) = \\epsilon _ { z , 1 } ^ N ( x , x ' ) + \\epsilon _ { z , 2 } ^ N ( x , x ' ) , \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} \\beta = ( p ( l + n - 1 ) , \\dots , p ( l + n - 1 ) ) - ( ( n p - p ) , \\dots , ( n p - p ) ) = ( p l , \\dots , p l ) , \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} \\exp ( \\beta V p _ { \\beta , \\Lambda , \\mu , \\lambda } ^ { ' } ) = \\Xi _ { \\Lambda } ( \\beta , \\mu , \\lambda ) ^ { ' } = \\int d ^ { 2 } z { \\rm { T r } } _ { { \\cal H } ^ { ' } } \\exp ( - \\beta ( H _ { \\Lambda , \\mu , \\lambda } ) ^ { ' } ( z ) ) \\ , \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} u _ i ( r ) = A _ i r ^ { \\mu _ i ^ + } + \\bar { u } _ i ( r ) , \\end{align*}"}
-{"id": "4646.png", "formula": "\\begin{align*} J ( \\xi ) + J ( \\eta ) - J ( \\zeta ) = O ( e ^ { - | \\xi | } + e ^ { - | \\eta | } ) , \\xi \\eta > 0 . \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} d _ p ( \\textbf { x } , \\textbf { y } ) = d _ H ( \\textbf { x } , \\textbf { y } ) + L . \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} \\frac { \\partial u } { \\partial t } = \\frac 1 2 \\Delta u + \\sqrt \\lambda u \\diamond \\frac { \\partial ^ { \\ell + 1 } W } { \\partial t \\partial x _ 1 \\dots \\partial x _ \\ell } \\ , , \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} \\rho \\left ( A \\times \\left \\{ 0 \\right \\} \\right ) = \\hat { \\rho } ( A ) = \\frac { 1 } { Z _ \\sigma } \\int _ A e ^ { \\frac { 2 } { \\sigma ^ 2 } ( \\frac { 1 } { 2 } | x | ^ 2 - \\frac { 1 } { 4 } | x | ^ 4 ) } \\ , d x \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} & \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\mathcal { F } ^ { - 1 } [ \\mathcal { G } _ { \\sigma , \\nu } ( t , \\xi ) ] \\| _ { 2 } = C t ^ { - \\gamma _ { \\sigma , k } - \\ell } \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} S _ k ( x ) = \\frac { 1 } { 2 \\pi i } \\int _ { c - i T } ^ { c + i T } \\widetilde { F } _ k ( s ) \\frac { x ^ s } { s } d s + O \\ ( \\frac { x \\log x } { T } + 1 \\ ) . \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} \\begin{pmatrix} I _ { p ' } & * & 0 \\\\ 0 & a _ \\sigma ( 0 ) & 0 \\\\ 0 & * & I _ { q ' } \\end{pmatrix} \\\\ \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\mathcal { F } ^ k : = \\prod _ { i = 0 } ^ { k } \\prod _ { ( r _ { k + 1 } , \\dots , r _ n ) \\in \\{ 0 , 1 \\} ^ { n - k } } F _ i ( x _ 0 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ k , x _ { k + 1 } ^ { r _ { k + 1 } } , \\dots , x _ { n } ^ { r _ { n } } ) . \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} \\vartheta _ L = \\sum _ { r = 0 } ^ \\infty n _ L ( r ) q ^ r \\ ; \\ ; n _ L ( r ) = \\{ x \\in L : x \\cdot x = 2 r \\} , \\end{align*}"}
-{"id": "8528.png", "formula": "\\begin{align*} h _ { \\sigma _ i } ( g _ j ) = \\begin{cases} g _ j & \\mbox { i f } j \\neq i , \\\\ g _ { i + 1 } g _ { i } ^ { - 1 } g _ { i - 1 } & \\mbox { i f } j = i \\neq 1 , \\\\ g _ 2 g _ 1 ^ { - 1 } & \\mbox { i f } j = i = 1 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} W \\ ; : \\ ; y ^ 2 + a _ 1 ' x y + a _ 3 ' y = x ^ 3 + a _ 2 ' x ^ 2 + a _ 4 ' x + a _ 6 ' , \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} \\left ( \\Phi ( t ) ( \\Phi ( s ) u ) \\right ) ( \\tau ) & = U ( \\tau , t ) ( \\Phi ( s ) u ) ( t ) = U ( \\tau , t ) U ( t , s ) u ( s ) \\\\ & = U ( \\tau , s ) u ( s ) = ( \\Phi ( s ) u ) ( \\tau ) . \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{align*} \\frac { 1 } { N } \\sum _ { m = 0 } ^ { N - 1 } \\int \\limits _ { e ^ m \\leq r \\leq e ^ { m + 1 } } | F _ { A } | ^ 2 r V ^ { - 1 } d v \\leq c _ 3 e ^ { 2 } . \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} & \\sigma ( H ^ { \\epsilon , \\kappa } ) \\cap \\left [ a _ 0 , b _ N \\right ] \\subset \\bigcup _ { k = 0 } ^ N [ a _ k , b _ k ] \\ , , \\quad { \\rm d i m } \\big ( { \\rm R a n } E _ { [ a _ k , b _ k ] } ( H ^ { \\epsilon , \\kappa } ) \\big ) = + \\infty \\ , , \\\\ & b _ k - a _ k \\leq C _ 0 \\ , \\kappa \\epsilon + C _ 1 \\ , \\epsilon ^ { 4 / 3 } \\ , , \\ ; 0 \\leq k \\leq N \\ , , { \\rm a n d } a _ { k + 1 } - b _ k \\geq \\frac { 1 } { C _ 2 } \\epsilon \\ , , \\ ; 0 \\leq k \\leq N - 1 \\ , . \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} V _ t ( k _ t ) = & \\sup _ { k _ { t + 1 } \\in \\hat { \\Gamma } ( k _ t ) } \\Big [ U _ t ( f ( k _ t ) - k _ { t + 1 } ) + \\beta _ { t + 1 } V _ { t + 1 } ( k _ { t + 1 } ) \\Big ] , \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} \\Phi _ { q } = \\pi ^ * \\phi - q \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} \\int _ 0 ^ \\zeta F _ N ^ * \\eta = \\int _ 0 ^ \\zeta ( Z _ u + X Y _ u ) ( u , u ^ N ) \\ , d u \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} - \\Delta \\varphi + \\omega \\varphi - \\varphi \\ , \\log | \\varphi | ^ 2 = 0 , x \\in { \\mathbb { R } } ^ N , \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} x \\mathbf { 1 } + c _ { 1 } + \\cdots + c _ { l } = c _ { 1 } ' + \\cdots + c _ { l } ' \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} p ( G ) ( h ) : = \\sum _ { \\phi : E \\to [ k ] } \\prod _ { v \\in V } h ( \\phi ( \\delta ( v ) ) ) , \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} \\begin{aligned} \\bold { p r } X & = \\xi ^ i \\left ( \\partial _ { x ^ i } + \\sum _ I \\left ( D _ { i ; I } - \\partial _ { x ^ i } \\right ) \\right ) + \\sum _ { \\alpha , J _ 1 , J _ 2 } ( D _ { J _ 1 } S _ { J _ 2 } Q ^ { \\alpha } ) \\partial _ { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } \\\\ & = \\xi ^ i D _ i + \\sum _ { \\alpha , J _ 1 , J _ 2 } \\left ( D _ { J _ 1 } S _ { J _ 2 } Q ^ { \\alpha } \\right ) \\partial _ { u _ { J _ 1 ; J _ 2 } ^ { \\alpha } } , \\end{aligned} \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} \\cos a - \\cos b = 2 \\sin \\frac { b + a } { 2 } \\sin \\frac { b - a } { 2 } , \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} u _ 1 ( x ) & = 2 g ' ( x ) \\varphi ( x ) + g ( x ) \\varphi ' ( x ) , & u _ 2 ( x ) & = x u _ 1 ( x ) , \\\\ v _ 1 ( x ) & = g ( x ) \\varphi ( x ) , & v _ 2 ( x ) & = x v _ 1 ( x ) . \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\textrm { l n } \\ , ( \\varphi _ t ( \\omega , x ) ) ^ { ( d ) } = \\textrm { l n } \\ , ( \\varphi _ 0 ( \\omega , x ) ) ^ { ( d ) } + \\int _ 0 ^ t \\left ( 1 - | \\varphi _ s ( \\omega , x ) | ^ 2 \\right ) \\ , d s \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} \\max _ { m \\in \\mathcal { M } _ n } \\mathbb { P } _ { P ^ { ( \\mathcal { C } ^ \\star _ n ) } } \\big ( \\tilde { M } \\neq m | M = m \\big ) & \\leq \\epsilon \\\\ \\max _ { m \\in \\mathcal { M } _ n } \\mathsf { D } \\Big ( P ^ { ( \\mathcal { C } ^ \\star _ n ) } _ { \\mathbf { Z } | M = m } \\Big | \\Big | P ^ { ( \\mathcal { C } ^ \\star _ n ) } _ \\mathbf { Z } \\Big ) & \\leq \\epsilon . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} \\operatorname { d i a m } ( G _ { 1 , 2 } ) & \\leq \\lbrack 2 { \\textstyle \\sum _ { x \\in X _ { 1 } } } f ( x ) + 1 ] + 1 + [ 2 { \\textstyle \\sum _ { x \\in X _ { 2 } } } f ( x ) + 2 ] - 2 \\\\ & = 2 { \\textstyle \\sum _ { x \\in X _ { 1 } \\cup X _ { 2 } } } f ( x ) + 2 . \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n ^ 2 } \\log b _ n = \\lim _ { n \\to \\infty } \\frac { 1 } { n ^ 2 } \\log c _ n = 0 \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} \\langle \\Gamma _ H F , G \\rangle _ T = \\langle F , \\Gamma _ H ^ * G \\rangle _ T , F , G \\in L ^ 1 _ T \\mathcal H , \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} A ^ 2 _ { i j } = \\sum _ { k = 1 } ^ 4 A _ { i k } A _ { k j } = | \\{ k : v _ i \\sim v _ k v _ k \\sim v _ j \\} | = \\begin{cases} 3 & i = j , \\\\ 2 & i \\neq j . \\end{cases} \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} \\mu ( U , f ) = \\sup _ { z \\in U } f ^ \\# ( z ) \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} [ m _ 1 , m _ 2 ] = [ m _ 1 , m _ 3 ] = [ m _ 2 , m _ 3 ] = [ m _ 1 , m _ 2 , m _ 3 ] . \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} \\nabla _ { \\nu } = z ^ 2 \\frac { \\partial ^ 2 } { \\partial z ^ 2 } + z \\frac { \\partial } { \\partial z } + z ^ 2 - \\nu ^ 2 . \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} _ \\omega = \\sum _ { i = 1 } ^ N \\frac { \\partial } { \\partial \\xi _ i } \\frac { \\partial } { \\partial \\theta _ i } , \\ ; \\ ; \\ ; \\theta _ i = \\frac { \\partial } { \\partial \\xi _ i } . \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} A ^ h : = \\frac { 1 } { h } ( R ^ h - I ) \\ , . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} \\frac { \\partial ^ { 2 } \\tau } { \\partial x ^ { 2 } } ( x _ * , y _ * ) = 2 + O ( \\varepsilon ) , \\ \\ \\ \\frac { \\partial ^ { 2 } \\tau } { \\partial y ^ { 2 } } ( x _ * , y _ * ) = 2 + O ( \\varepsilon ) , \\ \\ \\ \\frac { \\partial ^ { 2 } \\tau } { \\partial x \\partial y } ( x _ * , y _ * ) = O ( \\varepsilon ) . \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} n ( \\ell ) & = \\frac { \\ell _ n H _ 2 ( k _ n / \\ell _ n ) } { \\frac { 1 } { 2 } \\log ( 1 + k _ n P ) } \\\\ & = \\theta _ n n , \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} T \\overline \\partial _ { b } ^ \\ast = \\overline \\partial _ { b } ^ \\ast T ~ ~ \\Omega ^ { 0 , q } ( X ) , \\forall q = 1 , \\ldots , n - 1 \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} d i m ( B u n _ { \\mathcal { G } _ { X , x , \\theta } } ) = d i m ( G ) ( g - 1 ) + d i m ( G / B ) . \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\| x _ { k _ { j } } - x _ { k _ { n + 1 } } \\| & \\geq | < z ^ { * } _ { i } , x _ { k _ { j } } - x _ { k _ { n + 1 } } > | \\\\ & \\geq 1 - | < z ^ { * } _ { i } , x _ { k _ { n + 1 } } > | - | < z ^ { * } _ { i } , x _ { k _ { j } } - z _ { i } > | \\\\ & \\geq 1 - c - \\| x _ { k _ { j } } - z _ { i } \\| \\\\ & \\geq 1 - c - ( 1 + c - \\epsilon ) = \\epsilon - 2 c \\\\ & \\geq \\epsilon - \\delta \\\\ \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} u ( x ) & \\geq M \\int _ { \\Omega } { u ( z ) f _ 0 ( z ) \\ , d z } = \\frac { a _ { N , s } } { 2 } \\iint _ Q \\frac { ( u ( x ) - u ( y ) ) ( w ( x ) - w ( y ) ) } { | x - y | ^ { N + 2 s } } \\ , d x d y \\\\ & = M \\int _ { \\Omega } { w ( z ) f ( z ) \\ , d z } \\geq C _ 0 \\int _ { \\Omega } { f ( z ) \\overline { \\xi _ 0 } ( z ) \\ , d z } \\geq \\lambda w ( x ) , \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} \\sqrt { n } ( \\hat { \\theta } - \\theta _ { n } ) & = \\sqrt { n } ( \\hat { \\theta } - \\theta _ { 0 } ) + \\sqrt { n } ( \\theta _ { 0 } - \\theta _ { n } ) = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } \\zeta ( W _ { i } ) + o _ { p } ( 1 ) + \\mu \\bar { \\sigma } ^ { 2 } \\\\ & = \\frac { 1 } { \\sqrt { n } } \\sum _ { i = 1 } ^ { n } \\{ \\zeta ( W _ { i } ) - E _ { F _ { n } } [ \\zeta ( W ) ] \\} + \\mu \\bar { \\sigma } ^ { 2 } \\overset { d } { \\longrightarrow } N ( \\mu \\bar { \\sigma } ^ { 2 } , V ) . Q . E . D . \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} 1 _ { C \\left ( v \\right ) } = \\frac { 1 } { n } \\sum _ { j = 0 } ^ { n - 1 } \\left . v ^ j \\right \\rangle \\left \\langle v ^ j \\right . . \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} \\int _ 1 ^ R | \\nabla ^ { 0 , 1 } \\nabla u | ^ 2 + { \\rm R i c } ( \\nabla u , \\nabla u ) & = \\frac { 1 } { 2 } \\left ( \\oint _ R - \\oint _ 1 \\right ) ( \\langle \\nabla _ { \\partial _ r } \\nabla u , \\nabla u \\rangle + \\langle J \\nabla _ { J \\partial _ r } \\nabla u , \\nabla u \\rangle ) . \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } u = f \\geq 0 \\quad \\Omega , u = 0 \\R ^ N \\setminus \\Omega . \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} M _ h ( x ) = \\sum ^ r _ { i = 0 } R ( i ) P _ { h } ( \\varepsilon _ i ) x ^ { \\varepsilon _ i / 3 } \\sum _ { \\substack { n \\le x \\\\ ( n , h ) = 1 } } \\theta ( n ) a ( n ) P _ n ( \\varepsilon _ i ) G _ { h n } ( \\varepsilon _ i ) \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | B _ n ( z ) | ^ { 1 / n } = | \\Phi ( z ) | \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} M ( \\lambda , \\vec \\gamma ) = M ( 0 ) + \\lambda M ' ( 0 , \\vec \\gamma ) + \\dfrac { \\lambda ^ 2 } { 2 } M '' ( 0 , \\vec \\gamma ) + \\dfrac { \\lambda ^ 3 } { 6 } M ''' ( \\overline { \\lambda } , \\vec \\gamma ) \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} \\sum _ { n \\le x } a ( n ) = C _ 1 x + C _ 2 x ^ { 1 / 2 } + C _ 3 x ^ { 1 / 3 } + O \\left ( x ^ { 1 / 4 + \\varepsilon } \\right ) , \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} { } K _ \\gamma = \\overline { D \\big ( 0 , \\sqrt { 1 + \\gamma } \\big ) } \\setminus D ( a , \\sqrt { \\gamma } ) , \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{gather*} ( L _ 0 ) { _ { 2 1 } } \\equiv - \\frac { 1 - q _ 2 ^ 2 } { 4 } \\frac { u _ t } { u } = e _ 1 \\frac { 1 - q _ 2 ^ 2 } { 4 } + \\frac { q _ 1 q _ 2 } { 2 } , \\end{gather*}"}
-{"id": "7469.png", "formula": "\\begin{align*} I ( \\rho | e ^ { - \\Psi } ) = \\int _ { \\mathbb { S } ^ { 1 } } \\Big | \\nabla \\log \\frac { \\rho } { e ^ { - \\Psi } } \\Big | ^ 2 \\rho \\ , d \\omega , \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} \\C { T } _ 2 ^ * = ( { \\bf x } ^ * - { \\bf x } ^ I ) ( 1 ) + \\sum _ { j = 1 } ^ N \\omega _ j \\left ( \\dot { \\bf x } ^ I ( \\tau _ j ) - \\dot { \\bf x } ^ * ( \\tau _ j ) \\right ) . \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} d _ p = \\begin{cases} 0 & i f ~ i = 2 , \\\\ 2 & i f ~ i \\in \\{ 0 , 1 \\} . \\end{cases} \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} ( \\overline { A } _ 1 ) _ j ^ { j ' } ( l ) = \\begin{cases} - \\dfrac { 2 } { 3 } \\ , c _ 2 \\ , \\left ( \\dfrac { j - j ' } { j ' } \\right ) \\sqrt { \\lvert j - j ' \\rvert \\xi _ { j - j ' } } - 2 \\ , c _ 3 \\ , \\dfrac { 1 } { j ' ( j ' - j ) } \\ , \\sqrt { \\lvert j - j ' \\rvert \\xi _ { j - j ' } } \\mbox { i f } \\ , \\ , j - j ' \\in S , \\\\ [ 3 m m ] 0 \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} | \\lambda - \\bar { \\lambda } | \\le \\frac { \\delta ^ 2 \\| Z \\| _ F ^ 2 } { \\sum _ { i = 1 } ^ m | x ^ * A _ i x | ^ 2 } . \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} \\tilde { \\omega } ^ 2 = e ^ F \\omega ^ 2 , \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} \\partial _ t u ( t , x ) + \\partial _ x ( a ( x ) u ( t , x ) ) = 0 , ( t , x ) \\in [ 0 , T ] \\times D , \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} g ( t , x , z ) = f _ { \\mu , A } ^ { 1 - m } ( x ) z ^ { m - 1 } \\frac { A + A ^ t } { 2 } ( x - \\mu ) \\ , \\textrm { a n d } \\Lambda ( t , x , z ) = f _ { \\mu , A } ^ { 1 - m } ( x ) z ^ { m - 1 } T r \\left ( \\frac { A + A ^ t } { 2 } \\right ) . \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { N - 1 } S \\left ( n _ i \\right ) = \\sum _ { 0 \\leq k \\leq _ b n } \\prod _ { i = 0 } ^ { N - 1 } f ( n _ i , k _ i ) . \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} ( u _ { X X } + 1 ) ( u _ { Y Y } + u _ { t t } + u _ t + 1 ) - u _ { X Y } ^ 2 - u _ { X t } ^ 2 = e ^ F , \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} \\mathbf { B } = \\frac { 1 } { N } \\begin{bmatrix} 1 & 1 & \\cdots & 1 \\\\ j 2 \\pi \\frac { 0 } { N } & j 2 \\pi \\frac { 1 } { N } & \\cdots & j 2 \\pi \\frac { N - 1 } { N } \\\\ \\vdots & \\vdots & { } & \\vdots \\\\ \\left ( j 2 \\pi \\frac { 0 } { N } \\right ) ^ V & \\left ( j 2 \\pi \\frac { 1 } { N } \\right ) ^ V & \\cdots & \\left ( j 2 \\pi \\frac { N - 1 } { N } \\right ) ^ V \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} \\begin{array} { l l l l } Z ^ L _ j = & \\max \\ & c ' V ^ j \\lambda + d ' u & \\\\ & \\mbox { s . t . } \\ & 1 \\cdot \\lambda = 1 & \\\\ & & H V ^ j \\lambda + G u \\leq h & \\\\ & & \\lambda \\geq 0 & \\end{array} \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} F _ i = \\sum _ { k = 0 } ^ n x _ k \\frac { \\partial F _ i } { \\partial x _ k } \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} H = \\left [ \\begin{array} { c c c c c } s & 0 & \\lambda a _ { 2 } t & \\cdots & \\lambda a _ { n } t \\\\ 0 & t & \\lambda b _ { 2 } s & \\cdots & \\lambda b _ { n } s \\end{array} \\right ] . \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} \\mathbb { H } _ { \\mathbb { Q } _ \\ell } ( X _ \\eta ) : = ( p _ \\sigma ) _ \\ast i ^ \\ast j _ \\ast \\mathbb { Q } _ { \\ell , X _ \\eta } \\simeq i _ \\sigma ^ \\ast ( j _ \\eta ) _ \\ast ( p _ \\eta ) _ \\ast \\mathbb { Q } _ { \\ell , X _ \\eta } \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ Y \\leq ( n p ) ^ { 1 / ( 4 ( r - 1 ) ) } p ^ { - 1 } \\right ] & \\leq \\exp \\left ( - \\frac { ( 1 + o ( 1 ) ) ( \\mathbb { E } [ Y ] ) ^ 2 } { 2 \\mathbb { E } [ Y ] } \\right ) \\\\ & \\stackrel { \\eqref { e x p e c t a t i o n Y } } { \\leq } \\exp \\left ( - \\Omega \\left ( ( n p ) ^ { 1 / ( 2 ( r - 1 ) ) } p ^ { - 1 } \\right ) \\right ) \\stackrel { n p = \\omega ( 1 ) } { = } \\exp \\left ( - \\omega ( p ^ { - 1 } ) \\right ) . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} \\xi _ 1 = \\hat { u } + ( \\pi _ L ) , \\xi _ 2 = \\hat { r } + ( \\pi _ L ) , \\xi _ 3 = \\hat { s } + ( \\pi _ L ) , \\xi _ 4 = \\hat { t } + ( \\pi _ L ) . \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} K ( 0 , r , p ) = 2 4 ( 1 - r ^ 2 ) + 2 \\sqrt { 2 5 - 4 0 r p + 1 6 r ^ 2 } . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} p _ { K _ { 3 , 3 , 3 } } ( x ) = \\frac { 1 9 9 2 } { 2 6 1 2 5 } x + \\frac { 1 1 7 2 4 } { 2 6 1 2 5 } x ^ 2 + \\frac { 1 0 9 5 1 } { 2 6 1 2 5 } x ^ 3 + \\frac { 1 4 5 8 } { 2 6 1 2 5 } x ^ 4 , \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} \\begin{array} { c c c } ( a - b ) ( a + b - c ) ^ 2 & = & 0 \\\\ ( c - a ) ( a - b + c ) ^ 2 & = & 0 \\\\ ( c - b ) ( a - b - c ) ^ 2 & = & 0 \\end{array} . \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} ( 2 + _ S 3 ) \\times _ S ( 2 + _ S 3 ) = ( 2 \\times _ S 2 ) + _ S ( 2 \\times _ S 3 ) + _ S ( 3 \\times _ S 2 ) + _ S ( 3 \\times _ S 3 ) \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} D _ h ( X ) = \\sum _ { n \\leq X } \\tau ( n - h ) \\tau ( n ) \\tau ( n + h ) , \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} \\hat { P } ^ { ( \\mathcal { C } _ n ) } ( i , j | m , \\mathbf { s } ) = \\frac { p ^ n _ { S | U , V } \\big ( \\mathbf { s } \\big | \\mathbf { u } ( i ) , \\mathbf { v } ( i , j , m ) \\big ) } { \\sum _ { ( i ' , j ' ) \\in \\mathcal { I } _ n \\times \\mathcal { J } _ n } p ^ n _ { S | U , V } \\big ( \\mathbf { s } \\big | \\mathbf { u } ( i ' ) , \\mathbf { v } ( i ' , j ' , m ) \\big ) } , \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N { \\mathsf { R } } ^ { \\lambda } _ { { \\bar { \\mathcal { N } } } _ i } \\geq ( N - 1 ) \\mathsf { R } ^ { \\lambda } _ { \\mathcal { N } _ { [ 1 : N ] } } . \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} e ^ { 2 } c \\left ( c - 2 \\right ) - b ^ { 2 } c \\left ( c + 2 \\right ) = \\frac { 4 c } { c ^ { 2 } - 4 } \\left ( c + 2 \\right ) e _ { 1 } ^ { 2 } - \\left ( c - 2 \\right ) b _ { 1 } ^ { 2 } = 4 . \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} | f ( t , x , u _ 1 ) - f ( t , x , u _ 2 ) | \\leq L | u _ 1 - u _ 2 | , \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} d ^ * ( A B ) = d ^ * ( A ) < d ^ * ( A ) + d ^ * ( B ) < 1 , \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} H : = \\mathrm { i m } ( \\pi _ 1 ( p ) ) \\cap \\mathrm { i m } ( \\pi _ 1 ( f ) ) \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} \\chi = \\frac { \\partial } { \\partial \\eta ^ \\sigma } + \\eta ^ \\sigma V , \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} \\big ( H ( b , V ) - z \\big ) & \\begin{pmatrix} I & 0 \\\\ - ( H _ + - z ) ^ { - 1 } U & ( H _ + - z ) ^ { - 1 } \\end{pmatrix} \\\\ & = \\begin{pmatrix} H _ - - z - \\overline { U } ( H _ + - z ) ^ { - 1 } U & \\overline { U } ( H _ + - z ) ^ { - 1 } \\\\ 0 & I \\end{pmatrix} , \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\phi = a ( t , n ) u + b ( t , n ) . \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} \\alpha \\Delta _ { \\alpha u } F ( X ) & = y \\Delta _ { y u } F ( X ) + z \\Delta _ { z r u } F ( X ) \\rho \\\\ & = y \\Delta _ { y u } f ( X ) + ( y \\Delta _ { y u } g ( X ) + z \\Delta _ { z r u } f ( X ) + z r \\Delta _ { z r u } g ( X ) ) \\rho . \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} \\zeta ( f * \\widetilde { f } ) & = \\int _ { \\R } \\tau \\left ( f ( s ) \\alpha _ s ( \\widetilde { f } ( - s ) ) \\right ) d s \\\\ & = \\int _ { \\R } \\tau \\left ( f ( s ) f ( s ) ^ * \\right ) \\geq 0 \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} \\partial _ i ( b _ 3 - m _ 3 ) [ \\hat { \\imath } ] & = \\Upsilon ' ( 0 ) \\{ M _ x [ \\partial _ i ( g ( a _ 1 - 1 ) - g ( 0 ) ) [ \\hat { \\imath } ] ] - M _ { \\varphi , x } [ \\partial _ i ( g ( a _ 1 - 1 ) - g ( 0 ) ) [ \\hat { \\imath } ] ] \\} \\\\ & + \\partial _ i \\{ \\Upsilon _ { \\geq 2 } [ M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] ] - M _ { \\varphi } [ \\Upsilon _ { \\geq 2 } [ M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] ] ] \\} [ \\hat { \\imath } ] \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\pi ^ b _ { 0 a ( n - a ) } = \\frac { \\sum _ { i = b } ^ { n - b } z _ i } { n + 1 - 2 b } \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} f _ j ( x _ j ) = \\sqrt { 3 } \\cdot x _ j . \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l l } \\displaystyle - \\Delta _ p \\ , u = f ( u ) & \\textup { i n } & \\Omega _ { a , b } , \\\\ \\displaystyle u = 0 & \\textup { o n } & \\partial \\Omega _ { a , b } \\end{array} \\right . \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} \\phi _ { N } ( \\tau ) = e _ N ( \\tau ) - e _ N ( 1 ) w _ { N } ( \\tau ) , \\mbox { w h e r e } w _ N ( \\tau ) = \\frac { ( 1 + \\tau ) P _ N ' ( \\tau ) } { N ( N + 1 ) } . \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\int _ { \\R ^ N } \\int _ 0 ^ 1 \\frac { \\partial _ 1 u ( y ) } { | y - s t e _ 1 - x | ^ { N + 2 \\sigma } } 1 _ { \\{ | y - s t e _ 1 - x | \\geq \\varepsilon \\} } ( y ) \\ d s d y = \\int _ { \\R ^ N } \\frac { \\partial _ 1 u ( y ) } { | y - x | ^ { N + 2 \\sigma } } 1 _ { \\{ | y - x | \\geq \\varepsilon \\} } ( y ) \\ d y \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} \\mathcal { P } _ i = \\left \\lbrace \\mathbf { P } _ i = ( P _ i ^ { ( 1 ) } , \\ldots , P _ i ^ { ( M ) } ) : P _ i ^ { ( k ) } \\in \\{ p _ i ^ { ( 1 ) } , \\ldots , p _ i ^ { ( M ) } \\} , \\sum _ { j = 1 } ^ M \\alpha _ i ( j ) P _ i ^ { ( j ) } \\leq \\overline { P } _ i \\right \\rbrace . \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} \\underline { \\theta } : = \\widetilde { g } _ { 0 1 } { _ { | \\mathcal { I } ^ 1 } } , \\ ; \\underline { \\Theta } _ { a b } : = \\widetilde { g } _ { a b } { _ { | \\mathcal { I } ^ 1 } } , \\ ; \\underline { \\phi } : = \\Phi _ { | \\mathcal { I } ^ 1 } , \\ ; \\textbf { \\underline { f } } : = \\rho _ { | \\mathcal { I } ^ 1 } , \\ ; \\underline { \\psi } _ { \\mu \\nu } : = \\frac { \\partial \\widetilde { g } _ { \\mu \\nu } } { \\partial y ^ 1 } . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} \\tau _ o ^ { - 1 } ( t _ o I ^ { - 1 } ) \\beta ^ { - 1 } ( H _ o ) = \\beta ^ { - 1 } ( t _ o I _ o ^ { - 1 } H _ o ) \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} F _ { \\Omega } ( x ) = \\int _ 0 ^ x f _ \\Omega ( \\lambda ) d \\lambda \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} a \\cdot ( v _ 0 , v _ 1 ) = ( v _ 0 + \\omega ( a , v _ 1 ) , v _ 1 ) \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} \\frac { - 2 } { \\left ( 1 + \\sqrt { 1 - \\frac { 4 } { \\nu ^ { 2 } } | \\xi | ^ { 2 ( 1 - 2 \\sigma ) } } \\right ) \\sqrt { 1 - \\frac { 4 } { \\nu ^ { 2 } } | \\xi | ^ { 2 ( 1 - 2 \\sigma ) } } } + 1 = O ( | \\xi | ^ { 2 ( 1 - 2 \\sigma ) } ) \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} x _ 1 \\vee x _ 2 \\vee \\cdots \\vee x _ n & = \\frac { 1 } { n ! } \\sum _ { \\sigma \\in S _ n } x _ { \\sigma ( 1 ) } \\otimes x _ { \\sigma ( 2 ) } \\otimes \\cdots \\otimes x _ { \\sigma ( n ) } ; \\\\ x _ 1 \\wedge x _ 2 \\wedge \\cdots \\wedge x _ n & = \\frac { 1 } { n ! } \\sum _ { \\sigma \\in S _ n } \\epsilon ( \\sigma ) x _ { \\sigma ( 1 ) } \\otimes x _ { \\sigma ( 2 ) } \\otimes \\cdots \\otimes x _ { \\sigma ( n ) } , \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} \\mu \\left \\{ x : S ^ { \\sigma } _ { \\tau _ { n } } f \\geq t , \\tau _ { n } \\leq \\epsilon t \\right \\} \\leq \\mu \\left \\{ x : \\epsilon t \\left \\Vert f \\right \\Vert \\geq t \\right \\} = 0 \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} c ( T ) = c _ 1 T ^ { \\frac { n _ 1 } d } + c _ 2 T ^ { \\frac { n _ 2 } d } + \\dots , \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} u \\left ( t \\right ) = U \\left ( t , s \\right ) u \\left ( s \\right ) + \\int _ s ^ t U \\left ( t , \\xi \\right ) f \\left ( \\xi \\right ) d \\xi , t \\geq s \\geq \\tau . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} \\frac { c ^ { 2 } - 4 } { c \\cdot \\overline { a } ^ { 2 } } \\left ( \\overline { b } ^ { 2 } a ^ { 2 } \\overline { \\alpha } ^ { 2 } - \\overline { a } ^ { 2 } b ^ { 2 } \\alpha ^ { 2 } \\right ) = 4 . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} \\mathcal { A } _ n = \\frac { \\pi ^ { 2 k } ( 2 n + \\theta ) ^ { 2 k } } { \\omega ^ { 2 k } } \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix} - \\mathcal { B } _ n ^ \\theta + \\mathcal { C } _ n , \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} A + B & : = \\bigcup _ { a \\in A , \\ : b \\in B } a + _ F b , \\\\ A \\times B & = \\{ a \\times _ F b \\mid a \\in A , \\ : b \\in B \\} . \\end{align*}"}
-{"id": "3933.png", "formula": "\\begin{align*} L P ( b p , a p ) = \\{ \\lambda \\in P ( b p , a p ) | \\lambda ~ \\textrm { i s $ p $ - L i m a } \\} , \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} [ x , T ^ e \\cdot \\prod _ { ( i , p ) = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\infty } ( 1 - a _ { i j } T ^ i ) ^ { p ^ j } ) = e [ x , T ) + \\sum _ { ( i , p ) = 1 } ^ { \\infty } \\sum _ { j = 0 } ^ { \\infty } p ^ j [ x , 1 - a _ { i j } T ^ i ) , \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} f _ { 2 m } = \\left ( \\prod _ { k = 1 } ^ m \\frac { \\sqrt { 2 k - 1 } } { \\sqrt { 2 k } } \\right ) f _ 0 , f _ { 2 m + 1 } = \\left ( \\prod _ { k = 1 } ^ m \\frac { \\sqrt { 2 k } } { \\sqrt { 2 k + 1 } } \\right ) f _ 1 , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} f _ { m } = \\exp \\left [ \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ m \\log \\left ( 1 - \\frac { a } { 2 k } \\right ) - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ m \\log \\left ( 1 + \\frac { b } { 2 k } \\right ) \\right ] . \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} \\rho _ F ( u ) = | \\{ 0 \\le n \\le u - 1 : F ( n ) \\equiv 0 \\bmod u \\} | . \\end{align*}"}
-{"id": "8877.png", "formula": "\\begin{align*} \\epsilon ( v ^ { t + 1 } ) = \\epsilon ( ( y ' ) ^ t ) + ( 2 ^ { t + 1 } - 1 ) \\epsilon ( v ) - ( 2 ^ { t } - 1 ) , \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{gather*} A = \\int _ M ( j ^ \\infty \\phi ) ^ \\ast ( a ) , \\end{gather*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\omega _ { \\beta , \\mu , \\Lambda } ( A ) = { \\rm { T r } } _ { { \\cal H } _ { \\Lambda } } ( \\rho _ { \\Lambda } A ) \\ , \\ { \\rm { f o r \\ a l l } } \\ A \\in \\mathcal { B } ( \\mathcal { H } _ { \\Lambda } ) \\ . \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} { \\overline { R } } ^ { \\mathrm { L } } _ \\mathrm { U L A } = { \\log _ 2 } \\left ( 1 + \\frac { { N ^ 2 \\beta } r ^ { - { \\overline { \\alpha } } } } { { { \\lambda \\bar { G } \\Lambda _ \\mathrm { U L A } + \\frac { N _ o } { P _ t } } } } \\right ) , \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} M ( \\lambda , \\vec \\gamma ) \\leq \\dfrac { \\nu _ 0 ^ 2 \\lambda ^ 2 } { 2 } \\enspace , \\nu _ 0 ^ 2 = p + \\dfrac { 1 6 \\mathfrak g C _ { V , f } ^ 3 } { \\sqrt { n h ^ { 3 d } } } \\enspace , \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} d ( q ) = \\sup \\left \\{ s \\in \\mathbb { R } ^ + : \\sum _ { k = 1 } ^ \\infty \\sum _ { \\underline { a } \\in \\Lambda ^ k } \\phi ^ s ( \\underline { a } ) ^ { 1 - q } \\mu ( [ \\underline { a } ] ) ^ q < \\infty \\right \\} . \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} \\iota _ { X _ s } \\omega _ s = - \\alpha . \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} d t = \\frac { d q } { p + \\alpha \\sin { ( w t ) } q ^ 2 p } = \\frac { d \\gamma } { q + \\alpha \\sin { ( w t ) } p ^ 2 q } \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} \\ < a ( \\hbar ) u , b ( \\hbar ) v \\ > = a ( \\hbar ) b ( - \\hbar ) \\ < u , v \\ > , \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} \\tau ( \\omega ) : = \\inf \\left \\{ t \\geq 0 : \\left | \\varphi _ t ( \\omega , x ) - \\varphi _ t ( \\omega , y ) \\right | \\leq 4 \\right \\} . \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} \\widetilde { W } _ { N _ 1 } = \\widetilde { W } ( : , [ 1 \\ , 3 \\ , 4 \\ , 2 \\ , 5 \\ , 6 ] ) , \\widetilde { W } _ { N _ 2 } = \\widetilde { W } ( : , [ 2 \\ , 3 \\ , 4 \\ , 1 \\ , 5 \\ , 6 ] ) , \\\\ \\widetilde { W } _ { N _ 3 } = \\widetilde { W } ( : , [ 1 \\ , 5 \\ , 6 \\ , 2 \\ , 3 \\ , 4 ] ) , \\widetilde { W } _ { N _ 4 } = \\widetilde { W } ( : , [ 2 \\ , 5 \\ , 6 \\ , 1 \\ , 3 \\ , 4 ] ) . \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} \\upsilon = \\left ( \\frac { n _ 1 - 1 } { 2 } + \\nu _ 1 , \\frac { n _ 1 - 3 } { 2 } + \\nu _ 1 , \\dots , - \\frac { n _ 1 - 1 } { 2 } + \\nu _ 1 , \\frac { n _ 2 - 1 } { 2 } + \\nu _ 2 , \\dots , - \\frac { n _ 2 - 1 } { 2 } + \\nu _ 2 , \\dots \\right ) . \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\lambda \\left ( A _ { \\alpha } \\right ) = \\max _ { \\left \\Vert \\mathbf { x } \\right \\Vert _ { 2 } = 1 } \\left \\langle A _ { \\alpha } \\mathbf { x } , \\mathbf { x } \\right \\rangle \\lambda _ { \\min } \\left ( A _ { \\alpha } \\right ) = \\min _ { \\left \\Vert \\mathbf { x } \\right \\Vert _ { 2 } = 1 } \\left \\langle A _ { \\alpha } \\mathbf { x } , \\mathbf { x } \\right \\rangle . \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} F ( \\lambda _ j | \\lambda _ p , \\lambda _ h ) = \\frac { \\lambda _ j - \\tilde { \\lambda } _ j } { \\lambda _ { j + 1 } - \\lambda _ j } . \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} L = \\sup _ { x \\in \\mathcal { F } } \\| x \\| _ { L ^ 1 } . \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} \\alpha ^ 2 & = - a c - b \\alpha + a \\beta \\\\ \\alpha \\beta & = - a d \\\\ \\beta ^ 2 & = - b d - d \\alpha + c \\beta . \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} & M _ x [ \\alpha _ { 1 , 2 } ] = M _ x [ a _ { 0 , 2 } ] + M _ x [ a _ { 0 , 1 } \\ , ( \\beta _ 1 ) _ x ] = - 2 c _ 6 \\ , M _ x [ \\overline { v } _ x ^ 2 ] - 1 2 c _ 7 M _ x [ \\overline { v } ^ 2 ] + \\frac { 4 } { 3 } c _ 2 ^ 2 M _ x [ \\overline { v } _ x ^ 2 ] + 4 c _ 2 c _ 3 M _ x [ \\overline { v } ^ 2 ] , \\\\ [ 2 m m ] & M _ x [ \\alpha _ { 0 , 2 } ] = M _ x [ a _ { 0 , 1 } \\ , ( \\beta _ 1 ) _ { x x } ] = - 4 c _ 1 c _ 2 M _ x [ \\overline { v } _ { x x } ^ 2 ] - 1 2 c _ 1 c _ 3 M _ x [ \\overline { v } _ x ^ 2 ] . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} \\pi _ s ( \\lambda ) : = \\frac { q ( q ^ { 2 s + 1 } + q ^ { - 2 s - 1 } ) } { ( q - q ^ { - 1 } ) \\sqrt { [ 2 ] _ q } } \\ , I _ s , & & \\pi _ s ( \\tilde \\lambda ) : = q ^ { - 2 s } I _ s . \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} f ( z ) : = \\ln p ( z ) , \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} { { \\bar { R } } _ 1 ^ { \\mathrm { L } } } & = { { \\log } _ 2 } \\left ( 1 + e ^ { \\mathbb { E } \\left [ { \\ln \\gamma _ o } \\right ] } \\right ) , \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} \\begin{aligned} \\int \\Big ( | \\nabla v _ \\lambda | ^ 2 + 2 \\lambda ^ { \\frac 1 2 } \\mbox { I m } [ ( \\partial _ r u ) \\overline u ] - ( \\partial _ r ( r V ) ) | u | ^ 2 - 2 \\varepsilon \\mbox { I m } [ ( r \\partial _ r u ) \\overline { u } ] \\Big ) d x \\\\ = \\mbox { R e } \\int \\Big ( f ( 2 r \\partial _ r \\overline u + ( n - 1 ) \\overline u ) \\Big ) d x . \\end{aligned} \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } = D _ { J _ 1 } S _ { J _ 2 } \\phi ^ { \\alpha } \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} \\Im Z ( E ) \\geq 0 , \\ , \\ , \\ , \\ , \\Im Z ( E ) = 0 \\ , \\Rightarrow \\ , \\Re Z ( E ) < 0 . \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} C = \\overline { C ^ o } \\subset \\overline { C \\cap \\xi ( G ) } \\subset D . \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} \\limsup _ { n } \\| T x _ { n } \\| & = \\limsup _ { n } | < T ^ { * } y ^ { * } _ { n } , x _ { n } > | \\\\ & \\leq \\limsup _ { n } \\sup _ { x ^ { * } \\in A } | < x ^ { * } , x _ { n } > | \\leq \\eta _ { p } ( A ) \\\\ & \\leq C \\cdot w k _ { X ^ { * } } ( A ) \\leq C \\cdot w k _ { X ^ { * } } ( T ^ { * } ) , \\\\ \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{align*} \\varphi _ \\beta ' ( r ) = \\frac { 1 } { r } \\frac { 1 } { 2 \\beta - 1 } \\ , \\varphi _ \\beta ^ 2 ( \\beta r ) - \\frac { 1 } { r } \\frac { 1 } { 2 \\beta - 1 } \\ , \\varphi _ \\beta ( r ) , \\beta \\in [ 0 , 1 ] \\setminus \\Big \\{ \\frac { 1 } { 2 } \\Big \\} , \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} A _ { p - 1 } ^ { 1 \\rightarrow m } ( x ) & = 1 + \\sum _ { k > 0 } [ D ( 1 , p - 1 + k ) : D ( m , p - 1 ) ] x ^ k & \\\\ & = 1 + \\sum _ { k > 0 } [ D ( 1 , p + k ) : D ( m , p ) ] x ^ k - ( 1 - \\delta _ { m , 2 } ) \\sum _ { k > 0 } [ D ( 1 , p - 1 + k ) : D ( m , p + 1 ) ] x ^ k & \\\\ & = A _ p ^ { 1 \\rightarrow m } ( x ) - ( 1 - \\delta _ { m , 2 } ) x ^ 2 A _ { p + 1 } ^ { 1 \\rightarrow m } ( x ) . \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{gather*} \\widehat { M } = \\begin{pmatrix} \\tilde { \\alpha } & \\tilde { \\alpha } \\\\ \\alpha & \\alpha \\end{pmatrix} , \\tilde { \\alpha } = \\frac { 1 - q _ 2 } { 2 ( 1 + q _ 2 ) } ( - q _ 1 + ( 1 + q _ 2 ) e _ 1 ) , \\alpha = \\frac { 1 + q _ 2 } { 2 ( 1 - q _ 2 ) } ( q _ 1 + ( 1 - q _ 2 ) e _ 1 ) . \\end{gather*}"}
-{"id": "9466.png", "formula": "\\begin{align*} \\alpha ^ { \\dagger } \\ : = \\ \\psi ( \\alpha ) , \\alpha ' \\ : = \\ \\alpha + \\psi ( \\alpha ) . \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} H _ { \\Lambda , \\mu } = H _ { 0 , \\Lambda , \\mu } + V _ { \\Lambda } \\ , \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} g _ k = - \\varepsilon \\beta _ 1 \\ , \\partial _ x e ^ { \\mathrm { i } k x } + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} \\sup _ { x } \\left \\vert \\hat { \\gamma } _ { 2 \\ell } ( x ) - \\gamma _ { 2 0 } ( x ) \\right \\vert & = O _ { p } ( n ^ { - d _ { 1 } ( 2 \\xi _ { 1 } - 1 ) / ( 2 \\xi _ { 1 } + 1 ) } \\ln ( n ) ) , \\left \\vert \\hat { \\gamma } _ { 3 \\ell } - \\gamma _ { 3 0 } \\right \\vert = O _ { p } ( n ^ { - d _ { 1 } } ) , \\\\ \\left \\Vert \\hat { \\gamma } _ { 2 \\ell } - \\gamma _ { 2 0 } \\right \\Vert & = O _ { p } ( n ^ { - d _ { 1 } 2 \\xi _ { 1 } / ( 2 \\xi _ { 1 } + 1 ) } \\ln ( n ) ) , ( \\ell = 1 , . . . , L ) . \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} f _ 1 f _ 2 & = x ^ { d _ 1 + d _ 2 } + x ^ { d _ 1 } \\tilde { f _ 2 } + x ^ { d _ 2 } \\tilde { f _ 1 } \\\\ & = ( x ^ { d _ 1 } + x ^ { d _ 1 - d _ 2 } \\tilde { f _ 2 } + \\tilde { f _ 1 } ) x ^ { d _ 2 } \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} K _ { \\hat X } = \\pi ^ * K _ { X } + ( A ( \\nu _ E ) - 1 ) E . \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} \\varphi ( z ) \\left ( \\phi ( \\omega ( z ) ) - 1 \\right ) = c _ { 1 } d _ { 0 } \\omega _ { 1 } z + \\left ( c _ { 1 } d _ { 1 } \\omega _ { 1 } + d _ { 0 } ( c _ { 1 } \\omega _ { 2 } + c _ { 2 } \\omega _ { 1 } ^ { 2 } ) \\right ) z ^ { 2 } + \\dots \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} y ' + f y = g \\end{align*}"}
-{"id": "1351.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": "7474.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { d } { d s } \\Big | _ { s = 0 } H ( \\gamma ( s ) | e ^ { - \\Psi } ) = \\int _ { \\mathbb { S } ^ { 1 } } f e ^ { - \\Psi } d \\omega = \\int _ { \\mathbb { S } ^ { 1 } } ( \\rho - e ^ { - \\Psi } ) d \\omega = 0 , \\\\ & \\frac { d } { d s } \\Big | _ { s = 0 } I ( \\gamma ( s ) | e ^ { - \\Psi } ) = 0 , \\end{aligned} \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} P _ { \\rm { t o t } } = \\sum _ { n = 1 } ^ { N } \\frac { p _ { n } } { \\xi } + \\sum _ { n = 1 } ^ { N } \\sum _ { k = 1 } ^ { K } x _ k \\frac { p _ { k , n } } { \\xi } + \\sum _ { k = 1 } ^ { K } x _ k \\frac { q _ { k } } { \\xi } + P _ \\mathrm { c } . \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} V _ z ( E ) \\cap ( y _ 2 + t / 2 , y _ 2 + t ) = \\emptyset \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\sum _ { a = b } ^ { p ' } \\binom { a } b ^ 2 ( H _ a - H _ { a - b } ) & = \\sum _ { a = b } ^ { p - 1 } \\binom { a } b ^ 2 \\sum _ { j = 0 } ^ { b - 1 } \\frac { 1 } { a - j } = \\sum _ { a = 1 } ^ { p - 1 } \\binom { a } b ^ 2 \\sum _ { j = 0 , j \\not = a } ^ { b - 1 } \\frac { 1 } { a - j } \\\\ & = \\frac 1 { b ! ^ 2 } \\sum _ { a = 1 } ^ { p - 1 } \\sum _ { i = 1 } ^ { 2 b - 1 } \\alpha _ i \\ , a ^ i = \\sum _ { i = 1 } ^ { 2 b - 1 } \\frac { \\alpha _ i } { b ! ^ 2 } \\sum _ { a = 1 } ^ { p - 1 } a ^ i \\equiv _ p 0 , \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} \\Pi _ 1 F _ A = F _ A ^ B + \\Pi _ 1 F _ A ^ Z . \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} & \\| S ( ( 2 \\rho ) ^ \\theta ) \\tilde { u } ( \\tau ) ) \\| _ { L ^ \\infty ( { \\bf R } ^ N ) } \\ge C G ( 0 , ( 2 \\rho ) ^ \\theta + \\rho ^ \\theta ) \\int _ { B ( z , \\rho ) } u ( y , \\tau ) \\ , d y \\\\ & \\qquad \\ge C [ ( 2 \\rho ) ^ \\theta + \\rho ^ \\theta ) ] ^ { - \\frac { N } { \\theta } } G ( 0 , 1 ) \\int _ { B ( z , \\rho ) } u ( y , \\tau ) \\ , d y \\\\ & \\qquad \\ge C \\rho ^ { - N } \\int _ { B ( z , \\rho ) } u ( y , \\tau ) \\ , d y \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} i \\beta \\alpha _ 3 \\psi = \\psi \\ , , \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{gather*} ( S , S ) = 0 \\Leftrightarrow \\{ L , L \\} \\simeq 0 . \\end{gather*}"}
-{"id": "6443.png", "formula": "\\begin{align*} T ( u ) : = ( \\lambda _ 1 K F _ 1 ( u ) , \\lambda _ 2 K F _ 2 ( u ) , \\ldots , \\lambda _ n K F _ n ( u ) ) , \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { 2 , ' } = \\| f \\| ^ 2 _ { 2 , \\pitchfork ' } = \\| f \\| ^ 2 _ { 2 , ' } = \\int ^ 1 _ 0 | f ^ { ( r ) } ( x ) | ^ 2 \\ , { \\rm d } x . \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} \\tilde \\Phi _ t \\circ \\pi ( x ) = \\pi \\circ \\Phi _ t ( x ) \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} b _ { n _ k } b _ N ^ { - 1 } = \\sum _ { j = 1 } ^ k c _ j b _ { n _ k } b _ { n _ j } ^ { - 1 } a _ { n _ j } \\end{align*}"}
-{"id": "3110.png", "formula": "\\begin{align*} \\tilde Q : = \\tilde { q } _ 3 ^ { r _ 3 } \\dotsb \\tilde q _ D ^ { r _ D } = p _ m ^ { - r } Q . \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} R _ n : = ( L _ n \\tilde { \\Pi } _ n ^ { \\perp } - \\Pi _ n ^ { \\perp } L _ n ) \\textbf { T } _ n \\Pi _ n \\mathcal { F } ( U _ n ) , Q ' _ n : = - \\Pi _ n ( L _ n \\textbf { T } _ n - \\mathrm { I } ) \\Pi _ n \\mathcal { F } ( U _ n ) . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} \\phi ^ * ( \\gamma . p ) = - \\gamma . q + \\sup _ { \\Omega } - \\Phi _ q \\circ \\gamma ^ { - 1 } = - \\gamma . q + \\sup _ { \\Omega } - \\Phi _ q = - \\gamma . ( q + \\sup _ { \\Omega } \\Phi _ q ) = \\gamma . \\phi ^ * ( p ) . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} T ^ * B u n _ { \\mathcal { G } _ { X , x , \\theta } } ( S ) = \\{ ( E , s ) \\mid E \\in B u n _ { \\mathcal { G } _ { X , x , \\theta } } ( S ) , s \\in H ^ { 0 } ( X \\times S , a d ( E ) ^ * \\otimes \\Omega ^ 1 _ { X \\times S / S } ) \\} . \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} X ( s + z ) = X ( s ) ( Q | s | ) ^ { - 2 z } \\left \\{ 1 + O \\left ( | z | ( | s | + | z | ) ^ { - 1 } \\right ) \\right \\} \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} \\psi = Q ^ k f _ k . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } { e ^ { - i \\theta k } \\ , \\ , { h } ^ 0 ( \\theta ) } d \\theta = 0 , k = 0 , 1 , \\dots \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} \\| f \\| _ 1 ^ 2 + \\frac { 1 } { \\gamma _ i } \\| f \\| _ 2 ^ 2 = \\| f \\| _ { 1 + k _ { \\gamma _ i } } ^ 2 = \\langle f , 1 \\rangle _ 1 ^ 2 + \\| f - \\langle f , 1 \\rangle _ 1 \\| _ { k _ { \\gamma _ i } } ^ 2 = \\frac { 1 } { \\gamma _ i } \\| f \\| _ k ^ 2 . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} p _ { n } ( x , t ) : = \\sum _ { i = 1 } ^ { n } P _ { n , i } ( t ) w _ { i } ( x ) , q _ { n } ( x , t ) : = \\sum _ { i = 1 } ^ { n } Q _ { n , i } ( t ) w _ { i } ( x ) , r _ { n } ( x , t ) : = \\sum _ { i = 1 } ^ { n } R _ { n , i } ( t ) w _ { i } ( x ) , \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\hat { \\psi } : = \\mathcal { D } _ { \\omega } ^ { - 1 } \\{ g _ 1 + M _ 1 ( \\varphi ) M _ { \\varphi } [ \\hat { \\eta } ] + M _ 2 ( \\varphi ) g _ 2 + M _ 3 ( \\varphi ) g _ 3 - M _ 2 ( \\varphi ) [ \\partial _ { \\psi } \\theta _ 0 ] ^ T M _ { \\varphi } [ g _ 2 ] \\} . \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} E ( u ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ N } | \\nabla u | ^ 2 d x - \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ N } | u | ^ 2 \\log | u | ^ 2 d x . \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} \\int _ { C _ 1 } \\rho ^ * \\omega = \\lim _ { \\varepsilon \\rightarrow 0 } \\int _ { - \\pi } ^ { \\pi } \\int _ { \\epsilon } ^ { 1 } \\rho ^ * \\left ( \\frac { 1 } { h } d h \\wedge d \\theta \\right ) = \\lim _ { \\varepsilon \\rightarrow 0 } \\int _ { - \\pi } ^ { \\pi } \\int _ { \\epsilon } ^ { 1 } \\frac { 1 } { - h } d ( - h ) \\wedge d \\theta = \\int _ { C _ 1 } \\omega . \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} \\int d P _ { \\mathsf { B } _ n , 2 } = 2 ^ { - n ( R _ 1 + R _ 2 ) } \\sum _ { ( i , j ) \\in \\mathcal { I } _ n \\times \\mathcal { J } _ n } \\mathbb { P } _ { p _ { W | U , V } ^ n } \\Big ( \\big ( \\mathbf { U } ( i ) , \\mathbf { V } ( i , j ) , \\mathbf { W } \\big ) \\notin \\mathcal { A } _ { \\epsilon _ 1 , \\epsilon _ 2 } \\Big | \\mathbf { U } = \\mathbf { U } ( i ) , \\mathbf { V } = \\mathbf { V } ( i , j ) \\Big ) , \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} { { \\left \\Vert \\eta \\right \\Vert _ { H _ { G } ^ { p } } : = } } \\left [ { { \\mathbb { E } [ \\left \\{ { { \\int _ { 0 } ^ { T } | \\eta _ { t } | ^ { 2 } d t ] } } \\right \\} ^ { p / 2 } } } \\right ] ^ { 1 / p } . \\ \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} a \\left ( 0 \\right ) = a \\left ( 1 \\right ) = 1 , \\thinspace \\thinspace a \\left ( 2 \\right ) = 2 . \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} D ^ q ( \\mu ) = d ( q ) . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} 1 _ { A [ v ] } = \\sum _ { \\iota \\in I } \\left . e _ \\iota \\left ( v \\right ) \\right \\rangle \\left \\langle e _ \\iota \\left ( v \\right ) \\right . . \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} \\rho ( { a + b \\brack a } _ v ) = q ^ { p a b } { a + b \\choose a } . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} y _ n [ n - 1 ] = y _ n \\alpha _ 1 ' . \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\varphi ( t , q , \\lambda ) : = f _ 2 \\big ( t , \\hat x ( t ) , q , \\lambda \\big ) . \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} S _ J \\widetilde { u } _ n = \\widetilde { u } _ { n + J } \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} 1 + \\frac { 1 } { b } \\left ( ( 1 + i \\tan \\beta ) \\left ( \\frac { z D _ { q } \\mathcal { F } ( z ) } { \\mathcal { F } ( z ) } \\right ) - i \\tan \\beta - 1 \\right ) = p _ { k , \\alpha } ( w ( z ) ) . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} ( - \\Delta _ \\Omega ) ^ { \\alpha / 2 } \\varphi ( x ) = c _ { d , \\alpha } \\mbox { P . V . } \\int _ \\Omega [ \\varphi ( x ) - \\varphi ( y ) ] J ( x , y ) \\d y + c _ { d , \\alpha } \\ , \\kappa ( x ) \\varphi ( x ) , \\mbox { f o r } x \\in \\Omega \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{align*} \\lvert A \\rvert _ s ^ 2 : = \\sum _ { i \\in \\mathbb { Z } ^ b } \\langle i \\rangle ^ { 2 \\ , s } \\left ( \\sup _ { i _ 1 - i _ 2 = i } \\lvert A _ { i _ 1 } ^ { i _ 2 } \\rvert \\right ) ^ 2 . \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} F _ A & = F _ A ^ Z + F _ A ^ B , & \\nabla ^ q F _ A & = ( \\nabla ^ q F _ A ) ^ Z + ( \\nabla ^ q F _ A ) ^ B , \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | P _ n ( z ) | ^ { 1 / n } = | \\Phi ( z ) | , z \\in \\Omega , \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + a x + b , a = - \\frac { c _ 4 } { 4 8 } , b = - \\frac { c _ 6 } { 8 6 4 } \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} u _ 2 = - \\frac { 1 } { \\lambda a ^ { 1 2 } } \\left [ u _ 1 ^ { ( 4 ) } + \\lambda a _ { 1 1 } u _ 1 \\right ] , \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{gather*} F _ { 6 } ( t ) = \\lim _ { x \\to + \\infty } F ( x , t ; \\beta = 6 ) . \\end{gather*}"}
-{"id": "2866.png", "formula": "\\begin{align*} k ^ \\prime ( u , v ) = \\int _ { \\Omega } | u ( t ) - x ( t ) | ^ { p - 2 } ( u ( t ) - x ( t ) ) v ( t ) \\ , \\mu ( d t ) . \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} \\tau _ { \\le i } B \\to ( \\tau _ { \\le i } B ) ( d ) = ( \\tau _ { \\le i - d } ( B ) ) ( d ) . \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} u _ t = u _ 0 + \\int _ { 0 } ^ { t } \\zeta _ s d s + \\int _ { 0 } ^ { t } v _ s d B _ { s } + \\frac { 1 } { 2 } \\int _ { 0 } ^ { t } w _ s d \\langle B \\rangle _ { s } . \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{align*} 0 & = 3 [ 9 A ] - [ 3 A ] - [ 3 C ] - [ 3 B ] \\\\ & = [ 9 B ] - [ 9 A ] + 9 [ \\vartheta _ 9 ] \\\\ & = 3 [ 1 8 A ] - [ 6 A ] - [ 6 B ] - [ 6 C ] \\\\ & = [ 2 7 B C ] - [ 2 7 A ] - 3 / 2 [ \\vartheta _ 9 ] - 9 / 2 [ \\vartheta _ { 8 1 } ] \\\\ & = [ 3 6 B C ] - [ 3 6 A ] + 3 [ \\vartheta _ 4 ] + 3 [ \\vartheta _ { 3 6 } ] \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} I = \\sum _ { n = 0 } ^ \\infty I _ { 2 ^ { 2 ^ n } } \\stackrel { \\smash { \\textrm { ( \\ref { e q : I s } ) } } } { \\le } \\sum _ { n = 0 } ^ \\infty C k ^ 3 \\log ( | x | + | \\ell | ) \\frac { n } { 2 ^ n } \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} q _ \\sharp 1 _ { \\mathbb { G } _ { m , S } } = 1 _ S \\bigoplus 1 _ S ( 1 ) [ 1 ] \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} \\xi ^ i = \\frac { \\operatorname { d } \\ ! \\widetilde { x } ^ i } { \\operatorname { d } \\ ! \\varepsilon } \\Big | _ { \\varepsilon = e } , \\phi ^ { \\alpha } = \\frac { \\operatorname { d } \\ ! \\widetilde { u } ^ { \\alpha } } { \\operatorname { d } \\ ! \\varepsilon } \\Big | _ { \\varepsilon = e } . \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} \\begin{aligned} b _ 3 - m _ 3 & = \\Upsilon ' ( 0 ) \\{ M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] - M _ { \\varphi , x } [ g ( a _ 1 - 1 ) - g ( 0 ) ] \\} \\\\ [ 1 . 5 m m ] & + \\Upsilon _ { \\geq 2 } [ M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] ] - M _ { \\varphi } [ \\Upsilon _ { \\geq 2 } [ M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] ] ] . \\end{aligned} \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} \\begin{gathered} G _ i ( y ) = \\sum _ { j = 0 } ^ { \\lfloor \\frac { n + 1 - i } k \\rfloor } y ^ j e ' _ { n + 1 - k j - i } = \\sum _ { j = 0 } ^ { \\lfloor \\frac { n + 1 - i } k \\rfloor } y ^ j [ e _ { n + 1 - k j - i } + r e _ { n - k j - i } ] \\\\ = r g _ i ( y ) + \\begin{cases} y g _ { k - 1 } ( y ) & i = 0 ; \\\\ g _ { i - 1 } ( y ) & i \\geq 1 . \\end{cases} \\end{gathered} \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} \\phi ^ h ( x ) = \\left ( \\Phi _ { 1 2 } ( x _ 1 ) x _ 2 + \\Phi _ { 1 3 } ( x _ 1 ) x _ 3 \\ , , \\frac { 1 } { h } \\int _ 0 ^ { x _ 1 } \\Phi _ { 2 1 } ( s ) \\dd s + \\Phi _ { 2 3 } ( x _ 1 ) x _ 3 \\ , , \\frac { 1 } { h } \\int _ 0 ^ { x _ 1 } \\Phi _ { 3 1 } ( s ) \\dd s + \\Phi _ { 3 2 } ( x _ 1 ) x _ 2 \\right ) , \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} \\alpha _ j ( z , \\zeta ) & = \\alpha _ j ( y ^ i + \\eta ^ \\sigma \\alpha _ i , \\eta ^ \\sigma a _ \\sigma , \\eta ^ { \\rho \\neq \\sigma } + \\eta ^ \\sigma a _ { \\rho \\neq \\sigma } ) \\\\ & = \\alpha _ j ( y , 0 , \\widehat { \\eta } ) + \\eta ^ \\sigma \\kappa \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} \\gamma = \\frac { | h ^ { \\rm ( a , b ) } | ^ 2 \\frac { P _ { \\rm f } } { d ^ { \\alpha } _ { \\rm a , b } } } { | h ^ { \\rm ( j , b ) } | ^ 2 \\frac { P _ { \\rm j } } { d ^ { \\alpha } _ { \\rm j , b } } + \\sigma _ { \\rm b } ^ 2 } . \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} \\bold { D } _ F ( P ) : = \\frac { \\operatorname { d } } { \\operatorname { d } \\ ! \\varepsilon } \\Big | _ { \\varepsilon = 0 } F ( x , [ u + \\varepsilon P ( x , [ u ] ) ] ) . \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} \\mathcal { N } _ s u ( x ) : = a _ { N , s } \\int _ \\Omega \\frac { u ( x ) - u ( y ) } { | x - y | ^ { N + 2 s } } \\ , d y , x \\in \\R ^ N \\setminus \\overline { \\Omega } . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} \\phi _ a ^ + ( \\xi ) = \\sum _ { n = 1 } ^ { \\infty } \\xi ^ n n \\frac { \\partial } { \\partial p ^ \\lambda _ { n , a } } , \\phi _ a ^ - ( \\xi ) = \\sum _ { n = 0 } ^ { \\infty } \\frac { p _ { n , a } ^ \\lambda } { \\xi ^ n } , \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{gather*} \\phi _ t | _ { \\{ \\beta = 0 \\} } = \\Psi _ t - \\frac 1 J ( X _ \\alpha X _ t + Y _ \\alpha Y _ t ) \\Psi _ \\alpha - \\frac 1 J ( Y _ \\alpha X _ t - X _ \\alpha Y _ t ) \\Theta _ \\alpha , \\\\ \\frac 1 2 | \\nabla \\phi | ^ 2 | _ { \\{ \\beta = 0 \\} } = \\frac { 1 } { 2 J } ( \\Psi _ \\alpha ^ 2 + \\Theta _ \\alpha ^ 2 ) . \\end{gather*}"}
-{"id": "6736.png", "formula": "\\begin{align*} P \\pm Q = \\pm \\frac { 2 c } { c - 2 } Q ^ { - 1 } + \\frac { 2 c } { c + 2 } P ^ { - 1 } . \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { \\ell = 1 } ^ { L } \\sum _ { i \\in I _ { \\ell } } [ g ( W _ { i } , \\hat { \\gamma } _ { \\ell } , \\theta _ { 0 } ) + \\phi ( W _ { i } , \\hat { \\gamma } _ { \\ell } , \\hat { \\alpha } _ { \\ell } , \\tilde { \\theta } _ { \\ell } ) ] = \\hat { \\psi } ( \\theta _ { 0 } ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\psi ( W _ { i } , \\gamma _ { 0 } , \\alpha _ { 0 } , \\theta _ { 0 } ) + o _ { p } ( n ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} \\mathcal L ^ { \\nabla } _ v D \\pi \\ , : = \\nabla ^ H _ { \\frac { \\partial } { \\partial t } } \\ ; D ( \\pi \\circ \\Phi ) \\ ; \\big | _ 0 \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} \\| g _ k \\| _ q \\ ; \\le \\ ; \\| T \\sigma ^ \\ast ( g _ { k - 1 } ) \\| _ q \\ ; \\le \\ ; \\| T \\| _ q \\ , \\| g _ { k - 1 } \\| _ q ^ q \\ ; \\le \\ ; c \\cdot C ^ { ( 1 + \\frac { k - 1 } { 2 } ) q } \\ ; \\le \\ ; C ^ { \\frac { 3 } { 2 } - q + q + \\frac { k - 1 } { 2 } } \\ ; = \\ ; C ^ { 1 + \\frac { k } { 2 } } \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} \\Phi ^ - ( x ) & : = \\Phi ^ + ( - x ) , \\\\ \\Phi ^ \\circ ( x ) & : = \\Phi ^ + ( L / 2 + x ) , \\\\ \\Phi ( x ) & : = \\Phi ^ \\circ ( x ) + \\Phi ^ \\circ ( - x ) , \\\\ F & : = \\Phi \\ast \\Phi . \\end{align*}"}
-{"id": "3574.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\tilde { J } _ { 1 } ( t ) g \\right \\| _ { 2 } + \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\tilde { J } _ { 2 } ( t ) g \\right \\| _ { 2 } & \\le C ( 1 + t ) ^ { - \\frac { n } { 2 } - ( \\ell + k ) } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla _ { x } ^ { \\ell + k } g \\| _ { 2 } , \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} \\| E _ { \\rho } f \\| _ { L ^ { 2 } ( 0 , T ; H ^ { k } ( \\Omega ) ) } \\leq \\frac { C } { \\sigma ^ { 2 - k } } \\| f \\| _ { L ^ { 2 } ( 0 , T ; H ^ { 1 } ( \\Omega ) ) } , \\ , \\ , \\ , \\ , \\ , k = 1 , \\ , 2 . \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} w ( t , \\alpha , \\beta ) = z ( t , \\alpha , \\beta ) - ( \\alpha + i \\beta ) , \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} { \\left | A ( e ^ { i \\theta } ) g _ 0 ( \\theta ) + \\sum \\limits _ { j = 0 } ^ { \\infty } ( ( \\bold { B } ^ 0 ) ^ { - 1 } \\bold { R } ^ 0 \\bold { a } ) _ j e ^ { i j \\theta } \\right | ^ 2 } = \\gamma _ 1 { ( f _ 0 ( \\theta ) + g _ 0 ( \\theta ) ) ^ 2 } \\left ( f _ 0 ( \\theta ) \\right ) ^ { \\beta - 1 } , \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} F = \\{ ( k , x ) : 0 \\leq k \\leq N - 1 \\textrm { a n d } x \\in B _ R \\cap W _ { b , b ^ { - 1 } , e } \\} . \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{gather*} S [ \\phi ] = \\int _ M ( j ^ \\infty \\phi ) ^ \\ast ( L ) \\end{gather*}"}
-{"id": "7658.png", "formula": "\\begin{align*} \\begin{aligned} | \\langle f , \\phi _ n ^ { ( k ) } \\rangle | ^ 2 & \\leq C \\sigma _ { 2 \\alpha , \\gamma } ( n ) ^ { 2 ( k - 1 ) } \\left ( \\sum _ { m = 1 } ^ \\infty \\frac { | f _ m | } { | z _ n - \\mu _ m | m ^ \\alpha n ^ \\alpha } \\right ) ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} I \\left ( \\alpha \\right ) = \\left [ \\mathbb { Z } _ { K } ^ { + } : \\mathbb { Z } \\left [ \\alpha \\right ] ^ { + } \\right ] = \\left \\vert I \\left ( x _ { 2 } , . . . , x _ { n } \\right ) \\right \\vert \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = s } \\Phi _ t ( x ) = v ( \\Phi _ s ( x ) ) \\ ; . \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} j ( w ) ^ { o ( w ) } = j ( w _ 0 ( S _ 1 ) ) ^ { d _ 1 } \\cdots j ( w _ 0 ( S _ l ) ) ^ { d _ l } , \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} W ( \\mathbb { R } ^ N ) = \\big \\{ u \\in H ^ { 1 } ( \\mathbb { R } ^ N ) : | u | ^ 2 \\log | u | ^ 2 \\in L ^ { 1 } ( \\mathbb { R } ^ N ) \\big \\} , \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} R _ n ^ \\rho = ( 1 + R _ { n - 1 } ^ \\rho ) \\Lambda _ n ^ \\rho , n \\ge 1 , ~ ~ R _ 0 ^ \\rho = \\omega _ 0 / ( 1 - \\omega _ 0 ) \\rho , \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} y ^ 2 + a x y + b y = x ^ 3 , a , b \\in \\Z _ \\ell , \\Delta _ m = b ^ 3 ( a ^ 3 - 2 7 b ) \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} Z = \\left ( B _ { s _ 1 } - \\frac { s _ 1 } t B _ t + \\frac { s _ 1 } t ( x - z ) , \\dots , B _ { s _ n } - \\frac { s _ n } t B _ t + \\frac { s _ n } t ( x - z ) \\right ) . \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} d _ { S ( G , t ) } ( x ^ t , w ) = & \\min _ { P _ i \\in \\mathcal { P } ( x , z _ j ) } \\left \\{ d _ { S ( G , t - j ) } \\left ( \\left ( z _ j ^ { ( i ) } \\right ) ^ { t - j } , z _ { j + 1 } \\cdots z _ t \\right ) \\right \\} + \\\\ & + ( 2 ^ { t - j + 1 } - 1 ) d _ G ( x , z _ j ) - ( 2 ^ { t - j } - 1 ) . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} - \\Delta _ { q ( x ) } \\underline { v } = \\left \\{ \\begin{array} { l l } \\lambda ^ { \\sigma } w _ { 2 } ^ { \\beta _ { 2 } ( x ) } & \\Omega \\backslash \\overline { \\Omega } _ { \\delta } \\\\ - w _ { 2 } ^ { \\beta _ { 2 } ( x ) } & \\Omega _ { \\delta } \\end{array} \\right . , \\underline { v } = 0 \\partial \\Omega , \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} g = \\nabla d \\phi _ i . \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} \\alpha = \\xi , 2 \\xi - 2 c \\xi ^ { 2 } + \\xi ^ { 3 } \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} V ( x ) = x ^ 4 + \\frac { 2 ( b ^ 3 + 1 ) } { b } | x | ^ 3 + \\frac { b ^ 6 + 4 b ^ 3 + 1 } { b ^ 2 } x ^ 2 + 2 ( b ^ 3 - 1 ) | x | , b \\ne 0 , \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} \\phi ( \\omega ) : = a _ { j k } + b _ { l j k } \\cdot \\omega + q _ { j k } ( \\omega ) , l \\in \\mathbb { Z } ^ { \\nu } , \\ , j , k \\in S ^ c , \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} & \\Delta ( E _ i ) = E _ i \\otimes 1 + K _ i \\otimes E _ i , & & \\Delta ( K _ i ) = K _ i \\otimes K _ i , \\\\ & \\Delta ( F _ i ) = F _ i \\otimes K ' _ i + 1 \\otimes F _ i , & & \\Delta ( K ' _ i ) = K ' _ i \\otimes K ' _ i , \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} \\phi ( z ) = - \\Psi ( 0 ) + \\frac { 1 } { z } \\sum _ m \\Phi \\left ( \\frac { m } { z } \\right ) \\ll 1 \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} T ( a , c ) = \\sum _ { m = 1 } ^ \\infty \\psi _ m ( a ) e \\left ( \\frac { \\overline a } { c } \\ell _ m \\right ) \\int g ( x ) k _ m ( x ) d x \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} x = u ^ 2 x ' + r , y = u ^ 3 y ' u = t ^ 3 , r = ( a _ 0 + y _ 1 t ) t ^ 5 \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} U = p ^ { 2 } + q ^ { 2 } = \\pm \\varepsilon , V = p ^ { 2 } + 2 p q - q ^ { 2 } = \\pm \\varepsilon , Z = - p ^ { 2 } + 2 p q + q ^ { 2 } = \\pm \\varepsilon , \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} ( B ^ { - 1 } b _ 3 ) ( \\vartheta ) = m _ 3 \\ , \\rho ( \\vartheta ) , m _ 3 \\in \\mathbb { R } \\Longrightarrow b _ 3 ( \\varphi ) = m _ 3 ( 1 + \\omega \\cdot \\partial _ { \\varphi } \\alpha ( \\varphi ) ) . \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} \\dot y = \\lambda \\varphi ( t , y , \\lambda ) , \\lambda \\geq 0 , \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} b = 2 \\Re \\left [ R - \\P [ R \\bar Y ] \\right ] , \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } ^ { c } D ^ { q } \\mathbf { x } ( t ) = - A \\mathbf { x } + T ^ { R } \\mathbf { g } ^ { R } ( \\mathbf { x } , \\mathbf { y } ) - T ^ { I } \\mathbf { g } ^ { I } ( \\mathbf { x } , \\mathbf { y } ) + \\mathbf { I } ^ { R } \\\\ ^ { c } D ^ { q } \\mathbf { y } ( t ) = - A \\mathbf { y } + T ^ { R } \\mathbf { g } ^ { I } ( \\mathbf { x } , \\mathbf { y } ) + T ^ { I } \\mathbf { g } ^ { R } ( \\mathbf { x } , \\mathbf { y } ) + \\mathbf { I } ^ { I } . \\end{array} \\right . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\dot \\zeta = T _ \\rho ' ( g + \\Delta z ) ( h + \\Delta \\zeta ) H _ 0 ^ 1 ( \\Omega ) , I \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{gather*} \\mathcal { P } _ a = \\int _ S \\omega ^ { b c } \\wedge h _ { a b c } , \\mathcal { M } _ { a b } = \\int _ S e ^ c \\wedge h _ { a b c } + \\frac 1 2 \\big ( x _ b \\omega ^ { d c } \\wedge h _ { a d c } - x _ a \\omega ^ { d c } \\wedge h _ { b d c } \\big ) . \\end{gather*}"}
-{"id": "5373.png", "formula": "\\begin{align*} \\mathcal { L } _ 3 = \\Pi _ S ^ { \\perp } ( \\omega \\cdot \\partial _ { \\varphi } + m _ 3 \\partial _ { x x x } + \\varepsilon \\mathfrak { B } _ 1 + \\varepsilon ^ 2 \\mathfrak { B } _ 2 + \\tilde { d } _ 1 \\partial _ x + \\tilde { d } _ 0 ) \\Pi _ S ^ { \\perp } + \\tilde { \\mathcal { R } } _ * \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} d \\bar X ( s ) = \\left ( - \\frac { 1 } { 1 + \\beta } e ^ { \\mu ( s - t ) } \\pi ( s ) \\bar X ( s ) - \\frac { \\alpha } { 1 + \\beta } \\right ) d s + \\nu _ 0 \\bar X ( s ) d W _ 0 ( s ) \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} x ( n + 1 ) = \\lambda ^ { - 1 } C ( n ) x ( n ) , \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} \\begin{aligned} \\tilde { \\lambda } ( s ) & = \\tilde { \\lambda } _ 0 + \\tilde { \\lambda } _ 1 s + \\tilde { \\lambda } _ 2 s ^ 2 + \\cdots \\\\ \\phi ( \\xi ; s ) & = \\phi _ 0 ( \\xi ) + \\phi _ 1 ( \\xi ) s + \\phi _ 2 ( \\xi ) s ^ 2 + \\cdots , \\end{aligned} \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} \\Theta _ { - L , L } = e ^ { - 2 L D } - R _ { - L , L } - \\Omega _ L \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} \\left ( A ( e ^ { i \\theta } ) - { h } ( \\theta ) \\right ) ^ { < \\alpha - 1 > } f ( \\theta ) - \\left ( { h } ( \\theta ) \\right ) ^ { < \\alpha - 1 > } g ( \\theta ) = \\overline { C ( e ^ { i \\theta } ) } , \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} E _ j = - \\frac 1 \\pi \\int \\xi ^ { 2 j + 2 } \\ln | T ( \\xi / 2 ) | d \\xi \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} \\| \\partial _ { t } ^ { \\ell } \\nabla ^ { k } _ { x } u ( t ) \\| _ { 2 } \\le \\begin{cases} & C ( 1 + t ) ^ { - \\frac { n } { 4 ( 1 - \\sigma ) } - \\ell - \\frac { k } { 2 ( 1 - \\sigma ) } } ( \\sigma \\in ( 0 , \\frac { 1 } { 2 } ) ) , \\\\ & C ( 1 + t ) ^ { - \\frac { n } { 2 } - ( \\ell + k ) } ( \\sigma = \\frac { 1 } { 2 } ) , \\\\ & C ( 1 + t ) ^ { - \\frac { n } { 4 \\sigma } - \\frac { \\ell + k } { 2 \\sigma } } ( \\sigma \\in ( \\frac { 1 } { 2 } , 1 ] ) \\end{cases} \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\frac { \\eta } { 2 } & > - b ( x ) \\cdot D \\varphi ( x ) + f ( x ) > - b ( x ) \\cdot D \\psi ( x ) + g ( x ) - \\frac { \\eta } { 4 } \\\\ & = - b ( x ) \\cdot D \\varphi ( x ) + \\tilde G ( x ) - \\tilde G ( 0 ) - \\frac { \\eta } { 4 } \\\\ & > - b ( x ) \\cdot D \\varphi ( x ) + G ( x , 0 ) - G ( 0 , 0 ) - \\frac { \\eta } { 2 } . \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} h _ 1 - \\int _ \\mathbb { R } h _ 1 \\ , d \\nu = \\sum _ { k = 1 } ^ \\infty a _ k H _ k \\ , h _ 2 - \\int _ \\mathbb { R } h _ 2 \\ , d \\nu = \\sum _ { k = 1 } ^ \\infty b _ k H _ k \\ , \\ , , \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} f _ { n + 1 } ( z ) = \\frac { 1 } { z } \\cdot \\frac { f _ n ( z ) - f _ n ( 0 ) } { 1 - \\overline { f _ n ( 0 ) } f _ n ( z ) } , n = 0 , 1 , 2 , \\ldots , \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} \\max _ { 1 \\leq j \\leq ( 1 + \\delta ) k _ n } \\binom { \\ell _ n } { j } & \\leq \\binom { \\ell _ n } { ( 1 + \\delta ) k _ n } \\\\ & \\leq \\exp ( \\ell _ n H _ 2 ( ( 1 + \\delta ) \\alpha _ n ) ) . \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} \\langle I ' ( u ) , v \\rangle & = ( a + b \\| u \\| _ { E ^ { \\alpha , p } } ^ p ) ^ { p - 1 } \\int _ 0 ^ T \\phi _ p ( { _ 0 D _ t ^ \\alpha } u ( t ) ) { _ 0 D _ t ^ \\alpha } v ( t ) d t \\\\ & \\ \\ \\ \\ - \\int _ 0 ^ T f ( t , u ( t ) ) v ( t ) d t , \\ \\ \\forall u , v \\in E _ 0 ^ { \\alpha , p } , \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} A _ j = \\{ ( x , t ) \\in U ^ j : v ( x , t ) < k _ j \\} = \\{ ( x , t ) \\in U ^ j : u ( x , t ) < k _ j \\} . \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\| ( f ^ n ) ' ( z ) \\| = \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log ( f ^ n ) ^ \\# ( z ) = \\frac { \\log | \\lambda | } { p } . \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} c = \\frac { 2 d ^ 2 u ^ 2 - 2 d ^ 2 r ^ 2 - 4 0 0 8 } { d ^ 2 u ^ 2 - d ^ 2 r ^ 2 - 1 9 9 6 } . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} f ( y ) = \\frac { \\Gamma ( \\tau - 1 ) } { h } \\ ; y \\bigl ( 1 + \\delta ( y ) + o ( 1 ) \\bigr ) , \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} v _ k ( t ) = \\left \\{ \\begin{array} { l l l } \\displaystyle 0 & \\textup { i f } & t \\in ( 0 , 1 ) \\backslash ( t _ 0 - \\gamma , t _ 0 + \\gamma ) , \\\\ \\displaystyle \\xi _ k & \\textup { i f } & t \\in ( t _ 0 - \\frac { \\gamma } { 2 } , t _ 0 + \\frac { \\gamma } { 2 } ) , \\\\ \\displaystyle \\frac { 2 \\xi _ k } { \\gamma } ( \\gamma - | t - t _ 0 | ) & \\textup { i f } & t \\in ( t _ 0 - \\gamma , t _ 0 + \\gamma ) \\backslash ( t _ 0 - \\frac { \\gamma } { 2 } , t _ 0 + \\frac { \\gamma } { 2 } ) \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} & \\Delta v + k _ 2 ^ 2 v = 0 \\ \\mbox { i n } \\ \\mathbb { R } ^ 2 \\setminus \\overline { \\Omega _ 2 } , \\Delta v + k _ 1 ^ 2 v = 0 \\ \\mbox { i n } \\ { \\Omega _ 2 } , \\\\ & v ^ { - } = v ^ { + } , S _ 2 \\frac { \\partial v ^ - } { \\partial n } = S _ 1 \\frac { \\partial v ^ + } { \\partial n } \\mbox { o n } \\partial \\Omega \\\\ & \\mbox { o u t g o i n g r a d i a t i o n c o n d i t i o n w i t h } k _ 2 \\mbox { i n } \\mathbb { R } ^ 2 \\setminus \\overline { \\Omega _ 2 } . \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} \\lambda _ { t o p } & \\geq \\lim _ { m \\rightarrow \\infty } \\frac { 1 } { m } \\log | D \\varphi _ m ( \\omega , x ) v | = \\lim _ { m \\rightarrow \\infty } \\frac { 1 } { m } \\int _ 0 ^ m ( 1 - | \\varphi _ s ( \\omega , x ) | ^ 2 ) \\ , d s . \\end{align*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\lambda \\left ( A _ { \\alpha } \\right ) = \\sqrt { \\frac { 1 } { n } \\sum _ { u \\in V \\left ( G \\right ) } d _ { G } ^ { 2 } \\left ( u \\right ) } \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} R _ { a _ 1 , a _ 2 , a _ 3 } = \\{ ( x _ 1 , x _ 2 , x _ 3 ) \\in \\R ^ 3 : 0 < x _ 1 < a _ 1 , 0 < x _ 2 < a _ 2 , 0 < x _ 3 < ( a _ 1 a _ 2 ) ^ { - 1 } \\} . \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} Z _ 0 - \\bar Z _ 0 = P \\left \\{ F ( X - \\bar X ) + G ( v - \\bar v ) \\right \\} . \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} \\mathrm { c m } ( z , x ) = \\frac { [ x , b ( z ) ] } { | | x | | } = \\frac { [ x , y - x ] } { | | y - x | | _ a | | x | | } = \\frac { [ x , y ] } { | | y - x | | _ a | | x | | } , \\ \\mathrm { a n d } \\\\ \\mathrm { c m } ( z , y ) = \\frac { [ y , b ( z ) ] } { | | y | | } = \\frac { [ y , y - x ] } { | | y - x | | _ a | | y | | } = \\frac { [ x , y ] } { | | y - x | | _ a | | x | | } . \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} [ X _ 1 , X _ 2 ] : = \\left ( \\bold { p r } X _ 1 ( Q _ 2 ^ { \\alpha } ) - \\bold { p r } X _ 2 ( Q _ 1 ^ { \\alpha } ) \\right ) \\partial _ { u ^ { \\alpha } } . \\end{align*}"}
-{"id": "2088.png", "formula": "\\begin{align*} W ' \\ ; : \\ ; y '^ 2 + \\frac { a } { u _ 1 \\pi ^ \\alpha } x ' y ' + y ' = x '^ 3 \\upsilon _ F ( \\Delta ( W ' ) ) = \\upsilon _ F ( \\Delta _ m ) - 1 2 \\alpha = 0 . \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} W _ i ( s ) = \\begin{pmatrix} s ^ { \\mu _ i ^ + } & s ^ { \\mu _ i ^ - } \\\\ \\mu _ i ^ + s ^ { \\mu _ i ^ + - 1 } & \\mu _ i ^ - s ^ { \\mu _ i ^ - - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} h ^ * ( f \\cup g ) ( [ a _ 1 | \\dots | a _ { n + p } ] ) & = ( - 1 ) ^ { ( n + p - 1 ) w ( a _ 1 ) + \\dots + w ( a _ { n + p - 1 } ) } f \\cup g ( [ a _ 1 | \\dots | a _ { n + p } ] ) \\\\ & = ( - 1 ) ^ { n p + ( n + p - 1 ) w ( a _ 1 ) + \\dots + w ( a _ { n + p - 1 } ) } f ( [ a _ 1 | \\dots | a _ n ] ) g ( [ a _ { n + 1 } | \\dots | a _ { n + p } ] ) . \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} U ( g , A , \\epsilon ) : = \\{ h \\in N ( V , \\| \\cdot \\| ) \\colon \\sup _ { x \\in A } \\| h ( x ) - g ( x ) \\| < \\epsilon \\} , \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } E _ { 3 } ( t ) g \\right \\| _ { 2 } & \\le C t ^ { 1 - \\ell } ( 1 + t ) ^ { - \\frac { n } { 2 } - k } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla ^ { ( k + \\ell - 1 ) _ { + } } _ { x } g \\| _ { 2 } , \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} \\left \\{ \\frac { \\delta L } { \\delta u ^ { \\alpha } } = 0 \\right \\} \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} | G _ n ( w ) ^ \\mathrm { a b } | = | \\mathrm { R e s } ( f _ w ( x ) , x ^ n - 1 ) | = \\prod _ { \\lambda ^ n = 1 } | f _ w ( \\lambda ) | . \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} \\dim \\langle P _ I , F _ J \\rangle = \\begin{cases} a _ { I } - 1 & \\mbox { i f } J = \\emptyset \\\\ \\geq a _ { I \\cup J } & \\mbox { i f } J \\neq \\emptyset , J \\subsetneq [ k ] \\setminus I \\\\ d & \\mbox { i f } J = [ k ] \\setminus I \\end{cases} \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} \\mathcal { L } _ { V } \\left ( t , x _ { t } , i \\right ) \\leq - x ^ { 2 } + \\left \\vert b \\left ( t \\right ) \\right \\vert y ^ { 2 } , \\ ; i = 1 , 2 . \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} \\langle G ' ( u ) , u \\rangle & = b p ( p - 1 ) M _ u ^ { p - 2 } \\| u \\| _ { E ^ { \\alpha , p } } ^ { 2 p } + p M _ u ^ { p - 1 } \\| u \\| _ { E ^ { \\alpha , p } } ^ p \\\\ & \\ \\ \\ \\ - \\int _ { 0 } ^ { T } f ' _ 2 ( t , u ( t ) ) u ^ 2 ( t ) d t - \\int _ { 0 } ^ { T } f ( t , u ( t ) ) u ( t ) d t \\\\ & < M _ u ^ { p - 2 } \\| u \\| _ { E ^ { \\alpha , p } } ^ p ( b p ^ 2 \\| u \\| _ { E ^ { \\alpha , p } } ^ p + a p ) - p ^ 2 \\int _ { 0 } ^ { T } f ( t , u ( t ) ) u ( t ) d t \\\\ & = a ( p - p ^ 2 ) M _ u ^ { p - 2 } \\| u \\| _ { E ^ { \\alpha , p } } ^ p \\leq 0 , \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} \\begin{aligned} T _ k g _ k ^ \\pm & = ( \\mu _ { 2 k - 1 } + d _ k \\pm d _ k \\tau _ k ) g _ k ^ \\pm , \\\\ \\tau _ k & = \\sqrt { 1 - t _ k ^ 2 } , g _ k ^ \\pm = \\begin{pmatrix} 1 \\\\ G _ k ^ \\pm \\end{pmatrix} , G _ k = \\left ( \\frac { 1 + \\tau _ k } { 1 - \\tau _ k } \\right ) ^ \\frac 1 2 . \\end{aligned} \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} \\scriptstyle f _ N ^ { - 1 } H _ n ( N ) = \\begin{cases} \\scriptstyle H _ n ( N - 1 ) \\cup H _ h ( N - 1 ) & \\scriptstyle \\\\ \\scriptstyle H _ n ( N - 1 ) \\cup H _ h ( N - 1 ) \\cup H _ u ( N - 1 ) & \\scriptstyle \\end{cases} \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\Xi _ j : = \\left \\{ z \\notin \\Pi \\ , : \\Re z \\in [ \\mu _ { j - 1 } + \\frac \\kappa 2 ( j - 1 ) ^ { \\gamma - 1 } , \\mu _ { j + 1 } - \\frac \\kappa 2 j ^ { \\gamma - 1 } ] \\right \\} , j \\in \\N ; \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} \\Lambda ^ { \\ast } = \\{ k _ { j } = \\frac { 2 \\pi } { V ^ { { \\alpha _ { j } } } } n _ { j } : n _ { j } \\in \\mathbb { Z } \\} _ { j = 1 } ^ { d = 3 } \\ \\ \\ { \\rm { t h e n } } \\ \\ \\ \\varepsilon _ { k } = \\sum _ { j = 1 } ^ { d } { k _ { j } ^ 2 } \\ . \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} \\bar \\nabla ^ { H } _ { \\frac { \\partial } { \\partial t } } D ( \\pi \\circ \\Phi ) \\ ; \\big | _ 0 w = & \\pi ^ * \\bar \\nabla ^ { ( \\pi \\circ \\Phi ) ^ { - 1 } T Y } _ { w } D ( \\pi \\circ \\Phi ) \\frac { \\partial } { \\partial t } \\Big | _ 0 \\\\ = & \\pi ^ * \\bar \\nabla ^ { \\pi ^ { - 1 } T Y } _ { w } D \\pi v . \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} B _ h ^ { \\rm B D M } ( ( { \\bf u } _ h , p _ h ) , ( { \\bf v } _ h , q _ h ) ) : = a _ h ( { \\bf u } _ h , { \\bf v } _ h ) + b _ h ^ { \\rm B D M } ( p _ h , { \\bf v } _ h ) - b _ h ^ { \\rm B D M } ( q _ h , { \\bf u } _ h ) , \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} \\Lambda _ j ^ * ( q ) = \\frac { \\tau ( ( q , D ) ) } { 2 ^ j \\tau ( q ) } \\sum _ { m n = q } \\chi ( m ) \\sum _ { a + b = j } \\binom { j } { a } ( - 1 ) ^ b \\Lambda _ a ( m ) \\Lambda _ b ( n ) . \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} V ^ { \\ast } \\Psi A ^ { \\dagger } & = V ^ { \\ast } A ^ { \\dagger } + ( I + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } F _ { S _ { A } } K ^ { \\ast } A ^ { \\dagger } , \\\\ V ^ { \\ast } \\Psi K S _ { A } ^ { \\dagger } H & = ( S _ { A } - I ) S _ { A } ^ { \\dagger } H + ( I + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } F _ { S _ { A } } K ^ { \\ast } K S _ { A } ^ { \\dagger } H . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} \\lim _ { x \\to \\pm \\infty } \\frac { \\alpha ( x , y ) } { x } = 0 . \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{gather*} \\hat { B } _ 0 = - \\frac { x } { 2 } \\sigma _ 3 - \\begin{pmatrix} 0 & u \\\\ u & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "6634.png", "formula": "\\begin{align*} \\int _ { M } X ( b ) d M = \\frac { n } { 2 } \\int _ { M } \\left \\langle { \\stackrel { \\circ } { B } } , \\mathcal { L } _ { X } g \\right \\rangle d M - n \\int _ { \\Sigma } { \\stackrel { \\circ } { B } } ( X , \\nu ) d \\Sigma , \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} \\alpha _ { n } ( a ) & : = a \\ln ( n / k ) \\bigg ( \\frac { [ n x ] + 1 } { n } - x \\bigg ) ^ { p } , \\beta _ { n } ( a ) : = a \\ln ( n / k ) \\bigg ( \\frac { [ n ( x + \\epsilon ) ] + 1 } { n } - x \\bigg ) ^ { p } , \\\\ \\tau _ { n } ( a ) & : = a \\ln ( n / k ) \\bigg ( \\frac { [ n x ] } { n } - x \\bigg ) ^ { p } , \\mu _ { n } ( a ) : = a \\ln ( n / k ) \\bigg ( \\frac { [ n ( x + \\epsilon ) ] } { n } - x \\bigg ) ^ { p } \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\gamma = \\pm \\frac { e ^ { 2 t } + 2 C _ 2 } { \\sqrt { - \\alpha \\sin { ( w t ) } e ^ { 4 t } + 4 \\alpha \\sin { ( w t ) } e ^ { 2 t } C _ 2 - 4 \\alpha \\sin { ( w t ) } C _ 2 ^ 2 + 4 e ^ { 2 t } C _ 1 } } \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} u _ { t } ( x ) = ( \\zeta ( x ^ { 1 } + t ) \\eta ( x ^ { 2 } ) , \\zeta ' ( x ^ { 1 } + t ) \\eta ( x ^ { 2 } ) ) \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} f _ j = b \\partial _ \\alpha \\P [ R _ j \\bar R _ \\alpha ] - \\P \\left [ b R _ { \\alpha , j } \\bar R _ \\alpha \\right ] - \\P \\left [ R _ j \\partial _ \\alpha \\bar \\P ( b \\bar R _ \\alpha ) \\right ] . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} p _ { \\ell } ^ { ( a ) } ( \\nu , B ) : = Q _ { \\ell } \\bigl ( \\mu ^ { ( a ) } ( B ) \\bigr ) + Q _ { \\ell } ( \\nu ^ { ( a - 1 ) } ) - 2 Q _ { \\ell } ( \\nu ^ { ( a ) } ) + Q _ { \\ell } ( \\nu ^ { ( a + 1 ) } ) , \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\wedge _ { i } b _ 2 ^ i = \\pm \\nu ( \\mathcal { E } _ B ) ^ { - 1 } \\nu ( \\mathcal { E } _ 1 ) ^ { - 1 } \\nu ( \\mathcal { E } _ 3 ) ^ { - 1 } M \\wedge N \\wedge P \\wedge Q . \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{gather*} i _ X \\omega _ 2 = \\delta \\alpha _ 2 + \\delta _ Q \\alpha ' _ 1 + d \\alpha ' _ 2 \\end{gather*}"}
-{"id": "8305.png", "formula": "\\begin{align*} | \\nabla \\varphi | ^ 2 = \\alpha ^ { - 2 } F ^ { - 2 } | \\nabla F | ^ 2 , \\qquad \\Delta \\varphi = - \\alpha ^ { - 1 } \\Big ( F ^ { - 2 } | \\nabla F | ^ 2 + F ^ { - 1 } \\Delta F \\Big ) , \\end{align*}"}
-{"id": "1373.png", "formula": "\\begin{align*} \\dim ( Q P _ k ) _ n \\geqslant N ( k , n ) + \\sum _ { t = 2 } ^ p \\binom k t + ( k - 3 ) \\binom { k } 2 \\sum _ { u = 2 } ^ { q } \\binom k u . \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} B _ { k } ^ { i } = \\left \\{ A \\in \\Gamma _ { + } : \\sigma _ 1 ( A ) \\in \\{ k ^ { - 1 } , ( k - 1 ) ^ { - 1 } \\} \\setminus \\{ \\infty \\} \\sigma _ j ( A ) = i j \\in \\{ 2 , 3 \\} \\right \\} \\setminus \\{ I \\} . \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} \\alpha & = \\alpha ( P _ { i + 1 } ^ { ( 1 ) } , P _ { i + 1 } ^ { ( 2 ) } ) + \\alpha ( P _ i ^ { ( 2 ) } , P _ { i + 1 } ) \\\\ & = ( 1 - \\chi ( c - x ' ) - \\chi ( x ' - 1 - d ) + \\chi ( c - d - 1 ) ) \\\\ & + ( - \\chi ( c - b - 1 ) + \\chi ( c - x ) + \\chi ( d - x + 1 ) ) \\\\ & = 1 + \\chi ( c - d - 1 ) - \\chi ( c - b - 1 ) . \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} d ( E , x ) = \\frac { \\tau } { 2 } \\| E - r I \\| _ F ^ 2 + \\frac { 1 - \\tau } { 2 } \\| x \\| _ 2 ^ 2 , \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} \\bold { p r } X = \\sum _ { \\alpha , J } \\left ( S _ J Q ^ { \\alpha } _ n \\right ) \\frac { \\partial } { \\partial u _ { n + J } ^ { \\alpha } } . \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} \\infty > u ( x , t ) = \\int _ { { \\bf R } ^ N } G ( x - y , t ) \\ , d \\mu ( y ) + \\int _ 0 ^ t \\int _ { { \\bf R } ^ N } G ( x - y , t - s ) u ( y , s ) ^ p \\ , d y \\ , d s \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} \\lim _ { x \\to \\pm \\infty } \\nabla \\beta ( x , y ) = ( 0 , 1 ) . \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} k _ 1 = p \\pi - i \\kappa , \\ p \\in \\mathbb { N } , \\kappa \\ge 0 . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} a _ { \\iota } = \\left \\{ \\begin{aligned} 1 \\ ; \\ ; \\ ; & \\mbox { i f $ x _ { \\iota } ' \\not = 0 $ } \\\\ - 1 \\ ; \\ ; \\ ; & \\mbox { i f $ y _ { \\iota } ' \\not = 0 $ } \\\\ 0 \\ ; \\ ; \\ ; & \\mbox { i f $ x _ { \\iota } ' = y _ { \\iota } ' = 0 $ } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} C _ { i j } ^ { i i } + C _ { i j } ^ { j j } & = 0 , \\ \\mbox { i f } \\ i < j , \\\\ C _ { i j } ^ { i j } + C _ { j k } ^ { j k } & = C _ { i k } ^ { i k } , \\ \\mbox { i f } \\ i < j < k \\ \\mbox { a n d } \\ i \\ne k , \\\\ C _ { i j } ^ { i j } + C _ { j i } ^ { j i } & = 0 , \\ \\mbox { i f } \\ i < j < i , \\\\ C _ { i i } ^ { i i } & = C _ { x x } ^ { i i } , \\ \\mbox { f o r a l l } \\ x \\in X . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} ( n _ 2 + n _ 3 + n _ 4 ) r _ 1 + ( n _ 1 + n _ 3 + n _ 4 ) r _ 2 + ( n _ 1 + n _ 2 + n _ 4 ) r _ 3 + ( n _ 1 + n _ 2 + n _ 3 ) r _ 4 = 0 \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\mu _ k = \\left ( \\frac { k \\pi } { 2 l } \\right ) ^ 2 , k \\in \\N _ 0 , \\psi _ k ( x ) = \\begin{cases} \\displaystyle \\frac { 1 } { \\sqrt { 2 l } } , & k = 0 , \\\\ [ 4 m m ] \\displaystyle \\frac 1 { \\sqrt { l } } \\cos ( \\sqrt { \\mu _ k } ( x + l ) ) , & k \\in \\N , \\end{cases} \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} & ( M _ { p k } ^ { p - 1 } - M _ p ^ { p - 1 } ) \\int _ 0 ^ T \\phi _ p ( { _ 0 D _ t ^ \\alpha } u ( t ) ) ( { _ 0 D _ t ^ \\alpha } u _ k ( t ) - { _ 0 D _ t ^ \\alpha } u ( t ) ) d t \\\\ & = ( M _ { p k } ^ { p - 1 } - M _ p ^ { p - 1 } ) \\langle I _ 1 ' ( u ) , u _ k - u \\rangle \\\\ & \\rightarrow 0 \\ \\ \\mbox { a s } \\ k \\rightarrow \\infty , \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} _ { p } D _ { \\tau } ( 1 / \\lambda ) = _ { p } D _ { \\tau } ( \\lambda ) . \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} R = \\frac { \\log _ { n ^ 2 } S } { C } , \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\frac { ( - q ; q ) _ n } { ( q ; q ) _ n } = 1 + \\sum _ { \\pi \\in \\mathcal { U } _ { n } } 2 ^ { \\nu _ d ( \\pi ) } q ^ { | \\pi | } . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} g _ i ( s _ j ) \\begin{cases} < 0 & \\frac { j - i } k \\in \\cup _ { m = 0 } ^ { \\infty } ( 2 m , 2 m + 1 ) ; \\\\ = 0 & \\frac { j - i } k \\in \\{ 0 , 1 , 2 , \\dots \\} ; \\\\ > 0 & \\frac { j - i } k \\in ( - \\infty , 0 ) \\cup \\cup _ { m = 0 } ^ { \\infty } ( 2 m + 1 , 2 m + 2 ) . \\end{cases} \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} _ i ( 1 _ n ^ { * 3 } \\otimes 1 _ 1 ) = \\tfrac { 1 } { 8 } ( & 1 _ n ^ { \\ast 3 } \\otimes 1 _ 1 + 3 \\cdot 1 _ n ^ { \\ast 2 } i _ n ^ { \\ast } \\otimes i _ 1 - 3 \\cdot 1 _ n ^ { \\ast } i _ n ^ { \\ast 2 } \\otimes 1 _ 1 - i _ n ^ { \\ast 3 } \\otimes i _ 1 ) . \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} N _ { ( 2 , 5 ) } ( n ) & = ~ 8 \\sigma ( \\frac { n } { 2 } ) - 3 2 \\sigma ( \\frac { n } { 8 } ) + 8 \\sigma ( \\frac { n } { 5 } ) - 3 2 \\sigma ( \\frac { n } { 2 0 } ) + 6 4 \\ , W _ { ( 2 , 5 ) } ( n ) + 1 0 2 4 \\ , W _ { ( 2 , 5 ) } ( \\frac { n } { 4 } ) \\\\ & - 2 5 6 \\ , \\biggl ( W _ { ( 5 , 8 ) } ( n ) + W _ { ( 1 , 1 0 ) } ( \\frac { n } { 2 } ) \\biggr ) , \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} m _ K \\otimes m _ K ( K ( I ' \\times J ' ) ) = m _ H ( M D ) = m _ H ( \\Gamma D ) = m _ K \\otimes m _ K ( G ( I ' \\times J ' ) ) . \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} \\tilde { a } ( s , t ) = 1 - t \\kappa ( s ) + t ^ 2 K ( s ) \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} f ( u , v ) = \\frac { 1 } { 2 \\pi } \\ 1 _ { [ 0 , 1 / 2 ] \\times [ 0 , 1 / 2 ] } ( u , v ) ( v t ) ^ { - 1 / 2 } ( 1 - v - t ) ^ { - 3 / 2 } \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} \\operatorname { D i v } ^ { \\vartriangle } P = \\operatorname { D i v } ^ { \\vartriangle } R + Q ^ { \\alpha } F _ { \\alpha } . \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} \\omega _ i = \\frac { 2 } { ( 1 - \\tau _ i ^ 2 ) P _ N ' ( \\tau _ i ) ^ 2 } . \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} D _ t ^ \\delta v - \\partial ^ 2 v / \\partial x ^ 2 = 0 \\ ( x , t ) \\in ( 0 , \\pi ) \\times ( 0 , T ] \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} \\alpha ( D ) = \\sum _ { p \\mid D } \\frac { \\log { p } } { p - 1 } \\le \\log { | D | } . \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} U = \\pm u _ { m } = \\pm u _ { n } ^ { \\prime } \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} A ^ { \\widetilde { K } } ( \\widetilde { G } ) = \\begin{pmatrix} M _ 1 & \\phantom { - } M _ 2 \\cr M _ 2 & - M _ 1 \\end{pmatrix} \\quad A ^ { K , \\omega } ( G ) = M _ 1 + i M _ 2 \\ , , \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} w _ { S ^ n } ^ { ( \\vec { \\alpha _ 1 } , \\ldots , \\vec { \\alpha _ l } ) } ( \\vec { s _ 1 } , \\ldots , \\vec { s _ d } ) = \\vec { g } . \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} \\mbox { f o r s o m e } \\ \\ n _ 0 , \\ \\ g _ { n _ 0 } = g _ { n _ 0 + 1 } = \\ldots = 0 , \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} \\langle V ( x ) | W ( x ) \\rangle = \\langle ( d e ^ { i \\theta } V ) ( e ^ { i \\theta } \\circ x ) | ( d e ^ { i \\theta } W ) ( e ^ { i \\theta } \\circ x ) \\rangle , \\forall x \\in \\Omega , \\theta \\in [ 0 , 2 \\pi ) . \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} d _ { G , p } ^ { p } ( u , v ) = \\mathbb { E } [ \\sup _ { s \\in \\lbrack 0 , T ] } | u _ { s } - v _ { s } | ^ { p } + ( \\int _ { 0 } ^ { T } | \\mathcal { D } _ { x } ( u _ { s } - v _ { s } ) | ^ { 2 } d s ) ^ { \\frac { p } { 2 } } + \\int _ { 0 } ^ { T } | { \\mathcal { A } } _ { G } u _ { s } - { \\mathcal { A } } _ { G } v _ { s } | ^ { p } d s ] . \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} t I \\bar { I } = \\ < t , s , r , \\frac { s + \\sqrt { \\Delta } } { 2 } \\ > , \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} \\mathcal { A } _ 4 : = \\{ ( j _ 1 , j _ 2 , j _ 3 , j _ 4 ) \\in \\mathbb { Z } ^ 4 \\setminus \\{ \\textbf { 0 } \\} \\ , : \\ , & j _ 1 + j _ 2 + j _ 3 + j _ 4 = 0 , j _ 1 ^ 3 + j _ 2 ^ 3 + j _ 3 ^ 3 + j _ 4 ^ 3 \\neq 0 , \\\\ & \\mbox { a n d a t m o s t o n e a m o n g } \\ , \\ , j _ 1 , j _ 2 , j _ 3 , j _ 4 \\ , \\ , \\mbox { o u t s i d e } \\ , \\ , S \\} . \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{gather*} \\Psi ^ { ( 6 ) } _ 0 ( x , t ) = \\Psi ^ { ( 3 ) } _ 0 ( x , t ) S ^ { ( 3 ) } _ 0 S ^ { ( 4 ) } _ 0 = \\Psi ^ { ( 3 ) } _ 0 ( x , t ) \\begin{pmatrix} 1 & 0 \\\\ a & 1 \\end{pmatrix} \\begin{pmatrix} 1 & - a \\\\ 0 & 1 \\end{pmatrix} \\\\ \\hphantom { \\Psi ^ { ( 6 ) } _ 0 ( x , t ) } { } = \\Psi ^ { ( 3 ) } _ 0 ( x , t ) \\begin{pmatrix} 1 & - a \\\\ a & 1 - a ^ 2 \\end{pmatrix} . \\end{gather*}"}
-{"id": "7311.png", "formula": "\\begin{align*} E [ g ( W , \\theta _ { 0 } , \\tilde { \\gamma } ) ] & = E [ c ^ { \\prime } p ( Z ) \\{ \\bar { \\gamma } ( Z ) - \\gamma _ { 0 } ( Z ) \\} ] = - E [ E [ \\alpha _ { 0 } ( X ) | Z ] \\{ \\bar { \\gamma } ( Z ) - \\gamma _ { 0 } ( Z ) \\} ] \\\\ & = E [ \\alpha _ { 0 } ( X ) \\{ Y - \\bar { \\gamma } ( Z ) \\} ] = - c ^ { \\prime } \\Pi ^ { - 1 } E [ a ( X ) \\{ Y - \\bar { \\gamma } ( Z ) \\} ] = 0 . Q . E . D . \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} \\Delta ( \\mathbf { c } ) = { \\rm d i a g } ( \\mathbf { c } ) = ( c _ i \\delta _ { i , j } ) _ { 1 \\leq i \\leq n , 1 \\leq j \\leq n } , \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\beta _ { i + 1 } \\boldsymbol { p } _ { i + 1 } = A \\boldsymbol { p } _ { i } - \\alpha _ { i } \\boldsymbol { p } _ i - \\beta _ { i } \\boldsymbol { p } _ { i - 1 } , \\textrm { w i t h } \\beta _ { 1 } \\boldsymbol { p } _ { 0 } = 0 . \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} \\mathbf { X } = \\left ( \\begin{array} { c } \\mathbf { A } \\\\ \\mathbf { B } \\end{array} \\right ) . \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} ( \\pi ^ { \\flat p } x _ 0 , 1 + \\pi ^ { \\flat p ^ 2 } x _ 1 , \\dots ) = p + [ \\pi ^ \\flat ] ^ p x = \\xi = \\xi ^ \\prime a = ( \\xi _ 0 ^ \\prime , \\xi _ 1 ^ \\prime , \\dots ) ( a _ 0 , a _ 1 , \\dots ) = ( \\xi _ 0 ^ \\prime a _ 0 , \\xi _ 0 ^ { \\prime p } a _ 1 + \\xi _ 1 ^ \\prime a _ 0 ^ p , \\dots ) \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} \\left ( u _ j ^ 0 , v _ j ^ 0 \\right ) = \\frac { 1 } { \\Delta x } \\left ( \\int _ { x _ { j - 1 / 2 } } ^ { x _ { j + 1 / 2 } } u _ 0 ( x ) \\ , d x , \\int _ { x _ { j - 1 / 2 } } ^ { x _ { j + 1 / 2 } } v _ 0 ( x ) d x \\right ) j = 1 , \\dots , N _ x . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\mu _ k ^ { ( b ) } ( z ^ k ) = \\P ( Z ^ k = z ^ k ) = \\P ( [ X ^ k ] _ b = z ^ k ) . \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} [ H , \\hat { Q } _ n ] = 0 , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L _ { z _ { 2 } , p ( x ) } ( u ) = \\widetilde { f } ( x , z _ { 1 } , z _ { 2 } ) & \\Omega , \\\\ L _ { z _ { 1 } , q ( x ) } ( v ) = \\widetilde { g } ( x , z _ { 1 } , z _ { 2 } ) & \\Omega , \\\\ u , v > 0 & \\Omega , \\\\ u , v = 0 & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} \\left ( \\mathcal { S } _ n ^ \\ast \\right ) ^ c \\subseteq \\bigcup _ { k = 1 } ^ { z _ n } \\left \\{ 0 < N _ k < \\max _ { 1 \\le j \\le n } D _ j \\right \\} . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} \\Upsilon & = \\frac { 1 } { \\phi } \\frac { 1 } { N } \\mathrm { t r } \\mathbf { R } _ i ^ { - 1 } , \\\\ \\Psi & = \\frac { \\psi } { \\frac { N } { K } \\phi ^ 2 - \\psi } \\frac { 1 } { K } \\frac { 1 } { N } \\mathrm { t r } \\mathbf { R } _ i ^ { - 1 } \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow \\infty } n ^ { s / d } a _ n \\big ( I _ d : H ^ { s , r } ( \\Bbb T ^ d ) \\rightarrow L _ 2 ( \\Bbb T ^ d ) \\big ) = ( \\mathrm { v o l } ( B _ r ^ d ) ) ^ { s / d } \\asymp _ { s , r } d ^ { - s / r } . \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} \\hat { u } = \\frac { \\sigma ( t ^ 3 ) } { t ^ 3 } \\pmod { t } \\hat { r } = \\frac { \\sigma ( r ) - r } { t ^ 6 } \\pmod { t } . \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\tilde { P } ( \\tilde { \\mathcal { C } } _ n , w , i , \\tilde { \\mathbf { u } } _ 1 , \\mathbf { y } _ 2 | \\mathbf { u } _ 0 , \\mathbf { u } _ 2 ) = \\tilde { \\lambda } ( \\tilde { \\mathcal { C } } _ n ) \\tilde { P } ^ { ( \\tilde { \\mathcal { C } } _ n ) } ( w , i , \\tilde { \\mathbf { u } } _ 1 , \\mathbf { y } _ 2 | \\mathbf { u } _ 0 , \\mathbf { u } _ 2 ) . \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} \\| T ^ { l } { \\tt Q } _ { n } f \\| = | T ^ { l } { \\tt Q } _ { n } f - T ^ { l } { \\tt Q } _ { n + 1 } | = q ^ { l - ( n + 1 ) d } < 1 . \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} 2 \\lambda h _ { 2 , 0 } ^ 0 = - \\frac { 4 \\pi ^ 2 } { L _ 1 ^ 2 } \\left ( 1 + \\gamma \\hat { \\tilde { V } } _ { 2 , 0 } \\right ) h _ { 2 , 0 } ^ 0 - \\frac { 8 \\pi ^ 2 L _ 2 } { L _ 1 } \\gamma \\hat { \\tilde { V } } _ { 1 , 0 } , \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} \\| P _ N ' \\| _ { \\C { L } ^ 2 ( \\Omega ) } = \\sqrt { N ( N + 1 ) } . \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} \\gamma = \\frac { \\mu _ { g \\cap h } ( W | _ { ( g + h ) } ) } { \\mu _ { g + h } ( W | _ { ( g ) \\cap ( h ) } ) } . \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} F ( s ; a , z ) : = \\sum _ { n = 1 } ^ { \\infty } \\frac { ( z \\lambda _ a ( n ) ) ^ { \\omega ( n ) } } { n ^ s } = \\prod _ p \\ ( 1 + \\frac { z \\lambda _ a ( p ) } { p ^ s } \\ ) , ( \\Re ( s ) > 1 ) , \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} g ( x ) : = \\int _ { - \\infty } ^ \\infty e ( x - \\xi ) d W _ \\xi \\ , \\ , . \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} \\mathcal { A } ^ { \\vartriangle } = \\{ F _ { \\alpha } ( n , [ u ] ) = 0 \\} . \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} \\sum _ { h } W \\left ( \\frac { h } { H } \\right ) \\sum _ { X \\leq n \\leq 2 X } a ( n ) \\tau ( n + h ) \\tau ( n - h ) & = H \\widehat { W } ( 1 ) \\sum _ { X \\leq n \\leq 2 X } a ( n ) \\sum _ { d } \\frac { r _ d ( 2 n ) } { d ^ 2 } ( \\log n + 2 \\gamma - 2 \\log d ) ^ 2 \\\\ & + O \\left ( X ^ { \\varepsilon } \\left ( \\frac { H ^ 2 } { X ^ { 1 / 2 } } + H X ^ { 1 / 4 } + ( X H ) ^ { 1 / 2 } + \\frac { X } { H ^ { 1 / 2 } } \\right ) \\| a \\| _ 2 \\right ) . \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} k _ u ( { \\bf x } , { \\bf y } ) = \\prod _ { j \\in u } k ( x _ j , y _ j ) \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} \\left \\| h / \\sqrt { m } \\right \\| _ { K _ s } = 1 , \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{align*} \\exp \\biggl ( \\int _ 0 ^ { 2 \\pi } \\log ( 1 - | f _ N ( e ^ { i \\theta } ) | ^ 2 ) \\ , \\frac { d \\theta } { 2 \\pi } \\biggr ) = \\prod _ { j = N } ^ \\infty ( 1 - | \\alpha _ j | ^ 2 ) \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} \\sqrt { n } \\int \\left \\Vert \\hat { \\Delta } _ { \\ell 2 } ( w ) \\right \\Vert F _ { 0 } ( d w ) & \\leq \\sqrt { n } \\left \\Vert \\hat { \\alpha } _ { 2 \\ell } - \\alpha _ { 2 0 } \\right \\Vert \\left \\Vert H ( \\hat { \\gamma } _ { 1 \\ell } ) - \\hat { \\gamma } _ { 2 \\ell } - H ( \\gamma _ { 1 0 } ) + \\gamma _ { 2 0 } \\right \\Vert \\\\ & = O _ { p } ( \\sqrt { n } [ n ^ { - d _ { 1 } ( 2 \\xi _ { 1 } - 1 ) / ( 2 \\xi _ { 1 } + 1 ) } \\ln ( n ) ] n ^ { - d _ { 1 } 2 \\xi _ { 1 } / ( 2 \\xi _ { 1 } + 1 ) } \\ln ( n ) ) = o _ { p } ^ { { } } ( 1 ) . \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\chi : B u n _ { \\mathcal { G } _ { X , x , \\theta } } \\rightarrow \\bigoplus _ { i = 1 } ^ l H ^ { 0 } ( X , K _ X ^ { \\otimes d _ i } ( C x ) ) \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} L ( f ) ( x , y ) = L ( f | _ x ^ y ) ( x , y ) . \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\mathcal { H } ^ \\mathsf { R o b } _ { \\hbar , \\kappa , K } = - a _ { \\hbar , \\kappa , K } ^ { - 1 } ( \\tau ) \\partial _ { \\tau } \\left ( a _ { \\hbar , \\kappa , K } ( \\tau ) \\partial _ { \\tau } \\right ) = - \\partial ^ 2 _ { \\tau } + \\frac { \\hbar ^ 2 \\kappa - 2 \\hbar ^ 4 K \\tau } { a _ { \\hbar , \\kappa , K } ( \\tau ) } \\partial _ { \\tau } \\ , , \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} J _ { \\psi \\circ \\phi } = J _ \\psi \\cdot J _ \\phi \\ ; . \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\Gamma _ { \\lambda , \\ , \\underline { \\beta } } & = \\bigcup _ { \\ell \\in \\mathbb N } \\Gamma _ { \\lambda , \\ , \\underline { \\beta } } ( \\ell ) \\subset \\mathbb N \\times \\mathbb Z ^ N , \\\\ \\ \\ \\ \\Gamma _ { \\lambda , \\ , \\underline { \\beta } } ( \\ell ) & = \\{ ( \\ell , \\nu ( s / s _ 0 ^ \\ell ) ) ; s \\in \\mathrm { H } ^ 0 ( G / P , \\mathcal L _ \\lambda ^ { \\otimes \\ell } ) \\} . \\end{align*}"}
-{"id": "510.png", "formula": "\\begin{align*} \\bold { p r } X ( F _ { \\alpha } ) = \\sum _ { \\beta , J _ 1 , J _ 2 } K ^ { \\beta } _ { \\alpha ; J _ 1 , J _ 2 } ( D _ { J _ 1 } S _ { J _ 2 } F _ { \\beta } ) . \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\| u \\| _ { C ^ { k , \\alpha } _ \\nu ( U ) } = \\sup _ { x \\in U } \\left ( r ( x ) ^ { k + \\alpha - \\nu } [ \\nabla ^ k u ] _ { C ^ { 0 , \\alpha } ( U \\cap B ( x , \\frac { 1 } { 2 } r ( x ) ) ) } \\right ) + \\sum _ { j = 0 } ^ k \\| r ^ { j - \\nu } \\nabla ^ j u \\| _ { L ^ \\infty ( U ) } \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ \\infty \\varphi _ k ( x ) = 1 \\quad x \\in \\mathbb { R } ^ d . \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{gather*} q _ { 1 t } = - \\frac { 2 } { 3 } \\frac { u _ t } { u } q _ 1 q _ 2 + \\frac { 1 } { 3 } q ^ 2 _ 1 + \\frac { 2 } { 3 } \\left ( t - \\frac { u ^ 2 _ t } { u ^ 2 } \\right ) + \\frac { 2 } { 3 } \\left ( \\frac { u _ t } { u } \\right ) _ t q _ 2 . \\end{gather*}"}
-{"id": "6434.png", "formula": "\\begin{align*} { n \\choose 3 } + n { n - 1 \\choose 3 } = ( n - 2 ) { n \\choose 3 } . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} \\phi _ { \\sigma } = \\phi _ { \\sigma } ( \\xi ) = \\sqrt { 1 - \\frac { \\nu ^ { 2 } | \\xi | ^ { 4 \\sigma - 2 } } { 4 } } , \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{align*} V ( \\lambda ) _ { < _ { \\mathrm { o r } } \\underline { \\mathbf m } } = ( U ( \\mathfrak n ^ - _ P ) _ { < _ { \\mathrm { o r } } \\underline { \\mathbf m } } ) \\cdot v _ \\lambda , V ( \\lambda ) _ { \\le _ { \\mathrm { o r } } \\underline { \\mathbf m } } = ( U ( \\mathfrak n ^ - _ P ) _ { \\le _ { \\mathrm { o r } } \\underline { \\mathbf m } } ) \\cdot v _ \\lambda . \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} \\mathcal { A } f ( x ) = \\frac { \\lambda ^ 3 ( \\sigma _ 1 ^ 2 + \\sigma _ { - 1 } ^ 2 ) f '' ( x ) } { 2 } + \\left [ c - \\lambda H _ 1 \\left ( \\frac { x } { \\lambda } \\right ) 1 _ { \\{ x \\ge 0 \\} } + \\lambda H _ { - 1 } \\left ( \\frac { x } { \\lambda } \\right ) 1 _ { \\{ x < 0 \\} } \\right ] f ' ( x ) . \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} ( \\varphi ^ * R _ { - L , L } e ^ { 2 L D } \\varphi ) _ { n , m } = \\sqrt { \\frac { m ! } { n ! } \\frac { 2 ^ n } { 2 ^ m } } \\frac { e ^ { L m } } { e ^ { L n } } K _ N ^ h ( n , m ) \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} \\sum _ { k = p } ^ { q } F _ { b n + k } ^ { \\left ( b \\right ) } = F _ { b n + q + 2 } ^ { \\left ( b \\right ) } - F _ { b n + p + 1 } ^ { \\left ( b \\right ) } . \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} 2 { \\rm R i c } _ g ( \\nu , \\nu ) - 2 \\varepsilon ( n - 1 ) = \\frac { n - 2 } { n - 1 } ( H ^ 2 - H _ { \\varepsilon } ^ 2 ) \\ , . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} g _ { 2 } ( x ) = g _ { 1 } ( a ) + ( g _ { 1 } ( b ) - g _ { 1 } ( a ) ) \\cdot g _ { 1 } \\Big ( \\frac { x - a } { b - a } \\Big ) = \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} I = \\left [ \\begin{array} { c } \\ast \\\\ 0 \\end{array} \\right ] , J = \\left [ \\begin{array} { c c } \\ast & \\ast \\end{array} \\right ] . \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} k ! F _ k ( s ) = F ^ k _ a ( s ) + \\sum _ { m = 0 } ^ { k - 2 } F ^ m _ a ( s ) F _ { \\boldsymbol { n _ m } } ( s ; a ) , \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} P ( T _ 1 ^ \\alpha < x ) = P ( T _ 2 ^ \\alpha < x ) = ( x - x ^ 2 / 2 ) ^ { 1 / \\alpha } . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{align*} d e t ( M - T I d ) = d e t ( M _ 1 - T I d ) \\cdots d e t ( M _ r - T I d ) \\end{align*}"}
-{"id": "2769.png", "formula": "\\begin{align*} { n + m \\choose n } = ( n + m ) ! \\left ( \\frac 1 { n ! } \\times \\frac 1 { m ! } \\right ) . \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} F ( z _ { Q _ { 1 } } ) = F ( z _ { Q _ { 2 } } ) \\sim i \\cdot 1 7 . 8 8 8 5 4 3 8 2 0 0 0 0 . \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} s _ n = \\frac { p ^ { \\lfloor \\frac { n + 3 } { 3 } \\rfloor } - 1 + \\left ( s _ 1 - 1 \\right ) \\left ( p ^ { \\lfloor \\frac { n + 2 } { 3 } \\rfloor } - 1 \\right ) + \\left ( s _ 2 - s _ 1 \\right ) \\left ( p ^ { \\lfloor \\frac { n + 1 } { 3 } \\rfloor } - 1 \\right ) } { p - 1 } \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} \\beta _ { n , \\lambda , q } = & \\lambda ^ n \\int _ { \\mathbb { Z } _ p } \\left ( \\frac { [ x + y ] _ q } { \\lambda } \\right ) _ n d \\mu _ { q } ( x ) \\\\ = & \\lambda ^ n \\sum _ { m = 0 } ^ n S _ 1 ( n , m ) \\lambda ^ { - m } \\int _ { \\mathbb { Z } _ p } [ x + y ] _ q ^ m d \\mu _ { q } ( y ) \\\\ = & \\sum _ { m = 0 } ^ n S _ 1 ( n , m ) \\lambda ^ { n - m } \\beta _ { m , q } ( x ) . \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\tilde T _ a ^ * \\Gamma _ { i j } { } ^ k = a ^ \\varepsilon \\varepsilon = \\pm 1 + \\pm 1 + \\pm 1 \\in \\{ \\pm 1 , \\pm 3 \\} \\ , . \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } K _ { G \\left ( x , b ; c \\right ) } ^ { L } \\left ( u \\right ) d u & = \\frac { \\gamma \\left ( x / b + 1 , c / b \\right ) } { \\Gamma \\left ( x / b + 1 \\right ) } \\\\ \\int _ { 0 } ^ { \\infty } K _ { G \\left ( x , b ; c \\right ) } ^ { U } \\left ( u \\right ) d u & = \\frac { \\Gamma \\left ( x / b + 1 , c / b \\right ) } { \\Gamma \\left ( x / b + 1 \\right ) } . \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} u ( t , x ) & = \\omega ( t , x ) + \\int _ 0 ^ t \\int _ { \\R } G _ { t - \\theta } ( x - \\eta ) b ( u ( \\theta , \\eta ) ) d \\eta d \\theta \\\\ & + \\int _ 0 ^ t \\int _ { \\R } G _ { t - \\theta } ( x - \\eta ) \\sigma W ( d \\theta , d \\eta ) . \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} \\sigma ^ a : = ( \\mathcal L ^ { \\bar \\nabla } _ v D \\pi ) ^ a = d ( D \\pi v ) ^ a + \\bar \\Gamma ^ a _ { b c } ( D \\pi v ) ^ c d \\pi ^ b \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} \\sum _ { \\iota \\in I } a _ { \\iota } ( g a ^ * _ \\iota ) = \\left \\{ \\begin{array} { c l } 1 _ { M ( \\widetilde { A } ) } & g \\in G \\\\ 0 & g \\in G \\end{array} \\right . \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} - b _ h ( { \\bf q } _ h , { \\bf u } ) + c _ h ( { \\bf p } , { \\bf q } _ h ) = 0 , \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} \\lambda _ { 1 , s } = \\inf _ { u \\in \\operatorname { D o m } ( Q ^ D _ s ) } Q ^ D _ s [ u ] \\geq \\inf _ { u \\in \\operatorname { D o m } ( Q ^ N _ s ) } Q ^ N _ s [ u ] = \\lambda _ { 1 , 1 } ^ { s } , \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} \\alpha = \\frac { 1 } { d } \\left ( a + x \\xi + y \\xi ^ { 2 } + z \\xi ^ { 3 } \\right ) , a , x , y , z \\in \\mathbb { Z } _ { M } . \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} \\hat { L } \\left ( [ f , g ] \\right ) ( x , y ) & = L \\left ( [ f , g ] | _ { x } ^ { y } \\right ) ( x , y ) = L \\left ( [ f | _ x ^ y , g | _ { x } ^ { y } ] \\right ) ( x , y ) \\\\ & = \\left ( [ L ( f | _ x ^ y ) , g | _ { x } ^ { y } ] + [ f | _ x ^ y , L ( g | _ { x } ^ { y } ) ] \\right ) ( x , y ) \\\\ & = \\left ( [ L ( f | _ x ^ y ) , g ] + [ f , L ( g | _ { x } ^ { y } ) ] \\right ) ( x , y ) . \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} ( 1 + d ) ^ { - 1 / 2 } & = a _ { 3 ^ d } ( I _ d : \\ W _ 2 ^ { \\infty } ( \\Bbb T ^ d ) \\rightarrow L _ 2 ( \\Bbb T ^ d ) ) \\\\ & \\le a _ { n } ( I _ d : \\ W _ 2 ^ { \\infty } ( \\Bbb T ^ d ) \\rightarrow L _ 2 ( \\Bbb T ^ d ) ) \\\\ & \\le ( 2 + m ) ^ { - 1 / 2 } \\le ( 2 + d / 2 ) ^ { - 1 / 2 } , \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} \\lambda _ 0 ( n ) = ( 1 \\star \\chi ) ( n ) = \\sum _ { d \\mid n } \\chi ( d ) . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} 1 \\wedge ( \\xi + n ) \\wedge ( \\xi + n ) ^ 2 = 1 \\wedge \\xi \\wedge \\xi ^ 2 , \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} K = \\sum _ { j = 0 } ^ { \\infty } \\frac 1 { 2 ^ { j + 1 } - 1 } , K ' = \\sum _ { j = 1 } ^ { \\infty } \\frac { j } { 2 ^ { j + 1 } - 1 } . \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} \\rho ( \\lambda ) = \\frac { p _ 0 ' ( \\lambda ) } { 2 \\pi } + \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) \\rho ( \\nu ) + \\frac { K ( \\lambda - \\lambda _ p ) } { 2 \\pi L } + \\frac { 1 } { 2 4 L ^ 2 } \\sum _ { i a } \\frac { s _ a K ' ( \\lambda - \\lambda _ { i a } ) } { 2 \\pi \\rho ( \\lambda _ { i a } ) } \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} A _ i ^ { ( j j ) } Z _ { j k } - Z _ { k j } ^ T A _ i ^ { ( k k ) } = 0 , A _ i ^ { ( k k ) } Z _ { k j } - Z _ { j k } ^ T A _ i ^ { ( j j ) } = 0 , \\mbox { f o r $ i = 1 , \\dots , m $ } \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - q ^ 2 } = 1 + q ^ 2 + q ^ 4 + q ^ 6 + q ^ 8 + q ^ { 1 0 } + \\dots . \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\frac { u _ i ( x + c ( t ) , t ) } { e ^ { - \\mu x } } = 1 , \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} \\frac { \\gamma \\left ( a , a \\right ) } { \\Gamma \\left ( a \\right ) } = \\frac { 1 } { 2 } + \\frac { 1 } { \\sqrt { 2 \\pi } } \\left \\{ \\frac { 1 } { 3 a ^ { 1 / 2 } } + \\frac { 1 } { 5 4 0 a ^ { 3 / 2 } } + O \\left ( a ^ { - 5 / 2 } \\right ) \\right \\} a \\rightarrow \\infty . \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{gather*} L = R \\psi ^ { \\sigma _ 3 } \\hat { L } _ 0 \\psi ^ { - \\sigma _ 3 } R ^ { - 1 } + R _ x R ^ { - 1 } + \\left ( \\frac { x ^ 2 } { 2 } - \\frac { t } { 2 } \\right ) I \\equiv x ^ 3 J + x ^ 2 { L } _ 2 + x { L } _ 1 + { L } _ 0 , \\end{gather*}"}
-{"id": "1.png", "formula": "\\begin{align*} \\tilde \\kappa = L / \\mu , \\kappa = \\nu / \\mu , \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} \\dim _ { \\mathbb { R } } \\mathcal { W } ( r , c , c ' ) = 4 ( c ^ 2 + r c + c ^ { \\prime 2 } + c c ' ) - 4 ( c ^ 2 - c ^ { \\prime 2 } ) = 4 ( r c + c c ' ) \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} \\langle \\alpha | \\beta \\rangle = d e t _ { \\mathsf { N } } | | \\mathcal { M } _ { a , b } ^ { \\left ( \\alpha , \\beta \\right ) } | | \\mathcal { M } _ { a , b } ^ { \\left ( \\alpha , \\beta \\right ) } \\equiv \\sum _ { h = 0 } ^ { p - 1 } \\alpha _ { a } ^ { ( h ) } \\beta _ { a } ^ { ( h ) } ( X _ { a } ^ { ( h ) } ) ^ { ( b - 1 ) } . \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} \\Psi \\ : = \\ \\psi ( \\Gamma ^ { \\neq } ) \\ = \\ \\{ \\gamma ^ { \\dagger } : \\gamma \\in \\Gamma ^ { \\neq } \\} \\ = \\ \\{ \\gamma ^ { \\dagger } : \\gamma \\in \\Gamma ^ { > } \\} . \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} g ( x _ 1 , x _ 3 , x _ 5 ) = \\sum _ { x _ 2 } f _ 1 ( x _ 1 , x _ 2 , x _ 5 ) f _ 2 ( x _ 2 , x _ 3 ) , \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } e ^ { - i \\theta k } \\left ( \\left ( \\sum _ { j = 0 } ^ { \\infty } a _ j e ^ { i j \\theta } \\right ) - \\left ( \\sum _ { j = 0 } ^ { \\infty } \\overline { c } _ j e ^ { - i j \\theta } \\right ) ^ { < \\frac { 1 } { \\alpha - 1 } > } \\left ( f ( \\theta ) \\right ) ^ { \\frac { - 1 } { \\alpha - 1 } } \\right ) d \\theta = 0 , k = 0 , 1 , \\dots \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} \\| v ^ h _ t - u _ t \\| _ { l _ 2 ( \\mathbb { G } _ h ) } & = \\| \\mathfrak { I } u ^ h _ t - \\mathfrak { I } u _ t \\| _ { l _ 2 ( \\mathbb { G } _ h ) } \\\\ & \\leq N \\| u ^ h _ t - u _ t \\| _ { H ^ { m - 3 } } , \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} ( \\lambda - \\beta ) ^ { n - 2 } \\left ( \\lambda ^ { 2 } - ( \\alpha + \\beta ) \\lambda + \\alpha \\beta - \\gamma \\right ) = 0 , \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} \\widehat { P } = 1 - P , \\ \\ \\widehat { f } ( P ) = - f ( 1 - P ) , \\ \\ \\widehat { M _ j } = M _ { 3 - j } , \\ \\ \\ \\widehat { D _ j } = D _ { 3 - j } , \\ \\ \\ j = 1 , 2 , \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\gamma = \\varepsilon ^ { 2 + a } = \\varepsilon ^ { 2 \\ , b } , b : = 1 + ( a / 2 ) . \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} M = \\begin{pmatrix} A & C \\\\ B & D \\\\ \\end{pmatrix} , \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\mathcal { E } | _ { \\mathrm { T N } _ k \\setminus K } = l _ 1 \\oplus l _ 2 \\oplus \\ldots \\oplus l _ n . \\end{align*}"}
