diff --git "a/data_tmp/process_28/tokenized_finally.jsonl" "b/data_tmp/process_28/tokenized_finally.jsonl"
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@@ -1,9965 +0,0 @@
-{"id": "3963.png", "formula": "\\begin{align*} \\lambda ^ { \\min } ( u ^ { - } ( r ^ { - 1 } ) v ) = \\lambda ^ { \\max } ( u ^ { - } ( r ^ { - 1 } ) v ) = \\lambda ^ { \\max } ( v ) . \\end{align*}"}
-{"id": "8411.png", "formula": "\\begin{align*} \\forall ~ \\overline { n } \\in G _ { k - 1 } ( \\mathbb { N } ) , ~ w ^ { * } - \\underset { k \\in \\mathbb { M } _ 1 } { l i m } ~ f ( \\overline { n } , k ) = g ( \\overline { n } ) \\in J _ p ^ { * * } \\end{align*}"}
-{"id": "3136.png", "formula": "\\begin{align*} Q ^ T L _ 9 ( \\lambda ) Q = \\left [ \\begin{array} { c c c | c } - P _ 4 & \\lambda P _ 4 & 0 & 0 \\\\ \\lambda P _ 4 & \\lambda P _ 3 - P _ 2 & \\lambda P _ 2 & - I _ n \\\\ 0 & \\lambda P _ 2 & \\lambda P _ 1 + P _ 0 & \\lambda I _ n \\\\ \\hline 0 & - I _ n & \\lambda I _ n & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "1319.png", "formula": "\\begin{align*} T _ { \\ell } ( \\alpha ) = \\sum _ { n = 1 } ^ { N } e ( n ^ { \\ell } \\alpha ) \\ , , \\end{align*}"}
-{"id": "3278.png", "formula": "\\begin{align*} \\begin{gathered} \\hat { R } ( v _ { 1 } \\otimes v _ { 1 } ) = q ^ { 2 } v _ { 1 } \\otimes v _ { 1 } , \\quad \\hat { R } ( v _ { 0 } \\otimes v _ { 1 } ) = v _ { 1 } \\otimes v _ { 0 } , \\quad \\hat { R } ( v _ { - 1 } \\otimes v _ { 1 } ) = q ^ { - 2 } v _ { 1 } \\otimes v _ { - 1 } , \\\\ \\hat { R } ( v _ { - 1 } \\otimes v _ { 0 } ) = v _ { 0 } \\otimes v _ { - 1 } , \\quad \\hat { R } ( v _ { - 1 } \\otimes v _ { - 1 } ) = q ^ { 2 } v _ { - 1 } \\otimes v _ { - 1 } . \\end{gathered} \\end{align*}"}
-{"id": "124.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 - 4 t ) ^ { \\tfrac { x + y } { 2 } } d \\mu _ { - 1 } ( y ) & = \\frac { 2 } { 1 + \\sqrt { 1 - 4 t } } \\sqrt { ( 1 - 4 t ) ^ x } \\\\ & = \\sum _ { n = 0 } ^ \\infty C _ n ( x ) t ^ n . \\end{align*}"}
-{"id": "7195.png", "formula": "\\begin{align*} v ' + \\frac { v ^ 2 } { 2 } = \\frac { C _ 1 } { ( 1 + t ) ^ 2 } , v ( 0 ) = 0 . \\end{align*}"}
-{"id": "4440.png", "formula": "\\begin{align*} \\tau _ 1 < \\min \\biggl ( \\frac { 2 \\alpha } { 5 \\alpha + 3 d } \\ , , \\ , \\frac { \\alpha - d } { 2 \\alpha } \\ , , \\ , \\frac { 4 \\beta ^ * } { 4 \\beta ^ * + 3 d } \\biggr ) , \\tau _ 2 : = \\min \\biggl ( 1 - \\frac { d / 4 } { 1 + \\lfloor d / 4 \\rfloor } , 1 - \\frac { d } { 2 \\beta } \\biggr ) \\end{align*}"}
-{"id": "3334.png", "formula": "\\begin{align*} \\langle z , \\Gamma _ { i } ^ { ( k ) } v \\rangle _ { k - 1 } = \\langle z \\wedge w _ { i } , v \\rangle _ { k } , v \\in \\Lambda _ { q } ^ { k } ( \\mathfrak { u } _ { + } ) , \\ z \\in \\Lambda _ { q } ^ { k - 1 } ( \\mathfrak { u } _ { - } ) . \\end{align*}"}
-{"id": "8224.png", "formula": "\\begin{align*} E W _ n - i \\gamma W _ n - \\Omega \\bar { W } _ n = \\bar { W } _ { n - 1 } + \\epsilon ^ 2 \\bar { W } _ { n + 1 } - 2 \\epsilon \\bar { W } _ n + 6 \\epsilon ^ { 2 | n | } | W _ n | ^ 2 \\bar { W } _ n + 2 \\epsilon ^ { 2 | n | } W _ n ^ 3 , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "2859.png", "formula": "\\begin{align*} K ^ n ( C ) = M ^ n K ^ { \\bullet } ( C ) \\oplus N ^ n K ^ { \\bullet } ( C ) \\end{align*}"}
-{"id": "8422.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\textbf { u } + \\sum _ { j = 1 } ^ { d } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } + F _ { P } = 0 , \\\\ & \\nabla \\cdot v = 0 , \\end{cases} \\end{align*}"}
-{"id": "7330.png", "formula": "\\begin{align*} \\psi ( v _ 1 ) = w _ { - 1 } , \\psi ( v _ 0 ) = - w _ { 0 } , \\psi ( v _ { - 1 } ) = q ^ 2 w _ 1 . \\end{align*}"}
-{"id": "1051.png", "formula": "\\begin{align*} \\lambda | A | \\leq \\sum _ { j = 0 } ^ \\infty \\int _ { \\R ^ 2 } | ( Q ^ j \\ast 1 _ E ) ( x ) | 1 _ A ( x ) \\ , d x \\leq C | E | ^ { \\frac { 1 1 } { 1 2 } } | A | ^ \\frac 3 4 , \\end{align*}"}
-{"id": "6146.png", "formula": "\\begin{align*} \\omega _ G = \\omega - \\mu \\end{align*}"}
-{"id": "2792.png", "formula": "\\begin{align*} \\log \\biggl ( \\frac { ( d - 1 ) ^ { d - 1 } } { d ^ { d / 2 } } \\biggr ) & = \\left ( 1 + 2 x / n \\right ) \\log \\left ( 1 + \\frac { 2 x } { n } \\right ) - \\left ( 1 + x / n \\right ) \\log \\left ( 2 + \\frac { 2 x } { n } \\right ) \\\\ & = - \\log 2 + \\left ( 1 - \\log 2 \\right ) \\frac { x } { n } + \\frac { 3 } { 2 } \\left ( \\frac { x } { n } \\right ) ^ { 2 } + O \\left ( \\left ( \\frac { x } { n } \\right ) ^ { 3 } \\right ) \\end{align*}"}
-{"id": "2439.png", "formula": "\\begin{align*} \\Vert \\Box _ { k } ^ { \\alpha _ { 2 } } F _ { k , N } \\Vert _ { M _ { 2 } } = \\Vert F _ { k , N } \\Vert _ { M _ { 2 } } = \\Vert F _ { k , N } \\Vert _ { L ^ { p _ 2 } } \\sim 2 ^ { j n ( \\alpha _ { 2 } - \\alpha _ { 1 } ) / p _ { 2 } } 2 ^ { j n \\alpha _ { 1 } ( 1 - 1 / p _ { 2 } ) } \\end{align*}"}
-{"id": "1265.png", "formula": "\\begin{align*} \\psi = - \\mathcal G _ 0 v ^ * \\phi + c _ 0 ( 1 , 0 ) ^ T = i \\alpha \\cdot \\nabla G _ 0 v ^ * \\phi - 2 m G _ 0 I _ 1 v ^ * \\phi + c _ 0 ( 1 , 0 ) ^ T , \\end{align*}"}
-{"id": "9258.png", "formula": "\\begin{align*} s u p _ { u \\in \\mathcal { A } } J _ { \\tilde { P } } ( u ) = J _ { \\tilde { P } } ( u ^ { \\ast } ) , \\end{align*}"}
-{"id": "146.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { C _ E } } = \\| S ( x ) \\| _ { \\widehat { E } } . \\end{align*}"}
-{"id": "6487.png", "formula": "\\begin{align*} A ( \\xi ) = \\int _ { \\mathcal { S } ^ { n - 1 } } | \\xi \\cdot \\theta | ^ { 2 } a ( \\theta ) d \\theta , \\end{align*}"}
-{"id": "2832.png", "formula": "\\begin{align*} A _ k = \\bigcap _ { i = 0 } ^ { [ T _ k - S _ k ] } \\left \\{ \\max _ { 0 \\le u \\le L _ i ^ k } X _ { r : n } ( a _ { i , u } ^ k ) \\le y _ i ^ k - \\theta _ i ^ k / y _ i ^ k \\right \\} . \\end{align*}"}
-{"id": "3273.png", "formula": "\\begin{align*} \\Delta ( \\mathfrak { l } ) = \\{ \\pm \\alpha _ 1 \\} , \\Delta ( \\mathfrak { u } _ + ) = \\{ \\alpha _ 2 , \\ \\alpha _ 1 + \\alpha _ 2 , \\ 2 \\alpha _ 1 + \\alpha _ 2 \\} . \\end{align*}"}
-{"id": "2834.png", "formula": "\\begin{align*} T _ k = \\exp ( k ^ { 1 / p } ) , S _ k = T _ k \\exp \\left ( - ( 1 + 2 \\varepsilon ) ^ 2 h _ p ( T _ k ) \\right ) . \\end{align*}"}
-{"id": "8107.png", "formula": "\\begin{align*} C _ { k , l , N } : = \\inf \\{ { \\mathcal R } _ { k , l , N } ( M ) : \\ , \\mbox { { \\sl $ M $ i s m e a s u r a b l e w i t h $ 0 < \\mu _ l ( M ) < + \\infty $ . } } \\} \\end{align*}"}
-{"id": "1451.png", "formula": "\\begin{align*} \\Omega _ k : = \\bigcup _ { j \\in J _ k } Q ^ k _ j . \\end{align*}"}
-{"id": "3044.png", "formula": "\\begin{align*} d ( u , v ) = 2 ^ { 1 - \\min ( u ( | u \\wedge v | + 1 ) , v ( u \\wedge v | + 1 ) ) - \\sum _ { j = 1 } ^ { | u \\wedge v | } u ( j ) } , \\end{align*}"}
-{"id": "8799.png", "formula": "\\begin{align*} - \\int _ { \\partial \\Sigma } \\Pi ( N , N ) d s = \\sum _ { i = 1 } ^ { n + 1 } I ( f _ i , f _ i ) > - \\frac { m } { n + 1 } \\int _ { \\partial \\Sigma } \\Pi ( N , N ) d s \\ , . \\end{align*}"}
-{"id": "4891.png", "formula": "\\begin{align*} \\delta ( x - [ r _ 0 ] ) \\equiv \\delta ( \\sum _ { k = 1 } ^ \\nu p ^ k [ \\phi ^ { - k } ( r _ k ) ] ) \\bmod I ^ \\nu \\equiv \\delta ( p \\Big ( \\sum _ { k = 1 } ^ \\nu p ^ { k - 1 } [ \\phi ^ { - k } ( r _ k ) ] \\Big ) ) \\bmod I ^ \\nu \\end{align*}"}
-{"id": "3999.png", "formula": "\\begin{align*} b _ { H , a } = q ^ t + ( 1 - b _ { H , t } ) q ^ { t - a } - c _ H q ^ { d - a } , \\end{align*}"}
-{"id": "3514.png", "formula": "\\begin{align*} Z '' ( t ) = A ( t ) Z ' ( t ) + B ( t ) Z ( t ) , \\end{align*}"}
-{"id": "1010.png", "formula": "\\begin{align*} N & = 3 , & U & = 1 . 3 , & \\mu & = 0 . 5 , & t & = - 3 . 7 , \\\\ \\phi _ 1 & = \\pi / 3 , & \\theta _ 1 & = \\pi / 6 , & \\phi _ 2 & = \\pi / 4 , & \\theta _ 2 & = \\pi / 7 , \\end{align*}"}
-{"id": "2856.png", "formula": "\\begin{align*} U \\circ \\mathcal { B } _ { E _ 1 } = \\mathcal { B } _ { E _ 1 } ^ { \\mathbb { K } } . \\end{align*}"}
-{"id": "6383.png", "formula": "\\begin{align*} \\mu _ { ( k ) } = C ( \\varepsilon ) \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon ) ^ { n } } { n ! } \\mu _ { ( n p + k ) } ^ { G } , \\end{align*}"}
-{"id": "9758.png", "formula": "\\begin{align*} \\lambda _ { i } ^ { * } g _ { i } ( \\vec { z } ^ { * } , \\bar { \\vec { x } } ) = 0 , \\ i = 1 , \\ldots , m . \\end{align*}"}
-{"id": "8292.png", "formula": "\\begin{align*} \\frac { 1 } { S _ { N , p } } \\sum _ s A ( s ) { m \\choose p s } \\frac { ( p s ) ! } { ( p ! ) ^ s s ! } = 1 , \\end{align*}"}
-{"id": "8682.png", "formula": "\\begin{align*} \\sum _ { m \\ge 1 } \\sup _ { | a | _ U = 1 } | \\nabla _ k \\nabla _ { a } ^ G P _ t [ \\Phi _ m ] ( x ) | ^ 2 \\le \\frac { c } { t ^ 4 } | k | _ K ^ 2 \\ , \\| \\Phi \\| _ { \\infty } ^ 2 . \\end{align*}"}
-{"id": "6221.png", "formula": "\\begin{align*} \\mathbb { G } _ { j } ( \\mathcal { P } _ { 1 } , \\ \\mathcal { P } _ { 2 } , \\ \\mathsf { Q } ) & = \\mathbb { G } _ { j } ( - \\tilde { K } ^ { - 1 } ( 2 s _ { 0 } M + D ) , \\ - \\tilde { K } ^ { - 1 } M , \\ \\tilde { K } ^ { - 1 } F ) , \\\\ & = \\mathbb { G } _ { j } ( - \\tilde { K } ^ { - 1 } ( ( 2 s _ { 0 } + \\alpha ) M + \\beta K ) , \\ - \\tilde { K } ^ { - 1 } M , \\ \\tilde { K } ^ { - 1 } F ) , \\\\ & = \\mathbb { K } _ { j } ( \\mathcal { P } _ 1 , \\ \\mathsf { Q } ) , \\end{align*}"}
-{"id": "5319.png", "formula": "\\begin{align*} \\lambda _ { \\{ l ; k \\} } ( u ) = \\left ( u + \\omega + \\eta \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) \\left ( u - \\omega + \\eta \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) + \\eta ^ { - 2 } \\end{align*}"}
-{"id": "5103.png", "formula": "\\begin{align*} H _ { 1 } = M + \\sum _ { i = 1 } ^ { M } h _ { i - 1 , i } , h _ { i - 1 , i } = \\sum _ { a = 1 } ^ { r } ( \\beta _ { a , i - 1 } ^ { * } - \\beta _ { a , i } ^ { * } ) \\beta _ { a , i } q ^ { 2 \\sum _ { p = a + 1 } ^ { r } N _ { a , i } } , \\end{align*}"}
-{"id": "2304.png", "formula": "\\begin{align*} \\begin{aligned} \\| ( e _ n ) _ { n = 0 } ^ m \\| _ { L ^ \\infty ( X ) } & \\le C \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ 1 ( X ) } + C \\delta , \\end{aligned} \\end{align*}"}
-{"id": "2024.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 [ f ( x + x z ) - f ( x ) - f ' ( x ) x z ] \\nu _ U ( \\d z ) = x ^ \\beta \\int _ { - 1 } ^ 1 [ ( 1 + z ) ^ \\beta - 1 - \\beta z ] \\nu _ U ( \\d z ) + o ( x ^ { \\beta } ) . \\end{align*}"}
-{"id": "2855.png", "formula": "\\begin{align*} E _ 1 ( A ) = \\bigoplus _ { n \\geq 1 } ( E _ 1 ( n ) \\otimes A ^ { \\otimes n } ) _ { \\Sigma _ n } . \\end{align*}"}
-{"id": "402.png", "formula": "\\begin{align*} W _ \\phi ( w _ 1 , w _ 2 ) & = \\frac { 2 ^ { w _ 2 - 3 } } { \\pi ^ { \\frac { 1 + w _ 1 + w _ 2 } { 2 } } } \\Gamma \\left ( 1 - w _ 2 \\right ) \\cos \\left ( \\frac { \\pi } { 2 } ( 1 - w _ 2 ) \\right ) \\\\ & \\times \\Gamma \\left ( \\frac { - 1 + w _ 1 + 3 w _ 2 + 2 i t _ \\phi } { 4 } \\right ) \\Gamma \\left ( \\frac { - 1 + w _ 1 + 3 w _ 2 - 2 i t _ \\phi } { 4 } \\right ) \\end{align*}"}
-{"id": "7342.png", "formula": "\\begin{align*} \\langle w _ 0 \\wedge w _ 1 , v _ 1 \\wedge v _ 0 \\rangle = \\langle w _ { - 1 } \\wedge w _ 0 , v _ 0 \\wedge v _ { - 1 } \\rangle = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } , \\langle w _ { - 1 } \\wedge w _ 1 , v _ 1 \\wedge v _ { - 1 } \\rangle = \\frac { 1 } { [ 2 ] _ { q ^ 2 } } , \\end{align*}"}
-{"id": "8368.png", "formula": "\\begin{align*} f v f = f p v f = g q ( 1 - p ) p v f = 0 . \\end{align*}"}
-{"id": "5269.png", "formula": "\\begin{align*} \\vert I \\vert : = \\vert \\int _ { T } ^ { 2 T } Z ( t ) d t \\vert = \\int _ { T } ^ { 2 T } \\vert Z ( t ) \\vert d t . \\end{align*}"}
-{"id": "5623.png", "formula": "\\begin{align*} w = 0 1 2 2 , ~ \\overline { w } = 2 2 1 0 \\end{align*}"}
-{"id": "5538.png", "formula": "\\begin{align*} V _ t = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} V \\end{align*}"}
-{"id": "4591.png", "formula": "\\begin{align*} \\mathsf { W } _ S ^ n = \\mathsf { W } _ { C _ 1 } ^ n + \\hdots + \\mathsf { W } _ { C _ k } ^ n . \\end{align*}"}
-{"id": "8939.png", "formula": "\\begin{align*} \\int _ a ^ b f ( s ) ^ p d s = & - p \\int _ a ^ b s \\varphi ' ( s ) \\underline g ( \\varphi ( s ) ) f ( s ) ^ { p - 1 } d s + b f ( b ) ^ p - a f ( a ) ^ p . \\end{align*}"}
-{"id": "6535.png", "formula": "\\begin{align*} E _ m \\le C \\sum _ { n = k } ^ { m - 1 } a _ n E _ n + C F _ m , m = k , \\dotsc , N , \\end{align*}"}
-{"id": "9363.png", "formula": "\\begin{align*} \\frac { h _ 0 } { t + 1 } + \\frac { C } { 2 } & = \\frac { h _ 0 } { \\sqrt { t + 1 } } \\left ( \\frac { 1 } { \\sqrt { t + 1 } } + \\frac { C } { 2 h _ 0 } \\sqrt { t + 1 } \\right ) \\\\ & \\leq \\frac { h _ 0 } { \\sqrt { t + 1 } } \\left ( \\frac { 1 } { \\sqrt { t + 1 } } + \\sqrt { \\frac { C } { 2 h _ 0 } } \\right ) \\\\ & \\leq \\frac { h _ 0 } { \\sqrt { t + 1 } } \\left ( \\frac { 1 } { \\sqrt { t + 1 } } + 1 \\right ) \\\\ & \\leq \\frac { 2 h _ 0 } { \\sqrt { t + 1 } } . \\end{align*}"}
-{"id": "9763.png", "formula": "\\begin{align*} \\mathbb { A } : = \\{ \\mathcal { E } \\subseteq \\{ 1 , \\ldots , m \\} \\ | \\ \\exists \\bar { \\vec { x } } \\in \\mathcal { X } : \\mathcal { A } ( \\bar { \\vec { x } } ) = \\mathcal { E } , \\ \\mathcal { R } ( \\mathcal { E } ) \\ \\textrm { i s f u l l - d i m e n s i o n a l s e t } \\} , \\end{align*}"}
-{"id": "5460.png", "formula": "\\begin{align*} \\vert T _ H ( N , y ; \\alpha ) \\vert \\le \\sum _ { n = N - H } ^ { N + y } t _ H ( n - N ) \\ll H ( H + y + 1 ) \\ll H ^ 2 . \\end{align*}"}
-{"id": "5664.png", "formula": "\\begin{align*} \\mu ( E ) : = \\int v \\left ( F ( v , E ) - M ( v ) \\right ) \\d v , \\end{align*}"}
-{"id": "3392.png", "formula": "\\begin{align*} \\pi ( ( \\nabla - \\nu _ 0 ( z ) \\mathrm { i d } ) ( v ) ) & = ( \\nabla _ { \\nu ( w ) - \\nu _ 0 ( z ) } ) ( \\pi ( v ) ) \\\\ & = \\left ( c _ 0 w + ( c _ 0 b _ 1 ( w ) + c _ 0 a _ 0 ( w ) ) w ^ 2 + ( ) \\right ) d w / w ^ { m r - r + 1 } . \\end{align*}"}
-{"id": "1.png", "formula": "\\begin{align*} X _ { q _ 0 , \\gamma } = \\{ \\nu \\in L _ { q _ 0 , } ^ 1 , \\lim _ { j \\to \\infty } j ^ { \\gamma } | \\hat { \\nu } _ j | = 0 \\} \\end{align*}"}
-{"id": "7097.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\hat { \\mu } ^ x = D _ 1 . \\end{align*}"}
-{"id": "2974.png", "formula": "\\begin{align*} \\Theta ^ 1 ( x ) = \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ f _ i ^ x , \\alpha _ j ] , \\end{align*}"}
-{"id": "722.png", "formula": "\\begin{align*} \\hat { h } _ { X , f } ( P ) = \\lim _ { n \\to \\infty } \\frac { h _ { X } ( f ^ { n } ( P ) ) } { \\delta _ { f } ^ { n } } \\end{align*}"}
-{"id": "6979.png", "formula": "\\begin{align*} \\alpha _ n = \\delta _ H \\phi , ~ \\lambda _ n = \\delta _ L \\psi ~ \\mbox { a n d } ~ \\mu _ n = - \\delta _ L \\phi + \\delta _ v \\psi . \\end{align*}"}
-{"id": "8620.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & \\pi _ { 1 } ^ { 1 } \\rho _ { 2 } & 0 \\\\ 0 & 0 & \\rho _ { 1 } \\\\ 0 & 0 & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "2951.png", "formula": "\\begin{align*} & \\alpha ^ * ( \\mu _ 1 ) : = \\inf \\{ \\sigma > 0 \\mid g ( \\sigma , \\mu _ 1 ) > 0 \\} , \\\\ & \\beta ^ * ( \\epsilon ) : = \\inf \\{ \\sigma > 0 \\mid h ( \\sigma , \\epsilon ) > 0 \\} . \\end{align*}"}
-{"id": "9430.png", "formula": "\\begin{align*} E ^ p _ q ( f ) = E ^ p _ q [ f ] , \\end{align*}"}
-{"id": "5927.png", "formula": "\\begin{align*} { \\psi } = G _ { \\lambda } g \\in L ^ p ( \\R ^ d _ v ; H ^ { 2 / 3 } _ p ( \\R ^ d _ x ) ) \\ ; \\ ; \\ ; \\ ; \\| G _ { \\lambda } g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { 2 / 3 } _ p ( \\R ^ d _ x ) ) } \\ ; \\le \\ ; \\Big ( \\frac { \\lambda + 1 } { \\lambda } \\Big ) ^ { 2 / 5 } \\ , c _ 1 \\ , \\| g \\| _ { L ^ p ( \\R ^ { 2 d } ) } . \\end{align*}"}
-{"id": "2875.png", "formula": "\\begin{align*} X \\wedge Y = ( X _ - \\otimes Y _ - ) _ + ; \\end{align*}"}
-{"id": "1074.png", "formula": "\\begin{align*} e ( R ' ) \\leq \\left ( k - \\alpha - \\frac { 7 } { 1 6 } \\right ) \\frac { n ^ 2 } { 2 } = \\left ( k - \\frac { 1 1 } { 1 6 } \\right ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "1483.png", "formula": "\\begin{align*} \\tan ^ { - 1 } ( m _ f ( Q ) ) = m _ { \\tan ^ { - 1 } f } ( Q ) , \\end{align*}"}
-{"id": "7295.png", "formula": "\\begin{align*} E _ k \\triangleright [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q & = ( E _ k \\triangleright E _ \\xi ) E _ { \\xi ^ \\prime } ^ * + ( K _ k \\triangleright E _ \\xi ) ( E _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) \\\\ & - q ^ { - ( \\xi , \\xi ^ \\prime ) } ( E _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) E _ \\xi - q ^ { - ( \\xi , \\xi ^ \\prime ) } ( K _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) ( E _ k \\triangleright E _ \\xi ) . \\end{align*}"}
-{"id": "2244.png", "formula": "\\begin{align*} H = - \\int _ { - \\infty } ^ { \\infty } p ( x ) \\log p ( x ) d x . \\end{align*}"}
-{"id": "5119.png", "formula": "\\begin{align*} \\tilde { L } ^ { ( i ) } ( u ; s ) = \\frac { 1 } { 1 - s u } \\begin{pmatrix} 1 & 0 \\\\ 0 & ( - s u ) ^ { - 1 } \\end{pmatrix} L ^ { ( i ) } ( - s u ; s ) \\begin{pmatrix} 1 & 0 \\\\ 0 & u \\end{pmatrix} , \\end{align*}"}
-{"id": "1565.png", "formula": "\\begin{align*} F \\left ( 0 , S \\right ) = S \\end{align*}"}
-{"id": "6883.png", "formula": "\\begin{align*} X _ n \\to \\cdots \\to X _ 1 \\to X _ 0 = X \\end{align*}"}
-{"id": "7103.png", "formula": "\\begin{align*} \\nu _ { \\beta , n } = \\sum _ { | u | = n } e ^ { \\beta ( m _ n - V ( u ) ) } \\delta _ u \\nu _ { \\beta , \\infty } = \\sum _ { n \\in \\N } Z _ \\infty ^ \\beta e ^ { - \\beta \\xi ^ \\beta _ n } \\delta _ { u ^ { ( n ) } } . \\end{align*}"}
-{"id": "8617.png", "formula": "\\begin{align*} \\pi _ { b e _ { k } } \\beta \\psi ( n ) & = \\pi _ { b e _ { k } } \\beta \\left ( - n _ { 1 } - c _ { k } ^ { - 1 } i _ { 2 } \\sigma _ { 1 } \\overline { \\pi } _ { 3 } ( n _ { 2 } ) - c _ { k } ^ { - 1 } \\sigma _ { 3 } \\overline { \\pi } _ { 4 } ( n _ { 3 } ) - i _ { 3 } \\sigma _ { 2 } \\overline { \\pi } _ { 5 } ( n _ { 4 } ) \\right ) \\end{align*}"}
-{"id": "3042.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { \\sum _ { | u | = n } e ^ { - \\beta V ( u ) } } \\sum _ { | u | = n } e ^ { - \\beta V ( u ) } \\delta _ { H _ n ( u ) } = \\sum _ { k \\in \\N } p _ k \\delta _ { \\mathbf { e } _ k } , \\end{align*}"}
-{"id": "4452.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\bigl | \\mathbb { E } _ f ( \\hat { H } _ n ^ w ) - H ( f ) \\bigr | = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { 2 ( \\lfloor d / 4 \\rfloor + 1 ) } { d } } } { n ^ { \\frac { 2 ( \\lfloor d / 4 \\rfloor + 1 ) } { d } } } \\ , , \\ , \\frac { k ^ { \\frac { \\beta } { d } } } { n ^ { \\frac { \\beta } { d } } } \\biggr \\} \\biggr ) , \\end{align*}"}
-{"id": "4836.png", "formula": "\\begin{align*} \\sum _ { \\nu } s _ { \\nu / \\lambda } ( \\rho ) s _ { \\nu / \\mu } ( \\rho ' ) = H ( \\rho ; \\rho ' ) \\sum _ { \\tau } s _ { \\lambda / \\tau } ( \\rho ' ) s _ { \\mu / \\tau } ( \\rho ) , \\end{align*}"}
-{"id": "5783.png", "formula": "\\begin{align*} \\tau _ { k } ( u | v ) = ( k + h ^ { \\vee } - r _ { s } ) ( u | v ) \\end{align*}"}
-{"id": "8788.png", "formula": "\\begin{align*} T _ i \\sum _ \\mu f _ \\mu & = \\sum _ { \\mu : \\ \\mu _ i < \\mu _ { i + 1 } } \\left ( t f _ { s _ i \\mu } + ( t - 1 ) f _ \\mu \\right ) + \\sum _ { \\mu : \\ \\mu _ i = \\mu _ { i + 1 } } t f _ \\mu + \\sum _ { \\mu : \\ \\mu _ i > \\mu _ { i + 1 } } f _ { s _ i \\mu } \\\\ & = \\sum _ { \\mu : \\ \\mu _ i < \\mu _ { i + 1 } } t f _ { s _ i \\mu } + \\sum _ { \\mu : \\ \\mu _ i \\le \\mu _ { i + 1 } } t f _ \\mu = t \\sum _ \\mu f _ \\mu . \\end{align*}"}
-{"id": "5044.png", "formula": "\\begin{align*} \\int _ G \\lambda ( ( g ^ { - 1 } \\cdot f _ 1 ) f _ 2 ) \\ , d \\eta ( g ) = \\lambda ( f _ 1 ) \\lambda ( f _ 2 ) . \\end{align*}"}
-{"id": "90.png", "formula": "\\begin{align*} 2 5 \\epsilon n s ^ { n - 1 } = \\pm 1 2 5 - \\sum _ { j = 2 } ^ { n } \\binom { n } { j } ( 2 5 \\epsilon ) ^ j s ^ { n - j } \\end{align*}"}
-{"id": "5142.png", "formula": "\\begin{align*} h ^ { r ^ { k _ { r } } , \\ldots , a ^ { k _ { a } - 1 } , \\ldots , 1 ^ { k _ { 1 } } } _ { \\vec { w } } ( x _ { 1 } , \\ldots , x _ { k - 1 } ) = h ^ { r ^ { k _ { r } } , \\ldots , a ^ { k _ { a } - 1 } , \\ldots , 1 ^ { k _ { 1 } } } _ { \\vec { z } ( \\ell ( t ) , \\ldots , \\ell ( 0 ) ) } ( x _ { 1 } , \\ldots , x _ { k - 1 } ) . \\end{align*}"}
-{"id": "761.png", "formula": "\\begin{gather*} q : = q _ 1 \\otimes \\cdots \\otimes q _ l , \\end{gather*}"}
-{"id": "3648.png", "formula": "\\begin{align*} u _ { n + 1 } = \\frac { 8 u _ n } { 1 + u _ { n - 2 } } , \\end{align*}"}
-{"id": "3171.png", "formula": "\\begin{align*} & \\acute { p } ( 1 ) | 0 \\rangle \\langle 0 | ^ { \\mathfrak { B } } + \\acute { p } ( 2 ) | 1 \\rangle \\langle 1 | ^ { \\mathfrak { B } } \\\\ & = \\acute { p } ( 1 ) | 1 \\rangle \\langle 1 | ^ { \\mathfrak { B } } + \\acute { p } ( 2 ) | 2 \\rangle \\langle 2 | ^ { \\mathfrak { B } } \\end{align*}"}
-{"id": "984.png", "formula": "\\begin{align*} H _ { 1 , 1 } = U ( N _ { a , 1 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } - N _ { b , 1 } ) + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) \\end{align*}"}
-{"id": "9930.png", "formula": "\\begin{align*} \\begin{aligned} & x _ 0 ^ 2 & & + & & x _ 1 ^ 2 & & + & & x _ 2 ^ 2 & & + & & x _ 3 ^ 2 & = & 0 \\\\ & x _ 0 ^ 2 & & + & b ^ 4 & x _ 1 ^ 2 & & + & b ^ 2 & x _ 2 ^ 2 & & + & b ^ 2 & x _ 3 ^ 2 & = & 0 \\\\ & x _ 0 ^ 2 & & - & b ^ 2 & x _ 1 ^ 2 & & & & & & & & & = & 0 \\\\ & x _ 0 ^ 2 & & - & b ^ 2 & x _ 1 ^ 2 & & & & & & & & & = & 0 . \\end{aligned} \\end{align*}"}
-{"id": "5892.png", "formula": "\\begin{align*} \\left ( a - \\lambda \\right ) B ^ { t } - \\xi _ { , t } ^ { t } B ^ { t } = 0 . \\end{align*}"}
-{"id": "1437.png", "formula": "\\begin{align*} G ( z ) , W _ { 2 i } ( z ) \\ ( i = 0 , \\ldots , n - 1 ) \\end{align*}"}
-{"id": "72.png", "formula": "\\begin{align*} \\Sigma _ A ' ( y ) + \\frac { p } { p - 1 } \\Sigma _ A ( y ) ^ { ( p - 1 ) / p } & = \\frac { p } { p - 1 } ( 1 - y ) \\left [ A ^ { ( p - 1 ) / p } - A ( 1 - y ) ^ { ( 2 - p ) / ( p - 1 ) } \\right ] \\\\ & \\ge \\frac { p } { p - 1 } ( 1 - y ) \\left [ A ^ { ( p - 1 ) / p } - A \\right ] \\\\ & \\ge \\frac { p } { p - 1 } ( 1 - y ) a ^ { 2 - p } = \\psi ' ( y ) + \\frac { p } { p - 1 } \\psi ( y ) ^ { ( p - 1 ) / p } \\end{align*}"}
-{"id": "8641.png", "formula": "\\begin{align*} U = L ^ { 2 } \\left ( \\left [ 0 , 1 \\right ] \\right ) , \\ ; \\ ; \\dd ( \\Lambda ) = H _ { 0 } ^ { 1 } \\left ( \\left [ 0 , 1 \\right ] \\right ) \\cap H _ { } ^ { 2 } \\left ( \\left [ 0 , 1 \\right ] \\right ) , \\ ; \\ ; \\dd ( \\Lambda ^ { 1 / 2 } ) = H _ { 0 } ^ { 1 } \\left ( \\left [ 0 , 1 \\right ] \\right ) = V \\end{align*}"}
-{"id": "3250.png", "formula": "\\begin{align*} K _ { i } ^ { * } = K _ { i } , E _ { i } ^ { * } = K _ { i } F _ { i } , F _ { i } ^ { * } = E _ { i } K _ { i } ^ { - 1 } . \\end{align*}"}
-{"id": "4242.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 + t ) ^ { \\tfrac { x + y } { 2 } } d \\mu _ { - 1 } ( y ) & = \\frac { 2 } { 1 + \\sqrt { 1 + t } } \\sqrt { ( 1 + t ) ^ x } \\\\ & = \\sum _ { n = 0 } ^ \\infty C h _ { n , \\frac { 1 } { 2 } } ( x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "1229.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { m ( E _ n ) } w = \\frac { 1 } { 2 ^ n \\varphi ( t _ n ) } . \\end{align*}"}
-{"id": "4530.png", "formula": "\\begin{align*} T _ { 1 3 } : = \\biggl | \\int _ { \\mathcal { X } _ n } \\int _ { B _ x ^ c \\bigl ( \\frac { r _ { n , 1 } d _ n } { f ( x ) ^ { 1 / d } } \\bigr ) } f ( x ) & f ( y ) \\log f ( y ) \\\\ & \\int _ { \\tilde { u } _ { n , x , y } } ^ \\infty \\log \\bigl ( u f ( x ) \\bigr ) \\ , d ( \\tilde { F } _ { n , x } - F _ { n , x } ^ - ) ( u ) \\ , d y \\ , d x \\biggr | . \\end{align*}"}
-{"id": "1511.png", "formula": "\\begin{align*} \\bar e _ 0 = e _ 0 ( 1 + e _ 0 ) ^ { - 1 } \\end{align*}"}
-{"id": "426.png", "formula": "\\begin{align*} W ^ { 1 , p } _ H ( M ) = \\{ f \\in L ^ p ( M ) : X f \\in L ^ p ( M ) \\ ; \\forall X \\in C ^ \\infty ( M , H ) \\} . \\end{align*}"}
-{"id": "2806.png", "formula": "\\begin{align*} x ( t ) = \\sum _ { k = 1 } ^ { r } d _ k e ^ { \\left ( 2 \\pi \\imath f _ k - \\tau _ k \\right ) t } , \\end{align*}"}
-{"id": "4117.png", "formula": "\\begin{align*} C \\ : : \\ : ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q - z ^ { p + 2 q } x ^ { - p } = 0 . \\end{align*}"}
-{"id": "868.png", "formula": "\\begin{align*} \\operatorname * { l e v } \\nolimits _ { \\varphi , \\le } ( t ) = t k + A \\forall \\ , \\ t \\in { \\mathbb { R } } . \\end{align*}"}
-{"id": "8951.png", "formula": "\\begin{align*} U ( x , t ) = r ( x , t ) + i s ( x , t ) , \\end{align*}"}
-{"id": "7945.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 u = \\lambda u , & { \\rm i n } \\ \\Omega , \\\\ \\Delta u = 0 , & { \\rm o n } \\ \\partial \\Omega , \\\\ \\frac { \\partial \\Delta u } { \\partial \\nu } = 0 , & { \\rm o n } \\ \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "2288.png", "formula": "\\begin{align*} \\big \\| \\big ( ( A _ m - A _ n ) v _ n \\big ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } ^ p \\le \\sum _ { n = k } ^ { m - 1 } \\big ( \\| A _ m - A _ n \\| ^ p - \\| A _ m - A _ { n + 1 } \\| ^ p \\big ) E _ n . \\end{align*}"}
-{"id": "5258.png", "formula": "\\begin{align*} A = \\operatorname * { b d } A - \\mathbb { R } _ { + } k \\end{align*}"}
-{"id": "1676.png", "formula": "\\begin{align*} \\| D _ v { \\psi } \\| _ { W ^ { 1 , p } ( \\R ^ { 2 d } ) } \\le c ( \\lambda ) \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ { p } ( \\R ^ d _ x ) ) } , \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "7638.png", "formula": "\\begin{align*} \\ \\left \\{ \\begin{aligned} & u _ { t } = \\triangle ^ { \\alpha / 2 } u + f ( u ) , \\\\ & u ( 0 ) = u _ { 0 } \\geq 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4847.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) = \\iint \\frac { ( z - w ) h ^ { \\rm g e o } _ { 1 2 } ( z , w ) } { ( z ^ 2 - 1 ) w ( z w - 1 ) } \\frac { z - c } { z ( 1 - c w ) } \\frac { \\dd z } { z ^ u } \\frac { \\dd w } { w ^ v } = - K _ { 2 1 } ^ { \\rm g e o } ( j , v ; i , u ) , \\end{align*}"}
-{"id": "156.png", "formula": "\\begin{align*} \\bigr [ v , \\partial ( e ) \\bigl ] = \\begin{cases} 1 & v = t ( e ) t ( e ) \\neq s ( e _ ) , \\\\ - 1 & v = s ( e ) t ( e ) \\neq s ( e _ ) , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1947.png", "formula": "\\begin{align*} & F _ i ( Y ) = 2 p _ i Y _ i - q _ i ^ 2 Y _ i ^ 2 - E ( Y ) , i = 1 , 2 , \\ , \\ , { \\rm a n d } \\\\ & E ( Y ) = n _ 1 q _ 1 ^ 2 Y _ 1 ^ 2 + n _ 2 q _ 2 ^ 2 Y _ 2 ^ 2 . \\end{align*}"}
-{"id": "7124.png", "formula": "\\begin{align*} X \\odot Y = x _ 1 y _ 1 + x _ 2 y _ 2 - x _ 3 y _ 3 \\end{align*}"}
-{"id": "3103.png", "formula": "\\begin{align*} P ( \\lambda ) = ( \\Lambda _ s ( \\lambda ) ^ T \\otimes I _ n ) ( \\lambda B + A ) ( \\Lambda _ { s } ( \\lambda ) \\otimes I _ n ) \\in \\mathbb { F } [ \\lambda ] ^ { n \\times n } \\end{align*}"}
-{"id": "8396.png", "formula": "\\begin{align*} B \\lim _ \\lambda x _ \\lambda = w ^ * \\lim _ \\lambda x _ \\lambda = x \\end{align*}"}
-{"id": "944.png", "formula": "\\begin{align*} a : = \\limsup _ { d \\to \\infty } \\frac { \\sum _ { j = 1 } ^ d \\gamma _ j } { \\log ( d + 1 ) } < \\infty . \\end{align*}"}
-{"id": "4507.png", "formula": "\\begin{align*} V ( f ) = \\mathbb { E } _ f [ \\{ \\log f ( X _ 1 ) + H ( f ) \\} ^ 2 ] \\geq A _ { d , \\theta } ^ 2 \\mathbb { P } _ f ( X _ 1 \\in S _ { d , \\theta } ) \\geq A _ { d , \\theta } ^ 2 e ^ { - 4 A _ { d , \\theta } } V _ d r _ { d , \\theta } ^ d , \\end{align*}"}
-{"id": "8092.png", "formula": "\\begin{align*} P _ { r _ j } = \\frac { B _ { s _ i , r _ j } P _ { s _ i } } { p _ { t h } ^ { s _ i } P _ { s _ i } ^ 2 - A _ { s _ i , r _ j } } \\triangleq f \\left ( P _ { s _ i } \\right ) \\end{align*}"}
-{"id": "5701.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ l \\frac { 1 } { w ( Q _ 0 ^ { ( m ) } ) ^ \\frac { 1 } { p } } \\lesssim _ { p , w } \\frac { 1 } { w ( Q _ 0 ) ^ \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "6251.png", "formula": "\\begin{align*} \\frac { d } { d D } \\left ( D ^ 2 \\bar { \\lambda } _ 1 ( n , D , K ) \\right ) \\begin{cases} < 0 , & K > 0 , \\ D \\in ( 0 , \\frac { \\pi } { \\sqrt { K } } ) ; \\\\ = 0 , & K = 0 ; \\\\ > 0 , & K < 0 . \\end{cases} \\end{align*}"}
-{"id": "9373.png", "formula": "\\begin{align*} a , d \\in { \\mathbb { R } } , b , c \\in { i \\mathbb { R } } , \\mathcal { P T } q _ 1 = q _ 1 , \\mathcal { P T } q _ 2 = - q _ 2 , \\end{align*}"}
-{"id": "7613.png", "formula": "\\begin{align*} \\lambda _ 1 > \\lambda _ 1 ( K _ { 2 , n - 2 } ) = \\sqrt { 2 n - 4 } . \\end{align*}"}
-{"id": "4037.png", "formula": "\\begin{align*} N ( \\underline { x } , \\underline { y } ) = \\begin{cases} s u p \\{ n : x _ i = y _ i \\ \\forall \\ | i | < n \\} & \\underline { x } , \\underline { y } \\in \\Sigma _ A \\\\ s u p \\{ n : x _ i = y _ i \\ \\forall \\ 0 \\leq i < n \\} & \\underline { x } , \\underline { y } \\in \\Sigma _ A ^ + . \\end{cases} \\end{align*}"}
-{"id": "1467.png", "formula": "\\begin{align*} b _ Q : = \\frac { ( M \\chi _ Q ) ^ { \\alpha / n + \\varepsilon } } { \\ell ( Q ) ^ \\alpha } , \\end{align*}"}
-{"id": "1391.png", "formula": "\\begin{align*} \\kappa F _ { 1 } ( c , \\delta _ 0 ) + ( 1 - \\kappa ) F _ { 2 } ( c , \\delta _ 0 ) , \\kappa = P ( \\hat \\tau = 0 ) \\end{align*}"}
-{"id": "8339.png", "formula": "\\begin{align*} \\sigma _ i ( w ) = \\sigma _ i ( l ( y ) ) = l ( \\sigma _ i ( y ) ) = l ( y + \\mu _ i ) = l ( y ) + l ( \\mu _ i ) = w + l ( \\mu _ i ) . \\end{align*}"}
-{"id": "5150.png", "formula": "\\begin{align*} { } _ { [ 1 , M ] } \\langle \\mathrm { v a c } | \\tilde { A } ( w ) = ( 1 + w ) ^ { M - 1 } { } _ { [ 1 , M ] } \\langle \\mathrm { v a c } | . \\end{align*}"}
-{"id": "1971.png", "formula": "\\begin{align*} y _ i y _ j = y _ k y _ \\ell , \\forall \\ ; i + j = k + \\ell . \\end{align*}"}
-{"id": "2929.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V ^ * _ \\pi ( z _ 1 ; X ) V ^ * _ \\pi ( z _ 2 ; X ) \\cdots V ^ * _ \\pi ( z _ m ; X ) \\cdot 1 \\cr & = [ Z ^ { \\lambda + \\delta } ] \\ \\prod _ { 1 \\le i < j \\le m } ( z _ i - z _ j ) \\ \\prod _ { \\ell = 1 } ^ m \\ , L ( z _ \\ell ; X ) \\ M _ { \\pi ' } ( Z ) \\cr & = [ s _ \\lambda ( Z ) ] \\ L ( X Z ) \\ , M _ { \\pi ' } ( Z ) \\ , . \\end{align*}"}
-{"id": "6255.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\dd u } { \\dd t } = \\Delta u & \\Omega \\times \\mathbb R _ + ; \\\\ u = 0 & \\dd \\Omega \\times \\mathbb R _ + ; \\\\ u ( x , 0 ) = \\phi _ 1 . \\end{cases} \\end{align*}"}
-{"id": "250.png", "formula": "\\begin{align*} \\tilde { a } ( \\delta ) & : = d ^ { m / 2 } \\sup _ { x : \\| x \\| \\leq g ( \\delta ) } \\max \\biggl \\{ \\max _ { r = 1 , \\ldots , m } q _ r ( \\| x \\| ) \\ , , \\ , d ^ { 1 / 2 } e ^ { \\| x \\| } q _ { m + 1 } ( \\| x \\| + 1 ) \\biggr \\} \\\\ & = d ^ { m / 2 } \\max \\biggl \\{ \\max _ { r = 1 , \\ldots , m } q _ r \\bigl ( g ( \\delta ) \\bigr ) \\ , , \\ , d ^ { 1 / 2 } e ^ { g ( \\delta ) } q _ { m + 1 } \\bigl ( g ( \\delta ) + 1 \\bigr ) \\biggr \\} . \\end{align*}"}
-{"id": "3509.png", "formula": "\\begin{align*} f _ { ; i j } & = A _ { i j } f + B _ { i j k } X ^ k + C ^ { \\ell } _ { i j k } X ^ k _ { \\ ; ; \\ell } \\ ; , \\end{align*}"}
-{"id": "3793.png", "formula": "\\begin{align*} { \\cal F } = - \\lim _ { \\tau \\rightarrow 0 } \\frac { \\partial } { \\partial \\tau } \\max _ { { \\mathbb Q } , \\tilde { \\mathbb Q } } \\left \\{ { \\cal F } ^ { ( \\tau ) } \\right \\} \\end{align*}"}
-{"id": "9927.png", "formula": "\\begin{align*} x _ 0 ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 \\ ; = \\ ; 0 \\ ; = \\ ; x _ 3 ^ 2 + \\frac { 1 - \\gamma } { 1 + \\alpha } x _ 1 ^ 2 + \\frac { 1 + \\gamma } { 1 - \\beta } x _ 2 ^ 2 . \\end{align*}"}
-{"id": "3822.png", "formula": "\\begin{align*} \\alpha _ 1 & = \\frac { t } { ( - 1 ) ^ i ( t + 1 ) } & & \\alpha _ 2 = x _ i ^ { - 1 } & & \\beta _ 1 = \\frac { ( - 1 ) ^ { i - 1 } } { ( t + 1 ) x _ i ^ { \\lambda _ 1 + 1 } } & & \\beta _ 2 = \\frac { 1 } { ( t + 1 ) x _ i ^ { \\lambda _ 1 } } . \\end{align*}"}
-{"id": "660.png", "formula": "\\begin{align*} \\nu _ Y ( \\phi ) = \\int _ G \\nu ( g ^ { - 1 } \\cdot \\phi ) \\ , d \\eta ( g ) , \\textrm { f o r $ \\phi \\in C ( \\overline { Y } ) $ } . \\end{align*}"}
-{"id": "3875.png", "formula": "\\begin{align*} \\dfrac { \\partial } { \\partial x _ i } ( \\gamma ( \\tilde { r } ) ) = & \\ , \\gamma ^ { \\prime } ( \\tilde { r } ) \\dfrac { \\partial } { \\partial x _ i } \\sqrt { ( 2 \\xi _ 1 ( x ) - x _ 1 - a _ 1 ) ^ 2 + \\cdots + ( 2 \\xi _ { n + 1 } ( x ) - x _ { n + 1 } - a _ { n + 1 } ) ^ 2 } \\\\ = & \\ , \\dfrac { \\gamma ^ { \\prime } ( \\tilde { r } ) } { \\tilde { r } } \\left \\{ \\left ( \\sum _ { k } 2 \\dfrac { \\partial \\xi _ k ( x ) } { \\partial x _ i } ( \\tilde { x } - a ) _ k \\right ) - ( \\tilde { x } - a ) _ i \\right \\} . \\end{align*}"}
-{"id": "3046.png", "formula": "\\begin{align*} \\Gamma ( \\mathcal { O } ) = \\left \\{ u \\in \\mathcal { U } : C ( \\pi u , u ( | u | ) ) \\not \\subset \\mathcal { O } , \\exists j \\in \\N : C ( u , j ) \\subset \\mathcal { O } \\right \\} \\end{align*}"}
-{"id": "3335.png", "formula": "\\begin{align*} ( \\Gamma _ i ^ { ( k + 1 ) } v , v ^ \\prime ) _ { k } = ( v , \\Gamma _ i ^ { ( k + 1 ) * } v ^ \\prime ) _ { k + 1 } , v \\in \\Lambda _ q ^ { k + 1 } ( \\mathfrak { u } _ + ) , \\ v ^ \\prime \\in \\Lambda _ { q } ^ { k } ( \\mathfrak { u } _ + ) . \\end{align*}"}
-{"id": "7465.png", "formula": "\\begin{align*} \\Delta _ H ( V ) : = H ( V \\cup \\{ 0 \\} ) - H ( V ) . \\end{align*}"}
-{"id": "3749.png", "formula": "\\begin{align*} \\mu : = \\nu \\left \\{ \\left [ 0 , 1 \\right ] \\right \\} = \\int _ 0 ^ 1 e ^ { \\beta x } g ( x ) d x < \\infty . \\end{align*}"}
-{"id": "2543.png", "formula": "\\begin{align*} & \\frac 1 \\tau ( d p _ j ^ { n + 1 } \\wedge d q _ j ^ { n + 1 } - d p _ j ^ { n } \\wedge d q _ j ^ { n } ) - \\frac 1 h ( d p _ j ^ { n + \\frac 1 2 } \\wedge d v _ { j + 1 } ^ { n + \\frac 1 2 } - d p _ { j - 1 } ^ { n + \\frac 1 2 } \\wedge d v _ j ^ { n + \\frac 1 2 } ) \\\\ & - \\frac 1 h ( d q _ j ^ { n + \\frac 1 2 } \\wedge d w _ { j + 1 } ^ { n + \\frac 1 2 } - d q _ { j - 1 } ^ { n + \\frac 1 2 } \\wedge d w _ j ^ { n + \\frac 1 2 } ) = 0 , \\end{align*}"}
-{"id": "7680.png", "formula": "\\begin{align*} ( S _ { \\alpha } ( t ) u ) ( x ) = \\int _ { \\mathbb { R } ^ { d } } p ( t , x , y ) u ( y ) d y , \\end{align*}"}
-{"id": "187.png", "formula": "\\begin{align*} \\mathbb { E } ( \\hat { H } _ n ) & = \\int _ \\mathcal { X } f ( x ) \\int _ 0 ^ \\infty \\log u \\ , d F _ { n , x } ( u ) \\ , d x \\approx \\int _ \\mathcal { X } f ( x ) \\int _ 0 ^ \\infty \\log u \\ , d F _ x ( u ) \\ , d x \\\\ & = \\int _ { \\mathcal { X } } f ( x ) \\int _ 0 ^ \\infty \\log \\Bigl ( \\frac { t e ^ { - \\Psi ( k ) } } { f ( x ) } \\Bigr ) e ^ { - t } \\frac { t ^ { k - 1 } } { ( k - 1 ) ! } \\ , d t \\ , d x = H . \\end{align*}"}
-{"id": "4895.png", "formula": "\\begin{align*} f ( 2 ) = & ( q ^ 2 + 1 ) ( q - 1 ) + \\alpha q ( q ^ 2 + 1 ) ( q - 1 ) + g ( 2 ) , \\end{align*}"}
-{"id": "8837.png", "formula": "\\begin{align*} \\left ( \\theta ( \\tau ) \\theta ( 2 \\tau ) \\right ) ^ { k } = F _ { k , 2 } ( \\tau ) + { \\left ( \\theta ( \\tau ) \\theta ( 2 \\tau ) \\right ) ^ { k } \\sum _ { j = 1 } ^ { \\ell _ { 2 } } c _ { j , k , 2 } x _ { 2 } ^ { j } } . \\end{align*}"}
-{"id": "8158.png", "formula": "\\begin{align*} \\phi _ h ( S _ 2 ^ 2 ) = t + \\frac { t ( t - 1 ) } { \\varphi ( \\frac { n _ 2 } { ( h , n _ 2 ) } ) } \\mu ( \\frac { n _ 2 } { ( h , n _ 2 ) } ) . \\end{align*}"}
-{"id": "6152.png", "formula": "\\begin{align*} y _ i y _ j = \\left \\{ \\begin{array} { l r } y _ 0 y _ { i + j } & i + j \\leq m , \\\\ y _ m y _ { i + j - m } & i + j > m . \\end{array} \\right . \\end{align*}"}
-{"id": "4370.png", "formula": "\\begin{align*} e ^ { i T _ { c , h } ( f ) } = U _ { c , h } ( { \\rm E x p } ( f ) ) . \\end{align*}"}
-{"id": "7094.png", "formula": "\\begin{align*} \\mu _ n = \\sum _ { | u | = n } \\delta _ { u , V ( u ) - m _ n } , \\end{align*}"}
-{"id": "6481.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} L \\phi & = \\lambda \\phi \\Omega , \\\\ \\phi & = 0 \\quad \\ , \\ , \\ , \\ , \\mathbb { R } ^ { n } \\backslash \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "3097.png", "formula": "\\begin{align*} \\lambda B _ { i j } + A _ { i j } = \\lambda \\left ( \\beta _ { i j } P _ \\ell \\right ) + \\alpha _ { i j } P _ t , \\end{align*}"}
-{"id": "3467.png", "formula": "\\begin{align*} c = \\tilde { c } - c _ 0 . \\end{align*}"}
-{"id": "1462.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm B M O } _ X } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { \\| \\chi _ Q \\| _ X } \\| ( b - b _ Q ) \\chi _ Q \\| _ X . \\end{align*}"}
-{"id": "8740.png", "formula": "\\begin{align*} { \\widetilde Q } _ t ^ { 1 / 2 } ( K ) = I m \\mathcal L _ t = \\mathcal L _ t ( L ^ 2 ( 0 , t ; U ) ) \\end{align*}"}
-{"id": "5376.png", "formula": "\\begin{align*} c _ { d , d } { \\ , } c _ { r _ 0 , u _ 0 } ^ d + ( - 1 ) ^ { q _ 1 } { r _ 0 \\choose u _ 0 } \\gamma _ { r _ 0 } - ( 1 + \\delta _ { q _ 1 , q _ 2 } ) { \\ , } c _ { d , d } ^ { q _ 1 } { \\ , } c _ { q , q } ^ { q _ 2 + 1 } ~ = ~ 0 . \\end{align*}"}
-{"id": "7193.png", "formula": "\\begin{align*} ( 1 - t ^ 2 ) L ^ { '' } + 2 L + L L ' = 2 C _ 1 ( t - 1 ) , L ( 0 ) = L ( 1 ) = 0 . \\end{align*}"}
-{"id": "9642.png", "formula": "\\begin{align*} { \\cal F } _ { \\theta } = \\{ C ( \\omega _ 1 , \\ldots , \\omega _ { k - 1 } ) \\} _ { \\omega _ 1 , \\ldots , \\omega _ { k - 1 } } , \\end{align*}"}
-{"id": "19.png", "formula": "\\begin{align*} h | _ { \\Omega _ \\alpha } = e ^ { - \\varphi _ \\alpha } | \\cdot | ^ 2 \\end{align*}"}
-{"id": "8949.png", "formula": "\\begin{align*} U \\left ( x , 0 \\right ) = f \\left ( x \\right ) , - \\infty < x < \\infty \\end{align*}"}
-{"id": "8232.png", "formula": "\\begin{align*} E u _ n \\delta _ { n , 0 } = \\epsilon \\left ( \\bar { u } _ { n + 1 } - 2 \\bar { u } _ n + \\bar { u } _ { n - 1 } \\right ) + i \\gamma u _ n + \\Omega \\bar { u } _ n + 6 | u _ n | ^ 2 \\bar { u } _ n + 2 u _ n ^ 3 , \\end{align*}"}
-{"id": "3163.png", "formula": "\\begin{align*} & \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } I _ { q , s ^ n } ( \\gamma _ l ) \\\\ & \\geq \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } I _ { { q ' } , s ^ n } ( \\gamma _ l ) - f ( \\delta ) \\\\ & \\geq ( 1 - \\epsilon ) \\mu ( I _ { { q ' } , s ^ n } ) - f ( \\delta ) \\\\ & \\geq ( 1 - \\epsilon ) \\mu ( I _ { q , s ^ n } ) - ( 2 - \\epsilon ) f ( \\delta ) \\end{align*}"}
-{"id": "3576.png", "formula": "\\begin{align*} ( L U ) _ j = \\sum _ { k = 1 } ^ { n + 1 } L _ { j k } U ^ k , ( j = 1 , \\dots , n + 1 ) \\end{align*}"}
-{"id": "4454.png", "formula": "\\begin{align*} h _ x ^ { - 1 } ( s ) : = \\inf \\{ r > 0 : h _ x ( r ) \\geq s \\} = \\inf \\{ r > 0 : h _ x ( r ) = s \\} , \\end{align*}"}
-{"id": "1646.png", "formula": "\\begin{align*} H ^ { s } _ p ( \\R ^ d ) = \\{ f \\in L ^ p ( \\R ^ d ) \\ ; : \\ ; { \\cal F } ^ { - 1 } [ | \\cdot | ^ { s } \\ , { \\cal F } f ] \\in L ^ p ( \\R ^ d ) \\} \\end{align*}"}
-{"id": "5038.png", "formula": "\\begin{align*} \\nu _ Y ( \\phi ) = \\eta ( k _ \\phi ) = k _ \\phi ( e ) = \\nu ( \\phi ) , \\textrm { f o r a l l $ \\phi \\in C ( \\overline { Y } ) $ } . \\end{align*}"}
-{"id": "2590.png", "formula": "\\begin{align*} e ^ { - i t \\Delta } u ( t ) = t ^ { - \\frac { 1 } { p } } \\psi ( t , \\tfrac { x } { \\sqrt { t } } ) , \\end{align*}"}
-{"id": "1623.png", "formula": "\\begin{align*} A ^ { i j } \\xi _ { , i j } ^ { k } - 2 A ^ { i k } a _ { , i } + a B ^ { k } - \\xi _ { , i } ^ { k } B ^ { i } + \\xi ^ { i } B _ { , i } ^ { k } - \\lambda B ^ { k } = 0 \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "779.png", "formula": "\\begin{align*} I _ { k _ { 1 } , \\ldots , k _ { r } } = \\left \\{ \\vec { \\nu } = ( \\nu _ { 1 } , \\ldots , \\nu _ { k } ) \\in \\{ 1 , \\ldots , r \\} ^ { k } \\ , | \\ , \\# \\{ j \\ , | \\nu _ { j } = a \\} = k _ { a } \\ , ( 1 \\le \\forall { a } \\le r ) \\right \\} \\end{align*}"}
-{"id": "9590.png", "formula": "\\begin{align*} \\psi ( x , t ) = \\psi _ f ( x , t ) + \\frac { \\theta ( t - | x - y | ) } { 4 \\pi | x - y | } \\xi ( t - | x - y | ) - \\frac { m } { 4 \\pi } \\int _ 0 ^ t \\frac { \\theta ( s - | x - y | ) J _ 1 ( m \\sqrt { s ^ 2 - | x - y | ^ 2 } ) } { \\sqrt { s ^ 2 - | x - y | ^ 2 } } \\xi ( t - s ) d s . \\end{align*}"}
-{"id": "5354.png", "formula": "\\begin{align*} & \\langle \\chi _ j , \\chi _ { \\gamma } \\chi _ i \\rangle = \\frac { 1 } { | G | } \\sum _ { g \\in G } \\chi _ j ( g ) \\bar { \\chi _ { \\gamma } ( g ) \\chi _ i ( g ) } \\\\ & = \\frac { 1 } { | G | } \\sum _ { g \\in G } \\chi _ j ( g ) \\chi _ { \\gamma } ( g ) \\bar { \\chi _ i ( g ) } = \\langle \\chi _ j \\chi _ { \\gamma } , \\chi _ i \\rangle \\end{align*}"}
-{"id": "9112.png", "formula": "\\begin{align*} = ( a '' + b ' ) c ' + ( a ' + b '' ) c '' = a '' c ' + a ' c '' + b ' c ' + b '' c '' . \\end{align*}"}
-{"id": "7843.png", "formula": "\\begin{align*} \\{ , \\} = 0 . \\end{align*}"}
-{"id": "4313.png", "formula": "\\begin{align*} m ( K _ { X / Y } + L ) = K _ { X / Y } + L _ m \\end{align*}"}
-{"id": "9164.png", "formula": "\\begin{align*} \\mathcal { P } _ { s } & = \\{ m \\in \\mathbb N ^ { s } / \\sum _ { j = 1 } ^ { s } j m _ j = s \\} , \\end{align*}"}
-{"id": "2230.png", "formula": "\\begin{align*} | \\mathcal { U } _ { ( n , d ) } | & \\leq d \\cdot N ( n , ( d / 2 , d / 2 ) ) \\\\ & = d \\cdot N ( d / 2 , d / 2 ) ^ { 1 + o ( 1 ) } \\cdot \\log n \\\\ & = d \\cdot 2 ^ { H _ { 2 } ( 1 / 2 ) d + o ( d ) } \\cdot \\log n \\\\ & = d \\cdot 2 ^ { d + o ( d ) } \\cdot \\log n \\end{align*}"}
-{"id": "3213.png", "formula": "\\begin{align*} \\sum _ { 2 \\leq | \\gamma | < n } \\sum _ { | \\beta | \\geq 1 } \\max _ { j , k } \\sup _ { U _ j \\cap U _ k ^ * } \\left | F _ { k j , \\gamma , \\beta } ^ \\lambda \\right | \\cdot X ^ \\gamma \\cdot \\prod _ { \\lambda = 1 } ^ r \\left ( X ^ \\lambda + A ( X ) \\right ) ^ { \\beta _ \\lambda } . \\end{align*}"}
-{"id": "9027.png", "formula": "\\begin{align*} \\begin{pmatrix} \\Phi _ { k + 1 } ( z ) \\\\ \\Phi _ { k + 1 } ^ * ( z ) \\end{pmatrix} = & \\begin{pmatrix} z & - \\overline { \\alpha } _ k \\\\ - \\alpha _ k z & 1 \\end{pmatrix} \\begin{pmatrix} \\Phi _ { k } ( z ) \\\\ \\Phi _ { k } ^ * ( z ) \\end{pmatrix} \\end{align*}"}
-{"id": "1020.png", "formula": "\\begin{align*} u ' ( t ) = \\varphi ^ { - 1 } \\left [ t \\lambda \\frac { \\varphi ( b u ( 0 ) ) - \\varphi ( u ( 0 ) ) } { T } + \\varphi ( u ( 0 ) ) \\right ] \\end{align*}"}
-{"id": "5096.png", "formula": "\\begin{align*} \\mathbb { T } ^ { [ 1 , M ] } ( z ) = L ^ { ( 1 ) } ( z ) \\mathbb { T } ^ { [ 2 , M ] } ( z ) = \\mathbb { T } ^ { [ 1 , M - 1 ] } ( z ) L ^ { ( M ) } ( z ) , \\end{align*}"}
-{"id": "6918.png", "formula": "\\begin{align*} \\lambda _ 0 + \\lambda _ 0 ^ 2 = \\frac { ( R + d ^ 2 ) ^ 2 } { 4 d ^ 2 } - \\dfrac { 1 } { 4 } . \\end{align*}"}
-{"id": "5549.png", "formula": "\\begin{align*} b _ { 0 } ( x ) = A x ^ { 2 } + B x + C , b _ { 1 } ( x ) = \\frac { D } { \\rho ( x , t ) ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "3442.png", "formula": "\\begin{align*} y ( t ) = \\Gamma ^ * ( s , T ) u _ 0 ( \\xi ( T ) ) + \\int _ t ^ T f ( \\theta , \\xi ( \\theta ) , y ( \\theta ) , z ( \\theta ) ) d \\theta - \\int _ t ^ T z ( \\theta ) d w ( \\theta ) , s \\le t \\le T , \\end{align*}"}
-{"id": "5414.png", "formula": "\\begin{align*} \\gamma _ d ( 2 \\mathcal E _ s ) = 1 - \\mathsf P ( h _ s ( \\xi ) > 2 ) \\ge 1 - \\frac { \\mathsf { E } h _ s ( \\xi ) ^ 2 } { 4 } \\ge 3 / 4 > \\Psi ( 1 ) . \\end{align*}"}
-{"id": "430.png", "formula": "\\begin{align*} M = \\R ^ d \\quad \\mbox { a n d } X _ 0 , X _ 1 , \\dots , X _ m \\in C _ b ^ \\infty ( \\R ^ d , \\R ^ d ) \\ ; , \\end{align*}"}
-{"id": "9639.png", "formula": "\\begin{align*} A _ { j i } = \\left ( \\begin{array} { c c c c c c } A ^ { ( 1 ) } _ { j i } & * & \\ldots & * \\\\ 0 & A ^ { ( 2 ) } _ { j i } & * & \\vdots \\\\ \\vdots & { } & \\ddots & * \\\\ 0 & \\ldots & 0 & A ^ { ( r ) } _ { j i } \\end{array} \\right ) \\ , \\end{align*}"}
-{"id": "5959.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial z _ i } ( z ) & = 2 a f ^ { a - 1 } ( z ) z _ i \\ , , \\frac { \\partial ^ 2 g } { \\partial z _ i \\ , \\partial z _ j } ( z ) = 4 a ( a - 1 ) f ^ { a - 2 } ( z ) z _ i z _ j + 2 a f ^ { a - 1 } ( z ) \\delta _ { i , j } \\ , , \\end{align*}"}
-{"id": "2860.png", "formula": "\\begin{align*} R e s ^ n ( C ) = M ^ n R e s ^ { \\bullet } ( C ) \\wedge F ^ c ( N ^ n K ^ { \\bullet } ( C ) ) \\end{align*}"}
-{"id": "8606.png", "formula": "\\begin{align*} \\Delta _ { k } ' & = \\displaystyle \\sum _ { s a _ { 1 } , b _ { 1 } } \\displaystyle \\sum _ { v \\in L ( k ) } a _ { 1 } ^ { \\ast } v ^ { - 1 } ( ^ { \\ast } b _ { 1 } ) [ b _ { 1 } v a _ { 1 } ] \\\\ & = \\displaystyle \\sum _ { s a _ { 1 } , b _ { 1 } } \\displaystyle \\sum _ { v , r \\in L ( k ) } r ^ { \\ast } ( v ^ { - 1 } ) a _ { 1 } ^ { \\ast } r ( ^ { \\ast } b _ { 1 } ) [ b _ { 1 } v a _ { 1 } ] \\end{align*}"}
-{"id": "1546.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( \\mu , \\nu ) = \\underset { ( \\phi , \\psi ) \\in C _ W } { \\sup } \\int _ { \\Omega } \\phi ( x ) d \\mu ( x ) + \\int _ { \\Omega } \\psi ( y ) d \\nu ( y ) , \\end{align*}"}
-{"id": "6390.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\log \\frac { 2 \\pi \\sigma ^ { 2 } } { C ^ { 2 } ( \\varepsilon ) } + \\frac { \\mu _ { ( 2 ) } } { 2 \\sigma ^ { 2 } } + \\varepsilon \\mu _ { ( p ) } . \\end{align*}"}
-{"id": "7581.png", "formula": "\\begin{align*} \\mathcal A = \\mathcal A _ { 1 1 } \\cup \\mathcal A _ { 1 2 } , \\mathcal A _ { 1 1 } \\cap \\mathcal A _ { 1 2 } = \\emptyset , \\end{align*}"}
-{"id": "695.png", "formula": "\\begin{align*} \\alpha _ { f } ( P ) = \\lim _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n } ( P ) ) ^ { 1 / n } \\end{align*}"}
-{"id": "6362.png", "formula": "\\begin{align*} b = t p _ 0 + 2 s p _ n + 2 p _ { n + 1 } . \\end{align*}"}
-{"id": "1887.png", "formula": "\\begin{align*} z \\ge & 1 - 2 \\epsilon _ 0 + s \\epsilon _ 0 - \\frac 1 2 \\sqrt { 3 ( 1 - 2 \\epsilon _ 0 ) ( 1 - 3 \\epsilon _ 0 ) } \\\\ = & \\frac { 1 + \\sqrt { 2 } } 3 + \\frac { 2 - \\sqrt { 2 } } 6 s - \\frac { \\sqrt { 4 + 2 \\sqrt { 2 } } } 4 . \\end{align*}"}
-{"id": "4041.png", "formula": "\\begin{align*} h ( \\underline { a } ) = h ^ { + } ( \\underline { a } ) \\cap h ^ { - } ( \\underline { a } ) . \\end{align*}"}
-{"id": "7321.png", "formula": "\\begin{align*} \\hat { R } F ( v _ { 0 } \\otimes v _ { 0 } ) = [ 2 ] ^ { 1 / 2 } ( v _ { 0 } \\otimes v _ { - 1 } + \\hat { R } ( v _ { 0 } \\otimes v _ { - 1 } ) ) . \\end{align*}"}
-{"id": "6176.png", "formula": "\\begin{align*} \\nu ' ( A ) = \\nu _ { U L } ( ( A \\times \\R ) \\cap \\{ | z | > 1 \\} ) , \\end{align*}"}
-{"id": "9875.png", "formula": "\\begin{align*} & U ^ { \\epsilon } ( t , x ) = \\int _ 0 ^ 1 G _ t ( x , y ) \\eta ( y ) d y + \\sqrt { \\epsilon } \\int _ 0 ^ t \\int _ 0 ^ 1 G _ { t - s } ( x , y ) \\sigma ( s , U ( s ) ) ( y ) W ( d y , d s ) \\\\ & - \\int _ 0 ^ t \\int _ 0 ^ 1 \\partial _ y G _ { t - s } ( x , y ) g ( s , U ( s ) ) ( y ) d y d s + \\int _ 0 ^ t \\int _ 0 ^ 1 G _ { t - s } ( x , y ) f ( s , U ( s ) ) ( y ) d y d s . \\end{align*}"}
-{"id": "7739.png", "formula": "\\begin{align*} 4 e ^ { - 2 \\cdot 2 ^ n } - 8 e ^ { - 3 \\cdot 2 ^ n } - 1 6 e ^ { - 4 \\cdot 2 ^ n } = 4 e ^ { - 2 \\cdot 2 ^ n } \\left [ 1 - 2 e ^ { - 2 ^ n } - 4 e ^ { - 2 \\cdot 2 ^ n } \\right ] \\ge 2 e ^ { - 2 \\cdot 2 ^ n } > e ^ { - 2 ^ { n + 1 } } , \\end{align*}"}
-{"id": "3087.png", "formula": "\\begin{align*} a ( w , v ) = a ( u _ * , v ) - \\lambda _ * b ( u _ * , v ) \\quad \\forall v \\in V . \\end{align*}"}
-{"id": "9009.png", "formula": "\\begin{align*} y ( t ) = y ( 0 ) + c ( ~ ^ { A B } ~ _ { 0 } I ^ \\alpha ~ ^ { A B } I _ b ^ \\alpha y ) ( t ) . \\end{align*}"}
-{"id": "5697.png", "formula": "\\begin{align*} | m _ f ( Q ) | \\leq \\left ( f \\cdot \\chi _ Q \\right ) ^ * ( \\lambda | Q | ) , \\lim _ { \\ell ( Q ) \\to 0 , Q \\ni x } m _ f ( Q ) = f ( x ) ( { \\rm a . e . } \\ x \\in \\R ^ n ) . \\end{align*}"}
-{"id": "8233.png", "formula": "\\begin{align*} a _ 0 = \\epsilon ^ 2 A _ 0 , a _ n = \\epsilon ^ { | n | } A _ { | n | } , \\pm n \\in \\mathbb { N } \\end{align*}"}
-{"id": "7653.png", "formula": "\\begin{align*} \\| u _ { k } \\| _ { L ^ { q } } ^ { q } & = \\frac { \\omega _ { d } } { \\nu ( d , \\alpha ) ^ { q } } r _ { k } ^ { d } \\phi _ { k } ^ { q } \\\\ & = \\frac { \\omega _ { d } } { \\nu ( d , \\alpha ) ^ { q } } ( \\varepsilon \\phi _ { k } ^ { - q / d } k ^ { - \\alpha q / d } ) ^ { d } \\phi _ { k } ^ { q } \\\\ & = \\frac { \\omega _ { d } } { \\nu ( d , \\alpha ) ^ { q } } \\varepsilon ^ { d } k ^ { - \\alpha q } , \\end{align*}"}
-{"id": "1616.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } ( t ) ( F _ { x x } - F _ { x } ) + ( p ( t ) - x q ( t ) ) F _ { x } - F _ { t } = 0 , \\end{align*}"}
-{"id": "3908.png", "formula": "\\begin{align*} u ( t ) = \\varphi ^ { - 1 } ( - Q _ { \\varphi } ( K ( N _ { f } ( u ) ) ) ) + H \\left ( \\varphi ^ { - 1 } \\left [ K ( N _ { f } ( u ) ) - Q _ { \\varphi } ( K ( N _ { f } ( u ) ) ) \\right ] \\right ) ( t ) \\end{align*}"}
-{"id": "7719.png", "formula": "\\begin{align*} \\sum _ { k = N _ \\delta } ^ { n _ \\delta ( \\omega ) } \\sigma _ k \\xi _ { k + 1 } ( \\omega ) > \\delta . \\end{align*}"}
-{"id": "8543.png", "formula": "\\begin{align*} \\pi _ { b r } ' \\xi _ { b r } ' = i d _ { N _ { \\sigma ( b ) } } \\end{align*}"}
-{"id": "4320.png", "formula": "\\begin{align*} \\vert \\xi \\vert ^ 2 = \\sup _ { \\Vert u \\Vert _ y \\leq 1 } | \\langle \\xi , u \\rangle | ^ 2 . \\end{align*}"}
-{"id": "7774.png", "formula": "\\begin{align*} \\beta _ { i , d ^ { ( p ) } _ i } = \\frac { \\prod _ { j \\ne 0 } ^ n d ^ { ( p ) } _ j } { \\prod _ { j \\ne i } ^ n | d ^ { ( p ) } _ i - d ^ { ( p ) } _ j | } . \\end{align*}"}
-{"id": "1834.png", "formula": "\\begin{align*} L ( i , p ) = \\lim \\rho _ { p } ^ { - 1 } E _ i \\end{align*}"}
-{"id": "5610.png", "formula": "\\begin{align*} R _ { \\textit { H L } } ( n ) = \\sum _ { m _ 1 + m _ 2 ^ 2 = n } \\Lambda ( m _ 1 ) , \\end{align*}"}
-{"id": "3360.png", "formula": "\\begin{align*} \\| M \\| _ { r \\ast } & : = \\max _ { \\| X \\| _ r \\leq 1 } \\langle M , X \\rangle = \\max _ { \\sum _ { i = 1 } ^ r s _ i ^ 2 \\leq 1 } \\left [ \\sum _ { i = 1 } ^ { r } \\sigma _ i ( M ) s _ i + s _ r \\sum _ { i = r + 1 } ^ { n } \\sigma _ i ( M ) \\right ] . \\end{align*}"}
-{"id": "7761.png", "formula": "\\begin{align*} \\mathbb P \\left [ T _ N \\le 0 \\right ] = \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "9308.png", "formula": "\\begin{align*} d \\int _ D p ( t , x , z ) d x & = - \\int _ D \\frac { \\partial } { \\partial x } p ( t , x , z ) d x d t - \\frac { a _ 0 ( t , z ) } { b _ 0 ( t , z ) } \\int _ D p ( t , x , z ) d x d B ( t ) \\\\ & = [ p ( t , x , z ) ] _ { x \\in \\partial D } d t - \\frac { a _ 0 ( t , z ) } { b _ 0 ( t , z ) } \\int _ D p ( t , x , z ) d x d B ( t ) \\end{align*}"}
-{"id": "2964.png", "formula": "\\begin{align*} \\lambda _ t ( x , \\lambda _ t ( y , z ) ) = \\lambda _ t ( \\lambda _ t ( x , y ) , z ) + \\lambda _ t ( y , \\lambda _ t ( x , z ) ) . \\end{align*}"}
-{"id": "8808.png", "formula": "\\begin{align*} I ( u ^ + , u ^ + ) = n H \\int _ { \\Sigma } u ^ + d v o l _ { \\Sigma } \\ , . \\end{align*}"}
-{"id": "4520.png", "formula": "\\begin{align*} S _ 2 & = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ \\frac { a _ n } { n - 1 } ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log ^ 2 u _ { x , s } \\ , d s \\ , d x = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "6542.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\sum \\limits ^ { k } _ { i = 0 } \\delta _ i e _ { n - i } + A ( t _ n ) e _ n = \\sum \\limits ^ { k - 1 } _ { i = 0 } \\gamma _ i b _ { n - i - 1 } - d _ n , n = k , \\dotsc , N , \\end{align*}"}
-{"id": "1761.png", "formula": "\\begin{align*} w s _ \\alpha z _ \\mu t ( z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu ) & = \\Pi ^ S ( w s _ \\alpha z _ \\mu t ( z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu ) ) \\\\ & = \\lfloor w s _ \\alpha \\rfloor z _ { z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu } t ( z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu ) & ( & \\mbox { b y L e m m a \\ref { 2 . 2 . 6 } ( 3 ) } ) , \\end{align*}"}
-{"id": "8250.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial x } { \\partial s _ 2 } = \\Im ( f _ 1 ( 0 , 0 , s _ 3 ) + f _ 2 ( 0 , 0 , s _ 3 ) ) \\\\ \\frac { \\partial y } { \\partial s _ 2 } = \\Re ( f _ 2 ( 0 , 0 , s _ 3 ) - f _ 1 ( 0 , 0 , s _ 3 ) ) \\\\ \\frac { \\partial t } { \\partial s _ 2 } = \\Im g ( 0 , 0 , s _ 3 ) \\end{cases} \\end{align*}"}
-{"id": "5420.png", "formula": "\\begin{align*} K ^ n ( x , A ) = P r [ f ^ { ( n ) } ( x , \\xi ^ { ( n ) } ) \\in A ] = \\int _ A f ^ { ( n ) } ( x , \\omega ^ n ) \\lambda ^ n ( d \\omega ^ n ) . \\end{align*}"}
-{"id": "742.png", "formula": "\\begin{align*} N ( w ) : = \\{ \\alpha \\in \\Phi ^ + \\mid w \\alpha \\in - \\Phi ^ + \\} . \\end{align*}"}
-{"id": "6107.png", "formula": "\\begin{align*} z ' = \\alpha z ( 1 + z \\beta ( z ) ) \\end{align*}"}
-{"id": "5174.png", "formula": "\\begin{gather*} [ \\delta ( p ) ] = [ \\delta ( \\tau _ 1 ) ] + \\cdots + [ \\delta ( \\tau _ k ) ] = [ \\delta ( q ) ] . \\end{gather*}"}
-{"id": "9762.png", "formula": "\\begin{align*} \\mathcal { R } ( \\mathcal { E } ) : = \\{ \\vec { x } \\in \\mathcal { X } \\ | \\ \\mathcal { A } ( \\vec { x } ) = { \\mathcal { E } } \\} \\end{align*}"}
-{"id": "2380.png", "formula": "\\begin{align*} R _ { \\gamma f } ( x ) : = P _ { \\gamma f } ^ 2 ( x ) = 2 { \\rm { p r o x } } _ { \\gamma f } ( x ) - x . \\end{align*}"}
-{"id": "9412.png", "formula": "\\begin{align*} E ^ W = \\frac { T ^ W _ m } { T ^ W _ P } , \\end{align*}"}
-{"id": "3073.png", "formula": "\\begin{align*} \\lambda _ { n , \\beta } = W _ { n , \\beta } ^ { - 2 } \\sum _ { | u | = | v | = n } e ^ { \\beta ( 2 m _ n - V ( u ) - V ( v ) ) } \\delta _ { | u \\wedge v | } , \\end{align*}"}
-{"id": "2330.png", "formula": "\\begin{align*} l ( x ) = \\prod _ { i = 1 } ^ d ( x - z _ i ) , \\end{align*}"}
-{"id": "4532.png", "formula": "\\begin{align*} U _ { 1 2 } : = \\biggl | \\int _ { \\mathcal { X } } f ( x ) \\int _ { u _ n ^ * ( x ) } ^ \\infty \\log \\bigl ( u f ( x ) \\bigr ) \\ , d ( F _ { n , x } ^ - - F _ { n , x } ) ( u ) \\ , d x \\biggr | = o ( n ^ { - ( 3 - \\epsilon ) } ) . \\end{align*}"}
-{"id": "6587.png", "formula": "\\begin{align*} p _ { C } ^ { \\alpha } ( x ) & : = \\alpha r _ { C } ^ * ( x ) + \\tfrac { 1 - \\alpha } { 2 } \\| x \\| ^ 2 , \\\\ P _ { C } ^ { \\alpha } ( x ) & : = \\nabla p _ { C } ^ { \\alpha } ( x ) = \\alpha \\Pi _ { C } ( x ) + ( 1 - \\alpha ) x \\\\ R _ C ( x ) & : = 2 \\Pi _ C ( x ) - x . \\end{align*}"}
-{"id": "928.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\| U ^ { * } _ { 1 } \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| V ^ { * } _ { 1 } \\| _ { S _ { \\widehat { p } _ { 2 } } } . \\end{align*}"}
-{"id": "3501.png", "formula": "\\begin{align*} ( \\bar { g } , \\bar { \\pi } ) & = ( g , \\pi ) \\mbox { i n } B _ R \\\\ ( \\bar { g } , \\bar { \\pi } ) & = ( g ^ { \\theta } , \\pi ^ { \\theta } ) \\mbox { i n } M \\setminus B _ { 2 R } \\end{align*}"}
-{"id": "4701.png", "formula": "\\begin{align*} d _ k ( x , x ' ) & = \\sum _ { i = 0 } ^ r ( \\max ( s _ i , s ' _ i ) - s _ i ) \\\\ & = \\sum _ { i = 0 } ^ r ( \\max ( s _ i , s ' _ i ) - s ' _ i ) \\\\ & = \\sum _ { i = 0 } ^ { r } \\abs { s _ i - s ' _ i } = d _ { \\ell _ 1 } ( \\sigma _ k ( z ) , \\sigma _ k ( z ' ) ) . \\end{align*}"}
-{"id": "5686.png", "formula": "\\begin{align*} e _ 0 ( e _ 0 + 1 ) \\left ( e _ 0 + \\frac { \\alpha } { \\alpha + \\gamma } \\right ) \\left ( e _ 0 + \\frac { \\alpha + \\gamma + q - 1 } { q - 1 } \\right ) = 0 \\ ; . \\end{align*}"}
-{"id": "2373.png", "formula": "\\begin{align*} \\langle \\nabla f _ 1 ( x ) , x \\rangle - f _ 1 ( x ) = \\langle P x + q , x \\rangle - ( \\tfrac { 1 } { 2 } \\langle P x , x \\rangle + \\langle q , x \\rangle ) = \\tfrac { 1 } { 2 } \\langle P x , x \\rangle . \\end{align*}"}
-{"id": "967.png", "formula": "\\begin{align*} u = S ' _ q u + S '' _ q u + H _ q u \\ ; , \\ ; u \\in L ^ { 2 } _ { 0 , q } ( M ) \\ ; . \\end{align*}"}
-{"id": "9926.png", "formula": "\\begin{align*} E ' \\ ; = \\ ; E \\ , \\cup \\ , \\{ ( 1 , 0 , 0 , 0 ) , ( 0 , 1 , 0 , 0 ) , ( 0 , 0 , 1 , 0 ) , ( 0 , 0 , 0 , 1 ) \\} , \\end{align*}"}
-{"id": "2931.png", "formula": "\\begin{align*} ( s _ \\mu [ s _ \\nu ] ( Z ) ) ' \\ = \\ \\begin{cases} \\ s _ \\mu [ s _ { \\nu ' } ] ( Z ) & \\mbox { i f $ | \\nu | $ i s e v e n } ; \\cr \\ s _ { \\mu ' } [ s _ { \\nu ' } ] ( Z ) & \\mbox { i f $ | \\nu | $ i s o d d } . \\cr \\end{cases} \\end{align*}"}
-{"id": "3688.png", "formula": "\\begin{align*} \\frac { d Z } { d t } = \\int \\nabla \\cdot \\left ( \\frac { 1 } { 2 } q ^ 2 \\nabla H _ { \\mu } \\right ) d A = 0 ~ . \\end{align*}"}
-{"id": "2944.png", "formula": "\\begin{align*} | g ( r ) | = | \\langle \\widehat f ( T _ { r , \\lambda } ) e ^ \\lambda _ { i _ 0 } , e ^ \\lambda _ { j _ 0 } \\rangle _ { H ( K , \\lambda ) } | \\leq \\| \\widehat f ( T _ { r , \\lambda } ) \\| _ { H S } \\leq C _ \\lambda e ^ { - \\theta ( r ) } . \\end{align*}"}
-{"id": "8980.png", "formula": "\\begin{align*} \\mathcal K \\ , : = \\ , \\{ \\varphi \\in W ^ { s , p } _ 0 ( \\Omega ) \\ , : \\ , 0 \\leq \\varphi \\leq v \\ , \\ , \\ , \\ , \\Omega \\} \\ , . \\end{align*}"}
-{"id": "2301.png", "formula": "\\begin{align*} \\begin{aligned} J _ m & \\le C ( \\tilde \\lambda + r ) \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( D ) } \\\\ & + \\varepsilon \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( D ) } + C _ \\varepsilon \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( X ) } + C \\delta . \\end{aligned} \\end{align*}"}
-{"id": "1164.png", "formula": "\\begin{align*} v ( x ) = A _ { 1 } ( t , z ) ( x - 1 ) + B _ { 1 } ( t , z ) , \\end{align*}"}
-{"id": "1175.png", "formula": "\\begin{align*} \\rho = \\sum _ { j = 1 } ^ N m _ j \\delta _ { x _ j } , 0 < x _ 1 < \\dots < x _ N < 1 , \\end{align*}"}
-{"id": "6463.png", "formula": "\\begin{align*} ( 1 - q ) \\zeta _ { 2 } ( q ) \\leq 4 + 9 \\frac { n + 2 } { \\beta _ { 0 } } = : c _ { 2 } = c _ { 2 } ( n , \\beta _ { 0 } ) , \\end{align*}"}
-{"id": "1889.png", "formula": "\\begin{align*} f ( x ) = \\frac 1 { 2 \\pi } \\int _ { \\R } e ^ { - i u x } \\big ( \\phi ( u ) \\big ) ^ { 1 / K } \\d u . \\end{align*}"}
-{"id": "9353.png", "formula": "\\begin{align*} \\hat { \\pi } ( t ) = - \\frac { \\alpha _ 0 ( t ) \\mathbb { E } [ U ' ( X ( t ) ) | \\mathcal { R } _ t ] } { \\beta _ 0 ^ 2 ( t ) \\mathbb { E } [ U '' ( X ( t ) ) | \\mathcal { R } _ t ] } . \\end{align*}"}
-{"id": "9237.png", "formula": "\\begin{align*} H = H ( t , x , Y ( t , x , z ) , u ( t , z ) , \\hat { p } ( t , x ) , \\hat { q } ( t , x ) , \\hat { r } ( t , x , z , \\cdot ) ) , \\end{align*}"}
-{"id": "2149.png", "formula": "\\begin{align*} h _ { \\alpha , m } ( t ) + m ( h _ { \\alpha , m } * g _ { \\alpha } ) ( t ) = m g _ { \\alpha } ( t ) , t > 0 , \\ , \\ , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "341.png", "formula": "\\begin{align*} g A g ^ t = M _ { ( 1 i _ 0 ) } M _ { ( 2 j _ 0 ) } A M _ { ( 2 j _ 0 ) } ^ t M _ { ( 1 i _ 0 ) } ^ t = M _ { ( 2 j _ 0 ) } M _ { ( 1 i _ 0 ) } A M _ { ( 2 j _ 0 ) } M _ { ( 1 i _ 0 ) } \\end{align*}"}
-{"id": "7608.png", "formula": "\\begin{align*} \\lambda _ 1 \\mathbf { v } _ u = \\sum _ { w \\sim u } \\mathbf { v } _ w . \\end{align*}"}
-{"id": "3065.png", "formula": "\\begin{align*} W _ { n , \\beta } = n ^ { 3 \\beta / 2 } e ^ { n \\kappa ( \\beta ) } W _ n ^ { ( \\beta ) } = \\sum _ { | v | = n } e ^ { \\beta ( m _ n - V ( v ) ) } . \\end{align*}"}
-{"id": "8355.png", "formula": "\\begin{align*} \\phi ( x ) = \\sum _ { \\lambda \\in \\Lambda } \\phi _ \\lambda ( p _ \\lambda x p _ \\lambda ) , \\ \\ \\ x \\in M ^ + . \\end{align*}"}
-{"id": "6629.png", "formula": "\\begin{align*} \\partial _ x \\psi ( \\Phi _ t ( y ) , t ) = 0 \\end{align*}"}
-{"id": "9424.png", "formula": "\\begin{align*} v _ i ( y ) = \\sum _ { x \\in g _ i ^ { - 1 } ( y ) } w _ i ( x ) . \\end{align*}"}
-{"id": "9815.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m } f _ { i } ^ { \\ast } \\prec { \\bf 1 } _ { Q _ { m } } \\end{align*}"}
-{"id": "2274.png", "formula": "\\begin{align*} \\frac 1 \\tau \\sum _ { j = 0 } ^ k \\delta _ j v _ { n - j } + A ( s ) v _ n = f _ n , k \\le n \\le N , \\end{align*}"}
-{"id": "1193.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 ( n ) = \\{ w \\in \\mathcal { A } ( n ) ~ | ~ w = 0 1 2 0 2 1 \\{ 0 1 2 \\} * 1 2 0 2 1 0 \\} \\subseteq \\mathcal { A } ( n ) \\end{align*}"}
-{"id": "9134.png", "formula": "\\begin{align*} & A ( t ) ^ { \\frac { 1 } { 2 } } u ( t ) - A ( s ) ^ { \\frac { 1 } { 2 } } u ( s ) \\\\ & = A ( t ) ^ { \\frac { 1 } { 2 } } e ^ { - ( t - s ) A ( t ) } u ( s ) - A ( s ) ^ { \\frac { 1 } { 2 } } u ( s ) \\\\ & + A ( t ) ^ { \\frac { 1 } { 2 } } \\int _ { s } ^ { t } { e ^ { - ( t - r ) A ( t ) } ( \\mathcal { A } ( t ) - \\mathcal { A } ( r ) ) u ( r ) d r } \\\\ & + A ( t ) ^ { \\frac { 1 } { 2 } } \\int _ { s } ^ { t } { e ^ { - ( t - r ) A ( t ) } [ ( - B ( r ) + I ) A ( r ) u ( r ) - P ( r ) u ( r ) + f ( r ) ] } d r . \\end{align*}"}
-{"id": "9333.png", "formula": "\\begin{align*} & \\int _ D U ' ( Y ( T , x , z ) ) Y ( T , x , z ) d x = \\tilde { p } ( 0 , z ) \\exp \\big ( - \\int _ 0 ^ T \\{ \\Phi ( s , z ) + \\frac { b _ 0 ( s , z ) } { \\sigma _ 0 ( s , z ) } \\} d B ( s ) \\\\ & + \\frac { 1 } { 2 } \\int _ 0 ^ T \\{ \\Phi ^ 2 ( s , z ) - \\frac { b _ 0 ^ 2 ( s , z ) } { \\sigma _ 0 ^ 2 ( s , z ) } \\} d s \\big ) = : \\tilde { p } ( 0 , z ) \\Gamma ( T , z ) . \\end{align*}"}
-{"id": "9389.png", "formula": "\\begin{align*} q ( x ) = Z \\chi _ { [ - \\rho , \\rho ] } ( x ) = \\left \\{ \\begin{array} { l } Z , x \\in [ - \\rho , \\rho ] \\\\ 0 , x \\in \\mathbb { R } \\setminus [ - \\rho , \\rho ] \\end{array} \\right . Z \\in \\mathbb { R } , \\rho > 0 . \\end{align*}"}
-{"id": "1606.png", "formula": "\\begin{align*} C ^ { x } \\left ( x \\right ) = \\sigma \\left ( x \\right ) \\int \\frac { m } { \\sigma \\left ( x \\right ) } d x + c \\sigma \\left ( x \\right ) \\end{align*}"}
-{"id": "832.png", "formula": "\\begin{align*} \\vec { z } ( \\ell ( t ) , \\ldots , \\ell ( 0 ) ) _ { i } = \\left \\{ \\begin{array} { l l } z _ { i } & ( 1 \\le i < \\ell ( 0 ) , i \\not = \\ell ( t ) , \\ldots , \\ell ( 1 ) ) \\\\ z _ { \\ell ( s - 1 ) } & ( i = \\ell ( s ) , 1 \\le s \\le t ) \\\\ z _ { i + 1 } & ( \\ell ( 0 ) \\le i \\le k - 1 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8961.png", "formula": "\\begin{align*} y ^ { \\Delta } ( t ) = p ( t ) y ( t ) \\end{align*}"}
-{"id": "8779.png", "formula": "\\begin{align*} T _ 0 T _ 1 T _ 0 T _ 1 & = T _ 1 T _ 0 T _ 1 T _ 0 , \\\\ T _ i T _ { i + 1 } T _ i & = T _ { i + 1 } T _ i T _ { i + 1 } , \\\\ T _ n T _ { n - 1 } T _ n T _ { n - 1 } & = T _ { n - 1 } T _ n T _ { n - 1 } T _ n , \\end{align*}"}
-{"id": "5765.png", "formula": "\\begin{align*} R ( \\lambda , \\mu ) = \\begin{pmatrix} f ( \\lambda , \\mu ) & 0 & 0 & 0 \\\\ 0 & 1 & g ( \\lambda , \\mu ) & 0 \\\\ 0 & g ( \\lambda , \\mu ) & 1 & 0 \\\\ 0 & 0 & 0 & f ( \\lambda , \\mu ) \\end{pmatrix} . \\end{align*}"}
-{"id": "6166.png", "formula": "\\begin{align*} V _ 1 ( V _ 2 V _ 1 ) ^ * \\phi ( t ^ { - 1 } D ) f ( V _ 2 V _ 1 ) V _ 1 ^ * & = V _ 1 V _ 1 ^ * V _ 2 ^ * \\phi ( t ^ { - 1 } D ) V _ 2 f V _ 1 V _ 1 ^ * \\\\ & = V _ 1 V _ 1 ^ * V _ 2 ^ * \\phi ( t ^ { - 1 } D ) V _ 2 f \\\\ & \\varpropto ( - 1 ) ^ { \\partial \\phi \\cdot \\partial f } V _ 1 V _ 1 ^ * f V _ 2 ^ * \\phi ( t ^ { - 1 } D ) V _ 2 \\\\ & = ( - 1 ) ^ { \\partial \\phi \\cdot \\partial f } f V _ 2 ^ * \\phi ( t ^ { - 1 } D ) V _ 2 \\\\ & \\varpropto V _ 2 ^ * \\phi ( t ^ { - 1 } D ) f V _ 2 \\ , . \\end{align*}"}
-{"id": "2347.png", "formula": "\\begin{align*} \\theta _ 1 \\cdot \\frac { a _ 1 + a _ n } { 2 } + \\theta _ 2 \\cdot \\frac { a _ 2 - a _ 1 } { 2 } + \\dots + \\theta _ n \\cdot \\frac { a _ n - a _ { n - 1 } } { 2 } = a _ i . \\end{align*}"}
-{"id": "4058.png", "formula": "\\begin{align*} 3 p - 2 q \\leq ( \\alpha - 4 ) ( p - q ) + 3 p - 2 q = 1 + n + k \\leq 1 + \\dfrac { p } { 2 } + q , \\end{align*}"}
-{"id": "5131.png", "formula": "\\begin{align*} \\prod _ { 1 \\le i \\le m } ^ { \\curvearrowleft } ( f ( w , z _ { i } ) Y _ { i - 1 } ( w , z _ { i } ) ) u ( b , a ^ { m } ) = q ^ { m } u ( a ^ { m } , b ) + \\sum _ { \\ell = 1 } ^ { m } g ( w , z _ { \\ell } ) Z _ { \\ell } ^ { [ 1 , m ] } ( \\vec { z } ) u ( b , a ^ { m } ) \\end{align*}"}
-{"id": "401.png", "formula": "\\begin{align*} G _ { \\phi , p } ( x ) = & L _ p ( 1 + x , \\phi ) L _ p ( 1 + 3 x , \\phi ) \\\\ & \\times \\left ( 1 + \\frac { \\rho _ \\phi ( p ^ 2 ) - \\rho _ \\phi ( p ) ^ 2 } { p ^ { 2 + 4 x } } + O \\left ( \\frac { ( 1 + | \\rho _ \\phi ( p ) | ) ^ { 3 } } { p ^ { 3 + 7 x } } \\right ) \\right ) . \\end{align*}"}
-{"id": "6297.png", "formula": "\\begin{align*} \\| f \\| _ { B _ { p , q } ^ s } = \\left ( \\sum _ { j = 0 } ^ { \\infty } 2 ^ { j s q } \\| \\triangle _ j f \\| _ { L ^ p } ^ q \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "2992.png", "formula": "\\begin{align*} f _ 0 ^ { \\lambda _ i ( x , y ) } = - f _ i ^ { \\lambda _ 0 ( x , y ) } - \\sum _ { m + n = i , ~ m , n > 0 } f _ m ^ { \\lambda _ n ( x , y ) } + \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ x , f _ n ^ { y } ] + [ f _ 0 ^ x , f _ i ^ { y } ] + [ f _ i ^ x , f _ 0 ^ { y } ] . \\end{align*}"}
-{"id": "2775.png", "formula": "\\begin{align*} \\limsup \\lambda _ i ( \\alpha ) = \\mu ( \\alpha ) . \\end{align*}"}
-{"id": "5853.png", "formula": "\\begin{align*} 2 f _ { I , t } + c = 0 , \\end{align*}"}
-{"id": "9141.png", "formula": "\\begin{align*} \\frac { k } { m } \\leq \\frac { \\alpha _ u + ( m - \\alpha _ u ) / 2 } { m } = \\frac { 1 } { 2 } + \\frac { \\alpha _ u } { 2 m } \\leq \\frac { 1 } { 2 } + \\frac { m / s } { 2 m } = \\frac { 1 } { 2 } + \\frac { 1 } { 2 s } = \\frac { s + 1 } { 2 s } . \\end{align*}"}
-{"id": "7942.png", "formula": "\\begin{align*} \\alpha ( i , j ) + \\alpha ( j , i ) = \\lim _ { m \\to \\infty } \\alpha _ m ( i , j ) + \\lim _ { m \\to \\infty } \\alpha _ m ( j , i ) = \\lim _ { m \\to \\infty } ( \\alpha _ m ( i , j ) + \\alpha _ m ( j , i ) ) = 1 , \\end{align*}"}
-{"id": "7627.png", "formula": "\\begin{align*} \\lambda _ 1 = \\frac { n } { 2 } + c _ 1 \\sqrt { n } , \\end{align*}"}
-{"id": "8220.png", "formula": "\\begin{align*} E ^ 2 = ( \\Omega + 8 A ^ 2 ) ^ 2 \\left [ 1 - \\frac { \\gamma ^ 2 } { ( \\Omega + 4 A ^ 2 ) ^ 2 } \\right ] . \\end{align*}"}
-{"id": "7466.png", "formula": "\\begin{align*} | H ( V ) | = | a _ 1 H _ 1 ( V ) + a _ 2 H _ 2 ( V ) | \\leq ( | a _ 1 | c _ 1 + | a _ 2 | c _ 2 ) \\cdot ( V ) , \\end{align*}"}
-{"id": "5365.png", "formula": "\\begin{align*} P ( P ( x , y ) , z ) + P ( P ( y , z ) , x ) + P ( P ( z , x ) , y ) ~ = ~ 0 . \\end{align*}"}
-{"id": "705.png", "formula": "\\begin{align*} & h _ { E } = h _ { H } \\circ f - \\left < A \\vec { c } , { \\bf h } _ { \\vec { D } } \\right > \\\\ & { \\bf h } _ { \\vec { E } } = \\left ( \\begin{array} { c } h _ { E _ { 1 } } \\\\ h _ { E _ { 2 } } \\\\ \\vdots \\\\ h _ { E _ { r } } \\end{array} \\right ) = { \\bf h } _ { \\vec { D } } \\circ f - { } ^ { \\rm t } A { \\bf h } _ { \\vec { D } } \\ . \\end{align*}"}
-{"id": "4892.png", "formula": "\\begin{align*} \\delta ( x - [ r _ 0 ] ) & \\equiv \\phi ( \\sum _ { k = 1 } ^ \\nu p ^ { k - 1 } [ \\phi ^ { - k } ( r _ k ) ] ) \\bmod I ^ \\nu \\\\ & \\equiv \\sum _ { k = 1 } ^ \\nu p ^ { k - 1 } [ \\phi ^ { - ( k - 1 ) } ( r _ k ) ] ) \\bmod I ^ \\nu \\\\ & \\equiv \\sum _ { k = 0 } ^ { \\nu - 1 } p ^ { k } [ \\phi ^ { - k } ( r _ { k + 1 } ) ] ) \\bmod I ^ \\nu \\end{align*}"}
-{"id": "4774.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c t + a ) ^ 2 - t ^ 2 } , a = c o n s t , \\ ; c = c o n s t \\neq 0 , \\ ; c ^ 2 \\neq \\kappa ^ 2 , \\end{align*}"}
-{"id": "7351.png", "formula": "\\begin{align*} ( F ^ * v _ { - 1 } , v _ 0 ) = ( E K ^ { - 1 } v _ { - 1 } , v _ 0 ) = [ 2 ] ^ { 1 / 2 } q ^ 2 ( v _ 0 , v _ 0 ) . \\end{align*}"}
-{"id": "4342.png", "formula": "\\begin{align*} \\mathcal { R } ( a ) < \\infty \\ ; \\ ; \\limsup _ { r \\to \\mathcal { R } ( a ) } \\left ( | f ( r ) | + | g ( r ) | \\right ) = \\infty \\ . \\end{align*}"}
-{"id": "2068.png", "formula": "\\begin{align*} \\hat { H } ( s ) = ( \\hat { C } _ { p } + s \\hat { C } _ { v } ) \\hat { X } ( s ) , \\end{align*}"}
-{"id": "2630.png", "formula": "\\begin{align*} F _ m ( \\lambda ) + G _ m ( \\lambda ) = d _ m ( \\lambda ) ( d _ m ( \\lambda ) ) ^ * , \\ ; d _ m ( \\lambda ) = \\sum _ { u = 0 } ^ \\infty d _ m ( u ) e ^ { - i u \\lambda } , \\end{align*}"}
-{"id": "1070.png", "formula": "\\begin{align*} \\binom { N } { 2 } = ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } \\leq k \\left ( k - \\alpha - \\frac { 3 } { 8 } + \\frac { 1 } { 2 n } + \\frac { 1 } { n ^ 2 } \\right ) \\frac { n ^ 2 } { 2 } \\end{align*}"}
-{"id": "4602.png", "formula": "\\begin{align*} g A g ^ t = M _ { ( 1 i _ 0 ) } M _ { ( 2 j _ 0 ) } A M _ { ( 2 j _ 0 ) } ^ t M _ { ( 1 i _ 0 ) } ^ t = M _ { ( 2 j _ 0 ) } M _ { ( 1 i _ 0 ) } A M _ { ( 2 j _ 0 ) } M _ { ( 1 i _ 0 ) } \\end{align*}"}
-{"id": "2064.png", "formula": "\\begin{align*} & \\hat { M } = V ^ { T } M V , \\ \\hat { D } = V ^ { T } D V , \\ \\hat { K } = V ^ { T } K V , \\ \\hat { F } = V ^ { T } F , \\\\ & \\hat { C } _ { p } = C _ { p } V \\ \\ \\hat { C } _ { v } = C _ { v } V . \\end{align*}"}
-{"id": "6246.png", "formula": "\\begin{align*} & M \\ddot { x } ( t ) + D \\dot { x } ( t ) + K x ( t ) = F u ( t ) , \\\\ & y ( t ) = C _ p x ( t ) , \\end{align*}"}
-{"id": "2960.png", "formula": "\\begin{align*} g ( \\sigma , 1 / 2 ) = H _ 2 ( \\sigma ) + \\sigma \\log ( 2 ^ { \\gamma - 1 } - 1 ) - \\gamma \\frac { \\delta - 1 } { \\delta } + 1 . \\end{align*}"}
-{"id": "950.png", "formula": "\\begin{align*} \\bar \\partial _ M f = \\sum _ { j = 1 } ^ { m - 1 } \\sideset { } { ' } \\sum _ { \\vert J \\vert = q } ( \\overline { L _ j } f _ J ) \\overline { \\omega _ j } \\wedge \\overline { \\omega _ J } + \\sideset { } { ' } \\sum _ { \\vert J \\vert = q } f _ J \\bar \\partial _ M \\overline { \\omega _ J } . \\end{align*}"}
-{"id": "1762.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta u _ { i } ^ { \\varepsilon } = f _ { i } ( x , u _ { i } ^ { \\varepsilon } ) - \\frac { 1 } { \\varepsilon } u _ { i } ^ { \\varepsilon } \\sum \\limits _ { j \\neq i } ( u _ { j } ^ { \\varepsilon } ) ^ { \\beta } ( x ) & \\Omega , \\\\ u _ { i } ( x ) = \\phi _ { i } ( x ) & \\partial \\Omega , \\\\ i = 1 , \\cdots , m . \\end{array} \\right . \\end{align*}"}
-{"id": "7583.png", "formula": "\\begin{align*} \\mathcal A _ { 1 2 } = \\mathcal A _ { 2 1 } \\cup \\mathcal A _ { 2 2 } , \\mathcal A _ { 2 1 } \\cap \\mathcal A _ { 2 2 } = \\emptyset , \\end{align*}"}
-{"id": "2661.png", "formula": "\\begin{align*} \\hat F _ { P I } : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f \\left ( \\hat p _ k ( X _ i ) \\right ) . \\end{align*}"}
-{"id": "6635.png", "formula": "\\begin{align*} L ( s , V ) = \\varepsilon ( s , V ) L ( 1 - s , V ^ * ) \\end{align*}"}
-{"id": "149.png", "formula": "\\begin{align*} ( \\widehat { C _ E } , \\| \\cdot \\| _ { \\widehat { C _ E } } ) = \\overline { ( C _ E , \\| \\cdot \\| _ { C _ { \\widehat { E } } } ) } ^ { \\| \\cdot \\| _ { C _ { \\widehat { E } } } } \\subseteq ( C _ { \\widehat { E } } , \\| \\cdot \\| _ { C _ { \\widehat { E } } } ) \\end{align*}"}
-{"id": "6690.png", "formula": "\\begin{align*} ( A ) ~ ~ c _ 1 ( E ) . H = \\frac { 3 r } { 2 } H ^ 2 \\mbox { a n d } ( B ) ~ ~ c _ 2 ( E ) = \\frac { 1 } { 2 } c _ 1 ( E ) ^ 2 - ( H ^ 2 - 1 ) r , \\end{align*}"}
-{"id": "9432.png", "formula": "\\begin{align*} \\bigl ( d _ 1 \\sim _ { \\tau ( f ) } d _ 2 \\bigr ) \\Longleftrightarrow \\bigl ( f ( d _ 1 ) = f ( d _ 2 ) \\bigr ) . \\end{align*}"}
-{"id": "8613.png", "formula": "\\begin{align*} \\psi \\overline { \\overline { N } } ( a ) & = \\alpha \\xi _ { e _ { k } a } \\\\ \\pi _ { b e _ { k } } \\beta \\psi & = \\overline { \\overline { N } } ( b ) \\end{align*}"}
-{"id": "7915.png", "formula": "\\begin{align*} \\mathrm { d i s t } _ { g _ j } ( p , q ) \\le L _ { g _ j } ( \\gamma ) \\le C _ \\beta ^ 2 L _ { g _ 1 } ( \\gamma ) = C _ \\beta ^ 2 \\mathrm { d i s t } _ { g _ 1 } ( p , q ) \\le C _ \\beta ^ 2 \\mathrm { d i a m } ( M , g _ 1 ) . \\end{align*}"}
-{"id": "4535.png", "formula": "\\begin{align*} | U _ 2 | \\leq U _ { 2 1 } + U _ { 2 2 } = O \\biggl ( \\frac { k ^ { 1 / 2 } } { n } \\max \\biggl \\{ \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "583.png", "formula": "\\begin{align*} \\alpha _ 3 ( x ) = ( \\pi ( x ) , \\pi ( \\delta ( x ) ) , \\pi ( \\delta ( \\delta ( x - [ \\pi ( x ) ] ) + ( - 1 ) ^ p [ \\pi ( \\delta ( x ) ) ] ) ) ) \\end{align*}"}
-{"id": "8059.png", "formula": "\\begin{align*} i c ( n - 1 ) & = 2 \\sum _ { i , j \\in I } w ( i j ) + \\sum _ { i \\in I , j \\in I ^ { c } } w ( i j ) = 2 \\sum _ { i , j \\in I } w ( i j ) - 2 \\sum _ { i \\in I , j \\in I ^ { c } } w ( i j ) \\le 2 \\sum _ { i , j \\in I } w ( i j ) - 2 f ( i ) \\\\ & = 2 \\sum _ { i , j \\in I } w ( i j ) + c i ( n - 1 ) - a _ { i } , \\end{align*}"}
-{"id": "2830.png", "formula": "\\begin{align*} B _ k = \\{ \\xi _ p ( T _ k ) \\le S _ k \\} = \\left \\{ \\sup _ { S _ k < s \\le T _ k } \\frac { X _ { r : n } ( s ) } { f _ p ( s ) } < 1 \\right \\} . \\end{align*}"}
-{"id": "8146.png", "formula": "\\begin{align*} r : = \\frac { N p } { N - p + a p } . \\end{align*}"}
-{"id": "1225.png", "formula": "\\begin{align*} \\phi _ \\mathcal { M } ( t ) = \\frac { t } { W ( t ) } \\Big { / } \\varphi ^ { - 1 } \\left ( \\frac { 1 } { W ( t ) } \\right ) , \\ \\ \\ t \\in ( 0 , \\gamma ) . \\end{align*}"}
-{"id": "4439.png", "formula": "\\begin{align*} \\mathcal { W } ^ { ( k ) } : = \\biggl \\{ w \\in \\mathbb { R } ^ k : & \\sum _ { j = 1 } ^ k w _ j \\frac { \\Gamma ( j + 2 \\ell / d ) } { \\Gamma ( j ) } = 0 \\ , \\ , \\ell = 1 , \\ldots , \\lfloor d / 4 \\rfloor \\\\ & \\sum _ { j = 1 } ^ k w _ j = 1 \\ , \\ , \\ , w _ j = 0 \\ , \\ , \\ , j \\notin \\{ \\lfloor k / d \\rfloor , \\lfloor 2 k / d \\rfloor , \\ldots , k \\} \\biggr \\} . \\end{align*}"}
-{"id": "2414.png", "formula": "\\begin{align*} \\begin{cases} \\frac { 1 } { p _ { 2 } } \\leq \\frac { 1 } { p _ { 1 } } \\\\ s _ { 2 } + R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ { 1 } , \\alpha _ { 2 } ) + \\frac { n ( 1 - \\alpha _ { 1 } \\vee \\alpha _ { 2 } ) } { q _ { 2 } } < s _ { 1 } + \\frac { n ( 1 - \\alpha _ { 1 } \\vee \\alpha _ { 2 } ) } { q _ { 1 } } \\\\ \\frac { 1 } { q _ { 2 } } > \\frac { 1 } { q _ { 1 } } \\end{cases} \\end{align*}"}
-{"id": "2709.png", "formula": "\\begin{align*} \\varphi _ l ( x ) : = \\sup \\left \\{ u _ l ( x ) , \\ \\textrm { w h e r e } t \\to u _ t \\textrm { i s a s u b g e o d e s i c o f } \\ \\varphi _ 0 , \\varphi _ 1 \\right \\} , \\ \\ l \\in [ 0 , 1 ] , x \\in X . \\end{align*}"}
-{"id": "924.png", "formula": "\\begin{align*} \\textup { T r } ^ { p } ( A \\Sigma _ { Z } A ^ { T } ) & = \\sum _ { k } \\left ( \\sum _ { i } a ^ { 2 } _ { k i } \\sigma _ { i } \\right ) ^ { p } \\\\ & \\geq \\sum _ { k } \\sum _ { i } a ^ { 2 } _ { k i } \\sigma ^ { p } _ { i } \\\\ & = \\sum _ { i } \\sigma ^ { p } _ { i } \\\\ & = \\textup { T r } ^ { p } ( \\Sigma _ { Z } ) = \\| Z \\| ^ { p } _ { S _ { p } } . \\end{align*}"}
-{"id": "8367.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| e _ n x _ i v _ i e _ n \\| = 0 , \\ \\ \\ 1 \\leq i \\leq k . \\end{align*}"}
-{"id": "5741.png", "formula": "\\begin{align*} Q _ { L } ^ { S T } ( \\omega ) = S ( \\hat { \\delta } _ { 0 } ) ^ { \\mathrm { T } } I _ { L } ( \\hat \\delta _ 0 ) ^ { - 1 } S ( \\hat \\delta _ { 0 } ) . \\end{align*}"}
-{"id": "5992.png", "formula": "\\begin{align*} s _ \\beta x & = s _ \\gamma t ( a ' \\gamma ^ \\lor ) w z _ \\mu t ( \\mu ) \\\\ & = w s _ { w ^ { - 1 } \\gamma } z _ \\mu t ( a ' z _ \\mu ^ { - 1 } w ^ { - 1 } \\gamma ^ \\lor + \\mu ) \\\\ & = w s _ { \\alpha } z _ \\mu t ( a ' z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu ) . \\end{align*}"}
-{"id": "5122.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\left \\{ ( 1 - q ^ { 2 } ) ^ { - k } ( - s ) ^ { - \\sum _ { i = 1 } ^ { k } x _ { i } } \\prod _ { i = 1 } ^ { k } ( 1 + z _ { i } ) \\ , F _ { \\vec { x } } ( - z _ { 1 } / s , \\ldots , - z _ { k } / s ) \\right \\} . \\end{align*}"}
-{"id": "1186.png", "formula": "\\begin{align*} b _ { x x x } & = 0 , x \\neq x _ i , \\\\ z \\dot m _ i & = \\frac 1 2 [ b _ { x x } ] ( x _ i ) + z m _ i \\langle b _ x \\rangle ( x _ i ) \\\\ - z m _ i \\dot x _ i & = \\frac 1 2 [ b _ x ] ( x _ i ) + z m _ i b ( x _ i ) , \\end{align*}"}
-{"id": "8675.png", "formula": "\\begin{gather*} \\lim _ { x \\to 0 } \\sup _ { y \\in H } \\ , \\| \\nabla ^ K \\nabla _ { \\xi } ^ G R _ t [ \\Phi ] ( x + y ) - \\nabla \\nabla _ { \\xi } ^ G R _ t [ \\Phi ] ( y ) \\| _ { L ( K , J ) } \\\\ = \\lim _ { x \\to 0 } \\sup _ { y \\in H } \\sup _ { | k | _ K = 1 } \\ , | \\nabla _ k \\nabla _ { \\xi } ^ G R _ t [ \\Phi ] ( x + y ) - \\nabla _ k \\nabla _ { \\xi } ^ G R _ t [ \\Phi ] ( y ) | _ { J } = 0 , \\ ; \\ ; \\xi \\in U . \\end{gather*}"}
-{"id": "8173.png", "formula": "\\begin{align*} r _ { n , k } & = \\binom { 5 n } { k - 1 } \\frac { ( k - 1 ) ! } { n ! } B _ { n , k } ( 1 ! ( C _ 0 + C _ 1 ) , 2 ! ( C _ 1 + C _ 2 ) , \\dots ) , \\intertext { w h i c h b y m e a n s o f \\cite [ E x a m p l e ~ 3 . 2 ] { W W 0 9 } c a n b e w r i t t e n a s } r _ { n , k } & = \\binom { 5 n } { k - 1 } \\sum _ { j = 0 } ^ k \\frac { ( - 1 ) ^ { k - j } ( 2 j - k ) } { n k } \\binom { k } { j } \\binom { 2 ( n + j ) - k - 1 } { n - 1 } . \\end{align*}"}
-{"id": "8662.png", "formula": "\\begin{align*} \\nabla _ a ^ G F ( x ) = \\nabla _ { G a } F ( x ) \\in J , \\ ; \\ ; \\ ; \\ ; a \\in U , \\ ; x \\in H . \\end{align*}"}
-{"id": "6372.png", "formula": "\\begin{align*} u ( x ' , z ) = U _ a ( z ) \\Phi ( x ' , z ) . \\end{align*}"}
-{"id": "7229.png", "formula": "\\begin{align*} \\left \\{ \\bigoplus _ { \\lambda = 1 } ^ r L _ \\lambda \\left | L _ \\lambda \\in \\mathcal { P } ( \\widetilde { Y } ) , \\ d \\left ( \\mathbb { I } _ { \\widetilde { Y } } ^ { ( 1 ) } , \\ \\textstyle \\bigotimes _ { \\lambda = 1 } ^ r L _ \\lambda ^ { a _ \\lambda } \\right ) \\geq \\frac { 1 } { \\left ( 2 | a | \\right ) ^ A } \\ { \\rm f o r } \\ a = ( a _ \\lambda ) _ \\lambda \\in \\mathbb { Z } ^ r \\ { \\rm w i t h } \\ | a | \\geq 1 \\right \\} \\right . \\end{align*}"}
-{"id": "1884.png", "formula": "\\begin{align*} \\frac 3 8 I = & 3 ( K _ { 1 3 } - K _ { 1 2 } ) ( 1 - K _ { 1 2 } - K _ { 1 3 } ) + 3 ( 1 - 2 K _ { 1 2 } - K _ { 1 3 } ) K _ { 1 3 } \\\\ & + 2 ( K _ { 1 3 } - K _ { 1 2 } ) ^ 2 - 2 ( K _ { 1 3 } - K _ { 1 2 } ) ( 1 - K _ { 1 2 } - 2 K _ { 1 3 } ) - ( 1 - K _ { 1 2 } - 2 K _ { 1 3 } ) ^ 2 \\\\ = & - 1 + K _ { 1 2 } + 8 K _ { 1 3 } + 2 K _ { 1 2 } ^ 2 - 4 K _ { 1 3 } ^ 2 - 1 6 K _ { 1 2 } K _ { 1 3 } \\\\ = & - 4 z ^ 2 + 8 ( 1 + s m - 2 m ) z + [ - 1 + ( 1 - 8 s ) m + 2 ( 1 + 8 s - 2 s ^ 2 ) m ^ 2 ] \\\\ \\end{align*}"}
-{"id": "5720.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } m _ f ( 2 ^ l Q ) = 0 . \\end{align*}"}
-{"id": "4092.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 2 } { x z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ 2 z ^ { 4 } } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ { 4 } z ^ { 2 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ { 5 } z } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 4 } { x ^ { 5 } z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 4 } { x ^ { 7 } z } , \\end{gather*}"}
-{"id": "550.png", "formula": "\\begin{align*} f _ { \\alpha } ( z ) = f _ { \\alpha } \\left ( \\theta \\right ) + \\frac { \\sigma _ { \\alpha } ^ 2 } { 2 } ( z - \\theta ) ^ 2 + \\mathcal { O } \\left ( ( z - \\theta ) ^ 3 \\right ) , \\\\ f _ { \\alpha } ( z ) = f _ { \\alpha } \\left ( - \\theta \\right ) - \\frac { \\sigma _ { \\alpha } ^ 2 } { 2 } ( z + \\theta ) ^ 2 + \\mathcal { O } \\left ( ( z + \\theta ) ^ 3 \\right ) . \\end{align*}"}
-{"id": "8639.png", "formula": "\\begin{align*} \\sum \\limits _ { i = 1 } ^ d ( r _ i + s _ i ) X _ i = Z _ 1 ^ 2 h _ 1 + Z _ 2 ^ 2 h _ 2 + Z _ 1 Z _ 2 h _ 3 , \\end{align*}"}
-{"id": "6400.png", "formula": "\\begin{align*} \\gamma _ { q , p } ( q , 2 ) = \\beta _ { q , p } \\left ( \\frac { ( 4 q - 1 ) ! ! } { 6 } \\varepsilon _ { q } ^ { 4 } \\sigma ^ { 4 q } + \\frac { ( 2 q + 2 p - 1 ) ! ! } { 2 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } ^ { 2 } \\sigma ^ { 2 q + 2 p } \\right ) , \\end{align*}"}
-{"id": "693.png", "formula": "\\begin{align*} \\beta _ n ( A _ x ) = \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\Big ( \\sum _ { s \\in G } \\chi _ A ( s \\cdot x ) \\ , p ^ { * k } ( s ) \\Big ) \\end{align*}"}
-{"id": "698.png", "formula": "\\begin{align*} \\hat { h } _ { X , f } ( P ) = \\lim _ { n \\to \\infty } \\frac { h _ { H } ( f ^ { n } ( P ) ) } { \\delta _ { f } ^ { n } } \\end{align*}"}
-{"id": "8684.png", "formula": "\\begin{gather*} \\| T _ { x , k } ( \\Phi ) \\| _ { L _ 2 ( J , U ) } ^ 2 = \\sum _ { m \\ge 1 } | T _ { x , k } ( \\Phi ) ( f _ m ) | ^ 2 _ U = \\sum _ { m \\ge 1 } \\sup _ { | a | _ U = 1 } | \\nabla _ k \\nabla _ { a } ^ G P _ t [ \\Phi _ m ] ( x ) | ^ 2 . \\end{gather*}"}
-{"id": "3574.png", "formula": "\\begin{align*} ( L U ) _ j = \\sum _ { k = 1 } ^ { n + 1 } \\sum _ { | \\beta | = 0 } ^ 4 b _ { j k } ^ { \\beta } \\partial ^ { \\beta } U ^ k = : \\sum _ { k = 1 } ^ { n + 1 } L _ { j k } U ^ k , \\end{align*}"}
-{"id": "3254.png", "formula": "\\begin{align*} q ^ { - ( \\alpha _ t , \\xi ) } [ E _ \\xi , F _ t ] ^ * K _ t = q ^ { - ( \\alpha _ t , \\xi ) } ( q ^ { ( \\alpha _ t , \\xi ) } E _ \\xi E _ t ^ * - E _ t ^ * E _ \\xi ) ^ * = E _ t E _ \\xi ^ * - q ^ { - ( \\alpha _ t , \\xi ) } E _ \\xi ^ * E _ t . \\end{align*}"}
-{"id": "2536.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac 1 T \\int _ 0 ^ T \\mathbb { E } f ( U ( t ) ) d t = \\int _ { \\mathcal { S } } f d \\mu _ h , \\quad \\forall ~ f \\in B _ b ( \\mathcal { S } ) , \\quad \\ ; \\ ; L ^ 2 ( \\mathcal { S } , \\mu _ h ) , \\end{align*}"}
-{"id": "2474.png", "formula": "\\begin{align*} \\varepsilon _ K ( V _ 1 \\oplus \\cdots \\oplus V _ k , \\psi ) = \\varepsilon _ K ( V _ 1 , \\psi ) \\ldots \\varepsilon _ K ( V _ k , \\psi ) \\end{align*}"}
-{"id": "4188.png", "formula": "\\begin{align*} \\mu _ { B _ 1 \\cup \\cdots \\cup B _ m } = \\sum _ { k \\geq 0 } ( - 1 ) ^ k \\ ! \\ ! \\sum _ { 1 \\leq j _ 0 < \\cdots < j _ k \\leq m } \\mu _ { B _ { j _ 0 } \\cap \\cdots \\cap B _ { j _ k } } . \\end{align*}"}
-{"id": "1209.png", "formula": "\\begin{align*} \\begin{cases} Y _ 0 ^ 2 = p _ 1 ( Z ) p _ 2 ( Z ) \\\\ Y _ 1 ^ 2 = p _ 1 ( Z + 1 ) p _ 2 ( Z + 1 ) \\\\ Y _ 2 ^ 2 = p _ 1 ( Z + 2 ) p _ 2 ( Z + 2 ) \\\\ \\ \\ \\ \\ \\ \\vdots & \\\\ Y _ { m - 1 } ^ 2 = p _ 1 ( Z + m - 1 ) p _ 2 ( Z + m - 1 ) . \\end{cases} \\end{align*}"}
-{"id": "6392.png", "formula": "\\begin{align*} \\beta _ { 4 } = \\left ( 1 - 3 \\varepsilon \\sigma ^ { p } - \\frac { \\varepsilon ^ { 2 } 7 ! ! } { 2 } \\sigma ^ { 8 } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "1844.png", "formula": "\\begin{align*} t ( t + 1 ) \\iint _ R \\{ v \\} v ^ { - 3 / 2 } \\{ u \\} u ^ { - 3 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 2 } d u d v = \\sum _ { ( m , n ) \\in R ( t ) } \\left ( \\widetilde { F } _ { m , n } ( t ) - \\widetilde { G } _ { m , n } ( t ) \\right ) + O ( 1 ) , \\end{align*}"}
-{"id": "9661.png", "formula": "\\begin{align*} \\limsup _ { t \\to \\infty } \\{ t - \\tau ( t ) \\} = \\infty . \\end{align*}"}
-{"id": "3189.png", "formula": "\\begin{align*} \\left [ \\delta \\left \\{ \\left ( U _ j , \\sum _ { \\lambda = 1 } ^ r \\sum _ { | \\beta | = n + 1 } a _ { j , \\beta } ^ \\lambda \\cdot e _ { j , \\beta } ^ * \\otimes e _ j ^ \\beta \\right ) \\right \\} \\right ] = u _ n ( Y , X ; \\{ w _ j \\} ) - u _ n ( Y , X ; \\{ \\widehat { w } _ j \\} ) \\end{align*}"}
-{"id": "1720.png", "formula": "\\begin{gather*} \\Phi _ t ^ { \\mp } : = ( \\mp \\cosh t ) \\Phi _ I + ( \\sinh t ) \\Phi _ J - \\Phi _ K \\end{gather*}"}
-{"id": "3813.png", "formula": "\\begin{align*} \\langle \\mu ; n - 1 \\mid e ^ { H _ + [ s ( t ) ] } \\mid \\lambda ; n \\rangle = \\det _ { 1 \\leq p , q \\leq n } h _ { \\lambda _ q - \\mu _ p - q + p } [ x \\mid y ] = s _ { \\lambda / \\mu } ( x | t x ) , \\end{align*}"}
-{"id": "9364.png", "formula": "\\begin{align*} H = - \\frac { d ^ 2 } { d x ^ 2 } + V ( x ) \\end{align*}"}
-{"id": "4584.png", "formula": "\\begin{align*} \\tilde { a } ( \\delta ) : = d ^ { m / 2 } C _ m ' D ^ m a ( \\delta / 4 ) ^ { m ^ 2 + m + 1 } . \\end{align*}"}
-{"id": "3015.png", "formula": "\\begin{align*} { { \\bf { h } } ^ { [ 2 1 ] } } ( n ) { \\bf { V } } _ 1 ^ { [ 1 ] } ( n ) = { { \\bf { h } } ^ { [ 2 1 ] } } ( 1 ) , { { \\bf { h } } ^ { [ 1 1 ] } } ( n ) { \\bf { V } } _ 2 ^ { [ 1 ] } ( n ) = { { \\bf { h } } ^ { [ 1 1 ] } } ( 6 ) . \\end{align*}"}
-{"id": "1618.png", "formula": "\\begin{align*} \\ln F \\left ( t , x \\right ) = - e ^ { - m t } \\ln \\left ( 1 + \\varepsilon e ^ { x - x _ { 0 } } \\right ) + e ^ { - m t } \\left ( x - x _ { 0 } \\right ) + \\frac { c } { m } e ^ { - m t } - \\frac { 1 } { 4 m } e ^ { - 2 m t } , \\end{align*}"}
-{"id": "9362.png", "formula": "\\begin{align*} \\tilde { g } _ t \\leq \\left \\{ \\begin{aligned} \\frac { h _ 0 } { t + 1 } + \\frac { C } { 2 } & t + 1 \\leq \\frac { 2 h _ 0 } { C } \\ , , \\\\ \\sqrt { \\frac { 2 h _ 0 C } { t + 1 } } & \\ , . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1348.png", "formula": "\\begin{align*} S ( \\hat \\delta _ 0 ) = n ^ { - 1 / 2 } \\sum _ { t = 1 } ^ { n } \\left ( y _ t - m _ t \\pi _ t ( \\hat \\delta _ 0 ) \\right ) \\left ( \\sum _ { j = 0 } ^ { \\infty } \\tau _ { j } ( \\omega ) e _ { t - J _ L - j } ( \\hat \\delta _ 0 ) \\right ) \\end{align*}"}
-{"id": "4903.png", "formula": "\\begin{align*} \\xi _ { \\kappa } ( g ^ * ) = g . \\end{align*}"}
-{"id": "7151.png", "formula": "\\begin{align*} F ^ a _ { i j } = 0 i < n ; F ^ a _ { i j } = - a _ j \\phi i = n . \\end{align*}"}
-{"id": "8936.png", "formula": "\\begin{align*} s = n \\omega _ n \\int _ 0 ^ { F ( s ) } ( \\sinh r ) ^ { n - 1 } d r \\end{align*}"}
-{"id": "3125.png", "formula": "\\begin{align*} \\begin{bmatrix} - P ^ { - 1 } & 0 \\\\ \\lambda ^ t I _ n & I _ n \\end{bmatrix} \\begin{bmatrix} - P & \\lambda ^ t P \\\\ \\lambda ^ t P & Q ( \\lambda ) \\end{bmatrix} \\begin{bmatrix} I _ n & - \\lambda ^ t I _ n \\\\ 0 & I _ n \\end{bmatrix} = \\begin{bmatrix} I _ n & 0 \\\\ 0 & P ( \\lambda ) \\end{bmatrix} , \\end{align*}"}
-{"id": "6939.png", "formula": "\\begin{align*} L _ \\pi ( Z ) = L _ \\pi ( 1 ; Z ) = \\prod _ { T \\in { \\cal T } ^ \\pi } ( 1 - Z ^ T ) = \\prod _ { i _ 1 , i _ 2 , \\ldots , i _ m \\geq 0 } \\ ( 1 - z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\cdots z _ m ^ { i _ m } ) ^ { c ^ \\pi _ { ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) } } \\ , . \\end{align*}"}
-{"id": "6930.png", "formula": "\\begin{align*} & \\prod _ { i , j \\geq 0 } \\ , z ^ { 2 i + j + 1 } \\ , s _ { ( 1 ^ { 2 i + 1 } ) } ( X ) \\ , s _ { ( j ) } ( X ) = \\prod _ { a , b \\geq 0 } \\ , z ^ { a + b + 1 } \\ , s _ { ( a + 1 , 1 ^ b ) } ( X ) \\cr \\mbox { a n d } \\cr & \\prod _ { i , j \\geq 0 : ( i , j ) \\neq ( 0 , 0 ) } \\ , z ^ { 2 i + j } \\ , s _ { ( 1 ^ { 2 i } ) } ( X ) \\ , s _ { ( j ) } ( X ) = \\prod _ { a , b \\geq 0 } \\ , z ^ { a + b + 1 } \\ , s _ { ( a + 1 , 1 ^ b ) } ( X ) \\end{align*}"}
-{"id": "6019.png", "formula": "\\begin{gather*} \\Phi = \\mathrm { R e } \\Psi + \\varepsilon \\ , \\mathbb { S } ^ { \\flat } \\wedge \\mathbb { K } \\in \\Gamma \\big ( \\Lambda ^ 3 \\mathcal { V } \\big ) \\end{gather*}"}
-{"id": "9667.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\tau ( t ) } { t } = 0 . \\end{align*}"}
-{"id": "4498.png", "formula": "\\begin{align*} \\mathbb { E } _ f \\bigl \\{ ( \\hat { H } _ n ^ w - H _ n ^ * ) ^ 2 \\bigr \\} = _ f ( \\hat { H } _ n ^ w - H _ n ^ * ) + ( \\mathbb { E } _ f \\hat { H } _ n ^ w - H ( f ) ) ^ 2 , \\end{align*}"}
-{"id": "6935.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) = L ^ \\perp _ \\pi ( X ) \\ , s _ \\lambda ( X ) \\quad \\mbox { a n d } s ^ { * ( \\pi ) } _ \\lambda ( X ) = ( - 1 ) ^ { | \\lambda | } \\ , L ^ \\perp _ { \\pi } ( X ) \\ , s _ { \\lambda ' } ( X ) \\ , , \\end{align*}"}
-{"id": "9014.png", "formula": "\\begin{align*} f ( x , y , z ) = z ^ d + \\sum _ { i = 0 } ^ { d - 1 } a _ i ( x , y ) z ^ { i } , \\end{align*}"}
-{"id": "1763.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } \\Delta u _ { i } ^ { \\varepsilon } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { \\varepsilon } \\sum \\limits _ { j \\neq i } H ( u _ { j } ^ { \\varepsilon } ) ( x ) & \\Omega , \\\\ u _ { i } ^ { \\varepsilon } ( x ) = \\phi _ { i } ( x ) & ( \\partial \\Omega ) _ 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5139.png", "formula": "\\begin{align*} \\phi ( \\sigma _ { k - 1 } \\cdots \\sigma _ { \\ell } ; \\vec { z } ) u ( r ^ { k _ { r } } , \\ldots , 1 ^ { k _ { 1 } } ) = \\prod _ { \\begin{subarray} { c } i \\in J _ { p } \\\\ i > \\ell \\end{subarray} } \\frac { f ( z _ { i } , z _ { \\ell } ) } { f ( z _ { \\ell } , z _ { i } ) } \\prod _ { i \\in J _ { p + 1 } \\cup \\cdots \\cup J _ { 1 } } ^ { \\curvearrowleft } \\ ! \\ ! \\ ! Y _ { i - 1 } ( z _ { \\ell } , z _ { i } ) \\ , u ( r ^ { k _ { r } } , \\ldots , 1 ^ { k _ { 1 } } ) . \\end{align*}"}
-{"id": "116.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 - 4 t ) ^ { \\tfrac { x } { 2 } } d \\mu _ { - 1 } ( x ) = \\sum _ { n = 0 } ^ \\infty C h _ { n , \\frac { 1 } { 2 } } \\frac { ( - 4 ) ^ n t ^ n } { n ! } . \\end{align*}"}
-{"id": "3121.png", "formula": "\\begin{align*} P ( \\lambda ) : = ( \\widehat { \\Lambda } _ t ( \\lambda ) ^ T \\otimes I _ n ) ( \\lambda B + A ) ( \\widehat { \\Lambda } _ t ( \\lambda ) \\otimes I _ n ) \\end{align*}"}
-{"id": "9498.png", "formula": "\\begin{align*} \\langle F , G \\rangle & = \\sum _ { \\mu _ 1 < \\ldots < \\mu _ k } F _ { \\mu _ 1 \\ldots \\mu _ k } G ^ { \\mu _ 1 \\ldots \\mu _ k } \\\\ & = \\frac { 1 } { k ! } \\sum _ { \\mu _ 1 , \\ldots , \\mu _ k } F _ { \\mu _ 1 \\ldots \\mu _ k } G ^ { \\mu _ 1 \\ldots \\mu _ k } , \\end{align*}"}
-{"id": "4716.png", "formula": "\\begin{align*} P = \\{ x \\in \\mathbb { R } ^ n \\ , | \\ \\pi ^ { \\top } x \\leq \\pi _ 0 , \\ ( y ^ i ) ^ { \\top } A x \\geq ( y ^ i ) ^ { \\top } b , \\ i \\in [ p ' ] \\} \\end{align*}"}
-{"id": "8480.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } - \\psi ' ( t ) ( \\tilde { \\textbf { u } } ^ { \\varepsilon } , \\phi ) _ { 0 } ~ d t + \\sum _ { j = 1 } ^ { d } \\int _ { 0 } ^ { T } \\psi ( t ) ( A _ { 0 } ^ { - 1 } J _ { \\varepsilon } A _ { 0 } A _ { j } ( J _ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\bar { \\textbf { u } } ) ) \\partial _ { x _ { j } } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } , \\phi ) _ { 0 } ~ d t = 0 . \\end{align*}"}
-{"id": "3139.png", "formula": "\\begin{align*} z = \\begin{bmatrix} \\Lambda _ s ( \\lambda _ 0 ) \\otimes I _ n \\\\ * \\end{bmatrix} x , \\end{align*}"}
-{"id": "1485.png", "formula": "\\begin{align*} \\max _ { \\substack { R \\in \\mathcal { D } ( Q _ 0 ) : \\\\ R ^ { ( 1 ) } = Q ( x ) } } | m _ { f _ 1 } ( R ) | > ( f _ 1 \\cdot \\chi _ { Q _ 0 } ) ^ * ( \\lambda _ w | Q _ 0 | ) . \\end{align*}"}
-{"id": "7873.png", "formula": "\\begin{align*} q _ n ( t , x , y ) : = \\int _ 0 ^ t \\int _ { \\R ^ d } q _ 0 ( t - s , x , z ) q _ { n - 1 } ( s , z , y ) \\ , d z \\ , d s , ( t , x , y ) \\in ( 0 , \\infty ) \\times \\R ^ d \\times \\R ^ d \\ , . \\end{align*}"}
-{"id": "1893.png", "formula": "\\begin{align*} \\hat { m } _ { \\eta } : = \\min \\Big \\{ \\min \\{ u : | \\hat { \\phi } ( u ) | \\leq ( K n ) ^ { - \\frac { 1 } { 2 } } + \\sqrt { \\eta / K } ( n / \\log n ) ^ { - \\frac { 1 } { 2 } } \\} , n ^ { \\frac { 1 } { K } } \\Big \\} . \\end{align*}"}
-{"id": "4596.png", "formula": "\\begin{align*} f ( X ) = f ( [ X _ { i j } ] _ { 1 \\le i \\le r , 1 \\le j \\le n } ) . \\end{align*}"}
-{"id": "5700.png", "formula": "\\begin{align*} \\frac { \\sigma ( Q _ 0 ) } { | Q _ 0 | } \\| \\chi _ { Q _ 0 } \\| _ { \\mathcal { M } ^ p _ q ( d x , w ) } \\lesssim _ { [ \\sigma ] _ { A _ \\infty } } \\| \\chi _ { Q _ 0 \\setminus \\Omega _ 1 } \\cdot \\sigma \\| _ { \\mathcal { M } ^ p _ q ( d x , w ) } = \\| \\chi _ { Q _ 0 \\setminus \\Omega _ 1 } \\| _ { \\mathcal { M } ^ p _ q ( d x , \\sigma ) } . \\end{align*}"}
-{"id": "6533.png", "formula": "\\begin{align*} \\big \\| \\big ( ( A _ m - A _ n ) v _ n \\big ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } ^ p \\le \\sum _ { n = k } ^ { m - 1 } \\big ( \\| A _ m - A _ n \\| ^ p - \\| A _ m - A _ { n + 1 } \\| ^ p \\big ) E _ n . \\end{align*}"}
-{"id": "9896.png", "formula": "\\begin{align*} \\sigma ( M ) = \\sigma ( B ) \\cup \\{ p _ i ^ { [ n _ i - 1 ] } \\mid i = 1 , 2 \\ldots , t \\} , \\end{align*}"}
-{"id": "2726.png", "formula": "\\begin{align*} \\begin{aligned} & - L \\langle x _ j - \\frac { 1 } { L } g _ j , y + \\nu ^ \\ast - \\sum _ { i \\in I } \\alpha _ i ^ \\ast ( x _ i - \\frac { 1 } { L } g _ i ) \\rangle + ( f _ j - \\frac { 1 } { 2 L } \\| g _ j \\| ^ 2 ) \\\\ & \\leq - L \\langle x _ k - \\frac { 1 } { L } g _ k , y + \\nu ^ \\ast - \\sum _ { i \\in I } \\alpha _ i ^ \\ast ( x _ i - \\frac { 1 } { L } g _ i ) \\rangle + ( f _ k - \\frac { 1 } { 2 L } \\| g _ k \\| ^ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "170.png", "formula": "\\begin{align*} H _ { \\mu _ n } ( \\xi | \\xi _ { - \\infty } ^ { - 1 } ( { \\tau } ) ) = h _ { \\mu _ n } ( T ) . \\end{align*}"}
-{"id": "9474.png", "formula": "\\begin{align*} d F ( s , t ) & = \\eta ' ( s ) ( G ( s , t ) - s ) \\ , d s + ( 1 - \\eta ( s ) ) \\ , d s + \\eta ( s ) d G ( s , t ) \\end{align*}"}
-{"id": "4337.png", "formula": "\\begin{align*} \\psi \\left ( 1 - \\frac { f ( r , a ) } { a } , a \\right ) : = \\frac { | f ' ( r , a ) | ^ p } { a ^ p } \\ , r \\in [ 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "4577.png", "formula": "\\begin{align*} | g _ J ^ * ( x ) | \\leq \\sum _ { i = 1 } ^ { \\mathrm { c a r d } ( J ) } ( i - 1 ) ! S \\bigl ( \\mathrm { c a r d } ( J ) , i \\bigr ) a ( f ( x ) ) ^ i \\leq \\frac { 1 } { 2 } m ^ { m + 1 } m ! a ( f ( x ) ) ^ m . \\end{align*}"}
-{"id": "2106.png", "formula": "\\begin{align*} P = P ( x , r ) = p _ { \\mu m } x ^ { \\mu } r ^ { m } , \\end{align*}"}
-{"id": "1697.png", "formula": "\\begin{align*} \\theta _ t ^ h ( z ) = \\frac { \\phi _ t ( z + h e ) - \\phi _ t ( z ) } { h } \\ , , \\xi _ t ^ h ( z ) = \\frac { \\gamma \\big ( \\phi _ t ( z + h e ) \\big ) - \\gamma \\big ( \\phi _ t ( z ) \\big ) } { h } \\ , , \\end{align*}"}
-{"id": "9018.png", "formula": "\\begin{align*} f ( x , x _ n ) = a _ d ( x ) x _ n ^ d + a _ { d - 1 } ( x ) x _ n ^ { d - 1 } + \\cdots + a _ 0 ( x ) . \\end{align*}"}
-{"id": "946.png", "formula": "\\begin{align*} ( \\zeta _ 1 , \\dots , \\zeta _ n ) \\in T ^ { \\mathbb { C } } _ p M \\Leftrightarrow \\sum _ { j = 1 } ^ n \\dfrac { \\partial \\rho _ i } { \\partial z _ j } ( p ) \\zeta _ j = 0 , 1 \\leq i \\leq k . \\end{align*}"}
-{"id": "8608.png", "formula": "\\begin{align*} \\pi _ { b t } \\overline { \\gamma } \\xi _ { s a } ( n ) & = c _ { k } \\displaystyle \\sum _ { v \\in L ( k ) } v ^ { \\ast } ( t s ) [ b v a ] n \\\\ & = c _ { k } \\left [ b \\displaystyle \\sum _ { v \\in L ( k ) } v ^ { \\ast } ( t s ) v a \\right ] n \\\\ & = c _ { k } [ b t s a ] n \\\\ & = c _ { k } b t s a n \\\\ & = c _ { k } \\pi _ { b t } \\beta \\alpha \\xi _ { s a } ( n ) \\end{align*}"}
-{"id": "4933.png", "formula": "\\begin{align*} | \\langle a _ k , x _ 0 \\rangle | = | \\langle a _ k , \\tilde { x } \\rangle | , \\ , \\ , \\forall \\ , \\ , k \\in S _ { l _ 0 } \\cup S _ { j _ 0 } . \\end{align*}"}
-{"id": "9731.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { x ' ( t ) } { g ( G ^ { - 1 } ( t ) ) } = - ( a - b ) ^ { - \\beta / ( \\beta - 1 ) } \\left ( a - b ( 1 - q ) ^ { - \\beta / ( \\beta - 1 ) } \\right ) . \\end{align*}"}
-{"id": "2324.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { 4 , q } ( \\R ^ d ) ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { 4 , q } ( \\R ^ d ) ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "7141.png", "formula": "\\begin{align*} | G _ { i n } ( x , y ) | \\le \\frac { C _ 0 x _ n y _ n ^ 2 } { | x - y | ^ { n - 2 } \\cdot | x - y ^ * | ^ { 3 } } + C _ 0 1 _ { n = 2 } \\log ( 2 + \\frac { y _ n } { | x - y | } ) . \\end{align*}"}
-{"id": "5065.png", "formula": "\\begin{align*} F A _ { x _ o } A _ { x _ o } ^ { - 1 } \\supset \\big \\{ g \\in G \\ , : \\ , \\lambda ( ( F A \\cap g A ) _ { x _ o } ) > 0 \\big \\} = \\big \\{ g \\in G \\ , : \\ , \\nu ( F A \\cap g A ) > 0 \\big \\} . \\end{align*}"}
-{"id": "3352.png", "formula": "\\begin{align*} g ( x ) ( \\alpha ) = \\begin{cases} x ( \\rho ^ { - 1 } ( \\alpha ) ) & \\\\ p & \\end{cases} \\end{align*}"}
-{"id": "959.png", "formula": "\\begin{align*} ( u , v ) _ { L ^ 2 _ { ( 0 , q ) } ( M ) } = ( u , j _ q v ) _ { L ^ 2 _ { ( 0 , q ) } ( M ) } = ( ( j _ q ) ^ * u , v ) _ { g r a p h } \\ ; , \\end{align*}"}
-{"id": "9135.png", "formula": "\\begin{align*} u ' ( t ) + A ( t ) B ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) , \\ \\ u ( 0 ) = 0 \\end{align*}"}
-{"id": "9857.png", "formula": "\\begin{align*} \\bar v ( \\bar \\Xi ) = & \\frac { 1 } { 2 W } \\log _ 2 \\left ( \\frac { \\beta ( \\bar \\Xi ) } { ( 2 \\ln 2 ) W } \\right ) - \\frac { 1 } { ( 2 \\ln 2 ) W } + \\frac { 1 } { \\beta ( \\bar \\Xi ) } > 0 , \\end{align*}"}
-{"id": "7144.png", "formula": "\\begin{align*} - \\Delta v + \\nabla p = f + \\nabla \\cdot F , \\qquad { \\rm d i v } \\ , v = 0 \\mbox { i n } \\ , \\ , \\R ^ n _ + , \\end{align*}"}
-{"id": "2845.png", "formula": "\\begin{align*} P _ { \\infty } = M o r _ { P r o p } ( P _ { \\infty } , E n d _ X \\otimes \\Omega _ { \\bullet } ) , \\end{align*}"}
-{"id": "7217.png", "formula": "\\begin{align*} c _ { v } ^ { \\prime } \\left ( 0 \\right ) = \\lim _ { t \\rightarrow 0 ^ { + } } \\left ( \\Uparrow _ { c \\left ( 0 \\right ) } ^ { c \\left ( t \\right ) } \\right ) _ { X } , \\end{align*}"}
-{"id": "9049.png", "formula": "\\begin{align*} l _ k ^ { ( N ) } : = \\left \\{ \\begin{array} { l l } - \\upsilon - ( k - r ) ^ { \\alpha _ + } , & k \\leq \\lfloor N / 2 \\rfloor , \\\\ - \\upsilon - ( N - k ) ^ { \\alpha _ + } - \\frac { 3 } { 4 } \\log N , & \\lfloor N / 2 \\rfloor < k \\leq N . \\end{array} \\right . \\end{align*}"}
-{"id": "3590.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\left ( \\Pi _ { g } \\circ D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi _ { ( g , \\pi ) } ) ^ * ( f , X ) - \\Pi _ g ( \\psi , V ) \\right ) \\cdot _ g ( u , Y ) \\ , d \\mu _ g = 0 . \\end{align*}"}
-{"id": "3548.png", "formula": "\\begin{align*} \\bar { \\mu } = \\chi \\mu _ 1 + ( 1 - \\chi ) \\mu _ 2 + \\psi _ 0 R ^ { - 1 - q _ 0 } . \\end{align*}"}
-{"id": "4570.png", "formula": "\\begin{align*} | c ( t ) ^ { - 1 } - 1 | & = \\biggl | \\int _ { \\mathcal { X } } \\biggl ( \\frac { 2 } { 1 + e ^ { - 2 t g } } - 1 - t g \\biggr ) f \\biggr | \\\\ & \\leq \\int _ { A _ t } \\biggl | \\frac { e ^ { - 2 t g } - 1 + 2 t g + t g ( e ^ { - 2 t g } - 1 ) } { 1 + e ^ { - 2 t g } } \\biggr | f + \\int _ { A _ t ^ c } ( 1 + t | g | ) f \\\\ & \\leq \\frac { 1 6 } { 3 } t ^ 2 \\int _ { A _ t } g ^ 2 f + 7 2 t ^ 2 \\int _ { A _ t ^ c } g ^ 2 f \\leq 7 2 t ^ 2 \\int _ { \\mathcal { X } } g ^ 2 f . \\end{align*}"}
-{"id": "9715.png", "formula": "\\begin{align*} x _ { U , \\epsilon } ( t ) = G _ 0 ^ { - 1 } ( \\lambda ( \\epsilon ) t ) , t \\geq T _ 1 ( \\epsilon ) . \\end{align*}"}
-{"id": "3732.png", "formula": "\\begin{align*} | N ( x ) \\cap ( S _ 0 \\setminus ( S \\cup T ) ) | & = | N ( x ) \\cap S _ 0 | - | N ( x ) \\cap S _ 0 \\cap ( S \\cup T ) | \\\\ & = 2 ^ { 2 \\nu - 4 } - | N ( x ) \\cap S _ 0 \\cap S | - | N ( x ) \\cap S _ 0 \\cap T | \\\\ & = 2 ^ { 2 \\nu - 4 } - | N ( x ) \\cap S | \\\\ & = 2 ^ { 2 \\nu - 4 } - 4 . \\end{align*}"}
-{"id": "53.png", "formula": "\\begin{align*} & \\psi ' ( y ) + \\frac { p } { p - 1 } \\psi ( y ) ^ { ( p - 1 ) / p } ( y ) = \\frac { p } { p - 1 } a ^ { 2 - p } ( 1 - y ) \\ , y \\in ( 0 , 1 ) \\ , \\\\ & \\psi ( 0 ) = 0 \\ . \\end{align*}"}
-{"id": "979.png", "formula": "\\begin{align*} \\forall i = 1 , \\ldots , n ; j = 1 , \\ldots , k \\quad \\mbox { w e h a v e } w _ i + 1 \\ne c _ j ; \\end{align*}"}
-{"id": "7826.png", "formula": "\\begin{align*} Y _ { t } = g ( \\eta _ { T } ) + \\int _ t ^ T E ' [ f ( s , \\eta _ { s } , y ' _ { s } , z ' _ { s } , y _ { s } , z _ { s } ) ] d s - \\int _ t ^ T Z _ { s } d B _ { s } ^ { H } , \\ \\ 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "4899.png", "formula": "\\begin{align*} \\delta _ { G _ i } : = \\deg _ z P _ { G _ i } ( z , y ) \\leqslant A \\Delta _ G + k ^ i \\delta _ G . \\end{align*}"}
-{"id": "3295.png", "formula": "\\begin{align*} F w _ { - 1 } = - [ 2 ] ^ { 1 / 2 } w _ 0 , F w _ 0 = - [ 2 ] ^ { 1 / 2 } q ^ 2 w _ 1 , F w _ 1 = 0 . \\end{align*}"}
-{"id": "7829.png", "formula": "\\begin{align*} E \\big [ f ( t , x , v ( t , \\eta _ { t } ) , v ( t , x ) , v _ { x } ( t , x ) \\sigma _ { t } ) \\big ] = E ' \\big [ f ( t , x , v ( t , \\eta ' _ { t } ) , v ( t , x ) , v _ { x } ( t , x ) \\sigma _ { t } ) \\big ] . \\end{align*}"}
-{"id": "9583.png", "formula": "\\begin{align*} \\Vert \\Psi \\Vert _ { \\cal X } ^ 2 : = \\norm { \\psi } _ { H ^ 2 ( \\R ^ 3 ) } ^ 2 + \\norm { \\pi } _ { H ^ 1 ( \\R ^ 3 ) } ^ 2 . \\end{align*}"}
-{"id": "2928.png", "formula": "\\begin{align*} M _ { \\pi ' } ( Z ) = M _ { \\pi ' } ( 1 ; Z ) = \\prod _ { T \\in { \\cal T } ^ { \\pi ' } } ( 1 - Z ^ T ) ^ { - 1 } = \\prod _ { i _ 1 , i _ 2 , \\ldots , i _ m \\geq 0 } \\ c ^ { \\pi ' } _ { ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) } \\ ( 1 - z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\cdots z _ m ^ { i _ m } ) \\ , . \\end{align*}"}
-{"id": "4215.png", "formula": "\\begin{align*} \\alpha ( y _ k - x _ k ) + \\beta ( y _ k - x ^ * ) = \\beta ( z _ k - x ^ * ) . \\end{align*}"}
-{"id": "7006.png", "formula": "\\begin{align*} \\int _ X u ( t , x ) d P = \\int _ { D ( t , p ) } u ( t , x ) d P + \\int _ { X \\setminus D ( t , p ) } u ( t , x ) d P < \\max _ { y \\in B ( t , p ) } u ( t , y ) \\end{align*}"}
-{"id": "3703.png", "formula": "\\begin{align*} \\frac { d } { d t } \\vec { x } = \\nabla Z \\times \\nabla H ~ , \\end{align*}"}
-{"id": "5104.png", "formula": "\\begin{align*} & h _ { 1 2 } ( | m _ { 1 } , \\ldots , m _ { r } \\rangle \\otimes | n _ { 1 } , \\ldots , n _ { r } \\rangle ) = \\sum _ { a = 1 } ^ { r } q ^ { 2 \\sum _ { p = a + 1 } ^ { r } n _ { p } } ( 1 - q ^ { 2 n _ { a } } ) \\\\ & { } \\times \\left ( | \\ldots , m _ { a } + 1 , \\ldots \\rangle \\otimes | \\ldots , n _ { a } - 1 , \\ldots \\rangle - | \\ldots , m _ { a } , \\ldots \\rangle \\otimes | \\ldots , n _ { a } , \\ldots \\rangle \\right ) . \\end{align*}"}
-{"id": "6795.png", "formula": "\\begin{align*} \\dfrac { \\partial A _ i } { \\partial t _ j } = \\dfrac { [ A _ i , A _ j ] } { t _ i - t _ j } , \\ ; i \\neq j , \\displaystyle \\sum _ { i = 1 } ^ p \\dfrac { \\partial A _ i } { \\partial t _ j } = 0 . \\end{align*}"}
-{"id": "9851.png", "formula": "\\begin{align*} x ( t ) \\otimes ( \\delta ( t ) - \\alpha \\theta ( t - \\tau ) ) & = \\theta ( t ) \\otimes n _ { \\rm R } ( t ) , \\\\ y ( t ) & = h _ { \\rm R D } ( t ) \\otimes x ( t ) + n _ { \\rm D } ( t ) . \\end{align*}"}
-{"id": "444.png", "formula": "\\begin{align*} C ( y , z ) = \\Big \\{ \\sigma _ k ( z ' ) ~ \\Big | ~ & z ' \\in \\Z _ q ^ { n - k } , \\mu _ k ( z ) = \\mu _ k ( z ' ) , \\\\ & \\phi _ k ^ { - 1 } ( y , z ' ) \\in C \\Big \\} \\end{align*}"}
-{"id": "9893.png", "formula": "\\begin{align*} F _ r \\left ( \\frac { k } { 4 d } \\right ) = \\frac { 2 \\sqrt { x } } { k } \\int _ 0 ^ { \\infty } f \\left ( \\frac { r ^ 2 + \\frac { 3 x w ^ 2 } { k ^ 2 } } { 4 } \\right ) \\cos \\left ( \\frac { 2 \\pi w \\sqrt { x } } { 4 d } \\right ) d w . \\end{align*}"}
-{"id": "6801.png", "formula": "\\begin{align*} \\Vert f \\Vert _ { k , \\alpha } : = \\Vert f \\Vert _ \\infty + \\sup _ { x \\neq y } \\frac { \\vert f ^ { ( k ) } ( y ) - f ^ { ( k ) } ( x ) \\vert } { \\vert y - x \\vert ^ \\alpha } < + \\infty . \\end{align*}"}
-{"id": "9287.png", "formula": "\\begin{align*} J ( \\pi ) = \\mathbb { E } [ U ( X ^ { \\pi } ( T , Z ) ) ] . \\end{align*}"}
-{"id": "2654.png", "formula": "\\begin{align*} T ( j , k ) = 2 \\cdot T ( j - 1 , k ) + T ( j - 1 , k - 1 ) \\end{align*}"}
-{"id": "7272.png", "formula": "\\begin{align*} T _ { j k } w _ k = w _ j + \\sum _ { | \\alpha | \\geq n + 1 } f _ { k j , \\alpha } \\cdot w _ j ^ \\alpha . \\end{align*}"}
-{"id": "3327.png", "formula": "\\begin{align*} \\Gamma _ i \\Gamma _ j ^ * = \\sum _ { k , l } c _ { i j } ^ { k l } \\Gamma _ k ^ * \\Gamma _ l , i \\neq j . \\end{align*}"}
-{"id": "9315.png", "formula": "\\begin{align*} M ( t , z ) = \\exp ( \\int _ 0 ^ t \\Phi ( s , z ) d B ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ t \\Phi ^ 2 ( s , z ) d s ) . \\end{align*}"}
-{"id": "7491.png", "formula": "\\begin{align*} \\left \\{ R = n / 2 \\right \\} \\subseteq \\{ Q _ 1 = n \\} \\cap \\left ( \\bigcap _ { \\substack { 0 \\leq j \\leq \\kappa ' \\\\ j \\neq 1 } } \\{ Q _ j = 0 \\} \\right ) . \\end{align*}"}
-{"id": "2523.png", "formula": "\\begin{align*} \\vec { c } \\otimes \\pi \\longmapsto \\sigma ( \\pi , \\vec { c } \\ , ) \\oplus 0 \\longmapsto \\sigma ( \\pi , \\vec { c } \\ , ) + 0 = \\sigma ( \\pi , \\vec { c } \\ , ) \\end{align*}"}
-{"id": "8180.png", "formula": "\\begin{align*} \\d = ( a _ 2 - a _ 3 ) x _ 2 x _ 3 \\frac { \\partial } { \\partial x _ 1 } + ( a _ 3 - a _ 1 ) x _ 3 x _ 1 \\frac { \\partial } { \\partial x _ 2 } + ( a _ 1 - a _ 2 ) x _ 1 x _ 2 \\frac { \\partial } { \\partial x _ 3 } , \\end{align*}"}
-{"id": "1176.png", "formula": "\\begin{align*} - D ^ 2 _ x v = z \\rho ( x ) v , 0 < x < 1 , v _ x ( 0 ) - h v ( 0 ) = 0 , v _ x ( 1 ) + H v ( 1 ) = 0 , \\end{align*}"}
-{"id": "3943.png", "formula": "\\begin{align*} ( x \\cdot y ) _ z = \\frac { 1 } { 2 } \\left ( d ( z , x ) + d ( z , y ) - d ( x , y ) \\right ) . \\end{align*}"}
-{"id": "5782.png", "formula": "\\begin{align*} [ e _ { \\gamma } , [ u , v ] ] = [ [ e _ { \\gamma } , u ] , v ] - ( - 1 ) ^ { p ( u ) p ( v ) } [ [ e _ { \\gamma } , v ] , u ] . \\end{align*}"}
-{"id": "7842.png", "formula": "\\begin{align*} \\{ C _ { i j } , D _ { k l } \\} = - \\delta _ { i l } \\delta _ { j k } . \\end{align*}"}
-{"id": "1057.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ N g _ n & \\le N \\sum _ { \\substack { m \\ge 2 , j \\ge 1 \\\\ m ^ j \\le N } } \\frac 1 m \\le N \\sum _ { m = 2 } ^ N \\frac { 2 \\log N } m = \\mathcal O ( N ( \\log N ) ^ 2 ) = \\mathcal O ( N ^ { 1 + \\varepsilon } ) \\end{align*}"}
-{"id": "5924.png", "formula": "\\begin{align*} Q _ t = \\int _ 0 ^ t e ^ { s A } Q e ^ { s A ^ * } \\dd s = \\int _ 0 ^ t e ^ { s A } \\begin{pmatrix} 0 & 0 \\\\ 0 & \\mathbb { I } _ { \\R ^ d } \\end{pmatrix} e ^ { s A ^ * } \\dd s \\ , = \\begin{pmatrix} \\frac { 1 } { 3 } t ^ 3 \\mathbb { I } _ { \\R ^ d } & \\frac { 1 } { 2 } t ^ 2 \\mathbb { I } _ { \\R ^ d } \\\\ [ 3 p t ] \\frac { 1 } { 2 } t ^ 2 \\mathbb { I } _ { \\R ^ d } & t \\mathbb { I } _ { \\R ^ d } \\end{pmatrix} \\ , \\end{align*}"}
-{"id": "6821.png", "formula": "\\begin{align*} \\Omega _ m ^ + = \\left ( L _ m \\cdot \\left ( Y ^ - G \\right ) \\right ) G ^ { - 1 } \\left ( Y ^ - \\right ) ^ { - 1 } = \\Omega _ m ^ - + Y ^ - L _ m \\left ( G \\right ) G ^ { - 1 } \\left ( Y ^ - \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "9062.png", "formula": "\\begin{align*} G _ 0 v ( d , \\delta , E ) & : = \\left \\{ \\forall j \\in [ | r , N | ] , \\ , l _ j ^ { ( N ) } - \\delta \\leq d W _ { j - r + E _ { j - r } } \\leq u _ j ^ { ( N ) } + \\delta \\right \\} . \\end{align*}"}
-{"id": "3240.png", "formula": "\\begin{align*} \\pi _ { \\mathrm { \\dim } \\left ( S \\right ) } \\circ \\left ( p _ { k } ^ { N } \\right ) ^ { \\alpha } = \\left ( p _ { j \\left ( k \\right ) } ^ { S } \\right ) ^ { \\alpha } , \\end{align*}"}
-{"id": "6532.png", "formula": "\\begin{align*} & { } \\big \\| \\big ( ( A _ m - A _ n ) v _ n \\big ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } ^ p = \\tau \\sum _ { n = k } ^ m \\| ( A _ m - A _ n ) v _ n \\| _ { X } ^ p \\\\ & { } \\le \\tau \\sum _ { n = k } ^ m \\| A _ m - A _ n \\| ^ p \\| v _ n \\| _ { D } ^ p = \\sum _ { n = k } ^ m \\| A _ m - A _ n \\| ^ p ( E _ n - E _ { n - 1 } ) , \\end{align*}"}
-{"id": "5536.png", "formula": "\\begin{align*} - v _ { x x } = z \\rho ( x ) v , 0 < x < 1 , v _ x ( 0 ) - h v ( 0 ) = 0 , v _ x ( 1 ) + H v ( 1 ) = 0 , \\end{align*}"}
-{"id": "7576.png", "formula": "\\begin{align*} p _ 1 = p _ 1 ( x _ 0 ) : = \\mathbb P \\left \\{ \\omega \\in \\Omega : \\chi ( \\omega ) \\ge 1 - \\frac { l + A } { 2 l } \\right \\} , K _ 1 = K _ 1 ( x _ 0 ) : = \\left [ \\frac { 2 ( u _ l - x _ 0 ) } { l + A } \\right ] + 1 . \\end{align*}"}
-{"id": "9035.png", "formula": "\\begin{align*} A _ j ( \\theta ) & : = \\psi _ j ( \\theta ) - ( j + 1 ) \\theta . \\end{align*}"}
-{"id": "8333.png", "formula": "\\begin{align*} z ^ { 3 } _ { 3 , 2 } - z _ { 3 , 2 } = \\frac { \\omega ^ { 2 } } { \\left ( T + 1 \\right ) ^ { 2 } } + \\frac { \\omega ^ { 2 } } { T + 1 } + 2 T + \\omega ^ { 2 } + \\omega + 2 = r _ { 3 } ( T ) . \\end{align*}"}
-{"id": "2270.png", "formula": "\\begin{align*} T _ { u , r } ^ V : = \\{ v \\in V : \\min _ { 0 \\le t \\le T } \\| v - u ( t ) \\| _ V \\le r \\} , \\end{align*}"}
-{"id": "4882.png", "formula": "\\begin{align*} \\delta ( p [ a ] ) = \\frac { 1 } { p } ( \\phi ( p [ a ] ) - ( p [ a ] ) ^ p ) = \\frac { 1 } { p } ( p [ a ^ p ] - p ^ p [ a ] ^ p ) = [ a ^ p ] - p ^ { p - 1 } [ a ] ^ p . \\end{align*}"}
-{"id": "1426.png", "formula": "\\begin{align*} S ' = \\{ k \\in \\C \\backslash \\{ - h ^ { \\vee } \\} \\mid ( k + h ^ { \\vee } ) ^ { - 1 } \\in \\bigcap _ { n \\geq 0 } U ( n ) \\} \\end{align*}"}
-{"id": "3361.png", "formula": "\\begin{align*} f ^ \\ast ( D ) & = \\frac { 1 } { 2 } \\| N + D \\| _ r ^ 2 - \\frac { 1 } { 2 } \\| N \\| _ F ^ 2 , \\\\ f ^ { \\ast \\ast } ( M ) & = \\frac { 1 } { 2 } \\| M \\| _ { r \\ast } ^ 2 - \\langle N , M \\rangle + \\frac { 1 } { 2 } \\| N \\| _ F ^ 2 \\end{align*}"}
-{"id": "4028.png", "formula": "\\begin{align*} J _ { K , N } \\Bigl ( e ^ { \\frac { 2 \\pi \\sqrt { - 1 } } { 3 } } \\Bigr ) = \\left \\{ \\begin{array} { r l } 0 & ( N = 6 l ) , \\\\ 1 & ( N = 6 l + 1 ) , \\\\ 1 & ( N = 6 l + 2 ) , \\\\ 0 & ( N = 6 l + 3 ) , \\\\ - 1 & ( N = 6 l + 4 ) , \\\\ - 1 & ( N = 6 l + 5 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "532.png", "formula": "\\begin{align*} I ^ { \\rm e x p } _ { 1 1 } ( i , x ; j , y ) : = \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\dd z \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\dd w \\frac { ( z - w ) e ^ { - x z - y w } } { 4 z w ( z + w ) } \\frac { ( 1 + 2 z ) ^ { n _ i } ( 1 + 2 w ) ^ { n _ j } } { ( 1 - 2 z ) ^ { m _ i } ( 1 - 2 w ) ^ { m _ j } } ( 2 z + 2 \\alpha - 1 ) ( 2 w + 2 \\alpha - 1 ) ; \\end{align*}"}
-{"id": "7577.png", "formula": "\\begin{align*} B = B ( x _ 0 ) : = \\max _ { x \\in [ v _ l , x _ 0 ] } F ( x ) < l \\end{align*}"}
-{"id": "4659.png", "formula": "\\begin{align*} & Z ( f , L ; s , \\phi ) \\\\ & = \\int _ { G _ + / \\Gamma } \\chi ( g ) ^ s \\phi ( g ) \\sum _ { x \\in L } f ( g \\cdot x ) d g - \\int _ { G _ + / \\Gamma } \\chi ( g ) ^ s \\phi ( g ) \\sum _ { x \\in L _ 0 } f ( g \\cdot x ) d g . \\end{align*}"}
-{"id": "1104.png", "formula": "\\begin{align*} | \\langle a _ k , \\tilde { x } \\rangle | = | \\langle a _ k , x _ 0 \\rangle | , \\forall k . \\end{align*}"}
-{"id": "3347.png", "formula": "\\begin{align*} \\mathsf { g } + \\mathsf { h } & = [ \\mathtt { g } \\cdot \\mathtt { h } ] , \\\\ \\lambda \\mathsf { g } & = [ \\mathtt { g } ^ { ( \\lambda ) } ] . \\\\ \\end{align*}"}
-{"id": "1873.png", "formula": "\\begin{align*} 3 \\eta ^ 2 + 1 & + 2 \\gamma ( \\sqrt { 3 } \\eta - 1 ) \\\\ > & 3 \\eta ^ 2 + 1 + 2 ( 1 + \\sqrt { 3 } ) \\cdot ( \\sqrt { 3 } \\eta - 1 ) \\\\ = & 3 \\eta ^ 2 + ( 6 + 2 \\sqrt { 3 } ) \\eta - ( 1 + 2 \\sqrt { 3 } ) \\\\ = & 3 \\Big ( \\eta - \\frac { \\sqrt { 5 + 4 \\sqrt { 3 } } - ( 1 + \\sqrt { 3 } ) } { \\sqrt { 3 } } \\Big ) \\cdot \\Big ( \\eta + \\frac { \\sqrt { 5 + 4 \\sqrt { 3 } } + ( 1 + \\sqrt { 3 } ) } { \\sqrt { 3 } } \\Big ) . \\end{align*}"}
-{"id": "4466.png", "formula": "\\begin{align*} \\sup _ { I \\in \\mathcal { I } } \\lim _ { n \\rightarrow \\infty } \\max _ { \\lambda \\in I } n \\mathbb { E } _ { f _ { n ^ { - 1 / 2 } , g _ \\lambda } } \\bigl [ \\bigl \\{ \\hat { H } _ n ^ w - H ( f _ { n ^ { - 1 / 2 } , g _ \\lambda } ) \\bigr \\} ^ 2 \\bigr ] = V ( f ) . \\end{align*}"}
-{"id": "2995.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\ell a _ i \\equiv 0 \\mod d \\end{align*}"}
-{"id": "3842.png", "formula": "\\begin{align*} d _ { k + 1 } ( x ) = d _ k ( x ) - x ^ { k - 2 } n _ k ( x ) + x ^ { 2 k - 1 } . \\end{align*}"}
-{"id": "1463.png", "formula": "\\begin{align*} \\tan ^ { - 1 } ( m _ f ( Q ) ) = m _ { \\tan ^ { - 1 } f } ( Q ) , \\end{align*}"}
-{"id": "2192.png", "formula": "\\begin{align*} W ( s ) = \\int _ { B _ { 1 } } \\phi ( s ) \\psi ^ { 2 } ( x ) w ^ { 2 } ( s , x ) d x , \\end{align*}"}
-{"id": "2470.png", "formula": "\\begin{align*} z _ j = \\eta _ j ( \\Phi _ K ) \\cdot \\mu _ { \\tau _ j } \\end{align*}"}
-{"id": "589.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\alpha \\sigma _ 1 \\partial _ { \\sigma _ 1 } \\eta = \\partial _ { \\sigma _ 1 } \\psi , \\\\ - \\alpha \\sigma _ 3 \\partial _ { \\sigma _ 3 } \\eta = \\partial _ { \\sigma _ 3 } \\psi , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7948.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\Delta ^ 2 u \\Delta ^ 2 \\varphi d x = \\lambda \\int _ { \\Omega } \\Delta u \\Delta \\varphi \\ , , \\ \\ \\ \\forall u , \\varphi \\in H ^ 4 ( \\Omega ) \\cap H ^ 2 _ 0 ( \\Omega ) , \\end{align*}"}
-{"id": "1302.png", "formula": "\\begin{align*} T _ H ( N ; y , \\alpha ) & = e ( N \\alpha ) \\sum _ { n = - H } ^ y ( H - | n | ) e ( n \\alpha ) \\\\ & = \\frac { e ( ( N + y + 1 ) \\alpha ) } { 1 - e ( \\alpha ) } \\cdot ( y - H ) + \\frac { e ( ( N + 1 ) \\alpha ) } { ( 1 - e ( \\alpha ) ) ^ 2 } \\cdot \\bigl ( e ( y \\alpha ) - 2 + e ( - H \\alpha ) \\bigr ) . \\end{align*}"}
-{"id": "6184.png", "formula": "\\begin{align*} M _ m ( t ) = g _ m ( V _ t ^ n ) - \\int _ 0 ^ t \\mathcal { A } _ n g _ m ( V _ { s } ^ n ) \\d s . \\end{align*}"}
-{"id": "1843.png", "formula": "\\begin{align*} \\mathbb { E } | \\zeta ( 1 / 2 + i X _ t ) | ^ 2 = t ( t + 1 ) \\iint _ R \\{ v \\} v ^ { - 3 / 2 } \\{ u \\} u ^ { - 3 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 2 } d u d v + O ( 1 ) , \\end{align*}"}
-{"id": "4038.png", "formula": "\\begin{align*} W ^ s _ { l o c } ( \\underline { a } ) & = \\{ \\underline { b } \\in \\Sigma _ { A ^ 1 } : d ( \\sigma ^ n \\underline { a } , \\sigma ^ n \\underline { b } ) \\leq 1 / 3 \\ \\forall \\ n \\geq 0 \\} \\\\ W ^ u _ { l o c } ( \\underline { a } ) & = \\{ \\underline { b } \\in \\Sigma _ { A ^ 1 } : d ( \\sigma ^ n \\underline { a } , \\sigma ^ n \\underline { b } ) \\leq 1 / 3 \\ \\forall \\ n \\leq 0 \\} . \\end{align*}"}
-{"id": "8731.png", "formula": "\\begin{align*} \\int _ 0 ^ \\tau e ^ { - s A } G B ( s , X ^ { x } _ s ) \\ , d s = v ( 0 , x ) - e ^ { - \\tau A } v ( \\tau , X _ \\tau ^ { x } ) + \\int _ 0 ^ \\tau e ^ { - s A } \\nabla ^ G v ( s , X _ s ^ x ) \\ ; d W _ s \\end{align*}"}
-{"id": "6158.png", "formula": "\\begin{align*} g ( { \\bf x } ^ { \\bf s } ) ^ v = g ( { \\bf x } ^ { v \\bf s } ) = _ { \\Bbbk ^ \\times } x _ { \\pi ( 1 ) } ^ { v s _ 1 } \\cdots x _ { \\pi ( n ) } ^ { v s _ n } = _ { \\Bbbk ^ \\times } ( { \\bf x } ^ { \\pi _ g ( { \\bf s } ) } ) ^ v , \\end{align*}"}
-{"id": "4314.png", "formula": "\\begin{align*} e ^ { \\varphi _ { X / Y } ( x _ 0 ) } = \\sup _ { \\Vert u \\Vert _ { y _ 0 } \\leq 1 } | F _ u ( x _ 0 ) | ^ 2 \\end{align*}"}
-{"id": "9996.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": "9850.png", "formula": "\\begin{align*} \\frac { S _ { y s } ( f ) } { S _ { s s } ( f ) } = H _ { \\rm S D } ( f ) + \\frac { H _ { \\rm R D } ( f ) H _ { \\rm S R } ( f ) \\Theta ( f ) } { 1 - \\hat \\alpha ( f ) \\Theta ( f ) } , \\end{align*}"}
-{"id": "2617.png", "formula": "\\begin{align*} S _ { a , b } = 2 u a b p ^ { - k } , & & T _ { a , a } = u a ^ 2 p ^ { - k } \\end{align*}"}
-{"id": "7053.png", "formula": "\\begin{align*} { { \\bf { h } } ^ { [ 2 1 ] } } ( n ) { \\bf { V } } _ 1 ^ { [ 1 ] } ( n ) = { { \\bf { h } } ^ { [ 2 1 ] } } ( 1 ) , { { \\bf { h } } ^ { [ 1 1 ] } } ( n ) { \\bf { V } } _ 2 ^ { [ 1 ] } ( n ) = { { \\bf { h } } ^ { [ 1 1 ] } } ( 6 ) . \\end{align*}"}
-{"id": "7986.png", "formula": "\\begin{align*} i i i + \\sum ( 4 + n ) \\ , i _ n ^ * + 7 \\ , i i i ^ * + 8 \\ , i i ^ * = 2 0 . \\end{align*}"}
-{"id": "4009.png", "formula": "\\begin{align*} q ^ { t + r _ t + r _ d } - 1 = | W ^ * | & = \\sum _ { X \\in S T } | X ^ * | \\cr & \\leq | V ( t , q ) ^ * | + ( | S T | - 1 ) \\cdot | V ( r _ t + r _ d , q ) ^ * | = q ^ { t + r _ t + r _ d } - 1 . \\end{align*}"}
-{"id": "5573.png", "formula": "\\begin{align*} D _ x V = \\begin{bmatrix} 0 & 1 \\\\ - z \\rho & 0 \\end{bmatrix} V D _ t V = \\begin{bmatrix} - \\frac 1 2 D _ x ( b ) + \\beta & b \\\\ - \\frac 1 2 D ^ 2 _ x ( b ) - z \\rho b & \\frac 1 2 D _ x ( b ) + \\beta \\end{bmatrix} V . \\end{align*}"}
-{"id": "7317.png", "formula": "\\begin{align*} \\hat { R } F ( v _ { 1 } \\otimes v _ { 1 } ) & = [ 2 ] ^ { 1 / 2 } ( q ^ { - 2 } v _ { 1 } \\otimes v _ { 0 } + \\hat { R } ( v _ { 1 } \\otimes v _ { 0 } ) ) , \\\\ F \\hat { R } ( v _ { 1 } \\otimes v _ { 1 } ) & = [ 2 ] ^ { 1 / 2 } ( v _ { 0 } \\otimes v _ { 1 } + q ^ { 2 } v _ { 1 } \\otimes v _ { 0 } ) . \\end{align*}"}
-{"id": "7209.png", "formula": "\\begin{align*} P _ { \\alpha } ^ { S } = P _ { \\beta } ^ { S } \\circ \\Phi _ { \\beta , \\alpha } \\end{align*}"}
-{"id": "855.png", "formula": "\\begin{align*} & \\vec { z } ( \\ell ( t ) , \\ldots , \\ell ( 1 ) ; w ) _ { i } = \\left \\{ \\begin{array} { l l } z _ { i } & ( i \\not = \\ell ( t ) , \\ldots , \\ell ( 1 ) ) \\\\ z _ { \\ell ( s - 1 ) } & ( i = \\ell ( s ) , 2 \\le s \\le t ) \\\\ w & ( i = \\ell ( 1 ) ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5085.png", "formula": "\\begin{align*} h _ { \\vec { z } } ^ { \\vec { \\mu } } ( \\vec { x } ) = \\prod _ { 1 \\le i < j \\le k } f ( z _ { i } , z _ { j } ) \\sum _ { \\tau \\in \\mathfrak { S } _ { k } } \\prod _ { i = 1 } ^ { k } \\left ( \\frac { z _ { \\tau ^ { - 1 } ( i ) } } { 1 + z _ { \\tau ^ { - 1 } ( i ) } } \\right ) ^ { x _ { i } } \\phi ( \\tau ; \\vec { z } ) ( u _ { \\mu _ { 1 } } \\otimes \\cdots \\otimes u _ { \\mu _ { k } } ) . \\end{align*}"}
-{"id": "5532.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\varphi ( | x ( k ) | ) v ( k ) \\chi _ { E _ j } ( k ) \\le \\sum _ { k = j + 1 } ^ \\infty \\varphi ( | x ( k ) | ) v ( k ) \\to 0 \\ \\ \\ \\ \\ \\ j \\to \\infty . \\end{align*}"}
-{"id": "7726.png", "formula": "\\begin{align*} F ( M \\sigma _ n ) = f ( M \\sigma _ n ) - M \\sigma _ n \\ge \\kappa M \\sigma _ n \\ge \\sigma _ n \\ge \\sigma _ { n + 1 } . \\end{align*}"}
-{"id": "307.png", "formula": "\\begin{align*} A : = ( n - 2 ) \\begin{pmatrix} j ^ { - 1 } p _ { n , x , u } ^ { ( j ) } ( 1 - p _ { n , x , u } ^ { ( j ) } ) & j ^ { - 1 / 2 } l ^ { - 1 / 2 } ( p _ \\cap - p _ { n , x , u } ^ { ( j ) } p _ { n , y , v } ^ { ( l ) } ) \\\\ j ^ { - 1 / 2 } l ^ { - 1 / 2 } ( p _ \\cap - p _ { n , x , u } ^ { ( j ) } p _ { n , y , v } ^ { ( l ) } ) & l ^ { - 1 } p _ { n , y , v } ^ { ( l ) } ( 1 - p _ { n , y , v } ^ { ( l ) } ) \\end{pmatrix} . \\end{align*}"}
-{"id": "7171.png", "formula": "\\begin{align*} v _ i ( x ) & = \\int _ { \\R ^ n } ( U _ { i j } ( x - y ) - U _ { i j } ( x ) ) f _ j ( y ) d y \\\\ & = \\sum _ { j , k = 1 } ^ n \\Phi ^ { j k } _ i ( x ) \\hat b _ { j k } + \\sum _ { j , k = 1 } ^ n \\int _ { \\R ^ n } ( U _ { i j } ( x - y ) - U _ { i j } ( x ) + \\Phi ^ { j k } _ i ( x ) y _ k ) f _ j ( y ) d y , \\end{align*}"}
-{"id": "5887.png", "formula": "\\begin{align*} L _ { \\xi ^ { i } } A ^ { i j } = ( \\lambda - a ) A ^ { i j } , \\end{align*}"}
-{"id": "254.png", "formula": "\\begin{align*} a ( \\delta ) : = A _ m \\max \\biggl \\{ 1 \\ , , \\ , \\log ^ { 2 ( m + 1 ) } \\Bigl ( \\frac { 1 } { \\delta } \\Bigr ) \\biggr \\} , \\end{align*}"}
-{"id": "7676.png", "formula": "\\begin{align*} \\| u _ { 0 } \\| _ { L ^ { q } } \\leq \\sum _ { n = 1 } ^ { \\infty } \\| u _ { n } \\| _ { L ^ { q } } = \\frac { 1 } { \\nu ( d , \\alpha ) } \\omega _ { d } ^ { 1 / q } \\sum _ { n = 1 } ^ { \\infty } n ^ { - \\alpha } < \\infty . \\end{align*}"}
-{"id": "5911.png", "formula": "\\begin{align*} B ^ { s } _ { p , p } ( \\R ^ d ) = W ^ { s , p } ( \\R ^ d ) \\end{align*}"}
-{"id": "4013.png", "formula": "\\begin{align*} i \\geq n _ { d } - q ^ { ( k - 1 ) { d } + r _ d } = \\ell q ^ { d } - q ^ { ( k - 1 ) { d } + r _ d } = \\ell - q ^ { r _ d } . \\end{align*}"}
-{"id": "1871.png", "formula": "\\begin{align*} I I \\ge & ( 1 + \\eta ) ( 1 - \\eta ) - 2 \\eta \\cdot \\frac 2 3 \\cdot \\frac 1 { 1 + 1 } \\cdot \\frac { \\frac 3 2 ( 1 - \\eta ) \\cdot 4 } { \\sqrt { 3 } + 1 } \\\\ = & ( 1 - \\eta ) \\Big ( 1 + \\eta - \\frac 4 { 1 + \\sqrt { 3 } } \\eta \\Big ) \\ge 0 , \\end{align*}"}
-{"id": "1929.png", "formula": "\\begin{align*} & \\left ( \\prod _ { i = 1 } ^ m Y _ i ^ { - \\frac { 2 n _ i } { n } } \\right ) ^ { - 1 } \\frac { d \\bar { \\lambda } ( Y ( u ) ) } { d u } \\\\ = & \\frac { 2 } { n } \\left ( 2 \\sum _ { i = 1 } ^ m n _ i p _ i Y _ i - \\frac { 1 } { 2 } E ( Y ) \\right ) ^ 2 - \\frac { 1 } { 2 } E ( Y ) ^ 2 - \\sum _ { i = 1 } ^ m n _ i Y _ i ^ 2 ( 2 p _ i - q _ i ^ 2 Y _ i ) ^ 2 . \\end{align*}"}
-{"id": "2535.png", "formula": "\\begin{align*} X _ k ( p _ * , q _ * ) = \\sqrt { \\frac { \\eta _ k } M } \\begin{pmatrix} e _ k ( x _ 1 ) \\\\ \\vdots \\\\ e _ k ( x _ M ) \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{pmatrix} , \\ ; [ X _ 0 , X _ k ] ( p _ * , q _ * ) = \\sqrt { \\frac { \\eta _ k } M } \\begin{pmatrix} - \\hat { E } \\begin{pmatrix} e _ k ( x _ 1 ) \\\\ \\vdots \\\\ e _ k ( x _ M ) \\end{pmatrix} \\\\ ( \\frac 1 { h ^ 2 } A + \\frac 1 M I ) \\begin{pmatrix} e _ k ( x _ 1 ) \\\\ \\vdots \\\\ e _ k ( x _ M ) \\end{pmatrix} \\end{pmatrix} \\end{align*}"}
-{"id": "9.png", "formula": "\\begin{align*} \\frac { \\sin \\varphi _ q ^ k } { \\mu ( x ^ k _ q ) } & = \\frac { 1 } { q } \\left ( \\frac { \\sin \\left ( \\frac { \\mu \\left ( { k } / { q } \\right ) } { q } \\right ) } { \\frac { \\mu \\left ( { k } / { q } \\right ) } { q } } + \\frac { \\beta ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) \\right ) \\end{align*}"}
-{"id": "6956.png", "formula": "\\begin{align*} L ^ \\perp _ { \\pi / ( ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ { m - 1 } ) ) } ( z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\cdots z _ { m - 1 } ^ { i _ { m - 1 } } ) \\ , M ( z _ m ) = M ( z _ m ) \\prod _ { i _ m \\ge 0 } \\ , L ^ \\perp _ { \\pi / ( ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) ) } ( z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\cdots z _ { m } ^ { i _ { m } } ) \\end{align*}"}
-{"id": "582.png", "formula": "\\begin{align*} \\delta ( x - [ r _ 0 ] ) + [ r _ 1 ] & \\equiv 2 [ \\phi ^ { - 1 } ( r _ 2 ) ] \\bmod I ^ 2 \\\\ ( \\ref { a = b } ) \\Rightarrow \\delta ( \\delta ( x - [ r _ 0 ] ) + [ r _ 1 ] ) & \\equiv [ r _ 2 ] - 2 [ r _ 2 ] \\bmod I \\\\ p = 2 \\in I \\Rightarrow \\delta ( \\delta ( x - [ r _ 0 ] ) + [ r _ 1 ] ) & \\equiv [ r _ 2 ] \\bmod I \\end{align*}"}
-{"id": "2492.png", "formula": "\\begin{gather*} z _ 1 ( u _ k , t ) : = x ( u _ k , t ) , t \\geqslant 0 , 1 \\leqslant k \\leqslant n , \\\\ z _ { i + 1 } ( u _ k , t ) : = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k ( z _ i ( u _ j , t ) - z _ i ( u _ j , t \\wedge \\sigma _ i ) ) \\cdot \\ 1 _ { A _ { k j } ^ i } , t \\geqslant 0 , \\\\ 1 \\leqslant k \\leqslant n , 1 \\leqslant i \\leqslant n - 1 . \\end{gather*}"}
-{"id": "2018.png", "formula": "\\begin{align*} g ( x ) = x f ' ( x ) = \\gamma \\alpha ( \\log | x | ) ^ { \\alpha - 1 } \\exp \\{ \\gamma ( \\log | x | ) ^ \\alpha \\} , \\end{align*}"}
-{"id": "5180.png", "formula": "\\begin{gather*} R _ \\ell = C _ \\ell \\oplus \\varphi _ { \\ell - s } ( R _ { \\ell - s } ) . \\end{gather*}"}
-{"id": "2687.png", "formula": "\\begin{align*} \\theta _ { V _ { \\theta } } ^ n = { \\bf 1 } _ { \\{ V _ { \\theta } = 0 \\} } \\theta ^ n . \\end{align*}"}
-{"id": "1973.png", "formula": "\\begin{align*} s _ i = \\begin{cases} a _ i m + b & { } \\\\ a _ i m + ( m - b ) & { } . \\end{cases} \\end{align*}"}
-{"id": "2749.png", "formula": "\\begin{align*} \\sigma ( f ) = \\int f \\ , \\mu . \\end{align*}"}
-{"id": "9788.png", "formula": "\\begin{align*} m ( \\mathbf { y } | M ) = \\frac { \\sum _ { \\alpha \\in A _ { n } } \\lambda _ { n , \\alpha } \\int _ { p \\in \\mathcal { P } _ { n , \\alpha } : p \\in B } \\prod _ { i = 1 } ^ { n } p ( y _ { i } ) d \\Pi _ { n , \\alpha } ( p ) } { \\Pi _ { n } ( B | y _ { 1 } , \\ldots , y _ { n } ) } . \\end{align*}"}
-{"id": "5704.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } : = \\inf \\left \\{ \\lambda > 0 \\ , : \\ , \\int _ { \\mathbb { R } ^ n } \\left | \\frac { f ( x ) } { \\lambda } \\right | ^ { p ( x ) } \\ , d x \\le 1 \\right \\} . \\end{align*}"}
-{"id": "6063.png", "formula": "\\begin{align*} \\widehat { \\Phi } ( \\xi ) + { \\sum _ { j = 1 } ^ \\infty } \\widehat { \\Psi } ( 2 ^ { - j } \\xi ) = 1 . \\end{align*}"}
-{"id": "3077.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { k \\to \\infty } \\lim _ { n \\to \\infty } \\rho _ { n , k } = D \\end{align*}"}
-{"id": "5112.png", "formula": "\\begin{align*} \\psi _ { \\vec { z } } ^ { 1 ^ { k } } ( \\vec { x } ) = ( 1 - q ^ { 2 } ) ^ { - k } \\prod _ { i = 1 } ^ { k } \\frac { z _ { i } ^ { M ' - 1 } } { ( 1 + z _ { i } ) ^ { M } } \\langle \\prod _ { 1 \\le i \\le k } C ^ { [ M ' , M ] } ( z _ { i } ) \\prod _ { 1 \\le i \\le k } \\beta _ { x _ { i } } ^ { * } \\rangle _ { [ M ' , M ] } \\ , ( u _ { 1 } \\otimes \\cdots \\otimes u _ { 1 } ) . \\end{align*}"}
-{"id": "3998.png", "formula": "\\begin{align*} \\begin{cases} & \\mbox { $ n \\geq d _ i + d _ j $ i f $ n _ { d _ i } $ + $ n _ { d _ j } \\geq 2 $ a n d $ i \\not = j $ } ; \\mbox { a n d } \\\\ & \\mbox { $ n \\geq 2 d _ i $ i f $ n _ { d _ i } \\geq 2 $ } . \\end{cases} \\end{align*}"}
-{"id": "161.png", "formula": "\\begin{align*} H _ \\nu ( \\zeta | ( \\zeta _ { - n } ^ { - 1 } ( \\tau ) ) = \\infty . \\end{align*}"}
-{"id": "1607.png", "formula": "\\begin{align*} Z ^ { 1 } = e ^ { m t } K ^ { 1 } ~ , ~ Z ^ { 2 } = e ^ { - m t } \\left ( K ^ { 1 } + m \\int \\frac { d x } { \\sigma \\left ( x \\right ) } F \\partial _ { F } \\right ) \\end{align*}"}
-{"id": "1561.png", "formula": "\\begin{align*} [ v _ 0 v _ 1 \\dots v _ i ] = [ \\{ v _ 0 , \\dots , v _ i \\} \\ , , \\ , v _ 0 < v _ 1 < \\dots < v _ i ] \\ . \\end{align*}"}
-{"id": "1151.png", "formula": "\\begin{align*} b ( z ) = b _ 0 + \\sum _ { k = 1 } ^ { M } \\frac { b _ { - 1 } ^ { ( k ) } } { z + \\epsilon _ k } , \\epsilon _ k \\neq \\epsilon _ j , k \\neq j \\epsilon _ k > 0 \\end{align*}"}
-{"id": "2224.png", "formula": "\\begin{align*} u ( x _ { 0 } , t ) = \\int _ { 0 } ^ { t } \\mu ( t - s ) v ( x _ { 0 } , s ) d s = 0 \\end{align*}"}
-{"id": "6669.png", "formula": "\\begin{gather*} \\left < \\beta _ 1 - \\beta _ 2 \\right > _ 0 = 0 , \\\\ \\left < \\beta _ 1 - \\beta _ 2 \\right > _ \\cdot \\in C ( [ 0 ; + \\infty ) ) , \\end{gather*}"}
-{"id": "6625.png", "formula": "\\begin{align*} \\mu _ t = P _ t \\ , \\nu \\ + \\ \\int _ 0 ^ t P _ { t - s } F _ f ( \\mu _ s ) \\ , d s \\mbox { f o r a l l } \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "1280.png", "formula": "\\begin{align*} X A ( q ) X ^ { - 1 } = E , X A ^ * ( q ) X ^ { - 1 } = E ^ * , \\end{align*}"}
-{"id": "2941.png", "formula": "\\begin{align*} \\hat { f } ( \\lambda \\omega ) = \\mathcal F ( R f ( \\omega , \\cdot ) ) ( \\lambda ) , \\end{align*}"}
-{"id": "462.png", "formula": "\\begin{align*} \\alpha = \\inf \\{ \\pi ^ { \\top } x \\ , | \\ W ^ { i _ 0 } \\} . \\end{align*}"}
-{"id": "1823.png", "formula": "\\begin{align*} \\mu _ s = \\frac { | d z | ^ 2 } { | z | ^ 2 } ( \\log | z | ) ^ 2 \\left ( \\frac { \\log | t | } { \\pi \\log | z | } \\sin \\frac { \\pi \\log | z | } { \\log | t | } \\right ) ^ 2 . \\end{align*}"}
-{"id": "8229.png", "formula": "\\begin{align*} \\mathcal { H } '' _ E ( \\sigma \\Phi ) = 0 , \\end{align*}"}
-{"id": "2045.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd r } d \\left ( \\eta _ i ( r , \\tfrac { d _ 0 } { 2 } ) , \\eta _ i ( r , - \\tfrac { d _ 0 } { 2 } ) \\right ) \\big | _ { r = 0 } & = \\frac { \\dd } { \\dd r } \\int _ { - d _ 0 / 2 } ^ { d _ 0 / 2 } \\langle \\frac { \\dd \\eta _ i } { \\dd s } , \\frac { \\dd \\eta _ i } { \\dd s } \\rangle ^ { \\frac 1 2 } d s = \\langle J _ i , e _ n \\rangle | _ { - d _ 0 / 2 } ^ { d _ 0 / 2 } - \\int _ { - d _ 0 / 2 } ^ { d _ 0 / 2 } \\langle J _ i , \\nabla _ { e _ n } e _ n \\rangle = 0 . \\end{align*}"}
-{"id": "1810.png", "formula": "\\begin{align*} L _ { j } ( s _ { j } ) = 2 \\pi ^ { 2 } s _ { j } ( 1 + s _ { j } e ( s _ { j } ) ) \\end{align*}"}
-{"id": "5557.png", "formula": "\\begin{align*} K \\beta = \\begin{cases} \\frac { 1 } { 2 } b _ { x } ( 1 ) + H b ( 1 ) & \\ , 0 \\leq H < \\infty , \\\\ - \\frac { 1 } { 2 } b _ { x } ( 1 ) - \\frac { 1 } { 2 H } b _ { x x } ( 1 ) & \\ , 0 < H \\leq \\infty , \\end{cases} \\end{align*}"}
-{"id": "4884.png", "formula": "\\begin{align*} \\pi ( \\delta ( x ) ) = \\pi ( [ r _ 1 ] + y ) = \\pi ( [ r _ 1 ] ) + \\pi ( y ) = \\pi ( [ r _ 1 ] ) = r _ 1 \\end{align*}"}
-{"id": "2766.png", "formula": "\\begin{align*} F ( X _ { T } ) = F ( X _ { 0 } ) + \\int _ 0 ^ T \\Delta _ { t } F ( X _ { t } ) d t + \\int _ 0 ^ T \\Delta _ { x } F ( X _ { t } ) \\psi ( t ) d t + \\int _ 0 ^ T \\Delta _ { x } F ( X _ { t } ) \\varphi ( t ) \\circ d B ^ { H } ( t ) . \\end{align*}"}
-{"id": "4101.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p - q - r } y ^ { q } ( c x + b y ) ^ { r } } { z ^ { p } } \\end{align*}"}
-{"id": "2649.png", "formula": "\\begin{align*} E = & \\{ \\{ ( 1 , k ) , ( 2 , k ) \\} : k \\in [ n ] \\} \\\\ & \\bigcup \\ , \\{ \\{ ( 1 , k ) , ( 1 , k + 1 ) \\} : k \\in [ n - 1 ] \\} \\\\ & \\bigcup \\ , \\{ \\{ ( 2 , k ) , ( 2 , k + 1 ) \\} : k \\in [ n - 1 ] \\} \\ , . \\end{align*}"}
-{"id": "8602.png", "formula": "\\begin{align*} 0 = i \\Pi _ { 2 } ( w ) + j ' \\sigma _ { 2 } \\Pi _ { 3 } ( w ) = \\Pi _ { 2 } ( w ) + \\sigma _ { 2 } \\Pi _ { 3 } ( w ) \\end{align*}"}
-{"id": "6912.png", "formula": "\\begin{align*} \\biggl ( \\frac { ( d - 1 ) ^ { d - 1 } } { d ^ { d / 2 } } \\biggr ) ^ { n } = 2 ^ { - n } \\left ( \\frac { e } { 2 } \\right ) ^ { x } \\exp \\left ( \\frac { 3 x ^ { 2 } } { 2 n } + O \\left ( \\frac { x ^ { 3 } } { n ^ { 2 } } \\right ) \\right ) . \\end{align*}"}
-{"id": "5636.png", "formula": "\\begin{align*} d _ E ( \\mu , \\nu ) = \\langle \\nabla E ( \\mu ) - \\nabla E ( \\nu ) , \\mu - \\nu \\rangle . \\end{align*}"}
-{"id": "8150.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\lambda _ i = \\sum _ { s \\in S } \\psi _ { i } ( s ) , & \\mbox { $ i = 1 , \\ldots , m $ ; } \\\\ \\mu _ { h 1 } + \\mu _ { h 2 } = \\sum _ { s \\in S } \\chi _ { _ h } ( s ) , & h = 1 , 2 , \\ldots , [ \\frac { n - 1 } { 2 } ] ; \\\\ \\mu _ { h 1 } ^ 2 + \\mu _ { h 2 } ^ 2 = \\sum _ { s _ 1 , s _ 2 \\in S } \\chi _ { _ h } ( s _ 1 s _ 2 ) , & h = 1 , 2 , \\ldots , [ \\frac { n - 1 } { 2 } ] . \\end{array} \\right . \\end{align*}"}
-{"id": "881.png", "formula": "\\begin{align*} \\operatorname * { d o m } \\varphi _ { A , k } = { \\mathbb { R } } k + A = { \\mathbb { R } } k + \\operatorname * { i n t } A \\not = \\emptyset \\mbox { i s a n o p e n s e t } , \\end{align*}"}
-{"id": "6180.png", "formula": "\\begin{align*} \\mathcal { A } _ n f ( x ) = \\begin{cases} \\mathcal { A } f ( x ) , & | x | < n , \\\\ 0 , & | x | \\geq n . \\end{cases} \\end{align*}"}
-{"id": "3465.png", "formula": "\\begin{align*} \\Omega \\ni x \\mapsto ( x _ 1 , x _ 2 , x _ 3 + \\bar { \\eta } ( x , t ) ( 1 + x _ 3 / b ( x _ 1 , x _ 2 ) ) ) = \\Theta ( x , t ) = ( y _ 1 , y _ 2 , y _ 3 ) \\in \\Omega ( t ) . \\end{align*}"}
-{"id": "6154.png", "formula": "\\begin{align*} M = \\left ( m \\Z ^ n + g \\Z ( \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } { \\bf e } _ i ) \\right ) \\cap H _ v . \\end{align*}"}
-{"id": "2816.png", "formula": "\\begin{align*} X _ { 1 : n } ( t ) = \\min _ { 1 \\le j \\le n } X _ j ( t ) \\le X _ { 2 : n } ( t ) \\le \\ldots \\le X _ { n - 1 : n } ( t ) \\le \\max _ { 1 \\le j \\le n } X _ j ( t ) = X _ { n : n } ( t ) . \\end{align*}"}
-{"id": "9138.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } v ' ( t ) + B ( t ) A ( t ) v ( t ) - B ' ( t ) B ( t ) ^ { - 1 } v ( t ) + B ( t ) P ( t ) B ( t ) ^ { - 1 } v ( t ) = B ( t ) f ( t ) \\\\ v ( 0 ) = B ( 0 ) u _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "4542.png", "formula": "\\begin{align*} W _ { 3 1 } = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { 1 / 2 + 2 \\beta / d } } { n ^ { 1 + 2 \\beta / d } } \\ , , \\ , \\frac { \\log n } { n k ^ { 1 / 2 } } \\ , , \\ , \\frac { k ^ { - 1 / 2 + \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "4699.png", "formula": "\\begin{align*} \\sigma _ k ( z ) = ( s _ 0 , s _ 1 , \\dots , s _ r ) , \\\\ \\sigma _ k ( z ' ) = ( s ' _ 0 , s ' _ 1 , \\dots , s ' _ r ) . \\end{align*}"}
-{"id": "4460.png", "formula": "\\begin{align*} \\mathbb { E } _ f ( \\hat { H } _ n ) - H = o \\Bigl ( \\frac { k ^ { 1 - \\epsilon } } { n ^ { 1 - \\epsilon } } \\Bigr ) , \\end{align*}"}
-{"id": "6072.png", "formula": "\\begin{align*} \\langle \\Lambda , c \\rangle = ( \\Lambda | \\nu ( c ) ) = ( \\Lambda | \\delta ) = ( w \\Lambda | w \\delta ) = ( w \\Lambda | \\delta ) = ( t _ { w _ 0 \\lambda } \\Lambda _ 0 | \\delta ) = ( \\Lambda _ 0 | \\delta ) = 1 \\end{align*}"}
-{"id": "1786.png", "formula": "\\begin{align*} M : = \\{ \\nu ( \\varphi ^ i ( a _ j ) ) \\ | \\ i \\in \\Z , j = 1 , \\ldots , n \\} \\subseteq \\R \\end{align*}"}
-{"id": "4446.png", "formula": "\\begin{align*} p _ { n , x , u } : = \\int _ { B _ x ( r _ { n , u } ) } f ( y ) \\ , d y { \\rm a n d } r _ { n , u } : = \\biggl \\{ \\frac { e ^ { \\Psi ( k ) } u } { V _ d ( n - 1 ) } \\biggr \\} ^ { 1 / d } . \\end{align*}"}
-{"id": "7980.png", "formula": "\\begin{align*} s \\mapsto t = s ^ 2 , \\end{align*}"}
-{"id": "4627.png", "formula": "\\begin{align*} \\lVert f \\rVert _ { L ^ { \\infty } } = \\max _ { \\omega , \\varphi , \\vartheta } | f ( \\omega , \\varphi , \\vartheta ) | . \\end{align*}"}
-{"id": "8713.png", "formula": "\\begin{gather*} v ^ n ( t , x ) - v ^ { n + p } ( t , x ) = \\int _ { t } ^ { T } \\int _ { H } e ^ { - ( s - t ) { A } } \\left [ \\nabla ^ G v ^ n ( s , z + e ^ { ( s - t ) { A } } x ) \\ , B ^ n ( s , z + e ^ { ( s - t ) { A } } x ) \\right . \\\\ \\left . - \\nabla ^ G v ^ { n + p } ( s , z + e ^ { ( s - t ) { A } } x ) \\ , B ^ { n + p } ( s , z + e ^ { ( s - t ) { A } } x ) \\right ] \\mu _ { s - t } ( d z ) \\ , d s \\\\ + \\int _ t ^ T \\int _ { H } e ^ { - ( s - t ) { A } } \\left ( G B ^ n ( s , z + e ^ { ( s - t ) { A } } x ) - G B ^ { n + p } ( s , z + e ^ { ( s - t ) { A } } x ) \\right ) \\mu _ { s - t } ( d z ) . \\end{gather*}"}
-{"id": "5578.png", "formula": "\\begin{align*} - \\Psi _ { \\zeta \\zeta } + \\frac 1 4 \\Psi = z m \\Psi , \\Psi _ t = ( u - \\frac { 1 } { z } ) \\Psi _ { \\zeta } - \\frac { u _ { \\zeta } } { 2 } \\Psi , \\end{align*}"}
-{"id": "8093.png", "formula": "\\begin{align*} P _ { s _ i } ^ { m w t p } & = \\sqrt { \\frac { A _ { s _ i , r _ j } \\left ( 2 + C _ { s _ i , r _ j } + \\sqrt { C _ { s _ i , r _ j } ^ 2 + 8 C _ { s _ i , r _ j } } \\right ) } { 2 p _ { t h } ^ { s _ i } } } , \\\\ P _ { r _ j } ^ { m w t p } & = f \\left ( P _ { s _ i } ^ { m w t p } \\right ) . \\end{align*}"}
-{"id": "2095.png", "formula": "\\begin{align*} \\tilde { f } ( H ( s ) ) = f ( \\tilde { H } ( s ) ) = \\tilde { \\hat { H } } ( s ) , \\end{align*}"}
-{"id": "3386.png", "formula": "\\begin{align*} \\int _ { u } ^ \\infty \\frac { 1 } { \\beta + x } e ^ { - x } d x = e ^ { \\beta } \\hat { \\Gamma } ( 0 , u + \\beta ) \\end{align*}"}
-{"id": "9975.png", "formula": "\\begin{align*} \\Lambda ^ { ( n + 1 ) } ( s _ 0 ) = \\max _ { a \\in \\mathcal { A } ( s _ 0 ) } \\left [ g ( s _ 0 , a ) + \\tau \\sum _ { \\mathbf { H } ' } \\mathrm { P r } ( \\mathbf { H } ' | \\mathbf { H _ 0 } ) h ^ { ( n ) } ( s ' ) \\right ] , \\end{align*}"}
-{"id": "2991.png", "formula": "\\begin{align*} \\delta _ v ( \\Theta ( L ) ( x , y , z ) ) ( a ) = \\sum _ { i + j = N + 1 ; ~ i , j > 0 } - f _ 0 ^ { \\lambda _ i ( x , \\lambda _ j ( y , z ) ) } + f _ 0 ^ { \\lambda _ i ( y , \\lambda _ j ( x , z ) ) } + f _ 0 ^ { \\lambda _ i ( \\lambda _ j ( x , y ) , z ) } ( a ) \\end{align*}"}
-{"id": "1236.png", "formula": "\\begin{align*} S ( x ) = 0 \\ \\ \\ \\ \\ \\ \\ x \\in ( \\lambda _ { \\varphi , w } ) _ a . \\end{align*}"}
-{"id": "7909.png", "formula": "\\begin{align*} R _ g = W _ g + \\frac 1 2 B _ g \\textcircled { $ \\wedge $ } g + \\frac S { 2 4 } g \\textcircled { $ \\wedge $ } g , \\end{align*}"}
-{"id": "5375.png", "formula": "\\begin{align*} ( - 1 ) ^ { u - i } { u \\choose i } { \\ , } c _ { r , u } c _ { d , d } ^ u + \\delta _ { r , r _ 0 } \\delta _ { u , \\frac { q q _ 2 } { d } } { \\ , } \\chi _ { \\{ \\frac { q _ 1 q _ 2 } { d } , \\frac { q _ 2 ^ 2 } { d } \\} } ( i ) { \\ , } c _ { q , q _ 1 } ^ { q _ 2 + 1 } ~ = ~ 0 . \\end{align*}"}
-{"id": "4448.png", "formula": "\\begin{align*} \\mathbb { E } ( \\hat { H } _ n ) & = \\int _ \\mathcal { X } f ( x ) \\int _ 0 ^ \\infty \\log u \\ , d F _ { n , x } ( u ) \\ , d x \\approx \\int _ \\mathcal { X } f ( x ) \\int _ 0 ^ \\infty \\log u \\ , d F _ x ( u ) \\ , d x \\\\ & = \\int _ { \\mathcal { X } } f ( x ) \\int _ 0 ^ \\infty \\log \\Bigl ( \\frac { t e ^ { - \\Psi ( k ) } } { f ( x ) } \\Bigr ) e ^ { - t } \\frac { t ^ { k - 1 } } { ( k - 1 ) ! } \\ , d t \\ , d x = H . \\end{align*}"}
-{"id": "4638.png", "formula": "\\begin{align*} F ( x ) = x ^ 9 + A _ 6 x ^ 3 + A _ 8 x \\end{align*}"}
-{"id": "766.png", "formula": "\\begin{gather*} [ \\delta ( \\gamma ) ] = [ \\delta ( \\alpha ) ] + [ \\delta ( \\beta ) ] . \\end{gather*}"}
-{"id": "4417.png", "formula": "\\begin{align*} C _ 1 ' \\leq \\| g _ n \\| _ { \\widehat { E } } \\leq \\| | g _ n | + | g _ m | \\| _ { \\widehat { E } } = \\| g _ n - g _ m \\| _ { \\widehat { E } } \\to 0 , \\end{align*}"}
-{"id": "3845.png", "formula": "\\begin{align*} E _ R ( n ) \\sim \\sum _ { k = 1 } ^ \\infty \\frac { 9 } { 4 ^ { 1 - k } + 4 + 4 ^ k } \\approx 1 . 6 2 2 9 7 . \\end{align*}"}
-{"id": "8096.png", "formula": "\\begin{align*} C _ { k , l , N } ^ { r a d } : = ( N \\omega _ N ) ^ { ( l - k + 1 ) / ( l + N ) } \\cdot ( l + N ) ^ { ( k + N - 1 ) / ( l + N ) } . \\end{align*}"}
-{"id": "1049.png", "formula": "\\begin{align*} | A | = \\int _ A \\ , d x \\leq \\frac 1 \\lambda \\int _ { \\R ^ 2 } \\left | ( \\Phi _ 1 \\ast 1 _ E ) ( x ) \\right | 1 _ A ( x ) \\ , d x \\leq \\frac 1 \\lambda \\sum _ { j = 0 } ^ \\infty \\int _ { \\R ^ 2 } \\left | ( Q ^ j \\ast 1 _ E ) ( x ) \\right | 1 _ A ( x ) \\ , d x . \\end{align*}"}
-{"id": "6354.png", "formula": "\\begin{align*} L _ a U = | y | ^ a \\bigg ( \\frac { U _ a } { r } T ( x , r ) + G ( X ) \\bigg ) B _ 1 \\setminus \\mathcal { P } \\end{align*}"}
-{"id": "342.png", "formula": "\\begin{align*} g A g ^ t = M _ { ( 2 j _ 0 ) } M _ { ( 1 2 ) } A M _ { ( 1 2 ) } M _ { ( 2 j _ 0 ) } \\end{align*}"}
-{"id": "1805.png", "formula": "\\begin{align*} \\frac { 1 } { C } \\Big ( { \\sum _ { j = 1 } ^ \\infty } | c _ { j } | ^ { 2 } \\Big ) ^ { p / 2 } \\le \\int _ { 0 } ^ { 1 } \\Big | { \\sum _ { j = 1 } ^ \\infty } c _ { j } r _ { j } ( z ) \\Big | ^ { p } d z \\le C \\Big ( { \\sum _ { j = 1 } ^ \\infty } | c _ { j } | ^ { 2 } \\Big ) ^ { p / 2 } . \\end{align*}"}
-{"id": "3686.png", "formula": "\\begin{align*} \\frac { d H } { d t } = \\int \\left [ H _ { \\zeta } \\frac { \\partial \\zeta } { \\partial t } + H _ { \\mu } \\frac { \\partial \\mu } { \\partial t } \\right ] d A = 0 ~ , \\end{align*}"}
-{"id": "261.png", "formula": "\\begin{align*} S _ 3 & = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ 0 ^ { \\frac { a _ n } { n - 1 } } \\mathrm { B } _ { k , n - k } ( s ) \\biggl \\{ \\log ^ 2 u _ { x , s } - \\log ^ 2 \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) \\biggr \\} \\ , d s \\ , d x \\\\ & = O \\biggl \\{ \\max \\biggl ( \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\log n \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\biggr \\} . \\end{align*}"}
-{"id": "7963.png", "formula": "\\begin{align*} \\mu ( H \\setminus V ) = 0 . \\end{align*}"}
-{"id": "6847.png", "formula": "\\begin{align*} \\tfrac { 1 } { q _ 1 ' } = \\tfrac { p } { \\tilde q } + \\tfrac { 1 } { q _ 1 } , \\tfrac { 1 } { r _ 1 ' } = \\tfrac { p } { \\tilde r } + \\tfrac { 1 } { r _ 1 } , \\tfrac { 1 } { \\tilde q } - \\tfrac { 1 } { q _ 2 } = \\tfrac { d } { r _ 2 } - \\tfrac { d } { \\tilde r } = | s _ c | . \\end{align*}"}
-{"id": "9924.png", "formula": "\\begin{align*} X _ { \\pm } v _ { \\pm } & = \\sum _ { i = 0 } ^ n \\left ( \\kappa \\sqrt { \\pm 1 } \\ , ( q ^ { n / 2 - i } \\mp q ^ { - n } q ^ { i - n / 2 } ) \\lambda _ i v _ i \\ , - \\ , s ^ { - 1 } ( \\pm 1 ) [ i ] \\l _ i v _ { i - 1 } \\right ) \\\\ & = \\sum _ { i = 0 } ^ n \\left ( - q ^ { - n / 2 } \\sqrt { \\pm 1 } \\ , [ n - i ] \\lambda _ i \\ , \\mp \\ , s ^ { - 1 } [ i + 1 ] \\l _ { i + 1 } \\right ) v _ i \\\\ & = 0 \\end{align*}"}
-{"id": "6318.png", "formula": "\\begin{align*} \\| F _ { k , N } \\| _ { M _ 1 } & = \\left ( \\sum _ { l \\in \\widetilde { \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } } \\| f _ l ^ { \\alpha _ 1 } \\| ^ { q } _ { L ^ { p _ 1 } } \\right ) ^ { \\frac { 1 } { q } } \\\\ & \\sim | \\widetilde { \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } | ^ { 1 / q } 2 ^ { j n \\alpha _ 1 ( 1 - 1 / p _ 1 ) } \\\\ & \\sim 2 ^ { j n ( \\alpha _ 2 - \\alpha _ 1 ) / q } 2 ^ { j n \\alpha _ 1 ( 1 - 1 / p _ 1 ) } . \\end{align*}"}
-{"id": "8097.png", "formula": "\\begin{align*} d \\mu _ { l } ( x ) = | x | ^ { l } \\ , d x . \\end{align*}"}
-{"id": "4841.png", "formula": "\\begin{align*} \\sum _ { \\mu } s _ { \\kappa / \\mu } ( \\rho _ 2 ) s _ { \\nu / \\mu } ( \\rho _ 1 ) \\mathcal { U } ^ { \\llcorner } _ { \\rho _ 1 , \\rho _ 2 } ( \\pi \\vert \\nu , \\mu , \\kappa ) = \\frac { s _ { \\pi / \\kappa } ( \\rho _ 1 ) s _ { \\pi / \\nu } ( \\rho _ 2 ) } { H ( \\rho _ 2 ; \\rho _ 1 ) } . \\end{align*}"}
-{"id": "4223.png", "formula": "\\begin{align*} \\frac { 2 } { e ^ t + 1 } e ^ { x t } = & \\sum _ { n = 0 } ^ \\infty C h _ { n , \\lambda } ( x ) \\frac { 1 } { n ! } \\Big ( e ^ { \\frac { 1 } { \\lambda } t } - 1 \\Big ) ^ n \\\\ = & \\sum _ { n = 0 } ^ \\infty C h _ { n , \\lambda } ( x ) \\sum _ { m = n } ^ \\infty S _ 2 ( m , n ) \\lambda ^ { - m } \\frac { t ^ m } { m ! } , \\end{align*}"}
-{"id": "4519.png", "formula": "\\begin{align*} S _ 1 = \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\int _ 0 ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log ^ 2 u _ { x , s } \\ , d s \\ , d x = O \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "126.png", "formula": "\\begin{align*} \\frac { 1 } { n + 1 } { 2 n \\choose n } \\frac { ( - 1 ) ^ { n - 1 } n + 1 } { 4 ^ n ( 2 n - 1 ) } = \\frac { 1 } { n ! } \\sum _ { m = 0 } ^ n \\Big ( \\frac { 1 } { 2 } \\Big ) ^ m S _ 1 ( n , m ) \\end{align*}"}
-{"id": "6111.png", "formula": "\\begin{align*} f _ { k } = \\sum _ { \\alpha \\geq 2 k } s ^ { \\alpha } f _ { k , \\alpha } , \\ f _ { k , \\alpha } \\in \\mathcal { U } ^ { \\alpha } \\end{align*}"}
-{"id": "1464.png", "formula": "\\begin{align*} f - m _ f ( Q _ 0 ) = g _ 1 + \\sum _ { j \\in J _ 1 } \\alpha _ { j , 1 } \\chi _ { Q ^ 1 _ j } + \\sum _ { j \\in J _ 1 } ( f - m _ f ( Q ^ 1 _ j ) ) \\chi _ { Q ^ 1 _ j } , \\end{align*}"}
-{"id": "5171.png", "formula": "\\begin{gather*} r _ { \\ell } \\cdot r _ 1 = r _ { \\ell 1 } + r _ { \\ell + 1 } + 1 . \\end{gather*}"}
-{"id": "8362.png", "formula": "\\begin{align*} E _ { N _ 0 } ( \\sum _ { g \\in G } x _ g g ) = \\sum _ { g \\in G } x _ g z _ g g \\end{align*}"}
-{"id": "9901.png", "formula": "\\begin{align*} [ x _ 0 , x _ 1 ] & \\ ; = \\ ; \\phantom { - } 0 & \\{ x _ 0 , x _ 1 \\} & \\ ; = \\ ; 2 x _ 0 x _ 1 \\ ; = \\ ; [ x _ 2 , x _ 3 ] \\\\ [ x _ 0 , x _ 2 ] & \\ ; = \\ ; \\phantom { - } b ^ 2 \\{ x _ 1 , x _ 3 \\} & \\{ x _ 0 , x _ 2 \\} & \\ ; = \\ ; [ x _ 3 , x _ 1 ] \\\\ [ x _ 0 , x _ 3 ] & \\ ; = \\ ; - b ^ 2 \\{ x _ 1 , x _ 2 \\} & \\{ x _ 0 , x _ 3 \\} & \\ ; = \\ ; [ x _ 1 , x _ 2 ] \\end{align*}"}
-{"id": "4639.png", "formula": "\\begin{align*} \\mathcal { C } _ i : ( x - y ) ^ 2 + \\alpha _ i ( x + y ) + \\beta _ i = 0 , \\end{align*}"}
-{"id": "169.png", "formula": "\\begin{align*} H _ { \\mu _ n } ( \\xi | \\xi _ { - \\infty } ^ { - 1 } ( T ) ) = H _ { \\mu _ n } ( \\xi | \\xi _ { - n } ^ { - 1 } ( T ) ) = H _ \\nu ( \\zeta | \\zeta _ { - n } ^ { - 1 } ( \\tau ) ) = \\infty . \\end{align*}"}
-{"id": "3357.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\| \\cdot \\| _ { r \\ast } ^ 2 = \\left ( \\frac { 1 } { 2 } \\| \\cdot \\| _ { r } ^ 2 \\right ) ^ \\ast . \\end{align*}"}
-{"id": "2857.png", "formula": "\\begin{align*} K ^ n ( C ) = \\underbrace { \\overline { F ^ c } \\circ \\dots \\circ \\overline { F ^ c } } _ { n } ( U ( C ) ) . \\end{align*}"}
-{"id": "9185.png", "formula": "\\begin{align*} \\mathcal A f _ u ( x ) = ( F ( u ) + ( R ( u ) x ) f _ u ( x ) , x \\in D , \\end{align*}"}
-{"id": "7364.png", "formula": "\\begin{align*} ( X \\triangleright T ) ^ * = S ( X ) ^ * \\triangleright T ^ * , X \\in U _ q ( \\mathfrak { l } ) , \\ T \\in \\mathrm { E n d } ( \\Lambda _ q ( \\mathfrak { u } _ + ) ) . \\end{align*}"}
-{"id": "5945.png", "formula": "\\begin{align*} H _ t : = W _ t - \\int _ 0 ^ t F ( L _ r ) \\ , \\dd r \\ , , \\ ; \\ ; t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6272.png", "formula": "\\begin{align*} 0 \\geq \\langle \\nabla _ k \\nabla _ k \\nabla w , e _ n \\rangle , k = 1 , \\ldots n - 1 . \\end{align*}"}
-{"id": "9191.png", "formula": "\\begin{align*} B _ { t , i } & = \\int _ 0 ^ t b _ i ( \\widetilde X _ { s _ - } ) d s , \\\\ C _ { t , i j } & = \\int _ 0 ^ t c _ { i j } ( \\widetilde X _ { s - } ) d s , \\\\ \\nu ( \\omega ; d t , d \\xi ) & = K ( \\widetilde X _ t , d \\xi ) d t . \\end{align*}"}
-{"id": "3848.png", "formula": "\\begin{align*} \\gamma ( p , n ) = \\max \\left \\{ \\frac { n - 1 } { 2 } \\left ( \\frac { 1 } 2 - \\frac 1 p \\right ) \\ , , \\ , \\frac { n - 1 } { 2 } - \\frac { n } { p } \\right \\} \\ , . \\end{align*}"}
-{"id": "7394.png", "formula": "\\begin{align*} c ( [ M \\times T ^ n _ R \\times T ^ n ] , F ) & \\leq 2 | | H | | _ { C ^ 0 } + \\max \\{ 0 , \\lambda ( m + n ) \\} \\\\ & < 2 ( \\sum _ { i = 1 } ^ n ( R _ i - \\epsilon ) \\cdot | e _ i | + | | G | | _ { C ^ 0 } ) + \\max \\{ 0 , \\lambda ( m + n ) \\} , \\end{align*}"}
-{"id": "6948.png", "formula": "\\begin{align*} ( s _ \\mu [ s _ \\nu ] ( Z ) ) ' \\ = \\ \\begin{cases} \\ s _ \\mu [ s _ { \\nu ' } ] ( Z ) & \\mbox { i f $ | \\nu | $ i s e v e n } ; \\cr \\ s _ { \\mu ' } [ s _ { \\nu ' } ] ( Z ) & \\mbox { i f $ | \\nu | $ i s o d d } . \\cr \\end{cases} \\end{align*}"}
-{"id": "4408.png", "formula": "\\begin{align*} \\| x ^ * \\| = \\sup _ { \\| x \\| _ X \\leq 1 } | x ^ * ( x ) | = \\sup _ { \\| x \\| ^ * \\leq 1 } | x ^ * ( x ) | = \\sup _ { \\| x \\| _ { \\widehat { X } } \\leq 1 } | x ^ * ( x ) | . \\end{align*}"}
-{"id": "9682.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\tau ( t ) = + \\infty . \\end{align*}"}
-{"id": "9240.png", "formula": "\\begin{align*} I _ 1 & = \\mathbb { E } [ \\int _ 0 ^ T \\int _ D \\{ H ( t , x ) - \\widehat { H } ( t , x ) - \\widehat { p } ( t , x ) ( A _ u Y ( t , x ) - A _ { \\hat { u } } \\hat { Y } ( t , x ) + \\widetilde { a } ( t , x ) ) - \\widehat { q } ( t , x ) \\widetilde { b } ( t , x ) \\\\ & - \\int _ { \\mathbb { R } } \\hat { r } ( t , x , \\zeta ) \\tilde { c } ( t , x , \\zeta ) \\nu ( d \\zeta ) \\} d x d t ] . \\end{align*}"}
-{"id": "9883.png", "formula": "\\begin{align*} \\lim _ { C \\rightarrow \\infty } \\sup _ { { \\epsilon } \\in ( 0 , 1 ) } P \\big ( \\sup _ { t \\leq T } | V ^ { \\beta ( { \\epsilon } ) , u ^ { \\epsilon } } _ { \\xi ^ { \\epsilon } } ( t , . ) | _ { 2 } \\geq C \\big ) = 0 . \\end{align*}"}
-{"id": "1152.png", "formula": "\\begin{align*} \\begin{gathered} D _ t \\rho = \\sum _ { j = 1 } ^ N \\dot m _ j \\delta _ { x _ j } - m _ j \\dot x _ j \\delta _ { x _ j } ' , \\\\ z m _ j \\dot x _ j = - \\frac 1 2 [ b _ x ] ( x _ j ) - z m _ j b ( x _ j ) , \\\\ z \\dot m _ j = \\frac 1 2 [ b _ { x x } ] ( x _ j ) + z m _ j \\langle b _ x \\rangle ( x _ j ) , \\end{gathered} \\end{align*}"}
-{"id": "5023.png", "formula": "\\begin{align*} U _ A : = \\big \\{ g \\in G \\ , : \\ , g \\cdot A \\in U \\big \\} = A , \\textrm { f o r a l l $ A \\subset G $ } . \\end{align*}"}
-{"id": "9635.png", "formula": "\\begin{align*} { \\rm c o } _ { - } ( M ) \\ = \\ \\Bigl \\{ \\ , x - y \\ \\Bigl | \\ x \\in { \\rm c o } \\ , ( M ) , \\ y \\ge 0 \\Bigr \\} \\ ; { \\rm c o } _ { + } ( M ) \\ = \\ \\Bigl \\{ \\ , x + y \\ \\Bigl | \\ x \\in { \\rm c o } \\ , ( M ) , \\ y \\ge 0 \\Bigr \\} \\ , . \\end{align*}"}
-{"id": "5390.png", "formula": "\\begin{align*} e ( G ) \\leq ( k - 1 ) e ( R ' ) + \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } \\leq \\bigg ( ( k - 1 ) ( k - \\alpha - \\tfrac 1 4 ) + ( k - 2 \\alpha + 5 \\alpha ^ 2 ) \\bigg ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "7772.png", "formula": "\\begin{align*} \\beta _ { 0 , j } ( M ) = \\begin{cases} 1 & j = 0 \\\\ 0 & j \\ne 0 \\end{cases} \\beta _ { 1 , j } ( M ) = \\begin{cases} r & j = e \\\\ 0 & j \\ne e \\end{cases} . \\end{align*}"}
-{"id": "738.png", "formula": "\\begin{align*} & c _ { 0 } = b _ { 0 } \\\\ & c _ { n } = c _ { n - 1 } + C _ { 3 } \\sqrt [ ] { c _ { n - 1 } } \\ \\ \\ \\ . \\end{align*}"}
-{"id": "8659.png", "formula": "\\begin{align*} R _ { \\tau } \\left [ \\Phi \\right ] \\left ( x \\right ) = R _ { \\tau } \\Phi \\left ( x \\right ) = \\mathbb { E } \\Phi \\left ( X _ \\tau ^ { 0 , x } \\right ) , \\Phi \\in B _ b ( H , J ) , \\ ; \\ ; x \\in H , \\ ; \\tau \\ge 0 , \\end{align*}"}
-{"id": "7841.png", "formula": "\\begin{align*} \\{ C _ { i j } , D _ { k l } \\} = \\delta _ { i l } \\delta _ { j k } . \\end{align*}"}
-{"id": "1885.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial m } ( \\frac 3 8 I ) = & 8 z ( s - 2 ) + 1 - 8 s + 4 m ( 1 + 8 s - 2 s ^ 2 ) \\\\ = & - ( 1 6 z - 1 - 4 m ) - 8 s ( 1 + m s - z - 4 m ) . \\end{align*}"}
-{"id": "8579.png", "formula": "\\begin{align*} \\psi _ { k } \\overline { N } ( ^ { \\ast } b ) = \\overline { \\widehat { N } } ( \\widehat { \\varphi } ( ^ { \\ast } b ) ) \\end{align*}"}
-{"id": "8983.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | w ( x ) - w ( y ) | ^ { p - 2 } \\ , ( w ( x ) - w ( y ) ) \\ , ( \\varphi _ h ( x ) - \\varphi _ h ( y ) ) } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y - \\int _ \\Omega \\ , f ( x ) \\cdot \\Phi _ { k } ' ( w ) \\varphi _ h \\geq 0 . \\end{align*}"}
-{"id": "1706.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } \\frac { \\ln \\P \\{ \\max _ { 1 \\le n \\le N } | S _ n | \\le f _ N \\} } { f _ N ^ { - 1 / H } N } = 0 . \\end{align*}"}
-{"id": "3.png", "formula": "\\begin{align*} \\hat F ( X , 0 ) & = ( X , 0 ) & X & \\in \\R . \\end{align*}"}
-{"id": "4894.png", "formula": "\\begin{align*} w ( a , b ) w ( c , d ) = w ( a + c , b + d + ( a \\pi ) c ) . \\end{align*}"}
-{"id": "8282.png", "formula": "\\begin{align*} \\pi ( x ) = \\gamma _ v , \\ , \\ , \\ , x _ { v - 1 } < x \\leq x _ v \\end{align*}"}
-{"id": "7131.png", "formula": "\\begin{align*} z _ j ( t ) = P _ R ( X _ j ( t ) ) = P _ R \\Big ( Q \\Big ( t + \\frac { 2 \\pi j } { n } \\Big ) \\Big ) = q \\Big ( t + \\frac { 2 \\pi j } { n } \\Big ) , 0 \\leq j \\leq n - 1 , \\end{align*}"}
-{"id": "1165.png", "formula": "\\begin{align*} \\dot { A } _ { 1 } & = ( \\beta + \\frac { 1 } { 2 } b _ { x } ( 1 ) ) A _ { 1 } - \\frac { 1 } { 2 } b _ { x x } ( 1 ) B _ { 1 } = ( \\beta + \\frac { 1 } { 2 } b _ { x } ( 1 ) ) A _ { 1 } + H ( b _ x ( 1 ) + H b ( 1 ) ) B _ { 1 } , \\\\ \\dot { B } _ { 1 } & = ( \\beta - \\frac { 1 } { 2 } b _ { x } ( 1 ) ) B _ { 1 } + b ( 1 ) A _ { 1 } = ( \\beta - \\frac { 1 } { 2 } b _ { x } ( 1 ) ) B _ { 1 } - \\frac { 1 } { H ^ 2 } ( H b _ { x } ( 1 ) + \\frac { 1 } { 2 } b _ { x x } ( 1 ) ) A _ { 1 } . \\end{align*}"}
-{"id": "8791.png", "formula": "\\begin{align*} K ^ { ( 2 r + 1 , r _ L ) } _ 0 ( x ) = \\sum _ { i = - r } ^ { r } E ^ { ( i i ) } + \\frac { q - x ^ 2 } { h _ 0 ( a , c , x ) } \\Big ( \\sum _ { r _ L < i \\leq r } t _ 0 E ^ { ( - i , - i ) } + E ^ { ( r + 1 - i , r + 1 - i ) } \\\\ { } - \\sum _ { r _ L < i \\leq r } q ^ { - i } E ^ { ( - i , r + 1 - i ) } + q ^ { r + 1 - i } t _ 0 E ^ { ( r + 1 - i , - i ) } \\Big ) , \\end{align*}"}
-{"id": "9086.png", "formula": "\\begin{align*} M ^ { ( 2 ) } _ { N } ( s ) = - \\frac { \\sqrt { N ( N + \\alpha ) } } { s } \\int _ { 0 } ^ { \\infty } e ^ { - ( 1 - s ) x } x ^ { \\alpha } L ^ { ( \\alpha ) } _ { N } ( x ) L ^ { ( \\alpha ) } _ { N - 1 } ( x ) \\ , \\mathrm { d } x \\ . \\end{align*}"}
-{"id": "1653.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\| f ( \\cdot , v ) \\| _ { H ^ { s } _ p } ^ p \\ , \\dd v = \\int _ { \\R ^ d } \\dd v \\int _ { \\R ^ d } | { \\cal F } ^ { - 1 } _ x [ ( 1 + | \\cdot | ^ { s } ) { \\cal F } _ x f ( \\cdot , v ) ] ( x ) | ^ p \\dd x < \\infty \\end{align*}"}
-{"id": "7196.png", "formula": "\\begin{align*} v ' ( 0 ) = - g ' ( \\frac \\pi 2 ) = C _ 1 , v ^ { '' } ( 0 ) = g ^ { '' } ( \\frac \\pi 2 ) = - 2 C _ 1 , \\end{align*}"}
-{"id": "1920.png", "formula": "\\begin{align*} g _ { h , \\vec { b } } ( \\tau ) = \\sum _ { \\alpha , \\beta = 1 } ^ r h _ { \\alpha \\beta } ( \\tau ) \\ , \\sigma _ \\alpha ( \\cdot ) \\otimes \\sigma _ \\beta ( \\cdot ) + \\sum _ { i = 1 } ^ m \\ , b _ i ( \\tau ) \\ , \\pi _ i ^ * g _ i , \\end{align*}"}
-{"id": "8797.png", "formula": "\\begin{align*} \\tilde { R } _ { \\bar { 0 } , 0 } ^ { \\tau _ 0 } ( w u ) R _ { \\bar { 0 } , 0 } ^ { \\tau _ 0 } ( w u ) = \\mathbb { I } . \\end{align*}"}
-{"id": "1428.png", "formula": "\\begin{align*} L _ { - 1 } ( A x _ { \\alpha } ) & = ( L _ { - 1 } A ) x _ { \\alpha } + A ( L _ { - 1 } x _ { \\alpha } ) , \\\\ T ( A \\psi _ { \\alpha } ) & = ( T A ) \\psi _ { \\alpha } + A ( T \\psi _ { \\alpha } ) \\end{align*}"}
-{"id": "8890.png", "formula": "\\begin{align*} \\bigl ( \\tfrac { 1 } { 2 } - \\tfrac { 1 } { p } \\bigr ) \\int _ { \\R ^ N } \\abs { D _ { A } u } ^ 2 + \\abs { u } ^ 2 = \\mathcal { I } _ { A } ( u ) , \\end{align*}"}
-{"id": "6007.png", "formula": "\\begin{gather*} ( \\epsilon _ { \\times } ) _ { A B C D E F G } : = \\tfrac { 1 } { 4 2 } { \\Phi } _ { K [ A B } { \\Phi } ^ K { } _ { C D } { \\Phi } _ { E F G ] } . \\end{gather*}"}
-{"id": "6850.png", "formula": "\\begin{align*} S _ I ( u ) : = \\| u \\| _ { S ( I ) } . \\end{align*}"}
-{"id": "5168.png", "formula": "\\begin{gather*} q : = q _ 1 \\otimes \\cdots \\otimes q _ l , \\end{gather*}"}
-{"id": "6835.png", "formula": "\\begin{align*} [ e ^ { i t \\Delta } f ] ( x ) = ( 4 \\pi i t ) ^ { - \\frac { d } { 2 } } \\int _ { \\R ^ d } e ^ { \\frac { i | x - y | ^ 2 } { 4 t } } f ( y ) \\ , d y , t \\neq 0 . \\end{align*}"}
-{"id": "6872.png", "formula": "\\begin{align*} \\int _ { | x | \\geq \\tilde C ( \\eta ) \\sqrt { t } } \\bigl | | x | ^ { | s _ c | } e ^ { - i t \\Delta } \\Psi ( t ) \\bigr | ^ 2 \\ , d x & = \\lim _ { n \\to \\infty } \\int _ { | x | \\geq \\tilde C ( \\eta ) \\sqrt { t } } \\bigl | | x | ^ { | s _ c | } e ^ { - i t \\Delta } v ^ { [ t _ n ] } ( t ) \\bigr | ^ 2 \\ , d x \\\\ & = \\lim _ { n \\to \\infty } \\int _ { | x | \\geq \\tilde C ( \\eta ) \\sqrt { t _ n t } } \\bigl | | x | ^ { | s _ c | } e ^ { - i t _ n t \\Delta } v ( t _ n t ) \\bigr | ^ 2 \\ , d x \\leq \\eta \\end{align*}"}
-{"id": "9036.png", "formula": "\\begin{align*} & \\sup _ { 0 \\leq j \\leq k - 1 } \\ , \\sup _ { \\theta \\in [ 0 , 2 \\pi ) } | \\psi _ { j } ( \\theta ) - \\psi _ { j } ( 2 \\pi k ^ { - D } \\lfloor k ^ D \\theta / 2 \\pi \\rfloor ) | \\\\ = & \\sup _ { 0 \\leq j \\leq k - 1 } \\ , \\sup _ { 0 \\leq s \\leq k ^ D - 1 } \\psi _ j ( 2 ( s + 1 ) \\pi k ^ { - D } ) - \\psi _ j ( 2 s \\pi k ^ { - D } ) . \\end{align*}"}
-{"id": "1955.png", "formula": "\\begin{align*} \\frac { d { \\hat a } } { d { \\hat u } } = \\frac { 1 } { \\hat { a } } \\sum _ { \\alpha = 1 } ^ r \\sum _ { i = 1 } ^ m n _ i { \\hat Y } _ i ^ 2 \\left ( ( H Q ) _ { \\alpha i } \\right ) ^ 2 . \\end{align*}"}
-{"id": "4047.png", "formula": "\\begin{align*} \\beta ^ 2 \\gamma = 2 + 2 \\beta + \\gamma \\iff \\gamma ( \\beta - 1 ) = 2 . \\end{align*}"}
-{"id": "227.png", "formula": "\\begin{align*} y = y _ { x , z } : = x + \\frac { r _ { n , 1 } } { f ( x ) ^ { 1 / d } } z , \\end{align*}"}
-{"id": "6343.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } u ( x ) : = c _ { n , s } \\ , { \\rm P V } \\int _ { \\R ^ n } \\frac { u ( x ) - u ( x + z ) } { | z | ^ { n + 2 s } } \\ , \\d z . \\end{align*}"}
-{"id": "7268.png", "formula": "\\begin{align*} M \\cdot \\left ( - 1 - R ( X ^ 1 + X ^ 2 + \\cdots + X ^ r + r A ( X ) ) + \\prod _ { \\lambda = 1 } ^ r \\frac { 1 } { 1 - R ( X ^ \\lambda + A ( X ) ) } \\right ) . \\end{align*}"}
-{"id": "2825.png", "formula": "\\begin{align*} a _ { i , u } = a _ i + u q _ i , \\quad 0 \\le u \\le L _ i , L _ i = [ 1 / q _ i ] , q _ i = \\theta _ i y _ i ^ { - \\frac { 2 } { \\alpha } } . \\end{align*}"}
-{"id": "2634.png", "formula": "\\begin{align*} h _ m ^ l ( F , G ) = ( b _ m ( \\lambda ) ) ^ { \\top } C _ m ^ l ( F ) ( \\lambda ) , \\end{align*}"}
-{"id": "3922.png", "formula": "\\begin{align*} w _ { \\rm r e g } ( \\eta ) = u _ { \\rm r e g } ( 1 - \\eta ) \\ , . \\end{align*}"}
-{"id": "8914.png", "formula": "\\begin{align*} \\abs { A ( \\sigma _ H ( x ) ) } ^ 2 = \\abs { A ( x ) } ^ 2 , \\end{align*}"}
-{"id": "2836.png", "formula": "\\begin{align*} H o m _ { \\mathcal { C } ^ I } ( X , Y ) = \\int _ { i \\in I } H o m _ { \\mathcal { C } } ( X ( i ) , Y ( i ) ) . \\end{align*}"}
-{"id": "8690.png", "formula": "\\begin{gather*} | G B ( s , x ) - G B ( s , x + h ) | _ K = | B ( s , x ) - B ( s , x + h ) | _ U \\le C | h | _ H ^ { \\alpha } \\end{gather*}"}
-{"id": "9272.png", "formula": "\\begin{align*} & \\tilde { p } ( t , z ) = \\tilde { p } ( 0 , z ) \\exp ( \\int _ 0 ^ t ( b _ 0 ( s , z ) \\pi ( s , z ) - \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) d B ( s ) \\\\ & - \\frac { 1 } { 2 } \\int _ 0 ^ t ( b _ 0 ( s , z ) \\pi ( s , z ) - \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) ^ { 2 } d s ) , \\end{align*}"}
-{"id": "5518.png", "formula": "\\begin{align*} \\| \\chi _ { ( 0 , a ) } \\| _ \\mathcal { M } = \\frac { a } { W ( a ) } \\Big { / } \\varphi ^ { - 1 } \\left ( \\frac { 1 } { W ( a ) } \\right ) . \\end{align*}"}
-{"id": "4003.png", "formula": "\\begin{align*} \\alpha _ 0 = \\theta _ { n - t } - \\theta _ { n - t - a } . \\end{align*}"}
-{"id": "8977.png", "formula": "\\begin{align*} \\mathcal S _ \\varphi \\ , : = { \\rm s u p p } \\ , \\varphi \\mathcal Q _ \\varphi \\ , : = \\ , \\mathbb { R } ^ { 2 N } \\setminus \\big ( \\mathcal S _ \\varphi ^ c \\times \\mathcal S _ \\varphi ^ c \\big ) \\ , . \\end{align*}"}
-{"id": "5786.png", "formula": "\\begin{align*} F _ { p } C _ { k } = \\{ A \\in C _ { k } \\mid \\deg A \\geq p \\} \\end{align*}"}
-{"id": "9244.png", "formula": "\\begin{align*} & d X ( t ) = \\alpha ( X ( t ) , R ( t ) , u ( t , Z ) ) d t + \\beta ( X ( t ) , R ( t ) , u ( t , Z ) ) d v ( t ) \\\\ & + \\int _ { \\mathbb { R } } \\gamma ( X ( t ) , R ( t ) , u ( t , Z ) , \\zeta ) \\tilde { N } ( d t , d \\zeta ) ; t \\in [ 0 , T ] , \\\\ & X ( 0 ) F ( \\cdot ) , \\mathbb { E } [ \\phi ( X ( 0 ) ) ] = \\int _ { \\mathbb { R } } \\phi ( x ) F ( x ) d x ; \\phi \\in C _ 0 ( \\mathbb { R } ) . \\end{align*}"}
-{"id": "2001.png", "formula": "\\begin{align*} \\mathcal { A } _ n f ( x ) = \\begin{cases} \\mathcal { A } f ( x ) , & | x | < n , \\\\ 0 , & | x | \\geq n . \\end{cases} \\end{align*}"}
-{"id": "39.png", "formula": "\\begin{align*} \\vert \\xi \\vert ^ 2 = \\sup _ { \\Vert u \\Vert _ { y _ 0 } \\leq 1 } | \\langle \\xi , u \\rangle | ^ 2 \\end{align*}"}
-{"id": "8055.png", "formula": "\\begin{align*} \\mathrm { B s } ( | D | _ { G } ) = \\bigcap _ { \\widetilde { D } _ { \\Gamma } \\in | D | _ { G } } \\mathrm { S u p p } ( \\widetilde { D } _ { \\Gamma } ) . \\end{align*}"}
-{"id": "6347.png", "formula": "\\begin{align*} r : = ( y ^ 2 + d ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "3513.png", "formula": "\\begin{align*} X ^ i _ { ; j k } & = \\frac { 1 } { 2 } g ^ { i \\ell } \\left [ ( \\mathcal D _ g X ) _ { \\ell j ; k } + ( \\mathcal D _ g X ) _ { k \\ell ; j } - ( \\mathcal D _ g X ) _ { j k ; \\ell } ) \\right ] - \\frac { 1 } { 2 } g ^ { i \\ell } \\widetilde { C } ^ { p } _ { \\ell j k } X _ { p } \\\\ & = \\widetilde { A } ^ i _ { j k } f + \\widetilde { B } ^ { i \\ell } _ { j k } f _ { ; \\ell } - \\frac { 1 } { 2 } g ^ { i \\ell } \\widetilde { C } ^ { p } _ { \\ell j k } X _ { p } , \\end{align*}"}
-{"id": "9332.png", "formula": "\\begin{align*} M ( t , z ) = \\exp ( \\int _ 0 ^ t \\Phi ( s , z ) d B ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ t \\Phi ^ 2 ( s , z ) d s ) . \\end{align*}"}
-{"id": "581.png", "formula": "\\begin{align*} \\alpha _ 2 ( x ) = ( \\pi ( x ) , \\pi ( \\delta ( x ) ) \\end{align*}"}
-{"id": "1501.png", "formula": "\\begin{align*} w \\cdot x = \\zeta x , \\end{align*}"}
-{"id": "4169.png", "formula": "\\begin{align*} z f ' ( z ) + z ^ 2 f '' ( z ) ( \\lambda - \\mu + 2 \\lambda \\mu ) + \\lambda \\mu z ^ 3 f ''' ( z ) = p ( z ) G _ k ( z ) . \\end{align*}"}
-{"id": "5856.png", "formula": "\\begin{align*} \\frac { T _ { I , t t } } { T _ { I } } = m _ { I } ~ , ~ 2 f _ { I , c } + c - m _ { I } = 0 . \\end{align*}"}
-{"id": "8689.png", "formula": "\\begin{align*} u ( t , x ) = & \\int _ t ^ T R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } G B ( s , \\cdot ) \\right ] ( x ) \\ , d s + \\int _ t ^ T R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } \\nabla ^ G u ( s , \\cdot ) B ( s , \\cdot ) \\right ] ( x ) \\ , d s , \\end{align*}"}
-{"id": "5028.png", "formula": "\\begin{align*} \\eta ( U _ { x _ o } ) = \\eta ( T \\chi _ U ) = \\nu ( \\chi _ U ) = \\nu ( U ) . \\end{align*}"}
-{"id": "2104.png", "formula": "\\begin{align*} U _ a = \\frac { | y | ^ { 2 s } } { 2 ^ { s } ( r - d ) ^ s } , \\end{align*}"}
-{"id": "4299.png", "formula": "\\begin{align*} K _ { X / Y } = K _ X - p ^ \\star K _ Y \\end{align*}"}
-{"id": "8043.png", "formula": "\\begin{align*} a _ 1 ^ { x _ 1 } \\cdots a _ n ^ { x _ n } \\cdot a _ 1 ^ { y _ 1 } \\cdots a _ n ^ { y _ n } = a _ 1 ^ { q _ 1 } \\cdots a _ n ^ { q _ n } . \\end{align*}"}
-{"id": "8317.png", "formula": "\\begin{align*} f ( X ) = X ^ { p ^ n } + a _ { n - 1 } X ^ { p ^ { n - 1 } } + \\cdots + a _ 2 X ^ { p ^ 2 } + a _ 1 X ^ p + a _ 0 X \\in k [ X ] , a _ 0 \\neq 0 . \\end{align*}"}
-{"id": "7687.png", "formula": "\\begin{align*} | f ( x ) - K | = K - f ( x ) < K - x = | x - K | . \\end{align*}"}
-{"id": "5509.png", "formula": "\\begin{align*} ( \\Lambda _ { \\varphi , w } ) _ a = ( \\Lambda _ { \\varphi , w } ) _ b = \\{ f \\in L ^ 0 : \\forall \\lambda , \\ \\rho ( \\lambda f ) < \\infty \\} . \\end{align*}"}
-{"id": "5522.png", "formula": "\\begin{align*} \\alpha ( x ) = \\alpha _ { \\varphi , w } ( x ) = \\sum _ { k = 1 } ^ { \\infty } \\varphi ( x ^ * ( k ) ) w ( k ) , \\end{align*}"}
-{"id": "5227.png", "formula": "\\begin{align*} \\alpha ( \\overline { L } ) = - ( \\mathcal { L } _ T \\eta ) ( \\overline { L } ) = - \\left ( T \\eta ( \\overline { L } ) - \\eta ( [ T , \\overline { L } ] ) \\right ) = \\eta ( [ T , \\overline { L } ] ) \\ ; \\end{align*}"}
-{"id": "4093.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z ^ 2 } { x ^ { 4 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z } { x ^ { 3 } } , \\\\ \\end{gather*}"}
-{"id": "2179.png", "formula": "\\begin{align*} \\mathcal { E } ( \\tilde { u } , - \\psi ^ { 2 } \\tilde { u } ^ { - q } ) = \\frac { 1 } { 2 } + , \\end{align*}"}
-{"id": "2499.png", "formula": "\\begin{gather*} \\left < \\beta _ 1 - \\beta _ 2 \\right > _ 0 = 0 , \\\\ \\left < \\beta _ 1 - \\beta _ 2 \\right > _ \\cdot \\in C ( [ 0 ; + \\infty ) ) , \\end{gather*}"}
-{"id": "8881.png", "formula": "\\begin{align*} w = D u [ y ] + \\lambda i u . \\end{align*}"}
-{"id": "8313.png", "formula": "\\begin{align*} y _ { \\mu } : = ( \\mu y ) ^ { p ^ { n - 1 } } + ( \\mu y ) ^ { p ^ { n - 2 } } + \\cdots + ( \\mu y ) ^ p + ( \\mu p ) , \\end{align*}"}
-{"id": "7160.png", "formula": "\\begin{align*} - \\Delta w + ( w \\cdot \\nabla ) w + \\nabla \\pi = f + \\nabla \\cdot F , \\qquad { \\rm d i v } \\ , w = 0 \\mbox { i n } \\ , \\ , \\R ^ n _ + , \\end{align*}"}
-{"id": "4809.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\alpha \\sigma _ 1 \\partial _ { \\sigma _ 1 } \\eta = \\partial _ { \\sigma _ 1 } \\psi , \\\\ - \\alpha \\sigma _ 3 \\partial _ { \\sigma _ 3 } \\eta = \\partial _ { \\sigma _ 3 } \\psi , \\end{aligned} \\right . \\end{align*}"}
-{"id": "6983.png", "formula": "\\begin{align*} \\delta _ H \\Theta ^ 1 - \\delta _ L \\Theta ( A ) & = 0 ; \\\\ \\delta _ H \\Theta ^ 2 + \\delta _ L \\Theta ^ 1 & = 0 ; \\\\ \\delta _ v \\Theta ( L ) - \\delta _ L \\Theta ^ 2 & = 0 . \\end{align*}"}
-{"id": "2871.png", "formula": "\\begin{align*} \\mathcal { B } _ { E _ 1 } ^ { \\mathbb { K } } ( \\overline { \\Omega } ( C ) ) = \\overline { \\Omega } ( \\mathcal { B } _ { E _ 1 } ^ { \\mathbb { K } } ( C ) ) \\end{align*}"}
-{"id": "8846.png", "formula": "\\begin{align*} \\phi _ i = h u _ i - \\sum _ { j = 1 } ^ { k } c _ { i j } u _ j \\end{align*}"}
-{"id": "8752.png", "formula": "\\begin{align*} L _ i = ( 1 - t ) \\check { R } ' _ i ( 1 ) . \\end{align*}"}
-{"id": "958.png", "formula": "\\begin{align*} ( G _ q ) | _ { \\mathcal { H } _ { q } ^ { \\perp } } = j _ q \\circ ( j _ q ) ^ * \\ ; . \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "412.png", "formula": "\\begin{align*} \\big ( f _ { 2 j } ( x _ { t + m } ) , f _ { 2 j - 1 } ( x _ { t + m } ) \\big ) & = ( 0 , 2 ) , \\\\ \\big ( f _ { 2 j } ( x _ { t + m + 1 } ) , f _ { 2 j - 1 } ( x _ { t + m + 1 } ) \\big ) & = ( 0 , 3 ) , \\end{align*}"}
-{"id": "6666.png", "formula": "\\begin{gather*} z _ { i + 1 } ( u _ k , t ) = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k ( z _ i ( u _ j , t ) - z _ i ( u _ j , t \\wedge \\sigma _ i ) ) \\cdot \\ 1 _ { A _ { k j } ^ i } = \\\\ = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k z _ i ( u _ j , t ) \\cdot \\ 1 _ { A _ { k j } ^ i } - \\sum _ { j = 1 } ^ k z _ i ( u _ j , t \\wedge \\sigma _ i ) \\cdot \\ 1 _ { A _ { k j } ^ i } \\end{gather*}"}
-{"id": "4499.png", "formula": "\\begin{align*} \\beta _ 3 ( f ) : = \\mathbb { E } _ f \\bigl \\{ \\bigl | \\log f ( X _ 1 ) + H ( f ) \\bigr | ^ 3 \\bigr \\} = \\int _ { \\mathcal { X } } f ( x ) | \\log f ( x ) + H ( f ) | ^ 3 \\ , d x . \\end{align*}"}
-{"id": "6317.png", "formula": "\\begin{align*} \\Vert \\Box _ { k } ^ { \\alpha _ { 2 } } F _ { k , N } \\Vert _ { M _ { 2 } } = \\Vert F _ { k , N } \\Vert _ { M _ { 2 } } = \\Vert F _ { k , N } \\Vert _ { L ^ { p _ 2 } } \\sim 2 ^ { j n ( \\alpha _ { 2 } - \\alpha _ { 1 } ) / p _ { 2 } } 2 ^ { j n \\alpha _ { 1 } ( 1 - 1 / p _ { 2 } ) } \\end{align*}"}
-{"id": "1860.png", "formula": "\\begin{align*} S _ t = \\big \\{ u ' \\in W _ u \\colon 2 ^ { t - 1 } \\cdot 4 p ^ 2 | V | \\le | N ( u , u ' ) | < 2 ^ t \\cdot 4 p ^ 2 | V | \\big \\} \\end{align*}"}
-{"id": "212.png", "formula": "\\begin{align*} R _ 1 = \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\int _ 0 ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log u _ { x , s } \\ , d s \\ , d x = o \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\end{align*}"}
-{"id": "5963.png", "formula": "\\begin{align*} \\dd f _ n ^ 2 & = 2 f _ n \\dd f _ n \\\\ & = - 2 f _ n b \\cdot D f _ n \\ , \\dd t - 2 f _ n D _ { v } f _ n \\circ \\dd W _ t + 2 f _ n R _ n \\ , \\dd t \\ , . \\end{align*}"}
-{"id": "7496.png", "formula": "\\begin{align*} \\{ R = ( n - 1 ) / 2 \\} \\subseteq \\{ Q = 1 \\} \\end{align*}"}
-{"id": "7191.png", "formula": "\\begin{align*} f ^ { ' } \\sin \\varphi = B f + 2 B - 2 C _ 1 \\cos \\varphi + C _ 2 . \\end{align*}"}
-{"id": "5848.png", "formula": "\\begin{align*} L _ { Y _ { I } } C _ { x } - m Y _ { I } = 0 , \\end{align*}"}
-{"id": "4083.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } y ^ { q } ( a x + b y + c z ) ^ r } { z ^ { p + q + r } } , \\end{align*}"}
-{"id": "2079.png", "formula": "\\begin{align*} \\mathcal { K } _ { o l d } P _ { o l d } = & \\ \\mathcal { K } _ { o l d } ( I + ( s ^ { 2 } _ { n e w } - s ^ { 2 } _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } M + ( s _ { n e w } - s _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } D ) \\\\ & \\ \\cdot ( I + ( s ^ { 2 } _ { n e w } - s ^ { 2 } _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } M + ( s _ { n e w } - s _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } D ) ^ { - 1 } P _ { o l d } \\\\ = & \\ \\mathcal { K } _ { n e w } P _ { n e w } , \\end{align*}"}
-{"id": "9842.png", "formula": "\\begin{align*} \\Theta ( f ) = \\frac { \\Xi ( f ) } { 1 + \\hat \\alpha ( f ) \\Xi ( f ) } , \\forall f \\in \\Omega . \\end{align*}"}
-{"id": "3986.png", "formula": "\\begin{align*} E _ { 0 } = \\{ \\vect { 0 } \\} E _ i : = \\{ c _ 1 \\vect { e } _ 1 + \\cdots + c _ i \\vect { e } _ i : c _ 1 , \\dots , c _ i \\in \\Z _ { \\geq 0 } \\} , \\end{align*}"}
-{"id": "2786.png", "formula": "\\begin{gather*} - \\frac { i \\omega } { 2 h } = \\frac { 1 } { \\sqrt { \\lambda } } \\sum _ { n = 1 } ^ { \\infty } \\omega _ n ( - \\lambda ) ^ n , \\ ; \\ ; - \\frac { i \\tau } { h } = \\frac { 1 } { \\sqrt { \\lambda } } \\sum _ { n = 0 } ^ { \\infty } \\tau _ n ( - \\lambda ) ^ n , \\ ; \\ ; - \\frac { i \\sigma } { h } = \\frac { 1 } { \\sqrt { \\lambda } } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ n ( - \\lambda ) ^ n . \\end{gather*}"}
-{"id": "4646.png", "formula": "\\begin{align*} \\xi ( z + 1 ) E ( z , g ) = \\xi ( 1 - z ) E ( - z , g ) ; \\xi ( z ) = \\pi ^ { - \\frac { z } { 2 } } \\Gamma \\left ( \\frac { z } { 2 } \\right ) \\zeta ( z ) \\end{align*}"}
-{"id": "9945.png", "formula": "\\begin{align*} S : = \\inf _ { u \\in X _ 0 \\setminus \\{ 0 \\} } \\frac { \\left \\| u \\right \\| ^ 2 } { \\left \\| u \\right \\| _ { L ^ { 2 ^ * } ( \\Omega ) } ^ 2 } = \\inf _ { u \\in X _ 0 \\setminus \\{ 0 \\} } \\frac { \\left \\| u \\right \\| ^ 2 } { \\left \\| u \\right \\| _ { L ^ { 2 ^ * } ( \\R ^ n ) } ^ 2 } . \\end{align*}"}
-{"id": "1659.png", "formula": "\\begin{gather*} \\lambda \\| { \\psi } \\| _ { L ^ p ( \\R ^ { 2 d } ) } ^ p + \\frac { ( p - 1 ) } { 2 } \\sum _ { k = 1 } ^ d \\int _ { \\R ^ { 2 d } } | { \\psi } | ^ { p - 2 } | \\partial _ { v _ k } { \\psi } | ^ 2 \\ , \\dd z + \\int _ { \\R ^ { 2 d } } ( v \\cdot D _ x { \\psi } ) | { \\psi } | ^ { p - 2 } { \\psi } \\ , \\dd z \\\\ = \\int _ { \\R ^ { 2 d } } g | { \\psi } | ^ { p - 2 } { \\psi } \\ , \\dd z \\end{gather*}"}
-{"id": "285.png", "formula": "\\begin{align*} \\mu & : = \\mathbb { E } ( Y _ i ) = \\begin{pmatrix} p _ { n , x , u } \\\\ p _ { n , y , v } \\end{pmatrix} \\\\ V & : = \\mathrm { C o v } ( Y _ i ) = \\begin{pmatrix} p _ { n , x , u } ( 1 - p _ { n , x , u } ) & p _ \\cap - p _ { n , x , u } p _ { n , y , v } \\\\ p _ \\cap - p _ { n , x , u } p _ { n , y , v } & p _ { n , y , v } ( 1 - p _ { n , y , v } ) \\end{pmatrix} , \\end{align*}"}
-{"id": "4662.png", "formula": "\\begin{align*} F _ \\phi ( u ; z ) = \\sum _ { n \\geq 1 } \\frac { \\eta _ { \\frac { z } { 2 } } ( n ) \\rho _ \\phi ( n ) } { n ^ u } , \\end{align*}"}
-{"id": "2394.png", "formula": "\\begin{align*} N = \\Pi _ { D _ 0 } \\end{align*}"}
-{"id": "552.png", "formula": "\\begin{align*} \\| | X \\| | = \\| X \\| + \\max _ { 1 \\le j \\le n } | [ T _ j , X ] | _ { \\mathcal J } . \\end{align*}"}
-{"id": "9753.png", "formula": "\\begin{align*} \\mathcal { Z } ^ { * } ( \\vec { x } ) : = \\{ \\bar { \\vec { z } } \\in \\mathcal { Z } ( \\vec { x } ) \\ | \\ f ( \\bar { \\vec { z } } , \\vec { x } ) \\leq f ( \\vec { z } , \\vec { x } ) \\ \\forall \\vec { z } \\in \\mathcal { Z } ( \\vec { x } ) \\} . \\end{align*}"}
-{"id": "9694.png", "formula": "\\begin{gather*} g ( 0 ) = 0 ; \\\\ g ( x ) > 0 x > 0 ; \\\\ ; \\\\ g \\circ G ^ { - 1 } \\in _ \\infty ( - 1 ) , G ^ { - 1 } \\in _ \\infty ( 0 ) . \\end{gather*}"}
-{"id": "7011.png", "formula": "\\begin{align*} \\int _ X u ( t , x ) d P _ n & = \\sum _ { \\{ i \\in I \\mid x ^ i \\in U ( t ) \\} } \\alpha ^ i u ( t , x ^ i ) + \\sum _ { \\{ i \\in I \\mid x ^ i \\in X \\setminus U ( t ) \\} } \\alpha ^ i u ( t , y ^ i _ n ) \\\\ & > \\sum _ { i \\in I } \\alpha ^ i u ( t , x ^ i ) = \\int _ X u ( t , x ) d Q . \\end{align*}"}
-{"id": "9543.png", "formula": "\\begin{align*} \\begin{cases} V ( k ) = \\lefteqn { - H ( P _ { 1 } ( k + 1 ) ) ^ { - 1 } G _ { u } ( P _ { 1 } ( k + 1 ) ) ^ { T } , } \\\\ \\mathbb { V } ( k ) = V ( k ) + \\tilde { V } ( k ) = \\tilde { H } ( P _ { 1 } ( k + 1 ) , Q _ { 1 } ( k + 1 ) ) ^ { - 1 } \\tilde { G } _ { u } ( P _ { 1 } ( k + 1 ) , Q _ { 1 } ( k + 1 ) ) ^ { T } , \\end{cases} \\end{align*}"}
-{"id": "8952.png", "formula": "\\begin{align*} s _ { t } - r _ { x x } - q ( r ^ { 2 } + s ^ { 2 } ) r = 0 , \\end{align*}"}
-{"id": "6388.png", "formula": "\\begin{align*} H = - \\int _ { - \\infty } ^ { \\infty } p ( x ) \\log p ( x ) d x . \\end{align*}"}
-{"id": "2226.png", "formula": "\\begin{align*} & H ' ( u ( t ) ) \\frac { d } { d t } ( k * u ) ( t ) = \\frac { d } { d t } ( k * H ( u ) ) ( t ) + ( - H ( u ( t ) ) + H ' ( u ( t ) ) u ( t ) ) k ( t ) \\\\ & \\qquad + \\int _ { 0 } ^ { t } ( H ( u ( t - s ) ) - H ( u ( t ) ) - H ' ( u ( t ) ) [ u ( t - s ) - u ( t ) ] ) ( - \\dot { k } ( s ) ) d s , \\end{align*}"}
-{"id": "3790.png", "formula": "\\begin{align*} { \\left [ { \\tilde { \\bf { H } } } \\right ] _ { n m } } & \\simeq \\sum \\limits _ { l \\in { F _ { r , n } } \\cap { F _ { t , m } } } \\ ! \\ ! { { c _ l } \\ , { e ^ { - j 2 \\pi { d _ l } / { \\lambda _ { \\rm c } } } } } \\\\ { \\left [ { \\hat { { \\bf { H } } } } \\right ] _ { n m } } & \\simeq \\left \\{ \\begin{array} { l l l } { c _ 0 } \\ , { e ^ { - j 2 \\pi { d _ 0 } / { \\lambda _ { \\rm c } } } } & { } & { T ( n , m ) = 1 } \\\\ 0 & { } & { \\rm o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "3476.png", "formula": "\\begin{align*} J ( U ) = \\int _ \\Omega | \\nabla U | ^ 2 d x + 2 \\int _ D V ^ + d x ' \\end{align*}"}
-{"id": "1397.png", "formula": "\\begin{align*} B _ { n } ( u ) : = \\sum _ { t = 1 } ^ { n } m _ { t } \\left ( b ( x _ { t } ^ { T } \\beta ^ \\prime + x _ t ^ T u / \\sqrt { n } ) - b ( x _ { t } ^ { T } \\beta ^ { \\prime } ) - \\dot { b } ( x _ { t } ^ { T } \\beta ^ { \\prime } ) x _ { t } ^ { T } u / \\sqrt { n } \\right ) \\end{align*}"}
-{"id": "8957.png", "formula": "\\begin{align*} U _ { j } ( x , 0 ) = \\alpha _ { j } \\sqrt { 2 / q } e ^ { i \\left ( \\frac { S } { 2 } ( x - x _ { j } ) \\right ) } \\left ( \\alpha _ { j } ( x - x _ { j } ) \\right ) , j = 1 , 2 . \\end{align*}"}
-{"id": "3469.png", "formula": "\\begin{align*} G ^ { 2 } : = A K \\partial _ { 3 } u _ { 1 } + B K \\partial _ { 3 } u _ { 2 } + ( 1 - K ) \\partial _ { 3 } u _ { 3 } , \\end{align*}"}
-{"id": "8918.png", "formula": "\\begin{align*} \\mathcal { I } _ A ( t \\tilde { v } ) = \\Tilde { \\mathcal { I } } _ { \\abs { A } ^ 2 } ( t \\tilde { v } ) , \\end{align*}"}
-{"id": "7630.png", "formula": "\\begin{align*} \\displaystyle \\min _ { r \\in \\mathcal { R } } r c ( \\overline { u } , \\overline { \\sigma } ) = \\sum _ { i \\in \\mathcal { N } } \\sum _ { j \\in \\mathcal { N } } ( c _ { i j } - \\overline { u } _ i ^ { } ) x _ { r i j } - \\overline { \\sigma } \\end{align*}"}
-{"id": "8436.png", "formula": "\\begin{align*} [ A _ { 0 } ( \\textbf { u } ) , \\textbf { P } ] = \\Bigg [ \\left ( \\begin{array} { c c c c } \\frac { f ( \\textbf { u } ) } { \\rho } & 0 \\\\ 0 & I d \\\\ \\end{array} \\right ) , \\left ( \\begin{array} { c c c c } 1 & 0 \\\\ 0 & \\mathbb { P } \\\\ \\end{array} \\right ) \\Bigg ] = 0 . \\end{align*}"}
-{"id": "9527.png", "formula": "\\begin{align*} \\delta ( C _ 4 \\wedge H _ 3 \\wedge G _ 3 ) & = C _ 4 \\wedge d \\delta B _ 2 \\wedge G _ 3 \\\\ & = d \\left ( C _ 4 \\wedge \\delta B _ 2 \\wedge G _ 3 \\right ) - G _ 5 \\wedge \\delta B _ 2 \\wedge G _ 3 \\\\ & = d ( \\ldots ) - \\delta B _ 2 \\wedge G _ 5 \\wedge G _ 3 . \\end{align*}"}
-{"id": "3583.png", "formula": "\\begin{align*} & \\| \\Phi ^ W _ { ( g , \\pi ) } ( g , \\pi ) + ( \\psi , V ) - \\Phi ^ W _ { ( g , \\pi ) } ( g + h _ 0 , \\pi + w _ 0 ) \\| _ { \\mathcal B _ 0 \\times \\mathcal B _ 1 } \\\\ & = \\| Q ^ W _ { ( g , \\pi ) } ( h _ 0 , w _ 0 ) \\| _ { \\mathcal B _ 0 \\times \\mathcal B _ 1 } \\le D \\| ( h _ 0 , w _ 0 ) \\| _ { \\mathcal B _ 2 \\times \\mathcal B _ 2 } ^ 2 \\le D C ^ 2 \\| ( \\psi , V ) \\| _ { \\mathcal B _ 0 \\times \\mathcal B _ 1 } ^ 2 . \\end{align*}"}
-{"id": "1951.png", "formula": "\\begin{align*} Q ^ { ( j ) } \\doteqdot \\sum _ { \\alpha = 1 } ^ { r } \\ , q _ { \\alpha j } \\ , e _ { \\alpha } , \\ , \\ , \\ , 1 \\leq j \\leq m , \\end{align*}"}
-{"id": "8614.png", "formula": "\\begin{align*} \\sigma _ { 1 } ( \\alpha ( w ) + \\operatorname { k e r } \\beta \\cap \\operatorname { i m } \\alpha ) & = \\sigma _ { 1 } ( \\alpha ( x _ { 1 } ) + \\alpha ( x _ { 2 } ) + \\operatorname { k e r } \\beta \\cap \\operatorname { i m } \\alpha ) \\\\ & = \\sigma _ { 1 } ( \\alpha ( x _ { 2 } ) + \\operatorname { k e r } \\beta \\cap \\operatorname { i m } \\alpha ) \\end{align*}"}
-{"id": "4922.png", "formula": "\\begin{align*} W ( t ) = \\prod _ { i = 1 } ^ n ( 1 + t + \\ldots + t ^ { e _ i } ) , \\end{align*}"}
-{"id": "9805.png", "formula": "\\begin{align*} B F _ { k l } = \\frac { B F _ { k , } } { B F _ { l , } } , \\end{align*}"}
-{"id": "7671.png", "formula": "\\begin{align*} u _ { t } = \\Delta ^ { \\alpha / 2 } u + \\tilde { f } ( u ) \\end{align*}"}
-{"id": "2969.png", "formula": "\\begin{align*} \\delta _ H \\Theta ( A ) = 0 . \\end{align*}"}
-{"id": "393.png", "formula": "\\begin{align*} Z ( f , L ; s , \\phi ) & = \\int _ { G _ + / \\Gamma } \\chi ( g ) ^ s \\phi \\left ( g \\right ) \\sum _ { x \\in L \\setminus L _ 0 } f ( g \\cdot x ) d g \\\\ Z ( f , \\hat { L } ; s , \\phi ) & = \\int _ { G _ + / \\Gamma } \\chi ( g ) ^ s \\phi \\left ( g \\right ) \\sum _ { x \\in \\hat { L } \\setminus \\hat { L } _ 0 } f ( g \\cdot x ) d g . \\end{align*}"}
-{"id": "7444.png", "formula": "\\begin{align*} A _ f \\xi _ r = A _ g \\xi _ s + A _ h \\xi _ t . \\end{align*}"}
-{"id": "4888.png", "formula": "\\begin{align*} r _ \\nu = \\pi ( \\delta ( \\cdots \\delta ( \\delta ( x - [ r _ 0 ] ) - [ r _ 1 ] ) \\cdots - [ r _ { \\nu - 1 } ] ) ) \\nu = 1 , \\dots , n - 1 \\end{align*}"}
-{"id": "9347.png", "formula": "\\begin{align*} \\begin{cases} d u ( t , x , z ) = ( - x ^ 2 u ( t , x , z ) - x v ( t , x , z ) ) d t + v ( t , x , z ) d G ( t ) \\\\ u ( T , x , z ) = h ( T , x , z ) k ( T , x , z ) = y ( T , x , z ) p ( T , x , z ) k ( T , x , z ) = y ( T , x , z ) U ( x ) \\mathbb { E } [ \\delta _ { Z } ( z ) | \\mathcal { F } ^ { G } _ T ] k ( T , x , z ) . \\end{cases} \\end{align*}"}
-{"id": "7625.png", "formula": "\\begin{align*} \\lambda _ 1 = \\sum _ { y \\sim x } \\mathbf { v } _ y \\leq d _ x , \\end{align*}"}
-{"id": "4692.png", "formula": "\\begin{align*} \\big ( f _ { 2 j } ( x _ { t + m } ) , f _ { 2 j - 1 } ( x _ { t + m } ) \\big ) & = ( 0 , m - 1 ) , \\\\ \\big ( f _ { 2 j } ( x _ { t + m + 1 } ) , f _ { 2 j - 1 } ( x _ { t + m + 1 } ) \\big ) & = ( 1 , 0 ) , \\end{align*}"}
-{"id": "8657.png", "formula": "\\begin{align*} b \\left ( \\xi , y \\right ) & = 5 6 \\sqrt [ 4 ] { \\sin \\xi \\ , y ^ 3 } \\ , I _ { \\left \\lbrace \\vert y \\vert < 2 T ^ 8 \\right \\rbrace } + y \\ , I _ { \\left \\lbrace \\vert y \\vert < 2 T ^ 8 \\right \\rbrace } + 5 6 \\sqrt [ 4 ] { 8 T ^ { 2 4 } \\sin \\xi } \\ , I _ { \\left \\lbrace \\vert y \\vert \\geq 2 T ^ 8 \\right \\rbrace } + 2 T ^ 8 \\ , I _ { \\left \\lbrace \\vert y \\vert \\geq 2 T ^ 8 \\right \\rbrace } , \\end{align*}"}
-{"id": "3039.png", "formula": "\\begin{align*} \\gamma _ n = \\sum _ { | u | = n } \\delta _ { V ( u ) - m _ n } . \\end{align*}"}
-{"id": "8621.png", "formula": "\\begin{align*} \\left ( \\overline { u } + \\overline { \\rho ( u ) } \\right ) \\left ( \\overline { N } ( ^ { \\ast } b ) ( n ) \\right ) & = \\left ( - \\underline { u } _ { 1 } \\pi _ { 1 } p \\xi _ { b e _ { k } } ( n ) - \\pi _ { 1 } ^ { 1 } \\rho _ { 2 } \\gamma \\xi _ { b e _ { k } } ( n ) , - u _ { 2 } \\gamma \\xi _ { b e _ { k } } ( n ) , 0 \\right ) \\\\ & = ( - \\pi _ { 1 } ^ { 1 } u _ { 1 } p \\xi _ { b e _ { k } } ( n ) - \\pi _ { 1 } ^ { 1 } \\rho _ { 2 } \\gamma \\xi _ { b e _ { k } } ( n ) , - u _ { 2 } \\gamma \\xi _ { b e _ { k } } ( n ) , 0 ) \\\\ \\end{align*}"}
-{"id": "3454.png", "formula": "\\begin{align*} \\hat y ( t ) = \\Gamma ^ { * } ( t , \\tau ) u ( [ s + \\alpha ] \\wedge \\tau , \\xi ( [ s + \\alpha ] \\wedge \\tau ) ) + \\int _ t ^ { s + \\alpha } I _ { [ s , \\tau ] } ( \\theta ) f ( \\theta , \\xi ( \\theta ) , \\hat y ( \\theta ) , \\hat z ( \\theta ) ) d \\theta - \\end{align*}"}
-{"id": "1912.png", "formula": "\\begin{align*} I _ { n , m } : = [ - L _ n - 1 2 m - 4 , - L _ n - 1 2 m + 4 ] \\cup [ L _ n + 1 2 m - 4 , L _ n + 1 2 m + 4 ] , \\end{align*}"}
-{"id": "4923.png", "formula": "\\begin{align*} \\frac { 1 } { | W | } \\prod _ { i = 1 } ^ n ( h + e _ i + 1 ) , \\end{align*}"}
-{"id": "4618.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { 8 } f _ t ( p ' ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ { p ' } | } , \\end{align*}"}
-{"id": "8677.png", "formula": "\\begin{align*} \\lim _ { z \\to 0 } \\sup _ { x \\in H } \\sup _ { | k | _ K = 1 } | \\Gamma _ { t , x , k , \\xi } - \\Gamma _ { t , x + z , k , \\xi } | ^ 2 _ J = 0 . \\end{align*}"}
-{"id": "2406.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\tau ^ { - 1 } \\langle \\nabla f ( x + \\tau ( y - x ) ) & - \\nabla f ( x ) , ( x + \\tau ( y - x ) ) - x \\rangle d \\tau \\\\ & \\geq - \\int _ 0 ^ 1 \\tau ^ { - 1 } \\langle M \\tau ( x - y ) , \\tau ( x - y ) \\rangle d \\tau \\\\ & = - \\langle M ( x - y ) , x - y \\rangle \\int _ 0 ^ 1 \\tau d \\tau \\\\ & = - \\tfrac { 1 } { 2 } \\langle M ( x - y ) , x - y \\rangle . \\end{align*}"}
-{"id": "2033.png", "formula": "\\begin{align*} 0 = \\frac { d } { d c } \\left ( ( \\tilde { L } _ c - \\tilde { \\lambda } _ c ) \\phi _ c \\right ) \\big | _ { c = 1 } & = \\left ( \\frac { d \\tilde { L } _ c } { d c } - \\frac { d \\tilde { \\lambda } _ c } { d c } \\right ) \\phi _ c \\big | _ { c = 1 } + \\left ( \\tilde { L } _ c - \\tilde { \\lambda } _ c \\right ) \\frac { d \\phi _ c } { d c } \\Big | _ { c = 1 } . \\end{align*}"}
-{"id": "3994.png", "formula": "\\begin{align*} \\pi _ { p } ( v _ { i } ) = u ^ { - } ( \\vect { 0 } , \\dots , - \\varphi ( s _ { i } ) ^ { - 1 } , \\dots , \\vect { 0 } ) \\pi _ { p } ( v _ { i - 1 } ) = \\pi _ { p } ( u ^ { - } ( 0 , \\dots , - \\varphi ( s _ { i } ) ^ { - 1 } , \\dots , 0 ) v _ { i - 1 } ) , \\end{align*}"}
-{"id": "9252.png", "formula": "\\begin{align*} K _ t = M _ t ^ { - 1 } & = \\exp \\Big ( \\int _ 0 ^ t h ( X ( s , Z ) ) d w ( s ) + \\frac { 1 } { 2 } \\int _ 0 ^ t h ^ 2 ( X ( s , Z ) ) d s \\Big ) \\\\ & = \\exp \\Big ( \\int _ 0 ^ t h ( X ( s , Z ) ) d R ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ t h ^ 2 ( X ( s , Z ) ) d s \\Big ) . \\end{align*}"}
-{"id": "6627.png", "formula": "\\begin{align*} \\Omega _ z ^ T : = \\{ y \\in [ 0 , 1 ] \\ , : \\ , \\tau _ \\partial ( y ) < T \\Phi _ { \\tau _ \\partial ( y ) } ( y ) = z \\} . \\end{align*}"}
-{"id": "6517.png", "formula": "\\begin{align*} \\lambda = \\max _ { \\substack { x \\in \\bar \\varOmega \\\\ * [ 2 p t ] t \\in [ 0 , T ] } } \\frac { | a ( x , t ) + \\i \\ , b ( x , t ) | } { a ( x , t ) } < \\frac { 1 } { \\cos \\alpha _ k } , \\end{align*}"}
-{"id": "163.png", "formula": "\\begin{align*} \\tau ^ { - i } A _ i = \\tau ^ { - i } C _ { m _ i ( 0 ) } \\cap \\tau ^ { - i - 1 } C _ { m _ i ( 1 ) } \\cap \\dots \\cap \\tau ^ { - i - 2 ^ { k _ i } } C _ { m _ i ( 2 ^ { k _ i } ) } , \\ \\ 1 \\le i \\le n . \\end{align*}"}
-{"id": "7085.png", "formula": "\\begin{align*} \\forall u \\in \\mathcal { U } , \\mu ( B ( u ) ) = \\nu ( B ( u ) ) . \\end{align*}"}
-{"id": "7271.png", "formula": "\\begin{align*} ( \\pi | _ { \\widetilde { V } _ j } ) ^ * u _ j = \\frac { 1 } { d } \\sum _ { \\nu = 1 } ^ d i _ \\nu ^ * \\widetilde { u } _ j , \\end{align*}"}
-{"id": "9915.png", "formula": "\\begin{align*} \\xi _ 3 ^ 2 \\eta _ 4 ^ 2 E F & \\ ; = \\ ; ( \\xi _ 1 \\eta _ 4 K + \\xi _ 3 \\eta _ 1 K ' ) ( \\xi _ 2 \\eta _ 4 K + \\xi _ 3 \\eta _ 2 K ' ) \\\\ & \\ ; = \\ ; \\xi _ 1 \\xi _ 2 \\eta _ 4 ^ 2 K ^ 2 + \\xi _ 3 \\eta _ 4 ( \\xi _ 1 \\eta _ 2 + \\xi _ 2 \\eta _ 1 ) K K ' + \\xi _ 3 ^ 2 \\eta _ 1 \\eta _ 2 K ^ { ' 2 } \\\\ & \\ ; = \\ ; - \\kappa ^ { 2 } \\xi _ 3 ^ 2 \\eta _ 4 ^ 2 K ^ 2 + \\xi _ 3 \\eta _ 4 ( \\xi _ 1 \\eta _ 2 + \\xi _ 2 \\eta _ 1 ) K K ' - \\kappa ^ { 2 } \\xi _ 3 ^ 2 \\eta _ 4 ^ 2 K ^ { ' 2 } \\end{align*}"}
-{"id": "4157.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } S _ f ^ \\nu ( n ) S _ g ^ \\nu ( n ) = c _ { f , \\overline { g } } X ^ { 2 \\kappa ( f ) + \\frac { 3 } { 2 } - 2 \\nu } + O ( X ^ { 2 \\kappa ( f ) + 1 - 2 \\nu } \\log ^ 2 X ) \\end{align*}"}
-{"id": "1427.png", "formula": "\\begin{align*} J ^ { u } _ { ( 0 ) } \\psi _ { \\alpha } = \\sum _ { \\beta \\in [ \\alpha ] } c _ { \\beta , u } ^ { \\alpha } \\psi _ { \\beta } , J ^ { u } _ { ( n ) } \\psi _ { \\alpha } = 0 \\quad \\mathrm { f o r } \\ n \\geq 1 \\end{align*}"}
-{"id": "7385.png", "formula": "\\begin{align*} & \\lambda _ t \\overset { \\mathbf { . . } } { = } \\hat { \\lambda } _ { t } , \\frac { 1 } { n } ( h ^ { t + 1 } ) ^ * q ^ { r + 1 } \\overset { \\mathbf { . . } } { = } \\hat { \\lambda } _ { r + 1 } \\breve { E } _ { r , t } , \\\\ & \\frac { 1 } { n } ( h ^ { r + 1 } ) ^ * q ^ { t + 1 } \\overset { \\mathbf { . . } } { = } \\hat { \\lambda } _ { t + 1 } \\breve { E } _ { r , t } , \\end{align*}"}
-{"id": "6980.png", "formula": "\\begin{align*} \\tilde { \\alpha _ t } ( \\Phi _ t a , \\Phi _ t b ) & = \\Phi _ t \\alpha _ t ( a , b ) , \\\\ \\tilde { \\mu _ t } ( \\Psi _ t x , \\Phi _ t a ) & = \\Phi _ t \\mu _ t ( x , a ) , \\\\ \\tilde { \\lambda _ t } ( \\Psi _ t x , \\Psi _ t y ) & = \\Psi _ t \\lambda _ t ( x , y ) . \\end{align*}"}
-{"id": "5788.png", "formula": "\\begin{align*} \\C \\ \\backslash \\bigcap _ { n \\geq 0 } U ( n ) = \\bigcup _ { n \\geq 0 } ( \\C \\backslash U ( n ) ) \\end{align*}"}
-{"id": "6085.png", "formula": "\\begin{align*} F _ { m , n } ( t ) & = \\frac { 1 } { \\sqrt { n + 1 } } \\frac { 1 } { \\sqrt { m + 1 } } \\left ( 1 + i \\log \\frac { m + 1 } { n + 1 } \\right ) ^ { - t } \\\\ G _ { m , n } ( t ) & = \\frac { 1 } { \\sqrt { m + 1 } } \\int _ n ^ { n + 1 } v ^ { - 1 / 2 } \\left ( 1 + i \\log \\frac { m + 1 } { v } \\right ) ^ { - t } d v . \\end{align*}"}
-{"id": "3312.png", "formula": "\\begin{align*} ( F ^ * v _ 0 , v _ 1 ) = ( E K ^ { - 1 } v _ 0 , v _ 1 ) = [ 2 ] ^ { 1 / 2 } ( v _ 1 , v _ 1 ) . \\end{align*}"}
-{"id": "1956.png", "formula": "\\begin{align*} h ( Q ^ { ( i ) } , Q ^ { ( i ) } ) = \\sum _ { \\tilde { \\alpha } , \\tilde { \\beta } = 1 } ^ r \\ , q _ { \\tilde { \\alpha } i } q _ { \\tilde { \\beta } i } h _ { \\tilde { \\alpha } \\tilde { \\beta } } , \\end{align*}"}
-{"id": "2675.png", "formula": "\\begin{align*} \\int _ X e ^ { \\varphi _ j } d \\mu = e ^ { \\sup _ X \\varphi _ j } \\int _ X e ^ { \\psi _ j } d \\mu \\geq c e ^ { \\sup _ X \\varphi _ j } \\end{align*}"}
-{"id": "5114.png", "formula": "\\begin{align*} F _ { \\vec { x } } ( u _ { 1 } , \\ldots , u _ { k } ) = \\frac { ( 1 - q ^ 2 ) ^ { k } } { \\prod _ { i = 1 } ^ { k } ( 1 - s u _ { i } ) } \\sum _ { \\sigma \\in \\mathcal { S } _ { k } } \\prod _ { 1 \\le i < j \\le k } \\frac { u _ { \\sigma ( i ) } - q ^ { 2 } u _ { \\sigma ( j ) } } { u _ { \\sigma ( i ) } - u _ { \\sigma ( j ) } } \\prod _ { i = 1 } ^ { k } \\left ( \\frac { u _ { i } - s } { 1 - s u _ { i } } \\right ) ^ { x _ { i } } \\end{align*}"}
-{"id": "4728.png", "formula": "\\begin{align*} G ^ + = \\{ x \\in \\mathbb { R } ^ 2 \\ , | \\ \\begin{bmatrix} 0 & 0 \\\\ \\beta _ 2 v _ { 1 2 } & \\beta _ 2 v _ { 2 2 } \\\\ \\beta _ 1 v _ { 1 1 } & \\beta _ 1 v _ { 2 1 } \\end{bmatrix} \\begin{bmatrix} x _ 1 \\\\ x _ 2 \\end{bmatrix} \\succeq _ { \\mathcal { L } ^ 3 } \\begin{bmatrix} - \\eta \\\\ \\beta _ 2 \\alpha _ 2 \\\\ \\beta _ 1 \\alpha _ 1 \\end{bmatrix} \\} , \\end{align*}"}
-{"id": "643.png", "formula": "\\begin{align*} \\sum _ { s } p ( s ) \\nu ( s ^ { - 1 } \\cdot f ) = \\nu ( f ) , \\textrm { f o r a l l $ f \\in C ( \\overline { Y } ) $ } , \\end{align*}"}
-{"id": "3338.png", "formula": "\\begin{align*} c ( [ \\hat { W } ] , F ) & \\leq c ( [ \\hat { W } ] , L ) + | | \\bar { H } | | _ { C ^ 0 } \\\\ & \\leq ( | | H | | _ { C ^ 0 } + \\max \\{ 0 , \\lambda w \\} ) + | | H | | _ { C ^ 0 } \\\\ & = 2 | | H | | _ { C ^ 0 } + \\max \\{ 0 , \\lambda w \\} . \\end{align*}"}
-{"id": "8459.png", "formula": "\\begin{align*} \\partial _ { t } { \\textbf { u } } + \\sum _ { j = 1 } ^ { d } A _ { j } ( \\textbf { u } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\textbf { u } + ( 0 , \\nabla P ) ^ { T } = 0 . \\end{align*}"}
-{"id": "895.png", "formula": "\\begin{align*} \\chi ( s ) ^ { - 1 / 2 } = O \\left ( t ^ { \\sigma / 2 - 1 / 4 } \\right ) , 0 \\leq \\sigma \\leq 1 + \\delta . \\end{align*}"}
-{"id": "337.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } x & 0 \\\\ 0 & y \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & x \\\\ - x & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & - x ^ { - 1 } y \\\\ 1 & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "7642.png", "formula": "\\begin{align*} \\ \\left \\{ \\begin{aligned} & u _ { t } + ( - \\triangle ) ^ { \\alpha / 2 } u = \\frac { 1 } { \\Gamma ( 1 - \\beta ) } \\int _ { 0 } ^ { t } ( t - s ) ^ { - \\beta } | u | ^ { p - 1 } u ( s ) d s , \\ x \\in \\mathbb { R } ^ { d } , \\ t > 0 , \\\\ & u ( 0 ) = u _ { 0 } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2291.png", "formula": "\\begin{align*} \\begin{aligned} & { } \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\\\ & { } \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "8326.png", "formula": "\\begin{align*} \\sigma ( y ) = \\sigma \\big ( \\sum _ { i = 1 } ^ n \\mu _ i y _ i \\big ) = \\sum _ { i = 1 } ^ n \\mu _ i ( y _ i + \\nu _ i ) = y + \\sum _ { i = 1 } ^ n \\nu _ i \\mu _ i . \\end{align*}"}
-{"id": "5458.png", "formula": "\\begin{align*} t _ H ( m ) = H - \\vert m \\vert \\textrm { a n d } T _ H ( N , y ; \\alpha ) = \\sum _ { n = N - H } ^ { N + y } t _ H ( n - N ) e ( n \\alpha ) . \\end{align*}"}
-{"id": "7304.png", "formula": "\\begin{align*} D ^ 2 = \\sum _ { i , j } E _ { \\xi _ i } F _ { \\xi _ j } \\otimes \\gamma _ - ( w _ i ) \\gamma _ + ( v _ j ) + \\sum _ { i , j } F _ { \\xi _ i } E _ { \\xi _ j } \\otimes \\gamma _ + ( v _ i ) \\gamma _ - ( w _ j ) . \\end{align*}"}
-{"id": "7731.png", "formula": "\\begin{align*} & \\xi _ i , \\ , \\ , i \\in \\mathbf N , \\ , \\ , \\\\ & \\end{align*}"}
-{"id": "445.png", "formula": "\\begin{align*} d _ k ( c ' , c '' ) = d _ { \\ell _ 1 } ( \\sigma _ k ( z ' ) , \\sigma _ k ( z '' ) ) \\leq t . \\end{align*}"}
-{"id": "1398.png", "formula": "\\begin{align*} A _ { n } ( u ) : = u ^ T \\frac { 1 } { \\sqrt { n } } \\sum _ { t = 1 } ^ { n } ( y _ { t } - m _ { t } \\dot { b } ( x _ { t } ^ { T } \\beta ^ { \\prime } ) ) x _ { t } = u ^ T U _ n . \\end{align*}"}
-{"id": "3962.png", "formula": "\\begin{align*} \\lambda ^ { \\max } ( u ( r ) v ) = \\lambda ^ { \\max } ( \\sigma ( r ) u ( - r ) u ^ { - } ( r ^ { - 1 } ) v ) & = - \\lambda ^ { \\min } ( u ( - r ) u ^ { - } ( r ^ { - 1 } ) v ) \\\\ & = - \\lambda ^ { \\min } ( u ^ { - } ( r ^ { - 1 } ) v ) \\geq - \\lambda ^ { \\max } ( u ^ { - } ( r ^ { - 1 } ) v ) = - \\lambda ^ { \\max } ( v ) . \\end{align*}"}
-{"id": "8576.png", "formula": "\\begin{align*} \\overline { p } & = D _ { 1 } ^ { - 1 } p D : N _ { o u t } \\rightarrow \\operatorname { k e r } ( \\hat { \\gamma } ) \\\\ \\overline { \\sigma } _ { 2 } & = C _ { 2 } \\sigma _ { 2 } \\underline { C } ^ { - 1 } : \\frac { \\operatorname { k e r } ( \\hat { \\alpha } ) } { \\operatorname { i m } ( \\hat { \\gamma } ) } \\rightarrow \\operatorname { k e r } ( \\hat { \\alpha } ) \\end{align*}"}
-{"id": "5937.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\dd x \\int _ { \\R ^ d } | \\partial _ { v _ j } D _ x { \\psi } ( x , v ) | ^ p \\ , \\dd v = \\int _ { \\R ^ d } \\| \\partial _ { v _ j } D _ x { \\psi } ( x , \\cdot ) \\| _ { L ^ p ( \\R ^ { d } ) } ^ p \\dd x \\le C ( \\lambda ) \\ , \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ { p } ( \\R ^ d _ x ) ) } ^ p . \\end{align*}"}
-{"id": "4388.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } r _ n = \\infty , \\ \\ \\lim _ { n \\to \\infty } \\varepsilon _ n = 0 , \\end{align*}"}
-{"id": "8098.png", "formula": "\\begin{align*} \\mu _ l \\left ( B _ { R } \\right ) = \\mu _ l \\left ( M \\right ) = \\int _ M d \\mu _ l ( x ) . \\end{align*}"}
-{"id": "8649.png", "formula": "\\begin{align*} e ^ { t A } \\left ( \\begin{array} [ c ] { c } y \\\\ z \\end{array} \\right ) = \\left ( \\begin{array} [ c ] { c c } \\cos \\sqrt { \\Lambda } t & \\frac { 1 } { \\sqrt { \\Lambda } } \\sin \\sqrt { \\Lambda } t \\\\ - \\sqrt { \\Lambda } \\sin \\sqrt { \\Lambda } t & \\cos \\sqrt { \\Lambda } t \\end{array} \\right ) \\left ( \\begin{array} [ c ] { c } y \\\\ z \\end{array} \\right ) , t \\in \\mathbb { R } , \\ ; \\ ; \\left ( \\begin{array} [ c ] { c } y \\\\ z \\end{array} \\right ) \\in H . \\end{align*}"}
-{"id": "7890.png", "formula": "\\begin{align*} t _ { 1 , 2 } = \\frac { ( B - A ) \\times ( A - I ) \\pm \\left \\| B - A \\right \\| r } { ( B - A ) \\times ( E _ 1 - S _ 1 ) } \\end{align*}"}
-{"id": "6738.png", "formula": "\\begin{align*} P _ { [ \\theta , \\psi ] } ( \\varphi ) : = \\Big \\{ \\lim _ { C \\to + \\infty } P _ { \\theta } ( \\min ( \\psi + C , \\varphi ) ) \\Big \\} ^ * . \\end{align*}"}
-{"id": "6973.png", "formula": "\\begin{align*} \\mu _ t ( x , \\alpha _ t ( a , b ) ) = \\alpha _ t ( a , \\mu _ t ( x , b ) ) + \\alpha _ t ( \\mu _ t ( x , a ) , b ) , \\end{align*}"}
-{"id": "563.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\| ( I - A _ n ) R \\| | = 0 \\end{align*}"}
-{"id": "8744.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } e ^ { \\left ( T - s \\right ) A } G u ( s ) d s = \\int _ { 0 } ^ { T } e ^ { \\left ( T - s \\right ) A } \\left ( \\begin{array} [ c ] { c } 0 \\\\ \\psi _ 2 ( s ) \\end{array} \\right ) d s + \\int _ { 0 } ^ { T } e ^ { \\left ( T - s \\right ) A } G \\psi _ 1 ' ( s ) d s . \\end{align*}"}
-{"id": "248.png", "formula": "\\begin{align*} \\biggl | \\frac { \\partial f ^ t ( x ) } { \\partial x ^ t } \\biggr | = f ( x ) \\prod _ { j = 1 } ^ d | H _ { t _ j } ( x _ j ) | \\leq f ( x ) \\prod _ { j = 1 } ^ d p _ { t _ j } ( \\| x \\| ) \\leq f ( x ) q _ { | t | } ( \\| x \\| ) , \\end{align*}"}
-{"id": "8118.png", "formula": "\\begin{align*} J ^ { \\prime } ( 0 ) = k ( - 1 + s _ 1 ) = 0 \\ \\mbox { a n d } \\ J ^ { \\prime \\prime } ( 0 ) = k ( 2 k - 2 + s _ 2 ) = 2 k ( k - 1 - l ) . \\end{align*}"}
-{"id": "7234.png", "formula": "\\begin{align*} e _ j ^ \\alpha = \\sum _ { | \\beta | = n } \\tau ^ \\alpha _ { j k , \\beta } \\cdot e _ k ^ \\beta . \\end{align*}"}
-{"id": "9739.png", "formula": "\\begin{align*} x _ { L , \\epsilon } ( t ) = \\Lambda _ { 1 } ( 1 - \\epsilon ) G ^ { - 1 } ( t ) , t \\geq T _ 8 ( \\epsilon ) . \\end{align*}"}
-{"id": "921.png", "formula": "\\begin{align*} g \\left ( \\sum ^ { n } _ { i = 1 } t _ { i } x _ { i } \\right ) \\geq \\sum ^ { n } _ { i = 1 } t _ { i } g ( x _ { i } ) . \\end{align*}"}
-{"id": "3747.png", "formula": "\\begin{align*} C ( z ) = C _ 0 ( z + \\beta ) - C _ 0 ( z ) . \\end{align*}"}
-{"id": "3694.png", "formula": "\\begin{align*} \\frac { \\partial \\mu } { \\partial t } = \\{ \\mu , Z , H \\} _ { \\mu } ~ , \\end{align*}"}
-{"id": "7516.png", "formula": "\\begin{align*} \\mu _ 0 : = \\inf \\left \\{ x \\in [ 0 , c ] \\left | f ( x ) = \\max _ { s \\in [ 0 , \\infty ) } f ( s ) \\right \\} \\right . ; \\end{align*}"}
-{"id": "697.png", "formula": "\\begin{align*} \\hat { h } _ { X , f } ( P ) = \\lim _ { n \\to \\infty } \\frac { h _ { X } ( f ^ { n } ( P ) ) } { \\delta _ { f } ^ { n } } \\end{align*}"}
-{"id": "1266.png", "formula": "\\begin{align*} T Q \\phi & = ( U + v \\mathcal G _ 0 v ^ * ) \\phi = U \\phi + v ( c ( 1 , 0 ) ^ T - \\psi ) = v \\psi + c v ( 1 , 0 ) ^ T - v \\psi = c v ( 1 , 0 ) ^ T . \\end{align*}"}
-{"id": "4621.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { R } ( ( x w ^ * ) * R ( w w ^ * ) ^ * ) & = \\mathcal { R } ( ( w w ^ * ) * R ( w w ^ * ) ^ * ) , \\\\ \\mathcal { R } ( \\mathcal { F } ( ( x w ^ * ) * R ( w w ^ * ) ^ * ) ) & = \\mathcal { R } ( \\mathcal { F } ( ( w w ^ * ) * R ( w w ^ * ) ^ * ) ) . \\end{aligned} \\end{align*}"}
-{"id": "7837.png", "formula": "\\begin{align*} g _ t A g _ t ^ { - 1 } , g _ t B g _ t ^ { - 1 } = \\left [ \\begin{smallmatrix} * & t * & 0 & 0 \\\\ 0 & * & 0 & 0 \\\\ t * & t ^ 2 * & * & t * \\\\ 0 & t * & 0 & * \\end{smallmatrix} \\right ] , \\ \\ g _ t a = \\left [ \\begin{smallmatrix} * \\\\ 0 \\\\ t * \\\\ 0 \\end{smallmatrix} \\right ] , \\ \\ b g _ t ^ { - 1 } = \\left [ \\begin{smallmatrix} * & t * & 0 & 0 \\end{smallmatrix} \\right ] . \\end{align*}"}
-{"id": "953.png", "formula": "\\begin{align*} L ^ { 2 } _ { ( 0 , q ) } ( M ) = \\ker ( \\overline { \\partial } _ { M } ) \\oplus \\ker ( \\overline { \\partial } _ { M } ^ { * } ) \\oplus \\mathcal { H } _ { q } ( M ) \\ ; , \\end{align*}"}
-{"id": "4110.png", "formula": "\\begin{align*} y ^ q ( b y + c z ) ^ r - z ^ { p + q + r } = 0 \\end{align*}"}
-{"id": "5134.png", "formula": "\\begin{align*} & q ^ { m + 1 } \\ , u ( a ^ { m } , b ) + \\sum _ { \\ell = 2 } ^ { m + 1 } g ( w , z _ { \\ell } ) Z _ { \\ell } ^ { [ 1 , m + 1 ] } ( \\vec { z } ) u ( b , a ^ { m + 1 } ) \\\\ & + \\left \\{ g ( w , z _ { 1 } ) \\prod _ { i = 2 } ^ { m + 1 } f ( z _ { i } , w ) - \\sum _ { \\ell = 2 } ^ { m + 1 } g ( w , z _ { \\ell } ) g ( z _ { \\ell } , z _ { 1 } ) \\prod _ { \\begin{subarray} { c } i = 2 \\\\ i \\not = \\ell \\end{subarray} } ^ { m + 1 } f ( z _ { i } , z _ { \\ell } ) \\right \\} u ( b , a ^ { m + 1 } ) . \\end{align*}"}
-{"id": "4551.png", "formula": "\\begin{align*} \\biggl | \\frac { d } { d r } I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { r ^ 2 } { 4 } \\biggr ) \\biggr | = \\frac { ( 1 - r ^ 2 / 4 ) ^ \\frac { d - 1 } { 2 } } { \\mathrm { B } _ { ( d + 1 ) / 2 , 1 / 2 } } \\leq \\frac { 1 } { \\mathrm { B } _ { ( d + 1 ) / 2 , 1 / 2 } } . \\end{align*}"}
-{"id": "5642.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ 2 ^ 2 ( \\nu ^ i , \\mu ^ k ) & = 2 \\sum _ { i = 1 } ^ n \\underset { f \\in Z } { \\sup } \\ \\left \\{ \\int _ { \\Omega } f d \\mu ^ k + \\int _ { \\Omega } S f ( x ) d \\nu ^ i ( x ) \\right \\} \\le M , \\end{align*}"}
-{"id": "4792.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( \\pm a t + c ) ^ 2 - t ^ 2 } , c = c o n s t , \\end{align*}"}
-{"id": "4763.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } '' : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "521.png", "formula": "\\begin{align*} \\rho _ { \\circ } ^ + & = ( c , a _ { j _ 1 + 1 } , \\dots , a _ { i _ 1 } ) , \\\\ \\rho _ k ^ + & = ( a _ { i _ k + 1 } , \\dots , a _ { i _ { k + 1 } } ) k = 1 , \\dots , n - 1 , \\\\ \\rho _ k ^ - & = ( a _ { j _ { k + 1 } + 1 } , \\dots , a _ { j _ k } ) k = 1 , \\dots , n - 1 , \\\\ \\rho _ n ^ - & = ( a _ { 1 } , \\dots , a _ { j _ n } ) . \\end{align*}"}
-{"id": "3624.png", "formula": "\\begin{align*} u _ { n + 1 } = \\frac { r u _ n } { 1 + u _ { n - k + 1 } } , \\end{align*}"}
-{"id": "6797.png", "formula": "\\begin{align*} B _ n = \\sum _ { i = 1 } ^ p t _ i ^ { n + 1 } A _ i \\qquad \\ n = 0 , 1 , \\ldots \\end{align*}"}
-{"id": "2195.png", "formula": "\\begin{align*} \\| G _ { m } \\| _ { L ^ { 1 } ( [ 0 , t _ { * } ] ) } = ( g _ { 1 - \\alpha , m } * W ) ( t _ { * } ) + \\int _ { 0 } ^ { t _ { * } } F _ { m } ( s ) d s . \\end{align*}"}
-{"id": "955.png", "formula": "\\begin{align*} Q _ { q } ( f , f ) = ( \\Box _ { M , q } f , f ) _ { L ^ { 2 } _ { ( 0 , q ) } ( M ) } \\ ; , \\ ; f \\in d o m ( \\Box _ { M , q } ) \\ ; . \\end{align*}"}
-{"id": "9307.png", "formula": "\\begin{align*} \\sup _ { u ( . , . , z ) \\in \\mathcal { A } } j ( u ) ( z ) = j ( u ^ { \\star } ) ( z ) . \\end{align*}"}
-{"id": "9966.png", "formula": "\\begin{align*} \\mathbf { y } _ t = \\mathbf { H } _ t \\mathbf { W } _ t \\mathbf { x } _ t + \\mathbf { n } _ t , \\end{align*}"}
-{"id": "1359.png", "formula": "\\begin{align*} I _ { n } ( \\delta _ { 0 } ) = \\sum _ { t = 1 } ^ { n } \\sigma ^ 2 _ t ( \\delta _ { 0 } ) \\begin{bmatrix} x _ { t } x _ { t } ^ \\mathrm { T } & x _ { t } ( m \\pi ) _ { t - J _ \\phi } ^ \\mathrm { T } & 0 \\\\ ( m \\pi ) _ { t - J _ \\phi } x _ { t } ^ \\mathrm { T } & A _ { \\phi \\phi , t } & A _ { \\phi \\theta , t } \\\\ 0 & A _ { \\theta \\phi , t } & A _ { \\theta \\theta , t } \\end{bmatrix} \\end{align*}"}
-{"id": "2843.png", "formula": "\\begin{align*} M C _ { \\bullet } ( g ) = M C ( g \\hat { \\otimes } \\Omega _ { \\bullet } ) , \\end{align*}"}
-{"id": "3384.png", "formula": "\\begin{align*} \\frac { f ( \\gamma ) } { 1 - F ( \\gamma _ { \\max } ) } = \\frac { 1 } { \\bar { \\sigma } _ h ^ 2 } { e ^ { - \\frac { \\gamma } { \\bar { \\sigma } _ h ^ 2 } + \\frac { \\gamma _ { \\max } } { \\bar { \\sigma } _ h ^ 2 } } } . \\end{align*}"}
-{"id": "21.png", "formula": "\\begin{align*} K _ { X / Y } = K _ X - p ^ \\star K _ Y \\end{align*}"}
-{"id": "3689.png", "formula": "\\begin{align*} \\mathcal { A } _ q ( . ) = \\nabla \\cdot ( q \\nabla ( . ) ) ~ , \\end{align*}"}
-{"id": "4501.png", "formula": "\\begin{align*} h _ { n , L } ( x ) : = \\left \\{ \\begin{array} { l l } 0 & \\mbox { i f $ | x | > z _ { q / 2 } ( 1 + \\epsilon _ n ) + 1 / L $ } \\\\ L \\{ z _ { q / 2 } ( 1 + \\epsilon _ n ) + 1 / L - | x | \\} & \\mbox { i f $ 0 < | x | - z _ { q / 2 } ( 1 + \\epsilon _ n ) \\leq 1 / L $ } \\\\ 1 & \\mbox { i f $ | x | \\leq z _ { q / 2 } ( 1 + \\epsilon _ n ) $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "6422.png", "formula": "\\begin{align*} s _ { \\alpha , m } ( t ) + m ( s _ { \\alpha , m } * g _ { \\alpha } ) ( t ) = 1 , t > 0 , \\ , \\ , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "4658.png", "formula": "\\begin{align*} Z ^ + ( f , L ; s , \\phi ) & = \\int _ { G _ + / \\Gamma , \\chi ( g ) \\geq 1 } \\chi ( g ) ^ s \\phi ( g ) \\sum _ { x \\in L \\setminus L _ 0 } f ( g \\cdot x ) d g , \\\\ Z ^ + ( f , \\hat { L } ; s , \\phi ) & = \\int _ { G _ + / \\Gamma , \\chi ( g ) \\geq 1 } \\chi ( g ) ^ s \\phi ( g ) \\sum _ { x \\in \\hat { L } \\setminus \\hat { L } _ 0 } f ( g \\cdot x ) d g . \\end{align*}"}
-{"id": "162.png", "formula": "\\begin{align*} A _ i = C _ { m _ i ( 0 ) } \\cap \\tau ^ { - 1 } C _ { m _ i ( 1 ) } \\cap \\dots \\cap \\tau ^ { - 2 ^ { k _ i } } C _ { m _ i ( 2 ^ { k _ i } ) } , \\end{align*}"}
-{"id": "7098.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( M _ n - m _ n , \\sum _ { | u | = n } \\delta _ { u , V ( u ) - M _ n } \\right ) = \\left ( \\log ( \\mathbf { e } / Z _ \\infty ) , \\sum _ { d \\in \\mathcal { P } ( D _ n ) } \\delta _ { u ^ { ( n ) } , \\zeta _ n + d } \\right ) , \\end{align*}"}
-{"id": "8219.png", "formula": "\\begin{align*} \\sin ( 2 \\theta ) = \\frac { \\gamma } { 4 A ^ 2 + \\Omega } , \\cos ( 2 \\theta ) = \\frac { E } { 8 A ^ 2 + \\Omega } , \\end{align*}"}
-{"id": "3052.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } f _ \\mu ( u _ n ) = x \\end{align*}"}
-{"id": "4702.png", "formula": "\\begin{align*} C ( y , z ) = \\Big \\{ \\sigma _ k ( z ' ) ~ \\Big | ~ & z ' \\in \\Z _ q ^ { n - k } , \\mu _ k ( z ) = \\mu _ k ( z ' ) , \\\\ & \\phi _ k ^ { - 1 } ( y , z ' ) \\in C \\Big \\} \\end{align*}"}
-{"id": "1783.png", "formula": "\\begin{align*} ( \\sum \\limits _ { x \\in \\R } a _ x t ^ x ) ( \\sum \\limits _ { x \\in \\R } b _ x t ^ x ) : = \\sum \\limits _ { x \\in \\R } ( \\sum \\limits _ { y + z = x } a _ y b _ z ) t ^ x \\end{align*}"}
-{"id": "8503.png", "formula": "\\begin{align*} E _ i ^ * = E _ i , \\ \\ \\ F _ i ^ * = F _ i , \\ \\ \\ ( q ^ h ) ^ * = q ^ { - h } . \\end{align*}"}
-{"id": "8851.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\lambda _ i ^ 2 } { k ^ { \\frac { 4 } { n } } } = \\frac { n } { n + 4 } \\frac { 1 6 \\pi ^ 4 } { ( \\omega _ n \\mathrm { v o l } \\ , \\Omega ) ^ \\frac { 4 } { n } } , \\end{align*}"}
-{"id": "1218.png", "formula": "\\begin{align*} ( \\Lambda _ { \\varphi , w } ) _ a = ( \\Lambda _ { \\varphi , w } ) _ b = \\{ f \\in L ^ 0 : \\forall \\lambda , \\ \\rho ( \\lambda f ) < \\infty \\} . \\end{align*}"}
-{"id": "2107.png", "formula": "\\begin{align*} \\| P \\| : = \\max \\{ | p _ { \\mu m } | \\} . \\end{align*}"}
-{"id": "1297.png", "formula": "\\begin{align*} t _ H ( m ) = H - \\vert m \\vert \\textrm { a n d } T _ H ( N , y ; \\alpha ) = \\sum _ { n = N - H } ^ { N + y } t _ H ( n - N ) e ( n \\alpha ) . \\end{align*}"}
-{"id": "5587.png", "formula": "\\begin{align*} G _ 0 f ( x ) & = - \\frac { 1 } { 2 \\pi } \\int _ { \\R ^ 2 } \\log | x - y | f ( y ) \\ , d y , \\\\ G _ 1 f ( x ) & = \\int _ { \\R ^ 2 } | x - y | ^ 2 f ( y ) \\ , d y , \\\\ G _ 2 f ( x ) & = \\frac 1 { 8 \\pi } \\int _ { \\R ^ 2 } | x - y | ^ 2 \\log | x - y | f ( x ) \\ , d y . \\end{align*}"}
-{"id": "7535.png", "formula": "\\begin{align*} \\alpha _ n \\in ( a , b ) = ( \\alpha - l , \\alpha + l \\nu ) \\subset ( \\delta , 1 - \\delta ) , 1 - \\alpha _ n \\geq 1 - \\alpha - l \\nu , \\end{align*}"}
-{"id": "2679.png", "formula": "\\begin{align*} \\int _ X ( u _ { t } - u _ 0 ) \\theta _ { u _ 0 } ^ n = \\int _ X ( u _ t - u ) \\theta _ { u _ 0 } ^ n \\leq t \\int _ X \\chi \\theta _ { u _ 0 } ^ n . \\end{align*}"}
-{"id": "2455.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 2 ( 1 / p _ 1 - 1 / p _ 2 ) } = 2 ^ { j \\widetilde { A _ 1 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j \\widetilde { A _ 2 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j \\widetilde { A _ 3 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "9552.png", "formula": "\\begin{align*} \\mathcal { W } _ { i } = \\{ \\pi ^ { - 1 } ( F ) \\mid F \\in \\mathcal { Y } _ i \\} , 1 \\le i \\le k . \\end{align*}"}
-{"id": "5810.png", "formula": "\\begin{align*} b ' _ { i j } = \\begin{cases} - b _ { i j } & i = k j = k \\\\ b _ { i j } & b _ { i k } b _ { k j } \\leq 0 \\\\ b _ { i j } + | b _ { i k } | b _ { k j } & b _ { i k } b _ { k j } > 0 . \\end{cases} \\end{align*}"}
-{"id": "122.png", "formula": "\\begin{align*} ( 1 + t ) ^ { \\tfrac { x } { 2 } } & = \\sum _ { m = 0 } ^ \\infty \\Big ( \\frac { x } { 2 } \\Big ) ^ m \\frac { \\big ( \\log ( 1 + t ) \\big ) ^ m } { m ! } = \\sum _ { m = 0 } ^ \\infty \\Big ( \\frac { x } { 2 } \\Big ) ^ m \\sum _ { n = m } ^ \\infty S _ 1 ( n , m ) \\frac { t ^ n } { n ! } \\\\ & = \\sum _ { n = 0 } ^ \\infty \\left ( \\sum _ { m = 0 } ^ n \\Big ( \\frac { x } { 2 } \\Big ) ^ m S _ 1 ( n , m ) \\right ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "7639.png", "formula": "\\begin{align*} ( \\mathcal { F } ( - \\triangle ) ^ { \\alpha / 2 } u ) ( \\xi ) = | \\xi | ^ { \\alpha } \\mathcal { F } ( u ) ( \\xi ) , \\end{align*}"}
-{"id": "2644.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { h ( m , n ) } C _ { m } ^ { l } ( F ^ { 0 } ) ( \\lambda ) ( C _ { m } ^ { l } ( F ^ { 0 } ) ( \\lambda ) ) ^ { * } = d _ { m } ^ { 0 } ( \\lambda ) ^ { \\top } \\left \\{ ( \\beta _ { m k } ^ { 2 } + \\gamma _ { m _ { 1 } k } ( \\lambda ) + \\gamma _ { m _ { 2 } k } ( \\lambda ) ) \\delta _ k ^ l \\right \\} _ { k , l = 1 } ^ { \\infty } \\overline { d _ { m } ^ { 0 } ( \\lambda ) } , \\end{align*}"}
-{"id": "7439.png", "formula": "\\begin{align*} B _ 5 : = x \\overline { v ^ { - 1 } } B _ + B _ 6 : = \\overline { u } B _ - \\end{align*}"}
-{"id": "3155.png", "formula": "\\begin{align*} \\lVert \\sum _ { l = 1 } ^ { L _ { n } } \\frac { 1 } { L _ { n } } Q _ { t ^ n } ( x _ { j , l } ) - \\Theta _ { t ^ n } \\rVert _ 1 \\leq 2 ^ { - \\sqrt { n } \\frac { 1 } { 1 6 } \\hat { \\beta } ( \\alpha ) } \\end{align*}"}
-{"id": "4015.png", "formula": "\\begin{align*} \\sum _ { i = \\delta } ^ { \\ell } \\alpha _ i ( i - \\delta ) ( i - \\ell ) = 2 y + x - ( \\delta + \\ell ) x + \\delta \\ell z . \\end{align*}"}
-{"id": "6816.png", "formula": "\\begin{align*} L _ m \\cdot B _ n & = \\ \\sum _ { k = 0 } ^ { m } \\left [ B _ k , B _ { m + n - k } \\right ] + n B _ { m + n } \\\\ L _ { - m } \\cdot B _ n & = - \\sum _ { k = - m + 1 } ^ { - 1 } \\left [ B _ k , B _ { - m + n - k } \\right ] + n B _ { - m + n } \\end{align*}"}
-{"id": "2510.png", "formula": "\\begin{align*} ( \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } F ) ( x ) = ( - 2 \\lambda ) F ( x ) , ~ ( \\forall ) x \\in U _ { x _ e } . \\end{align*}"}
-{"id": "8404.png", "formula": "\\begin{align*} w ^ * \\lim _ \\mu x _ \\mu = 0 \\ \\ \\ \\mathrm { a n d } \\ \\ \\ w ^ * \\lim _ \\mu \\phi ( x _ \\mu ) = y \\in W ^ * ( X _ 0 ) \\end{align*}"}
-{"id": "9819.png", "formula": "\\begin{align*} L _ i \\ , { } _ \\lambda \\ , v _ j = ( \\partial + a \\lambda + b ) v _ { i + j } \\ \\ \\ \\mbox { f o r } \\ \\ \\ i , j \\in \\Z . \\end{align*}"}
-{"id": "386.png", "formula": "\\begin{align*} \\langle x , y \\rangle = x _ 4 y _ 1 - \\frac { 1 } { 3 } x _ 3 y _ 2 + \\frac { 1 } { 3 } x _ 2 y _ 3 - x _ 1 y _ 4 . \\end{align*}"}
-{"id": "1914.png", "formula": "\\begin{align*} \\| f _ { j } - f _ { k } \\| ^ 2 \\geq C m _ n m _ n ^ { - 2 \\beta } , \\ j \\not = k \\end{align*}"}
-{"id": "4503.png", "formula": "\\begin{align*} h _ x ( \\| x \\| + r ) = \\int _ { B _ x ( \\| x \\| + r ) } f ( y ) \\ , d y \\geq \\int _ { B _ 0 ( r ) } f ( y ) \\ , d y \\geq 1 - \\frac { \\mu _ \\alpha ( f ) } { r ^ \\alpha } . \\end{align*}"}
-{"id": "7076.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( \\gamma _ n , Z _ n \\right ) = \\left ( \\gamma _ \\infty , Z _ \\infty \\right ) \\end{align*}"}
-{"id": "9454.png", "formula": "\\begin{align*} T P _ F = & \\ ( P _ F + P _ { \\partial ^ + _ R F } + P _ { X \\setminus N _ R ^ + F } ) T P _ F \\\\ = & \\ P _ F T P _ F + P _ { \\partial ^ + _ R F } T ( P _ { N _ { R } ^ - F } + P _ { \\partial ^ - _ R F } ) + 0 \\\\ = & \\ P _ F T P _ F + 0 + P _ { \\partial ^ + _ R F } T P _ { \\partial ^ - _ R F } \\ ; , \\end{align*}"}
-{"id": "2683.png", "formula": "\\begin{align*} g ( 0 ) = F ( \\varphi ) \\geq F ( \\varphi _ t ) \\geq g ( t ) , \\ \\forall t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "8443.png", "formula": "\\begin{align*} J _ { \\varepsilon } \\textbf { w } ( x ) = ( j _ { \\varepsilon } * \\textbf { w } ) ( x ) = \\int _ { \\mathbb { R } ^ { d } } \\Phi \\Bigg ( \\frac { x - y } { \\varepsilon } \\Bigg ) \\textbf { w } ( y ) ~ d y . \\end{align*}"}
-{"id": "6962.png", "formula": "\\begin{align*} a ( t , u ) = t \\{ ( 1 + u ) ^ { \\gamma } - 1 - u ^ { \\gamma } \\} + ( 1 + u ^ { \\gamma } ) = : t p ( u ) + q ( u ) . \\end{align*}"}
-{"id": "3340.png", "formula": "\\begin{align*} c ( [ \\hat { W } ] , F ) & \\leq c ( [ \\hat { W } ] , L ^ 1 ) + | | \\bar { H } | | _ { C ^ 0 } \\\\ & = c ( [ \\hat { W } ] , H ) + | | H | | _ { C ^ 0 } \\\\ & < 2 ( \\sum _ { i = 1 } ^ n ( R _ i - \\epsilon ) \\cdot | e _ i | + | | G | | _ { C ^ 0 } ) . \\end{align*}"}
-{"id": "2747.png", "formula": "\\begin{align*} w ^ { L K } : = \\tilde b ^ L _ Q \\star v ^ { Q P } \\star \\tilde a ^ K _ P . \\end{align*}"}
-{"id": "2520.png", "formula": "\\begin{align*} ( A ) ~ ~ c _ 1 ( E ) . H = \\frac { 3 r } { 2 } H ^ 2 \\mbox { a n d } ( B ) ~ ~ c _ 2 ( E ) = \\frac { 1 } { 2 } c _ 1 ( E ) ^ 2 - ( H ^ 2 - 1 ) r , \\end{align*}"}
-{"id": "9888.png", "formula": "\\begin{align*} c _ n = \\Big \\{ \\begin{array} { c c } 1 x _ n \\geq \\tau _ n & \\\\ 0 x _ n < \\tau _ n & \\end{array} , \\end{align*}"}
-{"id": "9481.png", "formula": "\\begin{align*} p ( \\rho , \\phi ) = ( r _ { + } ( r _ { + } / r _ { - } ) ^ { \\rho } \\cos \\phi , r _ { + } ( r _ { + } / r _ { - } ) ^ { \\rho } \\sin \\phi ) \\end{align*}"}
-{"id": "2349.png", "formula": "\\begin{align*} \\mu = \\sum _ x \\deg ( x ) \\sum _ { n = 1 } ^ \\infty q ^ { - n \\deg ( x ) } \\delta ( g ^ n _ x ) , \\end{align*}"}
-{"id": "5791.png", "formula": "\\begin{align*} L _ { - 1 } ( A x _ { \\alpha } ) & = ( L _ { - 1 } A ) x _ { \\alpha } + A ( L _ { - 1 } x _ { \\alpha } ) , \\\\ T ( A \\psi _ { \\alpha } ) & = ( T A ) \\psi _ { \\alpha } + A ( T \\psi _ { \\alpha } ) \\end{align*}"}
-{"id": "1160.png", "formula": "\\begin{align*} & b _ { 0 , x x } = - 2 \\int \\rho _ { x } b _ { - 1 } d x - 4 \\int \\rho b _ { - 1 , x } d x + C _ 1 = - 2 \\rho b _ { - 1 } - 2 \\int \\rho b _ { - 1 , x } d x + C _ { 1 } , \\\\ & b _ { 0 , x } = - 2 \\int \\biggl ( \\rho b _ { - 1 } + \\int \\rho b _ { - 1 , x } d x \\biggr ) d x + C _ { 1 } x + C _ { 2 } : = f ( x ) + C _ { 1 } x + C _ { 2 } , \\\\ & b _ { 0 } = \\int f ( x ) d x + \\frac { 1 } { 2 } C _ { 1 } x ^ { 2 } + C _ { 2 } x + C _ { 3 } . \\end{align*}"}
-{"id": "3541.png", "formula": "\\begin{align*} g ^ R ( \\partial _ { x ^ i } , \\partial _ { x ^ j } ) ( x ) & = R ^ { - 2 } g ( ( F _ R ) _ * ( \\partial _ { x ^ i } ) , ( F _ R ) _ * ( \\partial _ { x ^ j } ) ) ( y ) = g ( \\partial _ { y ^ i } , \\partial _ { y ^ j } ) ( y ) \\\\ \\pi ^ R ( \\partial _ { x ^ i } , \\partial _ { x ^ j } ) ( x ) & = R ^ { - 1 } \\pi ( ( F _ R ) _ * ( \\partial _ { x ^ i } ) , ( F _ R ) _ * ( \\partial _ { x ^ j } ) ) ( y ) = R \\pi ( \\partial _ { y ^ i } , \\partial _ { y ^ j } ) ( y ) . \\end{align*}"}
-{"id": "3018.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ i ] } } ( n ) = { \\bf { V } } _ 1 ^ { [ i ] } ( n ) { { \\bf { u } } ^ { [ i ] } } + { \\bf { V } } _ 2 ^ { [ i ] } ( n ) { { \\bf { v } } ^ { [ i ] } } , \\end{align*}"}
-{"id": "5484.png", "formula": "\\begin{align*} \\min \\limits _ { 0 \\le i \\le N } P _ { i } ( A _ { i } ) \\le 0 . 7 1 \\vee \\frac { \\sum \\limits _ { i = 1 } ^ { N } { } ( P _ { i } \\| P _ { 0 } ) } { N \\log ( N + 1 ) } . \\end{align*}"}
-{"id": "9078.png", "formula": "\\begin{align*} \\mathrm { T r } \\left ( U \\int _ { - \\infty } ^ { 0 } \\Lambda e ^ { s \\Lambda } s ^ { k } \\ , \\mathrm { d } s \\ , U ^ { \\dagger } \\right ) = \\mathrm { T r } \\left ( \\int _ { - \\infty } ^ { 0 } \\Lambda e ^ { s \\Lambda } s ^ { k } \\ , \\mathrm { d } s \\ , \\right ) \\ , \\end{align*}"}
-{"id": "4645.png", "formula": "\\begin{align*} E ( z , g ) = \\sum _ { \\gamma \\in \\Gamma / \\Gamma _ \\infty } t ( g \\gamma ) ^ { z + 1 } \\end{align*}"}
-{"id": "7064.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( { t _ 3 } ) = { h ^ { [ j 1 ] } } ( { t _ 3 } ) u _ 3 ^ { [ 1 ] } + { h ^ { [ j 2 ] } } ( { t _ 3 } ) \\frac { { { h ^ { [ 3 2 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 3 2 ] } } ( { t _ 3 } ) } } u _ 1 ^ { [ 2 ] } + { h ^ { [ j 3 ] } } ( { t _ 3 } ) \\frac { { { h ^ { [ 3 3 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 3 3 ] } } ( { t _ 3 } ) } } u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "680.png", "formula": "\\begin{align*} E _ n = ( F _ n ^ { - 1 } s _ n ^ { - 1 } B ^ c ) ^ c = \\bigcap _ { f \\in F _ n } f ^ { - 1 } s _ n ^ { - 1 } B = \\big \\{ y \\in Y \\ , : \\ , s _ n F _ n \\subset B _ y \\big \\} \\subset Y \\end{align*}"}
-{"id": "8640.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\frac { d ^ { 2 } y } { d \\tau ^ { 2 } } ( \\tau ) = - \\Lambda y ( \\tau ) + B ( t , y ( \\tau ) , \\frac { d y } { d \\tau } ( \\tau ) ) + \\dot { W } ( \\tau ) , \\\\ y \\left ( 0 \\right ) = x _ { 0 } , \\\\ \\frac { d y } { d \\tau } ( 0 ) = x _ { 1 } , \\ ; \\ ; \\ ; \\tau \\in ( 0 , T ] , \\end{array} \\right . \\end{align*}"}
-{"id": "9726.png", "formula": "\\begin{align*} x _ 2 ( \\epsilon ) = g ( \\delta / 2 ) e ^ { c _ 2 \\int _ 0 ^ { T _ 4 ( \\epsilon ) } \\frac { 1 } { \\sigma ( s ) } \\ , d s } . \\end{align*}"}
-{"id": "1605.png", "formula": "\\begin{align*} C ^ { x } = - m + c _ { 1 } e ^ { - x } , \\end{align*}"}
-{"id": "644.png", "formula": "\\begin{align*} A B = \\bigcup _ { a \\in A } a B = \\big \\{ a \\cdot b \\ , : \\ , a \\in A , \\ : b \\in B \\big \\} \\subset Y , \\end{align*}"}
-{"id": "2569.png", "formula": "\\begin{align*} \\tfrac { 1 } { q _ 1 ' } = \\tfrac { p } { \\tilde q } + \\tfrac { 1 } { q _ 1 } , \\tfrac { 1 } { r _ 1 ' } = \\tfrac { p } { \\tilde r } + \\tfrac { 1 } { r _ 1 } , \\tfrac { 1 } { \\tilde q } - \\tfrac { 1 } { q _ 2 } = \\tfrac { d } { r _ 2 } - \\tfrac { d } { \\tilde r } = | s _ c | . \\end{align*}"}
-{"id": "7237.png", "formula": "\\begin{align*} f _ { j , \\alpha } - \\sum _ { | \\beta | = n + 1 } T _ { j k } f _ { k , \\beta } \\tau ^ \\beta _ { k j , \\alpha } = f _ { k j , \\alpha } \\end{align*}"}
-{"id": "2108.png", "formula": "\\begin{align*} L _ { - a } v _ { \\pm } = ( 2 f \\pm 2 M ) \\mathcal { H } ^ { n } \\llcorner \\{ y = 0 \\} . \\end{align*}"}
-{"id": "6736.png", "formula": "\\begin{align*} b ^ { ( n ) } _ \\ell ( x _ \\ell ) \\quad = \\begin{cases} 0 & | x _ \\ell | < k ^ { ( n ) } _ \\ell \\ , , \\\\ \\frac { x _ \\ell - k ^ { ( n ) } _ \\ell } { 2 k ^ { ( n ) } _ \\ell - x _ \\ell } & k ^ { ( n ) } _ \\ell \\leq | x _ \\ell | \\leq 2 k ^ { ( n ) } _ \\ell \\ , , \\\\ \\infty & 2 k ^ { ( n ) } _ \\ell < | x _ \\ell | \\ , , \\end{cases} \\end{align*}"}
-{"id": "7435.png", "formula": "\\begin{align*} \\deg _ { X _ f } \\left ( \\tilde { p } _ { \\vec { i } _ { ( u , v ) } } \\circ \\tilde { \\psi } _ { \\vec { i } _ { ( u , v ) } } \\circ \\tilde { \\chi } _ { \\vec { i } _ { ( u , v ) } } \\right ) ^ * ( X _ g ) = - \\delta _ { f g } . \\end{align*}"}
-{"id": "2005.png", "formula": "\\begin{align*} M _ m ( t ) = g _ m ( V _ t ^ n ) - \\int _ 0 ^ t \\mathcal { A } _ n g _ m ( V _ { s } ^ n ) \\d s . \\end{align*}"}
-{"id": "6869.png", "formula": "\\begin{align*} e ^ { - i t _ 0 \\Delta } u ( t _ 0 , x ) = e ^ { i x \\xi ( t _ 0 ) } h ( t _ 0 ) ^ { \\frac 2 { p } } \\psi ( t _ 0 , h ( t _ 0 ) x ) , \\end{align*}"}
-{"id": "140.png", "formula": "\\begin{align*} 0 \\leq ( \\abs { f } - f _ n ) ( t ) = \\left . \\begin{cases} 0 & \\ , \\abs { f } ( t ) \\leq g _ n ' ( t ) \\\\ \\abs { f } ( t ) - g _ n ' ( t ) & \\ , \\abs { f } ( t ) > g _ n ' ( t ) \\end{cases} \\right \\} \\ , \\leq \\abs { g } ( t ) - g _ n ' ( t ) . \\end{align*}"}
-{"id": "6413.png", "formula": "\\begin{align*} & C ^ { - 1 } \\Lambda ^ { - 1 } \\int _ { B } \\int _ { B } ( v ( x ) - v ( y ) ) ^ { 2 } k _ { 0 } ( x , y ) d x d y \\leq c _ { n , \\beta } \\int _ { B } \\int _ { B } \\frac { ( v ( x ) - v ( y ) ) ^ { 2 } } { | x - y | ^ { n + 2 \\beta } } d x d y \\\\ & \\leq C \\Lambda \\int _ { B } \\int _ { B } ( v ( x ) - v ( y ) ) ^ { 2 } k _ { 0 } ( x , y ) d x d y , B = B _ { \\rho } ( x _ { 0 } ) . \\end{align*}"}
-{"id": "71.png", "formula": "\\begin{align*} \\varphi ' ( Y _ a , a ) = 0 \\ , \\varphi ' ( y , a ) ( y - Y _ a ) < 0 \\ , y \\in ( 0 , 1 ) \\setminus \\{ Y _ a \\} \\ . \\end{align*}"}
-{"id": "4590.png", "formula": "\\begin{align*} ( \\mathsf W ^ k ) _ { i j } = \\sum _ { \\substack { p : \\ , i \\to j \\\\ \\ell ( p ) = k } } p \\ , , i , j = 1 , . . . , N , \\end{align*}"}
-{"id": "1403.png", "formula": "\\begin{align*} \\mu _ k ^ * ( X ' _ i ) = \\begin{cases} X _ i X _ k ^ { [ b _ { i k } ] _ + } ( 1 + X _ k ) ^ { - b _ { i k } } & i \\neq k \\\\ X _ k ^ { - 1 } & i = k , \\end{cases} \\end{align*}"}
-{"id": "8340.png", "formula": "\\begin{gather*} \\sigma _ i ( w ) = w + \\gamma _ i , 1 \\leq i \\leq n \\iff l ( \\mu _ i ) = \\gamma _ i , 1 \\leq i \\leq n \\\\ \\iff M \\left [ \\begin{array} { c } A _ 0 \\\\ A _ 1 \\\\ \\vdots \\\\ A _ { n - 2 } \\\\ A _ { n - 1 } \\end{array} \\right ] = \\left [ \\begin{array} { c } \\gamma _ 1 \\\\ \\gamma _ 2 \\\\ \\vdots \\\\ \\gamma _ { n - 1 } \\\\ \\gamma _ n \\end{array} \\right ] = \\left [ \\begin{array} { c } \\gamma _ 1 \\\\ \\vdots \\\\ \\gamma _ { m } \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array} \\right ] , \\end{gather*}"}
-{"id": "5137.png", "formula": "\\begin{align*} K _ { t } ( w ; z _ { \\ell ( 1 ) } , \\ldots , z _ { \\ell ( t ) } ) = \\frac { g ( w , z _ { \\ell ( 1 ) } ) g ( w , z _ { \\ell ( t ) } ) } { g ( z _ { \\ell ( t ) } , z _ { \\ell ( 1 ) } ) } \\prod _ { s = 1 } ^ { t - 1 } g ( z _ { \\ell ( s + 1 ) } , z _ { \\ell ( s ) } ) \\end{align*}"}
-{"id": "2257.png", "formula": "\\begin{align*} \\gamma _ { q , p } ( p , 2 ) = \\beta _ { q , p } \\left ( \\frac { ( 4 p - 1 ) ! ! } { 6 } \\varepsilon _ { p } ^ { 4 } \\sigma ^ { 4 p } + \\frac { ( 2 q + 2 p - 1 ) ! ! } { 2 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } ^ { 2 } \\sigma ^ { 2 q + 2 p } \\right ) , \\end{align*}"}
-{"id": "1246.png", "formula": "\\begin{align*} \\beta = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) , \\alpha _ 1 = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ) , \\alpha _ 2 = \\left ( \\begin{array} { c c } 0 & - i \\\\ i & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "5910.png", "formula": "\\begin{align*} H ^ { k } _ p ( \\R ^ d ) = W ^ { k , p } ( \\R ^ d ) \\end{align*}"}
-{"id": "464.png", "formula": "\\begin{align*} 0 = \\pi ^ { \\top } d _ 0 = \\sum _ { i = 1 } ^ m ( y ^ i ) ^ { \\top } A ^ i d _ 0 . \\end{align*}"}
-{"id": "3862.png", "formula": "\\begin{align*} \\begin{cases} { \\rm ( a ) } \\ i + j = k k \\le n , \\\\ { \\rm ( b ) } \\ ( 2 n + 1 - i ^ - ) + ( 2 n + 1 - j ^ - ) = 2 n + 1 - k ^ - , \\ k \\le n \\min \\{ i , j \\} \\le n , \\\\ { \\rm ( c ) } \\ i ^ - + j ^ - = k ^ - , \\ k \\ge n + 2 \\max \\{ i , j \\} \\ge n + 2 , \\\\ { \\rm ( d ) } \\ ( 2 n + 2 - i ) + ( 2 n + 2 - j ) = 2 n + 2 - k k \\ge n + 2 . \\end{cases} \\end{align*}"}
-{"id": "6172.png", "formula": "\\begin{align*} \\eta _ t = L _ t - \\sum _ { s \\leq t } \\frac { \\Delta U _ s \\Delta L _ s } { 1 + \\Delta U _ s } - \\sigma _ { U L } t , \\end{align*}"}
-{"id": "7533.png", "formula": "\\begin{align*} \\gamma = \\max \\{ b , 1 - a \\} = \\max \\{ \\alpha + l \\nu , 1 - \\alpha + l \\} , \\end{align*}"}
-{"id": "9991.png", "formula": "\\begin{align*} F _ k ( \\alpha ^ { ( j ) } ) = \\alpha ^ { ( j ) } \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\tilde { p } ^ { ( j ) } | H _ { \\tilde { i } k } | ^ 2 } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\ ! + \\ ! ( \\ ! 1 \\ ! - \\ ! \\alpha ^ { ( j ) } \\ ! ) \\sum _ { i = 1 } ^ { 2 } \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { p _ { i } ^ { ( j ) } } { \\sigma ^ 2 _ n } \\ ! \\Big ) , \\end{align*}"}
-{"id": "1384.png", "formula": "\\begin{align*} \\pi ( 1 , X _ i ) = \\int \\frac { e ^ { X _ i ^ T \\beta + \\sqrt { \\tau } z } } { 1 + e ^ { X _ i ^ T \\beta + \\sqrt { \\tau } z } } \\phi ( z ) d z = \\int \\frac { e ^ { X _ i ^ T \\beta _ 0 + \\sqrt { \\tau _ 0 } z } } { 1 + e ^ { X _ i ^ T \\beta _ 0 + \\sqrt { \\tau _ 0 } z } } \\phi ( z ) d z = \\pi ^ 0 ( 1 , X _ i ) . \\end{align*}"}
-{"id": "7973.png", "formula": "\\begin{align*} \\varphi ( x _ 1 , x _ 2 ) = \\alpha _ 1 \\varphi _ 1 ( x _ 1 ) + \\alpha _ 2 \\varphi _ 2 ( x _ 2 ) , \\end{align*}"}
-{"id": "4451.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\biggl | \\mathbb { E } _ f ( \\hat { H } _ n ) - H + \\frac { \\Gamma ( k + 2 / d ) } { 2 ( d + 2 ) V _ d ^ { 2 / d } \\Gamma ( k ) n ^ { 2 / d } } \\int _ \\mathcal { X } \\frac { \\Delta f ( x ) } { f ( x ) ^ { 2 / d } } \\ , d x \\biggr | = o \\Bigl ( \\frac { k ^ { 2 / d } } { n ^ { 2 / d } } \\Bigr ) . \\end{align*}"}
-{"id": "4303.png", "formula": "\\begin{align*} p _ { \\star } ( K _ { X / Y } + L ) _ { y } = H ^ { 0 } ( X _ { y } , K _ { X _ { y } } + L _ { y } ) \\end{align*}"}
-{"id": "9372.png", "formula": "\\begin{align*} \\mathcal { D } ( S _ { m i n } ) = \\left \\{ f \\in { W } _ 2 ^ 2 ( \\mathbb { R } \\backslash \\{ 0 \\} ) : \\begin{array} { c c } f _ r ( 0 ) = 0 & f _ s ( 0 ) = ( q _ 2 , f ) \\\\ f _ r ' ( 0 ) = 0 & f _ s ' ( 0 ) = ( q _ 1 , f ) \\end{array} \\right \\} . \\end{align*}"}
-{"id": "8542.png", "formula": "\\begin{align*} \\pi _ { b r _ { 1 } } ' \\xi _ { b r } ' = \\delta _ { r _ { 1 } , r } i d _ { N _ { \\sigma ( b ) } } \\end{align*}"}
-{"id": "4984.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } 1 & & \\\\ & \\ddots & \\\\ & & 1 \\\\ & & \\end{array} \\right ) . \\end{align*}"}
-{"id": "910.png", "formula": "\\begin{align*} \\vert \\sum _ { 3 \\sqrt { T / \\pi } < n \\leq C T / \\pi } \\frac { 1 } { n ^ { { 1 } / { 2 } } } \\int _ { T } ^ { 2 T } e ^ { i F ( n , t ) } d t \\vert = O \\left ( \\sum _ { 3 \\sqrt { T / \\pi } < n \\leq C T / \\pi } \\frac { 1 } { n ^ { 1 / 2 } } \\right ) = O \\left ( T ^ { 1 / 2 } \\right ) . \\end{align*}"}
-{"id": "9848.png", "formula": "\\begin{align*} x ( t ) \\otimes ( \\delta ( t ) - \\alpha \\theta ( t - \\tau ) ) = \\theta ( t ) \\otimes h _ { \\rm S R } ( t ) \\otimes s ( t ) , \\end{align*}"}
-{"id": "5859.png", "formula": "\\begin{align*} L _ { K _ { 1 } } C _ { x } = 0 ~ , ~ C ^ { x } Y _ { 1 } = 0 . \\end{align*}"}
-{"id": "377.png", "formula": "\\begin{align*} g \\cdot f ( x , y ) = f ( ( x , y ) g ^ t ) . \\end{align*}"}
-{"id": "4366.png", "formula": "\\begin{align*} L _ 0 ^ { c , h } \\Psi _ { c , h } = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; L ^ { c , h } _ n \\Psi _ { c , h } = 0 \\ ; \\ ; \\ ; \\textrm { f o r a l l } \\ ; \\ ; n > 0 \\end{align*}"}
-{"id": "5088.png", "formula": "\\begin{align*} L ( z ) \\otimes L ( w ) = \\sum _ { a , b = 0 } ^ { r } \\sum _ { c , d = 0 } ^ { r } \\left ( L ( z ) _ { a b } L ( w ) _ { c d } \\right ) \\otimes E _ { a b } \\otimes E _ { c d } . \\end{align*}"}
-{"id": "6562.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\Big \\| ( v _ n - v _ { n - 1 } ) _ { n = 0 } ^ N \\Big \\| _ { L ^ p ( X ) } & \\le \\sum _ { m = 0 } ^ N | \\chi _ m | \\ , \\| ( \\eta _ { n - m } \\dot v _ { n - m } ) _ { n = 0 } ^ N \\| _ { L ^ p ( X ) } \\\\ & \\le \\sum _ { m = 0 } ^ N | \\chi _ m | \\ , \\| ( \\dot v _ { n } ) _ { n = 0 } ^ N \\| _ { L ^ p ( X ) } \\\\ & \\le C \\big \\| ( \\dot v _ n ) _ { n = 0 } ^ N \\big \\| _ { L ^ p ( X ) } . \\end{align*}"}
-{"id": "7318.png", "formula": "\\begin{align*} \\hat { R } ( v _ { 1 } \\otimes v _ { 0 } ) = v _ { 0 } \\otimes v _ { 1 } + ( q ^ { 2 } - q ^ { - 2 } ) v _ { 1 } \\otimes v _ { 0 } . \\end{align*}"}
-{"id": "3383.png", "formula": "\\begin{align*} G & = C \\int _ { \\frac { \\gamma _ { \\max } } { \\sigma _ h ^ 2 } } ^ \\infty \\log \\left ( 1 + \\bar { \\sigma } _ h ^ 2 P _ { \\max } \\gamma \\right ) \\gamma ^ { N - 1 } e ^ { - \\gamma } d \\gamma \\end{align*}"}
-{"id": "7398.png", "formula": "\\begin{align*} & C _ { B P S } ^ P ( M \\times I _ R ^ n \\times T ^ n , X \\times T ^ n ; \\mathbf { l } ^ \\ast , ( 0 _ M , e ) ) \\\\ & = C _ { B P S } ^ P ( M \\times I _ R ^ n \\times T ^ n , X \\times T ^ n ; K \\mathbf { l } ^ \\ast , ( 0 _ M , e ) ) \\leq \\sum _ { i = 1 } ^ n R _ i \\cdot | e _ i | . \\\\ \\end{align*}"}
-{"id": "7020.png", "formula": "\\begin{align*} ( i ^ { k + 1 } z ^ { 2 ^ { k + 1 } - 1 } ) \\begin{pmatrix} P _ k ( z ^ 2 ) / \\sqrt { 2 ^ { k + 1 } } \\\\ Q _ k ( z ^ 2 ) / \\sqrt { 2 ^ { k + 1 } } \\end{pmatrix} = g ( z ^ { 2 ^ k } ) \\cdots g ( z ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7664.png", "formula": "\\begin{align*} \\| u _ { 0 } \\| _ { L ^ { 1 } } \\leq \\sum _ { n = 1 } ^ { \\infty } \\| u _ { n } \\| _ { L ^ { 1 } } = \\omega _ { d } \\sum _ { n = 1 } ^ { \\infty } n ^ { - \\alpha } < \\infty . \\end{align*}"}
-{"id": "7660.png", "formula": "\\begin{align*} f ( s ) \\leq c ( 1 + s ^ { p } ) , \\ \\mbox { w h e r e } \\ p = 1 + \\alpha q / d . \\end{align*}"}
-{"id": "194.png", "formula": "\\begin{align*} h _ x ^ { - 1 } ( s ) = \\begin{cases} s / 2 , & \\mbox { i f } s \\leq 2 x \\\\ s - x , & \\mbox { i f } 2 x < s \\leq 1 . \\end{cases} \\end{align*}"}
-{"id": "7430.png", "formula": "\\begin{align*} w ' : = \\left ( s _ { w ^ { - 1 } ( n ) } s _ { w ^ { - 1 } ( n ) + 1 } \\dots s _ { n - 1 } \\right ) ^ { - 1 } w , \\end{align*}"}
-{"id": "8023.png", "formula": "\\begin{align*} \\phi ( x _ j ) = \\mu _ j x _ j , \\phi ( y _ j ) = \\nu _ j y _ j \\end{align*}"}
-{"id": "6406.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { ( m ) } = \\frac { \\mu _ { ( m ) } ^ { G } + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } \\mu _ { ( k + m ) } ^ { G } } { 1 + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } \\mu _ { ( k ) } ^ { G } } = \\frac { ( m - 1 ) ! ! \\sigma ^ { m } + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } ( k + m - 1 ) ! ! \\sigma ^ { k + m } } { 1 + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } ( k - 1 ) ! ! \\sigma ^ { k } } . \\end{align*}"}
-{"id": "2670.png", "formula": "\\begin{align*} { \\rm V o l } ( \\{ \\theta \\} ) : = \\int _ { { \\rm A m p } ( \\{ \\theta \\} ) } \\theta _ { V _ \\theta } ^ n . \\end{align*}"}
-{"id": "7162.png", "formula": "\\begin{align*} \\Phi ^ { j k } ( x ) = \\begin{cases} ( 1 - n x _ j ^ 2 ) x ( k = j ) , \\\\ u ^ { j k } + v ^ { j k } ( k \\neq j ) , \\end{cases} \\end{align*}"}
-{"id": "5564.png", "formula": "\\begin{align*} D ( - \\lambda ) = W ( \\hat c _ 0 , c _ 0 ) \\prod _ { j = 1 } ^ { \\infty } \\left ( 1 + \\frac { \\lambda } { z _ j } \\right ) . \\end{align*}"}
-{"id": "4811.png", "formula": "\\begin{align*} L _ 1 ^ + ( \\Lambda _ R ( \\chi ^ N ) ) = & X _ 1 ^ + ( I _ R ^ N ) - C _ 1 ^ + ( \\Lambda _ R ( \\chi ^ N ) ) + \\delta \\Delta _ 1 ( \\Lambda _ R ( \\chi ^ N ) ) , \\\\ S _ 1 ( \\Lambda _ R ( \\chi ^ N ) ) + L _ 1 ^ - ( \\Lambda _ R ( \\chi ^ N ) ) = & X _ 1 ^ - ( I _ R ^ N ) + C _ 1 ^ - ( \\Lambda _ R ( \\chi ^ N ) ) + \\delta \\Delta _ 1 ( \\Lambda _ R ( \\chi ^ N ) ) . \\end{align*}"}
-{"id": "9483.png", "formula": "\\begin{align*} \\Phi ( \\phi , \\theta , x _ { i } , y _ { i } , \\rho ) = ( \\theta , x _ { i } , y _ { i } , e ^ { \\rho } \\cos \\phi , e ^ { \\rho } \\sin \\phi ) \\end{align*}"}
-{"id": "1177.png", "formula": "\\begin{align*} D _ x f = f _ x + \\sum _ { j = 1 } ^ N [ f ] ( x _ j ) \\delta _ { x _ j } , \\end{align*}"}
-{"id": "5224.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { q } \\sum _ { j , k = 1 } ^ { n } \\frac { \\partial ^ { 2 } h _ { A } ( z ) } { \\partial z _ { j } \\partial \\overline { z _ { k } } } ( t ^ { s } ) _ { j } \\overline { ( t ^ { s } ) _ { k } } \\geq A \\ ; , \\ ; t ^ { 1 } , \\hdots , t ^ { q } \\ ; \\ ; \\mathbb { C } ^ { n } \\end{align*}"}
-{"id": "1726.png", "formula": "\\begin{gather*} \\mathcal { C } L _ 4 : = \\mathcal { S } / A _ 0 \\cong \\mathcal { S } \\times _ { A } ( A / A _ 0 ) , \\end{gather*}"}
-{"id": "638.png", "formula": "\\begin{align*} \\sigma _ 1 * \\sigma _ k = \\frac { 1 } { 2 r } \\sigma _ { k - 1 } + \\big ( 1 - \\frac { 1 } { 2 r } \\big ) \\sigma _ { k + 1 } , \\textrm { f o r a l l $ k \\geq 1 $ } . \\end{align*}"}
-{"id": "8187.png", "formula": "\\begin{align*} N ( z _ 1 , z _ 2 ) = \\prod _ { i = 1 } ^ 3 ( z _ 1 - a _ i z _ 2 ) \\in A [ z _ 1 , z _ 2 ] . \\end{align*}"}
-{"id": "2072.png", "formula": "\\begin{align*} \\mathbb { G } _ { j } ( \\mathcal { P } _ { 1 } , \\ \\mathcal { P } _ { 2 } , \\ \\mathsf { Q } ) & = \\mathbb { G } _ { j } ( - \\tilde { K } ^ { - 1 } ( 2 s _ { 0 } M + D ) , \\ - \\tilde { K } ^ { - 1 } M , \\ \\tilde { K } ^ { - 1 } F ) , \\\\ & = \\mathbb { G } _ { j } ( - \\tilde { K } ^ { - 1 } ( ( 2 s _ { 0 } + \\alpha ) M + \\beta K ) , \\ - \\tilde { K } ^ { - 1 } M , \\ \\tilde { K } ^ { - 1 } F ) , \\\\ & = \\mathbb { K } _ { j } ( \\mathcal { P } _ 1 , \\ \\mathsf { Q } ) , \\end{align*}"}
-{"id": "465.png", "formula": "\\begin{align*} ( y ^ i ) ^ { \\top } A ^ i = \\lambda _ i \\pi ^ { \\top } . \\end{align*}"}
-{"id": "7997.png", "formula": "\\begin{align*} a _ { 2 1 1 0 } = a _ { 0 2 2 0 } a _ { 2 0 0 2 } ^ { 2 } . \\end{align*}"}
-{"id": "8410.png", "formula": "\\begin{align*} | | \\sum \\limits _ { i = 1 } ^ { n } x _ i ^ { * } | | _ { J _ p ^ { * } } ^ q \\geq c \\sum \\limits _ { i = 1 } ^ { n } | | x _ i ^ { * } | | _ { J _ p ^ { * } } ^ q . \\end{align*}"}
-{"id": "9411.png", "formula": "\\begin{align*} E ^ S = \\frac { m \\cdot T ^ S _ m } { P \\cdot T ^ S _ P } . \\end{align*}"}
-{"id": "3670.png", "formula": "\\begin{align*} Z = \\frac { 1 } { 2 } \\int h q ^ 2 d A ~ . \\end{align*}"}
-{"id": "7114.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\limsup _ { n \\to \\infty } \\P \\left ( \\rho _ { n , k } ( f ) - \\rho _ { n , n - k } ( f ) > 0 \\right ) = 0 . \\end{align*}"}
-{"id": "7506.png", "formula": "\\begin{align*} x _ { n + 1 } = \\frac { A x _ n } { 1 + B x _ n ^ { \\gamma } } , A > 1 , ~ B > 0 , ~ \\gamma > 1 , , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 , \\end{align*}"}
-{"id": "9269.png", "formula": "\\begin{align*} \\int _ D Y ( t , x , z ) q ( t , x , z ) d x = - \\frac { a _ 0 ( t , z ) } { b _ 0 ( t , z ) } \\int _ D Y ( t , x , z ) p ( t , x , z ) d x . \\end{align*}"}
-{"id": "284.png", "formula": "\\begin{align*} F _ { n , x , y } ^ { ( 1 ) } ( u , v ) : = \\mathbb { P } ( N _ 1 + N _ 3 \\geq k , N _ 2 + N _ 3 \\geq k ) , \\end{align*}"}
-{"id": "4069.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 4 , 4 u + 2 , 4 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 4 , 4 u + 2 , 4 v + 3 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "3911.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = \\lambda N _ { f } ( u ) & & \\\\ u ( T ) = u ' ( 0 ) = u ' ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3557.png", "formula": "\\begin{align*} T ^ R ( v , w ) = ( v , R w ) + \\theta _ 0 ^ R . \\end{align*}"}
-{"id": "2619.png", "formula": "\\begin{align*} S _ { ( a _ 1 , a _ 2 ) , ( b _ 1 , b _ 2 ) } = ( a _ 1 b _ 2 + b _ 1 a _ 2 ) 2 ^ { - k } , & & T _ { ( a , b ) , ( a , b ) } = a b 2 ^ { - k } \\end{align*}"}
-{"id": "4369.png", "formula": "\\begin{align*} U _ { c , h } ( \\gamma ) T _ { c , h } ( f ) U _ { c , h } ( \\gamma ) ^ * = T _ { c , h } ( \\gamma _ * f ) + r ( c , f , \\gamma ) I \\end{align*}"}
-{"id": "1582.png", "formula": "\\begin{align*} X = \\left ( 2 \\psi _ { I } \\int T ^ { I } \\left ( t \\right ) d t \\right ) + T ^ { I } \\left ( t \\right ) Y _ { I } \\partial _ { x } + \\left ( a \\left ( x , t \\right ) F \\right ) \\partial _ { F } \\end{align*}"}
-{"id": "1954.png", "formula": "\\begin{align*} \\frac { d H } { d { \\hat u } } = \\frac { 1 } { \\hat a } ( H ( \\hat { a } ^ 2 V ) H ) . \\end{align*}"}
-{"id": "1514.png", "formula": "\\begin{align*} e _ 0 = \\frac { B + \\alpha + \\gamma + q - 1 } { 1 - q } . \\end{align*}"}
-{"id": "6583.png", "formula": "\\begin{align*} r _ { C } ( x ) : = \\iota _ { C } ( x ) + \\tfrac { 1 } { 2 } \\| x \\| ^ 2 . \\end{align*}"}
-{"id": "8002.png", "formula": "\\begin{align*} \\gamma _ p ( \\partial Q ) = \\liminf _ { \\epsilon \\rightarrow + 0 } \\frac { \\gamma _ p ( ( Q + \\epsilon B _ 2 ^ n ) \\backslash Q ) } { \\epsilon } . \\end{align*}"}
-{"id": "8417.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\rho + \\nabla \\cdot ( \\rho v ) = 0 , \\\\ & \\partial _ { t } { v } + { v } \\cdot \\nabla { v } + \\nabla \\phi ( \\rho ) + { \\nabla { P } } = 0 , \\\\ & \\nabla \\cdot v = 0 , \\end{cases} \\end{align*}"}
-{"id": "3938.png", "formula": "\\begin{align*} r a n k ( H H ^ t ) & = r a n k ( K { K } ^ t ) \\\\ & = n - k - m \\\\ & = n - k - \\dim ( H u l l ( C ) ) \\\\ & = n - k - \\dim ( H u l l ( C ^ { \\bot } ) ) , \\end{align*}"}
-{"id": "225.png", "formula": "\\begin{align*} F _ { n , x } ^ - ( u ) & : = \\sum _ { j = k } ^ { n - 2 } \\binom { n - 2 } { j } p _ { n , x , u } ^ j ( 1 - p _ { n , x , u } ) ^ { n - 2 - j } , \\\\ \\tilde { F } _ { n , x } ( u ) & : = \\sum _ { j = k - 1 } ^ { n - 2 } \\binom { n - 2 } { j } p _ { n , x , u } ^ j ( 1 - p _ { n , x , u } ) ^ { n - 2 - j } , \\end{align*}"}
-{"id": "8493.png", "formula": "\\begin{align*} \\Lambda _ h \\psi ( x ) = \\sum _ { s \\in S } h _ s \\psi ( x . s ) \\ \\ \\ ( x \\in G ) \\end{align*}"}
-{"id": "7811.png", "formula": "\\begin{align*} G = \\left \\vert \\overset { \\circ } { \\mathrm { R m } _ { \\Sigma } } \\right \\vert ^ { 2 } S ^ { - 2 } + O \\left ( f ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "5387.png", "formula": "\\begin{align*} R _ 3 ( P _ n ) = \\begin{cases} 2 n - 2 & n \\ , , \\\\ 2 n - 1 & n \\ , . \\end{cases} \\end{align*}"}
-{"id": "6876.png", "formula": "\\begin{align*} \\tilde u ( t ) : = M ( - t ) u ( t ) . \\end{align*}"}
-{"id": "8049.png", "formula": "\\begin{align*} Z _ { \\overrightarrow { \\Gamma _ { 1 } } \\cdot \\overrightarrow { \\Gamma _ { 2 } } } = Z _ { \\overrightarrow { \\Gamma _ { 1 } } } \\cdot Z _ { \\overrightarrow { \\Gamma _ { 2 } } } . \\end{align*}"}
-{"id": "315.png", "formula": "\\begin{align*} \\mathrm { c a r d } \\bigl ( \\mathcal { P } _ i ( J ) \\bigr ) = \\frac { 1 } { i ! } \\sum _ { \\ell = 0 } ^ i ( - 1 ) ^ { i - \\ell } \\binom { i } { \\ell } \\ell ^ { \\mathrm { c a r d } ( J ) } = : S \\bigl ( \\mathrm { c a r d } ( J ) , i \\bigr ) , \\end{align*}"}
-{"id": "7839.png", "formula": "\\begin{align*} \\overline { \\mathcal N } ( \\lambda ) = \\bigsqcup _ { \\kappa \\le \\lambda } \\mathcal N ( \\kappa ) \\end{align*}"}
-{"id": "4513.png", "formula": "\\begin{align*} \\sup _ { x \\in \\mathbb { R } ^ d } \\max _ { r = 1 , \\ldots , m } \\frac { \\| f ^ { ( r ) } ( x ) \\| } { f ( x ) } \\leq d ^ { m / 2 } \\sup _ { x \\in \\mathbb { R } ^ d } \\max _ { r = 1 , \\ldots , m } \\frac { q _ r ( \\| x \\| ) } { ( 1 + \\| x \\| ^ 2 / \\rho ) ^ r } = : A _ { d , m , \\rho } ^ { ( 2 ) } , \\end{align*}"}
-{"id": "3667.png", "formula": "\\begin{align*} \\zeta = \\hat { k } \\cdot \\nabla \\times \\vec { v } = \\frac { \\partial v } { \\partial x } - \\frac { \\partial u } { \\partial y } ~ , \\end{align*}"}
-{"id": "525.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = I ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) + R ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) \\end{align*}"}
-{"id": "5445.png", "formula": "\\begin{align*} K _ 0 = \\begin{bmatrix} q & 1 & - 1 & 0 & 0 \\\\ 0 & 0 & q & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & J ( 3 r , 2 r ) \\\\ 0 & 0 & 0 & J ( r , 2 r ) & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "2764.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( X _ { t } ) \\circ d X ( t ) : = \\lim _ { n \\rightarrow \\infty } \\sum _ { i = 1 } ^ { n } F ( X _ { \\frac { t _ { i - 1 } + t _ { i } } { 2 } } ) ( X ( t _ { i } ) - X ( t _ { i - 1 } ) ) \\end{align*}"}
-{"id": "5822.png", "formula": "\\begin{gather*} b ^ - = \\big ( \\prod _ { j = n + 1 } ^ { K } F _ { i _ { j } } H _ { \\omega ^ { \\vee } _ { i _ { j } } } ( X _ { j } ^ { - 1 } ) \\big ) F _ n \\dots F _ 3 F _ 2 F _ 1 \\\\ = F _ { i _ { n + 1 } } H _ { \\omega ^ { \\vee } _ { i _ { n + 1 } } } ( X _ { n + 1 } ^ { - 1 } ) F _ { i _ { n + 2 } } H _ { \\omega ^ { \\vee } _ { i _ { n + 2 } } } ( X _ { n + 2 } ^ { - 1 } ) \\dots \\\\ F _ { i _ { K - 1 } } H _ { \\omega ^ { \\vee } _ { i _ { K - 1 } } } ( X _ { K - 1 } ^ { - 1 } ) F _ { i _ K } H _ { \\omega ^ { \\vee } _ { i _ K } } ( X _ { K } ^ { - 1 } ) F _ n \\dots F _ 3 F _ 2 F _ 1 . \\end{gather*}"}
-{"id": "6545.png", "formula": "\\begin{align*} J _ m : = \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = k } ^ m \\big \\| _ { L ^ p ( D ) } , \\end{align*}"}
-{"id": "8446.png", "formula": "\\begin{align*} A _ { 1 } ( \\textbf { u } ) = \\left ( \\begin{array} { c c c } v _ { 1 } & 0 & 0 \\\\ f ( \\textbf { u } ) & v _ { 1 } & 0 \\\\ 0 & 0 & v _ { 1 } \\end{array} \\right ) , \\ A _ { 2 } ( \\textbf { u } ) = \\left ( \\begin{array} { c c c } v _ { 2 } & 0 & 0 \\\\ 0 & v _ { 2 } & 0 \\\\ f ( \\textbf { u } ) & 0 & v _ { 2 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "665.png", "formula": "\\begin{align*} \\int _ G \\nu ( g B \\cap C ) \\ , d \\eta ( g ) = \\nu ( B ) \\ , \\nu ( C ) . \\end{align*}"}
-{"id": "1080.png", "formula": "\\begin{align*} ( 1 - \\delta ) \\binom { N } { 2 } = ( 1 - \\delta ) ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } \\leq k ( k - \\alpha - x ) \\frac { n ^ 2 } { 2 } \\end{align*}"}
-{"id": "5002.png", "formula": "\\begin{align*} & \\vec { { E ' } } = \\left ( \\begin{array} { c } E _ { 1 } ' \\\\ E _ { 2 } ' \\\\ \\vdots \\\\ E _ { s } ' \\end{array} \\right ) = { p } _ { * } \\vec { F } - { } ^ { \\rm t } B \\vec { D } . \\end{align*}"}
-{"id": "5706.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm B M O } _ X } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { \\| \\chi _ Q \\| _ X } \\| ( b - b _ Q ) \\chi _ Q \\| _ X . \\end{align*}"}
-{"id": "574.png", "formula": "\\begin{align*} a \\equiv b \\bmod I ^ n \\Rightarrow \\delta ( a ) \\equiv \\delta ( b ) \\bmod I ^ { n - 1 } \\end{align*}"}
-{"id": "5728.png", "formula": "\\begin{align*} f - m _ f ( Q _ 0 ) = g _ 1 + \\sum _ { j \\in J _ 1 } \\alpha _ { j , 1 } \\chi _ { Q ^ 1 _ j } + \\sum _ { j \\in J _ 1 } ( f - m _ f ( Q ^ 1 _ j ) ) \\chi _ { Q ^ 1 _ j } , \\end{align*}"}
-{"id": "2770.png", "formula": "\\begin{align*} \\begin{cases} d Y ( t ) = - f ( B ^ { H } _ { t } , Y ( t ) , Z ( t ) ) d t - Z ( t ) \\diamond d B ^ { H } ( t ) , \\ \\ 0 \\leq t \\leq T , \\\\ Y ( T ) = g ( B ^ { H } _ { T } ) . \\end{cases} \\end{align*}"}
-{"id": "8821.png", "formula": "\\begin{align*} A ^ { g } _ { \\infty } = \\C [ x _ { 1 , n } , x _ { 2 , n } , \\cdots , x _ { k , n } ] _ { } / ( P ^ { g } _ { 1 , n } , \\cdots , P ^ { g } _ { k , n } ) \\ ; \\ ; \\ ; n \\in \\frac { \\alpha _ i } { m } + \\mathbb { Z } , n \\leq 0 . \\end{align*}"}
-{"id": "4813.png", "formula": "\\begin{align*} \\frac { \\partial x _ 1 ( t ; z ) } { \\partial z } \\bigg | _ { z = z ^ * , t = t ^ * } = 0 . \\end{align*}"}
-{"id": "3551.png", "formula": "\\begin{align*} ( 2 \\psi ^ R , V ^ R ) = - \\Phi ( \\gamma ^ R , \\tau ^ R ) + \\chi \\Phi ( g _ 1 ^ R , \\pi _ 1 ^ R ) + ( 1 - \\chi ) \\Phi ( g _ 2 ^ R , \\pi _ 2 ^ R ) + ( 2 \\psi _ 0 R ^ { - 1 - q _ 0 } , 0 ) \\end{align*}"}
-{"id": "9695.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { x ( t ) } { G ^ { - 1 } ( t ) } = 1 . \\end{align*}"}
-{"id": "1800.png", "formula": "\\begin{align*} P ^ { \\alpha } ( t , x ) = \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i x \\cdot \\xi } e ^ { - t | \\xi | ^ \\alpha } d \\xi . \\end{align*}"}
-{"id": "9226.png", "formula": "\\begin{align*} \\delta ( t , x , z ) = \\frac { 1 } { 2 K } d i s t ( u ( t , x , z ) , \\partial \\mathbb { U } ) \\wedge 1 > 0 \\end{align*}"}
-{"id": "2689.png", "formula": "\\begin{align*} \\theta _ { \\varphi _ { \\beta } } ^ n = e ^ { \\beta \\varphi _ { \\beta } } \\theta _ + ^ n . \\end{align*}"}
-{"id": "3957.png", "formula": "\\begin{align*} u ( r ) = \\sigma ( r ) u ( - r ) u ^ { - } ( r ^ { - 1 } ) , \\ , \\forall r \\neq 0 . \\end{align*}"}
-{"id": "7336.png", "formula": "\\begin{align*} \\begin{gathered} w _ { - 1 } \\wedge w _ { - 1 } = 0 , w _ { 0 } \\wedge w _ { - 1 } = - q ^ { 2 } w _ { - 1 } \\wedge w _ { 0 } , w _ { 0 } \\wedge w _ { 0 } = - q ( q - q ^ { - 1 } ) w _ { - 1 } \\wedge w _ { 1 } , \\\\ w _ { 1 } \\wedge w _ { - 1 } = - w _ { - 1 } \\wedge w _ { 1 } , w _ { 1 } \\wedge w _ { 0 } = - q ^ { 2 } w _ { 0 } \\wedge w _ { 1 } , w _ { 1 } \\wedge w _ { 1 } = 0 . \\end{gathered} \\end{align*}"}
-{"id": "4827.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\| ( I - A _ n ) K \\| | = 0 . \\end{align*}"}
-{"id": "6682.png", "formula": "\\begin{align*} \\mathcal { L } _ { X } I _ 1 = \\dots = \\mathcal { L } _ { X } I _ k = 0 , \\mathcal { L } _ { X } D _ 1 = ( - \\lambda ) ( D _ 1 - d _ 1 ) , \\dots , \\mathcal { L } _ { X } D _ p = ( - \\lambda ) ( D _ p - d _ p ) , \\end{align*}"}
-{"id": "8358.png", "formula": "\\begin{align*} \\sigma ^ { \\tilde { \\phi } } _ t ( x z _ g g ) = \\sigma ^ \\phi _ t ( x ) z _ g \\alpha _ g ( u _ { g , t } ) g \\in I _ g g , \\ \\ \\ x \\in M , \\ \\ \\ g \\in G , \\ \\ \\ t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "5998.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\frac { \\ln \\P \\{ \\max _ { 1 \\le n \\le N _ j } | S _ n | \\le f _ { N _ j } \\} } { f _ { N _ j } ^ { - 1 / H } N _ j } = 0 . \\end{align*}"}
-{"id": "9733.png", "formula": "\\begin{align*} f _ n ( \\lambda _ n ) & = a \\lambda _ n ^ \\beta - \\lambda _ n - b \\lambda _ n ( 1 - q ) ^ { - \\beta / ( \\beta - 1 ) } = \\lambda _ n \\left \\{ \\left ( a - b ( 1 - q ) ^ { - \\beta / ( \\beta - 1 ) } \\right ) \\lambda _ n ^ { \\beta - 1 } - 1 \\right \\} \\\\ & = \\lambda _ n ( ( \\lambda _ n / \\Lambda ) ^ { \\beta - 1 } - 1 ) < 0 . \\end{align*}"}
-{"id": "7682.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\left \\{ f ( x _ n ) + \\sigma _ n \\xi _ { n + 1 } , 0 \\right \\} , x _ 0 > 0 , n = 0 , 1 , \\dots . \\end{align*}"}
-{"id": "6618.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } z ^ { - n } \\mu ^ \\natural | _ { W ^ \\natural _ { i + n \\deg ( z ) } } = \\mu _ 0 ^ \\natural | _ { W ^ \\natural _ i } . \\end{align*}"}
-{"id": "3443.png", "formula": "\\begin{align*} d Y ( t ) = - \\tilde G ( t , \\kappa ( t ) , Y ( t ) , Z ( t ) ) d t + \\langle Z ( t ) , d W ( t ) ( t ) \\rangle , Y ( T ) = \\zeta = \\langle \\eta ( T ) , u _ 0 ( \\xi ( T ) ) \\rangle , \\end{align*}"}
-{"id": "4873.png", "formula": "\\begin{align*} f _ { \\kappa } ( z ) = f _ { \\kappa } ( \\theta ) + \\frac { \\sigma ^ 3 } { 3 } ( z - \\theta ) ^ 3 + \\mathcal { O } \\big ( ( z - \\theta ) ^ 4 \\big ) . \\end{align*}"}
-{"id": "567.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } | [ ( I - A _ n ) R , \\tau ] | _ { \\mathcal I } = 0 \\end{align*}"}
-{"id": "2208.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\lambda _ { k } k ^ { - \\frac { 2 \\beta } { n } } = C _ { 0 } , \\end{align*}"}
-{"id": "9529.png", "formula": "\\begin{align*} F _ 3 & = \\left ( \\begin{array} { c } H _ 3 \\\\ G _ 3 \\end{array} \\right ) \\\\ \\mathcal { M } & = e ^ \\phi \\left ( \\begin{array} { c c } C _ 0 ^ 2 + e ^ { - 2 \\phi } & - C _ 0 \\\\ - C _ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2125.png", "formula": "\\begin{align*} f : = - \\Delta _ { x ' } u . \\end{align*}"}
-{"id": "3589.png", "formula": "\\begin{align*} \\mathcal { G } ( f , X ) & = \\int _ { \\Omega } \\left [ \\frac { 1 } { 2 } \\rho _ g \\left | ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) \\right | _ g ^ 2 - \\Pi _ g ( \\psi , V ) \\cdot _ g ( f , X ) \\right ] d \\mu _ g . \\end{align*}"}
-{"id": "7200.png", "formula": "\\begin{align*} f ( x ' ) : = u ( x ) \\cdot n ( x ) = 0 , x _ 3 = \\gamma ( x ' ) \\end{align*}"}
-{"id": "3993.png", "formula": "\\begin{align*} \\pi _ { p } ( v _ { i } ) _ { i - 1 } = \\pi _ { p } ( ( v _ { i } ) _ { i - 1 } ) = \\pi _ { p } ( v _ { i - 1 } ) . \\end{align*}"}
-{"id": "2972.png", "formula": "\\begin{align*} \\sum _ { i + j = n } [ f _ i ^ x , \\alpha _ j ] = 0 , \\end{align*}"}
-{"id": "8087.png", "formula": "\\begin{align*} \\frac { A _ { s _ i , r _ j } } { P _ { s _ i } ^ 2 } + \\frac { B _ { s _ i , r _ j } } { P _ { s _ i } P _ { r _ j } } = p _ { t h } ^ { s _ i } . \\end{align*}"}
-{"id": "7277.png", "formula": "\\begin{align*} w _ k ^ \\mu = \\frac { \\sum _ { \\lambda = 1 } ^ r ( S _ { j k } ) _ \\lambda ^ \\mu \\cdot w _ j ^ \\lambda } { 1 + \\sum _ { \\lambda = 1 } ^ r a _ { j k , \\lambda } w _ j ^ \\lambda } = \\sum _ { \\lambda = 1 } ^ r ( S _ { j k } ) _ \\lambda ^ \\mu \\cdot w _ j ^ \\lambda - \\sum _ { \\lambda = 1 } ^ r \\sum _ { \\nu = 1 } ^ r ( S _ { j k } ) _ \\lambda ^ \\mu \\cdot a _ { j k , \\nu } w _ j ^ \\lambda w _ j ^ \\nu + O ( | w _ j | ^ 3 ) , \\end{align*}"}
-{"id": "2488.png", "formula": "\\begin{align*} T ^ w = ( \\C ^ \\times ) ^ { r _ 1 } \\times \\cdots \\times ( \\C ^ \\times ) ^ { r _ m } \\end{align*}"}
-{"id": "3167.png", "formula": "\\begin{align*} C _ s ( \\{ ( W _ t ^ { 0 } , { V } _ t ^ { 0 } ) : t \\in \\theta \\} ) = 0 \\end{align*}"}
-{"id": "4886.png", "formula": "\\begin{align*} \\delta ( x - [ r _ 0 ] ) + [ r _ 1 ] & \\equiv 2 [ \\phi ^ { - 1 } ( r _ 2 ) ] \\bmod I ^ 2 \\\\ ( \\ref { a = b } ) \\Rightarrow \\delta ( \\delta ( x - [ r _ 0 ] ) + [ r _ 1 ] ) & \\equiv [ r _ 2 ] - 2 [ r _ 2 ] \\bmod I \\\\ p = 2 \\in I \\Rightarrow \\delta ( \\delta ( x - [ r _ 0 ] ) + [ r _ 1 ] ) & \\equiv [ r _ 2 ] \\bmod I \\end{align*}"}
-{"id": "6575.png", "formula": "\\begin{align*} X ^ \\star : = \\{ x ~ | ~ \\nabla F ( x ) = 0 \\} . \\end{align*}"}
-{"id": "4293.png", "formula": "\\begin{align*} ( n , A ) ( n , g f ) = n ( n , A , g f ) \\end{align*}"}
-{"id": "7052.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ 1 ] } } ( n ) = { \\bf { V } } _ 1 ^ { [ 1 ] } ( n ) { { \\bf { u } } ^ { [ 1 ] } } + { \\bf { V } } _ 2 ^ { [ 1 ] } ( n ) { { \\bf { v } } ^ { [ 1 ] } } , { { \\bf { X } } ^ { [ 2 ] } } ( n ) = { \\bf { V } } _ 1 ^ { [ 2 ] } ( n ) { { \\bf { u } } ^ { [ 2 ] } } + { \\bf { V } } _ 2 ^ { [ 2 ] } ( n ) { { \\bf { v } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "7615.png", "formula": "\\begin{align*} \\sum _ { z \\in L } \\lambda _ 1 \\mathbf { v } _ z = \\sum _ { z \\in L } \\sum _ { y \\sim z } \\mathbf { v } _ y = \\sum _ { z \\in L } \\left ( \\sum _ { \\substack { y \\sim z \\\\ y \\in S } } \\mathbf { v } _ y + \\sum _ { \\substack { y \\sim z \\\\ y \\in L } } \\mathbf { v } _ y \\right ) \\leq \\epsilon e ( S , L ) + 2 e ( L ) \\leq \\epsilon ( 2 n - 4 ) + \\frac { 1 8 \\sqrt { 2 n - 4 } } { \\epsilon } . \\end{align*}"}
-{"id": "348.png", "formula": "\\begin{align*} { [ } f , g ] : = f \\partial _ \\theta g - f \\partial _ \\theta g \\ ; \\ ; \\ ; \\ ; \\textrm { a n d } \\ ; \\ ; \\ ; ( f , g ) : = \\int _ { - \\pi } ^ { \\pi } \\left ( \\frac { d } { d \\theta } f ( e ^ { i \\theta } ) + \\left ( \\frac { d } { d \\theta } \\right ) ^ 3 f ( e ^ { i \\theta } ) \\right ) g ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } \\end{align*}"}
-{"id": "59.png", "formula": "\\begin{align*} \\frac { d } { d r } \\left ( e ^ r | f ' ( r ) | ^ { p - 2 } f ' ( r ) \\right ) = e ^ r \\left [ | f ' ( r ) | ^ { p - 2 } f ' ( r ) + | f ' ( r ) | ^ { p - 1 } - f ( r ) \\right ] \\ , r > 0 \\ . \\end{align*}"}
-{"id": "4574.png", "formula": "\\begin{align*} \\mathcal { P } _ 2 ( J ) = \\Bigl \\{ \\bigl \\{ \\{ 1 , 1 \\} , \\{ 2 \\} \\bigr \\} , \\bigl \\{ \\{ 1 , 2 \\} , \\{ 1 \\} \\bigr \\} , \\bigl \\{ \\{ 1 , 2 \\} , \\{ 1 \\} \\bigr \\} \\Bigr \\} . \\end{align*}"}
-{"id": "4927.png", "formula": "\\begin{align*} | A x | = | A x _ 0 | + e , \\end{align*}"}
-{"id": "9869.png", "formula": "\\begin{align*} B _ { q , p } ( H _ 1 ) \\leq n \\left ( \\frac { 1 - \\delta } { { 1 } / { n } - \\delta } \\right ) ^ { 1 - \\delta } \\omega _ n ^ { 1 - \\frac { 1 } { n } } \\left ( \\frac { 1 } { ( n + 1 ) ! } \\right ) ^ { \\frac { 1 } { n } - \\delta } , \\ , \\ , \\delta = \\frac { 1 } { p } + \\frac { 1 } { q } \\geq 0 \\end{align*}"}
-{"id": "9368.png", "formula": "\\begin{align*} \\mathcal { D } ( H _ { \\mathbf { T } } ) = \\{ f \\in { W } _ 2 ^ 2 ( \\mathbb { R } \\backslash \\{ 0 \\} ) \\ : \\ ( \\mathbf { T } \\Gamma _ 0 - \\Gamma _ 1 ) f = 0 \\} . \\end{align*}"}
-{"id": "7964.png", "formula": "\\begin{align*} \\mu ( B _ H ( R _ 0 ) ) = 1 , \\end{align*}"}
-{"id": "1351.png", "formula": "\\begin{align*} I _ { L } ( \\hat \\beta _ 0 ) = n ^ { - 1 } \\sum _ { t = 1 } ^ n \\sigma _ { t } ^ 2 ( \\hat \\beta _ 0 ) \\cdot \\textrm { d i a g } \\left ( \\sigma _ { t - j _ 1 } ^ { 2 - 2 \\gamma } ( \\hat \\beta _ 0 ) , \\ldots , \\sigma _ { t - j _ L } ^ { 2 - 2 \\gamma } ( \\hat \\beta _ 0 ) \\right ) . \\end{align*}"}
-{"id": "6751.png", "formula": "\\begin{align*} \\int _ X ( u _ { t } - u _ 0 ) \\theta _ { u _ 0 } ^ n = \\int _ X ( u _ t - u ) \\theta _ { u _ 0 } ^ n \\leq t \\int _ X \\chi \\theta _ { u _ 0 } ^ n . \\end{align*}"}
-{"id": "2085.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + K ) X ^ { ( j ) } ( s _ { i } ) = M \\tilde { V } _ { j } + \\eta _ { j i } \\ i = 1 , \\ldots , l . \\end{align*}"}
-{"id": "7280.png", "formula": "\\begin{align*} I = ( x ^ { 1 2 } , \\ x ^ 6 y ^ 2 z ^ 3 , \\ x ^ 3 y ^ 2 z ^ 7 , \\ x y ^ 7 z ^ 3 , \\ x y ^ 5 z ^ 5 , \\ x y z ^ 9 , \\ y ^ { 1 2 } , \\ z ^ { 1 2 } ) . \\end{align*}"}
-{"id": "8725.png", "formula": "\\begin{align*} \\widetilde Y _ { \\tau } ^ { t , x } = \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } G B ( s , X ^ { t , x } _ s ) \\ , d s + \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } \\widetilde Z _ s ^ { t , x } \\ , B ( s , X ^ { t , x } _ s ) \\ , d s - \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } Z ^ { t , x } _ { s } \\ ; d \\widetilde W _ s , \\end{align*}"}
-{"id": "2352.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\mu _ t + \\frac { \\partial } { \\partial x } ( v [ \\mu _ t ] \\ , \\mu _ t ) = F _ f ( \\mu _ t ) . \\end{align*}"}
-{"id": "6985.png", "formula": "\\begin{align*} \\sum _ { i + j = n } f _ i ^ { \\lambda _ j ( x , y ) } = \\sum _ { i + j = n } [ f _ i ^ x , f _ j ^ y ] . \\end{align*}"}
-{"id": "5116.png", "formula": "\\begin{align*} F _ { \\vec { x } } ( u _ { 1 } , \\ldots , u _ { k } ) = \\langle \\prod _ { 1 \\le i \\le k } \\tilde { C } ^ { [ 0 , M ] } ( u _ { i } ; s ) \\prod _ { 1 \\le i \\le k } \\beta _ { x _ { i } } ^ { * } \\rangle _ { [ 0 , M ] } , \\end{align*}"}
-{"id": "3873.png", "formula": "\\begin{align*} \\nabla g ( x ) = \\nabla ( \\gamma ( r ) ( x - a ) ) + \\nabla ( \\gamma ( \\tilde { r } ) ( i _ x ( \\tilde { x } - a ) ) ) . \\end{align*}"}
-{"id": "6377.png", "formula": "\\begin{align*} \\mathcal { K } ^ + : = \\mathcal { K } \\cap \\{ x _ n \\geq 0 \\} \\mathcal { K } ^ - : = \\mathcal { K } \\cap \\{ x _ n < 0 \\} . \\end{align*}"}
-{"id": "1378.png", "formula": "\\begin{align*} \\nu ( h ) = \\sup _ { t } \\sup _ { A \\in \\emph { F } _ { - \\infty } ^ { t } , B \\in \\emph { F } _ { t + h } ^ \\infty } | P ( A B ) - P ( A ) P ( B ) | \\to 0 \\end{align*}"}
-{"id": "536.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 1 1 } ( i , u ; j , v ) = \\iint _ { \\mathcal { C } ^ 2 } \\frac { ( z - w ) ( z - c ) ( w - c ) } { ( z ^ 2 - 1 ) ( w ^ 2 - 1 ) ( z w - 1 ) } \\frac { ( z - \\sqrt { q } ) ^ { n _ i } ( w - \\sqrt { q } ) ^ { n _ j } } { ( 1 - z \\sqrt { q } ) ^ { m _ i } ( 1 - w \\sqrt { q } ) ^ { m _ j } } \\frac { \\dd z \\dd w } { z ^ { n _ i + 1 } w ^ { n _ j + 1 } z ^ u w ^ v } , \\end{align*}"}
-{"id": "3102.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) = \\left [ \\begin{array} { c | c } \\lambda B + A & L _ { s } ( \\lambda ) ^ { T } \\otimes I _ { n } \\\\ \\hline \\sigma L _ { s } ( \\lambda ) \\otimes I _ { n } & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "288.png", "formula": "\\begin{align*} \\mu _ d \\bigl ( B _ x ( r _ { n , v } ) \\cap B _ y ( r _ { n , v } ) \\bigr ) & = V _ d r _ { n , v } ^ d I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { \\| x - y \\| ^ 2 } { 4 r _ { n , v } ^ 2 } \\biggr ) \\\\ & = \\frac { v e ^ { \\Psi ( k ) } } { n - 1 } I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { \\| z \\| ^ 2 } { 4 \\{ v f ( x ) \\} ^ { 2 / d } } \\biggr ) \\end{align*}"}
-{"id": "1073.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } \\leq \\left ( k - 2 \\alpha + 5 \\alpha ^ 2 \\right ) \\frac { n ^ 2 } { 2 } = \\left ( k - \\frac { 3 } { 1 6 } \\right ) \\frac { n ^ 2 } { 2 } \\ , , \\end{align*}"}
-{"id": "3507.png", "formula": "\\begin{align*} & D { \\Phi } _ { ( g , \\pi ) } ^ W ( h , w ) = D \\Phi | _ { ( g , \\pi ) } ( h , w ) + ( 0 , \\tfrac { 1 } { 2 } h \\cdot _ g ( \\textup { d i v } _ g \\pi + W ) ) . \\end{align*}"}
-{"id": "7284.png", "formula": "\\begin{align*} I = ( x ^ { 7 } , \\ x ^ 5 y z , \\ x y ^ 3 z ^ 3 , \\ y ^ { 7 } , \\ z ^ { 8 } ) . \\end{align*}"}
-{"id": "4981.png", "formula": "\\begin{align*} \\overline { \\alpha } _ { f } ( P ) = \\limsup _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n k } ( P ) ) ^ { 1 / n k } = \\overline { \\alpha } _ { f ^ { k } } ( P ) ^ { 1 / k } \\end{align*}"}
-{"id": "2255.png", "formula": "\\begin{align*} \\gamma _ { q , p } ( 2 , 2 ) = \\beta _ { q , p } \\left ( \\frac { ( 3 p + 1 ) ! ! } { 1 2 } \\varepsilon _ { p } ^ { 3 } \\sigma ^ { 3 p } + \\frac { ( 2 q + 2 p + 1 ) ! ! } { 4 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } \\sigma ^ { 2 q + p } \\right ) , \\end{align*}"}
-{"id": "4952.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ p ( B _ r ( x _ 0 ) ) } = \\| \\Psi _ \\lambda \\ast f \\| _ { L ^ p ( B _ r ( x _ 0 ) ) } \\leq \\| \\Psi _ \\lambda \\| _ { L ^ { t } ( B _ { r + r _ 0 } ( x _ 0 - y _ 0 ) ) } \\| f \\| _ { p ' } \\end{align*}"}
-{"id": "1886.png", "formula": "\\begin{align*} \\frac 3 8 I \\Big | _ { m = \\epsilon _ 0 } = & - 4 z ^ 2 + 8 ( 1 + s \\epsilon _ 0 - 2 \\epsilon _ 0 ) z \\\\ & + [ - 1 + ( 1 - 8 s ) \\epsilon _ 0 + 2 ( 1 + 8 s - 2 s ^ 2 ) \\epsilon _ 0 ^ 2 ] \\ge 0 , \\end{align*}"}
-{"id": "4493.png", "formula": "\\begin{align*} \\Sigma : = \\begin{pmatrix} 1 & \\alpha _ z \\\\ \\alpha _ z & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "7113.png", "formula": "\\begin{align*} \\displaystyle \\lim _ { k \\to \\infty } \\lim _ { n \\to \\infty } \\rho _ { n , k } = D \\end{align*}"}
-{"id": "7034.png", "formula": "\\begin{align*} \\limsup _ { x \\downarrow 0 } \\frac { A ( x ) } { 1 + \\sqrt { U ( x ) \\overline { \\Pi } ^ - ( x ) } } = \\infty . \\end{align*}"}
-{"id": "7773.png", "formula": "\\begin{align*} \\beta _ { 0 , j } ( D ) = \\begin{cases} 1 & j = 0 \\\\ 0 & j \\ne 0 \\end{cases} \\beta _ { 1 , j } ( D ) = \\begin{cases} r & j = e \\\\ 0 & j \\ne e \\end{cases} . \\end{align*}"}
-{"id": "7483.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ { \\kappa ' k } ( j - k ) Q _ j ^ { ( k ) } = 0 . \\end{align*}"}
-{"id": "3389.png", "formula": "\\begin{align*} F _ \\beta ( \\tau ) \\ , \\ , = \\ , \\ , \\Phi _ \\beta \\left ( \\log \\tau \\right ) \\ , \\hbox { a n d } \\tau \\ , \\ , F _ \\beta ' ( \\tau ) \\ , \\ , = \\ , \\ , { \\Phi _ \\beta ' \\left ( \\log \\tau \\right ) } \\ , . \\end{align*}"}
-{"id": "3418.png", "formula": "\\begin{align*} d y ( t ) = - f ( t , y ( t ) , z ( t ) ) d t - z d w ( t ) , y ( T ) = \\zeta , \\end{align*}"}
-{"id": "9720.png", "formula": "\\begin{align*} x _ 2 ( \\epsilon ) = g ( \\delta ( \\epsilon ) / 2 ) e ^ { c _ 2 \\int _ 0 ^ { T _ 4 ( \\epsilon ) } \\frac { 1 } { \\sigma ( s ) } \\ , d s } . \\end{align*}"}
-{"id": "2090.png", "formula": "\\begin{align*} & \\hat { M } = \\tilde { V } ^ { T } M \\tilde { V } , \\ \\hat { D } = \\tilde { V } ^ { T } D \\tilde { V } , \\ \\hat { K } = \\tilde { V } ^ { T } ( K + Z ) \\tilde { V } = \\tilde { \\hat { K } } + \\tilde { V } ^ { T } Z \\tilde { V } , \\\\ & \\hat { F } = \\tilde { V } ^ { T } F , \\ \\hat { C } _ { p } = C _ { p } \\tilde { V } , \\ \\ C _ { v } = C _ { v } \\tilde { V } . \\end{align*}"}
-{"id": "9345.png", "formula": "\\begin{align*} f ( t , x , z ) = - x ^ 2 u ( t , x , z ) - k ( t , x , z ) x y ( t , x , z ) q ( t , x , z ) \\end{align*}"}
-{"id": "6552.png", "formula": "\\begin{align*} \\lambda _ 3 = 1 4 . 4 5 0 8 7 , \\lambda _ 4 = 3 . 4 9 0 4 0 , \\lambda _ 5 = 1 . 6 2 8 9 2 9 7 9 , \\lambda _ 6 = 1 . 0 5 0 5 1 3 . \\end{align*}"}
-{"id": "8171.png", "formula": "\\begin{align*} \\phi _ n = \\sum _ { k = 1 } ^ { n } \\binom { ( \\alpha + \\beta ) n } { k - 1 } \\frac { ( k - 1 ) ! } { n ! } B _ { n , k } ( 1 ! \\theta _ 1 , 2 ! \\theta _ 2 , \\dots ) , \\end{align*}"}
-{"id": "2613.png", "formula": "\\begin{align*} \\delta _ v ( c ) = \\delta _ v ( \\eta ) - \\delta _ v ( l ) = \\alpha _ \\omega \\end{align*}"}
-{"id": "6028.png", "formula": "\\begin{gather*} \\gamma : = \\frac { r ^ 2 - 1 } { r ^ 2 + 1 } \\varphi + \\frac { s ^ 2 + 1 } { s ^ 2 - 1 } \\psi - 3 \\lambda + \\upsilon . \\end{gather*}"}
-{"id": "5900.png", "formula": "\\begin{align*} F ( x , v ) = \\varphi ( v ) G ( x ) \\ , , \\end{align*}"}
-{"id": "620.png", "formula": "\\begin{align*} \\gamma _ N : = \\left \\{ \\begin{array} { l l } 1 , & N = 3 , \\\\ - \\frac { \\pi } { \\Gamma \\left ( \\frac { N - 2 } { 2 } \\right ) } \\left ( \\frac 1 2 y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) ^ { \\frac { N - 2 } { 2 } } Y _ { \\frac { N - 2 } { 2 } } \\left ( y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) , & N \\geq 4 . \\end{array} \\right . \\end{align*}"}
-{"id": "819.png", "formula": "\\begin{align*} E _ { \\vec { z } } ^ { \\vec { \\mu } } ( \\vec { x } , \\vec { \\nu } ) = \\lim _ { \\begin{subarray} { c } M \\to \\infty \\\\ M ' \\to - \\infty \\end{subarray} } \\prod _ { i = 1 } ^ { k } \\frac { z _ { i } ^ { M ' - 1 } } { ( 1 + z _ { i } ) ^ { M } } \\langle \\prod _ { 1 \\le i \\le k } ^ { \\curvearrowright } C _ { \\mu _ { i } } ^ { [ M ' , M ] } ( z _ { i } ) \\prod _ { 1 \\le i \\le k } \\beta _ { \\nu _ { i } , x _ { i } } ^ { * } \\rangle _ { [ M ' , M ] } . \\end{align*}"}
-{"id": "7698.png", "formula": "\\begin{align*} \\sigma : = \\min \\left \\{ \\min _ { x \\in [ \\varepsilon _ 0 , K - \\delta ] } ( f ( x ) - x ) , \\min _ { x \\in [ K + \\delta , 2 K - \\varepsilon _ 0 ] } ( x - f ( x ) ) \\right \\} > 0 \\end{align*}"}
-{"id": "2779.png", "formula": "\\begin{align*} \\lim \\limits _ { j \\to \\infty } \\lambda _ { k ' ( j ) } ( \\gamma ' ) = \\lim \\limits _ { j \\to \\infty } \\lambda _ { k ( j ) } ( \\gamma ' ) = \\mu ( \\gamma ' ) \\end{align*}"}
-{"id": "6910.png", "formula": "\\begin{align*} \\frac { 1 } { ( d - 2 ) ^ { 3 / 2 } } \\biggl ( \\frac { \\hat { d } } { ( d - 2 ) ^ { d / 2 - 1 } } \\biggr ) ^ { n } = \\hat { d } ^ { n } \\left ( \\frac { n } { 2 x } \\right ) ^ { 3 / 2 + x } . \\end{align*}"}
-{"id": "102.png", "formula": "\\begin{align*} E _ n ( x ) = \\sum _ { l = 0 } ^ n { n \\choose l } x ^ { n - l } E _ l . \\end{align*}"}
-{"id": "6102.png", "formula": "\\begin{align*} z w ( 1 + z f ( z ) + w a ( w ) + w z e ( z , w ) ) ( 1 + w g ( w ) + z b ( z ) + w z e ( z , w ) ) = z w . \\end{align*}"}
-{"id": "6933.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V _ \\pi ( z _ 1 ; X ) V _ \\pi ( z _ 2 ; X ) \\cdots V _ \\pi ( z _ m ; X ) \\cdot 1 \\ , ; \\\\ \\cr s ^ { * ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V ^ * _ \\pi ( z _ 1 ; X ) V ^ * _ \\pi ( z _ 2 ; X ) \\cdots V ^ * _ \\pi ( z _ m ; X ) \\cdot 1 \\ , . \\end{align*}"}
-{"id": "708.png", "formula": "\\begin{align*} \\frac { h _ { H } ( f ^ { n } ( P ) ) } { n ^ { r - 1 } \\delta ^ { n } } \\leq C _ { 3 } \\left ( \\sqrt [ ] { h _ { H } ( P ) } + \\sum _ { k = 1 } ^ { n - 1 } \\sqrt [ ] { \\frac { h _ { H } ( f ^ { k } ( P ) ) } { k ^ { r - 1 } \\delta ^ { k } } } + h _ { H } ( P ) \\right ) . \\end{align*}"}