-{"id": "389.png", "formula": "\\begin{align*} H _ { m _ { N } ^ { k } } \\left ( P _ { p ^ { N } } \\right ) \\geq \\frac { 1 } { k } \\sum _ { i = 0 } ^ { k - 1 } H _ { T _ { p } ^ { i } . m _ { N } } \\left ( P _ { p ^ { N } } \\right ) . \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} \\operatorname { d e t } \\left ( \\mathbf { R _ e } ^ { m m s e } \\right ) = \\frac { \\operatorname { d e t } \\left ( \\mathbf { R } _ { f u l l } ^ { ( g ) } \\right ) } { \\operatorname { d e t } \\left ( \\mathbf { I } _ { T D } + \\sum _ { l = 0 } ^ { L _ g - 1 } \\mathbf { R } ^ { ( g ) } _ { c o d e } ( l ) \\otimes \\mathbf { S N R } ^ { ( g ) } _ { m i m o } ( l ) \\right ) } \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} L ^ { * } ( \\hat { T } , 0 ) = \\pm \\frac { h _ T R _ T } { w _ T } \\frac { [ \\mathbb { I I I } ^ 1 ( T ) ] } { [ H ^ { 1 } ( K , \\hat { T } ) ] } \\prod _ { p \\notin S _ { \\infty } } [ H ^ 0 ( \\hat { \\mathbb { Z } } , H ^ 1 ( I _ p , \\hat { T } ) ) ] . \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} P _ i ( l ) = \\frac { g _ l ( Y _ i ) P _ { i | i - 1 } ( l ) } { \\sum _ { s = 1 } ^ 2 g _ s ( Y _ i ) P _ { i | i - 1 } ( s ) } . \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} n p ( n ) = \\sum _ { k = 1 } ^ { n } \\sigma _ { 1 } ( k ) p ( n - k ) , \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} \\tilde \\psi ( x ) = \\begin{cases} \\psi ( x ) \\ \\ \\ & x \\in \\Omega ( r ^ 2 ) \\setminus \\overline B _ r , \\\\ 0 \\ \\ \\ & x \\in \\Omega ( r ^ 2 ) \\cap \\overline B _ r , \\end{cases} \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} h : = h _ 1 + \\cdots + h _ b = | B A ( S , M ) | . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} _ { \\tau } ( \\xi _ { a } ^ { ( 0 ) } ) = 0 , \\forall a \\in \\{ 1 , . . . , \\mathsf { N } \\} , \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} D ( p , \\tilde p ) = \\sum _ { i = 1 } ^ m p _ i \\log ( p _ i / \\tilde p _ i ) . \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} \\Psi ( z , f ) = q ^ { \\rho _ { f } } \\prod _ { n = 1 } ^ { \\infty } ( 1 - q ^ { n } ) ^ { c _ { f } ( n ^ { 2 } ) } , ( q = e ^ { 2 \\pi i z } ) , \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} \\mathcal { R } = \\frac { \\partial } { \\partial t } + p \\frac { \\partial } { \\partial q } - \\left ( \\omega ( t ) ^ 2 q - \\frac { k } { q ^ 3 } \\right ) \\frac { \\partial } { \\partial p } , \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} ( f g ) | _ { x } ^ { y } ( u , v ) & = ( f g ) ( u , v ) = ( f | _ u ^ v g | _ u ^ v ) ( u , v ) = ( ( f | _ x ^ y ) | _ u ^ v ( g | _ { x } ^ { y } ) | _ u ^ v ) ( u , v ) \\\\ & = ( f | _ x ^ y g | _ { x } ^ { y } ) ( u , v ) . \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} \\| P \\psi \\| _ { L ^ 2 ( \\R ^ 3 ) } ^ 2 = 2 \\| \\psi \\| ^ 2 _ { L ^ 2 ( \\R ^ 3 _ + ) } . \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} \\begin{bmatrix} \\xi + \\eta & - 2 \\eta & - 2 \\tanh \\xi \\\\ - \\xi & 0 & \\tanh ( \\xi + \\eta ) \\\\ - \\tanh \\eta & \\tanh ( \\xi + \\eta ) & 0 \\end{bmatrix} \\begin{bmatrix} A ^ h ( \\xi , \\eta ) \\\\ B ^ h ( \\xi , \\eta ) \\\\ C ^ h ( \\xi , \\eta ) \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ i \\xi \\eta \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} F ( x _ 1 , \\ldots , x _ k ) = \\frac { 1 } { 2 } \\sum _ { i = 1 } ^ k x _ i v _ i . \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} V ( x ) = \\ell + \\sum _ { \\sigma = 1 } ^ k \\frac { 1 } { 2 \\left | x - \\nu _ \\sigma \\right | } , \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} T _ { N , z } f ( x , t ) = \\left ( f \\ast \\nu \\ast J _ { N , z } \\right ) ( x , t ) \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} S _ { \\N } ^ { } : = \\{ ( i , i + 1 ) : i \\in \\N \\} \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} J = | 1 + W _ \\alpha | ^ 2 , F = \\P _ h \\left [ \\frac { Q _ \\alpha - \\bar Q _ \\alpha } { J } \\right ] . \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} X = C _ c \\times l ^ 2 _ \\tau L ^ \\infty \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} x ^ 2 + y ^ 2 - 5 0 1 = c ( x - 1 ) ( y - 1 ) , \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} \\langle x , u _ i \\rangle - \\eta _ i - \\sum _ { j = 1 } ^ n \\nu _ i \\langle u _ j \\ | \\ u _ i \\rangle \\leq 0 \\ i = 1 , \\dots , n , \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} J ^ * ( u , v ) = \\int _ 0 ^ \\infty k ^ * \\left ( \\frac { T } { x } \\right ) h \\left ( \\frac { x } { u } \\right ) h \\left ( \\frac { x } { v } \\right ) \\frac { d x } { x } \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} y ^ 2 + a x y + b y = x ^ 3 , \\Delta _ m = b ^ 3 ( a ^ 3 - 2 7 b ) \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} S _ \\sigma : = S \\cup \\big ( \\bigcup _ { ( l , a ) \\in I _ \\varsigma } \\overline \\Omega _ { l , a } \\big ) \\cup \\big ( \\bigcup _ { ( l , a ) \\in I _ \\sigma } \\delta _ { l , a } \\cup \\Upsilon _ { l , a } \\big ) \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{gather*} \\alpha | _ { \\Sigma ^ \\infty } = \\beta | _ { \\Sigma ^ \\infty } . \\end{gather*}"}
-{"id": "9112.png", "formula": "\\begin{align*} H ( k ) = \\{ f \\in H \\mid \\xi ( f ) = 0 \\} \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{align*} \\widehat { a d ( E ) } _ x ^ { * } \\simeq \\widehat { \\mathfrak { p } } ^ { \\vee } = \\frac { 1 } { z } ( \\mathfrak { n } + z \\mathfrak { g } [ [ z ] ] ) . \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} \\partial ^ { { \\rm E n d } ( H ) } \\theta = [ \\partial ^ h , \\theta ] . \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} \\partial _ t u + ( - \\Delta ) ^ { \\frac { \\theta } { 2 } } u = u ^ p , x \\in { \\bf R } ^ N , \\ , \\ , t > 0 , \\\\ \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} \\big | e ^ { - z } \\big | = e ^ { - x } \\leq \\exp \\Big ( - \\frac { | z | } { 1 + x ^ { - \\epsilon } } \\Big ) . \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } N ( a _ 1 , a _ 2 , a _ 3 , a _ 4 ; n ) q ^ n = \\varphi ( q ^ { a _ 1 } ) \\varphi ( q ^ { a _ 2 } ) \\varphi ( q ^ { a _ 3 } ) \\varphi ( q ^ { a _ 4 } ) . \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} ( f ( z ) \\mid U _ { \\ell ^ { m _ \\ell } } ) \\mid \\gamma = \\sum _ { \\substack { n = n _ 0 \\\\ n \\equiv 0 \\pmod { \\ell ^ { m _ \\ell } } } } ^ { \\infty } a _ 0 ( n ) q _ { 2 4 \\ell ^ { m _ \\ell } } ^ { n } \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} H ^ { ( 1 ) } = - \\Delta _ x - V ^ { ( 1 ) } , \\ \\ \\ \\ H ^ { ( 2 ) } = - \\Delta _ x - V ^ { ( 2 ) } . \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} \\mu ^ { * } \\lambda ( 1 9 ) = \\lambda _ B . \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} M _ k = P ^ H \\frac { 1 } { 2 \\pi i } \\int _ { \\gamma } z ^ k A ^ { - 1 } ( z ) Q d z , \\ k = 0 , \\cdots , 2 m - 1 , \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\widetilde { \\varphi } _ * ( \\widetilde { a } _ j ) = d + \\sum _ { i = 1 } ^ { g - 1 } ( B _ \\infty ( \\varphi ) ) _ { i , j } \\widetilde { b } _ i \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{align*} Y _ T ( x ) = \\frac { 1 } { d ( T ) } \\int _ 0 ^ { T x } \\Phi \\big [ g ( y ) \\big ] \\ , d y , x \\in \\R _ + , \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} \\left | A ( e ^ { i \\theta } ) - { h } ^ 0 ( \\theta ) \\right | ^ { \\alpha } = \\gamma _ 1 \\left ( f _ 0 ( \\theta ) \\right ) ^ { \\beta - 1 } , \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} \\alpha = ( \\underbrace { 1 , \\ldots , 1 } _ n , \\underbrace { r _ n } _ { \\neq 1 } , r _ { n + 1 } , r _ { n + 2 } \\ldots ) \\mapsto \\textstyle { \\int } \\alpha = ( \\underbrace { 0 , \\ldots , 0 } _ { n } , r _ n - 1 , r _ { n + 1 } , r _ { n + 2 } , \\ldots ) : \\Gamma _ { \\log } \\to \\Gamma _ { \\log } ^ { \\neq } \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { k = 0 } ^ 3 ( - 1 ) ^ k \\begin{bmatrix} 3 \\\\ k \\end{bmatrix} _ { t _ { m + n } } E _ { m + n } ^ k E _ { m + n - 1 } E _ { m + n } ^ { 3 - k } = \\Phi _ { m + n } \\sum _ { k = 0 } ^ 3 ( - 1 ) ^ { \\frac { k ( k + 1 ) } { 2 } } \\left \\{ \\begin{matrix} 3 \\\\ k \\end{matrix} \\right \\} _ { q _ { m + n } } e _ { m + n } ^ k e _ { m + n - 1 } e _ { m + n } ^ { 3 - k } = 0 , \\end{aligned} \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to + 0 } \\lim _ { V \\to \\infty } \\omega ^ { 0 } _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\eta _ { \\Lambda } ( b _ { { 0 } } ^ { * } ) ) = \\sqrt { \\rho _ { { 0 } } } \\exp ( i \\phi ) \\ , \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} F _ a ( s ) : = \\sum _ { p } \\frac { \\chi _ 0 ( p ) + \\chi _ a ( p ) } { p ^ s } = : F ( s , \\chi _ 0 ) + F ( s , \\chi _ a ) . \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} \\delta E ^ { ( 1 ) } = \\sum _ { i a } \\frac { \\partial E } { \\partial k _ { i a } } \\delta k _ { i a } = \\sum _ { i a } \\tilde { \\epsilon } ( \\lambda _ { i a } ) N _ { i a } \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} J e ^ { - \\beta A } = e ^ { - \\beta A } J . \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} \\gamma ( s , \\pi , \\tau , \\psi ) = \\prod _ { i = 1 } ^ k \\gamma ( s , \\pi _ i , \\tau , \\psi ) . \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} \\mathcal { R } _ { \\Phi } h = \\sum _ { j \\in S } \\int _ 0 ^ 1 ( h , g _ j ( \\tau ) ) _ { L ^ 2 ( \\mathbb { T } ) } \\chi _ j ( \\tau ) \\ , d \\tau + \\sum _ { j \\in S } ( h , \\psi _ j ) _ { L ^ 2 ( \\mathbb { T } ) } e ^ { \\mathrm { i } j x } \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} H = \\sum _ { j = 1 } ^ L \\left [ S ^ x _ j S ^ x _ { j + 1 } + S ^ y _ j S ^ y _ { j + 1 } + \\Delta ( S ^ z _ j S ^ z _ { j + 1 } - 1 / 4 ) \\right ] \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} I ( \\alpha ) = \\left [ \\mathbb { Z } _ { K } ^ { + } : \\mathbb { Z } _ { M } \\left [ \\alpha \\right ] ^ { + } \\right ] \\left [ \\mathbb { Z } _ { M } \\left [ \\alpha \\right ] ^ { + } : \\mathbb { Z } \\left [ \\alpha \\right ] ^ { + } \\right ] . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} \\Omega _ n = \\left \\{ \\max \\left \\{ \\sum _ { k = 0 } ^ \\infty \\left | \\sum _ { t = 0 } ^ k f _ n ( t ) - f ( t ) \\right | , \\sum _ { k = 0 } ^ \\infty \\left | f _ n ^ \\ast ( k ) - f ^ \\ast ( k ) \\right | \\right \\} \\le n ^ { - \\varepsilon } \\right \\} , \\end{align*}"}
-{"id": "8173.png", "formula": "\\begin{align*} \\mathcal { N } ( T ^ * ) = ( \\overline { T X } ) ^ \\perp = ( T X ) ^ \\perp \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\overline { T ^ * X ^ * } \\subseteq \\mathcal { N } ( T ) ^ \\perp . \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} - \\phi _ n '' ( z ) + z ^ 2 \\phi _ n ( z ) = ( 1 + 2 n ) \\phi _ n ( x ) , n \\in \\mathbb { N } _ 0 , \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} a _ k = b _ m \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} \\partial _ i b _ 3 [ \\hat { \\imath } ] = \\Upsilon ' ( M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] ) \\ , M _ x [ g ' ( a _ 1 - 1 ) \\ , \\partial _ i a _ 1 [ \\hat { \\imath } ] ] . \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} B = 2 \\prod _ { p > 2 } \\Bigl ( 1 - \\frac { 1 } { p - 1 } \\Bigr ) \\Bigl ( 1 - \\frac { 1 } { p } \\Bigr ) ^ { - 1 } \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} \\Phi ^ { - 1 } \\bigl ( T ^ { \\infty } _ r ( b ) \\bigr ) = ( \\nu , \\widetilde { J } ) \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} b & ( x , y , z ) - b \\left ( \\frac { x + y } { 2 } , \\frac { x + y } { 2 } , z \\right ) \\\\ & = \\left ( \\rho ^ { x / 2 } - \\rho ^ { y / 2 } \\right ) ^ 2 - \\left ( \\rho ^ { ( n - x ) / 2 } - \\rho ^ { ( n - y ) / 2 } \\right ) ^ 2 \\\\ & = ( 1 - r ^ { n - x - y } ) \\left ( \\rho ^ { x / 2 } - \\rho ^ { y / 2 } \\right ) ^ 2 \\geq 0 , \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} C ( T ) = H o m _ { \\C ^ * } ( T , \\C ^ * ) \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} L = v _ { 0 , 0 } \\left ( u _ { 1 , 0 } - u _ { 0 , 1 } - \\frac { a ( m ) - b ( n ) } { u _ { 0 , 0 } - u _ { 1 , 1 } } \\right ) , \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} g ( u ) = \\mathrm { L e f t } ( u ) - \\mathrm { R i g h t } ( u ) . \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { \\varphi ( n ) } = - \\frac { A } { 3 } \\left ( \\log x + \\gamma - B - \\frac { 8 } { 3 } \\log 2 \\right ) + O \\left ( x ^ { - 1 } ( \\log x ) ^ { 5 / 3 } \\right ) , \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} \\Delta _ { \\kappa } ( \\pi ) = \\max ( \\kappa _ n ) - \\min ( \\kappa _ n ) . \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{align*} \\# = 2 ^ { ( h + 1 ) / 2 } > ( h + 2 ) ^ c \\geq | \\mathcal { M } _ { h + 1 , c } | . \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ { p ( x ) } u = \\lambda u ^ { \\gamma ( x ) } & \\Omega , \\\\ u > 0 & \\Omega , \\\\ u = 0 & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} \\lambda ( p ) c ( \\Delta ^ { p m } ) & = \\chi ( p ^ 2 ) p ^ { 2 k - 3 } c ( \\Delta ) + \\chi ( p ) p ^ { k - 2 } \\sum _ { \\{ \\Delta : \\Omega \\} = ( 1 , p ) } c ( \\Omega ) + c ( \\Delta ^ { p ^ 2 } ) . \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} H ^ r = \\tau \\circ \\psi ( U ^ r ) . \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{align*} | k + p , n \\rangle = | k , n \\rangle . \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { 2 } ( K _ { 0 2 } w , w ) _ { L ^ 2 ( \\mathbb { T } ) } = \\varepsilon ^ { - 2 b } ( ( \\nabla \\mathcal { H } ) ( T _ { \\delta } ) , T _ 2 [ w , w ] ) _ { L ^ 2 ( \\mathbb { T } ) } + \\frac { \\varepsilon ^ { - 2 b } } { 2 } ( ( \\partial _ u \\nabla \\mathcal { H } ) ( T _ { \\delta } ) [ T _ 1 w ] , T _ 1 w ) _ { L ^ 2 ( \\mathbb { T } ) } . \\end{aligned} \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} & \\zeta = 1 U ( s ) \\\\ & | \\nabla \\zeta | \\le \\frac C { S - s } . \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\frac { \\partial g _ { \\alpha \\beta } } { \\partial x ^ { \\gamma } } = \\frac { \\partial g ( \\partial _ { \\alpha } , \\partial _ { \\beta } ) } { \\partial x ^ { \\gamma } } = 0 \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} v _ { n } | k , n \\rangle = q ^ { k } | k , n \\rangle \\forall ( n , k ) \\in \\{ 1 , . . . , \\mathsf { N } \\} \\times \\{ - l , . . . , l \\} . \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} \\gamma ' ( d ) = \\frac { \\mu ( D ) } { \\zeta _ D ( 2 ) } \\frac { ( d , w ) } { d D } \\chi \\left ( \\frac { w } { ( d , w ) } \\right ) \\xi \\left ( \\frac { w } { ( d , w ) } \\right ) . \\end{align*}"}
-{"id": "6957.png", "formula": "\\begin{align*} X ( s + z ) = X ( s ) ( Q | s | ) ^ { - 2 z } \\{ 1 + z ^ 2 \\eta ( s , z ) \\} , \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\begin{bmatrix} d _ { ( m + 1 ) p } ( x ) & d _ { ( m + 1 ) p + 1 } ( x ) & \\cdots & d _ { ( m + 1 ) p + m } ( x ) \\end{bmatrix} ^ { T } = K ^ { p } \\begin{bmatrix} d _ 0 ( x ) & d _ 1 ( x ) & \\cdots & d _ m ( x ) \\end{bmatrix} ^ T , \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} \\operatorname { D i v } P = D _ i P ^ i . \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} \\Phi _ { n + 1 } ( z ) = z \\Phi _ n ( z ) - \\overline { \\alpha } _ n \\Phi _ n ^ * ( z ) \\\\ \\Phi _ { n + 1 } ^ * ( z ) = \\Phi _ n ^ * ( z ) - \\alpha _ n z \\Phi _ n ( z ) , \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} H _ 1 ( A ^ \\dagger ( x ) \\psi _ n ^ { ( 2 ) } ( x ) ) = A ^ \\dagger ( x ) A ( x ) A ^ \\dagger ( x ) \\psi _ n ^ { ( 2 ) } ( x ) = E _ n ^ { ( 2 ) } ( A ^ \\dagger ( x ) \\psi _ n ^ { ( 2 ) } ( x ) ) \\ ; . \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{gather*} u ( t ) = \\pm \\sqrt { - \\frac { t } { 2 } } + O \\left ( \\frac { 1 } { t } \\right ) , t \\to - \\infty . \\end{gather*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\frac { \\partial q _ 2 } { \\partial w } ( w _ 0 ) d w = \\frac { \\partial q _ 2 } { \\partial w } ( g ( w _ 0 ) ) d w = - \\frac { \\partial q _ 2 } { \\partial w } ( w _ 0 ) d w \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\begin{aligned} - y '' + V ( x ) y & = \\lambda y \\\\ \\alpha _ 1 y ( 0 ) + \\alpha _ 2 y ' ( 0 ) & = 0 \\\\ \\beta _ 1 y ( s ) + \\beta _ 2 y ' ( s ) & = 0 . \\end{aligned} \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} d e t ( M - T I d ) = \\Sigma _ { j = 0 } ^ { 2 n } F _ j ( M ) T ^ { 2 n - j } \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} & [ B _ { m _ k } , B _ { m _ { k - 1 } } , \\cdots , B _ 1 ] \\cdot M = 0 , \\\\ & [ B _ { m _ k } , B _ { m _ { k - 1 } } , \\cdots , B _ 1 ] ( I + \\widetilde { M } ) + [ A _ { m _ k } , A _ { m _ { k - 1 } } , \\cdots , A _ 1 ] = 0 , \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} ( u ^ { k } _ { t } , \\zeta ) = ( u _ { 0 } , \\zeta ) + \\int _ { 0 } ^ { t } \\big [ ( u ^ { k } _ { s } , L ^ { 0 } _ { s } \\zeta ) + ( f _ { s } , \\zeta ) \\big ] \\ , d s + F ^ { k } _ { t } , \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} \\rho _ { \\Lambda } = \\frac { \\exp ( - \\beta ( H _ { \\Lambda } - \\mu N _ { \\Lambda } ) ) } { \\Xi _ { \\Lambda } ( \\mu , \\beta ) } \\ . \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} k x = 2 q ^ { 2 } + p ^ { 2 } - 2 c q p , \\ k y = q p - 2 c q ^ { 2 } , \\ k z = q ^ { 2 } . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} a = c _ 1 + c _ 2 ( - 1 ) ^ n , b = 0 , \\xi = \\left ( - c _ 1 + c _ 2 ( - 1 ) ^ n \\right ) t + c _ 3 ( n ) . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} A ( \\Omega ) = \\overline { \\mathrm { s p a n } \\{ x ^ * a y \\mid a \\in A \\ ; \\mathrm { a n d } \\ ; x , y \\in A ^ { \\alpha } ( \\Omega ) \\} } \\ , . \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} \\nu ( A _ { x _ o } B ) = \\mu ( A ) + \\nu ( B ) < 1 . \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} \\Sigma _ 1 ( s ) \\bar v ( s ) + \\Lambda _ 1 ( s ) \\bar X ( s ) + r ( s ) + \\bar r ( s ) + ( B ^ T ( s ) + \\bar { B } ^ T ( s ) ) \\phi ( s ) = 0 \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{gather*} 2 i _ Q \\omega _ 1 \\simeq \\delta ( L _ 1 + i _ Q \\theta _ 1 ) = \\delta L ' _ 1 . \\end{gather*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\lim _ { t \\to + \\infty } U _ t ( \\hat { x } ) = \\frac { \\gamma } { 1 - \\gamma } \\left [ \\left ( \\hat { \\phi } + \\frac { \\eta } { \\gamma } \\ , \\hat { x } \\right ) ^ { 1 - \\gamma } - 1 \\right ] = U ( \\hat { x } ) , \\ \\hat { x } \\in \\hat { X } . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} N \\Lambda ^ r \\Gamma ( P \\stackrel { \\varphi } { \\longrightarrow } Q ) _ n = \\begin{cases} P ^ { \\otimes r } & , \\\\ Q \\otimes P ^ { \\otimes ( r - 1 ) } & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} \\tilde { \\epsilon } ( \\lambda ) = \\epsilon _ 0 ( \\lambda ) - \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\nu \\ , \\epsilon _ 0 ' ( \\nu ) F ( \\nu | \\lambda ) . \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} \\begin{array} { c c c } ( b - c ) ( a ^ 2 + b ^ 2 - c ^ 2 - d ^ 2 ) & = & 0 \\\\ ( b + c ) ( a ^ 2 + b ^ 2 - a d + b c ) & = & 0 \\\\ d ( a ^ 2 + b ^ 2 - a d + b c ) ^ 2 & = & 0 \\\\ a ( c ^ 2 + d ^ 2 - a d + c b ) & = & 0 \\\\ ( b + c ) ( c ^ 2 + d ^ 2 - a d + b c ) & = & 0 \\\\ a c + b d & = & 0 \\\\ a + d & \\neq & 0 \\end{array} . \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} A _ { ( m + 1 ) p _ n - r _ n } = A _ { ( m + 1 ) p _ n + m - r _ n } - x ^ { 2 r _ n } A _ { ( m + 1 ) p _ n + r _ n } , \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} & p ^ { - 1 } ( p ( \\Gamma g ) ) \\cap \\Omega = \\{ \\Gamma c _ 1 g , \\ldots , \\Gamma c _ n g \\} , \\\\ & p ^ { - 1 } ( p ( \\Gamma c _ i ^ { - 1 } g ) ) \\cap \\Omega = \\{ \\Gamma c _ 1 c _ i ^ { - 1 } g , \\ldots , \\Gamma c _ n c _ i ^ { - 1 } g \\} , \\\\ & p ^ { - 1 } ( p ( \\Gamma c _ i g ) ) \\cap \\Omega = \\{ \\Gamma c _ 1 c _ i g , \\ldots , \\Gamma c _ n c _ i g \\} . \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} \\ell \\cdot ( x - x _ 0 ) = D _ { x _ 0 } \\ell \\cdot ( \\phi ( x ) - \\phi ( x _ 0 ) ) + o ( \\| x - x _ 0 \\| ) . \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rho , \\phi } ( T _ 0 > t ) = \\mathbb { E } _ { \\rho , \\phi } \\left [ W _ p ( \\Theta _ t ) { \\bf 1 } _ { \\{ T _ 0 < t \\} } \\right ] . \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} \\mathcal { T } = \\sum _ { j = 1 } ^ t \\mathcal { C } _ j \\bullet _ 1 \\mathbf { A } _ j \\bullet _ 2 \\mathbf { B } _ j , \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} l _ \\sigma ( v ^ { 2 k + 1 } ; 0 , 0 ) & = l _ \\sigma ( v ^ { 2 k } ; 0 , 0 ) l _ \\sigma ( v ; 0 , 0 ) = ( ( u v ) ^ { - 2 k } ; 0 , 2 k ) ( v ^ { - 1 } B ; 0 , 1 ) \\\\ & = ( ( u v ) ^ { - 2 k } \\theta ( 0 , 2 k ) ( v ) ^ { - 1 } \\theta ( 0 , 2 k ) ( B ) ; 0 , 2 k + 1 ) \\\\ & = ( ( u v ) ^ { - 2 k } v ^ { - 1 } B ; 0 , s ) = ( ( u v ) ^ { - s } ( u B ) ^ { \\delta _ s } , 0 , s ) . \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} \\tilde T _ 4 ( z ) = \\frac { i } { 4 \\pi } \\int _ { \\R ^ 3 } \\frac { 4 z + \\xi _ 1 + \\xi _ 1 } { ( 2 z + \\xi _ 1 ) ( 2 z + \\xi _ 2 ) ( 2 z + \\eta _ 1 ) ( 2 z + \\eta _ 2 ) } \\overline { \\hat u ( \\eta _ 1 + \\eta _ 2 - \\xi _ 1 ) \\hat u ( \\xi _ 1 ) } \\hat u ( \\eta _ 1 ) \\hat u ( \\eta _ 2 ) d \\xi _ 1 d \\eta _ 1 d \\eta _ 2 . \\end{align*}"}
-{"id": "8083.png", "formula": "\\begin{align*} Z _ { i j } & = U _ { i R , j R } - U _ { n + 1 - i L , j R } & & = \\delta _ { i j } - F ( \\lambda _ { j R } | \\lambda _ { i R } ) + F ( \\lambda _ { j R } | \\lambda _ { n + 1 - i L } ) , \\\\ Y _ { i j } & = U _ { i R , j R } + U _ { n + 1 - i L , j R } & & = \\delta _ { i j } - F ( \\lambda _ { j R } | \\lambda _ { i R } ) - F ( \\lambda _ { j R } | \\lambda _ { n + 1 - i L } ) . \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} | & ( - \\Delta ) ^ { \\sigma + m } f ( x ) | = \\frac { c _ { N , \\sigma } } { 2 } \\Bigg | \\int _ { \\R ^ N } \\frac { 2 \\phi ( x ) - \\phi ( x + y ) - \\phi ( x - y ) } { | y | ^ { N + 2 \\sigma } } \\ d y \\Bigg | \\\\ & \\leq C \\int _ { B } \\int _ 0 ^ 1 \\int _ 0 ^ 1 \\frac { | H _ { \\phi } ( x + ( t - \\tau ) y ) | } { | y | ^ { N + 2 \\sigma - 2 } } \\ d \\tau d t d y + \\Bigg | \\int _ { \\R ^ N \\setminus B } \\frac { 2 \\phi ( x ) - \\phi ( x + y ) - \\phi ( x - y ) } { | y | ^ { N + 2 \\sigma } } \\ d y \\Bigg | = : f _ 1 + f _ 2 . \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} \\lim _ { y \\rightarrow x } \\gamma ( x , y ) = \\lim _ { y \\rightarrow - x } \\gamma ( x , y ) = 1 . \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} \\pi _ \\sigma ( z _ 1 , \\ldots , z _ n ) = z _ \\sigma . \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} Q _ { 0 } \\left ( x , y , z \\right ) = u Q _ { 2 } \\left ( x , y , z \\right ) - v Q _ { 1 } \\left ( x , y , z \\right ) = 0 . \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{align*} M _ { a } ^ { s G } ( \\lambda ) \\sigma _ { a } ^ { x } = [ L _ { a , 2 \\mathsf { M } + 1 } ( \\frac { \\lambda q ^ { - 1 / 2 } } { \\xi _ { 2 \\mathsf { M } + 1 } } ) \\sigma _ { a } ^ { x } ] [ \\tilde { L } _ { a , 2 \\mathsf { M } } ( \\frac { \\lambda q ^ { - 1 / 2 } } { \\xi _ { 2 \\mathsf { M } } } ) \\sigma _ { a } ^ { x } ] \\cdots \\lbrack \\tilde { L } _ { a , 2 } ( \\frac { \\lambda q ^ { - 1 / 2 } } { \\xi _ { 2 } } ) \\sigma _ { a } ^ { x } ] [ L _ { a , 1 } ( \\frac { \\lambda q ^ { - 1 / 2 } } { \\xi _ { 1 } } ) \\sigma _ { a } ^ { x } ] , \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} \\begin{cases} \\chi '' ( z ) | p | ^ 2 - \\bar { c } ( | p | ) \\chi ' ( z ) + g ( \\chi ( z ) ) = 0 & z \\in ( 0 , 1 ) \\\\ \\chi ( 1 ) = \\chi ( 0 ) + 1 & \\\\ \\chi ' ( 0 ) = \\chi ' ( 1 ) . \\end{cases} \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} \\widehat f ( \\l ) = \\lim _ { T \\to \\infty } \\frac { 1 } { 2 T } \\int _ { - T } ^ { T } f ( x ) { \\rm e } ^ { - i \\l x } \\dd x \\ , , \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} 2 c _ 2 = c _ { 1 } ^ { 2 } + x ( 4 - c _ { 1 } ^ { 2 } ) , \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { q ^ n } { 1 + q ^ n } \\frac { \\tiny 1 } { ( q ; q ) _ { n - 1 } } = \\sum _ { n \\geq 1 } \\frac { q ^ { n ( n + 1 ) / 2 } } { ( q ^ 2 ; q ^ 2 ) _ n ( q ^ { n + 1 } ; q ) _ \\infty } . \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} N _ { 1 } & = \\Sigma , \\\\ N _ { 2 } & = F _ { S _ { A } } K ^ { \\ast } \\Sigma - S _ { A } ^ { \\dagger } H \\Phi , \\\\ N _ { 3 } & = \\Sigma H ^ { \\ast } E _ { S _ { A } } - \\Psi K S _ { A } ^ { \\dagger } , \\\\ N _ { 4 } & = S _ { A } ^ { \\dagger } - S _ { A } ^ { \\dagger } H \\Phi H ^ { \\ast } E _ { S _ { A } } - F _ { S _ { A } } K ^ { \\ast } \\Psi K S _ { A } ^ { \\dagger } + F _ { S _ { A } } K ^ { \\ast } \\Sigma H ^ { \\ast } E _ { S _ { A } } . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} \\left ( x ; \\emptyset \\right ) = \\mathcal { B } ( 1 , 0 ) = 0 \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} \\| f ( x ) - f ( y ) \\| = | \\phi ( x - y ) | \\| z \\| \\leq \\| x - y \\| \\| \\phi \\| _ * \\| z \\| = \\| x - y \\| \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} d ( M ) : = \\sum _ { \\lambda \\in \\Lambda ( M ) } \\left ( m _ { a } ( M , \\lambda ) - m _ { g } ( M , \\lambda ) \\right ) = \\sum _ { \\lambda \\in \\Lambda ( M ) } d ( M , \\lambda ) . \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} { \\gamma _ o } = \\frac { { { P _ t } G _ { } ^ 2 L \\left ( r \\right ) } } { { \\sum \\nolimits _ { i \\in \\Phi / o } { { P _ t } { G _ i } L \\left ( \\left | X _ i \\right | \\right ) } + { \\sigma _ o ^ 2 } } } , \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} \\frac { \\omega ^ { 2 k } \\| B \\| _ 2 } { \\pi ^ { 2 k } ( 2 m + 1 ) m ^ { 2 k - 2 } } < \\frac { 1 } { 4 } b c = d i r . \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} E _ 1 : Y ^ 2 = X ^ 3 + 2 w X ^ 2 + u ^ p X E _ 2 : Y ^ 2 = X ^ 3 + 2 w X ^ 2 - \\ell v ^ { 2 p } X . \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} U _ { \\textrm { M L } } ( m ) = \\{ U _ { m } ( n ' ) | U _ { m } ( n ' ) \\geq U _ { m } ( n ) , \\forall n \\in \\mathcal { N } \\} . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} \\tilde \\ell _ n ( G ) = \\ell _ n ( G ) + p _ n ( G ) . \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} \\tau _ { g } ( \\eta ( { \\sigma } ) ) = \\lambda \\ , R \\ , { n } \\ . \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} W = - \\tilde R ( 1 ) \\left ( L ( 1 , \\chi ) \\log { N } \\right ) ^ 2 E + O ( T ^ { - 1 / 4 } ) . \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} & \\omega \\geq \\left | \\left | \\begin{bmatrix} \\left ( L _ y \\left \\lbrace F _ { P i } \\right \\rbrace \\right ) ^ \\intercal & \\left ( L _ y \\left \\lbrace F _ { V i } \\right \\rbrace \\right ) ^ \\intercal \\end{bmatrix} ^ \\intercal \\right | \\right | _ 2 \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} \\lim _ { \\Lambda } \\omega _ { \\beta , \\mu , \\Lambda } ^ { 0 } ( \\frac { 1 } { V } \\int _ { \\Lambda } d x b ^ * ( x ) \\ \\frac { 1 } { V } \\int _ { \\Lambda } d x b ( x ) ) = \\lim _ { \\Lambda } \\omega _ { \\beta , \\mu , \\Lambda } ^ { 0 } ( \\frac { b ^ { * } _ { 0 } b _ { 0 } } { V } ) = \\rho _ { 0 } ( \\beta ) \\ , \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\textrm { d i a m } ( A ) = 0 \\right ) > 0 . \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} D _ { 0 + } ^ { q } ( u ( x , t ) - u ( x , 0 ) ) = u _ { x x } + r ( t ) f ( x , t ) , ( x , t ) \\in \\Omega _ { T } , \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} \\Big | \\int _ r ^ R \\int _ { \\mathbb { R } ^ { n + 1 } } & \\prod _ { i = 0 } ^ { n } F _ i ( x _ 0 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ n ) \\\\ & h _ t ( x _ 0 + \\dots + x _ n ) d x _ 0 \\dots d x _ n \\frac { d t } { t } \\Big | \\leq C \\Big ( \\log \\frac { R } { r } \\Big ) ^ { 1 - 2 ^ { - n + 1 } } , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} F _ B ( \\lambda | \\lambda ' ) = F ( \\lambda | \\lambda ' ) - \\frac { 1 } { 2 } Z ( \\lambda ) \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} E _ i F _ j - ( - 1 ) ^ { \\bar { i } \\bar { j } } F _ j E _ i = \\delta _ { i j } \\frac { K _ i - K _ i ^ { - 1 } } { t ^ { \\theta _ i } - t ^ { - \\theta _ i } } , 1 \\le i , j \\le m + n . \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} f ( s _ { \\sigma ( 1 ) } , \\ldots s _ { \\sigma ( n ) } ) = \\epsilon ( \\sigma ) \\Pi _ \\sigma f ( s _ 1 , \\ldots , s _ n ) , \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} C _ { k k } ^ { i i } = C _ { i i } ^ { i i } \\mbox { f o r } k < i , \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} \\langle \\mu _ 1 , \\theta \\rangle \\Rightarrow \\left \\{ \\begin{array} { l l } \\mu _ 1 ( T ) + \\overline { \\mu _ 1 ( T ) } = - 3 ^ k & \\theta = \\alpha _ 0 \\\\ \\mu _ 1 ( T ) \\overline { \\mu _ 1 ( T ) } = \\frac { 1 } { 4 } ( 3 ^ { 4 k + 1 } + 3 ^ { 2 k } ) & \\theta = \\mu _ 1 \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} A r e a ( B _ s ( y ) \\cap \\Gamma ) \\geq A r e a ( B _ { \\frac { s } { 2 } } ( z ) \\cap \\Gamma ) \\geq A r e a ( \\hat { B } _ { \\frac { s } { 2 } } ( z ) ) \\geq C _ 0 ( \\frac { s } { 2 } ) ^ 2 = \\frac { C _ 0 } { 4 } s ^ 2 . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\binom { n } { k } _ b { s _ b } ^ 2 ( k ) = s _ b ( n ) ( s _ b ( n ) + 1 ) 2 ^ { s _ b ( n ) - 2 } . \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} m = 2 ^ { a + 2 b } \\prod _ { i = 1 } ^ s ( 2 ^ { k _ i } - 1 ) \\prod _ { j = 1 } ^ t ( p _ j ^ { \\lambda _ j } - 1 ) \\ , , \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} \\Delta _ { T N } ( \\pi _ k ^ * f ) = \\pi _ k ^ * ( V ^ { - 1 } \\Delta _ { R ^ 3 } f ) , \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} h = \\sum \\limits _ { k = 1 } ^ { q } v _ k \\wedge w _ k \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} G ( x , \\xi ) = \\frac 1 { ( 2 \\pi ) ^ { d } } \\int \\widehat G ( q , p ) e ^ { i ( { p x - q \\xi } ) } d \\mu ( q , p , ) \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{gather*} \\frac { p } { q } \\big ( 1 + p u \\psi ^ 2 \\big ) = 1 . \\end{gather*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\frac { w ^ { n + 1 } _ j } { a _ j } = \\biggl ( 1 - \\frac { a _ j \\Delta t } { \\Delta x } \\biggr ) \\frac { w ^ { n } _ j } { a _ j } + \\frac { a _ j \\Delta t } { \\Delta x } \\frac { w _ { j - 1 } ^ n } { a _ j } : = \\mathcal { H } ( w ^ n _ { j - 1 } , w ^ n _ j ) , \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\mathcal { F } _ q ( \\mathcal { H } ) : = \\bigoplus _ { n = 0 } ^ \\infty \\mathcal { F } ^ { ( n ) } _ q ( \\mathcal { H } ) . \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} g & = \\alpha _ 1 ( n _ 1 g _ 1 ) \\alpha _ 2 ( n _ 2 g _ 2 ) \\alpha _ 3 ( n _ 1 g _ 1 ) ^ { - 1 } \\alpha _ 4 ( n _ 2 g _ 2 ) ^ { - 1 } \\\\ & = \\alpha _ 1 ( n _ 1 ) \\alpha _ 1 ( g _ 1 ) \\alpha _ 2 ( n _ 2 ) \\alpha _ 2 ( g _ 2 ) \\alpha _ 3 ( g _ 1 ) ^ { - 1 } \\alpha _ 3 ( n _ 1 ) ^ { - 1 } \\alpha _ 4 ( g _ 2 ) ^ { - 1 } \\alpha _ 4 ( n _ 2 ) ^ { - 1 } . \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} w _ { \\Delta x } ( t , x ) : = w ^ n _ j , ( t , x ) \\in [ t ^ n , t ^ { n + 1 } ) \\times [ x _ { j - 1 / 2 } , x _ { j + 1 / 2 } ) , \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} \\frac { [ E x t _ X ^ 1 ( j _ { * } \\hat { T } , \\mathbb { G } _ m ) ] } { [ E x t _ X ^ 2 ( j _ { * } \\hat { T } , \\mathbb { G } _ m ) ] } = \\frac { [ P ] [ Q ] } { [ \\mathbb { I I I } ^ 2 ( T ) ] [ R ] } . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} n : = | Q _ 0 | \\mbox { a n d } e ( Q ) : = | Q _ 1 | . \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} \\partial _ t \\psi = B ( z , u ) \\psi \\end{align*}"}
-{"id": "3635.png", "formula": "\\begin{align*} H ( 1 / w _ 1 , \\ldots , 1 / w _ k ) = ( \\pm 1 ) w _ 1 ^ { d _ 1 } \\ldots w _ k ^ { d _ k } H ( w _ 1 , \\ldots , w _ k ) \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} \\| u \\| _ { \\overline S } ^ 2 = - \\int _ X \\sum _ { k = 1 } ^ { n - 1 } \\langle \\nabla _ { e _ k } \\nabla _ { \\overline e _ k } u | u \\rangle d v _ X = \\sum _ { k = 1 } ^ { n - 1 } \\int _ X \\langle \\nabla _ { \\overline e _ k } u | \\nabla _ { \\overline e _ k } u \\rangle d v _ X \\geq 0 . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } u _ t - u _ { x x } & = & u ( r _ u - \\gamma _ u ( u + v ) ) + \\mu ( v - u ) \\R ^ { 2 } \\\\ v _ t - v _ { x x } & = & v ( r _ v - \\gamma _ v ( u + v ) ) + \\mu ( u - v ) \\R ^ { 2 } , \\end{array} \\right . \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } - p '' & = & p ( r _ u + \\beta - \\gamma _ u ( p + q ) ) + \\mu ( q - p ) \\\\ - q '' & = & q ( r _ v + \\beta - \\gamma _ v ( p + q ) ) + \\mu ( p - q ) \\end{array} \\right . \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} z : = \\bigg ( \\frac { r _ 1 } s \\bigg ) ^ { 1 / \\alpha } . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} \\sum _ d \\gamma ^ * ( d ) f ( d y ) = \\frac { \\xi ( w ) } { \\zeta ( 2 ) } \\sum _ { c \\mid w } \\chi ( c ) \\sum _ { ( d , D ) = 1 } f ( c d y ) d ^ { - 1 } \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{gather*} 4 A ( h ) C ( h ) - B ^ { 2 } ( h ) = 1 6 u \\left ( s - v \\right ) ^ { 2 } R ( h ) \\allowbreak { \\large , } \\\\ { \\large ( } C ( h ) D ^ { 2 } ( h ) + A ( h ) E ^ { 2 } ( h ) - B ( h ) D ( h ) E ( h ) { \\large ) } - F ( h ) { \\large ( } 4 A ( h ) C ( h ) - B ^ { 2 } ( h ) { \\large ) } = \\\\ = \\allowbreak 1 6 \\left ( s - v \\right ) ^ { 2 } u ^ { 2 } R ^ { 2 } ( h ) \\end{gather*}"}
-{"id": "840.png", "formula": "\\begin{align*} \\Gamma _ { m , b c } X = \\Gamma _ { b c } ( X - \\mathbb { P } _ { ( m ) } X \\mathbb { P } _ { ( m ) } ) , X \\in \\mathfrak { S } _ 2 ( \\mathcal { H } ) , \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} \\mathcal { V } = \\{ ( \\nu _ 1 ( a ) , \\ldots , \\nu _ K ( a ) ) : a \\in \\mathcal { P } \\} . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\omega _ H ( ( x - 1 ) ^ { \\beta p ^ { e - 1 } } g _ 1 ( x ) ) = \\omega _ H ( ( x - 1 ) ^ { \\beta p ^ { e - 1 } } g _ 2 ( x ) ) = \\beta + 1 . \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} Y ( t ) & = \\sum _ { i = 1 } ^ { K } \\widetilde { H } _ i ( t ) X _ i ( t ) + \\eta _ b ( t ) , \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} \\omega _ { \\beta } ( A ) = \\int _ { E _ { \\cal A } ^ { G } } d \\mu ( \\omega _ { \\beta } ^ { ' } ) \\omega _ { \\beta } ^ { ' } ( A ) \\ , \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} P _ { S , S _ 1 | V , Y } ( s , s _ 1 | v , y ) = P _ { S | V , Y } ( s | v , y ) P _ { S _ 1 | V , Y } ( s _ 1 | v , y ) , \\quad \\forall ( v , y , s , s _ 1 ) \\in \\mathcal { V } \\times \\mathcal { Y } \\times \\mathcal { S } \\times \\mathcal { S } _ 1 . \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} \\tau ( \\lambda ) - \\tau ^ { \\prime } ( \\lambda ) \\equiv \\left ( \\Lambda ^ { 2 } - X ^ { 2 } \\right ) \\sum _ { b = 1 } ^ { \\mathsf { N } } x _ { b } ^ { \\left ( \\tau , \\tau ^ { \\prime } \\right ) } \\Lambda ^ { b - 1 } , \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} \\tilde { \\mathbf { y } } _ i = \\bar { \\mathbf { x } } _ i + \\mathbf { H } ^ { - 1 } _ i \\mathbf { n } _ i . \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} x _ i ( n ) = \\sum \\limits ^ { K - 1 } _ { k = 0 } { \\sum \\limits ^ { M - 1 } _ { m = 0 } { d _ { i , k , m } g _ { k , m } ( n ) } } , \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\mathbb { P } ( | A _ n | > c ) = \\infty . \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} w ( x ) = \\left \\{ \\begin{array} { c } U ^ 1 ( x ) x \\in \\Omega _ 1 \\\\ U ^ 2 ( x ) x \\in \\Omega _ 2 \\end{array} \\right . . \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} \\lim _ { d ( x ) \\rightarrow 0 } N \\alpha ( x ) = L _ { 1 } \\in ( - 1 , 0 ) , \\lim _ { d ( x ) \\rightarrow 0 } N \\beta ( x ) = L _ { 2 } \\in ( - 1 , 0 ) , \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} \\mathcal { O P } _ k = \\mathcal { M P } _ k \\ominus \\mathcal { M P } _ { k - 1 } \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} \\eta ' ( s ) = \\xi ' ( s ) \\quad \\quad \\eta '' ( s ) \\geq \\xi '' ( s ) . \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} K ( \\xi , \\eta ) = \\int f _ N ( \\xi ) f _ N ( \\eta ) \\ , d N . \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{align*} \\mathbb { P } [ M _ k - m _ 0 \\geq \\lambda ] & \\stackrel { \\eqref { e x p b o u n d } } { \\leq } \\exp \\left ( \\frac { t ^ { 2 } } { 2 } \\frac { 1 } { 1 - t m / 3 } \\sum _ { j = 1 } ^ k \\sigma _ j ^ 2 - t \\lambda \\right ) \\\\ & \\stackrel { \\eqref { p a r a m c h o i c e } } { = } \\exp \\left ( - \\frac { \\lambda ^ 2 } { 2 \\left ( \\sum _ { i = 1 } ^ k \\sigma _ i ^ 2 + m \\lambda / 3 \\right ) } \\right ) . \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} \\mathrm { d } x \\left ( t \\right ) = f \\left ( t , x \\left ( t \\right ) , r \\left ( t \\right ) \\right ) \\mathrm { d } t + g \\left ( t , x \\left ( t - d \\left ( t , r ( t ) \\right ) \\right ) , r \\left ( t \\right ) \\right ) \\mathrm { d } w \\left ( t \\right ) , \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\Delta u = \\int _ 0 ^ 1 g ( s ) d s \\\\ u ( x , 0 ) = u _ 0 ( x ) . \\end{cases} \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} \\begin{aligned} & \\mu _ 1 : = 3 \\mu + 9 , \\qquad \\alpha : = 3 \\mu _ 1 + 1 , \\alpha _ 1 : = ( \\alpha - 3 \\mu ) / 2 , \\\\ & k : = 3 ( \\mu _ 1 + \\rho ^ { - 1 } ) + 1 , \\beta _ 1 : = 6 \\mu _ 1 + 3 \\rho ^ { - 1 } + 3 , 0 < \\rho < \\frac { 1 - 3 a } { C _ 1 ( 1 + a ) } . \\end{aligned} \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} \\max _ { \\substack { \\eta \\in X \\\\ \\eta ' ( 0 ) = - h ^ { 2 } } } \\frac { 1 } { 2 } \\tilde { D } ( \\eta ) = \\max _ { \\eta \\in \\Lambda } D ( \\eta ) . \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} U ^ + _ { \\mathbb { O } _ P } = \\bigoplus _ { a \\in \\N } \\mathbb { O } _ p \\cdot E ^ { ( a ) } , \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\mathcal { T _ S } ( x ) = \\{ y \\ , | \\ , ( \\nabla g ( x ) ) ^ T y \\leq 0 \\} . \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} F ^ { ( 5 ) } = \\sum _ { \\substack { j _ 1 + \\dots + j _ 5 = 0 , \\\\ a t \\ , \\ , m o s t \\ , \\ , o n e \\ , \\ , i n d e x \\ , \\ , o u t s i d e \\ , \\ , S } } F ^ { ( 5 ) } _ { j _ 1 , \\dots , j _ 5 } \\ , u _ { j _ 1 } \\dots u _ { j _ 5 } , F ^ { ( 5 ) } _ { j _ 1 , \\dots , j _ 5 } : = \\dfrac { H ^ { ( 4 ) } _ { 5 , \\ , j _ 1 , \\dots , j _ 5 } } { \\mathrm { i } ( j _ 1 ^ 3 + \\dots + j _ 5 ^ 3 ) } . \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{gather*} q _ 2 = - 1 + o ( 1 ) , \\alpha = o ( 1 ) , t \\to + \\infty . \\end{gather*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\lambda _ { \\phi } } p _ { \\beta , \\mu , \\Lambda , \\lambda _ { \\phi } } = - \\frac { \\bar { \\lambda } _ { \\phi } } { \\mu } \\ . \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} \\theta _ { a } = \\frac { ( q - 1 / q ) \\mathsf { A } _ { - } ( 1 / \\xi _ { a } ^ { \\left ( h _ { a } \\right ) } ) } { ( ( \\xi _ { a } ^ { ( h _ { a } ) } ) ^ { 2 } - 1 / ( \\xi _ { a } ^ { ( h _ { a } ) } ) ^ { 2 } ) } \\langle h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } | h _ { 1 } , . . . , h _ { a } , . . . , h _ { \\mathsf { N } } \\rangle . \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} \\mathcal { R } _ 2 : = \\mathcal { R } _ 2 - \\partial _ x ^ { - 1 } \\mathrm { R } _ 2 \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} M = 2 \\Re [ \\P , R ] \\bar Y _ \\alpha - 2 \\Re [ \\P , Y ] \\bar R _ \\alpha , \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} \\sum _ { b = 1 } ^ { \\mathsf { N } } \\mathcal { M } _ { a , b } ^ { \\left ( \\tau , \\tau ^ { \\prime } \\right ) } x _ { b } ^ { \\left ( \\tau , \\tau ^ { \\prime } \\right ) } = 0 \\forall a \\in \\{ 1 , . . . , \\mathsf { N } \\} , \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} \\widetilde J ( ( \\mu ^ { \\alpha _ { n _ k } } , \\rho ^ { \\alpha _ { n _ k } } ) , u ^ { \\alpha _ { n _ k } } ) \\ , = \\ , \\widetilde J \\left ( { \\cal S } _ { \\alpha _ { n _ k } } ( u ^ { \\alpha _ { n _ k } } ) , u ^ { \\alpha _ { n _ k } } \\right ) \\ , \\le \\ , \\widetilde J \\left ( { \\cal S } _ { \\alpha _ { n _ k } } ( \\bar u ) , \\bar u \\right ) \\ , , \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} Q _ { \\ell } ( \\xi ) = \\sum _ { j = 1 } ^ { \\infty } \\min ( \\ell , \\xi _ j ) \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} \\mathbf { C } = \\begin{bmatrix} \\mathbf { c } & { } & { } & { } \\\\ { } & \\mathbf { c } & { } & { } \\\\ { } & { } & \\ddots & { } \\\\ { } & { } & { } & \\mathbf { c } \\\\ \\end{bmatrix} _ { N \\times K } , \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} \\mathfrak { g } = Z _ { 0 } \\oplus Z _ { 1 } \\oplus Z _ { 2 } \\ , \\oplus \\cdots \\oplus \\ , Z _ { k } . \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} F _ 2 ^ { ( 2 , 3 ) } ( z ) = \\frac { 1 + z _ 2 + z _ 2 ^ 2 + z _ 1 z _ 2 ^ 2 + z _ 1 z _ 2 ^ 3 + z _ 1 z _ 2 ^ 4 } { ( 1 - z _ 1 ^ 2 z _ 2 ^ 3 ) ( 1 - z _ 2 ^ 3 ) } = \\frac { 1 + z _ 1 z _ 2 ^ 2 } { ( 1 - z _ 1 ^ 2 z _ 2 ^ 3 ) ( 1 - z _ 2 ) } \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} S _ { c } = \\{ \\pm 1 , \\pm \\sqrt { - 1 } , \\pm 1 \\pm \\sqrt { - 1 } , \\pm 2 \\pm \\sqrt { - 1 } , \\pm 1 \\pm \\sqrt { - 2 } , \\pm 1 \\pm \\sqrt { - 3 } , \\frac { \\pm 1 \\pm \\sqrt { - 3 } } { 2 } , \\frac { \\pm 3 \\pm \\sqrt { - 3 } } { 2 } \\} , \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} \\partial _ t n _ j = D _ j \\partial _ { x x } n _ j - M _ j \\partial _ x n _ j + f _ j ( \\vec { n } ) \\ , , j = 1 , 2 , \\dots , m , \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} L = - \\frac { ( u ' ) ^ 2 } { 2 } + \\exp ( u - u _ 1 ) . \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} \\mathcal I _ { s } ^ q ( \\mu ) = \\int _ { \\underline a \\in \\Sigma } \\left ( \\int _ { \\underline b \\in \\Sigma } \\frac { 1 } { d ( \\underline a , \\underline b ) ^ s } d \\mu ( \\underline b ) \\right ) ^ { q - 1 } d \\mu ( \\underline a ) \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} & x _ { 0 } = \\epsilon , x _ { 1 } = \\epsilon ( 2 k + 1 ) , \\ x _ { m + 2 } = 2 k x _ { m + 1 } - x _ { m } , \\ m \\geq 0 , \\\\ & y _ { 0 } = \\epsilon , y _ { 1 } = \\epsilon ( 2 k - 1 ) , \\ y _ { m + 2 } = 2 k y _ { m + 1 } - y _ { m } , \\ m \\geq 0 , \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} \\gamma = \\frac { 1 } { z } ( e + z \\tilde { \\gamma } ) , \\ , \\ e \\in \\mathfrak { n } _ { \\theta } \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} \\tilde { \\mathbf { P } } ^ { \\rm H } = \\tilde { \\mathbf { P } } . \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rm M D } ( u ) = P \\left ( \\frac { Z } { n } \\leq \\tau _ n | U = u , H _ 1 \\right ) > 1 - \\frac { \\epsilon } { 2 } \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} \\frac { 4 \\varepsilon } { \\eta } - \\frac { 1 } { 2 } + \\varepsilon \\le \\frac { 1 } { 2 + \\eta } - \\frac { 1 } { 2 } + \\frac { \\eta } { 8 + 4 \\eta } = - \\frac { \\eta } { 8 + 4 \\eta } \\le - \\varepsilon , \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} \\mu _ a = \\frac { \\lambda _ a } { \\ell } + \\frac { \\vartheta _ a } { 2 r } + O ( \\frac { \\ln ( r ) } { r ^ 2 } ) . \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} \\mu = \\sum _ { i = 1 } ^ n \\theta ^ i \\delta ^ i , \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} W _ \\ell ( N ) [ - k ] \\ : = \\ W _ { \\ell - k } ( N ) \\ , . \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} d \\mu _ { x } ^ { q , l , j } ( \\mathcal { A } ) = d \\mu _ { f _ q v _ { l , j } } ( \\mathcal { A } ) \\ \\mathrm { f o r } \\ \\mathrm { a . e . } \\ x . \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} \\sigma = \\varphi ( \\gamma ) \\gamma _ t ^ n = \\gamma ^ { - m } \\gamma _ t ^ n , m , n > 0 \\ , , \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} c _ 4 = u ^ 4 c ' _ 4 , c _ 6 = u ^ 6 c ' _ 6 , \\Delta = u ^ { 1 2 } \\Delta ' . \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\alpha ' \\beta ' = t + u \\alpha ' + v \\beta ' ( t , u , v \\in A ) , \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} \\sum _ { \\tau = 1 } ^ { t } X _ { \\tau } \\leq \\sqrt { 3 \\log ( 1 / \\delta ) \\sum _ { \\tau = 1 } ^ { t } \\sigma _ { \\tau } ^ 2 } . \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} \\omega _ { t , s } ^ n = c _ { t , s } e ^ { t F _ s } i ^ { n ^ 2 } \\Omega _ s \\wedge \\bar \\Omega _ s , \\ ; \\ , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ \\infty \\bigg ( \\frac 1 { 1 - z | q | ^ i } - 1 \\bigg ) = \\sum _ { i = 0 } ^ \\infty \\frac { z | q | ^ i } { 1 - z | q | ^ i } \\le \\frac { z } { 1 - z } \\sum _ { i = 0 } ^ \\infty | q | ^ i < \\infty . \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} \\Phi ( z ) \\left ( \\frac { z - a } { z } \\right ) ^ { \\frac { c } { 2 } \\sigma _ 3 } \\mathcal { P } ( z ) \\left ( \\frac { z - a } { z } \\right ) ^ { - \\frac { c } { 2 } \\sigma _ 3 } = \\left ( N ^ { c / 2 } \\eta ( z ) \\right ) ^ { \\sigma _ 3 } H ( z ) \\ , { \\cal F } ( \\zeta ( z ) ) \\left ( N ^ { c / 2 } \\eta ( z ) \\right ) ^ { - \\sigma _ 3 } \\Phi ( z ) . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} \\left \\langle D b ( x ) u , u \\right \\rangle & = - 2 \\left | \\left \\langle x , u \\right \\rangle \\right | ^ 2 + \\left ( 1 - | x | ^ 2 \\right ) | u | ^ 2 \\\\ & \\leq \\left ( 1 - | x | ^ 2 \\right ) | u | ^ 2 \\leq | u | ^ 2 \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} ( g , \\partial \\_ N [ g ] ) = \\chi ( N ) . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} V _ q = \\bigoplus _ { \\substack { \\lambda \\in X ^ \\ast ( T ) \\\\ \\langle \\sigma _ { F _ P } , \\lambda \\rangle \\geq q } } V _ \\lambda \\subset V , \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} \\left ( \\eta | _ { U } \\right ) \\left ( \\frac { \\partial } { \\partial t } \\right ) = 1 , \\iota _ { \\frac { \\partial } { \\partial t } } \\Omega = 0 \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} \\chi _ k : = - \\frac { 1 + \\beta _ x } { 1 + \\tau \\beta _ x } \\ , \\ , \\exp ( \\mathrm { i } k \\gamma ^ { \\tau } ( x + \\beta ( x ) ) ) , \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} \\begin{aligned} ( X _ 1 - i Y _ 1 ) ^ { - 1 } ( X _ 1 ^ t + i Y _ 1 ^ t ) ^ { - 1 } & = \\Big ( ( X _ 1 ^ t + i Y _ 1 ^ t ) ( X _ 1 - i Y _ 1 ) \\Big ) ^ { - 1 } \\\\ & = \\Big ( X _ 1 ^ t X _ 1 + Y _ 1 ^ t Y _ 1 + i ( Y _ 1 ^ t X _ 1 - X _ 1 ^ t Y _ 1 ) \\Big ) ^ { - 1 } = M _ 1 ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} u _ \\alpha = v _ \\beta , u _ \\beta = - v _ \\alpha . \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} \\bar { \\psi } _ { \\gamma } ( \\delta , \\alpha _ { 0 } , \\theta _ { 0 } ) = \\frac { d \\bar { \\psi } ( \\gamma _ { 0 } + \\tau \\delta , \\alpha _ { 0 } , \\theta _ { 0 } ) } { d \\tau } = 0 . \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ \\R \\d X \\ , \\phi _ { 2 \\tau _ 1 - e ^ { - 2 L } / 2 } ( u \\pm X ) e ^ { - 2 W X + \\sqrt 2 r X } & = e ^ { ( 4 \\tau _ 1 - e ^ { - 2 L } ) ( \\sqrt 2 r - 2 W ) ^ 2 / 4 \\mp ( \\sqrt 2 r - 2 W ) u } , \\\\ \\int _ \\R \\d Y \\ , \\phi _ { e ^ { 2 L } / 2 - 2 \\tau _ 2 } ( v \\pm Y ) e ^ { 2 Z Y - \\sqrt 2 r Y } & = e ^ { ( e ^ { 2 L } - 4 \\tau _ 2 ) ( \\sqrt 2 r - 2 Z ) ^ 2 / 4 \\pm ( \\sqrt 2 r - 2 Z ) v } . \\end{aligned} \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} m _ { 1 } : = \\int _ { \\R ^ { n } } u _ { 1 } ( y ) d y . \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\lim _ { n \\rightarrow \\infty } \\textrm { d i a m } ( \\varphi _ { t _ n } ( \\cdot , U ) ) = 0 \\right ) > 0 . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} & C \\psi = i \\beta \\alpha _ 2 \\overline { \\psi } , \\quad \\mbox { C h a r g e c o n j u g a t i o n o p e r a t o r , } \\\\ & T \\psi = - i \\gamma _ 5 \\alpha _ 2 \\overline { \\psi } , \\mbox { T i m e r e v e r s a l - s y m m e t r y o p e r a t o r , } \\\\ & C T \\psi = \\beta \\gamma _ 5 \\psi , \\mbox { C T - s y m m e t r y o p e r a t o r . } \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} \\o ( w , G ) = \\min _ { g \\in N _ w } o ( g ) . \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} ( x ^ \\ell + m ) ( x ^ { \\ell + 1 } - m ) ( x ^ { \\ell + 1 } - m x + m ) = x ^ { 3 \\ell + 2 } \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} P _ { R , 1 } = P _ { p t p } ( N _ { S } , K , p _ { S D } ) . \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} \\frac { 1 } { k ! } \\ ( ( \\log q T ) ^ k + \\sum _ { m = 0 } ^ { k - 2 } k ! C ^ k ( \\log q T ) ^ m \\ ) \\frac { x } { T } \\ll \\frac { 1 } { k ! } ( C k \\log q T ) ^ k \\frac { x } { T } . \\end{align*}"}
-{"id": "5900.png", "formula": "\\begin{align*} E _ { 2 } ^ { * } ( z ) = 1 - 2 4 \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { 1 } ( n ) e ( n z ) - \\frac { \\pi } { 3 y } , \\bigg ( \\sigma _ { 1 } ( n ) = \\sum _ { d \\mid N } d \\bigg ) , \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} \\langle \\tau | = \\sum _ { h _ { 1 } , . . . , h _ { \\mathsf { N } } = 0 } ^ { p - 1 } \\prod _ { a = 1 } ^ { \\mathsf { N } } \\hat { Q } _ { \\tau , a } ^ { ( h _ { a } ) } \\prod _ { 1 \\leq b < a \\leq \\mathsf { N } } ( X _ { a } ^ { ( h _ { a } ) } - X _ { b } ^ { ( h _ { b } ) } ) \\langle h _ { 1 } , . . . , h _ { \\mathsf { N } } | , \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} { R _ \\mathrm { s } } = { \\left [ { { { \\log } _ 2 } \\left ( { 1 + { \\gamma _ o } } \\right ) - { { \\log } _ 2 } \\left ( { 1 + { \\gamma _ { e ^ { * } } } } \\right ) } \\right ] ^ + } . \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} g ( e h ) = g ( e ) = g ( e ) g ( h ) = g ( e ) h \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} \\begin{aligned} v _ t & = \\big ( \\gamma ^ { - m } v _ x ^ n \\big ) _ x \\\\ \\gamma _ t & = v _ x , \\end{aligned} \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} E ( p ) = - \\nabla I ( p ) . \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} \\int _ { D _ 1 \\times D _ 2 } \\left | h \\right | \\ , \\mathrm d \\rho ^ { ( 2 ) } & \\leq c \\cdot E \\Bigl \\| \\sum _ { n = 1 } ^ m X _ n g _ n \\Bigr \\| _ { k _ 2 } \\leq c \\biggl ( E \\Bigl \\| \\sum _ { n = 1 } ^ m X _ n g _ n \\Bigr \\| _ { k _ 2 } ^ 2 \\biggr ) ^ { 1 / 2 } \\\\ & = c \\Bigl ( \\sum _ { n = 1 } ^ m \\| g _ n \\| _ { k _ 2 } ^ 2 \\Bigr ) ^ { 1 / 2 } = c \\left \\| h \\right \\| _ { K _ s } , \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ 3 \\left ( \\frac { [ H ^ 2 _ { e t } ( X , \\mathcal { F } _ i ) ] } { [ H ^ 1 _ { e t } ( X , \\mathcal { F } _ i ) ] } \\right ) ^ { ( - 1 ) ^ { i + 1 } } = \\frac { [ Q ] } { [ S ] } . \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} \\theta _ { ( i ) } \\begin{cases} \\leq & \\tilde { \\theta } _ { ( i ) } \\leq \\eta _ { ( i ) } , 1 \\leq i \\leq k _ 1 ; \\\\ = & \\tilde { \\theta } _ { ( i ) } , k _ 1 < i < k _ 2 ; \\\\ \\geq & \\tilde { \\theta } _ { ( i ) } \\geq \\eta _ { ( i ) } , k _ 2 \\leq i \\leq n . \\end{cases} \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} R _ i ( t ) = \\sum _ { k = - Q _ i ( 0 ) + 1 } ^ { N _ i ( t ) } 1 _ { \\{ d _ { i , k } \\le w _ { i , k } \\} } . \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\frac { W _ { m _ { 1 } + 1 } } { W _ { m _ { 1 } } } + \\frac { U _ { n + 1 - m _ { 1 } } } { U _ { n - m _ { 1 } } } = \\frac { V _ { n , 1 } } { \\sum _ { j = 0 } ^ { \\infty } \\left ( j + 1 \\right ) V _ { n + 1 , j + 1 } U _ { n } ^ { j } } . \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} i ^ { \\star } = \\frac { N + 2 } { 4 } , \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial y ^ 1 _ k } = \\frac { \\partial } { \\partial x ^ 1 } + \\sum _ l \\bigg ( \\frac { \\partial h ^ k _ l } { \\partial y ^ 1 _ k } \\bigg ) \\frac { \\partial } { \\partial x ^ l } + \\sum _ \\tau \\bigg ( \\frac { \\partial \\omega ^ k _ \\tau } { \\partial y ^ 1 _ k } \\bigg ) \\frac { \\partial } { \\partial \\xi ^ \\tau } , \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} = \\int ^ 1 _ 0 \\int _ { { \\mathbb R } } \\lambda ^ 2 e ^ { - ( 1 - s ) \\lambda ^ 2 } | \\widehat { f } ( \\lambda ) | ^ 2 d \\lambda d s , \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} F ( z ) = - \\frac { 1 } { 4 0 } \\cdot \\frac { E _ { 4 } ( z ) + 4 E _ { 4 } ( 2 z ) - 9 E _ { 4 } ( 3 z ) - 3 6 E _ { 4 } ( 6 z ) } { ( \\eta ( z ) \\eta ( 2 z ) \\eta ( 3 z ) \\eta ( 6 z ) ) ^ { 2 } } = q ^ { - 1 } - 4 - 8 3 q - 2 9 6 q ^ { 2 } + \\dots \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} & \\Psi _ 2 ( v _ { \\delta } ) = - c _ 1 \\ , \\partial _ x ( v _ { \\delta } ^ 2 ) - \\frac { c _ 2 } { 3 } \\partial _ { x x } [ ( \\partial _ x ^ { - 1 } v _ { \\delta } ) ^ 2 ] + \\frac { c _ 2 } { 3 } \\pi _ 0 [ v _ { \\delta } ^ 2 ] + c _ 3 \\pi _ 0 [ ( \\partial _ x ^ { - 1 } v _ { \\delta } ) ^ 2 ] \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} D _ t ^ \\delta v - \\partial ^ 2 v / \\partial x ^ 2 = 0 \\ ( x , t ) \\in ( 0 , \\pi ) \\times ( 0 , T ] \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\eta ( z ) : = \\frac { e ^ { - i c \\pi / 2 } } { N ^ { c / 2 } } \\left ( \\frac { a - z } { z } \\right ) ^ { \\frac { c } { 2 } } \\left ( \\frac { z \\ , \\zeta ( z ) } { z - \\beta } \\right ) ^ { c } \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} \\begin{aligned} & H _ 2 ( u ) : = \\frac { 1 } { 2 } \\int _ { \\mathbb { T } } u _ x ^ 2 \\ , d x , H _ 3 ( u ) : = \\int _ { \\mathbb { T } } c _ 1 \\ , u _ x ^ 3 + c _ 2 \\ , u _ x ^ 2 \\ , u + c _ 3 \\ , u ^ 3 \\ , d x , \\\\ & H _ 4 ( u ) : = \\int _ { \\mathbb { T } } c _ 4 \\ , u _ x ^ 4 + c _ 5 \\ , u _ x ^ 3 \\ , u + c _ 6 \\ , u _ x ^ 2 \\ , u ^ 2 + c _ 7 \\ , u ^ 4 \\ , d x , H _ { \\geq 5 } ( u ) : = \\int _ { \\mathbb { T } } f _ { \\geq 5 } ( x , u , u _ x ) \\ , d x . \\end{aligned} \\end{align*}"}
-{"id": "4417.png", "formula": "\\begin{align*} D ( \\tilde { \\eta } ) = D ( \\eta _ { 1 } ) \\geq D ( \\eta _ { 0 } ) = D ( \\eta ) \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} \\mathbf { y } _ i = \\mathbf { H } _ i \\mathbf { x } _ i + \\mathbf { n } _ i , \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} \\begin{array} { l l } 1 6 = 2 ^ 4 & \\pm \\sqrt { 2 } e _ i , \\ , i = 1 , \\ldots , 8 \\\\ 2 2 4 = 2 ^ 4 \\cdot 1 4 & \\frac { 1 } { \\sqrt { 2 } } \\sum _ { i = 1 } ^ 8 ( \\pm x _ i ) e _ i , \\ ; x \\in \\mathcal { H } _ 8 ( x ) = 4 , \\end{array} \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} \\overline { \\pi } _ s = \\prod _ { P \\in \\mathcal { P } _ X ( s ) } [ \\pi _ P ] . \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} G ( m , k , r ) = \\frac { F ( m , r ) F ( k , r ) } { F ( r , r ) } . \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} g . ( g ' , \\theta , r , s , i , j ) = ( g ' g ^ { - 1 } , g \\theta g ^ { - 1 } , r , s , g i , j g ^ { - 1 } ) \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} \\mbox { r e s i d u e o f $ E $ } = \\mbox { c o n s t a n t f u n c t i o n w i t h v a l u e $ Q _ n ^ { - 1 } $ . } \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} ( [ r ' , s ' ] ) _ { \\iota \\gamma } = \\left \\{ \\begin{aligned} 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; & \\mbox { i f $ \\iota < \\gamma $ } \\\\ ( r _ { \\iota \\iota } - r _ { \\gamma \\gamma } ) s _ { \\iota \\gamma } + \\sum _ { k > \\iota } r _ { \\iota k } s _ { k \\gamma } - \\sum _ { k < \\gamma } s _ { \\iota k } r _ { k \\gamma } \\ : \\ : \\ : & \\mbox { i f $ \\iota \\geq \\gamma $ } . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = s } \\Phi = v \\circ \\Phi _ s \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} c _ 4 = u ^ 2 c _ 4 ( W ' ) , c _ 6 = u ^ 3 c _ 6 ( W ' ) \\Delta _ m = u ^ 6 \\Delta ( W ' ) . \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\P ( Z ^ { \\ell _ 1 } = u ^ { \\ell _ 1 } , Z _ { \\ell _ 1 + g + 1 } ^ { \\ell _ 1 + g + \\ell _ 2 } = v ^ { \\ell _ 2 } ) \\leq \\P ( Z ^ { \\ell _ 1 } = u ^ { \\ell _ 1 } ) \\P ( Z _ { \\ell _ 1 + g + 1 } ^ { \\ell _ 1 + g + \\ell _ 2 } = v ^ { \\ell _ 2 } ) \\Psi ( b , g ) , \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} \\left ( \\mathrm { R e s } ( \\mathrm { d l o g } Y \\cdot g ^ { ( s ) } X ) \\right ) _ { s = 0 } ^ { \\infty } = & \\left ( \\mathrm { R e s } ( - i \\sum _ { k \\geq 1 } [ a ] ^ k T ^ { i k - 1 } \\cdot [ b ] ^ { p ^ s } T ^ { - l p ^ s } ) \\right ) _ { s = 0 } ^ { \\infty } \\\\ = & \\left \\{ \\begin{array} { c c } - i ( [ a ^ { l / i } b ] ^ { p ^ s } ) _ { s = 0 } ^ { \\infty } = - i g ( [ [ a ^ { l / i } b ] ] ) & \\textrm { i f } i \\mid l \\\\ 0 & \\textrm { i f } i \\nmid l . \\end{array} \\right . \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} H _ k ( z ) = I + { \\cal O } ( N ^ { - \\tau _ a } ) , z \\in \\overline { D _ \\beta } . \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} s _ t : = \\sqrt [ 3 ] { \\frac { 3 t } { 4 \\pi } } \\quad \\omega _ { p , t } : = s _ t ( - \\omega + p ) , \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} d \\bar X = \\left \\{ ( A + \\bar { A } ) \\bar X + ( B + \\bar { B } ) \\bar v \\right \\} d s + \\left \\{ ( F + \\bar { F } ) \\bar X + ( G + \\bar { G } ) \\bar v \\right \\} d W _ 0 , \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{align*} \\Sigma ^ \\gamma M = A ^ \\gamma \\otimes M . \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} \\bar { x } : = x - \\sum \\limits _ { i \\in N } \\tilde { \\nu } _ i u _ i \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} u _ i & = u _ { i - 1 } + \\tau [ ( L ^ h _ { i \\tau } + J ^ h ) u _ i + f _ { i \\tau } ] , \\ i = 1 , . . . , n \\\\ v _ 0 & = \\psi . \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} R ( z ) & = R _ 0 ( z ) - R _ 0 ( z ) V R _ 0 ( z ) + R _ 0 ( z ) V R ( z ) V R _ 0 ( z ) , \\\\ \\Gamma _ { H } & = \\Gamma _ { - \\Delta } - i \\Gamma _ { - \\Delta } V \\Gamma _ { - \\Delta } - \\Gamma _ { - \\Delta } V \\Gamma _ { H } V \\Gamma _ { - \\Delta } , \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } D ( F _ { n _ 1 } + \\ldots + F _ { n _ { k - 1 } } + F _ n ) = \\infty . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} f ( y ) = \\mathcal { N } ( y ) ^ { - 1 } \\sum _ { \\xi > > 0 } c _ p ( \\xi y d , f ) q ^ \\xi , \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} f ( t ) & = f ( 0 ) + \\int _ 0 ^ t \\left [ f ' ( 0 ) + \\int _ 0 ^ s f '' ( 0 ) d \\tau \\right ] d s \\\\ & = f ( 0 ) + t A + \\int _ 0 ^ t \\int _ 0 ^ s \\langle v \\nabla ^ 2 u | v \\rangle d \\tau d s \\\\ & \\geq f ( 0 ) + t A + \\int _ 0 ^ t \\int _ 0 ^ s - C _ { \\phi _ 0 } d \\tau d s \\\\ & = f ( 0 ) + t A - \\frac { t ^ 2 C _ { \\phi _ 0 } } 2 . \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} v ( n ) = ( 1 - \\epsilon ) \\frac { n } { 2 k _ n } \\log ( 1 + k _ n P ' ) . \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} g _ 1 = x ^ 8 + 8 x ^ 4 + 3 3 6 g _ 2 = x ^ 8 + 4 x ^ 6 + 2 8 x ^ 4 + 2 0 ; \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} \\hat { V } _ n ( p ) = \\begin{cases} \\displaystyle \\ \\ \\ \\hat { V } ( p ) & \\mbox { i f $ | p | \\le n $ } \\\\ \\ \\ \\ 0 & \\mbox { i f $ | p | > n $ } \\end{cases} . \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} \\phi ^ \\vdash ( x ) = \\overline { d } ( \\phi , d ( - , x ) ) . \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} p _ { \\beta , \\mu , \\Lambda , \\lambda } = \\frac { 1 } { \\beta V } \\ln \\Xi _ { \\Lambda } ( \\beta , \\mu , \\lambda ) \\ , \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} Q _ n = \\xi ( 2 ) \\dots \\xi ( n ) \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} I \\ , \\mathcal { M P } _ n = \\bigoplus _ { k = 0 } ^ n \\mathcal { F } ^ { ( k ) } _ q ( \\mathcal { H } ) , \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} [ X _ r , X _ j ] & = \\sum _ i f _ i \\bigg [ \\frac { \\partial } { \\partial x ^ i } , X _ j \\bigg ] + \\sum _ \\rho \\varphi _ \\rho \\bigg [ \\frac { \\partial } { \\partial \\xi ^ \\rho } , X _ j \\bigg ] - \\sum _ i X _ j f _ i \\frac { \\partial } { \\partial x ^ i } - \\sum _ \\rho X _ j \\varphi _ \\rho \\frac { \\partial } { \\partial \\xi ^ \\rho } = 0 . \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} \\mathbb C T _ x X = \\mathbb C T ( x ) \\oplus T _ x ^ { 1 , 0 } ( X ) \\oplus T _ x ^ { 0 , 1 } X . \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} = \\frac { 1 } { \\sqrt { | D _ { M } | } ^ { [ K : M ] } } \\cdot \\prod _ { 1 \\leq i _ { 1 } < i _ { 2 } \\leq m } \\ ; \\ ; \\prod _ { j _ { 1 } = 1 } ^ { k } \\prod _ { j _ { 2 } = 1 } ^ { k } \\left \\vert \\alpha ^ { ( i _ { 1 } , j _ { 1 } ) } - \\alpha ^ { ( i _ { 2 } , j _ { 2 } ) } \\right \\vert . \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{align*} y ( t ) = \\C x ( t ) , \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} 3 - \\frac { 2 a } { k - 1 } \\geq 3 - \\frac { 2 ( k - 1 ) / k } { k - 1 } = 3 - \\frac { 2 } { k } , \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} x _ i ^ * = \\frac { x _ i - x _ { m i n } } { x _ { m a x } - x _ { m i n } } \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{align*} \\chi _ \\Delta ( Q ) = \\chi _ { \\Delta _ 1 } ( Q ) \\chi _ { \\Delta _ 2 } ( Q ) . \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} Q ^ { \\alpha } ( x , n , [ u ] ) = \\phi ^ { \\alpha } - \\xi ^ i u ^ { \\alpha } _ { \\bold { 1 } _ i ; \\bold { 0 } } . \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} f = \\sum _ { u \\subseteq 1 : s } f _ u , \\end{align*}"}
-{"id": "8223.png", "formula": "\\begin{align*} A _ r ( r ) = A _ \\theta ( r ) = 0 , V _ r \\left ( r \\right ) = - \\frac { Z \\alpha _ g } { r } \\ ; , \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} \\sum _ { p \\in \\P } \\frac { | f ( p ) - 1 | } { p } < \\infty , \\sum _ { p \\in \\P } \\sum _ { \\nu = 2 } ^ { \\infty } \\frac { | f ( p ^ \\nu ) | } { p ^ \\nu } < \\infty . \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} P _ { Y | U } ( y | u ) = \\begin{cases} 1 & \\textnormal { i f } y = y ^ \\star , \\\\ 0 & \\textnormal { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { \\bold { 0 } ; J _ 2 + \\bold { 1 } _ k } = S _ k \\phi ^ { \\alpha } _ { \\bold { 0 ; } J _ 2 } \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} I ( W ) & = I ( W ) \\cap V ' \\oplus I ( W ) \\cap V '' \\\\ & = \\pi ' \\circ I ( W ) \\oplus \\pi '' \\circ I ( W ) \\\\ & \\subset V ' \\oplus S '' \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} B _ y = \\big \\{ g \\in G \\ , : \\ , g y \\in B \\big \\} \\subset G . \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} q = \\frac { \\sqrt { 2 } } { | W | } \\sqrt { C _ 1 y _ 1 ^ 2 + C _ 2 y _ 2 ^ 2 \\pm \\sqrt { 4 C _ 1 C _ 2 - k W ^ 2 y _ 1 y _ 2 } } \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} T ^ { s t } C T = D C , T C ^ { - 1 } T ^ { s t } = D C ^ { - 1 } \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} Q _ 6 = \\frac { 5 / 6 } { r ^ 3 } - \\frac { 7 } { r ^ 4 } + \\frac { 7 } { r ^ 5 } \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} D _ S : = \\mbox { d i a g } _ { i = 1 , \\dots , \\nu } \\{ \\overline { \\jmath } _ i \\} , v _ k : = D _ S ^ k \\ , \\vec { 1 } , U : = \\vec { 1 } ^ T \\ , \\vec { 1 } \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\phi ( x ) & = a _ 0 x ^ n + a _ 1 x ^ { n - 1 } + \\cdots + a _ { n - 1 } x + a _ n , \\\\ u ( x ) & = u _ 0 x ^ { n - 2 } + u _ 1 x ^ { n - 3 } + \\cdots + u _ { n - 3 } x + u _ { n - 2 } , \\\\ v ( x ) & = v _ 0 x ^ { n - 1 } + v _ 1 x ^ { n - 2 } + \\cdots + v _ { n - 2 } x + v _ { n - 1 } , \\\\ \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} G _ i = \\frac { 1 } { N ^ 2 } \\frac { { \\big [ 1 - \\cos ( N { \\mathcal K } _ 1 ( \\xi _ { r _ { i , o } } ) ) \\big ] \\big [ 1 - \\cos ( { N } { { \\mathcal K } _ 2 } ( \\varphi _ { t _ { i , o } } , { \\varphi _ { { t _ i } } } ) ) \\big ] } } { { \\big [ 1 - \\cos ( { \\mathcal K } _ 1 ( \\xi _ { r _ { i , o } } ) ) \\big ] \\big [ 1 - \\cos ( { { \\mathcal K } _ 2 } ( \\varphi _ { t _ { i , o } } , { \\varphi _ { { t _ i } } } ) ) \\big ] } } , \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} E _ 0 ^ { ( + ) } = - 2 b , \\psi _ 0 ^ { ( + ) } ( x ) = e ^ { - \\frac { 1 } { 4 } x ^ 4 - a | x | ^ 3 + b x ^ 2 } , \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} M _ k ( x ; \\mathbf { a } ) = \\frac { 1 } { \\phi ^ k ( q ) } \\frac { k } { k _ 1 ! k _ 2 ! \\cdots k _ l ! } \\frac { x ( \\log \\log x ) ^ { k - 1 } } { \\log x } \\left \\{ g \\ ( \\frac { \\phi ( q ) } { \\log \\log x } ; \\mathbf { k } \\ ) + O _ { A , q , l } \\ ( \\frac { k } { ( \\log \\log x ) ^ 2 } \\ ) \\right \\} , \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} \\| f \\| _ { 1 } = | P ( f ) | \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} g _ { k , m } ( n ) = g \\left ( ( n - m K ) _ N \\right ) e ^ { - j 2 \\pi \\frac { k } { K } n } , \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} & \\big \\{ c \\left ( ( x _ 1 , \\ldots , x _ { a - 1 } , x _ { a } = u ^ { \\ast } , x _ { a + 1 } , \\ldots , x _ t ) ; ( u , v ) \\right ) ~ : ~ ( x _ 1 , \\ldots , x _ { a - 1 } , x _ { a + 1 } , \\ldots , x _ t ) \\in [ r ] ^ { t - 1 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( { u } , { v } ) \\in [ r ] \\times [ s ] \\big \\} . \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} \\begin{array} { l l l l } Z ^ { B Z } = & \\max \\ & c ' z + d ' u & \\\\ & \\mbox { s . t . } \\ & A z \\leq b & \\\\ & & H z + G u \\leq h & \\end{array} \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { m + \\sigma } u ( x ) = ( - \\Delta ) ^ { \\sigma } \\ , [ \\ , ( - \\Delta ) ^ m u ( x ) \\ , ] = ( - \\Delta ) ^ m \\ , [ \\ , ( - \\Delta ) ^ { \\sigma } u ( x ) \\ , ] x \\in \\Omega . \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} m _ { N } = \\frac { 1 } { \\lvert M _ { N } \\rvert } \\sum _ { i = 0 } ^ { \\lvert M _ { N } \\rvert - 1 } \\delta _ { z _ { i } } \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} \\mathbf { S N R } ^ { t o t a l , ( g ) } _ { m i m o } = \\sum _ { \\left \\{ n \\ ; \\vert \\lambda _ n > 0 \\right \\} } \\lambda _ n \\mathbf { e } _ n \\mathbf { e } _ n ^ H \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} \\Phi ( t ) \\Phi ( s ) = \\Phi ( s ) \\Phi ( t ) = \\Phi ( s ) , t \\geq s \\geq 0 . \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} \\P ( | a _ { i 1 } z _ 1 + \\dots + a _ { i n } z _ n - u _ i | \\le \\frac { r \\kappa } { D \\sqrt { n } } ) = O ( \\frac { r \\kappa } { D \\sqrt { n } } ) , \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} \\overline \\Psi ( x _ 1 ) = \\int _ \\omega \\Psi ( x ) \\dd x ' \\ , , \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} Z ( t ) = \\sum _ { i = 1 } ^ { K } \\widetilde { G } _ i ( t ) X _ i ( t ) + \\eta _ e ( t ) , \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rm F A } ( { \\mathbf u } ) = P ( { \\mathbf X } \\in R _ { H _ 1 } ( n ) | \\boldsymbol { \\sigma } _ { \\rm j } ^ 2 + \\boldsymbol { \\sigma } _ { \\rm w } ^ 2 = { \\mathbf u } , H _ 0 ) . \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} | \\sum _ { k = 1 } ^ { k _ { j } } < P _ { k } x ^ { * } _ { n _ { j + 1 } } , \\pi _ { k } x _ { n _ { j + 1 } } > | < \\frac { \\epsilon _ { 0 } } { 4 } , j = 1 , 2 , . . . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} q : = ( 1 - \\frac { \\sigma } { 2 } ) ^ { - 1 } p : = \\frac { 2 - 2 \\sigma } { 1 - \\sigma ( 2 - \\sigma ) } . \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} L ( f ) ( x , x ) & = f ( x , x ) L ( e _ { x x } ) ( x , x ) - \\left ( L ( e _ { x x } ) f \\right ) ( x , x ) - \\left ( f L ( e _ { x x } ) \\right ) ( x , x ) \\\\ & = f | _ { x } ^ { x } ( x , x ) L ( e _ { x x } ) ( x , x ) - \\left ( L ( e _ { x x } ) f | _ { x } ^ { x } \\right ) ( x , x ) - \\left ( f | _ { x } ^ { x } L ( e _ { x x } ) \\right ) ( x , x ) \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} \\dim S _ { k , \\Lambda _ N } = \\left \\lfloor \\frac { N - 4 } { 6 } \\right \\rfloor + \\left \\{ \\begin{matrix} - 1 & \\textrm { i f } & N \\equiv 2 \\pmod 8 0 & \\textrm { i f } & N \\equiv 0 , 6 \\pmod 8 \\\\ 1 & \\textrm { i f } & N \\equiv 4 \\pmod 8 \\end{matrix} \\right . \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} w _ i ( q ) : = \\frac { 1 } { T } \\int _ 0 ^ T f _ 2 ^ i \\big ( t , \\hat x ( t ) , q , 0 \\big ) \\ , d t . \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} < { \\widetilde { \\widetilde { f } } } _ { n } , x _ { n } > = \\sum _ { k = 1 } ^ { \\infty } \\widetilde { f } _ { n } ( k ) x _ { n } ( k ) - \\sum _ { k \\in E _ { 2 } } \\widetilde { f } _ { n } ( k ) x _ { n } ( k ) > \\epsilon - \\frac { \\delta } { 2 } - \\frac { \\delta } { 2 ^ { 2 } } , \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} \\norm { \\dot z ( t _ 2 ) - \\dot z ( t _ 1 ) } _ { H _ 0 ^ 1 ( \\Omega ) } & = \\norm { T _ \\rho ( g ( t _ 2 ) + \\Delta z ( t _ 2 ) ) - T _ \\rho ( g ( t _ 1 ) + \\Delta z ( t _ 1 ) ) } _ { H _ 0 ^ 1 ( \\Omega ) } \\\\ & \\le L _ \\rho \\ , \\norm { g ( t _ 2 ) + \\Delta z ( t _ 2 ) - g ( t _ 1 ) - \\Delta z ( t _ 1 ) } _ { H ^ { - 1 } ( \\Omega ) } . \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} \\dim S _ { k , \\Lambda _ N } = \\frac { N + 2 } { 8 } - \\left ( \\frac { 1 } { 2 } + \\left \\{ \\frac { N - 2 } { 8 } \\right \\} \\right ) = \\left \\lfloor \\frac { N - 2 } { 8 } \\right \\rfloor . \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} u ( x , y ) = a ( x , y ) \\exp ( i \\phi ( x , y ) ) , \\end{align*}"}
-{"id": "4026.png", "formula": "\\begin{align*} f _ - ( r ) = - 4 \\sin ( \\pi r ^ 2 / 2 ) ^ 2 \\int _ 0 ^ { i \\infty } \\psi _ - ( z ) e ^ { \\pi i r ^ 2 z } \\ , d z , \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{gather*} \\delta _ X \\Phi = - \\{ \\Xi , \\Phi \\} , \\Phi = ( A , C , A ^ \\ast , C ^ \\ast ) . \\end{gather*}"}
-{"id": "8176.png", "formula": "\\begin{align*} l _ \\sigma ( w ( x , y ) , 0 , 0 ) = \\left ( x y ^ { - 1 } w ( x ^ { - 1 } , y ^ { - 1 } ) y x ^ { - 1 } , | w | _ x , | w | _ y \\right ) . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} \\widehat { \\cal F } ( \\zeta ) = { \\cal F } ( \\zeta ) F _ 1 ( \\zeta ) ^ { - 1 } . \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{align*} R _ { 1 2 } ( \\lambda / \\mu ) M _ { 1 } ( \\lambda ) M _ { 2 } ( \\mu ) = M _ { 2 } ( \\mu ) M _ { 1 } ( \\lambda ) R _ { 1 2 } ( \\lambda / \\mu ) , \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n s ^ i ( \\hat { x } _ t , \\theta _ t ) = \\hat { x } _ t . \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} - 2 \\ , \\ddot g _ \\rho + \\ddot { \\bar g } + 2 \\ , g _ \\rho - \\bar g + \\xi _ \\rho = 0 H ^ { - 1 } ( I ; L ^ 2 ( \\Omega ) ) , g _ \\rho ( 0 ) = 0 , 2 \\ , \\dot g _ \\rho ( T ) - \\dot { \\bar g } ( T ) = 0 . \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} f _ { 1 } & = ( f - \\{ ( v , 1 ) \\} ) \\cup \\{ ( v , 2 ) \\} \\\\ f _ { 2 } & = ( f _ { 1 } - \\{ ( w , f ( w ) ) \\} ) \\cup \\{ ( w , f ( w ) + 1 ) \\} . \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{align*} \\dfrac { b ^ { 2 } } { a ^ { 2 } } = \\dfrac { A + C - \\sqrt { ( A - C ) ^ { 2 } + B ^ { 2 } } } { A + C + \\sqrt { ( A - C ) ^ { 2 } + B ^ { 2 } } } \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} C ( n , k ) = k C ( n - 1 , k ) + ( 2 n - k ) C ( n - 1 , k - 1 ) \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} \\varepsilon ^ { - 2 b } H _ 2 \\circ A _ { \\varepsilon } = c o n s t + \\sum _ { j \\in S ^ { + } } j ^ 3 \\ , y _ j + \\frac { 1 } { 2 } \\int _ { \\mathbb { T } } z _ x ^ 2 \\ , d x , \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} N _ { M / \\mathbb { Q } } \\left ( F \\left ( u , v \\right ) \\right ) = \\pm \\frac { d ^ { 6 m } } { i _ { 0 } } , \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} f ^ * ( b _ 1 ) = f ^ * ( b _ 2 ) = \\dots = f ^ * ( b _ { k - 1 } ) = f ^ * ( b _ k ) < f ^ * ( b _ { k + 1 } ) , \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} - b ( x ) \\cdot D \\varphi ( x ) = - b ( x ) \\cdot D \\tilde \\psi ( x ) = - b ( x ) \\cdot D \\psi ( x ) . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } Z ( F _ { j , i } ) = Z ( F _ i ) . \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} \\eta ( k ; l ) & = \\sum _ { \\substack { m > 0 \\\\ 0 < a _ 1 \\leq \\cdots \\leq a _ { k - 1 } \\leq m \\\\ 0 < b _ 1 \\leq \\cdots \\leq b _ { l - 1 } \\leq m } } \\frac { 1 } { m ^ 2 a _ 1 \\cdots a _ { k - 1 } b _ 1 \\cdots b _ { l - 1 } } \\\\ & = \\zeta \\bigl ( ( \\underbrace { 1 , \\ldots , 1 } _ k ) ^ \\star \\circledast ( \\underbrace { 1 , \\ldots , 1 } _ l ) ^ \\star \\bigr ) \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} \\| P _ V ( x - y ) \\| ^ { 2 } = ( P _ V ( x - y ) ) \\cdot ( x - y ) . \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} U = \\{ ( \\mu , \\nu ) \\in \\mathbb { R } ^ 2 : \\mu > 0 , \\ ; | \\nu | < \\beta _ 1 \\sqrt { \\mu } \\} , \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} g ( W , \\gamma , \\theta ) & = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } D ( X _ { t } ) \\pi ( a ( X _ { t } , \\theta , \\gamma _ { 2 } , \\gamma _ { 3 } ) ) [ Y _ { 2 t } - \\Lambda ( a ( X _ { t } , \\theta , \\gamma _ { 2 } , \\gamma _ { 3 } ) ) ] , \\\\ \\pi ( a ) & = \\frac { \\Lambda _ { a } ( a ) } { \\Lambda ( a ) [ 1 - \\Lambda ( a ) ] } , \\Lambda _ { a } ( a ) = \\frac { d \\Lambda ( a ) } { d a } . \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} \\frac { 2 } { p ^ { 2 ^ n } } = \\frac { p ^ { 2 ^ n } - 1 } { p ^ { 2 ^ { n + 1 } } } + \\frac { p ^ { 2 ^ n } + 1 } { p ^ { 2 ^ { n + 1 } } } \\ \\ \\frac { p ^ { 2 ^ n } \\pm 1 } { p ^ { 2 ^ { n + 1 } } } = \\frac { p ^ { 2 ^ n } \\pm 1 } { 2 } \\frac { 2 } { p ^ { 2 ^ { n + 1 } } } . \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} \\varphi _ n ( x ) = \\pi ^ { - 1 / 4 } 2 ^ { - n / 2 } ( n ! ) ^ { - 1 / 2 } e ^ { - x ^ 2 / 2 } H _ n ( x ) \\end{align*}"}
-{"id": "5132.png", "formula": "\\begin{align*} \\nu ' ( A _ { x _ o } B ) = \\mu ( A ) + \\nu ' ( B ) < 1 , \\textrm { f o r $ \\kappa $ - a . e . $ \\nu ' $ } . \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} D _ k ^ { ( t ) } = \\left \\{ \\begin{matrix} D _ k & \\mid & t > 0 \\\\ \\C \\cdot 1 _ { D _ k } & \\mid & t = 0 \\end{matrix} \\right . \\ , . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} N _ { i a } = \\frac { \\Delta N _ i + s _ a \\Delta D _ i } { 2 } \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} \\langle d e ^ { i \\theta } U ( x _ 0 ) \\ , | \\ , d e ^ { i \\theta } V ( x _ 0 ) \\rangle = \\langle U ( x _ 0 ) \\ , | \\ , V ( x _ 0 ) \\rangle _ g , ~ \\forall ~ 0 \\leq \\theta < \\delta _ 1 + \\sigma . \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} Y _ 1 & = X _ 1 + \\sqrt { b } X _ 2 + Z _ 1 \\\\ Y _ 2 & = \\sqrt { a } X _ 1 + X _ 2 + Z _ 2 , \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} & \\P ( X _ t < N / 2 , \\ , \\forall t \\in [ 0 , T _ N ] | X _ 0 = 0 ) \\ge \\P \\left ( \\cap _ { k = 1 } ^ { T _ N } \\{ \\sigma _ k - \\sigma _ { k - 1 } < N / 2 \\} | X _ 0 = 0 \\right ) \\\\ & \\ge \\left ( 1 - c _ 1 ^ { - 1 } e ^ { - c _ 1 N } \\right ) ^ { T _ N } \\ge 1 - c _ 1 ^ { - 1 } T _ N e ^ { - c _ 1 N } = 1 - c _ 1 ^ { - 1 } e ^ { - \\alpha N } . \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} ( 2 - q ) a \\| w _ k \\| ^ { 2 } + ( 4 - q ) \\epsilon \\| w _ k \\| ^ { 4 } - ( 2 ^ * _ \\alpha - q ) \\int _ { \\Omega \\times \\{ 0 \\} } | w _ k ( z , 0 ) | ^ { 2 ^ * _ \\alpha } d z = o _ k ( 1 ) . \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} \\delta f ( u , v ) = f ( v ) - f ( u ) , u , v \\in M . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} A _ { \\varepsilon } ( \\theta , y , z ) : = \\varepsilon \\ , v _ { \\varepsilon } ( \\theta , y ) + \\varepsilon ^ b z , v _ { \\varepsilon } ( \\theta , y ) : = \\sum _ { j \\in S } \\sqrt { \\lvert j \\rvert } \\ , \\sqrt { \\xi _ j + \\varepsilon ^ { 2 ( b - 1 ) } y _ j } \\ , e ^ { i \\theta _ j } e ^ { i j x } . \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} \\frac { H _ { 1 } } { \\beta } = - \\Delta _ 1 + \\frac { 1 } { 2 } \\Delta _ 0 ^ 2 - \\frac { 1 } { 2 } \\Delta _ 0 , \\end{align*}"}
-{"id": "807.png", "formula": "\\begin{align*} S = \\frac { \\sum _ { j = 1 } ^ n x _ { j + 1 } ^ - \\left ( \\prod _ { \\ell = j + 2 } ^ { j - 1 } x _ { \\ell } \\right ) x _ j ^ + } { \\prod _ { j = 1 } ^ n x _ j } . \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} p ( t ) = \\begin{cases} \\overline { g } \\cdot \\alpha _ t ( g ) & \\mid t \\in [ - 2 r , 2 r ] \\\\ 0 & \\mid t \\not \\in [ - 2 r , 2 r ] . \\end{cases} \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} \\phi ( x , y ) = x ^ 3 + b x ^ 2 y + c x y ^ 2 + d y ^ 3 \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} \\partial _ t p = D ( p ) \\partial _ { x x } p - M ( p ) \\partial _ x p + f ( p , N ) \\ , , \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} \\epsilon ( 1 / 2 , \\pi _ 1 , \\pi _ 2 , \\psi ) = 1 . \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} \\frac { 1 } { B k } \\sum ^ { B k } _ { i = 1 } \\gamma ( X _ i ) \\leq \\Gamma . \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} ( \\theta _ { 1 / 2 } ^ { \\sigma _ { c } } , \\theta _ { 1 / 2 } ) = \\frac { \\pi } { 3 \\sqrt { N } } \\left ( \\frac { N } { c } + c \\right ) \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} Y = \\xi ^ i ( x , n , [ u ] ) \\partial _ { x ^ i } + \\sum _ { \\alpha , J _ 1 , J _ 2 } \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } ( x , n , [ u ] ) \\partial _ { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } , \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} a ^ { ( l , \\sigma ) } _ - ( y ) = & \\left ( \\Lambda \\left ( \\xi ^ { ( l , \\sigma ) } ( y ) \\right ) - \\left ( \\frac { 1 } { 2 } - 8 \\varepsilon \\right ) \\right ) \\cdot 8 L \\ ; , \\\\ a ^ { ( l , \\sigma ) } _ + ( y ) = & \\left ( \\Lambda \\left ( \\xi ^ { ( l , \\sigma ) } ( y ) \\right ) + \\left ( \\frac { 1 } { 2 } - 8 \\varepsilon \\right ) \\right ) \\cdot 8 L \\ ; . \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} A \\geq \\sum _ { I \\in S _ \\phi } \\frac { \\alpha _ I } { \\rho _ I } \\frac { y _ I ^ q } { [ ( \\beta + 1 ) - \\beta \\rho _ I ] ^ { q - 1 } } - \\sum _ { I \\in S _ \\phi } \\sum _ { \\substack { J \\in S _ \\phi \\\\ J ^ \\star = I } } \\mu ( J ) \\frac { y _ J ^ q } { ( \\beta + 1 ) ^ { q - 1 } } , \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} d s ^ 2 = \\sum _ { i = 1 } ^ g \\frac { d x _ i ^ 2 + d y _ i ^ 2 } { y _ i ^ 2 } , \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} \\alpha ^ { V } = - \\alpha _ i \\frac { \\partial } { \\partial p _ i } \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} [ \\lambda + a _ \\pi ( x , \\xi + i \\partial _ t ) ] u = 0 , t > 0 , b _ \\pi ( x , \\xi + i \\partial _ t ) u ( 0 ) = 0 \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} m _ K ( I ^ { - 1 } J ) \\leq \\nu ( A _ { x _ o } ^ { - 1 } C ) = \\mu ( A ) + \\nu ( C ) \\leq \\min ( 1 , m _ K ( I ) + m _ K ( J ) ) . \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} D _ { \\tau } ( 1 / \\lambda ) = O _ { C } O _ { R } ( D _ { \\tau } ( \\lambda ) ) , \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} \\dot y ( t ) = \\lambda h ( t , y ) + \\lambda ^ 2 \\phi ( t , y ( t ) ) \\int _ { - \\infty } ^ { 0 } y ( t - \\tau ) e ^ \\tau d \\tau , \\lambda \\geq 0 , \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} \\begin{aligned} \\lvert \\omega \\cdot l + m _ 3 ( j '^ 3 - j ^ 3 ) \\rvert & = \\lvert m _ 3 ( \\overline { \\omega } \\cdot l + j '^ 3 - j ^ 3 ) + ( \\omega - m _ 3 \\overline { \\omega } ) \\cdot l \\rvert \\geq \\\\ & \\geq \\lvert m _ 3 \\rvert \\ , \\lvert \\overline { \\omega } \\cdot l + j '^ 3 - j ^ 3 \\rvert - \\lvert \\omega - m _ 3 \\overline { \\omega } \\rvert \\ , \\lvert l \\rvert \\geq 1 / 2 , \\ , \\ , \\ , \\forall \\lvert l \\rvert \\le 1 \\end{aligned} \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} \\bar k ( s ) = \\frac { 1 } { 2 + \\epsilon } - \\frac { 2 + 2 \\epsilon } { 2 + \\epsilon } e ^ { \\mu ( s - t ) } \\left ( \\pi ( s ) \\bar X ( s ) + \\int _ t ^ s \\pi ( \\tau ) d \\tau \\right ) . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} \\alpha _ { 1 } ( \\underline { a } ) ^ { 2 } \\alpha _ { 2 } ( \\underline { a } ) ^ { s - 2 } \\leq \\prod _ { l = 1 } ^ { k } \\frac { \\alpha _ 1 ( a _ l ) ^ 2 } { \\alpha _ { 2 } ( a _ l ) ^ { 2 - s } } < \\gamma ^ { - k } \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{gather*} 2 7 f _ { t t t } + 3 P ( t ) f _ t + Q ( t ) f = 0 . \\end{gather*}"}
-{"id": "5548.png", "formula": "\\begin{align*} f _ 2 ( [ x , x ] , x \\partial ^ { 2 i - 1 } ) - 2 f _ 2 ( [ x , x \\partial ^ { 2 i - 1 } ] , x ) & \\\\ = 2 [ f _ 1 ( x ) , f _ 2 ( x , x \\partial ^ { 2 i - 1 } ) ] & - [ f _ 1 ( x \\partial ^ { 2 i - 1 } ) , f _ 2 ( x , x ) ] + d f _ 3 ( x , x , x \\partial ^ { 2 i - 1 } ) . \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} f ^ * \\circ f = ( d _ x ^ * \\circ d _ x ) B _ y + B _ x ( d _ y ^ * \\circ d _ y ) = B _ x ( 2 \\rho ) B _ y - x ( \\rho ) B _ x B _ y - B _ x B _ y ( y ^ { - 1 } \\rho ) \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } ( \\Phi ^ { \\prime } ( ( u _ { n } , v _ { n } ) ) - \\Phi ^ { \\prime } ( u _ { 0 } , v _ { 0 } ) , ( u _ { n } - u _ { 0 } , v _ { n } - v _ { 0 } ) = \\underset { n \\rightarrow + \\infty } { \\lim } ( f _ { n } , ( u _ { n } - u _ { 0 } , v _ { n } - v _ { 0 } ) ) = 0 . \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} \\eta ( z ) : = \\frac { 1 } { N ^ { c / 2 } } \\left ( \\frac { z \\ , \\zeta ( z ) } { z - \\beta } \\right ) ^ { c / 2 } , \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} \\lambda ( u , v ) = \\sum _ { c \\mid u v } \\chi ( c ) \\log { \\frac { c } { u } } \\log { \\frac { c } { v } } . \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} & \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\mathcal { F } ^ { - 1 } [ \\mathcal { H } _ { \\sigma , \\nu } ( t , \\xi ) ] \\| _ { 2 } = C t ^ { - \\tilde { \\gamma } _ { \\sigma , k } - \\ell } \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} V _ t : = \\{ x \\in V : \\ , \\exists y \\in V , \\ , y \\neq x ( x , y ) \\in E _ t \\} \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} \\| \\nabla P \\psi \\| ^ 2 _ { L ^ 2 ( \\R ^ 3 ) } = \\| \\alpha \\cdot D P \\psi \\| ^ 2 _ { L ^ 2 ( \\R ^ 3 ) } = 2 \\| \\alpha \\cdot D \\psi \\| ^ 2 _ { L ^ 2 ( \\R ^ 3 _ + ) } . \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} q & = ( \\xi ' ( q ) + h ^ { 2 } ) ( 1 - q ) ^ { 2 } \\\\ 1 & \\geq \\xi '' ( q ) ( 1 - q ) ^ { 2 } \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\sum _ { a = 0 } ^ c \\binom { p - 1 - a } { c - a } ^ 2 + \\binom { 2 c } c & \\equiv _ { p ^ 2 } \\sum _ { a = 0 } ^ c \\binom { c } a ^ 2 ( 1 - 2 p ( H _ c - H _ a ) ) + \\binom { 2 c } { c } \\\\ & = 2 \\binom { 2 c } c ( 1 - p H _ c ) + 2 p \\sum _ { a = 0 } ^ c \\binom { c } a ^ 2 H _ a \\\\ & = 2 \\binom { 2 c } c \\left ( 1 - p H _ c + p ( 2 H _ c - H _ { 2 c } ) \\right ) \\\\ & \\equiv _ { p ^ 2 } 2 \\binom { 2 c } c ( 1 - p H _ { 2 c } ) \\equiv _ { p ^ 2 } 2 \\binom { 2 c } c ( 1 - 1 ) = 0 , \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{gather*} F ( h ) \\leq F ( f ) \\leq c + \\epsilon ~ \\ ! , \\\\ d ( h , f ) : = \\sup _ { p \\in \\overline { B _ R } } \\| h ( p ) - f ( p ) \\| \\le \\epsilon ^ { 1 / 2 } ~ \\ ! , \\\\ F ( g ) > F ( h ) - \\epsilon ^ { 1 / 2 } d ( h , g ) ~ ~ \\forall g \\in \\Phi \\ \\hbox { w i t h } \\ g \\neq h ~ \\ ! . \\end{gather*}"}
-{"id": "7046.png", "formula": "\\begin{align*} \\lambda _ 4 ( u v ) = \\lambda ( v ) \\lambda _ 4 ( u ) + 6 \\lambda _ 2 ( u ) \\lambda _ 2 ( v ) + \\lambda ( u ) \\lambda _ 4 ( v ) . \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} \\mathrm { s n } ( x , y ) = \\frac { [ x , y ] } { | | y | | _ a | | x | | } . \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} F _ N ^ * \\eta = d Z _ N + X _ N d Y _ N . \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} F ( \\boldsymbol { n } s ; \\chi _ 0 ) : = \\prod _ { j = 1 } ^ l F ( n _ j s , \\chi _ 0 ) . \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} \\{ q _ s \\mid s \\succeq _ { \\lambda } r , \\ q _ s \\leq q _ r \\} & = \\{ q _ r - i e \\mid i \\in [ 0 , \\ , k ) \\} , \\\\ \\{ q _ s \\mid s \\succeq _ { \\lambda } r , \\ q _ s > q _ r \\} & = \\{ q _ r + i e \\mid i \\in [ 1 , \\ , l ] \\} , \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\begin{cases} x = u X \\\\ y = u \\sqrt { u } Y , \\ \\end{cases} \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} { \\cal F } \\ast { \\cal G } = R \\overset \\gets a _ * ( { \\cal F } \\boxtimes { \\cal G } ) . \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} b _ { s j } = \\frac { 1 } { \\omega } \\int _ 0 ^ \\omega q ( t ) e _ s ( t ) \\overline { e _ j ( t ) } \\ , d t = \\frac { 1 } { \\omega } \\sum _ { l \\in \\mathbb { Z } } q _ l \\int _ 0 ^ \\omega e ^ { i 2 \\pi l t / \\omega } e ^ { - i \\pi ( 2 s + \\theta ) t / \\omega } e ^ { i \\pi ( 2 j + \\theta ) t / \\omega } \\ , d t = q _ { s - j } . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\vdash n } \\sum _ { \\lambda _ i \\in \\lambda } \\Lambda ( \\lambda _ i ) = \\sum _ { \\lambda \\vdash n } \\sum _ { \\substack { \\lambda _ i \\in \\lambda \\\\ \\lambda _ i } } \\log ( \\lambda _ i ) = \\sum _ { k = 1 } ^ { n } p ( n - k ) \\log { k } . \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} D _ X ^ 2 . \\pi _ * \\beta ^ * c _ 2 ( V ) = \\pi _ * ( \\tilde { D } ^ 2 . \\beta ^ * c _ 2 ( V ) ) = D ^ 2 . c _ 2 ( V ) , \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} 1 0 8 \\lambda ( 1 9 ) = H _ n ( 1 9 ) + 1 4 H _ h ( 1 9 ) + 7 8 H _ u ( 1 9 ) , \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} e ^ { { - { i } \\tau \\kappa } } e ^ { - 2 \\pi i b \\cdot v } D _ { A _ R } e ^ { 2 \\pi i b \\cdot v } e ^ { { { i } \\tau \\kappa } } = \\hat { D } + c ( \\delta u ) , \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\mathbf { R } = \\mathbf { S } \\times \\mathbf { H } + \\mathbf { C } \\times \\mathbf { G } + \\mathbf { W } \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} \\sigma _ t ^ { \\hat \\psi } P _ { \\check \\varphi } = P _ { \\check \\varphi } \\sigma _ t ^ { \\hat \\psi } , & & \\sigma _ t ^ { \\hat \\psi } K _ { \\check \\varphi } = K _ { \\check \\varphi } \\sigma _ t ^ { \\hat \\psi } . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} \\bold { p r } X = \\xi ^ i D _ i + \\sum _ { \\alpha , J _ 1 , J _ 2 } \\left ( D _ { J _ 1 } Q _ { J _ 2 } ^ { \\alpha } \\right ) \\partial _ { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } , \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} \\lim _ { \\varepsilon \\to 0 } \\dfrac { \\lvert \\mathcal { C } _ { \\varepsilon } \\rvert } { \\lvert \\Omega _ { \\varepsilon } \\rvert } = 1 , \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{gather*} q _ { 2 t } = q _ 2 \\left ( \\frac { 2 } { 3 } \\alpha + \\frac { u _ t } { u } \\frac { 2 - q _ 2 } { 3 } \\right ) + \\frac { u _ t } { u } \\frac { 2 - q _ 2 } { 3 } . \\end{gather*}"}
-{"id": "3350.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\log \\left ( \\min \\left \\{ g _ { \\phi _ { n } } ^ { ' } ( 0 ) , \\frac { 1 } { \\alpha } \\cdot | | g _ { \\phi _ { n } } | | _ { 2 } \\right \\} \\right ) < \\infty . \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\| T x _ { n } \\| & = < T ^ { * } y ^ { * } _ { n } , x _ { n } > \\\\ & \\leq \\epsilon + | < x ^ { * } _ { n } , x _ { n } > | \\\\ & \\leq \\epsilon + \\sup _ { x ^ { * } \\in K } | < x ^ { * } , x _ { n } > | \\\\ & \\leq \\epsilon + c . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} \\theta ^ i u _ i ' \\left ( \\sigma ^ i ( \\theta ^ i , \\lambda ) \\right ) = \\lambda , \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} H _ { \\nu } \\left ( x + Q \\right ) \\geq \\log \\left ( \\frac { 1 } { 2 } p ^ { H _ { \\nu } ( Q ) } \\right ) = H _ { \\nu } ( Q ) + \\log \\left ( \\frac { 1 } { 2 } \\right ) . \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} \\Pr _ { r } ' [ T _ { \\mathrm { B a d } _ i } \\le 2 ^ { 1 8 n - 2 } \\mid T _ { \\mathrm { B a d } _ i } < T _ { \\mathrm { N i c e } } ] \\le \\Pr _ { o _ i } ' [ T _ { \\mathrm { B L } _ i } \\le 2 ^ { 1 8 n - 2 } \\mid T _ { \\mathrm { B L } _ i } < T _ { \\mathrm { G M P } _ i } ] = o ( 1 / n ) , \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{align*} x _ { } ^ { p } + y _ { } ^ { p } = k ( 1 + x _ { } ^ { p } y _ { } ^ { p } ) , k x _ { } ^ { p } = 1 - k ^ { ^ { \\prime } } s _ { } ^ { - p } , k y _ { } ^ { p } = 1 - k ^ { ^ { \\prime } } s _ { } ^ { p } , \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { h _ f ( n ) \\log n } { n ^ s } = - \\frac { \\log 2 } { 2 ^ { s - 1 } } \\cdot \\frac { S _ f ' ( 1 / 2 ^ s ) } { S _ f ( 1 / 2 ^ s ) ^ 2 } , \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} w _ i ( h ) = \\begin{cases} u _ i ^ + ( h ) - u _ i ^ + ( ( - 1 ) ^ i \\delta ) + d + \\delta ^ 2 \\ \\ \\ & h \\in \\bar J _ i \\setminus [ - \\delta , \\delta ] , \\\\ d + ( - 1 ) ^ i h \\delta \\ \\ \\ & h \\in \\bar J _ i \\cap [ - \\delta , \\delta ] , \\end{cases} \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} u ( I ) & = u ( f _ { i _ 1 } ) u ( f _ { i _ 2 } ) \\cdots u ( f _ { i _ m } ) ; ~ ~ u ( F ) = u ( g _ { j _ 1 } ) u ( g _ { j _ 2 } ) \\cdots u ( g _ { j _ m } ) . \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} P ( G , 1 ) = \\frac { 1 } { | E ( G ) | } \\sum _ { e \\in E ( G ) \\setminus B ( G ) } P ( G - e , 1 ) . \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} E _ 1 ^ { ( + ) } = - 6 b - c ^ 2 - \\frac { 6 a } { c } + \\frac { 2 } { c ^ 2 } , \\psi _ 1 ^ { ( + ) } ( x ) = e ^ { - \\frac { 1 } { 4 } x ^ 4 - a | x | ^ 3 + b x ^ 2 - c | x | } \\left ( | x | + \\frac { 1 } { c } \\right ) , \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} \\beta ( \\lambda ) = \\{ \\lambda _ { 1 } - 1 , \\lambda _ { 2 } - 2 , \\ldots \\} \\subseteq \\mathbb { Z } \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} \\hat { \\theta } ( x ^ n , y ^ n ) : = \\arg \\min _ { \\theta \\in \\Theta } \\left \\{ \\frac { 1 } { 2 n \\sigma ^ 2 } \\| Y - X \\theta \\| _ 2 ^ 2 + \\mu _ 1 \\| \\theta \\| _ { w , 1 } \\right \\} , \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} \\overline { D } ^ 1 ( \\mu ) = \\limsup _ { \\delta \\to 0 } \\frac { \\sum _ { Q \\in \\mathcal { M } _ \\delta } \\mu ( Q ) \\log \\mu ( Q ) } { \\log \\delta } \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} l _ \\sigma ( u ^ r ; 0 , 0 ) & = l _ \\sigma ( u ^ { - r } ; 0 , 0 ) ^ { - 1 } = ( ( B u ^ { - 1 } ) ^ { - r } B ^ r ; - r , 0 ) ^ { - 1 } = ( ( \\theta ( r , 0 ) ( ( B u ^ { - 1 } ) ^ { - r } B ^ r ) ) ^ { - 1 } ; r , 0 ) \\\\ & = ( ( ( \\theta ( r , 0 ) ( B ) \\theta ( r , 0 ) ( u ) ^ { - 1 } ) ^ { - r } \\theta ( r , 0 ) ( B ) ^ r ) ^ { - 1 } ; r , 0 ) \\\\ & = ( ( ( B ( B ^ r u B ^ { - r } ) ^ { - 1 } ) ^ { - r } B ^ r ) ^ { - 1 } ; r , 0 ) = ( ( B u ^ { - 1 } ) ^ { r } B ^ { - r } ; r , 0 ) . \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} \\psi ( t , \\alpha , \\beta ) = \\phi ( t , x ( t , \\alpha , \\beta ) , y ( t , \\alpha , \\beta ) ) . \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} | T ( \\xi ) | ^ 2 + | R ( \\xi ) | ^ 2 = 1 . \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} T _ m ( f ) ( t ) = f ( 0 ) - \\sum _ { j = 1 } ^ m \\frac { p _ j t ^ j } { j } = \\ln ( a _ 0 ) - \\sum _ { j = 1 } ^ m \\frac { p _ j t ^ j } { j } . \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\langle x , T ' \\rangle = c \\langle \\beta , T ' \\rangle + \\sum _ { ( i , p ) = 1 } \\langle c _ i [ T '^ { - i } ] , T ' \\rangle . \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{gather*} U ( t ) = - 2 \\omega - \\frac { 4 } { 1 - q ^ 2 _ 2 } \\alpha - \\frac { u _ t } { u } \\frac { 1 + q _ 2 } { 1 - q _ 2 } - \\frac { t ^ 2 } { 2 } . \\end{gather*}"}
-{"id": "4480.png", "formula": "\\begin{align*} Q ( v ) = \\int _ { \\mathbb { R } } v ^ 2 d x \\end{align*}"}
-{"id": "7322.png", "formula": "\\begin{align*} \\sqrt { n } \\int \\left \\Vert \\hat { \\Delta } _ { \\ell 2 } ( w ) \\right \\Vert F _ { 0 } ( d w ) & \\leq \\sqrt { n } \\left \\Vert \\hat { \\alpha } _ { 3 \\ell } - \\alpha _ { 3 0 } \\right \\Vert \\left \\Vert H ( \\hat { \\gamma } _ { 1 \\ell } ) - \\hat { \\gamma } _ { 3 \\ell } - H ( \\gamma _ { 1 0 } ) + \\gamma _ { 3 0 } \\right \\Vert \\\\ & = O _ { p } ( \\sqrt { n } [ n ^ { - d _ { 1 } ( 2 \\xi _ { 1 } - 1 ) / ( 2 \\xi _ { 1 } + 1 ) } \\ln ( n ) ] n ^ { - d _ { 1 } } ) = o _ { p } ^ { { } } ( 1 ) . \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} A _ i ^ { ( j k ) } G _ k = G _ j ^ T A _ i ^ { ( j k ) } , \\mbox { f o r } \\ ; i = 1 , 2 , \\dots , m , j , k = 1 , 2 , \\dots , s . \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} d x _ \\delta ( u _ j ) \\ne 0 , j = 1 , 2 , \\ldots . \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} \\widetilde { E } ( n ) = \\frac { 1 } { 2 \\delta } \\int _ { - \\delta } ^ { \\delta } \\widetilde { E } _ \\eta ( n ) \\mathrm { d } \\eta , \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} \\Big \\| \\sup _ { n \\ge 1 } | \\sum _ { k = 1 } ^ n a _ k k ^ { i \\cdot } | \\ , \\Big \\| _ { L ^ 2 [ 0 , 1 ] } \\le C \\Big ( \\sum _ { n \\ge 1 } n | a _ n | ^ 2 \\Big ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} t c ( \\delta u ) e ^ { - t c ( \\delta u ) ^ 2 } = [ - \\frac { i } { 4 \\pi } \\Big ( c ^ 1 \\partial _ { v _ 1 } + c ^ 2 \\partial _ { v _ 2 } + \\sin \\phi \\ , c ^ 3 \\partial _ { v _ 3 } \\Big ) + t c ^ 4 e ^ y V { i ( \\kappa - \\Lambda ) } ] e ^ { - t c ( \\delta u ) ^ 2 } . \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} s = ( m - 1 ) + \\left ( 1 + \\frac 2 n \\right ) ^ { - ( N + 1 ) } ( q - ( m - 1 ) ) , \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} H & = - \\Delta _ x - V \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} s ( n + 1 ) = \\frac { c _ { i _ { r r } } ( n ) } { \\big ( b _ { i } + \\frac { \\delta } { 2 } \\big ) } s ( n ) \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} \\mathrm { d } \\eta ( S , T ) = g \\big ( S , \\varphi ( T ) \\big ) = 0 . \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} \\mathcal { A } _ n = \\mathcal { B } _ n + \\mathcal { C } _ n + \\mathcal { D } _ n , \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} & \\frac { ( 1 - \\epsilon ' ) ( n - n _ 0 ) } { 2 ( 1 + \\delta _ n ) k _ n } \\log \\left ( 1 + ( 1 + \\delta _ n ) k _ n P \\right ) - ( 1 - \\epsilon ) B _ 1 ( n ) \\\\ & \\geq _ n \\left [ ( 1 - \\epsilon ' ) ^ 2 ( 1 - n _ 0 / n ) - ( 1 - \\epsilon ) \\right ] B _ 1 ( n ) \\\\ & = \\left [ ( 1 - \\epsilon ' ) ^ 3 - ( 1 - \\epsilon ) \\right ] B ( n ) , \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\rho ( G _ n ) = \\frac { 1 2 \\sum _ { i \\to j \\in G } R _ \\ast ( i \\to j ) R ^ \\ast ( i \\to j ) - 3 L _ n ( L _ n + 1 ) ^ 2 } { L _ n ^ 3 - L _ n } . \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} A _ { \\varepsilon } ( G _ { \\delta } ( \\psi , \\eta , w ) ) = \\varepsilon v _ { \\varepsilon } ( \\theta _ 0 ( \\psi ) , y _ { \\delta } ( \\psi ) + L _ 1 ( \\psi ) \\eta + L _ 2 ( \\psi ) w ) + \\varepsilon ^ b ( z _ 0 ( \\psi ) + w ) \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{align*} \\{ n > 0 \\mid \\nu [ S ^ n ] = 0 \\pi _ 0 \\Gamma \\} \\cup \\{ \\infty \\} , \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} g ' ( i ) & = - n i ( R ^ { 2 } - i ^ { 2 } ) ^ { ( n - 2 ) / 2 } \\leq 0 , \\\\ g '' ( i ) & = n ( R ^ { 2 } - i ^ { 2 } ) ^ { ( n - 4 ) / 2 } ( ( n - 1 ) i ^ { 2 } - R ^ { 2 } ) \\leq 0 . \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} G _ 1 ( s ; a , z ) & : = F ( s ; a , z ) ( L ( s , \\chi _ 0 ) ) ^ { - \\frac { z } { \\phi ( q ) } } \\prod _ { \\chi \\neq \\chi _ 0 } \\ ( L ( s , \\chi ) \\ ) ^ { - \\frac { \\bar { \\chi } ( a ) z } { \\phi ( q ) } } \\\\ & = \\prod _ p \\ ( 1 + \\frac { z \\lambda _ a ( p ) } { p ^ s } \\ ) \\ ( 1 - \\frac { \\chi _ 0 ( p ) } { p ^ s } \\ ) ^ { \\frac { z } { \\phi ( q ) } } \\prod _ { \\chi \\neq \\chi _ 0 } \\ ( 1 - \\frac { \\chi ( p ) } { p ^ s } \\ ) ^ { \\frac { \\bar { \\chi } ( a ) z } { \\phi ( q ) } } \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} \\sum _ { n \\geq 0 } E _ { n , k } ( x ) \\frac { z ^ { n } } { n ! } \\ = \\ \\left ( \\frac { 1 - x } { e ^ { k z ( x - 1 ) } - x } \\right ) ^ { \\frac { 1 } { k } } . \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} \\sum _ { n \\le x } t _ Q ( n ) \\sigma ( n ) = \\frac { \\pi ^ 2 } { 1 2 } \\left ( 2 \\prod _ { p \\in Q } \\left ( 1 - \\frac 1 { p } \\right ) \\left ( 1 - \\frac 1 { p ^ 2 } \\right ) - 1 \\right ) x ^ 2 + O \\left ( x ( \\log x ) ^ { 2 / 3 } \\right ) . \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} \\nu ( \\mathcal { E } ) _ { \\mathbb { R } } = \\frac { [ A _ { t o r } ] [ C _ { t o r } ] } { [ B _ { t o r } ] } = [ \\mathrm { c o k } ( \\psi _ { t o r } ) ] . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} \\Sigma _ 1 \\cap \\Sigma _ 2 = \\emptyset , \\overline { \\Sigma _ 1 } \\cup \\overline { \\Sigma _ 2 } = \\R ^ N \\setminus \\Omega , \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} G _ { \\bar \\mu } ( \\eta ) _ U \\big ( G _ { \\bar \\mu } ( \\kappa ^ { 0 , V } ) _ U \\big ) ^ { - 1 } = \\big ( G _ { \\bar \\mu } ( \\kappa ^ { 0 , W } ) _ U \\big ) ^ { - 1 } G _ { \\bar \\mu } ( \\eta ) _ U . \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} ( [ D _ j , D _ { \\bar k } ] u ) ( t ) = \\mathbb H ^ t ( i _ t ^ * [ [ d ^ E , \\delta _ { V _ j } ] , [ d ^ E , \\delta _ { \\overline { V _ k } } ] ] \\mathbf { u } ) = \\mathbb H ^ t ( i _ t ^ * [ [ d ^ E , \\delta _ { V _ j } ] , [ d ^ E , \\delta _ { \\overline { V _ k } } ] ] \\mathbf { u } ) . \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{gather*} r _ 2 = \\dfrac { q _ 2 ^ 2 - 1 } { 4 } \\big ( e _ 1 ^ 2 - e _ 2 \\big ) - \\frac { 1 } { 2 } e _ 1 q _ 1 q _ 2 + \\frac { 1 } { 2 } q _ 2 q _ 0 + \\frac { 1 } { 4 } q _ 1 ^ 2 , \\\\ r _ 1 = \\dfrac { q _ 2 ^ 2 - 1 } { 4 } ( e _ 3 - e _ 2 e _ 1 ) + \\frac { 1 } { 2 } e _ 2 q _ 1 q _ 2 - \\frac { 1 } { 2 } q _ 1 q _ 0 , \\end{gather*}"}
-{"id": "3802.png", "formula": "\\begin{align*} V ( x , \\mu ) = \\R ^ d , \\mu x , \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} u ( { t , x } ) = \\sum _ { n = 0 } ^ { \\infty } I _ n ( f _ n ( \\cdot , t , x ) ) \\ , , \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} \\Phi _ 0 ( z ) = 1 \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} \\beta \\left ( \\left \\{ b \\in B \\mid \\ ; L _ n ( b ) \\leq \\alpha ' n \\right \\} \\right ) = \\mu ^ { * n } \\left ( \\left \\{ \\varphi \\in \\widehat { \\mathcal { I } } _ g ^ { ( q ) } \\mid \\ ; \\log \\Vert \\rho _ q ( \\varphi ) \\cdot e \\Vert \\leq \\alpha ' n \\right \\} \\right ) . \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( a , b , 2 a ; u ) = \\frac { 1 } { ( 1 - ( u / 2 ) ) ^ b } { } _ 2 F _ 1 \\left ( \\frac { b } { 2 } , \\frac { b + 1 } { 2 } , a + \\frac { 1 } { 2 } ; \\frac { u ^ 2 } { ( 2 - u ) ^ 2 } \\right ) , | \\arg ( 1 - u ) | < \\pi . \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} R ( v ^ { k - q } z ^ q ) = M [ \\underbrace { v , \\dots , v } _ { ( k - q ) \\ , \\ , t i m e s } , \\underbrace { z , \\dots , z } _ { q \\ , \\ , t i m e s } ] , M = k - \\mbox { l i n e a r } . \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = A ( t ) x + c ( t ) + \\lambda f _ 1 ( t , x , y , \\lambda ) , \\\\ \\dot y = \\lambda f _ 2 ( t , x , y , \\lambda ) , \\end{array} \\right . \\lambda \\geq 0 , \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} G ( C , m ) = G \\star _ C ( C \\times \\Z ^ m ) \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} \\bar { V } ( h ) & : = \\bar { V } _ h + [ H _ 0 ^ { 3 / 2 } ( \\Gamma ^ 0 ) ] ^ d , \\\\ \\bar { Q } ( h ) & : = \\bar { Q } _ h + H ^ { 1 / 2 } _ 0 ( \\Gamma ^ 0 ) , \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} { { \\left \\Vert \\eta \\right \\Vert _ { M _ { G } ^ { p } } : = \\left \\{ \\mathbb { E } [ { { \\int _ { 0 } ^ { T } | \\eta _ { t } | ^ { p } d t ] } } \\right \\} ^ { 1 / p } } } , \\ \\ \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{align*} o \\left ( \\dfrac { v } { 2 } \\right ) = \\allowbreak \\dfrac { 1 } { 2 } \\left ( s - v \\right ) p _ { 1 } \\allowbreak , o \\left ( \\dfrac { s } { 2 } \\right ) = - \\allowbreak \\dfrac { 1 } { 2 } s ^ { 2 } \\left ( s - v \\right ) \\left ( s ^ { 2 } + t ^ { 2 } \\right ) \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\mathcal { A } _ i \\triangleq & \\left \\lbrace ( r _ i , P _ i ^ { ( 1 ) } , \\ldots , P _ i ^ { ( M ) } ) : r _ i \\in \\mathcal { R } _ i , P _ i ^ { ( k ) } \\in \\{ p _ i ^ { ( 1 ) } , \\ldots , p _ i ^ { ( M ) } \\} , \\sum _ { j = 1 } ^ M \\alpha _ i ( j ) \\beta _ i ^ { ( j ) } P _ i ^ { ( j ) } \\leq \\overline { P } _ i \\right \\rbrace \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} P _ C ( x ) = x - \\sum _ { i \\in I _ m } { \\nu } _ i u _ i . \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} D _ { i } ^ { ( N ) } = x _ i \\frac { \\partial } { \\partial x _ i } + \\beta \\sum _ { j \\neq i } \\frac { x _ i } { x _ i - x _ j } \\left ( 1 - K _ { i j } \\right ) . \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\omega } \\hat { w } = g _ 3 + \\partial _ x K _ { 1 1 } ( \\varphi ) \\hat { \\eta } , \\mathcal { L } _ { \\omega } : = \\omega \\cdot \\partial _ { \\varphi } - \\partial _ x K _ { 0 2 } ( \\varphi ) . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} F ^ { s + c } \\left ( \\sum _ { i = 0 } ^ { r + s + c - 1 } V ^ i [ \\pi a _ i ] \\right ) = \\sum _ { i = 0 } ^ { c - 1 } p ^ i [ \\pi a _ i ] ^ { p ^ { s + c - i } } + p ^ c F ^ s \\left ( \\sum _ { i = c } ^ { r + s + c - 1 } V ^ { i - c } [ \\pi a _ i ] \\right ) = 0 . \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{gather*} P ( t ) = 1 2 \\left ( \\frac { u ^ 2 } { \\omega } - \\omega \\right ) _ t - 4 t , Q ( t ) = \\frac { 2 } { 3 } P _ t ( t ) + \\frac { 2 } { 3 } . \\end{gather*}"}
-{"id": "5076.png", "formula": "\\begin{align*} 1 , 1 , 2 , 1 , 3 , 2 , 3 , 1 , 4 , 3 , 5 , \\dots \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} F _ 1 ( x ) ( t ) = \\alpha [ x ] v ( t ) + \\beta [ x ] w ( t ) , t \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} | s - t | \\le \\min \\left \\{ \\left ( \\frac { 2 C _ 0 C _ 1 } { A _ 1 ^ 2 } \\right ) ^ { \\frac { 1 } { H } } , \\left ( \\frac { 2 C _ 0 C _ 1 } { ( A _ 2 ( 2 H + 1 ) T ^ { 2 H } ) ^ 2 } \\right ) ^ { \\frac { 1 } { 2 - 2 H } } \\right \\} = : c _ 8 . \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} \\phi _ { \\beta , n } = \\frac { e ^ { - \\beta H _ n } \\Omega } { \\sqrt { Z _ { \\beta , n } } } . \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} \\operatorname { w i d t h } ( a _ { \\rho } ) \\ = \\ \\Gamma _ { \\infty } ^ { > v ( b - K ) } \\quad v ( b - K ) \\ = \\ \\Gamma _ { \\infty } ^ { < \\operatorname { w i d t h } ( a _ { \\rho } ) } \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { h _ f ( n ) } { n ^ s } = \\frac { 2 } { S _ f ( 1 / 2 ^ s ) } - 1 , \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} [ V : D ( m , n ) ] _ q : = 0 , \\mbox { i f $ n < 0 $ } . \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} \\rho \\ , u = U + \\gamma _ 3 \\beta ( U ) \\ , \\beta ^ { \\sharp \\top } . \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} f ( x ; \\rho G ) = \\rho \\int _ \\Theta f ( x ; \\theta ) d G ( \\theta ) . \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} & \\sum _ { j \\geq 0 } V _ { 1 , j } = 1 , \\\\ V _ { n , k } = \\sum _ { j = 0 } ^ { \\infty } \\binom { k + j } { j } & \\left ( \\left ( b \\right ) _ { n \\uparrow \\tau } \\right ) ^ { j } \\left ( a + b + \\tau \\left ( n - 1 \\right ) \\right ) ^ { k } V _ { n + 1 , k + j } . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} H ^ q _ { t , b , m } ( X ) = 0 , \\ : \\ : m > m _ 0 , \\ : \\ : | t | < \\delta _ 0 , \\ : \\ : q \\geq 1 . \\end{align*}"}
-{"id": "8088.png", "formula": "\\begin{align*} \\xi _ { i j } ( \\lambda ) = \\delta _ { i j } + \\sum _ k \\int _ { \\lambda _ { n + 1 - i L } } ^ { \\lambda _ { i R } } \\frac { d \\nu } { 2 \\pi } K ( \\lambda - \\nu ) \\xi _ { k j } ( \\nu ) \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} f ( x ) : = \\left ( \\begin{array} { l } a _ 1 x _ 1 + b _ 1 \\theta ( x _ 1 , x _ 2 ) + c _ 1 \\theta ( x _ 1 , x _ 4 ) + d _ 1 \\theta ( x _ 2 , x _ 3 ) \\\\ a _ 2 x _ 2 + b _ 2 \\theta ( x _ 1 , x _ 2 ) + c _ 2 \\theta ( x _ 1 , x _ 4 ) + d _ 2 \\theta ( x _ 2 , x _ 3 ) \\\\ a _ 3 x _ 3 + b _ 3 \\theta ( x _ 3 , x _ 4 ) + c _ 3 \\theta ( x _ 1 , x _ 4 ) + d _ 3 \\theta ( x _ 2 , x _ 3 ) \\\\ a _ 4 x _ 4 + b _ 4 \\theta ( x _ 3 , x _ 4 ) + c _ 4 \\theta ( x _ 1 , x _ 4 ) + d _ 4 \\theta ( x _ 2 , x _ 3 ) \\end{array} \\right ) \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} W \\eta ( k , x ) & = \\sqrt { \\frac { k } { N } } ( A \\delta _ { a ^ { N - k } \\beta ( x ) } + B \\delta _ { b ^ { N - k } \\alpha \\beta ( x ) } ) + \\sqrt { \\frac { N - k } { N } } ( A \\delta _ { a ^ { - k } \\beta ( x ) } + B \\delta _ { b ^ { - k } \\alpha \\beta ( x ) } ) \\\\ & = \\sqrt { \\frac { N - k } { N } } ( A \\delta _ { a ^ { ( N - k ) - N } \\beta ( x ) } + B \\delta _ { b ^ { - k } \\alpha \\beta ( x ) } ) \\\\ & + \\sqrt { \\frac { N - ( N - k ) } { N } } ( A \\delta _ { a ^ { N - k } \\beta ( x ) } + B \\delta _ { b ^ { N - k } \\alpha \\beta ( x ) } ) \\\\ & = \\eta ( N - k , \\beta ( x ) ) \\end{align*}"}
-{"id": "7977.png", "formula": "\\begin{align*} \\lambda _ 0 ( \\theta ) = \\lambda _ 0 ( - \\theta ) \\ , . \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; J _ 2 } = D _ i \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } - \\left ( S _ { J _ 2 } D _ i \\xi ^ j \\right ) u _ { J _ 1 + \\bold { 1 } _ j ; J _ 2 } ^ { \\alpha } , \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} \\ker \\big ( Q ' ( u ^ \\pm ) + s A \\big ) = \\{ 0 \\} \\quad \\mbox { f o r a n y } s \\in ( - \\eta ^ \\pm _ 1 , \\eta ^ \\pm _ 1 ) \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rm M D } ( { \\mathbf u } ) & = P ( { \\mathbf X } \\in R _ { H _ 0 } ( n ) | \\boldsymbol { \\sigma } _ { \\rm j } ^ 2 + \\boldsymbol { \\sigma } _ { \\rm w } ^ 2 = { \\mathbf u } , H _ 1 ) \\\\ & > 1 - \\frac { \\epsilon } { 2 } \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} \\frac { d X _ t ( s ) } { d t } = a ( X _ t ( s ) ) W + b ( X _ t ( s ) ) + \\frac { k } { 2 } \\frac { \\partial ^ 2 } { \\partial s ^ 2 } X _ t ( s ) , \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big ( Q _ H ( f , e ^ { i t A } f - f ) + Q _ H ( e ^ { i t A } f - f , f ) \\Big ) \\Big | _ { t = 0 } = Q _ { [ H , i A ] } ( f , f ) , \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} e u \\sigma ( e ) & = e ( \\lambda \\sigma ( e ) + u - \\lambda ) = \\lambda e \\sigma ( e ) + e u - \\lambda e = \\lambda e \\sigma ( e ) + \\lambda + u - 2 \\lambda e . \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} f ( x ; \\theta _ 1 , \\theta _ 2 ) = \\theta _ 1 \\exp ( - \\theta _ 1 x ) \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} \\Lambda _ Q = \\sum _ { i = 1 } ^ n \\frac { \\partial } { \\partial q ^ i } \\wedge \\frac { \\partial } { \\partial p _ i } \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} s \\ge 0 , \\ ; 0 \\le n \\le m \\implies \\nu ( s , n ) = \\begin{cases} 1 & s + n s - n m , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\chi _ 2 = \\max \\left ( P , \\left ( \\frac { \\sqrt { P } + \\sqrt { \\frac { \\pi } { 2 } } + \\sqrt { \\left ( \\sqrt { P } + \\sqrt { \\frac { \\pi } { 2 } } \\right ) ^ 2 - 2 } } { 2 } \\right ) ^ 2 \\right ) ; \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} f _ i ( x ) = \\sum _ { ( r , \\sigma ) \\in \\Gamma _ i } c _ { i r \\sigma } M _ { r \\sigma } ( x ) , \\end{align*}"}
-{"id": "7374.png", "formula": "\\begin{align*} \\mu _ a = \\frac { \\lambda _ a } { \\ell } + \\frac { \\vartheta _ a } { 2 r } + O ( \\frac { 1 } { r ^ 2 } ) . \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} \\omega _ { \\delta } ( x ) = \\frac { 1 } { \\delta } \\omega \\biggl ( \\frac { x } { \\delta } \\biggr ) , \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} \\begin{aligned} f ( x , u , u _ x ) : = \\ , & c _ 1 \\ , u _ x ^ 3 + c _ 2 \\ , u _ x ^ 2 \\ , u + c _ 3 \\ , u ^ 3 + c _ 4 \\ , u _ x ^ 4 + c _ 5 \\ , u _ x ^ 3 \\ , u + c _ 6 \\ , u _ x ^ 2 \\ , u ^ 2 + c _ 7 \\ , u ^ 4 + f _ { \\geq 5 } ( x , u , u _ x ) , \\end{aligned} \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{gather*} \\frac { d } { d t } \\log F _ { 6 } \\big ( 3 ^ { - 2 / 3 } t \\big ) = \\frac { 1 } { 1 2 } t ^ 2 - \\frac { \\sqrt { 2 } } { 3 } ( - t ) ^ { 1 / 2 } + \\frac { 1 } { 2 4 \\ , t } + O \\big ( | t | ^ { - \\frac { 5 } { 2 } } \\big ) \\mbox { a s \\ \\ $ t \\to - \\infty $ } , \\end{gather*}"}
-{"id": "6113.png", "formula": "\\begin{align*} \\frac { d } { d t } F ( \\phi + t v ) = \\int _ M v \\mu + \\int _ { M * } \\frac { d } { d t } ( \\phi + t v ) ^ * d \\nu . \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} Q _ 3 = \\frac { 2 / 3 } { r ^ 2 } \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} \\begin{aligned} S _ k : n \\mapsto n + \\bold { 1 } _ k , \\end{aligned} \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} E = \\sum _ e \\mathop { \\sum \\sum } _ { ( u , v ) = 1 } \\frac { \\rho ( e u ) \\rho ( e v ) } { e u v } g ( e u ) g ( e v ) \\lambda ( u v ) \\xi ( u v ) K ( u / v ) , \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} P _ { \\theta , n } & = \\left [ P _ { \\theta , n - 1 } \\ , p _ \\theta ( 2 , 2 ) + \\widetilde { P } _ { \\theta , n - 1 } \\ , p _ \\theta ( 1 , 2 ) \\right ] \\ , f _ { \\theta } ^ { ( 2 ) } ( Y _ n ) ; \\\\ \\widetilde { P } _ { \\theta , n } & = \\left [ P _ { \\theta , n - 1 } \\ , p _ \\theta ( 2 , 1 ) + \\widetilde { P } _ { \\theta , n - 1 } \\ , p _ \\theta ( 1 , 1 ) \\right ] \\ , f _ { \\theta } ^ { ( 1 ) } ( Y _ n ) \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} { u } _ i ( r ) = \\int { u } ( r , y ) \\phi _ i ( y ) \\ , d y = \\lambda _ i ^ { - \\frac { k + 2 } { 2 } } \\int ( ( - \\Delta _ y ) ^ { \\frac { k + 2 } { 2 } } { u } ( r , y ) ) \\phi _ i ( y ) \\ , d y \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} x ( 0 ) = \\int _ 0 ^ 1 A ( s ) \\textup d x ( s ) x ( 1 ) = \\int _ 0 ^ 1 B ( s ) \\textup d x ( s ) . \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} T & = T _ { a n , b n } \\sqcup \\left ( \\bigsqcup _ { k = 0 } ^ { a n - 1 } T _ { k , b n } \\right ) \\sqcup \\left ( \\bigsqcup _ { l = 0 } ^ { b n - 1 } T _ { a n , l } \\right ) \\\\ T & = \\overline { T _ { a n , b n } } \\supsetneq \\overline { T _ { a n - 1 , b n } } \\supsetneq \\dotsb \\supsetneq \\overline { T _ { 0 , b n } } = T _ { 0 , b n } \\\\ T & = \\overline { T _ { a n , b n } } \\supsetneq \\overline { T _ { a n , b n - 1 } } \\supsetneq \\dotsb \\supsetneq \\overline { T _ { a n , 0 } } = T _ { a n , 0 } \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} v \\left ( \\frac { P ( a _ { \\rho } ) } { Q ( a _ { \\rho } ) } - b \\right ) \\ = \\ v ( P ( a _ { \\rho } ) - b Q ( a _ { \\rho } ) ) - v ( Q ( a _ { \\rho } ) ) \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} A _ 0 ( s ) = A _ 1 ' ( s ) + A _ 1 ( s ) ^ 2 + a ( s ) \\ 1 \\ , , \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} [ ( x , y ) ] = \\frac { 1 } { 2 } [ ( x , y ) + ( y , x ) ] + \\frac { 1 } { 2 i } i [ ( x , y ) - ( y , x ) ] . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} J ( z , g ) : = j _ 1 ( z ) + j _ 2 ( z ( T ) ) + \\frac 1 2 \\| g \\| _ { H ^ 1 ( I ; L ^ 2 ( \\Omega ) ) } ^ 2 , \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} B = \\begin{pmatrix} \\mu _ { 1 1 } & \\mu _ { 1 2 } \\\\ \\mu _ { 2 1 } & \\mu _ { 2 2 } \\end{pmatrix} , \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} & ~ \\inf _ { x } ( \\max \\{ \\psi _ 1 ( x ) , \\psi _ 2 ( x ) \\} + ( \\phi ( x ) - X ( x , x ) ) ) \\\\ = & ~ \\phi \\circ ( \\psi _ 1 \\wedge \\psi _ 2 ) \\\\ = & ~ \\max \\{ \\phi \\circ \\psi _ 1 , \\phi \\circ \\psi _ 2 \\} \\\\ = & ~ \\max \\Big \\{ \\inf _ { x } ( \\psi _ 1 ( x ) + ( \\phi ( x ) - X ( x , x ) ) ) , \\inf _ { x } ( \\psi _ 2 ( x ) + ( \\phi ( x ) - X ( x , x ) ) ) \\Big \\} \\\\ = & ~ \\max \\{ \\lambda _ 1 + k + \\phi ( x _ 1 ) , \\lambda _ 2 + k + \\phi ( x _ 2 ) \\} \\\\ < & ~ \\delta _ 1 + \\delta _ 2 + k . \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 = 0 \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{align*} C _ { N + n } ^ * ( z ) = \\prod _ { j = 1 } ^ { k } \\left ( 1 - \\frac { z } { \\lambda _ { j , n } } \\right ) , \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} i _ { l , \\mathsf { k } } ( v _ \\mathsf { k } ) = \\mu ( \\norm { v _ \\mathsf { k } } ) \\mathrm { R } \\big ( \\delta _ \\mathsf { k } ( \\norm { v _ \\mathsf { k } } ) \\big ) v _ \\mathsf { k } . \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} & \\left [ F _ { \\Omega } ( x ) \\right ] ^ { N - 1 } = \\\\ & \\left [ \\frac { ( - 1 ) ^ { M } ( M - 1 ) ! } { \\mathcal { B } ( M , K ) \\prod _ { m = 1 } ^ { M } ( m - M - K ) } \\right ] ^ { N - 1 } \\sum _ { k = N - 1 } ^ { ( N - 1 ) ( M + K - 1 ) } q _ { k - ( N - 1 ) } \\frac { x ^ k } { ( 1 + x ) ^ { ( N - 1 ) ( M + K - 1 ) } } , \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} \\delta _ { \\ell } = k _ { \\ell } ^ { - \\frac { 1 } { 3 } } . \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{gather*} \\min ( | p _ { Y 1 } | , | p _ { Y 2 } | ) \\le | q | = 2 | w | + 7 . \\end{gather*}"}
-{"id": "3485.png", "formula": "\\begin{align*} \\log \\frac { f ( z ) } { z } = 2 \\sum _ { n = 1 } ^ { \\infty } \\gamma _ n z ^ n \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} \\mathbf X _ { ( \\chi _ i , [ g ] ) } = \\chi _ i ( g ) \\cdot \\frac { | [ g ] | } { | T h | } . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} \\Omega ( k _ 0 ) : = \\left \\{ \\mathbf { x } \\in \\ell _ + ^ n : 0 \\leq \\textstyle \\sum _ i x _ t ^ i \\leq f ( k _ t ) \\ ! - \\ ! k _ { t + 1 } , \\ t = 0 , 1 , \\ldots ; \\ \\ \\mathbf { k } \\in \\Pi ( k _ 0 ) \\right \\} . \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} H C = - C H , H T = T H \\mbox { a n d } H \\left ( C T \\right ) = - \\left ( C T \\right ) H \\ , . \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} u _ n ( t ) : = S ( t ) \\mu + \\int _ 0 ^ t S ( t - s ) u _ { n - 1 } ( s ) ^ p \\ , d s . \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} \\varphi ( - q ) ^ 2 - \\varphi ( - q ) = \\sum _ { N \\geq 1 } ( - 1 ) ^ N ( r _ 2 ( N ) - 2 \\chi ( N = \\square ) \\ , ) q ^ N , \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} ( f ( X ) - f ^ * ( X ) ) ( f ^ * ( X ) - Y ) = & ( f ( X ) - f ^ * ( X ) ) \\left ( \\left ( f ^ * ( X ) - f ( X ) \\right ) + \\left ( f ( X ) - Y \\right ) \\right ) \\\\ = & - ( f ( X ) - f ^ * ( X ) ) ^ 2 - \\Psi _ { f ^ * , f } ~ . \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} A u _ \\tau = Q ( u ) . \\end{align*}"}
-{"id": "550.png", "formula": "\\begin{align*} \\widehat { R } _ k ( x , n , [ u ] ) = \\int _ 0 ^ 1 R _ k ( \\varepsilon ; x , n , [ u ] ) \\operatorname { d } \\ ! \\varepsilon . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} \\sum _ r c _ { n , r } ( \\mu ) d _ r ^ { - 1 } & = \\sum _ { t _ 1 , \\ldots , t _ n } \\mu ( t _ 1 ) \\cdots \\mu ( t _ n ) \\Big ( \\sum _ r m _ { t _ 1 , \\ldots , t _ n } ^ r \\Big ) \\frac { 1 } { d _ { t _ 1 } \\cdots d _ { t _ n } } \\\\ & \\leq \\sum _ { t _ 1 , \\ldots , t _ n } \\mu ( t _ 1 ) \\cdots \\mu ( t _ n ) \\frac { \\dim ( U _ { t _ 1 } ) \\cdots \\dim ( U _ { t _ n } ) } { d _ { t _ 1 } \\cdots d _ { t _ n } } \\\\ & = \\Big ( \\sum _ t \\mu ( t ) \\ , \\frac { \\dim ( U _ t ) } { d _ t } \\Big ) ^ n . \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} \\beta = \\frac { 2 f } { \\lambda _ { 1 , \\sigma } } \\sqrt { \\frac { \\log n } { m } } , \\quad \\ ; \\ , m = \\left ( \\frac { 3 f } { \\lambda _ { 1 , \\sigma } } \\right ) ^ 2 s \\log n , \\ ; \\ , f \\in \\{ 0 . 5 , 1 , 1 . 5 , \\ldots , 4 \\} , \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} m _ K ( I ) = d ^ * _ G ( A ) \\leq d _ G ^ * ( A B ) \\leq m _ K ( I J ) = \\frac { 1 } { 2 } < d ^ * _ G ( A ) + d ^ * _ G ( B ) < 1 , \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} \\gamma _ { p , q } ( a - c , b - c ) = 2 ^ { 1 / p } \\left ( 1 + \\frac { 2 ^ { - 1 / q } } { 1 - 2 ^ { 1 - 1 / q } } \\right ) , \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} f _ + ( r ) = - 4 \\sin ( \\pi r ^ 2 / 2 ) ^ 2 \\int _ 0 ^ { i \\infty } \\psi _ + ( - 1 / z ) z ^ { n / 2 - 2 } e ^ { \\pi i r ^ 2 z } \\ , d z \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{gather*} I [ \\phi ] = \\int _ C ( j ^ { \\infty } \\phi ) ^ \\ast ( \\alpha ) . \\end{gather*}"}
-{"id": "4976.png", "formula": "\\begin{align*} \\left | \\frac { \\partial \\tilde { \\varphi } } { \\partial t } ( x , t ) \\right | & = \\left | \\frac { \\partial \\tilde { \\varphi } } { \\partial t } ( x , t ) - \\frac { \\partial \\tilde { \\varphi } } { \\partial t } ( y , t ) \\right | \\\\ & = | u ( x , t ) - u ( y , t ) | \\\\ & \\leq \\theta ( t ) \\\\ & \\leq C e ^ { - \\eta t } , \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{gather*} I _ 0 ( t ) = 2 r _ 0 + U - e _ 1 q _ 2 + 2 q _ 1 . \\end{gather*}"}
-{"id": "1388.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\int _ { { \\bf R } ^ N } u ( y , s _ j ) \\eta ( y ) \\ , d y = \\int _ { { \\bf R } ^ N } \\eta ( y ) \\ , d \\mu ' ( y ) \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} \\sum _ { A \\in \\mathfrak S ( m + n , m ) } | q | ^ { \\operatorname { i n v } ( A ) } = { m + n \\choose m } _ { | q | } . \\end{align*}"}
-{"id": "7243.png", "formula": "\\begin{align*} \\C _ { \\beta _ 1 - , \\beta _ 2 - } : = \\bigcap _ { 0 < \\alpha _ 1 < \\beta _ 1 } \\bigcap _ { 0 < \\alpha _ 2 < \\beta _ 2 } \\C _ { \\alpha _ 1 , \\alpha _ 2 } , \\C _ { \\beta _ 1 + , \\beta _ 2 + } : = \\bigcap _ { \\beta _ 1 < \\alpha _ 1 \\le 1 } \\bigcap _ { \\beta _ 2 < \\alpha _ 2 \\le 1 } \\C _ { \\alpha _ 1 , \\alpha _ 2 } . \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} [ ( L _ b - z ) S _ b ( \\cdot , x ' ; z ) ] ( x ) = \\delta ( x - x ' ) + T _ b ( x , x ' ; z ) \\ , , \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} f = \\bar { f } + \\sum _ { \\ell = 1 } ^ L f _ \\ell \\chi _ \\ell , \\ ; \\ , \\bar { f } \\in C ^ { k , \\alpha } _ \\nu ( M ) , \\ ; \\ , f _ \\ell \\in \\R , \\ ; \\ , \\int _ M f = 0 . \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} v _ 1 ' - v _ { - 1 } ' = \\exp ( v _ 2 - v ) - \\exp ( v - v _ { - 2 } ) , \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\| ( \\tilde { u } _ k , \\tilde { v } _ k ) \\| ^ p & = \\frac { p ( \\alpha + \\beta - q ) } { q ( \\alpha + \\beta - p ) } \\| ( u _ k , v _ k ) \\| ^ { q - p } \\int _ \\Omega ( \\lambda | \\tilde { u } _ k | ^ q + \\mu | \\tilde { v } _ k | ^ q ) d x + o _ k ( 1 ) \\\\ & = \\frac { p ( \\alpha + \\beta - q ) } { q ( \\alpha + \\beta - p ) } \\| ( u _ k , v _ k ) \\| ^ { q - p } \\int _ \\Omega ( \\lambda | \\tilde { u } | ^ q + \\mu | \\tilde { v } | ^ q ) d x + o _ k ( 1 ) . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} \\begin{pmatrix} I _ r & 0 & * \\\\ 0 & I _ s & * \\end{pmatrix} . \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} u ^ p + \\ell ( - v ^ 2 ) ^ p = w ^ 2 . \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} \\langle m _ { 1 } , m _ { 2 } \\rangle _ { B _ { 1 } } = \\langle \\phi ( m _ { 1 } ) , \\phi ( m _ { 2 } ) \\rangle _ { B _ { 2 } } . \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\sum | a _ p - a _ { p - 1 } | = a _ 0 + \\sum _ { 1 \\leq p \\leq k - 1 } | a _ p - a _ { p - 1 } | + a _ { k - 1 } \\geq \\max \\{ a _ 0 , \\cdots , a _ { k - 1 } \\} \\geq \\frac 1 k \\sum a _ p . \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} \\phi _ n ^ { ( \\epsilon ) } = \\begin{cases} \\sum _ { k = 0 } ^ n v _ k x ^ k & , \\\\ \\epsilon \\sum _ { k = 0 } ^ n ( - 1 ) ^ k v _ k x ^ k & . \\end{cases} \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} h ( p ^ 6 D ) & = h ( p ^ 4 D ) + p \\cdot \\big ( h ( D ) - h ( D / p ^ 2 ) \\big ) \\\\ \\hat h ( p ^ 6 D ) & = \\hat h ( p ^ 4 D ) + p \\cdot \\big ( \\hat h ( D ) - \\hat h ( D / p ^ 2 ) \\big ) , \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} a ( x , y , z ) = \\frac { | C _ G ( z ) | } { | C _ G ( y ) | } a ( z ^ { - 1 } , y , x ^ { - 1 } ) = \\frac { | C _ G ( z ) | } { | C _ G ( y ) | } a ( x ^ { - 1 } , z , y ) . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} | \\tau _ b ( x _ 0 ) ( z ) - h ( z ) | & \\geq | \\tau _ b ( x _ 0 ) ( z ) - \\tau _ b ( x _ n ) ( z ) | - | \\tau _ b ( x _ n ) ( z ) - h ( z ) | \\\\ & \\geq | d ( z , x _ 0 ) - d ( b , x _ 0 ) - d ( z , x _ n ) + d ( b , x _ n ) | - \\epsilon \\\\ & = | 1 - d ( z , x _ n ) + d ( b , x _ n ) | - \\epsilon \\\\ & \\geq 1 - d ( z , x _ n ) + d ( b , x _ n ) - \\epsilon . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} \\mathrm { R } ( \\theta ) \\ ! \\coloneqq \\ ! \\ ! \\begin{bmatrix} \\cos ( \\theta ) \\ ! \\ ! & - \\sin ( \\theta ) \\\\ \\sin ( \\theta ) \\ ! \\ ! & \\cos ( \\theta ) \\end{bmatrix} \\ ! , \\ ; \\ ; j \\ ! \\coloneqq \\ ! \\mathrm { R } ( \\pi / 2 ) , \\ ; \\ ; \\mathrm { r } ( \\theta ) \\ ! \\coloneqq \\ ! \\ ! \\begin{bmatrix} \\cos ( \\theta ) \\\\ \\sin ( \\theta ) \\end{bmatrix} \\ ! . \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} S _ { j , n } ( Q , S ) \\subset \\cup _ { k = 1 } ^ N B ( x _ k , r _ k + \\eta ) \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n 2 q ^ n } { 1 + q ^ n } \\frac { ( - q ; q ) _ { n - 1 } } { ( q ; q ) _ { n - 1 } } = 2 \\frac { ( q ; q ) _ \\infty } { ( - q ; q ) _ \\infty } \\left ( - 1 + \\sum _ { n \\geq 0 } ( - 1 ) ^ n q ^ { n ^ 2 } \\right ) . \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} & 4 ( s - 1 ) P = \\frac { 4 F _ { s - 1 } V _ s ( \\rho ) } { ( 1 - | x | ^ 2 ) | x - y | ^ 2 } \\Big [ ( s - 1 ) ( 1 - 2 x \\cdot y + | x | ^ 2 | y | ^ 2 ) ( 1 - | x | ^ 2 ) \\\\ & \\qquad \\qquad + ( 1 - | y | ^ 2 ) \\Big ( - \\frac { N } { 2 } ( | y | ^ 2 - 2 x \\cdot y + 1 ) ( 1 - | x | ^ 2 ) + ( s - 1 ) | x - y | ^ 2 \\\\ & \\qquad \\qquad - ( 2 s - 2 - N ) ( 1 - x \\cdot y ) ( 1 - | x | ^ 2 ) + ( s - 1 - \\frac { N } { 2 } ) ( 1 - | y | ^ 2 ) ( 1 - | x | ^ 2 ) \\Big ) \\Big ] . \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} f ( k , \\ell ) = 2 1 0 k ^ 2 \\log k + 1 0 \\ell ( k - 1 ) . \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\mathcal { B } ( x , y ) = { \\begin{cases} 1 & { } x < y , \\ \\ x , y \\in \\mathbb { N } \\cup \\{ 0 \\} \\\\ 1 / 2 & { } x = y \\ne 0 \\ x , y \\in \\mathbb { N } \\cup \\{ 0 \\} \\\\ 0 & { } x > y \\ { } x = y = 0 , x , y \\in \\mathbb { N } \\cup \\{ 0 \\} \\end{cases} } \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} \\widehat { F _ U } ( t ) = U \\widehat F ( U t ) = 1 2 0 U \\frac { \\big ( 2 \\sin \\big ( \\frac { U t } 2 \\big ) - U t \\cos \\big ( \\frac { U t } 2 \\big ) \\big ) ^ 2 } { ( U t ) ^ 6 } . \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} f _ { i } ( g ) = \\begin{cases} 0 & g \\notin \\mathfrak { t } \\\\ 0 & g \\in \\mathfrak { t } , \\ , \\ g \\in < h _ { j } > _ { j \\neq i } \\\\ c & g = c h _ i . \\end{cases} \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\begin{aligned} \\frac { 1 } { 2 } ( [ \\mathfrak { B } _ 1 , \\overline { A } _ 1 ] ) _ j ^ j ( 0 ) & = \\frac { 4 } { 3 } c _ 2 ^ 2 \\ , \\mathrm { i } \\ , \\sum _ { j _ 2 \\in S , j _ 2 + j \\in S ^ c } j _ 2 ^ 3 \\ , \\lvert j _ 2 \\rvert \\xi _ { j _ 2 } + 8 c _ 2 c _ 3 \\ , \\mathrm { i } \\sum _ { j _ 2 \\in S , j _ 2 + j \\in S ^ c } j _ 2 \\ , \\lvert j _ 2 \\rvert \\xi _ { j _ 2 } \\\\ & + 1 2 \\ , c _ 3 ^ 2 \\mathrm { i } \\sum _ { j _ 2 \\in S , j _ 2 + j \\in S ^ c } j _ 2 ^ { - 1 } \\ , \\lvert j _ 2 \\rvert \\ , \\xi _ { j _ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ R ^ a , S ^ a , V _ i , W _ i , J R ^ a , J S ^ a \\} ; \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} \\langle F , G \\rangle _ T = \\int _ { - T } ^ { T } \\left ( \\int _ { X } F ( t ) \\overline { G ( t ) } d \\mu \\right ) d t . \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { \\left \\lfloor { \\frac { y ^ { 1 / 2 } } { a } } \\right \\rfloor } ( y - a ^ { 2 } i ^ { 2 } ) ^ { n / 2 } \\leq \\frac { \\sqrt { \\pi } } { 2 a } \\frac { \\Gamma \\left ( \\frac { n + 2 } { 2 } \\right ) } { \\Gamma \\left ( \\frac { n + 3 } { 2 } \\right ) } y ^ { ( n + 1 ) / 2 } - \\frac 1 2 y ^ { n / 2 } + \\frac { ( 2 a n ) ^ { n / 2 } } { ( n + 2 ) ^ { ( n + 2 ) / 2 } } y ^ { n / 4 } . \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{gather*} I _ 0 = 0 . \\end{gather*}"}
-{"id": "9456.png", "formula": "\\begin{align*} A _ 0 & = 0 ; \\\\ A _ { j + 1 } & = \\inf \\{ t \\geq A _ j + \\eta : X _ t = n \\} , \\mbox { i f } j < m ; \\\\ A _ { m + 1 } & = t ^ * . \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ { n } \\binom { n } { k } H _ k = 2 ^ n \\left ( H _ n - \\sum _ { k = 1 } ^ { n } \\frac { 1 } { k 2 ^ k } \\right ) \\ ; , \\\\ & \\sum _ { k = 0 } ^ { n } \\binom { n } { k } ^ 2 H _ k = \\binom { 2 n } { n } ( 2 H _ n - H _ { 2 n } ) \\ ; . \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} C _ { r , m } ( p ^ { 1 + s } , p ^ { - 1 } ) = p ^ { - r ^ 2 \\binom { m } { 2 } + r ( m - 1 ) + s r ( m - 1 ) } C _ { r , m } ( p ^ { - 1 - s } , p ) . \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} u _ i = v _ i + v ' _ i \\in \\Phi ( A , \\sigma ) { \\rm a n d } w _ i = ( 1 + \\eta ) v _ i + \\eta v ' _ i \\in \\Phi ( A , \\sigma ) , \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} \\sum ^ r _ { i = 0 } R ( i ) P _ { h n q } ( \\varepsilon _ i ) \\bigl ( x ^ { 1 / 3 } / c \\bigr ) ^ { \\varepsilon _ i } \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} \\left ( \\frac { z } { z - \\beta } \\right ) ^ { c } & = e ^ { N \\phi _ { A } ( z ) } \\left ( \\frac { z - \\beta } { z - a } \\right ) ^ { c } \\frac { \\sqrt { 2 \\pi } ( a ^ 2 - 1 ) ^ c } { a \\Gamma ( c ) N ^ { \\frac { 1 } { 2 } - c } ( z - \\beta ) } , & a > 1 , \\\\ \\left ( \\frac { z } { z - a } \\right ) ^ { c } & = e ^ { N \\phi _ { A } ( z ) } \\frac { a ( 1 - a ^ 2 ) ^ { c - 1 } } { N ^ { 1 - c } \\Gamma ( c ) ( z - a ) } , & a < 1 . \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} E \\left \\{ \\bar { \\mathbf { d } } _ i \\bar { \\mathbf { d } } ^ { \\rm H } _ i \\right \\} = \\sum \\limits ^ { i } _ { i _ 1 = 0 } { \\hat { \\mathbf { P } } ^ { i _ 1 } \\left ( \\hat { \\mathbf { P } } ^ { \\rm H } \\right ) ^ { i _ 1 } } - \\sum \\limits ^ { i - 1 } _ { i _ 1 = 0 } { \\hat { \\mathbf { P } } ^ { i _ 1 } \\tilde { \\mathbf { P } } \\left ( \\hat { \\mathbf { P } } ^ { \\rm H } \\right ) ^ { i _ 1 } } \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} \\begin{array} { r c c l } \\Gamma _ { M , N } : & [ M , m _ 1 ; N , n _ 1 ] & \\longrightarrow & \\hom ( \\pi _ 1 ( M , m _ 1 ) , \\pi _ 1 ( N , n _ 1 ) ) \\\\ & \\alpha = [ f ] & \\longmapsto & \\alpha _ \\# : = f _ \\# . \\end{array} \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} x _ { 1 } = \\frac { \\lambda - \\delta ( x _ 0 ) } { 1 + \\delta ( x _ 0 ) } x _ 0 \\quad x _ { n + 1 } = \\frac { 1 + \\lambda } { 1 + \\delta ( x _ n ) } x _ n \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} \\| P _ j [ f g ] \\| _ { L ^ 2 } = \\| P _ j [ f g _ { \\leq j + 1 0 } ] \\| _ { L ^ 2 } + \\| P _ j [ f g _ { > j + 1 0 } ] \\| _ { L ^ 2 } . \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} _ { n } ( \\lambda ) = \\lambda \\alpha _ { n } - \\beta _ { n } / \\lambda , _ { n } ( \\lambda ) = \\gamma _ { n } / \\lambda - \\lambda \\delta _ { n } , \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} \\Omega ( \\xi , \\eta , \\zeta ) = 4 J ( \\eta ) \\xi + ( \\eta + J ( \\eta ) ) ^ 2 + S ( e ^ { \\xi } ) = - 4 J ( \\eta ) \\zeta + ( \\eta - J ( \\eta ) ) ^ 2 + S ( e ^ { - \\zeta } ) . \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} \\mathcal { D } _ { v _ 0 } = \\mathcal { D } _ { v _ 0 , \\phi } = \\{ \\phi ( \\theta \\cdot x - t ) , \\ ; x \\in B : \\| \\theta \\| _ 1 \\leq v _ 0 , \\ ; t \\in \\mathbb { R } \\} \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} f ( z ) = z + a _ 2 z ^ 2 + \\cdots , \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} \\mathcal { P } _ 1 ^ { \\mathrm { U L A } } \\left ( { x } \\right ) = \\exp \\left \\{ - 2 \\pi { \\lambda _ e } \\int _ 0 ^ \\infty \\int _ 0 ^ { 2 \\pi } { \\rm { \\mathbf { 1 } } } \\left ( { \\max \\{ { r _ e } , d \\} } < \\big ( \\frac { { P _ t } { G _ e ( \\varphi _ { t _ { e , o } } ) } \\beta } { x \\sigma _ e ^ 2 } \\big ) ^ { \\frac { 1 } { { \\alpha _ { \\mathrm { L o S } } } } } \\right ) \\frac { { { f _ { \\Pr } } ( { r _ e } ) } } { 2 \\pi } { r _ e } d { \\varphi _ { t _ { e , o } } } d { r _ e } \\right \\} \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} [ A , B ] = 0 , [ A ' , B ' ] = 0 , A F - F A ' = 0 , B F - F B ' = 0 , \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{align*} \\dot z ( t ) = A _ { - 1 } \\dot z ( t - 1 ) + L z _ t ( \\cdot ) + B u ( t ) , \\end{align*}"}
-{"id": "4066.png", "formula": "\\begin{align*} \\varphi _ \\beta ( r ) = \\int _ { S _ \\beta } \\varphi _ \\beta ^ 2 ( r s ) \\nu _ { \\beta } ( d s ) , r \\ge 0 , \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{gather*} \\alpha = \\dfrac { 1 + q _ 2 } { 2 ( 1 - q _ 2 ) } ( q _ 1 + e _ 1 ( 1 - q _ 2 ) ) , \\end{gather*}"}
-{"id": "4355.png", "formula": "\\begin{align*} \\sum _ { k = p _ { n } } ^ { q _ { n } } < P _ { k } x ^ { * } _ { n } , x ^ { ( n ) } _ { k } > = ( \\sum _ { k = p _ { n } } ^ { q _ { n } } \\| P _ { k } x ^ { * } _ { n } \\| ^ { p ^ { * } } ) ^ { \\frac { 1 } { p ^ { * } } } > \\epsilon _ { 0 } ^ { \\frac { 1 } { p ^ { * } } } . \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} { } J = \\int _ 0 ^ 1 { \\sin ( 3 \\ , \\pi \\ , t ) \\ , x ( t ) \\ , d t } \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} 1 + 1 = 1 + 0 = \\{ 1 \\} , \\\\ ( - 1 ) + ( - 1 ) = ( - 1 ) + 0 = \\{ - 1 \\} , \\\\ 1 + ( - 1 ) = \\{ - 1 , 0 , 1 \\} . \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { \\ell } { p } \\right ) ^ r \\left ( \\frac { 3 } { p } \\right ) ^ t = 1 . \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} \\langle \\chi , \\chi \\rangle = \\frac { 1 } { | N _ G ( P ) | } \\left ( ( 3 ^ k ( q - 1 ) ) ^ 2 ( 1 + q - 1 ) + 2 \\chi ( T ) \\overline { \\chi ( T ) } \\frac { q ( q - 1 ) } { 2 } + \\frac { q ^ 2 ( q - 1 ) } { 3 } \\cdot ( \\varepsilon \\cdot 3 ^ k ) ^ 2 ( 1 + \\omega \\overline { \\omega } + \\overline { \\omega } \\omega ) \\right ) = 1 . \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\mathbb { E } _ \\mu \\mathbb { P } _ { p _ { W | U , V } ^ n } \\Big ( \\big ( \\mathbf { U } ( i ) , \\mathbf { V } ( i , j ) , \\mathbf { W } \\big ) \\notin \\mathcal { A } _ { \\epsilon _ 1 , \\epsilon _ 2 } \\Big | \\mathbf { U } = \\mathbf { U } ( i ) , \\mathbf { V } = \\mathbf { V } ( i , j ) \\Big ) \\leq 2 ^ { - n \\beta ^ { ( 1 ) } _ { \\alpha , \\epsilon _ 1 } } + 2 ^ { - n \\beta ^ { ( 2 ) } _ { \\alpha , \\epsilon _ 2 } } , \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} v _ { 0 , S } = v _ { 0 , T } \\forall S \\neq T . \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} \\phi _ { \\omega } ( x ) : = e ^ { \\frac { \\omega + N } { 2 } } e ^ { - \\frac { 1 } { 2 } | x | ^ 2 } , x \\in { \\mathbb { R } } ^ N , \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} b ( \\alpha , \\lambda ) = \\langle \\check \\alpha , \\lambda \\rangle \\cdot q ( \\alpha ) , \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} \\delta _ { l j k } : = ( \\mathbb { M } + B ( j , k ) ) l \\neq 0 . \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} d ( \\omega ) \\leq \\frac { 1 } { 2 } \\| \\varphi \\| ^ 2 _ { L ^ 2 } \\leq \\frac { 1 } { 2 } \\liminf _ { n \\to \\infty } \\| v _ n \\| ^ 2 _ { L ^ 2 } = d ( \\omega ) , \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} \\widehat { \\mathfrak { p } } _ { \\theta } : = \\widehat { L i e ( \\mathcal { G } _ { X , x , \\theta } ) } \\mid _ x = e v ^ { - 1 } ( \\mathfrak { p } _ { \\theta } ) \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} b ' = - \\frac { \\tilde { c } _ 6 } { 8 6 4 } = - 6 \\tilde { c } _ 6 s ^ 2 \\ ; \\ ; s = \\frac { 1 } { 7 2 } \\in \\Q _ \\ell . \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 ( 1 - x ^ 2 ) \\ , e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( x ^ 2 - 1 \\right ) ^ 2 } \\ , d x \\leq \\int _ 0 ^ 1 ( 1 - x ^ 2 ) \\ , d x = \\left [ x - \\frac { 1 } { 3 } x ^ 3 \\right ] _ 0 ^ 1 = \\frac { 2 } { 3 } \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} \\mathbf { R } ^ { ( g ) } _ { c o d e } & = \\sum _ { \\left \\{ m \\ ; \\vert \\beta _ m > 0 \\right \\} } \\beta _ m \\boldsymbol { \\phi } _ m \\boldsymbol { \\phi } _ m ^ H \\\\ \\mathbf { S N R } ^ { t o t a l , ( g ) } _ { m i m o } & = \\boldsymbol { \\Gamma } \\operatorname { d i a g } \\left [ \\left \\{ \\lambda _ n \\right \\} _ { n = 1 } ^ { D } \\right ] \\boldsymbol { \\Gamma } ^ { - 1 } \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} | \\varepsilon _ n ( \\zeta ) | = 2 ^ { - \\frac { c } { 2 } } | \\zeta | ^ c \\left | \\widehat { \\varepsilon _ n } \\left ( \\frac { \\zeta ^ 2 } { 2 } \\right ) \\right | \\leq C \\left | \\frac { ( \\frac { c } { 2 } ) _ n ( \\frac { c + 1 } { 2 } ) _ n } { n ! ( \\zeta ^ 2 ) ^ { n } } \\right | . \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} g = - \\frac { | \\overline { g } | } { | \\gamma | } ( d x ^ 0 ) ^ 2 + g _ { i j } d x ^ i d x ^ j ; \\ ; \\overline { g } = ( g _ { i j } ) ; \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} \\Phi _ s ( y ) = \\Phi _ { s + t } ( z ) \\stackrel { \\eqref { e q : t h m a b o u t r e l a t i o n b e t w e e n b o x d i m e n s i o n a n d R o k h l i n d i m e n s i o n - t - 4 } } { \\in } \\Phi _ { \\left [ - \\left ( \\frac { 1 } { 2 } - 4 \\varepsilon \\right ) 8 L , \\left ( \\frac { 1 } { 2 } - 4 \\varepsilon \\right ) 8 L \\right ] } ( \\Delta ^ { ( l , \\sigma ) } _ 1 \\cap K ' ) \\ ; , \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} \\hat { \\theta } = \\arg \\min _ { \\theta \\in \\Theta } \\hat { \\psi } ( \\theta ) ^ { \\prime } \\hat { \\Upsilon } \\hat { \\psi } ( \\theta ) , \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} \\frac { d \\mu _ r } { d \\mu } ( n ) : = \\left ( [ n ] _ { | q | } ! \\right ) ^ { - 2 } r ^ { - n } . \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} g _ { \\ell , j } ( z ) \\equiv \\sum _ { n = 1 } ^ { \\infty } a ( \\ell ^ { m _ \\ell } n ) q ^ n - \\sum _ { n = 1 } ^ { \\infty } a ( \\ell ^ { m _ \\ell + 1 } n ) q ^ n \\equiv \\sum _ { \\substack { n = 1 \\\\ \\ell \\nmid n } } ^ { \\infty } a ( \\ell ^ { m _ \\ell } n ) q ^ n \\pmod { \\ell ^ j } . \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} A = - L , B = L \\quad \\mbox { w h i c h m e a n s } a = \\frac 1 { 1 + e ^ { 2 L } } , b = \\frac { e ^ { 2 L } } { 1 + e ^ { 2 L } } . \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} h = & \\left [ \\begin{array} { c c } f _ { 1 } - \\frac { a _ { 1 2 } + i _ 1 ( i _ 1 - i _ 2 ) - \\lambda ( a _ 1 - a _ 2 ) f _ { 1 } } { \\lambda ( a _ 1 - a _ 2 ) } & - \\frac { a _ { 1 2 } } { ( a _ 1 - a _ 2 ) } \\\\ f _ { 2 } & i _ 2 - \\lambda f _ 2 \\end{array} \\right ] , \\\\ h ' = & \\frac { a _ { 1 2 } + i _ 1 - \\lambda ( a _ 1 - a _ 2 ) f _ { 1 } } { \\lambda ( a _ 1 - a _ 2 ) } . \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} E _ C ( \\theta ) = - \\frac { 1 } { \\theta } \\ln \\left ( \\textsf { E } \\left \\{ e ^ { - \\theta R } \\right \\} \\right ) \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} \\theta _ { t + 1 } ^ i = \\frac { \\theta _ t ^ i \\delta ^ i } { \\sum _ j \\theta _ t ^ j \\delta ^ j } , & & i = 1 , \\ldots , n ; t = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} \\begin{array} { r l } Z ^ { I P } = \\max \\ & - x _ 1 + 2 x _ 2 + x _ 3 \\\\ \\mbox { s . t . } \\ & - x _ 1 + x _ 2 \\leq 0 . 5 , \\\\ & x _ 1 + x _ 2 \\leq 1 . 5 , \\\\ & x _ 3 \\leq 0 . 5 , \\\\ & x \\in \\{ 0 , 1 \\} ^ 3 . \\end{array} \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} b _ { n } = - q ^ { - 1 } a _ { n } , d _ { n } = - q ^ { - 1 } c _ { n } . \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\mu _ a ^ 0 = \\frac { \\lambda _ a } { \\ell } + \\frac { \\vartheta _ a } { 2 r } + O ( r ^ { - 2 } ) . \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} | \\Phi | & \\leq ( n / 4 ) ^ { - L } l ^ { 4 L } n ^ { d n } \\\\ & = ( \\sqrt { 2 } l ) ^ { 4 L } n ^ { d n - L } \\\\ & \\leq ( \\sqrt { 2 } l ) ^ { n / M } \\cdot n ^ { 1 + ( d - 1 / ( 4 M ) ) n } . \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} D _ { h } = - \\underset { \\varepsilon \\rightarrow 0 } { \\lim } { \\frac { \\ln [ N ( \\varepsilon ) ] } { \\ln ( \\varepsilon ) } } \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{gather*} \\delta _ Q \\alpha = \\pi ^ \\ast _ { \\infty } ( \\beta ) + d \\gamma , \\end{gather*}"}
-{"id": "5095.png", "formula": "\\begin{align*} \\binom { n } { k } _ b \\neq \\frac { ( n ! ) _ b } { ( k ! ) _ b ( n - k ) ! _ b } , \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} \\chi ( X _ { \\bar k } , { \\cal F } ) = ( C C { \\cal F } , T ^ * _ X X ) _ { T ^ * X } . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} A = \\left [ \\begin{array} { c c } A ' & 0 \\\\ 0 & a _ 2 \\end{array} \\right ] , B = \\left [ \\begin{array} { c c } B ' & 0 \\\\ 0 & b _ 2 \\end{array} \\right ] , F = \\left [ \\begin{array} { c } 1 \\\\ 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} c ( \\xi ) c ( \\eta ) = t ( \\xi \\bar { \\eta } + \\bar { \\xi } \\eta ) + c \\left ( \\frac { \\bar { \\gamma } \\bar { \\xi } \\bar { \\eta } } { t } \\right ) ; \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} V _ x : = \\{ v \\in V : ( \\pi ( \\xi ) - \\xi ( x ) ) ^ n v = 0 n \\in \\N \\} \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} D _ 1 C - C D _ 2 = \\hat { G } \\hat { B } ^ * , \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} \\binom { N } { k } \\left ( \\frac { - \\alpha } { ( \\theta + \\alpha ) _ { n \\uparrow } } \\right ) ^ { k } \\left ( \\frac { ( \\theta + \\alpha ) _ { n \\uparrow } } { ( \\theta ) _ { n \\uparrow } } \\right ) ^ { N } \\prod _ { i = 1 } ^ { k } ( 1 - \\alpha ) _ { m _ { i } - 1 \\uparrow } ( \\theta + \\alpha ) _ { n - m _ { i } \\uparrow } , \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} \\gamma ( 1 / 2 , \\pi , \\tau , \\psi ) = \\lim _ { s \\to 1 / 2 } \\frac { L ( 1 - s , \\pi , \\tau ) } { L ( s , \\pi , \\tau ) } \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} \\nabla _ { e _ 4 } s _ a = - i ( e ^ y \\lambda _ a + \\frac { m _ a } { 2 } ) s _ a + O ( e ^ { - y } ) . \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} \\dim ( S ) \\theta + c ' \\theta ' + \\theta _ { \\infty } r & = ( \\dim ( S ) - c ) \\theta > 0 \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} Y _ { t } = \\xi + \\int _ { t } ^ { T } g ( s , Y _ { s } , Z _ { s } ) d s - \\int _ { t } ^ { T } Z _ { s } d B _ { s } - ( K _ { T } - K _ { t } ) . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\sum _ { \\alpha , \\beta = 1 } ^ d n ^ \\alpha \\frac { \\partial n ^ \\beta } { \\partial x ^ \\alpha } \\nu ^ \\beta & = \\sum _ { \\alpha = 1 } ^ d n ^ \\alpha \\frac { \\partial ( n \\cdot \\nu ) } { \\partial x ^ \\alpha } - \\sum _ { \\alpha , \\beta = 1 } ^ d \\frac { \\partial \\nu ^ \\beta } { \\partial x ^ \\alpha } n ^ \\beta n ^ \\alpha \\\\ & = - \\sum _ { \\alpha , \\beta = 1 } ^ d \\frac { \\partial \\nu ^ \\beta } { \\partial x ^ \\alpha } n ^ \\beta n ^ \\alpha . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } A _ { 1 1 } h ' + A _ { 1 2 } v ' = 0 , \\\\ A _ { 2 1 } h ' + A _ { 2 2 } v ' = Q ' _ { 2 2 } ( u ^ \\pm ) v . \\end{array} \\right . \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} \\begin{array} { r l } Z ^ { I P } = \\max \\ \\ ; & c ' z + d ' u \\\\ \\mbox { s . t . } \\ \\ ; & z _ i \\leq z _ j , \\forall ( i , j ) \\in I , \\\\ & H z + G u \\leq h , \\\\ & z \\in \\{ 0 , 1 \\} ^ n . \\end{array} \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} \\alpha = \\mathbb { E } \\alpha + \\int ^ 1 _ 0 u ( t ) d w ( t ) , \\mathbb { E } \\alpha ^ 2 = ( \\mathbb { E } \\alpha ) ^ 2 + \\mathbb { E } \\int _ 0 ^ 1 u ( t ) ^ 2 d t . \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} e ( p e r m ( B _ \\alpha ) ) = b ( s ( \\alpha ) ) \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\chi _ 1 = \\sum _ j \\beta ^ j _ 1 \\frac { \\partial } { \\partial x ^ j } + \\sum _ \\rho b ^ \\rho _ 1 \\frac { \\partial } { \\partial \\xi ^ \\rho } . \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} \\int _ E h _ { T } \\dd \\| W \\| = \\int _ E g _ { \\| T \\| } \\dd \\| T \\| \\ ; , \\int _ E h _ { S } \\dd \\| W \\| = \\int _ E g _ { \\| S \\| } \\dd \\| S \\| \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} D _ n = \\begin{vmatrix} 2 & c _ 1 & c _ 2 & . . . & c _ n \\\\ c _ { - 1 } & 2 & c _ 1 & . . . & c _ n \\\\ . . . & . . . & . . . & . . . & . . . \\\\ c _ { - n } & c _ { - n + 1 } & c _ { - n + 2 } & . . . & 2 \\end{vmatrix} , n = 1 , 2 , 3 , . . . , \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} L ( e _ { i i } ) & = \\sum _ { x < i } C _ { x i } ^ { i i } e _ { x i } + \\sum _ { x \\in X } C _ { x x } ^ { i i } e _ { x x } + \\sum _ { y > i } C _ { i y } ^ { i i } e _ { i y } , \\\\ L ( e _ { i j } ) & = \\sum _ { x < i } C _ { x i } ^ { i i } e _ { x j } + C _ { i j } ^ { i j } e _ { i j } + \\sum _ { y > j } C _ { j y } ^ { j j } e _ { i y } , \\ \\mbox { i f } \\ i < j . \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} w _ k ^ { ( \\alpha ) } \\left ( { \\tau ^ { ( k ) } } \\right ) = { \\left ( { { \\tau _ k } - { \\tau ^ { ( k ) } } } \\right ) ^ { \\alpha - 1 / 2 } } { \\left ( { { \\tau ^ { ( k ) } } - { \\tau _ { k - 1 } } } \\right ) ^ { \\alpha - 1 / 2 } } , \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { q - 2 } \\alpha _ i ( x ) \\alpha _ i ( T ) \\alpha _ i ( z ) = \\sum \\limits _ { i = 0 } ^ { q - 2 } \\alpha _ i ( T ) = q - 1 . \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} A _ { ( m + 1 ) p _ n - \\frac { m + 1 } { 2 } } = A _ { ( m + 1 ) p _ n + \\frac { m - 1 } { 2 } } - x A _ { ( m + 1 ) p _ n + \\frac { m + 1 } { 2 } } ^ { m \\rightarrow m + 1 } ( x ) , \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} \\Big \\| \\sup _ { n \\ge 1 } | \\sum _ { k = 1 } ^ n a _ k k ^ { i \\cdot } | \\ , \\Big \\| _ { \\S ^ 2 } \\le C \\Big ( \\sum _ { n \\ge 0 } ( \\sum _ { k = 2 ^ n } ^ { 2 ^ { n + 1 } - 1 } | a _ k | ) ^ 2 \\Big ) ^ { 1 / 2 } \\ , . \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{align*} \\mathcal { H } _ { s } = \\sum _ { n = 1 } ^ { \\mathsf { N } - 1 } H _ { n , n + 1 } ^ { \\left ( s \\right ) } + \\frac { d } { d \\lambda } K _ { 1 , - } ^ { ( p ) } ( q ^ { s } ) + \\frac { _ { \\mathsf { 0 } } \\{ K _ { \\mathsf { 0 } , + } ^ { ( p ) } ( q ^ { s } ) H _ { \\mathsf { 0 } , \\mathsf { N } } ^ { \\left ( s \\right ) } \\} } { _ { \\mathsf { 0 } } \\{ K _ { \\mathsf { 0 } , + } ^ { ( p ) } ( q ^ { s } ) \\} } , \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} C ( t , s ) = 2 + \\frac { 2 } { \\min _ { r \\in [ s , t ] } | J _ { \\rho _ { r } } | } . \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} \\omega \\ne \\lambda \\omega _ { 1 } + ( 1 - \\lambda ) \\omega _ { 2 } \\mbox { w i t h } 0 < \\lambda < 1 \\ \\mbox { u n l e s s } \\ \\omega _ { 1 } = \\omega _ { 2 } = \\omega \\ . \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} \\xi : = \\begin{pmatrix} x _ 1 \\\\ x _ 2 \\end{pmatrix} , \\eta : = \\begin{pmatrix} y _ 1 \\\\ y _ 2 \\end{pmatrix} , A : = \\begin{pmatrix} 0 & 1 \\\\ - 1 & - \\alpha \\end{pmatrix} , c ( t ) : = \\begin{pmatrix} 0 \\\\ - \\delta ( t ) \\end{pmatrix} , \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} \\left ( x ; A \\cup B \\right ) = \\left ( x ; A \\right ) + \\left ( x ; B \\right ) - \\left ( x ; A \\cap B \\right ) \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} C ^ { 0 } \\simeq \\bigoplus _ { \\alpha \\in Q _ { 1 } , \\ , s ( \\alpha ) = j } \\mathbb { S } _ { 2 } ^ { - \\deg ( \\alpha ) } ( M ^ { t ( \\alpha ) } ) , C ^ { 1 } \\simeq \\bigoplus _ { \\alpha \\in Q _ { 1 } , \\ , t ( \\alpha ) = j } \\mathbb { S } _ { 2 } ^ { \\deg ( \\alpha ) - 1 } ( M ^ { s ( \\alpha ) } ) . \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} \\| u \\| _ { E ^ { \\alpha , p } } ^ p & = \\int _ 0 ^ T | _ 0 D _ t ^ \\alpha u ( t ) | ^ p d t = \\sum _ { i = 1 } ^ n | \\lambda _ i | ^ p \\int _ { \\mathbb { I } _ i } | _ 0 D _ t ^ \\alpha u _ i ( t ) | ^ p d t \\\\ & = \\sum _ { i = 1 } ^ n | \\lambda _ i | ^ p \\int _ 0 ^ T | _ 0 D _ t ^ \\alpha u _ i ( t ) | ^ p d t = \\sum _ { i = 1 } ^ n | \\lambda _ i | ^ p \\| u _ i \\| _ { E ^ { \\alpha , p } } ^ p \\\\ & = \\sum _ { i = 1 } ^ n | \\lambda _ i | ^ p , \\ \\ \\forall u \\in E _ n . \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} \\mathcal { N } i l p ( S ) = \\{ ( P , s ) \\mid P \\in B u n _ G ( S ) , s \\in H ^ 0 ( X \\times S , a d ( P ) \\otimes \\Omega ^ 1 _ { X \\times S / S } ) \\ , \\ \\} . \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} X _ { \\mathbb J } = X _ { \\{ j _ 1 , j _ 2 , \\ldots , j _ s \\} } = \\prod _ { j \\in \\mathbb N _ k \\setminus \\mathbb J } x _ j , \\ \\ \\mathbb J = \\{ j _ 1 , j _ 2 , \\ldots , j _ s \\} \\subset \\mathbb N _ k , \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{align*} \\Phi _ 1 ^ + \\sqcup \\Phi _ 2 ^ + \\sqcup \\cdots \\sqcup \\Phi _ j ^ + = \\{ \\beta _ { i _ j } , \\ldots , \\beta _ N \\} . \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\sup _ { x \\in B } \\inf _ { a \\in A ( \\omega ) } \\left | \\varphi _ t ( \\theta _ { - t } \\omega , x ) - a \\right | = 0 \\mathbb { P } \\textrm { - a l m o s t s u r e l y } . \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} I _ C ( x ) = \\sup _ { \\theta < 1 } \\left \\{ \\theta x + \\theta + \\log ( 1 - \\theta ) \\right \\} \\ \\ x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} x ( 0 ) = 0 . 1 , \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\gamma ^ { ( l , \\sigma ) } ( \\Phi _ t ( y ) ) = 0 \\ ; . \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} w ( x , \\tau ) = \\frac { e ^ { \\frac \\tau { m - 1 } } } k ( \\delta \\rho ^ 2 ) ^ { \\frac 1 { m - 1 } } u \\left ( x , s + \\frac { e ^ \\tau } { k ^ { m - 1 } } \\delta \\rho ^ 2 \\right ) \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} \\tilde { T } _ { 2 } & = \\breve { T } _ { 2 } + R _ { 3 } , \\breve { T } _ { 2 } = \\int \\alpha _ { 2 0 } ( x _ { t } , y _ { 2 t } ) H _ { p } ( \\gamma _ { 1 0 } ( x _ { t + 1 } ) ) [ \\hat { \\gamma } _ { 1 } ( x _ { t + 1 } ) - \\gamma _ { 1 0 } ( x _ { t + 1 } ) ] F _ { 0 } ( d w ) , \\\\ R _ { 3 } & = \\int \\alpha _ { 2 0 } ( x _ { t } , y _ { 2 t } ) H _ { p p } ( \\bar { \\gamma } _ { 1 } ( x _ { t + 1 } ) ) [ \\hat { \\gamma } _ { 1 } ( x _ { t + 1 } ) - \\gamma _ { 1 0 } ( x _ { t + 1 } ) ] ^ { 2 } F _ { 0 } ( d w ) , \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} \\langle E _ f , E _ { M , \\nu } \\rangle _ { \\Gamma \\backslash G } = 2 ^ { 1 - n } \\langle f , a ^ { - 2 \\rho } ( E _ { M , \\nu } ) _ N \\rangle _ A \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} C _ 0 : = \\frac { p - q } { p q p ^ * _ s } \\frac { ( p _ s ^ \\ast - q ) ^ { \\frac { p } { p - q } } } { ( p _ s ^ \\ast - p ) ^ { \\frac { q } { p - q } } } | \\Omega | ^ { \\frac { p ( p _ s ^ \\ast - q ) } { p _ s ^ \\ast ( p - q ) } } S ^ { - \\frac { q } { p - q } } , \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} \\omega ( \\delta { A } , \\delta ' \\ ! { A } ) = \\int _ \\Sigma \\ ! \\delta { A } \\wedge \\star ( \\delta ' F ) - \\delta ' \\ ! { A } \\wedge \\star ( \\delta F ) . \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} \\mathbb { E } [ \\xi ] = \\sup _ { Q \\in \\mathcal { P } } E _ { Q } [ \\xi ] \\ \\ \\xi \\in L _ { i p } ( \\Omega _ T ) . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} f = \\sum _ { \\alpha , \\beta \\in ( \\N _ 0 ) ^ m } c _ { \\alpha \\beta } \\ , h _ { \\alpha \\beta } . \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} [ \\lambda + a _ \\pi ( x , i \\partial _ t ) ] u = 0 , t > 0 , b _ \\pi ( x , i \\partial _ t ) u ( 0 ) = 0 \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} ( \\tilde { H } _ s v _ { l , j } ) ( x , n ) = e _ { l , j } ( x ) v _ { l , j } ( x , n ) , \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} \\mathcal L _ { t , x } ( U , V ) = - d \\omega _ 0 ( J _ t U , V ) , \\forall ~ U , V \\in H _ x X , \\forall ~ x \\in X . \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} R _ { ( c , d ) } ( n ) = & 1 2 \\sigma ( \\frac { n } { c } ) - 3 6 \\sigma ( \\frac { n } { 3 c } ) + 1 2 \\sigma ( \\frac { n } { d } ) - 3 6 \\sigma ( \\frac { n } { 3 d } ) + 1 4 4 \\ , W _ { ( c , d ) } ( n ) + 1 2 9 6 \\ , W _ { ( c , d ) } ( \\frac { n } { 3 } ) \\\\ & - 4 3 2 \\ , \\biggl ( W _ { ( 3 c , d ) } ( n ) + W _ { ( c , 3 d ) } ( n ) \\biggr ) . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} A _ { i j } \\equiv _ { p ^ 3 } \\frac { t } { i + j + 1 } \\binom { i + j } { i } \\binom { i + j + t } { i + j } . \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} \\min \\limits _ { \\stackrel { E _ i \\in Q _ k ^ { ( i ) } } { i = 1 , \\dots , m } } \\max \\limits _ { \\stackrel { \\| x _ j \\| \\leq \\eta , } { j = 1 , \\dots , L } } p ( E , x ) . \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\omega } \\hat { \\psi } = g _ 1 + M _ 1 ( \\varphi ) M _ { \\varphi } [ \\hat { \\eta } ] + M _ 2 ( \\varphi ) g _ 2 + M _ 3 ( \\varphi ) g _ 3 - M _ 2 ( \\varphi ) [ \\partial _ { \\psi } \\theta _ 0 ] ^ T M _ { \\varphi } [ g _ 2 ] , \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} e ( p e r m ( A _ \\alpha ) ) = a ( s ( \\alpha ) ) \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = - x + \\sin ( t ) - \\cos ( t ) + \\lambda f ( t , x , y , z ) , \\\\ \\dot y = \\lambda z \\big ( y + x \\sin ( t ) \\big ) , \\\\ \\dot z = - \\lambda y \\big ( y + x \\sin ( t ) \\big ) , \\\\ y ^ 2 + z ^ 2 = 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} a . \\psi ( e _ { t a } . m ) = a . \\psi ( m _ { t a } ) = V ( a ) ( \\phi _ { t a } ( m _ { t a } ) ) \\mbox { a n d } \\psi ( a . m _ { t a } ) = \\psi ( W ( a ) ( m _ { t a } ) ) = \\phi _ { h a } ( W ( a ) ( m _ { t a } ) ) , \\mbox { w h e r e } a \\in Q _ 1 . \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} \\phi ^ * ( p ) = L ( p ) + \\sup _ { x \\in \\Omega } \\Phi _ { L ( p ) } = L ( p ) + \\sup _ { x \\in \\Omega } ( q - q ) - \\Phi _ { q _ 0 } = L ( p ) + \\Phi _ { q _ 0 } ^ * ( L ( p ) - q _ 0 ) \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} \\frac { d \\alpha } { d s } = \\frac { d z } { d s } \\cdot \\frac { d \\beta } { d \\nu } . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\frac { f _ 1 ( x ) } { 1 + | x | ^ { 2 s + N } } \\ d x & \\leq \\int _ { \\R ^ N } \\int _ { \\{ | z | < 1 \\} } - \\ln | z | \\ d z | f ( y ) | \\ d y = - \\int _ { \\{ | z | < 1 \\} } \\ln ( | z | ) \\ d z \\| f ( y ) \\| _ { L ^ 1 ( \\R ^ N ) } < \\infty , \\\\ \\int _ { \\R ^ N } \\frac { f _ 2 ( x ) } { 1 + | x | ^ { 2 s + N } } \\ d x & = \\int _ { K } \\int _ { \\{ | z | \\geq 1 \\} } \\frac { \\ln ( | z | ) | z | ^ { 2 s - N } } { { 1 + | x + z | ^ { 2 s + N } } } \\ d z | f ( y ) | \\ d y \\leq M \\| f ( y ) \\| _ { L ^ 1 ( \\R ^ N ) } < \\infty , \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} \\overline { | \\sigma | } & = \\left \\{ ( z _ v ) _ v \\in | Z | ~ \\Bigl | ~ \\sum _ { v \\in \\sigma } z _ v = 1 \\right \\} \\\\ | \\sigma | & = \\left \\{ ( z _ v ) _ v \\in | Z | ~ \\Bigl | ~ \\sum _ { v \\in \\sigma } z _ v = 1 ~ ~ z _ v > 0 ~ v \\in \\sigma \\right \\} \\ ; . \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} \\lVert \\mathcal { F } ( U _ 0 ) \\rVert _ s = \\lVert \\mathcal { F } ( ( \\varphi , 0 , 0 ) , 0 ) \\rVert _ s = \\lVert X _ P ( i _ 0 ) \\rVert _ s \\le _ s \\varepsilon ^ { 6 - 2 b } . \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} W _ n W _ { n + i + j } - W _ { n + i } W _ { n + j } = ( - 1 ) \\cdot ( - c _ 2 ) ^ { n } \\cdot \\Delta \\cdot U _ i U _ j . \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{align*} \\varphi _ k ( x ) : = \\varphi _ 0 ( 2 ^ { - k } x ) - \\varphi _ 0 ( 2 ^ { - k + 1 } x ) x \\in \\mathbb { R } ^ d , \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} \\lambda u + \\overline G _ i ( h , u ' ) = 0 \\ \\ \\ J _ i i \\in \\{ 1 , 2 , 3 \\} , \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 0 } ^ { q - 2 } \\alpha _ i ( x ) \\alpha _ i ( y ) \\alpha _ i ( z ^ { - 1 } ) = \\sum \\limits _ { i = 0 } ^ { q - 2 } ( - 1 ) ^ i \\alpha _ i ( y z ^ { - 1 } ) = \\left \\{ \\begin{array} { l l } q - 1 & y z ^ { - 1 } \\in \\C ( J ) , \\C ( J T ) \\C ( J T ^ { - 1 } ) \\\\ 0 & o t h e r w i s e \\\\ \\end{array} \\right . ; \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} G = ( I \\ , | \\ , A ) = \\left ( \\begin{array} { l | l } 1 0 0 0 & 0 1 1 1 \\\\ 0 1 0 0 & 1 0 1 1 \\\\ 0 0 1 0 & 1 1 0 1 \\\\ 0 0 0 1 & 1 1 1 0 \\\\ \\end{array} \\right ) \\in \\mathbb { F } _ 2 ^ { 4 \\times 8 } , \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} \\langle ( v _ 1 , v _ 2 ) , ( v _ 1 ' , v _ 2 ' ) \\rangle = \\langle v _ 1 , v _ 2 ' \\rangle + \\langle v _ 2 , v _ 1 ' \\rangle + \\langle v _ 2 , L v _ 2 \\rangle . \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} U = V \\setminus \\overline { V _ 0 } . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} { \\bf f } ( { \\bf X } _ { i } ^ * , { \\bf U } _ { i } ^ * ) = { \\bf f } ( { \\bf x } ^ * ( \\tau _ i ) , { \\bf u } ^ * ( \\tau _ i ) ) = \\dot { \\bf x } ^ * ( \\tau _ i ) . \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} & g _ { 2 / 3 , 3 / 4 } ^ 2 ( w _ 2 ) = \\frac { n _ 0 } { 4 k _ { \\ell } } \\log \\left ( 1 + \\frac { w _ 2 P ' } { 3 } \\right ) \\\\ & - \\frac { n _ 0 } { 8 k _ { \\ell } } \\log \\left ( 1 + \\frac { 2 w _ 2 P ' } { 3 } \\right ) - \\frac { 3 \\ell } { 4 k _ { \\ell } } H _ 2 \\left ( \\frac { w _ 2 } { \\ell } \\right ) . \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} D _ N h ^ + _ \\gamma ( v ) = \\begin{cases} \\frac 1 \\gamma & v \\in [ L , L + \\gamma ] , \\\\ 0 & , \\end{cases} D _ N h ^ - _ \\gamma ( v ) = \\begin{cases} \\frac 1 \\gamma & v \\in [ U - \\gamma , U ] , \\\\ 0 & , \\end{cases} \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} v ( s ) = e ^ { \\mu ( s - t ) } \\left ( p ( s ) ( X ( s ) - \\bar X ( s ) ) + \\frac { 1 } { 1 + \\beta } \\pi ( s ) \\bar X ( s ) + \\frac { \\alpha } { ( 1 + \\beta ) ^ 2 } \\int _ t ^ s \\pi ( \\tau ) d \\tau \\right ) + \\frac { \\alpha } { 1 + \\beta } . \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} \\tau \\left ( \\alpha _ j ( s ) , \\beta _ j ( s ) ; \\xi _ 2 ^ { ( j ) } , \\xi _ 1 ^ { ( j ) } y _ j \\right ) = ( - 1 ) ^ { \\epsilon _ j } | \\xi _ 2 ^ { ( j ) } | ^ { \\alpha _ j ( s ) + \\beta _ j ( s ) - 1 } \\tau \\left ( \\alpha _ j ( s ) , \\beta _ j ( s ) ; 1 , d ^ { ( j ) } y _ j \\right ) , \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} \\rho < \\begin{cases} & \\frac { 1 } { 2 } \\left ( \\frac { \\nu } { 2 } \\right ) ^ { \\frac { 1 } { 1 - 2 \\sigma } } \\ \\ \\sigma \\in ( 0 , \\frac { 1 } { 2 } ) , \\\\ & \\frac { 1 } { 2 } \\ \\ \\sigma = \\frac { 1 } { 2 } , \\\\ & \\frac { 1 } { 2 } \\left ( \\frac { 2 } { \\nu } \\right ) ^ { \\frac { 1 } { 2 \\sigma - 1 } } \\ \\ \\sigma \\in ( \\frac { 1 } { 2 } , 1 ] . \\end{cases} \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} { \\sum \\limits _ { k = 1 } ^ { q } \\int \\limits _ \\gamma \\left | \\langle \\nabla g _ { k } ^ { H , \\beta } ( x ) , d x \\rangle \\right | ~ + ~ \\sum \\limits _ { k = 1 } ^ { q } \\int \\limits _ { \\widetilde { \\gamma } } \\left | \\langle \\nabla g _ { k } ^ { H , \\beta } ( y ) , d y \\rangle \\right | \\over \\parallel L _ n ( \\sigma ) - L _ n ( \\sigma ' ) \\parallel _ 1 + \\parallel L _ n ( \\tau ) - L _ n ( \\tau ' ) \\parallel _ 1 } ~ < 1 \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} Q _ { R , \\delta } : = \\int d { x } d t f _ { R } ( { x } ) f _ { d } ( t ) j ^ { 0 } ( { x } , t ) \\ , \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} Q _ { [ H , i A ] } ( f , g ) = \\left . \\frac { d } { d t } Q _ H ( e ^ { i t A } f , e ^ { i t A } g ) \\right | _ { t = 0 } \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{gather*} \\frac { d } { d t } \\log F _ { 6 } ( t ) = \\frac { 3 } { 4 } t ^ 2 - \\sqrt { 2 } ( - t ) ^ { 1 / 2 } + \\frac { 1 } { 2 4 t } + O \\big ( | t | ^ { - \\frac { 5 } { 2 } } \\big ) \\mbox { a s \\ \\ $ t \\to - \\infty $ } , \\end{gather*}"}
-{"id": "6478.png", "formula": "\\begin{align*} b ( x ) = \\sum _ { k \\in \\Lambda ^ { * } } b _ { k } \\phi _ { k } ^ { \\Lambda } ( x ) \\ . \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} d ^ * _ G ( A B ) \\leq d ^ * _ L ( A _ o B _ o ) \\leq d ^ * _ L ( \\tau _ o ^ { - 1 } ( I J ) ) = m _ K ( I J ) , \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} u _ i ^ + ( h _ i ) = \\lim _ { J _ i \\ni h \\to h _ i } u _ i ^ - ( h ) = v ^ { \\pm } ( x ) = d _ i \\ \\ \\ x \\in c _ i ( h _ i ) i \\in \\{ 1 , 2 , 3 \\} . \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} f g = T _ f g + T _ g f + \\Pi [ f , g ] . \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} \\pi ( \\textbf { x } ) = [ ( x _ 0 , x _ 1 ) , ( x _ 1 , x _ 2 ) , \\cdots , ( x _ { n - 2 } , x _ { n - 1 } ) , ( x _ { n - 1 } , x _ 0 ) ] . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} N _ h F _ 1 \\Gamma _ h = N _ h N _ v F _ 0 \\Gamma _ v \\Gamma _ h , \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} \\| w _ { 0 } + l _ { 0 } w _ { \\epsilon , \\eta } \\| ^ { 2 } & = \\| w _ { 0 } \\| ^ { 2 } + ( l _ { 0 } ) ^ { 2 } \\| w _ { \\epsilon , \\eta } \\| ^ { 2 } + 2 l _ { 0 } \\langle w _ { 0 } , w _ { \\epsilon , \\eta } \\rangle > \\| w _ { 0 } \\| ^ { 2 } + | c ^ { 2 } - \\| w _ { 0 } \\| ^ { 2 } | + 2 l _ { 0 } \\langle w _ { 0 } , w _ { \\epsilon , \\eta } \\rangle \\\\ & > c ^ { 2 } > \\left ( t ^ { - } \\left ( \\frac { w _ { 0 } + l _ { 0 } w _ { \\epsilon , \\eta } } { \\| w _ { 0 } + l _ { 0 } w _ { \\epsilon , \\eta } \\| } \\right ) \\right ) ^ { 2 } \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} F _ { \\lambda } ^ { \\theta } ( x ^ n , y ^ n ) : = d ^ n _ { \\lambda } ( p _ * , p _ { \\theta } ) - \\log \\frac { p _ * ( y ^ n | x ^ n ) } { p _ { \\theta } ( y ^ n | x ^ n ) } . \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} 0 = \\bold { D } _ F ^ { \\vartriangle } ( Q ) : = \\frac { \\operatorname { d } } { \\operatorname { d } \\ ! \\varepsilon } \\Big | _ { \\varepsilon = 0 } F ( n , [ u + \\varepsilon Q ( n , [ u ] ) ] ) , \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 + \\bold { 1 } _ k } = D _ { J _ 1 } \\left ( \\phi ^ { \\alpha } _ { \\bold { 0 } ; J _ 2 + \\bold { 1 } _ k } - \\xi ^ i u ^ { \\alpha } _ { \\bold { 1 } _ i ; J _ 2 + \\bold { 1 } _ k } \\right ) + \\xi ^ i u ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; J _ 2 + \\bold { 1 } _ k } . \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\| \\rho ( \\cdot , T ) \\| ^ 2 _ { L ^ 2 ( \\Omega ) } + \\int _ 0 ^ T \\int _ \\Omega h _ \\alpha \\rho ^ 2 d x \\ , d t + \\Gamma ( \\alpha + 1 ) \\int _ 0 ^ T \\int _ { x , y : \\ , [ x , y ] \\subset \\Omega } \\frac { ( \\varphi ( x ) - \\varphi ( y ) ) ^ 2 } { | x - y | ^ { d + \\alpha } } d x \\ , d y \\ , d t = \\frac { 1 } { 2 } \\| \\rho ^ { i n } \\| ^ 2 _ { L ^ 2 ( \\Omega ) } . \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} [ D ( m , s ) : D ( m + 1 , n ) ] _ q & = q ^ { 2 ( s - n ) } [ D ( m , s - m - 1 ) : D ( m + 1 , n - m - 1 ) ] _ q & \\\\ & + q ^ { 4 s _ 1 s _ 0 } [ D ( m + \\delta _ { s _ 1 , 1 } , s - 2 s _ 0 ) : D ( m + 1 , n ) ] _ q & \\\\ & + \\delta _ { n , m + 1 } \\delta _ { s _ 0 , 1 } \\delta _ { s _ 1 , 2 } q ^ { 3 m } + ( 1 - \\delta _ { 2 s _ 0 , m } ) \\delta _ { s _ 0 , n } \\delta _ { s _ 1 , 1 } q ^ { m + 2 s _ 0 } . \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} p ( G ) ( \\mathcal { H } ) = \\sum _ { \\phi : E \\to [ k ] } \\prod _ { v \\in V } h ^ { v } ( \\phi ( \\delta ( v ) ) ) . \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} V _ { \\Lambda } = \\frac { 1 } { V } \\sum _ { { k } , { p } , { q } } \\nu ( { p } ) b _ { { k } + { p } } ^ { * } b _ { { q } - { p } } ^ { * } b _ { { k } } b _ { { q } } \\ , \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} \\nabla ( A ) = A - Z A Z ^ * = G B ^ * . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} \\hat { u } = \\frac { \\sigma ( \\pi ^ \\alpha ) } { \\pi ^ \\alpha } = \\zeta _ 3 ^ \\alpha , \\hat { r } = \\hat { s } = \\hat { t } = 0 \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} b _ n = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } \\left ( - 1 \\right ) ^ k a _ k . \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} P ( \\mathcal { A } ) & = P ( \\tau _ n - \\delta - \\sigma _ { \\rm a } ^ 2 - \\sigma _ { \\rm w } ^ 2 < U < \\tau _ n + \\delta - \\sigma _ { \\rm w } ^ 2 ) \\\\ & \\leq \\frac { \\sigma _ { \\rm a } ^ 2 + 2 \\delta } { \\zeta } \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} M \\le \\lim _ { k \\to \\infty } \\int m _ k \\dd \\| T \\| = \\int m _ \\infty \\dd \\| T \\| \\le \\int g _ { \\| T \\| } \\dd \\| T \\| \\le M . \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} ( X _ t , Y _ t ) \\cdot ( - Y _ \\alpha , X _ \\alpha ) = U \\cdot ( - Y _ \\alpha , X _ \\alpha ) , \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} | \\partial _ l F ( t , s ) | , ~ | \\partial _ { l l ' } F ( t , s ) | \\leq C , l , l ' = 1 , 2 , ( t , s ) \\in [ 0 , 1 ] ^ 2 . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} \\mathcal { N } _ t \\leq \\| F _ k ^ 2 \\| _ { { 2 ^ { n - k } } } ^ { 2 ^ { n - k } } = \\| F _ k \\| _ { { 2 ^ { n - k + 1 } } } ^ { 2 ^ { n - k + 1 } } = 1 , \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} 3 m _ 3 w _ x + \\tilde { d } _ 1 + \\varepsilon ^ 2 c ( \\xi ) - m _ 1 = 0 \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} \\partial _ t N = \\sum _ { j = 1 } ^ { m } D _ j \\partial _ { x x } p _ j N - \\sum _ { j = 1 } ^ m M _ j \\partial _ x p _ j N + \\sum _ { j = 1 } ^ m f _ j ( \\vec { p } N ) \\ , . \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{align*} \\mathcal { U } _ { a , - } ( \\lambda ) = \\mathcal { U } _ { a , - } ^ { s G } ( \\lambda ) , \\mathcal { T } ( \\lambda ) = \\mathcal { T } ^ { s G } ( \\lambda ) . \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v = \\sum _ { i , j = 1 } ^ d \\partial _ { i j } ^ 2 \\big ( ( \\Phi \\Phi ^ t ) _ { i , j } ( t , x , v ) v \\big ) - d i v \\big ( g ( t , x , v ) v \\big ) + \\Lambda ( t , x , v ) v \\\\ v ( 0 , d x ) = v _ 0 ( d x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} f \\left ( \\frac 1 { k _ n } , 1 \\right ) & = \\frac { 1 } { 2 } \\log \\left ( 1 + \\frac { P ' } { 2 } \\right ) - \\frac { k _ n } { n } H _ 2 \\left ( \\frac { 1 } { k _ n } \\right ) \\\\ & - \\frac { 1 - \\epsilon } { 2 k _ n } \\log ( 1 + k _ n P ' ) . \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L u _ i ( x ) = \\lambda _ i f _ i ( x , u ( x ) ) , & x \\in \\Omega , i = 1 , 2 , \\ldots , n , \\\\ B u _ i ( x ) = 0 , & x \\in \\partial \\Omega , i = 1 , 2 , \\ldots , n , \\end{array} \\right . \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{gather*} \\lambda _ { \\pm } = \\frac { x ^ 2 - t } { 2 } \\pm \\mu , \\mu = \\sqrt { \\frac { x ^ 4 } { 4 } + r _ 2 x ^ 2 + r _ 1 x + r _ 0 } = \\frac { x ^ 2 } { 2 } + r _ 2 + \\frac { r _ 1 } { x } + + \\frac { r _ 0 - r _ 2 ^ 2 } { x ^ 2 } + \\cdots , \\ ! \\ ! \\ ! \\end{gather*}"}
-{"id": "3787.png", "formula": "\\begin{align*} b ( w ) = & ( i _ n ^ \\ast \\wedge j _ n ^ \\ast - 1 _ n ^ \\ast \\wedge k _ n ^ \\ast ) \\otimes ( i _ n ^ \\ast \\otimes 1 _ 1 - 1 _ n ^ \\ast \\otimes j _ 1 + j _ n ^ \\ast \\otimes k _ 1 - k _ n ^ \\ast \\otimes j _ 1 ) \\\\ - & ( 1 _ n ^ \\ast \\wedge j _ n ^ \\ast + i _ n ^ \\ast \\wedge k _ n ^ \\ast ) \\otimes ( 1 _ n ^ \\ast \\otimes 1 _ 1 + i _ n ^ \\ast \\otimes i _ 1 + j _ n ^ \\ast \\otimes j _ 1 + k _ n ^ \\ast \\otimes k _ 1 ) . \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} & \\langle c ^ p \\rangle \\leq [ a ^ i b ^ j z , H ] \\mbox { i f a n d o n l y i f } i = 0 \\mbox { a n d } j \\neq 0 \\\\ \\mbox { a n d } & \\langle c ^ p \\rangle \\leq [ c ^ i ( b d ) ^ j z , H ] \\mbox { i f a n d o n l y i f } i = j \\neq 0 . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} \\bigcap _ { i = 1 } ^ m \\{ v \\in V \\colon \\langle v , w _ i - f ( w _ i ) \\rangle \\leq \\langle w _ i , w _ i - f ( w _ i ) \\rangle \\} \\subseteq Q , \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} q _ { D _ { 2 6 - N } } ( \\varphi ( \\eta ) ) = - q _ { \\Lambda _ N } ( \\eta ) . \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\frac { d Q _ c } { d P } = \\frac { 1 } { D } \\left [ ( M ( P ) - c ) - \\frac { f ( P ) } { Q } \\right ] > \\frac { 1 } { D } \\left [ ( M ( P ) - \\widehat { c } \\ , ) - \\frac { f ( P ) } { Q } \\right ] = \\frac { d Q _ { \\widehat { c } } } { d P } \\ , , \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} f _ { \\Omega } ( x ) = \\frac { x ^ { M - 1 } ( 1 + x ) ^ { - M - K } } { \\mathcal { B } ( M , K ) } , \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} A ' = \\mathbb { F } _ { ( 1 ) } \\oplus \\left ( \\mathbb { F } _ { ( 2 ) } \\oplus \\mathbb { F } _ { ( 4 ) } \\oplus \\mathbb { F } _ { ( 6 ) } \\oplus \\cdots \\right ) . \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} V ( x ) = \\begin{cases} \\frac { F ( x ) } { \\abs { F ( x ) } } x \\in N \\setminus D , \\\\ \\psi ( \\phi ( x ) ) x \\in D . \\end{cases} \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\min _ { } \\left \\{ v ( c _ i ) + \\lceil \\log _ p \\left ( \\frac { r } { i } \\right ) \\rceil : ( i , p ) = 1 \\right \\} \\geq n . \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\begin{aligned} \\mathbb { B } : & = ( 2 4 c _ 1 ^ 2 - 1 2 c _ 4 ) D _ S ^ 6 \\{ \\mathrm { I } - 2 D _ { - 2 } U D _ S ^ 2 \\} + ( \\frac { 1 4 } { 3 } c _ 2 ^ 2 - 4 c _ 6 ) D _ S ^ 4 \\\\ & + ( 4 c _ 6 - \\frac { 1 6 } { 3 } c _ 2 ^ 2 ) \\{ D _ S ^ 4 U + D _ S ^ 2 U D _ S ^ 2 \\} - 6 c _ 3 ^ 2 \\mathrm { I } + 1 2 ( c _ 2 c _ 3 - c _ 7 ) D _ S ^ 2 \\\\ & + ( 2 4 c _ 7 - 1 6 c _ 2 c _ 3 ) D _ S ^ 2 U , \\end{aligned} \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} c = \\frac { 4 0 0 8 - 4 d ^ 2 u ^ 2 + 8 d ^ 2 r u - 5 d ^ 2 r ^ 2 } { 1 9 9 6 + 2 d ^ 2 r u - 2 d ^ 2 r ^ 2 } . \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\begin{bmatrix} e _ { ( m + 1 ) p } ( x ) & e _ { ( m + 1 ) p + 1 } ( x ) & \\cdots & e _ { ( m + 1 ) p + m } ( x ) \\end{bmatrix} ^ { T } = K ^ { p } \\begin{bmatrix} d _ 0 ( x ) & d _ 1 ( x ) & \\cdots & d _ m ( x ) \\end{bmatrix} ^ T . \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} A w ^ \\pm _ \\tau ( \\tau ) = 2 B ( u ^ \\pm , w ^ \\pm ( \\tau ) ) + Q ( w ^ \\pm ( \\tau ) ) . \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} 3 - \\frac { a } { k - 1 } \\geq 3 - \\frac { ( k - 1 ) / k } { k - 1 } = 3 - \\frac 1 { k } , \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} \\left | \\kappa ( e ^ l ) - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ { q } \\left | \\left \\langle L _ n ( \\tau ' ) - L _ n ( \\tau ) , \\nabla g _ { k } ^ { H , \\beta } ( L _ n ( \\tau ) ) \\right \\rangle \\right | - \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ { q } \\left | \\left \\langle L _ n ( \\sigma ' ) - L _ n ( \\sigma ) , \\nabla g _ { k } ^ { H , \\beta } ( L _ n ( \\sigma ) ) \\right \\rangle \\right | \\right | < C ^ \\prime \\varepsilon ^ 2 , \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} \\Pi ( k _ 0 ) : = \\left \\{ \\mathbf { k } \\in \\ell _ + : k _ { t + 1 } \\in \\Gamma ( k _ t ) , \\ t = 0 , 1 , \\ldots ; \\ k _ 0 \\ \\right \\} . \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} A f ( x ) = \\int e ^ { i 2 \\pi x \\cdot \\xi } a ( x , \\xi ) \\widehat { f } ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{align*} v _ { n } ( t ) = \\varphi _ { n } e _ { q } ( t , \\mu _ { n } ) + \\int \\limits _ { 0 } ^ { t } e _ { q , q } ( t - \\tau , \\mu _ { n } ) r ( \\tau ) f _ { n } ( \\tau ) d \\tau \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ g \\tau \\left ( \\alpha _ j , \\beta _ j , \\xi _ 2 ^ { ( j ) } , \\epsilon ^ { ( j ) } \\xi _ 1 ^ { ( j ) } y _ j \\right ) = \\prod _ { j = 1 } ^ g | \\epsilon ^ { ( j ) } | ^ { 1 - \\alpha _ j - \\beta _ j } \\cdot \\tau \\left ( \\alpha _ j , \\beta _ j , \\epsilon ^ { ( j ) } \\xi _ 2 ^ { ( j ) } , \\xi _ 1 ^ { ( j ) } y _ j \\right ) . \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} \\deg ( I d - G _ 1 , \\tilde \\Gamma , 0 ) = \\deg ( I d - G _ 0 , \\tilde \\Gamma , 0 ) . \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} \\aligned \\Theta ' & = \\inf _ { ( u _ 1 , . . . , u _ r ) \\in \\mathbb { D } \\setminus \\{ ( 0 , . . . , 0 ) \\} } \\max _ { t > 0 } J ( t u _ 1 , . . . , t u _ r ) \\\\ & = \\inf _ { ( u _ 1 , . . . , u _ r ) \\in \\mathbb { D } \\setminus \\{ ( 0 , . . . , 0 ) \\} } \\frac { 1 } { N } \\Big [ \\frac { \\int _ { \\R ^ N } E ( u _ 1 , . . . , u _ r ) } { \\int _ { \\R ^ N } F ( u _ 1 , . . . , u _ r ) } \\Big ] ^ { \\frac { N } { 2 } } , \\endaligned \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\frac { \\log ( 1 + \\lambda t ) } { \\lambda ( 1 + \\lambda t ) ^ { \\frac { 1 } { \\lambda } } - \\lambda } ( 1 + \\lambda t ) ^ { \\frac { x } { \\lambda } } = \\sum _ { n = 0 } ^ \\infty B _ { n , \\lambda } ( x ) \\frac { t ^ n } { n ! } ( \\textnormal { s e e } \\ , \\ , [ 7 ] ) . \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} Y ( z ) = e ^ { \\frac { N \\ell } { 2 } \\sigma _ 3 } \\left ( I + { \\cal O } \\left ( \\frac { 1 } { N ^ { L } } \\right ) \\right ) Z ^ \\infty ( z ) \\begin{bmatrix} 1 & 0 \\\\ \\displaystyle - \\star \\ , \\Big ( \\frac { z } { z - a } \\Big ) ^ { c } e ^ { N \\phi ( z ) } & 1 \\end{bmatrix} e ^ { \\frac { - N \\ell } { 2 } \\sigma _ 3 } e ^ { N g ( z ) \\sigma _ 3 } . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\lim _ { \\substack { m \\to \\infty \\\\ \\omega _ m \\to N } } \\frac { d _ { n , k } ^ { m , \\omega _ { m } } } { d ^ { m , \\omega _ { m } } } = V ^ { B B , \\alpha , \\theta } _ { n , k } ( N ) . \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} M ( x , z ) = \\sum _ { n \\geq 0 } M _ n ( x ) \\frac { z ^ n } { n ! } = \\sqrt { \\frac { x - 1 } { x - e ^ { 2 z ( x - 1 ) } } } . \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} v ( \\alpha , \\beta ) = \\frac { 1 } { \\sqrt { 2 \\pi } } \\int q ( \\xi , \\beta ) \\hat f ( \\xi ) e ^ { i \\alpha \\xi } \\ , d \\xi , \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} l _ i ( \\tau ) = \\prod ^ { N } _ { \\substack { j = 1 \\\\ j \\neq i } } \\frac { \\tau - \\tau _ j } { \\tau _ i - \\tau _ j } . \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} \\int _ { \\partial R } v \\ast d u = \\iint _ { R } v \\wedge \\ast u + \\iint _ R v \\cdot d \\ast d u . \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} \\widetilde { W } _ 1 ^ { \\lambda } ( X , Y ) = \\sum _ { ( i , k ) \\in \\mathfrak { l } _ 1 \\cap { I } _ { \\lambda } } c _ { i , k } X ^ i Y ^ k = 1 + ( m - 1 ) X ^ { \\beta } Y . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} v = \\frac { 1 - w } { \\left ( \\frac \\gamma 8 \\right ) ^ { \\frac { m + 1 } 2 } } . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\omega : \\ \\ & \\Gamma ( T M + L ) \\times \\Gamma ( T M + L ) \\rightarrow C ^ \\infty ( T M + L ) \\\\ & \\omega ( X + S , Y + T ) = \\omega _ M ( X , Y ) + \\omega _ L ( S , T ) \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{align*} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\mapsto \\begin{bmatrix} 0 & 1 \\\\ \\phi & 0 \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\begin{bmatrix} 0 & \\phi \\\\ 1 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} \\lambda u - \\cfrac { b \\cdot D u } { \\varepsilon } + G ( x , D u ) = 0 \\ \\ \\ \\Omega . \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\hat { \\omega } ^ 0 ( t ) : = - \\frac { t \\ , b ( x ) } { \\sigma } \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} \\mathbf { G } _ k = \\alpha \\mathbf { g } _ { r k } \\mathbf { g } _ { t k } ^ { \\dagger } \\ , , \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} \\begin{aligned} \\norm { \\Phi _ { n } } _ { L ^ { 2 } ( 0 , T ; H ^ { 3 } ) } ^ { 2 } \\leq C \\left ( \\norm { \\Xi _ { n } } _ { L ^ { 2 } ( 0 , T ; H ^ { 1 } ) } ^ { 2 } + \\norm { \\Phi _ { n } } _ { L ^ { 2 } ( 0 , T ; H ^ { 1 } ) } ^ { 2 } \\right ) \\leq D _ { 5 } \\norm { w } _ { L ^ { 2 } ( Q ) } ^ { 2 } , \\end{aligned} \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} F _ 2 ( x ) ( t ) = \\int _ 0 ^ 1 k ( t , s ) f ( s , x ( s ) ) \\textup d s , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} \\mathcal { U } ( k ) : = \\left \\{ z \\in \\underline { \\mathbb { R } } ^ n : z ^ i = w ^ i ( \\mathbf { x } ^ i ) , \\ i = 1 , \\ldots , n , \\ \\ \\mathbf { x } \\in \\Omega ( k ) \\right \\} . \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} \\nu ( s - m - 1 , n - m - 1 ) & = \\nu ( s - 1 , n - 1 ) - \\nu ( s - m - 1 , n + 2 r _ n - m - 1 ) , \\\\ \\nu ( s + m - 2 s _ 0 , n ) & = \\nu ( s + 2 m - 2 s _ 0 , n + m ) - \\nu ( s + m - 2 s _ 0 , n + 2 r _ n ) , \\\\ \\nu ( s - 2 s _ 0 + j - 1 , n - m - 1 ) & = \\nu ( s + m - 2 s _ 0 + j - 1 , n - 1 ) - \\nu ( s - 2 s _ 0 + j - 1 , n + 2 r _ n - m - 1 ) . \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} C _ { \\textnormal { f } , 0 } ^ { ( 2 ) } ( \\Lambda ) = 1 - \\Lambda , 0 \\leq \\Lambda \\leq 1 . \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } - u _ { t t } - u _ { x x } & = & u ( r _ u - \\gamma _ u ( u + v ) ) + \\mu v - \\mu u \\\\ - v _ { t t } - v _ { x x } & = & v ( r _ v - \\gamma _ v ( u + v ) ) + \\mu u - \\mu v , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} d _ y ^ * \\circ d _ y = \\rho B _ y - B _ y ( y ^ { - 1 } \\rho ) \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } \\Bigl ( \\int _ 0 ^ s P ( x u ) d u \\Bigr ) d F _ 1 ( x ) = \\int _ 0 ^ { \\infty } \\Bigl ( \\frac { 1 } { x } \\int _ { 0 } ^ { s x } P ( v ) d v \\Bigr ) d F _ 1 ( x ) \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} b = 2 \\Re \\left [ R - \\P [ R \\bar Y ] \\right ] , \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} | \\xi | ^ { k } e ^ { - \\frac { \\nu t | \\xi | ^ { 2 } } { 2 } } | \\cos ( t | \\xi | \\phi _ { \\sigma } ( \\xi ) ) - \\cos ( t | \\xi | ) | \\chi _ { L } \\le C e ^ { - ( 1 + t ) | \\xi | ^ { 2 \\sigma } } t | \\xi | ^ { k + 4 \\sigma - 1 } \\chi _ { L } , \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} \\int _ \\Omega \\left [ | \\nabla ( w _ 1 - w _ 2 ) | ^ 2 + \\left ( r ( x , 0 ) - \\frac 1 2 \\sum _ { i = 1 } ^ n \\frac { \\partial q _ i } { \\partial x _ i } ( x , 0 ) \\right ) ( w _ 1 - w _ 2 ) ^ 2 \\right ] \\ , d x = 0 \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} E _ c ( u ) = \\langle L u , u \\rangle _ { L ^ 2 } , \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} a _ { \\xi } ( y , s ) = c _ { \\xi } ( s ) \\cdot B _ { \\xi } ( y ; p ; s ) , \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} J ( \\xi ) = \\frac { 4 } { 3 } \\pi _ 1 ( \\xi ) + \\pi _ 7 ( \\xi ) - \\pi _ { 2 7 } ( \\xi ) \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} e _ { ( m + 1 ) p _ n + \\frac { m - 1 } { 2 } } ( x ) = e _ { ( m + 1 ) p _ n - \\frac { m + 1 } { 2 } } ( x ) + x e _ { ( m + 1 ) p _ n - \\frac { m - 1 } { 2 } } ( x ) . \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} \\beta _ p = \\frac { c \\beta _ c } { \\max ( 1 + \\alpha _ v , \\alpha _ p ^ { - 1 } ) } . \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} \\psi ( \\alpha ) \\star \\psi ( \\beta ) = \\lfloor \\phi ( p _ { i } ) ; h _ { i } \\rceil \\star \\lfloor \\phi ( p _ { j } ) ; h _ { j } \\rceil = \\lfloor \\gamma _ { 2 } \\phi ( p _ { i } ) \\phi ( p _ { j } ) ; h _ { i } + h _ { j } \\rceil = \\lfloor \\gamma _ { 2 } \\phi ( p _ { i } p _ { j } ) ; h _ { i } + h _ { j } \\rceil . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} q ( \\tau ) = \\sum _ { i = 1 } ^ N a _ i l _ i ( \\tau ) \\mbox { w h e r e } a _ i = \\left \\{ \\begin{array} { r l } 1 & \\mbox { i f } \\displaystyle { \\int _ { - 1 } ^ t } l _ i ( \\tau ) \\ ; d \\tau > 0 , \\\\ - 1 & \\mbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "6625.png", "formula": "\\begin{align*} C C R f _ * { \\cal F } = f _ * C C { \\cal F } \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\omega ( \\nabla _ X Y , Z ) = \\omega ( \\tilde \\nabla _ X Y , Z ) + \\frac { 1 } { 3 } ( \\tilde { \\nabla } _ X \\omega ) ( Y , Z ) + \\frac { 1 } { 3 } ( \\tilde { \\nabla } _ Y \\omega ) ( X , Z ) . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} T : = \\mathfrak { B } _ 2 + \\frac { 1 } { 2 } [ \\mathfrak { B } _ 1 , \\overline { A } _ 1 ] , R _ 4 : = \\frac { \\varepsilon ^ 2 } { 2 } [ \\mathfrak { B } _ 1 , \\tilde { A } _ 1 ] + \\tilde { R } _ 4 . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} 0 > \\gamma _ { i } ^ { + } \\geq \\gamma _ { i } ^ { - } \\geq - 1 , \\mbox { f o r } i = 1 , 2 . \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} \\zeta _ { \\mathrm { u } , i } = \\mathrm { t r } \\left [ \\left ( \\hat { \\mathbf { H } } _ i \\hat { \\mathbf { H } } _ i ^ { \\mathrm { H } } \\right ) ^ { - 1 } \\right ] . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { \\varphi ^ * ( n ) } { n ^ s } = \\left ( 1 - \\frac 1 { 2 ^ { s - 2 } } + \\frac 1 { 2 ^ { 2 s - 1 } } \\right ) \\left ( 1 - \\frac 1 { 2 ^ { s - 1 } } + \\frac 1 { 2 ^ { 2 s - 1 } } \\right ) ^ { - 1 } \\sum _ { n = 1 } ^ { \\infty } \\frac { \\varphi ^ * ( n ) } { n ^ s } ( \\Re s > 2 ) , \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} \\lim _ { \\substack { m \\to \\infty \\\\ \\frac { \\omega _ { m } } { \\log ( m ) } \\to \\gamma } } \\frac { d _ { n , k } ^ { m , \\omega _ { m } } } { d ^ { m , \\omega _ { m } } } = V ^ { 2 I B P , \\theta } _ { n , k } ( \\gamma ) . \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\nu ( A _ { x } ^ { - 1 } C ) = \\eta \\otimes \\eta ( G ( A \\times C ) ) = m _ K ( I ^ { - 1 } J ) = m _ M ( I _ o ^ { - 1 } J _ o ) , \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} Q _ { n + 1 } ( x , q ) = ( q + 2 n x ) Q _ n ( x , q ) + 2 x ( 1 - x ) \\frac { \\partial } { \\partial x } Q _ n ( x , q ) \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{gather*} \\sum _ { j = 1 } ^ n m _ j = 1 . \\end{gather*}"}
-{"id": "6079.png", "formula": "\\begin{align*} E _ 1 ^ { ( - ) } = - 6 a , \\psi _ 1 ^ { ( - ) } ( x ) = e ^ { - \\frac { 1 } { 3 } | x | ^ 3 + a x ^ 2 } x , \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} L ( x ^ n ) & = \\{ 3 , 4 , 5 , \\ldots , n - 4 \\} \\cup \\{ n - 2 , n \\} \\\\ L ( x ^ n ) & = \\{ 2 , 3 , 4 , 5 , \\ldots , n - 4 \\} \\cup \\{ n - 2 , n \\} \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\mathrm { e r f } ( x ) = \\frac { 2 } { \\sqrt { \\pi } } \\int _ { 0 } ^ { x } e ^ { - t ^ 2 } \\mathrm { d } t . \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} F _ n ^ { \\left ( 2 \\right ) } = 1 , \\ , \\ , \\forall n \\ge 0 , \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} \\liminf _ { m \\to \\infty } \\int _ { D ^ { ( s ) } } \\left | h / \\sqrt { m } \\right | \\ , \\mathrm d \\rho ^ { ( s ) } \\geq E ( \\lvert X \\rvert ) = \\sqrt { 2 / \\pi } . \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} \\mathcal { G } _ q ( \\mathcal { H } , r , \\alpha ) : = \\bigoplus _ { n = 0 } ^ \\infty \\mathcal { G } ^ { ( n ) } _ q ( \\mathcal { H } , r , \\alpha ) , \\mathcal { G } ^ { ( n ) } _ q ( \\mathcal { H } , r , \\alpha ) : = \\mathcal { H } ^ { \\otimes n } r ^ n ( [ n ] _ q ! ) ^ \\alpha . \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} ( \\tilde { A } _ 1 ) _ j ^ { j ' } ( l ) : = - \\frac { ( 2 c _ 2 \\ , j \\ , ( j - j ' ) ^ 2 + 6 \\ , c _ 3 \\ , j ) \\sqrt { \\lvert j - j ' \\rvert \\xi _ { j - j ' } } \\{ ( \\omega - \\overline { \\omega } ) \\cdot l + ( m _ 3 - 1 ) ( j '^ 3 - j ^ 3 ) \\} } { ( \\omega \\cdot l + m _ 3 ( j '^ 3 - j ^ 3 ) ) ( \\overline { \\omega } \\cdot l + j '^ 3 - j ^ 3 ) } \\end{align*}"}
-{"id": "8859.png", "formula": "\\begin{align*} I _ { u p p e r } ( f , w ) = \\log { \\left [ \\frac { | f ( w ) | } { ( 1 - | f ( w ) | ) ^ 2 } \\cdot \\frac { ( 1 - | w | ) ^ 2 } { | w | | f ' ( 0 ) | } \\right ] } \\leq 0 \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} \\int _ { C _ j } x \\ , d g _ k = - \\int _ { C _ j } g _ k \\ , d x \\ , \\approx \\ , \\delta _ { j , k } . \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\varpi = d \\tau + \\pi _ k ^ * \\omega , \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} \\int _ { x _ \\mu + \\delta _ 1 } ^ { x _ { \\frac 3 2 \\mu } } \\frac { Q ' \\ , \\dd x } { Q - \\mu } = \\log \\frac { \\mu } { 2 ( Q ( x _ \\mu + \\delta _ 1 ) - Q ( x _ \\mu ) ) } \\leq C \\log \\frac { \\mu } { a _ \\mu \\delta _ 1 } \\leq C \\log ( \\mu a _ \\mu ^ { - \\frac 2 3 } ) . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{align*} \\frac { \\partial \\lambda _ { i a } } { \\partial N _ { j b } } = \\frac { 1 } { L s _ a \\rho ( \\lambda _ { i a } ) } [ \\delta _ { i a , j b } - s _ a F ( \\lambda _ { i a } | \\lambda _ { j b } ) ] \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\Psi ( z ) = \\int _ { - \\infty } ^ \\infty \\Phi ( t ) \\cos ( t z ) d t \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} & \\sigma _ z ^ { - j - 1 } q _ j \\psi = ( - 1 ) ^ j \\sigma _ z ^ { - 1 } ( L \\sigma _ z ^ { - 1 } ) ^ j \\psi \\\\ & = ( - 1 ) ^ j \\sum _ { a + b + c = j } \\sigma _ z ^ { - 1 - a - 2 b - 3 c } p _ { a , b , c } ( L , [ L , c ( \\delta u ) ^ 2 ] , [ [ L , c ( \\delta u ) ^ 2 ] , c ( \\delta u ) ^ 2 ] ) \\psi , \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} A u ' = ( Q ' ( u ^ \\pm ) \\pm \\eta A ) u \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} \\pi _ l = \\pi _ { ( p ( l + n - 1 ) ^ p ) } ( x _ 1 , \\dots , x _ p ) = \\left ( e _ p ( x _ 1 , \\dots , x _ p ) ^ { p } \\right ) ^ { l + n - 1 } = ( x _ 1 \\cdots x _ p ) ^ { p ( l + n - 1 ) } . \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} \\partial _ { \\lambda } F ( \\lambda | \\lambda ' ) = L ( \\lambda | \\lambda ' ) - \\sum _ { i a } s _ a L ( \\lambda | \\lambda _ { i a } ) F ( \\lambda _ { i a } | \\lambda ' ) \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} \\psi _ { n , p } = e ^ { - \\imath E _ n t + \\imath p \\phi } | r | ^ { | p | / k } e ^ { - ( 1 / 2 ) \\omega r ^ 2 } L _ n ^ { | p | / k } ( \\omega r ^ 2 ) , ( n \\in \\mathbb { N } , p \\in \\mathbb { Z } ) \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\underline { Z } } \\exp \\left \\{ \\frac { 1 } { \\gamma } \\sum _ { i = 1 } ^ n Z _ i ( g ^ 2 ( X _ i ) - g ^ 2 ( X ^ { \\prime } _ i ) ) \\right \\} & = \\prod _ { i = 1 } ^ n \\mathbb { E } _ { Z _ i } \\exp \\left \\{ \\frac { 1 } { \\gamma } Z _ i ( g ^ 2 ( X _ i ) - g ^ 2 ( X ^ { \\prime } _ i ) ) \\right \\} . \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} G _ i : = \\frac { \\rho _ i L } { \\mu _ i \\tau _ 0 } G , i = 1 , 2 , \\mu : = \\frac { \\mu _ 2 } { \\mu _ 1 } \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} w ^ n _ { 1 , - Q ^ n ( 0 ) + 1 } & = t ^ n _ { - 1 , Q ^ n _ 1 ( 0 ) - R _ { 1 , - Q ^ n ( 0 ) + 1 } ( 0 ) } , \\\\ w ^ n _ { 1 , 1 } & = \\left ( t ^ n _ { - 1 , 1 + Q ^ n _ 1 ( 0 ) - R ^ n _ 1 ( t ^ n _ { 1 , 1 } - ) + R ^ n _ { - 1 } ( t ^ n _ { 1 , 1 } - ) } - t ^ n _ { 1 , 1 } \\right ) ^ + . \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} \\ell ( u ) = \\sum _ { n = 0 } ^ { \\infty } \\dfrac { 1 } { n ! } \\int _ { 0 } ^ { 1 } H _ { n } \\left ( \\dfrac { u } { \\sigma ( t ) } \\right ) p _ { \\sigma ( t ) ^ { 2 } } ( u ) H _ { n } \\left ( \\dfrac { x ( t ) } { \\sigma ( t ) } \\right ) \\mathrm { d } t \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} \\begin{array} { c c c } R = W + A \\odot g , & A = \\dfrac { 1 } { n - 2 } \\left ( r - \\dfrac { s \\ , g } { 2 ( n - 1 ) } \\right ) & \\end{array} \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} S _ { n , i } : = ( X _ n , Y _ n , Z _ n , X _ { n , i } , Y _ { n , i } ) \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\left ( \\partial _ { \\tau } + 1 - \\frac { \\kappa \\hbar ^ 2 } { 2 } \\right ) u ( \\cdot , 0 ) = 0 \\ , , \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} Z : K _ 0 ( C ) \\to \\C , \\ , \\ , Z ( E ) = - d ( E ) + \\sqrt { - 1 } \\ , r ( E ) \\end{align*}"}
-{"id": "7781.png", "formula": "\\begin{align*} ( f ^ { ( n ) } , g ^ { ( n ) } ) _ { \\mathbb { F } ^ { ( n ) } _ q ( \\mathcal { H } ) } : = ( P _ q ^ { ( n ) } f ^ { ( n ) } , P _ q ^ { ( n ) } g ^ { ( n ) } ) _ { \\mathcal { H } ^ { \\otimes n } } \\ , . \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} F ( X \\oplus Y ) = F ( X ) \\oplus \\operatorname { c r } _ 2 ( F ) ( X , Y ) \\oplus F ( Y ) , \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} \\Delta _ D ( f , g ) = - \\Delta ( h ( f _ 0 , g _ 0 ) ; f , g ) + \\delta ( f , g \\left | D _ f \\times D _ g \\right . ) \\rightarrow \\inf , \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} \\hat u ( t , r ) = t - \\gamma ( r ) , \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} \\lambda _ n = \\left ( \\frac { \\pi } { \\Omega _ \\beta } \\left ( n + \\frac 1 2 \\right ) \\right ) ^ \\frac { 2 \\beta } { \\beta + 2 } + \\frac { 1 } { \\Omega _ \\beta ' } \\left ( \\frac { \\pi } { \\Omega _ \\beta } n \\right ) ^ { - \\frac { 2 } { \\beta + 2 } } \\int _ \\R V ( x ) \\ ; \\dd x + o \\left ( n ^ { - \\frac { 2 } { \\beta + 2 } } \\right ) . \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\omega _ { \\beta , \\mu , \\Lambda , \\lambda } ( \\eta ( b _ { { 0 } } ^ { * } ) \\eta ( b _ { { 0 } } ) ) = \\omega _ { \\beta , \\mu , \\Lambda , - \\lambda } ( \\eta ( b _ { { 0 } } ^ { * } ) \\eta ( b _ { { 0 } } ) ) \\ . \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} f _ 2 '' ( B _ n ) & = \\frac { B _ n ^ { - \\frac 3 2 } } { { 4 \\sqrt { \\pi } } } e ^ { - \\left ( \\sqrt { B _ n } - \\sqrt { P } \\right ) ^ 2 } \\left ( 2 B _ n - 2 \\sqrt { P B _ n } + 1 \\right ) , \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} H _ B ^ \\omega u = - \\Delta u + 2 i A \\cdot \\nabla u + ( A ^ 2 + \\lambda V _ \\omega ) u , \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} \\mathbf { P } _ { \\rm w } = \\mathbf { P } _ { \\rm w } \\mathbf { A } ^ { - 1 } \\mathbf { Q } \\mathbf { P } ^ { - 1 } _ f \\mathbf { P } _ 2 . \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} \\Phi ^ { \\prime \\prime } _ w ( 1 ) = a \\| w \\| ^ 2 + 3 \\varepsilon \\| w \\| ^ 4 - ( q - 1 ) \\lambda \\int _ { \\Omega \\times \\{ 0 \\} } f ( z ) | w ( z , 0 ) | ^ q d z - { ( 2 ^ * _ \\alpha - 1 ) } \\int _ { \\Omega \\times \\{ 0 \\} } | w ( z , 0 ) | ^ { 2 ^ * _ \\alpha } d z . \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} \\begin{array} { l } u ^ { m _ t } ( s , y ) = \\int _ { \\mathcal { C } _ t ^ d } K ( y - X _ s ( \\omega ) ) \\exp \\left ( \\int _ 0 ^ s \\Lambda ( r , X _ r ( \\omega ) , u ^ { m _ t } ( r , X _ r ( \\omega ) ) ) d r \\right ) m _ t ( d \\omega ) \\ \\forall s \\in [ 0 , t ] \\ . \\end{array} \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} D = 1 + \\underset { N ' \\rightarrow \\infty } { \\lim } \\left \\{ \\frac { \\ln \\left [ N ' \\cdot \\sqrt { \\nu ^ { 2 } + \\frac { 1 } { { N ' } ^ 2 } } \\right ] } { \\ln ( 2 \\cdot N ' ) } \\right \\} = 2 \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} T ^ * _ E B u n _ G = H ^ 0 ( X \\times S , a d ( E ) \\otimes \\Omega ^ 1 _ { X \\times S / S } ) \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} A ^ a ( \\xi , 0 ) = \\frac { i } { \\Lambda ( \\xi ) ( e ^ { 2 \\xi } + 1 ) } \\left [ 2 J ( \\xi ) - \\xi J ' ( \\xi ) + J ( \\xi ) J ' ( \\xi ) \\right ] . \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} M ( r ) = \\sup \\left \\{ \\frac { | h ( x ) | } { \\log ( | x | ) } : | x | \\ge r \\right \\} \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} 2 = \\beta \\pi ^ 8 \\beta = 1 + \\pi ^ 6 + \\pi ^ 9 + \\pi ^ { 1 1 } + O ( \\pi ^ { 1 2 } ) . \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} B ^ s _ x \\otimes _ { R ^ s } R = B _ x . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} \\partial _ t z + A _ G ( z + u _ * ) z = 0 , z ( 0 ) = u ^ 0 - u _ * . \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{align*} d i m ( W ) = d i m ( G ) ( g - 1 ) + \\Sigma _ i ( d _ i - 1 ) = d i m ( G ) ( g - 1 ) + d i m ( G / B ) \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} \\operatorname { d i a m } ( G _ { X } ) & \\leq \\lbrack 1 + f ( a _ { 1 } ) ] + [ f ( a _ { 1 } ) + f ( a _ { 2 } ) ] + \\cdots + [ f ( a _ { t - 1 } ) + f ( a _ { t } ) ] + [ f ( a _ { t } ) + 1 ] \\\\ & = 2 { \\textstyle \\sum _ { i = 1 } ^ { t } } f ( a _ { i } ) + 2 . \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} f ( z ) \\mid _ { - 1 } A = \\sum _ { n = n _ 0 } ^ { \\infty } a _ 0 ( n ) q _ { 2 4 } ^ { n } \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} _ { \\tau } ( \\xi _ { n } ^ { ( 0 ) } ) \\left ( \\begin{array} { c } \\Psi _ { \\tau } ( h _ { 1 } , . . . , h _ { n } = 0 , . . . , h _ { \\mathsf { N } } ) \\\\ \\Psi _ { \\tau } ( h _ { 1 } , . . . , h _ { n } = 1 , . . . , h _ { \\mathsf { N } } ) \\\\ \\vdots \\\\ \\Psi _ { \\tau } ( h _ { 1 } , . . . , h _ { n } = p - 1 , . . . , h _ { \\mathsf { N } } ) \\end{array} \\right ) = \\left ( \\begin{array} { c } 0 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} u ( x , t _ j ) \\ge \\int _ { { \\bf R } ^ N } G ( x - y , t _ j - s _ { j ' } ) u ( y , s _ { j ' } ) \\ , d y , j = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} d \\omega = \\theta \\wedge \\omega \\end{align*}"}
-{"id": "8928.png", "formula": "\\begin{align*} \\{ 1 , \\dots , n \\} = [ 1 , n _ 1 ] \\cup [ n _ 1 + 1 , n _ 1 + n _ 2 ] \\cup \\dots [ \\sum _ { i = 1 } ^ { r - 1 } n _ i + 1 , n ] . \\end{align*}"}
-{"id": "8362.png", "formula": "\\begin{align*} \\vartheta ( - i \\nabla _ X ) \\vartheta ^ { - 1 } = + i \\nabla _ X . \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} d t = \\frac { d q } { \\gamma } = \\frac { d \\gamma } { \\frac { k } { q ^ 3 } - \\omega ^ 2 ( t ) q } . \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} \\begin{cases} u a ' _ 1 & = a _ 1 + 2 s \\\\ u ^ 2 a ' _ 2 & = a _ 2 - s a _ 1 + 3 r - s ^ 2 \\\\ u ^ 3 a ' _ 3 & = a _ 3 + r a _ 1 + 2 t \\\\ u ^ 4 a ' _ 4 & = a _ 4 - s a _ 3 + 2 r a _ 2 - ( t + r s ) a _ 1 + 3 r ^ 2 - 2 s t \\\\ u ^ 6 a ' _ 6 & = a _ 6 + r a _ 4 + r ^ 2 a _ 2 + r ^ 3 - t a _ 3 - t ^ 2 - r t a _ 1 . \\end{cases} \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} ( 1 - N ) - 2 4 \\sum _ { n > 0 } a ( - n ) ( \\sigma _ { 1 } ( n ) - N \\sigma _ { 1 } ( n / N ) ) = 0 . \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} \\widetilde { R } ^ { \\mathrm { L } } _ 1 = { \\log _ 2 } \\left ( 1 + \\frac { { G _ { } ^ S G _ { } \\beta } r ^ { - { \\overline { \\alpha } } } } { { { \\lambda \\widetilde { \\Lambda } + \\frac { N _ o } { \\mu P _ t } } } } \\right ) , \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} Z ( \\widetilde { G } ) = Z _ { 0 , D _ 0 } ( G ) ^ 2 + Z _ { 1 , D _ 0 } ( G ) ^ 2 \\ , . \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} E [ g ( W , \\gamma , \\theta _ { 0 } ) ] & = E [ E [ D Y | X ] \\{ \\gamma ( X ) - \\gamma _ { 0 } ( X ) \\} ] = E [ \\alpha _ { 0 } ( X ) \\gamma _ { 0 } ( X ) ^ { - 1 } \\{ \\gamma ( X ) - \\gamma _ { 0 } ( X ) \\} ] \\\\ & = E [ \\alpha _ { 0 } ( X ) \\{ \\gamma ( X ) P _ { 0 } ( X ) - 1 \\} ] = E [ \\alpha _ { 0 } ( X ) \\{ \\gamma ( X ) D - 1 \\} ] = - E [ \\alpha _ { 0 } ( X ) \\lambda ( W , \\gamma ) ] . \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{align*} \\Sigma _ i i m _ i = \\frac { 1 } { 2 } ( \\Sigma _ j \\tilde { m } _ j ^ 2 + \\tilde { m } _ j ) . \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} V _ i / I = \\overline W _ { i , 1 } = W _ { i , 1 } \\oplus \\cdots \\oplus W _ { i , 1 } . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} n ^ { - 1 / 2 } \\sum _ { r = \\lceil p n \\rceil } ^ n \\exp ( - n \\epsilon ( r / n - p ) ^ 2 ) & \\le n ^ { - 1 / 2 } \\sum _ { r = \\lceil p n \\rceil } ^ { \\lceil p n \\rceil + \\ell - 1 } \\sum _ { m = 0 } ^ \\infty \\exp ( - n \\epsilon ( ( r + m \\ell ) / n - p ) ^ 2 ) . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} a _ j = \\frac { 1 } { \\Delta x } \\int _ { x _ { j - 1 / 2 } } ^ { x _ { j + 1 / 2 } } a ( x ) \\ , d x , j = 1 , \\dots , N _ x , \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} \\begin{aligned} b _ 3 - m _ 3 = ( b _ 3 - 1 ) - M _ { \\varphi } [ b _ 3 - 1 ] \\stackrel { \\eqref { M i c h e l a } } { = } \\Upsilon [ M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] ] - M _ { \\varphi } [ \\Upsilon [ M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] ] ] . \\end{aligned} \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} f ( w ( 1 ) ) = \\mathbb { E } f ( w ( 1 ) ) + \\int ^ 1 _ 0 \\partial _ x P _ { 1 - t } f ( w ( t ) ) d w ( t ) . \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { N - 1 } S _ 2 \\left ( n _ i \\right ) = \\sum _ { 0 \\leq k \\leq _ b n } \\prod _ { i = 0 } ^ { N - 1 } \\binom { n _ i } { k _ i } f ( n _ i , k _ i ) = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } _ b \\prod _ { i = 0 } ^ { N - 1 } f ( n _ i , k _ i ) . \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} D ( \\eta ) = \\max _ { \\eta \\in \\Lambda } D ( \\eta ) . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\Big | \\int _ { \\mathbb { R } ^ { n + 1 } } \\prod _ { i = 0 } ^ { n } F _ i ( x _ 0 , \\dots , x _ { i - 1 } , x _ { i + 1 } , \\dots , x _ n ) \\varphi ( x _ 0 + \\dots + x _ n ) d x _ 0 \\dots d x _ n \\Big | \\leq \\| \\varphi \\| _ 1 \\| F _ 0 \\| _ { { 2 ^ n } } \\prod _ { i = 1 } ^ n \\| F _ i \\| _ { { 2 ^ { n - i + 1 } } } \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} \\begin{aligned} & A \\psi _ k = \\mu _ k \\psi _ k , \\| \\psi _ k \\| = 1 , \\\\ & \\exists \\gamma > 0 , \\ \\exists \\kappa > 0 , \\ \\exists N _ 0 > 0 , \\ \\forall k \\geq N _ 0 , \\mu _ { k + 1 } - \\mu _ k \\geq \\kappa k ^ { \\gamma - 1 } ; \\end{aligned} \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} V ( x ) : = e ^ { \\frac { 1 } { 2 } - \\frac { x ^ 2 } { 4 } } , x \\in \\mathbb { R } . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } g \\left ( f - P _ M f \\right ) ^ { < \\alpha - 1 > } d \\mu = 0 , g \\in M . \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\alpha ( 2 , 5 ) & = \\alpha ( 2 , 7 ) = 1 . \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\frac { \\varphi ( n ) } { n ^ s } = \\prod _ { p \\in \\P } \\sum _ { \\nu = 0 } ^ { \\infty } \\frac { \\varphi ( p ^ \\nu ) } { p ^ { \\nu s } } . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} ( \\bold { D } ^ { \\vartriangle } _ { \\bold { E } ^ { \\vartriangle } ( L _ n ) } ) ^ { \\ast } = \\bold { D } ^ { \\vartriangle } _ { \\bold { E } ^ { \\vartriangle } ( L _ n ) } . \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{align*} \\sum _ { k = \\ell + 1 } ^ n ( - 1 ) ^ k c _ { k i } \\Delta _ { I } ^ { E _ { \\ell } \\cup \\{ k \\} } = 0 \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{gather*} \\nu = r _ 1 - \\frac { q _ 2 } { 2 } , M = \\begin{pmatrix} r _ 2 ^ 2 - r _ 0 - \\dfrac { q _ 1 } { 2 } & \\tilde { \\alpha } \\\\ \\alpha & r _ 0 - r _ 2 ^ 2 + \\dfrac { q _ 1 } { 2 } \\end{pmatrix} . \\end{gather*}"}
-{"id": "4057.png", "formula": "\\begin{align*} T ( \\lambda ) & = 8 N ( \\lambda ) + 4 T _ { x _ { 1 } } ^ { + } ( \\lambda ) + 4 T _ { x _ { 2 } } ^ { + } ( \\lambda ) + 4 T _ { x _ { 3 } } ^ { + } ( \\lambda ) \\\\ & \\ \\ \\ + 2 \\left \\lfloor { \\frac { a _ { 1 } \\lambda ^ { 1 / 2 } } { \\pi } } \\right \\rfloor + 2 \\left \\lfloor { \\frac { a _ { 2 } \\lambda ^ { 1 / 2 } } { \\pi } } \\right \\rfloor + 2 \\left \\lfloor { \\frac { a _ { 3 } \\lambda ^ { 1 / 2 } } { \\pi } } \\right \\rfloor + 1 , \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\mathcal { R } ( U ) \\subseteq \\mathcal { R } ( A ) & \\Longleftrightarrow E _ { A } U = 0 , \\mathcal { R } ( V ) \\subseteq \\mathcal { R } ( A ^ { \\ast } ) \\Longleftrightarrow V ^ { \\ast } F _ { A } = 0 , \\\\ \\mathcal { R } ( U ^ { \\ast } ) \\subseteq \\mathcal { R } ( S _ { A } ) & \\Longleftrightarrow U E _ { S _ { A } } = 0 , \\mathcal { R } ( V ^ { \\ast } ) \\subseteq \\mathcal { R } ( S _ { A } ^ { \\ast } ) \\Longleftrightarrow F _ { S _ { A } } V ^ { \\ast } = 0 . \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} \\| u \\| _ { X _ 0 } = \\left ( \\int _ Q \\frac { | u ( x ) - u ( y ) | ^ { p } } { | x - y | ^ { n + p s } } d x \\ , d y \\right ) ^ { 1 / p } . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} \\theta _ { 1 } ^ { ( 1 ) } = \\pm \\sqrt { \\frac { c + 2 } { c } } , \\ \\theta _ { 2 } ^ { ( 1 ) } = \\pm \\sqrt { \\frac { c - 2 } { c } } , \\ \\theta _ { 1 } ^ { ( 2 ) } = - \\theta _ { 1 } ^ { ( 1 ) } , \\ \\theta _ { 2 } ^ { ( 2 ) } = - \\theta _ { 2 } ^ { ( 1 ) } , \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} Q _ t \\gamma ( y ) : = \\sum _ { 0 \\le j < k \\le n } \\int _ { [ 0 , t ] ^ 2 } \\gamma ( B ^ j _ { 0 , t } ( s ) - B ^ k _ { 0 , t } ( r ) + \\frac s t y ^ j - \\frac r t y ^ k ) { | s - r | ^ { - \\alpha _ 0 } } d r d s . \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} \\sum _ { \\substack { a \\bmod q \\\\ ( a , q ) = 1 } } M ( q , a ) = \\gamma + B - \\sum _ { p | q } \\frac { 1 } { p } . \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} \\theta ^ { ( a ) } 1 _ n : = \\frac { \\theta ^ a 1 _ n } { [ a ] _ v ! } , \\vartheta ^ { ( a ) } 1 _ n : = \\frac { \\vartheta ^ a 1 _ n } { [ a ] _ v ! } , \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} ( - \\underbar { q } , & - \\underbar { p } ) \\cdot ( \\dot { p } , \\dot { q } ) \\\\ & = - \\underbar { p } \\underbar { q } ( 1 - \\underbar { q } ) - p \\frac { \\underbar { q } } { m } \\Big ( \\frac { 2 m } { 1 + m } - \\frac { m } { \\lambda } \\big ( 1 - \\frac { 2 } { 1 + m } \\big ) - \\underbar { q } \\Big ) \\\\ & \\ge - \\underbar { p } \\underbar { q } ( 1 - \\underbar { q } ) \\\\ & = : \\delta > 0 . \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} e _ { n + d } = a _ { n + d } + A _ { d - 1 } a _ { n + d - 1 } + \\dots + A _ 0 a _ n \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{gather*} \\operatorname { g h } ( C ^ \\ast ) = - 2 , \\operatorname { g h } ( A ^ \\ast ) = - 1 , \\qquad \\operatorname { g h } ( A ) = 0 , \\operatorname { g h } ( C ) = 1 . \\end{gather*}"}
-{"id": "6077.png", "formula": "\\begin{align*} E _ 0 ^ { ( + ) } = - 2 a , \\psi _ 0 ^ { ( + ) } ( x ) = e ^ { - \\frac { 1 } { 3 } | x | ^ 3 + a x ^ 2 } , \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} \\sum _ { \\ell = p + 1 } ^ \\infty \\psi ( 1 , \\ell ) \\left | \\mathcal { E } ( 1 , \\ell ) - \\mathcal { E } _ n ( 1 , \\ell ) \\right | & \\le \\sum _ { \\ell = 0 } ^ \\infty \\left | f _ n ^ \\ast ( \\ell ) - f ^ \\ast ( \\ell ) \\right | . \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} | | \\xi _ n - \\hat { \\xi } _ n | | _ { \\alpha } ^ { \\alpha } = \\left [ \\xi _ n , \\xi _ n - \\hat { \\xi } _ n \\right ] _ { \\alpha } - \\left [ \\hat { \\xi } _ n , \\xi _ n - \\hat { \\xi } _ n \\right ] _ { \\alpha } = \\left [ \\xi _ n , \\xi _ n - \\hat { \\xi } _ n \\right ] _ { \\alpha } . \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\kappa _ { a } ^ { \\left ( h \\right ) } = k ( \\zeta _ { a } ^ { ( h ) } ) , h \\in \\{ 0 , . . . , p - 1 \\} , a \\in \\{ 1 , . . . , 2 \\mathsf { N } \\} , \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} & \\big ( m - \\big ( \\sum _ { j = 1 } ^ { d } \\big ( \\frac { 1 } { 2 } \\big ) ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } \\big ) ^ { p / g ( { \\bf R } ) } _ + \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) \\leq C ( m , { \\bf R } , d ) \\\\ & \\qquad \\leq \\big ( m + \\big ( \\sum _ { j = 1 } ^ { d } \\big ( \\frac { 1 } { 2 } \\big ) ^ { 2 R _ j } \\big ) ^ { 1 / ( 2 p ) } \\big ) ^ { p / g ( { \\bf R } ) } \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} \\tilde { g } ^ { k \\overline { l } } \\tilde { g } ^ { i \\overline { j } } \\overline { e } _ { l } e _ { i } ( f ) e _ { k } \\overline { e } _ { j } ( f ) \\geq & ~ ~ \\frac { 1 } { n } \\left ( \\tilde { g } ^ { i \\overline { j } } e _ { i } \\overline { e } _ { j } ( f ) \\right ) ^ { 2 } \\\\ \\geq & \\left ( 1 - \\frac { \\epsilon } { 5 } \\right ) \\frac { ( L f ) ^ { 2 } } { n } - \\frac { C } { \\epsilon } | \\partial f | _ { \\tilde { g } } ^ { 2 } . \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} 0 = J _ 0 \\subsetneq J _ 1 \\subsetneq \\cdots \\subsetneq J _ { m - 1 } \\subsetneq J _ m = A , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} ( \\theta _ { 1 / 2 } , \\theta _ { 1 / 2 } ) = \\frac { \\pi ( N + 1 ) } { 3 \\sqrt { N } } ( \\theta _ { 3 / 2 } , \\theta _ { 3 / 2 } ) = \\frac { \\sqrt { N } ( N - 1 ) } { 6 } . \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} | A ( z ) | = \\sum _ { n \\le z } g ( n ) \\le 3 \\frac z { \\log z } \\sum _ { n \\in A ( z ) } \\frac { 1 } { n } . \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\begin{aligned} R ^ + & = \\begin{pmatrix} r _ n ^ + & r _ { n - 1 } ^ + & \\dots & r _ 1 ^ + \\end{pmatrix} \\\\ S ^ + ( \\lambda ) & = \\begin{pmatrix} \\mu _ 1 ^ + r _ n ^ + & \\mu _ 2 ^ + r _ { n - 1 } ^ + & \\dots & \\mu _ n ^ + r _ 1 ^ + \\end{pmatrix} . \\end{aligned} \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} f = \\sum _ { \\nu } g _ { \\nu } ( { } _ { ( \\alpha ) } a _ { s t } ) ( J _ { \\nu } | I _ { \\nu } ) _ { \\mathbf { \\Phi _ { \\phi _ 1 } } \\mathbf { \\Phi _ { \\phi _ 2 } } \\cdots \\mathbf { \\Phi _ { \\phi _ l } } } , \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} F ( \\phi ) = \\int _ X ( \\phi - \\phi _ 0 ) d \\mu _ X + \\int _ { X ^ * } ( \\phi ^ * - \\phi _ 0 ^ * ) d \\nu _ X \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} z L ^ { - 1 } \\partial _ x ^ { - 1 } y _ { \\rm o d d } = 2 y _ { \\rm e v e n } , - z \\partial _ x ^ { - 1 } y _ { \\rm e v e n } = 2 L y _ { \\rm o d d } , \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ { \\infty } \\sum _ { \\lambda \\vdash n } \\sum _ { \\substack { \\lambda _ i \\in \\lambda \\\\ \\lambda _ i } } a ( \\lambda _ i ) q ^ n = \\sum _ { n = 0 } ^ { \\infty } p ( n ) q ^ n \\sum _ { n = 1 } ^ { \\infty } a ( n ) q ^ n , \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\sigma _ 0 ( x _ 1 , x _ 2 ) & = \\left \\{ \\begin{array} { l l } 0 & \\textrm { i f } \\ - 1 < x _ 1 \\leq 0 \\\\ - 2 x _ 1 x _ 2 & \\textrm { i f } \\ 0 < x _ 1 < \\frac { 1 } { 2 } \\\\ - x _ 2 & \\textrm { i f } \\frac { 1 } { 2 } \\leq x _ 1 < 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{align*} x _ { 2 } ^ { 2 } = \\dfrac { a ^ { 4 } x _ { 1 } ^ { 2 } } { ( a ^ { 2 } - b ^ { 2 } ) x _ { 1 } ^ { 2 } + b ^ { 2 } } \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} \\sum _ { i j k \\ell } N _ { i j k \\ell } t _ { i j k \\ell } = \\frac 1 2 \\sum _ { i j k \\ell } N _ { i j } N _ { k \\ell } t _ { i j k \\ell } - \\frac 1 2 \\sum _ { i j } N _ { i j } t _ { i j i j } \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} i \\colon \\mathbb { T } ^ { \\nu } \\to \\mathbb { T } ^ { \\nu } \\times \\mathbb { R } ^ { \\nu } \\times H _ S ^ { \\perp } , \\varphi \\mapsto i ( \\varphi ) : = ( \\theta ( \\varphi ) , y ( \\varphi ) , z ( \\varphi ) ) \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} E [ \\hat { R } _ { 1 \\ell i } | \\mathcal { W } _ { \\ell } ^ { c } ] & = \\int g ( w , \\hat { \\gamma } _ { \\ell } , \\theta _ { 0 } ) F _ { 0 } ( d w ) , E [ \\hat { R } _ { 2 \\ell i } | \\mathcal { W } _ { \\ell } ^ { c } ] = \\int \\phi ( w , \\hat { \\gamma } _ { \\ell } , \\alpha _ { 0 } , \\theta _ { 0 } ) F _ { 0 } ( d w ) , \\\\ E [ \\hat { R } _ { 3 \\ell i } | \\mathcal { W } _ { \\ell } ^ { c } ] & = \\int \\phi ( w , \\gamma _ { 0 } , \\hat { \\alpha } _ { \\ell } , \\tilde { \\theta } _ { \\ell } ) F _ { 0 } ( d w ) = 0 , \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{gather*} i _ X \\omega = \\delta \\alpha + d \\alpha ' \\end{gather*}"}