-{"id": "703.png", "formula": "\\begin{align*} \\vec { E } = \\left ( \\begin{array} { c } E _ { 1 } \\\\ E _ { 2 } \\\\ \\vdots \\\\ E _ { r } \\end{array} \\right ) = f ^ { * } \\vec { D } - { } ^ { \\rm t } A \\vec { D } . \\end{align*}"}
-{"id": "2436.png", "formula": "\\begin{align*} \\left \\| \\Box _ k ^ { \\alpha _ 2 } | ~ M _ 1 \\rightarrow M _ 2 \\right \\| \\gtrsim \\frac { \\| \\Box _ k ^ { \\alpha _ 2 } f _ l ^ { \\alpha _ 1 } \\| _ { M _ 2 } } { \\| f _ l ^ { \\alpha _ 1 } \\| _ { M _ 1 } } \\sim \\frac { \\| f _ l ^ { \\alpha _ 1 } \\| _ { L ^ { p _ 2 } } } { \\| f _ l ^ { \\alpha _ 1 } \\| _ { L ^ { p _ 1 } } } \\sim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\\\ = & 2 ^ { j A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\end{align*}"}
-{"id": "1502.png", "formula": "\\begin{align*} \\tilde { \\alpha } = \\frac { ( \\alpha + \\gamma + q - 1 ) \\alpha } { \\alpha + \\gamma } , \\tilde { \\gamma } = \\frac { ( \\alpha + \\gamma + q - 1 ) \\gamma } { \\alpha + \\gamma } . \\end{align*}"}
-{"id": "9041.png", "formula": "\\begin{align*} T _ k ( \\theta ) : = \\sum _ { j = 0 } ^ { k - 1 } ( j + 1 ) \\gamma _ j e ^ { i \\psi _ j ( \\theta ) } \\left ( 1 - \\sqrt { \\frac { E _ j + \\Gamma _ j } { \\beta _ j } } \\right ) . \\end{align*}"}
-{"id": "7041.png", "formula": "\\begin{align*} d _ - ( t ) : = \\inf \\bigl \\{ x > 0 : \\overline { \\Pi } ^ - ( x ) \\le t ^ { - 1 } \\bigr \\} , t > 0 , \\end{align*}"}
-{"id": "1922.png", "formula": "\\begin{align*} & \\frac { d a } { d \\tau } = \\sum _ { i = 1 } ^ m n _ i q _ i ^ 2 \\frac { a ^ 2 ( \\tau ) } { b _ i ^ 2 ( \\tau ) } , \\\\ & \\frac { d b _ i } { d \\tau } = 2 p _ i - q _ i ^ 2 \\frac { a ( \\tau ) } { b _ i ( \\tau ) } , \\ , \\ , \\ , 1 \\leq i \\leq m . \\end{align*}"}
-{"id": "3368.png", "formula": "\\begin{align*} \\mathbb { E } [ E _ b ( T ) ] - \\mathbb { E } [ E _ b ( 0 ) ] = \\sum _ { t = 0 } ^ { T - 1 } \\mathbb { E } [ E _ s ( t ) - \\Delta t P ( t ) ] . \\end{align*}"}
-{"id": "7026.png", "formula": "\\begin{align*} P _ 2 ( t ) \\ , = \\ , \\prod _ { i = 1 } ^ { b _ 2 ( X ) } ( 1 - \\alpha _ i t ) \\end{align*}"}
-{"id": "4218.png", "formula": "\\begin{align*} \\langle G ( y _ k ) , z _ k - x ^ * \\rangle = \\frac { 1 } { 2 \\gamma } ( \\| z _ k - x ^ * \\| ^ 2 - \\| z _ { k + 1 } - x ^ * \\| ^ 2 ) + \\frac { \\gamma } { 2 } \\| G ( y _ k ) \\| ^ 2 . \\end{align*}"}
-{"id": "3423.png", "formula": "\\begin{align*} d y _ l ( t ) = - g _ l ( t , \\xi ( t ) , y ( t ) , z _ l ( t ) ) d t + \\langle z _ l ( t ) , d w ( t ) \\rangle , y ( T ) = \\Gamma ^ * ( s , T ) u _ { 0 } ( \\xi ( T ) ) , \\end{align*}"}
-{"id": "8405.png", "formula": "\\begin{align*} E _ M ( y g ^ { - 1 } ) & = w ^ * \\lim _ \\mu E _ M ( \\phi ( x _ \\mu ) g ^ { - 1 } ) = w ^ * \\lim _ \\mu \\phi ( x _ \\mu ) _ g \\\\ & = w ^ * \\lim _ \\mu \\theta ( E _ M ( x _ \\mu \\sigma ( z _ { g ^ { - 1 } } g ^ { - 1 } ) ) ) = 0 , \\end{align*}"}
-{"id": "3479.png", "formula": "\\begin{align*} \\Delta u ( x ) = \\chi _ { \\{ v > 0 \\} } ( x ' ) + \\chi _ { \\{ v = 0 \\} } ( x ' ) [ \\partial _ \\nu u ( x ' , 0 ) + \\partial _ \\nu u ( x ' , 1 ) ] \\ , \\ , \\ , \\ , \\ , \\ , \\Omega . \\end{align*}"}
-{"id": "4319.png", "formula": "\\begin{align*} \\Vert u \\Vert ^ { \\frac { 2 } { m } } _ y : = \\int _ { X _ y } \\vert u \\vert ^ { \\frac { 2 } { m } } e ^ { - \\varphi _ L } , \\end{align*}"}
-{"id": "3741.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty \\rho ( u ) d u = e ^ \\gamma , \\end{align*}"}
-{"id": "7426.png", "formula": "\\begin{align*} \\left ( e _ { i ( k - 1 ) } e _ { - i ( k - 1 ) } ^ { - 1 } \\dots e _ { i ( 1 ) } e _ { - i ( 1 ) } ^ { - 1 } x _ - \\right ) \\left ( e _ { i ( k ) } e _ { - i ( k ) } ^ { - 1 } \\dots e _ { i ( 1 ) } e _ { - i ( 1 ) } ^ { - 1 } x _ - \\right ) ^ { - 1 } = e _ { - i ( k ) } e _ { i ( k ) } ^ { - 1 } \\in B _ - s _ { i ( k ) } B _ - , \\end{align*}"}
-{"id": "5755.png", "formula": "\\begin{align*} C ( l ) = \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } e _ { t } ^ { P } ( \\hat { \\beta } _ 0 ) e _ { t - l } ^ { P } ( \\hat { \\beta } _ 0 ) , \\end{align*}"}
-{"id": "7286.png", "formula": "\\begin{gather*} K _ i K _ i ^ { - 1 } = K _ i ^ { - 1 } K _ i = 1 , \\ \\ K _ i K _ j = K _ j K _ i , \\\\ K _ i E _ j K _ i ^ { - 1 } = q _ i ^ { a _ { i j } } E _ j , \\ \\ K _ i F _ j K _ i ^ { - 1 } = q _ i ^ { - a _ { i j } } F _ j , \\\\ E _ i F _ j - F _ j E _ i = \\delta _ { i j } \\frac { K _ i - K _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } , \\end{gather*}"}
-{"id": "7904.png", "formula": "\\begin{align*} \\mathrm { S p e c } ( \\Delta _ g ) : 0 = \\lambda _ 0 < \\lambda _ 1 \\le \\lambda _ 2 \\le \\lambda _ 3 \\le \\cdots \\to \\infty . \\end{align*}"}
-{"id": "9804.png", "formula": "\\begin{align*} \\pi ( \\theta | \\mathbf { y } ) = \\frac { f ( \\mathbf { y } | \\boldsymbol { \\theta } ) \\pi ( \\boldsymbol { \\theta } ) n ^ { - p / 2 } | I ( \\boldsymbol { \\theta } ^ { * } ) | ^ { 1 / 2 } } { f ( y | \\boldsymbol { \\theta } ^ { * } ) \\pi ( \\boldsymbol { \\theta } ^ { * } ) ( 2 \\pi ) ^ { p / 2 } } \\{ 1 + O _ { p } ( n ^ { - 1 } ) \\} , \\end{align*}"}
-{"id": "2694.png", "formula": "\\begin{align*} \\theta _ { \\phi } ^ n = e ^ { \\phi } \\theta _ + ^ n . \\end{align*}"}
-{"id": "2652.png", "formula": "\\begin{align*} C ^ d _ n = C ^ { d - 2 } _ { n - 3 } + C ^ { d - ( m + 1 ) } _ { n - 4 } + C ^ { d - m } _ { n - 3 } \\ , , \\end{align*}"}
-{"id": "4900.png", "formula": "\\begin{align*} f _ { k , \\delta } ( z ) : = \\sum _ { \\mathcal { Q } \\in \\mathcal { Q } _ { \\delta } } \\mathcal { Q } ( z , 1 ) ^ { - k } , \\end{align*}"}
-{"id": "2452.png", "formula": "\\begin{align*} \\left \\Vert \\Box _ { k } ^ { \\alpha _ { 1 } } | ~ M _ { 1 } \\rightarrow M _ { 2 } \\right \\Vert \\gtrsim & \\frac { \\Vert \\Box _ { k } ^ { \\alpha _ { 1 } } f _ { k } ^ { \\alpha _ { 1 } } \\Vert _ { M _ { 2 } } } { \\Vert f _ { k } ^ { \\alpha _ { 1 } } \\Vert _ { M _ { 1 } } } \\gtrsim 2 ^ { j \\widetilde { A _ { 2 } } ( \\mathbf { p } , q , \\alpha _ { 1 } , \\alpha _ { 2 } ) } . \\end{align*}"}
-{"id": "5962.png", "formula": "\\begin{align*} f _ n ( t , \\zeta ) & + \\int _ 0 ^ t b ( \\zeta ) \\cdot D f _ n ( s , \\zeta ) \\ , \\dd s + \\int _ 0 ^ t D _ { v } f _ n ( t , \\zeta ) \\circ \\dd W _ s = \\int _ 0 ^ t R _ n ( s , \\zeta ) \\ , \\dd s \\ , , \\\\ R _ n ( s , \\zeta ) & = \\int _ { \\R ^ { 2 d } } \\big ( b ( \\zeta ) - b ( z ) \\big ) \\cdot D _ { z } f ( s , z ) \\ , \\rho _ n ( \\zeta - z ) \\ \\dd z \\ , . \\end{align*}"}
-{"id": "8091.png", "formula": "\\begin{align*} \\frac { A _ { s _ i , r _ j } } { P _ { s _ i } ^ 2 } + \\frac { B _ { s _ i , r _ j } } { P _ { s _ i } P _ { r _ j } } = p _ { t h } ^ { s _ i } . \\end{align*}"}
-{"id": "3937.png", "formula": "\\begin{align*} G = \\left ( \\begin{array} { c c c c } \\gamma _ { 1 } & \\gamma _ { 2 } & \\ldots & \\gamma _ { n } \\\\ \\gamma _ { 1 } w _ { 1 } & \\gamma _ { 2 } w _ { 2 } & \\ldots & \\gamma _ { n } w _ { n } \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ \\gamma _ { 1 } w _ { 1 } ^ { k - 1 } & \\gamma _ { 2 } w _ { 2 } ^ { k - 1 } & \\ldots & \\gamma _ { n } w _ { n } ^ { k - 1 } \\end{array} \\right ) . \\end{align*}"}
-{"id": "9742.png", "formula": "\\begin{align*} x _ { U , \\epsilon } ( t ) = G _ 0 ^ { - 1 } ( \\lambda ( \\epsilon ) t ) , t \\geq T _ 1 ( \\epsilon ) . \\end{align*}"}
-{"id": "2115.png", "formula": "\\begin{align*} A _ { \\sigma l } = ( l + 1 ) ( l + 2 + 2 \\sigma _ n ) p _ { \\sigma , l + 1 } + 2 s ( \\sigma _ n + 1 ) p _ { \\sigma + \\bar { n } , l } + ( \\sigma _ i + 1 ) ( \\sigma _ i + 2 ) p _ { \\sigma + 2 \\bar { \\imath } , l - 1 } + c _ { \\sigma l } ^ { \\mu m } p _ { \\mu m } , \\end{align*}"}
-{"id": "3989.png", "formula": "\\begin{align*} \\pi _ { p } ( v _ { i - 1 } ) = \\pi _ { p } ( ( v _ { i } ) _ { i - 1 } ) = \\pi _ { p } ( v _ { i } ) _ { i - 1 } . \\end{align*}"}
-{"id": "9911.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 0 & - q ^ 3 K & q F \\\\ q ^ { - 3 } K & 0 & - q ^ { - 1 } E \\\\ - q ^ { - 1 } F & q E & \\kappa K \\end{pmatrix} . \\end{align*}"}
-{"id": "1636.png", "formula": "\\begin{align*} \\begin{cases} \\dd X _ t & = V _ t \\dd t , \\dd V _ t = F \\left ( X _ t , V _ t \\right ) \\dd t + \\dd W _ { t } \\\\ X \\left ( 0 \\right ) & = x _ { 0 } , \\ ; \\ ; V \\left ( 0 \\right ) = v _ { 0 } . \\end{cases} \\end{align*}"}
-{"id": "1538.png", "formula": "\\begin{align*} Q ( \\lambda ) ( v ) = \\nabla _ v M ( v ) , \\int _ { \\R ^ d } \\lambda ( v ) \\ , d v = 0 . \\end{align*}"}
-{"id": "6803.png", "formula": "\\begin{align*} F ( x ) = \\frac { 1 } { 2 i \\pi } \\oint _ \\S \\frac { f ( \\xi ) } { \\xi - x } \\dd \\xi \\end{align*}"}
-{"id": "9253.png", "formula": "\\begin{align*} L _ { r , m } \\varphi ( x ) & = \\alpha ( x , r , m ) \\frac { \\partial \\varphi } { \\partial x } ( x ) + \\frac { 1 } { 2 } \\beta ^ 2 ( x , r , m ) \\frac { \\partial ^ { 2 } \\varphi } { \\partial x ^ 2 } \\\\ & + \\int _ { \\mathbb { R } } \\{ \\varphi ( x + \\gamma ( x , r , m , \\zeta ) ) - \\varphi ( x ) - \\nabla \\varphi ( x ) \\gamma ( x , r , m , \\zeta ) \\} \\nu ( d \\zeta ) , \\end{align*}"}
-{"id": "5480.png", "formula": "\\begin{align*} { \\tilde \\mu _ { i j } } \\left ( { { f _ { i j } } \\left ( x \\right ) } \\right ) + d _ { i j } ^ - = 1 , \\ , \\ , \\ , \\ , i = 1 , 2 , . . . m ; j = 1 , 2 , . . . , p _ m \\end{align*}"}
-{"id": "1621.png", "formula": "\\begin{align*} A ^ { i j } \\left ( x ^ { k } \\right ) u _ { , i j } - B ^ { k } ( x ) u _ { k } = 0 , \\end{align*}"}
-{"id": "3474.png", "formula": "\\begin{align*} \\Phi ( f ) = \\int _ \\Omega | \\nabla u _ f | ^ 2 d x , \\end{align*}"}
-{"id": "682.png", "formula": "\\begin{align*} E = \\big \\{ z \\in Z \\ , : \\ , \\phi ( z ) \\geq \\delta \\big \\} \\end{align*}"}
-{"id": "6097.png", "formula": "\\begin{align*} N ^ + ( v ) & : = \\big \\{ w \\in N _ H ( v ) \\colon w > v \\big \\} \\ , , \\\\ N ^ - ( v ) & : = \\big \\{ w \\in N _ H ( v ) \\colon w < v \\big \\} \\ , , \\\\ N ^ { < u } ( v ) & : = \\big \\{ w \\in N _ H ( v ) \\colon w < u \\big \\} \\ , . \\end{align*}"}
-{"id": "6875.png", "formula": "\\begin{align*} F ( u ) = \\mu | u | ^ p u . \\end{align*}"}
-{"id": "2605.png", "formula": "\\begin{align*} T \\circ W ( \\gamma ) ( a \\oplus b ) & = \\gamma ( a ) + \\gamma ( b ) + ( \\gamma ( a \\otimes b ) - \\gamma ( a ) - \\gamma ( b ) ) \\\\ & = \\gamma ( a \\oplus b ) , \\end{align*}"}
-{"id": "702.png", "formula": "\\begin{align*} & E = f ^ { * } H - \\left < A \\vec { c } , \\vec { D } \\right > \\\\ & E _ { i } = f ^ { * } D _ { i } - \\sum _ { k = 1 } ^ { r } a _ { k i } D _ { k } . \\end{align*}"}
-{"id": "8591.png", "formula": "\\begin{align*} \\Pi _ { 2 } ' \\left ( \\widetilde { \\psi } ( d _ { 1 } \\overline { N } ( ^ { \\ast } b ) ( d _ { 2 } n ) ) \\right ) & = - d _ { 1 } \\hat { \\psi } _ { i } \\gamma ' \\xi _ { b e _ { k } } ( d _ { 2 } n ) \\\\ & = - d _ { 1 } \\gamma ' \\hat { \\psi } _ { o } \\xi _ { b e _ { k } } ( d _ { 2 } n ) \\\\ & = - d _ { 1 } \\gamma ' \\xi _ { b e _ { k } } ( d _ { 2 } \\widetilde { \\psi } ( n ) ) \\\\ & = \\Pi _ { 2 } ' \\left ( d _ { 1 } ( ^ { \\ast } b ) d _ { 2 } \\cdot \\psi ( n ) \\right ) \\end{align*}"}
-{"id": "500.png", "formula": "\\begin{align*} f \\in \\mathcal { F } \\mapsto \\mathcal { S } ( f ) : = 2 \\pi \\Re ( a _ 1 ) + \\sum _ { j = 1 } ^ n ( V _ j ^ 2 - A _ j ) \\end{align*}"}
-{"id": "610.png", "formula": "\\begin{align*} K _ { \\kappa , N } \\left ( z , \\mathfrak { z } \\right ) = - \\frac { i ^ { \\kappa } ( \\kappa - 1 ) } { 2 ^ { \\kappa - 1 } \\pi } \\Psi _ { \\kappa , 0 , N } ^ { * } ( \\mathfrak { z } , z ) . \\end{align*}"}
-{"id": "1292.png", "formula": "\\begin{align*} \\max _ { y \\in [ - H , H ) } \\ \\Bigl \\vert \\sum _ { n = N - H } ^ { N + y } e ^ { - n / N } \\Bigl ( R ( n ) - ( 2 \\psi ( n ) - n ) \\Bigr ) \\Bigl ( 1 - \\frac { \\vert n - N \\vert } { H } \\Bigr ) \\Bigr \\vert \\ll N ( \\log N ) ^ 2 \\log ( 2 H ) \\end{align*}"}
-{"id": "548.png", "formula": "\\begin{align*} f _ { \\kappa } ( z ) = - h z + \\log ( 1 + 2 z ) - \\kappa \\log ( 1 - 2 z ) . \\end{align*}"}
-{"id": "8003.png", "formula": "\\begin{align*} J _ { a , p } = \\int _ 0 ^ { \\infty } t ^ { a } e ^ { - \\frac { t ^ p } { p } } d t . \\end{align*}"}
-{"id": "5322.png", "formula": "\\begin{align*} E _ { n , m } = t \\left ( \\lambda _ { \\{ l ; k \\} } ( u ) + \\omega ^ 2 - u ^ 2 - \\eta ^ { - 2 } - u \\eta N - \\frac { \\eta ^ 2 N ^ 2 } { 4 } \\right ) . \\end{align*}"}
-{"id": "8475.png", "formula": "\\begin{align*} = - \\left ( \\begin{array} { c } 0 \\\\ \\frac { ( \\mathbb { I } - \\mathbb { P } ) \\partial _ { t } v ^ { \\varepsilon } } { \\varepsilon } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "2031.png", "formula": "\\begin{align*} \\bar { w } '' = - ( \\bar { \\lambda } _ 2 - \\bar { \\lambda } _ 1 ) \\bar { w } < 0 . \\end{align*}"}
-{"id": "4212.png", "formula": "\\begin{align*} \\varphi ( x ^ { ( t ) } ) - \\min \\varphi = & \\left ( \\varphi ( x ^ { ( t ) } ) - \\varphi ( x ^ { ( t + 1 ) } ) \\right ) + \\left ( \\varphi ( x ^ { ( t + 1 ) } ) - \\min \\varphi \\right ) \\\\ \\geq & \\frac { L _ { \\min } } { 2 } \\| x ^ { ( t ) } - x ^ { ( t + 1 ) } \\| ^ 2 + \\left ( \\varphi ( x ^ { ( t + 1 ) } ) - \\min \\varphi \\right ) \\\\ \\geq & \\left ( \\frac { \\eta ^ 2 L _ { \\min } } { 4 p L ^ 2 + 4 L _ { \\max } ^ 2 } + 1 \\right ) \\left ( \\varphi ( x ^ { ( t + 1 ) } ) - \\min \\varphi \\right ) , \\end{align*}"}
-{"id": "2653.png", "formula": "\\begin{align*} C ^ d _ n = \\sum ^ n _ { k = 1 } C ^ d _ n ( k ) + C ^ d _ n ( \\emptyset ) = C ^ { d - 2 } _ { n - 3 } + C ^ { d - ( m + 1 ) } _ { n - 4 } + C ^ { d - m } _ { n - 3 } \\ , . \\end{align*}"}
-{"id": "5281.png", "formula": "\\begin{align*} \\vert \\int _ { T } ^ { 2 T } Z ( t ) d t \\vert = O ( T ^ { { 3 } / { 4 } + \\delta / 2 } ) = o ( T ) \\ \\ \\delta < 1 / 2 . \\end{align*}"}
-{"id": "1109.png", "formula": "\\begin{align*} \\| h ^ { ( 2 ) } \\| _ 1 & = \\| D ^ * _ { S _ 0 ^ c } h \\| _ 1 - \\| h ^ { ( 1 ) } \\| _ 1 \\leq k \\alpha - \\ell \\cdot \\frac { \\alpha } { t - 1 } = ( k ( t - 1 ) - \\ell ) \\frac { \\alpha } { t - 1 } , \\\\ \\| h ^ { ( 2 ) } \\| _ \\infty & \\leq \\frac { \\alpha } { t - 1 } . \\end{align*}"}
-{"id": "1230.png", "formula": "\\begin{align*} \\varphi ( t _ { n } ) \\int _ { E _ n } v \\le \\varphi ( t _ { n } ) \\int _ { 0 } ^ { m ( E _ n ) } w = 1 / 2 ^ n \\to 0 . \\end{align*}"}
-{"id": "3770.png", "formula": "\\begin{align*} e ^ { I ( \\beta , t ) } - 1 = I ( \\beta , t ) + O \\left ( I ^ 2 \\left ( \\beta , t \\right ) \\right ) \\end{align*}"}
-{"id": "7983.png", "formula": "\\begin{align*} s \\mapsto t = \\frac { \\psi _ 0 ( s ) } { \\psi _ 1 ( s ) } , \\end{align*}"}
-{"id": "6385.png", "formula": "\\begin{align*} 1 = C ( \\varepsilon ) \\left ( 1 - \\varepsilon \\mu _ { ( p ) } ^ { G } + \\frac { \\varepsilon ^ { 2 } } { 2 } \\mu _ { ( 2 p ) } ^ { G } - . . . \\right ) = C ( \\varepsilon ) \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon ) ^ { n } } { n ! } \\mu _ { ( n p ) } ^ { G } . \\end{align*}"}
-{"id": "5060.png", "formula": "\\begin{align*} f ( g ) = \\int _ Z \\phi ( g z ) \\ , d m ( z ) , \\textrm { f o r a l l $ g \\in G $ } . \\end{align*}"}
-{"id": "9065.png", "formula": "\\begin{align*} & \\P \\left ( G _ 0 v ( d , \\delta , 0 ) \\right ) = \\P \\left ( G _ 0 v ( d , 0 , 0 ) \\right ) + \\P \\left ( G _ 0 v ( d , \\delta , 0 ) ; \\ , \\exists j _ 0 \\in [ | r , N | ] , \\ , d W _ { j _ 0 - r } \\leq l _ { j _ 0 } ^ { ( N ) } \\ , d W _ { j _ 0 - r } \\geq u _ { j _ 0 } ^ { ( N ) } \\right ) . \\end{align*}"}
-{"id": "3571.png", "formula": "\\begin{align*} D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) = ( \\psi , V ) , \\end{align*}"}
-{"id": "8271.png", "formula": "\\begin{align*} \\left | r ^ 0 ( u ) - r ^ 0 ( 0 ) + \\frac { 1 } { 2 \\sqrt { 2 \\pi } } u ^ 2 \\right | = \\left | r ^ 0 ( x ) - r ^ 0 ( 0 ) - ( r ^ 0 ) ' ( 0 ) u - \\frac { 1 } { 2 } ( r ^ 0 ) '' ( 0 ) u ^ 2 \\right | \\lesssim | u | ^ 3 \\end{align*}"}
-{"id": "5997.png", "formula": "\\begin{align*} d ( r ) : = r ^ { 1 / H } L ( r ) ^ { 1 / H } . \\end{align*}"}
-{"id": "5152.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t m + \\underbrace { u \\cdot \\nabla m } _ { } + \\underbrace { \\nabla u ^ { T } \\cdot m } _ { } + \\underbrace { m ( u ) } _ { } + \\underbrace { \\rho \\nabla \\bar { \\rho } } _ { } = 0 , \\\\ \\partial _ t \\rho + ( \\rho u ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "2835.png", "formula": "\\begin{align*} T _ k = \\exp \\left ( k ^ { ( 1 + \\varepsilon ^ 2 ) / p } \\right ) , S _ k = T _ k \\exp \\left ( - ( 1 - \\varepsilon ) h _ p ( T _ k ) \\right ) . \\end{align*}"}
-{"id": "7359.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 3 ) } = \\left ( \\begin{array} { c c c } 0 & 0 & q ^ 2 \\end{array} \\right ) ^ T , \\Gamma _ { 0 } ^ { ( 3 ) } = \\left ( \\begin{array} { c c c } 0 & - q ^ 2 & 0 \\end{array} \\right ) ^ T , \\Gamma _ { - } ^ { ( 3 ) } = \\left ( \\begin{array} { c c c } q ^ 4 & 0 & 0 \\end{array} \\right ) ^ T . \\end{align*}"}
-{"id": "8343.png", "formula": "\\begin{align*} l ( \\beta ) = l ( \\sum _ { i = 1 } ^ n c _ i \\mu _ i ) = \\sum _ { i = 1 } ^ n l ( c _ i \\mu _ i ) = \\sum _ { i = 1 } ^ n c _ i l ( \\mu _ i ) = \\sum _ { i = 1 } ^ n c _ i \\gamma _ i = \\sum _ { i = 1 } ^ m c _ i \\gamma _ i . \\end{align*}"}
-{"id": "367.png", "formula": "\\begin{align*} \\mu ( \\bar { d } _ { \\mathcal { E } } ) = 1 - \\cos ^ { \\alpha } \\left ( \\pi \\bar { d } _ { \\mathcal { E } } / 2 \\right ) , \\end{align*}"}
-{"id": "565.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\| K ( I - A _ n ) \\| | = 0 . \\end{align*}"}
-{"id": "4856.png", "formula": "\\begin{align*} I ^ { \\rm e x p } _ { 1 1 } ( i , x ; j , y ) : = \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\dd z \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\dd w \\frac { ( z - w ) e ^ { - x z - y w } } { 4 z w ( z + w ) } \\frac { ( 1 + 2 z ) ^ { n _ i } ( 1 + 2 w ) ^ { n _ j } } { ( 1 - 2 z ) ^ { m _ i } ( 1 - 2 w ) ^ { m _ j } } ( 2 z + 2 \\alpha - 1 ) ( 2 w + 2 \\alpha - 1 ) ; \\end{align*}"}
-{"id": "9039.png", "formula": "\\begin{align*} \\sup _ { k \\geq 0 } \\sup _ { \\theta \\in [ 0 , 2 \\pi ) } \\left | \\sum _ { j = 0 } ^ { k - 1 } \\gamma _ j e ^ { i \\psi _ j ( \\theta ) } + \\sqrt { \\frac { 2 } { \\beta } } Z _ k ( \\theta ) \\right | < \\infty \\ . \\end{align*}"}
-{"id": "4539.png", "formula": "\\begin{align*} | W _ 2 | \\leq \\biggl | W _ { 2 1 } + \\frac { 1 } { n } \\biggr | + 2 | W _ { 2 2 } | + | W _ { 2 3 } | = O \\biggl ( \\frac { k ^ { 1 / 2 } } { n } \\max \\biggl \\{ \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "2398.png", "formula": "\\begin{align*} \\lambda _ { i } ( M ) = \\begin{cases} 0 & { \\hbox { f o r $ i $ s u c h t h a t } } \\lambda _ { i } ( N ) = 1 \\\\ \\alpha _ 1 ( 1 - \\alpha _ 1 ) & { \\hbox { f o r $ i $ s u c h t h a t } } \\lambda _ { i } ( N ) = 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "8041.png", "formula": "\\begin{align*} a _ 1 ^ { x _ 1 } a _ 2 ^ { x _ 2 } \\cdots { } & a _ { m - 1 } ^ { x _ { m - 1 } } a _ m ^ { x _ m } \\cdots a _ { n - 1 } ^ { x _ { n - 1 } } a _ n ^ { x _ n } \\cdot a _ 1 ^ { - 1 } \\\\ & = s _ { 1 , 2 } ^ { x _ { 1 , 2 } } s _ { 2 , 3 } ^ { x _ { 2 3 } } \\cdots s _ { m - 1 , m } ^ { x _ { m - 1 , m } } s _ { 1 , 3 } ^ { x _ { 1 , 3 } } \\cdots s _ { 1 , m - 1 } ^ { x _ { 1 , m - 1 } } s _ { 2 , m } ^ { x _ { 2 , m } } s _ { 1 , m } ^ { x _ { 1 , m } } \\cdot s _ { 1 , 2 } ^ { - 1 } . \\end{align*}"}
-{"id": "7870.png", "formula": "\\begin{align*} \\delta _ f ( t , x ; z ) : = f ( t , x + z ) + f ( t , x - z ) - 2 f ( t , x ) \\ , . \\end{align*}"}
-{"id": "9971.png", "formula": "\\begin{align*} & B _ { t \\ ! + \\ ! 1 \\ ! , k _ { t + 1 } } \\ ! = \\ ! B _ { t , k _ t } \\ ! + \\ ! T _ f ( E _ { k _ t } \\ ! - \\ ! \\alpha _ t \\tilde { p } _ { t } \\ ! - \\ ! ( 1 \\ ! - \\ ! \\alpha _ t ) \\sum _ { i = 1 } ^ 2 | w _ { t , k _ t i } | ^ 2 p _ { t , i } ) , \\\\ & B _ { t \\ ! + \\ ! 1 \\ ! , \\bar { k } _ { t + 1 } } \\ ! = \\ ! B _ { t , \\bar { k } _ t } \\ ! + \\ ! T _ f ( E _ { \\bar { k } _ t } \\ ! - \\ ! ( 1 \\ ! - \\ ! \\alpha _ t ) \\sum _ { i = 1 } ^ 2 | w _ { t , \\bar { k } _ t i } | ^ 2 p _ { t , i } ) , \\end{align*}"}
-{"id": "8770.png", "formula": "\\begin{align*} K _ { 0 } ( x _ 1 ) T ( w ; x _ 1 , . . , x _ n ) = T ( w ; 1 / x _ 1 , . . , x _ n ) K _ { 0 } ( x _ 1 ) . \\end{align*}"}
-{"id": "1718.png", "formula": "\\begin{gather*} \\begin{pmatrix} b & Z & 0 \\\\ X & B & - \\Sigma ^ { - 1 } Z ^ { \\top } \\\\ 0 & - X ^ { \\top } \\Sigma & - b \\end{pmatrix} , \\end{gather*}"}
-{"id": "5905.png", "formula": "\\begin{align*} v _ { t } ^ { \\prime } - F \\left ( x , v _ { t } \\right ) & = A \\beta \\left ( \\beta - 1 \\right ) t ^ { \\beta - 2 } - s i g n \\left ( x _ { t } \\right ) \\left \\vert x _ { t } \\right \\vert ^ { \\alpha } \\\\ & = A \\beta \\left ( \\beta - 1 \\right ) t ^ { \\beta - 2 } - s i g n \\left ( A \\right ) \\left \\vert A \\right \\vert ^ { \\alpha } t ^ { \\alpha \\beta } = 0 \\end{align*}"}
-{"id": "7637.png", "formula": "\\begin{align*} \\sum _ { s \\in \\mathcal { S } _ { i j } } \\lambda _ s \\leq ( q ( p - 1 ) + k ) \\cdot \\frac { 1 } { p ' } = \\frac { p ' - q } { p ' } . \\end{align*}"}
-{"id": "1119.png", "formula": "\\begin{align*} \\mathbf { G } _ \\mathrm { s } [ \\iota ] = \\mathbf { \\hat { G } } _ \\mathrm { s } [ \\iota ] + \\mathcal { E } _ \\mathrm { s } [ \\iota ] \\end{align*}"}
-{"id": "7358.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 2 ) } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ q ^ { - 2 } & 0 & 0 \\\\ 0 & 1 & 0 \\end{array} \\right ) , \\Gamma _ { 0 } ^ { ( 2 ) } = \\left ( \\begin{array} { c c c } - 1 & 0 & 0 \\\\ 0 & - q ( q - q ^ { - 1 } ) & 0 \\\\ 0 & 0 & q ^ { - 2 } \\end{array} \\right ) , \\Gamma _ { - } ^ { ( 2 ) } = \\left ( \\begin{array} { c c c } 0 & - 1 & 0 \\\\ 0 & 0 & - 1 \\\\ 0 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4885.png", "formula": "\\begin{align*} \\alpha _ 2 ( x ) = ( \\pi ( x ) , \\pi ( \\delta ( x ) ) \\end{align*}"}
-{"id": "6519.png", "formula": "\\begin{align*} \\frac 1 \\tau \\sum _ { j = 0 } ^ k \\delta _ j v _ { n - j } + A ( s ) v _ n = f _ n , k \\le n \\le N , \\end{align*}"}
-{"id": "8056.png", "formula": "\\begin{align*} \\rho ( w _ { i j } , c , a _ { i } ) = \\pi ^ { * } ( c D _ { 2 } ) - \\sum _ { i \\ge 3 } a _ { i } B _ { i } = ( c , a _ { i } ) . \\end{align*}"}
-{"id": "9351.png", "formula": "\\begin{align*} & u ( t , x , z ) = k ( t , x ) \\mathbb { E } _ Q [ y ( T , x , z ) U ( x , z ) \\mathbb { E } _ Q [ \\delta _ Z ( z ) | \\mathcal { F } ^ { G } _ T ] . \\end{align*}"}
-{"id": "9617.png", "formula": "\\begin{align*} \\lambda _ { j , j } ( 0 ) = \\lim _ { t \\to 0 } \\lambda _ { j , j } ( t ) = - \\frac { m \\zeta _ { 0 j } - \\dot \\zeta _ { 0 j } } { 4 \\pi } . \\end{align*}"}
-{"id": "6674.png", "formula": "\\begin{align*} \\mathcal { L } _ { X } D _ 1 = ( - \\lambda ) ( D _ 1 - d _ 1 ) , \\dots , \\mathcal { L } _ { X } D _ p = ( - \\lambda ) ( D _ p - d _ p ) . \\end{align*}"}
-{"id": "3329.png", "formula": "\\begin{align*} \\Gamma _ - \\Gamma _ 0 ^ * = t \\Gamma _ 0 ^ * \\Gamma _ - + t ^ \\prime \\Gamma _ + ^ * \\Gamma _ 0 , \\end{align*}"}
-{"id": "8398.png", "formula": "\\begin{align*} E _ M ( x \\alpha _ { g _ 0 ^ { - 1 } } ( z _ { g _ 0 } - p _ { g _ 0 } ) g _ 0 ^ { - 1 } ) = 0 . \\end{align*}"}
-{"id": "3005.png", "formula": "\\begin{align*} \\int _ X \\langle \\pi _ 1 , \\imath _ X ( x ) \\rangle d P = \\left \\langle \\pi _ 1 , \\int _ X \\imath _ X ( x ) d P \\right \\rangle \\ge \\left \\langle \\pi _ 1 , \\psi _ n \\right \\rangle = \\langle \\pi , \\psi _ n \\rangle & > \\langle \\pi , \\omega ( t ) \\rangle \\\\ & \\ge \\langle \\pi _ 1 , \\omega ( t ) \\rangle \\end{align*}"}
-{"id": "4604.png", "formula": "\\begin{align*} g A g ^ t = \\left [ \\begin{array} { c c } x J & 0 \\\\ 0 & A ' \\end{array} \\right ] . \\end{align*}"}
-{"id": "6322.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = & \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ { p _ 2 } } = \\| \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\Box _ l ^ { \\alpha _ 1 } \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ { p _ 2 } } \\\\ \\lesssim & \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| ^ { p _ 2 } _ { L ^ { p _ 2 } } \\right ) ^ { 1 / p _ 2 } \\lesssim \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "1473.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { k } { \\Phi _ { p , q , w } ( 2 ^ k Q ) } \\leq \\frac { C } { \\Phi _ { p , q , w } ( Q ) } ( Q \\in \\mathcal { Q } ) . \\end{align*}"}
-{"id": "3373.png", "formula": "\\begin{align*} P ^ * ( t ) = & \\begin{cases} 0 & \\textrm { f o r } \\ E _ b ( t ) < E _ { b , } ( t ) . \\\\ \\frac { V } { \\Delta t ( V \\zeta _ { \\max } + \\Delta t P _ { \\max } + E _ { \\min } - E _ b ( t ) ) } - \\frac { 1 } { \\gamma ( t ) } & \\textrm { f o r } \\ E _ { b , } ( t ) \\le E _ b ( t ) \\le E _ { b , } ( t ) \\\\ P _ { \\max } & \\textrm { f o r } \\ E _ b ( t ) > E _ { b , } ( t ) \\end{cases} \\end{align*}"}
-{"id": "3772.png", "formula": "\\begin{align*} 0 \\leq a _ 0 < a _ 1 \\cdots < a _ L = 1 \\end{align*}"}
-{"id": "1691.png", "formula": "\\begin{align*} U ( Z _ t ) = & \\ , U ( z ) + \\int _ 0 ^ t D U ( Z _ s ) { R \\ , } \\dd W _ s + \\lambda \\int _ 0 ^ t U ( Z _ s ) \\ , \\dd s - Z _ t + z + \\int _ 0 ^ t A Z _ s \\ , \\dd s + { R \\ , } W _ t \\end{align*}"}
-{"id": "5248.png", "formula": "\\begin{align*} A ( \\delta B ) + ( \\delta A ) B = 0 . \\end{align*}"}
-{"id": "8323.png", "formula": "\\begin{align*} u = \\sum _ { i = 1 } ^ r \\sum _ { j = 1 } ^ { \\beta _ i } \\frac { Q _ j ^ { ( i ) } ( T ) } { P _ i ( T ) ^ j } + R ( T ) , \\end{align*}"}
-{"id": "6894.png", "formula": "\\begin{align*} \\mathsf { P } _ { \\mathrm { S I R } } \\left ( \\lambda \\right ) = \\mathbb { P } \\left ( \\mathrm { S I R } _ { \\mathrm { U } _ { 0 } } > \\tau \\right ) , \\end{align*}"}
-{"id": "2041.png", "formula": "\\begin{align*} \\tfrac { \\partial } { \\partial s } \\eta ( r , s ) = \\gamma ' ( s ) + r J ' ( s ) + \\frac 1 2 r ^ 2 \\nabla _ r \\nabla _ r \\frac { \\dd \\eta } { \\dd s } \\big | _ { r = 0 } + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "6440.png", "formula": "\\begin{align*} & \\mathcal { E } ( \\tilde { u } , - \\psi ^ { 1 + q } \\tilde { u } ^ { - q } ) \\\\ = & \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } ( \\tilde { u } ( s , x ) - \\tilde { u } ( s , y ) ) ( \\psi ^ { 1 + q } ( y ) \\tilde { u } ^ { - q } ( s , y ) - \\psi ^ { 1 + q } ( x ) \\tilde { u } ^ { - q } ( s , x ) ) \\frac { k ( x , y ) } { 2 } d x d y \\\\ \\geq & \\frac { 1 } { 2 ( q - 1 ) } - \\frac { \\vartheta ( q ) } { 2 } , \\end{align*}"}
-{"id": "9067.png", "formula": "\\begin{align*} \\P ( G _ 0 v _ { 2 | | E | | ^ { 1 / 3 } _ 1 } ( d , 2 | | E | | _ 1 ^ { 1 / 3 } , 0 ) ) & = \\P ( G _ 0 v _ { 2 | | E | | ^ { 1 / 3 } _ 1 } ( d , 0 , 0 ) ) + \\mathcal { O } _ { d } ( | | E | | _ 1 ^ { 1 / 3 } N ^ { - 3 / 2 } ( 1 + \\upsilon - 2 | | E | | ^ { 1 / 3 } ) ^ 4 ) \\\\ & = \\P ( G _ 0 v _ { 2 | | E | | ^ { 1 / 3 } _ 1 } ( d , 0 , 0 ) ) + \\mathcal { O } _ { d , \\upsilon } ( | | E | | _ 1 ^ { 1 / 3 } N ^ { - 3 / 2 } ) . \\end{align*}"}
-{"id": "6952.png", "formula": "\\begin{align*} s _ \\lambda & = \\det \\left [ \\begin{array} { c } h ^ { ( j - 1 ) } _ { \\lambda _ i - i + 1 } \\end{array} \\right ] _ { 1 \\leq i , j \\leq \\ell ( \\lambda ) } \\equiv \\det \\left [ \\begin{array} { c } h ^ { ( 0 ) } _ { \\lambda _ i - i + j } \\end{array} \\right ] _ { 1 \\leq i , j \\leq \\ell ( \\lambda ) } \\ , . \\end{align*}"}
-{"id": "9238.png", "formula": "\\begin{align*} j ( u ) - j ( \\hat { u } ) = I _ 1 + I _ 2 , \\end{align*}"}
-{"id": "2246.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\log \\frac { 2 \\pi \\sigma ^ { 2 } } { C ^ { 2 } ( \\varepsilon ) } + \\frac { \\mu _ { ( 2 ) } } { 2 \\sigma ^ { 2 } } + \\varepsilon \\mu _ { ( p ) } . \\end{align*}"}
-{"id": "7153.png", "formula": "\\begin{align*} - \\Delta w + \\nabla q = - ( u \\cdot \\nabla ) w - ( w \\cdot \\nabla ) \\tilde { u } , \\qquad { \\rm d i v } \\ , w = 0 \\mbox { i n } \\ , \\ , \\R ^ n _ + , \\end{align*}"}
-{"id": "5426.png", "formula": "\\begin{align*} \\beta _ 0 = 1 / ( 3 ^ 5 \\cdot 2 ^ 9 ) \\end{align*}"}
-{"id": "5083.png", "formula": "\\begin{align*} \\phi ( 1 ; \\vec { z } ) = 1 , \\phi ( \\sigma _ { i } \\tau ; \\vec { z } ) = Y _ { i } ( z _ { \\tau ^ { - 1 } ( i ) } , z _ { \\tau ^ { - 1 } ( i + 1 ) } ) \\phi ( \\tau ; \\vec { z } ) , \\end{align*}"}
-{"id": "9820.png", "formula": "\\begin{align*} D _ { \\vec { a } } \\ , { } _ \\lambda \\ , ( L _ i ) = \\sum _ { c \\in \\Z } a _ c M _ { i + c } , \\ \\ \\ D _ { \\vec { a } } \\ , { } _ \\lambda \\ , ( M _ i ) = D _ { \\vec { a } } \\ , { } _ \\lambda \\ , ( Y _ i ) = 0 \\ \\ \\ \\mbox { f o r } \\ \\ \\ i \\in \\Z . \\end{align*}"}
-{"id": "2878.png", "formula": "\\begin{align*} H o m _ { d g } ( X , Y ) = H o m ( X _ - , Y _ - ) , \\end{align*}"}
-{"id": "2959.png", "formula": "\\begin{align*} h ( \\sigma , \\epsilon ) > h ( \\sigma , 0 ) = g ( \\sigma , 1 / 2 ) . \\end{align*}"}
-{"id": "3223.png", "formula": "\\begin{align*} ( z _ j , w _ j ) \\mapsto \\left [ e _ { j , 0 } ^ * ( z _ j ) + \\sum _ { \\lambda = 1 } ^ r w _ j ^ \\lambda \\cdot e _ { j , \\lambda } ^ * ( z _ j ) \\right ] \\end{align*}"}
-{"id": "2631.png", "formula": "\\begin{align*} \\Delta ( F , G ) = \\Delta ( h & ( F , G ) ; F , G ) = \\\\ = \\mathop { \\min } \\limits _ { h \\in L _ { 2 } ^ { - } ( F + G ) } \\Delta ( h ; F , G ) & = \\mathop { \\min } \\limits _ { \\hat { A } { \\zeta } } E | A { \\zeta } - \\hat { A } { \\zeta } | ^ { 2 } . \\end{align*}"}
-{"id": "1182.png", "formula": "\\begin{align*} D _ x V = \\begin{bmatrix} 0 & 1 \\\\ - z \\rho & 0 \\end{bmatrix} V D _ t V = \\begin{bmatrix} - \\frac 1 2 D _ x ( b ) + \\beta & b \\\\ - \\frac 1 2 D ^ 2 _ x ( b ) - z \\rho b & \\frac 1 2 D _ x ( b ) + \\beta \\end{bmatrix} V . \\end{align*}"}
-{"id": "9286.png", "formula": "\\begin{align*} \\begin{cases} d R ( t , Z ) = X ( t , Z ) d t + d w ( t ) ; 0 \\leq t \\leq T , \\\\ R ( 0 ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "5273.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { R } } \\chi ( s ) ^ { - 1 / 2 } \\zeta ( s ) d s = 0 . \\end{align*}"}
-{"id": "2629.png", "formula": "\\begin{align*} G _ m ( \\lambda ) = \\psi _ m ( \\lambda ) ( \\psi _ m ( \\lambda ) ) ^ * , \\ ; \\psi _ m ( \\lambda ) = \\sum _ { u = 0 } ^ \\infty \\psi _ m ( u ) e ^ { - i u \\lambda } , \\end{align*}"}
-{"id": "8942.png", "formula": "\\begin{align*} \\lim _ { s \\searrow 0 } s h ( s ) ^ p = \\lim _ { s \\nearrow + \\infty } s h ( s ) ^ p = 0 . \\end{align*}"}
-{"id": "263.png", "formula": "\\begin{align*} S _ 5 = \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\log ^ 2 f ( x ) \\ , d x = O \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "983.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\alpha _ i ^ 2 = \\sum _ { j = 1 } ^ m \\beta _ j ^ 2 = 1 . \\end{align*}"}
-{"id": "7166.png", "formula": "\\begin{align*} ( 1 - n ) b _ m + \\sum _ { l < n , ~ l \\neq m } b _ l = 0 . \\end{align*}"}
-{"id": "241.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\sup _ { q \\in ( 0 , 1 ) } \\sup _ { k \\in \\{ k _ 0 ^ * , \\ldots , k _ 1 ^ * \\} } \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\mathbb { P } _ f \\bigl ( I _ { n , q } \\ni H ( f ) \\bigr ) - ( 1 - q ) \\leq \\inf _ { L \\geq 1 } \\frac { 2 } { L ( 2 \\pi ) ^ { 1 / 2 } } = 0 . \\end{align*}"}
-{"id": "3214.png", "formula": "\\begin{align*} \\sum _ { 2 \\leq | \\gamma | < n } \\sum _ { | \\beta | \\geq 1 } A _ \\gamma R ^ { | \\beta | } \\cdot X ^ \\gamma \\cdot \\prod _ { \\lambda = 1 } ^ r \\left ( X ^ \\lambda + A ( X ) \\right ) ^ { \\beta _ \\lambda } = A ( X ) \\cdot \\left ( - 1 + \\prod _ { \\lambda = 1 } ^ r \\frac { 1 } { 1 - R ( X ^ \\lambda + A ( X ) ) } \\right ) . \\end{align*}"}
-{"id": "7408.png", "formula": "\\begin{align*} A _ f ( x ) : = \\Delta ^ { I ( f ) , J ( f ) } ( x ) . \\end{align*}"}
-{"id": "926.png", "formula": "\\begin{align*} & V = V _ { 1 } U _ { 2 } , \\ , X = U _ { 1 } V ^ { T } _ { 1 } , \\\\ & \\| V \\| _ { S _ { \\widehat { p } _ { 2 } } } = \\| U _ { 2 } \\| _ { S _ { q } } \\| V _ { 1 } \\| _ { S _ { p _ { 2 } } } , \\\\ & \\| U _ { 1 } \\| _ { S _ { p _ { 1 } } } \\leq \\| U \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| U _ { 2 } \\| _ { S _ { q } } \\end{align*}"}
-{"id": "7567.png", "formula": "\\begin{align*} x _ 2 = f ( x _ 1 ) + l \\chi _ 2 \\le f ( x _ 1 ) + l \\le f ( x _ 1 ) + x _ 1 - f ( x _ 1 ) - \\Delta _ l ( x _ 1 ) \\le x _ 0 - \\Delta _ l ( x _ 0 ) - \\Delta _ l ( x _ 1 ) \\le x _ 0 - 2 \\Delta _ l ( x _ 0 ) . \\end{align*}"}
-{"id": "739.png", "formula": "\\begin{align*} c _ { 1 } = c _ { 0 } + C _ { 3 } \\sqrt [ ] { c _ { 0 } } \\leq ( 1 + C _ { 3 } ) c _ { 0 } \\leq \\widetilde { C } c _ { 0 } . \\end{align*}"}
-{"id": "3910.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = \\lambda N _ { f } ( u ) + ( 1 - \\lambda ) Q ( N _ { f } ( u ) ) & & \\\\ u ( T ) = u ' ( 0 ) = u ' ( T ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6238.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + ( K + Z ) ) X ^ { ( j ) } ( s _ { i } ) = M \\tilde { V } _ { j } \\ \\ j = 1 , \\ \\ldots , \\ J - 1 \\ \\ i = 1 , \\ldots , l . \\end{align*}"}
-{"id": "2277.png", "formula": "\\begin{align*} \\begin{aligned} \\| B ( t , v ) - B ( t , w ) \\| _ { X } \\le \\tilde \\lambda \\| v - w \\| _ D + C _ { B } ( \\| v \\| _ D + \\| w \\| _ D ) \\| v - w \\| _ { W } . \\end{aligned} \\end{align*}"}
-{"id": "4881.png", "formula": "\\begin{align*} \\delta ( p a ) & \\stackrel { ( \\ref { p r o d } ) } { = } \\delta ( p ) \\phi ( a ) + p ^ p \\delta ( a ) \\stackrel { } { \\equiv } \\delta ( p ) \\phi ( a ) \\bmod I ^ { p - 1 } \\stackrel { ( \\ref { 1 } ) } { \\equiv } \\phi ( a ) \\bmod I ^ { p - 1 } \\end{align*}"}
-{"id": "2648.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { h ( m , n ) } C _ { m } ^ { l } ( F ^ { 0 } ) ( \\lambda ) ( C _ { m } ^ { l } ( F ^ { 0 } ) ( \\lambda ) ) ^ { * } = d _ { m } ^ { 0 } ( \\lambda ) ^ { \\top } ( \\vec { \\beta } \\cdot \\vec { \\beta } ^ { * } + \\Gamma _ { m _ { 1 } } ( \\lambda ) + \\Gamma _ { m _ { 2 } } ( \\lambda ) ) \\overline { d _ { m } ^ { 0 } ( \\lambda ) } . \\end{align*}"}
-{"id": "2080.png", "formula": "\\begin{align*} \\tilde { f } ( x ) = f ( \\tilde { x } ) \\ \\ \\tilde { x } \\ \\ \\ \\frac { | | \\tilde { x } - x | | } { | | x | | } = \\mathcal { O } ( \\epsilon _ { m a c h i n e } ) , \\end{align*}"}
-{"id": "3713.png", "formula": "\\begin{align*} h _ t = \\left ( \\begin{array} { c c } t & 0 \\\\ 0 & t ^ { * - 1 } \\end{array} \\right ) ( t \\in A ^ { \\times } ) , w = \\left ( \\begin{array} { c c } 0 & 1 \\\\ \\varepsilon 1 & 0 \\end{array} \\right ) , u _ s = \\left ( \\begin{array} { c c } 1 & s \\\\ 0 & 1 \\end{array} \\right ) ( s \\in A ^ { \\varepsilon s y m } ) \\end{align*}"}
-{"id": "3546.png", "formula": "\\begin{align*} \\bar { \\mu } - | \\bar { J } | _ { \\bar { g } } & \\ge \\chi ( \\mu _ 1 - | J _ 1 | _ { g _ 1 } ) + ( 1 - \\chi ) ( \\mu _ 2 - | J _ 2 | _ { g _ 2 } ) \\\\ & + \\left ( \\psi _ 0 - 2 C _ 1 ^ 2 \\chi ( 1 - \\chi ) R ^ { - q } \\right ) R ^ { - 1 - q _ 0 } , \\end{align*}"}
-{"id": "858.png", "formula": "\\begin{align*} \\textstyle \\pi _ { \\eta } ' \\colon X ' \\to Y ' , ( B _ { l , 1 } , B _ { l , 2 } , i _ { l } ) _ { 1 \\le l \\le e } \\mapsto ( \\bigoplus _ { l = 1 } ^ { e } B _ { l , 1 } , B _ { 2 } , \\bigoplus _ { l = 1 } ^ { e } i _ { l } , Z _ { 1 } , Z _ { 2 } , . . . , Z _ { e } ) , \\end{align*}"}
-{"id": "3190.png", "formula": "\\begin{align*} T _ { j k } w _ k = w _ j + \\sum _ { | \\alpha | \\geq n + 1 } f _ { k j , \\alpha } ( z _ j ) \\cdot w _ j ^ \\alpha . \\end{align*}"}
-{"id": "3540.png", "formula": "\\begin{align*} g ^ R = R ^ { - 2 } F _ R ^ * g , \\pi ^ R = R ^ { - 1 } F _ R ^ * \\pi . \\end{align*}"}
-{"id": "6442.png", "formula": "\\begin{align*} = \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } \\left ( \\left ( \\frac { \\tilde { u } ( s , x ) } { \\psi ( x ) } \\right ) ^ { 1 - q } - \\left ( \\frac { \\tilde { u } ( s , y ) } { \\psi ( y ) } \\right ) ^ { 1 - q } \\right ) k ( x , y ) d x d y . \\end{align*}"}
-{"id": "7652.png", "formula": "\\begin{align*} \\ \\left \\{ \\begin{aligned} & u _ { t } = \\triangle ^ { \\alpha / 2 } u + f ( u ) , \\\\ & u ( x , 0 ) = u _ { 0 } ( x ) , \\\\ & u | _ { \\partial B _ { R } } = 0 \\end{aligned} \\right . \\end{align*}"}
-{"id": "533.png", "formula": "\\begin{align*} I ^ { \\rm e x p } _ { 1 2 } ( i , x ; j , y ) : = \\int _ { \\mathcal { C } _ { a _ z } ^ { \\pi / 3 } } \\dd z \\int _ { \\mathcal { C } _ { a _ w } ^ { \\pi / 3 } } \\dd w \\frac { ( z - w ) e ^ { - x z - y w } } { 2 z ( z + w ) } \\frac { ( 1 + 2 z ) ^ { n _ i } } { ( 1 - 2 w ) ^ { n _ j } } \\frac { ( 1 + 2 w ) ^ { m _ j } } { ( 1 - 2 z ) ^ { m _ i } } \\frac { 2 \\alpha - 1 + 2 z } { 2 \\alpha - 1 - 2 w } , \\end{align*}"}
-{"id": "2293.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\sum \\limits ^ { k } _ { i = 0 } \\delta _ i u _ { n - i } ^ \\star + A ( t _ n ) u _ n ^ \\star = \\sum \\limits ^ { k - 1 } _ { i = 0 } \\gamma _ i B ( t _ { n - i - 1 } , u ^ \\star _ { n - i - 1 } ) + d _ n , k \\le n \\le N . \\\\ \\end{align*}"}
-{"id": "7205.png", "formula": "\\begin{align*} W _ n ^ { \\textrm { c a s c } } = \\sum _ { u _ 1 , . . , u _ n \\in [ \\ ! [ 1 , N ] \\ ! ] } A _ { u _ 1 } A _ { u _ 1 u _ 2 } . . . A _ { u _ 1 . . u _ n } . \\end{align*}"}
-{"id": "5643.png", "formula": "\\begin{align*} \\underset { k \\rightarrow \\infty } { \\liminf } \\ J _ { \\P _ n } ^ { \\gamma } ( \\mu ^ k ) & = \\underset { k \\rightarrow \\infty } { \\liminf } \\ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ 2 ^ 2 ( \\mu ^ k , \\nu _ i ) + \\gamma E ( \\mu ^ k ) \\\\ & \\ge \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ 2 ^ 2 ( \\mu , \\nu _ i ) + \\gamma E ( \\mu ) = J _ { \\P _ n } ^ { \\gamma } ( \\mu ) . \\end{align*}"}
-{"id": "542.png", "formula": "\\begin{align*} f ( z ) = \\frac { \\sigma ^ 3 } { 3 } z ^ 3 + \\mathcal { O } ( z ^ 4 ) . \\end{align*}"}
-{"id": "3616.png", "formula": "\\begin{align*} & \\| U ^ k \\| _ { C ^ { t _ k , \\alpha } _ { \\phi , \\varphi _ k } ( B _ { \\phi ( x ) / 2 } ( x ) ) } \\\\ & \\le C \\left ( \\sum _ { j = 1 } ^ { n + 1 } \\| ( L U ) _ j \\| _ { C ^ { - s _ j , \\alpha } _ { \\phi , \\phi ^ { t _ j + s _ j } \\varphi _ j } ( B _ { \\phi ( x ) } ( x ) ) } + \\sum _ { j = 1 } ^ { n + 1 } \\| U ^ j \\| _ { L _ { \\phi ^ { - n } \\varphi _ j ^ 2 } ^ 2 ( B _ { \\phi ( x ) } ( x ) ) } \\right ) , \\end{align*}"}
-{"id": "1382.png", "formula": "\\begin{align*} Q ( \\delta ) = \\sum _ { m = 1 } ^ M \\kappa _ m \\int _ { \\mathbb { R } ^ r } \\sum _ { j = 0 } ^ m \\pi ^ 0 ( j , x ) \\log \\pi ( j , x ) d F ( x ) \\end{align*}"}
-{"id": "6077.png", "formula": "\\begin{align*} & \\int _ 0 ^ 2 u ^ { - 1 / 2 } ( 1 + i \\log u ) ^ { - t } d u \\\\ & = \\frac { ( i / 2 ) ^ N } { ( t - 1 ) ( t - 2 ) \\cdots ( t - N ) } \\int _ 0 ^ 2 u ^ { - 1 / 2 } ( 1 + i \\log u ) ^ { - t + N } d u + O ( e ^ { - c t } ) . \\end{align*}"}
-{"id": "8857.png", "formula": "\\begin{align*} \\upsilon _ k \\geq r _ { k } ^ { \\eta } = \\frac { 1 } { k } \\Big ( 1 + \\frac { 2 } { n } \\Big ) \\sum _ { i = 1 } ^ k \\upsilon _ i - \\Big [ \\Big ( \\frac { 2 } { k n } \\sum _ { i = 1 } ^ k \\upsilon _ i \\Big ) ^ 2 - \\frac { 1 } { k } \\Big ( 1 + \\frac { 4 } { n } \\Big ) \\sum _ { j = 1 } ^ k \\Big ( \\upsilon _ j - \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\upsilon _ i \\Big ) ^ 2 \\Big ] ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "3182.png", "formula": "\\begin{align*} f _ { k j , n + 1 } = \\left ( \\begin{array} { c } f _ { k j , n + 1 } ^ 1 \\\\ f _ { k j , n + 1 } ^ 2 \\\\ \\vdots \\\\ f _ { k j , n + 1 } ^ r \\end{array} \\right ) : = \\sum _ { | \\alpha | = n + 1 } \\left ( \\begin{array} { c } f _ { k j , \\alpha } ^ 1 \\\\ f _ { k j , \\alpha } ^ 2 \\\\ \\vdots \\\\ f _ { k j , \\alpha } ^ r \\end{array} \\right ) \\cdot e _ j ^ \\alpha , \\end{align*}"}
-{"id": "8481.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } - \\psi ' ( t ) ( \\tilde { \\textbf { u } } ^ { \\star } , \\phi ) _ { 0 } ~ d t + \\sum _ { j = 1 } ^ { d } \\int _ { 0 } ^ { T } \\psi ( t ) ( A _ { j } ( \\tilde { \\textbf { u } } ^ { \\star } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } ^ { \\star } , \\phi ) _ { 0 } ~ d t = 0 . \\end{align*}"}
-{"id": "5262.png", "formula": "\\begin{align*} \\operatorname * { c o r e } A = \\operatorname * { i n t } A , \\end{align*}"}
-{"id": "4846.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 1 1 } ( i , u ; j , v ) = \\iint \\frac { ( z - w ) h ^ { \\rm g e o } _ { 1 1 } ( z , w ) } { ( z ^ 2 - 1 ) ( w ^ 2 - 1 ) ( z w - 1 ) } \\frac { z - c } { z } \\frac { w - c } { w } \\frac { \\dd z } { z ^ u } \\frac { \\dd w } { w ^ v } , \\end{align*}"}
-{"id": "9993.png", "formula": "\\begin{align*} \\alpha ' = & \\gamma \\alpha ^ { ( 1 ) } + ( 1 - \\gamma ) \\alpha ^ { ( 1 ) } , \\\\ \\tilde { p } ' = & \\frac { \\gamma \\alpha ^ { ( 1 ) } } { { \\alpha ' } } \\tilde { p } ^ { ( 1 ) } + \\frac { ( 1 - \\gamma ) \\alpha ^ { ( 2 ) } } { { \\alpha ' } } \\tilde { p } ^ { ( 2 ) } , \\\\ p _ i ' = & \\frac { \\gamma ( 1 - \\alpha ^ { ( 1 ) } ) } { { 1 - \\alpha ' } } { p } _ i ^ { ( 1 ) } + \\frac { ( 1 - \\gamma ) ( 1 - \\alpha ^ { ( 2 ) } ) } { { 1 - \\alpha ' } } { p } _ i ^ { ( 2 ) } , i = 1 , 2 , \\end{align*}"}
-{"id": "1669.png", "formula": "\\begin{align*} \\| D _ v { \\psi } \\| _ { W ^ { 1 , p } ( \\R ^ { 2 d } ) } \\le c ( \\lambda ) \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ { p } ( \\R ^ d _ x ) ) } , \\ ; \\ ; \\ ; \\lambda > 0 , \\ ; \\ ; \\ ; \\ , c = c ( \\lambda ) \\to 0 \\ ; \\ ; . \\end{align*}"}
-{"id": "349.png", "formula": "\\begin{align*} U _ { c , h } ( \\gamma ) T _ { c , h } ( f ) U _ { c , h } ( \\gamma ) ^ * = T _ { c , h } ( \\gamma _ * f ) + r ( c , f , \\gamma ) I \\end{align*}"}
-{"id": "2511.png", "formula": "\\begin{align*} X _ { 0 } ^ { \\lambda } = \\left \\| \\bigwedge _ { i = 1 } ^ { p } \\nabla D _ i \\right \\| _ { p } ^ { - 2 } \\cdot \\sum _ { i = 1 } ^ { p } ( - 1 ) ^ { n - i } ( - \\lambda ) ( D _ i - d _ i ) \\Theta _ i , \\end{align*}"}
-{"id": "4367.png", "formula": "\\begin{align*} c = 1 - \\frac { 6 } { m ( m + 1 ) } \\ ; \\ ; \\ ; \\ ; \\ ; \\textrm { a n d } \\ ; \\ ; \\ ; \\ ; \\ ; h = \\frac { ( ( m + 1 ) p - m q ) ^ 2 - 1 } { 4 m ( m + 1 ) } . \\end{align*}"}
-{"id": "6693.png", "formula": "\\begin{align*} f \\ast g = \\sum _ { r = s } ^ \\infty \\nu ^ r C _ r ( f , g ) , \\end{align*}"}
-{"id": "7789.png", "formula": "\\begin{align*} ^ 1 ( T ' , S _ i ) = ^ 1 ( T , S _ i ) - 1 > 0 . \\end{align*}"}
-{"id": "3996.png", "formula": "\\begin{align*} \\{ u ( \\varphi ( s ) ) : s \\in I \\} \\subset P ^ - ( A ) F \\xi ( s _ 0 ) \\cap U ^ + ( A ) = P ^ - ( A ) ( F \\cap H ) \\xi ( s _ 0 ) \\cap U ^ + ( A ) . \\end{align*}"}
-{"id": "9995.png", "formula": "\\begin{align*} \\alpha ' \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\tilde { p } ' | H _ { \\tilde { i } k } | ^ 2 } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\ ! + \\ ! ( \\ ! 1 \\ ! - \\ ! \\alpha ' \\ ! ) \\sum _ { i = 1 } ^ { 2 } \\log _ 2 \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { p _ { i } ' } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\le F _ k ( \\alpha ' ) . \\end{align*}"}
-{"id": "4655.png", "formula": "\\begin{align*} L _ 0 ( I ) & = \\bigsqcup _ { m = 1 } ^ \\infty \\bigsqcup _ { \\Gamma / \\Gamma \\cap N } \\gamma \\cdot ( 0 , 0 , 0 , m ) , \\\\ L _ 0 ( I I ) & = \\bigsqcup _ { m = 1 } ^ \\infty \\bigsqcup _ { n = 0 } ^ { m - 1 } \\bigsqcup _ { \\gamma \\in \\Gamma } \\gamma \\cdot ( 0 , 0 , m , n ) , \\\\ \\hat { L } _ 0 ( I I ) & = \\bigsqcup _ { m = 1 } ^ \\infty \\bigsqcup _ { n = 0 } ^ { 3 m - 1 } \\bigsqcup _ { \\gamma \\in \\Gamma } \\gamma \\cdot ( 0 , 0 , 3 m , n ) . \\end{align*}"}
-{"id": "166.png", "formula": "\\begin{align*} H _ \\nu \\left ( \\zeta | \\zeta _ { - n } ^ { - 1 } ( \\tau ) \\right ) = \\sum _ { A \\in \\zeta _ { - n } ^ { - 1 } ( \\tau ) } \\nu ( A ) H _ \\nu ( \\zeta | A ) . \\end{align*}"}
-{"id": "8894.png", "formula": "\\begin{align*} - \\Delta _ { A _ n } w _ k + w _ k = W _ n [ v _ k ] . \\end{align*}"}
-{"id": "8261.png", "formula": "\\begin{align*} \\frac { 1 } { \\sigma _ { \\lambda _ 0 } ^ { 2 } } = \\partial _ { \\rho } \\varphi _ { V ' } ( \\rho ' ( \\lambda _ 0 ) ) , - \\frac { m _ { 3 , \\lambda _ 0 } } { \\sigma _ { \\lambda _ 0 } ^ 6 } = \\partial _ { \\rho \\rho } \\varphi _ { V ' } ( \\rho ' ( \\lambda _ 0 ) ) , \\end{align*}"}
-{"id": "6210.png", "formula": "\\begin{align*} M \\ddot { x } ( t ) = & - D \\dot { x } ( t ) - K x ( t ) + F u ( t ) , \\\\ y ( t ) = & \\ C _ { p } x ( t ) + C _ { v } \\dot { x } ( t ) , \\end{align*}"}
-{"id": "1149.png", "formula": "\\begin{align*} a = - \\frac { 1 } { 2 } b _ { x } + \\beta , c = - \\frac { 1 } { 2 } b _ { x x } - z \\rho b , d = \\frac { 1 } { 2 } b _ { x } + \\beta , \\end{align*}"}
-{"id": "5419.png", "formula": "\\begin{align*} f ^ { ( n + 1 ) } ( x , \\omega ^ { n + 1 } ) = f ^ { ( 1 ) } ( f ^ { ( n ) } ( x , \\omega ^ n ) , \\omega _ { n + 1 } ) . \\end{align*}"}
-{"id": "6058.png", "formula": "\\begin{align*} \\int _ \\Omega \\int _ { \\Omega _ 1 \\setminus \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\overline { u } _ { i } ( x ) \\phi _ j ( y ) K ( x , y ) d y d x = \\int _ \\Omega \\int _ { \\Omega _ 1 \\setminus \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\underline { u } _ { i } ( x ) \\phi _ j ( y ) K ( x , y ) \\ , d y \\ , d x . \\end{align*}"}
-{"id": "5014.png", "formula": "\\begin{align*} \\beta _ n ( B ) = \\frac { | B \\cap F _ n | } { | F _ n | } , \\textrm { f o r $ B \\subset G $ } . \\end{align*}"}
-{"id": "1756.png", "formula": "\\begin{align*} \\lfloor w _ u \\rfloor = \\lfloor x _ 0 \\rfloor \\xleftarrow { z _ 1 \\gamma _ 1 ^ \\lor } \\cdots \\xleftarrow { z _ r \\gamma _ r ^ \\lor } \\lfloor x _ r \\rfloor = \\lfloor w _ { u + 1 } \\rfloor . \\end{align*}"}
-{"id": "8078.png", "formula": "\\begin{align*} \\tilde { Q } ( x , n \\alpha ) = \\begin{cases} ( x + \\zeta , ( n + 1 ) \\alpha ) & x \\in [ 0 , \\frac { 1 } { 2 } ) , \\\\ ( x + \\zeta , ( n - 1 ) \\alpha ) & x \\in [ \\frac { 1 } { 2 } , 1 ) . \\end{cases} \\end{align*}"}
-{"id": "5841.png", "formula": "\\begin{align*} d x ^ { 2 } = \\frac { 2 } { \\sigma ^ { 2 } \\left ( x \\right ) } d x ^ { 2 } \\end{align*}"}
-{"id": "2096.png", "formula": "\\begin{align*} \\| H ( s ) - \\tilde { H } ( s ) \\| _ { H _ { 2 } } = \\mathcal { O } ( \\| Z \\| ) . \\end{align*}"}
-{"id": "4720.png", "formula": "\\begin{align*} \\alpha : = \\inf \\{ \\pi ^ { \\top } x \\ , | \\ x \\in W \\} \\end{align*}"}
-{"id": "3359.png", "formula": "\\begin{align*} \\begin{aligned} x _ { k + 1 } & = A x _ k + B u _ k , \\\\ y _ k & = C x _ k + D u _ k , \\end{aligned} \\end{align*}"}
-{"id": "9685.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log ( t - \\tau ( t ) ) } { \\log t } = \\beta . \\end{align*}"}
-{"id": "6558.png", "formula": "\\begin{align*} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { - 1 , q } ( \\varOmega ) ) } + \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { 1 , q } ( \\varOmega ) ) } \\le C \\tau ^ k , \\\\ & \\max _ { k \\le n \\le N } \\| e _ n \\| _ { L ^ \\infty ( \\varOmega ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "932.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t u + u \\cdot \\nabla u = F _ 1 ( u , \\gamma ) , & ( t , x ) \\in \\mathbb { R } \\times \\mathbb { R } ^ d , \\\\ \\partial _ t \\gamma + u \\cdot \\nabla \\gamma = F _ 2 ( u , \\gamma ) , & ( t , x ) \\in \\mathbb { R } \\times \\mathbb { R } ^ d , \\\\ u ( 0 , x ) = u _ { 0 } ( x ) , & x \\in \\mathbb { R } ^ d , \\\\ \\gamma ( 0 , x ) = \\gamma _ { 0 } ( x ) , & x \\in \\mathbb { R } ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "1660.png", "formula": "\\begin{align*} P _ t g ( z ) = P _ t g ( x , v ) & = \\int _ { \\R ^ { 2 d } } g ( e ^ { t A } z + y ) N ( 0 , Q _ t ) \\ , \\dd y \\\\ & = \\int _ { \\R ^ { 2 d } } g ( x + t v + y _ 1 , v + y _ 2 ) N ( 0 , Q _ t ) \\ , \\dd y \\ , , \\ ; \\ ; g \\in C _ c ^ { \\infty } ( \\R ^ { 2 d } ) , \\ ; t \\ge 0 , \\end{align*}"}
-{"id": "2595.png", "formula": "\\begin{align*} \\norm { v } _ { S ( [ t _ n , \\infty ) ) } = \\norm { v ^ { [ t _ n ] } } _ { S ( [ 1 , \\infty ) ) } < \\infty , \\end{align*}"}
-{"id": "3210.png", "formula": "\\begin{align*} u _ k ^ \\alpha = \\prod _ { \\lambda = 1 } ^ r \\left ( \\sum _ { \\mu = 1 } ^ r ( T _ { k j } ) ^ \\lambda _ \\mu u _ j ^ \\mu + O ( | u _ j | ^ n ) \\right ) ^ { \\alpha _ \\lambda } = \\sum _ { | \\beta | = | \\alpha | } \\tau _ { k j , \\beta } ^ \\alpha \\cdot u _ j ^ \\beta + O ( | u _ j | ^ { n + 1 } ) \\end{align*}"}
-{"id": "5600.png", "formula": "\\begin{align*} \\textrm { k e r } _ { Q ( L ^ 2 \\times L ^ 2 ) } ( Q U Q ) = \\{ 0 \\} \\textrm { i f a n d o n l y i f } \\int _ { \\R ^ 2 } V _ { 1 1 } ( y ) \\ , d y \\neq 0 . \\end{align*}"}
-{"id": "6242.png", "formula": "\\begin{align*} Z & = - \\eta \\mathbf { X } ^ { T } ( \\mathbf { X } \\mathbf { X } ^ { T } ) ^ { - 1 } . \\end{align*}"}
-{"id": "2069.png", "formula": "\\begin{align*} \\hat { X } ( s ) = \\sum \\limits _ { j = 0 } ^ { \\infty } \\hat { X } ^ { ( j ) } ( s _ { 0 } ) ( s - s _ { 0 } ) ^ { j } . \\end{align*}"}
-{"id": "1421.png", "formula": "\\begin{align*} S ^ { \\alpha } ( z ) = \\sum _ { n \\in \\Z } S _ { n } ^ { \\alpha } z ^ { - n } \\end{align*}"}
-{"id": "2269.png", "formula": "\\begin{align*} T _ { u , r } ^ D : = \\{ v \\in D : \\min _ { 0 \\le t \\le T } \\| v - u ( t ) \\| _ W \\le r \\} , \\end{align*}"}
-{"id": "6725.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( X _ { t } ) \\circ d X ( t ) : = \\lim _ { n \\rightarrow \\infty } \\sum _ { i = 1 } ^ { n } F ( X _ { \\frac { t _ { i - 1 } + t _ { i } } { 2 } } ) ( X ( t _ { i } ) - X ( t _ { i - 1 } ) ) \\end{align*}"}
-{"id": "7558.png", "formula": "\\begin{align*} p _ K : = \\mathbb P \\left \\{ \\omega \\in \\Omega : \\chi _ i ( \\omega ) \\in \\left ( \\frac { b - f ( b ) + \\delta } { l } , 1 \\right ) ~ ~ i = 1 , \\dots , K \\right \\} = p _ 1 ^ K > 0 . \\end{align*}"}
-{"id": "322.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 + e ^ { - 2 t g ^ * ( x ) } } { 2 c ( t ) } f _ { t , g ^ * } ( x ) \\geq \\frac { f _ { t , g ^ * } ( x ) } { 4 } . \\end{align*}"}
-{"id": "6054.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\Delta \\overline { u } _ { i } = \\frac { 1 } { \\varepsilon } \\overline { u } _ { i } \\sum \\limits _ { j \\neq i } H ( \\underline { u } _ { j } ) ( x ) & \\Omega , \\\\ \\Delta \\underline { u } _ { i } = \\frac { 1 } { \\varepsilon } \\underline { u } _ { i } \\sum \\limits _ { j \\neq i } H ( \\overline { u } _ { j } ) ( x ) & \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "6947.png", "formula": "\\begin{align*} L _ \\pi ( Z ) = \\sum _ { k \\ge 0 } \\ , ( - 1 ) ^ k s _ { ( 1 ^ k ) } [ s _ \\pi ( Z ) ] = \\sum _ \\nu \\ , \\ell _ { \\pi \\nu } \\ , s _ \\nu ( Z ) \\quad \\mbox { a n d } M _ \\pi ( Z ) = \\sum _ { k \\ge 0 } \\ , s _ { ( k ) } [ s _ \\pi ( Z ) ] = \\sum _ \\nu \\ , m _ { \\pi \\nu } \\ , s _ \\nu ( Z ) \\ , . \\end{align*}"}
-{"id": "9250.png", "formula": "\\begin{align*} M _ t ( \\omega ) = M _ t ( \\omega , Z ) = \\exp \\Big ( - \\int _ 0 ^ t h ( X ( s , Z ) ) d w ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ t h ^ 2 ( X ( s , Z ) ) d s \\Big ) . \\end{align*}"}
-{"id": "596.png", "formula": "\\begin{align*} \\delta _ { G _ i } : = \\deg _ z P _ { G _ i } ( z , y ) \\leqslant A \\Delta _ G + k ^ i \\delta _ G . \\end{align*}"}
-{"id": "2169.png", "formula": "\\begin{align*} & \\int _ { B _ { 1 } } \\phi \\psi ^ { 1 + q } w ^ { 2 } d x + \\frac { 1 } { 2 } g _ { \\alpha } * ( \\ , \\phi ) \\leq q g _ { \\alpha } * \\int _ { B _ { 1 } } \\psi ^ { 1 + q } w ^ { 2 } g _ { 1 - \\alpha } \\phi d x \\\\ & + \\frac { \\vartheta ( q ) ( q - 1 ) } { 2 } g _ { \\alpha } * ( \\ , \\phi ) + \\int _ { 0 } ^ { s } g _ { \\alpha } ( s - \\sigma ) \\dot { \\phi } ( \\sigma ) \\left ( g _ { 1 - \\alpha } * \\int _ { B _ { 1 } } \\psi ^ { 1 + q } w ^ { 2 } d x \\right ) ( \\sigma ) d \\sigma . \\end{align*}"}
-{"id": "4376.png", "formula": "\\begin{align*} \\tilde { T } _ { \\alpha , \\beta } ( f ) v = T ( f ) v + \\alpha J ( f ) v + \\beta J ( f ' ) v + \\frac { \\alpha ^ 2 + \\beta ^ 2 } { 2 } \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } f \\ ; v \\end{align*}"}
-{"id": "6722.png", "formula": "\\begin{align*} D _ { s } ^ { H } F = \\sum \\limits _ { i = 1 } ^ { n } \\frac { \\partial f } { \\partial x _ { i } } \\left ( \\int _ 0 ^ T \\xi _ { 1 } ( t ) d B ^ { H } ( t ) , . . . , \\int _ 0 ^ T \\xi _ { n } ( t ) d B ^ { H } ( t ) \\right ) \\xi _ { i } ( s ) , \\ \\ s \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6868.png", "formula": "\\begin{align*} e ^ { - i t \\Delta } u ( t ) = t ^ { - \\frac { 1 } { p } } \\psi ( t , \\tfrac { x } { \\sqrt { t } } ) , \\end{align*}"}
-{"id": "2795.png", "formula": "\\begin{align*} F ( T ) = \\sum _ { \\{ j , k \\} \\in E ( T ) } \\phi ( j ) \\ , \\phi ( k ) \\end{align*}"}
-{"id": "2285.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ n ) _ { n = k } ^ m \\big \\| _ { L ^ p ( D ) } \\\\ & \\le C \\big \\| ( f _ n ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } + C \\big \\| \\big ( ( A _ m - A _ n ) v _ n \\big ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } \\\\ & + C \\Big ( \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "1963.png", "formula": "\\begin{align*} V _ 1 ( V _ 2 V _ 1 ) ^ * \\phi ( t ^ { - 1 } D ) f ( V _ 2 V _ 1 ) V _ 1 ^ * & = V _ 1 V _ 1 ^ * V _ 2 ^ * \\phi ( t ^ { - 1 } D ) V _ 2 f V _ 1 V _ 1 ^ * \\\\ & = V _ 1 V _ 1 ^ * V _ 2 ^ * \\phi ( t ^ { - 1 } D ) V _ 2 f \\\\ & \\varpropto ( - 1 ) ^ { \\partial \\phi \\cdot \\partial f } V _ 1 V _ 1 ^ * f V _ 2 ^ * \\phi ( t ^ { - 1 } D ) V _ 2 \\\\ & = ( - 1 ) ^ { \\partial \\phi \\cdot \\partial f } f V _ 2 ^ * \\phi ( t ^ { - 1 } D ) V _ 2 \\\\ & \\varpropto V _ 2 ^ * \\phi ( t ^ { - 1 } D ) f V _ 2 \\ , . \\end{align*}"}
-{"id": "3620.png", "formula": "\\begin{align*} \\| ( \\phi ( x ) ) ^ { s _ j + t _ k - | \\beta | } \\widetilde { b _ { j k } ^ { \\beta } } \\| _ { C ^ { - s _ j , \\alpha } ( B _ 1 ( 0 ) ) } = \\| b _ { j k } ^ { \\beta } \\| _ { C ^ { - s _ j , \\alpha } _ { \\phi , \\phi ^ { s _ j + t _ k - | \\beta | } } ( B _ { \\phi ( x ) } ( x ) ) } . \\end{align*}"}
-{"id": "8084.png", "formula": "\\begin{align*} p _ { e } = \\frac { A _ { s _ i , r _ j } } { P _ { s _ i } ^ 2 } + \\frac { B _ { s _ i , r _ j } } { P _ { s _ i } P _ { r _ j } } \\end{align*}"}
-{"id": "8441.png", "formula": "\\begin{align*} \\tilde { \\rho } _ { 0 } ( x ) = \\rho _ { 0 } ( x ) - \\bar { \\rho } , ~ ~ ~ ~ ~ \\tilde { v } _ { 0 } ( x ) = v _ { 0 } ( x ) , \\end{align*}"}
-{"id": "4611.png", "formula": "\\begin{align*} \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ p | } - f _ 8 ( p ) = q ^ 3 ( q ^ 2 - 1 ) ( q + \\sqrt { 3 q } + 1 ) \\left ( \\frac { ( q - \\sqrt { 3 q } + 1 ) r } { | G _ { p } | } - k \\right ) , \\end{align*}"}
-{"id": "1337.png", "formula": "\\begin{align*} a ( \\lambda ) = \\prod _ { j = 1 } ^ N \\alpha ( \\lambda , \\xi _ j ) , d ( \\lambda ) = \\prod _ { j = 1 } ^ N \\delta ( \\lambda , \\xi _ j ) . \\end{align*}"}
-{"id": "6480.png", "formula": "\\begin{align*} k ( x , y ) = \\frac { a ( ( x - y ) / | x - y | ) } { | x - y | ^ { n + 2 \\beta } } . \\end{align*}"}
-{"id": "68.png", "formula": "\\begin{align*} \\left ( \\psi ^ { 1 / p } \\right ) ' ( y ) + \\frac { 1 } { p - 1 } & = \\frac { 1 } { p } \\psi ( y ) ^ { - ( p - 1 ) / p } \\psi ' ( y ) + \\frac { 1 } { p - 1 } \\\\ & = \\frac { a ^ { 2 - p } } { p - 1 } ( 1 - y ) \\psi ( y ) ^ { - ( p - 1 ) / p } \\ge 0 \\ . \\end{align*}"}
-{"id": "108.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } f ( x ) d \\mu _ { - 1 } ( x ) = & \\lim _ { N \\rightarrow \\infty } \\sum _ { x = 0 } ^ { p ^ N - 1 } f ( x ) \\mu _ { - 1 } ( x + p ^ N \\mathbb { Z } _ p ) \\\\ = & \\lim _ { N \\rightarrow \\infty } \\sum _ { x = 0 } ^ { p ^ N - 1 } f ( x ) ( - 1 ) ^ x , ( \\textnormal { s e e } \\ , \\ , [ 8 ] ) . \\end{align*}"}
-{"id": "4623.png", "formula": "\\begin{align*} \\mu ( \\bar { d } _ { \\mathcal { E } } ) = { \\bar { d } _ { \\mathcal { E } } } ^ { \\alpha } , \\end{align*}"}
-{"id": "6591.png", "formula": "\\begin{align*} z ^ { k + 1 } = ( 1 - \\alpha ) z ^ k + \\alpha R _ { \\gamma g } R _ { \\gamma f } z ^ k \\end{align*}"}
-{"id": "4280.png", "formula": "\\begin{align*} \\mathbb E [ \\max _ { k = 1 , . . . , n } \\Delta _ k ] = n ^ h \\mathbb E [ \\max _ { k = 1 , . . . , n } \\Delta _ { \\frac k n } ] \\sim n ^ h \\mathbb E [ \\sup _ { [ 0 , 1 ] } \\Delta ] \\ , . \\end{align*}"}
-{"id": "6931.png", "formula": "\\begin{align*} \\{ X ^ { ( 3 ) } ( z ) , X ^ { ( 3 ) } ( w ) \\} = 0 \\ , . \\end{align*}"}
-{"id": "3961.png", "formula": "\\begin{align*} \\lambda ^ { \\max } ( u ^ { - } ( s ) v ) = \\lambda ^ { \\max } ( v ) . \\end{align*}"}
-{"id": "7420.png", "formula": "\\begin{align*} \\sigma ^ * ( A _ { \\sigma ( i ) } ) = A _ i \\sigma ^ * ( X _ { \\sigma ( i ) } ) = X _ i . \\end{align*}"}
-{"id": "4400.png", "formula": "\\begin{align*} H _ { \\mu _ n } ( \\xi | \\xi _ { - \\infty } ^ { - 1 } ( { \\tau } ) ) = h _ { \\mu _ n } ( T ) . \\end{align*}"}
-{"id": "3081.png", "formula": "\\begin{align*} a ( u _ i , v ) = \\lambda _ i b ( u _ i , v ) \\forall v \\in V . \\end{align*}"}
-{"id": "160.png", "formula": "\\begin{align*} | p _ i - q _ i | \\le c q _ i , \\ \\ i = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "4119.png", "formula": "\\begin{align*} \\omega = ( 2 i p \\sqrt { c } y + b ( p + q ) x + a ( p + 2 q ) x ^ 2 ) d x + ( 2 q i \\sqrt { c } x + 2 q x y ) d y \\end{align*}"}
-{"id": "850.png", "formula": "\\begin{align*} g ( z _ { 1 } , w ) \\prod _ { i = 2 } ^ { m + 1 } f ( z _ { i } , w ) + \\sum _ { \\ell = 2 } ^ { m + 1 } g ( z _ { \\ell } , w ) g ( z _ { 1 } , z _ { \\ell } ) \\prod _ { \\begin{subarray} { c } i = 2 \\\\ i \\not = \\ell \\end{subarray} } ^ { m + 1 } f ( z _ { i } , z _ { \\ell } ) = g ( z _ { 1 } , w ) \\prod _ { i = 2 } ^ { m + 1 } f ( z _ { i } , z _ { 1 } ) . \\end{align*}"}
-{"id": "6449.png", "formula": "\\begin{align*} \\| w \\| ^ { 2 } _ { L ^ { 2 \\kappa } ( [ t _ { 2 } - t _ { 1 } , t _ { 0 } - t _ { 1 } ] \\times \\rho ' B _ { 1 } ) } \\leq C ( n , \\Lambda , p , \\alpha , \\eta ) \\int _ { 0 } ^ { t _ { 0 } - t _ { 1 } } F ( s ) d s . \\end{align*}"}
-{"id": "3780.png", "formula": "\\begin{align*} { \\left [ { \\overline { \\bf { y } } } \\right ] _ i } = { a _ i } { \\lambda _ i } \\sum \\limits _ { j = 1 } ^ 4 { { V _ { i j } } { d _ j } } i = 1 , 2 , 3 , 4 \\end{align*}"}
-{"id": "5531.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\varphi _ * \\left ( \\frac { \\delta | \\eta ( k ) | } { v ( k ) } \\right ) v ( k ) \\le p _ { \\varphi _ * , w } ( \\delta \\eta ) + 1 < \\infty . \\end{align*}"}
-{"id": "5976.png", "formula": "\\begin{align*} N ( x ) = 2 b n - b ^ 2 - b > b ( n - 1 ) . \\end{align*}"}
-{"id": "6549.png", "formula": "\\begin{align*} \\begin{aligned} \\| ( e _ n ) _ { n = 0 } ^ m \\| _ { L ^ \\infty ( X ) } & \\le C \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ 1 ( X ) } + C \\delta , \\end{aligned} \\end{align*}"}
-{"id": "5350.png", "formula": "\\begin{align*} \\tilde { C } \\delta ^ { ( g ) } = ( n - \\chi _ { \\gamma } ( g ) ) \\cdot \\delta ^ { ( g ) } \\end{align*}"}
-{"id": "2887.png", "formula": "\\begin{align*} P o i s _ { n \\infty } ^ + = \\Omega ( P o i s _ n ^ * \\{ n \\} ) ^ + = ( \\mathcal { F } ( \\overline { P o i s _ n ^ * \\{ n + 1 \\} } ^ + ) , \\partial ^ + ) \\end{align*}"}
-{"id": "9621.png", "formula": "\\begin{align*} \\tilde U ( \\zeta ) = U ( \\zeta ) , | \\zeta | \\le \\Lambda ( \\Psi _ 0 ) , \\end{align*}"}
-{"id": "4634.png", "formula": "\\begin{align*} ( f \\circ 1 ) ( y ) : = \\bigvee \\limits _ { ( x , w ) \\in { A _ y } } { \\min \\{ f ( x ) , 1 ( w ) \\} = \\bigvee \\limits _ { ( x , w ) \\in { A _ y } } f ( x ) } \\end{align*}"}
-{"id": "9083.png", "formula": "\\begin{align*} \\psi ( x ) = \\int _ { 0 } ^ { \\infty } \\mathrm { s g n } ( x - y ) y ^ { ( \\alpha - 1 ) / 2 } e ^ { - y / 2 } L ^ { ( a ) } _ { N - 2 } ( y ) \\ , \\mathrm { d } y \\ . \\end{align*}"}
-{"id": "115.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 - 4 t ) ^ { \\tfrac { x } { 2 } } d \\mu _ { - 1 } ( x ) = & \\frac { 2 } { 1 + \\sqrt { 1 - 4 t } } = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n + 1 } { 2 n \\choose n } t ^ n \\\\ = & \\sum _ { n = 0 } ^ \\infty C _ n t ^ n . \\end{align*}"}
-{"id": "5440.png", "formula": "\\begin{align*} A ^ 2 + ( \\mu - \\lambda ) A + ( \\mu - k ) I = \\mu J , \\end{align*}"}
-{"id": "6445.png", "formula": "\\begin{align*} \\geq \\frac { c _ { n , \\beta } } { 2 C \\Lambda } \\int _ { \\rho ' B _ { 1 } } \\int _ { \\rho ' B _ { 1 } } \\frac { ( w ( s , x ) - w ( s , y ) ) ^ { 2 } } { | x - y | ^ { n + 2 \\beta } } d x d y . \\end{align*}"}
-{"id": "3676.png", "formula": "\\begin{align*} \\int \\left [ H _ j \\mathcal { A } H _ i - H _ i \\mathcal { A } H _ j \\right ] d A = 0 ~ , \\end{align*}"}
-{"id": "6279.png", "formula": "\\begin{align*} z _ i \\leftarrow f _ i ( z _ i ) , ~ i = 1 , \\dots , m . \\end{align*}"}
-{"id": "6195.png", "formula": "\\begin{align*} \\left | \\frac { f ( x u ) - f ( x ) } { x f ' ( x ) } \\right | = \\alpha ^ { - 1 } \\log | x | \\left [ 1 - \\left ( 1 - \\frac { \\log u ^ { - 1 } } { \\log | x | } \\right ) ^ \\alpha \\right ] \\leq 2 \\log u ^ { - 1 } . \\end{align*}"}
-{"id": "7508.png", "formula": "\\begin{align*} x _ { n + 1 } = f ( x _ n ) - \\alpha _ n ( f ( x _ n ) - x _ n ) = ( 1 - \\alpha _ n ) f ( x _ n ) + \\alpha _ n x _ n , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 . \\end{align*}"}
-{"id": "6757.png", "formula": "\\begin{align*} \\theta _ { \\max ( \\varphi , \\psi - t ) } ^ n \\geq { \\bf 1 } _ { \\{ \\varphi > \\psi - t \\} } \\theta _ { \\max ( \\varphi , \\psi - t ) } ^ n = { \\bf 1 } _ { \\{ \\varphi > \\psi - t \\} } \\theta _ { \\varphi } ^ n = \\theta _ { \\varphi } ^ n , \\end{align*}"}
-{"id": "4597.png", "formula": "\\begin{align*} K ( \\infty ) = \\bigcup _ { n \\in \\N } K ( n ) . \\end{align*}"}
-{"id": "4857.png", "formula": "\\begin{align*} I ^ { \\rm e x p } _ { 1 2 } ( i , x ; j , y ) : = \\int _ { \\mathcal { C } _ { a _ z } ^ { \\pi / 3 } } \\dd z \\int _ { \\mathcal { C } _ { a _ w } ^ { \\pi / 3 } } \\dd w \\frac { ( z - w ) e ^ { - x z - y w } } { 2 z ( z + w ) } \\frac { ( 1 + 2 z ) ^ { n _ i } } { ( 1 - 2 w ) ^ { n _ j } } \\frac { ( 1 + 2 w ) ^ { m _ j } } { ( 1 - 2 z ) ^ { m _ i } } \\frac { 2 \\alpha - 1 + 2 z } { 2 \\alpha - 1 - 2 w } , \\end{align*}"}
-{"id": "9177.png", "formula": "\\begin{align*} \\frac { d ^ { s + 1 } f ( u ( x ) ) } { d x ^ { s + 1 } } = \\sum _ { \\widehat m \\in \\mathcal { P } _ { s + 1 } } a _ { \\widehat m } f ^ { ( | \\widehat m | ) } ( u ( x ) ) D ^ { \\widehat m } u ( x ) , \\end{align*}"}
-{"id": "5041.png", "formula": "\\begin{align*} \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\phi ) \\psi ) \\ , d \\eta ( g ) = \\nu ( \\phi ) \\ , \\nu ( \\psi ) . \\end{align*}"}
-{"id": "7296.png", "formula": "\\begin{align*} E _ k \\triangleright [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q & = ( E _ k \\triangleright E _ \\xi ) E _ { \\xi ^ \\prime } ^ * - q ^ { - ( \\xi + \\alpha _ k , \\xi ^ \\prime ) } E _ { \\xi ^ \\prime } ^ * ( E _ k \\triangleright E _ \\xi ) \\\\ & + q ^ { ( \\xi , \\alpha _ k ) } E _ \\xi ( E _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) - q ^ { - ( \\xi , \\xi ^ \\prime ) } ( E _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) E _ \\xi . \\end{align*}"}
-{"id": "7392.png", "formula": "\\begin{align*} \\mathcal { A } _ L ( [ x , c _ x \\sharp A ] ) & = \\mathcal { A } _ H ( [ x , c _ x \\sharp A ] ) \\\\ & = H ( x ) - \\omega ( A ) \\\\ & = H ( x ) - \\lambda c _ 1 ( A ) , \\end{align*}"}
-{"id": "4880.png", "formula": "\\begin{align*} \\delta ( p ) & = \\delta ( p \\cdot 1 ) = \\delta ( p [ 1 ] ) = [ 1 ] - p ^ { p - 1 } [ 1 ] \\equiv [ 1 ] \\bmod I ^ { p - 1 } \\equiv 1 \\bmod I ^ { p - 1 } \\end{align*}"}
-{"id": "2925.png", "formula": "\\begin{align*} c ^ \\pi _ { ( 1 ^ { i _ 1 } ) ( 1 ^ { i _ 2 } ) \\cdots ( 1 ^ { i _ m } ) } = c ^ { \\pi ' } _ { ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) } \\ , . \\end{align*}"}
-{"id": "5592.png", "formula": "\\begin{align*} V = B ^ * \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) U \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) B = v ^ * U v , \\end{align*}"}
-{"id": "7563.png", "formula": "\\begin{align*} u _ l < a , v _ l > b , F ( u _ l ) = l , F ( v _ l ) = - l . \\end{align*}"}
-{"id": "4667.png", "formula": "\\begin{align*} \\Sigma _ 1 ( f _ t , w _ 1 , w _ 2 ) & = t ^ { - w _ 1 - w _ 2 } \\Sigma _ 1 ( f , w _ 1 , w _ 2 ) . \\end{align*}"}
-{"id": "4006.png", "formula": "\\begin{align*} \\ell = q ^ { r _ d } \\sum \\limits _ { i = 0 } ^ { k - 2 } q ^ { i d } \\Rightarrow \\ell q ^ { d } = \\frac { q ^ { d } ( q ^ { n - { d } } - q ^ { r _ d } ) } { q ^ { d } - 1 } . \\end{align*}"}
-{"id": "2350.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } z ^ { - n } \\mu ^ \\natural | _ { W ^ \\natural _ { i + n \\deg ( z ) } } = \\mu _ 0 ^ \\natural | _ { W ^ \\natural _ i } . \\end{align*}"}
-{"id": "5024.png", "formula": "\\begin{align*} A _ { x _ o } = \\big \\{ g \\in G \\ , : \\ , g \\cdot x _ o \\in A \\big \\} = A , \\end{align*}"}
-{"id": "3409.png", "formula": "\\begin{gather*} \\Psi ( x , y ) \\sim \\rho ( u ) x , ( x = y ^ u ) , \\end{gather*}"}
-{"id": "389.png", "formula": "\\begin{align*} I _ { x _ + } = \\left \\{ I , \\begin{pmatrix} 0 & 1 \\\\ - 1 & - 1 \\end{pmatrix} , \\begin{pmatrix} - 1 & - 1 \\\\ 1 & 0 \\end{pmatrix} \\right \\} . \\end{align*}"}
-{"id": "6061.png", "formula": "\\begin{align*} P ^ { \\alpha } ( t , x ) = \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i x \\cdot \\xi } e ^ { - t | \\xi | ^ \\alpha } d \\xi . \\end{align*}"}
-{"id": "5598.png", "formula": "\\begin{align*} \\mathcal R _ V ^ { \\pm } ( \\lambda ) = \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) - \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) V \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) + \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) V \\mathcal R _ V ^ { \\pm } ( \\lambda ) V \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) . \\end{align*}"}
-{"id": "11.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ m ( 1 - \\delta _ n ) | a _ n | + \\eta _ m \\sum _ { n = 1 } ^ m | a _ n | \\le \\| \\sum _ { n = 1 } ^ m a _ n x _ n \\| \\end{align*}"}
-{"id": "5772.png", "formula": "\\begin{align*} T ( \\lambda | j ) = \\begin{pmatrix} A _ j ( \\lambda ) & B _ j ( \\lambda ) \\\\ C _ j ( \\lambda ) & D _ j ( \\lambda ) \\end{pmatrix} , \\end{align*}"}
-{"id": "9961.png", "formula": "\\begin{align*} & 0 \\le p _ n \\le q _ n , p _ 1 = 0 , q _ 1 = 1 , \\\\ & q _ n p _ { n + 1 } - p _ n q _ { n + 1 } = ( - 1 ) ^ { n + 1 } , \\\\ & | q _ n \\alpha - p _ n | < q _ { n + 1 } ^ { - 1 } , \\\\ & q _ n < q _ { n + 1 } \\le C _ 1 q _ n \\end{align*}"}
-{"id": "5999.png", "formula": "\\begin{align*} P _ t ( x , y ) = \\pi ^ { - 1 / 2 } \\int _ 0 ^ { \\infty } e ^ { - u } T _ { t ^ 2 \\slash ( 4 u ) } ( x , y ) \\frac { d u } { \\sqrt { u } } \\end{align*}"}
-{"id": "9031.png", "formula": "\\begin{align*} \\log \\Phi ^ * _ k ( e ^ { i \\theta } ) = & \\sum _ { j = 0 } ^ { k - 1 } \\log \\left ( 1 - \\gamma _ { j } e ^ { i \\psi _ { j } ( \\theta ) } \\right ) , \\end{align*}"}
-{"id": "3734.png", "formula": "\\begin{align*} \\mathcal { C } _ 0 & = \\left \\{ i \\in [ t ] \\ , \\Big | \\ , \\{ | N ( x ) \\cap C _ i | \\ , | \\ , x \\in D \\} = \\{ 0 \\} \\right \\} , \\\\ \\mathcal { C } _ { \\frac { 1 } { 2 } } & = \\left \\{ i \\in [ t ] \\ , \\Bigg | \\ , \\{ | N ( x ) \\cap C _ i | \\ , | \\ , x \\in D \\} = \\left \\{ \\frac { 1 } { 2 } | C _ i | \\right \\} \\right \\} , \\\\ \\mathcal { C } _ 1 & = \\left \\{ i \\in [ t ] \\ , \\Big | \\ , \\{ | N ( x ) \\cap C _ i | \\ , | \\ , x \\in D \\} = \\{ | C _ i | \\} \\right \\} . \\end{align*}"}
-{"id": "3690.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\left ( \\begin{array} { c } \\zeta \\\\ \\mu \\\\ h \\end{array} \\right ) = { \\boldsymbol A } \\left ( \\begin{array} { c } H _ { \\zeta } \\\\ H _ { \\mu } \\\\ H _ h \\end{array} \\right ) ~ , \\end{align*}"}
-{"id": "3623.png", "formula": "\\begin{align*} u _ { n + 1 } = u _ n \\exp ( r - u _ { n - k + 1 } ) \\end{align*}"}
-{"id": "6013.png", "formula": "\\begin{gather*} \\Phi _ { \\upsilon } : = ( \\cos \\upsilon ) \\Phi _ I + ( \\sin \\upsilon ) \\Phi _ J + \\Phi _ K . \\end{gather*}"}
-{"id": "5483.png", "formula": "\\begin{align*} - \\sum \\limits _ { i = 1 } ^ { n } g ( X _ { i } , Y _ { i } ) & = - \\sum \\limits _ { i = 1 } ^ { n } \\frac { h ' _ i } { 2 } | f ( X _ i ) - f ^ * ( X _ i ) | + \\sum \\limits _ { i = 1 } ^ { n } \\frac { \\xi ^ { ( h ' ) } _ { i } } { 2 } | f ( X _ i ) - f ^ * ( X _ i ) | \\\\ & \\le - ( \\min \\limits _ { i } h ' _ { i } ) \\sum \\limits _ { i = 1 } ^ { n } \\frac { 1 } { 2 } | f ( X _ i ) - f ^ * ( X _ i ) | + \\sum \\limits _ { i = 1 } ^ { n } \\frac { \\xi ^ { ( h ' ) } _ { i } } { 2 } | f ( X _ i ) - f ^ * ( X _ i ) | . \\end{align*}"}
-{"id": "4678.png", "formula": "\\begin{align*} [ P , a ^ 1 ] \\cdots [ P , a ^ { 2 k + 1 } ] = & ( - 1 ) ^ k P a ^ 1 ( 1 - P ) a ^ 2 P \\cdots ( 1 - P ) a ^ { 2 k } P a ^ { 2 k + 1 } ( 1 - P ) \\\\ & \\quad - ( - 1 ) ^ k ( 1 - P ) a ^ 1 P a ^ 2 ( 1 - P ) \\cdots P a ^ { 2 k } ( 1 - P ) a ^ { 2 k + 1 } P \\\\ = & ( - 1 ) ^ k P a _ + ^ 1 ( 1 - P ) a ^ 2 P \\cdots ( 1 - P ) a _ - ^ { 2 k } P a _ + ^ { 2 k + 1 } ( 1 - P ) \\\\ & \\quad - ( - 1 ) ^ k ( 1 - P ) a _ - ^ 1 P a _ + ^ 2 ( 1 - P ) \\cdots P a ^ { 2 k } ( 1 - P ) a _ - ^ { 2 k + 1 } P . \\end{align*}"}
-{"id": "5565.png", "formula": "\\begin{align*} \\beta ( z ; \\epsilon ) = \\frac 1 2 \\sum _ { k = 1 } ^ M \\big [ \\frac { \\left ( \\mu _ k D ( - \\epsilon _ k ) - \\mu _ 0 D ( 0 ) \\right ) } { \\epsilon _ k } - \\mu _ k \\frac { D ( - \\epsilon _ k ) } { z + \\epsilon _ k } \\big ] , \\end{align*}"}
-{"id": "2788.png", "formula": "\\begin{gather*} \\delta U _ { - 1 } = [ B _ 0 ^ - , U _ { - 1 } ] , \\ ; \\ ; \\ ; \\delta U _ 0 = \\left ( [ B _ 1 ^ - , U _ { - 1 } ] + [ B _ 0 ^ - , U _ 0 ] \\right ) _ { \\oplus } , \\\\ \\delta U _ 1 = \\left ( [ B _ 2 ^ - , U _ { - 1 } ] + [ B _ 1 ^ - , U _ 0 ] + [ B _ 0 ^ - , U _ 1 ] \\right ) _ { \\oplus } . \\end{gather*}"}
-{"id": "82.png", "formula": "\\begin{align*} & k \\ < x , y \\ > / ( x y ^ 2 + y ^ 2 x + y x y + x ^ 3 , y x ^ 2 + x ^ 2 y + x y x + y ^ 3 ) = k \\ < x , y \\ > / ( ( x + y ) ^ 3 , ( x - y ) ^ 3 ) , \\\\ & k \\ < x , y \\ > / ( x y ^ 2 + y ^ 2 x + y x y - x ^ 3 , y x ^ 2 + x ^ 2 y + x y x - y ^ 3 ) = k \\ < x , y \\ > / ( ( x + \\sqrt { - 1 } y ) ^ 3 , ( x - \\sqrt { - 1 } y ) ^ 3 ) , \\end{align*}"}
-{"id": "7366.png", "formula": "\\begin{align*} K _ k \\triangleright ( \\Gamma _ i \\Gamma _ j ^ * ) & = ( K _ k \\triangleright \\gamma _ - ( w _ i ) ) ( K _ k \\triangleright \\gamma _ - ( w _ j ) ^ * ) \\\\ & = q ^ { ( \\alpha _ k , \\beta _ i - \\beta _ j ) } \\gamma _ - ( w _ i ) \\gamma _ - ( w _ j ) ^ * . \\end{align*}"}
-{"id": "755.png", "formula": "\\begin{gather*} \\sum _ { \\alpha ( 1 ) , \\ldots , \\alpha ( k ) = 1 } ^ n \\delta _ p ( \\alpha , \\beta ) u _ { \\alpha ( 1 ) i ( 1 ) } ^ { r _ 1 } \\cdots u _ { \\alpha ( k ) i ( k ) } ^ { r _ k } = \\sum _ { \\gamma ( 1 ) , \\ldots , \\gamma ( l ) = 1 } ^ n \\delta _ p ( i , \\gamma ) u _ { \\beta ( 1 ) \\gamma ( 1 ) } ^ { s _ 1 } \\cdots u _ { \\beta ( l ) \\gamma ( l ) } ^ { s _ l } . \\end{gather*}"}
-{"id": "7062.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( { t _ 2 } ) = { h ^ { [ j 1 ] } } ( { t _ 2 } ) u _ 2 ^ { [ 1 ] } + { h ^ { [ j 2 ] } } ( { t _ 2 } ) \\frac { { { h ^ { [ 2 2 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 2 2 ] } } ( { t _ 2 } ) } } u _ 1 ^ { [ 2 ] } + { h ^ { [ j 3 ] } } ( { t _ 2 } ) \\frac { { { h ^ { [ 2 3 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 2 3 ] } } ( { t _ 2 } ) } } u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "8455.png", "formula": "\\begin{align*} \\mathcal { U } ^ { M } = \\{ \\tilde { \\textbf { u } } ^ { \\varepsilon } \\in V ^ { s } : | | \\tilde { \\textbf { u } } ^ { \\varepsilon } | | _ { s } \\le M \\} . \\end{align*}"}
-{"id": "9698.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log g ( x ( t ) ) } { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } = - \\log \\left ( \\frac { a } { b } \\right ) . \\end{align*}"}
-{"id": "1314.png", "formula": "\\begin{align*} \\sum _ { n \\le N } R _ { \\textit { H L } } ( n ) \\frac { ( 1 - n / N ) ^ k } { \\Gamma ( k + 1 ) } & = \\frac { \\pi ^ { 1 / 2 } } 2 \\frac { N ^ { 3 / 2 } } { \\Gamma ( k + 5 / 2 ) } - \\frac 1 2 \\frac { N } { \\Gamma ( k + 2 ) } \\\\ & - \\frac { \\pi ^ { 1 / 2 } } 2 \\sum _ { \\rho } \\frac { \\Gamma ( \\rho ) } { \\Gamma ( k + 3 / 2 + \\rho ) } N ^ { 1 / 2 + \\rho } \\end{align*}"}
-{"id": "8169.png", "formula": "\\begin{align*} \\phi _ n = \\sum \\frac { f _ 1 ^ { k _ 1 } f _ 2 ^ { k _ 2 } \\cdots } { k _ 1 ! k _ 2 ! \\cdots } , \\end{align*}"}
-{"id": "1099.png", "formula": "\\begin{align*} | A x _ 0 | = | A \\tilde { x } | . \\end{align*}"}
-{"id": "9487.png", "formula": "\\begin{align*} \\psi ( z ) = ( \\log \\abs { z } / 2 \\pi , \\arg { z } / 2 \\pi ) \\end{align*}"}
-{"id": "6434.png", "formula": "\\begin{align*} & 0 \\leq \\psi \\leq 1 , \\psi = 1 \\rho ' B _ { 1 } , \\psi \\subset \\rho B _ { 1 } , \\\\ & \\qquad | D \\psi | \\leq 2 / ( \\sigma ( \\rho - \\rho ' ) ) . \\end{align*}"}
-{"id": "6836.png", "formula": "\\begin{align*} [ D ( t ) f ] ( x ) : = ( 2 i t ) ^ { - \\frac { d } { 2 } } f \\bigl ( \\tfrac { x } { 2 t } \\bigr ) , \\end{align*}"}
-{"id": "501.png", "formula": "\\begin{align*} \\mathcal { S } ( f ) = 2 \\pi \\Re ( a _ 1 ) + \\sum _ { j = 1 } ^ n ( V _ j ^ 2 - A _ j ) \\ge 0 \\end{align*}"}
-{"id": "990.png", "formula": "\\begin{align*} R _ { 1 2 } ( u - v ) L ^ X _ 1 ( u ) L ^ X _ 2 ( v ) = L ^ X _ 2 ( v ) L ^ X _ 1 ( u ) R _ { 1 2 } ( u - v ) , \\end{align*}"}
-{"id": "1453.png", "formula": "\\begin{align*} | m _ f ( Q ) | \\leq \\left ( f \\cdot \\chi _ Q \\right ) ^ * ( \\lambda | Q | ) , \\lim _ { \\ell ( Q ) \\to 0 , Q \\ni x } m _ f ( Q ) = f ( x ) ( { \\rm a . e . } \\ x \\in \\R ^ n ) . \\end{align*}"}
-{"id": "8789.png", "formula": "\\begin{align*} T _ n f _ { \\ldots , - \\lambda _ n } & = T _ n ^ 2 f _ { \\ldots , \\lambda _ { n } } = \\left ( t _ n + ( t _ n - 1 ) T _ i \\right ) f _ { \\ldots , \\lambda _ { n } } \\\\ & = t _ n f _ { \\ldots , \\lambda _ { n } } + ( t _ n - 1 ) f _ { \\ldots , - \\lambda _ { n } } , \\end{align*}"}
-{"id": "8345.png", "formula": "\\begin{gather*} f _ V ( z ) = f _ V ( l ( y ) + D ) = f _ V ( \\sum _ { i = 0 } ^ { n - 1 } A _ i y ^ { p ^ i } + D ) = \\sum _ { i = 0 } ^ { n - 1 } f _ V ( A _ i y ^ { p ^ i } ) + f _ V ( D ) = \\chi . \\end{gather*}"}
-{"id": "6727.png", "formula": "\\begin{align*} F ( X _ { T } ) = F ( X _ { 0 } ) + \\int _ 0 ^ T \\Delta _ { t } F ( X _ { t } ) d t + \\int _ 0 ^ T \\Delta _ { x } F ( X _ { t } ) \\psi ( t ) d t + \\int _ 0 ^ T \\Delta _ { x } F ( X _ { t } ) \\varphi ( t ) \\circ d B ^ { H } ( t ) . \\end{align*}"}
-{"id": "460.png", "formula": "\\begin{align*} w _ m & = \\sqrt { w _ 1 ^ 2 + w _ 2 ^ 2 + \\dots + w _ { m - 1 } ^ 2 } \\\\ \\gamma _ m & = \\sqrt { \\gamma _ 1 ^ 2 + \\gamma _ 2 ^ 2 + \\dots + \\gamma _ { m - 1 } ^ 2 } . \\end{align*}"}
-{"id": "2614.png", "formula": "\\begin{align*} \\delta _ h ( c ) & = \\delta _ h ( \\eta ) - ( \\delta _ h ( \\eta ) + \\beta _ \\omega ) \\\\ & = - \\beta _ \\omega . \\end{align*}"}
-{"id": "9316.png", "formula": "\\begin{align*} & \\int _ D U ' ( y ( T , x , z ) ) d x = \\tilde { p } ( 0 , z ) \\exp \\big ( - \\int _ 0 ^ T \\{ \\Phi ( s , z ) + \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } \\} d B ( s ) \\\\ & + \\frac { 1 } { 2 } \\int _ 0 ^ T \\{ \\Phi ^ 2 ( s , z ) - \\frac { a _ 0 ^ 2 ( s , z ) } { b _ 0 ^ 2 ( s , z ) } \\} d s \\big ) = : \\tilde { p } ( 0 , z ) \\Gamma ( T , z ) . \\end{align*}"}
-{"id": "3232.png", "formula": "\\begin{align*} P _ { \\alpha } ^ { S } = P _ { \\beta } ^ { S } \\circ \\Phi _ { \\beta , \\alpha } ^ { 1 } \\end{align*}"}
-{"id": "3587.png", "formula": "\\begin{align*} \\Pi _ { g _ 0 } \\circ D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) = \\Pi _ { g _ 0 } ( \\psi , V ) , \\end{align*}"}
-{"id": "8828.png", "formula": "\\begin{align*} L _ m \\rightarrow \\sum ^ k _ { i = 1 } \\sum _ { n < 0 } - n x _ { i , n } \\frac { \\partial } { \\partial x _ { i , n - m } } \\end{align*}"}
-{"id": "9829.png", "formula": "\\begin{align*} \\mbox { $ ( \\frac { 1 } { 2 } \\lambda - \\mu ) h _ { 0 , m } ( \\lambda + \\mu ) = - \\mu h _ { 0 , m } ( \\mu ) . $ } \\end{align*}"}
-{"id": "6301.png", "formula": "\\begin{align*} \\mathcal { M } _ { p } ( l _ { q _ { 1 } } ^ { s _ { 1 } , 1 } , l _ { q _ { 2 } } ^ { s _ { 2 } , 1 } ) = \\left \\{ \\{ a _ { j } \\} _ { j \\in \\mathbb { N } } : ~ \\Vert \\{ a _ { j } \\lambda _ { j } \\} \\Vert _ { l _ { q _ { 2 } } ^ { s _ { 2 } , 1 } } \\lesssim \\Vert \\{ \\lambda _ { j } \\} \\Vert _ { l _ { q _ { 1 } } ^ { s _ { 1 } , 1 } } \\{ \\lambda _ { j } \\} \\in l _ { q } ^ { s _ 1 , 1 } \\right \\} . \\end{align*}"}
-{"id": "6140.png", "formula": "\\begin{align*} z \\ge & 1 - 2 \\epsilon _ 0 + s \\epsilon _ 0 - \\frac 1 2 \\sqrt { 3 ( 1 - 2 \\epsilon _ 0 ) ( 1 - 3 \\epsilon _ 0 ) } \\\\ = & \\frac { 1 + \\sqrt { 2 } } 3 + \\frac { 2 - \\sqrt { 2 } } 6 s - \\frac { \\sqrt { 4 + 2 \\sqrt { 2 } } } 4 . \\end{align*}"}
-{"id": "4104.png", "formula": "\\begin{align*} C \\ : : \\ : y ^ q ( a x + b y + c z ) ^ r - x ^ { - p } z ^ { p + q + r } = 0 \\end{align*}"}
-{"id": "3510.png", "formula": "\\begin{align*} \\tfrac { 1 } { 2 } ( \\mathcal { D } _ g X ) ^ { i j } & = \\big ( \\tfrac { 2 } { n - 1 } ( \\mbox { t r } _ g \\pi ) g ^ { i j } - 2 \\pi ^ { i j } \\big ) f . \\end{align*}"}
-{"id": "3740.png", "formula": "\\begin{align*} C ( \\beta ) = \\int _ 0 ^ \\beta \\frac { e ^ t - 1 } { t } d t = \\int _ 0 ^ 1 \\frac { e ^ { \\beta t } - 1 } { t } d t . \\end{align*}"}
-{"id": "1881.png", "formula": "\\begin{align*} \\frac d { d t } \\Big [ ( \\kappa + \\delta t ) R \\Big ] = & ( \\kappa + \\delta t ) \\frac d { d t } R + \\delta R \\\\ = & ( \\kappa + \\delta t ) \\cdot 2 | R i c | ^ 2 + \\delta R \\\\ = & ( \\kappa + \\delta t ) \\cdot \\frac { R ^ 2 } 2 + \\delta R \\\\ \\end{align*}"}
-{"id": "366.png", "formula": "\\begin{align*} \\mu ( \\bar { d } _ { \\mathcal { E } } ) = { \\bar { d } _ { \\mathcal { E } } } ^ { \\alpha } , \\end{align*}"}
-{"id": "9740.png", "formula": "\\begin{align*} x _ { L , \\epsilon } ( T _ 8 ( \\epsilon ) ) = \\Lambda _ { 1 } ( 1 - \\epsilon ) G ^ { - 1 } ( T _ 8 ( \\epsilon ) ) < x ( T _ 8 ( \\epsilon ) ) . \\end{align*}"}
-{"id": "9375.png", "formula": "\\begin{align*} a , d \\in { \\mathbb { R } } , b = - { c } ^ * , \\mathcal { P } q _ 1 = q _ 1 , \\mathcal { P } q _ 2 = - q _ 2 , \\end{align*}"}
-{"id": "4483.png", "formula": "\\begin{align*} R _ 3 ' & = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ 0 ^ \\frac { a _ n } { n - 1 } \\log \\biggl ( \\frac { V _ d f ( x ) h _ x ^ { - 1 } ( s ) ^ d } { s } \\biggr ) \\mathrm { B } _ { k , n - k } ( s ) \\ , d s \\ , d x \\\\ & = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\biggr \\} \\biggr ) \\end{align*}"}
-{"id": "9452.png", "formula": "\\begin{align*} V _ t ( \\delta _ x ) : = \\begin{cases} \\delta _ { t ( x ) } & \\ x \\in \\mathrm { d o m } ( t ) \\\\ 0 & \\ x \\notin \\mathrm { d o m } ( t ) \\ ; . \\end{cases} \\end{align*}"}
-{"id": "4993.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n } ( P ) ) = & - \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { n - 1 - i } c _ { 1 } h _ { Z _ { 1 } } ( p ^ { - 1 } ( f ^ { i } ( P ) ) ) + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { n - 1 - i } h _ { E } ( p ^ { - 1 } ( f ^ { i } ( P ) ) ) \\\\ & + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { n - 1 - i } c _ { 1 } h _ { E _ { 1 } ' } ( f ^ { i } ( P ) ) + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { n - i } h _ { N } ( f ^ { i } ( P ) ) + \\delta ^ { n } h _ { H } ( P ) . \\end{align*}"}
-{"id": "7902.png", "formula": "\\begin{align*} I \\ , - \\ , r _ 1 \\ , \\xi \\ , - \\chi _ 1 \\ , s \\ , = \\ , 1 \\mbox { w h e r e } I : = l _ 1 \\cdot h . \\end{align*}"}
-{"id": "5790.png", "formula": "\\begin{align*} J ^ { u } _ { ( 0 ) } \\psi _ { \\alpha } = \\sum _ { \\beta \\in [ \\alpha ] } c _ { \\beta , u } ^ { \\alpha } \\psi _ { \\beta } , J ^ { u } _ { ( n ) } \\psi _ { \\alpha } = 0 \\quad \\mathrm { f o r } \\ n \\geq 1 \\end{align*}"}
-{"id": "2508.png", "formula": "\\begin{align*} \\{ x \\in \\operatorname { M r k } ( D ) : ( \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } F ) ( x ) = 0 \\} = \\Sigma ^ { D _ 1 , \\dots , D _ p } _ { d _ 1 , \\dots , d _ p } \\cap \\operatorname { M r k } ( D ) . \\end{align*}"}
-{"id": "2345.png", "formula": "\\begin{align*} x = a _ { k _ { 1 } } - a _ { k _ { 2 } } + a _ { k _ { 3 } } - \\dotsb - a _ { k _ { 2 l } } + a _ { k _ { 2 l + 1 } } . \\end{align*}"}
-{"id": "5462.png", "formula": "\\begin{align*} T _ H ( N , y ; \\alpha ) = H \\sum _ { n = N - H } ^ { N + y } e ( n \\alpha ) - \\sum _ { m = - y } ^ { H } m e ( ( N - m ) \\alpha ) = A - D , \\end{align*}"}
-{"id": "6747.png", "formula": "\\begin{align*} \\int _ X e ^ { \\varphi _ j } d \\mu = e ^ { \\sup _ X \\varphi _ j } \\int _ X e ^ { \\psi _ j } d \\mu \\geq c e ^ { \\sup _ X \\varphi _ j } \\end{align*}"}
-{"id": "2504.png", "formula": "\\begin{align*} \\mathcal { L } _ { X } D _ 1 = ( - \\lambda ) ( D _ 1 - d _ 1 ) , \\dots , \\mathcal { L } _ { X } D _ p = ( - \\lambda ) ( D _ p - d _ p ) . \\end{align*}"}
-{"id": "5581.png", "formula": "\\begin{align*} x = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\tanh { \\frac \\zeta 2 } , \\Psi = ( \\cosh \\frac \\zeta 2 ) \\ , v , \\end{align*}"}
-{"id": "1608.png", "formula": "\\begin{align*} Z ^ { 3 } = e ^ { 2 m t } \\left ( \\partial _ { t } + H \\right ) \\end{align*}"}
-{"id": "4517.png", "formula": "\\begin{align*} & \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\int _ 0 ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log u _ { x , s } \\ , d s \\ , d x \\\\ & \\leq C _ { d , f } \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\biggl \\{ \\log n + \\log \\biggl ( 1 + \\frac { \\| x \\| } { \\mu _ \\alpha ^ { 1 / \\alpha } ( f ) } \\biggr ) \\biggr \\} \\ , d x = O \\bigl ( \\max \\{ q _ n \\log n , q _ n ^ { 1 - \\epsilon } \\} \\bigr ) , \\end{align*}"}
-{"id": "2934.png", "formula": "\\begin{align*} s _ \\lambda = & \\ , \\det \\left [ \\begin{array} { c } h _ { \\lambda _ i - i + j } \\end{array} \\right ] _ { 1 \\leq i , j \\leq \\ell ( \\lambda ) } \\ , . \\end{align*}"}
-{"id": "8312.png", "formula": "\\begin{gather*} y + \\xi _ { \\sigma \\tau } = \\sigma \\tau ( y ) = \\sigma ( y + \\xi _ { \\tau } ) = \\sigma ( y ) + \\xi _ { \\tau } = y + \\xi _ { \\sigma } + \\xi _ { \\tau } , \\\\ \\intertext { s o t h a t } \\theta ( \\sigma \\tau ) = \\xi _ { \\sigma \\tau } = \\xi _ { \\sigma } + \\xi _ { \\tau } = \\theta ( \\sigma ) + \\theta ( \\tau ) . \\end{gather*}"}
-{"id": "8822.png", "formula": "\\begin{align*} T \\cdot x _ { i , n } = - ( n - 1 ) x _ { i , n - 1 } \\end{align*}"}
-{"id": "725.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n } ( x ) ) & = \\sum _ { k = 0 } ^ { n - 1 } \\delta _ { f } ^ { n - 1 - k } \\bigl ( h _ { H } ( f ^ { k + 1 } ( P ) ) - \\delta _ { f } h _ { H } ( f ^ { k } ( P ) ) \\bigr ) + \\delta _ { f } ^ { n } h _ { H } ( P ) \\\\ & = \\sum _ { k = 0 } ^ { n - 1 } \\delta _ { f } ^ { n - 1 - k } B ( f ^ { k } ( P ) ) + \\delta _ { f } ^ { n } h _ { H } ( P ) . \\end{align*}"}
-{"id": "9956.png", "formula": "\\begin{align*} c : = \\inf _ { \\gamma \\in \\Gamma } \\sup _ { t \\in [ 0 , 1 ] } \\mathcal E _ { \\lambda , \\mu } ( \\gamma ( t ) ) , \\end{align*}"}
-{"id": "1851.png", "formula": "\\begin{align*} \\sum _ { t < n < t ^ 4 } \\sum _ { m \\in R ( n , t ) } \\left ( \\widetilde { F } _ { m , n } ( t ) - \\widetilde { G } _ { m , n } ( t ) \\right ) = O ( 1 ) , \\end{align*}"}
-{"id": "9350.png", "formula": "\\begin{align*} \\Gamma ( t , x ) k ( t , x ) = 1 . \\end{align*}"}
-{"id": "1824.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { | d z | ^ 2 } { | z | ^ 2 } = \\frac { d r d \\theta } { r ^ 2 } , \\\\ \\mu _ s = \\frac { d r d \\theta } { r ^ 4 } \\left ( \\frac r { \\pi s } \\sin \\frac { \\pi s } r \\right ) ^ 2 . \\end{gathered} \\end{align*}"}
-{"id": "5318.png", "formula": "\\begin{align*} A ( u ) | \\phi _ { \\{ l ; k \\} } \\rangle & = ( u + \\omega + \\eta \\sum _ { i = 1 } ^ { n - 1 } l _ i ) ( u - \\omega + \\eta \\sum _ { i = 1 } ^ { m - 1 } k _ i ) | \\phi _ { \\{ l ; k \\} } \\rangle \\\\ B ( u ) | \\phi _ { \\{ l ; k \\} } \\rangle & = 0 \\\\ C ( u ) | \\phi _ { \\{ l ; k \\} } \\rangle & \\neq 0 \\\\ D ( u ) | \\phi _ { \\{ l ; k \\} } \\rangle & = \\eta ^ { - 2 } | \\phi _ { \\{ l ; k \\} } \\rangle . \\end{align*}"}
-{"id": "249.png", "formula": "\\begin{align*} \\max _ { r = 1 , \\ldots , m } \\frac { \\| f ^ { ( r ) } ( x ) \\| } { f ( x ) } \\leq d ^ { m / 2 } \\max _ { r = 1 , \\ldots , m } q _ r ( \\| x \\| ) . \\end{align*}"}
-{"id": "1063.png", "formula": "\\begin{align*} e ( G ) \\leq ( k - 1 ) e ( R ' ) + \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } \\leq \\bigg ( ( k - 1 ) ( k - \\alpha - \\tfrac 1 4 ) + ( k - 2 \\alpha + 5 \\alpha ^ 2 ) \\bigg ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "3355.png", "formula": "\\begin{align*} \\| M \\| _ F = \\| M \\| _ { n } \\geq \\dots \\geq \\| M \\| _ 1 \\end{align*}"}
-{"id": "8490.png", "formula": "\\begin{align*} A _ { j } ( \\textbf { u } ^ { \\varepsilon } ) = \\tilde { A } _ { j } ( \\textbf { u } ^ { \\varepsilon } ) + \\frac { { A } ^ { 0 } _ { j } } { \\varepsilon } \\end{align*}"}
-{"id": "2677.png", "formula": "\\begin{align*} \\frac { d } { d t } { \\rm I } ( u _ t ) = \\int _ X \\chi \\theta _ { u _ t } ^ n , \\ \\forall t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "5582.png", "formula": "\\begin{align*} i \\partial _ t \\psi ( x , t ) = ( D _ m + V ( x ) ) \\psi ( x , t ) , \\psi ( x , 0 ) = \\psi _ 0 ( x ) . \\end{align*}"}
-{"id": "3803.png", "formula": "\\begin{align*} \\| u _ { 0 , n } \\| _ { B ^ \\gamma _ { 2 , \\infty } } & \\le n ^ { - 1 } + \\| n ^ { - s } \\cos ( n x ) \\| _ { B ^ \\gamma _ { 2 , \\infty } } \\\\ & = n ^ { - 1 } + \\sup _ q n ^ { - s } 2 ^ { \\gamma q } \\| \\Delta _ q \\cos ( n x ) \\| _ { L ^ 2 } , \\end{align*}"}
-{"id": "6118.png", "formula": "\\begin{align*} u ( s _ { 1 } , \\dots , s _ { k } , z _ { k + 1 } , \\dots , z _ { 3 g - 3 + n } ) = \\psi ( z , \\bar z ) + \\sum _ { i = 1 } ^ { k } 2 \\pi s _ { i } + \\sum _ { i = 1 } ^ { k } s _ { i } ^ { 2 } \\phi _ { i } ( s , z , \\bar z ) , \\end{align*}"}
-{"id": "8073.png", "formula": "\\begin{align*} H ( a , b ) = \\lim _ { n \\to \\infty } H ( r _ n , s _ n ) = \\lim _ { n \\to \\infty } h ( r _ n , s _ n , x ) = h ( a , b , x ) , \\end{align*}"}
-{"id": "5950.png", "formula": "\\begin{align*} \\big | D _ v \\tilde u ( Z _ s ) - D _ v \\tilde u ( Y _ s ) \\big | & = \\Big | \\sum _ { i = 1 } ^ { 2 d } ( Z _ s - Y _ s ) ^ i \\int _ 0 ^ 1 D _ i D _ v \\tilde u ( r Z _ s + ( 1 - r ) Y _ s ) \\ , \\dd r \\Big | \\\\ & \\le | Z _ s - Y _ s | \\int _ 0 ^ 1 \\big | D D _ v \\tilde u \\big ( r Z _ s + ( 1 - r ) Y _ s \\big ) \\big | \\ , \\dd r \\ , . \\end{align*}"}
-{"id": "2067.png", "formula": "\\begin{align*} X ^ { ( 0 ) } ( s _ { 0 } ) = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } F , \\\\ X ^ { ( 1 ) } ( s _ { 0 } ) = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } ( - ( 2 s _ { 0 } M + D ) ) X ^ { ( 0 ) } ( s _ { 0 } ) , \\\\ X ^ { ( 2 ) } ( s _ { 0 } ) = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } [ - ( 2 s _ { 0 } M + D ) X ^ { ( 1 ) } ( s _ { 0 } ) - M X ^ { ( 0 ) } ( s _ { 0 } ) ] , \\\\ \\vdots \\\\ X ^ { ( j ) } ( s _ { 0 } ) = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } [ - ( 2 s _ { 0 } M + D ) X ^ { ( j - 1 ) } ( s _ { 0 } ) - M X ^ { ( j - 2 ) } ( s _ { 0 } ) ] . \\end{align*}"}
-{"id": "4335.png", "formula": "\\begin{align*} ( | f ' | ^ { p - 2 } f ' ) ' + f - | f ' | ^ { p - 1 } = 0 \\ , r \\in ( 0 , \\infty ) \\ , \\end{align*}"}
-{"id": "1093.png", "formula": "\\begin{align*} P r [ f ^ { ( 3 ) } ( x , \\xi ^ { ( 3 ) } ) \\in T _ 0 ] \\geq ( 4 / 3 ) \\cdot 1 5 ^ 3 \\cdot ( 1 / 6 0 ) ^ 3 = 1 / 4 8 \\end{align*}"}
-{"id": "6342.png", "formula": "\\begin{align*} \\big \\| \\{ \\langle k \\rangle ^ { \\frac { R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ 1 , \\alpha _ 2 ) } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } } \\} \\big \\| _ { l _ { r } ^ { s _ 2 - s _ 1 , \\alpha _ 1 \\vee \\alpha _ 2 } } = \\left ( \\sum _ { k \\in \\mathbb { Z } ^ n } \\langle k \\rangle ^ { r \\big [ \\frac { s _ 2 - s _ 1 } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } + \\frac { R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ 1 , \\alpha _ 2 ) } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } \\big ] } \\right ) ^ { 1 / r } . \\end{align*}"}
-{"id": "5043.png", "formula": "\\begin{align*} \\int _ G \\lambda ( g ^ { - 1 } B \\cap C ) \\ , d \\eta ( g ) = \\lambda ( B ) \\ , \\lambda ( C ) . \\end{align*}"}
-{"id": "13.png", "formula": "\\begin{align*} \\| x ^ * - y ^ * \\| & \\ge ( x ^ * - y ^ * ) ( z ) \\\\ & = x ^ * ( z ) - y ^ * ( z ) \\\\ & > 1 - \\eta + 1 - \\eta > 2 - \\delta . \\end{align*}"}
-{"id": "7701.png", "formula": "\\begin{align*} n _ { 2 k } = \\inf \\{ i > n _ { 2 k - 1 } : \\gamma _ i < 0 \\} - 1 , n _ { 2 k + 1 } = \\inf \\{ i > n _ { 2 k } : \\gamma _ i > 0 \\} - 1 . \\end{align*}"}
-{"id": "3585.png", "formula": "\\begin{align*} \\langle \\Pi _ g ( \\psi , V ) , ( \\psi , V ) ^ { \\perp } \\rangle _ { L _ { \\rho ^ { - 1 } _ g } ^ 2 ( \\Omega ) } = \\langle \\Pi _ g ( \\psi , V ) , ( \\psi , V ) ^ { \\perp } \\rangle _ { L ^ 2 ( \\Omega _ 0 ) } = 0 . \\end{align*}"}
-{"id": "8237.png", "formula": "\\begin{align*} \\psi ( t ) = e ^ { - i \\alpha ( t ) \\sigma } \\left [ \\Phi + \\phi ( t ) \\right ] , \\end{align*}"}
-{"id": "7734.png", "formula": "\\begin{align*} A : = \\{ \\omega \\in \\Omega : \\lim _ { n \\to \\infty } x _ n ( \\omega ) = 0 \\} . \\end{align*}"}
-{"id": "3181.png", "formula": "\\begin{align*} e _ j ^ \\alpha = \\sum _ { | \\beta | = n } \\tau ^ \\alpha _ { j k , \\beta } \\cdot e _ k ^ \\beta . \\end{align*}"}
-{"id": "2019.png", "formula": "\\begin{align*} I ( u ) = \\frac { f ( x u ) - f ( x ) } { x f ' ( x ) } = \\frac { 1 } { \\gamma \\alpha } ( \\log | x | ) ^ { 1 - \\alpha } \\left [ \\exp \\left \\{ \\gamma ( \\log | x | u ) ^ \\alpha - \\gamma ( \\log | x | ) ^ \\alpha \\right \\} - 1 \\right ] . \\end{align*}"}
-{"id": "902.png", "formula": "\\begin{align*} I = \\int _ { T } ^ { 2 T } \\chi ( \\tfrac { 1 } { 2 } + i t ) ^ { - 1 / 2 } \\zeta ( \\tfrac { 1 } { 2 } + i t ) d t \\end{align*}"}
-{"id": "5448.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & c ( i + r , 3 r ) & c ( i + 3 r , r ) \\\\ 0 & 0 & c ( i + r , r ) & c ( i + 3 r , 3 r ) \\\\ c ( i , r ) & c ( i + 2 r , 3 r ) & 0 & 0 \\\\ c ( i , 3 r ) & c ( i + 2 r , r ) & 0 & 0 \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "7213.png", "formula": "\\begin{align*} P _ { \\alpha } ^ { S } = P _ { \\beta } ^ { S } \\circ \\Phi _ { \\beta , \\alpha } ^ { 1 } \\end{align*}"}
-{"id": "5819.png", "formula": "\\begin{align*} \\begin{gathered} x _ { a 2 } , x _ { a 1 } , x _ { b 1 } , x _ { a 2 } \\\\ x _ { b 1 } , x _ { b 2 } , x _ { a 2 } , x _ { b 1 } \\\\ x _ { a 2 } , x _ { a 1 } , x _ { b 1 } , x _ { a 2 } \\end{gathered} \\end{align*}"}
-{"id": "8661.png", "formula": "\\begin{align*} P _ { \\tau } \\left [ \\phi \\right ] \\left ( x \\right ) = P _ { \\tau } \\phi \\left ( x \\right ) = \\mathbb { E } \\phi \\left ( X _ \\tau ^ { 0 , x } \\right ) , \\phi \\in B _ b ( H ) , \\ ; \\ ; \\tau \\ge 0 . \\end{align*}"}
-{"id": "706.png", "formula": "\\begin{align*} h _ { H } ( f ( P ) ) = h _ { E } ( P ) + \\left < A \\vec { c } , { \\bf h } _ { \\vec { D } } \\right > ( P ) . \\end{align*}"}
-{"id": "6684.png", "formula": "\\begin{align*} F ( x ( t ; \\overline { x } ) ) = \\exp ( - 2 \\lambda t ) \\cdot F ( \\overline { x } ) , ~ ( \\forall ) t \\in I _ { \\overline { x } } . \\end{align*}"}
-{"id": "9235.png", "formula": "\\begin{align*} j ( u ) - j ( \\hat { u } ) = \\mathbb { E } [ \\int _ 0 ^ T \\int _ D \\{ h - \\hat { h } \\} d x d t + \\int _ D \\{ k - \\hat { k } \\} d x ] , \\end{align*}"}
-{"id": "9662.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\tau ( t ) } { t } = q . \\end{align*}"}
-{"id": "6373.png", "formula": "\\begin{align*} v : = u - U _ a \\sum _ { m = 0 } ^ l P _ m ( x , r ) = U _ a \\big ( P _ { l + 1 } ( x , r ) + o ( | X | ^ { l + 1 } ) \\big ) \\end{align*}"}
-{"id": "3896.png", "formula": "\\begin{align*} X _ { \\rm i n t } = \\frac { \\mu } { \\gamma _ R } e _ 3 \\ ; . \\end{align*}"}
-{"id": "8433.png", "formula": "\\begin{align*} \\partial _ { t } \\textbf { u } + \\sum _ { j = 1 } ^ { d } \\textbf { P } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } = 0 , \\end{align*}"}
-{"id": "1071.png", "formula": "\\begin{align*} \\frac { e ( H ) } { v ( H ) } > \\frac { 1 } { N } \\left ( k - \\alpha - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } = \\frac { n } { 2 } \\left ( 1 - \\frac { 1 } { 4 ( k - \\alpha ) } \\right ) \\ , . \\end{align*}"}
-{"id": "8727.png", "formula": "\\begin{align*} X _ \\tau ^ { x } & = { e ^ { \\tau A } x + e ^ { \\tau A } v ( 0 , x ) - v ( \\tau , X _ \\tau ^ { x } ) + \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } \\widetilde Z _ s ^ x \\ ; d W _ s + \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } G d W _ s } \\\\ & = e ^ { \\tau A } x + e ^ { \\tau A } v ( 0 , x ) - v ( \\tau , X _ \\tau ^ { x } ) + \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } \\nabla ^ G v ( s , X _ s ^ x ) \\ ; d W _ s + \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } G d W _ s \\end{align*}"}
-{"id": "8199.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ 2 \\phi ( a _ { i j } ) \\Phi _ j ^ 2 = ( \\sum _ { j = 1 } ^ 3 a _ { i j } x _ j ^ 2 ) ^ p - \\phi ( a _ { i 3 } ) ( x _ 3 ^ p + p \\Delta _ 3 ) ^ 2 , \\ \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "3388.png", "formula": "\\begin{align*} 1 _ J = \\frac { 1 } { | W _ L | } \\sum \\limits _ { w \\in W _ L } R _ { T _ w } ^ L ( 1 ) \\end{align*}"}
-{"id": "2112.png", "formula": "\\begin{align*} ( l + 1 ) ( l + 2 + 2 \\sigma _ n ) \\bar { p } _ { \\sigma , l + 1 } + 2 s ( \\sigma _ n + 1 ) \\bar { p } _ { \\sigma + \\bar { n } , l } + ( \\sigma _ i + 1 ) ( \\sigma _ i + 2 ) \\bar { p } _ { \\sigma + 2 \\bar { \\imath } , l - 1 } = 0 \\forall ( \\sigma , l ) . \\end{align*}"}
-{"id": "5234.png", "formula": "\\begin{align*} \\alpha ( \\overline { L } ) ( z ) = d h ( \\overline { L } ) ( z ) + 2 e ^ { h } i \\partial \\overline { \\partial } \\rho ( - i T , \\overline { L } ) ( z ) \\ ; , \\ ; z \\in M \\ ; , \\ ; L \\in \\mathcal { N } _ { z } \\ ; . \\end{align*}"}
-{"id": "350.png", "formula": "\\begin{align*} e ^ { i T _ { c , h } ( f ) } = U _ { c , h } ( { \\rm E x p } ( f ) ) . \\end{align*}"}
-{"id": "9914.png", "formula": "\\begin{align*} C ' : & \\kappa ^ 2 K '^ 2 + E F \\ ; = \\ ; K \\ ; = \\ ; 0 \\hbox { a n d } \\\\ C : & \\kappa ^ 2 K ^ 2 + E F \\ ; = \\ ; K ' \\ ; = \\ ; 0 \\end{align*}"}
-{"id": "7207.png", "formula": "\\begin{align*} M _ n ( \\beta ) : = \\sum _ { i = 1 } ^ n \\prod _ { x \\in \\Pi _ n } \\Big ( 1 + B _ { i , x _ i } \\big ( T _ \\beta ( i , x _ i ) \\big ) \\Big ) , \\end{align*}"}
-{"id": "3116.png", "formula": "\\begin{align*} & B _ { 1 1 } = P _ 5 , \\\\ & A _ { 1 1 } = P _ 4 - ( B _ { 1 2 } - B _ { 1 2 } ^ T ) , \\\\ & B _ { 2 2 } = P _ 3 - ( B _ { 1 3 } - B _ { 3 1 } ^ T ) - ( A _ { 1 2 } - A _ { 1 2 } ^ T ) , \\\\ & A _ { 2 2 } = P _ 2 - ( B _ { 2 3 } - B _ { 2 3 } ^ T ) - ( A _ { 1 3 } - A _ { 1 3 } ^ T ) , \\\\ & B _ { 3 3 } = P _ 1 - ( A _ { 2 3 } - A _ { 2 3 } ^ T ) , \\mbox { a n d } \\\\ & A _ { 3 3 } = P _ 0 , \\end{align*}"}
-{"id": "7249.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } m _ j ^ 2 \\\\ m _ j ^ 3 \\\\ \\vdots \\\\ m _ j ^ r \\end{array} \\right ) - ( t _ { j k } ^ 1 ) ^ { - 1 } S _ { j k } \\left ( \\begin{array} { c } m _ k ^ 2 \\\\ m _ k ^ 3 \\\\ \\vdots \\\\ m _ k ^ r \\end{array} \\right ) = ( t _ { j k } ^ 1 ) ^ { - 1 } \\cdot \\left ( \\begin{array} { c } a _ { j k } ^ 2 \\\\ a _ { j k } ^ 3 \\\\ \\vdots \\\\ a _ { j k } ^ r \\end{array} \\right ) \\end{align*}"}
-{"id": "4278.png", "formula": "\\begin{align*} \\mathbb P ( Z _ n ^ * \\le - 1 ) = \\frac 1 2 \\mathbb P ( T _ 0 > n ) \\sim \\frac { \\gamma a _ n } { n } \\mathbb E \\left [ \\sup _ { t \\in [ 0 , 1 ] } \\Delta _ t \\right ] , \\end{align*}"}
-{"id": "2789.png", "formula": "\\begin{gather*} B _ 0 ^ - = \\begin{pmatrix} \\tfrac { 1 } { 2 } \\delta u & e _ 0 \\\\ 0 & - \\tfrac { 1 } { 2 } \\delta u \\end{pmatrix} . \\end{gather*}"}
-{"id": "783.png", "formula": "\\begin{align*} \\gamma ( \\vec { x } ) = \\sum _ { \\vec { \\nu } \\in I _ { k _ { 1 } , \\ldots , k _ { r } } } q ^ { t ( \\vec { \\nu } ) } \\gamma _ { \\vec { \\nu } } ( \\vec { x } ) \\ , u _ { \\nu _ { 1 } } \\otimes \\cdots \\otimes u _ { \\nu _ { k } } ( \\vec { x } \\in L _ { k } ^ { + } ) , \\end{align*}"}
-{"id": "9457.png", "formula": "\\begin{align*} d ( e ^ { f } \\lambda ) - e ^ { f } d \\lambda = e ^ { f } d f \\wedge \\lambda \\end{align*}"}
-{"id": "242.png", "formula": "\\begin{align*} h _ x ( \\| x \\| + r ) = \\int _ { B _ x ( \\| x \\| + r ) } f ( y ) \\ , d y \\geq \\int _ { B _ 0 ( r ) } f ( y ) \\ , d y \\geq 1 - \\frac { \\mu _ \\alpha ( f ) } { r ^ \\alpha } . \\end{align*}"}
-{"id": "9293.png", "formula": "\\begin{align*} \\| y ( t , z ) \\| ^ 2 _ { \\mathbf { H } ^ 1 ( \\mathbb { R } ^ + ) } = \\| y ( t , z ) \\| ^ 2 _ { \\mathbf { L } ^ 2 ( \\mathbb { R } ^ + ) } + \\| y ' ( t , z ) \\| ^ 2 _ { \\mathbf { L } ^ 2 ( \\mathbb { R } ^ + ) } \\end{align*}"}
-{"id": "6720.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\int _ 0 ^ t \\psi ( s ) d s + \\int _ 0 ^ t \\varphi ( s ) d W ( s ) , \\end{align*}"}
-{"id": "981.png", "formula": "\\begin{align*} D _ 2 ( v ) = \\sum _ { u \\in \\Gamma ( v ) } d ( u ) . \\end{align*}"}
-{"id": "4949.png", "formula": "\\begin{align*} \\gamma _ N : = \\left \\{ \\begin{array} { l l } 1 , & N = 3 , \\\\ - \\frac { \\pi } { \\Gamma \\left ( \\frac { N - 2 } { 2 } \\right ) } \\left ( \\frac 1 2 y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) ^ { \\frac { N - 2 } { 2 } } Y _ { \\frac { N - 2 } { 2 } } \\left ( y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) , & N \\geq 4 . \\end{array} \\right . \\end{align*}"}
-{"id": "7202.png", "formula": "\\begin{align*} \\rho _ l = \\frac { \\mathbb { E } \\left \\{ \\mathcal { I } ( t ) \\ , \\mathcal { I } ( \\tau ) \\right \\} - \\mathbb { E } \\left \\{ \\mathcal { I } ( t ) \\right \\} ^ 2 } { \\mathbb { E } \\left \\{ \\mathcal { I } ^ { \\ , 2 } ( t ) \\right \\} - \\mathbb { E } \\left \\{ \\mathcal { I } ( t ) \\right \\} ^ 2 } . \\end{align*}"}
-{"id": "8685.png", "formula": "\\begin{align*} \\sum _ { m \\ge 1 } \\sup _ { | a | _ U = 1 } | \\nabla _ k \\nabla _ { a } ^ G P _ t [ \\Phi _ m ] ( x ) | ^ 2 \\le \\frac { c } { t } | k | _ K ^ 2 \\ , \\| \\Phi \\| _ { C ^ 1 _ K ( H , J ) } ^ 2 . \\end{align*}"}
-{"id": "9699.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ( x ) } { e ^ { - 1 / x ^ \\alpha } } = 1 . \\end{align*}"}
-{"id": "4287.png", "formula": "\\begin{align*} & k \\ < x , y \\ > / ( x y ^ 2 + y ^ 2 x + y x y + x ^ 3 , y x ^ 2 + x ^ 2 y + x y x + y ^ 3 ) = k \\ < x , y \\ > / ( ( x + y ) ^ 3 , ( x - y ) ^ 3 ) , \\\\ & k \\ < x , y \\ > / ( x y ^ 2 + y ^ 2 x + y x y - x ^ 3 , y x ^ 2 + x ^ 2 y + x y x - y ^ 3 ) = k \\ < x , y \\ > / ( ( x + \\sqrt { - 1 } y ) ^ 3 , ( x - \\sqrt { - 1 } y ) ^ 3 ) , \\end{align*}"}
-{"id": "7763.png", "formula": "\\begin{align*} \\beta ( M ) = \\begin{bmatrix} \\beta _ { 0 , 0 } & \\beta _ { 1 , 1 } & \\dots & \\beta _ { p , p } \\\\ \\beta _ { 0 , 1 } & \\beta _ { 1 , 2 } & \\dots & \\beta _ { p , p + 1 } \\\\ \\vdots & \\ddots & & \\vdots \\end{bmatrix} . \\end{align*}"}
-{"id": "3298.png", "formula": "\\begin{align*} \\begin{gathered} w _ { - 1 } \\wedge w _ { - 1 } = 0 , w _ { 0 } \\wedge w _ { - 1 } = - q ^ { 2 } w _ { - 1 } \\wedge w _ { 0 } , w _ { 0 } \\wedge w _ { 0 } = - q ( q - q ^ { - 1 } ) w _ { - 1 } \\wedge w _ { 1 } , \\\\ w _ { 1 } \\wedge w _ { - 1 } = - w _ { - 1 } \\wedge w _ { 1 } , w _ { 1 } \\wedge w _ { 0 } = - q ^ { 2 } w _ { 0 } \\wedge w _ { 1 } , w _ { 1 } \\wedge w _ { 1 } = 0 . \\end{gathered} \\end{align*}"}
-{"id": "7645.png", "formula": "\\begin{align*} \\ \\left \\{ \\begin{aligned} & u _ { t } = \\triangle ^ { \\alpha / 2 } u , \\\\ & u ( x , 0 ) = u _ { 0 } ( x ) \\in L ^ { 1 } ( B _ { R } ) , \\\\ & u | _ { \\partial B _ { R } } = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "259.png", "formula": "\\begin{align*} S _ 2 & = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ \\frac { a _ n } { n - 1 } ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log ^ 2 u _ { x , s } \\ , d s \\ , d x = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "3445.png", "formula": "\\begin{align*} d \\xi ( t ) = a ( \\xi ( t ) ) d t + A ( \\xi ( t ) ) d w ( t ) , \\xi ( s ) = x , \\end{align*}"}
-{"id": "8252.png", "formula": "\\begin{align*} \\begin{cases} 2 e ^ { - 1 / | w | ^ 2 } \\left ( 2 \\frac { e ^ { - 1 / | w | ^ 2 } } { | w | ^ 6 } + | z | ^ 2 \\left ( \\frac { 1 } { | w | ^ 6 } - \\frac { 1 } { | w | ^ 4 } \\right ) \\right ) = 0 \\\\ | z | ^ 2 + 2 e ^ { - 1 / | w | ^ 2 } = 1 . \\end{cases} \\end{align*}"}
-{"id": "4849.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = I ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) + R ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) \\end{align*}"}
-{"id": "9501.png", "formula": "\\begin{align*} g ^ { \\mu \\nu } \\langle i _ { \\partial _ \\mu } G , i _ { \\partial _ \\nu } G \\rangle & = g ^ { \\mu \\nu } \\frac { 1 } { ( k - 1 ) ! } \\sum _ { \\mu _ 2 , \\ldots , \\mu _ k } G _ { \\mu \\mu _ 2 \\ldots \\mu _ k } G _ { \\nu } ^ { \\ , \\ , \\ , \\mu _ 2 \\ldots \\mu _ k } \\\\ & = \\frac { 1 } { ( k - 1 ) ! } \\sum _ { \\mu , \\mu _ 2 , \\ldots , \\mu _ k } G _ { \\mu \\mu _ 2 \\ldots \\mu _ k } G ^ { \\mu \\mu _ 2 \\ldots \\mu _ k } \\\\ & = \\frac { k ! } { ( k - 1 ) ! } | G | ^ 2 \\\\ & = k | G | ^ 2 . \\end{align*}"}
-{"id": "3877.png", "formula": "\\begin{align*} \\nabla g ( x ) \\cdot S = & \\ , n ( \\gamma ( r ) + \\gamma ( \\tilde { r } ) ) + r \\gamma ^ { \\prime } ( r ) ( 1 - | S ^ { \\perp } ( \\nabla r ) | ^ 2 ) \\\\ & \\ , + \\tilde { r } \\gamma ^ { \\prime } ( \\tilde { r } ) \\left ( 1 - \\left | S ^ { \\perp } \\left ( i _ x \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\right ) \\right ) \\right | ^ 2 \\right ) + \\gamma ( \\tilde { r } ) \\varepsilon _ 1 ( x , S ) - \\tilde { r } \\gamma ^ { \\prime } ( \\tilde { r } ) \\varepsilon _ 2 ( x , S ) , \\end{align*}"}
-{"id": "3285.png", "formula": "\\begin{align*} \\hat { R } ( v _ { 0 } \\otimes v _ { - 1 } ) = v _ { - 1 } \\otimes v _ { 0 } + ( q ^ { 2 } - q ^ { - 2 } ) v _ { 0 } \\otimes v _ { - 1 } . \\end{align*}"}
-{"id": "1539.png", "formula": "\\begin{align*} x ' = F ( x ) , x ( 0 ) = x _ 0 \\end{align*}"}
-{"id": "2362.png", "formula": "\\begin{align*} \\alpha : = \\big \\{ t _ j \\in [ 0 , T ] \\ ; \\ , : \\ ; \\ , 0 \\leqslant j \\leqslant N , \\ ; t _ 0 = 0 , \\ ; t _ { N } = T , \\ ; t _ j < t _ { j + 1 } \\big \\} , \\end{align*}"}
-{"id": "5317.png", "formula": "\\begin{align*} \\Gamma _ { i } = \\langle \\mu _ i , a \\rangle , \\overline { \\Gamma } _ { j } = \\langle \\nu _ j , b \\rangle , i = 1 , 2 , \\cdots , n , j = 1 , 2 , \\cdots , m \\end{align*}"}
-{"id": "5778.png", "formula": "\\begin{align*} \\C ^ { [ \\beta ] } = \\bigoplus _ { \\alpha \\in [ \\beta ] } \\C x _ { \\alpha } \\end{align*}"}
-{"id": "597.png", "formula": "\\begin{align*} f ( z ) ! = \\left \\{ \\begin{aligned} & \\prod _ { y = 1 } ^ z f ( z ) , & & z > 0 , \\\\ & 1 , & & z = 0 , \\\\ & \\frac 1 { \\prod _ { y = z + 1 } ^ 0 f ( z ) } , & & z < 0 , \\end{aligned} \\right . \\qquad f ( z ) ! = f ( z ) \\cdot f ( z - 1 ) ! . \\end{align*}"}
-{"id": "24.png", "formula": "\\begin{align*} u : H ^ { 0 } ( Y , K _ { Y } ) \\to H ^ { 0 } ( Y , p _ { \\star } ( K _ { X } + L ) ) = H ^ { 0 } ( X , K _ { X } + L ) \\end{align*}"}
-{"id": "7176.png", "formula": "\\begin{align*} b _ { j j } - b ' _ { j j } & = \\int _ { \\R ^ n } \\left \\{ \\partial _ j y _ j ( p ( \\chi - \\chi ' ) ) - \\partial _ n y _ n ( p ( \\chi - \\chi ' ) ) \\right \\} = 0 . \\end{align*}"}
-{"id": "1458.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } f ^ * ( R ) = 0 \\end{align*}"}
-{"id": "5340.png", "formula": "\\begin{align*} \\int _ 0 ^ T f ( t , u ( t ) , u ' ( t ) ) d t = 0 . \\end{align*}"}
-{"id": "2489.png", "formula": "\\begin{align*} | M ( G ) | \\leq p ^ { \\frac { 1 } { 2 } ( d ( G ) - 1 ) ( 2 n - m ) } ( = p ^ { \\frac { 1 } { 2 } ( d ( G ) - 1 ) ( n + k ) } ) , \\end{align*}"}
-{"id": "623.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ p ( B _ r ( x _ 0 ) ) } = \\| \\Psi _ \\lambda \\ast f \\| _ { L ^ p ( B _ r ( x _ 0 ) ) } \\leq \\| \\Psi _ \\lambda \\| _ { L ^ { t } ( B _ { r + r _ 0 } ( x _ 0 - y _ 0 ) ) } \\| f \\| _ { p ' } \\end{align*}"}
-{"id": "5167.png", "formula": "\\begin{gather*} p = p _ u ^ * p _ u , p _ u = s _ 0 \\otimes t _ 1 \\otimes s _ 1 \\otimes \\cdots \\otimes t _ l \\otimes s _ l . \\end{gather*}"}
-{"id": "2607.png", "formula": "\\begin{align*} | S ^ 2 ( A \\oplus B ) | = | S ^ 2 ( A ) | | S ^ 2 ( B ) | | A \\otimes B | . \\end{align*}"}
-{"id": "5256.png", "formula": "\\begin{align*} \\operatorname * { l e v } \\nolimits _ { \\varphi _ { A , k } , = } ( t ) = t k + \\operatorname * { b d } A \\forall \\ , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "7665.png", "formula": "\\begin{align*} \\int _ { B _ { R } } \\int _ { 0 } ^ { t } S _ { \\alpha } ( t - s ) f ( S _ { \\alpha } ( s ) v _ { 0 } ) d s & \\geq \\mu ( d , \\alpha ) c ( d , \\alpha , \\tau ) \\sum _ { k } f ( c ( d , \\alpha ) \\phi _ { k } ) ( \\frac { \\psi } { \\phi _ { k } } ) ^ { 1 + \\alpha / d } \\\\ & = c \\psi ^ { p } \\sum _ { k } f ( s _ { k } ) s _ { k } ^ { - p } , \\end{align*}"}
-{"id": "4200.png", "formula": "\\begin{align*} x _ { i j } ( r ) x _ { i j } ( s ) & = x _ { i j } ( r + s ) , & \\\\ [ x _ { i j } ( r ) , \\ , x _ { h k } ( s ) ] & = 1 , & h \\neq j , \\ k \\neq i , \\\\ [ x _ { i j } ( r ) , \\ , x _ { j k } ( s ) ] & = x _ { i k } ( r s ) . & \\end{align*}"}
-{"id": "2035.png", "formula": "\\begin{align*} \\frac { d \\lambda _ 1 ( \\tilde { L } _ c ) } { d c } \\biggr | _ { c = 1 } & = \\frac { d } { d c } ( c ^ 2 \\bar { \\lambda } _ 1 ( n , c D , K ) ) \\biggr | _ { c = 1 } \\\\ & = \\frac { 1 } { D } \\frac { d } { d D } ( D ^ 2 \\bar { \\lambda } _ 1 ( n , D , K ) ) . \\end{align*}"}
-{"id": "3940.png", "formula": "\\begin{align*} x x ^ t = c - 1 & = \\begin{cases} a \\ne - 1 & \\gcd ( q , c ) = 1 , \\\\ - 1 & \\gcd ( q , c ) \\ne 1 . \\end{cases} \\end{align*}"}
-{"id": "8856.png", "formula": "\\begin{align*} \\upsilon _ { k + 1 } \\leq R _ { k + 1 } ^ { \\eta } = \\frac { 1 } { 2 k } \\Big [ \\Big ( 2 + \\frac { 4 } { n } \\Big ) \\sum _ { i = 1 } ^ k \\upsilon _ i + \\sqrt { \\mathcal { D } } \\Big ] . \\end{align*}"}
-{"id": "7810.png", "formula": "\\begin{align*} P : = - 2 S R _ { i j k l } R _ { p i q k } R _ { p j q l } - \\frac { 1 } { 2 } S R _ { i j k l } R _ { i j p q } R _ { p q k l } + \\left \\vert \\mathrm { R m } \\right \\vert ^ { 2 } \\left \\vert \\mathrm { R i c } \\right \\vert ^ { 2 } . \\end{align*}"}
-{"id": "890.png", "formula": "\\begin{align*} \\int _ { { 1 } / { 2 } + i T } ^ { { 1 } / { 2 } + 2 i T } \\zeta ( s ) d s = ( \\int _ { { 1 } / { 2 } + i T } ^ { 2 + i T } + \\int _ { 2 + i T } ^ { 2 + 2 i T } + \\int _ { 2 + 2 i T } ^ { { 1 } / { 2 } + 2 i T } ) \\zeta ( s ) d s . \\end{align*}"}
-{"id": "9808.png", "formula": "\\begin{align*} \\sup _ { \\theta } \\Pi _ { l } ( A _ { \\epsilon _ { n } } ( \\theta ) ) = o ( n ^ { - d / 2 } ) . \\end{align*}"}
-{"id": "6960.png", "formula": "\\begin{align*} B ( s , \\epsilon ) = \\sum _ { m _ 1 \\in M _ { \\epsilon } } A ( s , m _ 1 ) . \\end{align*}"}
-{"id": "6096.png", "formula": "\\begin{align*} \\begin{aligned} a ' & \\le \\binom { | V | } { 2 } ^ { - 1 } \\bigg ( | W | \\binom { 2 p | V | } { 2 } + 2 ( c ' p ) ^ 2 | W | \\binom { | V | } { 2 } \\bigg ) \\\\ & \\le \\big ( 4 + 2 ( c ' ) ^ 2 \\big ) p ^ 2 | W | \\le 5 p ^ 2 | W | \\ , . \\end{aligned} \\end{align*}"}
-{"id": "6764.png", "formula": "\\begin{align*} \\theta ^ n = ( d d ^ c g ) ^ n \\geq e ^ { \\beta ( g + c ) } \\left [ \\theta _ + ^ n + \\varepsilon ( 1 - \\lambda ) ^ n \\omega ^ n \\right ] . \\end{align*}"}
-{"id": "5863.png", "formula": "\\begin{align*} x C _ { , x } ^ { x } - C ^ { x } + m x = 0 , \\end{align*}"}
-{"id": "6513.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\sum \\limits ^ k _ { i = 0 } \\delta _ i u _ { n - i } + A ( t _ n ) u _ n = \\sum \\limits ^ { k - 1 } _ { i = 0 } \\gamma _ i B ( t _ { n - i - 1 } , u _ { n - i - 1 } ) , n = k , \\dotsc , N . \\end{align*}"}
-{"id": "4074.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 3 } { x ^ 2 z ^ 2 } , f ( x , y , z ) = \\dfrac { y ^ 2 ( a x + b y + c z ) ^ 2 } { x z ^ { 3 } } , \\\\ f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 4 } { x ^ 2 z ^ { 3 } } , f ( x , y , z ) = \\dfrac { y ^ 2 ( a x + b y + c z ) ^ 3 } { x z ^ { 4 } } , \\\\ f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 6 } { x ^ 3 z ^ { 4 } } , f ( x , y , z ) = \\dfrac { y ^ 3 ( a x + b y + c z ) ^ 4 } { x z ^ { 6 } } , \\end{gather*}"}
-{"id": "4068.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 4 , 4 u + 1 , 4 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 4 , 4 u + 3 , 4 v + 3 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "7797.png", "formula": "\\begin{align*} e ( \\Delta _ { B _ 1 } \\cup \\Delta _ { B _ 2 } ) = & e ( \\Delta _ { B _ 1 } ) + e ( \\Delta _ { B _ 2 } ) - e ( \\Delta _ { B _ 1 \\cap B _ 2 } ) \\\\ = & 2 | B _ 1 | - 3 + 2 | B _ 2 | - 3 - e ( \\Delta _ { B _ 1 \\cap B _ 2 } ) \\\\ \\geq & 2 | B _ 1 | - 3 + 2 | B _ 2 | - 3 - ( 2 | B _ 1 \\cap B _ 2 | - 3 ) = 2 | B | - 3 . \\end{align*}"}
-{"id": "9206.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } } g ( z ) \\delta _ { Z } ( z ) d z = g ( Z ) a . s . \\end{align*}"}
-{"id": "6628.png", "formula": "\\begin{align*} \\int _ { \\tau _ \\partial ( y ) \\wedge T } ^ T \\bigg ( \\partial _ x \\psi ( \\Phi _ t ( y ) , t ) \\cdot v ( \\Phi _ t ( y ) ) + \\partial _ t \\psi ( \\Phi _ t ( y ) , t ) \\bigg ) \\ , d t \\ = 0 \\end{align*}"}
-{"id": "2053.png", "formula": "\\begin{align*} 0 = \\nabla _ { \\nu } \\| \\nabla w \\| ^ 2 | _ { x _ 0 } = - I I ( \\nabla w , \\nabla w ) | _ { x _ 0 } < 0 , \\end{align*}"}
-{"id": "4374.png", "formula": "\\begin{align*} e ^ { i J ( g ) } J ( f ) e ^ { - i J ( g ) } = J ( f ) + \\int _ { - \\pi } ^ { \\pi } ( \\partial _ \\theta g ) ( e ^ { i \\theta } ) f ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } \\ ; I \\end{align*}"}
-{"id": "6349.png", "formula": "\\begin{align*} U _ a = \\frac { | y | ^ { 2 s } } { 2 ^ { s } ( r - d ) ^ s } , \\end{align*}"}
-{"id": "4282.png", "formula": "\\begin{align*} \\quad \\lim _ { n \\rightarrow + \\infty } n ^ { \\frac 1 4 } \\mathbb P ( T _ 0 ^ { ( 1 ) } > n ) = \\frac 3 2 K _ p \\ , \\mathbb E \\left [ \\sup _ { t \\in [ 0 , 1 ] } \\Delta ^ { ( 0 ) } _ t \\right ] . \\end{align*}"}
-{"id": "77.png", "formula": "\\begin{align*} f ' ( r ) = - a \\psi \\left ( 1 - \\frac { f ( r ) } { a } \\right ) ^ { 1 / p } \\ , r \\in [ 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "7297.png", "formula": "\\begin{align*} E _ k \\triangleright [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q & = c _ { k , \\xi } ( E _ { \\xi + \\alpha _ k } E _ { \\xi ^ \\prime } ^ * - q ^ { - ( \\xi + \\alpha _ k , \\xi ^ \\prime ) } E _ { \\xi ^ \\prime } ^ * E _ { \\xi + \\alpha _ k } ) \\\\ & - c _ { k , \\xi ^ \\prime } ^ \\prime q ^ { ( \\xi - \\alpha _ k , \\alpha _ k ) } ( E _ \\xi E _ { \\xi ^ \\prime - \\alpha _ k } ^ * - q ^ { - ( \\xi , \\xi ^ \\prime + \\alpha _ k ) } E _ { \\xi ^ \\prime - \\alpha _ k } ^ * E _ \\xi ) . \\end{align*}"}
-{"id": "8934.png", "formula": "\\begin{align*} C ( n , m , p ) = \\begin{cases} \\big ( p p ' / ( n - 1 ) ^ 2 \\big ) ^ { m / 2 } & \\mbox { i f $ m $ i s e v e n } , \\\\ ( p / ( n - 1 ) ) \\big ( p p ' / ( n - 1 ) ^ 2 \\big ) ^ { ( m - 1 ) / 2 } & \\mbox { i f $ m $ i s o d d } , \\end{cases} \\end{align*}"}
-{"id": "1864.png", "formula": "\\begin{align*} - d ( \\phi ^ a \\cdot \\iota _ a \\Theta ) = - \\omega ^ a \\wedge \\iota _ a \\Theta \\ ; + \\ ; \\frac 1 2 f ^ c _ { a b } \\ , \\phi ^ a \\wedge \\phi ^ b \\cdot \\iota _ c \\Theta \\ ; - \\ ; \\phi ^ a \\wedge \\iota _ a \\Omega \\ ; + \\ ; \\phi ^ a \\wedge [ \\iota _ a \\Theta , \\Theta ] . \\end{align*}"}
-{"id": "2482.png", "formula": "\\begin{align*} z _ j & = \\omega _ { s _ j } ( \\Phi _ K ) \\cdot \\mu _ { \\tau _ j } \\\\ & = q _ K ^ { - s _ j } \\cdot \\mu _ { \\tau _ j } \\end{align*}"}
-{"id": "1516.png", "formula": "\\begin{align*} R g ( x ) : = \\sum _ { k = 0 } ^ { \\infty } \\frac { M ^ k g ( x ) } { ( 2 A ) ^ k } ( g \\in X ' ) , \\end{align*}"}
-{"id": "5380.png", "formula": "\\begin{align*} c _ { 1 , 1 } + c _ { 1 , 0 } ~ = ~ 1 . \\end{align*}"}
-{"id": "1892.png", "formula": "\\begin{align*} u _ n ^ { ( a ) } : = \\min \\{ u \\geq 0 : | \\phi ( u ) | = a n ^ { - \\frac { 1 } { 2 } } \\} . \\end{align*}"}
-{"id": "4306.png", "formula": "\\begin{align*} ( U ; z = ( z _ 1 , \\ldots , z _ { n + m } ) ) \\end{align*}"}
-{"id": "5895.png", "formula": "\\begin{align*} T ^ { I } \\left ( L _ { Y _ { I } ^ { \\alpha } } C ^ { \\alpha } + 2 \\psi _ { I } C ^ { \\alpha } \\right ) - T _ { , t } ^ { I } Y _ { I } ^ { \\alpha } - 2 g ^ { \\beta \\alpha } a _ { , \\beta } = 0 . \\end{align*}"}
-{"id": "1496.png", "formula": "\\begin{align*} \\begin{aligned} N ( x ) = | \\Phi | - | \\Phi _ x | & \\geq | \\Phi _ { S _ { n } } | - | \\Phi _ { ( S _ k ) ^ { b } \\times S _ { n - b k } } | \\\\ [ 1 e x ] & = n ( n - 1 ) - [ b k ( k - 1 ) + ( n - k b ) ( n - k b - 1 ) ] \\\\ [ 1 e x ] & = 2 k b n - k ^ 2 b ^ 2 - b k ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "9162.png", "formula": "\\begin{align*} \\det A ( a _ 1 , \\dots , a _ n ) & = \\det ( a _ { j + 1 } ^ { i + 1 } - a _ 1 ^ { i + 1 } ) _ { i , j = 1 } ^ { n - 1 } = \\det \\left ( \\left ( a _ { j + 1 } - a _ 1 \\right ) \\left ( \\sum _ { k = 0 } ^ { i } a _ { j + 1 } ^ { k } a _ 1 ^ { i - k } \\right ) \\right ) _ { i , j = 1 } ^ { n - 1 } \\\\ & = \\prod _ { j = 2 } ^ { n } ( a _ { j } - a _ 1 ) \\det \\left ( \\sum _ { k = 0 } ^ { i } a _ { j + 1 } ^ { k } a _ 1 ^ { i - k } \\right ) _ { i , j = 1 } ^ { n - 1 } . \\end{align*}"}
-{"id": "1033.png", "formula": "\\begin{align*} c _ { d , d } ^ d { \\ , } y ^ { d ^ 2 } + \\sum _ { k = 0 } ^ { d - 1 } { d \\choose k } ( c _ { d , d } { \\ , } x ^ d ) ^ k \\Big ( \\sum _ { j = 0 } ^ r c _ { d , j } { \\ , } x ^ j y ^ { d - j } \\Big ) ^ { d - k } ~ = ~ 0 . \\end{align*}"}
-{"id": "754.png", "formula": "\\begin{gather*} T _ p ( e _ { i ( 1 ) } \\otimes \\dotsm \\otimes e _ { i ( k ) } ) = \\sum _ { 1 \\leq j ( 1 ) , \\dotsc , j ( l ) \\leq n } \\delta _ p ( i , j ) \\cdot e _ { j ( 1 ) } \\otimes \\dotsm \\otimes e _ { j ( l ) } . \\end{gather*}"}
-{"id": "3819.png", "formula": "\\begin{align*} \\det _ { 1 \\leq i , j \\leq n } \\left \\{ ( 1 - x _ i z _ j ) ^ { - 1 } \\right \\} = \\frac { \\prod _ { i < j } ( x _ i - x _ j ) ( z _ i - z _ j ) } { \\prod _ { 1 \\leq i , j \\leq n } ( 1 - z _ i x _ j ) } \\end{align*}"}
-{"id": "9745.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ( x ) } { e ^ { - 1 / x ^ \\alpha } } = \\lim _ { x \\to 0 ^ + } \\frac { \\int _ 0 ^ x g ' ( s ) \\ , d s } { \\int _ 0 ^ x \\alpha s ^ { - \\alpha - 1 } e ^ { - 1 / s ^ \\alpha } \\ , d s } = 1 . \\end{align*}"}
-{"id": "6901.png", "formula": "\\begin{align*} \\Omega ( d \\Gamma _ { ( u , u _ y ) } ( [ f ] ) , ( \\delta u , \\delta u _ y ) ) = i \\ , ( [ f ] \\ , \\omega ( \\delta u , \\delta u _ y ) ) \\end{align*}"}
-{"id": "6547.png", "formula": "\\begin{align*} J _ m \\le C \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( X ) } + C \\delta . \\end{align*}"}
-{"id": "6183.png", "formula": "\\begin{align*} M _ { t \\wedge T _ n } & = f ( V _ { t \\wedge T _ n } ) - \\int _ 0 ^ { t \\wedge T _ n } \\mathcal { A } f ( V _ s ) \\d s \\\\ & = f ( V _ t ^ n ) - \\int _ 0 ^ { t \\wedge T _ n } \\mathcal { A } _ n f ( V _ s ) \\d s \\\\ & = f ( V _ t ^ n ) - \\int _ 0 ^ { t } \\mathcal { A } _ n f ( V _ s ^ n ) \\d s , \\end{align*}"}
-{"id": "9082.png", "formula": "\\begin{align*} d _ { N } = \\frac { 1 } { 4 } \\frac { \\Gamma ( N ) } { \\Gamma ( \\alpha + N - 1 ) } \\end{align*}"}
-{"id": "1296.png", "formula": "\\begin{align*} V ( \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } e ^ { - m / N } e ( m \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } e ^ { - m z } = \\frac 1 { e ^ z - 1 } . \\end{align*}"}
-{"id": "4395.png", "formula": "\\begin{align*} A \\cap A ' = \\emptyset A \\cap C _ { m ' ( 0 ) } \\cap \\tau ^ { - l ( A ) - 1 } C _ { m ' ( 1 ) } \\cap \\dots \\cap \\tau ^ { - 2 ^ { k ' } } C _ { m ' ( 2 ^ { k ' } ) } , \\end{align*}"}
-{"id": "545.png", "formula": "\\begin{align*} \\tilde { H } ( n , n ) \\overset { ( d ) } { = } \\max _ { x \\in \\Z _ { > 0 } } \\left \\lbrace \\sum _ { i = 1 } ^ x w _ { 1 i } + \\bar { H } ( n - 1 , n - x ) \\right \\rbrace , \\end{align*}"}
-{"id": "7077.png", "formula": "\\begin{align*} \\gamma _ \\infty = \\sum _ { n = 1 } ^ { \\infty } \\theta _ { \\xi _ n - \\log Z _ \\infty } D _ n . \\end{align*}"}
-{"id": "5367.png", "formula": "\\begin{align*} P ( x , y ) ~ = ~ A x y + B ( x + y ) + D , \\end{align*}"}
-{"id": "7721.png", "formula": "\\begin{align*} x _ { n _ \\delta ( \\omega ) + 1 } \\ge \\sum _ { i = N _ \\delta } ^ { n _ \\delta ( \\omega ) } \\sigma _ i \\xi _ { i + 1 } ( \\omega ) > \\delta . \\end{align*}"}
-{"id": "9544.png", "formula": "\\begin{align*} \\begin{cases} U ( k ) = \\lefteqn { - H _ { 1 } ( \\tilde { P } _ { 1 } ( k + 1 ) ) ^ { - 1 } G _ { \\nu } ( \\tilde { P } _ { 1 } ( k + 1 ) ) ^ { T } , } \\\\ \\mathbb { U } ( k ) = U ( k ) + \\tilde { U } ( k ) = \\tilde { H } _ { 1 } ( \\tilde { P } _ { 1 } ( k + 1 ) , \\tilde { Q } _ { 1 } ( k + 1 ) ) ^ { - 1 } \\tilde { G } _ { \\nu } ( \\tilde { P } _ { 1 } ( k + 1 ) , \\tilde { Q } _ { 1 } ( k + 1 ) ) ^ { T } , \\end{cases} \\end{align*}"}
-{"id": "1279.png", "formula": "\\begin{align*} A ^ 2 + ( \\mu - \\lambda ) A + ( \\mu - k ) I = \\mu J , \\end{align*}"}
-{"id": "9192.png", "formula": "\\begin{align*} \\int _ { \\mathbb R _ + } e ^ { s x } \\mu ( d x ) = h ( s ) \\end{align*}"}
-{"id": "4274.png", "formula": "\\begin{align*} D _ n : = \\int _ 0 ^ { \\sigma _ n / 2 } \\sqrt { \\log N ( n , t ) } \\dd t , \\end{align*}"}
-{"id": "6812.png", "formula": "\\begin{align*} G _ Y ( \\xi ) = \\left ( Y ^ - ( \\xi ) \\right ) ^ { - 1 } Y ^ + ( \\xi ) , \\end{align*}"}
-{"id": "1727.png", "formula": "\\begin{gather*} \\hat { \\Omega } = \\mathrm { d } \\gamma + \\tfrac { 1 } { 2 } [ \\gamma , \\gamma ] + \\tfrac { 1 } { 2 } [ \\theta , \\theta ] \\in \\Omega ^ 2 ( \\mathcal { S } , \\mathfrak { a } ) . \\end{gather*}"}
-{"id": "3933.png", "formula": "\\begin{align*} S _ { m , n } ( c , \\eta ) = \\mathcal { N } \\ , e ^ { i \\theta } \\ , ( 1 - \\eta ^ 2 ) ^ { m / 2 } \\ , \\sum _ { j = 0 } ^ \\infty b _ j \\ , ( 1 - \\eta ) ^ j \\ , , \\end{align*}"}
-{"id": "1044.png", "formula": "\\begin{align*} - \\Delta u - u = Q | u | ^ { p - 2 } u \\quad \\R ^ 2 . \\end{align*}"}
-{"id": "9245.png", "formula": "\\begin{align*} \\begin{cases} d R ( t ) = h ( X ( t ) ) d t + d w ( t ) ; t \\in [ 0 , T ] , \\\\ R ( 0 ) = 0 . \\end{cases} \\end{align*}"}
-{"id": "5054.png", "formula": "\\begin{align*} \\lambda = \\frac { 1 } { 2 } \\lambda _ { + } + \\frac { 1 } { 2 } \\lambda _ { - } ( T ) , \\textrm { w h e r e $ \\lambda _ { \\pm } ( \\cdot ) = 2 \\lambda ( \\cdot \\cap \\pm T ) $ } . \\end{align*}"}
-{"id": "5443.png", "formula": "\\begin{align*} K _ i = \\begin{bmatrix} 0 & J ( i + 2 r , 2 r ) & 0 & 0 \\\\ J ( i , 2 r ) & 0 & 0 & 0 \\\\ 0 & 0 & 0 & J ( i + 3 r , 2 r ) \\\\ 0 & 0 & J ( i + r , 2 r ) & 0 \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "5370.png", "formula": "\\begin{align*} P _ d ( x , y ) ^ d ~ = ~ c _ { d , d } ^ d ( x ^ { d ^ 2 } - y ^ { d ^ 2 } ) . \\end{align*}"}
-{"id": "8009.png", "formula": "\\begin{align*} p ( r ) = ( \\int _ { - \\sqrt { r ^ 2 + \\rho ^ 2 } } ^ { \\sqrt { r ^ 2 + \\rho ^ 2 } } ( 1 - \\frac { t ^ 2 } { r ^ 2 + \\rho ^ 2 } ) ^ { \\frac { n - 3 } { 2 } } d t ) ^ { - 1 } \\int _ { \\rho } ^ { \\sqrt { r ^ 2 + \\rho ^ 2 } } ( 1 - \\frac { t ^ 2 } { r ^ 2 + \\rho ^ 2 } ) ^ { \\frac { n - 3 } { 2 } } d t . \\end{align*}"}
-{"id": "4359.png", "formula": "\\begin{align*} G ' ( y ) & + \\frac { p \\kappa ^ { 1 / p } } { 1 - y } \\left [ \\varphi ( y , a ^ * ) ^ { ( p - 1 ) / p } - \\varphi ( y , a _ * ) ^ { ( p - 1 ) / p } \\right ] \\\\ & = p \\frac { G ( y ) } { 1 - y } + \\frac { p } { p - 1 } \\left [ ( a ^ * ) ^ { 2 - p } - ( a _ * ) ^ { 2 - p } \\right ] ( 1 - y ) ^ { 1 - p } \\end{align*}"}
-{"id": "4462.png", "formula": "\\begin{align*} T _ k ( r , s , t ) : = e ^ { \\alpha _ r ( s , t ) } \\sum _ { \\ell = 0 } ^ { L ( r , s , t ) } \\sum _ { i = 0 } ^ { I ( r , s ) - \\ell } \\sum _ { j = 0 } ^ { J ( r , t ) - \\ell } & \\frac { \\{ s - \\alpha _ r ( s , t ) \\} ^ i \\{ t - \\alpha _ r ( s , t ) \\} ^ j \\alpha _ r ^ \\ell ( s , t ) } { i ! j ! \\ell ! } \\\\ & - \\sum _ { i = 0 } ^ { I ( r , s ) } \\sum _ { j = 0 } ^ { J ( r , t ) } \\frac { s ^ i t ^ j } { i ! j ! } , \\end{align*}"}
-{"id": "6600.png", "formula": "\\begin{align*} N = \\Pi _ { D _ 0 } \\end{align*}"}
-{"id": "7059.png", "formula": "\\begin{align*} d _ \\Sigma ^ { { X _ L } } ( 2 , 2 ; \\frac { 2 } { { { T _ { A B } } } } ) = \\frac { { 4 A n + \\min ( 2 A , B ) { T _ { A B } } ( { T _ { A B } } - 1 ) } } { { { T _ { A B } } n + { T _ { A B } } ( { T _ { A B } } - 1 ) } } . \\end{align*}"}
-{"id": "4622.png", "formula": "\\begin{align*} \\bar { d } _ { \\mathcal { E } } = \\frac { | \\mathcal { E } _ m - \\Gamma _ m ^ * | } { d _ { m a x } } , \\end{align*}"}
-{"id": "808.png", "formula": "\\begin{align*} \\tau ( z ) \\tau ( w ) = \\tau ( w ) \\tau ( z ) . \\end{align*}"}
-{"id": "6274.png", "formula": "\\begin{align*} x '' ( s ) & = \\frac { d } { d s } x ' ( s ) = \\nabla _ { x ' ( s ) } x ' ( s ) = 0 , \\end{align*}"}
-{"id": "5162.png", "formula": "\\begin{gather*} \\sum _ { \\alpha ( 1 ) , \\ldots , \\alpha ( k ) = 1 } ^ n \\delta _ p ( \\alpha , \\beta ) u _ { \\alpha ( 1 ) i ( 1 ) } ^ { r _ 1 } \\cdots u _ { \\alpha ( k ) i ( k ) } ^ { r _ k } = \\sum _ { \\gamma ( 1 ) , \\ldots , \\gamma ( l ) = 1 } ^ n \\delta _ p ( i , \\gamma ) u _ { \\beta ( 1 ) \\gamma ( 1 ) } ^ { s _ 1 } \\cdots u _ { \\beta ( l ) \\gamma ( l ) } ^ { s _ l } . \\end{gather*}"}
-{"id": "9144.png", "formula": "\\begin{align*} k _ i \\leq \\ell _ i + r _ i + f _ i + ( r - r _ i - f _ i ) / 2 = \\ell _ i + r / 2 + r _ i / 2 + f _ i / 2 . \\end{align*}"}
-{"id": "9260.png", "formula": "\\begin{align*} d y ( t , x , Z ) & = L ^ * _ { R ( t ) , u ( t , Z ) } y ( t , x , Z ) d t + h ( x ) y ( t , x , Z ) d R ( t ) ; t \\geq 0 \\\\ y ( 0 , x , Z ) & = F ( x , Z ) . \\end{align*}"}
-{"id": "8231.png", "formula": "\\begin{align*} \\phi = \\tilde { \\phi } + \\rho , \\end{align*}"}
-{"id": "7.png", "formula": "\\begin{align*} \\varphi _ q ^ k & = \\frac { 1 } { q } \\mu \\left ( \\frac { k } { q } + \\frac { \\alpha ( k / q ) } { q ^ 2 } \\right ) \\left ( 1 + \\frac { \\beta ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) \\right ) . \\end{align*}"}
-{"id": "3725.png", "formula": "\\begin{align*} \\rho _ g ( f ) & = \\sum _ { y \\in M } K _ g ( \\cdot , y ) f ( y ) ; \\\\ \\widehat { \\rho } _ g ( \\Lambda ) & = \\sum _ { y \\in M } K _ g ( \\cdot , y ) \\Lambda ( y ) . \\end{align*}"}
-{"id": "164.png", "formula": "\\begin{align*} A \\cap A ' = \\emptyset A \\cap C _ { k ' } , \\end{align*}"}
-{"id": "8242.png", "formula": "\\begin{align*} \\dot { \\alpha } - E = \\frac { \\langle \\mathcal { H } '' _ E \\mathcal { S } \\sigma \\Phi , \\phi \\rangle _ { \\ell ^ 2 } + \\langle \\sigma \\Phi , N ( \\phi ) \\rangle _ { \\ell ^ 2 } } { \\| \\Phi \\| _ { \\ell ^ 2 } ^ 2 + \\langle \\Phi , \\phi \\rangle _ { \\ell ^ 2 } } . \\end{align*}"}
-{"id": "1507.png", "formula": "\\begin{align*} e _ 1 \\ e _ 0 \\ e _ 1 \\ e _ 0 \\ - \\ e _ 0 \\ e _ 1 \\ e _ 0 \\ e _ 1 \\ = \\ \\omega ( \\ e _ 0 ^ 2 \\ e _ 1 \\ e _ 0 \\ - \\ e _ 0 \\ e _ 1 \\ e _ 0 ^ 2 \\ ) \\ ; . \\end{align*}"}
-{"id": "7966.png", "formula": "\\begin{align*} \\tilde \\mu ( K _ j ) : = \\frac { \\tilde \\mu ( K ) ( \\mu ( K _ 1 ) + \\mu ( K _ 2 ) ) } { \\mu ( K ) + \\tilde \\mu ( K ) } \\quad j = 1 , 2 . \\end{align*}"}
-{"id": "8593.png", "formula": "\\begin{align*} d _ { 1 } a ^ { \\ast } d _ { 2 } \\widetilde { \\psi } ( w ) & = d _ { 1 } \\overline { N } ' ( a ^ { \\ast } ) ( d _ { 2 } \\widetilde { \\psi } ( w ) ) \\\\ & = d _ { 1 } c _ { k } ^ { - 1 } \\pi _ { e _ { k } a } ' i \\hat { \\psi } _ { i } \\Pi _ { 2 } ( d _ { 2 } w ) + d _ { 1 } c _ { k } ^ { - 1 } \\pi _ { e _ { k } a } j ' \\sigma _ { 2 } ' \\overline { \\psi } _ { 2 } ( \\Pi _ { 3 } ( d _ { 2 } w ) ) ) \\end{align*}"}
-{"id": "4265.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } \\frac n { \\mathbb E [ \\mathcal R _ n ] } \\mathbb P ( T _ 0 > n ) = \\gamma \\ , ; \\end{align*}"}
-{"id": "4690.png", "formula": "\\begin{align*} k ( x , y ) = \\sum _ { \\mathcal { Q } \\in \\mathcal { I } } \\lambda _ \\mathcal { Q } e _ \\mathcal { Q } ( x ) f _ \\mathcal { Q } ( y ) . \\end{align*}"}
-{"id": "5767.png", "formula": "\\begin{align*} f ( \\lambda , \\mu ) = \\frac { \\sinh ( \\lambda - \\mu + \\eta ) } { \\sinh ( \\lambda - \\mu ) } , g ( \\lambda , \\mu ) = \\frac { \\sinh \\eta } { \\sinh ( \\lambda - \\mu ) } \\end{align*}"}
-{"id": "651.png", "formula": "\\begin{align*} \\eta ( U _ { x _ o } ) = \\eta ( T \\chi _ U ) = \\nu ( \\chi _ U ) = \\nu ( U ) . \\end{align*}"}
-{"id": "8469.png", "formula": "\\begin{align*} \\rho _ { 0 } ^ { \\varepsilon } ( x ) = \\rho _ { 0 } ( 0 , x ) , ~ ~ ~ ~ v _ { 0 } ^ { \\varepsilon } ( 0 , x ) = v _ { 0 } ( x ) + \\varepsilon v _ { 0 } ^ { 1 } ( x ) , \\end{align*}"}
-{"id": "2120.png", "formula": "\\begin{align*} \\bar { u } _ j ( z _ 1 , z _ 2 ) : = | z _ 1 | ^ { - a } z _ 1 \\bar { Q } _ j ( z _ 1 ^ 2 , z _ 2 ^ 2 ) . \\end{align*}"}
-{"id": "2727.png", "formula": "\\begin{align*} \\textstyle P _ C ( - \\frac { 1 } { L } g _ i ) = 0 , \\end{align*}"}
-{"id": "6194.png", "formula": "\\begin{align*} g ( x ) = x f ' ( x ) = \\alpha ( \\log | x | ) ^ { \\alpha - 1 } \\end{align*}"}
-{"id": "2656.png", "formula": "\\begin{align*} C _ 2 ^ 2 & = 1 \\\\ & = 2 \\cdot 0 + 1 \\\\ & = 2 \\cdot T ( 1 , 2 ) + T ( 1 , 1 ) \\\\ & = T ( 2 , 2 ) \\end{align*}"}
-{"id": "6033.png", "formula": "\\begin{gather*} I ^ 2 = \\frac { ( k ^ 2 + 1 ) ^ 2 } { ( k ^ 2 - 9 ) ( \\tfrac { 1 } { 9 } - k ^ 2 ) } ; \\end{gather*}"}
-{"id": "417.png", "formula": "\\begin{align*} \\sum _ { k = 2 ^ { j } } ^ \\infty | a _ { k } | ^ 2 & \\leq 2 \\sum _ { n = j } ^ \\infty \\sum _ { k = 2 ^ { n - 1 } + 1 } ^ { 2 ^ { n + 1 } - 1 } | \\hat { w } _ { n , 2 } ( k ) a _ k | ^ 2 = 2 \\sum _ { n = j } ^ \\infty \\| w _ { n , 2 } * a \\| _ { L ^ 2 ( S ^ 1 ) } ^ 2 \\\\ & \\leq 2 \\sum _ { n = j } ^ \\infty \\| w _ { n , 2 } * a \\| _ { L ^ \\infty ( S ^ 1 ) } ^ 2 \\leq 2 \\sum _ { n = j } ^ \\infty 2 ^ { - n } \\| a \\| _ { C ^ { 1 / 2 , * } ( S ^ 1 ) } ^ 2 = 2 ^ { - j + 2 } \\| a \\| _ { C ^ { 1 / 2 , * } ( S ^ 1 ) } ^ 2 = O ( 2 ^ { - j } ) . \\end{align*}"}
-{"id": "1417.png", "formula": "\\begin{align*} u \\cdot v \\cdot x _ { \\alpha } - ( - 1 ) ^ { p ( u ) p ( v ) } v \\cdot u \\cdot x _ { \\alpha } = [ u , v ] \\cdot x _ { \\alpha } \\end{align*}"}
-{"id": "1742.png", "formula": "\\begin{align*} F ( 2 ^ { - j } \\sqrt { L } ) ( x , y ) = \\sum _ { k = 0 } ^ \\infty m ( 2 ^ { - j } \\sqrt { L } ) g _ { j , y , k } ( x ) . \\end{align*}"}
-{"id": "1191.png", "formula": "\\begin{align*} \\begin{array} { l c c l l c } \\mathcal { B } & = & \\{ & \\mathcal { B } ^ 0 = \\{ 2 1 0 2 0 1 2 0 2 1 2 0 1 0 2 0 1 2 , & 2 1 0 2 0 1 0 2 1 2 0 2 1 0 2 0 1 2 \\} , & \\\\ & & & \\mathcal { B } ^ 1 = \\{ 0 2 1 0 1 2 0 1 0 2 0 1 2 1 0 1 2 0 , & 0 2 1 0 1 2 1 0 2 0 1 0 2 1 0 1 2 0 \\} , & \\\\ & & & \\mathcal { B } ^ 2 = \\{ 1 0 2 1 2 0 1 2 1 0 1 2 0 2 1 2 0 1 , & 1 0 2 1 2 0 2 1 0 1 2 1 0 2 1 2 0 1 \\} & \\} . \\end{array} \\end{align*}"}
-{"id": "6796.png", "formula": "\\begin{align*} \\tilde L _ m = \\sum _ { i = 1 } ^ p t _ i ^ { m + 1 } \\frac { \\partial } { \\partial t _ i } \\qquad \\ m = - 1 , 0 , 1 , \\ldots \\end{align*}"}
-{"id": "141.png", "formula": "\\begin{align*} \\| \\tilde { f _ n } - f \\chi _ A \\| _ { \\widehat { E } } = \\| | \\tilde { f _ n } - f \\chi _ A | \\| _ { \\widehat { E } } \\leq \\| \\abs { f _ n - f } \\| _ { \\widehat { E } } \\to 0 . \\end{align*}"}
-{"id": "3416.png", "formula": "\\begin{align*} \\frac { \\partial u _ l } { \\partial s } + \\frac 1 2 T r A ^ * ( x ) \\nabla ^ 2 u _ l A ( x ) + \\langle a ( x ) , \\nabla u _ l \\rangle + g _ l ( x , u , A ^ * \\nabla u _ l ) = 0 , u _ l ( T , x ) = u _ { 0 l } ( x ) , \\end{align*}"}
-{"id": "844.png", "formula": "\\begin{align*} & ( - 1 ) ^ { t - 1 } \\prod _ { s = 1 } ^ { t - 1 } g ( z _ { \\ell ( s ) } , z _ { \\ell ( s + 1 ) } ) + \\frac { g ( w , z _ { \\ell ( t ) } ) } { g ( z _ { \\ell ( t ) } , z _ { \\ell ( 1 ) } ) } \\prod _ { s = 1 } ^ { t - 1 } g ( z _ { \\ell ( s + 1 ) } , z _ { \\ell ( s ) } ) \\\\ & = \\frac { g ( w , z _ { \\ell ( t ) } ) } { g ( w , z _ { \\ell ( 1 ) } ) } \\prod _ { s = 1 } ^ { t - 1 } g ( z _ { \\ell ( s + 1 ) } , z _ { \\ell ( s ) } ) . \\end{align*}"}
-{"id": "8712.png", "formula": "\\begin{align*} \\nabla ^ G v ( \\tau , \\Xi _ \\tau ^ { t , x } ) = Z _ \\tau ^ { t , x } , \\ ; \\ ; \\ , \\P . \\end{align*}"}
-{"id": "5821.png", "formula": "\\begin{align*} \\sum _ { j \\in I } B _ { i j } ( \\lambda _ j , \\mu _ j , \\nu _ j ) = 0 \\end{align*}"}
-{"id": "6839.png", "formula": "\\begin{align*} J _ { \\leq N } ^ s ( t ) : = e ^ { i t \\Delta } \\varphi ( \\tfrac { x } { N } ) | x | ^ s e ^ { - i t \\Delta } = M ( t ) P _ { \\leq \\frac { N } { | t | } } ( - 4 t ^ 2 \\Delta ) ^ { \\frac { s } { 2 } } M ( - t ) . \\end{align*}"}
-{"id": "9201.png", "formula": "\\begin{align*} ( A _ u \\psi , \\phi ) _ { L ^ 2 ( D ) } = ( \\psi , A _ u ^ * \\phi ) _ { L ^ 2 ( D ) } \\end{align*}"}
-{"id": "5086.png", "formula": "\\begin{align*} h _ { \\vec { z } } ^ { 1 ^ { k } } ( \\vec { x } ) = \\sum _ { \\sigma \\in \\mathfrak { S } _ { k } } \\prod _ { 1 \\le i < j \\le k } \\ ! \\frac { z _ { \\sigma ( i ) } - q ^ 2 z _ { \\sigma ( j ) } } { z _ { \\sigma ( i ) } - z _ { \\sigma ( j ) } } \\ , \\prod _ { i = 1 } ^ { k } \\left ( \\frac { z _ { \\sigma ( i ) } } { 1 + z _ { \\sigma ( i ) } } \\right ) ^ { x _ { i } } \\ , ( u _ { 1 } \\otimes \\cdots \\otimes u _ { 1 } ) . \\end{align*}"}
-{"id": "1972.png", "formula": "\\begin{align*} y _ i y _ j = \\left \\{ \\begin{array} { l r } y _ 0 y _ { i + j } & i + j \\leq m , \\\\ y _ m y _ { i + j - m } & i + j > m . \\end{array} \\right . \\end{align*}"}
-{"id": "6006.png", "formula": "\\begin{gather*} \\Phi ( x , y , z ) : = H _ { \\times } ( x \\times y , z ) . \\end{gather*}"}
-{"id": "7573.png", "formula": "\\begin{align*} K = \\left [ \\frac { a - u _ l } { \\tilde \\Delta _ l ( x _ 0 ) } \\right ] + 1 , \\end{align*}"}
-{"id": "9618.png", "formula": "\\begin{align*} \\lambda _ j ( t ) : = \\lim _ { x \\to y _ j } \\ , \\psi _ { f } ( x , t ) = \\lambda _ { j , r e g } ( t ) + \\sum \\limits _ { 1 \\le k \\le n } \\lambda _ { k , j } ( t ) \\in C ( [ 0 , \\infty ) ) . \\end{align*}"}
-{"id": "2647.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { h ( m , n ) } C _ { m } ^ { l } ( G ^ { 0 } ) ( \\lambda ) ( C _ { m } ^ { l } ( G ^ { 0 } ) ( \\lambda ) ) ^ { * } = d _ { m } ^ { 0 } ( \\lambda ) ^ { \\top } \\vec { \\alpha _ { m } } \\cdot \\vec { \\alpha _ { m } } ^ { * } \\overline { d _ { m } ^ { 0 } ( \\lambda ) } , \\end{align*}"}
-{"id": "3550.png", "formula": "\\begin{align*} ( \\gamma ^ R , \\tau ^ R ) = \\chi ( g _ 1 ^ R , \\pi _ 1 ^ R ) + ( 1 - \\chi ) ( g _ 2 ^ R , \\pi _ 2 ^ R ) . \\end{align*}"}
-{"id": "9211.png", "formula": "\\begin{align*} Y ( t , x ) = Y ( t , x , Z ) = Y ( t , x , z ) _ { z = Z } = \\int _ { \\mathbb { R } } Y ( t , x , z ) \\delta _ Z ( z ) d z \\end{align*}"}
-{"id": "9588.png", "formula": "\\begin{align*} \\ddot { \\psi } ( x , t ) = H _ y \\psi ( x , t ) , \\psi ( x , 0 ) = \\psi _ 0 ( x ) , \\quad \\dot \\psi ( x , 0 ) = \\pi _ 0 ( x ) , \\end{align*}"}
-{"id": "5297.png", "formula": "\\begin{align*} \\mathbb { Z } _ + \\ , : = \\ , \\{ 0 , 1 , 2 , \\ldots \\} . \\end{align*}"}
-{"id": "9151.png", "formula": "\\begin{align*} g ( s , t ) = \\frac { 1 } { s } + \\frac { d ^ 2 } { s t ( 2 d + 1 ) } = \\frac { 1 } { s } + \\frac { \\frac { d } { t } } { s ( 2 + \\frac { 1 } { d } ) } = \\frac { 1 } { s } + \\frac { s - 1 } { s ( 2 + \\frac { 1 } { d } ) } = \\frac { s + 1 + \\frac { 1 } { d } } { ( 2 + \\frac { 1 } { d } ) s } ~ . \\end{align*}"}
-{"id": "2408.png", "formula": "\\begin{align*} \\langle \\nabla f _ l ( y ) , x - y \\rangle = \\langle \\nabla f _ h ( y ) , x - y \\rangle _ H = \\langle H \\nabla f _ h ( y ) , x - y \\rangle \\end{align*}"}
-{"id": "5128.png", "formula": "\\begin{align*} \\vec { z } ( \\ell _ { 1 } , \\ell _ { 0 } ) = ( z _ { 1 } , \\ldots , \\underset { \\hbox { \\scriptsize $ \\ell _ { 1 } $ - t h } } { z _ { \\ell _ { 0 } } } , \\ldots , z _ { \\ell _ { 0 } - 1 } , z _ { \\ell _ { 0 } + 1 } , \\ldots , z _ { k } ) . \\end{align*}"}
-{"id": "3791.png", "formula": "\\begin{align*} { \\rm v e c } ( { { { \\bf { H } } } } ) = \\sum \\limits _ { n = 1 } ^ { { N _ { \\mathrm r } } } { \\sum \\limits _ { m = 1 } ^ { { N _ { \\mathrm t } } } { \\left ( { { { \\big [ { { \\bf { \\tilde H } } } \\big ] } _ { n m } } + { { \\big [ { \\hat { \\bf { H } } } \\big ] } _ { n m } } } \\right ) \\left ( { \\bf { u } } _ { \\mathrm t } ^ * ( { { \\phi _ m } } ) \\otimes { { \\bf { u } } _ { \\mathrm r } } ( { { \\theta _ n } } ) \\right ) } } \\end{align*}"}
-{"id": "7958.png", "formula": "\\begin{align*} K _ 1 r ^ n \\leq | | \\psi _ { \\lambda _ j } | | ^ 2 _ { L ^ 2 ( B _ { r } ( x ) ) } \\leq K _ 2 r ^ n , \\ ; \\ ; r = ( \\log \\lambda _ j ) ^ { - \\frac { 1 } { 2 n } + \\epsilon } , \\end{align*}"}
-{"id": "5423.png", "formula": "\\begin{align*} P r [ f ^ { ( 4 ) } ( x , \\xi ^ { ( 4 ) } ) \\in S _ 0 ] \\geq ( 1 / 1 2 9 6 ) \\cdot ( 1 / 2 ) = 1 / 2 5 9 2 \\end{align*}"}
-{"id": "5214.png", "formula": "\\begin{align*} f = \\bar \\partial _ M \\bar \\partial ^ * _ M G _ q f + \\bar \\partial ^ * _ M \\bar \\partial _ M G _ q f \\ ; . \\end{align*}"}
-{"id": "3173.png", "formula": "\\begin{align*} f _ { x _ i } ( n ) \\ ! = \\ ! \\frac { p } { N } \\ ! + \\ ! \\left ( 1 \\ ! - \\ ! p \\right ) \\ ! \\frac { 3 N \\left ( 2 n \\ ! - \\ ! 1 \\right ) \\ ! - \\ ! 6 n \\left ( n \\ ! - \\ ! 1 \\right ) \\ ! - \\ ! 3 } { N ( N ^ 2 - 1 ) } , n \\ ! \\leq \\ ! N \\end{align*}"}
-{"id": "4031.png", "formula": "\\begin{align*} R _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ { m } & = c l \\ C _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ m , \\ m = 1 , \\ldots , 4 \\\\ F _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } & = \\cup _ { m = 1 } ^ 4 \\partial C _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ m , \\end{align*}"}
-{"id": "1076.png", "formula": "\\begin{align*} e ( B ) \\leq \\left ( \\beta - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } + \\left ( k - \\alpha - \\beta \\right ) \\frac { n ^ 2 } { 2 } = \\bigg ( k - \\alpha - \\frac 1 4 \\bigg ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "1002.png", "formula": "\\begin{align*} E _ { n , m } = t \\left ( \\lambda _ { \\{ l ; k \\} } ( u ) + \\omega ^ 2 - u ^ 2 - \\eta ^ { - 2 } - u \\eta N - \\frac { \\eta ^ 2 N ^ 2 } { 4 } \\right ) . \\end{align*}"}
-{"id": "1590.png", "formula": "\\begin{align*} 2 f _ { I , t } + c = 0 , \\end{align*}"}
-{"id": "4533.png", "formula": "\\begin{align*} | U _ 1 | \\leq U _ { 1 1 } + U _ { 1 2 } = o \\biggl ( \\frac { k ^ { \\frac { 1 } { 2 } + \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { 1 + \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "1815.png", "formula": "\\begin{align*} \\phi ^ * ( d t / t ) = d z / z + d w / w . \\end{align*}"}
-{"id": "7038.png", "formula": "\\begin{align*} \\frac { t _ k A ( x _ k ) } { x _ k } = \\frac { t _ k } { s _ k } \\frac { s _ k A ( x _ k ) } { x _ k } \\to \\infty , \\end{align*}"}
-{"id": "5177.png", "formula": "\\begin{gather*} \\varphi ( a ) = a ( r _ 1 ^ s - 1 ) , \\psi ( a ) = a r _ 1 ^ s . \\end{gather*}"}
-{"id": "6865.png", "formula": "\\begin{align*} u _ n ^ J - z _ { n , J _ 0 } ^ J = \\sum _ { j = 1 } ^ { J _ 0 } v _ n ^ j + e ^ { i t \\Delta } W _ n ^ J , \\end{align*}"}
-{"id": "3634.png", "formula": "\\begin{align*} c ^ * = \\max \\left \\{ 0 , 1 - \\frac { 1 } { \\max \\{ | \\frac { \\partial f } { \\partial y } ( \\mathbf { K } ) | , | \\frac { \\partial f } { \\partial x } ( \\mathbf { K } ) | + \\frac { \\partial f } { \\partial y } ( \\mathbf { K } ) \\} } \\right \\} , \\end{align*}"}
-{"id": "404.png", "formula": "\\begin{align*} \\Sigma _ 1 ( f _ t , w _ 1 , w _ 2 ) & = t ^ { - w _ 1 - w _ 2 } \\Sigma _ 1 ( f , w _ 1 , w _ 2 ) . \\end{align*}"}
-{"id": "244.png", "formula": "\\begin{align*} A _ { d , \\theta } : = \\max \\biggl \\{ \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } | H ( f ) | \\ , , \\ , - \\frac { 1 } { 2 } \\log \\inf _ { f \\in \\mathcal { F } _ { d , \\theta } } \\| f \\| _ \\infty \\ , , \\ , 1 \\biggr \\} \\end{align*}"}
-{"id": "4371.png", "formula": "\\begin{align*} J _ n \\Omega = 0 \\ ; \\ ; \\ ; \\textrm { f o r e v e r y } \\ ; \\ ; \\ ; n \\geq 0 \\end{align*}"}
-{"id": "3215.png", "formula": "\\begin{align*} M \\cdot \\left ( - 1 - R ( X ^ 1 + X ^ 2 + \\cdots + X ^ r + r A ( X ) ) + \\prod _ { \\lambda = 1 } ^ r \\frac { 1 } { 1 - R ( X ^ \\lambda + A ( X ) ) } \\right ) . \\end{align*}"}
-{"id": "4302.png", "formula": "\\begin{align*} u : H ^ { 0 } ( Y , K _ { Y } ) \\to H ^ { 0 } ( Y , p _ { \\star } ( K _ { X } + L ) ) = H ^ { 0 } ( X , K _ { X } + L ) \\end{align*}"}
-{"id": "2571.png", "formula": "\\begin{align*} N : = L _ t ^ { q _ 1 ' , 2 } \\dot X ^ { | s _ c | , r _ 1 ' } \\end{align*}"}
-{"id": "7119.png", "formula": "\\begin{align*} \\frac { 1 } { | | \\beta - \\alpha | | _ g } \\sum _ { l = 1 } ^ n ( \\Omega _ g ( \\alpha , \\beta , l ) \\omega _ g ( \\alpha + [ l ] , \\beta + [ l ] , l ) & + \\Omega _ g ( \\alpha - [ l ] , \\beta - [ l ] , l ) \\omega _ g ( \\alpha , \\beta , l ) ) ( x ^ { \\alpha } \\otimes g ) \\epsilon _ { \\beta } ^ * \\\\ & = ( x ^ { \\alpha } \\otimes g ) \\epsilon _ { \\beta } ^ * \\end{align*}"}
-{"id": "6681.png", "formula": "\\begin{align*} X _ { 0 } ^ { \\lambda } = \\left \\| \\bigwedge _ { i = 1 } ^ { p } \\nabla D _ i \\right \\| _ { p } ^ { - 2 } \\cdot \\sum _ { i = 1 } ^ { p } ( - 1 ) ^ { n - i } ( - \\lambda ) ( D _ i - d _ i ) \\Theta _ i , \\end{align*}"}
-{"id": "1778.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } ( u ( x ) - v ( x + 1 ) ) ^ { \\prime \\prime } = 0 & ( - a , a ) , \\\\ ( v ( x ) - u ( x - 1 ) ) ^ { \\prime \\prime } = 0 & ( - a , a ) , \\\\ u , v \\ge 0 & ( - a , a ) , \\\\ u ( - a ) = \\phi ( - a ) v ( a ) = \\varphi ( a ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "9509.png", "formula": "\\begin{align*} \\delta ( C \\wedge G \\wedge G ) & = \\delta C \\wedge G \\wedge G + C \\wedge \\delta d C \\wedge G + C \\wedge G \\wedge \\delta d C \\\\ & = \\delta C \\wedge G \\wedge G + 2 C \\wedge d \\delta C \\wedge G \\\\ & = \\delta C \\wedge G \\wedge G - 2 d ( C \\wedge \\delta C \\wedge G ) + 2 d C \\wedge \\delta C \\wedge G \\\\ & = d ( \\ldots ) + 3 \\delta C \\wedge G \\wedge G . \\end{align*}"}
-{"id": "5656.png", "formula": "\\begin{align*} \\alpha = \\sup \\left \\{ \\beta \\leq 2 \\ , ; \\ , \\int _ { \\R ^ d } | v | ^ \\beta M ( v ) \\ , d v < \\infty \\right \\} . \\end{align*}"}
-{"id": "9533.png", "formula": "\\begin{align*} C _ 3 \\wedge G _ 4 \\wedge H _ 3 & = C _ 3 \\wedge G _ 4 \\wedge d B _ 2 \\\\ & = - d ( C _ 3 \\wedge G _ 4 \\wedge B _ 2 ) + G _ 4 \\wedge G _ 4 \\wedge B _ 2 . \\end{align*}"}
-{"id": "253.png", "formula": "\\begin{align*} a ( \\delta ) : = \\max \\{ 1 , A _ { d , m , \\rho } ^ { ( 1 ) } , A _ { d , m , \\rho } ^ { ( 2 ) } \\} , \\end{align*}"}
-{"id": "7450.png", "formula": "\\begin{align*} \\xi _ i = \\sum _ { j = 1 } ^ i ( - 1 ) ^ { i - j } \\Delta ^ { I _ i \\setminus \\{ j \\} , I _ { i - 1 } } \\left ( x \\overline { v ^ { - 1 } } \\right ) \\alpha _ j . \\end{align*}"}
-{"id": "5368.png", "formula": "\\begin{align*} P ( x , y ) ~ = ~ B x + C y + D . \\end{align*}"}
-{"id": "2578.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } S _ { I _ n \\cap [ t _ n , \\infty ) } ( u _ n ) = \\infty . \\end{align*}"}
-{"id": "4799.png", "formula": "\\begin{align*} \\langle H , H \\rangle = \\frac { \\left ( f f '' + ( f ' ) ^ 2 + 1 \\right ) ^ 2 - \\kappa ^ 2 ( f '^ 2 + 1 ) } { 4 f ^ 2 ( f '^ 2 + 1 ) } . \\end{align*}"}
-{"id": "6712.png", "formula": "\\begin{align*} \\Omega = \\nu ^ { - 1 } \\Omega _ { - 1 } + \\Omega _ 0 + \\nu \\Omega _ 1 + \\ldots \\end{align*}"}
-{"id": "2423.png", "formula": "\\begin{align*} \\mathcal { M } _ { p } ( l _ { q _ { 1 } } ^ { s _ { 1 } , 1 } , l _ { q _ { 2 } } ^ { s _ { 2 } , 1 } ) = \\left \\{ \\{ a _ { j } \\} _ { j \\in \\mathbb { N } } : ~ \\Vert \\{ a _ { j } \\lambda _ { j } \\} \\Vert _ { l _ { q _ { 2 } } ^ { s _ { 2 } , 1 } } \\lesssim \\Vert \\{ \\lambda _ { j } \\} \\Vert _ { l _ { q _ { 1 } } ^ { s _ { 1 } , 1 } } \\{ \\lambda _ { j } \\} \\in l _ { q } ^ { s _ 1 , 1 } \\right \\} . \\end{align*}"}
-{"id": "4960.png", "formula": "\\begin{align*} Q _ n \\in L ^ \\infty _ c ( \\R ^ N ) , Q _ n ( y ) = M _ n ^ { p - 2 } c _ n ^ 2 Q ( x _ n + c _ n y ) . \\end{align*}"}
-{"id": "2921.png", "formula": "\\begin{align*} L ^ \\perp _ \\sigma ( w ) \\cdot 1 = \\begin{cases} 1 - w & \\mbox { i f $ \\sigma = ( 0 ) $ } ; \\cr 1 & \\mbox { o t h e r w i s e } , \\cr \\end{cases} \\end{align*}"}
-{"id": "3610.png", "formula": "\\begin{align*} R \\int _ { A _ 1 } D \\Phi ( \\bar { g } ^ R , \\bar { \\pi } ^ R ) \\cdot ( x ^ k , 0 ) \\ , d x & = R \\left ( B ^ 2 _ { ( ( g ^ \\theta ) ^ R , ( \\pi ^ \\theta ) ^ R ) } ( x ^ k , 0 ) - B ^ 1 _ { ( g ^ R , \\pi ^ R ) } ( x ^ k , 0 ) \\right ) \\\\ & = R ^ { - 1 } \\left ( B ^ { 2 R } _ { ( g ^ \\theta , \\pi ^ \\theta ) } ( x ^ k , 0 ) - B ^ R _ { ( g , \\pi ) } ( x ^ k , 0 ) \\right ) , \\end{align*}"}
-{"id": "4134.png", "formula": "\\begin{align*} C \\ : : \\ : z ^ { - ( p + 2 q ) } ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q - x ^ { - p } = 0 , \\end{align*}"}
-{"id": "5550.png", "formula": "\\begin{align*} \\begin{bmatrix} \\frac 1 2 & - h & h ^ 2 \\\\ \\frac 1 2 + H + \\frac 1 2 H ^ 2 & H + H ^ 2 & H ^ 2 \\end{bmatrix} \\begin{bmatrix} \\alpha _ 1 \\\\ \\alpha _ 2 \\\\ \\alpha _ 3 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "9957.png", "formula": "\\begin{align*} & u _ j \\rightharpoonup u _ \\infty \\mbox { i n } L ^ { 2 ^ * } ( \\R ^ n ) \\\\ & u _ j \\to u _ \\infty \\mbox { i n } L ^ p ( \\R ^ n ) , p \\in [ 1 , 2 ^ * ) \\\\ & u _ j \\to u _ \\infty \\mbox { a . e . i n } \\R ^ n \\\\ & | u _ j | ^ { 2 ^ * - 2 } u _ j \\rightharpoonup | u _ \\infty | ^ { 2 ^ * - 2 } u _ \\infty \\mbox { i n } L ^ \\frac { 2 ^ * } { 2 ^ * - 1 } ( \\Omega ) \\end{align*}"}
-{"id": "7887.png", "formula": "\\begin{align*} P ^ \\kappa _ t f ( x ) = T _ t f ( x ) + R _ t f ( x ) . \\end{align*}"}
-{"id": "5201.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , \\widehat { V } \\in \\mathbb { R } ^ { n \\times d } : X = U \\widehat { V } ^ { T } } \\| U \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| \\widehat { V } \\| _ { S _ { \\widehat { p } _ { 2 } } } . \\end{align*}"}
-{"id": "2719.png", "formula": "\\begin{align*} v ( \\varphi , \\psi ) _ t : = \\left ( \\lim _ { l \\to + \\infty } u ^ { l } _ t \\right ) ^ * , \\ t \\in [ 0 , + \\infty ) . \\end{align*}"}
-{"id": "6139.png", "formula": "\\begin{align*} \\frac 3 8 I \\Big | _ { m = \\epsilon _ 0 } = & - 4 z ^ 2 + 8 ( 1 + s \\epsilon _ 0 - 2 \\epsilon _ 0 ) z \\\\ & + [ - 1 + ( 1 - 8 s ) \\epsilon _ 0 + 2 ( 1 + 8 s - 2 s ^ 2 ) \\epsilon _ 0 ^ 2 ] \\ge 0 , \\end{align*}"}
-{"id": "1925.png", "formula": "\\begin{align*} a ( u ) = a ( 0 ) \\exp \\left ( \\int _ 0 ^ u E ( Y ( \\zeta ) ) \\ , d \\zeta \\right ) . \\end{align*}"}
-{"id": "2998.png", "formula": "\\begin{align*} \\int _ X u ( t , x ) d Q = \\int _ { D ( t , p ) } u ( t , x ) d Q & = \\int _ { D ( t , p ) } \\max _ { y \\in B ( t , p ) } u ( t , y ) d Q = \\max _ { y \\in B ( t , p ) } u ( t , y ) . \\end{align*}"}
-{"id": "2241.png", "formula": "\\begin{align*} 1 = C ( \\varepsilon ) \\left ( 1 - \\varepsilon \\mu _ { ( p ) } ^ { G } + \\frac { \\varepsilon ^ { 2 } } { 2 } \\mu _ { ( 2 p ) } ^ { G } - . . . \\right ) = C ( \\varepsilon ) \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon ) ^ { n } } { n ! } \\mu _ { ( n p ) } ^ { G } . \\end{align*}"}
-{"id": "3535.png", "formula": "\\begin{align*} | \\chi J _ 1 + ( 1 - \\chi ) J _ 2 | _ { g } & \\le \\chi | J _ 1 | _ { g } + ( 1 - \\chi ) | J _ 2 | _ { g } \\\\ & \\le \\chi | J _ 1 | _ { g _ 1 } + ( 1 - \\chi ) | J _ 2 | _ { g _ 2 } \\\\ & + \\chi ( 1 - \\chi ) | g _ 1 - g _ 2 | _ { g _ 1 } | J _ 1 | _ { g _ 1 } + ( 1 - \\chi ) \\chi | g _ 1 - g _ 2 | _ { g _ 2 } | J _ 2 | _ { g _ 2 } . \\end{align*}"}
-{"id": "6717.png", "formula": "\\begin{align*} \\dim ( \\vec { c } \\ , ) = h ( \\vec { c } \\ , ) + k - 1 = \\sum _ { i = 1 } ^ { k } h ( c _ { i } ) + 2 k - 2 . \\end{align*}"}
-{"id": "369.png", "formula": "\\begin{align*} e ^ { \\Omega _ e } _ { L ^ { \\mathrm { p } } } = \\frac { \\lVert p _ { n u m } ^ { \\Omega _ e } - p _ { r e f } ^ { \\Omega _ e } \\rVert _ { L ^ { \\mathrm { p } } } } { \\lVert p _ { r e f } ^ { \\Omega _ e } \\rVert _ { L ^ { \\mathrm { p } } } } , \\end{align*}"}
-{"id": "4294.png", "formula": "\\begin{align*} 5 \\epsilon n s ^ { n - 1 } = \\pm 1 2 5 - \\sum _ { j = 2 } ^ { n } \\binom { n } { j } ( 5 \\epsilon ) ^ j s ^ { n - j } . \\end{align*}"}
-{"id": "7107.png", "formula": "\\begin{align*} \\forall t \\in ( 0 , 1 ) \\lim _ { n \\to \\infty } \\omega _ { n , \\beta } ( ( t , 1 ] ) = \\pi _ \\beta . \\end{align*}"}
-{"id": "8539.png", "formula": "\\begin{align*} \\pi _ { r a } & = \\pi _ { r a } ' \\pi _ { a } ' : N _ { i n } \\rightarrow N _ { \\tau ( a ) } \\\\ \\xi _ { r a } & = \\xi _ { a } ' \\xi _ { r a } ' : N _ { \\tau ( a ) } \\rightarrow N _ { i n } \\end{align*}"}
-{"id": "2370.png", "formula": "\\begin{align*} x ^ { k + 1 } & = x ^ k - \\alpha P ^ { - 1 } \\nabla F ( x ^ k ) . \\end{align*}"}
-{"id": "6044.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - \\Delta u _ { i } ^ { \\varepsilon } = f _ { i } ( x , u _ { i } ^ { \\varepsilon } ) - \\frac { 1 } { \\varepsilon } u _ { i } ^ { \\varepsilon } \\sum \\limits _ { j \\neq i } ( u _ { j } ^ { \\varepsilon } ) ^ { \\beta } ( x ) & \\Omega , \\\\ u _ { i } ( x ) = \\phi _ { i } ( x ) & \\partial \\Omega , \\\\ i = 1 , \\cdots , m . \\end{array} \\right . \\end{align*}"}
-{"id": "6134.png", "formula": "\\begin{align*} \\frac d { d t } \\Big [ ( \\kappa + \\delta t ) R \\Big ] = & ( \\kappa + \\delta t ) \\frac d { d t } R + \\delta R \\\\ = & ( \\kappa + \\delta t ) \\cdot 2 | R i c | ^ 2 + \\delta R \\\\ = & ( \\kappa + \\delta t ) \\cdot \\frac { R ^ 2 } 2 + \\delta R \\\\ \\end{align*}"}
-{"id": "1484.png", "formula": "\\begin{align*} f - m _ f ( Q _ 0 ) = g _ 1 + \\sum _ { j \\in J _ 1 } \\alpha _ { j , 1 } \\chi _ { Q ^ 1 _ j } + \\sum _ { j \\in J _ 1 } ( f - m _ f ( Q ^ 1 _ j ) ) \\chi _ { Q ^ 1 _ j } , \\end{align*}"}
-{"id": "9479.png", "formula": "\\begin{align*} \\lambda _ { 0 } = d \\theta + \\alpha _ { n } \\end{align*}"}
-{"id": "5661.png", "formula": "\\begin{align*} D = \\int \\lambda ( v ) \\otimes v \\ , d v , Q ( \\lambda ) = \\nabla _ v M ( v ) . \\end{align*}"}
-{"id": "6660.png", "formula": "\\begin{gather*} I _ k ^ n : = \\left [ \\frac { k - 1 } { n } ; \\frac { k } { n } \\right ) , 1 \\leqslant k \\leqslant n - 1 , n \\geqslant 2 , \\\\ I _ n ^ n : = \\left [ \\frac { n - 1 } { n } ; 1 \\right ] , n \\geqslant 1 . \\end{gather*}"}
-{"id": "1510.png", "formula": "\\begin{align*} \\check K ( x ) = ( 1 - x \\bar e _ 0 ) \\left ( 1 - \\frac 1 x \\bar e _ 0 \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "7822.png", "formula": "\\begin{align*} \\eta _ { t } = \\eta _ { 0 } + \\int _ 0 ^ t b _ { s } d s + \\int _ 0 ^ t \\sigma _ { s } d B _ { s } ^ { H } , \\end{align*}"}
-{"id": "4485.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 5 | S _ i | = O \\biggl \\{ \\max \\biggl ( \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\log n \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\biggr \\} \\end{align*}"}
-{"id": "8814.png", "formula": "\\begin{align*} Y ( a , z ) = e ^ { z T } ( a ) = \\sum _ { n \\geq 0 } \\frac { z ^ n } { n ! } T ^ n ( a ) \\end{align*}"}
-{"id": "8004.png", "formula": "\\begin{align*} \\int _ C ^ { \\infty } e ^ { \\frac { - t y ^ 2 } { 2 } } d y \\geq e ^ { \\frac { - ( t - 1 ) C ^ 2 } { 2 } } \\int _ C ^ { \\infty } e ^ { - \\frac { y ^ 2 } { 2 } } d y = C ' e ^ { - C '' t } , \\end{align*}"}
-{"id": "5027.png", "formula": "\\begin{align*} ( s \\cdot \\mu ) ( \\phi ) = \\mu ( s ^ { - 1 } \\cdot \\phi ) , \\textrm { f o r $ \\phi \\in C ( X ) $ } . \\end{align*}"}
-{"id": "649.png", "formula": "\\begin{align*} ( T \\phi ) ( g ) = \\phi ( g \\cdot x _ o ) , \\textrm { f o r $ \\phi \\in C ( X ) $ } . \\end{align*}"}
-{"id": "9560.png", "formula": "\\begin{align*} \\lambda _ { n } ^ { ( 2 ) } = \\sum _ { i \\neq n } \\frac { \\left | \\langle \\psi ^ { ( 0 ) } _ { i } , H ^ { ( 1 ) } \\psi ^ { ( 0 ) } _ { n } \\rangle \\right | ^ { 2 } } { \\lambda _ { n } ^ { ( 0 ) } - \\lambda _ { i } ^ { ( 0 ) } } + \\langle \\psi _ { n } ^ { ( 0 ) } , H ^ { ( 2 ) } \\psi _ { n } ^ { ( 0 ) } \\rangle \\end{align*}"}
-{"id": "1689.png", "formula": "\\begin{align*} { \\varphi } ( Z _ t ) = { \\varphi } ( Z _ s ) + \\int _ s ^ t \\big [ b ( Z _ r ) \\cdot D { \\varphi } ( Z _ r ) + \\frac { 1 } { 2 } \\Delta _ v { \\varphi } ( Z _ r ) \\big ] \\dd r + \\int _ s ^ t D _ v { \\varphi } ( Z _ r ) \\ , \\dd W _ r \\ , . \\end{align*}"}
-{"id": "3473.png", "formula": "\\begin{align*} \\mathcal { P } f ( x ) = \\sum _ { n \\in ( L _ 1 ^ { - 1 } \\mathbb { Z } ) \\times ( L _ 2 ^ { - 1 } \\mathbb { Z } ) } e ^ { 2 \\pi i n \\cdot x _ \\ast } e ^ { 2 \\pi \\abs { n } x _ 3 } \\hat { f } ( n ) , \\end{align*}"}
-{"id": "3760.png", "formula": "\\begin{align*} = \\lim _ { A \\rightarrow \\infty } \\frac { 1 } { 2 \\pi } \\int _ { - A } ^ { A } e ^ { i y t } \\left \\{ e ^ { C _ 0 ( \\beta - i \\alpha t ) - C _ 0 ( \\beta ) + i \\alpha t u } - P ( T _ \\beta = 0 ) e ^ { i \\alpha t u } \\right \\} d t , \\end{align*}"}
-{"id": "3277.png", "formula": "\\begin{align*} \\hat { R } _ { V , W } ( v _ { \\mathrm { l w } } \\otimes w ) = q ^ { ( \\mathrm { w t } ( v _ { \\mathrm { l w } } ) , \\mathrm { w t } ( w ) ) } w \\otimes v _ { \\mathrm { l w } } . \\end{align*}"}
-{"id": "277.png", "formula": "\\begin{align*} \\mathrm { B } _ { a , b , c } ( s , t ) : = \\frac { s ^ { a - 1 } t ^ { b - 1 } ( 1 - s - t ) ^ { c - 1 } } { \\mathrm { B } _ { a , b , c } } \\end{align*}"}
-{"id": "2379.png", "formula": "\\begin{align*} P _ { \\gamma f } ^ { \\alpha } ( x ) : = \\nabla p _ { \\gamma f } ^ { \\alpha } ( x ) = \\alpha { \\rm { p r o x } } _ { \\gamma f } ( x ) + ( 1 - \\alpha ) x . \\end{align*}"}
-{"id": "2156.png", "formula": "\\begin{align*} L u ( t , x ) & = \\int _ { \\mathbb { R } ^ { n } } [ \\tilde { u } ( s , y ) - \\tilde { u } ( s , z ) ] \\tilde { a } ( y , z ) r ^ { - n - 2 \\beta } \\tilde { k } _ { 0 } ( y , z ) r ^ { n } d z \\\\ & = r ^ { - 2 \\beta } L \\tilde { u } ( s , y ) . \\end{align*}"}
-{"id": "6503.png", "formula": "\\begin{align*} & \\int _ { I ' } \\mathcal { E } ( ( h _ { m } * u - u ) ( t , \\cdot ) , \\phi ( t , \\cdot ) ) d t \\\\ = & \\int _ { I ' } \\int _ { B } \\int _ { B } ( ( h _ { m } * V ) ( t , x , y ) - V ( t , x , y ) ) \\Phi ( t , x , y ) k ( x , y ) d x d y d t \\\\ & + 2 \\int _ { I ' } \\int _ { B } \\phi ( t , x ) \\int _ { B ^ { c } } ( ( h _ { m } * V ) ( t , x , y ) - V ( t , x , y ) ) k ( x , y ) d y d x d t \\\\ = : & _ { m } + _ { m } . \\end{align*}"}
-{"id": "9671.png", "formula": "\\begin{align*} y ' ( t ) = - ( a - b ) g ( y ( t ) ) , t > 0 ; y ( 0 ) = x _ 0 > 0 , \\end{align*}"}
-{"id": "2204.png", "formula": "\\begin{align*} \\frac { - \\int _ { B } \\psi ^ { 2 } \\tilde { u } ^ { - 1 } \\partial _ { t } ( g _ { 1 - \\alpha , m } * \\tilde { u } ) d x } { \\int _ { B } \\psi ^ { 2 } ( x ) d x } + \\frac { c _ { 2 } } { r ^ { 2 \\beta } \\mu _ { n } ( B ) } \\int _ { B } ( w - W ) ^ { 2 } \\psi ^ { 2 } d x \\leq \\frac { C _ { 2 } } { r ^ { 2 \\beta } } + S _ { m } ( t ) , \\end{align*}"}
-{"id": "9692.png", "formula": "\\begin{align*} g ( y ( t ) ) = g ( y ( 0 ) ) \\exp \\left ( - C \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s \\right ) , \\end{align*}"}
-{"id": "5516.png", "formula": "\\begin{align*} \\phi _ \\mathcal { M } ( t ) = \\frac { t } { W ( t ) } \\Big { / } \\varphi ^ { - 1 } \\left ( \\frac { 1 } { W ( t ) } \\right ) , \\ \\ \\ t \\in ( 0 , \\gamma ) . \\end{align*}"}
-{"id": "7084.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { | u | = n } f _ \\mu ( u ) = 0 . \\end{align*}"}
-{"id": "1090.png", "formula": "\\begin{align*} \\sup \\{ | | P ^ n ( x , \\cdot ) - P ^ n ( y , \\cdot ) | | : x , y \\in S \\} \\leq C \\rho ^ n , \\ ; n = 1 , 2 . . . \\end{align*}"}
-{"id": "5094.png", "formula": "\\begin{align*} A ^ { [ M ' , M ] } ( z ) = \\mathbb { T } ^ { [ M ' , M ] } ( z ) _ { 0 0 } , C _ { a } ^ { [ M ' , M ] } ( z ) = \\mathbb { T } ^ { [ M ' , M ] } ( z ) _ { a 0 } \\ , \\ , ( 1 \\le a \\le r ) . \\end{align*}"}
-{"id": "2650.png", "formula": "\\begin{align*} \\Delta ^ m _ 0 : = \\widehat { Y } ^ m _ 1 \\Delta ^ m _ { - 1 } : = K _ 1 \\ , \\end{align*}"}
-{"id": "1797.png", "formula": "\\begin{align*} \\mathcal { V } u ( t , x ) & = \\sum _ { j = 1 } ^ d V _ j ^ 2 u ( t , x ) \\\\ u ( 0 , x ) & = f ( x ) , \\end{align*}"}
-{"id": "1188.png", "formula": "\\begin{align*} m _ { \\zeta } = \\frac 1 2 ( u _ { \\zeta \\zeta } - u ) _ { \\zeta } . \\end{align*}"}
-{"id": "8337.png", "formula": "\\begin{align*} u _ 1 ( T ' ) = \\sum _ { j = 1 } ^ { r ' } \\frac { Q ' _ j ( T ' ) } { ( P ' _ j ( T ' ) ) ^ { e ' _ j } } + u ( \\nu ) . \\end{align*}"}
-{"id": "440.png", "formula": "\\begin{align*} A _ q ( k - 1 ) = \\begin{pmatrix} q - 1 & 1 & & & \\\\ q - 1 & & 1 & & \\\\ \\vdots & & & \\ddots & \\\\ q - 1 & & & & 1 \\\\ q - 1 & & & & \\end{pmatrix} . \\end{align*}"}
-{"id": "5048.png", "formula": "\\begin{align*} \\nu ( B ) = \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\chi _ B ) \\chi _ B ) \\ , d \\eta ( g ) = \\nu ( B ) ^ 2 , \\end{align*}"}
-{"id": "6415.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ { s } ( \\Omega ) } ^ { 2 } = \\| u \\| _ { L ^ { 2 } ( \\Omega ) } ^ { 2 } + c _ { n , s } \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) - u ( y ) | ^ { 2 } } { | x - y | ^ { n + 2 s } } d x d y . \\end{align*}"}
-{"id": "7048.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ 1 ] } } ( 1 ) = { { \\bf { u } } ^ { [ 1 ] } } , { { \\bf { X } } ^ { [ 2 ] } } ( 1 ) = { { \\bf { u } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "1910.png", "formula": "\\begin{align*} \\psi _ m ( u ) = \\psi ( u - L _ n - 1 2 m ) + \\overline { \\psi } ( u + L _ n + 1 2 m ) . \\end{align*}"}
-{"id": "7316.png", "formula": "\\begin{align*} \\begin{gathered} \\hat { R } ( v _ { 1 } \\otimes v _ { 1 } ) = q ^ { 2 } v _ { 1 } \\otimes v _ { 1 } , \\quad \\hat { R } ( v _ { 0 } \\otimes v _ { 1 } ) = v _ { 1 } \\otimes v _ { 0 } , \\quad \\hat { R } ( v _ { - 1 } \\otimes v _ { 1 } ) = q ^ { - 2 } v _ { 1 } \\otimes v _ { - 1 } , \\\\ \\hat { R } ( v _ { - 1 } \\otimes v _ { 0 } ) = v _ { 0 } \\otimes v _ { - 1 } , \\quad \\hat { R } ( v _ { - 1 } \\otimes v _ { - 1 } ) = q ^ { 2 } v _ { - 1 } \\otimes v _ { - 1 } . \\end{gathered} \\end{align*}"}
-{"id": "121.png", "formula": "\\begin{align*} \\frac { 2 } { e ^ t + 1 } & = \\sum _ { n = 0 } ^ \\infty C _ n \\frac { 1 } { 4 ^ n } \\big ( 1 - e ^ { 2 t } \\big ) ^ n \\\\ & = \\sum _ { n = 0 } ^ \\infty C _ n \\frac { ( - 1 ) ^ n } { 4 ^ n } \\big ( e ^ { 2 t } - 1 \\big ) ^ n \\\\ & = \\sum _ { n = 0 } ^ \\infty C _ n \\frac { ( - 1 ) ^ n } { 4 ^ n } n ! \\sum _ { m = n } ^ \\infty S _ 2 ( m , n ) \\frac { 2 ^ m t ^ m } { m ! } \\\\ & = \\sum _ { m = 0 } ^ \\infty \\left ( \\sum _ { n = 0 } ^ m C _ n ( - 1 ) ^ n 2 ^ { m - 2 n } n ! S _ 2 ( m , n ) \\right ) \\frac { t ^ m } { m ! } . \\end{align*}"}
-{"id": "9156.png", "formula": "\\begin{align*} k \\geq ( t - 1 ) m / p + \\frac { 1 } { 2 } ( m - ( t - 1 ) m / p ) = \\frac { m } { 2 } ( 1 + ( t - 1 ) / ( s t ) ) . \\end{align*}"}
-{"id": "5147.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\ell - 1 } f ( z _ { i } , z _ { \\ell } ) + \\sum _ { t = \\ell + 1 } ^ { m } g ( z _ { \\ell } , z _ { t } ) \\prod _ { \\begin{subarray} { c } i = 1 \\\\ i \\not = \\ell \\end{subarray} } ^ { t - 1 } f ( z _ { i } , z _ { \\ell } ) = \\prod _ { \\begin{subarray} { c } i = 1 \\\\ i \\not = \\ell \\end{subarray} } ^ { m } f ( z _ { i } , z _ { \\ell } ) \\end{align*}"}
-{"id": "4510.png", "formula": "\\begin{align*} \\max _ { r = 1 , \\ldots , m } \\frac { \\| f ^ { ( r ) } ( x ) \\| } { f ( x ) } \\leq d ^ { m / 2 } \\max _ { r = 1 , \\ldots , m } q _ r ( \\| x \\| ) . \\end{align*}"}
-{"id": "2666.png", "formula": "\\begin{align*} P _ { [ \\theta , \\psi ] } ( \\varphi ) : = \\Big \\{ \\lim _ { C \\to + \\infty } P _ { \\theta } ( \\min ( \\psi + C , \\varphi ) ) \\Big \\} ^ * . \\end{align*}"}
-{"id": "6226.png", "formula": "\\begin{align*} \\min \\left | \\left | I - \\mathcal { K } _ { i } ^ { ( z ) } P _ { i } ^ { ( z ) } \\right | \\right | _ { F } ^ { 2 } = \\min \\sum _ { j = 1 } ^ { n } \\left | \\left | e ^ { ( j ) } - \\mathcal { K } _ { i } ^ { ( z ) } p ^ { ( j ) } _ { i } \\right | \\right | ^ { 2 } _ 2 , \\end{align*}"}
-{"id": "6361.png", "formula": "\\begin{align*} h ( x , r ) : = \\sum _ { m = 0 } ^ k r ^ m h _ m ( x ) \\end{align*}"}
-{"id": "8299.png", "formula": "\\begin{align*} 2 g ( \\tilde { x } ) + 2 = \\sum _ { \\tilde { v } \\in T _ { \\tilde { x } } ( \\widetilde { \\Gamma } ) } ( d _ { \\tilde { v } } ( \\varphi ) - 1 ) \\end{align*}"}
-{"id": "5247.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } B _ { 2 } ^ { ( l , l ) } = B _ { l , 2 } & & \\\\ B _ { m , 1 } B _ { 2 } ^ { ( m , l ) } - B _ { 2 } ^ { ( m , l ) } B _ { l , 1 } + i _ { m } i _ { l } ^ { * } = 0 & & \\mbox { i f $ m \\neq l $ } \\end{array} \\right . \\end{align*}"}
-{"id": "2745.png", "formula": "\\begin{align*} a _ K ^ I \\tilde a _ P ^ K = a _ K ^ I \\star \\tilde a _ P ^ K = \\delta ^ I _ P , b _ L ^ J \\tilde b _ Q ^ L = b _ L ^ J \\star \\tilde b _ Q ^ L = \\delta ^ J _ Q . \\end{align*}"}
-{"id": "9326.png", "formula": "\\begin{align*} \\theta ( t , z ) = \\beta ( t , c _ 1 , z ) - \\beta ( t , c _ 2 , z ) . \\end{align*}"}
-{"id": "4858.png", "formula": "\\begin{align*} I ^ { \\rm e x p } _ { 2 2 } ( i , x ; j , y ) : = \\int _ { \\mathcal { C } _ { b _ z } ^ { \\pi / 3 } } \\dd z \\int _ { \\mathcal { C } _ { b _ w } ^ { \\pi / 3 } } \\dd w \\frac { ( z - w ) e ^ { - x z - y w } } { z + w } \\frac { ( 1 + 2 z ) ^ { m _ i } ( 1 + 2 w ) ^ { m _ j } } { ( 1 - 2 z ) ^ { n _ i } ( 1 - 2 w ) ^ { n _ j } } \\frac { 1 } { 2 \\alpha - 1 - 2 z } \\frac { 1 } { 2 \\alpha - 1 - 2 w } , \\end{align*}"}
-{"id": "8022.png", "formula": "\\begin{align*} \\phi ( z _ j ) = \\tau _ j \\mu _ j \\nu _ j z ' _ j . \\end{align*}"}
-{"id": "947.png", "formula": "\\begin{align*} [ X , \\overline { Y } ] _ p = \\lambda ( X , \\overline { Y } ) ( p ) T ( p ) \\ ; T ^ { 1 , 0 } _ p M \\oplus T ^ { 0 , 1 } _ p M \\ ; . \\end{align*}"}
-{"id": "8907.png", "formula": "\\begin{align*} R _ \\lambda ( - \\theta ) \\circ \\Hat { A } \\circ R _ \\lambda ( \\theta ) = \\Hat { A } + \\bigl ( 2 ( \\cos \\theta - 1 ) + ( ( \\cos \\theta - 1 ) ^ 2 + ( \\sin \\theta ) ^ 2 ) \\bigr ) \\Hat { A } \\circ P _ { W _ \\lambda } = \\Hat { A } . \\end{align*}"}
-{"id": "4879.png", "formula": "\\begin{align*} a \\equiv b + c \\bmod I ^ n c \\in I ^ { n - 1 } \\Rightarrow \\delta ( a ) \\equiv \\delta ( b ) + \\delta ( c ) \\bmod I ^ { n - 1 } \\end{align*}"}
-{"id": "5837.png", "formula": "\\begin{align*} T ^ { I } L _ { Y _ { I } } C _ { \\alpha } - T _ { , t } ^ { I } Y _ { I \\alpha } - 2 a _ { , \\alpha } = 0 . \\end{align*}"}
-{"id": "9423.png", "formula": "\\begin{align*} \\left \\lvert \\bigl ( \\phi ^ G _ G \\bigr ) ' ( x ) \\right \\lvert = w _ 1 ( x ) / w _ 2 ( \\phi ( x ) ) . \\end{align*}"}
-{"id": "7724.png", "formula": "\\begin{align*} ( 1 - \\varepsilon ) \\sum _ { i = 0 } ^ { i = J - 1 } \\sigma _ { N + i } \\ge ( 1 - \\varepsilon ) J \\sigma _ { N + J - 1 } \\ge M \\sigma _ { N + J - 1 } . \\end{align*}"}
-{"id": "2321.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { 1 , q } ( \\R ^ d ) ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { 1 , q } ( \\R ^ d ) ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "5053.png", "formula": "\\begin{align*} \\sigma _ k = \\frac { 1 } { 2 } ( \\delta _ { k } + \\delta _ { - k } ) , \\textrm { f o r $ k \\geq 1 $ } , \\end{align*}"}
-{"id": "5964.png", "formula": "\\begin{gather*} \\int _ { \\R ^ { 2 d } } g ( t , z ) \\eta ( v / m ) \\ , \\dd z + \\int _ 0 ^ t \\int _ { \\R ^ { 2 d } } F ( z ) \\cdot D _ v g ( s , z ) \\eta ( v / m ) \\ , \\dd z \\dd s \\\\ = \\frac { 1 } { 2 m ^ 2 } \\int _ 0 ^ t \\int _ { \\R ^ { 2 d } } g ( s , z ) \\Delta \\eta ( v / m ) \\ , \\dd z \\dd s \\ , . \\end{gather*}"}
-{"id": "6770.png", "formula": "\\begin{align*} \\int _ { U } \\theta _ { V _ { \\theta } } ^ n \\leq \\liminf _ { \\beta \\to + \\infty } \\int _ U \\theta _ { \\varphi _ { \\beta } } ^ n \\leq \\liminf _ { \\beta \\to + \\infty } \\int _ U e ^ { - \\beta \\delta } \\theta _ + ^ n = 0 . \\end{align*}"}
-{"id": "4732.png", "formula": "\\begin{align*} ( \\beta _ 1 v _ { 1 1 } + \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } + \\beta _ 2 v _ { 2 2 } ) x _ 2 & = \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 , \\\\ ( \\beta _ 1 v _ { 1 1 } - \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } - \\beta _ 2 v _ { 2 2 } ) x _ 2 & = \\beta _ 1 \\alpha _ 1 - \\beta _ 2 \\alpha _ 2 . \\end{align*}"}
-{"id": "8013.png", "formula": "\\begin{align*} \\{ a , x \\} = a \\partial ( x ) \\forall x \\in P . \\end{align*}"}
-{"id": "6022.png", "formula": "\\begin{align*} \\hat { \\Omega } _ { i j } { } ^ k { } _ l = \\hat { R } _ { i j } { } ^ k { } _ l + \\varepsilon \\hat { g } _ { j l } \\delta ^ k { } _ i - \\varepsilon \\hat { g } _ { i l } \\delta ^ k { } _ j + \\hat { K } _ { j l } \\hat { K } ^ { k } { } _ i - \\hat { K } _ { i l } \\hat { K } ^ k { } _ j - 2 \\ , \\hat { K } _ { i j } \\hat { K } ^ k { } _ { l } , \\end{align*}"}
-{"id": "211.png", "formula": "\\begin{align*} \\sup _ { k \\in \\{ 1 , \\ldots , k ^ * \\} } \\frac { k ^ { 2 / d } } { n ^ { 2 / d } } \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\int _ { \\mathcal { X } _ n ^ c } \\frac { \\Delta f ( x ) } { f ( x ) ^ { 2 / d } } \\ , d x = O \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\end{align*}"}
-{"id": "1923.png", "formula": "\\begin{align*} u = u ( \\tau ) \\doteqdot \\int _ { 0 } ^ \\tau \\frac { 1 } { a ( \\zeta ) } \\ , \\ , d \\zeta . \\end{align*}"}
-{"id": "7616.png", "formula": "\\begin{align*} \\sum _ { z \\in S } \\lambda _ 1 \\mathbf { v } _ z = \\sum _ { z \\in S } \\sum _ { y \\sim z } \\mathbf { v } _ y \\leq 2 \\epsilon e ( S ) + e ( S , L ) \\leq ( 6 n - 1 2 ) \\epsilon + ( 2 n - 4 ) . \\end{align*}"}
-{"id": "5658.png", "formula": "\\begin{align*} & M > 0 , M ( v ) = M ( - v ) \\mbox { f o r a l l } v \\in \\R ^ d , \\int _ { \\R ^ d } M ( v ) \\d v = 1 , \\\\ & | v | ^ { d + \\alpha } M ( v ) \\longrightarrow \\gamma > 0 \\ , , \\qquad \\mbox { a s } | v | \\rightarrow \\infty , \\mbox { w h e r e } 1 \\leq \\alpha < 2 , \\end{align*}"}
-{"id": "1991.png", "formula": "\\begin{align*} \\Sigma = \\begin{pmatrix} \\sigma _ U ^ 2 & \\sigma _ { U L } \\\\ \\sigma _ { U L } & \\sigma _ L ^ 2 \\end{pmatrix} \\in \\R ^ { 2 \\times 2 } \\end{align*}"}
-{"id": "3925.png", "formula": "\\begin{align*} z = ( n + m + 1 ) ( n - m ) \\ , , \\end{align*}"}
-{"id": "6967.png", "formula": "\\begin{align*} \\sigma u p ' ( u ) q ( u ) + ( 1 - \\sigma ) u p ( u ) q ' ( u ) = \\mu _ 1 \\gamma p ( u ) q ( u ) , \\end{align*}"}
-{"id": "4461.png", "formula": "\\begin{align*} \\alpha _ r ( s , t ) : = \\frac { 1 } { V _ d } \\mu _ d \\bigl ( B _ 0 ( s ^ { 1 / d } ) \\cap B _ { r ^ { 1 / d } e _ 1 } ( t ^ { 1 / d } ) \\bigr ) , \\end{align*}"}
-{"id": "4057.png", "formula": "\\begin{align*} 1 \\leq q < p < 2 p - q = 1 + k + n \\leq 6 , \\end{align*}"}
-{"id": "6655.png", "formula": "\\begin{align*} \\varepsilon _ K ( \\chi , \\psi ) = \\chi ( c ) \\frac { \\int _ { \\mathcal { O } ^ { \\times } } \\chi ^ { - 1 } ( u ) \\psi ( u / c ) \\ , d u } { | \\int _ { \\mathcal { O } ^ { \\times } } \\chi ^ { - 1 } ( u ) \\psi ( u / c ) \\ , d u | } \\end{align*}"}
-{"id": "8052.png", "formula": "\\begin{align*} \\pi ^ { * } ( D _ { 2 } ) = \\sum _ { i \\ge 2 } ^ { \\lfloor n / 2 \\rfloor } { i \\choose 2 } B _ { i } \\end{align*}"}
-{"id": "8905.png", "formula": "\\begin{align*} D u ( x ) [ \\Hat { A } ( P _ { W _ \\lambda } ( x ) ] = 0 . \\end{align*}"}
-{"id": "3584.png", "formula": "\\begin{align*} D \\Phi ^ W _ { ( g , \\pi ) } \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f _ m , X _ m ) = \\Phi ^ W _ { ( g , \\pi ) } ( g , \\pi ) + ( \\psi , V ) - \\Phi ^ W _ { ( g , \\pi ) } ( \\gamma _ m , \\tau _ m ) , \\end{align*}"}
-{"id": "9232.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial u } ( t , x , z ) = 0 ( t , x ) \\in [ 0 , T ] \\times D . \\end{align*}"}
-{"id": "4025.png", "formula": "\\begin{align*} S = \\{ x \\in { \\mathcal M } _ \\rho : A ( x - x ^ \\dag ) = 0 \\} . \\end{align*}"}
-{"id": "7934.png", "formula": "\\begin{align*} c _ { i , \\ell } = m d _ i + \\frac { m - 1 } { 2 } . \\end{align*}"}
-{"id": "5796.png", "formula": "\\begin{align*} \\gamma _ { i } = \\prod _ { j = 1 } ^ { i } ( 1 - 2 j ( 2 j - 1 ) \\gamma ^ { 2 } ) . \\end{align*}"}
-{"id": "2793.png", "formula": "\\begin{align*} \\biggl ( \\frac { ( d - 1 ) ^ { d - 1 } } { d ^ { d / 2 } } \\biggr ) ^ { n } = 2 ^ { - n } \\left ( \\frac { e } { 2 } \\right ) ^ { x } \\exp \\left ( \\frac { 3 x ^ { 2 } } { 2 n } + O \\left ( \\frac { x ^ { 3 } } { n ^ { 2 } } \\right ) \\right ) . \\end{align*}"}
-{"id": "2289.png", "formula": "\\begin{align*} \\big \\| \\big ( ( A _ m - A _ n ) v _ n \\big ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } ^ p \\le c _ \\star \\sum _ { n = k } ^ { m - 1 } \\| A _ { n + 1 } - A _ n \\| E _ n . \\end{align*}"}
-{"id": "2114.png", "formula": "\\begin{align*} h ^ { \\mu m } ( x , r ) : = r ^ m h ^ { \\mu } _ { m } ( x ) + m r ^ { m - 1 } h ^ { \\mu } _ { m - 1 } ( x ) , \\end{align*}"}
-{"id": "9080.png", "formula": "\\begin{align*} L ^ { ( \\alpha ) } _ { N } ( x ) = \\sum _ { i = 0 } ^ { N } ( - 1 ) ^ { i } \\binom { N + \\alpha } { N - i } \\frac { x ^ { i } } { i ! } \\ . \\end{align*}"}
-{"id": "3834.png", "formula": "\\begin{align*} \\norm { \\partial _ t ^ i D ^ j v } { H ^ { k - m , 2 k - 2 m } ( D _ T ) } & = \\sum _ { \\ell = 0 } ^ { k - m } \\norm { \\partial _ t ^ { \\ell + i } D ^ j v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 m - 2 \\ell } ( D ) ) } \\\\ & = \\sum _ { \\ell = i } ^ { k - m + i } \\norm { \\partial _ t ^ { \\ell } D ^ j v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 m - 2 \\ell + 2 i } ( D ) ) } . \\end{align*}"}
-{"id": "3503.png", "formula": "\\begin{align*} \\pi ^ { i j } = K ^ { i j } - ( \\textup { t r } _ g K ) g ^ { i j } . \\end{align*}"}
-{"id": "172.png", "formula": "\\begin{align*} \\rho _ { ( k ) , i } : = \\| X _ { ( k ) , i } - X _ i \\| \\end{align*}"}
-{"id": "9769.png", "formula": "\\begin{align*} \\boldsymbol { \\lambda } _ { \\mathcal { E } } ^ { * } = - ( \\vec { G } _ { \\mathcal { E } } \\vec { H } ^ { - 1 } \\vec { G } _ { \\mathcal { E } } ^ { \\top } ) ^ { - 1 } ( \\vec { W } _ { \\mathcal { E } } + \\vec { S } _ { \\mathcal { E } } \\vec { x } ) , \\forall \\vec { x } \\in \\mathcal { R } ( \\mathcal { E } ) , \\end{align*}"}
-{"id": "200.png", "formula": "\\begin{align*} \\alpha _ r ( s , t ) : = \\frac { 1 } { V _ d } \\mu _ d \\bigl ( B _ 0 ( s ^ { 1 / d } ) \\cap B _ { r ^ { 1 / d } e _ 1 } ( t ^ { 1 / d } ) \\bigr ) , \\end{align*}"}
-{"id": "4506.png", "formula": "\\begin{align*} r _ { d , \\theta } : = \\Bigl \\{ \\frac { 7 ( 1 - e ^ { - 2 A _ { d , \\theta } } ) } { 3 0 d ^ { 1 / 2 } a ( e ^ { - 4 A _ { d , \\theta } } ) } \\Bigr \\} ^ { 1 / ( \\beta \\wedge 1 ) } \\wedge \\frac { 1 } { 8 d ^ { 1 / 2 } a ( e ^ { - 4 A _ { d , \\theta } } ) \\} ^ { 1 / ( \\beta \\wedge 1 ) } } . \\end{align*}"}
-{"id": "2446.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = & \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ { \\infty } } \\lesssim \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ { \\infty } } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ { p _ 1 } } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "3113.png", "formula": "\\begin{align*} L _ 5 ^ \\prime ( \\lambda ) = \\begin{bmatrix} 0 & 0 & \\lambda I _ n & - I _ n & 0 \\\\ 0 & 0 & 0 & \\lambda I _ n & - I _ n \\\\ \\lambda I _ n & 0 & - \\lambda P _ 1 + P _ 0 & - \\lambda P _ 2 + P _ 1 & P _ 2 \\\\ - I _ n & \\lambda I _ n & - \\lambda P _ 2 + P _ 1 & - \\lambda P _ 3 + P _ 2 & P _ 3 \\\\ 0 & - I _ n & P _ 2 & P _ 3 & \\lambda P _ 5 + P _ 4 \\end{bmatrix} . \\end{align*}"}
-{"id": "2225.png", "formula": "\\begin{align*} \\| \\rho \\| _ { L ^ { 1 } ( ( 0 , T ] ) } \\leq \\frac { T ^ { 1 - \\alpha } } { \\Gamma ( 2 - \\alpha ) } \\| \\mu \\| _ { L ^ { 1 } ( ( 0 , T ] ) } = 0 , \\end{align*}"}
-{"id": "2130.png", "formula": "\\begin{align*} \\mathcal { K } : = \\{ | z | > | x ' | \\} , \\end{align*}"}
-{"id": "4744.png", "formula": "\\begin{align*} ( B ^ c _ \\varepsilon ( x ) ) ^ \\pm : = \\big \\{ y \\in B _ \\varepsilon ^ c ( x ) : \\ , \\pm \\xi \\cdot ( y - x ) \\le 0 \\big \\} , \\end{align*}"}
-{"id": "2503.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mathcal { L } _ { X } I _ 1 = \\dots = \\mathcal { L } _ { X } I _ k = 0 , \\\\ \\mathcal { L } _ { X } D _ 1 = h _ 1 , \\dots , \\mathcal { L } _ { X } D _ p = h _ p , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6145.png", "formula": "\\begin{align*} \\Omega _ G = \\Omega - \\chi ^ a \\cdot \\iota _ a \\Theta \\end{align*}"}
-{"id": "6115.png", "formula": "\\begin{align*} \\alpha _ j = d \\log z _ i = d ( - s _ i ^ { - 1 } + i \\theta _ i ) , 1 \\le i \\le k ; \\ \\alpha _ j = d z _ j , 3 g - 3 + n \\geq j > k \\end{align*}"}
-{"id": "3084.png", "formula": "\\begin{align*} V = \\mathcal { M } \\oplus \\operatorname { k e r } ( S ) . \\end{align*}"}
-{"id": "8911.png", "formula": "\\begin{align*} \\abs { A ( x ) } ^ 2 = \\sum _ { j = 1 } ^ k \\lambda _ j ^ 2 \\abs { P _ { W _ j } ( x ) } ^ 2 , \\end{align*}"}
-{"id": "1773.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\sum _ { i } \\Delta \\overline { u } _ { i } ( x ) \\ , d x = \\int _ { \\partial \\Omega } \\sum _ { i } \\frac { \\partial \\overline { u } _ { i } } { \\partial n } \\ , d s \\le \\int _ { \\partial \\Omega } \\sum _ { i } \\frac { \\partial \\underline { u } _ { i } } { \\partial n } \\ , d s = \\int _ { \\Omega } \\sum _ { i } \\Delta \\underline { u } _ { i } ( x ) \\ , d x . \\end{align*}"}
-{"id": "3377.png", "formula": "\\begin{align*} G = & \\log \\left ( 1 + P _ { \\max } \\gamma _ { \\max } \\right ) + e ^ { \\gamma _ o } \\hat { \\Gamma } ( 0 , \\gamma _ o ) \\end{align*}"}
-{"id": "1540.png", "formula": "\\begin{align*} ( T ( t ) f ) ( x ) : = h _ t ( x ) f ( \\varphi ( t , x ) ) , f \\in L ^ p _ \\rho ( \\Omega ) , \\ x \\in \\Omega , \\ t \\geq 0 \\end{align*}"}
-{"id": "9376.png", "formula": "\\begin{align*} \\det [ ( \\mathbf { T } \\Gamma _ 0 - \\Gamma _ 1 ) u _ \\lambda , ( \\mathbf { T } \\Gamma _ 0 - \\Gamma _ 1 ) v _ \\lambda ] = 0 . \\end{align*}"}
-{"id": "6593.png", "formula": "\\begin{align*} \\nabla F _ { \\gamma } ^ { \\rm { D R } } ( z ) & = ( 2 \\gamma ) ^ { - 1 } \\big ( \\nabla R _ { \\gamma f } ( z ) z + R _ { \\gamma f } - R _ { \\gamma f } - \\nabla R _ { \\gamma f } ( z ) R _ { \\gamma g } ( R _ { \\gamma f } ( z ) ) \\big ) \\\\ & = ( 2 \\gamma ) ^ { - 1 } \\nabla R _ { \\gamma f } ( z ) ( z - R _ { \\gamma g } R _ { \\gamma f } ( z ) ) . \\end{align*}"}
-{"id": "4233.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 + t ) ^ { \\tfrac { x } { 2 } } d \\mu _ { - 1 } ( x ) & = \\sum _ { m = 0 } ^ \\infty \\int _ { \\mathbb { Z } _ p } x ^ m d \\mu _ { - 1 } ( x ) \\frac { 1 } { 2 ^ m } \\frac { 1 } { m ! } \\Big ( \\log ( 1 + t ) \\Big ) ^ m \\\\ & = \\sum _ { m = 0 } ^ \\infty E _ m 2 ^ { - m } \\sum _ { n = m } ^ \\infty S _ 1 ( n , m ) \\frac { t ^ n } { n ! } \\\\ & = \\sum _ { n = 0 } ^ \\infty \\left ( \\sum _ { m = 0 } ^ n 2 ^ { - m } E _ m S _ 1 ( n , m ) \\right ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "3055.png", "formula": "\\begin{align*} e ^ { - V ( u ) } Z _ \\infty ^ u = \\sum _ { j \\in \\N } e ^ { - V ( u . j ) } Z _ \\infty ^ { u . j } . \\end{align*}"}
-{"id": "4756.png", "formula": "\\begin{align*} n _ 1 = n ( v ) ; n _ 2 = g ' ( u ) \\ , l ( v ) + f ' ( u ) \\ , e _ 4 . \\end{align*}"}
-{"id": "2028.png", "formula": "\\begin{align*} ( a _ 1 , \\ldots , a _ 4 , b _ 1 , \\ldots , b _ 4 ) \\mapsto \\left ( \\begin{pmatrix} a _ 1 & a _ 2 \\\\ - b _ 2 & b _ 1 \\end{pmatrix} , \\begin{pmatrix} a _ 3 & a _ 4 \\\\ - b _ 4 & b _ 3 \\end{pmatrix} \\right ) \\end{align*}"}
-{"id": "9708.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ' ( x ) } { x ^ { - 2 } e ^ { 1 / x } \\exp ( - e ^ { 1 / x } ) } = 1 , \\end{align*}"}
-{"id": "7116.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( y ) & = \\left \\{ | u | \\leq n : V ( u _ j ) \\geq f _ n ( j ) - y , j \\leq | u | \\right \\} , \\\\ \\bar { \\mathcal { A } } _ n ( y , h ) & = \\left \\{ | u | = n : u \\in \\mathcal { A } _ n ( y ) , V ( u ) - f _ n ( n ) + y \\in [ h - 1 , h ] \\right \\} \\\\ \\mathcal { B } _ n ( y , z ) & = \\left \\{ | u | \\leq n : \\xi ( u _ j ) \\leq z + ( V ( u _ j ) - f _ n ( j ) + y ) / 2 , j \\leq | u | \\right \\} . \\end{align*}"}
-{"id": "6313.png", "formula": "\\begin{align*} \\widehat { f _ k ^ { \\alpha } } = \\widehat { f } ( \\frac { \\xi - \\langle k \\rangle ^ { \\frac { \\alpha } { 1 - \\alpha } } k } { \\langle k \\rangle ^ { \\frac { \\alpha } { 1 - \\alpha } } } ) . \\end{align*}"}
-{"id": "3692.png", "formula": "\\begin{align*} \\frac { \\partial \\zeta } { \\partial t } = \\{ \\zeta , Z , H \\} _ { \\zeta } ~ . \\end{align*}"}
-{"id": "7079.png", "formula": "\\begin{align*} V ( u ) = \\sum _ { j = 1 } ^ { | u | } \\ell ^ { u _ { j - 1 } } _ { u ( j ) } , \\end{align*}"}
-{"id": "9664.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log g ( x ( t ) ) } { \\log t } = - \\frac { 1 } { \\log ( 1 / ( 1 - q ) ) } \\log \\left ( \\frac { a } { b } \\right ) . \\end{align*}"}
-{"id": "6229.png", "formula": "\\begin{align*} \\tilde { f } ( x ) = f ( \\tilde { x } ) \\ \\ \\tilde { x } \\ \\ \\ \\frac { | | \\tilde { x } - x | | } { | | x | | } = \\mathcal { O } ( \\epsilon _ { m a c h i n e } ) , \\end{align*}"}
-{"id": "1284.png", "formula": "\\begin{align*} K _ 0 = \\begin{bmatrix} q & 1 & - 1 & 0 & 0 \\\\ 0 & 0 & q & 0 & 0 \\\\ 0 & 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 0 & J ( 3 r , 2 r ) \\\\ 0 & 0 & 0 & J ( r , 2 r ) & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "1878.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } ^ 2 + 2 R _ { 1 3 4 2 } R _ { 1 4 2 3 } = & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( x + 2 y ) - ( x + 2 y ) ^ 2 ] \\\\ \\ge & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( K _ { 1 4 } - K _ { 1 3 } ) - ( K _ { 1 4 } - K _ { 1 3 } ) ^ 2 ] . \\\\ \\end{align*}"}
-{"id": "9674.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { x ( t ) } { G ^ { - 1 } ( t ) } = \\left ( a - b \\left ( \\frac { 1 } { 1 - q } \\right ) ^ { \\beta / ( \\beta - 1 ) } \\right ) ^ { - 1 / ( \\beta - 1 ) } = \\Lambda , \\end{align*}"}
-{"id": "7548.png", "formula": "\\begin{align*} x _ { n + 1 } = ( 1 + l \\xi ) f ( x _ n ) . \\end{align*}"}
-{"id": "2009.png", "formula": "\\begin{align*} \\int _ { | z + 1 | > e / | x | } [ f ( x + x z ) - f ( x ) - f ' ( x ) x z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) = \\int _ { | z + 1 | > e / | x | } [ \\log | 1 + z | - z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) , \\end{align*}"}
-{"id": "6756.png", "formula": "\\begin{align*} \\int _ X \\chi \\theta _ { \\varphi } ^ n = \\int _ X \\chi e ^ { \\varphi } d \\mu . \\end{align*}"}
-{"id": "7745.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { \\sum _ { i = 0 } ^ n \\sigma _ i [ \\xi _ { i + 1 } - \\mu _ { i + 1 } ] } { \\sqrt { \\sum _ { k = 0 } ^ { n } \\sigma ^ 2 _ k } } = \\infty , \\end{align*}"}
-{"id": "8190.png", "formula": "\\begin{align*} E ' _ c : = S p e c \\ A [ x , y ] / ( y ^ 2 - F ( c _ 1 , c _ 2 , x ) ) . \\end{align*}"}
-{"id": "7992.png", "formula": "\\begin{align*} K \\cdot H = K \\cdot F + K \\cdot L + K \\cdot \\sum E _ i + K \\cdot \\sum E _ i ' \\leq 2 + 0 + 1 + 1 = 4 . \\end{align*}"}
-{"id": "2254.png", "formula": "\\begin{align*} \\gamma _ { q , p } ( 0 , 2 ) = \\beta _ { q , p } \\left ( \\frac { ( 3 p - 1 ) ! ! } { 6 } \\varepsilon _ { p } ^ { 3 } \\sigma ^ { 3 p } + \\frac { ( 2 q + p - 1 ) ! ! } { 2 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } \\sigma ^ { 2 q + p } \\right ) , \\end{align*}"}
-{"id": "2381.png", "formula": "\\begin{align*} p _ { C } ^ { \\alpha } ( x ) & : = \\alpha r _ { C } ^ * ( x ) + \\tfrac { 1 - \\alpha } { 2 } \\| x \\| ^ 2 , \\\\ P _ { C } ^ { \\alpha } ( x ) & : = \\nabla p _ { C } ^ { \\alpha } ( x ) = \\alpha \\Pi _ { C } ( x ) + ( 1 - \\alpha ) x \\\\ R _ C ( x ) & : = 2 \\Pi _ C ( x ) - x . \\end{align*}"}
-{"id": "4735.png", "formula": "\\begin{align*} \\frac { \\beta _ 1 v _ { 1 1 } + \\beta _ 2 v _ { 1 2 } } { \\tau } x _ 1 + \\frac { \\beta _ 1 v _ { 2 1 } + \\beta _ 2 v _ { 2 2 } } { \\tau } x _ 2 = \\frac { \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 } { \\tau } + \\varepsilon \\end{align*}"}
-{"id": "6996.png", "formula": "\\begin{align*} \\delta _ H \\Theta ^ 2 ( x , y ) = - \\sum _ { i + j = N + 1 , ~ i , j > 0 } ~ ~ \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ { \\lambda _ j ( x , y ) } , \\alpha _ n ] - [ \\alpha _ i , f _ 0 ^ { \\lambda _ j ( x , y ) } ] - [ \\alpha _ 0 , [ f _ i ^ y , f _ j ^ x ] ] \\end{align*}"}
-{"id": "2008.png", "formula": "\\begin{align*} I _ 2 & = \\int _ { - 1 } ^ 1 [ f ( x + x z ) - f ( x ) - f ' ( x ) x z ] \\nu ^ 1 ( \\d z ) + \\int _ { \\R } [ f ( x + x z ) - f ( x ) ] \\nu ^ 2 ( \\d z ) \\\\ & = \\int _ { \\R } [ f ( x + x z ) - f ( x ) - f ' ( x ) x z I ( | z | \\leq 1 ) ] \\nu _ { U } ( \\d z ) + f ' ( x ) x \\int _ { - 1 } ^ 1 z \\nu ' ( \\d z ) , \\end{align*}"}
-{"id": "3463.png", "formula": "\\begin{align*} \\xi _ r ( x ) = x \\left ( \\frac { \\sigma ( r ) } { r } - \\int _ { r } ^ x \\frac { \\sigma ( z ) } { z ^ 2 } d z \\right ) . \\end{align*}"}
-{"id": "6603.png", "formula": "\\begin{align*} \\nabla F _ { \\alpha _ 1 , \\alpha _ 2 } ^ { \\rm { G A P } } ( x ) & = P x - P \\nabla p _ C ^ { \\alpha _ 2 } ( P x + q ) \\\\ & = P ( x - P _ C ^ { \\alpha _ 2 } P _ D ^ { \\alpha _ 1 } x ) . \\end{align*}"}
-{"id": "4471.png", "formula": "\\begin{align*} u _ { x , s } : = \\frac { V _ d ( n - 1 ) h _ x ^ { - 1 } ( s ) ^ d } { e ^ { \\Psi ( k ) } } . \\end{align*}"}
-{"id": "7104.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\nu _ { \\beta , n } = \\nu _ { \\beta , \\infty } \\end{align*}"}
-{"id": "4340.png", "formula": "\\begin{align*} \\frac { d } { d r } \\left ( e ^ r | f ' ( r , a ) | ^ { p - 2 } f ' ( r , a ) \\right ) = - e ^ r f ( r , a ) \\ , r \\in ( 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "4855.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = I ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) + \\bar { R } ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) \\end{align*}"}
-{"id": "2287.png", "formula": "\\begin{align*} & { } \\big \\| \\big ( ( A _ m - A _ n ) v _ n \\big ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } ^ p = \\tau \\sum _ { n = k } ^ m \\| ( A _ m - A _ n ) v _ n \\| _ { X } ^ p \\\\ & { } \\le \\tau \\sum _ { n = k } ^ m \\| A _ m - A _ n \\| ^ p \\| v _ n \\| _ { D } ^ p = \\sum _ { n = k } ^ m \\| A _ m - A _ n \\| ^ p ( E _ n - E _ { n - 1 } ) , \\end{align*}"}
-{"id": "6539.png", "formula": "\\begin{align*} \\big \\| ( d _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } \\le \\delta \\end{align*}"}
-{"id": "5675.png", "formula": "\\begin{align*} e _ i ^ 2 = \\omega e _ i + 1 \\ , e _ j \\ , e _ { j + 1 } \\ , e _ j = e _ { j + 1 } \\ , e _ j \\ , e _ { j + 1 } , \\end{align*}"}
-{"id": "6164.png", "formula": "\\begin{align*} u f ^ { * } = \\varpi ^ { a } \\Xi - \\varpi ^ { a } g ^ { * } . \\end{align*}"}
-{"id": "8717.png", "formula": "\\begin{align*} L ( t , x ) \\xi = \\int _ t ^ T \\int _ { H } e ^ { - ( s - t ) { A } } G B ( s , z + e ^ { ( s - t ) { A } } x ) \\left \\langle Q _ { s - t } ^ { - 1 / 2 } e ^ { ( s - t ) { A } } G \\xi , Q _ { s - t } ^ { - 1 / 2 } z \\right \\rangle \\mu _ { s - t } ( d z ) \\\\ + \\int _ { t } ^ { T } \\int _ { H } e ^ { - ( s - t ) { A } } L ( s , z + e ^ { ( s - t ) { A } } x ) B ( s , z + e ^ { ( s - t ) { A } } x ) \\left \\langle Q _ { s - t } ^ { - 1 / 2 } e ^ { ( s - t ) { A } } G \\xi , Q _ { s - t } ^ { - 1 / 2 } z \\right \\rangle \\mu _ { s - t } ( d z ) \\ , d s . \\end{align*}"}
-{"id": "5517.png", "formula": "\\begin{align*} P ( x ) = P ( x ^ 0 ) = \\int _ 0 ^ a \\varphi \\left ( \\frac { a } { W ( a ) } \\right ) w ( s ) \\ , d s = \\varphi \\left ( \\frac { a } { W ( a ) } \\right ) W ( a ) . \\end{align*}"}
-{"id": "9984.png", "formula": "\\begin{align*} F _ k ( \\alpha ) = \\max _ { \\tilde { p } \\ ! , p _ 1 \\ ! , p _ 2 \\ ! } \\alpha \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\tilde { p } | H _ { \\tilde { i } k } | ^ 2 } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\ ! + \\ ! ( \\ ! 1 \\ ! - \\ ! \\alpha \\ ! ) \\sum _ { i = 1 } ^ { 2 } \\log _ 2 \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { p _ { i } } { \\sigma ^ 2 _ n } \\ ! \\Big ) , \\end{align*}"}
-{"id": "1948.png", "formula": "\\begin{align*} & \\frac { d E ( Y ( u ) ) } { d u } = - 2 n _ 1 q _ 1 ^ 2 Y _ 1 ^ 2 F _ 1 ( Y ) - 2 n _ 2 q _ 2 ^ 2 Y _ 2 ^ 2 F _ 2 ( Y ) \\\\ \\sim & - 2 n _ 1 q _ 1 ^ 2 Y _ 1 ^ 2 \\cdot ( 2 p _ 1 Y _ 1 ) - 2 n _ 2 q _ 2 ^ 2 Y _ 2 ^ 2 \\cdot ( - n _ 1 q _ 1 ^ 2 Y _ 1 ^ 2 ) < 0 , \\end{align*}"}
-{"id": "3326.png", "formula": "\\begin{align*} ( X \\triangleright T ) ^ * = S ( X ) ^ * \\triangleright T ^ * , X \\in U _ q ( \\mathfrak { l } ) , \\ T \\in \\mathrm { E n d } ( \\Lambda _ q ( \\mathfrak { u } _ + ) ) . \\end{align*}"}
-{"id": "8391.png", "formula": "\\begin{align*} \\theta ( \\beta ( x ) ) \\theta ( v ) = \\theta ( v ) \\theta ( x ) , \\ \\ \\ x \\in M . \\end{align*}"}
-{"id": "7036.png", "formula": "\\begin{align*} X _ t = t \\gamma - t \\nu _ + ( d _ + ) + t \\nu _ - ( 0 ) + X _ t ^ { ( S , d _ + , + ) } + X _ t ^ { ( B , d _ + , + ) } + X _ t ^ { ( 0 , - ) } , \\end{align*}"}
-{"id": "7598.png", "formula": "\\begin{align*} P _ p ( x _ 0 ) \\ge \\prod _ { i = 1 } ^ { K _ 1 } \\lambda _ i \\end{align*}"}
-{"id": "4304.png", "formula": "\\begin{align*} \\Sigma = ( t _ m = 0 ) . \\end{align*}"}
-{"id": "5230.png", "formula": "\\begin{align*} 1 / C \\leq | ( X _ { \\varepsilon } ) _ { T } | \\leq C \\ ; , \\ ; \\ ; \\left | \\left ( [ X _ { \\varepsilon } , \\overline { Z } ] \\right ) _ { T } \\right | \\leq \\varepsilon \\ ; , \\ ; Z \\in T ^ { 1 , 0 } ( M ) \\ ; , \\ ; | Z | = 1 \\ ; , \\end{align*}"}
-{"id": "328.png", "formula": "\\begin{align*} 6 _ { 1 , 3 } & = , 6 _ { 2 , 3 } = , \\\\ 8 _ { 1 , 4 } & = , 1 0 _ { 1 , 3 } = . \\end{align*}"}
-{"id": "9324.png", "formula": "\\begin{align*} \\lim _ { y \\rightarrow x } Z ( t , y , z ) = \\beta ( t , x , z ) ; x \\in \\partial D = \\{ c _ 1 , c _ 2 \\} . \\end{align*}"}
-{"id": "6462.png", "formula": "\\begin{align*} & \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } ( w ^ { 2 } ( s , x ) + w ^ { 2 } ( s , y ) ) k ( x , y ) d x d y \\\\ & \\quad \\quad \\quad \\quad \\quad \\leq C ( \\delta , \\Lambda ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x . \\end{align*}"}
-{"id": "1263.png", "formula": "\\begin{align*} 0 & = Q T Q \\phi = ( 1 - P ) T Q \\phi = T \\phi - P T \\phi = U \\phi + v \\mathcal G _ 0 v ^ * \\phi - c _ 0 v ( 1 , 0 ) ^ T . \\end{align*}"}
-{"id": "6707.png", "formula": "\\begin{align*} \\mu = \\rho \\ , d z d \\bar z d \\theta d \\bar \\theta \\end{align*}"}
-{"id": "5898.png", "formula": "\\begin{align*} \\dd _ { t } f + \\left ( v \\cdot D _ { x } f + F \\cdot D _ { v } f \\right ) \\dd t + D _ { v } f \\circ \\dd W _ { t } = 0 , f \\big | _ { t = 0 } = f _ { 0 } \\end{align*}"}
-{"id": "2458.png", "formula": "\\begin{align*} 2 ^ { j n ( \\alpha _ 1 - \\alpha _ 2 ) ( 1 / q - 1 / 2 ) } = 2 ^ { j \\widetilde { A _ 2 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j \\widetilde { A _ 3 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\end{align*}"}
-{"id": "9357.png", "formula": "\\begin{align*} & d p ( t , X ( t ) ) = \\frac { 1 } { 2 } [ \\frac { \\partial p } { \\partial t } ( t , X ( t ) ) d t + \\frac { \\partial p } { \\partial x } ( t , X ( t ) ) \\beta _ 0 ( t ) \\pi ( t ) d v ( t ) + q ( t , X ( t ) ) d w ( t ) ] \\\\ \\end{align*}"}
-{"id": "2483.png", "formula": "\\begin{align*} \\beta _ j = ( d _ j - 1 ) \\dim V _ j ^ I + d _ j [ a ( V _ j ) + n ( \\psi ) \\dim ( V _ j ) ] \\end{align*}"}
-{"id": "7860.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } p ^ { \\kappa } ( t , x , z ) p ^ { \\kappa } ( s , z , y ) \\ , d z = p ^ { \\kappa } ( t + s , x , y ) \\ , . \\end{align*}"}
-{"id": "2283.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) , \\end{aligned} \\end{align*}"}
-{"id": "5155.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t \\theta ^ { n , 1 } _ i + u ^ n \\cdot \\nabla \\theta ^ { n , 1 } _ i = f ^ n _ i - f ^ \\infty _ i , \\\\ \\partial _ t \\sigma ^ { n , 1 } + u ^ n \\cdot \\nabla \\sigma ^ { n , 1 } = g ^ n - g ^ \\infty , \\\\ { \\theta ^ { n , 1 } _ i } | _ { t = 0 } = \\nabla ( u ^ n _ 0 ) _ { i } - \\nabla ( u ^ \\infty _ 0 ) _ { i } , \\\\ { \\sigma ^ { n , 1 } } | _ { t = 0 } = \\nabla \\gamma ^ n _ 0 - \\nabla \\gamma ^ \\infty _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "5360.png", "formula": "\\begin{align*} K ( \\gamma ) \\cong \\bigoplus _ { i = 2 } ^ { p ( n ) - p ( n - 1 ) } \\Z / q _ i \\Z \\end{align*}"}
-{"id": "5803.png", "formula": "\\begin{align*} h _ { i } = e _ { i , i } - e _ { i + 1 , i + 1 } \\end{align*}"}
-{"id": "4330.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ l D _ { n _ 0 , i } ( C _ { n , i } - C _ { 1 , i } ) . \\end{align*}"}
-{"id": "901.png", "formula": "\\begin{align*} Z ( t ) = 2 \\sum _ { n \\leq \\sqrt { t / ( 2 \\pi ) } } n ^ { - 1 / 2 } \\cos \\left ( t \\log \\frac { \\sqrt { t / ( 2 \\pi ) } } { n } - \\frac { t } { 2 } - \\frac { \\pi } { 8 } \\right ) + O \\left ( t ^ { - 1 / 4 } \\right ) . \\end{align*}"}
-{"id": "6828.png", "formula": "\\begin{align*} L _ m \\cdot B _ { n + k } - L _ n \\cdot B _ { m + k } = \\sum _ { l = n + 1 } ^ { n + k - 1 } \\left [ B _ l , B _ { m + n + k - l } \\right ] + ( n - m ) B _ { m + n + k } \\end{align*}"}
-{"id": "2842.png", "formula": "\\begin{align*} \\sum _ { k \\geq 1 } \\frac { 1 } { k ! } l _ k ( \\tau , . . . , \\tau ) = 0 . \\end{align*}"}
-{"id": "6374.png", "formula": "\\begin{align*} \\tilde { v } ( X ) : = \\frac { v ( \\l X ) } { \\l ^ { l + 1 + s } } . \\end{align*}"}
-{"id": "6508.png", "formula": "\\begin{align*} N = N ( r , s ) ^ { 1 + o ( 1 ) } \\cdot \\log n \\end{align*}"}
-{"id": "9460.png", "formula": "\\begin{align*} \\lambda = d \\theta + \\pi ^ { * } \\beta . \\end{align*}"}
-{"id": "7043.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\ell a _ i \\equiv 0 \\mod d \\end{align*}"}
-{"id": "9852.png", "formula": "\\begin{align*} & R _ { y y } ( t ) \\otimes ( \\delta ( - t ) - \\alpha \\theta ( - t - \\tau ) ) \\otimes ( \\delta ( t ) - \\alpha \\theta ( t - \\tau ) ) \\\\ = & R _ { n _ { \\rm R } n _ { \\rm R } } ( t ) \\otimes h _ { \\rm R D } ( - t ) \\otimes \\theta ( - t ) \\otimes h _ { \\rm R D } ( t ) \\otimes \\theta ( t ) \\\\ & + R _ { n _ { \\rm D } n _ { \\rm D } } ( t ) \\otimes ( \\delta ( - t ) - \\alpha \\theta ( - t - \\tau ) ) \\otimes ( \\delta ( t ) - \\alpha \\theta ( t - \\tau ) ) . \\end{align*}"}
-{"id": "714.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } 1 & & \\\\ & \\ddots & \\\\ & & 1 \\\\ & & \\end{array} \\right ) . \\end{align*}"}
-{"id": "6138.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial m } ( \\frac 3 8 I ) = & 8 z ( s - 2 ) + 1 - 8 s + 4 m ( 1 + 8 s - 2 s ^ 2 ) \\\\ = & - ( 1 6 z - 1 - 4 m ) - 8 s ( 1 + m s - z - 4 m ) . \\end{align*}"}
-{"id": "1291.png", "formula": "\\begin{align*} R ( n ) = \\sum _ { h _ 1 + h _ 2 = n } \\Lambda ( h _ 1 ) \\Lambda ( h _ 2 ) \\end{align*}"}
-{"id": "2097.png", "formula": "\\begin{align*} & M \\ddot { x } ( t ) + D \\dot { x } ( t ) + K x ( t ) = F u ( t ) , \\\\ & y ( t ) = C _ p x ( t ) , \\end{align*}"}
-{"id": "1625.png", "formula": "\\begin{align*} \\eta = a ( x ^ { i } ) u + b ( x ^ { i } ) ~ ~ , ~ \\xi ^ { k } = \\xi ^ { k } ( x ^ { i } ) . \\end{align*}"}
-{"id": "3837.png", "formula": "\\begin{align*} \\beta \\partial _ t e - e \\Delta e - \\Delta e = 0 \\quad \\partial _ n e = 0 \\Gamma _ T . \\end{align*}"}
-{"id": "7793.png", "formula": "\\begin{align*} ( I _ \\Gamma + ( \\Theta , w ) ) _ 2 = S _ 2 , \\end{align*}"}
-{"id": "2221.png", "formula": "\\begin{align*} \\mu ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\left ( \\frac { \\rho ( 0 ) } { t ^ { 1 - \\alpha } } + \\int _ { 0 } ^ { t } \\frac { \\rho ' ( s ) } { ( t - s ) ^ { 1 - \\alpha } } d s \\right ) , \\end{align*}"}
-{"id": "3493.png", "formula": "\\begin{align*} \\alpha \\int _ { \\{ \\hat { v } \\geq \\alpha \\} } \\hat { f } d x ' = \\alpha \\int _ { \\{ \\hat { v } \\geq \\alpha \\} } \\widetilde { f } d x ' = \\int _ { \\{ \\hat { v } \\geq \\alpha \\} } \\widetilde { f } \\hat { v } d x ' = \\int _ { \\{ \\hat { v } \\geq \\alpha \\} } \\hat { f } \\hat { v } d x ' \\geq \\alpha \\int _ { \\{ \\hat { v } \\geq \\alpha \\} } \\hat { f } d x ' , \\end{align*}"}
-{"id": "9890.png", "formula": "\\begin{align*} R ( x ; D ) = \\sup _ { y \\le x } \\sum _ { d \\le D } | r _ d ( y ) | \\ll A ( x ) \\log ^ { - 2 } x \\end{align*}"}
-{"id": "9025.png", "formula": "\\begin{align*} \\tilde \\xi _ { i } ( x , w + N ) = \\tilde \\xi _ { i } ( x , w ) . \\end{align*}"}
-{"id": "9649.png", "formula": "\\begin{align*} \\omega _ 1 ^ { ( j + 1 ) } = \\omega _ 2 ^ { ( j ) } , \\omega _ 2 ^ { ( j + 1 ) } = \\omega _ 3 ^ { ( j ) } \\end{align*}"}
-{"id": "5447.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & p & 0 & 0 \\\\ p & 0 & 0 & 0 \\\\ 0 & 0 & 0 & p \\\\ 0 & 0 & p & 0 \\end{bmatrix} , \\begin{bmatrix} 0 & p ^ 2 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & p \\\\ 0 & 0 & p & 0 \\end{bmatrix} , \\begin{bmatrix} 0 & p ^ 2 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & p ^ 2 \\\\ 0 & 0 & 1 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "3385.png", "formula": "\\begin{align*} G & = \\frac { 1 } { \\bar { \\sigma } _ h ^ 2 } e ^ { \\frac { \\gamma _ { \\max } } { \\bar { \\sigma } _ h ^ 2 } } \\int _ { \\gamma _ { \\max } } ^ \\infty \\log \\left ( { 1 + P _ { \\max } \\gamma } \\right ) e ^ { - \\frac { \\gamma } { \\bar { \\sigma } _ h ^ 2 } } d \\gamma . \\end{align*}"}
-{"id": "9895.png", "formula": "\\begin{align*} M = \\left ( \\begin{array} { c c c } M _ { 1 1 } & \\cdots & M _ { 1 t } \\\\ \\vdots & \\ddots & \\vdots \\\\ M _ { t 1 } & \\cdots & M _ { t t } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "8195.png", "formula": "\\begin{align*} Q = x _ 1 x _ 2 \\cdot N ( H _ 1 , H _ 2 ) \\cdot A _ { p - 1 } ( H _ 1 , H _ 2 ) \\in A [ x _ 1 , x _ 2 , x _ 3 ] \\end{align*}"}
-{"id": "9756.png", "formula": "\\begin{align*} V ( \\bar { \\vec { x } } ) = V _ { \\mathrm { D } } ( \\bar { \\vec { x } } ) . \\end{align*}"}
-{"id": "93.png", "formula": "\\begin{align*} \\frac { 1 } { r } = \\sum _ { i = 1 } ^ l D _ { r , i } C _ { r - s , i } . \\end{align*}"}
-{"id": "2296.png", "formula": "\\begin{align*} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\le C \\delta , \\\\ & \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ \\infty ( W ) } \\le C \\delta , \\end{align*}"}
-{"id": "3032.png", "formula": "\\begin{align*} { { \\bf { H } } ^ { [ 1 1 ] } } ( n ) { { \\bf { v } } _ i } = { \\bf { h } } _ i ^ { [ 1 1 ] } ( { t _ 2 } ) , \\end{align*}"}
-{"id": "3630.png", "formula": "\\begin{align*} u _ { n + 1 } = h ( ( 1 - c ) u _ n ) , c \\in [ 0 , 1 ) , \\end{align*}"}
-{"id": "9796.png", "formula": "\\begin{align*} N _ { g } ( \\epsilon ) = \\left \\{ f : \\int g ( x ) \\log \\frac { g ( x ) } { f ( x ) } d x < \\epsilon \\right \\} . \\end{align*}"}
-{"id": "2078.png", "formula": "\\begin{align*} \\mathcal { K } _ { n e w } = \\mathcal { K } _ { o l d } ( I + ( s ^ { 2 } _ { n e w } - s ^ { 2 } _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } M + ( s _ { n e w } - s _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } D ) . \\end{align*}"}
-{"id": "397.png", "formula": "\\begin{align*} \\Sigma _ 1 ( f , z _ 1 , z _ 2 ) & = \\int _ { 0 } ^ { \\infty } \\int _ 0 ^ \\infty ( f ( 0 , 0 , t , u ) + f ( 0 , 0 , t , - u ) ) t ^ { z _ 1 - 1 } u ^ { z _ 2 - 1 } d t d u \\\\ \\Sigma _ 2 ( f , z ) & = \\int _ 0 ^ \\infty f ( 0 , 0 , 0 , u ) u ^ { z - 1 } d u \\\\ \\Sigma _ 3 ( f , z , u ) & = \\int _ 0 ^ \\infty f ( 0 , 0 , t , u ) t ^ { z - 1 } d t . \\end{align*}"}
-{"id": "1937.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow \\infty } \\frac { a ( \\tau ) } { \\tau } = E ( \\xi ) , \\lim _ { \\tau \\rightarrow \\infty } \\frac { b _ { i } ( \\tau ) } { \\tau } = E ( \\xi ) \\xi _ i ^ { - 1 } . \\end{align*}"}
-{"id": "9140.png", "formula": "\\begin{align*} A ( t ) u ( t ) & = A ( t ) e ^ { - t A ( t ) } u _ { 0 } + A ( t ) \\int _ { 0 } ^ { t } { e ^ { - ( t - l ) A ( t ) } ( \\mathcal { A } ( t ) - \\mathcal { A } ( l ) ) u ( l ) d l } \\\\ & + A ( t ) \\int _ { 0 } ^ { t } { e ^ { - ( t - l ) A ( t ) } [ f ( l ) ] } d l \\end{align*}"}
-{"id": "3995.png", "formula": "\\begin{align*} \\xi ( s ) u ( \\varphi ( s ) ) \\xi ( s _ 0 ) ^ { - 1 } F = \\exp ( w ^ 0 ( s ) ) \\exp ( w ^ - ( s ) ) \\exp ( w ^ + ( s ) ) F , \\end{align*}"}
-{"id": "8207.png", "formula": "\\begin{align*} F ( T , U ) = a _ n ( T ) U ^ n + \\cdots + a _ 0 ( T ) \\in K [ T , U ] \\end{align*}"}
-{"id": "9712.png", "formula": "\\begin{align*} G _ 0 ( x ) = \\int _ x ^ { x ( 0 ) } \\frac { 1 } { g ( u ) } \\ , d u , x > 0 . \\end{align*}"}
-{"id": "1026.png", "formula": "\\begin{align*} P ( x , y ) ~ = ~ B x + C y , \\end{align*}"}
-{"id": "3128.png", "formula": "\\begin{align*} & P _ d = - M _ { 1 1 } ( \\lambda ) , \\\\ & \\lambda ^ t P _ d = M _ { 1 2 } ( \\lambda ) ( \\Lambda _ { t - 1 } ( \\lambda ) \\otimes I _ n ) = ( \\Lambda _ { t - 1 } ( \\lambda ) ^ T \\otimes I _ n ) M _ { 2 1 } ( \\lambda ) , \\\\ & Q ( \\lambda ) = ( \\Lambda _ { t - 1 } ( \\lambda ) ^ T \\otimes I _ n ) M _ { 2 2 } ( \\lambda ) ( \\Lambda _ { t - 1 } ( \\lambda ) \\otimes I _ n ) . \\end{align*}"}
-{"id": "3545.png", "formula": "\\begin{align*} \\Phi ^ V _ { ( g , \\pi ) } ( g + h , \\pi + w ) = \\Phi ^ V _ { ( g , \\pi ) } ( g , h ) + ( 2 \\psi , V ) , \\end{align*}"}
-{"id": "6137.png", "formula": "\\begin{align*} \\frac 3 8 I = & 3 ( K _ { 1 3 } - K _ { 1 2 } ) ( 1 - K _ { 1 2 } - K _ { 1 3 } ) + 3 ( 1 - 2 K _ { 1 2 } - K _ { 1 3 } ) K _ { 1 3 } \\\\ & + 2 ( K _ { 1 3 } - K _ { 1 2 } ) ^ 2 - 2 ( K _ { 1 3 } - K _ { 1 2 } ) ( 1 - K _ { 1 2 } - 2 K _ { 1 3 } ) - ( 1 - K _ { 1 2 } - 2 K _ { 1 3 } ) ^ 2 \\\\ = & - 1 + K _ { 1 2 } + 8 K _ { 1 3 } + 2 K _ { 1 2 } ^ 2 - 4 K _ { 1 3 } ^ 2 - 1 6 K _ { 1 2 } K _ { 1 3 } \\\\ = & - 4 z ^ 2 + 8 ( 1 + s m - 2 m ) z + [ - 1 + ( 1 - 8 s ) m + 2 ( 1 + 8 s - 2 s ^ 2 ) m ^ 2 ] \\\\ \\end{align*}"}
-{"id": "2463.png", "formula": "\\begin{align*} \\big \\| \\{ \\langle k \\rangle ^ { \\frac { R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ 1 , \\alpha _ 2 ) } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } } \\} \\big \\| _ { l _ { \\infty } ^ { s _ 2 - s _ 1 , \\alpha _ 1 \\vee \\alpha _ 2 } } = \\sup _ { k \\in \\mathbb { Z } ^ n } \\langle k \\rangle ^ { \\frac { s _ 2 - s _ 1 } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } } \\langle k \\rangle ^ { \\frac { R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ 1 , \\alpha _ 2 ) } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } } . \\end{align*}"}
-{"id": "7031.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } P ( X _ t > 0 ) = 1 \\end{align*}"}
-{"id": "9255.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\tilde { P } } [ \\varphi ( X ( t , z ) ) K _ t ( z ) | \\mathcal { R } _ t ] = \\int _ { \\mathbb { R } } \\varphi ( x ) y ( t , x , z ) d x \\end{align*}"}
-{"id": "9606.png", "formula": "\\begin{align*} M _ T : = \\max \\limits _ { t \\in [ 0 , T ] } \\Vert ( u ( t ) , \\dot u ( t ) ) \\Vert _ { \\cal X } , \\end{align*}"}
-{"id": "234.png", "formula": "\\begin{align*} ( w _ { j _ t } ) _ { t = 1 } ^ { \\lfloor d / 4 \\rfloor + 1 } : = ( A ^ { ( k ) } ) ^ { - 1 } e _ 1 \\end{align*}"}
-{"id": "1283.png", "formula": "\\begin{align*} K ^ * _ i = \\begin{bmatrix} 0 & 0 & \\alpha J ( i + r , 3 r ) & \\overline \\alpha J ( i + 3 r , r ) \\\\ 0 & 0 & \\overline \\alpha J ( i + r , r ) & \\alpha J ( i + 3 r , 3 r ) \\\\ \\overline \\alpha J ( i , r ) & \\alpha J ( i + 2 r , 3 r ) & 0 & 0 \\\\ \\alpha J ( i , 3 r ) & \\overline \\alpha J ( i + 2 r , r ) & 0 & 0 \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "2977.png", "formula": "\\begin{align*} \\delta _ L \\Theta ( A ) ( x ) = - [ x , \\Theta ( A ) ] = \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ \\alpha _ i , \\delta _ L ( \\alpha _ j ) ] \\end{align*}"}
-{"id": "7461.png", "formula": "\\begin{align*} x _ i = \\deg _ z \\langle X _ i , \\chi \\rangle . \\end{align*}"}
-{"id": "2152.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } - \\eta _ { t } \\left [ g _ { 1 - \\alpha } * ( u - u _ { 0 } ) \\right ] d x d t + \\int _ { 0 } ^ { T } \\mathcal { E } ( u , \\eta ) d t = ( \\geq \\leq ) \\int _ { 0 } ^ { T } \\int _ { \\Omega } f \\eta d x d t . \\end{align*}"}
-{"id": "3235.png", "formula": "\\begin{align*} \\left \\{ \\left ( c _ { e _ { i } } \\left ( r \\right ) , c _ { e _ { i } } \\left ( - r \\right ) \\right ) \\right \\} _ { i = 1 } ^ { \\mathrm { \\dim } \\left ( S \\right ) } \\end{align*}"}
-{"id": "853.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\ell - 1 } f ( z _ { i } , z _ { \\ell } ) + \\sum _ { t = \\ell + 1 } ^ { m } g ( z _ { \\ell } , z _ { t } ) \\prod _ { \\begin{subarray} { c } i = 1 \\\\ i \\not = \\ell \\end{subarray} } ^ { t - 1 } f ( z _ { i } , z _ { \\ell } ) = \\prod _ { \\begin{subarray} { c } i = 1 \\\\ i \\not = \\ell \\end{subarray} } ^ { m } f ( z _ { i } , z _ { \\ell } ) \\end{align*}"}
-{"id": "1190.png", "formula": "\\begin{align*} x = \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\tanh { \\frac \\zeta 2 } , \\Psi = ( \\cosh \\frac \\zeta 2 ) \\ , v , \\end{align*}"}
-{"id": "5570.png", "formula": "\\begin{align*} D ( - \\lambda ) = W ( \\hat c _ 0 , c _ 0 ) \\prod _ { j = 1 } ^ N \\left ( 1 + \\frac { \\lambda } { z _ j } \\right ) . \\end{align*}"}
-{"id": "7340.png", "formula": "\\begin{gather*} v _ 1 \\wedge v _ 0 = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } \\pi _ + ( V _ 1 ) , v _ 1 \\wedge v _ { - 1 } = \\frac { 1 } { [ 2 ] _ { q ^ 2 } } \\pi _ + ( V _ 0 ) , v _ 0 \\wedge v _ { - 1 } = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } \\pi _ + ( V _ { - 1 } ) , \\\\ w _ { - 1 } \\wedge w _ 0 = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } \\pi _ - ( W _ { - 1 } ) , w _ { - 1 } \\wedge w _ 1 = \\frac { 1 } { [ 2 ] _ { q ^ 2 } } \\pi _ - ( W _ 0 ) , w _ 0 \\wedge w _ 1 = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } \\pi _ - ( W _ 1 ) . \\end{gather*}"}
-{"id": "6264.png", "formula": "\\begin{align*} 0 = \\nabla _ { \\nu } Q ( x , y _ 0 ) | _ { x _ 0 } = - 2 m \\frac { \\bar { w } ' ( \\frac { d ( x _ 0 , y _ 0 ) } { 2 } ) } { \\bar { w } ( \\frac { d ( x _ 0 , y _ 0 ) } { 2 } ) } \\nabla _ { \\nu } d ( x , y _ 0 ) | _ { x = x _ 0 } . \\end{align*}"}
-{"id": "1551.png", "formula": "\\begin{align*} D _ E ( \\mu _ { \\P } ^ { \\gamma } , \\mu _ { \\P } ^ 0 ) = E ( \\mu _ { \\P } ^ { \\gamma } ) - E ( \\mu _ { \\P } ^ 0 ) - \\langle \\nabla E ( \\mu _ { \\P } ^ 0 ) , \\mu _ { \\P } ^ { \\gamma } - \\mu _ { \\P } ^ 0 \\rangle . \\end{align*}"}
-{"id": "2014.png", "formula": "\\begin{align*} \\left | \\frac { f ( x ( 1 + z ) ) - f ( x ) } { x f ' ( x ) } \\right | = \\left | \\int _ { | 1 + z | } ^ 1 \\frac { g ( x y ) } { g ( x ) } y ^ { - 1 } \\d y \\right | \\leq 2 \\int _ { | 1 + z | } ^ 1 y ^ { - 1 - \\varepsilon } \\d y \\leq \\frac { 2 } { \\varepsilon } | 1 + z | ^ { - \\varepsilon } , \\end{align*}"}
-{"id": "2449.png", "formula": "\\begin{align*} 2 ^ { j A _ 2 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 3 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "7556.png", "formula": "\\begin{align*} l > b - f ( b ) = - F ( b ) , \\end{align*}"}
-{"id": "6602.png", "formula": "\\begin{align*} F _ { \\alpha _ 1 , \\alpha _ 2 } ^ { \\rm { G A P } } ( x ) = \\tfrac { 1 } { 2 } \\langle P x , x \\rangle - p _ { C } ^ { \\alpha _ 2 } ( P _ D ^ { \\alpha _ 1 } x ) \\end{align*}"}
-{"id": "2980.png", "formula": "\\begin{align*} \\delta _ L \\Theta ^ 1 ( x , y ) & = [ x , \\Theta ^ 1 ( y ) ] - [ y , \\Theta ^ 1 ( x ) ] - \\Theta ^ 1 [ x , y ] \\\\ & = \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ f _ 0 ^ x , [ f _ i ^ y , \\alpha _ j ] ] - [ f _ 0 ^ y , [ f _ i ^ x , \\alpha _ j ] ] - [ f _ i ^ { [ x , y ] } , \\alpha _ j ] ~ ~ \\mbox { ( u s i n g e q u a t i o n \\eqref { c 3 } ) } . \\end{align*}"}
-{"id": "4434.png", "formula": "\\begin{align*} \\hat { H } _ n = \\hat { H } _ n ( X _ 1 , \\ldots , X _ n ) : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\log \\biggl ( \\frac { \\rho _ { ( k ) , i } ^ d V _ d ( n - 1 ) } { e ^ { \\Psi ( k ) } } \\biggr ) , \\end{align*}"}
-{"id": "8522.png", "formula": "\\begin{align*} f ^ { \\ast } ( z w ) = ( w ^ { - 1 } ) ^ { \\ast } ( f ^ { - 1 } z ) \\end{align*}"}
-{"id": "1313.png", "formula": "\\begin{align*} R _ { \\textit { H L } } ( n ) = \\sum _ { m _ 1 + m _ 2 ^ 2 = n } \\Lambda ( m _ 1 ) , \\end{align*}"}
-{"id": "7273.png", "formula": "\\begin{align*} T _ { j k } \\widetilde { w } _ k = \\widetilde { w } _ j + \\sum _ { | \\alpha | \\geq n + 1 } ( \\pi | _ { \\widetilde { U } _ j } ) ^ * f _ { k j , \\alpha } \\cdot \\widetilde { w } _ j ^ \\alpha \\end{align*}"}
-{"id": "8133.png", "formula": "\\begin{align*} \\int _ { u > t } | x | ^ { l } \\ , d x = \\int _ { u ^ { \\star } > t } | x | ^ { l } \\ , d x \\forall t \\in [ 0 , + \\infty ) , \\end{align*}"}
-{"id": "615.png", "formula": "\\begin{align*} \\xi _ { 2 - 2 k } \\left ( \\mathcal { G } _ { Q } \\right ) & = f _ { Q } , \\\\ \\mathcal { D } ^ { 2 k - 1 } \\left ( \\mathcal { G } _ { Q } \\right ) & = - \\frac { ( 2 k - 2 ) ! } { ( 4 \\pi ) ^ { 2 k - 1 } } f _ { Q } . \\end{align*}"}
-{"id": "1009.png", "formula": "\\begin{align*} Q _ { j } | \\psi _ { \\{ l ; k \\} } \\rangle & = l _ j | \\psi _ { \\{ l ; k \\} } \\rangle \\\\ \\overline { Q } _ { j } | \\psi _ { \\{ l ; k \\} } \\rangle & = k _ j | \\psi _ { \\{ l ; k \\} } \\rangle . \\end{align*}"}
-{"id": "245.png", "formula": "\\begin{align*} r _ { d , \\theta } : = \\Bigl \\{ \\frac { 7 ( 1 - e ^ { - 2 A _ { d , \\theta } } ) } { 3 0 d ^ { 1 / 2 } a ( e ^ { - 4 A _ { d , \\theta } } ) } \\Bigr \\} ^ { 1 / ( \\beta \\wedge 1 ) } \\wedge \\frac { 1 } { 8 d ^ { 1 / 2 } a ( e ^ { - 4 A _ { d , \\theta } } ) \\} ^ { 1 / ( \\beta \\wedge 1 ) } } . \\end{align*}"}
-{"id": "1322.png", "formula": "\\begin{align*} x ( k + 1 ) = A \\ , x ( k ) + E \\ , d ( k ) \\end{align*}"}
-{"id": "8898.png", "formula": "\\begin{align*} \\lambda _ k ( L _ { n _ \\ell } ) \\rightarrow \\lambda _ k ^ \\star : = \\limsup _ { n \\to + \\infty } \\lambda _ k ( L _ n ) \\end{align*}"}
-{"id": "6478.png", "formula": "\\begin{align*} \\frac { - \\int _ { B } \\psi ^ { 2 } \\tilde { u } ^ { - 1 } \\partial _ { t } ( g _ { 1 - \\alpha , m } * \\tilde { u } ) d x } { \\int _ { B } \\psi ^ { 2 } ( x ) d x } + \\frac { c _ { 2 } } { r ^ { 2 \\beta } \\mu _ { n } ( B ) } \\int _ { B } ( w - W ) ^ { 2 } \\psi ^ { 2 } d x \\leq \\frac { C _ { 2 } } { r ^ { 2 \\beta } } + S _ { m } ( t ) , \\end{align*}"}
-{"id": "4921.png", "formula": "\\begin{align*} W ( t ) : = \\sum _ { w \\in W } t ^ { | N ( w ) | } = \\sum _ { w \\in W } t ^ { \\ell ( w ) } . \\end{align*}"}
-{"id": "5652.png", "formula": "\\begin{align*} T _ h ( t ) v ( x ) = \\exp { \\left ( \\int _ { - t } ^ 0 h ( x e ^ s ) d s \\right ) } v ( x e ^ { - t } ) , t \\geq 0 , \\ x \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "6579.png", "formula": "\\begin{align*} \\langle \\nabla f _ 1 ( x ) , x \\rangle - f _ 1 ( x ) = \\langle P x + q , x \\rangle - ( \\tfrac { 1 } { 2 } \\langle P x , x \\rangle + \\langle q , x \\rangle ) = \\tfrac { 1 } { 2 } \\langle P x , x \\rangle . \\end{align*}"}
-{"id": "9502.png", "formula": "\\begin{align*} \\delta ( d C \\wedge * d C ) & = d \\delta C \\wedge * d C + d C \\wedge * d \\delta C \\\\ & = 2 d \\delta C \\wedge * d C \\\\ & = 2 d ( \\delta C \\wedge * d C ) - 2 ( - 1 ) ^ k \\delta C \\wedge d * d C . \\end{align*}"}
-{"id": "9613.png", "formula": "\\begin{align*} \\psi _ 0 = \\psi _ { 0 , r e g } + \\sum \\limits _ { 1 \\le k \\le n } \\zeta _ { 0 k } g _ k , \\pi _ 0 = \\pi _ { 0 , r e g } + \\sum \\limits _ { 1 \\le k \\le n } \\dot \\zeta _ { 0 k } g _ k . \\end{align*}"}
-{"id": "6450.png", "formula": "\\begin{align*} & \\quad \\quad \\quad \\quad 0 \\leq \\phi \\leq 1 , 0 \\leq - \\dot { \\phi } \\leq \\frac { 4 } { t _ { 2 } - t _ { 1 } } , \\\\ & \\phi = 1 [ 0 , t _ { 1 } - t _ { 0 } ] , \\phi = 0 [ t _ { 1 } - t _ { 0 } + ( t _ { 2 } - t _ { 1 } ) / 2 , t _ { 2 } - t _ { 0 } ] . \\end{align*}"}
-{"id": "7532.png", "formula": "\\begin{align*} N _ 1 : = \\left [ \\frac { c - \\mu _ 1 } { ( 1 - \\alpha - \\nu l ) d _ 1 } \\right ] + 1 , \\end{align*}"}
-{"id": "9072.png", "formula": "\\begin{align*} F _ { g } ^ { ( 2 ) } ( z ) = \\displaystyle \\frac { R _ { g } ( z ) } { y ( z ) ^ { ( 3 g - 1 ) / 2 } } , \\\\ \\end{align*}"}
-{"id": "6685.png", "formula": "\\begin{align*} \\{ x \\in \\operatorname { M r k } ( ( D , I ) ) : ( \\mathcal { L } _ { X _ { \\lambda } } F ) ( x ) = 0 \\} = \\Sigma ^ { D _ 1 , \\dots , D _ p } _ { d _ 1 , \\dots , d _ p } \\cap \\operatorname { M r k } ( ( D , I ) ) . \\end{align*}"}
-{"id": "6762.png", "formula": "\\begin{align*} \\theta _ { \\varphi _ { \\beta , \\varepsilon } } ^ n = e ^ { \\beta \\varphi _ { \\beta , \\varepsilon } } \\left [ ( 1 + \\varepsilon ) \\theta ^ n _ + + \\varepsilon \\omega ^ n \\right ] . \\end{align*}"}
-{"id": "5586.png", "formula": "\\begin{align*} g ^ \\pm ( z ) & = - \\frac 1 { 2 \\pi } \\big ( \\log ( z / 2 ) + \\gamma \\big ) \\pm \\frac { i } 4 \\\\ g _ 1 ^ \\pm ( z ) & = - \\frac { z ^ 2 } 4 g ^ \\pm ( z ) - \\frac { z ^ 2 } { 8 \\pi } \\end{align*}"}
-{"id": "1758.png", "formula": "\\begin{align*} \\underbrace { 0 = d _ { j _ 1 } = \\cdots = d _ { j _ { u _ 1 } } } _ { = \\sigma _ 0 } < \\underbrace { d _ { j _ { u _ 1 + 1 } } = \\cdots = d _ { j _ { u _ 2 } } } _ { = \\sigma _ 1 } < \\cdots < \\underbrace { d _ { j _ { u _ { s - 1 } + 1 } } = \\cdots = d _ { j _ r } } _ { = \\sigma _ { s - 1 } } < 1 = \\sigma _ { s } ; \\end{align*}"}
-{"id": "4137.png", "formula": "\\begin{align*} L ( s , S _ f ) & : = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) } { n ^ s } , \\\\ L ( s , S _ f \\times S _ f ) : = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) \\overline { S _ f ( n ) } } { n ^ s } , & L ( s , S _ f \\times \\overline { S _ f } ) : = \\sum _ { n \\geq 1 } \\frac { S _ f ( n ) S _ f ( n ) } { n ^ s } , \\end{align*}"}
-{"id": "3786.png", "formula": "\\begin{align*} \\mathbf { G } = \\boldsymbol { \\lambda } _ { { \\rm r } } \\boldsymbol { \\lambda } _ { { \\rm t } } ^ T \\end{align*}"}
-{"id": "3714.png", "formula": "\\begin{align*} \\langle u , v \\rangle = u v ^ { * } . \\end{align*}"}
-{"id": "7504.png", "formula": "\\begin{align*} x _ { n + 1 } = x _ n e ^ { r ( 1 - x _ n ) } , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 , \\end{align*}"}
-{"id": "8803.png", "formula": "\\begin{align*} d i v ( \\psi - u N ) = n + n H u \\ , . \\end{align*}"}
-{"id": "6186.png", "formula": "\\begin{align*} M _ m ( t ) \\to f ( V _ t ^ n ) - \\int _ 0 ^ t \\mathcal { A } _ n f ( V _ s ^ n ) \\d s = : M ( t ) , \\textrm { a . s . ~ a n d i n } L ^ 1 . \\end{align*}"}
-{"id": "3913.png", "formula": "\\begin{align*} \\int _ 0 ^ T f ( t , u ( t ) , u ' ( t ) ) d t = 0 . \\end{align*}"}
-{"id": "2847.png", "formula": "\\begin{align*} \\underline { P _ { \\infty } \\{ X \\} } ^ { \\psi } ( A ) = h o f i b ( \\underline { P _ { \\infty } \\{ X \\} } ( A ) \\rightarrow \\underline { P _ { \\infty } \\{ X \\} } ( \\mathbb { K } ) ) \\end{align*}"}
-{"id": "3402.png", "formula": "\\begin{align*} A ^ { ( i ) } _ j & : = { \\mathcal O } _ { \\tilde { D } _ i \\times T } [ W ^ { ( i ) } _ j ] \\big / \\big ( ( W ^ { ( i ) } _ j ) ^ { r ^ { ( i ) } _ j } - t ^ { r ^ { ( i ) } _ j } - z _ i \\big ) \\\\ A ^ { ( i ) } _ { j , k } & : = A ^ { ( i ) } _ j \\Big / \\Big ( z _ { i , 1 } \\cdots z _ { i , m _ i - 1 } \\big ( W ^ { ( i ) } _ j - \\zeta _ { r ^ { ( i ) } _ j } ^ k t \\big ) \\Big ) , \\end{align*}"}
-{"id": "8619.png", "formula": "\\begin{align*} & u _ { k } \\alpha = \\alpha _ { 1 } u _ { i n } ; u _ { o u t } \\beta = \\beta _ { 1 } u _ { k } \\\\ & u _ { i n } \\gamma = \\gamma _ { 1 } u _ { o u t } \\end{align*}"}
-{"id": "9356.png", "formula": "\\begin{align*} d p ( t , X ( t ) ) & = [ - \\pi ( t ) \\alpha _ 0 ( t ) \\frac { \\partial p } { \\partial x } ( t , X ( t ) ) - \\frac { 1 } { 2 } \\pi ^ 2 ( t ) \\beta _ 0 ^ 2 ( t ) \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } ( t , X ( t ) ) ] d t \\\\ & + q ( t , X ( t ) ) d w ( t ) \\end{align*}"}
-{"id": "9429.png", "formula": "\\begin{align*} \\begin{aligned} \\abs { f ' ( a ) } + \\abs { f ' ( b ) } & \\le \\abs { f ' ( c ) } + \\abs { f ' ( d ) } \\\\ \\abs { f ' ( c ) } & \\le \\abs { f ' ( a ) } + \\abs { f ' ( b ) } + \\abs { f ' ( d ) } \\\\ \\abs { f ' ( d ) } & \\le \\abs { f ' ( a ) } + \\abs { f ' ( b ) } + \\abs { f ' ( c ) } . \\end{aligned} \\end{align*}"}
-{"id": "7967.png", "formula": "\\begin{align*} D _ f ( P , Q ) = \\int _ { \\Omega } f \\left ( \\frac { p } { q } \\right ) q \\ , d \\mu . \\end{align*}"}
-{"id": "8193.png", "formula": "\\begin{align*} \\begin{array} { r c l } \\d x _ 1 & = & ( a _ 2 - a _ 3 ) x _ 2 x _ 3 , \\\\ \\d x _ 2 & = & ( a _ 3 - a _ 1 ) x _ 3 x _ 1 , \\\\ \\d x _ 3 & = & ( a _ 1 - a _ 2 ) x _ 1 x _ 2 . \\end{array} \\end{align*}"}
-{"id": "6296.png", "formula": "\\begin{align*} \\widehat { \\Delta _ j f } & = \\left ( \\varphi ( 2 ^ { - j } \\xi ) - \\varphi ( 2 ^ { - j + 1 } \\xi ) \\right ) \\widehat { f } ( \\xi ) , \\ j \\geq 0 , \\\\ \\widehat { \\Delta _ 0 f } & = \\varphi ( \\xi ) f ( \\xi ) . \\end{align*}"}
-{"id": "5030.png", "formula": "\\begin{align*} \\Big ( \\bigcap _ { l \\in L } l A B \\Big ) _ x = \\bigcap _ { l \\in L } l A B _ x , \\textrm { f o r a l l $ x \\in X $ } . \\end{align*}"}
-{"id": "4310.png", "formula": "\\begin{align*} \\eta : = \\frac { v _ { \\infty } } { p ^ \\star ( t ) } \\end{align*}"}
-{"id": "4206.png", "formula": "\\begin{align*} J ( x _ { k + 1 } ) - J ( x _ { k } ) = J ( x _ k + \\Delta _ k ) - J ( x _ { k } ) \\leq ( L - 1 ) \\| \\Delta _ k \\| ^ 2 < 0 , \\end{align*}"}
-{"id": "2540.png", "formula": "\\begin{align*} & \\| X - Y \\| ^ 2 = - \\frac { \\lambda \\tau } 8 \\Im \\left [ ( \\overline { X } - \\overline { Y } ) ^ T H ( X , Y , z ) \\right ] \\\\ \\leq & \\frac \\tau 8 \\left ( \\max _ { 1 \\le m \\le M } | X _ m + z _ m | | Y _ m + z _ m | \\right ) \\| X - Y \\| ^ 2 \\le \\frac { \\tau } 2 \\| X - Y \\| ^ 2 , \\end{align*}"}
-{"id": "2724.png", "formula": "\\begin{align*} \\begin{aligned} & \\theta _ 0 = 1 , \\\\ & \\theta _ i = \\frac { 1 + \\sqrt { 1 + 4 \\theta _ { i - 1 } ^ 2 } } { 2 } , i = 1 , \\dots , N - 1 , \\\\ & \\theta _ N = \\frac { 1 + \\sqrt { 1 + 8 \\theta _ { N - 1 } ^ 2 } } { 2 } , \\end{aligned} \\end{align*}"}
-{"id": "7953.png", "formula": "\\begin{align*} U _ l ( r ) = \\alpha j _ l ( \\sqrt [ 4 ] { \\lambda } r ) + \\beta i _ l ( \\sqrt [ 4 ] { \\lambda } r ) , \\end{align*}"}
-{"id": "2840.png", "formula": "\\begin{align*} \\epsilon ( i ) = i + \\sum _ { j _ 1 < j _ 2 , \\sigma ( j _ 1 ) > \\sigma ( j _ 2 ) } ( | x _ { j _ 1 } | | x _ { j _ 2 } | + 1 ) . \\end{align*}"}
-{"id": "281.png", "formula": "\\begin{align*} W _ { 3 1 } = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { 1 / 2 + 2 \\beta / d } } { n ^ { 1 + 2 \\beta / d } } \\ , , \\ , \\frac { \\log n } { n k ^ { 1 / 2 } } \\ , , \\ , \\frac { k ^ { - 1 / 2 + \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "4872.png", "formula": "\\begin{align*} f _ { \\kappa } ( z ) = - h z + \\log ( 1 + 2 z ) - \\kappa \\log ( 1 - 2 z ) . \\end{align*}"}
-{"id": "7095.png", "formula": "\\begin{align*} \\mu _ \\infty = \\sum _ { n = 1 } ^ { \\infty } \\sum _ { d \\in \\mathcal { P } ( D _ n ) } \\delta _ { u ^ { ( n ) } , \\xi _ n + d - \\log Z _ \\infty } . \\end{align*}"}
-{"id": "2862.png", "formula": "\\begin{align*} E ^ 0 f = F ^ c ( \\int _ { \\underline { n } \\in \\Delta } K ^ n ( C ) \\otimes \\overline { N ^ * } ( \\Delta ^ n ) \\underbrace { \\rightarrow } _ { ( 2 - 3 ) } \\overline { \\Omega } ( C ) ) . \\end{align*}"}
-{"id": "2714.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 ^ + } d ( s ) = - n \\int _ X d \\dot { \\varphi } _ { t } \\wedge d ^ c \\dot { \\varphi } _ { t } \\wedge \\theta _ { \\varphi _ t } ^ { n - 1 } + \\int _ X \\ddot { \\varphi } _ { t } \\theta _ { \\varphi _ t } ^ n . \\end{align*}"}
-{"id": "7852.png", "formula": "\\begin{align*} h ( x ) = K _ { \\nu - 1 } ( x ) / K _ { \\nu } ( x ) , \\end{align*}"}
-{"id": "3840.png", "formula": "\\begin{align*} R _ k ( x ) = \\frac { x ^ { k - 2 } \\left ( n _ k ( x ) - \\sqrt { 1 - 4 x } \\right ) } { 2 d _ k ( x ) } , \\end{align*}"}
-{"id": "4540.png", "formula": "\\begin{align*} \\int _ { \\mathcal { X } _ n ^ c \\times \\mathcal { X } } f ( x ) f ( y ) \\int _ { l _ x } ^ { v _ x } ( h _ u F ) ( u , v _ y ) \\ , d u \\ , d y \\ , d x = O \\biggl ( \\frac { k ^ { - \\frac { 1 } { 2 } + \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "7211.png", "formula": "\\begin{align*} \\Phi _ { \\beta , \\alpha } \\left ( \\partial \\mathcal { U } _ { \\alpha } ^ { S } \\left ( 3 \\right ) \\cap G _ { \\alpha } \\right ) = \\partial \\mathcal { U } _ { \\beta } ^ { S } \\left ( 3 \\right ) \\cap G _ { \\beta } . \\end{align*}"}
-{"id": "278.png", "formula": "\\begin{align*} | W _ 2 | \\leq \\biggl | W _ { 2 1 } + \\frac { 1 } { n } \\biggr | + 2 | W _ { 2 2 } | + | W _ { 2 3 } | = O \\biggl ( \\frac { k ^ { 1 / 2 } } { n } \\max \\biggl \\{ \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "9291.png", "formula": "\\begin{align*} ( L ^ * _ { \\pi ( t ) } y ) ( t , x , z ) = - \\pi ( t , z ) \\alpha ( t ) y ' ( t , x , z ) + \\frac { 1 } { 2 } \\pi ^ 2 ( t , z ) \\beta ^ 2 ( t ) y '' ( t , x , z ) , \\end{align*}"}
-{"id": "4791.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } '' _ b : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J , \\end{align*}"}
-{"id": "1531.png", "formula": "\\begin{align*} \\mathcal { F } \\big ( ( - \\Delta ) ^ { \\alpha / 2 } f \\big ) ( k ) : = | k | ^ \\alpha \\mathcal { F } ( f ) ( k ) , \\end{align*}"}
-{"id": "3153.png", "formula": "\\begin{align*} & \\overline { W } _ { s } ^ { \\otimes n } ( { p } ^ n ) - \\overline { W } _ { s } ^ { \\otimes n } ( { p ' } ^ n ) \\\\ & = \\left ( 1 - \\frac { 1 } { P ( \\mathsf { T } ^ n _ { p , \\delta } ) } \\right ) \\sum _ { a ^ n \\in \\mathsf { T } ^ n _ { p , \\delta } } p ^ n ( a ^ n ) \\overline { W } _ { s } ^ { \\otimes n } ( a ^ n ) + \\sum _ { a ^ n \\notin \\mathsf { T } ^ n _ { p , \\delta } } p ^ n ( a ^ n ) \\overline { W } _ s ^ { \\otimes n } ( a ^ n ) \\end{align*}"}
-{"id": "5485.png", "formula": "\\begin{align*} K L ( P _ b ^ n | | P _ { b ' } ^ n ) = \\frac { n } { N } h \\log \\left ( \\frac { 1 + h } { 1 - h } \\right ) \\rho _ h ( b , b ' ) . \\end{align*}"}
-{"id": "4402.png", "formula": "\\begin{align*} [ v , w ] : = \\kappa _ { v , w } = \\begin{cases} 1 & v = w , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "3222.png", "formula": "\\begin{align*} \\widetilde { S } _ { j k } : = \\left ( \\begin{array} { c | c c c } 1 & a _ { j k , 1 } & \\cdots & a _ { j k , r } \\\\ \\hline 0 & & & \\\\ \\vdots & & S _ { j k } & \\\\ 0 & & & \\end{array} \\right ) . \\end{align*}"}
-{"id": "4725.png", "formula": "\\begin{align*} \\beta : = & \\inf \\{ \\pi ^ { \\top } x \\ , | \\ \\ A ^ { i _ 0 } x \\succeq _ { \\mathcal { L } _ { m _ { i _ 0 } } } b ^ { i _ 0 } \\} , \\\\ & \\sup \\{ ( b ^ { i _ 0 } ) ^ { \\top } y ^ { i _ 0 } \\ , | \\ ( y ^ { i _ 0 } ) ^ { \\top } A ^ { i _ 0 } = \\pi ^ { \\top } , \\ y ^ { i _ 0 } \\in \\mathcal { L } ^ * _ { m _ { i _ 0 } } \\} . \\end{align*}"}
-{"id": "2893.png", "formula": "\\begin{align*} C H _ { P o i s _ n } ^ { ( \\bullet > 0 ) } ( A , A ) [ n ] \\stackrel { \\simeq } \\longrightarrow T _ A \\end{align*}"}
-{"id": "8739.png", "formula": "\\begin{gather*} w ( t ) = e ^ { t A } h + \\int _ 0 ^ t e ^ { ( t - s ) A } G u ( s ) d s \\end{gather*}"}
-{"id": "8088.png", "formula": "\\begin{align*} P _ { s _ i } ^ { g l m } = \\sqrt { \\frac { A _ { s _ i , r _ j } E _ { r _ j } + B _ { s _ i , r _ j } E _ { s _ i } } { E _ { r _ j } p _ { t h } ^ { s _ i } } } , \\ ; P _ { r _ j } ^ { g l m } = P _ { s _ i } ^ { g l m } \\frac { E _ { r _ j } } { E _ { s _ i } } \\end{align*}"}
-{"id": "6581.png", "formula": "\\begin{align*} r _ { \\gamma f } ( x ) : = \\gamma f ( x ) + \\tfrac { 1 } { 2 } \\| x \\| ^ 2 \\end{align*}"}
-{"id": "1671.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( \\R ^ d _ x ; H ^ { s } _ p ( \\R ^ d _ v ) ) } = \\Big ( \\int _ { \\R ^ d } \\| f ( x , \\cdot ) \\| _ { H ^ { s } _ p ( \\R ^ d ) } ^ p \\dd x \\Big ) ^ { 1 / p } . \\end{align*}"}
-{"id": "4918.png", "formula": "\\begin{align*} F \\boxtimes G = \\pi _ X ^ * F \\otimes _ { \\O _ X } ^ L \\pi _ Y ^ * G \\end{align*}"}
-{"id": "1112.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { M } \\lambda _ j \\beta _ j - d \\beta _ i & = D ^ * _ { S _ 0 } h + h ^ { ( 1 ) } + \\mu \\cdot h ^ { ( 2 ) } - d \\beta _ i \\\\ & = ( 1 - \\mu - d ) ( D ^ * _ { S _ 0 } h + h ^ { ( 1 ) } ) - d \\mu u _ i + \\mu D ^ * h . \\end{align*}"}
-{"id": "758.png", "formula": "\\begin{gather*} \\sum _ { i ( 1 ) , \\ldots , i ( k ) , j ( 1 ) , \\ldots , j ( l ) = 1 } a ( i ( 1 ) , \\ldots , i ( k ) , j ( 1 ) , \\ldots , j ( l ) ) S _ { j ( 1 ) } \\cdots S _ { j ( l ) } S _ { i ( k ) } ^ * \\cdots S _ { i ( 1 ) } ^ * \\end{gather*}"}
-{"id": "1652.png", "formula": "\\begin{align*} \\big ( L ^ p ( \\R ^ d ; A _ 0 ) , L ^ p ( \\R ^ d ; A _ 1 ) \\big ) _ { \\theta , q } = L ^ p ( \\R ^ d ; ( A _ 0 , A _ 1 ) _ { \\theta , q } ) \\ , . \\end{align*}"}
-{"id": "9684.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\tau ( t ) } { t } = 1 - e ^ { - \\lambda } . \\end{align*}"}
-{"id": "6639.png", "formula": "\\begin{align*} V = \\C ^ n = \\C e _ 0 + \\cdots + \\C e _ { n - 1 } \\end{align*}"}
-{"id": "6060.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } ( u ( x ) - v ( x + 1 ) ) ^ { \\prime \\prime } = 0 & ( - a , a ) , \\\\ ( v ( x ) - u ( x - 1 ) ) ^ { \\prime \\prime } = 0 & ( - a , a ) , \\\\ u , v \\ge 0 & ( - a , a ) , \\\\ u ( - a ) = \\phi ( - a ) v ( a ) = \\varphi ( a ) . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8832.png", "formula": "\\begin{align*} \\mathcal { R } _ { 1 } ( n ) & = 4 \\sum _ { d | n } \\left ( \\frac { - 4 } { d } \\right ) , \\\\ \\mathcal { R } _ { 2 } ( n ) & = 8 \\sum _ { d | n } d - 3 2 \\sum _ { d | \\frac { n } { 4 } } d , \\\\ \\mathcal { R } _ { 3 } ( n ) & = - 4 \\sum _ { d | n } \\left ( \\frac { - 4 } { d } \\right ) d ^ { 2 } + 1 6 \\sum _ { d | n } \\left ( \\frac { - 4 } { n / d } \\right ) d ^ { 2 } , \\\\ \\mathcal { R } _ { 4 } ( n ) & = 1 6 \\sum _ { d | n } d ^ { 3 } - 3 2 \\sum _ { d | \\frac { n } { 2 } } d ^ { 3 } + 2 5 6 \\sum _ { d | \\frac { n } { 4 } } d ^ { 3 } \\end{align*}"}
-{"id": "3412.png", "formula": "\\begin{gather*} \\# M = \\Psi _ P ( x , y ) + \\delta \\geq \\Psi _ P ( x , y ) , \\end{gather*}"}
-{"id": "1135.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } + \\sum _ { i = 1 } ^ { K } \\beta _ { k } } \\right ) ^ 2 \\geq 1 + \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } } \\end{align*}"}
-{"id": "6761.png", "formula": "\\begin{align*} \\theta _ { \\varphi _ { \\beta } } ^ n = e ^ { \\beta \\varphi _ { \\beta } } \\theta _ + ^ n . \\end{align*}"}
-{"id": "304.png", "formula": "\\begin{align*} F _ { n , x , y } ^ { ' , ( 1 ) } : = \\mathbb { P } ( N _ 1 + N _ 3 \\geq j , N _ 2 + N _ 3 \\geq l ) . \\end{align*}"}
-{"id": "4276.png", "formula": "\\begin{align*} a _ n : = \\left \\{ \\begin{array} { l l l } n ^ { 1 - \\frac { 1 } { 2 \\alpha } } & & S d = 1 , \\ \\alpha \\in ( 1 , 2 ] . \\\\ \\sqrt { n \\log n } & & S \\alpha = d \\in \\{ 1 , 2 \\} . \\\\ \\sqrt { n } & & S . \\end{array} \\right . \\end{align*}"}
-{"id": "8065.png", "formula": "\\begin{align*} v _ 1 = \\begin{bmatrix} 1 \\\\ i \\end{bmatrix} , v _ 2 = \\begin{bmatrix} 1 \\\\ - i \\end{bmatrix} , \\end{align*}"}
-{"id": "5621.png", "formula": "\\begin{align*} \\mathcal { A } _ 1 ( n ) = \\{ w \\in \\mathcal { A } ( n ) ~ | ~ w = 0 1 2 0 2 1 \\{ 0 1 2 \\} * 1 2 0 2 1 0 \\} \\subseteq \\mathcal { A } ( n ) \\end{align*}"}
-{"id": "66.png", "formula": "\\begin{align*} \\psi ' ( y _ a , a ) = 0 \\ , \\psi ' ( y , a ) ( y - y _ a ) < 0 \\ , y \\in ( 0 , 1 ) \\setminus \\{ y _ a \\} \\ . \\end{align*}"}
-{"id": "9878.png", "formula": "\\begin{align*} V ^ { 0 , v } _ { \\eta } ( t , x ) & = \\int _ 0 ^ 1 G _ t ( x , y ) \\eta ( y ) d y + \\int _ 0 ^ t \\int _ 0 ^ 1 G _ { t - s } ( x , y ) \\sigma ( s , V ^ { 0 , v } _ { \\eta } ( s ) ) ( y ) v ( s , y ) d y d s \\\\ & + \\int _ 0 ^ t \\int _ 0 ^ 1 G _ { t - s } ( x , y ) f ( s , V ^ { 0 , v } _ { \\eta } ( s ) ) ( y ) d y d s \\\\ & - \\int _ 0 ^ t \\int _ 0 ^ 1 \\partial _ y G _ { t - s } ( x , y ) g ( s , V ^ { 0 , v } _ { \\eta } ( s ) ) ( y ) d y d s . \\end{align*}"}
-{"id": "9193.png", "formula": "\\begin{align*} \\frac { 1 } { u } \\cdot \\frac { d \\phi } { d u } & = \\frac { 1 } { \\ln 2 } \\cdot \\frac { \\alpha ( v ) } { v } \\\\ \\frac { d ^ 2 \\phi } { d u ^ 2 } & = \\frac { 1 } { \\ln 2 } \\cdot \\frac { \\alpha ( v ) - v } { v ^ 3 } , \\end{align*}"}
-{"id": "2165.png", "formula": "\\begin{align*} g _ { \\alpha } * \\partial _ { s } ( g _ { 1 - \\alpha , m } * [ \\psi ^ { 1 + q } \\tilde { u } ^ { 1 - q } ] ) = h _ { m } * ( \\psi ^ { 1 + q } \\tilde { u } ^ { 1 - q } ) . \\end{align*}"}
-{"id": "1159.png", "formula": "\\begin{align*} \\begin{bmatrix} \\frac 1 2 & - h & h ^ 2 \\\\ \\frac 1 2 + H + \\frac 1 2 H ^ 2 & H + H ^ 2 & H ^ 2 \\end{bmatrix} \\begin{bmatrix} \\alpha _ 1 \\\\ \\alpha _ 2 \\\\ \\alpha _ 3 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "8266.png", "formula": "\\begin{align*} E _ \\lambda [ F ] = \\int _ \\R V ' ( u ) p _ { \\lambda } ( u ) \\dd u = \\int _ \\R ( V ' ( u ) - \\lambda ) p _ { \\lambda } ( u ) \\dd u + \\lambda = - \\int _ \\R \\partial _ u p _ { \\lambda } ( u ) \\dd u + \\lambda = \\lambda \\end{align*}"}
-{"id": "7431.png", "formula": "\\begin{align*} \\vec { i } _ { ( u , v ) } = \\left ( - \\vec { i } _ u \\right ) \\# \\vec { i } ^ \\circ _ { v ^ { - 1 } } , \\end{align*}"}
-{"id": "8325.png", "formula": "\\begin{align*} f ( y ) = f \\big ( \\sum _ { i = 1 } ^ n \\mu _ i y _ i \\big ) = \\sum _ { i = 1 } ^ n f ( \\mu _ i y _ i ) = \\sum _ { i = 1 } ^ n h _ i = u \\in k . \\end{align*}"}
-{"id": "3025.png", "formula": "\\begin{align*} { x ^ { [ 2 ] } } ( { t _ 3 } ) = \\frac { { { h ^ { [ 3 2 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 3 2 ] } } ( { t _ 3 } - 2 ) } } u _ 1 ^ { [ 2 ] } , { x ^ { [ 3 ] } } ( { t _ 3 } ) = \\frac { { { h ^ { [ 3 3 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 3 3 ] } } ( { t _ 3 } - 2 ) } } u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "3400.png", "formula": "\\begin{align*} \\dim \\mathbf { H } ^ 1 ( { \\mathcal F } ^ { \\bullet } _ x ) & = \\chi ( { \\mathcal F } ^ 1 _ x ) - \\chi ( { \\mathcal F } ^ 0 _ x ) + 2 \\\\ & = 2 r ^ 2 ( g - 1 ) + 2 + \\sum _ { i = 1 } ^ n \\sum _ { j ' < j } 2 m _ i r ^ { ( i ) } _ { j ' } r ^ { ( i ) } _ j + \\sum _ { i = 1 } ^ n \\sum _ { j = 0 } ^ { s _ i - 1 } m _ i ( r ^ { ( i ) } _ j ) ^ 2 - \\sum _ { i = 1 } ^ n \\sum _ { j = 0 } ^ { s _ i - 1 } m _ i r ^ { ( i ) } _ j \\\\ & = 2 r ^ 2 ( g - 1 ) + 2 + \\sum _ { i = 1 } ^ n m _ i r ( r - 1 ) \\end{align*}"}
-{"id": "8387.png", "formula": "\\begin{align*} Y = \\{ E _ M ( x g ^ { - 1 } ) g : x \\in X , \\ g \\in G \\} , \\end{align*}"}
-{"id": "422.png", "formula": "\\begin{align*} u _ l : = \\Phi _ { l } ^ { \\theta } u . \\end{align*}"}
-{"id": "3031.png", "formula": "\\begin{align*} \\det ( G ) = { D _ { 1 1 } } { g _ { 1 1 } } + { D _ { 1 2 } } { g _ { 1 2 } } + . . . + { D _ { 1 B } } { g _ { 1 B } } . \\end{align*}"}
-{"id": "1766.png", "formula": "\\begin{align*} \\underset { u _ { i } ( y ) > 0 } { \\underset { y \\rightarrow x } { \\lim } } \\ , \\nabla u _ { i } ( y ) = - \\underset { u _ { j } ( y ) > 0 } { \\underset { y \\rightarrow x } { \\lim } } \\ , \\nabla u _ { j } ( y ) . \\end{align*}"}
-{"id": "4358.png", "formula": "\\begin{align*} \\varphi ' ( Y _ { a _ * } + \\delta , a _ * ) < - 2 \\varepsilon < \\varphi ' ( Y _ { a _ * } , a _ * ) = 0 < 2 \\varepsilon < \\varphi ' ( Y _ { a _ * } - \\delta , a _ * ) \\end{align*}"}
-{"id": "2242.png", "formula": "\\begin{align*} \\mu _ { ( k ) } ^ { G } = \\sigma ^ { k } ( k - 1 ) ! ! \\end{align*}"}
-{"id": "6567.png", "formula": "\\begin{align*} \\| \\varDelta f ( w ) - \\varDelta f ( v ) \\| _ { X } & = \\| \\varDelta f ( w ) - \\varDelta f ( v ) \\| _ { L ^ 2 ( \\R ^ d ) } + \\| \\varDelta f ( w ) - \\varDelta f ( v ) \\| _ { L ^ q ( \\R ^ d ) } \\\\ & \\le C _ K ( \\| w - v \\| _ { H ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { W ^ { 2 , q } ( \\R ^ d ) } ) \\\\ & \\le C _ K ( \\| w - v \\| _ { H ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { W ^ { 2 , \\infty } ( \\R ^ d ) } ) = C _ K \\| w - v \\| _ { W } . \\end{align*}"}
-{"id": "6670.png", "formula": "\\begin{gather*} \\beta _ 1 ( t ) - \\beta _ 2 ( t ) = \\\\ = [ x ( u _ k , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ k , \\widetilde { \\sigma } _ 1 ) ] - [ x ( u _ j , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ j , \\widetilde { \\sigma } _ 1 ) ] = \\\\ = [ x ( u _ k , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ j , t + \\widetilde { \\sigma } _ 1 ) ] - ( k - j ) \\cdot \\varepsilon \\geqslant - ( k - j ) \\cdot \\varepsilon . \\end{gather*}"}
-{"id": "3839.png", "formula": "\\begin{align*} L ( x ) = \\frac 1 2 \\left ( 1 + \\frac { 1 } { \\sqrt { 1 - 4 x } } \\right ) . \\end{align*}"}
-{"id": "474.png", "formula": "\\begin{align*} H : = \\{ x \\in \\mathbb { R } ^ 2 \\ , | \\ ( \\beta _ 1 v _ { 1 1 } + \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } + \\beta _ 2 v _ { 2 2 } ) x _ 2 & \\geq \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 , \\\\ ( \\beta _ 1 v _ { 1 1 } - \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } - \\beta _ 2 v _ { 2 2 } ) x _ 2 & \\geq \\beta _ 1 \\alpha _ 1 - \\beta _ 2 \\alpha _ 2 \\} . \\end{align*}"}
-{"id": "2572.png", "formula": "\\begin{align*} S _ I ( u ) : = \\| u \\| _ { S ( I ) } . \\end{align*}"}
-{"id": "1196.png", "formula": "\\begin{align*} w = 2 1 1 2 = \\overline { w } . \\end{align*}"}
-{"id": "3978.png", "formula": "\\begin{align*} \\mathcal { A } _ j \\mathfrak { n } ( \\varphi ( s _ i ) ) w = \\mathfrak { n } ( \\varphi ( s _ i ) ) \\mathcal { A } _ j w + [ \\mathcal { A } _ j , \\mathfrak { n } ( \\varphi ( s _ i ) ) ] w = \\begin{cases} ( \\delta _ j + 1 ) w & j \\neq i \\\\ ( \\delta _ i + 2 ) w & j = i . \\end{cases} \\end{align*}"}
-{"id": "9595.png", "formula": "\\begin{align*} \\psi _ f ( x , t ) = \\psi _ { f , r e g } ( x , t ) + \\psi _ { f , y } ( x , t ) , \\end{align*}"}
-{"id": "1360.png", "formula": "\\begin{align*} \\hat { E } _ n = \\sum _ { t = 1 } ^ { n } \\sigma ^ 2 _ t ( \\hat { \\delta } _ { 0 } ) x _ { t } x _ { t } ^ { \\mathrm { T } } , \\end{align*}"}
-{"id": "4710.png", "formula": "\\begin{align*} G = ( g ^ G \\cup g ^ { - G } ) ^ k . \\end{align*}"}
-{"id": "6083.png", "formula": "\\begin{align*} & t ( t + 1 ) \\iint _ R \\{ v \\} v ^ { - 3 / 2 } \\{ u \\} u ^ { - 3 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 2 } d u d v \\\\ & = i t \\sum _ { ( m , n ) \\in R ( t ) } \\left ( D _ { m , n } ( t ) - E _ { m , n } ( t ) \\right ) + O ( 1 ) . \\end{align*}"}
-{"id": "1207.png", "formula": "\\begin{align*} I ( m ) ^ { ( m j ) } = ( I ( m ) ^ { ( m ) } ) ^ j . \\end{align*}"}
-{"id": "3710.png", "formula": "\\begin{align*} \\frac { d } { d t } \\vec { x } = \\nabla _ { 1 2 3 } Z _ { 1 2 3 } \\times \\nabla _ { 1 2 3 } H _ { 1 2 3 } - b \\nabla _ { 1 2 5 } Z _ { 1 2 5 } \\times \\nabla _ { 1 2 5 } H _ { 1 2 5 } + \\frac { 1 } { \\epsilon } [ \\vec { x } , H _ { 4 5 } ] _ { 4 5 } ~ , \\end{align*}"}
-{"id": "8216.png", "formula": "\\begin{align*} V _ n = \\bar { U } _ n = \\bar { U } _ { - n } = V _ { - n } , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "394.png", "formula": "\\begin{align*} \\check { f } _ { \\ell , \\pm } * \\phi = \\lambda ( f _ { \\ell , \\pm } , \\phi ) \\phi . \\end{align*}"}
-{"id": "9822.png", "formula": "\\begin{align*} M _ i \\ , { } _ \\lambda \\ , v = 0 , \\ \\ \\ \\ Y _ i \\ , { } _ \\lambda \\ , v = d c ^ i v . \\end{align*}"}
-{"id": "298.png", "formula": "\\begin{align*} G _ { n , x , y } ' ( u , v ) : = \\mathbb { P } ( M _ 1 \\geq j , M _ 2 \\geq l ) . \\end{align*}"}
-{"id": "4273.png", "formula": "\\begin{align*} \\sum _ { i , j = 1 } ^ n r ( i - j ) = n ^ { 2 H } \\ell ( n ) , \\end{align*}"}
-{"id": "1741.png", "formula": "\\begin{align*} ( J ) \\geq m \\left ( ( 1 - \\theta ) + \\kappa \\theta \\right ) P _ t ( x , z ) = m \\left ( 1 - \\theta ( 1 - \\kappa ) \\right ) P _ t ( x , z ) , \\end{align*}"}
-{"id": "9294.png", "formula": "\\begin{align*} 2 \\langle - L ^ * _ { \\pi ( t ) } y , y \\rangle = \\pi ^ 2 ( t , z ) \\beta ^ 2 ( t ) \\| y ' ( t , z ) \\| ^ 2 _ { \\mathbf { L } ^ 2 ( \\mathbb { R } ^ + ) } . \\end{align*}"}
-{"id": "788.png", "formula": "\\begin{align*} f ( z , w ) = \\frac { z - q ^ 2 w } { z - w } , g ( z , w ) = - \\frac { ( 1 - q ^ { 2 } ) z } { z - w } = f ( w , z ) - 1 . \\end{align*}"}
-{"id": "2963.png", "formula": "\\begin{align*} \\alpha _ t ( a , \\alpha _ t ( b , c ) ) = \\alpha _ t ( \\alpha _ t ( a , b ) , c ) , \\end{align*}"}
-{"id": "3958.png", "formula": "\\begin{align*} \\sigma ( r ) V ^ { \\lambda } ( A ) = V ^ { - \\lambda } ( A ) \\end{align*}"}
-{"id": "8378.png", "formula": "\\begin{align*} x _ 0 + \\sum _ { i = 1 } ^ k x _ i v _ i \\end{align*}"}
-{"id": "992.png", "formula": "\\begin{align*} A ( u ) & = ( u + \\omega + \\eta N _ { A } ) ( u - \\omega + \\eta N _ { B } ) + A B ^ \\dagger \\\\ B ( u ) & = ( u + \\omega + \\eta N _ { A } ) B + \\eta ^ { - 1 } A \\\\ C ( u ) & = ( u - \\omega + \\eta N _ { B } ) A ^ \\dagger + \\eta ^ { - 1 } B ^ \\dagger \\\\ D ( u ) & = A ^ \\dagger B + \\eta ^ { - 2 } . \\end{align*}"}
-{"id": "5277.png", "formula": "\\begin{align*} \\zeta ( s ) = O \\left ( t ^ { ( 1 - \\sigma ) / 2 } ( \\log t ) ^ 5 \\right ) , \\mbox { u n i f o r m l y i n } \\ 0 \\leq \\sigma \\leq 1 . \\end{align*}"}
-{"id": "3768.png", "formula": "\\begin{align*} \\lim _ { t \\rightarrow \\infty } \\int _ 0 ^ 1 e ^ { ( \\beta - i \\alpha t ) x } g ( x ) d x = 0 , \\end{align*}"}
-{"id": "4445.png", "formula": "\\begin{align*} F _ { n , x } ( u ) : = \\mathbb { P } ( \\xi _ i \\leq u | X _ i = x ) = \\sum _ { j = k } ^ { n - 1 } \\binom { n - 1 } { j } p _ { n , x , u } ^ j ( 1 - p _ { n , x , u } ) ^ { n - 1 - j } , \\end{align*}"}
-{"id": "8648.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( A \\right ) = V \\times U , A \\left ( \\begin{array} [ c ] { c } y \\\\ z \\end{array} \\right ) = \\left ( \\begin{array} [ c ] { c c } 0 & I \\\\ - \\Lambda & 0 \\end{array} \\right ) \\left ( \\begin{array} [ c ] { c } y \\\\ z \\end{array} \\right ) , \\left ( \\begin{array} [ c ] { c } y \\\\ z \\end{array} \\right ) \\in \\mathcal { D } \\left ( A \\right ) , \\end{align*}"}
-{"id": "9585.png", "formula": "\\begin{align*} D _ y = \\{ \\psi \\in L ^ 2 ( \\R ^ 3 ) : \\psi = \\psi _ { r e g } + \\xi g ( x - y ) , ~ ~ \\psi _ { r e g } \\in H ^ 2 ( \\R ^ 3 ) , ~ ~ \\psi _ { r e g } ( y ) = \\xi \\in \\C \\} , \\end{align*}"}
-{"id": "826.png", "formula": "\\begin{align*} \\tilde { C } ^ { [ 0 , M ] } ( u ; s ) & = ( - s u ) ^ { - 1 } ( - s ( 1 - s u ) ) ^ { - ( M + 1 ) } \\prod _ { 1 \\le i \\le k } K ^ { ( i ) } ( ( - s ) ^ { - ( M + 1 - i ) } ) \\\\ & \\times C ^ { [ 0 , M ] } ( - s u ; s ) \\prod _ { 1 \\le i \\le k } K ^ { ( i ) } ( ( - s ) ^ { M + 1 - i } ) , \\end{align*}"}
-{"id": "8671.png", "formula": "\\begin{gather*} a _ t = f - V _ { t ^ 2 } f , \\ ; \\ ; \\ ; b _ t = V _ { t ^ 2 } f \\end{gather*}"}
-{"id": "2810.png", "formula": "\\begin{align*} \\mathbb { E } _ { \\mathbb { P } } \\Psi _ g = \\det \\biggl ( 1 + ( g - 1 ) K \\chi _ { B } \\biggr ) , \\end{align*}"}
-{"id": "9730.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { x ( t ) } { G ^ { - 1 } ( t ) } = ( a - b ) ^ { - 1 / ( \\beta - 1 ) } , \\end{align*}"}
-{"id": "2686.png", "formula": "\\begin{align*} \\int _ { \\{ v < u \\} } ( e ^ { \\beta u } - e ^ { \\beta v } ) e ^ { - \\phi } d \\mu = 0 . \\end{align*}"}
-{"id": "8679.png", "formula": "\\begin{align*} \\nabla _ k \\nabla _ { \\xi } ^ G P _ t [ \\phi ] ( x ) = \\int _ H \\ < \\Gamma _ t G \\xi , Q _ t ^ { - \\frac { 1 } { 2 } } y \\ > \\nabla _ { e ^ { t A } k } \\phi ( e ^ { t A } x + y ) \\mu _ t ( d y ) . \\end{align*}"}
-{"id": "3207.png", "formula": "\\begin{align*} F _ { j , \\beta } ( z _ j ) - \\sum _ { | \\alpha | = 2 } T _ { j k } F _ { k , \\alpha } ( z _ k ) \\cdot \\tau _ { k j , \\beta } ^ \\alpha = - f _ { k j , \\beta } . \\end{align*}"}
-{"id": "2335.png", "formula": "\\begin{align*} x _ j = a + \\frac { b } { z _ j + c } , ~ z _ j = \\frac { b } { x _ j - a } - c . \\end{align*}"}
-{"id": "8342.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } A _ 0 \\\\ A _ 1 \\\\ \\vdots \\\\ A _ { n - 2 } \\\\ A _ { n - 1 } \\end{array} \\right ] = M ^ { - 1 } \\left [ \\begin{array} { c } \\gamma _ 1 \\\\ \\vdots \\\\ \\gamma _ { m } \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "2759.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\int _ 0 ^ t \\psi ( s ) d s + \\int _ 0 ^ t \\varphi ( s ) d W ( s ) , \\end{align*}"}
-{"id": "2760.png", "formula": "\\begin{align*} F ( X _ { t } ) = & \\ F ( X _ { 0 } ) + \\int _ 0 ^ t \\Delta _ { s } F ( X _ { s } ) d s + \\int _ 0 ^ t \\Delta _ { x } F ( X _ { s } ) \\psi ( s ) d s \\\\ & + \\int _ 0 ^ t \\Delta _ { x } F ( X _ { s } ) \\varphi ( s ) d W ( s ) + \\frac { 1 } { 2 } \\int _ 0 ^ t \\Delta _ { x x } F ( X _ { s } ) \\varphi ( s ) ^ { 2 } d s . \\end{align*}"}
-{"id": "4745.png", "formula": "\\begin{align*} \\pm \\int _ { ( B ^ c _ \\varepsilon ( x ) ) ^ \\pm } \\frac { \\xi \\cdot ( x - y ) } { \\ , | x - y | ^ { N + 2 s } \\ , } \\ , d y & = - \\int _ \\varepsilon ^ { + \\infty } \\frac { d \\rho } { \\ , \\rho ^ { 2 s } \\ , } \\int _ { \\partial B _ 1 ^ \\pm } ( \\pm \\xi \\cdot w ) \\ , d { \\mathcal H } ^ { N - 1 } \\end{align*}"}
-{"id": "7378.png", "formula": "\\begin{align*} h ^ { t + 1 } = \\beta _ 0 - ( A ^ * z ^ t + \\beta ^ t ) , & q ^ t = \\beta ^ t - \\beta _ 0 , \\\\ b ^ t = w - z ^ t , & m ^ t = - z ^ t . \\end{align*}"}
-{"id": "861.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k q _ { j } ( n ) V _ { n + j } = 0 \\end{align*}"}
-{"id": "255.png", "formula": "\\begin{align*} M _ { f , a , \\beta } ( x ) & \\leq \\max \\biggl \\{ \\max _ { r = 1 , \\ldots , m } \\frac { A _ r ' } { ( 1 - x ^ 2 ) ^ { 2 r } } \\ , , \\ , \\sup _ { y : 0 < | y - x | \\leq r _ a ( x ) } \\frac { A _ { m + 1 } ' f ( y ) } { ( 1 - y ^ 2 ) ^ { 2 ( m + 1 ) } f ( x ) } \\biggr \\} \\\\ & \\leq \\frac { B _ { m + 1 } ' } { ( 1 - x ^ 2 ) ^ { 2 ( m + 1 ) } } \\leq a \\bigl ( f ( x ) \\bigr ) , \\end{align*}"}
-{"id": "3554.png", "formula": "\\begin{align*} \\| ( g ^ { \\theta } - g _ { \\mathbb { E } } , \\pi ^ { \\theta } ) \\| _ { C ^ 2 _ { - q } ( \\mathbb { R } ^ 3 \\setminus B _ R ) \\times C ^ 1 _ { - 1 - q } ( \\mathbb { R } ^ 3 \\setminus B _ R ) } & \\le \\kappa \\\\ \\| ( \\mu ^ \\theta , J ^ \\theta ) \\| _ { C ^ 0 _ { - 3 - q _ 0 } ( \\mathbb { R } ^ 3 \\setminus B _ R ) } & \\le \\kappa \\end{align*}"}
-{"id": "683.png", "formula": "\\begin{align*} f ( g ) = \\int _ Z \\phi ( g z ) \\ , d m ( z ) , \\textrm { f o r a l l $ g \\in G $ } . \\end{align*}"}
-{"id": "1837.png", "formula": "\\begin{align*} g = e ^ { 2 s ^ 2 f } Z ( w ) ^ 2 \\left ( \\frac { d w ^ { 2 } } { w ^ 2 } + w ^ 2 s ^ 2 d \\theta ^ { 2 } \\right ) , \\ Z ( w ) = \\frac { \\pi / w } { \\sin ( \\pi / w ) } \\end{align*}"}
-{"id": "2961.png", "formula": "\\begin{align*} \\mu _ t ( x , \\alpha _ t ( a , b ) ) = \\alpha _ t ( a , \\mu _ t ( x , b ) ) + \\alpha _ t ( \\mu _ t ( x , a ) , b ) , \\end{align*}"}
-{"id": "6767.png", "formula": "\\begin{align*} \\varphi _ { \\beta } \\geq u _ \\beta : = \\left ( 1 - \\frac { 1 } { \\beta } \\right ) V _ { \\theta } + \\frac { 1 } { \\beta } \\phi - \\frac { n \\log \\beta } { \\beta } . \\end{align*}"}
-{"id": "3521.png", "formula": "\\begin{align*} \\int _ \\Omega \\xi | u | ^ 2 d ^ { - 4 j } \\rho \\ , d \\mu _ g & \\le \\left ( \\frac { 4 } { N } \\right ) ^ j \\int _ \\Omega \\xi | \\nabla ^ j u | ^ 2 \\rho \\ , d \\mu _ g \\\\ & + \\sum _ { i = 1 } ^ j \\left ( \\frac { 4 } { N } \\right ) ^ { j + 1 - i } \\sup _ \\Omega ( | \\nabla \\xi | d ^ { - 4 i + 2 } ) \\| \\nabla ^ { j - i } u \\| _ { L _ \\rho ^ 2 ( \\Omega ) } ^ 2 . \\end{align*}"}
-{"id": "3280.png", "formula": "\\begin{align*} \\hat { R } ( v _ { 1 } \\otimes v _ { 0 } ) = v _ { 0 } \\otimes v _ { 1 } + ( q ^ { 2 } - q ^ { - 2 } ) v _ { 1 } \\otimes v _ { 0 } . \\end{align*}"}
-{"id": "6334.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } \\| _ { M _ 2 } = \\sup _ { l \\in \\Gamma _ k ^ { \\alpha _ 2 , \\alpha _ 1 } } \\| \\Box _ l ^ { \\alpha _ 2 } \\Box _ k ^ { \\alpha _ 1 } f \\| _ { L ^ { p _ 2 } } . \\end{align*}"}
-{"id": "213.png", "formula": "\\begin{align*} \\mathbb { P } \\bigl ( \\mathrm { B } _ 1 \\geq a _ n / ( n - 1 ) \\bigr ) = \\mathbb { P } ( B _ 2 \\leq k - 1 ) \\leq \\exp \\biggl ( - \\frac { ( a _ n - k + 1 ) ^ 2 } { 2 a _ n } \\biggr ) = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "4029.png", "formula": "\\begin{align*} [ x _ { \\alpha _ 0 } ^ { i _ 0 } , \\ldots x _ { \\alpha _ r } ^ { i _ r } ] : = \\{ \\underline { a } \\in \\Sigma _ { A } : a _ 0 = x _ { \\alpha _ 0 } ^ { i _ 0 } , \\ldots a _ r = x _ { \\alpha _ r } ^ { i _ r } \\} \\end{align*}"}
-{"id": "1645.png", "formula": "\\begin{align*} f \\left ( t , x _ { t } , v _ { t } \\right ) = f _ { 0 } \\left ( x _ { 0 } , v _ { 0 } \\right ) \\end{align*}"}
-{"id": "9697.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log g ( x ( t ) ) } { \\log t } = - \\frac { 1 } { \\log ( 1 / ( 1 - q ) ) } \\log \\left ( \\frac { a } { b } \\right ) . \\end{align*}"}
-{"id": "8686.png", "formula": "\\begin{align*} \\Big ( \\sum _ { m \\ge 1 } \\sup _ { | a | _ U = 1 } | \\nabla _ k \\nabla _ { a } ^ G P _ t [ \\Phi _ m ] ( x ) | ^ 2 \\Big ) ^ { 1 / 2 } \\ , \\le \\frac { c } { t ^ { \\frac { 4 - 3 \\alpha } { 2 } } } | k | _ K \\ , \\| \\Phi \\| _ { \\big ( C _ b ( H , J ) , C ^ 1 _ K ( H , J ) \\big ) _ { \\alpha , \\infty } } . \\end{align*}"}
-{"id": "8796.png", "formula": "\\begin{align*} T ( u ) = _ { 0 } \\left ( M _ 0 ^ { ( 1 ) } ( u ) K _ 0 ( u ) M _ 0 ^ { ( 2 ) } \\left ( u \\right ) \\tilde { K } _ 0 ( u ) \\right ) , \\end{align*}"}
-{"id": "4595.png", "formula": "\\begin{align*} & \\Big | \\widehat { m _ n ( D ) } ( A ) - \\prod _ { i = 1 } ^ r \\prod _ { j = 1 } ^ n \\Theta ( a _ i x _ j ) \\Big | \\le 2 \\cdot ( 1 - \\prod _ { w = 0 } ^ { 2 r - 1 } ( 1 - q ^ { w - 2 n } ) ) , \\end{align*}"}
-{"id": "3147.png", "formula": "\\begin{align*} & P r \\biggl ( \\lVert L ^ { - 1 } \\sum _ { i = 1 } ^ { L } \\Pi \\cdot \\Pi _ { X _ i } \\cdot X _ i \\cdot \\Pi _ { X _ i } \\cdot \\Pi - \\omega \\rVert _ 1 \\allowdisplaybreaks \\\\ & \\leq 1 - p ( \\mathsf { M } ' ) + 4 \\sqrt { 1 - p ( \\mathsf { M } ' ) } + 4 2 \\sqrt [ 8 ] { \\epsilon } \\biggr ) \\allowdisplaybreaks \\\\ & \\geq 1 - 2 D \\exp \\left ( - p ( \\mathsf { M } ' ) \\frac { \\epsilon ^ 3 L d } { 2 \\ln 2 D } \\right ) \\end{align*}"}
-{"id": "3150.png", "formula": "\\begin{align*} & \\mathrm { t r } ( \\Pi _ { \\mathsf { V } ( \\cdot , p ) , \\alpha } ( t ^ n ) ) \\allowdisplaybreaks \\\\ & \\leq 2 ^ { n ( S ( \\mathsf { V } ( \\cdot , p ) \\mid q ) + \\delta ( \\alpha ) ) } \\allowdisplaybreaks \\\\ & = 2 ^ { n ( \\sum _ { t } q ( t ) \\mathsf { V } ( t , p ) + \\delta ( \\alpha ) ) } \\allowdisplaybreaks \\\\ & = 2 ^ { n ( \\sum _ { t } q ( t ) S ( V _ { t } ( p ) ) + \\delta ( \\alpha ) ) } \\end{align*}"}
-{"id": "2927.png", "formula": "\\begin{align*} M ^ \\perp _ \\sigma ( w ) \\cdot 1 = \\begin{cases} ( 1 - w ) ^ { - 1 } & \\mbox { i f $ \\sigma = ( 0 ) $ } ; \\cr 1 & \\mbox { o t h e r w i s e } , \\cr \\end{cases} \\end{align*}"}
-{"id": "1118.png", "formula": "\\begin{align*} \\mathbf { G } _ \\mathrm { s } [ 1 ] = \\mathbf { \\hat { G } } _ \\mathrm { s } [ 1 ] + \\mathcal { E } _ \\mathrm { s } [ 1 ] \\end{align*}"}
-{"id": "9102.png", "formula": "\\begin{align*} a \\ll b \\textrm { i f a n d o n l y i f } a + b = b . \\end{align*}"}
-{"id": "9923.png", "formula": "\\begin{align*} [ E , F ] v _ i & \\ ; = \\ ; \\pm \\big ( [ n - i ] E v _ { i + 1 } - [ i ] F v _ { i - 1 } \\big ) \\\\ & \\ ; = \\ ; \\pm \\big ( [ n - i ] [ i + 1 ] [ i ] [ n - i + 1 \\big ) v _ { i } \\\\ & \\ ; = \\ ; \\pm [ n - 2 i ] v _ { i } \\\\ & \\ ; = \\ ; \\frac { K ^ 2 - K '^ 2 } { q - q ^ { - 1 } } \\ , v _ i , \\end{align*}"}
-{"id": "3256.png", "formula": "\\begin{align*} [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q = E _ \\xi E _ { \\xi ^ \\prime } ^ * - q ^ { - ( \\xi , \\xi ^ \\prime ) } E _ { \\xi ^ \\prime } ^ * E _ \\xi . \\end{align*}"}
-{"id": "1318.png", "formula": "\\begin{align*} U ( \\alpha , H ) = \\sum _ { 1 \\leq m \\leq H } e ( m \\alpha ) . \\end{align*}"}
-{"id": "191.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\bigl | \\mathbb { E } _ f ( \\hat { H } _ n ^ w ) - H ( f ) \\bigr | = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { 2 ( \\lfloor d / 4 \\rfloor + 1 ) } { d } } } { n ^ { \\frac { 2 ( \\lfloor d / 4 \\rfloor + 1 ) } { d } } } \\ , , \\ , \\frac { k ^ { \\frac { \\beta } { d } } } { n ^ { \\frac { \\beta } { d } } } \\biggr \\} \\biggr ) , \\end{align*}"}
-{"id": "3555.png", "formula": "\\begin{align*} | B ^ R _ { ( g ^ \\theta , \\pi ^ \\theta ) } ( x ^ k , 0 ) - 1 6 \\pi \\mathcal C ^ \\theta _ k | & \\le \\kappa | \\theta | ^ 2 R ^ { - 1 } \\\\ | B ^ R _ { ( g ^ \\theta , \\pi ^ \\theta ) } ( 0 , x \\times \\frac { \\partial } { \\partial x ^ k } ) - 8 \\pi \\mathcal J ^ \\theta _ k | & \\le \\kappa | \\theta | ^ { 2 } R ^ { - 1 } . \\end{align*}"}
-{"id": "8492.png", "formula": "\\begin{align*} B : = B _ 0 \\cdot B _ 0 ^ { - 1 } , B _ 2 = B \\cdot B , B _ 3 = B \\cdot B \\cdot B , \\ldots \\end{align*}"}
-{"id": "4435.png", "formula": "\\begin{align*} \\hat { H } _ n ^ w : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ k w _ j \\log \\xi _ { ( j ) , i } , \\end{align*}"}
-{"id": "1468.png", "formula": "\\begin{align*} \\| b \\| _ { \\rm B M O } : = \\| b ^ \\sharp \\| _ { L ^ \\infty } < \\infty , \\end{align*}"}
-{"id": "8897.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\abs { D _ { A _ n } u _ n } ^ 2 + \\abs { u _ n } ^ 2 = 1 , \\end{align*}"}
-{"id": "5208.png", "formula": "\\begin{align*} \\bar \\partial _ M f = \\sum _ { j = 1 } ^ { m - 1 } \\sideset { } { ' } \\sum _ { \\vert J \\vert = q } ( \\overline { L _ j } f _ J ) \\overline { \\omega _ j } \\wedge \\overline { \\omega _ J } + \\sideset { } { ' } \\sum _ { \\vert J \\vert = q } f _ J \\bar \\partial _ M \\overline { \\omega _ J } . \\end{align*}"}
-{"id": "9704.png", "formula": "\\begin{gather*} \\lim _ { y \\to \\infty } \\frac { G ^ { - 1 } ( y ) } { 1 / \\log _ 2 y } = 1 , G ^ { - 1 } \\in _ \\infty ( 0 ) , \\\\ \\lim _ { y \\to \\infty } \\frac { \\Gamma ( y ) } { 1 / ( y \\log y ( \\log _ 2 y ) ^ 2 ) } = 1 , \\Gamma = g \\circ G ^ { - 1 } \\in _ \\infty ( - 1 ) . \\end{gather*}"}
-{"id": "2797.png", "formula": "\\begin{align*} \\frac { \\prod _ { i = 1 } ^ r \\sum _ { j \\in V ( S _ i ) } ( x _ j - s _ j ) } { ( n - 2 ) _ { n - r } } \\prod _ { j = 1 } ^ n \\ , ( x _ j - 1 ) _ { s _ j - 1 } , \\end{align*}"}
-{"id": "2940.png", "formula": "\\begin{align*} R f ( \\omega , t ) = \\int _ { H _ { \\omega , t } } f ( x ) d m ( x ) , \\end{align*}"}
-{"id": "9469.png", "formula": "\\begin{align*} E ( u ) = 2 \\pi \\lim _ { s \\to \\infty } e ^ { f ( \\gamma ( s ) ) } \\in [ 0 , + \\infty ] . \\end{align*}"}
-{"id": "1124.png", "formula": "\\begin{align*} \\mathcal { R } ^ \\mathrm { D L } _ k [ \\iota ] = T _ d \\log _ 2 \\left ( 1 + \\gamma _ k ^ { \\mathrm { D L } } [ \\iota ] \\right ) \\end{align*}"}
-{"id": "6225.png", "formula": "\\begin{align*} A M \\tilde { x } = b , \\ x = M \\tilde { x } \\ \\ \\ \\ M _ { 1 } A M _ { 2 } \\tilde { x } = M _ { 1 } b , \\ x = M _ { 2 } \\tilde { x } . \\end{align*}"}
-{"id": "9854.png", "formula": "\\begin{align*} & R _ { x x } ( t ) \\otimes ( \\delta ( - t ) - \\alpha \\theta ( - t - \\tau ) ) \\otimes ( \\delta ( t ) - \\alpha \\theta ( t - \\tau ) ) \\\\ = & R _ { s s } ( t ) \\otimes ( \\theta ( - t ) \\otimes h _ { \\rm S R } ( - t ) ) \\otimes ( \\theta ( t ) \\otimes h _ { \\rm S R } ( t ) ) \\\\ & + R _ { n _ { \\rm R } n _ { \\rm R } } ( t ) \\otimes \\theta ( - t ) \\otimes \\theta ( t ) . \\end{align*}"}
-{"id": "5563.png", "formula": "\\begin{align*} D ( - \\lambda ) = W ( \\hat c _ 0 , c _ 0 ) + \\sum _ { n = 1 } \\lambda ^ n \\int _ { 0 \\leq \\xi _ 1 \\leq \\xi _ 2 \\cdots \\xi _ n \\leq 1 } \\hat c _ 0 ( \\xi _ n ) \\rho ( \\xi _ n ) \\left ( \\prod _ { j = 1 } ^ { n - 1 } ( \\xi _ { j + 1 } - \\xi _ j ) \\rho ( \\xi _ j ) \\right ) c _ 0 ( \\xi _ 1 ) d \\xi _ 1 d \\xi _ 2 \\dots d \\xi _ n , \\end{align*}"}
-{"id": "3371.png", "formula": "\\begin{align*} A = \\Delta t P _ { \\max } + E _ { \\min } + V \\zeta _ { \\max } \\end{align*}"}
-{"id": "2592.png", "formula": "\\begin{align*} v ^ { [ t _ n ] } ( t , x ) : = t _ n ^ { \\frac 1 { p } } v ( t _ n t , \\sqrt { t _ n } x ) . \\end{align*}"}
-{"id": "6701.png", "formula": "\\begin{align*} f \\ast g = u ^ { - 1 } ( ( u f ) _ { K Q } \\star v ^ { Q P } \\star ( u g ) _ { P L } ) \\theta ^ K \\bar \\theta ^ L , \\end{align*}"}
-{"id": "5323.png", "formula": "\\begin{align*} \\eta ^ 2 \\left ( v _ i + \\omega + \\eta \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) \\left ( v _ i - \\omega + \\eta \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) & = \\prod _ { j \\neq i } ^ { N - r } \\frac { v _ i - v _ j - \\eta } { v _ i - v _ j + \\eta } , r < N . \\end{align*}"}
-{"id": "9266.png", "formula": "\\begin{align*} H ( t , x , y , \\varphi , \\pi , p , q , z ) = [ \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial x ^ 2 } \\varphi ( x ) + \\pi y a _ 0 ( t , z ) ] p + \\pi y b _ 0 ( t , z ) q , \\end{align*}"}
-{"id": "2444.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = & \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ { p _ 2 } } = \\| \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\Box _ l ^ { \\alpha _ 1 } \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ { p _ 2 } } \\\\ \\lesssim & \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| ^ { p _ 2 } _ { L ^ { p _ 2 } } \\right ) ^ { 1 / p _ 2 } \\lesssim \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "345.png", "formula": "\\begin{align*} \\pi _ I ( U ( \\gamma ) ) = U ^ \\pi ( \\gamma ) \\end{align*}"}
-{"id": "1036.png", "formula": "\\begin{align*} c _ { d , d } { \\ , } c _ { r _ 0 , u _ 0 } ^ d + ( - 1 ) ^ { q _ 1 } { r _ 0 \\choose u _ 0 } \\gamma _ { r _ 0 } - ( 1 + \\delta _ { q _ 1 , q _ 2 } ) { \\ , } c _ { d , d } ^ { q _ 1 } { \\ , } c _ { q , q } ^ { q _ 2 + 1 } ~ = ~ 0 . \\end{align*}"}
-{"id": "5832.png", "formula": "\\begin{align*} \\mathcal { L } _ { X ^ { \\left [ 2 \\right ] } } \\Theta = 0 . \\end{align*}"}
-{"id": "6913.png", "formula": "\\begin{align*} \\frac { 6 d ^ { 2 } - 1 4 d + 7 } { 4 \\left ( d - 1 \\right ) ^ { 2 } } & = \\frac { 3 } { 4 } + O \\left ( \\frac { x } { n } \\right ) , \\\\ \\frac { 1 } { 2 \\left ( d - 1 \\right ) ^ { 3 } } & = \\frac { 1 } { 2 } + O \\left ( \\frac { x } { n } \\right ) , \\\\ \\frac { \\left ( 2 d ^ { 2 } - 4 d + 1 \\right ) } { 4 \\left ( d - 1 \\right ) ^ { 4 } d ^ { 2 } } & = \\frac { 1 } { 1 6 } + O \\left ( \\frac { x } { n } \\right ) . \\end{align*}"}
-{"id": "5632.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( \\mu , \\nu ) = \\underset { \\pi } { \\inf } \\iint _ { \\Omega \\times \\Omega } \\vert x - y \\vert ^ 2 d \\pi ( x , y ) , \\end{align*}"}
-{"id": "8885.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\int _ { \\R ^ N } \\abs { D _ { A _ n } ( \\tau ^ { A _ n } _ { a _ n } u _ n ) } ^ 2 + \\abs { \\tau ^ { A _ n } _ { a _ n } u _ n } ^ 2 = \\liminf _ { n \\to \\infty } \\int _ { \\R ^ N } \\abs { D _ { A _ n } u _ n } ^ 2 + \\abs { u _ n } ^ 2 < \\infty . \\end{align*}"}
-{"id": "6599.png", "formula": "\\begin{align*} x ^ { k + 1 } = ( 1 - \\alpha ) x ^ k + \\alpha P _ { C } ^ { \\alpha _ 2 } P _ { D } ^ { \\alpha _ 1 } x ^ k . \\end{align*}"}
-{"id": "9056.png", "formula": "\\begin{align*} \\P \\left ( \\forall j \\leq N , \\ , W _ j \\leq f ( j ) + z \\right ) & \\ll \\sum _ { k = 0 } ^ N ( N - k + 1 ) ^ { - 1 / 2 } \\P \\left ( \\min _ { j \\in [ | 0 , k | ] } W _ j \\geq 0 , \\ , W _ k \\leq f ( k ) + z \\right ) . \\end{align*}"}
-{"id": "1201.png", "formula": "\\begin{align*} R [ M ( \\Lambda _ 2 ) ] = R [ x , x y , x y ^ 2 , x y ^ 3 , \\ldots ] \\end{align*}"}
-{"id": "2375.png", "formula": "\\begin{align*} r _ { \\gamma f } ( x ) : = \\gamma f ( x ) + \\tfrac { 1 } { 2 } \\| x \\| ^ 2 \\end{align*}"}
-{"id": "5304.png", "formula": "\\begin{align*} H _ { 1 , 1 } = U ( N _ { a , 1 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } - N _ { b , 1 } ) + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) \\end{align*}"}
-{"id": "4782.png", "formula": "\\begin{align*} \\mathcal { M } ' _ a : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "5064.png", "formula": "\\begin{align*} \\lambda ( F A _ { x _ o } ) = \\nu ( F A ) > 1 - \\nu ( A ) . \\end{align*}"}
-{"id": "2006.png", "formula": "\\begin{align*} | \\mathcal { A } _ n h ( x ) | \\leq \\eta _ m = \\eta _ m ^ n , \\end{align*}"}
-{"id": "4244.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty { 2 n \\choose n } \\frac { ( - 1 ) ^ { n - 1 } } { 4 ^ n ( 2 n - 1 ) } t ^ n = \\sum _ { n = 0 } ^ \\infty \\left ( \\frac { 1 } { n ! } \\sum _ { m = 0 } ^ n \\Big ( \\frac { 1 } { 2 } \\Big ) ^ m S _ 1 ( n , m ) \\right ) t ^ n \\end{align*}"}
-{"id": "5170.png", "formula": "\\begin{gather*} r _ { x } \\otimes r _ { y } = \\sum _ { x = v z , \\ , y = \\bar { z } w } r _ { v w } + r _ { v \\cdot w } , \\end{gather*}"}
-{"id": "9919.png", "formula": "\\begin{align*} C \\ ; : = \\ ; e f + \\frac { q ^ { - 1 } k + q k ^ { - 1 } } { ( q - q ^ { - 1 } ) ^ 2 } \\ ; = \\ ; f e + \\frac { q k + q ^ { - 1 } k ^ { - 1 } } { ( q - q ^ { - 1 } ) ^ 2 } \\end{align*}"}
-{"id": "5259.png", "formula": "\\begin{align*} \\operatorname * { d o m } \\varphi _ { A , k } = \\mathbb { R } k + \\operatorname * { b d } A . \\end{align*}"}
-{"id": "2707.png", "formula": "\\begin{align*} \\int _ { \\{ V _ { \\theta } - j < \\psi _ s \\leq v < \\psi < \\varphi + s \\} } \\theta _ { \\psi _ { t , j } } ^ n = 0 . \\end{align*}"}
-{"id": "3008.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in D _ n ^ { \\{ 0 , \\ , 1 \\} } } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L ( \\sigma ) } } & = \\sum _ { \\substack { \\{ \\sigma \\in D _ n ^ { \\{ 0 , \\ , 1 \\} } : \\\\ | \\sigma ( 3 ) | = n \\} } } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L ( \\sigma ) } } + \\sum _ { \\substack { \\{ \\sigma \\in D _ n ^ { \\{ 0 , \\ , 1 \\} } : \\\\ | \\sigma ( n ) | = n \\} } } { ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L ( \\sigma ) } } . \\end{align*}"}
-{"id": "2385.png", "formula": "\\begin{align*} z ^ { k + 1 } = ( 1 - \\alpha ) z ^ k + \\alpha R _ { \\gamma g } R _ { \\gamma f } z ^ k \\end{align*}"}
-{"id": "5883.png", "formula": "\\begin{align*} \\ln F \\left ( t , x \\right ) = \\frac { 2 e ^ { - m t } } { \\sqrt { 1 + \\varepsilon ^ { 2 } } } i \\arctan \\left ( \\frac { \\tan \\left ( \\frac { x } { 2 } \\right ) + \\varepsilon } { \\sqrt { 1 - \\varepsilon ^ { 2 } } } \\right ) + \\frac { c } { m } e ^ { - m t } - \\frac { 1 } { 4 m } e ^ { - 2 m t } \\end{align*}"}
-{"id": "6129.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } ^ 2 + 2 R _ { 1 3 4 2 } R _ { 1 4 2 3 } = & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( x + 2 y ) - ( x + 2 y ) ^ 2 ] \\\\ \\ge & - \\frac 1 2 ( x + 2 y ) ^ 2 . \\\\ \\end{align*}"}
-{"id": "85.png", "formula": "\\begin{align*} w = \\Lambda ( r _ 1 , f ) \\cdot \\theta ^ { a } ( \\Lambda ( r _ 2 , f ) ) ^ { \\epsilon } , \\end{align*}"}
-{"id": "495.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } r ^ { N + 2 s + k } D ^ k F _ N ( r ) & = \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\ , \\alpha _ { j , k } \\ , \\lim _ { r \\to \\infty } r ^ { N + 2 s + 2 j } F _ { N + 2 j } ( r ) \\\\ & = \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\alpha _ { j , k } \\ , \\ell _ { N + 2 j , \\ , 0 } . \\end{align*}"}
-{"id": "6224.png", "formula": "\\begin{align*} M A x = M b \\end{align*}"}
-{"id": "6247.png", "formula": "\\begin{align*} \\Gamma ( \\Omega ) = \\lambda _ 2 - \\lambda _ 1 \\ge \\bar { \\lambda } _ 2 ( n , D ) - \\bar { \\lambda } _ 1 ( n , D ) \\ \\mbox { i f } \\ D \\le \\frac { \\pi } { 2 } , \\end{align*}"}
-{"id": "2621.png", "formula": "\\begin{align*} { \\zeta ( t , x ) } = { \\sum _ { m = 0 } ^ { \\infty } } { \\sum _ { l = 1 } ^ { h ( m , n ) } } S _ m ^ l ( x ) \\zeta _ { m } ^ l ( t ) , \\end{align*}"}
-{"id": "5966.png", "formula": "\\begin{align*} [ v _ 0 v _ 1 \\dots v _ i ] = [ \\{ v _ 0 , \\dots , v _ i \\} \\ , , \\ , v _ 0 < v _ 1 < \\dots < v _ i ] \\ . \\end{align*}"}
-{"id": "79.png", "formula": "\\begin{align*} v _ t = a \\ , v _ { x x } + \\lambda \\ , v , \\ \\ t > 0 , \\ x > 0 , \\end{align*}"}
-{"id": "9271.png", "formula": "\\begin{align*} \\begin{cases} d \\tilde { p } ( t , z ) & = \\tilde { p } ( t , z ) [ \\pi ( t , z ) b _ 0 ( t , z ) - \\frac { a _ 0 ( t , z ) } { b _ 0 ( t , z ) } ] d B ( t ) \\\\ \\tilde { p } ( T , z ) & = \\int _ D \\frac { \\partial U } { \\partial y } ( x , Y ( T , x , z ) , z ) Y ( T , x , z ) d x \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] . \\end{cases} \\end{align*}"}
-{"id": "3422.png", "formula": "\\begin{align*} f ( t , y ( t ) , z ( t ) ) = \\end{align*}"}
-{"id": "6341.png", "formula": "\\begin{align*} \\big \\| \\{ \\langle k \\rangle ^ { \\frac { R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ 1 , \\alpha _ 2 ) } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } } \\} \\big \\| _ { l _ { \\infty } ^ { s _ 2 - s _ 1 , \\alpha _ 1 \\vee \\alpha _ 2 } } = \\sup _ { k \\in \\mathbb { Z } ^ n } \\langle k \\rangle ^ { \\frac { s _ 2 - s _ 1 } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } } \\langle k \\rangle ^ { \\frac { R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ 1 , \\alpha _ 2 ) } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } } . \\end{align*}"}
-{"id": "2138.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } ( u ( x , t ) - u _ { 0 } ( x ) ) + ( - \\Delta ) ^ { \\beta } u ( x , t ) & = f ( x , t ) \\Omega \\times [ 0 , T ] , \\\\ u ( x , t ) & = 0 \\quad \\quad \\quad \\ , \\mathbb { R } ^ { n } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = u _ { 0 } ( x ) \\quad \\ , \\ , \\Omega , \\ , t = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1849.png", "formula": "\\begin{align*} \\sum _ { ( m , n ) \\in R ( t ) } E _ { m , n } ( t ) & \\ll \\sum _ { 1 \\le n < t ^ 4 } \\int _ n ^ { n + 1 } \\frac { v - n } { v ^ { 3 / 2 } } \\int _ { \\abs { \\log \\frac { u } { v } } < 2 \\sqrt { \\frac { \\log t } { t } } } u ^ { - 1 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 1 } d u d v \\\\ & = \\sum _ { 1 \\le n < t ^ 4 } \\int _ n ^ { n + 1 } \\frac { v - n } { v } \\int _ { \\exp ( - 2 \\sqrt { \\frac { \\log t } { t } } ) } ^ { \\exp ( 2 \\sqrt { \\frac { \\log t } { t } } ) } u ^ { - 1 / 2 } \\left ( 1 + i \\log u \\right ) ^ { - t - 1 } d u d v . \\end{align*}"}
-{"id": "9022.png", "formula": "\\begin{align*} t ^ { ( d - 1 ) m _ 0 } f _ \\psi ( p , t ^ N , z ) = { \\tilde a } _ d ( p , t ^ N ) \\prod _ { i } ( t ^ { m _ 0 } z - t ^ { m _ i } u _ i ( p , t ) ) \\end{align*}"}
-{"id": "1992.png", "formula": "\\begin{align*} \\d V _ t = V _ { t - } \\d U _ t + \\d L _ t , t \\geq 0 . \\end{align*}"}
-{"id": "9037.png", "formula": "\\begin{align*} \\forall j \\geq 0 , \\ \\gamma _ j = & \\sqrt { \\frac { E _ j } { E _ j + \\Gamma _ j } } e ^ { i \\Theta _ j } \\ , \\end{align*}"}
-{"id": "1136.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { K } \\beta _ { i } \\right ) ^ 2 + M \\beta _ { k } \\sum _ { i = 1 } ^ { K } \\beta _ { i } \\geq \\left ( \\sum _ { i = 1 } ^ { K } \\beta _ { i } \\right ) ^ 2 \\end{align*}"}
-{"id": "1331.png", "formula": "\\begin{align*} R _ { 1 2 } ( \\lambda _ 1 , \\lambda _ 2 ) R _ { 1 3 } ( \\lambda _ 1 , \\lambda _ 3 ) R _ { 2 3 } ( \\lambda _ 2 , \\lambda _ 3 ) = R _ { 2 3 } ( \\lambda _ 2 , \\lambda _ 3 ) R _ { 1 3 } ( \\lambda _ 1 , \\lambda _ 3 ) R _ { 1 2 } ( \\lambda _ 1 , \\lambda _ 2 ) . \\end{align*}"}
-{"id": "4008.png", "formula": "\\begin{align*} n _ { d } ( q ^ { d } - 1 ) + c _ 0 q ^ { d } - 1 = q ^ { n } - 1 \\Rightarrow n _ { d } ( q ^ { d } - 1 ) = q ^ { d } ( q ^ { n - { d } } - c _ 0 ) . \\end{align*}"}
-{"id": "7236.png", "formula": "\\begin{align*} f _ { j } = \\left ( \\begin{array} { c } f _ { j } ^ 1 \\\\ f _ { j } ^ 2 \\\\ \\vdots \\\\ f _ { j } ^ r \\end{array} \\right ) = \\sum _ { | \\alpha | = n + 1 } \\left ( \\begin{array} { c } f _ { j , \\alpha } ^ 1 \\\\ f _ { j , \\alpha } ^ 2 \\\\ \\vdots \\\\ f _ { j , \\alpha } ^ r \\end{array} \\right ) \\cdot e _ j ^ \\alpha \\end{align*}"}
-{"id": "3904.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = f ( t , u , u ' ) & & \\\\ u ( 0 ) = 0 = u ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1752.png", "formula": "\\begin{align*} u = x _ 0 \\xleftarrow { \\gamma _ 1 ^ \\lor } x _ 1 \\xleftarrow { \\gamma _ 2 ^ \\lor } \\cdots \\xleftarrow { \\gamma _ r ^ \\lor } x _ r = v \\end{align*}"}
-{"id": "2464.png", "formula": "\\begin{align*} \\big \\| \\{ \\langle k \\rangle ^ { \\frac { R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ 1 , \\alpha _ 2 ) } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } } \\} \\big \\| _ { l _ { r } ^ { s _ 2 - s _ 1 , \\alpha _ 1 \\vee \\alpha _ 2 } } = \\left ( \\sum _ { k \\in \\mathbb { Z } ^ n } \\langle k \\rangle ^ { r \\big [ \\frac { s _ 2 - s _ 1 } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } + \\frac { R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ 1 , \\alpha _ 2 ) } { 1 - \\alpha _ 1 \\vee \\alpha _ 2 } \\big ] } \\right ) ^ { 1 / r } . \\end{align*}"}
-{"id": "6891.png", "formula": "\\begin{gather*} f ^ + ( u , \\xi ) = \\begin{cases} 1 & , \\\\ 0 & , \\end{cases} f ^ - ( u , \\xi ) = \\begin{cases} - 1 & , \\\\ 0 & . \\end{cases} \\end{gather*}"}
-{"id": "9416.png", "formula": "\\begin{align*} E = \\sum _ { i , j = 1 } ^ k V _ i V _ j \\Lambda _ { i j } . \\end{align*}"}
-{"id": "8655.png", "formula": "\\begin{align*} \\mathcal { D } \\left ( \\Lambda \\right ) = H _ { 0 } ^ { 1 } \\left ( \\left [ 0 , 1 \\right ] \\right ) , \\Lambda y = - \\frac { \\partial ^ { 2 } y } { \\partial \\xi ^ { 2 } } \\in H ^ { - 1 } ( [ 0 , 1 ] ) , y \\in \\mathcal { D } \\left ( \\Lambda \\right ) . \\end{align*}"}
-{"id": "887.png", "formula": "\\begin{align*} \\zeta ( s ) = \\chi ( s ) \\zeta ( 1 - s ) , \\end{align*}"}
-{"id": "2185.png", "formula": "\\begin{align*} ( 1 - q ) \\zeta _ { 1 } ( q ) = \\frac { 2 q } { 3 } \\geq \\frac { 2 } { 3 } \\frac { \\beta _ { 0 } } { n + 2 } = : c _ { 1 } = c _ { 1 } ( n , \\beta _ { 0 } ) , \\end{align*}"}
-{"id": "2020.png", "formula": "\\begin{align*} ( \\log | x | u ) ^ \\alpha - ( \\log | x | ) ^ \\alpha = ( \\log | x | ) ^ \\alpha \\left [ \\left ( 1 + \\frac { \\log u } { \\log | x | } \\right ) ^ \\alpha - 1 \\right ] \\leq \\alpha ( \\log | x | ) ^ { \\alpha - 1 } \\log u . \\end{align*}"}
-{"id": "4964.png", "formula": "\\begin{align*} \\delta _ { f } = \\lim _ { n \\to \\infty } \\rho ( ( f ^ { n } ) ^ { * } \\colon N ^ { 1 } ( X ) _ { { \\mathbb { R } } } \\longrightarrow N ^ { 1 } ( X ) _ { { \\mathbb { R } } } ) ^ { 1 / n } . \\end{align*}"}
-{"id": "408.png", "formula": "\\begin{align*} ( f \\circ g ) ( w ) = \\bigvee \\limits _ { ( l , k ) \\in A _ w } { \\min \\{ f ( { l } } ) , g ( { k } ) \\} \\ge \\min \\{ f ( u ) , g ( c ) \\} \\end{align*}"}
-{"id": "4558.png", "formula": "\\begin{align*} r _ { n , u } ^ { ( j ) } : = \\biggl \\{ \\frac { u e ^ { \\Psi ( j ) } } { V _ d ( n - 1 ) } \\biggr \\} ^ { 1 / d } , p _ { n , x , u } ^ { ( j ) } : = h _ x ( r _ { n , u } ^ { ( j ) } ) . \\end{align*}"}
-{"id": "5169.png", "formula": "\\begin{gather*} u _ k \\otimes u _ l = u _ { | k - l | } \\oplus u _ { | k - l | + 2 } \\oplus \\cdots \\oplus u _ { k + l } . \\end{gather*}"}
-{"id": "4066.png", "formula": "\\begin{align*} ( \\beta - 2 ) \\gamma = 2 \\beta . \\end{align*}"}
-{"id": "4750.png", "formula": "\\begin{align*} S _ 2 & = \\sum _ { k \\le 2 ( h - 1 ) \\le 2 k } ( - 1 ) ^ h \\ , \\alpha _ { h - 1 , k } \\ , r ^ { 2 h - k - 1 } \\ , F _ { N + 2 h } ( r ) \\\\ & = \\sum _ { k + 1 \\le 2 h \\le 2 ( k + 1 ) } ( - 1 ) ^ h \\ , \\alpha _ { h - 1 , k } \\ , r ^ { 2 h - k - 1 } \\ , F _ { N + 2 h } ( r ) . \\end{align*}"}
-{"id": "9809.png", "formula": "\\begin{align*} \\mathcal { M } _ { 1 } : \\ ; Z _ { i } = W _ { i j } \\boldsymbol { \\beta } + \\sigma \\varepsilon _ { i } , \\ ; \\ ; \\ ; \\mathcal { M } _ { 2 } : \\ ; Z _ { i } = W _ { i j } \\boldsymbol { \\beta } + r ( X _ { i } ) + \\sigma \\varepsilon _ { i } , \\end{align*}"}
-{"id": "5956.png", "formula": "\\begin{align*} \\dd \\gamma ( Z _ t ) = \\big ( \\lambda U ( Z _ t ) + A Z _ t \\big ) \\ , \\dd t + \\big ( D U ( Z _ t ) + \\mathbb { I } \\big ) { R \\ , } \\cdot \\dd W _ t \\end{align*}"}
-{"id": "2223.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } \\tilde { u } ( x , t ) & = - L ( \\mu * v ) + \\frac { d } { d t } ( g _ { \\alpha } * \\rho ) * g _ { 1 - \\alpha } \\cdot g \\\\ & = - L \\tilde { u } ( x , t ) + \\rho g . \\end{align*}"}
-{"id": "1906.png", "formula": "\\begin{align*} g _ 0 ( x ) = \\dfrac { 3 \\sin ^ 4 ( x ) } { 2 \\pi x ^ 4 } . \\end{align*}"}
-{"id": "3092.png", "formula": "\\begin{align*} P ( \\lambda ) = \\sum _ { k = 0 } ^ { d } P _ k \\lambda ^ k , \\mbox { w i t h } P _ 0 , P _ 1 , \\hdots , P _ d \\in \\mathbb { F } ^ { n \\times n } , \\end{align*}"}
-{"id": "3785.png", "formula": "\\begin{align*} y = a { \\lambda _ 1 } { V _ { 1 1 } } { d _ 1 } + a { \\lambda _ 1 } { V _ { 1 2 } } { d _ 2 } + n \\end{align*}"}
-{"id": "8452.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\varepsilon } = F ^ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\tilde { \\textbf { u } } ^ { \\varepsilon } | _ { t = 0 } = \\tilde { \\textbf { u } } ^ { \\varepsilon } _ { 0 } ( x ) , \\end{align*}"}
-{"id": "598.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ \\infty \\bigl ( 1 - X ^ { 2 i } \\bigr ) \\bigl ( 1 + X ^ { 2 i - 1 } Y ^ 2 \\bigr ) \\Bigl ( 1 + \\frac { X ^ { 2 i - 1 } } { Y ^ 2 } \\Bigr ) = \\sum _ { j = - \\infty } ^ \\infty X ^ { j ^ 2 } Y ^ { 2 j } . \\end{align*}"}
-{"id": "2556.png", "formula": "\\begin{align*} & \\widehat { P _ { \\le N } f } ( \\xi ) : = \\widehat { f _ { \\le N } } ( \\xi ) : = \\varphi ( \\tfrac { \\xi } { N } ) \\widehat { f } ( \\xi ) , \\widehat { P _ { > N } f } ( \\xi ) : = \\widehat { f _ { > N } } ( \\xi ) : = \\bigl ( 1 - \\varphi ( \\tfrac { \\xi } { N } ) \\bigr ) \\widehat { f } ( \\xi ) , \\\\ & \\widehat { P _ { N } f } ( \\xi ) : = \\widehat { f _ { N } } ( \\xi ) : = \\bigl ( \\varphi ( \\tfrac { \\xi } { N } ) - \\varphi ( \\tfrac { 2 \\xi } { N } ) \\bigr ) \\widehat { f } ( \\xi ) . \\end{align*}"}
-{"id": "810.png", "formula": "\\begin{align*} & h _ { 1 2 } ( | m _ { 1 } , \\ldots , m _ { r } \\rangle \\otimes | n _ { 1 } , \\ldots , n _ { r } \\rangle ) = \\sum _ { a = 1 } ^ { r } q ^ { 2 \\sum _ { p = a + 1 } ^ { r } n _ { p } } ( 1 - q ^ { 2 n _ { a } } ) \\\\ & { } \\times \\left ( | \\ldots , m _ { a } + 1 , \\ldots \\rangle \\otimes | \\ldots , n _ { a } - 1 , \\ldots \\rangle - | \\ldots , m _ { a } , \\ldots \\rangle \\otimes | \\ldots , n _ { a } , \\ldots \\rangle \\right ) . \\end{align*}"}
-{"id": "3157.png", "formula": "\\begin{align*} \\| W \\| _ { \\lozenge } : = \\sup _ { n \\in \\mathbb { N } } \\max _ { a \\in S ( \\mathbb { C } ^ n \\otimes H ' ) , \\| a \\| _ 1 = 1 } \\| ( \\mathrm { I } _ n \\otimes W ) ( a ) \\| _ 1 \\end{align*}"}
-{"id": "7509.png", "formula": "\\begin{align*} x _ { n + 1 } = F _ { \\alpha _ n } ( x _ n ) , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 , \\end{align*}"}
-{"id": "4380.png", "formula": "\\begin{align*} G _ 1 ( t ) = \\langle \\Psi _ { c , h } , \\ , e ^ { i T _ { c , h } ( f _ 1 ) } e ^ { i t T _ { c , h } ( f _ 2 ) } \\Psi _ { c , h } \\rangle , \\end{align*}"}
-{"id": "3451.png", "formula": "\\begin{align*} { \\cal M } _ m ( s , x , \\phi , p , q _ m ) = \\frac 1 2 T r A ^ * ( x ) q _ m A ( x ) + \\langle a ( x ) , p _ m \\rangle + \\end{align*}"}
-{"id": "7147.png", "formula": "\\begin{align*} - \\Delta u + ( u \\cdot \\nabla ) u + \\nabla p = f + \\nabla \\cdot F , \\qquad { \\rm d i v } \\ , u = 0 \\mbox { i n } \\ , \\ , \\R ^ n _ + , \\end{align*}"}
-{"id": "6589.png", "formula": "\\begin{align*} f ^ { \\gamma } ( x ) = \\gamma ^ { - 1 } F ( x ) = \\gamma ^ { - 1 } \\left ( \\tfrac { 1 } { 2 } \\| x \\| ^ 2 - r _ { \\gamma f } ^ { * } ( x ) \\right ) , \\end{align*}"}
-{"id": "7323.png", "formula": "\\begin{align*} \\hat { R } ( v _ { 0 } \\otimes v _ { - 1 } ) = v _ { - 1 } \\otimes v _ { 0 } + ( q ^ { 2 } - q ^ { - 2 } ) v _ { 0 } \\otimes v _ { - 1 } . \\end{align*}"}
-{"id": "3530.png", "formula": "\\begin{align*} \\Phi ( g + h , \\pi + w ) = \\Phi ( g , \\pi ) + ( 2 \\psi , V ) - \\left ( 0 , \\tfrac { 1 } { 2 } h \\cdot _ g \\left ( \\textup { d i v } _ g \\pi + W \\right ) \\right ) . \\end{align*}"}
-{"id": "9374.png", "formula": "\\begin{align*} \\Gamma _ j \\mathcal { P T } f = \\sigma _ 3 { \\mathcal { T } } \\Gamma _ j f , \\sigma _ 3 = \\left [ \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ] , j = 0 , 1 . \\end{align*}"}
-{"id": "4328.png", "formula": "\\begin{align*} \\frac { 1 } { r } = \\sum _ { i = 1 } ^ l D _ { r , i } C _ { r - s , i } . \\end{align*}"}
-{"id": "3802.png", "formula": "\\begin{align*} D _ 1 & \\le \\sum _ { q = 0 } ^ \\infty \\| \\Delta _ q ( u ( \\tau ) - u ( t ) ) \\| _ { L ^ 2 } \\sum _ { q = 0 } ^ \\infty \\| \\Delta _ q u ( t ) \\| _ { L ^ 2 } \\\\ & = \\| u ( \\tau ) - u ( t ) \\| _ { B ^ { 0 } _ { 2 , 1 } } \\| u ( t ) \\| _ { B ^ { 0 } _ { 2 , 1 } } . \\end{align*}"}
-{"id": "847.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\ell ( t ) - 1 } \\left ( \\frac { z _ { i } } { 1 + z _ { i } } \\right ) ^ { x _ { \\tau ' ( i ) } } \\prod _ { i = \\ell ( t ) + 1 } ^ { k } \\left ( \\frac { z _ { i } } { 1 + z _ { i } } \\right ) ^ { x _ { \\tau ' ( i - 1 ) } } = \\prod _ { i = 1 } ^ { k - 1 } \\left ( \\frac { w _ { i } } { 1 + w _ { i } } \\right ) ^ { x _ { \\tilde { \\tau } ( i ) } } . \\end{align*}"}
-{"id": "4718.png", "formula": "\\begin{align*} f _ { \\gamma } ( u ) + f _ { \\gamma } ( v ) = \\left \\lceil \\gamma ^ { \\top } u \\right \\rceil + \\left \\lceil \\gamma ^ { \\top } v \\right \\rceil = \\gamma ^ { \\top } u + \\gamma ^ { \\top } v + 1 = f _ { \\gamma } ( u + v ) , \\end{align*}"}
-{"id": "3011.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( 1 ) = { { \\bf { h } } ^ { [ j 1 ] } } ( 1 ) { { \\bf { u } } ^ { [ 1 ] } } + { { \\bf { h } } ^ { [ j 2 ] } } ( 1 ) { { \\bf { u } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "7600.png", "formula": "\\begin{align*} x _ 1 = x _ 0 + F ( x _ 0 ) + l \\chi _ 1 \\geq x _ 0 + F ( x _ 0 ) + l - \\frac { l + F ( x _ 0 ) } { 2 } = x _ 0 + \\frac { l + F ( x _ 0 ) } { 2 } , x _ 1 - x _ 0 \\geq \\varepsilon . \\end{align*}"}
-{"id": "5707.png", "formula": "\\begin{align*} \\tan ^ { - 1 } ( m _ f ( Q ) ) = m _ { \\tan ^ { - 1 } f } ( Q ) , \\end{align*}"}
-{"id": "3897.png", "formula": "\\begin{align*} z & = \\frac { 1 } { \\gamma _ R } \\Big ( \\left ( \\left ( \\alpha _ R ^ 2 + \\beta _ R ^ 2 \\right ) \\left ( \\gamma _ R - 2 \\alpha _ R \\right ) - 4 \\alpha _ R ^ 2 \\gamma _ R \\right ) x \\\\ & \\quad - \\left ( \\alpha _ R ^ 2 + \\beta _ R ^ 2 + 2 \\alpha _ R \\gamma _ R \\right ) y + \\mu \\Big ) \\ ; . \\end{align*}"}
-{"id": "1946.png", "formula": "\\begin{align*} \\frac { d Y _ 1 } { d u } = - Y _ 1 F _ 1 ( Y ) , \\frac { d Y _ 2 } { d u } = - Y _ 2 F _ 2 ( Y ) , \\end{align*}"}
-{"id": "4954.png", "formula": "\\begin{align*} { r _ Q } : = ( Q ) \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "4227.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } f ( x ) d \\mu _ { - 1 } ( x ) = & \\lim _ { N \\rightarrow \\infty } \\sum _ { x = 0 } ^ { p ^ N - 1 } f ( x ) \\mu _ { - 1 } ( x + p ^ N \\mathbb { Z } _ p ) \\\\ = & \\lim _ { N \\rightarrow \\infty } \\sum _ { x = 0 } ^ { p ^ N - 1 } f ( x ) ( - 1 ) ^ x , ( \\textnormal { s e e } \\ , \\ , [ 8 ] ) . \\end{align*}"}
-{"id": "4470.png", "formula": "\\begin{align*} \\delta _ n : = k c _ n ^ d \\log ^ 2 ( n - 1 ) / ( n - 1 ) \\end{align*}"}
-{"id": "6423.png", "formula": "\\begin{align*} h _ { \\alpha , m } ( t ) + m ( h _ { \\alpha , m } * g _ { \\alpha } ) ( t ) = m g _ { \\alpha } ( t ) , t > 0 , \\ , \\ , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "1839.png", "formula": "\\begin{align*} \\mathbb { E } ( e ^ { i u X _ t } ) = ( 1 - i u ) ^ { - t } . \\end{align*}"}
-{"id": "3885.png", "formula": "\\begin{align*} S _ { k + 1 } & : = \\frac { ( - 1 ) ^ { k + 1 } k } { ( k - 1 ) ! } \\sum _ { p _ 1 \\neq \\dotsb \\neq p _ k } \\Bigl [ \\oint _ M w ^ { p _ 1 } w ^ { p _ 2 } _ { n } \\dotsm w ^ { p _ { k } } _ n u _ n \\sigma _ { k - 1 } ( L ) d \\mu \\\\ & + \\frac { 1 } { k } \\oint _ M u w ^ { p _ 1 } _ n \\dotsm w ^ { p _ { k } } _ n \\sigma _ { k - 1 } ( L ) d \\mu \\Bigr ] . \\end{align*}"}
-{"id": "1813.png", "formula": "\\begin{align*} z w ( 1 + z f ( z ) + w a ( w ) + w z e ( z , w ) ) ( 1 + w g ( w ) + z b ( z ) + w z e ( z , w ) ) = z w . \\end{align*}"}
-{"id": "438.png", "formula": "\\begin{align*} z & = 0 ^ { m _ 0 } v _ 1 0 ^ { m _ 1 } v _ 2 \\dots v _ t 0 ^ { m _ t } , \\\\ z ' & = 0 ^ { m ' _ 0 } v _ 1 0 ^ { m ' _ 1 } v _ 2 \\dots v _ t 0 ^ { m ' _ t } , \\end{align*}"}
-{"id": "756.png", "formula": "\\begin{gather*} \\alpha ( S _ i ) = \\sum _ { j = 1 } ^ n S _ j \\otimes u _ { j i } . \\end{gather*}"}
-{"id": "633.png", "formula": "\\begin{align*} V = X _ A \\cup X _ B . \\end{align*}"}
-{"id": "8776.png", "formula": "\\begin{align*} L ^ { ( 2 r + 1 ) } = \\frac { 1 - t } { 2 } T ^ { ( 2 r + 1 ) } { } ' ( 1 ) = ( 1 - t ) \\left ( \\frac { 1 } { 2 } K _ 0 ^ { ( 2 r + 1 ) } { } ' ( 1 ) + \\sum _ { i = 1 } ^ { n - 1 } \\check { R } _ { i , i + 1 } ^ { ( 2 r + 1 ) } { } ' ( 1 ) - \\frac { 1 } { 2 } K _ n ^ { ( 2 r + 1 ) } { } ' ( 1 ) \\right ) . \\end{align*}"}
-{"id": "792.png", "formula": "\\begin{align*} h _ { \\vec { z } } ^ { 1 ^ { k } } ( \\vec { x } ) = \\sum _ { \\sigma \\in \\mathfrak { S } _ { k } } \\prod _ { 1 \\le i < j \\le k } \\ ! \\frac { z _ { \\sigma ( i ) } - q ^ 2 z _ { \\sigma ( j ) } } { z _ { \\sigma ( i ) } - z _ { \\sigma ( j ) } } \\ , \\prod _ { i = 1 } ^ { k } \\left ( \\frac { z _ { \\sigma ( i ) } } { 1 + z _ { \\sigma ( i ) } } \\right ) ^ { x _ { i } } \\ , ( u _ { 1 } \\otimes \\cdots \\otimes u _ { 1 } ) . \\end{align*}"}
-{"id": "7848.png", "formula": "\\begin{align*} \\gamma ( x ) = - C _ 0 ( x ) / A _ 0 ( x ) \\end{align*}"}
-{"id": "866.png", "formula": "\\begin{align*} \\sigma ( A ) = \\int _ { \\Omega \\times \\Omega } \\mathcal { H } ^ 1 ( A \\cap [ x , y ] ) \\mathrm { d } \\gamma ( x , y ) \\ ; \\ ; \\ ; \\mbox { f o r e v e r y B o r e l s e t } \\ ; A \\end{align*}"}
-{"id": "765.png", "formula": "\\begin{gather*} \\gamma \\leqslant \\alpha \\cdot \\beta \\leqslant ( r _ 1 ) ^ { \\delta ( \\alpha ) } ( r _ 1 ) ^ { \\delta ( \\beta ) } = ( r _ 1 ) ^ { \\delta ( \\alpha ) + \\delta ( \\beta ) } . \\end{gather*}"}
-{"id": "3424.png", "formula": "\\begin{align*} \\frac { \\partial u _ l } { \\partial s } + \\frac 1 2 T r A ^ * \\nabla ^ 2 u _ l A + \\langle a , \\nabla u _ l \\rangle + g _ l ( s , x , u , A ^ * \\nabla u _ l ) = 0 , u _ l ( T , x ) = u _ { 0 l } ( x ) . \\end{align*}"}
-{"id": "9510.png", "formula": "\\begin{align*} X _ 8 & = \\frac { 1 } { 1 9 2 } ( p _ 1 ^ 2 - 4 p _ 2 ) \\\\ T _ 2 & = \\left ( \\frac { 2 \\pi ^ 2 } { \\kappa _ { 1 1 } ^ 2 } \\right ) ^ { 1 / 3 } \\end{align*}"}
-{"id": "3019.png", "formula": "\\begin{align*} { { \\bf { Y } } ^ { [ 1 ] } } ( n ) - { { \\bf { Y } } ^ { [ 1 ] } } ( { t _ 2 } ) = { { \\bf { H } } ^ { [ 1 1 ] } } ( n ) { \\bf { V } } _ 1 ^ { [ 1 ] } ( n ) { { \\bf { u } } ^ { [ 1 ] } } + { { \\bf { H } } ^ { [ 1 2 ] } } ( n ) { \\bf { V } } _ 1 ^ { [ 2 ] } ( n ) { { \\bf { u } } ^ { [ 2 ] } } . \\end{align*}"}
-{"id": "5257.png", "formula": "\\begin{align*} \\operatorname * { l e v } \\nolimits _ { \\varphi _ { A , k } , < } ( t ) = t k + \\operatorname * { c o r e } A \\forall \\ , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3720.png", "formula": "\\begin{align*} \\psi ( 2 ^ { - 1 } ( x u y ^ * + y u x ^ * ) ) = \\psi ( 2 ^ { - 1 } x u y ^ * ) \\psi ( 2 ^ { - 1 } y u x ^ * ) = \\psi ( 2 ^ { - 1 } x u y ^ * ) \\psi ( 2 ^ { - 1 } ( x ( y u ) ^ * ) ^ * ) = \\psi ( x u y ^ * ) = \\chi ( x , y u ) . \\end{align*}"}
-{"id": "5995.png", "formula": "\\begin{align*} \\P \\{ \\max _ { n = 1 , \\ldots , N } | S _ n | \\le f _ N \\} , , \\end{align*}"}
-{"id": "1244.png", "formula": "\\begin{align*} i \\partial _ t \\psi ( x , t ) = ( D _ m + V ( x ) ) \\psi ( x , t ) , \\psi ( x , 0 ) = \\psi _ 0 ( x ) . \\end{align*}"}
-{"id": "1930.png", "formula": "\\begin{align*} & \\left | 2 \\sum _ { i = 1 } ^ m n _ i p _ i Y _ i - \\frac { 1 } { 2 } E ( Y ) \\right | \\leq \\left | \\sum _ { i = 1 } ^ m n _ i Y _ i ( 2 p _ i - q _ i ^ 2 Y _ i ) \\right | + \\frac { 1 } { 2 } E ( Y ) \\\\ \\leq & \\left ( \\sum _ { i = 1 } ^ m n _ i \\right ) ^ { 1 / 2 } \\left ( \\sum _ { i = 1 } ^ m n _ i Y _ i ^ 2 ( 2 p _ i - q _ i ^ 2 Y _ i ) ^ 2 \\right ) ^ { 1 / 2 } + \\frac { 1 } { 2 } E ( Y ) . \\end{align*}"}
-{"id": "4192.png", "formula": "\\begin{align*} \\widehat { G } ( x ) = \\prod _ { i = 1 } ^ n \\frac { 2 } { 1 + 4 \\pi ^ 2 x _ i ^ 2 } , \\end{align*}"}
-{"id": "7898.png", "formula": "\\begin{align*} T _ A ( H ^ { \\otimes n } ) ^ { { \\rm C H } } ( r , l , \\chi ) \\ , = \\ , \\Big ( r , \\ , l \\ , + \\ , r \\ , n \\ , h , \\ , \\chi \\ , + \\ , n \\ , l \\cdot h \\ , + \\ , r \\ , n ^ 2 \\ , \\frac { h ^ 2 } { 2 } \\Big ) . \\end{align*}"}
-{"id": "9442.png", "formula": "\\begin{align*} \\rho ^ { - 1 } _ { s , t } = \\exp \\left ( \\int _ s ^ t h ( X ^ { s , x } _ r ) d \\tilde w _ r - \\frac 1 2 \\int _ s ^ t | h ( X ^ { s , x } _ r ) | ^ 2 d r \\right ) . \\end{align*}"}
-{"id": "5093.png", "formula": "\\begin{align*} \\mathbb { T } ^ { [ M ' , M ] } ( z ) \\otimes \\mathbb { T } ^ { [ M ' , M ] } ( w ) = \\sum _ { a , b = 0 } ^ { r } \\sum _ { c , d = 0 } ^ { r } \\left ( \\mathbb { T } ^ { [ M ' , M ] } ( z ) _ { a b } \\mathbb { T } ^ { [ M ' , M ] } ( w ) _ { c d } \\right ) \\otimes E _ { a b } \\otimes E _ { c d } . \\end{align*}"}
-{"id": "4237.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } { \\frac { x } { 2 } \\choose n } d \\mu _ { - 1 } ( x ) = \\frac { 1 } { n ! } \\sum _ { m = 0 } ^ n 2 ^ { - m } E _ m S _ 1 ( n , m ) \\end{align*}"}
-{"id": "8328.png", "formula": "\\begin{gather*} z ^ { 2 7 } - z = \\frac { 1 } { ( T + 1 ) ^ { 5 4 } } + \\frac { 1 } { T + 1 } + T ^ { 9 } + T ^ { 3 } + T + \\omega + 1 = u ( T ) , \\end{gather*}"}
-{"id": "8916.png", "formula": "\\begin{align*} - \\Delta \\abs { u - u \\circ \\sigma _ H } + ( 1 + \\abs { A } ^ 2 ) \\abs { u - u \\circ \\sigma _ H } = 2 \\abs { u _ H } ^ { p - 1 } - \\abs { u } ^ { p - 1 } - \\abs { u \\circ \\sigma _ H } ^ { p - 1 } \\ge 0 , H . \\end{align*}"}
-{"id": "674.png", "formula": "\\begin{align*} \\mu _ i = \\beta _ i m _ i + \\gamma _ i \\lambda _ i , \\textrm { f o r $ i = 1 , 2 $ } , \\end{align*}"}
-{"id": "2596.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\langle e ^ { - i t \\Delta } u ( t ) , \\phi \\rangle = \\lim _ { t \\to 0 } \\langle e ^ { - i t \\Delta } u ( t ) , \\phi \\rangle = 0 \\end{align*}"}
-{"id": "8998.png", "formula": "\\begin{align*} [ L _ 1 ( s ) + ~ ^ { A B R } ~ _ { 0 } D ^ \\alpha L _ 2 ( s ) ] = 0 , ~ \\texttt { f o r a l l } ~ s \\in [ 0 , b ] , \\end{align*}"}
-{"id": "3188.png", "formula": "\\begin{align*} \\widehat { w } _ j = w _ j + \\sum _ { | \\alpha | \\geq 2 } a _ { j , \\alpha } ( z _ j ) \\cdot w _ j ^ \\alpha . \\end{align*}"}
-{"id": "5731.png", "formula": "\\begin{align*} f ( y _ { t } | W _ t ) = \\exp \\left \\{ y _ { t } W _ { t } - m _ { t } b ( W _ { t } ) + c ( y _ { t } ) \\right \\} . \\end{align*}"}
-{"id": "7740.png", "formula": "\\begin{align*} \\Omega _ i : = \\{ \\omega \\in \\bar \\Omega : \\ , \\ , i \\le N ( \\omega ) < i + 1 \\} . \\end{align*}"}
-{"id": "9335.png", "formula": "\\begin{align*} h ( t , x , z ) = y ( t , x , z ) p ( t , x , z ) \\end{align*}"}
-{"id": "1641.png", "formula": "\\begin{align*} F \\left ( x , v \\right ) & = \\pm \\theta \\left ( x , v \\right ) s i g n \\left ( x \\right ) \\left \\vert x \\right \\vert ^ { \\alpha } \\\\ \\alpha & \\in \\left ( \\frac { 1 } { 2 } , 1 \\right ) , \\theta \\in C _ { c } ^ { \\infty } \\left ( \\mathbb { R } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "2513.png", "formula": "\\begin{align*} X _ { 0 } ^ { \\lambda } = \\left \\| \\bigwedge _ { i = 1 } ^ { p } \\nabla D _ i \\wedge \\bigwedge _ { j = 1 } ^ { k } \\nabla I _ j \\right \\| _ { k + p } ^ { - 2 } \\cdot \\sum _ { i = 1 } ^ { p } ( - 1 ) ^ { n - i } ( - \\lambda ) ( D - d _ i ) \\Theta _ i , \\end{align*}"}
-{"id": "9050.png", "formula": "\\begin{align*} U _ k ^ { ( N ) } : = \\tau ^ { ( r ) } _ k + u _ k ^ { ( N ) } L _ k ^ { ( N ) } : = \\tau ^ { ( r ) } _ k + l _ k ^ { ( N ) } . \\end{align*}"}
-{"id": "8633.png", "formula": "\\begin{align*} c _ 0 ^ J \\alpha _ { i + 1 } = - c _ 1 ^ J \\alpha _ { i } - - c ^ J _ 2 \\alpha _ { i - 1 } - - c ^ J _ 3 \\alpha _ { i - 2 } - \\ldots - c ^ J _ { i - 1 } \\alpha _ { 1 } , ~ ~ 1 \\leq i \\leq k - 1 . \\end{align*}"}
-{"id": "6155.png", "formula": "\\begin{align*} s _ i = \\begin{cases} a _ i m + b & { } \\\\ a _ i m + ( m - b ) & { } . \\end{cases} \\end{align*}"}
-{"id": "4351.png", "formula": "\\begin{align*} \\psi ' ( y _ a , a ) = 0 \\ , \\psi ' ( y , a ) ( y - y _ a ) < 0 \\ , y \\in ( 0 , 1 ) \\setminus \\{ y _ a \\} \\ . \\end{align*}"}
-{"id": "8986.png", "formula": "\\begin{align*} ( ~ ~ ^ { A B R } ~ _ { a } D ^ \\alpha f ) ( t ) = \\frac { B ( \\alpha ) } { 1 - \\alpha } \\frac { d } { d t } \\int _ a ^ t f ( x ) E _ \\alpha [ - \\alpha \\frac { ( t - x ) ^ \\alpha } { 1 - \\alpha } ] d x . \\end{align*}"}
-{"id": "6486.png", "formula": "\\begin{align*} u ( x , t ) = \\sum _ { k = 1 } ^ { \\infty } E _ { \\alpha , 1 } ( - \\lambda _ { k } t ^ { \\alpha } ) ( u _ { 0 } , \\phi _ { k } ) \\phi _ { k } ( x ) \\end{align*}"}
-{"id": "834.png", "formula": "\\begin{align*} \\vec { z } ( \\ell _ { 1 } , \\ell _ { 0 } ) = ( z _ { 1 } , \\ldots , \\underset { \\hbox { \\scriptsize $ \\ell _ { 1 } $ - t h } } { z _ { \\ell _ { 0 } } } , \\ldots , z _ { \\ell _ { 0 } - 1 } , z _ { \\ell _ { 0 } + 1 } , \\ldots , z _ { k } ) . \\end{align*}"}
-{"id": "314.png", "formula": "\\begin{align*} g _ J ^ * ( x ) = \\sum _ { i = 1 } ^ { \\mathrm { c a r d } ( J ) } \\frac { ( - 1 ) ^ { i - 1 } ( i - 1 ) ! } { f ^ i } \\sum _ { \\{ P _ 1 , \\ldots , P _ i \\} \\in \\mathcal { P } _ i ( J ) } f _ { P _ 1 } \\ldots f _ { P _ i } . \\end{align*}"}
-{"id": "8346.png", "formula": "\\begin{align*} \\sigma ( f _ V ( z ) ) & = \\sigma ( f _ V ( l ( y ) + D ) ) = f _ V ( l ( y + \\mu ) + D ) = f _ V ( l ( y ) + l ( \\mu ) + D ) \\\\ & = f _ V ( l ( y ) + D ) + f _ V ( l ( \\mu ) ) = f _ V ( z ) + f _ V ( l ( \\mu ) ) . \\end{align*}"}
-{"id": "4196.png", "formula": "\\begin{align*} ( I - \\Delta ) ^ { ( n + 1 ) / 2 } h ( x ) = 0 \\end{align*}"}
-{"id": "6055.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\sum _ { i } \\Delta \\overline { u } _ { i } ( x ) \\ , d x = \\int _ { \\partial \\Omega } \\sum _ { i } \\frac { \\partial \\overline { u } _ { i } } { \\partial n } \\ , d s \\le \\int _ { \\partial \\Omega } \\sum _ { i } \\frac { \\partial \\underline { u } _ { i } } { \\partial n } \\ , d s = \\int _ { \\Omega } \\sum _ { i } \\Delta \\underline { u } _ { i } ( x ) \\ , d x . \\end{align*}"}
-{"id": "9922.png", "formula": "\\begin{align*} F E \\ ; + \\ ; & \\frac { 1 } { ( q - q ^ { - 1 } ) ^ 2 } \\big ( q K - q ^ { - 1 } \\l ^ { - 1 } K ' ) ( K - \\l K ' ) \\\\ & \\ ; = \\ ; F \\big ( E - \\kappa s ( K - \\l ^ { - 1 } K ' ) \\big ) \\ ; + \\ ; \\kappa \\big ( q K - q ^ { - 1 } \\l ^ { - 1 } K ' \\big ) \\big ( s F + \\kappa ( K - \\l K ' ) \\big ) , \\end{align*}"}
-{"id": "9139.png", "formula": "\\begin{align*} u ' ( t ) + B _ 0 ( t ) A ( t ) B _ 1 ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) , \\ \\ u ( 0 ) = u _ 0 . \\end{align*}"}
-{"id": "1988.png", "formula": "\\begin{align*} \\| u \\| _ { L _ 2 ( P ) } = \\| \\phi _ { P , * } u \\| _ { L _ 2 ( J \\times B ) } \\leq C \\| | \\nabla _ { g _ C } ( \\phi _ { P , * } u ) | _ { g _ C } \\| _ { L _ 2 ( J \\times B ) } = C \\| | \\nabla _ { g _ P } u | _ { g _ P } \\| _ { L _ 2 ( P ) } . \\end{align*}"}
-{"id": "3709.png", "formula": "\\begin{align*} H _ { 1 2 5 } = \\frac { 1 } { 2 } ( x ^ 2 _ 1 + 2 x ^ 2 _ 2 + x ^ 2 _ 5 ) , ~ Z _ { 1 2 5 } = w H _ { 1 2 5 } - \\frac { 1 } { 2 } ( x ^ 2 _ 1 + x ^ 2 _ 2 ) ~ . \\end{align*}"}
-{"id": "8089.png", "formula": "\\begin{align*} T _ { g , ( s _ i , r _ j ) } ^ { g l m } = \\frac { E _ { s _ i } } { P _ { s _ i } ^ { g l m } } = \\frac { E _ { r _ j } } { P _ { r _ j } ^ { g l m } } \\end{align*}"}
-{"id": "4516.png", "formula": "\\begin{align*} M _ { f , a , \\beta } ( x ) & \\leq \\max \\biggl \\{ \\max _ { r = 1 , \\ldots , m } \\frac { A _ r ' } { ( 1 - x ^ 2 ) ^ { 2 r } } \\ , , \\ , \\sup _ { y : 0 < | y - x | \\leq r _ a ( x ) } \\frac { A _ { m + 1 } ' f ( y ) } { ( 1 - y ^ 2 ) ^ { 2 ( m + 1 ) } f ( x ) } \\biggr \\} \\\\ & \\leq \\frac { B _ { m + 1 } ' } { ( 1 - x ^ 2 ) ^ { 2 ( m + 1 ) } } \\leq a \\bigl ( f ( x ) \\bigr ) , \\end{align*}"}
-{"id": "4381.png", "formula": "\\begin{align*} G _ 2 ( t ) = \\langle \\zeta _ L , e ^ { i T _ { c , h _ c } ( f _ 1 ) } e ^ { i t T _ { c , h _ c } ( f _ 2 ) } \\zeta _ R \\rangle \\end{align*}"}
-{"id": "8847.png", "formula": "\\begin{align*} \\mathrm { d i v } _ { \\eta } ( T ( \\nabla x ) ) : = & \\big ( \\mathrm { d i v } _ { \\eta } ( T ( \\nabla x _ 1 ) ) , \\ldots , \\mathrm { d i v } _ { \\eta } ( T ( \\nabla x _ m ) ) \\big ) \\\\ = & ( \\mathrm { d i v } ( T ( \\nabla x _ 1 ) ) - \\langle \\nabla \\eta , T ( \\nabla x _ 1 ) \\rangle , \\ldots , \\mathrm { d i v } ( T ( \\nabla x _ m ) ) - \\langle \\nabla \\eta , T ( \\nabla x _ m ) \\rangle ) \\\\ = & \\mathrm { d i v } ( T ( \\nabla x ) ) - \\mathrm { d } \\eta \\circ T ( \\nabla x ) . \\end{align*}"}
-{"id": "273.png", "formula": "\\begin{align*} U _ { 2 2 } & : = \\biggl | \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ \\frac { a _ n } { n - 1 } ^ 1 \\log \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } } \\biggr ) \\mathrm { B } _ { k , n - k - 1 } ( s ) \\biggl \\{ \\frac { ( n - 1 ) s - k } { n - k - 1 } \\biggr \\} \\ , d s \\ , d x \\biggr | \\\\ & = o ( n ^ { - ( 3 - \\epsilon ) } ) . \\end{align*}"}
-{"id": "3010.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ 1 ] } } ( 1 ) = { { \\bf { u } } ^ { [ 1 ] } } , { { \\bf { X } } ^ { [ 2 ] } } ( 1 ) = { { \\bf { u } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "7644.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } s ^ { - ( 1 + \\alpha / d ) } F ( s ) d s < \\infty , \\ \\mbox { w h e r e } \\ F ( s ) = \\sup _ { 1 \\leq t \\leq s } \\frac { f ( t ) } { t } . \\end{align*}"}
-{"id": "101.png", "formula": "\\begin{align*} \\frac { 2 } { e ^ t + 1 } e ^ { x t } = \\sum _ { n = 0 } ^ \\infty E _ n ( x ) \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 1 - 1 3 ] ) . \\end{align*}"}
-{"id": "6917.png", "formula": "\\begin{align*} Y _ { j , 1 } - Y _ { j , 2 } & = \\left ( \\phi ( b _ j ( T _ 1 ) ) - \\phi ( b _ j ( T _ 2 ) ) \\right ) \\ , \\biggl ( \\phi ( a _ j ) - \\frac { 1 } { n - j - 2 } \\sum _ { \\ell \\in U _ j ( T _ 1 , T _ 2 ) } \\phi ( \\ell ) \\biggr ) \\\\ & = \\frac { \\phi ( b _ j ( T _ 1 ) ) - \\phi ( b _ j ( T _ 2 ) ) } { n - j - 2 } \\ , \\sum _ { \\ell \\in U _ j ( T _ 1 , T _ 2 ) } \\ ( \\phi ( a _ j ) - \\phi ( \\ell ) \\ ) . \\end{align*}"}
-{"id": "8125.png", "formula": "\\begin{align*} { \\mathcal R } _ { k , l , 2 } \\geq \\left ( C _ { \\frac { 1 } { 3 } , 0 , 2 } ^ { r a d } \\right ) ^ { \\frac { 3 ( l + 1 - k ) } { l + 2 } } \\cdot \\left ( \\frac { 3 ( k + 1 ) } { 4 d _ 2 } \\right ) ^ { \\frac { k + 1 } { l + 2 } } \\cdot \\left ( \\frac { 4 } { 3 } \\right ) ^ { \\frac { 3 k - 2 l - 1 } { l + 2 } } = C _ { k , l , 2 } ^ { r a d } \\ , \\end{align*}"}
-{"id": "2099.png", "formula": "\\begin{align*} - \\lim _ { y \\to 0 } | y | ^ a \\tilde { v } ( x , y ) = ( - \\Delta ) ^ s v ( x ) \\qquad \\forall \\ , x \\in \\R ^ n . \\end{align*}"}
-{"id": "6781.png", "formula": "\\begin{align*} \\varphi _ l ( x ) : = \\sup \\left \\{ u _ l ( x ) , \\ \\textrm { w h e r e } t \\to u _ t \\textrm { i s a s u b g e o d e s i c o f } \\ \\varphi _ 0 , \\varphi _ 1 \\right \\} , \\ \\ l \\in [ 0 , 1 ] , x \\in X . \\end{align*}"}
-{"id": "7473.png", "formula": "\\begin{align*} h ( ( \\mathcal { P } \\cap B _ S ( 0 ) ) \\cup \\{ 0 \\} \\cup A , 0 ) = h _ { \\infty } , \\end{align*}"}
-{"id": "4815.png", "formula": "\\begin{align*} \\| | X \\| | = \\| X \\| + \\max _ { 1 \\le j \\le n } | [ T _ j , X ] | _ { \\mathcal J } . \\end{align*}"}
-{"id": "656.png", "formula": "\\begin{align*} \\int _ G \\nu ( g ^ { - 1 } B \\cap C ) \\ , d \\eta ( g ) = \\nu ( B ) \\ , \\nu ( C ) . \\end{align*}"}
-{"id": "7174.png", "formula": "\\begin{align*} w = v \\chi + \\hat { v } , q = p \\chi , \\end{align*}"}
-{"id": "4565.png", "formula": "\\begin{align*} F _ { n , x , y } ^ { ' , ( 1 ) } : = \\mathbb { P } ( N _ 1 + N _ 3 \\geq j , N _ 2 + N _ 3 \\geq l ) . \\end{align*}"}
-{"id": "100.png", "formula": "\\begin{align*} J ( x _ { k + 1 } ) - J ( x _ { k } ) = J ( x _ k + \\Delta _ k ) - J ( x _ { k } ) \\leq ( L - 1 ) \\| \\Delta _ k \\| ^ 2 < 0 , \\end{align*}"}
-{"id": "7808.png", "formula": "\\begin{align*} \\sup _ { \\Sigma \\left ( T \\right ) } \\left \\vert \\overset { \\circ } { \\mathrm { R m } _ { \\Sigma } } \\right \\vert ^ { 2 } S ^ { - 2 } = \\varepsilon . \\end{align*}"}
-{"id": "6144.png", "formula": "\\begin{align*} \\iota _ b \\iota _ a \\Omega = f _ { a b } ^ c \\ , \\iota _ c \\Theta \\ ; + \\ ; [ \\iota _ a \\Theta , \\iota _ b \\Theta ] . \\end{align*}"}
-{"id": "8200.png", "formula": "\\begin{align*} A _ { p - 1 } ( z _ 1 , z _ 2 ) ^ { - 1 } \\cdot F ( x , z _ 1 , z _ 2 ) ^ { \\frac { p - 1 } { 2 } } - x ^ { p - 1 } = \\sum _ { i = 0 } ^ { 2 p - 2 } R _ i ( z _ 1 , z _ 2 ) x ^ i , \\end{align*}"}
-{"id": "6452.png", "formula": "\\begin{align*} R _ { m } ( s ) = ( 1 - q ) \\phi \\left [ \\mathcal { E } ( h _ { m } * \\tilde { u } , \\psi ^ { 2 } \\tilde { u } ^ { - q } ) - \\mathcal { E } ( \\tilde { u } , \\psi ^ { 2 } \\tilde { u } ^ { - q } ) \\right ] . \\end{align*}"}
-{"id": "2206.png", "formula": "\\begin{align*} k ( x , y ) = \\frac { a ( ( x - y ) / | x - y | ) } { | x - y | ^ { n + 2 \\beta } } . \\end{align*}"}
-{"id": "7674.png", "formula": "\\begin{align*} u _ { 0 } = \\sum _ { n = 1 } ^ { \\infty } u _ { n } , \\ u _ { n } = \\frac { 1 } { \\nu ( d , \\alpha ) } s _ { n } \\chi _ { n ^ { - \\alpha q / d } s _ { n } ^ { - q / d } } ( x _ { n } ) , \\end{align*}"}
-{"id": "7766.png", "formula": "\\begin{align*} \\beta ( M ) = \\tfrac { 1 } { 3 } \\beta ( M ' ) + \\tfrac { 1 } { 3 } \\beta ( M '' ) . \\end{align*}"}
-{"id": "730.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } 1 & & \\\\ & \\ddots & \\\\ & & 1 \\\\ & & \\end{array} \\right ) . \\end{align*}"}
-{"id": "9494.png", "formula": "\\begin{align*} \\Phi _ { * } \\partial _ { \\rho } = x _ { n } \\ , \\partial _ { x _ { n } } + y _ { n } \\ , \\partial _ { y _ { n } } \\Phi _ { * } \\partial _ { \\phi } = x _ { n } \\ , \\partial _ { y _ { n } } - y _ { n } \\ , \\partial _ { x _ { n } } . \\end{align*}"}
-{"id": "8804.png", "formula": "\\begin{align*} L = n \\left ( A + \\int _ { \\Sigma } H u d v o l _ { \\Sigma } \\right ) \\ , . \\end{align*}"}
-{"id": "1349.png", "formula": "\\begin{align*} I _ { L } ( \\hat \\delta _ 0 ) = n ^ { - 1 } \\sum _ { t = 1 } ^ n \\sigma _ { t } ^ 2 ( \\hat \\beta _ 0 ) \\Gamma _ { t , L } ( \\hat \\delta _ 0 ) . \\end{align*}"}
-{"id": "7216.png", "formula": "\\begin{align*} \\left \\{ \\left ( c _ { e _ { i } } \\left ( r \\right ) , c _ { e _ { i } } \\left ( - r \\right ) \\right ) \\right \\} _ { i = 1 } ^ { \\mathrm { \\dim } \\left ( S \\right ) } \\end{align*}"}
-{"id": "2937.png", "formula": "\\begin{align*} e ^ { i q } \\ , z ^ { \\alpha _ 0 } & = e ^ { i q } \\ , e ^ { ( l n z ) \\alpha _ 0 } = e ^ { ( \\ln z ) \\alpha _ 0 } e ^ { - ( \\ln z ) \\alpha _ 0 } e ^ { i q } \\ , e ^ { ( \\ln z ) \\alpha _ 0 } \\cr & = z ^ { \\alpha _ 0 } \\ , e ^ { i q + [ i q , ( \\ln z ) \\alpha _ 0 ] + \\cdots } = z ^ { \\alpha _ 0 } \\ , e ^ { i q + i ( \\ln z ) [ q , \\alpha _ 0 ] + \\cdots } \\cr & = z ^ { \\alpha _ 0 } \\ , e ^ { i q - ( \\ln z ) } = ( 1 / z ) \\ , z ^ { \\alpha _ 0 } \\ , e ^ { i q } \\ , , \\end{align*}"}
-{"id": "877.png", "formula": "\\begin{align*} \\operatorname * { d o m } \\varphi _ { A , k } = \\mathbb { R } k + \\operatorname * { b d } A . \\end{align*}"}
-{"id": "4143.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\neq 0 } A _ f ( n ) \\sqrt { y } K _ { i t _ j } ( 2 \\pi i \\lvert y \\rvert n ) e ^ { 2 \\pi i n x } , \\end{align*}"}
-{"id": "7146.png", "formula": "\\begin{align*} b _ n = 0 , b _ j = \\int _ { \\R ^ n _ + } ( y _ n f _ j ( y ) - F _ { n j } ( y ) ) d y , ( j < n ) , \\end{align*}"}
-{"id": "9563.png", "formula": "\\begin{align*} \\langle \\cdot , \\cdot \\rangle _ { t } = \\langle \\cdot , \\cdot \\rangle + t \\langle \\cdot , G ^ { ( 1 ) } \\cdot \\rangle + t ^ { 2 } \\langle \\cdot , G ^ { ( 2 ) } \\cdot \\rangle \\end{align*}"}
-{"id": "2548.png", "formula": "\\begin{align*} \\left | \\mathbb { E } \\left [ \\frac 1 N \\sum _ { n = 0 } ^ { N - 1 } f ( U ^ n ) - \\hat f \\right ] \\right | \\le C _ h ( \\frac 1 T + \\tau ) . \\end{align*}"}
-{"id": "5373.png", "formula": "\\begin{align*} c _ { d , d } ^ d { \\ , } y ^ { d ^ 2 } + \\sum _ { k = 0 } ^ { d - 1 } { d \\choose k } ( c _ { d , d } { \\ , } x ^ d ) ^ k \\Big ( \\sum _ { j = 0 } ^ r c _ { d , j } { \\ , } x ^ j y ^ { d - j } \\Big ) ^ { d - k } ~ = ~ 0 . \\end{align*}"}
-{"id": "1148.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 1 \\\\ - z \\rho & 0 \\end{bmatrix} _ t - \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} _ x + \\big [ \\begin{bmatrix} 0 & 1 \\\\ - z \\rho & 0 \\end{bmatrix} , \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\big ] = 0 , \\end{align*}"}
-{"id": "7791.png", "formula": "\\begin{align*} \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "919.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 2 / 5 } } = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { S _ { 1 / 2 } } \\| V \\| _ { F } \\\\ = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\ ! \\left ( \\frac { 4 \\| U \\| ^ { 1 / 2 } _ { S _ { 1 / 2 } } \\ ! + \\| V \\| ^ { 2 } _ { F } } { 5 } \\right ) ^ { 5 / 2 } . \\end{align*}"}
-{"id": "8012.png", "formula": "\\begin{align*} & \\phi ( x _ j ) = \\mu _ j x ' _ j , \\phi ( y _ j ) = \\nu _ j y ' _ j , & \\mbox { i f } \\ ; \\ ; \\tau _ j = 1 , \\\\ & \\phi ( x _ j ) = \\mu _ j y ' _ j , \\phi ( y _ j ) = \\nu _ j x ' _ j , & \\mbox { i f } \\ ; \\ ; \\tau _ j = - 1 . \\end{align*}"}
-{"id": "676.png", "formula": "\\begin{align*} \\sigma _ k = \\frac { 1 } { 2 } ( \\delta _ { k } + \\delta _ { - k } ) , \\textrm { f o r $ k \\geq 1 $ } , \\end{align*}"}
-{"id": "4979.png", "formula": "\\begin{align*} \\frac { h _ { H } ( f ^ { n } ( P ) ) } { n ^ { r - 1 } \\delta ^ { n } } \\leq C _ { 2 } \\left ( \\sqrt [ ] { h _ { H } ( P ) } + \\sum _ { k = 1 } ^ { n - 1 } \\sqrt [ ] { C _ { 4 } h _ { H } ( P ) } \\frac { k ^ { 1 + ( r - 1 ) / 2 } } { \\delta ^ { 1 + k / 2 } } + h _ { H } ( P ) \\right ) . \\end{align*}"}
-{"id": "44.png", "formula": "\\begin{align*} \\nu _ 0 : = \\sup \\left \\{ r > 0 : \\frac { 1 } { | s _ { 0 } | ^ { 2 r } } \\in L ^ 1 _ { \\rm l o c } ( X ) \\right \\} \\end{align*}"}
-{"id": "3342.png", "formula": "\\begin{align*} S _ R & = ( \\{ - R _ 1 , R _ 1 \\} \\times [ - R _ 2 , R _ 2 ] \\times \\cdots \\times [ - R _ n , R _ n ] ) \\\\ & \\cup ( [ - R _ 1 , R _ 1 ] \\times \\{ - R _ 2 , R _ 2 \\} \\times \\cdots \\times [ - R _ n , R _ n ] ) \\\\ & \\cup \\cdots \\cup ( [ - R _ 1 , R _ 1 ] \\times [ - R _ 2 , R _ 2 ] \\times \\cdots \\times \\{ - R _ n , R _ n \\} ) . \\\\ \\end{align*}"}
-{"id": "9887.png", "formula": "\\begin{align*} \\mathrm { H _ 0 } : x _ n & = \\omega _ n , \\\\ \\mathrm { H _ 1 } : x _ n & = A _ n + \\omega _ n , n = 1 , 2 , . . . , N , \\end{align*}"}
-{"id": "3907.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = f ( t , u , u ' ) & & \\\\ u ( T ) = u ( 0 ) = u ' ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3537.png", "formula": "\\begin{align*} \\Phi _ { ( g , \\pi ) } ^ V ( g + h , \\pi + w ) = \\Phi _ { ( g , \\pi ) } ^ V ( g , \\pi ) + ( 2 \\psi , V ) . \\end{align*}"}
-{"id": "5371.png", "formula": "\\begin{align*} R _ d ( y ) ^ { d - 1 } R _ d ( z ) + S _ d ( z ) ^ d ~ = ~ 0 . \\end{align*}"}
-{"id": "1959.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\hat { Y } _ i \\left ( r \\sum _ { j = 1 } ^ m n _ j c _ { j } \\hat { Y } _ j ^ 2 + c _ { i } \\hat { Y } _ i ^ 2 \\right ) \\leq c _ 0 \\left ( \\sum _ { i = 1 } ^ m \\hat { Y } _ i \\right ) ^ 3 . \\end{align*}"}
-{"id": "5285.png", "formula": "\\begin{align*} \\zeta ( \\tfrac { 1 } { 2 } + i t ) = \\sum _ { n \\leq C T / \\pi } { n ^ { - \\tfrac { 1 } { 2 } - i t } } + O \\left ( \\frac { T ^ { 1 / 2 } } { \\vert t \\vert } \\right ) + O ( T ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "1633.png", "formula": "\\begin{align*} T ^ { I } L _ { Y _ { I } } C _ { a } - T _ { , t } ^ { I } Y _ { I \\alpha } - 2 a _ { , \\beta } = 0 . \\end{align*}"}
-{"id": "3075.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho _ { n , k _ n } = D _ 1 . \\end{align*}"}
-{"id": "6897.png", "formula": "\\begin{align*} \\mathcal { A } = \\lambda \\mathsf { P } _ { \\mathrm { S I R } } \\left ( \\lambda \\right ) \\mathrm { l o g } _ { 2 } \\left ( 1 + \\tau \\right ) . \\ : \\left [ \\mathrm { b i t s } / \\left ( \\mathrm { s \\cdot H z \\cdot m ^ { 2 } } \\right ) \\right ] \\end{align*}"}
-{"id": "8610.png", "formula": "\\begin{align*} - \\theta _ { 1 } \\overline { \\pi } _ { 1 } \\overline { p } \\xi _ { e _ { k } a } ( n ) & = - \\theta _ { 1 } \\left ( \\overline { p } \\xi _ { e _ { k } a } ( n ) + \\operatorname { k e r } ( \\alpha ) \\right ) = - \\alpha \\overline { p } \\xi _ { e _ { k } a } ( n ) \\\\ - \\theta _ { 2 } ^ { - 1 } ( \\overline { \\gamma } ' ( \\xi _ { e _ { k } a } ( n ) ) ) & = - \\theta _ { 2 } ^ { - 1 } ( c _ { k } \\beta \\alpha \\xi _ { e _ { k } a } ( n ) ) = - c _ { k } \\overline { \\pi } _ { 3 } \\alpha \\xi _ { e _ { k } a } ( n ) \\end{align*}"}
-{"id": "1065.png", "formula": "\\begin{align*} \\binom { N } { 2 } = ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } & \\leq k \\left ( k - \\alpha - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } \\\\ \\alpha ^ 2 - k \\alpha + \\frac { k } { 4 } - \\frac { k - \\alpha } { n } & \\leq 0 \\ , . \\end{align*}"}
-{"id": "5412.png", "formula": "\\begin{align*} \\| v _ { t a i l ( 2 \\kappa ) } \\| _ b = \\left ( \\sum _ { j \\ge 2 } \\| v _ { B _ j } \\| _ b ^ b \\right ) ^ { \\frac { 1 } { b } } . \\end{align*}"}
-{"id": "9894.png", "formula": "\\begin{align*} B _ 1 ( M , N ) = \\sum _ { N \\le n \\le N ' } \\left \\lvert \\sum _ { M < m \\le M ' } a _ { m n } \\mu ( m ) \\right \\rvert \\end{align*}"}
-{"id": "7876.png", "formula": "\\begin{align*} \\lim _ { h \\downarrow 0 } \\sup _ { x , y \\in \\R ^ d } | \\int ^ { t + h } _ t \\int _ { \\R ^ d } q _ 0 ( t + h - s , x , z ) q _ 0 ( s , z , y ) d z d s | = 0 . \\end{align*}"}
-{"id": "5733.png", "formula": "\\begin{align*} e _ { t } = \\sigma _ { t } ^ { - \\gamma } e _ { t } ^ { I } \\end{align*}"}
-{"id": "7517.png", "formula": "\\begin{align*} \\mu _ 2 : = f ( \\mu _ 0 ) \\geq f ( c ) > f ( K ) = K > c , f ( x ) \\leq \\mu _ 2 , x \\in [ 0 , \\infty ) ; \\end{align*}"}
-{"id": "9433.png", "formula": "\\begin{align*} n _ g \\ge n _ { g \\circ c } \\ge n _ { [ f \\circ c ] } = n _ { f \\circ c } = n _ f , \\end{align*}"}
-{"id": "9566.png", "formula": "\\begin{align*} \\psi _ { n } ^ { ( 1 ) } = \\sum _ { i \\neq n } \\frac { \\langle \\psi _ { i } ^ { ( 0 ) } , H ^ { ( 1 ) } \\psi _ { n } ^ { ( 0 ) } \\rangle } { \\lambda _ { n } ^ { ( 0 ) } - \\lambda _ { i } ^ { ( 0 ) } } \\psi _ { i } ^ { ( 0 ) } - \\frac { 1 } { 2 } \\langle \\psi _ { n } ^ { ( 0 ) } , G ^ { ( 1 ) } \\psi _ { n } ^ { ( 0 ) } \\rangle \\psi _ { n } ^ { ( 0 ) } \\end{align*}"}
-{"id": "4800.png", "formula": "\\begin{align*} \\frac { \\left ( f f '' + ( f ' ) ^ 2 + 1 \\right ) ^ 2 - \\kappa ^ 2 ( f '^ 2 + 1 ) } { 4 f ^ 2 ( f '^ 2 + 1 ) } = c . \\end{align*}"}
-{"id": "4525.png", "formula": "\\begin{align*} u _ n ^ * ( x ) : = u _ { x , a _ n / ( n - 1 ) } = \\frac { V _ d ( n - 1 ) h _ x ^ { - 1 } ( \\frac { a _ n } { n - 1 } ) ^ d } { e ^ { \\Psi ( k ) } } , \\end{align*}"}
-{"id": "1365.png", "formula": "\\begin{align*} Q _ { L } ^ { B L P } = n ( n + 2 ) \\sum _ { l = 1 } ^ { L } ( n - l ) ^ { - 1 } r ( l ) ^ 2 . \\end{align*}"}
-{"id": "9531.png", "formula": "\\begin{align*} \\mathrm { I m } \\ , \\tau ' & = \\frac { e ^ { - \\phi } } { | c \\tau + d | ^ 2 } \\\\ d \\tau ' & = \\frac { 1 } { ( c \\tau + d ) ^ 2 } d \\tau \\end{align*}"}
-{"id": "2173.png", "formula": "\\begin{align*} \\int _ { B _ { 1 } } \\phi \\psi ^ { 1 + q } w ^ { 2 } d x + \\frac { c _ { n , \\beta } } { 4 C \\Lambda } g _ { \\alpha } * \\int _ { \\rho ' B _ { 1 } } \\int _ { \\rho ' B _ { 1 } } \\frac { ( w ( s , x ) - w ( s , y ) ) ^ { 2 } } { | x - y | ^ { n + 2 \\beta } } \\phi d x d y \\leq g _ { \\alpha } * F . \\end{align*}"}
-{"id": "4114.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q } { z ^ { p + 2 q } } , \\end{align*}"}
-{"id": "3970.png", "formula": "\\begin{align*} ( u ' ( \\psi ( s ) ) v ) ^ { \\mu _ 0 } ( A ) = u ' ( \\psi ( s ) ) v ^ { \\mu _ 0 } ( A ) . \\end{align*}"}
-{"id": "4953.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { 2 ^ \\ast } ( B _ r ( x _ 0 ) ) } = \\| \\Psi _ \\lambda \\ast f \\| _ { L ^ { 2 ^ \\ast } ( B _ r ( x _ 0 ) ) } \\leq \\| \\Psi _ \\lambda \\| _ { L ^ { \\frac { N } { N - 2 } , w } ( B _ { r + r _ 0 } ( x _ 0 - y _ 0 ) ) } \\| f \\| _ { ( 2 ^ \\ast ) ' } , \\end{align*}"}
-{"id": "1593.png", "formula": "\\begin{align*} \\frac { T _ { I , t t } } { T _ { I } } = m _ { I } ~ , ~ 2 f _ { I , c } + c - m _ { I } = 0 . \\end{align*}"}
-{"id": "3200.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | \\geq 2 } \\sum _ { | \\gamma | \\geq 1 } \\sum _ { | \\beta | = | \\alpha | } T _ { j k } F _ { k j , \\alpha , \\gamma } \\cdot \\tau _ { k j , \\beta } ^ \\alpha \\cdot u _ j ^ \\beta \\cdot \\prod _ { \\lambda = 1 } ^ r \\left ( u _ j ^ \\lambda + \\sum _ { | \\delta | \\geq 2 } F _ { j , \\delta } ^ \\lambda \\cdot u _ j ^ \\delta \\right ) ^ { \\gamma _ \\lambda } , \\end{align*}"}
-{"id": "2066.png", "formula": "\\begin{align*} X ( s ) = \\sum \\limits _ { j = 0 } ^ { \\infty } X ^ { ( j ) } ( s _ { 0 } ) ( s - s _ { 0 } ) ^ { j } , \\end{align*}"}
-{"id": "6586.png", "formula": "\\begin{align*} R _ { \\gamma f } ( x ) : = P _ { \\gamma f } ^ 2 ( x ) = 2 { \\rm { p r o x } } _ { \\gamma f } ( x ) - x . \\end{align*}"}
-{"id": "8888.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\abs { u } ^ p \\leq \\liminf _ { \\ell \\to \\infty } \\int _ { \\R ^ N } \\abs { \\tau ^ { A _ { n _ \\ell } } _ { a _ { n _ \\ell } } u _ { n _ \\ell } } ^ p = \\liminf _ { \\ell \\to \\infty } \\int _ { \\R ^ N } \\abs { u _ { n _ \\ell } } ^ p , \\end{align*}"}
-{"id": "2667.png", "formula": "\\begin{align*} \\nu ( \\varphi , x ) = \\sup \\{ \\gamma \\geq 0 \\textup { s . t . } \\varphi ( z ) \\leq \\gamma \\log \\| z - x \\| + O ( 1 ) \\textup { o n } U \\} . \\end{align*}"}
-{"id": "514.png", "formula": "\\begin{align*} \\sum _ { \\nu ' } s _ { \\nu / \\lambda } ( \\rho ) = H ^ \\circ ( \\rho ) \\sum _ { \\kappa ' } s _ { \\lambda / \\kappa } ( \\rho ) , \\end{align*}"}
-{"id": "2804.png", "formula": "\\begin{align*} y _ k = \\exp ( 2 \\pi \\imath f _ { 1 k } - \\tau _ { 1 k } ) , ~ z _ k = \\exp ( 2 \\pi \\imath f _ { 2 k } - \\tau _ { 2 k } ) , \\mbox { a n d } w _ k = \\exp ( 2 \\pi \\imath f _ { 3 k } - \\tau _ { 3 k } ) \\end{align*}"}
-{"id": "3982.png", "formula": "\\begin{align*} \\pi _ { p } \\circ q ( \\vect { \\delta } ) = q ( \\vect { \\delta } ) \\circ \\pi _ { p } . \\end{align*}"}
-{"id": "6651.png", "formula": "\\begin{align*} & \\prod _ { k = 0 } ^ { d _ j - 2 } q ^ { - ( s _ j + k ) \\dim V _ j ^ I } \\cdot \\det ( - \\rho _ j ( \\Phi ) | V _ j ^ I ) \\\\ & = \\det ( - \\rho _ j ( \\Phi ) | V _ j ^ I ) ^ { d _ j - 1 } \\cdot q ^ { - s _ j ( d _ j - 1 ) \\dim V _ j ^ I } \\cdot q ^ { - ( 1 + 2 + \\ldots + d _ j - 2 ) \\dim V _ j ^ I } \\\\ & = \\det ( - \\rho _ j ( \\Phi ) | V _ j ^ I ) ^ { d _ j - 1 } \\cdot q ^ { - s _ j ( d _ j - 1 ) \\dim V _ j ^ I } \\cdot q ^ { - \\frac { 1 } { 2 } ( d _ j - 2 ) ( d _ j - 1 ) \\dim V _ j ^ I } \\end{align*}"}
-{"id": "4797.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a t ) ^ 2 + t ^ 2 } , c = c o n s t , \\end{align*}"}
-{"id": "1983.png", "formula": "\\begin{align*} \\Xi = C _ { f } f ^ { * } + g ^ { * } \\end{align*}"}
-{"id": "5685.png", "formula": "\\begin{align*} e _ 0 = \\frac { B + \\alpha + \\gamma + q - 1 } { 1 - q } . \\end{align*}"}
-{"id": "3440.png", "formula": "\\begin{align*} + \\int _ t ^ T E [ I _ { \\{ y _ j ^ 1 ( s ) > y _ j ^ 2 ( s ) \\} } \\sum _ { k = 1 } ^ { d _ 1 } | \\bar y _ k ( s ) | ^ 2 ] d s . \\end{align*}"}
-{"id": "7106.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\omega _ { n , \\beta } = ( 1 - \\pi _ \\beta ) \\delta _ 0 + \\pi _ \\beta \\delta _ 1 \\end{align*}"}
-{"id": "2022.png", "formula": "\\begin{align*} \\int _ { - 1 + x ^ { - 1 } } ^ 1 [ f ( x + x z ) - f ( x ) - f ' ( x ) x z ] \\nu _ U ( \\d z ) = x ^ \\beta \\int _ { - 1 + x ^ { - 1 } } ^ 1 [ ( 1 + z ) ^ \\beta - 1 - \\beta z ] \\nu _ U ( \\d z ) , \\end{align*}"}
-{"id": "4296.png", "formula": "\\begin{align*} K _ { X / Y } : = K _ X - p ^ \\star ( K _ Y ) . \\end{align*}"}
-{"id": "9768.png", "formula": "\\begin{align*} \\nabla _ { \\vec { x } } V ( \\vec { x } ) = - \\vec { S } _ { \\mathcal { E } } ^ { \\top } \\boldsymbol { \\lambda } _ { \\mathcal { E } } ^ { * } ( \\vec { x } ) , \\forall \\vec { x } \\in \\mathcal { R } ( \\mathcal { E } ) \\cap \\mathrm { i n t } ( \\mathcal { X } ) , \\end{align*}"}
-{"id": "9609.png", "formula": "\\begin{align*} \\Vert v _ \\tau ( t ) \\Vert ^ 2 = \\frac 1 { 4 \\pi } \\int _ 0 ^ { t - \\tau } | h ( t - r ) | ^ 2 d r + \\frac { m ^ 2 } { 4 \\pi } \\int _ 0 ^ { t - \\tau } r ^ 2 | p _ \\tau ( r , t ) | ^ 2 d r + \\frac { m } { 2 \\pi } \\int _ 0 ^ { t - \\tau } h ( t - r ) p _ \\tau ( r , t ) r \\ , d r , t \\in [ \\tau , T ] . \\end{align*}"}
-{"id": "8380.png", "formula": "\\begin{align*} \\| x j - g - \\sum _ { i = 1 } ^ k x _ i v _ i \\| < \\frac { 2 } { 3 } . \\end{align*}"}
-{"id": "1620.png", "formula": "\\begin{align*} \\ln F \\left ( t , x \\right ) = \\frac { 2 e ^ { - m t } } { \\sqrt { 1 + \\varepsilon ^ { 2 } } } i \\arctan \\left ( \\frac { \\tan \\left ( \\frac { x } { 2 } \\right ) + \\varepsilon } { \\sqrt { 1 - \\varepsilon ^ { 2 } } } \\right ) + \\frac { c } { m } e ^ { - m t } - \\frac { 1 } { 4 m } e ^ { - 2 m t } \\end{align*}"}
-{"id": "3259.png", "formula": "\\begin{align*} E _ k \\triangleright [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q & = c _ { k , \\xi } ( E _ { \\xi + \\alpha _ k } E _ { \\xi ^ \\prime } ^ * - q ^ { - ( \\xi + \\alpha _ k , \\xi ^ \\prime ) } E _ { \\xi ^ \\prime } ^ * E _ { \\xi + \\alpha _ k } ) \\\\ & - c _ { k , \\xi ^ \\prime } ^ \\prime q ^ { ( \\xi - \\alpha _ k , \\alpha _ k ) } ( E _ \\xi E _ { \\xi ^ \\prime - \\alpha _ k } ^ * - q ^ { - ( \\xi , \\xi ^ \\prime + \\alpha _ k ) } E _ { \\xi ^ \\prime - \\alpha _ k } ^ * E _ \\xi ) . \\end{align*}"}
-{"id": "6300.png", "formula": "\\begin{align*} \\mathcal { M } _ { p } ( l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } , l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } ) = \\left \\{ \\{ a _ { k } \\} _ { k \\in \\mathbb { Z } ^ { n } } : ~ \\Vert \\{ a _ { k } \\lambda _ { k } \\} \\Vert _ { l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } } \\lesssim \\Vert \\{ \\lambda _ { k } \\} \\Vert _ { l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } } \\{ \\lambda _ { k } \\} \\in l _ { q } ^ { s _ 1 , \\alpha } \\right \\} . \\end{align*}"}
-{"id": "5960.png", "formula": "\\begin{align*} \\theta _ t ^ h ( z ) = \\frac { \\phi _ t ( z + h e ) - \\phi _ t ( z ) } { h } \\ , , \\xi _ t ^ h ( z ) = \\frac { \\gamma \\big ( \\phi _ t ( z + h e ) \\big ) - \\gamma \\big ( \\phi _ t ( z ) \\big ) } { h } \\ , , \\end{align*}"}
-{"id": "6232.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + K ) X ^ { ( 0 ) } ( s _ { i } ) & = F + \\eta _ { 0 i } . \\end{align*}"}
-{"id": "6711.png", "formula": "\\begin{align*} \\nu ^ { - n } L _ f = A _ 0 + \\nu A _ 1 + \\ldots \\end{align*}"}
-{"id": "4642.png", "formula": "\\begin{align*} \\sum _ { \\pm m \\geq 1 } \\psi \\left ( \\frac { | m | } { X } \\right ) \\sum _ { i = 1 } ^ { h ( m ) } \\frac { \\phi \\left ( g _ { i , m } ^ { - 1 } \\right ) } { | \\Gamma ( i , m ) | } \\ll _ { \\epsilon , \\psi } X ^ { \\frac { 1 } { 8 } + \\epsilon } \\end{align*}"}
-{"id": "9396.png", "formula": "\\begin{align*} \\widetilde { W } _ \\lambda = 2 i k - \\frac { | Z | ^ 2 } { i k ^ 3 } \\left [ ( e ^ { i k \\rho } - 2 ) ^ 2 + 2 i k \\rho - 1 \\right ] , \\lambda = k ^ 2 , k \\in \\mathbb { C } _ + . \\end{align*}"}
-{"id": "9561.png", "formula": "\\begin{align*} \\Lambda _ { n i } ^ { ( 2 ) } = \\sum _ { k \\neq n } \\frac { P _ { n i } ^ { ( 1 ) } H ^ { ( 1 ) } P _ { k } ^ { ( 0 ) } H ^ { ( 1 ) } P _ { n i } ^ { ( 1 ) } } { \\lambda _ { n } ^ { ( 0 ) } - \\lambda _ { k } ^ { ( 0 ) } } + P _ { n i } ^ { ( 1 ) } H ^ { ( 2 ) } P _ { n i } ^ { ( 1 ) } \\end{align*}"}
-{"id": "4475.png", "formula": "\\begin{align*} - \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\log ( 1 - s ) \\mathrm { B } _ { k , n - k } ( s ) \\ , d s \\leq \\frac { n - 1 } { n - k - 1 } \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\mathrm { B } _ { k , n - k - 1 } ( s ) \\ , d s = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "9198.png", "formula": "\\begin{align*} Y ( t , x ) = \\theta ( t , x ) ; ( t , x ) \\in [ 0 , T ] \\times \\partial D . \\end{align*}"}
-{"id": "409.png", "formula": "\\begin{align*} ( f \\circ 1 ) ( y ) : = \\bigvee \\limits _ { ( x , w ) \\in { A _ y } } { \\min \\{ f ( x ) , 1 ( w ) \\} = \\bigvee \\limits _ { ( x , w ) \\in { A _ y } } f ( x ) } \\end{align*}"}
-{"id": "3353.png", "formula": "\\begin{align*} \\| M \\| _ { r \\ast } \\ ; = \\ ; \\| \\Sigma \\| _ { r \\ast } \\ ; = \\ ; \\max _ { \\| X \\| _ r \\leq 1 } \\langle \\Sigma , X \\rangle \\ ; = \\ ; \\max _ { \\sum _ { i = 1 } ^ r \\sigma _ i ^ 2 ( X ) = 1 } \\sum _ { i = 1 } ^ { n } \\sigma _ i ( M ) \\sigma _ i ( X ) \\ ; = \\ ; \\max _ { \\sum _ { i = 1 } ^ r \\sigma _ i ^ 2 ( X ) \\leq 1 } \\left [ \\sum _ { i = 1 } ^ { r } \\sigma _ i ( M ) \\sigma _ i ( X ) + \\sigma _ r ( X ) \\sum _ { i = r + 1 } ^ { n } \\sigma _ i ( M ) \\right ] . \\end{align*}"}
-{"id": "9759.png", "formula": "\\begin{align*} \\dfrac { \\partial f } { \\partial \\vec { z } ^ { * } } ( \\vec { z } ^ { * } ( \\bar { \\vec { x } } ) , \\bar { \\vec { x } } ) + [ \\boldsymbol { \\lambda } ^ { * } ( \\bar { \\vec { x } } ) ] ^ { \\top } \\dfrac { \\partial \\vec { g } } { \\partial \\vec { z } ^ { * } } ( \\vec { z } ^ { * } ( \\bar { \\vec { x } } ) , \\bar { \\vec { x } } ) = \\vec { 0 } _ { s } ^ { \\top } . \\end{align*}"}
-{"id": "3166.png", "formula": "\\begin{align*} & \\tau ( 1 \\mid 0 ) = 0 ~ ~ \\tau ( 2 \\mid 0 ) = 1 \\\\ & \\tau ( 1 \\mid 1 ) = 1 ~ ~ \\tau ( 2 \\mid 1 ) = 0 \\end{align*}"}
-{"id": "2123.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } \\frac { 1 } { p _ i } = \\infty . \\end{align*}"}
-{"id": "4817.png", "formula": "\\begin{align*} \\begin{aligned} n m a c ( T ) & = m a c ( T \\otimes I _ n ) = m a c ( P ( T \\otimes I _ n ) P ) \\\\ & + m a c ( ( I \\otimes I _ n - P ) ( T \\otimes I _ n ) ( I \\otimes I _ n - P ) ) \\\\ & \\ge m a c ( P ( T \\otimes I _ n ) P ) . \\end{aligned} \\end{align*}"}
-{"id": "3753.png", "formula": "\\begin{align*} f _ Y ( y ) = \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - y ^ 2 / 2 } + O \\left ( \\frac { 1 } { \\sqrt { E _ \\beta } } \\right ) \\end{align*}"}
-{"id": "2.png", "formula": "\\begin{align*} \\tilde { T } _ { q _ 0 } = b _ { q _ 0 } \\tilde { \\ell } _ * + \\tilde { T } _ { q _ 0 , R } \\end{align*}"}
-{"id": "8290.png", "formula": "\\begin{align*} A _ { G _ 1 } = A _ { G _ 2 } + M , \\end{align*}"}
-{"id": "3007.png", "formula": "\\begin{align*} L ( \\tilde \\sigma ) & = L ( \\sigma ) + m , \\ell ( \\tilde \\sigma ) = \\ell ( \\sigma ) + n - 1 \\\\ L ( \\hat \\sigma ) & = L ( \\sigma ) + m , \\ell ( \\hat \\sigma ) = \\ell ( \\sigma ) + n - 1 \\\\ L ( \\check \\sigma ) & = L ( \\sigma ) + 2 m , \\ : \\ : \\ell ( \\check \\sigma ) = \\ell ( \\sigma ) + 2 ( n - 1 ) , \\end{align*}"}
-{"id": "4890.png", "formula": "\\begin{align*} x - [ r _ 0 ] \\equiv \\sum _ { k = 1 } ^ \\nu p ^ k [ \\phi ^ { - k } ( r _ k ) ] \\bmod I ^ { \\nu + 1 } \\end{align*}"}
-{"id": "1594.png", "formula": "\\begin{align*} X _ { I } = 2 \\psi _ { I } \\int T _ { I } \\left ( t \\right ) d t ~ \\partial _ { t } + T _ { I } \\left ( t \\right ) Y _ { I } + \\alpha \\left ( t , x \\right ) F \\partial _ { F } \\end{align*}"}
-{"id": "5508.png", "formula": "\\begin{align*} \\| x \\| = \\| x \\| _ { \\varphi , w } = \\inf \\{ \\epsilon > 0 : \\rho ( x / \\epsilon ) \\le 1 \\} . \\end{align*}"}
-{"id": "2378.png", "formula": "\\begin{align*} p _ { \\gamma f } ^ { \\alpha } ( x ) : = \\alpha r _ { \\gamma f } ^ * ( x ) + \\tfrac { 1 - \\alpha } { 2 } \\| x \\| ^ 2 , \\end{align*}"}
-{"id": "6518.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v ' ( t ) + A ( s ) v ( t ) = 0 , t > 0 , \\\\ & v ( 0 ) = v _ 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "6870.png", "formula": "\\begin{align*} v ^ { [ t _ n ] } ( t , x ) : = t _ n ^ { \\frac 1 { p } } v ( t _ n t , \\sqrt { t _ n } x ) . \\end{align*}"}
-{"id": "2851.png", "formula": "\\begin{align*} \\delta _ V ^ 2 = d _ V \\circ \\delta _ V + \\delta _ V \\circ d _ V , \\end{align*}"}
-{"id": "3236.png", "formula": "\\begin{align*} c _ { v } ^ { \\prime } \\left ( 0 \\right ) = \\lim _ { t \\rightarrow 0 ^ { + } } \\left ( \\Uparrow _ { c \\left ( 0 \\right ) } ^ { c \\left ( t \\right ) } \\right ) _ { X } , \\end{align*}"}
-{"id": "2300.png", "formula": "\\begin{align*} J _ m : = \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = k } ^ m \\big \\| _ { L ^ p ( D ) } , \\end{align*}"}
-{"id": "8940.png", "formula": "\\begin{align*} \\lim _ { a \\searrow 0 } a f ( a ) ^ p = \\lim _ { b \\nearrow + \\infty } b f ( b ) ^ p = 0 . \\end{align*}"}
-{"id": "9488.png", "formula": "\\begin{align*} \\alpha _ { n - 1 } ( p ) ( v ) = \\frac { 1 } { 2 } d \\alpha _ { n - 1 } ( p , v ) = - \\frac { 1 } { 2 } d \\alpha _ { n - 1 } ( v , p ) = \\frac { 1 } { 2 } g _ { j _ { 0 } } ( v , j _ { 0 } p ) . \\end{align*}"}
-{"id": "1034.png", "formula": "\\begin{align*} c _ { d , d } ^ d { \\ , } y ^ { d ^ 2 } + \\sum _ { k = 0 } ^ { m } { d \\choose k } ( c _ { d , d } { \\ , } x ^ d ) ^ k \\Big ( \\sum _ { j = 0 } ^ r c _ { d , j } { \\ , } x ^ j y ^ { d - j } \\Big ) ^ { d - k } ~ = ~ 0 . \\end{align*}"}
-{"id": "8218.png", "formula": "\\begin{align*} ( E - i \\gamma ) U _ 0 - \\Omega \\bar { U } _ 0 = 6 | U _ 0 | ^ 2 \\bar { U } _ 0 + 2 U _ 0 ^ 3 . \\end{align*}"}
-{"id": "5246.png", "formula": "\\begin{align*} \\textstyle \\pi _ { \\eta } ' \\colon X ' \\to Y ' , ( B _ { l , 1 } , B _ { l , 2 } , i _ { l } ) _ { 1 \\le l \\le e } \\mapsto ( \\bigoplus _ { l = 1 } ^ { e } B _ { l , 1 } , B _ { 2 } , \\bigoplus _ { l = 1 } ^ { e } i _ { l } , Z _ { 1 } , Z _ { 2 } , . . . , Z _ { e } ) , \\end{align*}"}
-{"id": "6647.png", "formula": "\\begin{align*} \\det ( A \\oplus B ) = ( \\det A ) ( \\det B ) \\end{align*}"}
-{"id": "7518.png", "formula": "\\begin{align*} \\mu _ 1 : = f ( \\mu _ 2 ) . \\end{align*}"}
-{"id": "75.png", "formula": "\\begin{align*} \\varphi ( y , a ^ * ) ^ { ( p - 1 ) / p } - \\varphi ( y , a _ * ) ^ { ( p - 1 ) / p } & = \\frac { p - 1 } { p } \\ \\int _ { \\varphi ( y , a _ * ) } ^ { \\varphi ( y , a ^ * ) } z ^ { - 1 / p } \\ d z \\\\ & \\le \\frac { p - 1 } { p } \\varphi ( y , a _ * ) ^ { - 1 / p } G ( y ) \\\\ & \\le \\frac { ( p - 1 ) ( 2 p - 1 ) } { 2 p } G ( y ) \\ . \\end{align*}"}
-{"id": "1806.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { T } \\int _ { { \\mathbb R } ^ d } \\int _ { 0 } ^ { 1 } \\Big | { \\sum _ { j = 1 } ^ \\infty } r _ { j } ( z ) 2 ^ { j ( s + \\alpha / p ) } \\Delta _ { j } T ^ { \\alpha } f ( t , x ) \\Big | ^ { p } d z d x d t \\\\ & \\ll \\int _ { 0 } ^ { T } \\int _ { { \\mathbb R } ^ d } \\int _ { 0 } ^ { 1 } \\Big | { \\sum _ { j = 1 } ^ \\infty } r _ { j } ( z ) 2 ^ { j ( s + \\alpha / p ) } \\Delta _ { j } T ^ { \\alpha } \\Delta _ { j } f ( t , x ) \\Big | ^ { p } d z d x d t . \\end{align*}"}
-{"id": "2812.png", "formula": "\\begin{align*} \\Pi ( E _ { b } ) ( z , w ) & = \\frac { 1 } { \\pi } \\frac { \\varphi _ z ( b ) \\overline { \\varphi ' _ w ( b ) } - \\varphi ' _ z ( b ) \\overline { \\varphi _ w ( b ) } } { z - \\bar { w } } \\\\ & = \\frac { 1 } { \\pi } \\int _ 0 ^ { b } \\varphi _ { z } ( t ) \\overline { \\varphi _ { w } ( t ) } d t . \\end{align*}"}
-{"id": "1532.png", "formula": "\\begin{align*} \\mathcal { F } ( f ) : = \\int e ^ { - i k \\cdot x } f ( x ) \\d x , \\end{align*}"}
-{"id": "5955.png", "formula": "\\begin{align*} \\gamma ( x ) = x + U ( x ) , \\ ; \\ ; x \\in \\R ^ { 2 d } \\ , . \\end{align*}"}
-{"id": "7938.png", "formula": "\\begin{align*} \\alpha ( w _ i , w _ j ) + \\alpha ( w _ j , w _ i ) = \\frac { 1 } { m ^ 2 } \\sum _ { \\ell = 1 } ^ m \\sum _ { k = 1 } ^ m ( \\alpha _ H ( v _ { i , \\ell } , v _ { j , k } ) + \\alpha _ H ( v _ { j , k } , v _ { i , \\ell } ) = 1 \\end{align*}"}
-{"id": "8562.png", "formula": "\\begin{align*} \\varphi ( d _ { 1 } a ^ { \\ast } d _ { 2 } w ) & = d _ { 1 } a ^ { \\ast } d _ { 2 } w \\\\ & = d _ { 1 } \\overline { N } ( a ^ { \\ast } ) ( d _ { 2 } w ) \\\\ & = d _ { 1 } \\overline { N ' } ( a ^ { \\ast } ) \\psi ( d _ { 2 } w ) \\\\ & = d _ { 1 } a ^ { \\ast } d _ { 2 } \\varphi ( w ) \\end{align*}"}
-{"id": "2110.png", "formula": "\\begin{align*} A _ { \\sigma l } = ( l + 1 ) ( l + 2 + 2 \\sigma _ n ) p _ { \\sigma , l + 1 } + 2 s ( \\sigma _ n + 1 ) p _ { \\sigma + \\bar n , l } + ( \\sigma _ i + 1 ) ( \\sigma _ i + 2 ) p _ { \\sigma + 2 \\bar \\imath , l - 1 } + c _ { \\sigma l } ^ { \\mu m } p _ { \\mu m } . \\end{align*}"}
-{"id": "2417.png", "formula": "\\begin{align*} \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } = \\{ l \\in \\mathbb { Z } ^ n : ~ \\Box _ k ^ { \\alpha _ 2 } \\circ \\Box _ l ^ { \\alpha _ 1 } \\neq 0 \\} , ~ \\widetilde { \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } = \\{ l \\in \\mathbb { Z } ^ n : ~ \\Box _ k ^ { \\alpha _ 2 } \\circ \\Box _ l ^ { \\alpha _ 1 } = \\Box _ l ^ { \\alpha _ 1 } \\} . \\end{align*}"}
-{"id": "1493.png", "formula": "\\begin{align*} P ( X ) : = X ^ n + a _ 1 X ^ { n - b } + \\cdots + a _ k X ^ { n - b k } , a _ k \\neq 0 . \\end{align*}"}
-{"id": "7751.png", "formula": "\\begin{align*} & \\mathbb P [ T _ { N } = x ] = \\mathbb P [ T _ { N + 1 } + z _ { N } = x ] = \\int _ { - \\infty } ^ \\infty \\mathbb P \\left [ T _ { N + 1 } + z _ { N } = x \\biggl | T _ { N + 1 } = y \\right ] d F _ { N + 1 } ( y ) \\\\ & = \\int _ { - \\infty } ^ \\infty \\mathbb P \\left [ z _ { N } = x - y \\biggl | T _ { N + 1 } = y \\right ] d F _ { N + 1 } ( y ) = \\int _ { - \\infty } ^ \\infty \\mathbb P \\left [ z _ { N } = x - y \\right ] d F _ { N + 1 } ( y ) = 0 , \\end{align*}"}
-{"id": "4852.png", "formula": "\\begin{align*} R ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) = - \\int \\frac { h ^ { \\rm g e o } _ { 1 2 } ( z , 1 / z ) } { z ^ { u - v + 1 } } \\dd z . \\end{align*}"}
-{"id": "7743.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { S _ n } { D _ n } = \\infty , \\liminf _ { n \\to \\infty } \\frac { S _ n } { D _ n } = - \\infty , \\quad \\end{align*}"}
-{"id": "1576.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( x \\right ) F _ { , x x } + \\left ( \\kappa \\left ( x \\right ) \\left ( \\mu \\left ( x \\right ) - \\lambda \\left ( x \\right ) - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( x \\right ) \\right ) F _ { , x } - F _ { , t } = 0 \\end{align*}"}
-{"id": "8090.png", "formula": "\\begin{align*} \\omega _ { ( s _ i , r _ j ) } = \\frac { 1 } { T _ { g , ( s _ i , r _ j ) } ^ { g l m } } = \\sqrt { \\frac { A _ { s _ i , r _ j } E _ { r _ j } + B _ { s _ i , r _ j } E _ { s _ i } } { E _ { s _ i } ^ 2 E _ { r _ j } p _ { t h } ^ { s _ i } } } . \\end{align*}"}
-{"id": "3556.png", "formula": "\\begin{align*} \\mathcal { I } ^ { R } ( \\theta ) = ( E ^ \\theta - E , P ^ \\theta - P , R ^ { - 1 } ( \\mathcal C ^ \\theta - \\mathcal C ^ R ) , R ^ { - 1 } ( \\mathcal J ^ \\theta - \\mathcal J ^ R ) ) + \\mathcal { I } _ 0 ^ R ( \\theta ) , \\end{align*}"}
-{"id": "6380.png", "formula": "\\begin{align*} \\mu _ { ( k ) } = C ( \\varepsilon ) \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon ) ^ { n } } { n ! } ( n p + k - 1 ) ! ! \\sigma ^ { n p + k } , \\end{align*}"}
-{"id": "9096.png", "formula": "\\begin{align*} K _ { N } ( s ) = - ( 2 + \\alpha ) u _ { N - 1 } ' - ( s - 1 ) u _ { N - 1 } '' - \\int _ { 0 } ^ { \\infty } x ^ { 2 } e ^ { s x } x ^ { \\alpha } e ^ { - x } L _ { N - 1 } ( x ) L ' _ { N - 2 } ( x ) \\ , \\mathrm { d } x \\ . \\end{align*}"}
-{"id": "3449.png", "formula": "\\begin{align*} \\frac { \\partial V } { \\partial s } + \\frac 1 2 T r Q ^ * ( x , h ) \\nabla ^ 2 V Q ( x , h ) + \\langle q ( x , h ) , \\nabla V \\rangle + G ( h , x , V , Q ^ * \\nabla V ) = 0 , \\end{align*}"}
-{"id": "5092.png", "formula": "\\begin{align*} \\check { R } ( z / w ) [ \\mathbb { T } ^ { [ M ' , M ] } ( z ) \\otimes \\mathbb { T } ^ { [ M ' , M ] } ( w ) ] = [ \\mathbb { T } ^ { [ M ' , M ] } ( w ) \\otimes \\mathbb { T } ^ { [ M ' , M ] } ( z ) ] \\check { R } ( z / w ) , \\end{align*}"}
-{"id": "966.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { q } \\sum _ { j , k = 1 } ^ { n } \\frac { \\partial ^ { 2 } h _ { A } ( z ) } { \\partial z _ { j } \\partial \\overline { z _ { k } } } ( t ^ { s } ) _ { j } \\overline { ( t ^ { s } ) _ { k } } \\geq A \\ ; , \\ ; t ^ { 1 } , \\hdots , t ^ { q } \\ ; \\ ; \\mathbb { C } ^ { n } \\end{align*}"}
-{"id": "904.png", "formula": "\\begin{align*} \\zeta ( \\tfrac { 1 } { 2 } + i t ) = \\sum _ { n \\leq C T / \\pi } { n ^ { - \\tfrac { 1 } { 2 } - i t } } + O \\left ( \\frac { T ^ { 1 / 2 } } { \\vert t \\vert } \\right ) + O ( T ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "8153.png", "formula": "\\begin{align*} \\phi _ h ( a ^ k ) = e ^ { \\frac { 2 h k \\pi } { n } \\mathbf { i } } , \\mbox { w h e r e $ 0 \\leq h \\leq n - 1 $ . } \\end{align*}"}
-{"id": "5506.png", "formula": "\\begin{align*} \\left ( \\sum _ { i = 1 } ^ { K } \\beta _ { i } \\right ) ^ 2 + M \\beta _ { k } \\sum _ { i = 1 } ^ { K } \\beta _ { i } \\geq \\left ( \\sum _ { i = 1 } ^ { K } \\beta _ { i } \\right ) ^ 2 \\end{align*}"}
-{"id": "8402.png", "formula": "\\begin{align*} \\sigma ( m z _ g g ) = \\theta ^ { - 1 } ( m ) \\sigma ( z _ g g ) , \\ \\ \\ m \\in M . \\end{align*}"}
-{"id": "7388.png", "formula": "\\begin{align*} \\frac { \\norm { \\Delta _ { t , t } } ^ 2 } { 2 ( t + 1 ) } & \\leq \\sum _ { r = 0 } ^ { t - 1 } ( \\gamma ^ t _ r - \\hat { \\gamma } ^ t _ r ) ^ 2 \\norm { b ^ r } ^ 2 + \\norm { Z ' _ t } ^ 2 \\Big ( \\frac { 1 } { \\sqrt { n } } \\norm { q ^ t _ { \\perp } } - \\sigma _ { t } ^ { \\perp } \\Big ) ^ 2 \\\\ & + \\frac { 1 } { n ^ 2 } \\norm { q ^ t _ { \\perp } } ^ 2 \\norm { \\tilde { M } _ t \\bar { Z } ' _ t } ^ 2 + \\sum _ { j = 0 } ^ { t - 1 } \\norm { m ^ j } ^ 2 [ \\textbf { M } _ t ^ { - 1 } v ] _ { j + 1 } ^ 2 , \\end{align*}"}
-{"id": "771.png", "formula": "\\begin{gather*} R _ { \\ell + s } = C _ { \\ell + s } \\oplus \\varphi _ { \\ell } ( R _ \\ell ) . \\end{gather*}"}
-{"id": "7381.png", "formula": "\\begin{align*} & [ 0 | M _ { t - 1 } ] \\Lambda _ t \\mathbf { Q } _ { t } ^ { - 1 } Q ^ * _ t q ^ t _ { \\parallel } - \\lambda _ t m ^ { t - 1 } = \\sum _ { i = 1 } ^ { t - 1 } \\lambda _ { i } \\gamma ^ t _ { i } m ^ { i - 1 } - \\lambda _ t m ^ { t - 1 } \\\\ & = - M _ t \\mathbf { M } _ { t } ^ { - 1 } M _ t ^ * \\Big ( \\lambda _ t m ^ { t - 1 } - \\sum _ { i = 1 } ^ { t - 1 } \\lambda _ { i } \\gamma ^ t _ { i } m ^ { i - 1 } \\Big ) . \\end{align*}"}
-{"id": "8747.png", "formula": "\\begin{align*} \\mathcal { E } _ { C } \\left ( t , k \\right ) = | { Q } _ t ^ { - 1 / 2 } e ^ { t A } k | _ H \\end{align*}"}
-{"id": "8929.png", "formula": "\\begin{align*} - \\Delta _ { A _ * } w _ { A _ * } = - \\Delta w _ { A _ * } + \\abs { A _ * } ^ 2 w _ { A _ * } . \\end{align*}"}
-{"id": "3375.png", "formula": "\\begin{align*} P ^ s ( t ) = \\left [ \\frac { E _ b ( t ) - \\bar E _ b ^ e ( t - 1 ) } { \\Delta t } \\right ] ^ + \\end{align*}"}
-{"id": "3305.png", "formula": "\\begin{gather*} \\langle 1 , 1 \\rangle = 1 , \\langle w _ { i } , v _ { j } \\rangle = \\delta _ { i j } , \\\\ \\langle w _ 0 \\wedge w _ 1 , v _ 1 \\wedge v _ 0 \\rangle = \\langle w _ { - 1 } \\wedge w _ 0 , v _ 0 \\wedge v _ { - 1 } \\rangle = q ^ { - 2 } , \\langle w _ { - 1 } \\wedge w _ { 1 } , v _ { 1 } \\wedge v _ { - 1 } \\rangle = 1 , \\\\ \\langle w _ { - 1 } \\wedge w _ { 0 } \\wedge w _ { 1 } , v _ { 1 } \\wedge v _ { 0 } \\wedge v _ { - 1 } \\rangle = 1 . \\end{gather*}"}
-{"id": "5132.png", "formula": "\\begin{align*} q \\left \\{ q ^ { m } \\ , u ( a ^ { m } , b ) + \\sum _ { \\ell = 2 } ^ { m + 1 } g ( w , z _ { \\ell } ) Z _ { \\ell } ^ { [ 2 , m + 1 ] } ( \\vec { z } ) u ( a , b , a ^ { m } ) \\right \\} . \\end{align*}"}
-{"id": "8637.png", "formula": "\\begin{align*} f ' _ i = \\sum _ { j = 1 } ^ 3 y _ { i j } f ^ X _ j + \\sum _ { j = 1 } ^ 3 y _ { i j } f ^ { p ^ { 1 / 2 } } _ j + \\sum _ { j = 1 } ^ 3 y _ { i j } f ^ Z _ j . \\end{align*}"}
-{"id": "7784.png", "formula": "\\begin{align*} \\dim V = \\sum _ { i \\in I } v _ i \\alpha _ i . \\end{align*}"}
-{"id": "9637.png", "formula": "\\begin{align*} A _ { j i } \\ = \\ \\left ( \\begin{array} { c c } A _ { j i } ^ { ( 1 ) } & D _ { j i } \\\\ 0 & A _ { j i } ^ { ( 2 ) } \\ , . \\end{array} \\right ) \\end{align*}"}
-{"id": "6826.png", "formula": "\\begin{align*} L _ m \\cdot B _ { n + k } = \\sum _ { l = 0 } ^ { n + k - 1 } \\left [ B _ l , B _ { m + n + k - l } \\right ] + ( n + k ) B _ { m + n + k } \\ , , \\end{align*}"}
-{"id": "8678.png", "formula": "\\begin{gather*} \\nabla _ k P _ t [ \\phi ] ( x ) = \\lim _ { s \\to 0 } \\int _ H \\frac { \\phi ( e ^ { t A } x + s e ^ { t A } k + y ) - \\phi ( e ^ { t A } x + y ) } { s } \\mu _ t ( d y ) = \\int _ H \\nabla _ { e ^ { t A } k } \\ , \\phi ( e ^ { t A } x + y ) \\ , \\mu _ t ( d y ) \\\\ = \\int _ H \\langle \\nabla ^ { K } \\ , \\phi ( e ^ { t A } x + y ) , e ^ { t A } k \\rangle _ K \\ , \\mu _ t ( d y ) \\end{gather*}"}
-{"id": "6730.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( B ^ { H } _ { t } ) \\diamond d B ^ { H } ( t ) = \\int _ 0 ^ T F ( B ^ { H } _ { t } ) \\circ d B ^ { H } ( t ) - H \\int _ 0 ^ T \\Delta _ { x } F ( B ^ { H } _ { t } ) t ^ { 2 H - 1 } d t . \\end{align*}"}
-{"id": "2865.png", "formula": "\\begin{align*} \\int _ { \\underline { n } \\in \\Delta } ^ { E _ 1 - A l g ( C h _ { \\mathbb { K } } ) } K ^ n ( C ) \\otimes \\overline { N ^ * } ( \\Delta ^ n ) = \\int _ { \\underline { n } \\in \\Delta } ^ { C h _ { \\mathbb { K } } } K ^ n ( C ) \\otimes \\overline { N ^ * } ( \\Delta ^ n ) \\stackrel { \\cong } { \\rightarrow } \\overline { N ^ * } K ^ { \\bullet } ( C ) \\end{align*}"}
-{"id": "5521.png", "formula": "\\begin{align*} \\varphi ( t _ { n } ) \\int _ { E _ n } v \\le \\varphi ( t _ { n } ) \\int _ { 0 } ^ { m ( E _ n ) } w = 1 / 2 ^ n \\to 0 . \\end{align*}"}
-{"id": "2209.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } ( u _ { k } ( t ) - u _ { 0 , k } ) = - \\lambda _ { k } u _ { k } ( t ) + f _ { k } ( t ) , t > 0 . \\end{align*}"}
-{"id": "4831.png", "formula": "\\begin{align*} \\| T + { \\mathcal K } ( \\tau ; { \\mathcal I } \\| _ { { \\mathcal E } / { \\mathcal K } } = \\inf _ { K \\in { \\mathcal K } ( \\tau ; { \\mathcal I } ) } | \\| T - K \\| | \\ge \\inf _ { K \\in { \\mathcal K } ( \\tau ; { \\mathcal I } ) } \\| T - K \\| \\ge \\| p ( T ) \\| . \\end{align*}"}
-{"id": "6664.png", "formula": "\\begin{gather*} A _ { k j } ^ i : = \\{ \\sigma _ i < + \\infty \\} \\cap \\{ z _ i ( u _ k , \\sigma _ i ) - z _ i ( u _ { k - 1 } , \\sigma _ i ) = \\varepsilon , \\ldots , z _ i ( u _ { j + 1 } , \\sigma _ i ) - z _ i ( u _ j , \\sigma _ i ) = \\varepsilon , \\\\ z _ i ( u _ j , \\sigma _ i ) - z _ i ( u _ { j - 1 } , \\sigma _ i ) > \\varepsilon \\} , 2 \\leqslant j \\leqslant k - 1 , 3 \\leqslant k \\leqslant n , 1 \\leqslant i \\leqslant n - 1 , \\end{gather*}"}
-{"id": "7110.png", "formula": "\\begin{align*} \\rho _ { n , k } ( f ) \\leq \\rho _ { n , 0 } ( f ) = \\sum _ { | u | = n } f ( V ( u ) - M _ n ) . \\end{align*}"}
-{"id": "9443.png", "formula": "\\begin{align*} \\rho ^ { - 1 , \\xi } _ { s , t } = \\xi \\exp \\left ( \\int _ s ^ t h ( X ^ { s , x } _ r ) d \\tilde w _ r - \\frac 1 2 \\int _ s ^ t | h ( X ^ { s , x } _ r ) | ^ 2 \\ , d r \\right ) = \\xi \\rho ^ { - 1 } _ { s , t } . \\end{align*}"}
-{"id": "9152.png", "formula": "\\begin{align*} b & = \\xi _ 1 \\binom { p - 1 } { t - 1 } + \\sum _ { r = 2 } ^ s \\xi _ r \\binom { p - 1 } { t - 2 } \\binom { p - t + 1 } { ( r - 1 ) t + 1 } \\\\ & = \\binom { p - 1 } { t - 2 } \\left ( \\xi _ 1 \\frac { p - t + 1 } { t - 1 } + \\sum _ { r = 2 } ^ s \\xi _ r \\binom { p - t + 1 } { ( r - 1 ) t + 1 } \\right ) \\end{align*}"}
-{"id": "2248.png", "formula": "\\begin{align*} \\beta _ { 4 } = \\left ( 1 - 3 \\varepsilon \\sigma ^ { p } - \\frac { \\varepsilon ^ { 2 } 7 ! ! } { 2 } \\sigma ^ { 8 } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "1520.png", "formula": "\\begin{align*} \\phi _ V ( f _ V ) ( K ) = f _ K . \\end{align*}"}
-{"id": "2530.png", "formula": "\\begin{align*} d u _ j = \\mathbf { i } \\left [ \\frac { u _ { j + 1 } - 2 u _ j + u _ { j - 1 } } { h ^ 2 } + \\lambda | u _ j | ^ 2 u _ j \\right ] d t + \\mathbf { i } u _ j \\sum _ { k = 1 } ^ K \\sqrt { \\eta _ k } e _ k ( x _ j ) \\circ d \\beta _ k ( t ) \\end{align*}"}
-{"id": "5016.png", "formula": "\\begin{align*} \\beta _ n = \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\sigma _ k , \\textrm { f o r $ n \\geq 1 $ } , \\end{align*}"}
-{"id": "3417.png", "formula": "\\begin{align*} d y ( t ) = - g ( t , \\xi ( t ) , y ( t ) , z ( t ) ) d t - z d w ( t ) , y ( T ) = \\Gamma ^ * ( s , T ) u _ 0 ( \\xi ( T ) ) , \\end{align*}"}
-{"id": "4747.png", "formula": "\\begin{align*} u ( x , t ) = \\int _ { \\R ^ N } u _ 0 ( y ) \\ , K ( x - y , \\ , t ) \\ , d y \\mbox { f o r a l l $ ( x , t ) \\in \\mathbb R ^ N \\times ( 0 , T ) $ , } \\end{align*}"}
-{"id": "411.png", "formula": "\\begin{align*} \\big ( f _ { 2 j } ( x _ { t + m } ) , f _ { 2 j - 1 } ( x _ { t + m } ) \\big ) & = ( 0 , m - 1 ) , \\\\ \\big ( f _ { 2 j } ( x _ { t + m + 1 } ) , f _ { 2 j - 1 } ( x _ { t + m + 1 } ) \\big ) & = ( 1 , 0 ) , \\end{align*}"}
-{"id": "4566.png", "formula": "\\begin{align*} \\mu ' : = \\mathbb { E } ( Y _ i ' ) = \\begin{pmatrix} p _ { n , x , u } ^ { ( j ) } \\\\ p _ { n , y , v } ^ { ( l ) } \\end{pmatrix} \\end{align*}"}
-{"id": "3115.png", "formula": "\\begin{align*} Q ^ T L _ 5 ^ \\prime ( \\lambda ) Q = \\left [ \\begin{array} { c c c | c c } \\lambda P _ 5 + P _ 4 & P _ 3 & P _ 2 & - I _ n & 0 \\\\ P _ 3 & - \\lambda P _ 3 + P _ 2 & - \\lambda P _ 2 + P _ 1 & \\lambda I _ n & - I _ n \\\\ P _ 2 & - \\lambda P _ 2 + P _ 1 & - \\lambda P _ 1 + P _ 0 & 0 & \\lambda I _ n \\\\ \\hline - I _ n & \\lambda I _ n & 0 & 0 & 0 \\\\ 0 & - I _ n & \\lambda I _ n & 0 & 0 \\end{array} \\right ] , \\end{align*}"}
-{"id": "7357.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 1 ) } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\end{array} \\right ) , \\Gamma _ { 0 } ^ { ( 1 ) } = \\left ( \\begin{array} { c c c } 0 & 1 & 0 \\end{array} \\right ) , \\Gamma _ { - } ^ { ( 1 ) } = \\left ( \\begin{array} { c c c } 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "9105.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ n } x _ { \\sigma ( 1 ) } ^ { d _ 1 } \\ldots x _ { \\sigma ( j ) } ^ { d _ j } = \\sum _ { \\pi \\in S _ n } \\big ( x _ { \\pi ( 1 ) } \\ldots x _ { \\pi ( j ) } \\big ) ^ { d _ j } \\cdot \\sum _ { \\rho \\in S _ n } x _ { \\rho ( 1 ) } ^ { d _ 1 - d _ j } \\ldots x _ { \\rho ( j - 1 ) } ^ { d _ { j - 1 } - d _ j } . \\end{align*}"}
-{"id": "3242.png", "formula": "\\begin{align*} P | _ { \\tilde { B } _ { m _ { l } } \\left ( \\rho \\right ) } = \\mu _ { _ { m _ { l } } } ^ { - 1 } \\circ \\tilde { p } _ { m _ { l } } , \\end{align*}"}
-{"id": "9214.png", "formula": "\\begin{align*} \\sup _ { u \\in \\mathcal { A } } j ( u ) ( z ) = j ( u ^ { \\star } ) ( z ) . \\end{align*}"}
-{"id": "8646.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\frac { d ^ { 2 } y } { d \\tau ^ { 2 } } ( \\tau ) = - \\Lambda y ( \\tau ) + \\dot { W } ( \\tau ) , \\\\ y \\left ( 0 \\right ) = x _ { 0 } , \\ ; \\ ; \\ ; \\frac { d y } { d \\tau } ( 0 ) = x _ { 1 } . \\end{array} \\right . \\end{align*}"}
-{"id": "4556.png", "formula": "\\begin{align*} \\max _ { v \\in \\{ v _ x , v _ y \\} } | v f ( x ) - 1 - 3 k ^ { - 1 / 2 } \\log ^ { 1 / 2 } n | \\lesssim a ( f ( x ) \\wedge f ( y ) ) \\Bigl ( \\frac { k } { n f ( x ) } \\Bigr ) ^ { \\beta / d } = o ( k ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "9472.png", "formula": "\\begin{align*} F ( s , t ) = ( 1 - \\eta ( s ) ) s + \\eta ( s ) G ( s , t ) . \\end{align*}"}
-{"id": "5938.png", "formula": "\\begin{align*} \\lambda { \\psi } ( z ) - \\frac { 1 } { 2 } \\triangle _ v { \\psi } ( z ) - v \\cdot D _ x { \\psi } ( z ) - F ( z ) \\cdot D _ v { \\psi } ( z ) = g ( z ) , \\ ; \\ ; \\ ; z = ( x , v ) \\in \\R ^ { 2 d } , \\end{align*}"}
-{"id": "7562.png", "formula": "\\begin{align*} v _ l : = \\inf \\{ v > b : F ( v ) > - l \\} , \\end{align*}"}
-{"id": "3320.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 2 ) } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ q ^ { - 2 } & 0 & 0 \\\\ 0 & 1 & 0 \\end{array} \\right ) , \\Gamma _ { 0 } ^ { ( 2 ) } = \\left ( \\begin{array} { c c c } - 1 & 0 & 0 \\\\ 0 & - q ( q - q ^ { - 1 } ) & 0 \\\\ 0 & 0 & q ^ { - 2 } \\end{array} \\right ) , \\Gamma _ { - } ^ { ( 2 ) } = \\left ( \\begin{array} { c c c } 0 & - 1 & 0 \\\\ 0 & 0 & - 1 \\\\ 0 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "4102.png", "formula": "\\begin{align*} C \\ : : \\ : x ^ p y ^ q ( a x + b y + c z ) ^ r - z ^ { p + q + r } = 0 \\end{align*}"}
-{"id": "2202.png", "formula": "\\begin{align*} - \\int _ { B } \\psi ^ { 2 } \\tilde { u } ^ { - 1 } \\partial _ { t } ( g _ { 1 - \\alpha , m } * \\tilde { u } ) d x + \\frac { c _ { 2 } } { r ^ { 2 \\beta } } \\int _ { B } ( w - W ) ^ { 2 } \\psi ^ { 2 } d x \\leq \\frac { C _ { 1 } \\mu _ { n } ( B ) } { r ^ { 2 \\beta } } + R _ { m } ( t ) , \\end{align*}"}
-{"id": "5683.png", "formula": "\\begin{align*} & K ( x ) = 1 + k ( x ) \\ , \\Big ( b _ 0 + x \\ , b _ 0 ^ + + \\frac 1 x \\ , b _ 0 ^ - \\Big ) , \\\\ & \\mbox { w i t h } k ( x ) = \\frac { \\left ( x ^ 2 - 1 \\right ) \\left ( \\alpha + \\gamma \\right ) } { \\left ( \\gamma x + \\alpha \\right ) \\left ( ( \\alpha + \\gamma ) ( x - 1 ) + ( q - 1 ) x \\right ) } \\ , \\end{align*}"}
-{"id": "7746.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\left [ \\frac { \\sum _ { i = 0 } ^ n \\sigma _ i \\xi _ { i + 1 } } { \\sqrt { \\sum _ { k = 0 } ^ { n } \\sigma ^ 2 _ k } } - \\frac { \\sum _ { i = 0 } ^ n \\sigma _ i \\mu _ { i + 1 } } { \\sqrt { \\sum _ { k = 0 } ^ { n } \\sigma ^ 2 _ k } } \\right ] = \\infty . \\end{align*}"}
-{"id": "2922.png", "formula": "\\begin{align*} L _ \\pi ( Z ) = L _ \\pi ( 1 ; Z ) = \\prod _ { T \\in { \\cal T } ^ \\pi } ( 1 - Z ^ T ) = \\prod _ { i _ 1 , i _ 2 , \\ldots , i _ m \\geq 0 } \\ ( 1 - z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\cdots z _ m ^ { i _ m } ) ^ { c ^ \\pi _ { ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) } } \\ , . \\end{align*}"}
-{"id": "450.png", "formula": "\\begin{align*} y & = ( a ^ k b ^ k ) ^ { \\ell / 2 } a ^ { k - 1 } , \\\\ x & = ( a ^ k b ^ k ) ^ { \\ell / 2 } a ^ { k - 1 } b ( a ^ k b ^ k ) ^ { \\ell / 2 } a ^ { k - 1 } . \\end{align*}"}
-{"id": "8259.png", "formula": "\\begin{align*} v ^ n _ t = \\sum _ k n ^ { 1 / 2 } ( u _ { n ^ 2 t } ( k ) - \\rho ' ( \\lambda ) ) n ^ { - 1 } \\delta _ { n ^ { - 1 } k + c _ n t } . \\end{align*}"}
-{"id": "4098.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } y ^ { q } ( a x + b y + c z ) ^ r } { z ^ { p + q + r } } , \\end{align*}"}
-{"id": "4755.png", "formula": "\\begin{align*} \\mathcal { M } ' : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "2803.png", "formula": "\\begin{gather*} m ^ 0 ( A ) = \\int _ A D _ \\nu m ^ 0 \\ d \\nu \\end{gather*}"}
-{"id": "6026.png", "formula": "\\begin{gather*} I = - \\phi , J = \\pm \\phi , K = 0 . \\end{gather*}"}
-{"id": "807.png", "formula": "\\begin{align*} \\check { R } ( z / w ) [ \\mathbb { T } ^ { [ 1 , M ] } ( z ) \\otimes \\mathbb { T } ^ { [ 1 , M ] } ( w ) ] \\check { R } ( z / w ) ^ { - 1 } = \\mathbb { T } ^ { [ 1 , M ] } ( w ) \\otimes \\mathbb { T } ^ { [ 1 , M ] } ( z ) . \\end{align*}"}
-{"id": "1595.png", "formula": "\\begin{align*} a \\left ( t , x \\right ) = - \\frac { 1 } { 2 } \\int \\left ( T _ { I } L _ { Y _ { I } } C _ { x } - T _ { I , t } Y _ { 1 } \\right ) d x + f _ { I } \\left ( t \\right ) . \\end{align*}"}
-{"id": "5407.png", "formula": "\\begin{align*} ( 1 - \\delta ) \\binom { N } { 2 } = ( 1 - \\delta ) ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } \\leq k ( k - \\alpha - x ) \\frac { n ^ 2 } { 2 } \\end{align*}"}
-{"id": "2469.png", "formula": "\\begin{align*} V = \\C ^ n = \\C e _ 0 + \\cdots + \\C e _ { n - 1 } \\end{align*}"}
-{"id": "1221.png", "formula": "\\begin{align*} \\| x \\| _ \\mathcal { M } ^ 0 = \\| x \\| ^ 0 _ { \\mathcal { M } _ { \\varphi , w } } = \\inf _ { k > 0 } \\frac { 1 } { k } ( P ( k x ) + 1 ) < \\infty , \\end{align*}"}
-{"id": "5635.png", "formula": "\\begin{align*} J _ { \\P _ n } ^ { \\gamma } ( \\mu ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ 2 ^ 2 ( \\mu , \\nu _ i ) + \\gamma E ( \\mu ) . \\end{align*}"}
-{"id": "9329.png", "formula": "\\begin{align*} M ( t , z ) : = \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] \\end{align*}"}
-{"id": "1241.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\varphi ( | x ( k ) | ) v ( k ) \\chi _ { E _ j } ( k ) \\le \\sum _ { k = j + 1 } ^ \\infty \\varphi ( | x ( k ) | ) v ( k ) \\to 0 \\ \\ \\ \\ \\ \\ j \\to \\infty . \\end{align*}"}
-{"id": "8714.png", "formula": "\\begin{align*} L ( t , x ) = \\lim _ { n \\rightarrow \\infty } \\nabla ^ G v ^ n ( t , x ) . \\end{align*}"}
-{"id": "299.png", "formula": "\\begin{align*} \\Sigma ' : = \\begin{pmatrix} 1 & ( j / l ) ^ { 1 / 2 } \\alpha _ z ' \\\\ ( j / l ) ^ { 1 / 2 } \\alpha _ z ' & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "608.png", "formula": "\\begin{align*} \\Psi _ { \\kappa , n , N } ^ { \\operatorname { c } } ( z , \\mathfrak { z } ) = \\Psi _ { \\kappa , n , N } ( - z , \\overline { \\mathfrak { z } } ) . \\end{align*}"}
-{"id": "8310.png", "formula": "\\begin{gather*} f ( \\alpha + \\beta ) = f ( \\alpha ) + f ( \\beta ) = 0 + 0 = 0 , \\\\ f ( 0 ) = 0 , f ( - \\alpha ) = f ( ( p - 1 ) \\alpha ) = \\underbrace { f ( \\alpha ) + \\cdots + f ( \\alpha ) } _ { p - 1 } = 0 + \\cdots + 0 = 0 . \\end{gather*}"}
-{"id": "8532.png", "formula": "\\begin{align*} X _ { a ^ { \\ast } } ( P ) a ' = \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } Y _ { [ b r a ] } ( P ) b r a ' \\end{align*}"}
-{"id": "6425.png", "formula": "\\begin{align*} \\eta \\in H _ { e } ^ { 1 , \\beta } ( Q _ { T } ) : = W ^ { 1 , 2 } ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) \\cap L ^ { 2 } ( [ 0 , T ] ; H _ { e } ^ { \\beta } ( \\Omega ) ) \\cap L ^ { \\infty } ( [ 0 , T ] ; L ^ { \\infty } ( \\mathbb { R } ^ { n } ) ) \\end{align*}"}
-{"id": "9977.png", "formula": "\\begin{align*} \\max _ { \\alpha \\ ! , \\tilde { p } \\ ! , p _ 1 \\ ! , p _ 2 \\ ! } \\quad & \\alpha \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\tilde { p } | H _ { \\tilde { i } k } | ^ 2 } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\ ! + \\ ! ( \\ ! 1 \\ ! - \\ ! \\alpha \\ ! ) \\sum _ { i = 1 } ^ { 2 } \\log _ 2 \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { p _ { i } } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\end{align*}"}
-{"id": "4272.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow + \\infty } ( \\mathbb E [ Z _ n ^ * ] ) ^ { - 1 } \\mathbb E \\left [ ( Z _ { n + \\lfloor \\varepsilon n \\rfloor } ^ * - Z _ { n + \\lfloor \\varepsilon n \\rfloor } ) \\mathbf 1 _ { \\{ \\sup _ { k = 1 , \\ldots , \\lfloor \\varepsilon n \\rfloor } ( Z _ { k } - Z _ { k + 1 } ) > b _ { d n } \\} } \\right ] < \\rho - 1 , \\end{align*}"}
-{"id": "9465.png", "formula": "\\begin{align*} X _ { f } = j \\nabla f \\qquad \\nabla f = - j X _ { f } \\end{align*}"}
-{"id": "9764.png", "formula": "\\begin{align*} \\mathcal { X } = \\bigcup _ { \\mathcal { E } \\in \\mathbb { A } } \\mathrm { c l } ( \\mathcal { R } ( \\mathcal { E } ) ) , \\end{align*}"}
-{"id": "2850.png", "formula": "\\begin{align*} \\phi ( \\partial ( \\delta ) ) = d _ V \\circ \\delta _ V + \\delta _ V \\circ d _ V \\end{align*}"}
-{"id": "2128.png", "formula": "\\begin{align*} v : = u - U _ a \\sum _ { m = 0 } ^ l P _ m ( x , r ) = U _ a \\big ( P _ { l + 1 } ( x , r ) + o ( | X | ^ { l + 1 } ) \\big ) \\end{align*}"}
-{"id": "6800.png", "formula": "\\begin{align*} L _ m \\cdot B _ n & = \\ \\sum _ { k = 0 } ^ { n - 1 } \\left [ B _ k , B _ { m + n - k } \\right ] + n B _ { m + n } \\\\ L _ m \\cdot B _ { - n } & = - \\sum _ { k = - n } ^ { - 1 } \\left [ B _ k , B _ { m - n - k } \\right ] - n B _ { m - n } \\end{align*}"}
-{"id": "7413.png", "formula": "\\begin{align*} \\vec { i } : = ( i ( 1 ) , i ( 2 ) , \\dots , i ( l ) ) \\end{align*}"}
-{"id": "2911.png", "formula": "\\begin{align*} M _ { \\pi / \\kappa } ( z ) = M ^ \\perp _ { \\pi / \\kappa } ( z ) = 1 \\quad \\mbox { a n d } L _ { \\pi / \\kappa } ( z ) = L ^ \\perp _ { \\pi / \\kappa } ( z ) = 1 \\quad \\mbox { f o r a l l $ \\kappa \\not \\subseteq \\pi $ \\ , . } \\end{align*}"}
-{"id": "1385.png", "formula": "\\begin{align*} \\hat { q } _ { n , t } = \\frac { 1 } { \\sqrt { n } } \\dot { l } _ { t } ( \\hat \\delta ) \\end{align*}"}
-{"id": "8235.png", "formula": "\\begin{align*} \\Delta = \\Delta _ 0 + \\Delta _ 2 ( \\tilde { \\phi } ) + \\Delta _ 3 ( \\tilde { \\phi } ) + \\Delta _ 4 ( \\tilde { \\phi } ) , \\end{align*}"}
-{"id": "4688.png", "formula": "\\begin{align*} W ^ { 1 , p } _ H ( M ) = \\{ f \\in L ^ p ( M ) : X f \\in L ^ p ( M ) \\ ; \\forall X \\in C ^ \\infty ( M , H ) \\} . \\end{align*}"}
-{"id": "6953.png", "formula": "\\begin{align*} \\chi _ { \\mathfrak { g } } ( \\lambda ) & = \\det \\left [ \\begin{array} { c } h ^ { ( j - 1 ) } _ { \\lambda _ i - i + 1 } \\end{array} \\right ] \\ , , h ^ { ( r ) } _ a = \\left \\{ \\begin{array} { l l } J _ { a + r } + J _ { a - r } & r > 0 \\\\ J _ { a + r } & r \\leq 0 \\end{array} \\ , , \\right . \\end{align*}"}
-{"id": "5082.png", "formula": "\\begin{align*} f ( z , w ) = \\frac { z - q ^ 2 w } { z - w } , g ( z , w ) = - \\frac { ( 1 - q ^ { 2 } ) z } { z - w } = f ( w , z ) - 1 . \\end{align*}"}
-{"id": "5846.png", "formula": "\\begin{align*} T _ { 1 } L _ { Y _ { 1 } } C _ { x } + T _ { 2 } L _ { Y _ { 2 } } C _ { x } - T _ { 1 , t } Y _ { 1 } - T _ { 2 , t } Y _ { 2 } - 2 a _ { , x } = 0 \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "4677.png", "formula": "\\begin{align*} [ P , a ^ 1 ] \\cdots [ P , a ^ { 2 k } ] = & ( - 1 ) ^ k P a ^ 1 ( 1 - P ) a ^ 2 P \\cdots ( 1 - P ) a ^ { 2 k } P \\\\ & \\quad + ( - 1 ) ^ k ( 1 - P ) a ^ 1 P a ^ 2 ( 1 - P ) \\cdots P a ^ { 2 k } ( 1 - P ) \\\\ = & ( - 1 ) ^ k P a _ + ^ 1 ( 1 - P ) a _ - ^ 2 P \\cdots ( 1 - P ) a _ - ^ { 2 k } P \\\\ & \\quad + ( - 1 ) ^ k ( 1 - P ) a _ - ^ 1 P a _ + ^ 2 ( 1 - P ) \\cdots P a ^ { 2 k } _ + ( 1 - P ) . \\end{align*}"}
-{"id": "3691.png", "formula": "\\begin{align*} { \\boldsymbol A } = \\left ( \\begin{array} { c c c } - J _ q & \\mathcal { A } _ q & 0 \\\\ - \\mathcal { A } _ q & - J _ q & - \\nabla ^ 2 \\\\ 0 & \\nabla ^ 2 & 0 \\end{array} \\right ) ~ , \\end{align*}"}
-{"id": "785.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } : \\mathcal { F } ( L _ { k } ^ { + } , ( U ^ { \\otimes k } ) _ { k _ { 1 } , \\ldots , k _ { r } } ) \\to F ( \\mathcal { S } _ { k _ { 1 } , \\ldots , k _ { r } } ) , ( \\varphi ^ { - 1 } \\gamma ) ( \\vec { x } , \\vec { \\nu } ) = \\gamma _ { \\vec { \\nu } } ( \\vec { x } ) . \\end{align*}"}
-{"id": "3769.png", "formula": "\\begin{align*} I = I \\left ( \\beta , t \\right ) = \\int _ 0 ^ 1 e ^ { ( \\beta - i \\alpha t ) x } g ( x ) d x , \\end{align*}"}
-{"id": "3744.png", "formula": "\\begin{align*} d \\nu = g ( x ) d x \\end{align*}"}
-{"id": "2477.png", "formula": "\\begin{align*} \\det ( A \\oplus B ) = ( \\det A ) ( \\det B ) \\end{align*}"}
-{"id": "7589.png", "formula": "\\begin{align*} | l - F ( x _ 0 ) | = | F ( u _ l ) - F ( x _ 0 ) | \\le \\frac { l \\varepsilon } { 2 C } , F ( x _ 0 ) \\ge l - \\frac { l \\varepsilon } { 2 C } , \\end{align*}"}
-{"id": "8549.png", "formula": "\\begin{align*} \\overline { N } _ { k } & = \\frac { k e r ( \\gamma ) } { i m ( \\beta ) } \\oplus i m ( \\gamma ) \\oplus \\frac { k e r ( \\alpha ) } { i m ( \\gamma ) } \\oplus V _ { k } \\\\ \\overline { V } _ { k } & = \\frac { k e r ( \\beta ) } { k e r ( \\beta ) \\cap i m ( \\alpha ) } \\end{align*}"}
-{"id": "4910.png", "formula": "\\begin{align*} \\left ( 2 i \\mathfrak { z } _ 2 \\right ) ^ { \\kappa - 1 } \\Psi _ { \\kappa , - 1 , N } ( z , \\mathfrak { z } ) = 2 \\pi i H _ { \\kappa , N } ( z , \\mathfrak { z } ) - 2 \\pi i K _ { \\kappa , N } ( z , \\mathfrak { z } ) . \\end{align*}"}
-{"id": "9229.png", "formula": "\\begin{align*} I _ 1 = \\mathbb { E } [ \\int _ D \\int _ 0 ^ T \\{ \\frac { \\partial h } { \\partial y } ( t , x , z ) \\chi ( t , x , z ) + \\frac { \\partial h } { \\partial u } ( t , x , z ) \\beta ( t , x , z ) \\} \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] d t d x ] , \\end{align*}"}
-{"id": "1418.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in [ \\beta ] } ( c _ { \\gamma , u } ^ { \\rho } c _ { \\rho , v } ^ { \\alpha } - ( - 1 ) ^ { p ( u ) p ( v ) } c _ { \\gamma , v } ^ { \\rho } c _ { \\rho , u } ^ { \\alpha } ) = c _ { \\gamma , [ u , v ] } ^ { \\alpha } \\end{align*}"}
-{"id": "509.png", "formula": "\\begin{align*} [ \\tilde { r } , \\delta ] \\ = \\ \\tilde { r } \\circ \\delta - \\delta \\circ \\tilde { r } \\end{align*}"}
-{"id": "2680.png", "formula": "\\begin{align*} { \\rm I } ( \\varphi _ { t , j } ) - { \\rm I } ( \\varphi _ { 0 , j } ) = \\int _ { 0 } ^ t \\int _ X \\chi \\theta _ { \\varphi _ { s , j } } ^ n d s . \\end{align*}"}
-{"id": "8449.png", "formula": "\\begin{align*} V ^ { s } : = \\{ ( \\tilde { \\rho } , \\tilde { v } ) \\in H ^ { s } ( \\mathbb { R } ^ { d } ) : \\nabla \\cdot \\tilde { v } = 0 \\} , \\end{align*}"}
-{"id": "2000.png", "formula": "\\begin{align*} V _ t ^ n = V _ { t \\wedge T ^ n } \\end{align*}"}
-{"id": "3952.png", "formula": "\\begin{align*} \\begin{array} { c c c } \\mu ( \\pi ( N ( L , W ) ) ) > 0 & & \\mu ( \\pi ( S ( L , W ) ) ) = 0 . \\end{array} \\end{align*}"}
-{"id": "2484.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ m \\det ( - \\rho _ j ( \\Phi ) | V _ j ^ I ) ^ { d _ j - 1 } \\cdot q ^ { - \\frac { 1 } { 2 } ( d _ j - 2 ) ( d _ j - 1 ) \\dim V _ j ^ I } \\end{align*}"}
-{"id": "7591.png", "formula": "\\begin{align*} p _ 1 = p _ 1 ( x _ 0 ) = \\mathbb P \\left \\{ \\omega \\in \\Omega : \\chi ( \\omega ) > 1 - \\frac { l + F ( x _ 0 ) } { 2 l } \\right \\} , K _ 1 = K _ 1 ( x _ 0 ) : = \\left [ \\frac { 2 ( u _ l - x _ 0 ) } { l + F ( x _ 0 ) } \\right ] + 1 ; \\end{align*}"}
-{"id": "4754.png", "formula": "\\begin{align*} P _ { E D } ^ { L M P > 0 } : = & \\sum _ { t \\in \\mathcal { T } } \\sum _ { n \\in \\mathcal { N } _ t ^ + } \\left [ \\sum _ { i \\in \\mathcal { G } _ n } p _ { i t } + \\sum _ { i \\in \\mathcal { I } _ n } \\left ( M _ { i t } - m _ { i t } \\right ) \\right . \\\\ & \\left . + \\sum _ { i \\in \\mathcal { W } _ n } \\left ( W _ { i t } - w _ { i t } \\right ) + \\sum _ { i \\in \\mathcal { R } _ n } \\left ( R _ { i t } - r _ { i t } \\right ) \\right ] \\end{align*}"}
-{"id": "8478.png", "formula": "\\begin{align*} \\nabla \\cdot \\tilde { v } ^ { \\star } = 0 . \\end{align*}"}
-{"id": "5936.png", "formula": "\\begin{align*} D _ x ( \\partial _ { v _ j } \\psi ) = \\partial _ { v _ j } ( D _ x { \\psi } ) \\in L ^ p ( \\R ^ { 2 d } ) . \\end{align*}"}
-{"id": "9111.png", "formula": "\\begin{align*} a ' c ' + a '' c '' + b ' c '' + b '' c ' = ( a ' + b '' ) c ' + ( a '' + b ' ) c '' = \\end{align*}"}
-{"id": "9828.png", "formula": "\\begin{align*} \\mbox { $ ( \\frac { a } { 2 } \\lambda - \\mu + \\frac { b } { 2 } ) h _ { 0 , m } ( \\lambda + \\mu ) = - \\mu h _ { 0 , m } ( \\mu ) . $ } \\end{align*}"}
-{"id": "9208.png", "formula": "\\begin{align*} & \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] \\\\ = & \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } \\exp \\big [ \\int _ 0 ^ t \\int _ { \\mathbb { R } } i x \\psi ( s , \\zeta ) \\tilde { N } ( d s , d \\zeta ) + \\int _ 0 ^ t i x \\beta ( s ) d B ( s ) \\\\ & + \\int _ t ^ { T _ 0 } \\int _ { \\mathbb { R } } ( e ^ { i x \\psi ( s , \\zeta ) } - 1 - i x \\psi ( s , \\zeta ) ) \\nu ( d \\zeta ) d s - \\int _ t ^ { T _ 0 } \\frac { 1 } { 2 } x ^ 2 \\beta ^ 2 ( s ) d s - i x z \\big ] d x . \\end{align*}"}
-{"id": "1522.png", "formula": "\\begin{align*} \\alpha = \\sup \\left \\{ \\beta \\leq 2 \\ , ; \\ , \\int _ { \\R ^ d } | v | ^ \\beta M ( v ) \\ , d v < \\infty \\right \\} . \\end{align*}"}
-{"id": "6537.png", "formula": "\\begin{align*} \\big \\| \\big ( \\frac 1 \\tau \\sum _ { j = 0 } ^ k \\delta _ j v _ { n - j } \\big ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\le C \\big ( \\| ( f _ n ) _ { n = k } ^ N \\| _ { L ^ p ( D ) } + \\| ( v _ n ) _ { n = k } ^ N \\| _ { L ^ p ( D ) } \\big ) . \\end{align*}"}
-{"id": "2212.png", "formula": "\\begin{align*} u ( x , t ) = \\sum _ { k = 1 } ^ { \\infty } E _ { \\alpha , 1 } ( - \\lambda _ { k } t ^ { \\alpha } ) ( u _ { 0 } , \\phi _ { k } ) \\phi _ { k } ( x ) \\end{align*}"}
-{"id": "3561.png", "formula": "\\begin{align*} \\mathcal { E } ^ R ( \\theta ) : = \\Phi ^ { V ^ R } _ { ( \\gamma ^ R , \\tau ^ R ) } ( \\bar { g } ^ R , \\bar { \\pi } ^ R ) - \\Phi ^ { V ^ R } _ { ( \\gamma ^ R , \\tau ^ R ) } ( \\gamma ^ R , \\tau ^ R ) - ( 2 \\psi ^ R , V ^ R ) - ( \\lambda \\zeta R ^ { - 2 } ( \\log R ) ^ \\frac { 1 } { 2 } , 0 ) = 0 . \\end{align*}"}
-{"id": "2390.png", "formula": "\\begin{align*} F _ { \\rho } ^ { \\rm { A D M M } } ( v ) & = - F _ { \\gamma } ^ { \\rm { D R } } ( z ) . \\end{align*}"}
-{"id": "1543.png", "formula": "\\begin{align*} T _ h ( t ) v ( x ) = \\exp { \\left ( \\int _ { - t } ^ 0 h ( x e ^ s ) d s \\right ) } v ( x e ^ { - t } ) , t \\geq 0 , \\ x \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "2253.png", "formula": "\\begin{align*} C _ { 2 } ( \\vec { \\varepsilon } ) = \\left ( 1 - ( p - 1 ) ! ! \\varepsilon _ { p } \\sigma ^ { p } + \\frac { \\varepsilon _ { q } ^ { 2 } } { 2 } ( 2 q - 1 ) ! ! \\sigma ^ { 2 q } + \\frac { \\varepsilon _ { p } ^ { 2 } } { 2 } ( 2 p - 1 ) ! ! \\sigma ^ { 2 p } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "772.png", "formula": "\\begin{gather*} C _ { \\ell } = \\bigoplus _ { 0 \\leqslant k \\leqslant \\ell , \\ , [ k ] = [ \\ell ] } \\partial C _ { k } . \\end{gather*}"}
-{"id": "8935.png", "formula": "\\begin{align*} \\int _ { \\mathbb H ^ n } \\Phi ( | f | ) d V _ g = \\int _ { \\mathbb H ^ n } \\Phi ( f ^ \\sharp ) d V _ g . \\end{align*}"}
-{"id": "632.png", "formula": "\\begin{align*} \\| \\Psi _ \\lambda & \\ast ( Q f ) \\| _ s \\leq c _ 1 \\| \\ , | \\cdot | ^ { 2 - N } \\| _ { \\frac { N } { N - 2 } , w } \\| Q f \\| _ { t _ 1 } + c _ 2 \\| \\ , | \\cdot | ^ { \\frac { 1 - N } { 2 } } \\| _ { \\frac { 2 N } { N - 1 } , w } \\| Q f \\| _ { t _ 2 } \\\\ & \\leq \\Bigl \\{ c _ 1 \\| \\ , | \\cdot | ^ { 2 - N } \\| _ { \\frac { N } { N - 2 } , w } | \\Omega | ^ { \\frac 1 { t _ 1 } } + c _ 2 \\| \\ , | \\cdot | ^ { \\frac { 1 - N } { 2 } } \\| _ { \\frac { 2 N } { N - 1 } , w } | \\Omega | ^ { \\frac 1 { t _ 2 } } \\Bigr \\} \\| Q \\| _ \\infty \\| f \\| _ \\infty = : C ( \\lambda ) \\| f \\| _ \\infty , \\end{align*}"}
-{"id": "6461.png", "formula": "\\begin{align*} = \\phi \\int _ { \\rho ' B _ { 1 } } \\int _ { \\rho ' B _ { 1 } } \\left [ w ( s , x ) - w ( s , y ) \\right ] ^ { 2 } k ( x , y ) d x d y . \\end{align*}"}
-{"id": "8570.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } \\left ( Y _ { [ b w a ] } ( \\varphi ( P ) ) \\right ) n & = \\varphi ^ { - 1 } \\rho ^ { - 1 } \\left ( X _ { [ b w a ] ^ { \\ast } } ( \\rho \\varphi ( P ) ) \\right ) n \\\\ & = \\left ( \\rho ^ { - 1 } \\hat { \\varphi } ^ { - 1 } X _ { [ b w a ] ^ { \\ast } } ( \\hat { \\varphi } ( \\rho ( P ) ) ) \\right ) n \\end{align*}"}
-{"id": "1680.png", "formula": "\\begin{align*} X _ t & = x _ 0 + \\int _ 0 ^ t V _ s \\ , \\dd s = x _ 0 + t v _ 0 + \\int _ 0 ^ t ( t - s ) F ( X _ s , V _ s ) \\ , \\dd s + \\int _ 0 ^ t W _ s \\ , \\dd s \\ , , \\\\ V _ t & = v _ 0 + \\int _ 0 ^ t F ( X _ s , V _ s ) \\ , \\dd s + W _ t \\ , . \\end{align*}"}
-{"id": "5292.png", "formula": "\\begin{align*} \\vert \\int _ { T } ^ { 2 T } Z ( t ) d t \\vert = O ( T ^ { 3 / 4 } ) . \\end{align*}"}
-{"id": "2696.png", "formula": "\\begin{align*} \\theta _ { u _ { \\beta } } ^ n \\geq \\frac { 1 } { \\beta ^ n } \\theta _ { \\phi } ^ n = \\frac { 1 } { \\beta ^ n } e ^ { \\phi } \\theta _ { + } ^ n = e ^ { \\phi - n \\log \\beta } \\theta _ { + } ^ n \\geq e ^ { \\beta u _ { \\beta } } \\theta _ + ^ n . \\end{align*}"}
-{"id": "3591.png", "formula": "\\begin{align*} 0 & = \\left . \\frac { d } { d t } \\right | _ { t = 0 } \\mathcal { G } \\left ( ( f , X ) + t ( u , Y ) \\right ) \\\\ & = \\int _ { \\Omega } \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) \\cdot _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( u , Y ) - \\Pi _ g ( \\psi , V ) \\cdot _ g ( u , Y ) \\ , d \\mu _ g \\\\ & = \\int _ { \\Omega } \\left ( D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) - \\Pi _ g ( \\psi , V ) \\right ) \\cdot _ g ( u , Y ) \\ , d \\mu _ g \\ ; , \\end{align*}"}
-{"id": "6494.png", "formula": "\\begin{align*} \\tilde { u } \\in L ^ { \\infty } ( ( 0 , T ] ; H _ { e } ^ { 2 \\beta } ( \\Omega ) ) \\subset L ^ { 2 } ( ( 0 , T ] ; H _ { e } ^ { 2 \\beta } ( \\Omega ) ) , \\lim _ { t \\rightarrow 0 } \\| \\tilde { u } ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\Omega ) } = 0 . \\end{align*}"}
-{"id": "1255.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { c c } \\textrm { s i g n } ( \\lambda _ 1 ) & 0 \\\\ 0 & \\textrm { s i g n } ( \\lambda _ 2 ) \\end{array} \\right ) , \\ , \\ , \\ , \\ , \\ , v = \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) : = \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) B . \\end{align*}"}
-{"id": "235.png", "formula": "\\begin{align*} | A | = \\prod _ { 1 \\leq t _ 1 < t _ 2 \\leq d ' } d ^ { - 2 / d } ( t _ 2 ^ { 2 / d } - t _ 1 ^ { 2 / d } ) > 0 . \\end{align*}"}
-{"id": "7754.png", "formula": "\\begin{align*} f _ { - i } ( x ) = f _ { i } ( - x ) = f _ i ( x ) , \\forall x \\in \\mathbb R . \\end{align*}"}
-{"id": "6094.png", "formula": "\\begin{align*} a ( \\tilde U ) : = \\binom { | \\tilde U | } { 2 } ^ { - 1 } \\sum _ { \\{ u , u ' \\} \\in \\binom { \\tilde U } { 2 } } \\deg ( u , u ' ) = \\binom { | \\tilde U | } { 2 } ^ { - 1 } \\sum _ { v \\in V } \\binom { \\deg ( v , \\tilde U ) } { 2 } . \\end{align*}"}
-{"id": "5320.png", "formula": "\\begin{align*} \\lambda _ { \\{ l ; k \\} } ( u ) & = \\left ( u + \\omega + \\eta \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) \\left ( u - \\omega + \\eta \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) \\prod _ { j = 1 } ^ { N - r } \\frac { u - v _ j + \\eta } { u - v _ j } \\\\ & + \\eta ^ { - 2 } \\prod _ { j = 1 } ^ { N - r } \\frac { u - v _ j - \\eta } { u - v _ j } . \\end{align*}"}
-{"id": "3766.png", "formula": "\\begin{align*} = \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - y ^ 2 / 2 } + O \\left ( \\frac { 1 } { \\sqrt { \\log u } } \\frac { 1 } { u ^ { r / 2 } } \\right ) ; \\end{align*}"}
-{"id": "8230.png", "formula": "\\begin{align*} ( \\sigma \\Phi ) _ n : = ( U _ n , - \\bar { U } _ n , V _ n , - \\bar { V } _ n ) , n \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "5551.png", "formula": "\\begin{align*} & b _ { 0 , x x } = - 2 \\int \\rho _ { x } b _ { - 1 } d x - 4 \\int \\rho b _ { - 1 , x } d x + C _ 1 = - 2 \\rho b _ { - 1 } - 2 \\int \\rho b _ { - 1 , x } d x + C _ { 1 } , \\\\ & b _ { 0 , x } = - 2 \\int \\biggl ( \\rho b _ { - 1 } + \\int \\rho b _ { - 1 , x } d x \\biggr ) d x + C _ { 1 } x + C _ { 2 } : = f ( x ) + C _ { 1 } x + C _ { 2 } , \\\\ & b _ { 0 } = \\int f ( x ) d x + \\frac { 1 } { 2 } C _ { 1 } x ^ { 2 } + C _ { 2 } x + C _ { 3 } . \\end{align*}"}
-{"id": "9007.png", "formula": "\\begin{align*} y ( t ) = c _ 1 + \\frac { \\alpha t ^ { \\alpha - 1 } } { B ( \\alpha ) \\Gamma ( \\alpha ) } , \\end{align*}"}
-{"id": "2584.png", "formula": "\\begin{align*} e ^ { - i t _ n \\Delta } ( { u } _ n ^ J ( t _ n ) - u _ { n } ( t _ n ) ) & = \\sum _ { j = 1 } ^ J e ^ { - i t _ n \\Delta } v _ n ^ j ( t _ n ) - e ^ { i x \\xi _ n ^ j } \\psi ^ j _ { \\{ h _ n ^ j \\} } \\\\ & = \\sum _ { j = 1 } ^ J e ^ { i x \\xi _ n ^ j } \\bigl [ ( e ^ { - i ( h _ n ^ j ) ^ 2 t _ n \\Delta } \\Psi ^ j ( ( h _ n ^ j ) ^ 2 t _ n ) ) _ { \\{ h _ n ^ j \\} } - \\psi ^ j _ { \\{ h _ n ^ j \\} } \\bigr ] , \\end{align*}"}
-{"id": "9766.png", "formula": "\\begin{align*} \\nabla _ { \\vec { x } } V ( \\vec { x } ) = \\left [ \\dfrac { \\partial f } { \\partial \\vec { x } } ( \\vec { z } ^ { * } ( \\vec { x } ) , \\vec { x } ) \\right ] ^ { \\top } + \\sum \\limits _ { i \\in \\mathcal { E } } \\left [ \\dfrac { \\partial g _ { i } } { \\partial \\vec { x } } ( \\vec { z } ^ { * } ( \\vec { x } ) , \\vec { x } ) \\right ] ^ { \\top } \\lambda _ { i } ^ { * } ( \\vec { x } ) . \\end{align*}"}
-{"id": "816.png", "formula": "\\begin{align*} L ^ { ( i ) } ( z ) = \\begin{pmatrix} 1 + z q ^ { 2 N _ { i } } & \\beta _ { i } ^ { * } \\\\ z \\beta _ { i } & z \\end{pmatrix} . \\end{align*}"}
-{"id": "6743.png", "formula": "\\begin{align*} \\mathcal E ( X , \\theta ) : = \\bigcup _ { \\chi \\in \\mathcal W ^ - } \\mathcal E _ \\chi ( X , \\theta ) . \\end{align*}"}
-{"id": "2778.png", "formula": "\\begin{align*} \\lambda _ { n } ( \\alpha ) = 3 + [ 0 ; 3 , a _ { n - 2 } , \\ldots , a _ 1 ] + [ 0 ; 3 , a _ { n + 2 } , \\ldots ] . \\end{align*}"}
-{"id": "1867.png", "formula": "\\begin{align*} \\omega _ G = \\omega - \\mu \\end{align*}"}
-{"id": "9309.png", "formula": "\\begin{align*} \\begin{cases} d \\tilde { p } ( t , z ) & = - \\frac { a _ 0 ( t , z ) } { b _ 0 ( t , z ) } \\tilde { p } ( t , z ) d B ( t ) \\\\ \\tilde { p } ( T , z ) & = \\int _ D U ' ( y ( T , x , z ) ) d x \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] \\end{cases} \\end{align*}"}
-{"id": "5336.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = \\lambda Q ( N _ { f } ( u ) ) & & \\\\ \\int _ 0 ^ T f ( t , u ( t ) , u ' ( t ) ) d t = \\varphi ( b u ( 0 ) ) - \\varphi ( u ( 0 ) ) , \\ u ' ( 0 ) = u ( 0 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "2132.png", "formula": "\\begin{align*} \\mathcal { K } ^ + : = \\mathcal { K } \\cap \\{ x _ n \\geq 0 \\} \\mathcal { K } ^ - : = \\mathcal { K } \\cap \\{ x _ n < 0 \\} . \\end{align*}"}
-{"id": "8854.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\upsilon _ i ^ 2 } { k ^ { \\frac { 4 } { n } } } = \\frac { n } { n + 4 } \\frac { 1 6 \\pi ^ 4 } { ( \\omega _ n \\mathrm { v o l } \\ , \\Omega ) ^ \\frac { 4 } { n } } . \\end{align*}"}
-{"id": "2923.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V _ \\pi ( z _ 1 ; X ) V _ \\pi ( z _ 2 ; X ) \\cdots V _ \\pi ( z _ m ; X ) \\cdot 1 \\cr & = [ Z ^ { \\lambda + \\delta } ] \\ \\prod _ { 1 \\le i < j \\le m } ( z _ i - z _ j ) \\ \\prod _ { \\ell = 1 } ^ m \\ , M ( z _ \\ell ; X ) \\ L _ \\pi ( Z ) \\cr & = [ s _ \\lambda ( Z ) ] \\ M ( X , Z ) \\ , L _ \\pi ( Z ) \\ , \\end{align*}"}
-{"id": "73.png", "formula": "\\begin{align*} \\varphi ' ( Y _ { a _ * } + \\delta , a _ * ) < - 2 \\varepsilon < \\varphi ' ( Y _ { a _ * } , a _ * ) = 0 < 2 \\varepsilon < \\varphi ' ( Y _ { a _ * } - \\delta , a _ * ) \\end{align*}"}
-{"id": "8997.png", "formula": "\\begin{align*} [ L _ 1 ( s ) + ~ ^ { A B R } D _ b ^ \\alpha L _ 2 ( s ) ] = 0 , ~ \\texttt { f o r a l l } ~ s \\in [ 0 , b ] , \\end{align*}"}
-{"id": "1081.png", "formula": "\\begin{align*} \\frac { e ( H ) } { v ( H ) } > \\frac { 1 } { N } \\left ( k - \\alpha - \\frac { 7 } { 1 6 } \\right ) \\frac { n ^ 2 } { 2 } = \\frac { n } { 2 } \\left ( 1 - \\frac { 7 } { 1 6 ( k - \\alpha ) } \\right ) \\ , . \\end{align*}"}
-{"id": "8243.png", "formula": "\\begin{align*} \\frac { d \\vec { x } } { d t } = \\vec { f } ( \\vec { x } ) , x \\in X , \\end{align*}"}
-{"id": "1490.png", "formula": "\\begin{align*} \\rho ( \\prod _ { k } x _ { i _ k j _ k } ^ { \\delta _ k } ) = \\sigma ( \\prod _ { k } ( i _ k + 1 , j _ k ) ) ^ { - 1 } \\sigma ^ { - 1 } . \\end{align*}"}
-{"id": "696.png", "formula": "\\begin{align*} & \\overline { \\alpha } _ { f } ( P ) = \\limsup _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n } ( P ) ) ^ { 1 / n } \\\\ & \\underline { \\alpha } _ { f } ( P ) = \\liminf _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n } ( P ) ) ^ { 1 / n } . \\end{align*}"}
-{"id": "2010.png", "formula": "\\begin{align*} \\int _ { \\R } [ f ( x + x z ) - f ( x ) - f ' ( x ) x z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) = \\int _ { \\R } [ \\log | 1 + z | - z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) + o ( 1 ) . \\end{align*}"}
-{"id": "7403.png", "formula": "\\begin{align*} \\nu ( [ g , f ] ) & \\leq \\nu ( g [ f , h h _ 0 h ^ { - 1 } ] g ^ { - 1 } ) + \\nu ( [ f , h h _ 0 h ^ { - 1 } ] ^ { - 1 } ) \\\\ & = 2 \\nu ( [ f , h h _ 0 h ^ { - 1 } ] ) \\\\ & \\leq 2 ( \\nu ( f ( h h _ 0 h ^ { - 1 } ) f ^ { - 1 } ) + \\nu ( ( h h _ 0 h ^ { - 1 } ) ^ { - 1 } ) ) \\\\ & = 4 \\nu ( h h _ 0 h ^ { - 1 } ) = 4 \\nu ( h _ 0 ) . \\end{align*}"}
-{"id": "5789.png", "formula": "\\begin{align*} S ' = \\{ k \\in \\C \\backslash \\{ - h ^ { \\vee } \\} \\mid ( k + h ^ { \\vee } ) ^ { - 1 } \\in \\bigcap _ { n \\geq 0 } U ( n ) \\} \\end{align*}"}
-{"id": "1068.png", "formula": "\\begin{align*} \\binom { N } { 2 } = ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } \\leq k ( k - \\alpha - x ) \\frac { n ^ 2 } { 2 } \\end{align*}"}
-{"id": "9860.png", "formula": "\\begin{align*} \\| f \\mid W ^ 1 _ p ( \\Omega ) \\| = \\biggr ( \\int \\limits _ { \\Omega } | f ( x ) | ^ { p } \\ , d x \\biggr ) ^ { \\frac { 1 } { p } } + \\biggr ( \\int \\limits _ { \\Omega } | \\nabla f ( x ) | ^ { p } \\ , d x \\biggr ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "2411.png", "formula": "\\begin{align*} \\rho ( z , w , t ) = u + P ( z , w ) + Q ( z , w ) + v R ( z , w ) + u ^ 2 + v ^ 2 + o ( u ^ 2 , u v , v ^ 2 , u | ( z , w ) | ^ { 2 k } ) . \\end{align*}"}
-{"id": "1103.png", "formula": "\\begin{align*} | \\langle a _ k , x _ 0 \\rangle | = | \\langle a _ k , \\tilde { x } \\rangle | , \\ , \\ , \\forall \\ , \\ , k \\in S _ { l _ 0 } \\cup S _ { j _ 0 } . \\end{align*}"}
-{"id": "7594.png", "formula": "\\begin{align*} \\varepsilon _ 0 : = ( F ( x _ 0 ) + l ) / ( 2 l ) \\in ( 0 , 1 ) , \\varepsilon _ { i } : = \\frac { l + 2 F ( x _ 0 + ( i - 1 ) \\varepsilon ) - F ( x _ 0 ) } { 2 l } , i = 1 , \\dots , K _ 1 , \\end{align*}"}
-{"id": "4966.png", "formula": "\\begin{align*} & \\overline { \\alpha } _ { f } ( P ) = \\limsup _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n } ( P ) ) ^ { 1 / n } \\\\ & \\underline { \\alpha } _ { f } ( P ) = \\liminf _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n } ( P ) ) ^ { 1 / n } . \\end{align*}"}
-{"id": "6601.png", "formula": "\\begin{align*} S _ 1 x & = P _ { D } ^ { \\alpha _ 1 } x = ( 1 - \\alpha _ 1 ) x + \\alpha _ 1 \\Pi _ D = ( 1 - \\alpha _ 1 ) x + \\alpha _ 1 ( N x + d ) . \\end{align*}"}
-{"id": "5515.png", "formula": "\\begin{align*} S ( x ) = 0 \\ \\ \\ \\ \\ \\ \\ x \\in ( \\Lambda _ { \\varphi , w } ) _ a . \\end{align*}"}
-{"id": "1597.png", "formula": "\\begin{align*} \\sigma ^ { 2 } \\left ( x \\right ) = 2 \\kappa \\left ( \\left ( \\mu - \\lambda \\right ) - x \\right ) + \\kappa + c _ { 1 } e ^ { - 2 x } , \\end{align*}"}
-{"id": "1876.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } ^ 2 + 2 R _ { 1 3 4 2 } R _ { 1 4 2 3 } = & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( x + 2 y ) - ( x + 2 y ) ^ 2 ] \\\\ \\ge & - \\frac 1 2 ( x + 2 y ) ^ 2 . \\\\ \\end{align*}"}
-{"id": "6414.png", "formula": "\\begin{align*} H ^ { s } ( \\Omega ) = \\left \\{ u \\in L ^ { 2 } ( \\Omega ) \\ , : \\ , \\frac { | u ( x ) - u ( y ) | } { | x - y | ^ { s + n / 2 } } \\in L ^ { 2 } ( \\Omega \\times \\Omega ) \\right \\} , \\end{align*}"}
-{"id": "2597.png", "formula": "\\begin{align*} F ( u ) = \\mu | u | ^ p u . \\end{align*}"}
-{"id": "69.png", "formula": "\\begin{align*} \\varphi ( y ) = \\varphi ( y , a ) : = \\psi ( y , a ) ( 1 - y ) ^ { - p } \\ , y \\in [ 0 , 1 ) \\ , \\end{align*}"}
-{"id": "2946.png", "formula": "\\begin{align*} \\mathbf { P } \\mathbf { H } \\mathbf { Q } = \\begin{pmatrix} \\mathbf { H } _ P & \\mathbf { H } _ I \\end{pmatrix} = \\begin{pmatrix} \\mathbf { H } _ { P , 1 } & & \\mathbf { O } & \\mathbf { H } _ { I , 1 } \\\\ & \\ddots & & \\vdots \\\\ \\mathbf { O } & & \\mathbf { H } _ { P , K } & \\mathbf { H } _ { I , K } \\end{pmatrix} , \\end{align*}"}
-{"id": "8836.png", "formula": "\\begin{align*} E _ { k , \\chi _ { - 2 } } ^ { \\infty } ( \\tau ) & = 1 - \\frac { 2 k } { B _ { k , 8 } } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { k - 1 , \\chi _ { - 2 } } ^ { \\infty } ( n ) q ^ { n } \\intertext { a n d } E _ { k , \\chi _ { - 2 } } ^ { 0 } ( \\tau ) & = \\delta _ { k , 1 } - \\frac { 2 k } { B _ { k , 8 } } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { k - 1 , \\chi _ { - 2 } } ^ { 0 } ( n ) q ^ { n } , \\end{align*}"}
-{"id": "7301.png", "formula": "\\begin{align*} \\eth = - \\sum _ { \\xi _ i \\in \\Delta ( \\mathfrak { u } _ + ) } E _ { \\xi _ i } \\otimes \\gamma _ - ( w _ i ) \\in U ( \\mathfrak { g } ) \\otimes \\mathrm { C l } . \\end{align*}"}
-{"id": "6713.png", "formula": "\\begin{align*} \\phi ^ { \\Delta } _ { i } = \\sum _ { \\vec { c } \\in \\Delta } \\phi ^ { \\Delta , \\vec { c } } _ { i } . \\end{align*}"}
-{"id": "1466.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ 2 \\sum _ { Q \\in \\mathfrak { A } _ k ( Q _ 0 ) } \\alpha _ Q \\chi _ Q \\right | \\leq 2 \\sum _ { k = 1 } ^ 2 \\sum _ { Q \\in \\mathfrak { A } _ { k - 1 } ( Q _ 0 ) } \\omega _ { \\lambda _ w } ( f ; Q ) \\chi _ Q , \\end{align*}"}
-{"id": "3679.png", "formula": "\\begin{align*} \\int f ~ d f \\wedge d g = 0 , ~ \\int g ~ d f \\wedge d g = 0 ~ . \\end{align*}"}
-{"id": "1380.png", "formula": "\\begin{align*} E ( Q _ n ( \\delta ) ) = \\frac { 1 } { n } \\sum _ { t = 1 } ^ n \\sum _ { j = 0 } ^ { m _ t } \\pi ^ 0 _ { t } ( j ) \\log \\pi _ { t } ( j ) , \\delta \\in \\Theta . \\end{align*}"}
-{"id": "2864.png", "formula": "\\begin{align*} F ^ c ( C \\otimes D ) = \\bigoplus _ { n \\geq 1 } C ^ { \\otimes n } \\otimes D ^ { \\otimes n } \\rightarrow \\bigoplus _ { m , n \\geq 1 } C ^ { \\otimes m } \\otimes D ^ { \\otimes n } = F ^ c ( C ) \\otimes F ^ c ( D ) \\end{align*}"}
-{"id": "6236.png", "formula": "\\begin{align*} \\tilde { \\hat { M } } = \\tilde { V } ^ { T } M \\tilde { V } , \\ \\tilde { \\hat { D } } = \\tilde { V } ^ { T } D \\tilde { V } , \\ \\tilde { \\hat { K } } = \\tilde { V } ^ { T } K \\tilde { V } , \\\\ \\tilde { \\hat { F } } = \\tilde { V } ^ T F , \\ \\tilde { \\hat { C } } _ p = C _ p \\tilde { V } , \\ \\ \\tilde { \\hat { C } } _ v = C _ v \\tilde { V } , \\end{align*}"}
-{"id": "5121.png", "formula": "\\begin{align*} F _ { \\vec { x } } ( - z _ { 1 } / s , \\ldots , - z _ { k } / s ) = \\frac { ( - s ) ^ { \\sum _ { i = 1 } ^ { k } x _ { i } } } { \\prod _ { i = 1 } ^ { k } z _ { i } ( 1 + z _ { i } ) ^ { M + 1 } } \\langle \\prod _ { 1 \\le i \\le k } C ^ { [ 0 , M ] } ( u _ { i } ; s ) \\prod _ { 1 \\le i \\le k } \\beta _ { x _ { i } } ^ { * } \\rangle _ { [ 0 , M ] } . \\end{align*}"}
-{"id": "7460.png", "formula": "\\begin{align*} \\chi _ i ^ * ( X _ k ) : = \\left \\{ \\begin{array} { l l } z & \\\\ 1 & \\end{array} \\right . \\end{align*}"}
-{"id": "6697.png", "formula": "\\begin{align*} u = u _ { P Q } \\theta ^ P \\bar \\theta ^ Q \\in C ^ \\infty ( U \\times \\C ^ { 0 | d } ) [ \\nu ^ { - 1 } \\nu ] ] \\end{align*}"}
-{"id": "7032.png", "formula": "\\begin{align*} \\lim _ { x \\downarrow 0 } \\frac { A ( x ) } { \\sqrt { U ( x ) \\overline { \\Pi } ^ - ( x ) } } = \\infty . \\end{align*}"}
-{"id": "6815.png", "formula": "\\begin{align*} L _ m \\cdot B _ n & = \\ \\sum _ { k = 0 } ^ { n - 1 } \\left [ B _ k , B _ { m + n - k } \\right ] + n B _ { m + n } \\\\ L _ m \\cdot B _ { - n } & = - \\sum _ { k = - n } ^ { - 1 } \\left [ B _ k , B _ { m - n - k } \\right ] - n B _ { m - n } \\end{align*}"}
-{"id": "9738.png", "formula": "\\begin{align*} x _ { L , \\epsilon } ( t ) = \\lambda _ { n + 1 } ( 1 - \\epsilon ) G ^ { - 1 } ( t ) , t \\geq T _ 3 ( \\epsilon ) . \\end{align*}"}
-{"id": "3004.png", "formula": "\\begin{align*} s ( \\varphi , I _ \\Gamma ( t , \\pi ) ) = \\sup _ { P \\in \\Gamma ( t , \\pi ) } \\left \\langle \\varphi , \\int _ X \\imath _ X ( x ) d P \\right \\rangle = \\sup _ { P \\in \\Gamma ( t , \\pi ) } \\int _ X \\langle \\varphi , \\imath _ X ( x ) \\rangle d P \\end{align*}"}
-{"id": "871.png", "formula": "\\begin{align*} \\operatorname * { e p i } \\varphi _ { A , k } = \\{ ( y , t ) \\in Y \\times \\mathbb { R } \\mid y \\in t k + A \\} \\end{align*}"}
-{"id": "7246.png", "formula": "\\begin{align*} F _ { j , \\alpha } ( z _ j ) = \\left ( \\begin{array} { c } F _ { j , \\alpha } ^ 1 ( z _ j ) \\\\ F _ { j , \\alpha } ^ 2 ( z _ j ) \\\\ \\vdots \\\\ F _ { j , \\alpha } ^ r ( z _ j ) \\end{array} \\right ) \\end{align*}"}
-{"id": "2409.png", "formula": "\\begin{align*} \\| \\nabla f _ h ( x ) - \\nabla f _ h ( y ) \\| _ H ^ 2 & = \\langle \\nabla f _ h ( x ) - \\nabla f _ h ( y ) , \\nabla f _ h ( x ) - \\nabla f _ h ( y ) \\rangle _ H \\\\ & = \\langle L ^ { - 1 } ( \\nabla f ( x ) - \\nabla f ( y ) ) , L ^ { - 1 } ( \\nabla f ( x ) - \\nabla f ( y ) ) \\rangle _ L \\\\ & = \\langle \\nabla f ( x ) - \\nabla f ( y ) , \\nabla f ( x ) - \\nabla f ( y ) \\rangle _ { L ^ { - 1 } } \\\\ & = \\| \\nabla f ( x ) - \\nabla f ( y ) \\| _ { L ^ { - 1 } } ^ 2 . \\end{align*}"}
-{"id": "223.png", "formula": "\\begin{align*} R _ 4 ' = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\log \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) \\ , \\mathrm { B } _ { k , n - k } ( s ) \\ , d s \\ , d x = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "5213.png", "formula": "\\begin{align*} Q _ { q } ( f , f ) = ( \\Box _ { M , q } f , f ) _ { L ^ { 2 } _ { ( 0 , q ) } ( M ) } \\ ; , \\ ; f \\in d o m ( \\Box _ { M , q } ) \\ ; . \\end{align*}"}
-{"id": "2316.png", "formula": "\\begin{align*} \\theta _ s = \\inf _ { x \\in \\varOmega } \\arctan \\frac { a ( x , s ) } { | b ( x , s ) | } , \\end{align*}"}
-{"id": "553.png", "formula": "\\begin{align*} k _ { \\mathcal J } ( \\tau ) = \\liminf _ { A \\in { \\mathcal R } ^ + _ 1 ( { \\mathcal H } ) } \\max _ { 1 \\le j \\le n } | [ A , T _ j ] | \\end{align*}"}
-{"id": "9163.png", "formula": "\\begin{align*} \\det A ( a _ 1 , \\dots , a _ n ) = \\prod _ { j = 2 } ^ { n } \\prod _ { k = j + 1 } ^ { n } ( a _ k - a _ j ) \\prod _ { j = 2 } ^ { n } a _ j ^ 2 \\neq 0 . \\end{align*}"}
-{"id": "9282.png", "formula": "\\begin{align*} \\hat { \\pi } ( t , z ) = \\hat { \\pi } _ 1 ( t , z ) = \\frac { \\Phi _ 1 ( t , z ) } { b _ 0 ( t , z ) } + \\frac { a _ 0 ( t , z ) } { \\sigma ^ 2 _ 0 ( t , z ) } , \\end{align*}"}
-{"id": "3506.png", "formula": "\\begin{align*} \\Phi ^ W _ { ( g , \\pi ) } ( \\gamma , \\tau ) = \\Phi ( \\gamma , \\tau ) + ( 0 , \\tfrac { 1 } { 2 } \\gamma \\cdot _ g ( \\textup { d i v } _ g \\pi + W ) ) , \\end{align*}"}
-{"id": "9937.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\Delta u - \\lambda u = | u | ^ { 2 _ * - 2 } u & \\mbox { i n } \\Omega \\\\ u = 0 & \\mbox { o n } \\partial \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "1899.png", "formula": "\\begin{align*} B _ { c , h , \\tau } ( u ) : = A _ { c } \\cap \\Big \\{ \\forall k \\leq \\lceil \\tfrac { u } { h } \\rceil : | \\hat { \\phi } ( t _ k ) - \\phi ( t _ k ) | \\leq \\tau \\Big ( \\frac { \\log n } { n } \\Big ) ^ { 1 / 2 } \\Big \\} . \\end{align*}"}
-{"id": "8754.png", "formula": "\\begin{align*} & \\alpha = \\frac { - a c ( 1 - t ) } { ( 1 + a ) ( 1 + c ) } , \\gamma = \\frac { 1 - t } { ( 1 + a ) ( 1 + c ) } , \\\\ & \\beta = \\frac { - b d ( 1 - t ) } { ( 1 + b ) ( 1 + d ) } , \\delta = \\frac { 1 - t } { ( 1 + b ) ( 1 + d ) } . \\end{align*}"}
-{"id": "6136.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } ^ 2 + 2 R _ { 1 3 4 2 } R _ { 1 4 2 3 } = & x ^ 2 - 2 y ( x + y ) \\\\ = & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( x + 2 y ) - ( x + 2 y ) ^ 2 ] \\\\ \\ge & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( K _ { 1 4 } - K _ { 1 3 } ) - ( K _ { 1 4 } - K _ { 1 3 } ) ^ 2 ] \\\\ \\end{align*}"}
-{"id": "4347.png", "formula": "\\begin{align*} \\psi ' \\left ( 1 - \\frac { f ( r ) } { a } \\right ) = - \\frac { p } { ( p - 1 ) a ^ { p - 1 } } \\left ( | f ' | ^ { p - 2 } f ' \\right ) ' ( r ) \\ , r \\in [ 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "8215.png", "formula": "\\begin{align*} E U _ n = \\epsilon \\left ( \\bar { U } _ { n + 1 } - 2 \\bar { U } _ n + \\bar { U } _ { n - 1 } \\right ) + i \\gamma U _ n + \\Omega \\bar { U } _ n + 6 | U _ n | ^ 2 \\bar { U } _ n + 2 U _ n ^ 3 , \\end{align*}"}
-{"id": "8967.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | u ( x ) - u ( y ) | ^ { p - 2 } \\ , ( u ( x ) - u ( y ) ) \\ , ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y = \\int _ { \\Omega } \\frac { f ( x ) } { u ^ \\gamma } \\ , \\varphi \\ , d x , \\end{align*}"}
-{"id": "4631.png", "formula": "\\begin{align*} ( f \\circ 1 ) ( a ) : = \\bigvee \\limits _ { ( y , z ) \\in { A _ a } } { \\min \\{ f ( y ) , 1 ( z ) \\} } = \\bigvee \\limits _ { ( y , z ) \\in { A _ a } } f ( y ) \\end{align*}"}
-{"id": "4512.png", "formula": "\\begin{align*} \\nu = d 2 ^ { \\alpha / 2 - 1 } \\frac { \\Gamma \\bigl ( \\frac { \\alpha } { 2 } + \\frac { d } { 2 } \\bigr ) } { \\Gamma \\bigl ( 1 + \\frac { d } { 2 } \\bigr ) } \\frac { ( \\rho / 2 ) ^ { \\alpha / 2 } \\Gamma \\bigl ( \\frac { \\rho - \\alpha } { 2 } \\bigr ) } { \\Gamma \\bigl ( \\frac { \\rho } { 2 } \\bigr ) } . \\end{align*}"}
-{"id": "1487.png", "formula": "\\begin{align*} \\tilde B _ 0 = [ B _ 0 | C ] = \\left [ \\begin{array} { c c | c c } 0 & 1 & 1 & 0 \\\\ - 1 & 0 & 0 & 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "6444.png", "formula": "\\begin{align*} \\ , \\phi \\leq & 2 \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } \\phi w ^ { 2 } k ( x , y ) d x d y \\\\ & + 4 \\int _ { \\rho B _ { 1 } } \\int _ { \\mathbb { R } ^ { n } \\backslash ( \\rho B _ { 1 } ) } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } \\phi w ^ { 2 } k ( x , y ) d y d x \\\\ \\leq & C _ { 1 } ( n , \\Lambda , \\delta ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } \\phi w ^ { 2 } d x , \\end{align*}"}
-{"id": "6174.png", "formula": "\\begin{align*} \\mathcal { E } ( U ) _ { ( s , t ] } & = e ^ { U _ t - U _ s - \\sigma _ U ^ 2 ( t - s ) / 2 } \\prod _ { s < u \\leq t } ( 1 + \\Delta U _ u ) e ^ { - \\Delta U _ u } , \\\\ \\mathcal { E } ( U ) _ { ( s , t ) } & = e ^ { U _ { t - } - U _ s - \\sigma _ U ^ 2 ( t - s ) / 2 } \\prod _ { s < u < t } ( 1 + \\Delta U _ u ) e ^ { - \\Delta U _ u } . \\end{align*}"}
-{"id": "8722.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } d X _ \\tau ^ { x } = A X _ \\tau ^ { x } d \\tau + G B ( t , X _ \\tau ^ { x } ) d \\tau + G d W _ \\tau , \\tau \\in \\left [ 0 , T \\right ] , \\\\ X _ 0 ^ { x } = x , \\end{array} \\right . \\end{align*}"}
-{"id": "5668.png", "formula": "\\begin{align*} \\rho _ f : = \\int f \\d v . \\end{align*}"}
-{"id": "585.png", "formula": "\\begin{align*} x \\equiv \\sum _ { k = 0 } ^ { n - 1 } p ^ k [ \\phi ^ { - k } ( r _ k ) ] \\bmod I ^ n \\Big ( \\in \\mathbb { Z } R / I ^ n \\Big ) \\end{align*}"}
-{"id": "1286.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & p & 0 & 0 \\\\ p & 0 & 0 & 0 \\\\ 0 & 0 & 0 & p \\\\ 0 & 0 & p & 0 \\end{bmatrix} , \\begin{bmatrix} 0 & p ^ 2 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & p \\\\ 0 & 0 & p & 0 \\end{bmatrix} , \\begin{bmatrix} 0 & p ^ 2 & 0 & 0 \\\\ 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & p ^ 2 \\\\ 0 & 0 & 1 & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "6198.png", "formula": "\\begin{align*} I ( u ) = \\frac { f ( x u ) - f ( x ) } { x f ' ( x ) } = \\frac { 1 } { \\gamma \\alpha } ( \\log | x | ) ^ { 1 - \\alpha } \\left [ \\exp \\left \\{ \\gamma ( \\log | x | u ) ^ \\alpha - \\gamma ( \\log | x | ) ^ \\alpha \\right \\} - 1 \\right ] . \\end{align*}"}
-{"id": "344.png", "formula": "\\begin{align*} W _ I T _ { c , h } ( f ) W _ I ^ * = T _ { \\tilde { c } , \\tilde { h } } ( f ) \\end{align*}"}
-{"id": "8785.png", "formula": "\\begin{align*} E _ { \\lambda } = \\sum _ { \\mu \\leq \\lambda } c _ { \\lambda \\mu } ( q , t ) f _ { \\mu } , f _ { \\lambda } = \\sum _ { \\mu \\leq \\lambda } d _ { \\lambda \\mu } ( q , t ) E _ { \\mu } \\end{align*}"}
-{"id": "3967.png", "formula": "\\begin{align*} V ^ { \\lambda } ( A ) = \\bigoplus _ { n \\delta _ 1 + \\delta _ 2 = \\lambda } V ^ { \\delta _ 1 , \\delta _ 2 } . \\end{align*}"}
-{"id": "876.png", "formula": "\\begin{align*} A = \\operatorname * { b d } A - \\mathbb { R } _ { + } k \\end{align*}"}
-{"id": "5946.png", "formula": "\\begin{align*} V _ t = v _ 0 + \\int _ 0 ^ t F ( X _ s , V _ s ) \\ , \\dd s + W _ t \\ , \\end{align*}"}
-{"id": "833.png", "formula": "\\begin{align*} F ^ { 1 ^ { k } } _ { \\vec { z } } ( x _ { 1 } , \\ldots , x _ { k - 1 } , 1 ) = \\sum _ { \\ell = 1 } ^ { k } \\frac { z _ { \\ell } } { 1 + z _ { \\ell } } \\prod _ { 0 < i \\le k \\atop i \\not = \\ell } f ( z _ { i } , z _ { \\ell } ) F ^ { 1 ^ { k - 1 } } _ { \\vec { z } ( \\ell ) } ( x _ { 1 } , \\ldots , x _ { k - 1 } ) \\otimes u _ { 1 } , \\end{align*}"}
-{"id": "8948.png", "formula": "\\begin{align*} i U _ { t } + U _ { x x } + q \\left \\vert U \\right \\vert ^ { 2 } U = 0 , - \\infty < x < \\infty , t > 0 \\end{align*}"}
-{"id": "4515.png", "formula": "\\begin{align*} a ( \\delta ) : = A _ m \\max \\biggl \\{ 1 \\ , , \\ , \\log ^ { 2 ( m + 1 ) } \\Bigl ( \\frac { 1 } { \\delta } \\Bigr ) \\biggr \\} , \\end{align*}"}
-{"id": "2023.png", "formula": "\\begin{align*} \\left | \\int _ { - 1 } ^ { - 1 + x ^ { - 1 } } [ f ( x + x z ) - f ( x ) - f ' ( x ) x z ] \\nu _ U ( \\d z ) \\right | \\leq 3 x ^ \\beta \\nu _ U ( [ - 1 , - 1 + x ^ { - 1 } ] ) = o ( x ^ { \\beta } ) , \\end{align*}"}
-{"id": "9782.png", "formula": "\\begin{align*} \\Delta ^ { \\mathcal { F } } _ { \\partial } = \\Delta ^ { \\mathcal { F } } _ { \\overline { \\partial } } . \\end{align*}"}
-{"id": "8068.png", "formula": "\\begin{align*} \\iota ( z ) = \\begin{cases} 0 & \\abs { z } < 1 , \\\\ 1 & \\abs { z } > 1 , \\end{cases} \\end{align*}"}
-{"id": "4775.png", "formula": "\\begin{align*} \\frac { \\beta } { \\alpha } \\ , \\kappa = \\frac { f f '' + ( f ' ) ^ 2 + 1 } { \\sqrt { f '^ 2 + 1 } } , \\alpha \\neq 0 , \\ ; \\beta \\neq 0 . \\end{align*}"}
-{"id": "572.png", "formula": "\\begin{align*} \\delta ( x + y ) = \\delta ( x ) + \\delta ( y ) - \\sum _ { k = 1 } ^ { p - 1 } \\frac { 1 } { p } \\binom { p } { k } x ^ k y ^ { p - k } \\end{align*}"}
-{"id": "3197.png", "formula": "\\begin{align*} M _ j : = \\left ( \\begin{array} { c | c c c } 1 & 0 & \\cdots & 0 \\\\ \\hline - m _ { j } ^ 2 & 1 & & 0 \\\\ \\vdots & & \\ddots & \\\\ - m _ { j } ^ r & 0 & & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2970.png", "formula": "\\begin{align*} \\delta _ L \\Theta ( L ) = 0 . \\end{align*}"}
-{"id": "9518.png", "formula": "\\begin{align*} R _ E & = e ^ { \\phi / 2 } \\left ( R + \\frac { 1 } { 2 } ( n - 1 ) \\Delta \\phi - \\frac { 1 } { 1 6 } ( n - 1 ) ( n - 2 ) | d \\phi | ^ 2 \\right ) \\\\ & = e ^ { \\phi / 2 } \\left ( R + \\frac { 9 } { 2 } \\Delta \\phi - \\frac { 9 } { 2 } | d \\phi | ^ 2 \\right ) , \\end{align*}"}
-{"id": "4126.png", "formula": "\\begin{align*} \\gcd ( 2 q , - p ) + 2 = p + 2 q , \\end{align*} % \\end{align*}"}
-{"id": "2565.png", "formula": "\\begin{align*} \\| f \\| _ { L _ t ^ { q , \\alpha } ( I ) } : = \\bigl \\| \\lambda \\ , \\bigl | \\{ t \\in I : | f ( t ) | > \\lambda \\} \\bigr | ^ { \\frac { 1 } { q } } \\bigr \\| _ { L ^ \\alpha ( ( 0 , \\infty ) , \\frac { d \\lambda } { \\lambda } ) } . \\end{align*}"}
-{"id": "2574.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u = \\mu | u | ^ p u + e , u ( t _ 0 ) = u _ 0 , \\end{align*}"}
-{"id": "365.png", "formula": "\\begin{align*} \\bar { d } _ { \\mathcal { E } } = \\frac { | \\mathcal { E } _ m - \\Gamma _ m ^ * | } { d _ { m a x } } , \\end{align*}"}
-{"id": "1753.png", "formula": "\\begin{align*} \\lfloor r _ i u \\rfloor = \\lfloor r _ i x _ 0 \\rfloor \\xleftarrow { z _ 1 \\gamma _ 1 ^ \\lor } \\cdots \\xleftarrow { z _ { p - 1 } \\gamma _ { p - 1 } ^ \\lor } \\lfloor r _ i x _ { p - 1 } \\rfloor = x _ p \\xleftarrow { \\gamma _ { p + 1 } ^ \\lor } \\cdots \\xleftarrow { \\gamma _ r ^ \\lor } x _ r = v . \\end{align*}"}
-{"id": "6171.png", "formula": "\\begin{align*} \\d V _ t = V _ { t - } \\d U _ t + \\d L _ t , t \\geq 0 . \\end{align*}"}
-{"id": "7087.png", "formula": "\\begin{align*} f _ \\mu ( u ) = x + \\sum _ { j = 1 } ^ \\infty f _ \\mu ( u . j ) , \\end{align*}"}
-{"id": "1084.png", "formula": "\\begin{align*} g ( x ) = a x + b \\ ; \\ ; \\pmod { 1 } \\end{align*}"}
-{"id": "5326.png", "formula": "\\begin{align*} Q _ { i } = \\Gamma _ { i } ^ \\dagger \\Gamma _ { i } , \\overline { Q } _ { j } = \\overline { \\Gamma } _ { j } ^ \\dagger \\overline { \\Gamma } _ { j } , i = 1 , 2 , \\cdots , n - 1 j = 1 , 2 , \\cdots , m - 1 . \\end{align*}"}
-{"id": "272.png", "formula": "\\begin{align*} | U _ 1 | \\leq U _ { 1 1 } + U _ { 1 2 } = o \\biggl ( \\frac { k ^ { \\frac { 1 } { 2 } + \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { 1 + \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "9033.png", "formula": "\\begin{align*} \\psi _ { j + 1 } ( \\theta ) = \\psi _ j ( \\theta ) + \\theta - 2 \\left ( \\Im \\log \\left ( 1 - \\gamma _ j e ^ { i \\psi _ j ( \\theta ) } \\right ) - \\Im \\log \\left ( 1 - \\gamma _ j \\right ) \\right ) \\end{align*}"}
-{"id": "6410.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } ( u ( x , t ) - u _ { 0 } ( x ) ) + L u ( x , t ) & = f ( x , t ) \\Omega \\times [ 0 , T ] , \\\\ u ( x , t ) & = 0 \\quad \\quad \\quad \\ , \\mathbb { R } ^ { n } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = u _ { 0 } ( x ) \\quad \\ , \\ , \\Omega , \\ , t = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "691.png", "formula": "\\begin{align*} B : = ( A D ) ^ c = \\bigcap _ { a \\in A } a C \\end{align*}"}
-{"id": "9844.png", "formula": "\\begin{align*} \\Upsilon ( \\mu , \\lambda ) = \\max _ { P ( f ) \\ge 0 , \\bar \\Xi ( f ) \\ge 0 } & \\mathcal L ( P ( f ) , \\bar \\Xi ( f ) , \\mu , \\lambda ) . \\end{align*}"}
-{"id": "3063.png", "formula": "\\begin{align*} W _ n ^ { ( \\beta ) } = \\sum _ { | u | = n } e ^ { - \\beta V ( u ) - n \\kappa ( \\beta ) } . \\end{align*}"}
-{"id": "4567.png", "formula": "\\begin{align*} V ' : = \\mathrm { C o v } ( Y _ i ' ) = \\begin{pmatrix} p _ { n , x , u } ^ { ( j ) } ( 1 - p _ { n , x , u } ^ { ( j ) } ) & p _ \\cap - p _ { n , x , u } ^ { ( j ) } p _ { n , y , v } ^ { ( l ) } \\\\ p _ \\cap - p _ { n , x , u } ^ { ( j ) } p _ { n , y , v } ^ { ( l ) } & p _ { n , y , v } ^ { ( l ) } ( 1 - p _ { n , y , v } ^ { ( l ) } ) \\end{pmatrix} , \\end{align*}"}
-{"id": "631.png", "formula": "\\begin{align*} Q _ n \\in L ^ \\infty _ c ( \\R ^ N ) , Q _ n ( y ) = M _ n ^ { p - 2 } c _ n ^ 2 Q ( x _ n + c _ n y ) . \\end{align*}"}
-{"id": "5489.png", "formula": "\\begin{align*} \\mathbf { G } _ \\mathrm { s } [ \\iota ] = \\mathbf { \\hat { G } } _ \\mathrm { s } [ \\iota ] + \\mathcal { E } _ \\mathrm { s } [ \\iota ] \\end{align*}"}
-{"id": "3457.png", "formula": "\\begin{align*} \\beta = \\sum _ I \\varphi _ 1 ^ * ( \\delta _ i ) \\smile \\gamma _ i \\end{align*}"}
-{"id": "5498.png", "formula": "\\begin{align*} \\gamma _ k ^ \\mathrm { B } [ \\iota ] = \\frac { M \\sigma _ { \\mathrm { s } k } ^ 2 [ \\iota ] } { \\sum _ { i = 1 } ^ { K } \\beta _ { \\mathrm { s } i } + \\left ( \\rho _ \\mathrm { p } \\sum _ { i = 1 } ^ { K } \\beta _ { \\mathrm { d } i } + 1 \\right ) / \\rho _ \\mathrm { s } } \\end{align*}"}
-{"id": "2665.png", "formula": "\\begin{align*} u ^ { ( n ) } ( x ) \\quad = \\begin{cases} \\frac { \\sum _ { \\ell = 1 } ^ \\infty b ^ { ( n ) } _ \\ell ( x _ \\ell ) } { 1 + \\sum _ { \\ell = 1 } ^ \\infty b ^ { ( n ) } _ \\ell ( x _ \\ell ) } & \\sum _ { \\ell = 1 } ^ \\infty b ^ { ( n ) } _ \\ell ( x _ \\ell ) < \\infty \\ , , \\\\ 1 & \\sum _ { \\ell = 1 } ^ \\infty b ^ { ( n ) } _ \\ell ( x _ \\ell ) = \\infty \\ , . \\end{cases} \\end{align*}"}
-{"id": "3439.png", "formula": "\\begin{align*} E | \\bar y _ j ( t ) ^ + | ^ 2 = E | \\bar \\zeta _ j ^ + | ^ 2 - E [ \\int _ t ^ T 2 I _ { y _ j ^ 1 ( s ) > y ^ 2 _ j ( s ) } \\bar y _ j ( s ) [ f _ j ( s , y ^ 1 ( s ) , z ^ 1 ( s ) ) - \\end{align*}"}
-{"id": "948.png", "formula": "\\begin{align*} \\bar \\partial _ M f = ( \\bar \\partial \\widetilde { f } ) _ { t _ M } \\ ; . \\end{align*}"}
-{"id": "6540.png", "formula": "\\begin{align*} \\frac 1 \\tau \\ , \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\le \\delta , \\end{align*}"}
-{"id": "6546.png", "formula": "\\begin{align*} \\begin{aligned} J _ m & \\le C ( \\tilde \\lambda + r ) \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( D ) } \\\\ & + \\varepsilon \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( D ) } + C _ \\varepsilon \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( X ) } + C \\delta . \\end{aligned} \\end{align*}"}
-{"id": "1638.png", "formula": "\\begin{align*} \\lambda { \\psi } ( x , v ) - \\frac { 1 } { 2 } \\triangle _ v { \\psi } ( x , v ) - v \\cdot D _ x { \\psi } ( x , v ) - F ( x , v ) \\cdot D _ v { \\psi } ( x , v ) = g ( x , v ) \\ , , \\end{align*}"}
-{"id": "5682.png", "formula": "\\begin{align*} \\bar e _ 0 = e _ 0 ( 1 + e _ 0 ) ^ { - 1 } \\end{align*}"}
-{"id": "7289.png", "formula": "\\begin{align*} U _ q ( \\mathfrak { l } ) = \\{ \\textrm { s u b a l g e b r a o f $ U _ q ( \\mathfrak { g } ) $ g e n e r a t e d b y $ K _ i ^ { \\pm 1 } $ a n d $ E _ j , F _ j $ w i t h $ j \\in S $ } \\} . \\end{align*}"}
-{"id": "9599.png", "formula": "\\begin{align*} \\lambda _ { y } ( t ) = \\lim _ { x \\to y } \\ , \\psi _ { f , y } ( x , t ) = - \\frac { m ( \\xi _ { 0 } + t \\dot \\xi _ { 0 } ) - \\dot \\xi _ 0 } { 4 \\pi } + \\frac { m } { 4 \\pi } \\int _ 0 ^ t \\frac { J _ 1 ( m ( t - s ) ) } { ( t - s ) } ( \\xi _ { 0 } + s \\dot \\xi _ { 0 } ) d s , \\end{align*}"}
-{"id": "8673.png", "formula": "\\begin{align*} \\lim _ { z \\to 0 } \\ ; \\sup _ { y \\in H } \\ , \\sup _ { | k | _ K = 1 } | I _ { t , y } k - I _ { t , y + z } k | _ J = 0 . \\end{align*}"}
-{"id": "6932.png", "formula": "\\begin{align*} \\{ X ^ { * ( 3 ) } ( z ) , X ^ { * ( 3 ) } ( w ) \\} = 0 \\ , . \\end{align*}"}
-{"id": "3141.png", "formula": "\\begin{align*} \\begin{bmatrix} \\Lambda _ t ( \\lambda _ 0 ) \\otimes I _ n \\\\ \\widehat { N } _ t ( \\lambda _ 0 ) ( \\lambda _ 0 B + A ) ( \\Lambda _ t ( \\lambda _ 0 ) \\otimes I _ n ) \\end{bmatrix} x \\end{align*}"}
-{"id": "8984.png", "formula": "\\begin{align*} H ( x , y ) : = \\frac { g ( u ( x ) - w ( x ) ) - g ( u ( y ) - w ( y ) ) } { ( u ( x ) - w ( x ) - ( u ( y ) - w ( y ) ) } \\ , . \\end{align*}"}
-{"id": "1393.png", "formula": "\\begin{align*} \\mathrm { V a r } ( \\hat \\beta | \\hat \\tau > 0 ) = \\Sigma _ { \\beta \\beta } ( \\delta _ 0 ) - \\Sigma _ { \\beta \\tau } ( \\delta _ 0 ) \\Sigma ^ { - 1 } _ { \\tau \\tau } ( \\delta _ 0 ) \\Sigma _ { \\tau \\beta } ( \\delta _ 0 ) + \\Sigma _ { \\beta \\tau } ( \\delta _ 0 ) \\Sigma ^ { - 2 } _ { \\tau \\tau } ( \\delta _ 0 ) \\Sigma _ { \\tau \\beta } ( \\delta _ 0 ) \\mathrm { V a r } ( \\hat \\tau - \\tau _ 0 | \\hat \\tau > 0 ) \\end{align*}"}
-{"id": "5313.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": "7322.png", "formula": "\\begin{align*} F \\hat { R } ( v _ { 0 } \\otimes v _ { 0 } ) & = F ( v _ { 0 } \\otimes v _ { 0 } ) + q ^ { - 2 } ( q ^ { 2 } - q ^ { - 2 } ) F ( v _ { 1 } \\otimes v _ { - 1 } ) \\\\ & = [ 2 ] ^ { 1 / 2 } ( v _ { - 1 } \\otimes v _ { 0 } + ( q ^ { 2 } + 1 - q ^ { - 2 } ) v _ { 0 } \\otimes v _ { - 1 } ) . \\end{align*}"}
-{"id": "2413.png", "formula": "\\begin{align*} \\begin{cases} \\frac { 1 } { p _ { 2 } } \\leq \\frac { 1 } { p _ { 1 } } \\\\ s _ { 2 } + R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ { 1 } , \\alpha _ { 2 } ) \\leq s _ { 1 } \\\\ \\frac { 1 } { q _ { 2 } } \\leq \\frac { 1 } { q _ { 1 } } \\end{cases} , \\end{align*}"}
-{"id": "2320.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( H ^ 1 ( \\R ^ d ) ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( H ^ 1 ( \\R ^ d ) ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "6946.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V ^ * _ \\pi ( z _ 1 ; X ) V ^ * _ \\pi ( z _ 2 ; X ) \\cdots V ^ * _ \\pi ( z _ m ; X ) \\cdot 1 \\cr & = [ Z ^ { \\lambda + \\delta } ] \\ \\prod _ { 1 \\le i < j \\le m } ( z _ i - z _ j ) \\ \\prod _ { \\ell = 1 } ^ m \\ , L ( z _ \\ell ; X ) \\ M _ { \\pi ' } ( Z ) \\cr & = [ s _ \\lambda ( Z ) ] \\ L ( X Z ) \\ , M _ { \\pi ' } ( Z ) \\ , . \\end{align*}"}
-{"id": "1703.png", "formula": "\\begin{align*} 1 - d ( 0 , C ) \\leq 1 - d ( 0 , k A ) = 1 - k d ( 0 , A ) . \\end{align*}"}
-{"id": "6132.png", "formula": "\\begin{align*} \\frac 3 8 I = & 3 M ( z - m ) + 3 z ( M - m ) + [ 2 ( z - m ) ^ 2 - 2 ( z - m ) ( M - z ) - ( M - z ) ^ 2 ] \\\\ = & 3 M ( 1 - M - 2 m ) + 3 ( 1 - M - m ) ( M - m ) + 2 ( 1 - M - 2 m ) ^ 2 \\\\ & - 2 ( 1 - M - 2 m ) ( 2 M + m - 1 ) - ( 2 M + m - 1 ) ^ 2 \\\\ = & - 4 M ^ 2 + 3 + 8 m M - 1 5 m + 1 4 m ^ 2 \\\\ = & - 4 M ^ 2 + 3 + ( - m ) ( 1 5 - 8 M ) + 1 4 m ^ 2 \\\\ > & 7 \\epsilon , \\end{align*}"}
-{"id": "4534.png", "formula": "\\begin{align*} U _ { 2 2 } & : = \\biggl | \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ \\frac { a _ n } { n - 1 } ^ 1 \\log \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } } \\biggr ) \\mathrm { B } _ { k , n - k - 1 } ( s ) \\biggl \\{ \\frac { ( n - 1 ) s - k } { n - k - 1 } \\biggr \\} \\ , d s \\ , d x \\biggr | \\\\ & = o ( n ^ { - ( 3 - \\epsilon ) } ) . \\end{align*}"}
-{"id": "3960.png", "formula": "\\begin{align*} \\lambda ^ { \\min } ( u ( r ) v ) = \\lambda ^ { \\min } ( v ) . \\end{align*}"}
-{"id": "1908.png", "formula": "\\begin{align*} \\phi _ g ( u ) = \\phi _ { g , 0 } ( u ) + \\frac { c } { 2 } \\sum _ { k = 1 } ^ { \\infty } ( 1 + ( 8 k ) ) ^ { - \\beta } ( \\phi _ { g , 0 } ( u + 1 2 k ) + \\phi _ { g , 0 } ( u - 1 2 k ) ) . \\end{align*}"}
-{"id": "7199.png", "formula": "\\begin{align*} - \\Delta w + ( u \\cdot \\nabla ) w + \\nabla p = 0 , \\qquad { \\rm d i v } \\ , w = 0 \\mbox { i n } \\ , \\ , \\R ^ n . \\end{align*}"}
-{"id": "7877.png", "formula": "\\begin{align*} \\lim _ { h \\downarrow 0 } \\int ^ t _ 0 \\int _ { \\R ^ d } \\left ( q _ 0 ( t + h - s , x , z ) - q _ 0 ( t - s , x , z ) \\right ) q _ 0 ( s , z , y ) d z d s = 0 . \\end{align*}"}
-{"id": "5539.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 1 \\\\ - z \\rho & 0 \\end{bmatrix} _ t - \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} _ x + \\big [ \\begin{bmatrix} 0 & 1 \\\\ - z \\rho & 0 \\end{bmatrix} , \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} \\big ] = 0 , \\end{align*}"}
-{"id": "6741.png", "formula": "\\begin{align*} \\langle T _ 1 \\wedge \\ldots \\wedge T _ p \\rangle = \\langle \\theta _ { \\varphi _ 1 } \\wedge \\ldots \\wedge \\theta _ { \\varphi _ 1 } \\rangle . \\end{align*}"}
-{"id": "9154.png", "formula": "\\begin{align*} \\beta & = \\xi _ 1 ( p - t + 1 ) + \\sum _ { r = 2 } ^ s ( t - 1 ) \\xi _ r \\binom { p - t + 1 } { ( r - 1 ) t + 1 } \\\\ \\gamma & = ( p - t + 1 ) \\sum _ { r = 1 } ^ { s - 1 } \\xi _ { r + 1 } \\binom { p - t } { r t } . \\end{align*}"}
-{"id": "9114.png", "formula": "\\begin{align*} u ' ( t ) + B ( t ) A ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) , \\ u ( 0 ) = u _ 0 \\end{align*}"}
-{"id": "7187.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac { \\pi } { 2 } } H ( \\varphi ) \\sin \\varphi d \\varphi = h ( \\frac { \\pi } { 2 } ) = 0 . \\end{align*}"}
-{"id": "2267.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\delta ( \\zeta ) & { } = \\sum _ { \\ell = 1 } ^ k \\frac 1 \\ell ( 1 - \\zeta ) ^ \\ell = \\sum \\limits ^ k _ { i = 0 } \\delta _ i \\zeta ^ { i } , \\beta ( \\zeta ) = 1 , \\\\ \\gamma ( \\zeta ) & { } = \\frac 1 \\zeta \\big [ 1 - ( 1 - \\zeta ) ^ k \\big ] = \\sum _ { i = 0 } ^ { k - 1 } \\gamma _ i \\zeta ^ i . \\end{aligned} \\right . \\end{align*}"}
-{"id": "2238.png", "formula": "\\begin{align*} p ( x ) = \\frac { C ( \\varepsilon ) } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon x ^ { p } ) ^ { n } } { n ! } . \\end{align*}"}
-{"id": "7035.png", "formula": "\\begin{align*} V _ + ( x ) = \\int _ { 0 < y \\le x } y ^ 2 \\Pi ( \\mathrm { d } y ) \\mbox { a n d } V _ - ( x ) = \\int _ { - x \\le y < 0 } y ^ 2 \\Pi ( \\mathrm { d } y ) , x > 0 . \\end{align*}"}
-{"id": "7120.png", "formula": "\\begin{align*} [ f , g ] = f \\circ _ { \\tilde { \\phi } } g - ( - 1 ) ^ { ( m - 1 ) ( l - 1 ) } g \\circ _ { \\tilde { \\phi } } f \\end{align*}"}
-{"id": "9203.png", "formula": "\\begin{align*} u ( t , x , \\omega ) = u _ 1 ( t , x , Z , \\omega ) \\end{align*}"}
-{"id": "7008.png", "formula": "\\begin{align*} \\mathrm { g p h } \\ , B _ \\mathcal { R } ( \\cdot , p ) & = \\left \\{ ( t , P ) \\in T \\times \\Pi ( X ) \\mid \\theta _ p ( t , P ) \\le 0 \\right \\} \\in \\Sigma \\otimes \\mathrm { B o r e l } ( \\Pi ( X ) ) . \\end{align*}"}
-{"id": "2404.png", "formula": "\\begin{align*} f ( y ) - f ( x ) & = h ( 1 ) - h ( 0 ) = \\int _ { 0 } ^ { 1 } \\nabla h ( \\tau ) d \\tau = \\int _ { 0 } ^ 1 \\langle y - x , \\nabla f ( x + \\tau ( y - x ) ) \\rangle d \\tau \\end{align*}"}
-{"id": "2256.png", "formula": "\\begin{align*} \\gamma _ { q , p } ( q , 2 ) = \\beta _ { q , p } \\left ( \\frac { ( 4 q - 1 ) ! ! } { 6 } \\varepsilon _ { q } ^ { 4 } \\sigma ^ { 4 q } + \\frac { ( 2 q + 2 p - 1 ) ! ! } { 2 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } ^ { 2 } \\sigma ^ { 2 q + 2 p } \\right ) , \\end{align*}"}
-{"id": "4018.png", "formula": "\\begin{align*} \\dim W \\leq n - ( k - 1 ) { d } = { d } + r _ d . \\end{align*}"}
-{"id": "129.png", "formula": "\\begin{align*} \\| x \\| ^ * = \\inf \\Bigl \\{ \\sum _ { i = 1 } ^ n \\| x _ i \\| _ { X } : \\sum _ { i = 1 } ^ n x _ i = x , \\ x _ i \\in X , \\ n \\in \\mathbb { N } \\Bigr \\} . \\end{align*}"}
-{"id": "3864.png", "formula": "\\begin{align*} \\begin{aligned} \\dfrac { d ^ { B ^ { ( 1 ) } _ { n + 1 } } _ { k , l } ( z ) } { ( z - q ^ { \\mathtt { h } ^ \\vee } ) ^ { \\delta _ { k l } } } & = D _ { k , l } ( z ) \\times D _ { k , l ^ * } ( z ) = D _ { k , l } ( z ) \\times D _ { k ^ * , l } ( z ) \\\\ & = D _ { k ^ * , l ^ * } ( z ) \\times D _ { k , l ^ * } ( z ) = D _ { k ^ * , l ^ * } ( z ) \\times D _ { k ^ * , l } ( z ) \\end{aligned} \\end{align*}"}
-{"id": "5095.png", "formula": "\\begin{align*} & A ( z ) A ( w ) = A ( w ) A ( z ) , C _ { a } ( z ) C _ { a } ( w ) = C _ { a } ( w ) C _ { a } ( z ) ( 1 \\le a \\le r ) , \\\\ & C _ { a } ( z ) A ( w ) = f ( z , w ) A ( w ) C _ { a } ( z ) + g ( z , w ) A ( z ) C _ { a } ( w ) ( 1 \\le a \\le r ) , \\\\ & q ^ { 2 } C _ { b } ( z ) C _ { a } ( w ) = f ( z , w ) C _ { a } ( w ) C _ { b } ( z ) - g ( w , z ) C _ { a } ( z ) C _ { b } ( w ) ( 1 \\le b < a \\le r ) , \\end{align*}"}
-{"id": "622.png", "formula": "\\begin{align*} 0 \\le \\psi _ r ( \\lambda ) \\leq \\psi _ r \\left ( r ^ { - 2 } \\left ( y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) ^ 2 \\right ) = - \\frac { \\pi } { \\Gamma ( \\frac { N - 2 } { 2 } ) } \\left ( { \\textstyle \\frac 1 2 } y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) ^ { \\frac { N - 2 } { 2 } } Y _ { \\frac { N - 2 } { 2 } } \\left ( y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) = \\gamma _ N \\end{align*}"}
-{"id": "8050.png", "formula": "\\begin{align*} \\pi ( B _ { I } ) = \\{ ( p _ { 1 } , p _ { 2 } , \\cdots , p _ { n } ) \\ ; | \\ ; p _ { i } = p _ { j } \\mbox { f o r a l l } i , j \\in I \\} . \\end{align*}"}
-{"id": "2858.png", "formula": "\\begin{align*} T o t ( C ^ { \\bullet } ) = \\int _ { \\underline { n } \\in \\Delta } ( C ^ n ) ^ { \\Delta ^ n } . \\end{align*}"}
-{"id": "7804.png", "formula": "\\begin{gather*} 2 \\left \\langle \\nabla f , \\nabla \\left ( \\nabla ^ { k } \\mathrm { R m } \\right ) \\right \\rangle = - \\left ( k + 2 \\right ) \\nabla ^ { k } \\mathrm { R m } \\\\ + \\sum _ { j = 0 } ^ { k } \\nabla ^ { j } \\mathrm { R m } \\ast \\nabla ^ { k - j } \\mathrm { R m } + 2 \\Delta \\nabla ^ { k } \\mathrm { R m } . \\end{gather*}"}
-{"id": "4084.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { x y ( a x + b y + c z ) } { z ^ 3 } , f ( x , y , z ) = \\dfrac { x y ( a x + b y + c z ) ^ 2 } { z ^ 4 } , \\\\ f ( x , y , z ) = \\dfrac { x y ^ 2 ( a x + b y + c z ) ^ 3 } { z ^ 6 } , \\end{gather*}"}
-{"id": "7161.png", "formula": "\\begin{align*} w = 0 \\mbox { o n } \\ , \\ , \\partial \\R ^ n _ + = \\{ x _ n = 0 \\} , \\end{align*}"}
-{"id": "7689.png", "formula": "\\begin{align*} \\lim _ { i \\to \\infty } \\gamma _ i = 0 . \\end{align*}"}
-{"id": "5437.png", "formula": "\\begin{align*} \\begin{bmatrix} \\alpha \\overline \\alpha & \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & \\overline \\alpha ^ 2 \\\\ 0 & - \\alpha J ( T ^ { - r } , T ^ { - r } ) J ( T ^ { - 2 r } , T ^ { - 3 r } ) & \\alpha ^ 2 J ( T ^ { - 3 r } , T ^ { - 3 r } ) - \\overline \\alpha ^ 2 J ( T ^ { - r } , T ^ { - r } ) \\\\ 0 & \\overline \\alpha ^ 2 J ( T ^ { - 2 r } , T ^ { - r } ) - \\alpha ^ 2 J ( T ^ { - 2 r } , T ^ { - 3 r } ) & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "8072.png", "formula": "\\begin{align*} H ( a , b ) = h ( a , b , x ) = \\lim _ { n \\to \\infty } h ( a _ n , b _ n , x ) = \\lim _ { n \\to \\infty } H ( a _ n , b _ n ) , \\end{align*}"}
-{"id": "3801.png", "formula": "\\begin{align*} D _ 1 & \\le \\sum _ { q = 0 } ^ \\infty \\| \\Delta _ q ( u ( \\tau ) - u ( t ) ) \\| _ { L ^ 2 } \\sum _ { q = 0 } ^ \\infty \\| \\Delta _ q u ( t ) \\| _ { L ^ 2 } \\\\ & = \\| u ( \\tau ) - u ( t ) \\| _ { B ^ { 0 } _ { 2 , 1 } } \\| u ( t ) \\| _ { B ^ { 0 } _ { 2 , 1 } } . \\end{align*}"}
-{"id": "801.png", "formula": "\\begin{align*} & A ( z ) A ( w ) = A ( w ) A ( z ) , C _ { a } ( z ) C _ { a } ( w ) = C _ { a } ( w ) C _ { a } ( z ) ( 1 \\le a \\le r ) , \\\\ & C _ { a } ( z ) A ( w ) = f ( z , w ) A ( w ) C _ { a } ( z ) + g ( z , w ) A ( z ) C _ { a } ( w ) ( 1 \\le a \\le r ) , \\\\ & q ^ { 2 } C _ { b } ( z ) C _ { a } ( w ) = f ( z , w ) C _ { a } ( w ) C _ { b } ( z ) - g ( w , z ) C _ { a } ( z ) C _ { b } ( w ) ( 1 \\le b < a \\le r ) , \\end{align*}"}
-{"id": "7971.png", "formula": "\\begin{align*} \\varphi \\left ( \\frac { h _ { K _ 1 } ( u ) } { h _ { { + } _ { \\varphi } ( K _ 1 , \\dots , K _ m ) } ( u ) } , \\dots , \\frac { h _ { K _ m } ( u ) } { h _ { { + } _ { \\varphi } ( K _ 1 , \\dots , K _ m ) } ( u ) } \\right ) = 1 . \\end{align*}"}
-{"id": "9856.png", "formula": "\\begin{align*} \\bar \\Xi ^ \\star = \\arg \\max _ { \\bar \\Xi \\ge 0 } ~ v ( \\bar \\Xi ) . \\end{align*}"}
-{"id": "1110.png", "formula": "\\begin{align*} \\| u _ i \\| _ 2 \\leq \\sqrt { \\| u _ i \\| _ 0 } \\cdot \\| u _ i \\| _ \\infty & = \\sqrt { k ( t - 1 ) - \\ell } \\cdot \\| u _ i \\| _ \\infty \\\\ & \\leq \\sqrt { k ( t - 1 ) } \\cdot \\| u _ i \\| _ \\infty \\\\ & \\leq \\alpha \\sqrt { k / ( t - 1 ) } . \\end{align*}"}
-{"id": "3990.png", "formula": "\\begin{align*} \\pi _ { p } ( v _ { i } ) = \\pi _ { p } ( v ) _ { i } = \\pi _ { p } ( v ) . \\end{align*}"}
-{"id": "1672.png", "formula": "\\begin{gather*} D _ x G _ { \\lambda } : L ^ p \\big ( \\R ^ d _ v ; H ^ { s ' } _ { p } ( \\R ^ d _ x ) \\big ) \\to L ^ p ( \\R ^ { 2 d } ) = L ^ p \\big ( \\R ^ d _ x ; L ^ p ( \\R ^ d _ v ) \\big ) , \\\\ D _ x G _ { \\lambda } : L ^ p \\big ( \\R ^ d _ v ; H ^ { 1 } _ { p } ( \\R ^ d _ x ) \\big ) \\to L ^ p \\big ( \\R ^ d _ x ; H ^ { 2 } _ { p } ( \\R ^ d _ v ) \\big ) . \\end{gather*}"}
-{"id": "1287.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & c ( i + r , 3 r ) & c ( i + 3 r , r ) \\\\ 0 & 0 & c ( i + r , r ) & c ( i + 3 r , 3 r ) \\\\ c ( i , r ) & c ( i + 2 r , 3 r ) & 0 & 0 \\\\ c ( i , 3 r ) & c ( i + 2 r , r ) & 0 & 0 \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "9174.png", "formula": "\\begin{align*} w _ j ^ { n + 1 } = w _ j ^ { n } + \\frac { k } { h } \\sum _ { l = 1 } ^ { r } \\alpha _ l ( w ^ n _ { j + l } - w ^ n _ { j - l } ) , \\end{align*}"}
-{"id": "1829.png", "formula": "\\begin{align*} u ( s _ { 1 } , \\dots , s _ { k } , z _ { k + 1 } , \\dots , z _ { 3 g - 3 + n } ) = \\psi ( z , \\bar z ) + \\sum _ { i = 1 } ^ { k } 2 \\pi s _ { i } + \\sum _ { i = 1 } ^ { k } s _ { i } ^ { 2 } \\phi _ { i } ( s , z , \\bar z ) , \\end{align*}"}
-{"id": "817.png", "formula": "\\begin{align*} C ^ { [ M ' , M ] } ( z ) = \\begin{pmatrix} 0 & 1 \\end{pmatrix} L ^ { ( M ' ) } ( z ) L ^ { ( M ' + 1 ) } ( z ) \\cdots L ^ { ( M ) } ( z ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "6092.png", "formula": "\\begin{align*} S _ 2 ( t ) & \\ll \\sum _ { 2 \\log t < k - \\delta < \\sqrt { \\frac { t } { \\log t } } } \\left ( \\frac { \\exp ( i \\pi \\delta ) } { \\pi } \\frac { ( - 1 ) ^ k } { k - \\delta } + \\frac { 1 } { ( k - \\delta ) \\log t } + \\frac { 1 } { \\sqrt { t \\log t } } \\right ) \\\\ & \\quad + \\sum _ { \\sqrt { \\frac { t } { \\log t } } < k - \\delta < 2 \\sqrt { t \\log t } } \\left ( \\frac { 1 } { k - \\delta } + \\frac { 1 } { \\sqrt { t \\log t } } \\right ) = O ( \\log \\log t ) . \\end{align*}"}
-{"id": "9989.png", "formula": "\\begin{align*} f _ { k , \\alpha } ( \\tilde { p } ) = \\alpha \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\tilde { p } | H _ { \\tilde { i } k } | ^ 2 } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\ ! + \\ ! ( \\ ! 1 \\ ! - \\ ! \\alpha \\ ! ) \\left [ \\log _ 2 \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { C _ { 1 } - \\alpha | w _ { \\bar { k } 2 } | ^ 2 \\tilde { p } } { \\sigma ^ 2 _ n C _ 0 } \\ ! \\Big ) + \\log _ 2 \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\alpha | w _ { \\bar { k } 1 } | ^ 2 \\tilde { p } - C _ { 2 } } { \\sigma ^ 2 _ n C _ 0 } \\ ! \\Big ) \\right ] . \\end{align*}"}
-{"id": "2231.png", "formula": "\\begin{align*} U ( n , d , q ) \\leq d q ^ { d } ( l n \\frac { n } { d } + l n q ) = O ( d q ^ { d } \\log n ) . \\end{align*}"}
-{"id": "2365.png", "formula": "\\begin{align*} \\mu ^ { f _ n , k } _ \\tau & = Q ^ { u , f _ n } _ { \\tau - t ^ k _ j } \\ , \\mu ^ { f _ n , k } _ { t ^ k _ j } , \\\\ \\mu ^ { f , k } _ \\tau & = Q ^ { \\bar { u } , f } _ { \\tau - t ^ k _ j } \\ , \\mu ^ { f , k } _ { t ^ k _ j } , \\end{align*}"}
-{"id": "9476.png", "formula": "\\begin{align*} \\begin{bmatrix} A ( s , t ) & B ( s , t ) \\\\ B ( s , t ) & C ( s , t ) \\end{bmatrix} \\end{align*}"}
-{"id": "1486.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ 2 \\sum _ { Q \\in \\mathfrak { A } _ k ( Q _ 0 ) } \\alpha _ Q \\chi _ Q \\right | \\leq 2 \\sum _ { k = 1 } ^ 2 \\sum _ { Q \\in \\mathfrak { A } _ { k - 1 } ( Q _ 0 ) } \\omega _ { \\lambda _ w } ( f ; Q ) \\chi _ Q , \\end{align*}"}
-{"id": "1622.png", "formula": "\\begin{align*} A ^ { i j } ( a _ { i j } u + b _ { i j } ) - ( a _ { , i } u + b _ { , i } ) B ^ { i } = 0 , \\end{align*}"}
-{"id": "7294.png", "formula": "\\begin{align*} [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q = E _ \\xi E _ { \\xi ^ \\prime } ^ * - q ^ { - ( \\xi , \\xi ^ \\prime ) } E _ { \\xi ^ \\prime } ^ * E _ \\xi . \\end{align*}"}
-{"id": "5607.png", "formula": "\\begin{align*} \\widetilde { S } _ { \\ell } ( \\alpha ) = \\sum _ { n = 1 } ^ { \\infty } \\Lambda ( n ) e ^ { - n ^ { \\ell } / N } e ( n ^ { \\ell } \\alpha ) , \\end{align*}"}
-{"id": "5330.png", "formula": "\\begin{align*} N & = 3 , & U & = 1 . 3 , & \\mu & = 0 . 5 , & t & = - 3 . 7 , \\\\ \\phi _ 1 & = \\pi / 3 , & \\theta _ 1 & = \\pi / 6 , & \\phi _ 2 & = \\pi / 4 , & \\theta _ 2 & = \\pi / 7 , \\end{align*}"}
-{"id": "4717.png", "formula": "\\begin{align*} u _ j + v _ j \\neq 0 , \\ \\ \\gamma ^ { \\top } ( u + v ) = \\gamma ^ { \\top } u + \\gamma ^ { \\top } v \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "8463.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\star } + \\sum _ { j = 1 } ^ { d } A _ { j } ( \\tilde { \\textbf { u } } ^ { \\star } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } ^ { \\star } = ( 0 , - \\nabla P ^ { \\star } ) ^ { T } . \\end{align*}"}
-{"id": "9349.png", "formula": "\\begin{align*} \\Gamma ( t , x ) = \\exp \\{ x G ( t ) + \\frac { 1 } { 2 } x ^ 2 t \\} . \\end{align*}"}
-{"id": "3841.png", "formula": "\\begin{align*} d _ k ( x ) = \\sum _ { i = 0 } ^ { k - 3 } ( i + 1 ) x ^ i + \\sum _ { i = k - 2 } ^ { 2 k - 3 } ( 2 k + 2 - i ) x ^ i . \\end{align*}"}
-{"id": "6678.png", "formula": "\\begin{align*} \\{ x \\in \\operatorname { M r k } ( D ) : ( \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } F ) ( x ) = 0 \\} = \\Sigma ^ { D _ 1 , \\dots , D _ p } _ { d _ 1 , \\dots , d _ p } \\cap \\operatorname { M r k } ( D ) . \\end{align*}"}
-{"id": "8861.png", "formula": "\\begin{align*} Q _ T ( s ) = \\P _ T \\left ( \\left \\{ \\omega \\in \\Omega _ T \\ , | \\ , \\sigma _ T ( \\omega ) = s T \\right \\} \\right ) . \\end{align*}"}
-{"id": "6254.png", "formula": "\\begin{align*} \\Delta u = - \\lambda _ 1 u \\ \\ \\mbox { o n } \\ \\ \\Omega , \\ \\ \\ u | _ { \\dd \\Omega } = 0 , \\end{align*}"}
-{"id": "6170.png", "formula": "\\begin{align*} \\Sigma = \\begin{pmatrix} \\sigma _ U ^ 2 & \\sigma _ { U L } \\\\ \\sigma _ { U L } & \\sigma _ L ^ 2 \\end{pmatrix} \\in \\R ^ { 2 \\times 2 } \\end{align*}"}
-{"id": "4138.png", "formula": "\\begin{align*} \\mathcal { W } ( \\theta _ 1 , \\theta _ 2 ) = \\{ r e ^ { i \\theta } : r > 0 , \\theta \\in [ \\theta _ 1 , \\theta _ 2 ] \\} \\end{align*}"}
-{"id": "6785.png", "formula": "\\begin{align*} d ( s ) = \\frac { 1 } { s } \\left ( \\int _ X ( \\varphi _ { t + s } - \\varphi _ t ) d d ^ c \\dot { \\varphi } _ { t + s } \\wedge T _ s + \\int _ X ( \\dot { \\varphi } _ { t + s } - \\dot { \\varphi } _ t ) \\theta _ { \\varphi _ t } ^ n \\right ) . \\end{align*}"}
-{"id": "1434.png", "formula": "\\begin{align*} Q _ { i } \\cdot G = 0 \\mathrm { f o r } \\ i = 1 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "9793.png", "formula": "\\begin{align*} m ( \\mathbf { y } | M _ { k } ) = \\int f ( \\mathbf { y } | \\boldsymbol { \\theta } _ { k } , M _ { k } ) \\pi ( \\boldsymbol { \\theta } _ { k } | M _ { k } ) d \\boldsymbol { \\theta } _ { k } . \\end{align*}"}
-{"id": "8902.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\R ^ N } \\abs { D _ { A _ n } z _ { n } - D z } ^ 2 + \\abs { z _ { n } - z } ^ 2 = 0 . \\end{align*}"}
-{"id": "7241.png", "formula": "\\begin{align*} \\widehat { w } _ j = w _ j + \\sum _ { | \\alpha | \\geq 2 } a _ { j , \\alpha } ( z _ j ) \\cdot w _ j ^ \\alpha . \\end{align*}"}
-{"id": "668.png", "formula": "\\begin{align*} \\nu ( B ) \\nu ( C ) = \\eta ( f ) = \\eta ( \\chi _ { A ^ c } f ) \\leq \\eta ( A ^ c ) \\nu ( C ) = ( 1 - \\eta ( A ) ) \\nu ( C ) . \\end{align*}"}
-{"id": "8865.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac 1 T X _ T ( \\omega ) = x ( \\omega ) \\end{align*}"}
-{"id": "9490.png", "formula": "\\begin{align*} d \\beta ^ { n } & = e ^ { - 2 n \\rho } ( - 2 \\ , d \\rho \\wedge d \\theta - 2 \\ , d \\rho \\wedge \\alpha _ { n - 1 } ) \\wedge ( d \\alpha _ { n - 1 } ) ^ { n - 1 } \\\\ & = - 2 e ^ { - 2 n \\rho } \\ , d \\rho \\wedge d \\theta \\wedge ( d \\alpha _ { n - 1 } ) ^ { n - 1 } \\end{align*}"}
-{"id": "9020.png", "formula": "\\begin{align*} f _ \\psi ( p , y , z ) = f ( \\psi ( p , y ) , z ) = a _ d ( \\psi ( p , y ) ) z ^ d + a _ { d - 1 } ( \\psi ( p , y ) ) z ^ { d - 1 } + \\cdots + a _ 0 ( \\psi ( p , y ) ) . \\end{align*}"}
-{"id": "1588.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int L _ { Y _ { I } } C _ { x } d x + C ^ { x } Y _ { I } + 2 \\psi _ { I } = 0 \\end{align*}"}
-{"id": "6504.png", "formula": "\\begin{align*} | \\mathcal { U } _ { ( n , d ) } | & \\leq d \\cdot N ( n , ( d / 2 , d / 2 ) ) \\\\ & = d \\cdot N ( d / 2 , d / 2 ) ^ { 1 + o ( 1 ) } \\cdot \\log n \\\\ & = d \\cdot 2 ^ { H _ { 2 } ( 1 / 2 ) d + o ( d ) } \\cdot \\log n \\\\ & = d \\cdot 2 ^ { d + o ( d ) } \\cdot \\log n \\end{align*}"}
-{"id": "6436.png", "formula": "\\begin{align*} - \\frac { 1 } { 1 - q } \\int _ { B _ { 1 } } \\psi ^ { 1 + q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ^ { 1 - q } ) d x & - \\mathcal { E } ( h _ { m } * \\tilde { u } , \\psi ^ { 1 + q } \\tilde { u } ^ { - q } ) \\\\ & \\leq \\frac { - q } { 1 - q } \\int _ { B _ { 1 } } \\psi ^ { 1 + q } \\tilde { u } ^ { 1 - q } g _ { 1 - \\alpha , m } d x . \\end{align*}"}
-{"id": "9761.png", "formula": "\\begin{align*} \\mathcal { A } ( \\bar { \\vec { x } } ) : = \\{ i \\in \\{ 1 , \\ldots , m \\} \\ | \\ g _ i ( \\vec { z } ^ { * } ( \\bar { \\vec { x } } ) , \\bar { \\vec { x } } ) = 0 \\} . \\end{align*}"}
-{"id": "6590.png", "formula": "\\begin{align*} F _ { \\gamma } ^ { \\rm { F B } } ( x ) = \\gamma ^ { - 1 } \\left ( \\langle x - \\gamma \\nabla f ( x ) , x \\rangle - ( \\tfrac { 1 } { 2 } \\| x \\| ^ 2 - \\gamma f ( x ) ) - r _ { \\gamma g } ^ * ( x - \\gamma \\nabla f ( x ) ) \\right ) . \\end{align*}"}
-{"id": "8279.png", "formula": "\\begin{align*} \\widehat { \\rho } _ { j } ^ { t + 1 } = 1 - \\frac { 1 } { 1 + \\mathrm { e x p } \\left ( l _ { j \\leftarrow n } ^ { t } \\right ) } , \\end{align*}"}
-{"id": "9879.png", "formula": "\\begin{align*} h & ( t , x ) = \\int _ 0 ^ 1 G _ t ( x , y ) \\eta ( y ) d y - \\int _ 0 ^ t \\int _ 0 ^ 1 \\partial _ y G _ { t - s } ( x , y ) g ( s , h ( s ) ) ( y ) d y d s \\\\ & + \\int _ 0 ^ t \\int _ 0 ^ 1 G _ { t - s } ( x , y ) f ( s , h ( s ) ) ( y ) d y d s + \\int _ 0 ^ t \\int _ 0 ^ 1 G _ { t - s } ( x , y ) \\sigma ( s , h ( s ) ) v ( s , y ) d y d s . \\end{align*}"}
-{"id": "3100.png", "formula": "\\begin{align*} z = \\begin{bmatrix} N _ 1 ( \\lambda _ 0 ) ^ T \\\\ * \\end{bmatrix} x , \\end{align*}"}
-{"id": "6475.png", "formula": "\\begin{align*} R _ { m } ( t ) : = \\mathcal { E } ( h _ { m } * \\tilde { u } , \\psi ^ { 2 } \\tilde { u } ^ { - 1 } ) - \\mathcal { E } ( \\tilde { u } , \\psi ^ { 2 } \\tilde { u } ^ { - 1 } ) . \\end{align*}"}
-{"id": "9546.png", "formula": "\\begin{gather*} \\alpha _ 1 \\beta _ 2 = 0 , \\alpha _ 1 \\beta _ 3 = 0 , \\alpha _ 4 \\beta _ 1 = 0 , \\alpha _ 4 \\beta _ 2 = 0 , \\\\ - \\alpha _ 2 \\beta _ 1 = \\alpha _ 2 \\beta _ 2 = - \\alpha _ 3 \\beta _ 2 = - \\alpha _ 3 \\beta _ 1 = \\alpha _ 4 \\beta _ 3 , \\\\ \\alpha _ 1 \\beta _ 1 = - \\alpha _ 2 \\beta _ 3 = \\alpha _ 3 \\beta _ 2 = - \\alpha _ 3 \\beta _ 3 . \\end{gather*}"}
-{"id": "9729.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { h ( f ( t ) ) } { h ( g ( t ) ) } = L ^ \\gamma . \\end{align*}"}
-{"id": "4177.png", "formula": "\\begin{align*} \\sum _ { a \\in A } v _ a Z ( a , b ) = 1 b \\in A . \\end{align*}"}
-{"id": "4555.png", "formula": "\\begin{align*} \\mathcal { X } _ n ^ { ( 1 ) } : = \\{ x : f ( x ) \\geq k ^ \\frac { d } { 2 \\beta } \\delta _ n \\} \\ , , \\mathcal { X } _ n ^ { ( 2 ) } : = \\{ x : \\delta _ n \\leq f ( x ) < k ^ \\frac { d } { 2 \\beta } \\delta _ n \\} , \\end{align*}"}
-{"id": "6775.png", "formula": "\\begin{align*} \\theta _ { \\varphi _ j } ^ n = e ^ { \\varphi _ j } \\mu _ j . \\end{align*}"}
-{"id": "7404.png", "formula": "\\begin{align*} F ( m + n ) & = \\nu ( g _ 1 ^ { [ s _ 1 ( m + n ) ] } \\cdots g _ k ^ { [ s _ k ( m + n ) ] } ) \\\\ & \\leq \\nu ( g _ 1 ^ { [ s _ 1 m ] + [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k m ] + [ s _ k n ] } ) + \\nu ( ( g _ 1 ^ { [ s _ 1 m ] + [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k m ] + [ s _ k n ] } ) ^ { - 1 } ( g _ 1 ^ { [ s _ 1 ( m + n ) ] } \\cdots g _ k ^ { [ s _ k ( m + n ) ] } ) ) \\\\ & \\leq \\nu ( g _ 1 ^ { [ s _ 1 m ] + [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k m ] + [ s _ k n ] } ) + \\sum _ { i = 1 } ^ k \\nu ( g _ i ) . \\\\ \\end{align*}"}
-{"id": "6379.png", "formula": "\\begin{align*} p ( x ) = \\frac { C ( \\varepsilon ) } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } e ^ { - \\varepsilon x ^ { p } } \\end{align*}"}
-{"id": "9051.png", "formula": "\\begin{align*} E v ( k , z ) & : = \\left \\{ \\forall j \\in [ | k , N | ] , \\ , l _ j ^ { ( N ) } \\leq \\sqrt { \\frac { 1 } { 2 } } W _ { \\tau _ j ^ { ( k ) } } - \\tau ^ { ( k ) } _ j + z \\leq u _ j ^ { ( N ) } \\right \\} , \\\\ G E v ( k , z ) & : = \\left \\{ \\forall j \\in [ | k , N | ] , \\ , l _ j ^ { ( N ) } \\leq \\sqrt { \\frac { 1 } { 2 } } W _ { \\tau _ j ^ { ( k ) } } + z \\leq u _ j ^ { ( N ) } \\right \\} . \\end{align*}"}
-{"id": "6207.png", "formula": "\\begin{align*} \\| u \\| _ { L _ 2 ( P ) } = \\| \\phi _ { P , * } u \\| _ { L _ 2 ( J \\times B ) } \\leq C \\| | \\nabla _ { g _ C } ( \\phi _ { P , * } u ) | _ { g _ C } \\| _ { L _ 2 ( J \\times B ) } = C \\| | \\nabla _ { g _ P } u | _ { g _ P } \\| _ { L _ 2 ( P ) } . \\end{align*}"}
-{"id": "4924.png", "formula": "\\begin{align*} \\frac { 1 } { | W | } \\prod _ { i = 1 } ^ n ( h + e _ i - 1 ) , \\end{align*}"}
-{"id": "5613.png", "formula": "\\begin{align*} r ( n ) = \\sum _ { p _ { 1 } + p _ { 2 } ^ 2 + p _ { 3 } ^ { 2 } = n } \\log p _ { 1 } \\log p _ { 2 } \\log p _ { 3 } , \\end{align*}"}
-{"id": "690.png", "formula": "\\begin{align*} \\nu ( A D ) < \\frac { 1 } { 2 } , \\textrm { w h e r e $ A = \\bigcup _ { n } F _ n s _ n $ } . \\end{align*}"}
-{"id": "6126.png", "formula": "\\begin{align*} g = e ^ { 2 s ^ 2 f } Z ( w ) ^ 2 \\left ( \\frac { d w ^ { 2 } } { w ^ 2 } + w ^ 2 s ^ 2 d \\theta ^ { 2 } \\right ) , \\ Z ( w ) = \\frac { \\pi / w } { \\sin ( \\pi / w ) } \\end{align*}"}
-{"id": "9150.png", "formula": "\\begin{align*} g ( s , t ) = 1 - \\frac { d ^ 2 + d } { ( t + d ) ( 2 d + 1 ) } = \\frac { t } { t + d } + \\frac { d ^ 2 } { ( t + d ) ( 2 d + 1 ) } = \\frac { s + 1 + 1 / d } { ( 2 + 1 / d ) s } ~ . \\end{align*}"}
-{"id": "1336.png", "formula": "\\begin{align*} a ( \\lambda ) = \\prod _ { j = 1 } ^ K a _ j ( \\lambda ) , d ( \\lambda ) = \\prod _ { j = 1 } ^ K d _ j ( \\lambda ) . \\end{align*}"}
-{"id": "4825.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | ( I - A _ n ) [ K , \\tau ] | _ { \\mathcal I } = 0 \\end{align*}"}
-{"id": "7954.png", "formula": "\\begin{align*} { \\rm d e t } M ( \\lambda , \\sigma ) = 0 , \\end{align*}"}
-{"id": "6657.png", "formula": "\\begin{align*} \\mathcal { I } _ W ( T ) : = \\{ ( w , t ) : w \\in W , t \\in T , w t = t \\} \\end{align*}"}
-{"id": "7610.png", "formula": "\\begin{align*} \\lambda _ 1 ^ 2 = \\sum _ { y \\sim x } \\sum _ { z \\sim y } \\mathbf { v } _ z = \\sum _ { y \\sim x } \\sum _ { \\substack { z \\sim y \\\\ z \\in N ( x ) } } \\mathbf { v } _ z + \\sum _ { y \\sim x } \\sum _ { \\substack { z \\sim y \\\\ z \\not \\in N ( x ) } } \\mathbf { v } _ z \\leq 2 e \\left ( N ( x ) \\right ) + e \\left ( N ( x ) , V ( G ) \\setminus N ( x ) , \\right ) \\end{align*}"}
-{"id": "7917.png", "formula": "\\begin{align*} 6 \\beta \\int _ M \\eta ^ 2 u ^ { \\beta - 1 } | \\nabla u | ^ 2 + 1 2 \\int _ M \\nabla u \\cdot \\nabla \\eta \\eta u ^ { \\beta } + S _ 0 \\int _ M \\eta ^ 2 u ^ { \\beta + 1 } = \\int _ M S _ g u ^ 2 \\eta ^ 2 u ^ { \\beta + 1 } . \\end{align*}"}
-{"id": "8499.png", "formula": "\\begin{align*} ( z , x + a _ j y ) = ( z , x ^ \\prime + ( A + a _ j ) y ) = ( z ^ \\prime - m , x ^ \\prime + ( A + a _ j ) y ) \\end{align*}"}
-{"id": "2882.png", "formula": "\\begin{align*} s u E _ 1 - C o g ^ { c o n i l } ( u A s s - C o g ^ { c o n i l } ) = s u E _ 1 \\otimes u A s s - C o g ^ { c o n i l } \\end{align*}"}
-{"id": "976.png", "formula": "\\begin{align*} \\alpha ( \\overline { L } ) ( z ) = d h ( \\overline { L } ) ( z ) + 2 e ^ { h } i \\partial \\overline { \\partial } \\rho ( - i T , \\overline { L } ) ( z ) \\ ; , \\ ; z \\in M \\ ; , \\ ; L \\in \\mathcal { N } _ { z } \\ ; . \\end{align*}"}
-{"id": "1144.png", "formula": "\\begin{align*} - v _ { x x } = \\frac z T \\rho ( x ) v . \\end{align*}"}
-{"id": "9322.png", "formula": "\\begin{align*} & \\tilde { Z } ( 0 , z ) \\exp \\Big \\{ \\int _ 0 ^ T ( - \\pi ( s , z ) b _ 0 ( s , z ) + \\frac { 1 } { 2 } \\pi ^ 2 ( s , z ) \\sigma _ 0 ^ 2 ( s , z ) ) d s - \\int _ 0 ^ T \\pi ( s , z ) \\sigma _ 0 ( s , z ) d B ( s ) \\Big \\} \\\\ & = \\tilde { p } ( 0 , z ) \\exp \\Big \\{ - \\int _ 0 ^ T ( \\Phi ( s , z ) + \\frac { b _ 0 ( s , z ) } { \\sigma _ 0 ( s , z ) } ) d B ( s ) \\\\ & + \\frac { 1 } { 2 } \\int _ 0 ^ T ( \\Phi ^ 2 ( s , z ) - \\frac { b _ 0 ^ 2 ( s , z ) } { \\sigma _ 0 ^ 2 ( s , z ) } ) d s \\Big \\} . \\end{align*}"}
-{"id": "6938.png", "formula": "\\begin{align*} L ^ \\perp _ \\sigma ( w ) \\cdot 1 = \\begin{cases} 1 - w & \\mbox { i f $ \\sigma = ( 0 ) $ } ; \\cr 1 & \\mbox { o t h e r w i s e } , \\cr \\end{cases} \\end{align*}"}
-{"id": "3488.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta _ { x ' } v = f ( x ' ) + 2 \\partial _ \\nu u ( x ' , 0 ) & \\mbox { i n } D , \\\\ v = 0 & \\mbox { o n } \\partial D . \\end{cases} \\end{align*}"}
-{"id": "3238.png", "formula": "\\begin{align*} - 1 \\leq D _ { v _ { i } } \\mathrm { d i s t } _ { a _ { i } } ^ { X } \\left ( \\cdot \\right ) = - \\cos \\left ( \\sphericalangle \\left ( \\left ( \\Uparrow _ { p } ^ { a _ { i } } \\right ) _ { X } , v _ { i } \\right ) \\right ) \\leq - 1 + \\tau \\left ( \\eta \\right ) . \\label { c l o o o o s e a t p } \\end{align*}"}
-{"id": "575.png", "formula": "\\begin{align*} a \\equiv b + c \\bmod I ^ n c \\in I ^ { n - 1 } \\Rightarrow \\delta ( a ) \\equiv \\delta ( b ) + \\delta ( c ) \\bmod I ^ { n - 1 } \\end{align*}"}
-{"id": "5920.png", "formula": "\\begin{align*} \\lambda { \\psi } ( z ) - \\frac { 1 } { 2 } \\triangle _ v { \\psi } ( z ) - v \\cdot D _ x { \\psi } ( z ) = \\lambda { \\psi } ( z ) - { \\cal L } { \\psi } ( z ) = g ( z ) , \\ ; \\ ; \\ ; z \\in \\R ^ { 2 d } . \\end{align*}"}
-{"id": "2237.png", "formula": "\\begin{align*} C ( \\varepsilon ) = \\left ( \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon ) ^ { n } } { n ! } ( n p - 1 ) ! ! \\sigma ^ { n p } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "7596.png", "formula": "\\begin{align*} \\lambda _ i : = \\mathbb P \\{ \\omega \\in \\Omega : \\chi ( \\omega ) > 1 - \\varepsilon _ i \\} = \\int _ { \\max \\{ - 1 , 1 - \\varepsilon _ i \\} } ^ 1 \\phi ( t ) d t , \\end{align*}"}
-{"id": "805.png", "formula": "\\begin{align*} \\mathcal { F } ^ { * } = \\bigoplus _ { m _ { 1 } , \\ldots m _ { r } \\in \\mathbb { Z } _ { \\ge 0 } } \\mathbb { C } \\langle m _ { 1 } , \\ldots , m _ { r } | \\end{align*}"}
-{"id": "9795.png", "formula": "\\begin{align*} \\Pi _ { n } ( A ) = \\frac { \\int _ { A } f ( \\mathbf { y } ) \\Pi ( d f ) } { \\int _ { \\Omega } f ( \\mathbf { y } ) \\Pi ( d f ) } = \\frac { \\int _ { A } f ( \\mathbf { y } ) \\Pi ( d f ) } { m ( \\mathbf { y } ) } , \\end{align*}"}
-{"id": "8094.png", "formula": "\\begin{align*} \\omega _ { ( s _ i , r _ j ) } = \\frac { P _ { s _ i } ^ { m w t p } } { { E _ { s _ i } } } + \\frac { P _ { r _ j } ^ { m w t p } } { E _ { r _ j } } . \\end{align*}"}
-{"id": "4938.png", "formula": "\\begin{align*} h ^ { ( 1 ) } : = D ^ * _ { S _ 0 ^ c } h \\cdot I _ { \\{ i : | D ^ * _ { S _ 0 ^ c } h ( i ) | > \\alpha / ( t - 1 ) \\} } , h ^ { ( 2 ) } : = D ^ * _ { S _ 0 ^ c } h \\cdot I _ { \\{ i : | D ^ * _ { S _ 0 ^ c } h ( i ) | \\leq \\alpha / ( t - 1 ) \\} } . \\end{align*}"}
-{"id": "711.png", "formula": "\\begin{align*} \\overline { \\alpha } _ { f } ( P ) = \\limsup _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n k } ( P ) ) ^ { 1 / n k } = \\overline { \\alpha } _ { f ^ { k } } ( P ) ^ { 1 / k } \\end{align*}"}
-{"id": "5868.png", "formula": "\\begin{align*} C ^ { x } = - m + c _ { 1 } e ^ { - x } , \\end{align*}"}
-{"id": "4865.png", "formula": "\\begin{align*} f ( z ) = - h z + \\log ( 1 + 2 z ) - \\log ( 1 - 2 z ) . \\end{align*}"}
-{"id": "9631.png", "formula": "\\begin{align*} \\delta b = \\psi ( b ) \\delta \\end{align*}"}
-{"id": "7580.png", "formula": "\\begin{align*} \\mathbb P \\left [ \\mathcal A \\right ] = p _ 1 ^ { K _ 1 } . \\end{align*}"}
-{"id": "7300.png", "formula": "\\begin{align*} S ^ { - 1 } ( E _ { \\xi ^ \\prime } ^ * ) = ( S ^ { - 1 } ( S ^ 2 ( E _ { \\xi ^ \\prime } ) ) ) ^ * = q ^ { ( \\xi , 2 \\rho ) } S ^ { - 1 } ( E _ { \\xi ^ \\prime } ) ^ * . \\end{align*}"}
-{"id": "2127.png", "formula": "\\begin{align*} u ( x ' , z ) = U _ a ( z ) \\Phi ( x ' , z ) . \\end{align*}"}
-{"id": "5076.png", "formula": "\\begin{align*} ( \\mathcal { H } h ) ( \\vec { x } , \\vec { \\nu } ) = \\sum _ { \\begin{subarray} { c } ( \\vec { y } , \\vec { \\mu } ) \\in \\mathcal { S } _ { k _ { 1 } , \\ldots , k _ { r } } \\\\ ( \\vec { y } , \\vec { \\mu } ) \\not = ( \\vec { x } , \\vec { \\nu } ) \\end{subarray} } q ( \\vec { x } , \\vec { \\nu } | \\vec { y } , \\vec { \\mu } ) \\left \\{ h ( \\vec { y } , \\vec { \\mu } ) - h ( \\vec { x } , \\vec { \\nu } ) \\right \\} ( h \\in F ( \\mathcal { S } _ { k _ { 1 } , \\ldots , k _ { r } } ) ) , \\end{align*}"}
-{"id": "4425.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { C _ E } } = \\| S ( x ) \\| _ { \\widehat { E } } \\ \\ \\ \\ . \\end{align*}"}
-{"id": "7203.png", "formula": "\\begin{align*} \\sigma _ l \\ ! = \\ ! \\ ! \\sum \\nolimits _ { n , k } { \\ ! \\mathbb { E } \\ ! \\left \\{ \\beta _ n \\beta _ { n + k } \\right \\} \\ ! g ( d _ n ) g ( d _ { n + k } ) \\mathbb { P } ( { n \\ ! + \\ ! k } , \\tau ) f _ x ( n \\ ! ) . } \\end{align*}"}
-{"id": "4178.png", "formula": "\\begin{align*} d ( a , b ) + d ( b , c ) \\geq d ( a , c ) , d ( a , a ) = 0 \\end{align*}"}
-{"id": "9405.png", "formula": "\\begin{align*} A = \\begin{bmatrix} A _ { I I } & A _ { F I } ^ T & \\\\ A _ { F I } & A _ { F F } & A _ { R F } ^ T \\\\ & A _ { R F } & A _ { R R } \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "9477.png", "formula": "\\begin{align*} \\lambda _ { 0 } = d z + \\alpha _ { n } \\end{align*}"}
-{"id": "3544.png", "formula": "\\begin{align*} ( 2 \\psi , V ) = - \\Phi ( g , \\pi ) + \\chi \\Phi ( g _ 1 , \\pi _ 1 ) + ( 1 - \\chi ) \\Phi ( g _ 2 , \\pi _ 2 ) + ( 2 \\psi _ 0 R ^ { - 1 - q _ 0 } , 0 ) , \\end{align*}"}
-{"id": "7464.png", "formula": "\\begin{align*} Q ^ { ( k ) } ( V ) = \\sum _ { v \\in V } { d _ { i n } ( v ) \\choose 2 } = \\sum _ { j \\geq 0 } \\frac { j ( j - 1 ) } { 2 } Q ^ { ( k ) } _ j ( V ) \\end{align*}"}
-{"id": "4723.png", "formula": "\\begin{align*} 0 = \\pi ^ { \\top } d _ 0 = \\sum _ { i = 1 } ^ m ( y ^ i ) ^ { \\top } A ^ i d _ 0 . \\end{align*}"}
-{"id": "8580.png", "formula": "\\begin{align*} N _ { i n } ^ { 1 } & = \\displaystyle \\bigoplus _ { a \\in _ { k } T _ { 1 } } D _ { k } \\otimes _ { F } N _ { \\tau ( a ) } \\\\ N _ { o u t } ^ { 1 } & = \\displaystyle \\bigoplus _ { b \\in ( T _ { 1 } ) _ { k } } D _ { k } \\otimes _ { F } N _ { \\sigma ( b ) } \\end{align*}"}
-{"id": "8473.png", "formula": "\\begin{align*} \\mathcal { U } ^ { M } = \\{ \\tilde { \\textbf { u } } ^ { \\varepsilon } \\in H ^ { m } ( \\mathbb { R } ^ { d } ) : | | \\tilde { \\textbf { u } } ^ { \\varepsilon } | | _ { m } \\le M \\} . \\end{align*}"}
-{"id": "6014.png", "formula": "\\begin{gather*} \\Phi _ t ^ { \\mp } : = ( \\mp \\cosh t ) \\Phi _ I + ( \\sinh t ) \\Phi _ J - \\Phi _ K \\end{gather*}"}
-{"id": "3886.png", "formula": "\\begin{align*} S _ 0 - \\sum _ { i = 1 } ^ { k + 1 } S _ i = k ^ 2 ( k + 1 ) \\int _ X u \\ , \\sigma _ { k } ( D ^ 2 w ^ 1 , \\dotsc , D ^ 2 w ^ { k } ) d x + \\sum _ { i = 1 } ^ { 2 k + 1 } U _ i \\end{align*}"}
-{"id": "1847.png", "formula": "\\begin{align*} D _ { m , n } ( t ) & = \\frac { 1 } { \\sqrt { m + 1 } } \\int _ n ^ { n + 1 } \\frac { v - n } { v ^ { 3 / 2 } } \\left ( 1 + i \\log \\frac { m + 1 } { v } \\right ) ^ { - t - 1 } d v \\\\ E _ { m , n } ( t ) & = \\int _ n ^ { n + 1 } \\frac { v - n } { v ^ { 3 / 2 } } \\int _ m ^ { m + 1 } u ^ { - 1 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 1 } d u d v , \\end{align*}"}
-{"id": "2885.png", "formula": "\\begin{align*} C H _ { P o i s _ n } ^ { * } ( A , A ) \\ , : = \\ , H o m _ { \\Sigma } ( u { P o i s _ n } ^ * \\{ n \\} , E n d _ A ) [ - n ] \\end{align*}"}
-{"id": "4773.png", "formula": "\\begin{align*} H _ 0 = \\frac { 1 } { \\sqrt { \\varepsilon \\left ( ( f f '' + ( f ' ) ^ 2 + 1 ) ^ 2 - \\kappa ^ 2 ( f '^ 2 + 1 ) \\right ) } } \\left ( - \\kappa \\sqrt { f '^ 2 + 1 } \\ , n _ 1 - ( f f '' + ( f ' ) ^ 2 + 1 ) \\ , n _ 2 \\right ) . \\end{align*}"}
-{"id": "1734.png", "formula": "\\begin{gather*} \\gamma : = \\frac { r ^ 2 - 1 } { r ^ 2 + 1 } \\varphi + \\frac { s ^ 2 + 1 } { s ^ 2 - 1 } \\psi - 3 \\lambda + \\upsilon . \\end{gather*}"}
-{"id": "2769.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( B ^ { H } _ { t } ) \\diamond d B ^ { H } ( t ) = \\int _ 0 ^ T F ( B ^ { H } _ { t } ) \\circ d B ^ { H } ( t ) - H \\int _ 0 ^ T \\Delta _ { x } F ( B ^ { H } _ { t } ) t ^ { 2 H - 1 } d t . \\end{align*}"}
-{"id": "6489.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } u ( x , t ) + L u ( x , t ) & = \\rho ( t ) g ( x ) \\Omega \\times [ 0 , T ] , \\\\ u ( x , t ) & = 0 \\quad \\quad \\quad \\mathbb { R } ^ { n } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = 0 \\quad \\quad \\quad \\Omega , \\ , t = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2798.png", "formula": "\\begin{align*} Y _ { j , 1 } - Y _ { j , 2 } & = \\left ( \\phi ( b _ j ( T _ 1 ) ) - \\phi ( b _ j ( T _ 2 ) ) \\right ) \\ , \\biggl ( \\phi ( a _ j ) - \\frac { 1 } { n - j - 2 } \\sum _ { \\ell \\in U _ j ( T _ 1 , T _ 2 ) } \\phi ( \\ell ) \\biggr ) \\\\ & = \\frac { \\phi ( b _ j ( T _ 1 ) ) - \\phi ( b _ j ( T _ 2 ) ) } { n - j - 2 } \\ , \\sum _ { \\ell \\in U _ j ( T _ 1 , T _ 2 ) } \\ ( \\phi ( a _ j ) - \\phi ( \\ell ) \\ ) . \\end{align*}"}
-{"id": "9791.png", "formula": "\\begin{align*} \\log B F _ { k l } = \\log \\frac { p ^ { n } ( \\mathbf { y } | \\mathcal { F } _ { n , k } ) } { p ^ { n } ( \\mathbf { y } | \\mathcal { F } _ { n , l } ) } + \\log \\frac { \\Pi _ { n , k } ( \\mathcal { F } _ { n , k } ) } { \\Pi _ { n , l } ( \\mathcal { F } _ { n , l } ) } - \\log \\frac { \\Pi _ { n , k } ( \\mathcal { F } _ { n , k } | \\mathbf { y } ) } { \\Pi _ { n , l } ( \\mathcal { F } _ { n , l } | \\mathbf { y } ) } . \\end{align*}"}
-{"id": "6112.png", "formula": "\\begin{align*} \\mu _ s = \\frac { | d z | ^ 2 } { | z | ^ 2 } ( \\log | z | ) ^ 2 \\left ( \\frac { \\log | t | } { \\pi \\log | z | } \\sin \\frac { \\pi \\log | z | } { \\log | t | } \\right ) ^ 2 . \\end{align*}"}
-{"id": "9824.png", "formula": "\\begin{align*} M _ i \\ , { } _ \\lambda \\ , v _ m = Y _ i \\ , { } _ \\lambda \\ , v _ m = 0 . \\end{align*}"}
-{"id": "3569.png", "formula": "\\begin{align*} \\| ( f _ k , X _ k ) \\| _ { H ^ 2 _ { \\rho _ k } \\times H ^ 1 _ { \\rho _ k } } = 1 \\end{align*}"}
-{"id": "8598.png", "formula": "\\begin{align*} \\Pi _ { 2 } \\left ( \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } r ^ { - 1 } \\overline { N } ( ^ { \\ast } b ) \\left ( \\pi _ { b r } ( x ) \\right ) \\right ) = - \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } r ^ { - 1 } \\gamma ' \\xi _ { b e _ { k } } \\pi _ { b r } ( x ) \\end{align*}"}
-{"id": "1648.png", "formula": "\\begin{align*} B ^ { s } _ { p , p } ( \\R ^ d ) = W ^ { s , p } ( \\R ^ d ) \\end{align*}"}
-{"id": "4919.png", "formula": "\\begin{align*} M = \\oplus _ i L _ i . \\end{align*}"}
-{"id": "3438.png", "formula": "\\begin{align*} y ^ i ( t ) = \\zeta ^ i + \\int _ t ^ T f ^ i ( \\theta , y ^ i ( \\theta ) , z ^ i ( \\theta ) ) d \\theta - \\int _ t ^ T z ^ i ( \\theta ) d w ( \\theta ) , i = 1 , 2 \\end{align*}"}
-{"id": "76.png", "formula": "\\begin{align*} \\sigma _ \\varepsilon ' ( y ) + \\frac { p } { p - 1 } \\sigma _ \\varepsilon ( y ) ^ { ( p - 1 ) / p } & = \\frac { p } { p - 1 } ( 1 - y ) \\left [ \\frac { 1 } { 2 ^ { ( p - 1 ) / p } \\varepsilon ^ { 2 - p } } - \\frac { ( 1 - y ) ^ { ( 2 - p ) / ( p - 1 ) } } { 2 \\varepsilon ^ { p ( 2 - p ) / ( p - 1 ) } } \\right ] \\\\ & \\ge \\frac { p } { p - 1 } ( 1 - y ) \\frac { 2 ^ { 1 / p } - 1 } { 2 \\varepsilon ^ { 2 - p } } \\\\ & \\ge \\frac { p a ^ { 2 - p } } { p - 1 } ( 1 - y ) = \\psi ' ( y ) + \\frac { p } { p - 1 } \\psi ( y ) ^ { ( p - 1 ) / p } \\ , \\end{align*}"}
-{"id": "7343.png", "formula": "\\begin{gather*} \\langle 1 , 1 \\rangle = 1 , \\langle w _ { i } , v _ { j } \\rangle = \\delta _ { i j } , \\\\ \\langle w _ 0 \\wedge w _ 1 , v _ 1 \\wedge v _ 0 \\rangle = \\langle w _ { - 1 } \\wedge w _ 0 , v _ 0 \\wedge v _ { - 1 } \\rangle = q ^ { - 2 } , \\langle w _ { - 1 } \\wedge w _ { 1 } , v _ { 1 } \\wedge v _ { - 1 } \\rangle = 1 , \\\\ \\langle w _ { - 1 } \\wedge w _ { 0 } \\wedge w _ { 1 } , v _ { 1 } \\wedge v _ { 0 } \\wedge v _ { - 1 } \\rangle = 1 . \\end{gather*}"}
-{"id": "7164.png", "formula": "\\begin{align*} 0 = \\int _ { | x | = 1 } f ( x ) x _ j x _ k = a _ { j k } \\int _ { | x | = 1 } x _ j ^ 2 x _ k ^ 2 , \\end{align*}"}
-{"id": "9659.png", "formula": "\\begin{align*} x ' ( t ) & = - a g ( x ( t ) ) + b \\max _ { t - \\tau ( t ) \\leq s \\leq t } g ( x ( s ) ) , t \\geq 0 \\\\ x ( t ) & = \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] \\end{align*}"}
-{"id": "8750.png", "formula": "\\begin{align*} b ^ + & = \\displaystyle \\frac { t ( 1 - x ) } { t - x } , & b ^ - & = t ^ { - 1 } b ^ + , \\\\ c ^ + & = 1 - b ^ + , & c ^ - & = 1 - b ^ - . \\end{align*}"}
-{"id": "842.png", "formula": "\\begin{align*} & K _ { t } ( w ; z _ { \\ell ( 1 ) } , \\ldots , z _ { \\ell ( t ) } ) \\\\ & = \\sum _ { s = 0 } ^ { t - 2 } g ( w , z _ { \\ell ( 1 ) } ) g ( w , z _ { \\ell ( t ) } ) \\prod _ { i = 1 } ^ { s } g ( z _ { \\ell ( i ) } , z _ { \\ell ( i + 1 ) } ) \\prod _ { i = s + 2 } ^ { t - 1 } g ( z _ { \\ell ( i + 1 ) } , z _ { \\ell ( i ) } ) . \\end{align*}"}
-{"id": "1687.png", "formula": "\\begin{align*} \\big | D _ v \\tilde u ( Z _ s ) - D _ v \\tilde u ( Y _ s ) \\big | & = \\Big | \\sum _ { i = 1 } ^ { 2 d } ( Z _ s - Y _ s ) ^ i \\int _ 0 ^ 1 D _ i D _ v \\tilde u ( r Z _ s + ( 1 - r ) Y _ s ) \\ , \\dd r \\Big | \\\\ & \\le | Z _ s - Y _ s | \\int _ 0 ^ 1 \\big | D D _ v \\tilde u \\big ( r Z _ s + ( 1 - r ) Y _ s \\big ) \\big | \\ , \\dd r \\ , . \\end{align*}"}
-{"id": "3859.png", "formula": "\\begin{align*} \\frac { 2 n + 1 + a _ { i _ t } - b _ { i _ t } } { 2 } - \\frac { 2 n + 1 + a _ { i ^ \\vee _ t } - b _ { i ^ \\vee _ t } } { 2 } = 1 \\end{align*}"}
-{"id": "2908.png", "formula": "\\begin{align*} X ( z ) = V ( z ) e ^ { i q } z ^ { \\alpha _ 0 } \\quad \\mbox { a n d } X ^ * ( z ) = V ^ * ( z ) z ^ { - \\alpha _ 0 } e ^ { - i q } \\ , . \\end{align*}"}
-{"id": "3809.png", "formula": "\\begin{align*} H [ t ] : = \\sum _ { n = 1 } ^ \\infty t _ n J _ n e ^ { H [ t ] } : = \\sum _ { k = 1 } ^ \\infty \\frac { H [ t ] ^ k } { k ! } \\end{align*}"}
-{"id": "3800.png", "formula": "\\begin{align*} \\begin{cases} u _ { x x t } - u _ t + \\frac 9 2 u _ x u _ { x x } + \\frac 3 2 u u _ { x x x } - \\frac 3 2 u u _ x + u _ x = 0 \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "7654.png", "formula": "\\begin{align*} \\| u _ { k } \\| _ { L ^ { q } } = \\frac { \\omega _ { d } ^ { 1 / q } } { \\nu ( d , \\alpha ) } \\varepsilon ^ { d / q } k ^ { - \\alpha } , \\end{align*}"}
-{"id": "1436.png", "formula": "\\begin{align*} f _ { r e g } = e _ { - \\beta _ { 1 } } + \\cdots + e _ { - \\beta _ { n - 1 } } + e _ { - 2 \\beta _ { n } } \\end{align*}"}
-{"id": "4203.png", "formula": "\\begin{align*} T ( F ( u , c v ) ) = X _ { \\tilde u , \\ , \\tilde v c / a ^ { 4 m } } ( a ^ { 2 m } ) = Y _ { \\tilde u c / a ^ { 4 m } , \\ , \\tilde v } ( a ^ { 2 m } ) = T ( S ( u c , v ) ) . \\end{align*}"}
-{"id": "2420.png", "formula": "\\begin{align*} \\Vert \\{ \\lambda _ { k } \\} \\Vert _ { l _ { q } ^ { s , \\alpha } } = \\begin{cases} \\left ( \\sum _ { k \\in \\mathbb { Z } ^ { n } } ^ { { } } \\langle k \\rangle ^ { \\frac { s q } { 1 - \\alpha } } | \\lambda _ { k } | ^ { q } \\right ) ^ { \\frac { 1 } { q } } ~ & ~ 0 < q < \\infty , \\\\ \\sup _ { k \\in \\mathbb { Z } ^ { n } } \\left ( \\langle k \\rangle ^ { \\frac { s } { 1 - \\alpha } } | \\lambda _ { k } | \\right ) ~ & ~ q = \\infty . \\end{cases} \\end{align*}"}
-{"id": "390.png", "formula": "\\begin{align*} S = \\{ 0 \\} \\sqcup G ^ 1 \\cdot w _ 1 \\sqcup G ^ 1 \\cdot w _ 2 . \\end{align*}"}
-{"id": "9885.png", "formula": "\\begin{align*} b _ n = \\mathrm { s g n } ( x _ n - \\tau _ n ) , \\end{align*}"}
-{"id": "3746.png", "formula": "\\begin{align*} C _ 0 ( z ) = \\int _ 0 ^ 1 \\left ( e ^ { z x } - 1 \\right ) g ( x ) d x \\end{align*}"}
-{"id": "6969.png", "formula": "\\begin{align*} & g ( 0 , \\mu _ 1 ) = \\Bigl ( 1 - \\gamma \\frac { \\delta - 1 } { \\delta } \\Bigr ) H _ 2 ( \\mu _ 1 ) , \\\\ & h ( 0 , \\epsilon ) = \\Bigl ( 1 - \\gamma \\frac { \\delta - 1 } { \\delta } \\Bigr ) H _ 2 \\Bigl ( \\frac { 1 - \\epsilon } { 2 } \\Bigr ) . \\end{align*}"}
-{"id": "7279.png", "formula": "\\begin{align*} s \\colon N _ { Y / X } ^ * \\to N _ { Y / X } \\otimes N _ { Y / X } ^ * \\otimes N _ { Y / X } ^ * : \\ \\varepsilon _ j ^ \\nu \\mapsto \\frac { 1 } { 2 } \\sum _ { \\mu = 1 } ^ r \\varepsilon _ { j , \\mu } ^ * \\otimes \\left ( \\varepsilon _ j ^ \\mu \\otimes \\varepsilon _ j ^ \\nu + \\varepsilon _ j ^ \\nu \\otimes \\varepsilon _ j ^ \\mu \\right ) . \\end{align*}"}
-{"id": "4172.png", "formula": "\\begin{align*} \\frac { z ^ k f ' ( z ) + z ^ { k + 1 } f '' ( z ) ( \\lambda - \\mu + 2 \\lambda \\mu ) + \\lambda \\mu z ^ { k + 2 } f ''' ( z ) } { g _ k ( z ) } = \\varphi ( w ( z ) ) . \\end{align*}"}
-{"id": "6078.png", "formula": "\\begin{align*} \\mathbb { E } | \\zeta ( 1 / 2 + i X _ t ) | ^ 2 = t ( t + 1 ) \\iint _ R \\{ v \\} v ^ { - 3 / 2 } \\{ u \\} u ^ { - 3 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 2 } d u d v + O ( 1 ) , \\end{align*}"}
-{"id": "6196.png", "formula": "\\begin{align*} & \\int _ { | 1 + z | \\leq | x | ^ { - \\delta } } \\left | \\frac { f ( x ( 1 + z ) ) - f ( x ) } { x f ' ( x ) } - z I ( | z | \\leq 1 ) \\right | \\nu _ U ( \\d z ) \\\\ & \\leq ( \\alpha ^ { - 1 } \\log | x | + 1 ) \\nu _ U ( ( - 1 - | x | ^ { - \\delta } , - 1 + | x | ^ { - \\delta } ) ) \\to 0 . \\end{align*}"}
-{"id": "6833.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| u ( t ) - e ^ { i t \\Delta } u _ + \\| _ { \\dot H _ x ^ s } = 0 , \\end{align*}"}
-{"id": "3131.png", "formula": "\\begin{align*} \\begin{bmatrix} \\lambda P _ d + W _ 1 & - \\lambda W _ 1 + W _ 2 & - \\lambda W _ 2 + W _ 3 & \\cdots & \\lambda W _ { t - 1 } + W _ t & - \\lambda W _ t \\end{bmatrix} , \\end{align*}"}
-{"id": "9173.png", "formula": "\\begin{align*} w ' _ j = \\frac { \\sum _ { l = 1 } ^ { r } \\alpha _ { l } ( w _ { j + l } - w _ { j - l } ) } { h } , \\end{align*}"}
-{"id": "6387.png", "formula": "\\begin{align*} \\beta _ { p } = \\left ( 1 - \\varepsilon ( p - 1 ) ! ! \\sigma ^ { p } - \\frac { \\varepsilon ^ { 2 } } { 2 } ( 2 p - 1 ) ! ! \\sigma ^ { 2 p } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "6088.png", "formula": "\\begin{align*} & \\exp \\left ( - t ( i x + x ^ 2 / 2 ) - 2 \\pi i k n e ^ x \\right ) \\\\ & = - \\frac { 1 } { t ( i + x ) + 2 \\pi i k n e ^ x } \\frac { d } { d x } \\exp \\left ( - t ( i x + x ^ 2 / 2 ) - 2 \\pi i k n e ^ x \\right ) , \\end{align*}"}
-{"id": "996.png", "formula": "\\begin{align*} c _ 0 & = \\tau ( 0 ) = \\frac { H _ { n , m } } { t } + ( \\eta ^ 2 \\frac { N ^ 2 } { 4 } - \\omega ^ 2 + \\eta ^ { - 2 } ) I \\\\ c _ 1 & = \\frac { d } { d u } \\tau ( u ) | _ { u = 0 } = 2 \\eta N \\\\ c _ 2 & = \\frac { 1 } { 2 } \\frac { d ^ 2 } { d u ^ 2 } \\tau ( u ) | _ { u = 0 } = I , \\end{align*}"}
-{"id": "3638.png", "formula": "\\begin{align*} f ( \\mathbf { x } ) = f ( x _ 1 , \\dots , x _ k ) = \\frac { r x _ 1 } { 1 + x _ k } \\end{align*}"}
-{"id": "357.png", "formula": "\\begin{align*} e ^ { i \\alpha J ( g ) } \\tilde { T } _ { 0 , \\beta } ( f ) e ^ { - i \\alpha J ( g ) } = \\tilde { T } _ { \\alpha , \\beta } ( f ) . \\end{align*}"}
-{"id": "3892.png", "formula": "\\begin{align*} X _ L = \\frac { \\mu } { \\delta _ L } \\begin{bmatrix} - 1 \\\\ \\tau _ L \\\\ - \\sigma _ L \\end{bmatrix} \\ ; , X _ R = \\frac { \\mu } { \\delta _ R } \\begin{bmatrix} - 1 \\\\ \\tau _ R \\\\ - \\sigma _ R \\end{bmatrix} \\ ; . \\end{align*}"}
-{"id": "2424.png", "formula": "\\begin{align*} \\Vert \\{ a _ { k } \\} | ~ \\mathcal { M } _ { p } ( l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } , l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } ) \\Vert = \\Vert \\{ a _ { k } \\} \\Vert _ { l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } \\rightarrow l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } } = \\sup _ { \\Vert \\{ \\lambda _ { k } \\} \\Vert _ { l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } = 1 } } \\Vert \\{ a _ { k } \\lambda _ { k } \\} \\Vert _ { l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } } . \\end{align*}"}
-{"id": "8077.png", "formula": "\\begin{align*} Q ( x , y ) = \\begin{cases} ( x + \\zeta , y - \\alpha ) & x \\in [ 0 , \\frac { 1 } { 2 } ) , \\\\ ( x + \\zeta , y + \\alpha ) & x \\in [ \\frac { 1 } { 2 } , 1 ) . \\end{cases} \\end{align*}"}
-{"id": "6928.png", "formula": "\\begin{align*} M _ { \\pi / \\kappa } ( z ) = M ^ \\perp _ { \\pi / \\kappa } ( z ) = 1 \\quad \\mbox { a n d } L _ { \\pi / \\kappa } ( z ) = L ^ \\perp _ { \\pi / \\kappa } ( z ) = 1 \\quad \\mbox { f o r a l l $ \\kappa \\not \\subseteq \\pi $ \\ , . } \\end{align*}"}
-{"id": "8653.png", "formula": "\\begin{gather*} S _ { \\tau } = \\left ( \\begin{array} [ c ] { c } C _ { \\tau } \\\\ D _ { \\tau } \\end{array} \\right ) \\ ; \\ ; \\ ; \\ ; C _ { \\tau } = \\int _ { 0 } ^ { \\tau } \\frac { \\sin \\big ( \\sqrt { \\Lambda } ( \\tau - s ) \\big ) } { \\sqrt { \\Lambda } } d W _ s , \\ ; \\ ; \\ ; D _ { \\tau } = \\int _ { 0 } ^ { \\tau } { \\cos \\big ( \\sqrt { \\Lambda } ( \\tau - s ) \\big ) } d W _ s . \\end{gather*}"}
-{"id": "8255.png", "formula": "\\begin{align*} \\partial _ t Z ( t , x ) = \\nu \\Delta Z ( t , x ) + \\frac { \\lambda \\sqrt { D } } { \\nu } \\xi ( t , x ) , ( t , x ) \\in \\R _ + \\times \\R . \\end{align*}"}
-{"id": "782.png", "formula": "\\begin{align*} ( \\mathcal { H } h ) ( \\vec { x } , \\vec { \\nu } ) = \\sum _ { \\begin{subarray} { c } ( \\vec { y } , \\vec { \\mu } ) \\in \\mathcal { S } _ { k _ { 1 } , \\ldots , k _ { r } } \\\\ ( \\vec { y } , \\vec { \\mu } ) \\not = ( \\vec { x } , \\vec { \\nu } ) \\end{subarray} } q ( \\vec { x } , \\vec { \\nu } | \\vec { y } , \\vec { \\mu } ) \\left \\{ h ( \\vec { y } , \\vec { \\mu } ) - h ( \\vec { x } , \\vec { \\nu } ) \\right \\} ( h \\in F ( \\mathcal { S } _ { k _ { 1 } , \\ldots , k _ { r } } ) ) , \\end{align*}"}
-{"id": "454.png", "formula": "\\begin{align*} A ^ 1 x ^ + _ 1 - A ^ 1 x ^ - _ 1 + A ^ 2 x ^ + _ 1 - A ^ 2 x ^ - _ 1 & \\succeq _ { \\mathcal { L } ^ { 3 } } b \\\\ x _ 1 ^ + , x _ 1 ^ - , x _ 2 ^ + , x _ 2 ^ - & \\geq 0 \\\\ x _ j = x _ j ^ + - x _ j ^ - \\ j & \\in \\{ 1 , 2 \\} . \\end{align*}"}
-{"id": "8775.png", "formula": "\\begin{align*} ( \\mathbb { I } _ { 2 r + 1 } \\otimes U ^ { - 1 } ) R ^ { ( 2 r + 1 ) } _ { 2 , 1 } \\left ( t ^ { 2 r + 1 } / x \\right ) ^ { \\tau _ 1 } ( \\mathbb { I } _ 3 \\otimes U ) R _ { 1 , 2 } ^ { ( 2 r + 1 ) } ( x ) ^ { \\tau _ 1 } = \\frac { ( 1 - x ) \\left ( t ^ { 2 r + 1 } - x \\right ) } { ( t - x ) \\left ( t ^ { 2 r } - x \\right ) } \\mathbb { I } _ { 2 r + 1 } \\otimes \\mathbb { I } _ { 2 r + 1 } , \\end{align*}"}
-{"id": "1943.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\ , \\frac { b _ i } { a _ i + b _ i } > 2 . \\end{align*}"}
-{"id": "8483.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\star } + \\sum _ { j = 1 } ^ { d } A _ { j } ( \\tilde { \\textbf { u } } ^ { \\star } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } ^ { \\star } = ( 0 , - \\nabla P ^ { \\star } ) ^ { T } . \\end{align*}"}
-{"id": "8382.png", "formula": "\\begin{align*} \\omega \\circ E ( \\theta ( x _ \\lambda ) ^ * \\theta ( x _ \\lambda ) ) & = \\omega \\circ E ( \\theta ( x _ \\lambda ^ * x _ \\lambda ) ) = \\omega \\circ \\theta \\circ \\theta ^ { - 1 } \\circ E \\circ \\theta ( x _ \\lambda ^ * x _ \\lambda ) = \\omega \\circ \\theta \\circ E ( x _ \\lambda ^ * x _ \\lambda ) \\end{align*}"}
-{"id": "7864.png", "formula": "\\begin{align*} q ( t , x , y ) = q _ 0 ( t , x , y ) + \\int _ 0 ^ t \\int _ { \\R ^ d } q _ 0 ( t - s , x - z ) q ( s , z , y ) \\ , d z \\ , d s \\ , , \\end{align*}"}
-{"id": "1028.png", "formula": "\\begin{align*} P ( x , y ) ~ = ~ B x + C y + D . \\end{align*}"}
-{"id": "3425.png", "formula": "\\begin{align*} d \\xi ( t ) = a ( t , \\xi ( t ) ) d t + A ( t , \\xi ( t ) ) d w ( t ) , \\xi ( s ) = x \\in R ^ d , \\end{align*}"}
-{"id": "913.png", "formula": "\\begin{align*} \\min _ { X \\in \\mathbb { R } ^ { m \\times n } } \\| X \\| ^ { p } _ { S _ { p } } , \\ ; \\ ; \\textrm { s u b j e c t t o } \\ ; \\ ; \\mathcal { A } ( X ) = b \\end{align*}"}
-{"id": "4324.png", "formula": "\\begin{align*} z _ { n + 1 } ^ { b _ { n + 1 } } = \\frac { t } { \\prod _ { j = k + 1 } ^ n z _ j ^ { b _ j } } . \\end{align*}"}
-{"id": "3804.png", "formula": "\\begin{align*} \\| \\Delta _ q \\cos ( n x ) \\| _ { L ^ 2 } = \\frac \\pi 2 \\left ( \\sum _ { m = - \\infty } ^ { \\infty } \\varphi _ q ^ 2 ( m ) \\left | \\delta _ { n } ( m ) + \\delta _ { - n } ( m ) \\right | ^ 2 \\right ) ^ { 1 / 2 } = \\frac \\pi { 2 } \\left ( 2 \\varphi _ q ^ 2 ( n ) \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "5194.png", "formula": "\\begin{align*} \\sum ^ { n } _ { i = 1 } | x _ { i } y _ { i } | \\leq \\left ( \\sum ^ { n } _ { i = 1 } | x _ { i } | ^ { p } \\right ) ^ { 1 / p } \\left ( \\sum ^ { n } _ { i = 1 } | y _ { i } | ^ { q } \\right ) ^ { 1 / q } \\end{align*}"}
-{"id": "7737.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\left \\{ x _ n + x _ n ^ 2 - x _ n ^ 3 - x _ n ^ 4 + \\sigma _ n \\xi _ { n + 1 } , 0 \\right \\} , n = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "4103.png", "formula": "\\begin{align*} p + q + r = \\gcd ( q , p + r ) + \\gcd ( p , q + r ) + \\gcd ( r , p + q ) . \\end{align*}"}
-{"id": "3104.png", "formula": "\\begin{align*} \\sum _ { i + j = d + 2 - k } B _ { i j } + \\sum _ { i + j = d + 1 - k } A _ { i j } = P _ k , \\mbox { f o r $ k = 0 , 1 , \\hdots , d $ , } \\end{align*}"}
-{"id": "3633.png", "formula": "\\begin{align*} K = f ( K , K , \\dots , K ) , \\end{align*}"}
-{"id": "5369.png", "formula": "\\begin{align*} P ( x , y ) ~ = ~ \\sum _ { k = 0 } ^ d \\sum _ { j = 0 } ^ k c _ { k , j } { \\ , } x ^ j y ^ { k - j } \\end{align*}"}
-{"id": "9814.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { M } f _ { i } \\succ \\prod _ { i } ^ { M } g _ { i } . \\end{align*}"}
-{"id": "363.png", "formula": "\\begin{align*} W _ { h , I } T _ { c , 0 } ( f ) W _ { h , I } ^ * = T _ { c , h } ( f ) \\end{align*}"}
-{"id": "6566.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { 1 , q } ( \\R ^ d ) ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { 1 , q } ( \\R ^ d ) ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "5158.png", "formula": "\\begin{align*} \\tau _ t ( E ( x ) ) = E ( \\tau _ t ( x ) ) . \\end{align*}"}
-{"id": "9088.png", "formula": "\\begin{align*} Y ( s ) = - \\int _ { 0 } ^ { \\infty } e ^ { s x } x ^ { ( \\alpha + 1 ) / 2 } e ^ { - x / 2 } L ^ { ( \\alpha ) } _ { N - 1 } ( x ) \\psi ( x ) \\ , \\mathrm { d } x \\ . \\end{align*}"}
-{"id": "4832.png", "formula": "\\begin{align*} \\| ( x , ( y _ j ) _ { 1 \\le j \\le n } ) \\| = \\max \\left ( | x | _ 1 , \\sum _ { 1 \\le j \\le n } | y _ j | _ { { \\mathcal I } ^ * } \\right ) \\end{align*}"}
-{"id": "8607.png", "formula": "\\begin{align*} Y _ { [ a ^ { \\ast } w ( ^ { \\ast } b ) ] } ( \\mu _ { k } P ) & = \\rho ^ { - 1 } \\left ( X _ { [ a ^ { \\ast } w ( ^ { \\ast } b ) ] } ( \\rho ( \\mu _ { k } P ) ) \\right ) \\\\ & = \\rho ^ { - 1 } \\left ( X _ { [ a ^ { \\ast } w ( ^ { \\ast } b ) ] } ( \\rho ( \\Delta _ { k } ' ) ) \\right ) \\\\ & = \\rho ^ { - 1 } \\left ( X _ { [ a ^ { \\ast } w ( ^ { \\ast } b ) ] } ( \\rho ( \\Delta _ { k } ' ) ) \\right ) \\end{align*}"}
-{"id": "3975.png", "formula": "\\begin{align*} v = \\xi v ^ 0 ( A ) \\xi \\in P ^ { - } ( A ) . \\end{align*}"}
-{"id": "2334.png", "formula": "\\begin{align*} v ( z ) = v _ { a , b , c } ( z ) = ( z + c ) ^ d p \\Big ( a + \\frac { b } { z + c } \\Big ) \\end{align*}"}
-{"id": "9456.png", "formula": "\\begin{align*} ( e ^ { f } \\lambda ) \\wedge d ( e ^ { f } \\lambda ) ^ { n } = e ^ { ( n + 1 ) f } \\lambda \\wedge d \\lambda . \\end{align*}"}
-{"id": "5624.png", "formula": "\\begin{align*} w = 2 1 1 2 = \\overline { w } . \\end{align*}"}
-{"id": "2787.png", "formula": "\\begin{align*} \\Omega ( d \\Gamma _ { ( u , u _ y ) } ( [ f ] ) , ( \\delta u , \\delta u _ y ) ) = i \\ , ( [ f ] \\ , \\omega ( \\delta u , \\delta u _ y ) ) \\end{align*}"}
-{"id": "3020.png", "formula": "\\begin{align*} { { \\bf { Y } } ^ { [ 2 ] } } ( n ) - { { \\bf { Y } } ^ { [ 2 ] } } ( { t _ 1 } ) = { { \\bf { H } } ^ { [ 2 1 ] } } ( n ) { \\bf { V } } _ 2 ^ { [ 1 ] } ( n ) { { \\bf { v } } ^ { [ 1 ] } } + { { \\bf { H } } ^ { [ 2 2 ] } } ( n ) { \\bf { V } } _ 2 ^ { [ 2 ] } ( n ) { { \\bf { v } } ^ { [ 2 ] } } . \\end{align*}"}
-{"id": "4497.png", "formula": "\\begin{align*} \\mathbb { E } _ f ( \\hat { H } _ n ^ { w } ) - H ( f ) = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { 2 d ' } { d } } } { n ^ { \\frac { 2 d ' } { d } } } \\ , , \\ , \\frac { k ^ \\frac { \\beta } { d } } { n ^ \\frac { \\beta } { d } } \\biggr \\} \\biggr ) = o ( n ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "3251.png", "formula": "\\begin{align*} U _ q ( \\mathfrak { l } ) = \\{ \\textrm { s u b a l g e b r a o f $ U _ q ( \\mathfrak { g } ) $ g e n e r a t e d b y $ K _ i ^ { \\pm 1 } $ a n d $ E _ j , F _ j $ w i t h $ j \\in S $ } \\} . \\end{align*}"}
-{"id": "9486.png", "formula": "\\begin{align*} \\dot \\gamma ( s ) = 2 \\pi \\nabla G ( \\gamma ( s ) ) \\end{align*}"}
-{"id": "5068.png", "formula": "\\begin{align*} B : = ( A D ) ^ c = \\bigcap _ { a \\in A } a C \\end{align*}"}
-{"id": "3552.png", "formula": "\\begin{align*} \\Pi _ { g _ { \\mathbb { E } } } \\circ \\Phi ^ { V ^ R } _ { ( g _ { \\mathbb { E } } , 0 ) } ( \\gamma ^ R + h ^ R , \\tau ^ R + w ^ R ) = \\Pi _ { g _ { \\mathbb { E } } } \\circ \\Phi ^ { V ^ R } _ { ( g _ { \\mathbb { E } } , 0 ) } ( \\gamma ^ R , \\tau ^ R ) + \\Pi _ { g _ { \\mathbb { E } } } ( 2 \\psi ^ R , V ^ R ) \\end{align*}"}
-{"id": "2363.png", "formula": "\\begin{align*} \\alpha _ k : = \\big \\{ t ^ k _ j \\in [ 0 , T ] \\ ; \\ , : \\ ; \\ , 0 \\leqslant j \\leqslant N _ k , \\ ; t ^ k _ 0 = 0 , \\ ; t ^ k _ { N _ k } = T , \\ ; t ^ k _ j < t ^ k _ { j + 1 } \\big \\} . \\end{align*}"}
-{"id": "3905.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = \\lambda f ( t , u , u ' ) & & \\\\ u ( 0 ) = 0 = u ( T ) . \\end{array} \\right . \\end{align*}"}
-{"id": "7004.png", "formula": "\\begin{align*} f _ 0 ^ { \\lambda _ i ( x , y ) } = - f _ i ^ { \\lambda _ 0 ( x , y ) } - \\sum _ { m + n = i , ~ m , n > 0 } f _ m ^ { \\lambda _ n ( x , y ) } + \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ x , f _ n ^ { y } ] + [ f _ 0 ^ x , f _ i ^ { y } ] + [ f _ i ^ x , f _ 0 ^ { y } ] . \\end{align*}"}
-{"id": "3408.png", "formula": "\\begin{gather*} \\prod _ { j = 1 } ^ n x _ j ^ { k _ j } = 1 . \\end{gather*}"}
-{"id": "5109.png", "formula": "\\begin{align*} & C _ { a } ^ { [ 1 , M + 1 ] } ( z ) X | \\mathrm { v a c } \\rangle _ { [ 1 , M + 1 ] } = ( 1 + z ) C _ { a } ^ { [ 1 , M ] } ( z ) X | \\mathrm { v a c } \\rangle _ { [ 1 , M ] } , \\\\ & C _ { a } ^ { [ 0 , M ] } ( z ) X | \\mathrm { v a c } \\rangle _ { [ 0 , M ] } = z \\ , C _ { a } ^ { [ 1 , M ] } ( z ) X | \\mathrm { v a c } \\rangle _ { [ 1 , M ] } . \\end{align*}"}
-{"id": "7314.png", "formula": "\\begin{align*} \\hat { R } _ { V , W } ( v \\otimes w _ { \\mathrm { h w } } ) = q ^ { ( \\mathrm { w t } ( v ) , \\mathrm { w t } ( w _ { \\mathrm { h w } } ) ) } w _ { \\mathrm { h w } } \\otimes v . \\end{align*}"}
-{"id": "6304.png", "formula": "\\begin{align*} \\| f \\| _ { M _ { p , q } ^ { s , \\alpha _ 1 } } \\sim \\begin{cases} \\big \\| \\{ \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { p , q } ^ { 0 , \\alpha _ 1 } } \\} | ~ { l _ { q } ^ { s , \\alpha _ 2 } } \\big \\| , ~ & ~ \\alpha _ 2 < 1 \\\\ \\big \\| \\{ \\| \\Delta _ j f \\| _ { M _ { p , q } ^ { 0 , \\alpha _ 1 } } \\} | ~ { l _ { q } ^ { s , 1 } } \\big \\| , ~ & ~ \\alpha _ 2 = 1 . \\end{cases} \\end{align*}"}
-{"id": "2894.png", "formula": "\\begin{align*} C H _ { P o i s } ^ { * > 0 } ( A , A ) [ n ] = g _ { P o i s _ n ^ + , A } ^ { \\psi ^ + } \\stackrel { \\sim } { \\rightarrow } T _ { \\varphi ^ * A } . \\end{align*}"}
-{"id": "4709.png", "formula": "\\begin{align*} y & = ( a ^ k b ^ k ) ^ { ( \\ell - 1 ) / 2 } a ^ k b ^ { k - 1 } , \\\\ x & = ( a ^ k b ^ k ) ^ { ( \\ell - 1 ) / 2 } a ^ k b ^ { k - 1 } a ( a ^ k b ^ k ) ^ { ( \\ell - 1 ) / 2 } a ^ k b ^ { k - 1 } . \\end{align*}"}
-{"id": "7578.png", "formula": "\\begin{align*} p _ 2 = p _ 2 ( x _ 0 ) : = \\mathbb P \\left \\{ \\omega \\in \\Omega : \\chi ( \\omega ) \\le - 1 + \\frac { l - B } { 2 l } \\right \\} , K _ 2 = K _ 2 ( x _ 0 ) : = \\left [ \\frac { 2 ( x _ 0 - v _ l ) } { l - B } \\right ] + 1 . \\end{align*}"}
-{"id": "870.png", "formula": "\\begin{align*} \\varphi _ { A , k } ( y ) : = \\inf \\{ t \\in { \\mathbb { R } } \\mid y \\in t k + A \\} . \\end{align*}"}
-{"id": "176.png", "formula": "\\begin{align*} M _ { f , a , \\beta } ( x ) : = \\max \\biggl \\{ \\max _ { t = 1 , \\ldots , m } \\frac { \\| f ^ { ( t ) } ( x ) \\| } { f ( x ) } \\ , , \\ , \\sup _ { y \\in B _ x ^ \\circ ( r _ a ( x ) ) } \\frac { \\| f ^ { ( m ) } ( y ) - f ^ { ( m ) } ( x ) \\| } { f ( x ) \\| y - x \\| ^ { \\beta - m } } \\biggr \\} . \\end{align*}"}
-{"id": "8597.png", "formula": "\\begin{align*} \\Pi _ { 1 } \\left ( \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } r ^ { - 1 } \\overline { N } ( ^ { \\ast } b ) \\left ( \\pi _ { b r } ( x ) \\right ) \\right ) = - \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } r ^ { - 1 } \\pi _ { 1 } p \\xi _ { b e _ { k } } \\pi _ { b r } ( x ) \\end{align*}"}
-{"id": "2200.png", "formula": "\\begin{align*} - \\int _ { B } \\psi ^ { 2 } \\tilde { u } ^ { - 1 } \\partial _ { t } ( g _ { 1 - \\alpha , m } * \\tilde { u } ) d x + \\mathcal { E } ( \\tilde { u } , - \\psi ^ { 2 } \\tilde { u } ^ { - 1 } ) \\leq R _ { m } ( t ) , \\end{align*}"}
-{"id": "4749.png", "formula": "\\begin{align*} D ^ { k + 1 } F _ N ( r ) & = D \\Big [ \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\ , \\alpha _ { j , k } \\ , r ^ { 2 j - k } \\ , F _ { N + 2 j } ( r ) \\Big ] \\\\ & = \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\ , ( 2 j - k ) \\ , \\alpha _ { j , k } \\ , r ^ { 2 j - k - 1 } \\ , F _ { N + 2 j } ( r ) - \\ ! \\ ! \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\ , \\alpha _ { j , k } \\ , r ^ { 2 j - k + 1 } \\ , F _ { N + 2 j + 2 } ( r ) . \\\\ \\noalign { \\medskip } & = : S _ 1 + S _ 2 . \\end{align*}"}
-{"id": "6109.png", "formula": "\\begin{align*} g = \\frac { \\pi ^ 2 s ^ 2 } { \\sin ^ 2 ( \\frac { \\pi s } { s _ w } ) } \\left ( \\frac { d s _ w ^ 2 } { s _ w ^ 4 } + d \\theta _ w ^ 2 \\right ) = \\frac { \\pi ^ 2 s ^ 2 } { \\sin ^ 2 ( \\frac { \\pi } { 1 + \\rho _ z } ) } \\left ( \\frac { d \\rho _ z ^ 2 } { s ^ 2 ( 1 + \\rho _ z ) ^ 4 } + d \\theta _ z ^ 2 \\right ) . \\end{align*}"}
-{"id": "905.png", "formula": "\\begin{align*} \\int _ { T } ^ { 2 T } \\vert Z ( t ) \\vert d t = \\int _ { T } ^ { 2 T } \\vert \\zeta ( \\tfrac { 1 } { 2 } + i t ) \\vert d t \\geq \\vert \\int _ { T } ^ { 2 T } \\zeta ( \\tfrac { 1 } { 2 } + i t ) d t \\vert . \\end{align*}"}
-{"id": "997.png", "formula": "\\begin{align*} \\Gamma _ { i } = \\langle \\mu _ i , a \\rangle , \\overline { \\Gamma } _ { j } = \\langle \\nu _ j , b \\rangle , i = 1 , 2 , \\cdots , n , j = 1 , 2 , \\cdots , m \\end{align*}"}
-{"id": "2785.png", "formula": "\\begin{gather*} \\omega _ { - 1 } = 0 , \\ ; \\ ; \\omega _ 0 = u _ z = \\tfrac { 1 } { 2 } ( u _ x - i u _ y ) , \\ ; \\ ; \\omega _ 1 = u _ { z z z } - 2 ( u _ z ) ^ 3 , \\ ; \\ ; \\\\ \\omega _ 2 = u _ { z z z z z } - 1 0 u _ { z z z } ( u _ z ) ^ 3 - 1 0 ( u _ { z z } ) ^ 2 u _ z + 6 ( u _ z ) ^ 5 , \\ ; \\ldots \\end{gather*}"}
-{"id": "3820.png", "formula": "\\begin{align*} \\prod _ { i < j } ( x _ i - x _ j ) ^ { - 1 } \\left [ \\prod _ { k = 1 } ^ n ( z _ k ^ { - 1 } + t _ n x _ n ) \\det _ { 1 \\leq i , j \\leq n } \\left \\{ ( 1 - x _ i z _ j ) ^ { - 1 } \\right \\} \\right ] \\ , \\Big \\vert _ { \\boldsymbol { z } ^ { \\lambda } } = \\prod _ { k = 1 } ^ n ( x _ k + t _ n x _ n ) s _ { \\lambda - \\rho } ( \\boldsymbol { x } ) . \\end{align*}"}
-{"id": "4648.png", "formula": "\\begin{align*} g \\cdot f ( x , y ) = f ( ( x , y ) \\cdot g ^ t ) . \\end{align*}"}
-{"id": "5019.png", "formula": "\\begin{align*} \\beta _ n ( f ) = \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } p ^ { * k } ( f ) , \\textrm { f o r $ f \\in \\ell ^ \\infty ( G ) $ } . \\end{align*}"}
-{"id": "9032.png", "formula": "\\begin{align*} \\psi _ k ( \\theta ) & = ( k + 1 ) \\theta - 2 \\left ( \\Im \\log \\Phi _ { k } ^ * ( e ^ { i \\theta } ) - \\Im \\log \\Phi _ { k } ^ * ( 1 ) \\right ) \\\\ & = ( k + 1 ) \\theta - 2 \\sum _ { j = 0 } ^ { k - 1 } \\left ( \\Im \\log \\left ( 1 - \\gamma _ { j } e ^ { i \\psi _ j ( \\theta ) } \\right ) - \\Im \\log \\left ( 1 - \\gamma _ { j } \\right ) \\right ) . \\end{align*}"}
-{"id": "5847.png", "formula": "\\begin{align*} \\Delta a - C ^ { x } a _ { , x } - a _ { , t } = 0 . \\end{align*}"}
-{"id": "3086.png", "formula": "\\begin{align*} \\lambda _ 1 = \\mu _ 1 ^ { - 1 } = \\inf \\limits _ { v \\in V , | v | _ b \\neq 0 } \\frac { \\norm { v } _ a ^ 2 } { | v | _ b ^ 2 } \\end{align*}"}
-{"id": "5857.png", "formula": "\\begin{align*} X _ { I } = 2 \\psi _ { I } \\int T _ { I } \\left ( t \\right ) d t ~ \\partial _ { t } + T _ { I } \\left ( t \\right ) Y _ { I } + \\alpha \\left ( t , x \\right ) F \\partial _ { F } \\end{align*}"}
-{"id": "978.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ k c _ { \\ell } = n \\mbox { a n d } \\sum _ { \\ell = 1 } ^ k s _ { \\ell } = M . \\end{align*}"}
-{"id": "7349.png", "formula": "\\begin{align*} M ^ { ( 0 ) } = ( 1 ) , M ^ { ( 1 ) } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & q ^ 2 \\end{array} \\right ) , M ^ { ( 2 ) } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & q ^ 4 & 0 \\\\ 0 & 0 & q ^ 2 \\end{array} \\right ) , M ^ { ( 3 ) } = ( 1 ) . \\end{align*}"}
-{"id": "3169.png", "formula": "\\begin{align*} & \\frac { 1 } { \\log e } \\biggl ( - \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } q - \\frac { 1 } { 2 } + \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } ( 1 - q ) + \\frac { 1 } { 2 } + \\log q + 1 - \\log ( 1 - q ) - 1 \\biggr ) \\\\ & = \\frac { 1 } { 2 \\log e } \\left ( \\log q - \\log ( 1 - q ) \\right ) \\end{align*}"}
-{"id": "3024.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( { t _ 2 } ) = { h ^ { [ j 1 ] } } ( { t _ 2 } ) u _ 2 ^ { [ 1 ] } + { h ^ { [ j 2 ] } } ( { t _ 2 } ) \\frac { { { h ^ { [ 2 2 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 2 2 ] } } ( { t _ 2 } ) } } u _ 1 ^ { [ 2 ] } + { h ^ { [ j 3 ] } } ( { t _ 2 } ) \\frac { { { h ^ { [ 2 3 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 2 3 ] } } ( { t _ 2 } ) } } u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "9675.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { x ( t ) } { G ^ { - 1 } ( t ) } = : \\lambda \\in ( 0 , \\infty ) , \\end{align*}"}
-{"id": "5519.png", "formula": "\\begin{align*} 1 \\ge \\rho ( x ) = \\int _ 0 ^ \\gamma \\varphi ( x ^ * ) w \\ge \\int _ { \\{ s \\in [ 0 , \\gamma ) : \\ x ^ * ( s ) > \\lambda \\} } \\varphi ( \\lambda ) w = \\varphi ( \\lambda ) \\int _ 0 ^ \\beta w , \\end{align*}"}
-{"id": "6774.png", "formula": "\\begin{align*} \\mu _ j : = e ^ { - u _ j } \\theta _ u ^ n + e ^ { - v _ j } \\theta _ v ^ n \\end{align*}"}
-{"id": "168.png", "formula": "\\begin{align*} h _ \\mu ( T ) = h _ { \\nu } ( \\tau ) = H _ \\nu ( \\eta ) < \\infty . \\end{align*}"}
-{"id": "965.png", "formula": "\\begin{align*} \\sideset { } { ' } \\sum _ { \\vert K \\vert = q - 1 } \\sum _ { j , k = 1 } ^ n \\dfrac { \\partial ^ 2 h _ A } { \\partial z _ j \\partial \\overline { z _ k } } ( z ) w _ { j K } \\overline { w _ { k K } } \\geq A \\vert w \\vert ^ 2 \\ ; , \\ ; z \\in U _ { A } \\ ; , \\ ; w \\in \\Lambda ^ { 0 , q } _ { z } \\ ; . \\end{align*}"}
-{"id": "8276.png", "formula": "\\begin{align*} l _ { j \\rightarrow n } ^ t = \\log \\left ( \\frac { p _ { \\widehat { \\mathrm { r } } } ( \\widehat { r } _ j ^ t ; \\rho _ j = 1 ) } { p _ { \\widehat { \\mathrm { r } } } ( \\widehat { r } _ j ^ t ; \\rho _ j = 0 ) } \\right ) . \\end{align*}"}
-{"id": "7618.png", "formula": "\\begin{align*} \\lambda _ 1 ^ 2 = \\sum _ { y \\sim x } \\sum _ { z \\sim y } \\mathbf { v } _ z \\leq \\left ( \\sum _ { u v \\in E ( G ) } \\mathbf { v } _ u + \\mathbf { v } _ v \\right ) - \\sum _ { y \\sim x } \\mathbf { v } _ y = \\left ( \\sum _ { u v \\in E ( G ) } \\mathbf { v } _ u + \\mathbf { v } _ v \\right ) - \\lambda _ 1 . \\end{align*}"}
-{"id": "8126.png", "formula": "\\begin{align*} l _ * : = \\sup \\{ l : \\ , l > - N , \\ , C _ { k , l , N } = C _ { k , l , N } ^ { r a d } \\} . \\end{align*}"}
-{"id": "7629.png", "formula": "\\begin{align*} c _ { r } = \\sum _ { j = 0 } ^ { p - 1 } c _ { i _ { j } i _ { j + 1 } } , \\end{align*}"}
-{"id": "5889.png", "formula": "\\begin{align*} B ^ { t } = 1 ~ , ~ B ^ { \\alpha } = \\Gamma ^ { \\alpha } + C ^ { \\alpha } , \\end{align*}"}
-{"id": "6514.png", "formula": "\\begin{align*} T _ { u , r } ^ D : = \\{ v \\in D : \\min _ { 0 \\le t \\le T } \\| v - u ( t ) \\| _ W \\le r \\} , \\end{align*}"}
-{"id": "3662.png", "formula": "\\begin{align*} ( a \\otimes m ) ( x _ \\alpha \\otimes m _ \\beta ) = a ( m x _ \\alpha ) \\otimes m m _ \\beta . \\end{align*}"}
-{"id": "1882.png", "formula": "\\begin{align*} I \\ge & ( c _ 2 - c _ 1 ) c _ 3 + ( a _ 2 - a _ 1 ) a _ 3 \\\\ \\ge & \\frac 2 3 [ ( a _ 2 + c _ 2 ) - ( a _ 1 + c _ 1 ) ] \\\\ = & \\frac 8 3 ( K _ { 1 3 } - K _ { 1 2 } ) = \\frac 8 3 [ K _ { 1 3 } + s K _ { 1 2 } - ( 1 + s ) K _ { 1 2 } ] \\\\ \\ge & \\frac 8 3 [ K _ s - ( 1 + s ) \\epsilon _ 0 + ( 1 + s ) \\epsilon ] \\\\ \\ge & \\frac 8 3 [ \\frac { \\sqrt { 2 } } 2 - \\frac { \\sqrt { 4 + 2 \\sqrt { 2 } } } 4 + ( 1 + s ) \\epsilon ] > \\frac 8 3 \\epsilon . \\end{align*}"}
-{"id": "9658.png", "formula": "\\begin{align*} x ' ( t ) & = - a g ( x ( t ) ) + b g ( x ( t - \\tau ( t ) ) , t \\geq 0 \\\\ x ( t ) & = \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] \\end{align*}"}
-{"id": "4931.png", "formula": "\\begin{align*} | A x _ 0 | = | A \\hat x | . \\end{align*}"}
-{"id": "929.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , \\widehat { V } \\in \\mathbb { R } ^ { n \\times d } : X = U \\widehat { V } ^ { T } } \\| U \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| \\widehat { V } \\| _ { S _ { \\widehat { p } _ { 2 } } } . \\end{align*}"}
-{"id": "3263.png", "formula": "\\begin{align*} \\eth = - \\sum _ { \\xi _ i \\in \\Delta ( \\mathfrak { u } _ + ) } E _ { \\xi _ i } \\otimes \\gamma _ - ( w _ i ) \\in U ( \\mathfrak { g } ) \\otimes \\mathrm { C l } . \\end{align*}"}
-{"id": "7256.png", "formula": "\\begin{align*} \\sum _ { 2 \\leq | \\gamma | < n } f _ { k j , \\gamma } \\prod _ { \\lambda = 1 } ^ r \\left ( u _ j ^ \\lambda + \\sum _ { 2 \\leq | \\beta | < n } F _ { j , \\beta } ^ \\lambda \\cdot u _ j ^ \\beta \\right ) ^ { \\gamma _ \\lambda } . \\end{align*}"}
-{"id": "6231.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + K ) X ^ { ( 0 ) } ( s _ { i } ) = F , \\end{align*}"}
-{"id": "2537.png", "formula": "\\begin{align*} \\| U ^ { n + 1 } \\| ^ 2 = \\| U ^ n \\| ^ 2 = 1 , \\quad \\forall ~ n \\in \\N . \\end{align*}"}
-{"id": "9506.png", "formula": "\\begin{align*} S _ E & = \\frac { 1 } { 2 \\kappa _ { 1 1 } ^ 2 } \\int _ M \\mathrm { d v o l } _ g R \\\\ S _ M & = - \\frac { 1 } { 4 \\kappa _ { 1 1 } ^ 2 } \\int _ M \\mathrm { d v o l } _ g | G | ^ 2 \\\\ S _ { C S } & = - \\frac { 1 } { 1 2 \\kappa _ { 1 1 } ^ 2 } \\int _ M C \\wedge G \\wedge G . \\end{align*}"}
-{"id": "4099.png", "formula": "\\begin{align*} C \\ : : \\ : x ^ p y ^ q ( a x + b y + c z ) ^ r - z ^ { p + q + r } = 0 , \\end{align*}"}
-{"id": "5411.png", "formula": "\\begin{align*} \\sum _ { n = 2 } ^ N g _ n & \\le N \\sum _ { \\substack { m \\ge 2 , j \\ge 1 \\\\ m ^ j \\le N } } \\frac 1 m \\le N \\sum _ { m = 2 } ^ N \\frac { 2 \\log N } m = \\mathcal O ( N ( \\log N ) ^ 2 ) = \\mathcal O ( N ^ { 1 + \\varepsilon } ) \\end{align*}"}
-{"id": "2945.png", "formula": "\\begin{align*} \\sum _ { k : s _ k = s _ j } u _ k ^ 2 \\leq \\frac { R ^ 2 } { 2 } \\sum _ { k : s _ k = s _ { j + 1 } } u _ k ^ 2 , ~ ~ \\forall j \\in \\mathcal { J } . \\end{align*}"}
-{"id": "3492.png", "formula": "\\begin{align*} \\int _ D \\widetilde { f } \\hat { v } d x ' = \\int _ { \\{ \\hat { v } \\geq \\alpha \\} } \\widetilde { f } \\hat { v } d x ' + \\int _ { \\{ \\hat { v } < \\alpha \\} } \\widetilde { f } \\hat { v } d x ' = \\int _ { \\{ \\hat { v } \\geq \\alpha \\} } \\hat { f } \\hat { v } d x ' + \\int _ { \\{ \\hat { v } < \\alpha \\} } \\hat { f } \\hat { v } d x ' = \\int _ D \\hat { f } \\hat { v } d x ' . \\end{align*}"}
-{"id": "5804.png", "formula": "\\begin{align*} \\Pi = \\{ \\alpha _ { 1 } , \\ldots , \\alpha _ { n - 1 } \\} \\end{align*}"}
-{"id": "9267.png", "formula": "\\begin{align*} \\begin{cases} d p ( t , x , z ) & = - [ \\frac { 1 } { 2 } \\frac { \\partial ^ 2 } { \\partial x ^ 2 } p ( t , x , z ) + \\pi ( t , z ) \\{ a _ 0 ( t , z ) p ( t , x , z ) + b _ 0 ( t , z ) q ( t , x , z ) \\} ] d t + q ( t , x , z ) d B ( t ) ; t \\in [ 0 , T ] \\\\ p ( T , x , z ) & = \\frac { \\partial U } { \\partial y } ( x , Y ( T , x , z ) , z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] \\\\ p ( t , x , z ) & = 0 ; \\forall ( t , x , z ) \\in [ 0 , T ] \\times \\partial D \\times \\mathbb { R } . \\end{cases} \\end{align*}"}
-{"id": "2897.png", "formula": "\\begin{align*} \\chi ( u ) = \\sum _ { x \\in X _ \\chi } x + \\sum _ { y \\in Y _ \\chi } y \\psi ( u ) = \\sum _ { x \\in X _ \\chi } x + \\sum _ { y \\in Y _ \\psi } y . \\end{align*}"}
-{"id": "8867.png", "formula": "\\begin{align*} D _ A u = D u + i A u . \\end{align*}"}
-{"id": "5679.png", "formula": "\\begin{align*} \\frac { x _ 2 } { x _ 1 } \\left ( 1 - \\left ( \\frac { 1 } { x _ 2 } - 1 \\right ) e _ 0 \\right ) = \\frac 1 { x _ 1 } ( 1 + ( x _ 2 - 1 ) ( e _ 0 + 1 ) ) \\end{align*}"}
-{"id": "8421.png", "formula": "\\begin{align*} \\nabla \\cdot \\Bigg ( \\frac { \\nabla P } { \\rho } \\Bigg ) = - \\nabla v \\nabla v . \\end{align*}"}
-{"id": "8307.png", "formula": "\\begin{align*} D - \\iota ( D ) = ( D _ 1 - \\iota ( D _ 2 ) ) - \\iota ( D _ 1 - \\iota ( D _ 2 ) ) . \\end{align*}"}
-{"id": "1372.png", "formula": "\\begin{align*} \\sum _ { i \\geq 0 } \\C _ { i , A _ X \\tilde { P } } \\left ( \\frac { 1 } { m } \\right ) \\frac { z ^ i } { i ! } = \\exp \\left \\{ \\frac { 1 } { m } K _ { A _ X \\tilde { P } } ( z ) \\right \\} . \\end{align*}"}
-{"id": "1121.png", "formula": "\\begin{align*} \\mathcal { R } = \\frac { 1 } { \\mathcal { L } T _ \\mathrm { c } } \\sum _ { \\iota = 1 } ^ { \\mathcal { L } } \\sum _ { k = 1 } ^ { K } \\mathcal { R } _ k [ \\iota ] . \\end{align*}"}
-{"id": "4175.png", "formula": "\\begin{align*} ( 1 + | B | ) \\displaystyle \\sum ^ \\infty _ { n = 2 } n [ 1 + ( n - 1 ) ( \\lambda - \\mu + n \\lambda \\mu ) ] | a _ n | + ( 1 + | A | ) \\displaystyle \\sum ^ \\infty _ { n = 2 } | B _ n | \\leq A - B \\end{align*}"}
-{"id": "4062.png", "formula": "\\begin{align*} 2 ( \\alpha - 1 ) + ( 2 - \\alpha ) = 1 + 1 + k \\leq 2 + 1 , \\end{align*}"}
-{"id": "6609.png", "formula": "\\begin{align*} \\nabla h ( \\tau ) = \\langle y - x , \\nabla f ( x + \\tau ( y - x ) ) \\rangle \\end{align*}"}
-{"id": "6087.png", "formula": "\\begin{align*} & \\sum _ { - \\log t < k < \\log t } \\int _ { n \\exp ( - 2 \\sqrt { \\frac { \\log t } { t } } ) } ^ { n \\exp ( 2 \\sqrt { \\frac { \\log t } { t } } ) } x ^ { - 1 / 2 } \\exp \\left ( - \\frac { t } { 2 } \\log ^ 2 \\frac { x } { n } \\right ) \\exp \\left ( - t i \\log \\frac { x } { n } - 2 \\pi i k x \\right ) d x \\\\ & = \\sqrt { n } \\sum _ { - \\log t < k < \\log t } \\int _ { - 2 \\sqrt { \\frac { \\log t } { t } } } ^ { 2 \\sqrt { \\frac { \\log t } { t } } } \\exp ( x / 2 ) \\exp \\left ( - t ( i x + x ^ 2 / 2 ) - 2 \\pi i k n e ^ x \\right ) d x . \\end{align*}"}
-{"id": "5862.png", "formula": "\\begin{align*} \\ln F \\left ( t , x \\right ) = \\frac { \\mu } { 2 } \\left ( \\mu \\kappa t - 2 \\sqrt { 2 \\left ( \\mu - \\lambda - x \\right ) + 1 } \\right ) . \\end{align*}"}
-{"id": "1203.png", "formula": "\\begin{align*} \\frac { B } { A } = \\frac { \\sum _ { i = 1 } ^ n b _ i } { \\sum _ { i = 1 } ^ n a _ i } \\leq \\frac { \\beta \\sum _ { i = 1 } ^ n a _ i } { \\sum _ { i = 1 } ^ n a _ i } = \\beta . \\end{align*}"}
-{"id": "2721.png", "formula": "\\begin{align*} \\int _ X \\langle T _ 1 \\wedge \\cdots \\wedge T _ n \\rangle = \\int _ X \\langle T _ { 1 , \\min } \\wedge \\cdots \\wedge T _ { n , m i n } \\rangle . \\end{align*}"}
-{"id": "2091.png", "formula": "\\begin{align*} Z \\ X ^ { ( 0 ) } ( s _ { t _ { 0 } } ) = - \\eta _ { 0 t _ { 0 } } Z \\ X ^ { ( j ) } ( s _ { t _ { j } } ) = - \\eta _ { j t _ { j } } \\ j = 1 , \\ \\ldots , \\ J - 1 , \\end{align*}"}
-{"id": "5524.png", "formula": "\\begin{align*} p _ { \\varphi , w } ( x ) = \\inf \\left \\{ \\sum _ { k = 1 } ^ { \\infty } \\varphi \\left ( \\frac { | x ( k ) | } { v ( k ) } \\right ) v ( k ) : v \\prec w \\right \\} , \\end{align*}"}
-{"id": "1262.png", "formula": "\\begin{align*} \\textrm { k e r } _ { Q ( L ^ 2 \\times L ^ 2 ) } ( Q U Q ) = \\{ 0 \\} \\textrm { i f a n d o n l y i f } \\int _ { \\R ^ 2 } V _ { 1 1 } ( y ) \\ , d y \\neq 0 . \\end{align*}"}
-{"id": "4786.png", "formula": "\\begin{align*} H = - \\frac { \\kappa } { 2 f } \\ , n _ 1 + \\frac { f f '' + ( f ' ) ^ 2 - 1 } { 2 f \\sqrt { f '^ 2 - 1 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "2846.png", "formula": "\\begin{align*} \\pi _ 0 P _ { \\infty } \\{ X \\} = [ P _ { \\infty } , E n d _ X ] _ { H o ( P r o p ) } \\end{align*}"}
-{"id": "5470.png", "formula": "\\begin{align*} S & \\ll H \\exp ( - c _ 1 ( \\log N ) ^ { 3 / 5 } ( \\log \\log n ) ^ { - 1 / 5 } ) \\sum _ { n = N - H } ^ { N + H } n + H N \\\\ & \\ll H ^ 2 N \\exp ( - c _ 1 ( \\log N ) ^ { 3 / 5 } ( \\log \\log n ) ^ { - 1 / 5 } ) + H N , \\end{align*}"}
-{"id": "3893.png", "formula": "\\begin{align*} v = \\begin{bmatrix} 1 \\\\ - 2 \\alpha _ R \\\\ \\alpha _ R ^ 2 + \\beta _ R ^ 2 \\end{bmatrix} \\ ; , \\end{align*}"}
-{"id": "6368.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\infty } \\frac { 1 } { p _ i } = \\infty . \\end{align*}"}
-{"id": "9719.png", "formula": "\\begin{align*} 0 < B _ 1 ( \\epsilon ) & : = \\frac { 1 } { \\gamma } ( 1 - \\epsilon ) ^ 2 \\leq \\frac { \\log \\Gamma _ 1 ( x ) } { \\log x } \\\\ & \\qquad \\qquad \\leq \\frac { 1 } { \\gamma } ( 1 + \\epsilon ) ^ 2 = : B _ 2 ( \\epsilon ) , x \\in ( 0 , g ( \\delta ( \\epsilon ) / 2 ) ] , \\\\ g ( \\delta ( \\epsilon ) / 2 ) & < 1 , g ' ( x ) > 0 x \\in ( 0 , \\delta ( \\epsilon ) ) . \\end{align*}"}
-{"id": "2310.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k ( k - i ) ^ \\ell \\delta _ i = \\ell k ^ { \\ell - 1 } = \\ell \\sum _ { i = 0 } ^ { k - 1 } ( k - i - 1 ) ^ { \\ell - 1 } \\gamma _ i , \\ell = 0 , 1 , \\dotsc , k . \\end{align*}"}
-{"id": "6763.png", "formula": "\\begin{align*} v _ { \\lambda } : = ( 1 - \\lambda ) \\varphi _ { \\beta , \\varepsilon } + \\lambda \\psi \\leq 0 , \\end{align*}"}
-{"id": "3040.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( \\gamma _ n , Z _ n \\right ) = \\left ( \\gamma _ \\infty , Z _ \\infty \\right ) \\end{align*}"}
-{"id": "3860.png", "formula": "\\begin{align*} | a _ { i } - b _ { i } | = 1 , i \\in I _ { 2 n } , \\end{align*}"}
-{"id": "4456.png", "formula": "\\begin{align*} \\{ f _ \\Sigma ( \\cdot ) = | \\Sigma | ^ { - 1 / 2 } f _ 0 \\bigl ( \\Sigma ^ { - 1 / 2 } ( \\cdot - \\mu ) \\bigr ) : \\mu \\in \\mathbb { R } ^ d , \\Sigma = \\Sigma ^ T \\in \\mathbb { R } ^ { d \\times d } \\ \\} . \\end{align*}"}
-{"id": "3627.png", "formula": "\\begin{align*} \\phi ( x ) = c T + ( 1 - c ) x \\end{align*}"}
-{"id": "9182.png", "formula": "\\begin{align*} M _ n = 2 \\left ( \\sum _ { i = 1 } ^ p \\lambda _ i ( e _ { n - 1 } ^ { \\overline i } ) ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7088.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } f _ \\mu ( u _ n ) = x \\end{align*}"}
-{"id": "471.png", "formula": "\\begin{align*} G ^ - = \\{ x \\in \\mathbb { R } ^ 2 \\ , | \\ \\begin{bmatrix} 0 & 0 \\\\ - \\beta _ 2 v _ { 1 2 } & - \\beta _ 2 v _ { 2 2 } \\\\ - \\beta _ 1 v _ { 1 1 } & - \\beta _ 1 v _ { 2 1 } \\end{bmatrix} \\begin{bmatrix} x _ 1 \\\\ x _ 2 \\end{bmatrix} \\succeq _ { \\mathcal { L } ^ 3 } \\begin{bmatrix} - \\eta \\\\ - \\beta _ 2 \\alpha _ 2 \\\\ - \\beta _ 1 \\alpha _ 1 \\end{bmatrix} \\} . \\end{align*}"}
-{"id": "3309.png", "formula": "\\begin{align*} \\gamma _ { - } ( w _ { a } ) v = \\sum _ { i } \\frac { \\langle w _ { i } ^ { ( k - 1 ) } , \\gamma _ { - } ( w _ { a } ) v \\rangle } { \\langle w _ { i } ^ { ( k - 1 ) } , v _ { i } ^ { ( k - 1 ) } \\rangle } v _ { i } ^ { ( k - 1 ) } = \\sum _ { i } \\frac { \\langle w _ { i } ^ { ( k - 1 ) } \\wedge w _ { a } , v \\rangle } { \\langle w _ { i } ^ { ( k - 1 ) } , v _ { i } ^ { ( k - 1 ) } \\rangle } v _ { i } ^ { ( k - 1 ) } . \\end{align*}"}
-{"id": "9811.png", "formula": "\\begin{align*} h ^ p ( Z _ { p , N } ( K ) , y ) = \\frac { 1 } { N } \\sum _ { i = 1 } ^ N \\abs { \\langle x _ i , y \\rangle } ^ p , \\end{align*}"}
-{"id": "5295.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { b _ I } { k ^ 2 _ j } = m ( m - 1 + r ) , \\end{align*}"}
-{"id": "1400.png", "formula": "\\begin{align*} U _ { 1 , n } : = S _ { 1 , n } ( \\beta ^ \\prime ) - E \\left ( S _ { 1 , n } ( \\beta ^ \\prime ) \\right ) = \\frac { 1 } { \\sqrt { n } } \\sum _ { t = 1 } ^ { n } e _ { t , \\beta ^ \\prime } ^ 2 - E e _ { t , \\beta ' } ^ 2 ; U _ { 2 , n } : = \\Omega _ { 1 } ^ { - 1 } \\frac { 1 } { \\sqrt { n } } \\sum _ { t = 1 } ^ { n } e _ { t , \\beta ^ \\prime } x _ { n t } = \\frac { 1 } { \\sqrt { n } } \\sum _ { t = 1 } ^ { n } e _ { t , \\beta ^ \\prime } c _ { n t } \\end{align*}"}
-{"id": "3831.png", "formula": "\\begin{align*} \\norm { v } { H ^ { k , 2 k } ( D _ T ) } : = \\sum _ { \\ell = 0 } ^ k \\norm { \\partial _ t ^ \\ell v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 \\ell } ( D ) ) } . \\end{align*}"}
-{"id": "159.png", "formula": "\\begin{align*} | \\mu ( A ) - \\mu _ n ( A ) | \\le \\varepsilon _ n \\mu _ n ( A ) A \\in \\vee _ { i = 0 } ^ { r _ n } T ^ { - i } \\xi , \\ n \\ge 1 , \\end{align*}"}
-{"id": "2336.png", "formula": "\\begin{align*} \\frac { 1 } { N _ v ( z ) } = \\frac { d } { z + c } - \\frac { b } { ( z + c ) ^ 2 N ( x ) } ~ { \\rm f o r } ~ v ( z ) ~ { \\rm o f ~ ( \\ref { e q v a b c } ) ~ a n d ~ } x ~ { \\rm o f ~ ( \\ref { e q r r r } ) } . \\end{align*}"}
-{"id": "7529.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } x _ n = K , a . s . \\end{align*}"}
-{"id": "7285.png", "formula": "\\begin{align*} d : = \\frac { d _ 1 + \\cdots + d _ t } { t - 1 } , \\end{align*}"}
-{"id": "7163.png", "formula": "\\begin{align*} f ( x ) : = \\sum _ { k < j } a _ { j k } x _ j x _ k + \\sum _ { l < n } b _ l ( 1 - n x _ l ^ 2 ) = 0 , \\end{align*}"}
-{"id": "3428.png", "formula": "\\begin{align*} \\Gamma ^ { - 1 } ( s , t ) = I - \\int _ { s } ^ { t } \\Gamma ^ { - 1 } ( \\theta , t ) [ c ( \\theta , \\xi ( \\theta ) ) - C ^ 2 ( \\theta , \\xi ( \\theta ) ) ] d \\theta - \\int _ { s } ^ { t } \\Gamma ^ { - 1 } ( { t } ) C ( \\theta , \\xi ( \\theta ) ) d w ( \\theta ) \\end{align*}"}
-{"id": "6598.png", "formula": "\\begin{align*} \\begin{tabular} { l l } m i n i m i z e & $ \\hat { f } ( x ) + \\hat { g } ( z ) $ \\\\ s u b j e c t t o & $ A x + B z = c $ \\end{tabular} \\end{align*}"}
-{"id": "9053.png", "formula": "\\begin{align*} \\P _ N ( \\theta , \\theta ' ) = & \\P \\left ( E v ( r , 0 ) \\right ) ^ 2 \\\\ \\stackrel { E q . \\eqref { e q : g i r s a n o v 1 } } { \\ll _ v } & 2 ^ { 2 r - 2 N } \\left ( N ^ { \\frac { 3 } { 2 } } \\P \\left ( G E v ( r , 0 ) \\right ) \\right ) ^ 2 \\ . \\end{align*}"}
-{"id": "1804.png", "formula": "\\begin{align*} \\Delta _ j f ( x ) : = \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i x \\cdot \\xi } \\widehat { \\Psi } ( 2 ^ { - j } \\xi ) \\widehat { f } ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "8730.png", "formula": "\\begin{align*} v ( 0 , x ) & = e ^ { - \\tau A } \\widetilde Y _ \\tau ^ { x } + \\int _ 0 ^ \\tau e ^ { - s A } G B ( s , X ^ { x } _ s ) \\ , d s - \\int _ 0 ^ \\tau e ^ { - s A } \\widetilde Z ^ { x } _ { s } \\ ; d W _ s \\\\ & = e ^ { - \\tau A } v ( \\tau , X _ \\tau ^ { x } ) + \\int _ 0 ^ \\tau e ^ { - s A } G B ( s , X ^ { x } _ s ) \\ , d s - \\int _ 0 ^ \\tau e ^ { - s A } \\nabla ^ G v ( s , X _ s ^ x ) \\ ; d W _ s . \\end{align*}"}
-{"id": "9690.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { \\log g _ 1 ( x ) } { \\log x } = \\beta _ 1 , \\lim _ { x \\to 0 ^ + } \\frac { \\log g _ 2 ( x ) } { \\log x } = \\beta _ 2 . \\end{align*}"}
-{"id": "30.png", "formula": "\\begin{align*} v _ k = \\sum _ { p = 1 } ^ r \\lambda ^ p _ k u _ p \\end{align*}"}
-{"id": "775.png", "formula": "\\begin{gather*} R _ { \\ell + s } = \\partial C _ { \\ell + s } \\oplus \\varphi ( \\partial R _ { \\ell } ) \\oplus R _ \\ell = \\partial C _ { \\ell + s } \\oplus \\bigoplus _ { 0 \\leqslant k \\leqslant \\ell , [ k ] = [ \\ell ] } \\partial C _ { k } \\oplus \\varphi ( R _ { \\ell + s } ) . \\end{gather*}"}
-{"id": "8530.png", "formula": "\\begin{align*} \\rho ( b ' X _ { b ^ { \\ast } } ( P ) ) & = \\displaystyle \\sum _ { r \\in L ( k ) , a \\in _ { k } T } \\rho ( b ' r a ) \\rho ( Y _ { [ b r a ] } ( P ) ) \\\\ & = \\rho \\left ( \\displaystyle \\sum _ { r \\in L ( k ) , a \\in _ { k } T } b ' r a Y _ { [ b r a ] } ( P ) \\right ) \\end{align*}"}
-{"id": "684.png", "formula": "\\begin{align*} S = \\big \\{ g \\in G \\ , : \\ , \\nu ( g B ) \\geq \\nu ( B ) \\big \\} \\end{align*}"}
-{"id": "9982.png", "formula": "\\begin{align*} \\frac { | H _ { \\tilde { i } k } | ^ 2 } { \\sigma ^ 2 _ n + \\tilde { p } | H _ { \\tilde { i } k } | ^ 2 } - \\frac { ( 1 - \\alpha ) | w _ { \\bar { k } 2 } | ^ 2 } { \\sigma ^ 2 _ n C _ 0 + C _ 1 - \\alpha | w _ { \\bar { k } 2 } | ^ 2 \\tilde { p } } + \\frac { ( 1 - \\alpha ) | w _ { \\bar { k } 1 } | ^ 2 } { \\sigma ^ 2 _ n C _ 0 + C _ 2 + \\alpha | w _ { \\bar { k } 1 } | ^ 2 \\tilde { p } } = 0 , \\end{align*}"}
-{"id": "3853.png", "formula": "\\begin{align*} L _ j ( x , t , y ) = \\int ^ \\infty _ 0 d s & \\int ( 1 - \\chi _ { I _ j } ( \\tau ) ) | \\tau | ^ \\beta e ^ { 2 \\pi i t \\tau } e ^ { - \\varpi s } d \\tau \\\\ & \\times \\int _ { \\R ^ n } \\varphi _ j ( \\xi ) \\ ^ { \\chi _ 2 } \\bigl ( \\tau - i | \\xi | \\sinh s \\bigr ) \\frac { e ^ { 2 \\pi i ( x - y ) \\cdot \\xi } } { | \\xi | ^ \\varpi } d \\xi . \\end{align*}"}
-{"id": "2002.png", "formula": "\\begin{align*} \\begin{gathered} \\exists f , \\ , \\exists c , d > 0 , \\ , g \\geq 1 , \\ , C \\\\ \\mathcal { A } _ n f ( x ) \\leq - c g ( x ) + d I _ C ( x ) , \\ | x | < n , n \\in \\N . \\end{gathered} \\end{align*}"}
-{"id": "2915.png", "formula": "\\begin{align*} \\{ X ^ { * ( 3 ) } ( z ) , X ^ { * ( 3 ) } ( w ) \\} = 0 \\ , . \\end{align*}"}
-{"id": "4201.png", "formula": "\\begin{align*} X ( M e _ 1 r + M e _ 2 , \\ , M ^ * e _ 3 a ) = X ( M e _ 1 , \\ , M ^ * e _ 3 a r ) X ( M e _ 2 , \\ , M ^ * e _ 3 a ) \\ , \\end{align*}"}
-{"id": "6408.png", "formula": "\\begin{align*} g _ { \\gamma } ( t ) = \\frac { t ^ { \\gamma - 1 } } { \\Gamma ( \\gamma ) } , \\end{align*}"}
-{"id": "1866.png", "formula": "\\begin{align*} \\Omega _ G = \\Omega - \\chi ^ a \\cdot \\iota _ a \\Theta \\end{align*}"}
-{"id": "4253.png", "formula": "\\begin{align*} S _ q ( x ) = \\frac { \\sin \\left ( \\mu ( x ) / q \\right ) } { \\mu ( x ) / q } - 1 . \\end{align*}"}
-{"id": "1780.png", "formula": "\\begin{align*} a x _ 1 ^ 2 + b x _ 2 ^ 2 - a b x _ 3 ^ 2 = d \\end{align*}"}
-{"id": "4480.png", "formula": "\\begin{align*} R _ 5 = \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\{ \\log ( n - 1 ) - \\Psi ( n ) - \\log f ( x ) \\} \\ , d x = o \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\end{align*}"}
-{"id": "7462.png", "formula": "\\begin{align*} y _ j : = ( \\phi ^ * ( Y _ j ) ) ^ t ( x _ i ) . \\end{align*}"}
-{"id": "8600.png", "formula": "\\begin{align*} \\operatorname { i m } ( \\overline { \\alpha } ) = \\frac { \\operatorname { k e r ( \\gamma ) } } { \\operatorname { i m } ( \\beta ) } \\oplus \\operatorname { i m } ( \\gamma ) \\oplus \\{ 0 \\} \\oplus \\{ 0 \\} \\end{align*}"}
-{"id": "4477.png", "formula": "\\begin{align*} \\max ( | R _ { 3 1 } | , | R _ { 3 2 } | ) = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { \\beta } { d } } } { n ^ { \\frac { \\beta } { d } } } \\biggr \\} \\biggr ) , \\end{align*}"}
-{"id": "7552.png", "formula": "\\begin{align*} F ( x ) = f ( x ) - x , x \\in [ 0 , \\infty ) . \\end{align*}"}
-{"id": "6792.png", "formula": "\\begin{align*} P _ { [ \\theta _ 1 + \\theta _ 2 , \\varphi ] } ( V _ { \\theta _ 1 } + V _ { \\theta _ 2 } ) \\geq P _ { [ \\theta _ 1 , u ] } ( V _ { \\theta _ 1 } ) + P _ { [ \\theta _ 2 , v ] } ( V _ { \\theta _ 2 } ) = V _ { \\theta _ 1 } + V _ { \\theta _ 2 } . \\end{align*}"}
-{"id": "6526.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i - v _ { i - 1 } ) _ { i = 1 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 1 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "4198.png", "formula": "\\begin{align*} ( I - \\Delta ) ^ m f ( x ) = 0 \\R ^ n \\setminus A \\end{align*}"}
-{"id": "4932.png", "formula": "\\begin{align*} \\langle a _ k , \\eta _ { j _ 0 } \\rangle = 0 \\ , \\ , \\ , \\ , \\ , \\ , \\langle a _ k , \\eta _ { l _ 0 } \\rangle = 0 , \\forall k \\in S _ { l _ 0 } \\cup S _ { j _ 0 } . \\end{align*}"}
-{"id": "6188.png", "formula": "\\begin{align*} \\int _ { | z + 1 | > e / | x | } [ f ( x + x z ) - f ( x ) - f ' ( x ) x z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) = \\int _ { | z + 1 | > e / | x | } [ \\log | 1 + z | - z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) , \\end{align*}"}
-{"id": "4977.png", "formula": "\\begin{align*} \\frac { r ^ { 2 } \\| \\vec { c } \\| \\| A ^ { k } \\| } { k ^ { r - 1 } \\rho ( A ) ^ { k } } = \\frac { r ^ { 2 } \\| \\vec { c } \\| \\| A ^ { k } \\| } { k ^ { r - 1 } \\delta ^ { k } } \\end{align*}"}
-{"id": "6439.png", "formula": "\\begin{align*} g _ { \\alpha } * \\partial _ { s } ( g _ { 1 - \\alpha , m } * [ \\psi ^ { 1 + q } \\tilde { u } ^ { 1 - q } ] ) = h _ { m } * ( \\psi ^ { 1 + q } \\tilde { u } ^ { 1 - q } ) . \\end{align*}"}
-{"id": "4703.png", "formula": "\\begin{align*} d _ k ( c ' , c '' ) = d _ { \\ell _ 1 } ( \\sigma _ k ( z ' ) , \\sigma _ k ( z '' ) ) \\leq t . \\end{align*}"}
-{"id": "9301.png", "formula": "\\begin{align*} \\hat { \\pi } ( t , z ) = \\frac { \\alpha ( t ) \\int _ { \\mathbb { R } _ + } y ' ( t , x , z ) p ( t , x , z ) d x } { \\beta ^ 2 ( t ) \\int _ { \\mathbb { R } _ + } y '' ( t , x , z ) p ( t , x , z ) d x } . \\end{align*}"}
-{"id": "9179.png", "formula": "\\begin{align*} e _ n ( X ) = \\sum _ { i _ 1 < \\ldots < i _ n } \\lambda _ { i _ 1 } ( X ) \\lambda _ { i _ 2 } ( X ) \\ldots \\lambda _ { i _ n } ( X ) \\ / , \\ \\ \\ n = 1 , \\ldots , p , \\end{align*}"}
-{"id": "8688.png", "formula": "\\begin{align*} J = K = V \\times U . \\end{align*}"}
-{"id": "311.png", "formula": "\\begin{align*} D _ g : = \\max \\biggl \\{ 1 , \\sup _ { \\delta \\in ( 0 , \\| f \\| _ \\infty ) } \\frac { \\sup _ { x : f ( x ) \\geq \\delta } M _ { g } ^ * ( x ) } { a ( \\delta ) ^ { m + 1 } } \\biggr \\} . \\end{align*}"}
-{"id": "6765.png", "formula": "\\begin{align*} \\theta _ { \\varphi _ { \\beta } } ^ n = e ^ { \\beta \\varphi _ { \\beta } } \\theta _ + ^ n . \\end{align*}"}
-{"id": "5977.png", "formula": "\\begin{align*} N ( x ) = 2 k b n - k ^ 2 b ^ 2 - b k ^ 2 \\geq b ( 2 k n - k n - k ^ 2 ) > b ( n - 1 ) . \\end{align*}"}
-{"id": "3948.png", "formula": "\\begin{align*} \\left | ( \\sum _ { i = 1 } ^ l | B _ i | ) \\int _ { G / \\Gamma } f \\dd \\mu _ G \\right | \\leq \\epsilon | I | , \\end{align*}"}
-{"id": "1999.png", "formula": "\\begin{align*} \\mathcal { A } f ( x ) = ( \\gamma _ U x + \\gamma _ L ) f ' ( x ) + \\frac { 1 } { 2 } ( x ^ 2 \\sigma _ U ^ 2 + 2 x \\sigma _ { U L } + \\sigma _ L ^ 2 ) f '' ( x ) . \\end{align*}"}
-{"id": "5239.png", "formula": "\\begin{align*} D _ 2 ( v ) = \\sum _ { u \\in \\Gamma ( v ) } d ( u ) . \\end{align*}"}
-{"id": "3257.png", "formula": "\\begin{align*} E _ k \\triangleright [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q & = ( E _ k \\triangleright E _ \\xi ) E _ { \\xi ^ \\prime } ^ * + ( K _ k \\triangleright E _ \\xi ) ( E _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) \\\\ & - q ^ { - ( \\xi , \\xi ^ \\prime ) } ( E _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) E _ \\xi - q ^ { - ( \\xi , \\xi ^ \\prime ) } ( K _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) ( E _ k \\triangleright E _ \\xi ) . \\end{align*}"}
-{"id": "2693.png", "formula": "\\begin{align*} \\theta _ { \\varphi _ { \\beta } } ^ n = e ^ { \\beta \\varphi _ { \\beta } } \\theta _ + ^ n . \\end{align*}"}
-{"id": "2438.png", "formula": "\\begin{align*} F _ { k , N } = \\sum _ { l \\in \\widetilde { \\Gamma _ { k } ^ { \\alpha _ { 1 } , \\alpha _ { 2 } } } } T _ { N l } f _ { l } ^ { \\alpha _ { 1 } } , \\end{align*}"}
-{"id": "3202.png", "formula": "\\begin{align*} h _ { 2 , j k , \\alpha } ( z _ j ) = \\left ( \\begin{array} { c } h _ { 2 , j k , \\alpha } ^ 1 ( z _ j ) \\\\ h _ { 2 , j k , \\alpha } ^ 2 ( z _ j ) \\\\ \\vdots \\\\ h _ { 2 , j k , \\alpha } ^ r ( z _ j ) \\end{array} \\right ) \\end{align*}"}
-{"id": "9317.png", "formula": "\\begin{align*} Z ( t , x , z ) : = \\frac { 1 } { Y ( t , x , z ) } \\end{align*}"}
-{"id": "4527.png", "formula": "\\begin{align*} \\rho _ n : = \\bigl [ c _ n \\log ^ { 1 / d } ( n - 1 ) \\bigr ] ^ { - 1 } . \\end{align*}"}
-{"id": "2821.png", "formula": "\\begin{align*} s _ i = S + i ( 1 + \\varepsilon ) , t _ i = s _ i + 1 , x _ i = f _ p ( t _ i ) , I _ i = ( s _ i , t _ i ] . \\end{align*}"}
-{"id": "8568.png", "formula": "\\begin{align*} ( \\rho ^ { k } \\otimes ( ^ { k } \\rho ) ) \\Delta ( \\alpha ) \\Diamond \\rho ( z ) & = ( \\rho ^ { k } \\otimes ( ^ { k } \\rho ) ) ( 1 \\otimes \\alpha ) \\Diamond \\rho ( z ) \\\\ & = ( \\bar { e _ { k } } \\otimes p ( \\alpha ) ) \\Diamond \\rho ( z ) \\\\ & = c y c ( \\rho ( \\alpha ) \\rho ( z ) ) \\\\ & = \\rho ( c y c ( \\alpha z ) ) \\\\ & = \\rho ^ { k } ( \\Delta ( \\alpha ) \\Diamond z ) \\end{align*}"}
-{"id": "8541.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { r \\in L ( k ) , a \\in _ { k } T } \\xi _ { r a } \\pi _ { r a } = i d _ { N _ { i n } } \\end{align*}"}
-{"id": "3302.png", "formula": "\\begin{gather*} v _ 1 \\wedge v _ 0 = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } \\pi _ + ( V _ 1 ) , v _ 1 \\wedge v _ { - 1 } = \\frac { 1 } { [ 2 ] _ { q ^ 2 } } \\pi _ + ( V _ 0 ) , v _ 0 \\wedge v _ { - 1 } = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } \\pi _ + ( V _ { - 1 } ) , \\\\ w _ { - 1 } \\wedge w _ 0 = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } \\pi _ - ( W _ { - 1 } ) , w _ { - 1 } \\wedge w _ 1 = \\frac { 1 } { [ 2 ] _ { q ^ 2 } } \\pi _ - ( W _ 0 ) , w _ 0 \\wedge w _ 1 = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } \\pi _ - ( W _ 1 ) . \\end{gather*}"}
-{"id": "9703.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ( x ) } { \\exp ( - e ^ { 1 / x } ) } = 1 . \\end{align*}"}
-{"id": "4610.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { 8 } f _ t ( p ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ p | } . \\end{align*}"}
-{"id": "7566.png", "formula": "\\begin{align*} x _ 1 = f ( x _ 0 ) + l \\chi _ 1 \\le f ( x _ 0 ) + l \\le f ( x _ 0 ) + x _ 0 - f ( x _ 0 ) - \\Delta _ l ( x _ 0 ) = x _ 0 - \\Delta _ l ( x _ 0 ) . \\end{align*}"}
-{"id": "355.png", "formula": "\\begin{align*} e ^ { i J ( g ) } T ( f ) e ^ { - i J ( g ) } = T ( f ) + J ( ( \\partial _ \\theta g ) f ) + \\int _ { - \\pi } ^ { \\pi } \\frac { ( \\partial _ \\theta g ) ^ 2 ( e ^ { i \\theta } ) } { 2 } f ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } \\ ; I , \\end{align*}"}
-{"id": "6977.png", "formula": "\\begin{align*} \\delta _ L f ( x _ 1 , . . . , x _ { q + 1 } ) = & ( - 1 ) ^ { q + 1 } \\sum _ { i = 1 } ^ q ( - 1 ) ^ { i + 1 } [ x _ i , f ( x _ 1 , . . , \\hat { x _ i } , . . , x _ { q + 1 } ) ] + [ f ( x _ 1 , . . . , x _ q ) , x _ { q + 1 } ] \\\\ & + ( - 1 ) ^ { q + 1 } \\sum _ { 1 \\leq i < j \\leq q + 1 } f ( x _ 1 , . . , \\hat { x _ i } , . . , x _ { j - 1 } , [ x _ i , x _ j ] , x _ { j + 1 } , . . , x _ { q + 1 } ) . \\end{align*}"}
-{"id": "1769.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } \\Delta u _ { i } ^ { k + 1 } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { k + 1 } \\sum \\limits _ { j \\neq i } H ( u _ { j } ^ { k } ) ( x ) & \\Omega , \\\\ u _ { i } ^ { k + 1 } ( x ) = \\phi _ { i } ( x ) & ( \\Omega ) _ 1 \\setminus \\Omega . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4050.png", "formula": "\\begin{align*} q + r = m + n + l + k . \\end{align*}"}
-{"id": "9449.png", "formula": "\\begin{align*} \\mu ( a A ) & = \\lim _ { \\omega } \\frac { \\dim ( ( W _ i + a W _ i ) \\cap a A ) } { \\dim ( W _ i ) } \\ge \\lim _ { \\omega } \\frac { \\dim ( a W _ i \\cap a A ) } { \\dim ( W _ i ) } \\\\ & \\ge \\lim _ { \\omega } \\frac { \\dim ( a ( W _ i \\cap A ) ) } { \\dim ( W _ i ) } = \\lim _ { \\omega } \\frac { \\dim ( W _ i \\cap A ) } { \\dim ( W _ i ) } = \\mu ( A ) , \\end{align*}"}
-{"id": "2943.png", "formula": "\\begin{align*} V _ n ( z ) = \\log | g ( z ) | - U _ n ( z ) , \\ : \\ : \\ : \\ : z \\in \\mathbb H . \\end{align*}"}
-{"id": "9934.png", "formula": "\\begin{align*} ( 1 + b ^ 2 ) x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 = 0 . \\end{align*}"}
-{"id": "4360.png", "formula": "\\begin{align*} \\varphi ( y , a ^ * ) ^ { ( p - 1 ) / p } - \\varphi ( y , a _ * ) ^ { ( p - 1 ) / p } & = \\frac { p - 1 } { p } \\ \\int _ { \\varphi ( y , a _ * ) } ^ { \\varphi ( y , a ^ * ) } z ^ { - 1 / p } \\ d z \\\\ & \\le \\frac { p - 1 } { p } \\varphi ( y , a _ * ) ^ { - 1 / p } G ( y ) \\\\ & \\le \\frac { ( p - 1 ) ( 2 p - 1 ) } { 2 p } G ( y ) \\ . \\end{align*}"}
-{"id": "3824.png", "formula": "\\begin{align*} e ^ { H [ \\boldsymbol { t } ] } \\psi ( z ) e ^ { - H [ \\boldsymbol { t } ] } & = e ^ { - \\sum _ { q \\geq 1 } t _ q z ^ q } \\psi ( z ) \\\\ e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast ( z ) e ^ { - H [ \\boldsymbol { t } ] } & = e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\psi ^ \\ast ( z ) . \\end{align*}"}
-{"id": "1750.png", "formula": "\\begin{align*} \\langle \\alpha ^ \\dag , ( \\beta ^ \\dag ) ^ \\lor \\rangle ^ \\dag = \\langle \\iota ( ( \\beta ^ \\dag ) ^ \\lor ) , \\iota ^ * ( \\alpha ^ \\dag ) \\rangle , \\mbox { f o r } \\alpha ^ \\dag , \\beta ^ \\dag \\in \\Delta ^ \\dag ; \\end{align*}"}
-{"id": "6159.png", "formula": "\\begin{align*} g ( { \\bf x } ^ { \\bf s } ) = c ^ N k _ 2 ^ { s _ 2 } \\cdots k _ n ^ { s _ n } { \\bf x } ^ { \\bf s } . \\end{align*}"}
-{"id": "3198.png", "formula": "\\begin{align*} F _ { k , \\alpha } ( z _ k ( z _ j , w _ j ) ) = F _ { k , \\alpha } ( z _ k ( z _ j , 0 ) ) + \\sum _ { | \\gamma | \\geq 1 } F _ { k j , \\alpha , \\gamma } ( z _ j ) \\cdot w _ j ^ \\gamma . \\end{align*}"}
-{"id": "3000.png", "formula": "\\begin{align*} D _ \\mathcal { R } ( t , p ) = \\left \\{ P \\in B _ \\mathcal { R } ( t , p ) \\mid \\tilde { u } ( t , P ) = \\max _ { Q \\in B _ \\mathcal { R } ( t , p ) } \\tilde { u } ( t , Q ) \\right \\} . \\end{align*}"}
-{"id": "4740.png", "formula": "\\begin{align*} \\Big | \\int _ { B _ r } \\frac { \\ , 2 u ( x ) - u ( x + z ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z \\Big | & \\le N \\omega _ N \\sup _ { \\eta \\in B _ r ( x ) } | D ^ 2 u ( \\eta ) | \\ , \\int _ 0 ^ r \\frac { d r } { \\ , r ^ { 2 s - 1 } \\ , } \\\\ & = N \\omega _ N \\ , \\frac { \\ , r ^ { 2 - 2 s } \\ , } { \\ , 2 - 2 s \\ , } \\ , \\sup _ { \\eta \\in B _ r ( x ) } | D ^ 2 u ( \\eta ) | . \\end{align*}"}
-{"id": "2809.png", "formula": "\\begin{align*} \\Pi ( E ) ( x , x ) = \\frac { 1 } { 2 \\pi } \\frac { \\partial } { \\partial y } | E ( x + i y ) | ^ 2 \\Big | _ { y = 0 } ( x \\in \\R ) . \\end{align*}"}
-{"id": "392.png", "formula": "\\begin{align*} L _ 0 ( I ) & = \\bigsqcup _ { m = 1 } ^ \\infty \\bigsqcup _ { \\Gamma / \\Gamma \\cap N } \\gamma \\cdot ( 0 , 0 , 0 , m ) , \\\\ L _ 0 ( I I ) & = \\bigsqcup _ { m = 1 } ^ \\infty \\bigsqcup _ { n = 0 } ^ { m - 1 } \\bigsqcup _ { \\gamma \\in \\Gamma } \\gamma \\cdot ( 0 , 0 , m , n ) , \\\\ \\hat { L } _ 0 ( I I ) & = \\bigsqcup _ { m = 1 } ^ \\infty \\bigsqcup _ { n = 0 } ^ { 3 m - 1 } \\bigsqcup _ { \\gamma \\in \\Gamma } \\gamma \\cdot ( 0 , 0 , 3 m , n ) . \\end{align*}"}
-{"id": "9749.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { e ^ { 1 / g ^ { - 1 } ( x ) } } { \\log ( 1 / x ) } = 1 , \\lim _ { x \\to 0 ^ + } \\frac { 1 / g ^ { - 1 } ( x ) } { \\log _ 2 ( 1 / x ) } = 1 . \\end{align*}"}
-{"id": "9892.png", "formula": "\\begin{align*} \\sum _ { n \\le x } a _ n \\Lambda ( n ) = H A ( x ) \\left \\{ 1 + O ( ( \\log x ) ^ { - 1 } ) \\right \\} + O ( T ( x , D ) \\log x ) \\end{align*}"}
-{"id": "6119.png", "formula": "\\begin{align*} g ^ { - 1 } = \\begin{pmatrix} \\pi ^ { - 1 } s _ { i } ^ { - 3 } + O ( s _ { i } ^ { - 2 } ) & O ( 1 ) & O ( 1 ) \\\\ O ( 1 ) & \\pi ^ { - 1 } s _ { j } ^ { - 3 } + O ( s _ { j } ^ { - 2 } ) & O ( 1 ) \\\\ O ( 1 ) & O ( 1 ) & \\psi _ { z \\bar z } ^ { - 1 } + \\sum O ( s _ { i } ^ { 2 } ) \\end{pmatrix} . \\end{align*}"}
-{"id": "4226.png", "formula": "\\begin{align*} x ^ n = \\sum _ { l = 0 } ^ n S _ 2 ( n , l ) ( x ) _ l , ( \\textnormal { s e e } \\ , \\ , [ 7 , 8 , 9 ] ) . \\end{align*}"}
-{"id": "3920.png", "formula": "\\begin{align*} t ( 2 - t ) \\frac { d ^ 2 } { d t ^ 2 } u ( t ) + 2 ( m + 1 ) ( 1 - t ) \\ , \\frac { d } { d t } u ( t ) + \\left [ z - c ^ 2 ( 1 - t ) ^ 2 \\right ] u ( t ) = 0 \\ , . \\end{align*}"}
-{"id": "2698.png", "formula": "\\begin{align*} \\int _ { U } \\theta _ { V _ { \\theta } } ^ n \\leq \\liminf _ { \\beta \\to + \\infty } \\int _ U \\theta _ { \\varphi _ { \\beta } } ^ n \\leq \\liminf _ { \\beta \\to + \\infty } \\int _ U e ^ { - \\beta \\delta } \\theta _ + ^ n = 0 . \\end{align*}"}
-{"id": "6121.png", "formula": "\\begin{align*} u = \\phi ( z , \\bar z ) + 2 \\pi s + s ^ { 2 } \\psi ( s , z , \\bar z ) \\end{align*}"}
-{"id": "7785.png", "formula": "\\begin{align*} ( { } _ i \\tilde x , \\phi ' ) = \\theta \\circ ( { } _ i \\tilde x , \\phi ) . \\end{align*}"}
-{"id": "8733.png", "formula": "\\begin{gather*} X ^ 1 _ { \\tau } - X ^ 2 _ { \\tau } = e ^ { \\tau A } ( x _ 1 - x _ 2 ) + e ^ { \\tau A } [ v ^ { ( 0 ) } ( 0 , x _ 1 ) - v ^ { ( 0 ) } ( 0 , x _ 2 ) ] \\\\ - [ v ^ { ( 0 ) } ( \\tau , X _ \\tau ^ { 1 } ) - v ^ { ( 0 ) } ( \\tau , X _ \\tau ^ { 2 } ) ] + { \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } [ \\widetilde Z ^ { x _ 1 } _ s - \\widetilde Z ^ { x _ 2 } _ s ] \\ ; d W _ s , } \\ ; \\ ; \\tau \\in [ 0 , T _ 0 ] , \\end{gather*}"}
-{"id": "510.png", "formula": "\\begin{align*} K _ { \\rm A i } ( u , v ) = \\int _ { \\mathcal { C } _ { - 1 } ^ { 2 \\pi / 3 } } \\dd w \\int _ { \\mathcal { C } _ 1 ^ { \\pi / 3 } } \\dd z \\frac { e ^ { z ^ 3 / 3 - z u } } { e ^ { w ^ 3 / 3 - w v } } \\frac { 1 } { z - w } . \\end{align*}"}
-{"id": "5528.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ j } \\leq \\varphi ( t _ { n _ j } ) \\sum _ { k = 1 } ^ { m ( E _ j ) } w ( k ) \\leq \\frac { 1 } { 2 ^ { j - 2 } } . \\end{align*}"}
-{"id": "3162.png", "formula": "\\begin{align*} & P r \\left \\{ \\frac { 1 } { L } \\sum _ { l = 1 } ^ { L } I _ { q , s ^ n } ( G _ l ) < \\mu ( I _ { q , s ^ n } ) ~ \\forall q \\in P _ 0 ^ { K } ( \\theta ) ~ \\forall s ^ n \\in \\theta ^ n \\right \\} \\\\ & \\leq \\exp \\left ( | \\theta | \\log \\left ( \\frac { 2 | \\theta | } { \\delta } \\right ) + n \\log | \\theta | - \\frac { L \\epsilon ^ 2 \\mu _ { * } } { 3 c } \\right ) \\end{align*}"}
-{"id": "8136.png", "formula": "\\begin{align*} S _ { a , p , p ^ * , N } = S _ { a , p , p ^ * , N } ^ { r a d } , \\end{align*}"}
-{"id": "2007.png", "formula": "\\begin{align*} M _ m ( t ) \\to f ( V _ t ^ n ) - \\int _ 0 ^ t \\mathcal { A } _ n f ( V _ s ^ n ) \\d s = : M ( t ) , \\textrm { a . s . ~ a n d i n } L ^ 1 . \\end{align*}"}
-{"id": "1067.png", "formula": "\\begin{align*} e ( B ) \\leq \\ell \\frac { n ^ 2 } { 2 } + ( k - \\alpha - \\ell - x ) \\frac { n ^ 2 } { 2 } = ( k - \\alpha - x ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "5338.png", "formula": "\\begin{align*} u ' ( t ) = \\varphi ^ { - 1 } \\left [ t \\lambda \\frac { \\varphi ( b u ( 0 ) ) - \\varphi ( u ( 0 ) ) } { T } + \\varphi ( u ( 0 ) ) \\right ] \\end{align*}"}
-{"id": "3370.png", "formula": "\\begin{align*} X ( t + 1 ) = X ( t ) - \\Delta t P ( t ) + E _ s ( t ) . \\end{align*}"}
-{"id": "9781.png", "formula": "\\begin{align*} \\mathcal { D } ( \\overline { \\partial } _ { E , m } + \\overline { \\partial } _ { E , m } ^ t ) = \\bigoplus _ { q = 0 } ^ { m } \\mathcal { D } ( { \\overline { \\partial } _ { E , m , q } } ) \\cap \\mathcal { D } ( { \\overline { \\partial } ^ t _ { E , m , q - 1 } } ) \\end{align*}"}
-{"id": "6515.png", "formula": "\\begin{align*} T _ { u , r } ^ V : = \\{ v \\in V : \\min _ { 0 \\le t \\le T } \\| v - u ( t ) \\| _ V \\le r \\} , \\end{align*}"}
-{"id": "9668.png", "formula": "\\begin{align*} x ' ( t ) & = - a g ( x ( t ) ) + b g ( x ( t - \\tau ( t ) ) , t \\geq 0 \\\\ x ( t ) & = \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] \\end{align*}"}
-{"id": "8357.png", "formula": "\\begin{align*} \\sigma ^ { \\tilde { \\phi } } _ t ( x z _ g ) = \\sigma ^ \\phi _ t ( x z _ g ) = \\sigma ^ \\phi _ t ( x ) z _ g \\in I _ g . \\ \\ \\ x \\in M , \\ \\ \\ g \\in G \\ \\ \\ t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "9462.png", "formula": "\\begin{align*} A = \\sum _ { i } \\alpha _ { i } \\otimes X _ { i } \\end{align*}"}
-{"id": "8654.png", "formula": "\\begin{align*} X _ { t } = e ^ { t A } x + \\int _ { 0 } ^ { t } e ^ { \\left ( t - s \\right ) A } G B \\left ( s , X _ { s } \\right ) d s + \\int _ { 0 } ^ { t } e ^ { \\left ( t - s \\right ) A } G d W _ { s } , \\ ; \\ ; \\ ; t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "7528.png", "formula": "\\begin{align*} a : = \\alpha - l > 1 - \\frac { 1 } { M _ \\varepsilon } , 1 - \\frac { 1 } { M } < b : = \\alpha + l \\nu < 1 . \\end{align*}"}
-{"id": "1602.png", "formula": "\\begin{align*} F \\left ( 0 , x \\right ) = \\frac { 1 } { x } . \\end{align*}"}
-{"id": "7230.png", "formula": "\\begin{align*} E _ \\rho : = \\widetilde { Y } \\times \\mathbb { C } ^ r / \\sim _ \\rho , \\end{align*}"}
-{"id": "5631.png", "formula": "\\begin{align*} \\frac { B } { A } = \\frac { \\sum _ { i = 1 } ^ n b _ i } { \\sum _ { i = 1 } ^ n a _ i } \\leq \\frac { \\beta \\sum _ { i = 1 } ^ n a _ i } { \\sum _ { i = 1 } ^ n a _ i } = \\beta . \\end{align*}"}
-{"id": "6397.png", "formula": "\\begin{align*} C _ { 2 } ( \\vec { \\varepsilon } ) = \\left ( 1 - ( p - 1 ) ! ! \\varepsilon _ { p } \\sigma ^ { p } + \\frac { \\varepsilon _ { q } ^ { 2 } } { 2 } ( 2 q - 1 ) ! ! \\sigma ^ { 2 q } + \\frac { \\varepsilon _ { p } ^ { 2 } } { 2 } ( 2 p - 1 ) ! ! \\sigma ^ { 2 p } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "2459.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } f \\| _ { M _ 2 } = & \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 2 , \\alpha _ 1 } } \\| \\Box _ l ^ { \\alpha _ 2 } \\Box _ k ^ { \\alpha _ 1 } f \\| ^ q _ { L ^ { \\infty } } \\right ) ^ { 1 / q } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 2 / 2 } \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 2 , \\alpha _ 1 } } \\| \\Box _ l ^ { \\alpha _ 2 } \\Box _ k ^ { \\alpha _ 1 } f \\| ^ q _ { L ^ { 2 } } \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "8006.png", "formula": "\\begin{align*} \\xi _ 1 ( y ) = e ^ { \\frac { | y | ^ p } { p } } \\alpha | y | ^ { 1 - n } J _ { n - 1 } \\approx \\sqrt { \\frac { 2 \\pi } { p } } e ^ { \\frac { | y | ^ p } { p } } \\alpha | y | ^ { 1 - n } n ^ { \\frac { 1 } { p } - \\frac { 1 } { 2 } } e ^ { F ( ( n - 1 ) ^ { \\frac { 1 } { p } } ) } , \\ , \\ , \\ , \\ , \\ , n \\rightarrow \\infty , \\end{align*}"}
-{"id": "5415.png", "formula": "\\begin{align*} g ( x ) = a x + b \\ ; \\ ; \\pmod { 1 } \\end{align*}"}
-{"id": "2240.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } p ( x ) d x = C ( \\varepsilon ) \\int _ { \\infty } ^ { \\infty } \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } e ^ { - \\varepsilon x ^ { p } } d x = 1 . \\end{align*}"}
-{"id": "8369.png", "formula": "\\begin{align*} f v ^ * f = g v ^ * v \\phi ( ( 1 - q ) p ) v ^ * g \\phi ( ( 1 - q ) p ) = g v ^ * ( 1 - q ) p g \\phi ( ( 1 - q ) p ) = 0 \\end{align*}"}
-{"id": "7914.png", "formula": "\\begin{align*} \\int _ M S _ g ^ 4 u ^ 4 d v _ 0 = \\int _ M S _ g ^ 4 d v _ g \\le C _ 2 . \\end{align*}"}
-{"id": "5558.png", "formula": "\\begin{align*} \\dot A _ 1 + H \\dot B _ 1 & = ( K + 1 ) \\beta [ A _ 1 + H B _ 1 ] , \\\\ \\dot A _ 1 - H \\dot B _ 1 & = ( 1 - K ) \\beta [ A _ 1 - H B _ 1 ] + b _ x ( 1 ) [ A _ 1 + H B _ 1 ] , \\end{align*}"}
-{"id": "846.png", "formula": "\\begin{align*} \\tilde { \\tau } = \\tau ' \\prod _ { 0 \\le s < t } ^ { \\curvearrowleft } ( \\prod _ { \\begin{subarray} { c } i \\in J _ { p ( s ) } \\\\ i < \\ell ( s ) \\end{subarray} } ^ { \\curvearrowright } \\sigma _ { i - 1 } ) \\end{align*}"}
-{"id": "7118.png", "formula": "\\begin{align*} \\frac { 1 } { | | \\beta - \\alpha | | _ g } \\sum _ { l = 1 } ^ n ( \\Omega _ g ( \\alpha , \\beta , l ) \\omega _ g ( \\alpha + [ l ] , \\beta + [ l ] , l ) & + \\Omega _ g ( \\alpha - [ l ] , \\beta - [ l ] , l ) \\omega _ g ( \\alpha , \\beta , l ) ) ( x ^ { \\alpha } \\otimes g ) \\epsilon _ { \\beta } ^ * \\\\ & = ( x ^ { \\alpha } \\otimes g ) \\epsilon _ { \\beta } ^ * \\end{align*}"}
-{"id": "5434.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\\\ 0 & - 1 & \\alpha & 0 & \\overline \\alpha \\\\ 0 & - \\overline \\alpha & 0 & - \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & 0 \\\\ 0 & 0 & - \\overline \\alpha J ( T ^ { - r } , T ^ { - r } ) & 0 & - \\alpha J ( T ^ { - 3 r } , T ^ { - 3 r } ) \\\\ 0 & - \\alpha & 0 & - \\overline \\alpha J ( T ^ { - 2 r } , T ^ { - r } ) & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "2528.png", "formula": "\\begin{align*} \\lim _ { T \\to \\infty } \\frac 1 T \\int _ 0 ^ T \\mathbb { E } f ( U ( t ) ) d t = \\int _ { \\mathcal { M } _ 0 } f d \\mu _ h = \\int _ { \\mathcal { S } } f d \\mu _ h . \\end{align*}"}
-{"id": "3539.png", "formula": "\\begin{align*} ( \\bar { g } _ k , \\bar { \\pi } _ k ) & = ( g , \\pi ) \\ ; \\ ; \\ ; \\mbox { i n } B _ { R _ 1 } \\\\ ( \\bar { g } _ k , \\bar { \\pi } _ k ) & = ( g _ k , \\pi _ k ) \\mbox { i n } M \\setminus B _ { R _ 2 } . \\end{align*}"}
-{"id": "5286.png", "formula": "\\begin{align*} \\int _ { T } ^ { 2 T } \\vert Z ( t ) \\vert d t = \\int _ { T } ^ { 2 T } \\vert \\zeta ( \\tfrac { 1 } { 2 } + i t ) \\vert d t \\geq \\vert \\int _ { T } ^ { 2 T } \\zeta ( \\tfrac { 1 } { 2 } + i t ) d t \\vert . \\end{align*}"}
-{"id": "9210.png", "formula": "\\begin{align*} & \\mathbb { E } [ D _ { t , z } \\delta _ Z ( z ) | \\mathcal { F } _ t ] = \\\\ & \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } \\exp \\big [ \\int _ 0 ^ t \\int _ { \\mathbb { R } } i x \\psi ( s , \\zeta ) \\tilde { N } ( d s , d \\zeta ) + \\int _ 0 ^ t i x \\beta ( s ) d B ( s ) \\\\ & + \\int _ t ^ { T _ 0 } \\int _ { \\mathbb { R } } ( e ^ { i x \\psi ( s , \\zeta ) } - 1 - i x \\psi ( s , \\zeta ) ) \\nu ( d \\zeta ) d s - \\int _ t ^ { T _ 0 } \\frac { 1 } { 2 } x ^ 2 \\beta ^ 2 ( s ) d s - i x z \\big ] ( e ^ { i x \\psi ( t , z ) } - 1 ) d x . \\end{align*}"}
-{"id": "9614.png", "formula": "\\begin{align*} \\psi _ f ( x , t ) = \\psi _ { f , r e g } ( x , t ) + \\sum \\limits _ { 1 \\le k \\le n } \\psi _ { f , k } ( x , t ) , \\end{align*}"}
-{"id": "717.png", "formula": "\\begin{align*} & { \\bf h } _ { \\vec { D } } = \\left ( \\begin{array} { c } h _ { D _ { 1 } } \\\\ h _ { D _ { 2 } } \\\\ \\vdots \\\\ h _ { D _ { r } } \\end{array} \\right ) , \\ { \\bf h } _ { \\vec { F } } = \\left ( \\begin{array} { c } h _ { F _ { 1 } } \\\\ h _ { F _ { 2 } } \\\\ \\vdots \\\\ h _ { F _ { s } } \\end{array} \\right ) , \\ { \\bf h } _ { { p } _ { * } \\vec { F } } = \\left ( \\begin{array} { c } h _ { { p } _ { * } F _ { 1 } } \\\\ h _ { { p } _ { * } F _ { 2 } } \\\\ \\vdots \\\\ h _ { { p } _ { * } F _ { s } } \\end{array} \\right ) \\ . \\end{align*}"}
-{"id": "1060.png", "formula": "\\begin{align*} R _ 3 ( P _ n ) = \\begin{cases} 2 n - 2 & n \\ , , \\\\ 2 n - 1 & n \\ , . \\end{cases} \\end{align*}"}
-{"id": "2564.png", "formula": "\\begin{align*} \\norm { f } _ { \\dot { X } ^ { s , r } } : = \\norm { J ^ s ( t ) f } _ { L _ x ^ r ( \\mathbb { R } ^ d ) } \\sim \\norm { \\ , | t | ^ s | \\nabla | ^ s M ( - t ) f } _ { L _ x ^ r ( \\R ^ d ) } . \\end{align*}"}
-{"id": "5561.png", "formula": "\\begin{align*} c _ n ( x ) = \\int _ { 0 \\leq \\xi _ 1 \\leq \\xi _ 2 \\leq \\dots \\leq \\xi _ { n + 1 } = x } \\left [ \\prod _ { j = 1 } ^ n ( \\xi _ { j + 1 } - \\xi _ j ) \\rho ( \\xi _ j ) \\right ] c _ 0 ( \\xi _ 1 ) d \\xi _ 1 d \\xi _ 2 \\dots d \\xi _ n \\end{align*}"}
-{"id": "8267.png", "formula": "\\begin{align*} \\mathrm { v a r } _ \\lambda ( F ) = \\int _ \\R ( V ' ( u ) - \\lambda ) ^ 2 p _ { \\lambda } ( u ) \\dd u = \\int _ \\R ( V ' ( u ) - \\lambda ) ( - \\partial _ u p _ { \\lambda } ( u ) ) \\dd u = \\int _ \\R V '' ( u ) p _ { \\lambda } ( u ) \\dd u . \\end{align*}"}
-{"id": "2042.png", "formula": "\\begin{align*} \\| \\eta ' \\| ( r , s ) = \\left [ 1 + 2 r \\langle \\gamma ' ( s ) , J ' ( s ) \\rangle + r ^ 2 \\left ( \\| J ' ( s ) \\| ^ 2 + \\langle \\nabla _ r \\nabla _ r \\frac { \\dd \\eta } { \\dd s } \\big | _ { r = 0 } , e _ n \\rangle \\right ) + O ( r ^ 3 ) \\right ] ^ { 1 / 2 } . \\end{align*}"}
-{"id": "3201.png", "formula": "\\begin{align*} T _ { j k } w _ k = u _ j + \\sum _ { | \\beta | \\geq 2 } \\left ( \\sum _ { | \\alpha | = | \\beta | } T _ { j k } F _ { k , \\alpha } ( z _ k ( z _ j , 0 ) ) \\cdot \\tau _ { k j , \\beta } ^ \\alpha + h _ { 1 , j k , \\beta } ( z _ j ) \\right ) \\cdot u _ j ^ \\beta . \\end{align*}"}
-{"id": "9387.png", "formula": "\\begin{align*} { } \\widetilde { W } _ \\lambda = \\widetilde { \\Gamma } _ 1 u _ \\lambda = ( { q } , G \\ast { q } ) + 2 i k [ 1 + ( G \\ast { q } ) ( 0 ) ] [ 1 + ( G \\ast { { q } ^ * } ) ( 0 ) ] . \\end{align*}"}
-{"id": "1167.png", "formula": "\\begin{align*} \\dot A _ 1 + H \\dot B _ 1 & = ( K + 1 ) \\beta [ A _ 1 + H B _ 1 ] , \\\\ \\dot A _ 1 - H \\dot B _ 1 & = ( 1 - K ) \\beta [ A _ 1 - H B _ 1 ] + b _ x ( 1 ) [ A _ 1 + H B _ 1 ] , \\end{align*}"}
-{"id": "111.png", "formula": "\\begin{align*} \\frac { 2 } { 1 + \\sqrt { 1 - 4 t } } = \\sum _ { n = 0 } ^ \\infty n ! C _ n \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 9 - 1 3 ] ) . \\end{align*}"}
-{"id": "7421.png", "formula": "\\begin{align*} \\{ X _ i , X _ j \\} = \\epsilon _ { i j } X _ i X _ j . \\end{align*}"}
-{"id": "1114.png", "formula": "\\begin{align*} \\| h \\| _ 2 & = \\| D ^ * h \\| _ 2 \\\\ & = \\sqrt { \\| D ^ * _ { S _ 0 } h \\| _ 2 ^ 2 + \\| D ^ * _ { S _ 0 ^ c } h \\| _ 2 ^ 2 } \\\\ & \\leq \\sqrt { \\| D ^ * _ { S _ 0 } h \\| _ 2 ^ 2 + ( \\| D ^ * _ { S _ 0 } h \\| _ 2 + R ) ^ 2 } \\\\ & \\leq \\sqrt { 2 \\| D ^ * _ { S _ 0 } h \\| _ 2 ^ 2 } + R \\leq \\sqrt { 2 } z + R \\\\ & \\leq \\frac { \\sqrt { 2 ( 1 + \\delta ) } } { 1 - \\sqrt { t / ( t - 1 ) } \\delta } \\epsilon + \\left ( \\frac { \\sqrt { 2 } \\delta + \\sqrt { t ( \\sqrt { ( t - 1 ) / t } - \\delta ) \\delta } } { t ( \\sqrt { ( t - 1 ) / t } - \\delta ) } + 1 \\right ) R . \\end{align*}"}
-{"id": "8179.png", "formula": "\\begin{align*} f _ n & = \\sum _ { k = 1 } ^ { n } \\frac { ( k - 1 ) ! } { n ! } B _ { n , k } ( 1 ! \\psi _ 1 , 2 ! \\psi _ 2 , \\dots ) , \\\\ \\phi _ n & = \\sum _ { k = 1 } ^ { n } \\frac { k ! } { n ! } B _ { n , k } ( 1 ! \\psi _ 1 , 2 ! \\psi _ 2 , \\dots ) , \\\\ \\theta _ n & = \\sum _ { k = 1 } ^ { n } \\binom { - ( \\alpha + \\beta ) n + k } { k - 1 } \\frac { ( k - 1 ) ! } { n ! } B _ { n , k } ( 1 ! \\psi _ 1 , 2 ! \\psi _ 2 , \\dots ) . \\end{align*}"}
-{"id": "2682.png", "formula": "\\begin{align*} g ( t ) : = I ( \\varphi _ t ) - \\int _ X e ^ { \\varphi + t \\chi } d \\mu , \\ t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "1303.png", "formula": "\\begin{align*} \\widetilde { S } ( \\alpha ) - V ( \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } ( \\Lambda ( m ) - 1 ) e ^ { - m / N } e ( m \\alpha ) \\end{align*}"}
-{"id": "6394.png", "formula": "\\begin{align*} p ( x ) = \\frac { C ( \\vec { \\varepsilon } ) } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } e ^ { \\varepsilon _ { q } x ^ { q } - \\varepsilon _ { p } x ^ { p } } , \\end{align*}"}
-{"id": "1401.png", "formula": "\\begin{align*} b ' _ { i j } = \\begin{cases} - b _ { i j } & i = k j = k \\\\ b _ { i j } & b _ { i k } b _ { k j } \\leq 0 \\\\ b _ { i j } + | b _ { i k } | b _ { k j } & b _ { i k } b _ { k j } > 0 . \\end{cases} \\end{align*}"}
-{"id": "3983.png", "formula": "\\begin{align*} a _ i ( t ) w _ { r + j } = e ^ { ( \\delta _ i - 2 j ) t } w _ { { r + j } } \\delta _ { i } - 2 j \\leq \\delta _ { i } , \\forall \\ , j = 0 , \\dots , l - r . \\end{align*}"}
-{"id": "9786.png", "formula": "\\begin{align*} \\log B F _ { 1 2 } = \\log \\frac { \\{ | \\mathcal { M } _ { 1 } \\} } { \\{ | \\mathcal { M } _ { 2 } \\} } + \\log \\frac { \\{ | \\mathcal { M } _ { 1 } \\} } { \\{ | \\mathcal { M } _ { 2 } \\} } - \\log \\frac { \\{ | \\mathcal { M } _ { 1 } \\} } { \\{ | \\mathcal { M } _ { 2 } \\} } . \\end{align*}"}
-{"id": "8972.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | w ( x ) - w ( y ) | ^ { p - 2 } \\ , ( w ( x ) - w ( y ) ) \\ , ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y = \\int _ { \\Omega } \\frac { f _ n ( x ) } { ( u ^ + + 1 / n ) ^ \\gamma } \\ , \\varphi \\ , d x . \\end{align*}"}
-{"id": "3275.png", "formula": "\\begin{align*} \\hat { R } _ { V , W } ( v \\otimes w ) = q ^ { ( \\mathrm { w t } ( v ) , \\mathrm { w t } ( w ) ) } w \\otimes v + \\sum _ i w _ i \\otimes v _ i , \\end{align*}"}
-{"id": "143.png", "formula": "\\begin{align*} \\| f \\| _ { \\widehat { E } } = \\lim _ { n \\to \\infty } \\| \\tilde { f _ n } \\| _ { \\widehat { E } } = \\lim _ { n \\to \\infty } \\| \\tilde { f _ n } \\circ \\sigma ^ { - 1 } \\| _ { \\widehat { E } } = \\| \\mu ( f ) \\| _ { \\widehat { E } } . \\end{align*}"}
-{"id": "428.png", "formula": "\\begin{align*} k ( x , y ) = \\sum _ { \\mathcal { Q } \\in \\mathcal { I } } \\lambda _ \\mathcal { Q } e _ \\mathcal { Q } ( x ) f _ \\mathcal { Q } ( y ) . \\end{align*}"}
-{"id": "2557.png", "formula": "\\begin{align*} [ e ^ { i t \\Delta } f ] ( x ) = ( 4 \\pi i t ) ^ { - \\frac { d } { 2 } } \\int _ { \\R ^ d } e ^ { \\frac { i | x - y | ^ 2 } { 4 t } } f ( y ) \\ , d y , t \\neq 0 . \\end{align*}"}
-{"id": "265.png", "formula": "\\begin{align*} T _ { 1 2 } : = \\biggl | \\int _ { \\mathcal { X } _ n ^ c } \\int _ { \\mathcal { X } _ n } f ( x ) f ( y ) \\log f ( y ) \\int _ { \\tilde { u } _ { n , x , y } } ^ \\infty \\log ( u f ( x ) ) \\ , d ( \\tilde { F } _ { n , x } - F ^ - _ { n , x } ) ( u ) \\ , d y \\ , d x \\biggr | . \\end{align*}"}
-{"id": "733.png", "formula": "\\begin{align*} & { \\bf h } _ { \\vec { D } } = \\left ( \\begin{array} { c } h _ { D _ { 1 } } \\\\ h _ { D _ { 2 } } \\\\ \\vdots \\\\ h _ { D _ { r } } \\end{array} \\right ) , \\ { \\bf h } _ { \\vec { F } } = \\left ( \\begin{array} { c } h _ { F _ { 1 } } \\\\ h _ { F _ { 2 } } \\\\ \\vdots \\\\ h _ { F _ { s } } \\end{array} \\right ) , \\ { \\bf h } _ { { p } _ { * } \\vec { F } } = \\left ( \\begin{array} { c } h _ { { p } _ { * } F _ { 1 } } \\\\ h _ { { p } _ { * } F _ { 2 } } \\\\ \\vdots \\\\ h _ { { p } _ { * } F _ { s } } \\end{array} \\right ) \\ . \\end{align*}"}
-{"id": "5117.png", "formula": "\\begin{align*} L ^ { ( i ) } ( z ; s ) = \\begin{pmatrix} 1 + z q ^ { 2 N _ { i } } & \\beta _ { i } ^ { * } ( 1 - s ^ { 2 } q ^ { 2 N _ { i } } ) \\\\ z \\beta _ { i } & z + s ^ { 2 } q ^ { 2 N _ { i } } \\end{pmatrix} ( 0 \\le i \\le M ) . \\end{align*}"}
-{"id": "9910.png", "formula": "\\begin{align*} { \\bf g } \\ ; : = \\ ; \\begin{pmatrix} g _ 1 \\\\ g _ 2 \\\\ g _ 3 \\end{pmatrix} \\ ; = \\ ; \\begin{pmatrix} - q ^ { 3 } K F + q F K \\\\ q ^ { - 3 } K E - q ^ { - 1 } E K \\\\ q E F - q ^ { - 1 } F E + \\kappa K ^ 2 - \\kappa K '^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "42.png", "formula": "\\begin{align*} \\vert \\xi \\vert ^ 2 = \\sup _ { \\Vert u \\Vert _ y \\leq 1 } | \\langle \\xi , u \\rangle | ^ 2 . \\end{align*}"}
-{"id": "6522.png", "formula": "\\begin{align*} \\begin{aligned} \\| B ( t , v ) - B ( t , w ) \\| _ { X } \\le \\tilde \\lambda \\| v - w \\| _ D + C _ { B } ( \\| v \\| _ D + \\| w \\| _ D ) \\| v - w \\| _ { W } . \\end{aligned} \\end{align*}"}
-{"id": "555.png", "formula": "\\begin{align*} \\begin{aligned} & m a c ( ( P _ 1 \\oplus P _ 2 ) ( T \\otimes I _ { n + m } ) ( P _ 1 \\oplus P _ 2 ) ) \\\\ & = m a c ( P _ 1 ( T \\otimes I _ n ) P _ 1 ) + m a c ( P _ 2 ( T \\otimes I _ m ) P _ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "7254.png", "formula": "\\begin{align*} T _ { j k } w _ k = u _ j + \\sum _ { | \\beta | \\geq 2 } \\left ( \\sum _ { | \\alpha | = | \\beta | } T _ { j k } F _ { k , \\alpha } ( z _ k ( z _ j , 0 ) ) \\cdot \\tau _ { k j , \\beta } ^ \\alpha + h _ { 1 , j k , \\beta } ( z _ j ) \\right ) \\cdot u _ j ^ \\beta . \\end{align*}"}
-{"id": "7935.png", "formula": "\\begin{align*} c _ { i , 1 } + c _ { n + 1 - i , m } = m ( d _ i + d _ { n + 1 - i } ) + m - 1 = m ( n - 1 ) + m - 1 = m n - 1 , \\end{align*}"}
-{"id": "5761.png", "formula": "\\begin{align*} B _ { n } ^ \\dag ( u ) = \\frac { 1 } { 2 } u ^ { T } \\left ( \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } \\sigma _ { 0 , t } ^ 2 H _ { 0 , t } H _ { 0 , t } ^ { T } \\right ) u + E _ { n } ^ { \\dag } ( u ^ { \\ast } ) \\end{align*}"}
-{"id": "4216.png", "formula": "\\begin{align*} z _ { k + 1 } = \\frac { 1 } { 2 } ( 1 + \\sqrt { \\frac { L } { \\mu } } ) x _ { k + 1 } + \\frac { 1 } { 2 } ( 1 - \\sqrt { \\frac { L } { \\mu } } ) x _ k . \\end{align*}"}
-{"id": "5200.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\| U ^ { * } _ { 1 } \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| V ^ { * } _ { 1 } \\| _ { S _ { \\widehat { p } _ { 2 } } } . \\end{align*}"}
-{"id": "2044.png", "formula": "\\begin{align*} \\nabla _ r \\nabla _ r T | _ { r = 0 } = - \\| J ' ( s ) \\| ^ 2 e _ n . \\end{align*}"}
-{"id": "4613.png", "formula": "\\begin{align*} \\sum _ { t = 2 } ^ { 8 } f _ t ( p ' ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { 8 } . \\end{align*}"}
-{"id": "5312.png", "formula": "\\begin{align*} A ( u ) & = ( u + \\omega + \\eta N _ { A } ) ( u - \\omega + \\eta N _ { B } ) + A B ^ \\dagger \\\\ B ( u ) & = ( u + \\omega + \\eta N _ { A } ) B + \\eta ^ { - 1 } A \\\\ C ( u ) & = ( u - \\omega + \\eta N _ { B } ) A ^ \\dagger + \\eta ^ { - 1 } B ^ \\dagger \\\\ D ( u ) & = A ^ \\dagger B + \\eta ^ { - 2 } . \\end{align*}"}
-{"id": "3003.png", "formula": "\\begin{align*} \\left \\langle \\pi , \\int _ X \\imath _ X ( x ) d P \\right \\rangle & = \\lim _ { n \\to \\infty } \\left \\langle \\varphi _ n , \\int _ X \\imath _ X ( x ) d P \\right \\rangle = \\lim _ { n \\to \\infty } \\int _ X \\langle \\varphi _ n , \\imath _ X ( x ) \\rangle d P \\\\ & = \\int _ X \\lim _ { n \\to \\infty } \\langle \\varphi _ n , \\imath _ X ( x ) \\rangle d P = \\int _ X \\langle \\pi , \\imath _ X ( x ) \\rangle d P \\end{align*}"}
-{"id": "378.png", "formula": "\\begin{align*} \\hat { L } = \\left \\{ f \\in L : 3 | b , c \\right \\} . \\end{align*}"}
-{"id": "4544.png", "formula": "\\begin{align*} W _ 3 = W _ { 3 1 } + W _ { 3 2 } = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { 1 / 2 + 2 \\beta / d } } { n ^ { 1 + 2 \\beta / d } } \\ , , \\ , \\frac { \\log n } { n k ^ { 1 / 2 } } \\ , , \\ , \\frac { k ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "5091.png", "formula": "\\begin{align*} \\mathbb { T } ^ { [ M ' , M ] } ( z ) = L ^ { ( M ' ) } ( z ) L ^ { ( M ' + 1 ) } ( z ) \\cdots L ^ { ( M - 1 ) } ( z ) L ^ { ( M ) } ( z ) , \\end{align*}"}
-{"id": "3835.png", "formula": "\\begin{align*} \\norm { \\partial _ t ^ i D ^ j v } { H ^ { k - m , 2 k - 2 m } ( D _ T ) } & \\le \\sum _ { \\ell = 1 } ^ { k } \\norm { \\partial _ t ^ { \\ell } v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 m - 2 \\ell + 2 i + j } ( D ) ) } \\\\ & \\le \\sum _ { \\ell = 1 } ^ { k } \\norm { \\partial _ t ^ { \\ell } v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 \\ell } ( D ) ) } \\le | v | _ { H ^ { k , 2 k } ( D _ T ) } . \\end{align*}"}
-{"id": "7788.png", "formula": "\\begin{align*} X _ o = \\widehat X _ o \\cap \\pi _ 1 ( \\pi _ 3 \\pi _ 2 ) ^ { - 1 } \\pi ' _ 3 \\pi ' _ 2 ( \\pi ' _ 1 ) ^ { - 1 } \\widehat X ' _ o , \\end{align*}"}
-{"id": "8099.png", "formula": "\\begin{align*} u ^ { \\star } ( x ) = \\sup \\left \\{ t \\geq 0 : \\mu _ l \\left ( \\left \\{ | u ( x ) | > t \\right \\} \\right ) > \\mu _ l \\left ( B _ { \\left \\vert x \\right \\vert } \\right ) \\right \\} . \\end{align*}"}
-{"id": "3212.png", "formula": "\\begin{align*} \\delta ( \\{ ( U _ j , F _ { j , \\alpha } ^ \\lambda ) \\} ) = \\{ ( U _ { j k } , h _ { 1 , j k , \\alpha } ^ \\lambda - h _ { 2 , j k , \\alpha } ^ \\lambda ) \\} \\end{align*}"}
-{"id": "9845.png", "formula": "\\begin{align*} \\bar \\Xi ^ { \\star } & = \\left \\{ \\begin{array} { l l } \\arg \\max _ { \\bar \\Xi \\ge 0 } \\beta ( \\bar \\Xi ) , & { \\rm i f } ~ \\max _ { \\bar \\Xi \\ge 0 } \\beta ( \\bar \\Xi ) > ( 2 \\ln 2 ) W \\\\ 0 , & { \\rm o t h e r w i s e } , \\end{array} \\right . \\\\ P ^ \\star & = \\chi ( \\bar \\Xi ^ \\star ) . \\end{align*}"}
-{"id": "7458.png", "formula": "\\begin{align*} \\vec { i } _ w = ( 1 , 2 , 3 , 1 ) \\vec { i } ^ * _ w = ( 2 , 1 , 2 , 3 ) . \\end{align*}"}
-{"id": "4917.png", "formula": "\\begin{align*} | U _ l | & = | U _ { l - 1 } \\cup S _ { j _ l } | \\geq | U _ { l - 1 } | + | S _ { j _ l } | - | U _ { l - 1 } \\cap S _ { j _ l } | \\\\ & \\geq ( ( l - 1 ) \\rho - \\Delta _ { l - 2 } t ( \\rho ) ) | M | + \\rho | M | - | \\bigcup _ { i = 1 } ^ { l - 1 } { S _ { j _ i } \\cap S _ { j _ l } } | \\\\ & > ( l \\rho - \\Delta _ { l - 2 } t ( \\rho ) ) | M | - ( l - 1 ) t ( \\rho ) | M | = ( l \\rho - \\Delta _ { l - 1 } t ( \\rho ) ) | M | , \\end{align*}"}
-{"id": "6984.png", "formula": "\\begin{align*} \\sum _ { i + j = n } [ f _ i ^ x , \\alpha _ j ] = 0 , \\end{align*}"}
-{"id": "1696.png", "formula": "\\begin{align*} \\frac { \\partial g } { \\partial z _ i } ( z ) & = 2 a f ^ { a - 1 } ( z ) z _ i \\ , , \\frac { \\partial ^ 2 g } { \\partial z _ i \\ , \\partial z _ j } ( z ) = 4 a ( a - 1 ) f ^ { a - 2 } ( z ) z _ i z _ j + 2 a f ^ { a - 1 } ( z ) \\delta _ { i , j } \\ , , \\end{align*}"}
-{"id": "559.png", "formula": "\\begin{align*} \\begin{aligned} f m a c ( \\tau ) & = \\alpha ( \\tau ) \\circ K _ 0 ( \\psi ( \\tau ) ) \\\\ & = \\alpha ( { \\tilde \\tau } ) \\circ K _ 0 ( W \\circ \\psi ( \\tau ) ) \\\\ & = \\alpha ( { \\tilde \\tau } ) \\circ K _ 0 ( \\psi ( { \\tilde \\tau } ) \\circ W ) \\\\ & = \\alpha ( { \\tilde \\tau } ) \\circ K _ 0 ( \\psi ( { \\tilde \\tau } ) \\circ K _ 0 ( W ) \\\\ & = \\alpha ( { \\tilde \\tau } ) \\circ K _ 0 ( \\psi ( { \\tilde \\tau } ) = f m a c ( { \\tilde \\tau } ) . \\end{aligned} \\end{align*}"}
-{"id": "1677.png", "formula": "\\begin{align*} f - T _ { \\lambda } f = g . \\end{align*}"}
-{"id": "4257.png", "formula": "\\begin{align*} \\frac { \\sin \\varphi _ q ^ k } { \\mu ( x ^ k _ q ) } & = \\frac { 1 } { q } \\left ( \\frac { \\sin \\left ( \\frac { \\mu \\left ( { k } / { q } \\right ) } { q } \\right ) } { \\frac { \\mu \\left ( { k } / { q } \\right ) } { q } } + \\frac { \\beta ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) \\right ) \\end{align*}"}
-{"id": "266.png", "formula": "\\begin{align*} \\rho _ n : = \\bigl [ c _ n \\log ^ { 1 / d } ( n - 1 ) \\bigr ] ^ { - 1 } . \\end{align*}"}
-{"id": "3237.png", "formula": "\\begin{align*} \\left \\{ \\left ( a _ { i } , b _ { i } \\right ) \\right \\} _ { i = 1 } ^ { k } \\left ( k , \\tilde { \\delta } , r \\right ) N _ { X } p X \\end{align*}"}
-{"id": "5511.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\varphi , w } = \\{ x \\in L ^ 0 , \\ \\ \\ P ( \\lambda x ) < \\infty \\ \\ \\ \\ \\ \\lambda > 0 \\} . \\end{align*}"}
-{"id": "1812.png", "formula": "\\begin{align*} \\phi ^ * t = z w , \\ \\phi ^ * \\tau _ j ' = \\tau _ j , \\ \\phi ^ * z _ r ' = z _ r , \\end{align*}"}
-{"id": "6175.png", "formula": "\\begin{align*} \\xi _ t = - \\log \\mathcal { E } ( U ) _ t = - U _ t + \\frac { \\sigma _ U ^ 2 } { 2 } t + \\sum _ { s \\leq t } [ \\Delta U _ s - \\log ( 1 + \\Delta U _ s ) ] . \\end{align*}"}
-{"id": "275.png", "formula": "\\begin{align*} W _ 1 & = \\int _ { \\mathcal { X } \\times \\mathcal { X } } f ( x ) f ( y ) \\int _ { ( [ l _ x , v _ x ] \\times [ l _ y , v _ y ] ) ^ c } \\ ! \\ ! h ( u , v ) \\ , d ( F _ { n , x , y } - F _ { n , x } F _ { n , y } ) ( u , v ) \\ , d x \\ , d y \\\\ & = o ( n ^ { - ( 9 / 2 - \\epsilon ) } ) . \\end{align*}"}
-{"id": "2358.png", "formula": "\\begin{align*} \\dfrac { d } { d t } \\psi ( \\Phi _ t ( y ) , t ) = \\partial _ x \\psi ( \\Phi _ t ( y ) , t ) \\cdot v ( \\Phi _ t ( y ) ) + \\partial _ t \\psi ( \\Phi _ t ( y ) , t ) \\end{align*}"}
-{"id": "6783.png", "formula": "\\begin{align*} \\theta + d d ^ c \\lambda \\psi = \\lambda ( \\theta + d d ^ c \\psi ) + ( 1 - \\lambda ) \\theta \\geq \\lambda \\omega + ( 1 - \\lambda ) \\theta \\geq 0 , \\end{align*}"}
-{"id": "2585.png", "formula": "\\begin{align*} \\limsup _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\| ( i \\partial _ t + \\Delta ) u _ n ^ J - F ( u _ n ^ J ) \\| _ { N ( \\tilde I _ n ) } = 0 . \\end{align*}"}
-{"id": "4315.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { y _ 0 } ^ 2 : = \\int _ { X _ { y _ 0 } } | u | ^ 2 e ^ { - \\varphi _ L } . \\end{align*}"}
-{"id": "4796.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( \\pm a t + c ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "9629.png", "formula": "\\begin{align*} R _ \\phi [ \\pi _ \\phi ( u ) \\pi _ \\phi ( b _ 1 ) & \\pi _ \\phi ( u ) \\pi _ \\phi ( b _ 2 ) \\cdots \\pi _ \\phi ( u ) \\pi _ \\phi ( b _ n ) ] \\\\ & = R _ \\phi [ \\pi _ \\phi ( u b _ 1 u b _ 2 \\cdots u b _ n ) ] \\\\ & = \\phi ( u b _ 1 u b _ 2 \\cdots u b _ n ) \\\\ & = R [ \\beta ( u b _ 1 u b _ 2 \\cdots u b _ n ) ] \\\\ & = R [ U \\alpha ( b _ 1 ) U \\alpha ( b _ 2 ) \\cdots U \\alpha ( b _ n ) ] , \\end{align*}"}
-{"id": "4548.png", "formula": "\\begin{align*} B _ x ( r _ { n , u } ) \\cap B _ y ( r _ { n , v } ) = \\{ B _ x ( r _ { n , v } ) \\cap B _ y ( r _ { n , v } ) \\} \\cup [ \\{ B _ x ( r _ { n , u } ) \\setminus B _ x ( r _ { n , v } ) \\} \\cap B _ y ( r _ { n , v } ) ] , \\end{align*}"}
-{"id": "8564.png", "formula": "\\begin{align*} \\operatorname { k e r } ( \\alpha ) & = C ^ { - 1 } ( \\operatorname { k e r } ( \\hat { \\alpha } ) ) \\\\ \\operatorname { i m } ( \\alpha ) & = \\operatorname { i m } ( \\hat { \\alpha } ) \\end{align*}"}
-{"id": "6012.png", "formula": "\\begin{gather*} \\begin{pmatrix} b & Z & 0 \\\\ X & B & - \\Sigma ^ { - 1 } Z ^ { \\top } \\\\ 0 & - X ^ { \\top } \\Sigma & - b \\end{pmatrix} , \\end{gather*}"}
-{"id": "5730.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ 2 \\sum _ { Q \\in \\mathfrak { A } _ k ( Q _ 0 ) } \\alpha _ Q \\chi _ Q \\right | \\leq 2 \\sum _ { k = 1 } ^ 2 \\sum _ { Q \\in \\mathfrak { A } _ { k - 1 } ( Q _ 0 ) } \\omega _ { \\lambda _ w } ( f ; Q ) \\chi _ Q , \\end{align*}"}
-{"id": "9859.png", "formula": "\\begin{align*} \\int \\limits _ { \\Omega } \\nabla u ( x ) \\cdot \\nabla v ( x ) ~ d x = \\mu \\int \\limits _ { \\Omega } u ( x ) v ( x ) ~ d x \\end{align*}"}
-{"id": "6070.png", "formula": "\\begin{align*} Y _ \\alpha ^ { \\langle \\lambda , \\alpha ^ \\vee \\rangle + 1 } u = 0 \\end{align*}"}
-{"id": "8590.png", "formula": "\\begin{align*} \\Pi _ { 1 } ' \\left ( d _ { 1 } ( ^ { \\ast } b ) d _ { 2 } \\cdot \\psi ( n ) \\right ) & = - d _ { 1 } \\pi _ { 1 } ' p ' \\xi _ { b e _ { k } } ( d _ { 2 } \\psi ( n ) ) \\\\ & = - d _ { 1 } \\pi _ { 1 } ' p ' \\hat { \\psi } _ { o } \\xi _ { b e _ { k } } ( d _ { 2 } n ) \\end{align*}"}
-{"id": "9021.png", "formula": "\\begin{align*} t ^ { d m _ 0 } \\hat { f } _ \\psi ( \\hat { x } , t ^ N , z ) = \\hat { a } _ d ( \\hat { x } , t ^ N ) \\prod _ { i } ( t ^ { m _ 0 } z - t ^ { m _ i } \\hat { u } _ i ( \\hat { x } , t ) ) \\end{align*}"}
-{"id": "7235.png", "formula": "\\begin{align*} f _ { k j , n + 1 } = \\left ( \\begin{array} { c } f _ { k j , n + 1 } ^ 1 \\\\ f _ { k j , n + 1 } ^ 2 \\\\ \\vdots \\\\ f _ { k j , n + 1 } ^ r \\end{array} \\right ) : = \\sum _ { | \\alpha | = n + 1 } \\left ( \\begin{array} { c } f _ { k j , \\alpha } ^ 1 \\\\ f _ { k j , \\alpha } ^ 2 \\\\ \\vdots \\\\ f _ { k j , \\alpha } ^ r \\end{array} \\right ) \\cdot e _ j ^ \\alpha , \\end{align*}"}
-{"id": "9239.png", "formula": "\\begin{align*} I _ 1 = \\mathbb { E } [ \\int _ 0 ^ T ( \\int _ D \\{ h ( t , x ) - \\widehat { h } ( t , x ) \\} \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] d x ) d t ] , I _ 2 = \\mathbb { E } [ \\int _ D \\{ k ( x ) - \\hat { k } ( x ) \\} \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] d x ] . \\end{align*}"}
-{"id": "1143.png", "formula": "\\begin{align*} { \\tilde \\mu _ { i j } } \\left ( { { f _ { i j } } \\left ( x \\right ) } \\right ) + d _ { i j } ^ - = 1 , \\ , \\ , \\ , \\ , i = 1 , 2 , . . . m ; j = 1 , 2 , . . . , p _ m \\end{align*}"}
-{"id": "3362.png", "formula": "\\begin{align*} \\textstyle 1 = \\langle \\bar { M } , \\bar { M } \\rangle = \\sum _ { i } \\alpha _ i \\langle \\bar { M } , M _ i \\rangle \\leq \\sum _ { i } \\alpha _ i = 1 . \\end{align*}"}
-{"id": "3358.png", "formula": "\\begin{align*} \\chi _ { \\mathcal { C } } ( M ) : = \\begin{cases} 0 , & M \\in \\mathcal { C } \\\\ \\infty , & M \\notin \\mathcal { C } \\end{cases} \\end{align*}"}
-{"id": "2716.png", "formula": "\\begin{align*} \\omega _ { F S } + d d _ z ^ c G = ( \\pi _ 2 ) _ \\star ( \\Theta + d d _ { x , z } ^ c \\Phi ) ^ { n + 1 } , \\end{align*}"}
-{"id": "3934.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 | S _ { m , n } ( c , \\eta ) | ^ 2 \\ , d \\eta = 1 \\end{align*}"}
-{"id": "5548.png", "formula": "\\begin{align*} & \\frac 1 2 b _ { x x } ( 0 ) - h b _ x ( 0 ) + h ^ 2 b ( 0 ) = 0 , \\\\ & \\frac 1 2 b _ { x x } ( 1 ) + H b _ x ( 1 ) + H ^ 2 b ( 1 ) = 0 , \\end{align*}"}
-{"id": "208.png", "formula": "\\begin{align*} b _ 1 ( x ) = - \\frac { \\Delta f ( x ) } { 2 ( d + 2 ) V _ d ^ { 2 / d } f ( x ) ^ { 1 + 2 / d } } . \\end{align*}"}
-{"id": "1713.png", "formula": "\\begin{gather*} ( \\epsilon _ { \\times } ) _ { A B C D E F G } : = \\tfrac { 1 } { 4 2 } { \\Phi } _ { K [ A B } { \\Phi } ^ K { } _ { C D } { \\Phi } _ { E F G ] } . \\end{gather*}"}
-{"id": "5203.png", "formula": "\\begin{align*} T _ p ^ { \\mathbb { C } } M : = T _ p M \\cap J T _ p M \\ ; . \\end{align*}"}
-{"id": "8148.png", "formula": "\\begin{align*} \\lambda _ 1 ( \\Omega ) = \\min \\left \\{ \\frac { \\displaystyle \\int _ { \\Omega } | \\nabla \\varphi | ^ p d x } { \\displaystyle \\int _ { \\Omega } | x | ^ { - \\beta p } | \\varphi ( x ) | ^ p d x } : \\ \\varphi \\in W ^ { 1 , p } _ 0 ( \\Omega ) \\setminus \\{ 0 \\} \\right \\} \\ , . \\end{align*}"}
-{"id": "3244.png", "formula": "\\begin{align*} P _ { \\alpha } ^ { S } = P _ { \\beta } ^ { S } \\circ \\Phi _ { \\beta , \\alpha } , \\end{align*}"}
-{"id": "3380.png", "formula": "\\begin{align*} X _ { } = - V \\zeta _ { \\max } - \\Delta t P _ { \\max } . \\end{align*}"}
-{"id": "3043.png", "formula": "\\begin{align*} V ( u ) = \\sum _ { j = 1 } ^ { | u | } \\ell ^ { u _ { j - 1 } } _ { u ( j ) } , \\end{align*}"}
-{"id": "5472.png", "formula": "\\begin{align*} I _ C ^ m = ( I _ { C _ 1 } I _ { C _ 2 } ) ^ m = I _ { C _ 1 } ^ m \\cdot I _ { C _ 2 } ^ m = I _ { C _ 1 } ^ m \\cap I _ { C _ 2 } ^ m = I _ { C _ 1 } ^ { ( m ) } \\cap I _ { C _ 2 } ^ { ( m ) } = I _ C ^ { ( m ) } . \\end{align*}"}
-{"id": "1869.png", "formula": "\\begin{align*} & \\lambda _ 1 = \\frac { 1 - \\eta } 4 R - x , & \\lambda _ 2 = \\frac { 1 - \\eta } 4 R + x , \\\\ & \\lambda _ 3 = \\frac { 1 + \\eta } 4 R - y , & \\lambda _ 4 = \\frac { 1 + \\eta } 4 R + y . \\end{align*}"}
-{"id": "1619.png", "formula": "\\begin{align*} \\ln F \\left ( t , x \\right ) = e ^ { - m t } \\left ( x - x _ { 0 } \\right ) + \\frac { c } { m } e ^ { - m t } - \\frac { 1 } { 4 m } e ^ { - 2 m t } \\end{align*}"}
-{"id": "154.png", "formula": "\\begin{align*} g ( \\Gamma ) : = g ( G , w ) : = b _ 1 ( G ) + \\sum _ { v \\in V } w ( v ) , \\end{align*}"}
-{"id": "586.png", "formula": "\\begin{align*} x - [ r _ 0 ] \\equiv \\sum _ { k = 1 } ^ \\nu p ^ k [ \\phi ^ { - k } ( r _ k ) ] \\bmod I ^ { \\nu + 1 } \\end{align*}"}
-{"id": "8304.png", "formula": "\\begin{align*} \\vert E ( \\gamma ) \\vert = \\vert E ( G ) \\vert - \\sum _ { i = 1 } ^ m \\vert E ( G _ i ) \\vert \\end{align*}"}
-{"id": "6909.png", "formula": "\\begin{align*} \\tau ( G ) = O ( 1 ) \\biggl ( \\frac { ( d - 1 ) ^ { d - 1 } } { ( d ^ 2 - 2 d ) ^ { d / 2 - 1 } } \\biggr ) ^ { \\ ! n } \\frac { \\log n } { n d \\log d } \\end{align*}"}
-{"id": "7377.png", "formula": "\\begin{align*} b ^ t : = A q ^ t - \\lambda _ t m ^ { t - 1 } , & m ^ t : = g _ t ( b ^ t , w ) , \\\\ h ^ { t + 1 } : = A ^ * m ^ t - \\xi _ t q ^ t , & q ^ { t } : = f _ t ( h ^ t , \\beta _ 0 ) . \\end{align*}"}
-{"id": "9584.png", "formula": "\\begin{align*} \\Psi ( t ) = ( \\psi ( t ) , \\dot \\psi ( t ) ) \\in C ( \\R , { \\cal D } _ F ) . \\end{align*}"}
-{"id": "798.png", "formula": "\\begin{align*} \\check { R } ( z / w ) [ \\mathbb { T } ^ { [ M ' , M ] } ( z ) \\otimes \\mathbb { T } ^ { [ M ' , M ] } ( w ) ] = [ \\mathbb { T } ^ { [ M ' , M ] } ( w ) \\otimes \\mathbb { T } ^ { [ M ' , M ] } ( z ) ] \\check { R } ( z / w ) , \\end{align*}"}
-{"id": "7049.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( 1 ) = { { \\bf { h } } ^ { [ j 1 ] } } ( 1 ) { { \\bf { u } } ^ { [ 1 ] } } + { { \\bf { h } } ^ { [ j 2 ] } } ( 1 ) { { \\bf { u } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "8439.png", "formula": "\\begin{align*} \\tilde { \\textbf { u } } : = { \\textbf { u } } - \\bar { \\textbf { u } } = ( \\tilde { \\rho } , \\tilde { v } ) ^ { T } = ( \\tilde { \\rho } , { v } ) ^ { T } , \\end{align*}"}