-{"id": "9128.png", "formula": "\\begin{align*} \\| f \\| _ { 2 } ^ 2 = \\sum _ { h \\neq 0 } | \\hat { f } ( h ) | ^ 2 \\ , \\omega _ h . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} N _ i = L \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\lambda \\ , \\rho ( \\lambda ) , D _ i = L \\left \\{ \\int _ { - \\infty } ^ { \\lambda _ { i L } } - \\int _ { \\lambda _ { i R } } ^ { \\infty } \\right \\} d \\lambda \\ , \\rho ( \\lambda ) \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} \\frac { \\delta n \\lambda } { 1 6 \\sigma ^ 2 } \\| \\theta \\| _ { w ^ * , 1 } \\ ! + \\ ! \\frac { \\delta n } { 2 \\sigma ^ 2 } \\sum _ { j = 1 } ^ p \\frac { w _ j ^ { 2 } } { w _ j ^ * } | \\theta _ j | \\ ! + \\ ! \\frac { \\log 4 p } { \\beta \\delta } \\| \\theta \\| _ { w ^ * , 1 } \\ ! + \\ ! \\frac { \\log 2 } { \\beta } . \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} C _ 3 ( x , q ) & = 1 + 2 x q + 2 x q ^ 2 + x ^ 2 q ^ 3 , \\textup { r e s p . \\ } \\\\ C _ 4 ( x , q ) & = ( 1 + x q ^ 2 ) ( 1 + 3 x q + 4 x q ^ 2 + 3 x q ^ 3 + x ^ 2 q ^ 4 ) . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} \\frac { \\partial \\phi _ 2 ( t , q ) } { \\partial t } \\Bigr | _ { t = 1 } = \\sum _ { n = 0 } ^ { \\infty } q ^ n \\sum _ { \\lambda \\vdash n } f _ { } ( \\lambda ) = \\frac { 1 } { ( q ; q ) _ { \\infty } } \\langle f _ { } \\rangle _ q . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} k g ^ + = f ^ * j ^ * k g ^ - = ( j \\circ f ) ^ * k g ^ - . \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} G _ i ( s _ j ) \\begin{cases} < 0 & \\frac { j - i } k \\in \\cup _ { m = 0 } ^ { \\infty } [ 2 m , 2 m + 1 ) ; \\\\ > 0 & \\frac { j - i } k \\in ( - \\infty , 0 ) \\cup \\cup _ { m = 0 } ^ { \\infty } [ 2 m + 1 , 2 m + 2 ) . \\end{cases} \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\mathcal { B } \\left ( x - \\frac { q + r } { 2 } , \\frac { q - r } { 2 } \\right ) = 1 \\frac { 1 } { 2 } \\left ( \\frac { x - r } { | x - r | } - \\frac { x - q } { | x - q | } \\right ) = 1 \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} \\bar u ( x ) = u _ n ( x ) = U _ { \\max } \\qquad n > N . \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} u _ { i } = \\mathrm { I m } ( \\mu _ { i } ( \\sigma \\upsilon ) ) = \\mathrm { I m } ( \\mu _ { i } ( \\sigma \\nu ) ) \\stackrel { \\eqref { r e v e r s e _ f o r m u l a } } { = } \\sum _ { j = 1 } ^ { i } \\mathrm { I m } ( \\sigma \\nu ) _ j = \\sum _ { j = 1 } ^ { i } t _ { \\sigma ^ { - 1 } j } \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} s = 4 a r = 4 a ^ 2 + 2 b \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} D _ x = S ( \\sigma ^ { - 1 } ) \\circ D _ { X \\backslash \\{ x \\} } \\circ S ( \\sigma ) , \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} \\| f \\| ^ 2 _ { \\Lambda ^ H } = H ( 2 H - 1 ) \\int _ \\R \\int _ \\R f ( u ) f ( v ) \\vert u - v \\vert ^ { 2 H - 2 } \\ , d u \\ , d v \\ , . \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} Y : = c y - c - 2 y . \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} T \\gamma ( X _ H ^ { \\gamma } ) = \\left ( p _ i \\frac { \\partial H } { \\partial p _ i } - H \\right ) T \\gamma \\left ( \\frac { \\partial } { \\partial t } \\right ) + \\frac { \\partial H } { \\partial p _ i } T \\gamma \\left ( \\frac { \\partial } { \\partial q ^ i } \\right ) = X _ H \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} k F _ k ( s ) = F _ { k - 1 } ( s ) F _ a ( s ) - \\sum _ { n = 2 } ^ k ( - 1 ) ^ n F _ { k - n } ( s ) F _ { n } ( n s ; a ) , \\end{align*}"}
-{"id": "7525.png", "formula": "\\begin{align*} & \\int _ r ^ R \\Big ( 1 + \\alpha ^ { - 2 } \\sum _ { j = k } ^ { n } \\alpha _ j ^ 2 \\Big ) 2 \\pi ^ 2 t ^ 2 \\alpha ^ 2 \\eta ^ 2 \\ , G _ t ( \\eta , \\xi _ k , \\dots , \\xi _ n ) \\frac { d t } { t } \\\\ & + \\int _ r ^ R \\Big ( \\sum _ { j = k } ^ n 4 \\pi ^ 2 t ^ 2 \\alpha _ j ^ 2 \\xi _ j ( \\xi _ j + \\eta ) \\Big ) \\ , G _ t ( \\eta , \\xi _ k , \\dots , \\xi _ n ) \\frac { d t } { t } . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} A ' = \\mbox { d i a g } ( \\alpha _ 1 , \\ldots , \\alpha _ { c ' } ) , B ' = \\mbox { d i a g } ( \\beta _ 1 , \\ldots , \\beta _ { c ' } ) , \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} f ( \\gamma , \\rho ) = E _ 0 ( \\gamma , \\rho ) - \\gamma \\rho \\frac { k _ n } { n } v ( n ) - \\frac { k _ n } { n } H _ 2 ( \\gamma ) , \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} u _ 1 ( x ) & = 3 \\hat { \\varphi } ( x ) + ( x - \\lambda _ 0 ) \\hat { \\varphi } ' ( x ) , & u _ 2 ( x ) & = x u _ 1 ( x ) , \\\\ v _ 1 ( x ) & = ( x - \\lambda _ 0 ) \\hat { \\varphi } ( x ) , & v _ 2 ( x ) & = x v _ 1 ( x ) . \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} f _ { \\Omega ^ { ( N ) } } ( x ) = N f _ { \\Omega } ( x ) \\left [ F _ { \\Omega } ( x ) \\right ] ^ { N - 1 } \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} \\kappa = \\{ k \\in L ' _ \\Gamma : ( k , x ) = ( x , x ) x \\in L _ \\Gamma \\} . \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} H _ { \\Lambda } ^ { { B } } = - { B } \\ , n \\cdot \\sum _ { { x } \\in \\Lambda } { \\sigma } _ { { x } } \\ , \\ B > 0 \\ . \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{align*} C \\left ( \\int _ \\Omega \\overline { \\xi _ 0 } ^ 2 w ^ { q j } \\ , d x \\right ) ^ { 2 / q } & \\ , \\leq \\lambda _ 1 \\int _ \\Omega \\overline { \\chi } _ 1 \\overline { \\xi _ 0 } w ^ { 2 j - 1 } \\ , d x + \\widetilde { C } j \\int _ \\Omega \\overline { \\xi _ 0 } ^ 2 w ^ { 2 j } \\ , d x \\\\ & \\ , \\leq C j \\int _ \\Omega \\overline { \\xi _ 0 } ^ 2 w ^ { 2 j } \\ , d x . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\bold { p r } X ( L _ n ) = \\operatorname { D i v } ^ { \\vartriangle } R , \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} r ^ 2 \\dot { \\rho } ^ 2 + \\frac { \\omega ^ 2 } { \\rho ^ 2 } - \\rho ^ 2 = c \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} \\alpha _ { \\infty , t } ( y ^ { ( j ) } ) = \\alpha _ { \\infty , t } ( x ^ { ( j ) } e ) = \\alpha _ { \\infty , t } ( x ^ { ( j ) } ) e = e ^ { i p t } x ^ { ( j ) } e = e ^ { i p t } y ^ { ( j ) } . \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} \\hat { k } _ j = \\underset { k \\in \\left \\lbrace 1 , 2 , \\cdots , K \\right \\rbrace } { \\mathrm { a r g m i n } } \\sqrt { \\beta ^ 2 ( \\tau _ j - \\tau _ k ) ^ 2 + ( \\theta ^ \\mathrm { A } _ j - \\theta ^ \\mathrm { A } _ k ) ^ 2 + ( \\theta ^ \\mathrm { E } _ j - \\theta ^ \\mathrm { E } _ k ) ^ 2 } \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} F ( t , s ) = \\frac { c ^ 2 _ { h _ { t , s } } } { c _ { h ( t ) } c _ { h ( s ) } } , G ( t , s , H ) = \\frac { 1 } { 2 } \\left ( { | t | } ^ { H } + { | s | } ^ { H } - { | t - s | } ^ { H } \\right ) . \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} \\partial _ { \\tau } z = \\Pi _ S ^ { \\perp } \\{ b ( \\tau , x ) \\partial _ x z \\} , b ( \\tau , x ) = \\frac { \\beta ( x ) } { 1 + \\tau \\beta _ x ( x ) } = \\varepsilon \\beta _ 1 + O ( \\varepsilon ^ 2 ) . \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ 0 ^ { x _ \\frac \\mu 2 } | w u | ^ q \\ , \\dd x \\leq \\frac { C w ( x _ \\mu ) ^ q x _ \\mu } { ( \\mu - Q ( x _ { \\frac \\mu 2 } ) ) ^ { \\frac q 4 } } \\leq C w ( x _ \\mu ) ^ q x _ \\mu \\mu ^ { - \\frac q 4 } \\leq C w ( x _ \\mu ) ^ q a _ \\mu ^ { - 1 } \\mu ^ { 1 - \\frac q 4 } . \\end{aligned} \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\gamma ( a _ 1 ) & = ( a _ 1 \\varphi ( \\widetilde { a } _ 1 ) ^ { - 1 } ; \\widetilde { a } _ 1 ) = ( a _ 1 \\left ( \\varphi ( 1 , 0 ) \\varphi ( 0 , 1 ) \\right ) ^ { - 1 } ; 1 , 1 ) = ( b _ 1 ^ { - 1 } ; 1 , 1 ) = ( v ^ { - 1 } u ^ { - 1 } ; 1 , 1 ) , \\\\ \\gamma ( a _ 2 ) & = ( a _ 2 \\varphi ( \\widetilde { a } _ 2 ) ^ { - 1 } ; \\widetilde { a } _ 2 ) = ( a _ 2 \\varphi ( 0 , 1 ) ; 0 , - 1 ) = ( b _ 2 ^ { - 1 } ; 0 , - 1 ) = ( v ; 0 , - 1 ) , \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} \\widetilde { R } = \\frac { 1 } { \\ln 2 } \\int _ 0 ^ \\infty { \\frac { 1 } { { z } } ( 1 - \\widetilde { \\Xi } _ 1 ( z ) ) \\widetilde { \\Xi } _ 2 ( z ) { e ^ { - z \\sigma _ 0 ^ 2 } } d z } , \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} \\mathcal { H } _ 8 ^ \\perp = \\left \\{ y \\in \\mathbb { F } _ 2 ^ 8 : \\sum _ { i = 1 } ^ 8 x _ i y _ i = 0 \\ ; ( 2 ) \\ ; x \\in \\mathcal { H } _ 8 \\right \\} . \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} y ^ 2 + a x y + b y = x ^ 3 , a , b \\in \\Z _ \\ell , \\Delta _ m = b ^ 3 ( a ^ 3 - 2 7 b ) \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} X _ { F ^ { ( N ) } } ( u ) = \\Pi _ E X _ { F ^ { ( N ) } } ( \\Pi _ E u ) . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\underset { ( t , x ) \\in \\mathbb R \\times ( - a _ 0 , a _ 0 ) } \\sup ( u + v ) = \\nu , \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} w _ { a ^ { k + 1 } } \\triangleq \\log { 1 \\over \\P ( [ X _ { k + 1 } ] _ b = a _ { k + 1 } | [ X ^ k ] _ b = a ^ k ) } . \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} \\underset { t \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\int _ { { \\bf R } ^ N } u ( y , t ) \\eta ( y ) \\ , d y = \\int _ { { \\bf R } ^ N } \\eta ( y ) \\ , d \\mu ( y ) , \\eta \\in C _ 0 ( { \\bf R } ^ N ) . \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} ( 1 - \\epsilon ) B _ 1 ( n ) = ( 1 - \\epsilon ) \\frac { n } { 2 k _ n } \\log \\left ( 1 + k _ n P \\right ) . \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} \\begin{cases} M _ { R } = K , \\ldots , N _ { S } ; \\ , M _ { D } = 0 , \\ldots , N _ { S } ; \\\\ M _ { R D } = \\max ( 0 , M _ { R } + M _ { D } - N _ { S } ) , \\ldots , \\min ( M _ { R } , M _ { D } ) . \\end{cases} \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} F ^ * \\tau = 0 . \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} - a _ j + T _ { j , j } g _ j ^ { \\prime } ( z _ j ^ \\ast ) = \\alpha , \\quad \\forall ~ j = \\overline { 1 , n } \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} ( \\pi ^ * _ \\pm d _ { \\pm } ) \\omega _ \\pm : = \\pi ^ * _ { \\pm } ( d _ { \\pm } \\sigma _ \\pm ) , \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} m ( D , D _ 1 , Q ( z ) d z ^ 2 ) = \\int _ { \\partial \\tilde { D } _ 1 } q \\frac { \\partial q _ 1 } { \\partial n } \\ , d s = \\mbox { R e } \\left ( \\frac { 1 } { i } \\int _ { \\partial \\tilde { D } _ 1 } \\left ( x - x _ 1 \\right ) \\frac { \\partial x } { \\partial z } \\ , d z \\right ) . \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} ( | b | ^ { 2 } K \\alpha ^ { 2 } ) 1 + ( \\overline { a } b + a \\overline { b } K ) \\alpha - ( \\overline { d } e ) \\beta - ( d \\overline { e } ) \\overline { \\beta } - | e | ^ { 2 } | \\beta | ^ { 2 } = 0 . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} \\phi _ t \\ ; = \\ ; \\psi _ { \\theta } + \\psi _ { \\alpha _ 1 } \\circ \\tau + \\ldots + \\psi _ { \\alpha _ r } \\circ \\tau ^ r \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} \\langle ( h , h ' ) \\cdot u , ( h , h ' ) \\cdot v \\rangle = \\langle u , v \\rangle \\end{align*}"}
-{"id": "195.png", "formula": "\\begin{align*} S _ { \\alpha , \\beta } = \\Big [ \\Big ( \\frac { \\alpha } { \\beta } \\Big ) ^ { \\frac { \\beta } { \\alpha + \\beta } } + \\Big ( \\frac { \\beta } { \\alpha } \\Big ) ^ { \\frac { \\alpha } { \\alpha + \\beta } } \\big ] S . \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{align*} \\int \\limits _ { 0 } ^ { t } K ( t , \\tau ) r ( \\tau ) d \\tau + F ( t ) = E ( t ) \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} \\eta _ { n , r , d } : = \\frac { 2 ^ { d - 2 } ( d - 1 ) r } { n + 2 ^ { d - 2 } ( d - 1 ) r - 1 } . \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} & \\partial F / \\partial \\eta = f \\\\ & F ( x , \\psi , 0 ) = 0 . \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} W \\ ; : \\ ; y ^ 2 + a _ 1 ' y x + a _ 3 ' y = x ^ 3 + a _ 2 ' x ^ 2 + a _ 4 ' x + a _ 6 ' \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} \\mathcal { E } = \\mathcal { A } \\ast E _ { r } f = \\left \\{ \\mathcal { F } : \\mathcal { F } ( z ) = ( f \\ast E _ { r } f ) ( z ) = z + \\sum _ { n = 2 } ^ \\infty \\frac { ( - 1 ) ^ { n - 1 } a _ n } { ( 2 n - 1 ) ( n - 1 ) ! } z ^ { n } , f \\in \\mathcal { A } \\right \\} , \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} \\phi ( z ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( 2 ) } \\tilde \\Psi ( s ) \\zeta ( s ) z ^ { - s } d s \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} U : = \\Big { \\{ } h \\in N ( V , \\| \\cdot \\| ) \\colon \\sup _ { x \\in \\partial B _ R ( 0 ) } \\| f ( x ) - h ( x ) \\| < \\epsilon / 2 \\Big { \\} } \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} F ( z ) = \\int _ 0 ^ { 2 \\pi } \\frac { e ^ { i \\theta } + z } { e ^ { i \\theta } - z } \\ , d \\mu ( \\theta ) \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} A ( t ) = \\sum _ { n \\geq 0 } \\frac { a _ n } { n ! } t ^ n \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} & \\eta _ 1 - \\tfrac 1 2 \\eta _ 2 = 0 , \\\\ & \\tfrac 1 2 \\eta _ { k - 1 } - \\eta _ { k } + \\tfrac 1 2 \\eta _ { k + 1 } = 0 , k = 2 , 3 , \\ldots , \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} \\Phi ( r ^ { p _ k } h ) \\leq \\Phi ( r ^ { p _ { k _ 0 - 1 } } h ) ( x _ { p _ { k _ 0 - 1 } } ) \\prod _ { k = k _ 0 } ^ \\infty A _ j . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} g _ { \\varepsilon } ( p ) = p ^ { - 1 } + \\sum ^ r _ { j = 1 } p ^ { \\varepsilon _ j - \\varepsilon - 1 } + O \\bigl ( p ^ { - 2 } \\bigr ) \\ , \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\frac { 1 } { 1 - a _ p p ^ { s - k + 1 } + p ^ { 2 s - k + 1 } \\epsilon ( p ) } \\begin{pmatrix} - p ^ { - 1 - s } - p ^ { 1 - k + s } \\epsilon ( p ) + a _ p p ^ { - k } & p ^ { - k } ( 1 - p ) \\epsilon ( p ) \\\\ 1 - p ^ { - 1 } & - p ^ { - 1 - s } - p ^ { s - k + 1 } \\epsilon ( p ) + a _ p p ^ { - k + 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} \\tau _ M ( a ) = \\frac { 1 } { 2 M } \\int _ { - M } ^ M \\tau ( \\alpha _ t ( a ) ) ~ d t . \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} h ( a ; z ) : = \\frac { H ( 1 ; a , z ) } { \\Gamma \\ ( \\frac { z } { \\phi ( q ) } + 1 \\ ) } . \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} 0 = \\int _ { \\Omega _ { \\varphi _ i } } \\langle \\nabla v _ i , \\nabla \\zeta \\rangle - \\varepsilon _ i \\abs { \\nabla v _ i } ^ 2 u _ i \\cdot \\zeta d x . \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} & \\abs { \\frac { \\partial \\Pi _ i ^ k } { \\partial z _ l } ( x , y , z ) - \\frac { \\partial \\Pi _ i ^ k } { \\partial z _ l } ( 0 , y , z ) } = \\\\ & \\abs { \\frac { ( y \\times \\nu _ i ( x ) ) \\otimes ( y \\times \\nu _ i ( x ) ) \\Lambda _ j } { \\abs { y \\times \\nu _ i ( x ) } ^ 2 } - \\frac { ( y \\times \\nu _ i ( 0 ) ) \\otimes ( y \\times \\nu _ i ( 0 ) ) \\Lambda _ j } { \\abs { y \\times \\nu _ i ( 0 ) } ^ 2 } } . \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\delta ^ 2 \\| Z \\| _ F ^ 2 & \\ge \\sum _ { i = 1 } ^ m \\| A _ i Z - Z ^ T A _ i \\| _ F ^ 2 \\ge \\sum _ { i = 1 } ^ m | x ^ * ( A _ i Z - Z ^ T A _ i ) x | ^ 2 \\\\ & = \\sum _ { i = 1 } ^ m | \\lambda - \\bar { \\lambda } | \\ | x ^ * A _ i x | ^ 2 = | \\lambda - \\bar { \\lambda } | \\sum _ { i = 1 } ^ m | x ^ * A _ i x | ^ 2 . \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} \\lambda ( f ) = \\int _ G f ( g ) \\ , d \\lambda ( g ) , \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} A ( { \\cal F } ) = C C { \\cal F } . \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{align*} E [ \\phi ( W , \\gamma _ { 0 } , \\alpha , \\theta ) ] = 0 . \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} u = \\frac 1 j ( x _ \\alpha \\psi _ \\alpha + x _ \\beta \\psi _ \\beta , y _ \\alpha \\psi _ \\alpha + y _ \\beta \\psi _ \\beta ) , \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} f _ u ( z ) : = \\frac { c _ + } { 2 } \\phi _ u ^ 2 \\Big ( \\Big ( \\frac { z } { 2 } \\Big ) ^ { \\frac { 1 } { \\alpha - 1 } } \\Big ) , z \\ge 0 , \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} { \\mathbf m } = \\left ( e _ 1 \\otimes z ^ { a _ { 1 1 } } \\cdots e _ 1 \\otimes z ^ { a _ { 1 \\lambda _ 1 } } \\right ) \\otimes \\left ( e _ 2 \\otimes z ^ { a _ { 2 1 } } \\cdots e _ 2 \\otimes z ^ { a _ { 2 \\lambda _ 2 } } \\right ) \\otimes \\cdots \\otimes \\left ( e _ s \\otimes z ^ { a _ { s 1 } } \\cdots e _ s \\otimes z ^ { a _ { s \\lambda _ s } } \\right ) \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} \\| \\hat { p } ^ { ( k ) } ( \\cdot | Z ^ n ) - \\mu _ k ^ { ( b ) } \\| _ 1 & \\leq { 1 \\over k + g } \\sum _ { i = 1 } ^ { k + g } \\| \\hat { p } ^ { ( 1 ) } ( \\cdot | S ^ { ( i ) , t } ) - \\mu _ k ^ { ( b ) } ( \\cdot ) \\| _ 1 + 1 - { t ( k + g ) \\over n - k } + { k + g \\over n - k } \\\\ & = { 1 \\over k + g } \\sum _ { i = 1 } ^ { k + g } \\| \\hat { p } ^ { ( 1 ) } ( \\cdot | S ^ { ( i ) , t } ) - \\mu _ k ^ { ( b ) } ( \\cdot ) \\| _ 1 + { n - t ( k + g ) + g \\over n - k } . \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} \\left ( \\partial _ { \\tau } + 1 - \\frac { \\kappa \\hbar ^ 2 } { 2 } \\right ) u ( \\cdot , 0 ) = 0 \\ , . \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} f ( x ; G ) = \\sum _ { j = 1 } ^ m \\alpha _ j f ( x ; \\theta _ j ) . \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} Q _ { 2 } ( x , y , z ) = y ^ { 2 } - x z + 2 c y z + 2 z ^ { 2 } = 0 , \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} g ( z ) : = \\frac { G ( 1 ; z ) } { \\Gamma ( z + 1 ) } . \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} [ \\Lambda ^ r _ n \\circ \\Lambda ^ s _ n ] = P _ { r , s } ( [ \\Lambda _ n ^ 1 ] , \\ldots , [ \\Lambda _ n ^ { r s } ] ) \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} \\delta k _ { i a } = \\frac { L s _ a N _ { i a } } { 2 \\pi } \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} 1 - F ( k ) = \\mathcal { L } ( k ) k ^ { - \\gamma } , \\gamma > 1 , \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} w _ { i , k } ^ n = \\left ( t ^ n _ { - i , k + Q ^ n _ i ( 0 ) - R ^ n _ i ( t ^ n _ { i , k } - ) - Q ^ n _ { - i } ( 0 ) + R ^ n _ { - i } ( t ^ n _ { i , k } - ) } - t ^ n _ { i , k } \\right ) 1 _ { \\{ Q ^ n _ { - i } ( t ^ n _ { i , k } - ) = 0 \\} } . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} Q ( \\xi _ { a } ^ { ( h ) } ) = \\mathbb { C } _ { a , h } Q ( \\xi _ { a } ^ { ( 0 ) } ) , \\forall h \\in \\{ 1 , . . . , p - 1 \\} , \\forall a \\in \\{ 1 , . . . , \\mathsf { N } \\} , \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} a \\ , \\ , \\sharp ^ { \\epsilon } \\ , \\ , b \\ , = \\ , a \\ , \\sharp ^ { 0 } \\ , b \\ , + \\ , \\epsilon \\ , r _ \\epsilon ( a , b ) \\ , , \\forall ( a , b ) \\in S ^ m _ \\rho ( \\Xi ) \\times S ^ { m ^ \\prime } _ \\rho ( \\Xi ) \\ , . \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} a _ { \\xi } ( y , s ) = c _ { \\xi } ( s ) \\cdot B _ { \\xi } ( y ; p ; s ) , \\end{align*}"}
-{"id": "3911.png", "formula": "\\begin{align*} \\Lambda _ S ( - \\theta , \\mathbf { H } ) = \\log \\left ( \\pi e ^ { - \\theta j ( K - k ) c } + \\sum _ { \\ell = 1 } ^ L \\pi _ \\ell e ^ { - \\theta r _ \\ell } \\right ) , \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} i _ { [ a _ f , a _ g ] } \\mathrm { d } \\eta = & [ \\mathcal { L } _ { a _ f } , i _ { a _ g } ] \\mathrm { d } \\eta \\\\ = & \\mathrm { d } i _ { a _ f } i _ { a _ g } \\mathrm { d } \\eta + i _ { a _ f } \\mathrm { d } i _ { a _ g } \\mathrm { d } \\eta - i _ { a _ g } \\mathrm { d } i _ { a _ f } \\mathrm { d } \\eta - i _ { a _ g } i _ { a _ f } \\mathrm { d } \\mathrm { d } \\eta \\\\ = & \\mathrm { d } \\eta ( a _ f , a _ g ) \\\\ = & \\{ f , g \\} . \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\mathfrak { R } _ 1 : = \\Phi ^ { - 1 } \\mathcal { R } _ { \\mathrm { I } \\mathrm { I } } = - \\varepsilon ^ 2 \\Pi _ S ^ { \\perp } \\partial _ x \\mathcal { R } _ 2 + \\mathcal { R } _ * \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} \\{ I _ j \\} = \\bigcup _ { i = 1 } ^ n \\{ I _ { i L } + 1 / 2 , I _ { i L } + 3 / 2 , \\ldots , I _ { i R } - 1 / 2 \\} . \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} \\sum _ { ( i , j , k ) \\in \\Lambda } \\abs { \\langle v , \\widetilde \\psi _ { i , j , k } \\rangle } ^ \\tau = \\sum _ { ( i , j , k ) \\in \\Lambda ^ 0 } \\abs { \\langle v , \\widetilde \\psi _ { i , j , k } \\rangle } ^ \\tau + \\sum _ { ( i , j , k ) \\in \\Lambda \\backslash \\Lambda ^ 0 } \\abs { \\langle v , \\widetilde \\psi _ { i , j , k } \\rangle } ^ \\tau = : I + I \\ ! \\ ! I . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} \\| A \\| _ 2 \\ge \\frac { \\| A g \\| _ 2 } { \\| g \\| _ 2 } \\ge \\sqrt { \\frac { 2 } { 1 5 } } n ^ 2 = \\sqrt { \\frac { 2 } { 1 5 } } n \\| \\nabla ( A ) \\| _ 2 , \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} E ( b _ { t } - b _ { s } ) = \\lambda l \\int _ { s } ^ { t } \\nu ( r ) \\ , d r \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { b - 1 } F _ { b n + k } ^ { \\left ( b \\right ) } = 2 F _ { b n + \\left ( b - 1 \\right ) } ^ { \\left ( b \\right ) } + F _ { b n + \\left ( b - 2 \\right ) } ^ { \\left ( b \\right ) } - F _ { n } ^ { \\left ( b \\right ) } \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} \\begin{cases} \\aligned ( N \\Theta _ 1 ) ^ { 2 / N } ( d _ { 1 , n } + d _ { 2 , n } ) \\leq N J ( u _ n , v _ n ) & \\leq N ( c _ 1 + c _ 2 ) \\Theta _ 1 + o ( 1 ) , \\\\ d _ { 1 , n } ^ { p - 1 } + \\nu \\alpha d _ { 1 , n } ^ { \\alpha / 2 - 1 } d _ { 2 , n } ^ { \\beta / 2 } & \\geq ( N \\Theta _ 1 ) ^ { 2 / N } , \\\\ d _ { 2 , n } ^ { p - 1 } + \\nu \\beta d _ { 1 , n } ^ { \\alpha / 2 } d _ { 2 , n } ^ { \\beta / 2 - 1 } & \\geq ( N \\Theta _ 1 ) ^ { 2 / N } . \\endaligned \\end{cases} \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} r _ { } ( n ) = \\cosh ^ { - 1 } \\left ( \\frac { 1 } { \\sqrt { 3 } } \\cot \\left ( \\frac { n \\pi } { 6 n + 1 } \\right ) \\right ) . \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} D _ t \\left ( \\frac { u _ 1 + u } { 2 } \\right ) + ( S - \\operatorname { i d } ) \\left ( \\frac { 1 } { 2 } u ^ 2 + u _ 1 - 2 u + u _ { - 1 } \\right ) = F _ 1 , \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} \\pi ^ { - \\frac { s } { 2 } } \\Gamma \\left ( \\frac { s } { 2 } \\right ) \\zeta ( x , s ) = \\pi ^ { - \\frac { 1 - s } { 2 } } \\Gamma \\left ( \\frac { 1 - s } { 2 } \\right ) \\Psi ( - x , 1 - s ) . \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} h = \\sum _ { n = 1 } ^ m X _ n ^ { \\otimes s } , \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} \\frac { ( - q ; q ^ 3 ) _ \\infty ( - q ^ 2 ; q ^ 3 ) _ \\infty } { ( q ; q ^ 3 ) _ \\infty ( q ^ 2 ; q ^ 3 ) _ \\infty } = \\sum _ { n \\geq 0 } \\frac { q ^ { n ^ 2 } ( - q ; q ) _ { n } } { ( q ; q ) _ { n } ( q ; q ^ 2 ) _ { n + 1 } } \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\begin{aligned} \\sum _ { j = 0 } ^ { n } ( \\lambda - \\delta ( x _ j ) ) x _ j & = \\sum _ { j = 0 } ^ { n - 1 } ( \\lambda - \\delta ( x _ j ) ) x _ j + ( \\lambda - \\delta ( x _ n ) ) x _ n \\\\ & \\geq ( 1 + \\delta ( x _ { n - 1 } ) ) x _ n + ( \\lambda - \\delta ( x _ n ) ) x _ n \\\\ & = ( 1 + \\lambda + \\delta ( x _ { n - 1 } ) - \\delta ( x _ n ) ) x _ n \\\\ & \\geq ( 1 + \\lambda ) x _ n = ( 1 + \\delta ( x _ n ) ) x _ { n + 1 } . \\end{aligned} \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} A _ i = V ^ T D _ i V , i = 1 , \\dots , m , \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} b ^ * ( x ) = b ( 1 - x + \\delta ) . \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} \\operatorname { P } = r ^ { - l } \\sum \\limits _ { i \\leq l } a _ { i } ( r ) ( - r \\partial _ { r } ) ^ { i } \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} { \\widetilde { \\gamma } _ o } = \\frac { { { P _ S } G _ \\mathrm { M } ^ { S } G _ \\mathrm { M } L \\left ( r \\right ) } } { { \\sum \\nolimits _ { i \\in \\Phi / o } \\left ( P _ S G _ i ^ { S } + P _ A G _ i ^ { A } \\right ) L \\left ( \\left | X _ i \\right | \\right ) + { \\sigma _ o ^ 2 } } } . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} r ^ n _ { \\mathrm { S C } } = B ^ n _ { \\mathrm { S C } } \\log _ 2 \\left ( 1 + \\frac { p _ { n } g _ { n } } { B ^ n _ { \\mathrm { S C } } N _ 0 } \\right ) . \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} y \\mapsto f _ { \\beta , x } ( y ) : = \\frac 1 2 ( y - x ) ^ 2 + \\beta y ^ b \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{align*} u ( t ) = F _ { - 1 } \\dot z ( t - 1 ) + F z _ t ( \\cdot ) , \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} \\sum _ { n \\in \\Z } \\overline { v _ { l , j } ( x , n ) } v _ { l ^ \\prime , j ^ \\prime } ( x , n ) = 0 . \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} \\mathcal { S } = \\{ x \\in \\mathbb { R } ^ n \\ , | \\ , g ( x ) \\leq 0 \\} , \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} \\lambda x _ { u } = S _ { G } ( v , \\mathbf { x } ) . \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} G ( ( I - P _ h ) \\widetilde { A } { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ u _ k ; v _ 1 ] } , \\ldots , ( I - P _ h ) \\widetilde { A } { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ u _ k ; v _ p ] } ) = G \\Big ( \\widetilde { A } { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ u _ k ; v _ 1 ] } , \\ldots , \\widetilde { A } { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ u _ k ; v _ p ] } , \\frac { h } { \\| h \\| } \\Big ) . \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} \\lambda : = \\lambda ^ * + a \\eta , \\lambda _ k : = 2 ( g _ 1 + g _ 2 + \\ldots + g _ k ) + k a , \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} \\varphi ( \\lambda ) = \\sum _ { i = 1 } ^ m \\varphi _ i ( \\lambda ) = \\sum _ { i = 1 } ^ m \\min _ { x _ i \\in X _ i } L _ i ( x _ i , \\lambda ) , \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} \\frac { \\Delta \\rho } { \\rho } + \\frac { \\nabla \\Psi \\cdot \\nabla \\rho } { \\rho } + \\Delta \\Psi = \\Delta \\log \\frac { \\rho } { e ^ { - \\Psi } } + \\Big | \\nabla \\log \\frac { \\rho } { e ^ { - \\Psi } } \\Big | ^ { 2 } - \\nabla \\Psi \\cdot \\nabla \\log \\frac { \\rho } { e ^ { - \\Psi } } , \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} { \\bf F } ' _ { i j k } = & \\sum \\limits _ { l = 1 } ^ { 3 } \\sum \\limits _ { m = k } ^ { 2 } p _ { l m } { \\bf G } _ { i l } { \\bf v } _ { l m } { \\bf v } ^ { T } _ { l m } { \\bf G } ^ { T } _ { i l } + \\sigma ^ { 2 } { \\bf I } _ 2 - { \\bf T } _ { i j k } , \\\\ = & \\sum \\limits _ { l = 1 } ^ { 3 } \\sum \\limits _ { m = k } ^ { 2 } { \\bf G } _ { i l } \\mathbf { Q } _ { l m } { \\bf G } ^ { T } _ { i l } + \\sigma ^ { 2 } { \\bf I } _ 2 - { \\bf T } _ { i j k } , \\ \\forall \\{ i , j \\} , k . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} \\| \\Phi ( r ^ 2 h ) \\| _ { L ^ \\infty } \\leq \\Phi ( r ^ { p _ { k _ 0 - 1 } } h ) ( x _ { p _ { k _ 0 - 1 } } ) \\prod _ { k = k _ 0 } ^ \\infty A _ k . \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} { } _ 2 F _ 1 ( c , d , e ; u ) = \\frac { \\Gamma ( e ) } { \\Gamma ( c - d ) \\Gamma ( d ) } \\int _ 0 ^ 1 ( 1 - u z ) ^ { - c } z ^ { d - 1 } ( 1 - z ) ^ { e - d - 1 } d z , e > d > 0 , | u | < 1 , c \\in \\mathbb { R } , \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} p ( - 1 ) & = u [ 1 + \\cos ( 2 A ) ] > 0 \\\\ p ( \\cos ( 2 A ) ) & = \\cos ^ 2 ( 2 A ) - 1 < 0 \\\\ p ( 1 ) & = u ( \\cos ( 2 A ) - 1 ) < 0 \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} ( \\Box ^ { ( 0 ) } _ { t , b , m } - T ^ 2 ) N _ { t , m } u = ( I - S _ { t , m } ) u + m ^ 2 N _ { t , m } u . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} W = \\left ( \\sum _ { N < \\ell \\le N ^ 2 } - \\sum _ { N ^ 2 < \\ell \\le N ^ 3 } \\right ) \\frac { \\tilde \\lambda ( \\ell ) } { \\ell } \\sum _ { m _ 1 \\mid \\ell } \\sum _ { m _ 2 \\mid \\ell } \\mu ( m _ 1 ) \\mu ( m _ 2 ) \\frac { g ( m _ 1 ) } { \\xi ( m _ 1 ) } \\frac { g ( m _ 2 ) } { \\xi ( m _ 2 ) } K \\left ( \\frac { m _ 1 } { m _ 2 } \\right ) . \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} E ' : y ^ 2 + y = x ^ 3 - 8 6 4 1 1 6 6 6 7 x + 7 8 2 5 7 9 7 0 6 1 0 8 3 2 . \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} G _ 0 = \\langle N _ G ( Q ) | 1 < Q \\leq S \\rangle \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} E _ \\alpha ( z ) = \\sum ^ \\infty _ { n = 0 } \\frac { z ^ n } { \\Gamma ( \\alpha n + 1 ) } \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} f _ { \\mathbf { Z } | H _ 1 } ( \\mathbf { z } | H _ 1 ) & = \\frac { e ^ { \\frac { M ( \\sigma _ { \\rm w } ^ 2 + \\sigma _ { \\rm a } ^ 2 ) } { \\zeta } } } { \\pi ^ n } \\prod _ { m = 1 } ^ M \\int _ { \\sigma _ { \\rm w } ^ 2 + \\sigma _ { \\rm a } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { \\frac { n } { M } } \\\\ & \\qquad \\qquad \\qquad \\qquad \\cdot e ^ { - \\frac { z _ m } { v } } e ^ { - \\frac { v } { \\zeta } } d v . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} \\gamma = \\sum _ { k = 2 } ^ { n } T _ { 1 k } T _ { k 1 } g _ { 1 } ^ { \\prime } ( z _ { 1 } ^ { \\ast } ) g _ { k } ^ { \\prime } ( z _ { k } ^ { \\ast } ) , \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} \\Pr _ { o _ { n } } ' [ T _ { \\mathrm { N i c e } } \\ge \\tau ] = 1 - o ( 1 ) . \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} { \\mathsf { C } } ^ { \\rm { F D } } _ { \\mathcal { N } _ { \\mathcal { M } _ 1 } } + { \\mathsf { C } } ^ { \\rm { F D } } _ { \\mathcal { N } _ { \\mathcal { M } _ 2 } } = { \\mathsf { C } } ^ { \\rm { F D } } _ { \\mathcal { N } _ { \\mathcal { K } } } + { \\mathsf { C } } ^ { \\rm { F D } } _ { \\mathcal { N } _ { [ 1 : N ] \\backslash \\mathcal { K } } } \\geq \\mathsf { C } _ { \\mathcal { N } _ { [ 1 : N ] } } ^ { \\rm { F D } } . \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} \\big ( \\Lambda ^ { \\textup { d } , k } \\big ) ^ 2 \\lesssim m \\sum _ { l = 0 } ^ { m - 1 } \\mathcal { M } ^ { \\textup { d } } _ l , \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} \\begin{pmatrix} a _ { 1 1 } & c _ { 1 1 } & s _ { 1 1 } \\\\ a _ { 2 2 } & c _ { 2 2 } & s _ { 2 2 } \\\\ a _ { 1 2 } & c _ { 1 2 } & s _ { 1 2 } \\end{pmatrix} \\end{align*}"}
-{"id": "3401.png", "formula": "\\begin{align*} 0 = J _ 0 \\subset J _ 1 \\subset \\cdots \\subset J _ { n } \\subset J _ { n + 1 } = A . \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} \\phi _ 1 ^ * \\eta = \\eta _ 0 \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} \\bar { x } _ { i } ( n ) = x _ { i } ( n ) + w _ i ( n ) , \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\left ( x ; \\bigcup _ { j = 1 } ^ n S ^ { ( j ) } \\right ) = \\sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k + 1 } \\sum _ { 1 \\leq j _ { 1 } < \\cdots < j _ { k } \\leq n } \\left ( x ; \\bigcap _ { m = 1 } ^ k S ^ { ( j _ { m } ) } \\right ) \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} L ^ * = \\sum _ { i = 0 } ^ { N ' } \\Delta L ^ * _ { i } \\end{align*}"}
-{"id": "7161.png", "formula": "\\begin{align*} L ( 1 , \\chi ) = \\sum _ 1 ^ { \\infty } \\chi ( n ) n ^ { - 1 } \\end{align*}"}
-{"id": "5776.png", "formula": "\\begin{align*} Z ^ \\infty ( z ) = \\Phi ( z ) \\left ( \\frac { z - a } { z } \\right ) ^ { \\frac { c } { 2 } \\sigma _ 3 } \\mathcal { P } ( z ) \\left ( \\frac { z - a } { z } \\right ) ^ { - \\frac { c } { 2 } \\sigma _ 3 } , z \\in D _ \\beta \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} d u ^ h _ t & = \\left ( ( L ^ h _ t + I ^ h ) u ^ h _ t + f _ t \\right ) d t \\\\ u ^ h _ 0 & = \\psi . \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} \\max _ { x , \\alpha } \\{ \\alpha g ( x ) - g ( f _ d ( x ) ) \\ , | \\ , \\alpha \\nabla _ x g ( x ) - \\nabla _ x g ( f _ d ( x ) ) = 0 , \\alpha \\geq 0 \\} . \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} d ( V , W ) = \\inf \\left \\{ \\theta \\in \\left [ 0 , \\frac { \\pi } { 2 } \\right ] \\ , \\Big | \\ , W \\subset C ( V , \\theta ) , V \\subset C ( W , \\theta ) \\right \\} \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = s } \\Theta = \\tilde v \\circ \\Theta _ s \\ ; . \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} { \\mathcal B } ^ R _ N ( T ) = 2 N ^ { 1 / 6 } \\left ( B _ N \\left ( \\tfrac 1 2 ( 1 + T N ^ { - 1 / 3 } ) \\right ) - \\sqrt { N } \\right ) . \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} n ^ { g ( { \\bf R } ) } a _ n ( I _ d ) & \\leq \\frac { ( A ( m + 1 , { \\bf R } , d ) ) ^ { g ( { \\bf R } ) } } { ( 1 + m ^ { 2 p } ) ^ { \\frac { 1 } { 2 } } } = \\frac { { ( m + 1 + b _ { \\bf R } ) } ^ { p } ( \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) ) ^ { g ( { \\bf R } ) } } { ( 1 + m ^ { 2 p } ) ^ { \\frac { 1 } { 2 } } } \\\\ & \\leq \\frac { { ( m + 1 + b _ { \\bf R } ) } ^ { p } ( \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) ) ^ { g ( { \\bf R } ) } } { m ^ { p } } \\leq 2 ^ p ( \\mathrm { v o l } ( B _ { 2 { \\bf R } } ^ d ) ) ^ { g ( { \\bf R } ) } . \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ n ^ { ( 1 ) } & = b ( \\psi _ n , \\psi _ n ) , \\lambda _ n ^ { ( 2 ) } = \\sum _ { j = 1 , j \\neq n } ^ { \\infty } \\frac { | b ( \\psi _ n , \\psi _ j ) | ^ 2 } { \\mu _ n - \\mu _ j } , \\\\ \\phi _ n ^ { ( 1 ) } & = \\sum _ { j = 1 , j \\neq n } ^ { \\infty } \\frac { b ( \\psi _ n , \\psi _ j ) } { \\mu _ n - \\mu _ j } \\psi _ j . \\end{aligned} \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} \\mathbb { H } _ m = \\left ( \\begin{array} { c c } \\left [ - \\frac { d ^ { 2 } } { d r ^ { 2 } } + V _ 1 ( r ) \\right ] & 0 \\\\ 0 & \\left [ - \\frac { d ^ { 2 } } { d r ^ { 2 } } + V _ 2 ( r ) \\right ] \\end{array} \\right ) \\ ; , \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} | X | = | X _ p | + | X _ v | \\le 4 2 k ^ 2 \\log k + 1 6 8 k ^ 2 \\log k = 2 1 0 k ^ 2 \\log k \\le f ( k , \\ell ) . \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} u . a = ( - 1 ) ^ { [ a ] } a , \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } G _ { 2 k } ^ { ( 1 , 4 ) } ( x , y ) = \\frac { 1 } { ( x ; x ^ 2 y ) _ { \\infty } ( x ^ 4 y ; x ^ 4 y ^ 2 ) _ { \\infty } } . \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\begin{cases} - \\ , \\cfrac { 1 } { 2 } \\ , \\Delta u ^ \\varepsilon - \\cfrac { b \\cdot D u ^ \\varepsilon } { \\varepsilon } = f \\ \\ \\ & \\Omega , \\\\ u ^ \\varepsilon = g ^ \\varepsilon \\ \\ \\ & \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} - 1 2 i \\Omega L _ \\xi & = - ( 6 \\zeta \\eta + \\xi ^ 2 ) J ( \\xi ) J ( \\eta ) J ( \\zeta ) - \\xi \\zeta J ( \\xi ) ^ 2 J ( \\eta ) - \\xi \\eta J ( \\xi ) ^ 2 J ( \\zeta ) \\\\ & - \\zeta \\eta J ( \\xi ) ^ 3 - \\zeta ^ 2 J ( \\xi ) J ( \\eta ) ^ 2 - \\eta ^ 2 J ( \\xi ) J ( \\zeta ) ^ 2 . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} \\begin{array} { l l l l } Z ^ * = & \\max \\ & c ' z + d ' u & \\\\ & \\mbox { s . t . } \\ & A z \\leq b & \\\\ & & H z + G u \\leq h & \\\\ & & z \\in \\{ 0 , 1 \\} . \\end{array} \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} \\frac { z D _ { q } \\mathcal { F } ( z ) } { \\mathcal { F } ( z ) } = 1 + \\frac { 1 - [ 2 ] _ { q } } { 3 } a _ { 2 } z + \\left ( \\frac { [ 3 ] _ { q } - 1 } { 1 0 } a _ { 3 } + \\frac { 1 - [ 2 ] _ { q } } { 9 } a _ { 2 } ^ { 2 } \\right ) z ^ 2 + \\dots \\end{align*}"}
-{"id": "4005.png", "formula": "\\begin{align*} \\widehat { f } ( y ) = \\int _ { \\mathbb { R } ^ n } f ( x ) e ^ { - 2 \\pi i y \\cdot x } d x \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} Q ^ { \\alpha } ( n , [ u ] ) = \\sum _ { J } S _ { - J } B _ J ^ { \\alpha } ( n , [ u ] ) . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} ( \\mathcal { R } _ { H _ 2 } + \\mathcal { R } _ { H _ 3 } + \\mathcal { R } _ { H _ 4 } ) ( T _ { \\delta } ) = \\varepsilon \\mathcal { R } _ 1 + \\varepsilon ^ 2 \\mathcal { R } _ 2 + \\tilde { \\mathcal { R } } _ { > 2 } , \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} ~ \\int _ 0 ^ 1 f ( x ) \\ , d x = \\frac { \\sum _ { j = 0 } ^ { n - 1 } f ( \\frac { j } { n } ) } { n - 1 } . \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} D ( e _ { i i } ) & = \\sum _ { x < i } C _ { x i } ^ { i i } e _ { x i } + \\sum _ { y > i } C _ { i y } ^ { i i } e _ { i y } , \\\\ D ( e _ { i j } ) & = \\sum _ { x < i } C _ { x i } ^ { i i } e _ { x j } + C _ { i j } ^ { i j } e _ { i j } + \\sum _ { y > j } C _ { j y } ^ { j j } e _ { i y } , \\mbox { i f } i < j . \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} m _ r ( \\theta ) = - \\lfloor ( \\theta , r ) \\rfloor \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} m _ { N } ^ { k } = \\frac { 1 } { k } \\sum _ { i = 0 } ^ { k - 1 } ( T _ { p } ^ { i } ) . m _ { N } \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} w _ i = c _ i - c _ { i - 1 } + y _ i + x _ { i - 1 } \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} x = \\sum _ { i \\in I ^ { \\prime } } \\alpha _ i b _ i + \\sum _ { j \\in N } \\beta _ j u _ j \\alpha _ i \\geq 0 \\ \\ i \\in I ^ { \\prime } \\ \\ \\beta _ j \\geq 0 \\ \\ j \\in N , \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} - \\Delta F _ { N , \\frac { N } { 2 } } ( x ) = - \\kappa _ { N , \\frac { N } { 2 } } ( N - 2 ) | x | ^ { - 2 } = F _ { N , \\frac { N } { 2 } - 1 } ( x ) . \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} \\begin{cases} v _ t ^ \\varepsilon + A _ \\varepsilon ^ + v ^ \\varepsilon = Q _ \\varepsilon ( \\bar { \\lambda } ) f ( v ^ \\varepsilon + z ^ \\varepsilon ) : = H _ \\varepsilon ( v ^ \\varepsilon , z ^ \\varepsilon ) \\\\ z _ t ^ \\varepsilon + A _ \\varepsilon ^ - z ^ \\varepsilon = ( I - Q _ \\varepsilon ( \\bar { \\lambda } ) ) f ( v ^ \\varepsilon + z ^ \\varepsilon ) : = G _ \\varepsilon ( v ^ \\varepsilon , z ^ \\varepsilon ) . \\end{cases} \\end{align*}"}
-{"id": "4316.png", "formula": "\\begin{align*} \\begin{aligned} x _ { 2 1 } x _ { 2 2 } \\det ( A _ 1 A _ 0 ) \\cdot ( A _ 1 A _ 0 ) _ { 2 , 1 } & = x _ { 2 1 } x _ { 2 2 } a _ { 1 1 } a _ { 2 2 } ( x _ { 1 1 } x _ { 2 2 } - x _ { 1 2 } x _ { 2 1 } ) a _ { 2 2 } x _ { 2 1 } \\\\ & = x _ { 2 1 } ^ 2 x _ { 2 2 } a _ { 1 1 } a _ { 2 2 } ^ 2 ( x _ { 1 1 } x _ { 2 2 } - x _ { 1 2 } x _ { 2 1 } ) . \\\\ \\end{aligned} \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} y = T ^ e \\cdot \\prod _ { ( i , p ) = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\infty } ( 1 - a _ { i j } T ^ i ) ^ { p ^ j } \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} 1 + \\partial _ 2 \\sigma ( \\pm 1 , x _ 2 ) + h _ { 0 } ' ( \\partial _ 2 \\sigma ( \\pm 1 , x _ 2 ) + 1 ) = 1 + h _ 0 ' ( 1 ) \\ \\textrm { a . e . } x _ 2 \\in ( - 1 , 1 ) . \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} ( - \\Delta _ \\Omega ) ^ { \\alpha / 2 } \\varphi ( x ) : = \\sum _ { i = 1 } ^ { \\infty } \\lambda _ i ^ { \\alpha / 2 } e _ i ( x ) \\int _ { \\Omega } e _ i ( y ) \\varphi ( y ) d y \\ , . \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{gather*} F \\big ( 3 ^ { - 1 / 3 } x , 3 ^ { - 2 / 3 } t ; \\beta = 6 \\big ) = ( \\Psi ( x , t ) ) _ { 1 1 } , \\\\ \\Psi ( x , t ) = \\kappa e ^ { \\frac { x ^ 3 } { 6 } - \\frac { x t } { 2 } } R ( x , t ) e ^ { - \\frac { i \\pi } { 2 } \\sigma _ 3 } u ^ { - \\frac { 1 } { 2 } \\sigma _ 3 } \\Psi _ { 0 } ( x , t ) , \\end{gather*}"}
-{"id": "2328.png", "formula": "\\begin{gather*} \\psi ^ { 2 } = - \\frac { 1 } { u } . \\end{gather*}"}
-{"id": "9154.png", "formula": "\\begin{align*} \\left ( x ; A \\right ) = { \\begin{cases} \\text 1 & { } x \\in A \\\\ 0 & { } x \\notin A \\end{cases} } \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} L = v ( v _ 1 ' - v ' ) + \\exp ( v _ 2 - v ) . \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} & 2 r + ( x - \\lambda _ 0 ) ^ 2 f - ( x - \\lambda _ 0 ) r ' - 2 ( x - \\lambda _ 0 ) g \\\\ & = 2 ( x - \\lambda _ 0 ) \\hat { r } + ( x - \\lambda _ 0 ) ( r ' + 2 \\mu \\varphi ) + \\mu ( x - \\lambda _ 0 ) ^ 2 \\varphi ' - ( x - \\lambda _ 0 ) r ' - 2 ( x - \\lambda _ 0 ) ( \\hat { r } + \\mu \\varphi ) \\\\ & = ( \\mu ( x - \\lambda _ 0 ) ^ 2 ) \\varphi ' , \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} \\| F \\otimes G \\| ^ 2 _ { \\mathcal { G } _ q ( \\mathcal { H } _ + , r , \\alpha ) } & \\le C _ 2 \\| F \\| ^ 2 _ { \\mathcal { G } _ q ( \\mathcal { H } _ + , s , \\alpha ) } \\| G \\| ^ 2 _ { \\mathcal { G } _ q ( \\mathcal { H } _ + , s , \\alpha ) } \\\\ & \\qquad \\times \\left ( \\sum _ { n = 0 } ^ \\infty \\bigg ( \\frac { r _ 1 } s \\bigg ) ^ n \\sum _ { i = 0 } ^ n { n \\choose i } _ q ^ \\alpha \\right ) \\ , . \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{gather*} \\omega = \\delta \\Phi _ A ^ \\ast \\wedge \\delta \\Phi ^ A \\wedge d ^ n x \\end{gather*}"}
-{"id": "6114.png", "formula": "\\begin{align*} \\pi ^ * \\mu = ( d \\Phi _ { q _ 0 } ) ^ { - 1 } _ * \\pi ^ * \\nu \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\theta } = e v ^ { - 1 } ( P _ { \\theta } ) . \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} \\forall \\ , \\alpha \\in \\mathbb { N } W _ { ( \\alpha , \\alpha ) } ( n ) = W _ { ( 1 , 1 ) } ( \\frac { n } { \\alpha } ) = \\frac { 5 } { 1 2 } \\ , \\sigma _ { 3 } ( \\frac { n } { \\alpha } ) + ( \\frac { 1 } { 1 2 } - \\frac { 1 } { 2 \\ , \\alpha } n ) \\sigma ( \\frac { n } { \\alpha } ) . \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} S \\ : = \\ \\{ v ( s - a ' ) : a \\in K \\} \\ \\subseteq \\ ( \\Gamma ^ { < } ) ' \\subseteq \\Gamma _ { \\infty } . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} \\tau ( \\omega ) \\leq \\sum _ { k = 0 } ^ \\infty \\frac { \\textrm { l n } \\ , 2 } { 2 \\cdot 4 ^ k } = \\frac { 2 } { 3 } \\textrm { l n } \\ , 2 < \\frac { 1 } { 2 } \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} y ( n + 1 ) = \\left ( \\begin{array} { c c c } A _ 1 ( n ) & & 0 \\\\ 0 & & A _ 2 ( n ) \\end{array} \\right ) y ( n ) , \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\epsilon ( 1 / 2 , \\pi , \\tau ) = 1 . \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} A q _ i = \\tilde { b } _ { i 1 } q _ 1 + \\tilde { b } _ { i 2 } q _ 2 + \\cdots + \\tilde { b } _ { i s } q _ s , ~ i = 1 , 2 , \\cdots , s , \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{gather*} E _ { \\beta N } ^ { { \\rm s o f t } } \\bigl ( 0 ; ( t , \\infty ) \\bigr ) = \\int _ { - \\infty } ^ { t } \\cdots \\int _ { - \\infty } ^ { t } p ( \\lambda _ 1 , \\ldots , \\lambda _ N ) d \\lambda _ 1 \\cdots d \\lambda _ N . \\end{gather*}"}
-{"id": "2708.png", "formula": "\\begin{align*} \\zeta _ { G _ n } ( s ) = \\frac { n ^ { 2 s } } { \\Gamma ( s ) } \\int _ 0 ^ \\infty \\theta ^ { G _ n } ( n ^ 2 t ) t ^ { s - 1 } d t = \\frac { n ^ { 2 s } } { \\Gamma ( s ) } \\Big ( S _ 1 ( n ) + S _ 2 ( n ) + S _ 3 ( n ) + S _ 4 ( n ) + S _ 5 ( n ) \\Big ) , \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} W / \\Q _ \\ell \\ ; \\ ; : \\ ; \\ ; y ^ 2 = x ^ 3 + a x + b , a = - \\frac { c _ 4 } { 4 8 } , b = - \\frac { c _ 6 } { 8 6 4 } . \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ t \\P \\Big ( Z _ { ( k + g ) ( j - 1 ) + i } ^ { ( k + g ) ( j - 1 ) + i + k - 1 } = s _ j \\Big ) & = 2 ^ { - t ( \\hat { H } _ 1 ( s ^ t ) + D _ { \\rm K L } ( \\hat { p } ^ { ( 1 ) } ( \\cdot | s ^ t ) , \\mu _ k ^ { ( b ) } ) ) } . \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} | ( f ^ n ) ' ( z ) | = \\prod _ { j = 0 } ^ { n - 1 } \\left | f ' ( z _ { j } ) \\right | \\leq C ^ n \\prod _ { j = 0 } ^ { n - 1 } | z _ { j } | ^ { \\rho - 1 } | z _ { j } | = \\frac { C ^ n | z _ n | } { | z _ 0 | } \\prod _ { j = 0 } ^ { n - 1 } | z _ { j } | ^ { \\rho } . \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} H _ 3 ( u ) = z ( u _ 0 ) - \\int _ { u _ 0 } ^ u H _ 1 ( \\cdotp , \\zeta ' ) \\ , d H _ 2 ( \\cdotp , \\zeta ' ) , u \\in M . \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} \\lVert \\mathfrak { I } _ 0 \\rVert _ { s _ 0 + \\mu } ^ { L i p ( \\gamma ) } \\le \\varepsilon ^ { 6 - 2 b } \\gamma ^ { - 1 } , \\lVert Z \\rVert _ { s _ 0 + \\mu } ^ { L i p ( \\gamma ) } \\le \\varepsilon ^ { 6 - 2 b } , \\gamma = \\varepsilon ^ { 2 + a } , a \\in ( 0 , 1 / 6 ) , \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} a b = \\frac { 1 } { 2 } ( a ^ 2 + b ^ 2 - ( a - b ) ^ 2 ) , a , b \\in \\R , \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} V ^ { \\ast } \\Psi A ^ { \\dagger } & = V ^ { \\ast } A ^ { \\dagger } - ( S _ { A } - I ) ( I + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } F _ { S _ { A } } K ^ { \\ast } A ^ { \\dagger } , \\\\ V ^ { \\ast } \\Psi K S _ { A } ^ { \\dagger } H & = ( S _ { A } - I ) S _ { A } ^ { \\dagger } H - ( S _ { A } - I ) ( I + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } F _ { S _ { A } } K ^ { \\ast } K S _ { A } ^ { \\dagger } H . \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} \\phi _ 0 ( z ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( \\varepsilon ) } \\theta ( s ) z ^ { - s } d s \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} f _ n ( z ) = \\frac { \\alpha _ n + z f _ { n + 1 } ( z ) } { 1 + \\bar \\alpha _ n z f _ { n + 1 } ( z ) } , n = 0 , 1 , 2 , \\dots . \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} \\langle Q ' ( u , v ) , ( u , v ) \\rangle & = p \\| ( u , v ) \\| ^ p - q \\int _ \\Omega ( \\lambda | u | ^ { q } + \\mu | v | ^ { q } ) d x - 2 ( \\alpha + \\beta ) \\int _ \\Omega | u | ^ { \\alpha } | v | ^ \\beta d x \\\\ & = ( p - 1 ) \\| ( u , v ) \\| ^ p - ( q - 1 ) \\int _ \\Omega ( \\lambda | u | ^ { q } + \\mu | v | ^ q ) d x - 2 ( \\alpha + \\beta - 1 ) \\int _ \\Omega | u | ^ { \\alpha } | v | ^ \\beta d x = \\varphi _ { u , v } '' ( 1 ) \\neq 0 , \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} H _ { j , 4 } = - \\real \\left [ i ^ j \\sum _ { \\alpha _ 1 + \\alpha _ 2 + \\alpha _ 3 = j - 2 } ( - 1 ) ^ { \\alpha _ 1 } \\int u ^ { ( \\alpha _ 2 ) } u ^ { ( \\alpha _ 3 ) } \\overline { u ^ { ( \\alpha _ 1 ) } u } d x \\right ] . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } u ( t , x ) = \\Delta u ( t , x ) + a u ( t , x ) - b u ( t , x ) ^ \\gamma , \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} \\lim _ { x \\to - \\infty } v _ \\mu ( x + \\int _ 0 ^ t c _ \\mu ( s ) d s , t ) = u ^ + ( t ) , \\lim _ { x \\to \\infty } v _ \\mu ( x + \\int _ 0 ^ t c _ \\mu ( s ) d s , t ) = 0 \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} \\mathcal { R } _ H = \\frac { \\partial } { \\partial t } + \\left ( p + \\alpha \\sin { ( w t ) } q ^ 2 p \\right ) \\frac { \\partial } { \\partial q } - \\left ( q + \\alpha \\sin { ( w t ) } p ^ 2 q \\right ) \\frac { \\partial } { \\partial p } . \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\mathcal { L } _ { p , F ' } = \\dfrac { l _ \\lambda e ' T r ' _ 1 \\left ( \\left ( T r ' _ 2 ( \\mathcal { E ' } _ c ^ { \\chi \\chi _ { - 1 } } * \\Theta ' _ \\chi \\mid [ \\mathfrak { l ' } ^ 2 ] ) \\right ) \\mid \\Psi ' _ 2 \\right ) } { \\Delta ' ( X ' , Y ' ) \\mathcal { E ' } _ 2 ( X ' , Y ' ) H ' } , \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { N } D _ { i j } { \\bf X } _ { k j } ^ * = \\dot { \\bf x } ^ I ( \\tau _ i ) , 1 \\leq i \\leq N , \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} A _ { ( m + 1 ) p _ n - ( m - r _ n ) } = A _ { ( m + 1 ) p _ n + r _ n } - x ^ { 2 m - 2 r _ n } A _ { ( m + 1 ) p _ n + m - r _ n } . \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} e _ k ^ { ( a ) } : = \\sum _ { 1 \\leq i _ 1 < i _ 2 < \\cdots < i _ k \\leq m } x _ { i _ 1 } ^ { ( a ) } x _ { i _ 2 } ^ { ( a + 1 ) } \\cdots x _ { i _ k } ^ { ( a + k - 1 ) } . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} r _ { 4 } ( n ) = 8 \\sigma ( n ) - 3 2 \\sigma ( \\frac { n } { 4 } ) . \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} R ( \\psi ) w = \\sum _ { \\lvert j \\rvert \\le C } ( w , g _ j ( \\psi ) ) _ { L ^ 2 ( \\mathbb { T } ) } \\ , \\chi _ j ( \\psi ) . \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} \\frac { - 1 } { ( 2 k - 1 ) x ^ { 2 k - 1 } } - 2 & x < - 1 , \\\\ \\frac { - 1 } { ( 2 k - 1 ) x ^ { 2 k - 1 } } + 2 & x > 1 . \\\\ \\end{cases} \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\P ( \\{ M _ n \\le \\lambda ^ n \\} ) < \\infty \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\infty } V _ { 1 , j } ^ { \\mu } & = \\sum _ { j = 0 } ^ { \\infty } \\int _ { \\mathcal { V } _ { \\alpha , \\theta } } V _ { 1 , j } \\mu ( \\mathrm { d } V ) \\\\ & = \\int _ { \\mathcal { V } _ { \\alpha , \\theta } } \\sum _ { j = 0 } ^ { \\infty } V _ { 1 , j } \\mu ( \\mathrm { d } V ) \\\\ & = \\int _ { \\mathcal { V } _ { \\alpha , \\theta } } 1 \\mu ( \\mathrm { d } V ) = 1 . \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} E _ s ( u ) = - \\frac 1 { \\pi } \\int _ \\R ( 1 + \\xi ^ 2 ) ^ s \\ln | T ( \\xi / 2 ) | d \\xi , \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} \\phi _ n ( z ) = \\frac { 1 } { \\sqrt { 2 ^ n n ! \\sqrt { \\pi } } } H _ n ( z ) e ^ { - \\frac { z ^ 2 } { 2 } } , n \\in \\mathbb { N } _ 0 , \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} ( \\operatorname { D P [ 0 ] } + \\operatorname { Q } ) ( \\Xi _ 2 ) = - \\operatorname { Q } ( \\Xi _ 1 ) - \\sum \\limits _ { j \\geq 2 } \\operatorname { Q } _ { i _ { 1 } } ( \\Xi _ { \\bullet } ) \\cdots \\operatorname { Q } _ { i _ { j } } ( \\Xi _ { \\bullet } ) , \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} \\ell _ n ( G ; \\sigma ^ 2 ) = \\sum _ { i = 1 } ^ n \\log f ( x _ i ; G ; \\sigma ^ 2 ) . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} T _ f g = \\sum \\limits _ { j > 4 } f _ { < j - 4 } g _ j , \\Pi [ f , g ] = \\sum \\limits _ { \\substack { | j - k | \\leq 4 \\\\ j , k \\geq 0 } } f _ j g _ k , \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\varepsilon ( D ) = L ( 1 , \\chi ) \\log { | D | } \\to 0 . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} m = \\prod _ { \\rho = 1 } ^ n \\sum _ { j _ \\rho = 1 } ^ N \\sum _ { k _ \\rho = 1 } ^ N a _ { j _ \\rho } ( t _ \\rho ) a _ { k _ \\rho } ( \\tau _ \\rho ) a _ { j _ { \\rho } } c _ { j _ { \\rho } + k _ \\rho } \\phi ( N \\xi _ { \\rho } - j _ { \\rho } ) \\phi ( N \\eta _ \\rho - k _ \\rho ) , \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} \\max \\left \\{ y , \\frac { v _ 1 } 2 \\right \\} \\le v _ i \\le \\min \\{ x , 2 v _ 1 \\} ( i = 2 , 3 ) . \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} & \\hat { f } ( n ) = \\int _ 0 ^ 1 [ D _ \\mu ( x ) - x ] e ( - n x ) \\ , d x \\\\ & = \\frac { 1 } { - 2 \\pi i n } [ D _ \\mu ( x ) - x ] e ( - n x ) | _ 0 ^ 1 + \\frac { 1 } { 2 \\pi i n } \\int _ 0 ^ 1 e ( - n x ) \\ , d [ D _ \\mu ( x ) - x ] \\\\ & = \\frac { \\hat { \\mu } ( n ) } { 2 \\pi i n } , \\end{align*}"}
-{"id": "3951.png", "formula": "\\begin{align*} L ( s , \\lambda , \\xi ) = \\prod _ { i = 1 } ^ k \\prod _ { j = 1 } ^ l L ( s , \\delta _ i , \\delta _ j ' ) \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} \\left \\vert \\bar { \\psi } ( \\gamma ) \\right \\vert & = | E [ \\bar { \\alpha } ( X ) \\{ \\gamma ( X ) - \\gamma _ { 0 } ( X ) \\} ] - E [ \\alpha _ { 0 } ( X ) f _ { 0 } ( 0 | X ) \\{ \\gamma ( X ) - \\gamma _ { 0 } ( X ) \\} ] \\\\ & + E [ \\{ \\partial f ( \\delta ( X ) | X ) / \\partial u \\} \\{ \\gamma ( X ) - \\gamma _ { 0 } ( X ) \\} ^ { 2 } ] | \\\\ & \\leq C \\left \\Vert \\gamma - \\gamma _ { 0 } \\right \\Vert ^ { 2 } . Q . E . D . \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} L = - \\frac { v _ t v _ x } { 2 } - \\frac { v _ x ^ 3 } { 6 } + \\frac { v _ { x x } ^ 2 } { 2 } , \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { \\sqrt { \\eta '' } } \\right ) '' = - \\mu \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\nu _ k \\delta ( x - x _ k ) , \\{ \\nu _ k \\} \\in \\ell ^ 1 ( \\N ) , \\{ x _ k \\} \\subset ( - l , l ) . \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\begin{aligned} B ( j , k ) : = & - ( 2 4 c _ 4 - 4 8 c _ 1 ^ 2 + \\frac { 1 2 c _ 6 - 1 6 c _ 2 ^ 2 } { 3 ( j ^ 2 + k ^ 2 + j k ) } ) D _ S ^ 3 U D _ S ^ 3 \\\\ [ 2 m m ] & + ( \\frac { 1 6 c _ 2 ^ 2 } { 3 } - 4 c _ 6 + \\frac { ( 1 6 c _ 2 c _ 3 - 2 4 c _ 7 ) } { j ^ 2 + k ^ 2 + j k } ) D _ S U D _ S ^ 3 . \\end{aligned} \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} z ' ( u ) = z ( u _ 0 ) - \\int _ { u _ 0 } ^ u x ' d y , u \\in R \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} & p = \\frac { m - 1 + \\delta \\left ( 1 + \\frac 2 n \\right ) } { m - 1 + \\delta } \\\\ & q = \\frac { n ( m - 1 + \\delta \\left ( 1 + \\frac 2 n \\right ) } { 2 \\delta } . \\end{align*}"}
-{"id": "8561.png", "formula": "\\begin{align*} C _ \\mathsf { S e m i - D e t } = \\max _ { p _ { X | S } } \\min \\Big \\{ H ( Y | Z ) , H ( Y | S ) \\Big \\} , \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} ( n _ 2 n _ 3 + n _ 2 n _ 4 + n _ 3 n _ 4 ) r _ 1 + . . . + ( n _ 1 n _ 2 + n _ 1 n _ 3 + n _ 2 n _ 3 ) r _ 4 = 0 \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} u ( t , x ) = \\int _ 0 ^ t \\int _ { \\R } G _ { t - \\theta } ( x - \\eta ) W ( d \\theta , d \\eta ) , ( t , x ) \\in [ 0 , T ] \\times \\R , \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} \\tfrac 1 2 L _ { ( k , k + 1 ) } ( t ) : = \\tfrac 1 2 L _ { ( k - 1 , k ) } ( t ) - Y _ k ( t ) + Y _ k ( 0 ) + g _ k t + B _ k ( t ) . \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} | u _ { 1 } | = | 2 c + 1 | \\geq 2 | c | - 1 \\geq 1 = | u _ { 0 } | . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\mathcal { A } _ 2 ( w _ 2 ) = \\left \\{ A _ 2 : A _ 2 \\subseteq \\{ 1 , \\cdots , \\ell \\} \\backslash A ^ { \\ast } , | A _ 2 | = w _ 2 \\right \\} . \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} \\vec { X } _ { 3 1 } & = ( 1 , 0 , 0 ) , \\mu _ { 3 1 } = - \\frac { h } { \\lambda } , \\\\ \\vec { X } _ { 3 2 } & = ( 0 , 1 , 0 ) , \\mu _ { 3 2 } = 1 , \\\\ \\vec { X } _ { 3 3 } & = ( 0 , 0 , 1 ) , \\mu _ { 3 3 } = - A . \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} \\Pi _ S ^ { \\perp } ( \\mathcal { A } - \\mathrm { I } ) \\Pi _ S [ b _ 3 \\partial _ { y y y } + b _ 2 \\partial _ { y y } + b _ 1 \\partial _ y + b _ 0 ] = \\varepsilon ^ 2 \\Pi _ S ^ { \\perp } [ \\partial _ x ( \\beta _ 1 \\ , \\partial _ x ( \\alpha _ { 1 , 1 } \\ , \\cdot ) ) ] + o ( \\varepsilon ^ 2 ) \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} a + 0 & = a , \\\\ a + a & = \\{ 0 , a \\} , \\\\ \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} [ a , B ] + [ A , b ] = 0 \\Leftrightarrow B _ { 1 2 } ( a _ { 1 1 } - a _ { 2 2 } ) + b _ { 2 2 } - b _ { 1 1 } = 0 . \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} f : = \\sum _ { i = 1 } ^ n ( 2 x _ i - x _ { i - 1 } - x _ { i + 1 } + \\frac { 1 } { 2 } h ^ 2 ( x _ { i } + i h + 1 ) ^ 3 ) ^ 2 , \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{align*} u = \\dfrac { v t - w s } { 2 v - s } \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} L _ t \\le L _ s e ^ { - c _ + \\sum _ { u = s } ^ { t - 1 } \\Phi _ u ^ 2 } \\le e ^ { - c _ + \\sum _ { u = s } ^ { t - 1 } \\Phi _ u ^ 2 } . \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\bar \\lambda ( A ^ { \\nu } , ( A \\cup \\{ x \\} ) ^ { \\nu + \\mathbf { e _ i } } ) & = x ^ i , \\\\ \\bar \\lambda ( [ n ] ^ { \\mu } , \\hat { 1 } ) & = ( n + 1 ) ^ 1 . \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} R : = [ A _ m , A _ { m + 1 } ] . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} & e ( 4 , 5 ) = e _ 2 ^ { ( 4 ) } = q _ 1 ^ { ( 1 ) } q _ 2 ^ { ( 1 ) } + q _ 2 ^ { ( 3 ) } q _ 3 ^ { ( 3 ) } + q _ 1 ^ { ( 1 ) } q _ 3 ^ { ( 3 ) } , \\\\ & e ( 3 , 3 ) = e _ 1 ^ { ( 3 ) } = q _ 1 ^ { ( 3 ) } + q _ 2 ^ { ( 2 ) } + q _ 3 ^ { ( 1 ) } , \\\\ & e ( a , a + 2 ) = e _ 3 ^ { ( a ) } = q _ 1 ^ { ( i ) } q _ 2 ^ { ( i ) } q _ 3 ^ { ( i ) } , i \\equiv a \\mod 3 , \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} ( x ^ \\ell + m ) ( x ^ { \\ell + 2 } - m ) ( x ^ { \\ell + 2 } - m x ^ 2 + m ) = x ^ { 3 \\ell + 4 } \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} y = \\sum _ { n \\in \\mathbb { N } _ 0 } c _ n u _ n , c _ n = \\langle u _ n , y \\rangle _ { L ^ 2 } , \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} \\begin{aligned} R _ { - L , L } ( x , y ) & = e ^ { ( y ^ 2 - x ^ 2 ) / 2 + L - \\sqrt 2 r ( e ^ L y - e ^ { - L } x ) + \\sinh ( 2 L ) r ^ 2 } \\\\ & \\times \\int _ \\R \\d u \\int _ \\R \\d v \\sum _ { j = 0 } ^ 4 Q _ j \\left ( u , v , e ^ { - L } x - \\frac { ( 1 + e ^ { - 2 L } ) r } { \\sqrt 2 } , e ^ L y - \\frac { ( 1 + e ^ { 2 L } ) r } { \\sqrt 2 } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} E _ C ( \\theta ) = - \\frac { 1 } { \\theta } \\ln \\left ( \\mathbb { E } \\left \\{ e ^ { - \\theta R } \\right \\} \\right ) \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} ( \\overline { a } b - d \\overline { e } \\overline { K } ) \\alpha + ( a \\overline { b } - \\overline { d } e K ) \\overline { \\alpha } + ( | b | ^ { 2 } - | e | ^ { 2 } | K | ^ { 2 } ) | \\alpha | ^ { 2 } = 0 , \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} h ^ t x _ j ^ { ( 2 ) } = y _ j ^ { ( 2 ) } , \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} D ( F ( z , \\{ p _ n \\} ) ) = z \\frac { \\partial } { \\partial z } F ( z , \\{ p _ n \\} ) + \\beta z \\oint \\frac { d \\xi } { \\xi ^ 2 } \\frac { \\phi ^ - ( \\xi ) } { 1 - \\frac { z } { \\xi } } V ^ { - 1 } ( \\xi ) V ( x _ i ) F ( \\xi , \\{ p _ n \\} ) , \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} \\Lambda _ { \\xi , h } = \\left \\{ \\eta \\in X : \\eta \\geq \\xi , \\eta ' ( 0 ) = - h ^ { 2 } , \\eta '' \\geq 0 \\right \\} \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} { \\Lambda } ^ A ( x , y ) = e ^ { - i \\int _ { [ x , y ] } A } \\ , , \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} & \\sum \\bigl ( ( \\omega ; \\omega _ 1 , \\ldots , \\omega _ j , \\ldots , ) ; \\nu _ 1 , \\ldots , \\nu _ i , \\ldots , \\nu _ n \\bigr ) \\\\ & = \\sum \\bigl ( \\nu ; ( \\omega _ 1 ; \\omega _ { 1 , 1 } , \\ldots ) , \\ldots , ( \\omega _ i ; \\omega _ { i , 1 } , \\ldots , \\omega _ { i , j } , \\ldots ) , \\ldots , ( \\omega _ n ; \\omega _ { n , 1 } , \\ldots ) \\bigr ) \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} z ^ i w ^ j \\frac { \\partial ^ i } { \\partial z ^ i } \\frac { \\partial ^ j } { \\partial w ^ j } W _ \\pm ( z , w ) \\begin{cases} = 0 , & \\sqrt { z } / w \\leq C ^ { - \\varepsilon } , \\\\ \\ll C ^ { \\varepsilon ( i + j ) } , & \\mathrm { o t h e r w i s e } , \\\\ \\end{cases} \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} \\begin{aligned} ( f ^ n ) ^ \\# ( z ) & \\geq \\exp \\ ! \\left ( ( 1 + \\delta ( x _ { n - 1 } ) ) x _ n - n \\log ( 9 6 \\pi ) - x _ n - 4 \\pi \\log ( 2 | z | ) \\right ) \\\\ & = \\exp \\ ! \\left ( \\delta ( x _ { n - 1 } ) x _ n - n \\log ( 9 6 \\pi ) - 4 \\pi \\log ( 2 | z | ) \\right ) \\\\ & \\geq \\exp \\ ! \\left ( \\frac { x _ n } { \\log x _ n } - n \\log ( 9 6 \\pi ) - 4 \\pi \\log ( 2 | z | ) \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} \\kappa _ { \\lambda b } = \\check { \\rho } ( \\lambda ^ { - 1 } ) \\lambda \\kappa _ b . \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{align*} \\Sigma _ i d _ i = d i m ( G / B ) + l . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} J _ { \\lambda , \\mu } ( u , v ) & = \\left ( \\frac { 1 } { p } - \\frac { 1 } { p _ s ^ \\ast } \\right ) \\| ( u , v ) \\| ^ p - \\left ( \\frac { 1 } { q } - \\frac { 1 } { p _ s ^ \\ast } \\right ) \\int _ \\Omega ( \\lambda | u | ^ q + \\mu | v | ^ q ) d x \\\\ & = \\frac { s } { n } \\| ( u , v ) \\| ^ p - \\left ( \\frac { 1 } { q } - \\frac { 1 } { p _ s ^ \\ast } \\right ) \\int _ \\Omega ( \\lambda | u | ^ q + \\mu | v | ^ q ) d x . \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} \\gamma _ \\lambda ( { x } ) = \\frac { 1 } { \\lambda _ 1 ! \\cdots \\lambda _ s ! } \\sum _ { \\sigma \\in S _ N } \\sigma ( \\mathbf { m } ) . \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} g _ { \\lambda , \\rho } ^ 1 \\ ! ( w _ 1 ) = \\frac { n _ 0 } { 4 k _ { \\ell } } \\log \\left ( 1 + \\lambda \\rho ( 1 - \\lambda \\rho ) w _ 1 P ' \\right ) - \\frac { | A ^ { \\ast } | } { k _ { \\ell } } H _ 2 \\Big ( \\ ! \\frac { w _ 1 } { | A ^ { \\ast } | } \\ ! \\Big ) \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} \\Omega ( a ) = \\prod _ { i = 1 } ^ { K } \\left ( \\nu _ i ( a ) - \\delta _ i \\right ) . \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} ( r - 1 ) ( s - 1 ) = ( r - 1 ) \\left ( { n } / { r } - 1 \\right ) \\leq r \\left ( { n } / { r } - 1 \\right ) = n - r \\leq n - 1 . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} D _ { q } \\mathcal { F } ( z ) = 1 + \\sum _ { n = 2 } ^ \\infty \\frac { ( - 1 ) ^ { n - 1 } [ n ] _ { q } a _ { n } } { ( 2 n - 1 ) ( n - 1 ) ! } z ^ { n - 1 } . \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\sum _ { n = N _ 1 + 1 } ^ { \\infty } | \\langle f , \\phi _ n ^ { ( 1 ) } \\rangle | ^ 2 \\leq C \\sum _ { n = 1 } ^ { \\infty } \\left ( \\sum _ { m = 1 } ^ { \\infty } \\frac { | f _ m | } { m ^ \\alpha n ^ { \\alpha } | z _ n - \\mu _ m | } \\right ) ^ 2 = C \\| \\mathcal M \\tilde f \\| ^ 2 _ { \\ell ^ 2 ( \\N ) } \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} \\psi ( t , r , \\phi ) = R ( r ) e ^ { - \\imath ( \\epsilon t + p \\phi ) } , \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\eta } : = \\pi \\left ( H _ { \\eta } \\cap \\mathcal { U } ( K ) \\right ) , \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} \\lambda ( f ) = \\liminf _ { r \\to \\infty } \\frac { \\log \\log M ( r , f ) } { \\log r } . \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} \\phi = \\psi _ { R _ 2 } \\ , \\phi ^ \\circ \\ , \\psi _ { R _ 1 } \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{gather*} \\Omega [ \\phi ] = \\int _ N ( j ^ { \\infty } \\phi ) ^ \\ast ( L ' _ 1 ) \\end{gather*}"}
-{"id": "9600.png", "formula": "\\begin{align*} \\wedge _ { i } b _ 2 ^ i = \\pm \\nu ( \\mathcal { E } _ 2 ) ^ { - 1 } \\nu ( \\mathcal { E } _ A ) ^ { - 1 } \\nu ( \\mathcal { E } _ C ) ^ { - 1 } M ' \\wedge N ' \\wedge P ' \\wedge Q ' . \\end{align*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\Lambda _ { A ( R ) } = \\{ ( i , \\lambda ) \\ ; \\mbox { w i t h } \\ ; 1 \\le i \\le \\ell ( \\lambda ) \\mbox { a n d } \\lambda \\in \\Lambda ^ 0 \\} . \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} ( u | v ) = \\int _ X \\langle u | v \\rangle d v _ X , \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} \\frac { \\partial h } { \\partial t } = \\nu \\nabla ^ 2 h + \\frac { \\lambda } { 2 } ( \\nabla h ) ^ 2 + W . \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{gather*} \\frac { \\partial F } { \\partial t } + \\frac { 2 } { \\beta } \\frac { \\partial ^ 2 F } { \\partial x ^ 2 } + \\big ( t - x ^ 2 \\big ) \\frac { \\partial F } { \\partial x } = 0 , ( x , t ) \\in { \\mathbb R } ^ 2 \\end{gather*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\frac { _ { q } \\mathcal { U } _ { - } ( \\xi _ { a } ^ { ( h + 1 / 2 ) } ) } { ( \\ , ( \\xi _ { a } ^ { ( h + 3 / 2 ) } ) ^ { 2 } - 1 / ( \\xi _ { a } ^ { ( h + 3 / 2 ) } ) ^ { 2 } \\ , ) } = \\mathsf { D } _ { - } ( \\xi _ { a } ^ { ( h + 1 ) } ) \\mathsf { D } _ { - } ( 1 / \\xi _ { a } ^ { ( h ) } ) = \\mathsf { A } _ { - } ( \\xi _ { a } ^ { ( h + 1 ) } ) \\mathsf { A } _ { - } ( 1 / \\xi _ { a } ^ { ( h ) } ) , \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} \\left \\{ s \\in [ 0 , 1 ) : \\eta ( s ) = \\xi ( s ) \\right \\} . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} & \\ell _ { i } = \\ell _ { \\left \\lfloor \\frac { N } { 2 } \\right \\rfloor + i } = \\frac { 2 i } { N } , \\ i \\in \\left [ 1 : \\left \\lfloor \\frac { N } { 2 } \\right \\rfloor \\right ] , \\\\ & r _ { i } = r _ { \\left \\lfloor \\frac { N } { 2 } \\right \\rfloor + i } = \\frac { N - 2 i + 2 } { N } , \\ i \\in \\left [ 1 : \\left \\lfloor \\frac { N } { 2 } \\right \\rfloor \\right ] , \\\\ & N \\ \\ \\ell _ N = L , \\ r _ N = \\frac { 1 } { N } , \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} \\nu _ i \\geq 0 , \\ i = 1 , \\dots , n , \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} \\gamma : = \\exp \\sup _ { \\theta \\in \\mathbb P \\mathbb R ^ 1 } \\limsup _ { n \\to \\infty } \\frac { 1 } { n } \\log | \\{ c _ 1 \\cdots c _ n : \\theta \\in \\phi _ { c _ n \\cdots c _ 1 } ( \\mathcal Q _ 2 ) \\} | . \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} ( h , h ' ) \\cdot X = X , X \\in \\mathbb { X } . \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} V _ t \\ , \\rfloor \\ , \\omega _ t = \\lambda , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{align*} \\lvert T \\rvert _ s ^ { L i p ( \\gamma ) } = \\lVert p \\rVert _ s ^ { L i p ( \\gamma ) } . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} 0 = E _ 0 \\subset E _ 1 \\subset \\ldots \\subset E _ { n - 1 } \\subset E _ n = E \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} \\partial g ^ + _ \\gamma ( v ) = \\begin{cases} 0 & v < L , \\\\ \\frac 1 \\gamma ( v - L ) & v \\in [ L , L + \\gamma ] , \\\\ 1 & v > L + \\gamma . \\end{cases} \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} \\frac { d ^ 2 q } { d t ^ 2 } = \\frac { k } { q ^ 3 } - \\omega ^ 2 ( t ) q . \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\tilde { J } _ { 3 } ( t ) g \\right \\| _ { 2 } & \\le C t ^ { 1 - \\ell } ( 1 + t ) ^ { - \\frac { n } { 2 } - k } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla ^ { ( k + \\ell - 1 ) _ { + } } _ { x } g \\| _ { 2 } , \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\pi & ^ b _ { 0 b ( n - b ) } = \\pi ^ { b + 1 } _ { 0 b ( n - b ) } - \\frac { n - 1 - 2 b } { 2 } c _ b \\\\ & = z _ b - \\frac { n - 1 - 2 b } { n + 1 - 2 b } \\left ( z _ b - \\frac { 1 } { n - 1 - 2 b } \\sum _ { i = b + 1 } ^ { n - b - 1 } z _ i \\right ) \\\\ & = \\frac { \\sum _ { i = b } ^ { n - b } z _ i } { n + 1 - 2 b } , \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} \\begin{cases} ( - \\Delta _ p ) ^ s u = \\lambda | u | ^ { p - 2 } u & \\Omega , \\\\ u = 0 & \\Omega ^ c , \\end{cases} \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} ( x ; [ a , b ] ) = \\mathcal { B } _ { + + } \\left ( x - \\frac { b + a } { 2 } , \\frac { b - a } { 2 } \\right ) \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} s _ { n + 3 } = s _ { n + 2 } + p \\big ( s _ { n } - s _ { n - 1 } \\big ) \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} H ( x , y ) : = \\frac { 2 u ( x ) - u ( x + y ) - u ( x - y ) } { | y | ^ { N + 2 \\sigma } } , h _ t ( x , y ) : = \\frac { H ( x + t e _ 1 , y ) - H ( x , y ) } { t } , t \\in \\R \\backslash \\{ 0 \\} . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} \\omega \\cdot ( \\mathtt { l } ( j _ 1 ) + \\mathtt { l } ( j _ 2 ) + \\mathtt { l } ( j _ 3 ) ) = \\overline { \\omega } \\cdot ( \\mathtt { l } ( j _ 1 ) + \\mathtt { l } ( j _ 2 ) + \\mathtt { l } ( j _ 3 ) ) + ( \\omega - \\overline { \\omega } ) \\cdot ( \\mathtt { l } ( j _ 1 ) + \\mathtt { l } ( j _ 2 ) + \\mathtt { l } ( j _ 3 ) ) = j _ 1 ^ 3 + j _ 2 ^ 3 + j _ 3 ^ 3 + O ( \\varepsilon ^ 2 ) \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} w ( n + 1 ) = \\frac { 1 } { \\lambda _ { \\ell - 1 } } B _ { 0 } ( n ) w ( n ) \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} \\mathbf { x } _ i = \\mathbf { A } \\mathbf { d } _ i , \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} | \\Phi | & \\leq l ^ L q ^ { 3 l ^ 2 L ^ 2 - L ( n - 1 ) } | G | ^ d \\leq l ^ { n / ( 9 l ^ 2 ) } q ^ { 3 l ^ 2 \\cdot n ^ 2 / ( 8 1 l ^ 4 ) - n / ( 9 l ^ 2 ) \\cdot n / 2 } | G | ^ d \\\\ & \\leq q ^ n \\cdot q ^ { n ^ 2 ( 1 / ( 2 7 l ^ 2 ) - 1 / ( 1 8 l ^ 2 ) ) } | G | ^ d = q ^ { n - n ^ 2 / ( 5 4 l ^ 2 ) } | G | ^ d = ( q ^ { n ^ 2 } ) ^ { 1 / n - 1 / ( 5 4 l ^ 2 ) } | G | ^ d \\\\ & \\leq ( q ^ { n ^ 2 } ) ^ { 1 / ( 2 1 6 l ^ 2 ) - 1 / ( 5 4 l ^ 2 ) } | G | ^ d = ( q ^ { n ^ 2 } ) ^ { - 1 / ( 7 2 l ^ 2 ) } | G | ^ d \\leq | G | ^ { d - 1 / ( 7 2 l ^ 2 ) } , \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} R _ { n , \\omega } = ( R _ { n - 1 , \\omega } + \\omega _ { n , n } ) \\Lambda _ n , n \\ge 1 , ~ ~ R _ { 0 , \\omega } = \\omega _ 0 / ( 1 - \\omega _ 0 ) , \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} 0 = \\bold { D } _ Q ^ { \\ast } ( F ) + \\bold { D } _ F ^ { \\ast } ( Q ) . \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} X = Q ^ { \\alpha } ( n , T ( n , u _ n ) ) \\frac { \\partial u _ n ^ { \\beta } } { \\partial \\widetilde { u } _ n ^ { \\alpha } } \\frac { \\partial } { \\partial u _ n ^ { \\beta } } . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} | A _ { \\rm s t } | = | B _ { \\rm s t } | = \\sqrt { \\frac { \\lambda } { - ( c + d ) } } . \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} k e _ k = \\sum _ { n = 1 } ^ k ( - 1 ) ^ { n - 1 } e _ { k - n } p _ n . \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } f _ { n } ( k ) x _ { n } ( k ) > \\epsilon , n = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "8974.png", "formula": "\\begin{align*} \\partial ^ h _ { t ^ j } u : t \\to ( \\partial ^ h _ { t ^ j } u ) ( t ) = \\mathbb H ^ t ( i _ t ^ * [ \\partial ^ E , \\delta _ { V _ j } ] \\mathbf { u } ) , \\ \\ ( \\theta _ j ^ * u ) ( t ) = \\mathbb H ^ t ( i _ t ^ * [ \\partial ^ E , \\delta _ { \\overline { V _ j } } ] \\mathbf { u } ) , \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} \\frac { ( a ( 1 - b ) q ; q ) _ n } { ( a q ; q ) _ n } = 1 + \\sum _ { \\pi \\in \\mathcal { U } _ { n } } a ^ { \\nu ( \\pi ) } b ^ { \\nu _ d ( \\pi ) } q ^ { | \\pi | } , \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} & \\frac { E [ f ( \\mathbf { W } ^ * ( \\boldsymbol { \\mu } + \\theta \\cdot \\mathbf { e } ^ k ; 0 ) , \\mathbf { J } ( 0 ) , \\boldsymbol { \\mu } + \\theta \\cdot \\mathbf { e } ^ k ) ] - E [ f ( \\mathbf { W } ^ * ( \\boldsymbol { \\mu } ; 0 ) , \\mathbf { J } ( 0 ) , \\boldsymbol { \\mu } ) ] } { \\theta } \\\\ \\rightarrow ~ & E \\left [ 1 ( k = h ( \\mathbf { W } ^ * ( 0 ) , \\mathbf { J } ( 0 ) , \\boldsymbol { \\mu } ) ) \\left ( V ^ * _ k - \\frac { J _ { k } ( 0 ) } { \\mu _ k ^ 2 } \\right ) \\right ] . \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} \\alpha \\ < \\ v \\left ( s - \\frac { \\epsilon _ 1 ' } { 1 + \\epsilon _ 1 } \\right ) \\ = \\ \\beta . \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} \\| \\nabla \\beta \\| _ { k , \\ell } ^ 2 = \\sum _ { p = k } ^ { \\ell - 1 } \\int _ { A _ { p , p + 1 } } \\lambda ^ { ( 2 - 2 n ) p } | \\nabla \\beta | _ { \\omega _ C } ^ 2 . \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} \\left ( W _ { i j } \\right ) = \\left ( \\frac { \\partial ^ 2 L } { \\partial \\dot { q } ^ i \\partial \\dot { q } ^ j } \\right ) \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} H _ { \\Lambda } = { \\sum _ { k \\in \\Lambda ^ { * } } } \\varepsilon _ { k } b _ { k } ^ { * } b _ { k } + \\frac { a } { 2 V } { \\sum _ { k \\in \\Lambda ^ { * } } } b _ { k } ^ { * } b _ { k } ^ { * } b _ { k } b _ { k } \\ , \\ a > 0 \\ . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} S _ { t , m } & = ( N _ { m } \\Box ^ { ( 0 ) } _ { b , m } + S _ { m } ) S _ { t , m } \\\\ & = N _ { m } ( \\Box ^ { ( 0 ) } _ { b , m } - \\Box ^ { ( 0 ) } _ { t , b , m } ) S _ { t , m } + S _ { m } S _ { t , m } . \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} R _ { j } \\cap B _ { \\frac { 1 } { j } } ( x ) \\cap ( x + C ( V _ { x _ { 0 } } ^ { \\bot } , \\tilde { \\theta } ) ) = x . \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} \\sum _ { ( u , v ) \\in [ r ] \\times [ s ] } c ( x ; ( u , v ) ) = 0 . \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} G = \\left \\{ \\begin{pmatrix} \\alpha & 0 \\\\ 0 & \\alpha \\end{pmatrix} , \\ \\begin{pmatrix} \\alpha & 0 \\\\ 0 & - \\alpha \\end{pmatrix} , \\ \\begin{pmatrix} 0 & \\alpha \\\\ \\alpha & 0 \\end{pmatrix} , \\ \\begin{pmatrix} 0 & \\alpha \\\\ - \\alpha & 0 \\end{pmatrix} \\ \\middle | \\ \\alpha \\in C ^ \\times \\right \\} . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} P _ { M } ( N , K , \\mathbf { p } ) = \\sum _ { \\mathbf { M } } \\Pr [ \\mathbf { M } ] \\Pr [ \\mathrm { r a n k } \\ , \\mathbf { C } _ { 1 } , \\mathbf { C } _ { 2 } = K | \\mathbf { M } ] . \\end{align*}"}
-{"id": "8255.png", "formula": "\\begin{align*} \\overline { \\mathcal { M } _ { 1 } } : = \\mathcal { M } _ { 1 } ( \\phi _ { 1 } ) - \\mathcal { M } _ { 1 } ( \\phi _ { 2 } ) , \\overline { u } : = u _ { 1 } - u _ { 2 } , \\overline { h } : = h ( \\varphi _ { 1 } ) - h ( \\varphi _ { 2 } ) . \\end{align*}"}
-{"id": "5313.png", "formula": "\\begin{align*} \\begin{aligned} & \\alpha _ { 1 , 1 } = 2 \\ , c _ 2 \\overline { v } _ { x x } - 6 \\ , c _ 3 \\overline { v } , \\\\ & \\alpha _ { 1 , 2 } = L _ { \\overline { \\omega } } [ \\beta _ 2 ] + \\frac { 8 } { 3 } ( \\beta _ 2 ) _ { x x x } - \\frac { 4 1 } { 3 } \\ , \\partial _ x [ ( \\beta _ 1 ) _ x \\ , ( \\beta _ 1 ) _ { x x } ] + a _ { 0 , 2 } + a _ { 0 , 1 } \\ , ( \\beta _ 1 ) _ x , \\end{aligned} \\end{align*}"}
-{"id": "5109.png", "formula": "\\begin{align*} \\nu ( A _ { x _ o } ^ { - 1 } C ) \\geq \\eta \\otimes \\eta ( G ( A ' \\times C ' ) ) = m _ { K } ( I ^ { - 1 } J ) , \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} \\Pr _ { r } ' [ T _ { \\mathrm { N i c e } } < T _ { \\mathrm { B a d } _ i } ] \\le \\Pr _ { o _ i } ' [ T _ { \\mathrm { G M P } _ i } < T _ { \\mathrm { B L } _ i } ] + o ( 1 / n ) = o ( 1 / n ) , \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} d _ { S ( T , t ) } ( w , w ' ) = & d _ { S ( T , t - j ) } \\left ( \\left ( x ' \\right ) ^ { t - j } , x _ { j + 1 } \\cdots x _ t \\right ) + d _ { S ( T , t - j ) } \\left ( \\left ( y ' \\right ) ^ { t - j } , y _ { j + 1 } \\cdots y _ t \\right ) \\\\ & + ( 2 ^ { t - j + 1 } - 1 ) d _ T ( x , y ) - 2 ( 2 ^ { t - j } - 1 ) , \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} \\sum _ { n \\le x } ( - 1 ) ^ { n - 1 } \\frac 1 { \\psi ( n ) } = \\frac { C } { 5 } \\left ( \\log x + \\gamma + D + \\frac { 2 4 } { 5 } \\log 2 \\right ) + O \\left ( x ^ { - 1 } ( \\log x ) ^ { 2 / 3 } ( \\log \\log x ) ^ { 4 / 3 } \\right ) , \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} C ( h ) = \\prod \\limits _ { \\substack { p | h \\\\ p > 2 } } \\ , \\Bigl ( 1 - \\frac { 1 } { p - 1 } \\Bigr ) ^ { - 1 } \\ . \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} \\Xi ^ { \\prime } _ { \\Lambda } : = \\prod _ { { k } \\ne { 0 } } ( 1 - \\exp ( - \\beta ( \\epsilon _ { { k } } - \\mu ) ) ) ^ { - 1 } \\ , \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} [ ( c , e ) , ( c ' , e ' ) ] : = ( \\{ c \\} \\{ c ' \\} _ 1 \\mp \\{ c ' \\} \\{ c \\} _ 1 , \\{ e \\} \\{ c ' \\} _ 1 \\mp \\{ e ' \\} \\{ c \\} _ 1 + e e ' \\mp e ' e ) . \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} r ( t ) - \\widetilde { r } ( t ) = P ( t ) - \\widetilde { P } ( t ) + \\int \\limits _ { 0 } ^ { t } \\left [ Q ( t , \\tau ) - \\widetilde { Q } ( t , \\tau ) \\right ] r ( \\tau ) d \\tau + \\int \\limits _ { 0 } ^ { t } \\widetilde { Q } ( t , \\tau ) \\left [ r ( \\tau ) - \\widetilde { r } ( \\tau ) \\right ] d \\tau . \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} B _ E ^ { - 1 } ( \\theta + \\alpha ) \\tilde { A } _ { | s | , E } ( \\theta ) B _ E ( \\theta ) = R _ { \\rho _ { | s | } ( E ) } . \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { \\sigma ( n ) } { n ^ s } = \\left ( 1 - \\frac { 6 } { 2 ^ s } + \\frac { 4 } { 2 ^ { 2 s } } \\right ) \\zeta ( s ) \\zeta ( s - 1 ) ( \\Re s > 2 ) . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L u _ i ( x ) = f _ i ( x , u ( x ) ) , & x \\in \\Omega , i = 1 , 2 , \\ldots , n , \\\\ B u _ i ( x ) = 0 , & x \\in \\partial \\Omega , i = 1 , 2 , \\ldots , n , \\end{array} \\right . \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} b _ { n } ^ { p } + a _ { n } ^ { p } = 0 , \\forall n \\in \\{ 1 , . . . , \\mathsf { N } \\} . \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} \\xi = B ^ { 1 - \\eta } , \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} \\begin{aligned} ( B _ 5 ) _ j ^ j ( 0 ) & = \\frac { 4 } { 3 } c _ 2 ^ 2 \\mathrm { i } \\sum _ { j _ 2 \\in S , j _ 2 + j \\in S } j _ 2 ^ 3 \\ , \\lvert j _ 2 \\rvert \\xi _ { j _ 2 } + 8 c _ 2 c _ 3 \\mathrm { i } \\sum _ { j _ 2 \\in S , j _ 2 + j \\in S } j _ 2 \\ , \\lvert j _ 2 \\rvert \\xi _ { j _ 2 } \\\\ & + 1 2 c _ 3 ^ 2 \\mathrm { i } \\sum _ { j _ 2 \\in S , j _ 2 + j \\in S } j _ 2 ^ { - 1 } \\ , \\lvert j _ 2 \\rvert \\xi _ { j _ 2 } . \\end{aligned} \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} u _ 2 ^ { ( 4 ) } = - \\frac { 1 } { \\lambda a ^ { 1 2 } } \\left [ u _ 1 ^ { ( 8 ) } + \\lambda a _ { 1 1 } u _ 1 ^ { ( 4 ) } \\right ] . \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} A _ v ( n _ { i _ 0 } ) \\ = \\ \\sum _ { j _ 0 \\in J _ 0 } \\lambda _ { i _ 0 , j _ 0 } \\ , n _ { j _ 0 } . \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} G = \\left [ \\begin{array} { c c c c c } s & \\lambda a _ { 1 } t & \\lambda a _ { 2 } t & \\cdots & \\lambda a _ { n } t \\\\ t & \\lambda b _ { 1 } s & \\lambda b _ { 2 } s & \\cdots & \\lambda b _ { n } s \\end{array} \\right ] \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} R _ { ( 1 , 1 1 ) } ( n ) = & 1 2 \\sigma ( n ) - 3 6 \\sigma ( \\frac { n } { 3 } ) + 1 2 \\sigma ( \\frac { n } { 1 1 } ) - 3 6 \\sigma ( \\frac { n } { 3 3 } ) + 1 4 4 \\ , W _ { ( 1 , 1 1 ) } ( n ) \\\\ & + 1 2 9 6 \\ , W _ { ( 1 , 1 1 ) } ( \\frac { n } { 3 } ) - 4 3 2 \\ , \\biggl ( W _ { ( 3 , 1 1 ) } ( n ) + W _ { ( 1 , 3 3 ) } ( n ) \\biggr ) . \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} \\int _ { u } ^ { + \\infty } x ^ { a - 1 } e ^ { - b x } d x = b ^ { - a } \\Gamma ( a , b u ) . \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} g ( s , x , y ) = u ( s , x , y + h + h \\pi _ { s - } ) - u ( s , x , y + h \\pi _ { s - } ) \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} v _ { \\ell } = \\frac { k _ { \\ell } A _ { \\ell } B - 2 } { 2 B ( 1 + k _ { \\ell } A _ { \\ell } / a _ { \\ell } ) } \\ , . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} J ^ { L Q } _ { x , t } ( \\hat v ) = \\inf _ { v } J ^ { L Q } _ { x , t } ( v ) . \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{gather*} F \\big ( 3 ^ { - 1 / 3 } x , 3 ^ { - 2 / 3 } t ; \\beta = 6 \\big ) \\\\ \\qquad { } = - i \\kappa e ^ { \\frac { x ^ 3 } { 6 } - \\frac { x t } { 2 } } \\left [ u ^ { - \\frac { 1 } { 2 } } \\left ( \\frac { 1 + q _ 2 } { 2 } x - \\alpha \\right ) \\Psi _ { 0 1 1 } ( x , t ) + u ^ { \\frac { 1 } { 2 } } \\Psi _ { 0 2 1 } ( x , t ) \\right ] . \\end{gather*}"}
-{"id": "4297.png", "formula": "\\begin{align*} \\boxtimes _ { j \\in J } K _ j = \\boxtimes _ { i \\in I } \\boxtimes _ { j \\in J _ i } K _ j \\xrightarrow { \\ \\boxtimes _ { i \\in I } \\psi _ i \\ } \\boxtimes _ { i \\in I } \\Delta ^ { ( J _ i ) } _ * L _ i = \\Delta ^ { ( \\pi ) } _ * \\left ( \\boxtimes _ { i \\in I } L _ i \\right ) \\xrightarrow { \\ \\Delta ^ { ( \\pi ) } ( \\varphi ) \\ } \\Delta ^ { ( \\pi ) } _ * \\Delta ^ { ( I ) } _ * M = \\Delta ^ { ( J ) } _ * M , \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} M _ x [ ( B ^ { - 1 } - \\mathrm { I } ) b _ 1 ] - M _ { \\varphi , x } [ ( B ^ { - 1 } - \\mathrm { I } ) b _ 1 ] = \\mathcal { D } _ { \\omega } \\alpha \\ , M _ x [ b _ 1 ] - M _ { \\varphi , x } [ ( \\mathcal { D } _ { \\omega } \\alpha ) \\ , b _ 1 ] + M _ x [ \\mathtt { R } _ { \\tilde { \\alpha } } ] - M _ { \\varphi , x } [ \\mathtt { R } _ { \\tilde { \\alpha } } ] \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} E ' \\ ; : \\ ; y ^ 2 = x ^ 3 + a _ 2 ' x ^ 2 + a _ 4 ' x , a _ 2 ' = \\frac { a } { \\pi ^ { 2 \\alpha } } , a _ 4 ' = \\frac { b } { \\pi ^ { 4 \\alpha } } , \\alpha = \\frac { \\upsilon ( \\Delta _ m ) } { 3 } ; \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} W \\ ; : \\ ; y ^ 2 + \\frac { a } { u _ 1 } x y + \\ell ^ \\alpha y = x ^ 3 , \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} q = k _ 1 \\bar { q } _ 1 + k _ 2 \\bar { q } _ 2 + \\cdots + k _ s \\bar { q } _ k + p \\hat { \\beta } , \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} \\alpha = \\sum ^ \\infty _ { n = 0 } A _ n ( \\xi , \\ldots , \\xi ) , \\ A _ n \\in H ^ { \\otimes n } _ s . \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} u _ t + u _ { x x x } \\pm 2 ( | u | ^ 2 u ) _ x = 0 , u ( 0 ) = u _ 0 , \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} Z ( \\mathcal { T } ) = Z ( \\varphi ( \\mathcal { T } ) ) . \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\tilde { K } _ { 1 } ( t ) g \\right \\| _ { 2 } + \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\tilde { K } _ { 2 } ( t ) g \\right \\| _ { 2 } & \\le C ( 1 + t ) ^ { - \\frac { n } { 2 } - ( \\ell + k ) } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla _ { x } ^ { \\ell + k } g \\| _ { 2 } , \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { \\ell } { p } \\right ) ^ r \\left ( \\frac { 2 } { p } \\right ) ^ t = 1 \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} \\sum _ { i \\to j \\in G } \\mathcal { F } _ n ^ \\ast ( D _ i ) \\mathcal { F } _ n ^ \\ast ( D _ j ) = \\sum _ { k = 1 } ^ { L _ n } a _ k a _ { \\sigma _ G ( k ) } . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} \\{ f _ 1 , \\cdots , f _ p \\} = \\Pi ( d f _ 1 \\wedge \\cdots \\wedge d f _ { p - 1 } ) \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} \\sum _ { n = 2 0 } ^ \\infty \\frac { ( \\log n ) ^ { x - 1 } } { ( n \\log \\log n ) ^ 2 } \\approx \\int _ { t = 2 0 } ^ \\infty \\frac { ( \\log t ) ^ { x - 1 } } { ( t \\log \\log t ) ^ 2 } d t . \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} F ( m , k ) = \\prod _ { i = 0 } ^ { k - 1 } ( 2 ^ { m } - 2 ^ { i } ) = 2 ^ { m k } \\prod _ { i = 0 } ^ { k - 1 } ( 1 - 2 ^ { i - m } ) . \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} I ( U ' , V ' ; Y ) - I ( U ' , V ' ; S ) = I ( U , V , \\tilde { V } ; Y ) - I ( U , V , \\tilde { V } ; S ) \\stackrel { ( a ) } = I ( U , V ; Y ) - I ( U , V ; S ) \\geq R _ \\mathsf { A } ^ \\mathsf { A l t } , \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} \\chi = 2 p \\leq 1 - \\frac 1 k . \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\mathfrak S ^ n ( a ) : = \\{ ( x _ 1 , \\ldots , x _ n ) \\in \\R ^ n ; \\ 0 \\leq x _ j < a , \\ \\sum _ { j = 1 } ^ n x _ j < a \\} . \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} { \\bf A } _ i = { \\bf A } ( \\tau _ i ) , { \\bf B } _ i = { \\bf B } ( \\tau _ i ) , { \\bf Q } _ i = { \\bf Q } ( \\tau _ i ) , { \\bf S } _ i = { \\bf S } ( \\tau _ i ) , { \\bf R } _ i = { \\bf R } ( \\tau _ i ) . \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\phi _ { \\ell } = \\frac { n ( \\ell ) } { k _ { \\ell } } = \\frac { 2 \\ell H _ 2 ( \\alpha _ { \\ell } ) } { k _ { \\ell } \\log ( 1 + k _ { \\ell } P ' ) } , \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} \\mbox { R H S } & = t \\sum _ { i = 0 } ^ { r - 1 } \\sum _ { j = 0 } ^ { s - 1 } \\binom { i + j } { i } \\binom { i + j + t } { i + j } \\frac { 1 } { i + j + 1 } . \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{gather*} \\delta \\lambda = \\delta \\phi ^ a \\wedge \\frac { \\delta \\lambda } { \\delta \\phi ^ a } + d \\gamma . \\end{gather*}"}
-{"id": "8654.png", "formula": "\\begin{align*} \\widetilde { q } ( \\widetilde { C } ) = n ^ { \\widetilde { K } } ( \\widetilde { C } ) + \\ell _ { \\widetilde { D } _ 0 } ( \\widetilde { C } ) + 1 \\ , , \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} \\begin{cases} Q _ k ( 0 ) = & \\sum _ { n = 1 } ^ { N } 1 ( A ( - n ) + W ^ * _ { k } ( - n ) + J _ { k } ( n ) > 0 ) \\\\ D _ k ( 0 ) = & \\sum _ { n = 1 } ^ { N } 1 ( A ( - n ) + W ^ * _ { k } ( - n ) + J _ { k } ( n ) < 0 \\\\ & A ( - n ) + S ^ * ( - n ) > 0 ) . \\end{cases} \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{gather*} R ( x , t ) = \\begin{pmatrix} p ( t ) x - \\alpha ( t ) & - 1 \\\\ q ( t ) & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "2015.png", "formula": "\\begin{align*} e _ { E ' , p } ( \\phi ( P ) , \\phi ( Q ) ) = e _ { E , p } ( P , Q ) ^ { d ( \\phi ) } \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} \\Vert { x } \\Vert _ M = \\inf \\left \\{ \\rho > 0 \\ , : \\ , \\sum _ { i = 1 } ^ n M \\left ( \\frac { | x _ i | } { \\rho } \\right ) \\leq 1 \\right \\} . \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} ( \\alpha ) ~ s \\preccurlyeq t ; ( \\beta ) ~ s = s s ^ { - 1 } t ; ( \\gamma ) ~ s = t s ^ { - 1 } s , \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} ( U ^ * ) ^ { \\gamma } = ( U ^ { \\gamma } ) ^ { * } . \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} f \\left ( t \\right ) = t ^ { 4 } + a _ { 1 } t ^ { 3 } + a _ { 2 } t ^ { 2 } + a _ { 3 } t + a _ { 4 } \\in \\mathbb { Z } _ { M } \\left [ t \\right ] . \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} a ( 2 - q ) t _ { \\max } ^ { 2 } \\| w \\| ^ { 2 } + \\epsilon ( 4 - q ) t _ { \\max } ^ { 4 } \\| w \\| ^ { 4 } - ( { 2 ^ * _ \\alpha } - q ) t ^ { 2 ^ * _ \\alpha } _ { \\max } \\int _ { \\Omega \\times \\{ 0 \\} } { | w ( z , 0 ) | ^ { 2 ^ * _ \\alpha } } d z , = 0 . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\zeta _ 0 ( x ) ( t ) ( s ) = x ( t ) ( s ) h _ 0 ^ { 1 / 2 } ( s - t ) h _ 0 ^ { 1 / 2 } ( s ) \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\pi & ^ b _ { 0 a ( n - a ) } = \\pi ^ { b + 1 } _ { 0 a ( n - a ) } + c _ b \\\\ & = \\frac { \\sum _ { i = b } ^ { n - b } z _ i } { n - 1 - 2 b } + \\left ( \\frac { \\sum _ { i = b } ^ { n - b } z _ i } { n + 1 - 2 b } - \\frac { \\sum _ { i = b + 1 } ^ { n - b - 1 } z _ i } { n - 1 - 2 b } \\right ) \\\\ & = \\frac { \\sum _ { i = b } ^ { n - b } z _ i } { n + 1 - 2 b } , \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} \\mathrm { d l o g } Y = \\mathrm { d l o g } \\tilde { y } = - i \\sum _ { k \\geq 1 } [ a ] ^ k T ^ { i k - 1 } \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} R _ h ( t , s ) = F ( t , s ) \\ G ( t , s , h ( t ) + h ( s ) ) , \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} \\iota _ { X _ s ^ G } \\omega _ s = - \\tilde { \\alpha } . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} \\rho ( v ^ - ( \\cdot , - n ) , v ^ + ( \\cdot , - n ) ) & = \\rho ( v ^ - ( \\cdot , - ( n + \\tau ) + \\tau ) , v ^ + ( \\cdot , - ( n + \\tau ) + \\tau ) ) \\\\ & < \\rho ( v ^ - ( \\cdot , - ( n + \\tau ) ) , v ^ + ( \\cdot , - ( n + \\tau ) ) ) - \\delta \\\\ & < \\rho ( v ^ - ( \\cdot , - ( n + k \\tau ) ) , v ^ + ( \\cdot , - ( n + k \\tau ) ) ) - k \\delta \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} \\xi ^ * = \\xi , x ^ * = { \\bf i } \\ , x , \\eta ^ * = \\eta \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\frac 1 { \\sigma ( n ) } = E \\left ( \\log x + \\gamma + F \\right ) + O \\left ( x ^ { - 1 } ( \\log x ) ^ { 2 / 3 } ( \\log \\log x ) ^ { 4 / 3 } \\right ) , \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} \\frac { d \\langle n ( x ) , \\nu ( x ) \\rangle } { d v } : = \\sum _ { \\alpha = 1 } ^ 3 \\frac { \\partial \\langle n ( x ) , \\nu ( x ) \\rangle } { \\partial x ^ \\alpha } v ^ \\alpha = 0 . \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} W '' : y ''^ 2 + \\sigma ( a _ 1 ' ) x '' y '' + \\sigma ( a _ 3 ' ) y '' = x ''^ 3 + \\sigma ( a _ 2 ' ) x ''^ 2 + \\sigma ( a _ 4 ' ) x '' + \\sigma ( a _ 6 ' ) \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\mathcal { R } _ { M , j } : = s _ { j - 2 , j - 1 } \\circ ( \\mu _ { j } \\circ \\mu _ { j + 1 } \\cdots \\circ \\mu _ { j - 4 } \\circ \\mu _ { j - 3 } ) \\circ ( \\mu _ { j - 1 } \\circ \\mu _ { j - 2 } \\circ \\cdots \\circ \\mu _ { j + 1 } \\circ \\mu _ j ) . \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} \\sigma _ z = c ( \\delta u ) ^ 2 - z . \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} Q ( x ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ A ( x ) & 0 \\end{array} \\right ) \\ ; , Q ^ \\dagger ( x ) = \\left ( \\begin{array} { c c } 0 & A ^ \\dagger ( x ) \\\\ 0 & 0 \\end{array} \\right ) \\ ; , \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} \\hat { \\mathbf { h } } ^ { ( g ) } = \\boldsymbol { \\Upsilon } _ { U } ^ { ( g ) } \\hat { \\mathbf { c } } ^ { ( g ) } = \\boldsymbol { \\Upsilon } _ { U } ^ { ( g ) } \\left ( \\mathbf { W } _ { m m s e , D } ^ { ( g ) } \\right ) ^ H \\mathbf { y } ^ { ( g ) } = \\underbrace { \\boldsymbol { \\Upsilon } _ { U } ^ { ( g ) } \\left ( \\mathbf { W } _ { m m s e , D } ^ { ( g ) } \\right ) ^ H { \\left ( \\boldsymbol { \\Upsilon } _ S ^ { ( g ) } \\right ) ^ H } } _ { \\textrm { R e d u c e d R a n k W i e n e r F i l t e r } } \\mathbf { y } \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} \\max \\limits _ { f \\in D _ f } \\Delta \\left ( h ( f _ 0 ) ; f \\right ) = \\Delta \\left ( h ( f _ 0 ) ; f _ 0 \\right ) , \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\delta \\lambda _ { i a } = \\sum _ { j b } \\frac { \\delta _ { i a , j b } - s _ a F ( \\lambda _ { i a } | \\lambda _ { j b } ) } { L s _ a \\rho ( \\lambda _ { i a } ) } N _ { j b } . \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\left . \\frac { P _ N ( \\tau ) / ( \\tau - \\tau _ i ) } { P _ N ' ( \\tau ) } \\right | _ { \\tau = \\tau _ i } = \\frac { \\prod _ { j \\ne i } ( \\tau _ i - \\tau _ j ) } { \\prod _ { j \\ne i } ( \\tau _ i - \\tau _ j ) } = 1 . \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} \\scriptstyle | R ( E _ 2 ) | = 2 , | R ( E _ 3 ) | = 8 , | R ( E _ 4 ) | = 2 0 , | R ( E _ 5 ) | = 4 0 , | R ( E _ 6 ) | = 7 2 , | R ( E _ 7 ) | = 1 2 6 , | R ( E _ 8 ) | = 2 4 0 . \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} \\{ Z \\in \\mathbb { R } ^ { n \\times n } \\ ; | \\ ; A _ i Z \\approx Z ^ T A _ i , \\ ; \\mbox { f o r } i = 1 , \\dots , m \\} . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} H _ { 0 , \\Lambda , \\mu , \\lambda } = H _ { 0 , \\Lambda , \\mu } + H _ { \\Lambda } ^ { \\lambda } \\ . \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} \\beta _ n ( t ) = \\begin{cases} - ( - 1 ) ^ i \\nu \\ , \\cfrac { D H ( Y _ n ( t ) ) } { | D H ( Y _ n ( t ) ) | } \\ \\ \\ & t \\in ( 0 , s _ n ) , \\\\ \\alpha _ n ( t - s _ n ) \\ \\ \\ & t \\in ( s _ n , t _ n ) , \\end{cases} \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { S T B C } } = \\frac { \\gamma _ 0 } { M R _ c } \\frac { \\sum _ { m = 1 } ^ { M } | h _ m | ^ 2 } { \\sum _ { k = 1 } ^ { K } | g _ k | ^ 2 \\zeta _ k } . \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} & ( i ) \\ , \\ , \\{ j _ 2 \\neq - j _ 1 , j _ 3 = - j _ 1 , j _ 4 = - j _ 2 \\} ( i i ) \\ , \\ , \\{ j _ { 2 } \\neq - j _ 1 , j _ 3 \\neq - j _ 1 , j _ 3 = - j _ 2 , j _ 4 = - j _ 1 \\} \\\\ [ 2 m m ] & ( i i i ) \\ , \\ , \\{ j _ 1 + j _ 2 = 0 \\} . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} \\left ( \\prod _ { i = 1 } ^ 3 [ H o m _ { X } ( \\mathcal { F } _ i , \\mathbb { G } _ m ) _ { t o r } ] ^ { ( - 1 ) ^ { i + 1 } } \\right ) = \\nu ( \\mathcal { H } o m ) _ { \\mathbb { R } } [ Q ^ D ] . \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} \\langle I ' ( u ) , v \\rangle & = ( a + b \\| u \\| _ { E ^ { \\alpha , p } } ^ p ) ^ { p - 1 } \\int _ 0 ^ T \\phi _ p ( { _ 0 D _ t ^ \\alpha } u ( t ) ) { _ 0 D _ t ^ \\alpha } v ( t ) d t \\\\ & \\ \\ \\ \\ - \\int _ 0 ^ T f ( t , u ( t ) ) v ( t ) d t , \\ \\ \\forall u , v \\in E _ 0 ^ { \\alpha , p } , \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} L ( s ) = \\prod _ p ( 1 - \\lambda ( p ) p ^ { - s } + \\kappa ( p ) p ^ { - 2 s } ) ^ { - 1 } \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} { \\ddot { \\mathbf { x } } } ( t ) = \\mathbf { F } ( t , \\mathbf { x } , \\mathbf { \\dot { x } } ) , \\mathbf { { x } } \\in \\R ^ N , \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} \\hat { A } \\xi = \\int _ { - \\pi } ^ { \\pi } { h } ( \\theta ) d Z ^ { \\xi } ( \\theta ) . \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} q ( T ) ^ { - 1 } ( Z - I ) ^ { - 1 } \\begin{bmatrix} G _ q & G _ p \\end{bmatrix} q ( T ) ^ { - * } ( Z - I ) ^ { - 1 } \\begin{bmatrix} B _ q & e _ 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} a _ n ( t , x ; s ) = & f ( s + t , u ( x , s + t ; s , u _ { 0 n } ) ) \\\\ & + u ( x , s + t ; s , u _ { 0 } ) \\cdot \\int _ 0 ^ 1 f _ u ( s + t , r u ( x , s + t ; s , u _ { 0 n } ) + ( 1 - r ) u ( x , s + t ; s , u _ 0 ) ) d r . \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} \\beta _ \\ell = \\sup _ { h > 0 } \\{ \\ ; h : h ^ 2 \\mid \\Delta _ \\ell \\ ; \\ ; \\mathcal { P } _ { \\Delta _ \\ell / h ^ 2 } ( j _ E ) \\equiv 0 \\pmod { \\ell } \\} \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} e _ { ( m + 1 ) p _ n + m } ( x ) = e _ { ( m + 1 ) ( p _ n - 1 ) + 1 } ( x ) + x ^ { m - 1 } e _ { ( m + 1 ) ( p _ n - 1 ) + m } ( x ) . \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} g _ { j i k } = \\sum _ { l = 1 } ^ K \\left | \\mathbf { x } _ { j i k } ^ { \\mathrm { H } } \\mathbf { w } _ { j l } \\right | ^ 2 . \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} \\widetilde { g } _ { 1 / \\delta } ( \\xi ) : = \\sum _ { \\gamma \\in \\Gamma ^ * } g _ { 1 / \\delta } ( \\xi - \\gamma ) \\ , , \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} & g _ i \\left ( x \\right ) : = a _ i x _ i ^ 2 - b _ i & i = 1 , \\ldots , m \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} A _ s = \\{ b \\in B | \\forall \\ x \\in X , \\ b _ x \\in A _ x + R ( B _ x ) \\} , \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} w _ 0 ^ i ( \\mathbf { x } ^ i ) : = \\sum _ { t = 0 } ^ \\infty ( \\delta ^ i ) ^ t u _ i ( x _ t ^ i ) , & & i = 1 , \\ldots , n . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} \\int _ \\R h ( x ) \\ , d Z ( x ) = \\lim _ { n \\to + \\infty } \\int _ \\R h _ n ( x ) \\ , d Z ( x ) \\ , , \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} \\frac { d q } { d t } & = \\gamma , \\\\ \\frac { d \\gamma } { d t } & = \\frac { k } { q ^ 3 } - \\omega ^ 2 ( t ) q , \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} \\int _ M | \\stackrel { \\circ } { \\rm R i c } _ g | ^ 2 d M = \\frac { 1 } { 2 } \\sum \\limits _ { i = 1 } ^ l \\kappa _ i \\int _ { \\Sigma _ i } \\left ( R _ { \\gamma _ i } - \\varepsilon ( n - 2 ) ( n - 1 ) - \\frac { n - 2 } { n - 1 } H _ i ^ 2 \\right ) d { \\Sigma _ i } , \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} | | P - Q | | _ { \\mathsf { T V } } = \\sup _ { \\mathcal { A } \\in \\mathcal { F } } \\big | P ( \\mathcal { A } ) - Q ( \\mathcal { A } ) \\big | . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} \\sum _ { \\iota '' \\in I '' } \\widetilde { a } _ { \\iota '' } ( g \\widetilde { a } ^ * _ { \\iota '' } ) = \\left \\{ \\begin{array} { c l } 1 _ { M ( \\widetilde { A } ) } & g \\in G \\\\ 0 & g \\in G \\end{array} \\right . \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} H ( 1 ) \\cap H ( k _ j ) = \\{ 0 \\} \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} \\mathrm { d } ( f g ) = & f \\mathrm { d } g + g \\mathrm { d } f \\\\ = & f i _ { a _ g } \\mathrm { d } \\eta + g i _ { a _ f } \\mathrm { d } \\eta \\\\ = & i _ { ( f a _ g + g a _ f ) } \\mathrm { d } \\eta . \\\\ \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} N / I = V _ 1 / I \\times \\cdots \\times V _ s / I , \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ { \\infty } \\frac { \\ell } { ( \\ell + 1 ) ^ { m + 1 } } \\left [ \\frac { 1 } { \\ell ^ p } - \\frac { 1 } { ( \\ell + 1 ) ^ p } \\right ] \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{gather*} L = e ^ a \\wedge d \\omega ^ { b c } \\wedge h _ { a b c } + \\tfrac 1 2 \\omega ^ a { } _ { c } \\wedge \\omega ^ { c b } \\wedge H _ { a b } + e ^ \\ast _ a \\wedge \\big ( d c ^ a + c ^ { a b } h _ b \\big ) + \\omega ^ \\ast _ { a b } \\wedge d c ^ { a b } . \\end{gather*}"}