-{"id": "9727.png", "formula": "\\begin{align*} x _ { U , \\epsilon } ( t ) = g ^ { - 1 } \\left ( x _ 2 ( \\epsilon ) e ^ { - c _ 2 \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } \\right ) , t \\geq T _ 4 ( \\epsilon ) . \\end{align*}"}
-{"id": "5442.png", "formula": "\\begin{align*} ( X ^ { - 1 } P X ) A ( q ) ( X ^ { - 1 } P X ) ^ { - 1 } = A ^ * ( q ) . \\end{align*}"}
-{"id": "8536.png", "formula": "\\begin{align*} N _ { i n } = \\displaystyle \\bigoplus _ { a \\in _ { k } T } D _ { k } \\otimes _ { F } N _ { \\tau ( a ) } \\\\ N _ { o u t } = \\displaystyle \\bigoplus _ { b \\in T _ { k } } D _ { k } \\otimes _ { F } N _ { \\sigma ( b ) } \\end{align*}"}
-{"id": "3303.png", "formula": "\\begin{align*} \\pi _ + ( V _ 0 ) & = v _ 1 \\wedge v _ { - 1 } - v _ { - 1 } \\wedge v _ 1 - q ( q - q ^ { - 1 } ) v _ 0 \\wedge v _ 0 \\\\ & = 2 v _ 1 \\wedge v _ { - 1 } + ( q - q ^ { - 1 } ) ^ 2 v _ 1 \\wedge v _ { - 1 } = ( q ^ 2 + q ^ { - 2 } ) v _ 1 \\wedge v _ { - 1 } . \\end{align*}"}
-{"id": "4458.png", "formula": "\\begin{align*} \\mathbb { E } _ f ( \\hat { H } _ n ) - H = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\tilde { \\beta } / d } } { n ^ { \\tilde { \\beta } / d } } \\int _ { \\mathcal { X } _ n } \\frac { C _ { n , \\tilde { \\beta } } ( x ) } { f ( x ) ^ { \\tilde { \\beta } / d } } \\ , d x \\ , , \\ , q _ n ^ { 1 - \\epsilon } \\ , , \\ , q _ n \\log n \\ , , \\ , \\frac { 1 } { n } \\biggr \\} \\biggr ) , \\end{align*}"}
-{"id": "5.png", "formula": "\\begin{align*} S _ q ( x ) = \\frac { \\sin \\left ( \\mu ( x ) / q \\right ) } { \\mu ( x ) / q } - 1 . \\end{align*}"}
-{"id": "6748.png", "formula": "\\begin{align*} \\frac { d } { d t } { \\rm I } ( \\varphi _ t ) = \\int _ X \\chi \\theta _ { \\varphi _ t } ^ n , \\ \\forall t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3138.png", "formula": "\\begin{align*} \\lambda \\mathcal { L } ( \\lambda ) = \\lambda \\mathcal { L } _ 1 + \\mathcal { L } _ 2 \\mbox { w i t h } \\mathcal { L } _ 1 ^ * = \\sigma \\mathcal { L } _ 1 \\ , \\ , \\mbox { a n d } \\ , \\ , \\mathcal { L } _ 0 ^ * = \\sigma \\mathcal { L } _ 0 \\end{align*}"}
-{"id": "2198.png", "formula": "\\begin{align*} \\| w \\| _ { L ^ { 2 \\kappa } ( [ 0 , t _ { 1 } - t _ { 0 } ] \\times \\rho ' B _ { 1 } ) } ^ { 2 } \\leq C \\left ( ( g _ { 1 - \\alpha } * W ) ( t _ { * } ) + \\| F \\| _ { L ^ { 1 } ( [ 0 , t _ { 2 } - t _ { 0 } ] ) } \\right ) . \\end{align*}"}
-{"id": "8149.png", "formula": "\\begin{align*} \\\\ \\\\ \\left \\{ \\begin{array} { l l } - { \\rm d i v } ( | \\nabla v | ^ { p - 2 } | \\nabla v | ) = \\lambda | x | ^ { - \\beta p } | v | ^ { p - 2 } v & \\mbox { i n } \\Omega ^ { \\star } \\\\ v = 0 & \\mbox { o n } \\partial \\Omega ^ { \\star } . \\end{array} \\right . \\\\ \\end{align*}"}
-{"id": "7632.png", "formula": "\\begin{align*} a _ { r i } : = \\sum _ { j \\in \\mathcal { N } } \\bar { x } _ { r i j } , \\ i \\in \\mathcal { C } . \\end{align*}"}
-{"id": "6854.png", "formula": "\\begin{align*} K : = \\bigl \\{ \\bigl ( e ^ { - i x \\xi ( t ) } e ^ { - i t \\Delta } u ( t ) \\bigr ) _ { \\{ \\frac { 1 } { h ( t ) } \\} } : t \\in I \\bigr \\} \\end{align*}"}
-{"id": "3002.png", "formula": "\\begin{align*} \\int _ X u ( t , x ) d P _ n & = \\sum _ { \\{ i \\in I \\mid x ^ i \\in U ( t ) \\} } \\alpha ^ i u ( t , x ^ i ) + \\sum _ { \\{ i \\in I \\mid x ^ i \\in X \\setminus U ( t ) \\} } \\alpha ^ i u ( t , y ^ i _ n ) \\\\ & > \\sum _ { i \\in I } \\alpha ^ i u ( t , x ^ i ) = \\int _ X u ( t , x ) d Q . \\end{align*}"}
-{"id": "634.png", "formula": "\\begin{align*} \\partial V _ 2 = \\varnothing \\end{align*}"}
-{"id": "7324.png", "formula": "\\begin{align*} \\hat { R } F ( v _ { 1 } \\otimes v _ { 0 } ) & = [ 2 ] ^ { 1 / 2 } ( \\hat { R } ( v _ { 0 } \\otimes v _ { 0 } ) + \\hat { R } ( v _ { 1 } \\otimes v _ { - 1 } ) ) \\\\ & = [ 2 ] ^ { 1 / 2 } ( v _ { 0 } \\otimes v _ { 0 } + q ^ { - 2 } ( q ^ { 2 } - q ^ { - 2 } ) v _ { 1 } \\otimes v _ { - 1 } + \\hat { R } ( v _ { 1 } \\otimes v _ { - 1 } ) ) . \\end{align*}"}
-{"id": "7039.png", "formula": "\\begin{align*} X _ t = t A ( h ) + X _ t ^ { ( S , h ) } + X _ t ^ { ( B , h , + ) } - t h \\overline { \\Pi } ^ + ( h ) + X _ t ^ { ( B , h , - ) } + t h \\overline { \\Pi } ^ - ( h ) , t > 0 . \\end{align*}"}
-{"id": "4554.png", "formula": "\\begin{align*} ( 1 - p _ { n , x , u } - p _ { n , y , v } + p _ \\cap ) \\biggl \\| V ^ { - 1 / 2 } \\begin{pmatrix} p _ { n , x , u } \\\\ p _ { n , y , v } \\end{pmatrix} \\biggr \\| ^ { 3 } \\lesssim ( k / n ) ^ { 3 / 2 } . \\end{align*}"}
-{"id": "3381.png", "formula": "\\begin{align*} A = E _ { \\min } + V \\zeta _ { \\max } + \\Delta t P _ { \\max } \\end{align*}"}
-{"id": "8419.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\rho + \\nabla \\cdot ( \\rho v ) = 0 , \\\\ & \\partial _ { t } { v } + { v } \\cdot \\nabla { v } + \\frac { \\nabla { P } } { \\rho } = 0 , \\\\ & \\nabla \\cdot v = 0 , \\end{cases} \\end{align*}"}
-{"id": "8910.png", "formula": "\\begin{align*} - \\Delta u + ( 1 + \\abs { A } ^ 2 ) u = \\abs { u } ^ { p - 2 } u \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "1213.png", "formula": "\\begin{align*} \\min \\limits _ { 0 \\le i \\le N } P _ { i } ( A _ { i } ) \\le 0 . 7 1 \\vee \\frac { \\sum \\limits _ { i = 1 } ^ { N } { } ( P _ { i } \\| P _ { 0 } ) } { N \\log ( N + 1 ) } . \\end{align*}"}
-{"id": "3708.png", "formula": "\\begin{align*} Z _ { 1 2 3 } = w H _ { 1 2 3 } - \\frac { 1 } { 2 } ( x ^ 2 _ 1 + x ^ 2 _ 2 ) ~ . \\end{align*}"}
-{"id": "1549.png", "formula": "\\begin{align*} d _ E ( \\mu , \\nu ) = \\langle \\nabla E ( \\mu ) - \\nabla E ( \\nu ) , \\mu - \\nu \\rangle . \\end{align*}"}
-{"id": "4390.png", "formula": "\\begin{align*} | p _ i - q _ i | \\le c q _ i , \\ \\ i = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "3062.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( M _ n - m _ n , \\sum _ { | u | = n } \\delta _ { u , V ( u ) - M _ n } \\right ) = \\left ( \\log ( \\mathbf { e } / Z _ \\infty ) , \\sum _ { d \\in \\mathcal { P } ( D _ n ) } \\delta _ { u ^ { ( n ) } , \\zeta _ n + d } \\right ) , \\end{align*}"}
-{"id": "5699.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } m _ f ( 2 ^ l Q ) = 0 . \\end{align*}"}
-{"id": "4429.png", "formula": "\\begin{align*} \\| V a _ n - V a _ m \\| _ { \\widehat { C _ E } } = \\| V a _ n - V a _ m \\| _ { C _ { \\widehat { E } } } \\to 0 \\ \\ \\ \\ \\ \\ n , m \\to \\infty , \\end{align*}"}
-{"id": "1592.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( L _ { Y _ { I } } C _ { x } \\right ) _ { , x } + C ^ { x } L _ { Y _ { I } } C _ { x } + \\frac { 1 } { 2 } m \\int Y _ { I } d x = c \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "6059.png", "formula": "\\begin{align*} ( u ^ { \\varepsilon } ( x ) - v ^ { \\varepsilon } ( x + 1 ) ) ^ { \\prime \\prime } = 0 . \\end{align*}"}
-{"id": "6453.png", "formula": "\\begin{align*} \\mathcal { E } ( \\tilde { u } , - \\psi ^ { 2 } \\tilde { u } ^ { - q } ) = \\frac { 1 } { 2 } + , \\end{align*}"}
-{"id": "625.png", "formula": "\\begin{align*} { r _ Q } : = ( Q ) \\in ( 0 , \\infty ) . \\end{align*}"}
-{"id": "4067.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 3 , 3 u + 1 , 3 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 3 , 3 u + 2 , 3 v + 2 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "2808.png", "formula": "\\begin{align*} \\Pi ( E ) ( z , w ) = \\frac { E ( z ) \\overline { E ( w ) } - E ^ { \\# } ( z ) \\overline { E ^ { \\# } ( w ) } } { - 2 \\pi i ( z - \\bar { w } ) } , \\end{align*}"}
-{"id": "5875.png", "formula": "\\begin{align*} Z ^ { 4 } = e ^ { - 2 m t } \\left ( \\partial _ { t } - H - c K ^ { 1 } + \\left ( 2 m ^ { 2 } \\left ( \\int \\frac { d x } { \\sigma \\left ( x \\right ) } \\right ) ^ { 2 } + 4 m c \\left ( \\int \\frac { d x } { \\sigma \\left ( x \\right ) } \\right ) + 2 c ^ { 2 } - m \\right ) F \\partial _ { F } \\right ) \\end{align*}"}
-{"id": "413.png", "formula": "\\begin{align*} \\langle P [ P , a ] [ P , b ] e _ l , e _ l \\rangle _ { L ^ 2 ( S ^ 1 ) } & = \\begin{cases} - \\sum _ { k > l } a _ { k } b _ { - k } , & l \\geq 0 \\\\ 0 , & l < 0 \\end{cases} \\quad \\mbox { a n d } \\\\ \\langle [ P , a ] [ P , b ] e _ l , e _ l \\rangle _ { L ^ 2 ( S ^ 1 ) } & = \\begin{cases} - \\sum _ { k > l } a _ { k } b _ { - k } , & l \\geq 0 \\\\ - \\sum _ { k \\leq l } a _ { k } b _ { - k } , & l < 0 \\end{cases} . \\end{align*}"}
-{"id": "3806.png", "formula": "\\begin{align*} { \\rm E } \\left [ { \\sum \\limits _ { i = p } ^ r \\sum \\limits _ { i = 1 } ^ m { \\sum \\limits _ { j = 1 } ^ n { \\sum \\limits _ { k = 1 } ^ l { { \\xi ^ { t p } _ { i j k } } } { x ^ { p } _ { i j k } } } } } \\right ] = \\sum \\limits _ { i = p } ^ r \\sum \\limits _ { i = 1 } ^ m { \\sum \\limits _ { j = 1 } ^ n { \\sum \\limits _ { k = 1 } ^ l { { x ^ { p } _ { i j k } } } } } { \\rm E } \\left [ { { \\xi ^ { t p } _ { i j k } } } \\right ] , t = 1 , 2 , . . . , K . \\end{align*}"}
-{"id": "6417.png", "formula": "\\begin{align*} L _ { e } ^ { p } ( \\Omega ) : = \\left \\{ u \\in L _ { e } ^ { p } ( \\mathbb { R } ^ { n } ) \\ , : \\ , u = 0 \\mathbb { R } ^ { n } \\backslash \\Omega \\right \\} . \\end{align*}"}
-{"id": "373.png", "formula": "\\begin{align*} D _ 1 & \\le \\sum _ { q = 0 } ^ \\infty \\| \\Delta _ q ( u ( \\tau ) - u ( t ) ) \\| _ { L ^ 2 } \\sum _ { q = 0 } ^ \\infty \\| \\Delta _ q u ( t ) \\| _ { L ^ 2 } \\\\ & = \\| u ( \\tau ) - u ( t ) \\| _ { B ^ { 0 } _ { 2 , 1 } } \\| u ( t ) \\| _ { B ^ { 0 } _ { 2 , 1 } } . \\end{align*}"}
-{"id": "3452.png", "formula": "\\begin{align*} \\frac { \\partial u _ m } { \\partial s } + { \\cal M } _ m ( s , x , u , \\nabla u , \\nabla ^ 2 u _ m ) = 0 \\end{align*}"}
-{"id": "330.png", "formula": "\\begin{align*} \\mathsf { W } _ S ^ n = \\mathsf { W } _ { C _ 1 } ^ n + \\hdots + \\mathsf { W } _ { C _ k } ^ n . \\end{align*}"}
-{"id": "8496.png", "formula": "\\begin{align*} \\binom { d + 1 } { 2 } \\leq r < \\binom { d + 2 } { 2 } . \\end{align*}"}
-{"id": "2896.png", "formula": "\\begin{align*} \\sigma _ j = \\tau _ { j + 1 } - \\tau _ j , \\ ; \\ ; \\sigma _ j > 0 \\ ; j = 0 , \\ldots , m - 1 . \\end{align*}"}
-{"id": "442.png", "formula": "\\begin{align*} \\tilde { y } & = y = y ' , \\\\ \\mu _ k ( \\tilde { z } ) & = \\mu _ k ( z ) = \\mu _ k ( z ' ) , \\\\ \\sigma _ k ( \\tilde { z } ) & = ( \\max ( s _ 0 , s ' _ 0 ) , \\dots , \\max ( s _ r , s ' _ r ) ) . \\end{align*}"}
-{"id": "5402.png", "formula": "\\begin{align*} e ( G ) \\leq ( k - 1 ) e ( R ' ) + \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } \\leq \\bigg ( k ^ 2 - \\frac { 1 1 } { 1 6 } k + \\frac { 1 } { 2 } \\bigg ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "6906.png", "formula": "\\begin{align*} I = \\int _ { \\| y \\| \\geq 1 } \\frac { \\theta ( y ) } { \\| y \\| ^ { n + 1 } } d y . \\end{align*}"}
-{"id": "599.png", "formula": "\\begin{align*} f _ { k , \\delta } ( z ) : = \\sum _ { \\mathcal { Q } \\in \\mathcal { Q } _ { \\delta } } \\mathcal { Q } ( z , 1 ) ^ { - k } , \\end{align*}"}
-{"id": "2706.png", "formula": "\\begin{align*} \\int _ { \\{ V _ { \\theta } - j < \\psi _ t < \\psi < \\varphi + t \\} } \\theta _ { \\psi _ { t , j } } ^ n = 0 . \\end{align*}"}
-{"id": "5383.png", "formula": "\\begin{align*} J _ { i j } ^ 2 = - { \\rm I d } _ { \\mathbb { R } ^ N } . \\end{align*}"}
-{"id": "1424.png", "formula": "\\begin{align*} C _ { k } = \\bigoplus _ { n \\geq 0 } C _ { k } ( n ) . \\end{align*}"}
-{"id": "1047.png", "formula": "\\begin{align*} \\Phi ( x ) = \\left \\{ \\begin{aligned} & \\frac 1 { 2 \\sqrt { 2 \\pi } } | x | ^ { - \\frac 1 2 } \\ , e ^ { i | x | + i \\frac { \\pi } { 4 } } [ 1 + O ( | x | ^ { - 1 } ) ] & & \\\\ & \\frac 1 { 2 \\pi } \\log \\Bigl ( \\frac { 2 } { | x | } \\Bigr ) \\left [ 1 + O \\Bigl ( \\frac 1 { \\bigl | \\log | x | \\bigr | } \\Bigr ) \\right ] & & , \\end{aligned} \\right . \\end{align*}"}
-{"id": "2486.png", "formula": "\\begin{align*} r _ 1 t _ 1 + \\cdots r _ m t _ m = n . \\end{align*}"}
-{"id": "1626.png", "formula": "\\begin{align*} B ^ { t } = 1 ~ , ~ B ^ { \\alpha } = \\Gamma ^ { \\alpha } + C ^ { \\alpha } , \\end{align*}"}
-{"id": "2690.png", "formula": "\\begin{align*} \\theta _ { \\varphi _ { \\beta , \\varepsilon } } ^ n = e ^ { \\beta \\varphi _ { \\beta , \\varepsilon } } \\left [ ( 1 + \\varepsilon ) \\theta ^ n _ + + \\varepsilon \\omega ^ n \\right ] . \\end{align*}"}
-{"id": "2405.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\tau ^ { - 1 } \\langle \\nabla f ( x + \\tau ( y - x ) ) & - \\nabla f ( x ) , ( x + \\tau ( y - x ) ) - x \\rangle d \\tau \\\\ & \\leq \\int _ 0 ^ 1 \\tau ^ { - 1 } \\langle L \\tau ( x - y ) , \\tau ( x - y ) \\rangle d \\tau \\\\ & = \\langle L ( x - y ) , x - y \\rangle \\int _ 0 ^ 1 \\tau d \\tau \\\\ & = \\tfrac { 1 } { 2 } \\langle L ( x - y ) , x - y \\rangle . \\end{align*}"}
-{"id": "458.png", "formula": "\\begin{align*} u _ j + v _ j \\neq 0 , \\ \\ \\gamma ^ { \\top } ( u + v ) = \\gamma ^ { \\top } u + \\gamma ^ { \\top } v \\in \\mathbb { Z } . \\end{align*}"}
-{"id": "3101.png", "formula": "\\begin{align*} \\left \\{ \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ * \\end{bmatrix} h _ 1 ( \\lambda ) , \\hdots , \\begin{bmatrix} N _ 1 ( \\lambda ) ^ T \\\\ * \\end{bmatrix} h _ p ( \\lambda ) \\right \\} , \\end{align*}"}
-{"id": "7798.png", "formula": "\\begin{align*} h _ { a b } = \\frac { f _ { a b } } { \\left \\vert \\nabla f \\right \\vert } , \\end{align*}"}
-{"id": "767.png", "formula": "\\begin{gather*} [ \\delta ( p ) ] = [ \\delta ( \\tau _ 1 ) ] + \\cdots + [ \\delta ( \\tau _ k ) ] = [ \\delta ( q ) ] . \\end{gather*}"}
-{"id": "6034.png", "formula": "\\begin{align*} 1 - d ( 0 , C ) \\leq 1 - d ( 0 , k A ) = 1 - k d ( 0 , A ) . \\end{align*}"}
-{"id": "7206.png", "formula": "\\begin{align*} L _ m = \\{ x \\in \\Z ^ d ; P ( x _ m = x ) > 0 \\} . \\end{align*}"}
-{"id": "470.png", "formula": "\\begin{align*} G ^ - & : = \\{ y \\in \\mathbb { R } ^ 2 \\ , | \\ ( \\eta ) ^ 2 + [ \\beta _ 2 ( y _ 2 - \\alpha _ 2 ) ] ^ 2 \\leq [ \\beta _ 1 ( y _ 1 - \\alpha _ 1 ) ] ^ 2 , \\ \\beta _ 1 ( y _ 1 - \\alpha _ 1 ) \\leq 0 \\} . \\end{align*}"}
-{"id": "5300.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\gamma _ j < \\infty . \\end{align*}"}
-{"id": "2985.png", "formula": "\\begin{align*} \\delta _ v \\Theta ( L ) ( x , y , z ) ( a ) - \\delta _ L \\Theta ^ 2 ( x , y , z ) ( a ) = 0 . \\end{align*}"}
-{"id": "7499.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ { \\kappa ' } Q _ i = n = \\sum _ { i = 0 } ^ { \\kappa ' } i Q _ i , \\end{align*}"}
-{"id": "6640.png", "formula": "\\begin{align*} z _ j = \\eta _ j ( \\Phi _ K ) \\cdot \\mu _ { \\tau _ j } \\end{align*}"}
-{"id": "2606.png", "formula": "\\begin{align*} W \\circ T ( f \\oplus \\gamma _ A \\oplus \\gamma _ B ) ( a _ 1 \\otimes b _ 1 \\oplus a _ 2 \\oplus b _ 2 ) & = b _ { T ( ( f \\oplus \\gamma _ A \\oplus \\gamma _ B ) } ( a _ 1 \\otimes b _ 1 ) \\oplus \\gamma _ A ( a _ 2 ) \\oplus \\gamma _ B ( b _ 2 ) \\\\ & = b _ { T ( ( f \\oplus 0 \\oplus 0 ) } ( a _ 1 \\otimes b _ 1 ) \\oplus \\gamma _ A ( a _ 2 ) \\oplus \\gamma _ B ( b _ 2 ) \\\\ & = f ( a _ 1 \\otimes b _ 1 ) \\oplus \\gamma _ A ( a _ 2 ) \\oplus \\gamma _ B ( b _ 2 ) . \\end{align*}"}
-{"id": "611.png", "formula": "\\begin{align*} \\xi _ { 2 - 2 k , z } \\left ( \\mathbb { P } _ { 2 - 2 k , n , N } ( z , \\mathfrak { z } ) \\right ) & = ( 4 \\mathfrak { z } _ 2 ) ^ { 2 k - 1 } \\Psi _ { 2 k , - n - 1 , N } ( z , \\mathfrak { z } ) , \\\\ \\mathcal { D } _ z ^ { 2 k - 1 } \\left ( \\mathbb { P } _ { 2 - 2 k , n , N } ( z , \\mathfrak { z } ) \\right ) & = - ( 2 k - 2 ) ! \\left ( \\frac { \\mathfrak { z } _ 2 } { \\pi } \\right ) ^ { 2 k - 1 } \\Psi _ { 2 k , n + 1 - 2 k , N } ( z , \\mathfrak { z } ) . \\end{align*}"}
-{"id": "4612.png", "formula": "\\begin{align*} | G | = q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) \\leq q ^ 3 ( q ^ 2 - 1 ) ( q + \\sqrt { 3 q } + 1 ) \\cdot \\frac { ( q - \\sqrt { 3 q } + 1 ) r } { | G _ { p } | } = f ( p ) , \\end{align*}"}
-{"id": "7063.png", "formula": "\\begin{align*} { x ^ { [ 2 ] } } ( { t _ 3 } ) = \\frac { { { h ^ { [ 3 2 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 3 2 ] } } ( { t _ 3 } - 2 ) } } u _ 1 ^ { [ 2 ] } , { x ^ { [ 3 ] } } ( { t _ 3 } ) = \\frac { { { h ^ { [ 3 3 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 3 3 ] } } ( { t _ 3 } - 2 ) } } u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "7869.png", "formula": "\\begin{align*} \\frac { \\Phi ^ { - 1 } ( t ^ { - 1 } ) ^ d } { ( 1 + | x | \\Phi ( t ^ { - 1 } ) ) ^ { d + 2 } } \\le \\frac { \\Phi ^ { - 1 } ( t ^ { - 1 } ) ^ d } { ( | x | \\Phi ^ { - 1 } ( t ^ { - 1 } ) ) ^ { d + 2 } } = | x | ^ { - d } \\left ( \\frac { \\Phi ^ { - 1 } ( \\Phi ( | x | ^ { - 1 } ) ) } { \\Phi ^ { - 1 } ( \\frac { \\Phi ( | x | ^ { - 1 } ) } { t \\Phi ( | x | ^ { - 1 } ) } ) } \\right ) ^ 2 \\le | x | ^ { - d } ( t \\Phi ( | x | ^ { - 1 } ) ) . \\end{align*}"}
-{"id": "6878.png", "formula": "\\begin{align*} v ( t ) = - M ( t ) P _ { > A T ^ { \\frac 1 2 } t ^ { - 1 } } \\tilde u ( t ) . \\end{align*}"}
-{"id": "5739.png", "formula": "\\begin{align*} S ( \\hat \\delta _ 0 ) = n ^ { - 1 / 2 } \\sum _ { t = 1 } ^ { n } \\left ( y _ t - m _ t \\pi _ t ( \\hat \\delta _ 0 ) \\right ) \\left ( \\sum _ { j = 0 } ^ { \\infty } \\tau _ { j } ( \\omega ) e _ { t - J _ L - j } ( \\hat \\delta _ 0 ) \\right ) \\end{align*}"}
-{"id": "2397.png", "formula": "\\begin{align*} \\nabla F _ { \\alpha _ 1 , \\alpha _ 2 } ^ { \\rm { G A P } } ( x ) & = P x - P \\nabla p _ C ^ { \\alpha _ 2 } ( P x + q ) \\\\ & = P ( x - P _ C ^ { \\alpha _ 2 } P _ D ^ { \\alpha _ 1 } x ) . \\end{align*}"}
-{"id": "7809.png", "formula": "\\begin{align*} \\Delta _ { f } \\left \\vert \\mathrm { R m } \\right \\vert ^ { 2 } = 2 \\left \\vert \\nabla \\mathrm { R m } \\right \\vert ^ { 2 } + 2 \\left \\vert \\mathrm { R m } \\right \\vert ^ { 2 } - 8 R _ { i j k l } R _ { p i q k } R _ { p j q l } - 2 R _ { i j k l } R _ { i j p q } R _ { p q k l } . \\end{align*}"}
-{"id": "6456.png", "formula": "\\begin{align*} \\geq C ( \\delta , \\Lambda ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x . \\end{align*}"}
-{"id": "3604.png", "formula": "\\begin{align*} \\| f \\| _ { C ^ { k , \\alpha } _ { - q } ( \\mathbb { R } ^ 3 \\setminus B ) } = \\sum _ { | I | \\le k } \\sup _ { x \\in \\mathbb { R } ^ 3 \\setminus B } \\left | | x | ^ { | I | + q } ( \\partial ^ { I } f ) ( x ) \\right | + \\sum _ { | I | = k } \\left [ | x | ^ { k + q + \\alpha } ( \\partial ^ { I } f ) ( x ) \\right ] _ { \\alpha , \\mathbb { R } ^ 3 \\setminus B } . \\end{align*}"}
-{"id": "673.png", "formula": "\\begin{align*} \\mu _ i ( \\phi ) = \\eta _ i ( P \\phi ) , \\textrm { f o r $ \\phi \\in L ^ \\infty ( Z , m ) $ } . \\end{align*}"}
-{"id": "2889.png", "formula": "\\begin{align*} \\overline { P o i s _ n ^ * \\{ n + 1 \\} } ^ + ( r ) = \\overline { P o i s _ n ^ * \\{ n + 1 \\} } ( r ) \\end{align*}"}
-{"id": "5328.png", "formula": "\\begin{align*} [ Q _ { j } , C ( u ) ] = 0 , \\ ; \\ ; \\ ; \\ ; \\ [ \\overline { Q } _ { j } , C ( u ) ] = 0 . \\end{align*}"}
-{"id": "8191.png", "formula": "\\begin{align*} I = ( H _ 1 - z _ 1 , H _ 2 - z _ 2 ) \\subset A [ x _ 1 , x _ 2 , x _ 3 ] [ z _ 1 , z _ 2 ] \\end{align*}"}
-{"id": "9030.png", "formula": "\\begin{align*} \\Psi _ k ( \\theta ) = ( k + 1 ) \\theta - 2 \\Im \\log \\Phi _ { k } ^ * ( e ^ { i \\theta } ) = ( k + 1 ) \\theta - 2 \\sum _ { j = 0 } ^ { k - 1 } \\Im \\log \\left ( 1 - \\alpha _ { j } e ^ { i \\Psi _ { j } ( \\theta ) } \\right ) . \\end{align*}"}
-{"id": "5906.png", "formula": "\\begin{align*} \\beta = \\frac { 2 } { 1 - \\alpha } , A = \\pm \\left ( \\frac { 1 } { \\beta \\left ( \\beta - 1 \\right ) } \\right ) ^ { \\frac { 1 } { 1 - \\alpha } } = \\pm \\left ( \\frac { \\left ( 1 - \\alpha \\right ) ^ { 2 } } { 2 \\left ( 1 + \\alpha \\right ) } \\right ) ^ { \\frac { 1 } { 1 - \\alpha } } . \\end{align*}"}
-{"id": "670.png", "formula": "\\begin{align*} \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\phi _ 1 ) \\phi _ 2 ) \\ , d \\eta ( g ) = \\int _ G \\lambda ( ( g ^ { - 1 } \\cdot T \\phi _ 1 ) T \\phi _ 2 ) \\ , d \\eta ( g ) = \\lambda ( \\phi _ 1 ) \\ , \\lambda ( \\phi _ 2 ) = \\nu ( \\phi _ 1 ) \\nu ( \\phi _ 2 ) . \\end{align*}"}
-{"id": "6056.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\int _ { \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\overline { u } _ { i } ( x ) \\underline { u } _ { j } ( y ) K ( x , y ) \\ , d y d x = \\int _ { \\Omega } \\int _ { \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\underline { u } _ { j } ( x ) \\overline { u } _ { i } ( y ) K ( x , y ) \\ , d y \\ , d x . \\end{align*}"}
-{"id": "9949.png", "formula": "\\begin{align*} & \\frac { 1 } { 2 } \\int _ Q \\left ( | u _ j ( x ) - u _ j ( y ) | ^ 2 - | u _ \\infty ( x ) - u _ \\infty ( y ) | ^ 2 \\right ) K ( x - y ) d x d y \\\\ & \\geq \\int _ Q ( u _ \\infty ( x ) - u _ \\infty ( y ) ) ( u _ j ( x ) - u _ j ( y ) - u _ \\infty ( x ) + u _ \\infty ( y ) ) K ( x - y ) d x d y \\\\ & + \\frac { 1 } { 2 } \\int _ Q | u _ j ( x ) - u _ j ( y ) - u _ \\infty ( x ) + u _ \\infty ( y ) | ^ 2 K ( x - y ) d x d y . \\end{align*}"}
-{"id": "4790.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } '' _ a : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "1771.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\Delta u _ { i } ^ { 2 k + 3 } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { 2 k + 3 } \\sum \\limits _ { j \\neq i } H ( u _ { j } ^ { 2 k + 2 } ) ( x ) & \\Omega , \\\\ \\Delta u _ { i } ^ { 2 k + 1 } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { 2 k + 1 } \\sum \\limits _ { j \\neq i } H ( u _ { j } ^ { 2 k } ) ( x ) & \\Omega . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4576.png", "formula": "\\begin{align*} \\mathrm { c a r d } \\bigl ( \\mathcal { P } _ i ( J ) \\bigr ) = \\frac { 1 } { i ! } \\sum _ { \\ell = 0 } ^ i ( - 1 ) ^ { i - \\ell } \\binom { i } { \\ell } \\ell ^ { \\mathrm { c a r d } ( J ) } = : S \\bigl ( \\mathrm { c a r d } ( J ) , i \\bigr ) , \\end{align*}"}
-{"id": "4419.png", "formula": "\\begin{align*} \\| \\tilde { f _ n } - f \\chi _ A \\| _ { \\widehat { E } } = \\| | \\tilde { f _ n } - f \\chi _ A | \\| _ { \\widehat { E } } \\leq \\| \\abs { f _ n - f } \\| _ { \\widehat { E } } \\to 0 . \\end{align*}"}
-{"id": "8465.png", "formula": "\\begin{align*} f ( \\textbf { u } ) = \\bar { f } + \\beta ( \\rho , v ^ { 2 } ) , ~ ~ ~ ~ \\beta ( \\rho _ { 0 } , v _ { 0 } ^ { 2 } ) \\ge 0 , \\end{align*}"}
-{"id": "2589.png", "formula": "\\begin{align*} \\sup _ { 1 \\le j \\le J _ 1 } \\norm { v _ n ^ j } _ { W ( K _ n ^ m ) } = \\norm { v _ n ^ { j ( m , n ) } } _ { W ( K _ n ^ m ) } = m . \\end{align*}"}
-{"id": "2926.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V ^ * _ \\pi ( z _ 1 ; X ) V ^ * _ \\pi ( z _ 2 ; X ) \\cdots V ^ * _ \\pi ( z _ m ; X ) \\cdot 1 \\cr & = [ Z ^ { \\lambda + \\delta } ] \\ \\prod _ { 1 \\le i < j \\le m } ( z _ i - z _ j ) \\ \\prod _ { \\ell = 1 } ^ m \\ , L ( z _ \\ell ; X ) \\ L _ { \\pi ' } ( Z ) \\cr & = [ s _ \\lambda ( Z ) ] \\ L ( X Z ) \\ , L _ { \\pi ' } ( Z ) \\ , . \\end{align*}"}
-{"id": "7891.png", "formula": "\\begin{align*} \\left \\| P - I _ c \\right \\| ^ 2 = r ^ 2 \\end{align*}"}
-{"id": "2445.png", "formula": "\\begin{align*} 1 = 2 ^ { j A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 3 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "8780.png", "formula": "\\begin{align*} T _ i ( u ) = T _ i + \\frac { 1 } { [ u ] } , [ u ] = \\frac { 1 - t ^ { u } } { 1 - t } . \\end{align*}"}
-{"id": "3706.png", "formula": "\\begin{align*} Z = \\frac { 1 } { 2 } ( x ^ 2 _ 2 + x ^ 2 _ 3 + x ^ 2 _ 4 + x ^ 2 _ 5 ) ~ . \\end{align*}"}
-{"id": "9515.png", "formula": "\\begin{align*} 8 e ^ { - 2 \\phi } d \\delta \\phi \\wedge * d \\phi & = 8 d \\left ( \\delta \\phi e ^ { - 2 \\phi } * d \\phi \\right ) - 8 \\delta \\phi d \\left ( e ^ { - 2 \\phi } * d \\phi \\right ) . \\end{align*}"}
-{"id": "8955.png", "formula": "\\begin{align*} U ( x , t ) = \\alpha \\sqrt { 2 / q } e ^ { i \\left ( \\frac { S } { 2 } x - \\frac { 1 } { 4 } ( S ^ { 2 } - \\alpha ^ { 2 } ) t \\right ) } \\left ( \\alpha ( x - S t ) \\right ) \\end{align*}"}
-{"id": "384.png", "formula": "\\begin{align*} E ( z , g ) & = t ^ { z + 1 } + t ^ { 1 - z } \\frac { \\xi ( z ) } { \\xi ( z + 1 ) } \\\\ & \\qquad + \\frac { 4 t } { \\xi ( z + 1 ) } \\sum _ { m = 1 } ^ \\infty \\eta _ { \\frac { z } { 2 } } ( m ) K _ { \\frac { z } { 2 } } ( 2 \\pi m t ^ 2 ) \\cos 2 \\pi m u , \\\\ \\eta _ { \\frac { z } { 2 } } ( m ) & = \\sum _ { a b = m } \\left ( \\frac { a } { b } \\right ) ^ { \\frac { z } { 2 } } . \\end{align*}"}
-{"id": "9770.png", "formula": "\\begin{align*} \\nabla _ { \\vec { x } } V ( \\vec { x } ) = - \\vec { S } _ { \\mathcal { E } } ^ { \\top } ( \\vec { G } _ { \\mathcal { E } } \\vec { H } ^ { - 1 } \\vec { G } _ { \\mathcal { E } } ^ { \\top } ) ^ { - 1 } ( \\vec { W } _ { \\mathcal { E } } + \\vec { S } _ { \\mathcal { E } } \\vec { x } ) , \\forall \\vec { x } \\in \\mathcal { R } ( \\mathcal { E } ) \\cap \\mathrm { i n t } ( \\mathcal { X } ) . \\end{align*}"}
-{"id": "1534.png", "formula": "\\begin{align*} \\rho _ f : = \\int f \\d v . \\end{align*}"}
-{"id": "8331.png", "formula": "\\begin{align*} z ^ { 3 } _ { 1 , 2 } - z _ { 1 , 2 } = \\frac { 1 } { \\left ( T + 1 \\right ) ^ { 2 } } + \\frac { 1 } { T + 1 } = r _ { 1 } ( T ) . \\end{align*}"}
-{"id": "1224.png", "formula": "\\begin{align*} S ( x ) = 0 \\ \\ \\ \\ \\ \\ \\ x \\in ( \\Lambda _ { \\varphi , w } ) _ a . \\end{align*}"}
-{"id": "2038.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\dd u } { \\dd t } = \\Delta u & \\Omega \\times \\mathbb R _ + ; \\\\ u = 0 & \\dd \\Omega \\times \\mathbb R _ + ; \\\\ u ( x , 0 ) = \\phi _ 1 . \\end{cases} \\end{align*}"}
-{"id": "8270.png", "formula": "\\begin{align*} & \\mathbb { E } _ { { \\mu } _ { \\lambda _ 0 } } \\left [ \\left ( \\sum _ k n ^ { - 1 } | \\Delta _ n \\eta ( n ^ { - 1 } k + c _ n r ) | ^ 2 | V ' ( u ( k ) ) - \\varphi _ { V ' } ( \\rho ' ( \\lambda _ 0 ) ) | ^ 2 \\right ) ^ p \\right ] \\\\ & \\leqslant \\left ( \\sum _ k n ^ { - 1 } | \\Delta _ n \\eta ( n ^ { - 1 } k + c _ n r ) | ^ 2 \\right ) ^ p E _ { \\lambda _ 0 } [ ( V ' ( u ( 0 ) ) - \\varphi _ { V ' } ( \\rho ' ( \\lambda _ 0 ) ) ) ^ { 2 p } ] , \\end{align*}"}
-{"id": "2121.png", "formula": "\\begin{align*} b _ i : = - \\frac { ( j - i + 1 ) ( j - i + 1 - s ) } { i ( i + s ) } \\ , b _ { i - 1 } , \\end{align*}"}
-{"id": "2625.png", "formula": "\\begin{align*} \\theta _ { m } ^ { l } ( j ) = \\sum _ { k = 1 } ^ \\infty \\theta _ { m k } ^ { l } ( j ) \\widetilde { e } _ k , \\end{align*}"}
-{"id": "6636.png", "formula": "\\begin{align*} \\varepsilon ( s , V ) = \\prod \\varepsilon _ { E _ { \\nu } } ( s , V _ { \\nu } , \\psi _ { \\nu } ) . \\end{align*}"}
-{"id": "8715.png", "formula": "\\begin{align*} v ( t , x ) = \\int _ { t } ^ { T } R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } G B ( s , \\cdot ) + e ^ { - ( s - t ) { A } } L ( s , \\cdot ) B \\left ( s , \\cdot \\right ) \\right ] ( x ) d s . \\end{align*}"}
-{"id": "5684.png", "formula": "\\begin{align*} K ( x ) = \\frac { ( \\alpha + \\gamma + q - 1 ) ( \\frac { 1 } { x } - 1 ) + q - 1 } { ( \\alpha + \\gamma + q - 1 ) ( x - 1 ) + q - 1 } \\left ( \\frac { 1 - ( x - 1 ) e _ 0 } { 1 - ( \\frac { 1 } { x } - 1 ) e _ 0 } \\right ) , \\end{align*}"}
-{"id": "1768.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\Delta u _ { i } ^ { 0 } = 0 & \\Omega , \\\\ u _ { i } ^ { 0 } = \\phi _ { i } & ( \\Omega ) _ 1 \\setminus \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "2598.png", "formula": "\\begin{align*} \\tilde u ( t ) : = M ( - t ) u ( t ) . \\end{align*}"}
-{"id": "7903.png", "formula": "\\begin{align*} r _ 2 \\ ; = \\ ; r _ 1 , l _ 2 \\ ; = \\ ; \\lambda ^ * l _ 1 \\ , + \\ , r _ 1 \\ , n \\ , p _ 2 ^ 2 \\ , h , \\chi _ 2 \\ ; = \\ ; \\chi _ 1 \\ , + \\ , n \\ , p _ 2 ^ 2 \\ , I \\ , + \\ , r _ 1 \\ , n ^ 2 \\ , p _ 2 ^ 4 \\ , \\frac { h ^ 2 } { 2 } . \\end{align*}"}
-{"id": "184.png", "formula": "\\begin{align*} F _ { n , x } ( u ) : = \\mathbb { P } ( \\xi _ i \\leq u | X _ i = x ) = \\sum _ { j = k } ^ { n - 1 } \\binom { n - 1 } { j } p _ { n , x , u } ^ j ( 1 - p _ { n , x , u } ) ^ { n - 1 - j } , \\end{align*}"}
-{"id": "7840.png", "formula": "\\begin{gather*} t _ \\delta A _ { n - 1 } = \\rho _ 0 ( t ) A _ { n - 1 } \\rho _ { n - 1 } ( t ) ^ { - 1 } , A _ i = \\rho _ { i + 1 } ( t ) A _ i \\rho _ i ( t ) ^ { - 1 } ( i \\neq n - 1 ) , B _ i = \\rho _ i ( t ) B _ i \\rho _ i ( t ) ^ { - 1 } , \\\\ t _ i ^ { - 1 } a _ i = \\rho _ i ( t ) a _ i , t _ \\delta t _ 0 b _ { n - 1 } = b _ { n - 1 } \\rho _ { n - 1 } ( t ) ^ { - 1 } , t _ { i + 1 } b _ i = b _ i \\rho _ i ( t ) ^ { - 1 } ( i \\neq n - 1 ) , \\end{gather*}"}
-{"id": "9475.png", "formula": "\\begin{align*} \\gamma ' ( \\tau ) = \\nabla ^ { g } F ( \\gamma ( \\tau ) ) \\gamma ( 0 ) = ( s _ { 0 } , t _ { 0 } ) \\end{align*}"}
-{"id": "4607.png", "formula": "\\begin{align*} | R | = | ^ { 2 } G _ { 2 } ( q ) | & = q ^ { 3 } ( q ^ { 3 } + 1 ) ( q - 1 ) \\\\ & = | R _ { 2 } | \\cdot | R _ { 3 } | \\cdot | H _ { 1 } | \\cdot | H _ { 2 } | \\cdot | H _ { 3 } | \\cdot | H _ { 4 } | , \\end{align*}"}
-{"id": "5965.png", "formula": "\\begin{align*} \\lambda V _ n ( z ) - \\frac { 1 } { 2 } \\mathrm { T r } \\big ( Q D ^ 2 V _ n ( z ) \\big ) - \\langle A z , D V _ n ( z ) \\rangle & - \\langle B ( z ) , D V _ n ( z ) \\rangle \\\\ = B _ n ( z ) - B ( z ) & + \\langle B _ n ( z ) - B ( z ) , D _ v U _ n ( z ) \\rangle . \\end{align*}"}
-{"id": "1189.png", "formula": "\\begin{align*} - v _ { x x } = z \\rho v , v _ t = ( - \\frac 1 2 b _ x + \\beta ) v + b v _ x . \\end{align*}"}
-{"id": "4606.png", "formula": "\\begin{align*} x ( t _ { 1 } , u _ { 1 } , v _ { 1 } ) x ( t _ { 2 } , u _ { 2 } , v _ { 2 } ) = x ( t _ { 1 } + t _ { 2 } , & u _ { 1 } + u _ { 2 } - t _ { 1 } t _ { 2 } ^ { 3 \\theta } , \\\\ & v _ { 1 } + v _ { 2 } - t _ { 2 } u _ { 1 } + t _ { 1 } t _ { 2 } ^ { 3 \\theta + 1 } - t _ { 1 } ^ { 2 } t _ { 2 } ^ { 3 \\theta } ) . \\end{align*}"}
-{"id": "9990.png", "formula": "\\begin{align*} \\alpha _ { \\mathrm { m i n } } = \\max \\left \\{ 0 , \\frac { 1 } { E _ k } \\Big ( \\frac { C _ { 2 } } { | w _ { \\bar { k } 1 } | ^ 2 } - \\frac { B _ k } { T _ f } \\Big ) \\right \\} . \\end{align*}"}
-{"id": "6047.png", "formula": "\\begin{align*} H ( u _ { j } ^ { \\varepsilon } ) ( x ) = \\underset { y \\in B _ { 1 } ( x ) } { \\sup } u ( y ) . \\end{align*}"}
-{"id": "8106.png", "formula": "\\begin{align*} C _ { k , l , N } ^ { r a d } : = ( N \\omega _ N ) ^ { ( l - k + 1 ) / ( l + N ) } \\cdot ( l + N ) ^ { ( k + N - 1 ) / ( l + N ) } . \\end{align*}"}
-{"id": "2562.png", "formula": "\\begin{align*} [ e ^ { i t \\Delta } ( e ^ { i x \\xi _ 0 } f ) ] ( x ) = e ^ { - i t | \\xi _ 0 | ^ 2 + i x \\xi _ 0 } ( e ^ { i t \\Delta } f ) ( x - 2 t \\xi _ 0 ) \\end{align*}"}
-{"id": "2521.png", "formula": "\\begin{align*} h ^ 0 ( 2 E _ i + K - D ) = h ^ 0 ( 2 E _ j + K - D ) = h ^ 0 ( 2 E _ k + K - D ) = 0 . \\end{align*}"}
-{"id": "4806.png", "formula": "\\begin{align*} [ \\tilde { r } , \\delta ] \\ = \\ \\tilde { r } \\circ \\delta - \\delta \\circ \\tilde { r } \\end{align*}"}
-{"id": "3456.png", "formula": "\\begin{align*} \\upsilon ( \\theta ) = \\Gamma ^ { * } ( s , s + \\Delta s ) \\phi ( s + \\Delta s , \\xi _ { s , x } ( s + \\Delta s ) ) - \\Gamma ^ { * } ( s , \\theta ) \\phi ( \\theta , \\xi _ { s , x } ( \\theta ) ) + \\end{align*}"}
-{"id": "7690.png", "formula": "\\begin{align*} x _ { n + 1 } \\ge f ( x _ n ) + \\gamma _ { n + 1 } , n = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "7519.png", "formula": "\\begin{align*} \\mu _ 1 = f ( \\mu _ 2 ) < f ( K ) = K . \\end{align*}"}
-{"id": "9755.png", "formula": "\\begin{align*} L ( \\vec { z } , \\boldsymbol { \\lambda } , \\vec { x } ) : = f ( \\vec { z } , \\vec { x } ) + \\boldsymbol { \\lambda } ^ { \\top } \\vec { g } ( \\vec { z } , \\vec { x } ) , \\end{align*}"}
-{"id": "4967.png", "formula": "\\begin{align*} \\hat { h } _ { X , f } ( P ) = \\lim _ { n \\to \\infty } \\frac { h _ { X } ( f ^ { n } ( P ) ) } { \\delta _ { f } ^ { n } } \\end{align*}"}
-{"id": "221.png", "formula": "\\begin{align*} \\mathbb { E } _ f ( \\hat { H } _ n ) = H + \\log ( n - 1 ) - \\Psi ( n ) + \\sum _ { i = 1 } ^ 5 R _ i ' . \\end{align*}"}
-{"id": "7168.png", "formula": "\\begin{align*} - \\Delta v + \\nabla p = f , \\qquad { \\rm d i v } \\ , v = 0 \\mbox { i n } \\ , \\ , \\R ^ n \\end{align*}"}
-{"id": "4615.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { 8 } f _ t ( p ' ) = ( q - 1 ) ( q ^ 3 + 1 ) r . \\end{align*}"}
-{"id": "1498.png", "formula": "\\begin{align*} N ( x ) = 2 k b n - k ^ 2 b ^ 2 - b k ^ 2 \\geq b ( 2 k n - k n - k ^ 2 ) > b ( n - 1 ) . \\end{align*}"}
-{"id": "2717.png", "formula": "\\begin{align*} \\int _ { \\{ \\Phi < P _ { \\Theta } ( F ) \\} } ( \\Theta + d d ^ c \\Phi ) ^ { n + 1 } = 0 . \\end{align*}"}
-{"id": "5736.png", "formula": "\\begin{align*} \\frac { \\partial Z _ { t } } { \\partial \\psi } \\vert _ { \\delta _ 0 } = e _ { t - J _ L } ( \\delta _ 0 ) + \\sum _ { j \\in J _ \\phi \\bigcap J _ \\theta } \\omega _ { j } \\frac { \\partial Z _ { t - j } ( \\delta _ 0 ) } { \\partial \\psi } . \\end{align*}"}
-{"id": "6890.png", "formula": "\\begin{gather*} G ^ 2 ( x , \\xi ) = \\sum _ { k = 1 } ^ \\infty | g _ k ( x , \\xi ) | ^ 2 \\leq C ( 1 + | \\xi | ^ 2 ) , \\\\ \\sum _ { k = 1 } ^ \\infty | g _ k ( x , \\xi ) - g _ k ( y , \\zeta ) | ^ 2 \\leq C \\Big ( | x - y | ^ 2 + | \\xi - \\zeta | r ( | \\xi - \\zeta | ) \\Big ) \\end{gather*}"}
-{"id": "7263.png", "formula": "\\begin{align*} u _ k ^ \\alpha = \\prod _ { \\lambda = 1 } ^ r \\left ( \\sum _ { \\mu = 1 } ^ r ( T _ { k j } ) ^ \\lambda _ \\mu u _ j ^ \\mu + O ( | u _ j | ^ n ) \\right ) ^ { \\alpha _ \\lambda } = \\sum _ { | \\beta | = | \\alpha | } \\tau _ { k j , \\beta } ^ \\alpha \\cdot u _ j ^ \\beta + O ( | u _ j | ^ { n + 1 } ) \\end{align*}"}
-{"id": "7799.png", "formula": "\\begin{align*} \\left \\vert \\frac { d S } { d t } \\right \\vert = \\frac { \\left \\vert \\left \\langle \\nabla S , \\nabla f \\right \\rangle \\right \\vert } { \\left \\vert \\nabla f \\right \\vert ^ { 2 } } \\leq \\frac { c _ { 0 } } { t } . \\end{align*}"}
-{"id": "4902.png", "formula": "\\begin{align*} g ^ * ( z ) : = ( 2 i ) ^ { 1 - \\kappa } \\int _ { - \\overline { z } } ^ { i \\infty } g ^ { \\operatorname { c } } ( w ) ( w + z ) ^ { 1 - \\kappa } d w , \\end{align*}"}
-{"id": "6900.png", "formula": "\\begin{gather*} - \\frac { i \\omega } { 2 h } = \\frac { 1 } { \\sqrt { \\lambda } } \\sum _ { n = 1 } ^ { \\infty } \\omega _ n ( - \\lambda ) ^ n , \\ ; \\ ; - \\frac { i \\tau } { h } = \\frac { 1 } { \\sqrt { \\lambda } } \\sum _ { n = 0 } ^ { \\infty } \\tau _ n ( - \\lambda ) ^ n , \\ ; \\ ; - \\frac { i \\sigma } { h } = \\frac { 1 } { \\sqrt { \\lambda } } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ n ( - \\lambda ) ^ n . \\end{gather*}"}
-{"id": "3836.png", "formula": "\\begin{align*} \\beta \\partial _ t u - u \\Delta u & = 0 \\quad D _ T , \\\\ \\partial _ n u & = 0 \\quad \\Gamma _ T \\end{align*}"}
-{"id": "7410.png", "formula": "\\begin{align*} I ^ \\circ = I \\epsilon _ { i j } ^ \\circ = - \\epsilon _ { i j } . \\end{align*}"}
-{"id": "2448.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ 2 } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { 2 , 2 } ^ { 0 , \\alpha _ 1 } } . \\end{align*}"}
-{"id": "2730.png", "formula": "\\begin{align*} & L ( \\zeta _ { i _ 2 + 1 } - \\zeta _ { i _ 3 + 1 } ) \\frac { \\zeta _ j - \\zeta _ { i _ 1 + 1 } } { \\zeta _ j - \\zeta _ { j + 1 } } + L ( \\zeta _ { i _ 3 + 1 } - \\zeta _ { i _ 1 + 1 } ) \\frac { \\zeta _ j - \\zeta _ { i _ 2 + 1 } } { \\zeta _ j - \\zeta _ { j + 1 } } + L ( \\zeta _ { i _ 1 + 1 } - \\zeta _ { i _ 2 + 1 } ) \\frac { \\zeta _ j - \\zeta _ { i _ 3 + 1 } } { \\zeta _ j - \\zeta _ { j + 1 } } = 0 . \\end{align*}"}
-{"id": "6923.png", "formula": "\\begin{align*} M _ \\pi ( z ; X ) & = \\prod _ { T \\in { \\cal T } ^ \\pi } \\frac { 1 } { 1 - z \\ , X ^ T } = \\sum _ { r \\geq 0 } z ^ r \\ , s _ { ( r ) } [ s _ \\pi ] ( X ) \\ , ; \\\\ \\cr L _ \\pi ( z ; X ) & = \\prod _ { T \\in { \\cal T } ^ \\pi } ( 1 - z \\ , X ^ T ) = \\sum _ { r \\geq 0 } ( - 1 ) ^ r \\ , z ^ r \\ , s _ { ( 1 ^ r ) } [ s _ \\pi ] ( X ) \\ , , \\end{align*}"}
-{"id": "1684.png", "formula": "\\begin{align*} \\Phi ^ { a ' } = \\exp \\Big ( \\int _ 0 ^ T { \\langle } a ' F ( L _ s ) , \\dd W _ s { \\rangle } \\ , - \\ , \\frac { 1 } { 2 } \\int _ 0 ^ T | a ' F ( L _ s ) | ^ 2 \\ , \\dd s + \\frac { ( a ' ) ^ 2 - a ' } { 2 } \\int _ 0 ^ T | F ( L _ s ) | ^ 2 \\ , \\dd s \\Big ) \\ , , \\end{align*}"}
-{"id": "7744.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\frac { \\sum _ { i = 0 } ^ n \\sigma _ i \\xi _ { i + 1 } } { \\sqrt { \\sum _ { k = 0 } ^ { n } \\sigma ^ 2 _ k } } = \\infty , \\liminf _ { n \\to \\infty } \\frac { \\sum _ { i = 0 } ^ n \\sigma _ i \\xi _ { i + 1 } } { \\sqrt { \\sum _ { k = 0 } ^ { n } \\sigma ^ 2 _ k } } = - \\infty , \\quad , \\end{align*}"}
-{"id": "28.png", "formula": "\\begin{align*} ( U ; z = ( z _ 1 , \\ldots , z _ { n + m } ) ) \\end{align*}"}
-{"id": "9305.png", "formula": "\\begin{align*} j ( u ) ( z ) ) & = \\mathbb { E } [ \\int _ 0 ^ T ( \\int _ D f ( t , x , y ( t , x , z ) , u ( t , x , z ) , z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] d x ) d t \\\\ & + \\int _ D g ( x , y ( T , x , z ) , z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] ] d x . \\end{align*}"}
-{"id": "4444.png", "formula": "\\begin{align*} \\hat { H } _ n = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\log \\xi _ i , \\end{align*}"}
-{"id": "2841.png", "formula": "\\begin{align*} g = F _ 1 g \\supseteq F _ 2 g \\supseteq . . . \\supseteq F _ r g \\supseteq . . . \\end{align*}"}
-{"id": "655.png", "formula": "\\begin{align*} \\eta ( \\phi ) = \\int _ G \\phi ( g ) \\ , d \\eta ( g ) , \\end{align*}"}
-{"id": "7985.png", "formula": "\\begin{align*} a _ { 1 0 1 2 } = a _ { 0 1 3 0 } , \\end{align*}"}
-{"id": "9790.png", "formula": "\\begin{align*} m ( \\mathbf { y } | M _ { n , \\alpha } ) = \\frac { p ^ { n } ( \\mathbf { y } | \\mathcal { F } _ { n , \\alpha } ) \\Pi _ { n , \\alpha } ( \\mathcal { F } _ { n , \\alpha } ) } { \\Pi _ { n , \\alpha } ( \\mathcal { F } _ { n , \\alpha } | \\mathbf { y } ) } . \\end{align*}"}
-{"id": "3787.png", "formula": "\\begin{align*} \\mathbf { R } & = \\psi ^ { \\circ } \\mathbf { A } _ { { \\rm R } } \\end{align*}"}
-{"id": "9045.png", "formula": "\\begin{align*} A _ n ^ { ( p ) } ( \\theta ) : = \\sum _ { j = p } ^ { n - 1 } a _ j ( \\theta ) = A _ n ( \\theta ) - A _ p ( \\theta ) , \\ . \\end{align*}"}
-{"id": "2061.png", "formula": "\\begin{align*} M \\ddot { x } ( t ) = & - D \\dot { x } ( t ) - K x ( t ) + F u ( t ) , \\\\ y ( t ) = & \\ C _ { p } x ( t ) + C _ { v } \\dot { x } ( t ) , \\end{align*}"}
-{"id": "6972.png", "formula": "\\begin{align*} g ( \\sigma , 1 / 2 ) = H _ 2 ( \\sigma ) + \\sigma \\log ( 2 ^ { \\gamma - 1 } - 1 ) - \\gamma \\frac { \\delta - 1 } { \\delta } + 1 . \\end{align*}"}
-{"id": "2174.png", "formula": "\\begin{align*} \\| w \\| ^ { 2 } _ { L ^ { 2 } ( [ t _ { 2 } - t _ { 1 } , t _ { 0 } - t _ { 1 } ] ; H ^ { \\beta } ( \\rho ' B _ { 1 } ) ) } \\leq 4 C \\Lambda \\int _ { 0 } ^ { t _ { 0 } - t _ { 1 } } F ( s ) d s . \\end{align*}"}
-{"id": "5985.png", "formula": "\\begin{align*} \\lfloor r _ i u \\rfloor = \\lfloor r _ i x _ 0 \\rfloor \\xleftarrow { z _ 1 \\gamma _ 1 ^ \\lor } \\cdots \\xleftarrow { z _ { p - 1 } \\gamma _ { p - 1 } ^ \\lor } \\lfloor r _ i x _ { p - 1 } \\rfloor = x _ p \\xleftarrow { \\gamma _ { p + 1 } ^ \\lor } \\cdots \\xleftarrow { \\gamma _ r ^ \\lor } x _ r = v . \\end{align*}"}
-{"id": "6117.png", "formula": "\\begin{align*} g ^ { j \\bar j } = \\pi ^ { - 1 } s _ j ^ { - 3 } ( 1 + s _ j ^ 2 \\delta _ j ) , \\ 1 \\le j \\le k \\end{align*}"}
-{"id": "2082.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + K ) X ^ { ( 0 ) } ( s _ { i } ) = F , \\end{align*}"}
-{"id": "5039.png", "formula": "\\begin{align*} \\nu ( \\phi ) \\nu ( C ) = \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\phi ) \\chi _ C ) \\ , d \\eta ( g ) , \\textrm { f o r a l l $ \\phi \\in C ( \\overline { Y } ) $ } . \\end{align*}"}
-{"id": "4412.png", "formula": "\\begin{align*} \\mu ( x ) = \\mu ( x ) \\chi _ { [ 0 , \\tau ( r ) ) } + \\mu ( \\infty , x ) \\chi _ { [ \\tau ( r ) , \\infty ) } = \\mu ( x _ 0 ) + \\mu ( \\infty , x ) \\chi _ { [ \\tau ( r ) , \\infty ) } , \\end{align*}"}
-{"id": "4492.png", "formula": "\\begin{align*} G _ { n , x , y } ( u , v ) & : = \\mathbb { P } ( M _ 1 \\geq k , M _ 2 \\geq k ) , \\end{align*}"}
-{"id": "7122.png", "formula": "\\begin{align*} ( x _ 2 \\otimes g ) \\epsilon _ { 0 , 1 } ^ * \\smile ( x _ 1 \\otimes h ) \\epsilon _ { 1 , 0 } ^ * & = \\chi _ { g , 1 } ^ 1 q ^ { 1 - 1 } ( - 1 ) ^ 1 ( x _ 1 x _ 2 \\otimes g h ) \\epsilon _ { 1 , 1 } ^ * \\\\ & = - \\chi _ { g , 1 } ( x _ 1 x _ 2 \\otimes g h ) \\epsilon _ { 1 , 1 } ^ * \\\\ & = - ( x _ 1 x _ 2 \\otimes g h ) \\epsilon _ { 1 , 1 } ^ * \\end{align*}"}
-{"id": "1014.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = f ( t , u , u ' ) & & \\\\ u ' ( 0 ) = u ( 0 ) , \\ u ' ( T ) = b u ' ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7372.png", "formula": "\\begin{align*} \\langle z , \\Gamma _ { i } ^ { ( k ) } v \\rangle _ { k - 1 } = \\langle z \\wedge w _ { i } , v \\rangle _ { k } , v \\in \\Lambda _ { q } ^ { k } ( \\mathfrak { u } _ { + } ) , \\ z \\in \\Lambda _ { q } ^ { k - 1 } ( \\mathfrak { u } _ { - } ) . \\end{align*}"}
-{"id": "991.png", "formula": "\\begin{align*} T ( u ) & = \\ , L ^ { A } ( u + \\omega ) L ^ { B } ( u - \\omega ) \\\\ & = \\ , \\left ( \\begin{array} { c c } A ( u ) & B ( u ) \\\\ C ( u ) & D ( u ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "2989.png", "formula": "\\begin{align*} f _ 0 ^ { \\lambda _ i ( x , \\lambda _ j ( y , z ) ) } = & - f _ i ^ { \\lambda _ 0 ( x , \\lambda _ j ( y , z ) ) } - \\sum _ { m + n = i , ~ m , n > 0 } f _ m ^ { \\lambda _ n ( x , \\lambda _ j ( y , z ) ) } + \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ x , f _ n ^ { \\lambda _ j ( y , z ) } ] \\\\ & + [ f _ 0 ^ x , f _ i ^ { \\lambda _ j ( y , z ) } ] + [ f _ i ^ x , f _ 0 ^ { \\lambda _ j ( y , z ) } ] . \\end{align*}"}
-{"id": "1705.png", "formula": "\\begin{align*} \\P \\{ \\max _ { n = 1 , \\ldots , N } | S _ n | \\le f _ N \\} , , \\end{align*}"}
-{"id": "8139.png", "formula": "\\begin{align*} a _ 1 = a _ 1 ( p , q , N ) : = \\frac { N - 1 } { 1 + \\frac { q } { p ' } } - \\frac { N } { p } + 1 , \\end{align*}"}
-{"id": "1814.png", "formula": "\\begin{align*} \\begin{gathered} w \\left ( g ( w ) + a ( w ) + w g ( w ) a ( w ) \\right ) = 0 \\Longleftrightarrow a ( w ) = - ( 1 + w g ( w ) ) ^ { - 1 } g ( w ) \\\\ z \\left ( b ( z ) + f ( z ) + z b ( z ) f ( z ) \\right ) = 0 \\Longleftrightarrow b ( z ) = - ( 1 + z f ( z ) ) ^ { - 1 } f ( z ) \\\\ \\begin{aligned} z w \\big ( 2 e ( z , w ) + f ( z ) g ( w ) + & z f ( z ) e ( z , w ) + a ( w ) b ( z ) + w a ( w ) e ( z , w ) \\\\ + & w g ( w ) e ( z , w ) + z b ( z ) e ( z , w ) + z w e ^ 2 ( z , w ) ) \\big ) = 0 . \\end{aligned} \\end{gathered} \\end{align*}"}
-{"id": "5795.png", "formula": "\\begin{align*} b _ { i } ( z ) = \\sum _ { j = i } ^ { n } \\alpha _ { j } ( z ) \\end{align*}"}
-{"id": "7523.png", "formula": "\\begin{align*} \\mathbb P \\{ B _ i \\} = \\tau ^ j \\end{align*}"}
-{"id": "8309.png", "formula": "\\begin{align*} f ( X ) = \\sum _ { i = 0 } ^ n a _ i X ^ { p ^ i } , \\end{align*}"}
-{"id": "4672.png", "formula": "\\begin{align*} Y _ j ^ { ( 1 , i ) } & = \\begin{cases} 0 & \\mbox { i f } j = 0 \\\\ [ X _ j , X _ i ] & \\mbox { i f } 1 \\leq j \\leq m \\end{cases} \\qquad \\mbox { a n d } \\\\ \\tilde Y _ j ^ { ( 1 , i ) } & = \\begin{cases} [ X _ 0 , X _ i ] + \\frac { 1 } { 2 } \\sum _ { l = 1 } ^ m [ X _ l , [ X _ l , X _ i ] ] & \\mbox { i f } j = 0 \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\ ; . \\end{align*}"}
-{"id": "4207.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - h \\cdot \\nabla f ( x _ k ) , ~ k \\geq 0 , \\end{align*}"}
-{"id": "8869.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} - \\Delta v ( x ) + ( 1 + \\abs { A ( x ) } ^ 2 ) v ( x ) & = 0 & & , \\\\ v ( x ) & \\to 0 & & . \\end{aligned} \\right . \\end{align*}"}
-{"id": "8721.png", "formula": "\\begin{align*} \\Vert \\nabla ^ G u ( t , x ) - \\nabla ^ G v ( t , x ) \\Vert _ { L ( U , { K } ) } = \\Big \\Vert \\int _ t ^ T \\nabla ^ G R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } \\big ( \\nabla ^ G u ( s , \\cdot ) B ( s , \\cdot ) - \\nabla ^ G v ( s , \\cdot ) B ( s , \\cdot ) \\big ) \\right ] ( x ) \\ , d s \\Big \\Vert _ { L } , \\end{align*}"}
-{"id": "4312.png", "formula": "\\begin{align*} \\sigma _ j \\wedge p ^ \\star ( d t ) = \\sum _ { p \\geq 0 } a _ { j p } ( z ^ \\prime ) z _ { n + 1 } ^ p \\end{align*}"}
-{"id": "3825.png", "formula": "\\begin{align*} h _ k [ \\boldsymbol { t } ] & : = \\langle 1 \\mid e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast _ { k + 1 / 2 } \\mid 0 \\rangle = \\langle 1 \\mid e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast ( z ) \\mid 0 \\rangle \\mid _ { z ^ { k } } \\\\ & = \\left [ e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\langle 1 \\mid \\psi ^ \\ast ( z ) \\mid 0 \\rangle \\right ] \\mid _ { z ^ { k } } = \\left [ e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\right ] \\mid _ { z ^ { k } } . \\end{align*}"}
-{"id": "9256.png", "formula": "\\begin{align*} \\mathbb { E } [ \\varphi ( X ( t ) ) | \\mathcal { R } _ t ] = \\frac { \\mathbb { E } _ { \\tilde { P } } [ \\varphi ( X ( t ) ) K _ t | \\mathcal { R } _ t ] } { \\mathbb { E } _ { \\tilde { P } } [ K _ t | \\mathcal { R } _ t ] } . \\end{align*}"}
-{"id": "9830.png", "formula": "\\begin{align*} \\mbox { $ ( \\frac { a } { 2 } \\lambda - \\mu ) h _ { 0 , m } ( \\lambda + \\mu ) = - \\mu h _ { 0 , m } ( \\mu ) . $ } \\end{align*}"}
-{"id": "566.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | ( I - A _ n ) [ T , \\tau ] | _ { \\mathcal I } = 0 . \\end{align*}"}
-{"id": "6858.png", "formula": "\\begin{align*} \\liminf _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\norm { e ^ { i t \\Delta } W ^ J _ n } _ { L _ t ^ { q , \\infty } L _ x ^ r ( [ t _ n , \\infty ) \\times \\R ^ d ) } = 0 , \\end{align*}"}
-{"id": "6911.png", "formula": "\\begin{align*} \\log \\biggl ( \\frac { ( d - 1 ) ^ { d - 1 } } { d ^ { d / 2 } } \\biggr ) & = \\left ( 1 + 2 x / n \\right ) \\log \\left ( 1 + \\frac { 2 x } { n } \\right ) - \\left ( 1 + x / n \\right ) \\log \\left ( 2 + \\frac { 2 x } { n } \\right ) \\\\ & = - \\log 2 + \\left ( 1 - \\log 2 \\right ) \\frac { x } { n } + \\frac { 3 } { 2 } \\left ( \\frac { x } { n } \\right ) ^ { 2 } + O \\left ( \\left ( \\frac { x } { n } \\right ) ^ { 3 } \\right ) \\end{align*}"}
-{"id": "2012.png", "formula": "\\begin{align*} \\frac { f ( x u ) - f ( x ) } { x f ' ( x ) } = \\int _ 1 ^ u \\frac { g ( x y ) } { g ( x ) } y ^ { - 1 } \\d y \\to \\log u , \\end{align*}"}
-{"id": "3447.png", "formula": "\\begin{align*} d \\eta ( t ) = c ( \\xi ( t ) ) \\eta ( t ) d t + C ( \\xi ( t ) ) ( \\eta ( t ) , d w ( t ) ) , \\eta ( s ) = h , \\end{align*}"}
-{"id": "7679.png", "formula": "\\begin{align*} v _ { t } + ( - \\triangle ) ^ { \\alpha / 2 } v & = 2 C e ^ { 2 C t } u + e ^ { 2 C t } ( u _ { t } + ( - \\triangle ) ^ { \\alpha / 2 } u ) \\\\ & = 2 C e ^ { 2 C t } u + e ^ { 2 C t } 2 C u ^ { p } \\\\ & \\geq 2 C ( v + ( e ^ { 2 C t } ) ^ { 1 - p } v ^ { p } ) \\\\ & \\geq 2 C v + v ^ { p } > C ( v + v ^ { p } ) , \\end{align*}"}
-{"id": "423.png", "formula": "\\begin{align*} w ( \\bar { w } - \\bar { z } ) ( z - u ) ( \\bar { u } - \\bar { w } ) = 2 i ( \\mathrm { I m } ( u \\bar { w } ) + \\mathrm { I m } ( w \\bar { z } ) ) \\ . \\end{align*}"}
-{"id": "1235.png", "formula": "\\begin{align*} H ( x ) = \\sum _ { k = 1 } ^ \\infty x ( k ) h ( k ) , \\ \\ \\ \\ x \\in \\lambda _ { \\varphi , w } , \\end{align*}"}
-{"id": "1808.png", "formula": "\\begin{align*} I & : = \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { t } \\left ( \\sum _ { ( t - s ) 2 ^ { j \\alpha } \\le 1 } 2 ^ { j \\alpha / p } \\exp ( - c ( t - s ) 2 ^ { j \\alpha } ) \\Big ( \\int _ { { \\mathbb R } ^ d } | F _ { j } ( s , y ) | ^ { p } d y \\Big ) ^ { 1 / p } \\right ) ^ { p } d s d t \\\\ I I & : = \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { t } \\left ( \\sum _ { ( t - s ) 2 ^ { j \\alpha } > 1 } 2 ^ { j \\alpha / p } \\exp ( - c ( t - s ) 2 ^ { j \\alpha } ) \\Big ( \\int _ { { \\mathbb R } ^ d } | F _ { j } ( s , y ) | ^ { p } d y \\Big ) ^ { 1 / p } \\right ) ^ { p } d s d t . \\end{align*}"}
-{"id": "8114.png", "formula": "\\begin{align*} J ^ { \\prime \\prime } ( 0 ) \\geq 0 = J ' ( 0 ) . \\end{align*}"}
-{"id": "4685.png", "formula": "\\begin{align*} w ( \\bar { w } - \\bar { z } ) ( z - u ) ( \\bar { u } - \\bar { w } ) = 2 i ( \\mathrm { I m } ( u \\bar { w } ) + \\mathrm { I m } ( w \\bar { z } ) ) \\ . \\end{align*}"}
-{"id": "5588.png", "formula": "\\begin{align*} M _ { 1 1 } ( x , y ) = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ) , M _ { 2 2 } ( x , y ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "6050.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\Delta u _ { i } ^ { 0 } = 0 & \\Omega , \\\\ u _ { i } ^ { 0 } = \\phi _ { i } & ( \\Omega ) _ 1 \\setminus \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "6594.png", "formula": "\\begin{align*} f ( x ) = \\tfrac { 1 } { 2 } \\langle H x , x \\rangle + \\langle h , x \\rangle . \\end{align*}"}
-{"id": "7792.png", "formula": "\\begin{align*} c - d = 0 c + d = 0 . \\end{align*}"}
-{"id": "3057.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { | u | = n } \\nu ( B ( u ) ) = 0 \\end{align*}"}
-{"id": "6081.png", "formula": "\\begin{align*} \\widetilde { G } _ { m , n } ( t ) = \\frac { 1 } { \\sqrt { m + 1 } } \\int _ n ^ { n + 1 } v ^ { - 1 / 2 } \\exp \\left ( - t \\left ( i \\log { \\frac { m + 1 } { v } } + \\frac { 1 } { 2 } \\log ^ 2 { \\frac { m + 1 } { v } } \\right ) \\right ) d v . \\end{align*}"}
-{"id": "9751.png", "formula": "\\begin{align*} \\mathcal { X } : = \\{ \\vec { x } \\in \\mathbb { R } ^ { n } \\ | \\ \\exists \\vec { z } \\in \\mathbb { R } ^ { s } : \\vec { g } ( \\vec { z } , \\vec { x } ) \\leq \\vec { 0 } _ { m } \\} . \\end{align*}"}
-{"id": "7585.png", "formula": "\\begin{align*} x _ n \\ge x _ 0 + K _ 1 ( \\alpha ) \\frac { l + A ( \\alpha ) } { 2 } = \\alpha + \\left ( \\left [ \\frac { 2 ( u _ l - \\alpha ) } { l + A ( \\alpha ) ) } \\right ] + 1 \\right ) \\frac { l + A ( \\alpha ) } { 2 } \\ge \\alpha + u _ l - \\alpha = u _ l . \\end{align*}"}
-{"id": "3921.png", "formula": "\\begin{align*} u _ { \\rm r e g } ( t ) = \\sum _ { j = 0 } ^ \\infty b _ j \\ , t ^ j \\ , , \\end{align*}"}
-{"id": "2899.png", "formula": "\\begin{align*} & \\chi ( u ) = \\tilde { \\varepsilon } _ p ( u ) \\chi ( g ) + \\sum _ { i = 1 } ^ k \\varepsilon _ { q _ i } ( u ) \\chi ( q _ i ) \\\\ & \\psi ( u ) = \\tilde { \\varepsilon } _ p ( u ) \\psi ( g ) + \\sum _ { i = 1 } ^ k \\varepsilon _ { q _ i } ( u ) ( \\chi ( q _ i ) + 1 ) . \\end{align*}"}
-{"id": "2671.png", "formula": "\\begin{align*} \\mathcal E ( X , \\theta ) : = \\bigcup _ { \\chi \\in \\mathcal W ^ - } \\mathcal E _ \\chi ( X , \\theta ) . \\end{align*}"}
-{"id": "182.png", "formula": "\\begin{align*} I _ { n , q } : = \\bigl [ \\hat { H } _ n ^ w - n ^ { - 1 / 2 } z _ { q / 2 } ( \\hat { V } _ n ^ w ) ^ { 1 / 2 } , \\hat { H } _ n ^ w + n ^ { - 1 / 2 } z _ { q / 2 } ( \\hat { V } _ n ^ w ) ^ { 1 / 2 } \\bigr ] , \\end{align*}"}
-{"id": "3728.png", "formula": "\\begin{align*} W _ { w } ^ { ( F , i ) } ( f ) ( x ) = \\gamma ( 1 ) ^ { \\dim _ k ( X ) } \\dfrac { 1 } { \\vert F ^ { - } \\vert ^ { 1 / 2 } } \\sum _ { y \\in F ^ - } \\varphi ( j ( x w , y ) ) f ( y ) \\end{align*}"}
-{"id": "98.png", "formula": "\\begin{align*} F ( x ) = y , \\end{align*}"}
-{"id": "8516.png", "formula": "\\begin{align*} \\eta ^ { ( k ) } = a ( \\sigma ^ { I \\backslash \\{ i _ 1 \\} } ) ^ { - 1 } \\sigma ^ { I \\backslash \\{ i _ 1 , \\dots , i _ { k - 1 } \\} } w \\end{align*}"}
-{"id": "7672.png", "formula": "\\begin{align*} u ( t ) = S _ { \\alpha } ( t ) u _ { 0 } + \\int _ { 0 } ^ { t } S _ { \\alpha } ( t - s ) f ( u ( s ) ) d s , \\end{align*}"}
-{"id": "6723.png", "formula": "\\begin{align*} F \\diamond \\int _ 0 ^ T g ( t ) d B ^ { H } ( t ) = F \\int _ 0 ^ T g ( t ) d B ^ { H } ( t ) - \\langle D ^ { H } F , g \\rangle _ { T } , \\end{align*}"}
-{"id": "3321.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 3 ) } = \\left ( \\begin{array} { c c c } 0 & 0 & q ^ 2 \\end{array} \\right ) ^ T , \\Gamma _ { 0 } ^ { ( 3 ) } = \\left ( \\begin{array} { c c c } 0 & - q ^ 2 & 0 \\end{array} \\right ) ^ T , \\Gamma _ { - } ^ { ( 3 ) } = \\left ( \\begin{array} { c c c } q ^ 4 & 0 & 0 \\end{array} \\right ) ^ T . \\end{align*}"}
-{"id": "2710.png", "formula": "\\begin{align*} u _ t ( x ) : = \\max ( \\varphi _ 0 - C t , \\varphi _ 1 + C ( t - 1 ) ) , \\end{align*}"}
-{"id": "4937.png", "formula": "\\begin{align*} \\| D ^ * _ { T _ 0 ^ c } h \\| _ 1 & \\leq \\| D ^ * _ { T _ 0 } h \\| _ 1 + 2 \\| D ^ * _ { T _ 0 ^ c } x _ 0 \\| _ 1 + \\rho \\\\ & = \\| D ^ * _ { T _ 0 } h \\| _ 1 + 2 \\sigma _ k ( D ^ * x _ 0 ) _ 1 + \\rho . \\end{align*}"}
-{"id": "3613.png", "formula": "\\begin{align*} \\| \\varphi ( x ) \\widetilde { f } \\| _ { C ^ { k , \\alpha } ( B _ a ( 0 ) ) } & = \\| f \\| _ { C ^ { k , \\alpha } _ { \\phi , \\varphi } ( B _ { a \\phi ( x ) } ( x ) ) } \\\\ \\| \\varphi ( x ) \\widetilde { f } \\| _ { L ^ 2 ( B _ a ( 0 ) ) } & = \\| f \\| _ { L ^ 2 _ { \\phi ^ { - n } \\varphi ^ 2 } ( B _ { a \\phi ( x ) } ( x ) ) } . \\end{align*}"}
-{"id": "1251.png", "formula": "\\begin{align*} I _ 1 \\left ( \\begin{array} { c } a \\\\ b \\end{array} \\right ) = \\left ( \\begin{array} { c } a \\\\ 0 \\end{array} \\right ) , I _ 2 \\left ( \\begin{array} { c } a \\\\ b \\end{array} \\right ) = \\left ( \\begin{array} { c } 0 \\\\ b \\end{array} \\right ) . \\end{align*}"}
-{"id": "252.png", "formula": "\\begin{align*} \\sup _ { x \\in \\mathbb { R } ^ d } \\max _ { r = 1 , \\ldots , m } \\frac { \\| f ^ { ( r ) } ( x ) \\| } { f ( x ) } \\leq d ^ { m / 2 } \\sup _ { x \\in \\mathbb { R } ^ d } \\max _ { r = 1 , \\ldots , m } \\frac { q _ r ( \\| x \\| ) } { ( 1 + \\| x \\| ^ 2 / \\rho ) ^ r } = : A _ { d , m , \\rho } ^ { ( 2 ) } , \\end{align*}"}
-{"id": "204.png", "formula": "\\begin{align*} n ^ { 1 / 2 } \\{ \\hat { H } _ n ^ w - H ( f ) \\} = \\frac { 1 } { n ^ { 1 / 2 } } \\sum _ { i = 1 } ^ n \\tilde { \\psi } _ f ( X _ i ) + o _ p ( 1 ) \\end{align*}"}
-{"id": "5934.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( \\R ^ d _ x ; H ^ { s } _ p ( \\R ^ d _ v ) ) } = \\Big ( \\int _ { \\R ^ d } \\| f ( x , \\cdot ) \\| _ { H ^ { s } _ p ( \\R ^ d ) } ^ p \\dd x \\Big ) ^ { 1 / p } . \\end{align*}"}
-{"id": "2948.png", "formula": "\\begin{align*} B ( s , \\epsilon ) = \\sum _ { m _ 1 \\in M _ { \\epsilon } } A ( s , m _ 1 ) . \\end{align*}"}
-{"id": "2608.png", "formula": "\\begin{align*} \\delta _ h ( f ) ( x , y | | z ) = f ( y | | z ) - f ( x + y | | z ) + f ( x | | z ) \\\\ \\delta _ v ( f ) ( x | | y , z ) = f ( x | | z ) - f ( x | | y + z ) + f ( x | | y ) \\end{align*}"}
-{"id": "8184.png", "formula": "\\begin{align*} H _ 1 : = \\sum _ { i = 1 } ^ 3 a _ i x _ i ^ 2 , \\ \\ \\ H _ 2 : = \\sum _ { i = 1 } ^ 3 x _ i ^ 2 . \\end{align*}"}
-{"id": "3496.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 u _ 1 ( x ' , x _ n ) d x _ n = \\int _ 0 ^ 1 u _ 2 ( x ' , x _ n ) d x _ n = 0 \\end{align*}"}
-{"id": "5237.png", "formula": "\\begin{align*} \\forall i = 1 , \\ldots , n ; j = 1 , \\ldots , k \\quad \\mbox { w e h a v e } w _ i + 1 \\ne c _ j ; \\end{align*}"}
-{"id": "3717.png", "formula": "\\begin{align*} \\gamma ( u + u ^ { \\prime } , x ) = \\psi ( 2 ^ { - 1 } ( \\langle x u + x u ^ { \\prime } , x \\rangle ) ) \\\\ = \\psi ( 2 ^ { - 1 } \\langle x , x u \\rangle ) \\psi ( 2 ^ { - 1 } \\langle x , x u ^ { \\prime } \\rangle ) \\\\ = \\gamma ( ( u , x ) ) \\gamma ( ( u ^ { \\prime } , x ) ) . \\end{align*}"}
-{"id": "1693.png", "formula": "\\begin{align*} \\dd \\gamma ( Z _ t ) = \\big ( \\lambda U ( Z _ t ) + A Z _ t \\big ) \\ , \\dd t + \\big ( D U ( Z _ t ) + \\mathbb { I } \\big ) { R \\ , } \\cdot \\dd W _ t \\end{align*}"}
-{"id": "3632.png", "formula": "\\begin{align*} u _ { n + 1 } = ( 1 - c ) f ( u _ n , u _ { n - 1 } , \\dots , u _ { n - k + 1 } ) , c \\in [ 0 , 1 ) , \\end{align*}"}
-{"id": "7406.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\cdot \\nu ( g _ 1 ^ { [ \\lambda s _ 1 n ] } \\cdots g _ k ^ { [ \\lambda s _ k n ] } ) \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { p n } \\cdot \\nu ( g _ 1 ^ { [ q s _ 1 n ] } \\cdots g _ k ^ { [ q s _ k n ] } ) \\\\ & = \\lim _ { n \\to \\infty } \\frac { q } { p n } \\cdot \\nu ( g _ 1 ^ { [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k n ] } ) \\\\ & = \\lambda \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\cdot \\nu ( g _ 1 ^ { [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k n ] } ) . \\\\ \\end{align*}"}
-{"id": "293.png", "formula": "\\begin{align*} ( 1 - p _ { n , x , u } - p _ { n , y , v } + p _ \\cap ) \\biggl \\| V ^ { - 1 / 2 } \\begin{pmatrix} p _ { n , x , u } \\\\ p _ { n , y , v } \\end{pmatrix} \\biggr \\| ^ { 3 } \\lesssim ( k / n ) ^ { 3 / 2 } . \\end{align*}"}
-{"id": "6570.png", "formula": "\\begin{align*} \\rho ( z , t ) = u + P ( z ) + Q ( z ) + v R ( z ) + u ^ 2 + v ^ 2 + o ( u ^ 2 , u v , v ^ 2 , u | z | ^ { 2 k } ) , \\end{align*}"}
-{"id": "4557.png", "formula": "\\begin{align*} \\int _ { - 2 \\sqrt { \\log n } } ^ { 2 \\sqrt { \\log n } } \\int _ { - 2 \\sqrt { \\log n } } ^ { 2 \\sqrt { \\log n } } \\{ \\Phi _ \\Sigma ( s , t ) - \\Phi ( s ) \\Phi ( t ) \\} \\ , d s \\ , d t = \\alpha _ z + O ( n ^ { - 2 } ) \\end{align*}"}
-{"id": "4521.png", "formula": "\\begin{align*} & \\log ^ 2 u _ { x , s } - \\log ^ 2 \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) \\\\ & = \\biggl \\{ 2 \\log \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) + \\log \\biggl ( \\frac { V _ d f ( x ) h _ x ^ { - 1 } ( s ) ^ d } { s } \\biggr ) \\biggr \\} \\log \\biggl ( \\frac { V _ d f ( x ) h _ x ^ { - 1 } ( s ) ^ d } { s } \\biggr ) . \\end{align*}"}
-{"id": "2048.png", "formula": "\\begin{align*} \\nabla _ E Q = \\frac { \\nabla _ E w ( x ) - \\nabla _ E w ( y ) } { \\bar { w } } - \\frac { ( w ( x ) - w ( y ) ) } { \\bar { w } ^ 2 } ( \\nabla _ E \\bar { w } ) , \\end{align*}"}
-{"id": "1139.png", "formula": "\\begin{align*} \\begin{array} { l } F i n d \\ , \\ , x \\ , \\\\ s o \\ , a s \\ , t o \\ , s a t i s f y \\ , \\\\ \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , { f _ { i j } } \\left ( x \\right ) \\left ( \\begin{array} { l } \\le \\\\ \\cong \\\\ \\ge \\end{array} \\right ) f _ { i j } ^ * , \\ , \\ , i = 1 , 2 , . . . m ; \\ , j = 1 , 2 , . . . , p _ m ^ { } \\\\ s u b j e c t \\ , \\ , t o \\ , \\\\ \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , x \\in S \\end{array} \\end{align*}"}
-{"id": "513.png", "formula": "\\begin{align*} \\sum _ { \\nu } s _ { \\lambda / \\nu } ( \\rho ) s _ { \\nu / \\mu } ( \\rho ' ) = s _ { \\lambda / \\mu } ( \\rho , \\rho ' ) . \\end{align*}"}
-{"id": "3511.png", "formula": "\\begin{align*} \\Big [ ( \\mathcal D _ g X ) _ { i j ; k } & + ( \\mathcal D _ g X ) _ { k i ; j } - ( \\mathcal D _ g X ) _ { j k ; i } ) \\Big ] \\\\ & = ( X _ { i ; j k } + X _ { i ; k j } ) + ( X _ { j ; i k } - X _ { j ; k i } ) + ( X _ { k ; i j } - X _ { k ; j i } ) \\\\ & = 2 X _ { i ; j k } + ( R ^ { \\ell } _ { k j i } + R ^ { \\ell } _ { i k j } + R ^ { \\ell } _ { i j k } ) X _ { \\ell } , \\end{align*}"}
-{"id": "6399.png", "formula": "\\begin{align*} \\gamma _ { q , p } ( 2 , 2 ) = \\beta _ { q , p } \\left ( \\frac { ( 3 p + 1 ) ! ! } { 1 2 } \\varepsilon _ { p } ^ { 3 } \\sigma ^ { 3 p } + \\frac { ( 2 q + 2 p + 1 ) ! ! } { 4 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } \\sigma ^ { 2 q + p } \\right ) , \\end{align*}"}
-{"id": "5830.png", "formula": "\\begin{align*} X _ { s o l } = \\mbox { \\rm e } ^ { \\kappa t } S \\partial _ { S } + F \\partial _ { F } \\end{align*}"}
-{"id": "9207.png", "formula": "\\begin{align*} Z = Z ( T _ 0 ) ; Z ( t ) = \\int _ 0 ^ t \\beta ( s ) d B ( s ) + \\int _ 0 ^ t \\int _ { \\mathbb { R } } \\psi ( s , \\zeta ) \\tilde { N } ( d s , d \\zeta ) , \\mbox { f o r } t \\in [ 0 , T _ 0 ] \\end{align*}"}
-{"id": "2599.png", "formula": "\\begin{align*} F ( \\tilde u ) = F ( \\tilde u _ { \\leq A T ^ { \\frac 1 2 } t ^ { - 1 } } ) + [ F ( \\tilde u ) - F ( \\tilde u _ { \\leq A T ^ { \\frac 1 2 } t ^ { - 1 } } ) ] . \\end{align*}"}
-{"id": "5650.png", "formula": "\\begin{align*} | | T ( t ) | | = \\left \\Vert \\frac { \\rho _ { t , p } } { \\rho } \\right \\Vert _ \\infty , \\end{align*}"}
-{"id": "9582.png", "formula": "\\begin{align*} \\Vert \\Psi \\Vert _ { { \\cal D } _ F } ^ 2 : = \\norm { \\psi _ { r e g } } _ { H ^ 2 ( \\R ^ 3 ) } ^ 2 + \\norm { \\pi _ { r e g } } _ { H ^ 1 ( \\R ^ 3 ) } ^ 2 + \\sum \\limits _ { 1 \\le j \\le n } | \\zeta _ j | ^ 2 + \\sum \\limits _ { 1 \\le j \\le n } | \\eta _ j | ^ 2 . \\end{align*}"}
-{"id": "7930.png", "formula": "\\begin{align*} 2 \\sum _ { i = 1 } ^ { l } x _ { i } ^ 2 + 2 \\sum _ { i = 1 } ^ { l } y _ { i } ^ 2 = \\sum _ { i = 1 } ^ { l } d _ { i } ^ 2 + \\sum _ { i = 1 } ^ { l } s _ { i } ^ 2 \\end{align*}"}
-{"id": "9223.png", "formula": "\\begin{align*} \\frac { \\partial \\widehat { H } ( t , x ) } { \\partial y } = \\frac { \\partial H } { \\partial y } ( t , x , \\hat { Y } ( t , x , z ) , \\widehat { Y } ( t , \\cdot , z ) ( x ) , \\hat { u } ( t , x , z ) , \\hat { p } ( t , x , z ) , \\hat { q } ( t , x , z ) , \\hat { r } ( t , x , z , . ) ) . \\end{align*}"}
-{"id": "4463.png", "formula": "\\begin{align*} f _ { t , g } ( x ) : = \\frac { 2 c ( t ) } { 1 + e ^ { - 2 t g ( x ) } } f ( x ) , \\end{align*}"}
-{"id": "8034.png", "formula": "\\begin{align*} s _ { i , j } ^ x \\ ; \\ ! s _ { k , \\ell } ^ y = \\begin{cases} s _ { k , \\ell } ^ y \\ ; \\ ! s _ { i , j } ^ x & \\ ; i \\neq \\ell j \\neq k , \\\\ s _ { k , \\ell } ^ y \\ ; \\ ! s _ { i , j } ^ x \\ ; s _ { i , \\ell } ^ { x y } & \\ ; j = k , \\\\ s _ { k , \\ell } ^ y \\ ; \\ ! s _ { i , j } ^ x \\ ; s _ { k , j } ^ { - x y } & \\ ; i = \\ell , \\end{cases} \\end{align*}"}
-{"id": "1370.png", "formula": "\\begin{align*} B _ { n } ^ \\dag ( u ) = \\frac { 1 } { 2 } u ^ { T } \\left ( \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } \\sigma _ { 0 , t } ^ 2 H _ { 0 , t } H _ { 0 , t } ^ { T } \\right ) u + E _ { n } ^ { \\dag } ( u ^ { \\ast } ) \\end{align*}"}
-{"id": "467.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } x ^ { \\top } Q x + d ^ { \\top } x + s = 0 , \\end{align*}"}
-{"id": "3410.png", "formula": "\\begin{gather*} n + \\alpha = \\frac { n \\beta + \\gamma } { \\beta } . \\end{gather*}"}
-{"id": "2295.png", "formula": "\\begin{align*} \\frac 1 \\tau \\ , \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\le \\delta , \\end{align*}"}
-{"id": "8182.png", "formula": "\\begin{align*} \\d H _ 1 = \\d H _ 2 = 0 . \\end{align*}"}
-{"id": "8393.png", "formula": "\\begin{align*} \\theta ( v ) \\theta ( x ) \\theta ( v ) ^ * = \\theta ( v x ) \\theta ( v ) ^ * = \\theta ( v x v ^ * v ) \\theta ( v ) ^ * = \\theta ( v x v ^ * ) \\theta ( v ) \\theta ( v ) ^ * , \\ \\ \\ x \\in M , \\end{align*}"}
-{"id": "8848.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k ( \\upsilon _ { k + 1 } - \\upsilon _ i ) ^ 2 \\leq \\frac { 4 } { n } \\sum _ { i = 1 } ^ k ( \\upsilon _ { k + 1 } - \\upsilon _ i ) \\upsilon _ i , \\end{align*}"}
-{"id": "3649.png", "formula": "\\begin{align*} K = c _ K T _ K + ( 1 - c _ K ) \\frac { 8 \\cdot K } { 1 + K } . \\end{align*}"}
-{"id": "4325.png", "formula": "\\begin{align*} u ( d t ^ { \\otimes m } ) = \\sigma _ { u , m } \\left ( d z ^ \\prime \\wedge p ^ \\star ( d t ) \\right ) ^ { \\otimes m } \\end{align*}"}
-{"id": "7827.png", "formula": "\\begin{align*} Y _ { t } = g ( \\eta _ { T } ) + \\int _ t ^ T E ' [ f ( s , \\eta _ { s } , Y _ { s } ' , Y _ { s } , Z _ { s } ) ] d s - \\int _ t ^ T Z _ { s } d B _ { s } ^ { H } , \\ \\ 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "7055.png", "formula": "\\begin{align*} { { \\bf { Y } } ^ { [ j ] } } ( { t _ 1 } ) = { { \\bf { H } } ^ { [ j 1 ] } } ( { t _ 1 } ) { { \\bf { u } } ^ { [ 1 ] } } + { { \\bf { H } } ^ { [ j 2 ] } } ( { t _ 1 } ) { { \\bf { u } } ^ { [ 2 ] } } , { { \\bf { Y } } ^ { [ j ] } } ( { t _ 2 } ) = { { \\bf { H } } ^ { [ j 1 ] } } ( { t _ 2 } ) { { \\bf { v } } ^ { [ 1 ] } } + { { \\bf { H } } ^ { [ j 2 ] } } ( { t _ 2 } ) { { \\bf { v } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "3165.png", "formula": "\\begin{align*} & \\| V _ { t } ^ { 0 } ( a ) - V _ { t } ^ { \\lambda } ( a ) \\| _ { 1 } \\\\ & = \\| - \\lambda | t + a - 1 \\rangle \\langle t + a - 1 | ^ { \\mathfrak { E } } + \\lambda | t + 2 \\rangle \\langle t + 2 | ^ { \\mathfrak { E } } \\| _ { 1 } \\\\ & = 2 \\lambda \\end{align*}"}
-{"id": "6426.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } \\int _ { \\Omega } - \\eta _ { t } \\left [ g _ { 1 - \\alpha } * ( u - u _ { 0 } ) \\right ] d x d t + \\int _ { 0 } ^ { T } \\mathcal { E } ( u , \\eta ) d t = ( \\geq \\leq ) \\int _ { 0 } ^ { T } \\int _ { \\Omega } f \\eta d x d t . \\end{align*}"}
-{"id": "6356.png", "formula": "\\begin{align*} \\bar { P } ( X ) : = \\bar { p } _ { \\mu m } x ^ \\mu | z | ^ m \\end{align*}"}
-{"id": "6337.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } f \\| _ { M _ 2 } = & \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 2 , \\alpha _ 1 } } \\| \\Box _ l ^ { \\alpha _ 2 } \\Box _ k ^ { \\alpha _ 1 } f \\| ^ q _ { L ^ { \\infty } } \\right ) ^ { 1 / q } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 2 / 2 } \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 2 , \\alpha _ 1 } } \\| \\Box _ l ^ { \\alpha _ 2 } \\Box _ k ^ { \\alpha _ 1 } f \\| ^ q _ { L ^ { 2 } } \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "2672.png", "formula": "\\begin{align*} { \\rm I } ( \\varphi ) : = \\frac { 1 } { ( n + 1 ) { \\rm V o l } ( \\theta ) } \\sum _ { k = 0 } ^ n \\int _ X ( \\varphi - V _ { \\theta } ) \\langle \\theta _ { \\varphi } ^ k \\wedge \\theta _ { V _ { \\theta } } ^ { n - k } \\rangle . \\end{align*}"}
-{"id": "9418.png", "formula": "\\begin{align*} E ^ p ( f ) \\coloneqq \\begin{cases} \\norm { f ' } _ { p , G } & p > 1 \\\\ \\int _ G w ( x ) \\abs { f ' ( x ) } \\ , d x & p = 1 . \\end{cases} \\end{align*}"}
-{"id": "8913.png", "formula": "\\begin{align*} \\abs { P _ { W _ j } ( x ) } = \\abs { P _ { W _ j } ( \\sigma _ H ( x ) ) } . \\end{align*}"}
-{"id": "5205.png", "formula": "\\begin{align*} [ X , \\overline { Y } ] _ p = \\lambda ( X , \\overline { Y } ) ( p ) T ( p ) \\ ; T ^ { 1 , 0 } _ p M \\oplus T ^ { 0 , 1 } _ p M \\ ; . \\end{align*}"}
-{"id": "2718.png", "formula": "\\begin{align*} \\begin{cases} U _ 0 ( x , z ) : = \\varphi _ 0 ( x ) + A ( \\log ( | z | ^ 2 + 3 ) - \\log ( | z | ^ 2 + 1 ) - \\log 2 ) ; \\\\ U _ 1 ( x , z ) : = \\varphi _ 1 ( x ) + A ( \\log | z | ^ 2 - \\log ( | z | ^ 2 + 1 ) + \\log ( e ^ 2 + 1 ) - 2 ) . \\end{cases} \\end{align*}"}
-{"id": "5622.png", "formula": "\\begin{align*} \\mathcal { A } _ 2 ( n ) = \\{ w \\in \\mathcal { A } ( n ) ~ | ~ w = 0 1 2 1 0 2 \\{ 0 1 2 \\} * 2 0 1 2 1 0 \\} \\subseteq \\mathcal { A } ( n ) , \\end{align*}"}
-{"id": "9707.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ' ( x ) } { \\alpha x ^ { - ( \\alpha + 1 ) } e ^ { - 1 / x ^ \\alpha } } = 1 \\end{align*}"}
-{"id": "531.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = I ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) + \\bar { R } ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) \\end{align*}"}
-{"id": "1554.png", "formula": "\\begin{align*} J ' ( \\mu ; \\eta - \\mu ) : = \\underset { \\underset { t > 0 } { t \\to 0 } } { \\lim } \\frac { J ( \\mu + t ( \\eta - \\mu ) ) - J ( \\mu ) } { t } \\ge 0 . \\end{align*}"}
-{"id": "7402.png", "formula": "\\begin{align*} \\mathsf { g } + \\mathsf { h } & = [ \\mathtt { g } \\cdot \\mathtt { h } ] , \\\\ \\lambda \\mathsf { g } & = [ \\mathtt { g } ^ { ( \\lambda ) } ] . \\\\ \\end{align*}"}
-{"id": "4125.png", "formula": "\\begin{align*} y ^ { 2 ( p + q ) } - s ^ { - p } = \\prod _ { k = 0 } ^ { m - 1 } ( y ^ { 2 ( p + q ) / m } - s ^ { - p / m } e ^ { 2 \\pi i k / m } ) , \\end{align*}"}
-{"id": "7309.png", "formula": "\\begin{align*} D ^ 2 = \\sum _ { i , j } \\mathcal { E } _ i \\mathcal { E } _ j ^ * \\otimes \\gamma _ { - } ( w _ i ) \\gamma _ { - } ( w _ j ) ^ * + \\sum _ { i , j } \\mathcal { E } _ j ^ * \\mathcal { E } _ i \\otimes \\gamma _ { - } ( w _ j ) ^ * \\gamma _ { - } ( w _ i ) . \\end{align*}"}
-{"id": "4155.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } S _ f ^ \\nu ( n ) \\overline { S _ g ^ \\nu ( n ) } = c _ { f , g } X ^ { 2 \\kappa ( f ) + \\frac { 3 } { 2 } - 2 \\nu } + O ( X ^ { 2 \\kappa ( f ) + 1 - 2 \\nu } \\log ^ 2 X ) \\end{align*}"}
-{"id": "9142.png", "formula": "\\begin{align*} g ( 1 + \\frac { d } { t } , t ) \\leq \\frac { ( 2 d + 1 ) t + d ^ 2 } { ( t + d ) ( 2 d + 1 ) } = 1 - \\frac { d ^ 2 + d } { ( t + d ) ( 2 d + 1 ) } . \\end{align*}"}
-{"id": "3953.png", "formula": "\\begin{align*} \\mu _ { \\infty } ( \\pi ( N ( L , W ) ) ) > 0 \\mu _ { \\infty } ( \\pi ( S ( L , W ) ) ) = 0 . \\end{align*}"}
-{"id": "5944.png", "formula": "\\begin{align*} & \\lambda U ( z ) - \\frac { 1 } { 2 } \\mathrm { T r } \\big ( Q D ^ 2 U ( z ) \\big ) - \\langle A z , D U ( z ) \\rangle - \\langle B ( z ) , D U ( z ) \\rangle = B ( z ) , \\\\ & \\ ; \\ ; \\lambda U ( z ) - { \\mathcal L } U ( z ) = B ( z ) \\end{align*}"}
-{"id": "4065.png", "formula": "\\begin{align*} \\alpha \\beta \\gamma = \\alpha \\gamma + \\alpha \\beta + \\beta \\gamma . \\end{align*}"}
-{"id": "7913.png", "formula": "\\begin{align*} \\left ( \\int | W | _ g ^ 4 d v _ g \\right ) ^ { { 1 } / { 2 } } = \\left ( \\int _ M | W | _ g ^ 4 u ^ 4 \\right ) ^ { { 1 } / { 2 } } d v _ 0 \\leq C _ 0 \\int _ M | \\nabla | W | _ g u | ^ 2 _ 0 d v _ 0 + K _ s \\int _ M | W | _ g ^ 2 u ^ 2 d v _ 0 , \\end{align*}"}
-{"id": "7868.png", "formula": "\\begin{align*} c _ 2 = 4 \\frac { C _ 0 ( T ) } { \\delta _ 1 - \\beta _ 1 - \\beta _ 2 } B \\left ( ( \\gamma _ 1 + \\beta _ 1 ) / 2 + 1 - \\theta , { \\gamma _ 2 + \\beta _ 2 } / { 2 } + 1 - \\eta \\right ) . \\end{align*}"}
-{"id": "4373.png", "formula": "\\begin{align*} V = \\oplus _ { k = 0 } ^ \\infty V _ k \\ ; \\ ; \\ ; \\textrm { w h e r e } \\ ; \\ ; \\ ; V _ k = { \\rm K e r } ( L _ 0 - k I ) . \\end{align*}"}
-{"id": "9920.png", "formula": "\\begin{align*} k m _ i = \\pm q ^ { n - 2 i } m _ i , f m _ i = \\begin{cases} m _ { i + 1 } & \\\\ 0 & \\end{cases} e m _ i = \\begin{cases} \\pm [ i ] [ n + 1 - i ] m _ { i - 1 } & \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "4171.png", "formula": "\\begin{align*} | p _ n | = \\Big | \\tfrac { \\displaystyle p ^ { ( n ) } ( 0 ) } { \\displaystyle n ! } \\Big | \\leq | \\varphi ' ( 0 ) | ( n \\in \\mathbb { N } ) . \\end{align*}"}
-{"id": "1399.png", "formula": "\\begin{align*} E \\left ( e ^ { i s ^ { T } U _ { n } } \\right ) = \\exp \\left [ - \\frac { 1 } { 2 } s ^ T \\Omega _ { 2 } s \\right ] . \\end{align*}"}
-{"id": "5901.png", "formula": "\\begin{align*} \\lambda { \\psi } ( x , v ) - \\frac { 1 } { 2 } \\triangle _ v { \\psi } ( x , v ) - v \\cdot D _ x { \\psi } ( x , v ) - F ( x , v ) \\cdot D _ v { \\psi } ( x , v ) = g ( x , v ) \\ , , \\end{align*}"}
-{"id": "9217.png", "formula": "\\begin{align*} \\varphi \\mapsto \\ell ( \\varphi ) ( x ) : = H ( t , x , \\varphi ( x ) , \\varphi , u , z , p , q , r ) \\end{align*}"}
-{"id": "3899.png", "formula": "\\begin{align*} \\begin{aligned} \\alpha _ L & = 0 . 3 \\ ; , & \\beta _ L & = 4 \\ ; , & \\gamma _ L & = 0 . 0 5 \\ ; , \\\\ \\alpha _ R & = 0 . 0 2 \\ ; , & \\beta _ R & = 1 \\ ; , & \\gamma _ R & = 1 \\ ; . \\end{aligned} \\end{align*}"}
-{"id": "5265.png", "formula": "\\begin{align*} \\pi ( x ) = \\int _ { 2 } ^ { x } \\frac { 1 } { \\log u } d u + O \\left \\{ x \\exp \\left ( - C ( \\log x ) ^ { 3 / 5 } ( \\log \\log x ) ^ { - 1 / 5 } \\right ) \\right \\} . \\end{align*}"}
-{"id": "9017.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ d ( a _ d z - \\eta _ i ) = \\tilde f ( x , y , a _ d z ) = a _ d ^ { d - 1 } f ( x , y , z ) . \\end{align*}"}
-{"id": "1192.png", "formula": "\\begin{align*} k ^ { 1 / ( n - 1 ) } \\leq s = \\underset { m \\rightarrow \\infty } { \\lim } ( s _ { m } ) ^ { 1 / m } . \\end{align*}"}
-{"id": "8127.png", "formula": "\\begin{align*} l _ * = k \\frac { N } { N - 1 } \\mbox { i f } \\ k \\leq 0 . \\end{align*}"}
-{"id": "4128.png", "formula": "\\begin{align*} \\omega = 2 q y z d y - ( ( p + 2 q ) y ^ 2 + b ( p + q ) z ) d z . \\end{align*}"}
-{"id": "481.png", "formula": "\\begin{align*} \\int _ { B ^ c _ \\varepsilon ( x ) } \\frac { \\ , u ( x ) - u ( y ) \\ , } { | x - y | ^ { N + 2 s } } \\ , d y & = \\int _ { B _ r \\cap B ^ c _ \\varepsilon } \\frac { \\ , u ( x ) - u ( x + z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z + \\int _ { B ^ c _ r } \\frac { \\ , u ( x ) - u ( x + z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z \\\\ & = \\int _ { B _ r \\cap B ^ c _ \\varepsilon } \\frac { \\ , u ( x ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z + \\int _ { B ^ c _ r } \\frac { \\ , u ( x ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z , \\end{align*}"}
-{"id": "5723.png", "formula": "\\begin{align*} f = \\sum _ { j = 3 } ^ \\infty \\chi _ { [ j ! , j ! + 1 ] } , \\end{align*}"}
-{"id": "4986.png", "formula": "\\begin{align*} & E = g ^ { * } H - \\left < A \\vec { c } , \\vec { F } \\right > \\\\ & \\vec { { E ' } } = \\left ( \\begin{array} { c } E _ { 1 } ' \\\\ E _ { 2 } ' \\\\ \\vdots \\\\ E _ { s } ' \\end{array} \\right ) = { p } _ { * } \\vec { F } - { } ^ { \\rm t } B \\vec { D } . \\end{align*}"}
-{"id": "6447.png", "formula": "\\begin{align*} \\int _ { B _ { 1 } } \\phi \\psi ^ { 1 + q } w ^ { 2 } d x + \\frac { c _ { n , \\beta } } { 4 C \\Lambda } g _ { \\alpha } * \\int _ { \\rho ' B _ { 1 } } \\int _ { \\rho ' B _ { 1 } } \\frac { ( w ( s , x ) - w ( s , y ) ) ^ { 2 } } { | x - y | ^ { n + 2 \\beta } } \\phi d x d y \\leq g _ { \\alpha } * F . \\end{align*}"}
-{"id": "2735.png", "formula": "\\begin{align*} \\mu _ \\star = \\frac { 1 } { m ! \\ , \\nu ^ m } \\left ( \\omega _ { - 1 } \\right ) ^ m e ^ \\varkappa , \\end{align*}"}
-{"id": "9634.png", "formula": "\\begin{align*} L = \\begin{bmatrix} A & B \\\\ C & D \\end{bmatrix} \\\\ \\end{align*}"}
-{"id": "2402.png", "formula": "\\begin{align*} - ( 2 \\alpha - 1 ) \\| x - y \\| _ { P ^ 2 } ^ 2 & = - ( 2 \\alpha - 1 ) \\| ( P x + q ) - ( P y - q ) \\| ^ 2 \\\\ & \\leq \\langle P \\nabla f _ 2 ( P x + q ) - P \\nabla f _ 2 ( P y + q ) , x - y \\rangle \\\\ & \\leq ( 2 \\beta - 1 ) \\| ( P x + q ) - ( P y - q ) \\| ^ 2 \\\\ & = ( 2 \\beta - 1 ) \\| x - y \\| _ { P ^ 2 } ^ 2 \\end{align*}"}
-{"id": "35.png", "formula": "\\begin{align*} m ( K _ { X / Y } + L ) = K _ { X / Y } + L _ m \\end{align*}"}
-{"id": "22.png", "formula": "\\begin{align*} E : = { \\cup } _ { y \\in Y } H ^ 0 \\big ( X _ y , K _ { X _ y } + L | _ { X _ y } \\big ) \\end{align*}"}
-{"id": "3700.png", "formula": "\\begin{align*} \\frac { d x _ 3 } { d t } = - x _ 1 x _ 2 ~ , \\end{align*}"}
-{"id": "9257.png", "formula": "\\begin{align*} d y ( t , x , z ) & = L ^ * _ { R ( t ) , u ( t ) } y ( t , x , z ) d t + h ( x ) y ( t , x , z ) d R ( t ) ; t \\geq 0 \\\\ y ( 0 , x , z ) & = F ( x , z ) . \\end{align*}"}
-{"id": "2848.png", "formula": "\\begin{align*} g _ { P , X } ^ { \\psi } = H o m _ { \\Sigma } ( \\overline { C } , E n d _ X ) ^ { \\psi } \\cong D e r _ { \\psi } ( \\Omega ( C ) , E n d _ X ) \\end{align*}"}
-{"id": "5814.png", "formula": "\\begin{align*} S _ { C A , B , \\omega _ k } + S _ { A C , D , \\omega _ k ^ { \\vee } } = 0 . \\end{align*}"}
-{"id": "1015.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ) ) ' = f ( t , u ) & & \\\\ u ( T ) = b u ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6120.png", "formula": "\\begin{align*} \\tilde R _ { i \\bar i i \\bar i } = - \\frac { 3 \\pi } { 4 } s _ { i } ^ { - 1 } + O ( s _ { i } ) . \\end{align*}"}
-{"id": "1023.png", "formula": "\\begin{align*} & ( \\varphi ( u ' ) ) ' = N _ f ( u ) , \\ u ' ( T ) = b u ' ( 0 ) , \\ u ' ( T ) = b u ' ( 0 ) \\\\ & \\Leftrightarrow ( D _ { \\varphi } D ) ( u ) = N _ { f } ( u ) , \\ u \\in { \\rm d o m } ( D ) \\\\ & \\Leftrightarrow u = ( D _ { \\varphi } D ) ^ { - 1 } N _ { f } ( u ) , \\ u \\in C ^ { 1 } . \\end{align*}"}
-{"id": "9950.png", "formula": "\\begin{align*} \\liminf _ { j \\rightarrow \\infty } \\left ( \\int _ \\Omega | u _ j ( x ) | ^ { 2 ^ * } d x - \\int _ \\Omega | u _ \\infty ( x ) | ^ { 2 ^ * } d x \\right ) = \\liminf _ { j \\rightarrow \\infty } \\int _ \\Omega | u _ j ( x ) - u _ \\infty ( x ) | ^ { 2 ^ * } d x \\end{align*}"}
-{"id": "5717.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { k } { \\Phi _ { p , q , w } ( 2 ^ k Q ) } \\leq \\frac { C } { \\Phi _ { p , q , w } ( Q ) } ( Q \\in \\mathcal { Q } ) . \\end{align*}"}
-{"id": "1027.png", "formula": "\\begin{align*} P ( x , y ) ~ = ~ A x y + B ( x + y ) + D , \\end{align*}"}
-{"id": "7897.png", "formula": "\\begin{align*} \\left ( \\left \\| D - C \\right \\| ^ 2 - \\left ( \\frac { ( A - C ) \\times ( D - C ) } { b } \\right ) ^ 2 \\right ) t ^ 2 + 2 ( D - C ) \\cdot ( C - P ) t + \\left \\| C - P \\right \\| ^ 2 = 0 \\end{align*}"}
-{"id": "3107.png", "formula": "\\begin{align*} B _ { 1 1 } & = P _ 5 , \\\\ B _ { 1 2 } + B _ { 1 2 } ^ T + A _ { 1 1 } & = P _ 4 , \\\\ B _ { 1 3 } + B _ { 2 2 } + B _ { 1 3 } ^ T + A _ { 1 2 } + A _ { 1 2 } ^ T & = P _ 3 , \\\\ B _ { 2 3 } + B _ { 2 3 } ^ T + A _ { 1 3 } + A _ { 2 2 } + A _ { 1 3 } ^ T & = P _ 2 , \\\\ B _ { 3 3 } + A _ { 2 3 } + A _ { 2 3 } ^ T & = P _ 1 , \\mbox { a n d } \\\\ A _ { 3 3 } & = P _ 0 , \\end{align*}"}
-{"id": "6573.png", "formula": "\\begin{align*} F ( x ) : = \\tfrac { 1 } { 2 } \\langle P x , x \\rangle - f _ 2 ( \\nabla f _ 1 ( x ) ) . \\end{align*}"}
-{"id": "3494.png", "formula": "\\begin{align*} \\Delta _ { x ' } v = \\Delta u ( x ' ) - [ \\partial _ \\nu u ( x ' , 0 ) + \\partial _ \\nu u ( x ' , 1 ) ] \\ , \\ , \\ , D . \\end{align*}"}
-{"id": "7646.png", "formula": "\\begin{align*} u ( t , x ) = ( S _ { \\alpha } ( t ) u _ { 0 } ) ( x ) : = \\int _ { B _ { R } } p _ { D } ( t , x , y ) u _ { 0 } ( y ) d y , \\end{align*}"}
-{"id": "2728.png", "formula": "\\begin{align*} & \\nu ^ * = 0 , \\\\ & \\alpha ^ * _ j = \\begin{cases} 1 , & j = i , \\\\ 0 , & , \\\\ \\end{cases} \\forall j \\in I . \\end{align*}"}
-{"id": "3398.png", "formula": "\\begin{align*} u _ { \\alpha \\beta } & : = \\varphi _ { \\alpha } \\circ \\varphi _ { \\beta } ^ { - 1 } - \\mathrm { i d } _ { E | _ { U _ { \\alpha \\beta } \\otimes \\mathbb { C } [ \\epsilon ] } } , \\\\ v _ { \\alpha } & : = ( \\varphi _ { \\alpha } \\otimes \\mathrm { i d } ) \\circ \\nabla ^ v \\circ \\varphi _ { \\alpha } ^ { - 1 } - \\nabla | _ { U _ { \\alpha } } \\otimes \\mathrm { i d } _ { \\mathbb { C } [ \\epsilon ] } . \\end{align*}"}
-{"id": "3069.png", "formula": "\\begin{align*} \\omega _ { n , \\beta } = W _ { n , \\beta } ^ { - 2 } \\sum _ { | u | = | v | = n } e ^ { \\beta ( 2 m _ n - V ( u ) - V ( v ) ) } \\delta _ { | u \\wedge v | / n } . \\end{align*}"}
-{"id": "2949.png", "formula": "\\begin{align*} & a _ { 0 , j } = \\binom { \\gamma } { j } \\mathbb { I } [ 1 \\le j \\le \\gamma - 1 ] , \\\\ & a _ { 1 , j } = \\mathbb { I } [ j = 0 j = \\gamma ] . \\end{align*}"}
-{"id": "8048.png", "formula": "\\begin{align*} Z _ { \\overrightarrow { \\Gamma } } : = \\prod _ { e \\in E ( \\Gamma ) } ( X _ { t ( e ) } Y _ { h ( e ) } - X _ { h ( e ) } Y _ { t ( e ) } ) . \\end{align*}"}
-{"id": "8501.png", "formula": "\\begin{align*} o _ R ( G + Y _ j ' ) = \\nu ( 0 ) < o _ R ( H ^ { R _ j } + Y _ j ' ) . \\end{align*}"}
-{"id": "712.png", "formula": "\\begin{align*} Z _ { i } = p ^ { * } { p } _ { * } g ^ { * } D _ { i } - g ^ { * } D _ { i } \\end{align*}"}
-{"id": "7961.png", "formula": "\\begin{align*} \\mathcal H _ { g } ^ 1 \\Big ( \\{ \\psi _ { \\lambda } = 0 \\} \\Big ) \\leq \\sum _ { x _ i \\in \\mathcal I } \\mathcal H _ { g } ^ 1 \\Big ( \\{ \\psi _ { \\lambda } = 0 \\} \\cap B _ { r / 2 } ( x _ i ) \\Big ) & \\leq ( C _ 0 r ^ { - n } ) C _ 2 r ^ { n - 1 } { \\tilde \\lambda } ^ { \\frac { 3 } { 4 } - \\beta } \\\\ & = c _ 3 r ^ { 1 - 2 \\beta } \\lambda ^ { \\frac { 3 } { 4 } - \\beta } . \\end{align*}"}
-{"id": "2461.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } f \\| _ { M _ 2 } \\lesssim 2 ^ { j n \\alpha _ 2 / 2 } 2 ^ { j n ( \\alpha _ 1 - \\alpha _ 2 ) ( 1 / q - 1 / 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "5993.png", "formula": "\\begin{align*} w s _ \\alpha z _ \\mu t ( z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu ) & = \\Pi ^ S ( w s _ \\alpha z _ \\mu t ( z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu ) ) \\\\ & = \\lfloor w s _ \\alpha \\rfloor z _ { z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu } t ( z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu ) & ( & \\mbox { b y L e m m a \\ref { 2 . 2 . 6 } ( 3 ) } ) , \\end{align*}"}
-{"id": "9669.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { G ( x ( t ) ) } { t } = a - b . \\end{align*}"}
-{"id": "7023.png", "formula": "\\begin{align*} \\mathbb { E } \\ , T ^ \\ell ( \\omega ^ { 2 ^ k } ) \\cdots T ^ \\ell ( \\omega ) v = \\mathbb { E } \\ , ( S ^ \\ell ) ^ { k + 1 } v . \\end{align*}"}
-{"id": "8497.png", "formula": "\\begin{align*} W = \\{ ( p _ 1 , p _ 2 , \\dots p _ r , D ) \\in ( \\mathbb P ^ 2 ) ^ r \\times U _ d : p _ i \\in D \\} \\end{align*}"}
-{"id": "2396.png", "formula": "\\begin{align*} F _ { \\alpha _ 1 , \\alpha _ 2 } ^ { \\rm { G A P } } ( x ) = \\tfrac { 1 } { 2 } \\langle P x , x \\rangle - p _ { C } ^ { \\alpha _ 2 } ( P _ D ^ { \\alpha _ 1 } x ) \\end{align*}"}
-{"id": "1495.png", "formula": "\\begin{align*} Q ( Y ) : = Y ^ k + a _ 1 Y ^ { k - 1 } + \\cdots + a _ k . \\end{align*}"}
-{"id": "4589.png", "formula": "\\begin{align*} 6 _ { 1 , 3 } & = , 6 _ { 2 , 3 } = , \\\\ 8 _ { 1 , 4 } & = , 1 0 _ { 1 , 3 } = . \\end{align*}"}
-{"id": "1326.png", "formula": "\\begin{align*} T ( \\lambda ) = \\prod _ { j = 1 } ^ N L _ j ( \\lambda , \\xi _ j ) = \\prod _ { j = 1 } ^ N \\begin{pmatrix} ( L _ j ( \\lambda , \\xi _ j ) ) _ { 1 1 } & ( L _ j ( \\lambda , \\xi _ j ) ) _ { 1 2 } \\\\ ( L _ j ( \\lambda , \\xi _ j ) ) _ { 2 1 } & ( L _ j ( \\lambda , \\xi _ j ) ) _ { 2 2 } \\end{pmatrix} = \\begin{pmatrix} A ( \\lambda ) & B ( \\lambda ) \\\\ C ( \\lambda ) & D ( \\lambda ) \\end{pmatrix} . \\end{align*}"}
-{"id": "7944.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 u = \\lambda u , & { \\rm i n } \\ \\Omega , \\\\ ( 1 - \\sigma ) \\frac { \\partial ^ 2 u } { \\partial \\nu ^ 2 } + \\sigma \\Delta u = 0 , & { \\rm o n } \\ \\partial \\Omega , \\\\ \\frac { \\partial \\Delta u } { \\partial \\nu } + ( 1 - \\sigma ) { \\rm d i v } _ { \\partial \\Omega } \\left ( D ^ 2 u \\cdot \\nu \\right ) _ { \\partial \\Omega } = 0 , & { \\rm o n } \\ \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "5081.png", "formula": "\\begin{align*} & ( R _ { i } - 1 ) ( R _ { i } + q ^ { 2 } ) = 0 ( 1 \\le i < k ) , \\\\ & R _ { i } R _ { i + 1 } R _ { i } = R _ { i + 1 } R _ { i } R _ { i + 1 } ( 1 \\le i \\le k - 2 ) , \\\\ & R _ { i } R _ { j } = R _ { j } R _ { i } ( | i - j | \\ge 2 ) . \\end{align*}"}
-{"id": "5160.png", "formula": "\\begin{gather*} \\delta _ p ( i , j ) = \\begin{cases} 1 , & , \\\\ 0 , & . \\end{cases} \\end{gather*}"}
-{"id": "2102.png", "formula": "\\begin{align*} r : = ( y ^ 2 + d ^ 2 ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6161.png", "formula": "\\begin{align*} ( x _ i ^ { v - 1 } x _ j ) x _ i ^ v = r _ { i , j } x _ i ^ v ( x _ i ^ { v - 1 } x _ j ) \\end{align*}"}
-{"id": "8014.png", "formula": "\\begin{align*} x _ j ^ { d _ j } y _ j = y _ j x _ j ^ { d _ j } , \\ ; \\ ; y _ j ^ { d _ j } x _ j = x _ j y _ j ^ { d _ j } . \\end{align*}"}
-{"id": "602.png", "formula": "\\begin{align*} \\xi _ { \\kappa } ( g ^ * ) = g . \\end{align*}"}
-{"id": "3378.png", "formula": "\\begin{align*} \\int _ { \\gamma _ 0 } ^ \\infty \\left ( \\frac { 1 } { \\gamma _ 0 } - \\frac { 1 } { \\gamma } \\right ) f ( \\gamma ) d \\gamma = E _ b ( t ) . \\end{align*}"}
-{"id": "2741.png", "formula": "\\begin{align*} L _ f = u ^ { - 1 } \\left ( L ^ \\star _ { f _ K ^ I } \\theta ^ K \\delta _ I \\right ) u \\mbox { f o r } f = u ^ { - 1 } ( f _ K ^ I \\star u _ { I L } ) \\theta ^ K \\bar \\theta ^ L . \\end{align*}"}
-{"id": "9129.png", "formula": "\\begin{align*} u ( t ) & = e ^ { - ( t - s ) A ( t ) } u ( s ) + \\int _ { s } ^ { t } { e ^ { - ( t - r ) A ( t ) } ( \\mathcal { A } ( t ) - B ( r ) \\mathcal { A } ( r ) ) u ( r ) d r } \\\\ & + \\int _ { s } ^ { t } { e ^ { - ( t - r ) A ( t ) } [ - P ( r ) u ( r ) + f ( r ) ] } d r . \\end{align*}"}
-{"id": "7668.png", "formula": "\\begin{align*} ( S \\ast f ) ( t ) = \\int _ { 0 } ^ { t } S ( t - s ) f ( s ) d s \\end{align*}"}
-{"id": "2688.png", "formula": "\\begin{align*} \\theta _ { V _ { \\theta } } ^ n \\leq { \\bf 1 } _ { \\{ V _ { \\theta } = 0 \\} } \\theta ^ n . \\end{align*}"}
-{"id": "9663.png", "formula": "\\begin{align*} x ' ( t ) & = - a g ( x ( t ) ) + b g ( x ( t - \\tau ( t ) ) , t \\geq 0 \\\\ x ( t ) & = \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] \\end{align*}"}
-{"id": "3050.png", "formula": "\\begin{align*} \\forall u \\in \\mathcal { U } , n \\in \\N , \\mu ( C ( u , n ) ) = \\sum _ { j = n } ^ \\infty \\mu ( B ( u . j ) ) = \\sum _ { j = n } ^ \\infty \\nu ( B ( u . j ) ) = \\nu ( C ( u , n ) ) , \\end{align*}"}
-{"id": "8652.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } d X _ \\tau = A X _ \\tau d \\tau + G d W _ { \\tau } , \\tau \\in \\left [ 0 , T \\right ] . \\\\ X _ 0 = x \\in H , \\end{array} \\right . \\end{align*}"}
-{"id": "2306.png", "formula": "\\begin{align*} \\big \\| ( e _ n ) _ { n = 0 } ^ M \\big \\| _ { L ^ \\infty ( W ) } \\le C \\delta . \\end{align*}"}
-{"id": "5123.png", "formula": "\\begin{align*} F _ { \\vec { z } } ^ { \\vec { \\mu } } ( x _ { 1 } + c , \\ldots , x _ { k } + c ) = \\prod _ { i = 1 } ^ { k } \\left ( \\frac { z _ { i } } { 1 + z _ { i } } \\right ) ^ { c } F _ { \\vec { z } } ^ { \\vec { \\mu } } ( x _ { 1 } , \\ldots , x _ { k } ) ( \\vec { x } \\in L _ { + } , c \\in \\mathbb { Z } ) . \\end{align*}"}
-{"id": "4322.png", "formula": "\\begin{align*} \\nu _ 0 : = \\sup \\left \\{ r > 0 : \\frac { 1 } { | s _ { 0 } | ^ { 2 r } } \\in L ^ 1 _ { \\rm l o c } ( X ) \\right \\} \\end{align*}"}
-{"id": "2027.png", "formula": "\\begin{align*} V ( a _ 1 , a _ 2 , a _ 3 ) = \\begin{pmatrix} 0 & - a _ 1 & - a _ 2 & - a _ 3 \\\\ a _ 1 & 0 & - b _ 3 & b _ 2 \\\\ a _ 2 & b _ 3 & 0 & - b _ 1 \\\\ a _ 3 & - b _ 2 & b _ 1 & 0 \\end{pmatrix} \\end{align*}"}
-{"id": "5209.png", "formula": "\\begin{align*} ( f , g ) _ { L ^ { 2 } _ { ( 0 , q ) } ( M ) } : = \\int _ { M } ( f , g ) ( z ) d \\sigma ( z ) \\ ; , \\end{align*}"}
-{"id": "7040.png", "formula": "\\begin{align*} d _ + ( t ) : = \\inf \\bigl \\{ x > 0 : \\overline { \\Pi } ^ + ( x ) \\le t ^ { - 1 } \\bigr \\} , t > 0 , \\end{align*}"}
-{"id": "5688.png", "formula": "\\begin{align*} A = \\begin{pmatrix} A _ { 1 1 } & A _ { 1 2 } \\\\ 0 & A _ { 2 2 } \\end{pmatrix} E = \\begin{pmatrix} E _ { 1 } \\\\ 0 \\end{pmatrix} \\end{align*}"}
-{"id": "8862.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { a _ t } { t } = \\inf _ { t \\geq 1 } \\frac { a _ t + c } { t } . \\end{align*}"}
-{"id": "6665.png", "formula": "\\begin{gather*} z _ { i + 1 } ( u _ k , t ) = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k ( z _ i ( u _ j , t ) - z _ i ( u _ j , t \\wedge \\sigma _ i ) ) \\cdot \\ 1 _ { A _ { k j } ^ i } = \\\\ = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k ( z _ i ( u _ j , t ) - z _ i ( u _ j , t \\wedge \\sigma _ i ) ) \\cdot \\ 1 _ { A _ { k j } ^ i } \\cdot \\ 1 \\{ \\sigma _ i \\leqslant t \\} , t \\geqslant 0 , \\end{gather*}"}
-{"id": "8859.png", "formula": "\\begin{align*} M ( P ) ( y ) = \\sum _ { x \\in { \\cal X } } P ( x ) M ( x , y ) . \\end{align*}"}
-{"id": "9624.png", "formula": "\\begin{align*} \\lim _ { x \\to y _ j } \\ , \\left ( \\psi ( x , t ) - \\zeta _ j ( t ) g _ j \\right ) = \\tilde F _ j ( \\zeta ( t ) ) , \\end{align*}"}
-{"id": "6162.png", "formula": "\\begin{align*} ( \\alpha + s ) m - \\beta v = 1 . \\end{align*}"}
-{"id": "6049.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } \\Delta u _ { i } ^ { \\varepsilon } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { \\varepsilon } \\sum \\limits _ { j \\neq i } \\int _ { B _ { 1 } ( x ) } u _ { j } ^ { \\varepsilon } ( y ) \\ , d y & \\Omega , \\\\ u _ { i } ( x ) = \\phi _ { i } ( x ) & ( \\partial \\Omega ) _ 1 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4908.png", "formula": "\\begin{align*} \\Psi _ { \\kappa , n , N } ( z , \\mathfrak { z } ) : = \\sum _ { M \\in \\Gamma _ 0 ( N ) } \\frac { ( z - \\mathfrak { z } ) ^ n } { ( z - \\overline { \\mathfrak { z } } ) ^ { \\kappa + n } } \\bigg | _ { \\kappa , z } M . \\end{align*}"}
-{"id": "2332.png", "formula": "\\begin{align*} q ( x ) = \\sum ^ { w } _ { i = 0 } q _ i x ^ i = p _ d \\prod ^ w _ { j = 1 } ( x - x _ j ) , ~ ~ ~ p _ d \\ne 0 . \\end{align*}"}
-{"id": "5192.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 1 / 3 } } = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { d \\times d } , W \\in \\mathbb { R } ^ { n \\times d } : X = U V W ^ { T } } \\| U \\| _ { * } \\| V \\| _ { * } \\| W \\| _ { * } \\\\ = & \\min _ { U , V , W : X = U V W ^ { T } } \\ ! \\left ( \\frac { \\| U \\| _ { * } + \\| V \\| _ { * } + \\| W \\| _ { * } } { 3 } \\right ) ^ { 3 } . \\end{align*}"}
-{"id": "7923.png", "formula": "\\begin{align*} ( 1 - 3 m ) ^ { 2 } + ( 3 - 3 m ) ^ { 2 } + \\ldots + ( 3 m - 1 ) ^ { 2 } = \\sum _ { i = 1 } ^ { m } \\left [ ( x _ { i } - ( y _ { i } - h ) ) ^ { 2 } + x _ { i } ^ { 2 } + ( x _ { i } + ( y _ { i } - h ) ) ^ { 2 } \\right ] . \\end{align*}"}
-{"id": "9311.png", "formula": "\\begin{align*} \\int _ D U ' ( y ( T , x , z ) ) d x \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] = \\tilde { p } ( 0 , z ) \\exp ( - \\int _ 0 ^ T \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } d B ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ T ( \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) ^ { 2 } d s ) . \\end{align*}"}
-{"id": "9482.png", "formula": "\\begin{align*} F ( x _ { 1 } , y _ { 1 } , \\dots , x _ { n } , y _ { n } ) = \\begin{cases} G ( p ^ { - 1 } ( x _ { n } , y _ { n } ) ) & ( x _ { n } , y _ { n } ) \\ne 0 \\\\ - 1 & ( x _ { n } , y _ { n } ) = 0 \\end{cases} \\end{align*}"}
-{"id": "2933.png", "formula": "\\begin{align*} M _ { \\pi ' } ( Z ) & = \\sum _ { k \\ge 0 } \\ , s _ { ( k ) } [ s _ { \\pi ' } ] ( Z ) = \\sum _ \\nu m _ { \\pi ' \\nu ' } \\ , s _ { \\nu ' } ( Z ) \\cr & = \\sum _ { k \\ge 0 } \\ , ( \\ , s _ { ( 1 ^ k ) } [ s _ { \\pi } ] ( Z ) \\ , ) ' \\sum _ { k \\ge 0 } \\ , ( - 1 ) ^ { k | \\pi | } ( ( - 1 ) ^ k s _ { ( 1 ^ k ) } [ s _ { \\pi } ( Z ) ] ) ' \\cr & = \\sum _ \\nu ( - 1 ) ^ { | \\nu | } \\ , \\ell _ { \\pi \\nu } \\ , s _ { \\nu ' } ( Z ) \\end{align*}"}
-{"id": "3702.png", "formula": "\\begin{align*} \\frac { d x _ 5 } { d t } = - \\frac { x _ 4 } { \\epsilon } + b x _ 1 x _ 2 ~ . \\end{align*}"}
-{"id": "4694.png", "formula": "\\begin{align*} T _ { i , k } ( x ) = x _ 1 x _ 2 \\dots x _ { i + k } x _ { i + 1 } x _ { i + 2 } \\dots x _ n . \\end{align*}"}
-{"id": "1083.png", "formula": "\\begin{align*} \\tilde \\Delta _ r \\otimes \\mathbb { C } = \\mathbb { C } ^ { 2 m } \\otimes \\Delta _ r . \\end{align*}"}
-{"id": "9838.png", "formula": "\\begin{align*} Q ( f ) = \\left | \\frac { \\Theta ( f ) } { 1 - \\hat \\alpha ( f ) \\Theta ( f ) } \\right | ^ 2 \\left ( \\left | H _ { \\rm S R } ( f ) \\right | ^ 2 P ( f ) + N _ 0 \\right ) . \\end{align*}"}
-{"id": "2144.png", "formula": "\\begin{align*} & V _ { p } ( [ 0 , T ] ; \\Omega ) : = \\Big \\{ u \\in L ^ { 2 p } ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) \\cap L ^ { 2 } ( [ 0 , T ] ; H _ { e } ^ { \\beta } ( \\Omega ) ) \\\\ & g _ { 1 - \\alpha } * ( u - u _ { 0 } ) \\in C ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) , ( g _ { 1 - \\alpha } * ( u - u _ { 0 } ) ) | _ { t = 0 } = 0 \\Big \\} , \\end{align*}"}
-{"id": "3935.png", "formula": "\\begin{align*} \\mathcal { N } = \\left [ \\sum _ { k = 0 } ^ m ( - 1 ) ^ k \\ , 2 ^ { m + 1 - k } \\ , { m \\choose k } \\ , \\sum _ { l = 0 } ^ \\infty \\frac { \\sum _ { j = 0 } ^ l b _ j \\ , b _ { l - j } ^ * } { l + m + k + 1 } \\right ] ^ { - 1 / 2 } \\ , , \\end{align*}"}
-{"id": "364.png", "formula": "\\begin{align*} W _ { h _ c , I } T _ { c , 0 } ( f ) W _ { h _ c , I } ^ * = T _ { c , h _ c } ( f ) = K _ { h , I } T _ { c , h } ( f ) K _ { h , I } ^ * \\end{align*}"}
-{"id": "2479.png", "formula": "\\begin{align*} ( E _ j ) ^ I _ { N _ j } = V _ j ^ I \\otimes \\omega _ { s _ j + d _ j - 1 } \\end{align*}"}
-{"id": "2218.png", "formula": "\\begin{align*} \\mu ( t ) : = \\frac { d } { d t } ( g _ { \\alpha } * \\rho ) ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\frac { d } { d t } \\int _ { 0 } ^ { t } \\frac { \\rho ( s ) } { ( t - s ) ^ { 1 - \\alpha } } d x 0 < t \\leq T . \\end{align*}"}
-{"id": "7045.png", "formula": "\\begin{align*} { \\cal G } = \\left ( { \\cal G } _ X \\mid { \\cal G } _ Y \\right ) = \\left ( \\begin{array} { c c | c c c } I _ { \\kappa } & T _ b & 2 T _ 2 & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & 2 T _ 1 & 2 I _ { \\gamma - \\kappa } & { \\mathbf { 0 } } \\\\ \\hline { \\mathbf { 0 } } & S _ b & S _ q & R & I _ { \\delta } \\end{array} \\right ) , \\end{align*}"}
-{"id": "2109.png", "formula": "\\begin{align*} L _ a U = | y | ^ a \\bigg ( \\frac { U _ a } { r } T ( x , r ) + G ( X ) \\bigg ) B _ 1 \\setminus \\mathcal { P } \\end{align*}"}
-{"id": "5708.png", "formula": "\\begin{align*} f - m _ f ( Q _ 0 ) = g _ 1 + \\sum _ { j \\in J _ 1 } \\alpha _ { j , 1 } \\chi _ { Q ^ 1 _ j } + \\sum _ { j \\in J _ 1 } ( f - m _ f ( Q ^ 1 _ j ) ) \\chi _ { Q ^ 1 _ j } , \\end{align*}"}
-{"id": "1100.png", "formula": "\\begin{align*} \\frac { \\eta _ 1 - \\eta _ l } { c _ 1 - c _ l } = \\frac { \\eta _ 1 - \\eta _ j } { c _ 1 - c _ j } \\in D \\Sigma _ k ^ N \\backslash \\{ 0 \\} , \\ , \\ , j , l \\in [ 2 : p ] , \\ , \\ , j \\neq l \\end{align*}"}
-{"id": "576.png", "formula": "\\begin{align*} \\delta ( p ) & = \\delta ( p \\cdot 1 ) = \\delta ( p [ 1 ] ) = [ 1 ] - p ^ { p - 1 } [ 1 ] \\equiv [ 1 ] \\bmod I ^ { p - 1 } \\equiv 1 \\bmod I ^ { p - 1 } \\end{align*}"}
-{"id": "2702.png", "formula": "\\begin{align*} \\mu _ j : = e ^ { - u _ j } \\theta _ u ^ n + e ^ { - v _ j } \\theta _ v ^ n \\end{align*}"}
-{"id": "4675.png", "formula": "\\begin{align*} \\langle P [ P , a ] [ P , b ] e _ l , e _ l \\rangle _ { L ^ 2 ( S ^ 1 ) } & = \\begin{cases} - \\sum _ { k > l } a _ { k } b _ { - k } , & l \\geq 0 \\\\ 0 , & l < 0 \\end{cases} \\quad \\mbox { a n d } \\\\ \\langle [ P , a ] [ P , b ] e _ l , e _ l \\rangle _ { L ^ 2 ( S ^ 1 ) } & = \\begin{cases} - \\sum _ { k > l } a _ { k } b _ { - k } , & l \\geq 0 \\\\ - \\sum _ { k \\leq l } a _ { k } b _ { - k } , & l < 0 \\end{cases} . \\end{align*}"}
-{"id": "3818.png", "formula": "\\begin{align*} \\psi _ { - 1 / 2 } \\left | \\lambda \\right > & = ( - 1 ) ^ n \\psi _ { \\lambda _ 1 + n - 1 / 2 } ^ \\ast \\psi _ { \\lambda _ 2 + n - 3 / 2 } ^ \\ast \\cdots \\psi _ { \\lambda _ n + 1 / 2 } ^ \\ast \\psi _ { - 1 / 2 } \\left | 0 \\right > \\\\ & = ( - 1 ) ^ n \\psi ^ \\ast ( z _ 1 ) \\psi ^ \\ast ( z _ 2 ) \\cdots \\psi ^ \\ast ( z _ n ) \\left | - 1 \\right > \\ , \\Big \\vert _ { \\boldsymbol { z } ^ { \\lambda } } . \\end{align*}"}
-{"id": "2266.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 3 } & \\frac { \\partial u } { \\partial t } + \\varDelta ^ 2 u = \\varDelta f ( u ) \\ , \\ , \\ , & & \\ , \\ , & & \\R ^ d \\times ( 0 , T ) , \\\\ & u ( \\cdot , 0 ) = u _ 0 & & \\ , \\ , & & \\R ^ d , \\end{alignedat} \\right . \\end{align*}"}
-{"id": "3127.png", "formula": "\\begin{align*} Q ( \\lambda ) : = P ( \\lambda ) - \\lambda ^ d P _ d = \\lambda ^ { d - 1 } P _ { d - 1 } + \\cdots + \\lambda P _ 1 + P _ 0 . \\end{align*}"}
-{"id": "6553.png", "formula": "\\begin{align*} \\frac 1 \\tau \\ , \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\le C \\tau ^ k . \\end{align*}"}
-{"id": "6752.png", "formula": "\\begin{align*} { \\rm I } ( \\varphi _ { t , j } ) - { \\rm I } ( \\varphi _ { 0 , j } ) = \\int _ { 0 } ^ t \\int _ X \\chi \\theta _ { \\varphi _ { s , j } } ^ n d s . \\end{align*}"}
-{"id": "9500.png", "formula": "\\begin{align*} \\frac { \\delta | G | ^ 2 } { \\delta g ^ { \\mu \\nu } } & = \\frac { k } { k ! } \\sum _ { \\mu _ 2 , \\ldots , \\mu _ k , \\nu _ 2 , \\ldots , \\nu _ k } G _ { \\mu \\mu _ 2 \\ldots \\mu _ k } G _ { \\nu \\nu _ 2 \\ldots \\nu _ k } g ^ { \\mu _ 2 \\nu _ 2 } \\cdots g ^ { \\mu _ k \\nu _ k } \\\\ & = \\frac { 1 } { ( k - 1 ) ! } \\sum _ { \\mu _ 2 , \\ldots , \\mu _ k } G _ { \\mu \\mu _ 2 \\ldots \\mu _ k } G _ { \\nu } ^ { \\ , \\ , \\ , \\mu _ 2 \\ldots \\mu _ k } \\\\ & = \\langle i _ { \\partial _ \\mu } G , i _ { \\partial _ \\nu } G \\rangle . \\end{align*}"}
-{"id": "3736.png", "formula": "\\begin{align*} u \\rho ( u ) = \\int _ { u - 1 } ^ u \\rho ( t ) d t \\textrm { f o r a l l r e a l } u . \\end{align*}"}
-{"id": "6223.png", "formula": "\\begin{align*} \\mathbb { K } _ { k } ( A , \\ r _ { 0 } ) = \\{ r _ { 0 } , \\ A r _ { 0 } , \\ A ^ { 2 } r _ { 0 } , \\ \\ldots , \\ A ^ { k - 1 } r _ { 0 } \\} . \\end{align*}"}
-{"id": "7479.png", "formula": "\\begin{align*} b _ s ( d ) = \\underset { { \\Gamma _ s } } { \\int \\cdots \\int } { \\rm e x p } \\left [ - m \\left ( \\bigcup _ { i = 1 } ^ { s } B _ { \\| x _ i \\| } ( x _ i ) \\right ) \\right ] d x _ 1 \\dots d x _ s , \\end{align*}"}
-{"id": "3281.png", "formula": "\\begin{align*} \\hat { R } F ( v _ { 0 } \\otimes v _ { 1 } ) & = [ 2 ] ^ { 1 / 2 } ( q ^ { - 4 } v _ { 1 } \\otimes v _ { - 1 } + \\hat { R } ( v _ { 0 } \\otimes v _ { 0 } ) ) , \\\\ F \\hat { R } ( v _ { 0 } \\otimes v _ { 1 } ) & = [ 2 ] ^ { 1 / 2 } ( v _ { 0 } \\otimes v _ { 0 } + v _ { 1 } \\otimes v _ { - 1 } ) . \\end{align*}"}
-{"id": "4081.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z ^ 2 } { x ^ { 4 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z } { x ^ { 3 } } , \\\\ \\end{gather*}"}
-{"id": "7894.png", "formula": "\\begin{align*} \\begin{pmatrix} - y _ { A B } & x _ { A B } \\\\ - y _ { A C } & x _ { A C } \\end{pmatrix} \\begin{pmatrix} x _ I \\\\ y _ I \\end{pmatrix} = \\begin{pmatrix} B \\times A \\pm r c \\\\ C \\times A \\pm r a \\\\ \\end{pmatrix} \\end{align*}"}
-{"id": "9446.png", "formula": "\\begin{align*} L _ a = & \\ \\{ e _ i \\ | \\ i \\in I , \\phi ( i , 0 ) = a \\} \\\\ R _ a = & \\ \\{ e _ i \\ | \\ i \\in I , \\phi ( i , 1 ) = a \\} \\ ; . \\end{align*}"}
-{"id": "4636.png", "formula": "\\begin{align*} & ( x + y ) ^ 7 + A _ 1 ( x ^ 6 + x ^ 5 y + x ^ 4 y ^ 2 + x ^ 3 y ^ 3 + x ^ 2 y ^ 4 + x y ^ 5 + y ^ 6 ) + A _ 2 ( x ^ 5 + x ^ 4 y + x ^ 3 y ^ 2 + x ^ 2 y ^ 3 + x y ^ 4 + y ^ 5 ) \\\\ & + A _ 3 ( x ^ 4 + x ^ 3 y + x ^ 2 y ^ 2 + x y ^ 3 + y ^ 4 ) + A _ 4 ( x + y ) ^ 3 + A _ 5 ( x ^ 2 + x y + y ^ 2 ) + A _ 6 ( x + y ) + A _ 7 = 0 . \\end{align*}"}
-{"id": "6642.png", "formula": "\\begin{align*} a ( V ) = \\dim V - \\dim V ^ I + \\sum _ { k \\geq 1 } \\frac { 1 } { [ I : I _ k ] } \\cdot \\dim V / V ^ { I _ k } \\end{align*}"}
-{"id": "4864.png", "formula": "\\begin{align*} \\mathfrak { r } ( x , y ) : = \\big ( n h + n ^ { 1 / 3 } \\sigma x , n h + n ^ { 1 / 3 } \\sigma y \\big ) , \\end{align*}"}
-{"id": "2762.png", "formula": "\\begin{align*} F \\diamond \\int _ 0 ^ T g ( t ) d B ^ { H } ( t ) = F \\int _ 0 ^ T g ( t ) d B ^ { H } ( t ) - \\langle D ^ { H } F , g \\rangle _ { T } , \\end{align*}"}
-{"id": "969.png", "formula": "\\begin{align*} \\alpha ( \\overline { L } ) = - ( \\mathcal { L } _ T \\eta ) ( \\overline { L } ) = - \\left ( T \\eta ( \\overline { L } ) - \\eta ( [ T , \\overline { L } ] ) \\right ) = \\eta ( [ T , \\overline { L } ] ) \\ ; \\end{align*}"}
-{"id": "9779.png", "formula": "\\begin{align*} L ^ 2 \\Omega ^ { m , 0 } ( A _ h , E | _ { A _ h } , h | _ { A _ h } ) = L ^ 2 \\Omega ^ { m , 0 } ( A _ h , E | _ { A _ h } , g | _ { A _ h } ) = L ^ 2 \\Omega ^ { m , 0 } ( M , E , g ) . \\end{align*}"}
-{"id": "2616.png", "formula": "\\begin{align*} \\omega _ { p , k } ^ { \\epsilon } : & & c ( 1 , 1 ) = u p ^ { - k } . \\\\ \\omega _ { 2 , k } ^ { \\epsilon } : & & c ( 1 , 1 ) = u 2 ^ { - k - 1 } . \\\\ E _ k : & & c ( \\vec { e } _ i , \\vec { e } _ i ) = 0 , \\ c ( ( 1 , 1 ) , ( 1 , 1 ) ) = c ( \\vec { e } _ 1 , \\vec { e } _ 2 ) = 2 ^ { - k } . \\\\ E _ k : & & c ( \\vec { e } _ i , \\vec { e } _ i ) = 2 ^ { - k } , \\ c ( ( 1 , 1 ) , ( 1 , 1 ) ) = 2 c ( \\vec { e } _ 1 , \\vec { e } _ 1 ) + c ( \\vec { e } _ 1 , \\vec { e } _ 2 ) = 2 ^ { - k } . \\end{align*}"}
-{"id": "9528.png", "formula": "\\begin{align*} P & = \\frac { i } { 2 } e ^ { \\phi } G _ 1 + \\frac { 1 } { 2 } d \\phi \\\\ G ' _ 3 & = i e ^ { \\phi / 2 } ( \\tau H _ 3 - G _ 3 ) \\\\ & = - i e ^ { \\phi / 2 } \\tilde { G _ 3 } - e ^ { - \\phi / 2 } H _ 3 , \\end{align*}"}
-{"id": "7135.png", "formula": "\\begin{align*} x ^ * = ( x ' , - x _ n ) x = ( x ' , x _ n ) . \\end{align*}"}
-{"id": "4751.png", "formula": "\\begin{align*} \\ell _ { N , k } = ( - 1 ) ^ k \\ , \\ell _ { N , 0 } \\prod _ { j = 0 } ^ { k - 1 } ( N + 2 s + j ) ( k \\ge 1 ) . \\end{align*}"}
-{"id": "8525.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { t \\in L ( i ) } r ^ { \\ast } ( v t ) t ^ { - 1 } = r ^ { - 1 } v \\end{align*}"}
-{"id": "8872.png", "formula": "\\begin{align*} ( b , s ) \\star ( a , t ) = ( a + b , t + s + i A ( a ) [ b ] ) , \\end{align*}"}
-{"id": "964.png", "formula": "\\begin{align*} \\overline { \\partial } _ { M } ^ { * } T _ { q } u = ( - 1 ) ^ { q + 1 } T _ { ( q + 1 ) } ( \\overline { \\partial } _ { M } u ) + \\ ; . \\end{align*}"}
-{"id": "4021.png", "formula": "\\begin{align*} E ( x , y ) = \\frac { 1 } { 4 \\pi ^ 2 } \\int _ { [ - \\pi , \\pi ] ^ 2 } \\frac { 2 } { i \\sin { u } - \\sin { v } } e ^ { i ( u x + v y ) } d u d v . \\end{align*}"}
-{"id": "8793.png", "formula": "\\begin{align*} _ 1 ( R _ { 1 , 3 } \\left ( w / x \\right ) R _ { 3 , 2 } \\left ( w x \\right ) \\widetilde { R } _ { 1 , 2 } \\left ( w ^ 2 \\right ) ) = _ 1 ( \\widetilde { R } _ { 1 , 2 } \\left ( w ^ 2 \\right ) R _ { 3 , 2 } \\left ( w x \\right ) R _ { 1 , 3 } \\left ( w / x \\right ) ) . \\end{align*}"}
-{"id": "9653.png", "formula": "\\begin{align*} y ' ( t ) = - ( a - b ) g ( y ( t ) ) , t > 0 ; y ( 0 ) > 0 . \\end{align*}"}
-{"id": "414.png", "formula": "\\begin{align*} \\langle P [ P , a ] [ P , b ] e _ l , e _ l \\rangle _ { L ^ 2 ( S ^ 1 ) } & = - \\langle ( 1 - P ) b _ - e _ l , ( \\overline { a } ) _ - e _ l \\rangle _ { L ^ 2 ( S ^ 1 ) } \\\\ & = - \\left \\langle ( 1 - P ) \\sum _ { k = 1 } ^ \\infty b _ { - k } e _ { l - k } , \\sum _ { k = 1 } ^ \\infty \\overline { a _ { k } } e _ { l - k } \\right \\rangle _ { L ^ 2 ( S ^ 1 ) } = - \\sum _ { k > l } a _ { k } b _ { - k } . \\end{align*}"}
-{"id": "6153.png", "formula": "\\begin{align*} s _ i = \\begin{cases} a _ i m + b & { } \\\\ a _ i m + ( m - b ) & { } . \\end{cases} \\end{align*}"}
-{"id": "4633.png", "formula": "\\begin{align*} ( f \\circ g ) ( w ) = \\bigvee \\limits _ { ( l , k ) \\in A _ w } { \\min \\{ f ( { l } } ) , g ( { k } ) \\} \\ge \\min \\{ f ( u ) , g ( c ) \\} \\end{align*}"}
-{"id": "220.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } | \\mathbb { E } _ f ( \\hat { H } _ n ) - H | = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { \\beta } { d } } } { n ^ { \\frac { \\beta } { d } } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "5825.png", "formula": "\\begin{align*} c = \\prod _ { k = 1 } ^ n x _ { k , k - 1 } = x _ { n , n - 1 } \\cdots x _ { 2 1 } x _ { 1 0 } . \\end{align*}"}
-{"id": "7380.png", "formula": "\\begin{align*} \\Delta _ { 0 , 0 } & = \\Big ( \\frac { \\norm { q ^ 0 } } { \\sqrt { n } } - \\sigma _ 0 \\Big ) Z ' _ 0 , \\\\ \\Delta _ { 1 , 0 } & = \\Big [ \\Big ( \\frac { \\norm { m ^ 0 } } { \\sqrt { n } } - \\tau _ 0 \\Big ) \\mathsf { I } _ N - \\frac { \\norm { m ^ 0 } } { \\sqrt { n } } \\mathsf { P } ^ { \\parallel } _ { q ^ 0 } \\Big ] Z _ 0 \\\\ & + q ^ 0 \\Big ( \\frac { \\norm { q ^ 0 } ^ 2 } { n } \\Big ) ^ { - 1 } \\Big ( \\frac { ( b ^ 0 ) ^ * m ^ 0 } { n } - \\xi _ 0 \\frac { \\norm { q ^ 0 } ^ 2 } { n } \\Big ) , \\end{align*}"}
-{"id": "860.png", "formula": "\\begin{align*} A ( \\delta B ) + ( \\delta A ) B = 0 . \\end{align*}"}
-{"id": "382.png", "formula": "\\begin{align*} E ( z , g ) = \\sum _ { \\gamma \\in \\Gamma / \\Gamma _ \\infty } t ( g \\gamma ) ^ { z + 1 } \\end{align*}"}
-{"id": "8658.png", "formula": "\\begin{align*} \\Lambda e _ j = \\lambda _ j e _ j , \\ ; \\ ; \\lambda _ j > 0 , \\ ; \\ ; j \\ge 1 ; \\ ; \\ ; \\ ; \\sum _ { j \\ge 1 } \\lambda _ j ^ { - 1 } < \\infty . \\end{align*}"}
-{"id": "7783.png", "formula": "\\begin{align*} \\emptyset \\subsetneq X _ 0 \\subsetneq \\cdots \\subsetneq X _ d = X \\end{align*}"}
-{"id": "1358.png", "formula": "\\begin{align*} \\frac { \\partial Z _ { t } } { \\partial \\phi } = Y _ { t - J _ \\phi } + \\sum _ { j \\in J _ \\theta } \\theta _ { j } \\frac { \\partial e _ { t - j } ^ I } { \\partial \\phi } , \\frac { \\partial Z _ { t } } { \\partial \\theta } = e _ { t - J _ \\theta } + \\sum _ { j \\in J _ \\theta } \\theta _ { j } \\frac { \\partial e _ { t - j } ^ I } { \\partial \\theta } . \\end{align*}"}
-{"id": "2568.png", "formula": "\\begin{align*} \\tfrac { 2 } { \\tilde q } + \\tfrac { d } { \\tilde r } = \\tfrac { d } { 2 } - s _ c = \\tfrac { 2 } { p } . \\end{align*}"}
-{"id": "2800.png", "formula": "\\begin{gather*} G ^ 2 ( x , \\xi ) = \\sum _ { k = 1 } ^ \\infty | g _ k ( x , \\xi ) | ^ 2 \\leq C ( 1 + | \\xi | ^ 2 ) , \\\\ \\sum _ { k = 1 } ^ \\infty | g _ k ( x , \\xi ) - g _ k ( y , \\zeta ) | ^ 2 \\leq C \\Big ( | x - y | ^ 2 + | \\xi - \\zeta | r ( | \\xi - \\zeta | ) \\Big ) \\end{gather*}"}
-{"id": "7472.png", "formula": "\\begin{align*} \\| v - u \\| ^ 2 > \\| v \\| ^ 2 - \\| v _ s \\| ( \\| v \\| - \\| v _ s \\| ) = \\| v \\| ^ 2 - \\| v _ s \\| \\| v \\| + \\| v _ s \\| ^ 2 , \\end{align*}"}
-{"id": "9224.png", "formula": "\\begin{align*} \\int _ { D } \\widetilde { Y } ( t , x , z ) A _ { \\hat { u } } ^ { * } \\widehat { p } ( t , x , z ) d x = \\int _ { D } \\widehat { p } ( t , x , z ) A _ { \\hat { u } } \\widetilde { Y } ( t , x , z ) d x . \\end{align*}"}
-{"id": "5512.png", "formula": "\\begin{align*} \\| x \\| _ \\mathcal { M } ^ 0 = \\| x \\| ^ 0 _ { \\mathcal { M } _ { \\varphi , w } } = \\inf _ { k > 0 } \\frac { 1 } { k } ( P ( k x ) + 1 ) < \\infty , \\end{align*}"}
-{"id": "960.png", "formula": "\\begin{align*} ( u , v ) _ { L ^ 2 _ { ( 0 , q ) } ( M ) } = ( \\square _ q G _ q u , v ) _ { L ^ 2 _ { ( 0 , q ) } ( M ) } = ( G _ q u , v ) _ { g r a p h } \\ ; . \\end{align*}"}
-{"id": "4559.png", "formula": "\\begin{align*} G _ { n , x , y } ' ( u , v ) : = \\mathbb { P } ( M _ 1 \\geq j , M _ 2 \\geq l ) . \\end{align*}"}
-{"id": "590.png", "formula": "\\begin{align*} K ( x ) = \\frac { \\beta } { 4 } \\sigma ' _ 2 ( \\frac { x } { 2 } ) \\end{align*}"}
-{"id": "4344.png", "formula": "\\begin{align*} \\frac { d } { d r } \\left ( e ^ r | f ' ( r ) | ^ { p - 2 } f ' ( r ) \\right ) = e ^ r \\left [ | f ' ( r ) | ^ { p - 2 } f ' ( r ) + | f ' ( r ) | ^ { p - 1 } - f ( r ) \\right ] \\ , r > 0 \\ . \\end{align*}"}
-{"id": "5805.png", "formula": "\\begin{align*} f _ { s u b } : = e _ { - \\alpha _ { 2 } } + \\cdots + e _ { - \\alpha _ { n - 1 } } \\end{align*}"}
-{"id": "1128.png", "formula": "\\begin{align*} \\gamma _ k ^ \\mathrm { B } [ \\iota ] = \\frac { M \\sigma _ { \\mathrm { s } k } ^ 2 [ \\iota ] } { \\sum _ { i = 1 } ^ { K } \\beta _ { \\mathrm { s } i } + \\left ( \\rho _ \\mathrm { p } \\sum _ { i = 1 } ^ { K } \\beta _ { \\mathrm { d } i } + 1 \\right ) / \\rho _ \\mathrm { s } } \\end{align*}"}
-{"id": "2148.png", "formula": "\\begin{align*} s _ { \\alpha , m } ( t ) + m ( s _ { \\alpha , m } * g _ { \\alpha } ) ( t ) = 1 , t > 0 , \\ , \\ , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "7722.png", "formula": "\\begin{align*} \\mathbb P [ \\overline \\Omega ] = p > 0 , \\overline \\Omega : = \\{ \\omega \\in \\Omega : \\lim _ { n \\to \\infty } x _ n = 0 \\} . \\end{align*}"}
-{"id": "9400.png", "formula": "\\begin{align*} ( I m \\ \\lambda ) \\| f _ \\lambda \\| ^ 2 = ( \\Gamma _ 0 f _ \\lambda , ( I m \\ W _ \\lambda ) \\Gamma _ 0 f _ \\lambda ) , \\mbox { w h e r e } I m \\ W _ \\lambda = \\frac { W _ \\lambda - W _ \\lambda ^ \\dag } { 2 i } . \\end{align*}"}
-{"id": "6267.png", "formula": "\\begin{align*} 0 & = \\frac { \\nabla _ E w ( x _ 0 ) - \\nabla _ E w ( y _ 0 ) } { \\bar { w } } - \\frac { m } { \\bar { w } } ( \\nabla _ E \\bar { w } ) , \\\\ 0 & \\ge \\frac { \\nabla ^ 2 _ { E , E } w ( x _ 0 ) - \\nabla ^ 2 _ { E , E } w ( y _ 0 ) } { \\bar { w } } - \\frac { m } { \\bar { w } } \\nabla ^ 2 _ { E , E } \\bar { w } . \\end{align*}"}
-{"id": "5759.png", "formula": "\\begin{align*} Q _ { L } ^ { W } = \\hat { \\psi } ^ { \\mathrm { T } } \\hat \\Psi ^ { - 1 } \\hat \\psi \\end{align*}"}
-{"id": "4781.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { t ^ 2 - ( a - c t ) ^ 2 } , a = c o n s t , \\ ; c = c o n s t \\neq 0 , \\ ; c ^ 2 \\neq \\kappa ^ 2 , \\end{align*}"}
-{"id": "3580.png", "formula": "\\begin{align*} L ( f , X ) = \\rho _ g ^ { - 1 } D \\Phi ^ W _ { ( g , \\pi ) } \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) , \\end{align*}"}
-{"id": "1499.png", "formula": "\\begin{align*} N ( x ) \\geq 7 2 - 4 0 = 3 2 > 5 \\times 6 , \\end{align*}"}
-{"id": "9382.png", "formula": "\\begin{align*} \\widetilde { S } _ { m i n } { f } = - \\frac { d ^ 2 f } { d x ^ 2 } , \\mathcal { D } ( \\widetilde { S } _ { m i n } ) = \\left \\{ f \\in { W } _ 2 ^ 2 ( \\mathbb { R } \\backslash \\{ 0 \\} ) : \\begin{array} { c } f _ s ( 0 ) = f _ r ( 0 ) = 0 \\\\ f _ s ' ( 0 ) = ( q , f ) \\end{array} \\right \\} . \\end{align*}"}
-{"id": "4405.png", "formula": "\\begin{align*} \\| x ^ * \\| = \\sup _ { \\| z \\| _ X \\leq 1 } | x ^ * ( z ) | = \\sup _ { z \\in \\mbox { c o } ( B _ X ) } | x ^ * ( z ) | = \\sup _ { \\| z \\| _ { \\widehat { X } } \\leq 1 } | x ^ * ( z ) | . \\end{align*}"}
-{"id": "4336.png", "formula": "\\begin{align*} R ( a ) : = \\inf \\left \\{ r > 0 \\ : \\ f ( r , a ) = 0 \\right \\} \\in ( 0 , \\infty ] \\ . \\end{align*}"}
-{"id": "5491.png", "formula": "\\begin{align*} \\mathcal { R } = \\frac { 1 } { \\mathcal { L } T _ \\mathrm { c } } \\sum _ { \\iota = 1 } ^ { \\mathcal { L } } \\sum _ { k = 1 } ^ { K } \\mathcal { R } _ k [ \\iota ] . \\end{align*}"}
-{"id": "8883.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\norm { \\mathcal { I } ' _ { A _ n } ( u _ n ) } _ { \\left ( H ^ 1 _ { A _ n } ( \\R ^ N , \\mathbb { C } ) \\right ) ^ \\prime } = 0 \\ \\ \\limsup _ { n \\to \\infty } \\mathcal { I } _ { A _ n } ( u _ n ) < \\infty , \\end{align*}"}
-{"id": "7502.png", "formula": "\\begin{align*} x _ { n + 1 } = f ( x _ n ) , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 , \\end{align*}"}
-{"id": "9064.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\sup _ { r \\geq 1 } \\sup _ { N \\geq g ( d , r ) } \\left | \\frac { \\P \\left ( G _ 0 v ( d , \\delta , 0 ) \\right ) } { \\P \\left ( G _ 0 v ( d , 0 , 0 ) \\right ) } - 1 \\right | = \\lim _ { \\delta , | | E | | _ 1 \\to 0 } \\sup _ { r \\geq 1 } \\sup _ { N \\geq g ( d , r ) } \\left | \\frac { \\P \\left ( G _ 0 v ( d , \\delta , E ) \\right ) } { \\P \\left ( G _ 0 v ( d , 0 , 0 ) \\right ) } - 1 \\right | = 0 . \\end{align*}"}
-{"id": "1737.png", "formula": "\\begin{gather*} \\hat E _ X = E _ X + x ( x p - y ) \\partial _ a , \\hat E _ H = E _ H - \\partial _ a , \\hat E _ Y = E _ Y , \\partial _ a , \\end{gather*}"}
-{"id": "6474.png", "formula": "\\begin{align*} - \\int _ { B } \\psi ^ { 2 } \\tilde { u } ^ { - 1 } \\partial _ { t } ( g _ { 1 - \\alpha , m } * \\tilde { u } ) d x + \\mathcal { E } ( \\tilde { u } , - \\psi ^ { 2 } \\tilde { u } ^ { - 1 } ) \\leq R _ { m } ( t ) , \\end{align*}"}
-{"id": "4579.png", "formula": "\\begin{align*} ( q \\circ g ) _ J ( x ) = \\sum _ { i = 1 } ^ { \\mathrm { c a r d } ( J ) } q ^ { ( i ) } \\bigl ( g ( x ) \\bigr ) \\sum _ { \\{ P _ 1 , \\ldots , P _ i \\} \\in \\mathcal { P } _ i ( J ) } g _ { P _ 1 } \\ldots g _ { P _ i } ( x ) , \\end{align*}"}
-{"id": "324.png", "formula": "\\begin{align*} \\left | \\begin{matrix} x _ 1 & y _ 1 & z _ 1 \\\\ x _ 2 & y _ 2 & z _ 2 \\\\ x _ 3 & y _ 3 & z _ 3 \\end{matrix} \\right | = 0 . \\end{align*}"}
-{"id": "845.png", "formula": "\\begin{align*} \\phi ( \\sigma _ { k - 1 } \\cdots \\sigma _ { \\ell } ; \\vec { z } ) u ( r ^ { k _ { r } } , \\ldots , 1 ^ { k _ { 1 } } ) = \\prod _ { \\begin{subarray} { c } i \\in J _ { p } \\\\ i > \\ell \\end{subarray} } \\frac { f ( z _ { i } , z _ { \\ell } ) } { f ( z _ { \\ell } , z _ { i } ) } \\prod _ { i \\in J _ { p + 1 } \\cup \\cdots \\cup J _ { 1 } } ^ { \\curvearrowleft } \\ ! \\ ! \\ ! Y _ { i - 1 } ( z _ { \\ell } , z _ { i } ) \\ , u ( r ^ { k _ { r } } , \\ldots , 1 ^ { k _ { 1 } } ) . \\end{align*}"}
-{"id": "8301.png", "formula": "\\begin{align*} \\ker \\varphi ^ * = \\{ D _ { \\gamma } \\mid \\gamma \\subseteq \\gamma ( \\varphi ) \\} , \\end{align*}"}
-{"id": "3902.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = f ( t , u , u ' ) & & \\\\ l ( u , u ' ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "4821.png", "formula": "\\begin{align*} \\begin{aligned} K _ 0 ( i ( { \\tilde \\tau } ) ) \\circ \\alpha ( { \\tilde \\tau } ) ^ { - 1 } & = K _ 0 ( W ) \\circ K _ 0 ( i ( \\tau ) ) \\circ \\alpha ( \\tau ) ^ { - 1 } \\\\ & = K _ 0 ( i ( \\tau ) ) \\circ \\alpha ( \\tau ) ^ { - 1 } \\end{aligned} \\end{align*}"}
-{"id": "5702.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } f ^ * ( R ) = 0 \\end{align*}"}
-{"id": "475.png", "formula": "\\begin{align*} ( \\beta _ 1 v _ { 1 1 } + \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } + \\beta _ 2 v _ { 2 2 } ) x _ 2 = \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 \\end{align*}"}
-{"id": "6645.png", "formula": "\\begin{align*} \\varepsilon _ K ( V \\otimes \\omega _ s , \\psi ) = \\varepsilon _ K ( V , \\psi ) q ^ { - s [ a ( V ) + n ( \\psi ) \\dim V ] } \\end{align*}"}
-{"id": "6123.png", "formula": "\\begin{align*} L ( i , p ) = \\lim \\rho _ { p } ^ { - 1 } E _ i \\end{align*}"}
-{"id": "2776.png", "formula": "\\begin{align*} \\limsup \\lambda _ i ( \\alpha _ 0 ) = \\lambda _ 0 . \\end{align*}"}
-{"id": "3647.png", "formula": "\\begin{align*} c ^ * = 1 - \\frac { 1 } { \\max \\{ | 0 | , | e ^ { 1 . 5 } | + 0 \\} } = 1 - \\frac { 1 } { e ^ { 1 . 5 } } \\approx 0 . 7 8 . \\end{align*}"}
-{"id": "9965.png", "formula": "\\begin{align*} \\Psi ( t h ) & : = \\int _ { \\mathbb R ^ N } \\frac { 1 } { p ( x ) } ( \\left \\vert \\nabla t h \\right \\vert ^ { p ( x ) } + V ( x ) \\left \\vert t h \\right \\vert ^ { p ( x ) } ) \\ , d x - \\int _ { \\mathbb R ^ N } C _ { 1 } \\left \\vert t h \\right \\vert ^ { p ( x ) } [ \\ln ( e + \\left \\vert t h \\right \\vert ) ] ^ { a ( x ) } d x \\\\ & \\rightarrow - \\infty t \\rightarrow + \\infty . \\end{align*}"}
-{"id": "5026.png", "formula": "\\begin{align*} ( T \\phi ) ( g ) = \\phi ( g \\cdot x _ o ) , \\textrm { f o r $ \\phi \\in C ( X ) $ } . \\end{align*}"}
-{"id": "6750.png", "formula": "\\begin{align*} \\int _ X ( u _ { t } - u _ 0 ) \\theta _ { u _ t } ^ n = \\int _ X ( u + t \\chi - u _ 0 ) \\theta _ { u _ t } ^ n \\geq t \\int _ X \\chi \\theta _ { u _ t } ^ n . \\end{align*}"}
-{"id": "6041.png", "formula": "\\begin{align*} Q [ k - 1 ] & = i _ { k - 1 } - \\frac { k - 1 } { \\alpha } i _ { k - 1 } - \\frac { k } { n } i _ { k } . \\end{align*}"}
-{"id": "1900.png", "formula": "\\begin{align*} B _ { \\kappa } ( u ) ^ { c } : = \\Big \\{ \\exists v \\in [ 0 , u ] : | \\hat { \\phi } ( v ) - \\phi ( v ) | > \\kappa ( \\log n / n ) ^ { 1 / 2 } \\Big \\} . \\end{align*}"}
-{"id": "2942.png", "formula": "\\begin{align*} I = \\int _ { \\| y \\| \\geq 1 } \\frac { \\theta ( y ) } { \\| y \\| ^ { n + 1 } } d y . \\end{align*}"}
-{"id": "4396.png", "formula": "\\begin{align*} H _ \\nu \\left ( \\zeta | \\zeta _ { - n } ^ { - 1 } ( \\tau ) \\right ) = \\sum _ { A \\in \\zeta _ { - n } ^ { - 1 } ( \\tau ) } \\nu ( A ) H _ \\nu ( \\zeta | A ) . \\end{align*}"}
-{"id": "974.png", "formula": "\\begin{align*} \\left [ X _ { \\varepsilon } , \\overline { L _ { k } } \\right ] _ { T } ( p ) = - e ^ { g _ { \\varepsilon } ( p ) } \\left ( \\overline { L _ { k } } g _ { \\varepsilon } ( p ) - \\alpha ( \\overline { L _ { k } } ) ( p ) \\right ) + b _ { \\varepsilon , k } ( p ) \\left [ L _ { k } , \\overline { L _ { k } } \\right ] _ { T } ( p ) \\ ; \\end{align*}"}
-{"id": "6735.png", "formula": "\\begin{align*} \\Psi _ n ( h _ n ) \\quad & = \\quad \\sqrt { \\frac { n } { 2 } } \\left ( h _ n ( \\textbf { X } ^ { ( n ) } ( 1 ) ) - h _ n ( \\textbf { X } ^ { ( n ) } ( 0 ) ) \\right ) \\ , . \\end{align*}"}
-{"id": "6260.png", "formula": "\\begin{align*} \\nabla _ r T ( r , s ) \\big | _ { r = 0 } = J ' ( s ) . \\end{align*}"}
-{"id": "8706.png", "formula": "\\begin{align*} f ^ { k } \\left ( x \\right ) = \\int _ { \\mathbb { R } ^ { k } } \\rho _ { k } \\Big ( y - Q _ { k } x \\Big ) f \\Big ( \\sum _ { i = 1 } ^ { k } y _ { i } g _ { i } \\Big ) d y , \\end{align*}"}
-{"id": "2684.png", "formula": "\\begin{align*} \\int _ X \\chi \\theta _ { \\varphi } ^ n = \\int _ X \\chi e ^ { \\varphi } d \\mu . \\end{align*}"}
-{"id": "9559.png", "formula": "\\begin{align*} \\langle \\psi ^ { ( 0 ) } _ { i } , H ^ { ( 1 ) } \\psi ^ { ( 0 ) } _ { j } \\rangle = 0 ~ ~ \\forall i , j \\end{align*}"}
-{"id": "2395.png", "formula": "\\begin{align*} S _ 1 x & = P _ { D } ^ { \\alpha _ 1 } x = ( 1 - \\alpha _ 1 ) x + \\alpha _ 1 \\Pi _ D = ( 1 - \\alpha _ 1 ) x + \\alpha _ 1 ( N x + d ) . \\end{align*}"}
-{"id": "3594.png", "formula": "\\begin{align*} \\| ( L U ) _ j ^ { \\perp } \\| _ { C ^ { - s _ j , \\alpha } _ { \\phi , \\phi ^ { t _ j + s _ j } \\varphi _ j } ( \\Omega ) } \\le C \\| ( L U ) ^ { \\perp } \\| _ { L ^ 2 ( \\Omega ) } = C \\| ( L U ) ^ { \\perp } \\| _ { L ^ 2 ( \\Omega _ 0 ) } \\le C \\| L U \\| _ { L ^ 2 ( \\Omega _ 0 ) } . \\end{align*}"}
-{"id": "7756.png", "formula": "\\begin{align*} \\mathbb P \\left [ T _ N \\le x \\right ] = \\mathbb P \\left [ T _ N \\ge - x \\right ] . \\end{align*}"}
-{"id": "192.png", "formula": "\\begin{align*} \\frac { d F _ { n , x } ( u ) } { d u } = \\mathrm { B } _ { k , n - k } ( p _ { n , x , u } ) \\frac { \\partial p _ { n , x , u } } { \\partial u } , \\end{align*}"}
-{"id": "5798.png", "formula": "\\begin{align*} Q _ { n } \\cdot G = 0 . \\end{align*}"}
-{"id": "5734.png", "formula": "\\begin{align*} Z _ { t } = \\sum _ { j \\in J _ { \\phi } \\bigcap J _ { \\theta } } \\omega _ { j } Z _ { t - j } + \\sum _ { j \\in J _ { \\phi } \\bigcap J _ { \\theta } ^ C } \\psi _ { j } Z _ { t - j } + \\sum _ { j \\in J _ { \\phi } \\bigcup J _ { \\theta } } \\psi _ j e _ { t - j } \\end{align*}"}
-{"id": "5986.png", "formula": "\\begin{align*} u = x _ 0 \\xleftarrow { \\gamma _ 1 ^ \\lor } \\cdots \\xleftarrow { \\gamma _ { p - 1 } ^ \\lor } x _ { p - 1 } = \\lfloor r _ i x _ { p } \\rfloor \\xleftarrow { z _ { p + 1 } \\gamma _ { p + 1 } ^ \\lor } \\cdots \\xleftarrow { z _ r \\gamma _ r ^ \\lor } \\lfloor r _ i x _ r \\rfloor = \\lfloor r _ i v \\rfloor . \\end{align*}"}
-{"id": "1016.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = \\lambda N _ { f } ( u ) + ( 1 - \\lambda ) Q ( N _ { f } ( u ) ) & & \\\\ u ' ( 0 ) = u ( 0 ) , \\ u ' ( T ) = b u ' ( 0 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "4262.png", "formula": "\\begin{align*} \\{ T , { \\frak I } , T \\} = \\{ T , T , { \\frak I } \\} = 0 . \\end{align*}"}
-{"id": "5187.png", "formula": "\\begin{align*} \\| X \\| _ { * } & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { F } \\| V \\| _ { F } \\\\ & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\frac { \\| U \\| ^ { 2 } _ { F } + \\| V \\| ^ { 2 } _ { F } } { 2 } . \\end{align*}"}
-{"id": "5972.png", "formula": "\\begin{align*} P ( X ) : = X ^ n + a _ 1 X ^ { n - b } + \\cdots + a _ k X ^ { n - b k } , a _ k \\neq 0 . \\end{align*}"}
-{"id": "9962.png", "formula": "\\begin{align*} \\delta _ i : = \\frac { 5 C _ 1 X _ i } { 2 X _ { i + 1 } } + \\frac { 2 C _ 1 } { \\delta _ 0 X _ { i - 1 } X _ i ^ { \\gamma + 1 } } \\end{align*}"}
-{"id": "5721.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ l \\frac { 1 } { w ( Q _ 0 ^ { ( m ) } ) ^ \\frac { 1 } { p } } \\lesssim _ { p , w } \\frac { 1 } { w ( Q _ 0 ) ^ \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "96.png", "formula": "\\begin{align*} | \\langle a ( f _ { k _ m } ) , a ( f _ { k _ n } ) \\rangle | = | \\langle a ^ * a ( f _ { k _ m } ) , f _ { k _ n } \\rangle | < \\frac { \\varepsilon ^ 2 } { 3 \\cdot 2 ^ n } \\end{align*}"}
-{"id": "1736.png", "formula": "\\begin{gather*} E _ X : = - ( x p - y ) ^ 2 \\partial _ p , E _ H : = x \\partial _ x + y \\partial _ y , E _ Y : = \\frac { 1 } { x p - y } ( \\partial _ x + p \\partial _ y ) . \\end{gather*}"}
-{"id": "7592.png", "formula": "\\begin{align*} p _ 2 = p _ 2 ( x _ 0 ) = \\mathbb P \\left \\{ \\omega \\in \\Omega : \\chi ( \\omega ) < - 1 + \\frac { l - F ( x _ 0 ) } { 2 l } \\right \\} , K _ 2 = K _ 2 ( x _ 0 ) : = \\left [ \\frac { 2 ( x _ 0 - v _ l ) } { l - F ( x _ 0 ) } \\right ] + 1 . \\end{align*}"}
-{"id": "2957.png", "formula": "\\begin{align*} & g ( 0 , \\mu _ 1 ) = \\Bigl ( 1 - \\gamma \\frac { \\delta - 1 } { \\delta } \\Bigr ) H _ 2 ( \\mu _ 1 ) , \\\\ & h ( 0 , \\epsilon ) = \\Bigl ( 1 - \\gamma \\frac { \\delta - 1 } { \\delta } \\Bigr ) H _ 2 \\Bigl ( \\frac { 1 - \\epsilon } { 2 } \\Bigr ) . \\end{align*}"}
-{"id": "7628.png", "formula": "\\begin{align*} K \\sqrt { n } \\geq d _ y - \\lambda _ 1 \\mathbf { v } _ y = \\sum _ { z \\sim y } \\left ( 1 - \\mathbf { v } _ z \\right ) \\geq \\sum _ { z \\in C } \\left ( 1 - \\mathbf { v } _ z \\right ) \\geq 2 | C | \\epsilon . \\end{align*}"}
-{"id": "1698.png", "formula": "\\begin{align*} \\xi _ t ^ h & = e + \\frac { 1 } { h } \\Big [ U ( z + h e ) - U ( z ) \\Big ] \\\\ & + \\int _ 0 ^ t \\frac { \\lambda } { h } \\Big [ U \\big ( \\phi _ s ( z + h e ) \\big ) - U \\big ( \\phi _ s ( z ) \\big ) \\Big ] + A \\theta _ t ^ h ( z ) \\ \\dd s \\\\ & + \\frac { 1 } { h } \\int _ 0 ^ t \\Big [ D _ v U \\big ( \\phi _ s ( z + h e ) \\big ) - D _ v U \\big ( \\phi _ s ( z ) \\big ) \\Big ] R \\cdot \\dd W _ s \\ , . \\end{align*}"}
-{"id": "7145.png", "formula": "\\begin{align*} v = 0 \\mbox { o n } \\ , \\ , \\partial \\R ^ n _ + = \\{ x _ n = 0 \\} . \\end{align*}"}
-{"id": "7534.png", "formula": "\\begin{align*} N _ 2 : = \\left [ \\left . \\ln \\left ( \\max \\left \\{ \\frac { K - c } { \\varepsilon } , \\frac { \\mu _ 2 - K } { \\varepsilon } , 1 \\right \\} \\right ) \\right / ( - \\ln \\gamma ) \\right ] + 2 . \\end{align*}"}
-{"id": "1134.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { R } _ k [ \\iota ] & \\leq \\mathcal { R } _ k ^ { } [ \\iota ] \\\\ \\mathcal { R } _ k [ \\iota ] & \\leq \\mathcal { R } _ k ^ { } [ \\iota ] . \\end{aligned} \\end{align*}"}
-{"id": "6315.png", "formula": "\\begin{align*} \\Vert \\Box _ { k } ^ { \\alpha _ { 2 } } f _ { k } ^ { \\alpha _ { 2 } } \\Vert _ { M _ { 2 } } = \\Vert f _ { k } ^ { \\alpha _ { 2 } } \\Vert _ { M _ { 2 } } \\sim \\Vert f _ { k } ^ { \\alpha _ { 2 } } \\Vert _ { L ^ { p _ { 2 } } } \\sim 2 ^ { j n \\alpha _ { 2 } ( 1 - 1 / p _ { 2 } ) } \\end{align*}"}
-{"id": "515.png", "formula": "\\begin{align*} \\mathcal { V } \\big ( \\bar \\lambda , \\bar \\mu \\big ) = \\tau _ { \\lambda ^ { ( 1 ) } } ( \\rho _ { 0 } ^ + ) s _ { \\lambda ^ { ( 1 ) } / \\mu ^ { ( 1 ) } } ( \\rho _ 1 ^ - ) s _ { \\lambda ^ { ( 2 ) } / \\mu ^ { ( 1 ) } } ( \\rho _ 1 ^ + ) \\dots s _ { \\lambda ^ { ( n ) } / \\mu ^ { ( n - 1 ) } } ( \\rho _ { n - 1 } ^ + ) s _ { \\lambda ^ { ( n ) } } ( \\rho _ n ^ - ) , \\end{align*}"}
-{"id": "8567.png", "formula": "\\begin{align*} ( \\rho ^ { k } \\otimes ( ^ { k } \\rho ) ) \\Delta ( \\alpha ) \\Diamond \\rho ( z ) & = ( \\rho ^ { k } \\otimes ( ^ { k } \\rho ) ) ( 1 \\otimes \\alpha - \\alpha \\otimes 1 ) \\Diamond \\rho ( z ) \\\\ & = ( \\bar { e _ { k } } \\otimes \\rho ( \\alpha ) - \\rho ( \\alpha ) \\otimes \\bar { e _ { k } } ) \\Diamond \\rho ( z ) \\\\ & = c y c ( \\rho ( \\alpha ) \\rho ( z ) - \\rho ( z ) \\rho ( \\alpha ) ) \\\\ & = \\rho ( c y c ( \\alpha z - z \\alpha ) ) \\\\ & = \\rho ^ { k } ( \\Delta ( \\alpha ) \\Diamond z ) \\end{align*}"}
-{"id": "786.png", "formula": "\\begin{align*} R ( u _ { a } \\otimes u _ { b } ) = \\left \\{ \\begin{array} { l l } q \\ , u _ { b } \\otimes u _ { a } & ( a > b ) \\\\ u _ { a } \\otimes u _ { a } & ( a = b ) \\\\ ( 1 - q ^ { 2 } ) u _ { a } \\otimes u _ { b } + q \\ , u _ { b } \\otimes u _ { a } & ( a < b ) . \\end{array} \\right . \\end{align*}"}
-{"id": "7010.png", "formula": "\\begin{align*} \\Gamma ( t , p ) = \\left \\{ P \\in \\Pi ( X ) \\mid \\tilde { u } ( t , P ) \\ge \\max _ { Q \\in B _ \\mathcal { R } ( t , p ) } \\tilde { u } ( t , Q ) \\right \\} \\end{align*}"}
-{"id": "2160.png", "formula": "\\begin{align*} & 0 \\leq \\psi \\leq 1 , \\psi = 1 \\rho ' B _ { 1 } , \\psi \\subset \\rho B _ { 1 } , \\\\ & \\qquad | D \\psi | \\leq 2 / ( \\sigma ( \\rho - \\rho ' ) ) . \\end{align*}"}
-{"id": "7878.png", "formula": "\\begin{align*} \\phi _ y ( t , x , s ) : = \\int _ { \\R ^ d } p _ z ( t - s , x - z ) q ( s , z , y ) \\ , d z , x \\in \\R ^ d , \\ , 0 < s < t \\end{align*}"}
-{"id": "3697.png", "formula": "\\begin{align*} \\{ F , H \\} _ { \\mu h } = - \\int F _ { \\mu } \\nabla ^ 2 H _ h d A + \\int F _ h \\nabla ^ 2 H _ { \\mu } d A ~ . \\end{align*}"}
-{"id": "405.png", "formula": "\\begin{align*} & - \\int _ { G ^ + / \\Gamma , \\chi ( g ) \\leq 1 } \\chi ( g ) ^ s \\phi ( g ) \\left \\{ \\sum _ { x \\in L _ 0 } f ( g \\cdot x ) - \\chi ^ { - 1 } ( g ) \\sum _ { x \\in \\hat { L } _ 0 } \\hat { f } ( g ^ { \\iota } x ) \\right \\} d g \\\\ & = - \\int _ 0 ^ 1 t ^ { 1 2 s } \\int _ { G ^ 1 / \\Gamma } \\phi ( g _ 1 ) \\left \\{ \\sum _ { x \\in L _ 0 } f _ { t ^ 3 } ( g _ 1 \\cdot x ) - \\sum _ { x \\in \\hat { L } _ 0 } \\widehat { f _ { t ^ 3 } } ( g _ 1 \\cdot x ) \\right \\} d g _ 1 \\frac { d t } { t } . \\end{align*}"}
-{"id": "4950.png", "formula": "\\begin{align*} \\psi _ r ( \\lambda ) = e ^ { - \\sqrt { | \\lambda | } r } \\psi _ r ( \\lambda ) = \\cos \\left ( \\sqrt { \\lambda } \\ r \\right ) \\end{align*}"}
-{"id": "5223.png", "formula": "\\begin{align*} \\sideset { } { ' } \\sum _ { \\vert K \\vert = q - 1 } \\sum _ { j , k = 1 } ^ n \\dfrac { \\partial ^ 2 h _ A } { \\partial z _ j \\partial \\overline { z _ k } } ( z ) w _ { j K } \\overline { w _ { k K } } \\geq A \\vert w \\vert ^ 2 \\ ; , \\ ; z \\in U _ { A } \\ ; , \\ ; w \\in \\Lambda ^ { 0 , q } _ { z } \\ ; . \\end{align*}"}
-{"id": "5930.png", "formula": "\\begin{gather*} \\int _ { \\R ^ d } \\| D _ v { \\psi } ( \\cdot , v ) \\| _ { H ^ s _ p ( \\R ^ { d } ) } ^ p \\dd v = \\int _ { \\R ^ d } \\dd v \\int _ { \\R ^ d } | D _ v \\ , G _ { \\lambda } h _ s ( x , v ) | ^ p \\ , \\dd x \\\\ \\le \\frac { C } { ( \\lambda ) ^ { p / 2 } } \\| h _ s \\| _ { L ^ p ( \\R ^ { 2 d } ) } ^ p = \\frac { C } { ( \\lambda ) ^ { p / 2 } } \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ d _ x ) ) } ^ p \\end{gather*}"}
-{"id": "6898.png", "formula": "\\begin{align*} \\mathsf { P } _ { \\mathrm { S I R } , g _ { 2 } } \\left ( \\lambda _ { \\mathrm { B S } } \\right ) = & \\mathbb { P } \\left ( \\frac { P _ { \\mathrm { B S } } H _ { \\mathrm { U } _ { 0 } , \\mathrm { B S } _ { 0 } } g _ { 2 } \\left ( d _ { 0 } \\right ) } { \\underset { \\mathrm { B S } _ { i } \\in \\Pi _ { \\mathrm { B S } } ^ { \\dagger } } { \\sum } P _ { \\mathrm { B S } } H _ { \\mathrm { U } _ { 0 } , \\mathrm { B S } _ { i } } g _ { 2 } \\left ( d _ { i } \\right ) } > \\tau \\right ) . \\end{align*}"}
-{"id": "9131.png", "formula": "\\begin{align*} u ( s ) & = e ^ { - ( t - s ) A ( s ) } u ( t ) + \\int _ { s } ^ { t } e ^ { - ( r - s ) A ( s ) } ( \\mathcal { A } ( s ) + B ( r ) A ( r ) ) u ( r ) d r \\\\ & - \\int _ { s } ^ { t } e ^ { - ( r - s ) A ( s ) } [ f ( r ) - P ( r ) u ( r ) ] d r . \\end{align*}"}
-{"id": "232.png", "formula": "\\begin{align*} \\Sigma : = \\begin{pmatrix} 1 & \\alpha _ z \\\\ \\alpha _ z & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "2234.png", "formula": "\\begin{align*} N = N ( r , s ) ^ { 1 + o ( 1 ) } \\cdot \\log n \\end{align*}"}
-{"id": "849.png", "formula": "\\begin{align*} \\tilde { A } _ { a } ( z ) = q ^ { 2 \\sum _ { p = a } ^ { r } N _ { p , 1 } } A ^ { [ 2 , M ] } ( z ) . \\end{align*}"}
-{"id": "1634.png", "formula": "\\begin{align*} X = 2 \\psi _ { I } \\int T ^ { I } \\left ( t \\right ) \\partial _ { t } + T ^ { I } Y _ { I } ^ { \\alpha } \\partial _ { \\alpha } + \\left ( a \\left ( x ^ { \\beta } , t \\right ) \\right ) u \\partial _ { u } , \\end{align*}"}
-{"id": "1524.png", "formula": "\\begin{align*} & M > 0 , M ( v ) = M ( - v ) \\mbox { f o r a l l } v \\in \\R ^ d , \\int _ { \\R ^ d } M ( v ) \\d v = 1 , \\\\ & | v | ^ { d + \\alpha } M ( v ) \\longrightarrow \\gamma > 0 \\ , , \\qquad \\mbox { a s } | v | \\rightarrow \\infty , \\mbox { w h e r e } 1 \\leq \\alpha < 2 , \\end{align*}"}
-{"id": "9792.png", "formula": "\\begin{align*} \\Pi _ { n } ( \\boldsymbol { \\theta } | \\mathbf { y } ) = \\frac { \\pi ( \\boldsymbol { \\theta } ) f ( \\mathbf { y } | \\boldsymbol { \\theta } ) } { \\int _ { \\boldsymbol { \\Omega } } \\pi ( \\boldsymbol { \\theta } ) f ( \\mathbf { y } | \\boldsymbol { \\theta } ) d \\boldsymbol { \\theta } } = \\frac { \\pi ( \\boldsymbol { \\theta } ) f ( \\mathbf { y } | \\boldsymbol { \\theta } ) } { m ( \\mathbf { y } ) } \\end{align*}"}
-{"id": "5608.png", "formula": "\\begin{align*} S _ { \\ell } ( \\alpha ) = \\sum _ { n = 1 } ^ { N } \\Lambda ( n ) e ( n ^ { \\ell } \\alpha ) . \\end{align*}"}
-{"id": "8588.png", "formula": "\\begin{align*} \\widetilde { \\psi } ( d _ { 1 } ( ^ { \\ast } b ) d _ { 2 } \\cdot n ) & = \\widetilde { \\psi } ( d _ { 1 } \\overline { N } ( ^ { \\ast } b ) ( d _ { 2 } n ) ) \\\\ & = \\widetilde { \\psi } \\begin{pmatrix} - d _ { 1 } \\pi _ { 1 } p \\xi _ { b e _ { k } } ( d _ { 2 } n ) \\\\ - d _ { 1 } \\gamma ' \\xi _ { b e _ { k } } ( d _ { 2 } n ) \\\\ 0 \\\\ 0 \\end{pmatrix} \\end{align*}"}
-{"id": "7013.png", "formula": "\\begin{align*} s ( \\varphi , I _ \\Gamma ( t , \\pi ) ) = \\sup _ { P \\in \\Gamma ( t , \\pi ) } \\left \\langle \\varphi , \\int _ X \\imath _ X ( x ) d P \\right \\rangle = \\sup _ { P \\in \\Gamma ( t , \\pi ) } \\int _ X \\langle \\varphi , \\imath _ X ( x ) \\rangle d P \\end{align*}"}
-{"id": "6937.png", "formula": "\\begin{align*} s _ { ( i _ 1 ) } \\ , s _ { ( i _ 2 ) } \\ , \\cdots \\ , s _ { ( i _ m ) } = \\sum _ { \\pi } \\ , c ^ \\pi _ { ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) } \\ s _ \\pi \\ , , \\end{align*}"}
-{"id": "7074.png", "formula": "\\begin{align*} \\rho = \\sum _ { r \\in \\mathcal { P } ( \\rho ) } \\delta _ r . \\end{align*}"}
-{"id": "8226.png", "formula": "\\begin{align*} \\Delta : = \\Lambda _ E ( \\Phi + \\phi ) - \\Lambda _ E ( \\Phi ) . \\end{align*}"}
-{"id": "7940.png", "formula": "\\begin{align*} \\alpha ( w _ i , w _ j ) & = \\frac { 1 } { m ^ 2 } \\sum _ { \\ell = 1 } ^ m \\sum _ { k = 1 } ^ m \\alpha _ H ( v _ { i , \\ell } , v _ { j , k } ) \\\\ & = \\frac { 1 } { m ^ 2 } \\sum _ { \\ell = 1 } ^ m \\sum _ { k = 1 } ^ m \\left ( 1 - \\alpha _ H ( v _ { n + 1 - i , m + 1 - \\ell } , v _ { n + 1 - j , m + 1 - k } ) \\right ) \\\\ & = 1 - \\frac { 1 } { m ^ 2 } \\sum _ { \\ell = 1 } ^ m \\sum _ { k = 1 } ^ m \\alpha _ H ( v _ { n + 1 - i , m + 1 - \\ell } , v _ { n + 1 - j , m + 1 - k } ) \\\\ & = 1 - \\alpha ( w _ { n + 1 - i } , w _ { n + 1 - j } ) . \\end{align*}"}
-{"id": "2561.png", "formula": "\\begin{align*} J _ { \\leq N } ^ s ( t ) : = e ^ { i t \\Delta } \\varphi ( \\tfrac { x } { N } ) | x | ^ s e ^ { - i t \\Delta } = M ( t ) P _ { \\leq \\frac { N } { | t | } } ( - 4 t ^ 2 \\Delta ) ^ { \\frac { s } { 2 } } M ( - t ) . \\end{align*}"}
-{"id": "5967.png", "formula": "\\begin{align*} \\Delta _ { i , X } ^ + = \\partial _ { i + 1 , X } \\delta _ { i , X } , \\qquad \\Delta _ { i , X } ^ - = \\delta _ { i - 1 , X } \\partial _ { i , X } , \\qquad \\Delta _ { i , X } = \\Delta _ { i , X } ^ + + \\Delta _ { i , X } ^ - \\ , , \\end{align*}"}
-{"id": "2168.png", "formula": "\\begin{align*} = \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } \\left ( \\left ( \\frac { \\tilde { u } ( s , x ) } { \\psi ( x ) } \\right ) ^ { 1 - q } - \\left ( \\frac { \\tilde { u } ( s , y ) } { \\psi ( y ) } \\right ) ^ { 1 - q } \\right ) k ( x , y ) d x d y . \\end{align*}"}
-{"id": "3505.png", "formula": "\\begin{align*} D \\Phi | _ { ( g , \\pi ) } ( h , w ) & = \\Big ( L _ g h - 2 h _ { i j } \\pi _ \\ell ^ i \\pi ^ { j \\ell } - 2 \\pi ^ j _ k w ^ k _ j + \\tfrac { 2 } { n - 1 } \\mbox { t r } _ g \\pi ( h _ { i j } \\pi ^ { i j } + \\mbox { t r } _ g w ) , \\\\ & ( \\mbox { d i v } _ g w ) ^ i - \\tfrac { 1 } { 2 } \\pi ^ { j k } h _ { j k ; \\ell } g ^ { \\ell i } + \\pi ^ { j k } h ^ i _ { j ; k } + \\tfrac { 1 } { 2 } \\pi ^ { i j } ( \\mbox { t r } _ g h ) _ { , j } \\Big ) . \\end{align*}"}
-{"id": "8109.png", "formula": "\\begin{align*} C _ { k , l , N } ^ { r a d } = { \\mathcal R } _ { k , l , N } ( B _ 1 ) . \\end{align*}"}
-{"id": "4951.png", "formula": "\\begin{align*} 0 \\le \\psi _ r ( \\lambda ) \\leq \\psi _ r \\left ( r ^ { - 2 } \\left ( y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) ^ 2 \\right ) = - \\frac { \\pi } { \\Gamma ( \\frac { N - 2 } { 2 } ) } \\left ( { \\textstyle \\frac 1 2 } y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) ^ { \\frac { N - 2 } { 2 } } Y _ { \\frac { N - 2 } { 2 } } \\left ( y _ { \\frac { N - 4 } { 2 } } ^ { ( 1 ) } \\right ) = \\gamma _ N \\end{align*}"}
-{"id": "7714.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\{ f ( x _ n ) + \\sigma _ n \\xi _ { n + 1 } , \\ , 0 \\} , x _ 0 > 0 , n = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "2271.png", "formula": "\\begin{align*} u ' ( t ) + A u ( t ) = f ( t ) , 0 < t < T , u ( 0 ) = 0 , \\end{align*}"}
-{"id": "718.png", "formula": "\\begin{align*} a _ { n } = \\frac { h _ { H } ( f ^ { n } ( P ) ) } { R ^ { n } } \\ \\ \\ . \\end{align*}"}
-{"id": "3969.png", "formula": "\\begin{align*} a ( t ) w _ i = e ^ { ( ( r - 2 i ) ( m + n ) / 2 + \\delta ) t } w _ i , \\quad 1 \\leq i \\leq r . \\end{align*}"}
-{"id": "8407.png", "formula": "\\begin{align*} x ^ { \\beta } = \\sum _ j m _ j ^ * x \\alpha _ { h ^ { - 1 } } ( m _ j ) . \\end{align*}"}
-{"id": "1654.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ d _ x ) ) } = \\Big ( \\int _ { \\R ^ d } \\| f ( \\cdot , v ) \\| _ { H ^ { s } _ p } ^ p \\dd v \\Big ) ^ { 1 / p } . \\end{align*}"}
-{"id": "6038.png", "formula": "\\begin{align*} \\int _ { \\R ^ N _ + } \\frac { u ( y ) } { | y | ^ { N + 2 s } } d y = - f ( 0 ) . \\end{align*}"}
-{"id": "7276.png", "formula": "\\begin{align*} ( z _ j , w _ j ) \\mapsto \\left [ e _ { j , 0 } ^ * ( z _ j ) + \\sum _ { \\lambda = 1 } ^ r w _ j ^ \\lambda \\cdot e _ { j , \\lambda } ^ * ( z _ j ) \\right ] \\end{align*}"}
-{"id": "6568.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( H ^ 4 ( \\R ^ d ) ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( H ^ 4 ( \\R ^ d ) ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "6149.png", "formula": "\\begin{align*} C _ i = \\{ ( t : b _ i ) \\mid t \\in T \\} \\backslash \\{ 0 \\} . \\end{align*}"}
-{"id": "9048.png", "formula": "\\begin{align*} u _ k ^ { ( N ) } : = \\left \\{ \\begin{array} { l l } \\upsilon - ( k - r ) ^ { \\alpha _ - } , & k \\leq \\lfloor N / 2 \\rfloor , \\\\ \\upsilon - ( N - k ) ^ { \\alpha _ - } - \\frac { 3 } { 4 } \\log N , & \\lfloor N / 2 \\rfloor < k \\leq N , \\end{array} \\right . \\end{align*}"}
-{"id": "9189.png", "formula": "\\begin{align*} d { \\langle } M _ { t , i j } , M _ { t , k l } { \\rangle } = \\left ( ( X _ t ) _ { i k } \\delta _ { j l } + ( X _ t ) _ { i l } \\delta _ { j k } + ( X _ t ) _ { j k } \\delta _ { i l } + ( X _ t ) _ { j l } \\delta _ { i k } \\right ) d t . \\end{align*}"}
-{"id": "8703.png", "formula": "\\begin{align*} \\ < \\widetilde Y _ { } ^ { t , x } , W ^ { \\xi } \\ > _ { \\tau } = \\int _ 0 ^ \\tau \\langle \\xi , ( Z ^ { t , x } _ { s } ) ^ * e ^ { - s A ^ * } h \\rangle _ U \\ , d s = \\int _ 0 ^ \\tau \\langle e ^ { - s { A } } Z ^ { t , x } _ { s } \\xi , h \\rangle _ { { K } } d s , \\ ; \\ ; \\tau \\in [ 0 , T ] , \\ ; \\P \\end{align*}"}
-{"id": "5197.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\| U ^ { * } \\| _ { S _ { p _ { 1 } } } \\| V ^ { * } \\| _ { S _ { p _ { 2 } } } = \\| U ^ { * } _ { 1 } \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| V ^ { * } _ { 1 } \\| _ { S _ { \\widehat { p } _ { 1 } } } . \\end{align*}"}
-{"id": "2040.png", "formula": "\\begin{align*} \\dd _ t \\nabla \\omega = \\nabla \\dd _ t \\omega = \\nabla \\Delta \\omega + \\nabla \\| \\nabla \\omega \\| ^ 2 . \\end{align*}"}
-{"id": "728.png", "formula": "\\begin{align*} & g ^ { * } D _ { i } \\equiv \\sum _ { m = 1 } ^ { s } a _ { m i } F _ { m } \\ \\ ( i = 1 , \\dots r ) \\\\ & { p } _ { * } F _ { j } \\equiv \\sum _ { l = 1 } ^ { r } b _ { l j } D _ { l } \\ \\ ( j = 1 , \\dots , s ) \\end{align*}"}
-{"id": "818.png", "formula": "\\begin{align*} \\psi _ { \\vec { z } } ^ { 1 ^ { k } } ( \\vec { x } ) = ( 1 - q ^ { 2 } ) ^ { - k } \\prod _ { i = 1 } ^ { k } \\frac { z _ { i } ^ { M ' - 1 } } { ( 1 + z _ { i } ) ^ { M } } \\langle \\prod _ { 1 \\le i \\le k } C ^ { [ M ' , M ] } ( z _ { i } ) \\prod _ { 1 \\le i \\le k } \\beta _ { x _ { i } } ^ { * } \\rangle _ { [ M ' , M ] } \\ , ( u _ { 1 } \\otimes \\cdots \\otimes u _ { 1 } ) . \\end{align*}"}
-{"id": "6189.png", "formula": "\\begin{align*} \\int _ { \\R } [ f ( x + x z ) - f ( x ) - f ' ( x ) x z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) = \\int _ { \\R } [ \\log | 1 + z | - z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) + o ( 1 ) . \\end{align*}"}
-{"id": "9600.png", "formula": "\\begin{align*} \\lambda _ { y } ( 0 ) = \\lim _ { t \\to 0 } \\lambda _ { y } ( t ) = - \\frac { m \\xi _ { 0 } - \\dot \\xi _ { 0 } } { 4 \\pi } . \\end{align*}"}
-{"id": "4572.png", "formula": "\\begin{align*} D _ g : = \\max \\biggl \\{ 1 , \\sup _ { \\delta \\in ( 0 , \\| f \\| _ \\infty ) } \\frac { \\sup _ { x : f ( x ) \\geq \\delta } M _ { g } ^ * ( x ) } { a ( \\delta ) ^ { m + 1 } } \\biggr \\} . \\end{align*}"}
-{"id": "3444.png", "formula": "\\begin{align*} d Y ^ i ( t ) = - \\tilde G ^ i ( t , \\kappa ( t ) , Y ^ i ( t ) , Z ^ i ( t ) ) d t + \\langle Z ^ i ( t ) , d W ( t ) \\rangle , Y ^ i ( T ) = \\Upsilon ^ i = \\langle \\eta ( T ) , u ^ i _ 0 ( \\xi ( T ) ) \\rangle . \\end{align*}"}
-{"id": "9348.png", "formula": "\\begin{align*} & u ( t , x , z ) = \\frac { 1 } { \\Gamma ( t , x ) } \\mathbb { E } _ Q [ u ( T , x , z ) \\Gamma ( T , x ) | \\mathcal { F } ^ { G } _ t ] \\end{align*}"}
-{"id": "8356.png", "formula": "\\begin{align*} \\sigma ^ { \\tilde { \\phi } } _ t ( g ) = g u _ { g , t } = \\alpha _ g ( u _ { g , t } ) g , \\ \\ \\ t \\in \\mathbb { R } , \\ \\ \\ g \\in G . \\end{align*}"}
-{"id": "157.png", "formula": "\\begin{align*} \\Phi _ X = \\Phi _ { G _ X } \\cong { \\partial C _ 1 ( G _ X , \\Z ) \\over \\partial \\delta C _ 0 ( G _ X , \\Z ) } , \\end{align*}"}
-{"id": "9789.png", "formula": "\\begin{align*} B F _ { k l } = \\frac { m ( \\mathbf { y } | M _ { k } ) } { m ( \\mathbf { y } | M _ { l } ) } = \\frac { \\lambda _ { n , k } \\int \\prod _ { i = 1 } ^ { n } p ( y _ { i } ) \\Pi _ { n , k } ( p ) } { \\lambda _ { n , l } \\int \\prod _ { i = 1 } ^ { n } p ( y _ { i } ) \\Pi _ { n , l } ( p ) } . \\end{align*}"}
-{"id": "9383.png", "formula": "\\begin{align*} u _ { \\lambda } ( x ) = - ( G \\ast { q } ) ( x ) - 2 i k [ 1 + ( G \\ast { q } ) ( 0 ) ] G ( x ) ; \\end{align*}"}
-{"id": "2993.png", "formula": "\\begin{align*} \\sum _ { i + j = n } { - \\lambda _ i ( x , \\lambda _ j ( y , z ) ) + \\lambda _ i ( y , \\lambda _ j ( x , z ) ) + \\lambda _ i ( \\lambda _ j ( x , y ) , z ) } = 0 , \\end{align*}"}
-{"id": "130.png", "formula": "\\begin{align*} \\| x ^ * \\| = \\sup _ { \\| x \\| _ X \\leq 1 } | x ^ * ( x ) | = \\sup _ { \\| x \\| ^ * \\leq 1 } | x ^ * ( x ) | = \\sup _ { \\| x \\| _ { \\widehat { X } } \\leq 1 } | x ^ * ( x ) | . \\end{align*}"}
-{"id": "2428.png", "formula": "\\begin{align*} \\| \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ p } = \\| \\sum _ { k \\in \\Gamma _ l ^ { \\alpha _ 2 , \\alpha _ 1 } } \\Box _ k ^ { \\alpha _ 2 } \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ p } . \\end{align*}"}
-{"id": "9011.png", "formula": "\\begin{align*} C _ { i 1 } \\subsetneq C _ { i 2 } \\subsetneq \\cdots \\subsetneq C _ { i s _ i } = M _ i . \\end{align*}"}
-{"id": "1751.png", "formula": "\\begin{align*} \\varepsilon _ i ( \\eta ) = m ^ \\eta _ i \\ \\mbox { a n d } \\ \\varphi _ i ( \\eta ) = H _ i ^ \\eta ( 1 ) - m ^ \\eta _ i . \\end{align*}"}
-{"id": "8815.png", "formula": "\\begin{align*} & \\sum _ { i = 0 } ^ { \\infty } \\binom { m } { i } ( a _ { l + i } b ) _ { m + n - i } \\\\ & = \\sum _ { i = 0 } ^ { \\infty } \\binom { l } { i } ( - 1 ) ^ i \\big ( a _ { l + m - i } b _ { n + i } + ( - 1 ) ^ { l + 1 } b _ { l + n - i } a _ { m + i } \\big ) . \\end{align*}"}
-{"id": "1324.png", "formula": "\\begin{align*} \\delta = \\epsilon \\ , \\min \\left \\{ \\alpha , \\frac { \\alpha ^ \\ast - \\alpha } { 1 + \\epsilon } \\right \\} \\end{align*}"}
-{"id": "8660.png", "formula": "\\begin{align*} R _ { \\tau - t } \\left [ \\Phi \\right ] \\left ( x \\right ) = \\mathbb { E } \\Phi \\left ( X _ \\tau ^ { t , x } \\right ) , \\ ; \\ ; \\Phi \\in B _ b ( H , J ) , \\end{align*}"}
-{"id": "7017.png", "formula": "\\begin{align*} l ^ { x } ( u ) = l ^ { x } ( u ) \\ 1 _ { [ \\min _ { [ 0 ; 1 ] } x ; \\max _ { [ 0 ; 1 ] } x ] } ( u ) . \\end{align*}"}
-{"id": "1508.png", "formula": "\\begin{align*} \\frac { x _ 2 } { x _ 1 } \\left ( 1 - \\left ( \\frac { 1 } { x _ 2 } - 1 \\right ) e _ 0 \\right ) = \\frac 1 { x _ 1 } ( 1 + ( x _ 2 - 1 ) ( e _ 0 + 1 ) ) \\end{align*}"}
-{"id": "3096.png", "formula": "\\begin{align*} L _ k ( \\lambda ) : = \\begin{bmatrix} - 1 & \\lambda \\\\ & - 1 & \\lambda \\\\ & & \\ddots & \\ddots \\\\ & & & - 1 & \\lambda \\\\ \\end{bmatrix} \\in \\mathbb { F } [ \\lambda ] ^ { k \\times ( k + 1 ) } , \\end{align*}"}
-{"id": "34.png", "formula": "\\begin{align*} \\sigma _ j \\wedge p ^ \\star ( d t ) = \\sum _ { p \\geq 0 } a _ { j p } ( z ^ \\prime ) z _ { n + 1 } ^ p \\end{align*}"}
-{"id": "8595.png", "formula": "\\begin{align*} \\mu _ { k } ^ { 2 } M & = M \\oplus M e _ { k } M \\oplus M ^ { \\ast } e _ { k } ( ^ { \\ast } M ) \\\\ \\mu _ { k } ^ { 2 } P & = \\rho ( P ) + \\displaystyle \\sum _ { b t , s a } \\left ( [ b t s a ] [ ( s a ) ^ { \\ast } ( ^ { \\ast } b t ) ] + [ ( s a ) ^ { \\ast } ( ^ { \\ast } ( b t ) ) ] ( b t ) ( s a ) \\right ) \\end{align*}"}
-{"id": "4545.png", "formula": "\\begin{align*} F _ { n , x , y } ^ { ( 1 ) } ( u , v ) : = \\mathbb { P } ( N _ 1 + N _ 3 \\geq k , N _ 2 + N _ 3 \\geq k ) , \\end{align*}"}
-{"id": "7678.png", "formula": "\\begin{align*} \\| u ( t ) \\| _ { L ^ { q } } ^ { q } = \\int _ { \\mathbb { R } ^ { d } } | u ( t , x ) | ^ { q } d x \\geq \\nu ( d , \\alpha ) ^ { q } t ^ { q } \\omega _ { d } n ^ { \\alpha q } \\rightarrow \\infty \\end{align*}"}
-{"id": "7624.png", "formula": "\\begin{align*} 0 \\leq \\sum _ { y \\sim x } \\left ( d _ y - \\lambda _ 1 \\mathbf { v } _ y \\right ) = O \\left ( n ^ { 3 / 2 } \\right ) . \\end{align*}"}
-{"id": "1361.png", "formula": "\\begin{align*} \\hat { F } _ n ^ \\mathrm { T } = \\sum _ { t = 1 } ^ { n } \\sigma ^ 2 _ t ( \\hat { \\delta } _ { 0 } ) \\begin{bmatrix} x _ { t } ( m \\hat { \\pi } ) _ { t - J _ \\theta } ^ { \\mathrm { T } } & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "313.png", "formula": "\\begin{align*} \\mathcal { P } _ 2 ( J ) = \\Bigl \\{ \\bigl \\{ \\{ 1 , 1 \\} , \\{ 2 \\} \\bigr \\} , \\bigl \\{ \\{ 1 , 2 \\} , \\{ 1 \\} \\bigr \\} , \\bigl \\{ \\{ 1 , 2 \\} , \\{ 1 \\} \\bigr \\} \\Bigr \\} . \\end{align*}"}
-{"id": "5144.png", "formula": "\\begin{align*} g ( z _ { 1 } , w ) \\prod _ { i = 2 } ^ { m + 1 } f ( z _ { i } , w ) + \\sum _ { \\ell = 2 } ^ { m + 1 } g ( z _ { \\ell } , w ) g ( z _ { 1 } , z _ { \\ell } ) \\prod _ { \\begin{subarray} { c } i = 2 \\\\ i \\not = \\ell \\end{subarray} } ^ { m + 1 } f ( z _ { i } , z _ { \\ell } ) = g ( z _ { 1 } , w ) \\prod _ { i = 2 } ^ { m + 1 } f ( z _ { i } , z _ { 1 } ) . \\end{align*}"}
-{"id": "6877.png", "formula": "\\begin{align*} F ( \\tilde u ) = F ( \\tilde u _ { \\leq A T ^ { \\frac 1 2 } t ^ { - 1 } } ) + [ F ( \\tilde u ) - F ( \\tilde u _ { \\leq A T ^ { \\frac 1 2 } t ^ { - 1 } } ) ] . \\end{align*}"}
-{"id": "6784.png", "formula": "\\begin{align*} d ( s ) = \\frac { 1 } { s } \\left ( \\int _ X \\dot { \\varphi } _ { t + s } d d ^ c ( \\varphi _ { t + s } - \\varphi _ t ) \\wedge T _ s + \\int _ X ( \\dot { \\varphi } _ { t + s } - \\dot { \\varphi } _ t ) \\theta _ { \\varphi _ t } ^ n \\right ) . \\end{align*}"}
-{"id": "7093.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { | u | = n } \\nu ( B ( u ) ) = 0 \\end{align*}"}
-{"id": "3390.png", "formula": "\\begin{align*} g \\ , = \\ , u ^ { \\frac 2 { n - 2 } } g _ { \\R ^ n } . \\end{align*}"}
-{"id": "3707.png", "formula": "\\begin{align*} H _ { 1 2 3 } = \\frac { 1 } { 2 } ( x ^ 2 _ 1 + 2 x ^ 2 _ 2 + x ^ 2 _ 3 ) ~ , \\end{align*}"}
-{"id": "9264.png", "formula": "\\begin{align*} U ( x , y , z ) = U ( x , y , z , \\omega ) : D \\times ( 0 , \\infty ) \\times \\mathbb { R } \\times \\Omega \\rightarrow \\mathbb { R } \\end{align*}"}
-{"id": "614.png", "formula": "\\begin{align*} F _ { \\kappa , j , r } ( \\mathfrak { z } , z ) : = \\mathfrak { z } _ 2 ^ { - j } \\sum _ { m = 0 } ^ { \\infty } \\ ; \\sideset { } { ^ * } \\sum _ { \\mathfrak { b } \\subseteq \\mathcal { O } _ { \\Q ( \\mathfrak { z } ) } } \\frac { C _ { \\kappa } \\left ( \\mathfrak { b } , m \\right ) } { N ( \\mathfrak { b } ) ^ { \\frac { \\kappa } { 2 } - j } } ( 4 \\pi m ) ^ r e ^ { \\frac { 2 \\pi m \\mathfrak { z } _ 2 } { N ( \\mathfrak { b } ) } } e ^ { 2 \\pi i m z } , \\end{align*}"}
-{"id": "825.png", "formula": "\\begin{align*} \\tilde { L } ^ { ( i ) } ( u ; s ) = \\frac { 1 } { 1 - s u } \\begin{pmatrix} 1 & 0 \\\\ 0 & ( - s u ) ^ { - 1 } \\end{pmatrix} L ^ { ( i ) } ( - s u ; s ) \\begin{pmatrix} 1 & 0 \\\\ 0 & u \\end{pmatrix} , \\end{align*}"}
-{"id": "5204.png", "formula": "\\begin{align*} ( \\zeta _ 1 , \\dots , \\zeta _ n ) \\in T ^ { \\mathbb { C } } _ p M \\Leftrightarrow \\sum _ { j = 1 } ^ n \\dfrac { \\partial \\rho _ i } { \\partial z _ j } ( p ) \\zeta _ j = 0 , 1 \\leq i \\leq k . \\end{align*}"}
-{"id": "6916.png", "formula": "\\begin{align*} \\frac { \\prod _ { i = 1 } ^ r \\sum _ { j \\in V ( S _ i ) } ( x _ j - s _ j ) } { ( n - 2 ) _ { n - r } } \\prod _ { j = 1 } ^ n \\ , ( x _ j - 1 ) _ { s _ j - 1 } , \\end{align*}"}
-{"id": "3234.png", "formula": "\\begin{align*} \\left \\{ \\left ( c _ { e _ { i } } \\left ( r \\right ) , c _ { e _ { i } } \\left ( - r \\right ) \\right ) \\right \\} _ { i = 1 } ^ { \\mathrm { \\dim } \\left ( S \\right ) } \\end{align*}"}
-{"id": "5032.png", "formula": "\\begin{align*} \\eta ( \\phi ) = \\int _ G \\phi ( g ) \\ , d \\eta ( g ) , \\end{align*}"}
-{"id": "9867.png", "formula": "\\begin{align*} 0 = \\mu _ 0 ( \\widetilde { \\Omega } ) < \\mu _ { 1 } ( \\widetilde { \\Omega } ) \\leq \\mu _ { 2 } ( \\widetilde { \\Omega } ) \\leq . . . \\leq \\mu _ { n } ( \\widetilde { \\Omega } ) \\leq . . . \\ , , \\end{align*}"}
-{"id": "7299.png", "formula": "\\begin{align*} E _ k \\triangleright ( E _ \\eta ^ * E _ { \\eta ^ \\prime } ) = ( E _ k \\triangleright E _ \\eta ^ * ) E _ { \\eta ^ \\prime } + ( K _ k \\triangleright E _ \\eta ^ * ) ( E _ k \\triangleright E _ { \\eta ^ \\prime } ) . \\end{align*}"}
-{"id": "9908.png", "formula": "\\begin{align*} \\phi ( s ) : = K ' s ( K ' ) ^ { - 1 } . \\end{align*}"}
-{"id": "243.png", "formula": "\\begin{align*} s _ { x , y } : = s - \\frac { s ^ { 1 + 2 / d } \\Delta f ( x ) } { 2 ( d + 2 ) V _ d ^ { 2 / d } f ( x ) ^ { 1 + 2 / d } } + y . \\end{align*}"}
-{"id": "2493.png", "formula": "\\begin{gather*} A _ { k 1 } ^ i : = \\{ \\sigma _ i < + \\infty \\} \\cap \\{ z _ i ( u _ k , \\sigma _ i ) - z _ i ( u _ { k - 1 } , \\sigma _ i ) = \\varepsilon , \\ldots , z _ i ( u _ 3 , \\sigma _ i ) - z _ i ( u _ 2 , \\sigma _ i ) = \\varepsilon , \\\\ z _ i ( u _ 2 , \\sigma _ i ) - z _ i ( u _ 1 , \\sigma _ i ) = \\varepsilon \\} , 2 \\leqslant k \\leqslant n , 1 \\leqslant i \\leqslant n - 1 , \\end{gather*}"}
-{"id": "3979.png", "formula": "\\begin{align*} V ^ { \\sigma } ( A ) = \\bigoplus _ { \\delta _ 1 + \\cdots + \\delta _ k = \\sigma } V ( \\delta _ 1 , \\dots , \\delta _ k ) . \\end{align*}"}
-{"id": "1567.png", "formula": "\\begin{align*} X _ { s o l } = \\mbox { \\rm e } ^ { \\kappa t } S \\partial _ { S } + F \\partial _ { F } \\end{align*}"}
-{"id": "1970.png", "formula": "\\begin{align*} x _ j x _ i = q x _ i x _ j , \\forall \\ , 1 \\leq i < j \\leq n . \\end{align*}"}
-{"id": "1932.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m n _ i Y _ i ^ 2 ( u _ 0 ) ( 2 p _ i - q _ i ^ 2 Y _ i ( u _ 0 ) ) ^ 2 = \\frac { n - 1 } { 2 } E ( Y ( u _ 0 ) ) ^ 2 \\end{align*}"}
-{"id": "709.png", "formula": "\\begin{align*} \\frac { h _ { H } ( f ^ { n } ( P ) ) } { n ^ { r - 1 } \\delta ^ { n } } \\leq C _ { 2 } \\left ( \\sqrt [ ] { h _ { H } ( P ) } + \\sum _ { k = 1 } ^ { n - 1 } \\sqrt [ ] { C _ { 4 } h _ { H } ( P ) } \\frac { k ^ { 1 + ( r - 1 ) / 2 } } { \\delta ^ { 1 + k / 2 } } + h _ { H } ( P ) \\right ) . \\end{align*}"}
-{"id": "8855.png", "formula": "\\begin{align*} \\mathcal { D } = \\Big ( 2 + \\frac { 4 } { n } \\Big ) ^ 2 \\Big ( \\sum _ { i = 1 } ^ k \\upsilon _ i \\Big ) ^ 2 - 4 k \\Big ( 1 + \\frac { 4 } { n } \\Big ) \\sum _ { i = 1 } ^ k ( \\upsilon _ i ) ^ 2 \\geq 0 . \\end{align*}"}
-{"id": "3061.png", "formula": "\\begin{align*} \\lim _ { x \\to \\infty } \\hat { \\mu } ^ x = D _ 1 . \\end{align*}"}
-{"id": "4001.png", "formula": "\\begin{align*} ( i - q ^ { t - a } ) ( i - q ^ t ) \\begin{cases} = 0 \\ ; & \\mbox { i f } i = q ^ { t - a } , \\\\ < 0 \\ ; & \\mbox { i f } q ^ { t - a } < i < q ^ t , \\\\ = 0 \\ ; & \\mbox { i f } i = q ^ t . \\end{cases} \\end{align*}"}
-{"id": "3673.png", "formula": "\\begin{align*} \\frac { d H } { d t } = \\int \\left [ H _ { \\zeta } \\frac { \\partial \\zeta } { \\partial t } + H _ { \\mu } \\frac { \\partial \\mu } { \\partial t } + H _ h \\frac { \\partial h } { \\partial t } \\right ] d A = 0 ~ , \\end{align*}"}
-{"id": "7595.png", "formula": "\\begin{align*} \\delta _ 0 : = ( l - F ( x _ 0 ) ) / ( 2 l ) \\in ( 0 , 1 ) , \\delta _ { i } : = \\frac { l - 2 F ( x _ 0 + i \\varepsilon ) + F ( x _ 0 ) } { 2 l } , i = 1 , \\dots , K _ 2 , \\end{align*}"}
-{"id": "7239.png", "formula": "\\begin{align*} \\tau ^ \\alpha _ { k j , \\beta } = \\begin{cases} t _ { j k } ^ { - \\alpha } : = \\prod _ { \\lambda = 1 } ^ r ( t _ { j k } ^ \\lambda ) ^ { - \\alpha _ \\lambda } & ( \\beta = \\alpha ) \\\\ 0 & ( { \\rm o t h e r w i s e } ) , \\end{cases} \\end{align*}"}
-{"id": "8461.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } - \\psi ' ( t ) ( \\tilde { \\textbf { u } } ^ { \\star } , \\phi ) _ { 0 } ~ d t + \\sum _ { j = 1 } ^ { d } \\int _ { 0 } ^ { T } \\psi ( t ) ( \\textbf { P } A _ { j } ( \\tilde { \\textbf { u } } ^ { \\star } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } ^ { \\star } , \\phi ) _ { 0 } ~ d t = 0 . \\end{align*}"}
-{"id": "1328.png", "formula": "\\begin{align*} R ( \\lambda , \\mu ) = \\begin{pmatrix} f ( \\lambda , \\mu ) & 0 & 0 & 0 \\\\ 0 & 1 & g ( \\lambda , \\mu ) & 0 \\\\ 0 & g ( \\lambda , \\mu ) & 1 & 0 \\\\ 0 & 0 & 0 & f ( \\lambda , \\mu ) \\end{pmatrix} . \\end{align*}"}
-{"id": "2473.png", "formula": "\\begin{align*} ( V \\otimes \\omega _ s ) ^ { I _ k } = V ^ { I _ k } \\end{align*}"}
-{"id": "1113.png", "formula": "\\begin{align*} ( 2 d - 1 ) \\sum _ { 1 \\leq i < j \\leq M } \\lambda _ i \\lambda _ j \\| A D ( \\beta _ i - \\beta _ j ) \\| _ 2 ^ 2 = \\sum _ { i = 1 } ^ { M } \\lambda _ i \\| A D ( \\sum _ { j = 1 } ^ { M } \\lambda _ j \\beta _ j - d \\beta _ i ) \\| _ 2 ^ 2 - \\sum _ { i = 1 } ^ { M } \\lambda _ i ( 1 - d ) ^ 2 \\| A D \\beta _ i \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "8468.png", "formula": "\\begin{align*} A _ { 0 } ( J _ { \\varepsilon } \\textbf { u } ^ { \\varepsilon } ) \\partial _ { t } \\textbf { u } ^ { \\varepsilon } + \\sum _ { j = 1 } ^ { d } J _ { \\varepsilon } A _ { 0 } A _ { j } ( J _ { \\varepsilon } \\textbf { u } ^ { \\varepsilon } ) \\partial _ { x _ { j } } J _ { \\varepsilon } \\textbf { u } ^ { \\varepsilon } = - \\left ( \\begin{array} { c } 0 \\\\ \\frac { ( \\mathbb { I } - \\mathbb { P } ) v ^ { \\varepsilon } } { \\varepsilon } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "3066.png", "formula": "\\begin{align*} W _ { n , \\beta } ^ u = \\sum _ { | v | = n , v > u } e ^ { \\beta ( m _ n + V ( u ) - V ( v ) ) } , \\end{align*}"}
-{"id": "3047.png", "formula": "\\begin{align*} \\mathcal { O } = \\bigcup _ { u \\in \\Gamma ( \\mathcal { O } ) } C ( u , j _ u ) , \\end{align*}"}
-{"id": "6472.png", "formula": "\\begin{align*} \\| w \\| _ { L ^ { 2 \\kappa } ( [ 0 , t _ { 1 } - t _ { 0 } ] \\times \\rho ' B _ { 1 } ) } ^ { 2 } \\leq C \\left ( ( g _ { 1 - \\alpha } * W ) ( t _ { * } ) + \\| F \\| _ { L ^ { 1 } ( [ 0 , t _ { 2 } - t _ { 0 } ] ) } \\right ) . \\end{align*}"}
-{"id": "9784.png", "formula": "\\begin{align*} B F _ { 1 2 } = \\frac { m ( | \\mathcal { M } _ { 1 } ) } { m ( | \\mathcal { M } _ { 2 } ) } . \\end{align*}"}
-{"id": "6495.png", "formula": "\\begin{align*} \\mu ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\left ( \\frac { \\rho ( 0 ) } { t ^ { 1 - \\alpha } } + \\int _ { 0 } ^ { t } \\frac { \\rho ' ( s ) } { ( t - s ) ^ { 1 - \\alpha } } d s \\right ) , \\end{align*}"}
-{"id": "4536.png", "formula": "\\begin{align*} W _ 1 & = \\int _ { \\mathcal { X } \\times \\mathcal { X } } f ( x ) f ( y ) \\int _ { ( [ l _ x , v _ x ] \\times [ l _ y , v _ y ] ) ^ c } \\ ! \\ ! h ( u , v ) \\ , d ( F _ { n , x , y } - F _ { n , x } F _ { n , y } ) ( u , v ) \\ , d x \\ , d y \\\\ & = o ( n ^ { - ( 9 / 2 - \\epsilon ) } ) . \\end{align*}"}
-{"id": "865.png", "formula": "\\begin{align*} \\begin{cases} - \\nabla \\cdot ( \\sigma \\nabla u ) = f ^ + \\geq 0 & \\mbox { i n } \\Omega \\\\ u = 0 & \\mbox { o n } \\partial \\Omega , \\\\ | \\nabla u | \\leq 1 & \\mbox { i n } \\Omega , \\\\ | \\nabla u | = 1 & \\sigma - \\mbox { a . e . } \\end{cases} \\end{align*}"}
-{"id": "9612.png", "formula": "\\begin{align*} \\ddot \\psi _ f ( x , t ) = ( \\Delta - m ^ 2 ) \\psi _ f ( x , t ) \\end{align*}"}
-{"id": "8842.png", "formula": "\\begin{align*} \\upsilon _ { k + 1 } \\leq \\Big ( 1 + \\frac { 2 } { n } \\Big ) \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\upsilon _ i + \\Big [ \\Big ( \\frac { 2 } { n } \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\upsilon _ i \\Big ) ^ 2 - \\Big ( 1 + \\frac { 4 } { n } \\Big ) \\frac { 1 } { k } \\sum _ { j = 1 } ^ k \\Big ( \\upsilon _ j - \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\upsilon _ i \\Big ) ^ 2 \\Big ] ^ \\frac { 1 } { 2 } , \\end{align*}"}
-{"id": "8164.png", "formula": "\\begin{align*} \\tilde \\sigma _ m ( R _ U ( k ) ) ( v , \\xi ) = a _ m ( v , \\xi ) , v \\in V , \\xi \\in N ^ * _ v \\setminus \\{ 0 \\} . \\end{align*}"}
-{"id": "3035.png", "formula": "\\begin{align*} \\textbf { u } ^ { ( i ) } = ( u ^ { ( i ) } \\mid u '^ { ( i ) } ) = ( u _ { 0 + i } , u _ { 1 + i } , \\dots , u _ { \\alpha - 1 + i } \\mid u ' _ { 0 + i } , u ' _ { 1 + i } , \\dots , u ' _ { \\beta - 1 + i } ) \\end{align*}"}
-{"id": "7855.png", "formula": "\\begin{align*} 0 < \\kappa _ 0 \\le \\kappa ( x , z ) \\le \\kappa _ 1 \\ , , \\kappa ( x , z ) = \\kappa ( x , - z ) \\ , , \\end{align*}"}
-{"id": "6582.png", "formula": "\\begin{align*} { \\rm { p r o x } } _ { \\gamma f } = \\nabla r _ { \\gamma f } ^ * \\end{align*}"}
-{"id": "4904.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\varrho } ^ + ( z ) & : = \\sum _ { n \\gg - \\infty } c _ { \\mathcal { F } , \\varrho } ^ + ( n ) e ^ { \\frac { 2 \\pi i n z } { \\ell _ { \\varrho } } } , \\\\ \\mathcal { F } _ { \\varrho } ^ - ( z ) & : = c _ { \\mathcal { F } , \\varrho } ^ - ( 0 ) y ^ { 1 - \\kappa } + \\sum _ { \\substack { n \\ll \\infty \\\\ n \\neq 0 } } c _ { \\mathcal { F } , \\varrho } ^ - ( n ) \\Gamma \\left ( 1 - \\kappa , - \\frac { 4 \\pi n y } { \\ell _ { \\varrho } } \\right ) e ^ { \\frac { 2 \\pi i n z } { \\ell _ { \\varrho } } } , \\end{align*}"}
-{"id": "5427.png", "formula": "\\begin{align*} | | \\mu - \\nu | | = \\sup \\{ \\langle u , \\mu \\rangle - \\langle u , \\nu \\rangle : u \\in B [ S , { \\cal F } ] , | | u | | \\leq 1 \\} . \\end{align*}"}
-{"id": "1253.png", "formula": "\\begin{align*} \\int _ m ^ \\infty e ^ { - i t \\lambda } \\chi _ j ( \\lambda ) [ \\mathcal R _ 0 ^ + ( \\lambda ) - \\mathcal R _ 0 ^ - ( \\lambda ) ] \\ , d \\lambda & = \\int _ 0 ^ \\infty e ^ { - i t \\sqrt { z ^ 2 + m ^ 2 } } \\frac { z \\chi _ j ( z ) [ \\mathcal R _ 0 ^ + ( \\lambda ) - \\mathcal R _ 0 ^ - ( \\lambda ) ] } { \\sqrt { z ^ 2 + m ^ 2 } } \\ , d z . \\end{align*}"}
-{"id": "3783.png", "formula": "\\begin{align*} \\left [ { { { \\bf { V } } _ { { { \\mathrm { \\bf { B } } } } } } } \\right ] _ { \\ell _ i \\ell _ j } = \\left \\{ \\begin{array} { l } { \\left [ { { { \\bf { V } } _ { s } } } \\right ] _ { m n } } i = ( s - 1 ) N _ { \\mathrm s } + m , \\ j = ( s - 1 ) N _ { \\mathrm s } + n \\\\ 0 \\qquad \\ ; \\ ; { \\rm o t h e r w i s e } \\end{array} \\right . \\end{align*}"}
-{"id": "7900.png", "formula": "\\begin{align*} 1 \\ , = \\ , \\langle v _ 0 , w _ 0 \\rangle _ A \\ , = \\ , I \\ , - \\ , s _ 0 \\ , \\chi _ 0 \\ , - \\ , r _ 0 \\ , \\xi _ 0 , \\mbox { w h e r e } I \\ ; : = \\ ; l _ 0 \\cdot h _ 0 . \\end{align*}"}
-{"id": "1211.png", "formula": "\\begin{align*} P ' \\Bigl [ ( P - P ' _ { n } ) \\tilde { g } \\ge \\frac { t / 2 + c _ { 2 } P \\tilde { g } } { 1 + c _ { 2 } } \\Bigr ] \\ ! \\le \\ ! \\frac { P \\tilde { g } ^ 2 ( 1 + c _ { 2 } ) ^ 2 } { n ( t / 2 + c _ { 2 } P \\tilde { g } ) ^ 2 } \\ ! \\le \\ ! \\frac { B P \\tilde { g } ( 1 + c _ { 2 } ) ^ 2 } { 2 n t c _ { 2 } P \\tilde { g } } \\ ! = \\ ! \\frac { B ( 1 + c _ { 2 } ) ^ 2 } { 2 n t c _ { 2 } } . \\end{align*}"}
-{"id": "7553.png", "formula": "\\begin{align*} b = \\min \\left \\{ \\beta > 0 : F ( \\beta ) = \\min _ { x \\in [ 0 , a ] } F ( x ) \\right \\} . \\end{align*}"}
-{"id": "2182.png", "formula": "\\begin{align*} \\geq C ( \\delta , \\Lambda ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x . \\end{align*}"}
-{"id": "8395.png", "formula": "\\begin{align*} \\lim _ \\lambda \\omega _ \\xi \\circ E _ M ( ( x g - x _ \\lambda g ) ^ * ( x g - x _ \\lambda g ) ) = \\lim _ \\lambda \\langle \\alpha _ { g ^ { - 1 } } ( x - x _ \\lambda ) ^ * \\alpha _ { g ^ { - 1 } } ( x - x _ \\lambda ) \\xi , \\xi \\rangle = 0 \\end{align*}"}
-{"id": "362.png", "formula": "\\begin{align*} U _ { c , h } ( \\gamma ) = e ^ { i T _ { c , h } ( f _ 1 ) } \\ldots e ^ { i T _ { c , h } ( f _ n ) } . \\end{align*}"}
-{"id": "3805.png", "formula": "\\begin{align*} \\int _ 0 ^ { T ^ * } \\| u _ x \\| _ { L ^ \\infty } d \\tau = + \\infty . \\end{align*}"}
-{"id": "7415.png", "formula": "\\begin{align*} e _ i : = \\exp E _ { i , i + 1 } e _ { - i } : = \\exp E _ { i + 1 , i } \\end{align*}"}
-{"id": "2971.png", "formula": "\\begin{align*} \\delta _ H \\Theta ^ 1 - \\delta _ L \\Theta ( A ) & = 0 ; \\\\ \\delta _ H \\Theta ^ 2 + \\delta _ L \\Theta ^ 1 & = 0 ; \\\\ \\delta _ v \\Theta ( L ) - \\delta _ L \\Theta ^ 2 & = 0 . \\end{align*}"}
-{"id": "1574.png", "formula": "\\begin{align*} T ^ { I } L _ { Y _ { I } } C _ { \\alpha } - T _ { , t } ^ { I } Y _ { I \\alpha } - 2 a _ { , \\alpha } = 0 . \\end{align*}"}
-{"id": "2938.png", "formula": "\\begin{align*} e ^ { - i q } \\ , z ^ { - \\alpha _ 0 } = ( 1 / z ) \\ , z ^ { - \\alpha _ 0 } \\ , e ^ { - i q } \\ , \\end{align*}"}
-{"id": "6992.png", "formula": "\\begin{align*} \\delta _ L \\Theta ^ 1 ( x , y ) & = [ x , \\Theta ^ 1 ( y ) ] - [ y , \\Theta ^ 1 ( x ) ] - \\Theta ^ 1 [ x , y ] \\\\ & = \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ f _ 0 ^ x , [ f _ i ^ y , \\alpha _ j ] ] - [ f _ 0 ^ y , [ f _ i ^ x , \\alpha _ j ] ] - [ f _ i ^ { [ x , y ] } , \\alpha _ j ] ~ ~ \\mbox { ( u s i n g e q u a t i o n \\eqref { c 3 } ) } . \\end{align*}"}
-{"id": "4651.png", "formula": "\\begin{align*} \\langle x , y \\rangle = \\langle g \\cdot x , g ^ \\iota \\cdot y \\rangle . \\end{align*}"}
-{"id": "6089.png", "formula": "\\begin{align*} \\sum _ { 1 \\le n \\le t } \\left ( \\widetilde { F } _ { n , n } ( t ) - \\widetilde { G } _ { n , n } ( t ) \\right ) = \\log t + O ( 1 ) . \\end{align*}"}
-{"id": "3105.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) = \\left [ \\begin{array} { c | c } \\lambda B + A & L _ { s } ( \\lambda ) ^ { T } \\otimes I _ { n } \\\\ \\hline L _ { s } ( \\lambda ) \\otimes I _ { n } & 0 \\end{array} \\right ] , \\end{align*}"}
-{"id": "4389.png", "formula": "\\begin{align*} | \\mu ( A ) - \\mu _ n ( A ) | \\le \\varepsilon _ n \\mu _ n ( A ) A \\in \\vee _ { i = 0 } ^ { r _ n } T ^ { - i } \\xi , \\ n \\ge 1 , \\end{align*}"}
-{"id": "8409.png", "formula": "\\begin{align*} \\sum _ j \\phi ( m _ j ^ * ) p w _ h \\phi ( \\alpha _ { h ^ { - 1 } } ( m _ j ) ) & = \\sum _ j \\phi ( m _ j ^ * \\sigma ( p ) z _ h h \\alpha _ { h ^ { - 1 } } ( m _ j ) ) \\\\ & = \\phi ( \\sum _ j m _ j ^ * m _ j \\sigma ( p ) z _ h h ) = p w _ h \\ne 0 , \\end{align*}"}
-{"id": "3067.png", "formula": "\\begin{align*} \\nu _ { \\beta , n } = \\sum _ { | u | = n } e ^ { \\beta ( m _ n - V ( u ) ) } \\delta _ u \\nu _ { \\beta , \\infty } = \\sum _ { n \\in \\N } Z _ \\infty ^ \\beta e ^ { - \\beta \\xi ^ \\beta _ n } \\delta _ { u ^ { ( n ) } } . \\end{align*}"}
-{"id": "4186.png", "formula": "\\begin{align*} w _ { a _ 1 } = \\frac { 1 } { 2 } \\left ( 1 + \\tanh \\frac { a _ 2 - a _ 1 } { 2 } \\right ) , w _ { a _ N } = \\frac { 1 } { 2 } \\left ( 1 + \\tanh \\frac { a _ N - a _ { N - 1 } } { 2 } \\right ) . \\end{align*}"}
-{"id": "8316.png", "formula": "\\begin{align*} ( y _ 1 + \\xi y _ 2 ) ^ p - ( y _ 1 + \\xi y _ 2 ) = ( y _ 1 ^ p - y _ 1 ) + \\xi ( y _ 2 ^ p - y _ 2 ) = T + \\xi T ^ 2 , \\end{align*}"}
-{"id": "3276.png", "formula": "\\begin{align*} \\hat { R } _ { V , W } ( v \\otimes w _ { \\mathrm { h w } } ) = q ^ { ( \\mathrm { w t } ( v ) , \\mathrm { w t } ( w _ { \\mathrm { h w } } ) ) } w _ { \\mathrm { h w } } \\otimes v . \\end{align*}"}
-{"id": "9231.png", "formula": "\\begin{align*} \\mathbb { E } [ \\int _ D \\int _ 0 ^ T \\frac { \\partial H } { \\partial u } ( t , x , z ) \\beta ( t , x , z ) d t d x ] = 0 , \\end{align*}"}
-{"id": "171.png", "formula": "\\begin{align*} H = H ( X ) = H ( f ) : = - \\mathbb { E } \\{ \\log f ( X ) \\} = - \\int _ { \\mathcal { X } } f ( x ) \\log f ( x ) \\ , d x \\end{align*}"}
-{"id": "8189.png", "formula": "\\begin{align*} F : = ( ( a _ 2 - a _ 3 ) x ^ 2 + z _ 1 - a _ 2 z _ 2 ) ( ( a _ 3 - a _ 1 ) x ^ 2 - z _ 1 + a _ 1 z _ 2 ) . \\end{align*}"}
-{"id": "7549.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\bigl \\{ f ( x _ n ) + l \\chi _ { n + 1 } , 0 \\bigr \\} , x _ 0 > 0 , n \\in \\mathbb N . \\end{align*}"}
-{"id": "3972.png", "formula": "\\begin{align*} p _ W ( u ' ( \\psi ( s ) ) v ) = \\sum _ { k \\geq i } a _ k w _ k , a _ { k } \\in \\R . \\end{align*}"}
-{"id": "6287.png", "formula": "\\begin{align*} v ( z ) = v _ { a , b , c } ( z ) = ( z + c ) ^ d p \\Big ( a + \\frac { b } { z + c } \\Big ) \\end{align*}"}
-{"id": "3194.png", "formula": "\\begin{align*} T _ { j k } : = \\left ( \\begin{array} { c | c c c } t _ { j k } ^ 1 & 0 & \\cdots & 0 \\\\ \\hline 0 & & & \\\\ \\vdots & & S _ { j k } & \\\\ 0 & & & \\end{array} \\right ) . \\end{align*}"}
-{"id": "2739.png", "formula": "\\begin{align*} u _ { P Q } \\star v ^ { Q K } = \\delta _ P ^ K \\mbox { a n d } v ^ { L P } \\star u _ { P Q } = \\delta ^ L _ Q . \\end{align*}"}
-{"id": "2216.png", "formula": "\\begin{align*} u ( x , t ) = ( \\mu * v ) ( x , t ) = \\int _ { 0 } ^ { t } \\mu ( t - s ) v ( x , s ) d s ( 0 < t \\leq T ) , \\end{align*}"}
-{"id": "6550.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = 0 } ^ M \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = 0 } ^ M \\big \\| _ { L ^ p ( D ) } \\le C \\delta . \\end{aligned} \\end{align*}"}
-{"id": "9167.png", "formula": "\\begin{align*} \\sum _ { l = - r } ^ { r } \\frac { \\partial H } { \\partial u _ l } ( v , \\dots , v ) = 0 , \\end{align*}"}
-{"id": "2388.png", "formula": "\\begin{align*} f ( x ) = \\tfrac { 1 } { 2 } \\langle H x , x \\rangle + \\langle h , x \\rangle . \\end{align*}"}
-{"id": "1770.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta u _ { i } ^ { 1 } \\ge 0 & \\Omega , \\\\ u _ { i } ^ { 1 } ( x ) = u _ { i } ^ { 0 } ( x ) = \\phi _ { i } ( x ) & \\partial \\Omega . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8995.png", "formula": "\\begin{align*} ( \\textbf { E } ^ \\gamma _ { \\rho , \\mu , \\omega , b ^ - } \\varphi ) ( x ) = \\int _ x ^ b ( t - x ) ^ { \\mu - 1 } E _ { \\rho , \\mu } ^ \\gamma [ \\omega ( t - x ) ^ \\rho ] \\varphi ( t ) d t , ~ ~ x < b , \\end{align*}"}
-{"id": "2281.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i - v _ { i - 1 } ) _ { i = 1 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 1 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "9169.png", "formula": "\\begin{align*} 0 = \\sum _ { l = - r } ^ { r } l ^ k \\frac { \\partial H } { \\partial u _ l } ( v , \\dots , v ) , k = 0 , \\dots , 2 r + 1 , k \\neq 1 , \\end{align*}"}
-{"id": "9076.png", "formula": "\\begin{align*} ( 4 s ^ { 3 } - s ) M _ N ^ { ( 1 ) \\prime \\prime } + s ( 1 6 s + 2 ( \\alpha - 1 ) + 4 N ) M _ N ^ { ( 1 ) \\prime } + s ( 9 - \\alpha ^ { 2 } ) M _ N ^ { ( 1 ) } = ( 3 s + 3 ) M _ { N - 1 } ^ { ( 2 ) \\prime } - ( 3 \\alpha + 6 N - 6 ) M _ { N - 1 } ^ { ( 2 ) } \\ . \\end{align*}"}
-{"id": "7427.png", "formula": "\\begin{align*} \\overline { w } _ 0 e _ { i ( m ) } e _ { - i ( m ) } ^ { - 1 } \\dots e _ { i ( 1 ) } e _ { - i ( 1 ) } ^ { - 1 } = \\overline { u ^ c } \\overline { u } e _ { i ( m ) } e _ { - i ( m ) } ^ { - 1 } \\dots e _ { i ( 1 ) } e _ { - i ( 1 ) } ^ { - 1 } \\in B _ + u ^ c B _ + , \\end{align*}"}
-{"id": "2171.png", "formula": "\\begin{align*} \\geq \\frac { c _ { n , \\beta } } { 2 C \\Lambda } \\int _ { \\rho ' B _ { 1 } } \\int _ { \\rho ' B _ { 1 } } \\frac { ( w ( s , x ) - w ( s , y ) ) ^ { 2 } } { | x - y | ^ { n + 2 \\beta } } d x d y . \\end{align*}"}
-{"id": "2774.png", "formula": "\\begin{align*} \\biggl | \\alpha - \\frac { p _ n } { q _ n } \\biggr | = \\frac { 1 } { \\lambda _ { n + 1 } ( \\alpha ) q _ n ^ 2 } \\end{align*}"}
-{"id": "18.png", "formula": "\\begin{align*} K _ { X / Y } : = K _ X - p ^ \\star ( K _ Y ) . \\end{align*}"}
-{"id": "5595.png", "formula": "\\begin{align*} v M _ { 1 1 } v ^ * = \\| ( a , c ) \\| _ 2 ^ 2 P , \\end{align*}"}
-{"id": "9797.png", "formula": "\\begin{align*} Z _ { i } = r ( X _ { i j } ) + \\varepsilon _ { i } , \\end{align*}"}
-{"id": "4339.png", "formula": "\\begin{align*} ( | f ' | ^ { p - 2 } f ' ) ' + f - | f | ^ { p - 2 } f = 0 \\ , r \\in ( 0 , \\infty ) \\ , \\end{align*}"}
-{"id": "888.png", "formula": "\\begin{align*} \\vert I \\vert : = \\vert \\int _ { T } ^ { 2 T } Z ( t ) d t \\vert = \\int _ { T } ^ { 2 T } \\vert Z ( t ) \\vert d t . \\end{align*}"}
-{"id": "3796.png", "formula": "\\begin{align*} C ( X ) ( { \\mathbb F } _ q ) & = C ( X ) \\cap { \\mathbb A } ^ { n + 1 } ( { \\mathbb F } _ q ) = Z _ { \\overline { \\mathbb F } _ q } \\langle f _ i , g _ j \\rangle \\cap Z _ { \\overline { \\mathbb F } _ q } \\langle x _ k ^ q - x _ k : 0 \\leq k \\leq n \\rangle \\\\ & = Z _ { \\overline { \\mathbb F } _ q } ( J ) , \\qquad . \\end{align*}"}
-{"id": "8086.png", "formula": "\\begin{align*} P _ { s _ i } = E _ { s _ i } V , \\ ; P _ { r _ j } = E _ { r _ j } V , \\end{align*}"}
-{"id": "4965.png", "formula": "\\begin{align*} \\alpha _ { f } ( P ) = \\lim _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n } ( P ) ) ^ { 1 / n } \\end{align*}"}
-{"id": "3093.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) = \\lambda \\mathcal { L } _ 1 + \\mathcal { L } _ 0 \\mbox { w i t h } \\mathcal { L } _ 1 ^ T = \\sigma \\mathcal { L } _ 1 \\ , \\ , \\mbox { a n d } \\ , \\ , \\mathcal { L } _ 0 ^ T = \\sigma \\mathcal { L } _ 0 , \\end{align*}"}
-{"id": "9632.png", "formula": "\\begin{align*} X = \\left ( \\begin{array} { c c c c c c c c } 0 & b _ 1 \\\\ & 0 & b _ 2 \\\\ & & 0 & \\ddots \\\\ & & & 0 & b _ n \\\\ & & & & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "7021.png", "formula": "\\begin{align*} \\mathbb { E } \\ , \\pi ( g ( \\omega ^ { 2 ^ k } ) \\cdots g ( \\omega ) ) = \\mathbb { E } \\ , \\pi ( g ( \\omega ^ { 2 ^ k } ) ) \\cdots \\pi ( g ( \\omega ) ) \\rightarrow 0 . \\end{align*}"}
-{"id": "8062.png", "formula": "\\begin{align*} A ( 1 , x ) = \\begin{bmatrix} \\cos ( \\pi { x } ) & - \\sin ( \\pi { x } ) \\\\ \\sin ( \\pi { x } ) & \\cos ( \\pi { x } ) \\end{bmatrix} = \\mathrm { r o t } _ { \\pi { x } } . \\end{align*}"}
-{"id": "5784.png", "formula": "\\begin{align*} S ^ { \\alpha } ( z ) = \\sum _ { n \\in \\Z } S _ { n } ^ { \\alpha } z ^ { - n } \\end{align*}"}
-{"id": "9862.png", "formula": "\\begin{align*} 0 = \\mu _ 0 ( \\widetilde { \\Omega } ) < \\mu _ { 1 } ( \\widetilde { \\Omega } ) \\leq \\mu _ { 2 } ( \\widetilde { \\Omega } ) \\leq . . . \\leq \\mu _ { n } ( \\widetilde { \\Omega } ) \\leq . . . \\ , , \\end{align*}"}
-{"id": "8978.png", "formula": "\\begin{align*} g _ { k } ( s ) \\ , : = \\ , \\begin{cases} \\min \\{ s ^ { - \\beta } \\ , , \\ , k \\} \\qquad , \\\\ k \\qquad \\qquad \\quad \\qquad . \\end{cases} \\end{align*}"}
-{"id": "7058.png", "formula": "\\begin{align*} { { \\bf { Y } } ^ { [ 2 ] } } ( n ) - { { \\bf { Y } } ^ { [ 2 ] } } ( { t _ 1 } ) = { { \\bf { H } } ^ { [ 2 1 ] } } ( n ) { \\bf { V } } _ 2 ^ { [ 1 ] } ( n ) { { \\bf { v } } ^ { [ 1 ] } } + { { \\bf { H } } ^ { [ 2 2 ] } } ( n ) { \\bf { V } } _ 2 ^ { [ 2 ] } ( n ) { { \\bf { v } } ^ { [ 2 ] } } . \\end{align*}"}
-{"id": "2509.png", "formula": "\\begin{align*} F ( x ( t ; \\overline { x } ) ) = \\exp ( - 2 \\lambda t ) \\cdot F ( \\overline { x } ) , ~ ( \\forall ) t \\in [ 0 , \\infty ) . \\end{align*}"}
-{"id": "6010.png", "formula": "\\begin{gather*} \\epsilon _ { \\Phi } = - e ^ { 1 2 3 4 5 6 7 } . \\end{gather*}"}
-{"id": "9444.png", "formula": "\\begin{align*} ( t ) : = \\{ ( x , t ( x ) ) \\in X \\times X : x \\in A \\} \\end{align*}"}
-{"id": "3945.png", "formula": "\\begin{align*} \\abs { P [ a , b ] } = d ( a , x _ 1 ) + d ( x _ 1 , x _ 2 ) + d ( x _ 2 , b ) \\leq d ( a , u _ 1 ) + d ( u _ 1 , u _ 2 ) + d ( u _ 2 , b ) + 1 2 N \\leq d ( a , b ) + 1 2 N . \\end{align*}"}
-{"id": "3984.png", "formula": "\\begin{align*} a _ i ( t ) w _ { r - j } = e ^ { ( \\delta _ { i } + 2 j ) t } w _ { r - j } \\delta _ { i } + 2 j \\geq - \\delta _ { i } . \\end{align*}"}
-{"id": "5873.png", "formula": "\\begin{align*} Z ^ { 1 } = e ^ { m t } K _ { 1 } ~ , ~ Z ^ { 2 } = e ^ { - m t } \\left ( K ^ { 1 } + \\left ( m \\int \\frac { d x } { \\sigma \\left ( x \\right ) } + c \\right ) F \\partial _ { F } \\right ) \\end{align*}"}
-{"id": "8306.png", "formula": "\\begin{align*} \\psi _ * ^ { - 1 } H _ 1 ( \\Gamma , \\Z ) = \\Omega ( \\widetilde { \\Gamma } / \\Gamma ) ^ * + \\frac { 1 } { 2 } \\psi ^ * H _ 1 ( \\Gamma , \\Z ) . \\end{align*}"}
-{"id": "3006.png", "formula": "\\begin{align*} \\sum _ { \\substack { \\{ \\sigma \\in { D } ^ { I } _ { n } : \\\\ \\sigma ( a ) = n \\} } } ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L ( \\sigma ) } & = & \\sum _ { \\substack { \\{ \\sigma \\in { D } ^ { I } _ n : \\sigma ( a ) = n , \\\\ \\sigma ( a - 1 ) < \\sigma ( a + 1 ) \\} } } \\left ( ( - 1 ) ^ { \\ell ( \\sigma ) } x ^ { L ( \\sigma ) } + ( - 1 ) ^ { \\ell ( \\sigma ^ a ) } x ^ { L ( \\sigma ^ a ) } \\right ) = 0 . \\end{align*}"}
-{"id": "539.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ m } { m ! } b _ m = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { k = \\lceil m / 2 \\rceil } ^ m b _ m ^ k = \\sum _ { k = 0 } ^ { \\infty } \\sum _ { i = 0 } ^ k b _ { k + i } ^ k = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ k } { k ! } a _ k , \\end{align*}"}
-{"id": "6130.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } ^ 2 + 2 R _ { 1 3 4 2 } R _ { 1 4 2 3 } = & 3 x ^ 2 - 6 x y - 2 ( x - y ) ^ 2 \\\\ \\ge & - 2 ( K _ { 1 3 } - K _ { 1 2 } ) ^ 2 . \\\\ \\end{align*}"}
-{"id": "4729.png", "formula": "\\begin{align*} G ^ - & : = \\{ y \\in \\mathbb { R } ^ 2 \\ , | \\ ( \\eta ) ^ 2 + [ \\beta _ 2 ( y _ 2 - \\alpha _ 2 ) ] ^ 2 \\leq [ \\beta _ 1 ( y _ 1 - \\alpha _ 1 ) ] ^ 2 , \\ \\beta _ 1 ( y _ 1 - \\alpha _ 1 ) \\leq 0 \\} . \\end{align*}"}
-{"id": "2967.png", "formula": "\\begin{align*} \\alpha _ n = \\delta _ H \\phi , ~ \\lambda _ n = \\delta _ L \\psi ~ \\mbox { a n d } ~ \\mu _ n = - \\delta _ L \\phi + \\delta _ v \\psi . \\end{align*}"}
-{"id": "5278.png", "formula": "\\begin{align*} & = \\sum _ { n = 1 } ^ { \\infty } \\tfrac { 1 } { n ^ { 1 + \\delta } } \\int _ { T } ^ { 2 T } \\left ( { \\tfrac { t } { 2 \\pi } } \\right ) ^ { { 1 } / { 4 } + { \\delta } / { 2 } + { i t } / { 2 } } e ^ { - i ( { t } / { 2 } + { \\pi } / { 8 } ) } n ^ { - i t } \\left \\{ 1 + O ( { 1 } / { t } ) \\right \\} d t . \\end{align*}"}
-{"id": "1294.png", "formula": "\\begin{align*} \\widetilde { S } ( \\alpha ) = \\sum _ { n = 1 } ^ { \\infty } \\Lambda ( n ) e ^ { - n / N } e ( n \\alpha ) , \\end{align*}"}
-{"id": "2153.png", "formula": "\\begin{align*} & \\int _ { \\Omega } \\psi \\partial _ { t } \\left [ g _ { 1 - \\alpha , m } * ( u - u _ { 0 } ) \\right ] d x + \\mathcal { E } ( h _ { m } * u , \\psi ) \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = ( \\geq \\leq ) \\int _ { \\Omega } ( h _ { m } * f ) \\psi d x \\ , \\ , \\ , t \\in ( 0 , T ) , \\ , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "3869.png", "formula": "\\begin{align*} \\begin{array} { l l } \\mathbf n _ V = \\nu _ { \\partial \\Omega } & \\mbox { o n } \\partial M \\setminus \\partial B ^ + , \\\\ \\mathbf n _ V - ( \\mathbf n _ V \\cdot \\nu _ { \\partial \\Omega } ) \\nu _ { \\partial \\Omega } = - \\sigma \\mathbf n _ { B ^ + } & \\mbox { o n } \\partial B ^ + . \\end{array} \\end{align*}"}
-{"id": "2643.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { h ( m , n ) } C _ { m } ^ { l } ( G ^ { 0 } ) ( \\lambda ) ( C _ { m } ^ { l } ( G ^ { 0 } ) ( \\lambda ) ) ^ { * } = d _ { m } ^ { 0 } ( \\lambda ) ^ { \\top } \\left \\{ \\alpha _ { m k } ^ { 2 } \\delta _ { k l } \\right \\} _ { k , l = 1 } ^ { \\infty } \\overline { d _ { m } ^ { 0 } ( \\lambda ) } , \\end{align*}"}
-{"id": "5568.png", "formula": "\\begin{align*} D _ x f = f _ x + \\sum _ { j = 1 } ^ N [ f ] ( x _ j ) \\delta _ { x _ j } , \\end{align*}"}
-{"id": "3151.png", "formula": "\\begin{align*} & \\Pi _ { \\mathsf { V } , \\alpha } ( t ^ n , x ^ n ) V _ { { t ^ n } } ( x ^ n ) \\Pi _ { \\mathsf { V } , \\alpha } ( t ^ n , x ^ n ) \\allowdisplaybreaks \\\\ & \\leq 2 ^ { - n ( S ( \\mathsf { V } | r ) + \\delta ( \\alpha ) ' ) } \\Pi _ { \\mathsf { V } , \\alpha } ( t ^ n , x ^ n ) \\allowdisplaybreaks \\\\ & = 2 ^ { - n ( \\sum _ { t , x } r ( t , x ) S ( \\mathsf { V } ( t , x ) ) + \\delta ( \\alpha ) ' ) } \\Pi _ { \\mathsf { V } , \\alpha } ( t ^ n , x ^ n ) \\end{align*}"}
-{"id": "7493.png", "formula": "\\begin{align*} \\left \\{ R = ( n - 1 ) / 2 \\right \\} \\subseteq \\{ Q _ 0 = Q _ 2 = 1 \\} \\cap \\{ Q _ 1 = n - 2 \\} \\cap \\left ( \\bigcap _ { j = 3 } ^ { \\kappa ' } \\{ Q _ j = 0 \\} \\right ) . \\end{align*}"}
-{"id": "9074.png", "formula": "\\begin{align*} D ^ { ( \\beta ) } _ N ( k , \\alpha ) = \\begin{cases} \\displaystyle \\frac { \\partial ^ { k - 1 } M _ N ^ { ( \\beta ) } ( s ) } { \\partial s ^ { k - 1 } } \\biggl | _ { s \\to 0 ^ { - } } & \\ , \\\\ \\displaystyle \\frac { ( - 1 ) ^ { | k | } } { | k | ! } \\int _ { - \\infty } ^ { 0 } M _ N ^ { ( \\beta ) } ( s ) s ^ { | k | } \\ , \\mathrm { d } s & \\ . \\end{cases} \\end{align*}"}
-{"id": "2210.png", "formula": "\\begin{align*} u _ { k } ( t ) = E _ { \\alpha , 1 } ( - \\lambda _ { k } t ^ { \\alpha } ) u _ { 0 , k } + \\int _ { 0 } ^ { t } ( t - s ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\lambda _ { k } ( t - s ) ^ { \\alpha } ) f _ { k } ( s ) d s . \\end{align*}"}
-{"id": "2975.png", "formula": "\\begin{align*} \\Theta ^ 2 ( x , y ) = \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ f _ i ^ x , f _ j ^ y ] - f _ i ^ { \\lambda _ j ( x , y ) } . \\end{align*}"}
-{"id": "118.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } { \\frac { x } { 2 } \\choose n } d \\mu _ { - 1 } ( x ) = \\frac { 1 } { n ! } \\sum _ { m = 0 } ^ n 2 ^ { - m } E _ m S _ 1 ( n , m ) \\end{align*}"}
-{"id": "20.png", "formula": "\\begin{align*} \\partial h = h \\cdot \\Psi \\end{align*}"}
-{"id": "385.png", "formula": "\\begin{align*} g \\cdot f ( x , y ) = f ( ( x , y ) \\cdot g ^ t ) . \\end{align*}"}
-{"id": "9384.png", "formula": "\\begin{align*} \\widetilde { \\Gamma } _ 0 f = f _ r ( 0 ) , \\widetilde { \\Gamma } _ 1 f = f _ s ' ( 0 ) - ( q , f ) , f \\in \\mathcal { D } ( \\widetilde { S } _ { m a x } ) \\end{align*}"}
-{"id": "6416.png", "formula": "\\begin{align*} H _ { e } ^ { s } ( \\Omega ) : = \\left \\{ u \\in H ^ { s } ( \\mathbb { R } ^ { n } ) \\ , : \\ , u = 0 \\mathbb { R } ^ { n } \\backslash \\Omega \\right \\} , \\end{align*}"}
-{"id": "3124.png", "formula": "\\begin{align*} \\begin{bmatrix} - P ^ { - 1 } & 0 \\\\ \\lambda ^ t I _ n & I _ n \\end{bmatrix} \\mbox { a n d } \\begin{bmatrix} I _ n & - \\lambda ^ t I _ n \\\\ 0 & I _ n \\end{bmatrix} . \\end{align*}"}
-{"id": "998.png", "formula": "\\begin{align*} A ( u ) | \\phi _ { \\{ l ; k \\} } \\rangle & = ( u + \\omega + \\eta \\sum _ { i = 1 } ^ { n - 1 } l _ i ) ( u - \\omega + \\eta \\sum _ { i = 1 } ^ { m - 1 } k _ i ) | \\phi _ { \\{ l ; k \\} } \\rangle \\\\ B ( u ) | \\phi _ { \\{ l ; k \\} } \\rangle & = 0 \\\\ C ( u ) | \\phi _ { \\{ l ; k \\} } \\rangle & \\neq 0 \\\\ D ( u ) | \\phi _ { \\{ l ; k \\} } \\rangle & = \\eta ^ { - 2 } | \\phi _ { \\{ l ; k \\} } \\rangle . \\end{align*}"}
-{"id": "5858.png", "formula": "\\begin{align*} a \\left ( t , x \\right ) = - \\frac { 1 } { 2 } \\int \\left ( T _ { I } L _ { Y _ { I } } C _ { x } - T _ { I , t } Y _ { 1 } \\right ) d x + f _ { I } \\left ( t \\right ) . \\end{align*}"}
-{"id": "4874.png", "formula": "\\begin{align*} f _ { \\alpha } ( z ) = f _ { \\alpha } \\left ( \\theta \\right ) + \\frac { \\sigma _ { \\alpha } ^ 2 } { 2 } ( z - \\theta ) ^ 2 + \\mathcal { O } \\left ( ( z - \\theta ) ^ 3 \\right ) , \\\\ f _ { \\alpha } ( z ) = f _ { \\alpha } \\left ( - \\theta \\right ) - \\frac { \\sigma _ { \\alpha } ^ 2 } { 2 } ( z + \\theta ) ^ 2 + \\mathcal { O } \\left ( ( z + \\theta ) ^ 3 \\right ) . \\end{align*}"}
-{"id": "2353.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot { x } ( t ) = v ( x ( t ) ) , \\\\ x ( 0 ) = x _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "1315.png", "formula": "\\begin{align*} \\theta ( z ) = \\Bigl ( \\frac \\pi z \\Bigr ) ^ { 1 / 2 } \\theta \\Bigl ( \\frac { \\pi ^ 2 } z \\Bigr ) \\textrm { f o r } \\ \\Re ( z ) > 0 . \\end{align*}"}
-{"id": "8066.png", "formula": "\\begin{align*} N ( x , z ) = \\begin{cases} ( T ( x ) , e ^ { 2 i \\alpha _ x } z ) & x \\in X _ r , \\\\ ( T ( x ) , { e ^ { 4 i \\beta _ x } } / { z } ) & x \\in X _ f , \\end{cases} \\end{align*}"}
-{"id": "6966.png", "formula": "\\begin{align*} & g ( \\sigma , \\mu _ 1 ) = H _ 2 ( \\sigma ) - \\gamma \\frac { \\delta - 1 } { \\delta } H _ 2 ( \\mu _ 1 ) \\\\ & \\qquad \\qquad + \\inf _ { u > 0 } \\{ \\sigma \\log p ( u ) + ( 1 - \\sigma ) \\log q ( u ) - \\mu _ 1 \\gamma \\log u \\} . \\end{align*}"}
-{"id": "9254.png", "formula": "\\begin{align*} ( L \\varphi , \\psi ) _ { L ^ 2 ( \\mathbb { R } ) } = ( \\varphi , L ^ * \\psi ) _ { L ^ 2 ( \\mathbb { R } ) } \\end{align*}"}
-{"id": "3658.png", "formula": "\\begin{align*} { n \\choose n _ 0 } ^ 3 \\cdot \\beta ^ { m _ i } \\binom { n _ 0 ^ 2 } { m _ i } ^ 3 \\cdot p ^ { 3 m _ i } \\leq 2 ^ { 3 n } \\beta ^ { m _ i } \\left ( \\frac { e n _ 0 ^ 2 p } { m _ i } \\right ) ^ { 3 m _ i } \\leq 2 ^ { 3 n } \\beta ^ { m _ i } \\left ( \\frac { 2 e } { a } \\right ) ^ { 3 m _ i } = 2 ^ { 3 n - 3 m _ i } . \\end{align*}"}
-{"id": "5057.png", "formula": "\\begin{align*} E _ n = ( F _ n ^ { - 1 } s _ n ^ { - 1 } B ^ c ) ^ c = \\bigcap _ { f \\in F _ n } f ^ { - 1 } s _ n ^ { - 1 } B = \\big \\{ y \\in Y \\ , : \\ , s _ n F _ n \\subset B _ y \\big \\} \\subset Y \\end{align*}"}
-{"id": "7233.png", "formula": "\\begin{align*} w _ j = \\left ( \\begin{array} { c } w _ j ^ 1 \\\\ w _ j ^ 2 \\\\ \\vdots \\\\ w _ j ^ r \\end{array} \\right ) , \\ f _ { k j , \\alpha } = \\left ( \\begin{array} { c } f _ { k j , \\alpha } ^ 1 \\\\ f _ { k j , \\alpha } ^ 2 \\\\ \\vdots \\\\ f _ { k j , \\alpha } ^ r \\end{array} \\right ) . \\end{align*}"}
-{"id": "5628.png", "formula": "\\begin{align*} M ( \\Lambda _ 1 ) = \\left \\{ x , x y , x y ^ 2 , x y ^ 3 , \\ldots \\right \\} , \\end{align*}"}
-{"id": "637.png", "formula": "\\begin{align*} \\beta _ n ( B ) = \\frac { | B \\cap F _ n | } { | F _ n | } , \\textrm { f o r $ B \\subset G $ } . \\end{align*}"}
-{"id": "1598.png", "formula": "\\begin{align*} \\ln g \\left ( x \\right ) = \\mp \\kappa \\frac { \\left ( 2 \\left ( \\mu - \\lambda - x \\right ) + 1 \\right ) } { 3 } \\sigma \\end{align*}"}
-{"id": "6222.png", "formula": "\\begin{align*} \\mathbb { K } _ { j } ( \\mathcal { P } _ 1 , \\mathsf { Q } ) = \\{ \\mathsf { Q } , \\ \\mathcal { P } _ 1 \\mathsf { Q } , \\ \\mathcal { P } _ 1 ^ { 2 } \\mathsf { Q } , \\ \\ldots , \\ \\mathcal { P } _ 1 ^ { j - 1 } \\mathsf { Q } \\} . \\end{align*}"}
-{"id": "5191.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 2 / 5 } } = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { S _ { 1 / 2 } } \\| V \\| _ { F } \\\\ = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\ ! \\left ( \\frac { 4 \\| U \\| ^ { 1 / 2 } _ { S _ { 1 / 2 } } \\ ! + \\| V \\| ^ { 2 } _ { F } } { 5 } \\right ) ^ { 5 / 2 } . \\end{align*}"}
-{"id": "7588.png", "formula": "\\begin{align*} \\mathbb P \\left \\{ \\Omega ^ { ( 1 ) } _ \\varepsilon \\right \\} = \\int ^ { 1 } _ { - 1 + \\varepsilon / C } \\phi ( s ) d s = 1 - \\int _ { - 1 } ^ { - 1 + \\varepsilon / C } \\phi ( s ) d s \\ge 1 - \\varepsilon . \\end{align*}"}
-{"id": "7456.png", "formula": "\\begin{align*} e _ { \\pm i } ^ \\iota = e _ { \\pm i } a ^ \\iota = a ^ { - 1 } \\forall a \\in H . \\end{align*}"}
-{"id": "7245.png", "formula": "\\begin{align*} w _ j = u _ j + \\sum _ { | \\alpha | \\geq 2 } F _ { j , \\alpha } ( z _ j ) \\cdot u _ j ^ \\alpha , \\end{align*}"}
-{"id": "3586.png", "formula": "\\begin{align*} \\| ( \\psi , V ) \\| ^ 2 _ { L ^ 2 _ { \\rho ^ { - 1 } _ g } } & = \\| \\Pi _ g ( \\psi , V ) \\| ^ 2 _ { L ^ 2 _ { \\rho ^ { - 1 } _ g } } + \\| ( \\psi , V ) ^ { \\perp } \\| ^ 2 _ { L ^ 2 _ { \\rho ^ { - 1 } _ g } } . \\end{align*}"}
-{"id": "6276.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ { n + 1 } ^ 2 = \\tfrac { 1 } { K } . \\end{align*}"}
-{"id": "6357.png", "formula": "\\begin{align*} ( l + 1 ) ( l + 2 + 2 \\sigma _ n ) \\bar { p } _ { \\sigma , l + 1 } + 2 s ( \\sigma _ n + 1 ) \\bar { p } _ { \\sigma + \\bar { n } , l } + ( \\sigma _ i + 1 ) ( \\sigma _ i + 2 ) \\bar { p } _ { \\sigma + 2 \\bar { \\imath } , l - 1 } = 0 \\forall ( \\sigma , l ) . \\end{align*}"}
-{"id": "421.png", "formula": "\\begin{align*} a ^ k : = \\sum _ { j = 0 } ^ { k - 2 } \\Phi ^ \\theta _ j ( a ) \\ . \\end{align*}"}
-{"id": "9398.png", "formula": "\\begin{align*} \\widetilde { W } _ \\lambda = 2 i k - ( q , u _ \\lambda ) = 2 i k - \\frac { 4 \\textsf { R e } \\ c } { \\mu - i k } + \\frac { \\| q \\| ^ 2 } { ( \\mu - i k ) ^ 2 } \\end{align*}"}
-{"id": "5025.png", "formula": "\\begin{align*} A _ x = \\big \\{ g \\in G \\ , : \\ , g \\cdot x \\in A \\big \\} . \\end{align*}"}
-{"id": "6148.png", "formula": "\\begin{align*} { \\mathcal D } _ w ( T ) = \\{ d _ w ( Z , Z ' ) \\mid Z , Z ' \\subseteq T , \\ ; | Z | = | Z ' | = w \\} . \\end{align*}"}
-{"id": "6769.png", "formula": "\\begin{align*} \\theta _ { V _ { \\theta } } ^ n = \\lim _ { \\beta \\to + \\infty } \\theta _ { \\varphi _ { \\beta } } ^ n \\leq \\theta _ { + } ^ n . \\end{align*}"}
-{"id": "6199.png", "formula": "\\begin{align*} ( \\log | x | u ) ^ \\alpha - ( \\log | x | ) ^ \\alpha = ( \\log | x | ) ^ \\alpha \\left [ \\left ( 1 + \\frac { \\log u } { \\log | x | } \\right ) ^ \\alpha - 1 \\right ] \\leq \\alpha ( \\log | x | ) ^ { \\alpha - 1 } \\log u . \\end{align*}"}
-{"id": "7555.png", "formula": "\\begin{align*} x _ { n + 1 } ( \\omega ) = f ( x _ n ( \\omega ) ) + l \\chi _ { n + 1 } ( \\omega ) > f ( a ) - l = f ( a ) - a + a - l > l + a - l = a . \\end{align*}"}
-{"id": "8578.png", "formula": "\\begin{align*} \\overline { N } ( a ^ { \\ast } ) = \\overline { \\widehat { N } } ( \\widehat { \\varphi } ( a ^ { \\ast } ) ) \\psi _ { k } \\end{align*}"}
-{"id": "7996.png", "formula": "\\begin{gather*} a _ { 1 1 1 1 } = a _ { 2 0 1 1 } = a _ { 2 0 2 0 } = a _ { 2 1 0 1 } = a _ { 2 2 0 0 } = 0 , \\\\ a _ { 0 1 3 0 } = a _ { 0 3 1 0 } = a _ { 0 2 2 0 } , \\\\ a _ { 1 1 2 0 } = a _ { 0 1 3 0 } a _ { 2 0 0 2 } , a _ { 1 2 1 0 } = a _ { 0 3 1 0 } a _ { 2 0 0 2 } . \\end{gather*}"}
-{"id": "3833.png", "formula": "\\begin{align*} H ^ { 1 , 2 } ( D _ T ) : = H ^ 1 ( 0 , T ; L ^ 2 ( D ) ) \\cap L ^ 2 ( 0 , T ; H ^ 2 ( D ) ) . \\end{align*}"}
-{"id": "6613.png", "formula": "\\begin{align*} 0 & \\leq h ( x ) - h ( y ) - \\langle \\nabla h ( y ) , x - y \\rangle + \\tfrac { 1 } { 2 } ( \\| x \\| ^ 2 - \\| y \\| ^ 2 - 2 \\langle y , x - y \\rangle ) \\\\ & = h ( x ) - h ( y ) - \\langle \\nabla h ( y ) , x - y \\rangle + \\tfrac { 1 } { 2 } \\| x - y \\| ^ 2 \\leq \\| x - y \\| ^ 2 . \\end{align*}"}
-{"id": "5455.png", "formula": "\\begin{align*} \\widetilde { S } ( \\alpha ) = \\sum _ { n = 1 } ^ { \\infty } \\Lambda ( n ) e ^ { - n / N } e ( n \\alpha ) , \\end{align*}"}
-{"id": "9060.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { N / 2 } \\P \\left ( \\mathfrak { E } _ { B , N , N ' , y } , \\ , d W _ k \\leq - ( k ^ { \\frac { 9 } { 1 0 } } + B ^ { 9 / 5 } ) / 1 0 \\right ) \\ll _ d N ^ { - 3 / 2 } \\sum _ { k = 1 } ^ { \\infty } e ^ { - \\frac { k ^ { 0 . 2 } B ^ { 1 . 2 } } { 4 0 0 d ^ 2 } } \\leq \\frac { \\epsilon } { N ^ { \\frac { 3 } { 2 } } } \\end{align*}"}
-{"id": "5988.png", "formula": "\\begin{align*} \\lfloor w _ u \\rfloor = \\lfloor x _ 0 \\rfloor \\xleftarrow { z _ 1 \\gamma _ 1 ^ \\lor } \\cdots \\xleftarrow { z _ r \\gamma _ r ^ \\lor } \\lfloor x _ r \\rfloor = \\lfloor w _ { u + 1 } \\rfloor . \\end{align*}"}
-{"id": "264.png", "formula": "\\begin{align*} u _ n ^ * ( x ) : = u _ { x , a _ n / ( n - 1 ) } = \\frac { V _ d ( n - 1 ) h _ x ^ { - 1 } ( \\frac { a _ n } { n - 1 } ) ^ d } { e ^ { \\Psi ( k ) } } , \\end{align*}"}
-{"id": "8296.png", "formula": "\\begin{align*} d _ { \\tilde { x } } ( \\varphi ) : = \\sum _ { \\tilde { v } \\in T _ { \\tilde { x } } ( \\widetilde { \\Gamma } ) , \\varphi ( \\tilde { v } ) = v } d _ { \\tilde { v } } ( \\varphi ) \\end{align*}"}
-{"id": "45.png", "formula": "\\begin{align*} ( z _ 1 , \\dots , z _ { n + 1 } ) \\to t = z _ { k + 1 } ^ { b _ { k + 1 } } \\cdots z _ { n + 1 } ^ { b _ { n + 1 } } \\end{align*}"}
-{"id": "6634.png", "formula": "\\begin{align*} ( \\Phi \\# \\mu ) ( E ) : = \\mu \\bigl ( \\Phi ^ { - 1 } ( E ) \\bigr ) . \\end{align*}"}
-{"id": "5263.png", "formula": "\\begin{align*} \\operatorname * { d o m } \\varphi _ { A , k } = { \\mathbb { R } } k + A = { \\mathbb { R } } k + \\operatorname * { i n t } A \\not = \\emptyset \\mbox { i s a n o p e n s e t } , \\end{align*}"}
-{"id": "5978.png", "formula": "\\begin{align*} N ( x ) \\geq 7 2 - 4 0 = 3 2 > 5 \\times 6 , \\end{align*}"}
-{"id": "2914.png", "formula": "\\begin{align*} \\{ X ^ { ( 3 ) } ( z ) , X ^ { ( 3 ) } ( w ) \\} = 0 \\ , . \\end{align*}"}
-{"id": "7210.png", "formula": "\\begin{align*} \\Phi _ { \\beta , \\alpha } \\left ( \\partial \\mathcal { U } _ { \\alpha } ^ { S } \\left ( 3 \\right ) \\right ) = \\partial \\mathcal { U } _ { \\beta } ^ { S } \\left ( 3 \\right ) , \\end{align*}"}
-{"id": "4649.png", "formula": "\\begin{align*} \\langle x , y \\rangle = x _ 4 y _ 1 - \\frac { 1 } { 3 } x _ 3 y _ 2 + \\frac { 1 } { 3 } x _ 2 y _ 3 - x _ 1 y _ 4 . \\end{align*}"}
-{"id": "5181.png", "formula": "\\begin{gather*} R _ \\ell = C _ \\ell \\oplus \\varphi ( R _ \\ell ) , \\end{gather*}"}
-{"id": "9391.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\lambda _ 0 } \\| ( \\lambda - \\lambda _ 0 ) \\Xi ( \\lambda ) v \\| = \\infty . \\end{align*}"}
-{"id": "6787.png", "formula": "\\begin{align*} \\tilde \\varphi _ t ^ { \\delta } ( x ) : = \\int _ { B } ( \\tau ( x - \\zeta ) + \\varphi _ t ( x - \\zeta ) ) \\tilde \\rho _ { \\delta } ( \\zeta ) d V ( \\zeta ) - \\tau ( x ) . \\end{align*}"}
-{"id": "5591.png", "formula": "\\begin{align*} \\int _ m ^ \\infty e ^ { - i t \\lambda } \\chi _ j ( \\lambda ) [ \\mathcal R _ 0 ^ + ( \\lambda ) - \\mathcal R _ 0 ^ - ( \\lambda ) ] \\ , d \\lambda & = \\int _ 0 ^ \\infty e ^ { - i t \\sqrt { z ^ 2 + m ^ 2 } } \\frac { z \\chi _ j ( z ) [ \\mathcal R _ 0 ^ + ( \\lambda ) - \\mathcal R _ 0 ^ - ( \\lambda ) ] } { \\sqrt { z ^ 2 + m ^ 2 } } \\ , d z . \\end{align*}"}
-{"id": "4673.png", "formula": "\\begin{align*} \\left [ X _ j , Y _ { j _ 1 , \\dots , j _ k } ^ { ( n - 1 , i ) } \\right ] , \\tilde Y _ { j _ 1 , \\dots , j _ k } ^ { ( n - 1 , i ) } & \\in C _ { n + 1 } \\quad \\mbox { a s w e l l a s } \\\\ \\left [ X _ 0 , Y _ { j _ 1 , \\dots , j _ k } ^ { ( n - 1 , i ) } \\right ] + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ m \\left [ X _ j , \\left [ X _ j , Y _ { j _ 1 , \\dots , j _ k } ^ { ( n - 1 , i ) } \\right ] \\right ] & \\in C _ { n + 2 } \\end{align*}"}
-{"id": "3817.png", "formula": "\\begin{align*} ( - 1 ) ^ n \\sum _ { J \\subseteq I } ( - 1 ) ^ { | J | } ( 1 + t ) ^ { s ( \\lambda ; \\mu ) + 1 - | J | } x ^ { | \\lambda | - | \\mu | } & = ( - 1 ) ^ n ( 1 + t ) ^ { s ( \\lambda ; \\mu ) + 1 - | \\mathcal { I } | } x ^ { | \\lambda | - | \\mu | } \\left ( \\sum _ { J \\subseteq I } ( - 1 ) ^ { | J | } ( 1 + t ) ^ { | \\mathcal { I } | - | J | } \\right ) \\\\ & = ( - 1 ) ^ n t ^ { | \\mathcal { I } | } ( 1 + t ) ^ { s ( \\lambda ; \\mu ) + 1 - | \\mathcal { I } | } x ^ { | \\lambda | - | \\mu | } . \\end{align*}"}
-{"id": "2146.png", "formula": "\\begin{align*} B u = \\frac { d } { d t } ( g _ { 1 - \\alpha } * u ) , D ( B ) = \\{ u \\in L ^ { p } ( [ 0 , T ] ; X ) \\ , : \\ , g _ { 1 - \\alpha } * u \\in { _ { 0 } } W { ^ { 1 , p } } ( [ 0 , T ] ; X ) \\} . \\end{align*}"}
-{"id": "4681.png", "formula": "\\begin{align*} | I I | \\lesssim \\sum _ { n = N + 1 } ^ \\infty \\gamma ^ { - n \\alpha } = \\frac { \\gamma ^ { - ( N + 1 ) \\alpha } } { 1 - \\gamma ^ { - \\alpha } } = O ( h ^ \\alpha ) . \\end{align*}"}
-{"id": "1838.png", "formula": "\\begin{align*} \\gamma ( \\theta , s ) = ( w ( s , \\theta ) , \\theta ) = ( 2 + h ( s ) + s u ( \\theta , s ) , \\theta ) , \\end{align*}"}
-{"id": "9468.png", "formula": "\\begin{align*} \\dot \\gamma ( s ) & = 2 \\pi \\nabla f ( \\gamma ( s ) ) \\\\ \\dot \\theta ( s ) & = - 2 \\pi \\beta ( \\nabla f ( \\gamma ( s ) ) ) \\\\ \\dot a ( s ) & = 2 \\pi e ^ { f ( \\gamma ( s ) ) } \\end{align*}"}
-{"id": "7760.png", "formula": "\\begin{align*} \\mathbb P \\left [ T _ N \\le 0 \\right ] = \\mathbb P \\left [ T _ N \\ge 0 \\right ] = 1 - \\mathbb P \\left [ T _ N \\le 0 \\right ] \\end{align*}"}
-{"id": "8676.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\frac { \\nabla ^ G _ { \\xi } R _ t [ \\Phi ] ( x + s k ) - \\nabla ^ G _ { \\xi } R _ t [ \\Phi ] ( x ) } { s } = \\Gamma _ { t , x , k , \\xi } , \\ ; \\ ; k \\in K . \\end{align*}"}
-{"id": "4637.png", "formula": "\\begin{align*} F ( x ) = x ^ 9 + A _ 3 x ^ 6 + A _ 6 x ^ 3 \\end{align*}"}
-{"id": "6168.png", "formula": "\\begin{align*} \\lambda \\nabla ^ { L C , \\lambda } _ X ( Y ) = & \\lambda \\nabla ^ { L C } _ X ( Y ) + ( \\partial _ Y \\lambda ) X + ( \\partial _ X \\lambda ) Y - g ( X , Y ) Z \\\\ = & \\nabla ^ { L C } _ X ( \\lambda Y ) + g ( Y , Z ) X - g ( X , Y ) Z \\\\ = & \\nabla ^ { L C } _ X ( \\lambda Y ) + \\frac 1 4 \\big ( - ( Y Z + Z Y ) X - X ( Y Z + Z Y ) \\\\ & \\phantom { \\nabla ^ { L C } _ X ( \\lambda Y ) + \\frac 1 4 \\big ( } + ( X Y + Y X ) Z + Z ( X Y + Y X ) \\big ) \\\\ = & \\nabla ^ { L C } _ X ( \\lambda Y ) + \\frac 1 4 ( Y X Z + Z X Y - Y Z X - X Z Y ) \\end{align*}"}
-{"id": "5372.png", "formula": "\\begin{align*} P _ d ( x , y ) ~ = ~ c _ { d , d } { \\ , } x ^ d + \\sum _ { j = 0 } ^ r c _ { d , j } { \\ , } x ^ j y ^ { d - j } \\end{align*}"}
-{"id": "4669.png", "formula": "\\begin{align*} M = \\R ^ d \\quad \\mbox { a n d } X _ 0 , X _ 1 , \\dots , X _ m \\in C _ b ^ \\infty ( \\R ^ d , \\R ^ d ) \\ ; , \\end{align*}"}
-{"id": "9248.png", "formula": "\\begin{align*} \\sup _ { u \\in \\mathcal { A } _ { \\mathbb { H } } } J ( u ) = J ( u ^ { \\ast } ) . \\end{align*}"}
-{"id": "9089.png", "formula": "\\begin{align*} - T L ^ { ( \\alpha ) } _ { N } ( x ) = N L ^ { ( \\alpha ) } _ { N } ( x ) \\ , \\end{align*}"}
-{"id": "7024.png", "formula": "\\begin{align*} \\mathbb { E } \\ , \\| A ( \\omega ) \\| _ { \\ell ^ 2 } ^ 2 & = | c | ^ 2 \\cdot \\mathbb { E } \\ , \\| A ( \\omega ^ 2 ) \\| _ { \\ell ^ 2 } ^ 2 + \\mathbb { E } \\ , \\| B ( \\omega ^ 2 ) \\| _ { \\ell ^ 2 } ^ 2 \\\\ & = | c | ^ 2 \\cdot \\mathbb { E } \\ , \\| A ( \\omega ) \\| _ { \\ell ^ 2 } ^ 2 + \\mathbb { E } \\ , \\| B ( \\omega ) \\| _ { \\ell ^ 2 } ^ 2 . \\end{align*}"}
-{"id": "5042.png", "formula": "\\begin{align*} \\int _ G \\nu ( g B \\cap C ) \\ , d \\eta ( g ) = \\nu ( B ) \\ , \\nu ( C ) . \\end{align*}"}
-{"id": "4202.png", "formula": "\\begin{align*} ( w ^ t \\cdot v ) \\cdot u = \\sum _ { p < q } u _ { p q } , u _ { p q } = ( e _ p v _ q - e _ q v _ p ) \\cdot ( u _ p w _ q - u _ q w _ p ) \\in { } \\ ! R ^ n , \\end{align*}"}
-{"id": "689.png", "formula": "\\begin{align*} A _ { x _ o } A _ { x _ o } ^ { - 1 } F ^ { - 1 } \\supset \\big \\{ g \\in G \\ , : \\ , \\lambda ( ( A \\cap g F A ) _ { x _ o } ) > 0 \\big \\} = \\big \\{ g \\in G \\ , : \\ , \\nu ( A \\cap g F A ) > 0 \\big \\} . \\end{align*}"}
-{"id": "4682.png", "formula": "\\begin{align*} [ c _ \\omega ] . x _ { c , d } = - \\frac { \\omega _ \\gamma \\circ C ( c \\cdot d ) } { \\log ( \\gamma ) } . \\end{align*}"}
-{"id": "3287.png", "formula": "\\begin{align*} F \\hat { R } ( v _ { 1 } \\otimes v _ { 0 } ) & = F ( v _ { 0 } \\otimes v _ { 1 } ) + ( q ^ { 2 } - q ^ { - 2 } ) F ( v _ { 1 } \\otimes v _ { 0 } ) \\\\ & = [ 2 ] ^ { 1 / 2 } ( ( q ^ { 2 } - q ^ { - 2 } ) v _ { 1 } \\otimes v _ { - 1 } + ( q ^ { 2 } + 1 - q ^ { - 2 } ) v _ { 0 } \\otimes v _ { 0 } + q ^ { - 2 } v _ { - 1 } \\otimes v _ { 1 } ) . \\end{align*}"}
-{"id": "9776.png", "formula": "\\begin{align*} \\partial ^ t _ { p , q , \\max / \\min } = - * \\overline { \\partial } _ { m - q , m - p - 1 , \\max / \\min } * \\ \\ \\overline { \\partial } ^ t _ { p , q , \\max / \\min } = - * \\partial _ { m - q - 1 , m - p , \\max / \\min } * \\end{align*}"}
-{"id": "2298.png", "formula": "\\begin{align*} \\| ( e _ n ) _ { n = 0 } ^ { M - 1 } \\| _ { L ^ \\infty ( W ) } \\le r . \\end{align*}"}
-{"id": "1006.png", "formula": "\\begin{align*} Q _ { i } = \\Gamma _ { i } ^ \\dagger \\Gamma _ { i } , \\overline { Q } _ { j } = \\overline { \\Gamma } _ { j } ^ \\dagger \\overline { \\Gamma } _ { j } , i = 1 , 2 , \\cdots , n - 1 j = 1 , 2 , \\cdots , m - 1 . \\end{align*}"}
-{"id": "5940.png", "formula": "\\begin{align*} f - T _ { \\lambda } f = g . \\end{align*}"}
-{"id": "2416.png", "formula": "\\begin{align*} \\Lambda _ k ^ { \\alpha , \\ast } = \\{ l \\in \\mathbb { Z } ^ n : ~ \\Box _ l ^ { \\alpha } \\circ \\Box _ m ^ { \\alpha } \\neq 0 ~ ~ ~ m \\in \\Lambda _ k ^ { \\alpha } \\} . \\end{align*}"}
-{"id": "8971.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | w _ k ( x ) - w _ k ( y ) | ^ { p - 2 } \\ , ( w _ k ( x ) - w _ k ( y ) ) \\ , ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y = \\int _ { \\Omega } \\frac { f _ n ( x ) } { ( u ^ + _ k + 1 / n ) ^ \\gamma } \\ , \\varphi \\ , d x . \\end{align*}"}
-{"id": "7333.png", "formula": "\\begin{align*} F w _ { - 1 } = - [ 2 ] ^ { 1 / 2 } w _ 0 , F w _ 0 = - [ 2 ] ^ { 1 / 2 } q ^ 2 w _ 1 , F w _ 1 = 0 . \\end{align*}"}
-{"id": "6799.png", "formula": "\\begin{align*} Y ^ + ( \\xi ) = Y ^ - ( \\xi ) G ( \\xi ) . \\end{align*}"}
-{"id": "3164.png", "formula": "\\begin{align*} & \\| W _ { t } ^ { 0 } ( a ) - W _ { t } ^ { \\lambda } ( a ) \\| _ { 1 } \\\\ & = \\| \\lambda | t + a - 1 \\rangle \\langle t + a - 1 | ^ { \\mathfrak { B } } - \\lambda | t + 2 \\rangle \\langle t + 2 | ^ { \\mathfrak { B } } \\| _ { 1 } \\\\ & = 2 \\lambda \\end{align*}"}
-{"id": "3626.png", "formula": "\\begin{align*} u _ { n + 1 } = h \\left ( c T + ( 1 - c ) u _ n \\right ) , T \\ge 0 , \\ ; c \\in [ 0 , 1 ) , \\end{align*}"}
-{"id": "8060.png", "formula": "\\begin{gather*} A ( 1 , x ) = A _ 0 ( x ) , \\\\ A ( 0 , x ) = I , \\\\ A ( n , x ) = A _ 0 ( T ^ { n - 1 } ( x ) ) \\cdots A _ 0 ( T ( x ) ) A _ 0 ( x ) , \\\\ A ( - n , x ) = A _ 0 ( T ^ { - n } ( x ) ) ^ { - 1 } \\cdots A _ 0 ( T ^ { - 1 } ( x ) ) ^ { - 1 } , \\end{gather*}"}
-{"id": "7525.png", "formula": "\\begin{align*} \\mathbb P [ B _ n \\ , \\ , ] = 1 . \\end{align*}"}
-{"id": "8431.png", "formula": "\\begin{align*} \\textbf { P } ( A _ { 0 } \\partial _ { t } \\textbf { u } + \\sum _ { j = 1 } ^ { d } A _ { 0 } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } ) = 0 . \\end{align*}"}
-{"id": "2742.png", "formula": "\\begin{align*} f \\ast g = u ^ { - 1 } ( ( u f ) _ { K Q } \\star v ^ { Q P } \\star ( u g ) _ { P L } ) \\theta ^ K \\bar \\theta ^ L , \\end{align*}"}
-{"id": "1731.png", "formula": "\\begin{gather*} \\xi ^ a = \\mu _ b \\phi ^ { b a } , \\\\ I _ { a b } = 3 \\mu ^ c \\mu _ { [ c } \\phi _ { a b ] } = - \\varepsilon \\phi _ { a b } - 2 \\mu _ { [ a } \\xi _ { b ] } , \\\\ J _ { a b } = 3 \\mu ^ c \\phi _ { [ c a } \\theta _ { b ] } = \\mu ^ c ( \\ast \\phi ) _ { c a b } , \\\\ K _ { a b } = - 2 \\mu ^ c \\mu _ { [ a } \\phi _ { b ] c } = 2 \\mu _ { [ a } \\xi _ { b ] } . \\end{gather*}"}
-{"id": "9413.png", "formula": "\\begin{align*} a ( \\vec { x } ) = \\left \\{ \\begin{array} { l l } 1 0 0 0 , & \\sum _ { i = 1 } ^ 3 \\lfloor \\frac { x _ i n } { 7 } \\rfloor \\equiv 0 \\pmod { 2 } \\\\ 0 . 1 , & \\sum _ { i = 1 } ^ 3 \\lfloor \\frac { x _ i n } { 7 } \\rfloor \\equiv 1 \\pmod { 2 } \\\\ \\end{array} \\right . , \\end{align*}"}
-{"id": "2952.png", "formula": "\\begin{align*} h ( \\sigma , \\epsilon ) = \\max _ { \\mu _ 1 \\in \\bar { M } _ { \\epsilon } } g ( \\sigma , \\mu _ 1 ) . \\end{align*}"}
-{"id": "2757.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ B ^ { H } ( t ) B ^ { H } ( s ) \\big ] = \\frac { 1 } { 2 } ( t ^ { 2 H } + s ^ { 2 H } - | t - s | ^ { 2 H } ) . \\end{align*}"}
-{"id": "1861.png", "formula": "\\begin{align*} \\begin{aligned} a ' & \\le \\binom { | V | } { 2 } ^ { - 1 } \\bigg ( | W | \\binom { 2 p | V | } { 2 } + 2 ( c ' p ) ^ 2 | W | \\binom { | V | } { 2 } \\bigg ) \\\\ & \\le \\big ( 4 + 2 ( c ' ) ^ 2 \\big ) p ^ 2 | W | \\le 5 p ^ 2 | W | \\ , . \\end{aligned} \\end{align*}"}
-{"id": "8123.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\mathbb { R } ^ N } | x | ^ k | \\nabla u _ n | \\ , d x = P _ { \\mu _ k } ( M ) \\end{align*}"}
-{"id": "6459.png", "formula": "\\begin{align*} ( 1 - q ) \\zeta _ { 1 } ( q ) = \\frac { 2 q } { 3 } \\geq \\frac { 2 } { 3 } \\frac { \\beta _ { 0 } } { n + 2 } = : c _ { 1 } = c _ { 1 } ( n , \\beta _ { 0 } ) , \\end{align*}"}
-{"id": "1668.png", "formula": "\\begin{gather*} \\| D _ v { \\psi } ( \\cdot , v ) \\| _ { H ^ s _ p ( \\R ^ { d } ) } ^ p = \\int _ { \\R ^ d } | D _ v \\ , G _ { \\lambda } h _ s ( x , v ) | ^ p \\ , \\dd x \\ , . \\end{gather*}"}
-{"id": "3365.png", "formula": "\\begin{align*} F ( D , r ) & : = g ^ \\ast ( - D ) + \\frac { 1 } { 2 } \\| N + D \\| _ r ^ 2 + \\frac { 1 } { 2 } \\| N \\| _ F ^ 2 . \\end{align*}"}
-{"id": "5949.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\Big \\| \\big [ D U ( Z _ s ^ z ) - D U ( Y _ s ^ y ) \\big ] { R \\ , } \\Big \\| ^ 2 _ { H S } \\ , \\dd s = \\int _ 0 ^ t \\big | Z _ s ^ z - Y _ s ^ y \\big | ^ 2 \\ , \\dd A _ s \\end{align*}"}
-{"id": "9548.png", "formula": "\\begin{align*} d ( \\pi ( g ) , \\pi ( h ) ) & = d ( f _ { h s } ( x _ 0 ) , f _ h ( x _ 0 ) ) \\\\ & \\leq d ( f _ h ( f _ s ( x _ 0 ) ) , f _ h ( x _ 0 ) ) + B \\\\ & \\leq \\ell ( d ( f _ s ( x _ 0 ) , x _ 0 ) ) + B \\\\ & \\leq \\ell ( \\lambda ) + B . \\end{align*}"}
-{"id": "8742.png", "formula": "\\begin{gather*} Q _ t = \\int _ { 0 } ^ { t } e ^ { \\left ( t - s \\right ) A } \\left ( \\begin{array} [ c ] { c c } 0 & 0 \\\\ 0 & \\Lambda ^ { - 1 } \\end{array} \\right ) e ^ { \\left ( t - s \\right ) A ^ * } d s . \\end{gather*}"}
-{"id": "6822.png", "formula": "\\begin{align*} \\left [ \\partial _ x , L _ m \\right ] = 0 , \\left [ L _ m , L _ n \\right ] = ( n - m ) L _ { m + n } ( m , n \\in \\Z ) . \\end{align*}"}
-{"id": "7571.png", "formula": "\\begin{align*} x _ 1 = f ( x _ 0 ) + l \\chi _ 1 \\ge f ( x _ 0 ) - l \\ge f ( x _ 0 ) + \\tilde \\Delta _ l ( x _ 0 ) - f ( x _ 0 ) + x _ 0 = x _ 0 + \\tilde \\Delta _ l ( x _ 0 ) . \\end{align*}"}
-{"id": "6726.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\int _ 0 ^ t \\psi ( s ) d s + \\int _ 0 ^ t \\varphi ( s ) \\circ d B ^ { H } ( s ) , \\end{align*}"}
-{"id": "8601.png", "formula": "\\begin{align*} \\operatorname { k e r } ( \\overline { \\beta } ) = \\frac { \\operatorname { k e r } ( \\gamma ) } { \\operatorname { i m } ( \\beta ) } \\oplus \\{ 0 \\} \\oplus \\{ 0 \\} \\oplus V _ { k } \\end{align*}"}
-{"id": "4364.png", "formula": "\\begin{align*} W _ I T _ { c , h } ( f ) W _ I ^ * = T _ { \\tilde { c } , \\tilde { h } } ( f ) \\end{align*}"}
-{"id": "8668.png", "formula": "\\begin{gather*} [ f ] _ { \\alpha , Q } = \\sup _ { h ' , \\ ; h \\in H , h - h ' \\not = 0 } \\ ; { | f ( Q ^ { 1 / 2 } h ) - f ( Q ^ { 1 / 2 } h ' ) | } \\ , \\cdot { | h - h ' | _ H ^ { - \\alpha } } < \\infty \\end{gather*}"}
-{"id": "3176.png", "formula": "\\begin{align*} \\left \\{ \\bigoplus _ { \\lambda = 1 } ^ r L _ \\lambda \\left | L _ \\lambda \\in \\mathcal { P } ( \\widetilde { Y } ) , \\ d \\left ( \\mathbb { I } _ { \\widetilde { Y } } ^ { ( 1 ) } , \\ \\textstyle \\bigotimes _ { \\lambda = 1 } ^ r L _ \\lambda ^ { a _ \\lambda } \\right ) \\geq \\frac { 1 } { \\left ( 2 | a | \\right ) ^ A } \\ { \\rm f o r } \\ a = ( a _ \\lambda ) _ \\lambda \\in \\mathbb { Z } ^ r \\ { \\rm w i t h } \\ | a | \\geq 1 \\right \\} \\right . \\end{align*}"}
-{"id": "6650.png", "formula": "\\begin{align*} \\omega _ s ( \\Phi ) = | | \\varpi _ K | | ^ s = q _ K ^ { - s } \\end{align*}"}
-{"id": "675.png", "formula": "\\begin{align*} ( 1 - \\alpha _ 1 \\beta _ 1 - \\alpha _ 2 \\beta _ 2 ) m = \\alpha _ 1 \\gamma _ 1 \\lambda _ 1 + \\alpha _ 2 \\gamma _ 2 \\lambda _ 2 . \\end{align*}"}
-{"id": "3592.png", "formula": "\\begin{align*} \\Pi _ { g _ 0 } \\circ D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) = \\Pi _ { g _ 0 } ( \\psi , V ) , \\end{align*}"}
-{"id": "7183.png", "formula": "\\begin{align*} ( h ' + h \\cot \\varphi ) ' = g ( h ' + h \\cot \\varphi ) . \\end{align*}"}
-{"id": "4398.png", "formula": "\\begin{align*} h _ \\mu ( T ) = h _ { \\nu } ( \\tau ) = H _ \\nu ( \\eta ) < \\infty . \\end{align*}"}
-{"id": "6418.png", "formula": "\\begin{align*} & V _ { p } ( [ 0 , T ] ; \\Omega ) : = \\Big \\{ u \\in L ^ { 2 p } ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) \\cap L ^ { 2 } ( [ 0 , T ] ; H _ { e } ^ { \\beta } ( \\Omega ) ) \\\\ & g _ { 1 - \\alpha } * ( u - u _ { 0 } ) \\in C ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) , ( g _ { 1 - \\alpha } * ( u - u _ { 0 } ) ) | _ { t = 0 } = 0 \\Big \\} , \\end{align*}"}
-{"id": "9055.png", "formula": "\\begin{align*} \\lim _ { N \\to \\infty } \\sqrt { N } \\P \\left ( \\forall j \\in \\{ 1 , 2 , \\dots , N \\} , \\ , W _ j \\geq 0 \\right ) = c _ 0 . \\end{align*}"}
-{"id": "2524.png", "formula": "\\begin{align*} \\Lambda & = \\langle a , a + d , a + 2 d , \\ldots \\rangle . \\\\ \\intertext { S i n c e f o r $ j \\geq a $ w e h a v e t h a t $ a + j \\cdot d = d \\cdot a + a + ( j - a ) \\cdot d $ , t h e s e m i g r o u p i s g e n e r a t e d b y t h e f i n i t e a r i t h m e t i c p r o g r e s s i o n } \\Lambda & = \\langle a , a + d , a + 2 d , \\ldots , a + ( a - 1 ) d \\ , \\rangle . \\end{align*}"}
-{"id": "5456.png", "formula": "\\begin{align*} z = \\frac 1 N - 2 \\pi i \\alpha , \\end{align*}"}
-{"id": "3742.png", "formula": "\\begin{align*} f ( u ) = e ^ { - \\gamma } \\rho ( u ) . \\end{align*}"}
-{"id": "7673.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } s ^ { - ( 1 + \\alpha / d ) } F ( s ) d s < \\infty , \\ w h e r e \\ F ( s ) = \\sup _ { 1 \\leq t \\leq s } \\frac { f ( t ) } { t } . \\end{align*}"}
-{"id": "5289.png", "formula": "\\begin{align*} F ' ( n , t ) = \\tfrac { 1 } { 2 } \\log \\left ( \\tfrac { t } { 2 \\pi n ^ { 2 } } \\right ) , \\ F '' ( n , t ) = \\tfrac { 1 } { 2 t } > \\tfrac { 1 } { 4 T } . \\end{align*}"}
-{"id": "3517.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ 2 _ { \\rho _ g } ( \\Omega , g ) } = \\left ( \\int _ { \\Omega } | u | ^ 2 \\rho _ g \\ , d \\mu _ g \\right ) ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "774.png", "formula": "\\begin{gather*} R _ \\ell = C _ \\ell \\oplus \\varphi ( R _ \\ell ) , \\end{gather*}"}
-{"id": "9268.png", "formula": "\\begin{align*} \\int _ D Y ( t , x , z ) [ a _ 0 ( t , z ) p ( t , x , z ) + b _ 0 ( t , z ) q ( t , x , z ) ] d x = 0 . \\end{align*}"}
-{"id": "7937.png", "formula": "\\begin{align*} \\alpha ( w _ i , w _ j ) = \\frac { 1 } { m ^ 2 } \\sum _ { \\ell = 1 } ^ m \\sum _ { k = 1 } ^ m \\alpha _ H ( v _ { i , \\ell } , v _ { j , k } ) \\end{align*}"}
-{"id": "4814.png", "formula": "\\begin{align*} { \\mathcal E } ( \\tau ; { \\mathcal J } ) = \\{ X \\in { \\mathcal B } ( { \\mathcal H } ) \\mid [ X , T _ j ] \\in { \\mathcal J } , \\ 1 \\le j \\le n \\} \\end{align*}"}
-{"id": "2136.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } ( u ( x , t ) - u _ { 0 } ( x ) ) + L u ( x , t ) & = f ( x , t ) \\Omega \\times [ 0 , T ] , \\\\ u ( x , t ) & = 0 \\quad \\quad \\quad \\ , \\mathbb { R } ^ { n } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = u _ { 0 } ( x ) \\quad \\ , \\ , \\Omega , \\ , t = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1506.png", "formula": "\\begin{align*} \\left ( 1 - \\left ( \\frac { 1 } { x } - 1 \\right ) e _ 0 \\right ) ^ { - 1 } = x \\ , \\Big ( 1 - ( 1 - x ) ( e _ 0 + 1 ) \\Big ) ^ { - 1 } = ( y + 1 ) \\sum _ { n = 0 } ^ \\infty ( - y ) ^ n ( e _ 0 + 1 ) ^ n \\ ; , \\end{align*}"}
-{"id": "6402.png", "formula": "\\begin{align*} \\beta _ { q , p } = \\left ( \\frac { 1 } { C _ { 2 } ( \\vec { \\varepsilon } ) } - \\frac { ( 3 p - 1 ) ! ! } { 6 } \\varepsilon _ { p } ^ { 3 } \\sigma ^ { 3 p } - \\frac { ( 2 q + p - 1 ) ! ! } { 2 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } \\sigma ^ { 2 q + p } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "2900.png", "formula": "\\begin{align*} \\chi ( g ) - \\psi ( g ) = \\tilde { \\varepsilon } _ p ( u ) ( \\chi ( g ) - \\psi ( g ) ) - \\tilde { \\varepsilon } _ q ( u ) \\tilde { \\varepsilon } _ q ( u ) ( \\chi ( g ) - \\psi ( g ) + 1 ) = 0 . \\end{align*}"}
-{"id": "4624.png", "formula": "\\begin{align*} \\mu ( \\bar { d } _ { \\mathcal { E } } ) = 1 - \\cos ^ { \\alpha } \\left ( \\pi \\bar { d } _ { \\mathcal { E } } / 2 \\right ) , \\end{align*}"}
-{"id": "3396.png", "formula": "\\begin{align*} ( \\theta \\otimes \\mathrm { i d } ) \\circ \\nabla & = \\nabla ' \\circ \\theta \\\\ \\theta | _ { m _ i t _ i \\times S } ( l ^ { ( i ) } _ j ) & = l '^ { ( i ) } _ j \\otimes { \\mathcal L } \\quad ( ) \\end{align*}"}
-{"id": "2273.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & v ' ( t ) + A ( s ) v ( t ) = 0 , t > 0 , \\\\ & v ( 0 ) = v _ 0 . \\end{aligned} \\right . \\end{align*}"}
-{"id": "9610.png", "formula": "\\begin{align*} [ 0 , T ] = [ 0 , t _ 0 ] \\cup [ t _ 0 , 2 t _ 0 ] \\cup . . . \\cup [ n t _ 0 , T ] \\end{align*}"}
-{"id": "2729.png", "formula": "\\begin{align*} \\alpha ^ * = ( \\underbrace { 0 , \\dots , 0 } _ { m } , \\alpha ^ * _ m , 1 - \\alpha ^ * _ m , \\underbrace { 0 , \\dots , 0 } _ { N - m } ) \\end{align*}"}
-{"id": "1179.png", "formula": "\\begin{align*} D ( - \\lambda ) = W ( \\hat c _ 0 , c _ 0 ) \\prod _ { j = 1 } ^ N \\left ( 1 + \\frac { \\lambda } { z _ j } \\right ) . \\end{align*}"}
-{"id": "3159.png", "formula": "\\begin{align*} & C _ s ( \\{ ( W _ t , { V } _ t ) : t \\in \\theta \\} ; c r ) \\\\ & = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\max _ { \\Lambda _ n } \\Bigl ( \\inf _ { B _ q \\in C o n v ( ( B _ t ) _ { t \\in \\theta } ) } \\chi ( p _ U ; B _ q ^ { \\otimes n } ) - \\max _ { t ^ n \\in \\theta ^ n } \\chi ( p _ U ; Z _ { t ^ n } ) \\Bigr ) \\end{align*}"}
-{"id": "1183.png", "formula": "\\begin{align*} v _ { x x } = 0 , x \\neq x _ j , [ v _ x ] ( x _ j ) = - z m _ j v ( x _ j ) , \\end{align*}"}
-{"id": "6832.png", "formula": "\\begin{align*} & M ( u ) = \\int _ { \\R ^ d } | u ( t , x ) | ^ 2 \\ , d x , \\\\ & E ( u ) = \\int _ { \\R ^ d } \\tfrac 1 2 | \\nabla u ( t , x ) | ^ 2 + \\tfrac { \\mu } { p + 2 } | u ( t , x ) | ^ { p + 2 } \\ , d x , \\end{align*}"}
-{"id": "6734.png", "formula": "\\begin{align*} n ( \\hat F _ n ( P ) - F _ { 2 : n } ) & = H _ 1 + \\sum _ { i = 2 } ^ n 1 _ { E _ i } \\left ( H _ i - H _ i ' \\right ) , \\end{align*}"}
-{"id": "5011.png", "formula": "\\begin{align*} & b _ { 0 } = a _ { 0 } \\\\ & b _ { n } = C \\left ( b _ { 0 } + \\sqrt [ ] { b _ { 0 } } + \\cdots + \\sqrt [ ] { b _ { n - 1 } } \\right ) \\ \\ \\ \\ . \\end{align*}"}
-{"id": "4321.png", "formula": "\\begin{align*} L _ m : = L + ( m - 1 ) ( K _ { X / Y } + L ) \\end{align*}"}
-{"id": "2155.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } ( u ( t , x ) - u _ { 0 } ( x ) ) = r ^ { - 2 \\beta } \\partial _ { s } ^ { \\alpha } ( \\tilde { u } ( s , y ) - \\tilde { u } _ { 0 } ( y ) ) \\end{align*}"}
-{"id": "2580.png", "formula": "\\begin{align*} \\liminf _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\norm { e ^ { i t \\Delta } W ^ J _ n } _ { L _ t ^ { q , \\infty } L _ x ^ r ( [ t _ n , \\infty ) \\times \\R ^ d ) } = 0 , \\end{align*}"}
-{"id": "1772.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\Delta \\overline { u } _ { i } = \\frac { 1 } { \\varepsilon } \\overline { u } _ { i } \\sum \\limits _ { j \\neq i } H ( \\underline { u } _ { j } ) ( x ) & \\Omega , \\\\ \\Delta \\underline { u } _ { i } = \\frac { 1 } { \\varepsilon } \\underline { u } _ { i } \\sum \\limits _ { j \\neq i } H ( \\overline { u } _ { j } ) ( x ) & \\Omega . \\end{array} \\right . \\end{align*}"}
-{"id": "3977.png", "formula": "\\begin{align*} [ \\mathcal { A } _ i , \\mathfrak { n } ( \\varphi ( s _ i ) ) ] = 2 \\mathfrak { n } ( \\varphi ( s _ i ) ) [ \\mathcal { A } _ j , \\mathfrak { n } ( \\varphi ( s _ i ) ) ] = \\mathfrak { n } ( \\varphi ( s _ i ) ) . \\end{align*}"}
-{"id": "7428.png", "formula": "\\begin{align*} \\overline { u } e _ { i ( m ) } e _ { - i ( m ) } ^ { - 1 } \\dots e _ { i ( 1 ) } e _ { - i ( 1 ) } ^ { - 1 } = \\overline { s _ { i ( 1 ) } \\dots s _ { i ( m - 1 ) } } e _ { i ( m ) } ^ { - 1 } e _ { i ( m - 1 ) } e _ { - i ( m - 1 ) } ^ { - 1 } \\dots e _ { i ( 1 ) } e _ { - i ( 1 ) } ^ { - 1 } . \\end{align*}"}
-{"id": "7531.png", "formula": "\\begin{align*} d _ 1 : = \\min _ { x \\in [ \\mu _ 1 , c ] } ( f ( x ) - x ) > 0 , \\end{align*}"}
-{"id": "6528.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) , \\end{aligned} \\end{align*}"}
-{"id": "3095.png", "formula": "\\begin{align*} \\mathcal { N } _ l ( P ) : = & \\{ y ( \\lambda ) ^ T \\in \\mathbb { F } ( \\lambda ) ^ { 1 \\times m } \\mbox { s u c h t h a t } y ( \\lambda ) ^ T P ( \\lambda ) = 0 \\} , \\\\ \\mathcal { N } _ r ( P ) : = & \\{ x ( \\lambda ) \\in \\mathbb { F } ( \\lambda ) ^ { n \\times 1 } \\mbox { s u c h t h a t } P ( \\lambda ) x ( \\lambda ) = 0 \\} . \\end{align*}"}
-{"id": "1744.png", "formula": "\\begin{align*} G _ \\infty ^ + ( x , y ) = \\frac { k _ N ^ s } { 2 } | x - y | ^ { 2 s - N } \\int _ 0 ^ { \\psi _ \\infty ^ + ( x , y ) } \\frac { t ^ { s - 1 } } { ( t + 1 ) ^ \\frac { N } { 2 } } d t , \\end{align*}"}
-{"id": "4249.png", "formula": "\\begin{align*} X _ { q _ 0 , \\gamma } = \\{ \\nu \\in L _ { q _ 0 , } ^ 1 , \\lim _ { j \\to \\infty } j ^ { \\gamma } | \\hat { \\nu } _ j | = 0 \\} \\end{align*}"}
-{"id": "333.png", "formula": "\\begin{align*} | 2 | = q ^ { - \\ell _ 0 } . \\end{align*}"}
-{"id": "5737.png", "formula": "\\begin{align*} \\frac { \\partial Z _ { t } } { \\partial \\psi } \\vert _ { \\delta _ 0 } = \\sum _ { j = 0 } ^ { \\infty } \\tau _ { j } ( \\omega ) e _ { t - J _ L - j } ( \\delta _ { 0 } ) . \\end{align*}"}
-{"id": "9409.png", "formula": "\\begin{align*} A u = f \\end{align*}"}
-{"id": "4132.png", "formula": "\\begin{gather*} ( p , q ) = ( - 4 - 6 u , 1 + 6 u ) , ( p , q ) = ( - 3 - 6 u , - 1 + 6 u ) , \\\\ ( p , q ) = ( - 5 - 6 u , 2 + 6 u ) , ( p , q ) = ( - 1 - 6 u , - 2 + 6 u ) , \\\\ ( p , q ) = ( - 3 - 2 u , 1 + 2 u ) , \\end{gather*}"}
-{"id": "1809.png", "formula": "\\begin{align*} z _ j = \\exp ( - s _ j ^ { - 1 } + i \\theta _ j ) \\end{align*}"}
-{"id": "8174.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ n \\tbinom { \\lambda - 1 } { k - 1 } ( k - 1 ) ! B _ { n , k } ( 1 ! z _ 1 , 2 ! z _ 2 , \\dots ) & = \\sum _ { k = 1 } ^ n \\tbinom { \\lambda + a n - 1 } { k - 1 } ( k - 1 ) ! B _ { n , k } ( 1 ! x _ 1 , 2 ! x _ 2 , \\dots ) \\\\ \\intertext { f o r a n y $ \\lambda \\in \\mathbb { C } $ , a n d } \\sum _ { k = 1 } ^ n \\tbinom { - 1 } { k - 1 } ( k - 1 ) ! B _ { n , k } ( 1 ! z _ 1 , 2 ! z _ 2 , \\dots ) & = \\sum _ { k = 1 } ^ n \\tbinom { a n - 1 } { k - 1 } ( k - 1 ) ! B _ { n , k } ( 1 ! x _ 1 , 2 ! x _ 2 , \\dots ) . \\end{align*}"}
-{"id": "1295.png", "formula": "\\begin{align*} z = \\frac 1 N - 2 \\pi i \\alpha , \\end{align*}"}
-{"id": "5510.png", "formula": "\\begin{align*} P ( x ) = P _ { \\varphi , w } ( x ) = \\inf \\left \\{ \\int _ I \\varphi \\left ( | x | / { v } \\right ) v : v \\prec w , v \\ge 0 \\right \\} , \\ \\ \\ \\ x \\in L ^ 0 , \\\\ \\end{align*}"}
-{"id": "9445.png", "formula": "\\begin{align*} \\frac { | \\partial _ { R _ 0 } \\widetilde { F } | } { | \\widetilde { F } | } & = \\frac { \\sum _ { j \\in J _ 0 } | \\partial _ { R _ 0 } F _ j | + \\sum _ { i \\in I _ 0 '' } | \\partial _ { R _ 0 } F _ { 0 , i } | } { | \\widetilde { F } | } \\\\ & = \\frac { \\sum _ { i \\in I _ 0 '' } | \\partial _ { R _ 0 } F _ { 0 , i } | } { | \\widetilde { F } | } \\le \\frac { | \\partial _ { R _ 0 } F _ 0 | } { | F _ 0 | } \\le \\varepsilon _ 0 , \\end{align*}"}
-{"id": "6090.png", "formula": "\\begin{align*} \\sum _ { 1 \\le n \\le t } \\widetilde { G } _ { n , n } ( t ) & = \\sum _ { 1 \\le n \\le t } \\frac { 1 } { \\sqrt { n + 1 } } \\int _ n ^ { n + 1 } v ^ { - 1 / 2 } \\exp \\left ( - t \\left ( i \\log { \\frac { n + 1 } { v } } + \\frac { 1 } { 2 } \\log ^ 2 { \\frac { n + 1 } { v } } \\right ) \\right ) d v \\\\ & = \\sum _ { 1 \\le n \\le t } \\int _ 1 ^ { 1 + 1 / n } v ^ { - 3 / 2 } \\exp \\left ( - t \\left ( i \\log v + \\frac { 1 } { 2 } \\log ^ 2 v \\right ) \\right ) d v . \\end{align*}"}
-{"id": "6971.png", "formula": "\\begin{align*} h ( \\sigma , \\epsilon ) > h ( \\sigma , 0 ) = g ( \\sigma , 1 / 2 ) . \\end{align*}"}
-{"id": "5344.png", "formula": "\\begin{align*} \\frac { 1 } { | G | } \\prod _ { i = 1 } ^ { \\ell } ( n - \\chi _ { \\gamma } ( c _ i ) ) = | K ( \\gamma ) | \\end{align*}"}
-{"id": "3253.png", "formula": "\\begin{align*} [ E _ \\xi , F _ t ] ^ * K _ t = ( K _ t E _ \\xi F _ { t } - K _ t F _ t E _ \\xi ) ^ * = ( q ^ { ( \\alpha _ t , \\xi ) } E _ \\xi K _ t F _ t - K _ t F _ t E _ \\xi ) ^ * . \\end{align*}"}
-{"id": "9097.png", "formula": "\\begin{align*} & ( \\alpha - 3 ) u _ { N - 1 } ' ( s ) - ( 2 s + 1 ) u _ { N - 1 } '' ( s ) + 2 K _ { N } ( s ) \\\\ & = ( \\alpha - 3 ) u _ { N - 1 } ' ( s ) + ( s - 1 ) u _ { N - 1 } '' ( s ) - 3 s u _ { N - 1 } '' ( s ) + 2 K _ { N } ( s ) \\\\ & = - 5 u _ { N - 1 } ' ( s ) - 3 s u _ { N - 1 } '' ( s ) - \\xi _ { N } ( s ) \\ , \\end{align*}"}
-{"id": "5703.png", "formula": "\\begin{align*} f = \\sum _ { j = 3 } ^ \\infty \\chi _ { [ j ! , j ! + 1 ] } , \\end{align*}"}
-{"id": "7693.png", "formula": "\\begin{align*} f ( x ) + \\gamma _ n > f ( x ) - x + x - \\delta _ 0 > \\delta _ 0 + x - \\delta _ 0 = x \\geq \\varepsilon _ 0 , \\end{align*}"}
-{"id": "8005.png", "formula": "\\begin{align*} \\xi ( y ) = \\varphi _ p ^ { - 1 } ( y ) \\int _ 0 ^ { \\infty } D ( y , t ) \\varphi _ p ( X ( y , t ) ) d t . \\end{align*}"}
-{"id": "1061.png", "formula": "\\begin{align*} e ( H ) \\leq \\begin{cases} \\left ( 1 - \\frac { 1 } { c } \\right ) \\frac { v ( H ) ^ 2 } { 2 } & v ( H ) \\leq n \\sqrt { \\frac { c } { c - 1 } } \\ , , \\\\ \\frac { n ^ 2 } { 2 } & n \\sqrt { \\frac { c } { c - 1 } } < v ( H ) \\leq \\frac { 5 n } { 4 } \\ , , \\\\ \\frac { n } { 2 } \\left ( v ( H ) - \\frac { n } { 4 } \\right ) & \\frac { 5 n } { 4 } < v ( H ) \\ , . \\end{cases} \\end{align*}"}
-{"id": "7311.png", "formula": "\\begin{align*} \\Delta ( \\mathfrak { l } ) = \\{ \\pm \\alpha _ 1 \\} , \\Delta ( \\mathfrak { u } _ + ) = \\{ \\alpha _ 2 , \\ \\alpha _ 1 + \\alpha _ 2 , \\ 2 \\alpha _ 1 + \\alpha _ 2 \\} . \\end{align*}"}
-{"id": "3308.png", "formula": "\\begin{align*} v = \\sum _ { i } \\frac { \\langle w _ { i } ^ { ( k ) } , v \\rangle } { \\langle w _ { i } ^ { ( k ) } , v _ { i } ^ { ( k ) } \\rangle } v _ { i } ^ { ( k ) } . \\end{align*}"}
-{"id": "5507.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { U L } = & \\lim _ { \\rho \\to 0 } \\tilde { \\gamma } _ k ^ \\mathrm { U L } \\\\ \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { D L } = & \\lim _ { \\rho \\to 0 } \\tilde { \\gamma } _ k ^ \\mathrm { D L } . \\end{align*}"}
-{"id": "3534.png", "formula": "\\begin{align*} | J + V | _ g & = | \\chi J _ 1 + ( 1 - \\chi ) J _ 2 | _ g \\\\ & \\le \\chi | J _ 1 | _ { g _ 1 } + ( 1 - \\chi ) | J _ 2 | _ { g _ 2 } + \\chi ( 1 - \\chi ) ( | g _ 1 - g _ 2 | _ { g _ 1 } | J _ 1 | _ { g _ 1 } + | g _ 1 - g _ 2 | _ { g _ 2 } | J _ 2 | _ { g _ 2 } ) , \\end{align*}"}
-{"id": "2366.png", "formula": "\\begin{align*} ( \\Phi \\# \\mu ) ( E ) : = \\mu \\bigl ( \\Phi ^ { - 1 } ( E ) \\bigr ) . \\end{align*}"}
-{"id": "9608.png", "formula": "\\begin{align*} p _ \\tau ( r , t ) = \\int _ \\tau ^ t \\frac { \\theta ( t - s - r ) J _ 1 ( m \\sqrt { ( t - s ) ^ 2 - r ^ 2 } ) } { \\sqrt { ( t - s ) ^ 2 - r ^ 2 } } h ( s ) d s , r \\ge 0 . \\end{align*}"}
-{"id": "4676.png", "formula": "\\begin{align*} \\langle P [ P , a ] [ P , b ] e _ l , e _ l \\rangle _ { L ^ 2 ( S ^ 1 ) } & = - \\langle ( 1 - P ) b _ - e _ l , ( \\overline { a } ) _ - e _ l \\rangle _ { L ^ 2 ( S ^ 1 ) } \\\\ & = - \\left \\langle ( 1 - P ) \\sum _ { k = 1 } ^ \\infty b _ { - k } e _ { l - k } , \\sum _ { k = 1 } ^ \\infty \\overline { a _ { k } } e _ { l - k } \\right \\rangle _ { L ^ 2 ( S ^ 1 ) } = - \\sum _ { k > l } a _ { k } b _ { - k } . \\end{align*}"}
-{"id": "9906.png", "formula": "\\begin{align*} K E & = q E K , & K F & = q ^ { - 1 } F K , & K K ' & = K ' K , \\\\ K ' E & = q ^ { - 1 } E K ' & K ' F & = q F K ' , & [ E , F ] & = \\frac { K ^ 2 - K ' \\ , ^ 2 } { q - q ^ { - 1 } } . \\end{align*}"}
-{"id": "6551.png", "formula": "\\begin{align*} \\big \\| ( e _ n ) _ { n = 0 } ^ M \\big \\| _ { L ^ \\infty ( W ) } \\le C \\delta . \\end{align*}"}
-{"id": "7820.png", "formula": "\\begin{align*} \\mathbb { D } _ { t } ^ { H } \\Phi = \\int _ 0 ^ T \\phi ( t - s ) D _ { s } ^ { H } \\Phi d s , \\ \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "2936.png", "formula": "\\begin{align*} \\chi _ { \\mathfrak { g } } ( \\lambda ) & = \\det \\left [ \\begin{array} { c } h ^ { ( j - 1 ) } _ { \\lambda _ i - i + 1 } \\end{array} \\right ] \\ , , h ^ { ( r ) } _ a = \\left \\{ \\begin{array} { l l } J _ { a + r } + J _ { a - r } & r > 0 \\\\ J _ { a + r } & r \\leq 0 \\end{array} \\ , , \\right . \\end{align*}"}
-{"id": "120.png", "formula": "\\begin{align*} & C h _ { n , \\frac { 1 } { 2 } } = \\sum _ { m = 0 } ^ n { n \\choose m } C _ m C h _ { n - m , \\frac { 1 } { 2 } } ( - 1 ) ^ { m } \\frac { ( m + 1 ) ! } { 4 ^ m ( 2 m - 1 ) } , \\\\ \\textnormal { a n d } & \\\\ & C h _ { 0 , \\frac { 1 } { 2 } } = 1 . \\end{align*}"}
-{"id": "2522.png", "formula": "\\begin{align*} \\phi ^ { \\Delta } _ { i } = \\sum _ { \\vec { c } \\in \\Delta } \\phi ^ { \\Delta , \\vec { c } } _ { i } . \\end{align*}"}
-{"id": "193.png", "formula": "\\begin{align*} h _ x ^ { - 1 } ( s ) : = \\inf \\{ r > 0 : h _ x ( r ) \\geq s \\} = \\inf \\{ r > 0 : h _ x ( r ) = s \\} , \\end{align*}"}
-{"id": "5885.png", "formula": "\\begin{align*} A ^ { i j } ( a _ { i j } u + b _ { i j } ) - ( a _ { , i } u + b _ { , i } ) B ^ { i } = 0 , \\end{align*}"}
-{"id": "4329.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ l D _ { r , i } \\left ( C _ { r - s , i } - C _ { 1 , i } \\right ) \\end{align*}"}
-{"id": "6656.png", "formula": "\\begin{align*} r _ 1 t _ 1 + \\cdots r _ m t _ m = n . \\end{align*}"}
-{"id": "1944.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\lambda _ i \\cdot \\nabla F _ i ( \\xi ) = - ( \\delta _ 1 , \\cdots , \\delta _ m ) \\end{align*}"}
-{"id": "5718.png", "formula": "\\begin{align*} | m _ f ( Q ) | \\leq \\left ( f \\cdot \\chi _ Q \\right ) ^ * ( \\lambda | Q | ) , \\lim _ { \\ell ( Q ) \\to 0 , Q \\ni x } m _ f ( Q ) = f ( x ) ( { \\rm a . e . } \\ x \\in \\R ^ n ) . \\end{align*}"}
-{"id": "9771.png", "formula": "\\begin{align*} \\nabla _ { \\vec { x } } V ( \\vec { x } ) = \\vec { 0 } _ { n } , \\forall \\vec { x } \\in \\mathcal { R } ( \\emptyset ) \\cap \\mathrm { i n t } ( \\mathcal { X } ) . \\end{align*}"}
-{"id": "9503.png", "formula": "\\begin{align*} H ^ \\phi ( \\partial _ \\mu , \\partial _ \\nu ) & = \\nabla _ \\mu \\partial _ \\nu \\phi \\\\ \\Delta \\phi & = \\nabla ^ \\mu \\partial _ \\mu \\phi . \\end{align*}"}
-{"id": "2575.png", "formula": "\\begin{align*} \\liminf _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\norm { e ^ { i t \\Delta } W ^ J _ n } _ { L _ t ^ { q , \\infty } L _ x ^ r ( \\R \\times \\R ^ d ) } = 0 \\end{align*}"}
-{"id": "9688.png", "formula": "\\begin{gather*} \\lim _ { t \\to \\infty } \\frac { \\sigma ( t ) } { t \\log \\varphi ( t ) } = 1 , \\\\ \\lim _ { t \\to \\infty } \\frac { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } { \\log t / \\log \\varphi ( t ) } = 1 . \\end{gather*}"}
-{"id": "8909.png", "formula": "\\begin{align*} D u ( x ) [ \\Hat { A } ( P _ { W _ \\lambda } ( x ) ) ] = 0 . \\end{align*}"}
-{"id": "8160.png", "formula": "\\begin{align*} p _ i = \\frac { ( t + w _ h ) ( t - w _ h ) } { t - w _ h ^ 2 } . \\end{align*}"}
-{"id": "4529.png", "formula": "\\begin{align*} \\{ y \\in \\mathcal { X } _ n : \\| x - y \\| < r _ { n , u _ n ^ * ( x ) } + r _ { n , u _ n ^ * ( y ) } \\} = \\emptyset \\end{align*}"}
-{"id": "8958.png", "formula": "\\begin{align*} U ( x , 0 ) = A e ^ { - x ^ { 2 } } \\end{align*}"}
-{"id": "7178.png", "formula": "\\begin{align*} - \\Delta u + ( u \\cdot \\nabla ) u + \\nabla p = f , \\qquad { \\rm d i v } \\ , u = 0 \\mbox { i n } \\ , \\ , \\Omega . \\end{align*}"}
-{"id": "4156.png", "formula": "\\begin{align*} c _ { f , g } = \\frac { 1 } { 4 \\pi ^ 2 ( 2 \\kappa + \\frac { 3 } { 2 } - 2 \\nu ) } \\sum _ { n \\geq 1 } \\frac { a _ f ( n ) \\overline { a _ g ( n ) } } { n ^ { 2 \\kappa + \\frac { 3 } { 2 } - 2 \\nu } } , \\end{align*}"}
-{"id": "8471.png", "formula": "\\begin{align*} \\tilde { \\rho } ^ { \\varepsilon } _ { 0 } ( x ) = \\rho ^ { \\varepsilon } _ { 0 } ( x ) - \\bar { \\rho } , ~ ~ ~ ~ ~ \\tilde { v } ^ { \\varepsilon } _ { 0 } ( x ) = v _ { 0 } ^ { \\varepsilon } ( x ) , \\end{align*}"}
-{"id": "9603.png", "formula": "\\begin{align*} ( | k | ^ 2 + m ^ 2 ) \\ , \\tilde { u _ 2 } ( k , t ) = \\int _ 0 ^ t \\frac { \\sin ( ( t - s ) \\sqrt { | k | ^ 2 + m ^ 2 } ) } { \\sqrt { | k | ^ 2 + m ^ 2 } } h ( s ) d s , \\sqrt { | k | ^ 2 + m ^ 2 } \\ , \\tilde { \\dot u _ 2 } ( k , t ) = \\int _ 0 ^ t \\frac { \\cos ( ( t - s ) \\sqrt { | k | ^ 2 + m ^ 2 } ) } { \\sqrt { | k | ^ 2 + m ^ 2 } } h ( s ) d s . \\end{align*}"}
-{"id": "1094.png", "formula": "\\begin{align*} \\alpha _ 0 = 3 8 3 / 3 8 4 \\end{align*}"}
-{"id": "127.png", "formula": "\\begin{align*} \\| x ^ * \\| = \\sup _ { \\| z \\| _ X \\leq 1 } | x ^ * ( z ) | = \\sup _ { z \\in \\mbox { c o } ( B _ X ) } | x ^ * ( z ) | = \\sup _ { \\| z \\| _ { \\widehat { X } } \\leq 1 } | x ^ * ( z ) | . \\end{align*}"}
-{"id": "6306.png", "formula": "\\begin{align*} \\| \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ p } = \\| \\sum _ { k \\in \\Gamma _ l ^ { \\alpha _ 2 , \\alpha _ 1 } } \\Box _ k ^ { \\alpha _ 2 } \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ p } . \\end{align*}"}
-{"id": "4748.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u & \\\\ u ( x , 0 ) = u _ 0 ( x ) & \\end{cases} \\end{align*}"}
-{"id": "8428.png", "formula": "\\begin{align*} A _ { 0 } ( \\textbf { u } ) F _ { P } = d i a g \\Bigg ( \\frac { f ( \\textbf { u } ) } { \\rho } , 1 , 1 , \\cdots , 1 \\Bigg ) \\cdot ( 0 , \\nabla P ) ^ { T } = ( 0 , \\nabla P ) ^ { T } = F _ { P } . \\end{align*}"}
-{"id": "7572.png", "formula": "\\begin{align*} x _ 2 = f ( x _ 1 ) + l \\chi _ 2 \\ge f ( x _ 1 ) - l \\ge f ( x _ 1 ) + \\tilde \\Delta _ l ( x _ 1 ) - f ( x _ 1 ) + x _ 1 \\ge x _ 0 + \\tilde \\Delta _ l ( x _ 0 ) + \\tilde \\Delta _ l ( x _ 1 ) \\ge x _ 0 + 2 \\tilde \\Delta _ l ( x _ 0 ) . \\end{align*}"}
-{"id": "1818.png", "formula": "\\begin{align*} z ' = \\alpha z ( 1 + z \\beta ( z ) ) \\end{align*}"}
-{"id": "3693.png", "formula": "\\begin{align*} \\{ F , Z , H \\} _ { \\zeta } = \\int F _ { \\zeta } J ( H _ { \\zeta } , Z _ { \\zeta } ) d A ~ , \\end{align*}"}
-{"id": "5907.png", "formula": "\\begin{align*} \\Lambda _ { t } = \\left \\{ \\left ( A \\left ( t - t _ { 0 } \\right ) ^ { \\beta } , A \\beta \\left ( t - t _ { 0 } \\right ) ^ { \\beta - 1 } \\right ) \\in \\mathbb { R } ^ { 2 } : t _ { 0 } \\in \\left [ 0 , t \\right ] \\right \\} . \\end{align*}"}
-{"id": "6887.png", "formula": "\\begin{align*} \\limsup \\lambda _ i ( \\alpha ) = \\lambda _ 0 \\end{align*}"}
-{"id": "7824.png", "formula": "\\begin{align*} E ' [ f ( s , Y ' _ { s } , Z ' _ { s } ) ] = E [ f ( s , Y _ { s } , Z _ { s } ) ] , \\ \\ E ' [ f ( s , \\eta _ { s } , Y _ { s } , Z _ { s } ) ] = f ( s , \\eta _ { s } , Y _ { s } , Z _ { s } ) . \\end{align*}"}
-{"id": "432.png", "formula": "\\begin{align*} \\mathcal { B } _ n = \\left \\{ Y _ { j _ 1 , \\dots , j _ k } ^ { ( n , i ) } \\colon 1 \\leq k \\leq n , 0 \\leq j _ 1 , \\dots , j _ k \\leq m , 1 \\leq i \\leq m \\right \\} & \\subset C _ { n + 1 } \\quad \\mbox { a n d } \\\\ \\tilde { \\mathcal { B } } _ n = \\left \\{ \\tilde Y _ { j _ 1 , \\dots , j _ k } ^ { ( n , i ) } \\colon 1 \\leq k \\leq n , 0 \\leq j _ 1 , \\dots , j _ k \\leq m , 1 \\leq i \\leq m \\right \\} & \\subset C _ { n + 2 } \\end{align*}"}
-{"id": "7054.png", "formula": "\\begin{align*} { { \\bf { h } } ^ { [ 2 2 ] } } ( n ) { \\bf { V } } _ 1 ^ { [ 2 ] } ( n ) = { { \\bf { h } } ^ { [ 2 2 ] } } ( 1 ) , { { \\bf { h } } ^ { [ 1 2 ] } } ( n ) { \\bf { V } } _ 2 ^ { [ 2 ] } ( n ) = { { \\bf { h } } ^ { [ 1 2 ] } } ( 6 ) . \\end{align*}"}
-{"id": "1791.png", "formula": "\\begin{align*} Q [ k - 1 ] & = i _ { k - 1 } - \\frac { k - 1 } { \\alpha } i _ { k - 1 } - \\frac { k } { n } i _ { k } . \\end{align*}"}
-{"id": "8875.png", "formula": "\\begin{align*} \\mathcal { Q } _ A ( v ) = \\bigl ( \\tfrac { 2 p } { p - 2 } \\mathcal { I } _ A ( v ) \\bigr ) ^ { \\frac { p - 2 } { p } } . \\end{align*}"}
-{"id": "6754.png", "formula": "\\begin{align*} g ( t ) : = I ( \\varphi _ t ) - \\int _ X e ^ { \\varphi + t \\chi } d \\mu , \\ t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "5055.png", "formula": "\\begin{align*} \\sum _ { g \\in G } p ^ { * n } ( g ) \\frac { d g \\nu } { d \\nu } ( y ) = 1 , \\textrm { f o r $ \\nu $ - a . e . $ y $ } . \\end{align*}"}
-{"id": "8692.png", "formula": "\\begin{gather*} \\sup _ { s \\in [ 0 , T ] } \\| \\Phi ( s , \\cdot ) \\| _ { C ^ { \\alpha } _ b ( H , K ) } \\le C _ T \\sup _ { t \\in [ 0 , T ] } \\| B ( t , \\cdot ) \\| _ { C ^ { \\alpha } _ b ( H , U ) } \\ ; \\ ; \\\\ w ( t , x ) = \\int _ t ^ T R _ { s - t } [ \\Phi ( s , \\cdot ) ] ( x ) \\ , d s , \\ ; \\ ; \\ ; t \\in [ 0 , T ] , \\ ; x \\in H . \\end{gather*}"}
-{"id": "7884.png", "formula": "\\begin{align*} \\partial _ i \\phi _ y ( t , \\cdot , s ) ( x ) = \\int _ { \\R ^ d } \\partial _ i p _ z ( t - s , \\cdot ) ( x - z ) q ( s , z , y ) \\ , d z \\ , . \\end{align*}"}
-{"id": "10.png", "formula": "\\begin{align*} \\alpha ( x ) & = \\sum _ { k \\geq 1 } 2 i \\alpha _ k \\sin 2 \\pi k x , & \\beta ( x ) & = \\sum _ { k \\geq 0 } 2 \\beta _ k \\cos 2 \\pi k x \\end{align*}"}
-{"id": "5217.png", "formula": "\\begin{align*} ( u , v ) _ { L ^ 2 _ { ( 0 , q ) } ( M ) } = ( u , j _ q v ) _ { L ^ 2 _ { ( 0 , q ) } ( M ) } = ( ( j _ q ) ^ * u , v ) _ { g r a p h } \\ ; , \\end{align*}"}
-{"id": "1206.png", "formula": "\\begin{align*} I ( m ) = ( x ( y ^ m - z ^ m ) , y ( z ^ m - x ^ m ) , z ( x ^ m - y ^ m ) ) \\subset R = k [ x , y , z ] . \\end{align*}"}
-{"id": "9744.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { e ^ { - 1 / x ^ \\alpha } } { x ^ { \\alpha + 1 } / G ( x ) } = \\frac { 1 } { \\alpha } , \\end{align*}"}
-{"id": "2117.png", "formula": "\\begin{align*} b = t p _ 0 + 2 s p _ n + 2 p _ { n + 1 } . \\end{align*}"}
-{"id": "4086.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ p y ^ q ( b y + c z ) ^ r } { z ^ { p + q + r } } \\end{align*}"}
-{"id": "9410.png", "formula": "\\begin{align*} E ^ S = \\frac { T ^ S _ 1 } { P \\cdot T ^ S _ P } . \\end{align*}"}
-{"id": "3337.png", "formula": "\\begin{align*} \\mathcal { A } _ L ( [ x , c _ x \\sharp A ] ) & = \\mathcal { A } _ H ( [ x , c _ x \\sharp A ] ) \\\\ & = H ( x ) - \\omega ( A ) \\\\ & = H ( x ) - \\lambda c _ 1 ( A ) , \\end{align*}"}
-{"id": "224.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 5 | S _ i | = O \\biggl \\{ \\max \\biggl ( \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\log n \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\biggr \\} \\end{align*}"}
-{"id": "4733.png", "formula": "\\begin{align*} H : = \\{ x \\in \\mathbb { R } ^ 2 \\ , | \\ ( \\beta _ 1 v _ { 1 1 } + \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } + \\beta _ 2 v _ { 2 2 } ) x _ 2 & \\geq \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 , \\\\ ( \\beta _ 1 v _ { 1 1 } - \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } - \\beta _ 2 v _ { 2 2 } ) x _ 2 & \\geq \\beta _ 1 \\alpha _ 1 - \\beta _ 2 \\alpha _ 2 \\} . \\end{align*}"}
-{"id": "5148.png", "formula": "\\begin{align*} C ^ { ( m _ { r } , m _ { r - 1 } , \\ldots , m _ { a + 1 } ) } ( \\vec { z } ) = \\prod _ { a + 1 \\le p \\le r } ^ { \\curvearrowleft } \\prod _ { i = m _ { r } + \\cdots + m _ { p + 1 } + 1 } ^ { m _ { r } + \\cdots + m _ { p } } C _ { p } ( z _ { i } ) \\end{align*}"}
-{"id": "7091.png", "formula": "\\begin{align*} e ^ { - V ( u ) } Z _ \\infty ^ u = \\sum _ { j \\in \\N } e ^ { - V ( u . j ) } Z _ \\infty ^ { u . j } . \\end{align*}"}
-{"id": "2054.png", "formula": "\\begin{align*} 0 & \\geq \\nabla _ k \\nabla _ k \\| \\nabla w \\| ^ 2 = 2 \\left ( \\langle \\nabla _ k \\nabla _ k \\nabla w , \\nabla w \\rangle + \\| \\nabla _ k \\nabla w \\| ^ 2 \\right ) \\\\ & \\geq 2 \\langle \\nabla _ k \\nabla _ k \\nabla w , \\nabla w \\rangle = 2 \\| \\nabla w \\| \\langle \\nabla _ k \\nabla _ k \\nabla w , e _ n \\rangle . \\end{align*}"}
-{"id": "2447.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } = 2 ^ { j A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 2 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "6124.png", "formula": "\\begin{align*} \\gamma _ s ( \\theta ) = ( w ( s , \\theta ) , \\theta ) = ( 2 + g ( s ) + g ' ( s , \\theta ) , \\theta ) \\end{align*}"}
-{"id": "8390.png", "formula": "\\begin{align*} \\beta ( x ) v = v x v ^ * v = v x z _ 1 = v z _ 1 x = v x , \\ \\ x \\in M , \\end{align*}"}
-{"id": "8460.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } - \\psi ' ( t ) ( \\tilde { \\textbf { u } } ^ { \\varepsilon } , \\phi ) _ { 0 } ~ d t + \\sum _ { j = 1 } ^ { d } \\int _ { 0 } ^ { T } \\psi ( t ) ( \\textbf { P } J _ { \\varepsilon } A _ { j } ( J _ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\bar { \\textbf { u } } ) ) \\partial _ { x _ { j } } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } , \\phi ) _ { 0 } ~ d t = 0 . \\end{align*}"}
-{"id": "2704.png", "formula": "\\begin{align*} U _ { k , l } = \\{ \\varphi _ k < f \\} \\cap \\{ \\varphi > V _ { \\theta } - l \\} . \\end{align*}"}
-{"id": "9040.png", "formula": "\\begin{align*} S _ k ( \\theta ) : = & \\sum _ { j = 0 } ^ { k - 1 } ( j + 1 ) \\gamma _ j ^ 2 e ^ { 2 i \\psi _ j ( \\theta ) } \\ . \\end{align*}"}
-{"id": "604.png", "formula": "\\begin{align*} \\quad \\mathcal { F } ( z ) = \\left ( z - \\overline { \\mathfrak { z } } \\right ) ^ { - \\kappa } \\left ( \\sum _ { n \\gg - \\infty } a _ { \\mathcal { F } , \\mathfrak { z } } ^ + ( n ) X _ { \\mathfrak { z } } ^ n ( z ) + \\sum _ { n \\ll \\infty } a _ { \\mathcal { F } , \\mathfrak { z } } ^ - ( n ) \\beta _ 0 \\left ( 1 - r _ { \\mathfrak { z } } ^ 2 ( z ) ; 1 - \\kappa , - n \\right ) X _ { \\mathfrak { z } } ^ n ( z ) \\right ) . \\end{align*}"}
-{"id": "387.png", "formula": "\\begin{align*} g ^ \\iota = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} ( g ^ { - 1 } ) ^ t \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "8794.png", "formula": "\\begin{align*} & \\overline { M } ( w ; x _ 1 , . . , x _ n ) = M ^ { ( 2 ) } ( w ; x _ 1 , . . , x _ n ) K _ n ( 1 / w ) M ^ { ( 1 ) } ( w ; x _ 1 , . . , x _ n ) , \\\\ & T ( w ; x _ 1 , . . , x _ n ) = _ 0 ( \\overline { M } ( w ; x _ 1 , . . , x _ n ) \\widetilde { K } _ 0 ( w ) ) . \\end{align*}"}
-{"id": "4854.png", "formula": "\\begin{align*} \\hat { R } ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) = - \\int \\frac { h ^ { \\rm g e o } _ { 1 2 } ( z , 1 / z ) } { z ^ { u - v + 1 } } \\dd z - c ^ { w } \\int \\frac { 1 - c z } { ( z ^ 2 - 1 ) z } h ^ { \\rm g e o } _ { 1 2 } ( z , 1 / c ) \\frac { \\dd z } { z ^ u } , \\end{align*}"}
-{"id": "5620.png", "formula": "\\begin{align*} k ^ { 1 / ( n - 1 ) } \\leq s = \\underset { m \\rightarrow \\infty } { \\lim } ( s _ { m } ) ^ { 1 / m } . \\end{align*}"}
-{"id": "5644.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\liminf } \\ \\mathcal { W } _ 2 ( \\delta _ { \\mu ^ n } , \\P ) & \\geq \\int \\underset { n \\rightarrow + \\infty } { \\liminf } \\ W _ 2 ^ 2 ( \\mu _ n , \\nu ) d \\P ( \\nu ) & \\mbox { b y F a t o u ' s L e m m a } & \\\\ & \\geq \\int W _ 2 ^ 2 ( \\mu , \\nu ) d \\P ( \\nu ) = \\mathcal { W } _ 2 ^ 2 ( \\delta _ { \\mu } , \\P ) \\qquad & & \\end{align*}"}
-{"id": "7201.png", "formula": "\\begin{align*} f _ { x _ i } ( n ) \\ ! = \\ ! \\frac { p } { N } \\ ! + \\ ! \\left ( 1 \\ ! - \\ ! p \\right ) \\ ! \\frac { 3 N \\left ( 2 n \\ ! - \\ ! 1 \\right ) \\ ! - \\ ! 6 n \\left ( n \\ ! - \\ ! 1 \\right ) \\ ! - \\ ! 3 } { N ( N ^ 2 - 1 ) } , n \\ ! \\leq \\ ! N \\end{align*}"}
-{"id": "4032.png", "formula": "\\begin{align*} \\mathcal { J } ( x ) & = \\{ C _ { x ^ i _ { \\alpha } } : x \\in C _ { x ^ i _ { \\alpha } } \\} \\\\ \\mathcal { J } ^ { * } ( x ) & = \\{ C _ { x ^ i _ { \\beta } } : \\exists C _ { x ^ i _ { \\alpha } } \\in \\mathcal { J } ( x ) , C _ { x ^ i _ { \\alpha } } \\cap C _ { x ^ i _ { \\beta } } \\neq \\emptyset \\} \\\\ P ( x ) & = \\bigcap \\{ i n t \\ , R _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ m : C _ { x ^ i _ { \\alpha } } \\in \\mathcal { J } ( x ) , C _ { x ^ i _ { \\beta } } \\in \\mathcal { J } ^ { * } ( x ) , x \\in C _ { x ^ i _ { \\alpha } : x ^ i _ { \\beta } } ^ m \\} . \\end{align*}"}
-{"id": "3247.png", "formula": "\\begin{align*} \\mathcal { E } _ { k } = \\left ( 1 + \\eta \\right ) ^ { 2 \\left ( k - 1 \\right ) } \\varepsilon + \\frac { 2 } { \\rho } \\xi \\left ( k - 1 \\right ) \\left ( 1 + \\eta \\right ) ^ { 2 \\left ( k - 1 \\right ) } . \\end{align*}"}
-{"id": "3332.png", "formula": "\\begin{align*} \\frac { c _ { 2 } } { c _ { 1 } } q ^ { - 3 } ( q - q ^ { - 1 } ) = \\frac { c _ { 1 } } { c _ { 0 } } t ^ { \\prime } , - \\frac { c _ { 2 } } { c _ { 1 } } q ^ { - 2 } = \\frac { c _ { 1 } } { c _ { 0 } } t . \\end{align*}"}
-{"id": "175.png", "formula": "\\begin{align*} \\mu _ \\alpha ( f ) : = \\int _ { \\mathbb { R } ^ d } \\| x \\| ^ \\alpha f ( x ) \\ , d x . \\end{align*}"}
-{"id": "6616.png", "formula": "\\begin{align*} \\frac { d } { d t } \\circ \\sigma = p t ^ { p - 1 } \\sigma \\circ \\frac { d } { d t } . \\end{align*}"}
-{"id": "4385.png", "formula": "\\begin{align*} v _ t = a \\ , v _ { x x } + \\lambda \\ , v , \\ \\ t > 0 , \\ x > 0 , \\end{align*}"}
-{"id": "3414.png", "formula": "\\begin{align*} d \\xi ( t ) = a ( \\xi ( t ) , u ( t , \\xi ( t ) ) ) d t + A ( \\xi ( t ) , u ( t , \\xi ( t ) ) ) d w ( t ) , \\xi ( s ) = x , \\end{align*}"}
-{"id": "2077.png", "formula": "\\begin{align*} \\min \\left | \\left | I - \\mathcal { K } _ { i } ^ { ( z ) } P _ { i } ^ { ( z ) } \\right | \\right | _ { F } ^ { 2 } = \\min \\sum _ { j = 1 } ^ { n } \\left | \\left | e ^ { ( j ) } - \\mathcal { K } _ { i } ^ { ( z ) } p ^ { ( j ) } _ { i } \\right | \\right | ^ { 2 } _ 2 , \\end{align*}"}
-{"id": "9108.png", "formula": "\\begin{align*} \\nu ( \\sup \\{ a _ 1 , a _ 2 \\} ) = \\nu ( a _ 1 + a _ 2 ) = \\nu ( a _ 1 ) + \\nu ( a _ 2 ) = 2 \\nu ( a _ 1 ) = 4 a _ 1 = 2 a _ 1 = \\nu ( a _ 1 ) = \\sup \\{ \\nu ( a _ 1 ) , \\nu ( a _ 2 ) \\} . \\end{align*}"}
-{"id": "9346.png", "formula": "\\begin{align*} v ( t , x , z ) = k ( t , x , z ) y ( t , x , z ) q ( t , x , z ) \\end{align*}"}
-{"id": "7893.png", "formula": "\\begin{align*} \\frac { ( B - A ) \\times ( A - I ) } { c } & = \\pm r \\\\ \\frac { ( C - A ) \\times ( C - I ) } { a } & = \\pm r \\end{align*}"}
-{"id": "8438.png", "formula": "\\begin{align*} \\tilde { \\rho } : = \\rho - \\bar { \\rho } , \\end{align*}"}
-{"id": "5546.png", "formula": "\\begin{align*} \\begin{gathered} \\begin{bmatrix} - h & 1 \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} V ( x = 0 ) = 0 , \\\\ \\begin{bmatrix} H & \\ , \\ , 1 \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} V ( x = 1 ) = 0 , \\end{gathered} \\end{align*}"}
-{"id": "4002.png", "formula": "\\begin{align*} b _ { H , a } = 0 \\ ; \\Rightarrow \\ ; b _ { H , t } = 1 . \\end{align*}"}
-{"id": "1204.png", "formula": "\\begin{align*} I _ X ^ { ( m ) } = \\bigoplus _ { t \\geq 0 } H ^ 0 ( \\mathcal I _ X ^ m ( t ) ) . \\end{align*}"}
-{"id": "3260.png", "formula": "\\begin{align*} [ E _ { \\xi + \\alpha _ k } , E _ { \\xi ^ \\prime } ^ * ] _ q = c _ { k , \\xi } ^ { - 1 } E _ k \\triangleright [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q + c _ { k , \\xi } ^ { - 1 } c _ { k , \\xi ^ \\prime } ^ \\prime q ^ { ( \\xi - \\alpha _ k , \\alpha _ k ) } ( E _ \\xi E _ { \\xi ^ \\prime - \\alpha _ k } ^ * - q ^ { - ( \\xi , \\xi ^ \\prime + \\alpha _ k ) } E _ { \\xi ^ \\prime - \\alpha _ k } ^ * E _ \\xi ) . \\end{align*}"}
-{"id": "4481.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } | \\mathbb { E } _ f ( \\hat { H } _ n ) - H | = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { \\beta } { d } } } { n ^ { \\frac { \\beta } { d } } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "1058.png", "formula": "\\begin{align*} \\Gamma = \\langle a , b , c | a ^ 2 , b ^ 2 , c ^ 2 , \\tau ^ n ( w _ 1 ) , \\tau ^ n ( w _ 2 ) , \\tau ^ n ( w _ 3 ) , \\tau ^ n ( w _ 4 ) n \\geq 0 \\rangle \\end{align*}"}
-{"id": "3641.png", "formula": "\\begin{align*} \\left | g ( { \\bf x } ) - K _ 1 \\right | & = \\left | c _ { K _ 1 } T _ { K _ 1 } + ( 1 - c _ { K _ 1 } ) f ( { \\bf x } ) - K _ 1 \\right | \\\\ & = \\left | ( 1 - c _ { K _ 1 } ) [ f ( { \\bf x } ) - f ( \\mathbf { K _ 1 } ) ] \\right | \\\\ & \\leq L ( 1 - c _ { K _ 1 } ) \\| { \\bf x } - \\mathbf { K _ 1 } \\| , ~ { \\bf x } \\in \\R ^ k _ + , \\end{align*}"}
-{"id": "4269.png", "formula": "\\begin{align*} \\kappa : = \\limsup _ { n \\rightarrow + \\infty } ( \\mathbb E [ Z _ n ^ * ] ) ^ { - 1 } \\Vert Z ^ * _ { n + \\lfloor \\varepsilon n \\rfloor } - Z _ { n + \\lfloor \\varepsilon n \\rfloor } \\Vert _ p < + \\infty \\ , . \\end{align*}"}
-{"id": "4414.png", "formula": "\\begin{align*} \\| f _ n \\| _ { L _ 1 + L _ \\infty } = \\| \\widehat { i d } ( f _ n ) - \\widehat { i d } ( f ) \\| _ { L _ 1 + L _ \\infty } \\leq C \\| f _ n - f \\| _ { \\widehat { E } } \\to 0 . \\end{align*}"}
-{"id": "1840.png", "formula": "\\begin{align*} \\sum _ { a < n < b } g ( n ) \\exp ( 2 \\pi i f ( n ) ) & = \\sum _ { \\alpha - \\epsilon < m < \\beta + \\epsilon } \\int _ a ^ b g ( x ) \\exp ( 2 \\pi i ( f ( x ) - m x ) ) d x \\\\ & \\quad + O ( G ( \\epsilon ^ { - 1 } + \\log ( \\beta - \\alpha + 2 ) ) ) , \\end{align*}"}
-{"id": "8069.png", "formula": "\\begin{align*} C ( T ( x ) ) ^ { - 1 } A ( 1 , x ) C ( x ) = \\lambda ( x ) I . \\end{align*}"}
-{"id": "6611.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\tau ^ { - 1 } \\langle \\nabla f ( x + \\tau ( y - x ) ) & - \\nabla f ( x ) , ( x + \\tau ( y - x ) ) - x \\rangle d \\tau \\\\ & \\leq \\int _ 0 ^ 1 \\tau ^ { - 1 } \\langle L \\tau ( x - y ) , \\tau ( x - y ) \\rangle d \\tau \\\\ & = \\langle L ( x - y ) , x - y \\rangle \\int _ 0 ^ 1 \\tau d \\tau \\\\ & = \\tfrac { 1 } { 2 } \\langle L ( x - y ) , x - y \\rangle . \\end{align*}"}
-{"id": "2932.png", "formula": "\\begin{align*} L _ { \\pi ' } ( Z ) & = \\sum _ { k \\ge 0 } \\ , ( - 1 ) ^ k s _ { ( 1 ^ k ) } [ s _ { \\pi ' } ] ( Z ) = \\sum _ \\nu \\ell _ { \\pi ' \\nu ' } \\ , s _ { \\nu ' } ( Z ) \\cr & = \\sum _ { k \\ge 0 } \\ , ( - 1 ) ^ k ( \\ , s _ { ( 1 ^ k ) } [ s _ { \\pi } ] ( Z ) \\ , ) ' = \\sum _ \\nu \\ell _ { \\pi \\nu } \\ , s _ { \\nu ' } ( Z ) \\end{align*}"}
-{"id": "4133.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { z ^ { - ( p + 2 q ) } ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q } { x ^ { - p } } , \\end{align*}"}
-{"id": "5349.png", "formula": "\\begin{align*} \\delta ^ { ( g ) } = ( \\chi _ 0 ( g ) , . . . , \\chi _ { \\ell } ( g ) ) ^ t \\end{align*}"}
-{"id": "593.png", "formula": "\\begin{align*} \\frac { \\partial x _ 1 ( t ; z ) } { \\partial z } \\bigg | _ { z = z ^ * , t = t ^ * } = 0 . \\end{align*}"}
-{"id": "2272.png", "formula": "\\begin{align*} \\lambda = \\max _ { \\substack { x \\in \\bar \\varOmega \\\\ * [ 2 p t ] t \\in [ 0 , T ] } } \\frac { | a ( x , t ) + \\i \\ , b ( x , t ) | } { a ( x , t ) } < \\frac { 1 } { \\cos \\alpha _ k } , \\end{align*}"}
-{"id": "9394.png", "formula": "\\begin{align*} \\| \\Xi ( \\lambda ) { u } _ { \\lambda ^ * } \\| = \\frac { \\| \\gamma ( \\lambda ) \\gamma ( \\lambda ^ * ) ^ \\dag { u } _ { \\lambda ^ * } \\| } { | a - \\widetilde { W } _ { \\lambda } | } = \\frac { I m \\ \\widetilde { W } _ \\lambda } { I m \\ \\lambda } \\frac { \\| u _ \\lambda \\| } { | a - \\widetilde { W } _ { \\lambda } | } . \\end{align*}"}
-{"id": "3261.png", "formula": "\\begin{align*} E _ k \\triangleright ( E _ \\eta ^ * E _ { \\eta ^ \\prime } ) = ( E _ k \\triangleright E _ \\eta ^ * ) E _ { \\eta ^ \\prime } + ( K _ k \\triangleright E _ \\eta ^ * ) ( E _ k \\triangleright E _ { \\eta ^ \\prime } ) . \\end{align*}"}
-{"id": "1863.png", "formula": "\\begin{align*} \\Omega _ Q = \\Omega \\ ; - \\ ; d ( \\phi ^ a \\cdot \\iota _ a \\Theta ) \\ ; - \\ ; [ \\phi ^ a \\cdot \\iota _ a \\Theta \\wedge \\Theta ] \\ ; + \\ ; \\frac 1 2 [ \\phi ^ a \\cdot \\iota _ a \\Theta \\wedge \\phi ^ b \\cdot \\iota _ b \\Theta ] \\end{align*}"}
-{"id": "3992.png", "formula": "\\begin{align*} 0 \\geq \\sigma ( c _ { 1 } \\vect { e } _ { 1 } + \\cdots + c _ { i - 1 } \\vect { e } _ { i - 1 } + ( c _ { i } + j ) \\vect { e } _ { i } ) = ( \\lambda - c _ { i } + j ) ( - ( k + 1 ) ) . \\end{align*}"}
-{"id": "6100.png", "formula": "\\begin{align*} \\phi ( G _ i ) = G _ { i } ' . \\end{align*}"}
-{"id": "1700.png", "formula": "\\begin{align*} \\dd f _ n ^ 2 & = 2 f _ n \\dd f _ n \\\\ & = - 2 f _ n b \\cdot D f _ n \\ , \\dd t - 2 f _ n D _ { v } f _ n \\circ \\dd W _ t + 2 f _ n R _ n \\ , \\dd t \\ , . \\end{align*}"}
-{"id": "8526.png", "formula": "\\begin{align*} X _ { [ b r a ] ^ { \\ast } } ( \\rho ( P ) ) & = \\displaystyle \\sum _ { u \\in N _ { b } } \\displaystyle \\sum _ { c \\in \\mathcal { C } ( u ) , r _ { c } = r , a _ { c } = a } f _ { u } \\rho ( z _ { c } ) s _ { c } \\\\ & = \\rho \\left ( \\displaystyle \\sum _ { u \\in N ( b ) } \\displaystyle \\sum _ { c \\in \\mathcal { C } ( u ) , r _ { c } = r , a _ { c } = a } f _ { u } z _ { c } s _ { c } \\right ) \\end{align*}"}
-{"id": "1747.png", "formula": "\\begin{align*} \\int _ { \\R ^ N _ + } \\frac { u ( y ) } { | y | ^ { N + 2 s } } d y = - f ( 0 ) . \\end{align*}"}
-{"id": "1126.png", "formula": "\\begin{align*} \\mathbf { R } ^ \\mathrm { C } [ \\iota ] = \\sqrt { \\rho _ \\mathrm { s } } \\mathbf { G } _ \\mathrm { s } [ \\iota ] \\mathbf { S } ^ \\mathrm { C } [ \\iota ] + \\mathbf { L } ^ \\mathrm { C } [ \\iota ] + \\mathbf { N } ^ \\mathrm { C } [ \\iota ] \\end{align*}"}
-{"id": "9983.png", "formula": "\\begin{align*} p _ 1 ^ * = & \\frac { C _ { 1 } - \\alpha | w _ { \\bar { k } 2 } | ^ 2 \\tilde { p } ^ * } { C _ 0 } , \\\\ p _ 2 ^ * = & \\frac { \\alpha | w _ { \\bar { k } 1 } | ^ 2 \\tilde { p } ^ * - C _ { 2 } } { C _ 0 } . \\end{align*}"}
-{"id": "2780.png", "formula": "\\begin{align*} \\mathsf { P } _ { \\mathrm { S I R } } \\left ( \\lambda \\right ) = \\mathbb { P } \\left ( \\mathrm { S I R } _ { \\mathrm { U } _ { 0 } } > \\tau \\right ) , \\end{align*}"}
-{"id": "8132.png", "formula": "\\begin{align*} R _ { k , l , N } ( \\Omega ) = R _ { \\widetilde { k } , \\widetilde { l } , N } ( \\widetilde { \\Omega } ) , \\mbox { w h e r e $ \\widetilde { \\Omega } : = \\{ y = \\frac { x } { | x | ^ 2 } : \\ , x \\in \\Omega \\} . $ } \\end{align*}"}
-{"id": "7993.png", "formula": "\\begin{align*} F = \\sum _ { i = 1 } ^ m r _ i E _ i + F ' , \\end{align*}"}
-{"id": "167.png", "formula": "\\begin{align*} H _ \\nu ( \\zeta | A ) \\ge & \\sum _ { k \\ge 1 } \\mu ( C _ k | A ) H _ \\nu ( \\zeta | C _ k \\cap A ) \\ge \\sum _ { k : 2 ^ k > l ( A ) } \\nu ( C _ k | A ) H _ \\nu ( \\zeta | C _ k \\cap A ) \\\\ = \\sum _ { k : 2 ^ k > l ( A ) } & \\frac { 1 } { 2 ^ k } H _ \\nu \\left ( \\eta _ { - 2 ^ k } ^ { - l ( A ) - 1 } ( \\tau ) \\right ) = \\sum _ { k : 2 ^ k > l ( A ) } \\frac { 1 } { 2 ^ k } ( 2 ^ k - l ( A ) ) H _ \\nu ( \\eta ) = \\infty . \\end{align*}"}
-{"id": "496.png", "formula": "\\begin{align*} f ( z ) = z + \\frac { a _ 1 } { z } + \\dots \\end{align*}"}
-{"id": "5146.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m } C _ { a } ( z _ { i } ) \\cdot \\beta _ { a , 1 } ^ { * } \\equiv \\beta _ { a , 1 } ^ { * } \\prod _ { i = 1 } ^ { m } C _ { a } ( z _ { i } ) + ( 1 - q ^ 2 ) \\sum _ { \\ell = 1 } ^ { m } z _ { \\ell } \\prod _ { i = 1 } ^ { \\ell - 1 } C _ { a } ( z _ { i } ) \\cdot \\tilde { A } _ { a } ( z _ { \\ell } ) \\prod _ { i = \\ell + 1 } ^ { m } C _ { a } ( z _ { i } ) . \\end{align*}"}
-{"id": "9058.png", "formula": "\\begin{align*} l _ k ^ { ( N ) } : = \\left \\{ \\begin{array} { l l } - \\upsilon - ( k - r ) ^ { \\alpha _ + } , & r \\leq k \\leq \\lfloor N / 2 \\rfloor , \\\\ - \\upsilon - ( N - k ) ^ { \\alpha _ + } - \\frac { 3 } { 4 } \\log N , & \\lfloor N / 2 \\rfloor < k \\leq N . \\end{array} \\right . \\end{align*}"}
-{"id": "3393.png", "formula": "\\begin{align*} \\bar { \\pi } _ k & : = ( \\pi | _ { \\hat { V } _ k \\otimes \\mathbb { C } [ [ w ] ] } ) \\otimes \\mathrm { i d } _ { \\mathbb { C } [ w ] / ( w ^ { m r - r + 1 } ) } \\colon V _ k \\otimes _ { \\mathbb { C } [ z ] / ( z ^ m ) } \\mathbb { C } [ w ] / ( w ^ { m r - r + 1 } ) \\longrightarrow ( w ^ k ) / ( w ^ { k + m r - r + 1 } ) \\\\ \\bar { \\nabla } _ k & : = \\nabla | _ { z ^ m = 0 } \\colon V _ k \\longrightarrow V _ k \\otimes \\frac { d z } { z ^ m } . \\end{align*}"}
-{"id": "2282.png", "formula": "\\begin{align*} \\frac 1 \\tau \\sum _ { j = 0 } ^ k \\delta _ j v _ { n - j } + A ( t _ n ) v _ n = f _ n , k \\le n \\le N , \\end{align*}"}
-{"id": "1056.png", "formula": "\\begin{align*} F ( s ) & = \\sum _ { j = 1 } ^ \\infty \\sum _ { n = 2 } ^ \\infty n ^ { j - 1 } ( n ^ j ) ^ { - s } = \\sum _ { n = 2 } ^ \\infty g _ n n ^ { - s } , \\end{align*}"}
-{"id": "9954.png", "formula": "\\begin{align*} \\mathcal E _ { \\lambda , \\mu } ( t \\bar { u } ) & = \\frac { 1 } { 2 } \\left \\| \\bar { u } \\right \\| ^ 2 t ^ 2 - \\frac { \\mu } { 2 ^ * } \\int _ \\Omega | \\bar { u } | ^ { 2 ^ * } d x \\ ; t ^ { 2 ^ * } - \\lambda \\int _ \\Omega G ( t \\bar { u } ) d x \\\\ & \\leq \\left ( \\frac { 1 } { 2 } \\left \\| \\bar { u } \\right \\| ^ 2 - \\lambda M \\left \\| \\bar { u } \\right \\| ^ 2 _ { L ^ 2 ( \\Omega ) } \\right ) t ^ 2 - \\frac { \\mu } { 2 ^ * } \\int _ \\Omega | \\bar { u } | ^ { 2 ^ * } d x \\ ; t ^ { 2 ^ * } \\\\ & < 0 \\end{align*}"}
-{"id": "3271.png", "formula": "\\begin{align*} D ^ 2 = \\sum _ { i , j } \\mathcal { E } _ i \\mathcal { E } _ j ^ * \\otimes \\gamma _ { - } ( w _ i ) \\gamma _ { - } ( w _ j ) ^ * + \\sum _ { i , j } \\mathcal { E } _ j ^ * \\mathcal { E } _ i \\otimes \\gamma _ { - } ( w _ j ) ^ * \\gamma _ { - } ( w _ i ) . \\end{align*}"}
-{"id": "9227.png", "formula": "\\begin{align*} \\beta ( t , x , z ) = \\delta ( t , x , z ) \\beta _ 0 ( t , x , z ) . \\end{align*}"}
-{"id": "6288.png", "formula": "\\begin{align*} x _ j = a + \\frac { b } { z _ j + c } , ~ z _ j = \\frac { b } { x _ j - a } - c . \\end{align*}"}
-{"id": "9180.png", "formula": "\\begin{align*} d X _ t = g ( X _ t ) d W _ t h ( X _ t ) + h ( X _ t ) d W _ t ^ T g ( X _ t ) + b ( X _ t ) d t \\ / , \\end{align*}"}
-{"id": "3518.png", "formula": "\\begin{align*} \\| u \\| ^ 2 _ { H ^ k _ { \\rho _ g } ( \\Omega , g ) } = \\sum _ { j = 0 } ^ k \\| \\nabla ^ j _ g u \\| ^ 2 _ { L ^ 2 _ { \\rho _ g } ( \\Omega , g ) } = \\sum _ { j = 0 } ^ k \\int _ { \\Omega } | \\nabla _ g ^ { j } u | ^ 2 \\rho _ g \\ , d \\mu _ g . \\end{align*}"}
-{"id": "134.png", "formula": "\\begin{align*} \\mu ( x ) = \\mu ( x ) \\chi _ { [ 0 , \\tau ( r ) ) } + \\mu ( \\infty , x ) \\chi _ { [ \\tau ( r ) , \\infty ) } = \\mu ( x _ 0 ) + \\mu ( \\infty , x ) \\chi _ { [ \\tau ( r ) , \\infty ) } , \\end{align*}"}
-{"id": "543.png", "formula": "\\begin{align*} E _ 2 = n ^ { - 1 / 3 } \\int _ { n ^ { 1 / 3 } \\mathcal { C } _ { 0 \\not \\in } ^ { \\rm l o c a l } } \\int _ { n ^ { 1 / 3 } \\mathcal { C } _ { 0 \\not \\in } ^ { \\rm l o c a l } } \\frac { ( \\tilde z - \\tilde w ) e ^ { \\frac { \\sigma ^ 3 } { 3 } \\tilde z ^ 3 + \\frac { \\sigma ^ 3 } { 3 } \\tilde w ^ 3 - \\sigma x \\tilde z - \\sigma y \\tilde w } } { 4 \\tilde z \\tilde w ( \\tilde z + \\tilde w ) } \\frac { 4 ( \\tilde z + \\tilde w ) } { 2 \\alpha - 1 } \\mathrm { d } \\tilde z \\mathrm { d } \\tilde w . \\end{align*}"}
-{"id": "8061.png", "formula": "\\begin{align*} f _ x ( y ) = \\begin{cases} y + \\dfrac { \\alpha _ x } { \\pi } & x \\in X _ r , \\\\ \\dfrac { 2 \\beta _ x } { \\pi } - y & x \\in X _ f , \\end{cases} \\ g _ x ( a ) = \\begin{cases} a & x \\in X _ r , \\\\ a + 1 & x \\in X _ f . \\end{cases} \\end{align*}"}
-{"id": "4713.png", "formula": "\\begin{align*} A ^ 1 x ^ + _ 1 - A ^ 1 x ^ - _ 1 + A ^ 2 x ^ + _ 1 - A ^ 2 x ^ - _ 1 & \\succeq _ { \\mathcal { L } ^ { 3 } } b \\\\ x _ 1 ^ + , x _ 1 ^ - , x _ 2 ^ + , x _ 2 ^ - & \\geq 0 \\\\ x _ j = x _ j ^ + - x _ j ^ - \\ j & \\in \\{ 1 , 2 \\} . \\end{align*}"}
-{"id": "7757.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb P \\left [ T _ N ( n ) \\le x \\right ] = \\mathbb P \\left [ T _ N \\le x \\right ] . \\end{align*}"}
-{"id": "7599.png", "formula": "\\begin{align*} P _ e ( x _ 0 ) \\ge \\prod _ { i = 1 } ^ { K _ 1 } \\mu _ i , \\end{align*}"}
-{"id": "7865.png", "formula": "\\begin{align*} r ( t , x ) = \\Phi ^ { - 1 } ( t ^ { - 1 } ) ^ d \\wedge \\frac { t \\Phi ( | x | ^ { - 1 } ) } { | x | ^ d } \\end{align*}"}
-{"id": "7634.png", "formula": "\\begin{align*} e ^ { s } _ i = \\left \\{ \\begin{array} { r l } + 1 , & \\mbox { i f } i \\mbox { i s v i s i t e d b y p a t h } s \\mbox { , b u t i t i s n o t t h e l a s t n o d e o f } s , \\\\ 0 , & \\mbox { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
-{"id": "2319.png", "formula": "\\begin{align*} E ( z , x , y ) = \\frac { c _ d } { z ^ d } \\bigg ( 1 + \\frac { | x - y | ^ 2 } { z ^ 2 } \\bigg ) ^ { - \\frac { d + 1 } { 2 } } \\ , , \\end{align*}"}
-{"id": "6827.png", "formula": "\\begin{align*} L _ n \\cdot B _ { m + k } = \\sum _ { l = 0 } ^ { n } \\left [ B _ l , B _ { m + n + k - l } \\right ] + ( m + k ) B _ { m + n + k } \\ , . \\end{align*}"}
-{"id": "9437.png", "formula": "\\begin{align*} n _ { [ f ] } ^ w ( e _ 0 ) \\le n _ { f _ 0 } ^ w ( y ) \\le n _ { f _ 0 } ^ { w _ 0 } ( y ) + k \\epsilon = K + k \\epsilon , \\end{align*}"}
-{"id": "8642.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\overset { \\cdot } { w } \\left ( t \\right ) = A w \\left ( t \\right ) + G u ( t ) , \\\\ w \\left ( 0 \\right ) = h \\in K , \\end{array} \\right . \\end{align*}"}
-{"id": "1383.png", "formula": "\\begin{align*} \\Omega _ { 1 , 2 } = \\underset { n \\to \\infty } \\lim \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } \\sum _ { s = 1 } ^ { n } \\mathrm { C o v } ( \\dot { l } _ { t } ( \\delta _ 0 ) , \\dot { l } _ { s } ( \\delta _ 0 ) ) \\end{align*}"}
-{"id": "2376.png", "formula": "\\begin{align*} { \\rm { p r o x } } _ { \\gamma f } = \\nabla r _ { \\gamma f } ^ * \\end{align*}"}
-{"id": "8029.png", "formula": "\\begin{align*} \\chi _ p ( M ) = \\dim H ^ 0 ( U , M ) - \\dim H ^ 1 _ p ( U , V ) + \\dim H ^ 2 _ c ( U , M ) \\end{align*}"}
-{"id": "4180.png", "formula": "\\begin{align*} d _ H ( A , B ) = \\max \\bigl \\{ \\sup _ { a \\in A } d ( a , B ) , \\sup _ { b \\in B } d ( b , A ) \\bigr \\} . \\end{align*}"}
-{"id": "80.png", "formula": "\\begin{align*} v ( t , 0 ) = g ( t ) , \\ \\ t > 0 , \\end{align*}"}
-{"id": "5915.png", "formula": "\\begin{align*} \\big ( L ^ p ( \\R ^ d ; A _ 0 ) , L ^ p ( \\R ^ d ; A _ 1 ) \\big ) _ { \\theta , q } = L ^ p ( \\R ^ d ; ( A _ 0 , A _ 1 ) _ { \\theta , q } ) \\ , . \\end{align*}"}
-{"id": "9799.png", "formula": "\\begin{align*} Z _ { i } = X _ { i j } \\boldsymbol { \\beta } + \\varepsilon _ { i } , \\ ; \\ ; \\ ; \\varepsilon _ { i } \\sim q ( \\cdot ) \\end{align*}"}
-{"id": "1242.png", "formula": "\\begin{align*} \\varphi ( t _ { n _ j } ) \\sum _ { k = 1 } ^ { \\infty } v ( k ) \\chi _ { E _ j } ( k ) \\le \\varphi ( t _ { n _ j } ) \\sum _ { k = 1 } ^ { m ( E _ j ) } w ( k ) \\le 1 / 2 ^ { j - 2 } \\to 0 . \\end{align*}"}
-{"id": "1617.png", "formula": "\\begin{align*} \\left ( \\kappa \\left ( \\mu - \\lambda - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } - \\frac { 1 } { 2 } \\sigma \\sigma _ { , x } \\right ) = \\sigma \\int \\frac { m } { \\sigma } d x + c \\sigma , \\end{align*}"}
-{"id": "760.png", "formula": "\\begin{gather*} p = p _ u ^ * p _ u , p _ u = s _ 0 \\otimes t _ 1 \\otimes s _ 1 \\otimes \\cdots \\otimes t _ l \\otimes s _ l . \\end{gather*}"}
-{"id": "6273.png", "formula": "\\begin{align*} & x ( s ) : = \\exp _ { x _ 0 } ( s \\ , e _ n ) \\\\ & y ( s ) : = \\exp _ { x _ 0 } ( - s \\ , e _ n ) \\\\ & g ( s ) : = Q ( x ( s ) , y ( s ) ) . \\end{align*}"}
-{"id": "7536.png", "formula": "\\begin{align*} \\alpha _ { k - 1 } ( \\omega ) = \\alpha + l \\xi _ { k } ( \\omega ) \\in ( a _ 1 , \\alpha + l \\nu ) , k = n + 1 , n + 2 , \\dots , n + N _ 2 , \\end{align*}"}
-{"id": "624.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ { 2 ^ \\ast } ( B _ r ( x _ 0 ) ) } = \\| \\Psi _ \\lambda \\ast f \\| _ { L ^ { 2 ^ \\ast } ( B _ r ( x _ 0 ) ) } \\leq \\| \\Psi _ \\lambda \\| _ { L ^ { \\frac { N } { N - 2 } , w } ( B _ { r + r _ 0 } ( x _ 0 - y _ 0 ) ) } \\| f \\| _ { ( 2 ^ \\ast ) ' } , \\end{align*}"}
-{"id": "8683.png", "formula": "\\begin{gather*} \\Phi _ l ( y ) = \\langle \\Phi ( y ) , l \\rangle _ J , \\ ; \\ ; \\ ; l \\in J . \\end{gather*}"}
-{"id": "4316.png", "formula": "\\begin{align*} e ^ { \\varphi _ { X / Y } ( x _ 0 ) } = \\sum _ { j = 1 } ^ N | F _ j ( x _ 0 ) | ^ 2 \\end{align*}"}
-{"id": "9216.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial y } ( t , x , z ) = \\frac { \\partial H } { \\partial y } ( t , x , y , Y ( t , . , z ) , u ( t , x , z ) , z , p ( t , x , z ) , q ( t , x , z ) , r ( t , x , z , . ) ) | _ { y = Y ( t , x , z ) } . \\end{align*}"}
-{"id": "6772.png", "formula": "\\begin{align*} \\theta _ { \\varphi } ^ n = e ^ { \\varphi - u } \\theta _ { u } ^ n + e ^ { \\varphi - v } \\theta _ v ^ n . \\end{align*}"}
-{"id": "573.png", "formula": "\\begin{align*} \\delta ( x y ) = \\delta ( x ) \\phi ( y ) + x ^ p \\delta ( y ) \\end{align*}"}
-{"id": "6253.png", "formula": "\\begin{align*} \\frac { 1 } { D } \\frac { d } { d D } \\left ( D ^ 2 ( \\bar { \\lambda } _ 2 ( n , D , K ) - \\bar { \\lambda } _ 1 ( n , D , K ) ) \\right ) = \\frac { d \\lambda _ 2 ( \\tilde { L } _ c ) } { d c } \\biggr | _ { c = 1 } - \\frac { d \\lambda _ 1 ( \\tilde { L } _ c ) } { d c } \\biggr | _ { c = 1 } . \\end{align*}"}
-{"id": "236.png", "formula": "\\begin{align*} \\mathbb { E } _ f ( \\hat { H } _ n ^ { w } ) - H ( f ) = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { 2 d ' } { d } } } { n ^ { \\frac { 2 d ' } { d } } } \\ , , \\ , \\frac { k ^ \\frac { \\beta } { d } } { n ^ \\frac { \\beta } { d } } \\biggr \\} \\biggr ) = o ( n ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "5948.png", "formula": "\\begin{align*} A _ t = \\int _ 0 ^ t 1 _ { \\{ Z _ s \\not = Y _ s \\} } \\ ; \\frac { \\| [ D U ( Z _ s ) - D U ( Y _ s ) ] { R \\ , } \\Big \\| ^ 2 _ { H S } } { | Z _ s - Y _ s | ^ 2 } \\ \\dd s \\ , , \\end{align*}"}
-{"id": "5077.png", "formula": "\\begin{align*} \\gamma ( \\vec { x } ) = \\sum _ { \\vec { \\nu } \\in I _ { k _ { 1 } , \\ldots , k _ { r } } } q ^ { t ( \\vec { \\nu } ) } \\gamma _ { \\vec { \\nu } } ( \\vec { x } ) \\ , u _ { \\nu _ { 1 } } \\otimes \\cdots \\otimes u _ { \\nu _ { k } } ( \\vec { x } \\in L _ { k } ^ { + } ) , \\end{align*}"}
-{"id": "518.png", "formula": "\\begin{align*} \\mathcal { U } ^ { \\llcorner } _ { \\rho _ 1 , \\rho _ 2 } ( \\pi \\vert \\nu , \\mu , \\kappa ) = \\mathcal { U } ^ { \\llcorner } _ { \\rho _ 1 , \\rho _ 2 } ( \\pi \\vert \\nu , \\kappa ) = \\frac { s _ { \\pi / \\nu } ( \\rho _ 2 ) s _ { \\pi / \\kappa } ( \\rho _ 1 ) } { \\sum _ { \\lambda } s _ { \\lambda / \\nu } ( \\rho _ 2 ) s _ { \\lambda / \\kappa } ( \\rho _ 1 ) } . \\end{align*}"}
-{"id": "1111.png", "formula": "\\begin{align*} \\| u _ i \\| _ 2 & \\leq \\alpha \\sqrt { k / ( t - 1 ) } \\\\ & \\leq \\frac { \\| D ^ * _ { S _ 0 } h \\| _ 2 } { \\sqrt { t - 1 } } + \\frac { 2 \\sigma _ k ( D ^ * x _ 0 ) _ 1 + \\rho } { \\sqrt { k ( t - 1 ) } } \\\\ & \\leq \\frac { \\| D ^ * _ { S _ 0 } h + h ^ { ( 1 ) } \\| _ 2 } { \\sqrt { t - 1 } } + \\frac { 2 \\sigma _ k ( D ^ * x _ 0 ) _ 1 + \\rho } { \\sqrt { k ( t - 1 ) } } \\\\ & = \\frac { z + R } { \\sqrt { t - 1 } } , \\end{align*}"}
-{"id": "2135.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } f ( t ) : = \\frac { d } { d t } ( g _ { 1 - \\alpha } * f ( \\cdot ) ) ( t ) , \\end{align*}"}
-{"id": "1874.png", "formula": "\\begin{align*} 3 \\eta ^ 2 + 1 + 2 \\gamma ( \\sqrt { 3 } \\eta - 1 ) \\ge & 3 \\eta ^ 2 + 1 + 2 \\cdot \\frac { 1 + \\sqrt { 3 } } { \\sqrt { 3 } } \\cdot ( \\sqrt { 3 } \\eta - 1 ) \\\\ = & 3 ( \\eta - \\frac 1 3 ) \\cdot ( \\eta + \\frac { 2 + \\sqrt { 3 } } { \\sqrt { 3 } } ) \\ge C ( \\eta _ 0 ) . \\end{align*}"}
-{"id": "9451.png", "formula": "\\begin{align*} I ( H , \\varnothing ) = & ~ \\mathrm { s p a n } ( \\{ \\mu \\nu ^ * \\colon \\mu \\nu r ( \\mu ) = r ( \\nu ) \\in H \\} ) \\\\ = & ~ \\mathrm { s p a n } \\left ( \\left \\{ \\big ( \\widehat { s ( \\mu ) } \\cdot \\mu \\cdot \\widehat { r ( \\mu ) } ) ( \\widehat { r ( \\nu ) } \\cdot \\nu ^ * \\cdot \\widehat { s ( \\nu ) } \\big ) \\colon r ( \\mu ) = r ( \\nu ) \\in H \\right \\} \\right ) \\ ; , \\end{align*}"}
-{"id": "7832.png", "formula": "\\begin{align*} ( X _ k , C _ k , D _ k ) = \\left ( \\left [ \\begin{smallmatrix} X _ k ^ { ( 1 ) } & 0 \\\\ 0 & X _ k ^ { ( 2 ) } \\end{smallmatrix} \\right ] , \\left [ \\begin{smallmatrix} C _ k ^ { ( 1 1 ) } & C _ k ^ { ( 1 2 ) } \\\\ C _ k ^ { ( 2 1 ) } & C _ k ^ { ( 2 2 ) } \\end{smallmatrix} \\right ] , \\left [ \\begin{smallmatrix} D _ k ^ { ( 1 1 ) } & D _ k ^ { ( 1 2 ) } \\\\ D _ k ^ { ( 2 1 ) } & D _ k ^ { ( 2 2 ) } \\end{smallmatrix} \\right ] \\right ) . \\end{align*}"}
-{"id": "54.png", "formula": "\\begin{align*} ( | f ' | ^ { p - 2 } f ' ) ' + f - | f | ^ { p - 2 } f = 0 \\ , r \\in ( 0 , \\infty ) \\ , \\end{align*}"}
-{"id": "7292.png", "formula": "\\begin{align*} q ^ { - ( \\alpha _ t , \\xi ) } [ E _ \\xi , F _ t ] ^ * K _ t = q ^ { - ( \\alpha _ t , \\xi ) } ( q ^ { ( \\alpha _ t , \\xi ) } E _ \\xi E _ t ^ * - E _ t ^ * E _ \\xi ) ^ * = E _ t E _ \\xi ^ * - q ^ { - ( \\alpha _ t , \\xi ) } E _ \\xi ^ * E _ t . \\end{align*}"}
-{"id": "3178.png", "formula": "\\begin{align*} \\sum _ { \\mu = 1 } ^ r ( T _ { j k } ) ^ \\lambda _ \\mu \\cdot w _ k ^ \\mu = w _ j ^ \\lambda + \\sum _ { | \\alpha | \\geq 2 } f _ { k j , \\alpha } ^ \\lambda ( z _ j ) \\cdot w _ j ^ \\alpha , \\end{align*}"}
-{"id": "7497.png", "formula": "\\begin{align*} n = \\sum _ { j = 0 } ^ { \\kappa ' } j Q _ j \\geq a Q _ a + b Q _ b \\geq a t _ { a } + b t _ { b } > a \\left ( \\frac { n } { a + 1 } - 1 \\right ) + b \\left ( \\frac { n } { b + 1 } - 1 \\right ) \\end{align*}"}
-{"id": "4776.png", "formula": "\\begin{align*} f f '' + ( f ' ) ^ 2 + 1 = c \\sqrt { f '^ 2 + 1 } . \\end{align*}"}
-{"id": "5888.png", "formula": "\\begin{align*} \\eta = a ( x ^ { i } ) u + b ( x ^ { i } ) ~ ~ , ~ \\xi ^ { k } = \\xi ^ { k } ( x ^ { i } ) . \\end{align*}"}
-{"id": "6173.png", "formula": "\\begin{align*} \\mathcal { E } ( U ) _ t = e ^ { U _ t - \\sigma _ U ^ 2 t / 2 } \\prod _ { s \\leq t } ( 1 + \\Delta U _ s ) e ^ { - \\Delta U _ s } . \\end{align*}"}
-{"id": "5469.png", "formula": "\\begin{align*} S & \\ll H ( \\log N ) ^ 2 \\sum _ { n = N - H } ^ { N + H } n ^ { 1 / 2 } + H N \\ll H ( \\log N ) ^ 2 \\Bigl ( ( N + H ) ^ { 3 / 2 } - ( N - H ) ^ { 3 / 2 } \\Bigr ) + H N \\\\ & \\ll H ^ 2 N ^ { 1 / 2 } ( \\log N ) ^ 2 + H N . \\end{align*}"}
-{"id": "3058.png", "formula": "\\begin{align*} \\mu _ n = \\sum _ { | u | = n } \\delta _ { u , V ( u ) - m _ n } , \\end{align*}"}
-{"id": "4947.png", "formula": "\\begin{align*} - \\Delta u - \\lambda u = Q ( x ) | u | ^ { p - 2 } u , x \\in \\R ^ N , \\end{align*}"}
-{"id": "4309.png", "formula": "\\begin{align*} v _ k \\to v _ \\infty : = \\sum _ { p = 1 } ^ r \\lambda ^ p _ \\infty u _ p \\end{align*}"}