-{"id": "9274.png", "formula": "\\begin{align*} U _ { T / R } = \\mathcal { Z } _ T ( \\mathfrak { t } _ 0 ) ^ c , \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} z \\mapsto h ( z ) = \\chi ( z ) G _ t ( z - y ) \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} \\begin{array} { l l l l } Z ( V ) = & \\max \\ & c ' V \\lambda + d ' u & \\\\ & \\mbox { s . t . } \\ & A V \\lambda \\leq b & ( \\alpha ) \\\\ & & H V \\lambda + G u \\leq h & ( \\pi ) . \\end{array} \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} & [ \\alpha , \\beta ] : = \\alpha \\circ \\beta - ( - 1 ) ^ { | \\alpha | \\cdot | \\beta | } \\beta \\circ \\alpha , \\\\ & ( \\alpha \\circ \\beta ) ( a _ 0 , \\ldots , a _ { | \\alpha | + | \\beta | } ) : = \\sum _ { r = 0 } ^ { | \\alpha | } ( - 1 ) ^ { r | \\beta | } \\alpha ( a _ 0 , \\ldots , a _ { r - 1 } , \\beta ( a _ { r } , \\ldots , a _ { r + | \\beta | } ) , a _ { r + | \\beta | + 1 } , \\ldots , a _ { | \\alpha | + | \\beta | } ) . \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} \\hat { x } : = \\begin{bmatrix} V _ { d 1 } & V _ { d 2 } & \\ldots & V _ { d n } & V _ { q 1 } & V _ { q 2 } & \\ldots & V _ { q n } \\end{bmatrix} ^ \\intercal . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} g ( \\mu _ { k + 1 } ) - g ( \\mu _ k ) = \\frac { \\pi } { 2 } ( 1 + o ( 1 ) ) , k \\to \\infty . \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} T ^ { * } G = ( P _ W ^ { * } T ^ { * } ) G + ( ( 1 - P _ W ^ { * } ) T ^ { * } ) G \\subseteq G + N + ( ( 1 - P _ W ^ { * } ) T ^ { * } ) G . \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{align*} \\mathbb { C } _ { a , p - 1 } Q ( \\xi _ { a } ^ { ( 0 ) } ) = \\prod _ { b = 1 } ^ { ( p - 1 ) \\mathsf { N } } \\frac { X _ { a } ^ { ( p - 1 ) } - w _ { b } } { w _ { ( p - 1 ) \\mathsf { N } + 1 } - w _ { b } } Q ( \\xi _ { ( p - 1 ) \\mathsf { N } + 1 } ) + \\sum _ { n = 1 } ^ { \\mathsf { N } } \\mathbb { D } _ { a , n } Q ( \\xi _ { n } ^ { ( 0 ) } ) , \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} p ( \\beta , \\mu , \\lambda ) ^ { ' } = \\lim _ { V \\to \\infty } p _ { \\beta , \\mu , \\Lambda , \\lambda } ^ { ' } \\ . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} \\alpha ^ 2 & = \\ell - b \\alpha + a \\beta \\\\ \\alpha \\beta & = m \\\\ \\beta ^ 2 & = n - d \\alpha + c \\beta , \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} \\dim \\langle P _ I , F _ J \\rangle = \\begin{cases} a _ { I } - 1 & \\mbox { i f } J = \\emptyset \\\\ \\geq a _ { I \\cup J } & \\mbox { i f } J \\neq \\emptyset \\end{cases} \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\begin{aligned} & E _ 0 = e _ 0 , F _ 0 = f _ 0 , K _ 0 = k _ 0 , \\ \\ E ' _ 0 = e ' _ 0 , F ' _ 0 = f ' _ 0 , K ' _ 0 = k ' _ 0 . \\end{aligned} \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} \\big [ r _ \\epsilon ( a , b ) \\big ] ( X ) \\ , = \\ , - \\frac { 4 i } { \\pi ^ { 4 } } \\int _ { \\Xi \\times \\Xi } \\ , d Y d Z \\ , e ^ { - 2 i \\sigma ( Y , Z ) } \\left ( \\int _ 0 ^ 1 e ^ { - 4 i t F _ { \\epsilon } ( x , y , z ) } d t \\right ) \\times \\\\ \\quad \\quad \\times F ^ \\circ _ { \\epsilon } ( x , y , z ) \\ , \\Big ( \\partial _ \\xi a ( X - Y ) \\wedge \\partial _ \\xi b ( X - Z ) \\Big ) \\ , . \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} \\sum _ { k = 2 } ^ { p - 1 } \\frac { 1 } { k ^ 2 } \\sum _ { m = k } ^ { p - 1 } \\frac { ( - 1 ) ^ { m - k } } { \\binom { m } { k } } \\equiv _ { p } - 2 q _ p ( 2 ) . \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} A \\mathcal { B } = \\mathcal { B } A \\ , , \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} | u - u ^ I | _ { \\C { H } ^ 1 ( \\Omega ) } \\le ( c / N ) ^ { p - 3 / 2 } \\| u \\| _ { \\C { H } ^ p ( \\Omega ) } , p = \\min \\{ \\eta , N + 1 \\} , \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} \\Pr \\left ( \\sum _ { t = 1 } ^ T X _ t > \\max \\{ 2 \\sigma , 3 b \\sqrt { \\log ( 1 / \\delta ) } \\} \\sqrt { \\log ( 1 / \\delta ) } \\right ) \\leq 4 \\delta \\log T . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} U ^ r = \\left \\{ \\prod _ { ( i , p ) = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\infty } ( 1 - a _ { i j } T ^ i ) ^ { p ^ j } : i p ^ j < r \\implies a _ { i j } = 0 \\right \\} \\subseteq U ^ 1 . \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} \\Gamma ^ \\pm _ N \\ = \\ \\bigcup _ { i = 0 } ^ N \\Lambda _ i ^ \\pm \\cup \\bigcup _ { i = 1 } ^ N S _ i ^ \\pm \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} \\int _ { - \\pi } ^ { \\pi } { e ^ { - i \\theta k } \\ , \\ , { h } ( \\theta ) } d \\theta = 0 , k = 0 , 1 , \\dots . \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\pi \\in \\P _ e , \\\\ | \\pi | = N } } ( - 1 ) ^ { \\nu _ e ( \\pi ) + 1 } = \\chi ( N \\not = \\triangle \\ , ) . \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} u ( r , y ) = \\sum _ { i = 0 } ^ \\infty u _ i ( r ) \\phi _ i ( y ) , \\ ; \\ , { f } ( r , y ) = \\sum _ { i = 0 } ^ \\infty { f } _ i ( r ) \\phi _ i ( y ) , \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} - D _ f \\sum _ { d = 1 } ^ { \\infty } h _ { 1 / f } ( d ) \\log d + O \\left ( \\sum _ { d > x } | h _ { 1 / f } ( d ) | \\log d \\right ) + O \\left ( x ^ { - 1 } R _ { 1 / f } ( x ) \\sum _ { d \\le x } d | h _ { 1 / f } ( d ) | \\right ) . \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} H _ { \\Lambda } = - \\sum _ { { x } , { y } \\in \\Lambda ; \\| { x } - { y } \\| = 1 } { \\sigma } _ { { x } } \\cdot { \\sigma } _ { { y } } \\ , \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} E _ n = \\omega \\left ( 2 n + \\frac { | p | } { k } + 1 \\right ) . \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} & \\langle I ' ( u _ k ) - I ' ( u ) , u _ k - u \\rangle \\\\ & = \\langle I ' ( u _ k ) , u _ k - u \\rangle - \\langle I ' ( u ) , u _ k - u \\rangle \\\\ & \\leq \\| I ' ( u _ k ) \\| _ { ( E _ 0 ^ { \\alpha , p } ) ^ * } \\| u _ k - u \\| _ { E ^ { \\alpha , p } } - \\langle I ' ( u ) , u _ k - u \\rangle \\\\ & \\rightarrow 0 \\ \\ \\mbox { a s } \\ k \\rightarrow \\infty . \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} m _ \\lambda = \\begin{cases} ( n + 1 ) ^ 2 & \\textit { i f } \\ ; \\ , k = 0 , \\ ; n _ 1 = n _ 2 = n , \\\\ 2 ( n + 1 ) ^ 2 & \\textit { i f } \\ ; \\ , k \\neq 0 , \\ ; n _ 1 = n _ 2 = n , \\\\ 2 ( n _ 1 + 1 ) ( n _ 2 + 1 ) & \\textit { i f } \\ ; \\ , k = 0 , \\ ; n _ 1 \\neq n _ 2 , \\\\ 4 ( n _ 1 + 1 ) ( n _ 2 + 1 ) & \\textit { i f } \\ ; \\ , k \\neq 0 , \\ ; n _ 1 \\neq n _ 2 . \\end{cases} \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} - b \\cdot D \\varphi = - b \\cdot D \\tilde \\psi \\zeta _ { 2 r } - \\tilde \\psi b \\cdot D \\zeta _ { 2 r } \\ \\ \\ \\Omega ( r ^ 2 ) \\setminus \\partial B _ r . \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} a _ { \\xi } ( y , s ) = c _ { \\xi } ( s ) \\cdot B _ { \\xi } ( y ; p ; s ) , \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} I _ { \\mathcal { O } / M } ( \\alpha ) = \\left [ \\mathcal { O } ^ { + } : \\mathbb { Z } _ { M } [ \\alpha ] ^ { + } \\right ] = \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} \\Phi _ q ^ * ( 0 ) = - \\sup _ { \\gamma \\in \\Pi } \\Phi _ q \\circ \\gamma ^ { - 1 } ( \\tilde x ) \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{gather*} F _ { \\beta } ( t ) = \\lim _ { x \\to \\infty } F ( x , t ; \\beta ) , \\end{gather*}"}
-{"id": "3031.png", "formula": "\\begin{gather*} q = \\int _ S \\tilde { F } . \\end{gather*}"}
-{"id": "5634.png", "formula": "\\begin{align*} p _ { k + 1 } = \\frac { i } 2 p ' _ k + r _ k u \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} P ^ * ( \\omega + i \\partial \\bar \\partial u ) = \\Phi ^ * \\omega _ { C _ x } + ( P ^ * \\omega - \\omega _ { C _ x } ) + i \\partial \\bar \\partial ( \\bar { u } \\circ P ) \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} e _ { x y } ( u , v ) = \\begin{cases} 1 , & \\mbox { i f } ( u , v ) = ( x , y ) , \\\\ 0 , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} c ( x ) = \\sum _ { j = 0 } ^ { p ^ k - 1 } x ^ { j p ^ { e - k } } b ( x ) . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} X _ n : = \\left | \\rho ( G _ n ) - \\mathbb { E } ^ \\prime \\left [ \\rho ( G _ n ) \\right ] \\right | , \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} [ \\nabla ^ * \\nabla , \\nabla ] = - 2 l ( e ^ b ) ( F _ { a b } + { \\cal R } _ { a b } ) \\nabla _ a . \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} [ [ \\alpha , \\beta ] ] = \\mathcal { L } _ { \\Pi ( \\alpha ) } \\beta - \\imath _ { \\Pi ( \\beta ) } d \\alpha \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} \\mathcal { A } _ 3 : = \\{ ( j _ 1 , j _ 2 , j _ 3 ) \\in \\mathbb { Z } ^ 3 \\setminus \\{ \\textbf { 0 } \\} : j _ 1 + j _ 2 + j _ 3 = 0 \\ , \\ , \\mbox { a n d a t l e a s t } \\ , \\ , 2 \\ , \\ , \\mbox { i n d i c e s a m o n g } \\ , \\ , j _ 1 , j _ 2 , j _ 3 \\ , \\ , \\mbox { b e l o n g t o } \\ , \\ , S \\} . \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{gather*} ( 1 - x ^ m ) A _ { ( m + 1 ) p _ n + \\frac { m + 1 } { 2 } } = A _ { ( m + 1 ) p _ n - \\frac { m - 1 } { 2 } } ^ { m \\rightarrow m + 1 } ( x ) + x ^ { m - 1 } A _ { ( m + 1 ) p _ n - \\frac { m + 1 } { 2 } } , \\end{gather*}"}
-{"id": "6827.png", "formula": "\\begin{align*} M _ { k } ( \\Gamma _ 0 ( N ) , \\chi ) = E _ { k } ( \\Gamma _ 0 ( N ) , \\chi ) \\oplus S _ { k } ( \\Gamma _ 0 ( N ) , \\chi ) . \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} c _ 1 = \\alpha m , c _ 2 = - m \\left ( V ( q ) - V ' ( q ) + m \\alpha S \\right ) . \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} [ [ d ^ E , \\delta _ { V _ j } ] , [ d ^ E , \\delta _ { \\overline { V _ k } } ] ] = [ d ^ E , \\delta _ { [ V _ j , \\overline { V _ k } ] } ] + \\Theta ^ E ( V _ j , \\overline { V _ k } ) , \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} Q ( f ) = \\int _ { \\R ^ d } [ \\sigma ( v , v ' ) M ( v ) f ( v ' ) - \\sigma ( v ' , v ) M ( v ' ) f ( v ) ] \\d v ' \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} \\phi ( X , Y ) = ( \\xi ( X + 1 ) ^ m - ( Y + 1 ) ) G ( X , Y ) \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} \\tilde { f } ( a ) \\times \\tilde { g } ( a ) = \\mathit { i } ( f ( a ) ) \\times \\mathit { i } ( g ( a ) ) = \\mathit { i } ( f ( a ) \\times g ( a ) ) = \\mathit { i } ( ( f \\times g ) ( a ) ) = \\mathit { i } \\circ ( f \\times g ) ( a ) . \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} B C ( J ) : = \\left \\{ v \\in D ( J _ { \\rm m a x } ) : W _ { \\infty } ( v , f ) = 0 \\mbox { \\rm f o r s o m e \\ ; } f \\in D ( J _ { \\rm m a x } ) \\right \\} , \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} P _ { V } ( K _ { n } = k ) = V _ { n , k } d ^ { n , k } , \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} b = - \\frac { 2 \\tilde { c } _ 6 } { 2 7 } \\equiv \\frac { 2 } { 2 7 } \\equiv 6 \\pmod { 8 } . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} \\tilde { f } ( 1 ) = 1 , \\tilde { f } ( 0 ) = 0 , \\tilde { f } ( a \\times b ) = \\tilde { f } ( a ) \\times \\tilde { f } ( b ) \\textrm { f o r } a \\in \\tilde { R } _ 1 ^ \\times , b \\in \\tilde { R } _ 1 \\end{align*}"}
-{"id": "5391.png", "formula": "\\begin{align*} \\begin{aligned} c ( \\xi ) & = \\sum _ { k \\in S ^ + } ( - 4 c _ 6 \\ , k ^ 3 - 2 4 c _ 7 k + \\frac { 1 6 } { 3 } c _ 2 ^ 2 \\ , k ^ 3 + 1 6 c _ 2 c _ 3 k ) \\ , \\xi _ k \\\\ [ 2 m m ] & = ( \\frac { 1 6 } { 3 } c _ 2 ^ 2 - 4 c _ 6 ) v _ 3 \\cdot \\xi + ( 1 6 c _ 2 c _ 3 - 2 4 c _ 7 ) v _ 1 \\cdot \\xi , \\end{aligned} \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} \\tilde { Q } \\left | \\{ \\lambda _ j \\} \\right \\rangle = Q _ n \\left | { \\{ \\lambda _ j \\} } \\right \\rangle , Q _ n = \\sum _ j \\lambda _ j ^ 2 \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} \\alpha ( P ^ { ( 1 ) } , P ^ { ( 2 ) } ) & = \\sum _ { i = a } ^ b \\sum _ { j = b + 1 } ^ c \\alpha ( q _ i , q _ j ) \\\\ & = \\chi ( 0 ) - \\chi ( a - b - 1 ) - \\chi ( b - c ) + \\chi ( a - c - 1 ) . \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} U _ { 1 + 2 ^ { p _ 1 - p _ 0 } } = 2 ^ { p _ 0 - p _ 1 } U _ 2 \\\\ + 4 ( 1 - 2 ^ { p _ 0 - p _ 1 } ) U _ 1 + 1 - 2 ^ { p _ 0 - p _ 1 } = 4 ( 1 - 2 ^ { p _ 0 - p _ 1 } ) + 1 . \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} \\| u ( t , x ) - u ( s , y ) \\| _ { \\mathbb L ^ 2 } = \\sqrt d \\| u _ 1 ( t , x ) - u _ 1 ( s , y ) \\| _ { \\mathbb L ^ 2 } , ( t , x ) , ( s , y ) \\in [ 0 , T ] \\times \\R . \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} \\omega ( t , x ) = \\begin{cases} \\frac 1 { \\sqrt { 4 \\pi t } } \\int _ { \\R } \\exp ( - \\frac { | x - \\eta | ^ 2 } { 4 t } ) u _ 0 ( \\eta ) d \\eta , & ; \\\\ \\frac 1 2 \\left ( u _ 0 ( x + t ) - u _ 0 ( x - t ) \\right ) + \\frac 1 2 \\int _ { x - t } ^ { x + t } v _ 0 ( \\eta ) d \\eta , & . \\end{cases} \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} P _ { Y | S } ( 0 | s ' ) = \\gamma ' ( 1 - \\alpha ) + ( 1 - \\gamma ' ) \\alpha . \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} \\Delta ( \\xi , \\eta , \\zeta ) = \\omega ( \\xi ) + \\omega ( \\eta ) + \\omega ( \\zeta ) . \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot x = A ( t ) x + c ( t ) , \\\\ \\dot y = \\lambda f _ 2 \\big ( t , \\hat x ( t ) , y , \\lambda \\big ) , \\end{array} \\right . \\ , \\ , \\lambda \\geq 0 \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} \\{ \\alpha _ { n , a } , \\alpha _ { m , b } \\} = n \\delta _ { a , b } \\delta _ { n + m , 0 } , \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} E _ 1 ^ { ( - ) } = - 6 b , \\psi _ 1 ^ { ( - ) } ( x ) = e ^ { - \\frac { 1 } { 4 } x ^ 4 - a | x | ^ 3 + b x ^ 2 } x , \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} f ( x ) & : = \\ \\left \\lbrace \\begin{aligned} & a - C \\int _ D g ( y ) | x - y | ^ { - N - 2 s } \\ d y & & \\ x \\in A , \\\\ & a C \\int _ A \\psi ( y ) | x - y | ^ { - N - 2 s } \\ d y - ( - \\Delta ) ^ s g ( x ) & & \\ x \\in D , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} \\lambda _ { \\underbrace { 0 0 \\ldots 0 } _ { \\frac { N - 1 } { 2 } } \\underbrace { 1 1 \\ldots 1 } _ { \\frac { N - 1 } { 2 } } \\underbrace { 1 } _ { 1 } } = \\lambda _ { \\underbrace { 1 1 \\ldots 1 } _ { \\frac { N - 1 } { 2 } } \\underbrace { 0 0 \\ldots 0 } _ { \\frac { N - 1 } { 2 } } \\underbrace { 1 } _ { 1 } } = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} 0 & \\geq \\| x - f ( w ) \\| ^ 2 - \\| x - w \\| ^ 2 \\\\ & = 2 \\langle x , w - f ( w ) \\rangle + \\| f ( w ) \\| ^ 2 - \\| w \\| ^ 2 \\\\ & > 2 \\langle w , w - f ( w ) \\rangle + \\| f ( w ) \\| ^ 2 - \\| w \\| ^ 2 \\\\ & = \\| w - f ( w ) \\| ^ 2 , \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} \\frac { 1 + q } { 1 + q ^ n } = \\frac { ( - q ; q ) _ { n - 1 } } { ( - q ^ 2 ; q ) _ { n - 1 } } , \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} N _ { ( 1 , 1 4 ) } ( n ) & = ~ 8 \\sigma ( n ) - 3 2 \\sigma ( \\frac { n } { 4 } ) + 8 \\sigma ( \\frac { n } { 1 4 } ) - 3 2 \\sigma ( \\frac { n } { 5 6 } ) + 6 4 \\ , W _ { ( 1 , 1 4 ) } ( n ) + 1 0 2 4 \\ , W _ { ( 1 , 1 4 ) } ( \\frac { n } { 4 } ) \\\\ & - 2 5 6 \\ , \\biggl ( W _ { ( 2 , 7 ) } ( \\frac { n } { 2 } ) + W _ { ( 1 , 5 6 ) } ( n ) \\biggr ) , \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} \\Box ^ { ( q ) } _ { t , b , m } : = \\Box ^ { ( q ) } _ { t , b } \\Big | _ { \\Omega ^ { 0 , q } _ { t , m } ( X ) } : \\Omega ^ { 0 , q } _ { t , m } ( X ) \\rightarrow \\Omega ^ { 0 , q } _ { t , m } ( X ) , \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} p _ G ( x ) = \\frac { 1 } { | E ( G ) | } \\sum _ { e \\in E ( G ) } p _ { G - e } ( x ) . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} L u _ i ( t , x ) = b _ i ( u ( t , x ) ) + \\sum _ { j = 1 } ^ d \\sigma _ { i , j } \\frac { \\partial ^ 2 W _ j } { \\partial t \\partial x } ( 0 , T ] \\times \\R , \\end{align*}"}
-{"id": "3143.png", "formula": "\\begin{align*} h ( k , \\ell ) = \\psi ( k , \\ell ) \\mathcal { E } ( k , \\ell ) , k , \\ , \\ell = 1 , \\ , 2 , \\ , \\dots . \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} B ^ h ( \\xi , \\eta ) = - \\frac { i } 2 \\left ( \\xi - \\frac { ( \\eta - J ( \\eta ) ) ^ 2 } { 4 J ( \\eta ) } \\right ) + S ( d ^ 2 \\rho ^ { - 1 } ) . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} u = T u : = ( K _ 1 F _ 1 u , K _ 2 F _ 2 u , \\ldots , K _ n F _ n u ) \\mbox { f o r $ u \\in D \\subset X $ . } \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} R _ \\mathsf { A } \\big ( p _ { \\tilde { U } , \\tilde { V } , \\tilde { X } | S } \\big ) = \\min \\Big \\{ I ( U , V ; Y ) - I ( U , V ; Z ) , I ( U , V ; Y ) - I ( U , V ; S ) \\Big \\} \\geq \\tilde { R } _ \\mathsf { A } \\left ( p _ { U , V , X | S } \\right ) . \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} \\left ( L - \\frac { \\partial } { \\partial t } \\right ) \\tilde { \\varphi } = n - \\sum _ { i } \\tilde { g } ^ { i \\overline { i } } - \\frac { \\partial \\varphi } { \\partial t } + \\int _ { M } \\frac { \\partial \\varphi } { \\partial t } ~ \\omega ^ { n } . \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} \\hat { \\gamma } _ { 2 2 } = - \\gamma + { 1 \\over 6 } , \\ \\ \\hat { \\gamma } _ { 3 2 } = \\gamma . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} \\Lambda = \\Lambda _ A = \\Lambda _ B = \\Lambda _ H . \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{gather*} Q = ( L , \\ , \\cdot \\ , ) . \\end{gather*}"}
-{"id": "6912.png", "formula": "\\begin{align*} S _ n ( \\mathbb { Z } ^ s , w ) = \\bigcup _ q \\{ \\Box _ q \\in \\mathcal { Q } _ q : w _ q ( \\Box _ q ) \\leq n \\} . \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} \\tilde { P } ( \\mu ) = K ( \\mu , \\eta _ { \\mu } ) . \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} d ( M , \\lambda ) : = m _ { a } ( M , \\lambda ) - m _ { g } ( M , \\lambda ) . \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} d _ { \\gamma } : = 2 ( p ! ) ^ { - 1 } ( h ^ { ( p ) } _ { r } ( x ) + \\gamma ) d ' _ { \\gamma } : = 2 ( p ! ) ^ { - 1 } ( h ^ { ( p ) } _ { l } ( x ) + ( - 1 ) ^ { p } \\gamma ) . \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} \\beta _ { a b c } = \\rho ^ { a } - \\rho ^ { n - a } + \\rho ^ { b } - \\rho ^ { n - b } + \\rho ^ { c } - \\rho ^ { n - c } , \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} H _ { 1 } = \\sum _ i D _ i + \\frac { \\beta } { 2 } \\sum _ { a , b } \\mathbb { E } ^ { a b } \\mathbb { E } ^ { b a } - \\frac { \\beta } { 2 } s p _ 0 , \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} d ^ n _ { 0 . 5 } ( p , r ) = - 2 \\log \\int q _ * ( x ^ n ) \\sqrt { p ( y ^ n | x ^ n ) r ( y ^ n | x ^ n ) } d x ^ n d y ^ n . \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} R ( 2 ; 0 , 0 ) = \\frac { \\log _ 4 S ( 2 ; 0 , 0 ) } { 1 6 } \\approx 0 . 2 5 5 3 . \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} W ^ + = W ^ - , g ^ { W ^ + } = g ^ { W ^ - } , \\nabla ^ { W ^ + } = \\nabla ^ { W ^ - } , \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} \\begin{aligned} & L ^ { ( 1 ) } U = D _ 1 U _ { x x } + c U _ x + A _ 1 U , \\\\ & L ^ { ( 2 ) } V = D _ 2 V _ { x x } + c V _ { x } + \\partial _ V R _ 2 ( 0 , 0 ) V . \\end{aligned} \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} k ^ * ( y ) = \\frac { \\xi ( w ) } { \\zeta ( 2 ) } \\sum _ { c \\mid w } \\chi ( c ) \\sum _ { ( d , D ) = 1 } \\phi ( c d y ) d ^ { - 1 } . \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} k _ q & \\geq \\sum _ { I \\in S _ \\phi } y _ I ^ { p - q } \\Biggl \\{ \\frac { ( \\mu ( I ) y _ I ) ^ q } { ( \\mu ( I ) \\tau _ I ) ^ { q - 1 } } - \\sum _ { \\substack { J \\in S _ \\phi \\\\ J ^ \\star = I } } \\frac { ( \\mu ( J ) y _ J ) ^ q } { ( ( \\beta + 1 ) \\mu ( J ) ) ^ { q - 1 } } \\Biggr \\} = \\\\ & = \\sum _ { I \\in S _ \\phi } \\mu ( I ) \\frac { y _ I ^ p } { \\tau _ I ^ { q - 1 } } - \\sum _ { I \\in S _ \\phi } y _ I ^ { p - q } \\sum _ { \\substack { J \\in S _ \\phi \\\\ J ^ \\star = I } } \\frac { y _ J ^ q } { ( \\beta + 1 ) ^ { q - 1 } } \\mu ( J ) . \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} \\bar u ( x ) = u _ n ( x ) = U _ { \\min } \\qquad n > N . \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} \\begin{aligned} 0 & = \\bold { E } ^ { \\vartriangle } ( L ( m , n , [ u ] ) ) \\\\ & = u _ { - 1 , 0 } - u _ { 0 , 1 } - \\frac { a ( m - 1 ) - b ( n ) } { u _ { 0 , 0 } - u _ { - 1 , 1 } } - \\left ( u _ { 0 , - 1 } - u _ { 1 , 0 } - \\frac { a ( m ) - b ( n - 1 ) } { u _ { 1 , - 1 } - u _ { 0 , 0 } } \\right ) . \\end{aligned} \\end{align*}"}
-{"id": "1504.png", "formula": "\\begin{align*} V = | x | ^ { - 1 } | x | V \\in M ^ { \\frac { n } { 2 } , \\sigma _ 1 } \\mbox { f o r s o m e } \\ \\frac { n - 1 } { 2 } < \\sigma _ 1 < \\frac { n } { 2 } \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{gather*} \\vec \\mu = R _ 0 \\vec y , R _ 0 = \\begin{pmatrix} ( - t ) ^ { - 1 / 4 } & 0 & 0 \\\\ 0 & ( - t ) ^ { - 1 / 4 } & 0 \\\\ 0 & 0 & ( - t ) ^ { 1 / 4 } \\\\ \\end{pmatrix} \\begin{pmatrix} 1 & 1 & 1 \\\\ 1 & - 1 & - 1 \\\\ 0 & \\sqrt 2 & - \\sqrt 2 \\end{pmatrix} , \\end{gather*}"}
-{"id": "3554.png", "formula": "\\begin{align*} u ( t ) = \\begin{cases} & \\tilde { K } _ { 1 } ( t ) u _ { 0 } + \\tilde { K } _ { 2 } ( t ) u _ { 0 } + \\tilde { K } _ { 3 } ( t ) u _ { 1 } \\ \\ 0 < \\nu < 2 , \\\\ & E _ { 1 } ( t ) u _ { 0 } + E _ { 2 } ( t ) u _ { 0 } + E _ { 3 } ( t ) u _ { 1 } \\ \\ \\nu = 2 , \\\\ & \\tilde { J } _ { 1 } ( t ) u _ { 0 } + \\tilde { J } _ { 2 } ( t ) u _ { 0 } + \\tilde { J } _ { 3 } ( t ) u _ { 1 } \\ \\ \\nu > 2 . \\end{cases} \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} ( A + U V ^ { \\ast } ) \\widehat { A } & = \\big ( A A ^ { \\dagger } + U E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ^ { - 1 } , \\\\ \\widehat { A } ( A + U V ^ { \\ast } ) & = \\big ( I + A ^ { \\dagger } U F _ { S _ { A } } U ^ { \\ast } ( A ^ { \\dagger } ) ^ { \\ast } \\big ) ^ { - 1 } \\big ( A ^ { \\dagger } A + A ^ { \\dagger } U F _ { S _ { A } } V ^ { \\ast } \\big ) . \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} \\int _ { x _ \\mu + \\delta _ 1 } ^ { x _ { \\frac 3 2 \\mu } } | w u | ^ q \\ , \\dd x \\leq C w ( x _ \\mu ) ^ q a _ \\mu ^ { - 1 } \\begin{cases} \\mu ^ { 1 - \\frac q 4 } , & 1 \\leq q < 4 , \\\\ [ 1 m m ] \\log ( \\mu a _ \\mu ^ { - \\frac 2 3 } ) , & q = 4 , \\\\ [ 1 m m ] a _ \\mu ^ { \\frac 2 3 - \\frac { q } { 6 } } , & q > 4 . \\end{cases} \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} c ( x ) = \\sum _ { j = 0 } ^ \\beta ( - 1 ) ^ { \\beta - j } { \\beta \\choose j } x ^ { j p ^ { e - 1 } } g ( x ) . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} S ^ * ( X , Y ) = \\sum _ { X < \\ell \\le Y } c ^ * ( \\ell ) ^ 2 \\ell ^ { - 1 } \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} X = \\frac { \\partial } { \\partial y ^ 1 _ { k + 1 } } + \\sum _ j \\bigg ( g ^ k _ j + \\frac { \\partial a _ j ^ k } { \\partial y ^ 1 _ k } \\bigg ) \\frac { \\partial } { \\partial y ^ j _ { k + 1 } } + \\sum _ \\rho \\bigg ( \\gamma ^ k _ \\rho + \\frac { \\partial \\beta ^ k _ \\rho } { \\partial y ^ 1 _ k } \\bigg ) \\frac { \\partial } { \\partial \\eta ^ \\rho _ { k + 1 } } + Z , \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} \\upsilon _ { e } ( y _ e ) = \\left \\{ \\begin{array} { l l } e ^ { J _ e } , & \\\\ e ^ { - J _ e } , & \\end{array} \\right . \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u - \\frac { \\lambda } { | x | ^ 2 } u = u ^ { 2 ^ * - 1 } + \\nu \\alpha u ^ { \\alpha - 1 } v ^ { \\beta } , \\ ; \\ ; x \\in \\R ^ N , \\\\ - \\Delta v - \\frac { \\lambda } { | x | ^ 2 } v = v ^ { 2 ^ * - 1 } + \\nu \\beta u ^ { \\alpha } v ^ { \\beta - 1 } , \\ ; \\ ; x \\in \\R ^ N . \\end{cases} \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} [ [ x , y ] ] _ c = \\frac { 1 } { 2 } ( x \\circ y - y \\circ x ) \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} \\vartheta _ { \\mathsf { E } _ 8 } ( z ) & = E _ 4 ( z ) = 1 + 2 4 0 \\sum _ { r = 1 } ^ \\infty \\sigma _ 3 ( r ) q ^ r \\\\ & = 1 + 2 4 0 \\cdot 1 \\cdot q + 2 4 0 \\cdot 9 \\cdot q ^ 2 + 2 4 0 \\cdot 2 8 \\cdot q ^ 3 + \\cdots \\\\ & = 1 + 2 4 0 \\cdot q + 2 1 6 0 \\cdot q ^ 2 + 6 7 2 0 \\cdot q ^ 3 + \\cdots \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} N _ E = 3 \\cdot 5 ^ 2 \\cdot 7 ^ 2 , N _ { E ' } = 3 \\cdot 5 ^ 2 \\cdot 7 ^ 2 \\cdot 1 3 , \\Delta _ m = - 3 ^ 5 \\cdot 5 ^ 8 \\cdot 7 ^ 2 , \\Delta _ m ' = - 3 ^ 2 \\cdot 5 ^ 8 \\cdot 7 ^ 2 \\cdot 1 3 ^ { 1 7 } . \\end{align*}"}
-{"id": "2018.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { 3 } { p } \\right ) ^ r = 1 \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\left | \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) } { | C _ G ( x ) | | C _ G ( y ) | } \\right | _ 3 \\geq \\frac { q } { 3 } = 3 ^ { 2 k } . \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} \\beta _ { \\ell , j } ^ { 1 } + \\beta _ { \\ell , j } ^ { 2 } + \\beta _ { \\ell , j } ^ { 3 } + \\gamma _ { \\ell , j } ^ { 1 } + \\gamma _ { \\ell , j } ^ { 2 } + \\gamma _ { \\ell , j } ^ { 3 } = n - b , \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} \\lambda x _ { k } = \\alpha d _ { G } \\left ( k \\right ) x _ { k } + \\left ( 1 - \\alpha \\right ) \\sum _ { \\left \\{ i , k \\right \\} \\in E \\left ( G \\right ) } x _ { i } , 1 < k \\leq n . \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} \\mu ( i , j ) & : = [ n ^ 2 - ( i - 1 ) n - ( j - 1 ) ] ! ^ { \\frac { n ^ 2 } { n ^ 2 - ( i - 1 ) n - ( j - 1 ) } } , \\\\ \\nu ( j ) & : = [ n ^ 2 - ( j - 1 ) n ] ! ^ { \\frac { n ^ 2 } { n ^ 2 - ( j - 1 ) n } } . \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} D ^ q ( \\mu ) = d = \\dim _ \\mathrm { H } F . \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} M = \\langle r _ 1 , r _ 2 , \\dots \\rangle , \\ \\ r _ n = \\frac { p _ 1 \\dots p _ n } { 2 ^ { k _ n } } . \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} \\left | \\frac { \\xi _ q ( \\varepsilon _ b z + m _ b ) } { \\xi ( z + m _ b + j _ b ) } \\right | \\leq C \\begin{cases} ( 1 + | t | ) ^ { - 0 . 0 3 } , & \\mbox { u n l e s s $ j _ b = 0 $ } , \\\\ ( 1 + | t | ) ^ { 0 . 0 1 / n ^ 4 } , & \\mbox { $ j _ b $ } = 0 , m _ b \\neq 1 \\\\ ( 1 + | t | ) ^ { 0 . 0 1 / n ^ 4 } \\log ( 5 + | t | ) , & \\mbox { $ j _ b $ } = 0 , m _ b = 1 \\\\ \\end{cases} \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} f \\circ _ { x } g = f ^ { \\sigma } \\ , { { _ { x } \\circ _ { \\ast _ { Y } } } } \\ , \\ , g , \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} v \\left ( \\frac { \\epsilon _ 1 ' } { 1 + \\epsilon _ 1 } - \\frac { \\epsilon _ 0 ' } { 1 + \\epsilon _ 0 } \\right ) = v \\left ( \\frac { \\delta ' } { 1 + \\delta } \\right ) = \\alpha . \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} \\varepsilon ( \\mathcal { A } ^ T - \\mathrm { I } ) \\alpha _ { k , 1 } ( \\varphi , y ) = - \\varepsilon ^ 2 \\ , \\partial _ y ( \\alpha _ { k , 1 } ) ( \\varphi , y ) \\ , \\beta _ 1 ( \\varphi , y ) + \\mathtt { R } _ { \\tilde { \\beta } } , \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} \\binom { ( i + j ) p } { r + i p } \\equiv _ { p ^ 3 } \\binom { i + j } { i } \\binom { p } { r } j \\left ( 1 - p \\left ( ( i + j - 1 ) H _ { r - 1 } + \\frac { i } { r } \\right ) \\right ) , \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} \\nu ( E ) = \\int \\limits _ { \\mathbb { R } ^ { 2 n } } \\chi _ { E } \\left ( w , \\varphi \\left ( w \\right ) \\right ) \\eta \\left ( w \\right ) d w , \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{gather*} w = - u _ t , \\delta = - \\frac { t } { 2 } - u ^ 2 . \\end{gather*}"}
-{"id": "7262.png", "formula": "\\begin{align*} \\sigma ^ { 2 } _ { b } ( \\Delta _ { g _ { _ { M } } } ) = \\sigma ^ { 1 } _ { b } ( \\operatorname { D P [ 0 ] } ) \\circ \\sigma ^ { 1 } _ { b } ( \\operatorname { D P [ 0 ] } ) \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} F | _ k \\ ( \\begin{smallmatrix} A & B \\\\ C & D \\end{smallmatrix} \\ ) ( Z ) : = \\mu ( \\ ( \\begin{smallmatrix} A & B \\\\ C & D \\end{smallmatrix} \\ ) ) ^ k \\det ( C Z + D ) ^ { - k } F ( ( A Z + B ) ( C Z + D ) ^ { - 1 } ) , \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} \\frac { u ' _ 1 + u ' } { 2 } + \\frac { u _ 1 ^ 2 - u ^ 2 } { 2 } + u _ 2 - 3 u _ 1 + 3 u - u _ { - 1 } = 0 . \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} \\left ( x ; \\bigcup _ { j = 1 } ^ n S ^ { ( j ) } \\right ) = \\left ( x ; S _ { \\le m } \\right ) + \\left ( x ; S _ { > m } \\right ) = \\sum _ { m = 1 } ^ { n } \\left ( x ; S _ { = m } \\right ) \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} \\begin{aligned} \\{ { \\eta } ( { \\xi } _ { 1 } ) + { \\eta } ( { \\xi } _ { 2 } ) + \\cdots + { \\eta } ( { \\xi } _ { k } ) \\} & = \\{ \\{ \\xi _ { 1 } - p _ { 1 } \\eta \\} + \\{ \\xi _ { 2 } - p _ { 2 } \\eta \\} + \\cdots + \\{ \\xi _ { k } - p _ { k } \\eta \\} \\} \\\\ & = \\{ \\xi _ { 1 } + \\xi _ { 2 } + \\cdots + \\xi _ { k } - ( p _ { 1 } + p _ { 2 } + \\dots + p _ { k } ) \\eta \\} \\\\ & = \\{ \\xi _ { 1 } + \\xi _ { 2 } + \\cdots + \\xi _ { k } \\} , \\end{aligned} \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} \\P ( y \\le - t ) = \\P ( z \\le e ^ { - t } ) \\le \\P ( z \\le 1 - \\min \\{ t , 1 \\} / 2 ) & \\le \\P ( | z - 1 | \\ge \\min \\{ t , 1 \\} / 2 ) \\\\ & \\le 2 e ^ { - \\frac { c '' } { K ^ 2 } n t ^ 2 } \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} U _ { T / R } = \\left \\{ \\mathfrak { t } \\in \\operatorname { M a x } T \\ | \\ R _ { \\mathfrak { t } \\cap R } \\right \\} . \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} { \\cal X } : = H ^ 1 ( 0 , T ; L ^ 2 ( \\Omega ) ) \\cap L ^ \\infty ( Q ) \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} p ( \\lambda ) : & = \\lambda ^ 6 \\{ 2 4 c _ 1 ^ 2 - 1 2 c _ 4 \\} + \\lambda ^ 4 \\{ ( \\frac { 1 4 } { 3 } - \\frac { 1 6 \\ , \\nu } { 3 } ) c _ 2 ^ 2 - 4 ( 1 - \\nu ) c _ 6 \\} \\\\ & + \\lambda ^ 2 \\{ ( 1 2 - 1 6 \\nu ) c _ 2 c _ 3 - 1 2 ( 1 - 2 \\nu ) c _ 7 \\} - 6 c _ 3 ^ 2 . \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{gather*} q _ 2 ( t ) = - 1 + o ( 1 ) , \\alpha = o ( 1 ) , t \\to + \\infty . \\end{gather*}"}
-{"id": "4455.png", "formula": "\\begin{align*} v ( s ) = - \\Sigma _ 0 ( s ) ^ { - 1 } \\Lambda _ 0 ( s ) ( X ( s ) - \\bar X ( s ) ) - \\Sigma _ 1 ( s ) ^ { - 1 } ( \\Lambda _ 1 ( s ) \\bar X ( s ) + r ( s ) + \\bar r ( s ) + ( B ^ T ( s ) + \\bar { B } ^ T ( s ) ) \\phi ( s ) ) . \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} 2 \\mbox { R e } \\langle H u , i A ( \\delta A ^ 2 + 1 ) ^ { - 1 } u \\rangle = 2 \\varepsilon \\langle u , A ( \\delta A ^ 2 + 1 ) ^ { - 1 } u \\rangle + \\mbox { I m } \\langle A f , ( \\delta A ^ 2 + 1 ) u \\rangle . \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} V ( x ) = x ^ 4 - s | x | ^ 3 + r x ^ 2 - q | x | = \\begin{cases} x ^ 4 + s x ^ 3 + r x ^ 2 + q x & , \\\\ x ^ 4 - s x ^ 3 + r x ^ 2 - q x & , \\end{cases} \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{align*} \\chi _ \\alpha ^ { ( N ) } = ( i _ ! \\circ \\pi ^ \\ast ) ( \\alpha ) \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} X _ { \\mu , \\nu } ^ { \\rm r e g } : = \\widetilde { X } _ { \\mu , \\nu } ^ 2 + ( \\sqrt \\mu ) ^ q ( P _ { \\mu , \\nu } ^ 2 + F _ { \\mu , \\nu } ^ 2 ) . \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} \\frac { n ( n + 1 ) } { 3 } & \\left ( \\rho ^ n + \\frac { \\rho ^ { n - 1 } } { 2 } \\right ) \\leq n \\rho ^ n + \\frac { n ( n - 1 ) } { 2 } \\rho ^ { n - 1 } \\\\ & \\leq \\sum _ { i = 0 } ^ { n } i \\rho ^ i = \\frac { n } { 3 } \\sum _ { i = 0 } ^ { n } \\rho ^ i \\leq \\frac { n ( n + 1 ) } { 3 } , \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} Q _ { s , t } ( a , 0 , \\ell ) = Q _ { s , t } ( 0 , b , \\ell ) = \\begin{cases} 1 & \\ell = 0 , \\\\ 0 & \\ell \\neq 0 , \\end{cases} \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} L ( s ) = \\zeta ( s ) L ( s , \\chi ) = \\sum _ n \\lambda ( n ) n ^ { - s } \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} \\mu _ k = \\tilde { c } _ * \\sum _ { j = 1 } ^ k j ^ { - 1 } \\geq \\tilde { c } _ * \\log k - c . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} \\pi _ 0 ( Z _ Q ( \\sigma , y ) ) = \\pi _ 0 ( Z _ { Q \\cap Z _ G ( \\sigma _ 0 ) } ( y ) ) \\to \\pi _ 0 ( Z _ { Z _ G ( \\sigma _ 0 ) } ( y ) ) = \\pi _ 0 ( Z _ G ( \\sigma , y ) ) \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} f ( w ( 1 ) ) - \\mathbb { E } f ( w ( 1 ) ) = \\int ^ 1 _ 0 u ( t ) d w ( t ) . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} T Z T = T , Z T Z = Z , ( T Z ) ^ { \\ast } = T Z , ( Z T ) ^ { \\ast } = Z T . \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\frac { 1 - F _ \\delta ( k ) } { 1 - F ( k ) } = 1 , \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} U ^ { 1 - } = U ^ { 2 + } , \\ S _ 1 \\frac { \\partial U ^ { 1 - } } { \\partial n } = S _ 2 \\frac { \\partial U ^ { 2 + } } { \\partial n } \\mbox { o n } \\partial \\Omega , \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} t I \\bar { I } & = \\ < t , t \\tau , t \\bar { \\tau } , t \\tau \\bar { \\tau } \\ > \\\\ & = \\ < t , \\frac { s + \\sqrt { \\Delta } } { - 2 } , \\frac { s - \\sqrt { \\Delta } } { - 2 } , \\frac { s ^ 2 - \\Delta } { 2 t } \\ > \\\\ & = \\ < t , s , r , \\frac { s + \\sqrt { \\Delta } } { 2 } \\ > \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} \\lvert R _ 6 \\rvert _ { s _ 0 + \\beta } ^ { L i p ( \\gamma ) } \\le C \\varepsilon ^ { 7 - 2 b } \\gamma ^ { - 1 } = C \\varepsilon ^ { 3 - 2 a } , \\lvert R _ 6 \\rvert _ { s _ 0 + \\beta } ^ { L i p ( \\gamma ) } \\gamma ^ { - 1 } \\le C \\varepsilon ^ { 1 - 3 a } . \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} R ( 3 ; 0 , 0 ) = \\frac { \\log _ 9 S ( 3 ; 0 , 0 ) } { 8 1 } \\approx 0 . 2 8 2 3 . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} t ^ j ( z ) = \\iota _ j ^ * t ( z ) + \\sum \\limits _ \\chi t ^ { j , \\chi } ( z ) . \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l c l } [ A , A ^ { \\dagger } ] + [ B , B ^ { \\dagger } ] + I I ^ { \\dagger } - J ^ { \\dagger } J + F F ^ { \\dagger } - G ^ { \\dagger } G & = & 0 \\\\ \\mbox { } [ A ^ { \\prime } , A ^ { \\prime \\dagger } ] + [ B ' , B ^ { \\prime \\dagger } ] - F ^ { \\dagger } F + G G ^ { \\dagger } & = & 0 \\end{array} \\right . , \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} | \\Lambda ( C ) | = | \\Lambda ( C ) \\cap \\Lambda ( A ) | + | \\Lambda ( C ) \\backslash \\Lambda ( A ) | = | S _ { 1 } | + | S _ { 2 } | . \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} f ( x _ 1 ; G _ k ) = \\frac { 0 . 5 } { \\sqrt { 2 \\pi } } ( 2 k ) + \\frac { 0 . 5 } { \\sqrt { 2 \\pi } } \\exp ( - \\frac { x _ 1 ^ 2 } { 2 } ) \\geq \\frac { k } { { 2 \\pi } } \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} \\overline { 1 } ^ { \\{ t = 2 \\} } = \\overline { 3 } ^ { \\{ 2 \\} } = 1 ~ ~ ~ ~ \\overline { 2 } ^ { \\{ 2 \\} } = 2 . \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} P _ 0 = \\sum _ { n < 0 } a ( n ) q ^ { n } \\in q ^ { - 1 } \\Q [ q ^ { - 1 } ] \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} S ^ { k } _ { s } : = \\{ \\phi \\in C ^ k ( \\R ^ N ) \\ ; : \\ ; \\sup _ { x \\in \\R ^ N } ( 1 + | x | ^ { N + 2 s } ) \\sum _ { | \\alpha | \\leq k } | \\partial ^ { \\alpha } \\phi ( x ) | < \\infty \\} \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} G ( s ) = A ( s ) + X ( s ) B ( s ) + X ( s ) R ( s ) + R f ( 1 - s ) ( 1 - s ) ^ { - 2 } \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{gather*} [ X , Q ] = 0 . \\end{gather*}"}
-{"id": "8734.png", "formula": "\\begin{align*} M ( t , i ) - M ( ( t , i ) - 1 ) = \\frac { X ( t , i ) - \\pi ( t ) } { 1 - \\pi ( t ) } - \\frac { X ( t - 1 , i ) - \\pi ( t - 1 ) } { 1 - \\pi ( t - 1 ) } . \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} 3 2 ( 1 4 - N ) \\lambda ( N ) = 1 6 H _ h ( N ) + \\tau ( N ) ( \\mu ( N ) - \\mu ( N + 8 ) ) H _ u ( N ) . \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} Z ^ 2 = f _ { 4 , 4 } ( ( x _ 0 : x _ 1 ) : ( x _ 2 : x _ 3 ) ) f _ { 4 , 4 } ( ( y _ 0 : y _ 1 ) : ( y _ 2 : y _ 3 ) ) . \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} ( \\alpha , \\beta ) _ E = \\int _ { [ X ] } e _ T ( E ) ^ { - 1 } \\alpha \\cup \\beta \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} \\mbox { L H S } = \\sum _ { a = 0 } ^ { p - 1 } \\sum _ { b = 0 } ^ a \\sum _ { c = 0 } ^ b \\binom { b } c ^ 2 \\binom { a } b ^ 2 = \\sum _ { a = 0 } ^ { p - 1 } \\sum _ { b = 0 } ^ a \\binom { 2 b } b \\binom { a } b ^ 2 = \\sum _ { b = 0 } ^ { p - 1 } \\binom { 2 b } b \\sum _ { a = b } ^ { p - 1 } \\binom { a } b ^ 2 . \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} h _ { t } = g _ { \\sigma _ { n } } h _ { \\sigma _ { n } - } \\quad t \\in [ \\sigma _ { n } , \\sigma _ { n + 1 } ) , \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} Q _ { 1 } = 6 x ^ { 2 } - 1 0 x y + 5 y ^ { 2 } , Q _ { 2 } = 4 2 x ^ 2 - 3 4 x y + 7 y ^ 2 , \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} g _ { \\ell , j } ( z ) : = f _ { m _ \\ell } ( z ) \\cdot F _ { \\ell } ( z ) ^ { \\ell ^ \\beta } \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} h _ { m ^ \\epsilon } ( \\xi ) \\ , : = \\ , m ^ \\epsilon _ 1 \\xi _ 1 ^ 2 \\ , + \\ , m ^ \\epsilon _ 2 \\xi _ 2 ^ 2 \\ , , \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} - a _ 0 z ^ \\ast + ( T _ 0 + T _ 1 + T _ 2 ) g _ 0 ( z ^ \\ast ) = 0 . \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\begin{aligned} \\Big ( \\tilde { \\Omega } ^ { ( 2 ) } _ { \\mathcal { P } } ( t _ * ) \\tilde { v } , \\tilde { v } \\Big ) _ { \\mathbb { C } ^ n } & = 2 \\Big ( ( X _ 2 ( t _ * ) ^ t Y _ 2 ' ( t _ * ) - Y _ 2 ^ t ( t _ * ) X _ 2 ' ( t _ * ) ) M _ 2 ( t _ * ) ^ 2 \\Big { \\{ } X _ 2 ( t _ * ) ^ t v _ 1 + Y _ 2 ( t _ * ) ^ t v _ 2 \\Big { \\} } , \\\\ & M _ 2 ( t _ * ) ^ 2 \\Big { \\{ } X _ 2 ( t _ * ) ^ t v _ 1 + Y _ 2 ( t _ * ) ^ t v _ 2 \\Big { \\} } \\Big ) _ { \\mathbb { C } ^ n } . \\end{aligned} \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\| P _ n \\| _ E ^ { 1 / n } \\le 1 \\mbox { a n d } \\lim _ { n \\rightarrow \\infty } | c _ { n , n } | ^ { 1 / n } = 1 / \\textup { c a p } ( E ) , \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} \\mathcal { D } ( v ) : = \\left \\{ f \\in D ( J _ { \\rm m a x } ) : W _ { \\infty } ( v , f ) = 0 \\right \\} , \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} \\binom { k _ n } { \\gamma k _ n } \\leq \\exp ( k _ n H _ 2 ( \\gamma ) ) , \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} \\int _ \\Omega \\left ( D W ^ h ( x , \\nabla _ h y ^ h ) : \\nabla _ h \\phi - h ^ 3 ( f _ 2 \\phi _ 2 + f _ 3 \\phi _ 3 ) \\right ) \\dd x = 0 \\ , , \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} K _ { a , \\pm } ( \\lambda ) = K _ { a } ( \\lambda q ^ { ( 1 \\pm 1 ) / 2 } ; \\zeta _ { \\pm } , \\kappa _ { \\pm } , \\tau _ { \\pm } ) = \\left ( \\begin{array} { c c } a _ { \\pm } \\left ( \\lambda \\right ) & b _ { \\pm } \\left ( \\lambda \\right ) \\\\ c _ { \\pm } \\left ( \\lambda \\right ) & d _ { \\pm } \\left ( \\lambda \\right ) \\end{array} \\right ) _ { a } , \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} { P \\sp { ( \\alpha , \\alpha ) } _ k } ' = \\frac { k + 2 \\alpha + 1 } { 2 } P \\sp { ( \\alpha + 1 , \\alpha + 1 ) } _ { k - 1 } , \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} \\theta ^ 0 : = 1 \\theta ^ { k + 1 } : = ( 1 - \\alpha ^ k ) \\theta ^ k k \\ge 0 . \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} \\mathbb { H } ^ * ( \\mathbb { Z } ^ s , w ) = \\mathbb { S } ^ * ( \\mathbb { Z } ^ s , w ) . \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} g _ 0 = - \\frac { | \\overline { g } _ 0 | } { | \\gamma | } ( d x ^ 0 ) ^ 2 + \\overline { g } _ { 0 i j } d x ^ i d x ^ j . \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{gather*} H ^ { p , q } ( J ^ \\infty E ; \\delta / d ) = 0 \\mbox { f o r } p > 0 \\mbox { a n d } H ^ { 0 , q } ( J ^ \\infty E ; \\delta / d ) \\simeq \\Lambda ^ q ( M ) / d \\Lambda ^ { q - 1 } ( M ) . \\end{gather*}"}
-{"id": "5458.png", "formula": "\\begin{align*} \\nu \\ll \\| T _ i \\| \\qquad \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} \\phi _ 1 : U ^ 1 = 1 + T k [ [ T ] ] \\to & \\prod _ { ( i , p ) = 1 } W ( k ) \\\\ \\prod _ { ( i , p ) = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\infty } ( 1 - a _ { i j } T ^ i ) ^ { p ^ j } \\mapsto & \\left ( \\sum _ { j = 0 } ^ { \\infty } p ^ j \\sum _ { l | i } l [ a _ { l j } ] ^ { i / l } \\right ) _ i \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} \\Phi ( x ) = \\sup _ { p ( x _ i ) \\leq \\Phi ( x _ i ) } p ( x ) \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} X : = \\begin{pmatrix} 0 & 1 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix} , Y : = \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix} , \\quad Z : = \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} G _ { p ' } = G / \\langle \\ , g \\in G \\ , | \\ , g \\mbox { \\em i s o f f i n i t e $ p $ - - p o w e r o r d e r } \\ , \\rangle . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} \\frac { 1 } { \\mu } \\leq \\prod \\limits _ { j = 0 } ^ { n - 1 } \\Big | \\frac { c _ { i _ { r r } } ( j ) } { h _ { i } ( j ) + \\Delta _ { i } ( j ) } \\Big | \\leq \\mu \\textnormal { f o r a n y $ n \\geq 0 $ } . \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} \\mathsf { E } ^ { ( a ) } \\mathsf { E ^ { ( b ) } } = \\left \\{ \\begin{array} { c c } q ^ { p a b } { a + b \\choose a } \\mathsf { E } ^ { ( a + b ) } & p \\neq 2 , \\\\ & \\\\ { a + b \\choose a } \\mathsf { E } ^ { ( a + b ) } & p = 2 . \\end{array} \\right . \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} - b \\cdot D \\psi = - f + \\bar f \\ \\ \\ \\Omega _ 1 . \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { d \\psi _ 1 } { d x } = & - i z \\psi _ 1 + \\psi _ 2 \\\\ \\frac { d \\psi _ 2 } { d x } = & i z \\psi _ 2 + u \\psi _ 1 \\end{aligned} \\right . \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} u ( I ) & = u ( f _ { i _ 1 } ) u ( f _ { i _ 2 } ) \\cdots u ( f _ { i _ m } ) ; \\\\ u ( F ) & = u ( g _ { j _ 1 } ) u ( g _ { j _ 2 } ) \\cdots u ( g _ { j _ m } ) . \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} \\mathrm { c m } ( x , y ) \\cdot \\mathrm { c m } ( y , x ) = \\frac { [ y , b ( x ) ] } { | | y | | } . \\frac { [ x , b ( y ) ] } { | | x | | } = - \\frac { ( \\alpha \\beta ) ( \\lambda \\sigma ) [ x , y ] ^ 2 } { | | x | | . | | y | | } , \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { T } ^ d } \\prod _ { l = 1 , l \\neq j } ^ d K _ n ( y _ l ) | f ( x - y ) | \\ , d y = \\int _ { \\mathbb { T } ^ d } ( \\prod _ { l = 1 , l \\neq j } ^ d K _ n ) ( x - y ) | f ( y ) | \\ , d y \\\\ & = \\int _ { \\mathbb { T } ^ { d - 1 } } ( \\prod _ { l = 1 , l \\neq j } ^ d K _ n ) ( \\tilde { x } _ j - \\tilde { y } _ j ) [ \\int _ \\mathbb { T } | f ( y _ 1 , \\cdots , y _ j , \\cdots , y _ d ) | d y _ j ] \\ , d \\tilde { y } _ j \\\\ & = \\int _ { \\mathbb { T } ^ { d - 1 } } K _ n ^ { d - 1 } ( y ) | f | _ j ( \\tilde { x } _ j - y ) | \\ , d y . \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} b _ { n } = - q ^ { 2 j _ { n } - 1 } a _ { n } , d _ { n } = - q ^ { 2 j _ { n } - 1 } c _ { n } , \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} S : = \\{ v ( s - a ^ { \\dagger } ) : a \\in K ^ { \\times } \\} \\subseteq \\Psi ^ { \\downarrow } . \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} \\mathcal { B } ( x , y ) = \\frac { 1 } { 2 } \\left ( \\frac { x + y } { | x + y | } - \\frac { x - y } { | x - y | } \\right ) = \\frac { 1 } { 2 } \\left ( ( x + y ) - ( x - y ) \\right ) = { \\begin{cases} ( y ) & { } | x | < | y | \\\\ ( y ) / 2 & { } | x | = | y | , y \\ne 0 \\\\ 0 & { } \\end{cases} } \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} \\bar \\nabla ^ { H } _ { \\frac { \\partial } { \\partial t } } D ( \\pi \\circ \\Phi ) \\Big | _ 0 \\ ! w & = \\pi ^ * \\bar \\nabla ^ { ( \\pi \\circ \\Phi ) ^ { - 1 } T Y } _ { \\frac { \\partial } { \\partial t } } D ( \\pi \\circ \\Phi ) w \\Big | _ 0 \\ ! - D ( \\pi \\circ \\Phi ) \\ ; \\bar \\nabla ^ { T ( X \\times \\mathbb R ) } _ { \\frac { \\partial } { \\partial t } } w \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} M ( p _ 0 , p _ 1 , \\ldots , p _ n ) = M ( q _ 0 , q _ 1 , \\ldots , q _ n ) . \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} I _ 1 = - \\Big | \\nabla \\log \\frac { \\rho } { e ^ { - \\Psi } } \\Big | ^ { 2 } , \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} f ^ { ( j ) } ( 0 ) = - ( j - 1 ) ! \\sum _ { i = 1 } ^ d \\zeta _ i ^ { - j } . \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} { \\bf a } _ { 2 L _ n } = \\left ( \\underbrace { D _ { \\phi ( n ) } , \\ , \\dots , D _ { \\phi ( n ) } } _ { D _ { \\phi ( n ) } } , D _ { \\phi ( n - 1 ) } , \\ , \\dots \\dots , \\underbrace { D _ { \\phi ( N _ 1 ) } , \\ , \\dots , D _ { \\phi ( 1 ) } } _ { N _ 1 } \\right ) . \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} d \\bar Y = - \\biggl ( A ^ T \\bar Y + C ^ T \\bar Z + F ^ T \\bar Z _ 0 + ( Q + \\bar Q ) \\bar X + ( S ^ T + \\bar S _ 2 ^ T ) \\bar v + q \\biggr ) d s + \\bar Z _ 0 d W _ 0 , \\ \\ \\ \\bar Y ( T ) = ( H + \\bar H ) \\bar \\xi ( T ) \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} \\rho ( s ) = \\left | \\left | \\frac { d } { d s } b \\left ( \\gamma _ { \\partial B } ( s ) \\right ) \\right | \\right | . \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} \\theta ( z ) \\ = \\ \\exp ( \\sum z _ j N _ j ) \\cdot F \\ , , \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} | f ( t ) - T _ m ( f ) ( t ) | \\leq \\left | \\sum _ { j = m + 1 } ^ { \\infty } \\frac { p _ j t ^ j } { j } \\right | \\leq \\frac { 1 } { m + 1 } \\sum _ { j = m + 1 } ^ { \\infty } | p _ j t ^ j | . \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} ( \\alpha ' _ { n } ( b ) , \\beta ' _ { n } ( b ) , \\tau ' _ { n } ( b ) , \\mu ' _ { n } ( b ) ) & : = ( z ^ { ( 1 ) } _ { n } ( b ) , z ^ { ( 2 ) } _ { n } ( b ) , z ^ { ( 3 ) } _ { n } ( b ) , z ^ { ( 4 ) } _ { n } ( b ) ) \\ , \\\\ ( \\alpha ' _ { n } ( b ) , \\beta ' _ { n } ( b ) , \\tau ' _ { n } ( b ) , \\mu ' _ { n } ( b ) ) & : = ( z ^ { ( 3 ) } _ { n } ( b ) , z ^ { ( 4 ) } _ { n } ( b ) , z ^ { ( 1 ) } _ { n } ( b ) , z ^ { ( 2 ) } _ { n } ( b ) ) \\ \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} \\sum _ { n = N _ { 1 } } ^ { \\infty } | \\widetilde { f } _ { n } ( k ) | \\leq \\sum _ { n = N _ { 1 } } ^ { \\infty } | f _ { n } ( k ) | \\leq 1 , k = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{align*} \\{ F ( u ) , G ( u ) \\} : = \\Omega ( X _ F , X _ G ) = \\int _ { \\mathbb { T } } \\nabla F ( u ) \\ , \\partial _ x \\nabla G ( u ) \\ , d x , \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} | D H ( x ) | = 2 | x | \\ \\ \\ x \\in B _ { \\kappa } , \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} \\sum _ { h \\leq H } \\left ( D _ h ( X ) - c _ h X ( \\log X ) ^ 3 \\right ) = o ( H X ( \\log X ) ^ 3 ) \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} I _ { 4 , r } = \\sum _ { \\xi = 0 } ^ { r } \\frac { r ! } { ( r - \\xi ) ! } & \\left [ \\frac { ( - 1 ) ^ { r - \\xi - 1 } } { g _ 0 ^ { r - \\xi } } e ^ { 1 / g _ 0 } \\textrm { E i } \\left ( \\frac { - 1 } { g _ 0 } \\right ) + \\right . \\\\ & \\left . \\sum _ { \\nu = 1 } ^ { r - \\xi } \\frac { ( - 1 ) ^ { r - \\xi - \\nu } ( \\nu - 1 ) ! } { g _ 0 ^ { r - \\xi - \\nu } } \\right ] \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} \\lambda _ { t o p } & = \\frac { 1 } { Z _ \\sigma } \\int _ { - \\infty } ^ { \\infty } ( 1 - x ^ 2 ) \\ , e ^ { \\frac { 2 } { \\sigma ^ 2 } ( \\frac { 1 } { 2 } x ^ 2 - \\frac { 1 } { 4 } x ^ 4 ) } \\ , d x \\\\ & = \\frac { 2 } { Z _ \\sigma } e ^ { \\frac { 1 } { 2 \\sigma ^ 2 } } \\int _ 0 ^ \\infty ( 1 - x ^ 2 ) \\ , e ^ { - \\frac { 1 } { 2 \\sigma ^ 2 } \\left ( x ^ 2 - 1 \\right ) ^ 2 } \\ , d x . \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} \\beta = \\sum _ { i = 0 } ^ \\infty \\sum _ { j = 0 } ^ { j _ i } ( \\beta _ { i , j } ^ + + \\beta _ { i , j } ^ - ) , \\ ; \\ , \\beta _ { i , j } ^ + = i \\partial \\bar \\partial ( r ^ { d _ { i , j } ^ + } \\phi _ { i , j } ) , \\ ; \\ , \\beta _ { i , j } ^ - = i \\partial \\bar \\partial ( r ^ { d _ { i } ^ - } \\phi _ { i , j } ) , \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} A \\boxtimes A ' = \\sum _ { a , a ' } m _ a m ' _ { a ' } \\cdot C _ a \\times C ' _ { a ' } \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} \\Lambda ^ 8 + \\lambda [ a ^ { 1 1 } + a ^ { 2 2 } ] \\Lambda ^ 4 + \\lambda ^ 2 \\left [ a ^ { 1 1 } a ^ { 2 2 } - a ^ { 1 2 } a ^ { 2 1 } \\right ] = 0 . \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{gather*} r _ 2 = - \\frac { t } { 2 } + \\frac { 1 } { 2 } I _ 2 \\nu = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } I _ 1 . \\end{gather*}"}
-{"id": "8881.png", "formula": "\\begin{align*} ^ { c } D ^ { q } z _ { k } ( t ) = - a _ { k } z _ { k } ( t ) + \\sum _ { j = 1 } ^ { n } T _ { k j } g _ { j } ( z _ { j } ( t ) ) + I _ { k } , \\forall k = \\overline { 1 , n } , ~ \\forall ~ t > 0 \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} \\sum _ { i j k \\ell } N _ { i j k \\ell } t _ { i j k \\ell } = \\frac 1 { 2 d } \\sum _ { i j k \\ell } X _ { i j } X _ { k \\ell } A _ { i k } A _ { j \\ell } + \\nu _ 1 + \\nu _ 2 + \\xi _ n , \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} D ^ { - 1 } = ( 1 + \\tilde { D } ) ^ { - 1 } = 1 - \\tilde { D } + \\tilde { D } ^ 2 - \\cdots \\ = 1 - \\tilde { D } + \\textrm { h i g h e r t e r m s } \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} F _ { 2 } ^ { \\mu } ( \\phi _ { I } , \\Pi _ { I } , x ) = F _ { 1 } ^ { \\mu } ( \\phi _ { I } , \\Phi _ { I } , x ) + \\Phi _ { J } \\Pi _ { J } ^ { \\mu } . \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} V _ z ( E ) \\cap ( x _ 2 - \\delta t / 4 , x _ 2 + \\delta t / 4 ) = \\emptyset \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{gather*} \\frac { \\delta \\lambda } { \\delta \\phi ^ a } = ( - \\partial ) _ I \\frac { \\partial \\lambda } { \\partial \\phi ^ a _ I } , \\end{gather*}"}
-{"id": "5515.png", "formula": "\\begin{align*} \\frac { \\tilde { \\beta } _ { t + \\tau } } { \\tilde { \\beta } _ t } \\ , \\tilde { U } ( b ) = \\frac { \\tilde { \\beta } _ { t + \\tau ' } } { \\tilde { \\beta } _ t } \\ , \\tilde { U } ( c ) . \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s - 1 } | x | ^ { 2 s - N - 2 } = ( - \\Delta ) ^ { \\frac { N } { 2 } } ( - \\Delta ) ^ { \\frac { 2 s - N - 2 } { 2 } } | x | ^ { 2 s - N - 2 } = ( - \\Delta ) ^ { \\frac { N } { 2 } } 1 = 0 \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} | ( F ^ n ) ' ( z ) | \\geq \\frac { 1 } { ( 9 6 \\pi ) ^ n } \\prod _ { j = 0 } ^ { n - 1 } h ( x _ j ) \\quad \\ z \\in V _ n . \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} \\lambda ^ * = ( \\lambda ^ * _ k ) _ { k = 1 } ^ \\infty , \\lambda ^ * _ k : = 2 ( g _ 1 + g _ 2 + \\ldots + g _ k ) , \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} Q _ 5 = - \\frac { 2 } { r ^ 3 } + \\frac { 1 4 / 5 } { r ^ 4 } \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\mathbb { E } \\int _ { \\mathbb { R } ^ d } \\log ^ { + } \\left \\| D \\varphi _ 1 ( \\omega , x ) \\right \\| \\ , d \\rho ( x ) \\leq \\int _ { \\mathbb { R } ^ d } \\log ^ { + } \\left ( e ^ 1 \\right ) \\ , d \\rho ( x ) = 1 < \\infty . \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} y _ { \\rm o d d } = \\sum _ { m \\in \\mathbb { N } } \\frac { ( - 1 ) ^ m } { \\sqrt { 2 m } } B _ m u _ { 2 m + 1 } { \\rm a n d } y _ { \\rm e v e n } = \\sum _ { m \\in \\mathbb { N } } \\frac { ( - 1 ) ^ { m - 1 } } { \\sqrt { 2 m - 1 } } A _ m u _ { 2 m } \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} \\begin{aligned} S ( U , f ^ { n + k } ) & \\geq \\frac 1 3 \\frac { \\log M ^ { n - 1 } ( R _ 2 , f ) } { \\log R _ 2 } \\frac { \\log R _ 2 } { \\log r _ 2 } - 3 \\frac { \\log ^ + M ^ { n - 1 } ( R _ 1 , f ) } { \\log R _ 1 } \\frac { \\log R _ 1 } { \\log r _ 1 } . \\end{aligned} \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} ( A _ 1 ) _ j ^ { j ' } ( l ) = \\begin{cases} - \\dfrac { ( \\mathfrak { B } _ 1 ) _ j ^ { j ' } ( l ) } { i ( \\omega \\cdot l + m _ 3 ( j '^ 3 - j ^ 3 ) ) } \\mbox { i f } \\ , \\ , \\overline { \\omega } \\cdot l + j '^ 3 - j ^ 3 \\neq 0 , j , \\ , j ' \\in S ^ c , \\ , l \\in \\mathbb { Z } ^ { \\nu } \\\\ [ 3 m m ] 0 \\mbox { o t h e r w i s e } \\end{cases} \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} \\det ( \\rho _ { s , T \\Gamma } ) = \\det ( T \\rho _ { s , \\Gamma } ) = \\det ( T ) ^ 2 \\det ( \\rho _ { s , \\Gamma } ) \\ , . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} D _ { 1 } ( \\sum _ { k = 1 } ^ { n } | \\alpha _ { k } | ^ { p } ) ^ { \\frac { 1 } { p } } \\leq \\| \\sum _ { k = 1 } ^ { n } \\alpha _ { k } ( x _ { 2 k - 1 } - x _ { 2 k } ) \\| \\leq D _ { 2 } ( \\sum _ { k = 1 } ^ { n } | \\alpha _ { k } | ^ { p } ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\mathsf { D } _ { - } ( \\zeta _ { a } ^ { ( h ) } ) = \\kappa _ { a } ^ { \\left ( h \\right ) } \\mathsf { A } _ { - } ( q / \\zeta _ { a } ^ { ( h ) } ) \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} \\operatorname { D i v } ^ { \\vartriangle } P ( n , [ u ] ) = \\sum _ { \\alpha , J } B ^ { \\alpha } _ J ( S _ J F _ { \\alpha } ) . \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\mathcal H ^ j : = ( \\mathcal H ^ j , - 1 z ) , \\mathcal H ^ j _ + : = ( \\mathcal H ^ j , - 1 z ) _ + , \\mathcal H ^ j _ - : = ( \\mathcal H ^ j , - 1 z ) _ - . \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{gather*} V ^ \\bullet : = H _ z ^ \\bullet ( B _ { x _ 1 } B _ { x _ 2 } \\dots B _ { x _ m } ) \\end{gather*}"}
-{"id": "13.png", "formula": "\\begin{align*} P ( 0 , x _ 0 ; t , y ) \\le C \\Big ( \\frac { 1 } { v ( \\lfloor t / 2 \\rfloor ) } + e ^ { - c _ \\star \\sum _ { u = \\lfloor t / 2 \\rfloor } ^ { t - 1 } \\Phi ^ 2 _ u } \\Big ) . \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} a ^ + ( \\varphi ) f ^ { ( n ) } & : = \\varphi \\otimes f ^ { ( n ) } , \\\\ a ^ - ( \\varphi ) f _ 1 \\otimes \\dots \\otimes f _ n & : = \\sum _ { i = 1 } ^ n q ^ { i - 1 } ( \\varphi , f _ i ) _ \\mathcal { H } \\ , f _ 1 \\otimes \\dots \\otimes \\check f _ { i } \\otimes \\dots \\otimes f _ n \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} u ' = \\frac { \\operatorname { d } \\ ! u ( t , n ) } { \\operatorname { d } \\ ! t } , u '' = \\frac { \\operatorname { d } \\ ! ^ 2 u ( t , n ) } { \\operatorname { d } \\ ! t ^ 2 } . \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} \\sum _ { a = b } ^ { p - 1 } \\binom { a } b ^ 2 ( H _ a - H _ { a - b } ) \\equiv _ p 0 . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} q _ { \\infty } \\equiv \\sum _ { a = 1 } ^ { ( p - 1 ) \\mathsf { N } + 1 } \\prod _ { b = 1 , b \\neq a } ^ { ( p - 1 ) \\mathsf { N } + 1 } \\frac { Q ( \\xi _ { a } ) } { w _ { a } - w _ { b } } , q _ { 0 } \\equiv Q ( i q ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} W ^ { \\star } ( z , w ) = \\int _ { 0 } ^ { \\infty } W ( y ) J _ { \\nu } ( 4 \\pi \\sqrt { y w + z } ) \\mathrm { d } y . \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { I } { I + i \\choose i } 2 ^ { I - i } = 4 ^ I . \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} U ( s ) : = ( P _ { t - s } f ) \\ , \\Gamma ^ { \\mathrm { h o r i } } \\log ( P _ { t - s } f ) \\quad V ( s ) : = ( P _ { t - s } f ) \\ , \\Gamma ^ { \\mathrm { e l l i } } ( \\log ( P _ { t - s } f ) ) . \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} \\vert l _ { k - 1 , k } ( z ) \\vert = \\prod _ { j = 0 } ^ { k - 2 } \\frac { \\vert z - e _ i \\vert } { \\vert e _ { k - 1 } - e _ i \\vert } \\le 1 , \\ \\ \\ k \\ge 2 . \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} d \\eta = \\sum _ { j = 1 } ^ n d x _ j \\wedge d y _ j \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} [ n ] _ q : = & 1 + q + q ^ 2 + \\dots + q ^ { n - 1 } = \\frac { 1 - q ^ n } { 1 - q } , \\\\ [ n ] _ q ! : = & [ 1 ] _ q [ 2 ] _ q \\dotsm [ n ] _ q = \\frac { ( 1 - q ) ( 1 - q ^ 2 ) \\dotsm ( 1 - q ^ n ) } { ( 1 - q ) ^ n } , [ 0 ] _ q ! : = 1 , \\\\ { n \\choose i } _ q : = & \\frac { [ n ] _ q ! } { [ i ] _ q ! [ n - i ] _ q ! } \\ , , i = 0 , 1 , \\dots , n . \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} \\mathcal { R } = \\mathbb { F } [ V ] [ Q ] / ( Q ^ 3 ) , \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\mu _ k - \\mu _ j \\geq \\frac { \\kappa } { \\gamma } \\begin{cases} ( k - 1 ) ^ \\gamma - ( j - 1 ) ^ \\gamma , & \\gamma \\geq 1 , \\\\ [ 2 m m ] k ^ \\gamma - j ^ \\gamma , & 0 < \\gamma < 1 . \\end{cases} \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} K ( c , r , 1 ) = 6 ( 1 - r ^ 2 ) ( 4 - c ^ 2 ) - ( c ^ 3 - 2 c - 1 0 ) + ( 4 - c ^ 2 ) ( 3 c r ^ 2 - 2 - 2 c ) . \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} E : = E _ C : = \\mbox { s p a n } \\left \\{ e ^ { \\mathrm { i } \\ , j \\ , x } : 0 < \\lvert j \\rvert \\le C \\right \\} , C > 0 , \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} W = \\left ( \\sum _ { N < \\ell \\le N ^ 2 } - \\sum _ { N ^ 2 < \\ell \\le N ^ 3 } \\right ) \\frac { \\tilde \\lambda ( \\ell ) } { \\ell } \\left ( \\sum _ { m \\mid \\ell } \\mu ( m ) \\frac { g ( m ) } { \\xi ( m ) } \\right ) ^ 2 \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} d ( h _ { i , j } ) & = \\sum _ { k = 1 } ^ { i - 1 } h _ { k , j } h _ { i - k , j + k } , \\\\ d ( w _ { i , j } ) & = \\sum _ { k = 1 } ^ { i - 1 } \\left ( w _ { k , j } h _ { i - k , j + k } + h _ { k , j } w _ { i - k , j + k } \\right ) \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} \\int _ { E _ 3 } \\frac { d t _ 1 } { ( 1 - t _ 1 ) ^ 2 } \\frac { d t _ 2 } { t _ 2 } \\frac { d t _ 3 } { t _ 3 } = \\sum _ { k = 1 } ^ { \\infty } \\frac { k } { k ^ 3 } = \\zeta ( 2 ) . \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{align*} M ( P ) = M ( P ' ) \\cup M ( R ) M ( Q ) = M ( Q ' ) \\cup M ( R ) . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} d \\mu ( z ) = \\rho ( z ) d \\ell ( z ) = \\frac { 1 } { 2 \\pi } \\Big | a - \\frac { 1 } { z } \\Big | d \\ell ( z ) , z \\in { \\cal S } , \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} \\mathcal { C } = \\mathcal { C } [ - 1 ] ^ { \\bot } = { } ^ { \\bot } \\mathcal { C } [ 1 ] . \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} & \\int _ 0 ^ t \\frac { 1 } { \\Delta J _ 0 ( s ) } \\int _ 0 ^ { \\Delta J _ 0 ( s ) } f ( W ( s - ) + x ) d x d J _ 0 ( s ) \\\\ & = c _ 0 \\int _ 0 ^ t f ( W ( s ) ) d s + \\sum _ { 0 < s \\le t } \\int _ 0 ^ { \\Delta J _ 0 ( s ) } f ( W ( s - ) + x ) d x \\ , \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} \\mathbf { R } ^ { ( g ) } _ { s u m } \\triangleq \\sum _ { l = 0 } ^ { L _ g - 1 } \\rho _ l ^ { ( g ) } \\mathbf { R } _ l ^ { ( g ) } . \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} 0 < g _ { 1 } ( x ) < 1 x \\in ( 0 , 1 ) , \\ g _ { 1 } ( 0 ) = 0 g _ { 1 } ( 1 ) = 1 . \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} J _ { l o w e r } ( D , D _ 1 , u , v ) = \\frac { e ^ { - 4 d _ { D _ 1 } ( u , v ) } - 1 } { e ^ { - 4 d _ D ( u , v ) } - 1 } \\cdot \\frac { \\lambda _ { D } ( u ) } { \\lambda _ { D _ 1 } ( u ) } \\ \\ \\ \\ \\ J _ { u p p e r } ( D , D _ 1 , u , v ) = \\frac { e ^ { 4 d _ D ( u , v ) } - 1 } { e ^ { 4 d _ { D _ 1 } ( u , v ) } - 1 } \\cdot \\frac { \\lambda _ { D _ 1 } ( u ) } { \\lambda _ { D } ( u ) } . \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} \\Omega = \\frac { 1 } { 2 } \\sum _ { j \\neq 0 } \\frac { 1 } { \\mathrm { i } j } \\ , d u _ j \\wedge d u _ { - j } , \\Omega ( u , v ) = \\sum _ { j \\neq 0 } \\frac { 1 } { \\mathrm { i } j } u _ j \\ , v _ { - j } , \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} \\| f \\| _ { 1 + k _ { \\gamma } } ^ 2 = | \\hat { f } ( 0 ) | ^ 2 + \\sum _ { h \\neq 0 } | \\hat { f } ( h ) | ^ 2 \\ , \\frac { \\omega _ h } { \\gamma } , \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ n \\theta ^ j \\frac { \\partial } { \\partial { \\theta ^ j } } \\left [ \\frac { \\partial { U ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ i } } \\right ] = \\frac { \\lambda ( \\hat { x } , \\theta ) } { \\theta ^ i } \\left [ \\sum _ { j = 1 } ^ n \\theta ^ j \\frac { \\partial { s ^ i ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ j } } \\right ] = 0 . \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{align*} \\phi ( G ) = \\phi _ 2 ( G ) \\geq \\phi _ 3 ( G ) \\geq \\phi _ 4 ( G ) \\geq \\cdots \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} \\beta ( \\alpha ) = \\limsup _ { n \\rightarrow \\infty } \\frac { \\ln { q _ { n + 1 } } } { q _ n } , \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} r _ d \\phi _ d \\phi _ { m + a } & = d * ( m + a ) = d * m + d * a \\\\ & = 0 + d * a = d * a = r _ d \\phi _ d \\phi _ a \\\\ \\Rightarrow \\phi _ d \\phi _ { m + a } & = \\phi _ d \\phi _ a \\\\ \\Rightarrow \\phi _ { m + a } & = \\phi _ a \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} ( A _ { 2 2 } - A _ { 2 1 } A _ { 1 1 } ^ { - 1 } A _ { 1 2 } ) v ^ \\pm _ \\tau ( \\tau ) = Q ' _ { 2 2 } ( u ^ \\pm ) v ^ \\pm ( \\tau ) + Q \\big ( v ^ \\pm ( \\tau ) - A _ { 1 1 } ^ { - 1 } A _ { 1 2 } v ^ \\pm ( \\tau ) \\big ) . \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\frac 1 { f ( n ) } = D _ f ( \\log x + E _ f ) + O \\left ( x ^ { - 1 } R _ { 1 / f } ( x ) \\right ) , \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ J k _ n ^ { ( j ) } v ^ { ( j ) } ( n ) \\leq \\\\ & ( 1 - \\epsilon ) \\left [ \\frac { n } { 2 } \\log \\left ( \\sum _ { j = 1 } ^ J k _ n ^ { ( j ) } \\right ) - \\sum _ { j = 1 } ^ J \\beta ^ { ( j ) } \\ell _ n H _ 2 \\left ( \\alpha _ n ^ { ( j ) } \\right ) \\right ] \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{align*} \\dd r _ r = \\left ( A r _ t + b ( r _ t ) \\right ) \\dd t + a ( r _ t ) \\dd W _ t + \\int _ E \\gamma ( r _ { t - } , x ) \\left ( \\mu ( \\dd t , \\dd x ) - p ( \\dd t , \\dd x ) \\right ) , \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} \\int _ { M } X ( b ) d M = \\frac { n } { 2 } \\displaystyle \\int _ { M } \\left \\langle { \\stackrel { \\circ } { B } } , \\mathcal { L } _ { X } g \\right \\rangle d M . \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} \\theta _ 1 = & \\frac { \\sqrt { - 1 } } { 2 } t r ( \\pi _ 1 \\wedge d \\pi _ 1 ^ { \\dagger } + \\pi _ 2 \\wedge d \\pi _ 2 ^ { \\dagger } + \\pi _ 3 \\wedge d \\pi _ 3 ^ { \\dagger } + \\pi _ 4 \\wedge d \\pi _ 4 ^ { \\dagger } \\\\ & + \\pi _ 5 ' \\wedge d \\pi _ 5 ^ { \\prime \\dagger } + \\pi _ 6 ' \\wedge d \\pi _ 6 ^ { \\prime \\dagger } + \\pi _ 7 \\wedge d \\pi _ 7 ^ { \\dagger } + \\pi _ 8 \\wedge d \\pi _ 8 ^ { \\dagger } ) . \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} \\Delta _ { i , a } ^ n Y : = \\sum _ { j = 0 } ^ q a _ j Y _ { \\frac { i + j } { n } } . \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\pi \\in \\mathcal { D } _ o , \\\\ | \\pi | = N } } ( - 1 ) ^ { \\nu _ e ( \\pi ) } = \\chi ( N = \\square \\ , ) . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} Q _ H ( f , g ) = \\langle f , ( - \\Delta + V ) g \\rangle : = \\langle \\nabla f , \\nabla g \\rangle + \\int V f \\bar { g } d x , \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} \\Omega _ { \\delta } = \\left \\{ x \\in \\Omega : d \\left ( x , \\partial \\Omega \\right ) < \\delta \\right \\} , \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\phi ( t , q ) & : = ( 1 + t ^ { a ( 1 ) } q + t ^ { 2 a ( 1 ) } q ^ 2 + \\ldots ) ( 1 + t ^ { a ( 2 ) } q ^ 2 + t ^ { 2 a ( 2 ) } q ^ 4 + \\ldots ) \\times \\cdots \\\\ & \\times ( 1 + t ^ { a ( n ) } q ^ n + t ^ { 2 a ( n ) } q ^ { 2 n } + \\ldots ) \\times \\cdots . \\end{align*}"}
-{"id": "4875.png", "formula": "\\begin{align*} A _ { i j } & : = t p \\sum _ { m = 0 } ^ { p - 1 } \\sum _ { k = m } ^ { p - 1 } \\binom { k + ( i + j ) p } { m + i p } \\binom { k + ( i + j + t ) p } { k + ( i + j ) p } \\frac { 1 } { k + ( i + j ) p + 1 } , \\\\ B _ { i j } & : = \\frac { t p } { ( i + j + 1 ) p + 1 } \\binom { ( i + j + t + 1 ) p } { ( i + j + 1 ) p } \\sum _ { m = 1 } ^ { p - 1 } \\binom { ( i + j + 1 ) p } { m + i p } , \\\\ C _ { i j } & : = t p \\sum _ { m = 0 } ^ { p - 1 } \\sum _ { k = 1 } ^ { m - 1 } \\binom { k + ( i + j + 1 ) p } { m + i p } \\binom { k + ( i + j + t + 1 ) p } { k + ( i + j + 1 ) p } \\frac { 1 } { k + 1 + ( i + j + 1 ) p } . \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} | x _ j + y _ j | ^ { 2 R _ j } & \\le \\big ( ( | x _ j | + | y _ j | ) ^ { R _ j / p } \\big ) ^ { 2 p } \\le ( | x _ j | ^ { R _ j / p } + | y _ j | ^ { R _ j / p } ) ^ { 2 p } . \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} \\begin{cases} u _ t - \\bar F ( \\nabla u , \\nabla ^ 2 u ) = 0 \\R ^ n \\times ( 0 , + \\infty ) \\\\ u ( x , 0 ) = u _ 0 ( x ) \\end{cases} \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} 2 g _ n - 2 = & - 2 p ^ n + \\sum _ { j = 1 } ^ n \\varphi ( p ^ j ) ( 1 + d p ^ { j - 1 } ) = - 2 p ^ n + p ^ n - 1 + d \\frac { p ^ { 2 n } - 1 } { p + 1 } \\\\ = & \\frac { d } { p + 1 } p ^ { 2 n } - p ^ n - \\frac { p + 1 + d } { p + 1 } . \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{align*} \\Sigma _ i i m _ i = \\frac { 1 } { 2 } ( \\Sigma _ j \\tilde { m } _ j ^ 2 + \\tilde { m } _ j ) \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\mathcal { D } _ { \\omega } : = \\omega \\cdot \\partial _ { \\varphi } . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} N _ { l / Y } = \\bigoplus _ { \\chi \\in C ( T ' ) } N _ { \\chi } \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} K ( \\xi , \\eta ) = - \\frac 1 2 \\left ( ( \\xi + \\eta ) + ( \\xi - \\eta ) \\tanh ( \\xi - \\eta ) \\right ) . \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{align*} S _ { n } ^ { + } = u _ { n } ^ { - 1 } \\left ( v _ { n } - 1 / v _ { n } \\right ) / 2 i , S _ { n } ^ { - } = u _ { n } \\left ( v _ { n } / q - q / v _ { n } \\right ) / 2 i , \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} \\mathcal { J } ( u , \\tau ) : = J _ { r } ( S _ { 1 } ( u ) , u , \\tau ) . \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{gather*} \\delta _ Q C ^ \\ast = d A ^ \\ast , \\delta _ Q A ^ \\ast = d A , \\delta _ Q A = d C , \\delta _ Q C = 0 . \\end{gather*}"}
-{"id": "8948.png", "formula": "\\begin{align*} { \\mathcal M } \\ , = \\ , { \\mathcal M } _ { D - D _ 0 } \\cap \\mathrm { A t } _ { D _ 0 } \\ , , { \\mathcal L } \\ , = \\ , { \\mathcal L } _ { D - D _ 0 } \\cap E _ P ( { \\mathfrak g } / { \\mathfrak p } ) \\ , . \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} j _ * \\lim F ^ { \\alpha } & = j \\circ \\lim F ^ { \\alpha } \\\\ & = \\coprod _ { a \\in A } j \\circ \\lim F ^ { \\alpha } ( a ) \\\\ & = \\coprod _ a \\lim j \\circ F ^ { \\alpha } ( a ) \\\\ & = \\lim j _ * F ^ { \\alpha } \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} ( \\beta F ) _ i \\b A : = \\sum _ j F _ { i + j } \\cap \\beta ^ j \\b A . \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} \\delta ( x \\otimes y ) : = \\delta ( x ) \\otimes y + ( - 1 ) ^ { d e g x } x \\otimes \\partial ( y ) . \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{gather*} p = \\frac { 1 + q _ 2 } { 2 } , q = \\frac { 1 - q ^ 2 _ 2 } { 4 } , \\end{gather*}"}
-{"id": "884.png", "formula": "\\begin{align*} | Z ^ t _ i | & = \\big ( d ^ t ( a _ i + \\xi _ i , a ' _ i + \\xi _ i ) + | a _ i - a ' _ i | \\big ) \\ , \\big | d ^ t ( a _ i + \\xi _ i , a ' _ i + \\xi _ i ) - | a _ i - a ' _ i | \\big | \\\\ & \\lesssim ( \\delta + | t | ) ( | a _ i - a ' _ i | + \\delta + | t | ) , \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} A _ G ( u ) w : = - \\partial _ x ( a _ G ( u ) \\partial _ x w ) , u \\in U ^ \\alpha , w \\in H ^ 2 _ N \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} 2 \\imath \\psi _ { t } + \\psi _ { r r } + \\frac { 1 } { r } \\psi _ { r } + \\frac { 1 } { k ^ 2 r ^ 2 } \\psi _ { \\phi \\phi } = 0 . \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\sum _ { q = m } ^ { 0 } { m \\choose q } \\frac { \\Gamma ( 2 + q ) } { ( m + 1 ) ^ { 1 + q } } = \\sum _ { q = 0 } ^ { m } { m \\choose q } \\frac { \\Gamma ( 2 + q ) } { ( m + 1 ) ^ { 1 + q } } = 1 . \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\Gamma _ 2 ^ { \\mathrm { h o r i } } ( f , f ) & : = \\frac { 1 } { 2 } ( L \\Gamma ^ { \\mathrm { h o r i } } ( f , f ) - 2 \\Gamma ^ { \\mathrm { h o r i } } ( f , L f ) ) , \\\\ \\Gamma _ 2 ^ { \\mathrm { v e r t } } ( f , f ) & : = \\frac { 1 } { 2 } ( L \\Gamma ^ { \\mathrm { v e r t } } ( f , f ) - 2 \\Gamma ^ { \\mathrm { v e r t } } ( f , L f ) ) , \\\\ \\Gamma _ 2 ^ { \\mathrm { m i x } } ( f , f ) & : = \\frac { 1 } { 2 } ( L \\Gamma ^ { \\mathrm { e l l i } } ( f , f ) - 2 \\Gamma ^ { \\mathrm { e l l i } } ( f , L f ) ) . \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} g ( z ) = z + \\sum _ { n = 2 } ^ { \\infty } b _ n z ^ n . \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} \\hat { u } = \\frac { \\sigma ( t ) } { t } \\pmod { t } \\hat { r } = \\frac { \\sigma ( r ) - r } { t ^ 2 } \\pmod { t } . \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} \\nabla f = - \\frac { \\nabla \\cos \\alpha } { \\cos ^ { 2 } \\alpha } = \\frac { \\sin \\alpha \\nabla \\alpha } { \\cos ^ { 2 } \\alpha } = \\frac { \\sin \\alpha } { \\cos \\alpha } f \\nabla \\alpha . \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} ( U _ { R } ^ { - 1 } \\psi ) ( x , n ) & = \\int _ { 0 } ^ { 1 } e ^ { - 2 \\pi i \\beta n } \\sum _ { p \\in \\Z } \\psi ( \\beta , p ) e ^ { - 2 \\pi i p ( x + n \\alpha ) } \\ d \\beta . \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} L ( x ^ { 1 0 } ) & = \\{ 2 , 3 , 4 , 5 , 6 , 7 , 8 , 1 0 \\} \\quad \\mbox { i f $ p > 2 $ , } \\\\ L ( x ^ { 1 0 } ) & = \\{ 2 , 3 , 4 , 5 , 6 , 8 , 1 0 \\} \\quad \\quad \\mbox { i f $ p = 2 $ . } \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\gamma ' ( d ) = \\frac { D } { d } \\sum _ { ( D , w ) D \\mid c d } \\chi ( w / ( c d , w ) ) ( c d , w ) \\mu ( c ) c ^ { - 2 } . \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} A \\to B = A / J \\to k = A / \\mathfrak { m } . \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} X = \\left ( c + 2 \\right ) e _ { 1 } ^ { 2 } Y = \\left ( c - 2 \\right ) b _ { 1 } ^ { 2 } . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} f ( r ) = g \\ ! \\left ( \\frac { 1 + \\sqrt { ( q - 1 ) ( q r - 1 ) } } { q } \\right ) + ( q - 1 ) \\ , g \\ ! \\left ( \\frac { 1 - \\frac { 1 + \\sqrt { ( q - 1 ) ( q r - 1 ) } } { q } } { q - 1 } \\right ) \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} f ( x ; B ) = \\sup _ { \\theta \\in B } f ( x ; \\theta ) . \\end{align*}"}
-{"id": "6345.png", "formula": "\\begin{align*} \\exists c _ 1 , \\ldots , c _ \\ell \\textrm { s u c h t h a t } \\left ( \\sum a _ i \\right ) \\times \\left ( \\sum b _ i \\right ) = \\sum c _ i . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} E _ { r } f ( z ) = \\frac { \\sqrt { \\pi z } } { 2 } e r f ( \\sqrt { z } ) = z + \\sum _ { n = 2 } ^ \\infty \\frac { ( - 1 ) ^ { n - 1 } } { ( 2 n - 1 ) ( n - 1 ) ! } z ^ { n } . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} { \\hat { \\bf { C } } } _ X = \\begin{bmatrix} C _ X & \\tilde { C } _ X \\\\ \\tilde { C } ^ { * } _ X & C _ X \\end{bmatrix} , \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} q ( \\lambda ) = b ' ( \\chi _ 1 ( \\lambda ) , \\chi _ 2 ( \\lambda ) ) . \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} D \\pi _ x v ( x ) = \\tilde v \\circ \\pi ( x ) \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} & 2 \\rho \\ , \\partial _ t \\mu + \\mu \\ , \\partial _ t \\rho - \\Delta \\mu = 0 \\\\ & - \\Delta \\rho + W ' ( \\rho ) = \\mu \\ , . \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} \\hat { \\theta } _ { i } ^ { \\prime } = p _ { i } ^ { \\prime } \\alpha + \\xi _ { i } ^ { \\prime } , \\ \\forall \\ 1 \\leq i \\leq \\bar { k } , \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} R ^ m ( \\boldsymbol { x } ) = \\sum _ { k = 1 } ^ { m } \\sigma ^ { - 1 } _ k ( f , u _ k ) _ { \\ell _ 2 ( X _ N ) } u _ k ( \\boldsymbol { x } ) . \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} \\Diamond ( p , q ) \\ \\not = \\ 0 \\quad \\hbox { i m p l i e s } 0 \\ , \\le \\ , p , q \\ , \\le \\ , n \\ , . \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} \\nu _ 1 & = - \\frac { 1 } { 2 d } \\sum _ { i j } A _ { i i } A _ { j j } \\pi _ i \\pi _ j , \\\\ \\nu _ 2 & = - \\frac { 1 } { 2 d ^ 2 } \\sum _ { i j k \\ell } A _ { i k } ^ 2 A _ { j \\ell } ^ 2 \\pi _ i \\pi _ j \\pi _ k \\pi _ \\ell , \\mbox { a n d } \\\\ \\xi _ n & = O ( n ^ { - 1 / 2 } ) \\sum _ { i j } | X _ { i j } | + O ( n ^ { - 1 } ) \\left ( \\sum _ { i j } | X _ { i j } | \\right ) ^ 2 . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} \\Phi _ \\Gamma ( x _ - , x _ + ) = ( x _ - , \\gamma ( x _ - ) + x _ + ) \\ , , \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} E ( c , \\emptyset , \\{ 1 , 4 \\} ) \\ = \\ \\lambda \\ , \\Delta _ 4 \\ , \\Delta _ 3 \\ , - \\ , \\lambda \\ , \\Delta _ 1 \\ , \\Delta _ 6 \\ , + \\ , \\Delta _ 1 \\ , \\Delta _ 3 \\ = \\ 0 . \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} \\tau _ { \\infty } \\equiv \\frac { \\kappa _ { + } \\kappa _ { - } ( e ^ { \\tau _ { + } - \\tau _ { - } } \\prod _ { b = 1 } ^ { \\mathsf { N } } \\delta _ { b } \\gamma _ { b } + e ^ { \\tau _ { - } - \\tau _ { + } } \\prod _ { b = 1 } ^ { \\mathsf { N } } \\alpha _ { b } \\beta _ { b } ) } { \\left ( \\zeta _ { + } - 1 / \\zeta _ { + } \\right ) \\left ( \\zeta _ { - } - 1 / \\zeta _ { - } \\right ) } . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} u _ { \\infty , n } = 1 + d p ^ { n - 1 } . \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} \\mu _ j ^ { \\infty } : = \\mathrm { i } ( - m _ 3 j ^ 3 + m _ 1 j ) + r _ j ^ { \\infty } \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} x ( t ) = ( A \\mathbf { 1 } _ { [ 0 ; t ] } , \\xi ) , \\ t \\in [ 0 ; 1 ] . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} \\alpha _ { j } = \\sum _ { l = 1 } ^ { r } p _ { j l } \\beta _ { l } + \\xi _ { j } , \\ \\forall 1 \\leq j \\leq m . \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { N - 1 } \\binom { x _ i + y _ i ; r } { n _ i } = \\sum _ { 0 \\leq k \\leq _ b n } \\prod _ { i = 0 } ^ { N - 1 } \\binom { x _ i ; r _ i } { k _ i } \\binom { y _ i ; r _ i } { n _ i - k _ i } . \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} \\chi = \\frac { \\partial } { \\partial \\eta ^ \\sigma _ { k + 1 } } + \\sum _ j \\bigg ( \\gamma ^ k _ j + \\frac { \\partial a ^ k _ j } { \\partial \\eta ^ \\sigma _ k } \\bigg ) \\frac { \\partial } { \\partial y ^ j _ { k + 1 } } + \\sum _ \\rho \\bigg ( g ^ k _ \\rho + \\frac { \\partial \\beta ^ k _ \\rho } { \\partial \\eta ^ \\sigma _ k } \\bigg ) \\frac { \\partial } { \\partial \\eta ^ \\rho _ { k + 1 } } + Z , \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\eta \\wedge ( d \\eta ) ^ n = n ! \\ , d x _ 1 \\wedge d y _ 1 \\wedge \\cdots \\wedge d x _ n \\wedge d y _ n \\wedge d z \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} f _ { ( L ) } ( x ) = L \\left ( F ( x ) \\right ) ^ { L - 1 } f ( x ) . \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} \\left < \\prod _ { a = 1 } ^ r G ^ { \\lambda _ a } \\left ( \\prod _ { i = 1 } ^ k x _ i \\right ) ^ { 1 + \\# \\{ \\} } \\right > \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} \\Phi ( f _ 1 , d _ 1 , d _ 2 ) = 2 + \\frac { f _ 1 ( d _ 1 - d _ 2 ) } { d _ 2 } = 2 + f _ 1 \\left ( \\frac { d _ 1 } { d _ 2 } - 1 \\right ) . \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} \\varphi ( - q ) ^ 2 - \\varphi ( - q ) = \\sum _ { x , y \\in \\mathbb { Z } } ( - 1 ) ^ { x + y } q ^ { x ^ 2 + y ^ 2 } - \\sum _ { n = - \\infty } ^ \\infty ( - 1 ) ^ n q ^ { n ^ 2 } . \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} \\dim ( Q P _ k ) _ { k - 1 } & \\geqslant \\dim Q P _ k ( \\omega _ { ( k , 1 ) } ) + \\dim Q P _ k ( \\bar \\omega _ { ( k , 1 ) } ) \\\\ & > \\dim Q P _ k ( \\omega _ { ( k , 1 ) } ) = k = c ( k , 1 ) . \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} \\pi _ 1 ( C ) = \\langle a _ 1 , b _ 1 , \\ldots , a _ g , b _ g | a _ 1 b _ 1 a _ 1 ^ { - 1 } b _ 1 ^ { - 1 } \\cdot \\ldots \\cdot a _ g b _ g a _ g ^ { - 1 } b _ g ^ { - 1 } \\rangle , \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} & T _ n T _ m f ( x ) = \\sum _ { j = 0 } ^ { n - 1 } [ T _ m f ( \\frac { x + j } { n } ) - T _ m f ( \\frac { j } { n } ) ] \\\\ & = \\sum _ { j = 0 } ^ { n - 1 } \\sum _ { k = 0 } ^ { m - 1 } \\{ [ f ( \\frac { \\frac { x + j } { n } + k } { m } ) - f ( \\frac { k } { m } ) ] - [ f ( \\frac { \\frac { j } { n } + k } { m } ) - f ( \\frac { k } { m } ) ] \\} \\\\ & = \\sum _ { j = 0 } ^ { n - 1 } \\sum _ { k = 0 } ^ { m - 1 } [ f ( \\frac { x + j + n k } { m n } ) - f ( \\frac { j + n k } { m n } ) ] \\\\ & = \\sum _ { j = 0 } ^ { m n - 1 } [ f ( \\frac { x + j } { m n } ) - f ( \\frac { j } { m n } ) ] = T _ { n m } f ( x ) . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} \\hat { \\alpha } _ { j l } = ( p _ { j } ) \\alpha + \\xi _ { j l } , \\ \\forall \\ 1 \\leq j \\leq k \\ \\ 0 \\leq l \\leq | p _ { j } | - 1 , \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} \\omega _ { \\beta , \\mu } ( \\prod _ { i = 1 } ^ { m } \\eta ( A _ { i } ) B ) = \\\\ \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu , \\Lambda } ( \\prod _ { i = 1 } ^ { m } \\eta _ { \\Lambda } ( A _ { i } ) B ) = \\lim _ { V \\to \\infty } \\omega _ { \\beta , \\mu } ( \\prod _ { i = 1 } ^ { m } \\eta _ { \\Lambda } ) ( A _ { i } ) B ) \\ . \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} \\widehat { \\delta } ( \\phi , \\{ \\xi \\} ) = \\sup \\{ \\widehat { \\psi } ( \\phi ) \\mid \\psi \\in \\mathcal { R } X , \\widehat { \\psi } ( \\xi ) = 0 \\} = \\widehat { \\xi } ( \\phi ) . \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} Y ( t ) = T _ q ( t ) y _ 0 + \\int _ 0 ^ t T _ q ( t - \\tau ) F _ q ( Y ( \\tau ) ) \\dd \\tau , t \\geq 0 . \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} a : = 2 \\Im \\P [ R \\bar { R } _ { \\alpha } ] . \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} P _ { s - 1 } ( x , y ) = \\frac { 2 k _ { N , 1 } ( s - 1 ) s } { k _ { N , s - 1 } k _ { N , s } } \\int _ { \\partial B } M _ { s - 1 } ( x , \\theta ) M _ { s } ( y , \\theta ) \\ d \\theta \\quad \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} L _ 2 ^ { ( 2 , 3 ) } ( q ) = \\frac { 1 + q ^ 3 } { ( 1 - q ) ( 1 - q ^ 5 ) } . \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\sum _ f ( n ^ K ( \\partial f ) + 1 ) = \\sum _ f ( n ^ { \\widetilde { K } } ( \\partial \\widetilde { f } ) + 1 ) + \\ell = \\ell \\ , , \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { 2 \\pi } e ^ { - i j \\theta } \\Phi _ n ( e ^ { i \\theta } ) \\ , d \\mu ( \\theta ) = 0 , j = 0 , 1 , 2 , \\dots , n - 1 . \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{align*} u ( t , x ) = \\sum _ { j \\in \\mathbb { Z } \\setminus \\{ 0 \\} } u _ j \\ , e ^ { \\mathrm { i } \\ , j ^ 3 \\ , t } \\ , e ^ { \\mathrm { i } \\ , j \\ , x } . \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} & e ^ { - \\frac { \\Delta } { c _ 2 } } \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { \\frac { n } { M } } e ^ { - \\frac { z ^ { ( 0 ) } } { v } } e ^ { - \\frac { v } { \\zeta } } d v \\\\ & \\qquad \\qquad = \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { \\frac { n } { M } } e ^ { - \\frac { ( z ^ { ( 0 ) } + \\Delta ) } { v } } e ^ { - \\frac { v } { \\zeta } } d v . \\end{align*}"}
-{"id": "9512.png", "formula": "\\begin{align*} \\mathcal S _ { 0 } = \\{ \\{ 0 \\} , \\{ 0 , 1 \\} \\} , \\ \\mathcal S _ { 1 } = \\{ \\{ 1 \\} , \\{ 1 , 2 \\} \\} , \\ \\mathcal S _ { 2 } = \\{ \\{ 2 \\} , \\{ 2 , 0 \\} \\} . \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\begin{array} { l l l l } v ^ * = & \\max \\ & c ' z + d ' u & \\\\ & \\mbox { s . t . } \\ & A z \\leq b & \\\\ & & H z + G u \\leq h & \\\\ & & z _ i = z _ j & \\forall i , j \\in I ^ q , \\ ; \\forall q = 1 , \\ldots , k . \\end{array} \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} E \\Big ( \\int ^ t _ 0 \\int _ { { \\mathbb R } } p _ { t - s } ( u - v ) W ( d v , d s ) \\Big ) ^ 2 = \\\\ = \\int ^ t _ 0 \\int _ { { \\mathbb R } } p _ { t - s } ( u - v ) ^ 2 d v d s . \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} \\bold { p r } X ( L ) + L ( D _ i \\xi ^ i ) = \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 . \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} \\omega ( x ) = ( \\omega _ 1 ( x ) , \\omega _ 2 ( x ) , \\ldots , \\omega _ i ( x ) , \\ldots ) , \\ \\ \\sigma ( x ) = ( \\nu _ 1 ( x ) , \\nu _ 2 ( x ) , \\ldots , \\nu _ k ( x ) ) , \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} g = \\alpha _ 1 ( n _ { \\iota ( 1 ) } g _ { \\iota ( 1 ) } ) ^ { \\epsilon _ 1 } \\cdots \\alpha _ l ( n _ { \\iota ( l ) } g _ { \\iota ( l ) } ) ^ { \\epsilon _ l } = t _ 1 \\cdots t _ l , \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{gather*} i _ Y \\omega = \\delta H , H = \\xi ^ a c ^ \\ast _ a + \\xi ^ { a b } c _ { a b } ^ \\ast . \\end{gather*}"}
-{"id": "6540.png", "formula": "\\begin{align*} \\lambda z _ { k } = \\alpha \\left ( n - n _ { k } \\right ) z _ { k } + \\left ( 1 - \\alpha \\right ) \\sum _ { i \\in \\left [ r \\right ] \\backslash \\left \\{ k \\right \\} } n _ { i } z _ { i } , 1 \\leq k \\leq r . \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} 2 \\frac { d c _ n } { d t } = n \\sqrt { n + 1 } c _ { n + 1 } - ( n - 2 ) \\sqrt { n } c _ { n - 1 } , n \\in \\mathbb { N } _ 0 . \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} F _ n ^ { \\left ( 3 \\right ) } = F _ { 0 } ^ { s _ 3 \\left ( n , 0 \\right ) } F _ { 1 } ^ { s _ 3 \\left ( n , 1 \\right ) } F _ { 2 } ^ { s _ 3 \\left ( n , 2 \\right ) } = 2 ^ { s _ 3 \\left ( n , 2 \\right ) } \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} { Y _ i ^ { ( k ) } } ( t ) = { R _ { m _ k , i } } ( t ) - \\frac { { { R _ { m _ k , i } } \\left ( \\tau ^ { ( k ) } \\right ) } } { \\hat G _ { m _ k + 1 , k } ^ { \\left ( \\alpha _ i ^ { ( k ) , * } \\right ) } \\left ( { { \\tau ^ { ( k ) } } } \\right ) } \\hat G _ { m _ k + 1 , k } ^ { \\left ( \\alpha _ i ^ { ( k ) , * } \\right ) } \\left ( { { t } } \\right ) \\forall i . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} \\upsilon ( \\Delta _ m ) = 3 , \\upsilon ( a _ 2 ) \\ge 1 , \\upsilon ( a _ 4 ) = 1 , \\upsilon ( a _ 6 ) \\ge 2 , \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} \\frac { ( c ^ 3 - 7 c - 3 ) \\zeta _ 1 ( c ) } { 3 c ^ 2 \\left ( c ^ 3 + 2 c + 6 \\right ) ^ 2 } = 0 \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} V ( k , \\theta ) = \\sup _ { y \\in \\Gamma ( k ) } \\Big [ U ( f ( k ) - y , \\theta ) + \\mu ( \\theta ) V ( y , F ( \\theta ) ) \\Big ] , \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} U ^ { ( L ) } \\mathcal { B } _ { - } ( \\lambda ) = \\Delta _ { \\mathcal { B } _ { - } } ( \\lambda ) U ^ { ( L ) } , \\mathcal { B } _ { - } ( \\lambda ) U ^ { ( R ) } = U ^ { ( R ) } \\Delta _ { \\mathcal { B } _ { - } } ( \\lambda ) , \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} \\left \\| { A } _ N \\xi - \\hat { A } \\xi \\right \\| _ \\alpha ^ \\alpha = \\int _ { - \\pi } ^ { \\pi } \\left | \\left ( \\overline { C ( e ^ { i \\theta } } ) \\right ) ^ { < \\frac { 1 } { \\alpha - 1 } > } ( f ( \\theta ) ) ^ { \\frac { - 1 } { \\alpha - 1 } } \\right | ^ { \\alpha } f ( \\theta ) d \\theta . \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} \\mathcal { B } = \\{ T ^ { d - 1 } { \\tt Q } _ { 0 } , \\dots , T { \\tt Q } _ { 0 } , { \\tt Q } _ { 0 } ; T ^ { d - 1 } { \\tt Q } _ { 1 } , \\dots , T { \\tt Q } _ { 1 } , { \\tt Q } _ { 1 } ; \\dots \\} \\end{align*}"}
-{"id": "3176.png", "formula": "\\begin{align*} A _ n & = \\left \\{ z _ n \\le z + 1 \\right \\} . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} X _ 1 = \\bar { \\hat { u } } ^ 2 X + \\bar { \\hat { r } } , Y _ 1 = \\bar { \\hat { u } } ^ 3 Y + \\bar { \\hat { s } } \\bar { \\hat { u } } ^ 2 X + \\bar { \\hat { t } } . \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} \\| \\sigma ^ \\ast ( x ) \\| _ q \\ ; = \\ ; \\sup \\{ \\ , | x _ \\nu ^ q | \\ , | \\zeta | ^ { \\nu q } : \\ ; \\nu \\le m \\ , \\} \\ ; \\leq \\ ; \\sup \\{ \\ , \\bigl ( | x _ \\nu | \\ , | \\zeta | ^ { \\nu q } \\bigr ) ^ q : \\ ; \\nu \\le m \\ , \\} \\ ; = \\ ; \\| x \\| _ q ^ q \\ , . \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} Z : = { \\rm G a l } ( H _ { A _ { \\infty _ { 1 } } } / H _ { \\mathcal { O } _ { K } } ) \\cong { \\rm K e r } \\left ( { \\sf C l } _ { A _ { \\infty _ { 1 } } } \\longrightarrow { \\sf C l } _ { \\mathcal { O } _ { K } } \\right ) , \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\forall \\alpha \\in \\mathbb { N } W _ { ( \\alpha , \\alpha ) } ( n ) = & \\frac { 5 } { 1 2 } \\sigma _ { 3 } ( \\frac { n } { \\alpha } ) + ( \\frac { 1 } { 1 2 } - \\frac { 1 } { 2 \\alpha } n ) \\sigma ( \\frac { n } { \\alpha } ) , \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\left \\{ \\ ! \\ ! \\ ! \\begin{array} { l } ^ { c } D ^ { q } z _ { 1 } ( t ) = - a z _ { 1 } ( t ) + T _ { 1 1 } g _ { 1 } ( z _ { 1 } ( t ) ) + T _ { 1 2 } g _ { 2 } ( z _ { 2 } ( t ) ) + I _ { 1 } \\\\ ^ { c } D ^ { q } z _ { 2 } ( t ) = - b z _ { 2 } ( t ) + T _ { 2 1 } g _ { 1 } ( z _ { 1 } ( t ) ) + T _ { 2 2 } g _ { 2 } ( z _ { 2 } ( t ) ) + I _ { 2 } \\end{array} \\right . \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} z c _ 1 = \\sqrt { 2 } A _ 1 , \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} \\left ( \\frac { \\partial \\varphi } { \\partial t } \\right ) _ { \\overline { k } } = \\tilde { g } ^ { i \\overline { i } } \\left ( \\overline { e } _ { k } e _ { i } \\overline { e } _ { i } ( \\varphi ) - \\overline { e } _ { k } [ e _ { i } , \\overline { e } _ { i } ] ^ { ( 0 , 1 ) } ( \\varphi ) \\right ) - F _ { \\overline { k } } . \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} \\Lambda _ S ( - \\theta , \\mathbf { F } ) = \\log \\left ( \\pi \\sum _ { s = 1 } ^ S w _ s e ^ { - \\theta r _ s } + \\sum _ { j = 1 } ^ J \\pi _ \\ell e ^ { - \\theta r _ j } \\right ) . \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} [ F _ { 2 } ^ { 1 } ( 0 ) ] _ { I } = \\begin{cases} s , & I = \\{ 0 , 1 \\} \\\\ - \\lambda a _ { i } , & I = \\{ 1 , i \\} , i \\ge 2 \\\\ 0 , & \\mbox { o t h e r w i s e } \\end{cases} . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} \\int _ { B _ { 1 0 s } ( y ) \\cap \\Sigma _ i } | { \\textbf A } _ i | ^ 2 = \\nu _ i ( B _ { 1 0 s } ( y ) ) < \\varepsilon . \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} q ^ 2 ( \\tau ) - 1 = r ( \\tau ) P _ N ( \\tau ) , \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} \\frac 1 d \\sum _ { i = 1 } ^ d \\sum _ { j < p } \\mu ^ 2 ( j ) \\chi ^ { i } ( j ) \\overline { \\chi } ^ i ( m ) \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} R _ n = r ^ n _ { \\mathrm { S C } } + \\sum _ { k = 1 } ^ K x _ k r _ { k , n } , \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{gather*} \\xi ^ { a } ( x ) = \\zeta ^ a + \\zeta ^ { a b } x _ b , \\xi ^ { a b } ( x ) = \\zeta ^ { a b } \\end{gather*}"}
-{"id": "8838.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\ln \\gamma _ o \\right ] = & \\underbrace { \\mathbb { E } \\left [ \\ln \\left ( { { P _ t } G _ \\mathrm { M } ^ 2 \\beta { r ^ { - { \\alpha _ o } } } } \\right ) \\right ] } _ { Z _ 1 } + \\\\ & \\underbrace { \\mathbb { E } \\left [ \\ln \\left ( \\frac { 1 } { \\sum \\nolimits _ { i \\in \\Phi / { o } } { { P _ t } { G _ i } \\beta { { \\left | { { X _ { i , o } } } \\right | } ^ { - { \\alpha _ i } } } } + { N _ o } } \\right ) \\right ] } _ { Z _ 2 } . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} F _ N ^ * \\eta / d u = Z _ u + Z _ v N u ^ { N - 1 } + X Y _ u + X Y _ v N u ^ { N - 1 } = Z _ u + X Y _ u . \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} j _ \\nu ( z ) = \\sqrt { \\frac { \\pi } { 2 z } } J _ { \\nu + \\frac { 1 } { 2 } } ( z ) \\ ; , \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{gather*} \\frac { u _ t ( t ) } { u ( t ) } = - \\frac { 1 } { 2 ( - t ) } - \\frac { 3 } { 8 } ( - t ) ^ { - 4 } - \\frac { 1 1 1 } { 3 2 } ( - t ) ^ { - 7 } - \\frac { 1 5 0 9 } { 1 6 } ( - t ) ^ { - 1 0 } \\\\ \\hphantom { \\frac { u _ t ( t ) } { u ( t ) } = } { } - \\frac { 2 6 1 7 5 9 9 } { 5 1 2 } ( - t ) ^ { - 1 3 } - \\frac { 9 4 4 6 9 5 9 8 3 } { 2 0 4 8 } ( - t ) ^ { - 1 6 } - \\frac { 1 2 7 7 5 6 2 3 3 3 0 9 } { 2 0 4 8 } ( - t ) ^ { - 1 9 } + \\cdots . \\end{gather*}"}
-{"id": "9541.png", "formula": "\\begin{align*} F _ { 2 } ^ { \\mu } ( \\phi _ { I } , \\Pi _ { I } , x ) = \\phi _ { J } \\ , \\Pi _ { J } ^ { \\mu } + \\epsilon \\ , j ^ { \\mu } ( \\phi _ { I } , \\pi _ { I } , x ) . \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} \\frac { 1 } { b _ N } = \\sum _ { j = 1 } ^ k c _ j \\frac { a _ { n _ j } } { b _ { n _ j } } \\in M . \\end{align*}"}
-{"id": "7733.png", "formula": "\\begin{align*} \\lim _ { d ( x ) \\rightarrow 0 } N \\alpha ( x ) = L _ { 1 } \\in ( - 1 , 0 ) , \\lim _ { d ( x ) \\rightarrow 0 } N \\beta ( x ) = L _ { 2 } \\in ( - 1 , 0 ) , \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} g ( t ) = ( t - s ) ^ { \\frac 1 { m - 1 } } \\Lambda ( t ) = \\ln \\left ( ( t - s ) \\frac { k ^ { m - 1 } } { \\delta \\rho ^ 2 } \\right ) . \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n } \\frac { z ^ k } { k ! } = e ^ z \\frac { \\Gamma ( n + 1 , z ) } { n ! } . \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} H = \\left [ \\begin{array} { c c } 0 & - h _ { 1 2 } \\\\ h _ { 2 1 } & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\leq C \\int _ { \\mathbb { R } ^ { n } } \\left \\{ \\sum _ { j = 1 } ^ { k } \\left ( \\frac { \\lambda _ { j } | B _ j | ^ { \\frac { \\alpha } { n } } \\chi _ { B _ j } ( x ) } { \\| \\chi _ { B _ j } \\| _ { p ( . ) } } \\right ) ^ { p _ 0 } \\right \\} ^ { \\frac { q _ 0 } { p _ 0 } } w ( x ) d x \\end{align*}"}
-{"id": "5137.png", "formula": "\\begin{align*} \\lambda ( G _ o B ) \\geq \\lambda ( \\tau ^ { - 1 } ( \\tau ( G _ o ) J ) ) + \\lambda ( G _ o ) > \\lambda ( \\tau ^ { - 1 } ( J ) ) + \\frac { 1 } { [ G : G _ o ] } = m _ K ( J ) + \\frac { 1 } { [ G : G _ o ] } , \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} ( T \\widetilde { f } - T f ) ^ 2 & = ( T \\widetilde { f } - T f _ 1 ) ^ 2 + [ 2 T \\widetilde { f } - T f _ 1 - T f ] ( T f _ 1 - T f ) \\\\ & \\leq ( T \\widetilde { f } - T f _ 1 ) ^ 2 + 4 B _ n | T f _ 1 - T f | \\\\ & \\leq ( \\widetilde { f } - f _ 1 ) ^ 2 + 4 B _ n | f _ 1 - f | \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} x \\xi = q \\xi x , x \\eta = q ^ { - 1 } \\eta x , \\eta \\xi = - q ^ 2 \\xi \\eta + q ^ { 1 / 2 } ( 1 - q ) x ^ 2 , \\xi ^ 2 = 0 = \\eta ^ 2 . \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} \\inf _ \\gamma I ( \\gamma ) = - \\inf _ \\phi I ( \\phi ) . \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} y ^ m = x ^ 0 + ( - 1 ) ^ m x ^ 1 , \\ ; y ^ a = x ^ a ; a = 2 , . . . , n ; m = 0 , 1 ; \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} t _ { n , p } = \\frac { \\int | F _ A | ^ 2 \\chi _ n r _ n ^ { 2 p } r ^ { - 1 } V ^ { - 1 } d v } { \\int | F _ A | ^ 2 r _ n ^ { 2 p } r ^ { - 1 } V ^ { - 1 } d v } . \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rm F A } ( u ) = P \\left ( \\frac { Z } { n } \\geq \\tau _ n | \\sigma _ { \\rm j } ^ 2 = u , H _ 0 \\right ) . \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{gather*} \\Lambda ^ \\ast ( J ^ \\infty E ) = \\lim _ { \\longrightarrow } \\Lambda ^ \\ast ( J ^ k E ) . \\end{gather*}"}
-{"id": "5547.png", "formula": "\\begin{align*} f _ 2 ( x , x \\partial ^ { 2 i - 1 } ) = - \\partial ^ { \\otimes 2 i - 3 } + 2 \\gamma \\circ x + b x \\partial ^ { \\otimes 2 i - 3 } + d \\rho \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} W _ t + F ( 1 + W _ \\alpha ) = 0 . \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} - \\frac { \\partial } { \\partial t } K ( t , x ) & = \\sum _ { i = 1 } ^ { d } ( - \\frac { \\partial } { \\partial t } K _ i ( t , x _ i ) ) \\prod _ { l \\neq i } K _ l ( t , x _ l ) \\\\ & = 2 d K ( t , x ) - \\sum _ { i = 1 } ^ { d } ( w _ { i , m } ^ { - 1 } K _ i ( t , x _ i + 1 ) + w _ { i , m - 1 } K _ i ( t , x _ i - 1 ) ) \\prod _ { l \\neq i } K _ l ( t , x _ l ) \\\\ & = \\Delta K ( t , x ) . \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} g _ k ^ { ( m ) } & : = \\begin{cases} g _ k , \\ k = 1 , \\ldots , m ; \\\\ b _ m , \\ k = m + 1 , \\ldots , m ^ 2 , \\end{cases} \\\\ b _ m & : = - \\frac { m ^ 2 } { 2 ( m ^ 2 - m ) } a - \\frac { g _ 1 + \\ldots + g _ m } { m ^ 2 - m } . \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} s _ { 4 } ( n ) = 1 2 \\sigma ( n ) - 3 6 \\sigma ( \\frac { n } { 3 } ) . \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} \\partial _ Y R ( 0 , 0 ) = \\begin{pmatrix} A _ 1 & \\partial _ V R _ 1 ( 0 , 0 ) \\\\ 0 & \\partial _ V R _ 2 ( 0 , 0 ) \\end{pmatrix} , L ^ - = \\begin{pmatrix} L ^ { ( 1 ) } & \\partial _ V R _ 1 ( 0 , 0 ) \\\\ 0 & L ^ { ( 2 ) } \\end{pmatrix} \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} \\eta : v \\mapsto v \\lrcorner ~ \\theta : = \\theta _ v , \\ v \\in T _ B . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} \\sum _ { n } \\overline { v _ { l , j } ( x + p \\alpha , n - p ) } v _ { l ^ \\prime , j } ( x , n ) = 0 . \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} r _ n ( i , j ) : = \\left ( B ^ h _ { \\frac { i + k } { n } } - B ^ h _ { \\frac { i } { n } } , B ^ h _ { \\frac { j + k } { n } } - B ^ h _ { \\frac { j } { n } } \\right ) , k = 1 , 2 . \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} f ( a ) = \\prod 1 _ { \\alpha _ i > 1 / T } \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} G _ { 2 n - 1 } ^ { ( k , \\ell ) } ( x , y ) = \\prod _ { i = 1 } ^ { 2 n - 1 } \\frac { 1 } { 1 - x ^ { a ^ { ( \\ell , k ) } _ i } y ^ { a _ { i - 1 } ^ { ( k , \\ell ) } } } . \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { \\sigma ^ * ( n ) } { n ^ s } = \\left ( 1 - \\frac { 6 } { 2 ^ s } + \\frac { 6 } { 2 ^ { 2 s } } \\right ) \\left ( 1 - \\frac 2 { 2 ^ { 2 s } } \\right ) ^ { - 1 } \\frac { \\zeta ( s ) \\zeta ( s - 1 ) } { \\zeta ( 2 s - 1 ) } ( \\Re s > 2 ) . \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} \\Phi ( \\psi _ 1 ) ( x ) & : = \\int _ { \\pi _ k ^ { - 1 } ( x ) } | \\psi _ 1 | ^ 2 d \\tau & & & Q ( \\psi _ 1 , \\psi _ 2 ) ( x ) & : = \\int _ { \\pi _ k ^ { - 1 } ( x ) } \\langle \\psi _ 1 , \\psi _ 2 \\rangle d \\tau . \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} \\Theta = \\sum \\limits _ { \\ell , n \\in \\left \\{ { { } , { } } \\right \\} } { \\rm { \\mathbf { 1 } } } \\left ( { \\max \\{ { r _ e } , d \\} } < \\big ( \\frac { { P _ t } { G _ \\ell } { G _ n ^ e } \\beta } { x \\sigma _ e ^ 2 } \\big ) ^ { \\frac { 1 } { { \\alpha _ { \\mathrm { L o S } } } } } \\right ) { { { \\Pr } _ { \\ell n } } } , \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} ( \\tilde { c } _ 4 - 3 \\tilde { \\Delta } ^ { 1 / 3 } ) ( \\tilde { c } _ 4 ^ 2 + 3 \\tilde { c } _ 4 \\tilde { \\Delta } ^ { 1 / 3 } + ( 3 \\tilde { \\Delta } ^ { 1 / 3 } ) ^ 2 ) = ( 2 ^ { n - 6 } \\tilde { c } _ 6 ) ^ 2 \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} c _ 0 f ( W ( s ) ) & + \\int _ { ( 0 , \\infty ) } ( F ( W ( s ) + x ) - F ( W ( s ) ) \\nu _ 0 ( d x ) \\\\ & = c _ 0 f ( W ( s ) ) + \\int _ { ( 0 , \\infty ) } \\int _ 0 ^ x f ( W ( s ) + y ) d y \\nu _ 0 ( d x ) \\\\ & = c _ 0 f ( W ( s ) ) + \\int _ 0 ^ \\infty f ( W ( s ) + y ) \\nu _ 0 ( y , \\infty ) d y \\\\ & = \\rho _ 0 g ( W ( s ) ) \\ . \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 } \\frac { 1 } { t } \\int _ { \\R ^ N \\backslash U } \\frac { u ( x + t e _ 1 ) - u ( x ) } { | y | ^ { N + 2 \\sigma } } \\ d y = \\int _ { \\R ^ N \\backslash U } \\frac { \\partial _ 1 u ( x ) } { | y | ^ { N + 2 \\sigma } } \\ d y , \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} K ( \\mu , \\eta ) = \\tilde { D } ( \\eta ) . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} \\mathcal H ^ k _ E : = \\oplus _ { p + q = k } \\mathcal H ^ { p , q } _ E , \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} s \\underbrace { \\langle A _ { 2 1 } y , v \\rangle } _ { = 0 } + s \\langle A _ { 1 2 } z , v \\rangle + s \\langle A _ { 2 2 } v , v \\rangle + \\langle T _ { 2 2 } v , v \\rangle = 0 . \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} Q _ H ( e ^ { i t A } f , e ^ { i t A } f ) = Q _ H ( f , f ) + \\int _ 0 ^ t Q _ { [ H , i A ] } ( e ^ { i s A } f , e ^ { i s A } f ) d s . \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} S \\frac { \\operatorname { d } \\ ! \\widetilde { u } } { \\operatorname { d } \\ ! \\widetilde { t } } = \\frac { \\operatorname { d } ( S \\widetilde { u } ) } { \\operatorname { d } ( S \\widetilde { t } ) } \\neq \\frac { \\operatorname { d } ( S \\widetilde { u } ) } { \\operatorname { d } \\ ! \\widetilde { t } } . \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} \\Im Z ( E ) \\geq 0 , \\ , \\ , \\ , \\ , \\Im Z ( E ) = 0 \\ , \\Rightarrow \\ , \\Re Z ( E ) \\leq 0 . \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} \\begin{aligned} \\int _ { 0 } ^ { s } \\Bigl ( \\int _ { - \\infty } ^ { \\infty } P ( x u ) d F _ 1 ( x ) \\Bigr ) d u \\ge \\sum _ { m = 1 } ^ { \\infty } \\int _ { A _ m } ^ { A _ { m + 1 } } \\frac { 1 } { x } \\Bigl ( \\int _ { 0 } ^ { s } P ( v ) d v \\Bigr ) d F _ 1 ( x ) \\ge \\\\ \\ge \\sum _ { m = 1 } ^ { \\infty } \\frac { 1 } { A _ { m + 1 } } \\int _ { 0 } ^ { A _ m s } P ( v ) d v \\Bigl ( F _ 1 ( A _ { m + 1 } ) - F _ 1 ( A _ m ) \\Bigr ) \\ge \\\\ \\end{aligned} \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} X ^ * = - X \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\varphi = - \\Delta \\varphi - | \\nabla \\varphi | ^ 2 , \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} \\hat \\phi ^ h ( x ) = \\Big ( \\Phi ^ h _ { 1 2 } ( x _ 1 ) x _ 2 + \\Phi ^ h _ { 1 3 } ( x _ 1 ) x _ 3 \\ , , - \\frac { 1 } { h } & \\int _ 0 ^ { x _ 1 } \\Phi _ { 1 2 } ^ h ( s ) \\dd s + \\Phi _ { 2 3 } ^ h ( x _ 1 ) x _ 3 \\ , , \\\\ & - \\frac { 1 } { h } \\int _ 0 ^ { x _ 1 } \\Phi _ { 1 3 } ^ h ( s ) \\dd s - \\Phi _ { 2 3 } ^ h ( x _ 1 ) x _ 2 \\Big ) \\ , , \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\C } 2 ^ { \\nu _ d ( \\pi ) } q ^ { | \\pi | } = \\sum _ { \\pi \\in \\R } \\mu ( \\pi ) q ^ { | \\pi | } = \\sum _ { \\pi \\in \\Q } \\eta ( \\pi ) q ^ { | \\pi | } . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{gather*} [ Q , Q ] = 2 Q ^ 2 = 0 , \\operatorname { g h } ( Q ) = 1 . \\end{gather*}"}
-{"id": "2054.png", "formula": "\\begin{align*} u = x _ 0 ^ 3 + a x _ 0 + b . \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{align*} \\chi _ \\Delta \\left ( \\varphi Q _ 1 \\right ) = \\chi _ \\Delta \\left ( \\varphi Q _ 2 \\right ) \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} \\P ( [ X ^ n ] _ b = u ^ n ) = \\P ( [ X ^ k ] _ b = u ^ k ) \\prod _ { i = k + 1 } ^ n \\P ( [ X _ i ] _ b = u _ i | [ X _ { i - k } ^ { i - 1 } ] _ b = u _ { i - k } ^ { i - 1 } ) , \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} N _ 1 = \\begin{bmatrix} c _ { 1 1 } & - c ^ 2 _ { 1 1 } \\\\ 1 & - c _ { 1 1 } \\end{bmatrix} , N _ 2 = \\begin{bmatrix} 0 & c _ { 1 2 } - c ^ 2 _ { 1 1 } \\\\ 0 & 0 \\end{bmatrix} , N _ 3 = \\begin{bmatrix} 0 & 0 \\\\ c _ { 2 1 } - 1 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} Z _ t & = A , \\pi ^ { ( t ) } ( X _ t , X _ t + 1 ) = 1 - \\theta , \\\\ Z _ t & = B , \\pi ^ { ( t ) } ( X _ t , X _ t + 1 ) = 1 + \\theta . \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} T _ { \\alpha , m } f ( x ) = \\int _ { \\mathbb { R } ^ { n } } \\left \\vert x - A _ { 1 } y \\right \\vert ^ { - \\alpha _ { 1 } } . . . \\left \\vert x - A _ { m } y \\right \\vert ^ { - \\alpha _ { m } } f ( y ) d y , \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} \\left \\langle j _ { n } - 1 , n \\right \\vert \\left ( L _ { a , n } ( \\lambda ) \\right ) _ { 1 1 } = _ { n } ( \\lambda q ^ { j _ { n } - 1 } ) \\left \\langle j _ { n } - 1 , n \\right \\vert \\left \\langle j _ { n } - 1 , n \\right \\vert \\left ( L _ { a , n } ( \\lambda ) \\right ) _ { 2 2 } = _ { n } ( \\lambda q ^ { 1 - j _ { n } } ) \\left \\langle j _ { n } - 1 , n \\right \\vert \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} & a _ 0 = \\varepsilon a _ { 0 , 1 } + \\varepsilon ^ 2 a _ { 0 , 2 } + \\mathtt { R } _ { a _ 0 } , a _ 1 - 1 = \\varepsilon a _ { 1 , 1 } + \\varepsilon ^ 2 a _ { 1 , 2 } + \\mathtt { R } _ { a _ 1 } , \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} F = \\left ( \\prod _ { k = 1 } ^ { \\infty } c _ { \\phi _ { k } } \\right ) \\cdot G . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} t ^ { j , \\chi } ( z ) = \\sum \\limits _ { j ' \\neq j , d } \\iota _ j ^ * ( _ + ) ^ { v i r } _ * \\left [ \\displaystyle \\frac { Q ^ d _ { ( e , s _ 1 ) } } { - z - \\psi _ + + \\chi } \\left \\{ ( _ - ) ^ * \\left [ F ^ { j ' } ( w ) - t ^ { j ' , \\chi } ( w ) \\right ] \\right \\} _ { w = \\psi _ - - \\chi } \\right ] \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{gather*} ( 1 - x ^ m ) A _ { ( m + 1 ) p _ n + \\frac { m - 1 } { 2 } } = A _ { ( m + 1 ) p _ n - \\frac { m + 1 } { 2 } } ^ { m \\rightarrow m + 1 } ( x ) + x A _ { ( m + 1 ) p _ n - \\frac { m - 1 } { 2 } } . \\end{gather*}"}