-{"id": "4404.png", "formula": "\\begin{align*} \\Phi _ X = \\Phi _ { G _ X } \\cong { \\partial C _ 1 ( G _ X , \\Z ) \\over \\partial \\delta C _ 0 ( G _ X , \\Z ) } , \\end{align*}"}
-{"id": "40.png", "formula": "\\begin{align*} \\Theta _ { X / Y } \\geq [ \\Sigma _ p ] : = \\sum _ { i \\in I _ h } ( e _ i - 1 ) [ W _ i ] \\end{align*}"}
-{"id": "4387.png", "formula": "\\begin{align*} v _ t = a \\ , v _ { x x } + \\lambda \\ , v , \\ \\ t > 0 , \\ x < 0 , \\end{align*}"}
-{"id": "707.png", "formula": "\\begin{align*} \\frac { r ^ { 2 } \\| \\vec { c } \\| \\| A ^ { k } \\| } { k ^ { r - 1 } \\rho ( A ) ^ { k } } = \\frac { r ^ { 2 } \\| \\vec { c } \\| \\| A ^ { k } \\| } { k ^ { r - 1 } \\delta ^ { k } } \\end{align*}"}
-{"id": "4261.png", "formula": "\\begin{align*} f _ A ( x ) & = x ^ { \\odot n } \\oplus A _ { i _ 1 i _ 1 } x ^ { \\odot n - 1 } \\oplus \\dots \\oplus A _ { i _ 1 i _ 1 } \\odot \\dots \\odot A _ { i _ n i _ n } \\enspace . \\end{align*}"}
-{"id": "3085.png", "formula": "\\begin{align*} b ( u _ i , u _ j ) = \\delta _ { i j } , \\forall i , j = 1 , 2 , \\dots , \\end{align*}"}
-{"id": "4090.png", "formula": "\\begin{align*} d \\alpha \\cdot X _ 1 = X \\circ \\alpha . \\end{align*}"}
-{"id": "1087.png", "formula": "\\begin{align*} f ^ { ( 1 ) } ( x , \\omega ) = g ( x ) + \\omega \\pmod { 1 } . \\end{align*}"}
-{"id": "5343.png", "formula": "\\begin{align*} \\chi _ { \\gamma } \\cdot \\chi _ i = \\sum _ { j = 0 } ^ { \\ell } m _ { i j } \\chi _ j \\end{align*}"}
-{"id": "8903.png", "formula": "\\begin{align*} u \\circ R = u . \\end{align*}"}
-{"id": "3858.png", "formula": "\\begin{align*} \\prod _ { k = 0 } ^ { 2 n } ( i _ 1 \\ i _ 2 \\ i _ 3 \\cdots i _ n ) ^ { k \\vee } \\end{align*}"}
-{"id": "8039.png", "formula": "\\begin{align*} & & & & a _ k & = s _ { i , j } & & k = i + \\sum _ { \\ell = 1 } ^ { j - i - 1 } ( m - \\ell ) . & & \\end{align*}"}
-{"id": "32.png", "formula": "\\begin{align*} \\eta : = \\frac { v _ { \\infty } } { p ^ \\star ( t ) } \\end{align*}"}
-{"id": "941.png", "formula": "\\begin{align*} \\psi ( x ) \\ , : = \\ , 1 - | 2 x - 1 | , x \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "6000.png", "formula": "\\begin{align*} ( J ) \\geq m \\left ( ( 1 - \\theta ) + \\kappa \\theta \\right ) P _ t ( x , z ) = m \\left ( 1 - \\theta ( 1 - \\kappa ) \\right ) P _ t ( x , z ) , \\end{align*}"}
-{"id": "938.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { b _ I } { k ^ 2 _ j } = m ( m - 1 + r ) , \\end{align*}"}
-{"id": "9044.png", "formula": "\\begin{align*} A _ j ^ { J , C } : = \\tau ( b ^ j ) + a _ j ^ { J , C } \\ . \\end{align*}"}
-{"id": "7047.png", "formula": "\\begin{align*} { { \\bf { Y } } ^ { [ j ] } } ( n ) = \\sum \\limits _ { i = 1 } ^ { M } { { { \\bf { H } } ^ { [ j i ] } } ( n ) { { \\bf { X } } ^ { [ i ] } } ( n ) } + { { \\bf { Z } } ^ { [ j ] } } ( n ) , \\ ; j \\in \\{ 1 , 2 , . . . , N \\} , \\end{align*}"}
-{"id": "4730.png", "formula": "\\begin{align*} G ^ - = \\{ x \\in \\mathbb { R } ^ 2 \\ , | \\ \\begin{bmatrix} 0 & 0 \\\\ - \\beta _ 2 v _ { 1 2 } & - \\beta _ 2 v _ { 2 2 } \\\\ - \\beta _ 1 v _ { 1 1 } & - \\beta _ 1 v _ { 2 1 } \\end{bmatrix} \\begin{bmatrix} x _ 1 \\\\ x _ 2 \\end{bmatrix} \\succeq _ { \\mathcal { L } ^ 3 } \\begin{bmatrix} - \\eta \\\\ - \\beta _ 2 \\alpha _ 2 \\\\ - \\beta _ 1 \\alpha _ 1 \\end{bmatrix} \\} . \\end{align*}"}
-{"id": "7424.png", "formula": "\\begin{align*} \\Delta ^ { I , J \\cup \\{ k \\} } \\Delta ^ { I \\cup \\{ i \\} , J \\cup \\{ j , l \\} } = \\Delta ^ { I , J \\cup \\{ j \\} } \\Delta ^ { I , J \\cup \\{ l \\} } + \\Delta ^ { I \\cup \\{ i \\} , J \\cup \\{ j , k \\} } \\Delta ^ { I \\cup \\{ i \\} , J \\cup \\{ k , l \\} } . \\end{align*}"}
-{"id": "3543.png", "formula": "\\begin{align*} & | \\partial ^ I _ x g _ { i j } ^ R ( x ) | = | R ^ { | I | } \\partial ^ I _ y g _ { i j } ( y ) | \\le C R ^ { - q } \\\\ & | \\partial ^ I _ x \\pi ^ R _ { i j } ( x ) | = | R ^ { 1 + | I | } \\partial ^ I _ y \\pi _ { i j } ( y ) | \\le C R ^ { - q } , \\end{align*}"}
-{"id": "727.png", "formula": "\\begin{align*} Z _ { i } = p ^ { * } { p } _ { * } g ^ { * } D _ { i } - g ^ { * } D _ { i } \\end{align*}"}
-{"id": "564.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\| ( I - A _ n ) K \\| | = 0 . \\end{align*}"}
-{"id": "9898.png", "formula": "\\begin{align*} \\sigma ( \\mathcal { Q } ( K _ { a , b } ) ) = \\left \\{ ( 2 n - a - 4 ) ^ { [ b - 1 ] } , ( 2 n - b - 4 ) ^ { [ a - 1 ] } , \\frac { 5 n - 8 \\pm \\sqrt { 9 n ^ { 2 } - 3 2 a b } } { 2 } \\ \\right \\} . \\end{align*}"}
-{"id": "4185.png", "formula": "\\begin{align*} w _ { a _ i } = \\frac { 1 } { 2 } \\left ( \\tanh \\frac { a _ i - a _ { i - 1 } } { 2 } + \\tanh \\frac { a _ { i + 1 } - a _ i } { 2 } \\right ) \\end{align*}"}
-{"id": "940.png", "formula": "\\begin{align*} \\mathbb { Z } _ + \\ , : = \\ , \\{ 0 , 1 , 2 , \\ldots \\} . \\end{align*}"}
-{"id": "7417.png", "formula": "\\begin{align*} \\overline { s } _ i : = \\varphi _ i \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} , \\end{align*}"}
-{"id": "9321.png", "formula": "\\begin{align*} & \\tilde { Z } ( t , z ) = \\tilde { Z } ( 0 , z ) \\exp \\Big \\{ \\int _ 0 ^ t ( - \\pi ( s , z ) b _ 0 ( s , z ) + \\frac { 1 } { 2 } \\pi ^ 2 ( s , z ) \\sigma _ 0 ^ 2 ( s , z ) ) d s \\\\ & - \\int _ 0 ^ t \\pi ( s , z ) \\sigma _ 0 ( s , z ) d B ( s ) \\Big \\} . \\end{align*}"}
-{"id": "9903.png", "formula": "\\begin{align*} \\{ \\alpha , \\beta , \\gamma \\} \\cap \\{ 0 , 1 , - 1 \\} = \\varnothing . \\end{align*}"}
-{"id": "688.png", "formula": "\\begin{align*} F A _ { x _ o } A _ { x _ o } ^ { - 1 } \\supset \\big \\{ g \\in G \\ , : \\ , \\lambda ( ( F A \\cap g A ) _ { x _ o } ) > 0 \\big \\} = \\big \\{ g \\in G \\ , : \\ , \\nu ( F A \\cap g A ) > 0 \\big \\} . \\end{align*}"}
-{"id": "5861.png", "formula": "\\begin{align*} \\ln g \\left ( x \\right ) = \\mp \\kappa \\frac { \\left ( 2 \\left ( \\mu - \\lambda - x \\right ) + 1 \\right ) } { 3 } \\sigma \\end{align*}"}
-{"id": "6560.png", "formula": "\\begin{align*} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\varOmega ) ) } + \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { 2 , q } ( \\varOmega ) ) } \\le C \\tau ^ k , \\\\ & \\max _ { k \\le n \\le N } \\| e _ n \\| _ { W ^ { 1 , \\infty } ( \\varOmega ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "8151.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\lambda _ i = \\sum _ { s \\in S } \\psi _ { i } ( s ) = \\psi _ i ( S ) , & i = 1 , \\ldots , m ; \\\\ \\mu _ { h 1 } + \\mu _ { h 2 } = \\chi _ { _ h } ( S _ 1 ) , & h = 1 , 2 , \\ldots , [ \\frac { n - 1 } { 2 } ] ; \\\\ \\mu _ { h 1 } ^ 2 + \\mu _ { h 2 } ^ 2 = \\chi _ { _ h } ( S _ 1 ^ 2 ) + \\chi _ { _ h } ( S _ 2 ^ 2 ) , & h = 1 , 2 , \\ldots , [ \\frac { n - 1 } { 2 } ] . \\\\ \\end{array} \\right . \\\\ \\end{align*}"}
-{"id": "5487.png", "formula": "\\begin{align*} \\sup \\limits _ { g ' \\in \\mathcal { G } _ { f ^ { * } } } \\left ( \\sum \\limits _ { i = 1 } ^ { n } \\xi _ { i } p ( g ' ( X _ { i } ) ) - \\frac { h } { 1 2 } p ( g ' ( X _ i ) ) \\right ) \\end{align*}"}
-{"id": "2312.png", "formula": "\\begin{align*} \\frac 1 \\tau \\ , \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { - 1 , q } ( \\varOmega ) ) } + \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { 1 , q } ( \\varOmega ) ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "7941.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k } d _ i ^ { ( m ) } \\geq \\sum _ { i = 1 } ^ k d _ i \\geq \\binom { k } { 2 } \\end{align*}"}
-{"id": "4822.png", "formula": "\\begin{align*} \\begin{aligned} f m a c ( \\tau ) & = \\alpha ( \\tau ) \\circ K _ 0 ( \\psi ( \\tau ) ) \\\\ & = \\alpha ( { \\tilde \\tau } ) \\circ K _ 0 ( W \\circ \\psi ( \\tau ) ) \\\\ & = \\alpha ( { \\tilde \\tau } ) \\circ K _ 0 ( \\psi ( { \\tilde \\tau } ) \\circ W ) \\\\ & = \\alpha ( { \\tilde \\tau } ) \\circ K _ 0 ( \\psi ( { \\tilde \\tau } ) \\circ K _ 0 ( W ) \\\\ & = \\alpha ( { \\tilde \\tau } ) \\circ K _ 0 ( \\psi ( { \\tilde \\tau } ) = f m a c ( { \\tilde \\tau } ) . \\end{aligned} \\end{align*}"}
-{"id": "9765.png", "formula": "\\begin{align*} \\nabla _ { \\vec { x } } V ( \\vec { x } ) = \\left [ \\dfrac { \\partial f } { \\partial \\vec { x } } ( \\vec { z } ^ { * } ( \\vec { x } ) , \\vec { x } ) \\right ] ^ { \\top } + \\left [ \\dfrac { \\partial \\vec { g } } { \\partial \\vec { x } } ( \\vec { z } ^ { * } ( \\vec { x } ) , \\vec { x } ) \\right ] ^ { \\top } \\boldsymbol { \\lambda } ^ { * } ( \\vec { x } ) , \\end{align*}"}
-{"id": "3883.png", "formula": "\\begin{align*} \\sigma _ { k } ( \\overbrace { B , \\dotsc , B } ^ { l } , C , \\dotsc , C ) & : = \\sigma _ { k } ( \\overbrace { B , \\dotsc , B } ^ { l } , \\overbrace { C , \\dotsc , C } ^ { k - l } ) , \\\\ T _ { k } ( \\overbrace { B , \\dotsc , B } ^ { l } , C , \\dotsc , C ) _ { i j } & : = T _ { k } ( \\overbrace { B , \\dotsc , B } ^ { l } , \\overbrace { C , \\dotsc , C } ^ { k - l } ) _ { i j } . \\end{align*}"}
-{"id": "9188.png", "formula": "\\begin{align*} d X _ t = d M _ t + \\alpha { I d t } , \\end{align*}"}
-{"id": "429.png", "formula": "\\begin{align*} G = ( g ^ G \\cup g ^ { - G } ) ^ k . \\end{align*}"}
-{"id": "5793.png", "formula": "\\begin{align*} L _ { - 1 } x _ { \\alpha } = T \\psi _ { \\alpha } \\end{align*}"}
-{"id": "3175.png", "formula": "\\begin{align*} \\sigma _ l \\ ! = \\ ! \\ ! \\sum \\nolimits _ { n , k } { \\ ! \\mathbb { E } \\ ! \\left \\{ \\beta _ n \\beta _ { n + k } \\right \\} \\ ! g ( d _ n ) g ( d _ { n + k } ) \\mathbb { P } ( { n \\ ! + \\ ! k } , \\tau ) f _ x ( n \\ ! ) . } \\end{align*}"}
-{"id": "5115.png", "formula": "\\begin{align*} \\tilde { L } ( u ; s ) = \\frac { 1 } { 1 - s u } \\begin{pmatrix} 1 - s u q ^ { 2 N } & u \\beta ^ { * } ( 1 - s ^ 2 q ^ { 2 N } ) \\\\ \\beta & u - s q ^ { 2 N } \\end{pmatrix} \\end{align*}"}
-{"id": "8734.png", "formula": "\\begin{gather*} \\sum _ { j \\ge 1 } \\int _ 0 ^ { \\tau } | \\nabla ^ G _ { e _ j } v ^ { ( 0 ) } ( s , X _ s ^ 1 ) - \\nabla ^ G _ { e _ j } v ^ { ( 0 ) } ( s , X _ s ^ 2 ) | _ { K } ^ 2 d s = \\int _ 0 ^ { \\tau } \\sum _ { m \\ge 1 } | \\nabla ^ G v ^ { ( 0 ) } _ m ( s , X _ s ^ 1 ) - \\nabla ^ G v ^ { ( 0 ) } _ m ( s , X _ s ^ 2 ) | ^ 2 _ U d s \\\\ = \\int _ 0 ^ { \\tau } \\sum _ { m \\ge 1 } \\sup _ { | a | _ U = 1 } | \\nabla ^ G _ a v ^ { ( 0 ) } _ m ( s , X _ s ^ 1 ) - \\nabla ^ G _ a v ^ { ( 0 ) } _ m ( s , X _ s ^ 2 ) | ^ 2 d s . \\end{gather*}"}
-{"id": "9200.png", "formula": "\\begin{align*} ( \\psi , \\phi ) _ { L ^ 2 ( D ) } = \\int _ { D } \\psi ( x ) \\phi ( x ) d x \\end{align*}"}
-{"id": "8188.png", "formula": "\\begin{align*} \\omega _ c = i _ c ^ * \\frac { d x _ 1 } { ( a _ 2 - a _ 3 ) x _ 2 x _ 3 } = i _ c ^ * \\frac { d x _ 2 } { ( a _ 3 - a _ 1 ) x _ 3 x _ 1 } = i _ c ^ * \\frac { d x _ 3 } { ( a _ 1 - a _ 2 ) x _ 1 x _ 2 } , \\end{align*}"}
-{"id": "1252.png", "formula": "\\begin{align*} \\mathcal G _ 0 & = - i \\alpha \\cdot \\nabla G _ 0 + 2 m G _ 0 I _ { 1 } \\\\ \\mathcal G _ 1 & = - i \\alpha \\cdot \\nabla G _ 1 + 2 m G _ 1 I _ { 1 } - \\frac 2 m M _ { 1 1 } - \\frac 2 m M _ { 2 2 } \\\\ \\mathcal G _ 2 & = - i \\alpha \\cdot \\nabla G _ 2 + 2 m G _ 2 I _ 1 + \\frac 1 { 2 m } G _ 0 - \\frac 1 { 4 \\pi m } M _ { 1 1 } - \\frac 1 { 4 \\pi m } M _ { 2 2 } . \\end{align*}"}
-{"id": "5864.png", "formula": "\\begin{align*} C ^ { x } = - m \\ln x + c _ { 1 } x . \\end{align*}"}
-{"id": "7310.png", "formula": "\\begin{align*} D ^ 2 & \\sim \\sum _ { i , j } \\mathcal { E } _ i \\mathcal { E } _ j ^ * \\otimes ( \\gamma _ { - } ( w _ i ) \\gamma _ { - } ( w _ j ) ^ * + q ^ { - ( \\xi _ i , \\xi _ j ) } \\gamma _ { - } ( w _ j ) ^ * \\gamma _ { - } ( w _ i ) ) \\\\ & + \\sum _ { i , j , k , l } c _ { i , j } ^ { k , l } \\mathcal { E } _ k \\mathcal { E } _ l ^ * \\otimes \\gamma _ { - } ( w _ j ) ^ * \\gamma _ { - } ( w _ i ) . \\end{align*}"}
-{"id": "9574.png", "formula": "\\begin{align*} g _ { j k } : = \\left \\{ \\begin{array} { c } \\frac { e ^ { - m | y _ j - y _ k | } } { 4 \\pi | y _ j - y _ k | } , ~ ~ { \\rm i f } ~ ~ j \\not = k \\\\ \\\\ 0 , \\qquad ~ ~ { \\rm i f } ~ ~ j = k \\end{array} \\right . \\end{align*}"}
-{"id": "6289.png", "formula": "\\begin{align*} \\frac { 1 } { N _ v ( z ) } = \\frac { d } { z + c } - \\frac { b } { ( z + c ) ^ 2 N ( x ) } ~ { \\rm f o r } ~ v ( z ) ~ { \\rm o f ~ ( \\ref { e q v a b c } ) ~ a n d ~ } x ~ { \\rm o f ~ ( \\ref { e q r r r } ) } . \\end{align*}"}
-{"id": "5015.png", "formula": "\\begin{align*} \\sigma _ 1 * \\sigma _ k = \\frac { 1 } { 2 r } \\sigma _ { k - 1 } + \\big ( 1 - \\frac { 1 } { 2 r } \\big ) \\sigma _ { k + 1 } , \\textrm { f o r a l l $ k \\geq 1 $ } . \\end{align*}"}
-{"id": "5649.png", "formula": "\\begin{align*} ( T ( t ) f ) ( x ) : = h _ t ( x ) f ( \\varphi ( t , x ) ) , f \\in L ^ p _ \\rho ( \\Omega ) , \\ x \\in \\Omega , \\ t \\geq 0 \\end{align*}"}
-{"id": "2219.png", "formula": "\\begin{align*} \\tilde { u } ( x , t ) : = \\int _ { 0 } ^ { t } \\mu ( t - s ) v ( x , s ) d s , \\end{align*}"}
-{"id": "4714.png", "formula": "\\begin{align*} | \\sum _ { i = 1 } ^ { n + 1 } \\varepsilon _ i ( w ^ i ) ^ { \\top } z - \\sum _ { i = 1 } ^ { n + 1 } \\varepsilon _ i w _ 0 ^ i | < w _ 0 - w ^ { \\top } z . \\end{align*}"}
-{"id": "2089.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + ( K + Z ) ) X ^ { ( j ) } ( s _ { i } ) = M \\tilde { V } _ { j } \\ \\ j = 1 , \\ \\ldots , \\ J - 1 \\ \\ i = 1 , \\ldots , l . \\end{align*}"}
-{"id": "5554.png", "formula": "\\begin{align*} \\beta = \\begin{cases} \\frac { 1 } { 2 } b _ { x } ( 0 ) - h b ( 0 ) & 0 \\leq h < \\infty , \\\\ - \\frac { 1 } { 2 } b _ { x } ( 0 ) + \\frac { 1 } { 2 h } b _ { x x } ( 0 ) & 0 < h \\leq \\infty . \\end{cases} \\end{align*}"}
-{"id": "5333.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ) ) ' = f ( t , u ) & & \\\\ u ( T ) = b u ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "5389.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } \\leq \\left ( k - 2 \\alpha + 5 \\alpha ^ 2 \\right ) \\frac { n ^ 2 } { 2 } \\ , , \\end{align*}"}
-{"id": "770.png", "formula": "\\begin{gather*} \\varphi ( a ) = a ( r _ 1 ^ s - 1 ) , \\psi ( a ) = a r _ 1 ^ s . \\end{gather*}"}
-{"id": "4592.png", "formula": "\\begin{align*} X _ i ^ { ( n ) } , Y _ i ^ { ( n ) } , Z _ { i j } , i < j , n = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "6588.png", "formula": "\\begin{align*} ( x - \\gamma \\nabla f ( x ) ) = \\nabla \\left ( \\tfrac { 1 } { 2 } \\| x \\| ^ 2 - \\gamma f ( x ) \\right ) . \\end{align*}"}
-{"id": "5482.png", "formula": "\\begin{align*} P ' \\Bigl [ ( P - P ' _ { n } ) \\tilde { g } \\ge \\frac { t / 2 + c _ { 2 } P \\tilde { g } } { 1 + c _ { 2 } } \\Bigr ] \\ ! \\le \\ ! \\frac { P \\tilde { g } ^ 2 ( 1 + c _ { 2 } ) ^ 2 } { n ( t / 2 + c _ { 2 } P \\tilde { g } ) ^ 2 } \\ ! \\le \\ ! \\frac { B P \\tilde { g } ( 1 + c _ { 2 } ) ^ 2 } { 2 n t c _ { 2 } P \\tilde { g } } \\ ! = \\ ! \\frac { B ( 1 + c _ { 2 } ) ^ 2 } { 2 n t c _ { 2 } } . \\end{align*}"}
-{"id": "3525.png", "formula": "\\begin{align*} C ^ { - 1 } \\rho ( y ) & \\le \\rho ( x ) \\le C \\rho ( y ) \\\\ C ^ { - 1 } \\phi ( y ) & \\le \\phi ( x ) \\le C \\phi ( y ) . \\end{align*}"}
-{"id": "9314.png", "formula": "\\begin{align*} \\Phi ( t , z ) = \\frac { \\mathbb { E } [ D _ t \\delta _ Z ( z ) | \\mathcal { F } _ t ] } { \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] } \\end{align*}"}
-{"id": "934.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t \\theta ^ { n , 1 } _ i + u ^ n \\cdot \\nabla \\theta ^ { n , 1 } _ i = f ^ n _ i - f ^ \\infty _ i , \\\\ \\partial _ t \\sigma ^ { n , 1 } + u ^ n \\cdot \\nabla \\sigma ^ { n , 1 } = g ^ n - g ^ \\infty , \\\\ { \\theta ^ { n , 1 } _ i } | _ { t = 0 } = \\nabla ( u ^ n _ 0 ) _ { i } - \\nabla ( u ^ \\infty _ 0 ) _ { i } , \\\\ { \\sigma ^ { n , 1 } } | _ { t = 0 } = \\nabla \\gamma ^ n _ 0 - \\nabla \\gamma ^ \\infty _ 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7696.png", "formula": "\\begin{align*} \\sigma : = \\inf _ { x > 2 K - \\varepsilon _ 0 } ( x - f ( x ) ) > 0 . \\end{align*}"}
-{"id": "3195.png", "formula": "\\begin{align*} D _ { j k } = \\left ( \\begin{array} { c | c c c } t _ { j k } ^ 1 & 0 & \\cdots & 0 \\\\ \\hline a _ { j k } ^ 2 & & & \\\\ \\vdots & & S _ { j k } & \\\\ a _ { j k } ^ r & & & \\end{array} \\right ) , \\end{align*}"}
-{"id": "7881.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow s } \\phi _ y ( t , x , s ) = q ( s , x , y ) \\ , . \\end{align*}"}
-{"id": "4835.png", "formula": "\\begin{align*} K _ { 1 1 } ^ { \\rm G S E } ( x , y ) & = K ^ { 4 } _ { 2 1 } ( x , y ) , \\\\ K _ { 1 2 } ^ { \\rm G S E } ( x , y ) & = K ^ { 4 } _ { 2 2 } ( x , y ) , \\\\ K _ { 2 1 } ^ { \\rm G S E } ( x , y ) & = - K ^ { 4 } _ { 1 1 } ( x , y ) = - K ^ { 4 } _ { 2 2 } ( y , x ) = - K ^ { \\rm G S E } _ { 1 2 } ( y , x ) , \\\\ K _ { 2 2 } ^ { \\rm G S E } ( x , y ) & = - K ^ { 4 } _ { 1 2 } ( x , y ) . \\end{align*}"}
-{"id": "5870.png", "formula": "\\begin{align*} Z ^ { 1 } = e ^ { m t } K ^ { 1 } ~ , ~ Z ^ { 2 } = e ^ { - m t } \\left ( K ^ { 1 } + m \\int \\frac { d x } { \\sigma \\left ( x \\right ) } F \\partial _ { F } \\right ) \\end{align*}"}
-{"id": "1247.png", "formula": "\\begin{align*} \\mathcal R _ 0 ( \\lambda ) = ( D _ m + \\lambda ) R _ 0 ( \\lambda ^ 2 - m ^ 2 ) . \\end{align*}"}
-{"id": "8947.png", "formula": "\\begin{align*} \\int _ { \\mathbb H ^ n } | \\Delta _ g u _ R | ^ p d V _ g = \\int _ 0 ^ { + \\infty } f _ R ( s ) ^ p d s = 1 + \\ln ( R / s _ 0 ) + \\int _ 0 ^ 1 ( 1 - s ) ^ p d s . \\end{align*}"}
-{"id": "8521.png", "formula": "\\begin{align*} f ^ { \\ast } ( w ^ { - 1 } z ) = w ^ { \\ast } ( z f ^ { - 1 } ) \\end{align*}"}
-{"id": "7252.png", "formula": "\\begin{align*} h _ { 1 , j k , \\alpha } ( z _ j ) = \\left ( \\begin{array} { c } h _ { 1 , j k , \\alpha } ^ 1 ( z _ j ) \\\\ h _ { 1 , j k , \\alpha } ^ 2 ( z _ j ) \\\\ \\vdots \\\\ h _ { 1 , j k , \\alpha } ^ r ( z _ j ) \\end{array} \\right ) \\end{align*}"}
-{"id": "2086.png", "formula": "\\begin{align*} \\tilde { V } _ { j + 1 } = [ X ^ { ( j ) } ( s _ { t _ { j } } ) / | | X ^ { ( j ) } ( s _ { t _ { j } } ) | | ] , \\end{align*}"}
-{"id": "5933.png", "formula": "\\begin{align*} W ^ { \\gamma , p } ( \\R ^ { 2 d } ) = \\Big \\{ f \\in L ^ p ( \\R ^ { 2 d } ) : \\int _ { \\R ^ d } \\| f ( x , \\cdot ) \\| _ { W ^ { \\gamma , p } ( \\R ^ d ) ) } ^ p \\dd x + \\int _ { \\R ^ d } \\| f ( \\cdot , v ) \\| _ { W ^ { \\gamma , p } ( \\R ^ d ) ) } ^ p \\dd v < \\infty \\Big \\} , \\end{align*}"}
-{"id": "1277.png", "formula": "\\begin{align*} v _ p ( \\kappa ( \\mu _ L ) ) = v _ p ( \\kappa ( { \\mu _ L } _ { \\vert M _ 0 } ) ) + \\sum _ { i = 1 } ^ { \\frac { q - 5 } { 4 } } \\geq 2 t + { \\frac { q - 5 } { 4 } } 4 t = ( q - 3 ) t = v _ p ( \\kappa ( \\mu _ L ) ) , \\end{align*}"}
-{"id": "1801.png", "formula": "\\begin{align*} T ^ { \\alpha } f ( t , x ) = \\int _ 0 ^ t \\int _ { { \\mathbb R } ^ d } P ^ { \\alpha } ( t - s , x - y ) f ( s , y ) d y d s \\end{align*}"}
-{"id": "2512.png", "formula": "\\begin{align*} \\mathcal { L } _ { X } I _ 1 = \\dots = \\mathcal { L } _ { X } I _ k = 0 , \\mathcal { L } _ { X } D _ 1 = ( - \\lambda ) ( D _ 1 - d _ 1 ) , \\dots , \\mathcal { L } _ { X } D _ p = ( - \\lambda ) ( D _ p - d _ p ) , \\end{align*}"}
-{"id": "9336.png", "formula": "\\begin{align*} d h ( t , x , z ) & = y ( t , x , z ) [ - \\{ L ^ * _ { \\pi ( t ) } p ( t , x , z ) + x q ( t , x , z ) \\} d t + q ( t , x , z ) d G ( t ) ] \\\\ & + p ( t , x , z ) [ L ^ * _ { \\pi ( t ) } y ( t , x , z ) d t + x y ( t , x , z ) d G ( t ) ] \\\\ & + x y ( t , z ) q ( t , x , z ) d \\langle G \\rangle _ t \\\\ & = y ( t , x , z ) q ( t , x , z ) d G ( t ) + p ( t , x , z ) x y ( t , x , z ) d G ( t ) \\\\ & = d F ( t , x , z ) + x h ( t , x , z ) d G ( t ) \\end{align*}"}
-{"id": "2799.png", "formula": "\\begin{align*} \\lambda _ 0 + \\lambda _ 0 ^ 2 = \\frac { ( R + d ^ 2 ) ^ 2 } { 4 d ^ 2 } - \\dfrac { 1 } { 4 } . \\end{align*}"}
-{"id": "8330.png", "formula": "\\begin{align*} y _ { 1 } ^ { 3 } - y _ { 1 } & = r ( T ) , \\\\ y _ { 1 } & = y ^ { 9 } + y ^ { 3 } + y . \\\\ y _ { 2 } ^ { 3 } - y _ { 2 } & = \\omega r ( T ) , \\\\ y _ { 2 } & = ( \\omega y ) ^ { 9 } + ( \\omega y ) ^ { 3 } + \\omega y . \\\\ y _ { 3 } ^ { 3 } - y _ { 3 } & = \\omega ^ { 2 } r ( T ) , \\\\ y _ { 3 } & = ( \\omega ^ { 2 } y ) ^ { 9 } + ( \\omega ^ { 2 } y ) ^ { 3 } + \\omega ^ { 2 } y . \\end{align*}"}
-{"id": "4698.png", "formula": "\\begin{align*} A _ q ( k - 1 ) = \\begin{pmatrix} q - 1 & 1 & & & \\\\ q - 1 & & 1 & & \\\\ \\vdots & & & \\ddots & \\\\ q - 1 & & & & 1 \\\\ q - 1 & & & & \\end{pmatrix} . \\end{align*}"}
-{"id": "8908.png", "formula": "\\begin{align*} u ( R _ \\lambda ( \\theta ) ( x ) ) = u ( x ) . \\end{align*}"}
-{"id": "2213.png", "formula": "\\begin{align*} A ( \\xi ) = \\int _ { \\mathcal { S } ^ { n - 1 } } | \\xi \\cdot \\theta | ^ { 2 } a ( \\theta ) d \\theta , \\end{align*}"}
-{"id": "3629.png", "formula": "\\begin{align*} u _ { n + 1 } = c T + ( 1 - c ) f ( u _ n , u _ { n - 1 } , \\dots , u _ { n - k + 1 } ) , T \\ge 0 , \\ ; c \\in [ 0 , 1 ) . \\end{align*}"}
-{"id": "7143.png", "formula": "\\begin{align*} \\nabla ^ \\beta _ y Q _ s | _ { y _ n = 0 } = 0 , \\forall s \\ge 0 , \\forall \\beta . \\end{align*}"}
-{"id": "8130.png", "formula": "\\begin{align*} { \\mathcal R } _ { k ' , 0 , 1 } ( U ) \\geq { \\mathcal R } _ { k ' , 0 , 1 } ( ( 0 , 1 ) ) \\mbox { f o r a l l s m o o t h s e t s $ U \\subset \\mathbb { R } $ . } \\end{align*}"}
-{"id": "3946.png", "formula": "\\begin{align*} \\psi ( s ) = ( \\varphi _ 1 ( s ) - \\varphi _ 1 ( s _ 0 ) ) ^ { - 1 } ( \\varphi _ 2 ( s ) - \\varphi _ 2 ( s _ 0 ) ) \\end{align*}"}
-{"id": "968.png", "formula": "\\begin{align*} \\alpha = - \\mathcal { L } _ T \\eta \\ ; , \\end{align*}"}
-{"id": "1333.png", "formula": "\\begin{align*} \\tau ( \\mu | \\{ \\lambda \\} ) = a ( \\mu ) \\prod _ { a = 1 } ^ M f ( \\lambda _ a , \\mu ) + d ( \\mu ) \\prod _ { a = 1 } ^ M f ( \\mu , \\lambda _ a ) \\end{align*}"}
-{"id": "202.png", "formula": "\\begin{align*} f _ { t , g } ( x ) : = \\frac { 2 c ( t ) } { 1 + e ^ { - 2 t g ( x ) } } f ( x ) , \\end{align*}"}
-{"id": "2259.png", "formula": "\\begin{align*} \\Delta H _ { q , p } ( 2 ) = \\log \\left ( 1 \\pm C ( \\vec { \\varepsilon } ) \\left | \\Delta _ { 0 } ( 2 ) \\right | \\right ) \\pm \\frac { 1 } { 2 \\sigma ^ { 2 } } \\left | \\Delta \\mu _ { ( 2 ) } ( 2 ) \\right | \\pm \\varepsilon _ { q } \\left | \\Delta \\mu _ { ( q ) } ( 2 ) \\right | \\pm \\varepsilon _ { p } \\left | \\Delta \\mu _ { ( p ) } ( 2 ) \\right | . \\end{align*}"}
-{"id": "9687.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { t - \\tau ( t ) } { t / \\varphi ( t ) } = 1 . \\end{align*}"}
-{"id": "8512.png", "formula": "\\begin{align*} \\pi _ 1 & = \\min \\{ c _ { \\alpha _ 1 } + c _ { \\alpha _ 1 + \\alpha _ 2 } , c _ { \\alpha _ 1 } + c _ { \\alpha _ 2 } , c _ { \\alpha _ 1 + 2 \\alpha _ 2 } + c _ { \\alpha _ 2 } \\} , \\\\ \\pi _ 2 & = \\min \\{ 2 c _ { \\alpha _ 1 } + c _ { \\alpha _ 1 + \\alpha _ 2 } , 2 c _ { \\alpha _ 1 } + c _ { \\alpha _ 2 } , 2 c _ { \\alpha _ 1 + 2 \\alpha _ 2 } + c _ { \\alpha _ 2 } \\} . \\end{align*}"}
-{"id": "7663.png", "formula": "\\begin{align*} \\ \\left \\{ \\begin{aligned} & u _ { t } = \\triangle ^ { \\alpha / 2 } u + f ( u ) , \\\\ & u ( x , 0 ) = u _ { 0 } ( x ) , \\\\ & u | _ { \\partial B _ { R } } = 0 \\end{aligned} \\right . \\end{align*}"}
-{"id": "9236.png", "formula": "\\begin{align*} \\hat { H } = H ( t , x , \\hat { Y } ( t , x , z ) , \\hat { u } ( t , z ) , \\hat { p } ( t , x , z ) , \\hat { q } ( t , x , z ) , \\hat { r } ( t , x , z , \\cdot ) ) \\end{align*}"}
-{"id": "5773.png", "formula": "\\begin{align*} a ( \\lambda ) = \\prod _ { j = 1 } ^ K a _ j ( \\lambda ) , d ( \\lambda ) = \\prod _ { j = 1 } ^ K d _ j ( \\lambda ) . \\end{align*}"}
-{"id": "3289.png", "formula": "\\begin{align*} \\begin{gathered} v _ 1 \\wedge v _ 1 = 0 , v _ 0 \\wedge v _ 1 = - q ^ 2 v _ 1 \\wedge v _ 0 , v _ 0 \\wedge v _ 0 = - q ^ { - 1 } ( q - q ^ { - 1 } ) v _ 1 \\wedge v _ { - 1 } , \\\\ v _ { - 1 } \\wedge v _ 1 = - v _ 1 \\wedge v _ { - 1 } , v _ { - 1 } \\wedge v _ 0 = - q ^ 2 v _ 0 \\wedge v _ { - 1 } , v _ { - 1 } \\wedge v _ { - 1 } = 0 . \\end{gathered} \\end{align*}"}
-{"id": "9711.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log g ( x ( t ) ) } { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } = - \\log \\left ( \\frac { a } { b } \\right ) . \\end{align*}"}
-{"id": "9689.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log g _ 1 ( x _ 1 ( t ) ) } { \\log g _ 2 ( x _ 2 ( t ) ) } = 1 . \\end{align*}"}
-{"id": "2890.png", "formula": "\\begin{align*} \\overline { P o i s _ n ^ * \\{ n + 1 \\} } ^ + ( 1 ) = \\overline { P o i s _ n ^ * \\{ n + 1 \\} } ( 1 ) \\oplus \\mathbb { K } [ 1 ] = ( \\overline { P o i s _ n ^ * \\{ n \\} } ( 1 ) \\oplus \\mathbb { K } ) [ 1 ] , \\end{align*}"}
-{"id": "613.png", "formula": "\\begin{align*} \\beta _ n : = \\frac { 6 } { E _ 6 ( \\rho ) } \\sum _ { ( \\lambda ) } \\sum _ { ( c , d ) } \\frac { h _ { ( c , d ) } ( n ) } { \\lambda ^ 3 } e ^ { \\frac { \\pi n \\sqrt { 3 } } { \\lambda } } . \\end{align*}"}
-{"id": "2076.png", "formula": "\\begin{align*} A M \\tilde { x } = b , \\ x = M \\tilde { x } \\ \\ \\ \\ M _ { 1 } A M _ { 2 } \\tilde { x } = M _ { 1 } b , \\ x = M _ { 2 } \\tilde { x } . \\end{align*}"}
-{"id": "7338.png", "formula": "\\begin{gather*} V _ 1 = v _ 1 \\otimes v _ 0 - q ^ 2 v _ 0 \\otimes v _ 1 , V _ { - 1 } = v _ 0 \\otimes v _ { - 1 } - q ^ 2 v _ { - 1 } \\otimes v _ 0 , \\\\ V _ 0 = v _ 1 \\otimes v _ { - 1 } - v _ { - 1 } \\otimes v _ 1 - q ( q - q ^ { - 1 } ) v _ 0 \\otimes v _ 0 . \\end{gather*}"}
-{"id": "841.png", "formula": "\\begin{align*} & X _ { I V } = \\sum _ { t = 2 } ^ { a + 1 } \\sum _ { a + 1 \\ge p ( t ) > \\cdots > p ( 2 ) > p ( 1 ) = 1 } \\sum _ { \\ell ( t ) \\in J _ { p ( t ) } } \\cdots \\sum _ { \\ell ( 1 ) \\in J _ { p ( 1 ) } } K _ { t } ( w ; z _ { \\ell ( 1 ) } , \\ldots , z _ { \\ell ( t ) } ) \\\\ & { } \\times \\prod _ { 1 \\le s \\le t } ^ { \\curvearrowleft } Z _ { \\ell ( s ) } ^ { J _ { p ( s + 1 ) - 1 } \\cup \\cdots \\cup J _ { p ( s ) } } ( \\vec { z } ) u ( a + 2 , ( a + 1 ) ^ { k _ { a + 1 } } , \\ldots , 1 ^ { k _ { 1 } } ) , \\end{align*}"}
-{"id": "5207.png", "formula": "\\begin{align*} ( f , g ) ( z ) = \\sideset { } { ' } \\sum _ { | J | = q } f _ { J } ( z ) \\overline { g _ { J } ( z ) } \\ , . \\end{align*}"}
-{"id": "4191.png", "formula": "\\begin{align*} \\widehat { F } ( x ) = \\frac { n ! \\omega _ n } { ( 1 + 4 \\pi ^ 2 \\norm { x } _ 2 ^ 2 ) ^ { ( n + 1 ) / 2 } } , \\end{align*}"}
-{"id": "6859.png", "formula": "\\begin{align*} v _ n ^ j ( t , x ) : = e ^ { i x \\xi _ n ^ j } e ^ { - i t | \\xi _ n ^ j | ^ 2 } \\Psi ^ j _ { [ h _ n ^ j ] } ( t , x - 2 t \\xi _ n ^ j ) , \\end{align*}"}
-{"id": "279.png", "formula": "\\begin{align*} \\int _ { \\mathcal { X } _ n ^ c \\times \\mathcal { X } } f ( x ) f ( y ) \\int _ { l _ x } ^ { v _ x } ( h _ u F ) ( u , v _ y ) \\ , d u \\ , d y \\ , d x = O \\biggl ( \\frac { k ^ { - \\frac { 1 } { 2 } + \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "7154.png", "formula": "\\begin{align*} w = 0 \\mbox { o n } \\ , \\ , \\partial \\R ^ n _ + = \\{ x _ n = 0 \\} . \\end{align*}"}
-{"id": "3356.png", "formula": "\\begin{align*} \\| M \\| _ { r \\ast } \\ ; = \\ ; \\max _ { \\sum _ { i = 1 } ^ r s _ i ^ 2 = 1 } \\sum _ { i = 1 } ^ { n } \\sigma _ i ( M ) s _ i \\geq \\| M \\| _ F \\ ; \\geq \\ ; \\max _ { \\sum _ { i = 1 } ^ r s _ i ^ 2 = 1 } \\sum _ { i = 1 } ^ { r } \\sigma _ i ( M ) s _ i \\ ; = \\ ; \\| M \\| _ r . \\end{align*}"}
-{"id": "6217.png", "formula": "\\begin{align*} \\hat { H } ( s ) = ( \\hat { C } _ { p } + s \\hat { C } _ { v } ) \\hat { X } ( s ) , \\end{align*}"}
-{"id": "8893.png", "formula": "\\begin{align*} L _ n : H ^ 1 _ { A _ n } ( \\R ^ N , \\C ) \\to H ^ 1 _ { A _ n } ( \\R ^ N , \\C ) : v \\mapsto L _ n v : = ( - \\Delta _ { A _ n } + 1 ) ^ { - 1 } W _ n [ v ] , \\end{align*}"}
-{"id": "4765.png", "formula": "\\begin{align*} f f '' + ( f ' ) ^ 2 + 1 = \\pm 2 a f \\sqrt { f '^ 2 + 1 } , a = c o n s t \\neq 0 . \\end{align*}"}
-{"id": "8170.png", "formula": "\\begin{align*} \\phi _ n = [ t ^ n ] D _ { \\beta / \\alpha } ( t ) = \\dfrac 1 { 1 + ( \\alpha + \\beta ) n } \\ , [ t ^ n ] \\tilde { D } _ { \\beta / \\alpha } ^ { 1 + ( \\alpha + \\beta ) n } ( t ) , \\end{align*}"}
-{"id": "6661.png", "formula": "\\begin{gather*} { \\bf E } W _ 1 ( \\lambda , \\lambda ^ n ) \\leqslant \\sum _ { k = 1 } ^ n \\int \\limits _ { I _ k ^ n } \\sqrt { C _ 1 \\cdot \\left | u - \\dfrac { 2 k - 1 } { 2 n } \\right | + \\left | u - \\dfrac { 2 k - 1 } { 2 n } \\right | ^ 2 } \\ , \\mu ( d u ) \\leqslant \\\\ \\leqslant \\sum _ { k = 1 } ^ n p _ k ^ n \\cdot \\sqrt { C _ 1 \\cdot \\dfrac { 1 } { 2 n } + \\dfrac { 1 } { 4 n ^ 2 } } \\leqslant \\dfrac { K } { \\sqrt { n } } , \\end{gather*}"}
-{"id": "5391.png", "formula": "\\begin{align*} e ( B ) \\leq \\left ( \\beta - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } + \\left ( k - \\alpha - \\beta \\right ) \\frac { n ^ 2 } { 2 } = \\bigg ( k - \\alpha - \\frac 1 4 \\bigg ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "2615.png", "formula": "\\begin{align*} \\left ( \\frac { a } { p } \\right ) = \\begin{cases} 1 & a p a \\not \\equiv 0 \\pmod { p } , \\\\ - 1 & a p , \\\\ 0 & a \\equiv 0 \\pmod { p } . \\end{cases} \\end{align*}"}
-{"id": "7847.png", "formula": "\\begin{align*} \\beta _ { n - 1 } ( x ) = \\prod _ { i = 0 } ^ { n - 1 } \\phi _ i ( x ) . \\end{align*}"}
-{"id": "7922.png", "formula": "\\begin{align*} \\aligned \\Phi ( u _ j , - p _ 0 , B _ { 3 r , { r } / { 4 } } ) & \\le \\Phi ( u _ j , 1 + \\tilde \\beta _ 0 , B _ { \\tilde \\rho _ 0 + \\tilde \\delta _ 0 , \\tilde \\sigma _ 0 - \\tilde \\delta _ 0 } ) \\\\ & \\leq \\prod _ { k = 0 } ^ m ( C ' / \\tilde \\delta _ k ^ 2 ) ^ { 1 / | 1 + \\tilde \\beta _ k | } \\Phi ( u _ j , - \\bar p , B _ { 2 r , r } ) \\\\ & \\le ( C ' ) ^ { \\frac 2 { p _ 0 } } 2 ^ { \\frac c { p _ 0 } } r ^ { - 4 ( \\frac { 1 } { p _ 0 } - \\frac { 1 } { \\bar p } ) } \\Phi ( u _ j , - \\bar p , B _ { 2 r , r } ) . \\endaligned \\end{align*}"}
-{"id": "7747.png", "formula": "\\begin{align*} \\frac { \\sum _ { i = 1 } ^ n \\sigma _ i \\mu _ { i + 1 } } { \\sqrt { \\sum _ { k = 1 } ^ { n } \\sigma ^ 2 _ k } } \\ge - \\frac { \\sum _ { i = 1 } ^ n \\sigma _ i [ - \\mu _ { i + 1 } ^ - ] } { \\sqrt { \\sum _ { k = 1 } ^ { n } \\sigma ^ 2 _ k } } . \\end{align*}"}
-{"id": "3144.png", "formula": "\\begin{align*} P _ 2 ( t ) \\ , = \\ , \\prod _ { i = 1 } ^ { b _ 2 ( X ) } ( 1 - \\alpha _ i t ) \\end{align*}"}
-{"id": "4968.png", "formula": "\\begin{align*} \\hat { h } _ { X , f } ( P ) = \\lim _ { n \\to \\infty } \\frac { h _ { H } ( f ^ { n } ( P ) ) } { \\delta _ { f } ^ { n } } \\end{align*}"}
-{"id": "8488.png", "formula": "\\begin{align*} \\partial _ { t } \\textbf { u } ^ { \\varepsilon } + \\sum _ { j = 1 } ^ { d } A _ { j } ( \\textbf { u } ^ { \\varepsilon } ) \\partial _ { x _ { j } } \\textbf { u } ^ { \\varepsilon } = 0 , \\end{align*}"}
-{"id": "4030.png", "formula": "\\begin{align*} C _ { x _ { \\alpha _ 0 } ^ { i _ 0 } , \\ldots x _ { \\alpha _ r } ^ { i _ r } } = \\theta [ x _ { \\alpha _ 0 } ^ { i _ 0 } , \\ldots x _ { \\alpha _ r } ^ { i _ r } ] . \\end{align*}"}
-{"id": "7728.png", "formula": "\\begin{align*} \\sigma _ { n + 1 } = q ^ { 2 ^ { n + 1 } } = ( q ^ { 2 ^ n } ) ^ 2 < F ( q ^ { 2 ^ n } ) = F ( \\sigma _ n ) , \\end{align*}"}
-{"id": "7931.png", "formula": "\\begin{align*} \\sum _ { o r t h } = 2 \\left [ \\binom { g + 1 } { 3 } - \\binom { m _ { 1 } + 1 } { 3 } - \\binom { n _ { 1 } + 1 } { 3 } \\right ] . \\end{align*}"}
-{"id": "824.png", "formula": "\\begin{align*} \\left [ L ^ { ( i ) } ( z ; s ) , \\ , \\begin{pmatrix} 1 & 0 \\\\ 0 & \\alpha \\end{pmatrix} \\ ! K ^ { ( i ) } ( \\alpha ) \\right ] = 0 \\end{align*}"}
-{"id": "9417.png", "formula": "\\begin{align*} r _ 1 & = \\frac { R _ 2 R _ 3 } { R _ 1 + R _ 2 + R _ 3 } & R _ 1 & = \\frac { r _ 2 r _ 3 + r _ 1 r _ 3 + r _ 1 r _ 2 } { r _ 1 } \\\\ r _ 2 & = \\frac { R _ 1 R _ 3 } { R _ 1 + R _ 2 + R _ 3 } & R _ 2 & = \\frac { r _ 2 r _ 3 + r _ 1 r _ 3 + r _ 1 r _ 2 } { r _ 2 } \\\\ r _ 3 & = \\frac { R _ 1 R _ 2 } { R _ 1 + R _ 2 + R _ 3 } & R _ 3 & = \\frac { r _ 2 r _ 3 + r _ 1 r _ 3 + r _ 1 r _ 2 } { r _ 3 } \\end{align*}"}
-{"id": "9158.png", "formula": "\\begin{align*} v ' _ { j } ( t ) = \\frac { H ( v _ { j - r } ( t ) , \\dots , v _ { j + r } ( t ) ) } { h } \\end{align*}"}
-{"id": "8824.png", "formula": "\\begin{align*} Y _ g ( x _ { i _ 1 , n _ 1 } \\cdots x _ { i _ s , n _ s } , z ^ { 1 / m } ) : = \\prod ^ { s } _ { j = 1 } \\partial ^ { - n _ j } _ { z } Y ( x _ { i _ j , 0 } , z ^ { 1 / m } ) \\end{align*}"}
-{"id": "5672.png", "formula": "\\begin{align*} Q ( \\lambda ) ( v ) = \\nabla _ v M ( v ) , \\int _ { \\R ^ d } \\lambda ( v ) \\ , d v = 0 . \\end{align*}"}
-{"id": "4255.png", "formula": "\\begin{align*} \\varphi _ q ^ k & = \\frac { 1 } { q } \\mu \\left ( \\frac { k } { q } + \\frac { \\alpha ( k / q ) } { q ^ 2 } \\right ) \\left ( 1 + \\frac { \\beta ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) \\right ) . \\end{align*}"}
-{"id": "8504.png", "formula": "\\begin{align*} S = \\{ f _ { i _ 1 } f _ { i _ 2 } \\cdots f _ { i _ t } \\cdot 1 \\in U _ q ^ - ( \\mathfrak { g } ) : t \\ge 0 , \\ i _ k \\in I \\} . \\end{align*}"}
-{"id": "9145.png", "formula": "\\begin{align*} k _ i \\leq ( \\ell + \\ell _ i ) / 2 + ( r + r _ i ) / 2 = m / 2 + \\ell _ i / 2 + r _ i / 2 . \\end{align*}"}
-{"id": "5777.png", "formula": "\\begin{align*} \\Pi = \\Pi _ { 0 } \\sqcup \\Pi _ { \\frac { 1 } { 2 } } \\sqcup \\Pi _ { 1 } \\end{align*}"}
-{"id": "2249.png", "formula": "\\begin{align*} H ^ { G } = \\log \\sqrt { 2 \\pi } \\sigma + \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "260.png", "formula": "\\begin{align*} & \\log ^ 2 u _ { x , s } - \\log ^ 2 \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) \\\\ & = \\biggl \\{ 2 \\log \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) + \\log \\biggl ( \\frac { V _ d f ( x ) h _ x ^ { - 1 } ( s ) ^ d } { s } \\biggr ) \\biggr \\} \\log \\biggl ( \\frac { V _ d f ( x ) h _ x ^ { - 1 } ( s ) ^ d } { s } \\biggr ) . \\end{align*}"}
-{"id": "1470.png", "formula": "\\begin{align*} [ w ] _ { A _ \\infty } : = \\lim _ { r \\uparrow \\infty } [ w ] _ { A _ r } . \\end{align*}"}
-{"id": "1040.png", "formula": "\\begin{align*} c _ { 1 , 1 } + c _ { 1 , 0 } ~ = ~ 1 . \\end{align*}"}
-{"id": "5693.png", "formula": "\\begin{align*} \\| b \\| _ { \\rm B M O } : = \\| b ^ \\sharp \\| _ { L ^ \\infty } < \\infty . \\end{align*}"}
-{"id": "9866.png", "formula": "\\begin{align*} 0 = \\mu _ 0 ( \\widetilde { \\Omega } ) < \\mu _ { 1 } ( \\widetilde { \\Omega } ) \\leq \\mu _ { 2 } ( \\widetilde { \\Omega } ) \\leq . . . \\leq \\mu _ { n } ( \\widetilde { \\Omega } ) \\leq . . . \\ , . \\end{align*}"}
-{"id": "70.png", "formula": "\\begin{align*} \\varphi ( y ) + p \\int _ 0 ^ y \\Phi ( z ) \\ d z = \\frac { p a ^ { 2 - p } } { ( p - 1 ) ( 2 - p ) } \\left [ 1 - ( 1 - y ) ^ { 2 - p } \\right ] \\end{align*}"}
-{"id": "6963.png", "formula": "\\begin{align*} & \\alpha ^ * ( \\mu _ 1 ) : = \\inf \\{ \\sigma > 0 \\mid g ( \\sigma , \\mu _ 1 ) > 0 \\} , \\\\ & \\beta ^ * ( \\epsilon ) : = \\inf \\{ \\sigma > 0 \\mid h ( \\sigma , \\epsilon ) > 0 \\} . \\end{align*}"}
-{"id": "1699.png", "formula": "\\begin{align*} f _ n ( t , \\zeta ) & + \\int _ 0 ^ t b ( \\zeta ) \\cdot D f _ n ( s , \\zeta ) \\ , \\dd s + \\int _ 0 ^ t D _ { v } f _ n ( t , \\zeta ) \\circ \\dd W _ s = \\int _ 0 ^ t R _ n ( s , \\zeta ) \\ , \\dd s \\ , , \\\\ R _ n ( s , \\zeta ) & = \\int _ { \\R ^ { 2 d } } \\big ( b ( \\zeta ) - b ( z ) \\big ) \\cdot D _ { z } f ( s , z ) \\ , \\rho _ n ( \\zeta - z ) \\ \\dd z \\ , . \\end{align*}"}
-{"id": "7863.png", "formula": "\\begin{align*} p ^ { \\kappa } ( t , x , y ) = p _ y ( t , x - y ) + \\int _ 0 ^ t \\int _ { \\R ^ d } p _ z ( t - s , x - z ) q ( s , z , y ) \\ , d z \\ , d s \\ , , \\end{align*}"}
-{"id": "5750.png", "formula": "\\begin{align*} I _ { n } ( \\delta _ { 0 } ) = \\sum _ { t = 1 } ^ { n } \\sigma ^ 2 _ t ( \\delta _ { 0 } ) \\begin{bmatrix} x _ { t } x _ { t } ^ \\mathrm { T } & x _ { t } ( m \\pi ) _ { t - J _ \\phi } ^ \\mathrm { T } & 0 \\\\ ( m \\pi ) _ { t - J _ \\phi } x _ { t } ^ \\mathrm { T } & A _ { \\phi \\phi , t } & A _ { \\phi \\theta , t } \\\\ 0 & A _ { \\theta \\phi , t } & A _ { \\theta \\theta , t } \\end{bmatrix} \\end{align*}"}
-{"id": "5611.png", "formula": "\\begin{align*} \\sum _ { n \\le N } R _ { \\textit { H L } } ( n ) \\frac { ( 1 - n / N ) ^ k } { \\Gamma ( k + 1 ) } & = \\frac { \\pi ^ { 1 / 2 } } 2 \\frac { N ^ { 3 / 2 } } { \\Gamma ( k + 5 / 2 ) } - \\frac 1 2 \\frac { N } { \\Gamma ( k + 2 ) } \\\\ & - \\frac { \\pi ^ { 1 / 2 } } 2 \\sum _ { \\rho } \\frac { \\Gamma ( \\rho ) } { \\Gamma ( k + 3 / 2 + \\rho ) } N ^ { 1 / 2 + \\rho } \\end{align*}"}
-{"id": "9119.png", "formula": "\\begin{align*} u ' ( t ) + B A ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) , \\ \\ u ( 0 ) = u _ 0 . \\end{align*}"}
-{"id": "7302.png", "formula": "\\begin{align*} D = \\eth + \\eth ^ * \\in U ( \\mathfrak { g } ) \\otimes \\mathrm { C l } . \\end{align*}"}
-{"id": "4487.png", "formula": "\\begin{align*} \\mathbb { P } ( \\xi _ 1 \\leq u | X _ 1 = x , X _ 2 = y ) = \\left \\{ \\begin{array} { l l } F _ { n , x } ^ - ( u ) & \\mbox { i f $ \\| x - y \\| > r _ { n , u } $ } \\\\ \\tilde { F } _ { n , x } ( u ) & \\mbox { i f $ \\| x - y \\| \\leq r _ { n , u } $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "1765.png", "formula": "\\begin{align*} H ( u _ { j } ^ { \\varepsilon } ) ( x ) = \\underset { y \\in B _ { 1 } ( x ) } { \\sup } u ( y ) . \\end{align*}"}
-{"id": "3737.png", "formula": "\\begin{align*} \\rho ( u ) = \\lim _ { x \\rightarrow \\infty } \\frac { 1 } { x } \\psi ( x , x ^ { 1 / u } ) . \\end{align*}"}
-{"id": "9283.png", "formula": "\\begin{align*} \\Phi _ 1 ( t , z ) = \\frac { \\mathbb { E } [ D _ t \\delta _ Z ( z ) | \\mathcal { F } _ t ] } { \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] } . \\end{align*}"}
-{"id": "6721.png", "formula": "\\begin{align*} F ( X _ { t } ) = & \\ F ( X _ { 0 } ) + \\int _ 0 ^ t \\Delta _ { s } F ( X _ { s } ) d s + \\int _ 0 ^ t \\Delta _ { x } F ( X _ { s } ) \\psi ( s ) d s \\\\ & + \\int _ 0 ^ t \\Delta _ { x } F ( X _ { s } ) \\varphi ( s ) d W ( s ) + \\frac { 1 } { 2 } \\int _ 0 ^ t \\Delta _ { x x } F ( X _ { s } ) \\varphi ( s ) ^ { 2 } d s . \\end{align*}"}
-{"id": "6398.png", "formula": "\\begin{align*} \\gamma _ { q , p } ( 0 , 2 ) = \\beta _ { q , p } \\left ( \\frac { ( 3 p - 1 ) ! ! } { 6 } \\varepsilon _ { p } ^ { 3 } \\sigma ^ { 3 p } + \\frac { ( 2 q + p - 1 ) ! ! } { 2 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } \\sigma ^ { 2 q + p } \\right ) , \\end{align*}"}
-{"id": "5035.png", "formula": "\\begin{align*} f _ { s ^ { - 1 } \\cdot \\phi } ( g ) = f _ \\phi ( s g ) = ( s ^ { - 1 } \\cdot f _ \\phi ) ( g ) , \\textrm { f o r a l l $ g , s \\in G $ } . \\end{align*}"}
-{"id": "7029.png", "formula": "\\begin{align*} A ( x ) = \\nu ( x ) + x \\bigl ( \\overline { \\Pi } ^ + ( x ) - \\overline { \\Pi } ^ - ( x ) \\bigr ) , x > 0 . \\end{align*}"}
-{"id": "588.png", "formula": "\\begin{align*} \\delta ( x - [ r _ 0 ] ) & \\equiv \\phi ( \\sum _ { k = 1 } ^ \\nu p ^ { k - 1 } [ \\phi ^ { - k } ( r _ k ) ] ) \\bmod I ^ \\nu \\\\ & \\equiv \\sum _ { k = 1 } ^ \\nu p ^ { k - 1 } [ \\phi ^ { - ( k - 1 ) } ( r _ k ) ] ) \\bmod I ^ \\nu \\\\ & \\equiv \\sum _ { k = 0 } ^ { \\nu - 1 } p ^ { k } [ \\phi ^ { - k } ( r _ { k + 1 } ) ] ) \\bmod I ^ \\nu \\end{align*}"}
-{"id": "8994.png", "formula": "\\begin{align*} ( \\textbf { E } ^ \\gamma _ { \\rho , \\mu , \\omega , a ^ + } \\varphi ) ( x ) = \\int _ a ^ x ( x - t ) ^ { \\mu - 1 } E _ { \\rho , \\mu } ^ \\gamma [ \\omega ( x - t ) ^ \\rho ] \\varphi ( t ) d t , ~ ~ x > a . \\end{align*}"}
-{"id": "4758.png", "formula": "\\begin{align*} H = - \\frac { \\kappa } { 2 f } \\ , n _ 1 - \\frac { f f '' + ( f ' ) ^ 2 + 1 } { 2 f \\sqrt { f '^ 2 + 1 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "1446.png", "formula": "\\begin{align*} \\pi ( E ) = \\mathcal { E } , \\pi ( F ) = \\mathcal { F } \\end{align*}"}
-{"id": "5660.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\partial _ t \\rho + \\kappa ( - \\Delta ) ^ { \\alpha / 2 } \\rho + \\nabla _ x \\cdot ( D E \\rho ) & = & 0 & ( 0 , \\infty ) \\times \\R ^ d , \\\\ \\rho ( \\cdot , 0 ) & = & \\rho ^ { i n } & \\R ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "612.png", "formula": "\\begin{align*} p ( n ) = 2 \\pi ( 2 4 n - 1 ) ^ { - \\frac { 3 } { 4 } } \\sum _ { j = 1 } ^ { \\infty } \\frac { A _ j ( n ) } { j } I _ { \\frac { 3 } { 2 } } \\left ( \\frac { \\pi \\sqrt { 2 4 n - 1 } } { 6 j } \\right ) . \\end{align*}"}
-{"id": "1857.png", "formula": "\\begin{align*} S _ 2 ( t ) & \\ll \\sum _ { 2 \\log t < k - \\delta < \\sqrt { \\frac { t } { \\log t } } } \\left ( \\frac { \\exp ( i \\pi \\delta ) } { \\pi } \\frac { ( - 1 ) ^ k } { k - \\delta } + \\frac { 1 } { ( k - \\delta ) \\log t } + \\frac { 1 } { \\sqrt { t \\log t } } \\right ) \\\\ & \\quad + \\sum _ { \\sqrt { \\frac { t } { \\log t } } < k - \\delta < 2 \\sqrt { t \\log t } } \\left ( \\frac { 1 } { k - \\delta } + \\frac { 1 } { \\sqrt { t \\log t } } \\right ) = O ( \\log \\log t ) . \\end{align*}"}
-{"id": "7794.png", "formula": "\\begin{align*} \\Phi _ t ( m ) = \\begin{cases} m \\frac { x _ p } { x _ q } + t m , & \\mbox { i f $ x _ q $ d i v i d e s $ m $ a n d $ m \\frac { x _ p } { x _ q } \\not \\in \\overline { \\mathcal G } $ } , \\\\ m , & \\mbox { o t h e r w i s e } , \\end{cases} \\end{align*}"}
-{"id": "92.png", "formula": "\\begin{align*} \\frac { 1 } { r } = \\sum _ { i = 1 } ^ l \\langle e _ r , d _ i ^ * ( e _ r ) \\rangle \\langle e _ { r - s } , c _ i ( e _ { r - s } ) \\rangle . \\end{align*}"}
-{"id": "5655.png", "formula": "\\begin{align*} Q ( f ) : = \\int \\sigma ( v , v ' ) M ( v ) f ( v ' ) - \\sigma ( v ' , v ) M ( v ' ) f ( v ) \\d v ' . \\end{align*}"}
-{"id": "9598.png", "formula": "\\begin{align*} \\ddot \\eta _ y ( x , t ) = ( \\Delta - m ^ 2 ) \\eta _ y ( x , t ) - ( \\xi _ { 0 } + t \\dot \\xi _ { 0 } ) \\ , \\delta ( x - y ) \\end{align*}"}
-{"id": "8622.png", "formula": "\\begin{align*} \\overline { N ^ { 1 } } ( ^ { \\ast } b ) ( u _ { \\sigma ( b ) } ( n ) ) & = ( - \\pi _ { 1 } ^ { 1 } p _ { 1 } \\xi _ { b e _ { k } } ( u _ { \\sigma ( b ) } ( n ) ) , - \\gamma _ { 1 } \\xi _ { b e _ { k } } ( u _ { \\sigma ( b ) } ( n ) ) , 0 ) \\\\ & = ( - \\pi _ { 1 } ^ { 1 } p _ { 1 } u _ { o u t } ( \\xi _ { b e _ { k } } ( n ) ) , - \\gamma _ { 1 } u _ { o u t } ( \\xi _ { b e _ { k } } ( n ) ) , 0 ) \\\\ & = ( - \\pi _ { 1 } ^ { 1 } u _ { 1 } p \\xi _ { b e _ { k } } ( n ) - \\pi _ { 1 } ^ { 1 } \\rho _ { 2 } \\gamma \\xi _ { b e _ { k } } ( n ) , - u _ { 2 } \\gamma \\xi _ { b e _ { k } } ( n ) , 0 ) \\end{align*}"}
-{"id": "9558.png", "formula": "\\begin{align*} 0 = - \\sum _ { i > 0 } \\frac { \\left | \\langle \\psi ^ { ( 0 ) } _ { i } , H ^ { ( 1 ) } \\psi ^ { ( 0 ) } _ { 0 } \\rangle \\right | ^ { 2 } } { \\left | \\lambda _ { 0 } ^ { ( 0 ) } - \\lambda _ { i } ^ { ( 0 ) } \\right | } \\end{align*}"}
-{"id": "6915.png", "formula": "\\begin{align*} \\| \\phi \\| _ m = \\min _ { c \\in \\mathbb { R } } \\ , \\sum _ { j = 1 } ^ n | \\phi ( j ) - c | . \\end{align*}"}
-{"id": "2043.png", "formula": "\\begin{align*} \\nabla _ r T ( r , s ) \\big | _ { r = 0 } = J ' ( s ) . \\end{align*}"}
-{"id": "7950.png", "formula": "\\begin{align*} \\lim _ { \\sigma \\rightarrow 1 ^ - } \\lambda _ j ( \\sigma ) = 0 , \\end{align*}"}
-{"id": "3142.png", "formula": "\\begin{align*} \\begin{bmatrix} \\Lambda _ t ( \\lambda _ 0 ) \\otimes I _ n \\\\ \\widehat { N } _ t ( \\lambda _ 0 ) ( \\lambda _ 0 B + A ) ( \\Lambda _ t ( \\lambda _ 0 ) \\otimes I _ n ) \\end{bmatrix} x \\in \\mathcal { N } _ r ( \\mathcal { L } ( \\lambda _ 0 ) ) . \\end{align*}"}
-{"id": "4356.png", "formula": "\\begin{align*} \\varphi ' ( Y _ a , a ) = 0 \\ , \\varphi ' ( y , a ) ( y - Y _ a ) < 0 \\ , y \\in ( 0 , 1 ) \\setminus \\{ Y _ a \\} \\ . \\end{align*}"}
-{"id": "3301.png", "formula": "\\begin{gather*} W _ { - 1 } = w _ { - 1 } \\otimes w _ 0 - q ^ 2 w _ 0 \\otimes w _ { - 1 } , W _ 1 = w _ 0 \\otimes w _ 1 - q ^ 2 w _ 1 \\otimes w _ 0 , \\\\ W _ 0 = w _ { - 1 } \\otimes w _ 1 - w _ 1 \\otimes w _ { - 1 } - q ^ { - 1 } ( q - q ^ { - 1 } ) w _ 0 \\otimes w _ 0 . \\end{gather*}"}
-{"id": "3288.png", "formula": "\\begin{align*} \\hat { R } _ { 1 2 } \\hat { R } _ { 2 3 } \\hat { R } _ { 1 2 } = \\hat { R } _ { 2 3 } \\hat { R } _ { 1 2 } \\hat { R } _ { 2 3 } , \\end{align*}"}
-{"id": "8534.png", "formula": "\\begin{align*} X _ { b ^ { \\ast } } ( P ) = \\displaystyle \\sum _ { r \\in L ( k ) , a \\in _ { k } T } r a Y _ { [ b r a ] } ( P ) \\end{align*}"}
-{"id": "477.png", "formula": "\\begin{align*} \\alpha : = \\inf \\{ \\pi ^ { \\top } x \\ , | \\ W ^ { i _ 0 } \\} = \\inf \\{ \\pi ^ { \\top } x \\ , | \\ x \\in W \\} , \\end{align*}"}
-{"id": "6179.png", "formula": "\\begin{align*} V _ t ^ n = V _ { t \\wedge T ^ n } \\end{align*}"}
-{"id": "1171.png", "formula": "\\begin{align*} | c _ { n } ( x ) | & \\leq C x ^ n \\int _ { 0 \\leq \\xi _ 1 \\leq \\xi _ 2 \\leq \\dots \\leq \\xi _ { n + 1 } = x } \\left [ \\prod _ { j = 1 } ^ n \\rho ( \\xi _ j ) \\right ] d \\xi _ 1 d \\xi _ 2 \\dots d \\xi _ n \\\\ & \\leq C \\frac { x ^ n } { n ! } \\left ( \\int _ { [ 0 , x ] } \\rho ( \\xi ) d \\xi \\right ) ^ n \\stackrel { \\emph { d e f } } { = } C \\frac { x ^ n } { n ! } ( M ( x ) ) ^ n , \\end{align*}"}
-{"id": "9204.png", "formula": "\\begin{align*} J ( u ) = \\mathbb { E } [ \\int _ 0 ^ T ( \\int _ D h ( t , x , Y ( t , x ) , u ( t , x , Z ) , Z ) d x ) d t + \\int _ D k ( x , Y ( T , x ) , Z ) d x ] , \\end{align*}"}
-{"id": "5332.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = f ( t , u , u ' ) & & \\\\ u ' ( 0 ) = u ( 0 ) , \\ u ' ( T ) = b u ' ( 0 ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6299.png", "formula": "\\begin{align*} \\Vert \\{ \\lambda _ { j } \\} \\Vert _ { l _ { q } ^ { s , 1 } } = \\begin{cases} \\left ( \\sum _ { j \\in \\mathbb { N } } 2 ^ { j s } | a _ { j } | ^ { q } \\right ) ^ { \\frac { 1 } { q } } ~ & ~ 0 < q < \\infty , \\\\ \\sup _ { j \\in \\mathbb { N } } \\left ( 2 ^ { j s } | a _ { j } | \\right ) ~ & ~ q = \\infty \\end{cases} \\end{align*}"}
-{"id": "4105.png", "formula": "\\begin{align*} q + r = \\gcd ( - p , q ) + \\gcd ( - p , r ) + \\gcd ( q , p + q + r ) + \\gcd ( r , p + q + r ) . \\end{align*}"}
-{"id": "8515.png", "formula": "\\begin{gather*} s _ 4 s _ 3 s _ 5 s _ 2 s _ 3 s _ 1 s _ 2 s _ 4 s _ 3 s _ 5 = \\tau ^ { ( 1 ) } , \\\\ s _ 3 s _ 4 s _ 2 s _ 1 s _ 3 s _ 2 = \\sigma ^ { - 1 } \\sigma ^ { \\{ 1 , 2 , 3 , 5 \\} } s _ 2 s _ 1 s _ 3 s _ 5 s _ 2 s _ 3 = \\tau ^ { ( 2 ) } , \\\\ s _ 4 s _ 3 = \\sigma ^ { - 1 } \\sigma ^ { \\{ 1 , 3 , 5 \\} } s _ 3 s _ 5 = \\tau ^ { ( 3 ) } , \\ s _ 1 = \\tau ^ { ( 4 ) } , \\ \\ s _ 4 = \\tau ^ { ( 5 ) } . \\end{gather*}"}
-{"id": "7373.png", "formula": "\\begin{align*} ( \\Gamma _ i ^ { ( k + 1 ) } v , v ^ \\prime ) _ { k } = ( v , \\Gamma _ i ^ { ( k + 1 ) * } v ^ \\prime ) _ { k + 1 } , v \\in \\Lambda _ q ^ { k + 1 } ( \\mathfrak { u } _ + ) , \\ v ^ \\prime \\in \\Lambda _ { q } ^ { k } ( \\mathfrak { u } _ + ) . \\end{align*}"}
-{"id": "7776.png", "formula": "\\begin{align*} r _ j ^ i : = \\sum _ k v _ k \\otimes v ^ k \\end{align*}"}
-{"id": "4906.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\kappa , m , N } ( z ) : = { \\displaystyle { \\sum _ { M \\in \\Gamma _ { \\infty } \\backslash \\Gamma _ 0 ( N ) } } } e ^ { 2 \\pi i m z } \\bigg | _ { \\kappa } M \\in M _ { \\kappa } ^ ! ( N ) , \\end{align*}"}
-{"id": "5062.png", "formula": "\\begin{align*} V = \\{ g \\in G \\ , : \\ , f ( g ) \\geq 0 \\big \\} \\end{align*}"}
-{"id": "2542.png", "formula": "\\begin{align*} \\P _ { 3 } \\left ( y , B ( z , \\delta ) \\right ) : = \\P \\left ( U ^ 3 \\in B ( z , \\delta ) \\big { | } U ^ 0 = y \\right ) > 0 , \\end{align*}"}
-{"id": "753.png", "formula": "\\begin{gather*} \\delta _ p ( i , j ) = \\begin{cases} 1 , & , \\\\ 0 , & . \\end{cases} \\end{gather*}"}
-{"id": "9343.png", "formula": "\\begin{align*} k ( t , x , z ) = \\exp ( - x G ( t ) - \\frac { 1 } { 2 } x ^ 2 t ) \\end{align*}"}
-{"id": "5813.png", "formula": "\\begin{align*} \\sum _ { j \\in I } B _ { i j } ( \\lambda _ j , \\mu _ j , \\nu _ j ) = 0 \\end{align*}"}
-{"id": "2632.png", "formula": "\\begin{align*} h _ m ^ l ( F , G ) = A _ m ^ l ( \\lambda ) - ( b _ m ( \\lambda ) ) ^ { \\top } C _ m ^ l ( G ) ( \\lambda ) , \\end{align*}"}
-{"id": "112.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 + t ) ^ { \\tfrac { x } { 2 } } d \\mu _ { - 1 } ( x ) = \\frac { 2 } { 1 + \\sqrt { 1 + t } } = \\sum _ { n = 0 } ^ \\infty C h _ { n , \\frac { 1 } { 2 } } \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "9265.png", "formula": "\\begin{align*} \\sup _ { \\pi \\in \\mathcal { A } _ { \\mathbb { H } } } j ( \\pi ) = j ( \\hat { \\pi } ) . \\end{align*}"}
-{"id": "3344.png", "formula": "\\begin{align*} & \\mathrm { D } ^ { f } _ H ( K ) \\\\ & = \\{ h _ 0 \\in G ; \\forall g _ 1 , \\ldots , \\forall g _ k \\in G , \\exists h \\in G h h _ 0 h ^ { - 1 } K ( h h _ 0 h ^ { - 1 } ) ^ { - 1 } g _ 1 H g _ 1 ^ { - 1 } \\cup \\cdots \\cup g _ k H g _ k ^ { - 1 } \\} . \\\\ \\end{align*}"}
-{"id": "5226.png", "formula": "\\begin{align*} \\alpha = - \\mathcal { L } _ T \\eta \\ ; , \\end{align*}"}
-{"id": "5335.png", "formula": "\\begin{align*} u = M ( \\lambda , u ) . \\end{align*}"}
-{"id": "6351.png", "formula": "\\begin{align*} P = P ( x , r ) = p _ { \\mu m } x ^ { \\mu } r ^ { m } , \\end{align*}"}
-{"id": "6488.png", "formula": "\\begin{align*} \\| L u ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } ^ { 2 } = \\int _ { \\mathbb { R } ^ { n } } | L u ( x , t ) | ^ { 2 } d x = \\int _ { \\mathbb { R } ^ { n } } | A ( \\xi ) \\mathcal { F } ( u ) ( \\xi , t ) | ^ { 2 } d \\xi . \\end{align*}"}
-{"id": "8757.png", "formula": "\\begin{align*} & M ^ { ( 1 ) } ( w ) = R _ { 0 , n } ( w ) R _ { 0 , n - 1 } ( w ) \\dots R _ { 0 , 1 } ( w ) , \\\\ & M ^ { ( 2 ) } ( w ) = R _ { 1 , 0 } ( w ) R _ { 2 , 0 } ( w ) \\dots R _ { n , 0 } ( w ) , . \\end{align*}"}
-{"id": "9679.png", "formula": "\\begin{align*} x ' ( t ) & = - a g ( x ( t ) ) + b g ( x ( t - \\tau ( t ) ) , t \\geq 0 \\\\ x ( t ) & = \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] \\end{align*}"}
-{"id": "5266.png", "formula": "\\begin{align*} \\vert \\int _ { a } ^ { b } f ( x ) d x \\mid = \\int _ { a } ^ { b } \\mid f ( x ) \\mid d x . \\end{align*}"}
-{"id": "6187.png", "formula": "\\begin{align*} I _ 2 & = \\int _ { - 1 } ^ 1 [ f ( x + x z ) - f ( x ) - f ' ( x ) x z ] \\nu ^ 1 ( \\d z ) + \\int _ { \\R } [ f ( x + x z ) - f ( x ) ] \\nu ^ 2 ( \\d z ) \\\\ & = \\int _ { \\R } [ f ( x + x z ) - f ( x ) - f ' ( x ) x z I ( | z | \\leq 1 ) ] \\nu _ { U } ( \\d z ) + f ' ( x ) x \\int _ { - 1 } ^ 1 z \\nu ' ( \\d z ) , \\end{align*}"}
-{"id": "1767.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } \\Delta u _ { i } ^ { \\varepsilon } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { \\varepsilon } \\sum \\limits _ { j \\neq i } \\int _ { B _ { 1 } ( x ) } u _ { j } ^ { \\varepsilon } ( y ) \\ , d y & \\Omega , \\\\ u _ { i } ( x ) = \\phi _ { i } ( x ) & ( \\partial \\Omega ) _ 1 . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6683.png", "formula": "\\begin{align*} X _ { 0 } ^ { \\lambda } = \\left \\| \\bigwedge _ { i = 1 } ^ { p } \\nabla D _ i \\wedge \\bigwedge _ { j = 1 } ^ { k } \\nabla I _ j \\right \\| _ { k + p } ^ { - 2 } \\cdot \\sum _ { i = 1 } ^ { p } ( - 1 ) ^ { n - i } ( - \\lambda ) ( D - d _ i ) \\Theta _ i , \\end{align*}"}
-{"id": "2251.png", "formula": "\\begin{align*} \\mu _ { ( k ) } = C ( \\vec { \\varepsilon } ) \\int _ { - \\infty } ^ { \\infty } \\frac { e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } } { \\sqrt { 2 \\pi } \\sigma } x ^ { k } \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { n ! } \\left ( \\varepsilon _ { q } x ^ { q } - \\varepsilon _ { p } x ^ { p } \\right ) ^ { n } d x . \\end{align*}"}
-{"id": "915.png", "formula": "\\begin{align*} \\| X \\| _ { * } & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { F } \\| V \\| _ { F } \\\\ & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\frac { \\| U \\| ^ { 2 } _ { F } + \\| V \\| ^ { 2 } _ { F } } { 2 } . \\end{align*}"}
-{"id": "9594.png", "formula": "\\begin{align*} \\ddot { \\varphi } ( x , t ) = ( \\Delta - m ^ 2 ) \\varphi ( x , t ) + \\xi ( t ) \\delta ( x - y ) , \\varphi ( x , 0 ) = 0 , \\quad \\dot \\varphi ( x , 0 ) = 0 . \\end{align*}"}
-{"id": "8272.png", "formula": "\\begin{align*} N ^ { - 1 / 2 } \\left | r ^ 1 _ { \\lambda } ( u ) - r ^ 1 _ { \\lambda } ( 0 ) + \\frac { 1 } { 2 \\sqrt { 2 \\pi } } \\frac { m _ { 3 , \\lambda } } { \\sigma ^ 3 _ { \\lambda } } u \\right | = N ^ { - 1 / 2 } | r ^ 1 _ { \\lambda } ( u ) - r ^ 1 _ { \\lambda } ( 0 ) - ( r ^ 1 _ { \\lambda } ) ' ( 0 ) u | \\lesssim N ^ { - 1 / 2 } \\| ( r ^ 1 _ { \\lambda } ) '' \\| _ { \\infty } u ^ 2 , \\end{align*}"}
-{"id": "3459.png", "formula": "\\begin{align*} f ( H , D ) = f _ 0 ( H , D ) \\cdot D \\end{align*}"}
-{"id": "8868.png", "formula": "\\begin{align*} A ( x ) [ v ] = \\frac { 1 } { 2 } B [ x , v ] . \\end{align*}"}
-{"id": "6269.png", "formula": "\\begin{align*} 0 \\geq \\frac { \\Delta w ( x _ 0 ) - \\Delta w ( y _ 0 ) } { \\bar { w } } - \\frac { m } { \\bar { w } } \\sum _ { i = 1 } ^ n \\nabla ^ 2 _ { E _ i , E _ i } \\bar { w } . \\end{align*}"}
-{"id": "1469.png", "formula": "\\begin{align*} f ^ { \\sharp } ( x ) : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { | Q | } \\int _ Q | f ( y ) - f _ Q | d y \\times \\chi _ Q ( x ) , \\end{align*}"}
-{"id": "7978.png", "formula": "\\begin{align*} X : \\sum _ { i _ 0 + i _ 1 + i _ 2 + i _ 3 = 4 } a _ { i _ 0 i _ 1 i _ 2 i _ 3 } x _ { 0 } ^ { i _ 0 } x _ { 1 } ^ { i _ 1 } x _ { 2 } ^ { i _ 2 } x _ { 3 } ^ { i _ 3 } = 0 , a _ { i _ 0 i _ 1 0 0 } = 0 , \\end{align*}"}
-{"id": "189.png", "formula": "\\begin{align*} \\lambda _ 1 = - \\frac { 1 } { 2 ( d + 2 ) V _ d ^ { 2 / d } } \\int _ \\mathcal { X } \\frac { \\Delta f ( x ) } { f ( x ) ^ { 2 / d } } \\ , d x , \\end{align*}"}
-{"id": "6899.png", "formula": "\\begin{gather*} \\omega _ { - 1 } = 0 , \\ ; \\ ; \\omega _ 0 = u _ z = \\tfrac { 1 } { 2 } ( u _ x - i u _ y ) , \\ ; \\ ; \\omega _ 1 = u _ { z z z } - 2 ( u _ z ) ^ 3 , \\ ; \\ ; \\\\ \\omega _ 2 = u _ { z z z z z } - 1 0 u _ { z z z } ( u _ z ) ^ 3 - 1 0 ( u _ { z z } ) ^ 2 u _ z + 6 ( u _ z ) ^ 5 , \\ ; \\ldots \\end{gather*}"}
-{"id": "5947.png", "formula": "\\begin{align*} \\Phi ^ { a ' } = \\exp \\Big ( \\int _ 0 ^ T { \\langle } a ' F ( L _ s ) , \\dd W _ s { \\rangle } \\ , - \\ , \\frac { 1 } { 2 } \\int _ 0 ^ T | a ' F ( L _ s ) | ^ 2 \\ , \\dd s + \\frac { ( a ' ) ^ 2 - a ' } { 2 } \\int _ 0 ^ T | F ( L _ s ) | ^ 2 \\ , \\dd s \\Big ) \\ , , \\end{align*}"}
-{"id": "8021.png", "formula": "\\begin{align*} & \\phi ( x _ j ) = \\mu _ j x _ j , \\phi ( y _ j ) = \\nu _ j y _ j \\mbox { o r } \\\\ & \\phi ( x _ j ) = \\mu _ j y _ j , \\phi ( y _ j ) = \\nu _ j x _ j \\end{align*}"}
-{"id": "4059.png", "formula": "\\begin{align*} \\alpha ( p - q ) - p + 2 q = 1 + n + k \\leq 1 + n + q . \\end{align*}"}
-{"id": "7795.png", "formula": "\\begin{align*} \\textstyle \\varphi _ t ( \\overline { \\mathcal G } ) = \\{ \\varphi _ t ( m ) : m \\in \\overline { \\mathcal G } \\setminus \\overline { \\mathcal H } \\} \\cup \\{ m \\frac { x _ p } { x _ q } + t m : m \\in \\mathcal H \\} \\cup \\{ x _ p ^ 2 + 2 t x _ p x _ q + t ^ 2 x _ q ^ 2 \\} . \\end{align*}"}
-{"id": "1796.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r l } \\mathcal { V } u ( t , x ) & = \\sum _ { i = 1 } ^ { d } V _ i ^ 2 u ( t , x ) \\\\ u ( 0 , x ) & = f ( x ) . \\end{array} \\right . \\end{align*}"}
-{"id": "3129.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) = \\left [ \\begin{array} { c | c | c } - P _ d & M _ { 1 2 } ( \\lambda ) & 0 \\\\ \\hline \\sigma M _ { 1 2 } ( \\lambda ) ^ T & \\phantom { \\Big { ( } } M _ { 2 2 } ( \\lambda ) \\phantom { \\Big { ( } } & L _ { t - 1 } ( \\lambda ) ^ T \\otimes I _ n \\\\ \\hline 0 & \\sigma L _ { t - 1 } ( \\lambda ) \\otimes I _ n & 0 \\end{array} \\right ] , \\mbox { w i t h } t = \\frac { d } { 2 } , \\end{align*}"}
-{"id": "789.png", "formula": "\\begin{align*} \\phi ( 1 ; \\vec { z } ) = 1 , \\phi ( \\sigma _ { i } \\tau ; \\vec { z } ) = Y _ { i } ( z _ { \\tau ^ { - 1 } ( i ) } , z _ { \\tau ^ { - 1 } ( i + 1 ) } ) \\phi ( \\tau ; \\vec { z } ) , \\end{align*}"}
-{"id": "381.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty K _ \\nu ( x ) x ^ { s - 1 } d x & = 2 ^ { s - 2 } \\Gamma \\left ( \\frac { s + \\nu } { 2 } \\right ) \\Gamma \\left ( \\frac { s - \\nu } { 2 } \\right ) , \\Re s > | \\Re \\nu | , \\\\ \\int _ 0 ^ \\infty \\cos ( x ) x ^ { s - 1 } d x & = \\Gamma ( s ) \\cos \\left ( \\frac { \\pi s } { 2 } \\right ) , 0 < \\Re s < 1 . \\end{align*}"}
-{"id": "7012.png", "formula": "\\begin{align*} \\left \\langle \\pi , \\int _ X \\imath _ X ( x ) d P \\right \\rangle & = \\lim _ { n \\to \\infty } \\left \\langle \\varphi _ n , \\int _ X \\imath _ X ( x ) d P \\right \\rangle = \\lim _ { n \\to \\infty } \\int _ X \\langle \\varphi _ n , \\imath _ X ( x ) \\rangle d P \\\\ & = \\int _ X \\lim _ { n \\to \\infty } \\langle \\varphi _ n , \\imath _ X ( x ) \\rangle d P = \\int _ X \\langle \\pi , \\imath _ X ( x ) \\rangle d P \\end{align*}"}
-{"id": "5161.png", "formula": "\\begin{gather*} T _ p ( e _ { i ( 1 ) } \\otimes \\dotsm \\otimes e _ { i ( k ) } ) = \\sum _ { 1 \\leq j ( 1 ) , \\dotsc , j ( l ) \\leq n } \\delta _ p ( i , j ) \\cdot e _ { j ( 1 ) } \\otimes \\dotsm \\otimes e _ { j ( l ) } . \\end{gather*}"}
-{"id": "4911.png", "formula": "\\begin{align*} K _ { \\kappa , N } \\left ( z , \\mathfrak { z } \\right ) = - \\frac { i ^ { \\kappa } ( \\kappa - 1 ) } { 2 ^ { \\kappa - 1 } \\pi } \\Psi _ { \\kappa , 0 , N } ^ { * } ( \\mathfrak { z } , z ) . \\end{align*}"}
-{"id": "391.png", "formula": "\\begin{align*} L _ 0 & = \\{ 0 \\} \\sqcup L _ 0 ( I ) \\sqcup L _ 0 ( I I ) , \\hat { L } _ 0 & = \\{ 0 \\} \\sqcup L _ 0 ( I ) \\sqcup \\hat { L } _ 0 ( I I ) , \\end{align*}"}
-{"id": "6841.png", "formula": "\\begin{align*} \\bigl [ e ^ { i t \\Delta } \\bigl ( e ^ { i x \\xi _ 0 } f ( \\tfrac { \\cdot } { \\lambda } ) \\bigr ) \\bigr ] ( x ) = e ^ { - i t | \\xi _ 0 | ^ 2 + i x \\xi _ 0 } ( e ^ { i t \\lambda ^ { - 2 } \\Delta } f ) ( \\tfrac { x - 2 t \\xi _ 0 } { \\lambda } ) . \\end{align*}"}
-{"id": "7656.png", "formula": "\\begin{align*} u ( t ) = S _ { \\alpha } ( t ) u _ { 0 } + \\int _ { 0 } ^ { t } S _ { \\alpha } ( t - s ) f ( u ( s ) ) d s . \\end{align*}"}
-{"id": "1198.png", "formula": "\\begin{align*} b _ i ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) \\} | , \\\\ b _ { i p } ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) ~ | ~ w = \\overline { w } \\} | , \\\\ b _ { i n } ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) ~ | ~ w \\neq \\overline { w } \\} | . \\end{align*}"}
-{"id": "7416.png", "formula": "\\begin{align*} \\overline { s } _ i : = e _ i ^ { - 1 } e _ { - i } e _ i ^ { - 1 } . \\end{align*}"}
-{"id": "4719.png", "formula": "\\begin{align*} w _ m & = \\sqrt { w _ 1 ^ 2 + w _ 2 ^ 2 + \\dots + w _ { m - 1 } ^ 2 } \\\\ \\gamma _ m & = \\sqrt { \\gamma _ 1 ^ 2 + \\gamma _ 2 ^ 2 + \\dots + \\gamma _ { m - 1 } ^ 2 } . \\end{align*}"}
-{"id": "2651.png", "formula": "\\begin{align*} \\Sigma ( A , B ) : = \\left \\{ I \\in \\mbox { I n d } ( G ) : A \\subseteq I \\phantom { . } \\mathrm { a n d } \\phantom { . } B \\cap I = \\emptyset \\right \\} \\ , . \\end{align*}"}
-{"id": "5569.png", "formula": "\\begin{align*} D ( - \\lambda ) = W ( \\hat c _ 0 , c _ 0 ) + \\sum _ { n = 1 } ^ N \\lambda ^ n \\sum _ { 1 \\leq i _ 1 < i _ 2 < \\dots i _ n \\leq N } m _ { i _ 1 } m _ { i _ 2 } \\dots m _ { i _ n } ( x _ { i _ n } - x _ { i _ { n - 1 } } ) ( x _ { i _ { n - 1 } } - x _ { i _ { n - 2 } } ) \\dots ( x _ { i _ 2 } - x _ { i _ 1 } ) c _ 0 ( x _ { i _ 1 } ) \\hat c _ 0 ( x _ { i _ n } ) , \\end{align*}"}
-{"id": "7727.png", "formula": "\\begin{align*} \\sigma _ n = q ^ { 2 ^ n } , q \\in ( 0 , 1 ) . \\end{align*}"}
-{"id": "6393.png", "formula": "\\begin{align*} H ^ { G } = \\log \\sqrt { 2 \\pi } \\sigma + \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "2435.png", "formula": "\\begin{align*} \\widehat { f _ k ^ { \\alpha } } = \\widehat { f } ( \\frac { \\xi - \\langle k \\rangle ^ { \\frac { \\alpha } { 1 - \\alpha } } k } { \\langle k \\rangle ^ { \\frac { \\alpha } { 1 - \\alpha } } } ) . \\end{align*}"}
-{"id": "1827.png", "formula": "\\begin{align*} g _ { } = \\frac 3 4 \\sum \\limits _ { i = 1 } ^ k \\left ( \\frac { d s _ i ^ 2 } { s _ i ^ 2 } + s _ i ^ 2 d \\theta _ i ^ 2 \\right ) + h \\end{align*}"}
-{"id": "1476.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } m _ f ( 2 ^ l Q ) = 0 . \\end{align*}"}
-{"id": "2515.png", "formula": "\\begin{align*} \\{ x \\in \\operatorname { M r k } ( ( D , I ) ) : ( \\mathcal { L } _ { X _ { \\lambda } } F ) ( x ) = 0 \\} = \\Sigma ^ { D _ 1 , \\dots , D _ p } _ { d _ 1 , \\dots , d _ p } \\cap \\operatorname { M r k } ( ( D , I ) ) . \\end{align*}"}
-{"id": "5347.png", "formula": "\\begin{align*} | K ( \\Gamma ) | = \\frac { \\lambda _ 1 \\cdots \\lambda _ { \\ell } } { \\ell + 1 } \\end{align*}"}
-{"id": "1219.png", "formula": "\\begin{align*} P ( x ) = P _ { \\varphi , w } ( x ) = \\inf \\left \\{ \\int _ I \\varphi \\left ( | x | / { v } \\right ) v : v \\prec w , v \\ge 0 \\right \\} , \\ \\ \\ \\ x \\in L ^ 0 , \\\\ \\end{align*}"}
-{"id": "2465.png", "formula": "\\begin{align*} L ( s , V ) = \\varepsilon ( s , V ) L ( 1 - s , V ^ * ) \\end{align*}"}
-{"id": "4768.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a \\ , t ^ 2 ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "1993.png", "formula": "\\begin{align*} \\eta _ t = L _ t - \\sum _ { s \\leq t } \\frac { \\Delta U _ s \\Delta L _ s } { 1 + \\Delta U _ s } - \\sigma _ { U L } t , \\end{align*}"}
-{"id": "5667.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\alpha / 2 } f ( x ) = c _ { d , \\alpha } \\ , \\ , \\int _ { \\R ^ d } \\frac { f ( x ) - f ( y ) } { | x - y | ^ { d + \\alpha } } \\d y , \\end{align*}"}
-{"id": "925.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\| U ^ { * } \\| _ { S _ { p _ { 1 } } } \\| V ^ { * } \\| _ { S _ { p _ { 2 } } } = \\| U ^ { * } _ { 1 } \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| V ^ { * } _ { 1 } \\| _ { S _ { \\widehat { p } _ { 1 } } } . \\end{align*}"}
-{"id": "283.png", "formula": "\\begin{align*} W _ 3 = W _ { 3 1 } + W _ { 3 2 } = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { 1 / 2 + 2 \\beta / d } } { n ^ { 1 + 2 \\beta / d } } \\ , , \\ , \\frac { \\log n } { n k ^ { 1 / 2 } } \\ , , \\ , \\frac { k ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "3082.png", "formula": "\\begin{align*} a ( S u , v ) = b ( u , v ) \\forall v \\in V . \\end{align*}"}
-{"id": "4386.png", "formula": "\\begin{align*} v ( t , 0 ) = g ( t ) , \\ \\ t > 0 , \\end{align*}"}
-{"id": "5606.png", "formula": "\\begin{align*} \\begin{cases} Y _ 0 ^ 2 = p _ 1 ( Z ) p _ 2 ( Z ) \\\\ Y _ 1 ^ 2 = p _ 1 ( Z + 1 ) p _ 2 ( Z + 1 ) \\\\ Y _ 2 ^ 2 = p _ 1 ( Z + 2 ) p _ 2 ( Z + 2 ) \\\\ \\ \\ \\ \\ \\ \\vdots & \\\\ Y _ { m - 1 } ^ 2 = p _ 1 ( Z + m - 1 ) p _ 2 ( Z + m - 1 ) . \\end{cases} \\end{align*}"}
-{"id": "2034.png", "formula": "\\begin{align*} \\frac { d } { d D } \\left ( D ^ 2 \\bar { \\lambda } _ 1 ( n , D , K ) \\right ) \\begin{cases} < 0 , & K > 0 , \\ D \\in ( 0 , \\frac { \\pi } { \\sqrt { K } } ) ; \\\\ = 0 , & K = 0 ; \\\\ > 0 , & K < 0 . \\end{cases} \\end{align*}"}
-{"id": "7350.png", "formula": "\\begin{align*} ( F ^ * v _ 0 , v _ 1 ) = ( E K ^ { - 1 } v _ 0 , v _ 1 ) = [ 2 ] ^ { 1 / 2 } ( v _ 1 , v _ 1 ) . \\end{align*}"}
-{"id": "3876.png", "formula": "\\begin{align*} & \\tilde { \\varepsilon } _ 1 ( x ) = 2 Q ( x ) + \\{ \\tau ( \\xi ( x ) ) + Q ( x ) \\} \\circ \\nabla _ { \\partial \\Omega } ( \\tau - \\nu ) ( \\xi ( x ) ) \\circ ( \\tilde { x } - a ) , \\\\ & \\tilde { \\varepsilon } _ 2 ( x ) = - 2 Q ( x ) \\circ \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\otimes i _ x \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\right ) \\right ) . \\end{align*}"}
-{"id": "5567.png", "formula": "\\begin{align*} - D ^ 2 _ x v = z \\rho ( x ) v , 0 < x < 1 , v _ x ( 0 ) - h v ( 0 ) = 0 , v _ x ( 1 ) + H v ( 1 ) = 0 , \\end{align*}"}
-{"id": "9777.png", "formula": "\\begin{align*} ( \\overline { \\partial } _ { E , p } + \\overline { \\partial } ^ t _ { E , p } ) _ { \\max } ( \\phi _ i \\omega ) = \\phi _ i ( \\overline { \\partial } _ { E , p } + \\overline { \\partial } ^ t _ { E , p } ) \\omega + ( \\overline { \\partial } _ { \\max } \\phi _ i ) \\wedge \\omega - { \\rm I n t } ( \\overline { \\partial } _ { \\max } \\phi _ i ) \\omega \\end{align*}"}
-{"id": "6554.png", "formula": "\\begin{align*} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\le C \\tau ^ k , \\\\ & \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ \\infty ( W ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "7905.png", "formula": "\\begin{align*} C _ 0 : = \\max ( C _ s , \\frac { 6 K _ s } { S _ 0 } ) . \\end{align*}"}
-{"id": "8374.png", "formula": "\\begin{align*} u p x = u x p = \\phi ( x ) u p , \\ \\ \\ x \\in M , \\end{align*}"}
-{"id": "2191.png", "formula": "\\begin{align*} & - \\int _ { B _ { 1 } } \\partial _ { s } ( g _ { 1 - \\alpha , m } * [ \\phi \\psi ^ { 2 } w ^ { 2 } ] ) d x + \\frac { 1 } { 2 } c _ { 1 } \\\\ \\leq & \\int _ { 0 } ^ { s } \\dot { g } _ { 1 - \\alpha , m } ( s - \\tau ) ( \\phi ( s ) - \\phi ( \\tau ) ) \\left ( \\int _ { B _ { 1 } } \\psi ^ { 2 } \\tilde { u } ^ { 1 - q } d x \\right ) ( \\tau ) d \\tau \\\\ & + c _ { 4 } ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\phi ( s ) \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x + R _ { m } ( s ) , \\end{align*}"}
-{"id": "2987.png", "formula": "\\begin{align*} \\delta _ v ( \\Theta ( L ) ( x , y , z ) ) ( a ) = \\sum _ { i + j = N + 1 ; ~ i , j > 0 } - f _ 0 ^ { \\lambda _ i ( x , \\lambda _ j ( y , z ) ) } + f _ 0 ^ { \\lambda _ i ( y , \\lambda _ j ( x , z ) ) } + f _ 0 ^ { \\lambda _ i ( \\lambda _ j ( x , y ) , z ) } ( a ) . \\end{align*}"}
-{"id": "6302.png", "formula": "\\begin{align*} \\Vert \\{ a _ { k } \\} | ~ \\mathcal { M } _ { p } ( l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } , l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } ) \\Vert = \\Vert \\{ a _ { k } \\} \\Vert _ { l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } \\rightarrow l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } } = \\sup _ { \\Vert \\{ \\lambda _ { k } \\} \\Vert _ { l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } = 1 } } \\Vert \\{ a _ { k } \\lambda _ { k } \\} \\Vert _ { l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } } . \\end{align*}"}
-{"id": "8142.png", "formula": "\\begin{align*} \\left ( \\frac { 2 } { p } - 1 + a _ 4 \\right ) ^ 2 = \\frac { 1 6 } { 2 7 \\left ( \\frac { 1 } { p } - \\frac { 1 } { q } \\right ) \\left ( 1 - \\frac { q } { p } + q \\right ) ^ 2 } . \\end{align*}"}
-{"id": "5303.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n \\alpha _ i ^ 2 = \\sum _ { j = 1 } ^ m \\beta _ j ^ 2 = 1 . \\end{align*}"}
-{"id": "2147.png", "formula": "\\begin{align*} B _ { m } u = \\frac { d } { d t } ( g _ { 1 - \\alpha , m } * u ) , u \\in L ^ { p } ( [ 0 , T ] ; X ) , \\ , \\ , m \\in \\mathbb { N } , \\end{align*}"}
-{"id": "6778.png", "formula": "\\begin{align*} \\int _ { \\{ V _ { \\theta } - j < \\psi _ t < \\psi < \\varphi + t \\} } \\theta _ { \\psi _ { t , j } } ^ n = 0 . \\end{align*}"}
-{"id": "1055.png", "formula": "\\begin{align*} \\zeta _ { A , R } ( s ) : = \\sum _ { m = 1 } ^ \\infty a _ m ( A , R ) m ^ { - s } \\end{align*}"}
-{"id": "1223.png", "formula": "\\begin{align*} H ( x ) = \\int _ I x h , \\ \\ \\ \\ x \\in \\Lambda _ { \\varphi , w } , \\end{align*}"}
-{"id": "678.png", "formula": "\\begin{align*} \\sum _ { g \\in G } p ^ { * n } ( g ) \\frac { d g \\nu } { d \\nu } ( y ) = 1 , \\textrm { f o r $ \\nu $ - a . e . $ y $ } . \\end{align*}"}
-{"id": "2197.png", "formula": "\\begin{align*} F ( s ) = - \\dot { \\phi } ( s ) \\left ( g _ { 1 - \\alpha } * \\int _ { B _ { 1 } } \\psi ^ { 2 } w ^ { 2 } d x \\right ) ( s ) + c _ { 4 } ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x . \\end{align*}"}
-{"id": "7526.png", "formula": "\\begin{align*} \\mathcal N ( \\omega ) = \\min \\left \\{ n \\ge n _ 0 ( \\omega ) : B _ n ( \\omega ) \\ , \\ , \\right \\} \\end{align*}"}
-{"id": "7170.png", "formula": "\\begin{align*} v _ i ( x ) = \\int _ { \\R ^ n } U _ { i j } ( x - y ) f _ j ( y ) d y . \\end{align*}"}
-{"id": "744.png", "formula": "\\begin{align*} W ( t ) = \\prod _ { i = 1 } ^ n ( 1 + t + \\ldots + t ^ { e _ i } ) , \\end{align*}"}
-{"id": "5880.png", "formula": "\\begin{align*} \\left ( \\kappa \\left ( \\mu - \\lambda - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } - \\frac { 1 } { 2 } \\sigma \\sigma _ { , x } \\right ) = \\sigma \\int \\frac { m } { \\sigma } d x + c \\sigma , \\end{align*}"}
-{"id": "1210.png", "formula": "\\begin{align*} P ' \\Bigl [ ( P - ( 1 + c _ { 2 } ) P ' _ { n } ) \\tilde { g } \\ge t / 2 \\Bigr ] = P ' \\Bigl [ ( P - P ' _ { n } ) \\tilde { g } \\ge \\frac { t / 2 + c _ { 2 } P \\tilde { g } } { 1 + c _ { 2 } } \\Bigr ] . \\end{align*}"}
-{"id": "8322.png", "formula": "\\begin{align*} u = \\sum _ { i = 1 } ^ r \\frac { Q _ i ( T ) } { P _ i ( T ) ^ { \\alpha _ i } } + R ( T ) \\end{align*}"}
-{"id": "4543.png", "formula": "\\begin{align*} W _ { 3 2 } : = \\int _ { \\mathcal { X } _ n ^ c \\times \\mathcal { X } } f ( x ) f ( y ) \\int _ { l _ x } ^ { v _ x } \\int _ { l _ y } ^ { v _ y } ( h _ { u v } F ) ( u , v ) \\ , d u \\ , d v \\ , d x \\ , d y = O \\biggl ( \\frac { k ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "2047.png", "formula": "\\begin{align*} 0 = \\nabla _ { \\nu } Q ( x , y _ 0 ) | _ { x _ 0 } = - 2 m \\frac { \\bar { w } ' ( \\frac { d ( x _ 0 , y _ 0 ) } { 2 } ) } { \\bar { w } ( \\frac { d ( x _ 0 , y _ 0 ) } { 2 } ) } \\nabla _ { \\nu } d ( x , y _ 0 ) | _ { x = x _ 0 } . \\end{align*}"}
-{"id": "3230.png", "formula": "\\begin{align*} \\Phi _ { \\beta , \\alpha } \\left ( \\partial \\mathcal { U } _ { \\alpha } ^ { S } \\left ( 3 \\right ) \\cap G _ { \\alpha } \\right ) = \\partial \\mathcal { U } _ { \\beta } ^ { S } \\left ( 3 \\right ) \\cap G _ { \\beta } . \\end{align*}"}
-{"id": "1704.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { n ^ H \\ell ( 1 / n ) ^ { 1 / 2 } } \\sum _ { j = 1 } ^ { [ n t ] } \\xi _ j \\right ) _ { t \\geq 0 } \\Rightarrow ( W ^ H _ t ) _ { t \\geq 0 } \\end{align*}"}
-{"id": "148.png", "formula": "\\begin{align*} ( \\widehat { C _ E } , \\| \\cdot \\| _ { \\widehat { C _ E } } ) = ( C _ { \\widehat { E } } , \\| \\cdot \\| _ { C _ { \\widehat { E } } } ) \\end{align*}"}
-{"id": "5935.png", "formula": "\\begin{gather*} D _ x G _ { \\lambda } : L ^ p \\big ( \\R ^ d _ v ; H ^ { s ' } _ { p } ( \\R ^ d _ x ) \\big ) \\to L ^ p ( \\R ^ { 2 d } ) = L ^ p \\big ( \\R ^ d _ x ; L ^ p ( \\R ^ d _ v ) \\big ) , \\\\ D _ x G _ { \\lambda } : L ^ p \\big ( \\R ^ d _ v ; H ^ { 1 } _ { p } ( \\R ^ d _ x ) \\big ) \\to L ^ p \\big ( \\R ^ d _ x ; H ^ { 2 } _ { p } ( \\R ^ d _ v ) \\big ) . \\end{gather*}"}
-{"id": "9275.png", "formula": "\\begin{align*} & K ( z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] = \\tilde { p } ( 0 , z ) \\exp ( \\int _ 0 ^ T ( b _ 0 ( s , z ) \\pi ( s , z ) - \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) d B ( s ) \\\\ & - \\frac { 1 } { 2 } \\int _ 0 ^ T ( b _ 0 ( t , z ) \\pi ( s , z ) - \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) ^ { 2 } d s ) . \\end{align*}"}
-{"id": "9390.png", "formula": "\\begin{align*} \\Xi ( \\lambda ) = ( H _ a - \\lambda { I } ) ^ { - 1 } - ( \\widetilde { H } _ \\infty - \\lambda { I } ) ^ { - 1 } . \\end{align*}"}
-{"id": "4593.png", "formula": "\\begin{align*} J : = \\left [ \\begin{array} { c c } 0 & 1 \\\\ - 1 & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "4962.png", "formula": "\\begin{align*} V = X _ A \\cup X _ B . \\end{align*}"}
-{"id": "4801.png", "formula": "\\begin{align*} \\frac { \\left ( f f '' + ( f ' ) ^ 2 + 1 \\right ) ^ 2 - 4 c f ^ 2 ( f '^ 2 + 1 ) } { f '^ 2 + 1 } = \\kappa ^ 2 . \\end{align*}"}
-{"id": "1650.png", "formula": "\\begin{align*} ( H ^ { s _ 0 } _ p ( \\R ^ d ) , H ^ { s _ 1 } _ p ( \\R ^ d ) ) _ { \\theta , p } = B ^ { s } _ { p , p } ( \\R ^ d ) \\end{align*}"}
-{"id": "7090.png", "formula": "\\begin{align*} Z _ n = \\sum _ { | u | = k } e ^ { - V ( u ) } Z ^ u _ { n } + \\sum _ { | u | = k } V ( u ) e ^ { - V ( u ) } \\sum _ { | v | = n , v > u } e ^ { V ( u ) - V ( v ) } . \\end{align*}"}
-{"id": "9743.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { G ( x ) } { e ^ { 1 / x ^ \\alpha } x ^ { \\alpha + 1 } } = \\lim _ { y \\to \\infty } \\frac { y ^ { - 2 } } { - ( \\alpha + 1 ) y ^ { - \\alpha - 2 } + \\alpha y ^ { - 2 } } = \\frac { 1 } { \\alpha } . \\end{align*}"}
-{"id": "3711.png", "formula": "\\begin{align*} \\frac { d } { d t } F ( \\vec { x } ) = \\{ F , Z _ { 1 2 3 } , H _ { 1 2 3 } \\} _ { 1 2 3 } - b \\{ F , Z _ { 1 2 5 } , H _ { 1 2 5 } \\} _ { 1 2 5 } + \\frac { 1 } { \\epsilon } [ F , H _ { 4 5 } ] _ { 4 5 } ~ , \\end{align*}"}
-{"id": "1170.png", "formula": "\\begin{align*} c _ n ( x ) = \\int _ { 0 \\leq \\xi _ 1 \\leq \\xi _ 2 \\leq \\dots \\leq \\xi _ { n + 1 } = x } \\left [ \\prod _ { j = 1 } ^ n ( \\xi _ { j + 1 } - \\xi _ j ) \\rho ( \\xi _ j ) \\right ] c _ 0 ( \\xi _ 1 ) d \\xi _ 1 d \\xi _ 2 \\dots d \\xi _ n \\end{align*}"}
-{"id": "6370.png", "formula": "\\begin{align*} f : = - \\Delta _ { x ' } u . \\end{align*}"}
-{"id": "647.png", "formula": "\\begin{align*} A _ { x _ o } = \\big \\{ g \\in G \\ , : \\ , g \\cdot x _ o \\in A \\big \\} = A , \\end{align*}"}
-{"id": "2732.png", "formula": "\\begin{align*} & \\theta _ { i - 1 } ^ 2 = \\theta _ i ( \\theta _ i - 1 ) , i = 1 , \\dots , N - 1 , \\end{align*}"}
-{"id": "1410.png", "formula": "\\begin{align*} \\begin{gathered} x _ { a 2 } , x _ { a 1 } , x _ { b 1 } , x _ { a 2 } \\\\ x _ { b 1 } , x _ { b 2 } , x _ { a 2 } , x _ { b 1 } \\\\ x _ { a 2 } , x _ { a 1 } , x _ { b 1 } , x _ { a 2 } \\end{gathered} \\end{align*}"}
-{"id": "7155.png", "formula": "\\begin{align*} - \\Delta v + \\nabla \\pi + ( u \\cdot \\nabla ) v = \\nabla F , { \\rm d i v } \\ , v = 0 \\mbox { i n } \\ , \\ , \\R ^ n _ + , \\end{align*}"}
-{"id": "787.png", "formula": "\\begin{align*} & ( R _ { i } - 1 ) ( R _ { i } + q ^ { 2 } ) = 0 ( 1 \\le i < k ) , \\\\ & R _ { i } R _ { i + 1 } R _ { i } = R _ { i + 1 } R _ { i } R _ { i + 1 } ( 1 \\le i \\le k - 2 ) , \\\\ & R _ { i } R _ { j } = R _ { j } R _ { i } ( | i - j | \\ge 2 ) . \\end{align*}"}
-{"id": "8523.png", "formula": "\\begin{align*} z ^ { \\ast } ( w f ) = ( w ^ { - 1 } ) ^ { \\ast } ( f z ^ { - 1 } ) \\end{align*}"}
-{"id": "5647.png", "formula": "\\begin{align*} W _ 2 ( \\mu , \\mu ' ) = \\left ( \\int _ 0 ^ 1 \\vert F _ { \\mu } ^ - ( t ) - F _ { \\mu ' } ^ - ( t ) \\vert ^ 2 d t \\right ) ^ { 1 / 2 } = \\Vert F _ { \\mu } ^ - - F _ { \\mu ' } ^ - \\Vert _ { \\mathbb { L } _ 2 ( [ 0 , 1 ] ) } \\end{align*}"}
-{"id": "8234.png", "formula": "\\begin{align*} - \\Omega \\bar { A } _ n ^ { ( 0 ) } - i \\gamma A _ n ^ { ( 0 ) } - \\bar { A } _ { n - 1 } ^ { ( 0 ) } = E W _ n , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "1535.png", "formula": "\\begin{align*} E \\cdot \\nabla _ v F - Q ( F ) = 0 , \\int _ { \\R ^ d } F ( v , E ) \\ , d v = 1 . \\end{align*}"}
-{"id": "8735.png", "formula": "\\begin{gather*} X _ \\tau ^ { 1 } - X _ \\tau ^ { 2 } = e ^ { \\tau A } ( x _ 1 - x _ 2 ) + e ^ { \\tau A } [ v ^ { ( 1 ) } ( 0 , x _ 1 ) - v ^ { ( 1 ) } ( 0 , x _ 2 ) ] \\\\ - [ v ^ { ( 1 ) } ( \\tau , X _ \\tau ^ { 1 } ) - v ^ { ( 1 ) } ( \\tau , X _ \\tau ^ { 2 } ) ] + \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } [ \\nabla ^ G v ^ { ( 1 ) } ( s , X _ s ^ 1 ) - \\nabla ^ G v ^ { ( 1 ) } ( s , X _ s ^ 2 ) ] \\ ; d W _ s , \\ ; \\ ; \\tau \\in [ T _ 0 , ( 2 T _ 0 ) \\wedge T ] . \\end{gather*}"}
-{"id": "2348.png", "formula": "\\begin{align*} \\frac { d } { d t } \\circ \\sigma = p t ^ { p - 1 } \\sigma \\circ \\frac { d } { d t } . \\end{align*}"}
-{"id": "6596.png", "formula": "\\begin{align*} F _ { \\rho } ^ { \\rm { A D M M } } ( v ) & = - F _ { \\gamma } ^ { \\rm { D R } } ( z ) . \\end{align*}"}
-{"id": "6169.png", "formula": "\\begin{align*} \\nabla ^ \\lambda _ X ( \\lambda Y \\xi ) - & \\lambda Y \\nabla ^ \\lambda _ X \\xi = \\nabla _ X ( \\lambda Y \\xi ) + \\frac { 1 } { 4 \\lambda } ( Z X - X Z ) ( \\lambda Y \\xi ) \\\\ & \\phantom { \\lambda Y \\nabla ^ \\lambda _ X \\xi = } - \\lambda Y \\left ( \\nabla _ X \\xi + \\frac { 1 } { 4 \\lambda } ( Z X - X Z ) \\xi \\right ) \\\\ = & \\left ( \\nabla ^ { L C } _ X ( \\lambda Y ) \\right ) \\xi + \\frac 1 4 ( Z X Y - X Z Y - Y Z X + Y X Z ) \\xi \\\\ = & \\lambda \\left ( \\nabla ^ { L C , \\lambda } _ X ( Y ) \\right ) \\xi \\ , , \\end{align*}"}
-{"id": "2189.png", "formula": "\\begin{align*} ( 1 - q ) \\zeta _ { 2 } ( q ) \\leq 4 + 9 \\frac { n + 2 } { \\beta _ { 0 } } = : c _ { 2 } = c _ { 2 } ( n , \\beta _ { 0 } ) , \\end{align*}"}
-{"id": "5399.png", "formula": "\\begin{align*} e ( H ' ) & \\leq \\sum _ { i = 1 } ^ { \\frac { 1 } { 2 } v ( M ) } \\left ( d ( v _ i ) + d ^ { \\ast } ( u _ i ) \\right ) + \\binom { \\frac { 1 } { 2 } v ( M ) } { 2 } \\\\ & \\leq \\frac { v ( M ) } { 2 } \\left ( n + \\frac { v ( M ) } { 4 } \\right ) \\\\ & \\leq \\frac { 5 } { 8 } n ^ 2 - \\frac { 3 } { 2 } \\abs { A } n + \\frac { \\abs { A } ^ 2 } { 2 } \\ , , \\end{align*}"}
-{"id": "520.png", "formula": "\\begin{align*} \\mathcal { U } ^ { \\angle } _ { \\rho _ { \\circ } , \\rho _ 1 } ( \\pi \\vert \\kappa , \\mu ) = \\mathcal { U } ^ { \\angle } _ { \\rho _ { \\circ } , \\rho _ 1 } ( \\pi \\vert \\kappa ) = \\frac { 1 } { H ^ o ( \\rho _ 1 ) H ( \\rho _ 1 ; \\rho _ { \\circ } ) } \\frac { \\tau _ { \\pi } ( \\rho _ { \\circ } ) s _ { \\pi / \\kappa } ( \\rho _ 1 ) } { \\tau _ { \\kappa } ( \\rho _ { \\circ } , \\rho _ 1 ) } . \\end{align*}"}
-{"id": "5058.png", "formula": "\\begin{align*} T = \\big \\{ g \\in G \\ , : \\ , \\int _ Z \\phi ( g z ) \\ , d m ( z ) > 0 \\big \\} \\end{align*}"}
-{"id": "8611.png", "formula": "\\begin{align*} \\overline { \\Pi } _ { 1 } \\overline { \\overline { N } } ( a ) & = - \\alpha \\overline { p } \\xi _ { e _ { k } a } \\\\ \\overline { \\Pi } _ { 2 } \\overline { \\overline { N } } ( a ) & = - c _ { k } \\overline { \\pi } _ { 3 } \\alpha \\xi _ { e _ { k } a } \\\\ \\overline { \\Pi } _ { 3 } \\overline { \\overline { N } } ( a ) & = \\overline { \\Pi } _ { 4 } \\overline { \\overline { N } } ( a ) = 0 \\end{align*}"}
-{"id": "3973.png", "formula": "\\begin{align*} \\psi _ i ( s ) = \\varphi ^ { - 1 } _ 1 ( s ) \\varphi _ i ( s ) \\end{align*}"}
-{"id": "7418.png", "formula": "\\begin{align*} \\overline { w } : = \\overline { s } _ { i ( 1 ) } \\overline { s } _ { i ( 2 ) } \\dots \\overline { s } _ { i ( l ) } \\end{align*}"}
-{"id": "2399.png", "formula": "\\begin{align*} \\lambda _ { i } ( L ) = \\begin{cases} \\alpha _ 2 & { \\hbox { f o r $ i $ s u c h t h a t } } \\lambda _ { i } ( N ) = 1 \\\\ ( 1 - \\alpha _ 1 ) ( 1 + ( \\alpha _ 2 - 1 ) ( 1 - \\alpha _ 1 ) ) & { \\hbox { f o r $ i $ s u c h t h a t } } \\lambda _ { i } ( N ) = 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "7666.png", "formula": "\\begin{align*} \\big \\{ k : 1 \\leq \\phi _ { k } , \\mbox { a n d } \\ \\ \\phi _ { k + 1 } \\leq \\frac { 1 } { \\alpha } \\phi _ { \\xi _ { n } } \\big \\} = \\{ k : k _ { 0 } \\leq k \\leq k _ { n } \\} , \\end{align*}"}
-{"id": "8432.png", "formula": "\\begin{align*} \\Bigg ( \\frac { f ( \\textbf { u } ) } { \\rho } \\partial _ { t } \\rho , \\partial _ { t } v \\Bigg ) ^ { T } + \\sum _ { j = 1 } ^ { d } \\textbf { P } A _ { 0 } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } = 0 . \\end{align*}"}
-{"id": "1194.png", "formula": "\\begin{align*} \\mathcal { A } _ 2 ( n ) = \\{ w \\in \\mathcal { A } ( n ) ~ | ~ w = 0 1 2 1 0 2 \\{ 0 1 2 \\} * 2 0 1 2 1 0 \\} \\subseteq \\mathcal { A } ( n ) , \\end{align*}"}
-{"id": "8103.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } F ( v ) \\ , d y = \\int _ { \\mathbb { R } ^ N } F ( \\widetilde { v } ) \\ , d y \\end{align*}"}
-{"id": "5046.png", "formula": "\\begin{align*} \\| f \\| _ \\infty - \\eta ( f ) = \\eta ( \\| f \\| _ \\infty - f ) \\leq \\| \\| f \\| _ \\infty - f \\| _ \\infty \\leq \\| f \\| _ \\infty , \\end{align*}"}
-{"id": "1560.png", "formula": "\\begin{align*} W _ 2 ( \\mu , \\mu ' ) = \\left ( \\int _ 0 ^ 1 \\vert F _ { \\mu } ^ - ( t ) - F _ { \\mu ' } ^ - ( t ) \\vert ^ 2 d t \\right ) ^ { 1 / 2 } = \\Vert F _ { \\mu } ^ - - F _ { \\mu ' } ^ - \\Vert _ { \\mathbb { L } _ 2 ( [ 0 , 1 ] ) } \\end{align*}"}
-{"id": "1455.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } m _ f ( 2 ^ l Q ) = 0 . \\end{align*}"}
-{"id": "2432.png", "formula": "\\begin{align*} \\Lambda _ k ^ { \\alpha _ 1 , \\ast } \\cap \\Lambda _ l ^ { \\alpha _ 1 , \\ast } = \\emptyset \\end{align*}"}
-{"id": "4772.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { t ^ 2 - ( c \\pm a \\ , t ^ 2 ) ^ 2 } , a = c o n s t \\neq 0 , c = c o n s t , \\end{align*}"}
-{"id": "451.png", "formula": "\\begin{align*} y & = ( a ^ k b ^ k ) ^ { ( \\ell - 1 ) / 2 } a ^ k b ^ { k - 1 } , \\\\ x & = ( a ^ k b ^ k ) ^ { ( \\ell - 1 ) / 2 } a ^ k b ^ { k - 1 } a ( a ^ k b ^ k ) ^ { ( \\ell - 1 ) / 2 } a ^ k b ^ { k - 1 } . \\end{align*}"}
-{"id": "9785.png", "formula": "\\begin{align*} m ( | \\mathcal { M } _ { k } ) = \\frac { \\{ | \\mathcal { M } _ { k } \\} \\times \\{ | \\mathcal { M } _ { k } \\} } { \\{ | \\mathcal { M } _ { k } \\} } . \\end{align*}"}
-{"id": "3387.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } s ^ p f ( s ) = \\ell \\in \\R \\end{align*}"}
-{"id": "9556.png", "formula": "\\begin{align*} \\begin{aligned} & \\lambda _ { n } ^ { ( 1 ) } = \\langle \\psi ^ { ( 0 ) } _ { n } , H ^ { ( 1 ) } \\psi ^ { ( 0 ) } _ { n } \\rangle \\\\ & \\lambda _ { n } ^ { ( 2 ) } = \\sum _ { i \\neq n } \\frac { \\left | \\langle \\psi ^ { ( 0 ) } _ { i } , H ^ { ( 1 ) } \\psi ^ { ( 0 ) } _ { n } \\rangle \\right | ^ { 2 } } { \\lambda _ { n } ^ { ( 0 ) } - \\lambda _ { i } ^ { ( 0 ) } } \\end{aligned} \\end{align*}"}
-{"id": "9569.png", "formula": "\\begin{align*} \\Delta ^ { ( 1 ) } \\psi & = f ^ { ( 1 ) } \\left ( \\Delta ^ { ( 0 ) } \\psi \\right ) \\\\ \\Delta ^ { ( 2 ) } \\psi & = f ^ { ( 2 ) } \\left ( \\Delta ^ { ( 0 ) } \\psi \\right ) . \\end{align*}"}
-{"id": "4587.png", "formula": "\\begin{align*} & w ^ 2 + x z w + ( x ^ 3 + u ^ 2 x ^ 2 y + u x y ^ 2 + y ^ 3 ) w = \\\\ & z ^ 3 + ( u ^ 2 x ^ 2 + x y + y ^ 2 ) x ^ 2 z + ( u ^ 2 x ^ 4 + u x ^ 3 y + x ^ 2 y ^ 2 ) y ^ 2 . \\end{align*}"}
-{"id": "1431.png", "formula": "\\begin{align*} W ^ { u _ { i } } ( z ) = J ^ { u _ { i } } ( z ) + ( \\mathrm { l o w e r } \\ \\mathrm { t e r m s } ) \\end{align*}"}
-{"id": "7361.png", "formula": "\\begin{gather*} \\Gamma _ { + } ^ { ( 2 ) * } = \\left ( \\begin{array} { c c c } 0 & q ^ { - 2 } & 0 \\\\ 0 & 0 & q ^ { - 2 } \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\Gamma _ { 0 } ^ { ( 2 ) * } = \\left ( \\begin{array} { c c c } - 1 & 0 & 0 \\\\ 0 & - q ^ { - 3 } ( q - q ^ { - 1 } ) & 0 \\\\ 0 & 0 & q ^ { - 2 } \\end{array} \\right ) , \\\\ \\Gamma _ { - } ^ { ( 2 ) * } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ - q ^ { - 4 } & 0 & 0 \\\\ 0 & - q ^ { - 2 } & 0 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "1140.png", "formula": "\\begin{align*} f _ { i j } ^ { \\max } = \\max _ { x \\in S } f _ { i j } \\left ( x \\right ) , i = 1 , 2 , . . . m ; j = 1 , 2 , . . . , p _ { m } ^ { } \\end{align*}"}
-{"id": "479.png", "formula": "\\begin{align*} P _ { E D } ^ { D i s p a t c h } : = & \\sum _ { t \\in \\mathcal { T } } \\left [ \\sum _ { i \\in \\mathcal { G } } p _ { i t } + \\sum _ { i \\in \\mathcal { I } } ( M _ { i t } - m _ { i t } ) \\right . \\\\ & \\left . + \\sum _ { i \\in \\mathcal { W } } ( W _ { i t } - w _ { i t } ) + \\sum _ { i \\in \\mathcal { R } } ( R _ { i t } - r _ { i t } ) \\right ] . \\end{align*}"}
-{"id": "5172.png", "formula": "\\begin{gather*} \\gamma \\leqslant \\alpha \\cdot \\beta \\leqslant ( r _ 1 ) ^ { \\delta ( \\alpha ) } ( r _ 1 ) ^ { \\delta ( \\beta ) } = ( r _ 1 ) ^ { \\delta ( \\alpha ) + \\delta ( \\beta ) } . \\end{gather*}"}
-{"id": "3266.png", "formula": "\\begin{align*} D ^ 2 = \\sum _ { i , j } E _ { \\xi _ i } F _ { \\xi _ j } \\otimes \\gamma _ - ( w _ i ) \\gamma _ + ( v _ j ) + \\sum _ { i , j } F _ { \\xi _ i } E _ { \\xi _ j } \\otimes \\gamma _ + ( v _ i ) \\gamma _ - ( w _ j ) . \\end{align*}"}
-{"id": "6991.png", "formula": "\\begin{align*} f _ 0 ^ { \\lambda _ n ( x , y ) } + f _ n ^ { \\lambda _ 0 ( x , y ) } + \\sum _ { i + j = n , ~ i , j > 0 } f _ i ^ { \\lambda _ j ( x , y ) } = \\sum _ { i + j = n , ~ i , j > 0 } [ f _ i ^ x , f _ j ^ y ] + [ f _ 0 ^ x , f _ n ^ y ] + [ f _ n ^ x , f _ 0 ^ y ] . \\end{align*}"}
-{"id": "5066.png", "formula": "\\begin{align*} A _ { x _ o } A _ { x _ o } ^ { - 1 } F ^ { - 1 } \\supset \\big \\{ g \\in G \\ , : \\ , \\lambda ( ( A \\cap g F A ) _ { x _ o } ) > 0 \\big \\} = \\big \\{ g \\in G \\ , : \\ , \\nu ( A \\cap g F A ) > 0 \\big \\} . \\end{align*}"}
-{"id": "9464.png", "formula": "\\begin{align*} i _ { X _ { f } } \\omega = - d f \\qquad g _ { j } ( \\nabla f , \\cdot ) = d f . \\end{align*}"}
-{"id": "8162.png", "formula": "\\begin{align*} \\{ a ^ { u _ 2 - u _ 1 } , \\ldots , a ^ { u _ { p _ i - 1 } - u _ 1 } \\} = [ a ^ { \\frac { n } { p _ i } } ] \\setminus \\{ a ^ { k _ 1 \\frac { n } { p _ i } } \\} . \\end{align*}"}
-{"id": "8810.png", "formula": "\\begin{align*} n \\int _ { \\Sigma } H u \\ , d v o l _ { \\Sigma } = \\int _ { \\Sigma } | \\nabla _ { \\Sigma } u | ^ 2 - | A _ { \\Sigma } | ^ 2 u ^ 2 d v o l _ { \\Sigma } \\ , . \\end{align*}"}
-{"id": "7705.png", "formula": "\\begin{align*} x _ { 2 ( n + 1 ) } = \\frac { \\sqrt { 1 + \\varepsilon } } { 2 n } - \\frac { \\sqrt { 1 + \\varepsilon } } { 2 n } + \\frac 1 { [ 2 ( n + 1 ) ] ^ 4 } = \\frac 1 { [ 2 ( n + 1 ) ] ^ 4 } . \\end{align*}"}
-{"id": "7443.png", "formula": "\\begin{align*} b _ i : = \\sum _ { k = 1 } ^ n a _ k x _ { k i } . \\end{align*}"}
-{"id": "799.png", "formula": "\\begin{align*} \\mathbb { T } ^ { [ M ' , M ] } ( z ) \\otimes \\mathbb { T } ^ { [ M ' , M ] } ( w ) = \\sum _ { a , b = 0 } ^ { r } \\sum _ { c , d = 0 } ^ { r } \\left ( \\mathbb { T } ^ { [ M ' , M ] } ( z ) _ { a b } \\mathbb { T } ^ { [ M ' , M ] } ( w ) _ { c d } \\right ) \\otimes E _ { a b } \\otimes E _ { c d } . \\end{align*}"}
-{"id": "2074.png", "formula": "\\begin{align*} \\mathbb { K } _ { k } ( A , \\ r _ { 0 } ) = \\{ r _ { 0 } , \\ A r _ { 0 } , \\ A ^ { 2 } r _ { 0 } , \\ \\ldots , \\ A ^ { k - 1 } r _ { 0 } \\} . \\end{align*}"}
-{"id": "6895.png", "formula": "\\begin{align*} \\mathrm { S I R } _ { \\mathrm { U } _ { 0 } } = \\frac { P _ { \\mathrm { B S } } g _ { n } \\left ( d _ { 0 } \\right ) H _ { \\mathrm { U } _ { 0 } , \\mathrm { B S } _ { 0 } } } { \\underset { \\mathrm { B S } _ { i } \\in \\Pi _ { \\mathrm { B S } } ^ { \\dagger } } { \\sum } P _ { \\mathrm { B S } } g _ { n } \\left ( d _ { i } \\right ) H _ { \\mathrm { U } _ { 0 } , \\mathrm { B S } _ { i } } } , \\ : n \\in \\left \\{ 1 , 2 \\right \\} \\end{align*}"}
-{"id": "9461.png", "formula": "\\begin{align*} \\lambda \\wedge ( d \\lambda ) ^ { n } & = ( d \\theta + \\pi ^ { * } \\beta ) \\wedge \\pi ^ { * } \\omega ^ { n } \\\\ & = d \\theta \\wedge \\pi ^ { * } \\omega ^ { n } > 0 . \\end{align*}"}
-{"id": "3209.png", "formula": "\\begin{align*} F _ { j , \\beta } ( z _ j ) - \\sum _ { | \\alpha | = n } T _ { j k } F _ { k , \\alpha } ( z _ k ) \\cdot \\tau _ { k j , \\beta } ^ \\alpha = h _ { 1 , j k , \\beta } - h _ { 2 , j k , \\beta } . \\end{align*}"}
-{"id": "523.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) = \\iint \\frac { ( z - w ) h ^ { \\rm g e o } _ { 1 2 } ( z , w ) } { ( z ^ 2 - 1 ) w ( z w - 1 ) } \\frac { z - c } { z ( 1 - c w ) } \\frac { \\dd z } { z ^ u } \\frac { \\dd w } { w ^ v } = - K _ { 2 1 } ^ { \\rm g e o } ( j , v ; i , u ) , \\end{align*}"}
-{"id": "6082.png", "formula": "\\begin{align*} D _ { m , n } ( t ) & = \\frac { 1 } { \\sqrt { m + 1 } } \\int _ n ^ { n + 1 } \\frac { v - n } { v ^ { 3 / 2 } } \\left ( 1 + i \\log \\frac { m + 1 } { v } \\right ) ^ { - t - 1 } d v \\\\ E _ { m , n } ( t ) & = \\int _ n ^ { n + 1 } \\frac { v - n } { v ^ { 3 / 2 } } \\int _ m ^ { m + 1 } u ^ { - 1 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 1 } d u d v , \\end{align*}"}
-{"id": "8381.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| e _ n x _ i v _ i e _ n \\| = 0 , \\ \\ \\ 1 \\leq i \\leq k , \\end{align*}"}
-{"id": "4245.png", "formula": "\\begin{align*} \\frac { 1 } { n + 1 } { 2 n \\choose n } \\frac { ( - 1 ) ^ { n - 1 } n + 1 } { 4 ^ n ( 2 n - 1 ) } = \\frac { 1 } { n ! } \\sum _ { m = 0 } ^ n \\Big ( \\frac { 1 } { 2 } \\Big ) ^ m S _ 1 ( n , m ) \\end{align*}"}
-{"id": "8293.png", "formula": "\\begin{align*} T _ 2 ( n , d ) & = d \\Psi ( \\mathbf { m } ^ * ) + \\frac { \\sqrt { d } } { n } \\Big [ \\max _ { \\underline { \\sigma } \\in \\Sigma ( \\mathbf { m } ^ * ) } H _ f ( \\underline { \\sigma } ) \\Big ] \\\\ | T _ 2 ( n , d ) | & \\leq d \\| f \\| _ { \\infty } + \\sqrt { d } \\Big | \\frac { 1 } { n } \\max _ { \\underline { \\sigma } \\in \\Sigma ( \\mathbf { m ^ * } ) } H _ f ( \\underline { \\sigma } ) \\Big | . \\end{align*}"}
-{"id": "67.png", "formula": "\\begin{align*} \\lim _ { y \\to 1 } \\psi ( y , a ) = \\ell ( a ) \\ , \\end{align*}"}
-{"id": "125.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty { 2 n \\choose n } \\frac { ( - 1 ) ^ { n - 1 } } { 4 ^ n ( 2 n - 1 ) } t ^ n = \\sum _ { n = 0 } ^ \\infty \\left ( \\frac { 1 } { n ! } \\sum _ { m = 0 } ^ n \\Big ( \\frac { 1 } { 2 } \\Big ) ^ m S _ 1 ( n , m ) \\right ) t ^ n \\end{align*}"}
-{"id": "5492.png", "formula": "\\begin{align*} \\mathcal { R } _ k [ \\iota ] = \\min \\{ \\mathcal { R } ^ \\mathrm { U L } _ k [ \\iota ] , \\mathcal { R } ^ \\mathrm { D L } _ k [ \\iota ] \\} , \\end{align*}"}
-{"id": "1327.png", "formula": "\\begin{align*} R _ { 1 2 } ( \\lambda , \\mu ) T _ 1 ( \\lambda ) T _ 2 ( \\mu ) = T _ 2 ( \\mu ) T _ 1 ( \\lambda ) R _ { 1 2 } ( \\lambda , \\mu ) . \\end{align*}"}
-{"id": "5751.png", "formula": "\\begin{align*} \\hat { E } _ n = \\sum _ { t = 1 } ^ { n } \\sigma ^ 2 _ t ( \\hat { \\delta } _ { 0 } ) x _ { t } x _ { t } ^ { \\mathrm { T } } , \\end{align*}"}
-{"id": "1096.png", "formula": "\\begin{align*} | | \\mu - \\nu | | = \\sup \\{ \\langle u , \\mu \\rangle - \\langle u , \\nu \\rangle : u \\in B [ S , { \\cal F } ] , | | u | | \\leq 1 \\} . \\end{align*}"}
-{"id": "4708.png", "formula": "\\begin{align*} y & = ( a ^ k b ^ k ) ^ { \\ell / 2 } a ^ { k - 1 } , \\\\ x & = ( a ^ k b ^ k ) ^ { \\ell / 2 } a ^ { k - 1 } b ( a ^ k b ^ k ) ^ { \\ell / 2 } a ^ { k - 1 } . \\end{align*}"}
-{"id": "3705.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } ( x ^ 2 _ 1 + 2 x ^ 2 _ 2 + x ^ 2 _ 3 + x ^ 2 _ 4 + x ^ 2 _ 5 ) ~ , \\end{align*}"}
-{"id": "8.png", "formula": "\\begin{align*} \\sin \\varphi _ q ^ k & = \\sin \\left ( \\frac { 1 } { q } \\mu \\left ( \\frac { k } { q } + \\frac { \\alpha ( k / q ) } { q ^ 2 } \\right ) \\left ( 1 + \\frac { \\beta ( k / q ) } { q ^ 2 } \\right ) \\right ) \\left ( 1 + \\varepsilon O ( q ^ { - 4 } ) \\right ) . \\end{align*}"}
-{"id": "8629.png", "formula": "\\begin{align*} \\hbox { i . e . } \\ a _ { i , j + 1 } + a _ { i + 1 , j } + a _ { i + 1 , j + 1 } = 0 . \\end{align*}"}
-{"id": "7251.png", "formula": "\\begin{align*} F _ { k , \\alpha } ( z _ k ( z _ j , w _ j ) ) = F _ { k , \\alpha } ( z _ k ( z _ j , 0 ) ) + \\sum _ { | \\gamma | \\geq 1 } F _ { k j , \\alpha , \\gamma } ( z _ j ) \\cdot w _ j ^ \\gamma . \\end{align*}"}
-{"id": "813.png", "formula": "\\begin{align*} \\langle X \\rangle _ { [ M ' , M ] } = { } _ { [ M , M ' ] } \\langle \\mathrm { v a c } | X | \\mathrm { v a c } \\rangle _ { [ M ' , M ] } ( X \\in \\mathcal { B } ^ { [ M ' , M ] } ) . \\end{align*}"}
-{"id": "6270.png", "formula": "\\begin{align*} 0 = \\nabla _ { \\nu } \\| \\nabla w \\| ^ 2 | _ { x _ 0 } = - I I ( \\nabla w , \\nabla w ) | _ { x _ 0 } < 0 , \\end{align*}"}
-{"id": "5971.png", "formula": "\\begin{align*} P ( X ) = \\sum c _ i X ^ i , \\end{align*}"}
-{"id": "1459.png", "formula": "\\begin{align*} f = \\sum _ { j = 3 } ^ \\infty \\chi _ { [ j ! , j ! + 1 ] } , \\end{align*}"}
-{"id": "7949.png", "formula": "\\begin{align*} \\lambda _ j ( \\sigma ) = \\inf _ { \\substack { E \\leq H ^ 2 ( \\Omega ) \\\\ { \\rm d i m } E = j } } \\sup _ { 0 \\ne u \\in E } \\frac { \\int _ { \\Omega } ( 1 - \\sigma ) | D ^ 2 u | ^ 2 + \\sigma ( \\Delta u ) ^ 2 d x } { \\int _ { \\Omega } u ^ 2 d x } . \\end{align*}"}
-{"id": "5385.png", "formula": "\\begin{align*} \\Gamma = \\langle a , b , c | a ^ 2 , b ^ 2 , c ^ 2 , \\tau ^ n ( w _ 1 ) , \\tau ^ n ( w _ 2 ) , \\tau ^ n ( w _ 3 ) , \\tau ^ n ( w _ 4 ) n \\geq 0 \\rangle \\end{align*}"}
-{"id": "3657.png", "formula": "\\begin{align*} \\beta ^ { m _ i } \\binom { n _ 0 ^ 2 } { m _ i } ^ 3 , \\end{align*}"}
-{"id": "2016.png", "formula": "\\begin{align*} \\left | \\frac { f ( x u ) - f ( x ) } { x f ' ( x ) } \\right | = \\alpha ^ { - 1 } \\log | x | \\left [ 1 - \\left ( 1 - \\frac { \\log u ^ { - 1 } } { \\log | x | } \\right ) ^ \\alpha \\right ] \\leq 2 \\log u ^ { - 1 } . \\end{align*}"}
-{"id": "346.png", "formula": "\\begin{align*} L _ 0 ^ { c , h } \\Psi _ { c , h } = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; L ^ { c , h } _ n \\Psi _ { c , h } = 0 \\ ; \\ ; \\ ; \\textrm { f o r a l l } \\ ; \\ ; n > 0 \\end{align*}"}
-{"id": "6213.png", "formula": "\\begin{align*} & \\hat { M } = V ^ { T } M V , \\ \\hat { D } = V ^ { T } D V , \\ \\hat { K } = V ^ { T } K V , \\ \\hat { F } = V ^ { T } F , \\\\ & \\hat { C } _ { p } = C _ { p } V \\ \\ \\hat { C } _ { v } = C _ { v } V . \\end{align*}"}
-{"id": "8635.png", "formula": "\\begin{align*} | \\langle T _ j x ' , x ' \\rangle | = \\bigl | \\langle T _ j x , x \\rangle + \\frac { 1 } { 2 ^ { k + 1 } } \\langle T _ j u , u \\rangle \\bigr | = \\bigl | \\xi _ j + \\frac { 1 } { 2 ^ { k + 1 } } \\langle T _ j u , u \\rangle \\bigr | \\le \\frac { r } { 2 ^ { k + 2 } } \\end{align*}"}
-{"id": "9654.png", "formula": "\\begin{align*} \\begin{array} { r c l } x ' ( t ) & = & - a g ( x ( t ) ) + b g ( x ( t - \\tau ( t ) ) , t \\geq 0 \\\\ x ( t ) & = & \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] \\end{array} \\end{align*}"}
-{"id": "4939.png", "formula": "\\begin{align*} \\| h ^ { ( 2 ) } \\| _ 1 & = \\| D ^ * _ { S _ 0 ^ c } h \\| _ 1 - \\| h ^ { ( 1 ) } \\| _ 1 \\leq k \\alpha - \\ell \\cdot \\frac { \\alpha } { t - 1 } = ( k ( t - 1 ) - \\ell ) \\frac { \\alpha } { t - 1 } , \\\\ \\| h ^ { ( 2 ) } \\| _ \\infty & \\leq \\frac { \\alpha } { t - 1 } . \\end{align*}"}
-{"id": "3698.png", "formula": "\\begin{align*} \\frac { d x _ 1 } { d t } = - x _ 2 x _ 3 + b x _ 2 x _ 5 ~ , \\end{align*}"}
-{"id": "4552.png", "formula": "\\begin{align*} \\mu _ d \\bigl ( B _ x ( r _ { n , u } ) \\cap B _ y ( r _ { n , v } ) ^ c \\bigr ) & \\geq \\mu _ d \\bigl ( B _ x ( r _ { n , v } ) \\cap B _ y ( r _ { n , v } ) ^ c \\bigr ) \\\\ & = V _ d r _ { n , v } ^ d \\biggl \\{ 1 - I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { \\| x - y \\| ^ 2 } { 4 r _ { n , v } ^ 2 } \\biggr ) \\biggr \\} \\gtrsim \\frac { k \\| z \\| } { n f ( x ) } . \\end{align*}"}
-{"id": "1975.png", "formula": "\\begin{align*} s _ i = \\begin{cases} a _ i m + b & { } \\\\ a _ i m + ( m - b ) & { } . \\end{cases} \\end{align*}"}
-{"id": "8183.png", "formula": "\\begin{align*} H _ 1 = c _ 1 , \\ \\ \\ H _ 2 = c _ 2 , \\end{align*}"}
-{"id": "9118.png", "formula": "\\begin{align*} ( \\lambda B ^ { - 1 } + A ( t ) ) ^ { - 1 } = ( \\lambda + A ( t ) ) ^ { - 1 } + ( \\lambda B ^ { - 1 } + A ( t ) ) ^ { - 1 } ( \\lambda ( - B ^ { - 1 } + I ) ) ( \\lambda + A ( t ) ) ^ { - 1 } . \\end{align*}"}
-{"id": "9900.png", "formula": "\\begin{align*} B _ 1 = \\begin{pmatrix} \\frac { 2 \\Delta ^ { 2 } } { \\Delta + 1 } & \\frac { t _ { v _ { i } } \\Delta + \\Delta - 2 \\Delta ^ { 2 } } { \\Delta + 1 } \\\\ \\frac { t _ { v _ { i } } \\Delta + \\Delta - 2 \\Delta ^ { 2 } } { n - \\Delta - 1 } & \\frac { S - 2 t _ { v _ { i } } \\Delta + 2 \\Delta ( \\Delta - 1 ) } { n - \\Delta - 1 } \\end{pmatrix} . \\end{align*}"}
-{"id": "2818.png", "formula": "\\begin{align*} \\eta _ p ( t ) = \\inf \\{ s \\ge t : X _ { r : n } ( s ) \\ge f _ p ( s ) \\} . \\end{align*}"}
-{"id": "9626.png", "formula": "\\begin{align*} { \\cal H } _ { \\tilde F } ( \\Psi ( t ) ) = \\Vert \\dot \\psi ( t ) \\Vert ^ 2 + \\Vert \\nabla \\psi _ { r e g } ( t ) \\Vert ^ 2 + m ^ 2 \\Vert \\psi _ { r e g } ( t ) \\Vert ^ 2 + \\tilde U ( \\zeta ( t ) ) - { \\cal G } ( \\zeta ( t ) ) = c o n s t , t \\in [ 0 , \\tau ] . \\end{align*}"}
-{"id": "4195.png", "formula": "\\begin{align*} \\norm { f _ R } _ { H ^ { ( n + 1 ) / 2 } } ^ 2 = \\norm { f _ 0 } _ { H ^ { ( n + 1 ) / 2 } } ^ 2 + o ( 1 ) = n ! \\omega _ n + o ( 1 ) \\end{align*}"}
-{"id": "7065.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ i ] } } ( { t _ 1 } ) = \\sum \\limits _ { j = 1 } ^ N { { { \\bf { s } } ^ { [ j i ] } } } , \\end{align*}"}
-{"id": "7159.png", "formula": "\\begin{align*} { \\rm { d i v } } \\ , \\tilde { w } = ( u - \\tilde { b } _ n K _ n ) \\cdot \\nabla \\zeta \\quad \\mbox { i n } \\ , \\ , B _ { l _ 2 } ^ + \\setminus B _ { l _ 1 } , \\qquad \\tilde { w } = 0 \\quad \\mbox { o n } \\ , \\ , \\partial ( B _ { l _ 2 } ^ + \\setminus B _ { l _ 1 } ) . \\end{align*}"}
-{"id": "9127.png", "formula": "\\begin{align*} & | ( \\int _ 0 ^ \\tau A ^ { 1 / 2 } e ^ { - ( \\tau - s ) A } g ( s ) d s , x ) | \\\\ & = | \\int _ 0 ^ \\tau ( g ( s ) , { A ^ * } ^ { 1 / 2 } e ^ { - ( \\tau - s ) A ^ * } x ) | \\\\ & \\le ( \\int _ 0 ^ \\tau \\| g ( s ) \\| ^ 2 d s ) ^ { 1 / 2 } ( \\int _ 0 ^ \\tau \\| { A ^ * } ^ { 1 / 2 } e ^ { - ( \\tau - s ) A ^ * } x \\| ^ 2 d s ) ^ { 1 / 2 } \\\\ & \\le C \\| x \\| ( \\int _ 0 ^ \\tau \\| g ( s ) \\| ^ 2 d s ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "5395.png", "formula": "\\begin{align*} \\binom { N } { 2 } = ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } \\leq k ( k - \\alpha - x ) \\frac { n ^ 2 } { 2 } \\end{align*}"}
-{"id": "9019.png", "formula": "\\begin{align*} \\psi ( p , y ) = p + ( y ^ { c _ 1 } , \\ldots , y ^ { c _ { n - 1 } } ) \\end{align*}"}
-{"id": "8203.png", "formula": "\\begin{align*} F ( 0 , 0 ) = F ( t ( A ) , u ( A ) ) = 0 . \\end{align*}"}
-{"id": "7910.png", "formula": "\\begin{align*} | R _ g | ^ 2 = | W _ g | ^ 2 + 2 | B _ g | ^ 2 + \\frac 1 6 S _ g ^ 2 . \\end{align*}"}
-{"id": "8941.png", "formula": "\\begin{align*} \\varphi ( b ) \\leqslant & ( n - 1 ) ^ { - p / ( p - 1 ) } \\int _ b ^ { + \\infty } s ^ { - p / ( p - 1 ) } d s \\\\ \\leqslant & ( n - 1 ) ^ { - p / ( p - 1 ) } ( p - 1 ) b ^ { - 1 / ( p - 1 ) } , \\end{align*}"}
-{"id": "4240.png", "formula": "\\begin{align*} \\frac { 2 } { e ^ t + 1 } & = \\sum _ { n = 0 } ^ \\infty C _ n \\frac { 1 } { 4 ^ n } \\big ( 1 - e ^ { 2 t } \\big ) ^ n \\\\ & = \\sum _ { n = 0 } ^ \\infty C _ n \\frac { ( - 1 ) ^ n } { 4 ^ n } \\big ( e ^ { 2 t } - 1 \\big ) ^ n \\\\ & = \\sum _ { n = 0 } ^ \\infty C _ n \\frac { ( - 1 ) ^ n } { 4 ^ n } n ! \\sum _ { m = n } ^ \\infty S _ 2 ( m , n ) \\frac { 2 ^ m t ^ m } { m ! } \\\\ & = \\sum _ { m = 0 } ^ \\infty \\left ( \\sum _ { n = 0 } ^ m C _ n ( - 1 ) ^ n 2 ^ { m - 2 n } n ! S _ 2 ( m , n ) \\right ) \\frac { t ^ m } { m ! } . \\end{align*}"}
-{"id": "295.png", "formula": "\\begin{align*} \\max _ { v \\in \\{ v _ x , v _ y \\} } | v f ( x ) - 1 - 3 k ^ { - 1 / 2 } \\log ^ { 1 / 2 } n | \\lesssim a ( f ( x ) \\wedge f ( y ) ) \\Bigl ( \\frac { k } { n f ( x ) } \\Bigr ) ^ { \\beta / d } = o ( k ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "4789.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } ' : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "2626.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { l = 1 } ^ { h ( m , n ) } \\sum _ { j = 0 } ^ \\infty \\| \\vec a _ m ^ l ( j ) \\| < \\infty , \\sum _ { m = 0 } ^ { \\infty } \\sum _ { l = 1 } ^ { h ( m , n ) } \\sum _ { j = 0 } ^ \\infty ( j + 1 ) \\| \\vec a _ m ^ l ( j ) \\| ^ 2 < \\infty , \\end{align*}"}
-{"id": "1536.png", "formula": "\\begin{align*} F ( v , E ) = M ( v ) + E \\cdot \\lambda ( v ) + G ( v , E ) \\end{align*}"}
-{"id": "666.png", "formula": "\\begin{align*} \\int _ G \\lambda ( g ^ { - 1 } B \\cap C ) \\ , d \\eta ( g ) = \\lambda ( B ) \\ , \\lambda ( C ) . \\end{align*}"}
-{"id": "9623.png", "formula": "\\begin{align*} \\ddot { \\psi } ( t ) = ( \\Delta - m ^ 2 ) \\psi ( t ) + \\sum \\limits _ { 1 \\le j \\le n } \\zeta _ j ( t ) g _ j , \\psi ( 0 ) = \\psi _ 0 , \\quad \\dot \\psi ( 0 ) = \\pi _ 0 , \\end{align*}"}
-{"id": "7982.png", "formula": "\\begin{align*} [ x _ 0 : x _ 1 : x _ 2 : x _ 3 ] \\mapsto t = \\frac { x _ 0 } { x _ 1 } , \\end{align*}"}
-{"id": "5309.png", "formula": "\\begin{align*} R _ { 1 2 } ( u - v ) R _ { 1 3 } ( u ) R _ { 2 3 } ( v ) = R _ { 2 3 } ( v ) R _ { 1 3 } ( u ) R _ { 1 2 } ( u - v ) . \\end{align*}"}
-{"id": "97.png", "formula": "\\begin{align*} | \\langle b ^ * ( e _ { k _ m } ) , b ^ * ( e _ { k _ n } ) \\rangle | = | \\langle b b ^ * ( e _ { k _ m } ) , e _ { k _ n } \\rangle | < \\frac { \\varepsilon ^ 2 } { 3 \\cdot 2 ^ n } \\end{align*}"}
-{"id": "1989.png", "formula": "\\begin{align*} X : = X _ \\R \\oplus i X _ \\R . \\end{align*}"}
-{"id": "3579.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } \\| U ^ j \\| _ { C ^ { \\ell + t _ j , \\alpha } _ { \\phi , \\varphi _ j } ( \\Omega ) } \\le C \\left ( \\sum _ { j = 1 } ^ { n + 1 } \\| ( L U ) _ j \\| _ { C ^ { \\ell - s _ j , \\alpha } _ { \\phi , \\phi ^ { t _ j + s _ j } \\varphi _ j } ( \\Omega ) } + \\sum _ { j = 1 } ^ { n + 1 } \\| U ^ j \\| _ { L ^ 2 _ { \\phi ^ { - n } \\varphi _ j ^ 2 } ( \\Omega ) } \\right ) , \\end{align*}"}
-{"id": "7717.png", "formula": "\\begin{align*} \\mathbb P \\{ \\omega \\in \\Omega : \\lim _ { n \\to \\infty } x _ n ( \\omega ) = 0 \\} = 0 \\end{align*}"}
-{"id": "516.png", "formula": "\\begin{align*} \\tau _ { \\mu } ( c ) = \\sum _ { \\kappa ' } s _ { \\lambda / \\kappa } ( c ) = c ^ { \\mu _ 1 - \\mu _ 2 + \\mu _ 3 - \\mu _ 4 + \\dots } . \\end{align*}"}
-{"id": "4348.png", "formula": "\\begin{align*} \\psi ' ( 0 ) = \\frac { p a ^ { 2 - p } } { p - 1 } > 0 \\ . \\end{align*}"}
-{"id": "7522.png", "formula": "\\begin{align*} \\gamma = \\max \\left \\{ 1 - \\alpha + l , \\ , \\alpha + l \\right \\} \\end{align*}"}
-{"id": "8011.png", "formula": "\\begin{align*} D ( E , B ) = d _ n . \\end{align*}"}
-{"id": "6951.png", "formula": "\\begin{align*} s _ \\lambda = & \\ , \\det \\left [ \\begin{array} { c } h _ { \\lambda _ i - i + j } \\end{array} \\right ] _ { 1 \\leq i , j \\leq \\ell ( \\lambda ) } \\ , . \\end{align*}"}
-{"id": "2103.png", "formula": "\\begin{align*} U _ a : = \\bigg ( \\frac { r + d } { 2 } \\bigg ) ^ s . \\end{align*}"}
-{"id": "9832.png", "formula": "\\begin{align*} M _ i \\ , { } _ \\lambda \\ , v = d c ^ i v , \\end{align*}"}
-{"id": "137.png", "formula": "\\begin{align*} \\| f _ n - \\widehat { i d } ( f ) \\| _ { L _ 1 + L _ \\infty } = \\| \\widehat { i d } ( f _ n ) - \\widehat { i d } ( f ) \\| _ { L _ 1 + L _ \\infty } \\leq C \\| f _ n - f \\| _ { \\widehat { E } } \\to 0 , \\end{align*}"}
-{"id": "6649.png", "formula": "\\begin{align*} ( E _ j ) ^ I _ { N _ j } = V _ j ^ I \\otimes \\omega _ { s _ j + d _ j - 1 } \\end{align*}"}
-{"id": "529.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = I ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) + \\hat { R } ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) \\end{align*}"}
-{"id": "636.png", "formula": "\\begin{align*} \\forall \\ , s \\in G , \\varlimsup _ n \\ , \\frac { | s F _ n \\Delta F _ n | } { | F _ n | } = 0 , \\end{align*}"}
-{"id": "9538.png", "formula": "\\begin{align*} g _ M ( R ^ M ( e _ a ' , e _ b ' ) e _ c ' , e _ d ' ) \\eta ^ { a d } \\eta ^ { b c } & = e ^ { 4 \\phi / 3 } g _ N ^ \\phi ( R ^ { N \\phi } ( e _ a , e _ b ) e _ c , e _ d ) \\eta ^ { a d } \\eta ^ { b c } - \\frac { 3 } { 2 } e ^ { 8 \\phi / 3 } | G _ 2 | ^ 2 \\\\ & = R ^ { N \\phi } - \\frac { 3 } { 2 } e ^ { 8 \\phi / 3 } | G _ 2 | ^ 2 \\\\ & = e ^ { 2 \\phi / 3 } \\left ( R ^ N + 6 \\Delta \\phi - 8 | d \\phi | ^ 2 \\right ) - \\frac { 3 } { 2 } e ^ { 8 \\phi / 3 } | G _ 2 | ^ 2 . \\end{align*}"}
-{"id": "4438.png", "formula": "\\begin{align*} \\mathcal { F } _ { d , \\theta } : = \\biggl \\{ f \\in \\mathcal { F } _ d : \\mu _ \\alpha ( f ) \\leq \\nu , \\| f \\| _ \\infty \\leq \\gamma , \\sup _ { x : f ( x ) \\geq \\delta } M _ { f , a , \\beta } ( x ) \\leq a ( \\delta ) \\ \\forall \\delta > 0 \\biggr \\} . \\end{align*}"}
-{"id": "145.png", "formula": "\\begin{align*} V \\mu ( x ) = x r + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } . \\end{align*}"}
-{"id": "6098.png", "formula": "\\begin{align*} z _ j = \\exp ( - s _ j ^ { - 1 } + i \\theta _ j ) \\end{align*}"}
-{"id": "716.png", "formula": "\\begin{align*} & E = g ^ { * } H - \\left < A \\vec { c } , \\vec { F } \\right > \\\\ & \\vec { { E ' } } = \\left ( \\begin{array} { c } E _ { 1 } ' \\\\ E _ { 2 } ' \\\\ \\vdots \\\\ E _ { s } ' \\end{array} \\right ) = { p } _ { * } \\vec { F } - { } ^ { \\rm t } B \\vec { D } . \\end{align*}"}
-{"id": "839.png", "formula": "\\begin{align*} u ( a , b , a ^ { m } ) = q ^ { - 1 } \\left \\{ f ( z _ { \\ell } , z _ { 1 } ) Y _ { 0 } ( z _ { \\ell } , z _ { 1 } ) - g ( z _ { \\ell } , z _ { 1 } ) \\right \\} u ( b , a ^ { m + 1 } ) \\end{align*}"}
-{"id": "4671.png", "formula": "\\begin{align*} \\mathcal { B } _ n = \\left \\{ Y _ { j _ 1 , \\dots , j _ k } ^ { ( n , i ) } \\colon 1 \\leq k \\leq n , 0 \\leq j _ 1 , \\dots , j _ k \\leq m , 1 \\leq i \\leq m \\right \\} & \\subset C _ { n + 1 } \\quad \\mbox { a n d } \\\\ \\tilde { \\mathcal { B } } _ n = \\left \\{ \\tilde Y _ { j _ 1 , \\dots , j _ k } ^ { ( n , i ) } \\colon 1 \\leq k \\leq n , 0 \\leq j _ 1 , \\dots , j _ k \\leq m , 1 \\leq i \\leq m \\right \\} & \\subset C _ { n + 2 } \\end{align*}"}
-{"id": "2498.png", "formula": "\\begin{gather*} \\beta _ 1 ( t ) : = x ( u _ k , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ k , \\widetilde { \\sigma } _ 1 ) , t \\geqslant 0 , \\\\ \\beta _ 2 ( t ) : = x ( u _ j , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ j , \\widetilde { \\sigma } _ 1 ) , t \\geqslant 0 . \\end{gather*}"}
-{"id": "988.png", "formula": "\\begin{align*} H _ { 4 , 1 } & = U ( N _ { a , 1 } + N _ { a , 2 } + N _ { a , 3 } + N _ { a , 4 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } + N _ { a , 2 } + N _ { a , 3 } + N _ { a , 4 } - N _ { b , 1 } ) \\\\ & + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) + t _ { 2 , 1 } ( a _ { 2 } b _ { 1 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 1 } ) \\\\ & + t _ { 3 , 1 } ( a _ { 3 } b _ { 1 } ^ \\dagger + a _ { 3 } ^ \\dagger b _ { 1 } ) + t _ { 4 , 1 } ( a _ { 4 } b _ { 1 } ^ \\dagger + a _ { 4 } ^ \\dagger b _ { 1 } ) \\end{align*}"}
-{"id": "1011.png", "formula": "\\begin{align*} \\alpha _ 1 & = \\sin \\phi _ 1 \\cos \\theta _ 1 , & \\alpha _ 2 & = \\sin \\phi _ 1 \\sin \\theta _ 1 , & \\alpha _ 3 & = \\cos \\phi _ 1 , \\\\ \\beta _ 1 & = \\sin \\phi _ 1 \\cos \\theta _ 1 , & \\beta _ 2 & = \\sin \\phi _ 2 \\sin \\theta _ 2 , & \\beta _ 3 & = \\cos \\phi _ 2 . \\end{align*}"}
-{"id": "961.png", "formula": "\\begin{align*} \\| j _ { q } u \\| ^ { 2 } _ { L ^ { 2 } _ { ( 0 , q ) } ( M ) } = \\| u \\| _ { L ^ { 2 } _ { ( 0 , q ) } ( M ) } ^ { 2 } \\leq \\varepsilon \\| u \\| _ { g r a p h } ^ { 2 } + C _ { \\varepsilon } \\| u \\| _ { W ^ { - 1 } _ { ( 0 , q ) } ( M ) } ^ { 2 } \\ ; . \\end{align*}"}
-{"id": "3681.png", "formula": "\\begin{align*} \\frac { d H } { d t } = \\int H _ { \\mu } \\frac { \\partial \\mu } { \\partial t } d A = 0 ~ . \\\\ \\end{align*}"}
-{"id": "3854.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\tilde \\chi _ 0 ( \\tau ) = & | \\tau | ^ \\beta \\chi _ 0 ( \\tau ) \\\\ \\tilde { \\tilde { \\chi } } ( s ) = & \\tilde \\chi ( s ) s ^ { - \\frac 1 2 } \\ , . \\end{aligned} \\right . \\end{align*}"}
-{"id": "7641.png", "formula": "\\begin{align*} \\ \\left \\{ \\begin{aligned} & u _ { t } = - ( - \\frac { \\triangle } { 2 } ) ^ { \\alpha / 2 } u + f ( u ) , \\\\ & u ( 0 ) = u _ { 0 } , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4384.png", "formula": "\\begin{align*} W _ { h _ c , I } T _ { c , 0 } ( f ) W _ { h _ c , I } ^ * = T _ { c , h _ c } ( f ) = K _ { h , I } T _ { c , h } ( f ) K _ { h , I } ^ * \\end{align*}"}
-{"id": "3343.png", "formula": "\\begin{align*} & C _ { B P S } ^ P ( M \\times I _ R ^ n \\times T ^ n , X \\times T ^ n ; \\mathbf { l } ^ \\ast , ( 0 _ M , e ) ) \\\\ & = C _ { B P S } ^ P ( M \\times I _ R ^ n \\times T ^ n , X \\times T ^ n ; K \\mathbf { l } ^ \\ast , ( 0 _ M , e ) ) \\leq \\sum _ { i = 1 } ^ n R _ i \\cdot | e _ i | . \\\\ \\end{align*}"}
-{"id": "7306.png", "formula": "\\begin{align*} S _ q ( V ) = T ( V ) / \\langle \\ker ( \\sigma _ { V V } + \\mathrm { i d } ) , \\Lambda _ q ( V ) = T ( V ) / \\langle \\ker ( \\sigma _ { V V } - \\mathrm { i d } ) . \\end{align*}"}
-{"id": "1686.png", "formula": "\\begin{align*} \\int _ 0 ^ t \\Big \\| \\big [ D U ( Z _ s ^ z ) - D U ( Y _ s ^ y ) \\big ] { R \\ , } \\Big \\| ^ 2 _ { H S } \\ , \\dd s = \\int _ 0 ^ t \\big | Z _ s ^ z - Y _ s ^ y \\big | ^ 2 \\ , \\dd A _ s \\end{align*}"}
-{"id": "3217.png", "formula": "\\begin{align*} \\varepsilon ^ { - 1 } _ n + \\varepsilon ^ { - 1 } _ m = \\frac { 1 } { K } \\left ( d ( \\mathbb { I } _ Y ^ { ( 1 ) } , N _ { \\alpha ^ { ( n ) } } ) + d ( \\mathbb { I } _ Y ^ { ( 1 ) } , N _ { \\alpha ^ { ( m ) } } ) \\right ) \\geq \\frac { 1 } { K } d ( N _ { \\alpha ^ { ( n ) } } , N _ { \\alpha ^ { ( m ) } } ) = \\frac { 1 } { K } d ( \\mathbb { I } _ Y ^ { ( 1 ) } , N _ { \\alpha ^ { ( n ) } - \\alpha ^ { ( m ) } } ) , \\end{align*}"}
-{"id": "8850.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\lambda _ i } { k ^ { \\frac { 2 } { n } } } = \\frac { n } { n + 2 } \\frac { 4 \\pi ^ 2 } { ( \\omega _ n \\mathrm { v o l } \\ , \\Omega ) ^ \\frac { 2 } { n } } \\end{align*}"}
-{"id": "6240.png", "formula": "\\begin{align*} Z \\ X ^ { ( 0 ) } ( s _ { t _ { 0 } } ) = - \\eta _ { 0 t _ { 0 } } Z \\ X ^ { ( j ) } ( s _ { t _ { j } } ) = - \\eta _ { j t _ { j } } \\ j = 1 , \\ \\ldots , \\ J - 1 , \\end{align*}"}
-{"id": "6458.png", "formula": "\\begin{align*} \\geq & \\zeta _ { 1 } ( q ) \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } \\left [ \\psi ( x ) w ( s , x ) - \\psi ( y ) w ( s , y ) \\right ] ^ { 2 } k ( x , y ) d x d y \\\\ & - \\zeta _ { 2 } ( q ) \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } ( w ^ { 2 } ( s , x ) + w ^ { 2 } ( s , y ) ) k ( x , y ) d x d y , \\end{align*}"}
-{"id": "4973.png", "formula": "\\begin{align*} \\vec { E } = \\left ( \\begin{array} { c } E _ { 1 } \\\\ E _ { 2 } \\\\ \\vdots \\\\ E _ { r } \\end{array} \\right ) = f ^ { * } \\vec { D } - { } ^ { \\rm t } A \\vec { D } . \\end{align*}"}
-{"id": "5894.png", "formula": "\\begin{align*} g ^ { \\beta \\gamma } \\xi _ { , \\beta \\gamma } ^ { \\alpha } + \\xi ^ { \\gamma } \\Gamma _ { , \\gamma } ^ { \\alpha } - \\xi _ { , \\gamma } ^ { \\alpha } \\Gamma ^ { \\gamma } + \\left ( a - \\lambda \\right ) \\Gamma ^ { \\alpha } = g ^ { \\beta \\gamma } \\left ( L _ { \\xi ^ { \\alpha } } \\Gamma _ { \\beta \\gamma } ^ { \\alpha } \\right ) \\end{align*}"}
-{"id": "8943.png", "formula": "\\begin{align*} \\int _ a ^ b h ( s ) ^ p d s = & b h ( b ) ^ p - a h ( a ) ^ p + p \\int _ a ^ b h ( s ) ^ { p - 1 } \\frac { s ^ 2 f ^ { * * } ( s ) } { [ n \\omega _ n ( \\sinh F ( s ) ) ^ { n - 1 } ] ^ 2 } d s \\\\ \\leqslant & p \\Big ( \\int _ a ^ b h ( s ) ^ p d s \\Big ) ^ { ( p - 1 ) / p } \\Big ( \\int _ a ^ b \\Big [ \\frac { s ^ 2 f ^ { * * } ( s ) } { [ n \\omega _ n ( \\sinh F ( s ) ) ^ { n - 1 } ] ^ 2 } \\Big ] ^ p d s \\Big ) ^ { 1 / p } \\\\ & + b h ( b ) ^ p - a h ( a ) ^ p . \\end{align*}"}
-{"id": "5236.png", "formula": "\\begin{align*} \\sum _ { \\ell = 1 } ^ k c _ { \\ell } = n \\mbox { a n d } \\sum _ { \\ell = 1 } ^ k s _ { \\ell } = M . \\end{align*}"}
-{"id": "1197.png", "formula": "\\begin{align*} a _ i ( n ) & = | \\mathcal { A } _ i ( n ) | , \\\\ a _ { i p } ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) ~ | ~ w = \\overline { w } \\} | , \\\\ a _ { i n } ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) ~ | ~ w \\neq \\overline { w } \\} | . \\end{align*}"}
-{"id": "7485.png", "formula": "\\begin{align*} Q ^ { ( k ) } = \\sum _ { j = 0 } ^ { \\kappa ' k } { j \\choose 2 } Q _ j ^ { ( k ) } , \\end{align*}"}
-{"id": "8741.png", "formula": "\\begin{gather*} \\widetilde Q _ t = \\int _ { 0 } ^ { t } e ^ { \\left ( t - s \\right ) A } \\left ( \\begin{array} [ c ] { c c } 0 & 0 \\\\ 0 & I _ U \\end{array} \\right ) e ^ { \\left ( t - s \\right ) A ^ * } d s \\end{gather*}"}
-{"id": "8223.png", "formula": "\\begin{align*} U _ n = \\epsilon ^ { | n | } W _ { | n | } , \\pm n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "4213.png", "formula": "\\begin{align*} \\varphi ( y ) & = \\ell ( y ; y ) \\geq \\ell ( p ( y ) ; y ) \\\\ & = g ( p ( y ) ) + f ( e ( y ) + \\nabla e ( y ) ( p ( y ) - y ) ) + \\frac { L } { 2 } \\| p ( y ) - y \\| ^ 2 \\\\ & \\geq ( g ( y ) + \\langle z , p ( y ) - y \\rangle ) + ( f ( e ( y ) ) + \\langle w , \\nabla e ( y ) ( p ( y ) - y ) \\rangle ) + \\frac { L } { 2 } \\| p ( y ) - y \\| ^ 2 \\\\ & = \\varphi ( y ) + \\langle z + \\nabla e ( y ) ^ T w , p ( y ) - y \\rangle + \\frac { L } { 2 } \\| p ( y ) - y \\| ^ 2 , \\end{align*}"}
-{"id": "3756.png", "formula": "\\begin{align*} \\varphi ( t ) : = E e ^ { i t Y } = P ( T _ \\beta = 0 ) e ^ { - i t \\alpha u } + \\int _ { - \\infty } ^ \\infty e ^ { i t y } f _ Y ( y ) d y . \\end{align*}"}
-{"id": "7227.png", "formula": "\\begin{align*} P = \\left \\{ \\begin{array} { c c } p _ { G } & \\mathrm { o n } \\ , \\ , \\ , \\tilde { W } \\\\ p _ { O } & \\mathrm { o n } \\ , \\ , \\ , \\tilde { O } \\setminus \\tilde { V } , \\end{array} \\right . \\end{align*}"}
-{"id": "9109.png", "formula": "\\begin{align*} \\begin{array} { l c l } \\nu ( \\sup \\{ a _ 1 , a _ 2 , \\ldots , a _ n \\} ) & = & \\nu ( \\sup \\{ a _ 1 , \\sup \\{ a _ 2 , \\ldots , a _ n \\} \\} ) \\\\ & = & \\sup \\{ \\nu ( a _ 1 ) , \\nu ( \\sup \\{ a _ 2 , \\ldots , a _ n \\} ) \\} \\\\ & = & \\sup \\{ \\nu ( a _ 1 ) , \\sup \\{ \\nu ( a _ 2 ) , \\ldots , \\nu ( a _ n ) \\} \\} \\\\ & = & \\sup \\{ \\nu ( a _ 1 ) , \\nu ( a _ 2 ) , \\ldots , \\nu ( a _ n ) \\} . \\\\ \\end{array} \\end{align*}"}
-{"id": "1918.png", "formula": "\\begin{align*} m _ 1 : = \\min \\{ u > 0 : | { \\phi } ( u ) | = ( n K ) ^ { - \\frac { 1 } { 2 } } \\} . \\end{align*}"}
-{"id": "6227.png", "formula": "\\begin{align*} \\mathcal { K } _ { n e w } = \\mathcal { K } _ { o l d } ( I + ( s ^ { 2 } _ { n e w } - s ^ { 2 } _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } M + ( s _ { n e w } - s _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } D ) . \\end{align*}"}
-{"id": "2997.png", "formula": "\\begin{align*} \\int _ X u ( t , x ) d P = \\int _ { D ( t , p ) } u ( t , x ) d P + \\int _ { X \\setminus D ( t , p ) } u ( t , x ) d P < \\max _ { y \\in B ( t , p ) } u ( t , y ) \\end{align*}"}
-{"id": "9106.png", "formula": "\\begin{align*} \\nu ( \\sup \\{ a _ 1 , \\ldots , a _ n \\} ) = \\sup \\{ \\nu ( a _ 1 ) , \\ldots , \\nu ( a _ n ) \\} . \\end{align*}"}
-{"id": "5129.png", "formula": "\\begin{align*} Z _ { \\ell } ^ { J } ( \\vec { z } ) = \\prod _ { \\begin{subarray} { c } i \\in J \\\\ i < \\ell \\end{subarray} } f ( z _ { \\ell } , z _ { i } ) \\prod _ { \\begin{subarray} { c } i \\in J \\\\ i > \\ell \\end{subarray} } f ( z _ { i } , z _ { \\ell } ) \\prod _ { \\begin{subarray} { c } i \\in J \\\\ i < \\ell \\end{subarray} } ^ { \\curvearrowleft } Y _ { i - 1 } ( z _ { \\ell } , z _ { i } ) . \\end{align*}"}
-{"id": "7389.png", "formula": "\\begin{align*} P & \\Big ( \\Big \\lvert \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\phi _ b ( b ^ 0 _ i , . . . , b ^ { t } _ i , w _ { i } ) - \\mathbb { E } \\phi _ b \\Big \\lvert \\geq \\epsilon \\Big ) \\\\ & \\leq K t ^ 3 K _ { t - 1 } e ^ { - { \\kappa \\kappa _ { t - 1 } n \\epsilon ^ 2 } / { t ^ 7 } } . \\end{align*}"}
-{"id": "6053.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l } \\Delta u _ { i } ^ { 2 k + 3 } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { 2 k + 3 } \\sum \\limits _ { j \\neq i } H ( u _ { j } ^ { 2 k + 2 } ) ( x ) & \\Omega , \\\\ \\Delta u _ { i } ^ { 2 k + 1 } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { 2 k + 1 } \\sum \\limits _ { j \\neq i } H ( u _ { j } ^ { 2 k } ) ( x ) & \\Omega . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "1575.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma \\left ( S \\right ) ^ { 2 } S ^ { 2 } F _ { , S S } + \\kappa \\left ( S \\right ) \\left ( \\mu \\left ( S \\right ) - \\lambda \\left ( S \\right ) - \\log S \\right ) S F _ { , S } - F _ { , t } = 0 . \\end{align*}"}
-{"id": "9471.png", "formula": "\\begin{align*} \\eta ( t ) = \\begin{cases} 0 & s < - \\delta \\\\ 1 & s > - \\delta / 2 \\end{cases} \\end{align*}"}
-{"id": "2870.png", "formula": "\\begin{align*} \\overline { N ^ * } \\mathcal { B } _ { E _ 1 } ^ { \\mathbb { K } } ( K ^ { \\bullet } ( C ) ) = \\overline { N ^ * } K ^ { \\bullet } ( \\mathcal { B } _ { E _ 1 } ^ { \\mathbb { K } } ( C ) ) . \\end{align*}"}
-{"id": "4563.png", "formula": "\\begin{align*} W _ 2 ' = O \\biggl ( \\frac { k ^ { 1 / 2 } } { n } \\max \\biggl \\{ \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "1949.png", "formula": "\\begin{align*} & b _ { 1 1 } ( 0 ) + \\left ( 2 p _ 1 - q _ 1 ^ 2 \\frac { a ( 0 ) } { b _ { 1 1 } ( 0 ) } \\right ) \\tau \\leq b _ { 1 1 } ( \\tau ) \\leq b _ { 1 1 } ( 0 ) + \\frac { 2 n _ 1 } { n _ 1 + 1 } p _ 1 \\tau , \\tau \\leq 0 , \\\\ & b _ { 1 2 } ( 0 ) + 2 p _ 2 \\tau \\leq b _ { 1 2 } ( \\tau ) \\leq b _ { 1 2 } ( 0 ) + \\left ( 2 p _ 2 - q _ 2 ^ 2 \\frac { a ( 0 ) } { b _ { 1 2 } ( 0 ) } \\right ) \\tau , \\tau \\leq 0 . \\end{align*}"}
-{"id": "4868.png", "formula": "\\begin{align*} f ( y , x ) - f ( x , x ) & = \\frac { z } { n } \\partial _ x f ( x , x ) + \\frac { z ^ 2 } { 2 n ^ 2 } \\partial _ x ^ 2 f ( x , x ) + \\frac { z ^ 3 } { 6 n ^ { 3 } } \\partial _ x ^ 3 f ( x , x ) + o ( z ^ 3 n ^ { - 3 } ) , \\\\ f ( x , y ) - f ( x , x ) & = \\frac { z } { n } \\partial _ y f ( x , x ) + \\frac { z ^ 2 } { 2 n ^ 2 } \\partial _ y ^ 2 f ( x , x ) + \\frac { z ^ 3 } { 6 n ^ { 3 } } \\partial _ y ^ 3 f ( x , x ) + o ( z ^ 3 n ^ { - 3 } ) . \\end{align*}"}
-{"id": "8895.png", "formula": "\\begin{align*} L v = ( - \\Delta _ A + 1 ) ^ { - 1 } W [ v ] . \\end{align*}"}
-{"id": "7924.png", "formula": "\\begin{align*} 2 \\binom { 3 m + 1 } { 3 } = 3 \\sum _ { i = 1 } ^ { m } x _ { i } ^ { 2 } + 2 \\sum _ { i = 1 } ^ { m } y _ { i } ^ { 2 } - 4 h \\sum _ { i = 1 } ^ { m } y _ { i } + 2 m h ^ { 2 } . \\end{align*}"}
-{"id": "8839.png", "formula": "\\begin{align*} \\left ( \\theta ( \\tau ) \\theta ( 4 \\tau ) \\right ) ^ { k } = F _ { k , 4 } ( \\tau ) + \\left ( \\theta ( \\tau ) \\theta ( 4 \\tau ) \\right ) ^ { k } \\sum _ { j = 1 } ^ { \\ell _ { 4 } } c _ { j , k , 4 } x _ { 4 } ^ { j } . \\end{align*}"}
-{"id": "2184.png", "formula": "\\begin{align*} \\geq & \\zeta _ { 1 } ( q ) \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } \\left [ \\psi ( x ) w ( s , x ) - \\psi ( y ) w ( s , y ) \\right ] ^ { 2 } k ( x , y ) d x d y \\\\ & - \\zeta _ { 2 } ( q ) \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } ( w ^ { 2 } ( s , x ) + w ^ { 2 } ( s , y ) ) k ( x , y ) d x d y , \\end{align*}"}
-{"id": "308.png", "formula": "\\begin{align*} \\int _ { A _ t ^ c } f | \\log f | & \\leq ( 8 t ) ^ { 2 ( 1 - \\epsilon ) } \\int _ { \\mathcal { X } } f | g | ^ { 2 ( 1 - \\epsilon ) } | \\log f | \\\\ & \\leq ( 8 t ) ^ { 2 ( 1 - \\epsilon ) } \\Bigl \\{ \\int _ { \\mathcal { X } } g ^ 2 f \\Bigr \\} ^ { 1 - \\epsilon } \\Bigl \\{ \\int _ { \\mathcal { X } } f | \\log f | ^ { 1 / \\epsilon } \\Bigr \\} ^ \\epsilon = o ( t ) \\end{align*}"}
-{"id": "4489.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 | T _ i | = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { - \\frac { 1 } { 2 } + \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { 1 } { 2 } + \\frac { \\beta } { d } } } { n ^ { 1 + \\frac { \\beta } { d } } } \\log ^ { 2 + \\beta / d } n \\biggr \\} \\biggr ) \\end{align*}"}
-{"id": "5475.png", "formula": "\\begin{align*} \\mathop { \\min } \\limits _ { x _ { 3 } } F _ { 3 } \\left ( x \\right ) = \\mathop { \\min } \\limits _ { x _ { 3 } } \\left ( f _ { 3 1 } \\left ( x \\right ) , f _ { 3 2 } \\left ( x \\right ) , . . . , f _ { 3 p _ { 3 } } \\left ( x \\right ) \\right ) , \\end{align*}"}
-{"id": "455.png", "formula": "\\begin{align*} | \\sum _ { i = 1 } ^ { n + 1 } \\varepsilon _ i ( w ^ i ) ^ { \\top } z - \\sum _ { i = 1 } ^ { n + 1 } \\varepsilon _ i w _ 0 ^ i | < w _ 0 - w ^ { \\top } z . \\end{align*}"}
-{"id": "5860.png", "formula": "\\begin{align*} \\sigma ^ { 2 } \\left ( x \\right ) = 2 \\kappa \\left ( \\left ( \\mu - \\lambda \\right ) - x \\right ) + \\kappa + c _ { 1 } e ^ { - 2 x } , \\end{align*}"}
-{"id": "5843.png", "formula": "\\begin{align*} C _ { x } \\left ( x \\right ) = - \\frac { 2 \\left ( \\kappa \\left ( \\mu - \\lambda - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } - \\frac { 1 } { 2 } \\sigma \\sigma _ { , x } \\right ) } { \\sigma ^ { 2 } \\left ( x \\right ) } . \\end{align*}"}
-{"id": "463.png", "formula": "\\begin{align*} \\sup \\{ \\sum _ { i = 1 } ^ m ( b ^ i ) ^ { \\top } y ^ i \\ , | \\ \\sum _ { i = 1 } ^ m ( y ^ i ) ^ { \\top } A ^ i = \\pi ^ { \\top } , \\ y ^ i \\in \\mathcal { L } ^ * _ { m _ i } \\ \\forall i \\in [ m ] \\} \\end{align*}"}
-{"id": "6729.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( B ^ { H } _ { t } ) \\diamond d B ^ { H } ( t ) : = \\lim _ { n \\rightarrow 0 } \\sum _ { i = 1 } ^ { n } F ( B ^ { H } _ { t _ { i - 1 } } ) \\diamond ( B ^ { H } ( t _ { i } ) - B ^ { H } ( t _ { i - 1 } ) ) \\end{align*}"}
-{"id": "9099.png", "formula": "\\begin{align*} ( s ^ { 2 } - 1 / 4 ) Y '' + \\left ( 4 s + \\frac { \\alpha - 1 } { 2 } + N \\right ) Y ' + \\frac { 9 - \\alpha ^ { 2 } } { 4 } Y + \\frac { 1 } { 4 d _ { N } } ( 3 s ^ { 2 } M _ { N - 1 } ^ { ( 2 ) \\prime \\prime } + ( 1 1 s - 1 ) M _ { N - 1 } ^ { ( 2 ) \\prime } + 5 M _ { N - 1 } ^ { ( 2 ) } ) = 0 \\ . \\end{align*}"}
-{"id": "3426.png", "formula": "\\begin{align*} d \\eta ( t ) = c ( t , \\xi ( t ) ) \\eta ( t ) d t + C ( t , \\xi ( t ) ) ( \\eta ( t ) , d w ( t ) ) , \\eta ( s ) = h \\in R ^ { d _ 1 } . \\end{align*}"}
-{"id": "3936.png", "formula": "\\begin{align*} { G R S } _ { n , k } ( \\gamma , w ) : = \\left \\{ ( \\gamma _ { 1 } f ( w _ { 1 } ) , \\gamma _ 2 f ( w _ 2 ) , \\ldots , \\gamma _ { n } f ( v _ { n } ) ) \\mid f ( X ) \\in \\mathbb { F } _ \\ell [ X ] _ { k } \\right \\} \\end{align*}"}
-{"id": "1879.png", "formula": "\\begin{align*} \\frac 3 8 I = & 3 M ( z - m ) + 3 z ( M - m ) + [ 2 ( z - m ) ^ 2 - 2 ( z - m ) ( M - z ) - ( M - z ) ^ 2 ] \\\\ = & 3 M ( 1 - M - 2 m ) + 3 ( 1 - M - m ) ( M - m ) + 2 ( 1 - M - 2 m ) ^ 2 \\\\ & - 2 ( 1 - M - 2 m ) ( 2 M + m - 1 ) - ( 2 M + m - 1 ) ^ 2 \\\\ = & - 4 M ^ 2 + 3 + 8 m M - 1 5 m + 1 4 m ^ 2 \\\\ = & - 4 M ^ 2 + 3 + ( - m ) ( 1 5 - 8 M ) + 1 4 m ^ 2 \\\\ > & 7 \\epsilon , \\end{align*}"}
-{"id": "3912.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = Q ( N _ { f } ( u ) ) & & \\\\ u ( T ) = u ' ( 0 ) = u ' ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "3682.png", "formula": "\\begin{align*} \\frac { \\partial \\mu } { \\partial t } d x \\wedge d y = d H _ { \\mu } \\wedge d Z _ { \\zeta } = J ( q , \\gamma ) d x \\wedge d y ~ , \\end{align*}"}
-{"id": "9847.png", "formula": "\\begin{align*} y ( t ) & = h _ { \\rm S D } ( t ) \\otimes s ( t ) + h _ { \\rm R D } ( t ) \\otimes x ( t ) , \\\\ r ( t ) & = h _ { \\rm S R } ( t ) \\otimes s ( t ) + \\alpha x ( t - \\tau ) , \\\\ x ( t ) & = \\theta ( t ) \\otimes r ( t ) . \\end{align*}"}
-{"id": "1447.png", "formula": "\\begin{align*} b _ Q : = \\frac { ( M \\chi _ Q ) ^ { \\alpha / n + \\varepsilon } } { \\ell ( Q ) ^ \\alpha } , \\end{align*}"}
-{"id": "7257.png", "formula": "\\begin{align*} F _ { j , \\beta } ( z _ j ) - \\sum _ { | \\alpha | = | \\beta | } T _ { j k } F _ { k , \\alpha } ( z _ k ) \\cdot \\tau _ { k j , \\beta } ^ \\alpha = h _ { 1 , j k , \\beta } ( z _ j ) - h _ { 2 , j k , \\beta } ( z _ j ) \\end{align*}"}
-{"id": "4127.png", "formula": "\\begin{align*} ( y ^ 2 + b z ) ^ q - z ^ { p + 2 q } = 0 \\end{align*}"}
-{"id": "894.png", "formula": "\\begin{align*} \\chi ( s ) = \\left ( { t } / { 2 \\pi } \\right ) ^ { { 1 } / { 2 } - ( \\sigma + i t ) } e ^ { i ( t + { \\pi } / { 4 } ) } \\left ( 1 + O ( t ^ { - 1 } ) \\right ) , \\end{align*}"}
-{"id": "7550.png", "formula": "\\begin{align*} x _ { n + 1 } = \\frac { A x _ n ^ 2 } { B + x _ n } e ^ { r ( 1 - x _ n ) } , \\end{align*}"}
-{"id": "1877.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } ^ 2 + 2 R _ { 1 3 4 2 } R _ { 1 4 2 3 } = & 3 x ^ 2 - 6 x y - 2 ( x - y ) ^ 2 \\\\ \\ge & - 2 ( K _ { 1 3 } - K _ { 1 2 } ) ^ 2 . \\\\ \\end{align*}"}
-{"id": "9273.png", "formula": "\\begin{align*} & \\int _ D \\frac { \\partial U } { \\partial y } ( x , Y ( T , x , z ) , z ) Y ( T , x , z ) d x \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] = \\tilde { p } ( 0 , z ) \\exp ( \\int _ 0 ^ T ( b _ 0 ( t , z ) \\pi ( s , z ) - \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) d B ( s ) \\\\ & - \\frac { 1 } { 2 } \\int _ 0 ^ T ( b _ 0 ( s , z ) \\pi ( s , z ) - \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) ^ { 2 } d s ) . \\end{align*}"}
-{"id": "4350.png", "formula": "\\begin{align*} \\xi _ 2 ' ( y ) + \\frac { p } { p - 1 } \\xi _ 2 ( y ) ^ { ( p - 1 ) / p } & \\ge \\psi _ 1 ' ( y ) + M + \\frac { p } { p - 1 } \\psi _ 1 ( y ) ^ { ( p - 1 ) / p } \\\\ & \\ge M ( 1 - y ) + \\frac { p a _ 1 ^ { 2 - p } } { p - 1 } ( 1 - y ) = \\frac { p a _ 2 ^ { 2 - p } } { p - 1 } ( 1 - y ) \\\\ & = \\psi _ 2 ' ( y ) + \\frac { p } { p - 1 } \\psi _ 2 ( y ) ^ { ( p - 1 ) / p } \\ . \\end{align*}"}
-{"id": "7667.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } s _ { k } ^ { - p } f ( s _ { k } ) = \\infty . \\end{align*}"}
-{"id": "4538.png", "formula": "\\begin{align*} \\mathrm { B } _ { a , b , c } ( s , t ) : = \\frac { s ^ { a - 1 } t ^ { b - 1 } ( 1 - s - t ) ^ { c - 1 } } { \\mathrm { B } _ { a , b , c } } \\end{align*}"}
-{"id": "2167.png", "formula": "\\begin{align*} = \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } \\psi ( x ) \\psi ( y ) \\left ( \\left ( \\frac { \\tilde { u } ( s , x ) } { \\psi ( x ) } \\right ) ^ { \\frac { 1 - q } { 2 } } - \\left ( \\frac { \\tilde { u } ( s , y ) } { \\psi ( y ) } \\right ) ^ { \\frac { 1 - q } { 2 } } \\right ) ^ { 2 } k ( x , y ) d x d y , \\end{align*}"}
-{"id": "2137.png", "formula": "\\begin{align*} L u ( t , x ) = \\int _ { \\mathbb { R } ^ { n } } [ u ( t , x ) - u ( t , y ) ] k ( x , y ) d y . \\end{align*}"}
-{"id": "3931.png", "formula": "\\begin{align*} z ^ { ( i + 1 ) } = z ^ { ( i ) } - \\frac { \\mathcal { W } _ \\sigma ( z ^ { ( i ) } ) } { \\mathcal { W } _ \\sigma ^ { \\prime } ( z ^ { ( i ) } ) } \\end{align*}"}
-{"id": "6200.png", "formula": "\\begin{align*} I ( u ) & = \\frac { 1 } { \\gamma \\alpha } ( \\log | x | ) ^ { 1 - \\alpha } \\left [ 1 - \\exp \\left \\{ - \\gamma ( \\log | x | ) ^ \\alpha \\left [ 1 - \\left ( 1 + \\frac { \\log u } { \\log | x | } \\right ) ^ \\alpha \\right ] \\right \\} \\right ] \\\\ & \\leq \\frac { 1 } { \\gamma \\alpha } ( \\log | x | ) ^ { 1 - \\alpha } \\left [ 1 - \\exp \\left \\{ 2 \\gamma \\alpha ( \\log | x | ) ^ { \\alpha - 1 } \\log u \\right \\} \\right ] \\\\ & \\leq - 2 \\log u , \\end{align*}"}
-{"id": "433.png", "formula": "\\begin{align*} Y _ j ^ { ( 1 , i ) } & = \\begin{cases} 0 & \\mbox { i f } j = 0 \\\\ [ X _ j , X _ i ] & \\mbox { i f } 1 \\leq j \\leq m \\end{cases} \\qquad \\mbox { a n d } \\\\ \\tilde Y _ j ^ { ( 1 , i ) } & = \\begin{cases} [ X _ 0 , X _ i ] + \\frac { 1 } { 2 } \\sum _ { l = 1 } ^ m [ X _ l , [ X _ l , X _ i ] ] & \\mbox { i f } j = 0 \\\\ 0 & \\mbox { o t h e r w i s e } \\end{cases} \\ ; . \\end{align*}"}
-{"id": "8849.png", "formula": "\\begin{align*} \\frac { F _ { k + l } } { ( k + l ) ^ { \\frac { 4 } { n } } } = \\Big ( 1 + \\frac { 2 } { n } \\Big ) \\left ( \\frac { \\frac { 1 } { k + l } \\sum _ { i = 1 } ^ { k + l } \\upsilon _ i } { ( k + l ) ^ { \\frac { 2 } { n } } } \\right ) ^ 2 - \\frac { \\frac { 1 } { k + l } \\sum _ { i = 1 } ^ { k + l } \\upsilon _ i ^ 2 } { ( k + l ) ^ { \\frac { 4 } { n } } } . \\end{align*}"}
-{"id": "7057.png", "formula": "\\begin{align*} { { \\bf { Y } } ^ { [ 1 ] } } ( n ) - { { \\bf { Y } } ^ { [ 1 ] } } ( { t _ 2 } ) = { { \\bf { H } } ^ { [ 1 1 ] } } ( n ) { \\bf { V } } _ 1 ^ { [ 1 ] } ( n ) { { \\bf { u } } ^ { [ 1 ] } } + { { \\bf { H } } ^ { [ 1 2 ] } } ( n ) { \\bf { V } } _ 1 ^ { [ 2 ] } ( n ) { { \\bf { u } } ^ { [ 2 ] } } . \\end{align*}"}
-{"id": "8533.png", "formula": "\\begin{align*} X _ { b ^ { \\ast } } ( P ) = e _ { k } X _ { b ^ { \\ast } } ( P ) = \\displaystyle \\sum _ { r \\in L ( k ) , a \\in _ { k } T } e _ { k } r a Y _ { [ b r a ] } ( P ) \\end{align*}"}
-{"id": "1288.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & d + D & b + D \\\\ 0 & 0 & c + D & a + D \\\\ a & b & 0 & 0 \\\\ c & d & 0 & 0 \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "3887.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { 2 k + 1 } U _ i = \\tilde U _ 1 + \\sum _ { i = 1 } ^ k U _ { 2 i + 1 } + \\sum _ { i = 1 } ^ { 2 k - 2 } W _ i + \\sum _ { i = 1 } ^ { k - 1 } V _ i . \\end{align*}"}
-{"id": "5067.png", "formula": "\\begin{align*} \\nu ( A D ) < \\frac { 1 } { 2 } , \\textrm { w h e r e $ A = \\bigcup _ { n } F _ n s _ n $ } . \\end{align*}"}
-{"id": "6739.png", "formula": "\\begin{align*} \\nu ( \\varphi , x ) = \\sup \\{ \\gamma \\geq 0 \\textup { s . t . } \\varphi ( z ) \\leq \\gamma \\log \\| z - x \\| + O ( 1 ) \\textup { o n } U \\} . \\end{align*}"}
-{"id": "8376.png", "formula": "\\begin{align*} w ^ * u x = w ^ * \\phi ( x ) u = x w ^ * u , \\ \\ \\ x \\in M , \\end{align*}"}
-{"id": "7068.png", "formula": "\\begin{align*} d _ \\Sigma ^ { { X _ L } } ( M , N ; \\frac { 2 } { { { T _ c } } } ) = \\frac { { M N ( N - 1 ) n + { T _ c } ( { T _ c } - 1 ) } } { { { T _ c } n + { T _ c } ( { T _ c } - 1 ) } } , \\end{align*}"}
-{"id": "9276.png", "formula": "\\begin{align*} M ( t , z ) : = \\mathbb { E } [ K ( z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] | \\mathcal { F } _ t ] ; 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "7651.png", "formula": "\\begin{align*} \\lim _ { s \\rightarrow \\infty } \\sup s ^ { - ( 1 + \\alpha q / d ) } f ( s ) = \\infty , \\end{align*}"}
-{"id": "5215.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { g r a p h } ^ 2 : = Q _ { q } ( u , u ) = \\Vert \\bar \\partial _ M u \\Vert ^ 2 _ { L ^ 2 _ { ( 0 , q + 1 ) } ( M ) } + \\Vert \\bar \\partial ^ * _ M u \\Vert ^ 2 _ { L ^ 2 _ { ( 0 , q - 1 ) } ( M ) } \\ ; . \\end{align*}"}
-{"id": "3070.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\omega _ { n , \\beta } = ( 1 - \\pi _ \\beta ) \\delta _ 0 + \\pi _ \\beta \\delta _ 1 \\end{align*}"}
-{"id": "8385.png", "formula": "\\begin{align*} \\langle x , y \\rangle = \\phi ( y ^ * x ) , \\ \\ \\ x , y \\in A \\rtimes _ \\beta G . \\end{align*}"}
-{"id": "2790.png", "formula": "\\begin{align*} \\tau ( G ) = O ( 1 ) \\biggl ( \\frac { ( d - 1 ) ^ { d - 1 } } { ( d ^ 2 - 2 d ) ^ { d / 2 - 1 } } \\biggr ) ^ { \\ ! n } \\frac { \\log n } { n d \\log d } \\end{align*}"}
-{"id": "5311.png", "formula": "\\begin{align*} T ( u ) & = \\ , L ^ { A } ( u + \\omega ) L ^ { B } ( u - \\omega ) \\\\ & = \\ , \\left ( \\begin{array} { c c } A ( u ) & B ( u ) \\\\ C ( u ) & D ( u ) \\end{array} \\right ) . \\end{align*}"}
-{"id": "4034.png", "formula": "\\begin{align*} \\phi _ { \\mu , k } ^ { - } ( x ) = p r o j _ { D _ { k } } ( f ^ { - 1 } x ) . \\end{align*}"}
-{"id": "7640.png", "formula": "\\begin{align*} u _ { t } = A u + f ( u ) , \\ u ( 0 ) = u _ { 0 } , \\end{align*}"}
-{"id": "9806.png", "formula": "\\begin{align*} H _ { 0 } : p = p _ { \\theta } , \\ ; \\theta \\in \\Theta . \\end{align*}"}
-{"id": "1267.png", "formula": "\\begin{align*} \\delta _ { S ' } = \\frac { 1 } { 2 } ( T ^ 0 - \\delta _ 0 + \\alpha T ^ r + \\overline \\alpha T ^ { - r } ) , \\end{align*}"}
-{"id": "1714.png", "formula": "\\begin{gather*} \\Phi : = - e ^ { 1 4 7 } + \\sqrt { 2 } e ^ { 1 5 6 } + \\sqrt { 2 } e ^ { 2 3 7 } + e ^ { 2 4 5 } + e ^ { 3 4 6 } , \\end{gather*}"}
-{"id": "1612.png", "formula": "\\begin{align*} Z ^ { 4 } = e ^ { - 2 m t } \\left ( \\partial _ { t } - H - c K ^ { 1 } + \\left ( 2 m ^ { 2 } \\left ( \\int \\frac { d x } { \\sigma \\left ( x \\right ) } \\right ) ^ { 2 } + 4 m c \\left ( \\int \\frac { d x } { \\sigma \\left ( x \\right ) } \\right ) + 2 c ^ { 2 } - m \\right ) F \\partial _ { F } \\right ) \\end{align*}"}
-{"id": "9013.png", "formula": "\\begin{align*} f ( x _ 1 , \\ldots , x _ n ) = \\sum _ { u } f ^ { \\mathbf u } ( x _ n ) \\prod _ { i = 1 } ^ { n - 1 } ( x _ i - \\alpha _ i ) ^ { u _ i } \\end{align*}"}
-{"id": "4505.png", "formula": "\\begin{align*} A _ { d , \\theta } : = \\max \\biggl \\{ \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } | H ( f ) | \\ , , \\ , - \\frac { 1 } { 2 } \\log \\inf _ { f \\in \\mathcal { F } _ { d , \\theta } } \\| f \\| _ \\infty \\ , , \\ , 1 \\biggr \\} \\end{align*}"}
-{"id": "1419.png", "formula": "\\begin{align*} [ e _ { \\gamma } , [ u , v ] ] = [ [ e _ { \\gamma } , u ] , v ] - ( - 1 ) ^ { p ( u ) p ( v ) } [ [ e _ { \\gamma } , v ] , u ] . \\end{align*}"}
-{"id": "7495.png", "formula": "\\begin{align*} \\{ R = n / 2 \\} \\subseteq \\{ Q = 0 \\} \\end{align*}"}
-{"id": "5275.png", "formula": "\\begin{align*} \\chi ( s ) = \\left ( { t } / { 2 \\pi } \\right ) ^ { { 1 } / { 2 } - ( \\sigma + i t ) } e ^ { i ( t + { \\pi } / { 4 } ) } \\left ( 1 + O ( t ^ { - 1 } ) \\right ) , \\end{align*}"}
-{"id": "8482.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\star } + \\sum _ { j = 1 } ^ { d } \\textbf { P } ( A _ { j } ( \\tilde { \\textbf { u } } ^ { \\star } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } ^ { \\star } ) = 0 . \\end{align*}"}
-{"id": "2772.png", "formula": "\\begin{align*} f _ j = c _ j x _ 1 ^ { s _ { 1 j } } x _ 2 ^ { s _ { 2 j } } \\prod _ { \\lambda \\in k ^ \\times } ( x _ 1 ^ { w ' _ 2 } + \\lambda x _ 2 ^ { w ' _ 1 } ) ^ { t _ { \\lambda j } } + h _ j \\end{align*}"}
-{"id": "3487.png", "formula": "\\begin{align*} u ( x ' , x _ n ) = u ( x ' , 1 - x _ n ) . \\end{align*}"}
-{"id": "8037.png", "formula": "\\begin{align*} N _ { k , \\ell } = \\sum _ { d = 1 } ^ { \\ell - k } \\binom { \\ell - 1 - k } { d - 1 } 2 ^ d = 2 \\sum _ { d = 0 } ^ { \\ell - 1 - k } \\binom { \\ell - 1 - k } { d } 2 ^ d = 2 ( 1 + 2 ) ^ { \\ell - 1 - k } . \\end{align*}"}
-{"id": "6805.png", "formula": "\\begin{align*} C _ \\pm = \\pm \\frac 1 2 \\mathbf { 1 } - \\frac 1 2 H \\end{align*}"}
-{"id": "5639.png", "formula": "\\begin{align*} E ( \\mu ) = \\left \\{ \\begin{array} { l l } E _ { G } ( \\mu ) + \\Vert f \\Vert _ { H ^ { k } ( \\Omega ) } ^ { 2 } = \\int _ { \\Omega } G ( f ( x ) ) d x + \\Vert f \\Vert _ { H ^ { k } ( \\Omega ) } ^ { 2 } , & \\mbox { i f } \\ f = \\frac { d \\mu } { d x } \\ \\mbox { a n d } \\ f \\geq \\alpha , \\\\ + \\infty & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "5189.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 2 / 3 } } & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { S _ { 4 / 3 } } \\| V \\| _ { S _ { 4 / 3 } } \\\\ & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\left ( \\frac { \\| U \\| ^ { 4 / 3 } _ { S _ { 4 / 3 } } \\ ! + \\| V \\| ^ { 4 / 3 } _ { S _ { 4 / 3 } } } { 2 } \\right ) ^ { 3 / 2 } . \\end{align*}"}
-{"id": "7564.png", "formula": "\\begin{align*} \\Delta _ l ( y ) : = \\inf _ { x \\in [ b , y ] } \\{ x - f ( x ) - l \\} . \\end{align*}"}
-{"id": "3759.png", "formula": "\\begin{align*} f _ Y ( y ) = \\lim _ { A \\rightarrow \\infty } \\frac { 1 } { 2 \\pi } \\int _ { - A } ^ { A } e ^ { i y t } \\hat { f _ Y } ( t ) d t \\end{align*}"}
-{"id": "2371.png", "formula": "\\begin{align*} x ^ { k + 1 } & = x ^ k - \\alpha P ^ { - 1 } \\nabla F ( x ^ k ) \\\\ & = x ^ k - \\alpha P ^ { - 1 } P ( S _ 2 S _ 1 x ^ k - x ^ k ) \\\\ & = x ^ k - \\alpha ( S _ 2 S _ 1 x ^ k - x ^ k ) \\\\ & = ( 1 - \\alpha ) x ^ k + \\alpha S _ 2 S _ 1 x ^ k , \\end{align*}"}
-{"id": "17.png", "formula": "\\begin{align*} f _ A ( x ) & = x ^ { \\odot n } \\oplus A _ { i _ 1 i _ 1 } x ^ { \\odot n - 1 } \\oplus \\dots \\oplus A _ { i _ 1 i _ 1 } \\odot \\dots \\odot A _ { i _ n i _ n } \\enspace . \\end{align*}"}
-{"id": "9296.png", "formula": "\\begin{align*} a \\| y ( t , z ) \\| _ { 1 , \\mathbb { R } ^ + } \\leq | y ( t , z ) | _ { 1 , \\mathbb { R } ^ + } = \\| y ' ( t , z ) \\| _ { \\mathbf { L } ^ 2 ( \\mathbb { R } ^ + ) } \\leq b \\| y ( t , z ) \\| _ { 1 , \\mathbb { R } ^ + } \\end{align*}"}
-{"id": "5325.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ N ( N - r + 1 ) \\frac { ( r + n + m - 3 ) ! } { ( n + m - 3 ) ! r ! } = \\frac { ( N + n + m - 1 ) ! } { ( n + m - 1 ) ! N ! } \\end{align*}"}
-{"id": "4362.png", "formula": "\\begin{align*} f ' ( r ) = - a \\psi \\left ( 1 - \\frac { f ( r ) } { a } \\right ) ^ { 1 / p } \\ , r \\in [ 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "7303.png", "formula": "\\begin{align*} D = \\eth + \\eth ^ * = \\sum _ { i } E _ { \\xi _ i } \\otimes \\gamma _ - ( w _ i ) + \\sum _ { i } F _ { \\xi _ i } \\otimes \\gamma _ + ( v _ i ) . \\end{align*}"}
-{"id": "4144.png", "formula": "\\begin{align*} f ( z ) = \\sum _ { n \\geq 1 } A _ f ( n ) n ^ { \\frac { k - 1 } { 2 } } e ^ { 2 \\pi i n z } . \\end{align*}"}
-{"id": "2782.png", "formula": "\\begin{align*} \\mathsf { P } _ { \\mathrm { S I R } , g _ { 2 } } \\left ( \\lambda \\right ) = \\mathbb { E } _ { d _ { 0 } } \\left [ e ^ { - \\frac { 2 \\pi \\lambda \\tau \\left ( 1 + d _ { 0 } ^ { \\alpha } \\right ) } { \\left ( \\alpha - 2 \\right ) d _ { 0 } ^ { \\alpha - 2 } } H y F _ { 1 } \\left ( \\frac { 1 + \\tau \\left ( 1 + d _ { 0 } ^ { \\alpha } \\right ) } { d _ { 0 } ^ { \\alpha } } \\right ) } \\right ] , \\end{align*}"}
-{"id": "2367.png", "formula": "\\begin{align*} F ( x ) : = \\tfrac { 1 } { 2 } \\langle P x , x \\rangle - f _ 2 ( \\nabla f _ 1 ( x ) ) . \\end{align*}"}
-{"id": "7308.png", "formula": "\\begin{align*} T _ { i j } = \\gamma _ { - } ( w _ i ) \\gamma _ { - } ( w _ j ) ^ * + \\sum _ { k , l } b _ { k , l } ^ { i , j } \\gamma _ { - } ( w _ k ) ^ * \\gamma _ { - } ( w _ l ) , b _ { k , l } ^ { i , j } \\in \\mathbb { C } . \\end{align*}"}
-{"id": "8771.png", "formula": "\\begin{align*} K _ { n } ( x _ n ) T ( w ; x _ 1 , . . , x _ n ) = T ( w ; x _ 1 , . . , 1 / x _ n ) K _ { n } ( x _ n ) , \\end{align*}"}
-{"id": "331.png", "formula": "\\begin{align*} X _ i ^ { ( n ) } , Y _ i ^ { ( n ) } , Z _ { i j } , i < j , n = 1 , 2 , \\cdots \\end{align*}"}
-{"id": "8494.png", "formula": "\\begin{align*} \\hat { k } _ 1 ( \\nu ) = \\sum _ { | j | \\leq | W | } \\overline { \\sum _ { w \\in W } w \\nu _ 0 ( j a ) } \\sum _ { w \\in W } w \\nu ( j a ) \\end{align*}"}
-{"id": "8288.png", "formula": "\\begin{align*} I I \\lesssim \\frac { n ^ p } { n ^ { 3 ( p - 1 ) } } + \\frac { n ^ p } { n ^ { 3 ( p - 1 ) / 2 } } \\lesssim \\frac { n ^ { 3 / 2 } } { n ^ { p / 2 } } + o ( 1 ) = o _ n ( n ) . \\end{align*}"}
-{"id": "2383.png", "formula": "\\begin{align*} f ^ { \\gamma } ( x ) = \\gamma ^ { - 1 } F ( x ) = \\gamma ^ { - 1 } \\left ( \\tfrac { 1 } { 2 } \\| x \\| ^ 2 - r _ { \\gamma f } ^ { * } ( x ) \\right ) , \\end{align*}"}
-{"id": "7706.png", "formula": "\\begin{align*} f ( \\gamma _ { 2 n + 1 } ) = f ( \\beta _ { 2 n } ) = \\frac { \\sqrt { \\varepsilon } } { 2 n } < \\frac { \\sqrt { \\varepsilon + 1 } } { 2 n } - \\frac { 1 } { [ 2 ( n + 1 ) ] ^ 2 } = \\beta _ { 2 n + 1 } = \\gamma _ { 2 ( n + 1 ) } . \\end{align*}"}
-{"id": "3130.png", "formula": "\\begin{align*} & \\lambda ^ t P _ d = M _ { 1 2 } ( \\lambda ) ( \\Lambda _ { t - 1 } ( \\lambda ) \\otimes I _ n ) \\mbox { a n d } \\\\ & Q ( \\lambda ) = ( \\Lambda _ { t - 1 } ( \\lambda ) ^ T \\otimes I _ n ) M _ { 2 2 } ( \\lambda ) ( \\Lambda _ { t - 1 } ( \\lambda ) \\otimes I _ n ) , \\end{align*}"}
-{"id": "1960.png", "formula": "\\begin{align*} \\frac { d \\hat { E } } { d \\hat { u } } \\leq c _ 0 \\hat { E } ^ 3 - \\frac { 2 } { m } \\hat { E } ^ 2 = { \\hat E } ^ 2 \\left ( c _ 0 { \\hat E } - \\frac { 2 } { m } \\right ) . \\end{align*}"}
-{"id": "1807.png", "formula": "\\begin{align*} P _ { j } ^ { \\alpha } ( t , x ) & = 2 ^ { j \\alpha / p } \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i x \\cdot \\xi } \\widehat { \\Psi } ( 2 ^ { - j } \\xi ) e ^ { - t | \\xi | ^ \\alpha } d \\xi \\\\ & = 2 ^ { j \\alpha / p } 2 ^ { j d } \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i 2 ^ { j } x \\cdot \\xi } \\widehat { \\Psi } ( \\xi ) e ^ { - t 2 ^ { j \\alpha } | \\xi | ^ \\alpha } d \\xi . \\end{align*}"}
-{"id": "9439.png", "formula": "\\begin{align*} \\rho \\equiv \\rho _ { 0 T } = \\exp \\left ( - \\int _ 0 ^ T h ( X _ t ) d w ^ 2 _ t - \\frac 1 2 \\int _ 0 ^ T | h ( X _ t ) | ^ 2 d t \\right ) . \\end{align*}"}
-{"id": "4862.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k b _ { k + i } ^ k = \\frac { ( - 1 ) ^ k } { k ! } a _ k . \\end{align*}"}
-{"id": "7223.png", "formula": "\\begin{align*} P | _ { \\tilde { B } _ { m _ { l } } \\left ( \\rho \\right ) } = \\mu _ { _ { m _ { l } } } ^ { - 1 } \\circ \\tilde { p } _ { m _ { l } } , \\end{align*}"}
-{"id": "2451.png", "formula": "\\begin{align*} \\left \\| \\Box _ k ^ { \\alpha _ 1 } | ~ M _ 1 \\rightarrow M _ 2 \\right \\| \\gtrsim \\frac { \\| \\Box _ k ^ { \\alpha _ 1 } f _ l ^ { \\alpha _ 2 } \\| _ { M _ 2 } } { \\| f _ l ^ { \\alpha _ 2 } \\| _ { M _ 1 } } \\sim 2 ^ { j \\widetilde { A _ 1 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\end{align*}"}
-{"id": "9840.png", "formula": "\\begin{align*} C = \\max _ { P ( f ) \\ge 0 , \\Theta ( f ) } ~ ~ ~ & \\frac { 1 } { W } \\int _ { f \\in \\Omega } R ( f ) \\mathrm { d } f \\\\ \\mathrm { s . t . } ~ ~ ~ & ( \\ref { e q n : p o w e r : S } ) ~ { \\rm a n d } ~ ( \\ref { e q n : p o w e r : R } ) , \\end{align*}"}
-{"id": "4991.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\infty } \\left \\{ \\left ( \\widetilde { C } + C + c _ { 1 } C \\right ) \\frac { \\sqrt [ ] { h _ { H } ( f ^ { k ( n - 1 ) } ( P ) ) } } { \\rho _ { k } ^ { n } } + \\frac { C \\sqrt [ ] { \\gamma } } { \\rho _ { k } ^ { n } } \\right \\} \\\\ \\leq & \\sum _ { n = 1 } ^ { \\infty } \\left \\{ \\left ( \\widetilde { C } + C + c _ { 1 } C \\right ) \\frac { \\sqrt [ ] { \\widetilde { K } h _ { H } ( P ) } ( n - 1 ) \\rho _ { k } ^ { ( n - 1 ) / 2 } } { \\rho _ { k } ^ { n } } + \\frac { C \\sqrt [ ] { \\gamma } } { \\rho _ { k } ^ { n } } \\right \\} . \\end{align*}"}
-{"id": "139.png", "formula": "\\begin{align*} C _ 1 ' \\leq \\| g _ n \\| _ { \\widehat { E } } \\leq \\| | g _ n | + | g _ m | \\| _ { \\widehat { E } } = \\| g _ n - g _ m \\| _ { \\widehat { E } } \\to 0 , \\end{align*}"}
-{"id": "5253.png", "formula": "\\begin{align*} \\operatorname * { e p i } \\varphi _ { A , k } = \\{ ( y , t ) \\in Y \\times \\mathbb { R } \\mid y \\in t k + A \\} \\end{align*}"}
-{"id": "7367.png", "formula": "\\begin{align*} \\Gamma _ - \\Gamma _ 0 ^ * = t \\Gamma _ 0 ^ * \\Gamma _ - + t ^ \\prime \\Gamma _ + ^ * \\Gamma _ 0 , \\end{align*}"}
-{"id": "3628.png", "formula": "\\begin{align*} u _ { n + 1 } = c T + ( 1 - c ) h ( u _ n ) , T \\ge 0 , \\ ; c \\in [ 0 , 1 ) , \\end{align*}"}
-{"id": "2761.png", "formula": "\\begin{align*} D _ { s } ^ { H } F = \\sum \\limits _ { i = 1 } ^ { n } \\frac { \\partial f } { \\partial x _ { i } } \\left ( \\int _ 0 ^ T \\xi _ { 1 } ( t ) d B ^ { H } ( t ) , . . . , \\int _ 0 ^ T \\xi _ { n } ( t ) d B ^ { H } ( t ) \\right ) \\xi _ { i } ( s ) , \\ \\ s \\in [ 0 , T ] . \\end{align*}"}
-{"id": "3471.png", "formula": "\\begin{align*} \\mathcal { F } : = \\int _ \\Omega p F ^ 2 J , \\end{align*}"}
-{"id": "6142.png", "formula": "\\begin{align*} \\Omega _ Q = \\Omega \\ ; - \\ ; d ( \\phi ^ a \\cdot \\iota _ a \\Theta ) \\ ; - \\ ; [ \\phi ^ a \\cdot \\iota _ a \\Theta \\wedge \\Theta ] \\ ; + \\ ; \\frac 1 2 [ \\phi ^ a \\cdot \\iota _ a \\Theta \\wedge \\phi ^ b \\cdot \\iota _ b \\Theta ] \\end{align*}"}
-{"id": "3099.png", "formula": "\\begin{align*} P ( \\lambda ) : = N _ 2 ( \\lambda ) M ( \\lambda ) N _ 1 ( \\lambda ) ^ T , \\end{align*}"}
-{"id": "6873.png", "formula": "\\begin{align*} \\norm { v } _ { S ( [ t _ n , \\infty ) ) } = \\norm { v ^ { [ t _ n ] } } _ { S ( [ 1 , \\infty ) ) } < \\infty , \\end{align*}"}
-{"id": "650.png", "formula": "\\begin{align*} ( s \\cdot \\mu ) ( \\phi ) = \\mu ( s ^ { - 1 } \\cdot \\phi ) , \\textrm { f o r $ \\phi \\in C ( X ) $ } . \\end{align*}"}
-{"id": "1214.png", "formula": "\\begin{align*} K L ( P _ b ^ n | | P _ { b ' } ^ n ) = \\frac { n } { N } h \\log \\left ( \\frac { 1 + h } { 1 - h } \\right ) \\rho _ h ( b , b ' ) . \\end{align*}"}
-{"id": "7735.png", "formula": "\\begin{align*} x _ { n + 1 } \\ge \\sum _ { i = N _ \\alpha } ^ n \\sigma _ { i } \\xi _ { i + 1 } . \\end{align*}"}
-{"id": "9577.png", "formula": "\\begin{align*} D _ F = \\{ \\psi \\in L ^ 2 ( \\R ^ 3 ) : \\psi ( x ) = \\psi _ { r e g } ( x ) + \\sum \\limits _ { 1 \\le j \\le n } \\zeta _ j g _ j ( x ) , ~ ~ \\psi _ { r e g } \\in H ^ 2 ( \\R ^ 3 ) , ~ ~ \\zeta _ j \\in \\C , ~ ~ \\lim \\limits _ { x \\to y _ j } ( \\psi ( x ) - \\zeta _ j g _ j ( x ) ) = F _ j ( \\zeta ) \\} , \\end{align*}"}
-{"id": "1356.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\Omega } ( u ) = P ( \\chi ^ 2 _ 1 > u ) + \\frac { 1 } { 2 \\pi } e ^ { - \\frac { u } { 2 } } \\ln \\left . \\left [ \\frac { 1 + \\omega } { 1 - \\omega } \\right ] \\right \\vert _ { \\omega _ { \\mathcal { L } } } ^ { \\omega _ { \\mathcal { U } } } \\end{align*}"}
-{"id": "851.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m } C _ { a } ( z _ { i } ) \\cdot \\beta _ { a , 1 } ^ { * } \\equiv \\beta _ { a , 1 } ^ { * } \\prod _ { i = 1 } ^ { m } C _ { a } ( z _ { i } ) + ( 1 - q ^ 2 ) \\sum _ { \\ell = 1 } ^ { r } z _ { \\ell } \\prod _ { \\begin{subarray} { c } i = 1 \\\\ i \\not = \\ell \\end{subarray} } ^ { m } f ( z _ { i } , z _ { \\ell } ) \\cdot \\tilde { A } _ { a } ( z _ { \\ell } ) \\prod _ { \\begin{subarray} { c } i = 1 \\\\ i \\not = \\ell \\end{subarray} } ^ { m } C _ { a } ( z _ { i } ) \\end{align*}"}
-{"id": "2827.png", "formula": "\\begin{align*} \\sup _ { \\substack { 0 \\le u \\le L _ i \\\\ 0 \\le v \\le L _ j } } | r ( a _ { j , v } - a _ { i , u } ) | \\le \\sup _ { | s - s ' | \\ge j - i - 1 } | r ( s - s ' ) | = r ^ * ( j - i - 1 ) \\le r ^ * ( 1 ) < 1 . \\end{align*}"}
-{"id": "1941.png", "formula": "\\begin{align*} 0 & \\leq F _ i ( Y ( 0 ) ) - F _ i ( \\xi ) \\\\ & = ( 2 p _ i - ( n _ i + 1 ) q _ i ^ 2 ( Y _ i ( 0 ) + \\xi _ i ) ) \\cdot ( Y _ i ( 0 ) - \\xi _ i ) - \\sum _ { j \\neq i } n _ j q _ j ^ 2 ( Y _ j ( 0 ) - \\xi _ j ) ( Y _ j ( 0 ) + \\xi _ j ) . \\end{align*}"}
-{"id": "6944.png", "formula": "\\begin{align*} M ^ \\perp _ \\sigma ( w ) \\cdot 1 = \\begin{cases} ( 1 - w ) ^ { - 1 } & \\mbox { i f $ \\sigma = ( 0 ) $ } ; \\cr 1 & \\mbox { o t h e r w i s e } , \\cr \\end{cases} \\end{align*}"}
-{"id": "7188.png", "formula": "\\begin{align*} f = \\frac { g ^ 2 } { 2 } + p + C _ 1 \\end{align*}"}
-{"id": "1515.png", "formula": "\\begin{align*} e _ 0 ( e _ 0 + 1 ) \\left ( e _ 0 + \\frac { \\alpha } { \\alpha + \\gamma } \\right ) \\left ( e _ 0 + \\frac { \\alpha + \\gamma + q - 1 } { q - 1 } \\right ) = 0 \\ ; . \\end{align*}"}
-{"id": "3991.png", "formula": "\\begin{align*} u ( \\varphi ( s _ { i } ) ) \\pi _ { p } ( v _ { i } ) = u ( \\varphi ( s _ { i } ) ) \\pi _ { p } ( v ) = \\pi _ { p } ( u ( \\varphi ( s _ { i } ) ) v ) \\in V ^ { 0 - } ( A ) . \\end{align*}"}
-{"id": "4971.png", "formula": "\\begin{align*} f ^ { * } H \\equiv \\sum _ { j = 1 } ^ { r } \\sum _ { k = 1 } ^ { r } c _ { j } a _ { k j } D _ { k } = \\left < A \\left ( \\begin{array} { c } c _ { 1 } \\\\ c _ { 2 } \\\\ \\vdots \\\\ c _ { r } \\end{array} \\right ) , \\left ( \\begin{array} { c } D _ { 1 } \\\\ D _ { 2 } \\\\ \\vdots \\\\ D _ { r } \\end{array} \\right ) \\right > = \\left < A \\vec { c } , \\vec { D } \\right > . \\end{align*}"}
-{"id": "1174.png", "formula": "\\begin{align*} \\beta ( z ; \\epsilon ) = \\frac 1 2 \\sum _ { k = 1 } ^ M \\big [ \\frac { \\left ( \\mu _ k D ( - \\epsilon _ k ) - \\mu _ 0 D ( 0 ) \\right ) } { \\epsilon _ k } - \\mu _ k \\frac { D ( - \\epsilon _ k ) } { z + \\epsilon _ k } \\big ] , \\end{align*}"}
-{"id": "4812.png", "formula": "\\begin{align*} w _ 1 = & \\partial _ x \\sigma _ 1 , \\\\ w _ 3 = & \\partial _ x \\sigma _ 3 . \\end{align*}"}
-{"id": "6467.png", "formula": "\\begin{align*} G _ { m } ( s ) = \\partial _ { s } ^ { \\alpha } ( h _ { m } * W ) ( s ) + F _ { m } ( s ) \\geq 0 , s \\in ( 0 , t _ { 2 } - t _ { 0 } ) . \\end{align*}"}
-{"id": "3914.png", "formula": "\\begin{align*} & d e g _ { L S } ( I - ( S + Q N _ { f } + K S ) , \\Omega , 0 ) \\\\ & = d e g _ { B } \\left ( I - ( S + Q N _ { f } + K S ) \\left | _ { \\overline { \\Omega \\cap \\mathbb { R } ^ { 2 } } } \\right . , \\Omega \\cap \\mathbb { R } ^ { 2 } , 0 \\right ) \\\\ & = d e g _ { B } ( G , \\Omega \\cap \\mathbb { R } ^ { 2 } , 0 ) \\neq 0 . \\end{align*}"}
-{"id": "9212.png", "formula": "\\begin{align*} \\begin{cases} d Y ( t , x , z ) = [ A _ u Y ( t , x , z ) + a ( t , x , Y ( t , x , z ) , u ( t , x , z ) , z ) ] d t + b ( t , x , Y ( t , x , z ) , u ( t , x , z ) , z ) d B ( t ) \\\\ + \\int _ { \\mathbb { R } } c ( t , x , Y ( t , x , z ) , u ( t , x , z ) , z , \\zeta ) \\tilde { N } ( d t , d \\zeta ) ; ( t , x ) \\in ( 0 , T ) \\times D \\\\ Y ( 0 , x , z ) = \\xi ( x , z ) ; x \\in D \\\\ \\int _ { \\mathbb { R } } Y ( t , x , z ) \\delta _ Z ( z ) d z = \\theta ( t , x ) ; ( t , x ) \\in [ 0 , T ] \\times \\partial D . \\end{cases} \\end{align*}"}
-{"id": "5231.png", "formula": "\\begin{align*} X _ { \\varepsilon } = e ^ { g _ { \\varepsilon } } T + \\sum _ { j = 1 } ^ { m - 1 } \\left ( b _ { \\varepsilon , j } L _ { j } + \\overline { b _ { \\varepsilon , j } L _ { j } } \\right ) \\ ; , \\end{align*}"}
-{"id": "5084.png", "formula": "\\begin{align*} \\phi ( \\tau ' \\tau ; \\vec { z } ) = \\phi ( \\tau ' ; z _ { \\tau ^ { - 1 } ( 1 ) } , \\ldots , z _ { \\tau ^ { - 1 } ( k ) } ) \\phi ( \\tau ; \\vec { z } ) . \\end{align*}"}
-{"id": "1832.png", "formula": "\\begin{align*} u = \\phi ( z , \\bar z ) + 2 \\pi s + s ^ { 2 } \\psi ( s , z , \\bar z ) \\end{align*}"}
-{"id": "2431.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } T _ m f \\| _ { M _ 2 } = & \\| \\Box _ k ^ { \\alpha _ 1 } T _ m \\Box _ k ^ { \\alpha _ 1 , \\ast } f \\| _ { M _ 2 } \\\\ \\lesssim & \\| \\Box _ k ^ { \\alpha _ 1 } T _ m \\| _ { M _ 1 \\rightarrow M _ 2 } \\| \\Box _ k ^ { \\alpha _ 1 , \\ast } f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "5409.png", "formula": "\\begin{align*} \\zeta _ { A , R } ( s ) : = \\sum _ { m = 1 } ^ \\infty a _ m ( A , R ) m ^ { - s } \\end{align*}"}
-{"id": "359.png", "formula": "\\begin{align*} F ( t , c , h ) = \\frac { F ( t , c + c _ 0 , h + h _ 0 ) } { F ( t , c _ 0 , h _ 0 ) } . \\end{align*}"}
-{"id": "6958.png", "formula": "\\begin{align*} \\mathbf { P } \\mathbf { H } \\mathbf { Q } = \\begin{pmatrix} \\mathbf { H } _ P & \\mathbf { H } _ I \\end{pmatrix} = \\begin{pmatrix} \\mathbf { H } _ { P , 1 } & & \\mathbf { O } & \\mathbf { H } _ { I , 1 } \\\\ & \\ddots & & \\vdots \\\\ \\mathbf { O } & & \\mathbf { H } _ { P , K } & \\mathbf { H } _ { I , K } \\end{pmatrix} , \\end{align*}"}
-{"id": "201.png", "formula": "\\begin{align*} T _ k ( r , s , t ) : = e ^ { \\alpha _ r ( s , t ) } \\sum _ { \\ell = 0 } ^ { L ( r , s , t ) } \\sum _ { i = 0 } ^ { I ( r , s ) - \\ell } \\sum _ { j = 0 } ^ { J ( r , t ) - \\ell } & \\frac { \\{ s - \\alpha _ r ( s , t ) \\} ^ i \\{ t - \\alpha _ r ( s , t ) \\} ^ j \\alpha _ r ^ \\ell ( s , t ) } { i ! j ! \\ell ! } \\\\ & - \\sum _ { i = 0 } ^ { I ( r , s ) } \\sum _ { j = 0 } ^ { J ( r , t ) } \\frac { s ^ i t ^ j } { i ! j ! } , \\end{align*}"}
-{"id": "9205.png", "formula": "\\begin{align*} \\sup _ { u \\in \\mathcal { A } } J ( u ) = J ( u ^ { \\star } ) . \\end{align*}"}
-{"id": "2917.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ s _ \\lambda ( Z ) ] \\ M ( X Z ) \\ L _ \\pi ( Z ) \\ , ; \\\\ \\cr s ^ { * ( \\pi ) } _ \\lambda ( X ) & = \\begin{cases} [ s _ \\lambda ( Z ) ] \\ L ( X Z ) \\ L _ { \\pi ' } ( Z ) & \\mbox { i f $ | \\pi | $ i s e v e n } ; \\cr [ s _ \\lambda ( Z ) ] \\ L ( X Z ) \\ M _ { \\pi ' } ( Z ) ) & \\mbox { i f $ | \\pi | $ i s o d d } . \\cr \\end{cases} \\end{align*}"}
-{"id": "51.png", "formula": "\\begin{align*} R ( a ) : = \\inf \\left \\{ r > 0 \\ : \\ f ( r , a ) = 0 \\right \\} \\in ( 0 , \\infty ] \\ . \\end{align*}"}
-{"id": "6323.png", "formula": "\\begin{align*} 1 = 2 ^ { j A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 3 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "2386.png", "formula": "\\begin{align*} F _ { \\gamma } ^ { \\rm { D R } } ( z ) = ( 2 \\gamma ) ^ { - 1 } \\left ( \\langle R _ { \\gamma f } ( z ) , z \\rangle - p _ { \\gamma f } ^ 2 ( z ) - p _ { \\gamma g } ^ 2 ( R _ { \\gamma f } z ) \\right ) . \\end{align*}"}
-{"id": "3795.png", "formula": "\\begin{align*} \\frac { { \\partial I _ { { \\rm { a s y } } } \\left ( { { \\bf { x } } ; { \\bf { y } } } \\right ) } } { { \\partial \\gamma _ m } } = 0 , \\ \\frac { { \\partial I _ { { \\rm { a s y } } } \\left ( { { \\bf { x } } ; { \\bf { y } } } \\right ) } } { { \\partial \\psi _ n } } = 0 . \\end{align*}"}
-{"id": "2994.png", "formula": "\\begin{align*} \\sum _ i a _ i [ F _ { i } ] = 0 H _ 2 ( X ; \\Z _ d ) \\quad a _ i \\in \\Z _ d . \\end{align*}"}
-{"id": "7102.png", "formula": "\\begin{align*} W _ { n , \\beta } ^ u = \\sum _ { | v | = n , v > u } e ^ { \\beta ( m _ n + V ( u ) - V ( v ) ) } , \\end{align*}"}
-{"id": "7399.png", "formula": "\\begin{align*} & \\mathrm { D } ^ { f } _ H ( K ) \\\\ & = \\{ h _ 0 \\in G ; \\forall g _ 1 , \\ldots , \\forall g _ k \\in G , \\exists h \\in G h h _ 0 h ^ { - 1 } K ( h h _ 0 h ^ { - 1 } ) ^ { - 1 } g _ 1 H g _ 1 ^ { - 1 } \\cup \\cdots \\cup g _ k H g _ k ^ { - 1 } \\} . \\\\ \\end{align*}"}
-{"id": "1537.png", "formula": "\\begin{align*} E \\cdot \\nabla _ v \\left ( \\partial _ { v _ i } F \\right ) + \\nu \\left ( \\partial _ { v _ i } F \\right ) & = \\int \\sigma ( v , v ' ) F ( v ' ) \\d v ' \\ , \\partial _ { v _ i } M ( v ) \\\\ & + \\int \\partial _ { v _ i } \\sigma ( v , v ' ) F ( v ' ) \\d v ' M ( v ) - \\left ( \\partial _ { v _ i } \\nu \\right ) F . \\end{align*}"}
-{"id": "4794.png", "formula": "\\begin{align*} \\frac { t } { 2 } \\ , ( \\varphi ^ 2 ) ' + \\varphi ^ 2 + 1 = \\pm a \\sqrt { \\varphi ^ 2 + 1 } . \\end{align*}"}
-{"id": "2070.png", "formula": "\\begin{align*} \\mathcal { P } _ { 1 } = & - ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } ( 2 s _ { 0 } M + D ) , \\\\ \\mathcal { P } _ { 2 } = & - ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } M , \\\\ \\mathsf { Q } = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } F , \\end{align*}"}
-{"id": "7979.png", "formula": "\\begin{align*} g _ t ( 0 , x _ 2 , x _ 3 ) = t \\alpha ( x _ 2 , x _ 3 ) + \\beta ( x _ 2 , x _ 3 ) = 0 . \\end{align*}"}
-{"id": "1129.png", "formula": "\\begin{align*} \\gamma _ k ^ \\mathrm { C } [ \\iota ] = \\begin{cases} \\frac { M \\sigma _ { \\mathrm { s } k } ^ 2 [ \\iota ] } { \\sum _ { i = 1 } ^ { K } \\beta _ { \\mathrm { s } i } + 1 / \\rho _ \\mathrm { s } } , & \\\\ \\frac { M \\sigma _ { \\mathrm { s } k } ^ 2 [ \\iota ] } { \\sum _ { i = 1 } ^ { K } \\beta _ { \\mathrm { s } i } + \\left ( \\rho _ \\mathrm { d } \\beta _ \\mathrm { L I } + 1 \\right ) / \\rho _ \\mathrm { s } } , & . \\end{cases} \\end{align*}"}
-{"id": "5634.png", "formula": "\\begin{align*} E ( \\mu ) = \\left \\{ \\begin{array} { l l } \\displaystyle \\int _ { \\Omega } G \\left ( \\frac { d \\mu } { d \\lambda } ( x ) \\right ) d \\lambda ( x ) , & \\mbox { i f } \\ \\mu \\ll \\lambda \\\\ + \\infty & \\mbox { o t h e r w i s e , } \\end{array} \\right . \\end{align*}"}
-{"id": "6740.png", "formula": "\\begin{align*} \\mathcal I ( t \\varphi , x ) = \\{ f \\in \\mathcal O _ x \\textup { s . t . } \\int _ V | f | e ^ { - t \\varphi } \\omega ^ n < \\infty \\textup { f o r s o m e o p e n s e t } x \\in V \\subset X \\} . \\end{align*}"}
-{"id": "5694.png", "formula": "\\begin{align*} \\mathcal { D } _ k : = \\left \\{ Q \\in \\mathcal { D } ( Q _ 0 ) : \\frac { w ( Q ) } { | Q | } > a ^ k \\gamma _ 0 \\right \\} ( k \\in \\N ) . \\end{align*}"}
-{"id": "752.png", "formula": "\\begin{gather*} T u = v T . \\end{gather*}"}
-{"id": "6957.png", "formula": "\\begin{align*} \\sum _ { k : s _ k = s _ j } u _ k ^ 2 \\leq \\frac { R ^ 2 } { 2 } \\sum _ { k : s _ k = s _ { j + 1 } } u _ k ^ 2 , ~ ~ \\forall j \\in \\mathcal { J } . \\end{align*}"}
-{"id": "6067.png", "formula": "\\begin{align*} & \\int _ { 0 } ^ { T } \\int _ { { \\mathbb R } ^ d } \\int _ { 0 } ^ { 1 } \\Big | { \\sum _ { j = 1 } ^ \\infty } r _ { j } ( z ) 2 ^ { j ( s + \\alpha / p ) } \\Delta _ { j } T ^ { \\alpha } f ( t , x ) \\Big | ^ { p } d z d x d t \\\\ & \\ll \\int _ { 0 } ^ { T } \\int _ { { \\mathbb R } ^ d } \\int _ { 0 } ^ { 1 } \\Big | { \\sum _ { j = 1 } ^ \\infty } r _ { j } ( z ) 2 ^ { j ( s + \\alpha / p ) } \\Delta _ { j } T ^ { \\alpha } \\Delta _ { j } f ( t , x ) \\Big | ^ { p } d z d x d t . \\end{align*}"}
-{"id": "8829.png", "formula": "\\begin{align*} \\psi ( z ^ { 1 / m } ) = \\sum _ { n \\geq 0 } a _ { \\frac { 1 } { m } + n } z ^ { \\frac { 1 } { m } + n } \\end{align*}"}
-{"id": "6737.png", "formula": "\\begin{align*} u ^ { ( n ) } ( x ) \\quad = \\begin{cases} \\frac { \\sum _ { \\ell = 1 } ^ \\infty b ^ { ( n ) } _ \\ell ( x _ \\ell ) } { 1 + \\sum _ { \\ell = 1 } ^ \\infty b ^ { ( n ) } _ \\ell ( x _ \\ell ) } & \\sum _ { \\ell = 1 } ^ \\infty b ^ { ( n ) } _ \\ell ( x _ \\ell ) < \\infty \\ , , \\\\ 1 & \\sum _ { \\ell = 1 } ^ \\infty b ^ { ( n ) } _ \\ell ( x _ \\ell ) = \\infty \\ , . \\end{cases} \\end{align*}"}
-{"id": "9725.png", "formula": "\\begin{gather*} 0 < B _ 1 : = \\frac { 1 } { \\gamma } - \\frac { 1 } { 2 } \\frac { 1 } { \\gamma } \\leq \\frac { \\log \\Gamma _ 1 ( x ) } { \\log x } \\leq \\frac { 1 } { \\gamma } + \\frac { 1 } { 2 } \\frac { 1 } { \\gamma } = : B _ 2 , \\ , x \\in ( 0 , g ( \\delta / 2 ) ] , \\\\ g ( \\delta / 2 ) < 1 , g ' ( x ) > 0 x \\in ( 0 , \\delta ) . \\end{gather*}"}
-{"id": "7129.png", "formula": "\\begin{align*} z = P _ R ( X ) = \\frac { R x _ 1 + i R x _ 2 } { R + x _ 3 } . \\end{align*}"}
-{"id": "5173.png", "formula": "\\begin{gather*} [ \\delta ( \\gamma ) ] = [ \\delta ( \\alpha ) ] + [ \\delta ( \\beta ) ] . \\end{gather*}"}
-{"id": "9128.png", "formula": "\\begin{align*} D ( A ( t ) ^ { 1 / 2 } ) = \\mathcal { V } \\ \\ c _ 1 \\| A ( t ) ^ { 1 / 2 } v \\| \\le \\| v \\| _ { \\mathcal { V } } \\le c _ 2 \\| A ( t ) ^ { 1 / 2 } v \\| \\end{align*}"}
-{"id": "9705.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ' ( x ) } { x ^ { - 2 } e ^ { 1 / x } \\exp ( - e ^ { 1 / x } ) } = 1 . \\end{align*}"}
-{"id": "4629.png", "formula": "\\begin{align*} \\left | I ( w , m ) \\right | \\ ; & = \\ ; \\binom { 3 m + \\ell } { m } - \\binom { 3 m + \\ell } { < m } + \\sum \\limits _ { e = 1 } ^ m 4 e \\left [ \\binom { 3 m + \\ell } { m - e } - \\binom { 3 m + \\ell } { < m - e } \\right ] \\\\ \\ ; & = \\ ; \\binom { 3 m + \\ell } { m } - \\sum \\limits _ { e = 1 } ^ m \\left ( 2 e ^ 2 - 6 e + 1 \\right ) \\binom { 3 m + \\ell } { m - e } . \\end{align*}"}
-{"id": "4816.png", "formula": "\\begin{align*} k _ { \\mathcal J } ( \\tau ) = \\liminf _ { A \\in { \\mathcal R } ^ + _ 1 ( { \\mathcal H } ) } \\max _ { 1 \\le j \\le n } | [ A , T _ j ] | \\end{align*}"}
-{"id": "2877.png", "formula": "\\begin{align*} X \\otimes _ e Y = X _ + \\wedge Y ( = ( X \\otimes Y _ - ) _ + ) ; \\end{align*}"}
-{"id": "3180.png", "formula": "\\begin{align*} w _ j = \\left ( \\begin{array} { c } w _ j ^ 1 \\\\ w _ j ^ 2 \\\\ \\vdots \\\\ w _ j ^ r \\end{array} \\right ) , \\ f _ { k j , \\alpha } = \\left ( \\begin{array} { c } f _ { k j , \\alpha } ^ 1 \\\\ f _ { k j , \\alpha } ^ 2 \\\\ \\vdots \\\\ f _ { k j , \\alpha } ^ r \\end{array} \\right ) . \\end{align*}"}
-{"id": "9403.png", "formula": "\\begin{align*} \\mbox { i n d } _ { C ( \\lambda _ 0 ; \\varepsilon ) } ( \\mathbf { T } - W _ \\lambda ) = \\mbox { t r } _ { \\mathbb { C } ^ m } \\oint _ { C ( \\lambda _ 0 ; \\varepsilon ) } { W _ \\xi ' } ( W _ \\xi - \\mathbf { T } ) ^ { - 1 } d \\xi , 0 < \\varepsilon < \\varepsilon _ { 0 } . \\end{align*}"}
-{"id": "9540.png", "formula": "\\begin{align*} \\kappa _ { 1 1 } ^ 2 & = \\frac { 1 } { 4 \\pi } ( 2 \\pi l _ p ) ^ 9 \\\\ \\kappa _ { 1 0 } ^ 2 & = \\frac { 1 } { 4 \\pi } ( 2 \\pi l _ s ) ^ 8 \\end{align*}"}
-{"id": "251.png", "formula": "\\begin{align*} \\nu = d 2 ^ { \\alpha / 2 - 1 } \\frac { \\Gamma \\bigl ( \\frac { \\alpha } { 2 } + \\frac { d } { 2 } \\bigr ) } { \\Gamma \\bigl ( 1 + \\frac { d } { 2 } \\bigr ) } \\frac { ( \\rho / 2 ) ^ { \\alpha / 2 } \\Gamma \\bigl ( \\frac { \\rho - \\alpha } { 2 } \\bigr ) } { \\Gamma \\bigl ( \\frac { \\rho } { 2 } \\bigr ) } . \\end{align*}"}
-{"id": "1985.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n ( - 2 n - 2 k - 2 m - t + 2 i - 2 ) & = - n ( 2 n + 2 k + 2 m + t + 2 ) + 2 \\sum _ { i = 1 } ^ n i \\\\ & = - n ( n + 2 k + 2 m + t + 1 ) \\end{align*}"}
-{"id": "4167.png", "formula": "\\begin{align*} F ' ( z ) = f ' ( z ) + z f '' ( z ) ( \\lambda - \\mu + 2 \\lambda \\mu ) + \\lambda \\mu z ^ { 2 } f ''' ( z ) \\end{align*}"}
-{"id": "1338.png", "formula": "\\begin{align*} B _ j ( \\lambda ) = ( L _ j ( \\lambda , \\xi _ j ) ) _ { 1 2 } . \\end{align*}"}
-{"id": "879.png", "formula": "\\begin{align*} A = \\operatorname * { c l } ( \\operatorname * { i n t } A ) , \\end{align*}"}
-{"id": "6617.png", "formula": "\\begin{align*} \\mu = \\sum _ x \\deg ( x ) \\sum _ { n = 1 } ^ \\infty q ^ { - n \\deg ( x ) } \\delta ( g ^ n _ x ) , \\end{align*}"}
-{"id": "4955.png", "formula": "\\begin{align*} [ \\Psi _ \\lambda \\ast ( Q f ) ] ( x ) = \\int _ { \\Omega } \\Psi _ \\lambda ( x - y ) Q ( y ) f ( y ) \\ , d y \\end{align*}"}
-{"id": "4974.png", "formula": "\\begin{align*} { \\bf h } _ { \\vec { D } } = \\left ( \\begin{array} { c } h _ { D _ { 1 } } \\\\ h _ { D _ { 2 } } \\\\ \\vdots \\\\ h _ { D _ { r } } \\end{array} \\right ) . \\end{align*}"}
-{"id": "4840.png", "formula": "\\begin{align*} \\tau _ { \\mu } ( c ) = \\sum _ { \\kappa ' } s _ { \\lambda / \\kappa } ( c ) = c ^ { \\mu _ 1 - \\mu _ 2 + \\mu _ 3 - \\mu _ 4 + \\dots } . \\end{align*}"}
-{"id": "692.png", "formula": "\\begin{align*} \\nu ( K \\cup s K ) = \\nu ( K ) + \\nu ( s K ) - \\nu ( K \\cap s K ) = 2 \\nu ( K ) > 1 , \\end{align*}"}
-{"id": "8258.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } n ^ { 1 / 2 } \\Delta _ n [ V ' ( n ^ { - 1 / 2 } v ^ n _ t + \\rho ' ( \\lambda _ 0 ) ) ] & = \\frac { 1 } { 2 } n ^ { 1 / 2 } \\Delta _ n \\{ V ' ( \\rho ' ( \\lambda _ 0 ) ) + V '' ( \\rho ' ( \\lambda _ 0 ) ) ( n ^ { - 1 / 2 } v ^ n _ t ) + O ( n ^ { - 1 } ) \\} \\\\ & = \\frac { 1 } { 2 } V '' ( \\rho ' ( \\lambda _ 0 ) ) \\Delta _ n v ^ n _ t + O ( n ^ { - 1 / 2 } ) , \\end{align*}"}
-{"id": "5543.png", "formula": "\\begin{align*} \\begin{gathered} D _ t \\rho = \\sum _ { j = 1 } ^ N \\dot m _ j \\delta _ { x _ j } - m _ j \\dot x _ j \\delta _ { x _ j } ' , \\\\ z m _ j \\dot x _ j = - \\frac 1 2 [ b _ x ] ( x _ j ) - z m _ j b ( x _ j ) , \\\\ z \\dot m _ j = \\frac 1 2 [ b _ { x x } ] ( x _ j ) + z m _ j \\langle b _ x \\rangle ( x _ j ) , \\end{gathered} \\end{align*}"}
-{"id": "7560.png", "formula": "\\begin{align*} \\alpha : = \\int _ 0 ^ 1 x \\phi ( x ) ~ d x > 0 . \\end{align*}"}
-{"id": "9495.png", "formula": "\\begin{align*} \\mathrm { R i c } ( X , Y ) - \\frac { 1 } { 2 } g ( X , Y ) R & = 8 \\pi T ( X , Y ) \\\\ & = 2 \\left ( \\langle i _ X F , i _ Y F \\rangle - \\frac { 1 } { 2 } g ( X , Y ) | F | ^ 2 \\right ) , \\end{align*}"}
-{"id": "3192.png", "formula": "\\begin{align*} w _ j = u _ j + \\sum _ { | \\alpha | \\geq 2 } F _ { j , \\alpha } ( z _ j ) \\cdot u _ j ^ \\alpha , \\end{align*}"}
-{"id": "398.png", "formula": "\\begin{align*} J _ L ( f ) ( g ) & = \\sum _ { x \\in L _ 0 } f ( g \\cdot x ) , J _ { \\hat { L } } ( f ) ( g ) = \\sum _ { x \\in \\hat { L } _ 0 } f ( g \\cdot x ) . \\end{align*}"}
-{"id": "4039.png", "formula": "\\begin{align*} h ^ { s } ( \\underline { a } ) = \\bigcap _ { n \\geq 0 } ( \\phi _ { a [ o , n ] } ^ { + } ) ^ { - 1 } ( R _ { a _ n } ) = \\bigcap _ { n \\geq 0 } V _ { a [ 0 , n ] } \\end{align*}"}
-{"id": "4877.png", "formula": "\\begin{align*} \\delta ( x y ) = \\delta ( x ) \\phi ( y ) + x ^ p \\delta ( y ) \\end{align*}"}
-{"id": "8399.png", "formula": "\\begin{align*} M ( z _ { g _ 0 } - p _ { g _ 0 } ) = E _ M ( M z _ { g _ 0 } g _ 0 \\alpha _ { g _ 0 ^ { - 1 } } ( z _ { g _ 0 } - p _ { g _ 0 } ) g _ 0 ^ { - 1 } ) = \\{ 0 \\} , \\end{align*}"}
-{"id": "155.png", "formula": "\\begin{align*} [ v , w ] : = \\kappa _ { v , w } = \\begin{cases} 1 & v = w , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "1178.png", "formula": "\\begin{align*} D ( - \\lambda ) = W ( \\hat c _ 0 , c _ 0 ) + \\sum _ { n = 1 } ^ N \\lambda ^ n \\sum _ { 1 \\leq i _ 1 < i _ 2 < \\dots i _ n \\leq N } m _ { i _ 1 } m _ { i _ 2 } \\dots m _ { i _ n } ( x _ { i _ n } - x _ { i _ { n - 1 } } ) ( x _ { i _ { n - 1 } } - x _ { i _ { n - 2 } } ) \\dots ( x _ { i _ 2 } - x _ { i _ 1 } ) c _ 0 ( x _ { i _ 1 } ) \\hat c _ 0 ( x _ { i _ n } ) , \\end{align*}"}
-{"id": "1568.png", "formula": "\\begin{align*} X = \\xi ^ { i } \\left ( x ^ { k } , u \\right ) \\partial _ { i } + \\eta \\left ( x ^ { k } , u \\right ) \\partial _ { u } . \\end{align*}"}
-{"id": "3764.png", "formula": "\\begin{align*} e ^ { C _ 0 ( \\beta - i \\alpha t ) - C _ 0 ( \\beta ) + i \\alpha t u } = e ^ { - t ^ 2 / 2 } \\left ( 1 + \\alpha t ^ 3 O _ 3 ( 1 ) + \\alpha ^ 2 t ^ 4 O _ 4 ( 1 ) \\right ) + O \\left ( \\left [ \\alpha t ^ 3 + \\alpha ^ 2 t ^ 4 \\right ] ^ 2 \\right ) . \\end{align*}"}
-{"id": "4643.png", "formula": "\\begin{align*} \\phi ( g ) = 2 t \\sum _ { n = 1 } ^ { \\infty } \\rho _ \\phi ( n ) K _ { s - \\frac { 1 } { 2 } } \\left ( 2 \\pi n t ^ 2 \\right ) \\cos ( 2 \\pi n u ) \\end{align*}"}
-{"id": "1610.png", "formula": "\\begin{align*} Z ^ { 1 } = e ^ { m t } K _ { 1 } ~ , ~ Z ^ { 2 } = e ^ { - m t } \\left ( K ^ { 1 } + \\left ( m \\int \\frac { d x } { \\sigma \\left ( x \\right ) } + c \\right ) F \\partial _ { F } \\right ) \\end{align*}"}
-{"id": "3814.png", "formula": "\\begin{align*} e ^ { H [ s ( t ) ] } \\psi ( z ) e ^ { - H [ s ( t ) ] } & = e ^ { - \\sum _ { q \\geq 1 } s _ q ( t ) z ^ q } \\psi ( z ) \\\\ [ 5 p t ] e ^ { H [ s ( t ) ] } \\psi ^ \\ast ( z ) e ^ { - H [ s ( t ) ] } & = e ^ { \\sum _ { q \\geq 1 } s _ q ( t ) z ^ q } \\psi ^ \\ast ( z ) . \\end{align*}"}
-{"id": "4860.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 1 1 } ( i , u ; j , v ) = \\iint _ { \\mathcal { C } ^ 2 } \\frac { ( z - w ) ( z - c ) ( w - c ) } { ( z ^ 2 - 1 ) ( w ^ 2 - 1 ) ( z w - 1 ) } \\frac { ( z - \\sqrt { q } ) ^ { n _ i } ( w - \\sqrt { q } ) ^ { n _ j } } { ( 1 - z \\sqrt { q } ) ^ { m _ i } ( 1 - w \\sqrt { q } ) ^ { m _ j } } \\frac { \\dd z \\dd w } { z ^ { n _ i + 1 } w ^ { n _ j + 1 } z ^ u w ^ v } , \\end{align*}"}
-{"id": "5493.png", "formula": "\\begin{align*} \\mathcal { R } _ k ^ { \\mathrm { P L } } [ \\iota ] = \\tau _ \\mathrm { d } [ \\iota ] \\log _ 2 ( 1 + \\gamma _ k ^ { \\mathrm { P L } } [ \\iota ] ) \\end{align*}"}
-{"id": "1386.png", "formula": "\\begin{align*} \\bar \\kappa _ { 2 } = \\Phi ( - \\sqrt { n } \\tau _ 0 / \\sigma _ { \\tau } ( \\delta _ 0 ) ) ; \\sigma _ { \\tau } ^ 2 ( \\delta _ 0 ) = ( \\Omega _ { 1 , 1 } ^ { - 1 } \\Omega _ { 1 , 2 } \\Omega _ { 1 , 1 } ^ { - 1 } ) _ { \\tau \\tau } \\end{align*}"}
-{"id": "5403.png", "formula": "\\begin{align*} e ( B ) \\leq \\left ( \\beta - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } + \\left ( k - \\alpha - \\beta \\right ) \\frac { n ^ 2 } { 2 } = \\bigg ( k - \\alpha - \\frac 1 4 \\bigg ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "1841.png", "formula": "\\begin{align*} \\sum _ { a < n < b } g ( n ) \\exp ( 2 \\pi i f ( n ) ) & = \\sum _ { \\alpha - \\eta < m < \\beta + \\eta } \\int _ a ^ b g ( x ) \\exp ( 2 \\pi i ( f ( x ) - m x ) ) d x \\\\ & + O \\left ( G _ 1 \\bigg ( \\eta ^ { - 1 } + \\log \\Big ( 1 + \\frac { \\beta - \\alpha } { \\eta } \\Big ) \\bigg ) + G _ 2 ( \\beta - \\alpha + \\eta ) \\right ) , \\end{align*}"}
-{"id": "2603.png", "formula": "\\begin{align*} \\gamma ( a ) & = \\gamma ( - a ) \\\\ \\gamma ( a + b + c ) - \\gamma ( b + c ) - \\gamma ( a + c ) & - \\gamma ( a + b ) + \\gamma ( a ) + \\gamma ( b ) + \\gamma ( c ) = 0 , \\end{align*}"}
-{"id": "1199.png", "formula": "\\begin{align*} M ( \\Lambda ) = \\left \\{ x ^ a y ^ b : ( a , b ) \\in \\Lambda \\right \\} . \\end{align*}"}
-{"id": "2579.png", "formula": "\\begin{align*} e ^ { - i t _ n \\Delta } u _ n ( t _ n ) = \\sum _ { j = 1 } ^ J e ^ { i x \\xi _ n ^ j } \\psi ^ j _ { \\{ h _ n ^ j \\} } + W ^ J _ n \\end{align*}"}
-{"id": "3735.png", "formula": "\\begin{align*} \\ \\rho ( u ) = 0 \\textrm { f o r } u < 0 , \\ \\rho ( u ) = 1 \\textrm { f o r } 0 \\leq u \\leq 1 , \\end{align*}"}
-{"id": "3304.png", "formula": "\\begin{align*} \\langle w _ 0 \\wedge w _ 1 , v _ 1 \\wedge v _ 0 \\rangle = \\langle w _ { - 1 } \\wedge w _ 0 , v _ 0 \\wedge v _ { - 1 } \\rangle = \\frac { q ^ { - 2 } } { [ 2 ] _ { q ^ 2 } } , \\langle w _ { - 1 } \\wedge w _ 1 , v _ 1 \\wedge v _ { - 1 } \\rangle = \\frac { 1 } { [ 2 ] _ { q ^ 2 } } , \\end{align*}"}
-{"id": "5186.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { S _ { p _ { 1 } } } \\| V \\| _ { S _ { p _ { 2 } } } . \\end{align*}"}
-{"id": "724.png", "formula": "\\begin{align*} \\frac { h _ { H } ( f ^ { n } ( P ) ) } { \\delta ^ { n } } = & - \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { - 1 - i } c _ { 1 } h _ { Z _ { 1 } } ( p ^ { - 1 } ( f ^ { i } ( P ) ) ) + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { - 1 - i } h _ { E } ( p ^ { - 1 } ( f ^ { i } ( P ) ) ) \\\\ & + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { - 1 - i } c _ { 1 } h _ { E _ { 1 } ' } ( f ^ { i } ( P ) ) + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { - i } h _ { N } ( f ^ { i } ( P ) ) + h _ { H } ( P ) . \\end{align*}"}
-{"id": "9907.png", "formula": "\\begin{align*} e & : = \\frac { 1 } { \\sqrt { q } } E K ^ { - 1 } , & f & : = \\frac { 1 } { \\sqrt { q } } F ( K ' ) ^ { - 1 } , & k & : = K ( K ' ) ^ { - 1 } , & \\mbox { a n d } k ^ { - 1 } & . \\end{align*}"}
-{"id": "406.png", "formula": "\\begin{align*} ( f \\circ 1 ) ( a ) : = \\bigvee \\limits _ { ( y , z ) \\in { A _ a } } { \\min \\{ f ( y ) , 1 ( z ) \\} } = \\bigvee \\limits _ { ( y , z ) \\in { A _ a } } f ( y ) \\end{align*}"}
-{"id": "9422.png", "formula": "\\begin{align*} \\sum _ { x \\in \\phi ^ { - 1 } ( y ) } w _ 1 ( x ) = \\lambda w _ 2 ( y ) . \\end{align*}"}
-{"id": "6218.png", "formula": "\\begin{align*} \\hat { X } ( s ) = \\sum \\limits _ { j = 0 } ^ { \\infty } \\hat { X } ^ { ( j ) } ( s _ { 0 } ) ( s - s _ { 0 } ) ^ { j } . \\end{align*}"}
-{"id": "747.png", "formula": "\\begin{align*} F \\boxtimes G = \\pi _ X ^ * F \\otimes _ { \\O _ X } ^ L \\pi _ Y ^ * G \\end{align*}"}
-{"id": "5264.png", "formula": "\\begin{align*} Y = \\operatorname * { b d } A + \\mathbb { R } k . \\end{align*}"}
-{"id": "737.png", "formula": "\\begin{align*} b _ { n } = b _ { n - 1 } + C _ { 1 } \\sqrt [ ] { b _ { n - 1 } } \\left ( 1 + \\sqrt [ ] { 1 + \\frac { C _ { 2 } } { b _ { n - 1 } } } \\right ) \\leq b _ { n - 1 } + C _ { 1 } \\left ( 1 + \\sqrt [ ] { 1 + C _ { 2 } } \\right ) \\sqrt [ ] { b _ { n - 1 } } . \\end{align*}"}
-{"id": "291.png", "formula": "\\begin{align*} \\mu _ d \\bigl ( B _ x ( r _ { n , u } ) \\cap B _ y ( r _ { n , v } ) ^ c \\bigr ) & \\geq \\mu _ d \\bigl ( B _ x ( r _ { n , v } ) \\cap B _ y ( r _ { n , v } ) ^ c \\bigr ) \\\\ & = V _ d r _ { n , v } ^ d \\biggl \\{ 1 - I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { \\| x - y \\| ^ 2 } { 4 r _ { n , v } ^ 2 } \\biggr ) \\biggr \\} \\gtrsim \\frac { k \\| z \\| } { n f ( x ) } . \\end{align*}"}
-{"id": "7623.png", "formula": "\\begin{align*} \\sum _ { y \\sim x } \\lambda _ 1 \\mathbf { v } _ y = \\lambda _ 1 ^ 2 \\leq d n - ( n - 1 ) . \\end{align*}"}
-{"id": "3928.png", "formula": "\\begin{align*} \\limsup _ { k \\to \\infty } \\left ( | a _ { k , \\sigma } | \\right ) ^ { 1 / k } = 1 \\ , , \\limsup _ { j \\to \\infty } \\left ( | b _ j | \\right ) ^ { 1 / j } = 2 ^ { - 1 } \\ , , \\end{align*}"}
-{"id": "2601.png", "formula": "\\begin{align*} \\delta _ n ( f ) ( x _ 1 , x _ 2 , \\ldots , x _ { n + 1 } ) & = x _ 1 \\cdot f ( x _ 2 , \\ldots , x _ { n + 1 } ) \\\\ & + \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i } f ( x _ 1 , \\ldots , x _ { i - 1 } , x _ i x _ { i + 1 } , x _ { i + 2 } , \\ldots , x _ { n + 1 } ) \\\\ & + ( - 1 ) ^ { n + 1 } f ( x _ 1 , \\ldots , x _ { n } ) . \\end{align*}"}
-{"id": "920.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 1 / 3 } } = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { d \\times d } , W \\in \\mathbb { R } ^ { n \\times d } : X = U V W ^ { T } } \\| U \\| _ { * } \\| V \\| _ { * } \\| W \\| _ { * } \\\\ = & \\min _ { U , V , W : X = U V W ^ { T } } \\ ! \\left ( \\frac { \\| U \\| _ { * } + \\| V \\| _ { * } + \\| W \\| _ { * } } { 3 } \\right ) ^ { 3 } . \\end{align*}"}
-{"id": "5196.png", "formula": "\\begin{align*} \\textup { T r } ^ { p } ( A \\Sigma _ { Z } A ^ { T } ) & = \\sum _ { k } \\left ( \\sum _ { i } a ^ { 2 } _ { k i } \\sigma _ { i } \\right ) ^ { p } \\\\ & \\geq \\sum _ { k } \\sum _ { i } a ^ { 2 } _ { k i } \\sigma ^ { p } _ { i } \\\\ & = \\sum _ { i } \\sigma ^ { p } _ { i } \\\\ & = \\textup { T r } ^ { p } ( \\Sigma _ { Z } ) = \\| Z \\| ^ { p } _ { S _ { p } } . \\end{align*}"}
-{"id": "1340.png", "formula": "\\begin{align*} f ( y _ { t } | W _ t ) = \\exp \\left \\{ y _ { t } W _ { t } - m _ { t } b ( W _ { t } ) + c ( y _ { t } ) \\right \\} . \\end{align*}"}
-{"id": "3205.png", "formula": "\\begin{align*} \\delta \\left \\{ \\left ( U _ j , \\sum _ { \\lambda , | \\beta | = n } F _ { j , \\beta } ^ \\lambda \\cdot e _ { j , \\lambda } \\ * \\otimes e _ j ^ \\beta \\right ) \\right \\} = \\left \\{ \\left ( U _ { j k } , \\sum _ { \\lambda , | \\alpha | = n } \\left ( h ^ \\lambda _ { 1 , j k , \\alpha } - h ^ \\lambda _ { 2 , j k , \\alpha } \\right ) \\cdot e _ { j , \\lambda } ^ * \\otimes e _ j ^ \\alpha \\right ) \\right \\} \\end{align*}"}
-{"id": "4333.png", "formula": "\\begin{align*} \\partial _ t u - \\partial _ x \\left ( | \\partial _ x u | ^ { p - 2 } \\partial _ x u \\right ) + | \\partial _ x u | ^ { p - 1 } = 0 \\ , ( t , x ) \\in ( 0 , \\infty ) \\times \\mathbb { R } \\ , \\end{align*}"}
-{"id": "494.png", "formula": "\\begin{align*} \\ell _ { N , k } = ( - 1 ) ^ k \\ , \\ell _ { N , 0 } \\prod _ { j = 0 } ^ { k - 1 } ( N + 2 s + j ) ( k \\ge 1 ) . \\end{align*}"}
-{"id": "310.png", "formula": "\\begin{align*} M _ { g } ^ * ( x ) : = \\max \\biggl \\{ \\max _ { t = 1 , \\ldots , m } \\| g ^ { ( t ) } ( x ) \\| \\ , , \\ , \\sup _ { y \\in B _ x ^ \\circ ( r _ a ( x ) ) } \\frac { \\| g ^ { ( m ) } ( y ) - g ^ { ( m ) } ( x ) \\| } { \\| y - x \\| ^ { \\beta - m } } \\biggr \\} \\end{align*}"}
-{"id": "9079.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { 0 } \\lambda _ { j } e ^ { s \\lambda _ { j } } s ^ { k } \\ , \\mathrm { d } s = \\frac { 1 } { \\lambda _ { j } ^ { k } } ( - 1 ) ^ { k } k ! \\ . \\end{align*}"}
-{"id": "942.png", "formula": "\\begin{align*} \\mathrm { A P P } _ d ( f ) \\ , : = \\ , f . \\end{align*}"}
-{"id": "1220.png", "formula": "\\begin{align*} \\mathcal { M } _ { \\varphi , w } = \\{ x \\in L ^ 0 , \\ \\ \\ P ( \\lambda x ) < \\infty \\ \\ \\ \\ \\ \\lambda > 0 \\} . \\end{align*}"}
-{"id": "2953.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\binom { \\alpha n } { \\beta n } = \\alpha H _ 2 ( \\beta / \\alpha ) , \\end{align*}"}
-{"id": "1642.png", "formula": "\\begin{align*} v _ { t } ^ { \\prime } - F \\left ( x , v _ { t } \\right ) & = A \\beta \\left ( \\beta - 1 \\right ) t ^ { \\beta - 2 } - s i g n \\left ( x _ { t } \\right ) \\left \\vert x _ { t } \\right \\vert ^ { \\alpha } \\\\ & = A \\beta \\left ( \\beta - 1 \\right ) t ^ { \\beta - 2 } - s i g n \\left ( A \\right ) \\left \\vert A \\right \\vert ^ { \\alpha } t ^ { \\alpha \\beta } = 0 \\end{align*}"}
-{"id": "1781.png", "formula": "\\begin{align*} a = \\sum \\limits _ { x \\in \\R } a _ x t ^ x \\end{align*}"}
-{"id": "5690.png", "formula": "\\begin{align*} \\Gamma = \\begin{pmatrix} A _ { 1 1 } ^ 0 \\ , E _ 1 & \\dots & A _ { 1 1 } ^ { M - 1 } E _ 1 \\end{pmatrix} \\in \\R ^ { r \\times ( M m ) } \\end{align*}"}
-{"id": "3395.png", "formula": "\\begin{align*} w ^ l \\pi _ k ( e _ k ) & = ( \\phi _ { k + 1 } \\circ \\cdots \\circ \\phi _ { k + l } ) ( \\pi _ { k + l } ( e _ { k + l } ) ) = \\pi _ k ( e _ { k + l } ) \\quad ( k + l < r ) \\\\ w ^ { r - k + l } \\pi _ k ( e _ k ) & = ( \\phi _ { k - 1 } \\circ \\cdots \\circ \\phi _ { r - 1 } \\circ \\phi _ r \\circ ( \\mathrm { i d } \\otimes \\phi _ 1 ) \\circ \\cdots ( \\mathrm { i d } \\otimes \\phi _ l ) ) ( z \\otimes \\pi _ l ( e _ l ) ) \\\\ & = \\pi _ k ( z e _ l ) \\quad ( l < k ) . \\end{align*}"}
-{"id": "3888.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { k - 1 } V _ i = \\sum _ { i = 1 } ^ { k - 1 } \\left ( B _ i + C _ i \\right ) . \\end{align*}"}
-{"id": "6193.png", "formula": "\\begin{align*} \\left | \\frac { f ( x ( 1 + z ) ) - f ( x ) } { x f ' ( x ) } \\right | = \\left | \\int _ { | 1 + z | } ^ 1 \\frac { g ( x y ) } { g ( x ) } y ^ { - 1 } \\d y \\right | \\leq 2 \\int _ { | 1 + z | } ^ 1 y ^ { - 1 - \\varepsilon } \\d y \\leq \\frac { 2 } { \\varepsilon } | 1 + z | ^ { - \\varepsilon } , \\end{align*}"}
-{"id": "4541.png", "formula": "\\begin{align*} \\mathcal { X } _ n ' = \\mathcal { X } _ n \\cup \\{ y : \\| y - x \\| \\leq r _ { n , v _ x } + r _ { n , v _ y } \\ , \\ , \\ , \\ , x \\in \\mathcal { X } _ n \\} . \\end{align*}"}
-{"id": "2638.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { l = 1 } ^ { h ( m , n ) } \\left [ \\| \\mathbf { B _ m ^ * \\Phi _ m ^ * \\Phi _ m a _ m ^ l } \\| ^ 2 \\right ] & \\rightarrow i n f , \\\\ G _ m ( \\lambda ) = d _ m ( \\lambda ) ( d _ m ( \\lambda ) ) ^ * & - F _ m ( \\lambda ) \\in D _ G . \\end{align*}"}
-{"id": "5100.png", "formula": "\\begin{align*} \\tau ( z ) = \\mathrm { t r } _ { \\mathbb { C } ^ { r + 1 } } ( \\mathbb { T } ^ { [ 1 , M ] } ( z ) ) = \\sum _ { a = 0 } ^ { r } \\mathbb { T } ^ { [ 1 , M ] } ( z ) _ { a a } . \\end{align*}"}
-{"id": "3762.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { - R ( u ) } ^ { R ( u ) } e ^ { i y t } \\hat { f } _ Y ( t ) d t = \\frac { 1 } { 2 \\pi } \\int _ { - R ( u ) } ^ { R ( u ) } e ^ { i y t } \\left [ e ^ { C _ 0 ( \\beta - i \\alpha t ) - C _ 0 ( \\beta ) + i \\alpha t u } \\right ] d t . \\end{align*}"}
-{"id": "8834.png", "formula": "\\begin{align*} \\theta ( \\tau ) = \\sum _ { n = - \\infty } ^ { \\infty } q ^ { n ^ { 2 } } = \\frac { \\eta _ { 2 } ^ { 5 } } { \\eta _ { 1 } ^ { 2 } \\eta _ { 4 } ^ { 2 } } \\end{align*}"}
-{"id": "5583.png", "formula": "\\begin{align*} D _ m = - i \\alpha \\cdot \\nabla + m \\beta = - i \\sum _ { k = 1 } ^ { n } \\alpha _ k \\partial _ { k } + m \\beta \\end{align*}"}
-{"id": "1343.png", "formula": "\\begin{align*} Z _ { t } = \\sum _ { j \\in J _ { \\phi } \\bigcap J _ { \\theta } } \\omega _ { j } Z _ { t - j } + \\sum _ { j \\in J _ { \\phi } \\bigcap J _ { \\theta } ^ C } \\psi _ { j } Z _ { t - j } + \\sum _ { j \\in J _ { \\phi } \\bigcup J _ { \\theta } } \\psi _ j e _ { t - j } \\end{align*}"}
-{"id": "3411.png", "formula": "\\begin{gather*} n \\beta + \\gamma = u _ 1 ^ { k _ 1 } \\cdots u _ r ^ { k _ r } q _ { k _ 1 } ^ { n _ 1 } \\ldots q _ { k _ m } ^ { n _ m } . \\end{gather*}"}
-{"id": "8225.png", "formula": "\\begin{align*} E W _ n ^ { ( 0 ) } - i \\gamma W _ n ^ { ( 0 ) } - \\Omega \\bar { W } _ n ^ { ( 0 ) } = \\bar { W } _ { n - 1 } ^ { ( 0 ) } , n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "4705.png", "formula": "\\begin{align*} x & = \\prod _ { i = 1 } ^ { \\alpha _ 1 } ( a _ 1 ^ { p _ { 1 i } } a _ 2 ^ { q _ { 1 i } } ) a _ 3 ^ { q _ { 2 1 } } \\prod _ { i = 2 } ^ { \\alpha _ 2 } ( a _ 2 ^ { p _ { 2 i } } a _ 3 ^ { q _ { 2 i } } ) \\\\ & \\quad \\ \\dots a _ m ^ { q _ { ( m - 1 ) 1 } } \\prod _ { i = 2 } ^ { \\alpha _ { m - 1 } } ( a _ { m - 1 } ^ { p _ { ( m - 1 ) i } } a _ m ^ { q _ { ( m - 1 ) i } } ) , \\end{align*}"}
-{"id": "6942.png", "formula": "\\begin{align*} c ^ \\pi _ { ( 1 ^ { i _ 1 } ) ( 1 ^ { i _ 2 } ) \\cdots ( 1 ^ { i _ m } ) } = c ^ { \\pi ' } _ { ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) } \\ , . \\end{align*}"}
-{"id": "1482.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm B M O } _ X } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { \\| \\chi _ Q \\| _ X } \\| ( b - b _ Q ) \\chi _ Q \\| _ X . \\end{align*}"}
-{"id": "3014.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ 1 ] } } ( n ) = { \\bf { V } } _ 1 ^ { [ 1 ] } ( n ) { { \\bf { u } } ^ { [ 1 ] } } + { \\bf { V } } _ 2 ^ { [ 1 ] } ( n ) { { \\bf { v } } ^ { [ 1 ] } } , { { \\bf { X } } ^ { [ 2 ] } } ( n ) = { \\bf { V } } _ 1 ^ { [ 2 ] } ( n ) { { \\bf { u } } ^ { [ 2 ] } } + { \\bf { V } } _ 2 ^ { [ 2 ] } ( n ) { { \\bf { v } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "899.png", "formula": "\\begin{align*} \\int _ { T } ^ { 2 T } { \\left ( \\tfrac { t } { 2 \\pi } \\right ) } ^ { { 1 } / { 4 } + { \\delta } / { 2 } } e ^ { \\frac { i t } { 2 } \\log ( { t } / { 2 \\pi e n ^ { 2 } ) } } d t = O \\left ( T ^ { { 1 } / { 4 } + { \\delta } / { 2 } } T ^ { { 1 } / { 2 } } \\right ) = O \\left ( T ^ { { 3 } / { 4 } + { \\delta } / { 2 } } \\right ) . \\end{align*}"}
-{"id": "1503.png", "formula": "\\begin{align*} & K ( x ) = 1 + k ( x ) \\ , \\Big ( b _ 0 + x \\ , b _ 0 ^ + + \\frac 1 x \\ , b _ 0 ^ - \\Big ) , \\\\ & \\mbox { w i t h } k ( x ) = \\frac { \\left ( x ^ 2 - 1 \\right ) \\left ( \\alpha + \\gamma \\right ) } { \\left ( \\gamma x + \\alpha \\right ) \\left ( ( \\alpha + \\gamma ) ( x - 1 ) + ( q - 1 ) x \\right ) } \\ , . \\end{align*}"}
-{"id": "4443.png", "formula": "\\begin{align*} I _ { n , q } : = \\bigl [ \\hat { H } _ n ^ w - n ^ { - 1 / 2 } z _ { q / 2 } ( \\hat { V } _ n ^ w ) ^ { 1 / 2 } , \\hat { H } _ n ^ w + n ^ { - 1 / 2 } z _ { q / 2 } ( \\hat { V } _ n ^ w ) ^ { 1 / 2 } \\bigr ] , \\end{align*}"}
-{"id": "9281.png", "formula": "\\begin{align*} \\pi ( t , z ) = \\hat { \\pi } ( t , z ) = \\frac { \\Phi _ K ( t , z ) } { b _ 0 ( t , z ) } + \\frac { a _ 0 ( t , z ) } { \\sigma ^ 2 _ 0 ( t , z ) } . \\end{align*}"}
-{"id": "5866.png", "formula": "\\begin{align*} F \\left ( t , x \\right ) = x ^ { \\exp \\left ( - m t \\right ) } \\exp \\left ( - \\frac { 1 } { 4 m } e ^ { - 2 m t } - \\frac { c _ { 1 } } { m } e ^ { - m t } \\right ) ~ , ~ m \\neq 0 \\end{align*}"}
-{"id": "4969.png", "formula": "\\begin{align*} D = \\sum _ { i = 1 } ^ { m } a _ { i } H _ { i } \\end{align*}"}
-{"id": "4652.png", "formula": "\\begin{align*} I _ { x _ + } = \\left \\{ I , \\begin{pmatrix} 0 & 1 \\\\ - 1 & - 1 \\end{pmatrix} , \\begin{pmatrix} - 1 & - 1 \\\\ 1 & 0 \\end{pmatrix} \\right \\} . \\end{align*}"}
-{"id": "1301.png", "formula": "\\begin{align*} T _ H ( N , y ; \\alpha ) = H \\sum _ { n = N - H } ^ { N + y } e ( n \\alpha ) - \\sum _ { m = - y } ^ { H } m e ( ( N - m ) \\alpha ) = A - D , \\end{align*}"}
-{"id": "7926.png", "formula": "\\begin{align*} 2 \\binom { 3 m + 1 } { 3 } - 4 \\binom { m + 2 } { 3 } = 3 \\sum _ { i = 1 } ^ { m } x _ { i } ^ { 2 } + 2 ( m - 2 ) h ^ { 2 } . \\end{align*}"}
-{"id": "2434.png", "formula": "\\begin{align*} A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) = & n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) , \\\\ A _ 2 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) = & n \\alpha _ { 2 } ( 1 - 1 / p _ { 2 } ) - n \\alpha _ { 1 } ( 1 - 1 / p _ { 1 } ) - n ( \\alpha _ { 2 } - \\alpha _ { 1 } ) / q , \\\\ A _ 3 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) = & n ( \\alpha _ 2 - \\alpha _ 1 ) ( 1 / p _ 2 - 1 / q ) + n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) . \\end{align*}"}
-{"id": "6073.png", "formula": "\\begin{align*} m _ i + \\sum _ { s = j + 1 } ^ r \\ell _ { i + 1 , s } - \\sum _ { s = j } ^ r \\ell _ { i s } & = ( \\lambda _ i ^ { r + 1 } - \\lambda _ { i + 1 } ^ { r + 1 } ) + \\sum _ { s = j + 1 } ^ r ( \\lambda _ { i + 1 } ^ { s + 1 } - \\lambda _ { i + 1 } ^ s ) - \\sum _ { s = j } ^ r ( \\lambda _ { i } ^ { s + 1 } - \\lambda _ { i } ^ s ) \\\\ & = \\lambda _ i ^ { j } - \\lambda _ { i + 1 } ^ { j + 1 } . \\end{align*}"}
-{"id": "8571.png", "formula": "\\begin{align*} g & = \\displaystyle \\sum _ { s \\in L ( \\sigma ( b ) ) } ( s \\rho ( b w a ) ) ^ { \\ast } \\left ( \\rho ^ { k } \\left ( m _ { 1 } \\hdots m _ { l } \\Delta ( r ) \\Diamond z \\right ) \\right ) s \\\\ & = \\displaystyle \\sum _ { s \\in L ( \\sigma ( b ) ) } ( s \\rho ( b w a ) ) ^ { \\ast } \\left ( r \\rho ( c y c ( \\bar { e _ { k } } z m _ { 1 } \\hdots m _ { l } ) ) - \\rho ( c y c ( \\bar { e _ { k } } z m _ { 1 } \\hdots m _ { l } ) ) r \\right ) s \\end{align*}"}
-{"id": "6167.png", "formula": "\\begin{align*} [ L _ t , f _ r ] & = L _ t \\sum _ { z \\in Y } f ( z ) \\chi _ z - ( - 1 ) ^ { \\partial L \\partial f } \\sum _ { y \\in Y } f ( y ) \\chi _ y L _ t \\\\ & = \\sum _ { y , z \\in Y } \\chi _ y L _ t \\chi _ z ( f ( z ) - f ( y ) ) = \\sum _ { \\substack { y , z \\in Y \\\\ d ( y , z ) \\leq 2 r + p _ t } } \\chi _ y L _ t \\chi _ z ( f ( z ) - f ( y ) ) \\end{align*}"}
-{"id": "4850.png", "formula": "\\begin{align*} R ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = \\int \\frac { 1 - z ^ 2 } { z ^ 2 } \\frac { 1 } { 1 - c z } \\frac { 1 } { 1 - c / z } \\frac { h ^ { \\rm g e o } _ { 2 2 } ( z , 1 / z ) } { z ^ { u - v } } \\dd z . \\end{align*}"}
-{"id": "6133.png", "formula": "\\begin{align*} \\frac d { d t } ( a _ 1 + c _ 1 ) = & a _ 1 ^ 2 + 2 a _ 2 a _ 3 + c _ 1 ^ 2 + 2 c _ 2 c _ 3 \\\\ = & a _ 1 ( a _ 1 + a _ 2 + a _ 3 ) + c _ 1 ( c _ 1 + c _ 2 + c _ 3 ) + I \\\\ = & ( a _ 1 + c _ 1 ) \\cdot \\frac R 2 + I \\\\ = & ( \\kappa + \\delta t ) \\frac { R ^ 2 } 2 + I \\\\ \\end{align*}"}
-{"id": "7264.png", "formula": "\\begin{align*} \\delta \\left ( \\left \\{ \\left ( U _ j , \\sum _ { | \\alpha | = n , \\alpha _ 1 = 0 } F _ { j , \\alpha } ^ 1 \\cdot e _ { j , 1 } ^ * \\otimes e _ j ^ \\alpha \\right ) \\right \\} \\right ) = \\left \\{ \\left ( U _ { j k } , \\sum _ { | \\alpha | = n , \\alpha _ 1 = 0 } ( h _ { 1 , j k , \\alpha } ^ 1 - h _ { 2 , j k , \\alpha } ^ 1 ) \\cdot e _ { j , 1 } ^ * \\otimes e _ j ^ \\alpha \\right ) \\right \\} \\end{align*}"}
-{"id": "6052.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\Delta u _ { i } ^ { 1 } \\ge 0 & \\Omega , \\\\ u _ { i } ^ { 1 } ( x ) = u _ { i } ^ { 0 } ( x ) = \\phi _ { i } ( x ) & \\partial \\Omega . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "4562.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\ ! \\ ! \\ ! \\int _ 0 ^ { 1 - s } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\log \\Bigl ( \\ ! \\frac { ( n - 1 ) s } { e ^ { \\Psi ( j ) } } \\Bigr ) \\ ! \\log \\Bigl ( \\ ! \\frac { ( n - 1 ) t } { e ^ { \\Psi ( l ) } } \\Bigr ) \\ ! \\{ \\mathrm { B } _ { j , l , n - j - l - 1 } \\ ! ( s , t ) - & \\mathrm { B } _ { j , n - j } ( s ) \\mathrm { B } _ { l , n - l } ( t ) \\} d t d s \\\\ = & - \\frac { 1 } { n } + O ( n ^ { - 2 } ) \\end{align*}"}
-{"id": "6615.png", "formula": "\\begin{align*} \\| \\nabla f _ h ( x ) - \\nabla f _ h ( y ) \\| _ H ^ 2 & = \\langle \\nabla f _ h ( x ) - \\nabla f _ h ( y ) , \\nabla f _ h ( x ) - \\nabla f _ h ( y ) \\rangle _ H \\\\ & = \\langle L ^ { - 1 } ( \\nabla f ( x ) - \\nabla f ( y ) ) , L ^ { - 1 } ( \\nabla f ( x ) - \\nabla f ( y ) ) \\rangle _ L \\\\ & = \\langle \\nabla f ( x ) - \\nabla f ( y ) , \\nabla f ( x ) - \\nabla f ( y ) \\rangle _ { L ^ { - 1 } } \\\\ & = \\| \\nabla f ( x ) - \\nabla f ( y ) \\| _ { L ^ { - 1 } } ^ 2 . \\end{align*}"}
-{"id": "1260.png", "formula": "\\begin{align*} \\mathcal R _ V ^ { \\pm } ( \\lambda ) = \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) - \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) V \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) + \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) V \\mathcal R _ V ^ { \\pm } ( \\lambda ) V \\mathcal R _ 0 ^ { \\pm } ( \\lambda ) . \\end{align*}"}
-{"id": "4549.png", "formula": "\\begin{align*} \\mu _ d \\bigl ( B _ x ( r _ { n , v } ) \\cap B _ y ( r _ { n , v } ) \\bigr ) & = V _ d r _ { n , v } ^ d I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { \\| x - y \\| ^ 2 } { 4 r _ { n , v } ^ 2 } \\biggr ) \\\\ & = \\frac { v e ^ { \\Psi ( k ) } } { n - 1 } I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { \\| z \\| ^ 2 } { 4 \\{ v f ( x ) \\} ^ { 2 / d } } \\biggr ) \\end{align*}"}
-{"id": "8412.png", "formula": "\\begin{align*} \\forall ~ \\overline { n } , \\overline { m } \\in G _ { k - 1 } ( \\mathbb { M } _ 2 ) \\ \\ \\ | c _ { \\overline { n } } - c _ { \\overline { m } } | = | ( v ( \\overline { n } ) - v ( \\overline { m } ) ) - ( g ( \\overline { n } ) - g ( \\overline { m } ) ) | _ { J _ p ^ { * * } } < \\eta . \\end{align*}"}
-{"id": "757.png", "formula": "\\begin{gather*} \\sum _ k ( 1 \\otimes u _ { i k } ^ * ) \\bigg ( \\sum _ l S _ l \\otimes u _ { l k } \\bigg ) = S _ i \\otimes 1 \\end{gather*}"}
-{"id": "8083.png", "formula": "\\begin{align*} p _ { e } = \\frac { 3 } { 4 K ^ 2 \\bar { \\gamma } _ { s _ i , d } } \\left ( \\frac { 1 } { \\bar { \\gamma } _ { s _ i , r _ j } } + \\frac { 1 } { \\bar { \\gamma } _ { r _ j , d } } \\right ) \\end{align*}"}
-{"id": "5738.png", "formula": "\\begin{align*} S ( \\delta _ { 0 } ) = n ^ { - 1 / 2 } \\sum _ { t = 1 } ^ { n } \\left ( y _ { t } - m _ t \\pi _ { t } ( \\delta _ { 0 } ) \\right ) \\begin{bmatrix} x _ { t } \\\\ \\sum _ { j = 0 } ^ { \\infty } \\tau _ { j } ( \\omega ) e _ { t - J _ L - j } ( \\delta _ 0 ) \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "5199.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } & = \\| U ^ { * } \\| _ { S _ { p _ { 1 } } } \\| V ^ { * } \\| _ { S _ { p _ { 2 } } } \\\\ & \\leq \\| U _ { 1 } \\| _ { S _ { p _ { 1 } } } \\| V _ { 1 } \\| _ { S _ { p _ { 2 } } } \\\\ & \\leq \\| U \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| U _ { 2 } \\| _ { S _ { q } } \\| V _ { 1 } \\| _ { S _ { p _ { 2 } } } \\\\ & = \\| U \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| V \\| _ { S _ { \\widehat { p } _ { 2 } } } \\end{align*}"}
-{"id": "5742.png", "formula": "\\begin{align*} I _ { L } ( \\hat \\beta _ 0 ) = n ^ { - 1 } \\sum _ { t = 1 } ^ n \\sigma _ { t } ^ 2 ( \\hat \\beta _ 0 ) \\cdot \\textrm { d i a g } \\left ( \\sigma _ { t - j _ 1 } ^ { 2 - 2 \\gamma } ( \\hat \\beta _ 0 ) , \\ldots , \\sigma _ { t - j _ L } ^ { 2 - 2 \\gamma } ( \\hat \\beta _ 0 ) \\right ) . \\end{align*}"}
-{"id": "3446.png", "formula": "\\begin{align*} d y ( t ) = - \\Gamma ^ * ( t ) g ( [ \\Gamma ^ * ] ^ { - 1 } ( t ) y ( t ) , [ \\Gamma ^ * ] ^ { - 1 } ( t ) z ( t ) ) d t + z ( t ) d w ( t ) , \\end{align*}"}
-{"id": "5575.png", "formula": "\\begin{align*} D _ t V = \\begin{bmatrix} v _ t \\\\ v _ { t x } - \\sum _ { i = 1 } ^ N [ v _ x ] ( x _ i ) \\dot x _ i \\delta _ { x _ i } \\end{bmatrix} = \\begin{bmatrix} - \\frac 1 2 D _ x ( b ) + \\beta & b \\\\ - \\frac 1 2 D ^ 2 _ x ( b ) - z \\rho b & \\frac 1 2 D _ x ( b ) + \\beta \\end{bmatrix} V , \\end{align*}"}
-{"id": "3596.png", "formula": "\\begin{align*} \\Pi _ { g _ 0 } \\circ D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f _ 0 , X _ 0 ) = \\Pi _ { g _ 0 } ( \\psi , V ) \\ ; , \\end{align*}"}
-{"id": "8462.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\star } + \\sum _ { j = 1 } ^ { d } \\textbf { P } ( A _ { j } ( \\tilde { \\textbf { u } } ^ { \\star } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } ^ { \\star } ) = 0 \\end{align*}"}
-{"id": "8079.png", "formula": "\\begin{align*} T _ f ( x , y ) = ( \\sigma ( x ) , y + f ( x ) ) . \\end{align*}"}
-{"id": "701.png", "formula": "\\begin{align*} f ^ { * } H \\equiv \\sum _ { j = 1 } ^ { r } \\sum _ { k = 1 } ^ { r } c _ { j } a _ { k j } D _ { k } = \\left < A \\left ( \\begin{array} { c } c _ { 1 } \\\\ c _ { 2 } \\\\ \\vdots \\\\ c _ { r } \\end{array} \\right ) , \\left ( \\begin{array} { c } D _ { 1 } \\\\ D _ { 2 } \\\\ \\vdots \\\\ D _ { r } \\end{array} \\right ) \\right > = \\left < A \\vec { c } , \\vec { D } \\right > . \\end{align*}"}
-{"id": "3415.png", "formula": "\\begin{align*} d \\eta ( t ) = c ( \\xi ( t ) , u ( t , \\xi ( t ) ) ) \\eta ( t ) d t + C ( \\xi ( t ) , u ( t , \\xi ( t ) ) ) ( \\eta ( t ) , d w ( t ) ) , \\eta ( s ) = h . \\end{align*}"}
-{"id": "7180.png", "formula": "\\begin{align*} f ' ( 0 ) = g ( 0 ) = h ( 0 ) = 0 . \\end{align*}"}
-{"id": "685.png", "formula": "\\begin{align*} V = \\{ g \\in G \\ , : \\ , f ( g ) \\geq 0 \\big \\} \\end{align*}"}
-{"id": "6577.png", "formula": "\\begin{align*} x ^ { k + 1 } & = x ^ k - \\alpha P ^ { - 1 } \\nabla F ( x ^ k ) \\\\ & = x ^ k - \\alpha P ^ { - 1 } P ( S _ 2 S _ 1 x ^ k - x ^ k ) \\\\ & = x ^ k - \\alpha ( S _ 2 S _ 1 x ^ k - x ^ k ) \\\\ & = ( 1 - \\alpha ) x ^ k + \\alpha S _ 2 S _ 1 x ^ k , \\end{align*}"}
-{"id": "2403.png", "formula": "\\begin{align*} \\nabla h ( \\tau ) = \\langle y - x , \\nabla f ( x + \\tau ( y - x ) ) \\rangle \\end{align*}"}
-{"id": "3797.png", "formula": "\\begin{align*} S _ U & = \\bigcup _ { i = 1 } ^ s \\Omega _ { \\alpha _ i } = \\bigcup _ { i = 1 } ^ s \\big ( G ( \\ell , m ) \\cap Z _ { \\overline { \\mathbb F } _ q } ( x _ { \\beta } : \\beta \\in I ( \\ell , m ) , \\ ; \\beta \\not \\leq \\alpha _ i ) \\big ) \\\\ & = G ( \\ell , m ) \\cap \\bigcup _ { i = 1 } ^ s Z _ { \\overline { \\mathbb F } _ q } \\big ( x _ { \\beta } : \\beta \\in I ( \\ell , m ) , \\ ; \\beta \\not \\leq \\alpha _ i \\big ) . \\end{align*}"}
-{"id": "9094.png", "formula": "\\begin{align*} K _ { N } ( s ) = \\int _ { 0 } ^ { \\infty } x ^ { 2 } e ^ { s x } x ^ { \\alpha } e ^ { - x } L ' _ { N - 1 } ( x ) L _ { N - 2 } ( x ) \\ , \\mathrm { d } x \\ . \\end{align*}"}
-{"id": "4476.png", "formula": "\\begin{align*} R _ 2 = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ \\frac { a _ n } { n - 1 } ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log u _ { x , s } \\ , d s \\ , d x = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "4844.png", "formula": "\\begin{align*} \\mathcal { U } ^ { \\angle } _ { \\rho _ { \\circ } , \\rho _ 1 } ( \\pi \\vert \\kappa , \\mu ) = \\mathcal { U } ^ { \\angle } _ { \\rho _ { \\circ } , \\rho _ 1 } ( \\pi \\vert \\kappa ) = \\frac { 1 } { H ^ o ( \\rho _ 1 ) H ( \\rho _ 1 ; \\rho _ { \\circ } ) } \\frac { \\tau _ { \\pi } ( \\rho _ { \\circ } ) s _ { \\pi / \\kappa } ( \\rho _ 1 ) } { \\tau _ { \\kappa } ( \\rho _ { \\circ } , \\rho _ 1 ) } . \\end{align*}"}
-{"id": "8467.png", "formula": "\\begin{align*} v ^ { \\varepsilon } = \\mathbb { P } v ^ { \\varepsilon } + \\varepsilon \\nabla P ^ { \\varepsilon } . \\end{align*}"}
-{"id": "3265.png", "formula": "\\begin{align*} D = \\eth + \\eth ^ * = \\sum _ { i } E _ { \\xi _ i } \\otimes \\gamma _ - ( w _ i ) + \\sum _ { i } F _ { \\xi _ i } \\otimes \\gamma _ + ( v _ i ) . \\end{align*}"}
-{"id": "4808.png", "formula": "\\begin{gather*} d ' \\sim \\sigma d ' = \\tau d \\xi \\unrhd d \\end{gather*}"}
-{"id": "290.png", "formula": "\\begin{align*} \\biggl | \\frac { d } { d r } I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { r ^ 2 } { 4 } \\biggr ) \\biggr | = \\frac { ( 1 - r ^ 2 / 4 ) ^ \\frac { d - 1 } { 2 } } { \\mathrm { B } _ { ( d + 1 ) / 2 , 1 / 2 } } \\leq \\frac { 1 } { \\mathrm { B } _ { ( d + 1 ) / 2 , 1 / 2 } } . \\end{align*}"}
-{"id": "196.png", "formula": "\\begin{align*} C _ { n , \\beta } ( x ) : = \\left \\{ \\begin{array} { l l } \\sup _ { y \\in B _ x ^ \\circ ( r _ x ) } | f ( y ) - f ( x ) | / \\| y - x \\| ^ \\beta & \\mbox { i f $ \\beta \\leq 1 $ , } \\\\ \\sup _ { y \\in B _ x ^ \\circ ( r _ x ) } \\| \\dot { f } ( y ) - \\dot { f } ( x ) \\| / \\| y - x \\| ^ { \\beta - 1 } & \\mbox { i f $ \\beta > 1 $ , } \\end{array} \\right . \\end{align*}"}
-{"id": "4743.png", "formula": "\\begin{align*} \\int _ { B _ 1 ^ c } \\frac 1 { \\ , | z | ^ { N + 2 s } \\ , } \\int _ { B ^ c _ { R _ t } ( \\mp z ) } \\frac { \\ , 1 + | y | ^ { 2 s - \\sigma } \\ , } { \\ , | y \\pm z | ^ { N + 2 s } \\ , } \\ , d y \\ , d z & = \\int _ { B _ 1 ^ c } \\frac 1 { \\ , | z | ^ { N + 2 s } \\ , } \\int _ { B ^ c _ { R _ t } } \\frac { \\ , 1 + | w \\mp z | ^ { 2 s - \\sigma } \\ , } { | w | ^ { N + 2 s } } \\ , d w \\ , d z \\\\ & \\le \\int _ { B _ 1 ^ c } C \\ , \\frac { \\ , 1 + | z | ^ { 2 s - \\sigma } \\ , } { | z | ^ { N + 2 s } } \\ , d z < + \\infty . \\end{align*}"}
-{"id": "1161.png", "formula": "\\begin{align*} \\begin{bmatrix} \\frac 1 2 & - h & h ^ 2 \\\\ \\frac 1 2 + H + \\frac 1 2 H ^ 2 & H + H ^ 2 & H ^ 2 \\end{bmatrix} \\begin{bmatrix} C _ 1 \\\\ C _ 2 \\\\ C _ 3 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ C \\end{bmatrix} , \\end{align*}"}
-{"id": "9427.png", "formula": "\\begin{align*} T ( d _ K ) = \\sum \\bigl \\{ \\ , \\abs { f ' ( d _ G ) } \\mid d _ G , f ( d _ G ) = d _ K \\ , \\} . \\end{align*}"}
-{"id": "973.png", "formula": "\\begin{align*} X _ { \\varepsilon } = e ^ { g _ { \\varepsilon } } T + \\sum _ { j = 1 } ^ { m - 1 } \\left ( b _ { \\varepsilon , j } L _ { j } + \\overline { b _ { \\varepsilon , j } L _ { j } } \\right ) \\ ; , \\end{align*}"}
-{"id": "9289.png", "formula": "\\begin{align*} J _ { \\tilde { P } } ( \\pi ) = \\mathbb { E } _ { \\tilde { P } } [ \\int _ { \\mathbb { R _ { + } } } U ( x , Z ) y ( T , x , Z ) d x ] , \\end{align*}"}
-{"id": "8962.png", "formula": "\\begin{align*} y ^ { \\Delta } ( t ) = p ( t ) y ( t ) , ~ ~ ~ y ( t _ 0 ) = c _ 0 . \\end{align*}"}
-{"id": "226.png", "formula": "\\begin{align*} \\mathbb { P } ( \\xi _ 1 \\leq u | X _ 1 = x , X _ 2 = y ) = \\left \\{ \\begin{array} { l l } F _ { n , x } ^ - ( u ) & \\mbox { i f $ \\| x - y \\| > r _ { n , u } $ } \\\\ \\tilde { F } _ { n , x } ( u ) & \\mbox { i f $ \\| x - y \\| \\leq r _ { n , u } $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "2329.png", "formula": "\\begin{align*} W _ { p , l } ( x ) = \\frac { p ( x ) } { p _ n l ' ( x ) } , \\end{align*}"}
-{"id": "3608.png", "formula": "\\begin{align*} \\int _ { \\{ R _ 1 \\le | x | \\le R _ 2 \\} } D \\Phi | _ { ( g _ { \\mathbb { E } } , 0 ) } ( g - g _ { \\mathbb { E } } , \\pi ) \\cdot ( N , X ) \\ , d x = B ^ { R _ 2 } _ { ( g , \\pi ) } ( N , X ) - B ^ { R _ 1 } _ { ( g , \\pi ) } ( N , X ) . \\end{align*}"}
-{"id": "2753.png", "formula": "\\begin{align*} \\Omega = \\nu ^ { - 1 } \\Omega _ { - 1 } + \\Omega _ 0 + \\nu \\Omega _ 1 + \\ldots \\end{align*}"}
-{"id": "9912.png", "formula": "\\begin{align*} h _ 1 & \\ ; = \\ ; E K ( \\kappa K ^ 2 + \\kappa ^ { - 1 } E F ) - \\kappa q ^ 2 K '^ 2 E K , \\\\ h _ 2 & \\ ; = \\ ; F K ( \\kappa K ^ 2 + \\kappa ^ { - 1 } E F ) - \\kappa q ^ { - 2 } K '^ 2 F K \\hbox { a n d } \\\\ h _ 3 & \\ ; = \\ ; K ^ 2 ( \\kappa K ^ 2 + \\kappa ^ { - 1 } E F ) - \\kappa K '^ 2 K ^ 2 . \\end{align*}"}
-{"id": "3231.png", "formula": "\\begin{align*} P _ { \\gamma } ^ { S } = Q ^ { S } \\circ P _ { \\gamma } ^ { N } \\label { g e n r e s p E E E E q n } \\end{align*}"}
-{"id": "7018.png", "formula": "\\begin{align*} P _ { k + 1 } ( z ) = P _ k ( z ) + z ^ { 2 ^ k } Q _ k ( z ) , \\end{align*}"}
-{"id": "9520.png", "formula": "\\begin{align*} \\frac { 9 } { 2 } \\mathrm { d v o l } _ g e ^ { - 2 \\phi } \\Delta \\phi & = \\frac { 9 } { 2 } e ^ { - 2 \\phi } d * d \\phi \\\\ & = d \\left ( \\frac { 9 } { 2 } e ^ { - 2 \\phi } * d \\phi \\right ) + 9 e ^ { - 2 \\phi } d \\phi \\wedge * d \\phi \\\\ & = d ( \\ldots ) + 9 e ^ { - 2 \\phi } | d \\phi | ^ 2 \\mathrm { d v o l } _ g . \\end{align*}"}
-{"id": "2673.png", "formula": "\\begin{align*} \\theta _ { \\varphi } ^ n = e ^ { \\beta \\varphi } \\mu , \\ \\ \\varphi \\in \\mathcal E ^ 1 ( X , \\omega ) , \\end{align*}"}
-{"id": "994.png", "formula": "\\begin{align*} A ( u ) C ( v ) & = \\frac { u - v + \\eta } { u - v } C ( v ) A ( u ) - \\frac { \\eta } { u - v } C ( u ) A ( v ) \\\\ D ( u ) C ( u ) & = \\frac { u - v - \\eta } { u - v } C ( v ) D ( u ) + \\frac { \\eta } { u - v } C ( u ) D ( v ) . \\end{align*}"}
-{"id": "8181.png", "formula": "\\begin{align*} H _ 1 = \\sum _ { i = 1 } ^ 3 a _ i x _ i ^ 2 , \\ \\ \\ H _ 2 = \\sum _ { i = 1 } ^ 3 x _ i ^ 2 , \\end{align*}"}
-{"id": "4197.png", "formula": "\\begin{align*} h ( x ) = \\begin{cases} e ^ { x - a } & x < a , \\\\ 1 & a \\le x \\le b , \\\\ e ^ { b - x } & x > b . \\end{cases} \\end{align*}"}
-{"id": "9638.png", "formula": "\\begin{align*} \\sum _ { s = 1 } ^ { k } \\ \\Pi _ { \\ , \\alpha ^ { + } _ { s } } ^ { ( 1 ) } \\ D _ { i _ { s + 1 } i _ { s } } \\ \\Pi _ { \\ , \\alpha ^ { - } _ { s } } ^ { ( 2 ) } \\ , . \\end{align*}"}
-{"id": "3495.png", "formula": "\\begin{align*} I ( u ) = \\int _ \\Omega | \\nabla u | ^ 2 + 2 u \\ , d x . \\end{align*}"}
-{"id": "7080.png", "formula": "\\begin{align*} d ( u , v ) = 2 ^ { 1 - \\min ( u ( | u \\wedge v | + 1 ) , v ( u \\wedge v | + 1 ) ) - \\sum _ { j = 1 } ^ { | u \\wedge v | } u ( j ) } , \\end{align*}"}
-{"id": "8426.png", "formula": "\\begin{align*} A _ { 0 } ( \\textbf { u } ) = d i a g \\Bigg ( \\frac { f ( \\textbf { u } ) } { \\rho } , 1 , 1 , \\cdots , 1 \\Bigg ) . \\end{align*}"}
-{"id": "8681.png", "formula": "\\begin{align*} \\Big ( \\sum _ { m \\ge 1 } \\sup _ { | a | _ U = 1 } | \\nabla _ k \\nabla _ { a } ^ G P _ t [ \\Phi _ m ] ( x ) | ^ 2 \\Big ) ^ { 1 / 2 } \\le \\frac { C } { t ^ { \\frac { 4 - 3 \\alpha } { 2 } } } | k | _ K \\ , \\| \\Phi \\| _ { C ^ { \\alpha } _ b ( H , J ) } . \\end{align*}"}
-{"id": "6114.png", "formula": "\\begin{align*} \\frac 1 { \\pi s ^ 3 } \\int _ s ^ 1 ( 1 - \\cos 2 \\pi \\tau ) d \\tau = \\frac { 1 } { \\pi s ^ 3 } ( 1 - s + \\frac { \\sin { 2 \\pi s } } { 2 \\pi } ) . \\end{align*}"}
-{"id": "1776.png", "formula": "\\begin{align*} \\int _ \\Omega \\int _ { \\Omega _ 1 \\setminus \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\overline { u } _ { i } ( x ) \\phi _ j ( y ) K ( x , y ) d y d x = \\int _ \\Omega \\int _ { \\Omega _ 1 \\setminus \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\underline { u } _ { i } ( x ) \\phi _ j ( y ) K ( x , y ) \\ , d y \\ , d x . \\end{align*}"}
-{"id": "9539.png", "formula": "\\begin{align*} g _ M ( \\nabla _ { a ' } \\nabla _ { 1 0 ' } e _ { 1 0 } ' , e _ { d } ' ) \\eta ^ { a d } & = e ^ { 2 \\phi / 3 } \\left ( \\frac { 1 6 } { 9 } | d \\phi | ^ 2 - \\frac { 2 } { 3 } \\Delta \\phi \\right ) \\\\ g _ M ( \\nabla _ { 1 0 ' } \\nabla _ { a ' } e _ { 1 0 } ' , e ' _ { d } ) \\eta ^ { a d } & = - \\frac { 1 } { 2 } e ^ { 8 \\phi / 3 } | G _ 2 | ^ 2 \\\\ g _ M ( \\nabla _ { [ a ' , 1 0 ' ] } e _ { 1 0 } ' , e _ { d } ' ) \\eta ^ { a d } & = \\frac { 4 } { 9 } e ^ { 2 \\phi / 3 } | d \\phi | ^ 2 \\end{align*}"}
-{"id": "936.png", "formula": "\\begin{align*} R _ 1 = P ^ { ( 1 ) } _ 1 \\cup P ^ { ( 1 ) } _ 2 \\cup \\cdots P ^ { ( 1 ) } _ r , \\ R _ 2 = P ^ { ( 2 ) } _ 1 \\cup P ^ { ( 2 ) } _ 2 \\cup \\cdots P ^ { ( 2 ) } _ r \\end{align*}"}
-{"id": "2883.png", "formula": "\\begin{align*} S y m ( V ) = \\mathbb { K } \\oplus S y m ^ { \\geq 1 } ( V ) \\end{align*}"}
-{"id": "4770.png", "formula": "\\begin{align*} f f '' + ( f ' ) ^ 2 - 1 = \\pm 2 a f \\sqrt { f '^ 2 - 1 } , a = c o n s t \\neq 0 . \\end{align*}"}
-{"id": "741.png", "formula": "\\begin{align*} & b _ { 0 } = a _ { 0 } \\\\ & b _ { n } = C \\left ( b _ { 0 } + \\sqrt [ ] { b _ { 0 } } + \\cdots + \\sqrt [ ] { b _ { n - 1 } } \\right ) \\ \\ \\ \\ . \\end{align*}"}
-{"id": "4152.png", "formula": "\\begin{align*} \\alpha & = \\frac { 5 } { 1 4 } + \\epsilon , \\sigma _ 0 = 0 , \\beta = \\frac { 3 } { 2 } + \\epsilon , \\\\ \\gamma _ { 1 } & = 1 , R _ 1 \\sim \\langle f , f \\rangle , \\sigma _ 1 = \\frac { 6 } { 7 } + \\epsilon , \\eta _ { 1 } = \\frac { 9 } { 1 4 } + \\epsilon , \\\\ \\gamma _ { 2 } & = 1 , R _ 2 \\sim \\langle f , \\overline { f } \\rangle , \\sigma _ 2 = \\frac { 6 } { 7 } + \\epsilon , \\eta _ { 2 } = \\frac { 9 } { 1 4 } + \\epsilon . \\end{align*}"}
-{"id": "3482.png", "formula": "\\begin{align*} \\tilde { u } ( x ' , x _ n ) = \\begin{cases} u ( x ' , x _ n ) & \\mbox { i f } \\ , \\ , \\ , x _ n \\geq 0 , \\\\ - u ( x ' , - x _ n ) & \\mbox { i f } \\ , \\ , \\ , x _ n < 0 . \\end{cases} \\end{align*}"}
-{"id": "7137.png", "formula": "\\begin{align*} G _ { i j } ( x , y ) | _ { x _ n = 0 } = 0 . \\end{align*}"}
-{"id": "8665.png", "formula": "\\begin{align*} Q = \\left ( \\begin{array} [ c ] { c c } \\Lambda ^ { - 1 } & 0 \\\\ 0 & \\Lambda ^ { - 1 } \\end{array} \\right ) , \\ ; \\ ; \\ ; \\ ; Q h = \\Big ( \\begin{array} [ c ] { c } \\Lambda ^ { - 1 } h _ 1 \\\\ \\Lambda ^ { - 1 } h _ 2 \\end{array} \\Big ) , \\ ; \\ ; \\ ; h = ( h _ 1 , h _ 2 ) \\in H = U \\times V ' . \\end{align*}"}
-{"id": "7503.png", "formula": "\\begin{align*} x _ { n + 1 } = f ( x _ n ) - \\alpha ( f ( x _ n ) - x _ n ) = ( 1 - \\alpha ) f ( x _ n ) + \\alpha x _ n , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 . \\end{align*}"}
-{"id": "4759.png", "formula": "\\begin{align*} H = - \\frac { \\kappa } { 2 f } \\ , n _ 1 + \\frac { f f '' + ( f ' ) ^ 2 - 1 } { 2 f \\sqrt { f '^ 2 - 1 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "5831.png", "formula": "\\begin{align*} X = \\xi ^ { i } \\left ( x ^ { k } , u \\right ) \\partial _ { i } + \\eta \\left ( x ^ { k } , u \\right ) \\partial _ { u } . \\end{align*}"}
-{"id": "4241.png", "formula": "\\begin{align*} ( 1 + t ) ^ { \\tfrac { x } { 2 } } & = \\sum _ { m = 0 } ^ \\infty \\Big ( \\frac { x } { 2 } \\Big ) ^ m \\frac { \\big ( \\log ( 1 + t ) \\big ) ^ m } { m ! } = \\sum _ { m = 0 } ^ \\infty \\Big ( \\frac { x } { 2 } \\Big ) ^ m \\sum _ { n = m } ^ \\infty S _ 1 ( n , m ) \\frac { t ^ n } { n ! } \\\\ & = \\sum _ { n = 0 } ^ \\infty \\left ( \\sum _ { m = 0 } ^ n \\Big ( \\frac { x } { 2 } \\Big ) ^ m S _ 1 ( n , m ) \\right ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "1422.png", "formula": "\\begin{align*} \\int a ( z ) \\ d z = a _ { ( 0 ) } \\end{align*}"}
-{"id": "512.png", "formula": "\\begin{align*} \\sum _ { \\nu } s _ { \\nu / \\lambda } ( \\rho ) s _ { \\nu / \\mu } ( \\rho ' ) = H ( \\rho ; \\rho ' ) \\sum _ { \\tau } s _ { \\lambda / \\tau } ( \\rho ' ) s _ { \\mu / \\tau } ( \\rho ) , \\end{align*}"}
-{"id": "6441.png", "formula": "\\begin{align*} = \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } \\psi ( x ) \\psi ( y ) \\left ( \\left ( \\frac { \\tilde { u } ( s , x ) } { \\psi ( x ) } \\right ) ^ { \\frac { 1 - q } { 2 } } - \\left ( \\frac { \\tilde { u } ( s , y ) } { \\psi ( y ) } \\right ) ^ { \\frac { 1 - q } { 2 } } \\right ) ^ { 2 } k ( x , y ) d x d y , \\end{align*}"}
-{"id": "5553.png", "formula": "\\begin{align*} v ( x ) = \\begin{cases} h x + 1 & 0 \\leq h < \\infty , \\\\ x + \\frac { 1 } { h } & 0 < h \\leq \\infty , \\end{cases} \\end{align*}"}
-{"id": "8319.png", "formula": "\\begin{align*} z _ { i _ 0 } = - \\sum _ { i \\neq i _ 0 } \\alpha _ { i _ 0 } ^ { - 1 } \\alpha _ i z _ i + \\alpha _ { i _ 0 } ^ { - 1 } \\beta + \\alpha _ { i _ 0 } A \\in k ( z _ 1 , \\ldots , z _ { i _ 0 - 1 } , z _ { i _ 0 + 1 } , \\ldots , z _ n ) , \\end{align*}"}
-{"id": "8476.png", "formula": "\\begin{align*} + \\sum _ { j = 1 } ^ { d } ( A _ { 0 } ( A _ { 0 } ^ { - 1 } J _ { \\varepsilon } A _ { j } ( J _ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\bar { \\textbf { u } } ) ) ) ' J _ { \\varepsilon } \\textbf { w } ^ { \\varepsilon } \\partial _ { x _ { j } } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } , \\textbf { w } ^ { \\varepsilon } ) _ { 0 } . \\end{align*}"}
-{"id": "933.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t a + u \\cdot \\nabla a = R _ 1 ( t , x ) , \\\\ \\partial _ t b + u \\cdot \\nabla b = R _ 2 ( t , x ) , \\\\ a | _ { t = 0 } = a _ 0 ( x ) , \\\\ b | _ { t = 0 } = b _ 0 ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "6245.png", "formula": "\\begin{align*} \\| H ( s ) - \\tilde { H } ( s ) \\| _ { H _ { 2 } } = \\mathcal { O } ( \\| Z \\| ) . \\end{align*}"}
-{"id": "9988.png", "formula": "\\begin{align*} \\max _ { a \\in \\mathcal { A } ( s ) } \\left [ g ( s , a ) + \\tau \\sum _ { \\mathbf { H } ' } \\mathrm { P r } ( \\mathbf { H } ' | \\mathbf { H } ) h ^ { ( n ) } ( s ' ) \\right ] = g ( s , A _ 1 ^ * , A _ 2 ^ * ) + \\tau \\sum _ { \\mathbf { H } ' } \\mathrm { P r } ( \\mathbf { H } ' | \\mathbf { H } ) h ^ { ( n ) } ( B _ 1 ' , B _ 2 ' , \\mathbf { H } ' ) . \\end{align*}"}
-{"id": "83.png", "formula": "\\begin{align*} x y ^ 3 - y ^ 3 x & = - ( y x y + y ^ 2 x + a x ^ 3 ) y + y ( x y ^ 2 + y x y + a x ^ 3 ) = - a x ^ 3 y + a y x ^ 3 \\\\ & = a x ( y x ^ 2 + x y x + b y ^ 3 ) - a ( x y x + x ^ 2 y + b y ^ 3 ) x = a b x y ^ 3 - a b y ^ 3 x , \\end{align*}"}
-{"id": "1505.png", "formula": "\\begin{align*} \\check K ( x ) = ( 1 - ( x - 1 ) e _ 0 ) \\left ( 1 - \\left ( \\frac { 1 } { x } - 1 \\right ) e _ 0 \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "4644.png", "formula": "\\begin{align*} \\int _ 0 ^ \\infty K _ \\nu ( x ) x ^ { s - 1 } d x & = 2 ^ { s - 2 } \\Gamma \\left ( \\frac { s + \\nu } { 2 } \\right ) \\Gamma \\left ( \\frac { s - \\nu } { 2 } \\right ) , \\Re s > | \\Re \\nu | , \\\\ \\int _ 0 ^ \\infty \\cos ( x ) x ^ { s - 1 } d x & = \\Gamma ( s ) \\cos \\left ( \\frac { \\pi s } { 2 } \\right ) , 0 < \\Re s < 1 . \\end{align*}"}
-{"id": "4780.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c t + a ) ^ 2 + t ^ 2 } , a = c o n s t , \\ ; c = c o n s t \\neq 0 , \\ ; c ^ 2 \\neq \\kappa ^ 2 , \\end{align*}"}
-{"id": "4524.png", "formula": "\\begin{align*} S _ 5 = \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\log ^ 2 f ( x ) \\ , d x = O \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "5726.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm B M O } _ X } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { \\| \\chi _ Q \\| _ X } \\| ( b - b _ Q ) \\chi _ Q \\| _ X . \\end{align*}"}
-{"id": "8300.png", "formula": "\\begin{align*} 2 g ( \\tilde { x } ) = \\sum _ { \\tilde { v } \\in T _ { \\tilde { x } } ( \\widetilde { \\Gamma } ) } ( d _ { \\tilde { v } } ( \\varphi ) - 1 ) \\end{align*}"}
-{"id": "7099.png", "formula": "\\begin{align*} W _ n ^ { ( \\beta ) } = \\sum _ { | u | = n } e ^ { - \\beta V ( u ) - n \\kappa ( \\beta ) } . \\end{align*}"}
-{"id": "7158.png", "formula": "\\begin{align*} u = 0 \\mbox { o n } \\ , \\ , \\Sigma _ r . \\end{align*}"}
-{"id": "2627.png", "formula": "\\begin{align*} \\hat { A } { \\zeta } = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { l = 1 } ^ { h ( m , n ) } \\int _ { - \\pi } ^ { \\pi } ( h _ m ^ l ( \\lambda ) ) ^ { \\top } Z _ m ^ { l \\ ; \\zeta + \\theta } ( d \\lambda ) , \\end{align*}"}
-{"id": "9087.png", "formula": "\\begin{align*} L ^ { ( \\alpha ) } _ { N } ( c x ) = \\sum _ { r = 0 } ^ { N } \\binom { N + \\alpha } { N - r } c ^ { r } ( 1 - c ) ^ { N - r } L ^ { ( \\alpha ) } _ { r } ( x ) \\ , \\end{align*}"}
-{"id": "1268.png", "formula": "\\begin{align*} \\begin{bmatrix} q & - \\alpha J ( T ^ { - i - r } , T ^ { - 3 r } ) & 0 & - \\overline \\alpha J ( T ^ { - i - 3 r } , T ^ { - r } ) \\\\ - \\overline \\alpha J ( T ^ { - i } , T ^ { - r } ) & q & - \\alpha J ( T ^ { - i - 2 r } , T ^ { - 3 r } ) & 0 \\\\ 0 & - \\overline \\alpha J ( T ^ { - i - r } , T ^ { - r } ) & q & - \\alpha J ( T ^ { - i - 3 r } , T ^ { - 3 r } ) \\\\ - \\alpha J ( T ^ { - i } , T ^ { - 3 r } ) & 0 & - \\overline \\alpha J ( T ^ { - i - 2 r } , T ^ { - r } ) & q \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "5530.png", "formula": "\\begin{align*} m _ 2 = \\min \\{ j \\in \\mathbb { N } : W ( j ) > 2 ^ { k _ 2 - 2 } \\} . \\end{align*}"}
-{"id": "7332.png", "formula": "\\begin{align*} \\psi ( v _ 1 ) = \\alpha w _ { - 1 } , \\psi ( v _ 0 ) = \\beta w _ 0 , \\psi ( v _ { - 1 } ) = \\gamma w _ 1 . \\end{align*}"}
-{"id": "7148.png", "formula": "\\begin{align*} u = 0 \\mbox { o n } \\ , \\ , \\partial \\R ^ n _ + = \\{ x _ n = 0 \\} . \\end{align*}"}
-{"id": "1414.png", "formula": "\\begin{align*} \\Pi = \\Pi _ { 0 } \\sqcup \\Pi _ { \\frac { 1 } { 2 } } \\sqcup \\Pi _ { 1 } \\end{align*}"}
-{"id": "873.png", "formula": "\\begin{align*} \\operatorname * { l e v } \\nolimits _ { \\varphi _ { A , k } , < } ( t ) = t k + \\operatorname * { i n t } A \\forall \\ , t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "7003.png", "formula": "\\begin{align*} \\delta _ v ( \\Theta ( L ) ( x , y , z ) ) ( a ) = \\sum _ { i + j = N + 1 ; ~ i , j > 0 } - f _ 0 ^ { \\lambda _ i ( x , \\lambda _ j ( y , z ) ) } + f _ 0 ^ { \\lambda _ i ( y , \\lambda _ j ( x , z ) ) } + f _ 0 ^ { \\lambda _ i ( \\lambda _ j ( x , y ) , z ) } ( a ) \\end{align*}"}
-{"id": "5968.png", "formula": "\\begin{align*} A _ 0 = \\C [ a _ 0 ] \\end{align*}"}
-{"id": "8348.png", "formula": "\\begin{align*} 0 = \\vec 0 _ 2 = \\frac { 1 } { p } \\big ( \\vec 0 ^ { ( 2 ) } - \\vec 0 _ 1 ^ p \\big ) = \\sum _ { i = 1 } ^ n \\alpha _ { i 2 } \\mu _ i ^ p = \\sum _ { i = 1 } ^ n \\alpha _ { i 2 } ^ p \\mu _ i ^ p = \\Big ( \\sum _ { i = 1 } ^ n \\alpha _ { i 2 } \\mu _ i \\Big ) ^ p = 0 . \\end{align*}"}
-{"id": "7602.png", "formula": "\\begin{align*} P _ p ( x _ 0 ) \\ge h ^ { K _ 1 } \\prod _ { i = 1 } ^ { K _ 1 } \\varepsilon _ i , P _ e ( x _ 0 ) \\ge h ^ { K _ 2 } \\prod _ { i = 1 } ^ { K _ 2 } \\delta _ i . \\end{align*}"}
-{"id": "2426.png", "formula": "\\begin{align*} \\| f \\| _ { M _ { p , q } ^ { s , \\alpha _ 1 } } \\sim \\begin{cases} \\big \\| \\{ \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { p , q } ^ { 0 , \\alpha _ 1 } } \\} | ~ { l _ { q } ^ { s , \\alpha _ 2 } } \\big \\| , ~ & ~ \\alpha _ 2 < 1 \\\\ \\big \\| \\{ \\| \\Delta _ j f \\| _ { M _ { p , q } ^ { 0 , \\alpha _ 1 } } \\} | ~ { l _ { q } ^ { s , 1 } } \\big \\| , ~ & ~ \\alpha _ 2 = 1 . \\end{cases} \\end{align*}"}
-{"id": "7521.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } x _ n ( \\omega ) = K , x _ 0 > 0 . \\end{align*}"}
-{"id": "931.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t m + \\underbrace { u \\cdot \\nabla m } _ { } + \\underbrace { \\nabla u ^ { T } \\cdot m } _ { } + \\underbrace { m ( u ) } _ { } + \\underbrace { \\rho \\nabla \\bar { \\rho } } _ { } = 0 , \\\\ \\partial _ t \\rho + ( \\rho u ) = 0 , \\end{array} \\right . \\end{align*}"}
-{"id": "7943.png", "formula": "\\begin{align*} \\alpha ( i , j ) = \\lim _ { m \\to \\infty } \\alpha _ m ( i , j ) = \\lim _ { m \\to \\infty } \\alpha _ m ' ( \\rho ( i ) , \\rho ( j ) ) & = \\lim _ { m \\to \\infty } ( 1 - \\alpha _ m ( \\rho ( i ) , \\rho ( j ) ) ) \\\\ & = 1 - \\lim _ { m \\to \\infty } \\alpha _ m ( \\rho ( i ) , \\rho ( j ) ) \\\\ & = 1 - \\alpha ( \\rho ( i ) , \\rho ( j ) ) \\\\ & = \\alpha ' ( \\rho ( i ) , \\rho ( j ) ) . \\end{align*}"}
-{"id": "5537.png", "formula": "\\begin{align*} V _ x = \\begin{bmatrix} 0 & 1 \\\\ - z \\rho & 0 \\end{bmatrix} V , V = \\begin{bmatrix} v \\\\ v _ x \\end{bmatrix} , \\end{align*}"}
-{"id": "9735.png", "formula": "\\begin{align*} \\liminf _ { t \\to \\infty } \\frac { \\varphi ( t ) } { g ( G ^ { - 1 } ( t ) ) } & = b \\liminf _ { t \\to \\infty } \\frac { g ( x ( t - \\tau ( t ) ) ) } { g ( G ^ { - 1 } ( t ) ) } \\\\ & \\geq b \\liminf _ { t \\to \\infty } \\frac { g ( \\lambda ( 1 - \\epsilon ) ^ 2 \\{ ( 1 + \\epsilon ) ( 1 - q ) \\} ^ { - 1 / ( \\beta - 1 ) } G ^ { - 1 } ( t ) ) } { g ( G ^ { - 1 } ( t ) ) } \\\\ & = b \\lambda ^ \\beta ( 1 - \\epsilon ) ^ { 2 \\beta } \\{ ( 1 + \\epsilon ) ( 1 - q ) \\} ^ { - \\beta / ( \\beta - 1 ) } . \\end{align*}"}
-{"id": "6443.png", "formula": "\\begin{align*} & \\int _ { B _ { 1 } } \\phi \\psi ^ { 1 + q } w ^ { 2 } d x + \\frac { 1 } { 2 } g _ { \\alpha } * ( \\ , \\phi ) \\leq q g _ { \\alpha } * \\int _ { B _ { 1 } } \\psi ^ { 1 + q } w ^ { 2 } g _ { 1 - \\alpha } \\phi d x \\\\ & + \\frac { \\vartheta ( q ) ( q - 1 ) } { 2 } g _ { \\alpha } * ( \\ , \\phi ) + \\int _ { 0 } ^ { s } g _ { \\alpha } ( s - \\sigma ) \\dot { \\phi } ( \\sigma ) \\left ( g _ { 1 - \\alpha } * \\int _ { B _ { 1 } } \\psi ^ { 1 + q } w ^ { 2 } d x \\right ) ( \\sigma ) d \\sigma . \\end{align*}"}
-{"id": "5178.png", "formula": "\\begin{gather*} R _ { \\ell + s } = C _ { \\ell + s } \\oplus \\varphi _ { \\ell } ( R _ \\ell ) . \\end{gather*}"}
-{"id": "8827.png", "formula": "\\begin{align*} [ L _ m , L _ n ] = ( n - m ) L _ { n + m } \\end{align*}"}
-{"id": "3515.png", "formula": "\\begin{align*} & K _ 0 = \\mbox { s p a n } \\{ 1 , x ^ 1 , x ^ 2 , x ^ 3 \\} \\\\ & K _ 1 = \\mbox { s p a n } \\left \\{ \\frac { \\partial } { \\partial x ^ 1 } , \\frac { \\partial } { \\partial x ^ 2 } , \\frac { \\partial } { \\partial x ^ 3 } , x \\times \\frac { \\partial } { \\partial x ^ 1 } , x \\times \\frac { \\partial } { \\partial x ^ 2 } , x \\times \\frac { \\partial } { \\partial x ^ 3 } \\right \\} . \\end{align*}"}
-{"id": "7780.png", "formula": "\\begin{align*} \\Lambda _ M = \\mathbb { R } e _ 1 \\oplus \\mathbb { C } e _ 2 \\oplus \\mathbb { C } \\tau \\oplus \\mathbb { R } \\overline \\tau \\oplus \\mathbb { R } \\overline \\tau i \\oplus \\mathbb { C } \\tau \\overline \\tau \\oplus \\mathbb { R } \\overline \\tau i \\tau . \\end{align*}"}
-{"id": "9381.png", "formula": "\\begin{align*} a = ( { q } , G \\ast { q } ) + 2 i k [ 1 + ( G \\ast { q } ) ( 0 ) ] [ 1 + ( G \\ast { { q } ^ * } ) ( 0 ) ] , k \\in \\mathbb { C } _ + . \\end{align*}"}
-{"id": "6128.png", "formula": "\\begin{align*} I \\ge & \\frac 2 3 [ ( a _ 2 + c _ 2 ) - ( a _ 1 + c _ 1 ) ] \\\\ = & \\frac 8 3 ( K _ { 1 3 } - K _ { 1 2 } ) = \\frac 8 3 ( K _ { 1 2 } + K _ { 1 3 } - 2 K _ { 1 2 } ) \\\\ > & \\frac { 1 6 } 3 \\epsilon . \\end{align*}"}
-{"id": "9656.png", "formula": "\\begin{align*} G ( x ) = \\int _ x ^ 1 \\frac { 1 } { g ( u ) } \\ , d u , x > 0 . \\end{align*}"}
-{"id": "736.png", "formula": "\\begin{align*} & b _ { 0 } = a _ { 0 } \\\\ & b _ { n } = b _ { n - 1 } + C _ { 1 } \\left ( \\sqrt [ ] { b _ { n - 1 } } + \\sqrt [ ] { b _ { n - 1 } + C _ { 2 } } \\right ) \\ \\ \\ \\ . \\end{align*}"}
-{"id": "3441.png", "formula": "\\begin{align*} \\kappa ( t ) = \\kappa + \\int _ s ^ t q ( \\kappa ( \\theta ) ) d \\theta + \\int _ s ^ t Q ( \\kappa ( \\theta ) ) d W ( \\theta ) , s \\le t \\le T , \\end{align*}"}
-{"id": "6465.png", "formula": "\\begin{align*} & - \\int _ { B _ { 1 } } \\partial _ { s } ( g _ { 1 - \\alpha , m } * [ \\phi \\psi ^ { 2 } w ^ { 2 } ] ) d x + \\frac { 1 } { 2 } c _ { 1 } \\\\ \\leq & \\int _ { 0 } ^ { s } \\dot { g } _ { 1 - \\alpha , m } ( s - \\tau ) ( \\phi ( s ) - \\phi ( \\tau ) ) \\left ( \\int _ { B _ { 1 } } \\psi ^ { 2 } \\tilde { u } ^ { 1 - q } d x \\right ) ( \\tau ) d \\tau \\\\ & + c _ { 4 } ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\phi ( s ) \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x + R _ { m } ( s ) , \\end{align*}"}
-{"id": "8592.png", "formula": "\\begin{align*} \\widetilde { \\psi } ( d _ { 1 } a ^ { \\ast } d _ { 2 } w ) & = \\widetilde { \\psi } \\left ( d _ { 1 } c _ { k } ^ { - 1 } \\pi _ { e _ { k } a } i \\Pi _ { 2 } ( d _ { 2 } w ) + d _ { 1 } c _ { k } ^ { - 1 } \\pi _ { e _ { k } a } j ' \\sigma _ { 2 } \\Pi _ { 3 } ( d _ { 2 } w ) \\right ) \\\\ & = d _ { 1 } c _ { k } ^ { - 1 } \\psi \\left ( \\pi _ { e _ { k } a } i \\Pi _ { 2 } ( d _ { 2 } w ) \\right ) + d _ { 1 } c _ { k } ^ { - 1 } \\psi \\left ( \\pi _ { e _ { k } a } j ' \\sigma _ { 2 } \\Pi _ { 3 } ( d _ { 2 } w ) \\right ) \\\\ \\end{align*}"}
-{"id": "8379.png", "formula": "\\begin{align*} \\| x j - x _ 0 - \\sum _ { i = 1 } ^ k x _ i v _ i \\| < \\frac { 1 } { 3 } . \\end{align*}"}
-{"id": "7014.png", "formula": "\\begin{align*} \\int _ X \\langle \\pi _ 1 , \\imath _ X ( x ) \\rangle d P = \\left \\langle \\pi _ 1 , \\int _ X \\imath _ X ( x ) d P \\right \\rangle \\ge \\left \\langle \\pi _ 1 , \\psi _ n \\right \\rangle = \\langle \\pi , \\psi _ n \\rangle & > \\langle \\pi , \\omega ( t ) \\rangle \\\\ & \\ge \\langle \\pi _ 1 , \\omega ( t ) \\rangle \\end{align*}"}
-{"id": "9532.png", "formula": "\\begin{align*} g _ M ( \\bar { X } , \\bar { Y } ) & = e ^ { - 2 \\phi / 3 } g _ N ( X , Y ) + e ^ { 4 \\phi / 3 } C _ 1 ( X ) C _ 1 ( Y ) \\\\ g _ M ( \\bar { X } , \\bar { S } ) & = - e ^ { 4 \\phi / 3 } C _ 1 ( X ) \\alpha ^ { 1 0 } ( S ) \\\\ g _ M ( \\bar { S } , \\bar { S } ) & = e ^ { 4 \\phi / 3 } g _ { S ^ 1 } ( S , S ) . \\end{align*}"}
-{"id": "9813.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { N } f _ { i } \\prec \\prod _ { i = 1 } ^ { N } f _ { i } ^ { \\ast } . \\end{align*}"}
-{"id": "884.png", "formula": "\\begin{align*} \\pi ( x ) = \\int _ { 2 } ^ { x } \\frac { 1 } { \\log u } d u + O \\left \\{ x \\exp \\left ( - C ( \\log x ) ^ { 3 / 5 } ( \\log \\log x ) ^ { - 1 / 5 } \\right ) \\right \\} . \\end{align*}"}
-{"id": "6569.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { 4 , q } ( \\R ^ d ) ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ q ( \\R ^ d ) ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { 4 , q } ( \\R ^ d ) ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "9016.png", "formula": "\\begin{align*} \\tilde f ( x , y , \\tilde z ) = \\tilde z ^ d + \\sum _ { i = 0 } ^ { d - 1 } a _ i a _ d ^ { d - 1 - i } \\tilde z ^ { i } \\end{align*}"}
-{"id": "6692.png", "formula": "\\begin{align*} A = \\nu ^ r A _ r + \\nu ^ { r + 1 } A _ { r + 1 } + \\ldots , \\end{align*}"}
-{"id": "8244.png", "formula": "\\begin{align*} 2 \\Re L : = \\Re ( f _ 1 + f _ 2 ) \\frac { \\partial } { \\partial x } + \\Im ( f _ 1 - f _ 2 ) \\frac { \\partial } { \\partial y } + \\Re ( g _ 1 + g _ 2 ) \\frac { \\partial } { \\partial u } + \\Im ( g _ 1 - g _ 2 ) \\frac { \\partial } { \\partial v } \\end{align*}"}
-{"id": "5036.png", "formula": "\\begin{align*} \\nu _ Y = \\nu ( C ) \\nu _ C + \\nu ( C ^ c ) \\nu _ { C ^ c } , \\end{align*}"}
-{"id": "7694.png", "formula": "\\begin{align*} f ( x ) + \\gamma _ n < K + \\varepsilon _ 0 + \\delta _ 0 < K + 2 \\varepsilon _ 0 < K + ( 1 - \\lambda ) K = 2 K - \\lambda K < 2 K - \\varepsilon _ 0 . \\end{align*}"}
-{"id": "3227.png", "formula": "\\begin{align*} \\pi _ { 1 } = \\pi _ { 2 } \\circ \\Phi , \\end{align*}"}
-{"id": "2982.png", "formula": "\\begin{align*} \\delta _ H \\Theta ^ 2 ( x , y ) = - \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ \\alpha _ 0 , f _ i ^ { \\lambda _ j ( x , y ) } ] - \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ \\alpha _ 0 , [ f _ i ^ y , f _ j ^ x ] ] . \\end{align*}"}
-{"id": "4096.png", "formula": "\\begin{align*} H _ { \\alpha } = \\frac { y ^ 2 + a x ^ 2 + \\alpha x z + c z ^ 2 } { x z ^ 3 } \\end{align*}"}
-{"id": "2122.png", "formula": "\\begin{align*} t ^ { 2 s + 2 l } = \\sum _ { j = 0 } ^ { l } c _ j \\bar { w } _ j ( t ) . \\end{align*}"}
-{"id": "6857.png", "formula": "\\begin{align*} e ^ { - i t _ n \\Delta } u _ n ( t _ n ) = \\sum _ { j = 1 } ^ J e ^ { i x \\xi _ n ^ j } \\psi ^ j _ { \\{ h _ n ^ j \\} } + W ^ J _ n \\end{align*}"}
-{"id": "8214.png", "formula": "\\begin{align*} H _ E : = H - E Q , \\end{align*}"}
-{"id": "7484.png", "formula": "\\begin{align*} P ( \\Delta _ { Q ^ { ( k ) } } ( \\infty ) = k ^ 2 ) > 0 P ( \\Delta _ { Q ^ { ( k ) } } ( \\infty ) = k ^ 2 - 1 ) > 0 , \\end{align*}"}
-{"id": "8283.png", "formula": "\\begin{align*} \\P [ ( i , j ) \\in E _ n ] = \\begin{cases} a / n & \\mbox { i f $ \\{ i , j \\} \\subseteq S $ o r $ \\{ i , j \\} \\subseteq S ^ c $ , } \\\\ b / n & \\mbox { i f $ i \\in S , \\ , j \\in S ^ c $ o r $ i \\in S ^ c , j \\in S $ . } \\end{cases} \\end{align*}"}
-{"id": "661.png", "formula": "\\begin{align*} \\nu _ Y ( \\phi ) = \\eta ( k _ \\phi ) = k _ \\phi ( e ) = \\nu ( \\phi ) , \\textrm { f o r a l l $ \\phi \\in C ( \\overline { Y } ) $ } . \\end{align*}"}
-{"id": "5749.png", "formula": "\\begin{align*} \\frac { \\partial Z _ { t } } { \\partial \\phi } = Y _ { t - J _ \\phi } + \\sum _ { j \\in J _ \\theta } \\theta _ { j } \\frac { \\partial e _ { t - j } ^ I } { \\partial \\phi } , \\frac { \\partial Z _ { t } } { \\partial \\theta } = e _ { t - J _ \\theta } + \\sum _ { j \\in J _ \\theta } \\theta _ { j } \\frac { \\partial e _ { t - j } ^ I } { \\partial \\theta } . \\end{align*}"}
-{"id": "1749.png", "formula": "\\begin{align*} s _ i x = s _ { \\alpha _ i } x = x - \\langle x , \\alpha ^ \\lor _ i \\rangle \\alpha _ i , & \\ \\ \\ x \\in \\mathfrak { h } ^ * , \\\\ s _ i y = s _ { \\alpha ^ \\lor _ i } y = y - \\langle \\alpha _ i , y \\rangle \\alpha ^ \\lor _ i , & \\ \\ \\ y \\in \\mathfrak { h } ; \\end{align*}"}
-{"id": "8783.png", "formula": "\\begin{align*} f _ \\delta & : = E _ \\delta , \\\\ f _ { \\ldots , \\lambda _ i , \\lambda _ { i + 1 } , \\ldots } & : = T _ i ^ { - 1 } f _ { \\ldots , \\lambda _ { i + 1 } , \\lambda _ { i } , \\ldots } \\lambda _ i > \\lambda _ { i + 1 } , \\\\ f _ { \\ldots , \\lambda _ { n - 1 } , \\lambda _ n } & : = T _ n ^ { - 1 } f _ { \\ldots , \\lambda _ { n - 1 } , - \\lambda _ { n } } , \\lambda _ n > 0 . \\end{align*}"}
-{"id": "856.png", "formula": "\\begin{align*} { } _ { [ 1 , M ] } \\langle \\mathrm { v a c } | \\tilde { A } ( w ) = ( 1 + w ) ^ { M - 1 } { } _ { [ 1 , M ] } \\langle \\mathrm { v a c } | . \\end{align*}"}
-{"id": "1342.png", "formula": "\\begin{align*} e _ { t } = \\sigma _ { t } ^ { - \\gamma } e _ { t } ^ { I } \\end{align*}"}
-{"id": "7140.png", "formula": "\\begin{align*} | G _ { i j } ( x , y ) | \\le \\frac { C _ 0 x _ n y _ n } { | x - y | ^ { n - 2 } \\cdot | x - y ^ * | ^ { 2 } } + C _ 0 1 _ { n = 2 } \\log ( 2 + \\frac { y _ n } { | x - y | } ) . \\end{align*}"}
-{"id": "606.png", "formula": "\\begin{align*} F _ { \\kappa , m , N } : = \\mathcal { F } _ { \\kappa , m , N } \\Big | \\operatorname { p r } \\in H _ { \\kappa } ( N ) . \\end{align*}"}
-{"id": "8115.png", "formula": "\\begin{align*} \\mu _ l ( U ( t ) ) = \\mu _ l ( B _ 1 ) \\mbox { f o r $ | t | < t _ 0 $ } . \\end{align*}"}
-{"id": "8453.png", "formula": "\\begin{align*} F ^ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } ) = - \\sum _ { j = 1 } ^ { d } \\textbf { P } J _ { \\varepsilon } A _ { j } ( J _ { \\varepsilon } ( \\tilde { \\textbf { u } ^ { \\varepsilon } } + \\bar { \\textbf { u } } ) ) \\partial _ { x _ { j } } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } . \\end{align*}"}
-{"id": "5513.png", "formula": "\\begin{align*} \\| x \\| _ \\mathcal { M } = \\| x \\| _ { \\mathcal { M } _ { \\varphi , w } } = \\inf \\{ \\epsilon > 0 : P ( x / \\epsilon ) \\le 1 \\} . \\end{align*}"}
-{"id": "181.png", "formula": "\\begin{align*} \\tilde { V } _ n ^ w : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ k w _ j \\log ^ 2 \\xi _ { ( j ) , i } - ( \\hat { H } _ n ^ w ) ^ 2 . \\end{align*}"}
-{"id": "641.png", "formula": "\\begin{align*} \\lambda ( f ) = f ( e ) , \\textrm { f o r a l l $ f \\in H ^ \\infty _ r ( G , p ) $ } . \\end{align*}"}
-{"id": "6771.png", "formula": "\\begin{align*} \\theta _ { u _ j } ^ n = e ^ { u _ j - \\varphi _ j } \\theta _ { \\varphi _ j } ^ n + e ^ { u _ j - \\psi _ j } \\theta _ { \\psi _ j } ^ n . \\end{align*}"}
-{"id": "5101.png", "formula": "\\begin{align*} \\check { R } ( z / w ) [ \\mathbb { T } ^ { [ 1 , M ] } ( z ) \\otimes \\mathbb { T } ^ { [ 1 , M ] } ( w ) ] \\check { R } ( z / w ) ^ { - 1 } = \\mathbb { T } ^ { [ 1 , M ] } ( w ) \\otimes \\mathbb { T } ^ { [ 1 , M ] } ( z ) . \\end{align*}"}
-{"id": "4975.png", "formula": "\\begin{align*} & h _ { E } = h _ { H } \\circ f - \\left < A \\vec { c } , { \\bf h } _ { \\vec { D } } \\right > \\\\ & { \\bf h } _ { \\vec { E } } = \\left ( \\begin{array} { c } h _ { E _ { 1 } } \\\\ h _ { E _ { 2 } } \\\\ \\vdots \\\\ h _ { E _ { r } } \\end{array} \\right ) = { \\bf h } _ { \\vec { D } } \\circ f - { } ^ { \\rm t } A { \\bf h } _ { \\vec { D } } \\ . \\end{align*}"}
-{"id": "8585.png", "formula": "\\begin{align*} \\hat { \\gamma } _ { s a , b r } ( n ) & = \\displaystyle \\sum _ { w \\in L ( k ) } s ^ { \\ast } ( r ^ { - 1 } w ) \\displaystyle \\sum _ { b _ { 1 } h a _ { 1 } , r ' , r '' } w ^ { \\ast } ( r ' r '' ) C _ { r '' a , h a _ { 1 } } Y _ { [ b _ { 1 } h a _ { 1 } ] } ( P ) D _ { b _ { 1 } , b r ' } n \\\\ & = \\displaystyle \\sum _ { b _ { 1 } h a _ { 1 } , r ' , r '' } s ^ { \\ast } ( r ^ { - 1 } r ' r '' ) C _ { r '' a , h a _ { 1 } } Y _ { [ b _ { 1 } h a _ { 1 } ] } ( P ) D _ { b _ { 1 } , b r ' } n \\end{align*}"}
-{"id": "8921.png", "formula": "\\begin{align*} u ( x ) = \\frac { \\exp \\bigl ( - \\frac { \\abs { x - a } } { 2 } \\sqrt { 1 + \\frac { B ^ 2 } { 4 } \\abs { x - a } ^ 2 } \\bigr ) } { \\abs { x - a } ^ { 1 / 2 } ( 1 + \\frac { B ^ 2 } { 4 } \\abs { x - a } ^ 2 ) ^ { \\frac { 1 } { 4 } } ( \\sqrt { 1 + \\frac { B ^ 2 } { 4 } \\abs { x - a } ^ 2 } + \\frac { \\abs { B } } { 2 } \\abs { x - a } ) ^ \\frac { 1 } { \\abs { B } } } ( c + o ( 1 ) ) , \\end{align*}"}
-{"id": "1347.png", "formula": "\\begin{align*} S ( \\delta _ { 0 } ) = n ^ { - 1 / 2 } \\sum _ { t = 1 } ^ { n } \\left ( y _ { t } - m _ t \\pi _ { t } ( \\delta _ { 0 } ) \\right ) \\begin{bmatrix} x _ { t } \\\\ \\sum _ { j = 0 } ^ { \\infty } \\tau _ { j } ( \\omega ) e _ { t - J _ L - j } ( \\delta _ 0 ) \\\\ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "5413.png", "formula": "\\begin{align*} \\forall L \\subset \\R ^ N , \\ ; \\dim L \\ge N / 4 \\quad \\Rightarrow \\exists x \\in L \\ ; \\ ; | x | = 1 , \\| x \\| _ \\infty \\ge 1 / 2 . \\end{align*}"}
-{"id": "980.png", "formula": "\\begin{align*} \\forall i = 1 , \\ldots , n ; \\quad \\mbox { w e h a v e } w _ i + 1 \\ne n + k - 1 . \\end{align*}"}
-{"id": "3114.png", "formula": "\\begin{align*} P ^ T L _ 3 ^ \\prime ( \\lambda ) P = \\left [ \\begin{array} { c c c | c c } \\lambda P _ 5 - P _ 4 & \\lambda P _ 4 & 0 & - I _ n & 0 \\\\ \\lambda P _ 4 & \\lambda P _ 3 + P _ 2 & P _ 1 & \\lambda I _ n & - I _ n \\\\ 0 & P _ 1 & - \\lambda P _ 1 + P _ 0 & 0 & \\lambda I _ n \\\\ \\hline - I _ n & \\lambda I _ n & 0 & 0 & 0 \\\\ 0 & - I _ n & \\lambda I _ n & 0 & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "9870.png", "formula": "\\begin{align*} 0 = \\mu _ 0 ( H _ g ) < \\mu _ { 1 } ( H _ g ) \\leq \\mu _ { 2 } ( H _ g ) \\leq . . . \\leq \\mu _ { n } ( H _ g ) \\leq . . . \\ , , \\end{align*}"}
-{"id": "3026.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( { t _ 3 } ) = { h ^ { [ j 1 ] } } ( { t _ 3 } ) u _ 3 ^ { [ 1 ] } + { h ^ { [ j 2 ] } } ( { t _ 3 } ) \\frac { { { h ^ { [ 3 2 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 3 2 ] } } ( { t _ 3 } ) } } u _ 1 ^ { [ 2 ] } + { h ^ { [ j 3 ] } } ( { t _ 3 } ) \\frac { { { h ^ { [ 3 3 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 3 3 ] } } ( { t _ 3 } ) } } u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "1341.png", "formula": "\\begin{align*} Z _ { t } = \\sum _ { j \\in J _ { \\phi } } \\phi _ { j } \\left ( Z _ { t - j } + e _ { t - j } \\right ) + \\sum _ { j \\in J _ { \\theta } } \\theta _ j e _ { t - j } \\end{align*}"}
-{"id": "9125.png", "formula": "\\begin{align*} g ( s ) : = ( I - B ( s ) A ( s ) u ( s ) - P ( s ) u ( s ) + f ( s ) . \\end{align*}"}
-{"id": "62.png", "formula": "\\begin{align*} \\psi ' \\left ( 1 - \\frac { f ( r ) } { a } \\right ) = - \\frac { p } { ( p - 1 ) a ^ { p - 1 } } \\left ( | f ' | ^ { p - 2 } f ' \\right ) ' ( r ) \\ , r \\in [ 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "9262.png", "formula": "\\begin{align*} J ( \\pi ) : = \\mathbb { E } [ \\int _ { D } U ( x , Y ^ { \\pi } ( T , x , Z ) , Z ) d x ] = \\int _ { \\mathbb { R } } j ( \\pi ) d z ; \\end{align*}"}
-{"id": "3140.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) \\begin{bmatrix} \\Lambda _ t ( \\lambda ) \\otimes I _ n \\\\ \\widehat { N } _ t ( \\lambda ) ( \\lambda B + A ) ( \\Lambda _ t ( \\lambda ) \\otimes I _ n ) \\end{bmatrix} = e _ { t + 1 } \\otimes P ( \\lambda ) , \\end{align*}"}
-{"id": "4349.png", "formula": "\\begin{align*} ( \\xi _ 1 ' - \\xi _ 2 ' ) ( y _ \\delta ) & \\le \\frac { p } { p - 1 } \\xi _ 2 ( y _ \\delta ) ^ { ( p - 1 ) / p } - \\frac { p } { p - 1 } \\xi _ 1 ( y _ \\delta ) ^ { ( p - 1 ) / p } \\\\ & = \\frac { p } { p - 1 } \\xi _ 2 ( y _ \\delta ) ^ { ( p - 1 ) / p } - \\frac { p } { p - 1 } \\left ( \\xi _ 2 ( y _ \\delta ) + \\delta \\right ) ^ { ( p - 1 ) / p } < 0 \\ , \\end{align*}"}
-{"id": "325.png", "formula": "\\begin{align*} F ( q ) = \\begin{cases} q + 1 + 6 \\sqrt { q } , & q , \\\\ 2 ( q + 1 ) , & q . \\end{cases} \\end{align*}"}
-{"id": "1497.png", "formula": "\\begin{align*} N ( x ) = 2 b n - b ^ 2 - b > b ( n - 1 ) . \\end{align*}"}
-{"id": "7247.png", "formula": "\\begin{align*} T _ { j k } : = \\left ( \\begin{array} { c | c c c } t _ { j k } ^ 1 & 0 & \\cdots & 0 \\\\ \\hline 0 & & & \\\\ \\vdots & & S _ { j k } & \\\\ 0 & & & \\end{array} \\right ) . \\end{align*}"}
-{"id": "1673.png", "formula": "\\begin{align*} D _ x ( \\partial _ { v _ j } \\psi ) = \\partial _ { v _ j } ( D _ x { \\psi } ) \\in L ^ p ( \\R ^ { 2 d } ) . \\end{align*}"}
-{"id": "2622.png", "formula": "\\begin{align*} \\{ \\zeta _ { m } ^ { l } ( j , u ) = \\zeta _ { m } ^ { l } ( u + j T ) , u \\in [ 0 , T ) , j \\in \\mathbb Z \\} , \\end{align*}"}
-{"id": "7345.png", "formula": "\\begin{align*} \\begin{gathered} \\gamma _ { - } ( w _ { 1 } ) v _ { 1 } \\wedge v _ { 0 } \\wedge v _ { - 1 } = q ^ { 2 } v _ { 0 } \\wedge v _ { - 1 } , \\gamma _ { - } ( w _ { 1 } ) v _ { 1 } \\wedge v _ { 0 } \\wedge v _ { - 1 } = - q ^ { 2 } v _ { 1 } \\wedge v _ { - 1 } , \\\\ \\gamma _ { - } ( w _ { 1 } ) v _ { 1 } \\wedge v _ { 0 } \\wedge v _ { - 1 } = q ^ { 4 } v _ { 1 } \\wedge v _ { 0 } . \\end{gathered} \\end{align*}"}
-{"id": "8341.png", "formula": "\\begin{align*} B ( \\xi ) = \\left [ \\begin{array} { c } F ( i _ 2 \\mu _ 2 + \\cdots + i _ n \\mu _ n ) \\\\ F ( \\mu _ 2 ) \\\\ \\vdots \\\\ F ( \\mu _ { n - 1 } ) \\\\ F ( \\mu _ n ) \\end{array} \\right ] = \\left [ \\begin{array} { c } i _ 2 F ( \\mu _ 2 ) + \\cdots + i _ n F ( \\mu _ n ) \\\\ F ( \\mu _ 2 ) \\\\ \\vdots \\\\ F ( \\mu _ { n - 1 } ) \\\\ F ( \\mu _ n ) \\end{array} \\right ] . \\end{align*}"}
-{"id": "6580.png", "formula": "\\begin{align*} \\nabla F ( x ) & = \\nabla ^ 2 f _ 1 ( x ) x + \\nabla f _ 1 ( x ) - \\nabla f _ 1 ( x ) - \\nabla ^ 2 f _ 1 ( x ) \\nabla f _ 2 ( \\nabla f _ 1 ( x ) ) \\\\ & = \\nabla ^ 2 f _ 1 ( x ) ( x - \\nabla f _ 2 ( \\nabla f _ 1 ( x ) ) ) \\\\ & = \\nabla ^ 2 f _ 1 ( x ) ( x - S _ 2 S _ 1 x ) . \\end{align*}"}
-{"id": "9576.png", "formula": "\\begin{align*} \\psi ( x , t ) = \\psi _ f ( x , t ) + \\varphi ( x , t ) , \\end{align*}"}
-{"id": "1934.png", "formula": "\\begin{align*} - \\frac { 1 } { a ( \\tau ) } + \\frac { 1 } { a ( \\tau _ 1 ) } & \\leq \\sum _ { i = 1 } ^ m \\frac { ( n _ i + 1 ) ^ 2 q _ i ^ 2 } { 4 n _ i p _ i ^ 2 } \\left ( - \\frac { 1 } { \\tau + c _ i } + \\frac { 1 } { \\tau _ 1 + c _ i } \\right ) \\\\ & \\leq \\left ( \\sum _ { i = 1 } ^ m \\frac { ( n _ i + 1 ) ^ 2 q _ i ^ 2 } { 4 n _ i p _ i ^ 2 } \\right ) \\left ( \\frac { 1 } { \\tau _ 1 } \\right ) . \\end{align*}"}
-{"id": "1335.png", "formula": "\\begin{align*} T ( \\lambda | j ) = \\begin{pmatrix} A _ j ( \\lambda ) & B _ j ( \\lambda ) \\\\ C _ j ( \\lambda ) & D _ j ( \\lambda ) \\end{pmatrix} , \\end{align*}"}
-{"id": "2382.png", "formula": "\\begin{align*} ( x - \\gamma \\nabla f ( x ) ) = \\nabla \\left ( \\tfrac { 1 } { 2 } \\| x \\| ^ 2 - \\gamma f ( x ) \\right ) . \\end{align*}"}
-{"id": "1880.png", "formula": "\\begin{align*} \\frac d { d t } ( a _ 1 + c _ 1 ) = & a _ 1 ^ 2 + 2 a _ 2 a _ 3 + c _ 1 ^ 2 + 2 c _ 2 c _ 3 \\\\ = & a _ 1 ( a _ 1 + a _ 2 + a _ 3 ) + c _ 1 ( c _ 1 + c _ 2 + c _ 3 ) + I \\\\ = & ( a _ 1 + c _ 1 ) \\cdot \\frac R 2 + I \\\\ = & ( \\kappa + \\delta t ) \\frac { R ^ 2 } 2 + I \\\\ \\end{align*}"}
-{"id": "5164.png", "formula": "\\begin{gather*} \\sum _ k ( 1 \\otimes u _ { i k } ^ * ) \\bigg ( \\sum _ l S _ l \\otimes u _ { l k } \\bigg ) = S _ i \\otimes 1 \\end{gather*}"}
-{"id": "9101.png", "formula": "\\begin{align*} P _ { 2 } ( \\zeta ) & = 2 , & P _ { 3 } ( \\zeta ) & = 6 , \\\\ P _ { 4 } ( \\zeta ) & = 2 \\zeta + 2 2 , & P _ { 5 } ( \\zeta ) & = 3 0 \\zeta + 9 0 , \\\\ P _ { 6 } ( \\zeta ) & = 1 6 \\zeta ^ 2 + 3 1 0 \\zeta + 3 9 4 , & P _ { 7 } ( \\zeta ) & = 5 0 4 \\zeta ^ 2 + 2 7 3 0 \\zeta + 1 8 0 6 , \\\\ P _ { 8 } ( \\zeta ) & = 3 6 0 \\zeta ^ 3 + 9 4 2 2 \\zeta ^ 2 + 2 1 9 8 0 \\zeta + 8 5 5 8 , & P _ { 9 } ( \\zeta ) & = 1 8 2 6 4 \\zeta ^ 3 + 1 3 5 9 5 4 \\zeta ^ 2 + 1 6 7 0 7 6 \\zeta + 4 1 5 8 6 . \\end{align*}"}
-{"id": "81.png", "formula": "\\begin{align*} v _ t = a \\ , v _ { x x } + \\lambda \\ , v , \\ \\ t > 0 , \\ x < 0 , \\end{align*}"}
-{"id": "8157.png", "formula": "\\begin{align*} t + m _ 2 \\varphi ( n _ 2 ) = t ^ 2 . \\end{align*}"}
-{"id": "5071.png", "formula": "\\begin{align*} \\lim _ { \\begin{subarray} { c } M \\to \\infty \\\\ M ' \\to - \\infty \\end{subarray} } \\prod _ { i = 1 } ^ { k } \\frac { z _ { i } ^ { M ' - 1 } } { ( 1 + z _ { i } ) ^ { M } } \\langle \\prod _ { 1 \\le i \\le k } ^ { \\curvearrowright } C _ { \\mu _ { i } } ^ { [ M ' , M ] } ( z _ { i } ) \\prod _ { 1 \\le i \\le k } \\beta _ { \\nu _ { i } , x _ { i } } ^ { * } \\rangle _ { [ M ' , M ] } . \\end{align*}"}
-{"id": "6686.png", "formula": "\\begin{align*} F ( x ( t ; \\overline { x } ) ) = \\exp ( - 2 \\lambda t ) \\cdot F ( \\overline { x } ) , ~ ( \\forall ) t \\in [ 0 , \\infty ) . \\end{align*}"}
-{"id": "544.png", "formula": "\\begin{align*} f ( y , x ) - f ( x , x ) & = \\frac { z } { n } \\partial _ x f ( x , x ) + \\frac { z ^ 2 } { 2 n ^ 2 } \\partial _ x ^ 2 f ( x , x ) + \\frac { z ^ 3 } { 6 n ^ { 3 } } \\partial _ x ^ 3 f ( x , x ) + o ( z ^ 3 n ^ { - 3 } ) , \\\\ f ( x , y ) - f ( x , x ) & = \\frac { z } { n } \\partial _ y f ( x , x ) + \\frac { z ^ 2 } { 2 n ^ 2 } \\partial _ y ^ 2 f ( x , x ) + \\frac { z ^ 3 } { 6 n ^ { 3 } } \\partial _ y ^ 3 f ( x , x ) + o ( z ^ 3 n ^ { - 3 } ) . \\end{align*}"}
-{"id": "848.png", "formula": "\\begin{align*} h ^ { r ^ { k _ { r } } , \\ldots , a ^ { k _ { a } - 1 } , \\ldots , 1 ^ { k _ { 1 } } } _ { \\vec { w } } ( x _ { 1 } , \\ldots , x _ { k - 1 } ) = h ^ { r ^ { k _ { r } } , \\ldots , a ^ { k _ { a } - 1 } , \\ldots , 1 ^ { k _ { 1 } } } _ { \\vec { z } ( \\ell ( t ) , \\ldots , \\ell ( 0 ) ) } ( x _ { 1 } , \\ldots , x _ { k - 1 } ) . \\end{align*}"}
-{"id": "6319.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { 2 , 2 } ^ { 0 , \\alpha _ 2 } } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { 2 , 2 } ^ { 0 , \\alpha _ 1 } } = \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } \\Box _ k ^ { \\alpha _ 2 } f \\| ^ 2 _ { L ^ 2 } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "7779.png", "formula": "\\begin{align*} \\overline { \\tau } \\tau = 0 { \\tau } { \\overline { \\tau } } - i { \\tau } { \\overline { \\tau } } i = 0 . \\end{align*}"}
-{"id": "4430.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { C _ E } } = \\lim _ { n \\to \\infty } \\| V a _ n \\| _ { \\widehat { C _ E } } = \\lim _ { n \\to \\infty } \\| V a _ n \\| _ { C _ { \\widehat { E } } } = \\| x \\| _ { C _ { \\widehat { E } } } . \\end{align*}"}
-{"id": "6205.png", "formula": "\\begin{align*} \\mathcal { A } f ( x ) & = ( \\gamma _ U x + \\gamma _ L ) \\alpha x ^ { - \\alpha - 1 } - \\frac { 1 } { 2 } ( x ^ 2 \\sigma _ U ^ 2 + 2 x \\sigma _ { U L } + \\sigma _ L ^ 2 ) \\alpha ( \\alpha + 1 ) x ^ { - \\alpha - 2 } \\\\ & = \\alpha x ^ { - \\alpha } \\left ( \\gamma _ U - \\frac { ( \\alpha + 1 ) \\sigma _ U ^ 2 } { 2 } \\right ) + O ( x ^ { - \\alpha - 1 } ) . \\end{align*}"}
-{"id": "7156.png", "formula": "\\begin{align*} v = 0 \\mbox { o n } \\ , \\ , \\partial \\R ^ n _ + = \\{ x _ n = 0 \\} . \\end{align*}"}
-{"id": "4861.png", "formula": "\\begin{align*} \\int \\int f ( x , y ) \\delta ' ( x , y ) \\mathrm { d } x \\mathrm { d } y = \\int \\left . \\big ( \\partial _ y f ( x , y ) - \\partial _ x f ( x , y ) \\big ) \\right | _ { y = x } \\mathrm { d } x , \\end{align*}"}
-{"id": "4140.png", "formula": "\\begin{align*} \\sum _ { n \\geq 1 } a ( n ) ^ 2 v _ Y ( \\frac { n } { X } ) = V _ Y ( \\gamma _ { 1 } ) R _ 1 X ^ { \\gamma _ { 1 } } + O ( Y ^ { \\eta _ { 1 } + \\epsilon } X ^ { \\sigma _ 1 } ) . \\end{align*}"}
-{"id": "4220.png", "formula": "\\begin{align*} \\frac { 2 } { e ^ t + 1 } e ^ { x t } = \\sum _ { n = 0 } ^ \\infty E _ n ( x ) \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 1 - 1 3 ] ) . \\end{align*}"}
-{"id": "7451.png", "formula": "\\begin{align*} M _ { i j } : = ( - 1 ) ^ { i - j } \\Delta ^ { I _ i \\setminus \\{ j \\} , I _ { i - 1 } } ( x ) \\end{align*}"}
-{"id": "2720.png", "formula": "\\begin{align*} P _ { [ \\theta _ 1 + \\theta _ 2 , \\varphi ] } ( V _ { \\theta _ 1 } + V _ { \\theta _ 2 } ) \\geq P _ { [ \\theta _ 1 , u ] } ( V _ { \\theta _ 1 } ) + P _ { [ \\theta _ 2 , v ] } ( V _ { \\theta _ 2 } ) = V _ { \\theta _ 1 } + V _ { \\theta _ 2 } . \\end{align*}"}
-{"id": "3665.png", "formula": "\\begin{align*} 0 & = \\phi ( x y + y x - t ^ 2 ) \\\\ & = 2 ( a _ 1 b _ 1 x ^ 2 + a _ 2 b _ 2 y ^ 2 ) + ( a _ 1 b _ 2 + a _ 2 b _ 1 ) x y + ( a _ 2 b _ 1 + a _ 1 b _ 2 ) y x - c _ 3 ^ 2 t ^ 2 \\\\ & = 2 ( a _ 1 b _ 1 x ^ 2 + a _ 2 b _ 2 y ^ 2 ) + ( a _ 1 b _ 2 + a _ 2 b _ 1 - c _ 3 ^ 2 ) t ^ 2 . \\end{align*}"}
-{"id": "5393.png", "formula": "\\begin{align*} x n = \\sum _ { i = 1 } ^ c \\max \\{ v ( B _ i ) - n , 0 \\} \\ , . \\end{align*}"}
-{"id": "7764.png", "formula": "\\begin{align*} \\beta ( M ) = \\begin{bmatrix} 1 & 1 & - \\\\ - & 1 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "8394.png", "formula": "\\begin{align*} B { } \\lim _ \\lambda x _ \\lambda g = x g , \\ \\ \\ \\ g \\in G . \\end{align*}"}
-{"id": "5464.png", "formula": "\\begin{align*} \\widetilde { S } ( \\alpha ) - V ( \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } ( \\Lambda ( m ) - 1 ) e ^ { - m / N } e ( m \\alpha ) \\end{align*}"}
-{"id": "3704.png", "formula": "\\begin{align*} \\frac { \\partial F } { \\partial t } = \\nabla F \\cdot \\nabla Z \\times \\nabla H \\equiv \\{ F , Z , H \\} ~ . \\end{align*}"}
-{"id": "6861.png", "formula": "\\begin{align*} { u } ^ J _ n ( t ) : = \\sum _ { j = 1 } ^ J v _ n ^ j ( t ) + e ^ { i t \\Delta } W _ n ^ J , \\end{align*}"}
-{"id": "4695.png", "formula": "\\begin{align*} y & = y ' , \\\\ \\mu _ k ( z ) & = \\mu _ k ( z ' ) . \\end{align*}"}
-{"id": "5921.png", "formula": "\\begin{align*} \\| f \\| _ { X _ { p , s } } = \\| f \\| _ { W ^ { 1 , p } ( \\R ^ { 2 d } ) } + \\| D ^ 2 _ v f \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ { d } _ x ) ) } + \\| v \\cdot D _ x f \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ { d } _ x ) ) } . \\end{align*}"}
-{"id": "3122.png", "formula": "\\begin{align*} \\widetilde { P } ( \\lambda ) : = ( \\widetilde { \\Lambda } _ t ( \\lambda ) ^ T \\otimes I _ n ) ( \\lambda A + B ) ( \\widetilde { \\Lambda } _ t ( \\lambda ) \\otimes I _ n ) \\end{align*}"}
-{"id": "8064.png", "formula": "\\begin{align*} A ( 1 , x ) = \\begin{cases} \\ \\begin{bmatrix} \\cos ( \\pi \\alpha ) & - \\sin ( \\pi \\alpha ) \\\\ \\sin ( \\pi \\alpha ) & \\cos ( \\pi \\alpha ) \\end{bmatrix} & x _ 0 = 0 , \\\\ [ 2 0 p t ] \\ \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} & x _ 0 = 1 . \\end{cases} \\end{align*}"}
-{"id": "1589.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( L _ { Y _ { I } } C _ { x } \\right ) _ { , x } + C ^ { x } L _ { Y _ { I } } C _ { x } = c \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "945.png", "formula": "\\begin{align*} T _ p ^ { \\mathbb { C } } M : = T _ p M \\cap J T _ p M \\ ; . \\end{align*}"}
-{"id": "8217.png", "formula": "\\begin{align*} \\Lambda _ E : = H - E ( u _ 0 \\bar { v } _ 0 + \\bar { u } _ 0 v _ 0 ) . \\end{align*}"}
-{"id": "6333.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 2 ( 1 / p _ 1 - 1 / p _ 2 ) } = 2 ^ { j \\widetilde { A _ 1 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j \\widetilde { A _ 2 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j \\widetilde { A _ 3 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "8987.png", "formula": "\\begin{align*} ( ~ ^ { A B } _ { a } I ^ \\alpha f ) ( t ) = \\frac { 1 - \\alpha } { B ( \\alpha ) } f ( t ) + \\frac { \\alpha } { B ( \\alpha ) } ( ~ _ { a } I ^ \\alpha f ) ( t ) . \\end{align*}"}
-{"id": "3316.png", "formula": "\\begin{align*} F ( v _ 1 \\wedge v _ 0 ) = [ 2 ] ^ { 1 / 2 } q ^ { - 2 } v _ 1 \\wedge v _ { - 1 } , F ( v _ 1 \\wedge v _ { - 1 } ) = [ 2 ] ^ { 1 / 2 } q ^ 2 v _ 0 \\wedge v _ { - 1 } , F ( v _ 0 \\wedge v _ { - 1 } ) = 0 . \\end{align*}"}
-{"id": "7605.png", "formula": "\\begin{align*} P _ p ( x _ 0 ) \\ge \\left ( \\frac { \\varepsilon } { 2 l } \\right ) ^ { K _ 1 } \\prod _ { i = 1 } ^ { K _ 1 } \\left ( 1 + \\kappa ( i - 1 ) \\right ) , P _ e ( x _ 0 ) \\ge \\left ( \\frac { \\delta } { 2 l } \\right ) ^ { K _ 2 } \\prod _ { i = 1 } ^ { K _ 2 } \\left ( 1 + \\kappa ( i - 1 ) \\right ) . \\end{align*}"}
-{"id": "4054.png", "formula": "\\begin{align*} q + r = m + n + l + k \\leq r + \\dfrac { q } { 2 } , \\end{align*}"}
-{"id": "9948.png", "formula": "\\begin{align*} \\liminf _ { j \\rightarrow \\infty } ( \\widehat { \\mathcal E } _ \\mu ( u _ j ) - \\widehat { \\mathcal E } _ \\mu ( u _ \\infty ) ) \\geq 0 . \\end{align*}"}
-{"id": "6045.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } \\Delta u _ { i } ^ { \\varepsilon } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { \\varepsilon } \\sum \\limits _ { j \\neq i } H ( u _ { j } ^ { \\varepsilon } ) ( x ) & \\Omega , \\\\ u _ { i } ^ { \\varepsilon } ( x ) = \\phi _ { i } ( x ) & ( \\partial \\Omega ) _ 1 , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5476.png", "formula": "\\begin{align*} \\begin{array} { l } F i n d \\ , \\ , x \\ , \\\\ s o \\ , a s \\ , t o \\ , s a t i s f y \\ , \\\\ \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , { f _ { i j } } \\left ( x \\right ) \\left ( \\begin{array} { l } \\le \\\\ \\cong \\\\ \\ge \\end{array} \\right ) f _ { i j } ^ * , \\ , \\ , i = 1 , 2 , . . . m ; \\ , j = 1 , 2 , . . . , p _ m ^ { } \\\\ s u b j e c t \\ , \\ , t o \\ , \\\\ \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , \\ , x \\in S \\end{array} \\end{align*}"}
-{"id": "7704.png", "formula": "\\begin{align*} x _ { 2 n } = \\frac 1 { ( 2 n ) ^ 4 } , x _ { 2 n + 1 } = \\frac { 1 + \\varepsilon } { ( 2 n ) ^ 2 } . \\end{align*}"}
-{"id": "3981.png", "formula": "\\begin{align*} w _ r \\in V ( \\delta _ 1 , \\delta _ 2 , \\cdots , \\delta _ k ) \\delta _ i = l - 2 r . \\end{align*}"}
-{"id": "5780.png", "formula": "\\begin{align*} u \\cdot v \\cdot x _ { \\alpha } - ( - 1 ) ^ { p ( u ) p ( v ) } v \\cdot u \\cdot x _ { \\alpha } = [ u , v ] \\cdot x _ { \\alpha } \\end{align*}"}
-{"id": "2346.png", "formula": "\\begin{align*} T e _ { N + 1 } = \\frac { a _ 1 + a _ n } { 2 } , T e _ { N + 2 } = \\frac { a _ 2 - a _ 1 } { 2 } , \\dotsc , T e _ { N + n } = \\frac { a _ n - a _ { n - 1 } } { 2 } , \\end{align*}"}
-{"id": "1757.png", "formula": "\\begin{align*} d _ i = 0 \\ \\Leftrightarrow \\ i \\leq M . \\end{align*}"}
-{"id": "9299.png", "formula": "\\begin{align*} A _ { \\pi ( t , z ) } p ( t , x , z ) = \\pi ( t , z ) \\alpha ( t ) p ' ( t , x , z ) + \\frac { 1 } { 2 } \\pi ^ 2 ( t , z ) \\beta ^ 2 ( t ) p '' ( t , x , z ) . \\end{align*}"}
-{"id": "2133.png", "formula": "\\begin{align*} r _ * : = \\begin{cases} r _ { \\l _ k } & \\mathcal { R } _ { \\l _ k } \\cap \\{ r \\leq 2 \\l _ k \\} \\\\ r _ { 4 \\l _ k } & \\mathcal { R } _ { \\l _ k } \\cap \\{ r > 3 \\l _ k \\} \\end{cases} \\qquad U _ { a , * } : = \\begin{cases} U _ { a , \\l _ k } & \\mathcal { R } _ { \\l _ k } \\cap \\{ r \\leq 2 \\l _ k \\} \\\\ U _ { a , 4 \\l _ k } & \\mathcal { R } _ { \\l _ k } \\cap \\{ r > 3 \\l _ k \\} \\end{cases} \\end{align*}"}
-{"id": "9549.png", "formula": "\\begin{align*} d ( \\pi ( g ) , \\pi ( h ) ) & = d ( \\pi ( h s _ 1 \\dots s _ k s _ { k + 1 } ) , \\pi ( h ) ) \\\\ & \\leq d ( \\pi ( h s _ 1 \\dots s _ k s _ { k + 1 } ) , \\pi ( h s _ 1 \\dots s _ k ) ) + d ( \\pi ( h s _ 1 \\dots s _ k ) , \\pi ( h ) ) \\\\ & \\leq \\ell ( \\lambda ) + B + ( \\ell ( \\lambda ) + B ) k \\\\ & = ( \\ell ( \\lambda ) + B ) ( k + 1 ) . \\end{align*}"}
-{"id": "9706.png", "formula": "\\begin{align*} \\lim _ { y \\to 0 ^ + } \\frac { \\Gamma _ 1 ( y ) } { \\log ( 1 / y ) \\log _ 2 ( 1 / y ) ^ 2 y } = 1 , \\Gamma _ 1 \\in _ 0 ( 1 ) . \\end{align*}"}
-{"id": "329.png", "formula": "\\begin{align*} ( \\mathsf W ^ k ) _ { i j } = \\sum _ { \\substack { p : \\ , i \\to j \\\\ \\ell ( p ) = k } } p \\ , , i , j = 1 , . . . , N , \\end{align*}"}
-{"id": "1529.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\partial _ t \\rho + \\kappa ( - \\Delta ) ^ { 1 / 2 } \\rho + \\mbox { d i v } _ x ( \\mu ( E ) \\rho ) & = & 0 & ( 0 , \\infty ) \\times \\R ^ d \\times \\R ^ d , \\\\ \\rho ( \\cdot , 0 ) & = & \\rho ^ { i n } & \\R ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "9057.png", "formula": "\\begin{align*} u _ k ^ { ( N ) } : = \\left \\{ \\begin{array} { l l } \\upsilon - ( k - r ) ^ { \\alpha _ - } , & r \\leq k \\leq \\lfloor N / 2 \\rfloor , \\\\ \\upsilon - ( N - k ) ^ { \\alpha _ - } - \\frac { 3 } { 4 } \\log N , & \\lfloor N / 2 \\rfloor < k \\leq N , \\end{array} \\right . \\end{align*}"}
-{"id": "4211.png", "formula": "\\begin{align*} \\varphi ( x _ k ) - \\varphi ( x _ { k + 1 } ) & = \\varphi ( x _ k ) - \\min \\varphi - ( \\varphi ( x _ { k + 1 } ) - \\min \\varphi ) \\\\ & \\leq ( \\frac { 1 } { \\nu } - \\frac { 1 } { 2 L } ) \\| \\mathcal { R } _ { 1 / L } ( x _ { k - 1 } ) \\| ^ 2 - \\frac { 1 } { 2 L } \\sum _ { i = 1 } ^ \\infty \\| \\mathcal { R } _ { 1 / L } ( x _ { k + i } ) \\| ^ 2 , ~ k \\geq 1 . \\end{align*}"}
-{"id": "3609.png", "formula": "\\begin{align*} R \\int _ { A _ 1 } D \\Phi ( \\bar { g } ^ R , \\bar { \\pi } ^ R ) \\cdot ( 1 , 0 ) \\ , d x & = R \\left ( B ^ 2 _ { ( ( g ^ \\theta ) ^ R , ( \\pi ^ \\theta ) ^ R ) } ( 1 , 0 ) - B ^ 1 _ { ( g ^ R , \\pi ^ R ) } ( 1 , 0 ) \\right ) \\\\ & = B ^ { 2 R } _ { ( g ^ \\theta , \\pi ^ \\theta ) } ( 1 , 0 ) - B ^ R _ { ( g , \\pi ) } ( 1 , 0 ) , \\end{align*}"}
-{"id": "5769.png", "formula": "\\begin{align*} [ T _ { j k } ( \\lambda ) , T _ { j k } ( \\mu ) ] & = 0 , j , k = 1 , 2 , \\\\ A ( \\mu ) B ( \\lambda ) & = f ( \\lambda , \\mu ) B ( \\lambda ) A ( \\mu ) + g ( \\mu , \\lambda ) B ( \\mu ) A ( \\lambda ) , \\\\ B ( \\mu ) A ( \\lambda ) & = f ( \\lambda , \\mu ) A ( \\lambda ) B ( \\mu ) + g ( \\mu , \\lambda ) A ( \\mu ) B ( \\lambda ) , \\\\ D ( \\mu ) B ( \\lambda ) & = f ( \\mu , \\lambda ) B ( \\lambda ) D ( \\mu ) + g ( \\lambda , \\mu ) B ( \\mu ) D ( \\lambda ) , \\\\ & \\end{align*}"}
-{"id": "6817.png", "formula": "\\begin{align*} \\sum _ { k = p } ^ q \\left [ B _ k , B _ { p + q - k } \\right ] = \\sum _ { k = p + 1 } ^ { q - 1 } \\left [ B _ k , B _ { p + q - k } \\right ] = 0 \\end{align*}"}
-{"id": "3572.png", "formula": "\\begin{align*} L ( f , X ) : & = \\rho _ g ^ { - 1 } D \\Phi ^ W _ { ( g , \\pi ) } \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) . \\end{align*}"}
-{"id": "9772.png", "formula": "\\begin{align*} V ( \\vec { x } ) = - \\dfrac { 1 } { 2 } [ \\boldsymbol { \\lambda } ^ { * } ( \\vec { x } ) ] ^ { \\top } \\vec { G } \\vec { H } ^ { - 1 } \\vec { G } ^ { \\top } \\boldsymbol { \\lambda } ^ { * } ( \\vec { x } ) - [ \\boldsymbol { \\lambda } ^ { * } ( \\vec { x } ) ] ^ { \\top } ( \\vec { W } + \\vec { S } \\vec { x } ) . \\end{align*}"}
-{"id": "4263.png", "formula": "\\begin{align*} u \\in \\phi ( \\{ k ' \\} , \\overline { k } _ { 2 m + 2 } , \\overline { k } _ { 2 m + 3 } ) = \\phi ( \\{ k ' \\} , \\overline { k } _ { 2 m + 3 } , \\overline { k } _ { 2 m + 2 } ) . \\end{align*}"}
-{"id": "4810.png", "formula": "\\begin{align*} K ( x ) = \\frac { \\beta } { 4 } \\sigma ' _ 2 ( \\frac { x } { 2 } ) \\end{align*}"}
-{"id": "8305.png", "formula": "\\begin{align*} = g + \\vert V ( G ) \\vert - 1 + \\sum _ { i = 1 } ^ m \\left ( g ( G _ i ) + \\vert V ( G _ i ) \\vert - 1 \\right ) = g + m - h - 1 . \\end{align*}"}
-{"id": "7298.png", "formula": "\\begin{align*} [ E _ { \\xi + \\alpha _ k } , E _ { \\xi ^ \\prime } ^ * ] _ q = c _ { k , \\xi } ^ { - 1 } E _ k \\triangleright [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q + c _ { k , \\xi } ^ { - 1 } c _ { k , \\xi ^ \\prime } ^ \\prime q ^ { ( \\xi - \\alpha _ k , \\alpha _ k ) } ( E _ \\xi E _ { \\xi ^ \\prime - \\alpha _ k } ^ * - q ^ { - ( \\xi , \\xi ^ \\prime + \\alpha _ k ) } E _ { \\xi ^ \\prime - \\alpha _ k } ^ * E _ \\xi ) . \\end{align*}"}
-{"id": "4053.png", "formula": "\\begin{align*} q + r = m + n + l + k \\leq 2 p + q , \\end{align*}"}
-{"id": "7179.png", "formula": "\\begin{align*} - \\Delta u + ( u \\cdot \\nabla ) u + \\nabla p = 0 , \\mbox { i n } \\ , \\R ^ 3 _ + , \\end{align*}"}
-{"id": "7320.png", "formula": "\\begin{align*} \\hat { R } ( v _ { 0 } \\otimes v _ { 0 } ) = v _ { 0 } \\otimes v _ { 0 } + q ^ { - 2 } ( q ^ { 2 } - q ^ { - 2 } ) v _ { 1 } \\otimes v _ { - 1 } . \\end{align*}"}
-{"id": "1346.png", "formula": "\\begin{align*} \\frac { \\partial Z _ { t } } { \\partial \\psi } \\vert _ { \\delta _ 0 } = \\sum _ { j = 0 } ^ { \\infty } \\tau _ { j } ( \\omega ) e _ { t - J _ L - j } ( \\delta _ { 0 } ) . \\end{align*}"}
-{"id": "6135.png", "formula": "\\begin{align*} I \\ge & ( c _ 2 - c _ 1 ) c _ 3 + ( a _ 2 - a _ 1 ) a _ 3 \\\\ \\ge & \\frac 2 3 [ ( a _ 2 + c _ 2 ) - ( a _ 1 + c _ 1 ) ] \\\\ = & \\frac 8 3 ( K _ { 1 3 } - K _ { 1 2 } ) = \\frac 8 3 [ K _ { 1 3 } + s K _ { 1 2 } - ( 1 + s ) K _ { 1 2 } ] \\\\ \\ge & \\frac 8 3 [ K _ s - ( 1 + s ) \\epsilon _ 0 + ( 1 + s ) \\epsilon ] \\\\ \\ge & \\frac 8 3 [ \\frac { \\sqrt { 2 } } 2 - \\frac { \\sqrt { 4 + 2 \\sqrt { 2 } } } 4 + ( 1 + s ) \\epsilon ] > \\frac 8 3 \\epsilon . \\end{align*}"}
-{"id": "8566.png", "formula": "\\begin{align*} ( \\rho ^ { k } \\otimes ^ { k } \\rho ) \\Delta ( m _ { 1 } r m _ { 2 } ) & = ( \\rho ^ { k } \\otimes ( ^ { k } \\rho ) ) ( \\Delta ( m _ { 1 } ) r m _ { 2 } + m _ { 1 } \\Delta ( r m _ { 2 } ) ) \\\\ & = ( \\rho ^ { k } \\otimes ^ { k } \\rho ) ( 1 \\otimes m _ { 1 } r m _ { 2 } + m _ { 1 } \\otimes r m _ { 2 } ) \\\\ & = \\bar { e _ { k } } \\otimes \\rho ( m _ { 1 } r m _ { 2 } ) + m _ { 1 } \\bar { e _ { k } } \\otimes ( ^ { k } \\rho ( r m _ { 2 } ) ) \\\\ & = \\bar { e _ { k } } \\otimes \\rho ( m _ { 1 } r m _ { 2 } ) \\end{align*}"}
-{"id": "4989.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n k } ( P ) ) = & - c _ { 1 } h _ { Z _ { 1 } } ( p ^ { - 1 } f ^ { k ( n - 1 ) } ( P ) ) + h _ { E } ( p ^ { - 1 } f ^ { k ( n - 1 ) } ( P ) ) \\\\ & + \\rho _ { k } c _ { 1 } h _ { D _ { 1 } } ( f ^ { k ( n - 1 ) } ( P ) ) + c _ { 1 } h _ { { E _ { 1 } ' } } ( f ^ { k ( n - 1 ) } ( P ) ) \\\\ \\leq & \\rho _ { k } c _ { 1 } h _ { D _ { 1 } } ( f ^ { k ( n - 1 ) } ( P ) ) + C \\sqrt [ ] { h _ { H } ( f ^ { k ( n - 1 ) } ( P ) ) + \\gamma } \\\\ & + c _ { 1 } C \\sqrt [ ] { h _ { H } ( f ^ { k ( n - 1 ) } ( P ) ) } \\end{align*}"}
-{"id": "4734.png", "formula": "\\begin{align*} ( \\beta _ 1 v _ { 1 1 } + \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } + \\beta _ 2 v _ { 2 2 } ) x _ 2 = \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 \\end{align*}"}
-{"id": "15.png", "formula": "\\begin{align*} A & = \\ \\sum _ { i = 3 } ^ { n } \\sum _ { j = 3 } ^ { m } ( A _ { i , j } + A _ { i - 1 , j - 1 } - A _ { i - 1 , j } - A _ { i , j - 1 } ) L ^ { ( i , j ) } + \\\\ & \\sum _ { j = 3 } ^ { m } ( A _ { 2 , j } - A _ { 2 , j - 1 } ) L ^ { ( 2 , j ) } + \\sum _ { i = 3 } ^ { n } ( A _ { i , 2 } - A _ { i - 1 , 2 } ) L ^ { ( i , 2 ) } + A _ { 2 , 2 } ^ { ( 2 , 2 ) } L ^ { ( 2 , 2 ) } \\end{align*}"}
-{"id": "4208.png", "formula": "\\begin{align*} d ^ 2 ( x _ { k + 1 } , \\mathrm { A r g } \\min f ) & = \\| x _ { k + 1 } - x _ { k + 1 } ^ \\prime \\| ^ 2 \\leq \\| x _ { k + 1 } - x _ k ^ \\prime \\| ^ 2 \\\\ & = \\| x _ k - h \\cdot \\nabla f ( x _ k ) - x _ k ^ \\prime \\| ^ 2 \\\\ & = d ^ 2 ( x _ k , \\mathrm { A r g } \\min f ) - 2 h \\langle \\nabla f ( x _ k ) , x _ k - x _ k ^ \\prime \\rangle + h ^ 2 \\| \\nabla f ( x _ k ) \\| ^ 2 , \\end{align*}"}
-{"id": "9714.png", "formula": "\\begin{align*} 0 < \\lambda ( \\epsilon ) = \\frac { \\eta ( \\epsilon ) ( 1 - q - \\epsilon ) G _ 0 ( \\delta ( \\epsilon ) ) } { T _ 1 ( \\epsilon ) } . \\end{align*}"}
-{"id": "8239.png", "formula": "\\begin{align*} \\alpha = \\epsilon , t _ 0 \\lesssim 1 : C ( \\epsilon + \\delta ) \\leq \\nu \\end{align*}"}
-{"id": "6710.png", "formula": "\\begin{align*} p = p ( \\theta , \\zeta ) = p _ K ^ I \\theta ^ K \\zeta _ I , \\end{align*}"}
-{"id": "7275.png", "formula": "\\begin{align*} \\widetilde { S } _ { j k } : = \\left ( \\begin{array} { c | c c c } 1 & a _ { j k , 1 } & \\cdots & a _ { j k , r } \\\\ \\hline 0 & & & \\\\ \\vdots & & S _ { j k } & \\\\ 0 & & & \\end{array} \\right ) . \\end{align*}"}
-{"id": "84.png", "formula": "\\begin{align*} \\Lambda ( r , f ) = \\prod _ { i = 0 } ^ { r - 1 } x _ { i f } \\end{align*}"}
-{"id": "7875.png", "formula": "\\begin{align*} & \\sum _ { n = 0 } ^ { \\infty } | q _ n ( t , x , y ) | \\ , \\le \\ , \\sum _ { n = 0 } ^ { \\infty } \\gamma _ n \\big ( \\rho _ { ( n + 1 ) { \\beta _ 2 } } ^ 0 + \\rho ^ { \\beta _ 2 } _ { n { \\beta _ 2 } } \\big ) ( t , x - y ) \\\\ & \\le \\ , \\sum _ { n = 0 } ^ { \\infty } \\gamma _ n \\Phi ^ { - 1 } ( T ^ { - 1 } ) ^ { - ( n + 1 ) { \\beta _ 2 } } \\big ( \\rho _ { { \\beta _ 2 } } ^ 0 + \\rho _ 0 ^ { { \\beta _ 2 } } \\big ) ( t , x - y ) = C _ 2 \\big ( \\rho _ { { \\beta _ 2 } } ^ 0 + \\rho _ 0 ^ { { \\beta _ 2 } } \\big ) ( t , x - y ) \\ , . \\end{align*}"}
-{"id": "8194.png", "formula": "\\begin{align*} \\phi ( b ) = b ^ p + p \\d b \\end{align*}"}
-{"id": "4826.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\| ( I - A _ n ) R \\| | = 0 \\end{align*}"}
-{"id": "2142.png", "formula": "\\begin{align*} H _ { e } ^ { s } ( \\Omega ) : = \\left \\{ u \\in H ^ { s } ( \\mathbb { R } ^ { n } ) \\ , : \\ , u = 0 \\mathbb { R } ^ { n } \\backslash \\Omega \\right \\} , \\end{align*}"}
-{"id": "5467.png", "formula": "\\begin{align*} \\sum _ { n = N - H } ^ { N + y } e ^ { - n / N } t _ H ( n - N ) \\Bigl ( R ( n ) - ( 2 \\psi ( n ) - n ) \\Bigr ) \\ll H N ( \\log N ) ^ 2 \\log ( 2 H ) \\end{align*}"}
-{"id": "441.png", "formula": "\\begin{align*} \\sigma _ k ( z ) = ( s _ 0 , s _ 1 , \\dots , s _ r ) , \\\\ \\sigma _ k ( z ' ) = ( s ' _ 0 , s ' _ 1 , \\dots , s ' _ r ) . \\end{align*}"}
-{"id": "4252.png", "formula": "\\begin{align*} x _ q ^ k & = \\frac { k } { q } + \\frac { \\alpha ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) , \\\\ \\varphi _ q ^ k & = \\frac { \\mu ( x _ q ^ k ) } { q } \\left ( 1 + \\frac { \\beta _ 3 ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) \\right ) , \\end{align*}"}
-{"id": "5461.png", "formula": "\\begin{align*} T _ H ( N , y ; \\alpha ) = H \\sum _ { n = N - H } ^ { N + y } e ( n \\alpha ) - \\sum _ { m = 0 } ^ { y } m e ( ( N + m ) \\alpha ) - \\sum _ { m = 1 } ^ { H } m e ( ( N - m ) \\alpha ) = A - B - C , \\end{align*}"}
-{"id": "7422.png", "formula": "\\begin{align*} \\Delta ^ { I \\cup \\{ i \\} , J \\cup \\{ j \\} } \\Delta ^ { I \\cup \\{ k \\} , J \\cup \\{ l \\} } = \\Delta ^ { I \\cup \\{ i \\} , J \\cup \\{ l \\} } \\Delta ^ { I \\cup \\{ k \\} , J \\cup \\{ j \\} } + \\Delta ^ { I , J } \\Delta ^ { I \\cup \\{ i , k \\} , J \\cup \\{ j , l \\} } . \\end{align*}"}
-{"id": "87.png", "formula": "\\begin{align*} G \\cong \\begin{cases} B \\left ( \\frac { 2 ^ n - ( - 1 ) ^ n } { 3 } , 3 , 2 ^ { 2 n / 3 } , 1 \\right ) & \\mathrm { i f ~ ( A ) ~ d o e s ~ n o t ~ h o l d ~ a n d } ~ k \\not \\equiv 0 , l \\not \\equiv 0 , k \\not \\equiv l , \\\\ \\Z _ { 2 ^ n - ( - 1 ) ^ n } & \\mathrm { i f } ~ k \\equiv 0 , \\mathrm { o r } ~ l \\equiv 0 , \\mathrm { o r } ~ k \\equiv l , \\\\ \\Z _ { ( { 2 ^ { n / 3 } - ( - 1 ) ^ { n / 3 } } ) / { 3 } } * \\Z * \\Z & \\mathrm { i f ~ ( A ) ~ h o l d s } . \\end{cases} \\end{align*}"}
-{"id": "9344.png", "formula": "\\begin{align*} & d u ( t , x , z ) = - x ^ 2 u ( t , x , z ) d t + k ( t , x , z ) [ d F ( t , x , z ) - x y ( t , x , z ) q ( t , x , z ) d t ] \\\\ & = - x ^ 2 u ( t , x , z ) d t + k ( t , x , z ) y ( t , x , z ) q ( t , x , z ) d G ( t ) \\\\ & - k ( t , x , z ) x y ( t , x , z ) q ( t , x , z ) d t \\\\ & = f ( t , x , z ) d t + k ( t , x , z ) y ( t , x , z ) q ( t , x , z ) d G ( t ) \\end{align*}"}
-{"id": "3815.png", "formula": "\\begin{align*} \\frac { ( 1 + t z x ) } { ( 1 - z x ) } = 1 + ( 1 + t ) \\sum _ { \\ell \\geq 1 } ( z x ) ^ \\ell \\end{align*}"}
-{"id": "723.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n } ( P ) ) = & - \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { n - 1 - i } c _ { 1 } h _ { Z _ { 1 } } ( p ^ { - 1 } ( f ^ { i } ( P ) ) ) + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { n - 1 - i } h _ { E } ( p ^ { - 1 } ( f ^ { i } ( P ) ) ) \\\\ & + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { n - 1 - i } c _ { 1 } h _ { E _ { 1 } ' } ( f ^ { i } ( P ) ) + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { n - i } h _ { N } ( f ^ { i } ( P ) ) + \\delta ^ { n } h _ { H } ( P ) . \\end{align*}"}
-{"id": "9904.png", "formula": "\\begin{align*} k e = q ^ 2 e k , k f = q ^ { - 2 } f k , k k ^ { - 1 } = k ^ { - 1 } k \\ ; = 1 , \\hbox { a n d } [ e , f ] \\ ; = \\ ; \\frac { k - k ^ { - 1 } } { q - q ^ { - 1 } } , \\end{align*}"}
-{"id": "3575.png", "formula": "\\begin{align*} & s _ 1 = 0 , t _ 1 = 4 , s _ j = - 1 , t _ k = 3 ( j , k = 2 , \\dots , n + 1 ) . \\end{align*}"}
-{"id": "6248.png", "formula": "\\begin{align*} \\bar { w } '' = - ( \\bar { \\lambda } _ 2 - \\bar { \\lambda } _ 1 ) \\bar { w } < 0 . \\end{align*}"}
-{"id": "9565.png", "formula": "\\begin{align*} \\lambda _ { n } ^ { ( 1 ) } = \\langle \\psi _ { n } ^ { ( 0 ) } , H ^ { ( 1 ) } \\psi _ { n } ^ { ( 0 ) } \\rangle \\end{align*}"}
-{"id": "7313.png", "formula": "\\begin{align*} \\hat { R } _ { V , W } ( v \\otimes w ) = q ^ { ( \\mathrm { w t } ( v ) , \\mathrm { w t } ( w ) ) } w \\otimes v + \\sum _ i w _ i \\otimes v _ i , \\end{align*}"}
-{"id": "8033.png", "formula": "\\begin{align*} s _ { i , j } ^ { - 1 } & = s _ { i , j } ( - 1 ) , \\\\ [ s _ { i , j } , s _ { j , k } ] & = \\mathrlap { s _ { i , k } , } \\hphantom { s _ { i , k } ( - 1 ) } \\\\ [ s _ { j i } , s _ { k , j } ] & = s _ { i , k } ( - 1 ) \\ : i < j < k . \\end{align*}"}
-{"id": "7713.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\sum _ { k = 1 } ^ n \\sigma _ k \\xi _ { k + 1 } = \\infty . \\end{align*}"}
-{"id": "3528.png", "formula": "\\begin{align*} \\Phi ^ W _ { ( g , \\pi ) } ( \\gamma , \\tau ) = \\Phi ( \\gamma , \\tau ) + ( 0 , \\tfrac { 1 } { 2 } \\gamma \\cdot _ g ( \\textup { d i v } _ g \\pi + W ) ) \\end{align*}"}
-{"id": "3354.png", "formula": "\\begin{align*} \\| M \\| _ { 1 \\ast } \\ ; = \\ ; \\max _ { \\sum _ { i = 1 } ^ 1 s _ i ^ 2 = 1 } \\sum _ { i = 1 } ^ { n } \\sigma _ i ( M ) s _ i \\geq \\ ; \\dots \\ ; \\geq \\ ; \\max _ { \\sum _ { i = 1 } ^ n s _ i ^ 2 = 1 } \\sum _ { i = 1 } ^ n \\sigma _ i ( M ) s _ i = \\| M \\| _ { n \\ast } = \\| M \\| _ F . \\end{align*}"}
-{"id": "8212.png", "formula": "\\begin{align*} \\gamma _ 0 : = \\left \\{ \\begin{array} { l l } \\Omega - 4 \\epsilon , & \\Omega > 0 \\\\ | \\Omega | , & \\Omega < 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2984.png", "formula": "\\begin{align*} \\delta _ H \\Theta ^ 2 ( x , y ) = - \\sum _ { i + j = N + 1 , ~ i , j > 0 } ~ ~ \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ { \\lambda _ j ( x , y ) } , \\alpha _ n ] - [ \\alpha _ i , f _ 0 ^ { \\lambda _ j ( x , y ) } ] - [ \\alpha _ 0 , [ f _ i ^ y , f _ j ^ x ] ] \\end{align*}"}