-{"id": "4044.png", "formula": "\\begin{align*} a _ { 3 } = \\frac { 1 0 b p _ 1 } { ( [ 3 ] _ { q } - 1 ) ( 1 + i \\tan \\beta ) } \\left ( w _ { 2 } - \\left ( \\frac { p _ 2 } { p _ 1 } + \\frac { p _ 1 b } { ( 1 + i \\tan \\beta ) ( 1 - [ 2 ] _ { q } ) } \\right ) w _ { 1 } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} \\psi ( y ) \\leq \\psi \\circ f ( x ) + \\rho ( y , \\{ f ( x ) \\} ) \\leq \\delta ( x , \\{ x \\} ) + \\rho ( y , \\{ f ( x ) \\} ) = \\rho ( y , \\{ f ( x ) \\} ) , \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} \\partial _ \\xi ^ * = \\partial _ \\xi , \\partial _ x ^ * = { \\bf i } \\ , \\partial _ x , \\partial _ \\eta ^ * = \\partial _ \\eta - h \\partial _ \\xi , \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta \\varphi = \\mbox { d i v } A , & \\mbox { i n } \\ , \\Omega \\\\ \\varphi = 0 , & \\mbox { i n } \\ , \\Gamma . \\end{array} \\right . \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} \\min _ { \\sigma \\in \\mathcal { P } ( { \\bf D } _ n ) } \\sum _ { k = 1 } ^ { L _ n } a _ k a _ { \\sigma ^ \\ast ( k ) } . \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} \\Phi _ { n } ( x , t ) : = \\sum _ { i = 1 } ^ { n } \\gamma _ { n , i } ( t ) w _ { i } ( x ) , \\Xi _ { n } ( x , t ) : = \\sum _ { i = 1 } ^ { n } \\delta _ { n , i } ( t ) w _ { i } ( x ) , \\Sigma _ { n } ( x , t ) : = \\sum _ { i = 1 } ^ { n } \\eta _ { n , i } ( t ) w _ { i } ( x ) \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} A _ { k } ^ { i } = \\left \\{ A \\in \\Gamma _ { + } : \\sigma _ j ( A ) = k ^ { - 1 } j \\in \\{ 1 , 2 \\} \\sigma _ 3 ( A ) \\in \\{ i - 1 , i \\} \\setminus \\{ 0 \\} \\right \\} , \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} S _ { \\alpha , \\beta } : = \\inf \\limits _ { ( u , v ) \\in E \\backslash \\{ 0 \\} } \\frac { \\| ( u , v ) \\| ^ p } { \\left ( \\displaystyle \\int _ { \\Omega } | u | ^ { \\alpha } | v | ^ \\beta d x \\right ) ^ { \\frac { p } { \\alpha + \\beta } } } . \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } m \\ddot x ( t ) = - r ( t ) x ( t ) - y \\phi ( t , x , y ) , \\\\ m \\ddot y ( t ) = - r ( t ) y ( t ) + x \\phi ( t , x , y ) , \\\\ x ( t ) ^ 2 + y ( t ) ^ 2 = 1 , \\end{array} \\right . \\ , \\ , \\ , \\lambda > 0 , \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} \\hat { a } _ { n + d } + A _ { d - 1 } \\hat { a } _ { n + d - 1 } + \\dots + A _ 0 \\hat { a } _ n = 0 . \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} L ( e _ { i j } ) = [ L ( e _ { i i } ) , e _ { i j } ] + [ e _ { i i } , L ( e _ { i j } ) ] = L ( e _ { i i } ) e _ { i j } - e _ { i j } L ( e _ { i i } ) + e _ { i i } L ( e _ { i j } ) - L ( e _ { i j } ) e _ { i i } . \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} \\begin{array} { r l } & W = x ^ { a } , \\\\ & W = x _ { 1 } ^ { a _ { 1 } } x _ { 2 } + x _ { 2 } ^ { a _ { 2 } } x _ { 3 } + \\dots + x _ { n } ^ { a _ { n } } x _ { 1 } , \\\\ & W = x _ { 1 } ^ { a _ { 1 } } x _ { 2 } + x _ { 2 } ^ { a _ { 2 } } x _ { 3 } + \\dots + x _ { n } ^ { a _ { n } } . \\end{array} \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} & d ( \\varphi _ i ( x _ 0 ) , \\mathcal { Y } ( x _ 0 ) g ) = \\min _ { 1 \\leq j \\leq l } d ( \\varphi _ i ( x _ 0 ) , \\varphi _ j ( x _ 0 ) g ) \\\\ \\leq & \\min _ { 1 \\leq j \\leq l } ( d ( \\varphi _ i ( x _ 0 ) , \\ , \\varphi _ j ( x _ 0 ) g _ n ) + d ( \\varphi _ j ( x _ 0 ) g _ n , \\varphi _ j ( x _ 0 ) g ) ) \\\\ \\leq & d ( \\varphi _ i ( x _ 0 ) , \\mathcal { Y } ( x _ 0 ) g _ n ) + d ( g _ n , g ) \\\\ = & d ( \\varphi _ i ( x _ 0 ) , \\mathcal { Y } _ 0 ( x _ 0 g _ n ) ) + d ( g _ n , g ) . \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} \\overline { M ' \\setminus D } \\ , \\cap \\ , \\overline { D \\setminus M ' } = \\emptyset . \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} f \\left ( t \\right ) = t ^ { 4 } - 2 c t ^ { 3 } + 2 t ^ { 2 } + 2 c t + 1 , \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} \\sup _ { x \\in \\mathbb { G } _ h } | v ^ h _ t ( x ) - u _ t ( x ) | & = \\sup _ { x \\in \\mathbb { G } _ h } | \\mathfrak { I } u ^ h _ t ( x ) - u _ t ( x ) | \\\\ & = \\sup _ { x \\in \\mathbb { G } _ h } | \\mathfrak { K } u ^ h _ t ( x ) - \\mathfrak { K } u _ t ( x ) | \\\\ & \\leq N \\| u ^ h _ t - u _ t \\| _ { H ^ { m - 3 } } , \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{align*} \\psi ( 4 N c , d ) = \\exp \\left ( - 2 \\pi i \\hat v \\frac { c d } { \\hat h } \\right ) . \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} [ D ( 2 , 2 n + 1 ) : D ( 3 , s ) ] _ q & = q ^ { 4 n + 2 - 2 s } [ D ( 2 , 2 n - 2 ) : D ( 3 , s - 3 ) ] _ q & \\\\ & + q ^ { 4 n } [ D ( 2 , 2 n - 1 ) : D ( 3 , s ) ] _ q , \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} \\mu ( \\mathcal { B } ) = \\prod _ { j ' \\in \\mathcal { J } _ n } p ^ n _ { V | U } \\big ( \\mathbf { v } ( j ' ) \\big | \\mathbf { u } \\big ) . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} \\Delta _ n ( f , \\widetilde { f } ) = n [ P _ n ( T \\widetilde { f } | | f ^ { \\star } ) - P _ n ( f | | f ^ { \\star } ) ] - ( n / \\tau ) [ P ^ { \\prime } _ n ( T \\widetilde { f } | | f ^ { \\star } ) - P ^ { \\prime } _ n ( T f | | f ^ { \\star } ) ] . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} Y ^ \\varepsilon ( t , 0 ) = \\begin{cases} X ( t ) \\ \\ \\ & t \\in [ 0 , t _ 0 ] , \\\\ Y ^ \\varepsilon ( t - t _ 0 , X ( t _ 0 ) ) \\ \\ \\ & t \\in [ t _ 0 , \\tau ^ \\varepsilon ] . \\end{cases} \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} & 4 ( s - 1 ) P = 4 ( s - 1 ) \\frac { F _ s } { | x - y | ^ 2 } \\frac { \\rho ^ { s - 1 } } { ( \\rho + 1 ) ^ { \\frac { N } { 2 } - 1 } } + 2 V _ s ( \\rho ) \\nabla F _ s \\cdot \\nabla \\rho + F _ s V _ s ' ( \\rho ) | \\nabla \\rho | ^ 2 + F _ s V _ s ( \\rho ) \\Delta \\rho \\\\ & = F _ s \\Big [ \\frac { V _ s ( \\rho ) ( 4 ( s - 1 ) ( \\rho + 1 ) + 2 ( 2 s - N ) ( x - y ) \\cdot \\nabla \\rho + | x - y | ^ 2 \\Delta \\rho ) } { | x - y | ^ 2 } + V _ s ' ( \\rho ) | \\nabla \\rho | ^ 2 \\Big ] . \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} & \\sum _ { b = 1 } ^ { j - i + 1 } \\sum _ { k : M _ k \\in A _ { j + 1 } ^ { i + b } } \\lambda _ { k } | A - M _ { k } | \\lesssim \\sum _ { b = 1 } ^ { j - i + 1 } \\sum _ { k : M _ k \\in A _ { j + 1 } ^ { i + b } } \\lambda _ { k } ( i + b ) = \\sum _ { b = 1 } ^ { j - i + 1 } \\nu ( A _ { j + 1 } ^ { i + b } ) ( i + b ) \\\\ & \\lesssim \\sum _ { b = 1 } ^ { j - i + 1 } j ^ { 2 ( m _ 1 ' + m _ 2 ' - n ' ) } \\frac { i ^ { m _ 1 ' } } { ( i + b ) ^ { m _ 1 ' + 1 } } \\lesssim j ^ { - 2 } . \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} \\tau ^ S = \\bigcap _ { { \\rm p a r a m e t e r \\ i d e a l s \\ } I \\ { \\rm i n \\ } S } ( I : I ^ * ) \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} \\tilde { c } _ 4 = 1 + \\beta _ 2 2 ^ 2 + \\beta _ 3 2 ^ 3 + \\beta _ 4 2 ^ 4 + s ' 2 ^ 5 , \\beta _ i \\in \\{ 0 , 1 \\} , s ' \\in \\Z _ 2 \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} l _ { 2 j , 2 N + 1 } ( z ) & = \\frac { ( z - e _ { 2 N } ) ( z - e _ { 2 j + 1 } ) } { ( e _ { 2 j } - e _ { 2 N } ) ( e _ { 2 j } - e _ { 2 j + 1 } ) } \\prod _ { i = 0 , i \\not = j } ^ { N - 1 } \\frac { ( z ^ 2 - e _ i ^ 2 ) } { ( e _ j - e _ i ) } \\\\ & = \\frac { ( z - e _ { 2 N } ) ( z + e _ { 2 j } ) ( e _ { 2 j } + e _ { 2 N } ) } { 2 e _ { 2 j } ( e _ j - e _ N ) } l _ { j , N } ( z ^ 2 ) . \\\\ \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } u \\ , ( - \\Delta ) ^ s \\varphi \\ d x = \\int _ { \\R ^ N } - \\Delta u \\ , ( - \\Delta ) ^ { s - 1 } \\varphi \\ d x \\varphi \\in C ^ \\infty _ c ( \\R ^ N ) . \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} C _ { 2 k + 2 } ^ { k - r + 1 } = C _ { 2 k } ^ { k - r - 1 } + 2 C _ { 2 k } ^ { k - r } + C _ { 2 k } ^ { k - r + 1 } , \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} \\bar { z } : = x - \\sum _ { k \\in J } \\xi _ k e _ k , \\ \\xi _ k : = \\left \\{ \\begin{array} { l l } x _ k - \\delta _ k \\eta _ k & \\ \\delta _ k x _ k \\geq \\eta _ k , \\\\ 0 & . \\end{array} \\right . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} \\varphi ( q ^ { a _ 1 } ) \\varphi ( q ^ { a _ 2 } ) \\varphi ( q ^ { a _ 3 } ) \\varphi ( q ^ { a _ 4 } ) = & x _ 1 ( L ( q ) - 2 L ( q ^ 2 ) ) + x _ 2 ( L ( q ) - 4 L ( q ^ 4 ) ) \\\\ & + x _ 3 ( L ( q ) - 5 L ( q ^ 5 ) ) + x _ 4 ( L ( q ) - 8 L ( q ^ 8 ) ) \\\\ & + x _ 5 ( L ( q ) - 1 0 L ( q ^ { 1 0 } ) ) + x _ 6 ( L ( q ) - 2 0 L ( q ^ { 2 0 } ) ) \\\\ & + x _ 7 ( L ( q ) - 4 0 L ( q ^ { 4 0 } ) ) + y _ 1 A _ 1 ( q ) + y _ 2 A _ 2 ( q ) + y _ 3 A _ 3 ( q ) . \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} x _ 1 = \\log \\frac { 1 } { 1 - v _ 1 } , x _ 2 = \\log \\frac { v _ 2 } { v _ 1 } . \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} 2 \\bar { \\partial } ^ * \\bar { \\partial } X & = \\nabla ^ * \\nabla X - { \\rm R i c } \\ , X , \\\\ ( d d ^ * + d ^ * d ) \\psi & = \\nabla ^ * \\nabla \\psi + { \\rm R i c } \\ , \\psi . \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} A ( x ) = \\sum _ { \\substack { n \\le x \\\\ ( n , h ) = 1 } } \\psi ( n ) \\theta ( n ) a ( n ) \\ . \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\operatorname { D i v } P = \\operatorname { D i v } R + Q ^ { \\alpha } F _ { \\alpha } . \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} \\| \\tilde \\beta _ \\infty \\| _ { 1 , 2 } = 1 , \\ ; \\ , \\| \\tilde \\beta _ \\infty \\| _ { 0 , 1 } \\leq \\lambda ^ { d - 2 } , \\ ; \\ , \\| \\tilde \\beta _ \\infty \\| _ { 2 , 3 } \\leq \\lambda ^ { - ( d - 2 ) } . \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} a [ \\psi ] = \\| \\psi ' \\| ^ 2 + \\int _ \\R x ^ 2 | \\psi ( x ) | ^ 2 \\ , \\dd x . \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} X = Q ^ { \\alpha } _ n ( n , u _ n ) \\frac { \\partial } { \\partial u _ n ^ { \\alpha } } , \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} d X _ t & = \\left ( X _ t - \\left | X _ t \\right | ^ { 2 } X _ t \\right ) d t + \\sigma \\ , d W _ t \\textrm { o n } \\ ; \\mathbb { R } ^ n . \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\psi _ p ( s ) \\cdot ( 2 \\pi ) ^ { 2 g ( 1 - s ) } = \\lambda _ p ( 2 s - 1 ) , \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} \\| \\mathcal { L } _ { q , \\alpha } - \\mathcal { L } _ { p , \\alpha } \\| _ { \\mathcal { B } ( \\mathcal { E } _ { \\alpha } ) } & = \\sup _ { x \\in \\mathbb { R } } | \\partial _ { Y } R ( Y _ 0 ( x - q ) ) - \\partial _ { Y } R ( Y _ 0 ( x - p ) ) | \\leq C | q - p | , \\\\ \\| \\mathcal { L } _ { q } - \\mathcal { L } _ { p } \\| _ { \\mathcal { B } ( \\mathcal { E } _ { 0 } ) } & = \\sup _ { x \\in \\mathbb { R } } | \\partial _ { Y } R ( Y _ 0 ( x - q ) ) - \\partial _ { Y } R ( Y _ 0 ( x - p ) ) | \\leq C | q - p | , \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l } \\beta u _ t - \\kappa u _ { t t } - u _ { x x } & = & u ( r _ u - \\gamma _ u ( u + v ) ) + \\mu v - \\mu u \\R ^ 2 \\\\ \\beta v _ t - \\kappa v _ { t t } - v _ { x x } & = & v ( r _ v - \\gamma _ v ( u + v ) ) + \\mu u - \\mu v \\R ^ 2 , \\end{array} \\right . \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} I = \\left \\{ \\begin{array} { l l } ( v / 2 , s / 2 ) & v < s \\\\ ( s / 2 , v / 2 ) & s < v \\end{array} \\right . . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} \\varphi _ { \\alpha } ( t , u ) = \\sup _ { \\tau \\geq t } e ^ { - \\alpha ( \\tau - t ) } \\parallel U ( \\tau , t ) u ( t ) \\parallel . \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} \\Lambda _ { 2 N , 2 } = \\Lambda _ { N , 2 } , \\ \\ \\ N \\ge 1 \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ N \\frac { \\omega _ i E _ N ^ 2 ( \\tau _ i ) } { 1 - \\tau _ i ^ 2 } \\right ) ^ { 1 / 2 } \\le c \\left [ \\left ( \\int _ \\Omega \\frac { E _ N ^ 2 ( \\tau ) } { 1 - \\tau ^ 2 } \\ ; d \\tau \\right ) ^ { 1 / 2 } + N ^ { - 1 } | E _ N | _ { \\C { H } ^ 1 ( \\Omega ) } \\right ] . \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{align*} M ( v _ 1 v _ 2 , \\ldots , v _ i v _ { i + 1 } ) = M ( v _ { i + 1 } v _ { i + 2 } , \\ldots , v _ { 2 i } v _ { 2 i + 1 } ) . \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} \\eqref { e q : s u b r i n g r e c n } n = - 1 & \\iff r _ 2 = r _ 1 - 1 \\\\ \\eqref { e q : s u b r i n g r e c n } n = - 2 & \\iff r _ 1 = 1 . \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} \\forall _ { x _ 1 \\in \\SS _ 1 } \\ldots \\forall _ { x _ n \\in \\SS _ n } \\exists _ { y \\in \\SS } \\bigwedge ^ n _ { i = 1 } d _ i ( \\pi _ i ( y ) , x _ i ) \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} a _ n = i ^ n f _ n , n \\in \\mathbb { N } . \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{gather*} \\delta _ Q \\omega = d \\omega _ 1 , i _ Q \\omega = \\delta L + d \\theta _ 1 , \\end{gather*}"}
-{"id": "5151.png", "formula": "\\begin{align*} \\tau ( Q \\cap T ) V \\supset \\tau ( Q \\cap F _ n g _ n ) \\tau ( g _ n ) ^ { - 1 } t ^ { - 1 } = \\tau ( Q g ^ { - 1 } \\cap F _ n ) t ^ { - 1 } \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} u _ j : = v _ j = \\sqrt { I _ j } \\ , e ^ { \\mathrm { i } \\theta _ j } , I _ j = I _ { - j } , \\theta _ { - j } = - \\theta _ j j \\in S , \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( - \\Delta ) ^ s \\overline { \\xi _ 0 } & = 1 & & \\mbox { i n } \\Omega , \\\\ \\overline { \\xi _ 0 } & = 0 & & \\mbox { i n } \\Sigma _ 1 , \\\\ \\mathcal { N } _ s \\overline { \\xi _ 0 } & = 0 & & \\mbox { i n } \\Sigma _ 2 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} \\theta ( m + 1 , 0 ) ( u ) & = \\theta ( m , 0 ) ( \\theta ( 1 , 0 ) ( u ) ) = \\theta ( m , 0 ) ( B u B ^ { - 1 } ) = B ^ { m + 1 } u B ^ { - ( m + 1 ) } , \\\\ \\theta ( m + 1 , 0 ) ( v ) & = \\theta ( m , 0 ) ( \\theta ( 1 , 0 ) ( v ) ) = \\theta ( m , 0 ) ( B v u ^ { - 2 } B ^ { - 1 } ) = B ^ { m + 1 } v u ^ { - 2 ( m + 1 ) } B ^ { - ( m + 1 ) } , \\ ; \\\\ \\theta ( m + 1 , 0 ) ( B ) & = \\theta ( m , 0 ) ( \\theta ( 1 , 0 ) ( B ) ) = \\theta ( m , 0 ) ( B ) = B , \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} S _ k ( x ) = & \\frac { x } { \\log x } \\frac { ( \\log \\log x ) ^ k } { ( k - 1 ) ! } \\bigg \\{ 1 + \\frac { k - 1 } { \\log \\log x } g ' \\ ( 0 \\ ) + \\frac { ( k - 1 ) ( k - 2 ) } { ( \\log \\log x ) ^ 2 } \\widetilde { g } \\ ( \\frac { k - 3 } { \\log \\log x } \\ ) \\\\ & + O _ { A , q } \\ ( \\frac { k ^ 3 } { ( \\log \\log x ) ^ 4 } \\ ) \\bigg \\} , \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} G ( c , 1 , p ) = \\sqrt { \\psi _ 1 ( c , p ) + \\frac { 1 } { 2 } \\left ( c ^ 2 - 4 \\right ) \\left ( c ^ 3 + 2 c + 6 \\right ) \\left ( 6 c p ^ 2 - 2 c p - 2 p - 3 c \\right ) } \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} ( \\Box ^ { ( q ) } _ b u | u ) = \\| u \\| _ { \\overline S } ^ 2 - q i ( \\nabla _ { T } u | u ) + ( R _ \\ast u | u ) \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} \\frac { d } { d s _ 1 } \\mathrm { s n } ( x , y ) = \\frac { d } { d s } \\left ( \\frac { [ \\gamma _ { \\partial B } ( s ) , y ] } { | | y | | _ a } \\right ) = \\frac { [ b ( x ) , y ] } { | | y | | _ a | | b ( x ) | | } = \\mathrm { s n } ( b ( x ) , y ) \\ \\mathrm { a n d } \\\\ \\frac { d } { d s _ 2 } \\mathrm { c m } ( x , y ) = \\frac { d } { d s } \\left ( [ \\gamma _ { \\partial B } ( s ) , b ( x ) ] \\right ) = \\frac { [ b ( y ) , b ( x ) ] } { | | b ( y ) | | } = \\mathrm { c m } ( x , b ( y ) ) . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( - \\Delta ) ^ s \\overline { \\chi } _ 1 & = \\lambda _ 1 \\overline { \\chi } _ 1 & & \\mbox { i n } \\Omega , \\\\ \\overline { \\chi } _ 1 & = 0 & & \\mbox { i n } \\Sigma _ 1 , \\\\ \\mathcal { N } _ s \\overline { \\chi } _ 1 & = 0 & & \\mbox { i n } \\Sigma _ 2 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} \\gamma _ 5 \\left ( \\alpha \\cdot D + m \\beta \\right ) \\gamma _ 5 = \\alpha \\cdot D - m \\beta \\ , , \\gamma _ 5 \\mathcal { B } \\gamma _ 5 = - \\mathcal { B } \\ , , \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} \\sigma = \\sigma ( \\Gamma , k ) = - \\dfrac { 1 } { 4 } ( | \\Gamma | + k ^ 2 ) , \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\ell _ n ( G _ k ) \\geq \\log ( k ) - \\frac { 1 } { 2 } \\sum _ { i = 2 } ^ n x _ i ^ 2 - n \\log ( 2 \\pi ) . \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\mu _ { \\mathcal { U } ^ + } ( x ) = \\big ( \\mu _ U ( x ) \\big ) _ { U \\in { \\mathcal { U } ^ + } } . \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} \\Delta \\left ( h ( f _ 0 ) ; f \\right ) = \\left \\| { A } \\xi - \\hat { A } \\xi \\right \\| _ \\alpha ^ \\alpha = \\int _ { - \\pi } ^ { \\pi } \\left | \\left ( C ^ 0 ( e ^ { i \\theta } ) \\right ) ^ { < \\frac { 1 } { \\alpha - 1 } > } ( f _ 0 ( \\theta ) ) ^ { \\frac { - 1 } { \\alpha - 1 } } \\right | ^ { \\alpha } f ( \\theta ) d \\theta . \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{gather*} L = \\frac 1 2 F \\wedge \\tilde { F } + A ^ \\ast \\wedge d C , i _ Q \\omega \\simeq \\delta L . \\end{gather*}"}
-{"id": "9340.png", "formula": "\\begin{align*} ( H ( \\theta ) u ) _ n = u _ { n + 1 } + u _ { n - 1 } + v ( \\theta + n \\alpha ) u _ n . \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { F _ 2 ( y ^ G _ 1 , y ^ G _ 2 ) } \\ ! \\ ! \\ ! p _ { Y _ 1 , Y _ 2 } ( F _ 1 ( y ^ G _ 1 ) , \\ , \\cdot \\ , ) \\ ; \\frac { \\partial F _ 1 } { \\partial y ^ G _ 1 } ( y ^ G _ 1 ) = \\int _ { - \\infty } ^ { y ^ G _ 2 } \\ ! \\ ! p _ { Y ^ G _ 1 , Y ^ G _ 2 } ( y ^ G _ 1 , \\ , \\cdot \\ , ) \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} \\Big ( \\Gamma \\big ( 1 + \\frac { 1 } { 2 g ( { \\bf R } ) } \\big ) \\Big ) ^ { - g ( { \\bf R } ) } \\leq \\Big ( \\frac { 1 } { 2 e g ( { \\bf R } ) } \\Big ) ^ { - \\frac { 1 } { 2 } } = \\Big ( \\frac { 2 e d g ( { \\bf R } ) } { d } \\Big ) ^ { \\frac { 1 } { 2 } } , \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} P _ { { t } } = \\sum _ { n = 1 } ^ { N } \\frac { p _ { n } } { \\xi } + \\sum _ { n = 1 } ^ { N } \\sum _ { k = 1 } ^ { K } x _ k \\frac { p _ { k , n } } { \\xi } + \\sum _ { k = 1 } ^ { K } x _ k \\frac { q _ { k } } { \\xi } , \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} \\tau _ b ( z _ { n _ k } ) ( y ^ * ) & = d ( y ^ * , z _ { n _ k } ) - d ( z _ { n _ k } , b ) \\\\ & \\leq d ( y ^ * , y _ { n _ k } ) + d ( y _ { n _ k } , z _ { n _ k } ) - d ( z _ { n _ k } , b ) \\\\ & < \\epsilon - d ( y _ { n _ k } , b ) = \\epsilon - r , \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u - \\frac { \\lambda } { | x | ^ 2 } u = u ^ { 2 ^ * - 1 } + \\nu \\alpha u ^ { \\alpha - 1 } v ^ { \\beta } , \\\\ - \\Delta v - \\frac { \\lambda } { | x | ^ 2 } v = v ^ { 2 ^ * - 1 } + \\nu \\alpha u ^ { \\alpha } v ^ { \\beta - 1 } . \\end{cases} \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} M _ k ( x ; \\mathbf { a } ) = & \\frac { 1 } { \\phi ^ k ( q ) } \\frac { x } { \\log x } \\frac { ( \\log \\log x ) ^ { k - 1 } } { ( k - 1 ) ! } \\bigg \\{ 1 + \\frac { k - 1 } { \\log \\log x } C _ { a , q } \\\\ & + \\frac { ( k - 1 ) ( k - 2 ) \\phi ^ 2 ( q ) } { ( \\log \\log x ) ^ 2 } \\widetilde { h } \\ ( a ; \\frac { ( k - 3 ) \\phi ( q ) } { \\log \\log x } \\ ) + O _ { A , q } \\ ( \\frac { k ^ 3 } { ( \\log \\log x ) ^ 4 } \\ ) \\bigg \\} , \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} G ( s ) = L ( s ) + L ' ( s ) / \\log { N } \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} & \\left \\{ \\lim _ { m \\rightarrow \\infty } \\textrm { d i a m } \\left ( \\varphi _ { t _ m } ( \\cdot , V ( \\cdot ) ) \\right ) = 0 \\right \\} \\\\ & \\subset \\left \\{ \\lim _ { m \\rightarrow \\infty } \\textrm { d i a m } \\left ( \\varphi _ { t _ m } \\left ( \\cdot , B \\left ( x , \\frac { 1 } { k } \\right ) \\right ) \\right ) = 0 \\textrm { f o r s o m e } k \\in \\mathbb { N } \\right \\} . \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\sigma _ { \\omega , \\gamma } ( n ) : = \\begin{cases} n ^ { - \\omega - \\gamma + 1 } \\log e n , & \\omega \\leq 1 , \\\\ n ^ { - \\gamma } , & \\omega > 1 . \\end{cases} \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\mathrm { d } \\bar { X } ( t ) = b \\big ( R ( t , \\bar { X } ( t ) ) \\big ) \\ , \\mathrm { d } t + \\sigma \\big ( R ( t , \\bar { X } ( t ) ) \\big ) \\ , \\mathrm { d } B ( t ) \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} \\frac { \\tilde { \\beta } _ { t + \\tau } } { \\tilde { \\beta } _ { t + \\tau ' } } - \\frac { \\tilde { \\beta } _ { t ' + \\tau } } { \\tilde { \\beta } _ { t ' + \\tau ' } } = 0 . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} N ( \\nu _ k ) = \\frac 1 8 T ( \\nu _ k ) - \\frac 3 8 T _ { x _ 1 } ( \\nu _ k ) + \\frac 3 4 \\left \\lfloor \\frac { \\nu _ k ^ { 1 / 2 } } { \\pi } \\right \\rfloor + \\frac 1 4 . \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} \\varphi ( q ) = \\frac { \\eta ^ 5 ( 2 z ) } { \\eta ^ 2 ( z ) \\eta ^ 2 ( 4 z ) } . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} h ( g _ 1 ) - h ( g _ 2 ) = 0 + \\frac { C } { r } o ( r ) \\xrightarrow { r \\to \\infty } 0 \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} ( \\bold { D } _ { \\bold { E } ( L ) } ) ^ { \\ast } = \\bold { D } _ { \\bold { E } ( L ) } . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} m ( \\ell _ { 1 } , \\ell _ { 2 } , [ \\Pi ] , [ Q ] ) = \\left [ \\begin{array} { c c } \\ell _ { 1 } & - m _ { 2 } \\\\ \\ell _ { 2 } & m _ { 1 } \\end{array} \\right ] . \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{align*} \\Sigma _ { j = 1 } ^ n n ( \\mu , 2 j ) = \\Sigma _ { a = 1 } ^ r a \\# C _ a . \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} C _ \\mathsf { L N } ^ \\mathrm { E n c - D e c - C S I } = \\max _ { P _ { X | S } } \\min \\Big \\{ I ( X ; Y | S ) , I ( X ; Y | S ) - I ( X ; Z | S ) + H ( S | Z ) \\Big \\} , \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} \\prod _ { i = 0 } ^ { \\infty } x ^ { b ^ i k _ i } f ( k _ i , i ) = x ^ { \\sum _ { i = 0 } ^ { \\infty } b ^ i k _ i } \\prod _ { i = 0 } ^ { \\infty } f ( k _ i , i ) = x ^ k \\prod _ { i = 0 } ^ { \\infty } f ( k _ i , i ) \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{gather*} u ^ 2 + \\frac { \\omega } { u ^ 2 } = \\left ( \\frac { u _ t } { u } \\right ) _ t . \\end{gather*}"}
-{"id": "7937.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } [ x + y ] _ q ^ n d \\mu _ { q } ( y ) = \\beta _ { n , q } ( x ) , ( n \\geq 0 ) . \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} M ( b ; b ; z ) = e ^ { z } . \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} \\sum _ { \\ell \\le X } \\phi ( \\ell ) = P _ r ( \\log { x } ) + O ( X ^ { - 1 / 5 r } ) \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} ( s - 1 ) ( | x - y | ^ 2 - | x | ^ 2 | y | ^ 2 ( | x | ^ 2 - 2 x \\cdot y + | y | ^ 2 ) ) = ( s - 1 ) | x - y | ^ 2 ( 1 - | x | ^ 2 | y | ^ 2 ) . \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} \\eta _ { \\tau } ( 0 ) = \\eta ( 0 ) \\quad \\forall \\ , \\tau . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} W _ { + } ( z ) = W _ { - } ( z ) \\begin{bmatrix} 1 & - \\zeta ( z ) ^ { - c } e ^ { \\zeta ( z ) } \\\\ 0 & 1 \\end{bmatrix} , z \\in \\partial U \\cap D _ \\beta . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} \\mathbf { G } = \\begin{bmatrix} \\mathbf { G } _ 0 & \\mathbf { G } _ 1 & \\cdots & \\mathbf { G } _ K \\end{bmatrix} , \\normalsize \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} & \\alpha = \\alpha ( \\rho ) = 1 - \\exp ( - \\rho ^ { - 1 } ) , & & \\beta = \\beta ( \\rho ) = \\frac { 1 - \\exp ( - 1 - \\rho ^ { - 1 } ) } { 1 - \\exp ( - \\rho ^ { - 1 } ) } > 1 , \\\\ & N = N ( \\rho ) = \\lfloor ( 1 + \\rho ^ { - 1 } ) \\exp ( 1 + \\rho ^ { - 1 } ) \\rfloor , & & \\sigma = \\sigma ( \\rho ) = \\sum _ { i = 1 } ^ N \\frac { \\alpha ^ i } { i } . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} \\nabla ( A ) = A - Z A Z ^ * , Z = \\begin{bmatrix} 0 & \\\\ 1 & 0 & \\\\ & \\ddots & \\ddots \\\\ & & 1 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} \\phi _ 2 ( t , q ) = \\prod _ { n = 1 } ^ { \\infty } \\left ( 1 + \\frac { t ^ { a ( n ) } q ^ n } { 1 - q ^ n } \\right ) \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} \\tilde { } : W ( K ) / \\wp W ( K ) \\to & R \\\\ c \\beta + \\sum _ { ( i , p ) = 1 } c _ i [ T ] ^ { - i } \\pmod { \\wp W ( K ) } \\mapsto & c \\beta + \\sum _ { ( i , p ) = 1 } c _ i T ^ { - i } . \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} \\prod _ { j = 0 } ^ { k - 1 } \\vert e _ k - e _ j \\vert = \\max _ { z \\in K } \\prod _ { j = 0 } ^ { k - 1 } \\vert z - e _ j \\vert . \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} u = \\bar { u } + \\sum _ { \\ell = 1 } ^ L \\hat \\chi _ \\ell \\left ( \\frac { f _ \\ell } { 2 m } r ^ 2 + \\sum _ { i = 0 } ^ { I _ \\ell } u _ { \\ell , i } r ^ { \\mu _ { \\ell , i } ^ + } \\phi _ { \\ell , i } \\right ) \\circ \\Phi _ \\ell ^ { - 1 } , \\ ; \\ , u _ { \\ell , i } \\in \\R , \\\\ \\| \\bar { u } \\| _ { C ^ { k + 2 , \\alpha } _ { \\hat { \\nu } + 2 } ( M ) } + \\sum _ { \\ell = 1 } ^ L \\sum _ { i = 0 } ^ { I _ \\ell } | u _ { \\ell , i } | \\leq c \\left ( \\| \\bar { f } \\| _ { C ^ { k , \\alpha } _ \\nu ( M ) } + \\sum _ { \\ell = 1 } ^ L | f _ \\ell | \\right ) . \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\hat { f } ( \\xi ) : = c _ { n } \\int _ { \\R ^ { n } } e ^ { - i x \\cdot \\xi } f ( x ) d x \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } u ( x ) ( - \\Delta ) ^ s \\varphi ( x ) \\ d x = \\int _ { \\R ^ N } ( - \\Delta ) ^ \\sigma u ( x ) ( - \\Delta ) ^ { m } \\varphi ( x ) \\ d x = \\int _ { \\R ^ N } ( - \\Delta ) ^ { m } ( - \\Delta ) ^ \\sigma u ( x ) \\varphi ( x ) \\ d x . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} \\sigma _ A ( x , \\xi ) = \\xi ( x ) ^ { * } ( A \\xi ) ( x ) , \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} \\frac { d } { d \\tau } E [ g ( W , \\gamma ( F _ { \\tau } ) , \\theta ) ] = \\int \\phi ( w , \\gamma _ { 0 } , \\alpha _ { 0 } , \\theta ) H ( d w ) , E [ \\phi ( W , \\gamma _ { 0 } , \\alpha _ { 0 } , \\theta ) ] = 0 . \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} \\widetilde { \\gamma } _ j ^ m = { 2 ^ { \\widetilde { R } _ j ^ m } } - 1 . \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} B ^ s _ { \\tau } ( L _ \\tau ( \\Omega ) ) , \\frac { 1 } { \\tau } = \\frac { s } { d } + \\frac { 1 } { 2 } , s > 0 , \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} \\psi ( u ) = \\psi ( u _ { 1 , 2 } + \\cdots + u _ { n - 1 , n } ) . \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} k ( t , s ) = \\begin{cases} \\dfrac { \\cos [ \\omega ( \\tfrac { 1 } { 2 } - t + s ) ] } { 2 \\omega \\sin ( \\tfrac { 1 } { 2 } \\omega ) } , , \\\\ [ 5 m m ] \\dfrac { \\cos [ \\omega ( \\tfrac { 1 } { 2 } - s + t ) ] } { 2 \\omega \\sin ( \\tfrac { 1 } { 2 } \\omega ) } , , \\end{cases} \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} W _ { k } ( n ) = ( W _ { k } ( n - 1 ) + J _ { k } ( n - 1 ) - I ( n ) ) ^ + , \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} \\| f \\| _ 1 = \\left | \\xi ( f ) \\right | \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} s ( \\gamma + s \\gamma ^ { \\dagger } ) \\ = \\ \\gamma ^ { \\dagger } \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} \\tau ^ { 2 f } = 1 , \\sigma ^ 4 = 1 , \\tau \\sigma \\tau ^ { - 1 } = \\sigma ^ k \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} \\eta _ { \\mu } = - h ^ { 2 } t + \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { s } \\frac { 1 } { \\phi _ { \\mu } ^ { 2 } } d \\tau d s , \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\lim _ { m \\rightarrow \\infty } \\textrm { d i a m } \\left ( \\varphi _ { t _ m } ( \\cdot , S ( \\cdot , x ) ) \\right ) = 0 \\right ) > 0 . \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} H _ 1 = - \\frac { d ^ 2 } { d x ^ 2 } + V _ 1 ( x ) \\ ; , \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} \\sigma _ { X \\times C } ( x , y ) = ( X \\times C ) _ { ( x , y ) } = C _ y = \\sigma _ C ( y ) . \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} \\left ( \\sum _ { k \\in \\mathbb { Z } ^ d } \\left | \\langle f , \\widetilde { \\phi } _ k \\rangle \\right | ^ p \\right ) ^ { \\frac { 1 } { p } } + \\left ( \\sum _ { i = 1 } ^ { 2 ^ { d } - 1 } \\sum _ { j \\in \\mathbb { N } _ 0 } 2 ^ { j \\left ( s + d \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { p } \\right ) \\right ) q } \\left ( \\sum _ { k \\in \\mathbb { Z } ^ d } \\left | \\langle f , \\widetilde { \\psi } _ { i , j , k } \\rangle \\right | ^ p \\right ) ^ { \\frac { q } { p } } \\right ) ^ { \\frac { 1 } { q } } < \\infty , \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} F ( u ) = \\int _ { M } u d \\mu + \\int _ { M ^ * } \\left [ ( u + \\phi _ 0 ) ^ * - \\phi _ 0 ^ * \\right ] d \\nu \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{align*} \\gamma \\alpha ^ k - \\zeta ^ m \\delta \\alpha ^ { u m / v } = O ( | \\alpha | ^ { k ( 1 - \\varepsilon ) } ) + O ( | \\alpha | ^ { u m ( 1 - \\varepsilon ) / v } ) . \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\bold { p r } \\widetilde { X } ( \\widetilde { L } _ n ) - ( S - \\operatorname { i d } ) P _ 0 = Q ^ { \\alpha } ( n , \\widetilde { u } ) \\widetilde { F } _ { \\alpha } + ( S - \\operatorname { i d } ) \\widetilde { P } . \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} \\sum _ { \\pi \\in \\P } ( - 1 ) ^ { \\nu ( \\pi ) } q ^ { | \\pi | } = \\frac { ( q ^ 2 ; q ^ 2 ) _ \\infty } { ( - q ; q ^ 2 ) _ \\infty } - 1 = \\psi ( - q ) - 1 . \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} \\aligned \\det ( F ) = & \\Big \\{ ( p - 1 ) ^ 2 + ( \\nu ^ 2 \\alpha \\beta ( \\frac { \\alpha } { 2 } - 1 ) ( \\frac { \\beta } { 2 } - 1 ) - \\frac { 1 } { 4 } \\nu ^ 2 \\alpha ^ 2 \\beta ^ 2 ) ( \\frac { x _ 1 } { x _ 2 } ) ^ { \\alpha - p } \\\\ & + ( p - 1 ) \\nu \\beta ( \\frac { \\beta } { 2 } - 1 ) ( \\frac { x _ 1 } { x _ 2 } ) ^ { \\frac { \\alpha } { 2 } } + ( p - 1 ) \\nu \\alpha ( \\frac { \\alpha } { 2 } - 1 ) ( \\frac { x _ 2 } { x _ 1 } ) ^ { p - \\frac { \\alpha } { 2 } } \\Big \\} x _ 1 ^ { p - 2 } x _ 2 ^ { p - 2 } . \\endaligned \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} \\sigma _ { \\chi , \\psi } ( n ) : = \\sum _ { 1 \\leq m | n } \\psi ( m ) \\chi ( n / m ) m . \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} L ( x , n , [ u ] ) = \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 , \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} - ( - \\Delta | _ \\Omega ) ^ { \\alpha / 2 } \\varphi ( x ) : = c _ { d , \\alpha } \\mbox { P . V . } \\int _ { \\R ^ d } \\frac { \\varphi ( y ) { \\rm \\bf 1 } _ \\Omega ( y ) - \\varphi ( x ) } { | x - y | ^ { d + \\alpha } } \\d y \\ , , c _ { d , \\alpha } > 0 \\ , . \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\underline { \\varepsilon } | \\underline { X } } \\exp \\left \\{ \\frac { 1 } { \\gamma } \\sum _ { i = 1 } ^ n \\varepsilon _ i g ( X _ i ) \\right \\} & \\leq \\exp \\left \\{ \\frac { \\sigma ^ 2 \\sum _ { i = 1 } ^ n g ^ 2 ( X _ i ) } { 2 \\gamma ^ 2 ( 1 - \\eta K / \\gamma ) } \\right \\} \\\\ & = \\exp \\left \\{ \\frac { 1 } { \\gamma A } \\sum _ { i = 1 } ^ n g ^ 2 ( X _ i ) \\right \\} , \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} \\bar f _ 2 ( B _ n ) = \\left \\{ \\begin{array} { l l } f _ 2 ( B _ n ) , & { \\rm i f } ~ B _ n \\in [ 0 , \\tau _ 1 ] \\cup [ \\tau _ 2 , + \\infty ) \\\\ c B _ n + d , & { \\rm i f } ~ B _ n \\in ( \\tau _ 1 , \\tau _ 2 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\mathsf { s p } ( H ) = \\{ \\pm \\mu _ { n } ( m ) , \\ ; n \\geq 1 \\} \\ , . \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} V _ 1 = X _ { T _ i } ^ { - 1 } X _ \\tau ^ { \\vphantom { - 1 } } V _ 2 = X _ \\tau ^ { - 1 } X _ { T _ { i + 1 } } ^ { \\vphantom { - 1 } } \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} \\sum _ { \\mathcal { A } \\in \\mathcal { F } } \\left ( \\frac { q } { q + 1 } \\right ) ^ { | \\mathcal { A } | } \\leqslant \\sum _ { \\mathcal { A } \\in \\mathcal { F } } 2 ^ { - | \\mathcal { A } | / q } \\leqslant 2 ^ { n - \\binom { n } { k } / q } < 1 , \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{gather*} u ( t ) = \\sqrt { - \\frac { t } { 2 } } \\left ( 1 - \\frac { 1 } { 8 } ( - t ) ^ { - 3 } - \\frac { 7 3 } { 1 2 8 } ( - t ) ^ { - 6 } - \\frac { 1 0 6 5 7 } { 1 0 2 4 } ( - t ) ^ { - 9 } \\right . \\\\ \\left . \\hphantom { u ( t ) = } { } - \\frac { 1 3 9 1 2 2 7 7 } { 3 2 7 6 8 } ( - t ) ^ { - 1 2 } - \\frac { 8 0 4 5 8 8 3 9 4 3 } { 2 6 2 1 4 4 } ( - t ) ^ { - 1 5 } - \\frac { 1 4 5 1 8 4 5 1 3 9 0 3 4 9 } { 4 1 9 4 3 0 4 } ( - t ) ^ { - 1 8 } + \\cdots \\right ) , \\end{gather*}"}
-{"id": "5199.png", "formula": "\\begin{align*} \\sum _ { j _ 1 + j _ 2 + j _ 3 = 0 } \\mathrm { i } \\ , ( j _ 1 ^ 3 + j _ 2 ^ 3 + j _ 3 ^ 3 ) \\ , F _ { j _ 1 j _ 2 j _ 3 } ^ { ( 3 ) } \\ , u _ { j _ 1 } \\ , u _ { j _ 2 } \\ , u _ { j _ 3 } = \\sum _ { ( j _ 1 , j _ 2 , j _ 3 ) \\in \\mathcal { A } _ 3 } ( - \\mathrm { i } \\ , c _ 1 \\ , j _ 1 j _ 2 j _ 3 - c _ 2 \\ , j _ 1 j _ 2 + c _ 3 ) \\ , u _ { j _ 1 } u _ { j _ 2 } u _ { j _ 3 } \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } D _ t ^ \\delta u ( x , t ) = 0 \\quad x \\in \\Omega . \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} Y ( X Y + 2 q _ 4 Z ^ 2 ) = X ^ 3 + p _ 2 X ^ 2 Z + p _ 4 X Z ^ 2 + p _ 6 Z ^ 3 . \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\P ( X _ N ^ { R , { \\rm r e s c } } \\leq U ) = \\frac { F _ { \\rm G U E } ( \\min \\{ U , R \\} ) } { F _ { \\rm G U E } ( R ) } , \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} \\frac { n _ 0 ^ { ( j ' ) } } { 2 } \\log k _ n ^ { ( j ) } = \\frac { \\epsilon ' } { 2 J } \\frac { n } { 2 } \\log k _ n ^ { ( j ) } . \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} ( \\Box ^ { ( 0 ) } _ { t , b , m } - T ^ 2 ) N _ { t , m } u = ( I - S _ { t , m } ) u + m ^ 2 N _ { t , m } u . \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} ( \\beta ^ * F ) _ i \\Omega C : = \\sum _ j F _ { i - j } \\cap \\beta ^ * _ j \\Omega C , \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} A = \\{ i _ 1 , i _ 2 , \\dots , i _ m \\} \\subset \\{ 1 , 2 , \\dots , m + n \\} . \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} \\mathbf { S } _ { D , o p t } ^ { ( g ) } = \\underset { \\mathbf { S } _ D ^ { ( g ) } } { \\operatorname { a r g m i n } } \\operatorname { T r } \\left \\{ \\left ( \\boldsymbol { \\mathcal { F } } ^ { ( g ) } + \\mathbf { I } _ { T D } \\right ) ^ { - 1 } \\right \\} \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} \\begin{cases} f _ 1 ( x _ 1 , x _ 2 ) : & = x _ 1 ^ { p - 1 } + \\nu \\alpha x _ 1 ^ { \\frac { \\alpha } { 2 } - 1 } x _ 2 ^ { \\frac { \\beta } { 2 } } = 1 , \\\\ f _ 2 ( x _ 1 , x _ 2 ) : & = x _ 2 ^ { p - 1 } + \\nu \\beta x _ 1 ^ { \\frac { \\alpha } { 2 } } x _ 2 ^ { \\frac { \\beta } { 2 } - 1 } = 1 , \\end{cases} \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} R = \\left \\{ \\sum _ { i \\in \\Z } a _ i T ^ i : a _ i \\in W ( k ) , \\ \\lim _ { i \\to - \\infty } a _ i = 0 \\right \\} = \\underset { \\underset { i } { \\leftarrow } } \\lim \\ W ( k ) / V ^ i W ( k ) ( ( T ) ) \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{gather*} d = d x ^ j \\wedge \\left ( \\frac { \\partial } { \\partial x ^ j } + \\phi ^ a _ { I j } \\frac { \\partial } { \\partial \\phi ^ a _ { I } } \\right ) , \\delta = \\delta \\phi _ I ^ a \\wedge \\frac { \\partial _ l } { \\partial \\phi ^ a _ I } . \\end{gather*}"}
-{"id": "5994.png", "formula": "\\begin{align*} t _ { \\tau , a } ^ { ( h ) } = \\tau ( \\zeta _ { a } ^ { ( h - 1 ) } ) t _ { \\tau , a } ^ { ( h - 1 ) } - \\frac { _ { q } K _ { + } ( \\xi _ { a } ^ { ( h - 3 / 2 ) } ) _ { q } \\mathcal { U } _ { - } ( \\xi _ { a } ^ { ( h - 3 / 2 ) } ) } { ( \\xi _ { a } ^ { ( h - 1 / 2 ) } ) ^ { 2 } - 1 / ( \\xi _ { a } ^ { ( h - 1 / 2 ) } ) ^ { 2 } } t _ { \\tau , a } ^ { ( h - 2 ) } h \\in \\left \\{ 1 , . . . , p - 1 \\right \\} \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} F ( c , r , - 1 ) = 6 ( 4 - c ^ 2 ) ( 1 - r ^ 2 ) + c | c ^ 2 + 4 - ( 2 r - 3 r ^ 2 ) ( 4 - c ^ 2 ) | . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} \\Vert u - u _ \\tau \\Vert _ { \\tilde U ^ p } ^ p \\le \\sum 2 \\Vert u _ j \\Vert _ { \\tilde U ^ p } ^ p . \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} \\mu = \\sum _ { i \\in \\Lambda } p ( i ) \\ , \\mu \\circ T _ i ^ { - 1 } . \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\pi _ 1 ( \\lambda , x , y ) : = ( \\lambda , x ) \\\\ \\pi _ 2 ( \\lambda , x , y ) : = ( \\lambda , y ) \\end{array} \\right \\} \\quad \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} p _ { Y _ 1 } ( F _ 1 ( y ^ G _ 1 ) ) \\ ; \\frac { \\partial F _ 1 } { \\partial y ^ G _ 1 } ( y ^ G _ 1 ) = p _ { Y ^ G _ 1 } ( y ^ G _ 1 ) \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} \\lambda _ { t o p } = \\lim _ { m \\rightarrow \\infty } \\frac { 1 } { m } \\log | D \\varphi _ m ( \\omega , x ) v | . \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\hat { h } ( \\Delta ) = \\begin{cases} 3 h ( \\Delta ) & \\Delta > 0 \\\\ h ( \\Delta ) & \\Delta < 0 , \\end{cases} \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} P \\psi ( x , y , z ) = T ^ \\star \\widetilde { P } T \\psi ( x , y , z ) = \\begin{cases} \\psi ( x , y , z ) , & \\mbox { i f } z > 0 , \\\\ \\left ( \\mathcal { B } \\circ \\Pi \\right ) \\left ( { \\psi } \\circ S \\right ) ( x , y , z ) , & \\mbox { i f } z < 0 . \\end{cases} \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} \\nu ( s - m - 1 , n - m - 1 ) = \\nu ( s - 1 , n - 1 ) - \\nu ( s - m - 1 , n + 2 r _ n - m - 1 ) , \\\\ \\nu ( s - 2 s _ 0 , n ) = \\nu ( s + m - 2 s _ 0 , n + m ) - \\nu ( s - 2 s _ 0 , n + 2 r _ n ) . \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\{ \\mathbf { c } ^ { ( g ) } \\left ( \\mathbf { c } ^ { ( g ) } \\right ) ^ H \\right \\} = \\mathbf { I } _ { K _ g \\left ( \\sum _ { l = 0 } ^ { L _ g - 1 } r _ { g , l } \\right ) } . \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{gather*} v = - \\frac { 3 \\zeta \\lambda + 9 \\lambda _ t } { \\lambda ^ 2 } . \\end{gather*}"}
-{"id": "4760.png", "formula": "\\begin{align*} \\tilde h ( u , t ) : = h \\left ( u + t \\left ( \\int _ 0 ^ 1 \\frac { 1 } { g ( s ) } d s \\right ) ^ { - 1 } , t \\right ) , \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} m _ j = \\exp ( \\sum _ { s \\ge 3 } \\frac { \\epsilon _ s } { 2 s } ( - \\hat x ) ^ s ) M _ j \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} \\lim _ { x _ 1 \\to 1 } 1 + \\partial _ 2 \\sigma ( x _ 1 , 1 ) + h _ 0 ' ( \\partial _ 2 \\sigma ( x _ 1 , 1 ) + 1 ) & = \\lim _ { x _ 2 \\to 1 } 1 + \\partial _ { 2 } \\sigma ( 1 , x _ 2 ) + h _ 0 ' ( \\partial _ { 2 } \\sigma ( 1 , x _ 2 ) + 1 ) \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} S ( x , z ) = \\sqrt { \\frac { x - 1 } { x e ^ { 2 z } - e ^ { 2 x z } } } . \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} & g _ { 2 / 3 , 3 / 4 } ^ 1 ( w _ 1 ) = \\\\ & \\frac { ( 1 + \\epsilon ) \\phi _ { \\ell } } { 4 } \\log \\left ( 1 + \\frac { w _ 1 P ' } { 4 } \\right ) - \\frac { | A ^ { \\ast } | } { k _ { \\ell } } H _ 2 \\left ( \\frac { w _ 1 } { | A ^ { \\ast } | } \\right ) . \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} e ^ { - \\frac { \\Delta } { c _ 1 } } & \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 } ^ { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { z ^ { ( 0 ) } } { v } } e ^ { - \\frac { v } { \\zeta } } d v \\\\ & \\qquad \\qquad = \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 } ^ { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { ( z ^ { ( 0 ) } + \\Delta ) } { v } } e ^ { - \\frac { v } { \\zeta } } d v . \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} D _ { x _ 0 } \\ell = \\sum _ { k = 1 } ^ { N } \\lambda _ k \\ , d e _ { k } ( x _ { 0 } ) . \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} \\mathcal { Q } : = \\underline { Q u o t } _ { X / S } : ( S c h / S ) ^ { o } \\rightarrow ( S e t s ) \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} y ( t ) = T _ q ( t ) y ( 0 ) + \\int _ 0 ^ t T _ q ( t - \\tau ) F _ q ( y ( \\tau ) ) \\dd \\tau , t \\geq 0 . \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{align*} u = v + z , v : = \\Pi _ S u : = \\sum _ { j \\in S } u _ j \\ , e ^ { \\mathrm { i } \\ , j \\ , x } , z = \\Pi _ S ^ { \\perp } u : = \\sum _ { j \\in S ^ c } u _ j \\ , e ^ { \\mathrm { i } \\ , j \\ , x } , \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} \\begin{aligned} P u & = E u , P & = \\left ( \\begin{matrix} P _ 1 & h W \\\\ h W ^ * & P _ 2 \\end{matrix} \\right ) , \\end{aligned} \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} \\hat { \\beta } _ { 1 } = - \\beta + { 2 \\over 3 } , \\ \\hat { \\beta } _ { 2 1 } = - \\beta , \\ \\hat { \\beta } _ { 2 2 } = - \\beta + { 1 \\over 2 } , \\ \\hat { \\beta } _ { 3 1 } = \\beta + { 1 \\over 3 } , \\ \\hat { \\beta } _ { 3 2 } = \\beta + { 2 \\over 3 } , \\ \\hat { \\beta } _ { 3 3 } = \\beta . \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} \\mathcal L _ x ( U , \\overline V ) = - d \\omega _ 0 ( J U , \\overline V ) = - i d \\omega _ 0 ( U , \\overline V ) . \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + a x + b , a = - \\frac { c _ 4 } { 4 8 } , b = - \\frac { c _ 6 } { 8 6 4 } . \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} p _ 1 ( t , x , z ) & = G _ 1 ( f ( t , x ) - z ) = \\mu p _ 2 ( t , x , f ) - \\sigma _ 1 ^ c \\partial _ x ^ 2 f ( t , x ) , \\\\ [ 5 p t ] p _ 2 ( t , x , z ) & = G _ 2 ( f ( t , x ) + g ( t , x ) - z ) = \\sigma _ 2 ^ c \\partial _ x ^ 2 ( f + g ) ( t , x ) . \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} \\Upsilon _ 0 ^ { [ l ] } = \\Upsilon _ { 0 , f } ^ { [ l ] } + \\Upsilon _ { 0 , g } ^ { [ l ] } + \\Upsilon _ { 0 , h } ^ { [ l ] } \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} \\| T \\| _ { C ^ { k , \\alpha } _ { \\rm s c } ( B ) } = \\rho ^ { k + \\alpha } [ \\nabla ^ k T ] _ { C ^ { 0 , \\alpha } ( B ) } + \\sum _ { j = 0 } ^ k \\rho ^ j \\| \\nabla ^ j T \\| _ { L ^ \\infty ( B ) } , \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} y ^ 2 + a x y + b y = x ^ 3 , a , b \\in \\Z _ \\ell , \\Delta = b ^ 3 ( a ^ 3 - 2 7 b ) , \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} Z ( f h , x , y ) & = f ( x ) Z ( h , x , y ) + h ( x ) Z ( f , x , y ) \\\\ & + \\cdot \\Delta _ + ( f , x , y ) \\Delta _ + ( h , x , y ) + \\cdot \\Delta _ - ( f , x , y ) \\Delta _ - ( h , x , y ) \\ , , \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\alpha _ n = \\alpha \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} \\partial _ t u + ( - \\Delta ) ^ { \\frac { \\theta } { 2 } } u = 0 \\quad \\mbox { i n } \\quad { \\bf R } ^ N \\times ( 0 , \\infty ) , \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} K _ 1 ( \\xi , \\eta ) = ( 1 - \\tanh \\xi ) ( 1 - \\tanh \\eta ) K ( \\xi , \\eta ) . \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\sum _ { K \\in \\mathcal { T } } \\norm { \\nabla ( \\Pi _ h v _ { q _ h } ) } ^ 2 _ { 0 , K } \\le & c \\norm { v _ { q _ h } } ^ 2 _ { 1 , \\Omega } , \\\\ \\sum _ { K \\in \\mathcal { T } } \\frac { \\alpha _ v } { h _ K } \\norm { \\bar { \\Pi } _ h v _ { q _ h } - \\Pi _ h v _ { q _ h } } ^ 2 _ { 0 , \\partial K } \\le & c \\alpha _ v \\norm { v _ { q _ h } } ^ 2 _ { 1 , \\Omega } , \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} u ( x , 0 ) = \\varphi ( x ) , 0 \\leq x \\leq 1 , \\end{align*}"}
-{"id": "1366.png", "formula": "\\begin{align*} n = s ( 2 ^ d - 1 ) + 2 ^ d m , \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} \\frac { u ( z ) } { ( 1 + s ) d ( z ) } \\leq \\frac { u ( z _ s ) } { ( 1 - s ) d ( z ) } = \\frac { u ( z _ s ) - u ( z _ 1 ) } { ( 1 - s ) | z - z _ 1 | } = \\frac { u ( z _ s ) - u ( z _ 1 ) } { | z _ s - z _ 1 | } . \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} D _ \\Gamma ( \\gamma _ 1 , \\dots , \\gamma _ n ) ( f _ 1 , \\dots , f _ m ) = \\pi \\mu \\Big ( \\prod _ { ( i , j ) \\in \\Gamma } \\sum _ { k = 1 } ^ N \\frac { \\partial } { \\partial \\theta _ k ^ { ( i ) } } \\otimes \\frac { \\partial } { \\partial \\xi _ k ^ { ( j ) } } \\Big ) ( \\gamma _ 1 \\otimes \\dots \\otimes \\gamma _ n \\otimes f _ 1 \\otimes \\dots \\otimes f _ m ) \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} \\log \\frac { f ( z ) } { z } = 2 \\sum _ { n = 1 } ^ { \\infty } \\gamma _ n z ^ n \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} A u _ \\tau = Q ( u ) \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} M ( p _ 1 , p _ 2 , \\ldots , p _ n ) = M ( q _ 0 , q _ 1 , \\ldots , q _ { n - 1 } ) . \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} y _ { 2 } ( n + 1 ) = A _ { 2 } ( n ) y _ { 2 } ( n ) \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} H & = H _ 0 + d b + 2 c ( a , F _ 0 ) + c ( a , d ^ { \\theta _ 0 } a ) + \\frac { 1 } { 3 } c ( a , [ a , a ] ) , \\\\ \\theta & = \\theta _ 0 + a , \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} V _ y ( E ) = \\{ x _ 2 \\in \\R : ( x _ 1 , x _ 2 ) \\in E x _ 1 = y _ 1 \\} . \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\dot y = \\lambda f _ 2 \\big ( t , \\hat x ( t ) , y , \\lambda \\big ) . \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} T _ { T V } ( \\delta ) : = \\inf \\Big \\{ t \\ge 0 : \\max _ { x , y \\in V _ N } | | K _ { 0 , t } ( x , \\cdot ) - K _ { 0 , t } ( y , \\cdot ) | | _ { T V } < \\delta \\Big \\} \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} T _ \\varepsilon ( t ) u _ 0 ^ \\varepsilon = e ^ { - A t } u _ 0 ^ \\varepsilon + \\int _ 0 ^ t e ^ { - A ( t - s ) } f ( T _ \\varepsilon ( s ) u _ 0 ^ \\varepsilon ) \\ , d s , t \\geq 0 . \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\operatorname { F r a c } R = \\operatorname { F r a c } ( R _ { \\mathfrak { n } \\cap R } ) = \\operatorname { F r a c } ( S _ { \\mathfrak { n } } ) = \\operatorname { F r a c } S . \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} g ( t , \\omega ) & = - [ ( \\alpha \\cdot x - t ) _ { + } \\cos ( \\| \\omega \\| _ 1 t + b ( \\omega ) ) + \\\\ & ( - \\alpha \\cdot x - t ) _ { + } \\cos ( \\| \\omega \\| _ 1 t - b ( \\omega ) ) ] \\| \\omega \\| ^ 2 _ 1 | \\widetilde { f } ( \\omega ) | . \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\sup _ { x \\in B _ i } \\mathbf { d } _ i ( x , g _ i ( \\Phi _ i ( x ) ) ) = 0 . \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} \\mathcal { L } _ 4 = \\Pi _ S ^ { \\perp } ( \\mathcal { D } _ { \\omega } + m _ 3 \\partial _ { x x x } + \\tilde { d } _ 1 \\ , \\partial _ x + \\varepsilon ^ 2 T + R _ 4 ) \\Pi _ S ^ { \\perp } , \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} w ( z ) = z \\frac { \\lambda + z } { 1 + \\lambda z } ( 0 \\le \\lambda \\leq 1 ) \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} ( k - 1 ) x ^ { 2 } - ( k + 1 ) y ^ { 2 } = - 2 \\mu . \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} M = M _ 0 Z _ M \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} 2 \\frac { \\partial h } { \\partial z } \\ , d z = \\frac { \\partial h } { \\partial s } d s + i \\frac { \\partial h } { \\partial n } d s \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{gather*} \\Psi ( x , t ) : = e ^ { \\frac { x ^ 3 } { 6 } - \\frac { x t } { 2 } } \\kappa ( t ) R ( x , t ) \\psi ^ { \\sigma _ 3 } ( t ) \\Psi _ 0 ( x , t ) , \\end{gather*}"}
-{"id": "3145.png", "formula": "\\begin{align*} \\mathcal { F } _ n ^ \\ast ( k ) = F _ n ^ \\ast ( k ) + F _ n ^ \\ast ( k - 1 ) . \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} C ( F , \\Gamma , s ) = | s | \\| F ^ { - 1 } | _ { F ( \\Gamma ) } \\| _ { C ^ r } ( 1 + \\max \\{ \\| F \\| _ - ^ s , \\| F \\| _ - ^ { - 1 } \\} ) \\ , , \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} F ( s , \\chi _ a ) = \\sum _ { \\chi \\neq \\chi _ 0 } \\bar { \\chi } ( a ) \\log L ( s , \\chi ) + G _ a ( s ) , \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} & y _ { 2 \\ell } \\overline { w } _ j = y _ { \\ell + \\frac { 1 } { 2 } } \\overline { w } _ j = y _ { \\ell - \\frac { 1 } { 2 } } \\overline { w } _ j = 0 , & \\\\ & \\ y _ { 2 \\ell - 2 } ^ { ( \\xi _ { \\ell - 1 } - \\xi _ { \\ell } + \\delta _ { j , 2 } + 1 ) } \\overline { w } _ j = \\delta _ { \\xi _ \\ell , \\xi _ { \\ell - 1 } } \\delta _ { p , 1 } \\delta _ { j , 0 } \\ y _ { \\ell - \\frac { 3 } { 2 } } ^ { ( \\xi _ { \\ell } ) } \\overline { w } _ j = 0 . \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} \\mathcal { H } ^ { m } \\left ( f ^ { - 1 } \\left ( f ( A ) \\cap P ( z ) \\right ) \\right ) = \\mathcal { H } ^ { m } \\left ( \\left \\{ x \\in A : h _ I ( x ) = z \\right \\} \\right ) > 0 . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta _ { p ( x ) } u = h ( x ) & \\Omega \\\\ u = 0 & \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} \\pi _ s ( k ) \\xi _ j ^ s & = q ^ { - j } \\xi _ j ^ s , & \\pi _ s ( e ) \\xi _ j ^ s & = ( [ s + j ] _ q [ s - j + 1 ] _ q ) ^ { 1 / 2 } \\xi _ { j - 1 } ^ s , \\\\ \\pi _ s ( f ) \\xi _ j ^ s & = ( [ s - j ] _ q [ s + j + 1 ] _ q ) ^ { 1 / 2 } \\xi _ { j + 1 } ^ s . \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} & \\big \\{ c \\left ( ( x _ 1 , \\ldots , x _ { a - 1 } , x _ { a } = u ^ { \\ast } , x _ { a + 1 } , \\ldots , x _ t ) ; ( { u } , { v } ) \\right ) ~ : ~ ( x _ 1 , \\ldots , x _ { a - 1 } , x _ { a + 1 } , \\ldots , x _ t ) \\in [ r ] ^ { t - 1 } \\\\ & ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ( { u } , { v } ) \\in [ r ] \\times [ s ] ~ ~ ( { u } , { v } ) \\neq ( u ^ { \\ast } , v ^ { \\ast } ) \\big \\} , \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} \\mathbf { H } ^ { ' } = \\begin{bmatrix} 0 . 4 0 e ^ { i 1 . 3 9 7 2 } & 1 . 1 2 e ^ { i 0 . 7 7 3 7 } & 0 . 4 3 e ^ { i 1 . 2 8 7 4 } & 0 . 8 4 e ^ { i 0 . 3 0 6 7 } \\\\ 1 . 2 4 e ^ { - i 0 . 9 8 7 2 } & 1 . 7 0 e ^ { i 0 . 9 7 8 4 } & 0 . 8 3 e ^ { - i 0 . 2 1 5 6 } & 0 . 6 7 e ^ { - i 1 . 6 4 1 4 } \\end{bmatrix} . \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} S ( \\phi _ { u u } ) = 0 , \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} \\| F \\| _ { \\mathcal { F } _ q ( \\mathcal { H } ) } ^ 2 \\le \\sum _ { n = 0 } ^ \\infty \\| f ^ { ( n ) } \\| _ { \\mathcal { H } ^ { \\otimes n } } ^ 2 [ n ] _ q ! \\ , = \\| F \\| _ { \\mathcal { G } _ q ( \\mathcal { H } ) } ^ 2 . \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} = \\sum _ { \\substack { t _ 1 , \\ldots , t _ { \\nu } \\ge 0 \\\\ t _ 1 + 2 t _ 2 + \\cdots + \\nu t _ { \\nu } = \\nu } } ( - 1 ) ^ { t _ 1 + \\cdots + t _ { \\nu } } \\binom { t _ 1 + \\cdots + t _ { \\nu } } { t _ 1 , \\ldots , t _ { \\nu } } a _ 1 ^ { t _ 1 } \\cdots a _ { \\nu } ^ { t _ { \\nu } } \\end{align*}"}
-{"id": "6727.png", "formula": "\\begin{align*} f ( t ) = 1 - \\frac { \\log 1 0 2 4 + \\log ( t + 3 ) } { \\log 2 7 - \\log { 4 0 9 6 } + 2 \\log ( t - 2 ) } , \\ t > 2 . \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} \\hat { \\tau } ( \\omega ) = \\sup \\left \\{ t < s : \\left | \\varphi _ t ( \\omega , x ) - \\varphi _ t ( \\omega , y ) \\right | \\leq 4 \\right \\} . \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ m \\| q _ { n _ l } ( \\theta - \\theta _ j ) \\| \\geq \\frac { e ^ { ( \\delta _ c - \\epsilon ) q _ { n _ l } } } { q _ { n _ l + 1 } } . \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{gather*} \\phi ( 3 ; - 3 ) = ( 5 ; - 3 ) , \\phi ( 3 ; - 1 ) = ( 7 ; - 1 ) , \\phi ( 3 ; 1 ) = ( 9 ; 1 ) , \\phi ( 3 ; 3 ) = ( 2 ; 2 ) \\\\ \\phi ( 4 ; 0 ) = ( 2 ; 0 ) . \\end{gather*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\overline { \\alpha } = K \\alpha , \\ | \\alpha | ^ { 2 } = K \\alpha ^ { 2 } , \\ \\overline { \\beta } = L \\beta , \\ | \\beta | ^ { 2 } = L \\beta ^ { 2 } , \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} \\left ( \\int _ { C _ i } \\rho ^ * \\omega \\right ) = - \\left ( \\int _ { \\rho ( C _ i ) } \\omega \\right ) = + \\left ( \\int _ { C _ i } \\omega \\right ) . \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} \\frac { v _ 1 ' - v _ { - 1 } ' } { 2 } + v ' v '' + v '''' = 0 , \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\lim _ { m \\rightarrow \\infty } \\textrm { d i a m } \\left ( \\varphi _ { t _ m } ( \\cdot , V ( \\cdot ) ) \\right ) = 0 \\right ) > 0 . \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} U ( a , b ; z ) = \\frac { \\pi } { \\sin ( \\pi b ) } \\left ( \\frac { { } _ { 1 } F _ { 1 } ( a , b ; z ) } { \\Gamma ( 1 + a - b ) \\Gamma ( b ) } - \\frac { x ^ { 1 - b } { } _ { 1 } F _ { 1 } ( 1 + a - b , 2 - b ; z ) } { \\Gamma ( a ) \\Gamma ( 2 - b ) } \\right ) . \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} C _ { 0 } \\simeq \\bigoplus _ { \\alpha \\in Q _ { 1 } , \\ , t ( \\alpha ) = j } \\mathbb { S } _ { 2 } ^ { \\deg ( \\alpha ) } ( M ^ { s ( \\alpha ) } ) , C _ { 1 } \\simeq \\bigoplus _ { \\alpha \\in Q _ { 1 } , \\ , s ( \\alpha ) = j } \\mathbb { S } _ { 2 } ^ { 1 - \\deg ( \\alpha ) } ( M ^ { t ( \\alpha ) } ) . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( Y _ i - \\hat { f } ( X _ i ) ) ^ 2 + \\frac { _ n ( \\hat { f } ) } { n } \\leq \\inf _ { f \\in \\mathcal { F } } \\left \\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n ( Y _ i - f ( X _ i ) ) ^ 2 + \\frac { _ n ( f ) } { n } + A _ f \\right \\} , \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} { \\rm D e s } _ D \\ , \\sigma = \\left \\{ i \\ | \\ 0 \\leq i < n \\ { \\rm a n d } \\ \\sigma _ i > \\sigma _ { i + 1 } \\right \\} . \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{gather*} d ^ 2 = 0 , \\delta ^ 2 = 0 , d \\delta + \\delta d = 0 . \\end{gather*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\int _ { M } X ( b ) d M = - n \\int _ { \\partial M } { \\stackrel { \\circ } { B } } ( X , \\nu ) d \\Sigma . \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} J \\nabla _ { J \\partial _ r } \\nabla u & = r ^ { \\mu - 1 } ( \\mu \\phi J \\nabla _ { J \\partial _ r } \\partial _ r + J \\nabla _ { J \\partial _ r } \\nabla _ Y \\phi ) \\\\ & = - r ^ { \\mu - 2 } ( \\mu \\phi \\partial _ r + \\nabla _ Y \\phi ) . \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} A _ n ( x ) & = E _ n ^ { ( 1 , 2 , \\ldots , n ) } ( x ) , \\\\ B _ n ( x ) & = E _ n ^ { ( 2 , 4 , \\ldots , 2 n ) } ( x ) . \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} d _ i = \\lim _ { \\Omega _ i \\ni y \\to x } v _ i \\circ H ( y ) - a & \\leq \\lim _ { \\Omega _ i \\ni y \\to x } v ^ - ( y ) \\\\ & \\leq \\lim _ { \\Omega _ i \\ni y \\to x } v ^ + ( y ) \\leq \\lim _ { \\Omega _ i \\ni y \\to x } W _ i \\circ H ( y ) = d _ i . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} d _ { S ( T , t ) } ( v ^ t , w ' ) = d _ { S ( T , t - 1 ) } ( y ' \\cdots y ' , y _ { 2 } \\cdots y _ t ) + ( 2 ^ { t } - 1 ) d _ T ( v , y ) - ( 2 ^ { t - 1 } - 1 ) , \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} \\Omega ( \\xi , \\eta , \\zeta ) & = \\Delta ( \\xi , \\eta , \\zeta ) \\Delta ( \\xi , - \\eta , - \\zeta ) \\Delta ( \\xi , - \\eta , \\zeta ) \\Delta ( \\xi , \\eta , - \\zeta ) \\\\ & = J ( \\xi ) ^ 2 + J ( \\eta ) ^ 2 + J ( \\zeta ) ^ 2 - 2 J ( \\xi ) J ( \\eta ) - 2 J ( \\eta ) J ( \\zeta ) - 2 J ( \\zeta ) J ( \\xi ) , \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} \\mathbb { P } _ { p _ { U , V , W } ^ n } \\left ( 2 ^ { \\lambda \\sum _ { t = 1 } ^ n i _ { p _ { U , V , W } } ( U _ t , V _ t , p _ t ) } \\geq 2 ^ { n \\lambda ( I ( U , V ; W ) + \\epsilon _ 2 ) } \\right ) \\leq 2 ^ { n \\lambda \\big ( d _ { \\lambda + 1 } ( p _ { U , V , W } , p _ { U , V } p _ W ) - I ( U , V ; W ) - \\epsilon _ 2 \\big ) } . \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} \\operatorname { T r } \\left \\{ \\mathbf { n M S E } ^ { ( g ) } \\right \\} = \\sum _ { l = 0 } ^ { L _ g - 1 } \\sum _ { m = 1 } ^ { R _ l } \\sum _ { n = 1 } ^ { d _ l } \\frac { 1 } { \\beta _ m ^ l \\lambda _ n ^ l + 1 } + \\left ( K _ g \\sum _ { l = 0 } ^ { L _ g - 1 } r _ { g , l } - \\sum _ { l = 0 } ^ { L _ g - 1 } R _ l d _ l \\right ) \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} \\frac { d } { d t } | _ { t = 0 } \\log \\int _ M e ^ { - \\lambda ( \\phi - \\phi _ 0 + t v ) } \\mu _ 0 = \\int _ M \\frac { d } { d t } | _ { t = 0 } e ^ { - \\lambda ( \\phi - \\phi _ 0 + t v ) } \\mu _ 0 = \\int _ M v e ^ { - \\lambda ( \\phi - \\phi _ 0 ) } \\mu _ 0 \\end{align*}"}
-{"id": "7699.png", "formula": "\\begin{align*} \\| u _ k \\| ^ 2 = \\pi \\Omega _ \\beta ' \\mu _ k ^ \\frac { 2 - \\beta } { 2 \\beta } ( 1 + o ( 1 ) ) = \\pi \\Omega _ \\beta ' \\left ( \\frac { \\pi } { \\Omega _ \\beta } k \\right ) ^ \\frac { 2 - \\beta } { 2 + \\beta } ( 1 + o ( 1 ) ) , k \\to \\infty , \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} w ^ n \\equiv ( w ^ n _ 1 , w ^ n _ { - 1 } ) = ( \\Psi _ 1 ( x ^ n ) , \\Psi _ { - 1 } ( x ^ n ) ) , \\ \\mbox { a n d } \\ w \\equiv ( w _ 1 , w _ { - 1 } ) = ( \\Psi _ 1 ( x ) , \\Psi _ { - 1 } ( x ) ) . \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} _ { \\tau } ( \\zeta _ { a } ^ { ( 0 ) } ) = 0 , \\forall a \\in \\{ 1 , . . . , \\mathsf { N } \\} . \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} g _ u ' ( s ) & = \\langle I ' ( s u ) , u \\rangle \\\\ & = \\| u \\| _ { E ^ { \\alpha , p } } ^ p M _ { s u } ^ { p - 1 } s ^ { p - 1 } - \\int _ { 0 } ^ { T } f ( t , s u ( t ) ) u ( t ) d t \\\\ & = 0 , \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} P _ \\varphi ^ m ( \\theta _ 0 ( x ) ) = \\theta _ 0 ( x ) , \\textrm { ( f o r e v e r y $ m \\geq 0 $ ) } . \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} & \\sum _ { a = m _ 2 + 1 } ^ { n - m _ 1 - 1 } \\sum _ { k : M _ k \\in S _ { j + 1 , i } ^ { a } } \\lambda _ { k } | A - M _ { k } | \\lesssim \\sum _ { a = m _ 2 + 1 } ^ { n - m _ 1 - 1 } \\sum _ { k : M _ k \\in S _ { j + 1 , i } ^ { a } } \\lambda _ { k } i = \\sum _ { a = m _ 2 + 1 } ^ { n - m _ 1 - 1 } \\nu ( S _ { j + 1 , i } ^ { a } ) i \\\\ & \\lesssim \\sum _ { a = m _ 2 + 1 } ^ { n - m _ 1 - 1 } j ^ { 2 ( m _ 1 ' + a - n ' ) } i ^ { m _ 1 ' + a - n ' + 1 } \\lesssim j ^ { - 2 } \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} \\omega _ 2 ( \\cdot , \\cdot ) = \\omega _ 3 ( \\cdot , \\Gamma \\cdot ) . \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } g _ k = g _ { \\infty } , \\sum _ { k = 1 } ^ { \\infty } ( g _ k - g _ { \\infty } ) ^ 2 < \\infty . \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} V _ 1 ( x ) = W ^ 2 ( x ) - W ' ( x ) \\ ; , \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} q ( \\alpha _ j , \\beta _ j ; 1 , d ^ { ( j ) } y _ j ; 1 , y _ j ) = \\frac { p ( \\beta _ j , \\alpha _ j , 1 , d ^ { ( j ) } y _ j ) } { p ( \\alpha _ j , \\beta _ j , 1 , y _ j ) } . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { 2 n } } { 1 - q ^ { 2 n } } \\frac { ( - q ; q ^ 2 ) _ { n - 1 } } { ( q ^ 2 , q ^ 2 ) _ { n - 1 } } q ^ { n - 1 } = \\psi ( - q ) - \\frac { \\tiny 1 } { 1 + q } . \\end{align*}"}
-{"id": "1409.png", "formula": "\\begin{align*} a _ k \\ge \\beta ^ { p ^ k } , k = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} [ A , B ] + I J + F G = 0 , [ A ' , B ' ] - G F = 0 , \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} ( \\forall \\ , p _ k \\in \\mathcal { V } \\sp { \\alpha } _ k ) \\ ( \\forall \\ , x \\in B ^ d ) p _ k ( - x ) = ( - 1 ) ^ k p _ k ( x ) . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} \\beta _ k = \\left \\{ \\begin{array} { l l } \\nu _ k / \\det G _ { I , I } & \\ k \\in I , \\\\ 0 & \\ k \\notin I . \\end{array} \\right . \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} Q = \\alpha \\cdot P _ 1 + P _ 2 . \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} m _ 3 = 1 + \\varepsilon ^ 2 d ( \\xi ) + \\mathtt { r } _ { m _ 3 } \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} T = \\begin{pmatrix} a & \\alpha & b \\\\ \\gamma & e & \\beta \\\\ c & \\delta & d \\end{pmatrix} = ( t _ { i j } ) . \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} c _ 4 ( E ) = 2 ^ 4 \\cdot 7 , c _ 6 ( E ) = - 2 ^ 7 \\cdot 5 , \\Delta ( E ) = 2 ^ 6 \\cdot 3 ^ 2 . \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} A _ { ( m + 1 ) p _ n - r _ n } = A _ { ( m + 1 ) p _ n + m - r _ n } - x ^ { 2 r _ n - m } A _ { ( m + 1 ) p _ n + r _ n } - x ^ { 2 r _ n - m - 1 } A _ { ( m + 1 ) p _ n + r _ n - m - 1 } , \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} S = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} \\ ; \\ ; T = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} \\tilde { f } _ { \\tilde { v } } ( n ) = \\left \\{ \\begin{matrix} f _ { \\tilde { v } } ( n ) z ( n ) u ( n ) , & n \\in \\mathcal { L } , \\\\ 0 , & n \\in \\mathcal { N } \\backslash \\mathcal { L } , \\end{matrix} \\right . \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} d \\mu ^ X = \\iota _ X \\omega = \\omega ( X ) = \\sum _ { i = 1 } ^ 3 d x _ i \\wedge d y _ i ( \\sum _ { j = 1 } ^ 3 a _ j \\dfrac { \\partial } { \\partial x _ j } ) = - \\sum _ { i = 1 } ^ 3 d y _ i \\wedge d x _ i ( \\sum _ { j = 1 } ^ 3 a _ j \\dfrac { \\partial } { \\partial x _ j } ) = - \\sum _ { i = 1 } ^ 3 a _ i d y _ i . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} | f | _ { \\theta } : = \\sup \\left \\{ \\frac { V _ n ( f ) } { \\theta ^ n } : n \\in \\N \\right \\} \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} D ( J _ { \\rm m a x } ) : = \\{ f \\in \\ell ^ 2 ( \\mathbb { N } ) : J f \\in \\ell ^ 2 ( \\mathbb { N } ) \\} \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} Z ( t ) = Z ( \\epsilon , \\nu , f , F , t ) : = \\nu \\left \\{ ( x , l ) \\in \\Lambda : \\left | \\frac { 1 } { t } \\int _ 0 ^ t F \\circ \\Phi ^ s ( x ) d s - \\int F d \\mu \\right | > \\epsilon \\right \\} \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} \\eta ( k ; l ) = \\eta ( l ; k ) \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{align*} a _ k = \\gamma \\alpha ^ k + O ( \\alpha ^ { k ( 1 - \\varepsilon ) } ) , b _ m = \\delta \\beta ^ m + O ( \\beta ^ { m ( 1 - \\varepsilon ) } ) \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} \\lim _ { y \\rightarrow - x } \\frac { 1 + [ y , b ( x ) ] } { 1 + [ x , b ( y ) ] } = \\lim _ { y \\rightarrow - x } \\frac { [ b ( y ) , b ( x ) ] } { [ x , - y ] } = \\lim _ { y \\rightarrow - x } \\frac { [ - y , b ( x ) ] } { [ x , - b ( y ) ] } = 1 . \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{align*} \\mathcal { I } ^ j _ e = \\{ \\gamma \\in e + z \\mathfrak { g } l _ m [ [ z ] ] \\mid \\nu _ t ( \\gamma ) \\ , \\ \\} \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} W H ^ r ( P ) \\cong _ { G } \\bigoplus _ { \\substack { x \\in P / \\sim \\\\ \\rho ( x ) = r } } \\widetilde H ^ { r - 2 } ( ( \\hat 0 , x ) ; { \\bf k } ) \\bigl \\uparrow _ { G _ x } ^ G , \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} \\Gamma u ' = E u , \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} H _ { j , 6 } = \\real \\Big [ i ^ j \\sum _ { | \\alpha | = j - 4 } ( - 1 ) ^ { \\alpha _ 1 + \\alpha _ 2 } \\int u ^ { ( \\alpha _ 1 ) } u ^ { ( \\alpha _ 2 ) } u \\overline { u ^ { ( \\alpha _ 3 ) } u ^ { ( \\alpha _ 4 ) } u ^ { ( \\alpha _ 5 ) } } + u ^ { ( \\alpha _ 1 ) } u ^ { ( \\alpha _ 2 ) } \\bar u \\frac { d ^ { \\alpha _ 3 } } { d x ^ { \\alpha _ 3 } } \\Big ( u \\overline { u ^ { ( \\alpha _ 4 ) } u ^ { ( \\alpha _ 5 ) } } \\Big ) d x \\Big ] . \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} M _ { a } ( \\lambda ) = \\left ( \\begin{array} { c c } A ( \\lambda ) & B ( \\lambda ) \\\\ C ( \\lambda ) & D ( \\lambda ) \\end{array} \\right ) _ { a } \\equiv L _ { a , \\mathsf { N } } ( \\lambda q ^ { - 1 / 2 } ) \\cdots L _ { a , 1 } ( \\lambda q ^ { - 1 / 2 } ) \\in ( \\mathbb { C } ^ { 2 } \\otimes \\mathcal { H } ) , \\end{align*}"}
-{"id": "7011.png", "formula": "\\begin{align*} I \\left ( \\frac { u } { v } \\right ) = \\frac { L ^ 2 ( 1 , \\chi ) } { \\sqrt { u v } } \\int _ 0 ^ \\infty k \\left ( \\frac { T } { x } \\right ) h \\left ( \\frac { x } { u } \\right ) h \\left ( \\frac { x } { v } \\right ) \\frac { d x } { x } + O ( T ^ { - \\frac { 1 } { 9 } } ) \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} & X _ i = \\frac { \\partial } { \\partial x ^ i } + \\sum _ { j > r } a _ { j i } \\frac { \\partial } { \\partial x ^ j } + \\sum _ { \\rho > s } \\alpha _ { j \\rho } \\frac { \\partial } { \\partial \\theta ^ \\rho } \\\\ & \\chi _ \\sigma = \\frac { \\partial } { \\partial \\theta ^ \\sigma } + \\sum _ { i > r } \\beta _ { j i } \\frac { \\partial } { \\partial x ^ i } + \\sum _ { \\rho > s } b _ { j \\rho } \\frac { \\partial } { \\partial \\theta ^ \\rho } \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} X ( n ) = \\left \\{ \\begin{array} { r c l } A ( n - 1 ) \\ldots A ( 0 ) & \\textnormal { i f } & n \\geq 1 \\\\ \\\\ I & \\textnormal { i f } & n = 0 \\\\ \\\\ A ^ { - 1 } ( n ) \\ldots A ^ { - 1 } ( - 1 ) & \\textnormal { i f } & n \\leq - 1 . \\end{array} \\right . \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} M _ { \\varphi , x } [ b _ 1 ] - M _ x [ b _ 1 ] = ( M _ { \\varphi , x } [ \\alpha _ 1 ] - M _ x [ \\alpha _ 1 ] ) + ( M _ { \\varphi , x } [ \\alpha _ 1 \\beta _ x ] - M _ x [ \\alpha _ 1 \\beta _ x ] ) . \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\mu _ \\rho = \\left ( \\mu ( 1 - F ( K _ \\delta ) ) + \\sum _ { t = 1 } ^ { K _ \\delta } \\frac { f ^ \\ast _ \\delta ( t ) } { t } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} e ' - \\sigma ( e ' ) & = e - e \\sigma ( x ) \\sigma ( e ) - \\sigma ( e ) + e x \\sigma ( e ) = e - \\sigma ( e ) + e ( x - \\sigma ( x ) ) \\sigma ( e ) \\\\ & = e - \\sigma ( e ) + \\lambda ^ { - 1 } e u \\sigma ( e ) = e - \\sigma ( e ) + e \\sigma ( e ) + 1 + \\lambda ^ { - 1 } u - 2 e \\\\ & = 1 - e - \\sigma ( e ) + e \\sigma ( e ) + \\lambda ^ { - 1 } u = ( 1 - e ) ( 1 - \\sigma ( e ) ) + \\lambda ^ { - 1 } u = \\lambda ^ { - 1 } u . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} \\left ( ( r , c , s ) , ( r ' , c ' , s ' ) \\right ) = c . c ' - r s ' - s r ' . \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} H ^ i ( B ) : = ( \\tau _ { \\le i } B / \\tau _ { < i } B ) ( i ) . \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} d \\pi _ 1 \\wedge d \\pi _ 1 ^ { \\dagger } ( ( a _ 1 , b _ 1 , i _ 1 , j _ 1 , a ' _ 1 , b ' _ 1 , f _ 1 , g _ 1 ) , ( a _ 2 , b _ 2 , i _ 2 , j _ 2 , a ' _ 2 , b ' _ 2 , f _ 2 , g _ 2 ) ) = & a _ 1 a _ 2 ^ { \\dagger } - a _ 2 a _ 1 ^ { \\dagger } . \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} u ( t ) = K _ { 1 } ( t ) u _ { 0 } + K _ { 2 } ( t ) u _ { 0 } + K _ { 3 } ( t ) u _ { 1 } , \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} ( A _ { 2 2 } - A _ { 2 1 } A _ { 1 1 } ^ { - 1 } A _ { 1 2 } ) v ' = Q ' _ { 2 2 } ( u ^ \\pm ) v \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} i _ n ^ { ( 1 , 2 , \\ldots , n ) } ( t ) \\ = \\ ( t + 1 ) ^ n \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\mathbf { R } ^ { ( g ) } _ { \\boldsymbol { \\xi } } \\triangleq \\mathbb { E } \\left \\{ \\boldsymbol { \\xi } ^ { ( g ) } \\left ( \\boldsymbol { \\xi } ^ { ( g ) } \\right ) ^ H \\right \\} = \\mathbf { I } _ T \\otimes \\mathbf { R } ^ { ( g ) } _ { \\boldsymbol { \\eta } } \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} S = S _ 1 \\sqcup S _ 2 = \\{ 1 , 2 , \\ldots , n \\} \\sqcup \\{ 1 ' , 2 ' , \\ldots , n ' \\} , \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{gather*} \\| v _ n \\| ^ 2 _ { L ^ 2 } = \\| u _ n ( t _ n ) \\| ^ 2 _ { L ^ 2 } = \\| u _ { n , 0 } \\| ^ 2 _ { L ^ 2 } \\to \\| \\phi _ { \\omega } \\| ^ 2 _ { L ^ 2 } \\\\ E ( v _ n ) = E ( u _ n ( t _ n ) ) = E ( u _ { n , 0 } ) \\to E ( \\phi _ { \\omega } ) , \\end{gather*}"}
-{"id": "1116.png", "formula": "\\begin{align*} & m _ { \\lambda , \\rho } ( w _ 1 , w _ 2 ) = \\exp \\left ( \\frac { 1 - \\rho } { 2 } \\log ( 1 + \\lambda w _ 2 P ' ) - \\right . \\\\ & \\left . \\frac { 1 } { 2 } \\log \\left ( 1 + \\lambda ( 1 - \\lambda \\rho ) w _ 2 P ' + \\lambda \\rho ( 1 - \\lambda \\rho ) w _ 1 P ' \\right ) \\right ) , \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} \\begin{aligned} v _ t & = \\sigma ( \\gamma , \\gamma _ t ) _ x , \\\\ \\gamma _ t & = v _ x . \\end{aligned} \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} d = 1 + \\min \\{ | \\xi | , | \\eta | , | \\zeta | \\} , \\rho = 1 + \\max \\{ | \\xi | , | \\eta | , | \\zeta | \\} . \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} h = 1 _ V . \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} D _ i = \\partial _ { x ^ i } + \\frac { \\partial u ^ { \\alpha } } { \\partial x ^ i } \\partial _ { u ^ { \\alpha } } + \\cdots + \\sum _ { \\alpha , J _ 1 , J _ 2 } u ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; J _ 2 } \\partial _ { u ^ { \\alpha } _ { J _ 1 ; J _ 2 } } . \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{align*} u _ t + u _ { x x x } = 0 . \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} Q _ 4 = - \\frac { 1 / 2 } { r ^ 2 } + \\frac { 5 / 4 } { r ^ 3 } \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} \\operatorname { d i a m } ( G _ { 1 , 2 , 3 } ) & \\leq \\lbrack 2 { \\textstyle \\sum _ { x \\in X _ { 1 } } } f ( x ) + 1 ] + 1 + [ 2 { \\textstyle \\sum _ { x \\in X _ { 2 } } } f ( x ) + 2 ] + 1 + [ 2 { \\textstyle \\sum _ { x \\in X _ { 3 } } } f ( x ) + 1 ] - 4 \\\\ & = 2 { \\textstyle \\sum _ { x \\in X _ { 1 } \\cup X _ { 2 } \\cup X _ { 3 } } } f ( x ) + 2 \\\\ \\operatorname { r a d } ( G _ { 1 , 2 , 3 } ) & \\leq { \\textstyle \\sum _ { x \\in X _ { 1 } \\cup X _ { 2 } \\cup X _ { 3 } } } f ( x ) + 1 . \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { \\beta ( n ) } { n ^ s } = \\left ( 1 - \\frac { 2 } { 2 ^ s } - \\frac { 4 } { 2 ^ { 2 s } } \\right ) \\frac { \\zeta ( s - 1 ) \\zeta ( 2 s ) } { \\zeta ( s ) } ( \\Re s > 2 ) . \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} \\mathcal { C } _ p = \\sharp _ p ( T ^ { * } _ p M ) + < Z _ p > , \\forall p \\in M \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} \\Lambda _ 1 = \\left ( \\frac { p - q } { 2 ( \\alpha + \\beta - q ) } \\right ) ^ { \\frac { p } { \\alpha + \\beta - p } } \\left ( \\frac { \\alpha + \\beta - q } { \\alpha + \\beta - p } | \\Omega | ^ { \\frac { \\alpha + \\beta - q } { \\alpha + \\beta } } \\right ) ^ { - \\frac { p } { p - q } } S ^ { \\frac { \\alpha + \\beta } { \\alpha + \\beta - p } + \\frac { q } { p - q } } , \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} b ( \\lambda _ 1 , \\lambda _ 2 ) = b ' ( \\chi _ 1 ( \\lambda _ 1 ) , \\chi _ 2 ( \\lambda _ 2 ) ) + b ' ( \\chi _ 1 ( \\lambda _ 2 ) , \\chi _ 2 ( \\lambda _ 1 ) ) . \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} D _ i ^ { ( N ) } ( F ( x _ i , \\{ p _ n \\} ) ) = x _ i \\frac { \\partial } { \\partial x _ i } F ( x _ i , \\{ p _ n \\} ) + \\beta x _ i \\oint \\frac { d \\xi } { \\xi ^ 2 } \\frac { \\phi ^ - ( \\xi ) - 1 } { 1 - \\frac { x _ i } { \\xi } } \\left ( \\tilde { V } ' ( \\xi ) \\tilde { V } ( x _ i ) F \\right ) ( \\xi , \\{ p _ n \\} ) , \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} E _ { \\alpha } ( z ) : = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { \\Gamma ( \\alpha k + 1 ) } \\ , . \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} \\left ( \\log { \\frac { u } { v } } \\right ) ^ 2 = \\sum _ { q \\mid u } \\Lambda _ 2 ( q ) - 2 \\sum _ { q \\mid u } \\sum _ { r \\mid v } \\Lambda ( q ) \\Lambda ( r ) + \\sum _ { r \\mid v } \\Lambda _ 2 ( r ) . \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} [ 0 , 1 ] \\to \\Lambda , \\quad \\tau \\to \\eta _ { \\tau } = \\eta + \\tau ( \\tilde { \\eta } - \\eta ) \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} p _ { 2 } \\left ( \\frac { Q \\ell ^ { m _ { \\ell } } n + 1 } { 1 2 } \\right ) = a ( Q \\ell ^ { m _ { \\ell } } n ) \\equiv 0 \\pmod { \\ell ^ j } \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} | \\ ! | \\ ! | s _ \\ast ^ \\varepsilon | \\ ! | \\ ! | = \\sup _ { v ^ \\varepsilon \\in Y _ \\varepsilon } \\| s _ \\ast ^ \\varepsilon ( v ^ \\varepsilon ) \\| _ { X _ \\varepsilon ^ \\frac { 1 } { 2 } } \\leq C ( \\tau ( \\varepsilon ) + p ( \\varepsilon ) ^ { - \\frac { 1 } { 2 } } ) , \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} \\| y \\| : = \\max \\left ( \\| y \\| _ { \\omega , \\alpha } , \\| y \\| _ { 0 , 0 } , \\| v \\| _ { \\omega , 0 } \\right ) \\leq \\delta . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } \\ast & 0 \\\\ 0 & \\ast \\end{array} \\right ] , \\mbox { f o r $ A ( t ) $ , $ B ( t ) $ , $ A ' ( t ) $ , $ B ' ( t ) $ , $ F ( t ) $ a n d $ G ( t ) $ } , \\\\ I ( t ) = \\left [ \\begin{array} { c } 0 \\\\ \\ast \\end{array} \\right ] , J ( t ) = \\left [ \\begin{array} { c c } 0 & \\ast \\end{array} \\right ] \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} J _ { N , z } = \\delta \\otimes . . . \\otimes \\delta \\otimes \\left ( I _ { z } \\ast _ { \\mathbb { R } } \\widehat { \\phi _ { N } } \\right ) \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} X _ i = \\langle A _ i ^ n , u ^ n \\rangle \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} F _ q = \\begin{pmatrix} 0 & - \\sqrt q \\\\ { \\sqrt { q } } ^ { - 1 } & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} \\left ( { { { p _ 1 } } , { { p _ 2 } } } \\right ) = \\left \\{ \\begin{array} { l l } \\left ( p _ 1 ^ { ( 1 ) } , p _ 2 ^ { ( 1 ) } \\right ) & \\mathrm { i f } \\ ; { { { \\beta _ 1 } } } \\ge { { { \\beta _ 2 } } } , \\\\ \\left ( p _ 1 ^ { ( 2 ) } , p _ 2 ^ { ( 2 ) } \\right ) & \\mathrm { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} d ^ * _ G ( A B ) \\leq m _ K ( I + I ) = 2 m _ K ( I ) = d ^ * _ G ( A ) + d ^ * _ G ( B ) < 1 . \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{gather*} S ^ { ( 1 ) } _ 0 = \\begin{pmatrix} 1 & 0 \\\\ - i s _ 1 & 1 \\end{pmatrix} , S ^ { ( 2 ) } _ 0 = \\begin{pmatrix} 1 & i s _ 2 \\\\ 0 & 1 \\end{pmatrix} , S ^ { ( 3 ) } _ 0 = \\begin{pmatrix} 1 & 0 \\\\ - i s _ 3 & 1 \\end{pmatrix} , \\\\ S ^ { ( 4 ) } _ 0 = \\begin{pmatrix} 1 & - i s _ 1 \\\\ 0 & 1 \\end{pmatrix} , S ^ { ( 5 ) } _ 0 = \\begin{pmatrix} 1 & 0 \\\\ i s _ 2 & 1 \\end{pmatrix} , S ^ { ( 6 ) } _ 0 = \\begin{pmatrix} 1 & - i s _ 3 \\\\ 0 & 1 \\end{pmatrix} . \\end{gather*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\partial _ i a _ 1 [ \\hat { \\imath } ] = \\partial _ i \\Phi _ B ( T _ { \\delta } ) [ \\hat { \\imath } ] + 2 \\Phi _ B ( T _ { \\delta } ) \\partial _ i \\Phi _ B ( T _ { \\delta } ) [ \\hat { \\imath } ] + \\partial _ i \\tilde { q } [ \\hat { \\imath } ] . \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} \\begin{pmatrix} \\alpha ^ * & \\gamma \\\\ - q \\gamma ^ * & \\alpha \\end{pmatrix} \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} x ^ r + y ^ q = z ^ p , 1 / r + 1 / q + 1 / p < 1 , \\gcd ( x , y , z ) = 1 , \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} B _ { i j } & \\equiv _ { p ^ 3 } \\frac { t ( i + j + 1 ) p ^ 2 } { ( i + j + 1 ) p + 1 } \\binom { i + j + t + 1 } { i + j + 1 } \\sum _ { m = 1 } ^ { p - 1 } \\binom { p - 1 + ( i + j ) p } { m - 1 + i p } \\frac { 1 } { m + i p } \\\\ & \\equiv _ { p ^ 3 } t p ^ 2 ( i + j + 1 ) \\binom { i + j + t + 1 } { i + j + 1 } \\binom { i + j } { i } \\sum _ { m = 1 } ^ { p - 1 } \\binom { p - 1 } { m - 1 } \\frac { 1 } { m } \\\\ & \\equiv _ { p ^ 3 } 2 t p ^ 2 ( i + j + t + 1 ) \\binom { i + j + t } { i , j , t } q _ p ( 2 ) , \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} \\mathcal { S } = \\bigcup _ { \\substack { \\mathcal { I } \\subseteq [ 1 : n ] \\\\ | \\mathcal { I } | = k + 1 } } \\left ( \\mathcal { A } _ { n + 1 } \\bigcap _ { i \\in \\mathcal { I } } \\mathcal { A } _ i \\right ) . \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} \\mathcal I _ s ^ q ( \\mu ) \\ \\leq \\ \\left ( \\sum _ { n = 1 } ^ { \\infty } \\beta ^ n \\right ) ^ { q - 1 } \\mathcal I _ { s ' } ( \\mu ) ^ { \\beta , q } \\ < \\ \\infty , \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} c _ { 1 } + \\cdots + c _ { l } + \\mathbf 1 = c _ { 1 } ' + \\cdots + c _ { l } ' . \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} \\hat { M } _ { a } ( \\lambda ) = ( - 1 ) ^ { \\mathsf { N } } \\ , \\sigma _ { a } ^ { y } \\ , M _ { a } ^ { t _ { a } } ( 1 / \\lambda ) \\ , \\sigma _ { a } ^ { y } . \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} L ( s ) ^ { - 1 } = \\prod _ p ( 1 - \\lambda ( p ) p ^ { - s } + \\chi ( p ) p ^ { - 2 s } ) = \\sum _ m \\rho ( m ) m ^ { - s } , \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\tilde { V } _ t ( k _ t ) = \\sup _ { k _ { t + 1 } \\in \\hat { \\Gamma } ( k _ t ) } \\Big [ \\tilde { U } ( f ( k _ t ) - k _ { t + 1 } ) + \\tilde { \\beta } _ { t + 1 } \\ , \\tilde { V } _ { t + 1 } ( k _ { t + 1 } ) \\Big ] , & & t = 0 , 1 , \\ldots , \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} P ( - \\alpha , - \\beta ; z ) = \\frac { 1 } { P ( \\alpha , \\beta ; z ) } . \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} A _ w ( \\Sigma ) = \\int _ \\Sigma w ( p ) \\ d \\Sigma \\ ! ~ , \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rm F A } + \\mathbb { P } _ { \\rm M D } & = E _ U [ \\mathbb { P } _ { \\rm F A } ( { \\mathbf U } ) + \\mathbb { P } _ { \\rm M D } ( { \\mathbf U } ) ] \\\\ & \\geq E _ U [ \\mathbb { P } _ { \\rm F A } ( { \\mathbf U } ) + \\mathbb { P } _ { \\rm M D } ( { \\mathbf U } ) | { \\mathcal A } ^ c ] P ( { \\mathcal A } ^ c ) \\\\ & > 1 - \\epsilon . \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} w ( q ) = \\frac { 1 } { 2 \\pi } \\int _ 0 ^ { 2 \\pi } - \\left ( \\frac { 1 } { 2 } + q + 2 \\hat x ( t ) \\sin t \\right ) d t = \\pi \\left ( 2 q + \\frac { 1 1 } { 1 0 } \\right ) . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} \\frac { \\partial N _ { i a } } { \\partial \\lambda _ { j b } } = L s _ b \\rho ( \\lambda _ { j b } ) [ \\delta _ { i a , j b } - s _ a F ( \\lambda _ { j b } | \\lambda _ { i a } ) ] . \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} \\sum _ { m _ 1 + m _ 2 < p } \\binom { m _ 1 + m _ 2 } { m _ 1 } ^ 2 \\left ( H _ { m _ 1 + m _ 2 } - H _ { m _ 1 } \\right ) \\equiv _ { p } 0 \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} M = \\bigg \\langle m , \\dots , 2 m - 1 , \\frac { q } { p ^ { m + 1 } } , \\frac { q } { p ^ { m + 2 } } , \\dots \\bigg \\rangle . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} a \\& b = \\min \\{ a , b \\} \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} \\sum _ { \\lambda \\in S _ { 1 } } m _ { a } ( C , \\lambda ) + \\sum _ { \\lambda \\in S _ { 2 } } m _ { a } ( C , \\lambda ) = \\sum _ { \\lambda \\in \\Lambda ( A ) } m _ { a } ( C , \\lambda ) + \\sum _ { \\lambda \\in S _ { 2 } } m _ { a } ( C , \\lambda ) = n , \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} h ( z ) = \\left ( \\frac { z - a } { z \\zeta ( z ) } \\right ) ^ { c } \\left ( \\frac { a ( 1 - a ^ 2 ) ^ { c - 1 } } { N ^ { 1 - c } \\Gamma ( c ) } \\frac { 1 } { z - a } \\right ) - \\frac { 1 } { \\zeta ( z ) \\Gamma ( c ) } = { \\cal O } \\left ( \\frac { c } { N } \\right ) . \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} E ( v ) = \\int _ { \\mathbb { R } } \\left [ \\left ( \\partial _ x v \\right ) ^ 2 - v ^ 2 \\log | v | + \\frac { 1 } { 2 } v ^ 2 \\right ] d x \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} \\sum _ { t = 0 } ^ \\infty f _ \\rho ( t ) & = \\sum _ { t = 1 } ^ \\infty \\frac { \\mu _ \\rho f _ \\rho ^ \\ast ( t ) } { t } \\\\ & = \\mu _ \\rho \\left ( \\sum _ { t = 1 } ^ { K _ \\delta } \\frac { f _ \\rho ^ \\ast ( t ) } { t } + \\sum _ { t = K _ \\delta + 1 } ^ \\infty \\frac { f ^ \\ast ( t ) } { t } \\right ) \\\\ & = \\mu _ \\rho \\left ( \\sum _ { t = 1 } ^ { K _ \\delta } \\frac { f _ \\rho ^ \\ast ( t ) } { t } + \\mu ( 1 - F ( K _ \\delta ) ) \\right ) = 1 , \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} \\bold { p r } \\widetilde { X } ( \\widetilde { L } _ n ) = ( S - \\operatorname { i d } ) P _ 0 . \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} a _ \\beta ( V ) = \\sum _ { v \\in V } \\beta ^ { | v | } , V \\in { \\cal E } . \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 - { c _ 4 \\over 4 8 } x - { c _ 6 \\over 8 6 4 } . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} U ^ + : = \\left \\{ u ^ + _ t : = \\begin{pmatrix} 1 & 0 \\\\ t & 1 \\end{pmatrix} : t \\in \\mathbb { R } \\right \\} \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} \\mathcal { I } _ G = \\{ F : F \\subseteq E \\mbox { a n d } ( V , F ) \\ \\hbox { i s } \\ u v \\hbox { - s p a r s e } \\} \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} { \\cal W } _ { \\mu , \\Lambda , \\lambda } ( z ) : = \\Xi _ { \\Lambda } ( \\beta , \\mu , \\lambda ) ^ { - 1 } \\\\ { \\rm { T r } } _ { { \\cal H } ^ { ' } } \\langle z | \\exp ( - \\beta H _ { \\Lambda , \\mu , \\lambda } ) | z \\rangle \\ . \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} d \\omega = * _ { 3 } d V = - * _ { 3 } \\frac { k d r } { 2 r ^ 2 } + O ( r ^ { - 3 } ) . \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\lim } ( \\Phi ^ { \\prime } ( u _ { n } , v _ { n } ) , ( u _ { n } - u , v _ { n } - v ) ) = \\underset { n \\rightarrow + \\infty } { \\lim } ( \\Phi ^ { \\prime } ( u _ { n } , v _ { n } ) - \\Phi ^ { \\prime } ( u , v ) , ( u _ { n } - u , v _ { n } - v ) ) = 0 . \\end{align*}"}
-{"id": "4592.png", "formula": "\\begin{align*} \\partial _ \\beta ^ k p ( \\xi , \\beta ) = O ( | \\xi | ^ k e ^ { \\frac \\beta h \\langle h \\xi \\rangle } ) . \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} ( \\overline { A } _ 1 ) _ j ^ { j ' } ( l ) : = - \\frac { 2 j \\ , c _ 2 \\ , ( j - j ' ) ^ 2 \\ , \\sqrt { \\lvert j - j ' \\rvert \\xi _ { j - j ' } } + 6 j \\ , c _ 3 \\ , \\sqrt { \\lvert j - j ' \\rvert \\xi _ { j - j ' } } } { \\overline { \\omega } \\cdot l + j '^ 3 - j ^ 3 } \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} \\sum _ { k = p _ { n } } ^ { q _ { n } } x _ { n } ( k ) \\alpha _ { n } ( k ) > c . \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} T \\gamma ( \\mathcal { R } ^ { \\gamma } _ H ) = T \\gamma \\left ( \\frac { \\partial } { \\partial t } + \\sum _ { i = 1 } ^ n \\frac { \\partial H } { \\partial p _ i } \\frac { \\partial } { \\partial q _ i } \\right ) \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} \\tilde { } : \\widehat { K ^ * } \\cong T ^ { \\Z _ p } \\times \\left ( 1 + T k [ [ T ] ] \\right ) \\to & T ^ { \\Z _ p } \\times \\left ( 1 + T W ( k ) [ [ T ] ] \\right ) \\\\ T ^ e \\cdot \\prod _ { ( i , p ) = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\infty } ( 1 - a _ { i j } T ^ i ) ^ { p ^ j } \\mapsto & T ^ e \\cdot \\prod _ { ( i , p ) = 1 } ^ { \\infty } \\prod _ { j = 0 } ^ { \\infty } ( 1 - [ a _ { i j } ] T ^ i ) ^ { p ^ j } . \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} \\sum _ { m \\geq 0 } ( - 1 ) ^ m \\frac { ( p ( j + 2 \\nu ) + 1 ) _ m } { ( b _ j + 1 ) _ m } \\frac { v ^ { m + p ( j + 2 \\nu ) - 1 } } { m ! } & = v ^ { p ( j + 2 \\nu ) - 1 } { } _ 1 F _ 1 \\left ( b _ j - a _ j + 1 , b _ j + 1 , - v \\right ) \\\\ & = e ^ { - v } v ^ { p ( j + 2 \\nu ) - 1 } { } _ 1 F _ 1 \\left ( a _ j , b _ j + 1 , v \\right ) . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} \\sum _ { \\nu = 0 } ^ { \\infty } \\frac { \\varphi ( p ^ \\nu ) } { p ^ { \\nu s } } = \\left ( 1 - \\frac 1 { p ^ s } \\right ) \\left ( 1 - \\frac 1 { p ^ { s - 1 } } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} & \\left \\| \\nabla ^ { k } _ { x } \\left ( \\partial _ { t } K _ { 1 L } ( t ) g + \\nabla _ { x } \\mathcal { F } ^ { - 1 } \\left [ e ^ { - \\frac { \\nu t | \\xi | ^ { 2 \\sigma } } { 2 } } \\sin ( t | \\xi | ) \\chi _ { L } \\right ] \\ast g \\right ) \\right \\| _ { 2 } \\\\ & \\le C ( 1 + t ) ^ { - \\frac { n } { 2 \\sigma } ( \\frac { 1 } { r } - \\frac { 1 } { 2 } ) - 1 - \\frac { k - \\tilde { k } } { 2 \\sigma } } \\| \\nabla _ { x } ^ { \\tilde { k } } g \\| _ { r } , \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} \\cdot \\prod ^ { 2 p } _ { j = l + 1 } p _ { \\varepsilon _ 2 } ( ( ( I - P _ h + P _ h ) \\widetilde { A } { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ u _ k ; v _ j ] } , \\widetilde { \\xi } ) - ( h , \\widetilde { \\xi } ) - \\zeta ) / ( P _ h \\widetilde { A } { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ u _ k ; v _ i ] } , \\widetilde { \\xi } ) , \\ i = \\overline { 1 , 2 p } , \\ ( h , \\widetilde { \\xi } ) , \\ \\zeta \\Big \\} d \\vec { v } . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} ( C _ 1 ) = \\left ( \\int _ { C _ 1 } \\omega \\right ) = \\left ( \\lim _ { \\varepsilon \\rightarrow 0 } \\int _ { - \\pi } ^ { \\pi } \\int _ \\varepsilon ^ { 1 } \\frac { 1 } { h } d h \\wedge d \\theta \\right ) = ( \\lim _ { \\varepsilon \\rightarrow 0 } - 2 \\pi \\log | \\varepsilon | ) \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} f ^ \\# ( z ) = \\frac { | f ' ( z ) | } { 1 + | f ( z ) | ^ 2 } \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} ( F ^ n ) ' ( u ) = z \\frac { ( f ^ n ) ' ( z ) } { f ^ n ( z ) } . \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} \\sqrt { n } \\int \\left \\Vert \\hat { \\Delta } _ { \\ell 1 } ( w ) \\right \\Vert F _ { 0 } ( d w ) & \\leq \\sqrt { n } \\left \\Vert \\hat { \\alpha } _ { 1 \\ell } - \\alpha _ { 1 0 } \\right \\Vert \\left \\Vert \\hat { \\gamma } _ { 1 \\ell } - \\gamma _ { 1 0 } \\right \\Vert \\\\ & = O _ { p } ( \\sqrt { n } \\{ n ^ { - d _ { 1 } [ ( 2 \\xi _ { 1 } - 1 ) / ( 2 \\xi _ { 1 } + 1 ) ] 2 \\xi _ { 2 } / [ 2 \\xi _ { 2 } + 1 ] } [ \\ln ( n ) ] ^ { 2 } + n ^ { - \\xi _ { 3 } / ( 2 \\xi _ { 3 } + 1 ) } \\ln ( n ) + n ^ { - d _ { 1 } } \\} n ^ { - d _ { 1 } } ) \\\\ & = o _ { p } ^ { { } } ( 1 ) . \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} - b _ i + \\min & ~ b ^ T H _ i \\\\ & G ^ T H _ i = A ^ T G _ i \\\\ & H _ i \\geq 0 . \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} \\lambda & = 1 + \\frac { \\log P } { \\log L } \\quad \\\\ C ^ { - 1 } & = 2 m p P \\left ( \\max \\left \\{ 1 , 2 l \\right \\} \\right ) ^ { \\lambda - 1 } . \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} B ( n ) = \\frac { n } { 2 k _ n } \\log ( 1 + k _ n P ) - \\frac { H _ 2 ( \\alpha _ n ) } { \\alpha _ n } . \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} p ^ { 4 } - 2 c p ^ { 3 } q + 2 p ^ { 2 } q ^ { 2 } + 2 c p q ^ { 3 } + q ^ { 4 } = \\mu , \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} \\sum _ { t = 1 } ^ { z _ n } t N _ t \\ge \\frac { L _ n } { 2 } \\sum _ { t = 1 } ^ { z _ n - 1 } t N _ t < \\frac { L _ n } { 2 } . \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} \\cdot \\prod ^ { 2 p } _ { j = l + 1 } p _ { \\varepsilon _ 2 } ( ( \\widetilde { A } { 1 \\ ! \\ ! \\ , { \\rm I } } _ { [ u _ k ; v _ j ] } , \\widetilde { \\xi } ) - ( h , \\widetilde { \\xi } ) - \\zeta ) d \\vec { v } , \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} L _ i ( h ) = \\int _ 0 ^ { T _ i ( h ) } | \\dot X ( t , x ) | \\ , d t = \\int _ 0 ^ { T _ i ( h ) } | D H ( X ( t , x ) ) | \\ , d t , \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} d v _ X = \\omega _ 0 \\wedge \\frac { ( d \\omega _ 0 ) ^ { n - 1 } } { ( n - 1 ) ! } . \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\psi _ { N , v , h } \\left ( \\gamma \\right ) : = \\exp \\left ( - 2 \\pi i v \\frac { c d } { N h } \\right ) , \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} V _ n ^ { ( k ) } : = \\sum _ { i = 0 } ^ { n - k } \\left ( \\frac { k } { n } \\right ) ^ { 2 h ( i / n ) } , \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} u ^ h & \\rightharpoonup u H ^ 1 ( 0 , L ) \\ , ; \\\\ v _ i ^ h & \\to v _ i H ^ 1 ( 0 , L ) \\ , , v _ i \\in H ^ 2 ( 0 , L ) i = 2 , 3 \\ , ; \\\\ w ^ h & \\rightharpoonup w H ^ 1 ( 0 , L ) \\ , . \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} \\frac { 2 i } { n } = \\frac { 2 d i } { ( n - i ) d + i d } > \\eqref { e q n 1 } \\geq \\frac { 2 d i \\Phi } { ( n - i ) d + d i \\Phi } = \\frac { 2 i \\Phi } { n + ( \\Phi - 1 ) i } . \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} H ^ { ( 5 ) } _ 5 = \\sum _ { q = 2 } ^ 5 R ( v ^ { 5 - q } z ^ q ) . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} A ^ h = \\frac { 2 \\eta B ^ h } { \\xi + \\eta } + \\frac { 2 \\tanh \\xi C ^ h } { \\xi + \\eta } . \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} \\frac { ( - n ) _ k } { k ! } = ( - 1 ) ^ k \\binom { n } { k } \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} \\frac { s ( \\theta ) } { | s | ( \\theta ) } = e ^ { f ( \\theta + \\alpha ) - f ( \\theta ) } . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} \\tilde { H } _ c = ( U _ { R } ^ { - 1 } U _ { k _ 0 } U _ R ) \\tilde { H } _ s ( U _ { R } ^ { - 1 } U _ { k _ 0 } U _ R ) ^ { - 1 } . \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{align*} c _ 3 = c _ 7 = 2 c _ 1 ^ 2 - c _ 4 = 7 c _ 2 ^ 2 - 6 c _ 6 = 0 \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} = \\frac { 1 } { \\sqrt { | N _ { M / \\mathbb { Q } } ( D _ { \\mathcal { O } / M } ) | } } \\cdot \\prod _ { i = 1 } ^ { m } \\ ; \\ ; \\prod _ { 1 \\leq j _ { 1 } < j _ { 2 } \\leq k } \\left \\vert \\alpha ^ { ( i , j _ { 1 } ) } - \\alpha ^ { ( i , j _ { 2 } ) } \\right \\vert \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} f ( h v | X ) & = \\sum _ { k = 0 } ^ { 1 } v ^ { k } h ^ { k } \\frac { d ^ { k } f ( 0 | X ) } { d u ^ { k } } + h ^ { 2 } R ( v , X ) , \\\\ \\left \\vert E [ K _ { h } ( U ) | X ] - f ( 0 | X ) \\right \\vert & = \\left \\vert \\int K ( v ) [ f ( h v | X ) - f ( 0 | X ) ] d v \\right \\vert \\leq C h ^ { 2 } \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\lambda z _ { 1 } & = \\alpha n _ { 2 } z _ { 1 } + \\left ( 1 - \\alpha \\right ) n _ { 2 } z _ { 2 } , \\\\ \\lambda z _ { 2 } & = \\alpha n _ { 1 } x _ { 2 } + \\left ( 1 - \\alpha \\right ) n _ { 1 } z _ { 1 } . \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} u ^ h ( x _ 1 ) & = \\int _ \\omega \\frac { \\hat y _ 1 ^ h - x _ 1 } { h ^ 2 } \\dd x ' \\ , , v _ i ^ h ( x _ 1 ) = \\int _ \\omega \\frac { \\hat y _ i ^ h } { h } \\dd x ' \\ , \\ i = 2 , 3 \\ , , \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{align*} S ^ N ( \\mathbf { \\bar \\xi } ) : = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\delta _ { \\bar \\xi ^ { i , N } } . \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} \\alpha = ( \\underbrace { 1 , \\ldots , 1 } _ n , \\underbrace { r _ n } _ { \\neq 1 } , r _ { n + 1 } , r _ { n + 1 } \\ldots ) \\mapsto s ( \\alpha ) = ( \\underbrace { 1 , \\ldots , 1 } _ { n + 1 } , 0 , 0 , \\ldots ) : \\Gamma _ { \\log } \\to \\Psi _ { \\log } \\subseteq \\Gamma _ { \\log } \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} A _ \\infty \\cap A ' = \\{ x \\in A _ { \\infty } \\mid x a = a x ~ ~ a \\in A \\} . \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{align*} r _ 2 ( N ) = 4 \\prod _ { i \\geq 1 } ( 1 + v _ i ) \\prod _ { j \\geq 1 } \\frac { 1 + ( - 1 ) ^ { w _ j } } { \\tiny 2 } . \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} P \\left ( \\max _ { i = 1 , 2 , \\ldots , M } ~ \\sigma _ { \\rm j , i } ^ 2 > c \\right ) < \\frac { \\epsilon } { 4 } . \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} U ( \\beta , \\alpha + \\beta ; t ) : = \\frac { 1 } { \\Gamma ( \\beta ) } \\sigma ( t ; \\alpha , \\beta ) . \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} Z ( x , y ) = b _ 0 + b _ 1 x - b _ 2 x ^ 2 + O ( x ^ 3 ) + O ( y ) , \\ \\ \\ b _ 2 > 0 . \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} A f ( x ) = \\sum _ { [ \\xi ] \\in \\widehat { G } } d _ { \\xi } [ \\xi ( x ) \\sigma _ A ( x , \\xi ) \\widehat { f } ( \\xi ) ] . \\end{align*}"}
-{"id": "2102.png", "formula": "\\begin{align*} \\hat { r } = a _ 0 \\frac { \\sigma ( t ) ^ 5 - t ^ 5 } { t ^ 6 } + y _ 1 \\frac { \\sigma ( t ) ^ 6 - t ^ 6 } { t ^ 6 } \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} R _ { \\ast } U = - i \\sum _ { k = 1 } ^ { n - 1 } R ( e _ k , \\overline e _ k ) J U , ~ ~ U \\in \\Gamma ( H X ) . \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} E ^ { n } _ { N F } = E ^ n _ { N F , h i g h } + E ^ n _ { N F , l o w } , \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} F _ 1 = \\Q _ 2 ( t ) t ^ 4 + 1 2 t ^ 2 + 6 = 0 \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} w _ t = \\partial _ x L w - \\partial _ x N ( w ) , \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{align*} \\mathrm { \\cfrac { d } { d t } } \\ , H ( X ^ \\pm ( t ) ) = \\pm \\nu | D H ( X ^ \\pm ( t ) ) | \\ \\ \\ t > 0 , \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} - x / t ^ 2 \\cdot ( \\nabla u ) ( x / t ) = \\frac { \\lvert x \\rvert } { t ^ 3 } \\chi _ { \\lvert x \\rvert < t } \\geq \\frac { \\chi _ { t / 2 \\leq \\lvert x \\rvert < t } } { 2 t ^ 2 } \\geq \\frac { 1 } { 2 \\omega _ { + } ^ 2 } \\chi _ { B ( x _ 0 , \\omega _ { - } / 4 ) } \\geq 2 \\chi _ { B ( x _ 0 , \\omega _ { - } / 4 ) } \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} z ^ { \\alpha } : = \\prod _ { j = 1 } ^ g z _ j ^ { \\alpha _ j } , \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} \\int _ { x _ \\frac \\mu 2 } ^ { x _ \\mu - \\delta } \\frac { Q ' \\ , \\dd x } { \\mu - Q } = \\log \\frac { \\mu } { 2 ( Q ( x _ \\mu ) - Q ( x _ \\mu - \\delta ) ) } \\leq C \\log \\frac { \\mu } { a _ \\mu \\delta } \\leq C \\log ( \\mu a _ \\mu ^ { - \\frac 2 3 } ) . \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} g ( t ) : = \\left ( \\int _ { \\R } G _ t ( y ) d y \\right ) ^ p = \\begin{cases} 1 , & \\\\ t ^ p , & \\end{cases} \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\bold { p r } X ( L ) = \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 , \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\mathcal { B } ( x , y ) = \\delta _ { x 0 } = { \\begin{cases} 1 & { } x = 0 , x \\in \\mathbb { Z } , y \\in \\left . \\left ( 0 , \\frac { 1 } { 2 } \\right . \\right ] , y \\in \\mathbb { R } \\\\ 0 & { } x \\ne 0 \\ \\ x \\in \\mathbb { Z } , y \\in \\left . \\left ( 0 , \\frac { 1 } { 2 } \\right . \\right ] , y \\in \\mathbb { R } \\end{cases} } \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} \\left | \\frac { \\partial _ 1 u ( y ) } { | y - s t e _ 1 - x | ^ { N + 2 \\sigma } } 1 _ A ( y ) \\right | \\leq \\frac { | \\partial _ 1 u ( y ) | } { 1 + | y | ^ { N + 2 \\sigma } } \\frac { 1 + | y | ^ { N + 2 \\sigma } } { | y - s t e _ 1 - x | ^ { N + 2 \\sigma } } 1 _ A ( y ) \\leq K \\frac { | \\partial _ 1 u ( y ) | } { 1 + | y | ^ { N + 2 \\sigma } } = : f ( y ) , \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} d _ H ( x , y ) : = \\log \\left ( \\max _ i \\frac { x _ i } { y _ i } \\right ) - \\log \\left ( \\min _ j \\frac { x _ j } { y _ j } \\right ) \\mbox { f o r } x , y \\in \\R ^ n _ { > 0 } . \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c l l } \\displaystyle - \\Delta _ p \\ , u = g ( | x | ) f ( u ) & \\textup { i n } & \\Omega _ { a , b } , \\\\ \\displaystyle u = 0 & \\textup { o n } & \\partial \\Omega _ { a , b } \\end{array} \\right . \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} x = u ^ 2 x ' + r , y = u ^ 3 y ' + u ^ 2 s x ' + T , \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{gather*} \\pi _ m ( [ i , j ] ) = [ i , j ] - \\sum _ { k \\neq \\{ i , j \\} } m _ k \\partial _ 2 ( [ i , j , k ] ) , \\end{gather*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\dot u _ i ( t ) = u _ { i + 1 } ( t ) - 2 u _ i ( t ) + u _ { i - 1 } ( t ) + u _ i ( t ) \\big ( \\bar f _ T - M u _ i ( t ) \\big ) . \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} u ( x ) = 2 k _ { N , 1 } ( 1 - | x | ^ 2 ) ^ { s - 1 } \\int _ { \\partial B } \\frac { 1 - | x | ^ 2 } { | x - \\theta | ^ N } \\varphi ( \\theta ) \\ d \\theta = \\frac { 2 k _ { N , 1 } { s } } { k _ { N , s } } \\int _ { \\partial B } M _ s ( x , \\theta ) \\varphi ( \\theta ) \\ d \\theta , \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} P _ { n + 1 } ( x , y , q ) = ( 2 n x + q y ) P _ n ( x , y , q ) + 2 x ( 1 - x ) \\frac { \\partial } { \\partial x } P _ n ( x , y , q ) + 2 x ( 1 - y ) \\frac { \\partial } { \\partial y } P _ n ( x , y , q ) . \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial \\lambda _ { j b } } ( \\lambda | \\lambda ' ) = s _ b L ( \\lambda | \\lambda _ { j b } ) F ( \\lambda _ { j b } | \\lambda ' ) \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} u _ 1 ^ \\prime ( 0 ) = u _ 2 ^ \\prime ( 0 ) = u _ 1 ^ { \\prime \\prime \\prime } ( 0 ) = u _ 2 ^ { \\prime \\prime \\prime } ( 0 ) = 0 , \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} \\mathbb { E } \\left \\{ \\boldsymbol { \\eta } _ n ^ { ( g ) } \\left ( \\boldsymbol { \\eta } _ { n ' } ^ { ( g ) } \\right ) ^ H \\right \\} = \\mathbf { R } ^ { ( g ) } _ { \\boldsymbol { \\eta } } \\delta _ { n n ' } , \\textrm { w h e r e } \\mathbf { R } ^ { ( g ) } _ { \\boldsymbol { \\eta } } \\triangleq E _ s \\left ( \\sum _ { g ' \\neq g } \\gamma ^ { ( g ' ) } K _ { g ' } \\sum _ { l = 0 } ^ { L _ { g ' } - 1 } \\rho _ l ^ { ( g ' ) } \\mathbf { R } _ l ^ { ( g ' ) } \\right ) + N _ 0 \\mathbf { I } _ N , \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} t _ n \\leq \\rho \\sum _ { j = 0 } ^ { n - 1 } t _ { j } \\quad \\ n \\geq n _ 0 . \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + a _ 2 x ^ 2 + a _ 4 x + a _ 6 . \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\vec { 1 } \\cdot \\mathbb { M } ( \\lambda \\vec { 1 } ) ^ { - 1 } \\vec { 1 } = \\frac { \\nu } { a ( \\lambda ) + b ( \\lambda ) \\nu } . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} a _ 1 '' ( u ) = - \\frac { d w ' + \\sum _ { j = 2 } ^ n a _ j ' d b _ j ' } { d b _ 1 ' } , u \\in S _ 1 . \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} H ( 1 + k _ \\gamma ) = H \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\ , \\sum _ { k = 1 } ^ n \\big | X ^ { ( n ) } _ { ( k ) } ( s ) - \\bar { X } ^ { ( n ) } _ { ( k ) } ( s ) \\big | ^ p = W _ p \\big ( \\rho ^ { ( n ) } ( s ) , \\bar { \\rho } ^ { ( n ) } ( s ) \\big ) ^ p \\le \\frac { 1 } { n } \\ , \\sum _ { i = 1 } ^ n \\big | X ^ { ( n ) } _ i ( s ) - \\bar { X } ^ { ( n ) } _ i ( s ) \\big | ^ p \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } A _ { 1 1 } h ^ \\pm _ \\tau ( \\tau ) + A _ { 1 2 } v ^ \\pm _ \\tau ( \\tau ) = 0 , \\\\ A _ { 2 1 } h ^ \\pm _ \\tau ( \\tau ) + A _ { 2 2 } v ^ \\pm _ \\tau ( \\tau ) = Q ' _ { 2 2 } ( u ^ \\pm ) v ^ \\pm ( \\tau ) + Q ( h ^ \\pm ( \\tau ) + v ^ \\pm ( \\tau ) ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} x _ n = ( 1 + \\lambda + \\eta _ n ) ^ n , \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} \\nu _ i : = \\left \\{ \\begin{array} { l c l } \\sum _ { j \\in I } w _ j B _ I ^ { j } B _ I ^ { i } \\det G _ { I \\backslash j , I \\backslash i } & & | I | > 1 , \\\\ w _ i & & | I | = 1 \\end{array} \\right . i \\in I \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} b _ 1 B a _ 2 B ^ { - 1 } a _ 2 ^ { - 1 } & \\stackrel { } { = } B a _ 2 B ^ { - 1 } b _ 1 a _ 2 ^ { - 1 } = B ( a _ 2 B ^ { - 1 } a _ 2 ^ { - 1 } ) ( a _ 2 b _ 1 a _ 2 ^ { - 1 } ) , \\ ; \\\\ a _ 2 b _ 1 a _ 2 ^ { - 1 } & = a _ 2 B a _ 2 ^ { - 1 } B ^ { - 1 } a _ 2 B ^ { - 1 } a _ 2 ^ { - 1 } , \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} s ^ i ( \\hat { x } _ t , \\theta _ t ) & = \\left [ \\frac { ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } } { \\sum _ j ( \\theta _ t ^ j ) ^ { \\frac { 1 } { \\gamma } } } \\right ] \\ , \\hat { x } _ t + \\frac { \\gamma \\hat { \\phi } } { \\eta } \\left [ \\frac { ( \\theta _ t ^ i ) ^ { \\frac { 1 } { \\gamma } } } { \\sum _ j ( \\theta _ t ^ j ) ^ { \\frac { 1 } { \\gamma } } } - \\frac { \\phi ^ i } { \\hat { \\phi } } \\right ] , \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\underline { Z } } \\sup _ { g \\in \\mathcal { G } } \\left \\{ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n Z _ i ( g ^ 2 ( X _ i ) - g ^ 2 ( X ^ { \\prime } _ i ) ) - \\frac { \\gamma } { n } L ( g ) - \\frac { 1 } { 2 \\gamma } s ^ 2 ( g ) \\right \\} & \\\\ \\leq \\frac { \\gamma } { n } \\log \\mathbb { E } _ { \\underline { Z } } \\sup _ { g \\in \\mathcal { G } } \\exp \\left \\{ \\frac { 1 } { \\gamma } \\sum _ { i = 1 } ^ n Z _ i ( g ^ 2 ( X _ i ) - g ^ 2 ( X ^ { \\prime } _ i ) ) - L ( g ) - \\frac { n } { 2 \\gamma ^ 2 } s ^ 2 ( g ) \\right \\} . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} \\partial _ { \\lambda ' } F ( \\lambda | \\lambda ' ) = - L ( \\lambda | \\lambda ' ) . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} \\deg _ t = t \\frac { \\partial } { \\partial t } , \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{align*} v = \\sum _ { k \\in \\Gamma } \\langle \\mathcal { E } v , \\widetilde { \\phi } _ k \\rangle \\ , \\phi _ k + \\sum _ { ( i , j , k ) \\in \\Lambda } \\langle \\mathcal { E } v , \\widetilde { \\psi } _ { i , j , k } \\rangle \\ , \\psi _ { i , j , k } \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\delta ( \\rho V ) = 0 , \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} [ n ] _ q : = \\frac { q ^ n - q ^ { - n } } { q - q ^ { - 1 } } \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{align*} \\omega _ { n } : = \\prod _ { j = 0 } ^ { n } ( 1 - | \\alpha _ j | ^ 2 ) . \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} \\varphi _ 0 ( x ) = 1 | x | \\leq 1 \\quad \\varphi _ 0 ( x ) = 0 | x | \\geq 3 / 2 , \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} \\mathcal { A } _ { i } & = \\{ X _ { j } : { \\textstyle \\sum _ { x \\in X _ { j } } } f ( x ) = j , \\ j = 1 , 2 , 3 \\} \\\\ \\mathcal { A } _ { 4 } & = \\{ X _ { j } : { \\textstyle \\sum _ { x \\in X _ { j } } } f ( x ) \\geq 4 \\} \\\\ \\mathcal { A } & = { \\textstyle \\bigcup _ { i = 1 } ^ { 4 } } \\mathcal { A } _ { i } . \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} N _ { ( 1 , 1 0 ) } ( n ) & = 8 \\sigma ( n ) - 3 2 \\sigma ( \\frac { n } { 4 } ) + 8 \\sigma ( \\frac { n } { 1 0 } ) - 3 2 \\sigma ( \\frac { n } { 4 0 } ) + 6 4 \\ , W _ { ( 1 , 1 0 ) } ( n ) + 1 0 2 4 \\ , W _ { ( 1 , 1 0 ) } ( \\frac { n } { 4 } ) \\\\ & - 2 5 6 \\ , \\biggl ( W _ { ( 2 , 5 ) } ( \\frac { n } { 2 } ) + W _ { ( 1 , 4 0 ) } ( n ) \\biggr ) , \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} u _ t + u u _ x + u _ { x x x } = 0 \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} H ^ { \\epsilon , \\kappa } : = ( - i \\partial _ { x _ 1 } - A ^ { \\epsilon , \\kappa } _ 1 ) ^ 2 + ( - i \\partial _ { x _ 2 } - A ^ { \\epsilon , \\kappa } _ 2 ) ^ 2 + V \\ , . \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} \\mu \\otimes \\nu ( G ( A \\times C ) ) = \\int _ X \\Big ( \\int _ Y \\chi _ { G ( A \\times C ) } ( x , y ) \\ , d \\nu ( y ) \\Big ) \\ , d \\mu ( x ) = \\int _ X \\nu ( A _ x ^ { - 1 } C ) \\ , d \\mu ( x ) . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} d v ^ h _ t & = \\left ( ( L ^ h _ t + I ^ h ) v ^ h _ t + f _ t \\right ) d t \\\\ v ^ h _ 0 & = \\psi . \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} \\lambda \\mathbb { E } ^ { 0 ' } \\left [ \\left \\vert V \\circ H ^ - ( 0 ) \\right \\vert \\right ] = \\lambda ' \\mathbb { E } ^ { 0 } \\left [ \\left \\vert V ' \\circ H ^ + ( 0 ) \\right \\vert \\right ] \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} p = \\frac { 2 c ^ 4 + 2 c ^ 3 - 5 c ^ 2 - 2 c + 3 } { 3 c \\left ( c ^ 3 + 2 c + 6 \\right ) } . \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} d ( \\alpha \\wedge \\beta ) = d \\alpha \\wedge \\beta + ( - 1 ) ^ k \\alpha \\wedge d \\beta . \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} d _ { \\lambda } ( p _ * , p _ { \\theta } ) = - \\frac { 1 } { 1 - \\lambda } \\log Z ^ { \\lambda } _ { \\theta } . \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} & \\int _ { e ^ { - ( 2 + \\delta ) y } } ^ \\infty t ^ { a + 2 b + 3 c - \\frac { 3 } { 2 } - m - \\frac { \\epsilon } { 2 } } \\left [ e ^ { y } V ( \\kappa - \\Lambda ) \\right ] ^ { a + 3 b + 4 c - 2 m } e ^ { - t e ^ { 2 y } V ^ 2 ( \\kappa - \\Lambda ) ^ 2 } d t \\\\ & = O \\left ( e ^ { - ( j + c - 1 - \\epsilon ) y } \\right ) + O \\left ( e ^ { \\frac { \\delta } { 2 } y - ( j + c - 1 - \\epsilon ) ( 1 + \\frac { \\delta } { 2 } ) y } \\right ) , \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} H _ { \\overline { g } } : \\ ; \\partial _ 0 R _ { i j } - \\overline { \\nabla } _ i R _ { j 0 } - \\overline { \\nabla } _ j R _ { i 0 } = \\partial _ 0 \\Lambda _ { i j } - \\overline { \\nabla } _ i \\Lambda _ { j 0 } - \\overline { \\nabla } _ j \\Lambda _ { i 0 } ; \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} R _ 1 = G _ 1 - G _ 2 \\mu , S _ 1 = G _ 2 . \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} \\psi = \\sum _ { j \\in \\N } b ( j ) P ^ j f _ k . \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\mathrm { S L } _ 2 ( \\mathbb { Z } ) = \\left \\{ \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix} : a , b , c , d \\in \\mathbb { Z } , a d - b c = 1 \\right \\} , \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} E \\ ; : \\ ; y ^ 2 = x ^ 3 + a x ^ 2 + b x , \\Delta = \\Delta _ m = 2 ^ 4 b ^ 2 ( a ^ 2 - 4 b ) . \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} \\| \\bold { A } \\vec { \\varphi } \\| ^ 2 \\to \\max , \\| \\vec { \\varphi } \\| ^ 2 = \\sum \\limits _ { j = 0 } ^ { \\infty } | \\varphi _ { j } | ^ { 2 } = P , \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} \\mathcal { D } = \\mathfrak { B } \\sqcup & \\bigsqcup _ f \\{ \\frac { b X ^ i } { f ^ j } : \\ ( j , p ) = 1 , \\ j \\geq 1 , \\ 0 \\leq i < \\deg ( f ) , \\ b \\in \\mathcal { C } \\} \\\\ \\sqcup & \\{ b X ^ j , \\ ( j , p ) = 1 , \\ j \\geq 1 , \\ b \\in \\mathcal { C } \\} , \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} b _ 3 - 1 = \\Upsilon ( M _ x [ g ( a _ 1 - 1 ) - g ( 0 ) ] ) - \\Upsilon ( 0 ) . \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} \\nabla ^ { \\pi ^ \\Lambda _ * F , u } : = \\nabla ^ { \\pi ^ \\Lambda _ * F } + \\frac { 1 } { 2 } k \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} \\left ( \\mathsf { I d } - \\lambda _ { 1 } ( \\mathcal { Q } _ \\infty ) ( - \\Delta ) ^ { - 1 } \\right ) v = ( - \\Delta ) ^ { - 1 } \\lambda \\psi _ { 1 , \\infty } - ( - \\Delta ) ^ { - 1 } \\left ( - \\Delta - \\lambda _ { 1 } ( \\mathcal { Q } _ \\infty ) \\right ) w \\ , . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} \\begin{aligned} & A _ h ( x ; u , v ) = { \\mathfrak S } ( h ; u , v ) ( u v ) ^ { - 1 } B ( x ; u , v ) \\\\ & \\quad + O \\bigl ( \\tau ( h ) ( u v D ) ^ 6 x ^ { \\frac 3 4 } ( \\log x ) ^ 4 + \\tau ( h ) ( u v ) ^ { - 1 } x ( \\log x ) ^ { - 2 0 } \\bigr ) \\ . \\end{aligned} \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} \\psi \\left ( \\dots , x _ i ^ { ( a ) } , \\dots , x _ j ^ { ( b ) } \\dots \\right ) = \\psi \\left ( \\dots , x _ j ^ { ( b ) } , \\dots , x _ i ^ { ( a ) } , \\dots \\right ) , \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ n \\tilde { \\theta } _ 0 ^ i ( \\tilde { \\delta } ^ i ) ^ t \\right ) ^ \\gamma \\left ( \\hat { \\phi } + \\frac { \\eta } { \\gamma } \\ , \\hat { x } _ t \\right ) ^ { - \\gamma } = \\left ( \\sum _ { i = 1 } ^ n \\tilde { \\theta } _ 0 ^ i ( \\tilde { \\delta } ^ i ) ^ { t + 1 } \\right ) ^ \\gamma \\left ( \\hat { \\phi } + \\frac { \\eta } { \\gamma } \\ , \\hat { x } _ { t + 1 } \\right ) ^ { - \\gamma } f ' ( k _ { t + 1 } ) . \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} \\phi ^ { \\otimes } ( F ) : = \\sum _ { n = 0 } ^ \\infty a _ n F ^ { \\otimes n } \\in \\mathbb { F } _ q ( \\mathcal { H } _ - \\ , , - 2 ) _ { \\mathbb C } . \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{gather*} \\{ L ' _ 1 , L ' _ 1 \\} _ 1 = - i _ Q ^ 2 \\omega _ 1 \\simeq 0 . \\end{gather*}"}