-{"id": "580.png", "formula": "\\begin{align*} \\pi ( \\delta ( x ) ) = \\pi ( [ r _ 1 ] + y ) = \\pi ( [ r _ 1 ] ) + \\pi ( y ) = \\pi ( [ r _ 1 ] ) = r _ 1 \\end{align*}"}
-{"id": "8413.png", "formula": "\\begin{align*} \\hat { N } ( t ) = \\sum _ { r = 0 } ^ d E _ r \\circ E _ r . \\end{align*}"}
-{"id": "8880.png", "formula": "\\begin{align*} - \\Delta \\abs { u } + \\abs { u } = \\abs { u } ^ { p - 1 } . \\end{align*}"}
-{"id": "3068.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\nu _ { \\beta , n } = \\nu _ { \\beta , \\infty } \\end{align*}"}
-{"id": "1120.png", "formula": "\\begin{align*} \\mathbf { G } _ { \\mathrm { d } } [ \\iota ] = \\mathbf { \\hat { G } } _ { \\mathrm { d } } [ \\iota ] + \\mathcal { E } _ { \\mathrm { d } } [ \\iota ] \\end{align*}"}
-{"id": "7544.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\mathcal { N } _ 0 < \\infty y _ n \\in \\left ( 0 , \\frac { l } { 1 - \\gamma } + \\varepsilon \\right ) , \\ , n \\geq \\mathcal N _ 0 \\right ] = 1 , \\end{align*}"}
-{"id": "5594.png", "formula": "\\begin{align*} v M _ { 1 1 } v ^ * f ( x ) = \\left ( \\begin{array} { c } a ( x ) \\\\ c ( x ) \\end{array} \\right ) \\int _ { \\R ^ 2 } \\overline a ( y ) f _ 1 ( y ) + \\overline { c } ( y ) f _ 2 ( y ) \\ , d y . \\end{align*}"}
-{"id": "5918.png", "formula": "\\begin{align*} \\big ( L ^ p ( \\R ^ d ; ( B ^ { s _ 0 } _ { p , p } ( \\R ^ d ) ) , L ^ p ( \\R ^ d ; ( B ^ { s _ 1 } _ { p , p } ( \\R ^ d ) ) \\big ) _ { \\theta , p } = L ^ p ( \\R ^ d ; B ^ { s } _ { p , p } ( \\R ^ d ) ) , \\end{align*}"}
-{"id": "8878.png", "formula": "\\begin{align*} \\mathcal { E } ( R _ \\# B ) = \\mathcal { E } ( B ) . \\end{align*}"}
-{"id": "3219.png", "formula": "\\begin{align*} T _ { j k } w _ k = w _ j + \\sum _ { | \\alpha | \\geq n + 1 } f _ { k j , \\alpha } \\cdot w _ j ^ \\alpha . \\end{align*}"}
-{"id": "6791.png", "formula": "\\begin{align*} v ( \\varphi , \\psi ) _ t : = \\left ( \\lim _ { l \\to + \\infty } u ^ { l } _ t \\right ) ^ * , \\ t \\in [ 0 , + \\infty ) . \\end{align*}"}
-{"id": "2429.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } T _ m f \\| _ { L ^ { p _ 2 } } \\sim & \\| \\Box _ k ^ { \\alpha _ 2 } T _ m f \\| _ { M _ 2 } \\\\ = & \\| \\Box _ k ^ { \\alpha _ 2 } T _ m \\Box _ k ^ { \\alpha _ 2 , \\ast } f \\| _ { M _ 2 } \\\\ \\lesssim & \\| \\Box _ k ^ { \\alpha _ 2 } T _ m \\| _ { M _ 1 \\rightarrow M _ 2 } \\| \\Box _ k ^ { \\alpha _ 2 , \\ast } f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "8277.png", "formula": "\\begin{align*} l _ { j \\rightarrow n } ^ t = \\log \\frac { \\phi \\left ( \\widehat { r } _ j ^ t ; 0 , 0 . 5 + v _ r \\right ) } { \\phi \\left ( \\widehat { r } _ j ^ t ; 0 , v _ r \\right ) } , \\end{align*}"}
-{"id": "4848.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = \\iint \\frac { ( z - w ) h ^ { \\rm g e o } _ { 2 2 } ( z , w ) } { z w ( z w - 1 ) } \\frac { 1 } { 1 - c z } \\frac { 1 } { 1 - c w } \\frac { \\dd z } { z ^ u } \\frac { \\dd w } { w ^ v } , \\end{align*}"}
-{"id": "6631.png", "formula": "\\begin{align*} \\alpha _ k : = \\big \\{ t ^ k _ j \\in [ 0 , T ] \\ ; \\ , : \\ ; \\ , 0 \\leqslant j \\leqslant N _ k , \\ ; t ^ k _ 0 = 0 , \\ ; t ^ k _ { N _ k } = T , \\ ; t ^ k _ j < t ^ k _ { j + 1 } \\big \\} . \\end{align*}"}
-{"id": "4901.png", "formula": "\\begin{align*} \\Delta _ \\kappa = - \\xi _ { 2 - \\kappa } \\circ \\xi _ \\kappa , \\end{align*}"}
-{"id": "9126.png", "formula": "\\begin{align*} & \\left ( \\int _ 0 ^ \\tau \\| A e ^ { - t A } \\int _ 0 ^ \\tau e ^ { - ( \\tau - s ) A } g ( s ) d s \\| ^ 2 d t \\right ) ^ { 1 / 2 } \\\\ & = \\left ( \\int _ 0 ^ \\tau \\| A ^ { 1 / 2 } e ^ { - t A } \\int _ 0 ^ \\tau A ^ { 1 / 2 } e ^ { - ( \\tau - s ) A } g ( s ) d s \\| ^ 2 d t \\right ) ^ { 1 / 2 } \\\\ & \\le C \\| \\int _ 0 ^ \\tau A ^ { 1 / 2 } e ^ { - ( \\tau - s ) A } g ( s ) d s \\| . \\end{align*}"}
-{"id": "9604.png", "formula": "\\begin{align*} F ( z ) = \\int _ { - \\infty } ^ \\infty d r ( e ^ { i z \\sqrt { r ^ 2 + m ^ 2 } } - e ^ { i z r } ) . \\end{align*}"}
-{"id": "7845.png", "formula": "\\begin{align*} \\{ \\eta _ { i j } , u ^ { k l } \\} = \\delta _ { j k } u ^ { i l } , \\end{align*}"}
-{"id": "5397.png", "formula": "\\begin{align*} \\binom { N } { 2 } = ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } \\leq k \\left ( k - \\alpha - \\frac { 3 } { 8 } + \\frac { 1 } { 2 n } + \\frac { 1 } { n ^ 2 } \\right ) \\frac { n ^ 2 } { 2 } \\end{align*}"}
-{"id": "3659.png", "formula": "\\begin{align*} \\beta ^ { m _ 0 } \\binom { n _ 0 ^ 2 } { m _ 0 } ^ 3 \\end{align*}"}
-{"id": "1312.png", "formula": "\\begin{align*} R _ { G } ( n ) = \\sum _ { m _ 1 + m _ 2 = n } \\Lambda ( m _ 1 ) \\Lambda ( m _ 2 ) , \\end{align*}"}
-{"id": "6820.png", "formula": "\\begin{align*} - \\sum _ { k = - m + 1 } ^ { - 1 } \\left [ B _ k , B _ { - m + n - k } \\right ] = \\sum _ { k = 0 } ^ { n - 1 } \\left [ B _ k , B _ { - m + n - k } \\right ] \\ , . \\end{align*}"}
-{"id": "5979.png", "formula": "\\begin{align*} N ( x ) \\geq 7 2 - 2 4 = 4 8 > 6 \\times 6 , \\end{align*}"}
-{"id": "7547.png", "formula": "\\begin{align*} f ( x ) = \\left \\{ \\begin{array} { l l } 7 . 8 6 x - 2 3 . 3 1 x ^ 2 + 2 8 . 7 5 x ^ 3 - 1 3 . 3 0 x ^ 4 , & \\mbox { i f ~ ~ } x \\in [ 0 , 0 . 9 9 ] , \\\\ \\frac { 1 0 0 F ( 0 . 9 9 ) } { 1 0 0 x + 1 } , & \\mbox { i f ~ ~ } x \\in ( 0 . 9 9 , \\infty ) . \\end{array} \\right . \\end{align*}"}
-{"id": "6936.png", "formula": "\\begin{align*} s _ \\lambda ( X ) = M ^ \\perp _ \\pi ( X ) s ^ { ( \\pi ) } _ \\lambda ( X ) \\quad \\mbox { a n d } { s _ { \\lambda ' } ( X ) = ( - 1 ) ^ { | \\lambda | } { M } ^ \\perp _ \\pi ( X ) s ^ { * ( \\pi ) } _ { \\lambda } ( X ) } \\ , . \\end{align*}"}
-{"id": "8973.png", "formula": "\\begin{align*} 0 < \\gamma \\leq 1 , f \\geq 0 , f \\in L ^ { m } ( \\Omega ) , m : = \\frac { N p } { N ( p - 1 ) + s p + \\gamma ( N - s p ) } . \\end{align*}"}
-{"id": "7307.png", "formula": "\\begin{align*} \\eth = \\sum _ i S ^ { - 1 } ( E _ { \\xi _ i } ) \\otimes \\gamma _ { - } ( w _ i ) \\in U _ q ( \\mathfrak { g } ) \\otimes \\mathrm { C l } _ q . \\end{align*}"}
-{"id": "681.png", "formula": "\\begin{align*} T = \\big \\{ g \\in G \\ , : \\ , \\int _ Z \\phi ( g z ) \\ , d m ( z ) > 0 \\big \\} \\end{align*}"}
-{"id": "8920.png", "formula": "\\begin{align*} \\lim _ { \\abs { x } \\to + \\infty } \\frac { \\Phi _ { \\tau _ - } ( \\abs { x } ) } { \\Phi _ { \\tau _ + } ( \\abs { x } ) } = 1 . \\end{align*}"}
-{"id": "9117.png", "formula": "\\begin{align*} \\| ( \\lambda I - B A ) u \\| \\| u \\| & = \\| B ( \\lambda B ^ { - 1 } - A ) u \\| \\| u \\| \\\\ & \\geq \\frac { 1 } { \\| B ^ { - 1 } \\| _ { \\mathcal { L ( H ) } } } \\| ( \\lambda B ^ { - 1 } I - A ) u \\| \\| u \\| \\\\ & \\geq \\frac { 1 } { \\| B ^ { - 1 } \\| _ { \\mathcal { L ( H ) } } } | ( \\lambda B ^ { - 1 } u - A u , u ) | \\\\ & = \\frac { | ( B ^ { - 1 } u , u ) | } { \\| B ^ { - 1 } \\| _ { \\mathcal { L ( H ) } } } | \\lambda - \\frac { ( A u , u ) } { ( B ^ { - 1 } u , u ) } | . \\end{align*}"}
-{"id": "5996.png", "formula": "\\begin{align*} \\limsup _ { N \\to \\infty } \\frac { \\ln \\P \\{ \\max _ { 1 \\le n \\le N } | S _ n | \\le f _ N \\} } { f _ N ^ { - 1 / H } N } = 0 . \\end{align*}"}
-{"id": "8999.png", "formula": "\\begin{align*} \\int _ 0 ^ x ( x - t ) ^ { \\mu - 1 } E ^ \\gamma _ { \\rho , \\mu } ( \\lambda [ x - t ] ^ \\rho ) t ^ { \\nu - 1 } E ^ \\sigma _ { \\rho , \\nu } ( \\lambda t ^ \\rho ) d t = x ^ { \\mu + \\nu - 1 } E ^ { \\gamma + \\sigma } _ { \\rho , \\mu + \\nu } ( \\lambda x ^ \\rho ) . \\end{align*}"}
-{"id": "9176.png", "formula": "\\begin{align*} \\mathcal { P } _ { s , j } & = \\{ m \\in \\mathcal { P } _ s / m _ j \\neq 0 \\} . \\end{align*}"}
-{"id": "8466.png", "formula": "\\begin{align*} A _ { 0 } ( J _ { \\varepsilon } \\textbf { u } ^ { \\varepsilon } ) \\partial _ { t } \\textbf { u } ^ { \\varepsilon } + \\sum _ { j = 1 } ^ { d } J _ { \\varepsilon } A _ { 0 } A _ { j } ( J _ { \\varepsilon } \\textbf { u } ^ { \\varepsilon } ) \\partial _ { x _ { j } } J _ { \\varepsilon } \\textbf { u } ^ { \\varepsilon } + \\left ( \\begin{array} { c } 0 \\\\ \\nabla P ^ { \\varepsilon } \\\\ \\end{array} \\right ) = 0 , \\end{align*}"}
-{"id": "3599.png", "formula": "\\begin{align*} ( h , w ) = \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) \\end{align*}"}
-{"id": "2131.png", "formula": "\\begin{align*} \\begin{cases} L _ a U = | y | ^ a f & B _ 1 \\setminus \\mathcal { P } \\\\ U = 0 & \\mathcal { P } \\end{cases} \\end{align*}"}
-{"id": "4661.png", "formula": "\\begin{align*} J _ L ( f ) ( g ) & = \\sum _ { x \\in L _ 0 } f ( g \\cdot x ) , J _ { \\hat { L } } ( f ) ( g ) = \\sum _ { x \\in \\hat { L } _ 0 } f ( g \\cdot x ) . \\end{align*}"}
-{"id": "3520.png", "formula": "\\begin{align*} \\Delta \\rho = N ^ 2 d ^ { - 4 } \\rho ( 1 + N ^ { - 1 } d ^ 2 \\Delta d - 2 N ^ { - 1 } d ) . \\end{align*}"}
-{"id": "7352.png", "formula": "\\begin{align*} F V _ 1 = [ 2 ] ^ { 1 / 2 } V _ 0 , F V _ 0 = [ 2 ] ^ { 1 / 2 } V _ { - 1 } , F V _ { - 1 } = 0 . \\end{align*}"}
-{"id": "448.png", "formula": "\\begin{align*} x ' & = \\prod _ { i = 1 } ^ { \\beta _ 1 } ( a _ 1 ^ { e _ { 1 i } } a _ 2 ^ { f _ { 1 i } } ) a _ 3 ^ { f _ { 2 1 } } \\prod _ { i = 2 } ^ { \\beta _ 2 } ( a _ 2 ^ { e _ { 2 i } } a _ 3 ^ { f _ { 2 i } } ) \\\\ & \\quad \\ \\dots a _ m ^ { f _ { ( m - 1 ) 1 } } \\prod _ { i = 2 } ^ { \\beta _ { m - 1 } } ( a _ { m - 1 } ^ { e _ { ( m - 1 ) i } } a _ m ^ { f _ { ( m - 1 ) i } } ) , \\end{align*}"}
-{"id": "5357.png", "formula": "\\begin{align*} & U _ i ( y ) = \\sum _ { \\substack { z \\in Y _ { i + 1 } \\\\ y \\leq z } } z & D _ { i + 1 } ( y ) = \\sum _ { \\substack { x \\in Y _ { i } \\\\ x \\leq y } } x \\end{align*}"}
-{"id": "1865.png", "formula": "\\begin{align*} \\iota _ b \\iota _ a \\Omega = f _ { a b } ^ c \\ , \\iota _ c \\Theta \\ ; + \\ ; [ \\iota _ a \\Theta , \\iota _ b \\Theta ] . \\end{align*}"}
-{"id": "3915.png", "formula": "\\begin{align*} \\left ( ( u ' ) ^ { 3 } \\right ) ' = \\frac { e ^ { u ' } } { 2 } - 1 , u ( T ) = u ' ( 0 ) = u ' ( T ) , \\end{align*}"}
-{"id": "1375.png", "formula": "\\begin{align*} L ( \\theta ) : = \\int _ { \\R ^ { n } } \\prod _ { t = 1 } ^ { n } \\exp \\left \\{ y _ { t } W _ { t } - m _ t b ( W _ { t } ) + c ( y _ { t } ) \\right \\} g ( \\alpha ; \\tau , \\psi ) d \\alpha \\end{align*}"}
-{"id": "8318.png", "formula": "\\begin{align*} z ^ p - z = \\wp ( z ) = \\wp \\Big ( \\sum _ { i = 1 } ^ n \\alpha _ i z _ i \\Big ) = \\sum _ { i = 1 } ^ n \\alpha _ i \\wp ( z _ i ) = \\sum _ { i = 1 } ^ n \\alpha _ i \\gamma _ i . \\end{align*}"}
-{"id": "9113.png", "formula": "\\begin{align*} u ' ( t ) + \\mathcal { A } ( t ) u ( t ) = f ( t ) , \\ u ( 0 ) = u _ 0 \\in \\mathcal { H } \\end{align*}"}
-{"id": "7488.png", "formula": "\\begin{align*} P ( \\Delta _ { R ^ { ( k ) } } ( \\infty ) = k ) > 0 . \\end{align*}"}
-{"id": "1138.png", "formula": "\\begin{align*} \\mathop { \\min } \\limits _ { x _ { 3 } } F _ { 3 } \\left ( x \\right ) = \\mathop { \\min } \\limits _ { x _ { 3 } } \\left ( f _ { 3 1 } \\left ( x \\right ) , f _ { 3 2 } \\left ( x \\right ) , . . . , f _ { 3 p _ { 3 } } \\left ( x \\right ) \\right ) , \\end{align*}"}
-{"id": "9073.png", "formula": "\\begin{align*} M _ N ^ { ( \\beta ) } ( s ) = \\mathbb { E } \\left [ \\mathrm { T r } ( W _ N e ^ { s W _ N } ) \\right ] \\ , \\end{align*}"}
-{"id": "2907.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = s _ \\lambda / L _ { \\pi } ( X ) = L _ { \\pi } ^ \\perp ( X ) s _ \\lambda \\ , , s _ \\lambda ( X ) = s ^ { ( \\pi ) } _ \\lambda / M _ { \\pi } ( X ) = M _ { \\pi } ^ \\perp ( X ) s ^ { ( \\pi ) } _ \\lambda \\ , , \\end{align*}"}
-{"id": "864.png", "formula": "\\begin{align*} \\begin{cases} - \\nabla \\cdot ( \\sigma \\nabla u ) = f = f ^ + - f ^ - & \\mbox { i n } \\Omega \\\\ \\sigma \\nabla u \\cdot n = 0 & \\mbox { o n } \\partial \\Omega \\\\ | \\nabla u | \\leq 1 & \\mbox { i n } \\Omega , \\\\ | \\nabla u | = 1 & \\sigma - \\mbox { a . e . } \\end{cases} \\end{align*}"}
-{"id": "3607.png", "formula": "\\begin{align*} \\Phi ( g , \\pi ) = D \\Phi | _ { ( g _ \\mathbb { E } , 0 ) } ( g - g _ \\mathbb { E } , \\pi ) + Q _ { ( g _ \\mathbb { E } , 0 ) } ( g - g _ \\mathbb { E } , \\pi ) . \\end{align*}"}
-{"id": "8647.png", "formula": "\\begin{align*} K = V \\times U = { \\dd ( \\Lambda ^ { 1 / 2 } ) } \\times { U } \\end{align*}"}
-{"id": "3090.png", "formula": "\\begin{align*} a ( u _ i , v ) = \\lambda _ i b ( u _ i , v ) \\forall v \\in V . \\end{align*}"}
-{"id": "2410.png", "formula": "\\begin{align*} \\rho ( z , t ) = u + P ( z ) + Q ( z ) + v R ( z ) + u ^ 2 + v ^ 2 + o ( u ^ 2 , u v , v ^ 2 , u | z | ^ { 2 k } ) , \\end{align*}"}
-{"id": "3218.png", "formula": "\\begin{align*} ( \\pi | _ { \\widetilde { V } _ j } ) ^ * u _ j = \\frac { 1 } { d } \\sum _ { \\nu = 1 } ^ d i _ \\nu ^ * \\widetilde { u } _ j , \\end{align*}"}
-{"id": "7975.png", "formula": "\\begin{align*} \\mu ( p ) \\mu ( p _ { z _ 0 } ^ \\circ ) \\leq \\big [ \\mu ( \\gamma _ n ) \\big ] ^ 2 = ( 2 \\pi ) ^ n . \\end{align*}"}
-{"id": "7136.png", "formula": "\\begin{align*} \\vec G _ { j } ( x , y ) | _ { x _ n = 0 } = 0 . \\end{align*}"}
-{"id": "2387.png", "formula": "\\begin{align*} \\nabla F _ { \\gamma } ^ { \\rm { D R } } ( z ) & = ( 2 \\gamma ) ^ { - 1 } \\big ( \\nabla R _ { \\gamma f } ( z ) z + R _ { \\gamma f } - R _ { \\gamma f } - \\nabla R _ { \\gamma f } ( z ) R _ { \\gamma g } ( R _ { \\gamma f } ( z ) ) \\big ) \\\\ & = ( 2 \\gamma ) ^ { - 1 } \\nabla R _ { \\gamma f } ( z ) ( z - R _ { \\gamma g } R _ { \\gamma f } ( z ) ) . \\end{align*}"}
-{"id": "6881.png", "formula": "\\begin{align*} A _ k ^ n : = \\int _ { t _ { k - 1 } } ^ { t _ k } \\mu _ n ( \\bar { t } _ n ) { \\biggl \\langle } Y ^ n _ { \\bar { t } _ n } - Z _ { \\bar { t } _ n } , \\bigl ( \\sigma \\bigl ( Y ^ n _ t \\bigr ) - \\sigma \\bigl ( Y ^ n _ { \\bar { t } _ n } \\bigr ) \\bigr ) \\dot { w } ^ n _ t - \\frac { 1 } { 2 } ( \\nabla \\sigma ) \\sigma \\bigl ( Y ^ n _ { \\bar { t } _ n } \\bigr ) { \\biggr \\rangle } \\ , \\d t , \\end{align*}"}
-{"id": "3088.png", "formula": "\\begin{align*} \\left | \\frac { \\lambda _ 1 - \\lambda _ * } { \\lambda _ 1 } \\right | \\leq \\left | \\frac { \\lambda _ i - \\lambda _ * } { \\lambda _ i } \\right | \\forall i = 1 , 2 , \\dots . \\end{align*}"}
-{"id": "8414.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\rho + \\nabla \\cdot ( \\rho v ) = 0 , \\\\ & \\partial _ { t } { v } + { v } \\cdot \\nabla { v } + f ( \\rho , v ) \\nabla \\rho + \\nabla { P } = 0 , \\\\ & \\nabla \\cdot v = 0 , \\end{cases} \\end{align*}"}
-{"id": "335.png", "formula": "\\begin{align*} f ( X ) = f ( [ X _ { i j } ] _ { 1 \\le i \\le r , 1 \\le j \\le n } ) . \\end{align*}"}
-{"id": "2471.png", "formula": "\\begin{align*} \\varepsilon _ K ( s , V , \\psi ) = \\varepsilon _ K ( V \\otimes \\omega _ { s - 1 / 2 } , \\psi ) \\end{align*}"}
-{"id": "5794.png", "formula": "\\begin{align*} W ^ { u _ { i } } ( z ) = J ^ { u _ { i } } ( z ) + ( \\mathrm { l o w e r } \\ \\mathrm { t e r m s } ) \\end{align*}"}
-{"id": "4670.png", "formula": "\\begin{align*} \\bar X _ i ( z ) & = \\begin{cases} \\chi ( z ) \\dd \\theta _ { \\theta ^ { - 1 } ( z ) } \\left ( X _ i \\left ( \\theta ^ { - 1 } ( z ) \\right ) \\right ) & \\mbox { i f } z \\in V _ 0 \\\\ 0 & \\mbox { i f } z \\in \\R ^ d \\setminus V _ 0 \\end{cases} & & \\mbox { f o r } i \\in \\{ 0 , 1 , \\dots , m \\} \\ ; , \\\\ \\bar X _ { m + k } ( z ) & = \\rho ( z ) e _ k & & \\mbox { f o r } k \\in \\{ 1 , \\dots , d \\} \\ ; , \\end{align*}"}
-{"id": "891.png", "formula": "\\begin{align*} \\int _ { { 1 } / { 2 } + i T } ^ { 2 + i T } \\zeta ( s ) d s = \\int _ { { 1 } / { 2 } } ^ { 2 } \\zeta ( \\sigma + i T ) d \\sigma = O \\left ( T ^ { 1 / 4 + \\varepsilon } \\right ) . \\end{align*}"}
-{"id": "6516.png", "formula": "\\begin{align*} u ' ( t ) + A u ( t ) = f ( t ) , 0 < t < T , u ( 0 ) = 0 , \\end{align*}"}
-{"id": "584.png", "formula": "\\begin{align*} r _ \\nu = \\pi ( \\delta ( \\cdots \\delta ( \\delta ( x - [ r _ 0 ] ) - [ r _ 1 ] ) \\cdots - [ r _ { \\nu - 1 } ] ) ) \\nu = 1 , \\dots , n - 1 \\end{align*}"}
-{"id": "2794.png", "formula": "\\begin{align*} \\frac { 6 d ^ { 2 } - 1 4 d + 7 } { 4 \\left ( d - 1 \\right ) ^ { 2 } } & = \\frac { 3 } { 4 } + O \\left ( \\frac { x } { n } \\right ) , \\\\ \\frac { 1 } { 2 \\left ( d - 1 \\right ) ^ { 3 } } & = \\frac { 1 } { 2 } + O \\left ( \\frac { x } { n } \\right ) , \\\\ \\frac { \\left ( 2 d ^ { 2 } - 4 d + 1 \\right ) } { 4 \\left ( d - 1 \\right ) ^ { 4 } d ^ { 2 } } & = \\frac { 1 } { 1 6 } + O \\left ( \\frac { x } { n } \\right ) . \\end{align*}"}
-{"id": "6529.png", "formula": "\\begin{align*} \\frac 1 \\tau \\sum _ { j = 0 } ^ k \\delta _ j v _ { n - j } + A _ m v _ n = f _ n + ( A _ m - A _ n ) v _ n , \\end{align*}"}
-{"id": "473.png", "formula": "\\begin{align*} ( \\beta _ 1 v _ { 1 1 } + \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } + \\beta _ 2 v _ { 2 2 } ) x _ 2 & = \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 , \\\\ ( \\beta _ 1 v _ { 1 1 } - \\beta _ 2 v _ { 1 2 } ) x _ 1 + ( \\beta _ 1 v _ { 2 1 } - \\beta _ 2 v _ { 2 2 } ) x _ 2 & = \\beta _ 1 \\alpha _ 1 - \\beta _ 2 \\alpha _ 2 . \\end{align*}"}
-{"id": "5481.png", "formula": "\\begin{align*} P ' \\Bigl [ ( P - ( 1 + c _ { 2 } ) P ' _ { n } ) \\tilde { g } \\ge t / 2 \\Bigr ] = P ' \\Bigl [ ( P - P ' _ { n } ) \\tilde { g } \\ge \\frac { t / 2 + c _ { 2 } P \\tilde { g } } { 1 + c _ { 2 } } \\Bigr ] . \\end{align*}"}
-{"id": "6482.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\lambda _ { k } k ^ { - \\frac { 2 \\beta } { n } } = C _ { 0 } , \\end{align*}"}
-{"id": "9773.png", "formula": "\\begin{align*} \\vec { z } ^ { * } ( \\vec { x } ) = - \\vec { H } ^ { - 1 } \\vec { G } ^ { \\top } \\boldsymbol { \\lambda } ^ { * } ( \\vec { x } ) , \\end{align*}"}
-{"id": "4783.png", "formula": "\\begin{align*} n _ 1 = n ( v ) ; n _ 2 = g ' ( u ) \\ , l ( v ) + f ' ( u ) \\ , e _ 4 , \\end{align*}"}
-{"id": "5571.png", "formula": "\\begin{align*} D ( - \\epsilon ) = W ( \\hat c _ 0 , c _ 0 ) \\prod _ { j = 1 } ^ N \\left ( 1 + \\frac { \\epsilon } { z _ j } \\right ) . \\end{align*}"}
-{"id": "435.png", "formula": "\\begin{align*} \\left \\langle u , \\tilde c _ 1 u \\right \\rangle = \\sum _ { i = 1 } ^ m \\int _ 0 ^ 1 \\left \\langle u , \\sum _ { k = 1 } ^ n \\sum _ { j _ 1 , \\dots , j _ k = 0 } ^ m \\int _ 0 ^ t \\int _ 0 ^ { t _ 2 } \\dots \\int _ 0 ^ { t _ k } Y _ { j _ 1 , \\dots , j _ k } ^ { ( n , i ) } ( 0 ) \\dd B _ s ^ { j _ k } \\dd B _ { t _ k } ^ { j _ { k - 1 } } \\dots \\dd B _ { t _ 2 } ^ { j _ 1 } \\right \\rangle ^ 2 \\dd t \\ ; , \\end{align*}"}
-{"id": "8143.png", "formula": "\\begin{align*} a _ 2 ( p , q , 2 ) = 1 + 2 \\left ( \\frac { 1 } { q } - \\frac { 1 } { p } \\right ) < a _ 4 , \\end{align*}"}
-{"id": "5840.png", "formula": "\\begin{align*} \\Delta F + \\left ( \\kappa \\left ( \\mu - \\lambda - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } - \\frac { 1 } { 2 } \\sigma \\sigma _ { , x } \\right ) F _ { , x } - F _ { , t } = 0 , \\end{align*}"}
-{"id": "4135.png", "formula": "\\begin{align*} S _ f ( n ) : = \\sum _ { m \\leq n } A _ f ( m ) , \\end{align*}"}
-{"id": "3458.png", "formula": "\\begin{align*} f ( H , D ) = \\sum _ { i = 0 } ^ { 2 n + 1 } ( - 1 ) ^ i c _ i H ^ i D ^ { 2 n + 1 - i } \\ , . \\end{align*}"}
-{"id": "9116.png", "formula": "\\begin{align*} u ' ( t ) + A ( t ) u ( t ) = f ( t ) \\ ( t \\in [ 0 , \\tau ] ) , \\ u ( 0 ) = u _ 0 \\end{align*}"}
-{"id": "2294.png", "formula": "\\begin{align*} \\big \\| ( d _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } \\le \\delta \\end{align*}"}
-{"id": "3564.png", "formula": "\\begin{align*} \\int _ { A _ 1 } \\lambda \\zeta R ^ { - 2 } ( \\log R ) ^ \\frac { 1 } { 2 } \\ , d x = 1 6 \\pi \\tilde { \\lambda } R ^ { - 2 } ( \\log R ) ^ \\frac { 1 } { 2 } > 0 , \\end{align*}"}
-{"id": "5764.png", "formula": "\\begin{align*} R _ { 1 2 } ( \\lambda , \\mu ) T _ 1 ( \\lambda ) T _ 2 ( \\mu ) = T _ 2 ( \\mu ) T _ 1 ( \\lambda ) R _ { 1 2 } ( \\lambda , \\mu ) . \\end{align*}"}
-{"id": "4777.png", "formula": "\\begin{align*} \\frac { t } { 2 } \\ , ( \\varphi ^ 2 ) ' + \\varphi ^ 2 + 1 = c \\sqrt { \\varphi ^ 2 + 1 } . \\end{align*}"}
-{"id": "8680.png", "formula": "\\begin{gather*} \\nabla _ k \\nabla _ { \\xi } ^ G P _ t [ \\phi ] ( x ) = \\int _ H \\big ( \\ < \\Gamma _ t k , Q _ t ^ { - \\frac { 1 } { 2 } } y \\ > \\ , \\ < \\Gamma _ t G \\xi , Q _ t ^ { - \\frac { 1 } { 2 } } y \\ > - \\ < \\Gamma _ t k , \\Gamma _ t G \\xi \\ > \\big ) \\phi ( e ^ { t A } x + y ) \\mu _ t ( d y ) , \\end{gather*}"}
-{"id": "7128.png", "formula": "\\begin{align*} X _ j ( t ) = R _ \\omega ( t ) \\ , Q \\Big ( t + \\frac { 2 \\pi j } { n } \\Big ) , 0 \\leq j \\leq n - 1 , R _ \\omega ( t ) = \\begin{bmatrix} \\cos ( \\omega t ) & - \\sin ( \\omega t ) & 0 \\\\ \\sin ( \\omega t ) & \\cos ( \\omega t ) & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} , \\end{align*}"}
-{"id": "9146.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p k _ i \\leq \\sum _ { i = 1 } ^ p \\ell _ i + p r = t \\ell + p r = t \\ell + ( t + d ) r = t m + d r . \\end{align*}"}
-{"id": "5037.png", "formula": "\\begin{align*} \\nu _ Y ( \\phi ) = \\int _ G \\nu ( g ^ { - 1 } \\cdot \\phi ) \\ , d \\eta ( g ) , \\textrm { f o r $ \\phi \\in C ( \\overline { Y } ) $ } . \\end{align*}"}
-{"id": "5339.png", "formula": "\\begin{align*} & d e g _ { L S } ( I - ( P + Q N _ { f } - \\frac { B _ { \\varphi , b } P } { T } + H P ) , \\Omega , 0 ) \\\\ & = d e g _ { B } \\left ( I - ( P + Q N _ { f } - \\frac { B _ { \\varphi , b } P } { T } + H P ) \\left | _ { \\overline { \\Omega \\cap \\mathbb { R } ^ { 2 } } } \\right . , \\Omega \\cap \\mathbb { R } ^ { 2 } , 0 \\right ) \\\\ & = d e g _ { B } ( G , \\Omega \\cap \\mathbb { R } ^ { 2 } , 0 ) \\neq 0 . \\end{align*}"}
-{"id": "7368.png", "formula": "\\begin{align*} \\Gamma _ - ^ { ( k + 1 ) } \\Gamma _ { 0 } ^ { ( k + 1 ) * } = t \\Gamma _ 0 ^ { ( k ) * } \\Gamma _ - ^ { ( k ) } + t ^ \\prime \\Gamma _ + ^ { ( k ) * } \\Gamma _ 0 ^ { ( k ) } . \\end{align*}"}
-{"id": "5913.png", "formula": "\\begin{align*} ( H ^ { s _ 0 } _ p ( \\R ^ d ) , H ^ { s _ 1 } _ p ( \\R ^ d ) ) _ { \\theta , p } = B ^ { s } _ { p , p } ( \\R ^ d ) \\end{align*}"}
-{"id": "963.png", "formula": "\\begin{align*} \\overline { \\partial } _ { M } T _ { q } u = ( - 1 ) ^ { q } T _ { ( q - 1 ) } ( \\overline { \\partial } _ { M } ^ { * } u ) + \\ ; , \\end{align*}"}
-{"id": "3652.png", "formula": "\\begin{align*} | \\lambda | \\le 2 ( 1 - c ) \\max _ { i = 1 , \\dots , k } \\left | \\frac { \\partial f } { \\partial x _ i } ( \\mathbf { P } _ c ) \\right | ^ { \\frac { 1 } { i } } . \\end{align*}"}
-{"id": "3857.png", "formula": "\\begin{align*} \\mathsf { h } _ { B _ { n + 1 } } ^ \\vee = \\mathsf { h } _ { A _ { 2 n } } ^ \\vee = 2 n + 1 , \\ \\mathsf { h } _ { C _ n } ^ \\vee = \\dfrac { 1 } { 2 } \\mathsf { h } _ { D _ { n + 1 } } ^ \\vee = \\mathsf { h } ^ \\vee _ { A _ n } = n + 1 . \\end{align*}"}
-{"id": "3027.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ i ] } } ( { t _ 1 } ) = \\sum \\limits _ { j = 1 } ^ N { { { \\bf { s } } ^ { [ j i ] } } } , \\end{align*}"}
-{"id": "6561.png", "formula": "\\begin{align*} \\theta _ s = \\inf _ { x \\in \\varOmega } \\arctan \\frac { a ( x , s ) } { | b ( x , s ) | } , \\end{align*}"}
-{"id": "4625.png", "formula": "\\begin{align*} \\hat { \\ell } _ { \\bar { d } _ \\mathcal { E } } = \\hat { \\ell } _ { m i n } + \\left ( \\hat { \\ell } _ { m a x } - \\hat { \\ell } _ { m i n } \\right ) \\mu ( \\bar { d } _ { \\mathcal { E } } ) . \\end{align*}"}
-{"id": "2039.png", "formula": "\\begin{align*} Z ( x , y , t ) & : = \\left \\langle \\nabla \\log u ( y , t ) , \\gamma ' \\left ( \\tfrac { d } { 2 } \\right ) \\right \\rangle - \\left \\langle \\nabla \\log u ( x , t ) , \\gamma ' \\left ( - \\tfrac { d } { 2 } \\right ) \\right \\rangle - 2 \\psi \\left ( \\frac { d ( x , y ) } { 2 } , t \\right ) , \\end{align*}"}
-{"id": "6709.png", "formula": "\\begin{align*} \\sum _ I \\int ( - 1 ) ^ { | I | } f _ I ^ I \\ , \\mu _ \\star = \\sum _ { K , Q } \\int ( - 1 ) ^ { \\lambda ( K , Q ) } ( f _ K ^ P \\star u _ { P Q } ) \\tau _ { K ' Q ' } \\mu _ \\star . \\end{align*}"}
-{"id": "9597.png", "formula": "\\begin{align*} \\eta _ y ( x , t ) : = \\psi _ { f , y } ( x , t ) - ( \\xi _ { 0 } + t \\dot \\xi _ { 0 } ) \\ , g ( x - y ) \\end{align*}"}
-{"id": "8001.png", "formula": "\\begin{align*} \\zeta ^ { d - j } + a _ { d - 1 } ( 0 ' , \\overline w ) \\ , \\zeta ^ { d - j - 1 } + . . . + a _ j ( 0 ' , \\overline w ) = 0 . \\end{align*}"}
-{"id": "3420.png", "formula": "\\begin{align*} d Y ( t ) = - F ( t , Y ( t ) , Z ( t ) ) d t + \\langle Z ( t ) , d W ( t ) \\rangle , Y ( T ) = \\zeta = \\langle \\eta ( T ) , u _ 0 ( \\xi ( T ) ) \\rangle , \\end{align*}"}
-{"id": "5254.png", "formula": "\\begin{align*} \\varphi _ { A , k } ( y + t k ) = \\varphi _ { A , k } ( y ) + t \\forall \\ , y \\in Y , \\ , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "3137.png", "formula": "\\begin{align*} L _ 6 ( \\lambda ) = \\begin{bmatrix} 0 & - I _ n & \\lambda I _ n & 0 \\\\ - I _ n & \\lambda P _ 4 - P _ 3 & \\lambda P _ 3 & 0 \\\\ \\lambda I _ n & \\lambda P _ 3 & \\lambda P _ 2 - P _ 1 & P _ 0 \\\\ 0 & 0 & P _ 0 & - \\lambda P _ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "4064.png", "formula": "\\begin{align*} p = n \\gamma , p = m \\beta , p = l \\alpha \\end{align*}"}
-{"id": "8168.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ N \\| X _ k u \\| _ s ^ 2 \\leq C ( \\| \\Delta _ H u \\| _ s ^ 2 + \\| u \\| ^ 2 _ s ) . \\end{align*}"}
-{"id": "247.png", "formula": "\\begin{align*} \\mathbb { E } _ f \\log ^ 4 \\xi _ { ( j ) , 1 } = \\mathbb { E } _ f [ \\{ \\log ( \\xi _ { ( j ) , 1 } f ( X _ 1 ) ) - \\log f ( X _ 1 ) \\} ^ 4 ] \\rightarrow \\mathbb { E } _ f \\log ^ 4 f ( X _ 1 ) \\end{align*}"}
-{"id": "8364.png", "formula": "\\begin{align*} z _ e & = 1 , \\\\ z _ g & = \\alpha _ g ( z _ { g ^ { - 1 } } ) , \\ \\ \\ g \\in G , \\\\ z _ g z _ { g h } & = z _ g \\alpha _ g ( z _ h ) , \\ \\ \\ g , h \\in G . \\end{align*}"}
-{"id": "5250.png", "formula": "\\begin{align*} \\operatorname * { l e v } \\nolimits _ { \\varphi , \\le } ( t ) = t k + A \\forall \\ , \\ t \\in { \\mathbb { R } } . \\end{align*}"}
-{"id": "8291.png", "formula": "\\begin{align*} Z _ i = d - \\sum _ { l _ 2 , \\cdots , l _ p } A _ { G _ 2 } ( i , l _ 2 , \\cdots , l _ p ) + o ( 1 ) . \\end{align*}"}
-{"id": "1891.png", "formula": "\\begin{align*} \\hat f _ { m } ( x ) : = \\frac { 1 } { 2 \\pi } \\int _ { - m } ^ { m } e ^ { - i u x } \\hat \\phi _ { X } ( u ) \\d u . \\end{align*}"}
-{"id": "2501.png", "formula": "\\begin{gather*} \\sum _ { k = 1 } ^ n { \\bf E } \\sup _ { 0 \\leqslant t \\leqslant 1 } \\left | z _ i ( u _ k , t ) - z _ { i + 1 } ( u _ k , t ) \\right | \\leqslant \\sum _ { k = 1 } ^ n \\sum _ { l = 1 } ^ k \\sum _ { j = 1 } ^ l 4 ( l - j ) \\cdot \\sqrt { \\varepsilon } = \\\\ = \\sum _ { k = 1 } ^ n \\dfrac { 2 k ( k ^ 2 - 1 ) } { 3 } \\cdot \\sqrt { \\varepsilon } \\leqslant \\sum _ { k = 1 } ^ n \\dfrac { 2 k ^ 3 } { 3 } \\cdot \\sqrt { \\varepsilon } \\leqslant \\dfrac { 2 n ^ 4 } { 3 } \\cdot \\sqrt { \\varepsilon } . \\end{gather*}"}
-{"id": "8974.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | u _ n ( x ) - u _ n ( y ) | ^ { p - 2 } \\ , ( u _ n ( x ) - u _ n ( y ) ) \\ , ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y = \\int _ { \\Omega } \\frac { f _ n ( x ) } { ( u _ n + 1 / n ) ^ \\gamma } \\ , \\varphi \\ , d x , \\end{align*}"}
-{"id": "8351.png", "formula": "\\begin{align*} \\sum _ { j \\in J } m _ j ^ * z m _ j = \\sum _ { j \\in J } z m _ j ^ * m _ j = z , \\end{align*}"}
-{"id": "7341.png", "formula": "\\begin{align*} \\pi _ + ( V _ 0 ) & = v _ 1 \\wedge v _ { - 1 } - v _ { - 1 } \\wedge v _ 1 - q ( q - q ^ { - 1 } ) v _ 0 \\wedge v _ 0 \\\\ & = 2 v _ 1 \\wedge v _ { - 1 } + ( q - q ^ { - 1 } ) ^ 2 v _ 1 \\wedge v _ { - 1 } = ( q ^ 2 + q ^ { - 2 } ) v _ 1 \\wedge v _ { - 1 } . \\end{align*}"}
-{"id": "8166.png", "formula": "\\begin{align*} | ( P X u , T _ { 2 \\delta } u ) | & = | ( X u , P ^ * T _ { 2 \\delta } u ) | \\leq \\frac 1 2 ( \\| X u \\| ^ 2 + \\| P ^ * T _ { 2 \\delta } u \\| ^ 2 ) \\\\ & \\leq \\frac 1 2 \\| X u \\| ^ 2 + \\| T _ { 2 \\delta } P ^ * u \\| ^ 2 + \\| [ P ^ * , T _ { 2 \\delta } ] u \\| ^ 2 . \\end{align*}"}
-{"id": "3486.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u ( x ) = f ( x ' ) & \\mbox { i n } \\Omega , \\\\ u = 0 & \\mbox { o n } \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "5471.png", "formula": "\\begin{align*} I _ X ^ { ( m ) } = \\bigoplus _ { t \\geq 0 } H ^ 0 ( \\mathcal I _ X ^ m ( t ) ) . \\end{align*}"}
-{"id": "7686.png", "formula": "\\begin{align*} F ( x ) : = f ( x ) - x , x > 0 \\end{align*}"}
-{"id": "461.png", "formula": "\\begin{align*} \\alpha : = \\inf \\{ \\pi ^ { \\top } x \\ , | \\ x \\in W \\} \\end{align*}"}
-{"id": "2582.png", "formula": "\\begin{align*} e ^ { - i t \\Delta } v _ n ^ j ( t ) = e ^ { i x \\xi _ n ^ j } e ^ { - i t \\Delta } \\Psi ^ j _ { [ h _ n ^ j ] } ( t ) = e ^ { i x \\xi _ n ^ j } \\bigl ( e ^ { - i ( h _ n ^ j ) ^ 2 t \\Delta } \\Psi ^ j ( ( h _ n ^ j ) ^ 2 t ) \\bigr ) _ { \\{ h _ n ^ j \\} } . \\end{align*}"}
-{"id": "4251.png", "formula": "\\begin{align*} \\hat F ( X , 0 ) & = ( X , 0 ) & X & \\in \\R . \\end{align*}"}
-{"id": "7921.png", "formula": "\\begin{align*} \\phi = \\begin{cases} \\frac { 1 } { m _ 1 } , & , \\\\ - \\frac { 1 } { m _ 2 } , & , \\\\ 0 , & \\end{cases} \\end{align*}"}
-{"id": "8430.png", "formula": "\\begin{align*} \\textbf { P } = \\left ( \\begin{array} { c c c c } I d & 0 \\\\ 0 & \\mathbb { P } \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "1367.png", "formula": "\\begin{align*} \\underset { \\omega \\in \\Omega } \\sup ~ Q _ { L } ^ { L R } ( \\omega ) = \\underset { \\omega \\in \\Omega } \\sup ~ 2 \\left [ l _ n ( \\hat \\beta , \\hat \\psi , \\omega ) - l _ n ( \\hat \\beta _ 0 , 0 , \\omega ) \\right ] . \\end{align*}"}
-{"id": "1996.png", "formula": "\\begin{align*} \\xi _ t = - \\log \\mathcal { E } ( U ) _ t = - U _ t + \\frac { \\sigma _ U ^ 2 } { 2 } t + \\sum _ { s \\leq t } [ \\Delta U _ s - \\log ( 1 + \\Delta U _ s ) ] . \\end{align*}"}
-{"id": "4285.png", "formula": "\\begin{align*} \\| x ^ * - y ^ * \\| & \\ge ( x ^ * - y ^ * ) ( z ) \\\\ & = x ^ * ( z ) - y ^ * ( z ) \\\\ & > 1 - \\eta + 1 - \\eta > 2 - \\delta . \\end{align*}"}
-{"id": "6773.png", "formula": "\\begin{align*} \\theta _ { \\psi } ^ n = e ^ { \\psi - u } \\theta _ { u } ^ n + e ^ { \\psi - v } \\theta _ v ^ n . \\end{align*}"}
-{"id": "2573.png", "formula": "\\begin{align*} \\tilde u ( t ) : = M ( - t ) u ( t ) . \\end{align*}"}
-{"id": "9003.png", "formula": "\\begin{align*} ( ~ ^ { A B C } ~ _ { 0 } D ^ \\alpha f ) ( t ) = ( ~ ^ { A B R } ~ _ { 0 } D ^ \\alpha f ) ( t ) - \\frac { B ( \\alpha ) } { 1 - \\alpha } f ( 0 ) E _ \\alpha ( \\lambda t ^ \\alpha ) , ~ ~ \\lambda = \\frac { - \\alpha } { 1 - \\alpha } , \\end{align*}"}
-{"id": "5724.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } : = \\inf \\left \\{ \\lambda > 0 \\ , : \\ , \\int _ { \\mathbb { R } ^ n } \\left | \\frac { f ( x ) } { \\lambda } \\right | ^ { p ( x ) } \\ , d x \\le 1 \\right \\} . \\end{align*}"}
-{"id": "4913.png", "formula": "\\begin{align*} p ( n ) = 2 \\pi ( 2 4 n - 1 ) ^ { - \\frac { 3 } { 4 } } \\sum _ { j = 1 } ^ { \\infty } \\frac { A _ j ( n ) } { j } I _ { \\frac { 3 } { 2 } } \\left ( \\frac { \\pi \\sqrt { 2 4 n - 1 } } { 6 j } \\right ) . \\end{align*}"}
-{"id": "5190.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 2 / 3 } } = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { * } \\| V \\| _ { F } \\\\ = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\ ! \\left ( \\frac { 2 \\| U \\| _ { * } + \\| V \\| ^ { 2 } _ { F } } { 3 } \\right ) ^ { 3 / 2 } . \\end{align*}"}
-{"id": "811.png", "formula": "\\begin{align*} ( H _ { 1 } - M ) | \\mathbf { s } \\rangle = ( 1 - q ^ { 2 } ) \\sum _ { \\mathbf { s } ' \\in \\mathcal { S } _ { M } ^ { \\mathrm { p e r } } \\setminus \\{ \\mathbf { s } \\} } q ( \\mathbf { s } | \\mathbf { s } ' ) ( | \\mathbf { s } ' \\rangle - | \\mathbf { s } \\rangle ) , \\end{align*}"}
-{"id": "8732.png", "formula": "\\begin{align*} \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } G B ( s , X ^ { x } _ s ) \\ , d s = e ^ { \\tau A } v ( 0 , x ) - v ( \\tau , X _ \\tau ^ { x } ) + \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } \\nabla ^ G v ( s , X _ s ^ x ) \\ ; d W _ s . \\end{align*}"}
-{"id": "977.png", "formula": "\\begin{align*} \\alpha ( \\overline { L } ) ( z ) = d h ( \\overline { L } ) ( z ) \\ ; , \\ ; z \\in M \\ ; , \\ ; L \\in \\mathcal { N } _ { z } \\ ; , \\end{align*}"}
-{"id": "8698.png", "formula": "\\begin{align*} v ( \\tau , \\Xi _ \\tau ^ { t , x } ) = Y _ \\tau ^ { t , x } , \\ , \\ ; \\ ; \\ ; \\tau \\in [ t , T ] , \\ ; \\P - a . s . . \\end{align*}"}
-{"id": "3606.png", "formula": "\\begin{align*} E & = \\tfrac { 1 } { 1 6 \\pi } \\lim _ { r \\to \\infty } B ^ r _ { ( g , \\pi ) } ( 1 , 0 ) \\\\ P _ i & = \\tfrac { 1 } { 8 \\pi } \\lim _ { r \\to \\infty } B ^ r _ { ( g , \\pi ) } ( 0 , \\frac { \\partial } { \\partial x ^ i } ) \\\\ \\mathcal C _ i & = \\tfrac { 1 } { 1 6 \\pi } \\lim _ { r \\rightarrow \\infty } B ^ r _ { ( g , \\pi ) } ( x ^ i , 0 ) \\\\ \\mathcal { J } _ { k } & = \\tfrac { 1 } { 8 \\pi } \\lim _ { r \\rightarrow \\infty } B ^ r _ { ( g , \\pi ) } ( 0 , x \\times \\frac { \\partial } { \\partial x ^ k } ) , \\end{align*}"}
-{"id": "1548.png", "formula": "\\begin{align*} J _ { \\P _ n } ^ { \\gamma } ( \\mu ) = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ 2 ^ 2 ( \\mu , \\nu _ i ) + \\gamma E ( \\mu ) . \\end{align*}"}
-{"id": "1856.png", "formula": "\\begin{align*} & \\exp \\left ( - i t \\left ( ( k - \\delta ) x - \\frac { ( k - \\delta ) ^ 2 x ^ 2 } { 2 } \\right ) - t ( k - \\delta ) ^ 2 x ^ 2 \\right ) \\\\ & = - \\frac { 1 } { 2 t ( k - \\delta ) ^ 2 x + i t ( k - \\delta ) ( 1 - ( k - \\delta ) x ) } \\frac { d } { d x } \\exp \\left ( - i t \\left ( ( k - \\delta ) x - \\frac { ( k - \\delta ) ^ 2 x ^ 2 } { 2 } \\right ) - t ( k - \\delta ) ^ 2 x ^ 2 \\right ) \\end{align*}"}
-{"id": "2624.png", "formula": "\\begin{align*} \\zeta _ { m } ^ { l } ( j ) = \\sum _ { k = 1 } ^ \\infty \\zeta _ { m k } ^ { l } ( j ) \\widetilde { e } _ k , \\end{align*}"}
-{"id": "1450.png", "formula": "\\begin{align*} \\mathcal { D } _ k : = \\left \\{ Q \\in \\mathcal { D } ( Q _ 0 ) : \\frac { w ( Q ) } { | Q | } > a ^ k \\gamma _ 0 \\right \\} ( k \\in \\N ) . \\end{align*}"}
-{"id": "6016.png", "formula": "\\begin{gather*} M _ { \\xi } = M _ 5 \\cup M _ 4 \\cup M _ 5 ^ - . \\end{gather*}"}
-{"id": "7243.png", "formula": "\\begin{align*} T _ { j k } w _ k = w _ j + \\sum _ { | \\alpha | \\geq n + 1 } f _ { k j , \\alpha } ( z _ j ) \\cdot w _ j ^ \\alpha . \\end{align*}"}
-{"id": "6412.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } ( u ( x , t ) - u _ { 0 } ( x ) ) + ( - \\Delta ) ^ { \\beta } u ( x , t ) & = f ( x , t ) \\Omega \\times [ 0 , T ] , \\\\ u ( x , t ) & = 0 \\quad \\quad \\quad \\ , \\mathbb { R } ^ { n } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = u _ { 0 } ( x ) \\quad \\ , \\ , \\Omega , \\ , t = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "9709.png", "formula": "\\begin{align*} x ' ( t ) & = - a g ( x ( t ) ) + b \\max _ { t - \\tau ( t ) \\leq s \\leq t } g ( x ( s ) ) , t \\geq 0 \\\\ x ( t ) & = \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] \\end{align*}"}
-{"id": "2618.png", "formula": "\\begin{align*} S _ { a , b } = u a b 2 ^ { - k } , & & T _ { a , a } = u a ^ 2 2 ^ { - ( k + 1 ) } \\end{align*}"}
-{"id": "6277.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ n ^ 2 - x _ { n + 1 } ^ 2 = \\tfrac { 1 } { K } \\end{align*}"}
-{"id": "3645.png", "formula": "\\begin{align*} c ^ * = 1 - \\frac { 1 } { \\max \\{ | - 2 | , | - 1 | - 2 \\} } = 1 - \\frac { 1 } { 2 } = 0 . 5 . \\end{align*}"}
-{"id": "6430.png", "formula": "\\begin{align*} L u ( t , x ) & = \\int _ { \\mathbb { R } ^ { n } } [ \\tilde { u } ( s , y ) - \\tilde { u } ( s , z ) ] \\tilde { a } ( y , z ) r ^ { - n - 2 \\beta } \\tilde { k } _ { 0 } ( y , z ) r ^ { n } d z \\\\ & = r ^ { - 2 \\beta } L \\tilde { u } ( s , y ) . \\end{align*}"}
-{"id": "5990.png", "formula": "\\begin{align*} \\underbrace { 0 = d _ { j _ 1 } = \\cdots = d _ { j _ { u _ 1 } } } _ { = \\sigma _ 0 } < \\underbrace { d _ { j _ { u _ 1 + 1 } } = \\cdots = d _ { j _ { u _ 2 } } } _ { = \\sigma _ 1 } < \\cdots < \\underbrace { d _ { j _ { u _ { s - 1 } + 1 } } = \\cdots = d _ { j _ r } } _ { = \\sigma _ { s - 1 } } < 1 = \\sigma _ { s } ; \\end{align*}"}
-{"id": "4586.png", "formula": "\\begin{align*} F ( q ) = \\begin{cases} q + 1 + 6 \\sqrt { q } , & q , \\\\ 2 ( q + 1 ) , & q . \\end{cases} \\end{align*}"}
-{"id": "4537.png", "formula": "\\begin{align*} W _ 2 = \\int _ { \\mathcal { X } \\times \\mathcal { X } } f ( x ) f ( y ) \\int _ { l _ x } ^ { v _ x } \\int _ { l _ y } ^ { v _ y } h ( u , v ) \\ , d ( G _ { n , x , y } - F _ { n , x } F _ { n , y } ) ( u , v ) \\ , d x \\ , d y + \\frac { 1 } { n } . \\end{align*}"}
-{"id": "4837.png", "formula": "\\begin{align*} \\sum _ { \\nu } s _ { \\lambda / \\nu } ( \\rho ) s _ { \\nu / \\mu } ( \\rho ' ) = s _ { \\lambda / \\mu } ( \\rho , \\rho ' ) . \\end{align*}"}
-{"id": "427.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | \\leq N } \\frac { ( m + | \\alpha | - 1 ) ! } { ( m - 1 ) ! \\alpha ! } z ^ \\alpha \\bar { w } ^ \\alpha & = \\sum _ { k = 0 } ^ N \\frac { ( m + k - 1 ) ( m + k - 2 ) \\cdots ( k + 1 ) } { ( m - 1 ) ! } ( z \\bar { w } ) ^ k \\ . \\end{align*}"}
-{"id": "5917.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ d _ x ) ) } = \\Big ( \\int _ { \\R ^ d } \\| f ( \\cdot , v ) \\| _ { H ^ { s } _ p } ^ p \\dd v \\Big ) ^ { 1 / p } . \\end{align*}"}
-{"id": "1904.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\int _ { - m ^ * } ^ { m ^ * } \\frac { 1 } { | \\phi _ X ( u ) | ^ { 2 ( K - 1 ) } } \\d u \\leq 2 a C _ 2 n ^ { - \\frac { 2 \\beta - 1 } { 2 \\beta K } } = o \\big ( ( \\log n ) ^ { - \\frac { 2 \\beta - 1 } { \\rho } } \\big ) . \\end{align*}"}
-{"id": "1173.png", "formula": "\\begin{align*} D ( - \\lambda ) = W ( \\hat c _ 0 , c _ 0 ) \\prod _ { j = 1 } ^ { \\infty } \\left ( 1 + \\frac { \\lambda } { z _ j } \\right ) . \\end{align*}"}
-{"id": "9746.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ^ { - 1 } ( x ) } { \\log ( 1 / x ) ^ { - 1 / \\alpha } } = 1 . \\end{align*}"}
-{"id": "3810.png", "formula": "\\begin{align*} t _ q = \\frac { 1 } { q } \\sum _ { i = 1 } ^ n x _ i ^ q \\end{align*}"}
-{"id": "4106.png", "formula": "\\begin{align*} q + r = m + n + l + k , \\end{align*} % \\end{align*}"}
-{"id": "5665.png", "formula": "\\begin{align*} \\mathcal { F } \\big ( ( - \\Delta ) ^ { \\alpha / 2 } f \\big ) ( k ) : = | k | ^ \\alpha \\mathcal { F } ( f ) ( k ) , \\end{align*}"}
-{"id": "3315.png", "formula": "\\begin{align*} F V _ 1 & = F v _ 1 \\otimes K ^ { - 1 } v _ 0 + v _ 1 \\otimes F v _ 0 - q ^ 2 F v _ 0 \\otimes K ^ { - 1 } v _ 1 - q ^ 2 v _ 0 \\otimes F v _ 1 \\\\ & = [ 2 ] ^ { 1 / 2 } ( v _ 0 \\otimes v _ 0 + v _ 1 \\otimes v _ { - 1 } - v _ { - 1 } \\otimes v _ 1 - q ^ 2 v _ 0 \\otimes v _ 0 ) = [ 2 ] ^ { 1 / 2 } V _ 0 . \\end{align*}"}
-{"id": "456.png", "formula": "\\begin{align*} \\pi ^ { \\top } z < w _ 0 + \\sum _ { i = 1 } ^ { n + 1 } \\varepsilon _ i w ^ i _ 0 & \\leq \\inf \\{ w ^ { \\top } x \\ , | \\ x \\in C \\} + \\sum _ { i = 1 } ^ { n + 1 } \\inf \\{ ( \\varepsilon _ i w ^ i ) ^ { \\top } x \\ , | \\ x \\in C \\} \\\\ & \\leq \\inf \\{ ( w ^ { \\top } + \\sum _ { i = 1 } ^ { n + 1 } \\varepsilon _ i w ^ i ) ^ { \\top } x \\ , | \\ x \\in C \\} \\leq \\pi ^ { \\top } x \\ \\ \\forall x \\in C , \\end{align*}"}
-{"id": "9801.png", "formula": "\\begin{align*} m ( \\mathbf { y } ) = \\frac { f ( \\mathbf { y } | \\boldsymbol { \\theta } ) \\pi ( \\boldsymbol { \\theta } ) } { \\pi ( \\boldsymbol { \\theta } | \\mathbf { y } ) } . \\end{align*}"}
-{"id": "2495.png", "formula": "\\begin{gather*} z _ { i + 1 } ( u _ k , t ) = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k ( z _ i ( u _ j , t ) - z _ i ( u _ j , t \\wedge \\sigma _ i ) ) \\cdot \\ 1 _ { A _ { k j } ^ i } = \\\\ = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k ( z _ i ( u _ j , t ) - z _ i ( u _ j , t \\wedge \\sigma _ i ) ) \\cdot \\ 1 _ { A _ { k j } ^ i } \\cdot \\ 1 \\{ \\sigma _ i \\leqslant t \\} , t \\geqslant 0 , \\end{gather*}"}
-{"id": "5663.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\partial _ t \\rho + \\kappa ( - \\Delta ) ^ { 1 / 2 } \\rho + \\mbox { d i v } _ x ( \\mu ( E ) \\rho ) & = & 0 & ( 0 , \\infty ) \\times \\R ^ d \\times \\R ^ d , \\\\ \\rho ( \\cdot , 0 ) & = & \\rho ^ { i n } & \\R ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "2265.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 3 } & \\frac { \\partial u } { \\partial t } + ( - \\varDelta ) ^ { 1 / 2 } u = f ( u ) \\ , \\ , \\ , & & \\ , \\ , & & \\R ^ d \\times ( 0 , T ) , \\\\ & u ( \\cdot , 0 ) = u _ 0 & & \\ , \\ , & & \\R ^ d , \\end{alignedat} \\right . \\end{align*}"}
-{"id": "258.png", "formula": "\\begin{align*} S _ 1 = \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\int _ 0 ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log ^ 2 u _ { x , s } \\ , d s \\ , d x = O \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "2337.png", "formula": "\\begin{align*} u ( y ) = ( - 1 ) ^ d p ( \\sqrt { y } ) p ( - \\sqrt { y } ) = \\prod _ { j = 1 } ^ d ( y - x _ j ^ 2 ) . \\end{align*}"}
-{"id": "502.png", "formula": "\\begin{align*} A = 4 l r + 2 \\pi \\Re ( a _ 1 ) + o ( 1 ) \\end{align*}"}
-{"id": "7468.png", "formula": "\\begin{align*} s H _ D ( V ) = \\sum _ { v \\in V } h _ D ( V , v ) . \\end{align*}"}
-{"id": "5366.png", "formula": "\\begin{align*} P ( x , y ) ~ = ~ B x + C y , \\end{align*}"}
-{"id": "5676.png", "formula": "\\begin{align*} \\check K ( x ) = ( 1 - ( x - 1 ) e _ 0 ) \\left ( 1 - \\left ( \\frac { 1 } { x } - 1 \\right ) e _ 0 \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "395.png", "formula": "\\begin{align*} Z ^ + ( f , L ; s , \\phi ) & = \\int _ { G _ + / \\Gamma , \\chi ( g ) \\geq 1 } \\chi ( g ) ^ s \\phi ( g ) \\sum _ { x \\in L \\setminus L _ 0 } f ( g \\cdot x ) d g , \\\\ Z ^ + ( f , \\hat { L } ; s , \\phi ) & = \\int _ { G _ + / \\Gamma , \\chi ( g ) \\geq 1 } \\chi ( g ) ^ s \\phi ( g ) \\sum _ { x \\in \\hat { L } \\setminus \\hat { L } _ 0 } f ( g \\cdot x ) d g . \\end{align*}"}
-{"id": "7042.png", "formula": "\\begin{align*} \\sum _ i a _ i [ F _ { i } ] = 0 H _ 2 ( X ; \\Z _ d ) \\quad a _ i \\in \\Z _ d . \\end{align*}"}
-{"id": "1716.png", "formula": "\\begin{gather*} \\epsilon _ { \\Phi } = - e ^ { 1 2 3 4 5 6 7 } . \\end{gather*}"}
-{"id": "2868.png", "formula": "\\begin{align*} F ^ c \\circ \\mathcal { B } _ { E _ 1 } ^ { \\mathbb { K } } = \\mathcal { B } _ { E _ 1 } \\circ F ^ c . \\end{align*}"}
-{"id": "8176.png", "formula": "\\begin{align*} f _ n = \\sum _ { k = 1 } ^ { n } \\binom { - 1 } { k - 1 } \\frac { ( k - 1 ) ! } { n ! } B _ { n , k } ( 1 ! \\phi _ 1 , 2 ! \\phi _ 2 , \\dots ) . \\end{align*}"}
-{"id": "1405.png", "formula": "\\begin{align*} S _ { C A , B , \\omega _ k } + S _ { A C , D , \\omega _ k ^ { \\vee } } = 0 . \\end{align*}"}
-{"id": "1275.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & \\overline \\alpha \\alpha & \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & \\overline \\alpha ^ 2 \\\\ 0 & \\overline \\alpha J ( T ^ { - r } , T ^ { - r } ) & 0 & \\alpha J ( T ^ { - 3 r } , T ^ { - 3 r } ) \\\\ 0 & \\alpha ^ 2 & \\overline \\alpha J ( T ^ { - 2 r } , T ^ { - r } ) & \\overline \\alpha \\alpha \\end{bmatrix} \\end{align*}"}
-{"id": "7807.png", "formula": "\\begin{align*} T : = \\sup \\left \\{ t : \\sup _ { \\Sigma \\left ( r \\right ) } \\left \\vert \\overset { \\circ } { \\mathrm { R m } _ { \\Sigma } } \\right \\vert ^ { 2 } S ^ { - 2 } < \\varepsilon t _ { 0 } \\leq r \\leq t \\right \\} . \\end{align*}"}
-{"id": "6275.png", "formula": "\\begin{align*} \\nabla v _ { y _ 0 } = \\nabla v _ { x _ 0 } = \\varphi ' e _ n . \\end{align*}"}
-{"id": "5678.png", "formula": "\\begin{align*} e _ 1 \\ e _ 0 \\ e _ 1 \\ e _ 0 \\ - \\ e _ 0 \\ e _ 1 \\ e _ 0 \\ e _ 1 \\ = \\ \\omega ( \\ e _ 0 ^ 2 \\ e _ 1 \\ e _ 0 \\ - \\ e _ 0 \\ e _ 1 \\ e _ 0 ^ 2 \\ ) \\ ; . \\end{align*}"}
-{"id": "9774.png", "formula": "\\begin{align*} P _ { \\max } ^ * = P ^ t _ { \\min } \\ \\ P _ { \\min } ^ * = P ^ t _ { \\max } \\end{align*}"}
-{"id": "4804.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { \\left ( b \\pm \\frac { t } { 2 } \\sqrt { a ^ 2 + 4 c t ^ 2 } \\pm \\frac { a ^ 2 } { 4 \\sqrt { c } } \\ln | 2 \\sqrt { c } t + \\sqrt { a ^ 2 + 4 c t ^ 2 } | \\right ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "3677.png", "formula": "\\begin{align*} \\frac { d H } { d t } = \\int H _ { \\zeta } \\frac { \\partial \\zeta } { \\partial t } d A = 0 ~ , \\end{align*}"}
-{"id": "7853.png", "formula": "\\begin{align*} h _ 0 ( x ) = x ^ { - \\alpha } h ( x ) , \\end{align*}"}
-{"id": "1395.png", "formula": "\\begin{align*} g _ j ( z ) = \\frac { \\dot { b } ( a ^ \\ast + \\sigma ^ \\ast z ) ^ { j - 1 } \\ddot { b } ( a ^ \\ast + \\sigma ^ \\ast z ) \\phi ( z ) } { \\int \\dot { b } ( a ^ \\ast + \\sigma ^ \\ast z ) ^ { j - 1 } \\ddot { b } ( a ^ \\ast + \\sigma ^ \\ast z ) \\phi ( z ) d z } , j = 1 , \\ldots , M \\end{align*}"}
-{"id": "8626.png", "formula": "\\begin{align*} T = \\begin{pmatrix} 1 & 0 & 0 & 0 & . . . . \\ 0 & 0 \\\\ 1 & 1 & 0 & 0 & . . . . \\ 0 & 0 \\\\ 0 & 1 & 1 & 0 & . . . . \\ 0 & 0 \\\\ & \\ddots & \\ddots & \\ddots & \\ddots & \\\\ 0 & 0 & 0 & \\ldots & 1 & 0 \\\\ 0 & 0 & 0 & \\ldots & 1 & 1 \\end{pmatrix} \\end{align*}"}
-{"id": "9341.png", "formula": "\\begin{align*} & d u ( t , x , z ) = u ( t , x , z ) b ( t , x , z ) d G ( t ) + k ( t , x , z ) d F ( t , x , z ) + x u ( t , x , z ) d G ( t ) + u ( t , x , z ) b ( t , x , z ) x d t \\\\ & + k ( t , x , z ) b ( t , x , z ) y ( t , x , z ) q ( t , x , z ) d t \\\\ & = u ( t , x , z ) [ \\{ b ( t , x , z ) + x \\} d G ( t ) + b ( t , x , z ) x d t ] + k ( t , x , z ) [ d F ( t , x , z ) + b ( t , x , z ) y ( t , x , z ) q ( t , x , z ) d t ] \\end{align*}"}
-{"id": "9133.png", "formula": "\\begin{align*} [ e ^ { - ( t - s ) A ( s ) } u ( t ) - u ( t ) ] & = [ e ^ { - ( t - s ) A ( s ) } u ( t ) - e ^ { - ( t - s ) A ( t ) } u ( t ) ] \\\\ & + [ e ^ { - ( t - s ) A ( t ) } u ( t ) - u ( t ) ] \\end{align*}"}
-{"id": "8389.png", "formula": "\\begin{align*} \\theta ( m _ 1 x m _ 2 ) = \\theta ( m _ 1 ) \\theta ( x ) \\theta ( m _ 2 ) . \\end{align*}"}
-{"id": "3874.png", "formula": "\\begin{align*} \\nabla ( \\gamma ( \\tilde { r } ) ( i _ x ( \\tilde { x } - a ) ) ) = \\left ( \\dfrac { \\partial } { \\partial x _ i } ( \\gamma ( \\tilde { r } ) ) ( i _ x ( \\tilde { x } - a ) ) _ j \\right ) + \\left ( \\gamma ( \\tilde { r } ) \\dfrac { \\partial ( i _ x ( \\tilde { x } - a ) ) _ j } { \\partial x _ i } \\right ) = : ( M _ { i j } ) + ( N _ { i j } ) . \\end{align*}"}
-{"id": "8696.png", "formula": "\\begin{align*} v ( t , x ) = Y _ t ^ { t , x } \\in K , ( t , x ) \\in [ 0 , T ] \\times H . \\end{align*}"}
-{"id": "9440.png", "formula": "\\begin{align*} Z ^ { 0 , z } _ t = Z ^ { s , Z ^ { 0 , z } _ s } _ t . \\end{align*}"}
-{"id": "7649.png", "formula": "\\begin{align*} [ S _ { \\alpha } ( t ) \\chi _ { r } ] ( x ) & \\geq c _ { 1 } \\int _ { B _ { 1 } ( 2 \\tau ) } \\frac { 1 } { ( 1 + | w | ) ^ { d + \\alpha } } d w \\\\ & = c _ { 2 } ( d , \\alpha ) , \\end{align*}"}
-{"id": "7886.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } \\sup _ { x \\in \\R ^ d } | u ( t , x ) - u ( 0 , x ) | = 0 \\ , , \\end{align*}"}
-{"id": "4295.png", "formula": "\\begin{align*} 2 5 \\epsilon n s ^ { n - 1 } = \\pm 1 2 5 - \\sum _ { j = 2 } ^ { n } \\binom { n } { j } ( 2 5 \\epsilon ) ^ j s ^ { n - j } \\end{align*}"}
-{"id": "2529.png", "formula": "\\begin{align*} \\left | \\mathbb { E } \\left [ \\frac 1 N \\sum _ { n = 0 } ^ { N - 1 } f ( U ^ n ) - \\int _ { \\mathcal { S } } f d \\mu _ h \\right ] \\right | \\le C _ h ( \\frac 1 T + \\tau ) . \\end{align*}"}
-{"id": "7543.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\mathcal { N } _ 1 < \\infty x _ { \\mathcal { N } _ 1 } \\in [ c , 2 K - c ] \\right ] = 1 . \\end{align*}"}
-{"id": "5232.png", "formula": "\\begin{align*} \\left [ X _ { \\varepsilon } , \\overline { L _ { k } } \\right ] _ { T } ( p ) = - e ^ { g _ { \\varepsilon } ( p ) } \\left ( \\overline { L _ { k } } g _ { \\varepsilon } ( p ) - \\alpha ( \\overline { L _ { k } } ) ( p ) \\right ) + b _ { \\varepsilon , k } ( p ) \\left [ L _ { k } , \\overline { L _ { k } } \\right ] _ { T } ( p ) \\ ; \\end{align*}"}
-{"id": "6160.png", "formula": "\\begin{align*} g _ 2 ( { \\bf x } ^ { \\bf s } ) = c ^ N k _ 1 ^ { s _ 1 } \\cdots k _ n ^ { s _ n } g _ 1 ( { \\bf x } ^ { \\bf s } ) . \\end{align*}"}
-{"id": "1481.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm B M O } _ { L ^ { p ( \\cdot ) } } } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { \\| \\chi _ Q \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } } \\| ( b - b _ Q ) \\chi _ Q \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } . \\end{align*}"}
-{"id": "8236.png", "formula": "\\begin{align*} i \\frac { d } { d t } \\left ( u _ 0 \\bar { v } _ 0 + \\bar { u } _ 0 v _ 0 \\right ) = \\epsilon \\left [ \\bar { u } _ 0 ( u _ 1 + u _ { - 1 } ) - u _ 0 ( \\bar { u } _ 1 + \\bar { u } _ { - 1 } ) + \\bar { v } _ 0 ( v _ 1 + v _ { - 1 } ) - v _ 0 ( \\bar { v } _ 1 + \\bar { v } _ { - 1 } ) \\right ] . \\end{align*}"}
-{"id": "4897.png", "formula": "\\begin{align*} a _ 0 ( z ) f ( z ) + a _ 1 ( z ) f ( z ^ k ) + \\cdots + a _ d ( z ) f ( z ^ { k ^ d } ) = 0 , \\end{align*}"}
-{"id": "6654.png", "formula": "\\begin{align*} \\prod _ { j = 1 } ^ m \\det ( - \\rho _ j ( \\Phi ) | V _ j ^ I ) ^ { d _ j - 1 } \\cdot q ^ { - \\frac { 1 } { 2 } ( d _ j - 2 ) ( d _ j - 1 ) \\dim V _ j ^ I } \\end{align*}"}
-{"id": "7414.png", "formula": "\\begin{align*} \\vec { i } = ( i ( 1 ) , i ( 2 ) , \\dots , i ( l ) ) \\end{align*}"}
-{"id": "1276.png", "formula": "\\begin{align*} \\begin{bmatrix} \\alpha \\overline \\alpha & \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & \\overline \\alpha ^ 2 \\\\ 0 & - \\alpha J ( T ^ { - r } , T ^ { - r } ) J ( T ^ { - 2 r } , T ^ { - 3 r } ) & \\alpha ^ 2 J ( T ^ { - 3 r } , T ^ { - 3 r } ) - \\overline \\alpha ^ 2 J ( T ^ { - r } , T ^ { - r } ) \\\\ 0 & \\overline \\alpha ^ 2 J ( T ^ { - 2 r } , T ^ { - r } ) - \\alpha ^ 2 J ( T ^ { - 2 r } , T ^ { - 3 r } ) & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "2555.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\| u ( t ) - e ^ { i t \\Delta } u _ + \\| _ { \\dot H _ x ^ s } = 0 , \\end{align*}"}
-{"id": "6151.png", "formula": "\\begin{align*} y _ i y _ j = y _ k y _ \\ell , \\forall \\ ; i + j = k + \\ell . \\end{align*}"}
-{"id": "9023.png", "formula": "\\begin{align*} f _ p ^ { \\mathbf v } ( z ) = { \\tilde a } _ d ( p , 0 ) z ^ \\nu \\prod _ { m _ i = m _ 0 } ( z - u _ i ( p , 0 ) ) \\prod _ { m _ i < m _ 0 } u _ i ( p , 0 ) , \\end{align*}"}
-{"id": "3738.png", "formula": "\\begin{align*} \\rho ( u ) = \\sqrt { \\frac { \\beta ^ \\prime ( u ) } { 2 \\pi } } e ^ { \\gamma - u \\beta + C ( \\beta ) } \\left \\{ 1 + O \\left ( \\frac { 1 } { u } \\right ) \\right \\} , \\end{align*}"}
-{"id": "5229.png", "formula": "\\begin{align*} d h ( L ) ( z ) = \\alpha ( L ) ( z ) , L \\in \\mathcal { N } _ z , \\ ; z \\in M \\ ; . \\end{align*}"}
-{"id": "8025.png", "formula": "\\begin{align*} \\frac { m _ j } { d _ j } + \\frac { m _ { j k } } { d _ { j k } } = \\frac { m ' _ { j k } } { d ' _ { j k } } \\end{align*}"}
-{"id": "1448.png", "formula": "\\begin{align*} [ w ] _ { A _ \\infty } : = \\lim _ { r \\downarrow 0 } [ w ] _ { A _ r } . \\end{align*}"}
-{"id": "6571.png", "formula": "\\begin{align*} \\rho ( z , w , t ) = u + P ( z , w ) + Q ( z , w ) + v R ( z , w ) + u ^ 2 + v ^ 2 + o ( u ^ 2 , u v , v ^ 2 , u | ( z , w ) | ^ { 2 k } ) . \\end{align*}"}
-{"id": "7895.png", "formula": "\\begin{align*} \\left \\| \\ ; | ( C - B ) \\times ( B - I ) | - r b \\ ; \\right \\| = 0 . \\end{align*}"}
-{"id": "7916.png", "formula": "\\begin{align*} C _ g = \\int _ M S _ { g } ^ 2 d v _ { g } - \\frac { ( \\int _ M S _ { g } d v _ { g } ) ^ 2 } { \\int _ M d v _ { g } } . \\end{align*}"}
-{"id": "4961.png", "formula": "\\begin{align*} \\| \\Psi _ \\lambda & \\ast ( Q f ) \\| _ s \\leq c _ 1 \\| \\ , | \\cdot | ^ { 2 - N } \\| _ { \\frac { N } { N - 2 } , w } \\| Q f \\| _ { t _ 1 } + c _ 2 \\| \\ , | \\cdot | ^ { \\frac { 1 - N } { 2 } } \\| _ { \\frac { 2 N } { N - 1 } , w } \\| Q f \\| _ { t _ 2 } \\\\ & \\leq \\Bigl \\{ c _ 1 \\| \\ , | \\cdot | ^ { 2 - N } \\| _ { \\frac { N } { N - 2 } , w } | \\Omega | ^ { \\frac 1 { t _ 1 } } + c _ 2 \\| \\ , | \\cdot | ^ { \\frac { 1 - N } { 2 } } \\| _ { \\frac { 2 N } { N - 1 } , w } | \\Omega | ^ { \\frac 1 { t _ 2 } } \\Bigr \\} \\| Q \\| _ \\infty \\| f \\| _ \\infty = : C ( \\lambda ) \\| f \\| _ \\infty , \\end{align*}"}
-{"id": "3898.png", "formula": "\\begin{align*} \\Pi = \\left \\{ X _ R + c _ 1 \\big ( X _ R - X _ { \\rm i n t } \\big ) + c _ 2 v ~ \\middle | ~ c _ 1 , c _ 2 \\in \\mathbb { R } \\right \\} \\ ; . \\end{align*}"}
-{"id": "1329.png", "formula": "\\begin{align*} f ( \\lambda , \\mu ) = \\frac { \\lambda - \\mu + 1 } { \\lambda - \\mu } , g ( \\lambda , \\mu ) = \\frac { 1 } { \\lambda - \\mu } \\end{align*}"}
-{"id": "4420.png", "formula": "\\begin{align*} \\| f \\| _ { \\widehat { E } } & \\leq \\| f - g ( f \\chi _ A \\circ \\sigma _ 1 ) \\| _ { \\widehat { E } } + \\| g ( f \\chi _ A \\circ \\sigma _ 1 ) \\| _ { \\widehat { E } } \\leq \\epsilon + \\| f \\chi _ A \\circ \\sigma _ 1 \\| _ { \\widehat { E } } = \\epsilon + \\lim _ { n \\to \\infty } \\| \\tilde { f _ n } \\circ \\sigma _ 1 \\| _ { \\widehat { E } } \\\\ & = \\epsilon + \\lim _ { n \\to \\infty } \\| \\tilde { f _ n } \\| _ { \\widehat { E } } = \\epsilon + \\| f \\chi _ A \\| _ { \\widehat { E } } , \\end{align*}"}
-{"id": "2833.png", "formula": "\\begin{align*} M _ { k , t } = C _ { n , r } \\sum _ { \\substack { 0 \\le i \\le [ T _ k - S _ k ] \\\\ 0 \\le j \\le [ T _ t - S _ t ] } } \\sum _ { \\substack { 0 \\le u \\le L _ i ^ k \\\\ 0 \\le v \\le L _ j ^ t } } \\left ( \\hat y _ i ^ k \\hat y _ j ^ t \\right ) ^ { - ( n - r ) } \\left | \\tilde A _ { s _ { i , u } ^ k s _ { j , v } ^ t } ^ { ( r ) } \\right | \\exp \\left ( - \\frac { \\hat r \\left ( ( \\hat y _ i ^ k ) ^ 2 + ( \\hat y _ j ^ t ) ^ 2 \\right ) } { 2 ( 1 + | r ( s _ { i , u } ^ k - s _ { j , v } ^ t ) | ) } \\right ) , \\end{align*}"}
-{"id": "8841.png", "formula": "\\begin{align*} \\upsilon _ { k + 1 } \\leq \\frac { 1 } { k } \\Big ( 1 + \\frac { 4 } { n } \\Big ) \\sum _ { i = 1 } ^ k \\upsilon _ i , \\end{align*}"}
-{"id": "6027.png", "formula": "\\begin{gather*} \\alpha : = \\frac { 2 } { 3 } \\left ( - \\frac { \\varphi r \\ , d r } { ( r ^ 2 + 1 ) ^ 2 } + \\frac { \\psi s \\ , d s } { ( s ^ 2 - 1 ) ^ 2 } \\right ) . \\end{gather*}"}
-{"id": "3808.png", "formula": "\\begin{align*} \\psi ( z ) = \\sum _ { k \\in \\mathbb { Z } - \\frac { 1 } { 2 } } \\psi _ { - k } z ^ { k - \\frac { 1 } { 2 } } & \\psi ^ * ( z ) = \\sum _ { k \\in \\mathbb { Z } - \\frac { 1 } { 2 } } \\psi _ { k } ^ * z ^ { k + \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "131.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { X } } = \\sup _ { \\| x ^ * \\| \\leq 1 } | x ^ * ( x ) | = \\| x \\| ^ * . \\end{align*}"}
-{"id": "6001.png", "formula": "\\begin{align*} F ( 2 ^ { - j } \\sqrt { L } ) ( x , y ) = \\sum _ { k = 0 } ^ \\infty m ( 2 ^ { - j } \\sqrt { L } ) g _ { j , y , k } ( x ) . \\end{align*}"}
-{"id": "1022.png", "formula": "\\begin{align*} \\int _ 0 ^ T f ( t , u ( t ) , u ' ( t ) ) d t = 0 . \\end{align*}"}
-{"id": "5926.png", "formula": "\\begin{align*} \\| D _ v { \\psi } \\| _ { L ^ p ( \\R ^ { 2 d } ) } ^ p = \\int _ { \\R ^ d } \\| D _ v \\psi ( x , \\cdot ) \\| _ { L ^ p ( \\R ^ d ) } ^ p \\dd x \\le ( \\| { \\psi } \\| _ { L ^ p ( \\R ^ { 2 d } ) } ) ^ { p / 2 } \\ , ( \\| D _ v ^ 2 { \\psi } \\| _ { L ^ p ( \\R ^ { 2 d } ) } ) ^ { p / 2 } . \\end{align*}"}
-{"id": "2906.png", "formula": "\\begin{align*} M _ \\pi ( z ; X ) & = \\prod _ { T \\in { \\cal T } ^ \\pi } \\frac { 1 } { 1 - z \\ , X ^ T } = \\sum _ { r \\geq 0 } z ^ r \\ , s _ { ( r ) } [ s _ \\pi ] ( X ) \\ , ; \\\\ \\cr L _ \\pi ( z ; X ) & = \\prod _ { T \\in { \\cal T } ^ \\pi } ( 1 - z \\ , X ^ T ) = \\sum _ { r \\geq 0 } ( - 1 ) ^ r \\ , z ^ r \\ , s _ { ( 1 ^ r ) } [ s _ \\pi ] ( X ) \\ , , \\end{align*}"}
-{"id": "489.png", "formula": "\\begin{align*} \\pm \\int _ { ( B ^ c _ \\varepsilon ( x ) ) ^ \\pm } \\frac { \\xi \\cdot ( x - y ) } { \\ , | x - y | ^ { N + 2 s } \\ , } \\ , d y = + \\infty , \\end{align*}"}
-{"id": "1739.png", "formula": "\\begin{gather*} I ^ 2 = \\frac { ( k ^ 2 + 1 ) ^ 2 } { ( k ^ 2 - 9 ) ( \\tfrac { 1 } { 9 } - k ^ 2 ) } ; \\end{gather*}"}
-{"id": "3185.png", "formula": "\\begin{align*} \\widehat { w } _ j ^ \\lambda : = w _ j ^ \\lambda + \\sum _ { | \\alpha | = n + 1 } f _ { j , \\alpha } ^ \\lambda ( z _ j ) \\cdot w _ j ^ \\alpha . \\end{align*}"}
-{"id": "4372.png", "formula": "\\begin{align*} L _ n = \\frac { 1 } { 2 } : \\ ! J ^ 2 \\ ! : _ n \\ , \\equiv \\ , \\frac { 1 } { 2 } \\left ( \\sum _ { m = - \\infty } ^ { - 1 } J _ m J _ { n - m } + \\sum _ { m = 0 } ^ { \\infty } J _ { n - m } J _ m \\right ) \\end{align*}"}
-{"id": "1030.png", "formula": "\\begin{align*} P _ d ( x , y ) ^ d ~ = ~ c _ { d , d } ^ d ( x ^ { d ^ 2 } - y ^ { d ^ 2 } ) . \\end{align*}"}
-{"id": "1345.png", "formula": "\\begin{align*} \\frac { \\partial Z _ { t } } { \\partial \\psi } \\vert _ { \\delta _ 0 } = e _ { t - J _ L } ( \\delta _ 0 ) + \\sum _ { j \\in J _ \\phi \\bigcap J _ \\theta } \\omega _ { j } \\frac { \\partial Z _ { t - j } ( \\delta _ 0 ) } { \\partial \\psi } . \\end{align*}"}
-{"id": "9636.png", "formula": "\\begin{align*} P _ { j } \\ = \\ \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & I _ { d _ j ^ { ( 2 ) } } \\ , \\end{array} \\right ) \\end{align*}"}
-{"id": "5382.png", "formula": "\\begin{align*} P ( P ( x , x ) , z ) ~ = ~ 0 , \\end{align*}"}
-{"id": "9905.png", "formula": "\\begin{align*} q & = \\frac { 1 - i b } { 1 + i b } , & E & = \\frac { i } { 2 } ( 1 - i b ) ( x _ 2 + i x _ 3 ) , & K \\phantom { ' } & = x _ 0 + b x _ 1 , \\\\ \\kappa & = \\frac { 1 } { q ^ { - 1 } - q } , & F & = \\frac { i } { 2 } ( 1 + i b ) ( x _ 2 - i x _ 3 ) , & K ' & = x _ 0 - b x _ 1 . \\end{align*}"}
-{"id": "8128.png", "formula": "\\begin{align*} l _ * ( k , N ) = l ^ * ( k , N ) \\ \\mbox { w h e n } \\ k \\geq 0 . \\end{align*}"}
-{"id": "7768.png", "formula": "\\begin{align*} \\beta ( M ) = \\tfrac { 1 } { 3 } \\beta ( M ' ) + \\tfrac { 1 } { 3 } \\beta ( M '' ) \\end{align*}"}
-{"id": "8876.png", "formula": "\\begin{align*} R _ \\# B [ v , w ] = B [ R ( v ) , R ( w ) ] , \\end{align*}"}
-{"id": "4345.png", "formula": "\\begin{align*} \\psi ( y ) = \\psi ( y , a ) : = \\frac { 1 } { a ^ p } | f ' ( F ( a y ) ) | ^ p \\ , y \\in [ 0 , 1 ) \\ . \\end{align*}"}
-{"id": "8840.png", "formula": "\\begin{align*} { \\rm o r d } _ { 1 / 2 } \\left ( F _ { k , 2 } \\right ) & = \\begin{cases} \\frac { 1 } { 2 } & \\mbox { i f $ k $ i s o d d , } \\\\ 1 & \\mbox { i f $ k $ i s e v e n , } \\end{cases} \\\\ { \\rm o r d } _ { 1 / 2 } \\left ( F _ { k , 4 } \\right ) & = \\begin{cases} 2 & \\mbox { i f $ k $ i s o d d , } \\\\ 1 & \\mbox { i f $ k $ i s e v e n . } \\end{cases} \\end{align*}"}
-{"id": "1523.png", "formula": "\\begin{align*} Q ( F ) - E \\cdot \\nabla _ v F = 0 , \\int _ { \\R ^ d } F \\ , d v = 1 , \\end{align*}"}
-{"id": "2748.png", "formula": "\\begin{align*} \\mu = \\rho \\ , d z d \\bar z d \\theta d \\bar \\theta \\end{align*}"}
-{"id": "5377.png", "formula": "\\begin{align*} c _ { d , d } { \\ , } c _ { q , q _ 1 } ^ d ~ = ~ ( 1 + \\delta _ { q _ 1 , q _ 2 } ) { \\ , } c _ { q , q } { \\ , } c _ { d , d } ^ q \\end{align*}"}
-{"id": "8457.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\varepsilon } + J _ { \\varepsilon } \\textbf { P } T _ { i A } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } = 0 . \\end{align*}"}
-{"id": "3830.png", "formula": "\\begin{align*} | L _ 3 ( q ) | = \\dfrac { 1 } { ( 3 , q - 1 ) } q ^ 3 ( q ^ 2 - 1 ) ( q ^ 3 - 1 ) . \\end{align*}"}
-{"id": "9107.png", "formula": "\\begin{align*} \\nu ( \\sup \\{ a _ 1 , a _ 2 \\} ) = \\nu ( a _ 1 + a _ 2 ) = \\nu ( a _ 1 ) + \\nu ( a _ 2 ) = \\sup \\{ \\nu ( a _ 1 ) , \\nu ( a _ 2 ) \\} . \\end{align*}"}
-{"id": "8964.png", "formula": "\\begin{align*} W ( t ) = E _ u ( t , t _ 0 ) W ( t _ 0 ) \\end{align*}"}
-{"id": "9985.png", "formula": "\\begin{align*} \\lambda ^ { ( n ) } + h ^ { ( n ) } ( s ) = g ( s , a ^ { ( n ) } ( s ) ) + \\sum _ { s ' \\in \\mathcal { S } } p _ { s \\rightarrow s ' | a ^ { ( n ) } ( s ) } h ^ { ( n ) } ( s ' ) \\end{align*}"}
-{"id": "5762.png", "formula": "\\begin{align*} R _ { n } ( u ) - R _ { n } ^ { \\dag } ( u ) = \\frac { 1 } { 2 } u ^ { T } \\left ( \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } ( y _ { t } - m _ { t } \\pi _ { 0 , t } ) \\ddot { W } _ { 0 , t } \\right ) u + E _ { n } ( u ^ { \\ast } ) - E _ { n } ^ { \\dag } ( u ^ { \\ast } ) \\end{align*}"}
-{"id": "2494.png", "formula": "\\begin{gather*} A _ { k j } ^ i : = \\{ \\sigma _ i < + \\infty \\} \\cap \\{ z _ i ( u _ k , \\sigma _ i ) - z _ i ( u _ { k - 1 } , \\sigma _ i ) = \\varepsilon , \\ldots , z _ i ( u _ { j + 1 } , \\sigma _ i ) - z _ i ( u _ j , \\sigma _ i ) = \\varepsilon , \\\\ z _ i ( u _ j , \\sigma _ i ) - z _ i ( u _ { j - 1 } , \\sigma _ i ) > \\varepsilon \\} , 2 \\leqslant j \\leqslant k - 1 , 3 \\leqslant k \\leqslant n , 1 \\leqslant i \\leqslant n - 1 , \\end{gather*}"}
-{"id": "2245.png", "formula": "\\begin{align*} - \\log p ( x ) = \\frac { 1 } { 2 } \\log \\frac { 2 \\pi \\sigma ^ { 2 } } { C ^ { 2 } ( \\varepsilon ) } + \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } + \\varepsilon x ^ { p } . \\end{align*}"}
-{"id": "2744.png", "formula": "\\begin{align*} \\theta ^ I = a ^ I _ K \\eta ^ K \\mbox { a n d } \\bar \\theta ^ J = b _ L ^ J \\bar \\eta ^ L . \\end{align*}"}
-{"id": "8363.png", "formula": "\\begin{align*} z _ e & = 1 , \\\\ z _ g & = \\alpha _ g ( z _ { g ^ { - 1 } } ) , \\ \\ \\ g \\in G , \\\\ z _ { g h } & \\geq z _ g \\alpha _ g ( z _ h ) , \\ \\ \\ g , h \\in G . \\end{align*}"}
-{"id": "1526.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { r c l l } \\partial _ t \\rho + \\kappa ( - \\Delta ) ^ { \\alpha / 2 } \\rho + \\nabla _ x \\cdot ( D E \\rho ) & = & 0 & ( 0 , \\infty ) \\times \\R ^ d , \\\\ \\rho ( \\cdot , 0 ) & = & \\rho ^ { i n } & \\R ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "7643.png", "formula": "\\begin{align*} \\ \\left \\{ \\begin{aligned} & u _ { t } = - ( - \\triangle ) ^ { \\alpha } u + h ( t ) u ^ { v } , \\ ( x , t ) \\in \\mathbb { R } ^ { d } \\times ( 0 , T ) , \\\\ & u ( 0 ) = u _ { 0 } , \\ u _ { 0 } \\geq 0 , \\ u _ { 0 } \\not \\equiv 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1013.png", "formula": "\\begin{align*} | ( \\psi \\circ \\Psi ) ( x - x _ i ) | & = | ( \\psi \\circ \\Theta \\circ \\pi ) ( x - x _ i ) | \\\\ & = | ( \\psi \\circ \\Theta ) ( e _ { M _ 1 } ^ \\perp ( x - x _ i ) ) | \\\\ & \\leq \\| e _ { M _ 1 } ^ \\perp \\| _ { \\psi \\circ \\Theta } \\ , \\| x - x _ i \\| _ { \\psi \\circ \\Theta } \\to 0 i \\to \\infty . \\end{align*}"}
-{"id": "4007.png", "formula": "\\begin{align*} \\sum _ { i = a } ^ t n _ i ( q ^ i - 1 ) = c _ 0 q ^ { d } - 1 , \\end{align*}"}
-{"id": "9404.png", "formula": "\\begin{align*} A u = f , \\end{align*}"}
-{"id": "7513.png", "formula": "\\begin{align*} \\gamma = \\max \\{ b , 1 - a \\} . \\end{align*}"}
-{"id": "1369.png", "formula": "\\begin{align*} W _ { t } = - 0 . 5 + ( t / n ) + Z _ { t } \\end{align*}"}
-{"id": "9395.png", "formula": "\\begin{align*} \\widetilde { W } _ \\lambda = 2 i k - ( q , u _ \\lambda ) = 2 i k + ( q , G \\ast { q } ) , \\lambda = k ^ 2 , k \\in \\mathbb { C } _ + . \\end{align*}"}
-{"id": "7475.png", "formula": "\\begin{align*} P ( \\Delta _ { Q _ 0 ^ { ( k ) } } ( \\infty ) = 1 ) > 0 , P ( \\Delta _ { Q _ k ^ { ( k ) } } ( \\infty ) = - k ) > 0 , P ( \\Delta _ { Q _ { k + 1 } ^ { ( k ) } } ( \\infty ) = k ) > 0 , \\end{align*}"}
-{"id": "829.png", "formula": "\\begin{align*} F _ { \\vec { z } } ^ { \\vec { \\mu } } ( x _ { 1 } + c , \\ldots , x _ { k } + c ) = \\prod _ { i = 1 } ^ { k } \\left ( \\frac { z _ { i } } { 1 + z _ { i } } \\right ) ^ { c } F _ { \\vec { z } } ^ { \\vec { \\mu } } ( x _ { 1 } , \\ldots , x _ { k } ) ( \\vec { x } \\in L _ { + } , c \\in \\mathbb { Z } ) . \\end{align*}"}
-{"id": "8724.png", "formula": "\\begin{align*} \\widetilde Y _ { \\tau } ^ { t , x } = \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } G B ( s , X ^ { t , x } _ s ) \\ , d s - \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } \\widetilde Z ^ { t , x } _ { s } d W _ s , \\tau \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6584.png", "formula": "\\begin{align*} p _ { \\gamma f } ^ { \\alpha } ( x ) : = \\alpha r _ { \\gamma f } ^ * ( x ) + \\tfrac { 1 - \\alpha } { 2 } \\| x \\| ^ 2 , \\end{align*}"}
-{"id": "3715.png", "formula": "\\begin{align*} \\chi ( ( x a , y ) ) = \\psi ( \\langle x a , y \\rangle ) = \\psi ( \\langle x , y a ^ { * } \\rangle ) = \\chi ( ( x , y a ^ { * } ) ) . \\end{align*}"}
-{"id": "3241.png", "formula": "\\begin{align*} f _ { e _ { j } } \\left ( q \\right ) = g _ { a _ { i } } \\left ( q \\right ) = g _ { b _ { i } } \\left ( q \\right ) \\end{align*}"}
-{"id": "9219.png", "formula": "\\begin{align*} \\hat { H } ( t , x ) = H ( t , x , \\widehat { Y } ( t , x , z ) , \\widehat { Y } ( t , \\cdot , z ) ( x ) , \\widehat { u } ( t , x , z ) , \\widehat { p } ( t , x , z ) , \\widehat { q } ( t , x , z ) , \\widehat { r } ( t , x , z , . ) ) \\end{align*}"}
-{"id": "5029.png", "formula": "\\begin{align*} \\eta ( U _ { x _ o } ) = \\eta ( \\sup _ n T f _ n ) \\geq \\eta ( T f _ m ) = \\nu ( f _ m ) , \\textrm { f o r e v e r y $ m $ } . \\end{align*}"}
-{"id": "7363.png", "formula": "\\begin{align*} X \\triangleright \\gamma _ + ( v ) = \\gamma _ + ( X \\triangleright v ) , X \\triangleright \\gamma _ - ( w ) = \\gamma _ - ( X \\triangleright w ) . \\end{align*}"}
-{"id": "874.png", "formula": "\\begin{align*} \\operatorname * { l e v } \\nolimits _ { \\varphi _ { A , k } , = } ( t ) = t k + \\operatorname * { b d } A \\forall \\ , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "9513.png", "formula": "\\begin{align*} B _ 2 ' & = B _ 2 + d \\zeta _ 1 \\\\ C _ 1 ' & = C _ 1 + d \\Lambda _ 0 \\\\ C _ 3 ' & = C _ 3 + d \\Lambda _ 2 + H _ 3 \\Lambda _ 0 \\end{align*}"}
-{"id": "4787.png", "formula": "\\begin{align*} \\mathcal { M } '' : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J \\end{align*}"}
-{"id": "3748.png", "formula": "\\begin{align*} u = E T _ \\beta < K _ 2 \\frac { e ^ { \\beta } } { \\beta } \\Rightarrow u < \\frac { K K _ 2 u } { \\beta } \\Rightarrow 1 < \\frac { K K _ 2 } { \\beta } \\end{align*}"}
-{"id": "6177.png", "formula": "\\begin{align*} U _ t = & \\gamma _ U t + a _ { 1 1 } W _ t + a _ { 1 2 } \\widetilde W _ t , \\\\ L _ t = & \\gamma _ L t + a _ { 2 1 } W _ t + a _ { 2 2 } \\widetilde W _ t . \\end{align*}"}
-{"id": "5698.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\Phi _ { p _ 0 , q _ 0 , v _ 0 , w _ 0 } ( 2 ^ l Q ) = \\infty , ( Q \\in \\mathcal { Q } ) , \\Phi _ { p _ 0 , q _ 0 , v _ 0 , w _ 0 } ( Q ) : = v _ 0 ( Q ) ^ { \\frac { 1 } { p _ 0 } - \\frac { 1 } { q _ 0 } } w _ 0 ( Q ) ^ \\frac { 1 } { q _ 0 } . \\end{align*}"}
-{"id": "4464.png", "formula": "\\begin{align*} \\tilde { \\psi } _ f : = - \\log f - H ( f ) . \\end{align*}"}
-{"id": "1032.png", "formula": "\\begin{align*} P _ d ( x , y ) ~ = ~ c _ { d , d } { \\ , } x ^ d + \\sum _ { j = 0 } ^ r c _ { d , j } { \\ , } x ^ j y ^ { d - j } \\end{align*}"}
-{"id": "7859.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } p ^ { \\kappa } ( t , x , y ) \\ , d y = 1 \\ , . \\end{align*}"}
-{"id": "8932.png", "formula": "\\begin{align*} \\sum _ { \\ell \\ , : \\ , \\abs { \\lambda _ \\ell ( M ) - \\lambda _ j ( L ) } \\le \\varepsilon } \\bigl ( \\lambda _ \\ell ( M ) - \\lambda _ j ( L ) \\bigr ) = \\sum _ { \\ell \\ , : \\ , \\abs { \\lambda _ \\ell ( { L } ) - \\lambda _ j ( L ) } \\le \\varepsilon } e _ \\ell \\cdot ( M - L ) ( e _ \\ell ) + o \\bigl ( \\norm { M - L } \\bigr ) , \\end{align*}"}
-{"id": "8874.png", "formula": "\\begin{align*} \\mathcal { E } ( d A ) = \\bigl ( \\tfrac { 1 } { 2 } - \\tfrac { 1 } { p } \\bigr ) \\inf _ { v \\in H ^ 1 _ A ( \\R ^ N , \\mathbb { C } ) \\setminus \\{ 0 \\} } \\mathcal { Q } _ A ( v ) ^ \\frac { p } { p - 2 } , \\end{align*}"}
-{"id": "2752.png", "formula": "\\begin{align*} \\nu ^ { - n } L _ f = A _ 0 + \\nu A _ 1 + \\ldots \\end{align*}"}
-{"id": "2154.png", "formula": "\\begin{align*} \\eta ( x , t ) = \\int _ { t } ^ { T } h _ { m } ( \\sigma - t ) \\varphi ( \\sigma , x ) d \\sigma = \\int _ { 0 } ^ { T - t } h _ { m } ( \\sigma ) \\varphi ( \\sigma + t , x ) d \\sigma , \\end{align*}"}
-{"id": "6286.png", "formula": "\\begin{align*} \\frac { 1 } { N _ p ( x ) } = \\sum ^ n _ { j = 1 } \\frac { 1 } { x - x _ j } . \\end{align*}"}
-{"id": "8015.png", "formula": "\\begin{align*} f = \\sum g _ i h _ i , \\end{align*}"}
-{"id": "7512.png", "formula": "\\begin{align*} \\left | x _ { n + 1 } - K \\right | = \\left | F _ { \\alpha _ n } ( x _ { n } ) - K \\right | \\leq \\gamma \\left | x _ { n } - K \\right | , \\end{align*}"}
-{"id": "1853.png", "formula": "\\begin{align*} & \\exp \\left ( - t ( i x + x ^ 2 / 2 ) - 2 \\pi i k n e ^ x \\right ) \\\\ & = - \\frac { 1 } { t ( i + x ) + 2 \\pi i k n e ^ x } \\frac { d } { d x } \\exp \\left ( - t ( i x + x ^ 2 / 2 ) - 2 \\pi i k n e ^ x \\right ) , \\end{align*}"}
-{"id": "5714.png", "formula": "\\begin{align*} [ w ] _ { A _ \\infty } : = \\lim _ { r \\uparrow \\infty } [ w ] _ { A _ r } . \\end{align*}"}
-{"id": "4788.png", "formula": "\\begin{align*} H = \\frac { \\kappa } { 2 f } \\ , n _ 1 + \\frac { f f '' + ( f ' ) ^ 2 - 1 } { 2 f \\sqrt { 1 - f '^ 2 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "4753.png", "formula": "\\begin{align*} P _ { E D } ^ { D i s p a t c h } : = & \\sum _ { t \\in \\mathcal { T } } \\left [ \\sum _ { i \\in \\mathcal { G } } p _ { i t } + \\sum _ { i \\in \\mathcal { I } } ( M _ { i t } - m _ { i t } ) \\right . \\\\ & \\left . + \\sum _ { i \\in \\mathcal { W } } ( W _ { i t } - w _ { i t } ) + \\sum _ { i \\in \\mathcal { R } } ( R _ { i t } - r _ { i t } ) \\right ] . \\end{align*}"}
-{"id": "2232.png", "formula": "\\begin{align*} \\bigcap \\limits _ { i = 1 } ^ { r } B _ { i } \\not \\subseteq \\bigcup \\limits _ { i = 1 } ^ { s } A _ { j } . \\end{align*}"}
-{"id": "1995.png", "formula": "\\begin{align*} \\mathcal { E } ( U ) _ { ( s , t ] } & = e ^ { U _ t - U _ s - \\sigma _ U ^ 2 ( t - s ) / 2 } \\prod _ { s < u \\leq t } ( 1 + \\Delta U _ u ) e ^ { - \\Delta U _ u } , \\\\ \\mathcal { E } ( U ) _ { ( s , t ) } & = e ^ { U _ { t - } - U _ s - \\sigma _ U ^ 2 ( t - s ) / 2 } \\prod _ { s < u < t } ( 1 + \\Delta U _ u ) e ^ { - \\Delta U _ u } . \\end{align*}"}
-{"id": "5358.png", "formula": "\\begin{align*} \\alpha _ i ( t ) = \\prod _ { \\substack { 0 \\leq k \\leq n \\\\ p ( n - k ) - p ( n - k - 1 ) \\geq i } } ( t + k ) \\end{align*}"}
-{"id": "3868.png", "formula": "\\begin{align*} ( \\mathbf n _ V - ( \\mathbf n _ V \\cdot \\nu _ { \\partial \\Omega } ) \\nu _ { \\partial \\Omega } ) \\| \\delta V \\| = - \\sigma \\mathbf n _ { B ^ + } \\mathcal H ^ { n - 1 } \\lfloor _ { \\partial ^ * B ^ + } \\end{align*}"}
-{"id": "1438.png", "formula": "\\begin{align*} W _ { 2 i } ( z ) = \\sum _ { 1 \\leq j _ { 1 } < \\cdots < j _ { n - i } \\leq n } : b ^ { 2 } _ { j _ { 1 } } ( z ) \\cdots b ^ { 2 } _ { j _ { n - i } } ( z ) : + \\cdots \\end{align*}"}
-{"id": "4142.png", "formula": "\\begin{align*} S _ f ( X ) = & \\sum _ { n \\leq X } a ( n ) = Q ( X ) + O ( X ^ { \\frac { \\delta } { 2 } - \\frac { 1 } { 4 A } + 2 A ( w - \\frac { \\delta } { 2 } - \\frac { 1 } { 4 A } ) \\eta + \\epsilon } ) \\\\ & + O ( X ^ { q - \\frac { 1 } { 2 A } - \\eta } \\log ( X ) ^ { r - 1 } ) + O \\bigg ( \\sum _ { X \\leq n \\leq X ' } | a ( n ) | \\bigg ) \\end{align*}"}
-{"id": "3270.png", "formula": "\\begin{align*} T _ { i j } = \\gamma _ { - } ( w _ i ) \\gamma _ { - } ( w _ j ) ^ * + \\sum _ { k , l } b _ { k , l } ^ { i , j } \\gamma _ { - } ( w _ k ) ^ * \\gamma _ { - } ( w _ l ) , b _ { k , l } ^ { i , j } \\in \\mathbb { C } . \\end{align*}"}
-{"id": "6976.png", "formula": "\\begin{align*} \\lambda _ t ( x , \\lambda _ t ( y , z ) ) = \\lambda _ t ( \\lambda _ t ( x , y ) , z ) + \\lambda _ t ( y , \\lambda _ t ( x , z ) ) . \\end{align*}"}
-{"id": "8105.png", "formula": "\\begin{align*} { \\mathcal R } _ { k , l , N } ( M ) : = \\frac { P _ { \\mu _ k } ( M ) } { \\left ( \\mu _ l ( M ) \\right ) ^ { ( k + N - 1 ) / ( l + N ) } } . \\end{align*}"}
-{"id": "3642.png", "formula": "\\begin{align*} \\left | g ( { \\bf x } ) - K _ 1 \\right | & = ( 1 - c _ { K _ 1 } ) \\left | f ( { \\bf x } ) - f ( \\mathbf { K _ 1 } ) \\right | \\\\ & \\leq ( 1 - c _ { K _ 1 } ) L _ 2 \\| { \\bf x } - \\mathbf { K _ 1 } \\| \\mbox { f o r } \\| { \\bf x } - \\mathbf { K _ 1 } \\| \\leq \\max \\{ K , K _ 1 \\} , ~ { \\bf x } \\in \\R ^ k _ + . \\end{align*}"}
-{"id": "6125.png", "formula": "\\begin{align*} L _ \\gamma ( s ) = 2 \\pi ^ { 2 } s ( 1 + s e ( s ) ) \\end{align*}"}
-{"id": "7244.png", "formula": "\\begin{align*} \\widehat { T } _ { j k } \\widehat { w } _ k = \\widehat { w } _ j + \\sum _ { | \\alpha | \\geq n + 1 } M _ j \\cdot f _ { k j , \\alpha } \\cdot \\prod _ { \\lambda = 1 } ^ r \\left ( \\sum _ { \\mu = 1 } ^ r ( M _ j ^ { - 1 } ) ^ \\lambda _ \\mu \\cdot \\widehat { w } _ j ^ \\mu \\right ) ^ { \\alpha _ \\nu } = \\widehat { w } _ j + O ( | \\widehat { w } _ j | ^ { n + 1 } ) , \\end{align*}"}
-{"id": "9587.png", "formula": "\\begin{align*} H _ y \\psi : = ( \\Delta - m ^ 2 ) \\psi _ { r e g } , \\quad \\psi \\in D _ y . \\end{align*}"}
-{"id": "7835.png", "formula": "\\begin{align*} V = S ^ { \\bot } \\cap T \\oplus S ^ { \\bot } \\cap T ^ { \\bot } \\oplus S \\cap T \\oplus S \\cap T ^ { \\bot } . \\end{align*}"}
-{"id": "8472.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\varepsilon } = F ^ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ \\tilde { \\textbf { u } } ^ { \\varepsilon } | _ { t = 0 } = \\tilde { \\textbf { u } } ^ { \\varepsilon } _ { 0 } ( x ) , \\end{align*}"}
-{"id": "4805.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { \\left ( b \\pm \\frac { t } { 2 } \\sqrt { a ^ 2 - 4 c t ^ 2 } \\pm \\frac { a ^ 2 } { 4 \\sqrt { - c } } \\arcsin \\frac { 2 \\sqrt { - c } \\ , t } { a } \\right ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "8017.png", "formula": "\\begin{align*} y _ 1 ^ { h _ 1 d _ 1 } \\ldots y _ n ^ { h _ n d _ n } = _ { T ^ \\times } x _ 1 ^ { g _ 1 d _ 1 } \\ldots x _ n ^ { g _ n d _ n } . \\end{align*}"}
-{"id": "3480.png", "formula": "\\begin{align*} \\Phi ( f ) = \\int _ \\Omega f u _ f d x = \\int _ \\Omega | \\nabla u _ f | ^ 2 d x = \\sup _ { u \\in W _ 0 ^ { 1 , 2 } ( \\Omega ) } \\int _ \\Omega 2 f u - | \\nabla u | ^ 2 d x . \\end{align*}"}
-{"id": "9124.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } u ' ( t ) + B ( 0 ) A ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) + ( B ( 0 ) - B ( t ) ) A ( t ) v ( t ) \\\\ u ( 0 ) = u _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "2164.png", "formula": "\\begin{align*} \\int _ { B _ { 1 } } g _ { \\alpha } * \\left ( \\phi \\psi ^ { 1 + q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ^ { 1 - q } ) \\right ) d x & + ( 1 - q ) g _ { \\alpha } * \\left [ \\mathcal { E } ( h _ { m } * \\tilde { u } , \\psi ^ { 1 + q } \\tilde { u } ^ { - q } ) \\phi \\right ] \\\\ & \\leq q g _ { \\alpha } * \\int _ { B _ { 1 } } \\psi ^ { 1 + q } \\tilde { u } ^ { 1 - q } g _ { 1 - \\alpha , m } \\phi d x , \\end{align*}"}
-{"id": "577.png", "formula": "\\begin{align*} \\delta ( p a ) & \\stackrel { ( \\ref { p r o d } ) } { = } \\delta ( p ) \\phi ( a ) + p ^ p \\delta ( a ) \\stackrel { } { \\equiv } \\delta ( p ) \\phi ( a ) \\bmod I ^ { p - 1 } \\stackrel { ( \\ref { 1 } ) } { \\equiv } \\phi ( a ) \\bmod I ^ { p - 1 } \\end{align*}"}
-{"id": "8884.png", "formula": "\\begin{align*} \\int _ { \\R ^ N } \\abs { u _ n } ^ { p } & \\le C \\Bigl ( \\sup _ { y \\in \\R ^ N } \\int _ { B _ 1 ( y ) } \\abs { u _ n } ^ { p } \\Bigr ) ^ { 1 - \\frac { 2 } { p } } \\int _ { \\R ^ N } \\abs { D \\abs { u _ n } } ^ 2 + \\abs { u _ n } ^ 2 \\\\ & \\le C \\Bigl ( \\sup _ { y \\in \\R ^ N } \\int _ { B _ 1 ( y ) } \\abs { u _ n } ^ { p } \\Bigr ) ^ { 1 - \\frac { 2 } { p } } \\int _ { \\R ^ N } \\abs { D _ { A _ n } u _ n } ^ 2 + \\abs { u _ n } ^ 2 . \\end{align*}"}
-{"id": "639.png", "formula": "\\begin{align*} \\beta _ n = \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\sigma _ k , \\textrm { f o r $ n \\geq 1 $ } , \\end{align*}"}
-{"id": "6336.png", "formula": "\\begin{align*} 2 ^ { j n ( \\alpha _ 1 - \\alpha _ 2 ) ( 1 / q - 1 / 2 ) } = 2 ^ { j \\widetilde { A _ 2 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j \\widetilde { A _ 3 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\end{align*}"}
-{"id": "2051.png", "formula": "\\begin{align*} \\nabla \\omega ( y _ 0 ) = \\nabla \\omega ( x _ 0 ) = - \\frac { m } { 2 } \\bar { w } ' ( d _ 0 / 2 ) e _ n . \\end{align*}"}
-{"id": "507.png", "formula": "\\begin{gather*} [ \\lambda ] = \\big \\{ ( i , j ) \\in \\mathbb { Z } ^ 2 \\colon 1 \\leq i \\leq \\ell ( \\lambda ) , \\ 1 \\leq j \\leq \\lambda _ i \\big \\} . \\end{gather*}"}
-{"id": "4427.png", "formula": "\\begin{align*} ( \\widehat { C _ E } , \\| \\cdot \\| _ { \\widehat { C _ E } } ) = \\overline { ( C _ E , \\| \\cdot \\| _ { C _ { \\widehat { E } } } ) } ^ { \\| \\cdot \\| _ { C _ { \\widehat { E } } } } \\subseteq ( C _ { \\widehat { E } } , \\| \\cdot \\| _ { C _ { \\widehat { E } } } ) \\end{align*}"}
-{"id": "8502.png", "formula": "\\begin{align*} w _ 0 = ( s _ 1 s _ 2 s _ 3 \\cdots s _ n ) ( s _ 1 \\cdots s _ { n - 1 } ) \\cdots ( s _ 1 s _ 2 ) s _ 1 . \\end{align*}"}
-{"id": "8067.png", "formula": "\\begin{gather*} N ( x , 0 ) = \\begin{cases} ( T ( x ) , 0 ) & x \\in X _ r , \\\\ ( T ( x ) , \\infty ) & x \\in X _ f , \\end{cases} \\\\ N ( x , \\infty ) = \\begin{cases} ( T ( x ) , \\infty ) & x \\in X _ r , \\\\ ( T ( x ) , 0 ) & x \\in X _ r . \\end{cases} \\end{gather*}"}
-{"id": "3771.png", "formula": "\\begin{align*} 0 \\leq a _ 0 < a _ 1 < \\ldots < a _ L = 1 \\end{align*}"}
-{"id": "2343.png", "formula": "\\begin{align*} \\frac { a _ n } { 2 } = \\frac { a _ 1 } { 2 } + \\frac { a _ 2 - a _ 1 } { 2 } + \\dotsb + \\frac { a _ n - a _ { n - 1 } } { 2 } , \\end{align*}"}
-{"id": "7907.png", "formula": "\\begin{align*} a _ 0 & = \\int _ M d v _ g = \\mathrm { V o l } ( M ) , \\\\ a _ 1 & = \\frac { 1 } { 6 } \\int _ M S _ g d v _ g , \\\\ a _ 2 & = \\frac { 1 } { 1 8 0 } \\int _ M \\bigg ( | W | ^ 2 + | B | ^ 2 + \\frac { 2 9 } { 1 2 } S _ g ^ 2 \\bigg ) d v _ g , \\end{align*}"}
-{"id": "852.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m } C _ { a } ( z _ { i } ) \\cdot \\beta _ { a , 1 } ^ { * } \\equiv \\beta _ { a , 1 } ^ { * } \\prod _ { i = 1 } ^ { m } C _ { a } ( z _ { i } ) + ( 1 - q ^ 2 ) \\sum _ { \\ell = 1 } ^ { m } z _ { \\ell } \\prod _ { i = 1 } ^ { \\ell - 1 } C _ { a } ( z _ { i } ) \\cdot \\tilde { A } _ { a } ( z _ { \\ell } ) \\prod _ { i = \\ell + 1 } ^ { m } C _ { a } ( z _ { i } ) . \\end{align*}"}
-{"id": "9280.png", "formula": "\\begin{align*} & M ( 0 , z ) \\exp ( \\int _ 0 ^ t \\Phi _ K ( s , z ) d B ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ t \\Phi _ K ^ 2 ( s , z ) d s ) \\\\ & = \\tilde { p } ( 0 , z ) \\exp ( \\int _ 0 ^ t ( b _ 0 ( s , z ) \\pi ( s , z ) - \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) d B ( s ) \\\\ & - \\frac { 1 } { 2 } \\int _ 0 ^ t ( b _ 0 ( s , z ) \\pi ( s , z ) - \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) ^ { 2 } d s ) ; 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "8758.png", "formula": "\\begin{align*} & M ( w ) = M ^ { ( 1 ) } ( w ) K _ 0 ( w ) M ^ { ( 2 ) } ( w ) . \\end{align*}"}
-{"id": "2867.png", "formula": "\\begin{align*} F ^ c ( \\overline { N ^ * } K ^ { \\bullet } ( C ) ) = \\overline { N ^ * } R e s ^ { \\bullet } ( C ) \\end{align*}"}
-{"id": "3939.png", "formula": "\\begin{align*} r a n k ( H H ^ \\dagger ) & = r a n k ( \\overline { K } \\ , \\overline { K } ^ \\dagger ) \\\\ & = r a n k ( K { K } ^ \\dagger ) \\\\ & = n - k - m \\\\ & = n - k - \\dim ( H u l l _ h ( C ) ) \\\\ & = n - k - \\dim ( H u l l _ h ( C ^ { \\bot h } ) ) \\end{align*}"}
-{"id": "113.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } e ^ { x t } d \\mu _ { - 1 } ( x ) = \\frac { 2 } { e ^ t + 1 } \\sum _ { n = 0 } ^ \\infty E _ n \\frac { t ^ n } { n ! } \\end{align*}"}
-{"id": "1184.png", "formula": "\\begin{align*} D _ t V = \\begin{bmatrix} v _ t \\\\ v _ { t x } - \\sum _ { i = 1 } ^ N [ v _ x ] ( x _ i ) \\dot x _ i \\delta _ { x _ i } \\end{bmatrix} = \\begin{bmatrix} - \\frac 1 2 D _ x ( b ) + \\beta & b \\\\ - \\frac 1 2 D ^ 2 _ x ( b ) - z \\rho b & \\frac 1 2 D _ x ( b ) + \\beta \\end{bmatrix} V , \\end{align*}"}
-{"id": "6311.png", "formula": "\\begin{align*} & \\| \\{ a _ k \\| \\Box _ k ^ { \\alpha _ 1 } T _ m \\| _ { M _ 1 \\rightarrow M _ 2 } \\| \\Box _ k ^ { \\alpha _ 1 , \\ast } f _ k \\| _ { M _ 1 } \\} | ~ l _ { q _ 2 } ^ { s _ 2 , \\alpha _ 1 } ( E _ m ) \\| \\\\ \\lesssim & \\| \\{ a _ k \\| \\Box _ k ^ { \\alpha _ 1 } T _ m \\Box _ k ^ { \\alpha _ 1 , \\ast } f _ k \\| _ { M _ 2 } \\} | ~ l _ { q _ 2 } ^ { s _ 2 , \\alpha _ 1 } ( E _ m ) \\| . \\end{align*}"}
-{"id": "8944.png", "formula": "\\begin{align*} f _ R ( s ) = \\begin{cases} s _ 0 ^ { - 1 / p } & \\mbox { i f $ s \\in ( 0 , s _ 0 ) $ } , \\\\ s ^ { - 1 / p } & \\mbox { i f $ s \\in [ s _ 0 , R ) $ } , \\\\ R ^ { - 1 / p } \\max \\{ 2 - s / R , 0 \\} & \\mbox { i f $ s \\geqslant R $ } . \\end{cases} \\end{align*}"}
-{"id": "5845.png", "formula": "\\begin{align*} X = \\left ( 2 \\psi _ { I } \\int T ^ { I } \\left ( t \\right ) d t \\right ) + T ^ { I } \\left ( t \\right ) Y _ { I } \\partial _ { x } + \\left ( a \\left ( x , t \\right ) F \\right ) \\partial _ { F } \\end{align*}"}
-{"id": "7579.png", "formula": "\\begin{align*} \\Omega _ k : = \\left \\{ \\omega \\in \\Omega : \\chi _ k ( \\omega ) \\geq 1 - \\frac { l + A } { 2 l } \\right \\} , k = 1 , 2 , \\dots , K _ 1 , \\mathcal A : = \\bigcap _ { k = 1 } ^ { K _ 1 } \\Omega _ k . \\end{align*}"}
-{"id": "9428.png", "formula": "\\begin{align*} \\abs { f ' ( a ) } + \\abs { f ' ( b ) } + \\abs { f ' ( c ) } = \\abs { f ' ( d ) } . \\end{align*}"}
-{"id": "3083.png", "formula": "\\begin{align*} S u _ i = \\mu _ i u _ i , \\end{align*}"}
-{"id": "8557.png", "formula": "\\begin{align*} \\rho ( x ) n & = s ( x _ { 1 } ) x _ { 1 } \\hdots s ( x _ { l - 2 } ) x _ { l - 2 } n ' \\\\ & = s ( x _ { 1 } ) x _ { 1 } \\hdots s ( x _ { l - 2 } ) x _ { l - 2 } s ( x _ { l - 1 } ) x _ { l - 1 } s ( x _ { l } ) x _ { l } n \\\\ & = x n \\end{align*}"}
-{"id": "5504.png", "formula": "\\begin{align*} \\begin{aligned} \\mathcal { R } _ k [ \\iota ] & \\leq \\mathcal { R } _ k ^ { } [ \\iota ] \\\\ \\mathcal { R } _ k [ \\iota ] & \\leq \\mathcal { R } _ k ^ { } [ \\iota ] . \\end{aligned} \\end{align*}"}
-{"id": "7101.png", "formula": "\\begin{align*} W _ { n , \\beta } = n ^ { 3 \\beta / 2 } e ^ { n \\kappa ( \\beta ) } W _ n ^ { ( \\beta ) } = \\sum _ { | v | = n } e ^ { \\beta ( m _ n - V ( v ) ) } . \\end{align*}"}
-{"id": "540.png", "formula": "\\begin{align*} \\mathfrak { r } ( x , y ) : = \\big ( n h + n ^ { 1 / 3 } \\sigma x , n h + n ^ { 1 / 3 } \\sigma y \\big ) , \\end{align*}"}
-{"id": "5605.png", "formula": "\\begin{align*} \\chi ( p _ 1 ( a ) ) \\chi ( p _ 2 ( a ) ) = 1 , \\forall a \\in \\{ z , z + 1 , \\ldots , z + b - 1 \\} \\end{align*}"}
-{"id": "4580.png", "formula": "\\begin{align*} q ^ { ( i ) } ( y ) = ( 2 t ) ^ i \\sum _ { \\ell = 1 } ^ i ( - 1 ) ^ { i - \\ell } \\frac { a _ \\ell ^ { ( i ) } e ^ { - 2 t \\ell y } } { ( 1 + e ^ { - 2 t y } ) ^ { \\ell + 1 } } , \\end{align*}"}
-{"id": "5099.png", "formula": "\\begin{align*} \\mathcal { F } ^ { * } = \\bigoplus _ { m _ { 1 } , \\ldots m _ { r } \\in \\mathbb { Z } _ { \\ge 0 } } \\mathbb { C } \\langle m _ { 1 } , \\ldots , m _ { r } | \\end{align*}"}
-{"id": "2966.png", "formula": "\\begin{align*} \\tilde { \\alpha } _ t ( a , b ) & = \\Phi _ t ^ { - 1 } \\alpha _ t ( \\Phi _ t a , \\Phi _ t b ) , \\\\ \\tilde { \\mu } _ t ( x , a ) & = \\Phi _ t ^ { - 1 } \\mu _ t ( \\Psi _ t x , \\Phi _ t a ) , \\\\ \\tilde { \\lambda } _ t ( x , y ) & = \\Psi _ t ^ { - 1 } \\lambda _ t ( \\Psi _ t x , \\Psi _ t y ) . \\end{align*}"}
-{"id": "7880.png", "formula": "\\begin{align*} \\partial _ t \\phi _ y ( t , x , s ) = \\int _ { \\R ^ d } \\partial _ t p _ z ( t - s , x - z ) q ( s , z , y ) \\ , d z \\ , . \\end{align*}"}
-{"id": "1048.png", "formula": "\\begin{align*} \\Phi _ 1 : = 2 \\pi ( \\psi * \\Phi ) , \\quad \\Phi _ 2 = \\Phi - \\Phi _ 1 . \\end{align*}"}
-{"id": "7537.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\omega \\in \\Omega : | x _ { \\mathcal { N } _ 1 ( \\omega ) } ( \\omega ) - K | \\leq \\varepsilon \\mathcal { N } _ 1 : = \\mathcal { N } + N _ 1 + N _ 2 \\right ] = 1 . \\end{align*}"}
-{"id": "4363.png", "formula": "\\begin{align*} \\int _ { f ( r ) / a } ^ 1 \\frac { d z } { \\psi ( 1 - z ) ^ { 1 / p } } = r \\ , r \\in [ 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "5929.png", "formula": "\\begin{gather*} \\int _ { \\R ^ d } \\| D _ v ^ 2 { \\psi } ( \\cdot , v ) \\| _ { H ^ s _ p ( \\R ^ { d } ) } ^ p \\dd v = \\int _ { \\R ^ d } d v \\int _ { \\R ^ d } | D _ v ^ 2 \\ , G _ { \\lambda } h _ s ( x , v ) | ^ p \\ , \\dd x \\\\ \\le C \\| h _ s \\| _ { L ^ p ( \\R ^ { 2 d } ) } ^ p = C \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ d _ x ) ) } ^ p , \\end{gather*}"}
-{"id": "5776.png", "formula": "\\begin{align*} \\prod _ { k _ 1 \\in \\mathcal { J } _ 1 } \\cdots \\prod _ { k _ N \\in \\mathcal { J } _ N } B _ 1 ( \\lambda _ { k _ 1 } ) B _ 2 ( \\lambda _ { k _ 2 } ) \\cdots B _ N ( \\lambda _ { k _ N } ) = B _ { n _ 1 } ( \\lambda _ { 1 } ) B _ { n _ 2 } ( \\lambda _ { 2 } ) \\cdots B _ { n _ M } ( \\lambda _ M ) . \\end{align*}"}
-{"id": "8294.png", "formula": "\\begin{align*} 1 - \\frac { b _ 2 + b } { b _ 1 + b } = \\frac { b _ 1 - b _ 2 } { b _ 1 + b } = \\frac { b _ 1 + b ' } { b _ 1 + b } - \\frac { b _ 2 + b ' } { b _ 1 + b } . \\end{align*}"}
-{"id": "8210.png", "formula": "\\begin{align*} i \\frac { d u _ n } { d t } = \\frac { \\partial H } { \\partial \\bar { v } _ n } , i \\frac { d v _ n } { d t } = \\frac { \\partial H } { \\partial \\bar { u } _ n } , n \\in \\mathbb { Z } , \\end{align*}"}
-{"id": "2299.png", "formula": "\\begin{align*} \\| b _ \\ell \\| _ X \\le ( \\tilde \\lambda + C _ B r ) \\| e _ \\ell \\| _ D + 2 C _ B \\| u _ \\ell ^ \\star \\| _ D \\| e _ \\ell \\| _ W , \\ell = 0 , \\dotsc , M - 1 . \\end{align*}"}
-{"id": "9880.png", "formula": "\\begin{align*} \\lim _ { C \\rightarrow \\infty } \\sup _ { { \\epsilon } \\in ( 0 , 1 ) } P \\big ( \\sup _ { t \\leq T } | V ^ { \\beta ( { \\epsilon } ) , u ^ { \\epsilon } } _ { \\xi ^ { \\epsilon } } ( t , . ) | _ { 2 } \\geq C \\big ) = 0 . \\end{align*}"}
-{"id": "2017.png", "formula": "\\begin{align*} & \\int _ { | 1 + z | \\leq | x | ^ { - \\delta } } \\left | \\frac { f ( x ( 1 + z ) ) - f ( x ) } { x f ' ( x ) } - z I ( | z | \\leq 1 ) \\right | \\nu _ U ( \\d z ) \\\\ & \\leq ( \\alpha ^ { - 1 } \\log | x | + 1 ) \\nu _ U ( ( - 1 - | x | ^ { - \\delta } , - 1 + | x | ^ { - \\delta } ) ) \\to 0 . \\end{align*}"}
-{"id": "6807.png", "formula": "\\begin{align*} F ^ + - F ^ - = f \\S \\end{align*}"}
-{"id": "9071.png", "formula": "\\begin{align*} & \\P ( G _ 0 v ( d , 0 , 0 ) ) ( 1 - \\theta _ 1 ) \\\\ & \\leq \\P ( G _ 0 v ( d , 0 , E ) ) + \\mathcal { O } _ { d , \\upsilon } \\left ( \\sum _ { k = 2 } ^ { \\infty } N ^ { - 3 / 2 } ( 1 + k ^ 3 ) e ^ { - \\frac { ( k - 1 ) ^ 2 } { 4 d ^ 2 | | E | | _ 1 ^ { 1 / 3 } } } \\right ) . \\end{align*}"}
-{"id": "9183.png", "formula": "\\begin{align*} P _ t f ( x ) = \\int _ D f ( \\xi ) p _ t ( x , d \\xi ) . \\end{align*}"}
-{"id": "8104.png", "formula": "\\begin{align*} \\widetilde { F ( v ) } = F ( \\widetilde { v } ) . \\end{align*}"}
-{"id": "5541.png", "formula": "\\begin{align*} z \\rho _ t = \\frac { 1 } { 2 } b _ { x x x } + z \\rho _ { x } b + 2 z \\rho b _ { x } , \\end{align*}"}
-{"id": "699.png", "formula": "\\begin{align*} D = \\sum _ { i = 1 } ^ { m } a _ { i } H _ { i } \\end{align*}"}
-{"id": "9359.png", "formula": "\\begin{align*} p ( t , x ) & = \\mathbb { E } [ p ( T , x ) - \\frac { 1 } { 2 } \\int _ t ^ T \\frac { \\partial p } { \\partial s } ( s , x ) d s | \\mathcal { F } ^ { v , w } _ t ] \\\\ & = \\mathbb { E } [ p ( T , x ) - \\frac { 1 } { 2 } [ p ( s , x ) ] _ t ^ T | \\mathcal { F } ^ { v , w } _ t ] \\\\ & = \\mathbb { E } [ p ( T , x ) - \\frac { 1 } { 2 } p ( T , x ) + \\frac { 1 } { 2 } p ( t , x ) | \\mathcal { F } ^ { v , w } _ t ] . \\end{align*}"}
-{"id": "3538.png", "formula": "\\begin{align*} \\bar { \\mu } - | \\bar { J } | _ { \\bar { g } } \\ge \\mu + \\psi - | J + V | _ g = \\chi \\mu _ 1 + ( 1 - \\chi ) \\mu _ 2 + \\psi _ 0 - | \\chi J _ 1 + ( 1 - \\chi ) J _ 2 | _ g . \\end{align*}"}
-{"id": "8701.png", "formula": "\\begin{gather*} \\widetilde Y _ { \\tau } ^ { t , x } = \\int _ \\tau ^ T e ^ { - s { A } } G B ( s , \\Xi ^ { t , x } _ s ) \\ , d s + \\int _ \\tau ^ T e ^ { - s { A } } Z _ s ^ { t , x } \\ , B ( s , \\Xi ^ { t , x } _ s ) \\ , d s - \\int _ \\tau ^ T e ^ { - s { A } } Z ^ { t , x } _ { s } \\ ; d W _ s \\\\ = \\widetilde Y _ { 0 } ^ { t , x } - \\int _ 0 ^ \\tau e ^ { - s { A } } G B ( s , \\Xi ^ { t , x } _ s ) \\ , d s - \\int _ 0 ^ \\tau e ^ { - s { A } } Z _ s ^ { t , x } \\ , B ( s , \\Xi ^ { t , x } _ s ) \\ , d s + \\int _ 0 ^ \\tau e ^ { - s { A } } Z ^ { t , x } _ { s } \\ ; d W _ s \\end{gather*}"}
-{"id": "9137.png", "formula": "\\begin{align*} B ( t + h ) ^ { - 1 } - B ( t ) ^ { - 1 } = - B ( t + h ) ^ { - 1 } ( B ( t + h ) - B ( t ) ) B ( t ) ^ { - 1 } \\end{align*}"}
-{"id": "1902.png", "formula": "\\begin{align*} | \\hat { \\phi } _ X ( u ) - \\phi _ X ( u ) | \\leq & 2 | \\phi _ X ( u ) | + | | \\hat { \\phi } _ X ( u ) | - | \\phi _ X ( u ) | | \\\\ = & 2 | \\phi _ X ( u ) | + | | \\hat { \\phi } ( u ) | ^ { \\frac { 1 } { K } } - | \\phi ( u ) | ^ { \\frac { 1 } { K } } | \\leq 2 | \\phi _ X ( u ) | + | \\hat { \\phi } ( u ) - \\phi ( u ) | ^ { \\frac { 1 } { K } } , \\end{align*}"}
-{"id": "2955.png", "formula": "\\begin{align*} \\sigma u p ' ( u ) q ( u ) + ( 1 - \\sigma ) u p ( u ) q ' ( u ) = \\mu _ 1 \\gamma p ( u ) q ( u ) , \\end{align*}"}
-{"id": "6051.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l l l } \\Delta u _ { i } ^ { k + 1 } = \\frac { 1 } { \\varepsilon } u _ { i } ^ { k + 1 } \\sum \\limits _ { j \\neq i } H ( u _ { j } ^ { k } ) ( x ) & \\Omega , \\\\ u _ { i } ^ { k + 1 } ( x ) = \\phi _ { i } ( x ) & ( \\Omega ) _ 1 \\setminus \\Omega . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "7267.png", "formula": "\\begin{align*} \\sum _ { 2 \\leq | \\gamma | < n } \\sum _ { | \\beta | \\geq 1 } A _ \\gamma R ^ { | \\beta | } \\cdot X ^ \\gamma \\cdot \\prod _ { \\lambda = 1 } ^ r \\left ( X ^ \\lambda + A ( X ) \\right ) ^ { \\beta _ \\lambda } = A ( X ) \\cdot \\left ( - 1 + \\prod _ { \\lambda = 1 } ^ r \\frac { 1 } { 1 - R ( X ^ \\lambda + A ( X ) ) } \\right ) . \\end{align*}"}
-{"id": "5045.png", "formula": "\\begin{align*} \\nu ( B ) \\nu ( C ) = \\eta ( f ) = \\eta ( \\chi _ { A ^ c } f ) \\leq \\eta ( A ^ c ) \\nu ( C ) = ( 1 - \\eta ( A ) ) \\nu ( C ) . \\end{align*}"}
-{"id": "5646.png", "formula": "\\begin{align*} \\vert h _ { \\mu } ( \\nu ) - h _ { \\mu ' } ( \\nu ) \\vert = & \\ \\vert W _ 2 ( \\nu , \\mu ) - W _ 2 ( \\nu , \\mu ' ) \\vert \\ ( W _ 2 ( \\nu , \\mu ) + W _ 2 ( \\nu , \\mu ' ) ) \\le \\ W _ 2 ( \\mu , \\mu ' ) \\ 2 H ( \\nu ) . \\end{align*}"}
-{"id": "4514.png", "formula": "\\begin{align*} a ( \\delta ) : = \\max \\{ 1 , A _ { d , m , \\rho } ^ { ( 1 ) } , A _ { d , m , \\rho } ^ { ( 2 ) } \\} , \\end{align*}"}
-{"id": "7501.png", "formula": "\\begin{align*} \\{ Q _ 0 \\leq 1 \\} \\subseteq \\{ Q _ 1 \\geq n - 2 \\} \\cap \\{ Q _ 2 \\leq 1 \\} \\cap \\left ( \\bigcap _ { j = 3 } ^ { \\kappa ' } \\{ Q _ j = 0 \\} \\right ) . \\end{align*}"}
-{"id": "4983.png", "formula": "\\begin{align*} & A = ( a _ { m i } ) _ { m i } \\ \\ \\ s \\times r \\\\ & B = ( b _ { l j } ) _ { l j } \\ \\ r \\times s . \\end{align*}"}
-{"id": "472.png", "formula": "\\begin{align*} \\beta _ 1 y _ 1 + \\beta _ 2 y _ 2 & = \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 , \\\\ \\beta _ 1 y _ 1 - \\beta _ 2 y _ 2 & = \\beta _ 1 \\alpha _ 1 - \\beta _ 2 \\alpha _ 2 . \\end{align*}"}
-{"id": "6988.png", "formula": "\\begin{align*} \\Theta ( A ) = 1 / 2 \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ \\alpha _ i , \\alpha _ j ] . \\end{align*}"}
-{"id": "8058.png", "formula": "\\begin{align*} F = B _ { \\{ 1 , 2 \\} } \\cap B _ { \\{ 1 , 2 , 3 \\} } \\cap B _ { \\{ 4 , 5 \\} } \\cap B _ { \\{ 4 , 5 , 6 \\} } \\cap B _ { \\{ 1 , 2 , 3 , 4 , 5 , 6 \\} } \\cap B _ { \\{ 7 , 8 \\} } \\cap B _ { \\{ 7 , 8 , 9 \\} } \\cap B _ { \\{ 1 0 , 1 1 \\} } \\cap B _ { \\{ 1 0 , 1 1 , 1 2 \\} } . \\end{align*}"}
-{"id": "4353.png", "formula": "\\begin{align*} \\left ( \\psi ^ { 1 / p } \\right ) ' ( y ) + \\frac { 1 } { p - 1 } & = \\frac { 1 } { p } \\psi ( y ) ^ { - ( p - 1 ) / p } \\psi ' ( y ) + \\frac { 1 } { p - 1 } \\\\ & = \\frac { a ^ { 2 - p } } { p - 1 } ( 1 - y ) \\psi ( y ) ^ { - ( p - 1 ) / p } \\ge 0 \\ . \\end{align*}"}
-{"id": "3816.png", "formula": "\\begin{align*} \\left [ ( - 1 ) ^ { \\ell ( \\sigma _ J ) } \\mathbf { z } ^ { \\sigma _ J ( \\mu _ - ) } \\prod _ { 1 \\leq j \\leq n } \\frac { ( 1 + t z _ j x ) } { ( 1 - z _ j x ) } \\right ] \\Big \\vert _ { \\mathbf { z } ^ \\lambda } & = \\left [ ( - 1 ) ^ { \\ell ( \\sigma _ J ) } \\prod _ { 1 \\leq j \\leq n } \\frac { ( 1 + t z _ j x ) } { ( 1 - z _ j x ) } \\right ] \\Big \\vert _ { \\mathbf { z } ^ { \\lambda - \\sigma _ J ( \\mu _ - ) } } \\\\ & = ( - 1 ) ^ { | J | } ( 1 + t ) ^ { s ( \\lambda ; \\mu ) + 1 - | J | } x ^ { | \\lambda | - | \\mu | } , \\end{align*}"}
-{"id": "3617.png", "formula": "\\begin{align*} \\widetilde { ( L U ) _ j } & = \\sum _ { k = 1 } ^ { n + 1 } \\sum _ { | \\beta | = 0 } ^ { s _ j + t _ k } \\widetilde { b _ { j k } ^ { \\beta } } ( \\phi ( x ) ) ^ { - | \\beta | } \\partial _ z ^ { \\beta } \\widetilde { U ^ k } \\\\ & = \\sum _ { k = 1 } ^ { n + 1 } \\sum _ { | \\beta | = 0 } ^ { s _ j + t _ k } ( \\phi ( x ) ) ^ { - 4 + t _ k - | \\beta | } \\widetilde { b _ { j k } ^ { \\beta } } \\partial _ z ^ { \\beta } ( ( \\phi ( x ) ) ^ { 4 - t _ k } \\widetilde { U ^ k } ) . \\end{align*}"}
-{"id": "7957.png", "formula": "\\begin{align*} F ( \\lambda ) = \\lambda ^ { 5 / 4 } \\left ( j _ l ( \\sqrt [ 4 ] { \\lambda } ) i _ l ' ( \\sqrt [ 4 ] { \\lambda } ) - i _ l ( \\sqrt [ 4 ] { \\lambda } ) j _ l ' ( \\sqrt [ 4 ] { \\lambda } ) \\right ) . \\end{align*}"}
-{"id": "8636.png", "formula": "\\begin{align*} \\mathcal { Z } \\bigotimes _ S \\big ( S / ( Z _ 0 - 1 , Z _ 1 , \\ldots , Z _ g ) \\big ) = & \\\\ & 0 \\rightarrow Z _ g \\bigotimes _ Q Q [ Z _ 0 ] / ( Z _ 0 - 1 ) \\rightarrow \\cdots \\rightarrow Z _ 2 \\bigotimes Q [ Z _ 0 ] / ( Z _ 0 - 1 ) \\rightarrow & \\\\ & \\rightarrow Z _ 1 \\bigotimes _ Q Q [ Z _ 0 ] / ( Z _ 0 - 1 ) \\rightarrow Q [ Z _ 0 ] / ( Z _ 0 - 1 ) \\rightarrow 0 , \\end{align*}"}
-{"id": "2863.png", "formula": "\\begin{align*} T o t _ { E _ 1 - A s s ^ { o p } } ( C ^ { \\bullet } ) = \\int _ { \\underline { n } \\in \\Delta } ^ { E _ 1 - A l g ( d g C o g ^ { c o n i l } ) } ( C ^ n ) ^ { \\Delta ^ n } _ { E _ 1 - A s s ^ { o p } } \\end{align*}"}
-{"id": "3774.png", "formula": "\\begin{align*} f _ \\beta ( t ) = e ^ { \\beta t } f ( t ) / \\exp ( C _ g ( \\beta ) ) . \\end{align*}"}
-{"id": "3009.png", "formula": "\\begin{align*} { { \\bf { Y } } ^ { [ j ] } } ( n ) = \\sum \\limits _ { i = 1 } ^ { M } { { { \\bf { H } } ^ { [ j i ] } } ( n ) { { \\bf { X } } ^ { [ i ] } } ( n ) } + { { \\bf { Z } } ^ { [ j ] } } ( n ) , \\ ; j \\in \\{ 1 , 2 , . . . , N \\} , \\end{align*}"}
-{"id": "4928.png", "formula": "\\begin{align*} | A \\hat x | = | A x _ 0 | \\end{align*}"}
-{"id": "3374.png", "formula": "\\begin{align*} P ^ * ( t ) = \\frac { V _ { \\max } } { \\Delta t ( E _ { \\max } - E _ { c , \\max } - E _ b ( t ) ) } - \\frac { 1 } { \\gamma ( t ) } . \\end{align*}"}
-{"id": "8040.png", "formula": "\\begin{align*} a _ 1 ^ { x _ 1 } a _ 2 ^ { x _ 2 } \\cdots a _ { m - 1 } ^ { x _ { m - 1 } } a _ m ^ { x _ m } & \\cdots a _ { 2 m - 2 } ^ { x _ { 2 m - 2 } } \\cdots a _ { n - 2 } ^ { x _ { n - 2 } } a _ { n - 1 } ^ { x _ { n - 1 } } a _ n ^ { x _ n } \\\\ & = s _ { 1 , 2 } ^ { x _ { 1 , 2 } } s _ { 2 , 3 } ^ { x _ { 2 3 } } \\cdots s _ { m - 1 , m } ^ { x _ { m - 1 , m } } s _ { 1 , 3 } ^ { x _ { 1 , 3 } } \\cdots s _ { 1 , m - 1 } ^ { x _ { 1 , m - 1 } } s _ { 2 , m } ^ { x _ { 2 , m } } s _ { 1 , m } ^ { x _ { 1 , m } } . \\end{align*}"}
-{"id": "8278.png", "formula": "\\begin{align*} l _ { j \\leftarrow n } ^ t = \\log \\left ( \\frac { p } { 1 - p } \\right ) + \\sum _ { k \\in G _ n , \\ , k \\neq j } l _ { k \\rightarrow n } ^ t . \\end{align*}"}
-{"id": "8507.png", "formula": "\\begin{align*} F _ i ^ { ( c ) } : = \\frac { F _ i ^ c } { [ c ] ! } [ c ] ! : = \\prod _ { j = 1 } ^ c \\frac { q ^ { j } - q ^ { - j } } { q - q ^ { - 1 } } . \\end{align*}"}
-{"id": "9986.png", "formula": "\\begin{align*} a ^ { ( n + 1 ) } ( s ) \\ ! = \\ ! \\arg \\ ! \\max _ { a \\in \\mathcal { A } ( s ) } \\ ! \\left [ g ( s , a ) \\ ! + \\ ! \\sum _ { s ' \\in \\mathcal { S } } \\ ! p _ { s \\rightarrow s ' | a } h ^ { ( n ) } ( s ' ) \\right ] . \\end{align*}"}
-{"id": "9812.png", "formula": "\\begin{align*} f ( x ) = \\int _ 0 ^ { \\infty } 1 _ { \\{ f > t \\} } ( x ) d t . \\end{align*}"}
-{"id": "6339.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } f \\| _ { M _ 2 } \\lesssim 2 ^ { j n \\alpha _ 2 / 2 } 2 ^ { j n ( \\alpha _ 1 - \\alpha _ 2 ) ( 1 / q - 1 / 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "2979.png", "formula": "\\begin{align*} f _ 0 ^ { \\lambda _ n ( x , y ) } + f _ n ^ { \\lambda _ 0 ( x , y ) } + \\sum _ { i + j = n , ~ i , j > 0 } f _ i ^ { \\lambda _ j ( x , y ) } = \\sum _ { i + j = n , ~ i , j > 0 } [ f _ i ^ x , f _ j ^ y ] + [ f _ 0 ^ x , f _ n ^ y ] + [ f _ n ^ x , f _ 0 ^ y ] . \\end{align*}"}
-{"id": "5627.png", "formula": "\\begin{align*} M ( \\Lambda ) = \\left \\{ x ^ a y ^ b : ( a , b ) \\in \\Lambda \\right \\} . \\end{align*}"}
-{"id": "7866.png", "formula": "\\begin{align*} \\rho ( t , x ) = \\rho ^ { ( d ) } ( t , x ) : = { \\Phi \\left ( \\left ( \\frac { 1 } { \\Phi ^ { - 1 } ( t ^ { - 1 } ) } + | x | \\right ) ^ { - 1 } \\right ) } \\left ( \\frac { 1 } { \\Phi ^ { - 1 } ( t ^ { - 1 } ) } + | x | \\right ) ^ { - d } \\ , . \\end{align*}"}
-{"id": "449.png", "formula": "\\begin{align*} z & = ( v _ 1 v _ 2 \\dots v _ { k + m - 1 } v _ m ) ^ 2 v _ { m + 1 } \\dots v _ { k + m - 1 } \\\\ & = v _ 1 v _ 2 \\dots v _ { k + m - 1 } v _ m v _ 1 v _ 2 \\dots v _ { m - 1 } ( v _ m v _ { m + 1 } \\dots v _ { k + m - 1 } ) ^ 2 . \\end{align*}"}
-{"id": "9665.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log g ( x ( t ) ) } { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } = - \\log \\left ( \\frac { a } { b } \\right ) . \\end{align*}"}
-{"id": "94.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ l D _ { r , i } \\left ( C _ { r - s , i } - C _ { 1 , i } \\right ) \\end{align*}"}
-{"id": "2538.png", "formula": "\\begin{align*} X - Y = \\mathbf { i } \\frac { \\tau } { h ^ 2 } A \\frac { X - Y } 2 + \\frac { \\mathbf { i } \\lambda \\tau } 8 H ( X , Y , z ) + \\mathbf { i } Z ( \\frac { X - Y } 2 ) \\delta _ { n + 1 } \\beta , \\end{align*}"}
-{"id": "3578.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } \\| U ^ j \\| _ { C ^ { t _ j , \\alpha } _ { \\phi , \\varphi _ j } ( \\Omega ) } \\le C \\left ( \\sum _ { j = 1 } ^ { n + 1 } \\| ( L U ) _ j \\| _ { C ^ { - s _ j , \\alpha } _ { \\phi , \\phi ^ { t _ j + s _ j } \\varphi _ j } ( \\Omega ) } + \\sum _ { j = 1 } ^ { n + 1 } \\| U ^ j \\| _ { L ^ 2 _ { \\phi ^ { - n } \\varphi ^ 2 _ j } ( \\Omega ) } \\right ) \\end{align*}"}
-{"id": "9534.png", "formula": "\\begin{align*} \\omega ^ M _ { \\mu \\nu \\rho } & = g _ M ( \\nabla ^ M _ { \\mu ' } e _ { \\nu } ' , e _ { \\rho } ' ) \\\\ \\omega ^ N _ { \\mu \\nu \\rho } & = g _ N ( \\nabla ^ N _ \\mu e _ \\nu , e _ \\rho ) \\end{align*}"}
-{"id": "317.png", "formula": "\\begin{align*} D _ { g ^ * } \\leq 8 d ^ { 1 / 2 } \\Bigl ( \\frac { 7 1 } { 4 1 } \\Bigr ) ^ { m } m ! ( m + 1 ) ^ { m + 2 } = : D . \\end{align*}"}
-{"id": "7688.png", "formula": "\\begin{align*} | f ( x ) - K | = f ( x ) - K < x - K = | x - K | , \\end{align*}"}
-{"id": "6612.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\tau ^ { - 1 } \\langle \\nabla f ( x + \\tau ( y - x ) ) & - \\nabla f ( x ) , ( x + \\tau ( y - x ) ) - x \\rangle d \\tau \\\\ & \\geq - \\int _ 0 ^ 1 \\tau ^ { - 1 } \\langle M \\tau ( x - y ) , \\tau ( x - y ) \\rangle d \\tau \\\\ & = - \\langle M ( x - y ) , x - y \\rangle \\int _ 0 ^ 1 \\tau d \\tau \\\\ & = - \\tfrac { 1 } { 2 } \\langle M ( x - y ) , x - y \\rangle . \\end{align*}"}
-{"id": "999.png", "formula": "\\begin{align*} \\lambda _ { \\{ l ; k \\} } ( u ) = \\left ( u + \\omega + \\eta \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) \\left ( u - \\omega + \\eta \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) + \\eta ^ { - 2 } \\end{align*}"}
-{"id": "3761.png", "formula": "\\begin{align*} P \\left ( T _ \\beta = 0 \\right ) e ^ { i \\alpha t u } \\end{align*}"}
-{"id": "951.png", "formula": "\\begin{align*} ( f , g ) _ { L ^ { 2 } _ { ( 0 , q ) } ( M ) } : = \\int _ { M } ( f , g ) ( z ) d \\sigma ( z ) \\ ; , \\end{align*}"}
-{"id": "2913.png", "formula": "\\begin{align*} & \\prod _ { i , j \\geq 0 } \\ , z ^ { 2 i + j + 1 } \\ , s _ { ( 1 ^ { 2 i + 1 } ) } ( X ) \\ , s _ { ( j ) } ( X ) = \\prod _ { a , b \\geq 0 } \\ , z ^ { a + b + 1 } \\ , s _ { ( a + 1 , 1 ^ b ) } ( X ) \\cr \\mbox { a n d } \\cr & \\prod _ { i , j \\geq 0 : ( i , j ) \\neq ( 0 , 0 ) } \\ , z ^ { 2 i + j } \\ , s _ { ( 1 ^ { 2 i } ) } ( X ) \\ , s _ { ( j ) } ( X ) = \\prod _ { a , b \\geq 0 } \\ , z ^ { a + b + 1 } \\ , s _ { ( a + 1 , 1 ^ b ) } ( X ) \\end{align*}"}
-{"id": "484.png", "formula": "\\begin{align*} | u _ \\xi ( x ) | & = \\Big | u _ \\xi ( 0 ) + | x | \\int _ 0 ^ 1 u _ { \\xi \\xi } ( t x ) \\ , d t \\Big | \\\\ & \\le | u _ \\xi ( 0 ) | + | x | \\int _ 0 ^ 1 \\frac { C } { \\ , | t x | ^ \\alpha \\ , } \\ , d t \\\\ & \\le | D u ( 0 ) | + \\frac { C } { \\ , 1 - \\alpha \\ , } \\ , | x | ^ { 1 - \\alpha } . \\end{align*}"}
-{"id": "8816.png", "formula": "\\begin{align*} Y _ M ( T a , z ^ { 1 / m } ) = \\partial _ z Y _ M ( a , z ^ { 1 / m } ) \\end{align*}"}
-{"id": "7386.png", "formula": "\\begin{align*} \\tau _ 0 Z _ { 0 _ i } + [ \\Delta _ { 1 , 0 } ] _ i = \\frac { 1 } { \\sqrt { n } } \\norm { m ^ 0 } [ ( \\mathsf { I } - \\mathsf { P } ^ { \\parallel } _ { q ^ 0 } ) Z _ { 0 } ] _ i + u _ i , \\end{align*}"}
-{"id": "4410.png", "formula": "\\begin{align*} \\mu ( t , x ) = \\inf \\{ s \\geq 0 : \\ , \\tau ( e ^ { \\abs { x } } ( s , \\infty ) ) \\leq t \\} , t \\geq 0 , \\end{align*}"}
-{"id": "6896.png", "formula": "\\begin{align*} \\mathsf { P } _ { \\mathrm { S I R } , g _ { 2 } } \\left ( \\lambda \\right ) = \\mathbb { E } _ { d _ { 0 } } \\left [ e ^ { - \\frac { 2 \\pi \\lambda \\tau \\left ( 1 + d _ { 0 } ^ { \\alpha } \\right ) } { \\left ( \\alpha - 2 \\right ) d _ { 0 } ^ { \\alpha - 2 } } H y F _ { 1 } \\left ( \\frac { 1 + \\tau \\left ( 1 + d _ { 0 } ^ { \\alpha } \\right ) } { d _ { 0 } ^ { \\alpha } } \\right ) } \\right ] , \\end{align*}"}
-{"id": "6009.png", "formula": "\\begin{gather*} [ H _ { \\Phi } ] = \\begin{pmatrix} 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & I _ 2 & 0 \\\\ 0 & 0 & - 1 & 0 & 0 \\\\ 0 & I _ 2 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 \\end{pmatrix} , \\end{gather*}"}
-{"id": "2570.png", "formula": "\\begin{align*} S ^ { } : = L _ t ^ { \\tilde q , \\infty } L _ x ^ { \\tilde r } , S : = L _ t ^ { \\tilde q , 2 } L _ x ^ { \\tilde r } , W : = \\cap _ { j = 1 } ^ 2 L _ t ^ { q _ j , 2 } \\dot X ^ { | s _ c | , r _ j } \\end{align*}"}
-{"id": "3317.png", "formula": "\\begin{align*} T = \\left ( \\begin{array} { c c c c } 0 & T ^ { ( 1 ) } & 0 & 0 \\\\ 0 & 0 & T ^ { ( 2 ) } & 0 \\\\ 0 & 0 & 0 & T ^ { ( 3 ) } \\\\ 0 & 0 & 0 & 0 \\end{array} \\right ) , T ^ * = \\left ( \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\\\ T ^ { ( 1 ) * } & 0 & 0 & 0 \\\\ 0 & T ^ { ( 2 ) * } & 0 & 0 \\\\ 0 & 0 & T ^ { ( 3 ) * } & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "36.png", "formula": "\\begin{align*} e ^ { \\varphi _ { X / Y } ( x _ 0 ) } = \\sup _ { \\Vert u \\Vert _ { y _ 0 } \\leq 1 } | F _ u ( x _ 0 ) | ^ 2 \\end{align*}"}
-{"id": "4159.png", "formula": "\\begin{align*} | S ^ \\nu _ { h _ 1 } ( n ) | ^ 2 = | S ^ \\nu _ f ( n ) | ^ 2 + | S ^ \\nu _ g ( n ) | ^ 2 + 2 \\Re \\left ( S ^ \\nu _ f ( n ) \\overline { S ^ \\nu _ g ( n ) } \\right ) \\end{align*}"}
-{"id": "4073.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { x y ( a x + b y + c z ) } { z ^ 3 } , f ( x , y , z ) = \\dfrac { x y ( a x + b y + c z ) ^ 2 } { z ^ 4 } , \\\\ f ( x , y , z ) = \\dfrac { x y ^ 2 ( a x + b y + c z ) ^ 3 } { z ^ 6 } , \\end{gather*}"}
-{"id": "821.png", "formula": "\\begin{align*} \\tilde { L } ( u ; s ) = \\frac { 1 } { 1 - s u } \\begin{pmatrix} 1 - s u q ^ { 2 N } & u \\beta ^ { * } ( 1 - s ^ 2 q ^ { 2 N } ) \\\\ \\beta & u - s q ^ { 2 N } \\end{pmatrix} \\end{align*}"}
-{"id": "2377.png", "formula": "\\begin{align*} r _ { C } ( x ) : = \\iota _ { C } ( x ) + \\tfrac { 1 } { 2 } \\| x \\| ^ 2 . \\end{align*}"}
-{"id": "9458.png", "formula": "\\begin{align*} \\begin{gathered} \\pi _ { \\lambda } \\circ d u \\circ j = J \\circ \\pi _ { \\lambda } \\circ d u \\\\ u ^ { * } \\lambda \\circ j = d a \\end{gathered} \\end{align*}"}
-{"id": "9397.png", "formula": "\\begin{align*} u _ \\lambda = \\left ( 1 - \\frac { c } { \\mu ^ 2 + \\lambda } \\right ) e ^ { i k | x | } + \\frac { q ( x ) } { \\mu ^ 2 + \\lambda } , \\lambda = k ^ 2 . \\end{align*}"}
-{"id": "5922.png", "formula": "\\begin{gather*} \\lambda \\| { \\psi } \\| _ { L ^ p ( \\R ^ { 2 d } ) } ^ p + \\frac { ( p - 1 ) } { 2 } \\sum _ { k = 1 } ^ d \\int _ { \\R ^ { 2 d } } | { \\psi } | ^ { p - 2 } | \\partial _ { v _ k } { \\psi } | ^ 2 \\ , \\dd z + \\int _ { \\R ^ { 2 d } } ( v \\cdot D _ x { \\psi } ) | { \\psi } | ^ { p - 2 } { \\psi } \\ , \\dd z \\\\ = \\int _ { \\R ^ { 2 d } } g | { \\psi } | ^ { p - 2 } { \\psi } \\ , \\dd z \\end{gather*}"}
-{"id": "4914.png", "formula": "\\begin{align*} \\beta _ n : = \\frac { 6 } { E _ 6 ( \\rho ) } \\sum _ { ( \\lambda ) } \\sum _ { ( c , d ) } \\frac { h _ { ( c , d ) } ( n ) } { \\lambda ^ 3 } e ^ { \\frac { \\pi n \\sqrt { 3 } } { \\lambda } } . \\end{align*}"}
-{"id": "619.png", "formula": "\\begin{align*} Q ( x ) = \\beta ( x ) ( x , \\R ^ N \\ ! \\setminus \\ ! \\Omega ) ^ \\alpha \\end{align*}"}
-{"id": "1969.png", "formula": "\\begin{align*} C _ i = \\{ ( t : b _ i ) \\mid t \\in T \\} \\backslash \\{ 0 \\} . \\end{align*}"}
-{"id": "2126.png", "formula": "\\begin{align*} f ( z ) = \\frac { U _ a } { r } R ( x _ n , r ) + F ( z ) \\end{align*}"}
-{"id": "7347.png", "formula": "\\begin{align*} \\gamma _ { - } ( w _ { a } ) v = \\sum _ { i } \\frac { \\langle w _ { i } ^ { ( k - 1 ) } , \\gamma _ { - } ( w _ { a } ) v \\rangle } { \\langle w _ { i } ^ { ( k - 1 ) } , v _ { i } ^ { ( k - 1 ) } \\rangle } v _ { i } ^ { ( k - 1 ) } = \\sum _ { i } \\frac { \\langle w _ { i } ^ { ( k - 1 ) } \\wedge w _ { a } , v \\rangle } { \\langle w _ { i } ^ { ( k - 1 ) } , v _ { i } ^ { ( k - 1 ) } \\rangle } v _ { i } ^ { ( k - 1 ) } . \\end{align*}"}
-{"id": "4561.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\mu _ d \\bigl ( ( K + z ) \\cap L \\bigr ) \\ , d z = \\mu _ d ( K ) \\mu _ d ( L ) , \\end{align*}"}
-{"id": "6404.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { ( m ) } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\left ( x _ { t } \\right ) ^ { m } . \\end{align*}"}
-{"id": "3547.png", "formula": "\\begin{align*} \\| ( 2 \\psi , V ) \\| _ { \\mathcal B _ 0 \\times \\mathcal B _ 1 } & = \\| \\chi \\Phi ( g _ 1 , \\pi _ 1 ) + ( 1 - \\chi ) \\Phi ( g _ 2 , \\pi _ 2 ) - \\Phi ( g , \\pi ) \\| _ { \\mathcal B _ 0 \\times \\mathcal B _ 1 } + \\| 2 \\psi _ 0 \\| _ { \\mathcal { B } _ 0 } R ^ { - 1 - q _ 0 } \\\\ & \\le C R ^ { - \\min ( q , 1 + q _ 0 ) } . \\end{align*}"}
-{"id": "2239.png", "formula": "\\begin{align*} \\mu _ { ( k ) } = C ( \\varepsilon ) \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon ) ^ { n } } { n ! } \\mu _ { ( n p + k ) } ^ { G } , \\end{align*}"}
-{"id": "3246.png", "formula": "\\begin{align*} P = \\left \\{ \\begin{array} { c c } p _ { G } & \\mathrm { o n } \\ , \\ , \\ , \\tilde { W } \\\\ p _ { O } & \\mathrm { o n } \\ , \\ , \\ , \\tilde { O } \\setminus \\tilde { V } , \\end{array} \\right . \\end{align*}"}
-{"id": "2430.png", "formula": "\\begin{align*} \\Lambda _ k ^ { \\alpha _ 2 , \\ast } \\cap \\Lambda _ l ^ { \\alpha _ 2 , \\ast } = \\emptyset \\end{align*}"}
-{"id": "2968.png", "formula": "\\begin{align*} \\tilde { \\alpha _ t } ( \\Phi _ t a , \\Phi _ t b ) & = \\Phi _ t \\alpha _ t ( a , b ) , \\\\ \\tilde { \\mu _ t } ( \\Psi _ t x , \\Phi _ t a ) & = \\Phi _ t \\mu _ t ( x , a ) , \\\\ \\tilde { \\lambda _ t } ( \\Psi _ t x , \\Psi _ t y ) & = \\Psi _ t \\lambda _ t ( x , y ) . \\end{align*}"}
-{"id": "5625.png", "formula": "\\begin{align*} a _ i ( n ) & = | \\mathcal { A } _ i ( n ) | , \\\\ a _ { i p } ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) ~ | ~ w = \\overline { w } \\} | , \\\\ a _ { i n } ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) ~ | ~ w \\neq \\overline { w } \\} | . \\end{align*}"}
-{"id": "6715.png", "formula": "\\begin{align*} \\Lambda & = \\langle a , a + d , a + 2 d , \\ldots \\rangle . \\\\ \\intertext { S i n c e f o r $ j \\geq a $ w e h a v e t h a t $ a + j \\cdot d = d \\cdot a + a + ( j - a ) \\cdot d $ , t h e s e m i g r o u p i s g e n e r a t e d b y t h e f i n i t e a r i t h m e t i c p r o g r e s s i o n } \\Lambda & = \\langle a , a + d , a + 2 d , \\ldots , a + ( a - 1 ) d \\ , \\rangle . \\end{align*}"}
-{"id": "9767.png", "formula": "\\begin{align*} \\nabla _ { \\vec { x } } V ( \\vec { x } ) = - \\vec { S } ^ { \\top } \\boldsymbol { \\lambda } ^ { * } ( \\vec { x } ) , \\forall \\vec { x } \\in \\mathrm { i n t } ( \\mathcal { X } ) . \\end{align*}"}
-{"id": "3028.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ i ] } } ( n ) = \\sum \\limits _ { j = 1 } ^ N { \\sum \\limits _ { a = 1 } ^ A { s _ a ^ { [ j i ] } { \\bf { v } } _ a ^ { [ j i ] } ( n ) } } = \\sum \\limits _ { j = 1 } ^ N { { \\bf { V } } _ j ^ { [ i ] } ( n ) { { \\bf { s } } ^ { [ j i ] } } } , \\end{align*}"}
-{"id": "4271.png", "formula": "\\begin{align*} \\mathbb E [ Z ^ * _ { \\lfloor \\varepsilon n \\rfloor + 1 , n + \\lfloor \\varepsilon n \\rfloor } ] = \\mathbb E [ Z _ { \\lfloor \\varepsilon n \\rfloor } ] + \\mathbb E [ Z ^ * _ { \\lfloor \\varepsilon n \\rfloor + 1 , n + \\lfloor \\varepsilon n \\rfloor } - Z _ { \\lfloor \\varepsilon n \\rfloor } ] = 0 + \\mathbb E [ Z _ n ^ * ] = \\mathbb E [ Z _ n ^ * ] \\ , . \\end{align*}"}
-{"id": "9976.png", "formula": "\\begin{align*} h ^ { ( n + 1 ) } ( s ) = ( 1 - \\tau ) h ^ { ( n ) } ( s ) + \\max _ { a \\in \\mathcal { A } ( s ) } \\left [ g ( s , a ) + \\tau \\sum _ { \\mathbf { H } ' } \\mathrm { P r } ( \\mathbf { H } ' | \\mathbf { H } ) h ^ { ( n ) } ( s ' ) \\right ] - \\Lambda ^ { ( n + 1 ) } ( s _ 0 ) . \\end{align*}"}
-{"id": "6691.png", "formula": "\\begin{align*} h ^ 0 ( 2 E _ i + K - D ) = h ^ 0 ( 2 E _ j + K - D ) = h ^ 0 ( 2 E _ k + K - D ) = 0 . \\end{align*}"}
-{"id": "1037.png", "formula": "\\begin{align*} c _ { d , d } { \\ , } c _ { q , q _ 1 } ^ d ~ = ~ ( 1 + \\delta _ { q _ 1 , q _ 2 } ) { \\ , } c _ { q , q } { \\ , } c _ { d , d } ^ q \\end{align*}"}
-{"id": "5535.png", "formula": "\\begin{align*} - v _ { x x } = \\frac z T \\rho ( x ) v . \\end{align*}"}
-{"id": "9648.png", "formula": "\\begin{align*} \\{ C _ i \\} _ { i = 1 , \\ldots , 9 } \\end{align*}"}
-{"id": "6407.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { ( m ) } = C _ { ( 2 ) } ( \\vec { \\varepsilon } ) \\int _ { - \\infty } ^ { \\infty } p _ { G } ( x ) x ^ { m } \\left ( 1 + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } x ^ { k } + \\frac { 1 } { 2 } \\left ( \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } x ^ { k } \\right ) ^ { 2 } \\right ) d x , \\end{align*}"}
-{"id": "5097.png", "formula": "\\begin{align*} \\mathcal { F } = \\bigoplus _ { m _ { 1 } , \\ldots m _ { r } \\in \\mathbb { Z } _ { \\ge 0 } } \\mathbb { C } | m _ { 1 } , \\ldots , m _ { r } \\rangle \\end{align*}"}
-{"id": "4948.png", "formula": "\\begin{align*} Q ( x ) = \\beta ( x ) ( x , \\R ^ N \\ ! \\setminus \\ ! \\Omega ) ^ \\alpha \\end{align*}"}
-{"id": "2924.png", "formula": "\\begin{align*} \\prod _ { \\ell = 1 } ^ m \\ , M ( z _ \\ell ; X ) = \\prod _ { \\ell = 1 } ^ m \\prod _ { k \\geq 1 } ( 1 - z _ \\ell x _ k ) ^ { - 1 } = M ( X Z ) , \\end{align*}"}
-{"id": "7214.png", "formula": "\\begin{align*} P _ { \\alpha } ^ { S } = Q ^ { S } \\circ P _ { \\alpha } ^ { N } = Q ^ { S } \\circ P _ { \\beta } ^ { N } \\circ \\Phi _ { \\beta , \\alpha } ^ { 1 } = P _ { \\beta } ^ { S } \\circ \\Phi _ { \\beta , \\alpha } ^ { 1 } . \\end{align*}"}
-{"id": "6438.png", "formula": "\\begin{align*} \\int _ { B _ { 1 } } g _ { \\alpha } * \\left ( \\phi \\psi ^ { 1 + q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ^ { 1 - q } ) \\right ) d x & + ( 1 - q ) g _ { \\alpha } * \\left [ \\mathcal { E } ( h _ { m } * \\tilde { u } , \\psi ^ { 1 + q } \\tilde { u } ^ { - q } ) \\phi \\right ] \\\\ & \\leq q g _ { \\alpha } * \\int _ { B _ { 1 } } \\psi ^ { 1 + q } \\tilde { u } ^ { 1 - q } g _ { 1 - \\alpha , m } \\phi d x , \\end{align*}"}
-{"id": "537.png", "formula": "\\begin{align*} \\int \\int f ( x , y ) \\delta ' ( x , y ) \\mathrm { d } x \\mathrm { d } y = \\int \\left . \\big ( \\partial _ y f ( x , y ) - \\partial _ x f ( x , y ) \\big ) \\right | _ { y = x } \\mathrm { d } x , \\end{align*}"}
-{"id": "7514.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left | x _ n - K \\right | = 0 . \\end{align*}"}
-{"id": "5108.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { k } \\frac { z _ { i } ^ { M ' - 1 } } { ( 1 + z _ { i } ) ^ { M } } \\langle \\prod _ { 1 \\le i \\le k } ^ { \\curvearrowright } C _ { \\mu _ { i } } ^ { [ M , M ' ] } ( z _ { i } ) \\prod _ { 1 \\le i \\le k } \\beta _ { \\nu _ { i } , x _ { i } } ^ { * } \\rangle _ { [ M ' , M ] } . \\end{align*}"}
-{"id": "7998.png", "formula": "\\begin{align*} e ( X ) \\geq 2 + 6 \\cdot 3 + \\sum _ { i = 1 } ^ 4 e ( F _ i ) \\geq 2 8 , \\end{align*}"}
-{"id": "986.png", "formula": "\\begin{align*} H _ { 2 , 2 } & = U ( N _ { a , 1 } + N _ { a , 2 } - N _ { b , 1 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } + N _ { a , 2 } - N _ { b , 1 } - N _ { b , 2 } ) \\\\ & \\quad + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) + t _ { 1 , 2 } ( a _ { 1 } b _ { 2 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 2 } ) \\\\ & \\quad + t _ { 2 , 1 } ( a _ { 2 } b _ { 1 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 1 } ) + t _ { 2 , 2 } ( a _ { 2 } b _ { 2 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 2 } ) \\end{align*}"}
-{"id": "4482.png", "formula": "\\begin{align*} \\mathbb { E } _ f ( \\hat { H } _ n ) = H + \\log ( n - 1 ) - \\Psi ( n ) + \\sum _ { i = 1 } ^ 5 R _ i ' . \\end{align*}"}
-{"id": "7474.png", "formula": "\\begin{align*} Q _ j ^ { ( k ) } ( V ) = \\sum _ { i = j } ^ { \\kappa ' k } ( - 1 ) ^ { i - j } { i \\choose j } H _ { D _ i } ( V ) , \\end{align*}"}
-{"id": "1008.png", "formula": "\\begin{align*} [ Q _ { j } , C ( u ) ] = 0 , \\ ; \\ ; \\ ; \\ ; \\ [ \\overline { Q } _ { j } , C ( u ) ] = 0 . \\end{align*}"}
-{"id": "1760.png", "formula": "\\begin{align*} s _ \\beta x & = s _ \\gamma t ( a ' \\gamma ^ \\lor ) w z _ \\mu t ( \\mu ) \\\\ & = w s _ { w ^ { - 1 } \\gamma } z _ \\mu t ( a ' z _ \\mu ^ { - 1 } w ^ { - 1 } \\gamma ^ \\lor + \\mu ) \\\\ & = w s _ { \\alpha } z _ \\mu t ( a ' z _ \\mu ^ { - 1 } \\alpha ^ \\lor + \\mu ) . \\end{align*}"}
-{"id": "9992.png", "formula": "\\begin{align*} { } \\gamma F _ k ( \\alpha ^ { ( 1 ) } ) + ( 1 - \\gamma ) F _ k ( \\alpha ^ { ( 2 ) } ) \\le \\alpha ' \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\tilde { p } ' | H _ { \\tilde { i } k } | ^ 2 } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\ ! + \\ ! ( \\ ! 1 \\ ! - \\ ! \\alpha ' \\ ! ) \\sum _ { i = 1 } ^ { 2 } \\log _ 2 \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { p _ { i } ' } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\end{align*}"}
-{"id": "2214.png", "formula": "\\begin{align*} \\| L u ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\mathbb { R } ^ { n } ) } ^ { 2 } = \\int _ { \\mathbb { R } ^ { n } } | L u ( x , t ) | ^ { 2 } d x = \\int _ { \\mathbb { R } ^ { n } } | A ( \\xi ) \\mathcal { F } ( u ) ( \\xi , t ) | ^ { 2 } d \\xi . \\end{align*}"}
-{"id": "6105.png", "formula": "\\begin{align*} u ' q ' { \\frac { d \\bar z } { \\bar z } } \\cdot \\frac { d z } z = u ' q ' \\frac { d r } r d \\theta . \\end{align*}"}
-{"id": "3394.png", "formula": "\\begin{align*} ( w ^ r ) \\otimes L _ 0 \\xrightarrow { \\phi _ r } L _ { r - 1 } \\xrightarrow [ \\sim ] { \\psi _ { r - 1 } } ( w ) \\otimes L _ { r - 2 } & \\xrightarrow [ \\sim ] { \\mathrm { i d } _ { ( w ) } \\otimes \\psi _ { r - 2 } } ( w ^ 2 ) \\otimes L _ { r - 3 } \\\\ & \\xrightarrow [ \\sim ] { \\mathrm { i d } _ { ( w ^ 2 ) } \\otimes \\psi _ { r - 3 } } \\cdots \\xrightarrow [ \\sim ] { \\mathrm { i d } _ { ( w ^ { r - 2 } ) } \\otimes \\psi _ 1 } ( w ^ { r - 1 } ) \\otimes L _ 0 \\end{align*}"}
-{"id": "4737.png", "formula": "\\begin{align*} \\beta = \\begin{cases} \\alpha + 1 \\ \\ \\alpha \\in \\mathbb { Z } \\\\ \\lceil \\alpha \\rceil \\ \\ \\alpha \\notin \\mathbb { Z } . \\end{cases} \\end{align*}"}
-{"id": "8315.png", "formula": "\\begin{align*} y ^ { p ^ 2 } - y & = y _ 1 ^ { p ^ 2 } + \\mu ^ { p ^ 2 } y _ 2 ^ { p ^ 2 } - y _ 1 - \\mu y _ 2 = ( y _ 1 ^ { p ^ 2 } - y _ 1 ) + \\mu ( y _ 2 ^ { p ^ 2 } - y _ 2 ) \\\\ & = [ ( y _ 1 ^ p - y _ 1 ) ^ p + ( y _ 1 ^ p - y _ 1 ) ] + \\mu [ ( y _ 2 ^ p - y _ 2 ) ^ p + ( y _ 2 ^ p - y _ 2 ) ] \\\\ & = T ^ p + T + \\mu T ^ { 2 p } + \\mu T ^ 2 = T ^ p ( 1 + \\mu T ^ p ) + ( T + \\mu T ^ 2 ) = \\gamma , \\end{align*}"}
-{"id": "6308.png", "formula": "\\begin{align*} \\Lambda _ k ^ { \\alpha _ 2 , \\ast } \\cap \\Lambda _ l ^ { \\alpha _ 2 , \\ast } = \\emptyset \\end{align*}"}
-{"id": "6907.png", "formula": "\\begin{align*} V _ n ( z ) = \\log | g ( z ) | - U _ n ( z ) , \\ : \\ : \\ : \\ : z \\in \\mathbb H . \\end{align*}"}
-{"id": "4876.png", "formula": "\\begin{align*} \\delta ( x + y ) = \\delta ( x ) + \\delta ( y ) - \\sum _ { k = 1 } ^ { p - 1 } \\frac { 1 } { p } \\binom { p } { k } x ^ k y ^ { p - k } \\end{align*}"}
-{"id": "8687.png", "formula": "\\begin{gather*} \\nabla _ k \\nabla _ { } ^ G R _ t [ \\Phi ] ( x ) \\in L _ 2 ( K , U ) \\ ; \\ ; \\\\ \\| \\nabla _ k \\nabla _ { } ^ G R _ t [ \\Phi ] ( x ) \\| _ { L _ 2 ( K , U ) } = \\Big ( \\sum _ { m \\ge 1 } \\sup _ { | a | _ U = 1 } | \\nabla _ k \\nabla _ { a } ^ G P _ t [ \\langle \\Phi , f _ m \\rangle _ K ] ( x ) | ^ 2 \\Big ) ^ { 1 / 2 } \\\\ \\le \\frac { C } { t ^ { \\frac { 4 - 3 \\alpha } { 2 } } } | k | _ K \\ , \\| \\Phi \\| _ { C ^ { \\alpha } _ b ( H , K ) , } \\ ; \\ ; t \\in ( 0 , T ] ; \\end{gather*}"}
-{"id": "7710.png", "formula": "\\begin{align*} \\mathbb P \\left \\{ \\omega \\in \\Omega : \\xi _ i \\in [ 1 - \\varepsilon , 1 ] , \\ , \\ , i = n + 1 , \\dots , n + J \\right \\} = p _ \\varepsilon ^ J . \\end{align*}"}
-{"id": "1755.png", "formula": "\\begin{align*} w _ u = x _ 0 \\xleftarrow { \\gamma _ 1 ^ \\lor } \\cdots \\xleftarrow { \\gamma _ r ^ \\lor } x _ r = w _ { u + 1 } \\end{align*}"}
-{"id": "2822.png", "formula": "\\begin{align*} s _ { i , u } = s _ i + u q _ i , 0 \\le u \\le L _ i , L _ i = [ 1 / q _ i ] , q _ i = \\theta x _ i ^ { - \\frac { 2 } { \\alpha } } . \\end{align*}"}
-{"id": "459.png", "formula": "\\begin{align*} f _ { \\gamma } ( u ) + f _ { \\gamma } ( v ) = \\left \\lceil \\gamma ^ { \\top } u \\right \\rceil + \\left \\lceil \\gamma ^ { \\top } v \\right \\rceil = \\gamma ^ { \\top } u + \\gamma ^ { \\top } v + 1 = f _ { \\gamma } ( u + v ) , \\end{align*}"}
-{"id": "3722.png", "formula": "\\begin{align*} \\sum _ { x \\in K ^ { n } } \\psi ( \\langle x , x \\rangle ) = \\dfrac { \\alpha ( - ( - i d ) ) } { c } = \\dfrac { 1 } { c } \\end{align*}"}
-{"id": "8332.png", "formula": "\\begin{align*} z ^ { 3 } _ { 2 , 2 } - z _ { 2 , 2 } = \\frac { \\omega } { \\left ( T + 1 \\right ) ^ { 2 } } + \\frac { \\omega } { T + 1 } + \\omega ^ { 2 } + \\omega = r _ { 2 } ( T ) , \\end{align*}"}
-{"id": "65.png", "formula": "\\begin{align*} \\xi _ 2 ' ( y ) + \\frac { p } { p - 1 } \\xi _ 2 ( y ) ^ { ( p - 1 ) / p } & \\ge \\psi _ 1 ' ( y ) + M + \\frac { p } { p - 1 } \\psi _ 1 ( y ) ^ { ( p - 1 ) / p } \\\\ & \\ge M ( 1 - y ) + \\frac { p a _ 1 ^ { 2 - p } } { p - 1 } ( 1 - y ) = \\frac { p a _ 2 ^ { 2 - p } } { p - 1 } ( 1 - y ) \\\\ & = \\psi _ 2 ' ( y ) + \\frac { p } { p - 1 } \\psi _ 2 ( y ) ^ { ( p - 1 ) / p } \\ . \\end{align*}"}
-{"id": "483.png", "formula": "\\begin{align*} \\Big | \\int _ { B _ r } \\frac { \\ , 2 u ( x ) - u ( x + z ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z \\Big | & \\le N \\omega _ N \\sup _ { \\eta \\in B _ r ( x ) } | D ^ 2 u ( \\eta ) | \\ , \\int _ 0 ^ r \\frac { d r } { \\ , r ^ { 2 s - 1 } \\ , } \\\\ & = N \\omega _ N \\ , \\frac { \\ , r ^ { 2 - 2 s } \\ , } { \\ , 2 - 2 s \\ , } \\ , \\sup _ { \\eta \\in B _ r ( x ) } | D ^ 2 u ( \\eta ) | . \\end{align*}"}
-{"id": "7269.png", "formula": "\\begin{align*} \\varepsilon _ n ^ { - 1 } : = \\frac { 1 } { K } \\min _ { \\alpha \\in \\mathbb { Z } ^ r , \\ | \\alpha | = n } d ( \\mathbb { I } _ Y ^ { ( 1 ) } , N _ \\alpha ) \\end{align*}"}
-{"id": "9969.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 2 | w _ { t , k i } | ^ 2 p _ { t , i } \\le P _ { t , k } , k = 1 , 2 . \\end{align*}"}
-{"id": "8375.png", "formula": "\\begin{align*} u x = \\phi ( x ) u , \\ \\ \\ x \\in M . \\end{align*}"}
-{"id": "1231.png", "formula": "\\begin{align*} \\alpha ( x ) = \\alpha _ { \\varphi , w } ( x ) = \\sum _ { k = 1 } ^ { \\infty } \\varphi ( x ^ * ( k ) ) w ( k ) , \\end{align*}"}
-{"id": "7467.png", "formula": "\\begin{align*} \\Delta _ H ( V ) & = H ( V \\cup \\{ 0 \\} ) - H ( V ) = a _ 1 H _ 1 ( V \\cup \\{ 0 \\} ) + a _ 2 H _ 2 ( V \\cup \\{ 0 \\} ) - ( a _ 1 H _ 1 ( V ) + a _ 2 H _ 2 ( V ) ) \\\\ & = a _ 1 \\Delta _ { H _ 1 } ( V ) + a _ 2 \\Delta _ { H _ 2 } ( V ) \\end{align*}"}
-{"id": "4784.png", "formula": "\\begin{align*} H = - \\frac { \\kappa } { 2 f } \\ , n _ 1 - \\frac { f f '' + ( f ' ) ^ 2 + 1 } { 2 f \\sqrt { f '^ 2 + 1 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "7232.png", "formula": "\\begin{align*} T _ { j k } w _ k = w _ j + \\sum _ { | \\alpha | \\geq 2 } f _ { k j , \\alpha } ( z _ j ) \\cdot w _ j ^ \\alpha , \\end{align*}"}
-{"id": "8095.png", "formula": "\\begin{align*} \\wp _ { s _ i } = \\frac { 1 } { E _ { s _ i } \\sigma _ { s _ i , d } ^ 2 } , \\ ; \\ ; s _ i \\in S \\end{align*}"}
-{"id": "8303.png", "formula": "\\begin{align*} \\vert V ( G ) \\vert = \\sum _ { i = 1 } ^ m \\vert V ( G _ i ) \\vert , \\end{align*}"}
-{"id": "8989.png", "formula": "\\begin{align*} ( ~ ^ { A B } ~ _ { a } I ^ \\alpha ( L _ p ) = \\{ f : f = ~ ^ { A B } ~ _ { a } I ^ \\alpha \\varphi , ~ ~ \\varphi \\in L _ p ( a , b ) \\} . \\end{align*}"}
-{"id": "9081.png", "formula": "\\begin{align*} \\rho ^ { ( 1 ) } _ { N } ( x ) = \\rho ^ { ( 2 ) } _ { N - 1 } ( x ) - d _ { N } x ^ { ( \\alpha - 1 ) / 2 } e ^ { - x / 2 } L ^ { ( \\alpha ) } _ { N - 2 } ( x ) \\psi ( x ) \\ , \\end{align*}"}
-{"id": "4357.png", "formula": "\\begin{align*} \\Sigma _ A ' ( y ) + \\frac { p } { p - 1 } \\Sigma _ A ( y ) ^ { ( p - 1 ) / p } & = \\frac { p } { p - 1 } ( 1 - y ) \\left [ A ^ { ( p - 1 ) / p } - A ( 1 - y ) ^ { ( 2 - p ) / ( p - 1 ) } \\right ] \\\\ & \\ge \\frac { p } { p - 1 } ( 1 - y ) \\left [ A ^ { ( p - 1 ) / p } - A \\right ] \\\\ & \\ge \\frac { p } { p - 1 } ( 1 - y ) a ^ { 2 - p } = \\psi ' ( y ) + \\frac { p } { p - 1 } \\psi ( y ) ^ { ( p - 1 ) / p } \\end{align*}"}
-{"id": "8350.png", "formula": "\\begin{align*} y ^ \\beta = \\sum _ { j \\in J } m _ j ^ * y \\alpha _ { { g _ 0 } ^ { - 1 } } ( m _ j ) \\in X , \\end{align*}"}
-{"id": "6830.png", "formula": "\\begin{align*} \\sigma _ j = \\tau _ { j + 1 } - \\tau _ j , \\ ; \\ ; \\sigma _ j > 0 \\ ; j = 0 , \\ldots , m - 1 . \\end{align*}"}
-{"id": "7920.png", "formula": "\\begin{align*} r a n k ~ H _ 4 ( B ) - r a n k H _ 4 ( M ) + 1 - r a n k H _ 3 ( M ) ) + r a n k H _ 3 ( B ) = 0 . \\end{align*}"}
-{"id": "3968.png", "formula": "\\begin{align*} a _ 1 ( t ) w _ i = e ^ { ( r - 2 i ) t } w _ i , 0 \\leq i \\leq r . \\end{align*}"}
-{"id": "5909.png", "formula": "\\begin{align*} H ^ { s } _ p ( \\R ^ d ) = \\{ f \\in L ^ p ( \\R ^ d ) \\ ; : \\ ; { \\cal F } ^ { - 1 } [ | \\cdot | ^ { s } \\ , { \\cal F } f ] \\in L ^ p ( \\R ^ d ) \\} \\end{align*}"}
-{"id": "3348.png", "formula": "\\begin{align*} \\nu ( [ g , f ] ) & \\leq \\nu ( g [ f , h h _ 0 h ^ { - 1 } ] g ^ { - 1 } ) + \\nu ( [ f , h h _ 0 h ^ { - 1 } ] ^ { - 1 } ) \\\\ & = 2 \\nu ( [ f , h h _ 0 h ^ { - 1 } ] ) \\\\ & \\leq 2 ( \\nu ( f ( h h _ 0 h ^ { - 1 } ) f ^ { - 1 } ) + \\nu ( ( h h _ 0 h ^ { - 1 } ) ^ { - 1 } ) ) \\\\ & = 4 \\nu ( h h _ 0 h ^ { - 1 } ) = 4 \\nu ( h _ 0 ) . \\end{align*}"}
-{"id": "1911.png", "formula": "\\begin{align*} \\phi _ { \\mathcal { M } } ( u ) : = \\phi _ { 0 } ( u ) - \\sum _ { m \\in \\mathcal { M } } \\psi _ m ( u ) \\phi _ 0 ( u ) . \\end{align*}"}
-{"id": "8919.png", "formula": "\\begin{align*} - \\Delta \\Phi _ \\tau + V ( \\abs { x } ) \\Phi _ \\tau = \\Bigl ( \\frac { 2 \\tau ( V ( \\abs { x } ) ) ^ { 1 / 2 } + \\omega _ \\tau ( \\abs { x } ) } { \\abs { x } ^ { 1 + \\beta } } \\Bigr ) \\Phi _ \\tau , \\end{align*}"}
-{"id": "6506.png", "formula": "\\begin{align*} \\bigcap \\limits _ { i = 1 } ^ { r } B _ { i } \\not \\subseteq \\bigcup \\limits _ { i = 1 } ^ { s } A _ { j } . \\end{align*}"}
-{"id": "7150.png", "formula": "\\begin{align*} v ^ { k + 1 } = 0 \\mbox { o n } \\ , \\ , \\partial \\R ^ n _ + . \\end{align*}"}
-{"id": "8464.png", "formula": "\\begin{align*} l i m _ { ~ t \\rightarrow 0 ^ { + } ~ } | | \\tilde { \\textbf { u } } ^ { \\star } ( t ) | | _ { m } = | | \\tilde { \\textbf { u } } _ { 0 } | | _ { m } . \\end{align*}"}
-{"id": "1828.png", "formula": "\\begin{align*} g ^ { j \\bar j } = \\pi ^ { - 1 } s _ j ^ { - 3 } ( 1 + s _ j ^ 2 \\delta _ j ) , \\ 1 \\le j \\le k \\end{align*}"}
-{"id": "9367.png", "formula": "\\begin{align*} H _ { \\mathbf { T } } f = - \\frac { d ^ 2 f } { d x ^ 2 } + [ q _ 1 , q _ 2 ] \\Gamma _ 0 f = - \\{ f '' ( x ) \\} _ { x \\not = 0 } + f _ r ( 0 ) q _ 1 ( x ) - f _ r ' ( 0 ) q _ 2 ( x ) \\end{align*}"}
-{"id": "1122.png", "formula": "\\begin{align*} \\mathcal { R } _ k [ \\iota ] = \\min \\{ \\mathcal { R } ^ \\mathrm { U L } _ k [ \\iota ] , \\mathcal { R } ^ \\mathrm { D L } _ k [ \\iota ] \\} , \\end{align*}"}
-{"id": "2284.png", "formula": "\\begin{align*} \\frac 1 \\tau \\sum _ { j = 0 } ^ k \\delta _ j v _ { n - j } + A _ m v _ n = f _ n + ( A _ m - A _ n ) v _ n , \\end{align*}"}
-{"id": "4217.png", "formula": "\\begin{align*} z _ { k + 1 } - x ^ * = z _ k - x ^ * - \\gamma \\cdot G ( y _ k ) . \\end{align*}"}
-{"id": "3208.png", "formula": "\\begin{align*} \\delta \\left ( \\left \\{ \\left ( U _ j , \\sum _ { | \\alpha | = 2 , \\alpha _ 1 = 0 } F _ { j , \\alpha } ^ 1 \\cdot e _ { j , 1 } ^ * \\otimes e _ j ^ \\alpha \\right ) \\right \\} \\right ) = 0 \\end{align*}"}
-{"id": "3268.png", "formula": "\\begin{align*} S _ q ( V ) = T ( V ) / \\langle \\ker ( \\sigma _ { V V } + \\mathrm { i d } ) , \\Lambda _ q ( V ) = T ( V ) / \\langle \\ker ( \\sigma _ { V V } - \\mathrm { i d } ) . \\end{align*}"}
-{"id": "233.png", "formula": "\\begin{align*} A _ { l t } ^ { ( k ) } = \\Gamma ( j _ t ) ^ { - 1 } \\Gamma ( j _ t + 2 ( l - 1 ) / d ) k ^ { - 2 ( l - 1 ) / d } , \\end{align*}"}
-{"id": "5459.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { u } e ( m \\alpha ) \\ll \\min \\Bigl ( u ; \\frac { 1 } { \\Vert \\alpha \\Vert } \\Bigr ) . \\end{align*}"}
-{"id": "6758.png", "formula": "\\begin{align*} \\int _ { \\{ v < u \\} } ( e ^ { \\beta u } - e ^ { \\beta v } ) e ^ { - \\phi } d \\mu = 0 . \\end{align*}"}
-{"id": "5308.png", "formula": "\\begin{align*} H _ { 4 , 1 } & = U ( N _ { a , 1 } + N _ { a , 2 } + N _ { a , 3 } + N _ { a , 4 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } + N _ { a , 2 } + N _ { a , 3 } + N _ { a , 4 } - N _ { b , 1 } ) \\\\ & + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) + t _ { 2 , 1 } ( a _ { 2 } b _ { 1 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 1 } ) \\\\ & + t _ { 3 , 1 } ( a _ { 3 } b _ { 1 } ^ \\dagger + a _ { 3 } ^ \\dagger b _ { 1 } ) + t _ { 4 , 1 } ( a _ { 4 } b _ { 1 } ^ \\dagger + a _ { 4 } ^ \\dagger b _ { 1 } ) \\end{align*}"}
-{"id": "2874.png", "formula": "\\begin{align*} X \\vee Y = ( X _ - \\oplus Y _ - ) _ + ; \\end{align*}"}
-{"id": "4609.png", "formula": "\\begin{align*} f ( p ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ p | } , \\end{align*}"}
-{"id": "5242.png", "formula": "\\begin{align*} \\sigma ( A ) = \\int _ { \\Omega \\times \\Omega } \\mathcal { H } ^ 1 ( A \\cap [ x , y ] ) \\mathrm { d } \\gamma ( x , y ) \\ ; \\ ; \\ ; \\mbox { f o r e v e r y B o r e l s e t } \\ ; A \\end{align*}"}
-{"id": "9517.png", "formula": "\\begin{align*} e ^ { - 2 \\phi } \\delta ( H _ 3 \\wedge * H _ 3 ) & = 2 \\delta d B _ 2 \\wedge e ^ { - 2 \\phi } * H _ 3 \\\\ & = 2 d \\left ( \\delta B _ 2 \\wedge e ^ { - 2 \\phi } * H _ 3 \\right ) - 2 \\delta B _ 2 \\wedge d \\left ( e ^ { - 2 \\phi } * H _ 3 \\right ) . \\end{align*}"}
-{"id": "4736.png", "formula": "\\begin{align*} \\alpha : = \\inf \\{ \\pi ^ { \\top } x \\ , | \\ W ^ { i _ 0 } \\} = \\inf \\{ \\pi ^ { \\top } x \\ , | \\ x \\in W \\} , \\end{align*}"}
-{"id": "4916.png", "formula": "\\begin{align*} \\xi _ { 2 - 2 k } \\left ( \\mathcal { G } _ { Q } \\right ) & = f _ { Q } , \\\\ \\mathcal { D } ^ { 2 k - 1 } \\left ( \\mathcal { G } _ { Q } \\right ) & = - \\frac { ( 2 k - 2 ) ! } { ( 4 \\pi ) ^ { 2 k - 1 } } f _ { Q } . \\end{align*}"}
-{"id": "6994.png", "formula": "\\begin{align*} \\delta _ H \\Theta ^ 2 ( x , y ) = - \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ \\alpha _ 0 , f _ i ^ { \\lambda _ j ( x , y ) } ] - \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ \\alpha _ 0 , [ f _ i ^ y , f _ j ^ x ] ] . \\end{align*}"}
-{"id": "6557.png", "formula": "\\begin{align*} \\frac 1 \\tau \\ , \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { - 1 , q } ( \\varOmega ) ) } + \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { 1 , q } ( \\varOmega ) ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "6103.png", "formula": "\\begin{align*} \\begin{gathered} w \\left ( g ( w ) + a ( w ) + w g ( w ) a ( w ) \\right ) = 0 \\Longleftrightarrow a ( w ) = - ( 1 + w g ( w ) ) ^ { - 1 } g ( w ) \\\\ z \\left ( b ( z ) + f ( z ) + z b ( z ) f ( z ) \\right ) = 0 \\Longleftrightarrow b ( z ) = - ( 1 + z f ( z ) ) ^ { - 1 } f ( z ) \\\\ \\begin{aligned} z w \\big ( 2 e ( z , w ) + f ( z ) g ( w ) + & z f ( z ) e ( z , w ) + a ( w ) b ( z ) + w a ( w ) e ( z , w ) \\\\ + & w g ( w ) e ( z , w ) + z b ( z ) e ( z , w ) + z w e ^ 2 ( z , w ) ) \\big ) = 0 . \\end{aligned} \\end{gathered} \\end{align*}"}
-{"id": "4051.png", "formula": "\\begin{align*} q + r = m + n + l + k = q + r + 2 \\gcd ( q , r ) \\geq q + r + 2 , \\end{align*}"}
-{"id": "3527.png", "formula": "\\begin{align*} \\| ( \\psi , V ) \\| _ { \\mathcal { B } _ 0 \\times \\mathcal { B } _ 1 } & = \\| \\psi \\| _ { \\mathcal { B } _ 0 } + \\| V \\| _ { \\mathcal { B } _ 1 } \\\\ \\| ( h , w ) \\| _ { \\mathcal { B } _ 2 \\times \\mathcal { B } _ 2 } & = \\| h \\| _ { \\mathcal { B } _ 2 } + \\| w \\| _ { \\mathcal { B } _ 2 } \\\\ \\| ( f , X ) \\| _ { \\mathcal { B } _ 4 \\times \\mathcal { B } _ 3 } & = \\| f \\| _ { \\mathcal { B } _ 4 } + \\| X \\| _ { \\mathcal { B } _ 3 } . \\end{align*}"}
-{"id": "6925.png", "formula": "\\begin{align*} X ( z ) = V ( z ) e ^ { i q } z ^ { \\alpha _ 0 } \\quad \\mbox { a n d } X ^ * ( z ) = V ^ * ( z ) z ^ { - \\alpha _ 0 } e ^ { - i q } \\ , . \\end{align*}"}
-{"id": "276.png", "formula": "\\begin{align*} W _ 2 = \\int _ { \\mathcal { X } \\times \\mathcal { X } } f ( x ) f ( y ) \\int _ { l _ x } ^ { v _ x } \\int _ { l _ y } ^ { v _ y } h ( u , v ) \\ , d ( G _ { n , x , y } - F _ { n , x } F _ { n , y } ) ( u , v ) \\ , d x \\ , d y + \\frac { 1 } { n } . \\end{align*}"}
-{"id": "906.png", "formula": "\\begin{align*} Z ( t ) = \\chi ( \\tfrac { 1 } { 2 } + i t ) ^ { - 1 / 2 } \\sum _ { n \\leq C T / \\pi } { n ^ { - \\tfrac { 1 } { 2 } - i t } } + O \\left ( \\frac { T ^ { 1 / 2 } } { \\vert t \\vert } \\right ) + O ( T ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "7604.png", "formula": "\\begin{align*} P _ p ( x _ 0 ) \\ge h ^ { K _ 1 } \\left ( \\frac { \\varepsilon } l \\right ) ^ { K _ 1 } \\prod _ { i = 1 } ^ { K _ 1 } \\left ( 1 + \\kappa ( i - 1 ) \\right ) , P _ e ( x _ 0 ) \\ge h ^ { K _ 2 } \\left ( \\frac { \\delta } l \\right ) ^ { K _ 2 } \\prod _ { i = 1 } ^ { K _ 2 } \\left ( 1 + \\kappa ( i - 1 ) \\right ) . \\end{align*}"}
-{"id": "8185.png", "formula": "\\begin{align*} { \\mathbb A } ^ 2 = S p e c \\ A [ z _ 1 , z _ 2 ] , \\ \\ \\ { \\mathbb A } ^ 3 = S p e c \\ A [ x _ 1 , x _ 2 , x _ 3 ] \\end{align*}"}
-{"id": "4024.png", "formula": "\\begin{align*} \\Phi ( t ) : = \\frac { t ^ r } { \\varphi ( t ) } , t > 0 . \\end{align*}"}
-{"id": "6163.png", "formula": "\\begin{align*} \\Xi = C _ { f } f ^ { * } + g ^ { * } \\end{align*}"}
-{"id": "6919.png", "formula": "\\begin{align*} p _ n ( X ) = \\sum _ { k } x _ k { } ^ n \\ , , \\end{align*}"}
-{"id": "8524.png", "formula": "\\begin{align*} f ^ { \\ast } ( w ^ { - 1 } z ) & = f _ { 1 } ^ { \\ast } f _ { 2 } ^ { \\ast } ( w _ { 1 } ^ { - 1 } w _ { 2 } ^ { - 1 } z _ { 2 } z _ { 1 } ) \\\\ & = f _ { 1 } ^ { \\ast } ( w _ { 1 } ^ { - 1 } z _ { 1 } f _ { 2 } ^ { \\ast } ( w _ { 2 } ^ { - 1 } z _ { 2 } ) ) \\\\ & = f _ { 1 } ^ { \\ast } ( w _ { 1 } ^ { - 1 } z _ { 1 } w _ { 2 } ^ { \\ast } ( z _ { 2 } f _ { 2 } ^ { - 1 } ) ) \\\\ & = w _ { 1 } ^ { \\ast } ( z _ { 1 } f _ { 1 } ^ { - 1 } w _ { 2 } ^ { \\ast } ( z _ { 2 } f _ { 2 } ^ { - 1 } ) ) \\\\ & = ( w _ { 1 } w _ { 2 } ) ^ { \\ast } ( z _ { 1 } z _ { 2 } ( f _ { 1 } f _ { 2 } ) ^ { - 1 } ) \\\\ & = w ^ { \\ast } ( z f ^ { - 1 } ) \\end{align*}"}
-{"id": "7111.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\rho _ { n , k _ n } = D _ 1 . \\end{align*}"}
-{"id": "7266.png", "formula": "\\begin{align*} \\sum _ { 2 \\leq | \\gamma | < n } \\sum _ { | \\beta | \\geq 1 } \\max _ { j , k } \\sup _ { U _ j \\cap U _ k ^ * } \\left | F _ { k j , \\gamma , \\beta } ^ \\lambda \\right | \\cdot X ^ \\gamma \\cdot \\prod _ { \\lambda = 1 } ^ r \\left ( X ^ \\lambda + A ( X ) \\right ) ^ { \\beta _ \\lambda } . \\end{align*}"}
-{"id": "954.png", "formula": "\\begin{align*} \\Box _ { M , q } : L ^ 2 _ { ( 0 , q ) } ( M ) \\longrightarrow L ^ 2 _ { ( 0 , q ) } ( M ) , \\ ; \\Box _ { _ M , q } : = \\bar \\partial _ M \\bar \\partial ^ * _ M + \\bar \\partial ^ * _ M \\bar \\partial _ M . \\end{align*}"}
-{"id": "7334.png", "formula": "\\begin{align*} F \\psi ( v _ 1 ) = - [ 2 ] ^ { 1 / 2 } \\alpha w _ 0 , F \\psi ( v _ 0 ) = - [ 2 ] ^ { 1 / 2 } q ^ 2 \\beta w _ 1 , F \\psi ( v _ { - 1 } ) = 0 . \\end{align*}"}
-{"id": "8627.png", "formula": "\\begin{align*} a _ { i + 1 , n } = 0 = a _ { n , i + 1 } , \\end{align*}"}
-{"id": "1166.png", "formula": "\\begin{align*} K \\beta = \\begin{cases} \\frac { 1 } { 2 } b _ { x } ( 1 ) + H b ( 1 ) & \\ , 0 \\leq H < \\infty , \\\\ - \\frac { 1 } { 2 } b _ { x } ( 1 ) - \\frac { 1 } { 2 H } b _ { x x } ( 1 ) & \\ , 0 < H \\leq \\infty , \\end{cases} \\end{align*}"}
-{"id": "1041.png", "formula": "\\begin{align*} c _ { 1 , 1 } ( c _ { 1 , 1 } + c _ { 1 , 0 } ) + c _ { 1 , 0 } ~ = ~ 0 . \\end{align*}"}
-{"id": "2662.png", "formula": "\\begin{align*} n ( \\hat F _ n ( P ) - F _ { 2 : n } ) & = H _ 1 + \\sum _ { i = 2 } ^ n 1 _ { E _ i } \\left ( H _ i - H _ i ' \\right ) , \\end{align*}"}
-{"id": "1187.png", "formula": "\\begin{align*} - \\Psi _ { \\zeta \\zeta } + \\frac 1 4 \\Psi = z m \\Psi , \\Psi _ t = ( u - \\frac { 1 } { z } ) \\Psi _ { \\zeta } - \\frac { u _ { \\zeta } } { 2 } \\Psi , \\end{align*}"}
-{"id": "8674.png", "formula": "\\begin{align*} \\nabla _ k \\nabla _ { \\xi } ^ G R _ t [ \\Phi ] ( x ) = \\int _ H \\big ( \\ < \\Gamma _ t k , Q _ t ^ { - \\frac { 1 } { 2 } } y \\ > \\ , \\ < \\Gamma _ t G \\xi , Q _ t ^ { - \\frac { 1 } { 2 } } y \\ > - \\ < \\Gamma _ t k , \\Gamma _ t G \\xi \\ > \\big ) \\Phi ( e ^ { t A } x + y ) \\mu _ t ( d y ) , \\end{align*}"}
-{"id": "2902.png", "formula": "\\begin{align*} p _ n ( X ) = \\sum _ { k } x _ k { } ^ n \\ , , \\end{align*}"}
-{"id": "8448.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\sum _ { j = 1 } ^ { d } J _ { \\varepsilon } A _ { j } ( J _ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\bar { \\textbf { u } } ) ) \\partial _ { x _ { j } } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } + ( 0 , \\nabla P ^ { \\varepsilon } ) ^ { T } = 0 , \\\\ & \\nabla \\cdot \\tilde { v } ^ { \\varepsilon } = 0 , \\end{cases} \\end{align*}"}
-{"id": "6325.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } = 2 ^ { j A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 2 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "9061.png", "formula": "\\begin{align*} k _ i = \\left \\{ \\begin{array} { l l } i ^ { \\frac { 1 } { 1 0 } } , & i \\leq N ' , \\\\ ( N - i ) ^ { \\frac { 1 } { 1 0 } } + y , & N ' < i \\leq N , \\end{array} \\right . \\end{align*}"}
-{"id": "2920.png", "formula": "\\begin{align*} s _ { ( i _ 1 ) } \\ , s _ { ( i _ 2 ) } \\ , \\cdots \\ , s _ { ( i _ m ) } = \\sum _ { \\pi } \\ , c ^ \\pi _ { ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) } \\ s _ \\pi \\ , , \\end{align*}"}
-{"id": "4.png", "formula": "\\begin{align*} x _ q ^ k & = \\frac { k } { q } + \\frac { \\alpha ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) , \\\\ \\varphi _ q ^ k & = \\frac { \\mu ( x _ q ^ k ) } { q } \\left ( 1 + \\frac { \\beta _ 3 ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) \\right ) , \\end{align*}"}
-{"id": "9122.png", "formula": "\\begin{align*} u ' ( t ) + B ( t ) A ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) , \\ \\ u ( 0 ) = 0 \\end{align*}"}
-{"id": "8470.png", "formula": "\\begin{align*} \\tilde { \\textbf { u } } ^ { \\varepsilon } _ { 0 } = ( \\tilde { \\rho } ^ { \\varepsilon } _ { 0 } , \\tilde { { v } } ^ { \\varepsilon } _ { 0 } ) ^ { T } , \\end{align*}"}
-{"id": "4715.png", "formula": "\\begin{align*} \\pi ^ { \\top } z < w _ 0 + \\sum _ { i = 1 } ^ { n + 1 } \\varepsilon _ i w ^ i _ 0 & \\leq \\inf \\{ w ^ { \\top } x \\ , | \\ x \\in C \\} + \\sum _ { i = 1 } ^ { n + 1 } \\inf \\{ ( \\varepsilon _ i w ^ i ) ^ { \\top } x \\ , | \\ x \\in C \\} \\\\ & \\leq \\inf \\{ ( w ^ { \\top } + \\sum _ { i = 1 } ^ { n + 1 } \\varepsilon _ i w ^ i ) ^ { \\top } x \\ , | \\ x \\in C \\} \\leq \\pi ^ { \\top } x \\ \\ \\forall x \\in C , \\end{align*}"}
-{"id": "2422.png", "formula": "\\begin{align*} \\mathcal { M } _ { p } ( l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } , l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } ) = \\left \\{ \\{ a _ { k } \\} _ { k \\in \\mathbb { Z } ^ { n } } : ~ \\Vert \\{ a _ { k } \\lambda _ { k } \\} \\Vert _ { l _ { q _ { 2 } } ^ { s _ { 2 } , \\alpha } } \\lesssim \\Vert \\{ \\lambda _ { k } \\} \\Vert _ { l _ { q _ { 1 } } ^ { s _ { 1 } , \\alpha } } \\{ \\lambda _ { k } \\} \\in l _ { q } ^ { s _ 1 , \\alpha } \\right \\} . \\end{align*}"}
-{"id": "5219.png", "formula": "\\begin{align*} \\| j _ { q } u \\| ^ { 2 } _ { L ^ { 2 } _ { ( 0 , q ) } ( M ) } = \\| u \\| _ { L ^ { 2 } _ { ( 0 , q ) } ( M ) } ^ { 2 } \\leq \\varepsilon \\| u \\| _ { g r a p h } ^ { 2 } + C _ { \\varepsilon } \\| u \\| _ { W ^ { - 1 } _ { ( 0 , q ) } ( M ) } ^ { 2 } \\ ; . \\end{align*}"}
-{"id": "8280.png", "formula": "\\begin{align*} \\Gamma _ q & = \\{ \\gamma : \\gamma \\textrm { i s a } ( q - 1 ) \\times ( q - 1 ) \\textrm { s y m m e t r i c p o s i t i v e s e m i d e f i n i t e m a t r i x } \\} , \\\\ \\Pi & = \\{ \\pi : [ 0 , 1 ] \\to \\Gamma _ q : \\pi \\textrm { i s l e f t c o n t i n u o u s } , \\pi ( x ) \\leq \\pi ( x ' ) \\textrm { f o r } x \\leq x ' \\} , \\end{align*}"}
-{"id": "1435.png", "formula": "\\begin{align*} Q _ { n } \\cdot G = 0 . \\end{align*}"}
-{"id": "165.png", "formula": "\\begin{align*} A \\cap A ' = \\emptyset A \\cap C _ { m ' ( 0 ) } \\cap \\tau ^ { - l ( A ) - 1 } C _ { m ' ( 1 ) } \\cap \\dots \\cap \\tau ^ { - 2 ^ { k ' } } C _ { m ' ( 2 ^ { k ' } ) } , \\end{align*}"}
-{"id": "1439.png", "formula": "\\begin{align*} \\mu ( G ' ) = G , \\quad \\mu ( X ^ { i } ) = W _ { 2 n - 2 i } . \\end{align*}"}
-{"id": "5525.png", "formula": "\\begin{align*} \\mathfrak { m } _ { \\varphi _ , w } = \\{ x : p _ { \\varphi , w } ( \\delta x ) < \\infty , \\ , \\ , \\ , \\ , \\delta > 0 \\} . \\end{align*}"}
-{"id": "8030.png", "formula": "\\begin{align*} \\chi _ p ( U , M ) = \\chi _ c ( M ) + \\sum _ { s \\in S } \\dim H ^ 0 _ { D R } ( M _ s ) \\end{align*}"}
-{"id": "3650.png", "formula": "\\begin{align*} x = g ( x ) : = c T + ( 1 - c ) f ( x , \\dots , x ) , \\end{align*}"}
-{"id": "338.png", "formula": "\\begin{align*} \\widehat { \\sigma } ( x e _ { 1 1 } ) = \\widehat { \\tau _ { * } ( \\sigma ) } ( 2 ^ { - 1 } x J ) . \\end{align*}"}
-{"id": "4164.png", "formula": "\\begin{align*} & \\left ( \\frac { 1 } { 2 \\pi i } \\right ) ^ 2 \\iint \\limits _ { ( \\sigma ) ( \\gamma ) } L ^ \\nu ( s - w ) \\zeta ( w ) \\frac { \\Gamma ( w ) \\Gamma ( s - w ) } { \\Gamma ( s ) } V ( s ) X ^ s d w d s \\\\ & = \\left ( \\frac { 1 } { 2 \\pi i } \\right ) ^ 2 \\iint \\limits _ { ( \\epsilon ) ( \\gamma ) } L ^ \\nu ( s ) \\zeta ( w ) \\frac { \\Gamma ( w ) \\Gamma ( s ) } { \\Gamma ( s + w ) } V ( s + w ) X ^ { s + w } d w d s . \\end{align*}"}
-{"id": "7453.png", "formula": "\\begin{align*} ( \\xi _ 1 , \\dots , \\xi _ n ) = ( \\alpha _ 1 , \\dots , \\alpha _ n ) \\left ( D \\left [ x \\overline { v ^ { - 1 } } \\right ] _ - ^ { - 1 } \\right ) ^ t \\end{align*}"}
-{"id": "662.png", "formula": "\\begin{align*} \\nu ( \\phi ) \\nu ( C ) = \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\phi ) \\chi _ C ) \\ , d \\eta ( g ) , \\textrm { f o r a l l $ \\phi \\in C ( \\overline { Y } ) $ } . \\end{align*}"}
-{"id": "5850.png", "formula": "\\begin{align*} 0 = \\left ( \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( L _ { Y _ { I } } C _ { x } \\right ) _ { , x } + C ^ { x } L _ { Y _ { I } } C _ { x } \\right ) - \\frac { T _ { I , t } } { T _ { I } } \\left ( \\frac { 1 } { 2 } \\int L _ { Y _ { I } } C _ { x } + C ^ { x } Y _ { I } + 2 \\psi _ { I } + \\frac { 1 } { 2 } \\int Y _ { I } \\mbox { \\rm d } x \\right ) + 2 f _ { I , t } . \\end{align*}"}
-{"id": "5153.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t u + u \\cdot \\nabla u = F _ 1 ( u , \\gamma ) , & ( t , x ) \\in \\mathbb { R } \\times \\mathbb { R } ^ d , \\\\ \\partial _ t \\gamma + u \\cdot \\nabla \\gamma = F _ 2 ( u , \\gamma ) , & ( t , x ) \\in \\mathbb { R } \\times \\mathbb { R } ^ d , \\\\ u ( 0 , x ) = u _ { 0 } ( x ) , & x \\in \\mathbb { R } ^ d , \\\\ \\gamma ( 0 , x ) = \\gamma _ { 0 } ( x ) , & x \\in \\mathbb { R } ^ d , \\end{array} \\right . \\end{align*}"}
-{"id": "5047.png", "formula": "\\begin{align*} \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\phi _ 1 ) \\phi _ 2 ) \\ , d \\eta ( g ) = \\int _ G \\lambda ( ( g ^ { - 1 } \\cdot T \\phi _ 1 ) T \\phi _ 2 ) \\ , d \\eta ( g ) = \\lambda ( \\phi _ 1 ) \\ , \\lambda ( \\phi _ 2 ) = \\nu ( \\phi _ 1 ) \\nu ( \\phi _ 2 ) . \\end{align*}"}
-{"id": "4148.png", "formula": "\\begin{align*} D _ 1 ( s ) & : = \\sum _ { n \\geq 1 } \\frac { A ( 1 , n ) \\overline { A ( 1 , n ) } } { n ^ s } \\\\ D _ 2 ( s ) & : = \\sum _ { n \\geq 1 } \\frac { A ( 1 , n ) ^ 2 } { n ^ s } . \\end{align*}"}
-{"id": "6031.png", "formula": "\\begin{gather*} \\hat E _ X = E _ X + x ( x p - y ) \\partial _ a , \\hat E _ H = E _ H - \\partial _ a , \\hat E _ Y = E _ Y , \\partial _ a , \\end{gather*}"}
-{"id": "8877.png", "formula": "\\begin{align*} R _ \\# A ( x ) [ v ] = A ( R ( x ) ) [ R ( v ) ] , \\end{align*}"}
-{"id": "7762.png", "formula": "\\begin{align*} F = S \\longleftarrow \\begin{matrix} S ^ 1 ( - 1 ) \\\\ \\oplus \\\\ S ^ 1 ( - 2 ) \\end{matrix} \\longleftarrow S ^ 1 ( - 3 ) \\longleftarrow 0 . \\end{align*}"}
-{"id": "6731.png", "formula": "\\begin{align*} \\begin{cases} d Y ( t ) = - f ( B ^ { H } _ { t } , Y ( t ) , Z ( t ) ) d t - Z ( t ) \\diamond d B ^ { H } ( t ) , \\ \\ 0 \\leq t \\leq T , \\\\ Y ( T ) = g ( B ^ { H } _ { T } ) . \\end{cases} \\end{align*}"}
-{"id": "4778.png", "formula": "\\begin{align*} z ' + \\frac { 1 } { t } \\ , z = \\frac { c } { t } , \\end{align*}"}
-{"id": "7650.png", "formula": "\\begin{align*} u ( t ) = S _ { \\alpha } ( t ) u _ { 0 } + \\int _ { 0 } ^ { t } S _ { \\alpha } ( t - s ) f ( u ( s ) ) d s \\end{align*}"}
-{"id": "4656.png", "formula": "\\begin{align*} Z ( f , L ; s , \\phi ) & = \\int _ { G _ + / \\Gamma } \\chi ( g ) ^ s \\phi \\left ( g \\right ) \\sum _ { x \\in L \\setminus L _ 0 } f ( g \\cdot x ) d g \\\\ Z ( f , \\hat { L } ; s , \\phi ) & = \\int _ { G _ + / \\Gamma } \\chi ( g ) ^ s \\phi \\left ( g \\right ) \\sum _ { x \\in \\hat { L } \\setminus \\hat { L } _ 0 } f ( g \\cdot x ) d g . \\end{align*}"}
-{"id": "7019.png", "formula": "\\begin{align*} Q _ { k + 1 } ( z ) = P _ k ( z ) - z ^ { 2 ^ k } Q _ k ( z ) . \\end{align*}"}
-{"id": "33.png", "formula": "\\begin{align*} \\sum \\lambda ^ j _ \\infty \\sigma _ j \\wedge p ^ \\star ( d t ) = \\rho _ \\infty \\end{align*}"}
-{"id": "9190.png", "formula": "\\begin{align*} d X _ t = \\sqrt { X _ t } d W _ t + d W _ t ^ \\top \\sqrt { X _ t } + \\alpha I d t . \\end{align*}"}
-{"id": "7885.png", "formula": "\\begin{align*} p ^ { \\kappa } ( t , x , y ) : = p _ y ( t , x - y ) + \\phi _ y ( t , x ) = p _ y ( t , x - y ) + \\int _ 0 ^ t \\int _ { \\R ^ d } p _ z ( t - s , x - z ) q ( s , z , y ) \\ , d z \\ , d s \\ , . \\end{align*}"}
-{"id": "9537.png", "formula": "\\begin{align*} g _ M ( R ^ M ( \\bar { X } , \\bar { Y } ) \\bar { Z } , \\bar { W } ) & = g ^ \\phi _ N ( R ^ { N \\phi } ( X , Y ) Z , W ) + 2 g _ M ( A _ { \\bar { X } } \\bar { Y } , A _ { \\bar { Z } } \\bar { W } ) \\\\ & - g _ M ( A _ { \\bar { Y } } \\bar { Z } , A _ { \\bar { X } } \\bar { W } ) - g _ M ( A _ { \\bar { Z } } \\bar { X } , A _ { \\bar { Y } } \\bar { W } ) . \\end{align*}"}
-{"id": "5279.png", "formula": "\\begin{align*} { } & \\sum _ { n = 1 } ^ { \\infty } \\tfrac { 1 } { n ^ { 1 + \\delta } } \\int _ { T } ^ { 2 T } \\left ( { \\tfrac { t } { 2 \\pi } } \\right ) ^ { { 1 } / { 4 } + { \\delta } / { 2 } + { i t } / { 2 } } e ^ { - { i t } / { 2 } - i t \\log n } d t \\\\ { } & = \\sum _ { n = 1 } ^ { \\infty } \\tfrac { 1 } { n ^ { 1 + \\delta } } \\int _ { T } ^ { 2 T } \\left ( { \\tfrac { t } { 2 \\pi } } \\right ) ^ { { 1 } / { 4 } + { \\delta } / { 2 } } e ^ { \\frac { i t } { 2 } \\log ( { t } / { 2 \\pi e n ^ { 2 } ) } } d t . \\end{align*}"}
-{"id": "7918.png", "formula": "\\begin{align*} v _ j = C _ j u _ j \\end{align*}"}
-{"id": "8807.png", "formula": "\\begin{align*} I ( u ^ - , u ^ - ) = n H \\int _ { \\Sigma } u ^ - d v o l _ { \\Sigma } \\end{align*}"}
-{"id": "5433.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - 1 & \\alpha & 0 & \\overline \\alpha \\\\ 0 & q & - q \\alpha & 0 & - q \\overline \\alpha \\\\ 0 & - \\overline \\alpha & q & - \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & 0 \\\\ 0 & 0 & - \\overline \\alpha J ( T ^ { - r } , T ^ { - r } ) & q & - \\alpha J ( T ^ { - 3 r } , T ^ { - 3 r } ) \\\\ 0 & - \\alpha & 0 & - \\overline \\alpha J ( T ^ { - 2 r } , T ^ { - r } ) & q \\end{bmatrix} \\end{align*}"}
-{"id": "8540.png", "formula": "\\begin{align*} \\pi _ { r _ { 1 } a _ { 1 } } \\xi _ { r a } = \\delta _ { r _ { 1 } a _ { 1 } , r a } i d _ { N _ { \\tau ( a ) } } \\end{align*}"}
-{"id": "7369.png", "formula": "\\begin{align*} \\frac { c _ { k + 1 } } { c _ { k } } \\Gamma _ { - } ^ { ( k + 1 ) } \\Gamma _ { 0 } ^ { ( k + 1 ) * } = \\frac { c _ { k } } { c _ { k - 1 } } ( t \\Gamma _ { 0 } ^ { ( k ) * } \\Gamma _ { - } ^ { ( k ) } + t ^ { \\prime } \\Gamma _ { + } ^ { ( k ) * } \\Gamma _ { 0 } ^ { ( k ) } ) . \\end{align*}"}
-{"id": "6025.png", "formula": "\\begin{gather*} \\xi ^ a = \\mu _ b \\phi ^ { b a } , \\\\ I _ { a b } = 3 \\mu ^ c \\mu _ { [ c } \\phi _ { a b ] } = - \\varepsilon \\phi _ { a b } - 2 \\mu _ { [ a } \\xi _ { b ] } , \\\\ J _ { a b } = 3 \\mu ^ c \\phi _ { [ c a } \\theta _ { b ] } = \\mu ^ c ( \\ast \\phi ) _ { c a b } , \\\\ K _ { a b } = - 2 \\mu ^ c \\mu _ { [ a } \\phi _ { b ] c } = 2 \\mu _ { [ a } \\xi _ { b ] } . \\end{gather*}"}
-{"id": "4026.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } - u '' + c u = f \\mbox { i n } ( 0 , 1 ) \\\\ u ( 0 ) = g _ 0 , u ( 1 ) = g _ 1 \\end{array} \\right . \\end{align*}"}
-{"id": "949.png", "formula": "\\begin{align*} ( f , g ) ( z ) = \\sideset { } { ' } \\sum _ { | J | = q } f _ { J } ( z ) \\overline { g _ { J } ( z ) } \\ , . \\end{align*}"}
-{"id": "7988.png", "formula": "\\begin{align*} X : x _ 0 ^ 4 + x _ 1 x _ 2 ^ 3 + x _ 1 ^ 3 x _ 3 + x _ 0 x _ 2 x _ 3 ^ 2 = 0 . \\end{align*}"}
-{"id": "9202.png", "formula": "\\begin{align*} \\mathbb { H } = \\{ \\mathcal { H } _ t \\} _ { 0 \\leq t \\leq T } , \\mathcal { H } _ t = \\mathcal { F } _ t \\vee \\sigma ( Z ) \\end{align*}"}
-{"id": "2311.png", "formula": "\\begin{align*} \\tau d _ n = \\sum \\limits ^ k _ { i = 0 } \\delta _ i u ( t _ { n - i } ) + \\tau A ( t _ n ) u ( t _ n ) - \\tau \\sum \\limits ^ { k - 1 } _ { i = 0 } \\gamma _ i B \\big ( t _ { n - i - 1 } , u ( t _ { n - i - 1 } ) \\big ) , \\end{align*}"}
-{"id": "5049.png", "formula": "\\begin{align*} ( P \\phi ) ( g ) = \\int _ Z \\phi ( g \\cdot z ) \\ , d m ( z ) , \\end{align*}"}
-{"id": "5087.png", "formula": "\\begin{align*} & q ^ { N _ { a } } \\beta _ { a } = q ^ { - 1 } \\beta _ { a } q ^ { N _ { a } } , q ^ { N _ { a } } \\beta ^ { * } _ { a } = q \\ , \\beta _ { a } ^ { * } q ^ { N _ { a } } , \\\\ & \\beta _ { a } \\beta _ { a } ^ { * } = 1 - q ^ { 2 } q ^ { 2 N _ { a } } , \\beta _ { a } ^ { * } \\beta _ { a } = 1 - q ^ { 2 N _ { a } } \\end{align*}"}
-{"id": "1017.png", "formula": "\\begin{align*} u = M ( \\lambda , u ) . \\end{align*}"}
-{"id": "8177.png", "formula": "\\begin{align*} \\phi _ n = \\sum _ { k = 1 } ^ { n } \\binom { ( \\alpha + \\beta ) n } { k - 1 } \\frac { ( k - 1 ) ! } { n ! } B _ { n , k } ( 1 ! \\theta _ 1 , 2 ! \\theta _ 2 , \\dots ) , \\end{align*}"}
-{"id": "9936.png", "formula": "\\begin{align*} 2 ^ * : = \\frac { 2 n } { n - 2 s } \\end{align*}"}
-{"id": "5577.png", "formula": "\\begin{align*} b _ { x x x } & = 0 , x \\neq x _ i , \\\\ z \\dot m _ i & = \\frac 1 2 [ b _ { x x } ] ( x _ i ) + z m _ i \\langle b _ x \\rangle ( x _ i ) \\\\ - z m _ i \\dot x _ i & = \\frac 1 2 [ b _ x ] ( x _ i ) + z m _ i b ( x _ i ) , \\end{align*}"}
-{"id": "8889.png", "formula": "\\begin{align*} \\mathcal { I } _ { A } ( u ) = \\bigl ( \\tfrac { 1 } { 2 } - \\tfrac { 1 } { p } \\bigr ) \\int _ { \\R ^ N } \\abs { u } ^ p . \\end{align*}"}
-{"id": "1564.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } S ^ { 2 } F _ { , S S } + \\kappa \\left ( \\mu - \\lambda - \\log S \\right ) S F _ { , S } - F _ { , t } = 0 , \\end{align*}"}
-{"id": "9960.png", "formula": "\\begin{align*} \\gamma = \\frac { 1 + \\sqrt { 5 } } { 2 } \\cong 1 . 6 1 8 \\end{align*}"}
-{"id": "5802.png", "formula": "\\begin{align*} \\mu ( G ' ) = G , \\quad \\mu ( X ^ { i } ) = W _ { 2 n - 2 i } . \\end{align*}"}
-{"id": "9562.png", "formula": "\\begin{align*} \\begin{aligned} \\lambda _ { n i } ^ { ( 1 ) } & = \\langle \\psi ^ { ( 0 ) } _ { n } , \\Lambda _ { n } ^ { ( 1 ) } \\psi ^ { ( 0 ) } _ { n } \\rangle ~ ~ ~ ~ ~ ~ ~ ~ \\psi ^ { ( 0 ) } _ { n } \\in P _ { n i j } ^ { ( 2 ) } \\mathcal { H } \\subseteq P _ { n i } ^ { ( 1 ) } \\mathcal { H } \\\\ \\lambda _ { n i j } ^ { ( 2 ) } & = \\langle \\psi ^ { ( 0 ) } _ { n } , \\Lambda _ { n i } ^ { ( 2 ) } \\psi ^ { ( 0 ) } _ { n } \\rangle ~ ~ ~ ~ ~ ~ ~ ~ \\psi ^ { ( 0 ) } _ { n } \\in P _ { n i j } ^ { ( 2 ) } \\mathcal { H } \\end{aligned} \\end{align*}"}
-{"id": "9450.png", "formula": "\\begin{align*} \\mu ( A ) & = \\lim _ { \\omega } \\frac { \\dim ( ( a ^ { - 1 } W _ i + W _ i ) \\cap A ) } { \\dim ( W _ i ) } \\ge \\lim _ { \\omega } \\frac { \\dim ( a ^ { - 1 } W _ i \\cap A ) } { \\dim ( W _ i ) } \\\\ & = \\lim _ { \\omega } \\frac { \\dim ( a ( a ^ { - 1 } W _ i ) \\cap a A ) } { \\dim ( W _ i ) } = \\mu ( a A ) . \\end{align*}"}
-{"id": "6309.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } T _ m f \\| _ { M _ 2 } = & \\| \\Box _ k ^ { \\alpha _ 1 } T _ m \\Box _ k ^ { \\alpha _ 1 , \\ast } f \\| _ { M _ 2 } \\\\ \\lesssim & \\| \\Box _ k ^ { \\alpha _ 1 } T _ m \\| _ { M _ 1 \\rightarrow M _ 2 } \\| \\Box _ k ^ { \\alpha _ 1 , \\ast } f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "8774.png", "formula": "\\begin{align*} \\kappa _ 0 ^ { ( 2 r + 1 ) } ( x ) = \\frac { t ^ { 2 r } \\left ( t - x ^ 2 \\right ) h _ 0 \\left ( a , c , x / t ^ { r } \\right ) } { \\left ( t ^ { 2 r + 1 } - x ^ 2 \\right ) h _ 0 ( a , c , x ) } , \\kappa _ n ^ { ( 2 r + 1 ) } ( x ) = \\frac { \\left ( t - x ^ 2 \\right ) h _ n \\left ( b , d , t ^ r / x \\right ) } { \\left ( t ^ { 2 r + 1 } - x ^ 2 \\right ) h _ n \\left ( b , d , 1 / x \\right ) } . \\end{align*}"}
-{"id": "6048.png", "formula": "\\begin{align*} \\underset { u _ { i } ( y ) > 0 } { \\underset { y \\rightarrow x } { \\lim } } \\ , \\nabla u _ { i } ( y ) = - \\underset { u _ { j } ( y ) > 0 } { \\underset { y \\rightarrow x } { \\lim } } \\ , \\nabla u _ { j } ( y ) . \\end{align*}"}
-{"id": "8159.png", "formula": "\\begin{align*} \\phi _ h ( S _ 2 ^ 2 ) = \\frac { t ( p _ i - t ) } { p _ i - 1 } . \\end{align*}"}
-{"id": "822.png", "formula": "\\begin{align*} F _ { \\vec { x } } ( u _ { 1 } , \\ldots , u _ { k } ) = \\langle \\prod _ { 1 \\le i \\le k } \\tilde { C } ^ { [ 0 , M ] } ( u _ { i } ; s ) \\prod _ { 1 \\le i \\le k } \\beta _ { x _ { i } } ^ { * } \\rangle _ { [ 0 , M ] } , \\end{align*}"}
-{"id": "1237.png", "formula": "\\begin{align*} \\frac { 1 } { 2 ^ j } \\leq \\varphi ( t _ { n _ j } ) \\sum _ { k = 1 } ^ { m ( E _ j ) } w ( k ) \\leq \\frac { 1 } { 2 ^ { j - 2 } } . \\end{align*}"}
-{"id": "1527.png", "formula": "\\begin{align*} D = \\int \\lambda ( v ) \\otimes v \\ , d v , Q ( \\lambda ) = \\nabla _ v M ( v ) . \\end{align*}"}
-{"id": "2773.png", "formula": "\\begin{align*} X _ n \\to \\cdots \\to X _ 1 \\to X _ 0 = X \\end{align*}"}
-{"id": "7500.png", "formula": "\\begin{align*} Q _ 0 = \\sum _ { i = 2 } ^ { \\kappa ' } ( i - 1 ) Q _ i = Q _ 2 + 2 Q _ 3 + 3 Q _ 4 + \\cdots + ( \\kappa ' - 1 ) Q _ { \\kappa ' } . \\end{align*}"}
-{"id": "8790.png", "formula": "\\begin{align*} T _ n \\sum _ \\mu f _ \\mu & = \\sum _ { \\mu : \\ \\mu _ n < 0 } \\left ( t _ n f _ { s _ n \\mu } + ( t _ n - 1 ) f _ \\mu \\right ) + \\sum _ { \\mu : \\ \\mu _ n = 0 } t _ n f _ \\mu + \\sum _ { \\mu : \\ \\mu _ n > 0 } f _ { s _ n \\mu } \\\\ & = \\sum _ { \\mu : \\ \\mu _ n < 0 } t _ n f _ { s _ n \\mu } + \\sum _ { \\mu : \\ \\mu _ n \\le 0 } t _ n f _ \\mu = t _ n \\sum _ \\mu f _ \\mu . \\end{align*}"}
-{"id": "150.png", "formula": "\\begin{align*} \\| V a _ n - x \\| _ { C _ { \\widehat { E } } } = \\| V a _ n - V S ( x ) \\| _ { C _ { \\widehat { E } } } = \\| a _ n - S ( x ) \\| _ { \\widehat { E } } \\to 0 , \\ \\ \\ \\ n \\to \\infty . \\end{align*}"}
-{"id": "5310.png", "formula": "\\begin{align*} R _ { 1 2 } ( u - v ) L ^ X _ 1 ( u ) L ^ X _ 2 ( v ) = L ^ X _ 2 ( v ) L ^ X _ 1 ( u ) R _ { 1 2 } ( u - v ) , \\end{align*}"}
-{"id": "2844.png", "formula": "\\begin{align*} l _ k ^ { \\tau } ( x _ 1 , . . . , x _ k ) = \\sum _ { i \\geq 0 } \\frac { 1 } { i ! } l _ { k + i } ( \\underbrace { \\tau , . . . , \\tau } _ i , x _ 1 , . . . , x _ k ) \\end{align*}"}
-{"id": "7758.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\mathbb P \\left [ T _ N ( n ) \\ge - x \\right ] = \\lim _ { n \\to \\infty } \\left \\{ 1 - \\mathbb P \\left [ T _ N ( n ) \\le - x \\right ] \\right \\} = 1 - \\mathbb P \\left [ T _ N \\le - x \\right ] = \\mathbb P \\left [ T _ N \\ge - x \\right ] . \\end{align*}"}
-{"id": "9408.png", "formula": "\\begin{align*} A = \\begin{bmatrix} A _ { F F } & A _ { R F } ^ T \\\\ A _ { R F } & A _ { R R } \\end{bmatrix} , \\end{align*}"}
-{"id": "4907.png", "formula": "\\begin{align*} F _ { \\kappa , m , N } : = \\mathcal { F } _ { \\kappa , m , N } \\Big | \\operatorname { p r } \\in H _ { \\kappa } ( N ) . \\end{align*}"}
-{"id": "1396.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { t = 1 } ^ n \\ddot { l } _ t ( \\delta ) - \\frac { 1 } { n } \\sum _ { t = 1 } ^ n E ( \\ddot { l } _ t ( \\delta ) ) \\overset { p } \\to 0 , \\Vert \\delta - \\delta _ 0 \\Vert \\le \\epsilon \\end{align*}"}
-{"id": "8931.png", "formula": "\\begin{align*} \\lambda _ 1 \\Biggl ( \\begin{pmatrix} 0 & t \\\\ t & 0 \\end{pmatrix} \\Biggr ) = - \\abs { t } \\end{align*}"}
-{"id": "6653.png", "formula": "\\begin{align*} \\beta _ j = ( d _ j - 1 ) \\dim V _ j ^ I + d _ j [ a ( V _ j ) + n ( \\psi ) \\dim ( V _ j ) ] \\end{align*}"}
-{"id": "7437.png", "formula": "\\begin{align*} g B _ + = x B _ + = B _ 1 g B _ - = x b w _ 0 B _ + = y B _ + = B _ 2 . \\end{align*}"}
-{"id": "5353.png", "formula": "\\begin{align*} ( n - \\chi _ { \\gamma } ( c _ i ) ) ( n - \\chi _ { \\gamma } ( c _ j ) ) = | n - \\chi _ { \\gamma } ( c _ i ) | ^ 2 > 0 \\end{align*}"}
-{"id": "7951.png", "formula": "\\begin{align*} W _ l ( r ) = \\alpha j _ l ( \\sqrt [ 4 ] { \\mu } r ) + \\beta i _ l ( \\sqrt [ 4 ] { \\mu } r ) , \\end{align*}"}
-{"id": "2222.png", "formula": "\\begin{align*} \\| \\partial _ { t } ^ { \\alpha } ( \\tilde { u } ( \\cdot , t ) - \\tilde { u } ( \\cdot , 0 ) ) \\| _ { L ^ { 2 } ( \\Omega ) } & = \\| \\partial _ { t } ^ { \\alpha } \\tilde { u } ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\Omega ) } = \\| \\partial _ { t } ( g _ { 1 - \\alpha } * \\mu * v ) ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\Omega ) } \\\\ \\leq & \\| \\mu \\| _ { L ^ { 1 } ( 0 , T ) } ( \\| \\partial _ { t } ^ { \\alpha } ( v - g ) \\| _ { L ^ { \\infty } ( ( 0 , T ] ; L ^ { 2 } ( \\Omega ) ) } + g _ { 1 - \\alpha } ( t ) \\| g \\| _ { L ^ { 2 } ( \\Omega ) } ) . \\end{align*}"}
-{"id": "9516.png", "formula": "\\begin{align*} \\delta \\left ( G _ 2 \\wedge * G _ 2 \\right ) & = 2 \\delta d C _ 1 \\wedge * G _ 2 \\\\ & = 2 d ( \\delta C _ 1 \\wedge * G _ 2 ) + 2 \\delta C _ 1 \\wedge d * G _ 2 \\end{align*}"}
-{"id": "3248.png", "formula": "\\begin{gather*} K _ i K _ i ^ { - 1 } = K _ i ^ { - 1 } K _ i = 1 , \\ \\ K _ i K _ j = K _ j K _ i , \\\\ K _ i E _ j K _ i ^ { - 1 } = q _ i ^ { a _ { i j } } E _ j , \\ \\ K _ i F _ j K _ i ^ { - 1 } = q _ i ^ { - a _ { i j } } F _ j , \\\\ E _ i F _ j - F _ j E _ i = \\delta _ { i j } \\frac { K _ i - K _ i ^ { - 1 } } { q _ i - q _ i ^ { - 1 } } , \\end{gather*}"}
-{"id": "2262.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { ( m ) } = \\frac { \\mu _ { ( m ) } ^ { G } + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } \\mu _ { ( k + m ) } ^ { G } } { 1 + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } \\mu _ { ( k ) } ^ { G } } = \\frac { ( m - 1 ) ! ! \\sigma ^ { m } + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } ( k + m - 1 ) ! ! \\sigma ^ { k + m } } { 1 + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } ( k - 1 ) ! ! \\sigma ^ { k } } . \\end{align*}"}
-{"id": "5692.png", "formula": "\\begin{align*} [ w ] _ { A _ \\infty } : = \\lim _ { r \\downarrow 0 } [ w ] _ { A _ r } . \\end{align*}"}
-{"id": "2139.png", "formula": "\\begin{align*} & C ^ { - 1 } \\Lambda ^ { - 1 } \\int _ { B } \\int _ { B } ( v ( x ) - v ( y ) ) ^ { 2 } k _ { 0 } ( x , y ) d x d y \\leq c _ { n , \\beta } \\int _ { B } \\int _ { B } \\frac { ( v ( x ) - v ( y ) ) ^ { 2 } } { | x - y | ^ { n + 2 \\beta } } d x d y \\\\ & \\leq C \\Lambda \\int _ { B } \\int _ { B } ( v ( x ) - v ( y ) ) ^ { 2 } k _ { 0 } ( x , y ) d x d y , B = B _ { \\rho } ( x _ { 0 } ) . \\end{align*}"}
-{"id": "3211.png", "formula": "\\begin{align*} \\delta \\left ( \\left \\{ \\left ( U _ j , \\sum _ { | \\alpha | = n , \\alpha _ 1 = 0 } F _ { j , \\alpha } ^ 1 \\cdot e _ { j , 1 } ^ * \\otimes e _ j ^ \\alpha \\right ) \\right \\} \\right ) = \\left \\{ \\left ( U _ { j k } , \\sum _ { | \\alpha | = n , \\alpha _ 1 = 0 } ( h _ { 1 , j k , \\alpha } ^ 1 - h _ { 2 , j k , \\alpha } ^ 1 ) \\cdot e _ { j , 1 } ^ * \\otimes e _ j ^ \\alpha \\right ) \\right \\} \\end{align*}"}
-{"id": "1913.png", "formula": "\\begin{align*} J _ { n , m } : = \\big ( [ - L _ n - 1 2 m - 8 , - L _ n - 1 2 m + 8 ] \\cup [ L _ n + 1 2 m - 8 , L _ n + 1 2 m + 8 ] \\big ) \\setminus I _ { n , m } , \\end{align*}"}
-{"id": "3523.png", "formula": "\\begin{align*} | \\nabla ^ j ( u \\rho ^ { \\frac { 1 } { 2 } } ) | & \\le | \\nabla ^ j u | \\rho ^ { \\frac { 1 } { 2 } } + \\sum _ { i = 1 } ^ { j } C _ { i j } | \\nabla ^ { j - i } u | \\ ; | \\nabla ^ i ( \\rho ^ { \\frac { 1 } { 2 } } ) | \\\\ & \\le | \\nabla ^ j u | \\rho ^ { \\frac { 1 } { 2 } } + C \\sum _ { i = 1 } ^ { j } N ^ i | \\nabla ^ { j - i } u | d ^ { - 2 i } \\rho ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "4147.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } A _ f ( n ) = O \\left ( X ^ { \\frac { 1 } { 4 } + ( \\frac { 1 } { 2 } + \\epsilon ) \\eta } \\right ) + O \\left ( \\sum _ { X \\leq n \\leq X ' } | A _ f ( n ) | \\right ) , \\end{align*}"}
-{"id": "3325.png", "formula": "\\begin{align*} X \\triangleright \\gamma _ + ( v ) = \\gamma _ + ( X \\triangleright v ) , X \\triangleright \\gamma _ - ( w ) = \\gamma _ - ( X \\triangleright w ) . \\end{align*}"}
-{"id": "8361.png", "formula": "\\begin{align*} E _ { N _ 0 } ( g ) = g w _ g = \\alpha _ g ( w _ g ) g \\in I _ g g . \\end{align*}"}
-{"id": "492.png", "formula": "\\begin{align*} D ^ { k + 1 } F _ N ( r ) & = D \\Big [ \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\ , \\alpha _ { j , k } \\ , r ^ { 2 j - k } \\ , F _ { N + 2 j } ( r ) \\Big ] \\\\ & = \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\ , ( 2 j - k ) \\ , \\alpha _ { j , k } \\ , r ^ { 2 j - k - 1 } \\ , F _ { N + 2 j } ( r ) - \\ ! \\ ! \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\ , \\alpha _ { j , k } \\ , r ^ { 2 j - k + 1 } \\ , F _ { N + 2 j + 2 } ( r ) . \\\\ \\noalign { \\medskip } & = : S _ 1 + S _ 2 . \\end{align*}"}
-{"id": "5106.png", "formula": "\\begin{align*} & | \\mathrm { v a c } \\rangle _ { [ M ' , M ] } = | \\mathbf { 0 } \\rangle \\otimes | \\mathbf { 0 } \\rangle \\otimes \\cdots \\otimes | \\mathbf { 0 } \\rangle \\in \\mathcal { F } ^ { [ M ' , M ] } , \\\\ & { } _ { [ M ' , M ] } \\langle \\mathrm { v a c } | = \\langle \\mathbf { 0 } | \\otimes \\langle \\mathbf { 0 } | \\otimes \\cdots \\otimes \\langle \\mathbf { 0 } | \\in ( \\mathcal { F ^ { * } } ) ^ { [ M ' , M ] } . \\end{align*}"}
-{"id": "6143.png", "formula": "\\begin{align*} - d ( \\phi ^ a \\cdot \\iota _ a \\Theta ) = - \\omega ^ a \\wedge \\iota _ a \\Theta \\ ; + \\ ; \\frac 1 2 f ^ c _ { a b } \\ , \\phi ^ a \\wedge \\phi ^ b \\cdot \\iota _ c \\Theta \\ ; - \\ ; \\phi ^ a \\wedge \\iota _ a \\Omega \\ ; + \\ ; \\phi ^ a \\wedge [ \\iota _ a \\Theta , \\Theta ] . \\end{align*}"}
-{"id": "3037.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } Z _ n = Z _ \\infty \\lim _ { n \\to \\infty } W _ n = 0 \\end{align*}"}
-{"id": "1784.png", "formula": "\\begin{align*} f \\colon Y _ { E , S } \\to X _ { E , S } = Y _ { E , S } / \\varphi ^ { \\Z } \\end{align*}"}
-{"id": "8720.png", "formula": "\\begin{align*} u ( t , x ) - v ( t , x ) = \\int _ t ^ T R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } \\big ( \\nabla ^ G u ( s , \\cdot ) B ( s , \\cdot ) - \\nabla ^ G v ( s , \\cdot ) B ( s , \\cdot ) \\big ) \\right ] ( x ) \\ , d s \\end{align*}"}
-{"id": "7812.png", "formula": "\\begin{align*} w _ { n n } = \\frac { f _ { i } f _ { j } } { \\left \\vert \\nabla f \\right \\vert ^ { 2 } } \\left ( u _ { i j } S ^ { - 2 } - 4 u _ { i } S _ { j } S ^ { - 3 } + 6 S _ { i } S _ { j } S ^ { - 4 } u - 2 S _ { i j } S ^ { - 3 } u \\right ) . \\end{align*}"}
-{"id": "2866.png", "formula": "\\begin{align*} F ^ c ( \\int _ { \\underline { n } \\in \\Delta } ^ { C h _ { \\mathbb { K } } } K ^ n ( C ) \\otimes \\overline { N ^ * } ( \\Delta ^ n ) ) = \\int _ { \\underline { n } \\in \\Delta } ^ { d g C o g ^ { c o n i l } } F ^ c ( K ^ n ( C ) \\otimes \\overline { N ^ * } ( \\Delta ^ n ) ) ) \\end{align*}"}
-{"id": "8287.png", "formula": "\\begin{align*} \\partial _ { i _ 1 , \\cdots , i _ p } G & = \\frac { p ! } { n } \\ , \\ < f \\ > , \\\\ \\partial _ { i _ 1 , \\cdots , i _ p } ^ 2 G & = \\frac { \\beta ( p ! ) ^ 2 } { n } \\ , [ \\ < f ^ 2 \\ > - \\ < f \\ > ^ 2 ] , \\\\ \\partial _ { i _ 1 , \\cdots , i _ p } ^ 3 G & = \\frac { \\beta ^ 2 ( p ! ) ^ 3 } { n } \\ , [ \\ < f ^ 3 \\ > - 3 \\ < f ^ 2 \\ > \\ < f \\ > + 2 \\ < f \\ > ^ 3 ] . \\end{align*}"}
-{"id": "7259.png", "formula": "\\begin{align*} H _ { j k , n } : = \\sum _ { \\lambda = 1 } ^ r \\sum _ { | \\alpha | = n } \\left ( h _ { 1 , j k , \\alpha } - h _ { 2 , j k , \\alpha } \\right ) \\cdot e _ { j , \\lambda } ^ * \\otimes e _ j ^ \\alpha , \\end{align*}"}
-{"id": "1919.png", "formula": "\\begin{align*} F ^ \\alpha = d \\sigma _ { \\alpha } = \\sum _ { i = 1 } ^ m \\ , q _ { \\alpha i } \\ , \\omega _ i . \\end{align*}"}
-{"id": "6505.png", "formula": "\\begin{align*} U ( n , d , q ) \\leq d q ^ { d } ( l n \\frac { n } { d } + l n q ) = O ( d q ^ { d } \\log n ) . \\end{align*}"}
-{"id": "7831.png", "formula": "\\begin{align*} \\mathcal { M } ^ A _ { \\mathrm { C o u l o m b } } = \\mathcal { M } ^ B _ { \\mathrm { H i g g s } } , \\ \\ \\ \\mathcal { M } ^ A _ { \\mathrm { H i g g s } } = \\mathcal { M } ^ B _ { \\mathrm { C o u l o m b } } . \\end{align*}"}
-{"id": "2526.png", "formula": "\\begin{align*} \\dim ( \\vec { c } \\ , ) = h ( \\vec { c } \\ , ) + k - 1 = \\sum _ { i = 1 } ^ { k } h ( c _ { i } ) + 2 k - 2 . \\end{align*}"}
-{"id": "3491.png", "formula": "\\begin{align*} \\hat { f } = \\widetilde { f } = \\chi _ { \\widetilde { D } } = 1 , \\ , \\ , \\ , \\ , \\{ \\hat { v } < \\alpha \\} . \\end{align*}"}
-{"id": "3651.png", "formula": "\\begin{align*} g ( M ) = c T + ( 1 - c ) f ( M , \\dots , M ) \\le c T + ( 1 - c ) M \\le M . \\end{align*}"}
-{"id": "3775.png", "formula": "\\begin{align*} x f ( x ) = \\int _ 0 ^ 1 f ( x - z ) z g ( z ) \\ d z \\end{align*}"}
-{"id": "5031.png", "formula": "\\begin{align*} \\eta \\Big ( \\bigcap _ { l \\in L } l F A B _ { y _ o } \\Big ) \\geq \\nu ( B ) = \\eta ( B _ { y _ o } ) , \\end{align*}"}
-{"id": "5801.png", "formula": "\\begin{align*} W _ { 2 i } ( z ) = \\sum _ { 1 \\leq j _ { 1 } < \\cdots < j _ { n - i } \\leq n } : b ^ { 2 } _ { j _ { 1 } } ( z ) \\cdots b ^ { 2 } _ { j _ { n - i } } ( z ) : + \\cdots \\end{align*}"}
-{"id": "7675.png", "formula": "\\begin{align*} \\| u _ { n } \\| _ { L ^ { q } } ^ { q } = \\frac { 1 } { \\nu ( d , \\alpha ) ^ { q } } \\omega _ { d } n ^ { - \\alpha q } , \\end{align*}"}
-{"id": "8249.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial x } { \\partial s _ 1 } = \\Re ( f _ 1 ( 0 , 0 , s _ 3 ) + f _ 2 ( 0 , 0 , s _ 3 ) ) \\\\ \\frac { \\partial y } { \\partial s _ 1 } = \\Im ( f _ 1 ( 0 , 0 , s _ 3 ) - f _ 2 ( 0 , 0 , s _ 3 ) ) \\\\ \\frac { \\partial t } { \\partial s _ 1 } = \\Re g ( 0 , 0 , s _ 3 ) , \\end{cases} \\end{align*}"}
-{"id": "7169.png", "formula": "\\begin{align*} b _ { j k } = - \\int _ { \\R ^ n } y _ k f _ j ( y ) d y \\quad j \\neq k , b _ { j j } = \\int _ { \\R ^ n } ( y _ n f _ n ( y ) - y _ j f _ j ( y ) ) d y . \\end{align*}"}
-{"id": "5526.png", "formula": "\\begin{align*} H ( x ) = \\sum _ { k = 1 } ^ \\infty x ( k ) h ( k ) , \\ \\ \\ \\ x \\in \\lambda _ { \\varphi , w } , \\end{align*}"}
-{"id": "6964.png", "formula": "\\begin{align*} h ( \\sigma , \\epsilon ) = \\max _ { \\mu _ 1 \\in \\bar { M } _ { \\epsilon } } g ( \\sigma , \\mu _ 1 ) . \\end{align*}"}
-{"id": "3637.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } u _ n = K , \\end{align*}"}
-{"id": "7709.png", "formula": "\\begin{align*} \\mathbb P \\{ \\Omega _ \\varepsilon ( n ) \\} = p _ \\varepsilon > 0 , \\Omega _ \\varepsilon ( n ) : = \\{ \\omega \\in \\Omega : \\xi _ n ( \\omega ) \\in [ 1 - \\varepsilon , 1 ] \\} . \\end{align*}"}
-{"id": "7524.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\infty \\mathbb { P } [ B _ i ] = \\infty . \\end{align*}"}
-{"id": "837.png", "formula": "\\begin{align*} \\prod _ { 1 \\le i \\le m } ^ { \\curvearrowleft } ( f ( w , z _ { i } ) Y _ { i - 1 } ( w , z _ { i } ) ) u ( b , a ^ { m } ) = q ^ { m } u ( a ^ { m } , b ) + \\sum _ { \\ell = 1 } ^ { m } g ( w , z _ { \\ell } ) Z _ { \\ell } ^ { [ 1 , m ] } ( \\vec { z } ) u ( b , a ^ { m } ) \\end{align*}"}
-{"id": "7828.png", "formula": "\\begin{align*} \\begin{cases} & v _ { t } ( t , x ) + v _ { x } ( t , x ) b _ { t } + \\frac { 1 } { 2 } v _ { x x } ( t , x ) \\widetilde { \\sigma } _ { t } \\\\ & \\ \\ \\ \\ \\ \\ + E \\big [ f ( t , x , v ( t , \\eta _ { t } ) , v ( t , x ) , v _ { x } ( t , x ) \\sigma _ { t } ) \\big ] = 0 , \\ \\ \\ ( t , x ) \\in [ 0 , T ) \\times \\mathbb { R } ; \\\\ & v ( T , x ) = g ( x ) , \\ \\ x \\in \\mathbb { R } , \\end{cases} \\end{align*}"}
-{"id": "3755.png", "formula": "\\begin{align*} P \\left ( T _ \\beta = 0 \\right ) = e ^ { - \\int _ 0 ^ 1 e ^ { \\beta x } g ( x ) d x } . \\end{align*}"}
-{"id": "8406.png", "formula": "\\begin{align*} \\sum _ i v _ i y _ h \\phi ( z _ h h ) \\phi ( \\alpha _ { h ^ { - 1 } } ( \\sigma ( v _ i ^ * ) ) ) & = \\sum _ i v _ i p \\phi ( z _ h h \\alpha _ { h ^ { - 1 } } ( \\sigma ( v _ i ^ * ) ) ) \\\\ & = \\sum _ i v _ i p \\phi ( \\sigma ( v _ i ^ * ) z _ h h ) = \\sum _ i v _ i p v _ i ^ * w _ h . \\end{align*}"}
-{"id": "5497.png", "formula": "\\begin{align*} \\mathcal { R } _ k ^ \\mathrm { U L } [ \\iota ] = K \\log _ 2 ( 1 + \\gamma _ k ^ \\mathrm { B } [ \\iota ] ) + ( T _ \\mathrm { d } - K ) \\log _ 2 ( 1 + \\gamma _ k ^ \\mathrm { C } [ \\iota ] ) \\end{align*}"}
-{"id": "5324.png", "formula": "\\begin{align*} E _ { n , m } & = t \\left [ \\left ( \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) \\left ( \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) \\eta ^ 2 + \\omega \\eta \\left ( \\sum _ { i = 1 } ^ { m - 1 } k _ i - \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) - \\frac { \\eta ^ 2 N ^ 2 } { 4 } \\right ] \\\\ & = U \\left ( \\sum _ { i = 1 } ^ { n - 1 } l _ i - \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) ^ 2 + \\mu \\left ( \\sum _ { i = 1 } ^ { n - 1 } l _ i - \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) . \\end{align*}"}
-{"id": "645.png", "formula": "\\begin{align*} U = \\big \\{ A \\in 2 ^ G \\ , : \\ , e \\in A \\big \\} , \\end{align*}"}
-{"id": "1601.png", "formula": "\\begin{align*} C ^ { x } = - m \\ln x + c _ { 1 } x . \\end{align*}"}
-{"id": "2758.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( X _ { t } ) d X ( t ) : = \\lim _ { n \\rightarrow \\infty } \\sum _ { i = 1 } ^ { n } F ( X _ { t _ { i - 1 } } ) ( X ( t _ { i } ) - X ( t _ { i - 1 } ) ) \\end{align*}"}
-{"id": "6922.png", "formula": "\\begin{align*} s ^ O _ \\lambda ( X ) = { s _ \\lambda ^ { ( 2 ) } ( X ) } = s _ \\lambda / L _ { ( 2 ) } ( X ) \\quad \\mbox { a n d } s ^ { S p } _ \\lambda ( X ) = { s _ \\lambda ^ { ( 1 ^ 2 ) } ( X ) } = s _ \\lambda / L _ { ( 1 ^ 2 ) } ( X ) \\ , . \\end{align*}"}
-{"id": "731.png", "formula": "\\begin{align*} \\vec { D } = \\left ( \\begin{array} { c } D _ { 1 } \\\\ D _ { 2 } \\\\ \\vdots \\\\ D _ { r } \\end{array} \\right ) , \\vec { F } = \\left ( \\begin{array} { c } F _ { 1 } \\\\ F _ { 2 } \\\\ \\vdots \\\\ F _ { s } \\end{array} \\right ) , \\vec { e } = \\left ( \\begin{array} { c } 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array} \\right ) , \\vec { Z } = \\left ( \\begin{array} { c } Z _ { 1 } \\\\ Z _ { 2 } \\\\ \\vdots \\\\ Z _ { r } \\end{array} \\right ) . \\end{align*}"}
-{"id": "6411.png", "formula": "\\begin{align*} L u ( t , x ) = \\int _ { \\mathbb { R } ^ { n } } [ u ( t , x ) - u ( t , y ) ] k ( x , y ) d y . \\end{align*}"}
-{"id": "9507.png", "formula": "\\begin{align*} \\delta \\left ( R \\mathrm { d v o l } _ g - \\frac { 1 } { 2 } | G | ^ 2 \\mathrm { d v o l } _ g \\right ) & = \\biggl ( R _ { \\mu \\nu } - \\frac { 1 } { 2 } g _ { \\mu \\nu } R \\\\ & - \\frac { 1 } { 2 } \\langle i _ { \\partial _ \\mu } G , i _ { \\partial _ \\nu } G \\rangle + \\frac { 1 } { 4 } g _ { \\mu \\nu } | G | ^ 2 \\biggr ) \\mathrm { d v o l } _ g \\delta g ^ { \\mu \\nu } + \\mathrm { t d } . \\end{align*}"}
-{"id": "1825.png", "formula": "\\begin{align*} \\frac 1 { \\pi s ^ 3 } \\int _ s ^ 1 ( 1 - \\cos 2 \\pi \\tau ) d \\tau = \\frac { 1 } { \\pi s ^ 3 } ( 1 - s + \\frac { \\sin { 2 \\pi s } } { 2 \\pi } ) . \\end{align*}"}
-{"id": "8269.png", "formula": "\\begin{align*} \\varphi _ { V ' } ( \\rho ) & = \\int _ \\R V ' ( u ) p _ { h ' ( \\rho ) } ( u ) \\dd u = \\int _ \\R ( V ' ( u ) - h ' ( \\rho ) ) p _ { h ' ( \\rho ) } ( u ) \\dd u + h ' ( \\rho ) \\\\ & = - \\int _ \\R \\partial _ u p _ { h ' ( \\rho ) } ( u ) \\dd u + h ' ( \\rho ) = h ' ( \\rho ) . \\end{align*}"}
-{"id": "672.png", "formula": "\\begin{align*} ( P \\phi ) ( g ) = \\int _ Z \\phi ( g \\cdot z ) \\ , d m ( z ) , \\end{align*}"}
-{"id": "106.png", "formula": "\\begin{align*} ( x ) _ n = x ( x - 1 ) \\cdots ( x - n + 1 ) = \\prod _ { l = 1 } ^ { n - 1 } ( x - l ) = \\sum _ { l = 0 } ^ n S _ 1 ( n , l ) x ^ l , \\end{align*}"}
-{"id": "6432.png", "formula": "\\begin{align*} Q _ { - } ( t _ { 0 } , x _ { 0 } , r ) & = ( t _ { 0 } , t _ { 0 } + \\delta \\tau r ^ { 2 \\beta / \\alpha } ) \\times B ( x _ { 0 } , \\delta r ) , \\\\ Q _ { + } ( t _ { 0 } , x _ { 0 } , r ) & = ( t _ { 0 } + ( 2 - \\delta ) \\tau r ^ { 2 \\beta / \\alpha } , t _ { 0 } + 2 \\tau r ^ { 2 \\beta / \\alpha } ) \\times B ( x _ { 0 } , \\delta r ) . \\end{align*}"}
-{"id": "3034.png", "formula": "\\begin{align*} { \\cal G } = \\left ( { \\cal G } _ X \\mid { \\cal G } _ Y \\right ) = \\left ( \\begin{array} { c c | c c c } I _ { \\kappa } & T _ b & 2 T _ 2 & { \\mathbf { 0 } } & { \\mathbf { 0 } } \\\\ { \\mathbf { 0 } } & { \\mathbf { 0 } } & 2 T _ 1 & 2 I _ { \\gamma - \\kappa } & { \\mathbf { 0 } } \\\\ \\hline { \\mathbf { 0 } } & S _ b & S _ q & R & I _ { \\delta } \\end{array} \\right ) , \\end{align*}"}
-{"id": "1793.png", "formula": "\\begin{align*} w \\Lambda = t _ { w _ 0 \\lambda } \\Lambda _ 0 \\quad \\quad \\textup { ( $ w $ i n t h e a f f i n e W e y l g r o u p , $ \\Lambda $ d o m i n a n t ) } \\end{align*}"}
-{"id": "8858.png", "formula": "\\begin{align*} \\hat u _ i = \\dot b _ i \\ , , u _ i = R _ { j i } \\ , \\dot b _ j \\quad \\Longrightarrow \\quad \\hat u _ i = R _ { i j } \\ @ u _ j \\end{align*}"}
-{"id": "9354.png", "formula": "\\begin{align*} \\begin{cases} d p ( t , x ) = - [ A _ { \\hat { \\pi } ( t ) } p ( t , x ) + x q ( t , x ) ] d t + q ( t , x ) d G ( t ) ; 0 \\leq t \\leq T , \\\\ p ( T , x ) = U ( x ) . \\end{cases} \\end{align*}"}
-{"id": "9104.png", "formula": "\\begin{align*} \\sum _ { \\sigma \\in S _ n } x _ { \\sigma ( 1 ) } ^ { d _ 1 } \\ldots x _ { \\sigma ( j ) } ^ { d _ j } = \\Big ( \\sum _ { \\pi \\in S _ n } x _ { \\pi ( 1 ) } \\ldots x _ { \\pi ( j ) } \\Big ) ^ { d _ j } \\cdot \\sum _ { \\rho \\in S _ n } x _ { \\rho ( 1 ) } ^ { d _ 1 - d _ j } \\ldots x _ { \\rho ( j - 1 ) } ^ { d _ { j - 1 } - d _ j } , \\end{align*}"}
-{"id": "5364.png", "formula": "\\begin{align*} | ( \\psi \\circ \\Psi ) ( x - x _ i ) | & = | ( \\psi \\circ \\Theta \\circ \\pi ) ( x - x _ i ) | \\\\ & = | ( \\psi \\circ \\Theta ) ( e _ { M _ 1 } ^ \\perp ( x - x _ i ) ) | \\\\ & \\leq \\| e _ { M _ 1 } ^ \\perp \\| _ { \\psi \\circ \\Theta } \\ , \\| x - x _ i \\| _ { \\psi \\circ \\Theta } \\to 0 i \\to \\infty . \\end{align*}"}
-{"id": "8213.png", "formula": "\\begin{align*} u ( t ) = U e ^ { - i E t } , v ( t ) = V e ^ { - i E t } , \\end{align*}"}
-{"id": "6202.png", "formula": "\\begin{align*} \\left | \\int _ { - 1 } ^ { - 1 + x ^ { - 1 } } [ f ( x + x z ) - f ( x ) - f ' ( x ) x z ] \\nu _ U ( \\d z ) \\right | \\leq 3 x ^ \\beta \\nu _ U ( [ - 1 , - 1 + x ^ { - 1 } ] ) = o ( x ^ { \\beta } ) , \\end{align*}"}
-{"id": "4522.png", "formula": "\\begin{align*} S _ 3 & = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ 0 ^ { \\frac { a _ n } { n - 1 } } \\mathrm { B } _ { k , n - k } ( s ) \\biggl \\{ \\log ^ 2 u _ { x , s } - \\log ^ 2 \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) \\biggr \\} \\ , d s \\ , d x \\\\ & = O \\biggl \\{ \\max \\biggl ( \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\log n \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\biggr \\} . \\end{align*}"}
-{"id": "4318.png", "formula": "\\begin{align*} \\Theta _ { X / Y } \\geq [ \\Sigma _ p ] : = \\sum _ { i \\in I _ h } ( e _ i - 1 ) [ W _ i ] \\end{align*}"}
-{"id": "9858.png", "formula": "\\begin{align*} \\begin{cases} - \\operatorname { d i v } \\left ( \\nabla u \\right ) = \\mu u & \\ , \\ , \\ , \\ , \\ , \\Omega \\\\ \\frac { \\partial u } { \\partial n } = 0 & \\ , \\ , \\ , \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "2258.png", "formula": "\\begin{align*} \\beta _ { q , p } = \\left ( \\frac { 1 } { C _ { 2 } ( \\vec { \\varepsilon } ) } - \\frac { ( 3 p - 1 ) ! ! } { 6 } \\varepsilon _ { p } ^ { 3 } \\sigma ^ { 3 p } - \\frac { ( 2 q + p - 1 ) ! ! } { 2 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } \\sigma ^ { 2 q + p } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4288.png", "formula": "\\begin{align*} x y ^ 3 - y ^ 3 x & = - ( y x y + y ^ 2 x + a x ^ 3 ) y + y ( x y ^ 2 + y x y + a x ^ 3 ) = - a x ^ 3 y + a y x ^ 3 \\\\ & = a x ( y x ^ 2 + x y x + b y ^ 3 ) - a ( x y x + x ^ 2 y + b y ^ 3 ) x = a b x y ^ 3 - a b y ^ 3 x , \\end{align*}"}
-{"id": "5421.png", "formula": "\\begin{align*} \\sup \\{ | | P ^ n ( x , \\cdot ) - P ^ n ( y , \\cdot ) | | : x , y \\in S \\} \\leq C \\rho ^ n , \\ ; n = 1 , 2 . . . \\end{align*}"}
-{"id": "9288.png", "formula": "\\begin{align*} s u p _ { \\pi \\in \\mathcal { A } } J _ { \\tilde { P } } ( \\pi ) = J _ { \\tilde { P } } ( \\hat { \\pi } ) , \\end{align*}"}
-{"id": "8420.png", "formula": "\\begin{align*} \\Delta P + \\nabla \\cdot ( f ( \\rho , v ) \\nabla \\rho ) ) = - \\nabla v \\nabla v . \\end{align*}"}
-{"id": "5298.png", "formula": "\\begin{align*} \\psi ( x ) \\ , : = \\ , 1 - | 2 x - 1 | , x \\in [ 0 , 1 ] , \\end{align*}"}
-{"id": "4571.png", "formula": "\\begin{align*} M _ { g } ^ * ( x ) : = \\max \\biggl \\{ \\max _ { t = 1 , \\ldots , m } \\| g ^ { ( t ) } ( x ) \\| \\ , , \\ , \\sup _ { y \\in B _ x ^ \\circ ( r _ a ( x ) ) } \\frac { \\| g ^ { ( m ) } ( y ) - g ^ { ( m ) } ( x ) \\| } { \\| y - x \\| ^ { \\beta - m } } \\biggr \\} \\end{align*}"}
-{"id": "7655.png", "formula": "\\begin{align*} \\| u _ { 0 } \\| _ { L ^ { q } } \\leq \\sum _ { k = 1 } ^ { \\infty } \\| u _ { k } \\| _ { L ^ { q } } = \\frac { \\omega _ { d } ^ { 1 / q } } { \\nu ( d , \\alpha ) } \\varepsilon ^ { d / q } \\sum _ { k = 1 } ^ { \\infty } k ^ { - \\alpha } < \\infty . \\end{align*}"}
-{"id": "1692.png", "formula": "\\begin{align*} \\gamma ( x ) = x + U ( x ) , \\ ; \\ ; x \\in \\R ^ { 2 d } \\ , . \\end{align*}"}
-{"id": "6853.png", "formula": "\\begin{align*} \\liminf _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\norm { e ^ { i t \\Delta } W ^ J _ n } _ { L _ t ^ { q , \\infty } L _ x ^ r ( \\R \\times \\R ^ d ) } = 0 \\end{align*}"}
-{"id": "2442.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } \\lesssim & \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| ^ 2 _ { L ^ 2 } \\right ) ^ { 1 / 2 } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| ^ 2 _ { L ^ { p _ 1 } } \\right ) ^ { 1 / 2 } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "9630.png", "formula": "\\begin{align*} \\psi _ j ( w ) = \\int _ { \\mathbb { S } ^ 1 } \\phi _ j ( \\gamma ( w ) ) . \\end{align*}"}
-{"id": "4422.png", "formula": "\\begin{align*} 0 \\leq x & = x r + x r ^ \\perp \\leq r V \\mu ( x ) r + \\mu ( \\infty , x ) r ^ \\perp = \\sum _ { i = 1 } ^ n \\left ( r V f _ i r + C _ i r ^ \\perp \\right ) . \\end{align*}"}
-{"id": "2765.png", "formula": "\\begin{align*} X ( t ) = X ( 0 ) + \\int _ 0 ^ t \\psi ( s ) d s + \\int _ 0 ^ t \\varphi ( s ) \\circ d B ^ { H } ( s ) , \\end{align*}"}
-{"id": "7748.png", "formula": "\\begin{align*} \\frac { \\sum _ { i = 0 } ^ n \\sigma _ i [ - \\mu _ { i + 1 } ^ - ] } { \\sqrt { \\sum _ { k = 0 } ^ { n } \\sigma ^ 2 _ k } } \\le \\frac { \\sqrt { \\sum _ { k = 0 } ^ { n } \\sigma ^ 2 _ k } \\sqrt { \\sum _ { i = 0 } ^ n [ \\mu ^ - _ { i + 1 } ] ^ 2 } } { \\sqrt { \\sum _ { k = 0 } ^ { n } \\sigma ^ 2 _ k } } = \\sqrt { \\sum _ { i = 1 } ^ n [ \\mu ^ - _ { i + 1 } ] ^ 2 } \\le \\| \\mu ^ - \\| _ { { \\bf \\ell } _ 2 } . \\end{align*}"}
-{"id": "8122.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ N } | x | ^ l u ^ { \\frac { l + N } { k + N - 1 } } \\ , d x = \\frac { N - 1 } { k + N - 1 } \\int _ { \\mathbb { R } ^ N } | y | ^ { \\frac { l ( N - 1 ) - k N } { k + N - 1 } } v ^ { \\frac { l + N } { k + N - 1 } } \\ , d y . \\end{align*}"}
-{"id": "1819.png", "formula": "\\begin{align*} \\log z ' = \\log z + \\log \\alpha + \\log ( 1 + z \\beta ) . \\end{align*}"}
-{"id": "2544.png", "formula": "\\begin{align*} \\mathcal { L } : = b \\cdot \\nabla + \\frac 1 2 \\sigma \\sigma ^ T : \\nabla ^ 2 \\end{align*}"}
-{"id": "3427.png", "formula": "\\begin{align*} d \\Gamma ( t ) = c ( t , \\xi ( t ) ) \\Gamma ( t ) d t + C ( t , \\xi ( t ) ) ( \\Gamma ( t ) , d w ( t ) ) , \\Gamma ( s , s ) = I , \\end{align*}"}
-{"id": "7606.png", "formula": "\\begin{align*} f ( x ) : = \\frac { 4 x } { 2 + ( x - 3 ) ^ 2 } , x > 0 . \\end{align*}"}
-{"id": "5249.png", "formula": "\\begin{align*} \\varphi _ { A , k } ( y ) : = \\inf \\{ t \\in { \\mathbb { R } } \\mid y \\in t k + A \\} , \\end{align*}"}
-{"id": "4281.png", "formula": "\\begin{align*} \\lim _ { n \\rightarrow + \\infty } n ^ { - \\frac 3 4 } \\mathbb E \\left [ \\max _ { k = 1 , \\ldots , n } M _ k ^ { ( 1 ) } \\right ] = K _ p \\mathbb E \\left [ \\sup _ { t \\in [ 0 , 1 ] } \\Delta _ t ^ { ( 0 ) } \\right ] . \\end{align*}"}
-{"id": "6810.png", "formula": "\\begin{align*} \\lim _ { \\substack { x \\to \\xi \\\\ x \\in \\mathbb D ^ \\pm } } \\frac { \\dd Y _ \\infty } { \\dd x } = \\frac { \\dd \\left ( Y _ \\infty ^ \\pm \\right ) } { \\dd \\xi } . \\end{align*}"}
-{"id": "4354.png", "formula": "\\begin{align*} \\varphi ( y ) = \\varphi ( y , a ) : = \\psi ( y , a ) ( 1 - y ) ^ { - p } \\ , y \\in [ 0 , 1 ) \\ , \\end{align*}"}
-{"id": "5705.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm B M O } _ { L ^ { p ( \\cdot ) } } } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { \\| \\chi _ Q \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } } \\| ( b - b _ Q ) \\chi _ Q \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } . \\end{align*}"}
-{"id": "1942.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\ , \\frac { b _ i } { a _ i + b _ i } & = \\sum _ { i = 1 } ^ m \\ , \\frac { 2 n _ i q _ i ^ 2 \\xi _ i ^ 2 } { 2 ( p _ i \\xi _ i - ( n _ i + 1 ) q _ i ^ 2 \\xi _ i ^ 2 ) + 2 n _ i q _ i ^ 2 \\xi _ i ^ 2 } \\\\ & = \\sum _ { i = 1 } ^ m \\ , \\frac { 2 n _ i q _ i ^ 2 \\xi _ i ^ 2 } { ( \\sum _ { j = 1 } ^ m \\ , n _ j q _ j ^ 2 \\xi _ j ^ 2 ) - q _ i ^ 2 \\xi _ i ^ 2 } , \\end{align*}"}
-{"id": "5165.png", "formula": "\\begin{gather*} \\sum _ { i ( 1 ) , \\ldots , i ( k ) , j ( 1 ) , \\ldots , j ( l ) = 1 } a ( i ( 1 ) , \\ldots , i ( k ) , j ( 1 ) , \\ldots , j ( l ) ) S _ { j ( 1 ) } \\cdots S _ { j ( l ) } S _ { i ( k ) } ^ * \\cdots S _ { i ( 1 ) } ^ * \\end{gather*}"}
-{"id": "8928.png", "formula": "\\begin{align*} v - L _ { A _ * } ( v ) = ( - \\Delta _ { A _ * } + I ) ^ { - 1 } f . \\end{align*}"}
-{"id": "3224.png", "formula": "\\begin{align*} w _ k ^ \\mu = \\frac { \\sum _ { \\lambda = 1 } ^ r ( S _ { j k } ) _ \\lambda ^ \\mu \\cdot w _ j ^ \\lambda } { 1 + \\sum _ { \\lambda = 1 } ^ r a _ { j k , \\lambda } w _ j ^ \\lambda } = \\sum _ { \\lambda = 1 } ^ r ( S _ { j k } ) _ \\lambda ^ \\mu \\cdot w _ j ^ \\lambda - \\sum _ { \\lambda = 1 } ^ r \\sum _ { \\nu = 1 } ^ r ( S _ { j k } ) _ \\lambda ^ \\mu \\cdot a _ { j k , \\nu } w _ j ^ \\lambda w _ j ^ \\nu + O ( | w _ j | ^ 3 ) , \\end{align*}"}
-{"id": "7126.png", "formula": "\\begin{align*} X _ j ( t ) = Q \\Big ( t + \\frac { 2 \\pi j } { n } \\Big ) , 0 \\leq j \\leq n - 1 , \\end{align*}"}
-{"id": "6485.png", "formula": "\\begin{align*} u ( x , t ) = & \\sum _ { k = 1 } ^ { \\infty } E _ { \\alpha , 1 } ( - \\lambda _ { k } t ^ { \\alpha } ) u _ { 0 , k } \\phi _ { k } ( x ) \\\\ & + \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { t } ( t - s ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\lambda _ { k } ( t - s ) ^ { \\alpha } ) f _ { k } ( s ) d s \\phi _ { k } ( x ) , \\end{align*}"}
-{"id": "8773.png", "formula": "\\begin{align*} K ^ { ( 2 r + 1 ) } _ n ( x ) = \\sum _ { i = - r } ^ { r } E ^ { ( i i ) } - \\frac { 1 - x ^ 2 } { h _ n ( b , d , x ) } \\Big ( \\sum _ { 0 < i \\leq r } E ^ { ( - i , - i ) } + t _ n E ^ { ( r + 1 - i , r + 1 - i ) } \\\\ { } - \\sum _ { 0 < i \\leq r } E ^ { ( r + 1 - i , - i ) } + t _ n E ^ { ( - i , r + 1 - i ) } \\Big ) . \\end{align*}"}
-{"id": "5823.png", "formula": "\\begin{align*} \\tilde B _ 0 = [ B _ 0 | C ] = \\left [ \\begin{array} { c c | c c } 0 & 1 & 1 & 0 \\\\ - 1 & 0 & 0 & 1 \\end{array} \\right ] \\end{align*}"}
-{"id": "8707.png", "formula": "\\begin{align*} \\sup _ { ( s , x ) \\in [ 0 , T ] \\times H } \\Vert \\nabla B ^ n ( s , x ) \\Vert _ { L ( H , U ) } = c ( n ) \\end{align*}"}
-{"id": "4393.png", "formula": "\\begin{align*} \\tau ^ { - i } A _ i = \\tau ^ { - i } C _ { m _ i ( 0 ) } \\cap \\tau ^ { - i - 1 } C _ { m _ i ( 1 ) } \\cap \\dots \\cap \\tau ^ { - i - 2 ^ { k _ i } } C _ { m _ i ( 2 ^ { k _ i } ) } , \\ \\ 1 \\le i \\le n . \\end{align*}"}
-{"id": "8845.png", "formula": "\\begin{align*} \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\lambda _ i \\geq \\frac { n } { n + 2 } \\frac { 4 \\pi ^ 2 } { ( \\omega _ n \\mathrm { v o l } \\ , \\Omega ) ^ \\frac { 2 } { n } } k ^ { \\frac { 2 } { n } } . \\end{align*}"}
-{"id": "6235.png", "formula": "\\begin{align*} \\tilde { V } _ { j + 1 } = [ X ^ { ( j ) } ( s _ { t _ { j } } ) / | | X ^ { ( j ) } ( s _ { t _ { j } } ) | | ] , \\end{align*}"}
-{"id": "5110.png", "formula": "\\begin{align*} L ^ { ( i ) } ( z ) = \\begin{pmatrix} 1 + z q ^ { 2 N _ { i } } & \\beta _ { i } ^ { * } \\\\ z \\beta _ { i } & z \\end{pmatrix} . \\end{align*}"}
-{"id": "2315.png", "formula": "\\begin{align*} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ q ( \\varOmega ) ) } + \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { 2 , q } ( \\varOmega ) ) } \\le C \\tau ^ k , \\\\ & \\max _ { k \\le n \\le N } \\| e _ n \\| _ { W ^ { 1 , \\infty } ( \\varOmega ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "114.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 + t ) ^ { \\tfrac { x } { 2 } } d \\mu _ { - 1 } ( x ) & = \\sum _ { m = 0 } ^ \\infty \\int _ { \\mathbb { Z } _ p } x ^ m d \\mu _ { - 1 } ( x ) \\frac { 1 } { 2 ^ m } \\frac { 1 } { m ! } \\Big ( \\log ( 1 + t ) \\Big ) ^ m \\\\ & = \\sum _ { m = 0 } ^ \\infty E _ m 2 ^ { - m } \\sum _ { n = m } ^ \\infty S _ 1 ( n , m ) \\frac { t ^ n } { n ! } \\\\ & = \\sum _ { n = 0 } ^ \\infty \\left ( \\sum _ { m = 0 } ^ n 2 ^ { - m } E _ m S _ 1 ( n , m ) \\right ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "6867.png", "formula": "\\begin{align*} \\sup _ { 1 \\le j \\le J _ 1 } \\norm { v _ n ^ j } _ { W ( K _ n ^ m ) } = \\norm { v _ n ^ { j ( m , n ) } } _ { W ( K _ n ^ m ) } = m . \\end{align*}"}
-{"id": "4121.png", "formula": "\\begin{align*} C \\ : : \\ : ( y ^ 2 + a x ^ 2 + b x z ) ^ q - z ^ { p + 2 q } x ^ { - p } = 0 . \\end{align*}"}
-{"id": "4600.png", "formula": "\\begin{align*} x J = \\left [ \\begin{array} { c c } 0 & x \\\\ - x & 0 \\end{array} \\right ] = \\left [ \\begin{array} { c c } u & 0 \\\\ 0 & 1 \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & \\varpi ^ { - k } \\\\ - \\varpi ^ { - k } & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } u & 0 \\\\ 0 & 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "1078.png", "formula": "\\begin{align*} x n = \\sum _ { i = 1 } ^ c \\max \\{ v ( B _ i ) - n , 0 \\} \\end{align*}"}
-{"id": "3307.png", "formula": "\\begin{align*} \\begin{gathered} \\gamma _ { - } ( w _ { 1 } ) v _ { 1 } \\wedge v _ { 0 } \\wedge v _ { - 1 } = q ^ { 2 } v _ { 0 } \\wedge v _ { - 1 } , \\gamma _ { - } ( w _ { 1 } ) v _ { 1 } \\wedge v _ { 0 } \\wedge v _ { - 1 } = - q ^ { 2 } v _ { 1 } \\wedge v _ { - 1 } , \\\\ \\gamma _ { - } ( w _ { 1 } ) v _ { 1 } \\wedge v _ { 0 } \\wedge v _ { - 1 } = q ^ { 4 } v _ { 1 } \\wedge v _ { 0 } . \\end{gathered} \\end{align*}"}
-{"id": "8140.png", "formula": "\\begin{align*} a _ 2 = a _ 2 ( p , q , N ) : = 1 + N \\left ( \\frac { 1 } { q } - \\frac { 1 } { p } \\right ) , \\end{align*}"}
-{"id": "7419.png", "formula": "\\begin{align*} g ^ * : = \\overline { w } _ 0 \\left ( g ^ { - 1 } \\right ) ^ t \\overline { w } _ 0 ^ { - 1 } . \\end{align*}"}
-{"id": "2518.png", "formula": "\\begin{align*} X _ { 0 } ^ { \\lambda ; p ^ { \\prime } } = \\left \\| \\bigwedge _ { i = 1 } ^ { p } \\nabla D _ i \\wedge \\bigwedge _ { j = 1 } ^ { k } \\nabla I _ j \\right \\| _ { k + p } ^ { - 2 } \\cdot \\sum _ { i = 1 } ^ { p ^ { \\prime } } ( - 1 ) ^ { n - i } ( - \\lambda ) ( D - d _ i ) \\Theta _ i , \\end{align*}"}
-{"id": "2740.png", "formula": "\\begin{align*} ( f _ K ^ I ) \\mapsto \\left \\{ u ^ { - 1 } \\left ( L ^ \\star _ { f _ K ^ I } \\theta ^ K \\delta _ I \\right ) u \\right \\} 1 = u ^ { - 1 } ( f _ K ^ I \\star u _ { I L } ) \\theta ^ K \\bar \\theta ^ L . \\end{align*}"}
-{"id": "2901.png", "formula": "\\begin{align*} 3 \\geq { \\rm { d i m } } ( \\bar { L } _ { 2 5 } ^ { \\zeta ^ 7 } / \\bar { L } ^ { \\zeta ^ 7 } _ { 2 3 } ) = { \\rm { d i m } } ( \\bar { L _ 5 } ^ { \\zeta ^ 7 } ) = { \\rm { d i m } } ( a _ 7 k ) = a _ 7 , \\end{align*}"}
-{"id": "8479.png", "formula": "\\begin{align*} = - \\int _ { 0 } ^ { T } { \\psi ( t ) } \\Bigg ( \\frac { ( \\mathbb { I } - \\mathbb { P } ) v ^ { \\varepsilon } } { \\varepsilon } , v \\Bigg ) _ { 0 } ~ d t . \\end{align*}"}
-{"id": "9732.png", "formula": "\\begin{align*} a \\lambda _ { n + 1 } ^ \\beta = \\lambda _ { n + 1 } + b \\lambda _ n ^ \\beta ( 1 - q ) ^ { - \\beta / ( \\beta - 1 ) } , n \\geq 1 \\end{align*}"}
-{"id": "7603.png", "formula": "\\begin{align*} P _ p ( x _ 0 ) \\ge h ^ { K _ 1 } \\prod _ { i = 1 } ^ { K _ 1 } \\left ( \\varepsilon _ 0 + \\frac { \\kappa ( i - 1 ) \\varepsilon } l \\right ) , P _ e ( x _ 0 ) \\ge h ^ { K _ 2 } \\prod _ { i = 1 } ^ { K _ 2 } \\left ( \\delta _ 0 + \\frac { \\kappa ( i - 1 ) \\delta } l \\right ) , \\end{align*}"}
-{"id": "4040.png", "formula": "\\begin{align*} h ^ { u } ( \\underline { a } ) = \\bigcap _ { n \\geq 0 } ( \\phi _ { a [ - n , 0 ] } ^ { - } ) ^ { - 1 } ( R _ { a _ { - n } } ) = \\bigcap _ { n \\geq 0 } W _ { a [ - n , 0 ] } \\end{align*}"}
-{"id": "5881.png", "formula": "\\begin{align*} \\ln F \\left ( t , x \\right ) = - e ^ { - m t } \\ln \\left ( 1 + \\varepsilon e ^ { x - x _ { 0 } } \\right ) + e ^ { - m t } \\left ( x - x _ { 0 } \\right ) + \\frac { c } { m } e ^ { - m t } - \\frac { 1 } { 4 m } e ^ { - 2 m t } , \\end{align*}"}
-{"id": "3016.png", "formula": "\\begin{align*} { { \\bf { h } } ^ { [ 2 2 ] } } ( n ) { \\bf { V } } _ 1 ^ { [ 2 ] } ( n ) = { { \\bf { h } } ^ { [ 2 2 ] } } ( 1 ) , { { \\bf { h } } ^ { [ 1 2 ] } } ( n ) { \\bf { V } } _ 2 ^ { [ 2 ] } ( n ) = { { \\bf { h } } ^ { [ 1 2 ] } } ( 6 ) . \\end{align*}"}
-{"id": "4199.png", "formula": "\\begin{align*} x _ \\alpha ( r ) x _ \\alpha ( s ) & = x _ \\alpha ( r + s ) , & \\\\ [ x _ \\alpha ( r ) , x _ \\beta ( s ) ] & = x _ { \\alpha + \\beta } ( N _ { \\alpha , \\beta } \\cdot r s ) , & \\ \\alpha , \\beta \\in \\Phi , \\ \\alpha + \\beta \\in \\Phi , \\\\ [ x _ \\alpha ( r ) , x _ \\beta ( s ) ] & = 1 , & \\ \\alpha , \\beta \\in \\Phi , \\ \\alpha + \\beta \\not \\in \\Phi \\cup \\{ 0 \\} . \\end{align*}"}
-{"id": "1155.png", "formula": "\\begin{align*} \\begin{gathered} \\begin{bmatrix} - h & 1 \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} V ( x = 0 ) = 0 , \\\\ \\begin{bmatrix} H & \\ , \\ , 1 \\end{bmatrix} \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} V ( x = 1 ) = 0 , \\end{gathered} \\end{align*}"}
-{"id": "7742.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac 1 { D ^ 2 _ { n } } \\sum _ { k = 1 } ^ { n } \\int \\limits _ { x : | x | \\ge \\varepsilon D _ n } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ x d F _ k ( x ) = 0 . \\end{align*}"}
-{"id": "9365.png", "formula": "\\begin{align*} - \\{ f '' ( x ) \\} _ { x \\not = 0 } + [ \\delta , \\delta ' ] [ \\mathbf { T } \\Gamma _ 0 f - \\Gamma _ 1 f ] , \\end{align*}"}
-{"id": "2325.png", "formula": "\\begin{align*} p ( x ) = 0 ~ { \\rm f o r } ~ p ( x ) = \\sum ^ { d } _ { j = 0 } p _ j x ^ j = p _ d \\prod ^ d _ { i = 1 } ( x - x _ i ) , ~ ~ ~ p _ d \\ne 0 , \\end{align*}"}
-{"id": "4738.png", "formula": "\\begin{align*} \\int _ { B ^ c _ \\varepsilon ( x ) } \\frac { \\ , u ( x ) - u ( y ) \\ , } { | x - y | ^ { N + 2 s } } \\ , d y & = \\int _ { B _ r \\cap B ^ c _ \\varepsilon } \\frac { \\ , u ( x ) - u ( x + z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z + \\int _ { B ^ c _ r } \\frac { \\ , u ( x ) - u ( x + z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z \\\\ & = \\int _ { B _ r \\cap B ^ c _ \\varepsilon } \\frac { \\ , u ( x ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z + \\int _ { B ^ c _ r } \\frac { \\ , u ( x ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z , \\end{align*}"}
-{"id": "6927.png", "formula": "\\begin{align*} X ^ \\pi ( z ) = V _ \\pi ( z ) e ^ { i q } z ^ { \\alpha _ 0 } : = \\sum _ { n \\in { \\mathbb Z } } z ^ { n + { \\alpha _ 0 } } X ^ \\pi _ { - n } \\quad \\mbox { a n d } X ^ { * \\pi } ( z ) = V _ \\pi ^ * ( z ) z ^ { - \\alpha _ 0 } e ^ { - i q } : = \\sum _ { n \\in { \\mathbb Z } } z ^ { - n - { \\alpha _ 0 } } X ^ { * \\pi } _ { { n } } \\ , . \\end{align*}"}
-{"id": "9818.png", "formula": "\\begin{align*} ( \\R ^ n ) ^ N \\ni ( x _ 1 , \\ldots , x _ N ) \\mapsto V _ j \\left ( \\bigcap _ { i = 1 } ^ N B ( x _ i , r ) \\right ) \\end{align*}"}
-{"id": "7619.png", "formula": "\\begin{align*} ( 2 - 7 \\epsilon ) n \\leq \\sum _ { u v \\in E ( S , L ) } \\mathbf { v } _ u + \\mathbf { v } _ v = \\left ( \\sum _ { \\substack { u v \\in E ( S , L ) \\\\ u = x } } \\mathbf { v } _ u + \\mathbf { v } _ v \\right ) + \\left ( \\sum _ { \\substack { u v \\in E ( S , L ) \\\\ u \\not = x } } \\mathbf { v } _ u + \\mathbf { v } _ v \\right ) \\leq \\epsilon e ( S , L ) + d _ x + \\sum _ { \\substack { u v \\in E ( S , L ) \\\\ u \\not = x } } \\mathbf { v } _ u , \\end{align*}"}
-{"id": "6794.png", "formula": "\\begin{align*} \\lambda _ j \\mu ( X ) = \\int _ X \\langle T _ j ^ n \\rangle . \\end{align*}"}
-{"id": "6507.png", "formula": "\\begin{align*} N ( r , s ) : = \\frac { d { d \\choose r } } { \\log { d \\choose r } } . \\end{align*}"}
-{"id": "486.png", "formula": "\\begin{align*} \\int _ { B _ 1 ^ c } \\frac 1 { \\ , | z | ^ { N + 2 s } \\ , } \\int _ { B ^ c _ { R _ t } ( \\mp z ) } \\frac { \\ , 1 + | y | ^ { 2 s - \\sigma } \\ , } { \\ , | y \\pm z | ^ { N + 2 s } \\ , } \\ , d y \\ , d z & = \\int _ { B _ 1 ^ c } \\frac 1 { \\ , | z | ^ { N + 2 s } \\ , } \\int _ { B ^ c _ { R _ t } } \\frac { \\ , 1 + | w \\mp z | ^ { 2 s - \\sigma } \\ , } { | w | ^ { N + 2 s } } \\ , d w \\ , d z \\\\ & \\le \\int _ { B _ 1 ^ c } C \\ , \\frac { \\ , 1 + | z | ^ { 2 s - \\sigma } \\ , } { | z | ^ { N + 2 s } } \\ , d z < + \\infty . \\end{align*}"}
-{"id": "8870.png", "formula": "\\begin{align*} u ( x ) = O \\biggl ( \\exp \\Bigl ( - \\frac { \\abs { B \\times ( x - a ) } ^ 2 } { 4 \\abs { B } } \\Bigr ) \\biggr ) , \\end{align*}"}
-{"id": "4599.png", "formula": "\\begin{align*} \\widehat { \\sigma } ( x e _ { 1 1 } ) = \\widehat { \\tau _ { * } ( \\sigma ) } ( 2 ^ { - 1 } x J ) . \\end{align*}"}
-{"id": "8982.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | \\varphi _ { h , t } ( x ) - \\varphi _ { h , t } ( y ) | ^ { p - 2 } \\ , ( \\varphi _ { h , t } ( x ) - \\varphi _ { h , t } ( y ) ) \\ , ( \\varphi _ h ( x ) - \\varphi _ h ( y ) ) } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y - \\int _ \\Omega \\ , f ( x ) \\cdot \\Phi _ { k } ' ( \\varphi _ { h , t } ) \\varphi _ h \\\\ & \\geq - \\int _ \\Omega f ( x ) \\cdot | \\Phi _ { k } ' ( \\varphi _ { h , t } ) - \\Phi _ { k } ' ( w ) | | \\varphi _ { h } | \\ , d x \\ , . \\\\ \\end{align*}"}
-{"id": "8518.png", "formula": "\\begin{align*} w _ 0 = ( s _ 1 s _ 2 \\cdots s _ { n - 1 } s _ n s _ { n - 2 } \\cdots s _ 2 s _ 1 ) \\cdots ( s _ { n - 2 } s _ { n - 1 } s _ n s _ { n - 2 } ) s _ { n - 1 } s _ n \\end{align*}"}
-{"id": "6608.png", "formula": "\\begin{align*} - ( 2 \\alpha - 1 ) \\| x - y \\| _ { P ^ 2 } ^ 2 & = - ( 2 \\alpha - 1 ) \\| ( P x + q ) - ( P y - q ) \\| ^ 2 \\\\ & \\leq \\langle P \\nabla f _ 2 ( P x + q ) - P \\nabla f _ 2 ( P y + q ) , x - y \\rangle \\\\ & \\leq ( 2 \\beta - 1 ) \\| ( P x + q ) - ( P y - q ) \\| ^ 2 \\\\ & = ( 2 \\beta - 1 ) \\| x - y \\| _ { P ^ 2 } ^ 2 \\end{align*}"}
-{"id": "8896.png", "formula": "\\begin{align*} L _ n u _ n = \\lambda _ k ( L _ n ) u _ n \\end{align*}"}
-{"id": "2247.png", "formula": "\\begin{align*} \\Delta H _ { p } ( 1 ) = \\log \\left ( 1 \\pm C ( \\varepsilon ) \\left | \\Delta _ { 0 } ( 1 ) \\right | \\right ) \\pm \\frac { 1 } { 2 \\sigma ^ { 2 } } \\left | \\Delta \\mu _ { ( 2 ) } ( 1 ) \\right | \\pm \\varepsilon \\left | \\Delta \\mu _ { ( p ) } ( 1 ) \\right | . \\end{align*}"}
-{"id": "2815.png", "formula": "\\begin{align*} { \\displaystyle \\frac { \\partial } { \\partial y } \\log | E ( x + i y ) | } = p y + \\displaystyle \\frac { 1 } { \\pi } \\displaystyle \\int \\limits _ { - \\infty } ^ { \\infty } \\displaystyle \\frac { y } { ( t - x ) ^ 2 + y ^ 2 } d \\phi ( t ) \\end{align*}"}
-{"id": "9643.png", "formula": "\\begin{align*} \\omega _ 1 ^ { ( \\ell + 1 ) } = \\omega _ 2 ^ { ( \\ell ) } , \\omega _ 2 ^ { ( \\ell + 1 ) } = \\omega _ 3 ^ { ( \\ell ) } , \\ldots , \\omega _ { k - 2 } ^ { ( \\ell + 1 ) } = \\omega _ { k - 1 } ^ { ( \\ell + 1 ) } \\end{align*}"}
-{"id": "7551.png", "formula": "\\begin{align*} x _ { n + 1 } = \\frac { A x _ n } { B + ( x _ n - T ) ^ 2 } , \\end{align*}"}
-{"id": "407.png", "formula": "\\begin{align*} t : = \\bigvee \\limits _ { ( y , z ) \\in A _ a } { \\min \\{ ( f \\circ g ) ( y ) , h ( z ) \\} } \\end{align*}"}
-{"id": "2186.png", "formula": "\\begin{align*} & \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } \\left [ \\psi ( x ) w ( s , x ) - \\psi ( y ) w ( s , y ) \\right ] ^ { 2 } k ( x , y ) d x d y \\\\ & \\quad \\quad \\quad \\quad \\quad \\geq \\int _ { \\rho ' B _ { 1 } } \\int _ { \\rho ' B _ { 1 } } \\left [ w ( s , x ) - w ( s , y ) \\right ] ^ { 2 } k ( x , y ) d x d y , \\end{align*}"}
-{"id": "4016.png", "formula": "\\begin{align*} ( i - \\delta ) ( i - \\ell ) \\begin{cases} = 0 \\ ; & \\mbox { i f } i = \\delta , \\\\ < 0 \\ ; & \\mbox { i f } \\delta < i < \\ell , \\\\ = 0 \\ ; & \\mbox { i f } i = \\ell . \\end{cases} \\end{align*}"}
-{"id": "5928.png", "formula": "\\begin{align*} \\partial _ { x _ k } { \\psi } ( z ) = \\partial _ { x _ k } G _ { \\lambda } g ( z ) = \\int _ { 0 } ^ { + \\infty } e ^ { - \\lambda t } P _ t ( \\partial _ { x _ k } g ) ( z ) \\dd t \\ , . \\end{align*}"}
-{"id": "9826.png", "formula": "\\begin{align*} - \\mu g _ { 0 , m } ( \\mu ) = ( ( a - 1 ) \\lambda - \\mu + b ) g _ { 0 , m } ( \\lambda + \\mu ) . \\end{align*}"}
-{"id": "3684.png", "formula": "\\begin{align*} \\mathcal { A } ( . ) = \\nabla ^ 2 ( . ) ~ . \\end{align*}"}
-{"id": "3294.png", "formula": "\\begin{align*} \\psi ( v _ 1 ) = \\alpha w _ { - 1 } , \\psi ( v _ 0 ) = \\beta w _ 0 , \\psi ( v _ { - 1 } ) = \\gamma w _ 1 . \\end{align*}"}
-{"id": "4011.png", "formula": "\\begin{align*} \\sigma _ q ( d _ { s + 1 } , \\dim W ) = q ^ { \\dim W } + 1 > q ^ { d _ s } + 1 = \\sigma _ q ( d _ { s + 1 } , d _ s ) . \\end{align*}"}
-{"id": "5641.png", "formula": "\\begin{align*} J ' ( \\mu ; \\eta - \\mu ) : = \\underset { \\underset { t > 0 } { t \\to 0 } } { \\lim } \\frac { J ( \\mu + t ( \\eta - \\mu ) ) - J ( \\mu ) } { t } \\ge 0 . \\end{align*}"}
-{"id": "6806.png", "formula": "\\begin{align*} H ( f ) ( u ) = P . V . \\frac { 1 } { i \\pi } \\oint _ \\S \\frac { f ( \\xi ) } { \\xi - u } \\dd \\xi \\end{align*}"}
-{"id": "560.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\| K - A _ n K \\| | = \\lim _ { n \\to \\infty } | \\| K - K A _ n \\| | = 0 \\end{align*}"}
-{"id": "4867.png", "formula": "\\begin{align*} E _ 2 = n ^ { - 1 / 3 } \\int _ { n ^ { 1 / 3 } \\mathcal { C } _ { 0 \\not \\in } ^ { \\rm l o c a l } } \\int _ { n ^ { 1 / 3 } \\mathcal { C } _ { 0 \\not \\in } ^ { \\rm l o c a l } } \\frac { ( \\tilde z - \\tilde w ) e ^ { \\frac { \\sigma ^ 3 } { 3 } \\tilde z ^ 3 + \\frac { \\sigma ^ 3 } { 3 } \\tilde w ^ 3 - \\sigma x \\tilde z - \\sigma y \\tilde w } } { 4 \\tilde z \\tilde w ( \\tilde z + \\tilde w ) } \\frac { 4 ( \\tilde z + \\tilde w ) } { 2 \\alpha - 1 } \\mathrm { d } \\tilde z \\mathrm { d } \\tilde w . \\end{align*}"}
-{"id": "5130.png", "formula": "\\begin{align*} u ( \\nu _ { 1 } , \\ldots , \\nu _ { m } ) = u _ { \\nu _ { 1 } } \\otimes \\cdots \\otimes u _ { \\nu _ { m } } \\end{align*}"}
-{"id": "5417.png", "formula": "\\begin{align*} g _ b ( x ) = x / 2 + b \\ ; \\pmod { 1 } . \\end{align*}"}
-{"id": "5952.png", "formula": "\\begin{align*} { \\varphi } ( Z _ t ) = { \\varphi } ( Z _ s ) + \\int _ s ^ t \\big [ b ( Z _ r ) \\cdot D { \\varphi } ( Z _ r ) + \\frac { 1 } { 2 } \\Delta _ v { \\varphi } ( Z _ r ) \\big ] \\dd r + \\int _ s ^ t D _ v { \\varphi } ( Z _ r ) \\ , \\dd W _ r \\ , . \\end{align*}"}
-{"id": "8020.png", "formula": "\\begin{align*} \\deg \\theta ( x _ k ) = \\deg \\theta ( y _ k ) = 0 \\ ; \\ ; \\mbox { f o r } \\ ; \\ ; k < j \\mbox { a n d } \\deg \\theta ( x _ j ) > 0 \\ ; \\ ; \\mbox { o r } \\ ; \\ ; \\deg \\theta ( y _ j ) > 0 . \\end{align*}"}
-{"id": "1903.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\int _ { - m ^ * } ^ { m ^ * } \\frac { 1 } { | \\phi _ X ( u ) | ^ { 2 ( K - 1 ) } } \\d u \\leq & \\frac { 2 u _ n ^ { ( a ) } } { n } | \\phi _ X ( u _ n ^ { ( a ) } ) | ^ { - 2 ( K - 1 ) } \\leq 2 C _ 2 n ^ { \\frac { 1 } { 2 \\beta K } - 1 } a n ^ { \\frac { 2 ( K - 1 ) } { 2 K } } \\\\ = & 2 a C _ 2 n ^ { - \\frac { 2 \\beta - 1 } { 2 \\beta K } } \\leq 2 a C _ 2 n ^ { - \\frac { 2 \\beta - 1 } { 2 b K } } . \\end{align*}"}
-{"id": "29.png", "formula": "\\begin{align*} y _ k \\to y _ \\infty \\in ( t _ m = 0 ) \\end{align*}"}
-{"id": "8289.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\beta } \\frac { \\phi _ n ( \\beta ) } { \\beta } = - \\frac { 1 } { n \\beta ^ 2 } S ( \\mu _ { \\beta , n } ) , \\end{align*}"}
-{"id": "3328.png", "formula": "\\begin{align*} K _ k \\triangleright ( \\Gamma _ i \\Gamma _ j ^ * ) & = ( K _ k \\triangleright \\gamma _ - ( w _ i ) ) ( K _ k \\triangleright \\gamma _ - ( w _ j ) ^ * ) \\\\ & = q ^ { ( \\alpha _ k , \\beta _ i - \\beta _ j ) } \\gamma _ - ( w _ i ) \\gamma _ - ( w _ j ) ^ * . \\end{align*}"}
-{"id": "2736.png", "formula": "\\begin{align*} a \\star f = a f \\mbox { a n d } f \\star b = f b \\end{align*}"}
-{"id": "1803.png", "formula": "\\begin{align*} S _ 0 f ( x ) : = \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i x \\cdot \\xi } \\widehat { \\Phi } ( \\xi ) \\widehat { f } ( \\xi ) d \\xi \\end{align*}"}
-{"id": "6852.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u = \\mu | u | ^ p u + e , u ( t _ 0 ) = u _ 0 , \\end{align*}"}
-{"id": "6704.png", "formula": "\\begin{align*} a _ K ^ I \\tilde a _ P ^ K = a _ K ^ I \\star \\tilde a _ P ^ K = \\delta ^ I _ P , b _ L ^ J \\tilde b _ Q ^ L = b _ L ^ J \\star \\tilde b _ Q ^ L = \\delta ^ J _ Q . \\end{align*}"}
-{"id": "3597.png", "formula": "\\begin{align*} \\Pi _ { g _ 0 } \\circ D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f _ m , X _ m ) = \\Pi _ { g _ 0 } \\circ \\Phi ^ W _ { ( g , \\pi ) } ( g , \\pi ) + \\Pi _ { g _ 0 } ( \\psi , V ) - \\Pi _ { g _ 0 } \\circ \\Phi ^ W _ { ( g , \\pi ) } ( \\gamma _ m , \\tau _ m ) \\end{align*}"}
-{"id": "4946.png", "formula": "\\begin{align*} \\lambda < \\lambda _ { e s s } : = \\inf \\sigma _ { e s s } ( - \\Delta + V ) , \\end{align*}"}
-{"id": "136.png", "formula": "\\begin{align*} \\| f _ n \\| _ { L _ 1 + L _ \\infty } = \\| \\widehat { i d } ( f _ n ) - \\widehat { i d } ( f ) \\| _ { L _ 1 + L _ \\infty } \\leq C \\| f _ n - f \\| _ { \\widehat { E } } \\to 0 . \\end{align*}"}
-{"id": "6989.png", "formula": "\\begin{align*} \\delta _ L \\Theta ( A ) ( x ) = - [ x , \\Theta ( A ) ] = \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ \\alpha _ i , \\delta _ L ( \\alpha _ j ) ] \\end{align*}"}
-{"id": "4079.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ 2 } { x z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ 2 } { x ^ 3 z } \\\\ \\end{gather*}"}
-{"id": "7781.png", "formula": "\\begin{align*} \\begin{aligned} & r _ 1 = 0 \\ \\Leftrightarrow \\ { } _ 1 x _ 2 \\circ { } _ 2 x _ 1 = 0 \\ \\Leftrightarrow \\ { } _ 1 x _ 2 \\circ { } _ 2 x _ 1 ( 1 ) = 0 \\ \\Leftrightarrow \\ w _ 1 z _ 1 + w _ i z _ i = 0 , \\end{aligned} \\end{align*}"}
-{"id": "7736.png", "formula": "\\begin{align*} \\mathbb P \\{ \\omega \\in \\Omega : \\lim _ { n \\to \\infty } x _ n ( \\omega ) = K \\} = 1 . \\end{align*}"}
-{"id": "1581.png", "formula": "\\begin{align*} \\sigma ^ { 2 } \\left ( x \\right ) = \\left ( 4 \\int e ^ { 2 x } K \\left ( x \\right ) \\mbox { \\rm d } x + c _ { 1 } \\right ) e ^ { - 2 x } , \\end{align*}"}
-{"id": "4807.png", "formula": "\\begin{gather*} [ \\lambda ] = \\big \\{ ( i , j ) \\in \\mathbb { Z } ^ 2 \\colon 1 \\leq i \\leq \\ell ( \\lambda ) , \\ 1 \\leq j \\leq \\lambda _ i \\big \\} . \\end{gather*}"}
-{"id": "9123.png", "formula": "\\begin{align*} u ' ( t ) + B ( t ) A ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) , \\ \\ u ( 0 ) = u _ 0 \\end{align*}"}
-{"id": "8514.png", "formula": "\\begin{align*} \\begin{aligned} c ' _ { \\alpha _ 2 } + 2 c ' _ { \\alpha _ 1 + 2 \\alpha _ 2 } + c ' _ { \\alpha _ 1 + \\alpha _ 2 } & = \\# ) \\ + \\ 2 \\# \\Big ) = y , \\\\ c ' _ { \\alpha _ 1 } + c ' _ { \\alpha _ 1 + \\alpha _ 2 } + c ' _ { \\alpha _ 1 + 2 \\alpha _ 2 } & = \\# ( \\ + \\ \\# \\Big ( = x . \\end{aligned} \\end{align*}"}
-{"id": "8882.png", "formula": "\\begin{align*} L _ u w = \\lambda w , w \\in H ^ { 1 } ( \\mathbb { R } ^ N , \\C ) , \\end{align*}"}
-{"id": "2118.png", "formula": "\\begin{align*} D _ { \\l , x ' } : = \\{ ( x ' , z ) \\in \\R ^ { n + 1 } : | z | < \\l \\} . \\end{align*}"}
-{"id": "6961.png", "formula": "\\begin{align*} & a _ { 0 , j } = \\binom { \\gamma } { j } \\mathbb { I } [ 1 \\le j \\le \\gamma - 1 ] , \\\\ & a _ { 1 , j } = \\mathbb { I } [ j = 0 j = \\gamma ] . \\end{align*}"}
-{"id": "9263.png", "formula": "\\begin{align*} j ( \\pi ) = j ( \\pi , z ) = \\mathbb { E } [ \\int _ D U ( x , Y ( T , x , z ) , z ) d x \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] ] , \\end{align*}"}
-{"id": "8767.png", "formula": "\\begin{align*} ( \\mathbb { I } _ 3 \\otimes U ) R _ { 2 , 1 } \\left ( t ^ 3 / x \\right ) ^ { \\tau _ 1 } ( \\mathbb { I } _ 3 \\otimes U ^ { - 1 } ) R _ { 1 , 2 } ( x ) ^ { \\tau _ 1 } = - \\frac { ( x - 1 ) \\left ( t ^ 3 - x \\right ) } { ( t - x ) \\left ( t ^ 2 - x \\right ) } \\mathbb { I } _ 3 \\otimes \\mathbb { I } _ 3 . \\end{align*}"}
-{"id": "9601.png", "formula": "\\begin{align*} \\lambda ( t ) = \\lambda _ { r e g } ( t ) + \\lambda _ { y } ( t ) \\in C ( [ 0 , \\infty ) ) . \\end{align*}"}
-{"id": "8573.png", "formula": "\\begin{align*} H & = s '' b ' r ' r '' a ' C _ { r '' a ' , r a _ { 1 } } z ( b _ { 1 } r a _ { 1 } ) m _ { 1 } \\hdots m _ { l - 1 } s ' n \\\\ & = \\displaystyle \\sum _ { w _ { 1 } \\in L ( k ) } s '' b ' w _ { 1 } ^ { \\ast } ( r ' r '' ) w _ { 1 } a ' C _ { r '' a ' , r a _ { 1 } } z ( b _ { 1 } r a _ { 1 } ) m _ { 1 } \\hdots m _ { l - 1 } s ' n \\end{align*}"}
-{"id": "9865.png", "formula": "\\begin{align*} 0 = \\mu _ 0 ( \\widetilde { \\Omega } ) < \\mu _ { 1 } ( \\widetilde { \\Omega } ) \\leq \\mu _ { 2 } ( \\widetilde { \\Omega } ) \\leq . . . \\leq \\mu _ { n } ( \\widetilde { \\Omega } ) \\leq . . . \\ , , \\end{align*}"}
-{"id": "9063.png", "formula": "\\begin{align*} \\lim _ { \\delta \\to 0 } \\sup _ { N \\geq 2 r } \\left | \\frac { \\P \\left ( G _ 0 v ( d , \\delta , 0 ) \\right ) } { \\P \\left ( G _ 0 v ( d , 0 , 0 ) \\right ) } - 1 \\right | = \\lim _ { \\delta , | | E | | _ 1 \\to 0 } \\sup _ { N \\geq 2 r } \\left | \\frac { \\P \\left ( G _ 0 v ( d , \\delta , E ) \\right ) } { \\P \\left ( G _ 0 v ( d , 0 , 0 ) \\right ) } - 1 \\right | = 0 , \\end{align*}"}
-{"id": "2807.png", "formula": "\\begin{align*} \\mathbb { P } \\left \\{ \\left \\| \\sum _ { k = 1 } ^ { \\widehat { m } } \\mathcal { R } _ { a _ k } \\right \\| \\geq t \\right \\} \\leq 2 n _ 1 n _ 2 \\exp \\left ( \\frac { - t ^ 2 / 2 } { \\frac { 1 2 \\mu c _ s { \\widehat { m } } r } { n ^ 2 } + \\frac { 5 \\mu c _ s r } { n } t / 3 } \\right ) . \\end{align*}"}
-{"id": "6427.png", "formula": "\\begin{align*} & \\int _ { \\Omega } \\psi \\partial _ { t } \\left [ g _ { 1 - \\alpha , m } * ( u - u _ { 0 } ) \\right ] d x + \\mathcal { E } ( h _ { m } * u , \\psi ) \\\\ & \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad = ( \\geq \\leq ) \\int _ { \\Omega } ( h _ { m } * f ) \\psi d x \\ , \\ , \\ , t \\in ( 0 , T ) , \\ , m \\in \\mathbb { N } . \\end{align*}"}
-{"id": "669.png", "formula": "\\begin{align*} \\| f \\| _ \\infty - \\eta ( f ) = \\eta ( \\| f \\| _ \\infty - f ) \\leq \\| \\| f \\| _ \\infty - f \\| _ \\infty \\leq \\| f \\| _ \\infty , \\end{align*}"}
-{"id": "8347.png", "formula": "\\begin{align*} l ( \\mu ) = \\sum _ { i = 1 } ^ n c _ i l ( \\mu _ i ) = \\sum _ { i = 1 } ^ m c _ i \\gamma _ i \\in V , \\end{align*}"}
-{"id": "5941.png", "formula": "\\begin{align*} { \\psi } = { \\psi } _ { \\lambda } = G _ { \\lambda } ( \\mathbb { I } - T _ \\lambda ) ^ { - 1 } g , \\end{align*}"}
-{"id": "3531.png", "formula": "\\begin{align*} \\Phi ^ V _ { ( g , \\pi ) } ( g + h , \\pi + w ) = \\Phi _ { ( g , \\pi ) } ^ V ( g , \\pi ) + ( 2 \\psi , V ) . \\end{align*}"}
-{"id": "5007.png", "formula": "\\begin{align*} b _ { n } = b _ { n - 1 } + C _ { 1 } \\sqrt [ ] { b _ { n - 1 } } \\left ( 1 + \\sqrt [ ] { 1 + \\frac { C _ { 2 } } { b _ { n - 1 } } } \\right ) \\leq b _ { n - 1 } + C _ { 1 } \\left ( 1 + \\sqrt [ ] { 1 + C _ { 2 } } \\right ) \\sqrt [ ] { b _ { n - 1 } } . \\end{align*}"}
-{"id": "3611.png", "formula": "\\begin{align*} \\widetilde { f } ( z ) = F _ x ^ * ( f ) ( z ) = f \\circ F _ x ( z ) \\end{align*}"}
-{"id": "9157.png", "formula": "\\begin{align*} k / m \\geq \\frac { 1 } { 2 } ( 1 + ( t - 1 ) / ( s t ) ) \\longrightarrow \\frac { 1 } { 2 } ( 1 + ( 1 / s ) = ( s + 1 ) / 2 s \\end{align*}"}
-{"id": "9038.png", "formula": "\\begin{align*} \\sup _ { k \\geq 0 } \\sup _ { \\theta \\in [ 0 , 2 \\pi ) } \\left | \\sum _ { j = 0 } ^ { k - 1 } \\gamma _ j ^ 2 e ^ { 2 i \\psi _ j ( \\theta ) } \\right | < \\infty \\end{align*}"}
-{"id": "9551.png", "formula": "\\begin{align*} \\mathcal { W } _ { i ( n + 1 ) + j } = \\{ g _ F \\left ( g _ F ^ { - 1 } \\pi ^ { - 1 } ( F ) \\cap { A } \\right ) \\mid F \\in \\mathcal { F } _ i , A \\in \\mathcal { A } _ j \\} . \\end{align*}"}
-{"id": "4707.png", "formula": "\\begin{align*} z & = ( v _ 1 v _ 2 \\dots v _ { k + m - 1 } v _ m ) ^ 2 v _ { m + 1 } \\dots v _ { k + m - 1 } \\\\ & = v _ 1 v _ 2 \\dots v _ { k + m - 1 } v _ m v _ 1 v _ 2 \\dots v _ { m - 1 } ( v _ m v _ { m + 1 } \\dots v _ { k + m - 1 } ) ^ 2 . \\end{align*}"}
-{"id": "9438.png", "formula": "\\begin{align*} Z ^ { s , z } _ T = u ( s , z ) . \\end{align*}"}
-{"id": "4664.png", "formula": "\\begin{align*} G _ { \\phi , p } ( x ) = & L _ p ( 1 + x , \\phi ) L _ p ( 1 + 3 x , \\phi ) \\\\ & \\times \\left ( 1 + \\frac { \\rho _ \\phi ( p ^ 2 ) - \\rho _ \\phi ( p ) ^ 2 } { p ^ { 2 + 4 x } } + O \\left ( \\frac { ( 1 + | \\rho _ \\phi ( p ) | ) ^ { 3 } } { p ^ { 3 + 7 x } } \\right ) \\right ) . \\end{align*}"}
-{"id": "447.png", "formula": "\\begin{align*} x & = \\prod _ { i = 1 } ^ { \\alpha _ 1 } ( a _ 1 ^ { p _ { 1 i } } a _ 2 ^ { q _ { 1 i } } ) a _ 3 ^ { q _ { 2 1 } } \\prod _ { i = 2 } ^ { \\alpha _ 2 } ( a _ 2 ^ { p _ { 2 i } } a _ 3 ^ { q _ { 2 i } } ) \\\\ & \\quad \\ \\dots a _ m ^ { q _ { ( m - 1 ) 1 } } \\prod _ { i = 2 } ^ { \\alpha _ { m - 1 } } ( a _ { m - 1 } ^ { p _ { ( m - 1 ) i } } a _ m ^ { q _ { ( m - 1 ) i } } ) , \\end{align*}"}
-{"id": "5891.png", "formula": "\\begin{align*} g ^ { i j } \\xi _ { , i j } ^ { \\alpha } - 2 g ^ { i \\alpha } a _ { , i } + a B ^ { \\alpha } - \\xi _ { , i } ^ { \\alpha } B ^ { i } + \\xi ^ { i } B _ { , i } ^ { \\alpha } - \\lambda B ^ { \\alpha } = 0 \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "7767.png", "formula": "\\begin{align*} \\beta ( M ) = \\begin{bmatrix} 1 & - & - & - \\\\ - & 4 & 4 & 1 \\end{bmatrix} = \\frac { 1 } { 3 } \\begin{bmatrix} 1 & - & - & - \\\\ - & 6 & 8 & 3 \\end{bmatrix} + \\frac { 2 } { 3 } \\begin{bmatrix} 1 & - & - \\\\ - & 3 & 2 \\end{bmatrix} . \\end{align*}"}
-{"id": "5381.png", "formula": "\\begin{align*} c _ { 1 , 1 } ( c _ { 1 , 1 } + c _ { 1 , 0 } ) + c _ { 1 , 0 } ~ = ~ 0 . \\end{align*}"}
-{"id": "8082.png", "formula": "\\begin{align*} I ( y + \\alpha ) & = \\int _ { 0 } ^ { 1 } { h ( x , y + \\alpha ) } \\ d x = \\int _ 0 ^ 1 { \\tilde { h } _ y ( x , \\alpha ) } \\ d x = \\int _ 0 ^ 1 { \\tilde { h } _ y ( x , 0 ) } \\ d x \\\\ & = \\int _ { 0 } ^ { 1 } { h ( x , y ) } \\ d x = I ( y ) . \\end{align*}"}
-{"id": "9525.png", "formula": "\\begin{align*} \\delta ( C _ 4 \\wedge H _ 3 \\wedge G _ 3 ) & = C _ 4 \\wedge H _ 3 \\wedge d \\delta C _ 2 \\\\ & = - d ( C _ 4 \\wedge H _ 3 \\wedge \\delta C _ 2 ) + G _ 5 \\wedge H _ 3 \\wedge \\delta C _ 2 \\\\ & = d ( \\ldots ) + \\delta C _ 2 \\wedge G _ 5 \\wedge H _ 3 . \\end{align*}"}
-{"id": "2065.png", "formula": "\\begin{align*} H ( s ) = ( C _ { p } + s C _ { v } ) ( s ^ { 2 } M + s D + K ) ^ { - 1 } F = ( C _ { p } + s C _ { v } ) X ( s ) , \\end{align*}"}
-{"id": "5970.png", "formula": "\\begin{align*} V ( b ) = \\bigcap _ { b \\nmid d _ i } \\{ x \\mid Q ( x ) = 0 \\} . \\end{align*}"}
-{"id": "1004.png", "formula": "\\begin{align*} E _ { n , m } & = t \\left [ \\left ( \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) \\left ( \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) \\eta ^ 2 + \\omega \\eta \\left ( \\sum _ { i = 1 } ^ { m - 1 } k _ i - \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) - \\frac { \\eta ^ 2 N ^ 2 } { 4 } \\right ] \\\\ & = U \\left ( \\sum _ { i = 1 } ^ { n - 1 } l _ i - \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) ^ 2 + \\mu \\left ( \\sum _ { i = 1 } ^ { n - 1 } l _ i - \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) . \\end{align*}"}
-{"id": "6490.png", "formula": "\\begin{align*} u ( x , t ) = ( \\mu * v ) ( x , t ) = \\int _ { 0 } ^ { t } \\mu ( t - s ) v ( x , s ) d s ( 0 < t \\leq T ) , \\end{align*}"}
-{"id": "8605.png", "formula": "\\begin{align*} \\Delta _ { k } ' & = \\displaystyle \\sum _ { s a _ { 1 } , b _ { 1 } t } \\displaystyle \\sum _ { v , v _ { 1 } \\in L ( k ) } ( v _ { 1 } ^ { - 1 } ) ^ { \\ast } ( s ^ { - 1 } t ^ { - 1 } ) v ^ { \\ast } ( t s ) a _ { 1 } ^ { \\ast } v _ { 1 } ^ { - 1 } ( ^ { \\ast } b _ { 1 } ) [ b _ { 1 } v a _ { 1 } ] \\\\ & = \\displaystyle \\sum _ { s a _ { 1 } , b _ { 1 } } \\displaystyle \\sum _ { v , v _ { 1 } , t \\in L ( k ) } ( v _ { 1 } ^ { - 1 } ) ^ { \\ast } ( s ^ { - 1 } t ^ { - 1 } ) v ^ { \\ast } ( t s ) a _ { 1 } ^ { \\ast } v _ { 1 } ^ { - 1 } ( ^ { \\ast } b _ { 1 } ) [ b _ { 1 } v a _ { 1 } ] \\end{align*}"}
-{"id": "6266.png", "formula": "\\begin{align*} = \\frac { \\nabla ^ 2 _ { E , E } w ( x ) - \\nabla ^ 2 _ { E , E } w ( y ) } { \\bar { w } } - \\frac { 2 } { \\bar { w } } ( \\nabla _ E Q ) ( \\nabla _ E \\bar { w } ) - \\frac { Q } { \\bar { w } } \\nabla ^ 2 _ { E , E } \\bar { w } . \\end{align*}"}
-{"id": "7224.png", "formula": "\\begin{align*} P = Q \\circ R , \\end{align*}"}
-{"id": "4149.png", "formula": "\\begin{align*} D _ 1 ( s ) & = \\sum _ { n \\geq 1 } \\frac { A ( 1 , n ) \\overline { A ( 1 , n ) } } { n ^ s } = L ( s , f \\times f ) T _ 1 ( s ) \\\\ D _ 2 ( s ) & = \\sum _ { n \\geq 1 } \\frac { A ( 1 , n ) ^ 2 } { n ^ s } = L ( s , f \\times \\overline { f } ) T _ 2 ( s ) , \\end{align*}"}
-{"id": "7353.png", "formula": "\\begin{align*} F V _ 1 & = F v _ 1 \\otimes K ^ { - 1 } v _ 0 + v _ 1 \\otimes F v _ 0 - q ^ 2 F v _ 0 \\otimes K ^ { - 1 } v _ 1 - q ^ 2 v _ 0 \\otimes F v _ 1 \\\\ & = [ 2 ] ^ { 1 / 2 } ( v _ 0 \\otimes v _ 0 + v _ 1 \\otimes v _ { - 1 } - v _ { - 1 } \\otimes v _ 1 - q ^ 2 v _ 0 \\otimes v _ 0 ) = [ 2 ] ^ { 1 / 2 } V _ 0 . \\end{align*}"}
-{"id": "2591.png", "formula": "\\begin{align*} e ^ { - i t _ 0 \\Delta } u ( t _ 0 , x ) = e ^ { i x \\xi ( t _ 0 ) } h ( t _ 0 ) ^ { \\frac 2 { p } } \\psi ( t _ 0 , h ( t _ 0 ) x ) , \\end{align*}"}
-{"id": "95.png", "formula": "\\begin{align*} 0 = \\sum _ { i = 1 } ^ l D _ { n _ 0 , i } ( C _ { n , i } - C _ { 1 , i } ) . \\end{align*}"}
-{"id": "334.png", "formula": "\\begin{align*} & \\Big | \\widehat { m _ n ( D ) } ( A ) - \\prod _ { i = 1 } ^ r \\prod _ { j = 1 } ^ n \\Theta ( a _ i x _ j ) \\Big | \\le 2 \\cdot ( 1 - \\prod _ { w = 0 } ^ { 2 r - 1 } ( 1 - q ^ { w - 2 n } ) ) , \\end{align*}"}
-{"id": "6622.png", "formula": "\\begin{align*} \\tau _ \\partial ( x _ 0 ) : = \\sup I _ { x _ 0 } \\in [ 0 , \\infty ] , \\end{align*}"}
-{"id": "4771.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a \\ , t ^ 2 ) ^ 2 + t ^ 2 } , a = c o n s t \\neq 0 , c = c o n s t . \\end{align*}"}
-{"id": "3436.png", "formula": "\\begin{align*} \\kappa ( t ) = \\kappa + \\int _ s ^ t q ( \\theta , \\kappa ( \\theta ) ) d \\theta + \\int _ s ^ t Q ( \\theta , \\kappa ( \\theta ) ) d W ( \\theta ) , \\end{align*}"}
-{"id": "4959.png", "formula": "\\begin{align*} - \\Delta v _ n - \\lambda _ n c _ n ^ 2 v _ n = Q _ n v _ n ^ { p - 1 } + t _ n M _ n ^ { 1 - p } Q _ n \\end{align*}"}
-{"id": "2886.png", "formula": "\\begin{align*} C H _ { P o i s _ n } ^ { ( \\bullet > 0 ) } ( A , A ) = H o m _ { \\Sigma } ( { P o i s _ n } ^ * \\{ n \\} , E n d _ A ) [ - n ] \\end{align*}"}
-{"id": "7787.png", "formula": "\\begin{align*} f _ i : = \\bigsqcup _ k f _ { i ; k - 1 } ^ { - 1 } f _ { i ; k } , \\quad e _ i : = \\bigsqcup _ k f _ { i ; k + 1 } ^ { - 1 } f _ { i ; k } , \\end{align*}"}
-{"id": "4258.png", "formula": "\\begin{align*} \\alpha ( x ) & = \\sum _ { k \\geq 1 } 2 i \\alpha _ k \\sin 2 \\pi k x , & \\beta ( x ) & = \\sum _ { k \\geq 0 } 2 \\beta _ k \\cos 2 \\pi k x \\end{align*}"}
-{"id": "7248.png", "formula": "\\begin{align*} D _ { j k } = \\left ( \\begin{array} { c | c c c } t _ { j k } ^ 1 & 0 & \\cdots & 0 \\\\ \\hline a _ { j k } ^ 2 & & & \\\\ \\vdots & & S _ { j k } & \\\\ a _ { j k } ^ r & & & \\end{array} \\right ) , \\end{align*}"}
-{"id": "1272.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & - 1 & \\alpha & 0 & \\overline \\alpha \\\\ 0 & q & - q \\alpha & 0 & - q \\overline \\alpha \\\\ 0 & - \\overline \\alpha & q & - \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & 0 \\\\ 0 & 0 & - \\overline \\alpha J ( T ^ { - r } , T ^ { - r } ) & q & - \\alpha J ( T ^ { - 3 r } , T ^ { - 3 r } ) \\\\ 0 & - \\alpha & 0 & - \\overline \\alpha J ( T ^ { - 2 r } , T ^ { - r } ) & q \\end{bmatrix} \\end{align*}"}
-{"id": "9728.png", "formula": "\\begin{align*} g ( x _ { L , \\epsilon } ( t ) ) e ^ { c _ 1 \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } & = x _ 1 ( \\epsilon ) \\\\ & < \\min _ { T _ 7 ( \\epsilon ) \\leq s \\leq T _ 8 ( \\epsilon ) } g ( x ( s ) ) e ^ { c _ 1 \\int _ 0 ^ s \\frac { 1 } { \\sigma ( u ) } \\ , d u } \\\\ & \\leq g ( x ( t ) ) e ^ { c _ 1 \\int _ 0 ^ t \\frac { 1 } { \\sigma ( u ) } \\ , d u } . \\end{align*}"}
-{"id": "1240.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\varphi _ * \\left ( \\frac { \\delta | \\eta ( k ) | } { v ( k ) } \\right ) v ( k ) \\le p _ { \\varphi _ * , w } ( \\delta \\eta ) + 1 < \\infty . \\end{align*}"}
-{"id": "4379.png", "formula": "\\begin{align*} F ( t , c , h ) = \\frac { F ( t , c + c _ 0 , h + h _ 0 ) } { F ( t , c _ 0 , h _ 0 ) } . \\end{align*}"}
-{"id": "2055.png", "formula": "\\begin{align*} 0 \\geq \\langle \\nabla _ k \\nabla _ k \\nabla w , e _ n \\rangle , k = 1 , \\ldots n - 1 . \\end{align*}"}
-{"id": "6021.png", "formula": "\\begin{gather*} \\hat { \\Omega } = \\mathrm { d } \\gamma + \\tfrac { 1 } { 2 } [ \\gamma , \\gamma ] + \\tfrac { 1 } { 2 } [ \\theta , \\theta ] \\in \\Omega ^ 2 ( \\mathcal { S } , \\mathfrak { a } ) . \\end{gather*}"}
-{"id": "5815.png", "formula": "\\begin{align*} S _ { A C , D , \\omega _ k ^ { \\vee } } = ( \\lambda , 0 , \\mu , 0 ) \\end{align*}"}
-{"id": "6409.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } f ( t ) : = \\frac { d } { d t } ( g _ { 1 - \\alpha } * f ( \\cdot ) ) ( t ) , \\end{align*}"}
-{"id": "1628.png", "formula": "\\begin{align*} g ^ { i j } \\xi _ { , i j } ^ { \\alpha } - 2 g ^ { i \\alpha } a _ { , i } + a B ^ { \\alpha } - \\xi _ { , i } ^ { \\alpha } B ^ { i } + \\xi ^ { i } B _ { , i } ^ { \\alpha } - \\lambda B ^ { \\alpha } = 0 \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "5572.png", "formula": "\\begin{align*} \\begin{gathered} b _ { - 1 } ( x ) \\stackrel { } { = } \\frac { c _ 0 ( x ) \\hat c _ 0 ( x ) } { W ( c _ 0 , \\hat c _ 0 ) } , \\\\ \\\\ b _ 0 ( x ) \\stackrel { } { = } \\ , \\frac { - ( c _ 0 ( x ) \\hat c _ 1 ( x ) + c _ 1 ( x ) \\hat c _ 0 ( x ) ) } { W ( c _ 0 , \\hat c _ 0 ) } , \\end{gathered} \\end{align*}"}
-{"id": "4416.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\| f _ n - g _ n \\| _ { \\widehat { E } } < \\infty . \\end{align*}"}
-{"id": "1299.png", "formula": "\\begin{align*} \\vert T _ H ( N , y ; \\alpha ) \\vert \\le \\sum _ { n = N - H } ^ { N + y } t _ H ( n - N ) \\ll H ( H + y + 1 ) \\ll H ^ 2 . \\end{align*}"}
-{"id": "1282.png", "formula": "\\begin{align*} K _ i = \\begin{bmatrix} 0 & J ( i + 2 r , 2 r ) & 0 & 0 \\\\ J ( i , 2 r ) & 0 & 0 & 0 \\\\ 0 & 0 & 0 & J ( i + 3 r , 2 r ) \\\\ 0 & 0 & J ( i + r , 2 r ) & 0 \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "762.png", "formula": "\\begin{gather*} u _ k \\otimes u _ l = u _ { | k - l | } \\oplus u _ { | k - l | + 2 } \\oplus \\cdots \\oplus u _ { k + l } . \\end{gather*}"}
-{"id": "5681.png", "formula": "\\begin{align*} \\check K ( x ) = ( 1 - x \\bar e _ 0 ) \\left ( 1 - \\frac 1 x \\bar e _ 0 \\right ) ^ { - 1 } \\end{align*}"}
-{"id": "7078.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { \\sum _ { | u | = n } e ^ { - \\beta V ( u ) } } \\sum _ { | u | = n } e ^ { - \\beta V ( u ) } \\delta _ { H _ n ( u ) } = \\sum _ { k \\in \\N } p _ k \\delta _ { \\mathbf { e } _ k } , \\end{align*}"}
-{"id": "2903.png", "formula": "\\begin{align*} s _ \\lambda ( X ) = & \\ , \\sum _ { T \\in { \\mathcal T } ^ \\lambda } X ^ T \\ , . \\end{align*}"}
-{"id": "8631.png", "formula": "\\begin{align*} a _ { l , n - l + 1 } = ( - 1 ) ^ { n + 1 - 2 l } \\ a _ { n - l + 1 , l } . \\end{align*}"}
-{"id": "504.png", "formula": "\\begin{align*} \\frac { 3 m + \\ell - 3 | L | + 1 } { 3 m + \\ell - | L | + 1 } \\binom { 3 m + \\ell } { | L | } \\ ; = \\ ; \\binom { 3 m + \\ell } { | L | } - 2 \\binom { 3 m + \\ell } { | L | - 1 } \\ ; . \\end{align*}"}
-{"id": "312.png", "formula": "\\begin{align*} \\mathcal { J } _ 3 \\ ! = \\ ! \\bigl \\{ & \\emptyset , \\{ 1 \\} , \\{ 2 \\} , \\{ 1 , 1 \\} , \\{ 1 , 2 \\} , \\{ 2 , 1 \\} , \\{ 2 , 2 \\} , \\\\ & \\{ 1 , 1 , 1 \\} , \\ ! \\{ 1 , 1 , 2 \\} , \\ ! \\{ 1 , 2 , 1 \\} , \\ ! \\{ 1 , 2 , 2 \\} , \\ ! \\{ 2 , 1 , 1 \\} , \\ ! \\{ 2 , 1 , 2 \\} , \\ ! \\{ 2 , 2 , 1 \\} , \\ ! \\{ 2 , 2 , 2 \\} \\bigr \\} . \\end{align*}"}
-{"id": "7005.png", "formula": "\\begin{align*} \\sum _ { i + j = n } { - \\lambda _ i ( x , \\lambda _ j ( y , z ) ) + \\lambda _ i ( y , \\lambda _ j ( x , z ) ) + \\lambda _ i ( \\lambda _ j ( x , y ) , z ) } = 0 , \\end{align*}"}
-{"id": "7816.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\infty } \\left \\vert \\overset { \\circ } { \\mathrm { R m } _ { \\Sigma } } \\right \\vert \\left ( x \\right ) = 0 \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\lim _ { x \\rightarrow \\infty } \\left \\vert S ( x ) - \\frac { n - 1 } { 2 } \\right \\vert = 0 . \\end{align*}"}
-{"id": "2517.png", "formula": "\\begin{align*} & \\mathcal { L } _ { X } D _ 1 = ( - \\lambda ) ( D _ 1 - d _ 1 ) , \\dots , \\mathcal { L } _ { X } D _ { p ^ { \\prime } } = ( - \\lambda ) ( D _ { p ^ { \\prime } } - d _ { p ^ { \\prime } } ) , \\\\ & \\mathcal { L } _ { X } D _ { p ^ { \\prime } + 1 } = \\dots = \\mathcal { L } _ { X } D _ { p } = \\mathcal { L } _ { X } I _ 1 = \\dots = \\mathcal { L } _ { X } I _ k = 0 , \\end{align*}"}
-{"id": "4056.png", "formula": "\\begin{align*} ( \\alpha - 1 ) p + ( 2 - \\alpha ) q = 1 + n + k . \\end{align*}"}
-{"id": "700.png", "formula": "\\begin{align*} h _ { D } = \\sum _ { i = 1 } ^ { m } a _ { i } h _ { H _ { i } } . \\end{align*}"}
-{"id": "5159.png", "formula": "\\begin{gather*} T u = v T . \\end{gather*}"}
-{"id": "7305.png", "formula": "\\begin{align*} \\gamma _ - ( w ) \\gamma _ + ( v ) + \\gamma _ + ( v ) \\gamma _ - ( w ) = \\langle w , v \\rangle 1 , w \\in \\mathfrak { u } _ - , \\ v \\in \\mathfrak { u } _ + . \\end{align*}"}
-{"id": "4687.png", "formula": "\\begin{align*} X _ j : = \\frac { \\partial } { \\partial z _ j } + \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ d L _ { j k } z _ k \\frac { \\partial } { \\partial t } , j = 1 , 2 , \\cdots , d . \\end{align*}"}
-{"id": "3001.png", "formula": "\\begin{align*} \\Gamma ( t , p ) = \\left \\{ P \\in \\Pi ( X ) \\mid \\tilde { u } ( t , P ) \\ge \\max _ { Q \\in B _ \\mathcal { R } ( t , p ) } \\tilde { u } ( t , Q ) \\right \\} \\end{align*}"}
-{"id": "7802.png", "formula": "\\begin{align*} \\left \\vert \\left \\langle \\nabla \\ln S , \\nabla \\psi \\right \\rangle \\right \\vert = \\frac { e ^ { \\frac { f } { \\sqrt { T - \\sqrt { T } } } } } { \\sqrt { T - \\sqrt { T } } e ^ { \\sqrt { T - \\sqrt { T } } } } \\left \\vert \\left \\langle \\nabla S , \\nabla f \\right \\rangle \\right \\vert S ^ { - 1 } . \\end{align*}"}
-{"id": "4450.png", "formula": "\\begin{align*} \\lambda _ 1 = - \\frac { 1 } { 2 ( d + 2 ) V _ d ^ { 2 / d } } \\int _ \\mathcal { X } \\frac { \\Delta f ( x ) } { f ( x ) ^ { 2 / d } } \\ , d x , \\end{align*}"}
-{"id": "6104.png", "formula": "\\begin{align*} \\phi ^ * ( d t / t ) = d z / z + d w / w . \\end{align*}"}
-{"id": "6605.png", "formula": "\\begin{align*} \\lambda _ { i } ( L ) = \\begin{cases} \\alpha _ 2 & { \\hbox { f o r $ i $ s u c h t h a t } } \\lambda _ { i } ( N ) = 1 \\\\ ( 1 - \\alpha _ 1 ) ( 1 + ( \\alpha _ 2 - 1 ) ( 1 - \\alpha _ 1 ) ) & { \\hbox { f o r $ i $ s u c h t h a t } } \\lambda _ { i } ( N ) = 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "8349.png", "formula": "\\begin{align*} \\lim _ \\lambda \\omega ( E ( ( x - x _ \\lambda ) ^ * ( x - x _ \\lambda ) ) ) = 0 , \\ \\ \\ \\omega \\in A _ * , \\end{align*}"}
-{"id": "658.png", "formula": "\\begin{align*} f _ { s ^ { - 1 } \\cdot \\phi } ( g ) = f _ \\phi ( s g ) = ( s ^ { - 1 } \\cdot f _ \\phi ) ( g ) , \\textrm { f o r a l l $ g , s \\in G $ } . \\end{align*}"}
-{"id": "8575.png", "formula": "\\begin{align*} \\operatorname { k e r } ( \\gamma ) & = D ( \\operatorname { k e r } ( \\hat { \\gamma } ) ) \\\\ \\operatorname { i m } ( \\gamma ) & = C ^ { - 1 } ( \\operatorname { i m } ( \\hat { \\gamma } ) ) \\end{align*}"}
-{"id": "815.png", "formula": "\\begin{align*} & C _ { a } ^ { [ 1 , M + 1 ] } ( z ) X | \\mathrm { v a c } \\rangle _ { [ 1 , M + 1 ] } = ( 1 + z ) C _ { a } ^ { [ 1 , M ] } ( z ) X | \\mathrm { v a c } \\rangle _ { [ 1 , M ] } , \\\\ & C _ { a } ^ { [ 0 , M ] } ( z ) X | \\mathrm { v a c } \\rangle _ { [ 0 , M ] } = z \\ , C _ { a } ^ { [ 1 , M ] } ( z ) X | \\mathrm { v a c } \\rangle _ { [ 1 , M ] } . \\end{align*}"}
-{"id": "2532.png", "formula": "\\begin{align*} \\mathbf { i } Z ( U ) \\circ d \\beta ( t ) = & \\mathbf { i } \\sum _ { k = 1 } ^ K \\sqrt { \\eta _ k } E _ k U \\circ d \\beta _ k ( t ) = - \\frac 1 2 \\sum _ { k = 1 } ^ K \\eta _ k E _ k ^ 2 U d t + \\mathbf { i } \\sum _ { k = 1 } ^ K \\sqrt { \\eta _ k } E _ k U d \\beta _ k ( t ) \\\\ = : & - \\hat { E } U d t + \\mathbf { i } \\sum _ { k = 1 } ^ K \\sqrt { \\eta _ k } E _ k U d \\beta _ k ( t ) \\end{align*}"}
-{"id": "3730.png", "formula": "\\begin{align*} | N ( x ) \\cap O ( i , j , k ) | & = \\# \\{ y \\in O ( i , j , k ) \\ , | \\ , x ^ T K y = 1 \\} \\\\ & = \\begin{cases} | O ( i , j , k ) | & , \\\\ 0 & . \\end{cases} \\end{align*}"}
-{"id": "9717.png", "formula": "\\begin{align*} x _ { U , \\epsilon } ( t ) & \\leq x _ { U , \\epsilon } ( T _ 1 ( \\epsilon ) ) \\\\ & = G _ 0 ^ { - 1 } ( \\lambda ( \\epsilon ) T _ 1 ( \\epsilon ) ) \\\\ & = G _ 0 ^ { - 1 } ( \\eta ( \\epsilon ) ( 1 - q - \\epsilon ) G _ 0 ( \\delta ( \\epsilon ) ) ) . \\end{align*}"}
-{"id": "790.png", "formula": "\\begin{align*} \\phi ( \\tau ' \\tau ; \\vec { z } ) = \\phi ( \\tau ' ; z _ { \\tau ^ { - 1 } ( 1 ) } , \\ldots , z _ { \\tau ^ { - 1 } ( k ) } ) \\phi ( \\tau ; \\vec { z } ) . \\end{align*}"}
-{"id": "6595.png", "formula": "\\begin{align*} z ^ { k + 1 } & = ( 1 - \\alpha ) z ^ { k } + \\alpha R _ { \\gamma g } ( R _ { \\gamma f } ( z ^ k ) ) ) . \\end{align*}"}
-{"id": "7711.png", "formula": "\\begin{align*} B _ i : = \\{ \\omega \\in \\Omega : \\xi _ s \\in [ 1 - \\varepsilon , 1 ] , s = ( i - 1 ) J + 1 , \\ , ( i - 1 ) J + 2 , \\dots , \\ , i J \\} i \\ge i _ 0 : = \\frac L J + 1 . \\end{align*}"}
-{"id": "4063.png", "formula": "\\begin{align*} p = m + n + l , \\end{align*}"}
-{"id": "3064.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\sum _ { | u | = n } e ^ { - \\beta V ( u ) } \\delta _ u } { \\sum _ { | u | = n } e ^ { - \\beta V ( u ) } } = \\frac { \\nu _ \\beta } { W _ \\infty ( \\beta ) } \\end{align*}"}
-{"id": "6511.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 3 } & \\frac { \\partial u } { \\partial t } + \\varDelta ^ 2 u = \\varDelta f ( u ) \\ , \\ , \\ , & & \\ , \\ , & & \\R ^ d \\times ( 0 , T ) , \\\\ & u ( \\cdot , 0 ) = u _ 0 & & \\ , \\ , & & \\R ^ d , \\end{alignedat} \\right . \\end{align*}"}
-{"id": "5544.png", "formula": "\\begin{align*} \\dot x _ j & = - b _ 0 ( x _ j ) , & - \\epsilon [ b _ { 0 , x } ] ( x _ j ) & = [ b _ { - 1 , x } ] ( x _ j ) , & \\frac 1 2 [ b _ { - 1 , x } ] ( x _ j ) & = \\epsilon m _ i b _ { - 1 } ( x _ j ) \\\\ \\dot m _ j & = m _ j \\langle b _ { 0 , x } \\rangle ( x _ j ) , & - \\epsilon [ b _ { 0 , x x } ] ( x _ j ) & = [ b _ { - 1 , x x } ] ( x _ j ) , & \\frac 1 2 [ b _ { - 1 , x x } ] ( x _ j ) & = \\epsilon m _ j \\langle b _ { - 1 , x } \\rangle ( x _ j ) , \\end{align*}"}
-{"id": "7100.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\sum _ { | u | = n } e ^ { - \\beta V ( u ) } \\delta _ u } { \\sum _ { | u | = n } e ^ { - \\beta V ( u ) } } = \\frac { \\nu _ \\beta } { W _ \\infty ( \\beta ) } \\end{align*}"}
-{"id": "2777.png", "formula": "\\begin{align*} \\limsup \\lambda _ i ( \\alpha ) = \\lambda _ 0 \\end{align*}"}
-{"id": "721.png", "formula": "\\begin{align*} & \\sum _ { n = 1 } ^ { \\infty } \\left \\{ \\left ( \\widetilde { C } + C + c _ { 1 } C \\right ) \\frac { \\sqrt [ ] { h _ { H } ( f ^ { k ( n - 1 ) } ( P ) ) } } { \\rho _ { k } ^ { n } } + \\frac { C \\sqrt [ ] { \\gamma } } { \\rho _ { k } ^ { n } } \\right \\} \\\\ \\leq & \\sum _ { n = 1 } ^ { \\infty } \\left \\{ \\left ( \\widetilde { C } + C + c _ { 1 } C \\right ) \\frac { \\sqrt [ ] { \\widetilde { K } h _ { H } ( P ) } ( n - 1 ) \\rho _ { k } ^ { ( n - 1 ) / 2 } } { \\rho _ { k } ^ { n } } + \\frac { C \\sqrt [ ] { \\gamma } } { \\rho _ { k } ^ { n } } \\right \\} . \\end{align*}"}
-{"id": "1248.png", "formula": "\\begin{align*} g ^ \\pm ( z ) & = - \\frac 1 { 2 \\pi } \\big ( \\log ( z / 2 ) + \\gamma \\big ) \\pm \\frac { i } 4 \\\\ g _ 1 ^ \\pm ( z ) & = - \\frac { z ^ 2 } 4 g ^ \\pm ( z ) - \\frac { z ^ 2 } { 8 \\pi } \\end{align*}"}
-{"id": "2869.png", "formula": "\\begin{align*} \\overline { N ^ * } \\mathcal { B } _ { E _ 1 } ( R e s ^ { \\bullet } ( C ) ) = F ^ c ( \\overline { N ^ * } \\mathcal { B } _ { E _ 1 } ^ { \\mathbb { K } } ( K ^ { \\bullet } ( C ) ) ) \\end{align*}"}
-{"id": "6278.png", "formula": "\\begin{align*} p ( x ) = 0 ~ { \\rm f o r } ~ p ( x ) = \\sum ^ { d } _ { j = 0 } p _ j x ^ j = p _ d \\prod ^ d _ { i = 1 } ( x - x _ i ) , ~ ~ ~ p _ d \\ne 0 , \\end{align*}"}
-{"id": "4205.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - \\nabla J ( x _ k ) , k = 0 , \\ldots , \\end{align*}"}
-{"id": "778.png", "formula": "\\begin{align*} \\frac { 1 - q ^ { 2 n _ { a } } } { 1 - q ^ { 2 } } q ^ { 2 \\sum _ { p = a + 1 } ^ { r } n _ { p } } , \\end{align*}"}
-{"id": "3871.png", "formula": "\\begin{align*} Q ( x ) \\circ \\nu ( \\xi ( x ) ) = 0 \\end{align*}"}
-{"id": "198.png", "formula": "\\begin{align*} \\mathcal { X } _ n = \\biggl [ \\Bigl ( \\frac { k } { n } \\Bigr ) ^ { \\frac { 1 } { a } - \\tau } , ( 1 - \\tau ) \\log \\frac { n } { k } \\biggr ] \\end{align*}"}
-{"id": "5392.png", "formula": "\\begin{align*} \\binom { N } { 2 } = ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } & \\leq k \\left ( k - \\alpha - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } \\\\ \\alpha ^ 2 - k \\alpha + \\frac { k } { 4 } - \\frac { k - \\alpha } { n } & \\leq 0 \\ , . \\end{align*}"}
-{"id": "4546.png", "formula": "\\begin{align*} \\mu & : = \\mathbb { E } ( Y _ i ) = \\begin{pmatrix} p _ { n , x , u } \\\\ p _ { n , y , v } \\end{pmatrix} \\\\ V & : = \\mathrm { C o v } ( Y _ i ) = \\begin{pmatrix} p _ { n , x , u } ( 1 - p _ { n , x , u } ) & p _ \\cap - p _ { n , x , u } p _ { n , y , v } \\\\ p _ \\cap - p _ { n , x , u } p _ { n , y , v } & p _ { n , y , v } ( 1 - p _ { n , y , v } ) \\end{pmatrix} , \\end{align*}"}
-{"id": "7255.png", "formula": "\\begin{align*} h _ { 2 , j k , \\alpha } ( z _ j ) = \\left ( \\begin{array} { c } h _ { 2 , j k , \\alpha } ^ 1 ( z _ j ) \\\\ h _ { 2 , j k , \\alpha } ^ 2 ( z _ j ) \\\\ \\vdots \\\\ h _ { 2 , j k , \\alpha } ^ r ( z _ j ) \\end{array} \\right ) \\end{align*}"}
-{"id": "4769.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a \\ , t ^ 2 ) ^ 2 + t ^ 2 } , a = c o n s t \\neq 0 , c = c o n s t , \\end{align*}"}
-{"id": "117.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n 4 ^ n \\int _ { \\mathbb { Z } _ p } { \\frac { x } { 2 } \\choose n } d \\mu _ { - 1 } ( x ) t ^ n & = \\sum _ { n = 0 } ^ \\infty C _ n t ^ n \\\\ & = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n + 1 } { 2 n \\choose n } t ^ n . \\end{align*}"}
-{"id": "952.png", "formula": "\\begin{align*} \\bar \\partial _ M ^ * f = - \\sum _ { j = 1 } ^ { m - 1 } \\sideset { } { ' } \\sum _ { \\vert K \\vert = q - 1 } L _ j f _ { j K } \\overline { \\omega _ K } + \\ ; , \\end{align*}"}
-{"id": "6093.png", "formula": "\\begin{align*} C _ 4 ( G ) = \\sum _ { \\{ u , u ' \\} \\in \\binom { U } { 2 } } \\binom { \\deg ( u , u ' ) } { 2 } \\ , . \\end{align*}"}
-{"id": "1870.png", "formula": "\\begin{align*} | \\mathring { R i c } | ^ 2 = & \\sum _ i ( \\frac R 4 - \\lambda _ i ) ^ 2 = \\lambda _ 1 ^ 2 + \\lambda _ 2 ^ 2 + \\lambda _ 3 ^ 2 + \\lambda _ 4 ^ 2 - \\frac 1 4 R ^ 2 \\\\ = & ( \\frac { 1 - \\eta } 2 ) ^ 2 R ^ 2 - 2 \\lambda _ 1 \\lambda _ 2 + 2 ( \\frac { 1 + \\eta } 4 ) ^ 2 R ^ 2 + 2 y ^ 2 - \\frac 1 4 R ^ 2 \\\\ = & \\frac 1 8 \\cdot ( 3 \\eta ^ 2 - 2 \\eta + 1 ) R ^ 2 + 2 y ^ 2 - 2 \\lambda _ 1 \\lambda _ 2 . \\\\ \\end{align*}"}
-{"id": "5904.png", "formula": "\\begin{align*} F \\left ( x , v \\right ) & = \\pm \\theta \\left ( x , v \\right ) s i g n \\left ( x \\right ) \\left \\vert x \\right \\vert ^ { \\alpha } \\\\ \\alpha & \\in \\left ( \\frac { 1 } { 2 } , 1 \\right ) , \\theta \\in C _ { c } ^ { \\infty } \\left ( \\mathbb { R } ^ { 2 } \\right ) . \\end{align*}"}
-{"id": "9434.png", "formula": "\\begin{align*} w \\{ e _ 0 \\mapsto k \\} ( e ) \\coloneqq \\begin{cases} k & e = e _ 0 \\\\ w ( e ) & . \\end{cases} \\end{align*}"}
-{"id": "9891.png", "formula": "\\begin{align*} T ( x ; D ) = \\sum _ { \\ell \\le D } \\left \\lvert \\underset { x D ^ { - 1 } < z \\le x ^ 2 D ^ { - 2 } } { \\sum _ { \\ell m \\le x } } a _ { \\ell m } \\mu ( m ) \\right \\rvert \\end{align*}"}
-{"id": "9385.png", "formula": "\\begin{align*} \\widetilde { \\Gamma } _ 0 u _ \\lambda = 1 , \\widetilde { \\Gamma } _ 1 u _ \\lambda = ( { q } , G \\ast { q } ) + 2 i k [ 1 + ( G \\ast { q } ) ( 0 ) ] [ 1 + ( G \\ast { { q } ^ * } ) ( 0 ) ] , \\end{align*}"}
-{"id": "9967.png", "formula": "\\begin{align*} \\mathbf { W } _ t = \\mathbf { H } _ t ^ { - 1 } . \\end{align*}"}
-{"id": "4020.png", "formula": "\\begin{align*} \\langle n _ i ^ { \\pm } S , g \\rangle = \\int _ { B } \\partial _ i ^ { \\mp , h } g d V ^ h = \\int _ { B } \\partial _ i ^ { \\mp , h } f d V ^ h . \\end{align*}"}
-{"id": "1611.png", "formula": "\\begin{align*} Z ^ { 3 } = e ^ { 2 m t } \\left ( \\partial _ { t } + H + c K ^ { 1 } \\right ) \\end{align*}"}
-{"id": "6295.png", "formula": "\\begin{align*} \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } = \\{ l \\in \\mathbb { Z } ^ n : ~ \\Box _ k ^ { \\alpha _ 2 } \\circ \\Box _ l ^ { \\alpha _ 1 } \\neq 0 \\} , ~ \\widetilde { \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } = \\{ l \\in \\mathbb { Z } ^ n : ~ \\Box _ k ^ { \\alpha _ 2 } \\circ \\Box _ l ^ { \\alpha _ 1 } = \\Box _ l ^ { \\alpha _ 1 } \\} . \\end{align*}"}
-{"id": "7138.png", "formula": "\\begin{align*} \\tilde G _ { i j } ( x , y ) | _ { x _ n = 0 } = 0 \\quad ( i < n ) ; \\tilde G _ { n j } ( x , y ) | _ { x _ n = 0 } = 2 U _ { n j } ( x ' - y ' , - y _ n ) . \\end{align*}"}
-{"id": "5050.png", "formula": "\\begin{align*} \\mu _ i ( \\phi ) = \\eta _ i ( P \\phi ) , \\textrm { f o r $ \\phi \\in L ^ \\infty ( Z , m ) $ } . \\end{align*}"}
-{"id": "7683.png", "formula": "\\begin{align*} x _ { n + 1 } = f ( x _ n ) , x _ 0 > 0 , n = 0 , 1 , \\dots \\end{align*}"}
-{"id": "4887.png", "formula": "\\begin{align*} \\alpha _ 3 ( x ) = ( \\pi ( x ) , \\pi ( \\delta ( x ) ) , \\pi ( \\delta ( \\delta ( x - [ \\pi ( x ) ] ) + ( - 1 ) ^ p [ \\pi ( \\delta ( x ) ) ] ) ) ) \\end{align*}"}
-{"id": "5585.png", "formula": "\\begin{align*} \\mathcal R _ 0 ( \\lambda ) = ( D _ m + \\lambda ) R _ 0 ( \\lambda ^ 2 - m ^ 2 ) . \\end{align*}"}
-{"id": "7969.png", "formula": "\\begin{align*} { \\widetilde { + } _ { \\varphi } ( p _ 1 , \\cdots \\ ! , p _ m ) } \\leq { \\tau _ 0 } ^ { - 1 } \\cdot { \\sum _ { j = 1 } ^ m p _ j } , \\end{align*}"}
-{"id": "9379.png", "formula": "\\begin{align*} 2 d k ^ 2 + i k ( \\det { \\mathbf { T } } - 4 ) + 2 a = 0 , \\end{align*}"}
-{"id": "1254.png", "formula": "\\begin{align*} V = B ^ * \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) U \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) B = v ^ * U v , \\end{align*}"}
-{"id": "7397.png", "formula": "\\begin{align*} S _ R & = ( \\{ - R _ 1 , R _ 1 \\} \\times [ - R _ 2 , R _ 2 ] \\times \\cdots \\times [ - R _ n , R _ n ] ) \\\\ & \\cup ( [ - R _ 1 , R _ 1 ] \\times \\{ - R _ 2 , R _ 2 \\} \\times \\cdots \\times [ - R _ n , R _ n ] ) \\\\ & \\cup \\cdots \\cup ( [ - R _ 1 , R _ 1 ] \\times [ - R _ 2 , R _ 2 ] \\times \\cdots \\times \\{ - R _ n , R _ n \\} ) . \\\\ \\end{align*}"}
-{"id": "5321.png", "formula": "\\begin{align*} H _ { n , m } = t \\left ( \\tau ( u ) + \\omega ^ 2 - u ^ 2 - \\eta ^ { - 2 } - u \\tau ' ( 0 ) - \\frac { \\tau ' ( 0 ) ^ 2 } { 4 } \\right ) , \\end{align*}"}
-{"id": "1728.png", "formula": "\\begin{align*} \\hat { \\Omega } _ { i j } { } ^ k { } _ l = \\hat { R } _ { i j } { } ^ k { } _ l + \\varepsilon \\hat { g } _ { j l } \\delta ^ k { } _ i - \\varepsilon \\hat { g } _ { i l } \\delta ^ k { } _ j + \\hat { K } _ { j l } \\hat { K } ^ { k } { } _ i - \\hat { K } _ { i l } \\hat { K } ^ k { } _ j - 2 \\ , \\hat { K } _ { i j } \\hat { K } ^ k { } _ { l } , \\end{align*}"}
-{"id": "4035.png", "formula": "\\begin{align*} W ^ { \\sigma } ( A ; \\gamma ) & : = \\cup _ { x \\in A } W ^ { \\sigma } ( x ; \\gamma ) , \\sigma \\neq c \\\\ s a t ^ c ( A ) & : = \\cup _ { x \\in A } W ^ { c } ( x ; \\gamma ) . \\end{align*}"}
-{"id": "7335.png", "formula": "\\begin{align*} \\psi ( F v _ 1 ) = [ 2 ] ^ { 1 / 2 } \\beta w _ 0 , \\psi ( F v _ 0 ) = [ 2 ] ^ { 1 / 2 } \\gamma w _ 1 , \\psi ( F v _ { - 1 } ) = 0 . \\end{align*}"}
-{"id": "3191.png", "formula": "\\begin{align*} \\widehat { T } _ { j k } \\widehat { w } _ k = \\widehat { w } _ j + \\sum _ { | \\alpha | \\geq n + 1 } M _ j \\cdot f _ { k j , \\alpha } \\cdot \\prod _ { \\lambda = 1 } ^ r \\left ( \\sum _ { \\mu = 1 } ^ r ( M _ j ^ { - 1 } ) ^ \\lambda _ \\mu \\cdot \\widehat { w } _ j ^ \\mu \\right ) ^ { \\alpha _ \\nu } = \\widehat { w } _ j + O ( | \\widehat { w } _ j | ^ { n + 1 } ) , \\end{align*}"}
-{"id": "9760.png", "formula": "\\begin{align*} V ( \\bar { \\vec { x } } ) = L ( \\vec { z } ^ { * } ( \\bar { \\vec { x } } ) , \\boldsymbol { \\lambda } ^ { * } ( \\bar { \\vec { x } } ) , \\bar { \\vec { x } } ) = f ( \\vec { z } ^ { * } ( \\bar { \\vec { x } } ) , \\bar { \\vec { x } } ) + [ \\boldsymbol { \\lambda } ^ { * } ( \\bar { \\vec { x } } ) ] ^ { \\top } \\vec { g } ( \\vec { z } ^ { * } ( \\bar { \\vec { x } } ) , \\bar { \\vec { x } } ) . \\end{align*}"}
-{"id": "5925.png", "formula": "\\begin{align*} { \\psi } ( z ) = G _ { \\lambda } g ( z ) = \\int _ { 0 } ^ { + \\infty } e ^ { - \\lambda t } P _ t g ( z ) \\ , \\dd t \\ , . \\end{align*}"}
-{"id": "3179.png", "formula": "\\begin{align*} T _ { j k } w _ k = w _ j + \\sum _ { | \\alpha | \\geq 2 } f _ { k j , \\alpha } ( z _ j ) \\cdot w _ j ^ \\alpha , \\end{align*}"}
-{"id": "5479.png", "formula": "\\begin{align*} { \\tilde \\mu _ { i j } } \\left ( { { f _ { i j } } \\left ( x \\right ) } \\right ) + d _ { i j } ^ - - d _ { i j } ^ + = 1 \\ , \\ , \\ , \\ , , i = 1 , 2 , . . . m ; j = 1 , 2 , . . . , p _ m \\end{align*}"}
-{"id": "135.png", "formula": "\\begin{align*} V \\mu ( x ) = V \\mu ( x _ 0 ) + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } = x _ 0 + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } = \\abs { x } r + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } . \\end{align*}"}
-{"id": "3258.png", "formula": "\\begin{align*} E _ k \\triangleright [ E _ \\xi , E _ { \\xi ^ \\prime } ^ * ] _ q & = ( E _ k \\triangleright E _ \\xi ) E _ { \\xi ^ \\prime } ^ * - q ^ { - ( \\xi + \\alpha _ k , \\xi ^ \\prime ) } E _ { \\xi ^ \\prime } ^ * ( E _ k \\triangleright E _ \\xi ) \\\\ & + q ^ { ( \\xi , \\alpha _ k ) } E _ \\xi ( E _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) - q ^ { - ( \\xi , \\xi ^ \\prime ) } ( E _ k \\triangleright E _ { \\xi ^ \\prime } ^ * ) E _ \\xi . \\end{align*}"}
-{"id": "7648.png", "formula": "\\begin{align*} [ S _ { \\alpha } ( t ) \\chi _ { r } ] ( x ) & \\geq c _ { 1 } \\int _ { B _ { r } } \\frac { t } { ( t ^ { 1 / \\alpha } + | y - x | ) ^ { d + \\alpha } } d y \\\\ & \\geq c _ { 1 } \\int _ { B _ { r } ( ( r + t ^ { 1 / \\alpha } ) \\tau ) } \\frac { t } { ( t ^ { 1 / \\alpha } + | z | ) ^ { d + \\alpha } } d z \\\\ & = c _ { 1 } \\int _ { B _ { r t ^ { - 1 / \\alpha } } ( ( t ^ { - 1 / \\alpha } r + 1 ) \\tau ) } \\frac { 1 } { ( 1 + | w | ) ^ { d + \\alpha } } d w . \\end{align*}"}
-{"id": "1945.png", "formula": "\\begin{align*} \\lim _ { \\tau \\rightarrow - T _ 1 ^ + } \\frac { b _ i ( \\tau ) } { T _ 1 + \\tau } = \\lim _ { \\tau \\rightarrow - T _ 1 ^ + } \\left ( \\frac { b _ i ( \\tau ) } { a ( \\tau ) } \\cdot \\frac { a ( \\tau ) } { T _ 1 + \\tau } \\right ) = \\xi _ i ^ { - 1 } E ( \\xi ) . \\end{align*}"}
-{"id": "568.png", "formula": "\\begin{align*} \\| T + { \\mathcal K } ( \\tau ; { \\mathcal I } \\| _ { { \\mathcal E } / { \\mathcal K } } = \\inf _ { K \\in { \\mathcal K } ( \\tau ; { \\mathcal I } ) } | \\| T - K \\| | \\ge \\inf _ { K \\in { \\mathcal K } ( \\tau ; { \\mathcal I } ) } \\| T - K \\| \\ge \\| p ( T ) \\| . \\end{align*}"}
-{"id": "7253.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | \\geq 2 } \\sum _ { | \\gamma | \\geq 1 } \\sum _ { | \\beta | = | \\alpha | } T _ { j k } F _ { k j , \\alpha , \\gamma } \\cdot \\tau _ { k j , \\beta } ^ \\alpha \\cdot u _ j ^ \\beta \\cdot \\prod _ { \\lambda = 1 } ^ r \\left ( u _ j ^ \\lambda + \\sum _ { | \\delta | \\geq 2 } F _ { j , \\delta } ^ \\lambda \\cdot u _ j ^ \\delta \\right ) ^ { \\gamma _ \\lambda } , \\end{align*}"}
-{"id": "7190.png", "formula": "\\begin{align*} ( f ^ { ' } \\sin \\varphi ) ' = ( B f ) ' + 2 B ' + 2 C _ 1 \\sin \\varphi . \\end{align*}"}
-{"id": "3072.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\limsup _ { n \\to \\infty } \\P \\left ( \\Delta _ n ^ { k , t } > \\delta , S \\right ) = 0 . \\end{align*}"}
-{"id": "8365.png", "formula": "\\begin{align*} \\alpha _ { g ^ { - 1 } } ( z _ g z _ { g h } ) = \\alpha _ { g ^ { - 1 } } = z _ { g ^ { - 1 } } \\alpha _ { g ^ { - 1 } } ( z _ { g h } ) \\leq z _ h , \\end{align*}"}
-{"id": "4234.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 - 4 t ) ^ { \\tfrac { x } { 2 } } d \\mu _ { - 1 } ( x ) = & \\frac { 2 } { 1 + \\sqrt { 1 - 4 t } } = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n + 1 } { 2 n \\choose n } t ^ n \\\\ = & \\sum _ { n = 0 } ^ \\infty C _ n t ^ n . \\end{align*}"}
-{"id": "4693.png", "formula": "\\begin{align*} \\big ( f _ { 2 j } ( x _ { t + m } ) , f _ { 2 j - 1 } ( x _ { t + m } ) \\big ) & = ( 0 , 2 ) , \\\\ \\big ( f _ { 2 j } ( x _ { t + m + 1 } ) , f _ { 2 j - 1 } ( x _ { t + m + 1 } ) \\big ) & = ( 0 , 3 ) , \\end{align*}"}
-{"id": "1390.png", "formula": "\\begin{align*} K _ { S } : = \\underset { n \\to \\infty } \\lim \\left \\{ \\sum _ { h = 0 } ^ { ( n - 1 ) } \\left ( \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } E ( e _ { t , \\beta ' } e _ { t + h , \\beta ' } ^ { 2 } ) x _ { n t } \\right ) + \\sum _ { h = 1 } ^ { ( n - 1 ) } \\left ( \\frac { 1 } { n } \\sum _ { t = h + 1 } ^ { n } E ( e _ { t , \\beta ' } e _ { t - h , \\beta ' } ^ { 2 } ) x _ { n t } \\right ) \\right \\} \\end{align*}"}
-{"id": "2708.png", "formula": "\\begin{align*} \\int _ { \\{ V _ { \\theta } - j < \\psi _ s \\leq v < \\psi < \\varphi + s \\} } \\theta _ { v _ j } ^ n = 0 . \\end{align*}"}
-{"id": "4494.png", "formula": "\\begin{align*} A _ { l t } ^ { ( k ) } = \\Gamma ( j _ t ) ^ { - 1 } \\Gamma ( j _ t + 2 ( l - 1 ) / d ) k ^ { - 2 ( l - 1 ) / d } , \\end{align*}"}
-{"id": "3605.png", "formula": "\\begin{align*} B ^ r _ { ( g , \\pi ) } ( N , X ) = \\int _ { | x | = r } \\sum \\limits _ { i , j = 1 } ^ 3 \\left [ N \\left ( g _ { i j , i } - g _ { i i , j } \\right ) - \\left ( N _ { , i } g _ { i j } - N _ { , j } g _ { i i } \\right ) + X ^ i \\pi _ { i j } \\right ] \\nu _ 0 ^ j \\ , d \\sigma _ 0 . \\end{align*}"}
-{"id": "812.png", "formula": "\\begin{align*} & | \\mathrm { v a c } \\rangle _ { [ M ' , M ] } = | \\mathbf { 0 } \\rangle \\otimes | \\mathbf { 0 } \\rangle \\otimes \\cdots \\otimes | \\mathbf { 0 } \\rangle \\in \\mathcal { F } ^ { [ M ' , M ] } , \\\\ & { } _ { [ M ' , M ] } \\langle \\mathrm { v a c } | = \\langle \\mathbf { 0 } | \\otimes \\langle \\mathbf { 0 } | \\otimes \\cdots \\otimes \\langle \\mathbf { 0 } | \\in ( \\mathcal { F ^ { * } } ) ^ { [ M ' , M ] } . \\end{align*}"}
-{"id": "7801.png", "formula": "\\begin{align*} \\psi : = \\frac { e ^ { \\sqrt { T - \\sqrt { T } } } - e ^ { \\frac { f } { \\sqrt { T - \\sqrt { T } } } } } { e ^ { \\sqrt { T - \\sqrt { T } } } } . \\end{align*}"}
-{"id": "1364.png", "formula": "\\begin{align*} C ( l ) = \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } e _ { t } ^ { P } ( \\hat { \\beta } _ 0 ) e _ { t - l } ^ { P } ( \\hat { \\beta } _ 0 ) , \\end{align*}"}
-{"id": "6099.png", "formula": "\\begin{align*} L _ { j } ( s _ { j } ) = 2 \\pi ^ { 2 } s _ { j } ( 1 + s _ { j } e ( s _ { j } ) ) \\end{align*}"}
-{"id": "215.png", "formula": "\\begin{align*} R _ 2 = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ \\frac { a _ n } { n - 1 } ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log u _ { x , s } \\ , d s \\ , d x = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "8599.png", "formula": "\\begin{align*} k e r ( \\overline { \\alpha } ) = \\operatorname { i m } ( \\beta ) \\end{align*}"}
-{"id": "128.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { X } } = \\sup _ { \\| x ^ * \\| \\leq 1 } | x ^ * ( x ) | = \\| J x \\| _ { X ^ { * * } } . \\end{align*}"}
-{"id": "8205.png", "formula": "\\begin{align*} | F ( \\tau , 0 ) | _ v < 1 , | F ' _ U ( \\tau , 0 ) | _ v = 1 . \\end{align*}"}
-{"id": "5640.png", "formula": "\\begin{align*} E ( \\mu ) = \\left \\{ \\begin{array} { l l } \\frac { 1 } { 2 } \\int _ { \\Omega } | f ( x ) | ^ 2 d x + \\Vert f \\Vert _ { H ^ { k } ( \\Omega ) } ^ { 2 } , & \\mbox { i f } \\ f = \\frac { d \\mu } { d x } \\ \\mbox { a n d } \\ f \\geq \\alpha , \\\\ + \\infty & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "5775.png", "formula": "\\begin{align*} B _ j ( \\lambda ) = ( L _ j ( \\lambda , \\xi _ j ) ) _ { 1 2 } . \\end{align*}"}
-{"id": "1086.png", "formula": "\\begin{align*} g _ b ( x ) = x / 2 + b \\ ; \\pmod { 1 } . \\end{align*}"}
-{"id": "9564.png", "formula": "\\begin{align*} \\begin{aligned} & H ( t ) = H ^ { ( 0 ) } + t H ^ { ( 1 ) } + t ^ { 2 } H ^ { ( 1 ) } \\\\ & \\psi _ { n } ( t ) = \\psi ^ { ( 0 ) } _ { n } + t \\psi ^ { ( 1 ) } _ { n } + t ^ { 2 } \\psi _ { n } ^ { ( 2 ) } + . . . \\\\ & \\lambda _ { n } ( t ) = \\lambda ^ { ( 0 ) } _ { n } + t \\lambda ^ { ( 1 ) } _ { n } + t ^ { 2 } \\lambda _ { n } ^ { ( 2 ) } + . . . \\end{aligned} \\end{align*}"}
-{"id": "1492.png", "formula": "\\begin{align*} P ( X ) = \\sum c _ i X ^ i , \\end{align*}"}
-{"id": "4469.png", "formula": "\\begin{align*} b _ 1 ( x ) = - \\frac { \\Delta f ( x ) } { 2 ( d + 2 ) V _ d ^ { 2 / d } f ( x ) ^ { 1 + 2 / d } } . \\end{align*}"}
-{"id": "7083.png", "formula": "\\begin{align*} \\mathcal { O } = \\bigcup _ { u \\in \\Gamma ( \\mathcal { O } ) } C ( u , j _ u ) , \\end{align*}"}
-{"id": "3126.png", "formula": "\\begin{align*} ( \\widehat { \\Lambda } _ t ( \\lambda ) ^ T \\otimes I _ n ) ( \\lambda B + A ) ( \\widehat { \\Lambda } _ t ( \\lambda ) \\otimes I _ n ) = \\begin{bmatrix} - P & \\lambda ^ t P \\\\ \\lambda ^ t P & Q ( \\lambda ) \\end{bmatrix} , \\end{align*}"}
-{"id": "4912.png", "formula": "\\begin{align*} \\xi _ { 2 - 2 k , z } \\left ( \\mathbb { P } _ { 2 - 2 k , n , N } ( z , \\mathfrak { z } ) \\right ) & = ( 4 \\mathfrak { z } _ 2 ) ^ { 2 k - 1 } \\Psi _ { 2 k , - n - 1 , N } ( z , \\mathfrak { z } ) , \\\\ \\mathcal { D } _ z ^ { 2 k - 1 } \\left ( \\mathbb { P } _ { 2 - 2 k , n , N } ( z , \\mathfrak { z } ) \\right ) & = - ( 2 k - 2 ) ! \\left ( \\frac { \\mathfrak { z } _ 2 } { \\pi } \\right ) ^ { 2 k - 1 } \\Psi _ { 2 k , n + 1 - 2 k , N } ( z , \\mathfrak { z } ) . \\end{align*}"}
-{"id": "8835.png", "formula": "\\begin{align*} E _ { k } ( \\tau ) = 1 - \\frac { 2 k } { B _ { k } } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { k - 1 } ( n ) q ^ { n } . \\end{align*}"}
-{"id": "6520.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) , \\end{aligned} \\end{align*}"}
-{"id": "1743.png", "formula": "\\begin{align*} & \\psi _ t ( \\sqrt { L } ) f ( x ) = \\int \\Psi _ t ( x , y ) f ( y ) \\ , d \\mu ( y ) , \\cr & \\sup _ { x \\in X , \\ , t > 0 } \\ \\int | \\Psi _ t ( x , y ) | \\ , d \\mu ( y ) \\leq C . \\end{align*}"}
-{"id": "8819.png", "formula": "\\begin{align*} A _ { \\infty } = \\C [ x _ { 1 , n } , x _ { 2 , n } , \\cdots , x _ { k , n } ] _ { n \\leq 0 } / ( P _ { 1 , n } , \\cdots , P _ { r , n } ) \\end{align*}"}
-{"id": "8425.png", "formula": "\\begin{align*} A _ { j } ( \\textbf { u } ) = \\left ( \\begin{array} { c c c c c } v _ { j } & \\delta _ { 1 j } \\rho & \\delta _ { 2 j } \\rho & \\cdots & \\delta _ { d j } \\rho \\\\ \\delta _ { j 1 } f ( \\textbf { u } ) & v _ { j } & 0 & \\cdots & 0 \\\\ \\delta _ { j 2 } f ( \\textbf { u } ) & 0 & v _ { j } & \\cdots & 0 \\\\ \\cdots & \\cdots & \\cdots & v _ { j } & \\cdots \\\\ \\delta _ { j d } f ( \\textbf { u } ) & 0 & 0 & \\cdots & v _ { j } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "3776.png", "formula": "\\begin{align*} f ( u ) = \\sqrt { \\frac { \\beta ^ \\prime ( u ) } { 2 \\pi } } e ^ { C ( \\beta ) - u \\beta } \\left ( 1 + o \\left ( 1 \\right ) \\right ) \\end{align*}"}
-{"id": "132.png", "formula": "\\begin{align*} \\mu ( t , x ) = \\inf \\{ s \\geq 0 : \\ , \\tau ( e ^ { \\abs { x } } ( s , \\infty ) ) \\leq t \\} , t \\geq 0 , \\end{align*}"}
-{"id": "547.png", "formula": "\\begin{align*} \\bar { R } _ { 2 2 } ^ { \\rm e x p , n } ( x ; y ) = - \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\frac { e ^ { n f ( z ) - x \\sigma n ^ { 1 / 3 } z } } { 4 z } \\dd z + \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\frac { e ^ { n f ( z ) - y \\sigma n ^ { 1 / 3 } z } } { 4 z } \\dd z + \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\frac { 1 } { 2 z } e ^ { - \\sigma n ^ { 1 / 3 } \\vert x - y \\vert z } \\dd z \\ \\ - \\frac { 1 } { 4 } , \\end{align*}"}
-{"id": "5534.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\varphi _ * \\left ( \\frac { \\delta | \\eta ( k ) | } { v ( k ) } \\right ) v ( k ) \\chi _ { E _ j } ( k ) \\le \\sum _ { k = j + 1 } ^ \\infty \\varphi _ * \\left ( \\frac { \\delta | \\eta ( k ) | } { v ( k ) } \\right ) v ( k ) \\to 0 . \\end{align*}"}
-{"id": "6113.png", "formula": "\\begin{align*} \\begin{gathered} \\frac { | d z | ^ 2 } { | z | ^ 2 } = \\frac { d r d \\theta } { r ^ 2 } , \\\\ \\mu _ s = \\frac { d r d \\theta } { r ^ 4 } \\left ( \\frac r { \\pi s } \\sin \\frac { \\pi s } r \\right ) ^ 2 . \\end{gathered} \\end{align*}"}
-{"id": "5267.png", "formula": "\\begin{align*} \\pi ^ { - s / 2 } \\Gamma ( s / 2 ) \\zeta ( s ) = \\pi ^ { - ( 1 - s ) / 2 } \\Gamma ( ( 1 - s ) / 2 ) \\zeta ( 1 - s ) . \\end{align*}"}
-{"id": "6502.png", "formula": "\\begin{align*} \\varphi ( t ) \\frac { d } { d t } ( k * v ) ( t ) = \\frac { d } { d t } ( k * [ \\varphi v ] ) ( t ) + \\int _ { 0 } ^ { t } \\dot { k } ( t - \\sigma ) ( \\varphi ( t ) - \\varphi ( \\sigma ) ) v ( \\sigma ) d \\sigma , \\end{align*}"}
-{"id": "1708.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\frac { \\ln \\P \\{ \\max _ { 1 \\le n \\le N _ j } | S _ n | \\le f _ { N _ j } \\} } { f _ { N _ j } ^ { - 1 / H } N _ j } = 0 . \\end{align*}"}
-{"id": "1831.png", "formula": "\\begin{align*} \\tilde R _ { i \\bar i i \\bar i } = - \\frac { 3 \\pi } { 4 } s _ { i } ^ { - 1 } + O ( s _ { i } ) . \\end{align*}"}
-{"id": "2891.png", "formula": "\\begin{align*} g _ { P o i s _ n ^ + , A } ^ { \\psi ^ + } = H o m _ { \\Sigma } ( \\overline { P o i s _ n ^ * \\{ n \\} } \\oplus I , E n d _ A ) ^ { \\psi } = C o n v ( P o i s _ n ^ * \\{ n \\} , E n d _ A ) \\end{align*}"}
-{"id": "8764.png", "formula": "\\begin{align*} R _ { 1 , 2 } \\left ( y / x \\right ) R _ { 1 , 3 } \\left ( y / z \\right ) R _ { 2 , 3 } \\left ( x / z \\right ) = R _ { 2 , 3 } \\left ( x / z \\right ) R _ { 1 , 3 } \\left ( y / z \\right ) R _ { 1 , 2 } \\left ( y / x \\right ) . \\end{align*}"}
-{"id": "6641.png", "formula": "\\begin{align*} \\varepsilon _ K ( s , V , \\psi ) = \\varepsilon _ K ( V \\otimes \\omega _ { s - 1 / 2 } , \\psi ) \\end{align*}"}
-{"id": "5674.png", "formula": "\\begin{align*} & K ( x ) = 1 + k ( x ) \\ , \\Big ( b _ 0 + x \\ , b _ 0 ^ + + \\frac 1 x \\ , b _ 0 ^ - \\Big ) , \\\\ & \\mbox { w i t h } k ( x ) = \\frac { \\left ( x ^ 2 - 1 \\right ) \\left ( \\alpha + \\gamma \\right ) } { \\left ( \\gamma x + \\alpha \\right ) \\left ( ( \\alpha + \\gamma ) ( x - 1 ) + ( q - 1 ) x \\right ) } \\ , . \\end{align*}"}
-{"id": "9342.png", "formula": "\\begin{align*} b ( t , x , z ) = - x \\end{align*}"}
-{"id": "8644.png", "formula": "\\begin{align*} X _ \\tau ^ { 0 , x } = e ^ { \\tau A } x + e ^ { \\tau A } v ( 0 , x ) - v ( \\tau , X _ \\tau ^ { x } ) + \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } \\nabla ^ G v ( s , X _ s ^ x ) \\ ; d W _ s + \\int _ 0 ^ \\tau e ^ { ( \\tau - s ) A } G d W _ s , \\end{align*}"}
-{"id": "1005.png", "formula": "\\begin{align*} \\sum _ { r = 0 } ^ N ( N - r + 1 ) \\frac { ( r + n + m - 3 ) ! } { ( n + m - 3 ) ! r ! } = \\frac { ( N + n + m - 1 ) ! } { ( n + m - 1 ) ! N ! } \\end{align*}"}
-{"id": "3581.png", "formula": "\\begin{align*} ( h , w ) = \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) \\end{align*}"}
-{"id": "9545.png", "formula": "\\begin{align*} u _ 1 \\phi ( [ v , u _ 2 ] ) = v _ 1 \\phi ( [ u , v _ 2 ] ) \\end{align*}"}
-{"id": "1836.png", "formula": "\\begin{align*} L _ \\gamma ( s ) = 2 \\pi ^ { 2 } s ( 1 + s e ( s ) ) \\end{align*}"}
-{"id": "4111.png", "formula": "\\begin{align*} \\omega = ( b y z ( q + r ) + q c z ^ 2 ) d y - ( ( p + q + r ) b y ^ 2 + c ( p + q ) y z ) d z . \\end{align*}"}
-{"id": "287.png", "formula": "\\begin{align*} B _ x ( r _ { n , u } ) \\cap B _ y ( r _ { n , v } ) = \\{ B _ x ( r _ { n , v } ) \\cap B _ y ( r _ { n , v } ) \\} \\cup [ \\{ B _ x ( r _ { n , u } ) \\setminus B _ x ( r _ { n , v } ) \\} \\cap B _ y ( r _ { n , v } ) ] , \\end{align*}"}
-{"id": "8152.png", "formula": "\\begin{align*} x ^ 2 - \\chi _ { _ h } ( S _ 1 ) \\cdot x + \\frac { 1 } { 2 } [ \\chi _ { _ h } ( S _ 1 ) ^ 2 - ( \\chi _ { _ h } ( S _ 1 ^ 2 ) + \\chi _ { _ h } ( S _ 2 ^ 2 ) ] = 0 , \\end{align*}"}
-{"id": "3856.png", "formula": "\\begin{align*} \\R = \\Q ^ \\N _ C / \\end{align*}"}
-{"id": "4286.png", "formula": "\\begin{align*} \\| x - z \\| & \\le \\| x - T y \\| + \\| T y - z \\| \\\\ & \\le \\| T ( x - y ) \\| + \\sum _ { n = 1 } ^ N \\lambda _ n \\big | 1 - \\| T y _ n \\| \\big | \\\\ & < ( 1 + \\eta ) \\gamma + \\max _ { 1 \\le n \\le N } { \\big | 1 - \\| T y _ n \\| \\big | } \\\\ & \\le ( 1 + \\eta ) \\gamma + \\eta < \\alpha . \\end{align*}"}
-{"id": "1402.png", "formula": "\\begin{align*} \\mu _ k ^ * ( A ' _ i ) = \\begin{cases} A _ i & i \\neq k \\\\ A _ k ^ { - 1 } \\biggl ( \\prod _ { b _ { k j } > 0 } A _ j ^ { b _ { k j } } + \\prod _ { b _ { k j } < 0 } A _ j ^ { - b _ { k j } } \\biggr ) & i = k . \\end{cases} \\end{align*}"}
-{"id": "3145.png", "formula": "\\begin{align*} & \\hat { C } _ s ( \\{ ( \\overline { W } _ s , { V } _ t ) : s \\in \\overline { \\theta } , t \\in \\theta \\} ) \\\\ & = \\lim _ { n \\rightarrow \\infty } \\frac { 1 } { n } \\max _ { \\Lambda _ n } \\Bigl ( \\min _ { s \\in \\overline { \\theta } } \\chi ( p _ U ; B _ s ^ { \\otimes n } ) - \\max _ { t ^ n \\in \\theta ^ n } \\chi ( p _ U ; Z _ { t ^ n } ) \\Bigr ) \\end{align*}"}
-{"id": "8137.png", "formula": "\\begin{align*} \\left ( \\frac { N } { p } - 1 + a ^ * \\right ) ^ 2 = ( N - 1 ) \\left ( \\frac { 1 } { q - p } - \\frac { 1 } { q + p ' } \\right ) , \\end{align*}"}
-{"id": "6689.png", "formula": "\\begin{align*} D _ 1 ^ 2 = D _ 2 ^ 2 = - 2 , ~ D _ 1 - D _ 2 = H \\end{align*}"}
-{"id": "327.png", "formula": "\\begin{align*} n & = , n _ 1 = . \\end{align*}"}
-{"id": "664.png", "formula": "\\begin{align*} \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\phi ) \\psi ) \\ , d \\eta ( g ) = \\nu ( \\phi ) \\ , \\nu ( \\psi ) . \\end{align*}"}
-{"id": "490.png", "formula": "\\begin{align*} u ( x , t ) = \\int _ { \\R ^ N } u _ 0 ( y ) \\ , K ( x - y , \\ , t ) \\ , d y \\mbox { f o r a l l $ ( x , t ) \\in \\mathbb R ^ N \\times ( 0 , T ) $ , } \\end{align*}"}
-{"id": "9723.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } { \\log \\sigma ( t ) } = 0 . \\end{align*}"}
-{"id": "2563.png", "formula": "\\begin{align*} \\bigl [ e ^ { i t \\Delta } \\bigl ( e ^ { i x \\xi _ 0 } f ( \\tfrac { \\cdot } { \\lambda } ) \\bigr ) \\bigr ] ( x ) = e ^ { - i t | \\xi _ 0 | ^ 2 + i x \\xi _ 0 } ( e ^ { i t \\lambda ^ { - 2 } \\Delta } f ) ( \\tfrac { x - 2 t \\xi _ 0 } { \\lambda } ) . \\end{align*}"}
-{"id": "63.png", "formula": "\\begin{align*} \\psi ' ( 0 ) = \\frac { p a ^ { 2 - p } } { p - 1 } > 0 \\ . \\end{align*}"}
-{"id": "1472.png", "formula": "\\begin{align*} \\Omega _ k : = \\bigcup _ { j \\in J _ k } Q ^ k _ j . \\end{align*}"}
-{"id": "3696.png", "formula": "\\begin{align*} \\{ F , Z , H \\} _ { \\mu } = \\int F _ { \\mu } J ( Z _ { \\zeta } , H _ { \\mu } ) d A + c y c ( F , Z , H ) ~ . \\end{align*}"}
-{"id": "9589.png", "formula": "\\begin{align*} \\psi ( x , t ) = \\psi _ { r e g } ( x , t ) + \\xi ( t ) g ( x - y ) , \\quad \\psi _ { r e g } ( y , t ) = \\xi ( t ) \\end{align*}"}
-{"id": "104.png", "formula": "\\begin{align*} \\frac { 2 } { e ^ t + 1 } e ^ { x t } = & \\sum _ { n = 0 } ^ \\infty C h _ { n , \\lambda } ( x ) \\frac { 1 } { n ! } \\Big ( e ^ { \\frac { 1 } { \\lambda } t } - 1 \\Big ) ^ n \\\\ = & \\sum _ { n = 0 } ^ \\infty C h _ { n , \\lambda } ( x ) \\sum _ { m = n } ^ \\infty S _ 2 ( m , n ) \\lambda ^ { - m } \\frac { t ^ m } { m ! } , \\end{align*}"}
-{"id": "6733.png", "formula": "\\begin{align*} \\hat F _ { P I } : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n f \\left ( \\hat p _ k ( X _ i ) \\right ) . \\end{align*}"}
-{"id": "1541.png", "formula": "\\begin{align*} | | T ( t ) | | = \\left \\Vert \\frac { \\rho _ { t , p } } { \\rho } \\right \\Vert _ \\infty , \\end{align*}"}
-{"id": "376.png", "formula": "\\begin{align*} \\int _ 0 ^ { T ^ * } \\| u _ x \\| _ { L ^ \\infty } d \\tau = + \\infty . \\end{align*}"}
-{"id": "831.png", "formula": "\\begin{align*} h _ { z } ^ { a } ( 1 ) = \\frac { z } { 1 + z } u _ { a } = \\psi _ { z } ^ { a } ( 1 ) \\end{align*}"}
-{"id": "7471.png", "formula": "\\begin{align*} \\| v _ s \\| ( \\| v \\| - \\| v _ s \\| ) > 3 S _ j \\frac { 2 \\| v \\| } { 3 } = 2 S _ j \\| v \\| . \\end{align*}"}
-{"id": "5438.png", "formula": "\\begin{align*} v _ p ( \\kappa ( \\mu _ L ) ) = v _ p ( \\kappa ( { \\mu _ L } _ { \\vert M _ 0 } ) ) + \\sum _ { i = 1 } ^ { \\frac { q - 5 } { 4 } } \\geq 2 t + { \\frac { q - 5 } { 4 } } 4 t = ( q - 3 ) t = v _ p ( \\kappa ( \\mu _ L ) ) , \\end{align*}"}
-{"id": "7818.png", "formula": "\\begin{align*} Y _ { t } = \\xi + \\int _ t ^ T E ' [ f ( s , \\eta _ { s } , Y ' _ { s } , Z ' _ { s } , Y _ { s } , Z _ { s } ) ] d s - \\int _ t ^ T Z _ { s } d B _ { s } ^ { H } , \\ \\ 0 \\leq t \\leq T , \\end{align*}"}
-{"id": "230.png", "formula": "\\begin{align*} v _ x : = \\inf \\{ u \\geq 0 : ( n - 1 ) p _ { n , x , u } = a _ n ^ + \\} , l _ x : = \\inf \\{ u \\geq 0 : ( n - 1 ) p _ { n , x , u } = a _ n ^ - \\} , \\end{align*}"}
-{"id": "4474.png", "formula": "\\begin{align*} \\mathbb { P } \\bigl ( \\mathrm { B } _ 1 \\geq a _ n / ( n - 1 ) \\bigr ) = \\mathbb { P } ( B _ 2 \\leq k - 1 ) \\leq \\exp \\biggl ( - \\frac { ( a _ n - k + 1 ) ^ 2 } { 2 a _ n } \\biggr ) = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "7288.png", "formula": "\\begin{align*} K _ { i } ^ { * } = K _ { i } , E _ { i } ^ { * } = K _ { i } F _ { i } , F _ { i } ^ { * } = E _ { i } K _ { i } ^ { - 1 } . \\end{align*}"}
-{"id": "2460.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 2 / 2 } \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 2 , \\alpha _ 1 } } \\| \\Box _ l ^ { \\alpha _ 2 } \\Box _ k ^ { \\alpha _ 1 } f \\| ^ q _ { L ^ { 2 } } \\right ) ^ { 1 / q } \\lesssim 2 ^ { j n \\alpha _ 2 / 2 } 2 ^ { j n ( \\alpha _ 1 - \\alpha _ 2 ) ( 1 / q - 1 / 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "1407.png", "formula": "\\begin{align*} \\lambda _ k + \\mu _ { k ^ * } = 0 = \\lambda _ { k ^ * } + \\mu _ { k } \\end{align*}"}
-{"id": "9068.png", "formula": "\\begin{align*} & \\P ( G _ 0 v _ { | | E | | ^ { 1 / 3 } } ( d , 0 , E + 2 | E | ) ) \\\\ & \\leq \\sum _ { k = 1 } ^ { \\infty } \\P ( G _ 0 v _ { | | E | | _ 1 ^ { 1 / 3 } } ( d , k | | E | | _ 1 ^ { 1 / 3 } , E ) ) \\ , \\P \\left ( \\sup _ { j \\geq 0 } d | \\widetilde { W } _ { 2 | E | _ j } | \\geq | | E | | _ 1 ^ { 1 / 3 } ( k - 1 ) \\right ) . \\end{align*}"}
-{"id": "6352.png", "formula": "\\begin{align*} \\| P \\| : = \\max \\{ | p _ { \\mu m } | \\} . \\end{align*}"}
-{"id": "4518.png", "formula": "\\begin{align*} \\biggl | \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\log f ( x ) \\ , d x \\biggr | = O ( q _ n ^ { 1 - \\epsilon } ) , \\end{align*}"}
-{"id": "3407.png", "formula": "\\begin{align*} \\pi ' _ 0 ( \\nabla ( e _ 0 ) ) & = \\pi ' _ 0 \\Big ( e _ 1 \\otimes \\frac { d z } { 2 z ^ 2 } \\Big ) = ( w + w ^ 2 ) \\frac { d w } { w ^ 3 } = \\nu ' ( w ) \\pi ' _ 0 ( e _ 0 ) \\\\ \\pi ' _ 0 ( \\nabla ( e _ 1 ) ) & = \\pi ' _ 0 \\Big ( z e _ 0 \\otimes \\frac { d z } { 2 z ^ 2 } + e _ 1 \\otimes \\frac { d z } { 2 z } \\Big ) = w ^ 2 \\frac { d w } { w ^ 3 } + w \\frac { d w } { w } = \\nu ' ( w ) ( w + w ^ 2 ) = \\nu ' ( w ) \\pi ' _ 0 ( e _ 1 ) . \\end{align*}"}
-{"id": "1562.png", "formula": "\\begin{align*} \\Delta _ { i , X } ^ + = \\partial _ { i + 1 , X } \\delta _ { i , X } , \\qquad \\Delta _ { i , X } ^ - = \\delta _ { i - 1 , X } \\partial _ { i , X } , \\qquad \\Delta _ { i , X } = \\Delta _ { i , X } ^ + + \\Delta _ { i , X } ^ - \\ , , \\end{align*}"}
-{"id": "4500.png", "formula": "\\begin{align*} \\epsilon _ n = \\epsilon _ n ^ w ( d , \\theta ) : = \\frac { \\sup _ { k \\in \\{ 1 , \\ldots , k ^ * \\} } \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\Bigl ( 2 \\mathbb { E } _ f \\bigl [ \\{ \\tilde { V } _ n ^ w - V ( f ) \\} ^ 2 \\bigr ] \\Bigr ) ^ { 1 / 3 } } { \\inf _ { f \\in \\mathcal { F } _ { d , \\theta } } V ( f ) ^ { 2 / 3 } } , \\end{align*}"}
-{"id": "4920.png", "formula": "\\begin{align*} N ( w ) : = \\{ \\alpha \\in \\Phi ^ + \\mid w \\alpha \\in - \\Phi ^ + \\} . \\end{align*}"}
-{"id": "5747.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\Omega } ( u ) = P ( \\chi ^ 2 _ 1 > u ) + \\frac { 1 } { 2 \\pi } e ^ { - \\frac { u } { 2 } } \\ln \\left . \\left [ \\frac { 1 + \\omega } { 1 - \\omega } \\right ] \\right \\vert _ { \\omega _ { \\mathcal { L } } } ^ { \\omega _ { \\mathcal { U } } } \\end{align*}"}
-{"id": "8946.png", "formula": "\\begin{align*} \\int _ s ^ { 2 R } \\frac { g _ { R , 1 } ( t ) } t d t = & \\Big ( \\Big ( p ' - \\frac 3 2 \\Big ) R ^ { 1 - 1 / p } - \\frac { s _ 0 ^ { 1 - 1 / p } } { p - 1 } \\Big ) \\Big ( \\frac 1 s - \\frac 1 { 2 R } \\Big ) \\\\ & + 2 R ^ { - 1 / p } \\ln \\frac { 2 R } s - \\frac { R ^ { - 1 - 1 / p } ( 2 R - s ) } 2 . \\end{align*}"}
-{"id": "3080.png", "formula": "\\begin{align*} \\mathcal { A } _ n ( y ) & = \\left \\{ | u | \\leq n : V ( u _ j ) \\geq f _ n ( j ) - y , j \\leq | u | \\right \\} , \\\\ \\bar { \\mathcal { A } } _ n ( y , h ) & = \\left \\{ | u | = n : u \\in \\mathcal { A } _ n ( y ) , V ( u ) - f _ n ( n ) + y \\in [ h - 1 , h ] \\right \\} \\\\ \\mathcal { B } _ n ( y , z ) & = \\left \\{ | u | \\leq n : \\xi ( u _ j ) \\leq z + ( V ( u _ j ) - f _ n ( j ) + y ) / 2 , j \\leq | u | \\right \\} . \\end{align*}"}
-{"id": "7836.png", "formula": "\\begin{align*} A , B = \\left [ \\begin{smallmatrix} * & * & 0 & 0 \\\\ 0 & * & 0 & 0 \\\\ * & * & * & * \\\\ 0 & * & 0 & * \\end{smallmatrix} \\right ] , \\ \\ a = \\left [ \\begin{smallmatrix} * \\\\ 0 \\\\ * \\\\ 0 \\end{smallmatrix} \\right ] , \\ \\ b = \\left [ \\begin{smallmatrix} * & * & 0 & 0 \\end{smallmatrix} \\right ] . \\end{align*}"}
-{"id": "2176.png", "formula": "\\begin{align*} & \\quad \\quad \\quad \\quad 0 \\leq \\phi \\leq 1 , 0 \\leq - \\dot { \\phi } \\leq \\frac { 4 } { t _ { 2 } - t _ { 1 } } , \\\\ & \\phi = 1 [ 0 , t _ { 1 } - t _ { 0 } ] , \\phi = 0 [ t _ { 1 } - t _ { 0 } + ( t _ { 2 } - t _ { 1 } ) / 2 , t _ { 2 } - t _ { 0 } ] . \\end{align*}"}
-{"id": "4248.png", "formula": "\\begin{align*} \\nu ( x ) & = \\sum _ { k \\ge 0 } \\hat { \\nu } _ k \\cos ( 2 \\pi k x ) \\end{align*}"}
-{"id": "9006.png", "formula": "\\begin{align*} ( ~ ^ { A B } I _ b ^ \\alpha ~ ^ { A B C } D _ b ^ \\alpha f ) ( x ) = f ( x ) - f ( b ) \\end{align*}"}
-{"id": "1092.png", "formula": "\\begin{align*} P r [ f ^ { ( 4 ) } ( x , \\xi ^ { ( 4 ) } ) \\in S _ 0 ] \\geq ( 1 / 1 2 9 6 ) \\cdot ( 1 / 2 ) = 1 / 2 5 9 2 \\end{align*}"}
-{"id": "383.png", "formula": "\\begin{align*} \\xi ( z + 1 ) E ( z , g ) = \\xi ( 1 - z ) E ( - z , g ) ; \\xi ( z ) = \\pi ^ { - \\frac { z } { 2 } } \\Gamma \\left ( \\frac { z } { 2 } \\right ) \\zeta ( z ) \\end{align*}"}
-{"id": "7356.png", "formula": "\\begin{align*} T ^ { ( k ) * } = ( M ^ { ( k ) } ) ^ { - 1 } T ^ { ( k ) \\dagger } M ^ { ( k - 1 ) } , \\end{align*}"}
-{"id": "9833.png", "formula": "\\begin{align*} M _ i \\ , { } _ \\lambda \\ , v _ m = 0 . \\end{align*}"}
-{"id": "8753.png", "formula": "\\begin{align*} L _ 0 = \\tfrac 1 2 ( 1 - t ) K ' _ 0 ( 1 ) , L _ n = - \\tfrac 1 2 ( 1 - t ) K ' _ n ( 1 ) . \\end{align*}"}
-{"id": "943.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ \\infty \\gamma _ j < \\infty . \\end{align*}"}
-{"id": "2170.png", "formula": "\\begin{align*} \\ , \\phi \\leq & 2 \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } \\phi w ^ { 2 } k ( x , y ) d x d y \\\\ & + 4 \\int _ { \\rho B _ { 1 } } \\int _ { \\mathbb { R } ^ { n } \\backslash ( \\rho B _ { 1 } ) } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } \\phi w ^ { 2 } k ( x , y ) d y d x \\\\ \\leq & C _ { 1 } ( n , \\Lambda , \\delta ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } \\phi w ^ { 2 } d x , \\end{align*}"}
-{"id": "3767.png", "formula": "\\begin{align*} P \\left ( T _ \\beta = 0 \\right ) = O \\left ( e ^ { - K u } \\right ) \\end{align*}"}
-{"id": "6039.png", "formula": "\\begin{align*} f ( u ( x _ n ) ) = \\frac { 1 } { 2 } \\int _ { \\R ^ N } \\frac { 2 u ( x _ n ) - u ( x _ n + y ) - u ( x _ n - y ) } { | y | ^ { N + 2 s } } \\ ; d y . \\end{align*}"}
-{"id": "9877.png", "formula": "\\begin{align*} \\frac { \\partial V ^ { \\epsilon } } { \\partial t } ( t , x ) & = \\frac { \\partial ^ 2 V ^ { \\epsilon } } { \\partial x ^ 2 } ( t , x ) + \\sqrt { \\epsilon } \\sigma ( t , x , V ^ \\epsilon ( t , x ) ) \\frac { \\partial ^ 2 W } { \\partial t \\partial x } ( t , x ) + \\frac { \\partial } { \\partial x } g ( t , x , V ^ { \\epsilon } ( t , x ) ) \\\\ & + f ( t , x , V ^ \\epsilon ( t , x ) ) + \\sigma ( t , x , V ^ \\epsilon ( t , x ) ) v ( t , x ) , \\end{align*}"}
-{"id": "1583.png", "formula": "\\begin{align*} T _ { 1 } L _ { Y _ { 1 } } C _ { x } + T _ { 2 } L _ { Y _ { 2 } } C _ { x } - T _ { 1 , t } Y _ { 1 } - T _ { 2 , t } Y _ { 2 } - 2 a _ { , x } = 0 \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "5463.png", "formula": "\\begin{align*} T _ H ( N ; y , \\alpha ) & = e ( N \\alpha ) \\sum _ { n = - H } ^ y ( H - | n | ) e ( n \\alpha ) \\\\ & = \\frac { e ( ( N + y + 1 ) \\alpha ) } { 1 - e ( \\alpha ) } \\cdot ( y - H ) + \\frac { e ( ( N + 1 ) \\alpha ) } { ( 1 - e ( \\alpha ) ) ^ 2 } \\cdot \\bigl ( e ( y \\alpha ) - 2 + e ( - H \\alpha ) \\bigr ) . \\end{align*}"}
-{"id": "5478.png", "formula": "\\begin{align*} f _ { i j } ^ { * } = \\min _ { x \\in S } f _ { i j } \\left ( x \\right ) , i = 1 , 2 , . . . m ; j = 1 , 2 , . . . , p _ { m } ^ { } \\end{align*}"}
-{"id": "3866.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } g \\cdot \\ , d \\ , ( \\delta V ) + \\sigma \\int _ { B ^ + } { \\rm d i v } _ { \\partial \\Omega } \\ , g \\ , d \\mathcal H ^ { n } = 0 \\end{align*}"}
-{"id": "4853.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = I ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) + \\hat { R } ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) \\end{align*}"}
-{"id": "1097.png", "formula": "\\begin{align*} | A x | = | A x _ 0 | + e , \\end{align*}"}
-{"id": "1795.png", "formula": "\\begin{align*} m _ i + \\sum _ { s = j + 1 } ^ r \\ell _ { i + 1 , s } - \\sum _ { s = j } ^ r \\ell _ { i s } & = ( \\lambda _ i ^ { r + 1 } - \\lambda _ { i + 1 } ^ { r + 1 } ) + \\sum _ { s = j + 1 } ^ r ( \\lambda _ { i + 1 } ^ { s + 1 } - \\lambda _ { i + 1 } ^ s ) - \\sum _ { s = j } ^ r ( \\lambda _ { i } ^ { s + 1 } - \\lambda _ { i } ^ s ) \\\\ & = \\lambda _ i ^ { j } - \\lambda _ { i + 1 } ^ { j + 1 } . \\end{align*}"}
-{"id": "9747.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { G ( x ) } { \\exp ( e ^ { 1 / x } ) e ^ { - 1 / x } x ^ 2 } & = \\lim _ { y \\to \\infty } \\frac { \\frac { 1 } { g ( 1 / y ) } y ^ { - 2 } } { \\exp ( e ^ y ) y ^ { - 2 } + \\exp ( e ^ y ) \\left ( - 2 e ^ { - y } y ^ { - 3 } - e ^ { - y } y ^ { - 2 } \\right ) } \\\\ & = \\lim _ { y \\to \\infty } \\frac { 1 } { 1 - e ^ { - y } ( 2 y ^ { - 1 } + 1 ) } = 1 . \\end{align*}"}
-{"id": "2279.png", "formula": "\\begin{align*} \\begin{aligned} \\| v \\| _ { L ^ \\infty ( 0 , T ; W ) } \\le C _ { W } \\big ( \\| v ' \\| _ { L ^ p ( 0 , T ; X ) } + \\| v \\| _ { L ^ p ( 0 , T ; D ) } \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "8238.png", "formula": "\\begin{align*} \\mathcal { M } _ K : = \\left \\{ \\phi \\in \\ell ^ 2 _ c ( \\mathbb { Z } ) : \\| \\phi \\| _ { \\ell ^ 2 } \\leq K \\right \\} . \\end{align*}"}
-{"id": "3106.png", "formula": "\\begin{align*} \\lambda B + A = \\begin{bmatrix} \\lambda B _ { 1 1 } + A _ { 1 1 } & \\lambda B _ { 1 2 } + A _ { 1 2 } & \\lambda B _ { 1 3 } + A _ { 1 3 } \\\\ \\lambda B _ { 1 2 } ^ T + A _ { 1 2 } ^ T & \\lambda B _ { 2 2 } + A _ { 2 2 } & \\lambda B _ { 2 3 } + A _ { 2 3 } \\\\ \\lambda B _ { 1 3 } ^ T + A _ { 1 3 } ^ T & \\lambda B _ { 2 3 } ^ T + A _ { 2 3 } ^ T & \\lambda B _ { 3 3 } + A _ { 3 3 } \\end{bmatrix} . \\end{align*}"}
-{"id": "147.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { C _ E } } = \\| S ( x ) \\| _ { \\widehat { E } } \\ \\ \\ \\ . \\end{align*}"}
-{"id": "922.png", "formula": "\\begin{align*} \\sum ^ { n } _ { i = 1 } | x _ { i } y _ { i } | \\leq \\left ( \\sum ^ { n } _ { i = 1 } | x _ { i } | ^ { p } \\right ) ^ { 1 / p } \\left ( \\sum ^ { n } _ { i = 1 } | y _ { i } | ^ { q } \\right ) ^ { 1 / q } \\end{align*}"}
-{"id": "1142.png", "formula": "\\begin{align*} { \\tilde \\mu _ { i j } } \\left ( { { f _ { i j } } \\left ( x \\right ) } \\right ) + d _ { i j } ^ - - d _ { i j } ^ + = 1 \\ , \\ , \\ , \\ , , i = 1 , 2 , . . . m ; j = 1 , 2 , . . . , p _ m \\end{align*}"}
-{"id": "3056.png", "formula": "\\begin{align*} \\nu = \\lim _ { n \\to \\infty } \\sum _ { | u | = n } V ( u ) e ^ { - V ( u ) } \\delta _ { u } \\end{align*}"}
-{"id": "5444.png", "formula": "\\begin{align*} K ^ * _ i = \\begin{bmatrix} 0 & 0 & \\alpha J ( i + r , 3 r ) & \\overline \\alpha J ( i + 3 r , r ) \\\\ 0 & 0 & \\overline \\alpha J ( i + r , r ) & \\alpha J ( i + 3 r , 3 r ) \\\\ \\overline \\alpha J ( i , r ) & \\alpha J ( i + 2 r , 3 r ) & 0 & 0 \\\\ \\alpha J ( i , 3 r ) & \\overline \\alpha J ( i + 2 r , r ) & 0 & 0 \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "4433.png", "formula": "\\begin{align*} \\rho _ { ( k ) , i } : = \\| X _ { ( k ) , i } - X _ i \\| \\end{align*}"}
-{"id": "8817.png", "formula": "\\begin{align*} A = \\C [ x _ 1 , \\cdots , x _ k ] / ( P _ 1 , \\cdots , P _ r ) \\end{align*}"}
-{"id": "2674.png", "formula": "\\begin{align*} F ( \\varphi ) : = { \\rm I } ( \\varphi ) - L _ { \\mu } ( \\varphi ) , \\ \\ \\varphi \\in \\textup { P S H } ( X , \\omega ) , \\end{align*}"}
-{"id": "8976.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | u _ n ( x ) - u _ n ( y ) | ^ { p - 2 } \\ , ( u _ n ( x ) - u _ n ( y ) ) \\ , ( \\varphi ( x ) - \\varphi ( y ) ) } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y = \\int _ { \\Omega } \\frac { f _ n ( x ) } { ( u _ n + 1 / n ) ^ \\gamma } \\ , \\varphi \\ , d x , \\end{align*}"}
-{"id": "2743.png", "formula": "\\begin{align*} \\tilde u = e ^ { a + b } u \\end{align*}"}
-{"id": "7454.png", "formula": "\\begin{align*} ( \\eta _ 1 , \\dots , \\eta _ n ) = ( \\alpha _ 1 , \\dots , \\alpha _ n ) \\left ( D ' \\left [ \\overline { u } ^ { - 1 } x \\right ] _ - ^ { - 1 } \\overline { u } ^ { - 1 } \\right ) ^ t \\end{align*}"}
-{"id": "4479.png", "formula": "\\begin{align*} | R _ 4 | & \\leq \\biggl \\{ \\log \\Bigl ( \\frac { n - 1 } { e ^ { \\Psi ( k ) } } \\Bigr ) \\ ! + \\ ! \\int _ { \\mathcal { X } _ n } \\ ! \\ ! \\ ! f ( x ) \\biggl ( | \\log f ( x ) | + \\frac { a ( f ( x ) ) } { f ( x ) ^ { \\frac { 2 } { d } } V _ d ^ { \\frac { 2 } { d } } } \\biggr ) \\ , d x \\biggr \\} \\mathbb { P } \\Bigl ( \\mathrm { B } _ 1 \\geq \\frac { a _ n } { n - 1 } \\Bigr ) \\\\ & = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "1440.png", "formula": "\\begin{align*} h _ { i } = e _ { i , i } - e _ { i + 1 , i + 1 } \\end{align*}"}
-{"id": "8632.png", "formula": "\\begin{align*} \\beta _ { i + 1 } = \\frac { 1 } { c _ 0 } ( - c _ 1 \\beta _ i - c _ 2 \\beta _ { i - 1 } - \\ldots - c _ { i - 1 } \\beta _ 1 ) , ~ 1 \\leq i \\leq k - 2 . \\end{align*}"}
-{"id": "9337.png", "formula": "\\begin{align*} d F ( t , x , z ) = y ( t , x , z ) q ( t , x , z ) d G ( t ) \\end{align*}"}
-{"id": "5779.png", "formula": "\\begin{align*} u \\cdot x _ { \\alpha } = \\sum _ { \\gamma \\in [ \\beta ] } c _ { \\gamma , u } ^ { \\alpha } x _ { \\gamma } \\end{align*}"}
-{"id": "5090.png", "formula": "\\begin{align*} \\check { R } ( z ) & = ( z - q ^ { 2 } ) \\sum _ { a = 1 } ^ { r } E _ { a a } \\otimes E _ { a a } + ( 1 - q ^ { 2 } ) \\sum _ { 1 \\le a < b \\le r } \\left ( z E _ { a a } \\otimes E _ { b b } + E _ { b b } \\otimes E _ { a a } \\right ) \\\\ & + ( z - 1 ) \\sum _ { 1 \\le a < b \\le r } \\left ( E _ { a b } \\otimes E _ { b a } + q ^ { 2 } E _ { b a } \\otimes E _ { a b } \\right ) . \\end{align*}"}
-{"id": "6950.png", "formula": "\\begin{align*} M _ { \\pi ' } ( Z ) & = \\sum _ { k \\ge 0 } \\ , s _ { ( k ) } [ s _ { \\pi ' } ] ( Z ) = \\sum _ \\nu m _ { \\pi ' \\nu ' } \\ , s _ { \\nu ' } ( Z ) \\cr & = \\sum _ { k \\ge 0 } \\ , ( \\ , s _ { ( 1 ^ k ) } [ s _ { \\pi } ] ( Z ) \\ , ) ' \\sum _ { k \\ge 0 } \\ , ( - 1 ) ^ { k | \\pi | } ( ( - 1 ) ^ k s _ { ( 1 ^ k ) } [ s _ { \\pi } ( Z ) ] ) ' \\cr & = \\sum _ \\nu ( - 1 ) ^ { | \\nu | } \\ , \\ell _ { \\pi \\nu } \\ , s _ { \\nu ' } ( Z ) \\end{align*}"}
-{"id": "7697.png", "formula": "\\begin{align*} x _ { n _ 2 + 1 } = f ( x _ { n _ 2 } ) + \\gamma _ { n _ 2 + 1 } \\le - \\lambda x _ { n _ 2 } + K ( 1 + \\lambda ) + \\delta _ 0 \\le K ( 1 + \\lambda ) + \\varepsilon _ 0 < 2 K - \\varepsilon _ 0 , \\end{align*}"}
-{"id": "4078.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q } { z ^ { p + 2 q } } , \\end{align*}"}
-{"id": "4798.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { t ^ 2 - ( c \\pm a t ) ^ 2 } , c = c o n s t , \\end{align*}"}
-{"id": "1107.png", "formula": "\\begin{align*} \\| D ^ * _ { T _ 0 ^ c } h \\| _ 1 & \\leq \\| D ^ * _ { T _ 0 } h \\| _ 1 + 2 \\| D ^ * _ { T _ 0 ^ c } x _ 0 \\| _ 1 + \\rho \\\\ & = \\| D ^ * _ { T _ 0 } h \\| _ 1 + 2 \\sigma _ k ( D ^ * x _ 0 ) _ 1 + \\rho . \\end{align*}"}
-{"id": "4397.png", "formula": "\\begin{align*} H _ \\nu ( \\zeta | A ) \\ge & \\sum _ { k \\ge 1 } \\mu ( C _ k | A ) H _ \\nu ( \\zeta | C _ k \\cap A ) \\ge \\sum _ { k : 2 ^ k > l ( A ) } \\nu ( C _ k | A ) H _ \\nu ( \\zeta | C _ k \\cap A ) \\\\ = \\sum _ { k : 2 ^ k > l ( A ) } & \\frac { 1 } { 2 ^ k } H _ \\nu \\left ( \\eta _ { - 2 ^ k } ^ { - l ( A ) - 1 } ( \\tau ) \\right ) = \\sum _ { k : 2 ^ k > l ( A ) } \\frac { 1 } { 2 ^ k } ( 2 ^ k - l ( A ) ) H _ \\nu ( \\eta ) = \\infty . \\end{align*}"}
-{"id": "12.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { m + 1 } ( 1 - \\delta _ n ) | a _ n | + \\eta _ { m + 1 } \\sum _ { n = 1 } ^ { m + 1 } | a _ n | \\le \\| \\sum _ { n = 1 } ^ { m + 1 } a _ n x _ n \\| \\end{align*}"}
-{"id": "2898.png", "formula": "\\begin{align*} \\chi ( u ^ q ) = \\sum _ { x \\in X _ \\chi } x ^ q + \\sum _ { y \\in Y _ \\chi } y \\psi ( u ^ q ) = \\sum _ { x \\in X _ \\chi } x ^ q + \\sum _ { y \\in Y _ \\psi } y . \\end{align*}"}
-{"id": "1835.png", "formula": "\\begin{align*} \\gamma _ s ( \\theta ) = ( w ( s , \\theta ) , \\theta ) = ( 2 + g ( s ) + g ' ( s , \\theta ) , \\theta ) \\end{align*}"}
-{"id": "7337.png", "formula": "\\begin{align*} \\langle w , v \\rangle _ { \\Lambda } = \\langle \\pi _ { - } ^ { - 1 } ( w ) , \\pi _ { + } ^ { - 1 } ( v ) \\rangle _ { T } , w \\in \\Lambda ^ k _ q ( \\mathfrak { u } _ - ) , \\ v \\in \\Lambda ^ k _ q ( \\mathfrak { u } _ + ) . \\end{align*}"}
-{"id": "4915.png", "formula": "\\begin{align*} F _ { \\kappa , j , r } ( \\mathfrak { z } , z ) : = \\mathfrak { z } _ 2 ^ { - j } \\sum _ { m = 0 } ^ { \\infty } \\ ; \\sideset { } { ^ * } \\sum _ { \\mathfrak { b } \\subseteq \\mathcal { O } _ { \\Q ( \\mathfrak { z } ) } } \\frac { C _ { \\kappa } \\left ( \\mathfrak { b } , m \\right ) } { N ( \\mathfrak { b } ) ^ { \\frac { \\kappa } { 2 } - j } } ( 4 \\pi m ) ^ r e ^ { \\frac { 2 \\pi m \\mathfrak { z } _ 2 } { N ( \\mathfrak { b } ) } } e ^ { 2 \\pi i m z } , \\end{align*}"}
-{"id": "5494.png", "formula": "\\begin{align*} \\mathcal { R } ^ \\mathrm { D L } _ k [ \\iota ] = T _ d \\log _ 2 \\left ( 1 + \\gamma _ k ^ { \\mathrm { D L } } [ \\iota ] \\right ) \\end{align*}"}
-{"id": "7447.png", "formula": "\\begin{align*} ( \\eta _ 1 , \\dots , \\eta _ n ) = ( \\xi ' _ 1 , \\dots , \\xi ' _ n ) \\begin{pmatrix} X & \\cdots & 0 & 0 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & X & 1 & \\cdots & 0 \\\\ 0 & \\cdots & 0 & 1 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 & 0 & \\cdots & 1 \\end{pmatrix} = ( \\xi ' _ 1 , \\dots , \\xi ' _ n ) e ^ i h ^ i ( X ) . \\end{align*}"}
-{"id": "5243.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ k q _ { j } ( n ) V _ { n + j } = 0 \\end{align*}"}
-{"id": "185.png", "formula": "\\begin{align*} p _ { n , x , u } : = \\int _ { B _ x ( r _ { n , u } ) } f ( y ) \\ , d y { \\rm a n d } r _ { n , u } : = \\biggl \\{ \\frac { e ^ { \\Psi ( k ) } u } { V _ d ( n - 1 ) } \\biggr \\} ^ { 1 / d } . \\end{align*}"}
-{"id": "6043.png", "formula": "\\begin{align*} a x _ 1 ^ 2 + b x _ 2 ^ 2 - a b x _ 3 ^ 2 = d \\end{align*}"}
-{"id": "9863.png", "formula": "\\begin{align*} 0 = \\mu _ 0 ( H _ g ) < \\mu _ { 1 } ( H _ g ) \\leq \\mu _ { 2 } ( H _ g ) \\leq . . . \\leq \\mu _ { n } ( H _ g ) \\leq . . . \\ , , \\end{align*}"}
-{"id": "1208.png", "formula": "\\begin{align*} \\chi ( p _ 1 ( a ) ) \\chi ( p _ 2 ( a ) ) = 1 , \\forall a \\in \\{ z , z + 1 , \\ldots , z + b - 1 \\} \\end{align*}"}
-{"id": "326.png", "formula": "\\begin{align*} & w ^ 2 + x z w + ( x ^ 3 + u ^ 2 x ^ 2 y + u x y ^ 2 + y ^ 3 ) w = \\\\ & z ^ 3 + ( u ^ 2 x ^ 2 + x y + y ^ 2 ) x ^ 2 z + ( u ^ 2 x ^ 4 + u x ^ 3 y + x ^ 2 y ^ 2 ) y ^ 2 . \\end{align*}"}
-{"id": "9213.png", "formula": "\\begin{align*} j ( u ) ( z ) & : = \\mathbb { E } [ \\int _ 0 ^ T ( \\int _ { D } h ( t , Y ( t , x , z ) , u ( t , x , z ) , z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] d x ) d t \\\\ & + \\int _ D k ( x , Y ( T , x , z ) , z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] d x . \\end{align*}"}
-{"id": "1038.png", "formula": "\\begin{align*} c _ { d , d } ( c _ { q , q } ^ d + c _ { q , 0 } ^ d ) + 2 c _ { q , q } { \\ , } c _ { d , d } ^ q ~ = ~ 0 . \\end{align*}"}
-{"id": "4228.png", "formula": "\\begin{align*} I _ { - 1 } ( f _ 1 ) + I _ { - 1 } ( f ) = 2 f ( 0 ) , \\textnormal { w h e r e } \\ , \\ , f _ 1 ( x ) = f ( x + 1 ) . \\end{align*}"}
-{"id": "7771.png", "formula": "\\begin{align*} [ x , y ] = z , [ x , z ] = [ y , z ] = 0 . \\end{align*}"}
-{"id": "9222.png", "formula": "\\begin{align*} I _ 1 & = \\mathbb { E } [ \\int _ 0 ^ T \\int _ D \\{ H ( t , x ) - \\widehat { H } ( t , x ) - \\widehat { p } ( t , x ) [ A _ u Y ( t , x ) - A _ { \\hat { u } } \\widehat { Y } ( t , x ) + \\widetilde { a } ( t , x ) ] - \\widehat { q } ( t , x ) \\widetilde { b } ( t , x ) \\\\ & - \\int _ { \\mathbb { R } } \\hat { r } ( t , x , \\zeta ) \\tilde { c } ( t , x , \\zeta ) \\nu ( d \\zeta ) \\} d x d t ] . \\end{align*}"}
-{"id": "6460.png", "formula": "\\begin{align*} & \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } \\left [ \\psi ( x ) w ( s , x ) - \\psi ( y ) w ( s , y ) \\right ] ^ { 2 } k ( x , y ) d x d y \\\\ & \\quad \\quad \\quad \\quad \\quad \\geq \\int _ { \\rho ' B _ { 1 } } \\int _ { \\rho ' B _ { 1 } } \\left [ w ( s , x ) - w ( s , y ) \\right ] ^ { 2 } k ( x , y ) d x d y , \\end{align*}"}
-{"id": "8933.png", "formula": "\\begin{align*} f ( M ) = \\sum _ { \\ell \\ , : \\ , \\abs { \\lambda _ \\ell ( { L } ) - \\lambda _ j ( L ) } \\le \\varepsilon } e _ \\ell \\cdot ( M - L ) ( e _ \\ell ) + o \\bigl ( \\norm { M - L } \\bigr ) ; \\end{align*}"}
-{"id": "6420.png", "formula": "\\begin{align*} B u = \\frac { d } { d t } ( g _ { 1 - \\alpha } * u ) , D ( B ) = \\{ u \\in L ^ { p } ( [ 0 , T ] ; X ) \\ , : \\ , g _ { 1 - \\alpha } * u \\in { _ { 0 } } W { ^ { 1 , p } } ( [ 0 , T ] ; X ) \\} . \\end{align*}"}
-{"id": "4453.png", "formula": "\\begin{align*} \\frac { d F _ { n , x } ( u ) } { d u } = \\mathrm { B } _ { k , n - k } ( p _ { n , x , u } ) \\frac { \\partial p _ { n , x , u } } { \\partial u } , \\end{align*}"}
-{"id": "7541.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\mathcal N _ 1 < \\infty x _ { \\mathcal { N } _ 1 } \\in [ c , 2 K - c ] \\right ] = 1 . \\end{align*}"}
-{"id": "9197.png", "formula": "\\begin{align*} Y ( 0 , x ) = \\xi ( x ) , x \\in D \\end{align*}"}
-{"id": "2612.png", "formula": "\\begin{align*} \\delta _ v ( \\delta _ h ( \\eta ) + \\beta _ \\omega ) & = \\delta _ v ( \\delta _ h ( \\eta ) ) + \\delta _ v ( \\beta _ \\omega ) \\\\ & = \\delta _ h ( \\delta _ v ( \\eta ) ) + \\delta _ v ( \\beta _ \\omega ) \\\\ & = \\delta _ h ( \\alpha _ \\omega ) - \\delta _ h ( \\alpha _ \\omega ) = 0 , \\\\ \\end{align*}"}
-{"id": "3299.png", "formula": "\\begin{align*} \\langle w , v \\rangle _ { \\Lambda } = \\langle \\pi _ { - } ^ { - 1 } ( w ) , \\pi _ { + } ^ { - 1 } ( v ) \\rangle _ { T } , w \\in \\Lambda ^ k _ q ( \\mathfrak { u } _ - ) , \\ v \\in \\Lambda ^ k _ q ( \\mathfrak { u } _ + ) . \\end{align*}"}
-{"id": "5957.png", "formula": "\\begin{align*} \\gamma ( Z _ t ) - \\gamma ( Y _ t ) = & \\ , z - y + U ( z ) - U ( y ) + \\int _ 0 ^ t \\big [ D U ( Z _ s ) - D U ( Y _ s ) \\big ] R \\cdot \\dd W _ s \\\\ & + \\lambda \\int _ 0 ^ t \\big [ U ( Z _ s ) - U ( Y _ s ) \\big ] \\dd s + \\int _ 0 ^ t A ( Z _ s - Y _ s ) \\ , \\dd s \\ , . \\end{align*}"}
-{"id": "5844.png", "formula": "\\begin{align*} \\sigma ^ { 2 } \\left ( x \\right ) = \\left ( 4 \\int e ^ { 2 x } K \\left ( x \\right ) \\mbox { \\rm d } x + c _ { 1 } \\right ) e ^ { - 2 x } , \\end{align*}"}
-{"id": "875.png", "formula": "\\begin{align*} \\operatorname * { l e v } \\nolimits _ { \\varphi _ { A , k } , < } ( t ) = t k + \\operatorname * { c o r e } A \\forall \\ , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "2057.png", "formula": "\\begin{align*} x '' ( s ) & = \\frac { d } { d s } x ' ( s ) = \\nabla _ { x ' ( s ) } x ' ( s ) = 0 , \\end{align*}"}
-{"id": "8254.png", "formula": "\\begin{align*} \\partial _ t h ( t , x ) = \\nu \\Delta h ( t , x ) + \\lambda ( | \\nabla h ( t , x ) | ^ 2 - \\infty ) + \\sqrt { D } \\xi ( t , x ) , ( t , x ) \\in \\R _ + \\times \\R , \\end{align*}"}
-{"id": "4978.png", "formula": "\\begin{align*} \\frac { h _ { H } ( f ^ { n } ( P ) ) } { n ^ { r - 1 } \\delta ^ { n } } \\leq C _ { 3 } \\left ( \\sqrt [ ] { h _ { H } ( P ) } + \\sum _ { k = 1 } ^ { n - 1 } \\sqrt [ ] { \\frac { h _ { H } ( f ^ { k } ( P ) ) } { k ^ { r - 1 } \\delta ^ { k } } } + h _ { H } ( P ) \\right ) . \\end{align*}"}
-{"id": "8511.png", "formula": "\\begin{align*} c ' _ { \\alpha _ 1 } & = c _ { \\alpha _ 1 } + c _ { \\alpha _ 1 + \\alpha _ 2 } + c _ { \\alpha _ 1 + 2 \\alpha _ 2 } - \\pi _ 1 , \\\\ c ' _ { \\alpha _ 1 + \\alpha _ 2 } & = 2 \\pi _ 1 - \\pi _ 2 , \\\\ c ' _ { \\alpha _ 1 + 2 \\alpha _ 2 } & = \\pi _ 2 - \\pi _ 1 , \\\\ c ' _ { \\alpha _ 2 } & = c _ { \\alpha _ 1 + \\alpha _ 2 } + 2 c _ { \\alpha _ 1 + 2 \\alpha _ 2 } + c _ { \\alpha _ 2 } - \\pi _ 2 , \\end{align*}"}
-{"id": "3404.png", "formula": "\\begin{align*} W ^ { ( i ) } _ j = \\zeta _ { r ^ { ( i ) } _ j } ^ k \\left ( t + \\frac { z _ i } { r ^ { ( i ) } _ j t } + \\cdots \\right ) \\end{align*}"}
-{"id": "4291.png", "formula": "\\begin{align*} ( n , A ) ( n , f ) = n ( n , A , f ) \\end{align*}"}
-{"id": "9909.png", "formula": "\\begin{align*} [ K ' , E ] & \\ ; = \\ ; [ K ' , F ] \\ ; = \\ ; [ K ' , K ] \\ ; = \\ ; 0 , \\\\ \\ ; K E \\ ; = \\ ; q ^ 2 E K , K F & \\ ; = \\ ; q ^ { - 2 } F K , q E F - q ^ { - 1 } F E \\ ; = \\ ; \\frac { K ^ 2 - K '^ { 2 } } { q - q ^ { - 1 } } . \\end{align*}"}
-{"id": "6147.png", "formula": "\\begin{align*} x _ j x _ i = q x _ i x _ j , \\forall \\ ; 1 \\leq i < j \\leq n . \\end{align*}"}
-{"id": "8737.png", "formula": "\\begin{align*} e ^ { t A } \\left ( \\begin{array} [ c ] { c } y \\\\ z \\end{array} \\right ) = \\left ( \\begin{array} [ c ] { c c } \\cos \\sqrt { { \\Lambda } } t & \\frac { 1 } { \\sqrt { { \\Lambda } } } \\sin \\sqrt { { \\Lambda } } t \\\\ - \\sqrt { { \\Lambda } } \\sin \\sqrt { { \\Lambda } } t & \\cos \\sqrt { { S } } t \\end{array} \\right ) \\left ( \\begin{array} [ c ] { c } y \\\\ z \\end{array} \\right ) , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "6776.png", "formula": "\\begin{align*} U _ { k , l } = \\{ \\varphi _ k < f \\} \\cap \\{ \\varphi > V _ { \\theta } - l \\} . \\end{align*}"}
-{"id": "8899.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\int _ { \\R ^ N } \\abs { D _ { A _ n } \\Tilde { u } _ n - D U } ^ 2 + \\abs { \\Tilde { u } _ n - U } ^ 2 = 0 , \\end{align*}"}
-{"id": "4626.png", "formula": "\\begin{align*} e ^ { \\Omega _ e } _ { L ^ { \\mathrm { p } } } = \\frac { \\lVert p _ { n u m } ^ { \\Omega _ e } - p _ { r e f } ^ { \\Omega _ e } \\rVert _ { L ^ { \\mathrm { p } } } } { \\lVert p _ { r e f } ^ { \\Omega _ e } \\rVert _ { L ^ { \\mathrm { p } } } } , \\end{align*}"}
-{"id": "1961.png", "formula": "\\begin{align*} \\hat { a } ( 0 ) \\leq \\hat { a } ( \\tau ) = \\sum _ { \\alpha = 1 } ^ r h _ { \\alpha \\alpha } ( \\tau ) \\leq \\frac { 1 } { c _ 0 m } ( 2 p _ i \\tau + b _ i ( 0 ) ) i \\tau \\in [ 0 . \\tau _ * ) . \\end{align*}"}
-{"id": "4509.png", "formula": "\\begin{align*} \\biggl | \\frac { \\partial f ^ t ( x ) } { \\partial x ^ t } \\biggr | = f ( x ) \\prod _ { j = 1 } ^ d | H _ { t _ j } ( x _ j ) | \\leq f ( x ) \\prod _ { j = 1 } ^ d p _ { t _ j } ( \\| x \\| ) \\leq f ( x ) q _ { | t | } ( \\| x \\| ) , \\end{align*}"}
-{"id": "993.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": "4869.png", "formula": "\\begin{align*} \\tilde { H } ( n , n ) \\overset { ( d ) } { = } \\max _ { x \\in \\Z _ { > 0 } } \\left \\lbrace \\sum _ { i = 1 } ^ x w _ { 1 i } + \\bar { H } ( n - 1 , n - x ) \\right \\rbrace , \\end{align*}"}
-{"id": "2497.png", "formula": "\\begin{gather*} \\sum _ { k = 1 } ^ n { \\bf E } \\sup _ { 0 \\leqslant t \\leqslant 1 } \\left | z _ 1 ( u _ k , t ) - z _ 2 ( u _ k , t ) \\right | \\leqslant \\dfrac { 2 n ^ 3 } { 3 } \\cdot \\sqrt { \\varepsilon } , \\\\ \\sum _ { k = 1 } ^ n { \\bf E } \\sup _ { 0 \\leqslant t \\leqslant 1 } \\left | z _ i ( u _ k , t ) - z _ { i + 1 } ( u _ k , t ) \\right | \\leqslant \\dfrac { 2 n ^ 4 } { 3 } \\cdot \\sqrt { \\varepsilon } , 2 \\leqslant i \\leqslant n - 1 . \\end{gather*}"}
-{"id": "4332.png", "formula": "\\begin{align*} | \\langle b ^ * ( e _ { k _ m } ) , b ^ * ( e _ { k _ n } ) \\rangle | = | \\langle b b ^ * ( e _ { k _ m } ) , e _ { k _ n } \\rangle | < \\frac { \\varepsilon ^ 2 } { 3 \\cdot 2 ^ n } \\end{align*}"}
-{"id": "1826.png", "formula": "\\begin{align*} \\alpha _ j = d \\log z _ i = d ( - s _ i ^ { - 1 } + i \\theta _ i ) , 1 \\le i \\le k ; \\ \\alpha _ j = d z _ j , 3 g - 3 + n \\geq j > k \\end{align*}"}
-{"id": "9700.png", "formula": "\\begin{gather*} \\lim _ { y \\to \\infty } \\frac { G ^ { - 1 } ( y ) } { ( \\log y ) ^ { - 1 / \\alpha } } = 1 , G ^ { - 1 } \\in _ \\infty ( 0 ) , \\\\ \\lim _ { y \\to \\infty } \\frac { \\Gamma ( y ) } { \\log ^ { - ( \\alpha + 1 ) / \\alpha } ( y ) / y } = \\frac { 1 } { \\alpha } , \\Gamma = g \\circ G ^ { - 1 } \\in _ \\infty ( - 1 ) . \\end{gather*}"}
-{"id": "7621.png", "formula": "\\begin{align*} \\lambda _ 1 \\mathbf { v } _ v = \\sum _ { u \\sim v } \\mathbf { v } _ u \\leq 4 + \\sum _ { \\substack { u \\sim v \\\\ u \\in A } } \\mathbf { v } _ u \\leq 4 + \\sum _ { u \\in A } \\mathbf { v } _ u \\leq C \\epsilon \\sqrt { n } , \\end{align*}"}
-{"id": "8295.png", "formula": "\\begin{align*} \\deg ( D ) : = \\sum a _ i . \\end{align*}"}
-{"id": "7274.png", "formula": "\\begin{align*} u _ n ( \\widetilde { Y } , \\widetilde { V } ; \\{ \\widetilde { w } _ j \\} ) = ( \\pi | _ { \\widetilde { Y } } ) ^ * u _ n ( Y , X ; \\{ w _ j \\} ) \\end{align*}"}
-{"id": "5872.png", "formula": "\\begin{align*} Z ^ { 4 } = e ^ { - 2 m t } \\left ( \\partial _ { t } - H - m \\left ( 2 m \\left ( \\int \\frac { d x } { \\sigma \\left ( x \\right ) } \\right ) ^ { 2 } - 1 \\right ) F \\partial _ { F } \\right ) \\end{align*}"}
-{"id": "6824.png", "formula": "\\begin{align*} \\forall n \\geq 1 \\ \\forall m \\in \\Z L _ m \\cdot B _ n = \\sum _ { k = 0 } ^ { n - 1 } [ B _ k , B _ { m + n - k } ] + n B _ { m + n } \\end{align*}"}
-{"id": "3923.png", "formula": "\\begin{align*} ( k + \\sigma ) ( k - 1 + \\sigma ) \\ , a _ { k , \\sigma } = \\left [ ( k - 1 + 2 m + \\sigma ) ( k - 2 + \\sigma ) - z \\right ] \\ , a _ { k - 2 , \\sigma } \\ , . \\\\ \\end{align*}"}
-{"id": "2052.png", "formula": "\\begin{align*} 0 \\geq \\frac { \\Delta w ( x _ 0 ) - \\Delta w ( y _ 0 ) } { \\bar { w } } - \\frac { m } { \\bar { w } } \\sum _ { i = 1 } ^ n \\nabla ^ 2 _ { E _ i , E _ i } \\bar { w } . \\end{align*}"}
-{"id": "4878.png", "formula": "\\begin{align*} a \\equiv b \\bmod I ^ n \\Rightarrow \\delta ( a ) \\equiv \\delta ( b ) \\bmod I ^ { n - 1 } \\end{align*}"}
-{"id": "8308.png", "formula": "\\begin{align*} D - \\iota ( D ) = ( D + x ) - \\iota ( D + x ) , \\end{align*}"}
-{"id": "4760.png", "formula": "\\begin{align*} \\mathcal { M } '' : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "878.png", "formula": "\\begin{align*} Y = \\mathbb { R } k + A . \\end{align*}"}
-{"id": "4803.png", "formula": "\\begin{align*} z ' + \\frac { 1 } { t } \\ , z = \\pm \\frac { \\sqrt { a ^ 2 + 4 c t ^ 2 } } { t } . \\end{align*}"}
-{"id": "9328.png", "formula": "\\begin{align*} \\int _ D U ' ( Y ( T , x , z ) ) Y ( T , x , z ) d x \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] = \\tilde { p } ( 0 , z ) \\exp ( - \\int _ 0 ^ T \\frac { b _ 0 ( s , z ) } { \\sigma _ 0 ( s , z ) } d B ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ T ( \\frac { b _ 0 ( s , z ) } { \\sigma _ 0 ( s , z ) } ) ^ { 2 } d s ) . \\end{align*}"}
-{"id": "6604.png", "formula": "\\begin{align*} \\lambda _ { i } ( M ) = \\begin{cases} 0 & { \\hbox { f o r $ i $ s u c h t h a t } } \\lambda _ { i } ( N ) = 1 \\\\ \\alpha _ 1 ( 1 - \\alpha _ 1 ) & { \\hbox { f o r $ i $ s u c h t h a t } } \\lambda _ { i } ( N ) = 0 \\\\ \\end{cases} \\end{align*}"}
-{"id": "1674.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\dd x \\int _ { \\R ^ d } | \\partial _ { v _ j } D _ x { \\psi } ( x , v ) | ^ p \\ , \\dd v = \\int _ { \\R ^ d } \\| \\partial _ { v _ j } D _ x { \\psi } ( x , \\cdot ) \\| _ { L ^ p ( \\R ^ { d } ) } ^ p \\dd x \\le C ( \\lambda ) \\ , \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ { p } ( \\R ^ d _ x ) ) } ^ p . \\end{align*}"}
-{"id": "7803.png", "formula": "\\begin{align*} \\Delta w = \\Delta _ { \\Sigma } w + w _ { n n } + H w _ { n } , \\end{align*}"}
-{"id": "5290.png", "formula": "\\begin{align*} \\vert \\sum _ { n \\leq 3 \\sqrt { T / \\pi } } \\frac { 1 } { n ^ { { 1 } / { 2 } } } \\int _ { T } ^ { 2 T } e ^ { i F ( n , t ) } d t \\vert = O \\left ( \\sum _ { n \\leq { 3 \\sqrt { T / \\pi } } } \\frac { 1 } { n ^ { 1 / 2 } } T ^ { 1 / 2 } \\right ) = O \\left ( T ^ { 3 / 4 } \\right ) . \\end{align*}"}
-{"id": "3453.png", "formula": "\\begin{align*} { \\cal K } ^ { u , \\phi } _ m ( s , x ) = \\frac { \\partial \\phi _ m } { \\partial s } + [ { \\cal A } \\phi ] _ m ( s , x ) + g _ m ( s , x , u ( s , x ) , K ( u , \\nabla \\phi ) ( s , x ) ) \\rangle < 0 . \\end{align*}"}
-{"id": "4027.png", "formula": "\\begin{align*} J _ { K , N } \\left ( e ^ { \\frac { 2 \\pi \\sqrt { - 1 } } { 3 } } \\right ) = \\left \\{ \\begin{array} { r l } 0 & ( N = 6 l ) \\\\ 1 & ( N = 6 l + 1 ) \\\\ 1 & ( N = 6 l + 2 ) \\\\ 0 & ( N = 6 l + 3 ) \\\\ - 1 & ( N = 6 l + 4 ) \\\\ - 1 & ( N = 6 l + 5 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "5854.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int L _ { Y _ { I } } C _ { x } d x + C ^ { x } Y _ { I } + 2 \\psi _ { I } = 0 , \\end{align*}"}
-{"id": "6018.png", "formula": "\\begin{gather*} \\sigma \\stackrel { \\sigma } { = } 1 , \\mu ^ e \\stackrel { \\sigma } { = } 0 , \\rho \\stackrel { \\sigma } { = } - \\tfrac { 1 } { 2 } \\varepsilon , \\end{gather*}"}
-{"id": "2725.png", "formula": "\\begin{align*} \\nu ^ * = P _ C ( - y + \\sum _ { i \\in I } \\alpha ^ * _ i ( x _ i - \\frac { 1 } { L } g _ i ) ) , \\end{align*}"}
-{"id": "8795.png", "formula": "\\begin{align*} [ T ( u ) , T ( w ) ] = 0 , \\end{align*}"}
-{"id": "3499.png", "formula": "\\begin{align*} | \\bar { J } | _ { \\bar { g } } = | J + V | _ { g + h } \\le | J + V | _ g + \\tfrac 1 2 | h | _ g | J + V | _ g + O ( | h | _ g ^ 2 ) . \\end{align*}"}
-{"id": "9497.png", "formula": "\\begin{align*} \\delta R & = R _ { \\mu \\nu } \\delta g ^ { \\mu \\nu } + \\mathrm { t d } \\\\ \\delta \\mathrm { d v o l } _ g & = - \\frac { 1 } { 2 } g _ { \\mu \\nu } \\mathrm { d v o l } _ g \\delta g ^ { \\mu \\nu } . \\end{align*}"}
-{"id": "1662.png", "formula": "\\begin{align*} { \\psi } ( z ) = G _ { \\lambda } g ( z ) = \\int _ { 0 } ^ { + \\infty } e ^ { - \\lambda t } P _ t g ( z ) \\ , \\dd t \\ , . \\end{align*}"}
-{"id": "7546.png", "formula": "\\begin{align*} f ( x ) = \\left \\{ \\begin{array} { l l } F ( x ) , & \\mbox { i f ~ ~ } x \\in [ 0 , 0 . 9 9 ] , \\\\ \\frac { F ( 0 . 9 9 ) } { x + 0 . 0 1 } , & \\mbox { i f ~ ~ } x \\in ( 0 . 9 9 , \\infty ) . \\end{array} \\right . \\end{align*}"}
-{"id": "746.png", "formula": "\\begin{align*} \\frac { 1 } { | W | } \\prod _ { i = 1 } ^ n ( h + e _ i - 1 ) , \\end{align*}"}
-{"id": "238.png", "formula": "\\begin{align*} \\beta _ 3 ( f ) : = \\mathbb { E } _ f \\bigl \\{ \\bigl | \\log f ( X _ 1 ) + H ( f ) \\bigr | ^ 3 \\bigr \\} = \\int _ { \\mathcal { X } } f ( x ) | \\log f ( x ) + H ( f ) | ^ 3 \\ , d x . \\end{align*}"}
-{"id": "4219.png", "formula": "\\begin{align*} d ( A ^ T x , \\partial g ( \\bar { y } ) ) & \\geq \\gamma \\cdot d ( A ^ T x , \\partial g ( \\bar { y } ) \\cap R ( A ^ T ) ) = \\gamma \\cdot \\min _ { A ^ T u \\in \\partial g ( \\bar { y } ) } \\| A ^ T x - A ^ T u \\| \\\\ & = \\gamma \\cdot \\min _ { y \\in A ^ T X _ 0 } \\| A ^ T x - y \\| \\geq \\gamma \\cdot \\min _ { y \\in V _ 0 } \\| A ^ T x - y \\| = \\gamma \\cdot \\| A ^ T x - \\hat { y } \\| , \\end{align*}"}
-{"id": "4526.png", "formula": "\\begin{align*} T _ { 1 2 } : = \\biggl | \\int _ { \\mathcal { X } _ n ^ c } \\int _ { \\mathcal { X } _ n } f ( x ) f ( y ) \\log f ( y ) \\int _ { \\tilde { u } _ { n , x , y } } ^ \\infty \\log ( u f ( x ) ) \\ , d ( \\tilde { F } _ { n , x } - F ^ - _ { n , x } ) ( u ) \\ , d y \\ , d x \\biggr | . \\end{align*}"}
-{"id": "5193.png", "formula": "\\begin{align*} g \\left ( \\sum ^ { n } _ { i = 1 } t _ { i } x _ { i } \\right ) \\geq \\sum ^ { n } _ { i = 1 } t _ { i } g ( x _ { i } ) . \\end{align*}"}
-{"id": "7677.png", "formula": "\\begin{align*} S _ { \\alpha } ( s ) u _ { n } & = S _ { \\alpha } ( s ) \\big ( \\frac { 1 } { \\nu ( d , \\alpha ) } s _ { n } \\chi _ { n ^ { - \\alpha q / d } s _ { n } ^ { - q / d } } ( x _ { n } ) \\big ) \\\\ & = \\frac { s _ { n } } { \\nu ( d , \\alpha ) } S _ { \\alpha } ( s ) \\chi _ { n ^ { - \\alpha q / d } s _ { n } ^ { - q / d } } ( x _ { n } ) \\\\ & \\geq s _ { n } \\chi _ { n ^ { - \\alpha q / d } s _ { n } ^ { - q / d } + \\sqrt { s } } ( x _ { n } ) \\\\ & \\geq s _ { n } \\chi _ { n ^ { - \\alpha q / d } s _ { n } ^ { - q / d } } ( x _ { n } ) . \\end{align*}"}
-{"id": "2820.png", "formula": "\\begin{align*} \\rho _ { i j } = \\max \\left ( \\left | \\sigma _ { i j } ^ { ( 0 ) } \\right | , \\left | \\sigma _ { i j } ^ { ( 1 ) } \\right | \\right ) , A _ { i j } ^ { ( r ) } = \\int _ { \\sigma _ { i j } ^ { ( 0 ) } } ^ { \\sigma _ { i j } ^ { ( 1 ) } } \\frac { \\left ( 1 + | h | \\right ) ^ { ( n - r ) / 2 } } { ( 1 - h ^ 2 ) ^ { \\hat r / 2 } } \\ , d h . \\end{align*}"}
-{"id": "3964.png", "formula": "\\begin{align*} u ^ { - } ( r ^ { - 1 } ) v = ( u ^ { - } ( r ^ { - 1 } ) v ) ^ { \\max } = v ^ { \\max } v = u ^ { - } ( - r ^ { - 1 } ) v ^ { \\max } , \\end{align*}"}
-{"id": "3196.png", "formula": "\\begin{align*} \\left ( \\begin{array} { c } m _ j ^ 2 \\\\ m _ j ^ 3 \\\\ \\vdots \\\\ m _ j ^ r \\end{array} \\right ) - ( t _ { j k } ^ 1 ) ^ { - 1 } S _ { j k } \\left ( \\begin{array} { c } m _ k ^ 2 \\\\ m _ k ^ 3 \\\\ \\vdots \\\\ m _ k ^ r \\end{array} \\right ) = ( t _ { j k } ^ 1 ) ^ { - 1 } \\cdot \\left ( \\begin{array} { c } a _ { j k } ^ 2 \\\\ a _ { j k } ^ 3 \\\\ \\vdots \\\\ a _ { j k } ^ r \\end{array} \\right ) \\end{align*}"}
-{"id": "1586.png", "formula": "\\begin{align*} a \\left ( t , x \\right ) = - \\frac { 1 } { 2 } \\int \\left ( T _ { 1 } L _ { Y _ { 1 } } C _ { x } + T _ { 2 } L _ { Y _ { 2 } } C _ { x } - T _ { 1 , t } Y _ { 1 } - T _ { 2 , t } Y _ { 2 } \\right ) d x + f \\left ( t \\right ) \\end{align*}"}
-{"id": "5560.png", "formula": "\\begin{align*} \\dot { A } _ { 1 } & = ( 1 - K ) \\beta A _ { 1 } - \\frac { 1 } { 2 } b _ { x x } ( 1 ) B _ { 1 } , \\\\ \\dot { B } _ { 1 } & = ( 1 + K ) \\beta B _ { 1 } . \\end{align*}"}
-{"id": "4044.png", "formula": "\\begin{align*} p + r = \\alpha q , q + r = \\beta p , p + q = \\gamma r . \\end{align*}"}
-{"id": "5931.png", "formula": "\\begin{gather*} \\| D _ v { \\psi } ( \\cdot , v ) \\| _ { H ^ s _ p ( \\R ^ { d } ) } ^ p = \\int _ { \\R ^ d } | D _ v \\ , G _ { \\lambda } h _ s ( x , v ) | ^ p \\ , \\dd x \\ , . \\end{gather*}"}
-{"id": "4170.png", "formula": "\\begin{align*} n | a _ n | \\big [ 1 + ( n - 1 ) ( \\lambda - \\mu + n \\lambda \\mu ) \\big ] = B _ n + p _ { n - 1 } + p _ 1 B _ { n - 1 } + \\cdots + p _ { n - 2 } B _ 2 . \\end{align*}"}
-{"id": "7346.png", "formula": "\\begin{align*} v = \\sum _ { i } \\frac { \\langle w _ { i } ^ { ( k ) } , v \\rangle } { \\langle w _ { i } ^ { ( k ) } , v _ { i } ^ { ( k ) } \\rangle } v _ { i } ^ { ( k ) } . \\end{align*}"}
-{"id": "6888.png", "formula": "\\begin{align*} \\lambda _ { n } ( \\alpha ) = 3 + [ 0 ; 3 , a _ { n - 2 } , \\ldots , a _ 1 ] + [ 0 ; 3 , a _ { n + 2 } , \\ldots ] . \\end{align*}"}
-{"id": "5008.png", "formula": "\\begin{align*} & c _ { 0 } = b _ { 0 } \\\\ & c _ { n } = c _ { n - 1 } + C _ { 3 } \\sqrt [ ] { c _ { n - 1 } } \\ \\ \\ \\ . \\end{align*}"}
-{"id": "8186.png", "formula": "\\begin{align*} E _ c = S p e c \\ A [ x _ 1 , x _ 2 , x _ 3 ] / ( H _ 1 - c _ 1 , H _ 2 - c _ 2 ) \\end{align*}"}
-{"id": "2697.png", "formula": "\\begin{align*} \\theta _ { V _ { \\theta } } ^ n = \\lim _ { \\beta \\to + \\infty } \\theta _ { \\varphi _ { \\beta } } ^ n \\leq \\theta _ { + } ^ n . \\end{align*}"}
-{"id": "8956.png", "formula": "\\begin{align*} U ( x , 0 ) = U _ { 1 } ( x , 0 ) + U _ { 2 } ( x , 0 ) , \\end{align*}"}
-{"id": "8036.png", "formula": "\\begin{align*} \\omega = \\prod _ { \\nu = 1 } ^ d X _ { \\lambda _ { \\nu - 1 } , \\lambda _ \\nu } \\end{align*}"}
-{"id": "1388.png", "formula": "\\begin{align*} U _ { 1 , n } : = \\sqrt { n } \\left ( S _ { 1 , n } ( \\beta ^ \\prime ) - E ( S _ { 1 , n } ( \\beta ^ \\prime ) ) \\right ) = \\frac { 1 } { \\sqrt { n } } \\sum _ { t = 1 } ^ { n } e _ { t , \\beta ^ \\prime } ^ 2 - E e _ { t , \\beta ' } ^ 2 ; U _ { 2 , n } : = \\frac { 1 } { \\sqrt { n } } \\sum _ { t = 1 } ^ { n } e _ { t , \\beta ^ \\prime } c _ { n t } \\end{align*}"}
-{"id": "2087.png", "formula": "\\begin{align*} \\tilde { \\hat { M } } = \\tilde { V } ^ { T } M \\tilde { V } , \\ \\tilde { \\hat { D } } = \\tilde { V } ^ { T } D \\tilde { V } , \\ \\tilde { \\hat { K } } = \\tilde { V } ^ { T } K \\tilde { V } , \\\\ \\tilde { \\hat { F } } = \\tilde { V } ^ T F , \\ \\tilde { \\hat { C } } _ p = C _ p \\tilde { V } , \\ \\ \\tilde { \\hat { C } } _ v = C _ v \\tilde { V } , \\end{align*}"}
-{"id": "6464.png", "formula": "\\begin{align*} & ( 1 - q ) \\zeta _ { 2 } ( q ) \\phi \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } ( w ^ { 2 } ( s , x ) + w ^ { 2 } ( s , y ) ) k ( x , y ) d x d y \\\\ & \\leq c _ { 2 } C ( \\delta , \\Lambda ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\phi \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x = c _ { 3 } ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\phi \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x , \\end{align*}"}
-{"id": "6005.png", "formula": "\\begin{gather*} x \\times ( x \\times y ) : = - H _ { \\times } ( x , x ) y + H _ { \\times } ( x , y ) x . \\end{gather*}"}
-{"id": "3089.png", "formula": "\\begin{align*} b ( u , v ) = b _ 1 ( \\gamma _ 1 u , \\gamma _ 1 v ) + b _ 2 ( \\gamma _ 2 u , \\gamma _ 2 v ) \\end{align*}"}
-{"id": "6882.png", "formula": "\\begin{align*} f _ j = c _ j x _ 1 ^ { s _ { 1 j } } x _ 2 ^ { s _ { 2 j } } \\prod _ { \\lambda \\in k ^ \\times } ( x _ 1 ^ { w ' _ 2 } + \\lambda x _ 2 ^ { w ' _ 1 } ) ^ { t _ { \\lambda j } } + h _ j \\end{align*}"}
-{"id": "8080.png", "formula": "\\begin{align*} \\tilde { T } _ f ( x , n \\alpha ) = ( \\sigma ( x ) , n \\alpha + f ( x ) ) . \\end{align*}"}
-{"id": "197.png", "formula": "\\begin{align*} \\mathbb { E } _ f ( \\hat { H } _ n ) - H = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\tilde { \\beta } / d } } { n ^ { \\tilde { \\beta } / d } } \\int _ { \\mathcal { X } _ n } \\frac { C _ { n , \\tilde { \\beta } } ( x ) } { f ( x ) ^ { \\tilde { \\beta } / d } } \\ , d x \\ , , \\ , q _ n ^ { 1 - \\epsilon } \\ , , \\ , q _ n \\log n \\ , , \\ , \\frac { 1 } { n } \\biggr \\} \\biggr ) , \\end{align*}"}
-{"id": "7051.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( 6 ) = { { \\bf { h } } ^ { [ j 1 ] } } ( 6 ) { { \\bf { v } } ^ { [ 1 ] } } + { { \\bf { h } } ^ { [ j 2 ] } } ( 6 ) { { \\bf { v } } ^ { [ 2 ] } } . \\end{align*}"}
-{"id": "8552.png", "formula": "\\begin{align*} \\overline { N } _ { i n } & = \\displaystyle \\bigoplus _ { b \\in T _ { k } } D _ { k } \\otimes _ { F } N _ { \\tau ( ^ { \\ast } b ) } \\\\ & = \\displaystyle \\bigoplus _ { b \\in T _ { k } } D _ { k } \\otimes _ { F } N _ { \\sigma ( b ) } \\\\ & = N _ { o u t } \\end{align*}"}
-{"id": "4871.png", "formula": "\\begin{align*} \\bar { R } _ { 2 2 } ^ { \\rm e x p , n } ( x ; y ) = - \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\frac { e ^ { n f ( z ) - x \\sigma n ^ { 1 / 3 } z } } { 4 z } \\dd z + \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\frac { e ^ { n f ( z ) - y \\sigma n ^ { 1 / 3 } z } } { 4 z } \\dd z + \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\frac { 1 } { 2 z } e ^ { - \\sigma n ^ { 1 / 3 } \\vert x - y \\vert z } \\dd z \\ \\ - \\frac { 1 } { 4 } , \\end{align*}"}
-{"id": "6068.png", "formula": "\\begin{align*} P _ { j } ^ { \\alpha } ( t , x ) & = 2 ^ { j \\alpha / p } \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i x \\cdot \\xi } \\widehat { \\Psi } ( 2 ^ { - j } \\xi ) e ^ { - t | \\xi | ^ \\alpha } d \\xi \\\\ & = 2 ^ { j \\alpha / p } 2 ^ { j d } \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i 2 ^ { j } x \\cdot \\xi } \\widehat { \\Psi } ( \\xi ) e ^ { - t 2 ^ { j \\alpha } | \\xi | ^ \\alpha } d \\xi . \\end{align*}"}
-{"id": "1150.png", "formula": "\\begin{align*} z \\rho _ t = \\frac { 1 } { 2 } b _ { x x x } + z \\rho _ { x } b + 2 z \\rho b _ { x } , \\end{align*}"}
-{"id": "5006.png", "formula": "\\begin{align*} & b _ { 0 } = a _ { 0 } \\\\ & b _ { n } = b _ { n - 1 } + C _ { 1 } \\left ( \\sqrt [ ] { b _ { n - 1 } } + \\sqrt [ ] { b _ { n - 1 } + C _ { 2 } } \\right ) \\ \\ \\ \\ . \\end{align*}"}
-{"id": "2183.png", "formula": "\\begin{align*} & - \\int _ { B _ { 1 } } \\partial _ { s } ( g _ { 1 - \\alpha , m } * [ \\phi \\psi ^ { 2 } w ^ { 2 } ] ) d x + ( 1 - q ) \\frac { 1 } { 2 } \\phi \\cdot \\\\ \\leq & \\int _ { 0 } ^ { s } \\dot { g } _ { 1 - \\alpha , m } ( s - \\tau ) ( \\phi ( s ) - \\phi ( \\tau ) ) \\left ( \\int _ { B _ { 1 } } \\psi ^ { 2 } \\tilde { u } ^ { 1 - q } d x \\right ) ( \\tau ) d \\tau \\\\ & + C ( \\delta , \\Lambda ) ( 1 - q ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\phi ( s ) \\int _ { \\rho B _ { 1 } } w ^ { 2 } d x + R _ { m } ( s ) . \\end{align*}"}
-{"id": "5238.png", "formula": "\\begin{align*} \\forall i = 1 , \\ldots , n ; \\quad \\mbox { w e h a v e } w _ i + 1 \\ne n + k - 1 . \\end{align*}"}
-{"id": "5839.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( x \\right ) F _ { , x x } + \\left ( \\kappa \\left ( x \\right ) \\left ( \\mu \\left ( x \\right ) - \\lambda \\left ( x \\right ) - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( x \\right ) \\right ) F _ { , x } - F _ { , t } = 0 \\end{align*}"}
-{"id": "4793.png", "formula": "\\begin{align*} f f '' + ( f ' ) ^ 2 + 1 = \\pm a \\sqrt { f '^ 2 + 1 } . \\end{align*}"}
-{"id": "6241.png", "formula": "\\begin{align*} Z \\ \\mathbf { X } & = - \\eta , \\end{align*}"}
-{"id": "4055.png", "formula": "\\begin{align*} q + p = l + 2 . \\end{align*}"}
-{"id": "3941.png", "formula": "\\begin{align*} x x ^ \\dagger = c - 1 & = \\begin{cases} a \\ne - 1 & \\gcd ( q , c ) = 1 , \\\\ - 1 & \\gcd ( q , c ) \\ne 1 . \\end{cases} \\end{align*}"}
-{"id": "1715.png", "formula": "\\begin{gather*} [ H _ { \\Phi } ] = \\begin{pmatrix} 0 & 0 & 0 & 0 & 1 \\\\ 0 & 0 & 0 & I _ 2 & 0 \\\\ 0 & 0 & - 1 & 0 & 0 \\\\ 0 & I _ 2 & 0 & 0 & 0 \\\\ 1 & 0 & 0 & 0 & 0 \\end{pmatrix} , \\end{gather*}"}
-{"id": "3429.png", "formula": "\\begin{align*} d y ( t ) = - \\Gamma ^ * ( s , t ) g ( [ \\Gamma ^ * ] ^ { - 1 } ( s , t ) y ( t ) , [ \\Gamma ^ * ] ^ { - 1 } ( s , t ) z ( t ) ) d t + z ( t ) d w ( t ) , \\ , y ( T ) = \\zeta , \\end{align*}"}
-{"id": "4674.png", "formula": "\\begin{align*} \\left \\langle u , \\tilde c _ 1 u \\right \\rangle = \\sum _ { i = 1 } ^ m \\int _ 0 ^ 1 \\left \\langle u , \\sum _ { k = 1 } ^ n \\sum _ { j _ 1 , \\dots , j _ k = 0 } ^ m \\int _ 0 ^ t \\int _ 0 ^ { t _ 2 } \\dots \\int _ 0 ^ { t _ k } Y _ { j _ 1 , \\dots , j _ k } ^ { ( n , i ) } ( 0 ) \\dd B _ s ^ { j _ k } \\dd B _ { t _ k } ^ { j _ { k - 1 } } \\dots \\dd B _ { t _ 2 } ^ { j _ 1 } \\right \\rangle ^ 2 \\dd t \\ ; , \\end{align*}"}
-{"id": "1444.png", "formula": "\\begin{align*} [ \\alpha _ { 2 } ] = \\{ \\alpha _ { 1 } + \\alpha _ { 2 } , \\ \\alpha _ { 2 } \\} , [ \\alpha _ { i } ] = \\{ \\alpha _ { i } \\} \\quad \\mathrm { f o r } \\ i > 2 . \\end{align*}"}
-{"id": "2030.png", "formula": "\\begin{align*} \\Gamma ( \\Omega ) = \\lambda _ 2 - \\lambda _ 1 \\ge \\bar { \\lambda } _ 2 ( n , D ) - \\bar { \\lambda } _ 1 ( n , D ) \\ \\mbox { i f } \\ D \\le \\frac { \\pi } { 2 } , \\end{align*}"}
-{"id": "1115.png", "formula": "\\begin{align*} \\| v _ { t a i l ( 2 \\kappa ) } \\| _ b = \\left ( \\sum _ { j \\ge 2 } \\| v _ { B _ j } \\| _ b ^ b \\right ) ^ { \\frac { 1 } { b } } . \\end{align*}"}
-{"id": "8120.png", "formula": "\\begin{align*} C _ { 0 , l ' , N } = C _ { 0 , l ' , N } ^ { r a d } . \\end{align*}"}
-{"id": "3183.png", "formula": "\\begin{align*} f _ { j } = \\left ( \\begin{array} { c } f _ { j } ^ 1 \\\\ f _ { j } ^ 2 \\\\ \\vdots \\\\ f _ { j } ^ r \\end{array} \\right ) = \\sum _ { | \\alpha | = n + 1 } \\left ( \\begin{array} { c } f _ { j , \\alpha } ^ 1 \\\\ f _ { j , \\alpha } ^ 2 \\\\ \\vdots \\\\ f _ { j , \\alpha } ^ r \\end{array} \\right ) \\cdot e _ j ^ \\alpha \\end{align*}"}
-{"id": "8565.png", "formula": "\\begin{align*} \\operatorname { i m } ( \\beta ) & = D ( \\operatorname { i m } ( \\hat { \\beta } ) ) \\\\ \\operatorname { k e r } ( \\beta ) & = \\operatorname { k e r } ( \\hat { \\beta } ) \\end{align*}"}
-{"id": "6071.png", "formula": "\\begin{align*} w \\Lambda = t _ { w _ 0 \\lambda } \\Lambda _ 0 \\quad \\quad \\textup { ( $ w $ i n t h e a f f i n e W e y l g r o u p , $ \\Lambda $ d o m i n a n t ) } \\end{align*}"}
-{"id": "9968.png", "formula": "\\begin{align*} R _ { t , i } = \\log _ 2 ( 1 + \\frac { p _ { t , i } } { \\sigma ^ 2 _ n } ) \\end{align*}"}
-{"id": "9615.png", "formula": "\\begin{align*} \\lambda _ { j , r e g } : = \\psi _ { f , r e g } ( y _ j , t ) \\in C ( [ 0 , \\infty ) ) , \\qquad \\lambda _ { j , k } ( t ) = \\lim _ { x \\to y _ j } \\ , \\psi _ { f , k } ( x , t ) \\end{align*}"}
-{"id": "1441.png", "formula": "\\begin{align*} \\Pi = \\{ \\alpha _ { 1 } , \\ldots , \\alpha _ { n - 1 } \\} \\end{align*}"}
-{"id": "3403.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { m _ i r ^ { ( i ) } _ j - r ^ { ( i ) } _ j } c ^ { ( i ) } _ { j , l } \\cdot ( \\zeta _ { r ^ { ( i ) } _ j } ^ { l k } - \\zeta _ { r ^ { ( i ) } _ j } ^ { l k ' } ) ( b ^ { ( j ) } _ { i , q } ( u ) ) ^ { l - 1 } \\neq 0 \\end{align*}"}
-{"id": "4726.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } x ^ { \\top } Q x + d ^ { \\top } x + s = 0 , \\end{align*}"}
-{"id": "4406.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { X } } = \\sup _ { \\| x ^ * \\| \\leq 1 } | x ^ * ( x ) | = \\| J x \\| _ { X ^ { * * } } . \\end{align*}"}
-{"id": "594.png", "formula": "\\begin{align*} a _ 0 ( z ) f ( z ) + a _ 1 ( z ) f ( z ^ k ) + \\cdots + a _ d ( z ) f ( z ^ { k ^ d } ) = 0 , \\end{align*}"}
-{"id": "1323.png", "formula": "\\begin{align*} A = \\begin{pmatrix} A _ { 1 1 } & A _ { 1 2 } \\\\ 0 & A _ { 2 2 } \\end{pmatrix} E = \\begin{pmatrix} E _ { 1 } \\\\ 0 \\end{pmatrix} \\end{align*}"}
-{"id": "6365.png", "formula": "\\begin{align*} \\bar { u } _ j ( z _ 1 , z _ 2 ) : = | z _ 1 | ^ { - a } z _ 1 \\bar { Q } _ j ( z _ 1 ^ 2 , z _ 2 ^ 2 ) . \\end{align*}"}
-{"id": "9839.png", "formula": "\\begin{align*} & \\int _ { f \\in \\Omega } P ( f ) \\mathrm { d } f \\le \\bar { P } , \\\\ & \\int _ { f \\in \\Omega } Q ( f ) \\mathrm { d } f \\le \\bar { Q } . \\end{align*}"}
-{"id": "4300.png", "formula": "\\begin{align*} E : = { \\cup } _ { y \\in Y } H ^ 0 \\big ( X _ y , K _ { X _ y } + L | _ { X _ y } \\big ) \\end{align*}"}
-{"id": "3701.png", "formula": "\\begin{align*} \\frac { d x _ 4 } { d t } = - \\frac { x _ 5 } { \\epsilon } ~ , \\end{align*}"}
-{"id": "2838.png", "formula": "\\begin{align*} E n d _ X ( m , n ) = H o m _ { \\mathcal { C } } ( X ^ { \\otimes m } , X ^ { \\otimes n } ) . \\end{align*}"}
-{"id": "6572.png", "formula": "\\begin{align*} | \\beta \\tau | ^ { 2 k } - \\sum _ { i = 0 } ^ { 2 k - 1 } | \\beta | ^ i | \\tau | ^ { 4 k - i } \\lesssim \\delta . \\end{align*}"}
-{"id": "9130.png", "formula": "\\begin{align*} u ( t ) & = e ^ { - ( t - s ) A ( t ) } u ( s ) + \\int _ { s } ^ { t } { e ^ { - ( t - r ) A ( t ) } ( \\mathcal { A } ( t ) - \\mathcal { A } ( r ) ) u ( r ) d r } \\\\ & + \\int _ { s } ^ { t } { e ^ { - ( t - r ) A ( t ) } [ ( - B ( r ) + I ) A ( r ) u ( r ) - P ( r ) u ( r ) + f ( r ) ] } d r , \\end{align*}"}
-{"id": "8248.png", "formula": "\\begin{align*} L u = h , \\Gamma \\end{align*}"}
-{"id": "546.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\tilde { H } ( n , n ) } { n } = \\max _ { \\kappa \\in [ 0 , 1 ] } \\left \\lbrace \\frac { 1 - \\kappa } { \\alpha } + ( 1 + \\sqrt { \\kappa } ) ^ 2 \\right \\rbrace = \\begin{cases} 4 & \\mbox { i f } \\alpha \\in ( 1 / 2 , 1 ) , \\\\ \\frac { 1 } { \\alpha ( 1 - \\alpha ) } & \\mbox { i f } \\alpha \\leqslant 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "7354.png", "formula": "\\begin{align*} F ( v _ 1 \\wedge v _ 0 ) = [ 2 ] ^ { 1 / 2 } q ^ { - 2 } v _ 1 \\wedge v _ { - 1 } , F ( v _ 1 \\wedge v _ { - 1 } ) = [ 2 ] ^ { 1 / 2 } q ^ 2 v _ 0 \\wedge v _ { - 1 } , F ( v _ 0 \\wedge v _ { - 1 } ) = 0 . \\end{align*}"}
-{"id": "587.png", "formula": "\\begin{align*} \\delta ( x - [ r _ 0 ] ) \\equiv \\delta ( \\sum _ { k = 1 } ^ \\nu p ^ k [ \\phi ^ { - k } ( r _ k ) ] ) \\bmod I ^ \\nu \\equiv \\delta ( p \\Big ( \\sum _ { k = 1 } ^ \\nu p ^ { k - 1 } [ \\phi ^ { - k } ( r _ k ) ] \\Big ) ) \\bmod I ^ \\nu \\end{align*}"}
-{"id": "6292.png", "formula": "\\begin{align*} \\begin{cases} \\frac { 1 } { p _ { 2 } } \\leq \\frac { 1 } { p _ { 1 } } \\\\ s _ { 2 } + R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ { 1 } , \\alpha _ { 2 } ) + \\frac { n ( 1 - \\alpha _ { 1 } \\vee \\alpha _ { 2 } ) } { q _ { 2 } } < s _ { 1 } + \\frac { n ( 1 - \\alpha _ { 1 } \\vee \\alpha _ { 2 } ) } { q _ { 1 } } \\\\ \\frac { 1 } { q _ { 2 } } > \\frac { 1 } { q _ { 1 } } \\end{cases} \\end{align*}"}
-{"id": "6786.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 ^ + } d ( s ) = - n \\int _ X d \\dot { \\varphi } _ { t } \\wedge d ^ c \\dot { \\varphi } _ { t } \\wedge \\theta _ { \\varphi _ t } ^ { n - 1 } + \\int _ X \\ddot { \\varphi } _ { t } \\theta _ { \\varphi _ t } ^ n . \\end{align*}"}
-{"id": "9567.png", "formula": "\\begin{align*} \\lambda _ { n } ^ { ( 2 ) } = \\sum _ { i \\neq n } \\frac { \\langle \\psi _ { i } ^ { ( 0 ) } , H ^ { ( 1 ) } \\psi _ { n } ^ { ( 0 ) } \\rangle \\langle \\psi _ { n } ^ { ( 0 ) } , H ^ { ( 1 ) } \\psi _ { i } ^ { ( 0 ) } \\rangle } { \\lambda _ { n } ^ { ( 0 ) } - \\lambda _ { i } ^ { ( 0 ) } } + \\langle \\psi _ { n } ^ { ( 0 ) } , H ^ { ( 2 ) } \\psi _ { n } ^ { ( 0 ) } \\rangle \\end{align*}"}
-{"id": "4712.png", "formula": "\\begin{align*} ( A _ { 2 1 } + A _ { 3 1 } ) x _ 1 + ( A _ { 2 2 } + A _ { 3 2 } ) x _ 2 & = b _ 3 + b _ 2 \\\\ ( - A _ { 2 1 } + A _ { 3 1 } ) x _ 1 + ( - A _ { 2 2 } + A _ { 3 2 } ) x _ 2 & = b _ 3 - b _ 2 . \\end{align*}"}
-{"id": "5331.png", "formula": "\\begin{align*} \\alpha _ 1 & = \\sin \\phi _ 1 \\cos \\theta _ 1 , & \\alpha _ 2 & = \\sin \\phi _ 1 \\sin \\theta _ 1 , & \\alpha _ 3 & = \\cos \\phi _ 1 , \\\\ \\beta _ 1 & = \\sin \\phi _ 1 \\cos \\theta _ 1 , & \\beta _ 2 & = \\sin \\phi _ 2 \\sin \\theta _ 2 , & \\beta _ 3 & = \\cos \\phi _ 2 . \\end{align*}"}
-{"id": "5763.png", "formula": "\\begin{align*} T ( \\lambda ) = \\prod _ { j = 1 } ^ N L _ j ( \\lambda , \\xi _ j ) = \\prod _ { j = 1 } ^ N \\begin{pmatrix} ( L _ j ( \\lambda , \\xi _ j ) ) _ { 1 1 } & ( L _ j ( \\lambda , \\xi _ j ) ) _ { 1 2 } \\\\ ( L _ j ( \\lambda , \\xi _ j ) ) _ { 2 1 } & ( L _ j ( \\lambda , \\xi _ j ) ) _ { 2 2 } \\end{pmatrix} = \\begin{pmatrix} A ( \\lambda ) & B ( \\lambda ) \\\\ C ( \\lambda ) & D ( \\lambda ) \\end{pmatrix} . \\end{align*}"}
-{"id": "7777.png", "formula": "\\begin{align*} r _ i : = \\sum _ { j : { } _ j a _ i \\in A } r _ i ^ { j } . \\end{align*}"}
-{"id": "7755.png", "formula": "\\begin{align*} \\mathbb P \\left [ T _ N ( n ) \\le x \\right ] = \\mathbb P \\left [ T _ N ( n ) \\ge - x \\right ] . \\end{align*}"}
-{"id": "9722.png", "formula": "\\begin{align*} g ( x _ { L , \\epsilon } ( t ) ) e ^ { c _ 1 \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } & = x _ 1 ( \\epsilon ) \\\\ & < \\min _ { T _ 7 ( \\epsilon ) \\leq s \\leq T _ 8 ( \\epsilon ) } g ( x ( s ) ) e ^ { c _ 1 \\int _ 0 ^ s \\frac { 1 } { \\sigma ( u ) } \\ , d u } \\\\ & \\leq g ( x ( t ) ) e ^ { c _ 1 \\int _ 0 ^ t \\frac { 1 } { \\sigma ( u ) } \\ , d u } . \\end{align*}"}
-{"id": "9485.png", "formula": "\\begin{align*} G ( \\theta , p , \\rho ) = 2 ( F _ { 1 } ( \\rho - \\log r _ { 0 } , \\theta ) + \\log r _ { 0 } ) \\end{align*}"}
-{"id": "5576.png", "formula": "\\begin{align*} z m _ i \\dot x _ i = - \\left ( \\frac 1 2 [ b _ x ] ( x _ i ) + z m _ i b ( x _ i ) \\right ) . \\end{align*}"}
-{"id": "2805.png", "formula": "\\begin{align*} & i = i _ 1 + i _ 2 \\cdot n _ 1 + i _ 3 \\cdot n _ 1 n _ 2 , \\\\ & j = j _ 1 + j _ 2 \\cdot ( N _ 1 - n _ 1 + 1 ) + j _ 3 \\cdot ( N _ 1 - n _ 1 + 1 ) ( N _ 2 - n _ 2 + 1 ) , \\\\ & l _ k = i _ k + j _ k , ~ 1 \\leq k \\leq 3 . \\end{align*}"}
-{"id": "617.png", "formula": "\\begin{align*} \\lambda < \\lambda _ { e s s } : = \\inf \\sigma _ { e s s } ( - \\Delta + V ) , \\end{align*}"}
-{"id": "3926.png", "formula": "\\begin{align*} \\lambda _ { m , n } ( c ^ 2 = 0 ) = n ( n + 1 ) \\ , . \\end{align*}"}
-{"id": "8321.png", "formula": "\\begin{align*} v _ { \\P } ( u - f ( \\delta ) ) = \\min \\{ v _ { \\P } ( u ) , v _ { \\P } ( f ( \\delta ) ) \\} = v _ { \\P } ( u ) . \\end{align*}"}
-{"id": "6619.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\mu _ t + \\frac { \\partial } { \\partial x } ( v \\ , \\mu _ t ) = F _ f ( \\mu _ t ) . \\end{align*}"}
-{"id": "9001.png", "formula": "\\begin{align*} ( \\frac { d } { d z } ) ^ n [ z ^ { \\mu - 1 } E ^ \\gamma _ { \\alpha , \\mu } ( \\lambda z ^ \\alpha ) ] = z ^ { \\mu - n - 1 } E ^ \\gamma _ { \\alpha , \\mu - n } ( \\lambda z ^ \\alpha ) , \\end{align*}"}
-{"id": "9958.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\int _ \\Omega g ( u _ j ) u _ j d x = \\lim _ { j \\to \\infty } \\int _ \\Omega G ( u _ j ) d x = 0 \\end{align*}"}
-{"id": "7593.png", "formula": "\\begin{align*} \\varepsilon : = \\frac { l + F ( x _ 0 ) } { 2 } , ~ ~ K _ 1 : = \\left [ \\frac { u _ l - x _ 0 } { \\varepsilon } \\right ] + 1 , ~ ~ \\delta : = \\frac { l - F ( x _ 0 ) } { 2 } , K _ 2 : = \\left [ \\frac { x _ 0 - v _ l } { \\delta } \\right ] + 1 , \\end{align*}"}
-{"id": "3314.png", "formula": "\\begin{align*} F V _ 1 = [ 2 ] ^ { 1 / 2 } V _ 0 , F V _ 0 = [ 2 ] ^ { 1 / 2 } V _ { - 1 } , F V _ { - 1 } = 0 . \\end{align*}"}
-{"id": "8945.png", "formula": "\\begin{align*} v _ { R , 1 } ( s ) & \\leqslant \\frac 1 { ( n - 1 ) ^ 2 } \\int _ s ^ { + \\infty } \\frac { ( p ' + 1 / 2 ) R ^ { 1 - 1 / p } - s _ 0 ^ { 1 - 1 / p } / ( p - 1 ) } { t ^ 2 } d t \\\\ & = \\frac 1 { ( n - 1 ) ^ 2 } \\frac { ( p ' + 1 / 2 ) R ^ { 1 - 1 / p } - s _ 0 ^ { 1 - 1 / p } / ( p - 1 ) } { s } \\end{align*}"}
-{"id": "1156.png", "formula": "\\begin{align*} & \\frac 1 2 b _ { x x } ( 0 ) - h b _ x ( 0 ) + ( h ^ 2 + z \\rho ( 0 ) ) b ( 0 ) = 0 , \\\\ & \\frac 1 2 b _ { x x } ( 1 ) + H b _ x ( 1 ) + ( H ^ 2 + z \\rho ( 1 ) ) b ( 1 ) = 0 . \\end{align*}"}
-{"id": "8663.png", "formula": "\\begin{align*} [ F ] _ { \\alpha , K } = \\sup _ { x \\in H = U \\times V ' , \\ ; k \\in K , k \\not = 0 } \\frac { | F ( x + k ) - F ( x ) | _ J } { | k | _ K ^ { \\alpha } } < \\infty . \\end{align*}"}
-{"id": "5500.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to \\infty } \\gamma _ k ^ \\mathrm { B } [ \\iota ] = & \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } + \\sum _ { i = 1 } ^ { K } \\beta _ { i } } \\\\ \\lim _ { \\rho \\to \\infty } \\gamma _ k ^ \\mathrm { C } [ \\iota ] = & \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } } \\end{align*}"}
-{"id": "3828.png", "formula": "\\begin{align*} \\frac { ( n ! ) ^ { 2 n } } { n ^ { n ^ 2 } } \\le n ^ 2 \\binom { n ^ 2 } { k } \\frac { n ! ^ { 2 n - \\frac { k } { n } } e ^ { n ( 3 + \\frac { \\ln ( 2 \\pi n ) ^ 2 } { 4 } ) } } { ( n - \\frac { k } { n } ) ! ^ { 2 n } e ^ k } . \\end{align*}"}
-{"id": "6521.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\| A ( \\tau _ i ) - A ( \\tau _ { i - 1 } ) \\| _ { { \\mathcal L } ( D , X ) } \\le M _ 2 \\end{align*}"}
-{"id": "8695.png", "formula": "\\begin{align*} Y _ { \\tau } ^ { t , x } = \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } G B ( s , \\Xi ^ { t , x } _ s ) \\ , d s + \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } Z _ s ^ { t , x } \\ , B ( s , \\Xi ^ { t , x } _ s ) \\ , d s - \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } Z ^ { t , x } _ { s } \\ ; d W _ s , \\end{align*}"}
-{"id": "3890.png", "formula": "\\begin{align*} \\dot { X } = \\begin{cases} C _ L X + e _ 3 \\mu \\ ; , & x \\le 0 \\\\ C _ R X + e _ 3 \\mu \\ ; , & x \\ge 0 \\end{cases} \\ ; . \\end{align*}"}
-{"id": "5302.png", "formula": "\\begin{align*} H _ { n , m } & = U ( N _ { A } - N _ { B } ) ^ 2 + \\mu ( N _ { A } - N _ { B } ) + t \\left ( A ^ { \\dagger } B + A B ^ \\dagger \\right ) \\\\ & = U ( N _ { A } - N _ { B } ) ^ 2 + \\mu ( N _ { A } - N _ { B } ) + \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ m t _ { i , j } ( a _ { i } b _ { j } ^ \\dagger + a _ { i } ^ \\dagger b _ { j } ) , \\end{align*}"}
-{"id": "3430.png", "formula": "\\begin{align*} d y ( t ) = - f ( t , \\xi ( t ) , y ( t ) , z ( t ) ) d t + z ( t ) d w ( t ) , y ( T ) = \\zeta \\in R ^ { d _ 1 } . \\end{align*}"}
-{"id": "9259.png", "formula": "\\begin{align*} J _ { \\tilde { P } } ( u ) = \\mathbb { E } _ { \\tilde { P } } [ \\int _ 0 ^ T \\int _ { \\mathbb { R } } f ( x , u ( t , Z ) ) y ( t , x , Z ) d x d t + \\int _ { \\mathbb { R } } g ( x , Z ) y ( T , x , Z ) d x \\Big ] , \\end{align*}"}
-{"id": "7489.png", "formula": "\\begin{align*} r ( d , k ) = \\sum _ { s = 1 } ^ k \\sum _ { t = 1 } ^ k r _ d ( s , t ) , \\end{align*}"}
-{"id": "1702.png", "formula": "\\begin{align*} \\lambda V _ n ( z ) - \\frac { 1 } { 2 } \\mathrm { T r } \\big ( Q D ^ 2 V _ n ( z ) \\big ) - \\langle A z , D V _ n ( z ) \\rangle & - \\langle B ( z ) , D V _ n ( z ) \\rangle \\\\ = B _ n ( z ) - B ( z ) & + \\langle B _ n ( z ) - B ( z ) , D _ v U _ n ( z ) \\rangle . \\end{align*}"}
-{"id": "653.png", "formula": "\\begin{align*} \\Big ( \\bigcap _ { l \\in L } l A B \\Big ) _ x = \\bigcap _ { l \\in L } l A B _ x , \\textrm { f o r a l l $ x \\in X $ } . \\end{align*}"}
-{"id": "8672.png", "formula": "\\begin{align*} & \\sup _ { x \\in H } | \\nabla _ k R _ t [ \\Phi ] ( x ) \\vert _ J \\leq \\frac { c } { t ^ { \\frac { 3 } { 2 } } } \\Vert \\Phi \\Vert _ \\infty \\vert k \\vert _ K , \\ , k \\in K ; \\\\ & \\sup _ { x \\in H } \\vert \\nabla ^ G _ { a } R _ t [ \\Phi ] ( x ) \\vert _ J \\leq \\frac { c } { t ^ { \\frac { 1 } { 2 } } } \\Vert \\Phi \\Vert _ \\infty \\vert G a \\vert _ { K } = \\frac { c } { t ^ { \\frac { 1 } { 2 } } } \\Vert \\Phi \\Vert _ \\infty \\vert a \\vert _ { U } , \\ , a \\in U . \\end{align*}"}
-{"id": "9327.png", "formula": "\\begin{align*} \\tilde { p } ( t , z ) = \\tilde { p } ( 0 , z ) \\exp ( - \\int _ 0 ^ t \\frac { b _ 0 ( s , z ) } { \\sigma _ 0 ( s , z ) } d B ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ t ( \\frac { b _ 0 ( s , z ) } { \\sigma _ 0 ( s , z ) } ) ^ { 2 } d s ) , \\end{align*}"}
-{"id": "7681.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\left \\{ f ( x _ n ) + \\gamma _ { n + 1 } , 0 \\right \\} , x _ 0 > 0 , n = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "6949.png", "formula": "\\begin{align*} L _ { \\pi ' } ( Z ) & = \\sum _ { k \\ge 0 } \\ , ( - 1 ) ^ k s _ { ( 1 ^ k ) } [ s _ { \\pi ' } ] ( Z ) = \\sum _ \\nu \\ell _ { \\pi ' \\nu ' } \\ , s _ { \\nu ' } ( Z ) \\cr & = \\sum _ { k \\ge 0 } \\ , ( - 1 ) ^ k ( \\ , s _ { ( 1 ^ k ) } [ s _ { \\pi } ] ( Z ) \\ , ) ' = \\sum _ \\nu \\ell _ { \\pi \\nu } \\ , s _ { \\nu ' } ( Z ) \\end{align*}"}
-{"id": "1965.png", "formula": "\\begin{align*} \\lambda \\nabla ^ { L C , \\lambda } _ X ( Y ) = & \\lambda \\nabla ^ { L C } _ X ( Y ) + ( \\partial _ Y \\lambda ) X + ( \\partial _ X \\lambda ) Y - g ( X , Y ) Z \\\\ = & \\nabla ^ { L C } _ X ( \\lambda Y ) + g ( Y , Z ) X - g ( X , Y ) Z \\\\ = & \\nabla ^ { L C } _ X ( \\lambda Y ) + \\frac 1 4 \\big ( - ( Y Z + Z Y ) X - X ( Y Z + Z Y ) \\\\ & \\phantom { \\nabla ^ { L C } _ X ( \\lambda Y ) + \\frac 1 4 \\big ( } + ( X Y + Y X ) Z + Z ( X Y + Y X ) \\big ) \\\\ = & \\nabla ^ { L C } _ X ( \\lambda Y ) + \\frac 1 4 ( Y X Z + Z X Y - Y Z X - X Z Y ) \\end{align*}"}
-{"id": "1202.png", "formula": "\\begin{align*} x ^ A y ^ B = \\prod _ { i = 1 } ^ n x ^ { a _ i } y ^ { b _ i } = x ^ { \\sum _ { i = 1 } ^ n a _ i } y ^ { \\sum _ { i = 1 } ^ n b _ i } . \\end{align*}"}
-{"id": "4766.png", "formula": "\\begin{align*} \\frac { t } { 2 } \\ , ( \\varphi ^ 2 ) ' + \\varphi ^ 2 + 1 = \\pm 2 a t \\sqrt { \\varphi ^ 2 + 1 } . \\end{align*}"}
-{"id": "7844.png", "formula": "\\begin{gather*} \\{ u _ { i j } , u _ { k l } \\} = 0 , \\{ u _ { i j } , \\eta _ { k l } \\} = \\delta _ { j k } u _ { i l } , \\{ \\eta _ { i j } , \\eta _ { k l } \\} = \\delta _ { j k } \\eta _ { i l } - \\delta _ { l i } \\eta _ { k j } . \\end{gather*}"}
-{"id": "7539.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\left \\{ f ( x _ n ) - \\alpha ( f ( x _ n ) - x _ n ) + l \\xi _ { n + 1 } , 0 \\right \\} , x _ 0 > 0 , n \\in { \\mathbb N _ 0 } . \\end{align*}"}
-{"id": "6424.png", "formula": "\\begin{align*} \\mathcal { E } ( u , v ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } [ u ( t , x ) - u ( t , y ) ] [ v ( t , x ) - v ( t , y ) ] k ( x , y ) d x d y . \\end{align*}"}
-{"id": "1957.png", "formula": "\\begin{align*} \\frac { \\sum _ { \\alpha = 1 } ^ r ( ( H Q ) _ { \\alpha i } ) ^ 2 } { { \\hat a } ^ 2 } \\leq \\frac { ( \\| H \\| \\| Q ^ { ( i ) } \\| ) ^ 2 } { { \\hat a } ^ 2 } \\leq \\| Q ^ { ( i ) } \\| ^ 2 . \\end{align*}"}
-{"id": "8923.png", "formula": "\\begin{align*} \\abs { A ( x ) } ^ 2 = \\sum _ { j = 1 } ^ k \\lambda _ j ^ 2 \\abs { P _ { W _ j } ( x ) } ^ 2 , \\end{align*}"}
-{"id": "9718.png", "formula": "\\begin{align*} x _ { U , \\epsilon } ' ( t ) = - \\lambda ( \\epsilon ) g ( x _ { U , \\epsilon } ( t ) ) , t > T _ 2 ( \\epsilon ) . \\end{align*}"}
-{"id": "4972.png", "formula": "\\begin{align*} & E = f ^ { * } H - \\left < A \\vec { c } , \\vec { D } \\right > \\\\ & E _ { i } = f ^ { * } D _ { i } - \\sum _ { k = 1 } ^ { r } a _ { k i } D _ { k } . \\end{align*}"}
-{"id": "9944.png", "formula": "\\begin{align*} \\langle u , v \\rangle _ { X _ 0 } : = \\int _ Q \\left ( u ( x ) - u ( y ) \\right ) \\left ( v ( x ) - v ( y ) \\right ) K ( x - y ) d x d y . \\end{align*}"}
-{"id": "5584.png", "formula": "\\begin{align*} \\beta = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & - 1 \\end{array} \\right ) , \\alpha _ 1 = \\left ( \\begin{array} { c c } 0 & 1 \\\\ 1 & 0 \\end{array} \\right ) , \\alpha _ 2 = \\left ( \\begin{array} { c c } 0 & - i \\\\ i & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "3974.png", "formula": "\\begin{align*} \\varphi ( s _ 1 ) = [ \\varphi _ 1 ( s _ 1 ) ; \\vect { 0 } ; \\dots ; \\vect { 0 } ] \\psi ( s _ 1 ) = \\vect { 0 } . \\end{align*}"}
-{"id": "735.png", "formula": "\\begin{align*} \\frac { h _ { H } ( f ^ { n } ( P ) ) } { \\delta _ { f } ^ { n } } = h _ { H } ( P ) + \\sum _ { k = 0 } ^ { n - 1 } \\frac { h _ { E ' _ { 1 } } ( f ^ { k } ( P ) ) } { \\delta _ { f } ^ { k + 1 } } - \\sum _ { k = 0 } ^ { n - 1 } \\frac { h _ { Z _ { 1 } } ( p ^ { - 1 } f ^ { k } ( P ) ) } { \\delta _ { f } ^ { k + 1 } } . \\end{align*}"}
-{"id": "9496.png", "formula": "\\begin{align*} R ( X , Y ) Z & = \\nabla _ X \\nabla _ Y Z - \\nabla _ Y \\nabla _ X Z - \\nabla _ { [ X , Y ] } Z \\\\ \\mathrm { R i c } ( X , Y ) & = \\mathrm { t r } ( Z \\longmapsto R ( Z , X ) Y ) \\\\ R & = \\mathrm { t r } ( Z \\longmapsto \\mathrm { R i c } ( Z ) ) , \\end{align*}"}
-{"id": "7702.png", "formula": "\\begin{align*} \\beta _ { 2 k } : = \\sum _ { i = n _ { 2 k - 1 } + 1 } ^ { n _ { 2 k } } \\gamma _ i > 0 , \\beta _ { 2 k + 1 } : = - \\sum _ { i = n _ { 2 k } + 1 } ^ { n _ { 2 k + 1 } } \\gamma _ i > 0 . \\end{align*}"}
-{"id": "4210.png", "formula": "\\begin{align*} \\langle G _ { \\varphi } ( x ) , x - x _ p \\rangle & \\geq \\varphi ( x ^ + ) - \\min \\varphi + \\frac { t } { 2 } \\| G _ { \\varphi } ( x ) \\| ^ 2 \\\\ & \\geq \\varphi ( x ^ + ) - \\min \\varphi + \\frac { t \\epsilon } { 2 } ( \\varphi ( x ) - \\varphi ( x ^ + ) ) \\\\ & = \\frac { t \\epsilon } { 2 } ( \\varphi ( x ) - \\min \\varphi ) + ( 1 - \\frac { t \\epsilon } { 2 } ) ( \\varphi ( x ^ + ) - \\min \\varphi ) \\\\ & \\geq \\frac { t \\epsilon } { 2 } ( \\varphi ( x ) - \\min \\varphi ) , \\end{align*}"}
-{"id": "9979.png", "formula": "\\begin{align*} & ( 1 - \\alpha ) \\sum _ { i = 1 } ^ 2 | w _ { k i } | ^ 2 p _ { i } + \\alpha \\tilde { p } = A _ { k } , \\\\ & ( 1 - \\alpha ) \\sum _ { i = 1 } ^ 2 | w _ { \\bar { k } i } | ^ 2 p _ { i } = A _ { \\bar { k } } , \\end{align*}"}
-{"id": "7511.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } x _ n = K ; \\end{align*}"}
-{"id": "9323.png", "formula": "\\begin{align*} \\pi ( t , z ) = \\hat { \\pi } ( t , z ) = \\frac { b _ 0 ( t , z ) } { \\sigma ^ 2 ( t , z ) } . \\end{align*}"}
-{"id": "8416.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } S + \\nabla \\cdot ( S v _ { S } ) = \\Gamma _ { S } , \\\\ & \\partial _ { t } { v } _ { S } + { v } _ { S } \\cdot \\nabla { v } _ { S } + \\frac { \\gamma \\nabla S } { S } + \\nabla { P } = \\Gamma _ { v _ { S } } , \\\\ & \\partial _ { t } { v } _ { L } + { v } _ { L } \\cdot \\nabla { v } _ { L } + \\nabla { P } = \\Gamma _ { v _ { L } } , \\\\ & \\nabla \\cdot ( S v _ { S } + ( 1 - S ) v _ { L } ) = 0 , \\\\ & L = 1 - S , \\end{cases} \\end{align*}"}
-{"id": "5202.png", "formula": "\\begin{align*} \\| \\widehat { V } \\| _ { S _ { \\widehat { p } _ { 2 } } } = \\min _ { V \\in \\mathbb { R } ^ { d \\times d } , W \\in \\mathbb { R } ^ { n \\times d } : \\widehat { V } = ( V W ^ { T } ) ^ { T } } \\| V \\| _ { S _ { p _ { 2 } } } \\| W \\| _ { S _ { p _ { 3 } } } . \\end{align*}"}
-{"id": "5533.png", "formula": "\\begin{align*} \\varphi ( t _ { n _ j } ) \\sum _ { k = 1 } ^ { \\infty } v ( k ) \\chi _ { E _ j } ( k ) \\le \\varphi ( t _ { n _ j } ) \\sum _ { k = 1 } ^ { m ( E _ j ) } w ( k ) \\le 1 / 2 ^ { j - 2 } \\to 0 . \\end{align*}"}
-{"id": "4290.png", "formula": "\\begin{align*} w = \\Lambda ( r _ 1 , f ) \\cdot \\theta ^ { a } ( \\Lambda ( r _ 2 , f ) ) ^ { \\epsilon } , \\end{align*}"}
-{"id": "4472.png", "formula": "\\begin{align*} \\sup _ { k \\in \\{ 1 , \\ldots , k ^ * \\} } \\frac { k ^ { 2 / d } } { n ^ { 2 / d } } \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\int _ { \\mathcal { X } _ n ^ c } \\frac { \\Delta f ( x ) } { f ( x ) ^ { 2 / d } } \\ , d x = O \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\end{align*}"}
-{"id": "4049.png", "formula": "\\begin{align*} q + r = \\gcd ( p , q ) + \\gcd ( p , r ) + \\gcd ( q , q + r - p ) + \\gcd ( r , q + r - p ) . \\end{align*}"}
-{"id": "618.png", "formula": "\\begin{align*} - \\Delta u - \\lambda u = Q ( x ) | u | ^ { p - 2 } u , x \\in \\R ^ N , \\end{align*}"}
-{"id": "6182.png", "formula": "\\begin{align*} M _ t = f ( V _ t ) - \\int _ 0 ^ t \\mathcal { A } f ( V _ s ) \\d s , t \\geq 0 , \\end{align*}"}
-{"id": "246.png", "formula": "\\begin{align*} V ( f ) = \\mathbb { E } _ f [ \\{ \\log f ( X _ 1 ) + H ( f ) \\} ^ 2 ] \\geq A _ { d , \\theta } ^ 2 \\mathbb { P } _ f ( X _ 1 \\in S _ { d , \\theta } ) \\geq A _ { d , \\theta } ^ 2 e ^ { - 4 A _ { d , \\theta } } V _ d r _ { d , \\theta } ^ d , \\end{align*}"}
-{"id": "914.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { S _ { p _ { 1 } } } \\| V \\| _ { S _ { p _ { 2 } } } . \\end{align*}"}
-{"id": "7974.png", "formula": "\\begin{align*} \\alpha _ 1 \\varphi _ 1 \\left ( \\frac { p _ 1 ( x ) } { p _ 1 \\widetilde { + } _ { \\varphi } p _ 2 ( x ) } \\right ) + \\alpha _ 2 \\varphi _ 2 \\left ( \\frac { p _ 2 ( x ) } { p _ 1 \\widetilde { + } _ { \\varphi } p _ 2 ( x ) } \\right ) = 1 . \\end{align*}"}
-{"id": "2811.png", "formula": "\\begin{align*} E _ b ( z ) = \\varphi _ z ( b ) + i \\varphi ' _ { z } ( b ) , \\end{align*}"}
-{"id": "3917.png", "formula": "\\begin{align*} S _ { m , n } ( c , \\eta ) = ( 1 - \\eta ^ 2 ) ^ { m / 2 } \\ , w ( \\eta ) \\ , , \\end{align*}"}
-{"id": "5001.png", "formula": "\\begin{align*} \\vec { D } = \\left ( \\begin{array} { c } D _ { 1 } \\\\ D _ { 2 } \\\\ \\vdots \\\\ D _ { r } \\end{array} \\right ) , \\vec { F } = \\left ( \\begin{array} { c } F _ { 1 } \\\\ F _ { 2 } \\\\ \\vdots \\\\ F _ { s } \\end{array} \\right ) , \\vec { e } = \\left ( \\begin{array} { c } 1 \\\\ 0 \\\\ \\vdots \\\\ 0 \\end{array} \\right ) , \\vec { Z } = \\left ( \\begin{array} { c } Z _ { 1 } \\\\ Z _ { 2 } \\\\ \\vdots \\\\ Z _ { r } \\end{array} \\right ) . \\end{align*}"}
-{"id": "4145.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } | A _ f ( n ) | ^ 2 = c _ f X + O ( X ^ { \\frac { 3 } { 5 } + \\epsilon } ) \\end{align*}"}
-{"id": "6848.png", "formula": "\\begin{align*} S ^ { } : = L _ t ^ { \\tilde q , \\infty } L _ x ^ { \\tilde r } , S : = L _ t ^ { \\tilde q , 2 } L _ x ^ { \\tilde r } , W : = \\cap _ { j = 1 } ^ 2 L _ t ^ { q _ j , 2 } \\dot X ^ { | s _ c | , r _ j } \\end{align*}"}
-{"id": "9228.png", "formula": "\\begin{align*} \\chi ( t , x , z ) : = \\frac { d } { d a } Y ^ { u + a \\beta } ( t , x , z ) | _ { a = 0 } \\end{align*}"}
-{"id": "1732.png", "formula": "\\begin{gather*} I = - \\phi , J = \\pm \\phi , K = 0 . \\end{gather*}"}
-{"id": "2357.png", "formula": "\\begin{align*} \\mu _ t = P _ t \\ , \\nu \\ + \\ \\int _ 0 ^ t P _ { t - s } F _ f ( \\mu _ s ) \\ , d s \\mbox { f o r a l l } \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "1651.png", "formula": "\\begin{align*} ( B ^ { s _ 0 } _ { p , p } ( \\R ^ d ) , B ^ { s _ 1 } _ { p , p } ( \\R ^ d ) ) _ { \\theta , p } = B ^ { s } _ { p , p } ( \\R ^ d ) \\end{align*}"}
-{"id": "6905.png", "formula": "\\begin{align*} \\hat { f } ( \\lambda \\omega ) = \\mathcal F ( R f ( \\omega , \\cdot ) ) ( \\lambda ) , \\end{align*}"}
-{"id": "1270.png", "formula": "\\begin{align*} C = \\begin{bmatrix} c ( i , r ) & c ( i + 2 r , 3 r ) & \\cdot & \\cdot \\\\ c ( i , 3 r ) & c ( i + 2 r , r ) & \\cdot & \\cdot \\\\ \\cdot & \\cdot & c ( i + 3 r , 3 r ) & c ( i + 3 r , r ) \\\\ \\cdot & \\cdot & c ( i + r , r ) & c ( i + r , 3 r ) \\end{bmatrix} \\end{align*}"}
-{"id": "2188.png", "formula": "\\begin{align*} & \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } ( w ^ { 2 } ( s , x ) + w ^ { 2 } ( s , y ) ) k ( x , y ) d x d y \\\\ & \\quad \\quad \\quad \\quad \\quad \\leq C ( \\delta , \\Lambda ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x . \\end{align*}"}
-{"id": "5490.png", "formula": "\\begin{align*} \\mathbf { G } _ { \\mathrm { d } } [ \\iota ] = \\mathbf { \\hat { G } } _ { \\mathrm { d } } [ \\iota ] + \\mathcal { E } _ { \\mathrm { d } } [ \\iota ] \\end{align*}"}
-{"id": "6262.png", "formula": "\\begin{align*} \\frac { \\dd } { \\dd r } d \\left ( \\eta _ i ( r , \\tfrac { d _ 0 } { 2 } ) , \\eta _ i ( r , - \\tfrac { d _ 0 } { 2 } ) \\right ) \\big | _ { r = 0 } & = \\frac { \\dd } { \\dd r } \\int _ { - d _ 0 / 2 } ^ { d _ 0 / 2 } \\langle \\frac { \\dd \\eta _ i } { \\dd s } , \\frac { \\dd \\eta _ i } { \\dd s } \\rangle ^ { \\frac 1 2 } d s = \\langle J _ i , e _ n \\rangle | _ { - d _ 0 / 2 } ^ { d _ 0 / 2 } - \\int _ { - d _ 0 / 2 } ^ { d _ 0 / 2 } \\langle J _ i , \\nabla _ { e _ n } e _ n \\rangle = 0 . \\end{align*}"}
-{"id": "9849.png", "formula": "\\begin{align*} & R _ { y s } ( t ) \\otimes ( \\delta ( - t ) - \\alpha \\theta ( - t - \\tau ) ) \\\\ = & R _ { s s } ( t ) \\otimes ( h _ { \\rm S D } ( - t ) \\otimes ( \\delta ( - t ) - \\alpha \\theta ( - t - \\tau ) \\\\ & + h _ { \\rm R D } ( - t ) \\otimes \\theta ( t ) \\otimes h _ { \\rm S R } ( t ) ) , \\end{align*}"}
-{"id": "9591.png", "formula": "\\begin{align*} \\ddot { \\psi } _ f ( x , t ) = ( \\Delta - m ^ 2 ) \\psi _ f ( x , t ) , \\psi _ f ( x , 0 ) = \\psi _ 0 ( x ) , \\quad \\dot \\psi _ f ( x , 0 ) = \\pi _ 0 ( x ) , \\end{align*}"}
-{"id": "530.png", "formula": "\\begin{align*} \\hat { R } ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) = - \\int \\frac { h ^ { \\rm g e o } _ { 1 2 } ( z , 1 / z ) } { z ^ { u - v + 1 } } \\dd z - c ^ { w } \\int \\frac { 1 - c z } { ( z ^ 2 - 1 ) z } h ^ { \\rm g e o } _ { 1 2 } ( z , 1 / c ) \\frac { \\dd z } { z ^ u } , \\end{align*}"}
-{"id": "5457.png", "formula": "\\begin{align*} V ( \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } e ^ { - m / N } e ( m \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } e ^ { - m z } = \\frac 1 { e ^ z - 1 } . \\end{align*}"}
-{"id": "9218.png", "formula": "\\begin{align*} \\langle \\nabla _ { \\varphi } \\ell , \\psi \\rangle = \\nabla _ { \\varphi } \\ell ( \\psi ) = A _ u ( \\psi ) ( x ) p + \\psi ( x ) \\frac { \\partial H } { \\partial y } ( t , x , y , \\varphi , u , z , p , q , r ) | _ { y = \\varphi ( x ) } ; \\psi \\in \\mathcal { D } . \\end{align*}"}
-{"id": "3094.png", "formula": "\\begin{align*} \\begin{bmatrix} A _ { 1 1 } & \\cdots & A _ { 1 p } \\\\ \\vdots & \\ddots & \\vdots \\\\ A _ { p 1 } & \\cdots & A _ { p p } \\end{bmatrix} \\odot \\begin{bmatrix} B _ { 1 1 } & \\cdots & B _ { 1 p } \\\\ \\vdots & \\ddots & \\vdots \\\\ B _ { p 1 } & \\cdots & B _ { p p } \\end{bmatrix} : = \\begin{bmatrix} A _ { 1 1 } B _ { 1 1 } & \\cdots & A _ { 1 p } B _ { 1 p } \\\\ \\vdots & \\ddots & \\vdots \\\\ A _ { p 1 } B _ { p 1 } & \\cdots & A _ { p p } B _ { p p } \\end{bmatrix} . \\end{align*}"}
-{"id": "1106.png", "formula": "\\begin{align*} \\| D ^ * x _ 0 \\| _ 1 + \\rho \\geq \\| D ^ * \\hat { x } \\| _ 1 & = \\| D ^ * x _ 0 + D ^ * h \\| _ 1 \\\\ & = \\| D ^ * _ { T _ 0 } x _ 0 + D ^ * _ { T _ 0 } h + D ^ * _ { T _ 0 ^ c } x _ 0 + D ^ * _ { T _ 0 ^ c } h \\| _ 1 \\\\ & \\geq \\| D ^ * _ { T _ 0 } x _ 0 \\| _ 1 - \\| D ^ * _ { T _ 0 } h \\| _ 1 - \\| D ^ * _ { T _ 0 ^ c } x _ 0 \\| _ 1 + \\| D ^ * _ { T _ 0 ^ c } h \\| _ 1 , \\end{align*}"}
-{"id": "9504.png", "formula": "\\begin{align*} F \\cdot \\psi & = \\frac { 1 } { k ! } \\sum _ { a _ 1 , \\ldots , a _ k } F _ { a _ 1 \\ldots a _ k } \\Gamma ^ { a _ 1 \\ldots a _ k } \\psi \\\\ & = \\sum _ { a _ 1 < \\ldots < a _ k } F _ { a _ 1 \\ldots a _ k } \\Gamma ^ { a _ 1 } \\cdots \\Gamma ^ { a _ k } \\psi , \\end{align*}"}
-{"id": "5787.png", "formula": "\\begin{align*} C _ { k } = \\bigoplus _ { n \\geq 0 } C _ { k } ( n ) . \\end{align*}"}
-{"id": "9285.png", "formula": "\\begin{align*} \\begin{cases} d S ( t ) = S ( t ) [ \\alpha ( t ) d t + \\beta ( t ) d v ( t ) ] ; 0 \\leq t \\leq T , \\\\ S ( 0 ) > 0 , \\end{cases} \\end{align*}"}
-{"id": "5719.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\Phi _ { p _ 0 , q _ 0 , v _ 0 , w _ 0 } ( 2 ^ l Q ) = \\infty , ( Q \\in \\mathcal { Q } ) , \\Phi _ { p _ 0 , q _ 0 , v _ 0 , w _ 0 } ( Q ) : = v _ 0 ( Q ) ^ { \\frac { 1 } { p _ 0 } - \\frac { 1 } { q _ 0 } } w _ 0 ( Q ) ^ \\frac { 1 } { q _ 0 } . \\end{align*}"}
-{"id": "2151.png", "formula": "\\begin{align*} \\eta \\in H _ { e } ^ { 1 , \\beta } ( Q _ { T } ) : = W ^ { 1 , 2 } ( [ 0 , T ] ; L _ { e } ^ { 2 } ( \\Omega ) ) \\cap L ^ { 2 } ( [ 0 , T ] ; H _ { e } ^ { \\beta } ( \\Omega ) ) \\cap L ^ { \\infty } ( [ 0 , T ] ; L ^ { \\infty } ( \\mathbb { R } ^ { n } ) ) \\end{align*}"}
-{"id": "7384.png", "formula": "\\begin{align*} \\frac { 1 } { n } ( q ^ { 0 } ) ^ * q ^ { t + 1 } & \\overset { \\mathbf { . . } } { = } \\tilde { E } _ { 0 , t + 1 } , \\frac { 1 } { n } ( q ^ { r + 1 } ) ^ * q ^ { t + 1 } \\overset { \\mathbf { . . } } { = } \\tilde { E } _ { r + 1 , t + 1 } , \\\\ \\frac { 1 } { n } ( m ^ r ) ^ * m ^ t & \\doteq \\breve { E } _ { r , t } . \\end{align*}"}
-{"id": "6856.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } S _ { I _ n \\cap [ t _ n , \\infty ) } ( u _ n ) = \\infty . \\end{align*}"}
-{"id": "6904.png", "formula": "\\begin{align*} R f ( \\omega , t ) = \\int _ { H _ { \\omega , t } } f ( x ) d m ( x ) , \\end{align*}"}
-{"id": "7981.png", "formula": "\\begin{align*} \\psi : s \\mapsto \\psi ( s ) : = [ \\psi _ 0 ( s ) : \\psi _ 1 ( s ) : \\psi _ 2 ( s ) : \\psi _ 3 ( s ) ] , \\end{align*}"}
-{"id": "6630.png", "formula": "\\begin{align*} \\alpha : = \\big \\{ t _ j \\in [ 0 , T ] \\ ; \\ , : \\ ; \\ , 0 \\leqslant j \\leqslant N , \\ ; t _ 0 = 0 , \\ ; t _ { N } = T , \\ ; t _ j < t _ { j + 1 } \\big \\} , \\end{align*}"}
-{"id": "6165.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ n ( - 2 n - 2 k - 2 m - t + 2 i - 2 ) & = - n ( 2 n + 2 k + 2 m + t + 2 ) + 2 \\sum _ { i = 1 } ^ n i \\\\ & = - n ( n + 2 k + 2 m + t + 1 ) \\end{align*}"}
-{"id": "9034.png", "formula": "\\begin{align*} \\Im \\log \\left ( 1 - \\gamma _ j \\right ) - \\Im \\log \\left ( 1 - \\gamma _ j e ^ { i \\psi _ j ( \\theta ) } \\right ) & = \\Im \\sum _ { k = 1 } ^ \\infty \\frac { \\gamma _ j ^ k } { k } ( e ^ { i k \\psi _ j ( \\theta ) } - 1 ) . \\end{align*}"}
-{"id": "1875.png", "formula": "\\begin{align*} I \\ge & \\frac 2 3 [ ( a _ 2 + c _ 2 ) - ( a _ 1 + c _ 1 ) ] \\\\ = & \\frac 8 3 ( K _ { 1 3 } - K _ { 1 2 } ) = \\frac 8 3 ( K _ { 1 2 } + K _ { 1 3 } - 2 K _ { 1 2 } ) \\\\ > & \\frac { 1 6 } 3 \\epsilon . \\end{align*}"}
-{"id": "1098.png", "formula": "\\begin{align*} | A \\hat x | = | A x _ 0 | \\end{align*}"}
-{"id": "3894.png", "formula": "\\begin{align*} w = \\begin{bmatrix} \\gamma _ R ^ 2 \\\\ - \\gamma _ R \\\\ 1 \\end{bmatrix} \\ ; , \\end{align*}"}
-{"id": "551.png", "formula": "\\begin{align*} { \\mathcal E } ( \\tau ; { \\mathcal J } ) = \\{ X \\in { \\mathcal B } ( { \\mathcal H } ) \\mid [ X , T _ j ] \\in { \\mathcal J } , \\ 1 \\le j \\le n \\} \\end{align*}"}
-{"id": "353.png", "formula": "\\begin{align*} V = \\oplus _ { k = 0 } ^ \\infty V _ k \\ ; \\ ; \\ ; \\textrm { w h e r e } \\ ; \\ ; \\ ; V _ k = { \\rm K e r } ( L _ 0 - k I ) . \\end{align*}"}
-{"id": "5842.png", "formula": "\\begin{align*} C ^ { x } \\left ( x \\right ) = - \\left ( \\kappa \\left ( \\mu - \\lambda - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } - \\frac { 1 } { 2 } \\sigma \\sigma _ { , x } \\right ) \\end{align*}"}
-{"id": "4459.png", "formula": "\\begin{align*} \\mathcal { X } _ n = \\biggl [ \\Bigl ( \\frac { k } { n } \\Bigr ) ^ { \\frac { 1 } { a } - \\tau } , ( 1 - \\tau ) \\log \\frac { n } { k } \\biggr ] \\end{align*}"}
-{"id": "6237.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + ( K + Z ) ) X ^ { ( 0 ) } ( s _ { i } ) & = F \\ \\ i = 1 , \\ldots , l . \\end{align*}"}
-{"id": "749.png", "formula": "\\begin{align*} \\omega \\circ \\tau _ t ^ u = \\omega \\end{align*}"}
-{"id": "2852.png", "formula": "\\begin{align*} ( d _ V - \\delta _ V ) ^ 2 = d _ V ^ 2 + \\delta _ V ^ 2 - d _ V \\circ \\delta _ V - \\delta _ V \\circ d _ V = 0 . \\end{align*}"}
-{"id": "6671.png", "formula": "\\begin{gather*} \\sum _ { k = 1 } ^ n { \\bf E } \\sup _ { 0 \\leqslant t \\leqslant 1 } \\left | z _ i ( u _ k , t ) - z _ { i + 1 } ( u _ k , t ) \\right | \\leqslant \\sum _ { k = 1 } ^ n \\sum _ { l = 1 } ^ k \\sum _ { j = 1 } ^ l 4 ( l - j ) \\cdot \\sqrt { \\varepsilon } = \\\\ = \\sum _ { k = 1 } ^ n \\dfrac { 2 k ( k ^ 2 - 1 ) } { 3 } \\cdot \\sqrt { \\varepsilon } \\leqslant \\sum _ { k = 1 } ^ n \\dfrac { 2 k ^ 3 } { 3 } \\cdot \\sqrt { \\varepsilon } \\leqslant \\dfrac { 2 n ^ 4 } { 3 } \\cdot \\sqrt { \\varepsilon } . \\end{gather*}"}
-{"id": "1377.png", "formula": "\\begin{align*} l _ 1 ( \\delta ) = \\sum _ { t = 1 } ^ { n } \\log f ( y _ { t } | x _ { t } , \\delta ) = \\sum _ { t = 1 } ^ { n } \\log \\int _ { \\R } \\exp \\left ( y _ { t } W _ { t } - m _ t b ( W _ { t } ) + c ( y _ { t } ) \\right ) g ( \\alpha _ t ; \\tau ) d \\alpha _ t . \\end{align*}"}
-{"id": "8227.png", "formula": "\\begin{align*} \\Delta = N _ 1 ( \\phi ) + N _ 2 ( \\phi ) + N _ 3 ( \\phi ) + N _ 4 ( \\phi ) , \\end{align*}"}
-{"id": "7455.png", "formula": "\\begin{align*} \\left [ x \\overline { v ^ { - 1 } } \\right ] _ - = x \\overline { v ^ { - 1 } } \\left [ x \\overline { v ^ { - 1 } } \\right ] _ + ^ { - 1 } \\left [ x \\overline { v ^ { - 1 } } \\right ] _ 0 ^ { - 1 } , \\end{align*}"}
-{"id": "1432.png", "formula": "\\begin{align*} b _ { i } ( z ) = \\sum _ { j = i } ^ { n } \\alpha _ { j } ( z ) \\end{align*}"}
-{"id": "5184.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\left ( \\sum _ { i = 1 } ^ { n } \\sigma ^ { p } _ { i } ( X ) \\right ) ^ { 1 / p } , \\end{align*}"}
-{"id": "2828.png", "formula": "\\begin{align*} ( - \\tilde A _ { a _ { i , u } a _ { j , v } } ^ { ( r ) } ) ^ + = 0 , & j = i + 1 , 1 + v q _ j - u q _ i \\le s _ 0 , \\\\ | r ( a _ { j , v } - a _ { i , u } ) | \\le r ^ * ( s _ 0 ) < 1 , & \\quad j = i + 1 , 1 + v q _ j - u q _ i > s _ 0 \\end{align*}"}
-{"id": "6637.png", "formula": "\\begin{align*} \\beta _ j = ( d _ j - 1 ) \\dim V ^ I _ j + d _ j [ a ( V _ j ) + n ( \\psi ) \\dim ( V _ j ) ] \\end{align*}"}
-{"id": "5120.png", "formula": "\\begin{align*} \\tilde { C } ^ { [ 0 , M ] } ( u ; s ) & = ( - s u ) ^ { - 1 } ( - s ( 1 - s u ) ) ^ { - ( M + 1 ) } \\prod _ { 1 \\le i \\le k } K ^ { ( i ) } ( ( - s ) ^ { - ( M + 1 - i ) } ) \\\\ & \\times C ^ { [ 0 , M ] } ( - s u ; s ) \\prod _ { 1 \\le i \\le k } K ^ { ( i ) } ( ( - s ) ^ { M + 1 - i } ) , \\end{align*}"}
-{"id": "784.png", "formula": "\\begin{align*} t ( \\vec { \\nu } ) = \\# \\{ ( i , j ) \\ , | \\ , 1 \\le i < j \\le k \\ , \\ , \\mathrm { a n d } \\ , \\ , \\nu _ { i } > \\nu _ { j } \\} . \\end{align*}"}
-{"id": "3349.png", "formula": "\\begin{align*} F ( m + n ) & = \\nu ( g _ 1 ^ { [ s _ 1 ( m + n ) ] } \\cdots g _ k ^ { [ s _ k ( m + n ) ] } ) \\\\ & \\leq \\nu ( g _ 1 ^ { [ s _ 1 m ] + [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k m ] + [ s _ k n ] } ) + \\nu ( ( g _ 1 ^ { [ s _ 1 m ] + [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k m ] + [ s _ k n ] } ) ^ { - 1 } ( g _ 1 ^ { [ s _ 1 ( m + n ) ] } \\cdots g _ k ^ { [ s _ k ( m + n ) ] } ) ) \\\\ & \\leq \\nu ( g _ 1 ^ { [ s _ 1 m ] + [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k m ] + [ s _ k n ] } ) + \\sum _ { i = 1 } ^ k \\nu ( g _ i ) . \\\\ \\end{align*}"}
-{"id": "3379.png", "formula": "\\begin{align*} P ' ( t ) = \\frac { - V } { \\Delta t X ( t ) } - \\frac { 1 } { \\gamma ( t ) } . \\end{align*}"}
-{"id": "7806.png", "formula": "\\begin{align*} \\left \\langle U , V \\right \\rangle = O \\left ( S ^ { 2 } f ^ { - 1 } \\right ) , \\ \\left \\langle U , W \\right \\rangle = O \\left ( S ^ { 2 } f ^ { - 1 } \\right ) , \\ \\left \\langle V , W \\right \\rangle = O \\left ( S ^ { 2 } f ^ { - 1 } \\right ) \\end{align*}"}
-{"id": "5757.png", "formula": "\\begin{align*} Q _ { L } ^ { L R } = 2 \\left [ l _ n ( \\hat { \\delta } ) - l _ n ( \\hat { \\delta } _ { 0 } ) \\right ] . \\end{align*}"}
-{"id": "8838.png", "formula": "\\begin{align*} E _ { k , \\chi _ { - 4 } } ^ { \\infty } ( \\tau ) & = 1 - \\frac { 2 k } { B _ { k , 4 } } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { k - 1 , \\chi _ { - 4 } } ^ { \\infty } ( n ) q ^ { n } \\intertext { a n d } E _ { k , \\chi _ { - 4 } } ^ { 0 } ( \\tau ) & = \\delta _ { k , 1 } - \\frac { 2 k } { B _ { k , 4 } } \\sum _ { n = 1 } ^ { \\infty } \\sigma _ { k - 1 , \\chi _ { - 4 } } ^ { 0 } ( n ) q ^ { n } , \\end{align*}"}
-{"id": "9186.png", "formula": "\\begin{align*} R ( u ) = \\frac { \\partial \\psi ( t , u ) } { \\partial t } | _ { t = 0 } = - 2 u ^ 2 , \\end{align*}"}
-{"id": "4785.png", "formula": "\\begin{align*} \\mathcal { M } ' _ b : z ( u , v ) = f ( u ) \\ , l ( v ) + g ( u ) \\ , e _ 4 , u \\in I , \\ , v \\in J , \\end{align*}"}
-{"id": "7182.png", "formula": "\\begin{align*} f ' = g g ' + p ' , \\end{align*}"}
-{"id": "9554.png", "formula": "\\begin{align*} H \\psi ^ { ( 0 ) } _ { n } = \\lambda ^ { ( 0 ) } _ { n } \\psi ^ { ( 0 ) } _ { n } \\end{align*}"}
-{"id": "6451.png", "formula": "\\begin{align*} & - \\int _ { B _ { 1 } } \\partial _ { s } ( g _ { 1 - \\alpha , m } * ( \\phi \\psi ^ { 2 } \\tilde { u } ^ { 1 - q } ) ) d x + ( 1 - q ) \\phi \\mathcal { E } ( \\tilde { u } , - \\psi ^ { 2 } \\tilde { u } ^ { - q } ) \\\\ & \\quad \\quad \\quad \\leq \\int _ { 0 } ^ { s } \\dot { g } _ { 1 - \\alpha , m } ( s - \\tau ) ( \\phi ( s ) - \\phi ( \\tau ) ) \\left ( \\int _ { B _ { 1 } } \\psi ^ { 2 } \\tilde { u } ^ { 1 - q } d x \\right ) ( \\tau ) d \\tau + R _ { m } ( s ) , \\end{align*}"}
-{"id": "8763.png", "formula": "\\begin{align*} L = \\frac 1 2 ( 1 - t ) \\left . \\frac { \\dd } { \\dd x _ i } \\right | _ { \\{ x _ j = 1 \\} } S _ i ( x _ 1 , . . , x _ n ) . \\end{align*}"}
-{"id": "8366.png", "formula": "\\begin{align*} \\sigma ^ { - 1 } \\circ E \\circ \\sigma = E , \\end{align*}"}
-{"id": "8198.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ 3 \\phi ( a _ { i j } ) \\Phi _ j ^ 2 = ( \\sum _ { j = 1 } ^ 3 a _ { i j } x _ j ^ 2 ) ^ p , \\ \\ \\ i = 1 , 2 . \\end{align*}"}
-{"id": "5430.png", "formula": "\\begin{align*} B = \\begin{bmatrix} u _ { 1 1 } J ( T ^ { - i } , T ^ { - r } ) & u _ { 1 2 } J ( T ^ { - i - 2 r } , T ^ { - 3 r } ) & 0 & 0 \\\\ u _ { 2 1 } J ( T ^ { - i } , T ^ { - 3 r } ) & u _ { 2 2 } J ( T ^ { - i - 2 r } , T ^ { - r } ) & 0 & 0 \\\\ 0 & 0 & v _ { 1 1 } J ( T ^ { - i - 3 r } , T ^ { - 3 r } ) & v _ { 1 2 } J ( T ^ { - i - 3 r } , T ^ { - r } ) \\\\ 0 & 0 & v _ { 2 1 } J ( T ^ { - i - r } , T ^ { - r } ) & v _ { 2 2 } J ( T ^ { - i - r } , T ^ { - 3 r } ) , \\end{bmatrix} \\end{align*}"}
-{"id": "2746.png", "formula": "\\begin{align*} f ' _ { K L } = f _ { I J } a ^ I _ K b ^ J _ L = a ^ I _ K \\star f _ { I J } \\star b ^ J _ L . \\end{align*}"}
-{"id": "8953.png", "formula": "\\begin{align*} r _ { t } + s _ { x x } + q ( r ^ { 2 } + s ^ { 2 } ) s = 0 . \\end{align*}"}
-{"id": "8864.png", "formula": "\\begin{align*} s ( \\omega ) = \\lim _ { T \\to \\infty } \\frac 1 T S _ T ( \\omega ) \\end{align*}"}
-{"id": "5953.png", "formula": "\\begin{align*} U ( Z _ t ) & = U ( z ) + \\int _ 0 ^ t D U ( Z _ s ) { R \\ , } \\ , \\dd W _ s + \\int _ 0 ^ t { \\mathcal L } U ( Z _ s ) \\ , \\dd s \\\\ & = U ( z ) + \\int _ 0 ^ t D U ( Z _ s ) { R \\ , } \\dd W _ s + \\lambda \\int _ 0 ^ t U ( Z _ s ) \\ , \\dd s - \\int _ 0 ^ t B ( Z _ s ) \\ , \\dd s \\ , . \\end{align*}"}
-{"id": "5983.png", "formula": "\\begin{align*} \\varepsilon _ i ( \\eta ) = m ^ \\eta _ i \\ \\mbox { a n d } \\ \\varphi _ i ( \\eta ) = H _ i ^ \\eta ( 1 ) - m ^ \\eta _ i . \\end{align*}"}
-{"id": "5715.png", "formula": "\\begin{align*} \\mathcal { D } _ k : = \\left \\{ Q \\in \\mathcal { D } ( Q _ 0 ) : \\frac { w ( Q ) } { | Q | } > a ^ k \\gamma _ 0 \\right \\} ( k \\in \\N ) . \\end{align*}"}
-{"id": "3156.png", "formula": "\\begin{align*} & \\| \\frac { 1 } { J _ n \\cdot L _ n } \\sum _ { j = 1 } ^ { J _ n } \\sum _ { l = 1 } ^ { L _ n } V _ { t ^ n } ( \\pi ( x _ { j , l } ) ) - \\Theta _ { t ^ n , \\pi } \\| _ 1 \\\\ & \\leq 2 ^ { - \\sqrt { n } \\frac { 1 } { 3 2 } \\hat { \\beta } ( \\alpha ) } \\end{align*}"}
-{"id": "301.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\ ! \\ ! \\ ! \\int _ 0 ^ { 1 - s } \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\log \\Bigl ( \\ ! \\frac { ( n - 1 ) s } { e ^ { \\Psi ( j ) } } \\Bigr ) \\ ! \\log \\Bigl ( \\ ! \\frac { ( n - 1 ) t } { e ^ { \\Psi ( l ) } } \\Bigr ) \\ ! \\{ \\mathrm { B } _ { j , l , n - j - l - 1 } \\ ! ( s , t ) - & \\mathrm { B } _ { j , n - j } ( s ) \\mathrm { B } _ { l , n - l } ( t ) \\} d t d s \\\\ = & - \\frac { 1 } { n } + O ( n ^ { - 2 } ) \\end{align*}"}
-{"id": "8781.png", "formula": "\\begin{align*} T _ i ( u ) T _ { i + 1 } ( u + v ) T _ i ( v ) = T _ { i + 1 } ( v ) T _ i ( u + v ) T _ { i + 1 } ( u ) . \\end{align*}"}
-{"id": "867.png", "formula": "\\begin{align*} \\varphi _ { A , k } ( y ) : = \\inf \\{ t \\in { \\mathbb { R } } \\mid y \\in t k + A \\} , \\end{align*}"}
-{"id": "8284.png", "formula": "\\begin{align*} \\mathbf { D } ( \\xi ) = \\frac { \\xi } { n } v v ^ { { \\sf T } } + \\mathbf { U } \\ , , \\end{align*}"}
-{"id": "5897.png", "formula": "\\begin{align*} X = 2 \\psi _ { I } \\int T ^ { I } \\left ( t \\right ) \\partial _ { t } + T ^ { I } Y _ { I } ^ { \\alpha } \\partial _ { \\alpha } + \\left ( a \\left ( x ^ { \\beta } , t \\right ) \\right ) u \\partial _ { u } , \\end{align*}"}
-{"id": "487.png", "formula": "\\begin{align*} ( B ^ c _ \\varepsilon ( x ) ) ^ \\pm : = \\big \\{ y \\in B _ \\varepsilon ^ c ( x ) : \\ , \\pm \\xi \\cdot ( y - x ) \\le 0 \\big \\} , \\end{align*}"}
-{"id": "3944.png", "formula": "\\begin{align*} d = d ( y _ 1 , y _ 2 ) \\geq d ( y _ 1 , x _ 1 ) + d ( x _ 1 , x _ 2 ) + d ( x _ 2 , y _ 2 ) - 1 2 N ( 3 , 0 ) . \\end{align*}"}
-{"id": "3293.png", "formula": "\\begin{align*} K w _ { - 1 } = q ^ 2 w _ { - 1 } , K w _ 0 = w _ 0 , K w _ 1 = q ^ { - 2 } w _ 1 . \\end{align*}"}
-{"id": "1907.png", "formula": "\\begin{align*} g ( x ) = g _ 0 ( x ) + c \\sum _ { k = 1 } ^ { \\infty } ( 1 + ( 8 k ) ) ^ { - \\beta } \\cos ( 1 2 k x ) g _ 0 ( x ) . \\end{align*}"}
-{"id": "6893.png", "formula": "\\begin{gather*} m ^ 0 ( A ) = \\int _ A D _ \\nu m ^ 0 \\ d \\nu \\end{gather*}"}
-{"id": "6331.png", "formula": "\\begin{align*} G _ { k , N } = \\sum _ { l \\in \\widetilde { \\Gamma _ { k } ^ { \\alpha _ { 2 } , \\alpha _ { 1 } } } } T _ { N l } f _ { l } ^ { \\alpha _ { 2 } } \\end{align*}"}
-{"id": "9491.png", "formula": "\\begin{align*} \\Phi ^ { * } d \\theta = d \\theta \\Phi ^ { * } d x _ { i } = d x _ { i } \\Phi ^ { * } d y _ { i } = d y _ { i } \\end{align*}"}
-{"id": "7387.png", "formula": "\\begin{align*} \\Delta _ { t , t } = & \\ , \\sum _ { r = 0 } ^ { t - 1 } ( \\gamma ^ t _ r - \\hat { \\gamma } ^ { t } _ r ) b ^ r + Z ' _ t \\Big ( \\frac { 1 } { \\sqrt { n } } \\norm { q ^ t _ { \\perp } } - \\sigma _ { t } ^ { \\perp } \\Big ) \\\\ & - \\frac { 1 } { n } \\norm { q ^ t _ { \\perp } } \\tilde { M } _ t \\bar { Z } ' _ t + M _ t \\mathbf { M } _ t ^ { - 1 } v , \\end{align*}"}
-{"id": "1350.png", "formula": "\\begin{align*} Q _ { L } ^ { S T } ( \\omega ) = S ( \\hat { \\delta } _ { 0 } ) ^ { \\mathrm { T } } I _ { L } ( \\hat \\delta _ 0 ) ^ { - 1 } S ( \\hat \\delta _ { 0 } ) . \\end{align*}"}
-{"id": "2641.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { h ( m , n ) } C _ { m } ^ { l } ( G ^ { 0 } ) ( \\lambda ) ( C _ { m } ^ { l } ( G ^ { 0 } ) ( \\lambda ) ) ^ { * } = \\alpha _ { m } ^ { 2 } d _ { m } ^ { 0 } ( \\lambda ) ^ { \\top } \\overline { d _ { m } ^ { 0 } ( \\lambda ) } , \\end{align*}"}
-{"id": "4317.png", "formula": "\\begin{align*} \\vert \\xi \\vert ^ 2 = \\sup _ { \\Vert u \\Vert _ { y _ 0 } \\leq 1 } | \\langle \\xi , u \\rangle | ^ 2 \\end{align*}"}
-{"id": "5220.png", "formula": "\\begin{align*} T _ q ( \\sideset { } { ' } \\sum _ { \\vert J \\vert = q } \\ , u _ J \\overline { \\omega _ J } \\ , ) \\ ; \\ ; \\ ; \\ ; = \\sideset { } { ' } \\sum _ { | J | = q , | K | = ( m - 1 - q ) } \\varepsilon _ { ( 1 , \\dots , m - 1 ) } ^ { J K } u _ J \\overline { \\omega _ K } \\ ; , \\end{align*}"}
-{"id": "5125.png", "formula": "\\begin{align*} h _ { z } ^ { a } ( 1 ) = \\frac { z } { 1 + z } u _ { a } = \\psi _ { z } ^ { a } ( 1 ) \\end{align*}"}
-{"id": "8538.png", "formula": "\\begin{align*} \\pi _ { r a } ' \\xi _ { r a } ' = i d _ { N _ { \\tau ( a ) } } \\end{align*}"}
-{"id": "506.png", "formula": "\\begin{align*} & \\sum \\limits _ { k = 0 } ^ { m - 1 } k \\binom { n } { k } \\ ; = \\ ; \\frac { n } { 2 } \\binom { n } { < m } \\ ; - \\ ; \\frac { m } { 2 } \\binom { n } { m } \\\\ & \\sum \\limits _ { k = 0 } ^ { m - 1 } k ( k - 1 ) \\binom { n } { k } \\ ; = \\ ; \\frac { n ( n - 1 ) } { 4 } \\binom { n } { < m } \\ ; - \\ ; \\frac { m ( 2 m + n - 3 ) } { 4 } \\binom { n } { m } , \\end{align*}"}
-{"id": "4174.png", "formula": "\\begin{align*} \\varphi ( w ( z ) ) & = \\varphi \\Bigg ( \\tfrac { \\displaystyle h ( z ) - 1 } { \\displaystyle h ( z ) + 1 } \\Bigg ) \\\\ & = 1 + \\tfrac { \\displaystyle 1 } { \\displaystyle 2 } Q _ 1 d _ 1 z + \\tfrac { \\displaystyle 1 } { \\displaystyle 2 } Q _ 1 \\Big ( d _ 2 - \\tfrac { \\displaystyle d _ 1 ^ 2 } { \\displaystyle 2 } \\Big ) z ^ 2 + \\tfrac { \\displaystyle 1 } { \\displaystyle 4 } Q _ 2 d _ 1 ^ 2 z ^ 2 + \\cdots , \\end{align*}"}
-{"id": "800.png", "formula": "\\begin{align*} A ^ { [ M ' , M ] } ( z ) = \\mathbb { T } ^ { [ M ' , M ] } ( z ) _ { 0 0 } , C _ { a } ^ { [ M ' , M ] } ( z ) = \\mathbb { T } ^ { [ M ' , M ] } ( z ) _ { a 0 } \\ , \\ , ( 1 \\le a \\le r ) . \\end{align*}"}
-{"id": "6680.png", "formula": "\\begin{align*} ( \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } F ) ( x ) = ( - 2 \\lambda ) F ( x ) , ~ ( \\forall ) x \\in U _ { x _ e } . \\end{align*}"}
-{"id": "8044.png", "formula": "\\begin{align*} \\abs { \\hat w } _ k & \\leq C \\cdot \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\ ! \\sum _ { \\stackrel { ( e _ 1 , \\dots , e _ n ) \\in \\N ^ n } { \\sum _ i e _ i \\nu ( a _ { i } ) \\leq \\nu ( a _ k ) } } \\prod _ i \\abs { w } _ { i } ^ { e _ i } \\end{align*}"}
-{"id": "883.png", "formula": "\\begin{align*} \\phi ^ { m _ 1 } = \\iota _ a \\circ \\phi ^ { m _ 2 } . \\end{align*}"}
-{"id": "3098.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) = \\begin{bmatrix} M ( \\lambda ) & K _ 2 ( \\lambda ) ^ T \\\\ K _ 1 ( \\lambda ) & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "5352.png", "formula": "\\begin{align*} R \\cdot R ' = \\sum _ { i = 0 } ^ { \\ell } \\dim ( \\chi _ i ) ^ 2 = | G | \\end{align*}"}
-{"id": "6141.png", "formula": "\\begin{align*} z _ { i _ 1 } ( z _ { i _ 2 } \\cdots z _ { i _ K } ) z _ { i _ { K + 1 } } & = \\sum _ j z _ { i _ 1 } z _ { i _ 2 } \\cdots z _ { i _ K } z _ j z _ j ^ * z _ { i _ { K + 1 } } = \\sum _ j z _ { i _ 1 } z _ j ^ * z _ { i _ { K + 1 } } z _ { i _ 2 } \\cdots z _ { i _ K } z _ j \\\\ & = \\sum _ j z _ { i _ { K + 1 } } z _ { i _ 2 } \\cdots z _ { i _ K } z _ { i _ 1 } z _ j ^ * z _ j = z _ { i _ { K + 1 } } ( z _ { i _ 2 } \\cdots z _ { i _ K } ) z _ { i _ 1 } \\end{align*}"}
-{"id": "9810.png", "formula": "\\begin{align*} h ^ p ( Z _ p ( K ) , y ) = \\frac { 1 } { V _ n ( K ) } \\int _ { K } \\abs { \\langle x , y \\rangle } ^ p d x \\end{align*}"}
-{"id": "9195.png", "formula": "\\begin{align*} \\phi ' ( 1 ) & = \\lim _ { u \\to 1 } \\frac { \\phi ' ( u ) } { u } = \\lim _ { x \\to 0 } \\frac { \\alpha ( x ) } { x \\ln 2 } \\\\ & = \\frac { 1 } { \\ln 2 } \\cdot \\lim _ { x \\to 0 } \\alpha ' ( x ) = \\frac { 1 } { \\ln 2 } \\cdot \\lim _ { x \\to 0 } \\frac { 1 } { 1 - x ^ 2 } = \\frac { 1 } { \\ln 2 } \\end{align*}"}
-{"id": "5713.png", "formula": "\\begin{align*} f ^ { \\sharp } ( x ) : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { | Q | } \\int _ Q | f ( y ) - f _ Q | d y \\times \\chi _ Q ( x ) , \\end{align*}"}
-{"id": "8178.png", "formula": "\\begin{align*} f _ n = \\sum _ { k = 1 } ^ { n } \\binom { ( \\alpha + \\beta ) n - 1 } { k - 1 } \\frac { ( k - 1 ) ! } { n ! } B _ { n , k } ( 1 ! \\theta _ 1 , 2 ! \\theta _ 2 , \\dots ) \\end{align*}"}
-{"id": "3269.png", "formula": "\\begin{align*} \\eth = \\sum _ i S ^ { - 1 } ( E _ { \\xi _ i } ) \\otimes \\gamma _ { - } ( w _ i ) \\in U _ q ( \\mathfrak { g } ) \\otimes \\mathrm { C l } _ q . \\end{align*}"}
-{"id": "9155.png", "formula": "\\begin{align*} \\beta & = \\xi _ 1 ( p - t + 1 ) + \\sum _ { r = 2 } ^ { \\lceil s \\rceil - 1 } ( t - 1 ) \\xi _ r \\binom { p - t + 1 } { ( r - 1 ) t + 1 } + ( t - 1 ) \\xi _ { \\lceil s \\rceil } \\\\ \\gamma & = ( p - t + 1 ) \\left ( \\sum _ { r = 1 } ^ { \\lceil s \\rceil - 2 } \\xi _ { r + 1 } \\binom { p - t } { r t } + \\xi _ { \\lceil s \\rceil } \\right ) . \\end{align*}"}
-{"id": "7409.png", "formula": "\\begin{align*} e _ { \\pm i } ^ \\iota = e _ { \\pm i } a ^ \\iota = a ^ { - 1 } \\forall a \\in H . \\end{align*}"}
-{"id": "2986.png", "formula": "\\begin{align*} & \\delta _ L \\Theta ^ 2 ( x , y , z ) \\\\ = & - [ x , \\Theta ^ 2 ( y , z ) ] + [ y , \\Theta ^ 2 ( x , z ) ] - [ z , \\Theta ^ 2 ( x , y ) ] + \\Theta ^ 2 ( [ x , y ] , z ) - \\Theta ^ 2 ( x , [ y , z ] ) + \\Theta ^ 2 ( y , [ x , z ] ) . \\end{align*}"}
-{"id": "2801.png", "formula": "\\begin{gather*} f ^ + ( u , \\xi ) = \\begin{cases} 1 & , \\\\ 0 & , \\end{cases} f ^ - ( u , \\xi ) = \\begin{cases} - 1 & , \\\\ 0 & . \\end{cases} \\end{gather*}"}
-{"id": "6538.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\sum \\limits ^ { k } _ { i = 0 } \\delta _ i u _ { n - i } ^ \\star + A ( t _ n ) u _ n ^ \\star = \\sum \\limits ^ { k - 1 } _ { i = 0 } \\gamma _ i B ( t _ { n - i - 1 } , u ^ \\star _ { n - i - 1 } ) + d _ n , k \\le n \\le N . \\\\ \\end{align*}"}
-{"id": "6037.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\int _ { \\R ^ N \\setminus B _ R ^ + } \\Gamma _ R ^ + ( x , y ) g _ 2 ( y ) d y = 0 \\end{align*}"}
-{"id": "1883.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } ^ 2 + 2 R _ { 1 3 4 2 } R _ { 1 4 2 3 } = & x ^ 2 - 2 y ( x + y ) \\\\ = & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( x + 2 y ) - ( x + 2 y ) ^ 2 ] \\\\ \\ge & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( K _ { 1 4 } - K _ { 1 3 } ) - ( K _ { 1 4 } - K _ { 1 3 } ) ^ 2 ] \\\\ \\end{align*}"}
-{"id": "7896.png", "formula": "\\begin{align*} \\left \\| I - P \\right \\| ^ 2 = \\left \\| \\frac { ( A - C ) \\times ( C - I ) } { b } \\right \\| ^ 2 \\end{align*}"}
-{"id": "4490.png", "formula": "\\begin{align*} | U _ 1 | + | U _ 2 | = O \\biggl ( \\frac { k ^ { 1 / 2 } } { n } \\max \\biggl \\{ \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "8285.png", "formula": "\\begin{align*} \\sum _ { i , j } M _ { i , j } \\sigma _ i ^ * \\sigma _ j ^ * = \\sum _ i | f _ i | . \\end{align*}"}
-{"id": "1670.png", "formula": "\\begin{align*} W ^ { \\gamma , p } ( \\R ^ { 2 d } ) = \\Big \\{ f \\in L ^ p ( \\R ^ { 2 d } ) : \\int _ { \\R ^ d } \\| f ( x , \\cdot ) \\| _ { W ^ { \\gamma , p } ( \\R ^ d ) ) } ^ p \\dd x + \\int _ { \\R ^ d } \\| f ( \\cdot , v ) \\| _ { W ^ { \\gamma , p } ( \\R ^ d ) ) } ^ p \\dd v < \\infty \\Big \\} , \\end{align*}"}
-{"id": "6809.png", "formula": "\\begin{align*} A ( \\xi ) = Y _ \\infty ^ - ( \\xi ) \\frac { \\dd G ( \\xi ) } { \\dd \\xi } G ( \\xi ) ^ { - 1 } \\left ( Y _ \\infty ^ - ( \\xi ) \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "3867.png", "formula": "\\begin{align*} \\int _ { \\partial \\Omega } g \\cdot { \\mathbf n } _ V \\ , d \\| \\delta V \\| + \\sigma \\int _ { \\partial ^ * B ^ + } g \\cdot \\mathbf n _ { B ^ + } \\ , d \\mathcal H ^ { n - 1 } = 0 . \\end{align*}"}
-{"id": "1475.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\Phi _ { p _ 0 , q _ 0 , v _ 0 , w _ 0 } ( 2 ^ l Q ) = \\infty , ( Q \\in \\mathcal { Q } ) , \\Phi _ { p _ 0 , q _ 0 , v _ 0 , w _ 0 } ( Q ) : = v _ 0 ( Q ) ^ { \\frac { 1 } { p _ 0 } - \\frac { 1 } { q _ 0 } } w _ 0 ( Q ) ^ \\frac { 1 } { q _ 0 } . \\end{align*}"}
-{"id": "6802.png", "formula": "\\begin{align*} Y ^ + ( \\xi ) = Y ^ - ( \\xi ) G ( \\xi ) , \\end{align*}"}
-{"id": "3600.png", "formula": "\\begin{align*} ( \\phi , Z ) & = - \\Big [ \\rho _ 1 ^ { - 1 } ( D \\Phi ^ W _ { ( g _ 1 , \\pi _ 1 ) } - D \\Phi ^ W _ { ( g _ 2 , \\pi _ 2 ) } ) ( h _ 2 , w _ 2 ) + \\rho _ 1 ^ { - 1 } \\Big ( Q ^ W _ { ( g _ 1 , \\pi _ 1 ) } ( h _ 1 , w _ 1 ) - Q ^ W _ { ( g _ 2 , \\pi _ 2 ) } ( h _ 2 , w _ 2 ) \\Big ) \\\\ & + \\rho _ 1 ^ { - 1 } D \\Phi ^ W _ { ( g _ 1 , \\pi _ 1 ) } \\Big ( \\big ( \\rho _ 1 ( D \\Phi ^ W _ { ( g _ 1 , \\pi _ 1 ) } ) ^ * - \\rho _ 2 ( D \\Phi ^ W _ { ( g _ 2 , \\pi _ 2 ) } ) ^ * \\big ) ( f _ 2 , X _ 2 ) \\Big ) \\Big ] . \\end{align*}"}
-{"id": "88.png", "formula": "\\begin{align*} ( n , A ) ( n , g f ) = n ( n , A , g f ) \\end{align*}"}
-{"id": "7476.png", "formula": "\\begin{align*} P ( \\Delta _ { Q _ j ^ { ( k ) } } ( \\infty ) = 0 ) > 0 j \\notin \\{ 0 , k , k + 1 \\} . \\end{align*}"}
-{"id": "686.png", "formula": "\\begin{align*} U = \\big \\{ g \\in G \\ , : \\ , \\nu ( C \\cap g B ) > 0 \\big \\} \\end{align*}"}
-{"id": "8070.png", "formula": "\\begin{align*} H ( a , b ) = \\int _ X h ( a , b , x ) \\ d \\mu ( x ) ; \\end{align*}"}
-{"id": "7856.png", "formula": "\\begin{align*} \\int _ 0 ^ 1 \\frac { \\Phi ( r ) } { r } \\ , d r = C _ * < \\infty \\ , . \\end{align*}"}
-{"id": "2318.png", "formula": "\\begin{align*} \\| f ( w ) - f ( v ) \\| _ { X } & = \\| f ( w ) - f ( v ) \\| _ { L ^ 2 ( \\R ^ d ) } + \\| f ( w ) - f ( v ) \\| _ { L ^ q ( \\R ^ d ) } \\\\ & \\le C _ K \\big ( \\| w - v \\| _ { L ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { L ^ q ( \\R ^ d ) } \\big ) \\\\ & \\le C _ K \\Big ( \\| w - v \\| _ { L ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { L ^ 2 ( \\R ^ d ) } ^ { \\frac { 2 } { q } } \\| w - v \\| _ { L ^ \\infty ( \\R ^ d ) } ^ { 1 - \\frac { 2 } { q } } \\Big ) \\\\ & \\le C _ K \\big ( \\| w - v \\| _ { L ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { L ^ \\infty ( \\R ^ d ) } \\big ) = C _ K \\| w - v \\| _ { W } . \\end{align*}"}
-{"id": "5294.png", "formula": "\\begin{align*} \\frac { b - 1 - k ( r - 1 ) } { \\frac { v } { k } - 1 } = s k - k + 1 \\end{align*}"}
-{"id": "3988.png", "formula": "\\begin{align*} \\pi _ { p } ( v ' ) _ { j } = \\pi _ { p } ( v ' _ { j } ) , \\quad . \\end{align*}"}
-{"id": "3598.png", "formula": "\\begin{align*} \\Pi _ { g _ 0 } \\circ \\Phi ^ W _ { ( g , \\pi ) } ( g + h , \\pi + w ) = \\Pi _ { g _ 0 } \\Phi ^ W _ { ( g , \\pi ) } ( g , \\pi ) + \\Pi _ { g _ 0 } ( \\psi , V ) . \\end{align*}"}
-{"id": "989.png", "formula": "\\begin{align*} R _ { 1 2 } ( u - v ) R _ { 1 3 } ( u ) R _ { 2 3 } ( v ) = R _ { 2 3 } ( v ) R _ { 1 3 } ( u ) R _ { 1 2 } ( u - v ) . \\end{align*}"}
-{"id": "859.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } B _ { 2 } ^ { ( l , l ) } = B _ { l , 2 } & & \\\\ B _ { m , 1 } B _ { 2 } ^ { ( m , l ) } - B _ { 2 } ^ { ( m , l ) } B _ { l , 1 } + i _ { m } i _ { l } ^ { * } = 0 & & \\mbox { i f $ m \\neq l $ } \\end{array} \\right . \\end{align*}"}
-{"id": "923.png", "formula": "\\begin{align*} \\textup { T r } ^ { p } ( A \\Sigma _ { Z } A ^ { T } ) = \\sum _ { k } \\left ( \\sum _ { i } a ^ { 2 } _ { k i } \\sigma _ { i } \\right ) ^ { p } . \\end{align*}"}
-{"id": "207.png", "formula": "\\begin{align*} \\biggl | V _ d f ( x ) h _ x ^ { - 1 } ( s ) ^ d - \\sum _ { l = 0 } ^ { \\lceil \\beta / 2 \\rceil - 1 } b _ l ( x ) s ^ { 1 + 2 l / d } \\biggr | \\lesssim s \\biggl \\{ \\frac { a ( f ( x ) ) ^ { d / ( 2 \\wedge \\beta ) } s } { f ( x ) } \\biggr \\} ^ { \\beta / d } , \\end{align*}"}
-{"id": "9325.png", "formula": "\\begin{align*} \\begin{cases} d \\tilde { Z } ( t , z ) = [ \\theta ( t , z ) - \\pi ( t , z ) b _ 0 ( t , z ) + \\pi ^ 2 ( t , z ) \\sigma ^ 2 _ 0 ( t , z ) ] \\tilde { Z } ( t , x , z ) d t - \\pi ( t , z ) \\sigma _ 0 ( t , z ) \\tilde { Z } ( t , z ) d B ( t ) \\\\ \\tilde { Z } ( 0 , z ) = \\int _ D \\frac { 1 } { \\alpha ( x ) } d x , \\end{cases} \\end{align*}"}
-{"id": "2140.png", "formula": "\\begin{align*} H ^ { s } ( \\Omega ) = \\left \\{ u \\in L ^ { 2 } ( \\Omega ) \\ , : \\ , \\frac { | u ( x ) - u ( y ) | } { | x - y | ^ { s + n / 2 } } \\in L ^ { 2 } ( \\Omega \\times \\Omega ) \\right \\} , \\end{align*}"}
-{"id": "1850.png", "formula": "\\begin{align*} F _ { m , n } ( t ) & = \\frac { 1 } { \\sqrt { n + 1 } } \\frac { 1 } { \\sqrt { m + 1 } } \\left ( 1 + i \\log \\frac { m + 1 } { n + 1 } \\right ) ^ { - t } \\\\ G _ { m , n } ( t ) & = \\frac { 1 } { \\sqrt { m + 1 } } \\int _ n ^ { n + 1 } v ^ { - 1 / 2 } \\left ( 1 + i \\log \\frac { m + 1 } { v } \\right ) ^ { - t } d v . \\end{align*}"}
-{"id": "1069.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } & \\leq \\left ( ( 1 + x ) ^ 2 + ( k - 2 ) + ( 1 - \\alpha - x ) ^ 2 \\right ) \\frac { n ^ 2 } { 2 } \\\\ & = \\left ( k - 2 \\alpha + \\alpha ^ 2 + 2 \\alpha x + 2 x ^ 2 \\right ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "5428.png", "formula": "\\begin{align*} \\delta _ { S ' } = \\frac { 1 } { 2 } ( T ^ 0 - \\delta _ 0 + \\alpha T ^ r + \\overline \\alpha T ^ { - r } ) , \\end{align*}"}
-{"id": "6993.png", "formula": "\\begin{align*} - [ \\alpha _ j , [ f _ 0 ^ x , f _ i ^ y ] ] + [ \\alpha _ j , [ f _ 0 ^ y , f _ i ^ x ] ] & = [ \\alpha _ j , - [ f _ 0 ^ x , f _ i ^ y ] + [ f _ 0 ^ y , f _ i ^ x ] ] \\\\ & = [ \\alpha _ j , \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ x , f _ n ^ y ] - f _ 0 ^ { \\lambda _ i ( x , y ) } - f _ i ^ { [ x , y ] } - \\sum _ { m + n = i , ~ m , n > 0 } f _ m ^ { \\lambda _ n ( x , y ) } ] . \\end{align*}"}
-{"id": "4017.png", "formula": "\\begin{align*} R \\subseteq \\bigcap _ { H \\in \\mathcal { H } _ 0 } H = W , \\end{align*}"}
-{"id": "2549.png", "formula": "\\begin{align*} \\left | \\mathbb { E } \\left [ \\frac 1 N \\sum _ { n = 0 } ^ { N - 1 } f ( U ^ n ) - \\frac 1 T \\int _ 0 ^ T f ( U ( t ) ) d t \\right ] \\right | \\le C _ h ( B ( T ) + \\tau ) , \\end{align*}"}
-{"id": "2935.png", "formula": "\\begin{align*} s _ \\lambda & = \\det \\left [ \\begin{array} { c } h ^ { ( j - 1 ) } _ { \\lambda _ i - i + 1 } \\end{array} \\right ] _ { 1 \\leq i , j \\leq \\ell ( \\lambda ) } \\equiv \\det \\left [ \\begin{array} { c } h ^ { ( 0 ) } _ { \\lambda _ i - i + j } \\end{array} \\right ] _ { 1 \\leq i , j \\leq \\ell ( \\lambda ) } \\ , . \\end{align*}"}
-{"id": "1787.png", "formula": "\\begin{align*} \\nu ( \\sigma _ x ( a ) ) = x + \\nu ( a ) \\end{align*}"}
-{"id": "8110.png", "formula": "\\begin{align*} C _ { k , l , N } = \\inf \\left \\{ { \\mathcal Q } _ { k , l , N } ( u ) : \\ , u \\in C _ 0 ^ 1 ( \\mathbb { R } ^ N ) \\setminus \\{ 0 \\} \\right \\} . \\end{align*}"}
-{"id": "1657.png", "formula": "\\begin{align*} \\lambda { \\psi } ( z ) - \\frac { 1 } { 2 } \\triangle _ v { \\psi } ( z ) - v \\cdot D _ x { \\psi } ( z ) = \\lambda { \\psi } ( z ) - { \\cal L } { \\psi } ( z ) = g ( z ) , \\ ; \\ ; \\ ; z \\in \\R ^ { 2 d } . \\end{align*}"}
-{"id": "7575.png", "formula": "\\begin{align*} A = A ( x _ 0 ) : = \\min _ { x \\in [ x _ 0 , u _ l ] } F ( x ) > - l \\end{align*}"}
-{"id": "9946.png", "formula": "\\begin{align*} \\mathcal { E } _ { \\lambda , \\mu } ( u ) : = \\frac { 1 } { 2 } \\left \\| u \\right \\| ^ 2 - \\frac { \\mu } { 2 ^ * } \\int _ \\Omega | u | ^ { 2 ^ * } d x - \\lambda \\int _ \\Omega G ( u ) d x \\end{align*}"}
-{"id": "6823.png", "formula": "\\begin{align*} L _ m \\cdot \\left ( \\Omega _ n Y \\right ) - L _ n \\cdot \\left ( \\Omega _ m Y \\right ) = ( n - m ) \\Omega _ { m + n } Y \\end{align*}"}
-{"id": "4289.png", "formula": "\\begin{align*} \\Lambda ( r , f ) = \\prod _ { i = 0 } ^ { r - 1 } x _ { i f } \\end{align*}"}
-{"id": "6203.png", "formula": "\\begin{align*} \\int _ { - 1 } ^ 1 [ f ( x + x z ) - f ( x ) - f ' ( x ) x z ] \\nu _ U ( \\d z ) = x ^ \\beta \\int _ { - 1 } ^ 1 [ ( 1 + z ) ^ \\beta - 1 - \\beta z ] \\nu _ U ( \\d z ) + o ( x ^ { \\beta } ) . \\end{align*}"}
-{"id": "5033.png", "formula": "\\begin{align*} \\int _ G \\nu ( g ^ { - 1 } B \\cap C ) \\ , d \\eta ( g ) = \\nu ( B ) \\ , \\nu ( C ) . \\end{align*}"}
-{"id": "5596.png", "formula": "\\begin{align*} M _ 0 ^ { \\pm } ( z ) = g _ 1 ^ \\pm ( z ) v \\mathcal G _ 1 v ^ * + z ^ 2 v \\mathcal G _ 2 v ^ * + M _ 1 ^ { \\pm } ( z ) , \\end{align*}"}
-{"id": "3059.png", "formula": "\\begin{align*} \\mu _ \\infty = \\sum _ { n = 1 } ^ { \\infty } \\sum _ { d \\in \\mathcal { P } ( D _ n ) } \\delta _ { u ^ { ( n ) } , \\xi _ n + d - \\log Z _ \\infty } . \\end{align*}"}
-{"id": "8723.png", "formula": "\\begin{align*} X _ \\tau ^ { t , x } = e ^ { ( \\tau - t ) A } x + \\int _ t ^ \\tau e ^ { ( \\tau - s ) A } G B ( s , X _ s ) d s + \\int _ t ^ \\tau e ^ { ( \\tau - s ) A } G d W _ s , \\tau \\in \\left [ t , T \\right ] , \\ ; \\ ; X _ \\tau ^ { t , x } = x , \\ ; \\ ; \\tau \\le t . \\end{align*}"}
-{"id": "7326.png", "formula": "\\begin{align*} \\hat { R } _ { 1 2 } \\hat { R } _ { 2 3 } \\hat { R } _ { 1 2 } = \\hat { R } _ { 2 3 } \\hat { R } _ { 1 2 } \\hat { R } _ { 2 3 } , \\end{align*}"}
-{"id": "6843.png", "formula": "\\begin{align*} \\| f \\| _ { L _ t ^ { q , \\alpha } ( I ) } : = \\bigl \\| \\lambda \\ , \\bigl | \\{ t \\in I : | f ( t ) | > \\lambda \\} \\bigr | ^ { \\frac { 1 } { q } } \\bigr \\| _ { L ^ \\alpha ( ( 0 , \\infty ) , \\frac { d \\lambda } { \\lambda } ) } . \\end{align*}"}
-{"id": "3152.png", "formula": "\\begin{align*} & P r \\biggl ( \\lVert \\sum _ { l = 1 } ^ { L _ { n } } \\frac { 1 } { L _ { n } } Q _ { t ^ n } ( X _ { j , l } ) - \\Theta _ { t ^ n } \\rVert _ 1 \\leq 2 ^ { - \\sqrt { n } \\frac { 1 } { 1 6 } \\hat { \\beta } ( \\alpha ) } \\forall t ^ n ~ \\forall j \\biggr ) \\allowdisplaybreaks \\\\ & \\geq 1 - 2 ^ { n \\vartheta } \\end{align*}"}
-{"id": "5982.png", "formula": "\\begin{align*} \\langle \\alpha ^ \\dag , ( \\beta ^ \\dag ) ^ \\lor \\rangle ^ \\dag = \\langle \\iota ( ( \\beta ^ \\dag ) ^ \\lor ) , \\iota ^ * ( \\alpha ^ \\dag ) \\rangle , \\mbox { f o r } \\alpha ^ \\dag , \\beta ^ \\dag \\in \\Delta ^ \\dag ; \\end{align*}"}
-{"id": "2711.png", "formula": "\\begin{align*} \\theta + d d ^ c \\lambda \\psi = \\lambda ( \\theta + d d ^ c \\psi ) + ( 1 - \\lambda ) \\theta \\geq \\lambda \\omega + ( 1 - \\lambda ) \\theta \\geq 0 , \\end{align*}"}
-{"id": "1258.png", "formula": "\\begin{align*} M _ 0 ^ { \\pm } ( z ) = g _ 1 ^ \\pm ( z ) v \\mathcal G _ 1 v ^ * + z ^ 2 v \\mathcal G _ 2 v ^ * + M _ 1 ^ { \\pm } ( z ) , \\end{align*}"}
-{"id": "6353.png", "formula": "\\begin{align*} L _ { - a } v _ { \\pm } = ( 2 f \\pm 2 M ) \\mathcal { H } ^ { n } \\llcorner \\{ y = 0 \\} . \\end{align*}"}
-{"id": "488.png", "formula": "\\begin{align*} \\pm \\int _ { ( B ^ c _ \\varepsilon ( x ) ) ^ \\pm } \\frac { \\xi \\cdot ( x - y ) } { \\ , | x - y | ^ { N + 2 s } \\ , } \\ , d y & = - \\int _ \\varepsilon ^ { + \\infty } \\frac { d \\rho } { \\ , \\rho ^ { 2 s } \\ , } \\int _ { \\partial B _ 1 ^ \\pm } ( \\pm \\xi \\cdot w ) \\ , d { \\mathcal H } ^ { N - 1 } \\end{align*}"}
-{"id": "6940.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V _ \\pi ( z _ 1 ; X ) V _ \\pi ( z _ 2 ; X ) \\cdots V _ \\pi ( z _ m ; X ) \\cdot 1 \\cr & = [ Z ^ { \\lambda + \\delta } ] \\ \\prod _ { 1 \\le i < j \\le m } ( z _ i - z _ j ) \\ \\prod _ { \\ell = 1 } ^ m \\ , M ( z _ \\ell ; X ) \\ L _ \\pi ( Z ) \\cr & = [ s _ \\lambda ( Z ) ] \\ M ( X , Z ) \\ , L _ \\pi ( Z ) \\ , \\end{align*}"}
-{"id": "773.png", "formula": "\\begin{gather*} R _ \\ell = C _ \\ell \\oplus \\varphi _ { \\ell - s } ( R _ { \\ell - s } ) . \\end{gather*}"}
-{"id": "1366.png", "formula": "\\begin{align*} Q _ { L } ^ { L R } = 2 \\left [ l _ n ( \\hat { \\delta } ) - l _ n ( \\hat { \\delta } _ { 0 } ) \\right ] . \\end{align*}"}
-{"id": "1792.png", "formula": "\\begin{align*} Y _ \\alpha ^ { \\langle \\lambda , \\alpha ^ \\vee \\rangle + 1 } u = 0 \\end{align*}"}
-{"id": "9680.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log g ( x ( t ) ) } { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } = - \\log \\left ( \\frac { a } { b } \\right ) . \\end{align*}"}
-{"id": "6673.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\mathcal { L } _ { X } I _ 1 = \\dots = \\mathcal { L } _ { X } I _ k = 0 , \\\\ \\mathcal { L } _ { X } D _ 1 = h _ 1 , \\dots , \\mathcal { L } _ { X } D _ p = h _ p , \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "3558.png", "formula": "\\begin{align*} \\mathcal I ^ R \\circ T ^ R ( v , w ) = ( v , w ) + \\mathcal { I } _ 0 ^ R \\circ T ^ R ( v , w ) . \\end{align*}"}
-{"id": "2824.png", "formula": "\\begin{align*} K \\sum _ { i = 0 } ^ { \\infty } ( S + i ) ^ { - \\frac { 2 } { 1 + \\sqrt { r ^ * ( \\varepsilon ) } } } \\le K S ^ { - \\frac { 1 - \\sqrt { r ^ * ( \\varepsilon ) } } { 4 } } . \\end{align*}"}
-{"id": "3900.png", "formula": "\\begin{align*} g ( x ) = \\begin{bmatrix} x \\\\ ( \\gamma _ R - 2 \\alpha _ R ) x \\\\ - 2 \\alpha _ R ( 2 \\alpha _ R + 1 ) x + \\mu \\end{bmatrix} \\ ; , \\end{align*}"}
-{"id": "2326.png", "formula": "\\begin{align*} z _ i \\leftarrow f _ i ( z _ i ) , ~ i = 1 , \\dots , m . \\end{align*}"}
-{"id": "3255.png", "formula": "\\begin{align*} [ E _ 1 ^ { c _ 1 } \\cdots E _ r ^ { c _ r } , F _ t ] & = E _ 1 ^ { c _ 1 } \\cdots E _ { t - 1 } ^ { c _ { t - 1 } } [ E _ t , F _ t ] E _ { t + 1 } ^ { c _ { t + 1 } } \\cdots E _ r ^ { c _ r } \\\\ & = E _ 1 ^ { c _ 1 } \\cdots E _ { t - 1 } ^ { c _ { t - 1 } } \\frac { K _ t - K _ t ^ { - 1 } } { q _ t - q _ t ^ { - 1 } } E _ { t + 1 } ^ { c _ { t + 1 } } \\cdots E _ r ^ { c _ r } \\in U _ { q } ( \\mathfrak { l } ) . \\end{align*}"}
-{"id": "5359.png", "formula": "\\begin{align*} \\chi _ { ( n - 1 , 1 ) } \\cdot \\chi _ { \\lambda } = C ( \\lambda ) \\chi _ { \\lambda } + \\sum \\chi _ { \\mu } \\end{align*}"}
-{"id": "5886.png", "formula": "\\begin{align*} A ^ { i j } \\xi _ { , i j } ^ { k } - 2 A ^ { i k } a _ { , i } + a B ^ { k } - \\xi _ { , i } ^ { k } B ^ { i } + \\xi ^ { i } B _ { , i } ^ { k } - \\lambda B ^ { k } = 0 \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "8424.png", "formula": "\\begin{align*} A _ { 1 } ( \\textbf { u } ) = \\left ( \\begin{array} { c c c } v _ { 1 } & \\rho & 0 \\\\ f ( \\textbf { u } ) & v _ { 1 } & 0 \\\\ 0 & 0 & v _ { 1 } \\end{array} \\right ) , \\ A _ { 2 } ( \\textbf { u } ) = \\left ( \\begin{array} { c c c } v _ { 2 } & 0 & \\rho \\\\ 0 & v _ { 2 } & 0 \\\\ f ( \\textbf { u } ) & 0 & v _ { 2 } \\end{array} \\right ) , \\end{align*}"}
-{"id": "9547.png", "formula": "\\begin{align*} X = \\bigcup _ { \\stackrel { \\scriptstyle i = 1 , 2 } { \\scriptstyle \\gamma = A \\cup B } } U _ { i , \\gamma } , \\end{align*}"}
-{"id": "5829.png", "formula": "\\begin{align*} \\ln F \\left ( t , S \\right ) = e ^ { - \\kappa t } \\ln S + \\left ( 1 - e ^ { - \\kappa t } \\right ) a ^ { \\ast } + \\frac { \\sigma ^ { 2 } } { 4 \\kappa } \\left ( 1 - e ^ { - 2 \\kappa t } \\right ) \\end{align*}"}
-{"id": "5346.png", "formula": "\\begin{align*} q _ i = \\prod _ { \\substack { 1 \\leq j \\leq n \\\\ p ( j ) - p ( j - 1 ) \\geq i } } j \\end{align*}"}
-{"id": "9625.png", "formula": "\\begin{align*} \\ddot \\psi _ { r e g } ( x , t ) = ( \\Delta - m ^ 2 ) \\psi _ { r e g } ( x , t ) - \\sum \\limits _ { 1 \\le j \\le n } \\ddot \\zeta _ j ( t ) g _ j ( x ) \\end{align*}"}
-{"id": "5080.png", "formula": "\\begin{align*} R ( u _ { a } \\otimes u _ { b } ) = \\left \\{ \\begin{array} { l l } q \\ , u _ { b } \\otimes u _ { a } & ( a > b ) \\\\ u _ { a } \\otimes u _ { a } & ( a = b ) \\\\ ( 1 - q ^ { 2 } ) u _ { a } \\otimes u _ { b } + q \\ , u _ { b } \\otimes u _ { a } & ( a < b ) . \\end{array} \\right . \\end{align*}"}
-{"id": "7782.png", "formula": "\\begin{align*} \\begin{aligned} & r _ 2 = 0 \\Leftrightarrow { } _ 2 x _ 1 \\circ { } _ 1 x _ 2 - i { } _ 2 x _ 1 \\circ { } _ 1 x _ 2 i = 0 \\Leftrightarrow z _ 1 w _ 1 + z _ i w _ i = 0 , \\ ; z _ i w _ 1 - z _ 1 w _ i = 0 . \\end{aligned} \\end{align*}"}
-{"id": "1845.png", "formula": "\\begin{align*} \\widetilde { F } _ { m , n } ( t ) = \\frac { 1 } { \\sqrt { n + 1 } } \\frac { 1 } { \\sqrt { m + 1 } } \\exp \\left ( - t \\left ( i \\log { \\frac { m + 1 } { n + 1 } } + \\frac { 1 } { 2 } \\log ^ 2 { \\frac { m + 1 } { n + 1 } } \\right ) \\right ) \\end{align*}"}
-{"id": "2026.png", "formula": "\\begin{align*} \\mathcal { A } f ( x ) & = ( \\gamma _ U x + \\gamma _ L ) \\alpha x ^ { - \\alpha - 1 } - \\frac { 1 } { 2 } ( x ^ 2 \\sigma _ U ^ 2 + 2 x \\sigma _ { U L } + \\sigma _ L ^ 2 ) \\alpha ( \\alpha + 1 ) x ^ { - \\alpha - 2 } \\\\ & = \\alpha x ^ { - \\alpha } \\left ( \\gamma _ U - \\frac { ( \\alpha + 1 ) \\sigma _ U ^ 2 } { 2 } \\right ) + O ( x ^ { - \\alpha - 1 } ) . \\end{align*}"}
-{"id": "7991.png", "formula": "\\begin{align*} H = F + L + \\sum n _ i E _ i \\end{align*}"}
-{"id": "7240.png", "formula": "\\begin{align*} u _ n ( Y , X ; \\{ w _ j \\} ) = ( u ^ \\lambda _ \\alpha ( Y , X ; \\{ w _ j \\} ) ) _ { \\lambda , \\alpha } \\in \\bigoplus _ { \\lambda = 1 } ^ r \\bigoplus _ { | \\alpha | = n + 1 } H ^ 1 ( Y , N _ \\lambda \\otimes N _ \\alpha ^ { - 1 } ) \\end{align*}"}
-{"id": "2025.png", "formula": "\\begin{align*} \\int _ { | z | > 1 } [ f ( x + x z ) - f ( x ) ] \\nu _ U ( \\d z ) = x ^ \\beta \\int _ { | z | > 1 } [ | 1 + z | ^ \\beta - 1 ] \\nu _ U ( \\d z ) + o ( x ^ \\beta ) . \\end{align*}"}
-{"id": "8784.png", "formula": "\\begin{align*} t E _ { s _ i \\lambda } & = T _ i ( \\langle \\lambda \\rangle _ { i + 1 } - \\langle \\lambda \\rangle _ { i } ) E _ \\lambda , \\lambda _ i < \\lambda _ { i + 1 } , \\\\ t _ n E _ { s _ n \\lambda } & = \\left [ T _ n + \\frac { 1 - t _ n + t _ n ( 1 - t _ 0 ) y _ n ( \\lambda ) ^ { - 1 } } { t _ 0 t _ n y _ n ( \\lambda ) ^ { - 2 } - 1 } \\right ] E _ \\lambda , \\lambda _ n < 0 \\end{align*}"}
-{"id": "2861.png", "formula": "\\begin{align*} T o t ( R e s ^ { \\bullet } ( C ) ) = \\int _ { \\underline { n } \\in \\Delta } F ^ c ( K ^ n ( C ) \\otimes \\overline { N ^ * } ( \\Delta ^ n ) ) . \\end{align*}"}
-{"id": "4721.png", "formula": "\\begin{align*} \\alpha = \\inf \\{ \\pi ^ { \\top } x \\ , | \\ W ^ { i _ 0 } \\} . \\end{align*}"}
-{"id": "616.png", "formula": "\\begin{align*} - \\Delta u + V ( x ) u - Q ( x ) | u | ^ { p - 2 } u = \\lambda u , x \\in \\R ^ N , \\end{align*}"}
-{"id": "2440.png", "formula": "\\begin{align*} \\| F _ { k , N } \\| _ { M _ 1 } & = \\left ( \\sum _ { l \\in \\widetilde { \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } } \\| f _ l ^ { \\alpha _ 1 } \\| ^ { q } _ { L ^ { p _ 1 } } \\right ) ^ { \\frac { 1 } { q } } \\\\ & \\sim | \\widetilde { \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } | ^ { 1 / q } 2 ^ { j n \\alpha _ 1 ( 1 - 1 / p _ 1 ) } \\\\ & \\sim 2 ^ { j n ( \\alpha _ 2 - \\alpha _ 1 ) / q } 2 ^ { j n \\alpha _ 1 ( 1 - 1 / p _ 1 ) } . \\end{align*}"}
-{"id": "6548.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = 0 } ^ m \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = 0 } ^ m \\big \\| _ { L ^ p ( D ) } \\le C \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( X ) } + C \\delta . \\end{align*}"}
-{"id": "2392.png", "formula": "\\begin{align*} \\begin{tabular} { l l } m i n i m i z e & $ \\hat { f } ( x ) + \\hat { g } ( z ) $ \\\\ s u b j e c t t o & $ A x + B z = c $ \\end{tabular} \\end{align*}"}
-{"id": "8800.png", "formula": "\\begin{align*} c = \\frac { \\int _ { \\Sigma } f _ i d v o l _ { \\Sigma } } { \\int _ { \\Sigma } f d v o l _ { \\Sigma } } \\ , . \\end{align*}"}
-{"id": "199.png", "formula": "\\begin{align*} \\mathbb { E } _ f ( \\hat { H } _ n ) - H = o \\Bigl ( \\frac { k ^ { 1 - \\epsilon } } { n ^ { 1 - \\epsilon } } \\Bigr ) , \\end{align*}"}
-{"id": "9800.png", "formula": "\\begin{align*} d _ { H } ( p , q ) = \\left \\{ \\int \\left ( \\sqrt { p } - \\sqrt { q } \\right ) ^ { 2 } d \\mu \\right \\} ^ { 1 / 2 } . \\end{align*}"}
-{"id": "6716.png", "formula": "\\begin{align*} h ( n ) = \\begin{cases} \\frac { 2 n } { a b } - 2 & \\mbox { i f } n \\equiv 0 \\bmod { a b } , \\\\ \\frac { 2 ( n - a ) } { a b } - 1 & \\mbox { i f } n \\equiv a \\bmod { a b } , \\\\ \\frac { 2 ( n - b ) } { a b } - 1 & \\mbox { i f } n \\equiv b \\bmod { a b } , \\\\ \\frac { 2 ( n - a - b ) } { a b } & \\mbox { i f } n \\equiv a + b \\bmod { a b } . \\end{cases} \\end{align*}"}
-{"id": "5609.png", "formula": "\\begin{align*} R _ { G } ( n ) = \\sum _ { m _ 1 + m _ 2 = n } \\Lambda ( m _ 1 ) \\Lambda ( m _ 2 ) , \\end{align*}"}
-{"id": "4162.png", "formula": "\\begin{align*} L ( s , S _ f ^ \\nu ) & = \\sum _ { n \\geq 1 } \\sum _ { h \\geq 0 } \\frac { a _ f ( n ) } { n ^ \\nu ( n + h ) ^ s } \\\\ & = L ^ \\nu ( s , f ) + \\frac { 1 } { 2 \\pi i } \\int _ { ( \\gamma ) } L ^ \\nu ( s - w , f ) \\zeta ( w ) \\frac { \\Gamma ( w ) \\Gamma ( s - w ) } { \\Gamma ( s ) } d w , \\end{align*}"}
-{"id": "8755.png", "formula": "\\begin{align*} \\widetilde { R } ( x ) = \\left ( \\left ( R \\left ( x \\right ) ^ { \\tau _ 1 } \\right ) { } ^ { - 1 } \\right ) { } ^ { \\tau _ 1 } , \\end{align*}"}
-{"id": "9953.png", "formula": "\\begin{align*} \\mathcal E _ { \\lambda , \\mu } ( w _ { \\mu , \\lambda } ) : = \\frac { 1 } { 2 } \\| w _ { \\mu , \\lambda } \\| ^ 2 - \\widetilde { \\mathcal E } _ { \\lambda , \\mu } ( w _ { \\mu , \\lambda } ) < \\frac { \\varrho _ { 0 , \\mu , \\lambda } ^ 2 } { 2 } - \\widetilde { \\mathcal E } _ { \\lambda , \\mu } ( u ) \\end{align*}"}
-{"id": "9935.png", "formula": "\\begin{align*} 0 = E F + \\kappa ^ 2 K '^ 2 & = - \\frac { 1 } { 4 } ( 1 + b ^ 2 ) ( x _ 2 ^ 2 + x _ 3 ^ 2 ) + 4 \\kappa ^ 2 b ^ 2 x _ 1 ^ 2 \\\\ & = - \\frac { 1 } { 4 } ( 1 + b ^ 2 ) ( x _ 2 ^ 2 + x _ 3 ^ 2 ) - \\frac { ( 1 + b ^ 2 ) ^ 2 } { 4 } x _ 1 ^ 2 \\\\ & = - \\frac { 1 } { 4 } ( 1 + b ^ 2 ) \\big ( x _ 2 ^ 2 + x _ 3 ^ 2 + ( 1 + b ^ 2 ) x _ 1 ^ 2 \\big ) . \\end{align*}"}
-{"id": "2368.png", "formula": "\\begin{align*} \\nabla F ( x ) = P x - \\nabla ^ 2 f _ 1 ( x ) \\nabla f _ 2 ( \\nabla f _ 1 ( x ) ) = P x - P S _ 2 ( S _ 1 x ) = P ( x - S _ 2 S _ 1 x ) . \\end{align*}"}
-{"id": "9426.png", "formula": "\\begin{align*} \\Gamma _ 1 & = \\Sigma _ 1 ^ 0 \\supsetneq \\Sigma _ 1 ^ 1 \\supsetneq \\dots \\supsetneq \\Sigma _ 1 ^ n \\\\ \\Gamma _ 2 & = \\Sigma _ 2 ^ 0 \\supsetneq \\Sigma _ 2 ^ 1 \\supsetneq \\dots \\supsetneq \\Sigma _ 2 ^ n , \\end{align*}"}
-{"id": "214.png", "formula": "\\begin{align*} - \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\log ( 1 - s ) \\mathrm { B } _ { k , n - k } ( s ) \\ , d s \\leq \\frac { n - 1 } { n - k - 1 } \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\mathrm { B } _ { k , n - k - 1 } ( s ) \\ , d s = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "9521.png", "formula": "\\begin{align*} \\delta | d \\phi | ^ 2 _ E \\mathrm { d v o l } _ { g _ E } & = 2 d \\delta \\phi \\wedge * _ E d \\phi \\\\ & = d ( \\ldots ) - 2 \\delta \\phi ( d * _ E d \\phi ) \\\\ & = d ( \\ldots ) - 2 \\delta \\phi ( \\Delta _ E \\phi ) \\mathrm { d v o l } _ { g _ E } . \\end{align*}"}
-{"id": "5013.png", "formula": "\\begin{align*} \\forall \\ , s \\in G , \\varlimsup _ n \\ , \\frac { | s F _ n \\Delta F _ n | } { | F _ n | } = 0 , \\end{align*}"}
-{"id": "5388.png", "formula": "\\begin{align*} e ( H ) \\leq \\begin{cases} \\left ( 1 - \\frac { 1 } { c } \\right ) \\frac { v ( H ) ^ 2 } { 2 } & v ( H ) \\leq n \\sqrt { \\frac { c } { c - 1 } } \\ , , \\\\ \\frac { n ^ 2 } { 2 } & n \\sqrt { \\frac { c } { c - 1 } } < v ( H ) \\leq \\frac { 5 n } { 4 } \\ , , \\\\ \\frac { n } { 2 } \\left ( v ( H ) - \\frac { n } { 4 } \\right ) & \\frac { 5 n } { 4 } < v ( H ) \\ , . \\end{cases} \\end{align*}"}
-{"id": "5677.png", "formula": "\\begin{align*} \\left ( 1 - \\left ( \\frac { 1 } { x } - 1 \\right ) e _ 0 \\right ) ^ { - 1 } = x \\ , \\Big ( 1 - ( 1 - x ) ( e _ 0 + 1 ) \\Big ) ^ { - 1 } = ( y + 1 ) \\sum _ { n = 0 } ^ \\infty ( - y ) ^ n ( e _ 0 + 1 ) ^ n \\ ; , \\end{align*}"}
-{"id": "1153.png", "formula": "\\begin{align*} \\dot x _ j & = - b _ 0 ( x _ j ) , & - \\epsilon [ b _ { 0 , x } ] ( x _ j ) & = [ b _ { - 1 , x } ] ( x _ j ) , & \\frac 1 2 [ b _ { - 1 , x } ] ( x _ j ) & = \\epsilon m _ i b _ { - 1 } ( x _ j ) \\\\ \\dot m _ j & = m _ j \\langle b _ { 0 , x } \\rangle ( x _ j ) , & - \\epsilon [ b _ { 0 , x x } ] ( x _ j ) & = [ b _ { - 1 , x x } ] ( x _ j ) , & \\frac 1 2 [ b _ { - 1 , x x } ] ( x _ j ) & = \\epsilon m _ j \\langle b _ { - 1 , x } \\rangle ( x _ j ) , \\end{align*}"}
-{"id": "896.png", "formula": "\\begin{align*} \\zeta ( s ) = O \\left ( t ^ { ( 1 - \\sigma ) / 2 } ( \\log t ) ^ 5 \\right ) , \\mbox { u n i f o r m l y i n } \\ 0 \\leq \\sigma \\leq 1 . \\end{align*}"}
-{"id": "2973.png", "formula": "\\begin{align*} \\sum _ { i + j = n } f _ i ^ { \\lambda _ j ( x , y ) } = \\sum _ { i + j = n } [ f _ i ^ x , f _ j ^ y ] . \\end{align*}"}
-{"id": "8760.png", "formula": "\\begin{align*} L = \\frac { 1 - t } { 2 } T ' ( 1 ) = ( 1 - t ) \\left ( \\frac { 1 } { 2 } K _ 0 ' ( 1 ) + \\sum _ { i = 1 } ^ { n - 1 } \\check { R } _ { i , i + 1 } ' ( 1 ) - \\frac { 1 } { 2 } K _ n ' ( 1 ) \\right ) . \\end{align*}"}
-{"id": "8026.png", "formula": "\\begin{align*} c ^ { \\gcd ( L _ 1 , L ) } = c ^ { \\gcd ( L _ 1 , \\ldots , L _ n ) } , \\end{align*}"}
-{"id": "7723.png", "formula": "\\begin{align*} J = \\left [ \\frac M { 1 - \\varepsilon } \\right ] . \\end{align*}"}
-{"id": "1031.png", "formula": "\\begin{align*} R _ d ( y ) ^ { d - 1 } R _ d ( z ) + S _ d ( z ) ^ d ~ = ~ 0 . \\end{align*}"}
-{"id": "8718.png", "formula": "\\begin{align*} \\nabla ^ G _ { \\xi } v ( t , x ) = \\int _ { t } ^ { T } \\nabla ^ G _ { \\xi } R _ { s - t } \\Big [ e ^ { - ( s - t ) { A } } G B ( s , \\cdot ) + e ^ { - ( s - t ) { A } } L ( s , \\cdot ) B \\left ( s , \\cdot \\right ) \\big ] ( x ) d s , \\ ; \\ ; \\ ; ( t , x ) \\in [ 0 , T ] \\times H , \\end{align*}"}
-{"id": "1321.png", "formula": "\\begin{align*} \\Bigl \\vert \\sum _ { \\rho } z ^ { - \\rho } \\Gamma ( \\rho ) \\Bigr \\vert & \\ll N + \\vert z \\vert ^ { 1 / 2 } \\log ^ 2 ( 2 N \\vert y \\vert ) . \\end{align*}"}
-{"id": "1185.png", "formula": "\\begin{align*} z m _ i \\dot x _ i = - \\left ( \\frac 1 2 [ b _ x ] ( x _ i ) + z m _ i b ( x _ i ) \\right ) . \\end{align*}"}
-{"id": "9702.png", "formula": "\\begin{align*} \\lim _ { y \\to 0 ^ + } \\frac { \\Gamma _ 1 ( y ) } { \\alpha \\log ( 1 / y ) ^ { ( \\alpha + 1 ) / \\alpha } y } = 1 , \\Gamma _ 1 \\in _ 0 ( 1 ) . \\end{align*}"}
-{"id": "9803.png", "formula": "\\begin{align*} \\log B F _ { k l } = \\log \\frac { f _ { k } ( \\mathbf { y } | \\boldsymbol { \\theta } _ { k } ) } { f _ { l } ( \\mathbf { y } | \\boldsymbol { \\theta } _ { l } ) } + \\log \\frac { \\pi _ { k } ( \\boldsymbol { \\theta } _ { k } ) } { \\pi _ { l } ( \\boldsymbol { \\theta } _ { l } ) } - \\log \\frac { \\pi ( \\boldsymbol { \\theta } _ { k } | \\mathbf { y } ) } { \\pi ( \\boldsymbol { \\theta } _ { l } | \\mathbf { y } ) } . \\end{align*}"}
-{"id": "5912.png", "formula": "\\begin{align*} [ f ] _ { B ^ { s } _ { p , q } } = \\sum _ { i = 1 } ^ d \\Big ( \\int _ { \\R ^ d } \\frac { \\dd h } { | h | ^ { d + s q } } \\Big ( \\int _ { \\R ^ d } | \\partial _ { x _ i } f ( x + h ) - \\partial _ { x _ i } f ( x ) | ^ p \\dd x \\Big ) ^ { q / p } \\Big ) ^ { 1 / q } < \\infty \\ , . \\end{align*}"}
-{"id": "5384.png", "formula": "\\begin{align*} \\tilde \\Delta _ r \\otimes \\mathbb { C } = \\mathbb { C } ^ { 2 m } \\otimes \\Delta _ r . \\end{align*}"}
-{"id": "9876.png", "formula": "\\begin{align*} \\widehat { W } ( t ) = W ( t ) + { \\epsilon } ^ { - 1 / 2 } \\int _ 0 ^ t v ( s ) d s , \\end{align*}"}
-{"id": "2854.png", "formula": "\\begin{align*} \\eta ^ * = t r i v _ P : \\mathcal { C } \\leftrightarrows P - C o g ^ { c o n i l } ( \\mathcal { C } ) : \\Omega _ P \\end{align*}"}
-{"id": "5272.png", "formula": "\\begin{align*} \\int _ { { 1 } / { 2 } + i T } ^ { 2 + i T } \\zeta ( s ) d s = \\int _ { { 1 } / { 2 } } ^ { 2 } \\zeta ( \\sigma + i T ) d \\sigma = O \\left ( T ^ { 1 / 4 + \\varepsilon } \\right ) . \\end{align*}"}
-{"id": "3526.png", "formula": "\\begin{align*} \\| u \\| _ { C ^ { k , \\alpha } _ { \\phi , \\varphi } ( \\Omega ) } = \\sup _ { x \\in \\Omega } \\left ( \\sum _ { j = 0 } ^ k \\varphi ( x ) \\phi ^ j ( x ) \\| \\nabla _ g ^ j u \\| _ { C ^ 0 ( B _ { \\phi ( x ) } ( x ) ) } + \\varphi ( x ) \\phi ^ { k + \\alpha } ( x ) [ \\nabla ^ k _ g u ] _ { 0 , \\alpha ; B _ { \\phi ( x ) } ( x ) } \\right ) . \\end{align*}"}
-{"id": "8320.png", "formula": "\\begin{align*} u = \\frac { b _ { - \\lambda p ^ m } } { \\pi ^ { \\lambda p ^ m } } + \\frac { b _ { - \\lambda p ^ m + 1 } } { \\pi ^ { \\lambda p ^ m - 1 } } + \\cdots + \\frac { b _ { - 1 } } { \\pi } + b _ 0 + b _ 1 \\pi + \\cdots \\in k _ { \\P } \\cong k ( \\P ) ( ( \\pi ) ) . \\end{align*}"}
-{"id": "5124.png", "formula": "\\begin{align*} q \\ , F _ { \\ldots , z _ { i } , z _ { i + 1 } , \\ldots } ^ { \\ldots , \\mu _ { i } , \\mu _ { i + 1 } , \\ldots } = f ( z _ { i } , z _ { i + 1 } ) F _ { \\ldots , z _ { i + 1 } , z _ { i } , \\ldots } ^ { \\ldots , \\mu _ { i + 1 } , \\mu _ { i } , \\ldots } - g ( z _ { i + 1 } , z _ { i } ) F _ { \\ldots , z _ { i } , z _ { i + 1 } , \\ldots } ^ { \\ldots , \\mu _ { i + 1 } , \\mu _ { i } , \\ldots } , \\end{align*}"}
-{"id": "6844.png", "formula": "\\begin{align*} \\| u \\| _ { L _ t ^ { q , \\alpha } X ( I \\times \\R ^ d ) } : = \\bigl \\| \\ , \\| u ( t ) \\| _ { X } \\bigr \\| _ { L _ t ^ { q , \\alpha } ( I ) } < \\infty . \\end{align*}"}
-{"id": "7584.png", "formula": "\\begin{align*} \\mathcal A _ { k - 1 , 2 } = \\mathcal A _ { k 1 } \\cup \\mathcal A _ { k 2 } , \\mathcal A _ { k 1 } \\cap \\mathcal A _ { k 2 } = \\emptyset , \\end{align*}"}
-{"id": "4130.png", "formula": "\\begin{gather*} ( p , q ) = ( 2 - 6 u , 1 + 6 u ) , ( p , q ) = ( 4 - 6 u , - 1 + 6 u ) , \\\\ ( p , q ) = ( 1 - 6 u , 2 + 6 u ) , ( p , q ) = ( 5 - 6 u , - 2 + 6 u ) , \\\\ ( p , q ) = ( 1 - 2 u , 1 + 2 u ) , \\end{gather*}"}
-{"id": "524.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = \\iint \\frac { ( z - w ) h ^ { \\rm g e o } _ { 2 2 } ( z , w ) } { z w ( z w - 1 ) } \\frac { 1 } { 1 - c z } \\frac { 1 } { 1 - c w } \\frac { \\dd z } { z ^ u } \\frac { \\dd w } { w ^ v } , \\end{align*}"}
-{"id": "6829.png", "formula": "\\begin{align*} F \\simeq \\oplus _ { \\lambda \\in Q ' _ 0 } F _ \\lambda \\ , , F _ \\lambda : = p ^ * E _ \\lambda \\otimes q ^ * H _ \\lambda , \\end{align*}"}
-{"id": "4994.png", "formula": "\\begin{align*} \\frac { h _ { H } ( f ^ { n } ( P ) ) } { \\delta ^ { n } } = & - \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { - 1 - i } c _ { 1 } h _ { Z _ { 1 } } ( p ^ { - 1 } ( f ^ { i } ( P ) ) ) + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { - 1 - i } h _ { E } ( p ^ { - 1 } ( f ^ { i } ( P ) ) ) \\\\ & + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { - 1 - i } c _ { 1 } h _ { E _ { 1 } ' } ( f ^ { i } ( P ) ) + \\sum _ { i = 0 } ^ { n - 1 } \\delta ^ { - i } h _ { N } ( f ^ { i } ( P ) ) + h _ { H } ( P ) . \\end{align*}"}
-{"id": "6676.png", "formula": "\\begin{align*} \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } F & = \\sum _ { i = 1 } ^ { p } \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } ( D _ i - d _ i ) ^ 2 = \\sum _ { i = 1 } ^ { p } 2 ( D _ i - d _ i ) \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } ( D _ i - d _ i ) \\\\ & = \\sum _ { i = 1 } ^ { p } 2 ( D _ i - d _ i ) ( - \\lambda ) ( D _ i - d _ i ) = ( - 2 \\lambda ) \\sum _ { i = 1 } ^ { p } ( D _ i - d _ i ) ^ 2 \\\\ & = ( - 2 \\lambda ) F . \\end{align*}"}
-{"id": "1667.png", "formula": "\\begin{gather*} \\int _ { \\R ^ d } \\| D _ v { \\psi } ( \\cdot , v ) \\| _ { H ^ s _ p ( \\R ^ { d } ) } ^ p \\dd v = \\int _ { \\R ^ d } \\dd v \\int _ { \\R ^ d } | D _ v \\ , G _ { \\lambda } h _ s ( x , v ) | ^ p \\ , \\dd x \\\\ \\le \\frac { C } { ( \\lambda ) ^ { p / 2 } } \\| h _ s \\| _ { L ^ p ( \\R ^ { 2 d } ) } ^ p = \\frac { C } { ( \\lambda ) ^ { p / 2 } } \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ d _ x ) ) } ^ p \\end{gather*}"}
-{"id": "8528.png", "formula": "\\begin{align*} \\rho ( b ' X _ { b ^ { \\ast } } ( P ) ) = \\displaystyle \\sum _ { r \\in L ( k ) , a \\in _ { k } T } [ b ' r a ] X _ { [ b r a ] ^ { \\ast } } ( \\rho ( P ) ) \\end{align*}"}
-{"id": "1935.png", "formula": "\\begin{align*} \\left | \\frac { b _ i ( \\tau ) } { \\tau } - 2 p _ i \\right | & = \\left | \\frac { b _ i ( \\tau _ 0 ) - 2 p _ i \\tau _ 0 + \\int _ { \\tau _ 0 } ^ \\tau \\left ( \\frac { d b _ i ( s ) } { d s } - 2 p _ i \\right ) d s } { \\tau } \\right | \\\\ & \\leq \\frac { \\left | b _ i ( \\tau _ 0 ) - 2 p _ i \\tau _ 0 \\right | } { \\tau } + \\frac { \\int _ { \\tau _ 0 } ^ \\tau \\left | \\frac { d b _ i ( s ) } { d s } - 2 p _ i \\right | d s } { \\tau } \\\\ & \\leq \\varepsilon + \\varepsilon . \\end{align*}"}
-{"id": "2609.png", "formula": "\\begin{align*} \\tau : C ^ * ( A , N ) & \\to C ^ n ( \\operatorname { T o t } ^ * ( D ^ { * , * } ( A , N ) ) ) \\\\ \\alpha & \\mapsto \\oplus _ { p = 1 } ^ { n - 1 } \\alpha _ p , \\end{align*}"}
-{"id": "5514.png", "formula": "\\begin{align*} H ( x ) = \\int _ I x h , \\ \\ \\ \\ x \\in \\Lambda _ { \\varphi , w } , \\end{align*}"}
-{"id": "1127.png", "formula": "\\begin{align*} \\mathcal { R } _ k ^ \\mathrm { U L } [ \\iota ] = K \\log _ 2 ( 1 + \\gamma _ k ^ \\mathrm { B } [ \\iota ] ) + ( T _ \\mathrm { d } - K ) \\log _ 2 ( 1 + \\gamma _ k ^ \\mathrm { C } [ \\iota ] ) \\end{align*}"}
-{"id": "23.png", "formula": "\\begin{align*} Y _ { h _ L } = \\{ y \\in Y _ 0 ; h _ { y } \\not \\equiv + \\infty \\} . \\end{align*}"}
-{"id": "2250.png", "formula": "\\begin{align*} p ( x ) = \\frac { C ( \\vec { \\varepsilon } ) } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } e ^ { \\varepsilon _ { q } x ^ { q } - \\varepsilon _ { p } x ^ { p } } , \\end{align*}"}
-{"id": "7050.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ 1 ] } } ( 6 ) = { { \\bf { v } } ^ { [ 1 ] } } , { { \\bf { X } } ^ { [ 2 ] } } ( 6 ) = { { \\bf { v } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "7947.png", "formula": "\\begin{align*} \\int _ { \\Omega } ( 1 - \\sigma ) D ^ 2 u : D ^ 2 \\varphi + \\sigma \\Delta u \\Delta \\varphi d x = \\lambda \\int _ { \\Omega } u \\varphi d x \\ , , \\end{align*}"}
-{"id": "6032.png", "formula": "\\begin{gather*} g _ I : = \\left [ - \\tfrac { 3 } { 2 } \\big ( I ^ 2 + 1 \\big ) y ^ 2 + 2 I p ^ 2 - \\tfrac { 1 } { 2 } q ^ 2 \\right ] \\ ! d x ^ 2 - 4 I p \\ , d x \\ , d y + q \\ , d x \\ , d p \\\\ \\hphantom { g _ I : = } { } - 3 p \\ , d x \\ , d q - 3 \\ , d x \\ , d z - 3 I \\ , d y ^ 2 + 3 \\ , d y \\ , d q - 2 \\ , d p ^ 2 . \\end{gather*}"}
-{"id": "1001.png", "formula": "\\begin{align*} H _ { n , m } = t \\left ( \\tau ( u ) + \\omega ^ 2 - u ^ 2 - \\eta ^ { - 2 } - u \\tau ' ( 0 ) - \\frac { \\tau ' ( 0 ) ^ 2 } { 4 } \\right ) , \\end{align*}"}
-{"id": "1512.png", "formula": "\\begin{align*} & K ( x ) = 1 + k ( x ) \\ , \\Big ( b _ 0 + x \\ , b _ 0 ^ + + \\frac 1 x \\ , b _ 0 ^ - \\Big ) , \\\\ & \\mbox { w i t h } k ( x ) = \\frac { \\left ( x ^ 2 - 1 \\right ) \\left ( \\alpha + \\gamma \\right ) } { \\left ( \\gamma x + \\alpha \\right ) \\left ( ( \\alpha + \\gamma ) ( x - 1 ) + ( q - 1 ) x \\right ) } \\ , \\end{align*}"}
-{"id": "222.png", "formula": "\\begin{align*} R _ 3 ' & = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ 0 ^ \\frac { a _ n } { n - 1 } \\log \\biggl ( \\frac { V _ d f ( x ) h _ x ^ { - 1 } ( s ) ^ d } { s } \\biggr ) \\mathrm { B } _ { k , n - k } ( s ) \\ , d s \\ , d x \\\\ & = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\biggr \\} \\biggr ) \\end{align*}"}
-{"id": "6793.png", "formula": "\\begin{align*} \\int _ X \\langle T _ 1 \\wedge \\cdots \\wedge T _ n \\rangle = \\int _ X \\langle T _ { 1 , \\min } \\wedge \\cdots \\wedge T _ { n , m i n } \\rangle . \\end{align*}"}
-{"id": "3622.png", "formula": "\\begin{align*} u _ { n + 1 } = h ( u _ n ) , n = 0 , 1 , 2 , \\dots \\end{align*}"}
-{"id": "5135.png", "formula": "\\begin{align*} & X _ { I V } = \\sum _ { t = 2 } ^ { a + 1 } \\sum _ { a + 1 \\ge p ( t ) > \\cdots > p ( 2 ) > p ( 1 ) = 1 } \\sum _ { \\ell ( t ) \\in J _ { p ( t ) } } \\cdots \\sum _ { \\ell ( 1 ) \\in J _ { p ( 1 ) } } K _ { t } ( w ; z _ { \\ell ( 1 ) } , \\ldots , z _ { \\ell ( t ) } ) \\\\ & { } \\times \\prod _ { 1 \\le s \\le t } ^ { \\curvearrowleft } Z _ { \\ell ( s ) } ^ { J _ { p ( s + 1 ) - 1 } \\cup \\cdots \\cup J _ { p ( s ) } } ( \\vec { z } ) u ( a + 2 , ( a + 1 ) ^ { k _ { a + 1 } } , \\ldots , 1 ^ { k _ { 1 } } ) , \\end{align*}"}
-{"id": "5807.png", "formula": "\\begin{align*} [ \\alpha _ { 2 } ] = \\{ \\alpha _ { 1 } + \\alpha _ { 2 } , \\ \\alpha _ { 2 } \\} , [ \\alpha _ { i } ] = \\{ \\alpha _ { i } \\} \\quad \\mathrm { f o r } \\ i > 2 . \\end{align*}"}
-{"id": "1685.png", "formula": "\\begin{align*} A _ t = \\int _ 0 ^ t 1 _ { \\{ Z _ s \\not = Y _ s \\} } \\ ; \\frac { \\| [ D U ( Z _ s ) - D U ( Y _ s ) ] { R \\ , } \\Big \\| ^ 2 _ { H S } } { | Z _ s - Y _ s | ^ 2 } \\ \\dd s \\ , , \\end{align*}"}
-{"id": "6220.png", "formula": "\\begin{align*} X ^ { ( 0 ) } ( s _ { 0 } ) = & \\ \\mathsf { Q } , \\\\ X ^ { ( 1 ) } ( s _ { 0 } ) = & \\ \\mathcal { P } _ { 1 } X ^ { ( 0 ) } ( s _ { 0 } ) , \\mathrm { a n d } \\\\ X ^ { ( j ) } ( s _ { 0 } ) = & \\ \\mathcal { P } _ { 1 } X ^ { ( j - 1 ) } ( s _ { 0 } ) + \\mathcal { P } _ { 2 } X ^ { ( j - 2 ) } ( s _ { 0 } ) \\end{align*}"}
-{"id": "1579.png", "formula": "\\begin{align*} C ^ { x } \\left ( x \\right ) = - \\left ( \\kappa \\left ( \\mu - \\lambda - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } - \\frac { 1 } { 2 } \\sigma \\sigma _ { , x } \\right ) \\end{align*}"}
-{"id": "1788.png", "formula": "\\begin{align*} \\theta ( \\pi - [ \\sigma _ x ( a _ 1 ) ] ) = 0 \\end{align*}"}
-{"id": "8702.png", "formula": "\\begin{gather*} \\langle \\widetilde Y _ { \\tau } ^ { t , x } , h \\rangle = \\langle \\widetilde Y _ { 0 } ^ { t , x } , h \\rangle - \\int _ 0 ^ \\tau \\langle e ^ { - s { A } } G B ( s , \\Xi ^ { t , x } _ s ) , h \\rangle \\ , d s - \\int _ 0 ^ \\tau \\langle e ^ { - s { A } } Z _ s ^ { t , x } \\ , B ( s , \\Xi ^ { t , x } _ s ) , h \\rangle \\ , d s \\\\ + \\int _ 0 ^ \\tau \\langle d W _ s , ( Z ^ { t , x } _ { s } ) ^ * e ^ { - s A ^ * } h \\rangle _ U \\end{gather*}"}
-{"id": "3143.png", "formula": "\\begin{align*} z _ i : = \\begin{bmatrix} \\Lambda _ t ( \\lambda _ 0 ) \\otimes I _ n \\\\ \\widehat { N } _ t ( \\lambda _ 0 ) ( \\lambda _ 0 B + A ) ( \\Lambda _ t ( \\lambda _ 0 ) \\otimes I _ n ) \\end{bmatrix} x _ i \\in \\mathcal { N } _ r ( \\mathcal { L } ( \\lambda _ 0 ) ) . \\end{align*}"}
-{"id": "3901.png", "formula": "\\begin{align*} \\dot { X } = \\begin{cases} C _ L X + e _ 3 \\mu + e _ 1 x y \\ ; , & x \\le 0 \\\\ C _ R X + e _ 3 \\mu \\ ; , & x \\ge 0 \\end{cases} \\ ; . \\end{align*}"}
-{"id": "1577.png", "formula": "\\begin{align*} \\Delta F + \\left ( \\kappa \\left ( \\mu - \\lambda - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } - \\frac { 1 } { 2 } \\sigma \\sigma _ { , x } \\right ) F _ { , x } - F _ { , t } = 0 , \\end{align*}"}
-{"id": "7703.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\left \\{ \\sqrt { x _ n } + \\gamma _ { n + 1 } , 0 \\right \\} , x _ 1 = 1 . \\end{align*}"}
-{"id": "4731.png", "formula": "\\begin{align*} \\beta _ 1 y _ 1 + \\beta _ 2 y _ 2 & = \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 , \\\\ \\beta _ 1 y _ 1 - \\beta _ 2 y _ 2 & = \\beta _ 1 \\alpha _ 1 - \\beta _ 2 \\alpha _ 2 . \\end{align*}"}
-{"id": "5410.png", "formula": "\\begin{align*} F ( s ) & = \\sum _ { j = 1 } ^ \\infty \\sum _ { n = 2 } ^ \\infty n ^ { j - 1 } ( n ^ j ) ^ { - s } = \\sum _ { n = 2 } ^ \\infty g _ n n ^ { - s } , \\end{align*}"}
-{"id": "4108.png", "formula": "\\begin{align*} C \\ : : \\ : x ^ p y ^ q ( b y + c z ) ^ r - z ^ { p + q + r } = 0 \\end{align*}"}
-{"id": "3029.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( n ) - { y ^ { [ j ] } } ( { t _ 1 } ) = \\sum \\limits _ { i = 1 } ^ M { { { \\bf { h } } ^ { [ j i ] } } ( n ) { \\bf { V } } _ j ^ { [ i ] } ( n ) { { \\bf { s } } ^ { [ j i ] } } } - \\sum \\limits _ { i = 1 } ^ M { { { \\bf { h } } ^ { [ j i ] } } ( { t _ 1 } ) { { \\bf { s } } ^ { [ j i ] } } } . \\end{align*}"}
-{"id": "7936.png", "formula": "\\begin{align*} \\rho ( v _ { i , \\ell } ) = v _ { n + 1 - i , m + 1 - \\ell } \\end{align*}"}
-{"id": "8996.png", "formula": "\\begin{align*} ( ~ ^ { A B C } ~ D _ b ^ \\alpha f ) ( t ) = ( ~ ^ { A B R } D _ b ^ \\alpha f ) ( t ) - \\frac { B ( \\alpha ) } { 1 - \\alpha } f ( b ) E _ \\alpha ( - \\frac { \\alpha } { 1 - \\alpha } ( b - t ) ^ \\alpha ) \\end{align*}"}
-{"id": "5337.png", "formula": "\\begin{align*} Z ( \\lambda , u ) = P ( u ) + Q ( N _ { f } ( u ) ) - \\frac { B _ { \\varphi , b } ( P u ) } { T } + H \\left ( \\varphi ^ { - 1 } \\left [ \\lambda \\frac { \\Psi B _ { \\varphi , b } ( P u ) } { T } + \\varphi ( P u ) \\right ] \\right ) , \\end{align*}"}
-{"id": "3053.png", "formula": "\\begin{align*} Z ^ u _ n = \\sum _ { | v | = n , v > u } ( V ( v ) - V ( u ) ) e ^ { V ( u ) - V ( v ) } Z ^ u _ \\infty = \\liminf _ { n \\to \\infty } Z ^ u _ n . \\end{align*}"}
-{"id": "4447.png", "formula": "\\begin{align*} F _ x ( u ) : = \\exp \\{ - u f ( x ) e ^ { \\Psi ( k ) } \\} \\sum _ { j = k } ^ \\infty \\frac { 1 } { j ! } \\bigl \\{ u f ( x ) e ^ { \\Psi ( k ) } \\bigr \\} ^ j = e ^ { - \\lambda _ { x , u } } \\sum _ { j = k } ^ \\infty \\frac { \\lambda _ { x , u } ^ j } { j ! } , \\end{align*}"}
-{"id": "2036.png", "formula": "\\begin{align*} \\frac { 1 } { D } \\frac { d } { d D } \\left ( D ^ 2 ( \\bar { \\lambda } _ 2 ( n , D , K ) - \\bar { \\lambda } _ 1 ( n , D , K ) ) \\right ) = \\frac { d \\lambda _ 2 ( \\tilde { L } _ c ) } { d c } \\biggr | _ { c = 1 } - \\frac { d \\lambda _ 1 ( \\tilde { L } _ c ) } { d c } \\biggr | _ { c = 1 } . \\end{align*}"}
-{"id": "9147.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p k _ i \\leq t m + d ^ 2 m / ( 2 d + 1 ) = \\frac { t ( 2 d + 1 ) + d ^ 2 } { 2 d + 1 } m , \\end{align*}"}
-{"id": "1707.png", "formula": "\\begin{align*} d ( r ) : = r ^ { 1 / H } L ( r ) ^ { 1 / H } . \\end{align*}"}
-{"id": "9415.png", "formula": "\\begin{align*} I _ i = \\sum _ { \\vec e x _ i } I ( \\vec e ) . \\end{align*}"}
-{"id": "7515.png", "formula": "\\begin{align*} \\gamma = \\max \\{ \\alpha , 1 - \\alpha \\} . \\end{align*}"}
-{"id": "6838.png", "formula": "\\begin{align*} J ( t ) = e ^ { i t \\Delta } x e ^ { - i t \\Delta } = M ( t ) 2 i t \\nabla M ( - t ) . \\end{align*}"}
-{"id": "262.png", "formula": "\\begin{align*} S _ 4 = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log ^ 2 \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) \\ , d s \\ , d x = o ( n ^ { - ( 3 - \\epsilon ) } ) . \\end{align*}"}
-{"id": "1894.png", "formula": "\\begin{align*} { I } : = \\{ ( \\beta , b ) : \\overline { B } \\geq b \\geq \\beta > \\underline { B } \\} \\times ( 0 , \\overline { C } ] \\times [ \\underline { C ' } , \\infty ) \\times ( 0 , \\overline { C } _ X ] \\times ( 0 , \\overline { C } _ f ] . \\end{align*}"}
-{"id": "9691.png", "formula": "\\begin{align*} g ( x _ { L , \\epsilon } ( t ) ) & = x _ 1 ( \\epsilon ) \\exp \\left ( - C _ 1 ( \\epsilon ) \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s \\right ) , \\\\ g ( x _ { U , \\epsilon } ( t ) ) & = x _ 2 ( \\epsilon ) \\exp \\left ( - C _ 2 ( \\epsilon ) \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s \\right ) . \\end{align*}"}
-{"id": "9159.png", "formula": "\\begin{align*} \\det ( a _ { j } ^ { i } ) _ { i , j = 1 } ^ { n } = \\prod _ { j = 1 } ^ { n } \\prod _ { k = j + 1 } ^ { n } ( a _ k - a _ j ) \\prod _ { j = 1 } ^ { n } a _ j . \\end{align*}"}
-{"id": "1717.png", "formula": "\\begin{gather*} \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & \\Sigma & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "8616.png", "formula": "\\begin{align*} \\pi _ { b e _ { k } } \\beta \\psi = \\overline { \\overline { N } } ( b ) \\end{align*}"}
-{"id": "2918.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) = L ^ \\perp _ \\pi ( X ) \\ , s _ \\lambda ( X ) \\quad \\mbox { a n d } s ^ { * ( \\pi ) } _ \\lambda ( X ) = ( - 1 ) ^ { | \\lambda | } \\ , L ^ \\perp _ { \\pi } ( X ) \\ , s _ { \\lambda ' } ( X ) \\ , , \\end{align*}"}
-{"id": "3450.png", "formula": "\\begin{align*} \\frac { \\partial V } { \\partial s } + \\Psi ( s , z , V , \\nabla V , \\nabla ^ 2 V ) = 0 . V ( T , z ) = V _ 0 ( z ) , \\end{align*}"}
-{"id": "5580.png", "formula": "\\begin{align*} - v _ { x x } = z \\rho v , v _ t = ( - \\frac 1 2 b _ x + \\beta ) v + b v _ x . \\end{align*}"}
-{"id": "3485.png", "formula": "\\begin{align*} I _ 2 = \\int _ { \\tilde { \\Omega } } h ( x ) \\phi ( x ' , x _ n ) ( 1 - \\varphi _ \\delta ( x _ n ) ) d x \\to _ { \\delta \\to 0 } \\int _ { \\tilde { \\Omega } } h ( x ) \\phi ( x ' , x _ n ) d x . \\end{align*}"}
-{"id": "3109.png", "formula": "\\begin{align*} B _ { k k } : = P _ { d - 2 k + 2 } - \\sum _ { \\substack { i + j = 2 k \\\\ i < j } } \\left ( B _ { i j } + B _ { i j } ^ T \\right ) - \\sum _ { \\substack { i + j = 2 k - 1 \\\\ i < j } } \\left ( A _ { i j } + A _ { i j } ^ T \\right ) , \\end{align*}"}
-{"id": "7490.png", "formula": "\\begin{align*} r _ d ( s , t ) = \\frac { \\omega ( d ) } { 2 } \\sum _ { i = 0 } ^ { \\min \\{ s - 1 , t - 1 \\} } \\frac { ( s + t - i - 2 ) ! } { i ! ( s - i - 1 ) ! ( t - i - 1 ) ! } ( 2 \\omega ( d ) - 1 ) ^ i ( 1 - \\omega ( d ) ) ^ { s + t - 2 i - 2 } \\end{align*}"}
-{"id": "4163.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { ( \\sigma ) } L ( s , S _ f ^ \\nu ) V ( s ) X ^ s d s & = \\sum _ { n \\geq 1 } S _ f ^ \\nu ( n ) v ( \\tfrac { n } { X } ) \\\\ & = \\sum _ { n \\leq X } S _ f ^ \\nu ( n ) + \\sum _ { X < n \\leq X + X / Y } S _ f ^ \\nu ( n ) v ( \\tfrac { n } { X } ) \\\\ & = \\sum _ { n \\leq X } S _ f ^ \\nu ( n ) + O \\bigg ( \\frac { X } { Y } X ^ { \\kappa ( f ) + \\frac { 1 } { 3 } + \\frac { 1 } { 3 } \\alpha ( f ) + \\epsilon - \\nu } \\bigg ) , \\end{align*}"}
-{"id": "6069.png", "formula": "\\begin{align*} I & : = \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { t } \\left ( \\sum _ { ( t - s ) 2 ^ { j \\alpha } \\le 1 } 2 ^ { j \\alpha / p } \\exp ( - c ( t - s ) 2 ^ { j \\alpha } ) \\Big ( \\int _ { { \\mathbb R } ^ d } | F _ { j } ( s , y ) | ^ { p } d y \\Big ) ^ { 1 / p } \\right ) ^ { p } d s d t \\\\ I I & : = \\int _ { 0 } ^ { T } \\int _ { 0 } ^ { t } \\left ( \\sum _ { ( t - s ) 2 ^ { j \\alpha } > 1 } 2 ^ { j \\alpha / p } \\exp ( - c ( t - s ) 2 ^ { j \\alpha } ) \\Big ( \\int _ { { \\mathbb R } ^ d } | F _ { j } ( s , y ) | ^ { p } d y \\Big ) ^ { 1 / p } \\right ) ^ { p } d s d t . \\end{align*}"}
-{"id": "7659.png", "formula": "\\begin{align*} \\mathcal { F } [ v ] ( t ) = S _ { \\alpha } ( t ) u _ { 0 } + \\int _ { 0 } ^ { t } S _ { \\alpha } ( t - s ) f ( v ( s ) ) d s . \\end{align*}"}
-{"id": "9401.png", "formula": "\\begin{align*} \\gamma ( \\lambda ^ * ) ^ \\dag = \\Gamma _ 1 ( H _ { \\infty } - \\lambda { I } ) ^ { - 1 } , \\frac { d } { d \\lambda } W _ \\lambda = \\gamma ( \\lambda ^ * ) ^ \\dag \\gamma ( \\lambda ) \\end{align*}"}
-{"id": "5743.png", "formula": "\\begin{align*} Q _ { L } ^ { S T } ( 0 ) = S ( \\hat \\beta _ 0 ) ^ T I _ { L } ( \\hat \\beta _ 0 ) ^ { - 1 } S ( \\hat \\beta _ 0 ) = \\sum _ { l = 1 } ^ { L } \\hat { C } ^ 2 ( l ) / \\hat { B } ( l ) \\end{align*}"}
-{"id": "2839.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ k \\sum _ { \\sigma \\in S h ( i , k - i ) } ( - 1 ) ^ { \\epsilon ( i ) } l _ k ( l _ i ( x _ { \\sigma ( 1 ) } , . . . , x _ { \\sigma ( i ) } ) , x _ { \\sigma ( i + 1 ) } , . . . , x _ { \\sigma ( k ) } ) = 0 \\end{align*}"}
-{"id": "7167.png", "formula": "\\begin{align*} c _ { j k } = \\int _ { | x | = R } \\Phi ^ { j k } ( x - y ) \\cdot \\nu ( x ) d S _ x , \\forall | y | < R / 2 . \\end{align*}"}
-{"id": "9652.png", "formula": "\\begin{align*} x ' ( t ) = - a g ( x ( t ) ) + b \\max _ { t - \\tau ( t ) \\leq s \\leq t } g ( x ( s ) ) , t > 0 ; , x ( t ) = \\psi ( t ) , t \\leq 0 \\end{align*}"}
-{"id": "4292.png", "formula": "\\begin{align*} G \\cong \\begin{cases} B \\left ( \\frac { 2 ^ n - ( - 1 ) ^ n } { 3 } , 3 , 2 ^ { 2 n / 3 } , 1 \\right ) & \\mathrm { i f ~ ( A ) ~ d o e s ~ n o t ~ h o l d ~ a n d } ~ k \\not \\equiv 0 , l \\not \\equiv 0 , k \\not \\equiv l , \\\\ \\Z _ { 2 ^ n - ( - 1 ) ^ n } & \\mathrm { i f } ~ k \\equiv 0 , \\mathrm { o r } ~ l \\equiv 0 , \\mathrm { o r } ~ k \\equiv l , \\\\ \\Z _ { ( { 2 ^ { n / 3 } - ( - 1 ) ^ { n / 3 } } ) / { 3 } } * \\Z * \\Z & \\mathrm { i f ~ ( A ) ~ h o l d s } . \\end{cases} \\end{align*}"}
-{"id": "9371.png", "formula": "\\begin{align*} H _ \\infty = S _ { m a x } \\upharpoonright _ { \\mathcal { D } ( H _ { \\infty } ) } , \\mathcal { D } ( H _ { \\infty } ) = \\{ f \\in \\mathcal { D } ( S _ { m a x } ) \\ : \\ \\Gamma _ 0 f = 0 \\} . \\end{align*}"}
-{"id": "6864.png", "formula": "\\begin{align*} z _ { n , J _ 0 } ^ J ( t ) : = \\sum _ { j = J _ 0 + 1 } ^ J v _ n ^ j ( t ) . \\end{align*}"}
-{"id": "1591.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int L _ { Y _ { I } } C _ { x } d x + C ^ { x } Y _ { I } + 2 \\psi _ { I } = 0 , \\end{align*}"}
-{"id": "6978.png", "formula": "\\begin{align*} \\tilde { \\alpha } _ t ( a , b ) & = \\Phi _ t ^ { - 1 } \\alpha _ t ( \\Phi _ t a , \\Phi _ t b ) , \\\\ \\tilde { \\mu } _ t ( x , a ) & = \\Phi _ t ^ { - 1 } \\mu _ t ( \\Psi _ t x , \\Phi _ t a ) , \\\\ \\tilde { \\lambda } _ t ( x , y ) & = \\Psi _ t ^ { - 1 } \\lambda _ t ( \\Psi _ t x , \\Psi _ t y ) . \\end{align*}"}
-{"id": "6320.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } \\lesssim & \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| ^ 2 _ { L ^ 2 } \\right ) ^ { 1 / 2 } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| ^ 2 _ { L ^ { p _ 1 } } \\right ) ^ { 1 / 2 } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "2722.png", "formula": "\\begin{align*} \\lambda _ j \\mu ( X ) = \\int _ X \\langle T _ j ^ n \\rangle . \\end{align*}"}
-{"id": "6496.png", "formula": "\\begin{align*} \\| \\partial _ { t } ^ { \\alpha } ( \\tilde { u } ( \\cdot , t ) - \\tilde { u } ( \\cdot , 0 ) ) \\| _ { L ^ { 2 } ( \\Omega ) } & = \\| \\partial _ { t } ^ { \\alpha } \\tilde { u } ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\Omega ) } = \\| \\partial _ { t } ( g _ { 1 - \\alpha } * \\mu * v ) ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\Omega ) } \\\\ \\leq & \\| \\mu \\| _ { L ^ { 1 } ( 0 , T ) } ( \\| \\partial _ { t } ^ { \\alpha } ( v - g ) \\| _ { L ^ { \\infty } ( ( 0 , T ] ; L ^ { 2 } ( \\Omega ) ) } + g _ { 1 - \\alpha } ( t ) \\| g \\| _ { L ^ { 2 } ( \\Omega ) } ) . \\end{align*}"}
-{"id": "4150.png", "formula": "\\begin{align*} T _ 1 ( s ) & = \\prod _ p ( 1 - b _ p p ^ { - 2 s } + ( 2 b _ p - 2 ) p ^ { - 3 s } - b _ p p ^ { - 4 s } + p ^ { - 6 s } ) \\\\ T _ 2 ( s ) & = \\prod _ p \\left ( 1 - \\sum _ { i , j = 1 } ^ 3 \\alpha _ { i , p } ^ { - 1 } \\alpha _ { j , p } ^ { - 1 } p ^ { - 2 s } + \\frac { 2 b _ p - 2 } { p ^ { 3 s } } - \\sum _ { i , j = 1 } ^ 3 \\alpha _ { i , p } \\alpha _ { j , p } p ^ { - 4 s } + p ^ { - 6 s } \\right ) \\\\ b _ p & = \\sum _ { i = 1 } ^ 3 \\sum _ { j = 1 } ^ 3 \\frac { \\alpha _ { i , p } } { \\alpha _ { j , p } } . \\end{align*}"}
-{"id": "9414.png", "formula": "\\begin{align*} \\ell ( c ) = \\int _ { x \\in W } w ( x ) \\abs { c ' ( x ) } \\ , d x = \\int _ { y \\in K } n _ c ( y ) \\ , d y . \\end{align*}"}
-{"id": "8209.png", "formula": "\\begin{align*} { \\cal P } \\left [ \\begin{array} { c } u \\\\ v \\end{array} \\right ] = \\left [ \\begin{array} { c } v \\\\ u \\end{array} \\right ] , { \\cal T } \\left [ \\begin{array} { c } u ( t ) \\\\ v ( t ) \\end{array} \\right ] = \\left [ \\begin{array} { c } \\bar { u } ( - t ) \\\\ \\bar { v } ( - t ) \\end{array} \\right ] . \\end{align*}"}
-{"id": "5744.png", "formula": "\\begin{align*} \\underset { \\omega \\in \\Omega } \\sup ~ Q _ { L } ^ { S T } ( \\omega ) = \\underset { \\omega \\in \\Omega } \\sup S ( \\hat \\delta _ 0 ) I _ { L } ( \\hat \\delta _ 0 ) ^ { - 1 } S ( \\hat \\delta _ 0 ) . \\end{align*}"}
-{"id": "1226.png", "formula": "\\begin{align*} P ( x ) = P ( x ^ 0 ) = \\int _ 0 ^ a \\varphi \\left ( \\frac { a } { W ( a ) } \\right ) w ( s ) \\ , d s = \\varphi \\left ( \\frac { a } { W ( a ) } \\right ) W ( a ) . \\end{align*}"}
-{"id": "7559.png", "formula": "\\begin{align*} l < b - f ( b ) = - F ( b ) , \\end{align*}"}
-{"id": "8045.png", "formula": "\\begin{align*} a _ n ^ { q _ n ( \\vec x , \\vec y ) } = { a _ { n - 1 } } ^ { - q _ { n - 1 } ( \\vec x , \\vec y ) } \\cdots a _ 1 ^ { - q _ 1 ( \\vec x , \\vec y ) } \\cdot a _ 1 ^ { x _ 1 } \\cdots a _ n ^ { x _ n } \\cdot a _ 1 ^ { y _ 1 } \\cdots a _ n ^ { y _ n } . \\end{align*}"}
-{"id": "5126.png", "formula": "\\begin{align*} \\vec { z } ( \\ell ( t ) , \\ldots , \\ell ( 0 ) ) _ { i } = \\left \\{ \\begin{array} { l l } z _ { i } & ( 1 \\le i < \\ell ( 0 ) , i \\not = \\ell ( t ) , \\ldots , \\ell ( 1 ) ) \\\\ z _ { \\ell ( s - 1 ) } & ( i = \\ell ( s ) , 1 \\le s \\le t ) \\\\ z _ { i + 1 } & ( \\ell ( 0 ) \\le i \\le k - 1 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "600.png", "formula": "\\begin{align*} \\Delta _ \\kappa = - \\xi _ { 2 - \\kappa } \\circ \\xi _ \\kappa , \\end{align*}"}
-{"id": "9646.png", "formula": "\\begin{align*} \\gamma _ 0 ( \\omega _ 1 , \\omega _ 2 ) = - \\frac { \\omega _ 1 ^ 3 \\omega _ 2 ^ 2 ( \\omega _ 2 + 1 ) ^ 2 } { ( \\omega _ 1 + 1 ) ( 3 \\omega _ 1 \\omega _ 2 ^ 2 + 4 \\omega _ 1 \\omega _ 2 + \\omega _ 1 + 2 \\omega _ 2 + 1 ) } \\end{align*}"}
-{"id": "6201.png", "formula": "\\begin{align*} \\int _ { - 1 + x ^ { - 1 } } ^ 1 [ f ( x + x z ) - f ( x ) - f ' ( x ) x z ] \\nu _ U ( \\d z ) = x ^ \\beta \\int _ { - 1 + x ^ { - 1 } } ^ 1 [ ( 1 + z ) ^ \\beta - 1 - \\beta z ] \\nu _ U ( \\d z ) , \\end{align*}"}
-{"id": "9605.png", "formula": "\\begin{align*} e ^ { i z \\sqrt { r ^ 2 + m ^ 2 } } - e ^ { i z r } = e ^ { i z r } \\Big ( e ^ { i z r ( \\sqrt { 1 + ( \\frac { m } { r } ) ^ 2 } - 1 ) } - 1 \\Big ) = e ^ { i z r } \\Big ( \\frac { i z m ^ 2 } { 2 r } + R ( r , z ) \\Big ) , | r | \\ge 2 m ^ 2 + 1 , \\end{align*}"}
-{"id": "3621.png", "formula": "\\begin{align*} & \\sum _ { j = 1 } ^ { n + 1 } \\| \\varphi _ j ( x ) \\widetilde { U ^ j } \\| _ { C ^ { t _ j , \\alpha } ( B _ { \\frac { 1 } { 2 } } ( 0 ) ) } \\\\ & \\le C \\left ( \\sum _ { j = 1 } ^ { n + 1 } \\| ( \\phi ( x ) ) ^ { t _ j + s _ j } \\varphi _ j ( x ) \\widetilde { ( L U ) _ j } \\| _ { C ^ { - s _ j , \\alpha } ( B _ 1 ( 0 ) ) } + \\sum _ { j = 1 } ^ { n + 1 } \\| \\varphi _ j ( x ) \\widetilde { U ^ j } \\| _ { L ^ 2 ( B _ 1 ( 0 ) ) } \\right ) . \\end{align*}"}
-{"id": "2175.png", "formula": "\\begin{align*} \\| w \\| ^ { 2 } _ { L ^ { 2 \\kappa } ( [ t _ { 2 } - t _ { 1 } , t _ { 0 } - t _ { 1 } ] \\times \\rho ' B _ { 1 } ) } \\leq C ( n , \\Lambda , p , \\alpha , \\eta ) \\int _ { 0 } ^ { t _ { 0 } - t _ { 1 } } F ( s ) d s . \\end{align*}"}
-{"id": "8336.png", "formula": "\\begin{align*} y ^ p - y = u ( T ) , \\end{align*}"}
-{"id": "7662.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ { \\infty } s _ { k } ^ { - p } f ( s _ { k } ) = \\infty , \\end{align*}"}
-{"id": "2317.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\Big \\| ( v _ n - v _ { n - 1 } ) _ { n = 0 } ^ N \\Big \\| _ { L ^ p ( X ) } & \\le \\sum _ { m = 0 } ^ N | \\chi _ m | \\ , \\| ( \\eta _ { n - m } \\dot v _ { n - m } ) _ { n = 0 } ^ N \\| _ { L ^ p ( X ) } \\\\ & \\le \\sum _ { m = 0 } ^ N | \\chi _ m | \\ , \\| ( \\dot v _ { n } ) _ { n = 0 } ^ N \\| _ { L ^ p ( X ) } \\\\ & \\le C \\big \\| ( \\dot v _ n ) _ { n = 0 } ^ N \\big \\| _ { L ^ p ( X ) } . \\end{align*}"}
-{"id": "5425.png", "formula": "\\begin{align*} \\alpha _ 0 = 3 8 3 / 3 8 4 \\end{align*}"}
-{"id": "1443.png", "formula": "\\begin{align*} x = \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ { n - 1 } ( n - i ) ( i n - 2 ) h _ { i } . \\end{align*}"}
-{"id": "3612.png", "formula": "\\begin{align*} \\| f \\| _ { C ^ { k , \\alpha } _ { \\phi , \\varphi } ( B _ { a \\phi ( x ) } ( x ) ) } = \\sum _ { j = 0 } ^ k \\varphi ( x ) \\phi ^ j ( x ) \\| \\nabla ^ j f \\| _ { C ^ 0 ( B _ { a \\phi ( x ) } ( x ) ) } + \\varphi ( x ) \\phi ^ { k + \\alpha } ( x ) [ \\nabla ^ { k } f ] _ { 0 , \\alpha ; B _ { a \\phi ( x ) } ( x ) } . \\end{align*}"}
-{"id": "1334.png", "formula": "\\begin{align*} T ( \\lambda ) = T ( \\lambda | 1 ) T ( \\lambda | 2 ) \\cdots T ( \\lambda | K ) . \\end{align*}"}
-{"id": "7192.png", "formula": "\\begin{align*} f ^ { ' } \\sin \\varphi = B f + 2 B - 2 C _ 1 ( \\cos \\varphi - 1 ) . \\end{align*}"}
-{"id": "4929.png", "formula": "\\begin{align*} | A x _ 0 | = | A \\tilde { x } | . \\end{align*}"}
-{"id": "9716.png", "formula": "\\begin{align*} x _ { U , \\epsilon } ( t ) & \\geq x _ { U , \\epsilon } ( T _ 2 ( \\epsilon ) ) = x _ { U , \\epsilon } ( T _ 1 ( \\epsilon ) / ( 1 - q - \\epsilon ) ) \\\\ & = G _ 0 ^ { - 1 } ( \\lambda ( \\epsilon ) T _ 1 ( \\epsilon ) / ( 1 - q - \\epsilon ) ) \\\\ & = G _ 0 ^ { - 1 } ( \\eta ( \\epsilon ) G _ 0 ( \\delta ( \\epsilon ) ) ) . \\end{align*}"}
-{"id": "1205.png", "formula": "\\begin{align*} I _ C ^ m = ( I _ { C _ 1 } I _ { C _ 2 } ) ^ m = I _ { C _ 1 } ^ m \\cdot I _ { C _ 2 } ^ m = I _ { C _ 1 } ^ m \\cap I _ { C _ 2 } ^ m = I _ { C _ 1 } ^ { ( m ) } \\cap I _ { C _ 2 } ^ { ( m ) } = I _ C ^ { ( m ) } . \\end{align*}"}
-{"id": "4893.png", "formula": "\\begin{align*} w ( a , b ) : = \\left ( \\begin{array} { c c c c } 1 & 0 & 0 & 0 \\\\ a & 1 & 0 & 0 \\\\ b & a \\pi & 1 & 0 \\\\ a ^ { 2 } ( a \\pi ) + a b + b \\pi & a ( a \\pi ) + b & a & 1 \\\\ \\end{array} \\right ) , \\end{align*}"}
-{"id": "9972.png", "formula": "\\begin{align*} g ( s _ t , a _ t ) = \\max _ { k _ t \\ ! , \\alpha _ t \\ ! , \\tilde { p } _ t \\ ! , \\mathbf { \\emph { p } } _ t } \\alpha _ t \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\tilde { p } _ t | H _ { t , \\tilde { i } k _ t } | ^ 2 } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\ ! + \\ ! ( \\ ! 1 \\ ! - \\ ! \\alpha _ t \\ ! ) \\sum _ { i = 1 } ^ { 2 } \\log _ 2 \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { p _ { t , i } } { \\sigma ^ 2 _ n } \\ ! \\Big ) , \\end{align*}"}
-{"id": "7096.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( \\sum _ { | u | = n , u > v } \\delta _ { V ( u ) - V ( v ) - m _ n } , Z ^ v _ n \\right ) = \\left ( \\theta _ { - V ( v ) } \\mu ^ v _ \\infty , Z ^ v _ \\infty \\right ) \\end{align*}"}
-{"id": "229.png", "formula": "\\begin{align*} | U _ 1 | + | U _ 2 | = O \\biggl ( \\frac { k ^ { 1 / 2 } } { n } \\max \\biggl \\{ \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "7486.png", "formula": "\\begin{align*} q ( d , k ) = \\sum _ { j = 0 } ^ { \\kappa ' k } { j \\choose 2 } q _ j ( d , k ) . \\end{align*}"}
-{"id": "7507.png", "formula": "\\begin{align*} x _ { n + 1 } = \\frac { A x _ n } { ( 1 + B x _ n ) ^ { \\gamma } } , A > 1 , ~ B > 0 , ~ \\gamma > 1 , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 \\end{align*}"}
-{"id": "8135.png", "formula": "\\begin{align*} S _ { a , p , p , N } = S _ { a , p , p , N } ^ { r a d } = \\left ( \\frac { N } { p } - 1 + a \\right ) ^ p , \\end{align*}"}
-{"id": "3221.png", "formula": "\\begin{align*} u _ n ( \\widetilde { Y } , \\widetilde { V } ; \\{ \\widetilde { w } _ j \\} ) = ( \\pi | _ { \\widetilde { Y } } ) ^ * u _ n ( Y , X ; \\{ w _ j \\} ) \\end{align*}"}
-{"id": "9284.png", "formula": "\\begin{align*} \\begin{cases} d X ( t , Z ) = \\pi ( t , Z ) [ \\alpha ( t ) d t + \\beta ( t ) d v ( t ) ] ; 0 \\leq t \\leq T , \\\\ X ( 0 ) F \\end{cases} \\end{align*}"}
-{"id": "3929.png", "formula": "\\begin{align*} W \\left [ w _ \\sigma , \\ , w _ { \\rm r e g } \\right ] ( 0 ) = - \\ , \\delta _ { \\sigma , 0 } \\ , \\left ( \\sum _ { j = 0 } ^ \\infty j \\ , b _ j \\right ) - \\delta _ { \\sigma , 1 } \\ , \\left ( \\sum _ { j = 0 } ^ \\infty b _ j \\right ) \\ , . \\end{align*}"}
-{"id": "8038.png", "formula": "\\begin{align*} \\sum _ { 1 \\leq k < \\ell \\leq m } N _ { k , \\ell } = 2 \\ ! \\ ! \\ ! \\sum _ { 1 \\leq k < \\ell \\leq m } 3 ^ { \\ell - k - 1 } = 2 \\ ! \\sum _ { 1 < \\ell \\leq m } \\ , \\sum _ { 1 \\leq i < \\ell } 3 ^ { i - 1 } \\leq \\sum _ { 1 < \\ell \\leq m } 3 ^ { \\ell - 1 } \\leq 3 ^ { m } . \\end{align*}"}
-{"id": "4956.png", "formula": "\\begin{align*} u = \\Psi _ \\lambda \\ast \\left ( Q | u | ^ { p - 2 } u \\right ) , u \\in L ^ p _ { l o c } ( \\R ^ N ) \\end{align*}"}
-{"id": "3233.png", "formula": "\\begin{align*} P _ { \\alpha } ^ { S } = Q ^ { S } \\circ P _ { \\alpha } ^ { N } = Q ^ { S } \\circ P _ { \\beta } ^ { N } \\circ \\Phi _ { \\beta , \\alpha } ^ { 1 } = P _ { \\beta } ^ { S } \\circ \\Phi _ { \\beta , \\alpha } ^ { 1 } . \\end{align*}"}
-{"id": "9306.png", "formula": "\\begin{align*} J ( u ) = \\int _ { \\mathbb { R } } j ( u ) ( z ) d z . \\end{align*}"}
-{"id": "5316.png", "formula": "\\begin{align*} c _ 0 & = \\tau ( 0 ) = \\frac { H _ { n , m } } { t } + ( \\eta ^ 2 \\frac { N ^ 2 } { 4 } - \\omega ^ 2 + \\eta ^ { - 2 } ) I \\\\ c _ 1 & = \\frac { d } { d u } \\tau ( u ) | _ { u = 0 } = 2 \\eta N \\\\ c _ 2 & = \\frac { 1 } { 2 } \\frac { d ^ 2 } { d u ^ 2 } \\tau ( u ) | _ { u = 0 } = I , \\end{align*}"}
-{"id": "358.png", "formula": "\\begin{align*} F ( t , c , h ) = \\langle \\Psi _ { c , h } , \\ , e ^ { i t T _ { c , h } ( f _ 1 ) } \\ldots e ^ { i t T _ { c , h } ( f _ n ) } \\Psi _ { c , h } \\rangle \\end{align*}"}
-{"id": "5175.png", "formula": "\\begin{gather*} q = q \\cdot 1 ^ k \\leqslant ( r _ 1 ) ^ { \\delta ( q ) } ( r _ 1 ) ^ s \\cdots ( r _ 1 ) ^ s = ( r _ 1 ) ^ { \\delta ( q ) + k s } . \\end{gather*}"}
-{"id": "719.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n k } ( P ) ) = & - c _ { 1 } h _ { Z _ { 1 } } ( p ^ { - 1 } f ^ { k ( n - 1 ) } ( P ) ) + h _ { E } ( p ^ { - 1 } f ^ { k ( n - 1 ) } ( P ) ) \\\\ & + \\rho _ { k } c _ { 1 } h _ { D _ { 1 } } ( f ^ { k ( n - 1 ) } ( P ) ) + c _ { 1 } h _ { { E _ { 1 } ' } } ( f ^ { k ( n - 1 ) } ( P ) ) \\\\ \\leq & \\rho _ { k } c _ { 1 } h _ { D _ { 1 } } ( f ^ { k ( n - 1 ) } ( P ) ) + C \\sqrt [ ] { h _ { H } ( f ^ { k ( n - 1 ) } ( P ) ) + \\gamma } \\\\ & + c _ { 1 } C \\sqrt [ ] { h _ { H } ( f ^ { k ( n - 1 ) } ( P ) ) } \\end{align*}"}
-{"id": "8354.png", "formula": "\\begin{align*} \\omega \\circ E _ M ( \\sigma ( x _ \\lambda ) ^ * \\sigma ( x _ \\lambda ) ) = \\omega \\circ E _ M ( \\sigma ( x _ \\lambda ^ * x _ \\lambda ) ) = \\omega \\circ \\sigma ( E _ M ( x _ \\lambda ^ * x _ \\lambda ) ) \\end{align*}"}
-{"id": "4531.png", "formula": "\\begin{align*} | T _ 1 | \\leq T _ { 1 1 } + T _ { 1 2 } + T _ { 1 3 } = o \\biggl ( \\frac { k ^ { - \\frac { 1 } { 2 } + \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "679.png", "formula": "\\begin{align*} p ^ { * n } ( s ) \\frac { d s \\nu } { d \\nu } ( y ) \\leq \\sum _ { g \\in G } p ^ { * n } ( g ) \\frac { d g \\nu } { d \\nu } ( y ) = 1 , \\end{align*}"}
-{"id": "3595.png", "formula": "\\begin{align*} \\Pi _ { g _ 0 } \\circ \\Phi ^ W _ { ( g , \\pi ) } ( g + h , \\pi + w ) = \\Pi _ { g _ 0 } \\circ \\Phi ^ W _ { ( g , \\pi ) } ( g , \\pi ) + \\Pi _ { g _ 0 } ( \\psi , V ) \\end{align*}"}
-{"id": "5552.png", "formula": "\\begin{align*} \\begin{bmatrix} \\frac 1 2 & - h & h ^ 2 \\\\ \\frac 1 2 + H + \\frac 1 2 H ^ 2 & H + H ^ 2 & H ^ 2 \\end{bmatrix} \\begin{bmatrix} C _ 1 \\\\ C _ 2 \\\\ C _ 3 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ C \\end{bmatrix} , \\end{align*}"}
-{"id": "7888.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } \\sup _ { x \\in \\R ^ d } \\left | \\int _ { \\R ^ d } p _ y ( t , x - y ) f ( y ) \\ , d y - f ( x ) \\right | = 0 \\ , . \\end{align*}"}
-{"id": "3435.png", "formula": "\\begin{align*} d Y ( t ) = - \\tilde G ( t , \\kappa ( t ) , Y ( t ) , Z ( t ) ) d t + \\langle Z ( t ) , d W ( t ) \\rangle , Y ( T ) = \\langle \\eta ( T ) , u _ 0 ( \\xi ( T ) \\rangle , \\end{align*}"}
-{"id": "6310.png", "formula": "\\begin{align*} \\Lambda _ k ^ { \\alpha _ 1 , \\ast } \\cap \\Lambda _ l ^ { \\alpha _ 1 , \\ast } = \\emptyset \\end{align*}"}
-{"id": "7270.png", "formula": "\\begin{align*} \\varepsilon ^ { - 1 } _ n + \\varepsilon ^ { - 1 } _ m = \\frac { 1 } { K } \\left ( d ( \\mathbb { I } _ Y ^ { ( 1 ) } , N _ { \\alpha ^ { ( n ) } } ) + d ( \\mathbb { I } _ Y ^ { ( 1 ) } , N _ { \\alpha ^ { ( m ) } } ) \\right ) \\geq \\frac { 1 } { K } d ( N _ { \\alpha ^ { ( n ) } } , N _ { \\alpha ^ { ( m ) } } ) = \\frac { 1 } { K } d ( \\mathbb { I } _ Y ^ { ( 1 ) } , N _ { \\alpha ^ { ( n ) } - \\alpha ^ { ( m ) } } ) , \\end{align*}"}
-{"id": "2713.png", "formula": "\\begin{align*} d ( s ) = \\frac { 1 } { s } \\left ( \\int _ X ( \\varphi _ { t + s } - \\varphi _ t ) d d ^ c \\dot { \\varphi } _ { t + s } \\wedge T _ s + \\int _ X ( \\dot { \\varphi } _ { t + s } - \\dot { \\varphi } _ t ) \\theta _ { \\varphi _ t } ^ n \\right ) . \\end{align*}"}
-{"id": "49.png", "formula": "\\begin{align*} u _ s ( t , x ) = \\left ( ( 2 - p ) ( T - t ) _ + \\right ) ^ { 1 / ( 2 - p ) } f ( | x | ) \\ , ( t , x ) \\in ( 0 , \\infty ) \\times \\mathbb { R } \\ , \\end{align*}"}
-{"id": "5156.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t \\theta ^ { n , 2 } _ i + u ^ n \\cdot \\nabla \\theta ^ { n , 2 } _ i = f ^ \\infty _ i , \\\\ \\partial _ t \\sigma ^ { n , 2 } + u ^ n \\cdot \\nabla \\sigma ^ { n , 2 } = g ^ \\infty , \\\\ { \\theta ^ { n , 2 } _ i } | _ { t = 0 } = \\nabla ( u ^ \\infty _ 0 ) _ { i } , \\\\ { \\sigma ^ { n , 2 } } | _ { t = 0 } = \\nabla \\gamma ^ \\infty _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "5903.png", "formula": "\\begin{align*} D _ { t } f + v \\cdot D _ { x } f + F \\cdot D _ { v } f = 0 , f | _ { t = 0 } = f _ { 0 } \\end{align*}"}
-{"id": "8286.png", "formula": "\\begin{align*} | \\sum _ { i , j } M _ { i j } \\sigma _ i ^ * \\sigma _ j ^ * - \\sum _ { i , j } M _ { i , j } \\sigma _ i ^ { \\star } \\sigma _ j ^ { \\star } | = 2 \\ , | \\sum _ { i \\in W } \\sum _ { j \\in W ^ c } M _ { i , j } \\sigma _ i ^ * \\sigma _ j ^ * | \\leq 2 0 C ( \\xi ) | W | + 2 \\ , \\sum _ { i , j \\in W } | M _ { i , j } | . \\end{align*}"}
-{"id": "869.png", "formula": "\\begin{align*} \\varphi ( y ) = \\inf \\{ t \\in { \\mathbb { R } } \\mid y \\in t k + A \\} \\ ; \\ ; \\forall y \\in Y \\end{align*}"}
-{"id": "3723.png", "formula": "\\begin{align*} \\Theta \\circ \\sigma _ { \\beta } = \\pi _ { \\beta } \\circ \\Theta . \\end{align*}"}
-{"id": "1457.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ l \\frac { 1 } { w ( Q _ 0 ^ { ( m ) } ) ^ \\frac { 1 } { p } } \\lesssim _ { p , w } \\frac { 1 } { w ( Q _ 0 ) ^ \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "6484.png", "formula": "\\begin{align*} u _ { k } ( t ) = E _ { \\alpha , 1 } ( - \\lambda _ { k } t ^ { \\alpha } ) u _ { 0 , k } + \\int _ { 0 } ^ { t } ( t - s ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\lambda _ { k } ( t - s ) ^ { \\alpha } ) f _ { k } ( s ) d s . \\end{align*}"}
-{"id": "457.png", "formula": "\\begin{align*} P = \\{ x \\in \\mathbb { R } ^ n \\ , | \\ \\pi ^ { \\top } x \\leq \\pi _ 0 , \\ ( y ^ i ) ^ { \\top } A x \\geq ( y ^ i ) ^ { \\top } b , \\ i \\in [ p ' ] \\} \\end{align*}"}
-{"id": "6574.png", "formula": "\\begin{align*} \\nabla F ( x ) = P x - \\nabla ^ 2 f _ 1 ( x ) \\nabla f _ 2 ( \\nabla f _ 1 ( x ) ) = P x - P S _ 2 ( S _ 1 x ) = P ( x - S _ 2 S _ 1 x ) . \\end{align*}"}
-{"id": "4378.png", "formula": "\\begin{align*} F ( t , c , h ) = \\langle \\Psi _ { c , h } , \\ , e ^ { i t T _ { c , h } ( f _ 1 ) } \\ldots e ^ { i t T _ { c , h } ( f _ n ) } \\Psi _ { c , h } \\rangle \\end{align*}"}
-{"id": "5884.png", "formula": "\\begin{align*} A ^ { i j } \\left ( x ^ { k } \\right ) u _ { , i j } - B ^ { k } ( x ) u _ { k } = 0 , \\end{align*}"}
-{"id": "6075.png", "formula": "\\begin{align*} \\sum _ { a < n < b } g ( n ) \\exp ( 2 \\pi i f ( n ) ) & = \\sum _ { \\alpha - \\epsilon < m < \\beta + \\epsilon } \\int _ a ^ b g ( x ) \\exp ( 2 \\pi i ( f ( x ) - m x ) ) d x \\\\ & \\quad + O ( G ( \\epsilon ^ { - 1 } + \\log ( \\beta - \\alpha + 2 ) ) ) , \\end{align*}"}
-{"id": "5735.png", "formula": "\\begin{align*} l ( \\delta ) = \\sum _ { t = 1 } ^ n Y _ t W _ t ( \\delta ) - m _ t b ( W _ t ( \\delta ) ) + c ( y _ t ) . \\end{align*}"}
-{"id": "6902.png", "formula": "\\begin{gather*} \\delta U _ { - 1 } = [ B _ 0 ^ - , U _ { - 1 } ] , \\ ; \\ ; \\ ; \\delta U _ 0 = \\left ( [ B _ 1 ^ - , U _ { - 1 } ] + [ B _ 0 ^ - , U _ 0 ] \\right ) _ { \\oplus } , \\\\ \\delta U _ 1 = \\left ( [ B _ 2 ^ - , U _ { - 1 } ] + [ B _ 1 ^ - , U _ 0 ] + [ B _ 0 ^ - , U _ 1 ] \\right ) _ { \\oplus } . \\end{gather*}"}
-{"id": "103.png", "formula": "\\begin{align*} \\frac { 2 } { ( 1 + t ) ^ { \\lambda } + 1 } ( 1 + t ) ^ { \\lambda x } = \\sum _ { n = 0 } ^ \\infty C h _ { n , \\lambda } ( x ) \\frac { t ^ n } { n ! } , \\quad \\textnormal { w h e r e } \\ , \\ , \\lambda \\in \\mathbb { Z } _ p ( \\textnormal { s e e } \\ , \\ , [ 5 ] ) . \\end{align*}"}
-{"id": "6982.png", "formula": "\\begin{align*} \\delta _ L \\Theta ( L ) = 0 . \\end{align*}"}
-{"id": "671.png", "formula": "\\begin{align*} \\nu ( B ) = \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\chi _ B ) \\chi _ B ) \\ , d \\eta ( g ) = \\nu ( B ) ^ 2 , \\end{align*}"}
-{"id": "3906.png", "formula": "\\begin{align*} M ( \\lambda , u ) = H ( \\varphi ^ { - 1 } \\left [ \\lambda H ( N _ f ( u ) ) - Q _ { \\varphi } ( \\lambda H ( N _ f ( u ) ) ) \\right ] ) . \\end{align*}"}
-{"id": "9874.png", "formula": "\\begin{align*} \\frac { \\partial U ^ { \\epsilon } } { \\partial t } ( t , x ) & = \\frac { \\partial ^ 2 U ^ { \\epsilon } } { \\partial x ^ 2 } ( t , x ) + \\sqrt { \\epsilon } \\sigma ( t , x , U ^ \\epsilon ( t , x ) ) \\frac { \\partial ^ 2 W } { \\partial t \\partial x } ( t , x ) \\\\ & + \\frac { \\partial } { \\partial x } g ( t , x , U ^ { \\epsilon } ( t , x ) ) + f ( t , x , U ^ \\epsilon ( t , x ) ) , \\end{align*}"}
-{"id": "3614.png", "formula": "\\begin{align*} ( L U ) _ j = \\sum _ { k = 1 } ^ { n + 1 } \\sum _ { | \\beta | = 0 } ^ { s _ j + t _ k } b _ { j k } ^ { \\beta } \\partial _ y ^ { \\beta } U ^ k , \\end{align*}"}
-{"id": "985.png", "formula": "\\begin{align*} H _ { 2 , 1 } & = U ( N _ { a , 1 } + N _ { a , 2 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } + N _ { a , 2 } - N _ { b , 1 } ) \\\\ & \\quad + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) + t _ { 2 , 1 } ( a _ { 2 } b _ { 1 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 1 } ) \\end{align*}"}
-{"id": "5079.png", "formula": "\\begin{align*} \\varphi ^ { - 1 } : \\mathcal { F } ( L _ { k } ^ { + } , ( U ^ { \\otimes k } ) _ { k _ { 1 } , \\ldots , k _ { r } } ) \\to F ( \\mathcal { S } _ { k _ { 1 } , \\ldots , k _ { r } } ) , ( \\varphi ^ { - 1 } \\gamma ) ( \\vec { x } , \\vec { \\nu } ) = \\gamma _ { \\vec { \\nu } } ( \\vec { x } ) . \\end{align*}"}
-{"id": "646.png", "formula": "\\begin{align*} U _ A : = \\big \\{ g \\in G \\ , : \\ , g \\cdot A \\in U \\big \\} = A , \\textrm { f o r a l l $ A \\subset G $ } . \\end{align*}"}
-{"id": "1257.png", "formula": "\\begin{align*} v M _ { 1 1 } v ^ * = \\| ( a , c ) \\| _ 2 ^ 2 P , \\end{align*}"}
-{"id": "2374.png", "formula": "\\begin{align*} \\nabla F ( x ) & = \\nabla ^ 2 f _ 1 ( x ) x + \\nabla f _ 1 ( x ) - \\nabla f _ 1 ( x ) - \\nabla ^ 2 f _ 1 ( x ) \\nabla f _ 2 ( \\nabla f _ 1 ( x ) ) \\\\ & = \\nabla ^ 2 f _ 1 ( x ) ( x - \\nabla f _ 2 ( \\nabla f _ 1 ( x ) ) ) \\\\ & = \\nabla ^ 2 f _ 1 ( x ) ( x - S _ 2 S _ 1 x ) . \\end{align*}"}
-{"id": "5618.png", "formula": "\\begin{align*} \\Bigl \\vert \\sum _ { \\rho } z ^ { - \\rho } \\Gamma ( \\rho ) \\Bigr \\vert & \\ll N + \\vert z \\vert ^ { 1 / 2 } \\log ^ 2 ( 2 N \\vert y \\vert ) . \\end{align*}"}
-{"id": "9166.png", "formula": "\\begin{align*} \\Phi ' ( 0 ) = \\sum _ { l = - r } ^ { r } l \\frac { \\partial H } { \\partial u _ { l } } ( v , \\dots , v ) = a ( v ) , \\end{align*}"}
-{"id": "5477.png", "formula": "\\begin{align*} f _ { i j } ^ { \\max } = \\max _ { x \\in S } f _ { i j } \\left ( x \\right ) , i = 1 , 2 , . . . m ; j = 1 , 2 , . . . , p _ { m } ^ { } \\end{align*}"}
-{"id": "6845.png", "formula": "\\begin{align*} \\bigl ( \\tfrac 1 { q _ 1 } , \\tfrac 1 { r _ 1 } \\bigr ) = \\bigl ( \\tfrac 1 { q _ 2 } , \\tfrac 1 { r _ 2 } \\bigr ) : = \\bigl ( \\tfrac { d p - 2 } { 2 ( p + 2 ) } , \\tfrac { 2 d + 4 - d p } { 2 d ( p + 2 ) } \\bigr ) , \\quad \\bigl ( \\tfrac 1 { \\tilde { q } } , \\tfrac 1 { \\tilde { r } } \\bigr ) : = \\bigl ( \\tfrac { 4 - ( d - 1 ) p } { p ( p + 2 ) } , \\tfrac { 2 d p - 4 } { d p ( p + 2 ) } \\bigr ) . \\end{align*}"}
-{"id": "9928.png", "formula": "\\begin{align*} \\begin{aligned} & x _ 0 ^ 2 & & + & & x _ 1 ^ 2 & & + & & x _ 2 ^ 2 & & + & & x _ 3 ^ 2 & = & 0 \\\\ & x _ 0 ^ 2 & & - & \\beta \\gamma & x _ 1 ^ 2 & & - & \\gamma & x _ 2 ^ 2 & & + & \\beta & x _ 3 ^ 2 & = & 0 \\\\ & x _ 0 ^ 2 & & + & \\gamma & x _ 1 ^ 2 & & - & \\alpha \\gamma & x _ 2 ^ 2 & & - & \\alpha & x _ 3 ^ 2 & = & 0 \\\\ & x _ 0 ^ 2 & & - & \\beta & x _ 1 ^ 2 & & + & \\alpha & x _ 2 ^ 2 & & - & \\alpha & x _ 3 ^ 2 & = & 0 . \\end{aligned} \\end{align*}"}
-{"id": "836.png", "formula": "\\begin{align*} u ( \\nu _ { 1 } , \\ldots , \\nu _ { m } ) = u _ { \\nu _ { 1 } } \\otimes \\cdots \\otimes u _ { \\nu _ { m } } \\end{align*}"}
-{"id": "5916.png", "formula": "\\begin{align*} \\int _ { \\R ^ d } \\| f ( \\cdot , v ) \\| _ { H ^ { s } _ p } ^ p \\ , \\dd v = \\int _ { \\R ^ d } \\dd v \\int _ { \\R ^ d } | { \\cal F } ^ { - 1 } _ x [ ( 1 + | \\cdot | ^ { s } ) { \\cal F } _ x f ( \\cdot , v ) ] ( x ) | ^ p \\dd x < \\infty \\end{align*}"}
-{"id": "7067.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( n ) - { y ^ { [ j ] } } ( { t _ 1 } ) = \\sum \\limits _ { i = 1 } ^ M { { { \\bf { h } } ^ { [ j i ] } } ( n ) { \\bf { V } } _ j ^ { [ i ] } ( n ) { { \\bf { s } } ^ { [ j i ] } } } - \\sum \\limits _ { i = 1 } ^ M { { { \\bf { h } } ^ { [ j i ] } } ( { t _ 1 } ) { { \\bf { s } } ^ { [ j i ] } } } . \\end{align*}"}
-{"id": "6268.png", "formula": "\\begin{align*} \\nabla \\omega ( y _ 0 ) = \\nabla \\omega ( x _ 0 ) = - \\frac { m } { 2 } \\bar { w } ' ( d _ 0 / 2 ) e _ n . \\end{align*}"}
-{"id": "3460.png", "formula": "\\begin{align*} \\begin{cases} \\sigma \\in C ^ 3 ( [ 0 , \\infty ) ) \\\\ \\sigma \\end{cases} \\end{align*}"}
-{"id": "2305.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = 0 } ^ M \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = 0 } ^ M \\big \\| _ { L ^ p ( D ) } \\le C \\delta . \\end{aligned} \\end{align*}"}
-{"id": "9573.png", "formula": "\\begin{align*} g ( x ) = \\frac { e ^ { - m | x | } } { 4 \\pi | x | } . \\end{align*}"}
-{"id": "3245.png", "formula": "\\begin{align*} D \\psi _ { x } \\left ( v \\right ) = D \\pi _ { x } \\left ( v \\right ) + \\omega \\left ( x \\right ) D \\left ( p - \\pi \\right ) _ { x } \\left ( v \\right ) + \\left \\langle \\nabla \\omega , v \\right \\rangle \\left ( p - \\pi \\right ) \\left ( x \\right ) . \\end{align*}"}
-{"id": "2853.png", "formula": "\\begin{align*} \\epsilon ^ * = t r i v _ O : \\mathcal { C } \\rightarrow O - A l g ( \\mathcal { C } ) \\end{align*}"}
-{"id": "1986.png", "formula": "\\begin{align*} b Q = P ( b ) + \\partial R \\ , , \\end{align*}"}
-{"id": "4990.png", "formula": "\\begin{align*} h _ { N } = c _ { 1 } h _ { D _ { 1 } } - h _ { H } . \\end{align*}"}
-{"id": "3891.png", "formula": "\\begin{align*} C _ L = \\begin{bmatrix} \\tau _ L & 1 & 0 \\\\ - \\sigma _ L & 0 & 1 \\\\ \\delta _ L & 0 & 0 \\end{bmatrix} \\ ; , C _ R = \\begin{bmatrix} \\tau _ R & 1 & 0 \\\\ - \\sigma _ R & 0 & 1 \\\\ \\delta _ R & 0 & 0 \\end{bmatrix} \\ ; , \\end{align*}"}
-{"id": "6243.png", "formula": "\\begin{align*} \\tilde { V } ^ { T } Z \\tilde { V } = - \\tilde { V } ^ { T } \\eta \\mathbf { X } ^ { T } ( \\mathbf { X } \\mathbf { X } ^ { T } ) ^ { - 1 } \\tilde { V } . \\end{align*}"}
-{"id": "4279.png", "formula": "\\begin{align*} \\mathbb P \\left ( \\Delta _ 1 \\le - 2 a , \\ \\max _ { k = 1 , . . . , n } ( \\Delta _ { k + 1 } - \\Delta _ 1 ) \\le a \\right ) \\ge c _ a \\mathbb P \\left ( \\max _ { k = 1 , . . . , n } ( \\Delta _ { k + 1 } - \\Delta _ 1 ) \\le a \\right ) \\ , . \\end{align*}"}
-{"id": "5908.png", "formula": "\\begin{align*} f \\left ( t , x _ { t } , v _ { t } \\right ) = f _ { 0 } \\left ( x _ { 0 } , v _ { 0 } \\right ) \\end{align*}"}
-{"id": "8007.png", "formula": "\\begin{align*} ( | y | ^ 2 + t _ 1 ^ 2 + 2 t _ 1 | y | \\alpha ) ^ { \\frac { p } { 2 } } = | y | ^ p + 1 . \\end{align*}"}
-{"id": "6479.png", "formula": "\\begin{align*} & \\quad \\quad \\quad \\int _ { 0 } ^ { T } \\mathcal { E } ( u , u ^ { - } ) d t = \\int _ { 0 } ^ { T } \\mathcal { E } ( u ^ { + } , u ^ { - } ) d t - \\int _ { 0 } ^ { T } \\mathcal { E } ( u ^ { - } , u ^ { - } ) d t , \\\\ & \\int _ { 0 } ^ { T } \\mathcal { E } ( u ^ { - } , u ^ { - } ) d t = \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } ( u ^ { - } ( x , t ) - u ^ { - } ( y , t ) ) ^ { 2 } k ( x , y ) d x d y d t > 0 , \\end{align*}"}
-{"id": "5280.png", "formula": "\\begin{align*} \\int _ { T } ^ { 2 T } { \\left ( \\tfrac { t } { 2 \\pi } \\right ) } ^ { { 1 } / { 4 } + { \\delta } / { 2 } } e ^ { \\frac { i t } { 2 } \\log ( { t } / { 2 \\pi e n ^ { 2 } ) } } d t = O \\left ( T ^ { { 1 } / { 4 } + { \\delta } / { 2 } } T ^ { { 1 } / { 2 } } \\right ) = O \\left ( T ^ { { 3 } / { 4 } + { \\delta } / { 2 } } \\right ) . \\end{align*}"}
-{"id": "6623.png", "formula": "\\begin{align*} \\Phi _ t ( x _ 0 ) : = \\begin{cases} x ( t ; x _ 0 ) , & \\mbox { i f } \\ t \\in I _ { x _ 0 } , \\\\ x ( \\tau _ \\partial ( x _ 0 ) ; x _ 0 ) , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "4361.png", "formula": "\\begin{align*} \\sigma _ \\varepsilon ' ( y ) + \\frac { p } { p - 1 } \\sigma _ \\varepsilon ( y ) ^ { ( p - 1 ) / p } & = \\frac { p } { p - 1 } ( 1 - y ) \\left [ \\frac { 1 } { 2 ^ { ( p - 1 ) / p } \\varepsilon ^ { 2 - p } } - \\frac { ( 1 - y ) ^ { ( 2 - p ) / ( p - 1 ) } } { 2 \\varepsilon ^ { p ( 2 - p ) / ( p - 1 ) } } \\right ] \\\\ & \\ge \\frac { p } { p - 1 } ( 1 - y ) \\frac { 2 ^ { 1 / p } - 1 } { 2 \\varepsilon ^ { 2 - p } } \\\\ & \\ge \\frac { p a ^ { 2 - p } } { p - 1 } ( 1 - y ) = \\psi ' ( y ) + \\frac { p } { p - 1 } \\psi ( y ) ^ { ( p - 1 ) / p } \\ , \\end{align*}"}
-{"id": "3519.png", "formula": "\\begin{align*} \\nabla ( \\rho ^ { \\frac { 1 } { 2 } } ) ( x ) = \\left \\{ \\begin{array} { l l } \\frac { 1 } { 2 } N \\rho ^ { \\frac { 1 } { 2 } } d ^ { - 2 } \\nabla d & \\mbox { i f $ 0 < d ( x ) \\le r _ 1 $ } \\\\ \\frac { 1 } { 2 } N \\rho ^ { \\frac { 1 } { 2 } } d ^ { - 2 } \\left ( d ^ 2 ( \\log \\tilde { \\rho } ) ' ( d ) \\right ) \\nabla d & \\mbox { i f $ r _ 1 \\le d ( x ) \\le r _ 0 $ } \\\\ 0 & \\mbox { i f $ d ( x ) \\ge r _ 0 $ } \\end{array} \\right . . \\end{align*}"}
-{"id": "6632.png", "formula": "\\begin{align*} M ^ { ( k ) } : = \\max _ { j \\in \\{ 0 , \\ldots , N _ k - 1 \\} } t ^ k _ { j + 1 } - t ^ k _ j \\end{align*}"}
-{"id": "7220.png", "formula": "\\begin{align*} g _ { a } ( y ) = \\frac { 1 } { \\mathrm { v o l } ( B ( a , \\eta ) ) } \\int _ { z \\in B ( a , \\eta ) } \\mathrm { d i s t } ( y , z ) . \\end{align*}"}
-{"id": "3477.png", "formula": "\\begin{align*} J ( u ) = \\int _ \\Omega | \\nabla u | ^ 2 d x + 2 \\int _ D v ^ + d x ' \\end{align*}"}
-{"id": "4722.png", "formula": "\\begin{align*} \\sup \\{ \\sum _ { i = 1 } ^ m ( b ^ i ) ^ { \\top } y ^ i \\ , | \\ \\sum _ { i = 1 } ^ m ( y ^ i ) ^ { \\top } A ^ i = \\pi ^ { \\top } , \\ y ^ i \\in \\mathcal { L } ^ * _ { m _ i } \\ \\forall i \\in [ m ] \\} \\end{align*}"}
-{"id": "8531.png", "formula": "\\begin{align*} b ' X _ { b ^ { \\ast } } ( P ) = \\displaystyle \\sum _ { r \\in L ( k ) , a \\in _ { k } T } b ' r a Y _ { [ b r a ] } ( P ) \\end{align*}"}
-{"id": "2159.png", "formula": "\\begin{align*} \\int _ { B } \\varphi \\partial _ { s } \\left ( g _ { 1 - \\alpha , m } * \\tilde { u } \\right ) d x + \\mathcal { E } ( h _ { m } * \\tilde { u } , \\varphi ) \\geq 0 , s \\in ( 0 , t _ { 0 } - t _ { 1 } ) , m \\in \\mathbb { N } , \\end{align*}"}
-{"id": "2456.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } \\| _ { M _ 2 } = \\sup _ { l \\in \\Gamma _ k ^ { \\alpha _ 2 , \\alpha _ 1 } } \\| \\Box _ l ^ { \\alpha _ 2 } \\Box _ k ^ { \\alpha _ 1 } f \\| _ { L ^ { p _ 2 } } . \\end{align*}"}
-{"id": "2166.png", "formula": "\\begin{align*} & \\mathcal { E } ( \\tilde { u } , - \\psi ^ { 1 + q } \\tilde { u } ^ { - q } ) \\\\ = & \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } ( \\tilde { u } ( s , x ) - \\tilde { u } ( s , y ) ) ( \\psi ^ { 1 + q } ( y ) \\tilde { u } ^ { - q } ( s , y ) - \\psi ^ { 1 + q } ( x ) \\tilde { u } ^ { - q } ( s , x ) ) \\frac { k ( x , y ) } { 2 } d x d y \\\\ \\geq & \\frac { 1 } { 2 ( q - 1 ) } - \\frac { \\vartheta ( q ) } { 2 } , \\end{align*}"}
-{"id": "5287.png", "formula": "\\begin{align*} Z ( t ) = \\chi ( \\tfrac { 1 } { 2 } + i t ) ^ { - 1 / 2 } \\sum _ { n \\leq C T / \\pi } { n ^ { - \\tfrac { 1 } { 2 } - i t } } + O \\left ( \\frac { T ^ { 1 / 2 } } { \\vert t \\vert } \\right ) + O ( T ^ { - 1 / 2 } ) . \\end{align*}"}
-{"id": "2892.png", "formula": "\\begin{align*} C H _ { P o i s _ n } ^ { ( \\bullet > 1 ) } ( A , A ) [ n ] : = \\ker \\Big ( C H _ { P o i s _ n } ^ { ( \\bullet > 0 ) } ( A , A ) [ n ] \\twoheadrightarrow H o m ( A , A ) [ n ] \\Big ) \\end{align*}"}
-{"id": "6446.png", "formula": "\\begin{align*} F ( s ) = & \\frac { 1 } { 2 } C _ { 1 } ( n , \\Lambda , \\delta ) \\vartheta ( q ) ( q - 1 ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } \\phi w ^ { 2 } d x \\\\ & + q g _ { 1 - \\alpha } ( s ) \\phi ( s ) \\int _ { \\rho B _ { 1 } } \\psi ^ { 1 + q } w ^ { 2 } d x + \\dot { \\phi } ( s ) \\left ( g _ { 1 - \\alpha } * \\int _ { \\rho B _ { 1 } } \\psi ^ { 1 + q } w ^ { 2 } d x \\right ) ( s ) \\end{align*}"}
-{"id": "5770.png", "formula": "\\begin{align*} \\tau ( \\mu | \\{ \\lambda \\} ) = a ( \\mu ) \\prod _ { a = 1 } ^ M f ( \\lambda _ a , \\mu ) + d ( \\mu ) \\prod _ { a = 1 } ^ M f ( \\mu , \\lambda _ a ) \\end{align*}"}
-{"id": "6534.png", "formula": "\\begin{align*} \\big \\| \\big ( ( A _ m - A _ n ) v _ n \\big ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } ^ p \\le c _ \\star \\sum _ { n = k } ^ { m - 1 } \\| A _ { n + 1 } - A _ n \\| E _ n . \\end{align*}"}
-{"id": "9570.png", "formula": "\\begin{align*} f ^ { ( 1 ) } \\psi _ { n } ^ { ( 0 ) } = \\sum _ { \\substack { i \\\\ \\lambda ^ { ( 0 ) } _ { i } \\neq \\lambda _ { n } ^ { ( 0 ) } } } \\langle \\psi _ { n } ^ { ( 0 ) } , f ^ { ( 1 ) } \\psi _ { i } ^ { ( 0 ) } \\rangle \\psi _ { i } ^ { ( 0 ) } . \\end{align*}"}
-{"id": "86.png", "formula": "\\begin{align*} ( n , A ) ( n , f ) = n ( n , A , f ) \\end{align*}"}
-{"id": "8825.png", "formula": "\\begin{align*} Y _ g ( P _ { i , n } , z ^ { 1 / m } ) = Y _ g ( \\frac { T ^ { n } } { n ! } P _ { i , 0 } , z ^ { 1 / m } ) & = \\frac { \\partial ^ { n } _ z } { n ! } Y _ g ( P _ { i , 0 } , z ^ { 1 / m } ) \\\\ & = \\frac { \\partial ^ { n } _ z } { n ! } P _ i ( x _ 1 ( z ^ { 1 / m } ) , . . . , x _ k ( z ^ { 1 / m } ) ) \\end{align*}"}
-{"id": "4160.png", "formula": "\\begin{align*} | S ^ \\nu _ { h _ 2 } ( n ) | ^ 2 = | S ^ \\nu _ f ( n ) | ^ 2 + | S ^ \\nu _ g ( n ) | ^ 2 + 2 \\Im \\left ( S ^ \\nu _ f ( n ) \\overline { S ^ \\nu _ g ( n ) } \\right ) . \\end{align*}"}
-{"id": "1748.png", "formula": "\\begin{align*} f ( u ( x _ n ) ) = \\frac { 1 } { 2 } \\int _ { \\R ^ N } \\frac { 2 u ( x _ n ) - u ( x _ n + y ) - u ( x _ n - y ) } { | y | ^ { N + 2 s } } \\ ; d y . \\end{align*}"}
-{"id": "3172.png", "formula": "\\begin{align*} C _ s ( \\{ ( W _ t ^ { \\lambda } , { V } _ t ^ { \\lambda } ) : t \\in \\theta \\} ) = C _ s ( \\{ ( W _ t ^ { \\lambda } , { V } _ t ^ { \\lambda } ) : t \\in \\theta \\} , c r ) \\end{align*}"}
-{"id": "3133.png", "formula": "\\begin{align*} P ^ T L _ 5 ( \\lambda ) P = \\left [ \\begin{array} { c c c | c } - P _ 4 & \\lambda P _ 4 - P _ 3 & \\lambda P _ 3 & 0 \\\\ \\lambda P _ 4 - P _ 3 & \\lambda P _ 3 - P _ 2 & \\lambda P _ 2 & - I _ n \\\\ \\lambda P _ 3 & \\lambda P _ 2 & \\lambda P _ 1 + P _ 0 & \\lambda I _ n \\\\ \\hline 0 & - I _ n & \\lambda I _ n & 0 , \\end{array} \\right ] , \\end{align*}"}
-{"id": "5406.png", "formula": "\\begin{align*} e ( B ) \\leq \\ell \\frac { n ^ 2 } { 2 } + ( k - \\alpha - \\ell - x ) \\frac { n ^ 2 } { 2 } = ( k - \\alpha - x ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "2767.png", "formula": "\\begin{align*} D ^ { H } _ { s } F = \\Delta _ { x } F ( B ^ { H } _ { t } ) I _ { [ 0 , t ] } ( s ) , \\ \\ 0 \\leq s \\leq T . \\end{align*}"}
-{"id": "6219.png", "formula": "\\begin{align*} \\mathcal { P } _ { 1 } = & - ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } ( 2 s _ { 0 } M + D ) , \\\\ \\mathcal { P } _ { 2 } = & - ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } M , \\\\ \\mathsf { Q } = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } F , \\end{align*}"}
-{"id": "9754.png", "formula": "\\begin{align*} V _ { \\mathrm { D } } ( \\vec { x } ) : = \\ \\max \\limits _ { \\boldsymbol { \\lambda } \\geq \\vec { 0 } _ { m } } \\ \\inf \\limits _ { \\vec { z } } \\ L ( \\vec { z } , \\boldsymbol { \\lambda } , \\vec { x } ) , \\end{align*}"}
-{"id": "4823.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\| K - A _ n K \\| | = \\lim _ { n \\to \\infty } | \\| K - K A _ n \\| | = 0 \\end{align*}"}
-{"id": "8334.png", "formula": "\\begin{align*} ( \\wp \\circ h ) = ( h \\circ \\wp ) = g . \\end{align*}"}
-{"id": "380.png", "formula": "\\begin{align*} \\phi ( g ) = 2 t \\sum _ { n = 1 } ^ { \\infty } \\rho _ \\phi ( n ) K _ { s - \\frac { 1 } { 2 } } \\left ( 2 \\pi n t ^ 2 \\right ) \\cos ( 2 \\pi n u ) \\end{align*}"}
-{"id": "7834.png", "formula": "\\begin{align*} S : = \\sum _ k S ^ k , \\ \\ T : = \\bigcap _ k T ^ k . \\end{align*}"}
-{"id": "4696.png", "formula": "\\begin{align*} z & = 0 ^ { m _ 0 } v _ 1 0 ^ { m _ 1 } v _ 2 \\dots v _ t 0 ^ { m _ t } , \\\\ z ' & = 0 ^ { m ' _ 0 } v _ 1 0 ^ { m ' _ 1 } v _ 2 \\dots v _ t 0 ^ { m ' _ t } , \\end{align*}"}
-{"id": "231.png", "formula": "\\begin{align*} G _ { n , x , y } ( u , v ) & : = \\mathbb { P } ( M _ 1 \\geq k , M _ 2 \\geq k ) , \\end{align*}"}
-{"id": "8738.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} [ c ] { l } \\overset { \\cdot } { w } \\left ( t \\right ) = A w \\left ( t \\right ) + G u \\left ( t \\right ) , \\\\ w \\left ( 0 \\right ) = h \\in K , \\end{array} \\right . \\end{align*}"}
-{"id": "8147.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - { \\rm d i v } ( | \\nabla u | ^ { p - 2 } \\nabla u ) = \\lambda | x | ^ { - \\beta p } | u | ^ { p - 2 } u & \\mbox { i n } \\Omega \\\\ u = 0 & \\mbox { o n } \\partial \\Omega \\end{array} \\right . \\\\ \\\\ \\end{align*}"}
-{"id": "2275.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) , \\end{aligned} \\end{align*}"}
-{"id": "5626.png", "formula": "\\begin{align*} b _ i ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) \\} | , \\\\ b _ { i p } ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) ~ | ~ w = \\overline { w } \\} | , \\\\ b _ { i n } ( n ) & = | \\{ w \\in \\mathcal { A } _ i ( n ) ~ | ~ w \\neq \\overline { w } \\} | . \\end{align*}"}
-{"id": "9008.png", "formula": "\\begin{align*} y ( t ) = y ( 0 ) + \\frac { \\alpha t ^ { \\alpha - 1 } } { B ( \\alpha ) \\Gamma ( \\alpha ) } , \\end{align*}"}
-{"id": "8201.png", "formula": "\\begin{align*} \\frac { x _ 3 ^ { p - 1 } d x _ 3 + d ( \\overline { i } ^ * _ c \\overline { \\Delta } _ 3 ) } { ( a _ 1 - a _ 2 ) ^ p x _ 1 ^ p x _ 2 ^ p } = A _ { p - 1 } ( c _ 1 , c _ 2 ) ^ { - 1 } \\frac { d x _ 3 } { ( a _ 1 - a _ 2 ) x _ 2 x _ 3 } \\end{align*}"}
-{"id": "8451.png", "formula": "\\begin{align*} \\frac { d x } { d t } = F ( x ) , ~ ~ ~ ~ ~ x | _ { t = 0 } = x _ { 0 } \\in \\mathcal { U } , \\end{align*}"}
-{"id": "3674.png", "formula": "\\begin{align*} \\frac { d Z } { d t } = \\int \\left [ Z _ { \\zeta } \\frac { \\partial \\zeta } { \\partial t } + Z _ { h } \\frac { \\partial h } { \\partial t } \\right ] d A = 0 ~ . \\end{align*}"}
-{"id": "9370.png", "formula": "\\begin{align*} ( S _ { m a x } f , g ) - ( f , S _ { m a x } g ) = ( \\Gamma _ 1 f ) \\cdot \\Gamma _ 0 g - ( \\Gamma _ 0 f ) \\cdot \\Gamma _ 1 g , \\end{align*}"}
-{"id": "3123.png", "formula": "\\begin{align*} P ( \\lambda ) = \\lambda ^ d P + Q ( \\lambda ) \\mbox { a n d } \\widehat { P } ( \\lambda ) = \\begin{bmatrix} - P & \\lambda ^ t P \\\\ \\lambda ^ t P & Q ( \\lambda ) \\end{bmatrix} \\end{align*}"}
-{"id": "7956.png", "formula": "\\begin{align*} j _ l ( z ) i _ l '' ( z ) - i _ l ( z ) j _ l '' ( z ) = z ^ { 1 - N } \\left ( 2 z I _ { \\frac { N } { 2 } - 1 + l } ( z ) J _ { \\frac { N } { 2 } - 1 + l } ( z ) - ( N - 1 ) C _ l ^ + ( z ) \\right ) , \\end{align*}"}
-{"id": "5774.png", "formula": "\\begin{align*} a ( \\lambda ) = \\prod _ { j = 1 } ^ N \\alpha ( \\lambda , \\xi _ j ) , d ( \\lambda ) = \\prod _ { j = 1 } ^ N \\delta ( \\lambda , \\xi _ j ) . \\end{align*}"}
-{"id": "8474.png", "formula": "\\begin{align*} ( ( \\mathbb { I } - \\mathbb { P } ) D ^ { \\alpha } v ^ { \\varepsilon } , D ^ { \\alpha } v ^ { \\varepsilon } ) _ { 0 } = ( ( \\mathbb { I } - \\mathbb { P } ) D ^ { \\alpha } v ^ { \\varepsilon } , D ^ { \\alpha } ( ( \\mathbb { I } - \\mathbb { P } ) v ^ { \\varepsilon } + \\mathbb { P } v ^ { \\varepsilon } ) ) _ { 0 } = | | ( \\mathbb { I } - \\mathbb { P } ) v ^ { \\varepsilon } | | _ { 0 } ^ { 2 } . \\end{align*}"}
-{"id": "1316.png", "formula": "\\begin{align*} r ( n ) = \\sum _ { p _ { 1 } + p _ { 2 } ^ 2 + p _ { 3 } ^ { 2 } = n } \\log p _ { 1 } \\log p _ { 2 } \\log p _ { 3 } , \\end{align*}"}
-{"id": "2481.png", "formula": "\\begin{align*} & \\prod _ { k = 0 } ^ { d _ j - 2 } q ^ { - ( s _ j + k ) \\dim V _ j ^ I } \\cdot \\det ( - \\rho _ j ( \\Phi ) | V _ j ^ I ) \\\\ & = \\det ( - \\rho _ j ( \\Phi ) | V _ j ^ I ) ^ { d _ j - 1 } \\cdot q ^ { - s _ j ( d _ j - 1 ) \\dim V _ j ^ I } \\cdot q ^ { - ( 1 + 2 + \\ldots + d _ j - 2 ) \\dim V _ j ^ I } \\\\ & = \\det ( - \\rho _ j ( \\Phi ) | V _ j ^ I ) ^ { d _ j - 1 } \\cdot q ^ { - s _ j ( d _ j - 1 ) \\dim V _ j ^ I } \\cdot q ^ { - \\frac { 1 } { 2 } ( d _ j - 2 ) ( d _ j - 1 ) \\dim V _ j ^ I } \\end{align*}"}
-{"id": "9233.png", "formula": "\\begin{align*} \\mathcal { A } _ 0 = \\{ u \\in \\mathcal { A } ; u ( t , x ) = u ( t ) x \\} . \\end{align*}"}
-{"id": "3718.png", "formula": "\\begin{align*} \\gamma ( u , x a ) = \\psi ( 2 ^ { - 1 } \\langle x a u , x a \\rangle ) = \\psi ( 2 ^ { - 1 } \\langle x a u a ^ { * } , x \\rangle ) = \\gamma ( ( a u a ^ { * } , x ) ) . \\end{align*}"}
-{"id": "5902.png", "formula": "\\begin{align*} x ^ { \\prime } & = v , v ^ { \\prime } = F \\left ( x , v \\right ) \\\\ x \\left ( 0 \\right ) & = x _ { 0 } , v \\left ( 0 \\right ) = v _ { 0 } \\end{align*}"}
-{"id": "3985.png", "formula": "\\begin{align*} \\sigma ( \\vect { \\delta } - j \\vect { e } _ { i } ) = \\sigma ( \\vect { \\delta } ) + j ( k + 1 ) \\geq \\sigma ( \\vect { \\delta } ) - ( k + 1 ) \\delta _ { i } \\geq \\sigma ( \\vect { \\delta } ) ( 1 - ( k + 1 ) / k ) \\geq 0 . \\end{align*}"}
-{"id": "7281.png", "formula": "\\begin{align*} d : = \\frac { d _ 1 + \\cdots + d _ t } { t - 1 } . \\end{align*}"}
-{"id": "5792.png", "formula": "\\begin{align*} L _ { - 1 } A = T A \\end{align*}"}
-{"id": "9843.png", "formula": "\\begin{align*} \\Xi ( f ) = \\bar { \\Xi } ( f ) \\frac { H _ { \\rm S D } ( f ) H _ { \\rm R D } ^ \\dagger ( f ) H _ { \\rm S R } ^ \\dagger ( f ) } { \\left | H _ { \\rm S D } ( f ) H _ { \\rm R D } ^ \\dagger ( f ) H _ { \\rm S R } ^ \\dagger ( f ) \\right | } , \\end{align*}"}
-{"id": "4308.png", "formula": "\\begin{align*} v _ k = \\sum _ { p = 1 } ^ r \\lambda ^ p _ k u _ p \\end{align*}"}
-{"id": "5179.png", "formula": "\\begin{gather*} C _ { \\ell } = \\bigoplus _ { 0 \\leqslant k \\leqslant \\ell , \\ , [ k ] = [ \\ell ] } \\partial C _ { k } . \\end{gather*}"}
-{"id": "453.png", "formula": "\\begin{align*} ( A _ { 2 1 } + A _ { 3 1 } ) x _ 1 + ( A _ { 2 2 } + A _ { 3 2 } ) x _ 2 & = b _ 3 + b _ 2 \\\\ ( - A _ { 2 1 } + A _ { 3 1 } ) x _ 1 + ( - A _ { 2 2 } + A _ { 3 2 } ) x _ 2 & = b _ 3 - b _ 2 . \\end{align*}"}
-{"id": "2550.png", "formula": "\\begin{align*} u _ { 0 , 1 } ( x ) = & \\frac 1 { \\sqrt { 2 } } + \\frac { \\bf { i } } { \\sqrt { 2 } } , u _ { 0 , 2 } ( x ) = 1 , u _ { 0 , 3 } ( x ) = 2 x , \\\\ u _ { 0 , 4 } ( x ) = & \\left ( 1 - \\sqrt { \\frac { \\pi } { 2 } ( \\exp { \\frac 1 4 } - 1 ) } \\right ) ( 1 - \\exp { ( x ( 1 - x ) ) } ) , \\\\ u _ { 0 , 5 } ( x ) = & c _ * ( \\frac { x } { \\sqrt { 2 } } ) \\exp { ( { \\bf { i } } \\frac { x } 2 ) } \\end{align*}"}
-{"id": "6023.png", "formula": "\\begin{align*} \\hat { \\Omega } _ { k j } { } ^ k { } _ { l } = \\hat { R } _ { k j } { } ^ k { } _ { l } + 6 \\varepsilon \\hat { g } _ { j l } . \\end{align*}"}
-{"id": "9392.png", "formula": "\\begin{align*} \\| ( \\lambda - \\lambda _ 0 ) \\Xi ( \\lambda ) { u } _ { \\lambda ^ * } \\| = \\left | \\frac { \\lambda - \\lambda _ 0 } { a - \\widetilde { W } _ { \\lambda } } \\right | \\| \\gamma ( \\lambda ) \\gamma ( \\lambda ^ * ) ^ \\dag { u } _ { \\lambda ^ * } \\| . \\end{align*}"}
-{"id": "9453.png", "formula": "\\begin{align*} p ( T ) : = \\sup \\Big \\{ d ( x , y ) : x , y \\in X \\quad \\mathrm { a n d } T _ { x y } \\neq 0 \\Big \\} . \\end{align*}"}
-{"id": "8623.png", "formula": "\\begin{align*} u _ { \\tau ( a ) } \\overline { N } ( a ^ { \\ast } ) ( x , y , z ) & = c _ { k } ^ { - 1 } u _ { \\tau ( a ) } ( \\pi _ { e _ { k } a } i ( y ) + \\pi _ { e _ { k } a } j ' \\sigma _ { 2 } ( z ) ) \\\\ & = c _ { k } ^ { - 1 } ( \\pi ^ { 1 } _ { e _ { k } a } i _ { 1 } u _ { 2 } ( y ) + \\pi ^ { 1 } _ { e _ { k } a } j _ { 1 } ' u _ { 3 } \\sigma _ { 2 } ( z ) ) \\end{align*}"}
-{"id": "2669.png", "formula": "\\begin{align*} \\langle T _ 1 \\wedge \\ldots \\wedge T _ p \\rangle = \\langle \\theta _ { \\varphi _ 1 } \\wedge \\ldots \\wedge \\theta _ { \\varphi _ 1 } \\rangle . \\end{align*}"}
-{"id": "9816.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ { n } } \\ldots \\int _ { \\mathbb R ^ { n } } \\prod _ { k = 1 } ^ { M } F ^ { ( k ) } ( x _ { 1 } , \\ldots , x _ { \\ell } ) \\prod _ { i = 1 } ^ { N } f _ { i } ^ { \\ast } \\left ( \\sum _ { j = 1 } ^ { \\ell } a _ { i j } x _ { j } \\right ) d \\mu ( x _ { \\ell } ) \\ldots d \\mu ( x _ { 1 } ) . \\end{align*}"}
-{"id": "4465.png", "formula": "\\begin{align*} n ^ { 1 / 2 } \\{ \\hat { H } _ n ^ w - H ( f ) \\} = \\frac { 1 } { n ^ { 1 / 2 } } \\sum _ { i = 1 } ^ n \\tilde { \\psi } _ f ( X _ i ) + o _ p ( 1 ) \\end{align*}"}
-{"id": "7194.png", "formula": "\\begin{align*} v ' + \\frac { v ^ 2 } { 2 } = \\frac { C _ 1 } { ( 1 + t ) ^ 2 } + \\frac { C _ 3 } { ( 1 - t ^ 2 ) ^ 2 } , v ( 0 ) = 0 . \\end{align*}"}
-{"id": "2029.png", "formula": "\\begin{align*} f ( a _ 1 , \\ldots , a _ 4 , b _ 1 , \\ldots , b _ 4 ) = ( a _ 1 a _ 3 - a _ 2 b _ 4 , a _ 1 a _ 4 + a _ 2 b _ 3 , - a _ 4 b _ 2 + b _ 1 b _ 3 , a _ 3 b _ 2 + b _ 1 b _ 4 , a _ 1 b _ 1 + a _ 2 b _ 2 ) . \\end{align*}"}
-{"id": "1239.png", "formula": "\\begin{align*} m _ 2 = \\min \\{ j \\in \\mathbb { N } : W ( j ) > 2 ^ { k _ 2 - 2 } \\} . \\end{align*}"}
-{"id": "5562.png", "formula": "\\begin{align*} | c _ { n } ( x ) | & \\leq C x ^ n \\int _ { 0 \\leq \\xi _ 1 \\leq \\xi _ 2 \\leq \\dots \\leq \\xi _ { n + 1 } = x } \\left [ \\prod _ { j = 1 } ^ n \\rho ( \\xi _ j ) \\right ] d \\xi _ 1 d \\xi _ 2 \\dots d \\xi _ n \\\\ & \\leq C \\frac { x ^ n } { n ! } \\left ( \\int _ { [ 0 , x ] } \\rho ( \\xi ) d \\xi \\right ) ^ n \\stackrel { \\emph { d e f } } { = } C \\frac { x ^ n } { n ! } ( M ( x ) ) ^ n , \\end{align*}"}
-{"id": "3290.png", "formula": "\\begin{align*} v _ { 1 } \\wedge v _ { - 1 } + q ^ { 2 } v _ { - 1 } \\wedge v _ { 1 } - q ^ { 2 } v _ { 0 } \\wedge v _ { 0 } = 0 , v _ { 1 } \\wedge v _ { - 1 } + q ^ { - 4 } v _ { - 1 } \\wedge v _ { 1 } + q ^ { - 1 } ( q + q ^ { - 1 } ) v _ { 0 } \\wedge v _ { 0 } = 0 . \\end{align*}"}
-{"id": "1552.png", "formula": "\\begin{align*} E ( \\mu ) = \\left \\{ \\begin{array} { l l } E _ { G } ( \\mu ) + \\Vert f \\Vert _ { H ^ { k } ( \\Omega ) } ^ { 2 } = \\int _ { \\Omega } G ( f ( x ) ) d x + \\Vert f \\Vert _ { H ^ { k } ( \\Omega ) } ^ { 2 } , & \\mbox { i f } \\ f = \\frac { d \\mu } { d x } \\ \\mbox { a n d } \\ f \\geq \\alpha , \\\\ + \\infty & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "2331.png", "formula": "\\begin{align*} E _ { p , i } ( x ) = 0 ~ { \\rm i f } ~ p ( x ) = 0 ; ~ \\frac { 1 } { E _ { p , i } ( x ) } = \\frac { 1 } { N _ p ( x ) } - \\sum _ { j = 1 , j \\neq i } ^ d \\frac { 1 } { x - z _ j } ~ { \\rm o t h e r w i s e } ; \\end{align*}"}
-{"id": "3508.png", "formula": "\\begin{align*} ( D { \\Phi } ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) = D \\Phi | ^ * _ { ( g , \\pi ) } ( f , X ) + \\left ( \\tfrac { 1 } { 4 } [ X _ i ( \\textup { d i v } _ g \\pi + W ) _ j + X _ j ( \\textup { d i v } _ g \\pi + W ) _ i ] , 0 \\right ) , \\end{align*}"}
-{"id": "3161.png", "formula": "\\begin{align*} \\max _ { t ^ n \\in \\theta ^ n } \\frac { 1 } { \\left | \\Gamma \\right | } \\sum _ { \\gamma = 1 } ^ { \\left | \\Gamma \\right | } \\chi \\left ( R _ { u n i } ; Z _ { \\mathcal { C } ^ { \\gamma } , t ^ n } \\right ) \\leq \\epsilon _ { n } \\end{align*}"}
-{"id": "1488.png", "formula": "\\begin{align*} D ( \\beta ) = \\{ x \\in \\mathbb { R } ^ n : \\left < x , \\beta \\right > = 0 , \\ \\left < x , \\beta ' \\right > \\leq 0 \\beta ' \\subseteq \\beta \\} . \\end{align*}"}
-{"id": "4936.png", "formula": "\\begin{align*} \\| D ^ * x _ 0 \\| _ 1 + \\rho \\geq \\| D ^ * \\hat { x } \\| _ 1 & = \\| D ^ * x _ 0 + D ^ * h \\| _ 1 \\\\ & = \\| D ^ * _ { T _ 0 } x _ 0 + D ^ * _ { T _ 0 } h + D ^ * _ { T _ 0 ^ c } x _ 0 + D ^ * _ { T _ 0 ^ c } h \\| _ 1 \\\\ & \\geq \\| D ^ * _ { T _ 0 } x _ 0 \\| _ 1 - \\| D ^ * _ { T _ 0 } h \\| _ 1 - \\| D ^ * _ { T _ 0 ^ c } x _ 0 \\| _ 1 + \\| D ^ * _ { T _ 0 ^ c } h \\| _ 1 , \\end{align*}"}
-{"id": "3532.png", "formula": "\\begin{align*} \\bar { \\mu } = \\mu + \\psi \\mbox { a n d } \\bar { J } = Y - \\frac { 1 } { 2 } h \\cdot _ g Y . \\end{align*}"}
-{"id": "8042.png", "formula": "\\begin{align*} s _ { 1 , 2 } ^ { x _ { 1 , 2 } } s _ { 2 , 3 } ^ { x _ { 2 , 3 } } \\cdots { } & s _ { m - 1 , m } ^ { x _ { m - 1 , m } } s _ { 1 , 3 } ^ { x _ { 1 , 3 } } \\cdots s _ { 1 , m - 1 } ^ { x _ { 1 , m - 1 } } s _ { 2 , m } ^ { x _ { 2 , m } } s _ { 1 , m } ^ { x _ { 1 , m } } \\cdot s _ { 1 , 2 } ^ { - 1 } \\\\ & = s _ { 1 , 2 } ^ { x _ { 1 , 2 } } s _ { 2 , 3 } ^ { x _ { 2 , 3 } } \\cdots s _ { m - 1 , m } ^ { x _ { m - 1 , m } } s _ { 1 , 3 } ^ { x _ { 1 , 3 } } \\cdots s _ { 1 , m - 1 } ^ { x _ { 1 , m - 1 } } \\cdot s _ { 1 , 2 } ^ { - 1 } \\cdot s _ { 2 , m } ^ { x _ { 2 , m } } \\ , s _ { 1 , m } ^ { x _ { 1 , m } + x _ { 2 , m } } . \\end{align*}"}
-{"id": "4942.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { M } \\lambda _ j \\beta _ j - d \\beta _ i & = D ^ * _ { S _ 0 } h + h ^ { ( 1 ) } + \\mu \\cdot h ^ { ( 2 ) } - d \\beta _ i \\\\ & = ( 1 - \\mu - d ) ( D ^ * _ { S _ 0 } h + h ^ { ( 1 ) } ) - d \\mu u _ i + \\mu D ^ * h . \\end{align*}"}
-{"id": "1730.png", "formula": "\\begin{align*} \\hat { R } _ { i j } { } ^ k { } _ l \\hat { K } ^ l { } _ k = 2 \\ , \\hat { K } ^ l { } _ i \\hat { R } _ { k l } { } ^ k { } _ j \\end{align*}"}
-{"id": "5732.png", "formula": "\\begin{align*} Z _ { t } = \\sum _ { j \\in J _ { \\phi } } \\phi _ { j } \\left ( Z _ { t - j } + e _ { t - j } \\right ) + \\sum _ { j \\in J _ { \\theta } } \\theta _ j e _ { t - j } \\end{align*}"}
-{"id": "8811.png", "formula": "\\begin{align*} \\frac { \\int _ { \\Sigma } u \\ , d v o l _ { \\Sigma } } { A } = \\frac { L - n A } { n H A } \\ , . \\end{align*}"}
-{"id": "2819.png", "formula": "\\begin{align*} \\sigma _ { i l , j k } ^ { ( \\kappa ) } = \\sigma _ { i j } ^ { ( \\kappa ) } 1 _ { \\{ l = k \\} } \\end{align*}"}
-{"id": "9467.png", "formula": "\\begin{align*} \\lambda _ { f } = e ^ { \\pi ^ { * } f } \\lambda = e ^ { \\pi ^ { * } f } ( d \\theta + \\pi ^ { * } \\beta ) . \\end{align*}"}
-{"id": "7142.png", "formula": "\\begin{align*} G ( x , y ) = \\frac 1 { 4 \\pi | x - y | } - \\frac 1 { 4 \\pi | x - y ^ * | } . \\end{align*}"}
-{"id": "3286.png", "formula": "\\begin{align*} \\hat { R } F ( v _ { 1 } \\otimes v _ { 0 } ) & = [ 2 ] ^ { 1 / 2 } ( \\hat { R } ( v _ { 0 } \\otimes v _ { 0 } ) + \\hat { R } ( v _ { 1 } \\otimes v _ { - 1 } ) ) \\\\ & = [ 2 ] ^ { 1 / 2 } ( v _ { 0 } \\otimes v _ { 0 } + q ^ { - 2 } ( q ^ { 2 } - q ^ { - 2 } ) v _ { 1 } \\otimes v _ { - 1 } + \\hat { R } ( v _ { 1 } \\otimes v _ { - 1 } ) ) . \\end{align*}"}
-{"id": "6990.png", "formula": "\\begin{align*} \\delta _ H \\Theta ^ 2 ( x , y ) ( a , b ) + \\delta _ L \\Theta ^ 1 ( x , y ) ( a , b ) = 0 \\end{align*}"}
-{"id": "956.png", "formula": "\\begin{align*} f = \\bar \\partial _ M \\bar \\partial ^ * _ M G _ q f + \\bar \\partial ^ * _ M \\bar \\partial _ M G _ q f \\ ; . \\end{align*}"}
-{"id": "7429.png", "formula": "\\begin{align*} \\overline { s _ { i ( 1 ) } \\dots s _ { i ( m - 1 ) } } e _ { i ( m ) } ^ { - 1 } = b \\overline { s _ { i ( 1 ) } \\dots s _ { i ( m - 1 ) } } \\end{align*}"}
-{"id": "7086.png", "formula": "\\begin{align*} \\forall u \\in \\mathcal { U } , n \\in \\N , \\mu ( C ( u , n ) ) = \\sum _ { j = n } ^ \\infty \\mu ( B ( u . j ) ) = \\sum _ { j = n } ^ \\infty \\nu ( B ( u . j ) ) = \\nu ( C ( u , n ) ) , \\end{align*}"}
-{"id": "2976.png", "formula": "\\begin{align*} \\Theta ( A ) = 1 / 2 \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ \\alpha _ i , \\alpha _ j ] . \\end{align*}"}
-{"id": "2433.png", "formula": "\\begin{align*} & \\| \\{ a _ k \\| \\Box _ k ^ { \\alpha _ 1 } T _ m \\| _ { M _ 1 \\rightarrow M _ 2 } \\| \\Box _ k ^ { \\alpha _ 1 , \\ast } f _ k \\| _ { M _ 1 } \\} | ~ l _ { q _ 2 } ^ { s _ 2 , \\alpha _ 1 } ( E _ m ) \\| \\\\ \\lesssim & \\| \\{ a _ k \\| \\Box _ k ^ { \\alpha _ 1 } T _ m \\Box _ k ^ { \\alpha _ 1 , \\ast } f _ k \\| _ { M _ 2 } \\} | ~ l _ { q _ 2 } ^ { s _ 2 , \\alpha _ 1 } ( E _ m ) \\| . \\end{align*}"}
-{"id": "8968.png", "formula": "\\begin{align*} [ u ] _ { D ^ { s , p } ( \\mathbb { R } ^ N ) } : = \\left ( \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | u ( x ) - u ( y ) | ^ p } { | x - y | ^ { N + s p } } \\ , d x d y \\right ) ^ { 1 / p } \\end{align*}"}
-{"id": "8666.png", "formula": "\\begin{gather*} \\nabla _ { Q ^ { 1 / 2 } h } f ( x ) = \\langle D _ Q f ( x ) , h \\rangle _ H . \\end{gather*}"}
-{"id": "2355.png", "formula": "\\begin{align*} \\Phi _ t ( x _ 0 ) : = \\begin{cases} x ( t ; x _ 0 ) , & \\mbox { i f } \\ t \\in I _ { x _ 0 } , \\\\ x ( \\tau _ \\partial ( x _ 0 ) ; x _ 0 ) , & \\mbox { o t h e r w i s e } . \\end{cases} \\end{align*}"}
-{"id": "2567.png", "formula": "\\begin{align*} \\bigl ( \\tfrac 1 { q _ 1 } , \\tfrac 1 { r _ 1 } \\bigr ) = \\bigl ( \\tfrac 1 { q _ 2 } , \\tfrac 1 { r _ 2 } \\bigr ) : = \\bigl ( \\tfrac { d p - 2 } { 2 ( p + 2 ) } , \\tfrac { 2 d + 4 - d p } { 2 d ( p + 2 ) } \\bigr ) , \\quad \\bigl ( \\tfrac 1 { \\tilde { q } } , \\tfrac 1 { \\tilde { r } } \\bigr ) : = \\bigl ( \\tfrac { 4 - ( d - 1 ) p } { p ( p + 2 ) } , \\tfrac { 2 d p - 4 } { d p ( p + 2 ) } \\bigr ) . \\end{align*}"}
-{"id": "2290.png", "formula": "\\begin{align*} E _ m \\le C \\sum _ { n = k } ^ { m - 1 } a _ n E _ n + C F _ m , m = k , \\dotsc , N , \\end{align*}"}
-{"id": "2692.png", "formula": "\\begin{align*} \\theta ^ n = ( d d ^ c g ) ^ n \\geq e ^ { \\beta ( g + c ) } \\left [ \\theta _ + ^ n + \\varepsilon ( 1 - \\lambda ) ^ n \\omega ^ n \\right ] . \\end{align*}"}
-{"id": "2280.png", "formula": "\\begin{align*} \\tilde \\lambda = \\sup _ { t \\in [ 0 , T ] } \\sum _ { i , j = 1 } ^ d \\sup _ { x \\in \\varOmega } \\sup _ { \\substack { | \\xi - u ( x , t ) | \\le r \\\\ * [ 2 p t ] | \\vec \\eta - \\nabla u ( x , t ) | \\le r } } \\bigg | \\frac { \\partial g _ i ( \\xi , \\vec \\eta , x , t ) } { \\partial \\eta _ j } \\bigg | \\ , . \\end{align*}"}
-{"id": "1137.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { U L } = & \\lim _ { \\rho \\to 0 } \\tilde { \\gamma } _ k ^ \\mathrm { U L } \\\\ \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { D L } = & \\lim _ { \\rho \\to 0 } \\tilde { \\gamma } _ k ^ \\mathrm { D L } . \\end{align*}"}
-{"id": "8032.png", "formula": "\\begin{align*} d C \\cdot C ^ { - 1 } = C B C ^ { - 1 } - B ' . \\end{align*}"}
-{"id": "3149.png", "formula": "\\begin{align*} & \\lVert L ^ { - 1 } \\sum _ { i \\notin \\mathsf { M } ' } \\Pi \\cdot \\Pi _ { i } \\cdot \\rho _ i \\cdot \\Pi _ { i } \\cdot \\Pi \\rVert _ 1 \\allowdisplaybreaks \\\\ & = \\mathrm { t r } \\left ( L ^ { - 1 } \\sum _ { i \\notin \\mathsf { M } ' } \\Pi \\cdot \\Pi _ { i } \\cdot \\rho _ i \\cdot \\Pi _ { i } \\cdot \\Pi \\right ) \\allowdisplaybreaks \\\\ & \\leq 1 - p ( \\mathsf { M } ' ) + 2 \\sqrt { 1 - p ( \\mathsf { M } ' ) } + 2 0 \\sqrt [ 8 ] { \\epsilon } \\end{align*}"}
-{"id": "7833.png", "formula": "\\begin{align*} X _ { k + 1 } ^ { ( 1 ) } C _ k ^ { ( 1 2 ) } = C _ k ^ { ( 1 2 ) } X _ k ^ { ( 2 ) } , \\ \\ \\ X _ k ^ { ( 1 ) } D _ k ^ { ( 1 2 ) } = D _ k ^ { ( 1 2 ) } X _ { k + 1 } ^ { ( 2 ) } . \\end{align*}"}
-{"id": "1830.png", "formula": "\\begin{align*} g ^ { - 1 } = \\begin{pmatrix} \\pi ^ { - 1 } s _ { i } ^ { - 3 } + O ( s _ { i } ^ { - 2 } ) & O ( 1 ) & O ( 1 ) \\\\ O ( 1 ) & \\pi ^ { - 1 } s _ { j } ^ { - 3 } + O ( s _ { j } ^ { - 2 } ) & O ( 1 ) \\\\ O ( 1 ) & O ( 1 ) & \\psi _ { z \\bar z } ^ { - 1 } + \\sum O ( s _ { i } ^ { 2 } ) \\end{pmatrix} . \\end{align*}"}
-{"id": "9148.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p k _ i & \\leq p m / 2 + \\sum _ { i = 1 } ^ p \\ell _ i / 2 + \\sum _ { i = 1 } ^ p r _ i / 2 \\\\ & \\leq p m / 2 + t \\ell / 2 + ( t - 1 ) r / 2 \\\\ & = ( ( p + t ) m - r ) / 2 . \\end{align*}"}
-{"id": "6863.png", "formula": "\\begin{align*} \\limsup _ { J \\to \\infty } \\limsup _ { n \\to \\infty } \\| ( i \\partial _ t + \\Delta ) u _ n ^ J - F ( u _ n ^ J ) \\| _ { N ( \\tilde I _ n ) } = 0 . \\end{align*}"}
-{"id": "4275.png", "formula": "\\begin{align*} \\P ( Z _ n ^ * \\leq - a ) & = \\P ( \\forall k \\leq n \\ , : \\ , Z _ k + f ( k ) \\leq - a + f ( k ) ) \\\\ & \\geq \\P ( \\forall k \\leq n \\ , : \\ , Z _ k + f ( k ) \\leq a ) \\\\ & \\geq \\P ( Z _ n ^ * \\leq a ) \\ , \\exp ( - \\sqrt { 2 \\| f \\| _ { \\mathcal H } ^ 2 \\log ( 1 / \\P ( Z _ n ^ * \\leq a ) ) } - \\| f \\| _ { \\mathcal H } ^ 2 / 2 ) \\ , . \\end{align*}"}
-{"id": "1740.png", "formula": "\\begin{align*} P _ t ( x , y ) = \\pi ^ { - 1 / 2 } \\int _ 0 ^ { \\infty } e ^ { - u } T _ { t ^ 2 \\slash ( 4 u ) } ( x , y ) \\frac { d u } { \\sqrt { u } } \\end{align*}"}
-{"id": "8138.png", "formula": "\\begin{align*} a ^ * = a _ * . \\end{align*}"}
-{"id": "7328.png", "formula": "\\begin{align*} v _ { 1 } \\wedge v _ { - 1 } + q ^ { 2 } v _ { - 1 } \\wedge v _ { 1 } - q ^ { 2 } v _ { 0 } \\wedge v _ { 0 } = 0 , v _ { 1 } \\wedge v _ { - 1 } + q ^ { - 4 } v _ { - 1 } \\wedge v _ { 1 } + q ^ { - 1 } ( q + q ^ { - 1 } ) v _ { 0 } \\wedge v _ { 0 } = 0 . \\end{align*}"}
-{"id": "3752.png", "formula": "\\begin{align*} & \\left | \\left ( h _ 3 ( t + \\delta ) - h _ 3 ( t ) \\right ) / \\delta - d ( t ) \\right | \\\\ & \\leq \\left | \\sum _ { k = 3 } ^ { k _ 0 } p _ { \\beta } \\left ( \\left ( q ^ { * k } \\right ) ^ { \\prime } ( t _ k ) - \\left ( q ^ { * k } \\right ) ^ { \\prime } ( t ) \\right ) \\right | + \\left | \\sum _ { k \\geq k _ 0 } \\left ( \\left ( q ^ { * k } \\right ) ^ { \\prime } ( t _ k ) - ( q ^ { * k } ) ^ { \\prime } ( t ) \\right ) \\right | \\end{align*}"}
-{"id": "9366.png", "formula": "\\begin{align*} - \\{ f '' ( x ) \\} _ { x \\not = 0 } + [ \\delta , \\delta ' ] [ \\mathbf { T } \\Gamma _ 0 f - { \\Gamma } _ 1 f ] + [ q _ 1 , q _ 2 ] \\Gamma _ 0 f , \\end{align*}"}
-{"id": "7862.png", "formula": "\\begin{align*} p ^ { \\kappa } ( t , x , y ) \\ge c _ 5 \\begin{cases} \\Phi ^ { - 1 } ( t ^ { - 1 } ) ^ d & | x - y | \\le 3 \\Phi ^ { - 1 } ( t ^ { - 1 } ) ^ { - 1 } , \\\\ t j \\left ( | x - y | \\right ) & | x - y | > 3 \\Phi ^ { - 1 } ( t ^ { - 1 } ) ^ { - 1 } . \\end{cases} \\end{align*}"}
-{"id": "9358.png", "formula": "\\begin{align*} \\begin{cases} d p ( t , x ) = \\frac { 1 } { 2 } [ \\frac { \\partial p } { \\partial t } ( t , x ) d t + \\frac { \\partial p } { \\partial x } ( t , x ) \\beta _ 0 ( t ) \\pi ( t ) d v ( t ) + q ( t , x ) d w ( t ) ] \\\\ p ( T , x ) = U ( x ) \\end{cases} \\end{align*}"}
-{"id": "3045.png", "formula": "\\begin{align*} C ( u , n ) = B ( u ) \\backslash \\left ( \\cup _ { j = 1 } ^ { n - 1 } B ( u . j ) \\right ) = \\{ u \\} \\cup \\bigcup _ { j = n } ^ { \\infty } B ( u . j ) . \\end{align*}"}
-{"id": "7769.png", "formula": "\\begin{align*} \\beta ( M ^ { \\oplus 3 N } ) = \\begin{pmatrix} 2 N & 3 N & - \\\\ - & - & N \\end{pmatrix} + \\begin{pmatrix} N & - & - \\\\ - & 3 N & 2 N \\end{pmatrix} . \\end{align*}"}
-{"id": "4139.png", "formula": "\\begin{align*} & \\sum _ { n \\leq X } \\Re a ( n ) ^ 2 + \\Im a ( n ) ^ 2 = c _ { 1 } X ^ { \\gamma _ { 1 } } + O ( X ^ { \\eta _ { 1 } } ) , \\\\ & \\sum _ { n \\leq X } \\Re a ( n ) ^ 2 - \\Im a ( n ) ^ 2 + 2 i \\Re a ( n ) \\Im a ( n ) = c _ { 2 } X ^ { \\gamma _ { 2 } } + O ( X ^ { \\eta _ { 2 } } ) , \\end{align*}"}
-{"id": "9015.png", "formula": "\\begin{align*} f ( x , y , z ) = a _ d ( x , y ) z ^ d + \\sum _ { i = 0 } ^ { d - 1 } a _ i ( x , y ) z ^ { i } , \\end{align*}"}
-{"id": "4523.png", "formula": "\\begin{align*} S _ 4 = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log ^ 2 \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) \\ , d s \\ , d x = o ( n ^ { - ( 3 - \\epsilon ) } ) . \\end{align*}"}
-{"id": "886.png", "formula": "\\begin{align*} \\pi ^ { - s / 2 } \\Gamma ( s / 2 ) \\zeta ( s ) = \\pi ^ { - ( 1 - s ) / 2 } \\Gamma ( ( 1 - s ) / 2 ) \\zeta ( 1 - s ) . \\end{align*}"}
-{"id": "2129.png", "formula": "\\begin{align*} \\tilde { v } ( X ) : = \\frac { v ( \\l X ) } { \\l ^ { l + 1 + s } } . \\end{align*}"}
-{"id": "7480.png", "formula": "\\begin{align*} \\Gamma _ s = \\left \\{ ( x _ 1 , \\dots , x _ s ) : x _ i \\in \\mathbb { R } ^ d , \\| x _ i \\| < \\underset { 1 \\leq l \\leq s , l \\neq i } { \\min } \\| x _ i - x _ l \\| , 1 \\leq i \\leq s \\right \\} \\end{align*}"}
-{"id": "300.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ d } \\mu _ d \\bigl ( ( K + z ) \\cap L \\bigr ) \\ , d z = \\mu _ d ( K ) \\mu _ d ( L ) , \\end{align*}"}
-{"id": "8900.png", "formula": "\\begin{align*} - \\Delta _ { A _ n } ( \\Tilde { u } _ n - \\Tilde { v } _ n ) + ( \\Tilde { u } _ n - \\Tilde { v } _ n ) = W _ n [ \\Tilde { u } _ n - \\Tilde { v } _ n ] , \\end{align*}"}
-{"id": "7401.png", "formula": "\\begin{align*} | | \\mathtt { g } \\cdot \\mathtt { h } | | _ \\nu & \\leq | | \\mathtt { g } | | _ \\nu + | | \\mathtt { h } | | _ \\nu , \\\\ | | \\mathtt { h } \\cdot \\mathtt { g } \\cdot \\bar { \\mathtt { h } } | | _ \\nu & = | | \\mathtt { g } | | _ \\nu , \\\\ | | \\bar { \\mathtt { g } } | | _ \\nu & = | | \\mathtt { g } | | _ \\nu . \\\\ \\end{align*}"}
-{"id": "424.png", "formula": "\\begin{align*} \\lambda . ( t , z ) = ( \\lambda ^ 2 t , \\lambda z ) , \\end{align*}"}
-{"id": "1298.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ { u } e ( m \\alpha ) \\ll \\min \\Bigl ( u ; \\frac { 1 } { \\Vert \\alpha \\Vert } \\Bigr ) . \\end{align*}"}
-{"id": "8253.png", "formula": "\\begin{align*} \\mathrm { d } \\phi _ t ( i ) = \\left \\{ p V ' ( \\phi _ t ( i + 1 ) - \\phi _ t ( i ) ) - q V ' ( \\phi _ t ( i ) - \\phi _ t ( i - 1 ) ) \\right \\} \\mathrm { d } t + \\mathrm { d } W _ t ( i ) , ( t , i ) \\in \\R _ + \\times \\Z , \\end{align*}"}
-{"id": "5645.png", "formula": "\\begin{align*} \\gamma \\langle \\nabla E ( \\mu ) - & \\nabla E ( \\zeta ) , \\mu - \\zeta \\rangle \\le - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\int _ { \\Omega } \\left [ ( \\phi ^ { \\mu , \\nu _ i } ) ^ { c c } ( x ) - ( \\phi ^ { \\zeta , \\eta _ i } ) ^ { c c } ( x ) \\right ] d ( \\mu - \\zeta ) ( x ) \\\\ & = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\iint _ { \\Omega \\times \\Omega } \\left [ ( \\phi ^ { \\mu , \\nu _ i } ) ^ { c c } ( x ) - ( \\phi ^ { \\zeta , \\eta _ i } ) ^ { c c } ( x ) \\right ] d ( \\pi ^ { \\mu , \\nu _ i } - \\pi ^ { \\zeta , \\eta _ i } ) ( x , y ) , \\end{align*}"}
-{"id": "528.png", "formula": "\\begin{align*} R ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) = - \\int \\frac { h ^ { \\rm g e o } _ { 1 2 } ( z , 1 / z ) } { z ^ { u - v + 1 } } \\dd z . \\end{align*}"}
-{"id": "4941.png", "formula": "\\begin{align*} \\| u _ i \\| _ 2 & \\leq \\alpha \\sqrt { k / ( t - 1 ) } \\\\ & \\leq \\frac { \\| D ^ * _ { S _ 0 } h \\| _ 2 } { \\sqrt { t - 1 } } + \\frac { 2 \\sigma _ k ( D ^ * x _ 0 ) _ 1 + \\rho } { \\sqrt { k ( t - 1 ) } } \\\\ & \\leq \\frac { \\| D ^ * _ { S _ 0 } h + h ^ { ( 1 ) } \\| _ 2 } { \\sqrt { t - 1 } } + \\frac { 2 \\sigma _ k ( D ^ * x _ 0 ) _ 1 + \\rho } { \\sqrt { k ( t - 1 ) } } \\\\ & = \\frac { z + R } { \\sqrt { t - 1 } } , \\end{align*}"}
-{"id": "3481.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u ( x ) = h ( x ) & \\mbox { i n } \\Omega , \\\\ u = 0 & \\mbox { o n } \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "6934.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ s _ \\lambda ( Z ) ] \\ M ( X Z ) \\ L _ \\pi ( Z ) \\ , ; \\\\ \\cr s ^ { * ( \\pi ) } _ \\lambda ( X ) & = \\begin{cases} [ s _ \\lambda ( Z ) ] \\ L ( X Z ) \\ L _ { \\pi ' } ( Z ) & \\mbox { i f $ | \\pi | $ i s e v e n } ; \\cr [ s _ \\lambda ( Z ) ] \\ L ( X Z ) \\ M _ { \\pi ' } ( Z ) ) & \\mbox { i f $ | \\pi | $ i s o d d } . \\cr \\end{cases} \\end{align*}"}
-{"id": "2421.png", "formula": "\\begin{align*} \\Vert \\{ \\lambda _ { j } \\} \\Vert _ { l _ { q } ^ { s , 1 } } = \\begin{cases} \\left ( \\sum _ { j \\in \\mathbb { N } } 2 ^ { j s } | a _ { j } | ^ { q } \\right ) ^ { \\frac { 1 } { q } } ~ & ~ 0 < q < \\infty , \\\\ \\sup _ { j \\in \\mathbb { N } } \\left ( 2 ^ { j s } | a _ { j } | \\right ) ~ & ~ q = \\infty \\end{cases} \\end{align*}"}
-{"id": "8965.png", "formula": "\\begin{align*} \\Psi _ A ( t , t _ 0 ) = \\sum _ { k = 0 } ^ { m - 1 } P _ k \\varphi _ { k + 1 } ( t ) , \\end{align*}"}
-{"id": "1859.png", "formula": "\\begin{align*} a ( \\tilde U ) : = \\binom { | \\tilde U | } { 2 } ^ { - 1 } \\sum _ { \\{ u , u ' \\} \\in \\binom { \\tilde U } { 2 } } \\deg ( u , u ' ) = \\binom { | \\tilde U | } { 2 } ^ { - 1 } \\sum _ { v \\in V } \\binom { \\deg ( v , \\tilde U ) } { 2 } . \\end{align*}"}
-{"id": "7238.png", "formula": "\\begin{align*} \\widehat { w } _ j ^ \\lambda : = w _ j ^ \\lambda + \\sum _ { | \\alpha | = n + 1 } f _ { j , \\alpha } ^ \\lambda ( z _ j ) \\cdot w _ j ^ \\alpha . \\end{align*}"}
-{"id": "9572.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c } \\ddot \\psi ( x , t ) = ( \\Delta - m ^ 2 ) \\psi ( x , t ) + \\sum \\limits _ { 1 \\le j \\le n } \\zeta _ j ( t ) \\delta ( x - y _ j ) \\\\ \\\\ \\lim \\limits _ { x \\to y _ j } ( \\psi ( x , t ) - \\zeta _ j ( t ) g _ j ( x ) ) = F _ j ( \\zeta ( t ) ) , 1 \\le j \\le n \\end{array} \\right | y _ j \\in \\R ^ 3 , x \\in \\R ^ 3 , t \\in \\R , \\end{align*}"}
-{"id": "8154.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\phi _ h ( a ^ k ) = \\langle \\phi _ h , \\phi _ 0 \\rangle = 0 \\mbox { f o r $ 1 \\leq h \\leq n - 1 $ } . \\end{align*}"}
-{"id": "911.png", "formula": "\\begin{align*} \\vert \\int _ { T } ^ { 2 T } Z ( t ) d t \\vert = O ( T ^ { 3 / 4 } ) . \\end{align*}"}
-{"id": "7396.png", "formula": "\\begin{align*} & C _ { B P S } ^ P ( N , Y ; \\mathbf { l } ^ \\ast , \\alpha ) \\\\ & = \\inf \\{ K > 0 ; \\forall H \\in C ^ \\infty ( N ) H | _ Y \\geq K , \\\\ & \\exists \\mu \\in \\mathfrak { M } ( N , X _ H ) | \\langle \\mathbf { l } ^ \\ast , \\rho ( \\mu , X _ H ) \\rangle | \\geq \\mathbf { l } ^ \\ast ( \\alpha ) \\} . \\\\ \\end{align*}"}
-{"id": "2092.png", "formula": "\\begin{align*} Z \\ \\mathbf { X } & = - \\eta , \\end{align*}"}
-{"id": "2539.png", "formula": "\\begin{align*} \\Im \\left [ ( \\overline { X } - \\overline { Y } ) ^ T H ( X , Y , z ) \\right ] = \\Im \\left [ \\sum _ { m = 1 } ^ M ( X _ m + z _ m ) ( Y _ m + z _ m ) ( \\overline { X _ m } - \\overline { Y _ m } ) ^ 2 \\right ] \\end{align*}"}
-{"id": "2309.png", "formula": "\\begin{align*} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\le C \\tau ^ k , \\\\ & \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ \\infty ( W ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "3498.png", "formula": "\\begin{align*} \\Delta v = \\chi _ { \\{ v > 0 \\} } h ( x ' ) , \\end{align*}"}
-{"id": "2880.png", "formula": "\\begin{align*} \\epsilon \\circ e = i d _ { \\mathbb { K } } . \\end{align*}"}
-{"id": "7574.png", "formula": "\\begin{align*} \\beta _ l = \\inf \\{ b < x < a : F ( x ) > l \\} , \\alpha _ l = \\sup \\{ b < x < a : F ( x ) < - l \\} . \\end{align*}"}
-{"id": "7691.png", "formula": "\\begin{align*} x _ { n + 1 } \\ge f ( x _ n ) + \\sigma _ n \\xi _ { n + 1 } , n = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "3367.png", "formula": "\\begin{align*} \\bar { E } _ s - \\Delta t \\bar { P } = 0 . \\end{align*}"}
-{"id": "6563.png", "formula": "\\begin{align*} \\| f ( w ) - f ( v ) \\| _ { X } & = \\| f ( w ) - f ( v ) \\| _ { L ^ 2 ( \\R ^ d ) } + \\| f ( w ) - f ( v ) \\| _ { L ^ q ( \\R ^ d ) } \\\\ & \\le C _ K \\big ( \\| w - v \\| _ { L ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { L ^ q ( \\R ^ d ) } \\big ) \\\\ & \\le C _ K \\Big ( \\| w - v \\| _ { L ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { L ^ 2 ( \\R ^ d ) } ^ { \\frac { 2 } { q } } \\| w - v \\| _ { L ^ \\infty ( \\R ^ d ) } ^ { 1 - \\frac { 2 } { q } } \\Big ) \\\\ & \\le C _ K \\big ( \\| w - v \\| _ { L ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { L ^ \\infty ( \\R ^ d ) } \\big ) = C _ K \\| w - v \\| _ { W } . \\end{align*}"}
-{"id": "5768.png", "formula": "\\begin{align*} R _ { 1 2 } ( \\lambda _ 1 , \\lambda _ 2 ) R _ { 1 3 } ( \\lambda _ 1 , \\lambda _ 3 ) R _ { 2 3 } ( \\lambda _ 2 , \\lambda _ 3 ) = R _ { 2 3 } ( \\lambda _ 2 , \\lambda _ 3 ) R _ { 1 3 } ( \\lambda _ 1 , \\lambda _ 3 ) R _ { 1 2 } ( \\lambda _ 1 , \\lambda _ 2 ) . \\end{align*}"}
-{"id": "8297.png", "formula": "\\begin{align*} R _ { \\tilde { x } } = ( 2 - 2 \\tilde { g } ( \\tilde { x } ) ) - d _ { \\tilde { x } } ( \\varphi ) ( 2 - 2 g ( \\varphi ( \\tilde { x } ) ) ) - \\sum _ { \\tilde { v } \\in T _ { \\tilde { x } } ( \\widetilde { \\Gamma } ) } ( d _ { \\tilde { v } } ( \\varphi ) - 1 ) , \\end{align*}"}
-{"id": "5195.png", "formula": "\\begin{align*} \\textup { T r } ^ { p } ( A \\Sigma _ { Z } A ^ { T } ) = \\sum _ { k } \\left ( \\sum _ { i } a ^ { 2 } _ { k i } \\sigma _ { i } \\right ) ^ { p } . \\end{align*}"}
-{"id": "908.png", "formula": "\\begin{align*} F ' ( n , t ) = \\tfrac { 1 } { 2 } \\log \\left ( \\tfrac { t } { 2 \\pi n ^ { 2 } } \\right ) , \\ F '' ( n , t ) = \\tfrac { 1 } { 2 t } > \\tfrac { 1 } { 4 T } . \\end{align*}"}
-{"id": "119.png", "formula": "\\begin{align*} C h _ { n , \\frac { 1 } { 2 } } + \\sum _ { m = 0 } ^ n { n \\choose m } C _ m C h _ { n - m , \\frac { 1 } { 2 } } ( m + 1 ) ! ( - 1 ) ^ { m + 1 } \\frac { 1 } { 4 ^ m ( 2 m - 1 ) } = \\begin{cases} 2 \\textnormal { i f } \\ , \\ , n = 0 \\\\ 0 \\textnormal { i f } \\ , \\ , n > 0 . \\end{cases} \\end{align*}"}
-{"id": "3272.png", "formula": "\\begin{align*} D ^ 2 & \\sim \\sum _ { i , j } \\mathcal { E } _ i \\mathcal { E } _ j ^ * \\otimes ( \\gamma _ { - } ( w _ i ) \\gamma _ { - } ( w _ j ) ^ * + q ^ { - ( \\xi _ i , \\xi _ j ) } \\gamma _ { - } ( w _ j ) ^ * \\gamma _ { - } ( w _ i ) ) \\\\ & + \\sum _ { i , j , k , l } c _ { i , j } ^ { k , l } \\mathcal { E } _ k \\mathcal { E } _ l ^ * \\otimes \\gamma _ { - } ( w _ j ) ^ * \\gamma _ { - } ( w _ i ) . \\end{align*}"}
-{"id": "7821.png", "formula": "\\begin{align*} F ( t , X _ { t } ) = & F ( 0 , X _ { 0 } ) + \\int _ 0 ^ t \\frac { \\partial F } { \\partial s } ( s , X _ { s } ) d s + \\int _ 0 ^ t \\frac { \\partial F } { \\partial x } ( s , X _ { s } ) g _ { s } d s \\\\ & + \\int _ 0 ^ t \\frac { \\partial F } { \\partial x } ( s , X _ { s } ) f _ { s } d B _ { s } ^ { H } + \\frac { 1 } { 2 } \\int _ 0 ^ t \\frac { \\partial ^ { 2 } F } { \\partial x ^ { 2 } } ( s , X _ { s } ) \\bigg [ \\frac { d } { d s } \\| f \\| _ { s } ^ { 2 } \\bigg ] d s , \\ \\ t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "2093.png", "formula": "\\begin{align*} Z & = - \\eta \\mathbf { X } ^ { T } ( \\mathbf { X } \\mathbf { X } ^ { T } ) ^ { - 1 } . \\end{align*}"}
-{"id": "8844.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\Box ( | \\nabla f | ^ 2 ) = \\langle \\nabla ( \\Box f ) , \\nabla f \\rangle + R _ T ( \\nabla f , \\nabla f ) + \\langle \\nabla ^ 2 f , \\nabla ^ 2 f \\circ T \\rangle - \\langle \\nabla ^ 2 f , \\nabla _ { \\nabla f } T \\rangle \\end{align*}"}
-{"id": "2754.png", "formula": "\\begin{align*} Z _ t = x + \\int _ 0 ^ t \\sigma ( Z _ s ) \\dot { h } _ s \\ , \\d s + \\int _ 0 ^ t b ( Z _ s ) \\ , \\d s + \\psi _ t . \\end{align*}"}
-{"id": "830.png", "formula": "\\begin{align*} q \\ , F _ { \\ldots , z _ { i } , z _ { i + 1 } , \\ldots } ^ { \\ldots , \\mu _ { i } , \\mu _ { i + 1 } , \\ldots } = f ( z _ { i } , z _ { i + 1 } ) F _ { \\ldots , z _ { i + 1 } , z _ { i } , \\ldots } ^ { \\ldots , \\mu _ { i + 1 } , \\mu _ { i } , \\ldots } - g ( z _ { i + 1 } , z _ { i } ) F _ { \\ldots , z _ { i } , z _ { i + 1 } , \\ldots } ^ { \\ldots , \\mu _ { i + 1 } , \\mu _ { i } , \\ldots } , \\end{align*}"}
-{"id": "2502.png", "formula": "\\begin{gather*} W _ 1 ( \\Lambda , \\Lambda ^ n ) \\leqslant \\dfrac { K } { \\sqrt { n } } , \\\\ W _ 1 ( \\Lambda _ 0 ^ n , \\Lambda _ 0 ) = W _ 1 ( \\Lambda _ 0 , \\Lambda _ 0 ^ n ) \\leqslant \\dfrac { K } { \\sqrt { n } } . \\end{gather*}"}
-{"id": "8801.png", "formula": "\\begin{align*} \\left ( \\int _ { \\Sigma } f _ i d v o l _ { \\Sigma } \\right ) ^ 2 = \\left | \\int _ { \\Sigma } f _ i d v o l _ { \\Sigma } \\right | ^ 2 \\leq \\left ( \\int _ { \\Sigma } | f _ i | d v o l _ { \\Sigma } \\right ) ^ 2 \\leq A ^ 2 \\ , . \\end{align*}"}
-{"id": "439.png", "formula": "\\begin{align*} \\tilde { x } = \\phi _ k ^ { - 1 } ( y ' , \\mu _ k ( z ' ) 0 ^ { n - k - \\abs { \\mu _ k ( z ' ) } } ) , \\end{align*}"}
-{"id": "5017.png", "formula": "\\begin{align*} ( s \\cdot \\eta ) ( f ) = \\eta ( s ^ { - 1 } \\cdot f ) , \\textrm { f o r $ s \\in G $ a n d $ f \\in \\ell ^ \\infty ( G ) $ } . \\end{align*}"}
-{"id": "1981.png", "formula": "\\begin{align*} ( x _ i ^ { v - 1 } x _ j ) x _ i ^ v = r _ { i , j } x _ i ^ v ( x _ i ^ { v - 1 } x _ j ) \\end{align*}"}
-{"id": "4266.png", "formula": "\\begin{align*} \\mathbb P \\left ( Z ^ * _ n \\le - 1 \\right ) = \\frac 1 2 \\mathbb P ( T _ 0 > n ) \\sim \\frac { \\gamma \\ , \\mathbb E [ \\mathcal R _ n ] } { 2 n } \\ , . \\end{align*}"}
-{"id": "5260.png", "formula": "\\begin{align*} Y = \\mathbb { R } k + A . \\end{align*}"}
-{"id": "2437.png", "formula": "\\begin{align*} \\Vert \\Box _ { k } ^ { \\alpha _ { 2 } } f _ { k } ^ { \\alpha _ { 2 } } \\Vert _ { M _ { 2 } } = \\Vert f _ { k } ^ { \\alpha _ { 2 } } \\Vert _ { M _ { 2 } } \\sim \\Vert f _ { k } ^ { \\alpha _ { 2 } } \\Vert _ { L ^ { p _ { 2 } } } \\sim 2 ^ { j n \\alpha _ { 2 } ( 1 - 1 / p _ { 2 } ) } \\end{align*}"}
-{"id": "9918.png", "formula": "\\begin{align*} X ' & : = \\xi _ 3 \\eta _ 4 E - q \\ ; \\xi _ 1 \\eta _ 4 K - q \\ ; \\xi _ 3 \\eta _ 1 K ' , \\\\ Y ' & : = \\xi _ 3 \\eta _ 4 F - q ^ { - 1 } \\xi _ 2 \\eta _ 4 K - q ^ { - 1 } \\xi _ 3 \\eta _ 2 K ' , \\end{align*}"}
-{"id": "7157.png", "formula": "\\begin{align*} - \\Delta u + ( u \\cdot \\nabla ) u + \\nabla p = 0 , \\qquad { \\rm d i v } \\ , u = 0 \\mbox { i n } \\ , \\ , \\Omega _ r , \\end{align*}"}
-{"id": "7225.png", "formula": "\\begin{align*} P _ { \\alpha } ^ { S } = P _ { \\beta } ^ { S } \\circ \\Phi _ { \\beta , \\alpha } , \\end{align*}"}
-{"id": "6746.png", "formula": "\\begin{align*} F ( \\varphi ) : = { \\rm I } ( \\varphi ) - L _ { \\mu } ( \\varphi ) , \\ \\ \\varphi \\in \\textup { P S H } ( X , \\omega ) , \\end{align*}"}
-{"id": "4641.png", "formula": "\\begin{align*} \\hat { L } = \\left \\{ f \\in L : 3 | b , c \\right \\} . \\end{align*}"}
-{"id": "9168.png", "formula": "\\begin{align*} \\delta _ { k , 1 } a ( v ) = \\sum _ { l = - r } ^ { r } l ^ k \\frac { \\partial H } { \\partial u _ l } ( v , \\dots , v ) , k = 0 , \\dots , p , \\quad \\forall v . \\end{align*}"}
-{"id": "7927.png", "formula": "\\begin{align*} 2 \\binom { 3 m + 1 } { 3 } - 4 \\binom { m + 2 } { 3 } = 3 \\sum _ { i = 1 } ^ { m } x _ { i } ^ { 2 } + 2 ( m + 1 ) h ^ { 2 } . \\end{align*}"}
-{"id": "4700.png", "formula": "\\begin{align*} \\tilde { y } & = y = y ' , \\\\ \\mu _ k ( \\tilde { z } ) & = \\mu _ k ( z ) = \\mu _ k ( z ' ) , \\\\ \\sigma _ k ( \\tilde { z } ) & = ( \\max ( s _ 0 , s ' _ 0 ) , \\dots , \\max ( s _ r , s ' _ r ) ) . \\end{align*}"}
-{"id": "3229.png", "formula": "\\begin{align*} \\Phi _ { \\beta , \\alpha } \\left ( \\partial \\mathcal { U } _ { \\alpha } ^ { S } \\left ( 3 \\right ) \\right ) = \\partial \\mathcal { U } _ { \\beta } ^ { S } \\left ( 3 \\right ) , \\end{align*}"}
-{"id": "5501.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to \\infty } \\gamma _ k ^ \\mathrm { D L } [ \\iota ] = \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } } . \\end{align*}"}
-{"id": "7089.png", "formula": "\\begin{align*} Z ^ u _ n = \\sum _ { | v | = n , v > u } ( V ( v ) - V ( u ) ) e ^ { V ( u ) - V ( v ) } Z ^ u _ \\infty = \\liminf _ { n \\to \\infty } Z ^ u _ n . \\end{align*}"}
-{"id": "5255.png", "formula": "\\begin{align*} \\operatorname * { l e v } \\nolimits _ { \\varphi _ { A , k } , < } ( t ) = t k + \\operatorname * { i n t } A \\forall \\ , t \\in \\mathbb { R } , \\end{align*}"}
-{"id": "4957.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ \\infty ( \\overline \\Omega ) } + t \\leq \\kappa _ 4 + \\kappa _ 2 = : C \\end{align*}"}
-{"id": "6391.png", "formula": "\\begin{align*} \\Delta H _ { p } ( 1 ) = \\log \\left ( 1 \\pm C ( \\varepsilon ) \\left | \\Delta _ { 0 } ( 1 ) \\right | \\right ) \\pm \\frac { 1 } { 2 \\sigma ^ { 2 } } \\left | \\Delta \\mu _ { ( 2 ) } ( 1 ) \\right | \\pm \\varepsilon \\left | \\Delta \\mu _ { ( p ) } ( 1 ) \\right | . \\end{align*}"}
-{"id": "9596.png", "formula": "\\begin{align*} \\lambda _ { r e g } ( t ) : = \\lim \\limits _ { x \\to y } \\psi _ { f , r e g } ( x , t ) = \\psi _ { f , r e g } ( y , t ) \\in C ( [ 0 , \\infty ) ) . \\end{align*}"}
-{"id": "431.png", "formula": "\\begin{align*} \\bar X _ i ( z ) & = \\begin{cases} \\chi ( z ) \\dd \\theta _ { \\theta ^ { - 1 } ( z ) } \\left ( X _ i \\left ( \\theta ^ { - 1 } ( z ) \\right ) \\right ) & \\mbox { i f } z \\in V _ 0 \\\\ 0 & \\mbox { i f } z \\in \\R ^ d \\setminus V _ 0 \\end{cases} & & \\mbox { f o r } i \\in \\{ 0 , 1 , \\dots , m \\} \\ ; , \\\\ \\bar X _ { m + k } ( z ) & = \\rho ( z ) e _ k & & \\mbox { f o r } k \\in \\{ 1 , \\dots , d \\} \\ ; , \\end{align*}"}
-{"id": "2888.png", "formula": "\\begin{align*} \\overline { P o i s _ n ^ * \\{ n + 1 \\} } ^ + ( 1 ) = \\overline { P o i s _ n ^ * \\{ n + 1 \\} } ( 1 ) \\oplus \\mathbb { K } [ 1 ] = \\overline { P o i s _ n ^ * \\{ n + 1 \\} } ( 1 ) \\oplus \\mathbb { K } d \\end{align*}"}
-{"id": "6749.png", "formula": "\\begin{align*} \\frac { d } { d t } { \\rm I } ( u _ t ) = \\int _ X \\chi \\theta _ { u _ t } ^ n , \\ \\forall t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "8555.png", "formula": "\\begin{align*} \\rho ( x ) n & = s ( x _ { 1 } ) \\rho ( x _ { 1 } s ( x _ { 2 } ) x _ { 2 } ) n \\\\ & = s ( x _ { 1 } ) x _ { 1 } s ( x _ { 2 } ) x _ { 2 } n \\\\ & = x n \\end{align*}"}
-{"id": "5827.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } S ^ { 2 } F _ { , S S } + \\kappa \\left ( \\mu - \\lambda - \\log S \\right ) S F _ { , S } - F _ { , t } = 0 , \\end{align*}"}
-{"id": "7250.png", "formula": "\\begin{align*} M _ j : = \\left ( \\begin{array} { c | c c c } 1 & 0 & \\cdots & 0 \\\\ \\hline - m _ { j } ^ 2 & 1 & & 0 \\\\ \\vdots & & \\ddots & \\\\ - m _ { j } ^ r & 0 & & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "5182.png", "formula": "\\begin{gather*} R _ { \\ell + s } = \\partial C _ { \\ell + s } \\oplus \\varphi ( \\partial R _ { \\ell } ) \\oplus R _ \\ell = \\partial C _ { \\ell + s } \\oplus \\bigoplus _ { 0 \\leqslant k \\leqslant \\ell , [ k ] = [ \\ell ] } \\partial C _ { k } \\oplus \\varphi ( R _ { \\ell + s } ) . \\end{gather*}"}
-{"id": "7684.png", "formula": "\\begin{align*} \\mathbb P \\{ \\omega \\in \\Omega : \\lim _ { n \\to \\infty } x _ n ( \\omega ) = 0 \\} = 0 . \\end{align*}"}
-{"id": "7838.png", "formula": "\\begin{align*} y _ 1 - y _ 2 - \\xi ( w _ 1 - w _ 2 ) & = b ( B - w _ 2 ) a - b ( B - w _ 1 ) a - b a ( w _ 1 - w _ 2 ) = 0 , \\end{align*}"}
-{"id": "6492.png", "formula": "\\begin{align*} \\mu ( t ) : = \\frac { d } { d t } ( g _ { \\alpha } * \\rho ) ( t ) = \\frac { 1 } { \\Gamma ( \\alpha ) } \\frac { d } { d t } \\int _ { 0 } ^ { t } \\frac { \\rho ( s ) } { ( t - s ) ^ { 1 - \\alpha } } d x 0 < t \\leq T . \\end{align*}"}
-{"id": "415.png", "formula": "\\begin{align*} [ P , a ^ 1 ] \\cdots [ P , a ^ { 2 k } ] = & ( - 1 ) ^ k P a ^ 1 ( 1 - P ) a ^ 2 P \\cdots ( 1 - P ) a ^ { 2 k } P \\\\ & \\quad + ( - 1 ) ^ k ( 1 - P ) a ^ 1 P a ^ 2 ( 1 - P ) \\cdots P a ^ { 2 k } ( 1 - P ) \\\\ = & ( - 1 ) ^ k P a _ + ^ 1 ( 1 - P ) a _ - ^ 2 P \\cdots ( 1 - P ) a _ - ^ { 2 k } P \\\\ & \\quad + ( - 1 ) ^ k ( 1 - P ) a _ - ^ 1 P a _ + ^ 2 ( 1 - P ) \\cdots P a ^ { 2 k } _ + ( 1 - P ) . \\end{align*}"}
-{"id": "57.png", "formula": "\\begin{align*} \\mathcal { R } ( a ) < \\infty \\ ; \\ ; \\limsup _ { r \\to \\mathcal { R } ( a ) } \\left ( | f ( r ) | + | g ( r ) | \\right ) = \\infty \\ . \\end{align*}"}
-{"id": "8812.png", "formula": "\\begin{align*} L = n A + n H \\int _ { \\Sigma } u \\ , d v o l _ { \\Sigma } \\geq n A + \\frac { L } { n + 1 } \\left ( \\frac { \\int _ { \\Sigma } u \\ , d v o l _ { \\Sigma } } { A } \\right ) ^ 2 = n A + \\frac { L } { n + 1 } \\left ( \\frac { L - n A } { n H A } \\right ) ^ 2 \\ , . \\end{align*}"}
-{"id": "2506.png", "formula": "\\begin{align*} \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } F & = \\sum _ { i = 1 } ^ { p } \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } ( D _ i - d _ i ) ^ 2 = \\sum _ { i = 1 } ^ { p } 2 ( D _ i - d _ i ) \\mathcal { L } _ { X _ { 0 } ^ { \\lambda } } ( D _ i - d _ i ) \\\\ & = \\sum _ { i = 1 } ^ { p } 2 ( D _ i - d _ i ) ( - \\lambda ) ( D _ i - d _ i ) = ( - 2 \\lambda ) \\sum _ { i = 1 } ^ { p } ( D _ i - d _ i ) ^ 2 \\\\ & = ( - 2 \\lambda ) F . \\end{align*}"}
-{"id": "6624.png", "formula": "\\begin{align*} P _ t \\mu : = \\Phi _ t \\# \\mu = \\mu \\circ \\Phi _ t ^ { - 1 } ; \\end{align*}"}
-{"id": "3865.png", "formula": "\\begin{align*} R _ g = - n ( n + 1 ) + O ( r ^ { n + 2 } \\log r ) . \\end{align*}"}
-{"id": "5710.png", "formula": "\\begin{align*} \\left | \\sum _ { k = 1 } ^ 2 \\sum _ { Q \\in \\mathfrak { A } _ k ( Q _ 0 ) } \\alpha _ Q \\chi _ Q \\right | \\leq 2 \\sum _ { k = 1 } ^ 2 \\sum _ { Q \\in \\mathfrak { A } _ { k - 1 } ( Q _ 0 ) } \\omega _ { \\lambda _ w } ( f ; Q ) \\chi _ Q , \\end{align*}"}
-{"id": "41.png", "formula": "\\begin{align*} \\Vert u \\Vert ^ { \\frac { 2 } { m } } _ y : = \\int _ { X _ y } \\vert u \\vert ^ { \\frac { 2 } { m } } e ^ { - \\varphi _ L } , \\end{align*}"}
-{"id": "5816.png", "formula": "\\begin{align*} \\lambda _ k + \\mu _ { k ^ * } = 0 = \\lambda _ { k ^ * } + \\mu _ { k } \\end{align*}"}
-{"id": "1939.png", "formula": "\\begin{align*} \\operatorname { V o l } _ { \\tilde { g } ( ( \\tau ) } ( P ) = & \\ , \\ , \\frac { 1 } { \\tau ^ { ( 1 + \\sum _ { i = 1 } ^ m 2 n _ i ) / 2 } } \\operatorname { V o l } _ { g ( \\tau ) } ( P ) \\\\ \\sim & \\ , \\ , \\tau ^ { - ( 1 + \\sum _ { i = 1 } ^ m 2 n _ i ) / 2 } \\cdot \\tau ^ { \\sum _ { i = 1 } ^ m n _ i } \\operatorname { V o l } _ { g _ 0 } ( P ) \\\\ = & \\ , \\tau ^ { - 1 / 2 } \\operatorname { V o l } _ { g _ 0 } ( P ) \\rightarrow 0 \\end{align*}"}
-{"id": "862.png", "formula": "\\begin{align*} g ( z ) = \\ell _ 1 y _ { 1 , 1 } ( z ) + \\ldots + \\ell _ \\mu y _ { 1 , \\mu } ( z ) \\end{align*}"}
-{"id": "158.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } r _ n = \\infty , \\ \\ \\lim _ { n \\to \\infty } \\varepsilon _ n = 0 , \\end{align*}"}
-{"id": "2872.png", "formula": "\\begin{align*} ( - ) _ { - } : X \\in C h _ { \\mathbb { K } } ^ { p t } \\mapsto X _ { - } = k e r ( \\epsilon ) \\in C h _ { \\mathbb { K } } \\end{align*}"}
-{"id": "7685.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } \\sum _ { i = 1 } ^ n \\sigma _ i \\xi _ { i + 1 } = \\infty , \\liminf _ { n \\to \\infty } \\sum _ { i = 1 } ^ n \\sigma _ i \\xi _ { i + 1 } = \\infty \\end{align*}"}
-{"id": "7708.png", "formula": "\\begin{align*} \\mathbb P \\left \\{ \\omega \\in \\Omega : \\xi _ n \\in [ 1 - \\varepsilon , 1 ] , n = N _ J + 1 , \\dots , N _ J + J \\right \\} = 1 . \\end{align*}"}
-{"id": "3225.png", "formula": "\\begin{align*} - \\sum _ { p = 1 } ^ r \\sum _ { \\nu = 1 } ^ r a _ { j k , \\nu } \\cdot \\varepsilon _ { j , p } ^ * \\otimes \\varepsilon _ j ^ p \\cdot \\varepsilon _ j ^ \\nu . \\end{align*}"}
-{"id": "7407.png", "formula": "\\begin{align*} h ^ i ( X ) e _ { \\pm j } = e _ { \\pm j } h ^ i ( X ) . \\end{align*}"}
-{"id": "9607.png", "formula": "\\begin{align*} v _ \\tau ( x , t ) = \\frac { \\theta ( t - \\tau - | x | ) } { 4 \\pi | x | } h ( t - | x | ) - \\frac { m } { 4 \\pi } p _ \\tau ( | x | , t ) , t \\ge \\tau , \\end{align*}"}
-{"id": "627.png", "formula": "\\begin{align*} u = \\Psi _ \\lambda \\ast \\left ( Q | u | ^ { p - 2 } u \\right ) , u \\in L ^ p _ { l o c } ( \\R ^ N ) \\end{align*}"}
-{"id": "9508.png", "formula": "\\begin{align*} - \\frac { 9 } { 2 } R & = R - \\frac { 1 1 } { 2 } R = g ^ { \\mu \\nu } \\left ( R _ { \\mu \\nu } - \\frac { 1 } { 2 } g _ { \\mu \\nu } R \\right ) \\\\ & = \\frac { 1 } { 2 } g ^ { \\mu \\nu } \\langle i _ { \\partial _ \\mu } G , i _ { \\partial _ \\nu } G \\rangle - \\frac { 1 1 } { 4 } | G | ^ 2 \\\\ & = 2 | G | ^ 2 - \\frac { 1 1 } { 4 } | G | ^ 2 \\\\ & = - \\frac { 3 } { 4 } | G | ^ 2 . \\end{align*}"}
-{"id": "1530.png", "formula": "\\begin{align*} \\mu ( E ) : = \\int v \\left ( F ( v , E ) - M ( v ) \\right ) \\d v , \\end{align*}"}
-{"id": "8437.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\textbf { u } + \\sum _ { j = 1 } ^ { d } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } + ( 0 , \\nabla P ) ^ { T } = 0 , \\\\ & \\nabla \\cdot v = 0 , \\\\ & \\rho ( 0 , x ) = \\rho _ { 0 } ( x ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ v ( 0 , x ) = v _ { 0 } ( x ) ~ ~ ~ ~ \\nabla \\cdot v _ { 0 } ( x ) = 0 , \\end{cases} \\end{align*}"}
-{"id": "4779.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c t + a ) ^ 2 - t ^ 2 } . \\end{align*}"}
-{"id": "7185.png", "formula": "\\begin{align*} H ( \\frac { \\pi } { 2 } ) = h ' ( \\frac { \\pi } { 2 } ) = - \\frac { \\gamma } { 1 - \\gamma } h ( \\frac { \\pi } { 2 } ) . \\end{align*}"}
-{"id": "2895.png", "formula": "\\begin{align*} F \\simeq \\oplus _ { \\lambda \\in Q ' _ 0 } F _ \\lambda \\ , , F _ \\lambda : = p ^ * E _ \\lambda \\otimes q ^ * H _ \\lambda , \\end{align*}"}
-{"id": "3119.png", "formula": "\\begin{align*} \\mathcal { L } ( \\lambda ) = \\left [ \\begin{array} { c | c } \\lambda B + A & \\widehat { L } _ t ( \\lambda ) ^ T \\otimes I _ n \\\\ \\hline \\sigma \\widehat { L } _ t ( \\lambda ) \\otimes I _ n & \\phantom { \\Big { ( } } 0 \\phantom { \\Big { ( } } \\end{array} \\right ] \\end{align*}"}
-{"id": "527.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) = I ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) + R ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) \\end{align*}"}
-{"id": "1964.png", "formula": "\\begin{align*} [ L _ t , f _ r ] & = L _ t \\sum _ { z \\in Y } f ( z ) \\chi _ z - ( - 1 ) ^ { \\partial L \\partial f } \\sum _ { y \\in Y } f ( y ) \\chi _ y L _ t \\\\ & = \\sum _ { y , z \\in Y } \\chi _ y L _ t \\chi _ z ( f ( z ) - f ( y ) ) = \\sum _ { \\substack { y , z \\in Y \\\\ d ( y , z ) \\leq 2 r + p _ t } } \\chi _ y L _ t \\chi _ z ( f ( z ) - f ( y ) ) \\end{align*}"}
-{"id": "2354.png", "formula": "\\begin{align*} \\tau _ \\partial ( x _ 0 ) : = \\sup I _ { x _ 0 } \\in [ 0 , \\infty ] , \\end{align*}"}
-{"id": "4089.png", "formula": "\\begin{align*} d f _ q ( X ( q ) ) = 0 , \\quad \\forall q \\in C . \\end{align*}"}
-{"id": "1679.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & \\mathbb { I } \\\\ 0 & 0 \\end{pmatrix} \\ , { R \\ , } = \\begin{pmatrix} 0 \\\\ \\mathbb { I } \\end{pmatrix} , \\ ; \\ ; \\ ; \\ ; { R \\ , } { R \\ , } ^ * = Q = \\begin{pmatrix} 0 & 0 \\\\ 0 & \\mathbb { I } \\end{pmatrix} , \\ ; \\ ; \\ ; B = R F = \\begin{pmatrix} 0 \\\\ F \\end{pmatrix} : \\R ^ { 2 d } \\to \\R ^ { 2 d } \\ , . \\end{align*}"}
-{"id": "7854.png", "formula": "\\begin{align*} \\gamma ( x ) = x ^ { 2 \\alpha } , \\ , \\eta _ 0 ( x ) = \\tau x ^ { \\alpha - 1 } \\end{align*}"}
-{"id": "5465.png", "formula": "\\begin{align*} \\sum _ { n = N - H } ^ { N + H } t _ H ( n - N ) \\Bigl ( R ( n ) - ( 2 \\psi ( n ) - n ) \\Bigr ) \\ll H N \\Bigl ( \\log \\frac { 2 N } { H } \\Bigr ) ^ 2 + H ^ 2 ( \\log N ) ^ 2 \\log ( 2 H ) \\end{align*}"}
-{"id": "316.png", "formula": "\\begin{align*} | g _ J ^ * ( x ) | \\leq \\sum _ { i = 1 } ^ { \\mathrm { c a r d } ( J ) } ( i - 1 ) ! S \\bigl ( \\mathrm { c a r d } ( J ) , i \\bigr ) a ( f ( x ) ) ^ i \\leq \\frac { 1 } { 2 } m ^ { m + 1 } m ! a ( f ( x ) ) ^ m . \\end{align*}"}
-{"id": "3461.png", "formula": "\\begin{align*} H ( \\eta ) = \\mathrm { d i v } _ * \\left ( \\ \\frac { \\nabla _ * \\eta } { \\sqrt { 1 + | \\nabla _ * \\eta | ^ 2 } } \\right ) . \\end{align*}"}
-{"id": "9386.png", "formula": "\\begin{align*} H _ a = \\widetilde { S } _ { m a x } \\upharpoonright \\mathcal { D } ( H _ a ) , \\mathcal { D } ( H _ a ) = \\{ f \\in \\mathcal { D } ( \\widetilde { S } _ { m a x } ) \\ : \\ ( a \\widetilde { \\Gamma } _ 0 - \\widetilde { \\Gamma } _ 1 ) f = 0 \\} ; \\end{align*}"}
-{"id": "1158.png", "formula": "\\begin{align*} b _ { 0 } ( x ) = A x ^ { 2 } + B x + C , b _ { 1 } ( x ) = \\frac { D } { \\rho ( x , t ) ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "3382.png", "formula": "\\begin{align*} R _ { \\max } - R ( t ) = \\log \\left ( \\frac { 1 + P _ { \\max } \\gamma ( t ) } { 1 + P ^ s ( t ) \\gamma ( t ) } \\right ) \\end{align*}"}
-{"id": "3812.png", "formula": "\\begin{align*} s _ q ^ + ( t ) : = \\frac { 1 } { q } \\sum _ { j = 1 } ^ n ( 1 - ( - t _ j ) ^ { q } ) x _ j ^ q s _ q ^ - ( t ) : = \\frac { 1 } { q } \\sum _ { j = 1 } ^ n ( 1 - ( - t _ j ) ^ { q } ) x _ j ^ { - q } . \\end{align*}"}
-{"id": "9973.png", "formula": "\\begin{align*} & ( 1 - \\alpha _ t ) \\sum _ { i = 1 } ^ 2 | w _ { t , k _ t i } | ^ 2 p _ { t , i } + \\alpha _ t \\tilde { p } _ t \\le A _ { t , k _ t } , \\\\ & ( 1 - \\alpha _ t ) \\sum _ { i = 1 } ^ 2 | w _ { t , \\bar { k } _ t i } | ^ 2 p _ { t , i } \\le A _ { t , \\bar { k } _ t } , \\end{align*}"}
-{"id": "6995.png", "formula": "\\begin{align*} [ \\alpha _ 0 , f _ i ^ { \\lambda _ j ( x , y ) } ] = \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ { \\lambda _ j ( x , y ) } , \\alpha _ n ] - [ \\alpha _ i , f _ 0 ^ { \\lambda _ j ( x , y ) } ] . \\end{align*}"}
-{"id": "8535.png", "formula": "\\begin{align*} X _ { a ^ { \\ast } } ( P ) = \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } Y _ { [ b r a ] } ( P ) b r \\end{align*}"}
-{"id": "7851.png", "formula": "\\begin{align*} c _ { \\nu } ^ { ( i + 1 ) } ( x ) = c _ { \\nu + 1 } ^ { ( i ) } ( x ) \\left ( 1 + { \\cal O } ( x ^ { - 1 } ) \\right ) , \\ , x \\rightarrow + \\infty \\end{align*}"}
-{"id": "3666.png", "formula": "\\begin{align*} 0 = u _ d u _ { \\ell - 1 } - u _ d v _ { \\ell - 1 } = u _ d ( u _ { \\ell - 1 } - v _ { \\ell - 1 } ) . \\end{align*}"}
-{"id": "5687.png", "formula": "\\begin{align*} x ( k + 1 ) = A \\ , x ( k ) + E \\ , d ( k ) \\end{align*}"}
-{"id": "7278.png", "formula": "\\begin{align*} - \\sum _ { p = 1 } ^ r \\sum _ { \\nu = 1 } ^ r a _ { j k , \\nu } \\cdot \\varepsilon _ { j , p } ^ * \\otimes \\varepsilon _ j ^ p \\cdot \\varepsilon _ j ^ \\nu . \\end{align*}"}
-{"id": "5345.png", "formula": "\\begin{align*} K ( \\gamma ) \\cong \\bigoplus _ { i = 2 } ^ { p ( n ) - p ( n - 1 ) } \\Z / q _ i \\Z \\end{align*}"}
-{"id": "4996.png", "formula": "\\begin{align*} a _ { k } = \\frac { B ( f ^ { k } ( P ) ) } { \\sqrt [ ] { \\delta _ { f } + \\epsilon } ^ { k } } . \\end{align*}"}
-{"id": "2384.png", "formula": "\\begin{align*} F _ { \\gamma } ^ { \\rm { F B } } ( x ) = \\gamma ^ { - 1 } \\left ( \\langle x - \\gamma \\nabla f ( x ) , x \\rangle - ( \\tfrac { 1 } { 2 } \\| x \\| ^ 2 - \\gamma f ( x ) ) - r _ { \\gamma g } ^ * ( x - \\gamma \\nabla f ( x ) ) \\right ) . \\end{align*}"}
-{"id": "1695.png", "formula": "\\begin{align*} B _ t & = \\int _ 0 ^ t 1 _ { \\{ Z _ s \\not = Y _ s \\} } \\ ; \\frac { \\Big \\| [ D U ( Z _ s ) - D U ( Y _ s ) ] { R \\ , } \\Big \\| ^ 2 _ { H S } } { | \\gamma ( Z _ s ) - \\gamma ( Y _ s ) | ^ 2 } \\ \\dd s \\le C ^ 2 A _ t \\ , , \\end{align*}"}
-{"id": "5722.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } f ^ * ( R ) = 0 , \\end{align*}"}
-{"id": "2196.png", "formula": "\\begin{align*} \\left ( \\int _ { 0 } ^ { t _ { 1 } - t _ { 0 } } \\left ( \\int _ { B _ { 1 } } ( \\psi w ) ^ { 2 } d x \\right ) ^ { p } d s \\right ) ^ { 1 / p } \\leq C \\left ( ( g _ { 1 - \\alpha } * W ) ( t _ { * } ) + \\| F \\| _ { L ^ { 1 } ( [ 0 , t _ { 2 } - t _ { 0 } ] ) } \\right ) , \\end{align*}"}
-{"id": "1259.png", "formula": "\\begin{align*} Q v M _ { 1 1 } = M _ { 1 1 } v ^ * Q = 0 \\end{align*}"}
-{"id": "3091.png", "formula": "\\begin{align*} q _ 1 | K | \\leq | K ' _ i | \\leq q _ 2 | K | \\quad \\forall i = 1 , 2 , \\dots , n ( K ) . \\end{align*}"}
-{"id": "6316.png", "formula": "\\begin{align*} F _ { k , N } = \\sum _ { l \\in \\widetilde { \\Gamma _ { k } ^ { \\alpha _ { 1 } , \\alpha _ { 2 } } } } T _ { N l } f _ { l } ^ { \\alpha _ { 1 } } , \\end{align*}"}
-{"id": "2161.png", "formula": "\\begin{align*} - \\tilde { u } ^ { - q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ) & \\geq - \\frac { 1 } { 1 - q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ^ { 1 - q } ) + \\left ( \\frac { \\tilde { u } ^ { 1 - q } } { 1 - q } - \\tilde { u } ^ { 1 - q } \\right ) g _ { 1 - \\alpha , m } \\\\ & \\geq - \\frac { 1 } { 1 - q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ^ { 1 - q } ) + \\frac { q } { 1 - q } \\tilde { u } ^ { 1 - q } g _ { 1 - \\alpha , m } . \\end{align*}"}
-{"id": "6086.png", "formula": "\\begin{align*} \\sum _ { t < n < t ^ 4 } \\sum _ { m \\in R ( n , t ) } \\left ( \\widetilde { F } _ { m , n } ( t ) - \\widetilde { G } _ { m , n } ( t ) \\right ) = O ( 1 ) , \\end{align*}"}
-{"id": "2101.png", "formula": "\\begin{align*} \\mathcal { P } : = \\{ x _ n \\leq \\gamma ( x ' ) , \\ , y = 0 \\} , \\end{align*}"}
-{"id": "3765.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi } \\int _ { - R ( u ) } ^ { R ( u ) } e ^ { i y t } e ^ { - t ^ 2 / 2 } d t = \\frac { 1 } { \\sqrt { 2 \\pi } } e ^ { - y ^ 2 / 2 } + O \\left ( \\frac { 1 } { R ( u ) } e ^ { - R ^ 2 ( u ) / 2 } \\right ) \\end{align*}"}
-{"id": "2988.png", "formula": "\\begin{align*} f _ 0 ^ { \\lambda _ i ( x , y ) } = - f _ i ^ { \\lambda _ 0 ( x , y ) } - \\sum _ { m + n = i , ~ m , n > 0 } f _ m ^ { \\lambda _ n ( x , y ) } + \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ x , f _ n ^ { y } ] + [ f _ 0 ^ x , f _ i ^ { y } ] + [ f _ i ^ x , f _ 0 ^ { y } ] . \\end{align*}"}
-{"id": "763.png", "formula": "\\begin{gather*} r _ { x } \\otimes r _ { y } = \\sum _ { x = v z , \\ , y = \\bar { z } w } r _ { v w } + r _ { v \\cdot w } , \\end{gather*}"}
-{"id": "1513.png", "formula": "\\begin{align*} K ( x ) = \\frac { ( \\alpha + \\gamma + q - 1 ) ( \\frac { 1 } { x } - 1 ) + q - 1 } { ( \\alpha + \\gamma + q - 1 ) ( x - 1 ) + q - 1 } \\left ( \\frac { 1 - ( x - 1 ) e _ 0 } { 1 - ( \\frac { 1 } { x } - 1 ) e _ 0 } \\right ) , \\end{align*}"}
-{"id": "1271.png", "formula": "\\begin{align*} v _ p ( \\kappa ( B ) ) \\geq c ( i , r ) + c ( i + r , 3 r ) + c ( i + 2 r , r ) + c ( i + 3 r , 3 r ) = 4 t . \\end{align*}"}
-{"id": "4502.png", "formula": "\\begin{align*} \\limsup _ { n \\rightarrow \\infty } \\sup _ { q \\in ( 0 , 1 ) } \\sup _ { k \\in \\{ k _ 0 ^ * , \\ldots , k _ 1 ^ * \\} } \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\mathbb { P } _ f \\bigl ( I _ { n , q } \\ni H ( f ) \\bigr ) - ( 1 - q ) \\leq \\inf _ { L \\geq 1 } \\frac { 2 } { L ( 2 \\pi ) ^ { 1 / 2 } } = 0 . \\end{align*}"}
-{"id": "2116.png", "formula": "\\begin{align*} h ( x , r ) : = \\sum _ { m = 0 } ^ k r ^ m h _ m ( x ) \\end{align*}"}
-{"id": "1571.png", "formula": "\\begin{align*} X ^ { \\left [ 2 \\right ] } = X + \\eta _ { i } \\partial _ { u _ { , i } } + \\eta _ { i j } \\partial _ { u _ { , i j } } , \\end{align*}"}
-{"id": "2516.png", "formula": "\\begin{align*} F ( x ( t ; \\overline { x } ) ) = \\exp ( - 2 \\lambda t ) \\cdot F ( \\overline { x } ) , ~ ( \\forall ) t \\in [ 0 , \\infty ) . \\end{align*}"}
-{"id": "1799.png", "formula": "\\begin{align*} & Y _ 2 = \\prod _ { i = 1 } ^ n C [ d _ i , e _ i ] , & Y _ 3 = \\prod _ { i = 1 } ^ n C [ c _ i , d _ i ] , \\qquad & Y _ 4 = C [ \\{ c _ 1 , \\ldots , c _ n \\} ] , \\\\ & Y _ 5 = \\prod _ { i = 1 } ^ n C [ b _ i , e _ i ] , & Y _ 6 = \\prod _ { i = 1 } ^ n C [ c _ i , e _ i ] . \\qquad & \\end{align*}"}
-{"id": "3336.png", "formula": "\\begin{align*} & \\operatorname { i n d } _ { \\mathrm { M o r s e } } ( x ) + 2 c _ 1 ( A ) \\\\ & = ( w - \\operatorname { i n d } _ { \\mathrm { C Z } } ( [ x , c _ x ] ) ) + 2 c _ 1 ( A ) \\\\ & = w - \\operatorname { i n d } _ { \\mathrm { C Z } } ( [ x , c _ x \\sharp A ] ) \\\\ & = - w - ( - w ) = 0 . \\end{align*}"}
-{"id": "3639.png", "formula": "\\begin{align*} | f ( \\mathbf { x } ) - K | & = \\left | \\frac { r x _ 1 } { 1 + x _ k } - ( r - 1 ) \\right | = \\left | \\frac { r x _ 1 - ( r - 1 ) x _ k - ( r - 1 ) } { 1 + x _ k } \\right | \\\\ & = \\left | \\frac { r ( x _ 1 - ( r - 1 ) ) } { 1 + x _ k } - \\frac { ( r - 1 ) ( x _ k - ( r - 1 ) ) } { 1 + x _ k } \\right | \\\\ & \\leq r | x _ 1 - ( r - 1 ) | + ( r - 1 ) | x _ k - ( r - 1 ) | \\leq L \\| \\mathbf { x } - ( K , \\dots , K ) \\| , \\mathbf { x } \\in \\R ^ k _ + . \\end{align*}"}
-{"id": "6204.png", "formula": "\\begin{align*} \\int _ { | z | > 1 } [ f ( x + x z ) - f ( x ) ] \\nu _ U ( \\d z ) = x ^ \\beta \\int _ { | z | > 1 } [ | 1 + z | ^ \\beta - 1 ] \\nu _ U ( \\d z ) + o ( x ^ \\beta ) . \\end{align*}"}
-{"id": "6530.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ n ) _ { n = k } ^ m \\big \\| _ { L ^ p ( D ) } \\\\ & \\le C \\big \\| ( f _ n ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } + C \\big \\| \\big ( ( A _ m - A _ n ) v _ n \\big ) _ { n = k } ^ m \\big \\| _ { L ^ p ( X ) } \\\\ & + C \\Big ( \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "2401.png", "formula": "\\begin{align*} \\langle P \\nabla f _ 2 ( P x + q ) - P & \\nabla f _ 2 ( P y + q ) , x - y \\rangle \\\\ & = \\langle \\nabla f _ 2 ( P x + q ) - \\nabla f _ 2 ( P y + q ) , P ( x - y ) \\rangle \\\\ & = \\langle \\nabla f _ 2 ( P x + q ) - \\nabla f _ 2 ( P y + q ) , ( P x + q ) - ( P y + q ) ) \\rangle \\end{align*}"}
-{"id": "8510.png", "formula": "\\begin{align*} c ' _ { \\alpha _ 1 } & = \\max \\{ c _ { \\alpha _ 1 + \\alpha _ 2 } , c _ { \\alpha _ 1 } + c _ { \\alpha _ 1 + \\alpha _ 2 } - c _ { \\alpha _ 2 } \\} , \\\\ c ' _ { \\alpha _ 1 + \\alpha _ 2 } & = \\min \\{ c _ { \\alpha _ 1 } , c _ { \\alpha _ 2 } \\} , \\\\ c ' _ { \\alpha _ 2 } & = \\max \\{ c _ { \\alpha _ 1 + \\alpha _ 2 } , c _ { \\alpha _ 2 } + c _ { \\alpha _ 1 + \\alpha _ 2 } - c _ { \\alpha _ 1 } \\} . \\end{align*}"}
-{"id": "2983.png", "formula": "\\begin{align*} [ \\alpha _ 0 , f _ i ^ { \\lambda _ j ( x , y ) } ] = \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ { \\lambda _ j ( x , y ) } , \\alpha _ n ] - [ \\alpha _ i , f _ 0 ^ { \\lambda _ j ( x , y ) } ] . \\end{align*}"}
-{"id": "1643.png", "formula": "\\begin{align*} \\beta = \\frac { 2 } { 1 - \\alpha } , A = \\pm \\left ( \\frac { 1 } { \\beta \\left ( \\beta - 1 \\right ) } \\right ) ^ { \\frac { 1 } { 1 - \\alpha } } = \\pm \\left ( \\frac { \\left ( 1 - \\alpha \\right ) ^ { 2 } } { 2 \\left ( 1 + \\alpha \\right ) } \\right ) ^ { \\frac { 1 } { 1 - \\alpha } } . \\end{align*}"}
-{"id": "9298.png", "formula": "\\begin{align*} \\begin{cases} d p ( t , x , z ) = - [ A _ { \\pi ( t , z ) } p ( t , x , z ) + x q ( t , x , z ) ] d t + q ( t , x , z ) d R ( t ) ; 0 \\leq t \\leq T , \\\\ p ( T , x , z ) = U ( x , z ) \\mathbb { E } _ { Q } [ \\delta _ { Z } ( z ) | \\mathcal { R } _ T ] . \\end{cases} \\end{align*}"}
-{"id": "5221.png", "formula": "\\begin{align*} \\overline { \\partial } _ { M } T _ { q } u = ( - 1 ) ^ { q } T _ { ( q - 1 ) } ( \\overline { \\partial } _ { M } ^ { * } u ) + \\ ; , \\end{align*}"}
-{"id": "9536.png", "formula": "\\begin{align*} A _ { e _ a ' } e _ b ' & = \\frac { 1 } { 2 } e ^ { 4 \\phi / 3 } { G _ 2 } _ { a b } e _ { 1 0 } ' \\\\ A _ { e _ a ' } e _ { 1 0 } ' & = - \\frac { 1 } { 2 } e ^ { 4 \\phi / 3 } { G _ 2 } _ a ^ { \\ , \\ , \\ , c } e _ c ' . \\end{align*}"}
-{"id": "4619.png", "formula": "\\begin{align*} \\sum _ { { t = 3 } \\atop { t \\neq 7 } } ^ { 8 } f _ t ( p ' ) = q ^ 3 ( q ^ 2 - 1 ) ( q - \\sqrt { 3 q } + 1 ) \\left ( \\frac { ( q + \\sqrt { 3 q } + 1 ) r } { | G _ { p ' } | } - k _ 7 \\right ) . \\end{align*}"}
-{"id": "6233.png", "formula": "\\begin{align*} \\tilde { V } _ { 1 } = [ X ^ { ( 0 ) } ( s _ { t _ { 0 } } ) / | | X ^ { ( 0 ) } ( s _ { t _ { 0 } } ) | | ] , \\end{align*}"}
-{"id": "8769.png", "formula": "\\begin{align*} \\check { R } _ { i } ( x _ { i + 1 } / x _ i ) T ( w ; x _ 1 , . . , x _ i , x _ { i + 1 } , . . , x _ n ) = T ( w ; x _ 1 , . . , x _ { i + 1 } , x _ { i } , . . , x _ n ) \\check { R } _ { i } ( x _ { i + 1 } / x _ i ) . \\end{align*}"}
-{"id": "6110.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ { \\infty } f _ { i } , \\ f _ { i } = \\sum \\limits _ { \\alpha \\geq 2 i } s ^ { \\alpha } f _ { i , \\alpha } , \\end{align*}"}
-{"id": "4616.png", "formula": "\\begin{align*} \\sum _ { t = 3 } ^ { 8 } f _ t ( p ' ) = \\frac { q ^ 3 ( q - 1 ) ( q ^ 3 + 1 ) r } { | G _ { p ' } | } . \\end{align*}"}
-{"id": "8699.png", "formula": "\\begin{align*} \\nabla ^ G v ( \\tau , \\Xi _ \\tau ^ { t , x } ) = Z _ \\tau ^ { t , x } \\ , \\ , \\ ; L _ 2 ( U , K ) . \\end{align*}"}
-{"id": "5098.png", "formula": "\\begin{align*} & q ^ { N _ { a } } | m _ { 1 } , \\ldots , m _ { r } \\rangle = q ^ { m _ { a } } | m _ { 1 } , \\ldots , m _ { r } \\rangle , \\\\ & \\beta _ { a } ^ { * } | m _ { 1 } , \\ldots , m _ { r } \\rangle = | m _ { 1 } , \\ldots , m _ { a } + 1 , \\ldots , m _ { r } \\rangle , \\\\ & \\beta _ { a } | m _ { 1 } , \\ldots , m _ { r } \\rangle = ( 1 - q ^ { 2 m _ { a } } ) | m _ { 1 } , \\ldots , m _ { a } - 1 , \\ldots , m _ { r } \\rangle . \\end{align*}"}
-{"id": "8281.png", "formula": "\\begin{align*} U ( \\gamma ) _ { q , k } = U ( \\gamma ) _ { k , q } = \\frac { 1 } { q } - \\sum _ { k ' } \\gamma _ { k , k ' } , \\ , \\ , \\ , , \\ , U ( \\gamma ) _ { q , q } = - \\Big ( 1 - \\frac { 2 } { q } \\Big ) + \\sum _ { k , k ' } \\gamma _ { k k ' } . \\end{align*}"}
-{"id": "5503.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { D L } [ \\iota ] = \\frac { M K \\rho ^ 2 \\beta _ { k } ^ 4 } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } ^ 2 } \\end{align*}"}
-{"id": "3754.png", "formula": "\\begin{align*} f _ Y ( 0 ) = \\frac { 1 } { \\sqrt { 2 \\pi } } + O \\left ( \\frac { 1 } { E _ \\beta } \\right ) . \\end{align*}"}
-{"id": "2203.png", "formula": "\\begin{align*} W ( t ) : = \\frac { \\int _ { B } w ( t , x ) \\psi ^ { 2 } ( x ) d x } { \\int _ { B } \\psi ^ { 2 } ( x ) d x } , \\end{align*}"}
-{"id": "7700.png", "formula": "\\begin{align*} n _ 0 = \\inf \\{ i > 1 : \\gamma _ i < 0 \\} - 1 , n _ 1 = \\inf \\{ i > n _ 0 : \\gamma _ i > 0 \\} - 1 , \\end{align*}"}
-{"id": "6920.png", "formula": "\\begin{align*} s _ \\lambda ( X ) = & \\ , \\sum _ { T \\in { \\mathcal T } ^ \\lambda } X ^ T \\ , . \\end{align*}"}
-{"id": "3724.png", "formula": "\\begin{align*} \\theta ( \\beta . x ) = \\pi _ { \\beta } \\circ \\theta ( x ) . \\end{align*}"}
-{"id": "8901.png", "formula": "\\begin{align*} z _ n = \\frac { \\Tilde { u } _ n - \\Tilde { v } _ n } { \\norm { \\Tilde { u } _ n - \\Tilde { v } _ n } _ { H ^ 1 _ { A _ n } ( \\R ^ N , \\C ) } } \\end{align*}"}
-{"id": "6256.png", "formula": "\\begin{align*} Z ( x , y , t ) & : = \\left \\langle \\nabla \\log u ( y , t ) , \\gamma ' \\left ( \\tfrac { d } { 2 } \\right ) \\right \\rangle - \\left \\langle \\nabla \\log u ( x , t ) , \\gamma ' \\left ( - \\tfrac { d } { 2 } \\right ) \\right \\rangle - 2 \\psi \\left ( \\frac { d ( x , y ) } { 2 } , t \\right ) , \\end{align*}"}
-{"id": "2958.png", "formula": "\\begin{align*} g ( \\sigma ^ + , \\mu _ 1 ) = \\frac { \\gamma } { \\delta } H _ 2 ( \\mu _ 1 ) > 0 . \\end{align*}"}
-{"id": "497.png", "formula": "\\begin{align*} f \\in \\mathcal { F } \\mapsto \\mathcal { L } ( f ) : = \\Re ( a _ 1 ) \\end{align*}"}
-{"id": "7990.png", "formula": "\\begin{align*} \\varphi ( s ) & : = a _ { 2 0 2 0 } s ^ { 6 } + a _ { 2 0 1 1 } s ^ { 5 } + a _ { 1 1 2 0 } s ^ { 4 } + a _ { 2 0 0 2 } s ^ { 4 } \\\\ & + a _ { 1 1 1 1 } s ^ { 3 } + a _ { 0 2 2 0 } s ^ { 2 } + a _ { 1 1 0 2 } s ^ { 2 } + a _ { 0 2 1 1 } s + a _ { 0 2 0 2 } . \\end{align*}"}
-{"id": "802.png", "formula": "\\begin{align*} \\mathbb { T } ^ { [ 1 , M ] } ( z ) = L ^ { ( 1 ) } ( z ) \\mathbb { T } ^ { [ 2 , M ] } ( z ) = \\mathbb { T } ^ { [ 1 , M - 1 ] } ( z ) L ^ { ( M ) } ( z ) , \\end{align*}"}
-{"id": "648.png", "formula": "\\begin{align*} A _ x = \\big \\{ g \\in G \\ , : \\ , g \\cdot x \\in A \\big \\} . \\end{align*}"}
-{"id": "809.png", "formula": "\\begin{align*} H _ { 1 } = M + \\sum _ { i = 1 } ^ { M } h _ { i - 1 , i } , h _ { i - 1 , i } = \\sum _ { a = 1 } ^ { r } ( \\beta _ { a , i - 1 } ^ { * } - \\beta _ { a , i } ^ { * } ) \\beta _ { a , i } q ^ { 2 \\sum _ { p = a + 1 } ^ { r } N _ { a , i } } , \\end{align*}"}
-{"id": "4019.png", "formula": "\\begin{align*} S = h ^ { - 2 } s , \\end{align*}"}
-{"id": "7946.png", "formula": "\\begin{align*} \\begin{cases} \\Delta ^ 2 w = \\mu w , & { \\rm i n } \\ \\Omega , \\\\ w = 0 , & { \\rm o n } \\ \\partial \\Omega , \\\\ \\frac { \\partial w } { \\partial \\nu } = 0 , & { \\rm o n } \\ \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "7498.png", "formula": "\\begin{align*} n < \\left ( { a } + b \\right ) \\frac { { a } b + { a } + b + 1 } { { a } b - 1 } = ( { a } + b ) \\left ( 1 + \\frac { { a } + b + 2 } { { a } b - 1 } \\right ) . \\end{align*}"}
-{"id": "4352.png", "formula": "\\begin{align*} \\lim _ { y \\to 1 } \\psi ( y , a ) = \\ell ( a ) \\ , \\end{align*}"}
-{"id": "1635.png", "formula": "\\begin{align*} \\dd _ { t } f + \\left ( v \\cdot D _ { x } f + F \\cdot D _ { v } f \\right ) \\dd t + D _ { v } f \\circ \\dd W _ { t } = 0 , f \\big | _ { t = 0 } = f _ { 0 } \\end{align*}"}
-{"id": "1655.png", "formula": "\\begin{align*} \\big ( L ^ p ( \\R ^ d ; ( B ^ { s _ 0 } _ { p , p } ( \\R ^ d ) ) , L ^ p ( \\R ^ d ; ( B ^ { s _ 1 } _ { p , p } ( \\R ^ d ) ) \\big ) _ { \\theta , p } = L ^ p ( \\R ^ d ; B ^ { s } _ { p , p } ( \\R ^ d ) ) , \\end{align*}"}
-{"id": "5899.png", "formula": "\\begin{align*} \\begin{cases} \\dd X _ t & = V _ t \\dd t , \\dd V _ t = F \\left ( X _ t , V _ t \\right ) \\dd t + \\dd W _ { t } \\\\ X \\left ( 0 \\right ) & = x _ { 0 } , \\ ; \\ ; V \\left ( 0 \\right ) = v _ { 0 } . \\end{cases} \\end{align*}"}
-{"id": "6851.png", "formula": "\\begin{align*} \\tilde u ( t ) : = M ( - t ) u ( t ) . \\end{align*}"}
-{"id": "8612.png", "formula": "\\begin{align*} \\overline { \\overline { N } } ( b ) \\overline { J } _ { 1 } & = \\overline { \\overline { N } } ( b ) \\overline { J } _ { 4 } = 0 \\\\ \\overline { \\overline { N } } ( b ) \\overline { J } _ { 2 } & = c _ { k } ^ { - 1 } \\pi _ { b e _ { k } } \\overline { i } _ { 3 } \\theta _ { 2 } \\\\ \\overline { \\overline { N } } ( b ) \\overline { J } _ { 3 } & = c _ { k } ^ { - 1 } \\pi _ { b e _ { k } } \\overline { j } \\overline { \\sigma } \\theta _ { 3 } \\end{align*}"}
-{"id": "3794.png", "formula": "\\begin{align*} g \\left ( { { \\bf { B } } } \\right ) = - I _ { \\rm a s y } \\left ( { { \\bf { x } } ; { \\bf { y } } } \\right ) + \\kappa \\left [ { { \\rm { t r } } \\left ( { { \\bf { B } } { \\bf { B } } ^ H } \\right ) - P } \\right ] \\end{align*}"}
-{"id": "1563.png", "formula": "\\begin{align*} A _ 0 = \\C [ a _ 0 ] \\end{align*}"}
-{"id": "2664.png", "formula": "\\begin{align*} b ^ { ( n ) } _ \\ell ( x _ \\ell ) \\quad = \\begin{cases} 0 & | x _ \\ell | < k ^ { ( n ) } _ \\ell \\ , , \\\\ \\frac { x _ \\ell - k ^ { ( n ) } _ \\ell } { 2 k ^ { ( n ) } _ \\ell - x _ \\ell } & k ^ { ( n ) } _ \\ell \\leq | x _ \\ell | \\leq 2 k ^ { ( n ) } _ \\ell \\ , , \\\\ \\infty & 2 k ^ { ( n ) } _ \\ell < | x _ \\ell | \\ , , \\end{cases} \\end{align*}"}
-{"id": "3318.png", "formula": "\\begin{align*} T ^ { ( k ) * } = ( M ^ { ( k ) } ) ^ { - 1 } T ^ { ( k ) \\dagger } M ^ { ( k - 1 ) } , \\end{align*}"}
-{"id": "8650.png", "formula": "\\begin{align*} G u = \\left ( \\begin{array} [ c ] { c } 0 \\\\ u \\end{array} \\right ) = \\left ( \\begin{array} [ c ] { c } 0 \\\\ I \\end{array} \\right ) u , \\ ; \\ ; \\ ; u \\in U . \\end{align*}"}
-{"id": "4230.png", "formula": "\\begin{align*} \\frac { 2 } { 1 + \\sqrt { 1 - 4 t } } = \\sum _ { n = 0 } ^ \\infty n ! C _ n \\frac { t ^ n } { n ! } , ( \\textnormal { s e e } \\ , \\ , [ 9 - 1 3 ] ) . \\end{align*}"}
-{"id": "9580.png", "formula": "\\begin{align*} \\ddot \\psi ( x , t ) = H _ F \\psi ( x , t ) , x \\in \\R ^ 3 , t \\in \\R . \\end{align*}"}
-{"id": "1454.png", "formula": "\\begin{align*} \\lim _ { l \\to \\infty } \\Phi _ { p _ 0 , q _ 0 , v _ 0 , w _ 0 } ( 2 ^ l Q ) = \\infty , ( Q \\in \\mathcal { Q } ) , \\Phi _ { p _ 0 , q _ 0 , v _ 0 , w _ 0 } ( Q ) : = v _ 0 ( Q ) ^ { \\frac { 1 } { p _ 0 } - \\frac { 1 } { q _ 0 } } w _ 0 ( Q ) ^ \\frac { 1 } { q _ 0 } . \\end{align*}"}
-{"id": "863.png", "formula": "\\begin{align*} ( n + 2 ) ^ 3 U _ { n + 2 } - ( 3 4 n ^ 3 + 1 5 3 n ^ 2 + 2 3 1 n + 1 1 7 ) U _ { n + 1 } + ( n + 1 ) ^ 3 U _ n = 0 , \\end{align*}"}
-{"id": "9091.png", "formula": "\\begin{align*} ( s - 1 / 2 ) Y ' ( s ) = - \\frac { \\alpha + 3 } { 2 } Y ( s ) + 2 u _ { N - 1 } ' ( s ) + \\chi _ { N } ( s ) \\ , \\end{align*}"}
-{"id": "6477.png", "formula": "\\begin{align*} W ( t ) : = \\frac { \\int _ { B } w ( t , x ) \\psi ^ { 2 } ( x ) d x } { \\int _ { B } \\psi ^ { 2 } ( x ) d x } , \\end{align*}"}
-{"id": "9571.png", "formula": "\\begin{align*} \\langle \\psi _ { n } ^ { ( 0 ) } , ( f ^ { ( 1 ) } ) ^ { 2 } \\psi _ { n } ^ { ( 0 ) } \\rangle = \\sum _ { \\substack { i \\\\ \\lambda ^ { ( 0 ) } _ { i } \\neq \\lambda _ { n } ^ { ( 0 ) } } } \\left | \\langle \\psi _ { i } ^ { ( 0 ) } , f ^ { ( 1 ) } \\psi _ { n } ^ { ( 0 ) } \\rangle \\right | ^ { 2 } \\end{align*}"}
-{"id": "3434.png", "formula": "\\begin{align*} d \\kappa ( t ) = q ( t , \\kappa ( t ) ) d t + Q ( t , \\kappa ( t ) ) d W ( t ) , \\kappa ( s ) = \\kappa = ( x , h ) , \\end{align*}"}
-{"id": "1309.png", "formula": "\\begin{align*} S & \\ll H \\exp ( - c _ 1 ( \\log N ) ^ { 3 / 5 } ( \\log \\log n ) ^ { - 1 / 5 } ) \\sum _ { n = N - H } ^ { N + H } n + H N \\\\ & \\ll H ^ 2 N \\exp ( - c _ 1 ( \\log N ) ^ { 3 / 5 } ( \\log \\log n ) ^ { - 1 / 5 } ) + H N , \\end{align*}"}
-{"id": "8545.png", "formula": "\\begin{align*} \\pi _ { b _ { 1 } r _ { 1 } } \\xi _ { b r } = \\delta _ { b _ { 1 } r _ { 1 } , b r } i d _ { N _ { \\sigma ( b ) } } \\end{align*}"}
-{"id": "5141.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\ell ( t ) - 1 } \\left ( \\frac { z _ { i } } { 1 + z _ { i } } \\right ) ^ { x _ { \\tau ' ( i ) } } \\prod _ { i = \\ell ( t ) + 1 } ^ { k } \\left ( \\frac { z _ { i } } { 1 + z _ { i } } \\right ) ^ { x _ { \\tau ' ( i - 1 ) } } = \\prod _ { i = 1 } ^ { k - 1 } \\left ( \\frac { w _ { i } } { 1 + w _ { i } } \\right ) ^ { x _ { \\tilde { \\tau } ( i ) } } . \\end{align*}"}
-{"id": "4620.png", "formula": "\\begin{align*} \\begin{aligned} \\varphi ( ( w w ^ * ) * ( R ( w ) ^ * R ( w ) ) ) & = \\varphi ( w w ^ * ) \\varphi ( R ( w ) ^ * R ( w ) ) \\\\ & = \\| w \\| _ 2 ^ 2 \\| R ( w ) \\| _ 2 ^ 2 = \\| w \\| ^ 4 _ 2 \\end{aligned} \\end{align*}"}
-{"id": "7425.png", "formula": "\\begin{align*} \\left ( e _ { i ( k - 1 ) } e _ { - i ( k - 1 ) } ^ { - 1 } \\dots e _ { i ( 1 ) } e _ { - i ( 1 ) } ^ { - 1 } x _ - \\right ) \\left ( e _ { i ( k ) } e _ { - i ( k ) } ^ { - 1 } \\dots e _ { i ( 1 ) } e _ { - i ( 1 ) } ^ { - 1 } x _ - \\right ) ^ { - 1 } = e _ { - i ( k ) } e _ { i ( k ) } ^ { - 1 } \\in B _ + s _ { i ( k ) } B _ + , \\end{align*}"}
-{"id": "8643.png", "formula": "\\begin{align*} u ( t , x ) = & \\int _ t ^ T R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } G B ( s , \\cdot ) \\right ] ( x ) \\ , d s + \\int _ t ^ T R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } \\nabla ^ G u ( s , \\cdot ) B ( s , \\cdot ) \\right ] ( x ) \\ , d s , \\end{align*}"}
-{"id": "1042.png", "formula": "\\begin{align*} P ( P ( x , x ) , z ) ~ = ~ 0 , \\end{align*}"}
-{"id": "491.png", "formula": "\\begin{align*} \\begin{cases} u _ t = \\Delta u & \\\\ u ( x , 0 ) = u _ 0 ( x ) & \\end{cases} \\end{align*}"}
-{"id": "4762.png", "formula": "\\begin{align*} \\widetilde { \\mathcal { M } } ' : z ( u , v ) = g ( u ) \\ , e _ 1 + f ( u ) \\ , l ( v ) , u \\in I , \\ , v \\in J . \\end{align*}"}
-{"id": "1089.png", "formula": "\\begin{align*} K ^ n ( x , A ) = P r [ f ^ { ( n ) } ( x , \\xi ^ { ( n ) } ) \\in A ] = \\int _ A f ^ { ( n ) } ( x , \\omega ^ n ) \\lambda ^ n ( d \\omega ^ n ) . \\end{align*}"}