-{"id": "7142.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac { \\int _ { r T } ^ { T } f ( x \\Delta ( u _ t ) ) d t } { \\int _ { r T } ^ { T } \\Psi ( x \\Delta ( u _ t ) ) d t } = \\frac { \\mu ( f ) } { \\mu ( \\Psi ) } \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} \\beta = \\left ( \\begin{array} { c c } 1 _ 2 & 0 \\\\ 0 & - 1 _ 2 \\end{array} \\right ) , \\ ; \\gamma _ 5 = \\left ( \\begin{array} { c c } 0 & 1 _ 2 \\\\ 1 _ 2 & 0 \\end{array} \\right ) , \\ ; \\alpha _ k = \\left ( \\begin{array} { c c } 0 & \\sigma _ k \\\\ \\sigma _ k & 0 \\end{array} \\right ) \\mbox { f o r } k = 1 , 2 , 3 \\ , . \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} \\mathcal { I } ( M ) : = \\sum _ { \\lambda \\in \\Lambda ( M ) } \\left ( m _ { g } ( M , \\lambda ) - 1 \\right ) . \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} \\| ( \\Gamma _ { m , b c } X _ * ) \\mathbb { P } _ n \\| _ 2 = \\| \\Gamma _ { m , b c } ( X _ * \\mathbb { P } _ n - \\mathbb { P } _ n X _ * \\mathbb { P } _ n ) \\| _ 2 \\leq 2 d _ n \\| B \\mathbb { P } _ n - \\mathbb { P } _ n B \\mathbb { P } _ n \\| _ 2 , \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} 0 & = ( B F - F B ' ) v \\\\ & \\Rightarrow F ( B ' v ) = 0 \\mbox { , f o r a l l } v \\in \\ker ( F ) \\\\ & \\Rightarrow B ' ( v ) \\in \\ker ( F ) \\mbox { , f o r a l l } v \\in \\ker ( F ) \\\\ & \\Rightarrow B ' ( \\ker ( F ) ) \\subset \\ker ( F ) \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\vartheta _ { N } ^ n = \\left ( x _ 0 D _ 0 x _ 1 D _ 1 \\cdots x _ { N - 1 } D _ { N - 1 } \\right ) ^ n = ( x _ 0 D _ 0 ) ^ n ( x _ 1 D _ 1 ) ^ n \\cdots ( x _ { N - 1 } D _ { N - 1 } ) ^ n . \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} Z ( G ) ( ( z _ v ) _ { v \\in V } ) = \\sum _ { \\substack { I \\subseteq V \\\\ } } \\prod _ { v \\in I } z _ v . \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\mathrm { d i v } \\left ( \\frac { r _ n ^ { 2 p } } { 2 \\sqrt { V } } | F _ { 1 m } | ^ 2 \\hat { e } _ 1 \\right ) = r _ n ^ { 2 p } V ^ { - \\frac { 1 } { 2 } } \\frac { 1 } { 2 } \\hat e _ 1 | F _ A | ^ 2 & + ( p \\chi _ n + 1 ) \\frac { r _ n ^ { 2 p } | F _ { 1 m } | ^ 2 } { r V } \\\\ & + O \\left ( | F _ { 1 m } | ^ 2 r _ n ^ { 2 p } r ^ { - 2 } \\right ) . \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} r ( t ) = P ( t ) + \\int \\limits _ { 0 } ^ { t } Q ( t , \\tau ) r ( \\tau ) d \\tau \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} x ( u , t ) = \\int ^ t _ 0 \\int _ { { \\mathbb R } } p _ { t - s } ( u - v ) W ( d v , d s ) , \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} w _ 1 ' = & - ( u _ { l o } + U ) w _ 2 - U ^ 2 ( w _ 1 - w _ 2 ) \\\\ w _ 2 ' = & - 2 \\tau w _ 2 - U w _ 1 + u _ { l o } ( w _ 1 - 2 w _ 2 ) - U ^ 2 ( w _ 1 - w _ 2 ) \\end{aligned} \\right . \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} R _ { j } = \\alpha _ j { R _ { \\Sigma } } ( \\boldsymbol { \\alpha } ) . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} 0 = \\widetilde { F } _ { \\alpha } ( n , [ \\widetilde { u } ] ) = F _ { \\alpha } ( n , [ T ^ { - 1 } ( n , \\widetilde { u } _ n ) ] ) . \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} x '' ( t ) = - \\lambda f ( t , x ( t ) ) , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} \\langle \\phi , \\psi _ { x , y } \\rangle _ { \\mathcal { H } _ { p , \\varepsilon } } = \\int _ 0 ^ { t _ n } f ( \\frac s { t _ n } , B _ { 0 , t _ n } ( s ) + \\frac { s } { \\sqrt { t _ n t } } y ) d s . \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} F ( s ; a , z ) : = \\prod _ p \\ ( 1 + \\frac { z \\lambda _ a ( p ) } { \\phi ( q ) p ^ s } \\ ) , \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} L ( E ) = L ( \\alpha , A _ { d , E } ( \\cdot + i \\epsilon ) ) = L ( \\alpha , \\tilde { A } _ { | d | , E } ( \\cdot + i \\epsilon ) ) = 0 \\ \\ \\mathrm { f o r } \\ \\ | \\epsilon | \\leq \\frac { L ( \\lambda ) } { 2 \\pi } , \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} 4 ( \\tau _ 2 - \\tau _ 1 ) = \\frac { t _ 2 - t _ 1 } { ( 1 - t _ 1 ) ( 1 - t _ 2 ) } , \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\xi _ r : = \\xi \\phi ^ { - 1 } ( \\xi ) \\cdots \\phi ^ { - ( r - 1 ) } ( \\xi ) \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} \\varphi ( q ) : = \\sum _ { n = - \\infty } ^ \\infty q ^ { n ^ 2 } , \\psi ( q ) : = \\sum _ { n = 0 } ^ \\infty q ^ { n ( n + 1 ) / 2 } . \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{gather*} = x \\sum _ { j = 1 } ^ N \\frac { A _ j } { ( \\log x ) ^ { j - 1 / 2 } } \\sum _ { d \\le x / 2 } \\frac { h _ { 1 / \\tau } ( d ) } { d ( 1 - \\frac { \\log d } { \\log x } ) ^ { j - 1 / 2 } } + O \\left ( \\frac { x } { ( \\log x ) ^ { N + 1 / 2 } } \\sum _ { d \\le x / 2 } \\frac { | h _ { 1 / \\tau } ( d ) | } { d ( 1 - \\frac { \\log d } { \\log x } ) ^ { N + 1 / 2 } } \\right ) \\\\ + \\sum _ { x / 2 < d \\le x } h _ { 1 / \\tau } ( d ) . \\end{gather*}"}
-{"id": "7701.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty | w u | ^ q \\ , \\dd x = \\left ( \\int _ 0 ^ { x _ { \\frac \\mu 2 } } + \\int _ { x _ { \\frac \\mu 2 } } ^ { x _ \\mu - \\delta } + \\int _ { x _ \\mu - \\delta } ^ { x _ \\mu + \\delta _ 1 } + \\int _ { x _ \\mu + \\delta _ 1 } ^ { x _ { \\frac 3 2 \\mu } } + \\int _ { x _ { \\frac 3 2 \\mu } } ^ \\infty \\right ) | w u | ^ q \\ , \\dd x . \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} | \\mathrm { R e s } ( f ( x ) , g ( x ) ) | = \\prod _ { g ( \\lambda ) = 0 } | f ( \\lambda ) | \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} & \\bigg [ w _ 1 w _ 2 \\cdots w _ { n - 1 } y + w _ 1 w _ 2 \\cdots w _ n x + w _ n \\sum _ { j = 1 } ^ { n - 1 } \\big ( \\prod _ { \\substack { i = 1 \\\\ i \\neq j } } ^ { n - 1 } w _ i \\big ) k _ j \\bigg ] _ q \\\\ & = [ w _ 1 w _ 2 \\cdots w _ { n - 1 } ] _ q \\Big [ y + w _ n x + \\frac { w _ n } { w _ 1 } k _ 1 + \\cdots + \\frac { w _ n } { w _ { n - 1 } } k _ { n - 1 } \\Big ] _ { q ^ { w _ 1 w _ 2 \\cdots w _ { n - 1 } } } . \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} Q _ { - 1 } U = U + c _ a \\delta _ a + c _ b \\delta _ b , \\textrm { f o r s o m e } c _ a , c _ b \\in \\C . \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} T _ 2 ( z ) = \\int _ { x _ 1 < y _ 1 } u ( y _ 1 ) \\overline { v ( x _ 1 ) } e ^ { 2 i z ( y _ 1 - x _ 1 ) } d x _ 1 d y _ 1 \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} V _ { 2 1 3 } ( 4 2 1 3 5 ) = \\bigl \\{ \\{ 1 , 2 , 5 \\} , \\{ 1 , 3 , 5 \\} , \\{ 1 , 4 , 5 \\} , \\{ 2 , 3 , 4 \\} , \\{ 2 , 3 , 5 \\} \\bigr \\} . \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} \\| f ' ( z ) \\| = f ^ \\# ( z ) ( 1 + | z | ^ 2 ) = | f ' ( z ) | \\frac { 1 + | z | ^ 2 } { 1 + | f ( z ) | ^ 2 } . \\end{align*}"}
-{"id": "7614.png", "formula": "\\begin{align*} m ( L ' / L , V ) = \\displaystyle \\sum _ { \\psi \\in \\mathrm { G a l ( L ' / L ) } } m _ L ( V ) - m _ L ( V ( \\psi ) ) . \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} e ^ n _ j \\left ( x \\right ) = \\left \\{ \\begin{array} { c l } \\widetilde { e _ j } \\left ( \\widetilde { x } \\right ) & x \\in \\mathcal { U } ^ n _ j ~ \\& ~ x = \\widetilde { \\pi } ^ n \\left ( \\widetilde { x } \\right ) ~ \\& ~ \\widetilde { x } \\in \\widetilde { \\mathcal { U } } _ j \\\\ 0 & x \\notin \\mathcal { U } ^ n _ j \\end{array} \\right . \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} d _ { S ( G , t + 1 ) } ( x ^ { t + 1 } , y ^ { t + 1 } ) \\le & d _ { S ( G , t + 1 ) } ( x ^ { t + 1 } , x ( u _ 1 ) ^ t ) + \\sum _ { i = 0 } ^ { s - 2 } d _ { S ( G , t + 1 ) } ( u _ { i + 1 } ( u _ i ) ^ t , u _ { i + 1 } ( u _ { i + 2 } ) ^ t ) + \\\\ & + d _ { S ( G , t + 1 ) } ( y ( u _ { s - 1 } ) ^ t , y ^ { t + 1 } ) + s \\\\ = & 2 ^ t - 1 + \\sum _ { i = 0 } ^ { s - 2 } 2 \\left ( 2 ^ t - 1 \\right ) + 2 ^ t - 1 + s \\\\ = & 2 ^ t - 1 + 2 \\left ( 2 ^ t - 1 \\right ) ( d _ G ( x , y ) - 1 ) + 2 ^ t - 1 + d _ G ( x , y ) \\\\ = & ( 2 ^ { t + 1 } - 1 ) d _ G ( x , y ) . \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ n T ( n , k ) H _ { 2 k } & = \\sum _ { k = 0 } ^ n T ( n , k ) \\sum _ { j = 1 } ^ k \\frac 1 { 2 j - 1 } + \\frac { 1 } { 2 } \\sum _ { k = 0 } ^ n T ( n , k ) H _ { k } \\\\ & = \\sum _ { k = 0 } ^ n T ( n , k ) \\sum _ { j = 1 } ^ n \\frac 1 { 2 j - 1 } - \\sum _ { j = 0 } ^ { n - 1 } \\frac 1 { 2 j + 1 } \\sum _ { k = 0 } ^ j T ( n , k ) + H _ n \\\\ & = H _ { 2 n } - \\frac 1 2 H _ n - H _ { 2 n } + \\frac 5 2 H _ n - H _ { \\lfloor n / 2 \\rfloor } + H _ n \\\\ & = 3 H _ n - H _ { \\lfloor n / 2 \\rfloor } . \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} \\varphi ( \\lambda ) = \\min _ { x \\in X } L ( x , \\lambda ) , \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{gather*} | { \\rm e } ^ { i t \\lambda _ k } - { \\rm e } ^ { i t \\log _ 2 u _ k } | = | { \\rm e } ^ { i t \\log _ 2 ( 2 ^ { \\lambda _ k } ) } - { \\rm e } ^ { i t \\log _ 2 u _ k } | \\le \\frac { C | t | } { u _ k } \\le \\frac { C | t | } { 2 ^ n } \\ , . \\end{gather*}"}
-{"id": "8659.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\partial _ t v = \\Delta \\beta ( v ) + f ( v ) \\\\ v ( 0 , \\cdot ) = v _ 0 \\ , \\end{array} \\right . \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} | | ( \\alpha , \\beta ) | | _ { p , q } = \\left \\{ \\begin{array} { l l } \\left ( | \\alpha | ^ p + | \\beta | ^ p \\right ) ^ { 1 / p } \\ \\mathrm { i f } \\ \\alpha \\beta \\geq 0 \\\\ \\left ( | \\alpha | ^ q + | \\beta | ^ q \\right ) ^ { 1 / q } \\ \\mathrm { i f } \\ \\alpha \\beta \\leq 0 \\end{array} \\right . . \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} P _ 1 = W ( \\pi _ 1 ) \\colon W ( W ( k ) ) \\to W ( k ) \\\\ ( a _ i ) _ i \\mapsto ( \\pi _ 1 ( a _ i ) ) _ i , \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{align*} w : = \\frac { \\overline { \\chi } _ 1 } { \\overline { \\xi _ 0 } } . \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} T _ \\lambda : = \\left \\{ \\begin{array} { l l } \\exp ( \\delta \\lambda ^ { - ( p - 1 ) } ) & \\mbox { i f } A > N , \\\\ \\exp ( \\delta \\lambda ^ { - \\frac { p - 1 } { p } } ) & \\mbox { i f } A = N . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} & \\hat { \\mathbf { d } } _ { i , \\rm M F } = \\mathbf { A } ^ { \\rm H } \\mathbf { H } ^ { - 1 } _ i \\mathbf { y } _ i , \\\\ & \\hat { \\mathbf { d } } _ { i , \\rm Z F } = \\mathbf { A } ^ { - 1 } \\mathbf { H } ^ { - 1 } _ i \\mathbf { y } _ i , \\\\ & \\hat { \\mathbf { d } } _ { i , \\rm M M S E } = \\left ( \\mathbf { R } _ { i , \\rm w } + \\mathbf { A } ^ { \\rm H } \\mathbf { H } ^ { \\rm H } _ i \\mathbf { H } _ i \\mathbf { A } \\right ) ^ { - 1 } \\mathbf { A } ^ { \\rm H } \\mathbf { H } ^ { \\rm H } _ i \\mathbf { y } _ i , \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} \\exp ( T ) = \\frac { 1 } { 2 \\pi \\mathrm { i } } \\int _ \\Gamma e ^ z ( z I - T ) ^ { - 1 } \\ , \\mathrm { d } z , \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} ( U _ R \\psi ) ( x , n ) & = \\int _ { 0 } ^ { 1 } e ^ { 2 \\pi i \\beta n } \\sum _ { p \\in \\Z } \\psi ( \\beta , p ) e ^ { 2 \\pi i p ( x + n \\alpha ) } \\ d \\beta . \\\\ ( U _ { k } \\psi ) ( x , n ) & = e ^ { 2 \\pi i ( n k ( \\frac { n \\alpha } { 2 } + x ) ) } \\psi ( x , n ) . \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} & R _ 2 \\geq \\beta _ 2 , \\\\ & \\log \\frac { | \\sigma ^ { 2 } { \\bf I } _ 2 + \\sum _ { j = 2 } ^ { 3 } { \\bf G } _ { 1 j } { \\bf Q } _ j { \\bf G } _ { 1 j } ^ { T } | } { | \\sigma ^ { 2 } { \\bf I } _ 2 + { \\bf G } _ { 1 3 } { \\bf Q } _ 3 { \\bf G } _ { 1 3 } ^ { T } | } \\geq 2 \\beta _ 2 , \\\\ & \\log | \\sigma ^ { 2 } { \\bf I } _ 2 + \\sum _ { j = 2 } ^ { 3 } { \\bf G } _ { 1 j } { \\bf Q } _ j { \\bf G } _ { 1 j } ^ { T } | - \\log | \\sigma ^ { 2 } { \\bf I } _ 2 + { \\bf G } _ { 1 3 } { \\bf Q } _ 3 { \\bf G } _ { 1 3 } ^ { T } | \\geq 2 \\beta _ 2 . \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} \\hat { \\psi } _ { \\gamma } ( \\delta , \\alpha ) = \\frac { d } { d \\tau } \\frac { 1 } { n - n _ { \\ell } } \\sum _ { i \\notin I _ { \\ell } } \\psi ( W _ { i } , \\hat { \\gamma } _ { \\ell } + \\tau \\delta , \\alpha , \\tilde { \\theta } _ { \\ell } ) . \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} h ( \\theta ) & = \\frac { A ( e ^ { i \\theta } ) f ( \\theta ) - \\sum \\limits _ { j = 0 } ^ { \\infty } ( \\bold { B } ^ { - 1 } \\bold { R } \\bold { a } ) _ j e ^ { i j \\theta } } { f ( \\theta ) + g ( \\theta ) } = \\\\ & = A ( e ^ { i \\theta } ) - \\frac { A ( e ^ { i \\theta } ) g ( \\theta ) + \\sum \\limits _ { j = 0 } ^ { \\infty } ( \\bold { B } ^ { - 1 } \\bold { R } \\bold { a } ) _ j e ^ { i j \\theta } } { f ( \\theta ) + g ( \\theta ) } , \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} & d f \\big ( \\sum _ { \\sigma \\in S _ n } s g n ( \\sigma ) ^ { m - 1 } [ x _ { \\sigma ( 1 ) } | \\dots | x _ { \\sigma ( n ) } ] \\big ) \\\\ & = 2 \\sum _ { \\sigma \\in S _ n } s g n ( \\sigma ) ^ { m - 1 } f ( [ x _ { \\sigma ( 1 ) } | \\dots | x _ { \\sigma ( n - 1 ) } ] ) x _ { \\sigma ( n ) } . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} V _ { n , k } = \\sum _ { j = 0 } ^ { \\infty } \\binom { k + j } { j } \\left ( \\left ( \\theta + \\alpha \\right ) _ { n \\uparrow } \\right ) ^ { j } \\left ( \\theta + n \\right ) ^ { k } V _ { n + 1 , k + j } . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} \\mathbf { R } ^ { ( g ) } _ { c o d e } ( l ) & = \\sum _ { \\left \\{ m \\ ; \\vert \\beta _ m ^ l > 0 \\right \\} } \\beta _ m ^ l \\boldsymbol { \\phi } _ m ^ l \\left [ \\boldsymbol { \\phi } _ m ^ l \\right ] ^ H \\\\ \\mathbf { S N R } ^ { ( g ) } _ { m i m o } ( l ) & = \\boldsymbol { \\Gamma } _ l \\operatorname { d i a g } \\left [ \\left \\{ \\lambda _ n ^ l \\right \\} _ { n = 1 } ^ { D } \\right ] \\left ( \\boldsymbol { \\Gamma } _ l \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} \\partial _ t u = \\partial _ { x x } u + v g ( u ) , \\partial _ t v = - \\beta v g ( u ) , \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} \\rho \\ , \\ < n + \\gamma _ 1 \\beta ^ \\sharp , \\ v \\ > = 0 \\quad ( v \\in W ) . \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} \\biggl ( 1 + \\sum _ { i = 2 } ^ n C _ i \\biggr ) \\ , \\ , \\textrm { ` ` } \\ ! = \\ ! \\textrm { '' } \\ , \\ , e ^ { \\sum _ { i = 2 } ^ \\infty T _ i } . \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} y ^ 0 = \\tau : = \\rho \\sqrt { 1 + \\sum _ { i = 1 } ^ 3 ( V ^ i ) ^ 2 } \\mbox { a n d } y ^ i = \\rho V ^ i \\mbox { f o r } i = 1 , 2 , 3 . \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} ( A ( t ) , B ( t ) , I ( t ) , J ( t ) , A ' ( t ) , B ' ( t ) , F ( t ) , G ( t ) ) = ( h ( t ) , h ' ( t ) ) \\cdot X \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} \\vert \\vert \\vert \\tilde { w } \\vert \\vert \\vert _ 2 \\leq e ^ { - t \\frac { C } { 2 } } \\vert \\vert \\vert \\tilde { w } ^ 0 \\vert \\vert \\vert _ 2 , t \\geq 0 , \\tilde { w } ^ 0 = ( f ^ 0 + g ^ 0 , f ^ 0 , \\Gamma ^ 0 ) . \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\vec u = A \\cdot [ 1 , 0 , \\ldots , 0 ] ^ { \\top } , A ^ { - 1 } \\vec u = [ 1 , 0 , \\ldots , 0 ] \\enspace , \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} u ^ 4 a _ 4 ' = a _ 4 + 3 r ^ 2 - 2 s T \\equiv 4 t ^ { 1 2 } + ( 1 + \\beta + \\beta _ 2 - \\beta _ 3 ) t ^ { 2 0 } \\equiv ( \\beta + \\beta _ 2 - \\beta _ 3 ) t ^ { 2 0 } \\pmod { t ^ { 2 1 } } . \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} d \\alpha = D \\bigg ( \\int ^ \\infty _ 0 T _ t ( \\alpha - \\mathbb { E } \\alpha ) d t \\bigg ) = \\int ^ \\infty _ 0 D ( T _ t \\alpha ) d t , \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} p _ { i j } ( x , 0 ) \\equiv 1 i , j \\ \\ r ( x , 0 ) - \\frac 1 2 \\sum _ { i = 1 } ^ n \\frac { \\partial q _ i } { \\partial x _ i } ( x , 0 ) > 0 x \\in \\Omega . \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} H = \\frac { p ^ 2 } { 2 m } + V ( q ) + \\alpha S \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{gather*} b = \\frac { 2 } { 3 } e _ 1 . \\end{gather*}"}
-{"id": "4849.png", "formula": "\\begin{align*} \\tilde P _ { M , S ^ n , X _ 1 } = P _ M \\times \\tilde P _ S ^ n \\times P _ { X _ 1 | M , S ^ n } \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} \\delta _ 2 = \\frac { \\delta _ 1 \\tau _ 1 } { 2 M _ 1 } . \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} G _ \\mu ( \\kappa ^ { 0 , V } ) _ U \\geq \\bigoplus _ { t \\in I } m _ V ^ t \\sum _ { n = 0 } ^ N P _ \\mu ( \\kappa ^ { 0 , V } ) _ t \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} L ( E ) \\equiv L ( \\lambda ) = \\ln { \\frac { 1 + \\sqrt { 1 - 4 \\lambda _ 1 \\lambda _ 3 } } { \\max { ( \\lambda _ 1 + \\lambda _ 3 , \\lambda _ 2 ) } + \\sqrt { \\max { ( \\lambda _ 1 + \\lambda _ 3 , \\lambda _ 2 ) } ^ 2 - 4 \\lambda _ 1 \\lambda _ 3 } } } > 0 . \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\chi ^ 2 & = V - \\eta ^ \\sigma W = 0 \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} \\Omega _ X ( u , v ) = \\widetilde { b _ { 2 2 } } a _ { 2 2 } - \\widetilde { a _ { 2 2 } } b _ { 2 2 } \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } K _ { 3 } ( t ) g \\right \\| _ { 2 } \\le C ( 1 + t ) ^ { - \\frac { n } { 4 \\sigma } - \\frac { \\ell + k - 1 } { 2 \\sigma } } \\| g \\| _ { 1 } + C e ^ { - c t } \\| \\nabla ^ { ( k - 2 \\sigma ( 1 - \\ell ) ) _ { + } } _ { x } g \\| _ { 2 } , \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} \\frac { \\partial { U ( \\hat { x } , \\theta ) } } { \\partial { \\theta ^ i } } = u _ i ( s ^ i ( \\hat { x } , \\theta ) ) , \\ i \\in N , \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} \\left ( F _ { b n + q } ^ { \\left ( b \\right ) } \\right ) ^ { 2 } - F _ { b n + q + 1 } ^ { \\left ( b \\right ) } F _ { b n + q - 1 } ^ { \\left ( b \\right ) } = \\left ( - 1 \\right ) ^ { q } \\left ( F _ { n } ^ { \\left ( b \\right ) } \\right ) ^ { 2 } \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} & = \\sum _ { l = 0 } ^ { | k | - 1 } \\frac { 2 | k | - ( 2 l + 1 ) } { 2 k ^ 2 h ^ 2 } \\left [ \\phi ( x - ( l + 1 ) h ) - 2 \\phi ( x - l h ) + \\phi ( x - ( l - 1 ) h ) \\right ] \\\\ & = \\frac { 1 } { 2 k ^ 2 h ^ 2 } \\left ( \\phi ( x + k h ) + \\phi ( x + ( k + 1 ) h ) + ( 2 | k | - 1 ) \\phi ( x + h ) - ( 2 | k | + 1 ) \\phi ( x ) \\right ) , \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{align*} u ( t , x ) = \\sum _ { j \\in S ^ + } 2 \\ , \\sqrt { j \\ , \\xi _ j } \\ , \\cos ( \\omega _ j t + j x ) + o ( \\sqrt { \\lvert \\xi \\rvert } ) , \\omega _ j = j ^ 3 + O ( \\xi _ j ) \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} \\frac { d q } { d t } & = p ( 1 + \\alpha \\sin { ( w t ) } q ^ 2 ) , \\\\ \\frac { d \\gamma } { d t } & = q ( 1 + \\alpha \\sin { ( w t ) } p ^ 2 ) . \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} B _ { \\xi } ( y ; p ; s ) : = \\left [ y \\mapsto \\prod _ { j = 1 } ^ g \\tau ( A _ j ( s ) , B _ j ( s ) ; 1 , \\xi ^ { ( j ) } y _ j ) \\right ] , \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} Z - \\bar Z = P \\left \\{ C ( X - \\bar X ) + D ( v - \\bar v ) \\right \\} , \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} S ^ { s u m o } & \\le S ( 3 ; 0 , 0 ) ^ 9 S ( 3 ; 2 , 2 ) ^ 4 \\\\ & \\le S ( 3 ; 0 , 0 ) ^ 9 S _ U ( 3 ; 2 , 2 ) ^ 4 = : S _ U ^ { s u m o } \\\\ & \\approx ( 6 . 6 7 0 9 \\times 1 0 ^ { 2 1 } ) ^ 9 \\times ( 1 . 5 9 7 6 \\times 1 0 ^ { 1 1 } ) ^ 4 \\\\ & \\approx 1 . 7 0 4 5 \\times 1 0 ^ { 2 4 1 } . \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} & \\mathcal { L } _ 5 : = \\Phi _ 2 ^ { - 1 } \\mathcal { L } _ 4 \\Phi _ 2 = \\Pi _ S ^ { \\perp } ( \\mathcal { D } _ { \\omega } + m _ 3 \\partial _ { x x x } + ( \\tilde { d } _ 1 + \\varepsilon ^ 2 c ( \\xi ) ) \\ , \\partial _ x + R _ 5 ) \\Pi _ S ^ { \\perp } , \\\\ & R _ 5 : = ( \\Phi _ 2 ^ { - 1 } - \\mathrm { I } ) \\Pi _ S ^ { \\perp } ( \\tilde { d } _ 1 + \\varepsilon ^ 2 c ( \\xi ) ) \\partial _ x + \\Phi _ 2 ^ { - 1 } \\Pi _ S ^ { \\perp } \\tilde { R } _ 5 . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\Delta = \\{ \\vec { t } \\in [ 0 , 1 ] ^ n \\mid t _ 1 + t _ 2 + \\cdots + t _ n = 1 \\} \\ , . \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} \\tilde { \\mathcal { R } } _ * : = \\mathcal { T } ^ { - 1 } \\mathcal { R } _ * \\mathcal { T } + \\varepsilon ^ 2 \\Pi _ S ^ { \\perp } \\partial _ x ( \\mathcal { R } _ 2 - \\mathcal { T } ^ { - 1 } \\mathcal { R } _ 2 \\mathcal { T } ) + \\varepsilon ^ 2 \\Pi _ S ^ { \\perp } \\partial _ x ( \\overline { \\mathcal { R } } _ 2 - \\mathcal { R } _ 2 ) , \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} ( - 1 ) ^ { | \\sigma ^ \\ast | - | \\rho | - | \\nu | } \\prod _ { i \\in \\sigma } z _ i \\prod _ { i = n - k + 1 } ^ n z _ i ^ { - 1 } . \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} ~ D _ \\mu ( x ) - x = \\frac { 1 } { \\pi } \\sum _ { n = 1 } ^ \\infty \\Im ( \\frac { \\hat { \\mu } ( n ) } { n } [ e ( n x ) - 1 ] ) + \\lim _ { n \\to \\infty } \\frac { 1 } { 2 n } \\sum _ { j = 1 } ^ n \\hat { \\mu } ( j ) [ 1 - e ( j x ) ] . \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} \\bold { p r } ^ { ( k ) } \\ ! X = X + \\sum _ { \\alpha } \\sum _ { | J | = 1 } ^ k \\phi _ J ^ { \\alpha } \\frac { \\partial } { \\partial u _ J ^ { \\alpha } } , \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} \\Psi ( s ) : = s [ \\log ( e + s ) ] ^ { \\frac { N } { \\theta } } , \\rho ( s ) : = s ^ { - N } \\biggr [ \\log \\biggr ( e + \\frac { 1 } { s } \\biggr ) \\biggr ] ^ { - \\frac { N } { \\theta } } , \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} E ^ { n , ( 3 ) } = E ^ { n , ( 3 ) } _ { h i g h } + E ^ { n , ( 3 ) } _ { l o w } , \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} \\zeta _ 1 & = \\left ( \\frac { \\sqrt { P } - \\sqrt { P - 2 } } { 2 } \\right ) ^ 2 , \\\\ \\zeta _ 2 & = \\left ( \\frac { \\sqrt { P } + \\sqrt { P - 2 } } { 2 } \\right ) ^ 2 . \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} \\tilde { H } f : = Q _ H ( f , \\cdot ) , \\tilde { S } f : = Q _ { [ H , i A ] } ( f , \\cdot ) , \\tilde { S } ^ { \\prime } = Q _ { [ [ H , i A ] , i A ] } ( f , \\cdot ) f \\in { \\mathcal G } ^ 1 . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} \\widehat { F } & = ( I + F _ { S _ { A } } + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } ( I + F _ { S _ { A } } K ^ { \\ast } K ) \\\\ & = I - ( I + F _ { S _ { A } } + F _ { S _ { A } } K ^ { \\ast } K ) ^ { - 1 } F _ { S _ { A } } \\\\ & = I - F _ { S _ { A } } ( I + F _ { S _ { A } } + K ^ { \\ast } K F _ { S _ { A } } ) ^ { - 1 } . \\end{align*}"}
-{"id": "7128.png", "formula": "\\begin{align*} L _ { a , b } = a A + b B \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} X : = \\left \\{ v \\in H ^ 1 ( \\mathbb { R } ) : v ^ 2 \\log | v | \\in L ^ 1 ( \\mathbb { R } ) \\right \\} , \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} \\hat { L } = - \\frac { 1 } { 4 } \\partial _ k ^ 2 + k ^ 2 - \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} N \\le 2 1 - \\frac 1 2 \\chi _ { \\operatorname { t o p } } ( \\mathfrak { X } _ b ) = 2 0 - \\rho ( \\mathfrak { X } _ b ) + h ^ { 1 , 2 } ( \\mathfrak { X } _ b ) \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} \\aligned \\Big ( ( 2 ^ * - 2 ) \\int _ { \\R ^ N } | u _ j | ^ { 2 ^ * } & + \\sum _ { k \\neq j } | \\beta _ { j k } | \\alpha _ { j k } ( 2 - \\alpha _ { j k } ) | u _ j | ^ { \\alpha _ { j k } } | u _ k | ^ { \\alpha _ { k j } } \\Big ) L _ j \\\\ & - \\sum _ { i \\neq j } L _ i | \\beta _ { j i } | \\alpha _ { j i } \\alpha _ { i j } \\int _ { \\R ^ N } | u _ j | ^ { \\alpha _ { j k } } | u _ k | ^ { \\alpha _ { k j } } = 0 . \\endaligned \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} \\pi _ { \\omega } ( \\tau _ { g } ( A ) ) = U _ { g } \\pi _ { \\omega } U _ { g } ^ { \\ast } \\ \\mbox { w i t h } \\ U _ { g } \\Omega _ { \\omega } = \\Omega _ { \\omega } \\ . \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\tilde { A } _ { | c | , E } ( \\theta ) = \\frac { 1 } { \\sqrt { | c | ( \\theta ) | c | ( \\theta - \\alpha ) } } D _ { | c | , E } ( \\theta ) = \\frac { 1 } { \\sqrt { | c | ( \\theta ) | c | ( \\theta - \\alpha ) } } \\left ( \\begin{matrix} E - v ( \\theta ) \\ \\ & - | c | ( \\theta - \\alpha ) \\\\ | c | ( \\theta ) \\ \\ & 0 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\Delta u = - \\frac { \\Delta \\varphi + | \\nabla \\varphi | ^ 2 } { \\mathrm { e } ^ \\varphi } , \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} x ( n + 1 ) = \\lambda _ { \\ell } ^ { - 1 } A ( n ) x ( n ) \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} a _ n ( I _ d : \\ W _ 2 ^ { \\infty } ( \\Bbb T ^ d ) \\rightarrow L _ 2 ( \\Bbb T ^ d ) ) = ( 1 + m ) ^ { - 1 / 2 } . \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} ( \\mathcal { A } ^ { - 1 } h ) ( \\varphi , y ) : = ( 1 + \\tilde { \\beta } _ y ( \\varphi , y ) ) \\ , h ( \\varphi , y + \\tilde { \\beta } ( \\varphi , y ) ) , ( \\mathcal { A } ^ T h ) ( \\varphi , y ) = h ( \\varphi , y + \\tilde { \\beta } ( \\varphi , y ) ) \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} \\max \\{ p ( x ) \\mid x \\in \\overline { B ( x _ { 0 } , \\varepsilon ) } \\} = \\max \\{ p ( x ) \\mid x \\in B ( x _ { 0 } , \\varepsilon , \\delta , \\theta ) \\} . \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} \\tilde P _ { M , S ^ n , X _ 1 } = P _ M \\times \\tilde P _ S ^ n \\times P _ { X _ 1 | M , S ^ n } , \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} N _ { ( 1 , 1 ) } ( n ) & = 1 6 \\ , \\sigma ( n ) - 6 4 \\ , \\sigma ( \\frac { n } { 4 } ) + 6 4 \\ , W _ { ( 1 , 1 ) } ( n ) + 1 0 2 4 \\ , W _ { ( 1 , 1 ) } ( \\frac { n } { 4 } ) - 5 1 2 \\ , W _ { ( 1 , 4 ) } ( n ) \\\\ & = 1 6 \\ , \\sigma _ { 3 } ( n ) - 3 2 \\ , \\sigma _ { 3 } ( \\frac { n } { 2 } ) + 2 5 6 \\ , \\sigma _ { 3 } ( \\frac { n } { 4 } ) , \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} m _ { \\Gamma _ 2 } ( \\Gamma _ 2 q _ 0 B ) & = \\mu ( \\Gamma _ 1 \\backslash G \\times \\Gamma _ 2 q _ 0 B ) \\\\ & = m _ { \\Gamma _ 0 } ( \\psi ^ { - 1 } ( \\Gamma _ 1 \\backslash G \\times \\Gamma _ 2 q _ 0 B ) ) \\\\ & = \\sum _ { \\alpha } m _ { \\Gamma } ( \\Gamma \\gamma _ { \\alpha } B ) , \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} g ( x ) : = \\inf _ { U } f ( x ) - p ( x ) \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} \\left ( X + Y \\right ) ^ { s _ b ( n ) } = \\sum _ { k = 0 } ^ { n } \\binom { n } { k } _ b X ^ { s _ b ( k ) } Y ^ { s _ b ( n - k ) } . \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ y \\chi - | p | ^ 2 \\chi _ { z z } - 2 p \\cdot \\nabla _ { y } \\chi _ z + \\bar c ( p ) \\chi _ z ( y , z ) - g ( y , \\chi ( y , z ) ) = 0 & ( y , z ) \\in [ 0 , 1 ] ^ { n + 1 } \\\\ \\chi ( y , z + 1 ) = \\chi ( y , z ) + 1 & \\\\ \\chi ( \\cdot , z ) & \\end{cases} \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} \\tilde { \\mathbf { w } } _ { \\mathrm { a } , i k } = \\frac { 1 } { \\sqrt { \\tilde { \\zeta } _ { \\mathrm { a } , i } } } \\ ! \\left ( \\rho N \\mathbf { I } _ N \\ ! + \\ ! \\ ! \\sum _ { l = 1 } ^ K \\hat { \\mathbf { h } } _ { i i l } \\hat { \\mathbf { h } } _ { i i l } ^ { \\mathrm { H } } \\ ! + \\ ! \\ ! \\sum _ { l = 1 } ^ { K ' } \\hat { \\mathbf { h } } _ { i \\bar { i } l } \\hat { \\mathbf { h } } _ { i \\bar { i } l } ^ { \\mathrm { H } } \\right ) ^ { - 1 } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\hat { \\mathbf { h } } _ { i i k } \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\tau = t , u = r \\cos ( k \\phi ) , v = r \\sin ( k \\phi ) . \\end{align*}"}
-{"id": "783.png", "formula": "\\begin{align*} \\Sigma _ { i = 1 } ^ k m _ i \\leq \\Sigma _ { i = 1 } ^ k \\lambda _ i . \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} ( B w ) ( \\varphi , y ) : = w ( \\varphi + \\omega \\alpha ( \\varphi ) , y ) , ( B ^ { - 1 } h ) ( \\vartheta , y ) : = h ( \\vartheta + \\omega \\tilde { \\alpha } ( \\vartheta ) , y ) , \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} F ^ { ( 4 ) } _ { j _ 1 j _ 2 j _ 3 j _ 4 } : = \\begin{cases} \\dfrac { H ^ { ( 3 ) } _ { 4 , \\ , j _ 1 j _ 2 j _ 3 j _ 4 } } { \\mathrm { i } ( j _ 1 ^ 3 + j _ 2 ^ 3 + j _ 3 ^ 3 + j _ 4 ^ 3 ) } \\mbox { i f } \\ , \\ , ( j _ 1 , j _ 2 , j _ 3 , j _ 4 ) \\in \\mathcal { A } _ 4 , \\\\ [ 2 m m ] 0 \\mbox { o t h e r w i s e } , \\\\ \\end{cases} \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} \\frac { 1 } { z } ( \\mathfrak { n } + z \\mathfrak { g } [ [ z ] ] ) = \\widehat { \\mathfrak { p } } ^ { \\vee } . \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} \\nabla ^ { F , u } = \\nabla ^ F + \\frac { 1 } { 2 } \\omega ( F , g ^ F ) \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} { \\gamma _ { { e ^ * } } } = \\mathop { \\max } \\limits _ { e \\in { \\Phi _ e } } \\left \\{ { \\frac { { { P _ t } { G _ e } L \\left ( \\left | X _ e \\right | \\right ) } } { { { \\sigma _ e ^ 2 } } } } \\right \\} , \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} f ( t ) : = \\frac { c ( t - c ) } { ( c + t ) ^ 2 } \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} k ( t , \\epsilon _ j ) : = \\sum _ { 0 \\leq \\theta < \\epsilon _ j } \\dim \\ker \\big ( \\tilde { W } ( t ) - e ^ { i ( \\pi + \\theta ) } I \\big ) . \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} \\langle f '^ c , T r _ { G L _ 2 } ^ { S L _ 2 } ( v , \\psi ' ) g \\mathcal { E ' } \\rangle = \\ast \\Gamma _ \\infty \\left ( ( s - \\frac { 3 } { 4 } ) t + \\frac { k + l ' } { 2 } \\right ) { \\mathcal { D } \\left ( s - \\frac { 1 } { 2 } , f ' _ 1 - 1 , g \\right ) } . \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} f B ' + F b ' - b F - B f = 0 \\Leftrightarrow \\left \\{ \\begin{array} { l c l } b ' & = & b _ { 1 1 } + B _ { 1 2 } f _ { 2 } \\\\ b _ { 2 1 } & = & 0 \\end{array} \\right . \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} p ^ * ( y ^ * _ \\varnothing ) + q ^ * ( y ^ * _ \\varnothing ) & \\leqslant a ( 1 - \\delta ) + b = a + b - a \\delta \\\\ & \\leqslant a + b - \\delta ( a + b ) / ( 1 + C ) \\\\ & \\leqslant ( 1 + 2 \\mu ) \\bigl ( 1 - \\frac { \\delta } { 1 + C } \\bigr ) . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} [ X , Y ] = Y , \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} - | p | ^ 2 w '' ( z ) + \\bar c ( | p | ) ( w ' ( z ) + 1 ) - g ( w ( z ) + z ) = 0 . \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} \\mathbb { I } \\equiv \\sum _ { h _ { 1 } , . . . , h _ { \\mathsf { N } } = 0 } ^ { p - 1 } \\prod _ { 1 \\leq b < a \\leq \\mathsf { N } } ( X _ { a } ^ { ( h _ { a } ) } - X _ { a } ^ { ( h _ { a } ) } ) | h _ { 1 } , . . . , h _ { \\mathsf { N } } \\rangle \\langle h _ { 1 } , . . . , h _ { \\mathsf { N } } | . \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} B _ x B _ s B _ y = B _ x B _ y ( 1 ) \\oplus B _ x B _ y ( - 1 ) \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} M _ 1 ( \\varphi ) : = K _ { 2 0 } ( \\varphi ) + K _ { 1 1 } ^ T ( \\varphi ) \\mathcal { L } _ { \\omega } ^ { - 1 } \\partial _ x K _ { 1 1 } ( \\varphi ) , M _ 2 ( \\varphi ) : = M _ 1 ( \\varphi ) \\mathcal { D } _ { \\omega } ^ { - 1 } , M _ 3 ( \\varphi ) : = K _ { 1 1 } ^ T ( \\varphi ) \\mathcal { L } _ { \\omega } ^ { - 1 } . \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} u = T u + \\mu e . \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} T _ { 2 j } ( z ) = \\int \\limits _ { x _ 1 < y _ 1 < x _ 2 < y _ 2 < \\dots < x _ j < y _ { j } } \\prod _ { l = 1 } ^ j e ^ { 2 i z ( y _ { l } - x _ l ) } u ( y _ { l } ) \\overline { v ( x _ l ) } d x _ 1 d y _ 1 \\dots d x _ { j } d y _ j . \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} \\lim _ { t \\to 0 ^ + } D _ t ^ { \\delta } g ( t ) = 0 . \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} & 1 - \\int _ { - \\infty } ^ { 0 } \\int _ { - \\infty } ^ { 0 } p ( y _ n ^ { \\rm R } , y _ n ^ { \\rm I } ) \\mathrm { d } y _ n ^ { \\rm R } \\mathrm { d } y _ n ^ { \\rm I } \\\\ = & 1 - \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( z _ n ^ { \\rm R } + \\sqrt { P / 2 } \\big ) \\right ) \\left ( \\frac { 1 } { 2 } - \\frac { 1 } { 2 } { \\rm { e r f } } \\big ( z _ n ^ { \\rm I } + \\sqrt { P / 2 } \\big ) \\right ) . \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} { \\gamma _ { t , i k } } = \\frac { { P d _ { t , i k } ^ { - \\beta } { { \\left \\| { { h _ { t , i k } } } \\right \\| } ^ 2 } } } { { \\sum \\limits _ { j \\in { \\mathcal { M } \\mathord { \\left / { \\vphantom { \\mathcal { M } { { \\mathcal { M } _ i } } } } \\right . \\kern - \\nulldelimiterspace } { { \\mathcal { M } _ i } } } } { P d _ { t , i j } ^ { - \\beta } { { \\left \\| { { h _ { t , i j } } } \\right \\| } ^ 2 } + { \\sigma ^ 2 } } } } , \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} c _ { m , n } ^ { } = \\log _ 2 \\ ! \\left ( \\ ! 1 + \\frac { p _ n \\psi ( \\boldsymbol { y } _ m , \\boldsymbol { y } _ n ) g ( \\boldsymbol { y } _ m , \\boldsymbol { y } _ n ) } { w _ 1 N _ 0 } \\right ) , \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} n = x ^ 2 + y ^ 2 + z ^ k , \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{gather*} M ( h _ { + } ) = \\dfrac { 4 p _ { 1 } \\left ( s - v \\right ) ^ { 2 } } { ( s ^ { 2 } + t ^ { 2 } ) s ^ { 4 } } \\times \\\\ ( s - 2 h _ { + } ) ^ { 2 } \\allowbreak { \\large ( } 2 w s t - ( t ^ { 2 } - s ^ { 2 } ) v { \\large ) } ^ { 2 } \\end{gather*}"}
-{"id": "7232.png", "formula": "\\begin{align*} \\abs { F _ { n + i + j } ^ r } _ { 0 \\leq i , j \\leq r } = ( - 1 ) ^ { ( n + 1 ) { r + 1 \\choose 2 } } ( F _ 1 ^ r F _ 2 ^ { r - 1 } \\cdots F _ r ) ^ 2 \\cdot \\prod _ { i = 0 } ^ r { r \\choose i } . \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L z = \\lambda f ( x , z ) , & x \\in \\Omega , \\\\ B z = 0 , & x \\in \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} P _ n ( x ) T ^ n u ( x + n \\alpha ) = T ^ n u ( x + n \\alpha ) , \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} g _ y ( u , v ) = \\frac 1 2 \\ , \\frac { \\partial ^ 2 } { \\partial s \\ , \\partial t } \\ , \\big [ F ^ 2 ( y + s u + t v ) \\big ] _ { | \\ , s = t = 0 } \\ , . \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{align*} \\dot z ( t ) = ( A _ { - 1 } + B F _ { - 1 } ) \\dot z ( t - 1 ) + L z _ t + B u . \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} \\mathbb { P } \\left ( \\textrm { d i a m } ( A ) = 0 \\right ) = 1 . \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} \\tilde { g } ( x ) : = C \\int _ { \\R ^ { N } \\backslash B } \\frac { g ( y ) } { | x - y | ^ { N + 2 s } } d y \\geq 0 x \\in B , \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} \\sigma = \\sigma ( \\Gamma , k ) = - \\dfrac { 1 } { 4 } ( | \\Gamma | + k ^ 2 ) , \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} \\gamma _ { \\mathrm { S T B C } } = \\frac { \\gamma _ 0 } { M R _ c } \\mathbf { h } ^ { \\dagger } \\times \\mathbf { h } = \\frac { \\gamma _ 0 } { M R _ c } \\sum _ { m = 1 } ^ { M } | h _ m | ^ 2 \\end{align*}"}
-{"id": "704.png", "formula": "\\begin{align*} ( e v _ - ) ^ * F ^ { j ' } ( \\psi _ - - \\chi ) = & ( e v _ - ) ^ * F ^ { j ' } ( - \\chi ) \\\\ + & \\psi _ - ( e v _ - ) ^ * \\displaystyle \\frac { \\partial } { \\partial t } \\left . F ^ { j ' } ( t ) \\right | _ { t = - \\chi } + \\frac { ( \\psi _ - ) ^ 2 } { 2 ! } ( e v _ - ) ^ * \\frac { \\partial ^ 2 } { \\partial z ^ 2 } \\left . F ^ { j ' } ( t ) \\right | _ { t = - \\chi } + \\dotsc . \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} \\begin{aligned} U _ { 1 1 } ^ t U _ { 1 1 } + U _ { 2 1 } ^ t U _ { 2 1 } & = I \\\\ U _ { 1 1 } U _ { 1 1 } ^ t + U _ { 2 1 } U _ { 2 1 } ^ t & = I \\\\ U _ { 1 1 } ^ t U _ { 2 1 } - U _ { 2 1 } ^ t U _ { 1 1 } & = 0 \\\\ U _ { 1 1 } U _ { 2 1 } ^ t - U _ { 2 1 } U _ { 1 1 } ^ t & = 0 \\end{aligned} \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } L u _ i ( x ) = \\lambda _ i f _ i ( x , u ( x ) ) , & x \\in \\Omega , i = 1 , 2 , \\ldots , n , \\\\ B u _ i ( x ) = 0 , & x \\in \\partial \\Omega , i = 1 , 2 , \\ldots , n , \\end{array} \\right . \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} K _ 2 ( \\xi , \\eta ) = \\frac 1 2 ( K _ 1 ( \\xi . \\eta ) + K _ 1 ( - \\xi , - \\eta ) ) . \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} ( X + Y ) ^ { s _ 2 ( n ) } = \\sum _ { 0 \\leq k \\leq _ 2 n } X ^ { s _ 2 ( k ) } Y ^ { s _ 2 ( n - k ) } , \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} P ( \\lambda ) = \\lambda ^ { 2 } - ( \\alpha + \\beta ) \\lambda + \\alpha \\beta - \\gamma = 0 . \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} \\sum _ { ( u , v ) \\in [ r ] \\times [ s ] } c ( u ^ { \\ast } ; ( u , v ) ) = 0 . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} M : = \\bigotimes _ { n = 1 } ^ \\infty ( M _ n , \\omega _ n ) : = ( \\pi _ \\omega ( A ) ) '' . \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} ( T u ) ( X _ 1 , \\ldots , X _ q ) : = \\frac { \\partial } { \\partial \\theta } \\left ( ( e ^ { i \\theta } ) ^ \\ast u ( X _ 1 , \\ldots , X _ q ) \\right ) \\Big | _ { \\theta = 0 } \\ , , X _ 1 , \\ldots , X _ q \\in T _ x ^ { 1 , 0 } X . \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} \\ddot z ( t ) = - k z ( t ) + c ( t ) + f \\big ( r ( t ) / m \\big ) . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} 0 = \\dot { \\eta _ t } + d ( V _ t \\rfloor \\eta _ t ) + V _ t \\ , \\rfloor \\ , d \\eta _ t = ( \\eta - \\eta _ 0 + d h _ t ) + Y _ t \\ , \\rfloor \\ , d \\eta _ t . \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} a = 1 , b = \\ell , c = 1 , x = u , y = - v ^ 2 , z = w . \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dfrac { \\partial g _ { k } ^ { R } } { \\partial x } ( x _ k ^ \\ast , y _ k ^ \\ast ) = \\dfrac { \\partial g _ { k } ^ { I } } { \\partial y } ( x _ k ^ \\ast , y _ k ^ \\ast ) , \\\\ \\dfrac { \\partial g _ { k } ^ { R } } { \\partial y } ( x _ k ^ \\ast , y _ k ^ \\ast ) = - \\dfrac { \\partial g _ { k } ^ { I } } { \\partial x } ( x _ k ^ \\ast , y _ k ^ \\ast ) . \\end{array} \\right . \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} A ( f ) = \\left \\{ z \\in \\C \\colon l \\in \\N | f ^ { n } ( z ) | > M ^ { n - l } ( R , f ) n \\geq l \\right \\} , \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} \\hat { \\beta } _ 0 \\ ; = \\ ; \\hat { \\beta } _ 1 \\ ; = \\ ; 1 , \\hat { \\beta } _ { t + 1 } \\ ; = \\ ; \\hat { \\beta } _ t + \\frac { 1 } { \\hat { \\beta } _ t } . \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} a \\left ( y , \\gamma _ { \\partial B } ( 0 ) \\right ) = \\frac { 1 } { 2 } \\int _ 0 ^ { s _ y } [ \\gamma _ { \\partial B } ( s ) , \\gamma _ { \\partial B } ' ( s ) ] \\ d s , \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} \\left . \\begin{array} { l } \\mathrm { p r } _ 1 ( \\lambda , p , q ) : = ( \\lambda , p ) \\\\ \\mathrm { p r } _ 2 ( \\lambda , p , q ) : = ( \\lambda , q ) \\end{array} \\right \\} \\quad . \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} \\left \\| \\frac { z - w _ i } { \\| z - w _ i \\| } - \\frac { z } { \\| z \\| } \\right \\| & \\le \\frac { \\| w _ i \\| } { \\| z - w _ i \\| } + \\| z \\| \\left | \\frac { 1 } { \\| z - w _ i \\| } - \\frac { 1 } { \\| z \\| } \\right | \\\\ & \\le \\frac { R _ 0 } { \\| z \\| - R _ 0 } + \\frac { | \\| z \\| - \\| z - w _ i \\| | } { \\| z - w _ i \\| } \\\\ & \\le \\frac { R _ 0 } { \\| z \\| - R _ 0 } + \\frac { R _ 0 } { \\| z \\| - R _ 0 } \\\\ & < \\frac { 2 R _ 0 } { ( \\frac { 2 + \\delta } { \\delta } ) R _ 0 - R _ 0 } = \\delta . \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} [ a _ 1 , \\ldots , a _ { n - 1 } , b c ] _ n = & [ a _ 1 , \\ldots , a _ { n - 1 } , b ] _ n c \\pm b [ a _ 1 , \\ldots , a _ { n - 1 } , c ] _ n \\\\ & + \\hbar [ a _ 1 , \\ldots , a _ { n - 1 } , b , c ] _ { n + 1 } . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} D ^ a ( \\xi , - \\eta ) = - i \\xi + S ( d ^ 2 \\rho ^ { - 1 } ) , \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} \\frac { \\partial \\psi ^ * ( f ) } { \\partial u ^ a } = \\sum _ b \\frac { \\partial \\psi ^ * ( v ^ b ) } { \\partial u ^ a } \\ , \\psi ^ * \\bigg ( \\frac { \\partial f } { \\partial v ^ b } \\bigg ) \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\partial ( F _ { k + 1 } - F ' _ { k + 1 } ) = 0 . \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} \\Delta _ k ( x ; \\boldsymbol { a } ) = \\frac { 1 } { \\phi ^ k ( q ) } \\frac { 1 } { 2 \\pi i } \\int _ { c - i T } ^ { c + i T } \\ ( F _ k ( s ) - \\widetilde { F } _ k ( s ) \\ ) \\frac { x ^ s } { s } d s + O \\ ( \\frac { x \\log x } { T } + 1 \\ ) , \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{align*} \\begin{aligned} & \\lvert A \\rvert _ s ^ { \\sup } : = \\sup _ { \\omega \\in \\Omega _ 0 } \\lvert A ( \\omega ) \\rvert _ s , \\ , \\ , \\ , \\lvert A \\rvert _ s ^ { l i p } : = \\sup _ { \\omega _ 1 \\neq \\omega _ 2 } \\frac { \\lvert A ( \\omega _ 1 ) - A ( \\omega _ 2 ) \\rvert _ s } { \\lvert \\omega _ 1 - \\omega _ 2 \\rvert } , \\\\ & \\lvert A \\rvert _ s ^ { L i p ( \\gamma ) } : = \\lvert A \\rvert _ s ^ { \\sup } + \\gamma \\lvert A \\rvert _ s ^ { l i p } . \\end{aligned} \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} y ^ 2 = x ^ 3 + a _ 4 x + a _ 6 , a _ 4 = - \\frac { c _ 4 } { 4 8 } , a _ 6 = - \\frac { c _ 6 } { 8 6 4 } \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} \\widetilde { \\Psi } _ k = { \\psi } ^ 0 \\oplus \\left ( \\oplus _ { i = 1 } ^ { p } \\oplus _ { \\alpha \\in P ( i , k p - i ) } \\psi ^ i _ { \\alpha } \\right ) . \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} \\{ \\alpha , \\beta \\} = \\imath _ { \\Pi ( \\beta ) } d \\alpha + ( - 1 ) ^ p ( \\Pi d \\alpha ) \\beta . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} \\tilde u _ t ( x , t ) & = H \\tilde u ( x , t ) + \\tilde u ( x , t ) f ( t , u _ 2 ( x , t ) ) \\\\ & \\ge H \\tilde u ( x , t ) + \\tilde u ( x , t ) f ( t , \\tilde u ( x , t ) ) + \\delta _ \\epsilon \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N | r _ i | = p ( \\tau _ j ) . \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{gather*} \\hat { L } _ 0 = \\frac { x ^ 2 } { 2 } \\sigma _ 3 + x \\begin{pmatrix} 0 & u \\\\ u & 0 \\end{pmatrix} + \\begin{pmatrix} \\delta & w \\\\ - w & - \\delta \\end{pmatrix} \\end{gather*}"}
-{"id": "9108.png", "formula": "\\begin{align*} H ( 1 ) \\cap H ( k _ { \\gamma } ) = \\{ 0 \\} . \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { \\infty } k ^ { 1 + \\eta } f ( k ) < \\infty \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} \\mathbf { R } _ { \\mathbf { \\tilde { \\mathbf { G } } \\boldsymbol { \\psi } } } = \\left ( m _ 0 m _ 1 \\rho _ p \\right ) ^ { - 2 } \\mathbb { E } \\left \\{ \\mathbf { Z } \\boldsymbol { \\Psi } _ p ^ { \\dagger } \\boldsymbol { \\psi } \\boldsymbol { \\psi } ^ { \\dagger } \\boldsymbol { \\Psi } _ p \\mathbf { Z } ^ { \\dagger } \\right \\} . \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} \\psi ( a , b ; z ) = \\frac { 1 } { \\Gamma ( a ) } \\int _ { 0 } ^ { \\infty } e ^ { - z t } t ^ { a - 1 } ( 1 + t ) ^ { b - a - 1 } d t \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{gather*} i _ X \\omega _ 1 = \\delta \\alpha _ 1 + \\delta _ Q \\alpha ' + d \\alpha ' _ 1 \\end{gather*}"}
-{"id": "175.png", "formula": "\\begin{align*} V \\ , \\rfloor \\ , \\eta = h , V \\ , \\rfloor \\ , d \\eta = - d h + R ( h ) \\eta . \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} \\mathcal { H } = H \\circ \\Phi _ B = ( H _ 2 + H _ 3 ) \\circ \\Phi _ B + H _ 4 \\circ \\Phi _ B + H _ { \\geq 5 } \\circ \\Phi _ B , \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} h _ n ( k , \\ell ) = \\psi _ n ( k , \\ell ) \\mathcal { E } _ n ( k , \\ell ) . \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} \\begin{aligned} V > 0 , \\delta _ 0 ^ { - 1 } V \\geq - x \\cdot \\nabla V \\geq ( 1 + \\delta _ 0 ) V . \\end{aligned} \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} h _ f ( f _ 0 , g _ 0 ) = \\frac { \\left | A ( e ^ { i \\theta } ) g _ 0 ( \\theta ) + \\sum \\limits _ { j = 0 } ^ { \\infty } ( ( \\bold { B } ^ 0 ) ^ { - 1 } \\bold { R } ^ 0 \\bold { a } ) _ j e ^ { i j \\theta } \\right | ^ 2 } { ( f _ 0 ( \\theta ) + g _ 0 ( \\theta ) ) ^ 2 } , \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} \\left ( \\frac { a } { p } \\right ) = \\begin{cases} 1 a p , \\\\ - 1 a p , \\\\ 0 p . \\ \\end{cases} \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} M \\left ( \\int _ \\Omega | ( - \\Delta ) ^ { \\frac { \\alpha } { 2 } } u | ^ 2 d x \\right ) ( - \\Delta ) ^ { \\alpha } u = \\lambda f ( x ) | u | ^ { q - 2 } u + | u | ^ { 2 ^ * _ \\alpha - 2 } u \\ ; \\ ; \\ ; \\Omega , \\ ; u = 0 \\ ; \\textrm { i n } \\ ; \\mathbb R ^ n \\setminus \\Omega , \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} \\operatorname { C o v } ( \\mathcal { D } ^ { i , i + 1 } _ n , \\mathcal { D } ^ { j , j + 1 } _ n ) = \\begin{cases} \\alpha n + o ( n ) & i = j , \\\\ - \\frac { \\alpha n } 2 + o ( n ) & | i - j | = 1 , \\\\ o ( n ) & . \\end{cases} \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} h _ { + } ^ { 2 } = \\dfrac { ( s - 2 h _ { 0 } ) K } { 2 \\left ( s ^ { 2 } + t ^ { 2 } \\right ) \\left ( s - v \\right ) } \\end{align*}"}
-{"id": "1342.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": "2023.png", "formula": "\\begin{align*} \\beta _ \\ell = \\sup _ { h > 0 } \\{ \\ ; h : h ^ 2 \\mid \\Delta _ \\ell \\ ; \\ ; \\mathcal { P } _ { \\Delta _ \\ell / h ^ 2 } ( j _ E ) \\equiv 0 \\pmod { \\ell } \\} \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} - \\varepsilon < m - \\sum _ { j = 1 } ^ { n } \\left ( x ; S ^ { ( j ) } \\right ) < \\varepsilon \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} { \\bf v } ^ { ' } _ { j k } = \\sqrt p _ { j k } { \\bf v } _ { j k } . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} \\int \\limits _ { 0 } ^ { t } D _ { \\tau } ^ { q } K ( t , \\tau ) r ( \\tau ) d \\tau + r ( t ) \\underset { \\tau \\rightarrow t - 0 } { \\lim } I _ { \\tau } ^ { q - 1 } K ( t , \\tau ) + D _ { 0 + } ^ { q } ( F ( t ) - F ( 0 ) ) = D _ { 0 + } ^ { q } ( E ( t ) - E ( 0 ) ) . \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{align*} \\left ( \\sum _ { k \\in \\mathbb { Z } ^ d } \\left | \\langle f , \\widetilde { \\phi } _ k \\rangle \\right | ^ \\tau \\right ) ^ { \\frac { 1 } { \\tau } } + \\left ( \\sum _ { i = 1 } ^ { 2 ^ { d } - 1 } \\sum _ { j \\in \\mathbb { N } _ 0 } \\sum _ { k \\in \\mathbb { Z } ^ d } \\left | \\langle f , \\widetilde { \\psi } _ { i , j , k } \\rangle \\right | ^ \\tau \\right ) ^ { \\frac { 1 } { \\tau } } < \\infty \\ , , \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} N \\Lambda ^ r \\Gamma ( P \\stackrel { \\varphi } { \\longrightarrow } Q ) _ n = \\operatorname { c r } _ n ( \\Lambda ^ r ) ( P , \\dots , P ) \\oplus \\operatorname { c r } _ { n + 1 } ( \\Lambda ^ r ) ( Q , P , \\dots , P ) . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\langle \\omega _ 0 , T \\rangle = 1 , ~ \\langle \\omega _ 0 , H X \\rangle = 0 , ~ T \\rfloor d \\omega _ 0 = 0 . \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} u ^ \\varepsilon ( x ) \\leq C ( 1 + \\lambda ) \\cfrac { 2 \\sqrt { h } } { c _ 0 \\mu } + v ^ + ( x _ 2 ) + \\eta = C ( 1 + \\lambda ) \\cfrac { 2 \\sqrt { h } } { c _ 0 \\mu } + u _ 2 ^ + ( h ) + \\eta . \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} ( c + 2 ) Y ^ 2 - ( c - 2 ) X ^ 2 = - 1 9 9 6 c + 4 0 0 8 . \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} V ^ { 2 I B P , \\theta } _ { n , k } ( \\gamma ) & = \\frac { 1 } { k ! } \\bigg ( \\frac { \\gamma } { \\left ( \\theta + 1 \\right ) _ { n - 1 \\uparrow } } \\bigg ) ^ { k } \\exp \\bigg ( - \\sum _ { i = 1 } ^ { n } \\gamma \\frac { \\left ( \\theta \\right ) _ { i - 1 \\uparrow } } { \\left ( 1 + \\theta \\right ) _ { i - 1 \\uparrow } } \\bigg ) \\\\ & = \\frac { 1 } { k ! } \\bigg ( \\frac { \\gamma } { ( \\theta + 1 ) _ { n - 1 \\uparrow } } \\bigg ) ^ { k } \\exp \\bigg ( - \\gamma \\sum _ { i = 1 } ^ { n } \\frac { \\theta } { \\theta + i - 1 } \\bigg ) , \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} \\begin{aligned} ( \\mathcal { A } - \\mathrm { I } ) u ( \\varphi , x ) & = ( 1 + \\beta _ x ) u ( \\varphi , x + \\beta ) - u ( \\varphi , x ) = u ( \\varphi , x + \\beta ) - u ( \\varphi , x ) + \\beta _ x u ( \\varphi , x + \\beta ) = \\\\ & = u _ x ( \\varphi , x ) \\beta ( \\varphi , x ) + \\beta _ x ( \\varphi , x ) u ( \\varphi , x ) + O ( \\beta ^ 2 ) = \\\\ & = \\varepsilon \\partial _ x ( \\beta _ 1 ( \\varphi , x ) \\ , u ( \\varphi , x ) ) + O ( \\varepsilon ^ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} \\begin{aligned} \\Delta ( a b ) & = \\sum ( - 1 ) ^ { [ b _ { ( 1 ) } ) ] [ a _ { ( 2 ) } ] } a _ { ( 1 ) } b _ { ( 1 ) } \\otimes a _ { ( 2 ) } b _ { ( 2 ) } . \\end{aligned} \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} a ( t ) + \\int _ { - \\infty } ^ 0 e ^ { - \\frac { 1 } { 8 } z ^ 2 } w ( t , z ) d z = A , t \\in \\mathbb { R } ^ + , \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} & S ( 2 ; 1 , 1 ) = 1 2 \\le S _ U ( 2 ; 1 , 1 ) = 3 9 , \\\\ & S ( 2 ; 1 , 2 ) = 4 = S _ U ( 2 ; 1 , 2 ) . \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} C ' _ 1 = \\frac { 1 } { 2 } \\log \\Bigl ( 1 + \\frac { a P _ 1 } { P _ 2 + N } \\Bigr ) . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} \\nu _ { i } & \\left ( \\psi _ { i } , \\psi _ { - i } \\right ) = \\sum _ { a \\in \\mathcal { P } } \\left [ \\prod _ { j = 1 } ^ { K } \\psi _ { \\pi ( j ) } \\left ( a _ { \\pi ( j ) } \\right ) \\right ] \\nu _ { i } \\left ( a _ { i } , a _ { - i } \\right ) . \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} F ^ { ( 3 ) } ( u ) : = \\sum _ { j _ 1 + j _ 2 + j _ 3 = 0 } F ^ { ( 3 ) } _ { j _ 1 \\ , j _ 2 \\ , j _ 3 } \\ , u _ { j _ 1 } \\ , u _ { j _ 2 } \\ , u _ { j _ 3 } . \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} A : = \\pi ^ { - 1 } ( \\{ 1 , 2 , \\cdots , m \\} ) : = \\{ i _ 1 , i _ 2 , \\cdots , i _ m \\} , \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} p ^ { 4 } - 2 p ^ { 3 } q + 2 p ^ { 2 } q ^ { 2 } + 2 p q ^ { 3 } + q ^ { 4 } = \\mu , \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} C ( \\sigma _ i | \\sigma _ 1 , \\ldots , \\sigma _ { i - 1 } ) = C ( \\sigma _ i ) \\setminus \\bigcup _ { j = 1 } ^ { i - 1 } C ( \\sigma _ j ) . \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\delta ( x ) = \\max \\left \\{ \\varepsilon ( x ) , \\frac { 1 } { \\log x } \\right \\} . \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} M ^ j _ 0 = \\left [ \\begin{array} { c c c c } f _ j ( Y _ 0 | X _ 0 = 1 ) & 0 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & \\cdots & f _ j ( Y _ 0 | X _ 0 = d ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} g _ { \\ell , j } ( z ) \\equiv \\sum _ { \\substack { n = 1 \\\\ { \\ell \\nmid n } } } ^ { \\infty } a ( \\ell ^ { m _ \\ell } n ) q ^ n \\pmod { \\ell ^ j } . \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} S ^ \\flat ( X , Y ) = \\sum _ { \\substack { X < \\ell \\le Y \\\\ ( \\ell , q ) = 1 } } \\mu ( \\ell ) ^ 2 c ( \\ell ) ^ 2 \\ell ^ { - 1 } \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} \\begin{array} { r l } Z ^ { I P } = \\max \\ \\ ; & c ' z + d ' u \\\\ \\mbox { s . t . } \\ \\ ; & A z = b , \\\\ & H z + G u \\leq h , \\\\ & z \\in \\{ 0 , 1 \\} ^ n . \\end{array} \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{align*} \\alpha ( t , x ) : = \\begin{cases} \\dot Y ^ \\varepsilon ( t , x ) - \\cfrac { b ( Y ^ \\varepsilon ( t , x ) ) } { \\varepsilon } \\ \\ \\ & x \\in \\Omega , \\\\ 0 \\ \\ \\ & x \\in \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} \\sup _ { \\mathbf { \\hat { x } } \\in \\Psi ( k _ 0 ) , \\ , \\mathbf { k } \\in \\Pi ( k _ 0 ) } \\ & \\ \\sum _ { t = 0 } ^ \\infty \\tilde { \\beta } _ t \\ , \\tilde { U } ( \\hat { x } _ t ) \\\\ [ 5 p t ] & \\hat { x } _ t + k _ { t + 1 } \\leq f ( k _ t ) , & & t = 0 , 1 , \\ldots , \\\\ [ 5 p t ] & k _ 0 > 0 \\ , \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} | \\mathcal { D } ^ { i j } ( - J , - 1 ) | \\leq | X ( - J , - 1 ) | = - \\mathcal { C } ( - J , - 1 ) + 2 . \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} \\Phi _ q ( x ) = \\Phi _ { q + r } ( x ) + \\langle d r , x - x _ 0 \\rangle = \\Phi _ { q + r } ( x _ 0 ) + \\delta \\| x - x _ 0 \\| . \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} \\mathcal C : = - \\underset { x \\in \\mathbb R } { \\min } \\left ( r _ u ( x ) - \\gamma _ u ( x ) \\left ( \\frac { q ( x ) } { K } + C \\right ) - \\mu ( x ) , r _ v ( x ) - \\gamma _ v ( x ) \\left ( \\frac { p ( x ) } { K } + C \\right ) - \\mu ( x ) , 0 \\right ) \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} h _ f ( f _ 0 ) = \\left | \\left ( C ^ 0 ( e ^ { i \\theta } ) \\right ) ^ { < \\frac { 1 } { \\alpha - 1 } > } ( f _ 0 ( \\theta ) ) ^ { \\frac { - 1 } { \\alpha - 1 } } \\right | ^ { \\alpha } \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} v _ { f , \\mathsf { k } } & = r _ { f , \\mathsf { k } } i _ { f , \\mathsf { k } } , \\\\ \\tau _ { m , \\mathsf { k } } & = D _ \\mathsf { k } \\omega _ 0 + \\tau _ { e , \\mathsf { k } } , \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} f _ 1 = x ^ 4 + 1 2 x ^ 2 + 6 f _ 2 = x ^ 4 + 4 x ^ 2 + 6 ; \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} v _ { 0 , 1 } ( x ) = \\sin ( 2 \\pi x ) , u _ { 0 , 1 } ( x ) = \\cos ( 2 \\pi x ) , \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} \\mathbb { F } _ q ( \\mathcal { H } , r , \\alpha ) : = \\bigoplus _ { n = 0 } ^ \\infty \\mathbb { F } ^ { ( n ) } _ q ( \\mathcal { H } , r , \\alpha ) , ~ ~ \\mathbb { F } ^ { ( n ) } _ q ( \\mathcal { H } , r , \\alpha ) : = \\mathbb { F } _ q ^ { ( n ) } ( \\mathcal { H } ) r ^ n ( [ n ] _ { | q | } ! ) ^ \\alpha . \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} I ( \\xi u ) & = \\frac { 1 } { b p ^ 2 } ( a + b \\| \\xi u \\| _ { E ^ { \\alpha , p } } ^ p ) ^ p - \\int _ 0 ^ T F ( t , \\xi u ( t ) ) d t - \\frac { a ^ p } { b p ^ 2 } \\\\ & \\leq \\frac { 1 } { b p ^ 2 } ( a + b \\| \\xi u \\| _ { E ^ { \\alpha , p } } ^ p ) ^ p - c _ 1 \\int _ 0 ^ T | \\xi u ( t ) | ^ \\mu d t + c _ 2 T - \\frac { a ^ p } { b p ^ 2 } \\\\ & = \\frac { 1 } { b p ^ 2 } ( a + b \\xi ^ p \\| u \\| _ { E ^ { \\alpha , p } } ^ p ) ^ p - c _ 1 \\xi ^ \\mu \\| u \\| _ { L ^ \\mu } ^ \\mu + c _ 2 T - \\frac { a ^ p } { b p ^ 2 } \\\\ & \\rightarrow - \\infty \\ \\ \\mbox { a s } \\ \\xi \\rightarrow \\infty . \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} \\Psi _ x = \\left ( \\begin{matrix} - i z & u \\\\ - \\bar u & i z \\end{matrix} \\right ) \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} ( \\theta _ t \\omega ) ( s ) : = \\omega ( s + t ) - \\omega ( t ) . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} K _ 2 ( \\xi , \\eta ) = & \\ \\frac 1 2 ( \\tanh \\xi + \\tanh \\eta ) ( \\xi + \\eta ) - \\frac 1 2 ( 1 + \\tanh \\xi \\tanh \\eta ) ( \\xi - \\eta ) \\tanh ( \\xi - \\eta ) \\\\ = & \\ \\tanh \\xi \\tanh \\eta \\left ( \\frac { \\xi } { \\tanh \\xi } + \\frac { \\eta } { \\tanh \\eta } - ( \\xi - \\eta ) \\tanh ( \\xi - \\eta ) \\right ) . \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} P _ { M } ( N , K , \\mathbf { p } ) = \\sum _ { \\mathbf { M } } B ^ { * } ( \\mathbf { M } , N , \\mathbf { p } ) \\mathbb { P } ^ { * } ( \\mathbf { M } , K ) , \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} \\mathcal { D } _ \\Omega : = \\{ \\varphi \\in C _ 0 ^ \\infty ( \\overline \\Omega \\times [ 0 , \\infty ) ) : \\ , \\delta ( x ) ^ { - 2 } \\varphi ( x , t ) \\mbox { b o u n d e d } \\} \\ , , \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} F _ { 3 n } ^ { \\left ( 3 \\right ) } = F _ { n } ^ { \\left ( 3 \\right ) } , \\thinspace \\thinspace F _ { 3 n + 1 } ^ { \\left ( 3 \\right ) } = F _ { n } ^ { \\left ( 3 \\right ) } , \\thinspace \\thinspace F _ { 3 n + 2 } ^ { \\left ( 3 \\right ) } = 2 F _ { n } ^ { \\left ( 3 \\right ) } \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { 2 ^ { \\ell } - 1 } \\lambda _ { \\ell , j } = 1 , B _ { 0 , 0 } = \\sum _ { j = 0 } ^ { 2 ^ { \\ell } - 1 } \\lambda _ { \\ell , j } B _ { \\ell , j } \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} A _ { c , E } ( \\theta ) = \\frac { 1 } { c ( \\theta ) } D _ { c , E } ( \\theta ) = \\frac { 1 } { c ( \\theta ) } \\left ( \\begin{matrix} E - v ( \\theta ) \\ \\ & - \\tilde { c } ( \\theta - \\alpha ) \\\\ c ( \\theta ) \\ \\ & 0 \\end{matrix} \\right ) . \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{gather*} \\big \\{ J _ 1 ^ \\xi , J _ 1 ^ { \\xi ' } \\big \\} _ 1 = 0 , \\big \\{ J _ 2 ^ \\xi , J _ 2 ^ { \\xi ' } \\big \\} _ 2 = 0 , \\big \\{ J _ 3 ^ \\xi , J _ 3 ^ { \\xi ' } \\big \\} _ 3 = - \\big ( \\xi ^ a \\xi '^ { b c } + \\xi '^ { a } \\xi ^ { b c } \\big ) h _ { a b c } , \\end{gather*}"}
-{"id": "2766.png", "formula": "\\begin{align*} \\frac { d a } { d s } ( s _ y ) = \\frac { d } { d s } \\int _ 0 ^ { s _ y } [ \\gamma _ { \\partial B } ( s ) , \\gamma _ { \\partial B } ' ( s ) ] \\ d s = \\left [ y , \\frac { b ( y ) } { | | b ( y ) | | } \\right ] = \\frac { 1 } { | | b ( y ) | | } \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} s - \\frac { \\delta ' } { 1 + \\delta } \\ & = \\ s - ( ( 1 + \\epsilon ) ( 1 + u b ) ) ^ { \\dagger } \\\\ & = \\ s - \\frac { \\epsilon ' } { 1 + \\epsilon } - \\frac { ( u b ) ' } { 1 + u b } \\\\ & = \\ u b ' - \\frac { ( u b ) ' } { 1 + u b } \\\\ & = \\ \\frac { u ^ 2 b b ' - u ' b } { 1 + u b } . \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} I ( X _ n , X _ 1 ) & = H ( X _ n ) + H ( X _ 1 ) - H ( X _ n , X _ 1 ) \\\\ & = H ( X _ n ) - H ( X _ n | X _ 1 ) = H ( X _ n ) - H ( X _ { n - 1 } ) , \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{align*} \\sum _ { p \\leq x } \\frac { 1 } { p } = \\log \\log x + \\gamma + B + O \\ ( \\frac { 1 } { \\log x } \\ ) , \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} ( - | e | ^ { 2 } | K | ^ { 2 } | \\alpha | ^ { 4 } ) \\cdot 1 + ( \\overline { a } b ) \\alpha + ( a \\overline { b } ) \\overline { \\alpha } + ( | b | ^ { 2 } - \\overline { d } e K - d \\overline { e } \\overline { K } ) | \\alpha | ^ { 2 } = 0 . \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} \\ell E ^ k = d _ k \\ell _ k E ^ k \\cong d _ k \\ell _ k \\C ^ { r _ k } = \\ell \\C ^ { r _ k } , \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} \\mathrm { c m } ( \\alpha x - v , v ) = \\frac { [ v , b ( \\alpha x - v ) ] } { | | v | | } = \\frac { [ v , - x ] } { | | x | | _ a | | v | | } = - \\mathrm { s n } ( v , x ) , \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} h _ i = A _ i r ^ { \\mu _ i ^ + } \\phi _ i , \\ ; \\ , A _ i \\in \\R , \\ ; \\ ; | A _ i | \\leq \\left | \\int _ U ( \\Delta u ) r ^ { \\mu _ i ^ - } \\phi _ i \\right | + c \\left ( \\| u \\| _ { L ^ 1 ( \\partial U ) } + \\| \\partial _ r u \\| _ { L ^ 1 ( \\partial U ) } \\right ) , \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} u ( t ) = J _ { 1 } ( t ) u _ { 0 } + J _ { 2 } ( t ) u _ { 1 } + J _ { 3 } ( t ) u _ { 0 } + J _ { 4 } ( t ) u _ { 1 } , \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} C ( w , \\theta ) = \\{ v \\in \\mathbb { R } ^ { N } \\ , | \\ , v \\cdot w \\ge \\cos ( \\theta ) \\| v \\| \\} . \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} g = q - 3 q ^ 3 + 1 0 q ^ 5 - 4 q ^ 7 + 9 q ^ 9 + 2 0 q ^ { 1 1 } + O ( q ^ { 1 2 } ) \\in S _ 4 ( \\Gamma _ 0 ( 9 6 ) ) \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} C T \\frac 1 { ( x y z ) ^ { p - 1 } } \\prod _ { c y c } ( y + z ) ^ { p - 1 } & = C T \\frac 1 { ( x y z ) ^ { p - 1 } } \\prod _ { c y c } \\sum _ { i = 0 } ^ { p - 1 } \\binom { p - 1 } i y ^ i z ^ { p - 1 - i } = \\sum _ { j = 0 } ^ { p - 1 } \\binom { p - 1 } j ^ 3 \\\\ & \\equiv _ { p ^ 2 } \\sum _ { j = 0 } ^ { p - 1 } ( - 1 ) ^ j \\left [ 1 - 3 p H _ j \\right ] = 1 - \\frac { 3 p } 2 H _ { p ' } \\equiv _ { p ^ 2 } 1 + 3 p \\ , q _ p ( 2 ) . \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} \\lambda _ n ( f ) = \\frac { 1 } { | F _ n | } \\sum _ { g \\in F _ n } f ( g ) , \\textrm { f o r $ f \\in \\ell ^ \\infty ( G ) $ } . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} \\tilde { a } ( X ) = \\begin{pmatrix} ( R _ 3 + S _ 3 ) \\frac { f ^ 3 } { 3 } + S _ 3 \\mu \\frac { f ^ 2 g } { 2 } & S _ 3 \\mu \\left ( \\frac { f ^ 3 } { 3 } + \\frac { f ^ 2 g } { 2 } \\right ) \\\\ S _ 3 \\frac { g ^ 3 } { 3 } + ( R _ 3 + S _ 3 ) \\frac { f ^ 2 g } { 2 } + S _ 3 f g ^ 2 & S _ 3 \\frac { g ^ 3 } { 3 } + S _ 3 \\mu \\left ( \\frac { f ^ 2 g } { 2 } + f g ^ 2 \\right ) \\end{pmatrix} \\mbox { i n } \\ ; \\Omega = ( 0 , L ) \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} \\eta ^ { ( k ) } ( a ) ( \\vec { t } ) = t _ k ( 1 _ { D _ { 1 } } \\otimes 1 _ { D _ { 2 } } \\otimes \\cdots \\otimes a \\otimes \\cdots \\otimes 1 _ { D _ { n } } ) \\ , , \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} \\hat { f } ( h ) = \\int _ 0 ^ { 1 } f ( t ) e ^ { - 2 \\pi i h t } \\ , \\mathrm d t . \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to - \\infty } b ( t , \\alpha ) = 0 . \\end{align*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\frac { \\partial { U } ( \\hat { x } _ t , \\theta _ t ) } { \\partial { \\hat { x } } } = \\mu ( \\theta _ t ) \\ , \\frac { \\partial { U } ( \\hat { x } _ { t + 1 } , \\theta _ { t + 1 } ) } { \\partial { \\hat { x } } } \\ , f ' ( k _ { t + 1 } ) , \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} ( f ^ { ( n ) } , g ^ { ( n ) } ) _ { \\mathcal { F } ^ { ( n ) } _ q ( \\mathcal { H } ) } : = ( P _ q ^ { ( n ) } f ^ { ( n ) } , g ^ { ( n ) } ) _ { \\mathcal { H } ^ { \\otimes n } } \\ , . \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\ell _ \\mathsf { S e m } ( c _ n ) = \\max _ { p _ M \\in \\mathcal { P } ( \\mathcal { M } _ n ) } \\ell ( p _ M , c _ n ) . \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} ( T \\widetilde { f } - T f ) ( 2 y - T \\widetilde { f } - T f ) & \\leq 2 | T \\widetilde { f } - T f | | y | + 2 B _ n | T \\widetilde { f } - T f | \\\\ & = 2 | T \\widetilde { f } - T f | ( | y | - B _ n ) + 4 B _ n | T \\widetilde { f } - T f | \\\\ & \\leq 4 B _ n ( | y | - B _ n ) + 4 B _ n | \\widetilde { f } - f | . \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} l ( P _ 2 ) = & d _ { S ( G , t - j ) } ( x _ { j + 1 } \\cdots x _ t , ( u _ 1 ) ^ { t - j } ) + 2 ( 2 ^ { t - j } - 1 ) ( s - 1 ) + s \\\\ & + d _ { S ( G , t - j ) } ( ( u _ { s - 1 } ) ^ { t - j } , y _ { j + 1 } \\cdots y _ t ) . \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} \\int \\limits _ { \\Sigma } \\prod _ { l = 1 } ^ j e ^ { 2 i z ( y _ { l } - x _ l ) } u ( y _ { l } ) \\overline { v ( x _ { l } ) } d x _ 1 d y _ 1 \\dots d x _ { j } d y _ j . \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} A _ { \\alpha } \\left ( G \\right ) - A _ { \\beta } \\left ( G \\right ) = \\left ( \\alpha - \\beta \\right ) L \\left ( G \\right ) , \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} \\begin{pmatrix} 1 & a & c \\\\ 0 & 1 & b \\\\ 0 & 0 & 1 \\\\ \\end{pmatrix} \\equiv \\exp \\begin{pmatrix} 0 & x & z \\\\ 0 & 0 & y \\\\ 0 & 0 & 0 \\\\ \\end{pmatrix} = \\exp ( x X + y Y + z Z ) , \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} A u ' = Q ' ( u ^ \\pm ) u . \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} \\mathbf { S } _ N ^ { ( g ) } = \\left [ \\mathbf { v } _ 1 \\ , \\mathbf { v } _ 2 \\ , \\ldots \\ , \\mathbf { v } _ N \\right ] . \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} P _ d = \\Big \\{ x \\mapsto p ( \\| x \\| ^ 2 ) e ^ { - \\pi \\| x \\| ^ 2 } : p \\Big \\} , \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} \\gamma ^ { ( \\kappa ) } ( t _ \\ell , \\cdot ) \\to u ( t _ \\ell , \\cdot ) \\ ; \\ ; \\ ; \\ ; { \\mathcal A } _ s \\gamma ^ { ( \\kappa ) } \\to { \\mathcal A } _ s u = g \\ ; \\ ; \\ ; \\ ; \\kappa \\to \\infty . \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} U = ( K ' ) ^ o \\cap \\left ( x ^ { ( l , \\sigma ) } \\right ) ^ { - 1 } ( D _ 1 ^ o ) \\ ; , \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} \\bar u ( x ) \\in \\begin{cases} \\{ U _ { \\min } \\} & \\bar p ( x ) < 0 , \\\\ [ U _ { \\min } , U _ { \\max } ] & \\bar p ( x ) = 0 , \\\\ \\{ U _ { \\max } \\} & \\bar p ( x ) > 0 . \\end{cases} \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} \\delta _ r ( \\alpha _ 0 < \\cdots < \\alpha _ r ) = & \\\\ \\sum _ { i = 0 } ^ { r + 1 } ( - 1 ) ^ { i } & \\sum _ { \\alpha \\in ( \\alpha _ { i - 1 } , \\alpha _ i ) } ( \\alpha _ 0 < \\cdots < \\alpha _ { i - 1 } < \\alpha < \\alpha _ i < \\cdots < \\alpha _ r ) , \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} d _ { \\lambda } ( p , r ) = - \\frac { 1 } { 1 - \\lambda } \\log E _ { q _ * ( x ) p ( y | x ) } \\left ( \\frac { r ( y | x ) } { p ( y | x ) } \\right ) ^ { 1 - \\lambda } . \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} \\frac { a ^ n ( c / a ; q ) _ n } { ( c ; q ) _ n } \\ = \\ \\sum _ { m = 0 } ^ n \\frac { ( a ; q ) _ m ( q ^ { - n } ; q ) _ m } { ( c ; q ) _ m ( q ; q ) _ m } q ^ m \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} | \\overline G _ i ( h , 0 ) - G ( 0 , 0 ) | \\leq \\gamma + \\frac { C _ 2 } { T _ i ( h ) } \\int _ 0 ^ { T _ i ( h ) } | D H ( X ( t , x ) ) | \\ , d t = \\gamma + \\frac { C _ 2 L _ i ( h ) } { T _ i ( h ) } . \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} Z : = \\big \\{ ( 0 , \\hat x ( 0 ) , q ) \\in \\mathcal { U } : w ( q ) = 0 \\big \\} \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\begin{aligned} u ^ \\varepsilon ( z _ 1 ) & \\leq \\int _ 0 ^ { t _ 1 + t _ 2 } L \\big ( \\xi ^ \\varepsilon ( s , z _ 1 , \\gamma ) , \\gamma ( \\xi ^ \\varepsilon ( s , x , \\gamma ) ) \\big ) e ^ { - \\lambda s } \\ , d s + u ^ \\varepsilon ( z _ 2 ) e ^ { - \\lambda ( t _ 1 + t _ 2 ) } \\\\ & \\leq C ( 1 + \\lambda ) \\cfrac { 4 \\sqrt { h } } { c _ 0 \\mu } + u ^ \\varepsilon ( z _ 2 ) , \\end{aligned} \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} I _ { z } ( s ) = \\frac { 2 ^ { - \\frac { z } { 2 } } } { \\Gamma \\left ( \\frac { z } { 2 } \\right ) } \\left \\vert s \\right \\vert ^ { z - 1 } . \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} \\begin{bmatrix} x _ 1 ^ { ( 1 ) } & x _ 2 ^ { ( 2 ) } & x _ 3 ^ { ( 1 ) } \\\\ x _ 1 ^ { ( 2 ) } & x _ 2 ^ { ( 1 ) } & x _ 3 ^ { ( 2 ) } \\end{bmatrix} = \\begin{bmatrix} \\alpha & \\gamma & \\epsilon \\\\ \\beta & \\delta & \\zeta \\end{bmatrix} \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} G ( z , s ) = 2 \\zeta ( 2 s ) \\cdot E ( z , s ) , \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} \\sum _ { m _ 2 } \\lambda _ { 2 } ( m _ 2 ) e \\left ( \\frac { d m _ 2 } { c } \\right ) V _ { - \\eta H ' } \\left ( \\frac { n - m _ 2 } { H ' } \\right ) = \\frac { H ' } { c } \\sum _ { m _ 2 } \\lambda _ { 2 } ( m _ 2 ) e \\left ( - \\frac { \\bar { d } m _ 2 } { c } \\right ) V ^ { \\star } _ { - \\eta H ' } \\left ( \\frac { m _ 2 n } { c ^ 2 } , \\frac { m _ 2 H ' } { c ^ 2 } \\right ) , \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} \\left | \\left ( \\frac { \\lambda _ { + } \\lambda _ { - } } { \\lambda _ { + } - \\lambda _ { - } } - \\frac { 1 } { \\nu } | \\xi | ^ { 2 ( 1 - \\sigma ) } \\right ) \\chi _ { L } \\right | & \\le C | \\xi | ^ { 2 ( 2 - 3 \\sigma ) } \\chi _ { L } \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} H = \\sum _ { i = 1 } ^ { N } \\left ( x _ i \\frac { \\partial } { \\partial x _ i } \\right ) ^ 2 + \\beta \\sum _ { i < j } \\frac { x _ i + x _ j } { x _ i - x _ j } \\left ( x _ i \\frac { \\partial } { \\partial x _ i } - x _ j \\frac { \\partial } { \\partial x _ j } \\right ) - 2 \\beta \\sum _ { i < j } \\frac { x _ i x _ j } { \\left ( x _ i - x _ j \\right ) ^ 2 } \\left ( 1 - K _ { i j } \\right ) , \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} \\flat : T M \\rightarrow T ^ { * } M \\flat ( X ) = \\iota _ X \\Omega \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} \\delta ( \\pi ) & = \\sum _ { j = 1 } ^ { n - 1 } \\chi ( f _ { 2 j } > 1 ) , \\\\ \\gamma ( \\pi ) & = ( \\chi ( n ) + 2 \\cdot f _ n \\cdot \\chi ( n ) ) \\prod _ { 2 j + 1 < n } ( 2 f _ { 2 j + 1 } + 1 ) , \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} I ( u ) & = \\frac { 1 } { b p ^ 2 } \\left ( a + b \\int _ 0 ^ T | { _ 0 D _ t ^ \\alpha } u ( t ) | ^ p d t \\right ) ^ p - \\int _ 0 ^ T F ( t , u ( t ) ) d t - \\frac { a ^ p } { b p ^ 2 } \\\\ & = \\frac { 1 } { b p ^ 2 } ( a + b \\| u \\| _ { E ^ { \\alpha , p } } ^ p ) ^ p - \\int _ 0 ^ T F ( t , u ( t ) ) d t - \\frac { a ^ p } { b p ^ 2 } . \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} N _ 0 ^ { C _ 1 } \\varepsilon ^ { b _ * + 1 } \\gamma ^ { - 2 } = N _ 0 ^ { C _ 1 } \\varepsilon ^ { 1 - 3 a } = \\varepsilon ^ { 1 - 3 a - \\rho \\ , C _ 1 ( 1 + a ) } < \\delta _ 0 \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} \\gamma = \\pi _ + \\circ \\pi _ \\Gamma ^ { - 1 } : \\real ^ { d _ s } \\cap \\pi _ - ( \\Gamma ) \\to \\real ^ { d _ u } \\ , , \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ N \\frac { \\omega _ i w _ N ^ 2 ( \\tau _ i ) } { 1 - \\tau _ i ^ 2 } \\le c . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} \\alpha _ f ( a ) = \\int _ { - \\infty } ^ { \\infty } f ( t ) \\alpha _ t ( a ) ~ d t \\ , . \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} F \\big | _ { \\{ [ \\alpha , \\beta ] ; \\mu \\} , \\gamma } ( z ) = F ( z ) . \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} \\beta _ c \\ln \\beta _ c = \\beta _ c - { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} \\gamma _ { n } = a _ { n } c _ { n } / \\alpha _ { n } , \\delta _ { n } = b _ { n } d _ { n } / \\beta _ { n } , \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} \\underset { S \\rightarrow \\infty } { \\lim } D = 2 . \\end{align*}"}
-{"id": "7710.png", "formula": "\\begin{align*} a _ { \\mu } ^ { \\frac 1 3 - \\frac { 1 } { 3 q } } \\mu ^ { - \\frac 1 4 } + a _ { \\mu } ^ { \\frac 1 2 - \\frac 1 q } \\mu ^ { - \\frac 1 2 + \\frac 1 q } = a _ { \\mu } ^ { \\frac 1 2 - \\frac 1 q } \\mu ^ { - \\frac 1 2 + \\frac 1 q } \\left ( 1 + ( a _ \\mu ^ { \\frac 1 6 } \\mu ^ { - \\frac 1 4 } ) ^ \\frac { 4 - q } { q } \\right ) \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} | E _ N | _ { \\C { H } ^ 1 ( \\Omega ) } \\le ( c / N ) ^ { p - 1 } \\| u \\| _ { \\C { H } ^ p ( \\Omega ) } , \\mbox { w h e r e } p = \\min \\{ \\eta , N + 1 \\} . \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} Y _ n ( t ) = \\begin{cases} Z ^ \\varepsilon ( s _ n - t , x _ n ) \\ \\ \\ & t \\in [ 0 , s _ n ] , \\\\ X _ n ( t - s _ n ) \\ \\ \\ & t \\in [ s _ n , t _ n ] . \\end{cases} \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} ( \\bold { D } ^ { \\ast } _ Q ) _ { \\alpha \\beta } = \\sum _ J ( - D ) _ J \\cdot \\frac { \\partial Q ^ { \\beta } } { \\partial u _ J ^ { \\alpha } } . \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} | z _ 0 | _ { \\beta } = \\max \\{ | z _ 0 | _ 0 , | z _ 0 | _ { \\alpha } \\} \\leq \\delta _ 0 , \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ l \\kappa _ i \\int _ { \\Sigma _ i } \\left ( R _ { \\gamma _ i } - \\varepsilon ( n - 2 ) ( n - 1 ) - \\frac { n - 2 } { n - 1 } H _ i ^ 2 \\right ) d { \\Sigma _ i } \\geq 0 , \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} a ( r , s + t ) | _ { [ r , r + s ] \\times [ 0 , r ] } & = a ( r , s ) | _ { [ r , r + s ] \\times [ 0 , r ] } \\\\ a ( r + s , t ) | _ { [ r + s , r + s + t ] \\times [ 0 , r ] } & = a ( r , s + t ) | _ { [ r + s , r + s + t ] \\times [ 0 , r ] } , \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { D V ^ p } = \\sup \\left \\{ \\int f \\phi d t : \\Vert \\phi \\Vert _ { U ^ q } \\le 1 , \\phi \\in C ^ \\infty _ 0 \\right \\} \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} \\frac { u _ i ' ( \\tilde { x } _ t ^ i ) } { \\delta ^ i u _ i ' ( \\tilde { x } _ { t + 1 } ^ i ) } = \\frac { u _ j ' ( \\tilde { x } _ t ^ j ) } { \\delta ^ j u _ j ' ( \\tilde { x } _ { t + 1 } ^ j ) } , & & 1 \\leq j < i ; t = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} \\hat { \\Gamma } ( k _ t ) : = \\Big \\{ k _ { t + 1 } \\in \\Gamma ( k _ t ) : U _ t ( f ( k _ t ) - k _ { t + 1 } ) > - \\infty \\Big \\} . \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} L ( s , \\pi , \\tau ) = \\prod _ { i = 1 } ^ k \\prod _ { j = 1 } ^ l L ( s , \\delta _ i , \\delta _ j ' ) \\ \\ \\ \\ \\ \\ \\epsilon ( s , \\pi , \\tau , \\psi ) = \\prod _ { i = 1 } ^ k \\prod _ { j = 1 } ^ l \\epsilon ( s , \\delta _ i , \\delta _ j ' , \\psi ) . \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{align*} \\mathbf { M } _ { \\gamma - \\eta } \\left \\lbrace g y \\right \\rbrace : = L _ { y } \\left \\lbrace g \\ , x _ { \\gamma - \\eta } ^ { \\vphantom { \\intercal } } x _ { \\gamma - \\eta } ^ { \\intercal } \\right \\rbrace . \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 q ^ 2 ( \\tau ) \\ ; d \\tau = 2 . \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} \\zeta ( p , q ) = \\tilde \\zeta \\left ( e ^ { \\frac { 3 } { 4 } y ( q ) } d ( \\pi _ k ( p ) , \\pi _ k ( q ) ) \\right ) . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\sum _ { m _ 1 } \\lambda _ { 1 } ( m _ 1 ) e \\left ( \\frac { d m _ 1 } { c } \\right ) W _ { \\eta H } \\left ( \\frac { m _ 1 - n } { H } \\right ) = \\frac { H } { c } \\sum _ { m _ 1 } \\lambda _ { 1 } ( m _ 1 ) e \\left ( - \\frac { \\bar { d } m _ 1 } { c } \\right ) W ^ { \\star } _ { \\eta H } \\left ( \\frac { m _ 1 n } { c ^ 2 } , \\frac { m _ 1 H } { c ^ 2 } \\right ) , \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} \\Gamma \\left ( a + 1 , z \\right ) & = z ^ { a } \\exp \\left ( - z \\right ) \\left \\{ 1 + \\frac { a } { z } + \\frac { a \\left ( a - 1 \\right ) } { z ^ { 2 } } + \\cdots + \\frac { a \\left ( a - 1 \\right ) \\cdots 2 } { z ^ { a - 1 } } \\right \\} \\\\ & + a \\left ( a - 1 \\right ) \\cdots 2 \\cdot 1 \\cdot \\Gamma \\left ( 1 , z \\right ) , \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} P ( b ) & = \\left ( \\frac { 4 \\pi V c } { b } \\right ) ^ N \\prod _ { i = 1 } ^ N \\bigl ( { m _ i } ^ 2 K _ 2 ( m _ i b c ^ 2 ) \\bigr ) \\ , , \\\\ \\rho _ b & = \\frac { 1 } { P ( b ) } \\exp \\left ( - b c \\sum _ { i = 1 } ^ N \\sqrt { { p _ i } ^ 2 + { m _ i } ^ 2 c ^ 2 } \\right ) \\ , . \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} { \\bf u } _ k ( s , x ) = ( { \\bf \\Phi } ( { \\bf u } _ { k - 1 } ) ) ( s , x ) \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} - \\Delta u = \\lambda f ( x ) | u | ^ { q - 2 } u + | u | ^ { 2 ^ * - 2 } u , \\ ; \\ ; u \\in H ^ 1 _ 0 ( \\Omega ) , \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} & \\sum _ { k = 0 } ^ n T ( n , k ) = 1 \\ , , \\\\ & \\sum _ { k = 0 } ^ n T ( n , k ) H _ k = 2 H _ n \\ , . \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} \\ddot { \\rho } \\left ( r + \\frac { \\rho ^ 2 } { r } h _ 0 '' ( d ) \\right ) = \\frac { 1 } { r } \\left ( \\rho + \\frac { \\omega ^ 2 } { \\rho ^ 3 } \\right ) - \\dot { \\rho } + \\frac { \\rho } { r } \\left ( d - \\dot { \\rho } ^ 2 \\right ) h _ 0 '' ( d ) . \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} \\frac { d } { d s } \\mathrm { s n } \\left ( \\gamma _ { \\partial B } ( s ) , x _ 0 \\right ) = \\mathrm { s n } \\left ( b \\left ( \\gamma _ { \\partial B } ( s ) \\right ) , x _ 0 \\right ) . \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} \\operatorname { D i v } P = 0 \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} \\varphi ^ { ( i ) } ( a ) ( t + M \\Z ) = \\begin{cases} h _ 0 ( t + M \\Z ) \\alpha _ t ( a ) & \\mid i = 0 \\\\ h _ 1 ( t + M \\Z ) \\alpha _ { t - M / 2 } ( a ) & \\mid i = 1 \\end{cases} \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} G ( i \\tau ) = \\sum _ { j = 0 } ^ N ( - 1 ) ^ j \\tau ^ { - 2 j - 1 } G _ { 2 j } + o ( \\tau ^ { - 2 N - 1 } ) . \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} \\mathbb { E } \\left [ \\frac { 1 } { t } \\int _ 0 ^ t \\left ( 1 - | a ( \\theta _ s \\omega ) | ^ 2 \\right ) \\ , d s \\right ] & = \\frac { 1 } { t } \\int _ 0 ^ t \\mathbb { E } \\left [ 1 - | a ( \\theta _ s \\omega ) | ^ 2 \\right ] \\ , d s \\\\ & = \\frac { 1 } { Z _ \\sigma } \\int _ { \\mathbb { R } } ( 1 - y ^ 2 ) \\ , e ^ { \\frac { 2 } { \\sigma ^ 2 } ( \\frac { 1 } { 2 } y ^ 2 - \\frac { 1 } { 4 } y ^ 4 ) } \\ , d y \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} \\delta = \\sqrt { \\dfrac { 2 } { d } } , \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} | \\nabla ^ j _ { g _ { C _ \\ell } } \\Delta _ { \\Phi _ \\ell ^ * g } ( r ^ { \\mu _ { \\ell , i } ^ + } \\phi _ { \\ell , i } ) | _ { g _ { C _ \\ell } } = O ( r ^ { \\mu _ { \\ell , i } ^ + - 2 + Q _ { \\ell , i } - j } ) , \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} \\mathbf { y } ^ { ( g ) } \\triangleq \\underbrace { \\left ( \\mathbf { I } _ T \\otimes \\left [ \\mathbf { S } _ D ^ { ( g ) } \\right ] ^ H \\right ) } _ { \\triangleq \\left ( \\boldsymbol { \\Upsilon } _ S ^ { ( g ) } \\right ) ^ H } \\mathbf { y } , \\ ; g = 1 , \\ldots , G , \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} U _ { y _ o } = A _ { x _ o } B _ { y _ o } \\supset Q _ o \\cap T , \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\sup _ { n \\in \\N } \\sup _ { z \\in U } ( f ^ n ) ^ \\# ( z ) = \\infty \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} \\frac { d ^ 2 f ( \\gamma , 1 ) } { d \\gamma ^ 2 } & = \\frac { c _ n } { \\gamma ( 1 - \\gamma ) } - \\frac { ( k _ n P ' ) ^ 2 } { 2 ( 2 + \\gamma k _ n P ' ) ^ 2 } . \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} \\omega _ { \\infty , \\Lambda } = ( \\Omega _ { \\Lambda } , \\cdot \\Omega _ { \\Lambda } ) ) \\ , \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} Y ( z ) = e ^ { \\frac { N \\ell } { 2 } \\sigma _ 3 } \\left ( I + { \\cal O } \\left ( \\frac { 1 } { N ^ { L } } \\right ) \\right ) Z ^ \\infty ( z ) \\begin{bmatrix} 1 & 0 \\\\ \\displaystyle - \\star \\ , \\Big ( \\frac { z } { z - a } \\Big ) ^ { c } e ^ { N \\phi ( z ) } & 1 \\end{bmatrix} e ^ { \\frac { - N \\ell } { 2 } \\sigma _ 3 } e ^ { N g ( z ) \\sigma _ 3 } \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} Q = h ( P ) = - f ( P ) / D ( P ) \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} \\left [ - 2 L , 2 L \\right ] + \\left [ - L , L \\right ] = \\left [ - 3 L , 3 L \\right ] = \\left [ - \\left ( \\frac { 1 } { 2 } - 1 2 \\varepsilon \\right ) 8 L , \\left ( \\frac { 1 } { 2 } - 1 2 \\varepsilon \\right ) 8 L \\right ] \\ ; . \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} 2 \\int _ { 0 } ^ { + \\infty } \\Phi ( x ) \\cosh \\Big ( \\frac { x } { 2 } \\Big ) \\d x = \\frac { 1 } { T ^ { 1 / 4 } } \\int _ { 0 } ^ L \\Phi ^ + ( x ) e ^ { x / 2 } \\d x + T ^ { 1 / 4 } \\int _ { 0 } ^ L \\Phi ^ + ( x ) e ^ { - x / 2 } \\d x . \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} a \\left ( \\frac { \\log { y } } { \\log { N } } \\right ) = \\frac { 1 } { 2 \\pi i } \\int _ { ( 1 ) } f ( z ) y ^ { - z } z ^ { - 2 } d z , \\ y > 0 . \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} B _ \\epsilon ( \\theta ^ * ) = \\{ \\theta : \\rho ( \\theta , \\theta ^ * ) < \\epsilon \\} \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} \\P \\left [ \\left | G \\right | \\geq \\gamma ^ { - 1 + \\rho } \\right ] = 2 Q ( \\gamma ^ { - 1 + \\rho } ) \\leq 2 \\gamma ^ { 1 - \\rho } \\cdot \\exp \\left ( - \\frac { 1 } { 2 \\gamma ^ { 2 - 2 \\rho } } \\right ) . \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} \\int _ M \\lambda | \\stackrel { \\circ } { \\rm R i c } _ g | ^ 2 d M = - \\sum _ i \\kappa _ i \\int _ { \\Sigma _ i } \\left ( { \\rm R i c } _ { g } ( \\nu , \\nu ) - \\varepsilon ( n - 1 ) \\right ) d \\Sigma _ i , \\end{align*}"}
-{"id": "4635.png", "formula": "\\begin{gather*} B ^ h [ W , W ] = \\frac { 1 } { 2 \\pi } \\int B ^ h ( \\xi , \\eta ) \\hat W ( \\xi ) \\hat W ( \\eta ) e ^ { i ( \\xi + \\eta ) \\alpha } \\ , d \\xi d \\eta \\\\ B ^ a [ W , \\bar W ] = \\frac { 1 } { 2 \\pi } \\int B ^ a ( \\xi , \\eta ) \\hat W ( \\xi ) \\bar { \\hat W } ( \\eta ) e ^ { i ( \\xi - \\eta ) \\alpha } \\ , d \\xi d \\eta . \\end{gather*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\mathrm { c n } ( x , z ) ^ 2 + \\mathrm { c n } ( x , b ( z ) ) ^ 2 = [ x , b ( z ) ] . [ z , b ( x ) ] + \\frac { [ b ( z ) , b ( x ) ] } { | | b ( z ) | | } . \\frac { [ x , - z ] } { | | z | | _ a } . \\end{align*}"}
-{"id": "183.png", "formula": "\\begin{align*} \\langle J _ { \\lambda , \\mu } ' ( u , v ) , ( \\phi , \\psi ) \\rangle & = \\mathcal { A } ( u , \\phi ) + \\mathcal { A } ( v , \\psi ) - \\int _ \\Omega \\left ( \\lambda | u | ^ { q - 2 } u \\phi + \\mu | v | ^ { q - 2 } v \\psi \\right ) d x \\\\ & - \\frac { 2 \\alpha } { \\alpha + \\beta } \\int _ \\Omega | u | ^ { \\alpha - 2 } u | v | ^ \\beta \\phi d x - \\frac { 2 \\beta } { \\alpha + \\beta } \\int _ \\Omega | u | ^ { \\alpha } | v | ^ { \\beta - 2 } v \\psi d x \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{gather*} ( j ^ { \\infty } \\phi ) ^ \\ast ( \\partial _ I Q ^ a ) = 0 . \\end{gather*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\phi _ t \\ ; = \\ ; \\Delta _ 0 + \\Delta _ 1 \\circ \\tau + \\ldots + \\Delta _ s \\circ \\tau ^ s \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} \\begin{aligned} u ^ \\varepsilon ( y _ 3 ) & \\leq \\int _ 0 ^ { s _ 3 + s _ 4 } L \\big ( \\xi ^ \\varepsilon ( s , y _ 1 , - \\gamma ) , - \\gamma ( \\xi ^ \\varepsilon ( s , x , - \\gamma ) ) \\big ) e ^ { - \\lambda s } \\ , d s + u ^ \\varepsilon ( y _ 4 ) e ^ { - \\lambda ( s _ 3 + s _ 4 ) } \\\\ & \\leq C ( 1 + \\lambda ) \\cfrac { 4 \\sqrt { h } } { c _ 0 \\mu } + u ^ \\varepsilon ( y _ 4 ) . \\end{aligned} \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } w ( x ) = d ( x ) & d ( x ) < \\delta \\\\ \\delta \\leq w ( x ) \\leq C _ { \\delta } & d ( x ) \\geq \\delta , \\end{array} \\right . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} \\varphi ^ \\star ( \\eta ) - M _ 1 = \\kappa ' _ 1 e ^ { - \\eta } \\vec { X } _ { 1 1 } + \\kappa ' _ 2 e ^ { - \\frac { 1 - m + n } { m - n } \\eta } \\vec { X } _ { 1 2 } + \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } \\frac { ( - 1 ) ^ n q ^ { 2 n } } { 1 - q ^ { 4 n } } \\frac { ( - q ; q ^ 2 ) _ { n - 1 } } { ( q ^ 2 ; q ^ 2 ) _ { n - 1 } } q ^ { n - 1 } = \\frac { 1 } { 1 - q } \\sum _ { n \\geq 0 } ( - 1 ) ^ n q ^ { n ^ 2 } - \\frac { 1 } { 1 - q ^ 2 } . \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} T = \\{ 1 \\pm \\sqrt { - 1 } , \\pm \\sqrt { - 2 } , \\frac { 1 \\pm \\sqrt { - 7 } } { 2 } , 1 \\pm \\sqrt { - 2 } , \\pm \\sqrt { - 3 } , \\frac { 3 \\pm \\sqrt { - 3 } } { 2 } , \\frac { 1 \\pm \\sqrt { - 1 1 } } { 2 } \\} , \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} { 1 \\over n b } \\ell _ { \\rm L Z } ( v ^ n ) & \\leq { 1 \\over b } \\hat { H } _ k ( v ^ n ) + \\d . \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} T ^ { a b } _ { 0 } = & \\oint \\frac { d \\xi } { \\xi } \\phi _ a ^ - ( \\xi ) V _ a ^ { - 1 } ( \\xi ) V _ b ( \\xi ) , \\\\ T ^ { a b } _ { 1 } = - \\frac { 1 } { \\beta } & \\oint \\frac { d \\xi } { \\xi } \\phi _ a ^ - ( \\xi ) V _ a ^ { - 1 } ( \\xi ) \\phi _ b ^ + ( \\xi ) V _ b ( \\xi ) \\\\ & - \\sum _ c \\oint \\frac { d \\xi d \\eta } { \\xi \\eta } \\frac { \\frac { \\eta } { \\xi } } { 1 - \\frac { \\eta } { \\xi } } \\phi _ a ^ - ( \\eta ) V _ a ^ { - 1 } ( \\eta ) \\phi _ c ^ - ( \\xi ) V _ c ( \\eta ) V _ c ^ { - 1 } ( \\xi ) V _ b ( \\xi ) . \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} I _ j : = \\int _ 0 ^ { \\delta } | \\log \\sigma | ^ { k - j } \\sigma ^ m x ^ { - \\sigma } d \\sigma . \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} \\gamma ( s , \\delta _ 1 , \\delta _ 2 , \\psi ) = \\prod _ { i _ 1 = 1 } ^ { k _ 1 } \\prod _ { i _ 2 = 1 } ^ { k _ 2 } \\gamma ( s + \\frac { k _ 1 + k _ 2 } 2 + 1 - i _ 1 - i _ 2 , \\rho _ 1 , \\rho _ 2 , \\psi ) . \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} Q _ H ( e ^ { i t A } f - f , e ^ { i t A } f - f ) = O ( t ^ 2 ) , | t | \\le 1 . \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} - \\langle T _ { 2 2 } v , v \\rangle = s \\big \\langle \\big ( A _ { 2 2 } - A _ { 2 1 } ( \\tilde { A } _ { 1 1 } ) ^ { - 1 } A _ { 1 2 } \\big ) v , v \\big \\rangle . \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} \\deg ( I d - F _ 0 , \\Gamma , 0 ) = \\deg ( I d - F _ 0 , \\tilde \\Gamma , 0 ) = \\deg ( I d - G _ 1 , \\tilde \\Gamma , 0 ) . \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} f ^ \\ast ( k ) = \\frac { k f ( k ) } { \\zeta ( \\gamma ) } , \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow 0 } \\frac { W _ { \\frac { j } { k \\ell _ { \\lambda } } , j + \\frac { 1 } { 2 } } \\left ( x \\right ) } { W _ { \\frac { j } { k \\ell _ { \\lambda } } , j - \\frac { 1 } { 2 } } \\left ( x \\right ) } = C _ m \\ ; , \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} w ( q ) : = \\frac { 1 } { T } \\int _ 0 ^ T f _ 2 \\big ( t , \\hat x ( t ) , q , 0 \\big ) \\ , d t , q \\in M , \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{gather*} i _ Q \\delta _ Q \\omega = - \\delta \\delta _ Q L - d \\delta _ Q \\theta _ 1 , \\\\ i _ Q d \\omega _ 1 = - \\delta d L _ 1 - d ( i _ Q \\delta - \\delta i _ Q ) \\theta _ 1 , \\\\ d i _ Q \\omega _ 1 = d \\delta ( L _ 1 + i _ Q \\theta _ 1 ) - d i _ Q \\omega _ 1 , \\end{gather*}"}
-{"id": "7930.png", "formula": "\\begin{align*} \\Psi ( 0 ) = 0 , \\ \\lim _ { t \\rightarrow \\infty } \\Psi ( t ) = \\infty . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} \\lim _ { x \\to x _ \\mu } \\frac { x _ \\mu - x } { \\mu - Q ( x ) } = \\frac { 1 } { a _ \\mu } , \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} ( - 1 ) ^ { d _ x ( \\eta ) } & = ( - \\sigma _ { a _ * } \\sigma _ { b _ * } ) ( - \\sigma _ { b _ * } \\sigma _ { c _ * } ) ( - \\sigma _ { c _ * } \\sigma _ { d _ * } ) ( - \\sigma _ { d _ * } \\sigma _ { a _ * } ) = 1 . \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} E [ p ] E ' [ p ] \\Leftrightarrow \\left ( \\frac { \\ell } { p } \\right ) ^ r \\left ( \\frac { 2 } { p } \\right ) ^ t = 1 \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} E ( n ) : = \\sum _ { h \\asymp H } \\lambda _ { 1 } \\left ( n + h \\right ) \\lambda _ { 2 } \\left ( n - h \\right ) W \\left ( \\frac { h } { H } \\right ) . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{gather*} \\omega = \\delta A \\wedge \\delta A ^ \\ast + \\delta C \\wedge \\delta C ^ \\ast , \\operatorname { g h } ( \\omega ) = - 1 . \\end{gather*}"}
-{"id": "7700.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\infty } u ^ 2 \\ , \\dd x \\geq C \\int _ 0 ^ { x _ \\mu } \\frac { \\dd x } { ( \\mu - Q ) ^ \\frac 1 2 } \\geq \\frac { 1 } { a _ \\mu } \\int _ 0 ^ { x _ \\mu } \\frac { Q ' \\ , \\dd x } { ( \\mu - Q ) ^ \\frac 1 2 } = \\frac { \\mu ^ \\frac 1 2 } { a _ \\mu } . \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} \\bar \\lambda ( A ^ { \\nu } , ( A \\cup \\{ x \\} ) ^ { \\nu + \\mathbf { e _ i } } ) = x ^ i , \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} P : = \\frac { F _ s } { | x - y | ^ 2 } \\frac { \\rho ^ { s - 1 } } { ( \\rho + 1 ) ^ { \\frac { N } { 2 } - 1 } } + \\frac { 2 V _ s ( \\rho ) \\nabla F _ s \\cdot \\nabla \\rho + F _ s V _ s ' ( \\rho ) | \\nabla \\rho | ^ 2 + F _ s V _ s ( \\rho ) \\Delta \\rho } { 4 ( s - 1 ) } . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\lambda x ( t ) = \\int _ { \\overline { \\Omega } } k ( t , s ) f ( s , x ( s ) ) \\textup d s , t \\in \\overline { \\Omega } , \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} \\| i _ j \\| = \\sqrt { 2 / \\pi } . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} \\hat x _ n = \\arg \\min _ { s \\in \\mathcal { M } } | y _ n - \\sqrt { P } s | ^ 2 . \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} \\begin{aligned} \\log ( f ^ n ) ^ \\# ( z ) & \\leq \\log \\ ! \\left ( C ^ n \\prod _ { j = 0 } ^ { n - 1 } | z _ { j } | ^ { \\rho } \\right ) = n \\log C + \\rho \\sum _ { j = 0 } ^ { n - 1 } t _ { j } \\leq n \\log C + \\rho c _ 0 ( 1 + \\rho ) ^ { n - 1 } . \\end{aligned} \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ^ { - 1 } ( A + U V ^ { \\ast } ) & = A + U V ^ { \\ast } , \\\\ ( A + U V ^ { \\ast } ) \\big ( I + A ^ { \\dagger } U F _ { S _ { A } } U ^ { \\ast } ( A ^ { \\dagger } ) ^ { \\ast } \\big ) ^ { - 1 } & = A + U V ^ { \\ast } . \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { b - 1 } F _ { b n + k } ^ { \\left ( b \\right ) } & = \\sum _ { k = 0 } ^ { b - 1 } F _ { k } F _ { n } ^ { \\left ( b \\right ) } = F _ { n } ^ { \\left ( b \\right ) } \\left ( F _ { b + 1 } - 1 \\right ) \\\\ & = F _ { n } ^ { \\left ( b \\right ) } \\left ( F _ { b } + F _ { b - 1 } - 1 \\right ) = F _ { n } ^ { \\left ( b \\right ) } \\left ( F _ { b - 2 } + 2 F _ { b - 1 } - 1 \\right ) \\\\ & = F _ { b n + b - 2 } ^ { \\left ( b \\right ) } + 2 F _ { b n + b - 1 } ^ { \\left ( b \\right ) } - F _ { n } ^ { \\left ( b \\right ) } . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} \\max \\{ \\beta \\ , | \\ , x ^ T Q _ 1 x \\leq 1 , x ^ T Q _ 2 x = 1 , \\beta = x ^ T A ^ T Q _ 1 x \\} . \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} | \\phi ( t x _ n + r y _ n ) | = \\frac { n | r | } { 2 } \\leq 1 . \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} D ^ H : = \\sum d \\bar t ^ j \\otimes D _ { \\bar j } + \\sum d t ^ j \\otimes D _ j . \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} g ( W _ { i } , \\hat { \\gamma } _ { \\ell } , \\theta _ { 0 } ) + \\phi ( W _ { i } , \\hat { \\gamma } _ { \\ell } , \\hat { \\alpha } _ { \\ell } , \\tilde { \\theta } _ { \\ell } ) - \\psi ( W _ { i } , \\gamma _ { 0 } , \\alpha _ { 0 } , \\theta _ { 0 } ) = \\hat { R } _ { 1 \\ell i } + \\hat { R } _ { 2 \\ell i } + \\hat { R } _ { 3 \\ell i } + \\hat { \\Delta } _ { \\ell } ( W _ { i } ) . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} V _ 2 ( x ) = W ^ 2 ( x ) + W ' ( x ) \\ ; . \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} Z = \\sum _ { x _ 1 , \\ldots , x _ 5 } f _ 1 ( x _ 1 , x _ 2 , x _ 5 ) f _ 2 ( x _ 2 , x _ 3 ) f _ 3 ( x _ 3 , x _ 4 , x _ 5 ) . \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} \\tau ^ { 2 s + 1 } | T _ { 2 j } ( i \\tau ) | & \\lesssim \\sqrt { A _ p B _ p } \\Vert ( u , v ) \\Vert ^ 2 _ { H ^ s } \\Vert ( u , v ) \\Vert _ { l ^ 2 _ 1 ( D U ^ 2 ) } ^ { 2 j - 2 } \\\\ \\int _ 1 ^ { \\infty } \\tau ^ { 2 s } | T _ { 2 j } ( i \\tau ) | d \\tau & \\lesssim A _ i \\Vert ( u , v ) \\Vert ^ 2 _ { H ^ s } \\Vert ( u , v ) \\Vert _ { l ^ 2 _ 1 ( D U ^ 2 ) } ^ { 2 j - 2 } \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} \\delta \\hat { \\psi } ( p ) = \\sqrt { \\hat { \\psi } ( 0 ) ^ 2 - \\hat { \\psi } ( p ) ^ 2 } . \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} \\hat { \\pi } ( t _ 0 ) & = \\mathbb { P } [ \\mathrm { B i n } ( t _ 0 , p ) \\geq r ] = \\sum _ { j = r } ^ { t _ 0 } \\binom { t _ 0 } { j } p ^ { j } ( 1 - p ) ^ { t _ 0 - j } \\stackrel { \\eqref { t 0 s m a l l } } { = } ( 1 + o ( 1 ) ) \\frac { t _ 0 ^ r p ^ r } { r ! } \\\\ & = ( 1 + o ( 1 ) ) \\frac { t _ 0 ^ { r - 1 } p ^ r } { r ! } t _ 0 = ( 1 + o ( 1 ) ) \\frac { 1 } { r } \\frac { t _ 0 } { n } \\stackrel { n p = \\omega ( 1 ) } { = } o \\left ( t _ 0 p \\right ) \\stackrel { \\eqref { t 0 s m a l l } } { = } o ( 1 ) . \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{gather*} \\omega ( t ) = - \\frac { 1 } { 4 } ( - t ) ^ 2 - \\frac { 1 } { 8 } ( - t ) ^ { - 1 } - \\frac { 9 } { 6 4 } ( - t ) ^ { - 4 } - \\frac { 1 8 9 } { 1 2 8 } ( - t ) ^ { - 7 } - \\frac { 2 1 6 6 3 } { 5 1 2 } ( - t ) ^ { - 1 0 } \\\\ \\hphantom { \\omega ( t ) = } { } - \\frac { 4 8 2 5 9 7 1 } { 2 0 4 8 } ( - t ) ^ { - 1 3 } - \\frac { 3 5 4 0 3 1 1 7 3 9 } { 1 6 3 8 4 } ( - t ) ^ { - 1 6 } - \\frac { 2 4 1 9 8 0 2 9 7 1 1 1 } { 8 1 9 2 } ( - t ) ^ { - 1 9 } + \\cdots . \\end{gather*}"}
-{"id": "5671.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { k } p _ { j l } = 0 , \\ \\forall 1 \\leq l \\leq r . \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} & P = q _ { a - c } ^ { ( c + 1 ) } q _ { a - c + 1 } ^ { ( c + 1 ) } \\cdots q _ { b - c + 1 } ^ { ( c + 1 ) } , \\\\ & Q = q _ { 1 } ^ { ( c ) } q _ { 2 } ^ { ( c ) } \\cdots q _ { d - c + 1 } ^ { ( c ) } . \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} - \\Delta _ { p ( x ) } u _ { 0 } = \\left \\{ \\begin{array} { l l } 1 & \\Omega \\backslash \\overline { \\Omega } _ { \\delta } \\\\ - 1 & \\Omega _ { \\delta } \\end{array} \\right . , u _ { 0 } = 0 \\partial \\Omega \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} c = \\min \\left \\{ \\frac { b A } { 6 4 ( 1 + A a ) } , 1 \\right \\} . \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{align*} | V ( F ) | = k ( F ) + k \\leq \\frac { V ( F ) | } { 2 } + k . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} \\lambda = [ \\lambda _ { 0 0 0 } \\ \\ \\lambda _ { 0 0 1 } \\ \\ \\lambda _ { 0 1 0 } \\ \\ \\lambda _ { 0 1 1 } \\ \\ \\lambda _ { 1 0 0 } \\ \\ \\lambda _ { 1 0 1 } \\ \\ \\lambda _ { 1 1 0 } \\ \\ \\lambda _ { 1 1 1 } ] ^ T \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} r _ n = \\sqrt { P } x _ n + v _ n , \\end{align*}"}
-{"id": "6264.png", "formula": "\\begin{align*} [ \\Gamma : H \\Gamma ' ] = \\infty . \\end{align*}"}
-{"id": "8225.png", "formula": "\\begin{align*} \\sin \\left ( \\eta \\right ) = \\frac { Z \\alpha _ g } { j } \\ ; . \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} A _ { \\Lambda ' } ( \\underline \\omega _ \\Lambda ) = A & = \\{ \\omega \\in \\{ 0 , 1 \\} ^ { \\Lambda ' } : \\ ; \\omega \\eta _ { \\Lambda ^ c } \\in \\Omega _ { \\textrm { E P } } , \\ , \\omega _ { \\Lambda } = \\underline \\omega _ \\Lambda \\} , \\\\ B _ { \\Lambda ' } ( \\underline \\omega _ \\Lambda ) = B & = \\{ \\sigma \\in \\{ - 1 , 1 \\} ^ { \\Lambda ' _ * } : \\delta ( \\sigma _ { \\Lambda _ { m + 1 , * } } ) = \\underline \\omega _ { \\Lambda } \\} . \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} \\zeta = 1 \\quad \\mbox { i n } B ( 0 , \\rho / 2 ) , 0 \\le \\zeta \\le 1 \\quad \\mbox { i n } { \\bf R } ^ N , \\zeta = 0 \\quad \\mbox { o u t s i d e } B ( 0 , \\rho ) . \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\frac { d \\psi _ 1 } { d x } = & - i z \\psi _ 1 + \\psi _ 2 \\\\ \\frac { d \\psi _ 2 } { d x } = & i z \\psi _ 2 + u \\psi _ 1 \\end{aligned} \\right . \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} U _ { 2 ^ { p _ 0 } + \\cdots + 2 ^ { p _ s } } = 1 + 2 ^ { p _ 0 - p _ 1 } \\left [ ( 2 ^ { 2 + p _ 1 - p _ 0 } - 2 ) U _ { 2 ^ { p _ 1 } + \\cdots + 2 ^ { p _ s } } - 4 U _ { 2 ^ { p _ 2 } + \\cdots + 2 ^ { p _ s } } - 2 \\right ] . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} \\| T \\| ( F ) = 0 \\qquad \\textrm { a n d } \\int _ F g _ { \\| S \\| } \\dd \\| S \\| > 0 . \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} \\nu _ z ( F _ { 2 j } ( \\gamma ) ) = - 2 j + \\nu _ z ( F _ { 2 j } ( e + z \\tilde { \\gamma } ) ) \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} & \\mathcal G ^ + = \\left \\{ B \\in \\mathcal G : | \\{ x \\in B : u ( x ) > 1 \\} | > \\frac { \\tilde \\delta } { 8 c _ n } | B | \\right \\} \\\\ & \\mathcal G ^ - = \\left \\{ B \\in \\mathcal G : | \\{ x \\in B : u ( x ) > 1 \\} | \\le \\frac { \\tilde \\delta } { 8 c _ n } | B | \\right \\} . \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} f ( z ) = z ^ 2 \\int _ 0 ^ \\infty a \\left ( \\frac { \\log y } { \\log N } \\right ) y ^ { z - 1 } d y . \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} d _ { M ( ) } ( s ^ { - w ( m ) } m ) : = s ^ { - w ( m ) } d _ M ( m ) \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} \\begin{aligned} S ( U , f ^ { n + k } ) & \\geq \\left ( 4 \\frac { \\log ^ + M ^ { n - 1 } ( R _ 2 , f ) } { \\log R _ 2 } - 3 \\frac { \\log ^ + M ^ { n - 1 } ( R _ 1 , f ) } { \\log R _ 1 } \\right ) \\frac { \\log R _ 1 } { \\log r _ 1 } \\end{aligned} \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} \\partial _ t N = D \\ , \\partial _ { x x } N - M \\partial _ x N + \\sum _ { j = 1 } ^ m f _ j ( \\vec { p } N ) \\ , . \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( - \\Delta ) ^ s v & = g & & \\mbox { i n } \\Omega , \\\\ v & = 0 & & \\mbox { i n } \\Sigma _ 1 , \\\\ \\mathcal { N } _ s v & = 0 & & \\mbox { i n } \\Sigma _ 2 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} \\begin{cases} u ^ \\varepsilon _ t + A _ \\varepsilon u ^ \\varepsilon = f ( u ^ \\varepsilon ) , \\\\ u ^ \\varepsilon ( 0 ) = u _ 0 ^ \\varepsilon \\in X _ \\varepsilon ^ { \\frac { 1 } { 2 } } , \\varepsilon \\in [ 0 , \\varepsilon _ 0 ] . \\end{cases} \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { \\lambda ( w ' ) \\mbox { i n } P ( w ' ) _ { n + k } } \\mathcal { O } _ { w ' } \\right ) \\cap \\left ( \\bigcup _ { \\lambda ( z ) \\mbox { i n } P ( w ) _ { n + k } } \\mathcal { O } _ z \\right ) = \\varnothing \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} \\mu ^ { * n } \\star \\delta _ e \\left ( \\left \\{ v \\in \\mathbb { P } ( \\wedge ^ { g - 1 } \\C ^ { 2 g - 2 } ) \\mid \\frac { \\vert ( v , f ) \\vert } { \\Vert v \\Vert } \\leq \\delta \\right \\} \\right ) \\\\ = \\mu ^ { * n } \\left ( \\left \\{ \\varphi \\in \\widehat { \\mathcal { I } } _ g ^ { ( q ) } \\mid \\frac { \\vert ( \\rho _ q ( \\varphi ) e , f ) \\vert } { \\Vert \\rho _ q ( \\varphi ) e \\Vert } \\leq \\delta \\right \\} \\right ) . \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} - v ' - \\frac { v _ 1 + v } { ( u _ 1 - u ) ^ 2 } + \\frac { v + v _ { - 1 } } { ( u - u _ { - 1 } ) ^ 2 } = 0 . \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} E _ 2 ( z ) = \\frac { 1 } { 2 \\zeta ( 2 ) } \\sum _ { c \\in \\mathbb { Z } } \\sum _ { d \\in \\mathbb { Z } } \\frac { 1 } { ( c + d z ) ^ 2 } = 1 - 2 4 \\sum _ { n = 1 } ^ \\infty \\sigma _ 1 ( n ) e ^ { 2 \\pi i n z } , \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} M _ k ( x ; \\boldsymbol { a } ) = \\frac { 1 } { \\phi ^ k ( q ) } \\frac { 1 } { 2 \\pi i } \\int _ { c - i T } ^ { c + i T } F _ k ( s ) \\frac { x ^ s } { s } d s + O \\ ( \\frac { x \\log x } { T } + 1 \\ ) , \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} D _ i = \\frac { \\partial } { \\partial x ^ i } + \\sum _ { \\alpha , J } u _ { J + \\bold { 1 } _ i } ^ { \\alpha } \\frac { \\partial } { \\partial u _ J ^ { \\alpha } } , \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} \\{ P , g _ i \\} & = 0 , \\{ P , E \\} = 0 , \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} \\textbf { H } = \\beta \\frac { \\sin ^ 2 \\alpha } { \\cos ^ 2 \\alpha } \\textbf { V } , \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} V ( B ^ h ; k ) ^ n : = \\sum _ { i = 0 } ^ { n - k } \\left ( B ^ h _ { \\frac { i + k } { n } } - B ^ h _ { \\frac { i } { n } } \\right ) ^ 2 , S _ n ( B ^ h ) : = \\frac { V ( B ^ h ; 1 ) ^ n } { V ( B ^ h ; 2 ) ^ n } . \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} \\sum _ { t \\rhd r , t \\lhd s } w _ { i + 1 } ^ r w _ i ^ t w _ { i + 1 } ^ s & = \\sum _ { t \\rhd r , t \\lhd s } w _ { i + 1 } ^ s w _ i ^ t w _ { i + 1 } ^ r , \\\\ \\sum _ { t \\lhd r , t \\rhd s } w _ { i + 1 } ^ r w _ i ^ t w _ { i + 1 } ^ s & = \\sum _ { t \\lhd r , t \\rhd s } w _ { i + 1 } ^ s w _ i ^ t w _ { i + 1 } ^ r . \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} T _ { \\tau _ 1 , \\tau _ 2 } ( x , y ) = \\phi _ { 2 ( \\tau _ 2 - \\tau _ 1 ) } ( y - x ) - \\phi _ { 2 ( \\tau _ 2 - \\tau _ 1 ) } ( y + x ) \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ N \\frac { \\omega _ i E _ N ^ 2 ( \\tau _ i ) } { 1 - \\tau _ i ^ 2 } \\right ) ^ { 1 / 2 } \\le ( c / N ) | E _ N | _ { \\C { H } ^ 1 ( \\Omega ) } . \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ \\infty | y | ^ { r + 1 } f ( y ) \\ , \\d Q ( y ) < \\infty \\int _ { - \\infty } ^ \\infty | y | ^ { r + 1 } g _ l ( y ) \\ , \\d Q ( y ) < \\infty ~ ~ ~ l = 1 , 2 \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} \\phi _ C ( x , y ) = a p ^ { 2 i - j } x ^ 3 + b p ^ i x ^ 2 y + c p ^ j x y ^ 2 + d p ^ { 2 j - i } . \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } \\sum _ { l = 0 } ^ { m } b _ l \\ , e ^ { - k t _ l } = 0 & \\mbox { f o r a l l $ k \\in \\{ 0 , \\ldots , m - 1 \\} $ } , \\\\ \\\\ \\sum _ { l = 0 } ^ m b _ l \\ , e ^ { - m t _ l } = 1 \\ , \\ , . \\end{array} \\right . \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\underset { t \\to + 0 } { \\mbox { { \\rm e s s l i m } } } \\int _ { { \\bf R } ^ N } u ( y , t ) \\eta ( y ) \\ , d y = \\int _ { { \\bf R } ^ N } \\eta ( y ) \\ , d \\mu ' ( y ) . \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} \\mathcal { E } _ 1 = \\{ x \\ , | \\ , x ^ T Q _ 1 x \\leq 1 \\} , ~ ~ ~ \\mathcal { E } _ 2 = \\{ x \\ , | \\ , x ^ T Q _ 2 x \\leq 1 \\} , \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} \\alpha ( r q _ { k + 1 } + u q _ k ) ^ 2 - \\beta ( r p _ { k + 1 } + u p _ k ) ^ 2 = ( - 1 ) ^ k ( u ^ 2 t _ { k + 1 } + 2 r u s _ { k + 2 } - r ^ 2 t _ { k + 2 } ) . \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} \\lim _ n \\varphi ^ \\infty \\big ( j _ n ( x ) ^ * j _ n ( x ) \\big ) \\leq \\sum _ i c _ i \\lim _ n p _ \\mu ^ n ( 0 , s _ i ) = 0 . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} \\mathbb { H } ^ { \\textit { t o t } } ( \\Gamma , k ) = \\mathbb { H } ^ 0 ( \\Gamma , k ) \\oplus \\mathbb { H } ^ 1 ( \\Gamma , k ) [ - 1 ] . \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} \\mathbf { y } = \\sqrt { \\rho _ d \\beta _ K } { \\mathbf { G } } \\mathbf { u } + \\mathbf { z } . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} n ( \\ell ) = \\frac { \\ell H _ 2 ( k _ { \\ell } / \\ell ) } { \\frac { 1 } { 2 } \\log ( 1 + k _ { \\ell } P ) } . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} \\hat { \\mathbf { h } } _ { e f f , 2 } ^ { ( g ) } = \\left ( \\mathbf { X } ^ { ( g ) } \\otimes \\mathbf { S N R } ^ { t o t a l , ( g ) } _ { m i m o } \\right ) ^ H \\left ( \\mathbf { R } ^ { ( g ) } _ { c o d e } \\otimes \\mathbf { S N R } ^ { t o t a l , ( g ) } _ { m i m o } + \\mathbf { I } _ { T D } \\right ) ^ { - 1 } \\mathbf { y } ^ { ( g ) } \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\frac { \\partial \\phi } { \\partial W } ( W ^ * _ { k } ( n ) , \\mu _ k ) & = 1 ( W ^ * _ { k } ( n + 1 ) > 0 ) , \\\\ \\frac { \\partial \\phi } { \\partial \\mu } ( W ^ * _ { k } ( n ) , \\mu _ k ) & = - 1 ( W ^ * _ { k } ( n + 1 ) > 0 ) \\frac { J _ { k } ( - n ) } { \\mu _ k ^ 2 } . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} \\binom { k + ( i + j ) p } { m + i p } & \\equiv _ { p ^ 2 } \\binom { i + j } { j } \\left [ \\binom { k } { m } + j \\sum _ { l = 1 } ^ { m } \\binom { p } { l } \\binom { k } { m - l } + i \\sum _ { l = 1 } ^ { k - m } \\binom { p } { l } \\binom { k } { k - m - l } \\right ] \\\\ & = \\binom { i + j } { j } \\left [ ( 1 - j - i ) \\binom { k } { m } + j \\binom { p + k } { m } + i \\binom { p + k } { k - m } \\right ] \\\\ & \\equiv _ { p ^ 2 } \\binom { i + j } { i } \\binom { k } { m } ( 1 + p ( ( i + j ) H _ k - j H _ { k - m } - i H _ m ) ) . \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} \\left | \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) } { | C _ G ( x ) | | C _ G ( y ) | } \\right | _ 3 \\geq \\frac { q } { 9 } = 3 ^ { 2 k - 1 } . \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} P ( \\theta ) = \\sum _ { j = 0 } ^ k a _ k \\cos ( j \\theta ) + \\sum _ { j = 1 } ^ k b _ k \\sin ( j \\theta ) \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} \\gamma _ { t } ( s ) = ( 1 - s ) \\Omega _ { \\rho _ { t } } + s \\Omega _ { \\infty } . \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} \\underbrace { | \\alpha | ^ { 4 } } _ { \\in \\mathbb { Q } ( \\sqrt { - D } ) } = \\underbrace { A ^ { 2 } + C ^ { 2 } \\alpha ^ { 2 } + E ^ { 2 } \\overline { \\alpha } ^ { 2 } } _ { \\in \\mathbb { Q } ( \\sqrt { - D } ) } + 2 A C \\alpha + 2 A E \\overline { \\alpha } + 2 C E \\underbrace { ( A + C \\alpha + E \\overline { \\alpha } ) } _ { | \\alpha | ^ { 2 } } , \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} \\mathrm { N r } _ P ^ { P ' } ( \\Xi _ { U _ { P ' } ^ { a b } } ) = \\Pi _ P ^ { P ' } ( \\Xi _ { U _ { P ' } ^ { a b } } ) . \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} \\tau _ k ( \\omega ) : = \\inf \\left \\{ t \\geq 0 : \\left | \\varphi _ t ( \\omega , x ) - \\varphi _ t ( \\omega , y ) \\right | \\leq 2 ^ { k + 2 } \\right \\} . \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} \\sum _ { a = 0 } ^ { p - 1 } \\sum _ { b = 0 } ^ a \\sum _ { c = 0 } ^ b \\binom { b } c ^ 2 \\binom { a } b ^ 2 H _ a \\equiv _ p \\sum _ { a = 0 } ^ { p - 1 } \\sum _ { b = 0 } ^ a \\sum _ { c = 0 } ^ b \\binom { b } c ^ 2 \\binom { a } b ^ 2 H _ c . \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} \\chi + k - 2 = - \\frac 1 2 \\sum _ { j \\neq l } \\mathbb { E } [ \\mathcal { D } ^ { j l } ( 0 ) \\mathcal { D } ^ { j l } ( - J , - 1 ) ] = - \\frac { k ( k - 1 ) } 2 \\mathbb { E } [ \\mathcal { D } ^ { j l } ( 0 ) \\mathcal { D } ^ { j l } ( - J , - 1 ) ] \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} \\begin{array} { c c c } ~ ~ [ A , B ] + I J = 0 , & J F = 0 , & ~ ~ \\\\ ~ ~ [ A ' , B ' ] = 0 , & A F - F A ' = 0 , & B F - F B ' = 0 , \\end{array} \\end{align*}"}
-{"id": "453.png", "formula": "\\begin{align*} u _ 2 = \\frac { u _ 1 ^ 2 } { u } \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} u ' _ n = \\pm \\frac { \\epsilon } { 2 \\sqrt { c } } ( Q - \\frac { 2 c } { c - 2 } Q ^ { - 1 } ) , \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n f \\left ( \\mathcal { A } _ i \\right ) \\geq \\sum _ { j = 1 } ^ n f \\left ( \\mathcal { E } ^ { ( n ) } _ j \\right ) , \\end{align*}"}
-{"id": "6034.png", "formula": "\\begin{align*} \\overline { \\left \\vert a , n \\right \\rangle } = \\left ( \\begin{array} { c c c c c } 0 & \\cdots & 1 & \\cdots & 0 \\end{array} \\right ) ^ { t _ { 0 } } , a \\in \\{ 1 , . . . , 2 s + 1 \\} \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} D ( a b ) = D a \\cdot b + ( - 1 ) ^ { \\langle \\deg ( D ) , \\deg ( a ) \\rangle } a \\cdot D b \\ ; . \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} \\mathrm { d } X ^ { ( n ) } _ i ( t ) = b \\big ( F _ { \\rho ^ { ( n ) } ( t ) } \\big ( X ^ { ( n ) } _ i ( t ) \\big ) \\big ) \\ , \\mathrm { d } t + \\sigma \\big ( F _ { \\rho ^ { ( n ) } ( t ) } \\big ( X ^ { ( n ) } _ i ( t ) \\big ) \\big ) \\ , \\mathrm { d } B ^ { ( n ) } _ i ( t ) , i = 1 , \\ , 2 , \\ , \\ldots , \\ , n . \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{align*} \\mathbb { D } _ { b , n } = \\sum _ { h = 0 } ^ { p - 2 } \\prod _ { _ { c = 1 , c \\neq a ( n , h ) } } ^ { ( p - 1 ) \\mathsf { N } + 1 } \\frac { X _ { b } ^ { ( p - 1 ) } - w _ { c } } { w _ { n } - w _ { c } } \\mathbb { C } _ { n , h } . \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} V _ { n , k } \\prod _ { i = 1 } ^ { k } W _ { m _ { l } } U _ { n - m _ { l } } = \\sum _ { j = 0 } ^ { \\infty } \\binom { k + j } { j } U _ { n } ^ { j } W _ { 1 } ^ { j } \\sum _ { \\underline { z } \\in \\left \\{ 0 , 1 \\right \\} ^ { k } } V _ { n + 1 , k + j } \\prod _ { i = 1 } ^ { k } W _ { m _ { i } + z _ { i } } U _ { n + 1 - m _ { i } - z _ { i } } , \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{align*} M ( P ) = M ( P ' ) \\cup M ( R ) M ( Q ) = M ( Q ' ) \\cup M ( R ) . \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} E = L \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\lambda \\epsilon _ 0 ( \\lambda ) \\rho _ { \\infty } ( \\lambda ) - \\sum _ { i a } \\frac { s _ a \\epsilon ' ( \\lambda _ { i a } ) } { 2 4 L \\rho _ { \\infty } ( \\lambda _ { i a } ) } \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} F ( y ) = \\alpha ( y ) \\ , \\phi ( s ) , s = \\beta ( y ) / \\alpha ( y ) . \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} F _ \\infty = C _ \\infty \\oplus D _ \\infty \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} z _ t ^ i & = u _ i ( x _ t ^ i ) + \\delta ^ i z _ { t + 1 } ^ i , & t = 0 , 1 , \\ldots , \\\\ \\theta _ t ^ i \\delta ^ i & = \\mu _ t \\theta _ { t + 1 } ^ i , & t = 0 , 1 , \\ldots . \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} v _ { n } = q ^ { ( S _ { n } ^ { z } + p + 1 ) / 2 } \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} e ( p e r m _ m ( B ) ) = { \\binom n m } ^ 2 b ( m ) \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} \\phi _ t + \\frac 1 2 | \\nabla \\phi | ^ 2 + g y + p = 0 . \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} L : & = \\hat { D } ^ 2 + \\{ \\hat { D } , c ( \\delta u ) \\} \\\\ & = \\hat { D } ^ 2 + L _ 1 + c ( d \\delta u ) + d ^ * \\delta u , \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} f ^ { + } ( \\tau ) & = \\sum \\limits _ { r ( 2 N ) } \\sum \\limits _ { \\substack { D \\gg \\infty \\\\ D \\equiv - r ^ { 2 } ( 4 N ) } } c _ { f } ^ { + } ( D , r ) q ^ { D / 4 N } \\mathfrak { e } _ r , \\\\ f ^ { - } ( \\tau ) & = \\sum \\limits _ { r ( 2 N ) } \\bigg ( c _ { f } ^ { - } ( 0 , r ) v ^ { 1 - k } + \\sum \\limits _ { \\substack { D < 0 \\\\ D \\equiv - r ^ { 2 } ( 4 N ) } } c _ { f } ^ { - } ( D , r ) \\Gamma ( 1 - k , \\pi | D | v / N ) q ^ { D / 4 N } \\bigg ) \\mathfrak { e } _ r , \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} ( A + U V ^ { \\ast } ) ^ { \\dagger } = \\big ( I + A ^ { \\dagger } U F _ { S _ { A } } U ^ { \\ast } ( A ^ { \\dagger } ) ^ { \\ast } \\big ) ^ { - 1 } \\big ( A ^ { \\dagger } - A ^ { \\dagger } U S _ { A } ^ { \\dagger } V ^ { \\ast } A ^ { \\dagger } \\big ) \\big ( I + ( A ^ { \\dagger } ) ^ { \\ast } V E _ { S _ { A } } V ^ { \\ast } A ^ { \\dagger } \\big ) ^ { - 1 } , \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} & \\lim _ { t \\to \\infty } \\int _ { \\R ^ { n } } | \\eta | ^ { 2 ( k - 1 ) } e ^ { - \\nu | \\eta | ^ { 2 \\sigma } } \\cos ( 2 t ^ { 1 - \\frac { 1 } { 2 \\sigma } } | \\eta | ) d \\eta \\\\ & = C \\lim _ { t \\to \\infty } \\int _ { 0 } ^ { \\infty } \\tau ^ { 2 ( k - 1 ) + n - 1 } e ^ { - \\nu \\tau ^ { 2 \\sigma } } \\cos ( 2 t ^ { 1 - \\frac { 1 } { 2 \\sigma } } \\tau ) d \\tau = 0 \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} N _ { 0 0 } ( T ) = N ( T ) + O \\left ( T \\log { | D | } + ( L ( 1 , \\chi ) \\log { T } ) ^ { 1 / 4 } T \\log { T } \\right ) \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} ( A ( t ) , B ( t ) , I ( t ) , J ( t ) , A ' ( t ) , B ' ( t ) , F ( t ) , G ( t ) ) = ( h ( t ) , h ' ( t ) ) \\cdot X \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} \\operatorname { P } ( \\Xi ) = \\left ( ( \\operatorname { e x p _ { g _ { _ { \\mathfrak { X } } } } } ( \\mathcal { V } _ { \\Xi } ) \\circ \\Phi ) ^ { \\ast } ( \\omega ) , ( \\operatorname { e x p _ { g _ { _ { \\mathfrak { X } } } } } ( \\mathcal { V } _ { \\Xi } ) \\circ \\Phi ) ^ { \\ast } ( \\operatorname { I m } \\Omega ) \\right ) , \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} F ( z ) = \\frac { 1 + z f ( z ) } { 1 - z f ( z ) } . \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} \\pi = \\left ( 1 , \\left \\{ \\big ( ( 1 , 0 ) , G \\big ) , \\big ( ( 1 , 1 ) , H \\big ) , \\big ( ( 2 , 0 ) , \\gamma _ 2 \\big ) , \\big ( ( 3 , 0 ) , \\gamma _ 3 \\big ) , \\big ( ( 4 , 0 ) , \\gamma _ 4 \\big ) \\right \\} \\right ) . \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\begin{aligned} \\operatorname { D i v } R _ 1 + \\operatorname { D i v } ^ { \\vartriangle } R _ 2 & = Q ^ { \\alpha } F _ { \\alpha } ^ { \\ast } + Q ^ { \\alpha } _ { \\ast } F _ { \\alpha } \\\\ & = Q ^ { \\alpha } F _ { \\alpha } ^ { \\ast } - v ^ { \\alpha } \\bold { p r } X ( F _ { \\alpha } ) + \\operatorname { D i v } P _ 1 + \\operatorname { D i v } ^ { \\vartriangle } P _ 2 . \\end{aligned} \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} ( H u ) _ n = u _ { n + 1 } + u _ { n - 1 } + V ( \\theta + n \\omega ) u _ n \\end{align*}"}
-{"id": "3648.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } 0 & - v _ 2 ' & - v _ 3 ' \\\\ v _ 2 ' & 0 & - w \\\\ v _ 3 ' & w & 0 \\end{array} \\right ) \\ , . \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} \\langle H w , w \\rangle _ { L ^ 2 ( \\mathbb { R } ^ - ) } = - [ w ( 0 ) ] ^ 2 + \\int _ { - \\infty } ^ 0 \\left [ z ( \\partial _ z w ) ^ 2 + \\frac { 1 } { 4 } z w ^ 2 \\right ] d z \\leq - \\frac { 1 } { 2 } \\| w \\| _ { L ^ 2 ( \\mathbb { R } ^ - ) } ^ 2 . \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} P _ { i | i - 1 } ( 2 ) = P _ { i - 1 } ( 2 ) ( 1 - p ) + P _ { i - 1 } ( 1 ) q , P _ { i | i - 1 } ( 1 ) = P _ { i - 1 } ( 1 ) ( 1 - q ) + P _ { i - 1 } ( 2 ) p . \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} X _ { n , i } : = - \\frac { 1 } { \\sqrt n } \\sum _ { j = 1 } ^ n \\epsilon _ { i j } x _ j \\quad Y _ { n , i } : = - \\frac { 1 } { \\sqrt n } \\sum _ { j = 1 } ^ n \\epsilon _ { i j } y _ j . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} \\hat { u } = \\frac { \\sigma ( u ) } { u } = \\frac { u _ 0 \\zeta _ 4 ^ \\alpha \\pi ^ \\alpha } { u _ 0 \\pi ^ \\alpha } = \\zeta _ 4 ^ \\alpha \\hat { r } = \\hat { s } = \\hat { t } = 0 \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} F \\left ( u , v \\right ) = \\left ( u + 2 v \\right ) \\left ( u - 2 \\left ( c + 1 \\right ) v \\right ) \\left ( u + 2 \\left ( c - 1 \\right ) v \\right ) = \\varepsilon , \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} ( 1 8 c ^ 4 - 9 6 ) p ^ 2 - 8 ( 3 c ^ 4 - 1 2 c ^ 2 + 8 ) p + ( 1 2 c ^ 4 - 9 6 c ^ 2 + 1 6 0 ) = 0 . \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} \\iota _ { \\mathcal { R } } \\eta = 1 , \\iota _ { \\mathcal { R } } \\Omega = 0 . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} & \\mathbb { P } \\left ( { \\gamma } _ { \\mathrm { u } , i k } \\geq \\theta \\right ) \\\\ & \\approx \\mathbb { E } \\left [ \\mathbb { P } \\left ( \\left . R _ k \\leq \\frac { \\left ( 1 - \\tau _ { { i i k } } ^ 2 \\right ) ( 1 - \\beta ) N t ^ \\alpha } { \\left ( \\tau _ { i i k } ^ 2 + \\frac { 2 \\pi \\lambda t ^ { 2 } } { \\alpha - 2 } \\right ) \\theta } - t ^ \\alpha \\right \\vert r _ { i i k } = t \\right ) \\right ] . \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} \\gamma = \\frac { 1 } { 1 + \\epsilon } , \\ \\delta = \\frac { \\epsilon } { 1 + \\epsilon } \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} \\iota _ { \\sharp ( \\alpha ) } \\eta = 0 , \\iota _ { \\sharp ( \\alpha ) } d \\eta = - ( \\alpha - < \\alpha , \\mathcal { R } > \\eta ) \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} S = \\{ 0 , \\pm 1 , \\pm \\sqrt { - 1 } , \\pm 1 \\pm \\sqrt { - 1 } , \\pm \\sqrt { - 2 } , \\pm \\sqrt { - 3 } , \\pm \\omega , \\pm \\omega ^ { 2 } \\} , \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} \\P ( Z ^ { \\ell _ 1 } = u ^ { \\ell _ 1 } , Z _ { \\ell _ 1 + g + 1 } ^ { \\ell _ 1 + g + \\ell _ 2 } = v ^ { \\ell _ 2 } ) \\leq \\P ( Z ^ { \\ell _ 1 } = u ^ { \\ell _ 1 } ) \\P ( Z _ { \\ell _ 1 + g + 1 } ^ { \\ell _ 1 + g + \\ell _ 2 } = v ^ { \\ell _ 2 } ) \\tilde { \\Psi } ( g ) . \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} P \\left ( \\bigcap _ { i = 1 } ^ M \\left \\{ \\chi _ { \\frac { 2 n } { M } , i } ^ 2 \\in \\left ( 1 - \\frac { \\delta } { \\sigma _ { \\rm w } ^ 2 + c } , 1 + \\frac { \\delta } { \\sigma _ { \\rm w } ^ 2 + c } \\right ) \\right \\} \\right ) > 1 - \\frac { \\epsilon } { 2 } , \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} S _ { t + 1 } : = \\Big \\{ y \\in V : \\ , \\frac { \\pi ^ { ( t ) } ( S _ t , y ) } { \\pi ^ { ( t + 1 ) } ( y ) } > U _ { t + 1 } \\Big \\} . \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} { { \\dot x } _ 2 } ( t ) = u ( t ) , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} \\operatorname { A n n } ( A , A _ \\infty ) = \\{ x \\in A _ { \\infty } \\mid x A = A x = \\{ 0 \\} \\} . \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} M _ 0 ^ { \\lambda , m , n } = \\begin{pmatrix} 0 \\\\ 0 \\\\ a \\end{pmatrix} , M _ 1 ^ { \\lambda , m , n } = \\begin{pmatrix} 0 \\\\ 1 \\\\ c \\end{pmatrix} , M _ 2 ^ { \\lambda , m , n } = \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix} , M _ 3 ^ { \\lambda , m , n } = \\begin{pmatrix} 0 \\\\ 0 \\\\ 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} | b | ^ { 2 } \\frac { \\overline { a } b } { a \\overline { b } } \\ , \\alpha ^ { 2 } = | e | ^ { 2 } \\frac { \\overline { d } e } { d \\overline { e } } \\ , \\beta ^ { 2 } \\ \\Leftrightarrow \\ \\left ( \\frac { \\overline { a } b } { | a | } \\right ) ^ { 2 } \\alpha ^ { 2 } = \\left ( \\frac { \\overline { d } e } { | d | } \\right ) ^ { 2 } \\beta ^ { 2 } . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{align*} x _ { } \\equiv a _ { } / d _ { } , y _ { } \\equiv b _ { } / c _ { } , s _ { } \\equiv d _ { } / c _ { } , \\ , t _ { } \\equiv x _ { } y _ { } , k ^ { 2 } + ( k ^ { ^ { \\prime } } ) ^ { 2 } = 1 , \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} R _ \\ast ^ t u ( \\overline e _ { t , k _ 1 } , \\ldots , \\overline e _ { t , k _ q } ) = \\sum _ { j = 1 } ^ q u ( \\overline e _ { t , k _ 1 } , \\ldots , R ^ t _ \\ast \\overline e _ { t , k _ j } , \\ldots , \\overline e _ { t , k _ q } ) , \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} { } _ k { \\mathbf { P } } _ { O B } ^ { ( 1 ) } = \\frac { 1 } { 2 } ( { \\tau _ k } - { \\tau _ { k - 1 } } ) { \\mathbf { P } } _ { O B } ^ { ( 1 ) } \\forall k \\in \\mathbb { K } . \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} P _ { s - 1 } ( x , y ) & : = \\frac { ( 1 - | x | ^ 2 ) _ + ^ { s - 2 } ( 1 - | y | ^ 2 ) _ + ^ { s - 1 } ( 1 - | x | ^ 2 | y | ^ 2 ) } { [ x , y ] ^ { N } } \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{align*} \\bold { p r } X ( F _ { \\alpha } ) = 0 , \\mathcal { A } . \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } ( - 1 ) ^ { n - 1 } \\frac { \\psi ( n ) } { n ^ s } = \\frac { 2 ^ s - 5 } { 2 ^ s + 1 } \\cdot \\frac { \\zeta ( s ) \\zeta ( s - 1 ) } { \\zeta ( 2 s ) } ( \\Re s > 2 ) , \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} \\left < G ^ \\lambda G _ \\mu \\right > = \\delta ^ \\lambda _ \\mu \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} { } J = \\frac { 1 } { 2 } \\int _ 0 ^ 1 { { u ^ 2 } ( t ) \\ , d t } \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { i = 1 } ^ { \\bar { k } } \\xi _ { i } ^ { \\prime } \\right \\} \\in ( 0 , 1 ) \\backslash \\{ 1 / 2 \\} . \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} \\mathcal { D } _ { t } u ^ { n } _ t = \\eta _ { t } - \\alpha _ { t } ^ { n } , \\ \\mathcal { D } _ { x } u ^ { n } _ t = \\zeta _ { t } + \\frac { 1 } { 2 } w _ { t } \\lambda _ { t } ^ { n } , \\ \\mathcal { D } _ { x } ^ { 2 } u ^ { n } _ t = w _ { t } . \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} \\mathcal I _ { s } ^ q ( \\mu ) : = \\int _ { \\underline a \\in \\Sigma } \\left ( \\int _ { \\underline b \\in \\Sigma } \\frac { 1 } { d ( \\underline a , \\underline b ) ^ s } d \\mu ( \\underline b ) \\right ) ^ { q - 1 } d \\mu ( \\underline a ) . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} \\max \\limits _ { ( f , g ) \\in D _ f \\times D _ g } \\Delta \\left ( h ( f _ 0 , g _ 0 ) ; f , g \\right ) = \\Delta \\left ( h ( f _ 0 , g _ 0 ) ; f _ 0 , g _ 0 \\right ) , \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} \\delta \\psi ( x ) = \\sqrt { \\psi ( 0 ) ^ 2 - \\psi ( x ) ^ 2 } . \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} \\upsilon = \\left ( \\frac { n _ 1 - 1 } { 2 } + \\nu _ 1 , \\frac { n _ 1 - 3 } { 2 } + \\nu _ 1 , \\dots , - \\frac { n _ 1 - 1 } { 2 } + \\nu _ 1 , \\frac { n _ 2 - 1 } { 2 } + \\nu _ 2 , \\dots , - \\frac { n _ 2 - 1 } { 2 } + \\nu _ 2 , \\dots \\right ) . \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} \\pi _ { L / L ^ \\prime } ( \\Theta _ { \\mathfrak { m } } ( L / F , \\kappa ) ) = \\Theta _ { \\mathfrak { m } } ( L ^ \\prime / F , \\kappa ) , \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} P ( x , y , q ; z ) = e ^ { q z ( y - 1 ) } Q ( x , q ; z ) . \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} [ H , i A ] & = - 2 \\Delta - x \\cdot \\nabla V = 2 H - 2 V - x \\cdot \\nabla V , \\\\ [ [ H , i A ] , i A ] & = - 4 \\Delta + ( x \\cdot \\nabla ) ^ 2 V = 2 [ H , i A ] + 2 x \\cdot \\nabla V + ( x \\cdot \\nabla ) ^ 2 V \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} d i m ( \\chi ^ { - 1 } ( b ) ) = d i m ( B u n _ { \\mathcal { G } _ { X , x , \\theta } } ) , \\ , \\ \\forall b \\in B . \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} \\mathcal { R } _ { 0 } = \\sum _ { i = 1 } ^ { n _ t } \\log _ 2 ( 1 + \\gamma _ { 0 , i } ) . \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} \\operatorname { I m } \\Omega = \\operatorname { I m } ( d z _ 1 \\wedge \\dots \\wedge d z _ { n } ) & = \\operatorname { I m } d ( x _ 1 + i y _ 1 ) \\wedge \\dots \\wedge d ( x _ { n } + i y _ { n } ) \\\\ & = \\sum \\limits _ { \\abs { I } = o d d } c _ { I } d y _ { I } \\wedge d x _ { 1 } \\wedge \\dots \\wedge \\widehat { d x _ { I } } \\wedge \\dots d x _ { n } \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} \\exists \\beta > 1 , \\lim _ { x \\to \\infty } \\frac { Q ( x ) } { | x | ^ \\beta } = 1 . \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\widehat { A } ( s ) \\textup d e ( s ) = \\int _ 0 ^ 1 \\widehat { B } ( s ) \\textup d e ( s ) = 0 . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( x , \\tilde { \\alpha } ) ^ { - 1 } \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( x , \\tilde { \\beta } ) \\ , = \\ , \\widetilde { \\Lambda } ^ { \\epsilon , \\kappa } ( \\tilde { \\alpha } , \\tilde { \\beta } ) \\ , \\Omega ^ { \\kappa , \\epsilon } ( \\tilde { \\alpha } , x , \\tilde { \\beta } ) \\ , , \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} D _ { c , E } ( \\theta ) = M _ { c } ( \\theta + \\alpha ) D _ { | c | , E } ( \\theta ) M ^ { - 1 } _ { c } ( \\theta ) , \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} H _ h ( N ) = 2 ( 1 4 - N ) \\lambda ( N ) , \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} | \\nabla v _ \\lambda | ^ 2 = | \\nabla u - i \\lambda ^ { \\frac 1 2 } | x | ^ { - 1 } x u | ^ 2 = | \\nabla u | ^ 2 + \\lambda | u | ^ 2 - 2 \\lambda ^ { \\frac 1 2 } \\mbox { I m } [ ( \\partial _ r u ) \\overline u ] . \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} \\| u \\| _ { C ^ { k , \\alpha } _ \\nu ( M ) } = \\| u \\| _ { C ^ { k , \\alpha } ( \\hat { M } _ 0 ) } + \\sum _ { \\ell = 1 } ^ L \\| u \\circ \\Phi _ \\ell \\| _ { C ^ { k , \\alpha } _ { \\nu _ \\ell } ( U _ \\ell ) } , \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} C _ \\mathsf { S e m i - D e t } = \\max _ { P _ { X | S } } \\min \\Big \\{ H ( Y | Z ) , H ( Y | S ) \\Big \\} , \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} & e ^ { - 2 \\eta } \\left [ \\begin{pmatrix} p ( \\eta ) \\\\ q ( \\eta ) \\\\ r ( \\eta ) \\end{pmatrix} - M _ 0 \\right ] \\rightarrow { \\bar \\Gamma ( 0 ) ^ { 1 + m - n } a ^ { - n } } \\vec { X } _ { 0 2 } , \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} H ^ 0 = \\tau \\circ \\psi ( k ^ * U ^ 1 ) = \\{ \\chi \\in H : \\chi ( \\beta ) = 0 \\} . \\end{align*}"}
-{"id": "2810.png", "formula": "\\begin{align*} v _ 1 = ( 2 , - 2 , - 2 ) , v _ 2 = ( - 2 , 2 , - 2 ) , v _ 3 = ( - 2 , - 2 , 2 ) . \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\left ( p ^ 2 + \\frac { k } { q ^ 2 } \\right ) + \\frac { 1 } { 2 } \\omega ( t ) ^ 2 q ^ 2 \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} J _ 1 ^ { ( a ) } = \\begin{cases} - x _ 1 ^ { ( a + 1 ) } & a + 1 > i , \\\\ - k & a + 1 = i , \\\\ 0 & a + 1 < i . \\end{cases} \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} \\mu \\left \\{ x : S ^ { \\sigma } _ { \\tau _ { n } } f \\geq t \\right \\} = \\mu \\left \\{ x : S ^ { \\sigma } _ { \\tau _ { n } } f \\geq t , \\tau _ { n } \\leq \\epsilon t \\right \\} + \\mu \\left \\{ x : S ^ { \\sigma } _ { \\tau _ { n } } f \\geq t , \\tau _ { n } > \\epsilon t \\right \\} , \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} \\langle \\lfloor m ; g \\rceil , \\lfloor n ; h \\rceil \\rangle = \\left \\{ \\begin{array} { l l } \\langle m , n \\rangle _ { \\mathcal { Q } _ { W | _ { ( g ) } } } & g = - h , \\\\ 0 & \\end{array} \\right . \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} \\int _ M u ^ 2 \\leq c \\int _ \\Omega | \\nabla u | ^ 2 \\leq c \\int _ \\Omega | u ( f - \\bar { f } ) | , \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} u = \\left ( \\begin{pmatrix} 0 & a _ { 1 2 } \\\\ 0 & a _ { 2 2 } \\end{pmatrix} , \\begin{pmatrix} b _ { 1 1 } & b _ { 1 2 } \\\\ 0 & b _ { 2 2 } \\end{pmatrix} , \\begin{pmatrix} i _ 1 \\\\ i _ 2 \\end{pmatrix} , 0 , 0 , b _ { 1 1 } + f _ 2 , \\begin{pmatrix} f _ 1 \\\\ f _ 2 \\end{pmatrix} \\right ) \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} x ( t ) = \\int _ 0 ^ 1 ( 1 - t ) x ( s ) \\textup d A ( s ) + \\int _ 0 ^ 1 t x ( s ) \\textup d B ( s ) + \\lambda \\int _ 0 ^ 1 k ( t , s ) f ( s , x ( s ) ) \\textup d s , \\ t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} i _ { \\mathcal { R } } d \\eta = 0 , i _ { \\mathcal { R } } \\eta = 1 . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} G ^ { T } = \\{ g \\in G _ { W ^ { T } } ^ { m a x } \\mid g A h ^ { T } \\in \\Z h \\in G \\} . \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\mathfrak { b } = d \\rho ^ { 2 } + ( \\sinh \\rho ) ^ { 2 } h _ { 0 } , \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} D = \\frac { 1 } { 2 } K _ X ^ { [ n ] } + \\left ( \\frac { a } { 2 } + \\frac { n } { a H ^ 2 } \\right ) H ^ { [ n ] } - \\frac { 1 } { 2 } E \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} | \\alpha _ n | & \\ne 1 , n = 0 , 1 , 2 , \\dots N - 1 , \\\\ | \\alpha _ n | & < 1 , n = N , N + 1 , N + 2 , \\dots . \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} \\left ( F _ { b n + q } ^ { \\left ( b \\right ) } \\right ) ^ { 2 } - F _ { b n + q + r } ^ { \\left ( b \\right ) } F _ { b n + q - r } ^ { \\left ( b \\right ) } = \\left ( - 1 \\right ) ^ { n - r + 1 } \\left ( F _ { b n + r - 1 } ^ { \\left ( b \\right ) } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p , w } ( \\R ^ { N } ) } : = \\sup _ { A \\subset \\R ^ { N } , 0 < | A | < \\infty } | A | ^ { - \\frac { p - 1 } { p } } \\int _ { A } | f ( x ) | \\ d x < \\infty . \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } u = f \\quad u = 0 \\quad , \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } | A _ n | ^ { 1 / n } = 1 \\quad \\mbox { a . s . } , \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} \\varepsilon _ i ( b ) = \\max \\{ k \\mid e _ i ^ k ( b ) \\neq 0 \\} , \\qquad \\varphi _ i ( b ) = \\max \\{ k \\mid f _ i ^ k ( b ) \\neq 0 \\} , \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} E _ n ^ { ( s _ 1 , s _ 2 , \\ldots , s _ n ) } ( x ) = E _ n ^ { ( s _ n , s _ { n - 1 } , \\ldots , s _ 1 ) } ( x ) . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} E _ s = & \\ \\frac 1 \\pi \\int _ { - \\infty } ^ \\infty ( 1 + \\xi ^ 2 ) ^ { s } ( - \\real \\ln T ( \\xi / 2 ) ) d \\xi \\\\ = & \\ \\frac { 2 \\sin ( \\pi s ) } \\pi \\int _ 1 ^ \\infty ( \\tau ^ 2 - 1 ) ^ s \\Big [ \\real \\ln T ( i \\tau / 2 ) + \\sum _ { j = 0 } ^ N ( - 1 ) ^ j H _ { 2 j } \\tau ^ { - 2 j - 1 } \\Big ] d \\tau + \\sum _ { j = 0 } ^ N \\binom { s } { j } H _ { 2 j } \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} P ( T _ 1 ^ \\alpha + T _ 2 ^ \\alpha < x ) = \\frac { 2 } { \\pi } \\int _ 0 ^ { \\sqrt { 1 / 2 } } \\int _ 0 ^ { \\sqrt { 1 / 2 } } \\ 1 _ { z + y < x } ( 1 - z ^ 2 - y ^ 2 ) ^ { - 3 / 2 } d z d y . \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{gather*} J ^ \\infty E = \\lim _ { \\longleftarrow } J ^ k E . \\end{gather*}"}
-{"id": "647.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 0 } ^ { \\infty } a _ j \\int \\limits _ { - \\pi } ^ { \\pi } \\frac { e ^ { i ( j - k ) \\theta } f ( \\theta ) } { f ( \\theta ) + g ( \\theta ) } d \\theta - \\sum \\limits _ { j = 0 } ^ { \\infty } c _ j \\int \\limits _ { - \\pi } ^ { \\pi } \\frac { e ^ { i ( j - k ) \\theta } } { f ( \\theta ) + g ( \\theta ) } d \\theta = 0 , k = 0 , 1 , \\dots \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} ( A + B ) \\times C & = \\left ( \\sum a _ i + \\sum b _ j \\right ) \\times \\left ( \\sum c _ k \\right ) \\\\ & = \\sum a _ i c _ k + \\sum b _ j c _ k \\\\ & = ( A \\times C ) + ( B \\times C ) . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} F _ { q } ^ { 2 } - F _ { q + r } F _ { q - r } = \\left ( - 1 \\right ) ^ { q - r + 1 } F _ { r - 1 } ^ { 2 } , \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} \\mathcal { L } _ { \\Pi ( d f _ 1 \\wedge \\cdots \\wedge d f _ { p - 1 } ) } ( \\Pi \\beta ) = \\Pi ( \\mathcal { L } _ { \\Pi ( d f _ 1 \\wedge \\cdots \\wedge d f _ { p - 1 } ) } \\beta ) \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} \\tilde { \\mu } \\big ( \\tilde { \\mathcal { C } } _ V ^ { ( n ) } \\big ) = \\prod _ { \\mathbf { u } \\in \\mathcal { U } ^ n } \\prod _ { j \\in \\mathcal { J } _ n } p ^ n _ { V | U } \\big ( \\tilde { \\mathbf { v } } ( \\mathbf { u } , j ) \\big | \\mathbf { u } \\big ) , \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{gather*} c = - \\frac { 1 } { 3 } e _ 2 . \\end{gather*}"}
-{"id": "4951.png", "formula": "\\begin{align*} \\| \\Phi _ q ( y , z _ 0 ) - \\Phi _ q ( y , \\bar { z } _ 0 ) \\| = \\| T _ q ( \\cdot ) P _ q ^ s ( z _ 0 - \\bar { z } _ 0 ) \\| \\leq C | z - \\bar { z } _ 0 | _ { \\beta } . \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\bigl \\langle \\Xi ( X _ 1 ) , Y \\bigr \\rangle & = \\bigl \\langle \\Xi ( X _ 1 ) , [ Y _ 1 , b ] \\bigr \\rangle = - \\bigl \\langle \\Xi ( Y _ 1 ) , [ b , X _ 1 ] \\bigr \\rangle - \\bigl \\langle \\Xi ( b ) , [ X _ 1 , Y _ 1 ] \\bigr \\rangle = \\bigl \\langle \\Xi ( Y _ 1 ) , X \\bigr \\rangle \\ , , \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} f ( x ; G _ m ) = \\int f ( x ; \\theta ) d G _ m ( \\theta ) \\to \\int f ( x ; \\theta ) d G _ 0 ( \\theta ) = f ( x ; G _ 0 ) \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} \\lim _ { t \\to + 0 } \\| S ( t ) \\eta - \\eta \\| _ { L ^ \\infty ( { \\bf R } ^ N ) } = 0 , \\qquad \\eta \\in C _ 0 ( { \\bf R } ^ N ) . \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} w \\cdot ( x , g P ) = ( x , g w ^ { - 1 } P ) . \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{gather*} \\Psi ^ { ( k + 1 ) } _ 0 ( x ) = \\Psi ^ { ( k ) } _ 0 ( x ) S ^ { ( k ) } _ 0 , k = 1 , 2 , \\dots , 6 . \\end{gather*}"}
-{"id": "7681.png", "formula": "\\begin{align*} \\lim _ { x \\to + \\infty } \\frac { Q ( \\sigma x ) } { Q ( x ) } = \\sigma ^ \\beta \\lim _ { x \\to + \\infty } \\frac { Q ( \\sigma x ) } { ( \\sigma x ) ^ \\beta } \\frac { x ^ \\beta } { Q ( x ) } = \\sigma ^ \\beta . \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\mathbf { T } _ 0 : = ( D \\tilde { G } _ { \\delta } ) ( \\varphi , 0 , 0 ) \\circ \\mathbb { D } ^ { - 1 } \\circ ( D G _ { \\delta } ( \\varphi , 0 , 0 ) ) ^ { - 1 } \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{gather*} \\Big | \\sum _ { k = p } ^ q a _ k { \\rm e } ^ { i t \\lambda _ k } - \\sum _ { k = v _ { p ' } } ^ { v _ { q ' } } b _ k { \\rm e } ^ { i t \\log _ 2 u _ k } \\Big | \\\\ \\le \\sum _ { k \\ , : \\ , u _ k = u _ p } | a _ k | + \\sum _ { k \\ , : \\ , u _ k = u _ q } | a _ k | + \\sum _ { \\ell = p ' } ^ { q ' } \\sum _ { k \\ , : \\ , u _ k = v _ \\ell } | a _ k | \\ , | { \\rm e } ^ { i t \\lambda _ k } - { \\rm e } ^ { i t \\log _ 2 u _ k } | \\end{gather*}"}
-{"id": "4839.png", "formula": "\\begin{align*} F ( \\xi _ k ) = \\max _ { \\eta \\in [ 0 , a _ k ] } F ( \\eta ) . \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} X : = \\{ ( \\vec { g } , \\vec { z } ) \\in G ^ d \\times \\Delta ^ L \\mid & \\forall i \\not = j : d _ { \\vec { g } } ^ { ( \\alpha _ 1 , \\ldots , \\alpha _ l ) } ( z _ i , z _ j ) > 2 l + 2 \\\\ & \\forall i : | \\{ z _ i , \\nu _ 1 ( z _ i ) , \\ldots , \\nu _ l ( z _ i ) \\} | \\leq l \\} , \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} \\epsilon = \\frac { 2 } { 9 \\left [ \\frac { \\Delta { { \\overline { x } } _ { 1 , 2 } } } { s ( \\Delta { \\overline { x } _ { 1 , 2 } } ) } \\right ] ^ 2 } \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} \\operatorname { D i v } P = Q ^ { \\alpha } \\bold { E } _ { \\alpha } ( L ) . \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} a = \\left [ \\begin{array} { c c } a _ { 1 1 } & a _ { 1 2 } \\\\ 0 & a _ { 2 2 } \\end{array} \\right ] , & & b = \\left [ \\begin{array} { c c } b _ { 1 1 } & b _ { 1 2 } \\\\ 0 & b _ { 2 2 } \\end{array} \\right ] , & & i = \\left [ \\begin{array} { c } i _ { 1 } \\\\ i _ { 1 } \\end{array} \\right ] , \\\\ f = \\left [ \\begin{array} { c } f _ { 1 } \\\\ f _ { 2 } \\end{array} \\right ] , & & a ' = f _ 2 + a _ { 1 1 } , & & b ' = B _ { 1 2 } f _ 2 + b _ { 1 1 } . \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial x ( u , t ) } { \\partial t } = \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial u ^ 2 } x ( u , t ) + W , \\\\ x ( u , 0 ) = 0 , \\ t \\geq 0 , \\ u \\in { \\mathbb R } . \\end{cases} \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} \\log ( f ^ n ) ^ \\# ( z ) = O \\ ! \\left ( ( 1 + \\rho ) ^ { n } \\right ) \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} \\| . \\| _ { \\rho } \\geq \\gamma _ n \\| . \\| _ 2 \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\gamma _ n = 0 . \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} 0 = \\left \\vert \\frac { P } { \\sqrt { c } } \\right \\vert ^ { 2 } - \\left \\vert \\frac { Q } { \\sqrt { c } } \\right \\vert ^ { 2 } = ( \\overline { a } b ) \\alpha + ( a \\overline { b } ) \\overline { \\alpha } + | b | ^ { 2 } | \\alpha | ^ { 2 } - ( \\overline { d } e ) \\beta - ( d \\overline { e } ) \\overline { \\beta } - | e | ^ { 2 } | \\beta | ^ { 2 } . \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} & E _ { \\alpha + \\beta } = \\frac { E _ \\alpha E _ \\beta - q ^ { - ( \\alpha , \\beta ) } E _ \\beta E _ \\alpha } { q - q ^ { - 1 } } , \\\\ & F _ { \\alpha + \\beta } = \\frac { F _ \\beta F _ \\alpha - q ^ { ( \\alpha , \\beta ) } F _ \\alpha F _ \\beta } { q - q ^ { - 1 } } . \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} f _ { ( l ) } ( x ) = \\frac { L ! } { ( l - 1 ) ! ( L - l ) ! } \\left ( F ( x ) \\right ) ^ { l - 1 } \\left ( 1 - F ( x ) \\right ) ^ { L - l } f ( x ) \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} \\bar \\lambda ( \\c ) = \\bar \\lambda ( x _ 0 , x _ 1 ) \\bar \\lambda ( x _ 1 , x _ 2 ) \\cdots \\bar \\lambda ( x _ { \\ell - 1 } , x _ { \\ell } ) , \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{align*} D _ { [ 1 A ] } = ( - 1 ) ^ { n } \\left ( I _ { \\frac 1 2 } \\left ( \\pi \\sqrt { 3 n } \\right ) - \\sqrt { 2 } ( 3 n ) ^ { 1 / 4 } \\right ) . \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} C ^ { i i } _ { x x } = C ^ { i i } _ { y y } \\mbox { f o r a l l } x \\le y . \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} \\nu ( \\mathcal { B } _ n ) = \\prod _ { i \\in \\mathcal { I } _ b } p ^ n _ U \\big ( \\mathbf { u } ( i ) \\big ) \\prod _ { \\big ( \\hat { i } , j \\big ) \\in \\mathcal { I } _ n \\times \\mathcal { J } _ n } p ^ n _ { V | U } \\Big ( \\mathbf { v } \\big ( \\hat { i } , j \\big ) \\Big | \\mathbf { u } ( \\hat { i } ) \\Big ) . \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} \\bold { E } _ { \\alpha } ^ { \\vartriangle } = \\sum _ J S _ { - J } \\frac { \\partial } { \\partial u _ J ^ { \\alpha } } . \\end{align*}"}
-{"id": "5681.png", "formula": "\\begin{align*} \\left ( I _ { - } ^ { \\lambda , \\lambda ^ { \\prime } , \\xi , \\xi ^ { \\prime } , \\gamma } f \\right ) ( x ) = \\frac { x ^ { - \\lambda ^ { \\prime } } } { \\Gamma ( \\gamma ) } \\int _ { x } ^ { \\infty } ( t - x ) ^ { \\gamma - 1 } t ^ { - \\lambda } F _ { 3 } \\left ( \\lambda , \\lambda ^ { \\prime } , \\xi , \\xi ^ { \\prime } ; \\gamma ; 1 - \\frac { t } { x } , 1 - \\frac { x } { t } \\right ) f ( t ) \\ , \\mathrm { d } t . \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} p _ { G \\mathaccent \\cdot \\cup H } ( x ) = p _ G ( x ) p _ H ( x ) , \\end{align*}"}
-{"id": "8586.png", "formula": "\\begin{align*} e _ a ( \\mathcal { C } _ n ) & = \\frac { 1 } { \\mathcal { M } _ n } \\sum _ { m \\in \\mathcal { M } _ n } \\mathbb { P } _ { P ^ { ( \\mathcal { C } _ n ) } } \\big ( \\tilde { M } \\neq m | M = m \\big ) \\leq \\frac { \\epsilon } { 3 } \\\\ I _ { P ^ { ( \\mathcal { C } _ n ) } } ( M ; \\mathbf { Z } ) & = \\frac { 1 } { \\mathcal { M } _ n } \\sum _ { m \\in \\mathcal { M } _ n } \\mathsf { D } \\Big ( P ^ { ( \\mathcal { C } _ n ) } _ { \\mathbf { Z } | M = m } \\Big | \\Big | P ^ { ( \\mathcal { C } _ n ) } _ \\mathbf { Z } \\Big ) \\leq \\frac { \\epsilon } { 3 } . \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{align*} M _ \\ell ( t ) = \\begin{cases} 0 , & t = 0 , \\cr \\frac { 1 } { \\ell } e ^ { - \\frac { q } { ( \\ell t ) ^ { 2 / q } } } , & t \\in \\Big ( 0 , \\frac { 1 } { \\ell } \\left ( \\frac { 2 q } { q + 2 } \\right ) ^ { q / 2 } \\Big ) , \\cr \\frac { ( q + 2 ) ^ { 1 + q / 2 } } { 2 ^ { q / 2 } q ^ { 1 + q / 2 } } e ^ { - \\frac { q } { 2 } } t - \\frac { 2 } { e q \\ell } e ^ { - \\frac { q } { 2 } } , & t \\geq \\frac { 1 } { \\ell } \\left ( \\frac { 2 q } { q + 2 } \\right ) ^ { q / 2 } . \\end{cases} \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} \\phi ^ * ( p ) = \\sup _ { x \\in M } [ x , p ] - \\phi ( x ) . \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\iota _ { X _ H } \\omega _ Q = d H \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{gather*} F _ { z z z } = v F _ { z z } , \\end{gather*}"}
-{"id": "4893.png", "formula": "\\begin{align*} & \\sum _ { i _ 1 = 0 } ^ { r - 1 } \\sum _ { i _ 2 = 0 } ^ { s - 1 } \\binom { i _ 1 + i _ 2 } { i _ 1 } ^ 2 \\sum _ { m _ 1 + m _ 2 < p } \\binom { m _ 1 + m _ 2 } { m _ 1 } ^ 2 \\cdot \\left ( 1 + 2 p ( ( i _ 1 + i _ 2 ) H _ { m _ 1 + m _ 2 } - i _ 1 H _ { m _ 1 } - i _ 2 H _ { m _ 2 } ) \\right ) . \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{gather*} L _ 1 = C \\wedge d \\tilde { F } , d L _ 1 = \\delta _ Q L . \\end{gather*}"}
-{"id": "2651.png", "formula": "\\begin{align*} \\mathbb { E } \\| g - \\tilde { f } _ { m _ 0 , m _ 1 } \\| ^ 2 - \\| g - f \\| ^ 2 & = \\mathbb { E } \\| \\tilde { f } _ { m _ 0 , m _ 1 } - \\mathbb { E } \\tilde { f } _ { m _ 0 , m _ 1 } \\| ^ 2 + \\| g - \\mathbb { E } \\tilde { f } _ { m _ 0 , m _ 1 } \\| ^ 2 - \\| g - f \\| ^ 2 \\\\ & = \\mathbb { E } \\| \\tilde { f } _ { m _ 0 , m _ 1 } - \\mathbb { E } \\tilde { f } _ { m _ 0 , m _ 1 } \\| ^ 2 + \\langle f - \\mathbb { E } \\tilde { f } _ { m _ 0 , m _ 1 } , 2 g - f - \\mathbb { E } \\tilde { f } _ { m _ 0 , m _ 1 } \\rangle . \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u _ t + u \\cdot \\nabla u = \\nabla p - g \\j \\\\ & u = 0 \\\\ & u ( 0 , x ) = u _ 0 ( x ) . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } u = - \\Delta u , \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} \\ell _ n ( G ) = \\ell _ n ( G ; A _ 1 ) + \\ell _ n ( G ; A _ 1 ^ c A _ 2 ) + \\ell _ n ( G ; A _ 1 ^ c A _ 2 ^ c ) \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} L ( f ) ( x , y ) & = [ e _ { x x } , L ( f ) ] ( x , y ) \\\\ & = \\left ( [ f , L ( e _ { x x } ) ] + L ( [ e _ { x x } , f ] ) \\right ) ( x , y ) \\\\ & = \\left ( f L ( e _ { x x } ) \\right ) ( x , y ) - \\left ( L ( e _ { x x } ) f \\right ) ( x , y ) \\\\ & + L ( e _ { x x } f ) ( x , y ) - L ( f e _ { x x } ) ( x , y ) . \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } ^ c D ^ q z _ 1 ( t ) = - z _ 1 ( t ) + T _ 0 g ( z _ 1 ( t ) ) + T _ 1 g ( z _ 2 ( t ) ) + T _ 2 g ( z _ 3 ( t ) ) \\\\ ^ c D ^ q z _ 2 ( t ) = - z _ 2 ( t ) + T _ 2 g ( z _ { 1 } ( t ) ) + T _ 0 g ( z _ 2 ( t ) ) + T _ 1 g ( z _ { 3 } ( t ) ) \\\\ ^ c D ^ q z _ 3 ( t ) = - z _ 3 ( t ) + T _ 1 g ( z _ 1 ( t ) ) + T _ 2 g ( z _ { 2 } ( t ) ) + T _ 0 g ( z _ 3 ( t ) ) \\end{array} \\right . \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} C _ { G , S } ( q ) : = \\sum _ { n = 0 } ^ { \\infty } \\gamma _ { G , S } ( n ) q ^ n . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} e _ { ( m + 1 ) p _ n + r _ n } ( x ) = e _ { ( m + 1 ) p _ n - ( m - r _ n ) } ( x ) + x ^ { m - 2 r _ n } e _ { ( m + 1 ) p _ n - r _ n } ( x ) + x ^ { m - 2 r _ n - 1 } e _ { ( m + 1 ) p _ n - r _ n - 1 } ( x ) . \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} { \\cal F } ( \\zeta ) : = \\begin{bmatrix} 1 & \\displaystyle \\frac { - 1 } { 2 \\mathrm { i } \\pi } \\int _ { \\cal L } \\frac { e ^ { s } } { s ^ { c } ( s - \\zeta ) } d s \\\\ 0 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "8616.png", "formula": "\\begin{align*} I ( V ' ; Y | U ' ) - I ( V ' ; S | U ' ) = \\epsilon \\Big [ I ( V ; Y | U ) - I ( V ; S | U ) \\Big ] . \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} \\Delta ( E _ 1 ) & = \\i X _ 1 ( 1 + q X _ 2 ( 1 + q X _ 3 ( 1 + q X _ 4 ) ) ) , \\\\ \\Delta ( E _ 2 ) & = \\i X _ 6 ( 1 + q X _ 7 ( \\dots ( 1 + q X _ { 1 2 } ( 1 + q X _ { 1 3 } ) ) \\dots ) ) . \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} m : = \\inf \\{ d ( \\underline a , \\underline b ) : \\underline a , \\underline b \\in \\Sigma : \\underline a \\wedge \\underline b = \\phi \\} \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} p _ 0 ( \\lambda ) & = 2 \\arctan \\left [ \\frac { \\tanh ( \\lambda ) } { \\tan ( \\zeta / 2 ) } \\right ] , \\theta ( \\lambda ) = 2 \\arctan \\left [ \\frac { \\tanh ( \\lambda ) } { \\tan ( \\zeta ) } \\right ] \\intertext { a n d } \\epsilon _ 0 ( \\lambda ) & = \\frac { - 2 \\sin ^ 2 \\zeta } { \\cosh ( 2 \\lambda ) - \\cos \\zeta } . \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} \\chi ( z ) G _ t ( z - y ) = \\chi ( y ) H _ D ( z , y , t ) + \\int _ 0 ^ t \\int _ { \\Omega } H _ D ( z , w , t - s ) F ( w , s ) d w d s \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} f ( x ; G ) = \\int f ( x ; \\theta ) d G ( \\theta ) \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} | \\kappa _ { N , s } ^ { - 1 } F _ { N , s } \\ast f ( x ) | \\leq \\int _ { \\{ | z | < 1 \\} } | \\ln | z | \\ f ( y ) | \\ d y + \\int _ { \\{ | z | \\geq 1 \\} } | \\ln ( | z | ) | z | ^ { 2 s - N } f ( y ) | \\ d y = : f _ 1 ( x ) + f _ 2 ( x ) . \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} \\Phi ^ * ( \\mathrm { d } _ \\mathcal { B } \\alpha ) = \\mathrm { d } _ \\mathcal { A } \\big ( \\Phi ^ * ( \\alpha ) \\big ) \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} \\pi _ 1 ( M ) = \\pi _ 1 ( T ^ n ) \\rtimes _ { \\theta _ M } \\langle t \\rangle , \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} u _ 1 ^ \\prime ( 0 ) = u _ 1 ^ { \\prime \\prime \\prime } ( 0 ) = u _ 1 ^ { ( 5 ) } ( 0 ) = u _ 1 ^ { ( 7 ) } ( 0 ) = 0 . \\end{align*}"}
-{"id": "8694.png", "formula": "\\begin{align*} K _ \\delta = \\bigcap _ { \\theta \\in S ^ { n - 1 } } \\left \\{ x \\in \\R ^ n \\ , : \\ , | \\langle x , \\theta \\rangle | \\leq t _ \\theta \\right \\} , \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} M ( u ) ( x ) : = \\sup _ { r > 0 } \\frac { 1 } { | B ( x , r ) | } \\int _ { B ( x , r ) } | u ( y ) | \\dd y \\ , , \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} g \\left ( d \\Phi ( x ) \\right ) \\det \\left ( \\Phi _ { i j } ( x ) \\right ) = c f ( x ) , \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} N ( x , z ) = \\sum _ { n \\geq 0 } N _ n ( x ) \\frac { z ^ n } { n ! } = \\sqrt { \\frac { 1 - x } { 1 - x e ^ { 2 z ( 1 - x ) } } } . \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} X _ i ^ { Y _ 1 ^ { C _ 1 ( i , j ) } \\cdots Y _ n ^ { C _ n ( i , j ) } } = X _ j , ( i , j ) \\in R , \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} n ^ { - 1 / 2 } \\sum _ { r = \\lceil p n \\rceil } ^ n \\exp ( - n \\epsilon ( r / n - p ) ^ 2 ) \\le n ^ { - 1 / 2 } \\ell \\frac { 1 } { 1 - e ^ { - \\epsilon } } , \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} R _ 3 = \\frac { 1 } { 2 } \\log \\frac { | \\sigma ^ { 2 } { \\bf I } _ 2 + \\sum _ { j = 1 } ^ { 3 } { \\bf G } _ { 2 j } { \\bf Q } _ j { \\bf G } _ { 2 j } ^ { T } | } { | \\sigma ^ { 2 } { \\bf I } _ 2 + \\sum _ { j = 1 } ^ { 2 } { \\bf G } _ { 2 j } { \\bf Q } _ j { \\bf G } _ { 2 j } ^ { T } | } . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} & f ( \\gamma , 1 ) \\geq \\\\ & \\frac { 1 } { 2 } \\log \\left ( 1 + \\frac { \\gamma ' k _ n P ' } { 2 } \\right ) - \\frac { 1 - \\epsilon } { 2 } \\log ( 1 + k _ n P ' ) - \\frac { k _ n } { n } . \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} ( - 1 ) ^ { k - n + 1 } \\int _ { \\partial M / B } i ^ * \\omega = \\int _ { M / B } d ^ M \\omega - d ^ B \\int _ { M / B } \\omega , \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} L ^ { * } _ { t } + \\sum _ { i = 1 } ^ { n } M _ { \\mu _ { t } ^ { ( i ) } } + \\sum _ { j = 1 } ^ { m } F _ { \\nu _ { t } ^ { ( j ) } } , \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} f ( \\phi ) \\otimes \\max \\{ \\psi _ 1 , \\psi _ 2 \\} & = \\phi \\otimes ( \\max \\{ \\psi _ 1 , \\psi _ 2 \\} \\circ f ) \\\\ & = \\phi \\otimes ( \\max \\{ \\psi _ 1 \\circ f , \\psi _ 2 \\circ f \\} ) \\\\ & = \\max \\{ \\phi \\otimes ( \\psi _ 1 \\circ f ) , \\phi \\otimes ( \\psi _ 2 \\circ f ) \\} & ( \\phi ~ { \\rm i s ~ f l a t } ) \\\\ & = \\max \\{ f ( \\phi ) \\otimes \\psi _ 1 , f ( \\phi ) \\otimes \\psi _ 2 \\} , \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\frac 1 2 \\left ( I ( \\xi , \\eta ) + I ( - \\xi , - \\eta ) \\right ) = \\frac { \\xi } { \\tanh \\xi } + \\frac { \\eta } { \\tanh \\eta } - \\frac { \\xi - \\eta } { \\tanh ( \\xi - \\eta ) } . \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} \\partial _ { t } w _ { t } ( x ) = v _ { t } ( x ) , \\partial _ { t } v _ { t } ( x ) = D _ { 1 1 } w _ { t } ( x ) \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} \\phi ^ { \\alpha } _ { J _ 1 ; J _ 2 } = D _ { J _ 1 } \\left ( \\phi ^ { \\alpha } _ { \\bold { 0 } ; J _ 2 } - \\xi ^ i u ^ { \\alpha } _ { \\bold { 1 } _ i ; J _ 2 } \\right ) + \\xi ^ i u ^ { \\alpha } _ { J _ 1 + \\bold { 1 } _ i ; J _ 2 } \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} \\{ f , g \\} = \\rho ( a _ g ) \\cdot f = - \\rho ( a _ f ) \\cdot g \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} [ m _ 1 \\otimes \\dotsc \\otimes m _ k ] \\cdot [ n _ 1 \\otimes \\dotsc \\otimes n _ l ] = [ m _ 1 \\otimes \\dotsc \\otimes m _ k \\otimes n _ 1 \\otimes \\dotsc \\otimes n _ l ] . \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} P & \\Bigg ( \\bigcap _ { i = 1 } ^ M \\bigg \\{ X _ i \\in \\bigg ( ( \\sigma _ { \\rm w } ^ 2 + \\sigma _ { \\rm j , i } ^ 2 ) \\left ( 1 - \\frac { \\delta } { \\sigma _ { \\rm w } ^ 2 + c } \\right ) , \\\\ & \\qquad \\qquad ( \\sigma _ { \\rm w } ^ 2 + \\sigma _ { \\rm j , i } ^ 2 ) \\left ( 1 + \\frac { \\delta } { \\sigma _ { \\rm w } ^ 2 + c } \\right ) \\bigg ) \\bigg \\} \\Bigg ) > 1 - \\frac { \\epsilon } { 2 } . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} k ( y ) = \\sum _ d \\gamma ( d ) \\phi ( d y ) \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { S _ { G } ^ { p } } ^ { p } = \\mathbb { E } [ \\sup _ { s \\in \\lbrack 0 , T ] } | u ( s , \\omega ) | ^ { p } ] . \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} E [ \\psi ( W , \\gamma , \\alpha , \\theta _ { 0 } ) ] & = E [ g ( W , \\gamma , \\theta _ { 0 } ) ] + E [ \\alpha _ { 0 } ( X ) ] - \\int \\alpha _ { 0 } ( u ) \\gamma ( u ) d u \\\\ & = \\int \\alpha _ { 0 } ( u ) \\{ \\gamma ( u ) - \\gamma _ { 0 } ( u ) \\} d u + \\int \\alpha ( u ) \\{ \\gamma _ { 0 } ( u ) - \\gamma ( u ) \\} d u = 0 . Q . E . D . \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{gather*} x '' ( t ) + \\omega ^ 2 x ( t ) = \\lambda f ( t , x ( t ) ) , t \\in [ 0 , 1 ] , \\\\ x ( 0 ) = x ( 1 ) , x ' ( 0 ) = x ' ( 1 ) . \\end{gather*}"}
-{"id": "162.png", "formula": "\\begin{align*} \\gamma _ j : = \\{ t u _ j : t \\in [ 1 , 2 ] \\} \\subset \\c . \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{gather*} \\pi ( i _ 1 ) = \\theta ( 1 ) , \\ \\pi ( i _ 2 ) = \\theta ( 2 ) , \\dots , \\pi ( i _ m ) = \\theta ( m ) , \\\\ \\pi ( j _ 1 ) = m + \\nu ( 1 ) , \\ \\pi ( j _ 2 ) = m + \\nu ( 2 ) , \\dots , \\pi ( j _ n ) = m + \\nu ( n ) . \\end{gather*}"}
-{"id": "1024.png", "formula": "\\begin{align*} & \\lim _ { t \\to 0 } \\int _ { U } h _ t ( x , y ) \\ d y = \\int _ { U } \\partial _ 1 H ( x , y ) \\ d y \\\\ & \\lim _ { t \\to 0 } \\int _ { \\R ^ N \\backslash U } h _ t ( x , y ) \\ d y = \\int _ { \\R ^ N \\backslash U } \\partial _ 1 H ( x , y ) \\ d y . \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} S [ M ] = \\{ \\sigma \\in S _ n : \\mbox { $ \\sigma $ i s m o n o t o n e d e c r e a s i n g o n e a c h o f $ [ 1 , n _ 1 ] $ , $ [ n _ 1 + 1 , n _ 2 ] $ e t c . } \\} \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} \\tau \\sigma \\tau ^ { - 1 } = \\sigma ^ { k } , k \\in \\{ 0 , 1 , \\ldots , e - 1 \\} . \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} r _ { m , m } ( 2 ^ { - \\rho } A ) ^ { 2 ^ \\rho } = \\exp ( A + E ) , \\ ; \\norm { E } \\le u \\norm { A } , \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} f ( z ) \\log N = \\int _ \\beta ^ \\alpha a '' ( u ) N ^ { u z } d u . \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} \\Delta _ { 2 ^ { p _ 0 } + \\cdots + 2 ^ { p _ s } } & = \\sum _ { j = 1 } ^ s 2 ^ { j - p _ j + p _ 0 } ( 2 ^ { p _ j - p _ { j - 1 } } - 2 ) \\left [ ( 3 \\ 2 ^ { j - 1 } - 1 ) \\Delta _ { 2 ^ { p _ j } + \\cdots + 2 ^ { p _ s } } \\right . \\\\ & \\left . + ( 2 ^ { j - 2 } ( 2 ^ { p _ j - p _ { j - 1 } } - 2 ^ 2 ) + 1 ) ( 1 + 2 ^ { p _ { j + 1 } - p _ j } + \\cdots + 2 ^ { p _ s - p _ j } ) \\right ] \\ge 0 \\\\ \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} \\sum _ { \\overline { b } \\in \\mathcal { B } } \\overline { c _ { \\overline { b } } / p ^ m } \\overline { b } = 0 , \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} \\Lambda ( z ^ { ( 0 ) } ) & = \\frac { e ^ { \\frac { \\sigma _ { \\mathrm { a } } ^ 2 } { \\zeta } } \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 + \\sigma _ { \\mathrm { a } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { z ^ { ( 0 ) } } { v } } e ^ { - \\frac { v } { \\zeta } } d v } { \\int _ { \\sigma _ { \\mathrm { w } } ^ 2 } ^ { \\infty } \\left ( \\frac { 1 } { v } \\right ) ^ { n } e ^ { - \\frac { z ^ { ( 0 ) } } { v } } e ^ { - \\frac { v } { \\zeta } } d v } = \\gamma , \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} \\omega _ t = ( 1 - t ) \\omega _ 0 + t \\omega , t \\in [ 0 , 1 ] . \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} ( 4 \\tau ) ^ { \\frac { n + 1 } 2 } = e ^ { T N ^ { 2 / 3 } + T ^ 3 / 3 - T \\xi + o ( 1 ) } . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} u ( x ) = \\begin{cases} \\frac { \\gamma } { 1 - \\gamma } \\left [ \\left ( \\phi + \\frac { \\eta } { \\gamma } \\ , x \\right ) ^ { 1 - \\gamma } - 1 \\right ] , & 0 < \\gamma < + \\infty , \\ \\gamma \\neq 1 , \\\\ [ 5 p t ] \\log ( \\phi + \\eta \\ , x ) , & \\gamma = 1 , \\end{cases} \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} \\int _ { M } X ( R _ g ) d M = - \\frac { n } { n - 2 } \\int _ { M } \\left \\langle { \\stackrel { \\circ } { { \\rm R i c } _ g } } , \\mathcal { L } _ { X } g \\right \\rangle d M + \\frac { 2 n } { n - 2 } \\int _ { \\Sigma } { \\stackrel { \\circ } { { \\rm R i c } _ g } } ( X , \\nu ) d \\Sigma , \\end{align*}"}
-{"id": "8517.png", "formula": "\\begin{align*} \\mathbb { P } _ { \\rm M D } ( u ) = P \\left ( \\frac { Z } { n } \\leq \\tau _ n | \\sigma _ { \\rm j } ^ 2 = u , H _ 1 \\right ) > 1 - \\frac { \\epsilon } { 2 } \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} { \\cal F } ( \\zeta ) = \\widehat { \\cal F } ( \\zeta ) F _ k ( \\zeta ) \\cdots F _ 1 ( \\zeta ) , \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} b ( \\lambda _ 1 , \\lambda _ 2 ) = q ( \\lambda _ 1 + \\lambda _ 2 ) - q ( \\lambda _ 1 ) - q ( \\lambda _ 2 ) . \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} f = q - 8 q ^ 3 - 1 0 q ^ 5 - 1 6 q ^ 7 + 3 7 q ^ 9 + 4 0 q ^ { 1 1 } + O ( q ^ { 1 2 } ) \\in S _ 4 ( \\Gamma _ 0 ( 3 2 ) ) \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} S ( U , f ^ { n + k } ) \\geq S ( D ( 0 , r _ 2 ) \\backslash D ( 0 , r _ 1 ) , f ^ n ) = S ( r _ 2 , f ^ n ) - S ( r _ 1 , f ^ n ) , \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} \\mathcal { I } = \\sum \\nolimits _ { i \\in \\Phi / o } { { P _ t } { G _ i } L \\left ( \\left | X _ i \\right | \\right ) } . \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} \\theta ( \\ell ) = \\sum _ { m \\mid \\ell } \\mu ( m ) g ( m ) h ( \\ell / m ) . \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} \\frac { d } { d s } \\int _ { B _ s ( x _ 0 ) \\cap \\Sigma } f d \\mu = \\int _ { \\partial B _ { s } ( x _ 0 ) \\cap \\Sigma } \\frac { f } { | \\nabla r | } d \\sigma . \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} \\frac { z ^ { a } \\exp \\left ( - z \\right ) } { \\Gamma \\left ( a + 1 \\right ) } & = \\left \\{ \\frac { 1 + O \\left ( a ^ { - 1 } \\right ) } { \\sqrt { 2 \\pi } } \\right \\} a ^ { - 1 / 2 } \\exp \\left \\{ a \\ln \\left ( \\frac { e } { \\rho e ^ { 1 / \\rho } } \\right ) \\right \\} \\\\ & = O \\left [ a ^ { - 1 / 2 } \\exp \\left \\{ a \\ln \\left ( \\frac { e } { \\rho e ^ { 1 / \\rho } } \\right ) \\right \\} \\right ] , \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} H ^ { 2 } ( 0 ) = \\left [ \\begin{array} { c c c c c } 1 & 0 & 0 & \\cdots & 0 \\\\ 0 & t & b _ { 2 } & \\cdots & b _ { n } \\end{array} \\right ] . \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} x '' ( t ) = - \\lambda f ( t , x ( t ) ) , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} 3 r ^ 2 = 3 + 6 \\pi ^ 4 + 3 \\pi ^ 8 , 2 s t \\equiv 2 \\pi ^ 4 + 2 \\pi ^ 6 \\pmod { \\pi ^ { 1 6 } } \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} & \\sum _ { i _ 1 = 0 } ^ { r - 1 } \\sum _ { i _ 2 = 0 } ^ { s - 1 } \\sum _ { i _ 3 = 0 } ^ { t - 1 } \\binom { i _ 1 + i _ 2 + i _ 3 } { i _ 1 , i _ 2 , i _ 3 } ^ 2 \\sum _ { m _ 1 + m _ 2 + m _ 3 < p } \\binom { m _ 1 + m _ 2 + m _ 3 } { m _ 1 , m _ 2 , m _ 3 } ^ 2 \\\\ & \\qquad \\cdot \\left ( 1 + 2 p ( ( i _ 1 + i _ 2 + i _ 3 ) H _ { m _ 1 + m _ 2 + m _ 3 } - i _ 1 H _ { m _ 1 } - i _ 2 H _ { m _ 2 } - i _ 3 H _ { m _ 3 } ) \\right ) . \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} E _ 0 \\ : = \\ \\{ v \\in E \\ | \\ Q ^ * ( v , E ) = 0 \\} \\ , . \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} \\textrm { s i g n } \\left ( ( - 1 ) ^ k \\sin ( \\phi ) \\right ) = \\textrm { s i g n } \\left ( \\sin \\left ( \\frac { \\phi } { 2 } \\pm A \\right ) \\right ) . \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} \\operatorname { D i v } B ( x , n ) = L ( x , n , [ 0 ] ) . \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} N _ { | c | } ( E ) = 1 - 2 \\rho _ { | c | } ( E ) . \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} b ( x , y ) = - b ( y , x ) . \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} \\psi _ { r , x _ 0 } ( x ) = \\left \\{ \\begin{aligned} & \\gamma _ { N , s } ( r ^ 2 - | x - x _ 0 | ^ 2 ) ^ { s } , & & | x - x _ 0 | < r , \\\\ & 0 , & & | x - x _ 0 | \\geq r , \\end{aligned} \\right . \\ \\gamma _ { N , s } = \\frac { \\Gamma ( \\frac { N } { 2 } ) 4 ^ { - s } } { \\Gamma ( s + 1 ) \\Gamma ( \\frac { N } { 2 } + s ) } . \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} b ^ { 2 } \\alpha ^ { 2 } = e ^ { 2 } \\beta ^ { 2 } . \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} q _ { F } ( x ^ { * } _ { k _ { i } } - x ^ { * } _ { k _ { j } } ) & \\leq q _ { F } ( x ^ { * } _ { k _ { i } } - z ^ { * } _ { k _ { i } } ) + q _ { F } ( z ^ { * } _ { k _ { i } } - z ^ { * } _ { k _ { j } } ) + q _ { F } ( z ^ { * } _ { k _ { j } } - x ^ { * } _ { k _ { j } } ) \\\\ & \\leq 2 \\epsilon + q _ { F } ( z ^ { * } _ { k _ { i } } - z ^ { * } _ { k _ { j } } ) , i , j = 1 , 2 , . . . \\\\ \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} _ { } ( _ i ( 1 _ n ^ { * 3 } \\otimes 1 _ 1 ) ) = \\tfrac { 1 } { 1 6 } ( & 1 _ n ^ { \\ast 3 } \\otimes 1 _ 1 + 3 \\cdot 1 _ n ^ { \\ast 2 } i _ n ^ { \\ast } \\otimes i _ 1 - 3 \\cdot 1 _ n ^ { \\ast } i _ n ^ { \\ast 2 } \\otimes 1 _ 1 - i _ n ^ { \\ast 3 } \\otimes i _ 1 + \\\\ + & j _ n ^ { \\ast 3 } \\otimes j _ 1 + 3 \\cdot j _ n ^ { \\ast 2 } k _ n ^ { \\ast } \\otimes k _ 1 - 3 \\cdot j _ n ^ { \\ast } k _ n ^ { \\ast 2 } \\otimes j _ 1 - k _ n ^ { \\ast 3 } \\otimes k _ 1 ) . \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{gather*} d \\alpha | _ { { \\Sigma ^ \\infty } } = 0 . \\end{gather*}"}
-{"id": "2815.png", "formula": "\\begin{align*} H _ h ( N ) = 2 ( 1 4 - N ) \\lambda ( N ) + \\frac { \\tau ( N ) } { 1 6 } ( \\mu ( N + 8 ) - \\mu ( N ) ) H _ u ( N ) . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} | \\pi | \\ & : = \\pi , \\\\ s ( \\pi ) \\ & : = \\pi , \\\\ \\nu _ e ( \\pi ) \\ & : = \\pi , \\\\ \\nu _ o ( \\pi ) \\ & : = \\pi , \\\\ \\nu ( \\pi ) & : = \\nu _ e ( \\pi ) + \\nu _ o ( \\pi ) = \\pi , \\\\ \\nu _ { d } ( \\pi ) \\ & : = \\pi . \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} u _ t - \\Delta u ^ m = 0 \\quad \\Omega _ { T _ 0 } , \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} C ( q , a ) = \\phi ( q ) M ( q , a ) - \\gamma - B + \\sum _ { p | q } \\frac { 1 } { p } , \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} \\sum _ { t = k + 1 } ^ \\infty t N _ t < \\sum _ { t = 1 } ^ \\ell t N _ t \\sum _ { t = 1 } ^ { \\ell - 1 } t N _ t < \\sum _ { t = k } ^ \\infty t N _ t . \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} a ( x , y , z ) = \\frac { ( q - 1 ) } { 2 \\cdot 3 } \\left ( ( q - 1 ) + 1 + \\sum \\limits _ { \\theta \\in \\mathfrak { A } } \\frac { \\theta ( x ) \\theta ( y ) \\theta ( z ^ { - 1 } ) } { \\theta ( 1 ) } \\right ) , \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} \\det ( 1 - \\mathrm { F r } _ \\lambda X \\mid V _ f ) = 1 - T _ \\lambda ( f ) X + S _ \\lambda ( f ) X ^ 2 \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} M ( r , f ) = \\max _ { | z | = r } | f ( z ) | \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} { \\cal { \\bar I } } ( \\boldsymbol { x } ) = \\sum _ { j = 1 } ^ { d } { \\bar R } _ j ^ { m _ j } ( \\boldsymbol { x } ) W _ j ( \\boldsymbol { x } ) , \\boldsymbol { x } \\in \\Omega , \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} \\phi _ t ^ * \\omega _ t = \\omega _ 0 , t \\in [ 0 , 1 ] \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} A _ { n + 1 } ( z ) M _ n ( z ) = \\frac { d M _ n ( z ) } { d z } + M _ n ( z ) A _ n ( z ) . \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{align*} \\chi _ \\alpha ( f ^ \\ast \\rho ) = f ^ \\ast \\chi _ \\alpha ( \\rho ) \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} \\mathfrak { t } \\cap R = \\mathfrak { t } \\cap S \\cap R = \\mathfrak { n } \\cap R . \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} \\tilde { B } _ E ( \\theta ) = \\frac { 1 } { \\sqrt { 2 i } } B _ E ( \\theta ) \\left ( \\begin{matrix} i \\ \\ & - i \\\\ 1 \\ \\ & 1 \\end{matrix} \\right ) , \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} I _ 1 = \\begin{cases} C _ 1 ( t - s ) ^ H , & \\ ; \\\\ C _ 2 ( t - s ) ^ { 2 H + 1 } , & \\ . \\end{cases} \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} A _ { ( m + 1 ) p _ n - \\frac { m - 1 } { 2 } } = A _ { ( m + 1 ) p _ n + \\frac { m + 1 } { 2 } } - x ^ { m - 1 } A _ { ( m + 1 ) p _ n + \\frac { m - 1 } { 2 } } . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} \\int _ 1 ^ \\infty \\tau ^ { 2 s } \\left | \\real \\Big ( \\tilde T _ { 2 j } ( i \\tau / 2 ) + \\sum _ { l = 0 } ^ k ( - 1 ) ^ { j + l } i ^ { l - 1 } \\tau ^ { - ( 2 j - 1 + l ) } T _ { 2 j } ^ l \\Big ) \\right | d \\tau \\lesssim \\frac { 1 } { | \\sin ( \\pi s ) | } \\| ( u , v ) \\| _ { \\dot H ^ s } ^ 2 \\| ( u , v ) \\| _ { l ^ 2 _ 1 D U ^ 2 } ^ { 2 j - 2 } . \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} F _ { \\alpha _ { 1 } } ( \\sigma ) : = \\chi _ { ( 0 , \\infty ) } ( \\sigma ) L _ { \\alpha _ { 1 } } ^ { 0 } \\left ( \\sigma \\right ) e ^ { - \\frac { \\sigma } { 2 } } \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} \\vec { X } _ { 0 1 } & = \\Bigg ( 0 , 1 , - \\frac { \\lambda } { m - n } \\bigg ( \\frac { 1 } { 1 - A ^ { - 1 } } \\bigg ) \\Bigg ) , \\mu _ { 0 1 } = 1 , \\\\ \\vec { X } _ { 0 2 } & = \\Bigg ( 1 , a b , - \\frac { \\lambda a d } { m - n } \\bigg ( \\frac { 1 } { 1 - 2 A ^ { - 1 } } \\bigg ) \\Bigg ) , \\mu _ { 0 2 } = 2 , \\\\ \\vec { X } _ { 0 3 } & = ( 0 , 0 , 1 ) , \\mu _ { 0 3 } = A . \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} \\omega ^ 2 = \\frac { 4 b ^ 4 k ^ 2 a ^ 2 - a ^ 4 ( b ^ 2 + k ^ 2 ) ^ 2 } { ( b ^ 2 - k ^ 2 ) ^ 2 } . \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\eta ^ \\rho _ k } = \\frac { \\partial } { \\partial \\eta ^ \\rho _ { k + 1 } } + \\sum _ l \\bigg ( \\frac { \\partial a ^ k _ l } { \\partial \\eta ^ \\rho _ k } \\bigg ) \\frac { \\partial } { \\partial y ^ l _ { k + 1 } } + \\sum _ \\tau \\bigg ( \\frac { \\partial \\beta ^ k _ \\tau } { \\partial \\eta ^ \\rho _ k } \\bigg ) \\frac { \\partial } { \\partial \\eta ^ \\tau _ { k + 1 } } . \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} L = v ( v ' _ 1 - v ' ) - \\ln ( v _ 2 - v ) . \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\tilde { D } ( \\eta ) = \\frac { 1 } { 2 } \\tilde { D } ( \\tilde { \\eta } ) = D ( \\tilde { \\eta } ) . \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} L = - \\partial _ x ^ 2 + \\frac { 1 } { 4 } ( x ^ 2 - 6 ) . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} \\mathcal { F } ( U _ { n + 1 } ) = \\mathcal { F } ( U _ n ) + L _ n H _ { n + 1 } + Q _ n , \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} \\begin{aligned} \\phi ( \\omega ) : & = \\mathrm { i } \\omega \\cdot l + \\mu _ j ^ { \\infty } ( \\omega ) - \\mu _ k ^ { \\infty } ( \\omega ) = \\mathrm { i } \\omega \\cdot l - \\mathrm { i } m _ 3 ( \\omega ) ( j ^ 3 - k ^ 3 ) + \\mathrm { i } m _ 1 ( j - k ) + ( r _ j ^ { \\infty } - r _ k ^ { \\infty } ) ( \\omega ) \\end{aligned} \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} e _ { x x } f e _ { y y } = f ( x , y ) e _ { x y } \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\mu ( V , | \\lambda | , { \\rho } ) = - \\frac { | \\lambda | } { \\sqrt { { \\rho } - { \\rho _ { c } } ( \\beta ) } } \\\\ + \\alpha ( | \\lambda | , V ) \\ , \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} \\epsilon ( \\lambda ) = \\tilde { \\epsilon } ( \\lambda ) - \\sum _ { i a } s _ a \\tilde { \\epsilon } ( \\lambda _ { i a } ) F ( \\lambda | \\lambda _ { i a } ) . \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} \\left ( \\frac { 2 5 } { 4 } - \\frac { 9 } { 4 } - C \\epsilon \\right ) \\left \\| \\frac { \\phi _ n \\tilde \\mu _ a } { r } \\right \\| ^ 2 - C \\int \\frac { \\phi _ n ^ 2 } { ( r ^ 2 + 1 ) ^ 3 } d v = o ( n ^ { - 2 } ) . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} r = \\frac { \\sqrt { c ^ 3 + 2 c + 6 } } { \\sqrt { 3 } \\sqrt { c ^ 3 - 4 c } } . \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} \\phi ( t , q ) = \\prod _ { n = 1 } ^ { \\infty } \\left ( \\frac { 1 } { 1 - t ^ { a ( n ) } q ^ n } \\right ) \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon _ 4 = & 8 C _ 4 \\\\ \\epsilon _ 5 = & 8 0 \\kappa C _ 4 \\\\ \\epsilon _ 6 = & 5 2 8 \\kappa ^ 2 C _ 4 + 1 2 C _ 6 - 4 8 \\theta _ { 2 , 2 , 2 } \\\\ \\epsilon _ 7 = & 2 9 1 2 \\kappa ^ 3 C _ 4 + 1 6 8 \\kappa C _ 6 - 6 7 2 \\kappa \\theta _ { 2 , 2 , 2 } - 5 6 \\theta _ { 3 , 3 , 1 } \\end{aligned} \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} Q _ { s , t } ( a , b , \\ell ) = \\frac { { b \\choose \\ell } { s + t - b - 1 \\choose a - \\ell } } { { s + t - 1 \\choose a } } = \\frac { { a \\choose \\ell } { s + t - a - 1 \\choose b - \\ell } } { { s + t - 1 \\choose b } } . \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{gather*} \\| f _ \\varepsilon ( i t ) \\| _ { X _ 1 } \\le \\sum _ { n \\ge 1 } \\Big ( \\sum _ { n \\le \\lambda _ k < n + 1 } | a _ k | \\Big ) ^ { 2 / ( 2 - s ) } = 1 \\ , . \\end{gather*}"}
-{"id": "8791.png", "formula": "\\begin{align*} \\sum _ { n \\le x } \\frac 1 { \\kappa ^ * ( n ) } = \\frac { A \\zeta ( 3 / 2 ) } { \\zeta ( 3 ) } x ^ { 1 / 2 } + \\frac { B \\zeta ( 2 / 3 ) } { \\zeta ( 2 ) } x ^ { 1 / 3 } + O ( x ^ { 1 / 5 } ) , \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} \\textrm { s i g n } ( \\sin ( B \\pm A ) ) = \\textrm { s i g n } \\left ( \\sin B \\cos A \\pm \\cos B \\sin A \\right ) = \\textrm { s i g n } ( \\cos A ) . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} Y : = \\begin{pmatrix} y _ 1 & y _ 2 \\\\ \\sigma ( y _ 1 ) & \\sigma ( y _ 2 ) \\end{pmatrix} \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} 2 p \\hat { \\beta } ^ T A \\sum _ { i } k _ i \\bar { q } _ i - 2 p \\hat { \\beta } ^ T q & = 2 p \\hat { \\beta } ^ T ( A q - p A \\hat { \\beta } ) - 2 p \\lambda _ 0 \\hat { \\beta } ^ T q \\\\ & \\equiv 2 p \\hat { \\beta } ^ T ( A q - \\lambda _ 0 q ) \\\\ & \\equiv 0 ~ ( { \\rm m o d } ~ p ^ 2 ) . \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\left \\{ \\sum _ { j = 1 } ^ { k } \\sum _ { l = 0 } ^ { | p _ { j } | - 1 } \\xi _ { j l } \\right \\} \\in ( 0 , 1 ) \\backslash \\{ { 1 / 2 } \\} . \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} \\begin{aligned} ( \\phi _ 0 ' ( 1 ) , \\phi _ 0 ( 1 ) ) _ { \\mathbb { R } ^ n } & = ( \\phi _ 0 ' ( 1 ) , P _ { D _ 1 } \\phi _ 0 ( 1 ) + P _ { N _ 1 } \\phi _ 0 ( 1 ) + P _ { R _ 1 } \\phi _ 0 ( 1 ) ) _ { \\mathbb { R } ^ n } \\\\ & = ( P _ { N _ 1 } \\phi _ 0 ' ( 1 ) + P _ { R _ 1 } \\phi _ 0 ' ( 1 ) , \\phi _ 0 ( 1 ) ) _ { \\mathbb { R } ^ n } = 0 , \\end{aligned} \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} n = \\theta N \\min \\left \\{ 1 , \\left ( \\frac { r } { \\sigma } \\right ) ^ 2 \\right \\} ~ . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{align*} a _ k = b _ m \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} Q _ n : = Q ( U _ n , H _ { n + 1 } ) , Q ( U _ n , H ) : = \\mathcal { F } ( U _ n + H ) - \\mathcal { F } ( U _ n ) - L _ n H , H \\in E _ n \\times \\mathbb { R } ^ { \\nu } . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} F ( s ) = G ( s ) M ( s ) - 1 \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty \\| j _ n ( x ) - j _ { n + 1 } ( x ) \\| ^ 2 _ { \\varphi _ \\mu ^ \\infty } < \\infty . \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} \\alpha : = \\frac { \\gamma } { 2 ( 1 - \\delta ) } , \\ \\beta : = \\frac { \\delta } { 2 ( 1 - \\delta ) } . \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} \\iota ( p _ i ) = \\frac { x _ { i } ^ - x _ { i + 1 } } { x _ { i + 1 } ^ - x _ i } X _ { i + 1 ^ - , i } , \\iota ( q _ i ) = \\frac { x _ i x _ { i + 1 } ^ + } { x _ { i + 1 } x _ i ^ + } X _ { i + 1 , i ^ + } \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} d _ i = D _ i ^ { ( N ) } + \\beta \\sum _ { j < i } K _ { i j } , \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} \\mathcal { B } _ { - } ( \\lambda ^ { - 1 } ) = - \\frac { ( \\lambda ^ { 2 } q - 1 / \\left ( q \\lambda ^ { 2 } \\right ) ) } { ( \\lambda ^ { 2 } / q - q / \\lambda ^ { 2 } ) } \\mathcal { B } _ { - } ( \\lambda ) , \\mathcal { C } _ { - } ( \\lambda ^ { - 1 } ) = - \\frac { ( \\lambda ^ { 2 } q - 1 / \\left ( q \\lambda ^ { 2 } \\right ) ) } { ( \\lambda ^ { 2 } / q - q / \\lambda ^ { 2 } ) } \\mathcal { C } _ { - } ( \\lambda ) . \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} M _ { \\ell } ^ { - 1 } \\Big ( \\frac { 1 } { N } \\Big ) = \\frac { q ^ { q / 2 } } { \\ell \\left ( \\log \\frac { N } { \\ell } \\right ) ^ { q / 2 } } , \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} a = \\min \\left \\{ \\frac { \\epsilon } { 8 } \\log \\left ( 1 + \\frac { P ' } { 4 } \\right ) , 1 \\right \\} . \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} T _ \\hbar = \\delta h ^ { - \\frac 1 2 } = h ^ { - \\frac 1 4 } = \\hbar ^ { - 1 } \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} \\begin{pmatrix} A & B \\\\ 0 & C \\end{pmatrix} \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} D ^ a ( 0 , - \\eta ) = - \\frac { i } { \\Lambda ( \\eta ) ( e ^ { 2 \\eta } + 1 ) } \\left [ 2 J ( \\eta ) - \\eta J ' ( \\eta ) - J ( \\eta ) J ' ( \\eta ) \\right ] . \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} \\mathcal { R } ( u ) [ h ] = \\sum _ { \\lvert j \\rvert \\le C } ( h , g _ j ( u ) ) _ { L ^ 2 ( \\mathbb { T } ) } \\chi _ j ( u ) \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} \\hat { \\mathbf { H } } _ i = \\mathbf { R } _ i ^ \\frac { 1 } { 2 } \\hat { \\mathbf { X } } _ i . \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} h _ t ( x ) : = t ^ { - 1 } h ( t ^ { - 1 } x ) . \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} A _ { \\Lambda } = \\{ 0 , \\zeta , \\xi , \\zeta ' \\} . \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} \\frac { \\partial \\tilde { \\epsilon } } { \\partial \\lambda _ { j b } } ( \\lambda ) = - s _ b \\epsilon _ 0 ' ( \\lambda _ { j b } ) F ( \\lambda _ { j b } | \\lambda ) - \\sum _ i \\int _ { \\lambda _ { i L } } ^ { \\lambda _ { i R } } d \\nu \\ , \\epsilon _ 0 ' ( \\nu ) \\frac { \\partial F } { \\partial \\lambda _ { j b } } ( \\nu | \\lambda ) , \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} Q ( I , m ) : = \\{ S \\in Q ( I ) \\mid | S | = m \\} . \\end{align*} % \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} \\partial ^ { n - 1 } a = 2 \\Im \\sum \\limits _ { k = 0 } ^ { n - 1 } \\P [ R ^ { ( k ) } \\bar R ^ { ( n - k ) } ] . \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} x & = r \\sin ( \\alpha ) \\cos ( \\phi ) , \\\\ y & = r \\sin ( \\alpha ) \\sin ( \\phi ) , \\\\ z & = r \\cos ( \\alpha ) \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} Y _ { t } = \\xi + \\int _ { t } ^ { T } f ( s , Y _ { s } , Z _ { s } , \\eta _ { s } ) d s - \\int _ { t } ^ { T } Z _ { s } d B _ { s } - ( \\int _ t ^ T \\frac { 1 } { 2 } \\eta _ s d \\langle B \\rangle _ s - \\int _ t ^ T G ( \\eta _ s ) d s ) , \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{gather*} \\mu : = \\frac { \\mu _ + + \\mu _ - } { 2 } \\qquad \\chi : = \\frac { \\mu _ + - \\mu _ - } { 2 } . \\end{gather*}"}
-{"id": "6927.png", "formula": "\\begin{align*} A '' = \\bigoplus _ { k } \\left ( ( \\mathbb { F } _ { ( n _ k ) } \\oplus \\mathbb { F } _ { ( n _ k + 2 ) } \\right ) . \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} \\eta _ { \\mu } = c - h ^ { 2 } t + \\int _ { 0 } ^ { t } \\int _ { 0 } ^ { s } \\frac { 1 } { \\phi _ { \\mu } ^ { 2 } } \\ , d \\tau d s , \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} H = H _ k = \\left ( N ^ { c / 2 } \\eta \\right ) ^ { - \\sigma _ 3 } R _ k \\cdots R _ 1 \\left ( N ^ { c / 2 } \\eta \\right ) ^ { \\sigma _ 3 } F _ 1 ^ { - 1 } \\cdots F _ k ^ { - 1 } , \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} p = 0 \\ \\ \\Gamma ( t ) , \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} I _ \\ell : = \\{ c _ { \\ell } - n _ \\ell + 1 , \\ldots , c _ \\ell \\} . \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} \\begin{aligned} \\| \\langle D \\rangle ^ s T _ { \\langle D \\rangle ^ \\sigma u } f \\| _ { L ^ \\infty } & \\lesssim \\| f \\| _ { B ^ { s + \\sigma , \\infty } _ 2 } \\| u \\| _ { B ^ { 0 , \\infty } _ 2 } , \\sigma > 0 \\\\ \\| \\langle D \\rangle ^ s \\Pi [ f , \\langle D \\rangle ^ \\sigma u ] \\| _ { L ^ \\infty } & \\lesssim \\| f \\| _ { B ^ { s + \\sigma , \\infty } _ 2 } \\| u \\| _ { B ^ { 0 , \\infty } _ 2 } , \\sigma \\geq 0 . \\end{aligned} \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} c ' = 2 ^ { n \\left ( I ( V ; W | U ) - R _ 2 - \\epsilon ^ { ( 1 ) } _ { \\alpha , \\delta _ 1 } + \\epsilon ^ { ( 2 ) } _ { \\alpha , \\delta _ 2 } + 2 \\beta ^ { ( 2 ) } _ { \\alpha , \\delta _ 2 } + \\frac { \\delta _ 2 } { 2 } \\right ) } - 1 , \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} p ( F ) ( \\mathcal { H } ) : = \\sum _ { \\phi : E ( F ) \\to [ k ] } \\prod _ { v \\in V ( F ) } h ^ { v } ( \\phi ( \\delta ( v ) ) ) . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} 2 \\imath \\psi _ { t } + \\psi _ { u u } + \\psi _ { v v } - \\omega ^ 2 ( u ^ 2 + v ^ 2 ) \\psi = 0 . \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} \\nabla ^ { W ( a ) } _ { X } \\tilde w _ a = ( A ^ 0 ( X ) _ a ^ a + i \\mu _ a \\omega ( X ) ) \\tilde w _ a , \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} \\Bigr \\{ \\bigr [ \\ , 0 : 1 : 0 \\ , \\bigr | \\ , \\Delta _ { 4 5 6 } : 0 : \\Delta _ { 4 6 7 } : 0 \\ , \\bigr ] \\Bigr \\} \\Bigr \\{ \\bigr [ \\ , \\Delta _ 1 : \\Delta _ 2 : 0 \\ , \\bigr | \\ , 0 : \\Delta _ { 4 5 7 } : \\Delta _ { 4 6 7 } : 0 \\ , \\bigr ] \\ , \\Bigr | \\ , \\Delta _ 2 \\Delta _ { 4 5 7 } = \\Delta _ 1 \\Delta _ { 4 6 7 } \\ , \\Bigr \\} . \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} \\mathfrak l _ 0 = \\mathfrak t \\subset \\mathfrak l _ 1 \\subset \\mathfrak l _ 2 \\subset \\ldots \\subset \\mathfrak l _ n = \\mathfrak g . \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} \\alpha _ 1 ( ( ^ { g _ i } c ) \\otimes q ) = \\big ( g _ i \\ , c \\ , \\phi ( c ) ^ { - 1 } g _ i ^ { - 1 } ( U \\cap G _ 1 ) ^ \\prime \\big ) \\otimes q , \\qquad \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } \\tilde { J } _ { 3 } ( t ) g \\right \\| _ { 2 } & \\le C t ^ { 1 - \\ell } ( 1 + t ) ^ { - n ( \\frac { 1 } { r _ { 1 } } - \\frac { 1 } { 2 } ) - ( k - \\tilde { k } _ { 3 } ) } \\| \\nabla ^ { \\tilde { k } _ { 3 } } _ { x } g \\| _ { r _ { 1 } } \\\\ & + C e ^ { - c t } t ^ { ( 1 - \\ell ) - n ( \\frac { 1 } { r _ { 2 } } - \\frac { 1 } { 2 } ) - ( k - \\tilde { k } _ { 4 } ) } \\| \\nabla ^ { \\tilde { k } _ { 4 } } _ { x } g \\| _ { r _ { 2 } } , \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} a _ n ^ { ( \\ell ) } = \\ell a _ { n - 1 } ^ { ( \\ell ) } - a _ { n - 2 } ^ { ( \\ell ) } , \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} L ( x , t ) = - e ^ { i \\phi ( x ) } \\big ( i \\Delta \\phi ( x ) - | \\nabla \\phi ( x ) | ^ { 2 } + 2 y \\cdot \\nabla \\phi ( x ) + 2 A ( x ) \\cdot y - 2 A \\cdot \\nabla \\phi ( x ) + h ( x , t ) \\big ) . \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} X : = b ! \\left ( \\overset { \\infty } { \\sum _ { n = 0 } } \\frac { \\chi ( n ) } { n ! } - \\overset { b } { \\sum _ { n = 0 } } \\frac { \\chi ( n ) } { n ! } \\right ) = b ! \\overset { \\infty } { \\sum _ { n = b + 1 } } \\frac { \\chi ( n ) } { n ! } > 0 , \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} F _ k ( s ) - \\widetilde { F } _ k ( s ) = & \\frac { 1 } { k ! } \\ ( \\ ( F ( s , \\chi _ 0 ) + F ( s , \\chi _ a ) \\ ) ^ k - F ^ k ( s , \\chi _ 0 ) \\ ) \\\\ & + \\frac { 1 } { k ! } \\sum _ { m = 0 } ^ { k - 2 } \\ ( \\ ( F ( s , \\chi _ 0 ) + F ( s , \\chi _ a ) \\ ) ^ m F _ { \\boldsymbol { n _ m } } ( s ; a ) - F ^ m ( s , \\chi _ 0 ) F _ { \\boldsymbol { n _ m } } ( s ; \\chi _ 0 ) \\ ) . \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} [ 8 - 3 E + \\frac { 1 } { 2 } ( - 3 E ) ^ 2 - 2 c _ 2 ( X ) - \\frac { 9 } { 2 } E ^ 2 + \\dots ] [ 1 + \\frac { 1 } { 1 2 } c _ 2 ( X ) + \\dots ] = \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} \\begin{cases} \\dot { u } ^ 0 + ( \\lambda + V _ 0 ) u ^ 0 = f ( u ^ 0 ) , t > 0 , \\\\ u ^ 0 ( 0 ) = u ^ 0 _ 0 , \\end{cases} \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} g ( [ S , T ] , Z ) = g ( S , [ T , Z ] ) = 0 , \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} \\begin{array} { l l l l } L ( \\pi ) = & \\max \\ & c ' v - \\pi ' ( H v - h ) & \\\\ & \\mbox { s . t . } \\ & A v \\leq b , & \\\\ & & v \\in \\{ 0 , 1 \\} ^ n & \\end{array} \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} = 3 ^ a \\cdot \\frac { q - 1 } { s } ( \\delta _ { x , ( z y ^ { - 1 } ) } \\cdot | C _ { N _ G ( P ) } ( x ) | ) , \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} R '' + \\frac { 1 } { r } R ' + \\left ( 2 \\epsilon - \\frac { p ^ 2 } { k ^ 2 r ^ 2 } \\right ) R = 0 , \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} - a _ c ( f , g , 0 ) = \\begin{pmatrix} ( R _ 3 + S _ 3 \\mu ) \\frac { f ^ 3 } { 3 } + S _ 3 \\mu \\frac { f ^ 2 g } { 2 } & S _ 3 \\mu \\left ( \\frac { f ^ 3 } { 3 } + \\frac { f ^ 2 g } { 2 } \\right ) & - \\mu \\frac { f ^ 2 } { 2 } \\sigma ^ \\prime ( 0 ) \\\\ S _ 3 \\frac { g ^ 3 } { 3 } + ( R _ 3 + S _ 3 \\mu ) \\frac { f ^ 2 g } { 2 } + S _ 3 \\mu f g ^ 2 & S _ 3 \\frac { g ^ 3 } { 3 } + S _ 3 \\mu \\left ( \\frac { f ^ 2 g } { 2 } + f g ^ 2 \\right ) & - \\left ( \\mu f g + \\frac { g ^ 2 } { 2 } \\right ) \\sigma ^ \\prime ( 0 ) \\\\ 0 & 0 & D \\end{pmatrix} . \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} B _ 1 ( n ) = \\frac { n } { 2 k _ n } \\log ( 1 + k _ n P ) . \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} \\| \\hat { p } ^ { ( k ) } ( \\cdot | Z ^ n ) - \\mu _ k ^ { ( b ) } \\| _ 1 & \\leq \\sum _ { a ^ k } \\Big | { t \\over n - k } \\sum _ { i = 1 } ^ { k + g } \\hat { p } ^ { ( 1 ) } ( a ^ k | S ^ { ( i ) , t } ) - \\mu _ k ^ { ( b ) } ( a ^ k ) \\Big | + { k + g \\over n - k } . \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} Q _ 1 = 1 , ~ Q _ 2 = t , ~ Q _ 3 = u ' , \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { \\infty } \\min \\left \\{ 1 - \\frac { 1 } { g _ { \\phi _ { n } } ^ { ' } ( 0 ) } , 1 - \\frac { \\alpha } { | | g _ { \\phi _ { n } } | | _ { 2 } } \\right \\} < \\infty . \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} L _ \\xi e ^ { 2 \\xi } & = \\frac 1 4 i \\xi \\eta \\zeta ( \\xi - \\eta ) e ^ { 2 \\xi } + \\frac 1 { 1 6 } i \\xi ^ 2 \\eta ( J ( \\zeta ) ^ 2 - \\zeta ^ 2 ) J ( \\zeta ) ^ { - 1 } e ^ { 2 \\xi } + \\xi \\eta \\zeta E S ( 1 ) . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{gather*} \\hat { L } _ 0 = \\frac { x ^ 2 } { 2 } \\sigma _ 3 + x \\begin{pmatrix} 0 & u \\\\ u & 0 \\end{pmatrix} + \\begin{pmatrix} - \\dfrac { t } { 2 } - u ^ 2 & - u _ t \\\\ u _ t & \\dfrac { t } { 2 } + u ^ 2 \\end{pmatrix} \\end{gather*}"}
-{"id": "36.png", "formula": "\\begin{align*} \\frac { 1 } { \\sum _ { l = 0 } ^ t \\alpha _ t } \\delta _ { t } , \\ ; \\delta _ { t } \\ ; \\leq \\ ; \\beta _ { t + 1 } D + \\frac { 1 } { 2 } \\sum _ { l = 0 } ^ t \\frac { \\alpha _ l ^ 2 } { \\beta _ l } \\| g _ l \\| _ { * } ^ 2 . \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} \\phi _ { ( i ; I ) } ( X ^ { 2 ^ { d - 1 } - 1 } z ^ { 2 ^ { d - 1 } } ) = \\phi _ { ( i ; I ) } ( X ^ { 2 ^ { d - 1 } - 1 } ) f _ i ( z ^ { 2 ^ { d - 1 } } ) . \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } u \\ , ( - \\Delta ) ^ s \\varphi \\ d x = \\int _ { \\R ^ N } ( - \\Delta ) ^ { m } ( - \\Delta ) ^ \\sigma u \\ , \\varphi \\ d x \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} \\wp = F - \\mathrm { i d } : W _ n ( K ^ { \\mathrm { s e p } } ) \\to & W _ n ( K ^ { \\mathrm { s e p } } ) \\\\ x \\mapsto & F x - x , \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} F ( s , \\chi ) = \\log L ( s , \\chi ) + G ( s , \\chi ) , \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} \\sum _ k Z _ { i k } Y _ { j k } = \\delta _ { i j } \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} \\dot x ( t ) = - \\tan \\left ( { \\frac { \\pi } { 8 } { u ^ 3 } ( t ) + t } \\right ) , t \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "6730.png", "formula": "\\begin{align*} u _ m = \\frac { \\epsilon } { 2 \\sqrt { c } } ( P + \\frac { 2 c } { c + 1 } P ^ { - 1 } ) , \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} \\left \\langle h _ { 1 } , . . . , h _ { \\mathsf { N } } \\right \\vert \\mathcal { T } ( \\zeta _ { n } ^ { ( h _ { n } ) } ) | \\tau \\rangle = \\tau ( \\zeta _ { n } ^ { ( h _ { n } ) } ) \\langle h _ { 1 } , . . . , h _ { \\mathsf { N } } | \\tau \\rangle \\forall n \\in \\{ 1 , . . . , \\mathsf { N } \\} \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} H _ D = \\boldsymbol { \\sigma } \\cdot ( \\boldsymbol { p } - e \\boldsymbol { A } ( \\boldsymbol { x } ) ) \\ ; , \\end{align*}"}
-{"id": "1350.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": "5580.png", "formula": "\\begin{align*} H _ k = - \\frac { ( - 1 ) ^ k } \\pi \\int _ { \\R } \\xi ^ k \\ln | T ( - \\xi / 2 ) | \\ , d \\xi = - \\frac 1 \\pi \\int _ { \\R } \\xi ^ k \\real \\ln T ( \\xi / 2 ) \\ , d \\xi \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\lim _ { \\alpha \\to \\infty } \\tfrac { 2 \\nu { \\alpha \\omega , B } } { \\alpha } ( E ) = - \\mu _ { \\omega , B } ^ { - 1 } ( E ) . \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} \\lim _ { z \\rightarrow 0 } z ^ { 1 - a - b } \\cdot \\left ( z ^ { a + b - 1 } \\cdot U ( b , a + b ; z ) \\right ) = 0 . \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} \\lim _ { d ( x ) \\rightarrow 0 } N \\gamma ( x ) = L \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow 0 } ( h ( t ) , h ' ( t ) \\cdot ( X , z ) ) & = \\lim _ { t \\rightarrow 0 } t ^ { ( ( \\widetilde { c } - c ) \\theta + ( \\widetilde { c ' } - c ' ) \\theta ' ) } \\cdot Z \\\\ & = 0 \\in \\mathbb { X } \\times \\{ 0 \\} , \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} ( P _ 1 , P _ 2 ) _ { L ^ 2 ( \\mu ) } : = \\mu ( P _ 2 ^ * P _ 1 ) = ( P _ 1 \\Omega , P _ 2 \\Omega ) _ { \\mathcal { F } _ q ( \\mathcal { H } ) } , P _ 1 , P _ 2 \\in \\mathcal { P } . \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} | \\mathrm { c m } ( x , x + y ) | = \\frac { \\left | [ x + y , \\beta x - ( x + y ) ] \\right | } { | | \\beta x - ( x + y ) | | _ a | | x + y | | } = \\frac { \\beta \\left | [ y , x ] \\right | } { | | \\beta x - ( x + y ) | | _ a | | x + y | | } ; \\ \\mathrm { a n d } \\\\ \\left | \\mathrm { c m } ( y , x + y ) \\right | = \\frac { \\left | [ x + y , \\alpha y - ( x + y ) ] \\right | } { | | \\alpha y - ( x + y ) | | _ a | | x + y | | } = \\frac { \\alpha \\left | [ x , y ] \\right | } { | | \\alpha y - ( x + y ) | | _ a | | x + y | | } . \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} v ( x , t ) = \\frac M N u \\left ( 2 x , t _ 0 + \\left ( \\frac M N \\right ) ^ { m - 1 } t \\right ) , \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} c _ { c u b i c } ^ { \\ast } = \\frac { M _ 1 + M _ 2 } { 2 } + \\frac { 2 k D ( 1 - 2 \\widehat { p } ) } { M _ 2 - M _ 1 \\pm \\sqrt { ( M _ 2 - M _ 1 ) ^ 2 + 8 k D } } \\ , . \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} x _ 1 ^ 3 - 3 a x _ 1 ^ 2 - 2 b x _ 1 - c = - ( x _ 2 ^ 3 - 3 a x _ 2 ^ 2 - 2 b x _ 2 - c ) = \\frac { 1 } { x _ 1 - x _ 2 } , x _ 1 + x _ 2 = c x _ 1 x _ 2 . \\end{align*}"}
-{"id": "4428.png", "formula": "\\begin{align*} P ( \\mu _ { \\epsilon } ) - P ( \\mu ) & = \\int _ { 0 } ^ { 1 } ( \\xi '' - \\frac { 1 } { \\phi _ { \\epsilon } \\phi } ) ( \\phi _ { \\epsilon } - \\phi ) \\ , d s + h ^ { 2 } ( \\phi _ { \\epsilon } ( 0 ) - \\phi ( 0 ) ) \\\\ & = ( \\epsilon - \\phi _ { \\mu } ( 1 - \\epsilon ) ) \\left [ \\xi ' ( 1 - \\epsilon ) + h ^ { 2 } - \\int _ { 0 } ^ { 1 - \\epsilon } \\frac { 1 } { \\phi _ { \\epsilon } \\phi } \\right ] + \\int _ { 1 - \\epsilon } ^ { 1 } ( \\xi '' - \\frac { 1 } { \\phi _ { \\epsilon } \\phi } ) ( 1 - t - \\phi ) \\ , d s \\\\ & = ( i ) + ( i i ) . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} \\beta \\psi ( s ) \\in \\ker ( \\mathcal { B } - 1 _ 4 ) ^ \\perp = \\ker ( \\mathcal { B } + 1 _ 4 ) \\ , , \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} | z _ n | > \\prod _ { j = 0 } ^ { n - 1 } | z _ { j } | ^ { \\rho } , \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} \\left ( \\bigcup _ { s = s _ i } ^ { t _ i } S _ s \\right ) \\cap \\left ( \\bigcup _ { s = s _ j } ^ { t _ j } S _ s \\right ) , \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} R _ \\mathsf { R L N } \\left ( p _ X , p _ { A | S } , p _ { B | A } \\right ) = \\min \\Big \\{ I ( A ; S _ 1 | B ) - I ( A ; S _ 2 | B ) , I ( X ; Y ) - I ( A ; S | S _ 1 ) \\Big \\} , \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} u ( x , t ) = \\min \\left [ u _ { 0 } ( x ) + t \\int _ 0 ^ 1 g ( s ) d s , \\ u _ 0 ( \\bar x ) + t \\left ( \\int _ 0 ^ 1 \\frac { 1 } { g ( s ) } d s \\right ) ^ { - 1 } \\right ] \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} p ^ { \\min \\{ n _ c , n \\} } ( 2 g _ n - 2 ) = & p ^ { n } ( 2 g _ 0 - 2 ) + \\sum _ { x } [ k ( x ) k ' : k ' ] \\sum _ { j = 0 } ^ n \\varphi ( p ^ j ) u _ { x , j } . \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} ( S \\otimes _ { \\mathcal { O } } \\mathcal { O } G ) ( \\Delta _ { \\varphi _ x } ( Q ) ) & = S ( \\Delta _ { \\varphi _ x } ( Q ) ) \\otimes _ k ( \\mathcal { O } G ) ( \\Delta _ { \\varphi _ x } ( Q ) ) \\\\ & = \\bigoplus _ { t \\in [ C _ G ( Q ) / C _ N ( Q ) , z = x t ] } ( S ( \\Delta _ { \\varphi _ x } ( Q ) ) \\otimes _ k \\bar { N } _ { \\mathcal { O } N z } ^ { \\varphi _ x } ( Q ) ) . \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} m _ L ( V ) = \\mathrm { d i m } _ E ( V _ { I _ L } ) ^ { G _ L } . \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} & \\underset { \\tau \\to + 0 } { \\mbox { { \\rm e s s l i m i n f } } } \\int _ { B ( 0 , R ) } \\int _ { { \\bf R } ^ N } G ( x - y , t - \\tau ) u ( y , \\tau ) \\eta ( x ) \\ , d x \\ , d y \\\\ & \\qquad \\ge \\int _ { { \\bf R } ^ N } \\eta ( y ) \\ , d \\mu ' ( y ) - C \\| \\eta ( t ) - \\eta \\| _ { L ^ \\infty ( B ( 0 , R ) ) } \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} \\| f \\| _ { B ^ { s , p } _ q } ^ q = \\sum \\limits _ { j \\geq 0 } \\| \\langle D \\rangle ^ s f _ j \\| _ { L ^ p } ^ q , \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} d ^ n f ( s _ 1 , \\ldots , s _ { n + 1 } ) : = \\kappa _ { s _ 1 } f ( s _ 2 , \\ldots , s _ { n + 1 } ) & + \\sum _ { i = 1 } ^ n { ( - 1 ) ^ n f ( s _ 1 , \\ldots , s _ i + s _ { i + 1 } , \\ldots , s _ { n + 1 } ) } \\\\ & + ( - 1 ) ^ { n + 1 } f ( s _ 1 , \\ldots , s _ n ) \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} \\mathcal { E } ( n , m , \\ell ) & = \\Lambda \\left ( h _ { i , j } , w _ { i ^ { \\prime } , j ^ { \\prime } } : 1 \\leq i \\leq m , 1 \\leq i ^ { \\prime } \\leq \\ell , 0 \\leq j < n / 2 , 0 \\leq j ^ { \\prime } \\leq n / 2 \\right ) \\\\ & \\subseteq E _ 0 \\mathcal { K } ( n , m ) \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{gather*} u ( t , x , y + h \\pi _ { t } ) = \\int _ { 0 } ^ { \\sigma _ 1 } [ D ^ { 2 } _ { x } u ( s , x , y ) + f ( s , x , y ) ] \\ , d s \\\\ + u ( \\sigma _ 1 , x , y + h ) - u ( \\sigma _ 1 , x , y ) + \\int _ { \\sigma _ 1 } ^ { t } [ D ^ { 2 } _ { x } u ( s , x , y + h ) + f ( s , x , y ) ] \\ , d s . \\end{gather*}"}
-{"id": "5286.png", "formula": "\\begin{align*} \\nabla H _ { \\geq 5 } ( u ) = \\pi _ 0 [ ( \\partial _ u f ) ( x , u , u _ x ) ] - \\partial _ x \\{ ( \\partial _ { u _ x } f ) ( x , u , u _ x ) \\} . \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} \\int _ { \\omega _ k } | A _ k - D g _ k ( x ) | d x & \\leq \\sum _ { \\ell = 1 } ^ { N _ k } \\int _ { \\omega _ { k , \\ell } } \\left ( | A _ k - M _ { k , \\ell } | + | M _ { k , \\ell } - D g _ k ( x ) | \\right ) d x \\\\ & \\lesssim \\sum _ { \\ell = 1 } ^ { N _ k } \\int _ { \\omega _ { k , \\ell } } | A _ k - M _ { k , \\ell } | d x = \\sum _ { \\ell = 1 } ^ { N _ k } \\lambda _ { k , \\ell } | A _ k - M _ { k , \\ell } | | \\omega _ { k } | \\lesssim j ^ { - 2 } | \\omega _ { k } | , \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} \\frac { d \\mu _ r } { d \\mu } ( n + m ) & = \\left ( [ n + m ] _ { | q | } ! \\right ) ^ { - 2 } r ^ { - ( n + m ) } \\le \\left ( [ n ] _ { | q | } ! \\ , [ m ] _ { | q | } ! \\right ) ^ { - 2 } r ^ { - ( n + m ) } \\\\ & = \\frac { d \\mu _ r } { d \\mu } ( n ) \\frac { d \\mu _ r } { d \\mu } ( m ) . \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} M ( D , D _ 1 , S ) - M ( D , D , S ) = M ( D , D _ 1 , - p ) . \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} \\sum _ k 2 ^ { n - 2 k } { n - 2 \\choose k - 1 } { n - k - 1 \\choose n - 2 k } = { 2 n - 4 \\choose n - 2 } = \\frac { n } { 4 n - 6 } { 2 n - 2 \\choose n } . \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\Phi _ j ^ * \\eta = d z + y _ j d x _ j + \\sum _ { i \\neq j } x _ i d y _ i . \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} p e r m ( B _ \\gamma + C _ \\gamma ) = \\sum _ { \\alpha \\subset \\gamma } p e r m ( B _ \\alpha ) p e r m ( C _ { \\bar \\alpha } ) \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} r _ 1 + r _ 2 + r _ 3 + r _ 4 = 0 \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} \\iota ^ * _ \\pm \\circ \\lambda _ { S _ X } = \\lambda _ X \\circ \\iota ^ * _ \\pm . \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} \\left \\| \\partial _ { t } ^ { \\ell } \\nabla _ { x } ^ { k } E _ { 3 } ( t ) g \\right \\| _ { 2 } & \\le C t ^ { 1 - \\ell } ( 1 + t ) ^ { - n ( \\frac { 1 } { r _ { 1 } } - \\frac { 1 } { 2 } ) - ( k - \\tilde { k } _ { 3 } ) } \\| \\nabla ^ { \\tilde { k } _ { 3 } } _ { x } g \\| _ { r _ { 1 } } \\\\ & + C e ^ { - c t } t ^ { ( 1 - \\ell ) - n ( \\frac { 1 } { r _ { 2 } } - \\frac { 1 } { 2 } ) - ( k - \\tilde { k } _ { 4 } ) } \\| \\nabla ^ { \\tilde { k } _ { 4 } } _ { x } g \\| _ { r _ { 2 } } , \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} \\| P _ n \\| _ E \\le \\sum _ { k = 0 } ^ n | A _ k | \\| B _ k \\| _ E \\le ( n + 1 ) \\max _ { 0 \\le k \\le n } | A _ k | \\ , \\max _ { 0 \\le k \\le n } \\| B _ k \\| _ E , \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} v = 2 \\left ( \\frac { d _ 1 + 3 } { 3 } \\right ) . \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} \\int _ \\Omega p _ \\varepsilon | \\nabla u ^ \\varepsilon | ^ 2 \\ , d x & + \\int _ \\Omega ( \\lambda + V _ \\varepsilon ) u ^ \\varepsilon ( u ^ \\varepsilon - u ^ 0 ) \\ , d x - \\int _ \\Omega ( \\lambda + V _ 0 ) u ^ 0 ( P u ^ \\varepsilon - u ^ 0 ) \\ , d x \\\\ & = \\| u ^ \\varepsilon - u ^ 0 \\| _ { X _ \\varepsilon ^ \\frac { 1 } { 2 } } ^ 2 + \\int _ \\Omega ( V _ \\varepsilon - V _ 0 ) u ^ 0 ( u ^ \\varepsilon - u ^ 0 ) \\ , d x . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} \\mathbb { P } _ { Q ^ { ( \\mathsf { B } _ n ) } } \\Big ( \\mathbf { U } ( I , \\mathsf { B } _ n ) \\notin \\mathcal { T } _ \\epsilon ^ n ( Q _ U ) \\Big ) = 2 ^ { - n R _ 1 } \\sum _ { i \\in \\mathcal { I } _ n } \\mathbb { P } _ { Q ^ { ( \\mathsf { B } _ n ) } } \\Big ( \\mathbf { U } ( i , \\mathsf { B } _ n ) \\notin \\mathcal { T } _ \\epsilon ^ n ( Q _ U ) \\Big | I = i \\Big ) \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} E : y ^ 2 + x y = x ^ 3 - x ^ 2 - 1 9 1 5 1 5 2 x + 7 1 3 1 0 6 9 1 7 \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ s \\bigl ( \\| f _ j \\| ^ 2 _ { 1 , } + \\frac { 1 } { \\gamma _ j } \\| f _ j \\| ^ 2 _ { 2 , ' } \\bigr ) = 1 \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} \\int _ M u ^ 2 & \\leq c \\int _ M | \\nabla u | ^ 2 + c \\left ( \\bar { u } _ 0 ^ 2 + \\sum _ { \\ell = 1 } ^ L ( \\bar { u } _ \\ell ' ) ^ 2 + \\sum _ { \\ell = 1 } ^ L ( \\bar { u } _ \\ell '' ) ^ 2 \\right ) \\\\ & \\leq c \\int _ M | \\nabla u | ^ 2 + c \\left ( \\sum _ { \\ell = 1 } ^ L ( \\bar { u } _ \\ell ' - \\bar { u } _ \\ell '' ) ^ 2 + \\sum _ { \\ell = 1 } ^ L ( \\bar { u } _ \\ell '' - \\bar { u } _ 0 ) ^ 2 \\right ) . \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\begin{aligned} f ( a ) = \\sum _ { j = 2 } ^ { k _ 1 } \\left \\{ p _ { j } a + \\xi _ { j } \\right \\} + \\sum _ { j = k _ { 1 } + 1 } ^ { k } \\left \\{ p _ { j } a \\right \\} & = \\sum _ { j = 2 } ^ { k _ 1 } \\left ( p _ { j } a + \\xi _ { j } \\right ) + \\sum _ { j \\in \\mathcal { K } _ { 0 } ^ { + } } p _ { j } a + \\sum _ { j \\in \\mathcal { K } _ { 0 } ^ { - } } \\left ( 1 + p _ { j } a \\right ) \\\\ & = k _ { 0 } ^ { - } + \\sum _ { j = 2 } ^ { k } p _ { j } a + \\sum _ { j = 2 } ^ { k _ 1 } \\xi _ { j } , \\end{aligned} \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} 0 & = L \\left ( [ e _ { i i } , e _ { k l } ] \\right ) - \\delta _ { i k } L ( e _ { i l } ) + \\delta _ { i l } L ( e _ { k i } ) \\\\ & = L ( e _ { i i } ) e _ { k l } - e _ { k l } L ( e _ { i i } ) + e _ { i i } L ( e _ { k l } ) - L ( e _ { k l } ) e _ { i i } \\\\ & - \\delta _ { i k } L ( e _ { i l } ) + \\delta _ { i l } L ( e _ { k i } ) . \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} z y _ { \\rm o d d } = 2 \\partial _ x L y _ { \\rm e v e n } , - z y _ { \\rm e v e n } = 2 \\partial _ x L y _ { \\rm o d d } . \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} p \\sum _ { k = p - 1 } ^ { p - 1 } \\sum _ { m = 0 } ^ { k } \\cdots & = \\frac { 1 } { i + j + t + 1 } \\binom { ( i + j + t + 1 ) p } { ( i + j + 1 ) p } \\sum _ { m = 0 } ^ { k } \\binom { p - 1 + ( i + j ) p } { m + i p } \\\\ & \\equiv _ { p ^ 3 } \\frac { 1 } { i + j + 1 } \\binom { i + j + t } { i + j } \\binom { i + j } { i } 2 ^ { ( p - 1 ) ( i + j + 1 ) } \\\\ & \\equiv _ { p ^ 3 } \\binom { i + j + t } { i , j , t } \\left ( \\frac { 1 } { i + j + 1 } + p q _ p ( 2 ) + \\frac { ( i + j ) } { 2 } \\ , p ^ 2 q ^ 2 _ p ( 2 ) \\right ) \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} H _ { \\Lambda , \\mu , \\lambda } = H _ { \\Lambda , \\mu } + H _ { \\Lambda } ^ { \\lambda } \\ , \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} 2 ( 1 2 + ( 2 6 - N ) ( 2 5 - N ) ) \\lambda ( N ) = H _ n ( N ) + 2 ( 2 6 - N ) H _ u ( N ) . \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} & d ( \\varphi _ i ( x _ 0 ) , \\mathcal { Y } ( x _ 0 ) g ) \\leq d ( \\varphi _ i ( x _ 0 ) , \\mathcal { Y } _ 0 ( x _ 0 h _ n ) ) + d ( g _ n , g ) \\\\ = & d ( \\varphi _ i ( x _ 0 ) , \\mathcal { Y } ( x _ 0 ) h _ n ) + d ( g _ n , g ) \\\\ \\leq & d ( \\varphi _ i ( x _ 0 ) , \\varphi _ i ( x _ 0 ) h _ n ) + d ( g _ n , g ) \\\\ \\to & 0 & & \\ , \\ , n \\to \\infty . \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} e _ { ( m + 1 ) p _ n + r _ n } ( x ) = e _ { ( m + 1 ) p _ n - ( m - r _ n ) } ( x ) + x ^ { 2 m - 2 r _ n } e _ { ( m + 1 ) p _ n - r _ n } ( x ) + x ^ { m - 1 } e _ { ( m + 1 ) ( p _ n - 1 ) + r _ n } ( x ) . \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} \\langle \\xi ' ( 0 ) , \\omega \\rangle = - \\frac { p \\mathcal { A } ( u , \\omega _ 1 ) + p \\mathcal { A } ( v , \\omega _ 2 ) - K _ { \\lambda , \\mu } ( z , \\omega ) - 2 \\displaystyle \\int _ \\Omega ( \\alpha | u | ^ { \\alpha - 2 } u \\omega _ 1 | v | ^ \\beta + \\beta | u | ^ \\alpha | v | ^ { \\beta - 2 } v \\omega _ 2 ) d x } { ( p - q ) \\| ( u , v ) \\| ^ p - 2 ( \\alpha + \\beta - q ) \\displaystyle \\int _ { \\Omega } | u | ^ { \\alpha } | u | ^ { \\beta } d x } , \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} f _ { \\tau } \\in \\sum _ { i = 2 } ^ { l } C ( n _ { i - 1 } - n _ { 1 } , n _ { i } - n _ { 1 } , k _ { i } ) , & & f _ { \\tau } ' \\in \\sum _ { i = 2 } ^ { l } C ( n _ { i - 1 } - n _ { 1 } , n _ { i } - n _ { 1 } , k _ { i } ' ) , \\end{align*}"}
-{"id": "5451.png", "formula": "\\begin{align*} T : = \\int T _ { \\nu _ { \\gamma } } \\dd \\pi ( \\gamma ) \\ ; . \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} \\psi \\circ f \\leq \\phi & ~ \\Rightarrow ~ \\psi \\circ g \\circ \\eta _ X \\leq \\phi \\\\ & ~ \\Rightarrow ~ \\widehat { \\xi } \\circ \\eta _ X \\leq \\phi \\\\ & ~ \\Rightarrow ~ \\forall x \\in X , \\xi ( x ) \\leq \\phi ( x ) \\\\ & ~ \\Rightarrow ~ \\widehat { \\xi } ( \\phi ) = 0 . \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} i _ { p ^ n _ { U , W } } ( \\mathbf { u } , \\mathbf { w } ) & = \\sum _ { t = 1 } ^ n i _ { p _ { U , W } } ( u _ t , w _ t ) \\\\ i _ { p ^ n _ { U , V , W } } ( \\mathbf { u } , \\mathbf { v } , \\mathbf { w } ) & = \\sum _ { t = 1 } ^ n i _ { p _ { U , V , W } } ( u _ t , v _ t , w _ t ) , \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} \\mathcal { R } ^ { \\gamma } _ H = T _ { \\pi } \\circ \\mathcal { R } _ H \\circ \\gamma \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} w _ 0 = \\underbrace { s _ 1 } _ { \\tau _ 1 } \\underbrace { s _ 2 s _ 1 \\cdots } _ { \\tau _ 2 } \\ \\cdots \\ \\underbrace { s _ { j _ 1 } \\cdots s _ { j _ r } } _ { \\tau _ j } \\ \\cdots \\ \\underbrace { s _ { n _ 1 } \\cdots s _ { n _ t } } _ { \\tau _ n } . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} \\frac { d } { d t } \\Big | _ { t = 0 } \\Theta _ x & = D ( \\pi \\circ \\Phi _ x ) \\big | _ 0 \\ , \\frac { \\partial } { \\partial t } = D \\pi _ x v ( x ) , \\end{align*}"}
-{"id": "3089.png", "formula": "\\begin{align*} \\mu _ { 1 , c } ( x ) : = \\max \\{ x _ 2 , x _ 3 , c ( x _ 1 + x _ 2 + x _ 3 ) \\} , \\\\ \\mu _ { 2 , c } ( x ) : = \\max \\{ x _ 1 , x _ 3 , c ( x _ 1 + x _ 2 + x _ 3 ) \\} , \\\\ \\mu _ { 3 , c } ( x ) : = \\max \\{ x _ 1 , x _ 2 , c ( x _ 1 + x _ 2 + x _ 3 ) \\} . \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} N _ n : = N _ 0 ^ { \\chi ^ n } , n = 0 , 1 , 2 , \\dots , \\chi : = 3 / 2 , N _ 0 > 0 . \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} \\widetilde { \\lambda } ^ { \\epsilon , \\kappa } _ { \\gamma ^ * } \\ : = \\ \\big ( \\lambda ^ \\epsilon \\ , - \\epsilon a \\big ) \\ , \\sharp ^ { \\epsilon , \\kappa } \\ , \\tau _ { \\gamma ^ * } \\widetilde { \\chi } \\ ; . \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} Z ^ + = \\hat { I } ^ + _ 0 \\cap \\{ \\rho \\ge \\rho _ 0 \\} \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{gather*} \\alpha = \\frac { 3 } { 2 } \\frac { q _ { 2 t } } { q _ 2 } - \\frac { u _ t } { u } \\frac { ( 1 + q _ 2 ) ( 2 - q _ 2 ) } { 2 q _ 2 } , \\end{gather*}"}
-{"id": "8718.png", "formula": "\\begin{align*} \\sigma ( F _ { \\rho ^ { ( n ) } ( s ) } ( x ) ) ^ 2 \\ , \\rho ^ { ( n ) } ( s ) ( \\mathrm { d } x ) \\ ! = \\ ! 2 \\ , \\mathrm { d } \\Sigma _ n ( F _ { \\rho ^ { ( n ) } ( s ) } ( \\cdot ) ) \\ ! \\to \\ ! 2 \\ , \\mathrm { d } \\Sigma ( R ( s , \\cdot ) ) \\ ! = \\ ! \\sigma ( R ( s , x ) ) ^ 2 \\ , \\rho ( s ) ( \\mathrm { d } x ) , \\\\ s \\in [ 0 , t ] \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{align*} M _ { n - 1 } = \\bigg \\langle \\frac { a _ 1 } { p } , \\dots , \\frac { a _ { n - 1 } } { p ^ { n - 1 } } \\bigg \\rangle \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{align*} \\hat { b } _ { n , m } = \\bar { b } _ { - , n } ^ { p } - \\bar { b } _ { - , m } ^ { p } , \\forall n \\neq m \\in \\left \\{ 1 , . . . , N \\right \\} , \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} \\Delta ^ * ( \\hbar ) : = i ( \\Delta ) ( - \\hbar ) ^ t , \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} & \\left ( 1 + j ^ { - 2 } \\sum _ { k = 1 } ^ { \\ell - 1 } 2 ^ { n } C ( i + k ) ^ { - 2 } \\right ) \\left ( 1 + C \\left ( \\frac { i + \\ell + 2 } { i + \\ell + 1 } \\right ) ^ { n ' - m _ 2 ' } j ^ { - 2 } ( i + \\ell ) ^ { - 2 } \\right ) \\\\ & \\leq 1 + j ^ { - 2 } \\sum _ { k = 1 } ^ { \\ell - 1 } 2 ^ { n } C ( i + k ) ^ { - 2 } + 2 C 2 ^ { n - 1 } j ^ { - 2 } ( i + \\ell ) ^ { - 2 } \\leq 1 + j ^ { - 2 } \\sum _ { k = 1 } ^ { \\ell } 2 ^ { n } C ( i + k ) ^ { - 2 } . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} f ( z ) = ( \\log { N } ) ^ { - 1 } + z + \\dots \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} C _ 2 \\epsilon - C \\epsilon ^ \\frac { n - 2 \\alpha } { 2 } & = C _ 2 \\lambda ^ \\frac { 2 } { 2 - q } - C \\lambda ^ \\frac { n - 2 \\alpha } { 2 - q } = \\lambda ^ \\frac { 2 } { 2 - q } \\left ( C _ 2 - C \\lambda ^ \\frac { n - 2 \\alpha - 2 } { 2 - q } \\right ) . \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} R _ t = - B ( R ) _ x + \\Sigma ( R ) _ { x x } , R ( 0 , \\cdot ) = F _ \\lambda ( \\cdot ) , \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} T \\gamma \\left ( \\frac { \\partial } { \\partial S } \\right ) = \\frac { \\partial } { \\partial S } + \\frac { \\partial \\gamma } { \\partial S } \\frac { \\partial } { \\partial p } , T \\gamma \\left ( \\frac { \\partial } { \\partial q } \\right ) = \\frac { \\partial } { \\partial q } + \\frac { \\partial \\gamma } { \\partial q } \\frac { \\partial } { \\partial p } \\end{align*}"}