-{"id": "511.png", "formula": "\\begin{align*} K _ { 1 1 } ^ { \\rm G S E } ( x , y ) & = K ^ { 4 } _ { 2 1 } ( x , y ) , \\\\ K _ { 1 2 } ^ { \\rm G S E } ( x , y ) & = K ^ { 4 } _ { 2 2 } ( x , y ) , \\\\ K _ { 2 1 } ^ { \\rm G S E } ( x , y ) & = - K ^ { 4 } _ { 1 1 } ( x , y ) = - K ^ { 4 } _ { 2 2 } ( y , x ) = - K ^ { \\rm G S E } _ { 1 2 } ( y , x ) , \\\\ K _ { 2 2 } ^ { \\rm G S E } ( x , y ) & = - K ^ { 4 } _ { 1 2 } ( x , y ) . \\end{align*}"}
-{"id": "5113.png", "formula": "\\begin{align*} E _ { \\vec { z } } ^ { \\vec { \\mu } } ( \\vec { x } , \\vec { \\nu } ) = \\lim _ { \\begin{subarray} { c } M \\to \\infty \\\\ M ' \\to - \\infty \\end{subarray} } \\prod _ { i = 1 } ^ { k } \\frac { z _ { i } ^ { M ' - 1 } } { ( 1 + z _ { i } ) ^ { M } } \\langle \\prod _ { 1 \\le i \\le k } ^ { \\curvearrowright } C _ { \\mu _ { i } } ^ { [ M ' , M ] } ( z _ { i } ) \\prod _ { 1 \\le i \\le k } \\beta _ { \\nu _ { i } , x _ { i } } ^ { * } \\rangle _ { [ M ' , M ] } . \\end{align*}"}
-{"id": "14.png", "formula": "\\begin{align*} \\| x - z \\| & \\le \\| x - T y \\| + \\| T y - z \\| \\\\ & \\le \\| T ( x - y ) \\| + \\sum _ { n = 1 } ^ N \\lambda _ n \\big | 1 - \\| T y _ n \\| \\big | \\\\ & < ( 1 + \\eta ) \\gamma + \\max _ { 1 \\le n \\le N } { \\big | 1 - \\| T y _ n \\| \\big | } \\\\ & \\le ( 1 + \\eta ) \\gamma + \\eta < \\alpha . \\end{align*}"}
-{"id": "5252.png", "formula": "\\begin{align*} \\varphi _ { A , k } ( y ) : = \\inf \\{ t \\in { \\mathbb { R } } \\mid y \\in t k + A \\} . \\end{align*}"}
-{"id": "5176.png", "formula": "\\begin{gather*} R _ { \\ell } = \\bigoplus _ { 0 \\leqslant k \\leqslant \\ell , [ k ] = [ \\ell ] } \\partial R _ k , \\end{gather*}"}
-{"id": "3110.png", "formula": "\\begin{align*} A _ { k k } : = P _ { d - 2 k + 1 } - \\sum _ { \\substack { i + j = 2 k + 1 \\\\ i < j } } \\left ( B _ { i j } + B _ { i j } ^ T \\right ) - \\sum _ { \\substack { i + j = 2 k \\\\ i < j } } \\left ( A _ { i j } + A _ { i j } ^ T \\right ) , \\end{align*}"}
-{"id": "6209.png", "formula": "\\begin{align*} ( \\lambda - \\Delta ) ^ { - 1 } u = \\int _ 0 ^ \\infty e ^ { - \\lambda t } e ^ { t \\Delta } u \\ , d t . \\end{align*}"}
-{"id": "4739.png", "formula": "\\begin{align*} 2 \\int _ { B ^ c _ \\varepsilon ( x ) } \\frac { \\ , u ( x ) - u ( y ) \\ , } { | x - y | ^ { N + 2 s } } \\ , d y & = \\int _ { B _ r \\cap B ^ c _ \\varepsilon } \\frac { \\ , 2 u ( x ) - u ( x + z ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z \\\\ & + \\int _ { B ^ c _ r } \\frac { \\ , 2 u ( x ) - u ( x + z ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z . \\end{align*}"}
-{"id": "8979.png", "formula": "\\begin{align*} J _ k ( \\varphi ) \\ , : = \\ , \\frac { 1 } { p } \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | \\varphi ( x ) - \\varphi ( y ) | ^ p } { | x - y | ^ { N + p s } } \\ , d x \\ , d y - \\int _ { \\mathbb { R } ^ { N } } f ( x ) \\Phi _ { k } ( \\varphi ) \\ , d x \\qquad \\varphi \\in W ^ { s , p } _ 0 ( \\Omega ) \\ , . \\end{align*}"}
-{"id": "6702.png", "formula": "\\begin{align*} \\tilde u = e ^ { a + b } u \\end{align*}"}
-{"id": "4818.png", "formula": "\\begin{align*} \\begin{aligned} & m a c ( ( P _ 1 \\oplus P _ 2 ) ( T \\otimes I _ { n + m } ) ( P _ 1 \\oplus P _ 2 ) ) \\\\ & = m a c ( P _ 1 ( T \\otimes I _ n ) P _ 1 ) + m a c ( P _ 2 ( T \\otimes I _ m ) P _ 2 ) . \\end{aligned} \\end{align*}"}
-{"id": "6328.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ { \\infty } } \\\\ \\lesssim \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ { \\infty } } . \\end{align*}"}
-{"id": "7197.png", "formula": "\\begin{align*} f ( \\frac \\pi 2 ) = - g ' ( \\frac \\pi 2 ) , f ' ( \\frac \\pi 2 ) = - g '' ( \\frac \\pi 2 ) . \\end{align*}"}
-{"id": "3672.png", "formula": "\\begin{align*} \\Phi = \\frac { 1 } { 2 } v ^ 2 + g h ~ , \\end{align*}"}
-{"id": "1281.png", "formula": "\\begin{align*} ( X ^ { - 1 } P X ) A ( q ) ( X ^ { - 1 } P X ) ^ { - 1 } = A ^ * ( q ) . \\end{align*}"}
-{"id": "267.png", "formula": "\\begin{align*} \\sup _ { y \\in \\mathcal { X } _ n } r _ { n , u _ n ^ * ( y ) } = \\sup _ { y \\in \\mathcal { X } _ n } h _ y ^ { - 1 } \\Bigl ( \\frac { a _ n } { n - 1 } \\Bigr ) \\lesssim \\sup _ { y \\in \\mathcal { X } _ n } \\biggl \\{ \\frac { k \\log n } { n f ( y ) } \\biggr \\} ^ { 1 / d } \\ ! \\ ! \\ ! \\ ! \\leq \\biggl ( \\frac { k \\log n } { n \\delta _ n } \\biggr ) ^ { 1 / d } \\ ! \\ ! \\ ! = o ( \\rho _ n ) . \\end{align*}"}
-{"id": "6338.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 2 / 2 } \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 2 , \\alpha _ 1 } } \\| \\Box _ l ^ { \\alpha _ 2 } \\Box _ k ^ { \\alpha _ 1 } f \\| ^ q _ { L ^ { 2 } } \\right ) ^ { 1 / q } \\lesssim 2 ^ { j n \\alpha _ 2 / 2 } 2 ^ { j n ( \\alpha _ 1 - \\alpha _ 2 ) ( 1 / q - 1 / 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "4632.png", "formula": "\\begin{align*} t : = \\bigvee \\limits _ { ( y , z ) \\in A _ a } { \\min \\{ ( f \\circ g ) ( y ) , h ( z ) \\} } \\end{align*}"}
-{"id": "3726.png", "formula": "\\begin{align*} j ( x , y ) = i ( x , y ) - i ( y , x ) \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; x , y \\in F . \\end{align*}"}
-{"id": "9897.png", "formula": "\\begin{align*} \\sigma ( \\mathcal { D } ( K _ { a , b } ) ) = \\left \\{ ( - 2 ) ^ { [ n - 2 ] } , n - 2 \\pm \\sqrt { n ^ { 2 } - 3 a b } \\right \\} , \\end{align*}"}
-{"id": "3863.png", "formula": "\\begin{align*} d ^ { A ^ { ( 1 ) } _ { 2 n } } _ { k , l } ( z ) & = D _ { k , l } ( z ) \\times ( z + q ^ { \\mathtt { h } ^ \\vee } ) ^ { \\delta _ { l , k ^ * } } \\end{align*}"}
-{"id": "1701.png", "formula": "\\begin{gather*} \\int _ { \\R ^ { 2 d } } g ( t , z ) \\eta ( v / m ) \\ , \\dd z + \\int _ 0 ^ t \\int _ { \\R ^ { 2 d } } F ( z ) \\cdot D _ v g ( s , z ) \\eta ( v / m ) \\ , \\dd z \\dd s \\\\ = \\frac { 1 } { 2 m ^ 2 } \\int _ 0 ^ t \\int _ { \\R ^ { 2 d } } g ( s , z ) \\Delta \\eta ( v / m ) \\ , \\dd z \\dd s \\ , . \\end{gather*}"}
-{"id": "6536.png", "formula": "\\begin{align*} \\begin{aligned} & { } \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\\\ & { } \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "4194.png", "formula": "\\begin{align*} f _ R ( x ) = \\begin{cases} 1 & \\norm { x } _ 2 \\le R , \\\\ e ^ R e ^ { - \\norm { x } _ 2 } & \\norm { x } _ 2 \\ge \\sqrt { R } \\end{cases} \\end{align*}"}
-{"id": "6293.png", "formula": "\\begin{align*} \\Lambda _ k ^ { \\alpha } = \\{ l \\in \\mathbb { Z } ^ n : ~ \\Box _ l ^ { \\alpha } \\circ \\Box _ k ^ { \\alpha } \\neq 0 \\} \\end{align*}"}
-{"id": "796.png", "formula": "\\begin{align*} \\check { R } ( z ) & = ( z - q ^ { 2 } ) \\sum _ { a = 1 } ^ { r } E _ { a a } \\otimes E _ { a a } + ( 1 - q ^ { 2 } ) \\sum _ { 1 \\le a < b \\le r } \\left ( z E _ { a a } \\otimes E _ { b b } + E _ { b b } \\otimes E _ { a a } \\right ) \\\\ & + ( z - 1 ) \\sum _ { 1 \\le a < b \\le r } \\left ( E _ { a b } \\otimes E _ { b a } + q ^ { 2 } E _ { b a } \\otimes E _ { a b } \\right ) . \\end{align*}"}
-{"id": "8008.png", "formula": "\\begin{align*} \\xi ( y ) : = \\xi _ 1 ( y ) + \\xi _ 2 ( y ) \\gtrsim n ^ { \\frac { 1 } { p } - \\frac { 1 } { 2 } } \\left ( \\sqrt { \\frac { 2 \\pi } { p } } \\alpha + \\frac { C } { C _ 1 \\alpha \\sqrt { n } + 1 } \\right ) , \\end{align*}"}
-{"id": "2260.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { ( m ) } = \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\left ( x _ { t } \\right ) ^ { m } . \\end{align*}"}
-{"id": "8477.png", "formula": "\\begin{align*} \\partial _ { t } { \\textbf { u } } + \\sum _ { j = 1 } ^ { d } A _ { j } ( \\textbf { u } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\textbf { u } + ( 0 , \\nabla P ) ^ { T } = 0 . \\end{align*}"}
-{"id": "9650.png", "formula": "\\begin{align*} \\begin{array} { l l l } C _ 1 = C ( 1 / \\theta , 1 / \\theta , 1 / \\theta ) & C _ 2 = C ( 1 / \\theta , 1 / \\theta , 1 ) & C _ 3 = C ( 1 / \\theta , 1 / \\theta , \\theta ) \\\\ C _ 4 = C ( 1 / \\theta , 1 , 1 / \\theta ) & C _ 5 = C ( 1 / \\theta , 1 , 1 ) & C _ 6 = C ( 1 / \\theta , 1 , \\theta ) \\\\ \\ldots & \\ldots & \\ldots \\end{array} \\end{align*}"}
-{"id": "8705.png", "formula": "\\begin{align*} G B ( s , \\cdot ) = \\left ( \\begin{array} { l } 0 \\\\ B ( s , \\cdot ) \\end{array} \\right ) , \\end{align*}"}
-{"id": "8691.png", "formula": "\\begin{align*} \\Big ( \\sum _ { m \\ge 1 } \\sup _ { | a | _ U = 1 } | \\nabla _ k \\nabla _ { a } ^ G u _ m ( t , x ) | ^ 2 \\Big ) ^ { 1 / 2 } \\le C | k | _ K \\ , \\sup _ { t \\in [ 0 , T ] } \\| B ( t , \\cdot ) \\| _ { C ^ { \\alpha } _ b ( H , U ) } , \\ ; \\ ; \\ ; \\ ; \\ ; k \\in K , \\end{align*}"}
-{"id": "6621.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l } \\dot { x } ( t ) = v ( x ( t ) ) , \\\\ x ( 0 ) = x _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "6131.png", "formula": "\\begin{align*} R _ { 1 2 3 4 } ^ 2 + 2 R _ { 1 3 4 2 } R _ { 1 4 2 3 } = & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( x + 2 y ) - ( x + 2 y ) ^ 2 ] \\\\ \\ge & \\frac 1 3 [ 2 ( x - y ) ^ 2 + 2 ( x - y ) ( K _ { 1 4 } - K _ { 1 3 } ) - ( K _ { 1 4 } - K _ { 1 3 } ) ^ 2 ] . \\\\ \\end{align*}"}
-{"id": "4401.png", "formula": "\\begin{align*} g ( \\Gamma ) : = g ( G , w ) : = b _ 1 ( G ) + \\sum _ { v \\in V } w ( v ) , \\end{align*}"}
-{"id": "6929.png", "formula": "\\begin{align*} M _ { ( 0 ) } ( z ) = M ^ \\perp _ { ( 0 ) } ( z ) = \\frac { 1 } { 1 - z } \\quad \\mbox { a n d } L _ { ( 0 ) } ( z ) = L ^ \\perp _ { ( 0 ) } ( z ) = ( 1 - z ) \\ , . \\end{align*}"}
-{"id": "1580.png", "formula": "\\begin{align*} C _ { x } \\left ( x \\right ) = - \\frac { 2 \\left ( \\kappa \\left ( \\mu - \\lambda - x \\right ) - \\frac { 1 } { 2 } \\sigma ^ { 2 } - \\frac { 1 } { 2 } \\sigma \\sigma _ { , x } \\right ) } { \\sigma ^ { 2 } \\left ( x \\right ) } . \\end{align*}"}
-{"id": "5075.png", "formula": "\\begin{align*} \\mathcal { S } _ { k _ { 1 } , \\ldots , k _ { r } } = \\{ ( \\vec { x } , \\vec { \\nu } ) \\in L _ { k } ^ { + } \\times I _ { k _ { 1 } , \\ldots , k _ { r } } \\ , | \\ , \\hbox { F o r a n y $ 1 \\le i \\le k $ , i f $ x _ { i } = x _ { i + 1 } $ t h e n $ \\nu _ { i } \\le \\nu _ { i + 1 } $ . } \\} . \\end{align*}"}
-{"id": "2457.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } \\| _ { M _ 2 } \\lesssim & 2 ^ { j n ( \\alpha _ 2 - \\alpha _ 1 ) ( 1 - 1 / p _ 2 ) } \\| \\Box _ k ^ { \\alpha _ 1 } f \\| _ { L ^ { p _ 1 } } \\\\ \\lesssim & 2 ^ { j n ( \\alpha _ 2 - \\alpha _ 1 ) ( 1 - 1 / p _ 2 ) } \\| f \\| _ { M _ 1 } = 2 ^ { j \\widetilde { A _ 2 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "2655.png", "formula": "\\begin{align*} C _ 1 ^ 1 & = 2 \\\\ & = 2 ( 2 \\cdot 0 + 1 ) + 0 \\\\ & = 2 ( 2 \\cdot T ( 0 , 1 ) + T ( 0 , 0 ) ) + T ( 1 , 0 ) \\\\ & = 2 \\cdot T ( 1 , 1 ) + T ( 1 , 0 ) \\\\ & = T ( 2 , 1 ) \\end{align*}"}
-{"id": "8667.png", "formula": "\\begin{align*} [ f ] _ { \\alpha , K } = \\sup _ { k ' , \\ ; k \\in K , k - k ' \\not = 0 } \\frac { | f ( k ) - f ( k ' ) | _ J } { | k - k ' | _ K ^ { \\alpha } } < \\infty . \\end{align*}"}
-{"id": "1976.png", "formula": "\\begin{align*} M = \\left ( m \\Z ^ n + \\left ( \\frac { m } { g } \\right ) \\Z ( \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } { \\bf e } _ i ) \\right ) \\cap H _ v . \\end{align*}"}
-{"id": "7469.png", "formula": "\\begin{align*} \\| x - y \\| = \\| a ( x ' - y ' ) - ( b - a ) y ' \\| \\leq a \\| x ' - y ' \\| + ( b - a ) \\| y ' \\| < a + ( b - a ) = b = \\| y \\| \\end{align*}"}
-{"id": "843.png", "formula": "\\begin{align*} K _ { t } ( w ; z _ { \\ell ( 1 ) } , \\ldots , z _ { \\ell ( t ) } ) = \\frac { g ( w , z _ { \\ell ( 1 ) } ) g ( w , z _ { \\ell ( t ) } ) } { g ( z _ { \\ell ( t ) } , z _ { \\ell ( 1 ) } ) } \\prod _ { s = 1 } ^ { t - 1 } g ( z _ { \\ell ( s + 1 ) } , z _ { \\ell ( s ) } ) \\end{align*}"}
-{"id": "5566.png", "formula": "\\begin{align*} \\rho = \\sum _ { j = 1 } ^ N m _ j \\delta _ { x _ j } , 0 < x _ 1 < \\dots < x _ N < 1 , \\end{align*}"}
-{"id": "897.png", "formula": "\\begin{align*} & = \\sum _ { n = 1 } ^ { \\infty } \\tfrac { 1 } { n ^ { 1 + \\delta } } \\int _ { T } ^ { 2 T } \\left ( { \\tfrac { t } { 2 \\pi } } \\right ) ^ { { 1 } / { 4 } + { \\delta } / { 2 } + { i t } / { 2 } } e ^ { - i ( { t } / { 2 } + { \\pi } / { 8 } ) } n ^ { - i t } \\left \\{ 1 + O ( { 1 } / { t } ) \\right \\} d t . \\end{align*}"}
-{"id": "3788.png", "formula": "\\begin{align*} { \\bf { H } } = { c _ 0 } \\ , { e ^ { - j 2 \\pi { d _ 0 } / { \\lambda _ c } } } \\ , { { \\bf { u } } _ { \\mathrm r } } ( { { \\theta _ { 0 , a } } } ) \\ , { \\bf { u } } _ { \\mathrm t } ^ H ( { { \\phi _ { 0 , d } } } ) + \\sum \\limits _ { l = 1 } ^ L { { c _ l } \\ , { e ^ { - j 2 \\pi { d _ l } / { \\lambda _ { \\rm c } } } } \\ , { { \\bf { u } } _ { \\mathrm r } } ( { { \\theta _ { l , a } } } ) \\ , { \\bf { u } } _ { \\mathrm t } ^ H ( { { \\phi _ { l , d } } } ) } \\end{align*}"}
-{"id": "4399.png", "formula": "\\begin{align*} H _ { \\mu _ n } ( \\xi | \\xi _ { - \\infty } ^ { - 1 } ( T ) ) = H _ { \\mu _ n } ( \\xi | \\xi _ { - n } ^ { - 1 } ( T ) ) = H _ \\nu ( \\zeta | \\zeta _ { - n } ^ { - 1 } ( \\tau ) ) = \\infty . \\end{align*}"}
-{"id": "5691.png", "formula": "\\begin{align*} b _ Q : = \\frac { ( M \\chi _ Q ) ^ { \\alpha / n + \\varepsilon } } { \\ell ( Q ) ^ \\alpha } , \\end{align*}"}
-{"id": "5276.png", "formula": "\\begin{align*} \\chi ( s ) ^ { - 1 / 2 } = O \\left ( t ^ { \\sigma / 2 - 1 / 4 } \\right ) , 0 \\leq \\sigma \\leq 1 + \\delta . \\end{align*}"}
-{"id": "8063.png", "formula": "\\begin{align*} A ( 1 , x ) = \\begin{cases} \\ \\begin{bmatrix} \\cos ( \\pi \\alpha ) & - \\sin ( \\pi \\alpha ) \\\\ \\sin ( \\pi \\alpha ) & \\cos ( \\pi \\alpha ) \\end{bmatrix} & x \\in [ 0 , 1 - \\eta ) , \\\\ [ 2 0 p t ] \\ \\begin{bmatrix} 1 & 0 \\\\ 0 & - 1 \\end{bmatrix} & x \\in [ 1 - \\eta , 1 ) . \\end{cases} \\end{align*}"}
-{"id": "9938.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } - \\mathcal L _ K u - \\lambda u = | u | ^ { 2 ^ * - 2 } u + f ( x , u ) & \\mbox { i n } \\Omega \\\\ u = 0 & \\mbox { i n } \\R ^ n \\setminus \\Omega , \\end{array} \\right . \\end{align*}"}
-{"id": "9463.png", "formula": "\\begin{align*} d \\lambda = d ( \\pi ^ { * } \\beta ) = \\pi ^ { * } ( d \\beta ) = \\pi ^ { * } \\omega \\end{align*}"}
-{"id": "5271.png", "formula": "\\begin{align*} \\int _ { { 1 } / { 2 } + i T } ^ { { 1 } / { 2 } + 2 i T } \\zeta ( s ) d s = ( \\int _ { { 1 } / { 2 } + i T } ^ { 2 + i T } + \\int _ { 2 + i T } ^ { 2 + 2 i T } + \\int _ { 2 + 2 i T } ^ { { 1 } / { 2 } + 2 i T } ) \\zeta ( s ) d s . \\end{align*}"}
-{"id": "9802.png", "formula": "\\begin{align*} \\log m ( \\mathbf { y } ) = \\log f ( \\mathbf { y } | \\boldsymbol { \\theta } ) + \\log \\pi ( \\boldsymbol { \\theta } ) - \\log \\pi ( \\boldsymbol { \\theta } | \\mathbf { y } ) , \\end{align*}"}
-{"id": "7972.png", "formula": "\\begin{align*} V _ s ( K , L ) = \\frac { 1 } { n } \\int _ { S ^ { n - 1 } } h _ L ^ s ( u ) h _ K ^ { 1 - s } ( u ) \\ , d S _ K ( u ) , \\end{align*}"}
-{"id": "4431.png", "formula": "\\begin{align*} \\widehat { C _ p } = C _ 1 \\| x \\| _ { \\widehat { C _ p } } = \\| S ( x ) \\| _ 1 , \\ , x \\in \\widehat { C _ p } . \\end{align*}"}
-{"id": "8247.png", "formula": "\\begin{align*} L ( \\psi ) = 0 , \\end{align*}"}
-{"id": "6239.png", "formula": "\\begin{align*} & \\hat { M } = \\tilde { V } ^ { T } M \\tilde { V } , \\ \\hat { D } = \\tilde { V } ^ { T } D \\tilde { V } , \\ \\hat { K } = \\tilde { V } ^ { T } ( K + Z ) \\tilde { V } = \\tilde { \\hat { K } } + \\tilde { V } ^ { T } Z \\tilde { V } , \\\\ & \\hat { F } = \\tilde { V } ^ { T } F , \\ \\hat { C } _ { p } = C _ { p } \\tilde { V } , \\ \\ C _ { v } = C _ { v } \\tilde { V } . \\end{align*}"}
-{"id": "823.png", "formula": "\\begin{align*} L ^ { ( i ) } ( z ; s ) = \\begin{pmatrix} 1 + z q ^ { 2 N _ { i } } & \\beta _ { i } ^ { * } ( 1 - s ^ { 2 } q ^ { 2 N _ { i } } ) \\\\ z \\beta _ { i } & z + s ^ { 2 } q ^ { 2 N _ { i } } \\end{pmatrix} ( 0 \\le i \\le M ) . \\end{align*}"}
-{"id": "7692.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\left \\{ f ( x _ n ) + \\gamma _ { n + 1 } , 0 \\right \\} , n = 0 , 1 , \\dots , \\end{align*}"}
-{"id": "5089.png", "formula": "\\begin{align*} \\check { R } ( z / w ) [ L ( z ) \\otimes L ( w ) ] = [ L ( w ) \\otimes L ( z ) ] \\check { R } ( z / w ) , \\end{align*}"}
-{"id": "4926.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ \\infty \\bigl ( 1 - X ^ { 2 i } \\bigr ) \\bigl ( 1 + X ^ { 2 i - 1 } Y ^ 2 \\bigr ) \\Bigl ( 1 + \\frac { X ^ { 2 i - 1 } } { Y ^ 2 } \\Bigr ) = \\sum _ { j = - \\infty } ^ \\infty X ^ { j ^ 2 } Y ^ { 2 j } . \\end{align*}"}
-{"id": "4190.png", "formula": "\\begin{align*} \\norm { \\widehat { \\varphi _ k } } _ \\infty \\le \\norm { \\varphi _ k } _ 1 = \\int _ { \\R ^ n } \\frac { 1 } { \\widehat { F _ p } ( x / k ) } \\abs { \\widehat { f } ( x ) } \\ d x \\le \\int _ { \\R ^ n } \\frac { 1 } { \\widehat { F _ p } ( x ) } \\abs { \\widehat { f } ( x ) } \\ d x < \\infty \\end{align*}"}
-{"id": "7487.png", "formula": "\\begin{align*} P ( \\Delta _ { R ^ { ( k ) } } ( \\infty ) = 0 ) > 0 . \\end{align*}"}
-{"id": "8192.png", "formula": "\\begin{align*} \\pi ^ * ( \\frac { d x } { y } ) = i _ c ^ * \\frac { d x _ 3 } { ( a _ 1 - a _ 2 ) x _ 1 x _ 2 } = \\omega _ c . \\end{align*}"}
-{"id": "217.png", "formula": "\\begin{align*} R _ 4 = \\int _ { \\mathcal { X } _ n } \\ ! \\ ! f ( x ) \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\biggl \\{ \\log \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) - \\frac { V _ d ^ { - 2 / d } s ^ { 2 / d } \\Delta f ( x ) } { 2 ( d + 2 ) f ( x ) ^ { 1 + 2 / d } } \\biggr \\} \\mathrm { B } _ { k , n - k } ( s ) \\ , d s \\ , d x . \\end{align*}"}
-{"id": "7339.png", "formula": "\\begin{gather*} W _ { - 1 } = w _ { - 1 } \\otimes w _ 0 - q ^ 2 w _ 0 \\otimes w _ { - 1 } , W _ 1 = w _ 0 \\otimes w _ 1 - q ^ 2 w _ 1 \\otimes w _ 0 , \\\\ W _ 0 = w _ { - 1 } \\otimes w _ 1 - w _ 1 \\otimes w _ { - 1 } - q ^ { - 1 } ( q - q ^ { - 1 } ) w _ 0 \\otimes w _ 0 . \\end{gather*}"}
-{"id": "5495.png", "formula": "\\begin{align*} \\gamma _ k ^ \\mathrm { D L } [ \\iota ] = \\frac { M \\sigma _ { \\mathrm { d } k } ^ 4 [ \\iota ] } { ( \\beta _ { \\mathrm { d } k } + 1 / \\rho _ \\mathrm { d } ) \\sum _ { i = 1 } ^ { K } \\sigma _ { \\mathrm { d } i } ^ 2 [ \\iota ] } . \\end{align*}"}
-{"id": "9821.png", "formula": "\\begin{align*} M _ i \\ , { } _ \\lambda \\ , v = Y _ i \\ , { } _ \\lambda \\ , v = 0 . \\end{align*}"}
-{"id": "9447.png", "formula": "\\begin{align*} 1 & \\ge \\mu ( A _ 0 \\oplus g _ 1 A _ 1 \\oplus \\ldots g _ n A _ n \\oplus B _ 0 \\oplus h _ 1 B _ 1 \\oplus \\ldots \\oplus h _ m B _ m ) \\\\ & \\ge \\mu ( A _ 0 ) + \\sum _ { k = 1 } ^ n \\mu ( g _ k A _ k ) + \\mu ( B _ 0 ) + \\sum _ { l = 1 } ^ m \\mu ( h _ l B _ l ) \\\\ & \\ge \\sum _ { k = 0 } ^ n \\mu ( A _ k ) + \\sum _ { l = 0 } ^ m \\mu ( B _ l ) = 2 , \\end{align*}"}
-{"id": "2763.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( s ) \\diamond d B ^ { H } ( s ) : = \\lim _ { n \\rightarrow 0 } \\sum _ { i = 1 } ^ { n } F ( t _ { i - 1 } ) \\diamond ( B ^ { H } ( t _ { i } ) - B ^ { H } ( t _ { i - 1 } ) ) \\end{align*}"}
-{"id": "7874.png", "formula": "\\begin{align*} q ( t , x , y ) = q _ 0 ( t , x , y ) + \\int _ 0 ^ t \\int _ { \\R ^ d } q _ 0 ( t - s , x , z ) q ( s , z , y ) \\ , d z \\ , d s \\ , . \\end{align*}"}
-{"id": "5022.png", "formula": "\\begin{align*} U = \\big \\{ A \\in 2 ^ G \\ , : \\ , e \\in A \\big \\} , \\end{align*}"}
-{"id": "3341.png", "formula": "\\begin{align*} & C _ { B P S } ^ P ( N , Y ; \\mathbf { l } ^ \\ast , \\alpha ) \\\\ & = \\inf \\{ K > 0 ; \\forall H \\in C ^ \\infty ( N ) H | _ Y \\geq K , \\\\ & \\exists \\mu \\in \\mathfrak { M } ( N , X _ H ) | \\langle \\mathbf { l } ^ \\ast , \\rho ( \\mu , X _ H ) \\rangle | \\geq \\mathbf { l } ^ \\ast ( \\alpha ) \\} . \\\\ \\end{align*}"}
-{"id": "9075.png", "formula": "\\begin{align*} s ( 1 - s ^ { 2 } ) M _ N ^ { ( 2 ) \\prime \\prime } + ( 3 - 2 ( \\alpha + 2 N ) s - 5 s ^ { 2 } ) M _ N ^ { ( 2 ) \\prime } - ( 3 ( \\alpha + 2 N ) + 4 s - \\alpha ^ { 2 } s ) M _ N ^ { ( 2 ) } = 0 \\ . \\end{align*}"}
-{"id": "8108.png", "formula": "\\begin{align*} C _ { k , l , N } = \\inf \\{ { \\mathcal R } _ { k , l , N } ( \\Omega ) : \\ , \\Omega \\subset \\mathbb { R } ^ N , \\ , \\Omega \\mbox { s m o o t h } \\} . \\end{align*}"}
-{"id": "8748.png", "formula": "\\begin{align*} | { \\widetilde Q } _ t ^ { - 1 / 2 } e ^ { t A } k | _ K = | { Q } _ t ^ { - 1 / 2 } e ^ { t A } k | _ H , \\ ; \\ ; \\ ; k \\in K , \\ ; t > 0 . \\end{align*}"}
-{"id": "6753.png", "formula": "\\begin{align*} { \\rm I } ( \\varphi _ { t } ) - { \\rm I } ( \\varphi _ { 0 } ) = \\int _ { 0 } ^ t \\int _ X \\chi \\theta _ { \\varphi _ { s } } ^ n d s . \\end{align*}"}
-{"id": "292.png", "formula": "\\begin{align*} ( p _ { n , x , u } - p _ \\cap ) \\biggl \\| V ^ { - 1 / 2 } \\begin{pmatrix} 1 - p _ { n , x , u } \\\\ - p _ { n , y , v } \\end{pmatrix} \\biggr \\| ^ { 3 } \\leq ( p _ { n , x , u } - p _ \\cap ) p _ { n , y , v } ^ { 3 / 2 } | V | ^ { - 3 / 2 } \\lesssim \\Bigl ( \\frac { n } { k \\| z \\| } \\Bigr ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "5980.png", "formula": "\\begin{align*} w \\cdot x = \\zeta x , \\end{align*}"}
-{"id": "4680.png", "formula": "\\begin{align*} | I | \\lesssim \\sum _ { n = 1 } ^ N \\gamma ^ { - n \\alpha } \\gamma ^ n h = h \\sum _ { n = 1 } ^ N \\gamma ^ { n ( 1 - \\alpha ) } = h \\frac { \\gamma ^ { 1 - \\alpha } - \\gamma ^ { N ( 1 - \\alpha ) } } { 1 - \\gamma ^ { 1 - \\alpha } } = h \\cdot O ( h ^ { \\alpha - 1 } ) = O ( h ^ \\alpha ) . \\end{align*}"}
-{"id": "2412.png", "formula": "\\begin{align*} | \\beta \\tau | ^ { 2 k } - \\sum _ { i = 0 } ^ { 2 k - 1 } | \\beta | ^ i | \\tau | ^ { 4 k - i } \\lesssim \\delta . \\end{align*}"}
-{"id": "3264.png", "formula": "\\begin{align*} D = \\eth + \\eth ^ * \\in U ( \\mathfrak { g } ) \\otimes \\mathrm { C l } . \\end{align*}"}
-{"id": "4014.png", "formula": "\\begin{align*} x = \\sum _ { i = \\delta } ^ \\ell i \\alpha _ i , y = \\sum _ { i = \\delta } ^ { \\ell } { i \\choose 2 } \\alpha _ i , \\mbox { a n d } z = \\sum _ { i = \\delta } ^ { \\ell } \\alpha _ i . \\end{align*}"}
-{"id": "4095.png", "formula": "\\begin{align*} \\omega & = - z ( 3 a x ^ 2 + 3 y ^ 2 + c z ^ 2 ) d x + 4 x y z d y - x ( 3 a x ^ 2 + 3 y ^ 2 - c z ^ 2 ) d z , \\\\ \\eta & = z ^ 2 x d x - x ^ 2 z d z , \\end{align*}"}
-{"id": "6468.png", "formula": "\\begin{align*} 0 \\leq h _ { m } * W = g _ { \\alpha } * \\partial _ { s } ^ { \\alpha } ( h _ { m } * W ) \\leq g _ { \\alpha } * G _ { m } + g _ { \\alpha } * [ - F _ { m } ( s ) ] ^ { + } \\end{align*}"}
-{"id": "5398.png", "formula": "\\begin{align*} \\frac { e ( H ) } { v ( H ) } > \\frac { 1 } { N } \\left ( k - \\alpha - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } = \\frac { n } { 2 } \\left ( 1 - \\frac { 1 } { 4 ( k - \\alpha ) } \\right ) \\ , . \\end{align*}"}
-{"id": "1285.png", "formula": "\\begin{align*} K ^ * _ 0 = \\begin{bmatrix} q & 1 & - \\alpha & 0 & - \\overline \\alpha \\\\ 0 & 0 & q \\alpha & 0 & q \\overline \\alpha \\\\ 0 & \\overline \\alpha & 0 & \\alpha J ( 2 r , 3 r ) & 0 \\\\ 0 & 0 & \\overline \\alpha J ( r , r ) & 0 & \\alpha J ( 3 r , 3 r ) \\\\ 0 & \\alpha & 0 & \\overline \\alpha J ( 2 r , r ) & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "6708.png", "formula": "\\begin{align*} \\sigma ( f ) = \\int f \\ , \\mu . \\end{align*}"}
-{"id": "9095.png", "formula": "\\begin{align*} ( s ^ { 2 } - 1 / 4 ) Y '' + \\left ( 4 s + \\frac { \\alpha - 1 } { 2 } + N \\right ) Y ' + \\frac { 9 - \\alpha ^ { 2 } } { 4 } Y + ( \\alpha - 3 ) u _ { N - 1 } ' - ( 2 s + 1 ) u _ { N - 1 } '' + 2 K _ { N } = 0 \\ . \\end{align*}"}
-{"id": "898.png", "formula": "\\begin{align*} { } & \\sum _ { n = 1 } ^ { \\infty } \\tfrac { 1 } { n ^ { 1 + \\delta } } \\int _ { T } ^ { 2 T } \\left ( { \\tfrac { t } { 2 \\pi } } \\right ) ^ { { 1 } / { 4 } + { \\delta } / { 2 } + { i t } / { 2 } } e ^ { - { i t } / { 2 } - i t \\log n } d t \\\\ { } & = \\sum _ { n = 1 } ^ { \\infty } \\tfrac { 1 } { n ^ { 1 + \\delta } } \\int _ { T } ^ { 2 T } \\left ( { \\tfrac { t } { 2 \\pi } } \\right ) ^ { { 1 } / { 4 } + { \\delta } / { 2 } } e ^ { \\frac { i t } { 2 } \\log ( { t } / { 2 \\pi e n ^ { 2 } ) } } d t . \\end{align*}"}
-{"id": "3184.png", "formula": "\\begin{align*} f _ { j , \\alpha } - \\sum _ { | \\beta | = n + 1 } T _ { j k } f _ { k , \\beta } \\tau ^ \\beta _ { k j , \\alpha } = f _ { k j , \\alpha } \\end{align*}"}
-{"id": "5183.png", "formula": "\\begin{gather*} \\varphi ( c ) = \\sum _ { } \\alpha _ j r _ { x ^ j _ 1 \\dots x ^ j _ \\lambda 1 \\dots 1 } + m ' , \\end{gather*}"}
-{"id": "3406.png", "formula": "\\begin{align*} \\phi ' _ 1 ( w ) & = w \\\\ \\phi ' _ 2 ( z ) & = z + 2 w z , \\end{align*}"}
-{"id": "2286.png", "formula": "\\begin{align*} E _ \\ell : = \\big \\| ( v _ n ) _ { n = k } ^ \\ell \\big \\| _ { L ^ p ( D ) } ^ p = \\tau \\sum _ { n = k } ^ \\ell \\| v _ n \\| _ { D } ^ p , \\ell = k , \\dotsc , N , \\end{align*}"}
-{"id": "734.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n } ( P ) ) = \\delta _ { f } ^ { n } h _ { H } ( P ) + \\sum _ { k = 0 } ^ { n - 1 } \\delta _ { f } ^ { n - 1 - k } h _ { E ' _ { 1 } } ( f ^ { k } ( P ) ) - \\sum _ { k = 0 } ^ { n - 1 } \\delta _ { f } ^ { n - 1 - k } h _ { Z _ { 1 } } ( p ^ { - 1 } f ^ { k } ( P ) ) . \\end{align*}"}
-{"id": "9478.png", "formula": "\\begin{align*} \\alpha _ { n } = \\sum _ { i = 1 } ^ { n } x _ { i } \\ , d y _ { i } - y _ { i } \\ , d x _ { i } \\end{align*}"}
-{"id": "8222.png", "formula": "\\begin{align*} E U _ 0 = 2 \\epsilon \\left ( \\bar { U } _ { 1 } - \\bar { U } _ 0 \\right ) + i \\gamma U _ 0 + \\Omega \\bar { U } _ 0 + 6 | U _ 0 | ^ 2 \\bar { U } _ 0 + 2 U _ 0 ^ 3 , \\end{align*}"}
-{"id": "3671.png", "formula": "\\begin{align*} Z _ n = \\frac { 1 } { 2 + n } \\int _ A h q ^ { 2 + n } d A ~ , \\end{align*}"}
-{"id": "7379.png", "formula": "\\begin{align*} b ^ { t } + \\lambda _ t m ^ { t - 1 } = A q ^ t , h ^ { t + 1 } + \\xi _ t q ^ t = A ^ * m ^ t . \\end{align*}"}
-{"id": "6798.png", "formula": "\\begin{align*} \\tilde L _ m B _ n = \\sum _ { k = 1 } ^ m \\left [ B _ k , B _ { m + n - k } \\right ] + n B _ { m + n } n \\geq 0 , m \\geq - 1 . \\end{align*}"}
-{"id": "5291.png", "formula": "\\begin{align*} \\vert \\sum _ { 3 \\sqrt { T / \\pi } < n \\leq C T / \\pi } \\frac { 1 } { n ^ { { 1 } / { 2 } } } \\int _ { T } ^ { 2 T } e ^ { i F ( n , t ) } d t \\vert = O \\left ( \\sum _ { 3 \\sqrt { T / \\pi } < n \\leq C T / \\pi } \\frac { 1 } { n ^ { 1 / 2 } } \\right ) = O \\left ( T ^ { 1 / 2 } \\right ) . \\end{align*}"}
-{"id": "5806.png", "formula": "\\begin{align*} x = \\frac { 1 } { 2 n } \\sum _ { i = 1 } ^ { n - 1 } ( n - i ) ( i n - 2 ) h _ { i } . \\end{align*}"}
-{"id": "4665.png", "formula": "\\begin{align*} W _ \\phi ( w _ 1 , w _ 2 ) & = \\frac { 2 ^ { w _ 2 - 3 } } { \\pi ^ { \\frac { 1 + w _ 1 + w _ 2 } { 2 } } } \\Gamma \\left ( 1 - w _ 2 \\right ) \\cos \\left ( \\frac { \\pi } { 2 } ( 1 - w _ 2 ) \\right ) \\\\ & \\times \\Gamma \\left ( \\frac { - 1 + w _ 1 + 3 w _ 2 + 2 i t _ \\phi } { 4 } \\right ) \\Gamma \\left ( \\frac { - 1 + w _ 1 + 3 w _ 2 - 2 i t _ \\phi } { 4 } \\right ) \\end{align*}"}
-{"id": "6074.png", "formula": "\\begin{align*} \\mathbb { E } ( e ^ { i u X _ t } ) = ( 1 - i u ) ^ { - t } . \\end{align*}"}
-{"id": "52.png", "formula": "\\begin{align*} \\psi \\left ( 1 - \\frac { f ( r , a ) } { a } , a \\right ) : = \\frac { | f ' ( r , a ) | ^ p } { a ^ p } \\ , r \\in [ 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "8445.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\rho + v \\cdot \\nabla \\rho = 0 , \\\\ & \\partial _ { t } { v } + { v } \\cdot \\nabla { v } + f ( \\rho , v ) \\nabla \\rho + \\nabla { P } = 0 , \\\\ & \\nabla \\cdot v = 0 . \\end{cases} \\end{align*}"}
-{"id": "2586.png", "formula": "\\begin{align*} z _ { n , J _ 0 } ^ J ( t ) : = \\sum _ { j = J _ 0 + 1 } ^ J v _ n ^ j ( t ) . \\end{align*}"}
-{"id": "558.png", "formula": "\\begin{align*} \\begin{aligned} K _ 0 ( i ( { \\tilde \\tau } ) ) \\circ \\alpha ( { \\tilde \\tau } ) ^ { - 1 } & = K _ 0 ( W ) \\circ K _ 0 ( i ( \\tau ) ) \\circ \\alpha ( \\tau ) ^ { - 1 } \\\\ & = K _ 0 ( i ( \\tau ) ) \\circ \\alpha ( \\tau ) ^ { - 1 } \\end{aligned} \\end{align*}"}
-{"id": "9959.png", "formula": "\\begin{align*} c = \\left ( \\frac 1 2 - \\frac { \\mu } { 2 ^ * } \\right ) L = \\frac { n - \\mu ( n - 2 s ) } { 2 n } L , \\end{align*}"}
-{"id": "6527.png", "formula": "\\begin{align*} \\frac 1 \\tau \\sum _ { j = 0 } ^ k \\delta _ j v _ { n - j } + A ( t _ n ) v _ n = f _ n , k \\le n \\le N , \\end{align*}"}
-{"id": "1452.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { k } { \\Phi _ { p , q , w } ( 2 ^ k Q ) } \\leq \\frac { C } { \\Phi _ { p , q , w } ( Q ) } ( Q \\in \\mathcal { Q } ) . \\end{align*}"}
-{"id": "3475.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u _ f ( x ) = f ( x ) & \\mbox { i n } \\Omega , \\\\ u _ f = 0 & \\mbox { o n } \\partial \\Omega , \\end{cases} \\end{align*}"}
-{"id": "745.png", "formula": "\\begin{align*} \\frac { 1 } { | W | } \\prod _ { i = 1 } ^ n ( h + e _ i + 1 ) , \\end{align*}"}
-{"id": "9093.png", "formula": "\\begin{align*} - ( N - 1 ) Y ' ( s ) = \\chi _ { N } ( s ) + ( s + 1 / 2 ) \\chi _ { N } ' ( s ) + \\frac { - \\alpha + 1 } { 2 } \\chi _ { N } ( s ) + 2 K _ { N } ( s ) \\ , \\end{align*}"}
-{"id": "6150.png", "formula": "\\begin{align*} x _ j x _ i = q x _ i x _ j , \\forall \\ , 1 \\leq i < j \\leq n . \\end{align*}"}
-{"id": "3784.png", "formula": "\\begin{align*} { { \\bf { E } } _ { s } } = E \\left [ { \\left ( { { { \\bf { d } } _ { s } } - { { { \\bf { \\hat d } } } _ { s } } } \\right ) { { \\left ( { { { \\bf { d } } _ { s } } - { { { \\bf { \\hat d } } } _ { s } } } \\right ) } ^ H } } \\right ] . \\end{align*}"}
-{"id": "294.png", "formula": "\\begin{align*} \\mathcal { X } _ n ^ { ( 1 ) } : = \\{ x : f ( x ) \\geq k ^ \\frac { d } { 2 \\beta } \\delta _ n \\} \\ , , \\mathcal { X } _ n ^ { ( 2 ) } : = \\{ x : \\delta _ n \\leq f ( x ) < k ^ \\frac { d } { 2 \\beta } \\delta _ n \\} , \\end{align*}"}
-{"id": "1711.png", "formula": "\\begin{gather*} x \\times ( x \\times y ) : = - H _ { \\times } ( x , x ) y + H _ { \\times } ( x , y ) x . \\end{gather*}"}
-{"id": "1504.png", "formula": "\\begin{align*} e _ i ^ 2 = \\omega e _ i + 1 \\ , e _ j \\ , e _ { j + 1 } \\ , e _ j = e _ { j + 1 } \\ , e _ j \\ , e _ { j + 1 } , \\end{align*}"}
-{"id": "1019.png", "formula": "\\begin{align*} Z ( \\lambda , u ) = P ( u ) + Q ( N _ { f } ( u ) ) - \\frac { B _ { \\varphi , b } ( P u ) } { T } + H \\left ( \\varphi ^ { - 1 } \\left [ \\lambda \\frac { \\Psi B _ { \\varphi , b } ( P u ) } { T } + \\varphi ( P u ) \\right ] \\right ) , \\end{align*}"}
-{"id": "8165.png", "formula": "\\begin{align*} \\sigma _ m ( R _ U ( k ) ) ( x , \\xi ) = \\tilde \\sigma _ m ( R _ U ( k ) ) ( v , d s _ { v } ^ * ( \\xi ) ) , \\xi \\in \\mathcal F ^ * _ x \\setminus \\{ 0 \\} , \\end{align*}"}
-{"id": "2527.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d u = \\mathbf { i } \\big { ( } \\Delta u + \\lambda | u | ^ 2 u \\big { ) } d t + \\mathbf { i } u \\circ d W , \\\\ & u ( t , 0 ) = u ( t , 1 ) = 0 , ~ t \\geq 0 , \\\\ & u ( 0 , x ) = u _ 0 ( x ) , ~ x \\in [ 0 , 1 ] \\end{aligned} \\right . \\end{align*}"}
-{"id": "9836.png", "formula": "\\begin{align*} x ( t ) & = \\theta ( t ) \\otimes r ( t ) . \\end{align*}"}
-{"id": "4578.png", "formula": "\\begin{align*} D _ { g ^ * } \\leq 8 d ^ { 1 / 2 } \\Bigl ( \\frac { 7 1 } { 4 1 } \\Bigr ) ^ { m } m ! ( m + 1 ) ^ { m + 2 } = : D . \\end{align*}"}
-{"id": "6382.png", "formula": "\\begin{align*} p ( x ) = \\frac { C ( \\varepsilon ) } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon x ^ { p } ) ^ { n } } { n ! } . \\end{align*}"}
-{"id": "9627.png", "formula": "\\begin{align*} \\tilde U ( \\zeta ( t ) ) = U ( \\zeta ( t ) ) , t \\in [ 0 , \\tau ] . \\end{align*}"}
-{"id": "2663.png", "formula": "\\begin{align*} \\Psi _ n ( h _ n ) \\quad & = \\quad \\sqrt { \\frac { n } { 2 } } \\left ( h _ n ( \\textbf { X } ^ { ( n ) } ( 1 ) ) - h _ n ( \\textbf { X } ^ { ( n ) } ( 0 ) ) \\right ) \\ , . \\end{align*}"}
-{"id": "6471.png", "formula": "\\begin{align*} F ( s ) = - \\dot { \\phi } ( s ) \\left ( g _ { 1 - \\alpha } * \\int _ { B _ { 1 } } \\psi ^ { 2 } w ^ { 2 } d x \\right ) ( s ) + c _ { 4 } ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x . \\end{align*}"}
-{"id": "768.png", "formula": "\\begin{gather*} q = q \\cdot 1 ^ k \\leqslant ( r _ 1 ) ^ { \\delta ( q ) } ( r _ 1 ) ^ s \\cdots ( r _ 1 ) ^ s = ( r _ 1 ) ^ { \\delta ( q ) + k s } . \\end{gather*}"}
-{"id": "4875.png", "formula": "\\begin{align*} \\beta _ n ( r _ 0 , r _ 1 , \\dots , r _ { n - 1 } ) = \\sum _ { k = 0 } ^ { n - 1 } p ^ k [ \\phi ^ { - k } ( r _ k ) ] \\bmod I ^ n \\end{align*}"}
-{"id": "5877.png", "formula": "\\begin{align*} Z ^ { 3 } = 2 t \\partial _ { t } + H + c t K ^ { 1 } \\end{align*}"}
-{"id": "534.png", "formula": "\\begin{align*} I ^ { \\rm e x p } _ { 2 2 } ( i , x ; j , y ) : = \\int _ { \\mathcal { C } _ { b _ z } ^ { \\pi / 3 } } \\dd z \\int _ { \\mathcal { C } _ { b _ w } ^ { \\pi / 3 } } \\dd w \\frac { ( z - w ) e ^ { - x z - y w } } { z + w } \\frac { ( 1 + 2 z ) ^ { m _ i } ( 1 + 2 w ) ^ { m _ j } } { ( 1 - 2 z ) ^ { n _ i } ( 1 - 2 w ) ^ { n _ j } } \\frac { 1 } { 2 \\alpha - 1 - 2 z } \\frac { 1 } { 2 \\alpha - 1 - 2 w } , \\end{align*}"}
-{"id": "7729.png", "formula": "\\begin{align*} T _ N ( n ) : = \\sum _ { i = N } ^ { n } \\sigma _ i \\xi _ { i + 1 } , n \\ge N , \\end{align*}"}
-{"id": "2881.png", "formula": "\\begin{align*} E _ 1 - C o g ^ { c o n i l , p t } ( d g C o g ^ { c o n i l } ) = s u E _ 1 - C o g ^ { c o n i l , c o a u g } ( d g C o g ^ { c o n i l } ) . \\end{align*}"}
-{"id": "6706.png", "formula": "\\begin{align*} w ^ { L K } : = \\tilde b ^ L _ Q \\star v ^ { Q P } \\star \\tilde a ^ K _ P . \\end{align*}"}
-{"id": "4504.png", "formula": "\\begin{align*} s _ { x , y } : = s - \\frac { s ^ { 1 + 2 / d } \\Delta f ( x ) } { 2 ( d + 2 ) V _ d ^ { 2 / d } f ( x ) ^ { 1 + 2 / d } } + y . \\end{align*}"}
-{"id": "2003.png", "formula": "\\begin{align*} M _ t = f ( V _ t ) - \\int _ 0 ^ t \\mathcal { A } f ( V _ s ) \\d s , t \\geq 0 , \\end{align*}"}
-{"id": "9794.png", "formula": "\\begin{align*} Z _ { i } = X _ { i j } \\boldsymbol { \\beta } + \\varepsilon _ { i } \\end{align*}"}
-{"id": "1790.png", "formula": "\\begin{align*} P _ { G _ { d , n } } ( 1 ) = e ^ { \\left ( \\frac { 1 } { 2 } + o _ d ( 1 ) \\right ) \\frac { \\log ^ 2 d } { d } n } , \\end{align*}"}
-{"id": "1862.png", "formula": "\\begin{align*} N ^ + ( v ) & : = \\big \\{ w \\in N _ H ( v ) \\colon w > v \\big \\} \\ , , \\\\ N ^ - ( v ) & : = \\big \\{ w \\in N _ H ( v ) \\colon w < v \\big \\} \\ , , \\\\ N ^ { < u } ( v ) & : = \\big \\{ w \\in N _ H ( v ) \\colon w < u \\big \\} \\ , . \\end{align*}"}
-{"id": "4746.png", "formula": "\\begin{align*} \\pm \\int _ { ( B ^ c _ \\varepsilon ( x ) ) ^ \\pm } \\frac { \\xi \\cdot ( x - y ) } { \\ , | x - y | ^ { N + 2 s } \\ , } \\ , d y = + \\infty , \\end{align*}"}
-{"id": "2327.png", "formula": "\\begin{align*} ~ N _ p ( x ) = p ( x ) / p ' ( x ) , \\end{align*}"}
-{"id": "2587.png", "formula": "\\begin{align*} u _ n ^ J - z _ { n , J _ 0 } ^ J = \\sum _ { j = 1 } ^ { J _ 0 } v _ n ^ j + e ^ { i t \\Delta } W _ n ^ J , \\end{align*}"}
-{"id": "8500.png", "formula": "\\begin{align*} o _ L ( G ^ { L _ i } + X _ i ' ) = \\nu ( 0 ) < o _ L ( H + X _ i ' ) , \\end{align*}"}
-{"id": "9955.png", "formula": "\\begin{align*} \\widehat { \\mathcal E } _ \\mu ( u ) & : = \\frac { 1 } { 2 } \\left \\| u \\right \\| ^ 2 - \\frac { \\mu } { 2 ^ * } \\int _ \\Omega ( u ^ + ) ^ { 2 ^ * } d x , \\\\ \\widetilde { \\mathcal E } _ { \\lambda , \\mu } ( u ) & : = \\frac { \\mu } { 2 ^ * } \\int _ \\Omega ( u ^ + ) ^ { 2 ^ * } d x + \\frac { \\lambda } { r } \\int _ \\Omega ( u ^ + ) ^ r d x + \\lambda \\int _ \\Omega G ( u ^ + ) d x \\end{align*}"}
-{"id": "425.png", "formula": "\\begin{align*} X _ j : = \\frac { \\partial } { \\partial z _ j } + \\frac { 1 } { 2 } \\sum _ { k = 1 } ^ d L _ { j k } z _ k \\frac { \\partial } { \\partial t } , j = 1 , 2 , \\cdots , d . \\end{align*}"}
-{"id": "7813.png", "formula": "\\begin{align*} u _ { i j } f _ { i } f _ { j } = 5 u + O \\left ( S ^ { 3 } \\right ) \\end{align*}"}
-{"id": "6672.png", "formula": "\\begin{gather*} W _ 1 ( \\Lambda , \\Lambda ^ n ) \\leqslant \\dfrac { K } { \\sqrt { n } } , \\\\ W _ 1 ( \\Lambda _ 0 ^ n , \\Lambda _ 0 ) = W _ 1 ( \\Lambda _ 0 , \\Lambda _ 0 ^ n ) \\leqslant \\dfrac { K } { \\sqrt { n } } . \\end{gather*}"}
-{"id": "1794.png", "formula": "\\begin{align*} \\langle \\Lambda , c \\rangle = ( \\Lambda | \\nu ( c ) ) = ( \\Lambda | \\delta ) = ( w \\Lambda | w \\delta ) = ( w \\Lambda | \\delta ) = ( t _ { w _ 0 \\lambda } \\Lambda _ 0 | \\delta ) = ( \\Lambda _ 0 | \\delta ) = 1 \\end{align*}"}
-{"id": "9823.png", "formula": "\\begin{align*} L _ i \\ , { } _ { { } _ \\lambda } v _ m = \\begin{cases} ( \\partial + \\b ) v _ { i + m } & \\ \\mbox { i f } \\ \\ \\ ( a _ m , a _ { i + m } ) = ( 0 , 0 ) , \\\\ [ 4 p t ] ( \\partial + \\b + \\lambda ) v _ { i + m } & \\ \\mbox { i f } \\ \\ \\ ( a _ m , a _ { i + m } ) = ( 1 , 1 ) , \\\\ [ 4 p t ] v _ { i + m } & \\ \\mbox { i f } \\ \\ \\ ( a _ m , a _ { i + m } ) = ( 0 , 1 ) , \\\\ [ 4 p t ] ( \\partial + \\b ) ( \\partial + \\b + \\lambda ) v _ { i + m } & \\ \\mbox { i f } \\ \\ \\ ( a _ m , a _ { i + m } ) = ( 1 , 0 ) . \\end{cases} \\end{align*}"}
-{"id": "6084.png", "formula": "\\begin{align*} \\sum _ { ( m , n ) \\in R ( t ) } E _ { m , n } ( t ) & \\ll \\sum _ { 1 \\le n < t ^ 4 } \\int _ n ^ { n + 1 } \\frac { v - n } { v ^ { 3 / 2 } } \\int _ { \\abs { \\log \\frac { u } { v } } < 2 \\sqrt { \\frac { \\log t } { t } } } u ^ { - 1 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 1 } d u d v \\\\ & = \\sum _ { 1 \\le n < t ^ 4 } \\int _ n ^ { n + 1 } \\frac { v - n } { v } \\int _ { \\exp ( - 2 \\sqrt { \\frac { \\log t } { t } } ) } ^ { \\exp ( 2 \\sqrt { \\frac { \\log t } { t } } ) } u ^ { - 1 / 2 } \\left ( 1 + i \\log u \\right ) ^ { - t - 1 } d u d v . \\end{align*}"}
-{"id": "7173.png", "formula": "\\begin{align*} - \\Delta v + \\nabla p = f , \\qquad { \\rm d i v } \\ , v = 0 \\mbox { i n } \\ , \\ , \\Omega , \\end{align*}"}
-{"id": "561.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| ( I - A _ n ) K \\| = 0 . \\end{align*}"}
-{"id": "1480.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } : = \\inf \\left \\{ \\lambda > 0 \\ , : \\ , \\int _ { \\mathbb { R } ^ n } \\left | \\frac { f ( x ) } { \\lambda } \\right | ^ { p ( x ) } \\ , d x \\le 1 \\right \\} . \\end{align*}"}
-{"id": "6855.png", "formula": "\\begin{align*} L ( E ) = \\sup \\{ S _ { I } ( u ) \\} , \\end{align*}"}
-{"id": "1572.png", "formula": "\\begin{align*} \\Delta u - C ^ { \\alpha } u _ { , \\beta } - u _ { , t } = 0 , \\end{align*}"}
-{"id": "6500.png", "formula": "\\begin{align*} & H ' ( u ( t ) ) \\frac { d } { d t } ( k * u ) ( t ) = \\frac { d } { d t } ( k * H ( u ) ) ( t ) + ( - H ( u ( t ) ) + H ' ( u ( t ) ) u ( t ) ) k ( t ) \\\\ & \\qquad + \\int _ { 0 } ^ { t } ( H ( u ( t - s ) ) - H ( u ( t ) ) - H ' ( u ( t ) ) [ u ( t - s ) - u ( t ) ] ) ( - \\dot { k } ( s ) ) d s , \\end{align*}"}
-{"id": "1234.png", "formula": "\\begin{align*} \\mathfrak { m } _ { \\varphi _ , w } = \\{ x : p _ { \\varphi , w } ( \\delta x ) < \\infty , \\ , \\ , \\ , \\ , \\delta > 0 \\} . \\end{align*}"}
-{"id": "5648.png", "formula": "\\begin{align*} x ' = F ( x ) , x ( 0 ) = x _ 0 \\end{align*}"}
-{"id": "9932.png", "formula": "\\begin{align*} C ' & : E F + \\kappa ^ 2 K '^ 2 = K \\phantom { ' } = 0 , \\\\ C \\phantom { ' } & : E F + \\kappa ^ 2 K ^ 2 \\phantom { ' } = K ' = 0 , \\\\ L \\phantom { ' } & : \\qquad \\phantom { E F + \\kappa ^ 2 } K \\phantom { '^ 2 } = K ' = 0 . \\end{align*}"}
-{"id": "6046.png", "formula": "\\begin{align*} H ( u _ { j } ^ { \\varepsilon } ) ( x ) = \\int _ { B _ { 1 } ( x ) } u _ { j } ^ { \\varepsilon } ( y ) d y , \\end{align*}"}
-{"id": "9807.png", "formula": "\\begin{align*} G L R _ { n } = \\ell ( \\hat { \\mathbf { p } } _ { \\hat { \\boldsymbol { \\zeta } } } , \\hat { \\boldsymbol { \\zeta } } ) - \\ell ( \\mathbf { p } _ { \\hat { \\theta } _ { 0 } } , \\hat { \\boldsymbol { \\zeta } } _ { 0 } ) . \\end{align*}"}
-{"id": "3799.png", "formula": "\\begin{align*} d _ r ( C ( n , 2 n ) ) & = | G ( n , 2 n ) ( { \\mathbb F } _ q ) | - g _ r ( n , 2 n ) \\\\ & \\leq | G ( n , 2 n ) ( { \\mathbb F } _ q ) | - g ( V ) = | G ( n , 2 n ) ( { \\mathbb F } _ q ) | - | L ( n , 2 n ) ( { \\mathbb F } _ q ) | . \\end{align*}"}
-{"id": "89.png", "formula": "\\begin{align*} 5 \\epsilon n s ^ { n - 1 } = \\pm 1 2 5 - \\sum _ { j = 2 } ^ { n } \\binom { n } { j } ( 5 \\epsilon ) ^ j s ^ { n - j } . \\end{align*}"}
-{"id": "1353.png", "formula": "\\begin{align*} \\underset { \\omega \\in \\Omega } \\sup ~ Q _ { L } ^ { S T } ( \\omega ) = \\underset { \\omega \\in \\Omega } \\sup S ( \\hat \\delta _ 0 ) I _ { L } ( \\hat \\delta _ 0 ) ^ { - 1 } S ( \\hat \\delta _ 0 ) . \\end{align*}"}
-{"id": "2559.png", "formula": "\\begin{align*} e ^ { i t \\Delta } m ( x ) e ^ { - i t \\Delta } = M ( t ) m ( 2 i t \\nabla ) M ( - t ) \\end{align*}"}
-{"id": "5394.png", "formula": "\\begin{align*} e ( B ) \\leq \\ell \\frac { n ^ 2 } { 2 } + ( k - \\alpha - \\ell - x ) \\frac { n ^ 2 } { 2 } = ( k - \\alpha - x ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "9361.png", "formula": "\\begin{align*} & \\hat { \\pi } ( t , z ) = - \\frac { \\alpha _ 0 ( t ) \\mathbb { E } [ U ' ( X ( t ) ) | \\mathcal { R } _ t ] } { \\beta _ 0 ^ 2 ( t ) \\mathbb { E } [ U '' ( X ( t ) ) | \\mathcal { R } _ t ] } , \\end{align*}"}
-{"id": "838.png", "formula": "\\begin{align*} q \\left \\{ q ^ { m } \\ , u ( a ^ { m } , b ) + \\sum _ { \\ell = 2 } ^ { m + 1 } g ( w , z _ { \\ell } ) Z _ { \\ell } ^ { [ 2 , m + 1 ] } ( \\vec { z } ) u ( a , b , a ^ { m } ) \\right \\} . \\end{align*}"}
-{"id": "9872.png", "formula": "\\begin{align*} \\begin{cases} - \\operatorname { d i v } \\left ( | \\nabla u | ^ { p - 2 } \\nabla u \\right ) = \\mu _ p | u | ^ { p - 2 } u & \\ , \\ , \\ , \\ , \\ , \\Omega \\\\ \\frac { \\partial u } { \\partial n } = 0 & \\ , \\ , \\ , \\partial \\Omega . \\end{cases} \\end{align*}"}
-{"id": "743.png", "formula": "\\begin{align*} W ( t ) : = \\sum _ { w \\in W } t ^ { | N ( w ) | } = \\sum _ { w \\in W } t ^ { \\ell ( w ) } . \\end{align*}"}
-{"id": "3282.png", "formula": "\\begin{align*} \\hat { R } ( v _ { 0 } \\otimes v _ { 0 } ) = v _ { 0 } \\otimes v _ { 0 } + q ^ { - 2 } ( q ^ { 2 } - q ^ { - 2 } ) v _ { 1 } \\otimes v _ { - 1 } . \\end{align*}"}
-{"id": "9279.png", "formula": "\\begin{align*} M ( t , z ) = \\exp ( \\int _ 0 ^ t \\Phi _ K ( s , z ) d B ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ t \\Phi _ K ^ 2 ( s , z ) d s ) . \\end{align*}"}
-{"id": "2391.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} \\tfrac { 1 } { 2 } \\langle H x , x \\rangle + \\langle h , x \\rangle & { \\hbox { i f } } A x = b \\\\ \\infty & { \\hbox { e l s e } } \\end{cases} \\end{align*}"}
-{"id": "3405.png", "formula": "\\begin{align*} \\pi ' _ 0 ( f _ 0 ( w ) e _ 0 + f _ 1 ( w ) e _ 1 ) & = f _ 0 ( w ) + f _ 1 ( w ) ( w + w ^ 2 ) \\\\ \\pi ' _ 1 ( f _ 1 ( w ) e _ 1 + f _ 2 ( w ) z e _ 0 ) & = f _ 1 ( w ) ( w + w ^ 2 ) + f _ 2 ( w ) ( z + 2 z w ) . \\end{align*}"}
-{"id": "5991.png", "formula": "\\begin{align*} \\underbrace { 0 = d _ { 0 , 1 } = \\cdots d _ { 0 , t _ 0 } } _ { = \\tau _ 0 } < \\underbrace { d _ { 1 , 1 } = \\cdots = d _ { 1 , t _ 1 } } _ { = \\tau _ 1 } < \\cdots < \\underbrace { d _ { s - 1 , 1 } = \\cdots = d _ { s - 1 , t _ { s - 1 } } } _ { = \\tau _ { s - 1 } } < 1 = \\tau _ { s } = \\sigma _ s . \\end{align*}"}
-{"id": "1709.png", "formula": "\\begin{gather*} x \\times y : = x y + H ( x , y ) 1 ; \\end{gather*}"}
-{"id": "7374.png", "formula": "\\begin{align*} H ( t ) \\leq M ( t ) \\leq H ( C t ) \\textrm { f o r } t \\geq t _ 0 \\\\ H ( t ) = 0 \\textrm { f o r } t \\leq t _ 0 \\end{align*}"}
-{"id": "7658.png", "formula": "\\begin{align*} S _ { \\alpha } ( t ) u _ { 0 } = \\int _ { B _ { R } } p _ { D } ( t , \\cdot , y ) u _ { 0 } ( y ) d y < \\infty \\ \\ \\mbox { f o r } \\ \\ t > 0 . \\end{align*}"}
-{"id": "713.png", "formula": "\\begin{align*} & A = ( a _ { m i } ) _ { m i } \\ \\ \\ s \\times r \\\\ & B = ( b _ { l j } ) _ { l j } \\ \\ r \\times s . \\end{align*}"}
-{"id": "2912.png", "formula": "\\begin{align*} M _ { ( 0 ) } ( z ) = M ^ \\perp _ { ( 0 ) } ( z ) = \\frac { 1 } { 1 - z } \\quad \\mbox { a n d } L _ { ( 0 ) } ( z ) = L ^ \\perp _ { ( 0 ) } ( z ) = ( 1 - z ) \\ , . \\end{align*}"}
-{"id": "1066.png", "formula": "\\begin{align*} x n = \\sum _ { i = 1 } ^ c \\max \\{ v ( B _ i ) - n , 0 \\} \\ , . \\end{align*}"}
-{"id": "2145.png", "formula": "\\begin{align*} H _ { 0 } ^ { s } ( \\Omega ) = H _ { e } ^ { s } ( \\Omega ) s \\notin \\left \\{ \\frac { 1 } { 2 } , \\frac { 3 } { 2 } , \\frac { 5 } { 2 } , \\cdots \\right \\} . \\end{align*}"}
-{"id": "46.png", "formula": "\\begin{align*} z _ { n + 1 } ^ { b _ { n + 1 } } = \\frac { t } { \\prod _ { j = k + 1 } ^ n z _ j ^ { b _ j } } . \\end{align*}"}
-{"id": "5473.png", "formula": "\\begin{align*} I ( m ) = ( x ( y ^ m - z ^ m ) , y ( z ^ m - x ^ m ) , z ( x ^ m - y ^ m ) ) \\subset R = k [ x , y , z ] . \\end{align*}"}
-{"id": "4305.png", "formula": "\\begin{align*} p ^ \\star ( \\Sigma ) = \\sum _ { j \\in J _ v } b ^ j Z _ j + \\sum _ { j \\in J _ h } b ^ j Z _ j \\end{align*}"}
-{"id": "3419.png", "formula": "\\begin{align*} \\frac { \\partial \\Phi } { \\partial s } + \\frac 1 2 T r Q ^ * ( x , h ) \\nabla ^ 2 \\Phi Q ( x , h ) + \\langle q ( x , h ) , \\nabla \\Phi \\rangle + G ( s , h , x , \\Phi , Q ^ * \\nabla \\Phi ) = 0 , \\end{align*}"}
-{"id": "4004.png", "formula": "\\begin{align*} q ^ t + 1 = | S T | \\geq \\frac { \\left | \\sum \\limits _ { X \\in S T } | X ^ * | \\right | } { | V ( t , q ) ^ * | } = \\frac { | V ( n , q ) ^ * | - | Y ^ * | } { | V ( t , q ) ^ * | } = \\frac { ( q ^ n - 1 ) - ( q ^ { d } - 1 ) } { q ^ t - 1 } \\geq q ^ { d } , \\end{align*}"}
-{"id": "9431.png", "formula": "\\begin{align*} E ^ p _ r [ g \\circ f ] = E ^ p _ r ( g \\circ f ) = E ^ p _ q ( f ) E ^ q _ r ( g ) . \\end{align*}"}
-{"id": "595.png", "formula": "\\begin{align*} \\Delta _ { G _ i } : = \\deg _ y P _ { G _ i } ( z , y ) = \\deg _ y P _ G ( z , y ) = \\Delta _ G \\end{align*}"}
-{"id": "1274.png", "formula": "\\begin{align*} \\begin{bmatrix} - 1 & \\alpha & 0 & \\overline \\alpha \\\\ - \\overline \\alpha & 0 & - \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & 0 \\\\ 0 & - \\overline \\alpha J ( T ^ { - r } , T ^ { - r } ) & 0 & - \\alpha J ( T ^ { - 3 r } , T ^ { - 3 r } ) \\\\ - \\alpha & 0 & - \\overline \\alpha J ( T ^ { - 2 r } , T ^ { - r } ) & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "8831.png", "formula": "\\begin{align*} g \\cdot [ y , t _ y , p ( x _ { i , n } ) ] = [ g y , t _ y \\circ g ^ { - 1 } , p ( x _ { i , n } \\circ g ^ { - 1 } ) ] \\end{align*}"}
-{"id": "434.png", "formula": "\\begin{align*} \\left [ X _ j , Y _ { j _ 1 , \\dots , j _ k } ^ { ( n - 1 , i ) } \\right ] , \\tilde Y _ { j _ 1 , \\dots , j _ k } ^ { ( n - 1 , i ) } & \\in C _ { n + 1 } \\quad \\mbox { a s w e l l a s } \\\\ \\left [ X _ 0 , Y _ { j _ 1 , \\dots , j _ k } ^ { ( n - 1 , i ) } \\right ] + \\frac { 1 } { 2 } \\sum _ { j = 1 } ^ m \\left [ X _ j , \\left [ X _ j , Y _ { j _ 1 , \\dots , j _ k } ^ { ( n - 1 , i ) } \\right ] \\right ] & \\in C _ { n + 2 } \\end{align*}"}
-{"id": "6700.png", "formula": "\\begin{align*} L _ f = u ^ { - 1 } \\left ( L ^ \\star _ { f _ K ^ I } \\theta ^ K \\delta _ I \\right ) u \\mbox { f o r } f = u ^ { - 1 } ( f _ K ^ I \\star u _ { I L } ) \\theta ^ K \\bar \\theta ^ L . \\end{align*}"}
-{"id": "9377.png", "formula": "\\begin{align*} \\det ( \\mathbf { T } - W _ \\lambda ) = 0 , \\end{align*}"}
-{"id": "6156.png", "formula": "\\begin{align*} M = \\left ( m \\Z ^ n + \\left ( \\frac { m } { g } \\right ) \\Z ( \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } { \\bf e } _ i ) \\right ) \\cap H _ v . \\end{align*}"}
-{"id": "9136.png", "formula": "\\begin{align*} u ' ( t ) + A ( t ) B ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) , \\ \\ u ( 0 ) = u _ 0 \\end{align*}"}
-{"id": "3603.png", "formula": "\\begin{align*} \\Phi ^ W _ { ( g , \\pi ) } ( \\gamma + h , \\tau + w ) = \\Phi ^ W _ { ( g , \\pi ) } ( \\gamma , \\tau ) + D \\Phi ^ W _ { ( g , \\pi ) } | _ { ( \\gamma , \\tau ) } ( h , w ) + Q ^ W _ { ( g , \\pi ) , ( \\gamma , \\tau ) } ( h , w ) . \\end{align*}"}
-{"id": "8124.png", "formula": "\\begin{align*} { \\mathcal R } _ { \\frac { 1 } { 3 } , 0 , 2 } ( M ) \\geq C _ { \\frac { 1 } { 3 } , 0 , 2 } ^ { r a d } , \\end{align*}"}
-{"id": "9535.png", "formula": "\\begin{align*} \\omega ^ M _ { 1 0 1 0 c } & = - \\frac { 2 } { 3 } e ^ { \\phi / 3 } d \\phi _ c \\\\ \\omega ^ M _ { 1 0 b c } & = - \\frac { 1 } { 2 } e ^ { 4 \\phi / 3 } { G _ 2 } _ { b c } \\\\ \\omega ^ M _ { a 1 0 c } & = - \\frac { 1 } { 2 } e ^ { 4 \\phi / 3 } { G _ 2 } _ { a c } \\\\ \\omega ^ M _ { a b c } & = e ^ { \\phi / 3 } \\omega ^ N _ { a b c } - \\frac { 1 } { 3 } e ^ { \\phi / 3 } ( d \\phi _ b \\eta _ { a c } - d \\phi _ c \\eta _ { a b } ) . \\end{align*}"}
-{"id": "7463.png", "formula": "\\begin{align*} \\deg _ { X _ i } \\mathrm { D T } ^ * ( X _ j ) = - \\delta _ { i j } \\end{align*}"}
-{"id": "6620.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\mu _ t + \\frac { \\partial } { \\partial x } ( v [ \\mu _ t ] \\ , \\mu _ t ) = F _ f ( \\mu _ t ) . \\end{align*}"}
-{"id": "9121.png", "formula": "\\begin{align*} \\sigma ( t , \\xi ) : = \\left \\{ \\begin{array} { l c l } A ( 0 ) ( i \\xi + B ( 0 ) A ( 0 ) ) ^ { - 1 } & & t < 0 \\\\ A ( t ) ( i \\xi + B ( t ) A ( t ) ) ^ { - 1 } & & 0 \\le t \\le \\tau \\\\ A ( \\tau ) ( i \\xi + B ( \\tau ) A ( \\tau ) ) ^ { - 1 } & & t > \\tau . \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "6359.png", "formula": "\\begin{align*} h ^ { \\mu m } ( x , r ) : = r ^ m h ^ { \\mu } _ { m } ( x ) + m r ^ { m - 1 } h ^ { \\mu } _ { m - 1 } ( x ) , \\end{align*}"}
-{"id": "3311.png", "formula": "\\begin{align*} M ^ { ( 0 ) } = ( 1 ) , M ^ { ( 1 ) } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & q ^ 2 \\end{array} \\right ) , M ^ { ( 2 ) } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\\\ 0 & q ^ 4 & 0 \\\\ 0 & 0 & q ^ 2 \\end{array} \\right ) , M ^ { ( 3 ) } = ( 1 ) . \\end{align*}"}
-{"id": "9673.png", "formula": "\\begin{align*} \\Lambda = a - b \\left ( \\frac { 1 } { 1 - q } \\right ) ^ { - 1 / ( \\beta - 1 ) } . \\end{align*}"}
-{"id": "8664.png", "formula": "\\begin{gather*} \\| f \\| _ { C _ K ^ 1 ( H , J ) } = \\| \\nabla ^ K f \\| _ { \\infty } + \\| f \\| _ { \\infty } , \\ ; \\ ; f \\in C _ K ^ 1 ( H , J ) , \\end{gather*}"}
-{"id": "2180.png", "formula": "\\begin{align*} = \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\tilde { u } ( s , x ) - \\tilde { u } ( s , y ) ) ( \\psi ^ { 2 } ( y ) \\tilde { u } ^ { - q } ( s , y ) - \\psi ^ { 2 } ( x ) \\tilde { u } ^ { - q } ( s , x ) ) k ( x , y ) d x d y , \\end{align*}"}
-{"id": "9215.png", "formula": "\\begin{align*} & H ( t , x , y , \\varphi , u , z , p , q , r ) = H ( t , x , y , \\varphi , u , z , p , q , r , \\omega ) \\\\ & = \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] h ( t , x , y , u , z ) + [ A _ u ( \\varphi ) + a ( t , x , y , u , z ) ] p \\\\ & + b ( t , x , y , u , z ) q + \\int _ { \\mathbb { R } } c ( t , x , y , u , z , \\zeta ) r ( \\zeta ) \\nu ( d \\zeta ) . \\end{align*}"}
-{"id": "4260.png", "formula": "\\begin{align*} a _ k = A _ { i _ 1 i _ 1 } \\odot \\dots \\odot A _ { i _ k i _ k } \\enspace . \\end{align*}"}
-{"id": "2551.png", "formula": "\\begin{align*} Z ( U ^ 1 - U ^ 0 ) \\delta _ { 1 } \\beta = & \\begin{pmatrix} u _ 1 ^ 1 - u _ 1 ^ 0 & & \\\\ & \\ddots & \\\\ & & u _ M ^ 1 - u _ M ^ 0 \\end{pmatrix} E _ { M K } \\Lambda \\delta _ { 1 } \\beta \\\\ = & \\begin{pmatrix} \\sum _ { k = 1 } ^ K e _ k ( x _ 1 ) \\sqrt { \\eta _ k } \\delta _ 1 \\beta _ k & & \\\\ & \\ddots & \\\\ & & \\sum _ { k = 1 } ^ K e _ k ( x _ M ) \\sqrt { \\eta _ k } \\delta _ 1 \\beta _ k \\end{pmatrix} ( U ^ 1 - U ^ 0 ) \\\\ = : & G ( U ^ 1 - U ^ 0 ) , \\end{align*}"}
-{"id": "3204.png", "formula": "\\begin{align*} F _ { j , \\beta } ( z _ j ) - \\sum _ { | \\alpha | = | \\beta | } T _ { j k } F _ { k , \\alpha } ( z _ k ) \\cdot \\tau _ { k j , \\beta } ^ \\alpha = h _ { 1 , j k , \\beta } ( z _ j ) - h _ { 2 , j k , \\beta } ( z _ j ) \\end{align*}"}
-{"id": "3206.png", "formula": "\\begin{align*} H _ { j k , n } : = \\sum _ { \\lambda = 1 } ^ r \\sum _ { | \\alpha | = n } \\left ( h _ { 1 , j k , \\alpha } - h _ { 2 , j k , \\alpha } \\right ) \\cdot e _ { j , \\lambda } ^ * \\otimes e _ j ^ \\alpha , \\end{align*}"}
-{"id": "5890.png", "formula": "\\begin{align*} X = \\xi ^ { t } \\left ( t \\right ) \\partial _ { t } + \\xi ^ { \\alpha } \\left ( t , x ^ { \\beta } \\right ) \\partial _ { \\alpha } + \\left ( a ( x ^ { \\alpha } , t ) u + b ( x ^ { a } , t ) \\right ) \\partial _ { u } . \\end{align*}"}
-{"id": "4043.png", "formula": "\\begin{align*} p + q + r = m + n + l \\iff \\left \\{ \\begin{aligned} m = \\gcd ( q , p + r ) & = q , \\\\ n = \\gcd ( p , q + r ) & = p , \\\\ l = \\gcd ( r , p + q ) & = r . \\end{aligned} \\right . \\end{align*}"}
-{"id": "1195.png", "formula": "\\begin{align*} w = 0 1 2 2 , ~ \\overline { w } = 2 2 1 0 \\end{align*}"}
-{"id": "5431.png", "formula": "\\begin{align*} C = \\begin{bmatrix} c ( i , r ) & c ( i + 2 r , 3 r ) & \\cdot & \\cdot \\\\ c ( i , 3 r ) & c ( i + 2 r , r ) & \\cdot & \\cdot \\\\ \\cdot & \\cdot & c ( i + 3 r , 3 r ) & c ( i + 3 r , r ) \\\\ \\cdot & \\cdot & c ( i + r , r ) & c ( i + r , 3 r ) \\end{bmatrix} \\end{align*}"}
-{"id": "4581.png", "formula": "\\begin{align*} | ( q \\circ g ) _ J ( x ) | & \\leq q \\bigl ( g ( x ) \\bigr ) \\sum _ { i = 1 } ^ { s } 2 ^ { 2 i - 1 } i ^ i t ^ i S ( s , i ) a ( f ( x ) ) ^ { i ( m + 1 ) } D _ g ^ i \\\\ & \\leq q \\bigl ( g ( x ) \\bigr ) s ^ { s + 1 } 2 ^ { 2 s - 1 } \\max ( 1 , t ) ^ s B _ s a ( f ( x ) ) ^ { s ( m + 1 ) } D _ g ^ s , \\end{align*}"}
-{"id": "1072.png", "formula": "\\begin{align*} e ( H ' ) & \\leq \\sum _ { i = 1 } ^ { \\frac { 1 } { 2 } v ( M ) } \\left ( d ( v _ i ) + d ^ { \\ast } ( u _ i ) \\right ) + \\binom { \\frac { 1 } { 2 } v ( M ) } { 2 } \\\\ & \\leq \\frac { v ( M ) } { 2 } \\left ( n + \\frac { v ( M ) } { 4 } \\right ) \\\\ & \\leq \\frac { 5 } { 8 } n ^ 2 - \\frac { 3 } { 2 } \\abs { A } n + \\frac { \\abs { A } ^ 2 } { 2 } \\ , , \\end{align*}"}
-{"id": "7657.png", "formula": "\\begin{align*} S _ { \\alpha } ( s ) u _ { k } & = S _ { \\alpha } ( s ) \\big ( \\frac { 1 } { \\nu ( d , \\alpha ) } \\phi _ { k } \\chi _ { r _ { k } } \\big ) \\\\ & = \\frac { \\phi _ { k } } { \\nu ( d , \\alpha ) } S _ { \\alpha } ( s ) \\chi _ { r _ { k } } \\\\ & \\geq \\phi _ { k } \\chi _ { r _ { k } + t ^ { 1 / \\alpha } } \\\\ & \\geq \\phi _ { k } \\chi _ { r _ { k } } , \\end{align*}"}
-{"id": "2909.png", "formula": "\\begin{align*} X ( z ) = \\sum _ { n \\in { \\mathbb Z } } z ^ { n + { \\alpha _ 0 } } X _ n \\quad \\mbox { a n d } X ^ * ( z ) = \\sum _ { n \\in { \\mathbb Z } } z ^ { - n - { \\alpha _ 0 } } X ^ * _ { n } \\ , , \\end{align*}"}
-{"id": "1305.png", "formula": "\\begin{align*} \\sum _ { n = N - H } ^ { N + H } t _ H ( n - N ) n = H ^ 2 N \\end{align*}"}
-{"id": "9313.png", "formula": "\\begin{align*} \\begin{cases} d M ( t , z ) = \\mathbb { E } [ D _ t \\delta _ Z ( z ) | \\mathcal { F } _ t ] d B ( t ) = \\Phi ( t , z ) M ( t , z ) d B ( t ) \\\\ M ( 0 , z ) = 1 \\end{cases} \\end{align*}"}
-{"id": "9757.png", "formula": "\\begin{align*} \\nabla _ { \\vec { z } } f ( \\vec { z } ^ { * } , \\bar { \\vec { x } } ) + \\sum \\limits _ { i = 1 } ^ { m } \\lambda _ { i } ^ { * } \\nabla _ { \\vec { z } } g _ { i } ( \\vec { z } ^ { * } , \\bar { \\vec { x } } ) = \\vec { 0 } _ { s } , \\end{align*}"}
-{"id": "297.png", "formula": "\\begin{align*} r _ { n , u } ^ { ( j ) } : = \\biggl \\{ \\frac { u e ^ { \\Psi ( j ) } } { V _ d ( n - 1 ) } \\biggr \\} ^ { 1 / d } , p _ { n , x , u } ^ { ( j ) } : = h _ x ( r _ { n , u } ^ { ( j ) } ) . \\end{align*}"}
-{"id": "3036.png", "formula": "\\begin{align*} \\forall n \\geq 0 , W _ n = \\sum _ { | u | = n } e ^ { - V ( u ) } Z _ n = \\sum _ { | u | = n } V ( u ) e ^ { - V ( u ) } . \\end{align*}"}
-{"id": "3930.png", "formula": "\\begin{align*} \\mathcal { W } _ \\sigma ( z ) \\equiv - \\ , W \\left [ w _ \\sigma , \\ , w _ { \\rm r e g } \\right ] ( 0 ) = \\delta _ { \\sigma , 0 } \\ , \\left ( \\sum _ { j = 0 } ^ \\infty j \\ , b _ j ( z ) \\right ) + \\delta _ { \\sigma , 1 } \\ , \\left ( \\sum _ { j = 0 } ^ \\infty b _ j ( z ) \\right ) \\ , , \\end{align*}"}
-{"id": "3660.png", "formula": "\\begin{align*} \\frac { \\abs { E _ 0 } - n _ 0 s } { \\Theta ( n p ) } = \\Theta ( n ^ { 1 - \\gamma / 2 } ) \\end{align*}"}
-{"id": "8134.png", "formula": "\\begin{align*} S _ { a , p , q , N } = S _ { a , p , q , N } ^ { r a d } \\end{align*}"}
-{"id": "1141.png", "formula": "\\begin{align*} f _ { i j } ^ { * } = \\min _ { x \\in S } f _ { i j } \\left ( x \\right ) , i = 1 , 2 , . . . m ; j = 1 , 2 , . . . , p _ { m } ^ { } \\end{align*}"}
-{"id": "7732.png", "formula": "\\begin{align*} \\mathbb P [ T _ N \\le 0 ] = \\frac 1 2 . \\end{align*}"}
-{"id": "5614.png", "formula": "\\begin{align*} z = 1 / N - 2 \\pi i \\alpha \\end{align*}"}
-{"id": "6371.png", "formula": "\\begin{align*} f ( z ) = \\frac { U _ a } { r } R ( x _ n , r ) + F ( z ) \\end{align*}"}
-{"id": "5446.png", "formula": "\\begin{align*} K ^ * _ 0 = \\begin{bmatrix} q & 1 & - \\alpha & 0 & - \\overline \\alpha \\\\ 0 & 0 & q \\alpha & 0 & q \\overline \\alpha \\\\ 0 & \\overline \\alpha & 0 & \\alpha J ( 2 r , 3 r ) & 0 \\\\ 0 & 0 & \\overline \\alpha J ( r , r ) & 0 & \\alpha J ( 3 r , 3 r ) \\\\ 0 & \\alpha & 0 & \\overline \\alpha J ( 2 r , r ) & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "1130.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to \\infty } \\gamma _ k ^ \\mathrm { B } [ \\iota ] = & \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } + \\sum _ { i = 1 } ^ { K } \\beta _ { i } } \\\\ \\lim _ { \\rho \\to \\infty } \\gamma _ k ^ \\mathrm { C } [ \\iota ] = & \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } } \\end{align*}"}
-{"id": "6974.png", "formula": "\\begin{align*} \\mu _ t ( \\lambda _ t ( x , y ) , a ) = \\mu _ t ( x , \\mu _ t ( y , a ) ) - \\mu _ t ( y , \\mu _ t ( x , a ) ) . \\end{align*}"}
-{"id": "1950.png", "formula": "\\begin{align*} \\hat { u } = \\hat { u } ( \\tau ) = \\int _ 0 ^ \\tau \\frac { 1 } { \\hat { a } ( \\zeta ) } \\ , d \\zeta . \\end{align*}"}
-{"id": "2050.png", "formula": "\\begin{align*} 0 & = \\frac { \\nabla _ E w ( x _ 0 ) - \\nabla _ E w ( y _ 0 ) } { \\bar { w } } - \\frac { m } { \\bar { w } } ( \\nabla _ E \\bar { w } ) , \\\\ 0 & \\ge \\frac { \\nabla ^ 2 _ { E , E } w ( x _ 0 ) - \\nabla ^ 2 _ { E , E } w ( y _ 0 ) } { \\bar { w } } - \\frac { m } { \\bar { w } } \\nabla ^ 2 _ { E , E } \\bar { w } . \\end{align*}"}
-{"id": "7850.png", "formula": "\\begin{align*} \\gamma ( x ) = x ^ { 2 \\alpha } , \\ , \\eta _ 0 ( x ) = \\lambda x ^ { \\alpha - 1 } \\end{align*}"}
-{"id": "6597.png", "formula": "\\begin{align*} f ( x ) = \\begin{cases} \\tfrac { 1 } { 2 } \\langle H x , x \\rangle + \\langle h , x \\rangle & { \\hbox { i f } } A x = b \\\\ \\infty & { \\hbox { e l s e } } \\end{cases} \\end{align*}"}
-{"id": "6178.png", "formula": "\\begin{align*} \\mathcal { A } f ( x ) = ( \\gamma _ U x + \\gamma _ L ) f ' ( x ) + \\frac { 1 } { 2 } ( x ^ 2 \\sigma _ U ^ 2 + 2 x \\sigma _ { U L } + \\sigma _ L ^ 2 ) f '' ( x ) . \\end{align*}"}
-{"id": "9084.png", "formula": "\\begin{align*} M ^ { ( 2 ) } _ N ( s ) = N ( \\alpha + N ) _ 2 F _ 1 ( 1 - \\alpha - N , 1 - N ; 2 ; s ^ { 2 } ) ( 1 - s ) ^ { - ( \\alpha + 2 N ) } \\ . \\end{align*}"}
-{"id": "6945.png", "formula": "\\begin{align*} M _ { \\pi ' } ( Z ) = M _ { \\pi ' } ( 1 ; Z ) = \\prod _ { T \\in { \\cal T } ^ { \\pi ' } } ( 1 - Z ^ T ) ^ { - 1 } = \\prod _ { i _ 1 , i _ 2 , \\ldots , i _ m \\geq 0 } \\ c ^ { \\pi ' } _ { ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) } \\ ( 1 - z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\cdots z _ m ^ { i _ m } ) \\ , . \\end{align*}"}
-{"id": "4415.png", "formula": "\\begin{align*} \\| f _ n - \\widehat { i d } ( f ) \\| _ { L _ 1 + L _ \\infty } = \\| \\widehat { i d } ( f _ n ) - \\widehat { i d } ( f ) \\| _ { L _ 1 + L _ \\infty } \\leq C \\| f _ n - f \\| _ { \\widehat { E } } \\to 0 , \\end{align*}"}
-{"id": "5958.png", "formula": "\\begin{align*} B _ t & = \\int _ 0 ^ t 1 _ { \\{ Z _ s \\not = Y _ s \\} } \\ ; \\frac { \\Big \\| [ D U ( Z _ s ) - D U ( Y _ s ) ] { R \\ , } \\Big \\| ^ 2 _ { H S } } { | \\gamma ( Z _ s ) - \\gamma ( Y _ s ) | ^ 2 } \\ \\dd s \\le C ^ 2 A _ t \\ , , \\end{align*}"}
-{"id": "549.png", "formula": "\\begin{align*} f _ { \\kappa } ( z ) = f _ { \\kappa } ( \\theta ) + \\frac { \\sigma ^ 3 } { 3 } ( z - \\theta ) ^ 3 + \\mathcal { O } \\big ( ( z - \\theta ) ^ 4 \\big ) . \\end{align*}"}
-{"id": "7184.png", "formula": "\\begin{align*} f + g ' + g \\cot \\varphi = 0 . \\end{align*}"}
-{"id": "5865.png", "formula": "\\begin{align*} F \\left ( 0 , x \\right ) = \\frac { 1 } { x } . \\end{align*}"}
-{"id": "4407.png", "formula": "\\begin{align*} \\| x \\| ^ * = \\inf \\Bigl \\{ \\sum _ { i = 1 } ^ n \\| x _ i \\| _ { X } : \\sum _ { i = 1 } ^ n x _ i = x , \\ x _ i \\in X , \\ n \\in \\mathbb { N } \\Bigr \\} . \\end{align*}"}
-{"id": "6889.png", "formula": "\\begin{align*} \\lim \\limits _ { j \\to \\infty } \\lambda _ { k ' ( j ) } ( \\gamma ' ) = \\lim \\limits _ { j \\to \\infty } \\lambda _ { k ( j ) } ( \\gamma ' ) = \\mu ( \\gamma ' ) \\end{align*}"}
-{"id": "2193.png", "formula": "\\begin{align*} G _ { m } ( s ) = \\partial _ { s } ^ { \\alpha } ( h _ { m } * W ) ( s ) + F _ { m } ( s ) \\geq 0 , s \\in ( 0 , t _ { 2 } - t _ { 0 } ) . \\end{align*}"}
-{"id": "2962.png", "formula": "\\begin{align*} \\mu _ t ( \\lambda _ t ( x , y ) , a ) = \\mu _ t ( x , \\mu _ t ( y , a ) ) - \\mu _ t ( y , \\mu _ t ( x , a ) ) . \\end{align*}"}
-{"id": "4998.png", "formula": "\\begin{align*} & g ^ { * } D _ { i } \\equiv \\sum _ { m = 1 } ^ { s } a _ { m i } F _ { m } \\ \\ ( i = 1 , \\dots r ) \\\\ & { p } _ { * } F _ { j } \\equiv \\sum _ { l = 1 } ^ { r } b _ { l j } D _ { l } \\ \\ ( j = 1 , \\dots , s ) \\end{align*}"}
-{"id": "893.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { { 1 } / { 2 } + i T } ^ { { 1 } / { 2 } + 2 i T } \\chi ( s ) ^ { - 1 / 2 } \\zeta ( s ) d s & = \\frac { 1 } { 2 \\pi i } \\left ( \\int _ { { 1 } / { 2 } + i T } ^ { 1 + \\delta + i T } + \\int _ { 1 + \\delta + i T } ^ { 1 + \\delta + 2 i T } + \\int _ { 1 + \\delta + 2 i T } ^ { { 1 } / { 2 } + 2 i T } \\right ) \\chi ( s ) ^ { - 1 / 2 } \\zeta ( s ) d s \\\\ & = I _ { 1 } + I _ { 2 } + I _ { 3 } . \\end{align*}"}
-{"id": "3079.png", "formula": "\\begin{align*} \\xi ( u ) = \\log \\sum _ { | v | = | u | + 1 , v > u } ( 1 + ( V ( v ) - V ( u ) ) _ + ) e ^ { V ( u ) - V ( v ) } . \\end{align*}"}
-{"id": "8587.png", "formula": "\\begin{align*} p ' & = \\psi _ { 1 } p ( \\hat { \\psi } _ { o } ) ^ { - 1 } \\\\ \\sigma ' & = \\psi _ { 2 } \\sigma ( \\overline { \\psi } _ { 2 } ) ^ { - 1 } \\end{align*}"}
-{"id": "8359.png", "formula": "\\begin{align*} g x = \\alpha _ g ( x ) g , \\ \\ \\ x \\in M , \\end{align*}"}
-{"id": "554.png", "formula": "\\begin{align*} \\begin{aligned} n m a c ( T ) & = m a c ( T \\otimes I _ n ) = m a c ( P ( T \\otimes I _ n ) P ) \\\\ & + m a c ( ( I \\otimes I _ n - P ) ( T \\otimes I _ n ) ( I \\otimes I _ n - P ) ) \\\\ & \\ge m a c ( P ( T \\otimes I _ n ) P ) . \\end{aligned} \\end{align*}"}
-{"id": "804.png", "formula": "\\begin{align*} & q ^ { N _ { a } } | m _ { 1 } , \\ldots , m _ { r } \\rangle = q ^ { m _ { a } } | m _ { 1 } , \\ldots , m _ { r } \\rangle , \\\\ & \\beta _ { a } ^ { * } | m _ { 1 } , \\ldots , m _ { r } \\rangle = | m _ { 1 } , \\ldots , m _ { a } + 1 , \\ldots , m _ { r } \\rangle , \\\\ & \\beta _ { a } | m _ { 1 } , \\ldots , m _ { r } \\rangle = ( 1 - q ^ { 2 m _ { a } } ) | m _ { 1 } , \\ldots , m _ { a } - 1 , \\ldots , m _ { r } \\rangle . \\end{align*}"}
-{"id": "8386.png", "formula": "\\begin{align*} S _ 0 ( x ) = x ^ * , \\ \\ \\ x \\in A \\rtimes _ \\beta G , \\end{align*}"}
-{"id": "4277.png", "formula": "\\begin{align*} \\mathbb E [ \\mathcal V _ n ] = n + 2 \\sum _ { 1 \\leq i < j \\leq n } \\mathbb P ( Z _ { j - i } = 0 ) \\sim C ' \\frac { n ^ 2 } { a _ n } , \\end{align*}"}
-{"id": "4963.png", "formula": "\\begin{align*} \\partial V _ 2 = \\varnothing \\end{align*}"}
-{"id": "2356.png", "formula": "\\begin{align*} P _ t \\mu : = \\Phi _ t \\# \\mu = \\mu \\circ \\Phi _ t ^ { - 1 } ; \\end{align*}"}
-{"id": "7545.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\mathcal { N } < \\infty x _ n \\in \\left [ \\max \\left \\{ K - \\varepsilon _ 1 , 0 \\right \\} , K + \\varepsilon _ 1 \\right ] , \\ , n \\geq \\mathcal { N } \\right ] = 1 . \\end{align*}"}
-{"id": "6291.png", "formula": "\\begin{align*} \\begin{cases} \\frac { 1 } { p _ { 2 } } \\leq \\frac { 1 } { p _ { 1 } } \\\\ s _ { 2 } + R ( \\mathbf { p } , \\mathbf { q } , \\alpha _ { 1 } , \\alpha _ { 2 } ) \\leq s _ { 1 } \\\\ \\frac { 1 } { q _ { 2 } } \\leq \\frac { 1 } { q _ { 1 } } \\end{cases} , \\end{align*}"}
-{"id": "1558.png", "formula": "\\begin{align*} \\gamma \\langle \\nabla E ( \\mu ) - & \\nabla E ( \\zeta ) , \\mu - \\zeta \\rangle \\le - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\int _ { \\Omega } \\left [ ( \\phi ^ { \\mu , \\nu _ i } ) ^ { c c } ( x ) - ( \\phi ^ { \\zeta , \\eta _ i } ) ^ { c c } ( x ) \\right ] d ( \\mu - \\zeta ) ( x ) \\\\ & = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\iint _ { \\Omega \\times \\Omega } \\left [ ( \\phi ^ { \\mu , \\nu _ i } ) ^ { c c } ( x ) - ( \\phi ^ { \\zeta , \\eta _ i } ) ^ { c c } ( x ) \\right ] d ( \\pi ^ { \\mu , \\nu _ i } - \\pi ^ { \\zeta , \\eta _ i } ) ( x , y ) , \\end{align*}"}
-{"id": "5994.png", "formula": "\\begin{align*} \\left ( \\frac { 1 } { n ^ H \\ell ( 1 / n ) ^ { 1 / 2 } } \\sum _ { j = 1 } ^ { [ n t ] } \\xi _ j \\right ) _ { t \\geq 0 } \\Rightarrow ( W ^ H _ t ) _ { t \\geq 0 } \\end{align*}"}
-{"id": "4706.png", "formula": "\\begin{align*} x ' & = \\prod _ { i = 1 } ^ { \\beta _ 1 } ( a _ 1 ^ { e _ { 1 i } } a _ 2 ^ { f _ { 1 i } } ) a _ 3 ^ { f _ { 2 1 } } \\prod _ { i = 2 } ^ { \\beta _ 2 } ( a _ 2 ^ { e _ { 2 i } } a _ 3 ^ { f _ { 2 i } } ) \\\\ & \\quad \\ \\dots a _ m ^ { f _ { ( m - 1 ) 1 } } \\prod _ { i = 2 } ^ { \\beta _ { m - 1 } } ( a _ { m - 1 } ^ { e _ { ( m - 1 ) i } } a _ m ^ { f _ { ( m - 1 ) i } } ) , \\end{align*}"}
-{"id": "5052.png", "formula": "\\begin{align*} ( 1 - \\alpha _ 1 \\beta _ 1 - \\alpha _ 2 \\beta _ 2 ) m = \\alpha _ 1 \\gamma _ 1 \\lambda _ 1 + \\alpha _ 2 \\gamma _ 2 \\lambda _ 2 . \\end{align*}"}
-{"id": "6525.png", "formula": "\\begin{align*} \\tilde \\lambda = \\sup _ { t \\in [ 0 , T ] } \\sum _ { i , j = 1 } ^ d \\sup _ { x \\in \\varOmega } \\sup _ { \\substack { | \\xi - u ( x , t ) | \\le r \\\\ * [ 2 p t ] | \\vec \\eta - \\nabla u ( x , t ) | \\le r } } \\bigg | \\frac { \\partial g _ i ( \\xi , \\vec \\eta , x , t ) } { \\partial \\eta _ j } \\bigg | \\ , . \\end{align*}"}
-{"id": "9941.png", "formula": "\\begin{align*} \\liminf _ { t \\to 0 ^ + } \\frac { \\displaystyle \\int _ 0 ^ t g ( \\tau ) d \\tau } { t ^ 2 } = + \\infty . \\end{align*}"}
-{"id": "7081.png", "formula": "\\begin{align*} C ( u , n ) = B ( u ) \\backslash \\left ( \\cup _ { j = 1 } ^ { n - 1 } B ( u . j ) \\right ) = \\{ u \\} \\cup \\bigcup _ { j = n } ^ { \\infty } B ( u . j ) . \\end{align*}"}
-{"id": "8264.png", "formula": "\\begin{align*} \\frac { p _ { \\lambda } ( u ( 0 ) ) \\ldots p _ { \\lambda } ( u ( \\ell - 2 ) ) p _ { \\lambda } ( \\ell \\rho - \\sum _ { i = 0 } ^ { \\ell - 2 } u ( i ) ) } { p _ { \\lambda } ^ { \\ast \\ell } ( \\ell \\rho ) } = \\frac { p _ 0 ( u ( 0 ) ) \\ldots p _ 0 ( u ( \\ell - 2 ) ) p _ 0 ( \\ell \\rho - \\sum _ { i = 0 } ^ { \\ell - 2 } u ( i ) ) } { p _ 0 ^ { \\ast \\ell } ( \\ell \\rho ) } \\end{align*}"}
-{"id": "6811.png", "formula": "\\begin{align*} \\oint _ \\S A ( \\xi ) \\dd \\xi = 0 , \\end{align*}"}
-{"id": "8335.png", "formula": "\\begin{gather*} y ^ q - y = u = \\sum _ { i = 1 } ^ r \\frac { Q _ i ( T ) } { P _ i ( T ) ^ { \\alpha _ i } } + R ( T ) \\end{gather*}"}
-{"id": "601.png", "formula": "\\begin{align*} g ^ * ( z ) : = ( 2 i ) ^ { 1 - \\kappa } \\int _ { - \\overline { z } } ^ { i \\infty } g ^ { \\operatorname { c } } ( w ) ( w + z ) ^ { 1 - \\kappa } d w , \\end{align*}"}
-{"id": "9921.png", "formula": "\\begin{align*} \\Omega ( \\l ) \\ ; : = \\ ; & E F \\ ; + \\ ; \\frac { q ^ { - 1 } K ^ 2 + q K ^ { ' 2 } } { ( q - q ^ { - 1 } ) ^ 2 } \\ ; - \\ ; \\frac { q \\l + q ^ { - 1 } \\l ^ { - 1 } } { ( q - q ^ { - 1 } ) ^ 2 } \\ , K K ' \\\\ \\ ; = \\ ; & E F \\ ; + \\ ; \\kappa ^ 2 \\big ( q ^ { - 1 } K - q \\l K ' ) ( K - \\l ^ { - 1 } K ' ) \\\\ \\ ; = \\ ; & F E \\ ; + \\ ; \\kappa ^ 2 \\big ( q K - q ^ { - 1 } \\l ^ { - 1 } K ' ) ( K - \\l K ' ) . \\end{align*}"}
-{"id": "1557.png", "formula": "\\begin{align*} \\underset { n \\rightarrow + \\infty } { \\liminf } \\ \\mathcal { W } _ 2 ( \\delta _ { \\mu ^ n } , \\P ) & \\geq \\int \\underset { n \\rightarrow + \\infty } { \\liminf } \\ W _ 2 ^ 2 ( \\mu _ n , \\nu ) d \\P ( \\nu ) & \\mbox { b y F a t o u ' s L e m m a } & \\\\ & \\geq \\int W _ 2 ^ 2 ( \\mu , \\nu ) d \\P ( \\nu ) = \\mathcal { W } _ 2 ^ 2 ( \\delta _ { \\mu } , \\P ) \\qquad & & \\end{align*}"}
-{"id": "6986.png", "formula": "\\begin{align*} \\Theta ^ 1 ( x ) = \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ f _ i ^ x , \\alpha _ j ] , \\end{align*}"}
-{"id": "5449.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & d + D & b + D \\\\ 0 & 0 & c + D & a + D \\\\ a & b & 0 & 0 \\\\ c & d & 0 & 0 \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "2558.png", "formula": "\\begin{align*} [ D ( t ) f ] ( x ) : = ( 2 i t ) ^ { - \\frac { d } { 2 } } f \\bigl ( \\tfrac { x } { 2 t } \\bigr ) , \\end{align*}"}
-{"id": "3391.png", "formula": "\\begin{align*} F _ \\beta ( \\tau ) \\ , \\ , = \\ , \\ , \\Phi _ \\beta \\left ( \\log \\tau \\right ) \\ , \\hbox { a n d } \\tau \\ , \\ , F _ \\beta ' ( \\tau ) \\ , \\ , = \\ , \\ , { \\Phi _ \\beta ' \\left ( \\log \\tau \\right ) } \\ , . \\end{align*}"}
-{"id": "4070.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 6 , 6 u + 2 , 6 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 6 , 6 u + 4 , 6 v + 5 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "9611.png", "formula": "\\begin{align*} \\xi _ { \\epsilon } ( t ) = \\xi * \\rho _ { \\epsilon } ( t ) = \\int _ 0 ^ T \\rho _ { \\epsilon } ( s ) \\xi ( t - s ) d s \\in C ^ \\infty ( [ 0 , T ] ) , \\end{align*}"}
-{"id": "4355.png", "formula": "\\begin{align*} \\varphi ( y ) + p \\int _ 0 ^ y \\Phi ( z ) \\ d z = \\frac { p a ^ { 2 - p } } { ( p - 1 ) ( 2 - p ) } \\left [ 1 - ( 1 - y ) ^ { 2 - p } \\right ] \\end{align*}"}
-{"id": "7044.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\ell a _ i ( L _ i \\cdot P _ j ) \\equiv 0 \\mod d \\end{align*}"}
-{"id": "7994.png", "formula": "\\begin{align*} F = n _ 1 E _ 1 + n _ 2 E _ 2 + F ' \\end{align*}"}
-{"id": "6968.png", "formula": "\\begin{align*} & h ( \\sigma , \\epsilon ) = \\max _ { \\mu _ 1 \\in \\bar { M } _ { \\epsilon } } \\biggl [ H _ 2 ( \\sigma ) - \\gamma \\frac { \\delta - 1 } { \\delta } H _ 2 ( \\mu _ 1 ) \\\\ & \\quad \\qquad + \\inf _ { u > 0 } \\{ \\sigma \\log p ( u ) + ( 1 - \\sigma ) \\log q ( u ) - \\mu _ 1 \\gamma \\log u \\} \\biggr ] . \\end{align*}"}
-{"id": "8912.png", "formula": "\\begin{align*} u ( x ) = v ( \\abs { P _ { W _ 0 } ( x ) - a } , \\abs { P _ { W _ 1 } ( x ) } , \\dotsc , \\abs { P _ { W _ k } ( x ) } ) . \\end{align*}"}
-{"id": "5746.png", "formula": "\\begin{align*} S ( \\delta _ 0 ) = n ^ { - 1 / 2 } \\sum _ { t = 1 } ^ n ( y _ t - m _ t \\pi _ { t } ( \\beta _ 0 ) ) \\left ( \\sum _ { i = 0 } ^ { ( t - 2 ) } \\omega ^ { i } e _ { t - 1 - i } ^ P ( \\beta _ 0 ) \\right ) . \\end{align*}"}
-{"id": "410.png", "formula": "\\begin{align*} \\big ( f _ { 2 k } ( x _ { t + 1 } ) , f _ { 2 k - 1 } ( x _ { t + 1 } ) \\big ) & = ( 2 , m - 1 ) , \\\\ \\big ( f _ { 2 k } ( x _ { t + m } ) , f _ { 2 k - 1 } ( x _ { t + m } ) \\big ) & = ( 2 , 0 ) , \\\\ \\big ( f _ { 2 k } ( x _ { t + m + 1 } ) , f _ { 2 k - 1 } ( x _ { t + m + 1 } ) \\big ) & = ( 1 , m - 1 ) . \\\\ \\end{align*}"}
-{"id": "9199.png", "formula": "\\begin{align*} & ( Y ( t , \\cdot ) , \\phi ) _ { L ^ 2 ( D ) } = ( \\xi , \\phi ) _ { L ^ 2 ( D ) } + \\int _ 0 ^ t ( Y ( s , \\cdot ) , A _ u ^ * \\phi ) _ { L ^ 2 ( D ) } d s + \\int _ 0 ^ t ( a ( s , Y ( s , \\cdot ) , \\cdot ) , \\phi ) _ { L ^ 2 ( D ) } d s \\\\ & + \\int _ 0 ^ t ( b ( s , Y ( s , \\cdot ) , \\cdot ) , \\phi ) _ { L ^ 2 ( D ) } d B ( s ) + \\int _ 0 ^ t \\int _ { \\mathbb { R } } c ( s , Y ( s , \\cdot ) , \\zeta , \\cdot ) , \\phi ) _ { L ^ 2 ( D ) } \\tilde { N } ( d s , d \\zeta ) , \\end{align*}"}
-{"id": "7445.png", "formula": "\\begin{align*} \\eta : = A _ g \\xi _ s + A _ h \\xi _ t - A _ f \\xi _ r . \\end{align*}"}
-{"id": "4560.png", "formula": "\\begin{align*} \\Sigma ' : = \\begin{pmatrix} 1 & ( j / l ) ^ { 1 / 2 } \\alpha _ z ' \\\\ ( j / l ) ^ { 1 / 2 } \\alpha _ z ' & 1 \\end{pmatrix} , \\end{align*}"}
-{"id": "3030.png", "formula": "\\begin{align*} d _ \\Sigma ^ { { X _ L } } ( M , N ; \\frac { 2 } { { { T _ c } } } ) = \\frac { { M N ( N - 1 ) n + { T _ c } ( { T _ c } - 1 ) } } { { { T _ c } n + { T _ c } ( { T _ c } - 1 ) } } , \\end{align*}"}
-{"id": "1615.png", "formula": "\\begin{align*} Z ^ { 4 } = t ^ { 2 } \\partial _ { t } + H + \\frac { t } { 2 c } K ^ { 1 } - \\left ( \\frac { 1 } { 2 } \\int \\frac { d x } { \\sigma \\left ( x \\right ) } ^ { 2 } + \\frac { 1 } { 2 c } \\int \\frac { d x } { \\sigma \\left ( x \\right ) } - c t \\int \\frac { d x } { \\sigma \\left ( x \\right ) } + \\frac { c ^ { 2 } } { 2 } t ^ { 2 } \\right ) F \\partial _ { F } \\end{align*}"}
-{"id": "2264.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u ' ( t ) + A ( t ) u ( t ) = B ( t , u ( t ) ) , 0 < t < T , \\\\ & u ( 0 ) = u _ 0 ; \\end{aligned} \\right . \\end{align*}"}
-{"id": "8782.png", "formula": "\\begin{align*} Y _ i E _ \\lambda = y _ i ( \\lambda ) E _ \\lambda ( i = 1 , \\ldots , n ) , \\end{align*}"}
-{"id": "8076.png", "formula": "\\begin{align*} S _ B ( x , y ) = \\begin{cases} ( x + \\beta , k \\alpha - y ) & x \\in [ 1 - \\beta , 1 ) , \\\\ ( x + \\beta , ( k - 1 ) \\alpha - y ) & x \\in [ 0 , 1 - \\beta ) . \\end{cases} \\end{align*}"}
-{"id": "640.png", "formula": "\\begin{align*} ( s \\cdot \\eta ) ( f ) = \\eta ( s ^ { - 1 } \\cdot f ) , \\textrm { f o r $ s \\in G $ a n d $ f \\in \\ell ^ \\infty ( G ) $ } . \\end{align*}"}
-{"id": "8000.png", "formula": "\\begin{align*} 6 8 = \\Phi ( X ) \\leq 4 \\cdot ( 1 3 - 2 ) + 2 4 = 6 8 . \\end{align*}"}
-{"id": "6705.png", "formula": "\\begin{align*} f ' _ { K L } = f _ { I J } a ^ I _ K b ^ J _ L = a ^ I _ K \\star f _ { I J } \\star b ^ J _ L . \\end{align*}"}
-{"id": "5893.png", "formula": "\\begin{align*} g ^ { \\beta \\gamma } \\xi _ { , \\beta \\gamma } ^ { \\alpha } - \\xi _ { , \\gamma } ^ { \\alpha } \\Gamma ^ { \\gamma } + \\xi ^ { \\gamma } \\Gamma _ { , \\gamma } ^ { \\alpha } - \\xi _ { , \\gamma } ^ { \\alpha } C ^ { \\gamma } + \\xi ^ { \\gamma } C _ { , \\gamma } ^ { \\alpha } - 2 g ^ { \\beta \\alpha } a _ { , \\beta } + \\left ( a - \\lambda \\right ) \\Gamma ^ { \\alpha } + \\left ( a - \\lambda \\right ) C ^ { \\alpha } - \\xi _ { , t } ^ { \\alpha } = 0 . \\end{align*}"}
-{"id": "7133.png", "formula": "\\begin{align*} P _ { j , i } ( t ) = \\big [ R ^ 2 - \\vert z _ j ( t ) \\vert ^ 2 \\big ] \\big [ R ^ 2 - \\vert z _ i ( t ) \\vert ^ 2 \\big ] ^ 2 \\big [ R ^ 2 - \\bar { z } _ i ( t ) z _ j ( t ) \\big ] \\big [ z _ i ( t ) - z _ j ( t ) \\big ] , \\end{align*}"}
-{"id": "8263.png", "formula": "\\begin{align*} ( ( V ' ( v ( i + 1 ) ) - V ' ( v ( i - 1 ) ) ) \\partial _ i ) ^ { \\ast } & = \\partial _ i ^ { \\ast } ( V ' ( v ( i + 1 ) ) - V ' ( v ( i - 1 ) ) ) \\\\ & = ( - \\partial _ i - ( \\lambda - V ' ( v _ i ) ) ) ( V ' ( v ( i + 1 ) ) - V ' ( v ( i - 1 ) ) ) \\\\ & = - ( V ' ( v ( i + 1 ) ) - V ' ( v ( i - 1 ) ) ) \\partial _ i \\\\ & - ( \\lambda - V ' ( v ( i ) ) ) ( V ' ( v ( i + 1 ) ) - V ' ( v ( i - 1 ) ) ) . \\end{align*}"}
-{"id": "8509.png", "formula": "\\begin{align*} \\beta _ { k + 2 } & = w s _ { i _ k } s _ { i _ { k + 1 } } \\alpha _ { i _ k } = w s _ { i _ k } ( \\alpha _ { i _ k } + \\alpha _ { i _ { k + 1 } } ) = w \\alpha _ { i _ { k + 1 } } , \\\\ \\beta _ { k + 1 } & = w s _ { i _ k } \\alpha _ { i _ { k + 1 } } = w ( \\alpha _ { i _ k } + \\alpha _ { i _ { k + 1 } } ) = w s _ { i _ { k + 1 } } \\alpha _ { i _ k } , \\\\ \\beta _ k & = w \\alpha _ { i _ k } = w s _ { i _ { k + 1 } } ( \\alpha _ { i _ k } + \\alpha _ { i _ { k + 1 } } ) = w s _ { i _ { k + 1 } } s _ { i _ k } \\alpha _ { i _ { k + 1 } } . \\end{align*}"}
-{"id": "3376.png", "formula": "\\begin{align*} G & = C \\int _ { \\frac { \\gamma _ { \\max } } { \\bar { \\sigma } _ h ^ 2 } } ^ \\infty \\log \\left ( 1 + \\bar { \\sigma } _ h ^ 2 P _ { \\max } \\gamma \\right ) \\gamma ^ { N - 1 } e ^ { - \\gamma } d \\gamma \\end{align*}"}
-{"id": "9523.png", "formula": "\\begin{align*} B _ 2 ' & = B _ 2 + d \\zeta _ 1 \\\\ C _ 0 ' & = C _ 0 \\\\ C _ 2 ' & = C _ 2 + d \\Lambda _ 1 \\\\ C _ 4 ' & = C _ 4 + d \\Lambda _ 3 - \\frac { 1 } { 2 } H _ 3 \\wedge \\Lambda _ 1 + \\frac { 1 } { 2 } G _ 3 \\wedge \\zeta _ 1 \\end{align*}"}
-{"id": "3593.png", "formula": "\\begin{align*} L ( f , X ) & = \\rho _ g ^ { - 1 } D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) \\\\ U & = ( f , X ^ 1 , \\dots , X ^ n ) \\mbox { ( w i t h r e s p e c t t o a f i x e d c o o r d i n a t e c h a r t ) } \\end{align*}"}
-{"id": "2205.png", "formula": "\\begin{align*} & \\quad \\quad \\quad \\int _ { 0 } ^ { T } \\mathcal { E } ( u , u ^ { - } ) d t = \\int _ { 0 } ^ { T } \\mathcal { E } ( u ^ { + } , u ^ { - } ) d t - \\int _ { 0 } ^ { T } \\mathcal { E } ( u ^ { - } , u ^ { - } ) d t , \\\\ & \\int _ { 0 } ^ { T } \\mathcal { E } ( u ^ { - } , u ^ { - } ) d t = \\int _ { 0 } ^ { T } \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } ( u ^ { - } ( x , t ) - u ^ { - } ( y , t ) ) ^ { 2 } k ( x , y ) d x d y d t > 0 , \\end{align*}"}
-{"id": "3346.png", "formula": "\\begin{align*} | | \\mathtt { g } \\cdot \\mathtt { h } | | _ \\nu & \\leq | | \\mathtt { g } | | _ \\nu + | | \\mathtt { h } | | _ \\nu , \\\\ | | \\mathtt { h } \\cdot \\mathtt { g } \\cdot \\bar { \\mathtt { h } } | | _ \\nu & = | | \\mathtt { g } | | _ \\nu , \\\\ | | \\bar { \\mathtt { g } } | | _ \\nu & = | | \\mathtt { g } | | _ \\nu . \\\\ \\end{align*}"}
-{"id": "7108.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\limsup _ { n \\to \\infty } \\P \\left ( \\Delta _ n ^ { k , t } > \\delta , S \\right ) = 0 . \\end{align*}"}
-{"id": "3846.png", "formula": "\\begin{align*} E _ T ( n ) \\sim \\sum _ { k = 1 } ^ \\infty \\frac { 3 } { 4 ^ k + 2 } \\approx 0 . 7 2 7 6 4 9 . \\end{align*}"}
-{"id": "3880.png", "formula": "\\begin{align*} & | \\varepsilon _ i ( x ) | \\le C \\varkappa \\rho , \\mbox { f o r } \\ ; x \\in \\tilde { B } _ \\rho ( a ) , \\ ; i = 1 , 2 , \\\\ & | \\varepsilon _ 3 ( x ) | \\le C \\varkappa \\rho \\mbox { f o r } \\ ; x \\in B ^ + \\cap B _ \\rho ( a ) \\end{align*}"}
-{"id": "8546.png", "formula": "\\begin{align*} \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } \\xi _ { b r } \\pi _ { b r } = i d _ { N _ { o u t } } \\end{align*}"}
-{"id": "2660.png", "formula": "\\begin{align*} d ^ { m a x } _ n : = \\left \\{ \\begin{array} { l l } \\left \\lfloor \\dfrac { 3 n + 2 } { 4 } \\right \\rfloor & \\mbox { i f } m = 2 \\\\ n + 1 + ( m - 3 ) \\left \\lfloor \\dfrac { n + 2 } { 3 } \\right \\rfloor & \\mbox { o t h e r w i s e } \\\\ \\end{array} \\right . \\ , . \\end{align*}"}
-{"id": "8960.png", "formula": "\\begin{align*} \\mu ( t ) : = \\sigma ( t ) - t \\end{align*}"}
-{"id": "5760.png", "formula": "\\begin{align*} W _ { t } = - 0 . 5 + ( t / n ) + Z _ { t } \\end{align*}"}
-{"id": "4495.png", "formula": "\\begin{align*} ( w _ { j _ t } ) _ { t = 1 } ^ { \\lfloor d / 4 \\rfloor + 1 } : = ( A ^ { ( k ) } ) ^ { - 1 } e _ 1 \\end{align*}"}
-{"id": "1053.png", "formula": "\\begin{align*} I _ 3 ( x ) = \\frac { e ^ { \\frac { i \\pi } { 4 } } } { 2 \\sqrt { 2 \\pi } } \\int \\limits _ { D ( x ) } \\frac { e ^ { i | x - y | } } { | x - y | ^ { \\frac 1 2 } } \\Bigl ( 1 + \\delta ( | x - y | ) \\Bigr ) f ( y ) \\ , d y , \\end{align*}"}
-{"id": "5712.png", "formula": "\\begin{align*} \\| b \\| _ { \\rm B M O } : = \\| b ^ \\sharp \\| _ { L ^ \\infty } < \\infty , \\end{align*}"}
-{"id": "505.png", "formula": "\\begin{align*} \\left | I ( w , m ) \\right | \\ ; & = \\ ; \\binom { 3 m + \\ell } { m } - \\binom { 3 m + \\ell } { < m } + \\sum \\limits _ { e = 1 } ^ m 4 e \\left [ \\binom { 3 m + \\ell } { m - e } - \\binom { 3 m + \\ell } { < m - e } \\right ] \\\\ \\ ; & = \\ ; \\binom { 3 m + \\ell } { m } - \\sum \\limits _ { e = 1 } ^ m \\left ( 2 e ^ 2 - 6 e + 1 \\right ) \\binom { 3 m + \\ell } { m - e } . \\end{align*}"}
-{"id": "5523.png", "formula": "\\begin{align*} \\lambda _ { \\varphi , w } = \\{ x : \\alpha ( \\lambda x ) < \\infty , \\ , \\ , \\ , \\ , \\lambda > 0 \\} . \\end{align*}"}
-{"id": "6214.png", "formula": "\\begin{align*} H ( s ) = ( C _ { p } + s C _ { v } ) ( s ^ { 2 } M + s D + K ) ^ { - 1 } F = ( C _ { p } + s C _ { v } ) X ( s ) , \\end{align*}"}
-{"id": "5879.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } ( t ) ( F _ { x x } - F _ { x } ) + ( p ( t ) - x q ( t ) ) F _ { x } - F _ { t } = 0 , \\end{align*}"}
-{"id": "2965.png", "formula": "\\begin{align*} \\delta _ L f ( x _ 1 , . . . , x _ { q + 1 } ) = & ( - 1 ) ^ { q + 1 } \\sum _ { i = 1 } ^ q ( - 1 ) ^ { i + 1 } [ x _ i , f ( x _ 1 , . . , \\hat { x _ i } , . . , x _ { q + 1 } ) ] + [ f ( x _ 1 , . . . , x _ q ) , x _ { q + 1 } ] \\\\ & + ( - 1 ) ^ { q + 1 } \\sum _ { 1 \\leq i < j \\leq q + 1 } f ( x _ 1 , . . , \\hat { x _ i } , . . , x _ { j - 1 } , [ x _ i , x _ j ] , x _ { j + 1 } , . . , x _ { q + 1 } ) . \\end{align*}"}
-{"id": "6879.png", "formula": "\\begin{align*} Z _ t = x + \\int _ 0 ^ t \\sigma ( Z _ s ) \\dot { h } _ s \\ , \\d s + \\int _ 0 ^ t b ( Z _ s ) \\ , \\d s + \\psi _ t . \\end{align*}"}
-{"id": "3751.png", "formula": "\\begin{align*} h _ 3 = \\sum _ { k = 3 } ^ { \\infty } p _ k q ^ { * k } \\end{align*}"}
-{"id": "2604.png", "formula": "\\begin{align*} c ( a , b ) = \\frac { a b } { 2 n } , \\ \\ \\ \\ \\ \\omega ( a , b , c ) = \\begin{cases} \\frac { a } { 2 } , & b + c \\geq n , \\\\ 0 . & \\end{cases} \\end{align*}"}
-{"id": "820.png", "formula": "\\begin{align*} F _ { \\vec { x } } ( u _ { 1 } , \\ldots , u _ { k } ) = \\frac { ( 1 - q ^ 2 ) ^ { k } } { \\prod _ { i = 1 } ^ { k } ( 1 - s u _ { i } ) } \\sum _ { \\sigma \\in \\mathcal { S } _ { k } } \\prod _ { 1 \\le i < j \\le k } \\frac { u _ { \\sigma ( i ) } - q ^ { 2 } u _ { \\sigma ( j ) } } { u _ { \\sigma ( i ) } - u _ { \\sigma ( j ) } } \\prod _ { i = 1 } ^ { k } \\left ( \\frac { u _ { i } - s } { 1 - s u _ { i } } \\right ) ^ { x _ { i } } \\end{align*}"}
-{"id": "320.png", "formula": "\\begin{align*} | ( q \\circ g ) _ J ( x ) | & \\leq q \\bigl ( g ( x ) \\bigr ) \\sum _ { i = 1 } ^ { s } 2 ^ { 2 i - 1 } i ^ i t ^ i S ( s , i ) a ( f ( x ) ) ^ { i ( m + 1 ) } D _ g ^ i \\\\ & \\leq q \\bigl ( g ( x ) \\bigr ) s ^ { s + 1 } 2 ^ { 2 s - 1 } \\max ( 1 , t ) ^ s B _ s a ( f ( x ) ) ^ { s ( m + 1 ) } D _ g ^ s , \\end{align*}"}
-{"id": "9541.png", "formula": "\\begin{align*} \\Gamma _ a ' \\bar { \\delta } & = \\overline { \\Gamma _ a \\delta } \\\\ \\Gamma _ { 1 0 } ' \\bar { \\delta } & = \\overline { \\Gamma _ { 1 1 } \\delta } , \\end{align*}"}
-{"id": "6658.png", "formula": "\\begin{align*} T ^ w = ( \\C ^ \\times ) ^ { r _ 1 } \\times \\cdots \\times ( \\C ^ \\times ) ^ { r _ m } \\end{align*}"}
-{"id": "9916.png", "formula": "\\begin{align*} X & : = \\xi _ 3 \\eta _ 4 E - \\xi _ 1 \\eta _ 4 K - \\xi _ 3 \\eta _ 1 K ' = 0 , \\\\ Y & : = \\xi _ 3 \\eta _ 4 F - \\xi _ 2 \\eta _ 4 K - \\xi _ 3 \\eta _ 2 K ' = 0 . \\end{align*}"}
-{"id": "7208.png", "formula": "\\begin{align*} \\pi _ { 1 } = \\pi _ { 2 } \\circ \\Phi , \\end{align*}"}
-{"id": "2111.png", "formula": "\\begin{align*} \\bar { P } ( X ) : = \\bar { p } _ { \\mu m } x ^ \\mu | z | ^ m \\end{align*}"}
-{"id": "7189.png", "formula": "\\begin{align*} f ^ { '' } + f ' \\cot \\varphi = g f ' - f ^ 2 - 2 f + 2 C _ 1 . \\end{align*}"}
-{"id": "7749.png", "formula": "\\begin{align*} z _ i : = \\sigma _ { i - 1 } \\xi _ i , i \\in \\mathbb N . \\end{align*}"}
-{"id": "3636.png", "formula": "\\begin{align*} | u _ { n + l } - K | \\leq \\theta \\max _ { n - k + 1 \\leq j \\leq n } | u _ j - K | , ~ ~ l = 1 , \\dots , k . \\end{align*}"}
-{"id": "6036.png", "formula": "\\begin{align*} h _ R ( x ) = \\int _ { B _ R ^ + } G _ R ^ + ( x , y ) g _ 1 ( y ) d y + \\int _ { \\R ^ N \\setminus B _ R ^ + } \\Gamma _ R ^ + ( x , y ) g _ 2 ( y ) d y . \\end{align*}"}
-{"id": "8372.png", "formula": "\\begin{align*} y x = \\phi ( x ) y , \\ \\ \\ x \\in M . \\end{align*}"}
-{"id": "5638.png", "formula": "\\begin{align*} D _ E ( \\mu _ { \\P } ^ { \\gamma } , \\mu _ { \\P } ^ 0 ) = E ( \\mu _ { \\P } ^ { \\gamma } ) - E ( \\mu _ { \\P } ^ 0 ) - \\langle \\nabla E ( \\mu _ { \\P } ^ 0 ) , \\mu _ { \\P } ^ { \\gamma } - \\mu _ { \\P } ^ 0 \\rangle . \\end{align*}"}
-{"id": "1915.png", "formula": "\\begin{align*} \\phi _ j ( u ) = \\phi _ 0 ( u ) e ^ { i 2 \\pi / K } = | \\phi _ 0 ( u ) | e ^ { i 2 \\pi / K } . \\end{align*}"}
-{"id": "4802.png", "formula": "\\begin{align*} \\left ( \\frac { t } { 2 } \\ , ( \\varphi ^ 2 ) ' + \\varphi ^ 2 + 1 \\right ) ^ 2 - 4 c t ^ 2 ( \\varphi ^ 2 + 1 ) = a ^ 2 ( \\varphi ^ 2 + 1 ) . \\end{align*}"}
-{"id": "1300.png", "formula": "\\begin{align*} T _ H ( N , y ; \\alpha ) = H \\sum _ { n = N - H } ^ { N + y } e ( n \\alpha ) - \\sum _ { m = 0 } ^ { y } m e ( ( N + m ) \\alpha ) - \\sum _ { m = 1 } ^ { H } m e ( ( N - m ) \\alpha ) = A - B - C , \\end{align*}"}
-{"id": "9641.png", "formula": "\\begin{align*} u _ { j + k } = \\alpha _ { j , k - 1 } u _ { j + k - 1 } + \\ldots + \\alpha _ { j , 1 } u _ { j + 1 } + \\alpha _ { j , 0 } u _ j , \\end{align*}"}
-{"id": "7329.png", "formula": "\\begin{align*} \\langle X _ { ( 1 ) } w , X _ { ( 2 ) } v \\rangle = \\varepsilon ( X ) \\langle w , v \\rangle . \\end{align*}"}
-{"id": "5824.png", "formula": "\\begin{align*} D ( \\beta ) = \\{ x \\in \\mathbb { R } ^ n : \\left < x , \\beta \\right > = 0 , \\ \\left < x , \\beta ' \\right > \\leq 0 \\beta ' \\subseteq \\beta \\} . \\end{align*}"}
-{"id": "5545.png", "formula": "\\begin{align*} \\dot x _ j = - b _ 0 ( x _ j ) , \\dot m _ j = m _ j \\langle b _ { 0 , x } \\rangle ( x _ j ) . \\end{align*}"}
-{"id": "6307.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } T _ m f \\| _ { L ^ { p _ 2 } } \\sim & \\| \\Box _ k ^ { \\alpha _ 2 } T _ m f \\| _ { M _ 2 } \\\\ = & \\| \\Box _ k ^ { \\alpha _ 2 } T _ m \\Box _ k ^ { \\alpha _ 2 , \\ast } f \\| _ { M _ 2 } \\\\ \\lesssim & \\| \\Box _ k ^ { \\alpha _ 2 } T _ m \\| _ { M _ 1 \\rightarrow M _ 2 } \\| \\Box _ k ^ { \\alpha _ 2 , \\ast } f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "5424.png", "formula": "\\begin{align*} P r [ f ^ { ( 3 ) } ( x , \\xi ^ { ( 3 ) } ) \\in T _ 0 ] \\geq ( 4 / 3 ) \\cdot 1 5 ^ 3 \\cdot ( 1 / 6 0 ) ^ 3 = 1 / 4 8 \\end{align*}"}
-{"id": "9000.png", "formula": "\\begin{align*} \\int _ 0 ^ x E _ { \\alpha } ( \\lambda [ x - t ] ^ \\alpha ) t ^ { \\nu - 1 } E ^ \\sigma _ { \\alpha , \\nu } ( \\lambda t ^ \\alpha ) d t = x ^ { \\nu } E ^ { 1 + \\sigma } _ { \\alpha , 1 + \\nu } ( \\lambda x ^ \\alpha ) . \\end{align*}"}
-{"id": "1528.png", "formula": "\\begin{align*} Q ( F ) - E \\cdot \\nabla _ v F = 0 , \\int _ { \\R ^ d } F ( v , E ) \\ , d v = 1 . \\end{align*}"}
-{"id": "2105.png", "formula": "\\begin{align*} \\bar { U } _ a : = \\bigg ( \\frac { | z | + d } { 2 } \\bigg ) ^ s . \\end{align*}"}
-{"id": "205.png", "formula": "\\begin{align*} \\sup _ { I \\in \\mathcal { I } } \\lim _ { n \\rightarrow \\infty } \\max _ { \\lambda \\in I } n \\mathbb { E } _ { f _ { n ^ { - 1 / 2 } , g _ \\lambda } } \\bigl [ \\bigl \\{ \\hat { H } _ n ^ w - H ( f _ { n ^ { - 1 / 2 } , g _ \\lambda } ) \\bigr \\} ^ 2 \\bigr ] = V ( f ) . \\end{align*}"}
-{"id": "7510.png", "formula": "\\begin{align*} F _ { \\alpha } ( x ) : = \\alpha x + ( 1 - \\alpha ) f ( x ) . \\end{align*}"}
-{"id": "142.png", "formula": "\\begin{align*} \\| f \\| _ { \\widehat { E } } & \\leq \\| f - g ( f \\chi _ A \\circ \\sigma _ 1 ) \\| _ { \\widehat { E } } + \\| g ( f \\chi _ A \\circ \\sigma _ 1 ) \\| _ { \\widehat { E } } \\leq \\epsilon + \\| f \\chi _ A \\circ \\sigma _ 1 \\| _ { \\widehat { E } } = \\epsilon + \\lim _ { n \\to \\infty } \\| \\tilde { f _ n } \\circ \\sigma _ 1 \\| _ { \\widehat { E } } \\\\ & = \\epsilon + \\lim _ { n \\to \\infty } \\| \\tilde { f _ n } \\| _ { \\widehat { E } } = \\epsilon + \\| f \\chi _ A \\| _ { \\widehat { E } } , \\end{align*}"}
-{"id": "3041.png", "formula": "\\begin{align*} \\gamma _ \\infty = \\sum _ { n = 1 } ^ { \\infty } \\theta _ { \\xi _ n - \\log Z _ \\infty } D _ n . \\end{align*}"}
-{"id": "5051.png", "formula": "\\begin{align*} \\mu _ i = \\beta _ i m _ i + \\gamma _ i \\lambda _ i , \\textrm { f o r $ i = 1 , 2 $ } , \\end{align*}"}
-{"id": "3013.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( 6 ) = { { \\bf { h } } ^ { [ j 1 ] } } ( 6 ) { { \\bf { v } } ^ { [ 1 ] } } + { { \\bf { h } } ^ { [ j 2 ] } } ( 6 ) { { \\bf { v } } ^ { [ 2 ] } } . \\end{align*}"}
-{"id": "8915.png", "formula": "\\begin{align*} \\abs { u - u \\circ \\sigma _ H } = 2 u ^ H - u - u \\circ \\sigma _ H H . \\end{align*}"}
-{"id": "7720.png", "formula": "\\begin{align*} x _ { k + 1 } \\ge \\sum _ { i = N _ \\delta } ^ { k } \\sigma _ i \\xi _ { i + 1 } , \\end{align*}"}
-{"id": "9303.png", "formula": "\\begin{align*} \\hat { \\pi } ( t , z ) = - \\frac { \\alpha ( t ) \\mathbb { E } [ p ' ( t , X ( t ) , z ) | \\mathcal { R } _ t ] } { \\beta ^ 2 ( t ) \\mathbb { E } [ p '' ( t , X ( t ) , z ) | \\mathcal { R } _ t ] } , \\end{align*}"}
-{"id": "9339.png", "formula": "\\begin{align*} d ( h ( t , x , z ) k ( t , x , z ) ) & = h ( t , x , z ) k ( t , x , z ) b ( t , x , z ) d G ( t ) + k ( t , x , z ) [ d F ( t , x , z ) + x h ( t , x , z ) d G ( t ) ] \\\\ & + k ( t , x , z ) b ( t , x , z ) ( x h ( t , x , z ) + y ( t , x , z ) q ( t , x , z ) ) d t \\end{align*}"}
-{"id": "4129.png", "formula": "\\begin{align*} p + q = \\gcd ( 2 q , - p ) + \\gcd ( q , - 2 p ) . \\end{align*}"}
-{"id": "2443.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } = 2 ^ { j A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 2 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 3 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "4259.png", "formula": "\\begin{align*} A & = \\ \\sum _ { i = 3 } ^ { n } \\sum _ { j = 3 } ^ { m } ( A _ { i , j } + A _ { i - 1 , j - 1 } - A _ { i - 1 , j } - A _ { i , j - 1 } ) L ^ { ( i , j ) } + \\\\ & \\sum _ { j = 3 } ^ { m } ( A _ { 2 , j } - A _ { 2 , j - 1 } ) L ^ { ( 2 , j ) } + \\sum _ { i = 3 } ^ { n } ( A _ { i , 2 } - A _ { i - 1 , 2 } ) L ^ { ( i , 2 ) } + A _ { 2 , 2 } ^ { ( 2 , 2 ) } L ^ { ( 2 , 2 ) } \\end{align*}"}
-{"id": "9225.png", "formula": "\\begin{align*} \\nabla _ { \\hat { Y } } \\widehat { H } ( \\tilde { Y } ) = \\nabla _ { \\varphi } \\widehat { H } ( \\tilde { Y } ) | _ { \\varphi = \\hat { Y } } \\end{align*}"}
-{"id": "4741.png", "formula": "\\begin{align*} | u _ \\xi ( x ) | & = \\Big | u _ \\xi ( 0 ) + | x | \\int _ 0 ^ 1 u _ { \\xi \\xi } ( t x ) \\ , d t \\Big | \\\\ & \\le | u _ \\xi ( 0 ) | + | x | \\int _ 0 ^ 1 \\frac { C } { \\ , | t x | ^ \\alpha \\ , } \\ , d t \\\\ & \\le | D u ( 0 ) | + \\frac { C } { \\ , 1 - \\alpha \\ , } \\ , | x | ^ { 1 - \\alpha } . \\end{align*}"}
-{"id": "5072.png", "formula": "\\begin{align*} \\frac { 1 - q ^ { 2 n _ { a } } } { 1 - q ^ { 2 } } q ^ { 2 \\sum _ { p = a + 1 } ^ { r } n _ { p } } , \\end{align*}"}
-{"id": "9340.png", "formula": "\\begin{align*} u ( t , x , z ) = h ( t , x , z ) k ( t , x , z ) \\end{align*}"}
-{"id": "1101.png", "formula": "\\begin{align*} | A x _ 0 | = | A \\hat x | . \\end{align*}"}
-{"id": "7857.png", "formula": "\\begin{align*} \\rho ( t , x ) = \\rho ^ { ( d ) } ( t , x ) : = { \\Phi \\left ( \\left ( \\frac { 1 } { \\Phi ^ { - 1 } ( t ^ { - 1 } ) } + | x | \\right ) ^ { - 1 } \\right ) } \\left ( \\frac { 1 } { \\Phi ^ { - 1 } ( t ^ { - 1 } ) } + | x | \\right ) ^ { - d } \\ , . \\end{align*}"}
-{"id": "7448.png", "formula": "\\begin{align*} ( \\eta _ 1 , \\dots , \\eta _ n ) = ( \\xi ' _ 1 , \\dots , \\xi ' _ n ) \\begin{pmatrix} X ^ { - 1 } & \\cdots & 0 & 0 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & X & 0 & \\cdots & 0 \\\\ 0 & \\cdots & X & 1 & \\cdots & 0 \\\\ \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & \\cdots & 0 & 0 & \\cdots & 1 \\end{pmatrix} = ( \\xi ' _ 1 , \\dots , \\xi ' _ n ) e ^ { - i } h ^ { i } ( X ) . \\end{align*}"}
-{"id": "9402.png", "formula": "\\begin{align*} ( H _ { \\mathbf { T } } - \\lambda { I } ) ^ { - 1 } - ( H _ \\infty - \\lambda { I } ) ^ { - 1 } = \\gamma ( \\lambda ) ( \\mathbf { T } - W _ \\lambda ) ^ { - 1 } \\gamma ( \\lambda ^ * ) ^ \\dag \\end{align*}"}
-{"id": "8798.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { n + 1 } I ( f _ i , f _ i ) > - \\frac { m } { n + 1 } \\int _ { \\partial \\Sigma } \\Pi ( N , N ) d s \\end{align*}"}
-{"id": "9882.png", "formula": "\\begin{align*} \\lim _ { C \\rightarrow \\infty } \\sup _ { { \\epsilon } \\in ( 0 , 1 ) } P \\big ( \\sup _ { t \\leq T } | V ^ { \\beta ( { \\epsilon } ) , v ^ { \\epsilon } } _ { { \\eta ^ { \\epsilon } , n } } ( t , . ) | _ { 2 } \\geq C \\big ) = 0 . \\end{align*}"}
-{"id": "8589.png", "formula": "\\begin{align*} d _ { 1 } ( ^ { \\ast } b ) d _ { 2 } \\cdot \\widetilde { \\psi } ( n ) & = d _ { 1 } ( ^ { \\ast } b ) d _ { 2 } \\cdot \\psi ( n ) \\\\ & = \\begin{pmatrix} - d _ { 1 } \\pi _ { 1 } ' p ' \\xi _ { b e _ { k } } ( d _ { 2 } \\psi ( n ) ) \\\\ - d _ { 1 } \\gamma ' \\xi _ { b e _ { k } } ( d _ { 2 } \\psi ( n ) ) \\\\ 0 \\\\ 0 \\end{pmatrix} \\end{align*}"}
-{"id": "1544.png", "formula": "\\begin{align*} \\int _ y ^ { \\varphi ( t , y ) } \\frac { h ( 0 ) - \\left ( \\frac 1 p - 1 \\right ) F ' ( s ) } { F ( s ) } d s \\leq & - \\frac { h ( 0 ) } { r - 1 } \\left ( y ^ { 1 - r } - \\varphi ( t , y ) ^ { 1 - r } \\right ) + C \\\\ = & h ( 0 ) t + C . \\end{align*}"}
-{"id": "5185.png", "formula": "\\begin{align*} \\min _ { X \\in \\mathbb { R } ^ { m \\times n } } \\| X \\| ^ { p } _ { S _ { p } } , \\ ; \\ ; \\textrm { s u b j e c t t o } \\ ; \\ ; \\mathcal { A } ( X ) = b \\end{align*}"}
-{"id": "3413.png", "formula": "\\begin{align*} \\langle h , u ( s , x ) \\rangle = E _ { s , x } \\left [ \\langle \\eta ( T ) , u _ 0 ( \\xi ( T ) ) \\rangle + \\int _ s ^ T \\langle \\eta ( \\theta ) , g ( \\theta , \\xi ( \\theta ) , u ( \\theta , \\xi ( \\theta ) ) ) d \\theta \\right ] , \\end{align*}"}
-{"id": "6541.png", "formula": "\\begin{align*} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\le C \\delta , \\\\ & \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ \\infty ( W ) } \\le C \\delta , \\end{align*}"}
-{"id": "8809.png", "formula": "\\begin{align*} I ( \\tilde { u } , \\tilde { u } ) = - n a H \\int _ { \\Sigma } u d v o l _ { \\Sigma } \\ , . \\end{align*}"}
-{"id": "1465.png", "formula": "\\begin{align*} \\max _ { \\substack { R \\in \\mathcal { D } ( Q _ 0 ) : \\\\ R ^ { ( 1 ) } = Q ( x ) } } | m _ { f _ 1 } ( R ) | > ( f _ 1 \\cdot \\chi _ { Q _ 0 } ) ^ * ( \\lambda _ w | Q _ 0 | ) . \\end{align*}"}
-{"id": "6975.png", "formula": "\\begin{align*} \\alpha _ t ( a , \\alpha _ t ( b , c ) ) = \\alpha _ t ( \\alpha _ t ( a , b ) , c ) , \\end{align*}"}
-{"id": "375.png", "formula": "\\begin{align*} \\| \\Delta _ q \\cos ( n x ) \\| _ { L ^ 2 } = \\frac \\pi 2 \\left ( \\sum _ { m = - \\infty } ^ { \\infty } \\varphi _ q ^ 2 ( m ) \\left | \\delta _ { n } ( m ) + \\delta _ { - n } ( m ) \\right | ^ 2 \\right ) ^ { 1 / 2 } = \\frac \\pi { 2 } \\left ( 2 \\varphi _ q ^ 2 ( n ) \\right ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "482.png", "formula": "\\begin{align*} 2 \\int _ { B ^ c _ \\varepsilon ( x ) } \\frac { \\ , u ( x ) - u ( y ) \\ , } { | x - y | ^ { N + 2 s } } \\ , d y & = \\int _ { B _ r \\cap B ^ c _ \\varepsilon } \\frac { \\ , 2 u ( x ) - u ( x + z ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z \\\\ & + \\int _ { B ^ c _ r } \\frac { \\ , 2 u ( x ) - u ( x + z ) - u ( x - z ) \\ , } { | z | ^ { N + 2 s } } \\ , d z . \\end{align*}"}
-{"id": "3773.png", "formula": "\\begin{align*} = \\frac { 1 } { \\beta - i \\alpha t } \\left ( g ( a _ { j + 1 } ) e ^ { ( \\beta - i \\alpha t ) a _ { j + 1 } } - g ( a _ j ) e ^ { ( \\beta - i \\alpha t ) a _ j } \\right ) - \\frac { 1 } { \\beta - i \\alpha t } \\int _ { a _ j } ^ { a _ { j + 1 } } e ^ { ( \\beta - i \\alpha t ) x } g ^ { \\prime } ( x ) d x \\end{align*}"}
-{"id": "321.png", "formula": "\\begin{align*} \\biggl | \\frac { \\partial ^ { \\omega } f _ { t , g ^ * } ( x ) } { \\partial x ^ { \\omega } } \\biggr | & = \\biggl | 2 c ( t ) \\sum _ { \\nu : \\nu \\leq \\omega } \\binom { \\omega } { \\nu } \\frac { \\partial ^ \\nu q \\bigl ( g ^ * ( x ) \\bigr ) } { \\partial x ^ \\nu } \\frac { \\partial ^ { \\omega - \\nu } f ( x ) } { \\partial x ^ { \\omega - \\nu } } \\biggr | \\\\ & \\leq 2 ^ { 3 m - 1 } m ^ { m + 1 } B _ m D _ { g ^ * } ^ m a ( f ( x ) ) ^ { m ^ 2 + m } f _ { t , g ^ * } ( x ) . \\end{align*}"}
-{"id": "3333.png", "formula": "\\begin{align*} - \\frac { c _ { 3 } } { c _ { 2 } } q ^ { 1 0 } = \\frac { c _ { 2 } } { c _ { 1 } } ( t - t ^ \\prime q ^ { - 1 } ( q - q ^ { - 1 } ) ) , 0 = \\frac { c _ { 2 } } { c _ { 1 } } q ^ { - 4 } ( t q ( q - q ^ { - 1 } ) + t ^ \\prime ) . \\end{align*}"}
-{"id": "5361.png", "formula": "\\begin{align*} q _ i = \\prod _ { \\substack { 1 \\leq j \\leq n \\\\ p ( j ) - p ( j - 1 ) \\geq i } } j \\end{align*}"}
-{"id": "3323.png", "formula": "\\begin{gather*} \\Gamma _ { + } ^ { ( 2 ) * } = \\left ( \\begin{array} { c c c } 0 & q ^ { - 2 } & 0 \\\\ 0 & 0 & q ^ { - 2 } \\\\ 0 & 0 & 0 \\end{array} \\right ) , \\Gamma _ { 0 } ^ { ( 2 ) * } = \\left ( \\begin{array} { c c c } - 1 & 0 & 0 \\\\ 0 & - q ^ { - 3 } ( q - q ^ { - 1 } ) & 0 \\\\ 0 & 0 & q ^ { - 2 } \\end{array} \\right ) , \\\\ \\Gamma _ { - } ^ { ( 2 ) * } = \\left ( \\begin{array} { c c c } 0 & 0 & 0 \\\\ - q ^ { - 4 } & 0 & 0 \\\\ 0 & - q ^ { - 2 } & 0 \\end{array} \\right ) . \\end{gather*}"}
-{"id": "9026.png", "formula": "\\begin{align*} X _ n ( z ) = \\prod _ { j = 1 } ^ { n } ( 1 - z e ^ { i \\theta _ j } ) = \\prod _ { j = 1 } ^ n ( 1 - \\lambda _ j z ) , \\end{align*}"}
-{"id": "8495.png", "formula": "\\begin{align*} \\bar \\alpha : \\widehat \\O _ { V , p } = \\C [ [ x _ 1 , x _ 2 , x _ 3 ] ] / ( F _ i ) \\to \\C [ [ x _ 1 , x _ 2 , x _ 3 ] ] / ( F ^ \\prime _ i ) = \\widehat \\O _ { W , p } \\end{align*}"}
-{"id": "6369.png", "formula": "\\begin{align*} \\tilde { w } ( z ) : = \\frac { w _ { k + 1 } ( \\l z ) } { \\l ^ { 2 s + 2 k + 2 } } , \\end{align*}"}
-{"id": "8051.png", "formula": "\\begin{align*} \\{ ( p _ { 1 } , p _ { 2 } , \\cdots , p _ { n } ) \\ ; | \\ ; p _ { i } = p _ { j } \\mbox { f o r a l l } i , j \\in I \\mbox { o r } i , j \\in I ^ { c } \\} . \\end{align*}"}
-{"id": "1682.png", "formula": "\\begin{align*} H _ t : = W _ t - \\int _ 0 ^ t F ( L _ r ) \\ , \\dd r \\ , , \\ ; \\ ; t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "6448.png", "formula": "\\begin{align*} \\| w \\| ^ { 2 } _ { L ^ { 2 } ( [ t _ { 2 } - t _ { 1 } , t _ { 0 } - t _ { 1 } ] ; H ^ { \\beta } ( \\rho ' B _ { 1 } ) ) } \\leq 4 C \\Lambda \\int _ { 0 } ^ { t _ { 0 } - t _ { 1 } } F ( s ) d s . \\end{align*}"}
-{"id": "3849.png", "formula": "\\begin{align*} \\mathcal I _ j ( x , y ) = & \\iint a _ { 1 , j } ( \\tau , s , \\xi ) e ^ { 2 \\pi i \\phi _ 1 ( x , y , t ; , \\xi , \\tau , s , \\theta ) } d \\theta d s d \\tau d \\xi , \\\\ \\mathcal J _ j ( x , y ) = & \\iint a _ { 2 , j } ( \\tau , s , \\xi ) e ^ { 2 \\pi i \\phi _ 2 ( x , y , t ; , \\xi , \\tau , s , r ) } d r d s d \\tau d \\xi , \\end{align*}"}
-{"id": "1339.png", "formula": "\\begin{align*} \\prod _ { k _ 1 \\in \\mathcal { J } _ 1 } \\cdots \\prod _ { k _ N \\in \\mathcal { J } _ N } B _ 1 ( \\lambda _ { k _ 1 } ) B _ 2 ( \\lambda _ { k _ 2 } ) \\cdots B _ N ( \\lambda _ { k _ N } ) = B _ { n _ 1 } ( \\lambda _ { 1 } ) B _ { n _ 2 } ( \\lambda _ { 2 } ) \\cdots B _ { n _ M } ( \\lambda _ M ) . \\end{align*}"}
-{"id": "1021.png", "formula": "\\begin{align*} & d e g _ { L S } ( I - ( P + Q N _ { f } - \\frac { B _ { \\varphi , b } P } { T } + H P ) , \\Omega , 0 ) \\\\ & = d e g _ { B } \\left ( I - ( P + Q N _ { f } - \\frac { B _ { \\varphi , b } P } { T } + H P ) \\left | _ { \\overline { \\Omega \\cap \\mathbb { R } ^ { 2 } } } \\right . , \\Omega \\cap \\mathbb { R } ^ { 2 } , 0 \\right ) \\\\ & = d e g _ { B } ( G , \\Omega \\cap \\mathbb { R } ^ { 2 } , 0 ) \\neq 0 . \\end{align*}"}
-{"id": "7631.png", "formula": "\\begin{align*} c _ r : = \\sum _ { i \\in \\mathcal { N } } \\sum _ { j \\in \\mathcal { N } } c _ { i j } ^ { } \\bar { x } _ { r i j } , \\end{align*}"}
-{"id": "6122.png", "formula": "\\begin{align*} \\begin{pmatrix} s ^ 3 ( \\pi + a ' s ) & s ^ 3 b ' \\\\ s ^ 3 \\overline { b ' } & h ' \\end{pmatrix} \\end{align*}"}
-{"id": "1416.png", "formula": "\\begin{align*} u \\cdot x _ { \\alpha } = \\sum _ { \\gamma \\in [ \\beta ] } c _ { \\gamma , u } ^ { \\alpha } x _ { \\gamma } \\end{align*}"}
-{"id": "6259.png", "formula": "\\begin{align*} \\| \\eta ' \\| ( r , s ) = \\left [ 1 + 2 r \\langle \\gamma ' ( s ) , J ' ( s ) \\rangle + r ^ 2 \\left ( \\| J ' ( s ) \\| ^ 2 + \\langle \\nabla _ r \\nabla _ r \\frac { \\dd \\eta } { \\dd s } \\big | _ { r = 0 } , e _ n \\rangle \\right ) + O ( r ^ 3 ) \\right ] ^ { 1 / 2 } . \\end{align*}"}
-{"id": "3331.png", "formula": "\\begin{align*} \\frac { c _ { k + 1 } } { c _ { k } } \\Gamma _ { - } ^ { ( k + 1 ) } \\Gamma _ { 0 } ^ { ( k + 1 ) * } = \\frac { c _ { k } } { c _ { k - 1 } } ( t \\Gamma _ { 0 } ^ { ( k ) * } \\Gamma _ { - } ^ { ( k ) } + t ^ { \\prime } \\Gamma _ { + } ^ { ( k ) * } \\Gamma _ { 0 } ^ { ( k ) } ) . \\end{align*}"}
-{"id": "4116.png", "formula": "\\begin{align*} C \\ : : \\ : ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q - z ^ { p + 2 q } x ^ { - p } = 0 . \\end{align*}"}
-{"id": "930.png", "formula": "\\begin{align*} \\| \\widehat { V } \\| _ { S _ { \\widehat { p } _ { 2 } } } = \\min _ { V \\in \\mathbb { R } ^ { d \\times d } , W \\in \\mathbb { R } ^ { n \\times d } : \\widehat { V } = ( V W ^ { T } ) ^ { T } } \\| V \\| _ { S _ { p _ { 2 } } } \\| W \\| _ { S _ { p _ { 3 } } } . \\end{align*}"}
-{"id": "4112.png", "formula": "\\begin{gather*} \\pi ^ * _ { P _ 3 } C \\ : : \\ : t ^ q ( b t + c ) ^ r - z ^ p = 0 , \\\\ \\pi ^ * _ { P _ 3 } \\omega = - p t ( b t + c ) d z + z ( b t ( q + r ) + q c ) d t . \\end{gather*}"}
-{"id": "1833.png", "formula": "\\begin{align*} \\begin{pmatrix} s ^ 3 ( \\pi + a ' s ) & s ^ 3 b ' \\\\ s ^ 3 \\overline { b ' } & h ' \\end{pmatrix} \\end{align*}"}
-{"id": "1238.png", "formula": "\\begin{align*} m _ 1 = \\min \\{ j \\in \\mathbb { N } : W ( j ) > 2 ^ { k _ 1 - 1 } \\} . \\end{align*}"}
-{"id": "7025.png", "formula": "\\begin{align*} \\mathbb { E } \\ , ( \\omega ^ { 2 ^ { k + 1 } - 1 } ) ^ \\lambda \\pi ( g ( \\omega ^ { 2 ^ k } ) \\cdots g ( \\omega ) ) = \\mathbb { E } \\ , \\omega ^ { \\lambda 2 ^ k } \\pi ( g ( \\omega ^ { 2 ^ k } ) ) \\cdots \\omega ^ \\lambda \\pi ( g ( \\omega ) ) \\rightarrow 0 \\end{align*}"}
-{"id": "777.png", "formula": "\\begin{align*} \\lim _ { \\begin{subarray} { c } M \\to \\infty \\\\ M ' \\to - \\infty \\end{subarray} } \\prod _ { i = 1 } ^ { k } \\frac { z _ { i } ^ { M ' - 1 } } { ( 1 + z _ { i } ) ^ { M } } \\langle \\prod _ { 1 \\le i \\le k } ^ { \\curvearrowright } C _ { \\mu _ { i } } ^ { [ M ' , M ] } ( z _ { i } ) \\prod _ { 1 \\le i \\le k } \\beta _ { \\nu _ { i } , x _ { i } } ^ { * } \\rangle _ { [ M ' , M ] } . \\end{align*}"}
-{"id": "2645.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { h ( m , n ) } C _ { m } ^ { l } ( G ^ { 0 } ) ( \\lambda ) ( C _ { m } ^ { l } ( G ^ { 0 } ) ( \\lambda ) ) ^ { * } = \\alpha _ { m } ^ { 2 } d _ { m } ^ { 0 } ( \\lambda ) ^ { \\top } B _ { 1 } \\overline { d _ { m } ^ { 0 } ( \\lambda ) } , \\end{align*}"}
-{"id": "8016.png", "formula": "\\begin{align*} \\deg y _ j = - \\deg x _ j = e _ j , \\end{align*}"}
-{"id": "6303.png", "formula": "\\begin{align*} [ M _ { p _ 1 , q _ 1 } ^ { s _ 1 , \\alpha } , M _ { p _ 2 , q _ 2 } ^ { s _ 2 , \\alpha } ] _ { \\theta } = M _ { p _ { \\theta } , q _ { \\theta } } ^ { s _ { \\theta } , \\alpha } \\end{align*}"}
-{"id": "8937.png", "formula": "\\begin{align*} \\varphi ( s ) = ( n \\omega _ n ) ^ { - p / ( p - 1 ) } \\int _ s ^ { + \\infty } ( \\sinh F ( t ) ) ^ { - p ( n - 1 ) / ( p - 1 ) } d t \\end{align*}"}
-{"id": "5396.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } & \\leq \\left ( ( 1 + x ) ^ 2 + ( k - 2 ) + ( 1 - \\alpha - x ) ^ 2 \\right ) \\frac { n ^ 2 } { 2 } \\\\ & = \\left ( k - 2 \\alpha + \\alpha ^ 2 + 2 \\alpha x + 2 x ^ 2 \\right ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "7587.png", "formula": "\\begin{align*} \\Omega ^ { ( 1 ) } _ \\varepsilon : = \\left \\{ \\omega \\in \\Omega : \\chi _ 1 ( \\omega ) \\ge - 1 + \\frac { \\varepsilon } { C } \\right \\} . \\end{align*}"}
-{"id": "9103.png", "formula": "\\begin{align*} x ^ { i _ k - j _ k } + y ^ { i _ k - j _ k } = ( x + y ) ^ { i _ k - j _ k } . \\end{align*}"}
-{"id": "1555.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ 2 ^ 2 ( \\nu ^ i , \\mu ^ k ) & = 2 \\sum _ { i = 1 } ^ n \\underset { f \\in Z } { \\sup } \\ \\left \\{ \\int _ { \\Omega } f d \\mu ^ k + \\int _ { \\Omega } S f ( x ) d \\nu ^ i ( x ) \\right \\} \\le M , \\end{align*}"}
-{"id": "1967.png", "formula": "\\begin{align*} x _ j x _ i = q x _ i x _ j , \\forall \\ ; 1 \\leq i < j \\leq n . \\end{align*}"}
-{"id": "1025.png", "formula": "\\begin{align*} P ( P ( x , y ) , z ) + P ( P ( y , z ) , x ) + P ( P ( z , x ) , y ) ~ = ~ 0 . \\end{align*}"}
-{"id": "1445.png", "formula": "\\begin{align*} b _ { j } = \\sum _ { i = 1 } ^ { j } h _ { i } \\end{align*}"}
-{"id": "3345.png", "formula": "\\begin{align*} \\mathtt { g } \\cdot \\mathtt { h } & = g _ 1 ^ { s _ 1 } \\cdots g _ k ^ { s _ k } h _ 1 ^ { t _ 1 } \\cdots h _ l ^ { t _ l } , \\\\ \\bar { \\mathtt { g } } & = g _ k ^ { - s _ k } \\cdots g _ 1 ^ { - s _ 1 } , \\\\ \\mathtt { g } ^ { ( \\lambda ) } & = g _ 1 ^ { \\lambda s _ 1 } \\cdots g _ k ^ { \\lambda s _ k } . \\\\ \\end{align*}"}
-{"id": "4224.png", "formula": "\\begin{align*} E _ m ( x ) = \\sum _ { n = 0 } ^ m C h _ { n , \\lambda } ( x ) S _ 2 ( m , n ) \\lambda ^ { - m } , \\quad ( m \\geq 0 ) . \\end{align*}"}
-{"id": "9798.png", "formula": "\\begin{align*} Z _ { i } = X _ { i j } \\boldsymbol { \\beta } + r ( X _ { i j } ) + \\varepsilon _ { i } , \\end{align*}"}
-{"id": "6252.png", "formula": "\\begin{align*} \\frac { d \\lambda _ 1 ( \\tilde { L } _ c ) } { d c } \\biggr | _ { c = 1 } & = \\frac { d } { d c } ( c ^ 2 \\bar { \\lambda } _ 1 ( n , c D , K ) ) \\biggr | _ { c = 1 } \\\\ & = \\frac { 1 } { D } \\frac { d } { d D } ( D ^ 2 \\bar { \\lambda } _ 1 ( n , D , K ) ) . \\end{align*}"}
-{"id": "6228.png", "formula": "\\begin{align*} \\mathcal { K } _ { o l d } P _ { o l d } = & \\ \\mathcal { K } _ { o l d } ( I + ( s ^ { 2 } _ { n e w } - s ^ { 2 } _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } M + ( s _ { n e w } - s _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } D ) \\\\ & \\ \\cdot ( I + ( s ^ { 2 } _ { n e w } - s ^ { 2 } _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } M + ( s _ { n e w } - s _ { o l d } ) \\mathcal { K } _ { o l d } ^ { - 1 } D ) ^ { - 1 } P _ { o l d } \\\\ = & \\ \\mathcal { K } _ { n e w } P _ { n e w } , \\end{align*}"}
-{"id": "6837.png", "formula": "\\begin{align*} e ^ { i t \\Delta } m ( x ) e ^ { - i t \\Delta } = M ( t ) m ( 2 i t \\nabla ) M ( - t ) \\end{align*}"}
-{"id": "9980.png", "formula": "\\begin{align*} \\Lambda ^ * = \\max \\ ; \\lim _ { N \\rightarrow \\infty } \\mathbb { E } _ { \\mathbf { H } } \\ ! \\left [ \\ ! \\frac { 1 } { N } \\sum _ { t = 1 } ^ N g ( s _ t , a _ t ( s _ t ) ) \\ ! \\right ] = \\max \\ ; \\lim _ { N \\rightarrow \\infty } \\mathbb { E } _ { \\mathbf { H } } \\ ! \\left [ \\ ! \\frac { 1 } { N } \\sum _ { t = 1 } ^ N \\bar { g } ( s _ t , a _ t ( s _ t ) ) \\ ! \\right ] \\end{align*}"}
-{"id": "8520.png", "formula": "\\begin{align*} w _ 0 = ( s _ 1 s _ 2 \\dots s _ { n - 1 } s _ n s _ { n - 1 } \\cdots s _ 2 s _ 1 ) ( s _ 2 \\cdots s _ { n - 1 } s _ n s _ { n - 1 } \\cdots s _ 2 ) \\cdots ( s _ { n - 1 } s _ n s _ { n - 1 } ) s _ n \\end{align*}"}
-{"id": "4819.png", "formula": "\\begin{align*} m a c ( P ( T \\otimes I _ n ) P ) + m \\ m a c ( T ) = m a c ( Q ( T \\otimes I _ n ) Q ) + m \\ m a c ( T ) \\end{align*}"}
-{"id": "6367.png", "formula": "\\begin{align*} t ^ { 2 s + 2 l } = \\sum _ { j = 0 } ^ { l } c _ j \\bar { w } _ j ( t ) . \\end{align*}"}
-{"id": "2418.png", "formula": "\\begin{align*} \\widehat { \\Delta _ j f } & = \\left ( \\varphi ( 2 ^ { - j } \\xi ) - \\varphi ( 2 ^ { - j + 1 } \\xi ) \\right ) \\widehat { f } ( \\xi ) , \\ j \\geq 0 , \\\\ \\widehat { \\Delta _ 0 f } & = \\varphi ( \\xi ) f ( \\xi ) . \\end{align*}"}
-{"id": "7601.png", "formula": "\\begin{align*} { \\mathbb P } \\left \\{ x _ { K _ 1 } \\geq a \\right \\} = { \\mathbb P } \\left \\{ x _ { i } \\geq u _ l \\right \\} \\geq { \\mathbb P } \\left \\{ \\cap _ { j = 1 } ^ { i } \\Omega _ j \\right \\} = \\prod _ { i = 1 } ^ { i } \\lambda _ j \\ge \\prod _ { i = 1 } ^ { K _ 1 } \\lambda _ j . \\end{align*}"}
-{"id": "403.png", "formula": "\\begin{align*} \\Theta _ { \\psi } ^ { ( 2 ) } ( w ; \\phi ) & \\sim \\frac { 2 \\psi ( 1 ) } { \\xi ( 2 ) ( w - 1 ) } \\int _ 0 ^ \\infty \\int _ { - \\infty } ^ \\infty \\sum _ { \\ell , m = 1 } ^ { \\infty } \\rho _ \\phi ( \\ell m ) \\\\ & \\times K _ { s - \\frac { 1 } { 2 } } ( 2 \\pi \\ell m t ^ 2 ) f ( 0 , 0 , t ^ { - 1 } m , u ) \\cos ( 2 \\pi \\ell t ^ 3 u ) d u t d t . \\end{align*}"}
-{"id": "1729.png", "formula": "\\begin{align*} \\hat { \\Omega } _ { k j } { } ^ k { } _ { l } = \\hat { R } _ { k j } { } ^ k { } _ { l } + 6 \\varepsilon \\hat { g } _ { j l } . \\end{align*}"}
-{"id": "4598.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c c } x & 0 \\\\ 0 & y \\end{array} \\right ] = \\left [ \\begin{array} { c c } 0 & x \\\\ - x & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & - x ^ { - 1 } y \\\\ 1 & 0 \\end{array} \\right ] \\end{align*}"}
-{"id": "3296.png", "formula": "\\begin{align*} F \\psi ( v _ 1 ) = - [ 2 ] ^ { 1 / 2 } \\alpha w _ 0 , F \\psi ( v _ 0 ) = - [ 2 ] ^ { 1 / 2 } q ^ 2 \\beta w _ 1 , F \\psi ( v _ { - 1 } ) = 0 . \\end{align*}"}
-{"id": "8491.png", "formula": "\\begin{align*} A _ { 0 } ( \\textbf { u } ^ { \\varepsilon } ) = d i a g \\Bigg ( \\frac { f ( \\textbf { u } ^ { \\varepsilon } ) } { { \\rho } ^ { \\varepsilon } } , 1 , 1 , \\cdots , 1 \\Bigg ) . \\end{align*}"}
-{"id": "2268.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\sum \\limits ^ k _ { i = 0 } \\delta _ i u _ { n - i } + A ( t _ n ) u _ n = \\sum \\limits ^ { k - 1 } _ { i = 0 } \\gamma _ i B ( t _ { n - i - 1 } , u _ { n - i - 1 } ) , n = k , \\dotsc , N . \\end{align*}"}
-{"id": "6531.png", "formula": "\\begin{align*} E _ \\ell : = \\big \\| ( v _ n ) _ { n = k } ^ \\ell \\big \\| _ { L ^ p ( D ) } ^ p = \\tau \\sum _ { n = k } ^ \\ell \\| v _ n \\| _ { D } ^ p , \\ell = k , \\dotsc , N , \\end{align*}"}
-{"id": "9933.png", "formula": "\\begin{align*} x _ 0 ^ 2 - b ^ 2 x _ 1 ^ 2 = ( x _ 0 - b x _ 1 ) ( x _ 0 + b x _ 1 ) = K K ' = 0 . \\end{align*}"}
-{"id": "8547.png", "formula": "\\begin{align*} \\pi _ { e _ { k } a } ( r ^ { - 1 } m ) & = \\displaystyle \\sum _ { s \\in L ( k ) , a _ { 1 } \\in _ { k } T } \\pi _ { e _ { k } a } r ^ { - 1 } \\xi _ { s a _ { 1 } } \\pi _ { s a _ { 1 } } ( m ) \\\\ & = \\displaystyle \\sum _ { s , u \\in L ( k ) , a _ { 1 } \\in _ { k } T } \\pi _ { e _ { k } a } \\xi _ { u a _ { 1 } } \\left ( u ^ { \\ast } ( r ^ { - 1 } s ) \\pi _ { s a _ { 1 } } ( m ) \\right ) \\\\ & = \\displaystyle \\sum _ { s \\in L ( k ) } e _ { k } ^ { \\ast } ( r ^ { - 1 } s ) \\pi _ { s a } ( m ) \\\\ & = \\pi _ { r a } ( m ) \\end{align*}"}
-{"id": "5301.png", "formula": "\\begin{align*} a : = \\limsup _ { d \\to \\infty } \\frac { \\sum _ { j = 1 } ^ d \\gamma _ j } { \\log ( d + 1 ) } < \\infty . \\end{align*}"}
-{"id": "6652.png", "formula": "\\begin{align*} z _ j & = \\omega _ { s _ j } ( \\Phi _ K ) \\cdot \\mu _ { \\tau _ j } \\\\ & = q _ K ^ { - s _ j } \\cdot \\mu _ { \\tau _ j } \\end{align*}"}
-{"id": "1133.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { D L } [ \\iota ] = \\frac { M K \\rho ^ 2 \\beta _ { k } ^ 4 } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } ^ 2 } \\end{align*}"}
-{"id": "1570.png", "formula": "\\begin{align*} \\mathcal { L } _ { X ^ { \\left [ 2 \\right ] } } \\Theta = \\lambda \\Theta ~ , ~ { m o d } \\Theta = 0 , \\end{align*}"}
-{"id": "8618.png", "formula": "\\begin{align*} \\pi _ { b e _ { k } } \\beta ( - c _ { k } ^ { - 1 } i _ { 2 } \\sigma _ { 1 } \\overline { \\pi } _ { 3 } ( n _ { 2 } ) - c _ { k } ^ { - 1 } \\sigma _ { 3 } \\overline { \\pi } _ { 4 } ( n _ { 3 } ) ) & = \\pi _ { b e _ { k } } \\beta ( - c _ { k } ^ { - 1 } n _ { 2 } - c _ { k } ^ { - 1 } n _ { 3 } ) \\\\ & = - c _ { k } ^ { - 1 } b n _ { 2 } - c _ { k } ^ { - 1 } b n _ { 3 } \\end{align*}"}
-{"id": "3455.png", "formula": "\\begin{align*} \\tilde y ( t ) = \\Gamma ^ { * } ( s + \\Delta s ) \\phi ( s + \\Delta s , \\xi _ { s , x } ( s + \\Delta s ) ) + \\int _ t ^ { s + \\Delta s } f ( \\theta , \\xi _ { s , x } ( \\theta ) , \\tilde y ( \\theta ) , \\tilde z ( \\theta ) ) d \\theta - \\end{align*}"}
-{"id": "1125.png", "formula": "\\begin{align*} \\gamma _ k ^ \\mathrm { D L } [ \\iota ] = \\frac { M \\sigma _ { \\mathrm { d } k } ^ 4 [ \\iota ] } { ( \\beta _ { \\mathrm { d } k } + 1 / \\rho _ \\mathrm { d } ) \\sum _ { i = 1 } ^ { K } \\sigma _ { \\mathrm { d } i } ^ 2 [ \\iota ] } . \\end{align*}"}
-{"id": "9721.png", "formula": "\\begin{align*} x _ { U , \\epsilon } ( t ) = g ^ { - 1 } \\left ( x _ 2 ( \\epsilon ) e ^ { - c _ 2 \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } \\right ) , t \\geq T _ 4 ( \\epsilon ) . \\end{align*}"}
-{"id": "3821.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ n e _ i ( \\boldsymbol { x } ) ( t _ n x _ n ) ^ { n - i } s _ { \\lambda - \\rho } ( \\boldsymbol { x } ) = \\sum _ { i = 0 } ^ n ( t _ n x _ n ) ^ { n - i } \\sum _ { \\mu \\in ( \\lambda - \\rho ) \\otimes 1 ^ i } s _ { \\mu } ( \\boldsymbol { x } ) , \\end{align*}"}
-{"id": "1901.png", "formula": "\\begin{align*} \\hat { \\phi } _ X ( u ) = e ^ { \\frac { 1 } { K } \\hat { \\psi } ( u ) } = e ^ { \\frac { 1 } { K } \\Re \\hat { \\psi } ( u ) } e ^ { \\frac { 1 } { K } i \\Im \\hat { \\psi } ( u ) } = | \\hat { \\phi } ( u ) | ^ { \\frac { 1 } { K } } e ^ { \\frac { 1 } { K } i \\Im \\hat { \\psi } ( u ) } , \\end{align*}"}
-{"id": "5386.png", "formula": "\\begin{align*} R _ 2 ( P _ n ) = \\left \\lfloor \\frac { 3 n - 2 } { 2 } \\right \\rfloor . \\end{align*}"}
-{"id": "1525.png", "formula": "\\begin{align*} \\nu ( v ) = \\nu ( - v ) \\mbox { f o r a l l } v \\in \\R ^ d . \\end{align*}"}
-{"id": "8116.png", "formula": "\\begin{align*} s _ 1 = 1 \\ \\mbox { a n d } \\ s _ 2 = - 2 l . \\end{align*}"}
-{"id": "2514.png", "formula": "\\begin{align*} F ( x ( t ; \\overline { x } ) ) = \\exp ( - 2 \\lambda t ) \\cdot F ( \\overline { x } ) , ~ ( \\forall ) t \\in I _ { \\overline { x } } . \\end{align*}"}
-{"id": "3782.png", "formula": "\\begin{align*} { \\left [ { { { \\bf { \\Lambda } } _ { { { \\mathrm { \\bf { B } } } } } } } \\right ] _ { \\ell _ { j } \\ell _ { j } } } = \\left [ { { { \\bf { \\Lambda } } _ { s } } } \\right ] _ { i i } \\end{align*}"}
-{"id": "9917.png", "formula": "\\begin{align*} M _ \\ell & = M _ { p , p ' } = \\frac { S } { S X + S Y } , \\\\ M _ p & = \\frac { S } { S K ' + S X + S Y } , \\\\ M _ { p ' } & = \\frac { S } { S K + S X + S Y } . \\end{align*}"}
-{"id": "591.png", "formula": "\\begin{align*} L _ 1 ^ + ( \\Lambda _ R ( \\chi ^ N ) ) = & X _ 1 ^ + ( I _ R ^ N ) - C _ 1 ^ + ( \\Lambda _ R ( \\chi ^ N ) ) + \\delta \\Delta _ 1 ( \\Lambda _ R ( \\chi ^ N ) ) , \\\\ S _ 1 ( \\Lambda _ R ( \\chi ^ N ) ) + L _ 1 ^ - ( \\Lambda _ R ( \\chi ^ N ) ) = & X _ 1 ^ - ( I _ R ^ N ) + C _ 1 ^ - ( \\Lambda _ R ( \\chi ^ N ) ) + \\delta \\Delta _ 1 ( \\Lambda _ R ( \\chi ^ N ) ) . \\end{align*}"}
-{"id": "9480.png", "formula": "\\begin{align*} G ( \\rho , \\phi ) = \\begin{cases} F _ { 1 / 4 } ( \\rho , \\phi ) & \\rho > - 3 / 4 \\\\ - F _ { 1 / 4 } ( - \\rho - 1 , \\phi ) - 1 & \\rho < - 1 / 4 \\end{cases} \\end{align*}"}
-{"id": "5362.png", "formula": "\\begin{align*} \\beta _ i ( t ) = \\prod _ { \\substack { 0 \\leq k \\leq n \\\\ p ( n - k ) - p ( n - k - 1 ) \\geq i } } ( t - ( n - k ) ) \\end{align*}"}
-{"id": "3879.png", "formula": "\\begin{align*} \\varepsilon _ 3 ( x ) = \\langle \\tau ( x ) , \\nabla _ { \\partial \\Omega } ( \\tau - \\nu ) ( x ) , x - a \\rangle . \\end{align*}"}
-{"id": "399.png", "formula": "\\begin{align*} F _ \\phi ( u ; z ) = \\sum _ { n \\geq 1 } \\frac { \\eta _ { \\frac { z } { 2 } } ( n ) \\rho _ \\phi ( n ) } { n ^ u } , \\end{align*}"}
-{"id": "7265.png", "formula": "\\begin{align*} \\delta ( \\{ ( U _ j , F _ { j , \\alpha } ^ \\lambda ) \\} ) = \\{ ( U _ { j k } , h _ { 1 , j k , \\alpha } ^ \\lambda - h _ { 2 , j k , \\alpha } ^ \\lambda ) \\} \\end{align*}"}
-{"id": "5876.png", "formula": "\\begin{align*} Z ^ { 1 } = K ^ { 1 } ~ , ~ Z ^ { 2 } = t K ^ { 1 } - \\left ( \\int \\frac { d x } { \\sigma \\left ( x \\right ) } F - t \\right ) F \\partial _ { F } \\end{align*}"}
-{"id": "8206.png", "formula": "\\begin{align*} | f | = | a _ n | & = 1 ; \\\\ | \\alpha _ i - \\alpha _ j | & = 1 & & ( 1 \\le i < j \\le s ) ; \\\\ \\left | \\frac { f ^ { ( e _ i ) } ( \\alpha _ i ) } { e _ i ! } \\right | & = 1 & & ( 1 \\le i \\le s ) . \\end{align*}"}
-{"id": "8121.png", "formula": "\\begin{align*} v ( y ) : = u ( x ) \\mbox { w h e r e $ y : = x | x | ^ { \\frac { k } { N - 1 } } $ , ( $ x \\in \\mathbb { R } ^ N $ ) . } \\end{align*}"}
-{"id": "9994.png", "formula": "\\begin{align*} \\alpha ' \\tilde { p } ' = & { \\gamma \\alpha ^ { ( 1 ) } } \\tilde { p } ^ { ( 1 ) } + { ( 1 - \\gamma ) \\alpha ^ { ( 2 ) } } \\tilde { p } ^ { ( 2 ) } \\\\ { } \\le & \\gamma \\left ( \\frac { B _ k } { T _ f } + \\alpha ^ { ( 1 ) } E _ k \\right ) + ( 1 - \\gamma ) \\left ( \\frac { B _ k } { T _ f } + \\alpha ^ { ( 2 ) } E _ k \\right ) = \\frac { B _ k } { T _ f } + \\alpha ' E _ k , \\end{align*}"}
-{"id": "5078.png", "formula": "\\begin{align*} t ( \\vec { \\nu } ) = \\# \\{ ( i , j ) \\ , | \\ , 1 \\le i < j \\le k \\ , \\ , \\mathrm { a n d } \\ , \\ , \\nu _ { i } > \\nu _ { j } \\} . \\end{align*}"}
-{"id": "6101.png", "formula": "\\begin{align*} \\phi ^ * t = z w , \\ \\phi ^ * \\tau _ j ' = \\tau _ j , \\ \\phi ^ * z _ r ' = z _ r , \\end{align*}"}
-{"id": "4704.png", "formula": "\\begin{align*} \\phi _ k ( c ' ) & = ( y , z ' ) \\\\ \\phi _ k ( c '' ) & = ( y , z '' ) \\\\ \\mu _ k ( z ) & = \\mu _ k ( z ' ) = \\mu _ k ( z '' ) . \\end{align*}"}
-{"id": "522.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 1 1 } ( i , u ; j , v ) = \\iint \\frac { ( z - w ) h ^ { \\rm g e o } _ { 1 1 } ( z , w ) } { ( z ^ 2 - 1 ) ( w ^ 2 - 1 ) ( z w - 1 ) } \\frac { z - c } { z } \\frac { w - c } { w } \\frac { \\dd z } { z ^ u } \\frac { \\dd w } { w ^ v } , \\end{align*}"}
-{"id": "8327.png", "formula": "\\begin{gather*} w ^ p - w = u = \\sum _ { i = 1 } ^ r \\frac { Q _ i } { P _ i ^ { e _ i } } + f ( T ) = \\frac { Q } { P _ 1 ^ { e _ 1 } \\cdots P _ r ^ { e _ r } } + f ( T ) , \\end{gather*}"}
-{"id": "7614.png", "formula": "\\begin{align*} L : = \\{ \\mathbf { v } _ z \\in V ( G ) : \\mathbf { v } _ z > \\epsilon \\} \\end{align*}"}
-{"id": "6234.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + K ) X ^ { ( j ) } ( s _ { i } ) = M \\tilde { V } _ { j } + \\eta _ { j i } \\ i = 1 , \\ldots , l . \\end{align*}"}
-{"id": "776.png", "formula": "\\begin{gather*} \\varphi ( c ) = \\sum _ { } \\alpha _ j r _ { x ^ j _ 1 \\dots x ^ j _ \\lambda 1 \\dots 1 } + m ' , \\end{gather*}"}
-{"id": "2261.png", "formula": "\\begin{align*} p ( x ) = \\frac { C ( \\vec { \\varepsilon } ) } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } e ^ { \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } x ^ { k } } , \\end{align*}"}
-{"id": "9230.png", "formula": "\\begin{align*} I _ 2 = \\mathbb { E } [ \\int _ D \\frac { \\partial k } { \\partial y } ( x , Y ( T , x , z ) , z ) \\chi ( T , x , z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] d x ] = \\mathbb { E } [ \\int _ D p ( T , x , z ) \\chi ( T , x , z ) d x ] . \\end{align*}"}
-{"id": "3158.png", "formula": "\\begin{align*} \\frac { 1 } { J _ n } \\sum _ { j = 1 } ^ { J _ n } \\sum _ { x ^ n \\in \\mathsf { A } ^ n } E ( x ^ n \\mid j ) \\mathrm { t r } \\left ( \\overline { W } _ { s '' } ^ { \\otimes n } \\left ( \\rho _ { x ^ n } \\right ) D _ { j } ^ n \\right ) \\geq 1 - ( \\frac { 3 } { \\tau _ n } ) ^ { 2 { d ' } ^ 4 } 2 ^ { - n ^ { 1 / 1 6 } \\beta } \\geq 1 - 2 ^ { - \\frac { 1 } { 2 } n ^ { 1 / 1 6 } \\beta } \\end{align*}"}
-{"id": "4124.png", "formula": "\\begin{align*} \\omega '' = y p d s + 2 ( p + q ) s d y , C '' \\ : : \\ : y ^ { 2 ( p + q ) } - s ^ { - p } = 0 , \\end{align*}"}
-{"id": "7242.png", "formula": "\\begin{align*} \\left [ \\delta \\left \\{ \\left ( U _ j , \\sum _ { \\lambda = 1 } ^ r \\sum _ { | \\beta | = n + 1 } a _ { j , \\beta } ^ \\lambda \\cdot e _ { j , \\beta } ^ * \\otimes e _ j ^ \\beta \\right ) \\right \\} \\right ] = u _ n ( Y , X ; \\{ w _ j \\} ) - u _ n ( Y , X ; \\{ \\widehat { w } _ j \\} ) \\end{align*}"}
-{"id": "2646.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { h ( m , n ) } C _ { m } ^ { l } ( F ^ { 0 } ) ( \\lambda ) ( C _ { m } ^ { l } ( F ^ { 0 } ) ( \\lambda ) ) ^ { * } = ( \\beta _ { m } ^ { 2 } + \\gamma _ { m _ { 1 } } ( \\lambda ) + \\gamma _ { m _ { 2 } } ( \\lambda ) ) d _ { m } ^ { 0 } ( \\lambda ) ^ { \\top } B _ { 2 } \\overline { d _ { m } ^ { 0 } ( \\lambda ) } ; \\end{align*}"}
-{"id": "6921.png", "formula": "\\begin{align*} \\langle \\ , f / g \\ , \\ , | \\ , h \\rangle = \\langle f \\ , | \\ , g \\ ! \\cdot \\ ! h \\ , \\rangle \\ , , \\end{align*}"}
-{"id": "5486.png", "formula": "\\begin{align*} & \\sup \\limits _ { g ' \\in \\mathcal { G } _ { f ^ { * } } } \\left ( \\sum \\limits _ { i = 1 } ^ { n } \\xi _ { i } g ' ( X _ { i } ) - \\frac { h } { 3 } g ' ( X _ i ) \\right ) \\\\ & \\le \\gamma \\left ( 1 + \\frac { h } { 1 2 } \\right ) \\ ! + \\ ! \\sup \\limits _ { g ' \\in \\mathcal { G } _ { f ^ { * } } } \\left ( \\sum \\limits _ { i = 1 } ^ { n } \\xi _ { i } p ( g ' ( X _ { i } ) ) - \\frac { h } { 1 2 } p ( g ' ( X _ i ) ) \\right ) . \\end{align*}"}
-{"id": "8102.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { W ^ { 1 , p } ( \\Omega , d \\mu _ { l } ) } : = \\left \\Vert u \\right \\Vert _ { L ^ { p } ( \\Omega , d \\mu _ { l } ) } + \\left \\Vert \\nabla u \\right \\Vert _ { L ^ { p } ( \\Omega , d \\mu _ { l } ) } . \\end{align*}"}
-{"id": "6512.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\delta ( \\zeta ) & { } = \\sum _ { \\ell = 1 } ^ k \\frac 1 \\ell ( 1 - \\zeta ) ^ \\ell = \\sum \\limits ^ k _ { i = 0 } \\delta _ i \\zeta ^ { i } , \\beta ( \\zeta ) = 1 , \\\\ \\gamma ( \\zeta ) & { } = \\frac 1 \\zeta \\big [ 1 - ( 1 - \\zeta ) ^ k \\big ] = \\sum _ { i = 0 } ^ { k - 1 } \\gamma _ i \\zeta ^ i . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5981.png", "formula": "\\begin{align*} s _ i x = s _ { \\alpha _ i } x = x - \\langle x , \\alpha ^ \\lor _ i \\rangle \\alpha _ i , & \\ \\ \\ x \\in \\mathfrak { h } ^ * , \\\\ s _ i y = s _ { \\alpha ^ \\lor _ i } y = y - \\langle \\alpha _ i , y \\rangle \\alpha ^ \\lor _ i , & \\ \\ \\ y \\in \\mathfrak { h } ; \\end{align*}"}
-{"id": "5756.png", "formula": "\\begin{align*} Q _ { L } ^ { B L P } = n ( n + 2 ) \\sum _ { l = 1 } ^ { L } ( n - l ) ^ { - 1 } r ( l ) ^ 2 . \\end{align*}"}
-{"id": "5871.png", "formula": "\\begin{align*} Z ^ { 3 } = e ^ { 2 m t } \\left ( \\partial _ { t } + H \\right ) \\end{align*}"}
-{"id": "9160.png", "formula": "\\begin{align*} A ( a _ 1 , \\dots , a _ n ) _ { i , j } = \\begin{cases} 1 & i = 1 \\\\ a _ j ^ { i } & i \\geq 2 \\end{cases} , \\end{align*}"}
-{"id": "9024.png", "formula": "\\begin{align*} \\psi ( \\tau , y ) = p ( \\tau ) + ( y ^ { c _ 1 } , \\ldots , y ^ { c _ { n - 1 } } ) , \\end{align*}"}
-{"id": "8010.png", "formula": "\\begin{align*} \\deg x _ j = \\deg y _ j = \\chi _ j . \\end{align*}"}
-{"id": "5753.png", "formula": "\\begin{align*} \\hat { G } _ n ^ \\mathrm { T } = \\sum _ { t = 1 } ^ { n } \\sigma ^ 2 _ t ( \\hat { \\delta } _ { 0 } ) \\begin{bmatrix} A _ { \\hat { \\phi } \\hat { \\phi } , t } & A _ { \\hat { \\phi } \\hat { \\theta } , t } \\\\ A _ { \\hat { \\theta } \\hat { \\phi } , t } & A _ { \\hat { \\theta } \\hat { \\theta } , t } \\end{bmatrix} . \\end{align*}"}
-{"id": "5143.png", "formula": "\\begin{align*} \\tilde { A } _ { a } ( z ) = q ^ { 2 \\sum _ { p = a } ^ { r } N _ { p , 1 } } A ^ { [ 2 , M ] } ( z ) . \\end{align*}"}
-{"id": "7375.png", "formula": "\\begin{align*} z ^ t & = y - A \\beta ^ t + \\frac { z ^ { t - 1 } } { n } \\sum _ { i = 1 } ^ N \\eta _ { t - 1 } ^ { \\prime } ( [ A ^ * z ^ { t - 1 } + \\beta ^ { t - 1 } ] _ i ) , \\\\ \\beta ^ { t + 1 } & = \\eta _ t ( A ^ * z ^ t + \\beta ^ t ) , \\end{align*}"}
-{"id": "9028.png", "formula": "\\begin{align*} \\sup _ { z \\in \\mathbb { U } ' } \\Re ( \\log X _ n ( z ) ) - \\sup _ { z \\in \\mathbb { U } } \\Re ( \\log \\Phi ^ * _ { n - 1 } ( z ) ) & = \\sup _ { z \\in \\mathbb { U } ' } \\Re ( \\log X _ n ( z ) ) - \\Re \\log \\Phi ^ * _ { n - 1 } ( z _ 0 ) \\\\ & \\geq \\Re \\log X _ n ( z _ 0 ) - \\Re \\log \\Phi ^ * _ { n - 1 } ( z _ 0 ) = \\Re \\log ( 1 - U ) . \\end{align*}"}
-{"id": "7198.png", "formula": "\\begin{align*} - \\Delta v + ( u \\cdot \\nabla ) v + \\nabla p = f , \\qquad { \\rm d i v } \\ , v = 0 \\mbox { i n } \\ , \\ , \\R ^ n , \\end{align*}"}
-{"id": "9526.png", "formula": "\\begin{align*} e ^ { - 2 \\phi } \\delta ( H _ 3 \\wedge * H _ 3 ) & = 2 \\delta d B _ 2 \\wedge e ^ { - 2 \\phi } * H _ 3 \\\\ & = 2 d \\left ( \\delta B _ 2 \\wedge e ^ { - 2 \\phi } * H _ 3 \\right ) - 2 \\delta B _ 2 \\wedge d \\left ( e ^ { - 2 \\phi } * H _ 3 \\right ) . \\end{align*}"}
-{"id": "4122.png", "formula": "\\begin{align*} ( y ^ 2 + a x ^ 2 + b x ) ^ q - x ^ { - p } = 0 \\end{align*}"}
-{"id": "3078.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\limsup _ { n \\to \\infty } \\P \\left ( \\rho _ { n , k } ( f ) - \\rho _ { n , n - k } ( f ) > 0 \\right ) = 0 . \\end{align*}"}
-{"id": "6329.png", "formula": "\\begin{align*} \\left \\| \\Box _ k ^ { \\alpha _ 1 } | ~ M _ 1 \\rightarrow M _ 2 \\right \\| \\gtrsim \\frac { \\| \\Box _ k ^ { \\alpha _ 1 } f _ l ^ { \\alpha _ 2 } \\| _ { M _ 2 } } { \\| f _ l ^ { \\alpha _ 2 } \\| _ { M _ 1 } } \\sim 2 ^ { j \\widetilde { A _ 1 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\end{align*}"}
-{"id": "8765.png", "formula": "\\begin{align*} \\check { R } _ { 2 , 3 } \\left ( y / x \\right ) \\check { R } _ { 1 , 2 } \\left ( y / z \\right ) \\check { R } _ { 2 , 3 } \\left ( x / z \\right ) = \\check { R } _ { 1 , 2 } \\left ( x / z \\right ) \\check { R } _ { 2 , 3 } \\left ( y / z \\right ) \\check { R } _ { 1 , 2 } \\left ( y / x \\right ) , \\end{align*}"}
-{"id": "7906.png", "formula": "\\begin{align*} \\mathrm { T r } ( e ^ { - t \\Delta } ) = \\sum _ i e ^ { - t \\lambda _ i } \\sim { ( 4 \\pi t ) ^ { - \\frac { n } { 2 } } } ( a _ 0 + a _ 1 t + a _ 2 t ^ 2 + a _ 3 t ^ 3 + \\cdots ) , \\end{align*}"}
-{"id": "1719.png", "formula": "\\begin{gather*} \\Phi _ { \\upsilon } : = ( \\cos \\upsilon ) \\Phi _ I + ( \\sin \\upsilon ) \\Phi _ J + \\Phi _ K . \\end{gather*}"}
-{"id": "6886.png", "formula": "\\begin{align*} \\limsup \\lambda _ i ( \\alpha _ 0 ) = \\lambda _ 0 . \\end{align*}"}
-{"id": "6834.png", "formula": "\\begin{align*} & \\widehat { P _ { \\le N } f } ( \\xi ) : = \\widehat { f _ { \\le N } } ( \\xi ) : = \\varphi ( \\tfrac { \\xi } { N } ) \\widehat { f } ( \\xi ) , \\widehat { P _ { > N } f } ( \\xi ) : = \\widehat { f _ { > N } } ( \\xi ) : = \\bigl ( 1 - \\varphi ( \\tfrac { \\xi } { N } ) \\bigr ) \\widehat { f } ( \\xi ) , \\\\ & \\widehat { P _ { N } f } ( \\xi ) : = \\widehat { f _ { N } } ( \\xi ) : = \\bigl ( \\varphi ( \\tfrac { \\xi } { N } ) - \\varphi ( \\tfrac { 2 \\xi } { N } ) \\bigr ) \\widehat { f } ( \\xi ) . \\end{align*}"}
-{"id": "7883.png", "formula": "\\begin{align*} \\nabla _ x \\phi _ y ( t , x ) = \\int _ 0 ^ t \\int _ { \\R ^ d } \\nabla p _ z ( t - s , \\cdot ) ( x - z ) q ( s , z , y ) \\ , d z \\ , d s . \\end{align*}"}
-{"id": "7597.png", "formula": "\\begin{align*} \\mu _ i : = \\mathbb P \\{ \\omega \\in \\Omega : \\chi ( \\omega ) < - 1 + \\delta _ i \\} = \\int _ { - 1 } ^ { \\min \\{ - 1 + \\delta _ i , 1 \\} } \\phi ( t ) d t . \\end{align*}"}
-{"id": "1952.png", "formula": "\\begin{align*} v _ { \\alpha } \\doteqdot \\sum _ { i = 1 } ^ { m } \\ , q _ { \\alpha i } \\frac { \\sqrt { n _ i } } { b _ i } \\ , { \\tilde e } _ i , \\ , \\ , \\ , \\ , 1 \\leq \\alpha \\leq r , \\end{align*}"}
-{"id": "4158.png", "formula": "\\begin{align*} c _ { f , \\overline { g } } = \\frac { 1 } { 4 \\pi ^ 2 ( 2 \\kappa + \\frac { 3 } { 2 } - 2 \\nu ) } \\sum _ { n \\geq 1 } \\frac { a _ f ( n ) a _ g ( n ) } { n ^ { 2 \\kappa + \\frac { 3 } { 2 } - 2 \\nu } } . \\end{align*}"}
-{"id": "6819.png", "formula": "\\begin{align*} \\sum _ { k = - m + 1 } ^ { n - 1 } \\left [ B _ k , B _ { - m + n - k } \\right ] = 0 \\end{align*}"}
-{"id": "4036.png", "formula": "\\begin{align*} \\Sigma _ A & : = \\{ \\underline { x } \\in \\{ 1 , \\ldots , k \\} ^ { \\mathbb { Z } } : A _ { x _ n , x _ { n + 1 } } \\forall n \\in \\mathbb { Z } \\} \\\\ \\Sigma _ A ^ + & : = \\{ \\underline { x } \\in \\{ 1 , \\ldots , k \\} ^ { \\mathbb { N } } : A _ { x _ n , x _ { n + 1 } } \\forall n \\in \\mathbb { N } \\} . \\end{align*}"}
-{"id": "6330.png", "formula": "\\begin{align*} \\left \\Vert \\Box _ { k } ^ { \\alpha _ { 1 } } | ~ M _ { 1 } \\rightarrow M _ { 2 } \\right \\Vert \\gtrsim & \\frac { \\Vert \\Box _ { k } ^ { \\alpha _ { 1 } } f _ { k } ^ { \\alpha _ { 1 } } \\Vert _ { M _ { 2 } } } { \\Vert f _ { k } ^ { \\alpha _ { 1 } } \\Vert _ { M _ { 1 } } } \\gtrsim 2 ^ { j \\widetilde { A _ { 2 } } ( \\mathbf { p } , q , \\alpha _ { 1 } , \\alpha _ { 2 } ) } . \\end{align*}"}
-{"id": "9942.png", "formula": "\\begin{align*} \\| u \\| _ X : = \\| u \\| _ { L ^ 2 ( \\Omega ) } + \\left ( \\int _ Q | u ( x ) - u ( y ) | ^ 2 K ( x - y ) d x d y \\right ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "1982.png", "formula": "\\begin{align*} ( \\alpha + s ) m - \\beta v = 1 . \\end{align*}"}
-{"id": "1442.png", "formula": "\\begin{align*} f _ { s u b } : = e _ { - \\alpha _ { 2 } } + \\cdots + e _ { - \\alpha _ { n - 1 } } \\end{align*}"}
-{"id": "8553.png", "formula": "\\begin{align*} \\overline { N } ( a ^ { \\ast } ) J _ { 1 } & = 0 \\\\ \\overline { N } ( a ^ { \\ast } ) J _ { 2 } & = c _ { k } ^ { - 1 } \\pi _ { e _ { k } a } i \\\\ \\overline { N } ( a ^ { \\ast } ) J _ { 3 } & = c _ { k } ^ { - 1 } \\pi _ { e _ { k } a } j ' \\sigma _ { 2 } \\\\ \\overline { N } ( a ^ { \\ast } ) J _ { 4 } & = 0 \\end{align*}"}
-{"id": "4437.png", "formula": "\\begin{align*} M _ { f , a , \\beta } ( x ) : = \\max \\biggl \\{ \\max _ { t = 1 , \\ldots , m } \\frac { \\| f ^ { ( t ) } ( x ) \\| } { f ( x ) } \\ , , \\ , \\sup _ { y \\in B _ x ^ \\circ ( r _ a ( x ) ) } \\frac { \\| f ^ { ( m ) } ( y ) - f ^ { ( m ) } ( x ) \\| } { f ( x ) \\| y - x \\| ^ { \\beta - m } } \\biggr \\} . \\end{align*}"}
-{"id": "5529.png", "formula": "\\begin{align*} m _ 1 = \\min \\{ j \\in \\mathbb { N } : W ( j ) > 2 ^ { k _ 1 - 1 } \\} . \\end{align*}"}
-{"id": "1500.png", "formula": "\\begin{align*} N ( x ) \\geq 7 2 - 2 4 = 4 8 > 6 \\times 6 , \\end{align*}"}
-{"id": "3284.png", "formula": "\\begin{align*} F \\hat { R } ( v _ { 0 } \\otimes v _ { 0 } ) & = F ( v _ { 0 } \\otimes v _ { 0 } ) + q ^ { - 2 } ( q ^ { 2 } - q ^ { - 2 } ) F ( v _ { 1 } \\otimes v _ { - 1 } ) \\\\ & = [ 2 ] ^ { 1 / 2 } ( v _ { - 1 } \\otimes v _ { 0 } + ( q ^ { 2 } + 1 - q ^ { - 2 } ) v _ { 0 } \\otimes v _ { - 1 } ) . \\end{align*}"}
-{"id": "6871.png", "formula": "\\begin{align*} v ^ { [ t _ n ] } ( 1 , x ) = t _ n ^ { \\frac 1 { p } } v ( t _ n , \\sqrt { t _ n } x ) = [ e ^ { i \\Delta } \\psi ( t _ n ) ] ( x ) . \\end{align*}"}
-{"id": "1317.png", "formula": "\\begin{align*} z = 1 / N - 2 \\pi i \\alpha \\end{align*}"}
-{"id": "4184.png", "formula": "\\begin{align*} \\mu _ { [ a , b ] } = \\frac { 1 } { 2 } ( \\delta _ a + \\lambda _ { [ a , b ] } + \\delta _ b ) \\end{align*}"}
-{"id": "2554.png", "formula": "\\begin{align*} & M ( u ) = \\int _ { \\R ^ d } | u ( t , x ) | ^ 2 \\ , d x , \\\\ & E ( u ) = \\int _ { \\R ^ d } \\tfrac 1 2 | \\nabla u ( t , x ) | ^ 2 + \\tfrac { \\mu } { p + 2 } | u ( t , x ) | ^ { p + 2 } \\ , d x , \\end{align*}"}
-{"id": "9602.png", "formula": "\\begin{align*} \\ddot { u } ( x , t ) = ( \\Delta - m ^ 2 ) u ( x , t ) + h ( t ) \\ , g ( x - y ) , u ( x , 0 ) = u _ 0 ( x ) , \\dot u ( x , 0 ) = \\dot u _ 0 ( x ) \\end{align*}"}
-{"id": "6263.png", "formula": "\\begin{align*} \\lim _ { y \\to x } Q ( x , y ) = 2 \\frac { \\langle \\nabla w ( x ) , X \\rangle } { \\bar { w } ' ( 0 ) } , \\end{align*}"}
-{"id": "9657.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } G ( x ) = + \\infty . \\end{align*}"}
-{"id": "2837.png", "formula": "\\begin{align*} ( f _ 1 \\circ _ h f _ 2 ) \\circ _ v ( g _ 1 \\circ _ h g _ 2 ) = ( f _ 1 \\circ _ v g _ 1 ) \\circ _ h ( f _ 2 \\circ _ v g _ 2 ) \\end{align*}"}
-{"id": "6454.png", "formula": "\\begin{align*} = \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\tilde { u } ( s , x ) - \\tilde { u } ( s , y ) ) ( \\psi ^ { 2 } ( y ) \\tilde { u } ^ { - q } ( s , y ) - \\psi ^ { 2 } ( x ) \\tilde { u } ^ { - q } ( s , x ) ) k ( x , y ) d x d y , \\end{align*}"}
-{"id": "7955.png", "formula": "\\begin{align*} j _ l ( z ) i _ l ' ( z ) - i _ l ( z ) j _ l ' ( z ) = z ^ { 2 - N } C _ l ^ + ( z ) , \\end{align*}"}
-{"id": "2576.png", "formula": "\\begin{align*} K : = \\bigl \\{ \\bigl ( e ^ { - i x \\xi ( t ) } e ^ { - i t \\Delta } u ( t ) \\bigr ) _ { \\{ \\frac { 1 } { h ( t ) } \\} } : t \\in I \\bigr \\} \\end{align*}"}
-{"id": "5954.png", "formula": "\\begin{align*} U ( Z _ t ) = & \\ , U ( z ) + \\int _ 0 ^ t D U ( Z _ s ) { R \\ , } \\dd W _ s + \\lambda \\int _ 0 ^ t U ( Z _ s ) \\ , \\dd s - Z _ t + z + \\int _ 0 ^ t A Z _ s \\ , \\dd s + { R \\ , } W _ t \\end{align*}"}
-{"id": "7718.png", "formula": "\\begin{align*} \\mathbb P \\{ \\omega \\in \\Omega : \\lim _ { n \\to \\infty } x _ n ( \\omega ) = 0 \\} \\le \\frac 1 2 . \\end{align*}"}
-{"id": "7542.png", "formula": "\\begin{align*} r = \\left [ \\frac { 2 c } { l } \\right ] + 1 . \\end{align*}"}
-{"id": "9713.png", "formula": "\\begin{align*} \\Gamma ( x ) = g ( G _ 0 ^ { - 1 } ( x ) ) . \\end{align*}"}
-{"id": "5334.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = \\lambda N _ { f } ( u ) + ( 1 - \\lambda ) Q ( N _ { f } ( u ) ) & & \\\\ u ' ( 0 ) = u ( 0 ) , \\ u ' ( T ) = b u ' ( 0 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "748.png", "formula": "\\begin{align*} M = \\oplus _ i L _ i . \\end{align*}"}
-{"id": "1471.png", "formula": "\\begin{align*} \\mathcal { D } _ k : = \\left \\{ Q \\in \\mathcal { D } ( Q _ 0 ) : \\frac { w ( Q ) } { | Q | } > a ^ k \\gamma _ 0 \\right \\} ( k \\in \\N ) . \\end{align*}"}
-{"id": "7033.png", "formula": "\\begin{align*} \\sigma ^ { 2 } = 0 \\mbox { a n d } A ( x ) \\ge 0 \\mbox { f o r a l l s m a l l } x , \\end{align*}"}
-{"id": "6908.png", "formula": "\\begin{align*} | g ( r ) | = | \\langle \\widehat f ( T _ { r , \\lambda } ) e ^ \\lambda _ { i _ 0 } , e ^ \\lambda _ { j _ 0 } \\rangle _ { H ( K , \\lambda ) } | \\leq \\| \\widehat f ( T _ { r , \\lambda } ) \\| _ { H S } \\leq C _ \\lambda e ^ { - \\theta ( r ) } . \\end{align*}"}
-{"id": "3916.png", "formula": "\\begin{align*} \\frac { d } { d \\eta } \\left [ ( 1 - \\eta ^ 2 ) \\frac { d } { d \\eta } S _ { m , n } ( c , \\eta ) \\right ] + \\left ( \\lambda _ { m , n } - c ^ 2 \\eta ^ 2 - \\frac { m ^ 2 } { 1 - \\eta ^ 2 } \\right ) S _ { m , n } ( c , \\eta ) = 0 \\end{align*}"}
-{"id": "3160.png", "formula": "\\begin{align*} \\max _ { s \\in \\theta } \\frac { 1 } { \\left | \\Gamma \\right | } \\sum _ { \\gamma = 1 } ^ { \\left | \\Gamma \\right | } P _ e ( \\mathcal { C } _ n ^ { \\gamma } , t ^ n ) \\leq \\lambda _ n \\end{align*}"}
-{"id": "8506.png", "formula": "\\begin{align*} \\beta = \\sum _ { i \\in I } p _ i \\alpha _ i \\ \\ \\ \\ \\ \\ \\beta ' = \\sum _ { i \\in I } p ' _ i \\alpha _ i . \\end{align*}"}
-{"id": "4683.png", "formula": "\\begin{align*} a ^ k : = \\sum _ { j = 0 } ^ { k - 2 } \\Phi ^ \\theta _ j ( a ) \\ . \\end{align*}"}
-{"id": "2637.png", "formula": "\\begin{align*} G _ m ( \\lambda ) = \\psi _ m ( \\lambda ) ( \\psi _ m ( \\lambda ) ) ^ * \\in D _ { G } , \\end{align*}"}
-{"id": "2178.png", "formula": "\\begin{align*} R _ { m } ( s ) = ( 1 - q ) \\phi \\left [ \\mathcal { E } ( h _ { m } * \\tilde { u } , \\psi ^ { 2 } \\tilde { u } ^ { - q } ) - \\mathcal { E } ( \\tilde { u } , \\psi ^ { 2 } \\tilde { u } ^ { - q } ) \\right ] . \\end{align*}"}
-{"id": "8736.png", "formula": "\\begin{align*} K = \\mathcal { D } ( { \\Lambda } ^ { \\frac { 1 } { 2 } } ) \\times { U } = V \\times U \\end{align*}"}
-{"id": "9622.png", "formula": "\\begin{align*} \\psi ( x , t ) : = \\psi _ f ( x , t ) + \\sum \\limits _ { 1 \\le j \\le n } \\psi _ j ( x , t ) \\in D _ { \\tilde F } , t \\in [ 0 , \\tau ] \\end{align*}"}
-{"id": "6344.png", "formula": "\\begin{align*} - \\lim _ { y \\to 0 } | y | ^ a \\tilde { v } ( x , y ) = ( - \\Delta ) ^ s v ( x ) \\qquad \\forall \\ , x \\in \\R ^ n . \\end{align*}"}
-{"id": "1924.png", "formula": "\\begin{align*} & \\frac { d \\ln a } { d u } = E ( Y ) , \\\\ & \\frac { d Y _ i } { d u } = - Y _ i F _ i ( Y ) , 1 \\leq i \\leq m . \\end{align*}"}
-{"id": "3339.png", "formula": "\\begin{align*} c ( [ M \\times T ^ n _ R \\times T ^ n ] , F ) & \\leq 2 | | H | | _ { C ^ 0 } + \\max \\{ 0 , \\lambda ( m + n ) \\} \\\\ & < 2 ( \\sum _ { i = 1 } ^ n ( R _ i - \\epsilon ) \\cdot | e _ i | + | | G | | _ { C ^ 0 } ) + \\max \\{ 0 , \\lambda ( m + n ) \\} , \\end{align*}"}
-{"id": "3120.png", "formula": "\\begin{align*} \\lambda B + A = \\begin{bmatrix} M _ { 1 1 } ( \\lambda ) & M _ { 1 2 } ( \\lambda ) \\\\ M _ { 2 1 } ( \\lambda ) & M _ { 2 2 } ( \\lambda ) \\end{bmatrix} , \\end{align*}"}
-{"id": "4980.png", "formula": "\\begin{align*} \\overline { \\alpha } _ { f } ( P ) & = \\limsup _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n } ( P ) ) ^ { 1 / n } \\\\ & \\leq \\limsup _ { n \\to \\infty } \\left ( C h _ { X } ^ { + } ( P ) \\right ) ^ { 1 / n } ( \\delta _ { f } + \\epsilon ) \\\\ & = \\delta _ { f } + \\epsilon . \\end{align*}"}
-{"id": "9778.png", "formula": "\\begin{align*} ( \\overline { \\partial } _ { E , p } + \\overline { \\partial } ^ t _ { E , p } ) _ { \\min } ( \\phi _ i \\omega ) = \\phi _ i ( \\overline { \\partial } _ { E , p } + \\overline { \\partial } ^ t _ { E , p } ) \\omega + ( \\overline { \\partial } _ { \\min } \\phi _ i ) \\wedge \\omega - { \\rm I n t } ( \\overline { \\partial } _ { \\min } \\phi _ i ) \\omega . \\end{align*}"}
-{"id": "9489.png", "formula": "\\begin{align*} d \\beta = e ^ { - 2 \\rho } ( - 2 \\ , d \\rho \\wedge d \\theta - 2 \\ , d \\rho \\wedge \\alpha _ { n - 1 } + d \\alpha _ { n - 1 } ) \\end{align*}"}
-{"id": "2545.png", "formula": "\\begin{align*} U ^ { n + 1 } = U ^ n + \\tau \\Phi ( U ^ n , \\tau , h , \\delta _ { n + 1 } \\beta ) \\end{align*}"}
-{"id": "5579.png", "formula": "\\begin{align*} m _ { \\zeta } = \\frac 1 2 ( u _ { \\zeta \\zeta } - u ) _ { \\zeta } . \\end{align*}"}
-{"id": "8556.png", "formula": "\\begin{align*} \\rho ( x ) n & = \\rho ( s ( x _ { 1 } ) x _ { 1 } \\hdots s ( x _ { l - 2 } ) x _ { l - 2 } ) \\rho ( s ( x _ { l - 1 } ) x _ { l - 1 } s ( x _ { l } ) x _ { l } ) n \\\\ & = \\rho ( s ( x _ { 1 } ) x _ { 1 } \\hdots ( x _ { l - 2 } ) x _ { l - 2 } ) s ( x _ { l - 1 } ) \\rho ( x _ { l - 1 } s ( x _ { l } ) x _ { l } ) n \\\\ & = \\rho ( s ( x _ { 1 } ) x _ { 1 } \\hdots s ( x _ { l - 2 } ) x _ { l - 2 } ) s ( x _ { l - 1 } ) x _ { l - 1 } s ( x _ { l } ) x _ { l } n \\\\ & = \\rho ( s ( x _ { 1 } ) x _ { 1 } \\hdots s ( x _ { l - 2 } ) x _ { l - 2 } ) n ' \\end{align*}"}
-{"id": "5468.png", "formula": "\\begin{align*} S & : = \\sum _ { \\rho } \\frac { ( N + H ) ^ { \\rho + 2 } - 2 N ^ { \\rho + 2 } + ( N - H ) ^ { \\rho + 2 } } { \\rho ( \\rho + 1 ) ( \\rho + 2 ) } . \\end{align*}"}
-{"id": "8400.png", "formula": "\\begin{align*} \\phi ( m z _ g g ) = \\theta ( m z _ g ) \\phi ( z _ g g ) = \\theta ( m ) \\theta ( z _ g ) \\phi ( z _ g g ) , \\ \\ \\ m \\in M . \\end{align*}"}
-{"id": "5630.png", "formula": "\\begin{align*} x ^ A y ^ B = \\prod _ { i = 1 } ^ n x ^ { a _ i } y ^ { b _ i } = x ^ { \\sum _ { i = 1 } ^ n a _ i } y ^ { \\sum _ { i = 1 } ^ n b _ i } . \\end{align*}"}
-{"id": "6360.png", "formula": "\\begin{align*} A _ { \\sigma l } = ( l + 1 ) ( l + 2 + 2 \\sigma _ n ) p _ { \\sigma , l + 1 } + 2 s ( \\sigma _ n + 1 ) p _ { \\sigma + \\bar { n } , l } + ( \\sigma _ i + 1 ) ( \\sigma _ i + 2 ) p _ { \\sigma + 2 \\bar { \\imath } , l - 1 } + c _ { \\sigma l } ^ { \\mu m } p _ { \\mu m } , \\end{align*}"}
-{"id": "7105.png", "formula": "\\begin{align*} \\omega _ { n , \\beta } = W _ { n , \\beta } ^ { - 2 } \\sum _ { | u | = | v | = n } e ^ { \\beta ( 2 m _ n - V ( u ) - V ( v ) ) } \\delta _ { | u \\wedge v | / n } . \\end{align*}"}
-{"id": "8100.png", "formula": "\\begin{align*} P _ { \\mu _ { k } } ( M ) : = \\sup \\left \\{ \\int _ { M } \\mbox { d i v } \\ , \\left ( | x | ^ { k } \\mathbf { v } \\right ) \\ , d x : \\ , \\mathbf { v } \\in C _ { 0 } ^ { 1 } ( \\mathbb { R } ^ N , \\mathbb { R } ^ { N } ) , \\ , | \\mathbf { v } | \\leq 1 \\mbox { i n } \\ , M \\right \\} . \\end{align*}"}
-{"id": "446.png", "formula": "\\begin{align*} \\phi _ k ( c ' ) & = ( y , z ' ) \\\\ \\phi _ k ( c '' ) & = ( y , z '' ) \\\\ \\mu _ k ( z ) & = \\mu _ k ( z ' ) = \\mu _ k ( z '' ) . \\end{align*}"}
-{"id": "8360.png", "formula": "\\begin{align*} E _ { N _ 0 } ( g ) x = \\alpha _ g ( x ) E _ { N _ 0 } ( g ) = g x g ^ { - 1 } E _ { N _ 0 } ( g ) , \\ \\ \\ x \\in M , \\ \\ \\ g \\in G , \\end{align*}"}
-{"id": "2639.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\sum _ { l = 1 } ^ { h ( m , n ) } \\left [ \\| \\mathbf { B _ m ^ * \\Psi _ m ^ * \\Psi _ m a _ m ^ l } \\| ^ 2 \\right ] & \\rightarrow i n f , \\\\ F _ m ( \\lambda ) = d _ m ( \\lambda ) ( d _ m ( \\lambda ) ) ^ * & - G _ m ( \\lambda ) \\in D _ F . \\end{align*}"}
-{"id": "3582.png", "formula": "\\begin{align*} \\Phi ^ W _ { ( g , \\pi ) } ( g + h , \\pi + w ) = \\Phi ^ W _ { ( g , \\pi ) } ( g , \\pi ) + ( \\psi , V ) \\end{align*}"}
-{"id": "6401.png", "formula": "\\begin{align*} \\gamma _ { q , p } ( p , 2 ) = \\beta _ { q , p } \\left ( \\frac { ( 4 p - 1 ) ! ! } { 6 } \\varepsilon _ { p } ^ { 4 } \\sigma ^ { 4 p } + \\frac { ( 2 q + 2 p - 1 ) ! ! } { 2 } \\varepsilon _ { q } ^ { 2 } \\varepsilon _ { p } ^ { 2 } \\sigma ^ { 2 q + 2 p } \\right ) , \\end{align*}"}
-{"id": "8604.png", "formula": "\\begin{align*} \\operatorname { \\overline { \\gamma } } = c _ { k } \\beta \\alpha \\end{align*}"}
-{"id": "1491.png", "formula": "\\begin{align*} V ( b ) = \\bigcap _ { b \\nmid d _ i } \\{ x \\mid Q ( x ) = 0 \\} . \\end{align*}"}
-{"id": "6181.png", "formula": "\\begin{align*} \\begin{gathered} \\exists f , \\ , \\exists c , d > 0 , \\ , g \\geq 1 , \\ , C \\\\ \\mathcal { A } _ n f ( x ) \\leq - c g ( x ) + d I _ C ( x ) , \\ | x | < n , n \\in \\N . \\end{gathered} \\end{align*}"}
-{"id": "4468.png", "formula": "\\begin{align*} \\biggl | V _ d f ( x ) h _ x ^ { - 1 } ( s ) ^ d - \\sum _ { l = 0 } ^ { \\lceil \\beta / 2 \\rceil - 1 } b _ l ( x ) s ^ { 1 + 2 l / d } \\biggr | \\lesssim s \\biggl \\{ \\frac { a ( f ( x ) ) ^ { d / ( 2 \\wedge \\beta ) } s } { f ( x ) } \\biggr \\} ^ { \\beta / d } , \\end{align*}"}
-{"id": "5401.png", "formula": "\\begin{align*} e ( R ' ) \\leq \\left ( k - \\alpha - \\frac { 7 } { 1 6 } \\right ) \\frac { n ^ 2 } { 2 } = \\left ( k - \\frac { 1 1 } { 1 6 } \\right ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "6017.png", "formula": "\\begin{gather*} M _ 5 : = M _ 5 ^ + \\cup M _ 5 ^ - = \\{ x \\in M \\colon \\sigma _ x \\neq 0 \\} = M - \\Sigma \\end{gather*}"}
-{"id": "2849.png", "formula": "\\begin{align*} g _ { P , X } ^ { \\psi } = H o m _ { \\Sigma } ( \\overline { C } , E n d _ X ) ^ { \\psi } . \\end{align*}"}
-{"id": "7929.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { l } d _ { i } = \\sum _ { i = 1 } ^ { l } y _ { i } - \\sum _ { i = 1 } ^ { l } x _ { i } \\mbox { a n d } \\sum _ { i = 1 } ^ { l } s _ { i } = \\sum _ { i = 1 } ^ { l } y _ { i } + \\sum _ { i = 1 } ^ { l } x _ { i } . \\end{align*}"}
-{"id": "1888.png", "formula": "\\begin{align*} Y _ j = \\sum _ { k = 1 } ^ { K } X _ { j , k } , \\ \\ \\ j = 1 , \\ldots , n . \\end{align*}"}
-{"id": "2496.png", "formula": "\\begin{gather*} z _ { i + 1 } ( u _ k , t ) = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k ( z _ i ( u _ j , t ) - z _ i ( u _ j , t \\wedge \\sigma _ i ) ) \\cdot \\ 1 _ { A _ { k j } ^ i } = \\\\ = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k z _ i ( u _ j , t ) \\cdot \\ 1 _ { A _ { k j } ^ i } - \\sum _ { j = 1 } ^ k z _ i ( u _ j , t \\wedge \\sigma _ i ) \\cdot \\ 1 _ { A _ { k j } ^ i } \\end{gather*}"}
-{"id": "3226.png", "formula": "\\begin{align*} s \\colon N _ { Y / X } ^ * \\to N _ { Y / X } \\otimes N _ { Y / X } ^ * \\otimes N _ { Y / X } ^ * : \\ \\varepsilon _ j ^ \\nu \\mapsto \\frac { 1 } { 2 } \\sum _ { \\mu = 1 } ^ r \\varepsilon _ { j , \\mu } ^ * \\otimes \\left ( \\varepsilon _ j ^ \\mu \\otimes \\varepsilon _ j ^ \\nu + \\varepsilon _ j ^ \\nu \\otimes \\varepsilon _ j ^ \\mu \\right ) . \\end{align*}"}
-{"id": "3954.png", "formula": "\\begin{align*} A = \\left \\{ a ( t ) : = \\begin{bmatrix} e ^ t & \\\\ & e ^ { - t } \\end{bmatrix} : t \\in \\R \\right \\} , \\ U = \\left \\{ u ( s ) : = \\begin{bmatrix} 1 & s \\\\ 0 & 1 \\end{bmatrix} : s \\in \\R \\right \\} , \\ U ^ { - } = \\left \\{ u ^ { - } ( s ) : = \\begin{bmatrix} 1 & 0 \\\\ s & 1 \\end{bmatrix} : s \\in \\R \\right \\} . \\end{align*}"}
-{"id": "7538.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } x _ n = K , \\end{align*}"}
-{"id": "2172.png", "formula": "\\begin{align*} F ( s ) = & \\frac { 1 } { 2 } C _ { 1 } ( n , \\Lambda , \\delta ) \\vartheta ( q ) ( q - 1 ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\int _ { \\rho B _ { 1 } } \\phi w ^ { 2 } d x \\\\ & + q g _ { 1 - \\alpha } ( s ) \\phi ( s ) \\int _ { \\rho B _ { 1 } } \\psi ^ { 1 + q } w ^ { 2 } d x + \\dot { \\phi } ( s ) \\left ( g _ { 1 - \\alpha } * \\int _ { \\rho B _ { 1 } } \\psi ^ { 1 + q } w ^ { 2 } d x \\right ) ( s ) \\end{align*}"}
-{"id": "9330.png", "formula": "\\begin{align*} \\begin{cases} d M ( t , z ) = \\mathbb { E } [ D _ t \\delta _ Z ( z ) | \\mathcal { F } _ t ] d B ( t ) = \\Phi ( t , z ) M ( t , z ) d B ( t ) \\\\ M ( 0 , z ) = 1 \\end{cases} \\end{align*}"}
-{"id": "8704.png", "formula": "\\begin{align*} \\ < \\widetilde v ^ h ( \\cdot , \\Xi ^ { t , x } ) , W ^ { \\xi } \\ > _ { \\tau } = \\int _ 0 ^ \\tau \\nabla ^ G \\widetilde v ^ h ( s , \\Xi _ s ^ { t , x } ) \\xi \\ , d s = \\int _ 0 ^ \\tau \\langle \\nabla ^ G v ( s , \\Xi _ s ^ { t , x } ) \\xi , e ^ { - s { A } ^ * } h \\rangle \\ , d s , \\ ; \\ ; \\tau \\in [ 0 , T ] . \\end{align*}"}
-{"id": "5852.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( L _ { Y _ { I } } C _ { x } \\right ) _ { , x } + C ^ { x } L _ { Y _ { I } } C _ { x } = c \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "8852.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\lambda _ i } { k ^ { \\frac { 2 } { n } } } & = \\lim _ { k \\rightarrow \\infty } \\frac { \\Big ( \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\lambda _ i \\Big ) + \\frac { n ^ 2 H _ 0 ^ 2 + \\eta _ 0 ^ 2 + 2 \\bar { \\eta _ 0 } } { 4 } } { k ^ { \\frac { 2 } { n } } } \\\\ & = \\lim _ { k \\rightarrow \\infty } \\frac { \\frac { 1 } { k } \\sum _ { i = 1 } ^ k ( \\lambda _ i + \\frac { n ^ 2 H _ 0 ^ 2 + \\eta _ 0 ^ 2 + 2 \\bar { \\eta _ 0 } } { 4 } ) } { k ^ { \\frac { 2 } { n } } } . \\end{align*}"}
-{"id": "5151.png", "formula": "\\begin{align*} t ( \\vec { \\mu } ) - t ( \\vec { \\nu } ) = t ( r ^ { k _ { r } } , \\ldots , a ^ { k _ { 1 } - 1 } , \\ldots , 1 ^ { k _ { 1 } } ) - t ( \\nu _ { 1 } , \\ldots , \\nu _ { k - 1 } ) - \\sum _ { b = 1 } ^ { a - 1 } k _ { b } \\end{align*}"}
-{"id": "9616.png", "formula": "\\begin{align*} \\lambda _ { j , j } ( t ) = \\lim _ { x \\to y _ j } \\ , \\psi _ { f , j } ( x , t ) = - \\frac { m ( \\zeta _ { 0 j } + t \\dot \\zeta _ { 0 j } ) - \\dot \\zeta _ 0 } { 4 \\pi } + \\frac { m } { 4 \\pi } \\int _ 0 ^ t \\frac { J _ 1 ( m ( t - s ) ) } { ( t - s ) } ( \\zeta _ { 0 j } + s \\dot \\zeta _ { 0 j } ) d s . \\end{align*}"}
-{"id": "6970.png", "formula": "\\begin{align*} g ( \\sigma ^ + , \\mu _ 1 ) = \\frac { \\gamma } { \\delta } H _ 2 ( \\mu _ 1 ) > 0 . \\end{align*}"}
-{"id": "1163.png", "formula": "\\begin{align*} \\beta = \\begin{cases} \\frac { 1 } { 2 } b _ { x } ( 0 ) - h b ( 0 ) & 0 \\leq h < \\infty , \\\\ - \\frac { 1 } { 2 } b _ { x } ( 0 ) + \\frac { 1 } { 2 h } b _ { x x } ( 0 ) & 0 < h \\leq \\infty . \\end{cases} \\end{align*}"}
-{"id": "6524.png", "formula": "\\begin{align*} \\begin{aligned} \\| v \\| _ { L ^ \\infty ( 0 , T ; W ) } \\le C _ { W } \\big ( \\| v ' \\| _ { L ^ p ( 0 , T ; X ) } + \\| v \\| _ { L ^ p ( 0 , T ; D ) } \\big ) . \\end{aligned} \\end{align*}"}
-{"id": "5212.png", "formula": "\\begin{align*} \\Box _ { M , q } : L ^ 2 _ { ( 0 , q ) } ( M ) \\longrightarrow L ^ 2 _ { ( 0 , q ) } ( M ) , \\ ; \\Box _ { _ M , q } : = \\bar \\partial _ M \\bar \\partial ^ * _ M + \\bar \\partial ^ * _ M \\bar \\partial _ M . \\end{align*}"}
-{"id": "6405.png", "formula": "\\begin{align*} p ( x ) = \\frac { C ( \\vec { \\varepsilon } ) } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } e ^ { \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } x ^ { k } } , \\end{align*}"}
-{"id": "1330.png", "formula": "\\begin{align*} f ( \\lambda , \\mu ) = \\frac { \\sinh ( \\lambda - \\mu + \\eta ) } { \\sinh ( \\lambda - \\mu ) } , g ( \\lambda , \\mu ) = \\frac { \\sinh \\eta } { \\sinh ( \\lambda - \\mu ) } \\end{align*}"}
-{"id": "2768.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( B ^ { H } _ { t } ) \\diamond d B ^ { H } ( t ) : = \\lim _ { n \\rightarrow 0 } \\sum _ { i = 1 } ^ { n } F ( B ^ { H } _ { t _ { i - 1 } } ) \\diamond ( B ^ { H } ( t _ { i } ) - B ^ { H } ( t _ { i - 1 } ) ) \\end{align*}"}
-{"id": "1559.png", "formula": "\\begin{align*} \\vert h _ { \\mu } ( \\nu ) - h _ { \\mu ' } ( \\nu ) \\vert = & \\ \\vert W _ 2 ( \\nu , \\mu ) - W _ 2 ( \\nu , \\mu ' ) \\vert \\ ( W _ 2 ( \\nu , \\mu ) + W _ 2 ( \\nu , \\mu ' ) ) \\le \\ W _ 2 ( \\mu , \\mu ' ) \\ 2 H ( \\nu ) . \\end{align*}"}
-{"id": "8027.png", "formula": "\\begin{align*} 0 \\to H ^ 0 ( U , V ) \\to \\bigoplus _ { s \\in S } ( j _ * V ) _ s \\to H ^ 1 _ c ( U , j _ * V ) \\to H ^ 1 ( \\P ^ 1 , j _ * V ) \\to 0 \\\\ 0 \\to H ^ 1 ( \\P ^ 1 , j _ * V ) \\to H ^ 1 ( U , V ) \\to \\bigoplus _ { s \\in S } ( R ^ 1 j _ * V ) _ s \\to H ^ 2 ( \\P ^ 1 , j _ * V ) \\to 0 \\end{align*}"}
-{"id": "8574.png", "formula": "\\begin{align*} \\displaystyle \\varphi ^ { - 1 } ( C _ { r '' a , h a _ { 1 } } ) & = \\displaystyle \\sum _ { u \\in L ( \\sigma ( a ) ) } ( r '' ) ^ { \\ast } ( h u ) \\varphi ^ { - 1 } ( C _ { u a , a _ { 1 } } ) \\\\ \\varphi ^ { - 1 } ( D _ { b _ { 1 } , b r ' } ) & = \\displaystyle \\sum _ { v \\in L ( \\tau ( b _ { 1 } ) ) } ( r ' ) ^ { \\ast } ( v ) \\varphi ^ { - 1 } ( D _ { b _ { 1 } , b v } ) \\end{align*}"}
-{"id": "8054.png", "formula": "\\begin{align*} \\mathrm { S u p p } ( \\widetilde { D } _ { \\Gamma } ) = \\bigcup _ { I } B _ { I } \\end{align*}"}
-{"id": "2427.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { p , q } ^ { 0 , \\alpha _ 1 } } = \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } \\Box _ k ^ { \\alpha _ 2 } f \\| ^ q _ { L ^ p } \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "8397.png", "formula": "\\begin{align*} K _ g = \\{ m \\in M : m g \\in \\mathrm { A l g } ( Y , Y ^ * ) \\} \\subseteq M z _ g . \\end{align*}"}
-{"id": "3743.png", "formula": "\\begin{align*} f ( u ) = \\sqrt { \\frac { \\beta ^ \\prime ( u ) } { 2 \\pi } } e ^ { - u \\beta + C ( \\beta ) } \\left \\{ 1 + O \\left ( \\frac { 1 } { u } \\right ) \\right \\} . \\end{align*}"}
-{"id": "8489.png", "formula": "\\begin{align*} \\textbf { u } ^ { \\varepsilon } _ { 0 } = ( \\rho ^ { \\varepsilon } _ { 0 } , \\tilde { P } ^ { \\varepsilon } _ { 0 } , v ^ { \\varepsilon } _ { 0 } ) ^ { T } , \\end{align*}"}
-{"id": "4327.png", "formula": "\\begin{align*} \\frac { 1 } { r } = \\sum _ { i = 1 } ^ l \\langle e _ r , d _ i ^ * ( e _ r ) \\rangle \\langle e _ { r - s } , c _ i ( e _ { r - s } ) \\rangle . \\end{align*}"}
-{"id": "3947.png", "formula": "\\begin{align*} \\int f \\dd \\lambda ^ { J } _ t : = \\frac { 1 } { \\abs { J } } \\int _ { s \\in J } f ( z ( s ) a ( t ) u ( \\varphi ( s ) ) x ) \\dd s . \\end{align*}"}
-{"id": "379.png", "formula": "\\begin{align*} \\sum _ { \\pm m \\geq 1 } \\psi \\left ( \\frac { | m | } { X } \\right ) \\sum _ { i = 1 } ^ { h ( m ) } \\frac { \\phi \\left ( g _ { i , m } ^ { - 1 } \\right ) } { | \\Gamma ( i , m ) | } \\ll _ { \\epsilon , \\psi } X ^ { \\frac { 1 } { 8 } + \\epsilon } \\end{align*}"}
-{"id": "9940.png", "formula": "\\begin{align*} \\left \\| u \\right \\| _ { X _ { 0 , s } } & : = \\left ( \\int _ { \\R ^ n } \\frac { | u ( x ) - u ( y ) | ^ 2 } { | x - y | ^ { n + 2 s } } d x d y \\right ) ^ { 1 / 2 } , \\\\ c _ { 2 ^ * _ s } & : = \\sup _ { u \\in X _ { 0 , s } \\setminus \\{ 0 \\} } \\frac { \\left \\| u \\right \\| _ { L ^ { 2 ^ * } ( \\R ^ n ) } } { \\left \\| u \\right \\| _ { X _ { 0 , s } } } \\end{align*}"}
-{"id": "4365.png", "formula": "\\begin{align*} \\pi _ I ( U ( \\gamma ) ) = U ^ \\pi ( \\gamma ) \\end{align*}"}
-{"id": "8163.png", "formula": "\\begin{align*} \\mathcal { U } = \\big \\{ \\boldsymbol { u } \\in \\mathbb { R } ^ 3 \\ \\arrowvert \\ \\parallel \\boldsymbol { u } \\parallel \\leq u _ { m a x } \\big \\} , \\\\ \\end{align*}"}
-{"id": "2659.png", "formula": "\\begin{align*} d ^ { m i n } _ n : = \\left \\{ \\begin{array} { l l } \\left \\lfloor \\dfrac { 2 n + 2 } { 3 } \\right \\rfloor & \\mbox { i f } n = 3 k \\mbox { o r } n = 3 k + 1 \\\\ 2 \\left \\lfloor \\dfrac { n - 1 } { 3 } \\right \\rfloor + m & \\mbox { i f } n = 3 k + 2 \\\\ \\end{array} \\right . \\ , . \\end{align*}"}
-{"id": "3903.png", "formula": "\\begin{align*} G _ { h } ( s ) = 0 \\end{align*}"}
-{"id": "8302.png", "formula": "\\begin{align*} f = \\frac { 1 } { 2 } \\varphi ^ * ( d _ { \\gamma } - d _ { p } ) , \\end{align*}"}
-{"id": "2328.png", "formula": "\\begin{align*} z _ i \\leftarrow z _ i - W _ { p , l } ( z _ i ) , ~ i = 1 , \\dots , d , \\end{align*}"}
-{"id": "1898.png", "formula": "\\begin{align*} | \\hat { \\phi } ( u ) - \\hat { \\phi } ( u + h ) \\big | \\leq \\frac { h } { n } \\sum _ { j = 1 } ^ { n } | Y _ j | . \\end{align*}"}
-{"id": "1569.png", "formula": "\\begin{align*} \\mathcal { L } _ { X ^ { \\left [ 2 \\right ] } } \\Theta = 0 . \\end{align*}"}
-{"id": "3882.png", "formula": "\\begin{align*} \\begin{cases} \\sigma _ k ( D ^ 2 u ) = 0 , & , \\\\ u = f , & , \\end{cases} \\end{align*}"}
-{"id": "9710.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { G ( x ( t ) ) } { t } = a - b . \\end{align*}"}
-{"id": "2547.png", "formula": "\\begin{align*} \\left | \\mathbb { E } \\left [ \\frac 1 N \\sum _ { n = 0 } ^ { N - 1 } f ( U ^ n ) - \\hat f \\right ] \\right | \\le C _ h ( \\frac 1 T + \\tau ) . \\end{align*}"}
-{"id": "8963.png", "formula": "\\begin{align*} \\mathcal { V } _ k ( B ) : = \\sum _ { \\Lambda _ k \\in \\Omega _ k ^ { ( m ) } } \\det B ( \\Lambda _ k ) \\end{align*}"}
-{"id": "8111.png", "formula": "\\begin{align*} u ( x ) = \\int _ 0 ^ { \\infty } \\chi _ { \\{ u > t \\} } ( x ) \\ , d t , ( x \\in \\mathbb { R } ^ N ) . \\end{align*}"}
-{"id": "1604.png", "formula": "\\begin{align*} F \\left ( t , x \\right ) = x \\exp \\left ( \\left ( c _ { 1 } + \\frac { 1 } { 2 } \\right ) t \\right ) ~ , ~ m = 0 . \\end{align*}"}
-{"id": "9978.png", "formula": "\\begin{align*} \\max _ { \\tilde { p } \\ ! , p _ 1 \\ ! , p _ 2 \\ ! } \\quad & \\alpha \\ ! \\log _ 2 \\ ! \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { \\tilde { p } | H _ { \\tilde { i } k } | ^ 2 } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\ ! + \\ ! ( \\ ! 1 \\ ! - \\ ! \\alpha \\ ! ) \\sum _ { i = 1 } ^ { 2 } \\log _ 2 \\Big ( \\ ! 1 \\ ! + \\ ! \\frac { p _ { i } } { \\sigma ^ 2 _ n } \\ ! \\Big ) \\end{align*}"}
-{"id": "7015.png", "formula": "\\begin{align*} y ( t ) = \\begin{cases} w _ 1 ( t ) - \\frac { t } { s _ 1 } w _ 1 ( s _ 1 ) , & t \\in [ 0 ; s _ 1 ] \\\\ w _ 2 ( t - s _ 1 ) - \\frac { t - s _ 1 } { s _ 2 - s _ 1 } w _ 2 ( s _ 2 - s _ 1 ) , & t \\in [ s _ 1 ; s _ 2 ] \\\\ \\cdot \\\\ \\cdot \\\\ \\cdot \\\\ w _ { N + 1 } ( t - s _ N ) - \\frac { t - s _ N } { 1 - s _ N } w _ { N + 1 } ( 1 - s _ N ) , & t \\in [ s _ N ; 1 ] . \\end{cases} \\end{align*}"}
-{"id": "2978.png", "formula": "\\begin{align*} \\delta _ H \\Theta ^ 2 ( x , y ) ( a , b ) + \\delta _ L \\Theta ^ 1 ( x , y ) ( a , b ) = 0 \\end{align*}"}
-{"id": "1304.png", "formula": "\\begin{align*} \\sum _ { n = N - H } ^ { N + H } t _ H ( n - N ) \\Bigl ( R ( n ) - ( 2 \\psi ( n ) - n ) \\Bigr ) \\ll H N \\Bigl ( \\log \\frac { 2 N } { H } \\Bigr ) ^ 2 + H ^ 2 ( \\log N ) ^ 2 \\log ( 2 H ) \\end{align*}"}
-{"id": "6429.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } ( u ( t , x ) - u _ { 0 } ( x ) ) = r ^ { - 2 \\beta } \\partial _ { s } ^ { \\alpha } ( \\tilde { u } ( s , y ) - \\tilde { u } _ { 0 } ( y ) ) \\end{align*}"}
-{"id": "2004.png", "formula": "\\begin{align*} M _ { t \\wedge T _ n } & = f ( V _ { t \\wedge T _ n } ) - \\int _ 0 ^ { t \\wedge T _ n } \\mathcal { A } f ( V _ s ) \\d s \\\\ & = f ( V _ t ^ n ) - \\int _ 0 ^ { t \\wedge T _ n } \\mathcal { A } _ n f ( V _ s ) \\d s \\\\ & = f ( V _ t ^ n ) - \\int _ 0 ^ { t } \\mathcal { A } _ n f ( V _ s ^ n ) \\d s , \\end{align*}"}
-{"id": "9886.png", "formula": "\\begin{align*} \\mathrm { G L R T } : \\quad \\sum _ { n = 1 } ^ N b _ n \\alpha _ n \\substack { H _ 1 \\\\ > \\\\ < \\\\ H _ 0 } \\eta ^ { ' } , \\end{align*}"}
-{"id": "7362.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 3 ) ^ * } = \\left ( \\begin{array} { c c c } 0 & 0 & q ^ 4 \\end{array} \\right ) , \\Gamma _ { 0 } ^ { ( 3 ) ^ * } = \\left ( \\begin{array} { c c c } 0 & - q ^ 6 & 0 \\end{array} \\right ) , \\Gamma _ { - } ^ { ( 3 ) ^ * } = \\left ( \\begin{array} { c c c } q ^ 4 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "289.png", "formula": "\\begin{align*} \\alpha _ z = I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { \\| z \\| ^ 2 } { 4 } \\biggr ) . \\end{align*}"}
-{"id": "478.png", "formula": "\\begin{align*} \\beta = \\begin{cases} \\alpha + 1 \\ \\ \\alpha \\in \\mathbb { Z } \\\\ \\lceil \\alpha \\rceil \\ \\ \\alpha \\notin \\mathbb { Z } . \\end{cases} \\end{align*}"}
-{"id": "795.png", "formula": "\\begin{align*} \\check { R } ( z / w ) [ L ( z ) \\otimes L ( w ) ] = [ L ( w ) \\otimes L ( z ) ] \\check { R } ( z / w ) , \\end{align*}"}
-{"id": "179.png", "formula": "\\begin{align*} \\tau _ 1 < \\min \\biggl ( \\frac { 2 \\alpha } { 5 \\alpha + 3 d } \\ , , \\ , \\frac { \\alpha - d } { 2 \\alpha } \\ , , \\ , \\frac { 4 \\beta ^ * } { 4 \\beta ^ * + 3 d } \\biggr ) , \\tau _ 2 : = \\min \\biggl ( 1 - \\frac { d / 4 } { 1 + \\lfloor d / 4 \\rfloor } , 1 - \\frac { d } { 2 \\beta } \\biggr ) \\end{align*}"}
-{"id": "274.png", "formula": "\\begin{align*} | U _ 2 | \\leq U _ { 2 1 } + U _ { 2 2 } = O \\biggl ( \\frac { k ^ { 1 / 2 } } { n } \\max \\biggl \\{ \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "2037.png", "formula": "\\begin{align*} \\Delta u = - \\lambda _ 1 u \\ \\ \\mbox { o n } \\ \\ \\Omega , \\ \\ \\ u | _ { \\dd \\Omega } = 0 , \\end{align*}"}
-{"id": "469.png", "formula": "\\begin{align*} G ^ + = \\{ x \\in \\mathbb { R } ^ 2 \\ , | \\ \\begin{bmatrix} 0 & 0 \\\\ \\beta _ 2 v _ { 1 2 } & \\beta _ 2 v _ { 2 2 } \\\\ \\beta _ 1 v _ { 1 1 } & \\beta _ 1 v _ { 2 1 } \\end{bmatrix} \\begin{bmatrix} x _ 1 \\\\ x _ 2 \\end{bmatrix} \\succeq _ { \\mathcal { L } ^ 3 } \\begin{bmatrix} - \\eta \\\\ \\beta _ 2 \\alpha _ 2 \\\\ \\beta _ 1 \\alpha _ 1 \\end{bmatrix} \\} , \\end{align*}"}
-{"id": "4896.png", "formula": "\\begin{align*} g ( 2 ) = ( q ^ 2 + 1 ) ( q - 1 ) ( r - 1 - \\alpha q ) . \\end{align*}"}
-{"id": "4668.png", "formula": "\\begin{align*} & - \\int _ { G ^ + / \\Gamma , \\chi ( g ) \\leq 1 } \\chi ( g ) ^ s \\phi ( g ) \\left \\{ \\sum _ { x \\in L _ 0 } f ( g \\cdot x ) - \\chi ^ { - 1 } ( g ) \\sum _ { x \\in \\hat { L } _ 0 } \\hat { f } ( g ^ { \\iota } x ) \\right \\} d g \\\\ & = - \\int _ 0 ^ 1 t ^ { 1 2 s } \\int _ { G ^ 1 / \\Gamma } \\phi ( g _ 1 ) \\left \\{ \\sum _ { x \\in L _ 0 } f _ { t ^ 3 } ( g _ 1 \\cdot x ) - \\sum _ { x \\in \\hat { L } _ 0 } \\widehat { f _ { t ^ 3 } } ( g _ 1 \\cdot x ) \\right \\} d g _ 1 \\frac { d t } { t } . \\end{align*}"}
-{"id": "8268.png", "formula": "\\begin{align*} \\partial _ { \\rho } \\varphi _ { V ' } ( \\rho ) = \\sigma _ { h ' ( \\rho ) } ^ { - 2 } , \\partial _ { \\rho \\rho } \\varphi _ { V ' } ( \\rho ) = - \\frac { m _ { 3 , h ' ( \\rho ) } } { \\sigma _ { h ' ( \\rho ) } ^ 6 } . \\end{align*}"}
-{"id": "5432.png", "formula": "\\begin{align*} v _ p ( \\kappa ( B ) ) \\geq c ( i , r ) + c ( i + r , 3 r ) + c ( i + 2 r , r ) + c ( i + 3 r , 3 r ) = 4 t . \\end{align*}"}
-{"id": "8873.png", "formula": "\\begin{align*} \\mathcal { I } _ A ( u ) = \\mathcal { E } ( d A ) \\ \\mathcal { I } _ A ' ( u ) = 0 . \\end{align*}"}
-{"id": "9578.png", "formula": "\\begin{align*} \\psi _ { r e g } ( y _ j ) + \\sum \\limits _ { 1 \\le k \\le n } g _ { k j } \\zeta _ k = F _ j ( \\zeta ) . \\end{align*}"}
-{"id": "8966.png", "formula": "\\begin{align*} ( - 1 ) ^ k \\sum _ { 1 \\leq i _ 1 < i _ 2 \\cdot \\cdot \\cdot < i _ k \\leq m } \\alpha _ { i _ 1 } \\alpha _ { i _ 2 } \\cdot \\cdot \\cdot \\alpha _ { i _ k } = \\beta _ k \\end{align*}"}
-{"id": "7527.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } x _ n ( \\omega ) = K . \\end{align*}"}
-{"id": "9741.png", "formula": "\\begin{align*} 0 < \\lambda ( \\epsilon ) = \\frac { \\epsilon ( 1 - \\epsilon ) G _ 0 ( \\delta ( \\epsilon ) ) } { T _ 1 ( \\epsilon ) } . \\end{align*}"}
-{"id": "8581.png", "formula": "\\begin{align*} N _ { i n } & = \\displaystyle \\bigoplus _ { a \\in _ { k } T } D _ { k } \\otimes _ { F } N _ { \\tau ( a ) } \\\\ N _ { o u t } & = \\displaystyle \\bigoplus _ { b \\in T _ { k } } D _ { k } \\otimes _ { F } N _ { \\sigma ( b ) } \\end{align*}"}
-{"id": "9644.png", "formula": "\\begin{align*} \\lim _ { \\ell \\rightarrow \\infty } C _ { \\theta } ^ { ( \\ell ) } \\ldots C _ { \\theta } ^ { ( 0 ) } = O . \\end{align*}"}
-{"id": "9841.png", "formula": "\\begin{align*} R ( f ) = \\frac { 1 } { 2 } \\log _ 2 \\left ( 1 + \\frac { \\left | H _ { \\rm S D } ( f ) + \\frac { H _ { \\rm R D } ( f ) H _ { \\rm S R } ( f ) \\Theta ( f ) } { 1 - \\hat \\alpha ( f ) \\Theta ( f ) } \\right | ^ 2 P ( f ) } { \\left ( \\left | \\frac { H _ { \\rm R D } ( f ) \\Theta ( f ) } { 1 - \\hat \\alpha ( f ) \\Theta ( f ) } \\right | ^ 2 + 1 \\right ) N _ 0 } \\right ) . \\end{align*}"}
-{"id": "6914.png", "formula": "\\begin{align*} F ( T ) = \\sum _ { \\{ j , k \\} \\in E ( T ) } \\phi ( j ) \\ , \\phi ( k ) \\end{align*}"}
-{"id": "1256.png", "formula": "\\begin{align*} v M _ { 1 1 } v ^ * f ( x ) = \\left ( \\begin{array} { c } a ( x ) \\\\ c ( x ) \\end{array} \\right ) \\int _ { \\R ^ 2 } \\overline a ( y ) f _ 1 ( y ) + \\overline { c } ( y ) f _ 2 ( y ) \\ , d y . \\end{align*}"}
-{"id": "2657.png", "formula": "\\begin{align*} C _ 2 ^ 3 & = 2 \\\\ & = 2 \\cdot 1 + 0 \\\\ & = 2 \\cdot T ( 2 , 0 ) + T ( 2 , - 1 ) \\\\ & = T ( 3 , 0 ) \\end{align*}"}
-{"id": "4820.png", "formula": "\\begin{align*} m a c ( P ( T \\otimes I _ n ) P ) \\mid { \\mathbb R } \\backslash \\omega ( T ) = m a c ( Q ( T \\otimes I _ n ) Q ) \\mid { \\mathbb R } \\backslash \\omega ( T ) . \\end{align*}"}
-{"id": "6987.png", "formula": "\\begin{align*} \\Theta ^ 2 ( x , y ) = \\sum _ { i + j = N + 1 , ~ i , j > 0 } [ f _ i ^ x , f _ j ^ y ] - f _ i ^ { \\lambda _ j ( x , y ) } . \\end{align*}"}
-{"id": "2292.png", "formula": "\\begin{align*} \\big \\| \\big ( \\frac 1 \\tau \\sum _ { j = 0 } ^ k \\delta _ j v _ { n - j } \\big ) _ { n = k } ^ N \\big \\| _ { L ^ p ( D ) } \\le C \\big ( \\| ( f _ n ) _ { n = k } ^ N \\| _ { L ^ p ( D ) } + \\| ( v _ n ) _ { n = k } ^ N \\| _ { L ^ p ( D ) } \\big ) . \\end{align*}"}
-{"id": "3663.png", "formula": "\\begin{align*} g ( x ) = a x , g ( y ) = a y , g ( t ) = b t \\quad g ( x ) = a y , g ( y ) = a x , g ( t ) = b t . \\end{align*}"}
-{"id": "9837.png", "formula": "\\begin{align*} y ( t ) & = h _ { \\rm S D } ( t ) \\otimes s ( t ) + h _ { \\rm R D } ( t ) \\otimes x ( t ) + n _ { \\rm D } ( t ) , \\end{align*}"}
-{"id": "5422.png", "formula": "\\begin{align*} \\sup \\{ | | P ^ n ( x , \\cdot ) - \\mu | | : x \\in S \\} \\leq C \\rho ^ n , \\ ; n = 1 , 2 . . . \\ ; . \\end{align*}"}
-{"id": "2297.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\sum \\limits ^ { k } _ { i = 0 } \\delta _ i e _ { n - i } + A ( t _ n ) e _ n = \\sum \\limits ^ { k - 1 } _ { i = 0 } \\gamma _ i b _ { n - i - 1 } - d _ n , n = k , \\dotsc , N , \\end{align*}"}
-{"id": "9247.png", "formula": "\\begin{align*} J ( u ) = \\mathbb { E } [ \\int _ 0 ^ T f ( X ( t ) , u ( t ) ) d t + g ( X ( T ) ) ] . \\end{align*}"}
-{"id": "2094.png", "formula": "\\begin{align*} \\tilde { V } ^ { T } Z \\tilde { V } = - \\tilde { V } ^ { T } \\eta \\mathbf { X } ^ { T } ( \\mathbf { X } \\mathbf { X } ^ { T } ) ^ { - 1 } \\tilde { V } . \\end{align*}"}
-{"id": "1091.png", "formula": "\\begin{align*} \\sup \\{ | | P ^ n ( x , \\cdot ) - \\mu | | : x \\in S \\} \\leq C \\rho ^ n , \\ ; n = 1 , 2 . . . \\ ; . \\end{align*}"}
-{"id": "468.png", "formula": "\\begin{align*} [ \\beta _ 1 ( y _ 1 - \\alpha _ 1 ) ] ^ 2 - [ \\beta _ 2 ( y _ 2 - \\alpha _ 2 ) ] ^ 2 = \\pm \\eta ^ 2 , \\end{align*}"}
-{"id": "8891.png", "formula": "\\begin{align*} \\mathcal { E } ( d A ) = \\bigl ( \\tfrac { 1 } { 2 } - \\tfrac { 1 } { p } \\bigr ) \\inf _ { v \\in C ^ 1 _ c ( \\R ^ N , \\C ) \\setminus \\{ 0 \\} } \\bigl ( \\mathcal { Q } _ { A } ( v ) \\bigr ) ^ \\frac { p } { p - 2 } . \\end{align*}"}
-{"id": "6206.png", "formula": "\\begin{align*} \\begin{cases} & R ( 1 ) = 1 R ( 0 ) : = \\lim \\limits _ { t \\to 0 } R ( t ) = 0 ; \\\\ & \\int _ J d t / R ( t ) = \\infty ; \\\\ & \\| \\partial ^ k _ t R \\| _ { \\infty } < \\infty , k \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "5012.png", "formula": "\\begin{align*} A B = \\big \\{ a b \\ , : \\ , a \\in A , \\ : b \\in B \\big \\} . \\end{align*}"}
-{"id": "7432.png", "formula": "\\begin{align*} I = \\left \\{ u _ { < f } ^ { - 1 } ( 1 ) , \\dots , u _ { < f } ^ { - 1 } ( k ) \\right \\} J = \\left \\{ v _ { > f } ( 1 ) , \\dots , v _ { > f } ( k ) \\right \\} . \\end{align*}"}
-{"id": "9234.png", "formula": "\\begin{align*} s u p _ { u \\in \\mathcal { A } _ 0 } j ( u ) ( z ) = j ( u _ 0 ^ { \\ast } ) ( z ) . \\end{align*}"}
-{"id": "8035.png", "formula": "\\begin{align*} t _ 1 ^ { a _ 2 ^ k } ( a _ 1 ^ { x _ 1 } a _ 2 ^ { x _ 2 } a _ 3 ^ { x _ 3 } ) & = x _ 1 = t _ 1 \\\\ t _ 2 ^ { a _ 2 ^ k } ( a _ 1 ^ { x _ 1 } a _ 2 ^ { x _ 2 } a _ 3 ^ { x _ 3 } ) & = x _ 2 - 1 = t _ 2 - k \\\\ t _ 3 ^ { a _ 2 ^ k } ( a _ 1 ^ { x _ 1 } a _ 2 ^ { x _ 2 } a _ 3 ^ { x _ 3 } ) & = x _ 3 = t _ 3 \\\\ t _ 1 ^ { a _ 3 ^ k } ( a _ 1 ^ { x _ 1 } a _ 2 ^ { x _ 2 } a _ 3 ^ { x _ 3 } ) & = x _ 1 = t _ 1 \\\\ t _ 2 ^ { a _ 3 ^ k } ( a _ 1 ^ { x _ 1 } a _ 2 ^ { x _ 2 } a _ 3 ^ { x _ 3 } ) & = x _ 2 = t _ 2 \\\\ t _ 3 ^ { a _ 3 ^ k } ( a _ 1 ^ { x _ 1 } a _ 2 ^ { x _ 2 } a _ 3 ^ { x _ 3 } ) & = x _ 3 - 1 = t _ 3 - k \\end{align*}"}
-{"id": "8519.png", "formula": "\\begin{align*} \\begin{cases} \\beta _ { n - 1 , n - 1 } \\prec \\gamma _ { n - 1 , n } & n \\equiv 0 \\bmod 2 , \\\\ \\gamma _ { n - 1 , n } \\prec \\beta _ { n - 1 , n - 1 } & n \\equiv 1 \\bmod 2 . \\end{cases} \\end{align*}"}
-{"id": "9221.png", "formula": "\\begin{align*} I _ 1 = \\mathbb { E } [ \\int _ 0 ^ T ( \\int _ D \\{ h ( t , x ) - \\widehat { h } ( t , x ) \\} \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] d x ) d t ] , I _ 2 = \\mathbb { E } [ \\int _ D \\{ k ( x ) - \\hat { k } ( x ) \\} \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] d x ] . \\end{align*}"}
-{"id": "8197.png", "formula": "\\begin{align*} \\d c _ 1 = \\d c _ 2 = 0 \\ \\ \\ \\ \\ \\ N ( c _ 1 , c _ 2 ) \\cdot A _ { p - 1 } ( c _ 1 , c _ 2 ) \\in A ^ { \\times } \\end{align*}"}
-{"id": "2610.png", "formula": "\\begin{align*} s _ n : H ^ n _ { a b } ( A , N ) & \\to H ^ n ( A , N ) \\\\ \\oplus _ { p _ 1 , \\ldots , p _ r \\geq 1 : r + \\sum _ { i = 1 } ^ r p _ i = n + 1 } \\alpha _ { p _ 1 , \\ldots , p _ r } & \\mapsto \\alpha _ n . \\end{align*}"}
-{"id": "3193.png", "formula": "\\begin{align*} F _ { j , \\alpha } ( z _ j ) = \\left ( \\begin{array} { c } F _ { j , \\alpha } ^ 1 ( z _ j ) \\\\ F _ { j , \\alpha } ^ 2 ( z _ j ) \\\\ \\vdots \\\\ F _ { j , \\alpha } ^ r ( z _ j ) \\end{array} \\right ) \\end{align*}"}
-{"id": "5429.png", "formula": "\\begin{align*} \\begin{bmatrix} q & - \\alpha J ( T ^ { - i - r } , T ^ { - 3 r } ) & 0 & - \\overline \\alpha J ( T ^ { - i - 3 r } , T ^ { - r } ) \\\\ - \\overline \\alpha J ( T ^ { - i } , T ^ { - r } ) & q & - \\alpha J ( T ^ { - i - 2 r } , T ^ { - 3 r } ) & 0 \\\\ 0 & - \\overline \\alpha J ( T ^ { - i - r } , T ^ { - r } ) & q & - \\alpha J ( T ^ { - i - 3 r } , T ^ { - 3 r } ) \\\\ - \\alpha J ( T ^ { - i } , T ^ { - 3 r } ) & 0 & - \\overline \\alpha J ( T ^ { - i - 2 r } , T ^ { - r } ) & q \\\\ \\end{bmatrix} \\end{align*}"}
-{"id": "1387.png", "formula": "\\begin{align*} \\bar \\kappa _ { 1 } = \\Phi ( - \\sqrt { n } c _ S / \\sigma _ S ) . \\end{align*}"}
-{"id": "2531.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & d U = \\mathbf { i } \\left [ \\frac 1 { h ^ 2 } A U + \\lambda F ( U ) U \\right ] d t + \\mathbf { i } Z ( U ) \\circ d \\beta ( t ) , \\\\ & U ( 0 ) = c _ * \\left ( u _ 0 ( x _ 1 ) , \\cdots , u _ 0 ( x _ M ) \\right ) ^ T , \\end{aligned} \\right . \\end{align*}"}
-{"id": "7391.png", "formula": "\\begin{align*} & \\operatorname { i n d } _ { \\mathrm { M o r s e } } ( x ) + 2 c _ 1 ( A ) \\\\ & = ( w - \\operatorname { i n d } _ { \\mathrm { C Z } } ( [ x , c _ x ] ) ) + 2 c _ 1 ( A ) \\\\ & = w - \\operatorname { i n d } _ { \\mathrm { C Z } } ( [ x , c _ x \\sharp A ] ) \\\\ & = - w - ( - w ) = 0 . \\end{align*}"}
-{"id": "2263.png", "formula": "\\begin{align*} \\widehat { \\mu } _ { ( m ) } = C _ { ( 2 ) } ( \\vec { \\varepsilon } ) \\int _ { - \\infty } ^ { \\infty } p _ { G } ( x ) x ^ { m } \\left ( 1 + \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } x ^ { k } + \\frac { 1 } { 2 } \\left ( \\sum _ { k \\neq 2 } ^ { M } \\varepsilon _ { k } x ^ { k } \\right ) ^ { 2 } \\right ) d x , \\end{align*}"}
-{"id": "1117.png", "formula": "\\begin{align*} \\gamma _ d ( 2 \\mathcal E _ s ) = 1 - \\mathsf P ( h _ s ( \\xi ) > 2 ) \\ge 1 - \\frac { \\mathsf { E } h _ s ( \\xi ) ^ 2 } { 4 } \\ge 3 / 4 > \\Psi ( 1 ) . \\end{align*}"}
-{"id": "9913.png", "formula": "\\begin{align*} \\xi _ 3 \\eta _ 4 E \\ , - \\ , \\xi _ 1 \\eta _ 4 K \\ , - \\ , \\xi _ 3 \\eta _ 1 K ' \\ ; = \\ ; \\xi _ 3 \\eta _ 4 F \\ , - \\ , \\xi _ 2 \\eta _ 4 K \\ , - \\ , \\xi _ 3 \\eta _ 2 K ' \\ ; = \\ ; 0 . \\end{align*}"}
-{"id": "2910.png", "formula": "\\begin{align*} X ^ \\pi ( z ) = V _ \\pi ( z ) e ^ { i q } z ^ { \\alpha _ 0 } : = \\sum _ { n \\in { \\mathbb Z } } z ^ { n + { \\alpha _ 0 } } X ^ \\pi _ { - n } \\quad \\mbox { a n d } X ^ { * \\pi } ( z ) = V _ \\pi ^ * ( z ) z ^ { - \\alpha _ 0 } e ^ { - i q } : = \\sum _ { n \\in { \\mathbb Z } } z ^ { - n - { \\alpha _ 0 } } X ^ { * \\pi } _ { { n } } \\ , . \\end{align*}"}
-{"id": "3292.png", "formula": "\\begin{align*} \\psi ( v _ 1 ) = w _ { - 1 } , \\psi ( v _ 0 ) = - w _ { 0 } , \\psi ( v _ { - 1 } ) = q ^ 2 w _ 1 . \\end{align*}"}
-{"id": "8260.png", "formula": "\\begin{align*} \\partial _ t u = \\frac { 1 } { 2 \\sigma _ { \\lambda _ 0 } ^ { 2 } } \\Delta u - \\frac { m _ { 3 , \\lambda _ 0 } } { 2 \\sigma _ { \\lambda _ 0 } ^ 6 } \\nabla u ^ 2 + \\nabla \\xi , \\end{align*}"}
-{"id": "9787.png", "formula": "\\begin{align*} \\Pi _ { n } = \\sum _ { \\alpha \\in A _ { n } } \\lambda _ { n , \\alpha } \\Pi _ { n , \\alpha } . \\end{align*}"}
-{"id": "3448.png", "formula": "\\begin{align*} d y ( t ) = - f ( t , \\xi ( t ) , y ( t ) , z ( t ) ) d t + z ( t ) d w ( t ) , y ( T ) = \\Gamma ^ * ( s , T ) u _ 0 ( \\xi ( T ) ) , \\end{align*}"}
-{"id": "3146.png", "formula": "\\begin{align*} & P r \\biggl ( \\lVert L ^ { - 1 } \\sum _ { l = 1 } ^ { L } \\Pi \\cdot \\Pi _ { X _ l } \\cdot X _ l \\cdot \\Pi _ { X _ l } \\cdot \\Pi - \\rho \\rVert _ 1 \\leq \\epsilon + 4 \\sqrt { \\epsilon } + 2 4 \\sqrt [ 4 ] { \\epsilon } \\biggr ) \\allowdisplaybreaks \\\\ & \\geq 1 - 2 D \\exp \\left ( - \\frac { \\epsilon ^ 3 L d } { 2 \\ln 2 D } \\right ) \\end{align*}"}
-{"id": "318.png", "formula": "\\begin{align*} ( q \\circ g ) _ J ( x ) = \\sum _ { i = 1 } ^ { \\mathrm { c a r d } ( J ) } q ^ { ( i ) } \\bigl ( g ( x ) \\bigr ) \\sum _ { \\{ P _ 1 , \\ldots , P _ i \\} \\in \\mathcal { P } _ i ( J ) } g _ { P _ 1 } \\ldots g _ { P _ i } ( x ) , \\end{align*}"}
-{"id": "3421.png", "formula": "\\begin{align*} d y ( t ) = - f ( t , y ( t ) , z ( t ) ) d t + z ( t ) d w ( t ) , y ( T ) = \\Gamma ^ * ( s , T ) u _ 0 ( \\xi ( T ) ) , \\end{align*}"}
-{"id": "7984.png", "formula": "\\begin{align*} s \\mapsto \\psi ( s ) = \\left [ \\ , \\sum _ { i = 0 } ^ { k - 2 } a _ i s ^ { i + 2 } : \\sum _ { i = 0 } ^ { k - 2 } a _ i s ^ i : \\sum _ { i = 0 } ^ { k } b _ i s ^ i : \\sum _ { i = 0 } ^ { k } c _ i s ^ i \\ , \\right ] , \\end{align*}"}
-{"id": "1980.png", "formula": "\\begin{align*} g _ 2 ( { \\bf x } ^ { \\bf s } ) = c ^ N k _ 1 ^ { s _ 1 } \\cdots k _ n ^ { s _ n } g _ 1 ( { \\bf x } ^ { \\bf s } ) . \\end{align*}"}
-{"id": "3542.png", "formula": "\\begin{align*} \\Phi ( g ^ R , \\pi ^ R ) = R ^ 2 F _ R ^ * \\Phi ( g , \\pi ) . \\end{align*}"}
-{"id": "5009.png", "formula": "\\begin{align*} c _ { 1 } = c _ { 0 } + C _ { 3 } \\sqrt [ ] { c _ { 0 } } \\leq ( 1 + C _ { 3 } ) c _ { 0 } \\leq \\widetilde { C } c _ { 0 } . \\end{align*}"}
-{"id": "2419.png", "formula": "\\begin{align*} \\| f \\| _ { B _ { p , q } ^ s } = \\left ( \\sum _ { j = 0 } ^ { \\infty } 2 ^ { j s q } \\| \\triangle _ j f \\| _ { L ^ p } ^ q \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "6384.png", "formula": "\\begin{align*} \\int _ { - \\infty } ^ { \\infty } p ( x ) d x = C ( \\varepsilon ) \\int _ { \\infty } ^ { \\infty } \\frac { 1 } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } e ^ { - \\varepsilon x ^ { p } } d x = 1 . \\end{align*}"}
-{"id": "5659.png", "formula": "\\begin{align*} \\nu ( v ) = \\nu ( - v ) \\mbox { f o r a l l } v \\in \\R ^ d . \\end{align*}"}
-{"id": "1352.png", "formula": "\\begin{align*} Q _ { L } ^ { S T } ( 0 ) = S ( \\hat \\beta _ 0 ) ^ T I _ { L } ( \\hat \\beta _ 0 ) ^ { - 1 } S ( \\hat \\beta _ 0 ) = \\sum _ { l = 1 } ^ { L } \\hat { C } ^ 2 ( l ) / \\hat { B } ( l ) \\end{align*}"}
-{"id": "4343.png", "formula": "\\begin{align*} \\frac { d } { d r } \\left [ \\frac { p - 1 } { p } | g | ^ { p / ( p - 1 ) } + \\frac { 1 } { 2 } f ^ 2 \\right ] & = | g | ^ { ( 2 - p ) / ( p - 1 ) } g ( f - | g | ) - | g | ^ { ( 2 - p ) / ( p - 1 ) } g f \\\\ & = - | g | ^ { 1 / ( p - 1 ) } g \\\\ & \\le \\frac { p } { p - 1 } \\left [ \\frac { p - 1 } { p } | g | ^ { p / ( p - 1 ) } + \\frac { 1 } { 2 } f ^ 2 \\right ] \\ , \\end{align*}"}
-{"id": "7669.png", "formula": "\\begin{align*} \\int _ { 1 } ^ { \\infty } s ^ { - ( 1 + \\alpha / d ) } F ( s ) d s < \\infty , \\ w h e r e \\ F ( s ) = \\sup _ { 1 \\leq t \\leq s } \\frac { f ( t ) } { t } , \\end{align*}"}
-{"id": "6903.png", "formula": "\\begin{gather*} B _ 0 ^ - = \\begin{pmatrix} \\tfrac { 1 } { 2 } \\delta u & e _ 0 \\\\ 0 & - \\tfrac { 1 } { 2 } \\delta u \\end{pmatrix} . \\end{gather*}"}
-{"id": "8024.png", "formula": "\\begin{align*} & \\phi ( x _ j ) = \\mu _ j x _ j , \\phi ( y _ j ) = \\nu _ j y _ j , \\ ; \\ ; \\mbox { f o r a l l } \\ ; \\ ; j \\neq k , \\\\ & \\phi ( x _ k ) = \\mu _ k y _ k , \\phi ( y _ k ) = \\nu _ k x _ k \\end{align*}"}
-{"id": "8970.png", "formula": "\\begin{align*} \\left ( \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | w ( x ) - w ( y ) | ^ p } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y \\right ) ^ { \\frac { 1 } { p } } \\leq C n ^ { \\frac { \\gamma + 1 } { p - 1 } } , \\end{align*}"}
-{"id": "2534.png", "formula": "\\begin{align*} \\mathcal { S } _ 2 = & \\{ U = P + \\mathbf { i } Q \\in \\mathcal { S } : \\ ; P < 0 \\} , \\\\ \\mathcal { S } _ 3 = & \\{ U = P + \\mathbf { i } Q \\in \\mathcal { S } : \\ ; Q > 0 \\} , \\\\ \\mathcal { S } _ 4 = & \\{ U = P + \\mathbf { i } Q \\in \\mathcal { S } : \\ ; Q < 0 \\} . \\end{align*}"}
-{"id": "6431.png", "formula": "\\begin{align*} \\partial _ { s } ^ { \\alpha } ( \\tilde { u } - \\tilde { u } _ { 0 } ) + L \\tilde { u } = ( \\geq \\leq ) \\tilde { f } , s \\in ( 0 , t _ { 0 } ) , \\ , \\ , y \\in B ( 0 , 1 ) . \\end{align*}"}
-{"id": "6003.png", "formula": "\\begin{gather*} x \\times y : = x y + H ( x , y ) 1 ; \\end{gather*}"}
-{"id": "2302.png", "formula": "\\begin{align*} J _ m \\le C \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( X ) } + C \\delta . \\end{align*}"}
-{"id": "7181.png", "formula": "\\begin{align*} f ^ { '' } + f ' \\cot \\varphi = g f ' - ( f ^ 2 + g ^ 2 ) - 2 p , \\end{align*}"}
-{"id": "2519.png", "formula": "\\begin{align*} D _ 1 ^ 2 = D _ 2 ^ 2 = - 2 , ~ D _ 1 - D _ 2 = H \\end{align*}"}
-{"id": "8584.png", "formula": "\\begin{align*} W ( c _ { 1 } ) & = \\displaystyle \\sum _ { i } \\displaystyle \\sum _ { s \\in L ( \\sigma ( b ) ) } ( s \\rho ( b w a ) ) ^ { \\ast } \\left ( \\hat { \\Delta } ( \\rho ( c _ { i } ) ) \\Diamond t _ { i } z ( \\rho ( c _ { 1 } ) ) r _ { i } \\right ) s \\\\ & = \\displaystyle \\sum _ { i } \\displaystyle \\sum _ { s \\in L ( \\sigma ( b ) ) } ( s \\rho ( b w a ) ) ^ { \\ast } \\left ( \\rho ( c _ { i } ) t _ { i } z ( \\rho ( c _ { 1 } ) ) r _ { i } \\right ) s = 0 . \\end{align*}"}
-{"id": "8450.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\sum _ { j = 1 } ^ { d } \\textbf { P } J _ { \\varepsilon } A _ { j } ( J _ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\bar { \\textbf { u } } ) ) \\partial _ { x _ { j } } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } = 0 , \\end{align*}"}
-{"id": "4697.png", "formula": "\\begin{align*} \\tilde { x } = \\phi _ k ^ { - 1 } ( y ' , \\mu _ k ( z ' ) 0 ^ { n - k - \\abs { \\mu _ k ( z ' ) } } ) , \\end{align*}"}
-{"id": "3497.png", "formula": "\\begin{align*} \\Delta w = \\begin{cases} 0 & \\mbox { i n } ( \\{ v _ 1 > 0 \\} \\cup \\{ v _ 2 = 0 \\} ) \\times ( 0 , 1 ) , \\\\ 1 - 2 \\partial _ \\nu u _ 1 & \\mbox { i n } ( \\{ v _ 1 = 0 \\} \\setminus \\{ v _ 2 > 0 \\} ) \\times ( 0 , 1 ) , \\end{cases} \\end{align*}"}
-{"id": "2817.png", "formula": "\\begin{align*} \\xi _ p ( t ) = \\sup \\{ s : 0 \\le s \\le t , X _ { r : n } ( s ) \\ge f _ p ( s ) \\} . \\end{align*}"}
-{"id": "1777.png", "formula": "\\begin{align*} ( u ^ { \\varepsilon } ( x ) - v ^ { \\varepsilon } ( x + 1 ) ) ^ { \\prime \\prime } = 0 . \\end{align*}"}
-{"id": "4176.png", "formula": "\\begin{align*} \\sum _ { b \\in A } Z ( a , b ) w _ b = 1 a \\in A \\end{align*}"}
-{"id": "2791.png", "formula": "\\begin{align*} \\frac { 1 } { ( d - 2 ) ^ { 3 / 2 } } \\biggl ( \\frac { \\hat { d } } { ( d - 2 ) ^ { d / 2 - 1 } } \\biggr ) ^ { n } = \\hat { d } ^ { n } \\left ( \\frac { n } { 2 x } \\right ) ^ { 3 / 2 + x } . \\end{align*}"}
-{"id": "8603.png", "formula": "\\begin{align*} \\operatorname { i m } ( \\overline { \\beta } ) = \\operatorname { k e r } ( \\alpha ) \\end{align*}"}
-{"id": "1425.png", "formula": "\\begin{align*} \\C \\ \\backslash \\bigcap _ { n \\geq 0 } U ( n ) = \\bigcup _ { n \\geq 0 } ( \\C \\backslash U ( n ) ) \\end{align*}"}
-{"id": "7959.png", "formula": "\\begin{align*} \\mathcal H ^ { n - 1 } ( \\{ \\psi _ \\lambda = 0 \\} \\cap B _ { r } ( p ) ) & \\leq \\mathcal H ^ { n - 1 } ( \\{ \\psi _ \\lambda = 0 \\} \\cap Q _ r ( p ) ) \\\\ & = \\frac { 1 } { 2 r } \\mathcal H ^ { n } ( \\{ H = 0 \\} \\cap \\tilde Q _ r ( \\tilde p ) ) \\\\ & \\leq \\frac { \\kappa } { 2 r } ( 2 r ) ^ n N ( H , \\tilde Q _ r ( \\tilde p ) ) ^ { 2 \\alpha } \\\\ & = \\kappa ' r ^ { n - 1 } N ( H , \\tilde Q _ r ( \\tilde p ) ) ^ { 2 \\alpha } \\end{align*}"}
-{"id": "1131.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to \\infty } \\gamma _ k ^ \\mathrm { D L } [ \\iota ] = \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } } . \\end{align*}"}
-{"id": "6004.png", "formula": "\\begin{align*} H ( x \\times y , x ) = 0 \\textrm { a n d } H ( x \\times y , x \\times y ) = H ( x , x ) H ( y , y ) - H ( x , y ) ^ 2 \\end{align*}"}
-{"id": "3758.png", "formula": "\\begin{align*} \\hat { f _ Y } ( t ) = \\int _ { - \\infty } ^ { \\infty } e ^ { - i t y } f _ Y ( y ) d y = e ^ { C _ 0 ( \\beta - i \\alpha t ) - C _ 0 ( \\beta ) + i \\alpha t u } - P ( T _ \\beta = 0 ) e ^ { i \\alpha t u } . \\end{align*}"}
-{"id": "9274.png", "formula": "\\begin{align*} U ( x , y , z ) = k ( x , z ) \\ln ( y ) , \\end{align*}"}
-{"id": "1035.png", "formula": "\\begin{align*} ( - 1 ) ^ { u - i } { u \\choose i } { \\ , } c _ { r , u } c _ { d , d } ^ u + \\delta _ { r , r _ 0 } \\delta _ { u , \\frac { q q _ 2 } { d } } { \\ , } \\chi _ { \\{ \\frac { q _ 1 q _ 2 } { d } , \\frac { q _ 2 ^ 2 } { d } \\} } ( i ) { \\ , } c _ { q , q _ 1 } ^ { q _ 2 + 1 } ~ = ~ 0 . \\end{align*}"}
-{"id": "8906.png", "formula": "\\begin{align*} R _ \\lambda ( \\theta ) = ( I - P _ { W _ \\lambda } ) + \\bigl ( \\cos \\theta \\ , + \\sin \\theta \\ , \\Hat { A } / \\lambda \\bigr ) \\circ P _ { W _ \\lambda } \\end{align*}"}
-{"id": "3051.png", "formula": "\\begin{align*} f _ \\mu ( u ) = x + \\sum _ { j = 1 } ^ \\infty f _ \\mu ( u . j ) , \\end{align*}"}
-{"id": "9677.png", "formula": "\\begin{align*} x ' ( t ) & = - a g ( x ( t ) ) + b g ( x ( t - \\tau ( t ) ) , t \\geq 0 \\\\ x ( t ) & = \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] \\end{align*}"}
-{"id": "109.png", "formula": "\\begin{align*} I _ { - 1 } ( f _ 1 ) + I _ { - 1 } ( f ) = 2 f ( 0 ) , \\textnormal { w h e r e } \\ , \\ , f _ 1 ( x ) = f ( x + 1 ) . \\end{align*}"}
-{"id": "7889.png", "formula": "\\begin{align*} \\frac { ( B - A ) \\times ( A - I _ c ) } { \\left \\| B - A \\right \\| } = \\pm r . \\end{align*}"}
-{"id": "5018.png", "formula": "\\begin{align*} \\lambda ( f ) = f ( e ) , \\textrm { f o r a l l $ f \\in H ^ \\infty _ r ( G , p ) $ } . \\end{align*}"}
-{"id": "1816.png", "formula": "\\begin{align*} u ' q ' { \\frac { d \\bar z } { \\bar z } } \\cdot \\frac { d z } z = u ' q ' \\frac { d r } r d \\theta . \\end{align*}"}
-{"id": "3187.png", "formula": "\\begin{align*} u _ n ( Y , X ; \\{ w _ j \\} ) = ( u ^ \\lambda _ \\alpha ( Y , X ; \\{ w _ j \\} ) ) _ { \\lambda , \\alpha } \\in \\bigoplus _ { \\lambda = 1 } ^ r \\bigoplus _ { | \\alpha | = n + 1 } H ^ 1 ( Y , N _ \\lambda \\otimes N _ \\alpha ^ { - 1 } ) \\end{align*}"}
-{"id": "1725.png", "formula": "\\begin{gather*} \\Phi = \\mathrm { R e } \\Psi + \\varepsilon \\ , \\mathbb { S } ^ { \\flat } \\wedge \\mathbb { K } \\in \\Gamma \\big ( \\Lambda ^ 3 \\mathcal { V } \\big ) \\end{gather*}"}
-{"id": "7590.png", "formula": "\\begin{align*} x _ 1 = x _ 0 + F ( x _ 0 ) + l \\chi _ 1 \\ge x _ 0 + l - \\frac { l \\varepsilon } { 2 C } + l \\left ( - 1 + \\frac { \\varepsilon } { C } \\right ) = x _ 0 + \\frac { l \\varepsilon } { 2 C } > u _ l - \\delta + \\frac { l \\varepsilon } { 2 C } _ l \\geq u _ l . \\end{align*}"}
-{"id": "4071.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 6 , 6 u + 3 , 6 v + 1 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 6 , 6 u + 3 , 6 v + 5 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "9701.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ' ( x ) } { \\alpha x ^ { - \\alpha - 1 } e ^ { - 1 / x ^ \\alpha } } = 1 . \\end{align*}"}
-{"id": "3653.png", "formula": "\\begin{align*} A = \\max _ { P _ c \\in I } \\left \\{ \\max _ { i = 1 , \\dots , k } \\left | \\frac { \\partial f } { \\partial x _ i } ( \\mathbf { P } _ c ) \\right | ^ { \\frac { 1 } { i } } \\right \\} . \\end{align*}"}
-{"id": "1909.png", "formula": "\\begin{align*} f _ 0 ( x ) = c _ 1 g ( x ) + c _ 2 h ( x ) . \\end{align*}"}
-{"id": "5206.png", "formula": "\\begin{align*} \\bar \\partial _ M f = ( \\bar \\partial \\widetilde { f } ) _ { t _ M } \\ ; . \\end{align*}"}
-{"id": "2278.png", "formula": "\\begin{align*} \\begin{aligned} \\| v \\| _ { W } \\le \\varepsilon \\| v \\| _ { D } + C _ \\varepsilon \\| v \\| _ { X } \\quad \\forall \\ , v \\in D . \\end{aligned} \\end{align*}"}
-{"id": "9085.png", "formula": "\\begin{align*} \\frac { \\mathrm { d } } { \\mathrm { d } x } \\left ( x \\rho ^ { ( 2 ) } _ { N } ( x ) \\right ) = \\sqrt { N ( N + \\alpha ) } L ^ { ( \\alpha ) } _ { N } ( x ) L ^ { ( \\alpha ) } _ { N - 1 } ( x ) x ^ { \\alpha } e ^ { - x } \\ . \\end{align*}"}
-{"id": "5943.png", "formula": "\\begin{align*} X _ t & = x _ 0 + \\int _ 0 ^ t V _ s \\ , \\dd s = x _ 0 + t v _ 0 + \\int _ 0 ^ t ( t - s ) F ( X _ s , V _ s ) \\ , \\dd s + \\int _ 0 ^ t W _ s \\ , \\dd s \\ , , \\\\ V _ t & = v _ 0 + \\int _ 0 ^ t F ( X _ s , V _ s ) \\ , \\dd s + W _ t \\ , . \\end{align*}"}
-{"id": "7481.png", "formula": "\\begin{align*} ( V ) = \\sum _ { j = 0 } ^ { \\kappa ' k } Q _ j ^ { ( k ) } ( V ) . \\end{align*}"}
-{"id": "4075.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ p y ^ q ( b y + c z ) ^ r } { z ^ { p + q + r } } \\end{align*}"}
-{"id": "6638.png", "formula": "\\begin{align*} \\rho ( w ) N \\rho ( w ) ^ { - 1 } = | | w | | N . \\end{align*}"}
-{"id": "5740.png", "formula": "\\begin{align*} I _ { L } ( \\hat \\delta _ 0 ) = n ^ { - 1 } \\sum _ { t = 1 } ^ n \\sigma _ { t } ^ 2 ( \\hat \\beta _ 0 ) \\Gamma _ { t , L } ( \\hat \\delta _ 0 ) . \\end{align*}"}
-{"id": "9378.png", "formula": "\\begin{align*} W _ \\lambda = \\left [ \\begin{array} { c c } ( { q } _ 1 , G \\ast { q _ 1 } ) & ( { q } _ 1 , G \\ast { q _ 2 } ) \\\\ ( { q } _ 2 , G \\ast { q _ 1 } ) & ( { q } _ 2 , G \\ast { q _ 2 } ) \\end{array} \\right ] + \\left [ \\begin{array} { c c } r _ { 1 1 } & r _ { 1 2 } \\\\ r _ { 2 1 } & r _ { 2 2 } \\end{array} \\right ] , \\end{align*}"}
-{"id": "5633.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( \\mu , \\nu ) = \\underset { ( \\phi , \\psi ) \\in C _ W } { \\sup } \\int _ { \\Omega } \\phi ( x ) d \\mu ( x ) + \\int _ { \\Omega } \\psi ( y ) d \\nu ( y ) , \\end{align*}"}
-{"id": "2323.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( H ^ 4 ( \\R ^ d ) ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( H ^ 4 ( \\R ^ d ) ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "2507.png", "formula": "\\begin{align*} F ( x ( t ; \\overline { x } ) ) = \\exp ( - 2 \\lambda t ) \\cdot F ( \\overline { x } ) , ~ ( \\forall ) t \\in I _ { \\overline { x } } . \\end{align*}"}
-{"id": "1355.png", "formula": "\\begin{align*} S ( \\delta _ 0 ) = n ^ { - 1 / 2 } \\sum _ { t = 1 } ^ n ( y _ t - m _ t \\pi _ { t } ( \\beta _ 0 ) ) \\left ( \\sum _ { i = 0 } ^ { ( t - 2 ) } \\omega ^ { i } e _ { t - 1 - i } ^ P ( \\beta _ 0 ) \\right ) . \\end{align*}"}
-{"id": "903.png", "formula": "\\begin{align*} \\zeta ( s ) = \\sum _ { n \\leq x } { n ^ { - s } } + \\frac { x ^ { 1 - s } } { s - 1 } + O ( x ^ { - \\sigma } ) , \\end{align*}"}
-{"id": "5766.png", "formula": "\\begin{align*} f ( \\lambda , \\mu ) = \\frac { \\lambda - \\mu + 1 } { \\lambda - \\mu } , g ( \\lambda , \\mu ) = \\frac { 1 } { \\lambda - \\mu } \\end{align*}"}
-{"id": "319.png", "formula": "\\begin{align*} q ^ { ( i ) } ( y ) = ( 2 t ) ^ i \\sum _ { \\ell = 1 } ^ i ( - 1 ) ^ { i - \\ell } \\frac { a _ \\ell ^ { ( i ) } e ^ { - 2 t \\ell y } } { ( 1 + e ^ { - 2 t y } ) ^ { \\ell + 1 } } , \\end{align*}"}
-{"id": "280.png", "formula": "\\begin{align*} \\mathcal { X } _ n ' = \\mathcal { X } _ n \\cup \\{ y : \\| y - x \\| \\leq r _ { n , v _ x } + r _ { n , v _ y } \\ , \\ , \\ , \\ , x \\in \\mathcal { X } _ n \\} . \\end{align*}"}
-{"id": "3512.png", "formula": "\\begin{align*} ( f \\circ \\gamma ) '' ( t ) & = f _ { ; i j } | _ { \\gamma ( t ) } \\dot \\gamma ^ i ( t ) \\dot \\gamma ^ j ( t ) \\\\ \\left ( \\frac { D ^ 2 X ( \\gamma ( t ) ) } { d t ^ 2 } \\right ) ^ k & = ( \\nabla _ { \\dot \\gamma } \\nabla _ { \\dot \\gamma } X ) ^ k = X ^ k _ { \\ ; ; i j } | _ { \\gamma ( t ) } \\dot \\gamma ^ i ( t ) \\dot \\gamma ^ j ( t ) . \\end{align*}"}
-{"id": "6696.png", "formula": "\\begin{align*} \\omega = \\nu ^ { - 1 } \\omega _ { - 1 } + \\omega _ 0 + \\nu \\omega _ 1 + \\ldots \\end{align*}"}
-{"id": "6327.png", "formula": "\\begin{align*} 2 ^ { j A _ 2 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 3 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "5989.png", "formula": "\\begin{align*} d _ i = 0 \\ \\Leftrightarrow \\ i \\leq M . \\end{align*}"}
-{"id": "9242.png", "formula": "\\begin{align*} \\delta ( t , z ) = \\frac { 1 } { 2 K } d i s t ( ( u ( t , z ) , \\partial \\mathbb { U } ) \\wedge 1 > 0 \\end{align*}"}
-{"id": "8563.png", "formula": "\\begin{align*} \\varphi ( d _ { 1 } ( ^ { \\ast } b ) d _ { 2 } n ) & = \\psi ( d _ { 1 } ( ^ { \\ast } b ) d _ { 2 } n ) \\\\ & = \\psi ( d _ { 1 } \\overline { N } ( ^ { \\ast } b ) ( d _ { 2 } n ) ) \\\\ & = d _ { 1 } \\psi ( \\overline { N } ( ^ { \\ast } b ) ( d _ { 2 } n ) ) \\\\ & = d _ { 1 } \\overline { N ' } ( ^ { \\ast } b ) ( d _ { 2 } n ) \\\\ & = d _ { 1 } ( ^ { \\ast } b ) d _ { 2 } \\varphi ( n ) \\end{align*}"}
-{"id": "2487.png", "formula": "\\begin{align*} \\mathcal { I } _ W ( T ) : = \\{ ( w , t ) : w \\in W , t \\in T , w t = t \\} \\end{align*}"}
-{"id": "8161.png", "formula": "\\begin{align*} S _ 2 ^ 2 = t * [ 1 ] \\cup m _ 2 * [ a ^ { d _ 2 } ] \\end{align*}"}
-{"id": "2215.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } u ( x , t ) + L u ( x , t ) & = \\rho ( t ) g ( x ) \\Omega \\times [ 0 , T ] , \\\\ u ( x , t ) & = 0 \\quad \\quad \\quad \\mathbb { R } ^ { n } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = 0 \\quad \\quad \\quad \\Omega , \\ , t = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4935.png", "formula": "\\begin{align*} v = & \\sum _ { i = 1 } ^ { M } \\lambda _ i u _ i 0 \\leq \\lambda _ i \\leq 1 , \\ , \\sum _ { i = 1 } ^ { M } \\lambda _ i = 1 , \\\\ & u _ i \\in \\mathit { U } ( \\alpha , s , v ) . \\end{align*}"}
-{"id": "5133.png", "formula": "\\begin{align*} u ( a , b , a ^ { m } ) = q ^ { - 1 } \\left \\{ f ( z _ { \\ell } , z _ { 1 } ) Y _ { 0 } ( z _ { \\ell } , z _ { 1 } ) - g ( z _ { \\ell } , z _ { 1 } ) \\right \\} u ( b , a ^ { m + 1 } ) \\end{align*}"}
-{"id": "368.png", "formula": "\\begin{align*} \\hat { \\ell } _ { \\bar { d } _ \\mathcal { E } } = \\hat { \\ell } _ { m i n } + \\left ( \\hat { \\ell } _ { m a x } - \\hat { \\ell } _ { m i n } \\right ) \\mu ( \\bar { d } _ { \\mathcal { E } } ) . \\end{align*}"}
-{"id": "1332.png", "formula": "\\begin{align*} [ T _ { j k } ( \\lambda ) , T _ { j k } ( \\mu ) ] & = 0 , j , k = 1 , 2 , \\\\ A ( \\mu ) B ( \\lambda ) & = f ( \\lambda , \\mu ) B ( \\lambda ) A ( \\mu ) + g ( \\mu , \\lambda ) B ( \\mu ) A ( \\lambda ) , \\\\ B ( \\mu ) A ( \\lambda ) & = f ( \\lambda , \\mu ) A ( \\lambda ) B ( \\mu ) + g ( \\mu , \\lambda ) A ( \\mu ) B ( \\lambda ) , \\\\ D ( \\mu ) B ( \\lambda ) & = f ( \\mu , \\lambda ) B ( \\lambda ) D ( \\mu ) + g ( \\lambda , \\mu ) B ( \\mu ) D ( \\lambda ) , \\\\ & \\end{align*}"}
-{"id": "6314.png", "formula": "\\begin{align*} \\left \\| \\Box _ k ^ { \\alpha _ 2 } | ~ M _ 1 \\rightarrow M _ 2 \\right \\| \\gtrsim \\frac { \\| \\Box _ k ^ { \\alpha _ 2 } f _ l ^ { \\alpha _ 1 } \\| _ { M _ 2 } } { \\| f _ l ^ { \\alpha _ 1 } \\| _ { M _ 1 } } \\sim \\frac { \\| f _ l ^ { \\alpha _ 1 } \\| _ { L ^ { p _ 2 } } } { \\| f _ l ^ { \\alpha _ 1 } \\| _ { L ^ { p _ 1 } } } \\sim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\\\ = & 2 ^ { j A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\end{align*}"}
-{"id": "8823.png", "formula": "\\begin{align*} Y _ g ( x _ { i , 0 } , z ^ { 1 / m } ) = \\sum _ { n \\in \\frac { \\alpha _ i } { m } + \\mathbb { Z } , n \\leq 0 } x _ { i , n } z ^ { - n } \\\\ \\end{align*}"}
-{"id": "6280.png", "formula": "\\begin{align*} ~ N _ p ( x ) = p ( x ) / p ' ( x ) , \\end{align*}"}
-{"id": "9194.png", "formula": "\\begin{align*} \\frac { \\phi ' ( u ) } { u } & = \\frac { 1 } { \\ln 2 } \\cdot \\Bigl ( 1 + \\sum _ { n \\ge 1 } \\frac { v ^ { 2 n } } { 2 n + 1 } \\Bigr ) \\\\ \\phi '' ( u ) & = \\frac { 1 } { \\ln 2 } \\cdot \\Bigl ( \\frac { 1 } { 3 } + \\sum _ { n \\ge 1 } \\frac { v ^ { 2 n } } { 2 n + 3 } \\Bigr ) \\end{align*}"}
-{"id": "3490.png", "formula": "\\begin{align*} \\alpha = \\sup _ { \\widetilde { D } } \\hat { v } \\leq \\inf _ { D \\setminus \\widetilde { D } } \\hat { v } . \\end{align*}"}
-{"id": "3262.png", "formula": "\\begin{align*} S ^ { - 1 } ( E _ { \\xi ^ \\prime } ^ * ) = ( S ^ { - 1 } ( S ^ 2 ( E _ { \\xi ^ \\prime } ) ) ) ^ * = q ^ { ( \\xi , 2 \\rho ) } S ^ { - 1 } ( E _ { \\xi ^ \\prime } ) ^ * . \\end{align*}"}
-{"id": "6831.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u = \\mu | u | ^ { p } u . \\end{align*}"}
-{"id": "1585.png", "formula": "\\begin{align*} L _ { Y _ { I } } C _ { x } - m Y _ { I } = 0 , \\end{align*}"}
-{"id": "354.png", "formula": "\\begin{align*} e ^ { i J ( g ) } J ( f ) e ^ { - i J ( g ) } = J ( f ) + \\int _ { - \\pi } ^ { \\pi } ( \\partial _ \\theta g ) ( e ^ { i \\theta } ) f ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } \\ ; I \\end{align*}"}
-{"id": "5145.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { m } C _ { a } ( z _ { i } ) \\cdot \\beta _ { a , 1 } ^ { * } \\equiv \\beta _ { a , 1 } ^ { * } \\prod _ { i = 1 } ^ { m } C _ { a } ( z _ { i } ) + ( 1 - q ^ 2 ) \\sum _ { \\ell = 1 } ^ { r } z _ { \\ell } \\prod _ { \\begin{subarray} { c } i = 1 \\\\ i \\not = \\ell \\end{subarray} } ^ { m } f ( z _ { i } , z _ { \\ell } ) \\cdot \\tilde { A } _ { a } ( z _ { \\ell } ) \\prod _ { \\begin{subarray} { c } i = 1 \\\\ i \\not = \\ell \\end{subarray} } ^ { m } C _ { a } ( z _ { i } ) \\end{align*}"}
-{"id": "5836.png", "formula": "\\begin{align*} X _ { L } = \\left ( c _ { 1 } + 2 \\psi _ { I } \\int T ^ { I } \\left ( t \\right ) d t \\right ) + \\left ( T ^ { I } \\left ( t \\right ) Y _ { I } ^ { \\alpha } \\left ( x ^ { \\beta } \\right ) \\right ) \\partial _ { \\alpha } + \\left ( a \\left ( x ^ { \\beta } , t \\right ) u + b \\left ( x ^ { \\beta } , t \\right ) + c _ { 2 } u \\right ) \\partial _ { u } , \\end{align*}"}
-{"id": "9484.png", "formula": "\\begin{align*} \\Phi ^ { * } \\lambda _ { 0 } = e ^ { 2 \\rho } \\lambda . \\end{align*}"}
-{"id": "7520.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\left \\{ f ( x _ n ) - \\left ( \\alpha + l \\xi _ { n + 1 } \\right ) ( f ( x _ n ) - x _ n ) , 0 \\right \\} , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 . \\end{align*}"}
-{"id": "4297.png", "formula": "\\begin{align*} h | _ { \\Omega _ \\alpha } = e ^ { - \\varphi _ \\alpha } | \\cdot | ^ 2 \\end{align*}"}
-{"id": "8155.png", "formula": "\\begin{align*} c ( s , n ) = \\frac { \\varphi ( n ) } { \\varphi ( \\frac { n } { ( s , n ) } ) } \\mu ( \\frac { n } { ( s , n ) } ) \\end{align*}"}
-{"id": "2313.png", "formula": "\\begin{align*} & \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { - 1 , q } ( \\varOmega ) ) } + \\big \\| ( e _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( W ^ { 1 , q } ( \\varOmega ) ) } \\le C \\tau ^ k , \\\\ & \\max _ { k \\le n \\le N } \\| e _ n \\| _ { L ^ \\infty ( \\varOmega ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "635.png", "formula": "\\begin{align*} A B = \\big \\{ a b \\ , : \\ , a \\in A , \\ : b \\in B \\big \\} . \\end{align*}"}
-{"id": "2201.png", "formula": "\\begin{align*} R _ { m } ( t ) : = \\mathcal { E } ( h _ { m } * \\tilde { u } , \\psi ^ { 2 } \\tilde { u } ^ { - 1 } ) - \\mathcal { E } ( \\tilde { u } , \\psi ^ { 2 } \\tilde { u } ^ { - 1 } ) . \\end{align*}"}
-{"id": "5034.png", "formula": "\\begin{align*} f _ \\phi ( g ) = \\frac { \\nu ( ( g ^ { - 1 } \\cdot \\phi ) \\chi _ C ) } { \\nu ( C ) } , \\textrm { f o r $ g \\in G $ } . \\end{align*}"}
-{"id": "9899.png", "formula": "\\begin{align*} \\mathcal { D } ( G ) = \\begin{pmatrix} \\begin{matrix} 0 & 1 & 1 & \\dots & 1 \\\\ 1 & 0 & 2 & \\dots & 2 \\\\ 1 & 2 & 0 & \\dots & 2 \\\\ \\dots & \\dots & \\dots & \\dots & \\dots \\\\ 1 & 2 & 2 & \\dots & 0 \\end{matrix} & \\\\ & \\\\ \\end{pmatrix} . \\end{align*}"}
-{"id": "3199.png", "formula": "\\begin{align*} h _ { 1 , j k , \\alpha } ( z _ j ) = \\left ( \\begin{array} { c } h _ { 1 , j k , \\alpha } ^ 1 ( z _ j ) \\\\ h _ { 1 , j k , \\alpha } ^ 2 ( z _ j ) \\\\ \\vdots \\\\ h _ { 1 , j k , \\alpha } ^ r ( z _ j ) \\end{array} \\right ) \\end{align*}"}
-{"id": "9100.png", "formula": "\\begin{align*} & ( 4 s ^ { 2 } - 1 ) M _ { N } ^ { ( 1 ) \\prime \\prime } + ( 1 6 s + 2 ( \\alpha - 1 ) + 4 N ) M _ { N } ^ { ( 1 ) \\prime } + ( 9 - \\alpha ^ { 2 } ) M _ { N } ^ { ( 1 ) } \\\\ & + ( 1 - s ^ { 2 } ) M _ { N - 1 } ^ { ( 2 ) \\prime \\prime } - ( 5 s + 2 \\alpha - 1 + 4 N ) M _ { N - 1 } ^ { ( 2 ) \\prime } + ( \\alpha ^ { 2 } - 4 ) M _ { N - 1 } ^ { ( 2 ) } = 0 \\ . \\end{align*}"}
-{"id": "5105.png", "formula": "\\begin{align*} ( H _ { 1 } - M ) | \\mathbf { s } \\rangle = ( 1 - q ^ { 2 } ) \\sum _ { \\mathbf { s } ' \\in \\mathcal { S } _ { M } ^ { \\mathrm { p e r } } \\setminus \\{ \\mathbf { s } \\} } q ( \\mathbf { s } | \\mathbf { s } ' ) ( | \\mathbf { s } ' \\rangle - | \\mathbf { s } \\rangle ) , \\end{align*}"}
-{"id": "5452.png", "formula": "\\begin{align*} R ( n ) = \\sum _ { h _ 1 + h _ 2 = n } \\Lambda ( h _ 1 ) \\Lambda ( h _ 2 ) \\end{align*}"}
-{"id": "2813.png", "formula": "\\begin{align*} \\int \\limits _ { \\mathbb R } \\displaystyle \\frac { \\frac { \\partial } { \\partial y } | E ( x + i y ) | ^ 2 | _ { y = 0 } } { ( 1 + x ^ 2 ) | E ( x ) | ^ { 2 } } d x < + \\infty . \\end{align*}"}
-{"id": "5729.png", "formula": "\\begin{align*} \\max _ { \\substack { R \\in \\mathcal { D } ( Q _ 0 ) : \\\\ R ^ { ( 1 ) } = Q ( x ) } } | m _ { f _ 1 } ( R ) | > ( f _ 1 \\cdot \\chi _ { Q _ 0 } ) ^ * ( \\lambda _ w | Q _ 0 | ) . \\end{align*}"}
-{"id": "9931.png", "formula": "\\begin{align*} E = \\frac { i } { 2 } ( 1 - i b ) ( x _ 2 + i x _ 3 ) , F = \\frac { i } { 2 } ( 1 + i b ) ( x _ 2 - i x _ 3 ) , K = x _ 0 + b x _ 1 , K ' = x _ 0 - b x _ 1 . \\end{align*}"}
-{"id": "7790.png", "formula": "\\begin{align*} f _ i ^ * f _ j e _ i ^ * e _ j X = f _ j f _ i ^ * e _ i ^ * e _ j X = X . \\end{align*}"}
-{"id": "1157.png", "formula": "\\begin{align*} & \\frac 1 2 b _ { x x } ( 0 ) - h b _ x ( 0 ) + h ^ 2 b ( 0 ) = 0 , \\\\ & \\frac 1 2 b _ { x x } ( 1 ) + H b _ x ( 1 ) + H ^ 2 b ( 1 ) = 0 , \\end{align*}"}
-{"id": "7849.png", "formula": "\\begin{align*} h _ 0 ( x ) = x ^ { - \\alpha } h ( x ) , \\end{align*}"}
-{"id": "4999.png", "formula": "\\begin{align*} & A = ( a _ { m i } ) _ { m i } \\ \\ \\ s \\times r \\\\ & B = ( b _ { l j } ) _ { l j } \\ \\ r \\times s . \\end{align*}"}
-{"id": "7899.png", "formula": "\\begin{align*} I \\ , - \\ , r _ 1 \\ , \\xi _ 1 \\ , - \\ , \\chi _ 1 \\ , s _ 1 \\ ; = \\ ; 1 \\mbox { w h e r e } I \\ ; : = \\ ; l _ 1 \\cdot b _ 1 . \\end{align*}"}
-{"id": "4264.png", "formula": "\\begin{align*} { \\neg \\frak I } = \\bigoplus \\limits _ { r \\in { J } } { \\mathbb F } u _ r \\subset { \\mathcal I } . \\end{align*}"}
-{"id": "4076.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p - q - r } y ^ q ( a x + b y ) ^ r } { z ^ { p } } = \\dfrac { y ^ q ( a x + b y ) ^ r } { x ^ { - p ' } z ^ { p ' + q + r } } , p ' = p - q - r . \\end{align*}"}
-{"id": "5695.png", "formula": "\\begin{align*} \\Omega _ k : = \\bigcup _ { j \\in J _ k } Q ^ k _ j . \\end{align*}"}
-{"id": "1542.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\ , t } u ( t , x ) + x \\frac { \\partial } { \\partial \\ , x } u ( t , x ) = h ( x ) \\ , u ( t , x ) , \\ ; \\ ; t \\geq 0 , \\ , 0 < x < 1 \\end{align*}"}
-{"id": "7494.png", "formula": "\\begin{align*} R = ( n - 3 ) / 2 , Q _ 0 = 3 , Q _ 1 = n - 6 \\ { \\rm a n d } \\ Q _ 2 = 3 . \\end{align*}"}
-{"id": "2314.png", "formula": "\\begin{align*} \\frac 1 \\tau \\ , \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ q ( \\varOmega ) ) } + \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { 2 , q } ( \\varOmega ) ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "557.png", "formula": "\\begin{align*} m a c ( P ( T \\otimes I _ n ) P ) \\mid { \\mathbb R } \\backslash \\omega ( T ) = m a c ( Q ( T \\otimes I _ n ) Q ) \\mid { \\mathbb R } \\backslash \\omega ( T ) . \\end{align*}"}
-{"id": "203.png", "formula": "\\begin{align*} \\tilde { \\psi } _ f : = - \\log f - H ( f ) . \\end{align*}"}
-{"id": "5355.png", "formula": "\\begin{align*} g \\mapsto \\begin{pmatrix} e ^ { 2 \\pi i / m } & 0 \\\\ 0 & e ^ { - 2 \\pi i / m } \\end{pmatrix} \\end{align*}"}
-{"id": "343.png", "formula": "\\begin{align*} g A g ^ t = \\left [ \\begin{array} { c c } x J & 0 \\\\ 0 & A ' \\end{array} \\right ] . \\end{align*}"}
-{"id": "882.png", "formula": "\\begin{align*} Y = \\operatorname * { b d } A + \\mathbb { R } k . \\end{align*}"}
-{"id": "3619.png", "formula": "\\begin{align*} \\sum _ { j = 1 } ^ { n + 1 } \\| ( \\phi ( x ) ) ^ { 4 - t _ j } & \\widetilde { U ^ j } \\| _ { C ^ { t _ j , \\alpha } ( B _ { \\frac { 1 } { 2 } } ( 0 ) ) } \\\\ & \\le C \\left ( \\sum _ { j = 1 } ^ { n + 1 } \\| ( \\phi ( x ) ) ^ { 4 + s _ j } \\widetilde { L _ j U } \\| _ { C ^ { - s _ j , \\alpha } ( B _ 1 ( 0 ) ) } + \\sum _ { j = 1 } ^ { n + 1 } \\| ( \\phi ( x ) ) ^ { 4 - t _ j } \\widetilde { U ^ j } \\| _ { L ^ 2 ( B _ 1 ( 0 ) ) } \\right ) , \\end{align*}"}
-{"id": "8871.png", "formula": "\\begin{align*} D _ A \\circ \\tau ^ A _ a = \\tau ^ A _ a \\circ D _ A . \\end{align*}"}
-{"id": "6998.png", "formula": "\\begin{align*} & \\delta _ L \\Theta ^ 2 ( x , y , z ) \\\\ = & - [ x , \\Theta ^ 2 ( y , z ) ] + [ y , \\Theta ^ 2 ( x , z ) ] - [ z , \\Theta ^ 2 ( x , y ) ] + \\Theta ^ 2 ( [ x , y ] , z ) - \\Theta ^ 2 ( x , [ y , z ] ) + \\Theta ^ 2 ( y , [ x , z ] ) . \\end{align*}"}
-{"id": "340.png", "formula": "\\begin{align*} | x _ { 1 2 } | = \\max _ { 1 \\le i , j \\le 2 n } | x _ { i j } | . \\end{align*}"}
-{"id": "7433.png", "formula": "\\begin{align*} \\deg _ { X _ f } \\left ( \\psi _ { \\vec { i } _ { ( u , v ) } } \\circ \\chi _ { \\vec { i } _ { ( u , v ) } } \\right ) ^ * ( X _ g ) = \\deg _ { X _ f } \\left ( \\tilde { p } _ { \\vec { i } _ { ( u , v ) } } \\circ \\tilde { \\psi } _ { \\vec { i } _ { ( u , v ) } } \\circ \\tilde { \\chi } _ { \\vec { i } _ { ( u , v ) } } \\right ) ^ * ( X _ g ) \\end{align*}"}
-{"id": "6555.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k ( k - i ) ^ \\ell \\delta _ i = \\ell k ^ { \\ell - 1 } = \\ell \\sum _ { i = 0 } ^ { k - 1 } ( k - i - 1 ) ^ { \\ell - 1 } \\gamma _ i , \\ell = 0 , 1 , \\dotsc , k . \\end{align*}"}
-{"id": "917.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 2 / 3 } } & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { S _ { 4 / 3 } } \\| V \\| _ { S _ { 4 / 3 } } \\\\ & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\left ( \\frac { \\| U \\| ^ { 4 / 3 } _ { S _ { 4 / 3 } } \\ ! + \\| V \\| ^ { 4 / 3 } _ { S _ { 4 / 3 } } } { 2 } \\right ) ^ { 3 / 2 } . \\end{align*}"}
-{"id": "2344.png", "formula": "\\begin{align*} x = a _ 1 & + \\frac { a _ 2 - a _ 1 } { 2 } + \\dotsb + \\frac { a _ n - a _ { n - 1 } } { 2 } + \\\\ & + \\theta _ 2 \\cdot \\frac { a _ 2 - a _ 1 } { 2 } + \\dotsb + \\theta _ n \\cdot \\frac { a _ n - a _ { n - 1 } } { 2 } . \\end{align*}"}
-{"id": "7635.png", "formula": "\\begin{align*} \\omega _ { j } \\geq \\omega _ { i } + ( s _ i + t _ { i j } ) \\sum _ { s \\in \\mathcal { S } _ { i j } } \\lambda _ s - M ( 1 - \\sum _ { s \\in \\mathcal { S } _ { i j } } \\lambda _ s ) , \\ i = 0 , \\ldots , n , \\ j = 1 , \\ldots , n + 1 , \\end{align*}"}
-{"id": "9655.png", "formula": "\\begin{align*} \\begin{array} { r c l } x ' ( t ) & = & - a g ( x ( t ) ) + b \\sup _ { t - \\tau ( t ) \\leq s \\leq t } g ( x ( s ) ) , t \\geq 0 \\\\ x ( t ) & = & \\psi ( t ) , t \\in [ - \\bar { \\tau } , 0 ] . \\end{array} \\end{align*}"}
-{"id": "3712.png", "formula": "\\begin{align*} \\{ F , Z _ { 1 2 i } , H _ { 1 2 i } \\} _ { 1 2 i } = \\nabla _ { 1 2 i } F \\cdot \\nabla _ { 1 2 i } Z _ { 1 2 i } \\times \\nabla _ { 1 2 i } H _ { 1 2 i } ~ , ~ i = 3 , 5 ~ , \\end{align*}"}
-{"id": "1079.png", "formula": "\\begin{align*} e ( B ) \\leq \\ell \\frac { n ^ 2 } { 2 } + ( k - \\alpha - \\ell - x ) \\frac { n ^ 2 } { 2 } = ( k - \\alpha - x ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "9470.png", "formula": "\\begin{align*} \\lim _ { s _ { 0 } \\to - \\infty } \\varphi ( a ( s _ { 0 } ) ) e ^ { f ( \\gamma ( s _ { 0 } ) ) } = 0 \\end{align*}"}
-{"id": "1132.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { B } [ \\iota ] = \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { C } [ \\iota ] = M K \\rho ^ 2 \\beta _ { k } ^ 2 \\end{align*}"}
-{"id": "2472.png", "formula": "\\begin{align*} a ( V ) = \\dim V - \\dim V ^ I + \\sum _ { k \\geq 1 } \\frac { 1 } { [ I : I _ k ] } \\cdot \\dim V / V ^ { I _ k } \\end{align*}"}
-{"id": "210.png", "formula": "\\begin{align*} u _ { x , s } : = \\frac { V _ d ( n - 1 ) h _ x ^ { - 1 } ( s ) ^ d } { e ^ { \\Psi ( k ) } } . \\end{align*}"}
-{"id": "7750.png", "formula": "\\begin{align*} T _ { N } = \\sum _ { i = N } ^ { \\infty } z _ i = \\sum _ { i = N + 1 } ^ { \\infty } z _ i + z _ { N } = T _ { N + 1 } + z _ { N } , \\end{align*}"}
-{"id": "781.png", "formula": "\\begin{align*} \\mathcal { S } _ { k _ { 1 } , \\ldots , k _ { r } } = \\{ ( \\vec { x } , \\vec { \\nu } ) \\in L _ { k } ^ { + } \\times I _ { k _ { 1 } , \\ldots , k _ { r } } \\ , | \\ , \\hbox { F o r a n y $ 1 \\le i \\le k $ , i f $ x _ { i } = x _ { i + 1 } $ t h e n $ \\nu _ { i } \\le \\nu _ { i + 1 } $ . } \\} . \\end{align*}"}
-{"id": "6483.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } ( u _ { k } ( t ) - u _ { 0 , k } ) = - \\lambda _ { k } u _ { k } ( t ) + f _ { k } ( t ) , t > 0 . \\end{align*}"}
-{"id": "4085.png", "formula": "\\begin{gather*} ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 3 } { x ^ 2 z ^ 2 } , { f ( x , y , z ) = \\dfrac { y ^ 2 ( a x + b y + c z ) ^ 2 } { x z ^ { 3 } } } , \\\\ ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 4 } { x ^ 2 z ^ { 3 } } , ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ^ 2 ( a x + b y + c z ) ^ 3 } { x z ^ { 4 } } , \\\\ ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ( a x + b y + c z ) ^ 6 } { x ^ 3 z ^ { 4 } } , ^ { \\dagger } f ( x , y , z ) = \\dfrac { y ^ 3 ( a x + b y + c z ) ^ 4 } { x z ^ { 6 } } , \\end{gather*}"}
-{"id": "9170.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { r } l ^ { 2 k } ( \\frac { \\partial H } { \\partial u _ l } ( v , \\dots , v ) + \\frac { \\partial H } { \\partial u _ { - l } } ( v , \\dots , v ) ) = 0 , k = 1 , \\dots , r , \\end{align*}"}
-{"id": "9246.png", "formula": "\\begin{align*} \\mathbb { H } : = \\{ \\mathcal { H } _ t \\} _ { 0 \\leq t \\leq T } , \\mathcal { H } _ t = \\mathcal { R } _ t \\vee \\sigma ( Z ) , \\end{align*}"}
-{"id": "3971.png", "formula": "\\begin{align*} p _ W ( u ' ( \\psi ( s ) ) v ^ { \\mu _ 0 } ( A ) ) = a _ i w _ i , 0 \\leq i \\leq r , \\ 0 \\neq a _ { i } \\in \\R ; \\end{align*}"}
-{"id": "3699.png", "formula": "\\begin{align*} \\frac { d x _ 2 } { d t } = x _ 1 x _ 3 - b x _ 1 x _ 5 ~ , \\end{align*}"}
-{"id": "4061.png", "formula": "\\begin{align*} 5 q < 5 + \\dfrac { 5 } { 2 } p \\leq 5 + ( 1 + 3 q ) = 6 + 3 q , \\end{align*}"}
-{"id": "556.png", "formula": "\\begin{align*} m a c ( P ( T \\otimes I _ n ) P ) + m \\ m a c ( T ) = m a c ( Q ( T \\otimes I _ n ) Q ) + m \\ m a c ( T ) \\end{align*}"}
-{"id": "5136.png", "formula": "\\begin{align*} & K _ { t } ( w ; z _ { \\ell ( 1 ) } , \\ldots , z _ { \\ell ( t ) } ) \\\\ & = \\sum _ { s = 0 } ^ { t - 2 } g ( w , z _ { \\ell ( 1 ) } ) g ( w , z _ { \\ell ( t ) } ) \\prod _ { i = 1 } ^ { s } g ( z _ { \\ell ( i ) } , z _ { \\ell ( i + 1 ) } ) \\prod _ { i = s + 2 } ^ { t - 1 } g ( z _ { \\ell ( i + 1 ) } , z _ { \\ell ( i ) } ) . \\end{align*}"}
-{"id": "3203.png", "formula": "\\begin{align*} \\sum _ { 2 \\leq | \\gamma | < n } f _ { k j , \\gamma } \\prod _ { \\lambda = 1 } ^ r \\left ( u _ j ^ \\lambda + \\sum _ { 2 \\leq | \\beta | < n } F _ { j , \\beta } ^ \\lambda \\cdot u _ j ^ \\beta \\right ) ^ { \\gamma _ \\lambda } . \\end{align*}"}
-{"id": "5216.png", "formula": "\\begin{align*} ( G _ q ) | _ { \\mathcal { H } _ { q } ^ { \\perp } } = j _ q \\circ ( j _ q ) ^ * \\ ; . \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "8019.png", "formula": "\\begin{align*} \\deg \\phi ( x _ k ) = \\deg \\phi ( y _ k ) = 1 \\ ; \\ ; \\mbox { f o r } \\ ; \\ ; k < j \\mbox { a n d } \\deg \\phi ( x _ j ) > 1 \\ ; \\ ; \\mbox { o r } \\ ; \\ ; \\deg \\phi ( y _ j ) > 1 . \\end{align*}"}
-{"id": "2802.png", "formula": "\\begin{align*} \\begin{cases} d v = \\Phi ( v ) d W ( t ) \\\\ v ( \\cdot , s ) = v _ s ( \\cdot ) \\end{cases} \\end{align*}"}
-{"id": "6659.png", "formula": "\\begin{align*} | M ( G ) | \\leq p ^ { \\frac { 1 } { 2 } ( d ( G ) - 1 ) ( 2 n - m ) } ( = p ^ { \\frac { 1 } { 2 } ( d ( G ) - 1 ) ( n + k ) } ) , \\end{align*}"}
-{"id": "5441.png", "formula": "\\begin{align*} X A ( q ) X ^ { - 1 } = E , X A ^ * ( q ) X ^ { - 1 } = E ^ * , \\end{align*}"}
-{"id": "7030.png", "formula": "\\begin{align*} \\nu _ + ( h _ + ) : = \\int _ { ( h _ + , 1 ] } x \\Pi ( \\mathrm { d } x ) \\mbox { a n d } \\nu _ - ( h _ - ) : = \\int _ { ( h _ - , 1 ] } x \\Pi ^ { ( - ) } ( \\mathrm { d } x ) . \\end{align*}"}
-{"id": "3516.png", "formula": "\\begin{align*} \\rho _ g ( x ) = ( { \\tilde \\rho } \\circ d _ g ( x ) ) ^ N . \\end{align*}"}
-{"id": "3567.png", "formula": "\\begin{align*} \\mathcal G ( f , X ) = \\int _ { \\Omega } \\left ( \\frac 1 2 \\rho _ g \\left | ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) \\right | _ g ^ 2 - ( \\psi , V ) \\cdot _ g ( f , X ) \\right ) \\ ; d \\mu _ g . \\end{align*}"}
-{"id": "6435.png", "formula": "\\begin{align*} - \\tilde { u } ^ { - q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ) & \\geq - \\frac { 1 } { 1 - q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ^ { 1 - q } ) + \\left ( \\frac { \\tilde { u } ^ { 1 - q } } { 1 - q } - \\tilde { u } ^ { 1 - q } \\right ) g _ { 1 - \\alpha , m } \\\\ & \\geq - \\frac { 1 } { 1 - q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ^ { 1 - q } ) + \\frac { q } { 1 - q } \\tilde { u } ^ { 1 - q } g _ { 1 - \\alpha , m } . \\end{align*}"}
-{"id": "5974.png", "formula": "\\begin{align*} Q ( Y ) : = Y ^ k + a _ 1 Y ^ { k - 1 } + \\cdots + a _ k . \\end{align*}"}
-{"id": "654.png", "formula": "\\begin{align*} \\eta \\Big ( \\bigcap _ { l \\in L } l F A B _ { y _ o } \\Big ) \\geq \\nu ( B ) = \\eta ( B _ { y _ o } ) , \\end{align*}"}
-{"id": "8886.png", "formula": "\\begin{align*} \\int _ { B _ 1 ( 0 ) } \\abs { u } ^ { p } = \\lim _ { \\ell \\to \\infty } \\int _ { B _ 1 ( 0 ) } \\abs { \\tau ^ { A _ { n _ \\ell } } _ { a _ { n _ \\ell } } u _ { n _ \\ell } } ^ { p } \\ge \\liminf _ { n \\to \\infty } \\int _ { B _ 1 ( a _ n ) } \\abs { u _ n } ^ { p } > 0 , \\end{align*}"}
-{"id": "6108.png", "formula": "\\begin{align*} \\log z ' = \\log z + \\log \\alpha + \\log ( 1 + z \\beta ) . \\end{align*}"}
-{"id": "5745.png", "formula": "\\begin{align*} \\int _ { \\omega _ { \\mathcal { L } } } ^ { \\omega _ { \\mathcal { U } } } \\psi ( \\omega ) d \\omega = \\pi ^ { - 1 } e ^ { - \\frac { u } { 2 } } \\int _ { \\omega _ { \\mathcal { L } } } ^ { \\omega _ { \\mathcal { U } } } \\lambda ^ { 1 / 2 } ( \\omega ) d \\omega . \\end{align*}"}
-{"id": "6030.png", "formula": "\\begin{gather*} E _ X : = - ( x p - y ) ^ 2 \\partial _ p , E _ H : = x \\partial _ x + y \\partial _ y , E _ Y : = \\frac { 1 } { x p - y } ( \\partial _ x + p \\partial _ y ) . \\end{gather*}"}
-{"id": "5454.png", "formula": "\\begin{align*} \\Bigl \\vert \\sum _ { n = N - H } ^ { N + H } e ^ { - n / N } \\Bigl ( R ( n ) - ( 2 \\psi ( n ) - n ) \\Bigr ) \\Bigl ( 1 - \\frac { \\vert n - N \\vert } { H } \\Bigr ) \\Bigr \\vert \\ll N \\Bigl ( \\log \\frac { 2 N } { H } \\Bigr ) ^ 2 . \\end{align*}"}
-{"id": "7125.png", "formula": "\\begin{align*} d ( X , Y ) = \\sqrt { ( X - Y ) \\odot ( X - Y ) } \\end{align*}"}
-{"id": "592.png", "formula": "\\begin{align*} w _ 1 = & \\partial _ x \\sigma _ 1 , \\\\ w _ 3 = & \\partial _ x \\sigma _ 3 . \\end{align*}"}
-{"id": "6668.png", "formula": "\\begin{gather*} \\beta _ 1 ( t ) : = x ( u _ k , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ k , \\widetilde { \\sigma } _ 1 ) , t \\geqslant 0 , \\\\ \\beta _ 2 ( t ) : = x ( u _ j , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ j , \\widetilde { \\sigma } _ 1 ) , t \\geqslant 0 . \\end{gather*}"}
-{"id": "8926.png", "formula": "\\begin{align*} u _ A ( - x ) = u _ A ( x ) . \\end{align*}"}
-{"id": "7823.png", "formula": "\\begin{align*} Y _ { t } = \\xi + \\int _ t ^ T E ' [ f ( s , \\eta _ { s } , Y ' _ { s } , Z ' _ { s } , Y _ { s } , Z _ { s } ) ] d s - \\int _ t ^ T Z _ { s } d B _ { s } ^ { H } , \\ \\ 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "1666.png", "formula": "\\begin{gather*} \\int _ { \\R ^ d } \\| D _ v ^ 2 { \\psi } ( \\cdot , v ) \\| _ { H ^ s _ p ( \\R ^ { d } ) } ^ p \\dd v = \\int _ { \\R ^ d } d v \\int _ { \\R ^ d } | D _ v ^ 2 \\ , G _ { \\lambda } h _ s ( x , v ) | ^ p \\ , \\dd x \\\\ \\le C \\| h _ s \\| _ { L ^ p ( \\R ^ { 2 d } ) } ^ p = C \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ d _ x ) ) } ^ p , \\end{gather*}"}
-{"id": "8710.png", "formula": "\\begin{align*} v ^ n ( t , x ) = \\int _ t ^ T R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } G B ^ n ( s , \\cdot ) + e ^ { - ( s - t ) { A } } \\nabla ^ G v ^ n ( s , \\cdot ) \\ , B ^ n ( s , \\cdot ) \\right ] ( x ) \\ , d s , \\ ; t \\in \\left [ 0 , T \\right ] , \\ ; x \\in H . \\end{align*}"}
-{"id": "1630.png", "formula": "\\begin{align*} g ^ { \\beta \\gamma } \\xi _ { , \\beta \\gamma } ^ { \\alpha } - \\xi _ { , \\gamma } ^ { \\alpha } \\Gamma ^ { \\gamma } + \\xi ^ { \\gamma } \\Gamma _ { , \\gamma } ^ { \\alpha } - \\xi _ { , \\gamma } ^ { \\alpha } C ^ { \\gamma } + \\xi ^ { \\gamma } C _ { , \\gamma } ^ { \\alpha } - 2 g ^ { \\beta \\alpha } a _ { , \\beta } + \\left ( a - \\lambda \\right ) \\Gamma ^ { \\alpha } + \\left ( a - \\lambda \\right ) C ^ { \\alpha } - \\xi _ { , t } ^ { \\alpha } = 0 . \\end{align*}"}
-{"id": "6874.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\langle e ^ { - i t \\Delta } u ( t ) , \\phi \\rangle = \\lim _ { t \\to 0 } \\langle e ^ { - i t \\Delta } u ( t ) , \\phi \\rangle = 0 \\end{align*}"}
-{"id": "1172.png", "formula": "\\begin{align*} D ( - \\lambda ) = W ( \\hat c _ 0 , c _ 0 ) + \\sum _ { n = 1 } \\lambda ^ n \\int _ { 0 \\leq \\xi _ 1 \\leq \\xi _ 2 \\cdots \\xi _ n \\leq 1 } \\hat c _ 0 ( \\xi _ n ) \\rho ( \\xi _ n ) \\left ( \\prod _ { j = 1 } ^ { n - 1 } ( \\xi _ { j + 1 } - \\xi _ j ) \\rho ( \\xi _ j ) \\right ) c _ 0 ( \\xi _ 1 ) d \\xi _ 1 d \\xi _ 2 \\dots d \\xi _ n , \\end{align*}"}
-{"id": "6607.png", "formula": "\\begin{align*} \\langle P \\nabla f _ 2 ( P x + q ) - P & \\nabla f _ 2 ( P y + q ) , x - y \\rangle \\\\ & = \\langle \\nabla f _ 2 ( P x + q ) - \\nabla f _ 2 ( P y + q ) , P ( x - y ) \\rangle \\\\ & = \\langle \\nabla f _ 2 ( P x + q ) - \\nabla f _ 2 ( P y + q ) , ( P x + q ) - ( P y + q ) ) \\rangle \\end{align*}"}
-{"id": "1521.png", "formula": "\\begin{align*} Q ( f ) : = \\int \\sigma ( v , v ' ) M ( v ) f ( v ' ) - \\sigma ( v ' , v ) M ( v ' ) f ( v ) \\d v ' . \\end{align*}"}
-{"id": "1895.png", "formula": "\\begin{align*} { I } : = [ \\underline { B } , \\infty ) \\times ( 0 , \\overline { \\rho } ] \\times ( - \\infty , \\overline { B } ] \\times ( 0 , \\overline { c } ] \\times ( 0 , \\overline { C } ] \\times [ \\underline { C ' } , \\infty ) \\times ( 0 , \\overline { C } _ X ] \\times ( 0 , \\overline { C } _ f ] . \\end{align*}"}
-{"id": "4859.png", "formula": "\\begin{align*} \\hat { R } _ { 1 2 } ^ { \\rm e x p } ( i , x ; j , y ) = - \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\frac { ( 1 + 2 z ) ^ { n _ i } } { ( 1 + 2 z ) ^ { n _ j } } \\frac { ( 1 - 2 z ) ^ { m _ j } } { ( 1 - 2 z ) ^ { m _ i } } e ^ { - \\vert x - y \\vert z } \\dd z , \\end{align*}"}
-{"id": "5969.png", "formula": "\\begin{align*} \\sum _ { i \\geq 0 } \\C _ { i , A _ X \\tilde { P } } \\left ( \\frac { 1 } { m } \\right ) \\frac { z ^ i } { i ! } = \\exp \\left \\{ \\frac { 1 } { m } K _ { A _ X \\tilde { P } } ( z ) \\right \\} . \\end{align*}"}
-{"id": "8843.png", "formula": "\\begin{align*} \\upsilon _ { k + 1 } - \\upsilon _ k \\leq 2 \\Big [ \\Big ( \\frac { 2 } { n } \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\upsilon _ i \\Big ) ^ 2 - \\Big ( 1 + \\frac { 4 } { n } \\Big ) \\frac { 1 } { k } \\sum _ { j = 1 } ^ k \\Big ( \\upsilon _ j - \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\upsilon _ i \\Big ) ^ 2 \\Big ] ^ \\frac { 1 } { 2 } . \\end{align*}"}
-{"id": "7212.png", "formula": "\\begin{align*} P _ { \\gamma } ^ { S } = Q ^ { S } \\circ P _ { \\gamma } ^ { N } \\label { g e n r e s p E E E E q n } \\end{align*}"}
-{"id": "2756.png", "formula": "\\begin{align*} A _ k ^ n : = \\int _ { t _ { k - 1 } } ^ { t _ k } \\mu _ n ( \\bar { t } _ n ) { \\biggl \\langle } Y ^ n _ { \\bar { t } _ n } - Z _ { \\bar { t } _ n } , \\bigl ( \\sigma \\bigl ( Y ^ n _ t \\bigr ) - \\sigma \\bigl ( Y ^ n _ { \\bar { t } _ n } \\bigr ) \\bigr ) \\dot { w } ^ n _ t - \\frac { 1 } { 2 } ( \\nabla \\sigma ) \\sigma \\bigl ( Y ^ n _ { \\bar { t } _ n } \\bigr ) { \\biggr \\rangle } \\ , \\d t , \\end{align*}"}
-{"id": "3291.png", "formula": "\\begin{align*} \\langle X _ { ( 1 ) } w , X _ { ( 2 ) } v \\rangle = \\varepsilon ( X ) \\langle w , v \\rangle . \\end{align*}"}
-{"id": "4243.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 - 4 t ) ^ { \\tfrac { x + y } { 2 } } d \\mu _ { - 1 } ( y ) & = \\frac { 2 } { 1 + \\sqrt { 1 - 4 t } } \\sqrt { ( 1 - 4 t ) ^ x } \\\\ & = \\sum _ { n = 0 } ^ \\infty C _ n ( x ) t ^ n . \\end{align*}"}
-{"id": "6760.png", "formula": "\\begin{align*} \\theta _ { V _ { \\theta } } ^ n \\leq { \\bf 1 } _ { \\{ V _ { \\theta } = 0 \\} } \\theta ^ n . \\end{align*}"}
-{"id": "1077.png", "formula": "\\begin{align*} ( 1 - \\delta ) \\binom { N } { 2 } = ( 1 - \\delta ) ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } & \\leq k \\left ( k - \\alpha - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } \\\\ ( 1 - \\delta ) \\alpha ^ 2 - ( 1 - 2 \\delta ) k \\alpha + \\frac { k } { 4 } - ( 1 - \\delta ) \\frac { k - \\alpha } { n } - \\delta k ^ 2 & \\leq 0 \\ , . \\end{align*}"}
-{"id": "1936.png", "formula": "\\begin{align*} & \\left ( 2 p _ i - q _ i ^ 2 ( \\xi _ i + \\epsilon ) \\right ) ( \\tau - \\tau _ 0 ) \\leq b _ { i } ( \\tau ) - b _ { i } ( \\tau _ 0 ) \\leq ( 2 p _ i - q _ i ^ 2 ( \\xi _ i - \\epsilon ) ) ( \\tau - \\tau _ 0 ) , \\\\ & ( E ( \\xi ) - \\epsilon ) ( \\tau - \\tau _ 0 ) \\leq a ( \\tau ) - a ( \\tau _ 0 ) \\leq ( E ( \\xi ) + \\epsilon ) ( \\tau - \\tau _ 0 ) , \\end{align*}"}
-{"id": "9310.png", "formula": "\\begin{align*} \\tilde { p } ( t , z ) = \\tilde { p } ( 0 , z ) \\exp ( - \\int _ 0 ^ t \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } d B ( s ) - \\frac { 1 } { 2 } \\int _ 0 ^ t ( \\frac { a _ 0 ( s , z ) } { b _ 0 ( s , z ) } ) ^ { 2 } d s ) , \\end{align*}"}
-{"id": "7565.png", "formula": "\\begin{align*} \\Delta _ l ( b ) = b - f ( b ) - l > 0 , \\Delta _ l ( v _ l ) = 0 , \\end{align*}"}
-{"id": "5711.png", "formula": "\\begin{align*} b _ Q : = \\frac { ( M \\chi _ Q ) ^ { \\alpha / n + \\varepsilon } } { \\ell ( Q ) ^ \\alpha } , \\end{align*}"}
-{"id": "3716.png", "formula": "\\begin{align*} \\chi ( ( y , x ) ) = \\psi ( \\langle y , x \\rangle ) \\\\ = \\psi ( y x ^ * ) = \\psi ( ( x y ^ { * } ) ^ * ) = \\psi ( \\langle x , y \\rangle ) . \\end{align*}"}
-{"id": "8429.png", "formula": "\\begin{align*} A _ { 0 } ( \\textbf { u } ) \\partial _ { t } \\textbf { u } + \\sum _ { j = 1 } ^ { d } A _ { 0 } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } + F _ { P } = 0 . \\end{align*}"}
-{"id": "912.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } = \\left ( \\sum _ { i = 1 } ^ { n } \\sigma ^ { p } _ { i } ( X ) \\right ) ^ { 1 / p } , \\end{align*}"}
-{"id": "4426.png", "formula": "\\begin{align*} ( \\widehat { C _ E } , \\| \\cdot \\| _ { \\widehat { C _ E } } ) = ( C _ { \\widehat { E } } , \\| \\cdot \\| _ { C _ { \\widehat { E } } } ) \\end{align*}"}
-{"id": "2162.png", "formula": "\\begin{align*} - \\frac { 1 } { 1 - q } \\int _ { B _ { 1 } } \\psi ^ { 1 + q } \\partial _ { s } ( g _ { 1 - \\alpha , m } * \\tilde { u } ^ { 1 - q } ) d x & - \\mathcal { E } ( h _ { m } * \\tilde { u } , \\psi ^ { 1 + q } \\tilde { u } ^ { - q } ) \\\\ & \\leq \\frac { - q } { 1 - q } \\int _ { B _ { 1 } } \\psi ^ { 1 + q } \\tilde { u } ^ { 1 - q } g _ { 1 - \\alpha , m } d x . \\end{align*}"}
-{"id": "3433.png", "formula": "\\begin{align*} d y ( t ) = - f ( t , \\xi ( t ) , y ( t ) , z ( t ) ) d t + z ( t ) d w ( t ) , y ( T ) = \\Gamma ^ * ( s , T ) u _ 0 ( \\xi ( T ) ) , \\end{align*}"}
-{"id": "270.png", "formula": "\\begin{align*} | T _ 1 | \\leq T _ { 1 1 } + T _ { 1 2 } + T _ { 1 3 } = o \\biggl ( \\frac { k ^ { - \\frac { 1 } { 2 } + \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "7607.png", "formula": "\\begin{align*} f ( x ) : = \\frac { 4 x ^ 2 } { 2 + x } e ^ { 2 ( 1 - x ) } , x > 0 . \\end{align*}"}
-{"id": "8743.png", "formula": "\\begin{align*} \\mathcal { E } _ { C } \\left ( t , h \\right ) = | { \\widetilde Q } _ t ^ { - 1 / 2 } e ^ { t A } h | _ K , \\ ; \\ ; \\ ; h \\in K . \\end{align*}"}
-{"id": "4120.png", "formula": "\\begin{align*} \\omega = p v d u + q u d v C = \\{ v ^ q - u ^ { - p } = 0 \\} . \\end{align*}"}
-{"id": "8892.png", "formula": "\\begin{align*} \\lim _ { \\ell \\to + \\infty } \\mathcal { I } _ { A _ { n _ \\ell } } ( u _ { n _ \\ell } ) = \\liminf _ { n \\to \\infty } \\mathcal { I } _ { A _ n } ( u _ n ) \\ge \\mathcal { I } _ A ( u ) , \\end{align*}"}
-{"id": "5288.png", "formula": "\\begin{align*} I = \\int _ { T } ^ { 2 T } \\sum _ { n \\leq C T / \\pi } { n ^ { - \\tfrac { 1 } { 2 } - i t } } \\left ( \\tfrac { t } { 2 \\pi } \\right ) ^ { { i t } / { 2 } } e ^ { - i ( { t } / { 2 } + { \\pi } / { 8 } ) } \\left \\{ 1 + O \\left ( \\tfrac { 1 } { t } \\right ) \\right \\} d t + O ( T ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "4496.png", "formula": "\\begin{align*} | A | = \\prod _ { 1 \\leq t _ 1 < t _ 2 \\leq d ' } d ^ { - 2 / d } ( t _ 2 ^ { 2 / d } - t _ 1 ^ { 2 / d } ) > 0 . \\end{align*}"}
-{"id": "9881.png", "formula": "\\begin{align*} \\frac { \\partial u ^ { \\epsilon } } { \\partial t } ( t , x ) & = \\frac { \\partial ^ 2 u ^ { \\epsilon } } { \\partial x ^ 2 } ( t , x ) + \\sqrt { \\epsilon } \\sigma ( t , x , u ^ \\epsilon ( t , x ) ) \\frac { \\partial ^ 2 W } { \\partial t \\partial x } ( t , x ) \\\\ & + \\frac { \\partial } { \\partial x } g ( t , x , u ^ { \\epsilon } ( t , x ) ) + f ( t , x , u ^ \\epsilon ( t , x ) ) , \\end{align*}"}
-{"id": "9737.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { x ( t ) } { G ^ { - 1 } ( t ) } = \\Lambda _ 1 . \\end{align*}"}
-{"id": "9593.png", "formula": "\\begin{align*} \\varphi ( x , t ) : = \\frac { \\theta ( t - | x - y | ) } { 4 \\pi | x - y | } \\xi ( t - | x - y | ) - \\frac { m } { 4 \\pi } \\int _ 0 ^ t \\frac { \\theta ( s - | x - y | ) J _ 1 ( m \\sqrt { s ^ 2 - | x - y | ^ 2 } ) } { \\sqrt { s ^ 2 - | x - y | ^ 2 } } \\xi ( t - s ) d s . \\end{align*}"}
-{"id": "3664.png", "formula": "\\begin{align*} 0 & = \\phi ( x y + y x - t ^ 2 ) _ 1 \\\\ & = 2 \\left [ a _ 0 ( b _ 1 x + b _ 2 y + b _ 3 t ) + b _ 0 ( a _ 1 x + a _ 2 y + a _ 3 t ) - c _ 0 c _ 3 t \\right ] \\\\ & = 2 \\left [ ( a _ 0 b _ 1 + b _ 0 a _ 1 ) x + ( a _ 0 b _ 2 + b _ 0 a _ 2 ) y + ( a _ 0 b _ 3 + b _ 0 a _ 3 - c _ 0 c _ 3 ) t \\right ] . \\end{align*}"}
-{"id": "1460.png", "formula": "\\begin{align*} \\| f \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } : = \\inf \\left \\{ \\lambda > 0 \\ , : \\ , \\int _ { \\mathbb { R } ^ n } \\left | \\frac { f ( x ) } { \\lambda } \\right | ^ { p ( x ) } \\ , d x \\le 1 \\right \\} . \\end{align*}"}
-{"id": "3573.png", "formula": "\\begin{align*} U = ( U ^ 1 , U ^ 2 , \\dots , U ^ { n + 1 } ) = ( f , X ^ 1 , \\dots , X ^ n ) . \\end{align*}"}
-{"id": "7459.png", "formula": "\\begin{align*} u = \\begin{pmatrix} 1 & 2 & 3 & 4 \\\\ 4 & 2 & 1 & 3 \\end{pmatrix} v = \\begin{pmatrix} 1 & 2 & 3 & 4 \\\\ 3 & 1 & 4 & 2 \\end{pmatrix} \\end{align*}"}
-{"id": "2879.png", "formula": "\\begin{align*} \\mu ( e ( 1 _ { \\mathbb { K } } ) , - ) = \\mu ( - , e ( 1 _ { \\mathbb { K } } ) ) = i d _ A . \\end{align*}"}
-{"id": "2071.png", "formula": "\\begin{align*} X ^ { ( 0 ) } ( s _ { 0 } ) = & \\ \\mathsf { Q } , \\\\ X ^ { ( 1 ) } ( s _ { 0 } ) = & \\ \\mathcal { P } _ { 1 } X ^ { ( 0 ) } ( s _ { 0 } ) , \\mathrm { a n d } \\\\ X ^ { ( j ) } ( s _ { 0 } ) = & \\ \\mathcal { P } _ { 1 } X ^ { ( j - 1 ) } ( s _ { 0 } ) + \\mathcal { P } _ { 2 } X ^ { ( j - 2 ) } ( s _ { 0 } ) \\end{align*}"}
-{"id": "3850.png", "formula": "\\begin{align*} \\phi _ 1 ( x , y , t ; , \\xi , \\tau , s , \\theta ) = & ( x - y ) \\cdot \\xi + ( t - s ) \\tau + s | \\xi | \\sin \\theta - \\theta \\varpi / 2 \\pi , \\\\ \\phi _ 2 ( x , y , t ; , \\xi , \\tau , s , r ) = & ( x - y ) \\cdot \\xi + ( t - s ) \\tau + i ( | \\xi | s \\sinh r + \\varpi r / 2 \\pi ) . \\end{align*}"}
-{"id": "4236.png", "formula": "\\begin{align*} \\sum _ { n = 0 } ^ \\infty ( - 1 ) ^ n 4 ^ n \\int _ { \\mathbb { Z } _ p } { \\frac { x } { 2 } \\choose n } d \\mu _ { - 1 } ( x ) t ^ n & = \\sum _ { n = 0 } ^ \\infty C _ n t ^ n \\\\ & = \\sum _ { n = 0 } ^ \\infty \\frac { 1 } { n + 1 } { 2 n \\choose n } t ^ n . \\end{align*}"}
-{"id": "7965.png", "formula": "\\begin{align*} \\| f _ { \\ell + 1 } - f _ { \\ell } \\| _ { L ^ 2 ( \\Omega ) } ^ 2 + \\| f - f _ { \\ell + 1 } \\| _ { L ^ 2 ( \\Omega ) } ^ 2 - \\| f - f _ { \\ell } \\| _ { L ^ 2 ( \\Omega ) } ^ 2 = 0 \\end{align*}"}
-{"id": "2623.png", "formula": "\\begin{align*} \\{ \\theta _ { m } ^ { l } ( j , u ) = \\theta _ { m } ^ { l } ( u + j T ) , u \\in [ 0 , T ) , j \\in \\mathbb Z \\} \\end{align*}"}
-{"id": "2583.png", "formula": "\\begin{align*} { u } ^ J _ n ( t ) : = \\sum _ { j = 1 } ^ J v _ n ^ j ( t ) + e ^ { i t \\Delta } W _ n ^ J , \\end{align*}"}
-{"id": "5222.png", "formula": "\\begin{align*} \\overline { \\partial } _ { M } ^ { * } T _ { q } u = ( - 1 ) ^ { q + 1 } T _ { ( q + 1 ) } ( \\overline { \\partial } _ { M } u ) + \\ ; . \\end{align*}"}
-{"id": "5235.png", "formula": "\\begin{align*} \\alpha ( \\overline { L } ) ( z ) = d h ( \\overline { L } ) ( z ) \\ ; , \\ ; z \\in M \\ ; , \\ ; L \\in \\mathcal { N } _ { z } \\ ; , \\end{align*}"}
-{"id": "6378.png", "formula": "\\begin{align*} r _ * : = \\begin{cases} r _ { \\l _ k } & \\mathcal { R } _ { \\l _ k } \\cap \\{ r \\leq 2 \\l _ k \\} \\\\ r _ { 4 \\l _ k } & \\mathcal { R } _ { \\l _ k } \\cap \\{ r > 3 \\l _ k \\} \\end{cases} \\qquad U _ { a , * } : = \\begin{cases} U _ { a , \\l _ k } & \\mathcal { R } _ { \\l _ k } \\cap \\{ r \\leq 2 \\l _ k \\} \\\\ U _ { a , 4 \\l _ k } & \\mathcal { R } _ { \\l _ k } \\cap \\{ r > 3 \\l _ k \\} \\end{cases} \\end{align*}"}
-{"id": "4418.png", "formula": "\\begin{align*} 0 \\leq ( \\abs { f } - f _ n ) ( t ) = \\left . \\begin{cases} 0 & \\ , \\abs { f } ( t ) \\leq g _ n ' ( t ) \\\\ \\abs { f } ( t ) - g _ n ' ( t ) & \\ , \\abs { f } ( t ) > g _ n ' ( t ) \\end{cases} \\right \\} \\ , \\leq \\abs { g } ( t ) - g _ n ' ( t ) . \\end{align*}"}
-{"id": "6358.png", "formula": "\\begin{align*} ( l + 1 ) ( l + 2 + 2 \\sigma _ n ) p ' _ { \\sigma , l + 1 } + 2 s ( \\sigma _ n + 1 ) p ' _ { \\sigma + \\bar n , l } + ( \\sigma _ i + 1 ) ( \\sigma _ i + 2 ) p ' _ { \\sigma + 2 \\bar \\imath , l - 1 } + \\tilde { c } _ { \\sigma l } ^ { \\mu m } p ' _ { \\mu m } = 0 \\forall ( \\sigma , l ) \\end{align*}"}
-{"id": "178.png", "formula": "\\begin{align*} \\mathcal { W } ^ { ( k ) } : = \\biggl \\{ w \\in \\mathbb { R } ^ k : & \\sum _ { j = 1 } ^ k w _ j \\frac { \\Gamma ( j + 2 \\ell / d ) } { \\Gamma ( j ) } = 0 \\ , \\ , \\ell = 1 , \\ldots , \\lfloor d / 4 \\rfloor \\\\ & \\sum _ { j = 1 } ^ k w _ j = 1 \\ , \\ , \\ , w _ j = 0 \\ , \\ , \\ , j \\notin \\{ \\lfloor k / d \\rfloor , \\lfloor 2 k / d \\rfloor , \\ldots , k \\} \\biggr \\} . \\end{align*}"}
-{"id": "3319.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 1 ) } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\end{array} \\right ) , \\Gamma _ { 0 } ^ { ( 1 ) } = \\left ( \\begin{array} { c c c } 0 & 1 & 0 \\end{array} \\right ) , \\Gamma _ { - } ^ { ( 1 ) } = \\left ( \\begin{array} { c c c } 0 & 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "1614.png", "formula": "\\begin{align*} Z ^ { 3 } = 2 t \\partial _ { t } + H + c t K ^ { 1 } \\end{align*}"}
-{"id": "2252.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\log \\frac { 2 \\pi \\sigma ^ { 2 } } { C ^ { 2 } ( \\vec { \\varepsilon } ) } + \\frac { \\mu _ { ( 2 ) } } { 2 \\sigma ^ { 2 } } - \\varepsilon _ { q } \\mu _ { ( q ) } + \\varepsilon _ { p } \\mu _ { ( p ) } . \\end{align*}"}
-{"id": "6849.png", "formula": "\\begin{align*} N : = L _ t ^ { q _ 1 ' , 2 } \\dot X ^ { | s _ c | , r _ 1 ' } \\end{align*}"}
-{"id": "1854.png", "formula": "\\begin{align*} \\sum _ { 1 \\le n \\le t } \\left ( \\widetilde { F } _ { n , n } ( t ) - \\widetilde { G } _ { n , n } ( t ) \\right ) = \\log t + O ( 1 ) . \\end{align*}"}
-{"id": "5505.png", "formula": "\\begin{align*} \\left ( 1 + \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } + \\sum _ { i = 1 } ^ { K } \\beta _ { k } } \\right ) ^ 2 \\geq 1 + \\frac { M \\beta _ { k } } { \\sum _ { i = 1 } ^ { K } \\beta _ { i } } \\end{align*}"}
-{"id": "8202.png", "formula": "\\begin{align*} \\d c _ 1 = \\d c _ 2 = 0 , \\ \\ \\ N ( c _ 1 , c _ 2 ) A _ { p - 1 } ( c _ 1 , c _ 2 ) \\not \\equiv 0 \\ \\ \\ \\ \\ \\ p . \\end{align*}"}
-{"id": "5405.png", "formula": "\\begin{align*} x n = \\sum _ { i = 1 } ^ c \\max \\{ v ( B _ i ) - n , 0 \\} \\end{align*}"}
-{"id": "9511.png", "formula": "\\begin{align*} p _ 1 & = - \\frac { 1 } { 8 \\pi ^ 2 } \\mathrm { t r } \\ , \\Omega ^ 2 \\in \\Omega ^ 4 ( M ) \\\\ p _ 2 & = - \\frac { 1 } { 6 4 \\pi ^ 4 } \\mathrm { t r } \\ , \\Omega ^ 4 + \\frac { 1 } { 1 2 8 \\pi ^ 4 } ( \\mathrm { t r } \\ , \\Omega ^ 2 ) ^ 2 \\in \\Omega ^ 8 ( M ) . \\end{align*}"}
-{"id": "1363.png", "formula": "\\begin{align*} Q _ L ^ B ( \\hat \\delta _ 0 ) = \\begin{bmatrix} S _ { \\phi } ( \\hat \\delta _ { 0 } ) & S _ { \\theta } ( \\hat { \\delta } _ { 0 } ) \\end{bmatrix} [ \\hat { G } _ n - \\hat { F } _ n \\hat { E } _ n ^ { - 1 } \\hat { F } _ n ^ \\mathrm { T } ] ^ { - 1 } \\begin{bmatrix} S _ { \\phi } ( \\hat { \\delta } _ { 0 } ) & S _ { \\theta } ( \\hat \\delta _ { 0 } ) \\end{bmatrix} . \\end{align*}"}
-{"id": "3071.png", "formula": "\\begin{align*} \\forall t \\in ( 0 , 1 ) \\lim _ { n \\to \\infty } \\omega _ { n , \\beta } ( ( t , 1 ] ) = \\pi _ \\beta . \\end{align*}"}
-{"id": "5341.png", "formula": "\\begin{align*} & ( \\varphi ( u ' ) ) ' = N _ f ( u ) , \\ u ' ( T ) = b u ' ( 0 ) , \\ u ' ( T ) = b u ' ( 0 ) \\\\ & \\Leftrightarrow ( D _ { \\varphi } D ) ( u ) = N _ { f } ( u ) , \\ u \\in { \\rm d o m } ( D ) \\\\ & \\Leftrightarrow u = ( D _ { \\varphi } D ) ^ { - 1 } N _ { f } ( u ) , \\ u \\in C ^ { 1 } . \\end{align*}"}
-{"id": "7376.png", "formula": "\\begin{align*} \\tau _ t ^ 2 = \\sigma ^ 2 + \\sigma _ t ^ 2 , \\sigma _ t ^ 2 & = \\frac { 1 } { \\delta } \\mathbb { E } \\left [ \\left ( \\eta _ { t - 1 } ( \\beta + \\tau _ { t - 1 } Z ) - \\beta \\right ) ^ 2 \\right ] , \\end{align*}"}
-{"id": "4940.png", "formula": "\\begin{align*} \\| u _ i \\| _ 2 \\leq \\sqrt { \\| u _ i \\| _ 0 } \\cdot \\| u _ i \\| _ \\infty & = \\sqrt { k ( t - 1 ) - \\ell } \\cdot \\| u _ i \\| _ \\infty \\\\ & \\leq \\sqrt { k ( t - 1 ) } \\cdot \\| u _ i \\| _ \\infty \\\\ & \\leq \\alpha \\sqrt { k / ( t - 1 ) } . \\end{align*}"}
-{"id": "892.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { \\mathcal { R } } \\chi ( s ) ^ { - 1 / 2 } \\zeta ( s ) d s = 0 . \\end{align*}"}
-{"id": "7117.png", "formula": "\\begin{align*} \\delta ^ m ( ( x ^ { \\alpha } \\otimes g ) \\epsilon _ { \\beta } ^ * ) = \\sum _ { l = 1 } ^ n \\Omega _ g ( \\alpha , \\beta , l ) ( x ^ { \\alpha + [ l ] } \\otimes g ) \\epsilon _ { \\beta + [ l ] } ^ * . \\end{align*}"}
-{"id": "9748.png", "formula": "\\begin{align*} \\lim _ { x \\to 0 ^ + } \\frac { g ( x ) } { \\exp ( - e ^ { 1 / x } ) } = \\lim _ { x \\to 0 ^ + } \\frac { \\int _ 0 ^ x g ' ( s ) \\ , d s } { \\int _ 0 ^ x s ^ { - 2 } e ^ { 1 / s } \\exp ( - e ^ { 1 / s } ) \\ , d s } = 1 . \\end{align*}"}
-{"id": "2784.png", "formula": "\\begin{align*} \\mathsf { P } _ { \\mathrm { S I R } , g _ { 2 } } \\left ( \\lambda _ { \\mathrm { B S } } \\right ) = & \\mathbb { P } \\left ( \\frac { P _ { \\mathrm { B S } } H _ { \\mathrm { U } _ { 0 } , \\mathrm { B S } _ { 0 } } g _ { 2 } \\left ( d _ { 0 } \\right ) } { \\underset { \\mathrm { B S } _ { i } \\in \\Pi _ { \\mathrm { B S } } ^ { \\dagger } } { \\sum } P _ { \\mathrm { B S } } H _ { \\mathrm { U } _ { 0 } , \\mathrm { B S } _ { i } } g _ { 2 } \\left ( d _ { i } \\right ) } > \\tau \\right ) . \\end{align*}"}
-{"id": "7442.png", "formula": "\\begin{align*} | I ( w ) | = | J ( w ) | + 1 . \\end{align*}"}
-{"id": "3369.png", "formula": "\\begin{align*} X ( t ) = E _ b ( t ) - A \\end{align*}"}
-{"id": "3779.png", "formula": "\\begin{align*} { \\bf { y } } = { \\bf { H } } \\ , { { \\bf { U } } _ { { { \\mathrm { \\bf { B } } } } } } { { \\bf { \\Lambda } } _ { { { \\mathrm { \\bf { B } } } } } } { { \\bf { V } } _ { { { \\mathrm { \\bf { B } } } } } } \\ , { \\bf { d } } + { \\bf { n } } \\end{align*}"}
-{"id": "78.png", "formula": "\\begin{align*} \\int _ { f ( r ) / a } ^ 1 \\frac { d z } { \\psi ( 1 - z ) ^ { 1 / p } } = r \\ , r \\in [ 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "6.png", "formula": "\\begin{align*} \\mu ( x _ q ^ k ) = \\mu \\left ( \\frac { k } { q } + \\frac { \\alpha ( k / q ) } { q ^ 2 } \\right ) \\left ( 1 + \\varepsilon O ( q ^ { - 4 } ) \\right ) . \\end{align*}"}
-{"id": "4284.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ { m + 1 } ( 1 - \\delta _ n ) | a _ n | + \\eta _ { m + 1 } \\sum _ { n = 1 } ^ { m + 1 } | a _ n | \\le \\| \\sum _ { n = 1 } ^ { m + 1 } a _ n x _ n \\| \\end{align*}"}
-{"id": "9115.png", "formula": "\\begin{align*} u ' ( t ) + A ( t ) B ( t ) u ( t ) + P ( t ) u ( t ) = f ( t ) , \\ u ( 0 ) = u _ 0 , \\end{align*}"}
-{"id": "5590.png", "formula": "\\begin{align*} \\mathcal G _ 0 & = - i \\alpha \\cdot \\nabla G _ 0 + 2 m G _ 0 I _ { 1 } \\\\ \\mathcal G _ 1 & = - i \\alpha \\cdot \\nabla G _ 1 + 2 m G _ 1 I _ { 1 } - \\frac 2 m M _ { 1 1 } - \\frac 2 m M _ { 2 2 } \\\\ \\mathcal G _ 2 & = - i \\alpha \\cdot \\nabla G _ 2 + 2 m G _ 2 I _ 1 + \\frac 1 { 2 m } G _ 0 - \\frac 1 { 4 \\pi m } M _ { 1 1 } - \\frac 1 { 4 \\pi m } M _ { 2 2 } . \\end{align*}"}
-{"id": "1243.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\varphi _ * \\left ( \\frac { \\delta | \\eta ( k ) | } { v ( k ) } \\right ) v ( k ) \\chi _ { E _ j } ( k ) \\le \\sum _ { k = j + 1 } ^ \\infty \\varphi _ * \\left ( \\frac { \\delta | \\eta ( k ) | } { v ( k ) } \\right ) v ( k ) \\to 0 . \\end{align*}"}
-{"id": "9871.png", "formula": "\\begin{align*} 0 = \\mu _ 0 ( H _ g ) < \\mu _ { 1 } ( H _ g ) \\leq \\mu _ { 2 } ( H _ g ) \\leq . . . \\leq \\mu _ { n } ( H _ g ) \\leq . . . \\ , , \\end{align*}"}
-{"id": "1661.png", "formula": "\\begin{align*} Q _ t = \\int _ 0 ^ t e ^ { s A } Q e ^ { s A ^ * } \\dd s = \\int _ 0 ^ t e ^ { s A } \\begin{pmatrix} 0 & 0 \\\\ 0 & \\mathbb { I } _ { \\R ^ d } \\end{pmatrix} e ^ { s A ^ * } \\dd s \\ , = \\begin{pmatrix} \\frac { 1 } { 3 } t ^ 3 \\mathbb { I } _ { \\R ^ d } & \\frac { 1 } { 2 } t ^ 2 \\mathbb { I } _ { \\R ^ d } \\\\ [ 3 p t ] \\frac { 1 } { 2 } t ^ 2 \\mathbb { I } _ { \\R ^ d } & t \\mathbb { I } _ { \\R ^ d } \\end{pmatrix} \\ , \\end{align*}"}
-{"id": "2475.png", "formula": "\\begin{align*} \\varepsilon _ K ( V \\otimes \\omega _ s , \\psi ) = \\varepsilon _ K ( V , \\psi ) q ^ { - s [ a ( V ) + n ( \\psi ) \\dim V ] } \\end{align*}"}
-{"id": "5102.png", "formula": "\\begin{align*} \\tau ( z ) \\tau ( w ) = \\tau ( w ) \\tau ( z ) . \\end{align*}"}
-{"id": "8826.png", "formula": "\\begin{align*} \\rho ( z ) = a _ 1 z + a _ 2 z ^ 2 + \\cdots \\end{align*}"}
-{"id": "6375.png", "formula": "\\begin{align*} \\mathcal { K } : = \\{ | z | > | x ' | \\} , \\end{align*}"}
-{"id": "4346.png", "formula": "\\begin{align*} \\psi \\left ( 1 - \\frac { f ( r ) } { a } \\right ) = \\frac { | f ' ( r ) | ^ p } { a ^ p } \\ , r \\in [ 0 , R ( a ) ) \\ , \\end{align*}"}
-{"id": "306.png", "formula": "\\begin{align*} V ' : = \\mathrm { C o v } ( Y _ i ' ) = \\begin{pmatrix} p _ { n , x , u } ^ { ( j ) } ( 1 - p _ { n , x , u } ^ { ( j ) } ) & p _ \\cap - p _ { n , x , u } ^ { ( j ) } p _ { n , y , v } ^ { ( l ) } \\\\ p _ \\cap - p _ { n , x , u } ^ { ( j ) } p _ { n , y , v } ^ { ( l ) } & p _ { n , y , v } ^ { ( l ) } ( 1 - p _ { n , y , v } ^ { ( l ) } ) \\end{pmatrix} , \\end{align*}"}
-{"id": "3306.png", "formula": "\\begin{align*} \\langle w , \\gamma _ { - } ( z ) v \\rangle _ { k } = \\langle w \\wedge z , v \\rangle _ { k + 1 } , z \\in \\mathfrak { u } _ { - } , \\ w \\in \\Lambda _ { q } ^ { k } ( \\mathfrak { u } _ { - } ) , \\ v \\in \\Lambda _ { q } ^ { k + 1 } ( \\mathfrak { u } _ { + } ) . \\end{align*}"}
-{"id": "4757.png", "formula": "\\begin{align*} K = \\frac { f '' } { f } ; \\end{align*}"}
-{"id": "7287.png", "formula": "\\begin{gather*} \\Delta ( K _ i ) = K _ i \\otimes K _ i , \\Delta ( E _ i ) = E _ i \\otimes 1 + K _ i \\otimes E _ i , \\Delta ( F _ i ) = F _ i \\otimes K _ i ^ { - 1 } + 1 \\otimes F _ i , \\\\ S ( K _ { i } ) = K _ { i } ^ { - 1 } , S ( E _ { i } ) = - K _ { i } ^ { - 1 } E _ { i } , S ( F _ { i } ) = - F _ { i } K _ { i } , \\varepsilon ( K _ i ) = 1 , \\varepsilon ( E _ i ) = \\varepsilon ( F _ i ) = 0 . \\end{gather*}"}
-{"id": "8694.png", "formula": "\\begin{align*} B ( t , \\cdot ) \\in C _ b ( H , U ) , \\ ; \\ ; t \\in [ 0 , T ] , \\ ; \\ ; \\| B \\| _ { \\infty } = \\sup _ { t \\in [ 0 , T ] \\times H } | B ( t , x ) | _ U < \\infty \\end{align*}"}
-{"id": "9947.png", "formula": "\\begin{align*} h _ \\mu ( t ) : = \\frac { t - \\mu c _ { 2 ^ * } ^ { 2 ^ * } t ^ { 2 ^ * - 1 } } { a _ 1 c _ { 2 ^ * } | \\Omega | ^ \\frac { 2 ^ * - 1 } { 2 ^ * } + a _ 2 c _ { 2 ^ * } ^ q | \\Omega | ^ \\frac { 2 ^ * - q } { 2 ^ * } t ^ { q - 1 } } \\mbox { f o r a l l } t \\geq 0 . \\end{align*}"}
-{"id": "4166.png", "formula": "\\begin{align*} G _ k ( z ) = \\tfrac { \\displaystyle g _ k ( z ) } { \\displaystyle z ^ { k - 1 } } = z + \\sum ^ \\infty _ { n = 2 } B _ n z ^ n \\in \\mathcal { S ^ { * } } \\subset \\mathcal { S } . \\end{align*}"}
-{"id": "3478.png", "formula": "\\begin{align*} v ( x ' ) = \\int _ 0 ^ 1 u ( x ' , x _ n ) d x _ n . \\end{align*}"}
-{"id": "8172.png", "formula": "\\begin{align*} r _ { n , k } = \\binom { ( \\alpha + \\beta ) n } { k - 1 } \\frac { ( k - 1 ) ! } { n ! } B _ { n , k } ( 1 ! \\theta _ 1 , 2 ! \\theta _ 2 , \\dots ) , \\end{align*}"}
-{"id": "8925.png", "formula": "\\begin{align*} w ( x ) = \\exp \\Bigl ( - \\sum _ { j = 1 } ^ k \\frac { \\abs { \\lambda _ j } } { 2 } \\abs { P _ { W _ j } ( x ) } ^ 2 \\Bigr ) , \\end{align*}"}
-{"id": "7127.png", "formula": "\\begin{align*} A = \\int _ 0 ^ { 2 \\pi } \\big ( K ( t ) - U ( t ) \\big ) d t , \\end{align*}"}
-{"id": "6703.png", "formula": "\\begin{align*} \\theta ^ I = a ^ I _ K \\eta ^ K \\mbox { a n d } \\bar \\theta ^ J = b _ L ^ J \\bar \\eta ^ L . \\end{align*}"}
-{"id": "7932.png", "formula": "\\begin{align*} \\sum _ { i \\in J } d _ i \\geq \\binom { | J | } { 2 } \\end{align*}"}
-{"id": "4724.png", "formula": "\\begin{align*} ( y ^ i ) ^ { \\top } A ^ i = \\lambda _ i \\pi ^ { \\top } . \\end{align*}"}
-{"id": "4767.png", "formula": "\\begin{align*} z ' + \\frac { 1 } { t } \\ , z = \\pm 2 a . \\end{align*}"}
-{"id": "7082.png", "formula": "\\begin{align*} \\Gamma ( \\mathcal { O } ) = \\left \\{ u \\in \\mathcal { U } : C ( \\pi u , u ( | u | ) ) \\not \\subset \\mathcal { O } , \\exists j \\in \\N : C ( u , j ) \\subset \\mathcal { O } \\right \\} \\end{align*}"}
-{"id": "4583.png", "formula": "\\begin{align*} f ( x ) = \\frac { 1 + e ^ { - 2 t g ^ * ( x ) } } { 2 c ( t ) } f _ { t , g ^ * } ( x ) \\geq \\frac { f _ { t , g ^ * } ( x ) } { 4 } . \\end{align*}"}
-{"id": "1629.png", "formula": "\\begin{align*} \\left ( a - \\lambda \\right ) B ^ { t } - \\xi _ { , t } ^ { t } B ^ { t } = 0 . \\end{align*}"}
-{"id": "6421.png", "formula": "\\begin{align*} B _ { m } u = \\frac { d } { d t } ( g _ { 1 - \\alpha , m } * u ) , u \\in L ^ { p } ( [ 0 , T ] ; X ) , \\ , \\ , m \\in \\mathbb { N } , \\end{align*}"}
-{"id": "1782.png", "formula": "\\begin{align*} \\mathrm { s u p p } ( a ) : = \\{ x \\in \\R \\ | \\ a _ x \\neq 0 \\} \\end{align*}"}
-{"id": "8630.png", "formula": "\\begin{align*} a _ { i , n - l } = 0 = a _ { n - l , i } . \\end{align*}"}
-{"id": "916.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 1 / 2 } } & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { * } \\| V \\| _ { * } \\\\ & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\left ( \\frac { \\| U \\| _ { * } + \\| V \\| _ { * } } { 2 } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "7075.png", "formula": "\\begin{align*} \\gamma _ n = \\sum _ { | u | = n } \\delta _ { V ( u ) - m _ n } . \\end{align*}"}
-{"id": "9012.png", "formula": "\\begin{align*} \\alpha _ p = \\max \\{ s _ i - s _ i ' + \\lambda _ i + 2 : 1 \\le i \\le t \\} , \\end{align*}"}
-{"id": "2476.png", "formula": "\\begin{align*} \\varepsilon _ K ( ( V , N ) , \\psi ) : = \\varepsilon _ K ( V , \\psi ) \\det ( - \\Phi | V ^ I / V ^ I _ N ) \\end{align*}"}
-{"id": "5942.png", "formula": "\\begin{align*} A = \\begin{pmatrix} 0 & \\mathbb { I } \\\\ 0 & 0 \\end{pmatrix} \\ , { R \\ , } = \\begin{pmatrix} 0 \\\\ \\mathbb { I } \\end{pmatrix} , \\ ; \\ ; \\ ; \\ ; { R \\ , } { R \\ , } ^ * = Q = \\begin{pmatrix} 0 & 0 \\\\ 0 & \\mathbb { I } \\end{pmatrix} , \\ ; \\ ; \\ ; B = R F = \\begin{pmatrix} 0 \\\\ F \\end{pmatrix} : \\R ^ { 2 d } \\to \\R ^ { 2 d } \\ , . \\end{align*}"}
-{"id": "8693.png", "formula": "\\begin{gather*} \\nabla _ k \\nabla _ { } ^ G w ( t , x ) = \\int _ t ^ T \\nabla _ k \\nabla _ { } ^ G R _ { s - t } [ \\Phi ( s , \\cdot ) ] ( x ) d s . \\end{gather*}"}
-{"id": "9647.png", "formula": "\\begin{align*} \\gamma _ 1 ( \\omega _ 1 , \\omega _ 2 ) = \\frac { \\omega _ 2 ^ 2 ( \\omega _ 1 ^ 2 \\omega _ 2 ^ 2 + 4 \\omega _ 1 ^ 2 \\omega _ 2 + 2 \\omega _ 1 \\omega _ 2 + 3 \\omega _ 1 ^ 2 + 3 \\omega _ 1 + 1 ) } { ( \\omega _ 1 + 1 ) ( 3 \\omega _ 1 \\omega _ 2 ^ 2 + 4 \\omega _ 1 \\omega _ 2 + \\omega _ 1 + 2 \\omega _ 2 + 1 ) } . \\end{align*}"}
-{"id": "305.png", "formula": "\\begin{align*} \\mu ' : = \\mathbb { E } ( Y _ i ' ) = \\begin{pmatrix} p _ { n , x , u } ^ { ( j ) } \\\\ p _ { n , y , v } ^ { ( l ) } \\end{pmatrix} \\end{align*}"}
-{"id": "4449.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\biggl | \\mathbb { E } _ f ( \\hat { H } _ n ) - H - \\sum _ { l = 1 } ^ { \\lceil \\beta / 2 \\rceil - 1 } \\frac { \\Gamma ( k + 2 l / d ) \\Gamma ( n ) } { \\Gamma ( k ) \\Gamma ( n + 2 l / d ) } \\lambda _ l \\biggr | = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { \\beta } { d } } } { n ^ { \\frac { \\beta } { d } } } \\biggr \\} \\biggr ) \\end{align*}"}
-{"id": "8537.png", "formula": "\\begin{align*} \\pi _ { r _ { 1 } a } ' \\xi _ { r a } ' = \\delta _ { r , r _ { 1 } } i d _ { N _ { \\tau ( a ) } } \\end{align*}"}
-{"id": "6212.png", "formula": "\\begin{align*} & V ^ { T } ( M V \\ddot { \\hat { x } } ( t ) + D V \\dot { \\hat { x } } ( t ) + K V \\hat { x } ( t ) - F u ( t ) ) = 0 , \\\\ & \\hat { y } ( t ) = \\ C _ { p } V \\hat { x } ( t ) + C _ { v } V \\dot { \\hat { x } } ( t ) . \\end{align*}"}
-{"id": "769.png", "formula": "\\begin{gather*} R _ { \\ell } = \\bigoplus _ { 0 \\leqslant k \\leqslant \\ell , [ k ] = [ \\ell ] } \\partial R _ k , \\end{gather*}"}
-{"id": "7007.png", "formula": "\\begin{align*} \\int _ X u ( t , x ) d Q = \\int _ { D ( t , p ) } u ( t , x ) d Q & = \\int _ { D ( t , p ) } \\max _ { y \\in B ( t , p ) } u ( t , y ) d Q = \\max _ { y \\in B ( t , p ) } u ( t , y ) . \\end{align*}"}
-{"id": "5556.png", "formula": "\\begin{align*} \\dot { A } _ { 1 } & = ( \\beta + \\frac { 1 } { 2 } b _ { x } ( 1 ) ) A _ { 1 } - \\frac { 1 } { 2 } b _ { x x } ( 1 ) B _ { 1 } = ( \\beta + \\frac { 1 } { 2 } b _ { x } ( 1 ) ) A _ { 1 } + H ( b _ x ( 1 ) + H b ( 1 ) ) B _ { 1 } , \\\\ \\dot { B } _ { 1 } & = ( \\beta - \\frac { 1 } { 2 } b _ { x } ( 1 ) ) B _ { 1 } + b ( 1 ) A _ { 1 } = ( \\beta - \\frac { 1 } { 2 } b _ { x } ( 1 ) ) B _ { 1 } - \\frac { 1 } { H ^ 2 } ( H b _ { x } ( 1 ) + \\frac { 1 } { 2 } b _ { x x } ( 1 ) ) A _ { 1 } . \\end{align*}"}
-{"id": "3108.png", "formula": "\\begin{align*} & B _ { 1 1 } : = P _ 5 , \\\\ & A _ { 1 1 } : = P _ 4 - ( B _ { 1 2 } + B _ { 1 2 } ^ T ) , \\\\ & B _ { 2 2 } : = P _ 3 - ( B _ { 1 3 } + B _ { 3 1 } ^ T ) - ( A _ { 1 2 } + A _ { 1 2 } ^ T ) , \\\\ & A _ { 2 2 } : = P _ 2 - ( B _ { 2 3 } + B _ { 2 3 } ^ T ) - ( A _ { 1 3 } + A _ { 1 3 } ^ T ) , \\\\ & B _ { 3 3 } : = P _ 1 - ( A _ { 2 3 } + A _ { 2 3 } ^ T ) , \\mbox { a n d } \\\\ & A _ { 3 3 } : = P _ 0 . . \\end{align*}"}
-{"id": "1556.png", "formula": "\\begin{align*} \\underset { k \\rightarrow \\infty } { \\liminf } \\ J _ { \\P _ n } ^ { \\gamma } ( \\mu ^ k ) & = \\underset { k \\rightarrow \\infty } { \\liminf } \\ \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ 2 ^ 2 ( \\mu ^ k , \\nu _ i ) + \\gamma E ( \\mu ^ k ) \\\\ & \\ge \\frac { 1 } { n } \\sum _ { i = 1 } ^ n W _ 2 ^ 2 ( \\mu , \\nu _ i ) + \\gamma E ( \\mu ) = J _ { \\P _ n } ^ { \\gamma } ( \\mu ) . \\end{align*}"}
-{"id": "4045.png", "formula": "\\begin{align*} ( \\alpha + 1 ) q & = ( \\beta + 1 ) p ; \\\\ ( \\beta + 1 ) p & = ( \\gamma + 1 ) r ; \\\\ ( \\gamma \\beta - 1 ) p & = ( \\gamma + 1 ) q , \\end{align*}"}
-{"id": "972.png", "formula": "\\begin{align*} 1 / C \\leq | ( X _ { \\varepsilon } ) _ { T } | \\leq C \\ ; , \\ ; \\ ; \\left | \\left ( [ X _ { \\varepsilon } , \\overline { Z } ] \\right ) _ { T } \\right | \\leq \\varepsilon \\ ; , \\ ; Z \\in T ^ { 1 , 0 } ( M ) \\ ; , \\ ; | Z | = 1 \\ ; , \\end{align*}"}
-{"id": "9834.png", "formula": "\\begin{align*} M _ i \\ , { } _ \\lambda \\ , v _ m = d v _ { i + m } , \\end{align*}"}
-{"id": "3577.png", "formula": "\\begin{align*} & s _ 1 = 0 , t _ 1 = 4 \\\\ & s _ j = - 1 , t _ k = 3 , ( j , k = 2 , \\dots , n + 1 ) . \\end{align*}"}
-{"id": "9178.png", "formula": "\\begin{align*} \\left [ \\begin{array} { c } s \\\\ S _ 0 ^ { - 1 } ( \\widehat { m } ) \\end{array} \\right ] = \\frac { s ! } { ( \\widehat { m } _ 1 - 1 ) ! \\widehat { m } _ 2 ! \\cdots \\widehat { m } _ s ! } = \\frac { s ! } { \\widehat { m } _ 1 ! \\widehat { m } _ 2 ! \\cdots \\widehat { m } _ s ! \\widehat { m } _ { s + 1 } ! } \\widehat { m } _ 1 , \\end{align*}"}
-{"id": "8762.png", "formula": "\\begin{align*} \\left . \\frac { \\dd } { \\dd x _ i } \\right | _ { \\{ x _ j = 1 \\} } S _ i ( x _ 1 , . . , x _ n ) = \\left . \\frac { \\dd } { \\dd w } \\right | _ { w = 1 } T ( w ) + \\left . \\frac { \\dd } { \\dd x _ i } \\right | _ { x _ i = 1 } T ( 1 ; x _ i ) = T ' ( 1 ) , \\end{align*}"}
-{"id": "26.png", "formula": "\\begin{align*} \\Sigma = ( t _ m = 0 ) . \\end{align*}"}
-{"id": "4752.png", "formula": "\\begin{align*} \\lim _ { r \\to \\infty } r ^ { N + 2 s + k } D ^ k F _ N ( r ) & = \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\ , \\alpha _ { j , k } \\ , \\lim _ { r \\to \\infty } r ^ { N + 2 s + 2 j } F _ { N + 2 j } ( r ) \\\\ & = \\sum _ { k \\le 2 j \\le 2 k } ( - 1 ) ^ j \\alpha _ { j , k } \\ , \\ell _ { N + 2 j , \\ , 0 } . \\end{align*}"}
-{"id": "7056.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ i ] } } ( n ) = { \\bf { V } } _ 1 ^ { [ i ] } ( n ) { { \\bf { u } } ^ { [ i ] } } + { \\bf { V } } _ 2 ^ { [ i ] } ( n ) { { \\bf { v } } ^ { [ i ] } } , \\end{align*}"}
-{"id": "55.png", "formula": "\\begin{align*} \\frac { d } { d r } \\left ( e ^ r | f ' ( r , a ) | ^ { p - 2 } f ' ( r , a ) \\right ) = - e ^ r f ( r , a ) \\ , r \\in ( 0 , R ( a ) ) \\ . \\end{align*}"}
-{"id": "8820.png", "formula": "\\begin{align*} P _ { i , n } = \\frac { \\partial ^ n _ t } { n ! } P _ i ( x _ { 1 } ( t ) , x _ 2 ( t ) , \\cdots , x _ k ( t ) ) \\vert _ { t = 0 } \\end{align*}"}
-{"id": "1146.png", "formula": "\\begin{align*} V _ x = \\begin{bmatrix} 0 & 1 \\\\ - z \\rho & 0 \\end{bmatrix} V , V = \\begin{bmatrix} v \\\\ v _ x \\end{bmatrix} , \\end{align*}"}
-{"id": "5914.png", "formula": "\\begin{align*} ( B ^ { s _ 0 } _ { p , p } ( \\R ^ d ) , B ^ { s _ 1 } _ { p , p } ( \\R ^ d ) ) _ { \\theta , p } = B ^ { s } _ { p , p } ( \\R ^ d ) \\end{align*}"}
-{"id": "4094.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) z ^ 2 } { x ^ { 4 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z } { x ^ { 3 } } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 2 z } { x ^ { 5 } } . \\\\ \\end{gather*}"}
-{"id": "2533.png", "formula": "\\begin{align*} d \\phi = \\mathbf { i } \\left [ \\frac 1 { h ^ 2 } A \\phi + \\lambda F ( \\phi ) \\phi \\right ] d t + \\mathbf { i } Z ( \\phi ) d \\Psi ( t ) \\end{align*}"}
-{"id": "8245.png", "formula": "\\begin{align*} 2 \\Im L : = \\Im ( f _ 1 + f _ 2 ) \\frac { \\partial } { \\partial x } + \\Re ( f _ 2 - f _ 1 ) \\frac { \\partial } { \\partial y } + \\Im ( g _ 1 + g _ 2 ) \\frac { \\partial } { \\partial u } + \\Re ( g _ 2 - g _ 1 ) \\frac { \\partial } { \\partial v } , \\end{align*}"}
-{"id": "7670.png", "formula": "\\begin{align*} S _ { \\alpha } ( t ) \\chi _ { B _ { R } } & = \\int _ { B _ { R } } p _ { D } ( t , x , y ) \\chi _ { B _ { R } } ( y ) d y \\\\ & \\leq \\int _ { R ^ { d } } p _ { D } ( t , x , y ) \\chi _ { B _ { R } } ( y ) d y \\\\ & \\leq \\chi _ { B _ { R } } . \\end{align*}"}
-{"id": "962.png", "formula": "\\begin{align*} T _ q ( \\sideset { } { ' } \\sum _ { \\vert J \\vert = q } \\ , u _ J \\overline { \\omega _ J } \\ , ) \\ ; \\ ; \\ ; \\ ; = \\sideset { } { ' } \\sum _ { | J | = q , | K | = ( m - 1 - q ) } \\varepsilon _ { ( 1 , \\dots , m - 1 ) } ^ { J K } u _ J \\overline { \\omega _ K } \\ ; , \\end{align*}"}
-{"id": "3829.png", "formula": "\\begin{align*} \\mu ( S ) = \\left \\{ \\dfrac { q ^ 2 + q + 1 } { ( 3 , q - 1 ) } , \\dfrac { q ^ 2 - 1 } { ( 3 , q - 1 ) } , q - 1 , \\dfrac { p ( q - 1 ) } { ( 3 , q - 1 ) } \\right \\} . \\end{align*}"}
-{"id": "4368.png", "formula": "\\begin{align*} { [ } f , g ] : = f \\partial _ \\theta g - f \\partial _ \\theta g \\ ; \\ ; \\ ; \\ ; \\textrm { a n d } \\ ; \\ ; \\ ; ( f , g ) : = \\int _ { - \\pi } ^ { \\pi } \\left ( \\frac { d } { d \\theta } f ( e ^ { i \\theta } ) + \\left ( \\frac { d } { d \\theta } \\right ) ^ 3 f ( e ^ { i \\theta } ) \\right ) g ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } \\end{align*}"}
-{"id": "6335.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 1 } \\| _ { M _ 2 } \\lesssim & 2 ^ { j n ( \\alpha _ 2 - \\alpha _ 1 ) ( 1 - 1 / p _ 2 ) } \\| \\Box _ k ^ { \\alpha _ 1 } f \\| _ { L ^ { p _ 1 } } \\\\ \\lesssim & 2 ^ { j n ( \\alpha _ 2 - \\alpha _ 1 ) ( 1 - 1 / p _ 2 ) } \\| f \\| _ { M _ 1 } = 2 ^ { j \\widetilde { A _ 2 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "1846.png", "formula": "\\begin{align*} \\widetilde { G } _ { m , n } ( t ) = \\frac { 1 } { \\sqrt { m + 1 } } \\int _ n ^ { n + 1 } v ^ { - 1 / 2 } \\exp \\left ( - t \\left ( i \\log { \\frac { m + 1 } { v } } + \\frac { 1 } { 2 } \\log ^ 2 { \\frac { m + 1 } { v } } \\right ) \\right ) d v . \\end{align*}"}
-{"id": "7647.png", "formula": "\\begin{align*} [ S _ { \\alpha } ( t ) \\chi _ { r } ] ( x ) & = \\int _ { B _ { r } ( 0 ) } p _ { D } ( t , x , y ) d y \\\\ & \\geq c _ { 0 } \\int _ { B _ { r } } ( t ^ { - d / \\alpha } \\wedge \\frac { t } { | x - y | ^ { d + \\alpha } } ) d y . \\end{align*}"}
-{"id": "6066.png", "formula": "\\begin{align*} \\frac { 1 } { C } \\Big ( { \\sum _ { j = 1 } ^ \\infty } | c _ { j } | ^ { 2 } \\Big ) ^ { p / 2 } \\le \\int _ { 0 } ^ { 1 } \\Big | { \\sum _ { j = 1 } ^ \\infty } c _ { j } r _ { j } ( z ) \\Big | ^ { p } d z \\le C \\Big ( { \\sum _ { j = 1 } ^ \\infty } | c _ { j } | ^ { 2 } \\Big ) ^ { p / 2 } . \\end{align*}"}
-{"id": "1933.png", "formula": "\\begin{align*} & ( \\sqrt { n _ 1 } \\ , Y _ 1 ( u _ 0 ) ( 2 p _ 1 - q _ 1 ^ 2 Y _ 1 ( u _ 0 ) ) , \\cdots , \\sqrt { n _ m } \\ , Y _ m ( u _ 0 ) ( 2 p _ m - q _ m ^ 2 Y _ m ( u _ 0 ) ) ) \\\\ = & c ( \\sqrt { n _ 1 } , \\cdots , \\sqrt { n _ m } ) , \\end{align*}"}
-{"id": "2658.png", "formula": "\\begin{align*} & T ( n - d + 2 , 3 d - 2 n ) = \\\\ & 2 \\cdot T ( n - d + 1 , 3 d - 2 n ) + T ( n - d + 1 , 3 d - 2 n - 1 ) = \\\\ & 2 \\cdot T ( J ( d - 2 , n - 3 ) , K ( d - 2 , n - 3 ) ) + T ( J ( d - 3 , n - 4 ) , K ( d - 3 , n - 4 ) ) \\end{align*}"}
-{"id": "3134.png", "formula": "\\begin{align*} M _ { 1 2 } ( \\lambda ) = \\begin{bmatrix} \\lambda P _ 4 - P _ 3 & \\lambda P _ 3 \\end{bmatrix} \\mbox { a n d } M _ { 2 2 } ( \\lambda ) = \\begin{bmatrix} \\lambda P _ 3 - P _ 2 & \\lambda P _ 2 \\\\ \\lambda P _ 2 & \\lambda P _ 1 + P _ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "216.png", "formula": "\\begin{align*} \\max ( | R _ { 3 1 } | , | R _ { 3 2 } | ) = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { \\beta } { d } } } { n ^ { \\frac { \\beta } { d } } } \\biggr \\} \\biggr ) , \\end{align*}"}
-{"id": "6428.png", "formula": "\\begin{align*} \\eta ( x , t ) = \\int _ { t } ^ { T } h _ { m } ( \\sigma - t ) \\varphi ( \\sigma , x ) d \\sigma = \\int _ { 0 } ^ { T - t } h _ { m } ( \\sigma ) \\varphi ( \\sigma + t , x ) d \\sigma , \\end{align*}"}
-{"id": "1990.png", "formula": "\\begin{align*} ( \\lambda - \\Delta ) ^ { - 1 } u = \\int _ 0 ^ \\infty e ^ { - \\lambda t } e ^ { t \\Delta } u \\ , d t . \\end{align*}"}
-{"id": "3074.png", "formula": "\\begin{align*} \\rho _ { n , k } ( f ) \\leq \\rho _ { n , 0 } ( f ) = \\sum _ { | u | = n } f ( V ( u ) - M _ n ) . \\end{align*}"}
-{"id": "6095.png", "formula": "\\begin{align*} S _ t = \\big \\{ u ' \\in W _ u \\colon 2 ^ { t - 1 } \\cdot 4 p ^ 2 | V | \\le | N ( u , u ' ) | < 2 ^ t \\cdot 4 p ^ 2 | V | \\big \\} \\end{align*}"}
-{"id": "6955.png", "formula": "\\begin{align*} e ^ { - i q } \\ , z ^ { - \\alpha _ 0 } = ( 1 / z ) \\ , z ^ { - \\alpha _ 0 } \\ , e ^ { - i q } \\ , \\end{align*}"}
-{"id": "8609.png", "formula": "\\begin{align*} & \\overline { \\Pi } _ { 1 } \\overline { \\overline { N } } ( a ) = - \\theta _ { 1 } \\overline { \\pi } _ { 1 } \\overline { p } \\xi _ { e _ { k } a } \\\\ & \\overline { \\Pi } _ { 2 } \\overline { \\overline { N } } ( a ) = - \\theta _ { 2 } ^ { - 1 } \\overline { \\gamma } ' \\xi _ { e _ { k } } a \\\\ & \\overline { \\Pi } _ { 3 } \\overline { \\overline { N } } ( a ) = \\overline { \\Pi } _ { 4 } \\overline { \\overline { N } } ( a ) = 0 \\end{align*}"}
-{"id": "9369.png", "formula": "\\begin{align*} S _ { m a x } f = - \\frac { d ^ 2 f } { d x ^ 2 } + [ q _ 1 , q _ 2 ] \\Gamma _ 0 f , { f } \\in \\mathcal { D } ( S _ { m a x } ) = { W } _ 2 ^ 2 ( \\mathbb { R } \\backslash \\{ 0 \\} ) . \\end{align*}"}
-{"id": "5662.png", "formula": "\\begin{align*} Q ( F ) - E \\cdot \\nabla _ v F = 0 , \\int _ { \\R ^ d } F ( v , E ) \\ , d v = 1 . \\end{align*}"}
-{"id": "1600.png", "formula": "\\begin{align*} x C _ { , x } ^ { x } - C ^ { x } + m x = 0 , \\end{align*}"}
-{"id": "6510.png", "formula": "\\begin{align*} \\left \\{ \\begin{alignedat} { 3 } & \\frac { \\partial u } { \\partial t } + ( - \\varDelta ) ^ { 1 / 2 } u = f ( u ) \\ , \\ , \\ , & & \\ , \\ , & & \\R ^ d \\times ( 0 , T ) , \\\\ & u ( \\cdot , 0 ) = u _ 0 & & \\ , \\ , & & \\R ^ d , \\end{alignedat} \\right . \\end{align*}"}
-{"id": "9835.png", "formula": "\\begin{align*} r ( t ) & = h _ { \\rm S R } ( t ) \\otimes s ( t ) + \\alpha x ( t - \\tau ) + n _ { \\rm R } ( t ) , \\end{align*}"}
-{"id": "4653.png", "formula": "\\begin{align*} S = \\{ 0 \\} \\sqcup G ^ 1 \\cdot w _ 1 \\sqcup G ^ 1 \\cdot w _ 2 . \\end{align*}"}
-{"id": "6789.png", "formula": "\\begin{align*} \\int _ { \\{ \\Phi < P _ { \\Theta } ( F ) \\} } ( \\Theta + d d ^ c \\Phi ) ^ { n + 1 } = 0 . \\end{align*}"}
-{"id": "8778.png", "formula": "\\begin{align*} T _ 0 & = t _ 0 - \\frac { ( x _ 1 - a ) ( x _ 1 - c ) } { x _ 1 ^ 2 - q } ( 1 - s _ 0 ) , \\\\ T _ i & = t - \\frac { t x _ i - x _ { i + 1 } } { x _ i - x _ { i + 1 } } ( 1 - s _ i ) ( i = 1 , \\ldots , n - 1 ) , \\\\ T _ n & = t _ n - \\frac { ( b x _ n - 1 ) ( d x _ n - 1 ) } { 1 - x _ n ^ 2 } ( 1 - s _ n ) . \\end{align*}"}
-{"id": "1779.png", "formula": "\\begin{align*} s _ 1 ^ 2 + s _ 2 ^ 2 d + 2 f s _ 1 s _ 2 - a ( s _ 3 ^ 2 + s _ 4 ^ 2 d ) - 2 a f s _ 3 s _ 4 = 0 \\end{align*}"}
-{"id": "5799.png", "formula": "\\begin{align*} f _ { r e g } = e _ { - \\beta _ { 1 } } + \\cdots + e _ { - \\beta _ { n - 1 } } + e _ { - 2 \\beta _ { n } } \\end{align*}"}
-{"id": "4331.png", "formula": "\\begin{align*} | \\langle a ( f _ { k _ m } ) , a ( f _ { k _ n } ) \\rangle | = | \\langle a ^ * a ( f _ { k _ m } ) , f _ { k _ n } \\rangle | < \\frac { \\varepsilon ^ 2 } { 3 \\cdot 2 ^ n } \\end{align*}"}
-{"id": "9388.png", "formula": "\\begin{align*} u _ { \\lambda } ( x ) = \\left \\{ \\begin{array} { l } A _ k ( x ) e ^ { i k x } + B _ k ( x ) e ^ { - i k x } , x > 0 \\\\ C _ k ( x ) e ^ { i k x } + D _ k ( x ) e ^ { - i k x } , x < 0 \\end{array} \\right . , \\lambda = k ^ 2 , \\end{align*}"}
-{"id": "3643.png", "formula": "\\begin{align*} u _ { n + 1 } = \\exp ( 1 - u _ n ) \\exp ( 1 - u ^ 2 _ { n - 1 } ) , \\end{align*}"}
-{"id": "7582.png", "formula": "\\begin{align*} \\mathcal A _ { 1 1 } : = \\left \\{ \\omega \\in \\mathcal A : x _ 1 ( \\omega ) > u _ l \\right \\} , \\mathcal A _ { 1 2 } : = \\left \\{ \\omega \\in \\mathcal A : x _ 1 ( \\omega ) \\in \\left ( x _ 0 + \\frac { l + A } { 2 } , \\ , u _ l \\right ] \\right \\} . \\end{align*}"}
-{"id": "303.png", "formula": "\\begin{align*} W _ 3 ' = O \\biggl ( \\max \\biggl \\{ \\frac { \\log n } { n k ^ { 1 / 2 } } \\ , , \\ , \\frac { k ^ { \\frac { 1 } { 2 } + \\frac { 2 \\beta } { d } } } { n ^ { 1 + \\frac { 2 \\beta } { d } } } \\ , , \\ , \\frac { k ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "3283.png", "formula": "\\begin{align*} \\hat { R } F ( v _ { 0 } \\otimes v _ { 0 } ) = [ 2 ] ^ { 1 / 2 } ( v _ { 0 } \\otimes v _ { - 1 } + \\hat { R } ( v _ { 0 } \\otimes v _ { - 1 } ) ) . \\end{align*}"}
-{"id": "8777.png", "formula": "\\begin{align*} K ^ { ( 2 r + 1 ) } _ 0 ( x ) = \\sum _ { i = - r } ^ { r } E ^ { ( i i ) } + \\frac { q - x ^ 2 } { h _ 0 ( a , c , x ) } \\Big ( \\sum _ { 0 < i \\leq r } t _ 0 E ^ { ( - i , - i ) } + E ^ { ( r + 1 - i , r + 1 - i ) } \\\\ { } - \\sum _ { 0 < i \\leq r } q ^ { - i } E ^ { ( - i , r + 1 - i ) } + q ^ { r + 1 - i } t _ 0 E ^ { ( r + 1 - i , - i ) } \\Big ) . \\end{align*}"}
-{"id": "4268.png", "formula": "\\begin{align*} K : = \\limsup _ { n \\rightarrow + \\infty } n \\mathbb P ( - Z _ 1 > b _ n ) < + \\infty \\ , , \\end{align*}"}
-{"id": "9951.png", "formula": "\\begin{align*} \\lim _ { j \\to \\infty } \\int _ Q ( u _ \\infty ( x ) - u _ \\infty ( y ) ) ( u _ j ( x ) - u _ j ( y ) - u _ \\infty ( x ) + u _ \\infty ( y ) ) K ( x - y ) d x d y = 0 . \\end{align*}"}
-{"id": "2876.png", "formula": "\\begin{align*} H o m _ { p t } ( X , Y ) = H o m ( X _ - , Y _ - ) _ + ; \\end{align*}"}
-{"id": "9817.png", "formula": "\\begin{align*} \\int _ { \\mathbb R ^ { n } } \\ldots \\int _ { \\mathbb R ^ { n } } \\prod _ { k = 1 } ^ { M } F ^ { ( k ) } ( x _ { 1 } , \\ldots , x _ { \\ell } ) \\prod _ { i = 1 } ^ { N } f _ { i } ^ { \\ast } \\left ( \\sum _ { j = 1 } ^ { \\ell } a _ { i j } x _ { j } | \\theta \\right ) d \\mu ( x _ { \\ell } ) \\ldots d \\mu ( x _ { 1 } ) . \\end{align*}"}
-{"id": "9499.png", "formula": "\\begin{align*} * \\alpha ^ 0 & = - \\alpha ^ 1 \\wedge \\ldots \\wedge \\alpha ^ n \\\\ * \\alpha ^ i & = ( - 1 ) ^ i \\alpha ^ 0 \\wedge \\ldots \\widehat { \\alpha ^ i } \\wedge \\ldots \\alpha ^ n , \\end{align*}"}
-{"id": "7112.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\limsup _ { n \\to \\infty } \\P ( \\rho _ { n , k _ n } ( f ) - \\rho _ { n , k } ( f ) > \\epsilon ) = 0 \\epsilon > 0 , \\end{align*}"}
-{"id": "626.png", "formula": "\\begin{align*} [ \\Psi _ \\lambda \\ast ( Q f ) ] ( x ) = \\int _ { \\Omega } \\Psi _ \\lambda ( x - y ) Q ( y ) f ( y ) \\ , d y \\end{align*}"}
-{"id": "8772.png", "formula": "\\begin{align*} K ^ { ( 2 r + 1 ) } _ 0 ( x ) = \\sum _ { i = - r } ^ { r } E ^ { ( i i ) } + \\frac { 1 - x ^ 2 } { h _ 0 ( a , c , x ) } \\Big ( \\sum _ { 0 < i \\leq r } t _ 0 E ^ { ( - i , - i ) } + E ^ { ( r + 1 - i , r + 1 - i ) } \\\\ { } - \\sum _ { 0 < i \\leq r } E ^ { ( - i , r + 1 - i ) } + t _ 0 E ^ { ( r + 1 - i , - i ) } \\Big ) , \\end{align*}"}
-{"id": "3522.png", "formula": "\\begin{align*} \\| u \\rho ^ { \\frac { 1 } { 2 } } \\| ^ 2 _ { H ^ k } = \\sum _ { j = 0 } ^ k \\| \\nabla ^ j ( u \\rho ^ { \\frac { 1 } { 2 } } ) \\| ^ 2 _ { L ^ 2 } . \\end{align*}"}
-{"id": "939.png", "formula": "\\begin{align*} \\phi _ 0 ( x ) \\ , = \\ , 1 , \\qquad \\mbox { a n d } \\phi _ k ( x ) \\ , = \\ , \\sqrt { 2 } \\cos ( \\pi k x ) \\mbox { f o r } k \\in \\N . \\end{align*}"}
-{"id": "3484.png", "formula": "\\begin{align*} \\int _ { \\tilde { \\Omega } } \\nabla \\tilde { u } ( x ) \\nabla \\phi ( x ) d x = \\int _ { \\tilde { \\Omega } } \\nabla \\tilde { u } ( x ) \\nabla ( \\phi ( x ) \\varphi _ \\delta ( x _ n ) ) d x + \\int _ { \\tilde { \\Omega } } \\nabla \\tilde { u } ( x ) \\nabla ( \\phi ( x ) ( 1 - \\varphi _ \\delta ( x _ n ) ) ) d x = I _ 1 + I _ 2 \\end{align*}"}
-{"id": "1311.png", "formula": "\\begin{align*} S _ { \\ell } ( \\alpha ) = \\sum _ { n = 1 } ^ { N } \\Lambda ( n ) e ( n ^ { \\ell } \\alpha ) . \\end{align*}"}
-{"id": "7741.png", "formula": "\\begin{align*} \\Omega ( \\delta , \\alpha ) : = \\bigcup _ { i = 1 } ^ { \\bar N } \\Omega _ i = \\{ \\omega \\in \\bar \\Omega : N ( \\omega ) \\le \\bar N \\} = \\{ \\omega \\in \\bar \\Omega : x _ n ( \\omega ) < \\delta , \\ , \\ , n \\ge \\bar N \\} . \\end{align*}"}
-{"id": "151.png", "formula": "\\begin{align*} \\| V a _ n - V a _ m \\| _ { \\widehat { C _ E } } = \\| V a _ n - V a _ m \\| _ { C _ { \\widehat { E } } } \\to 0 \\ \\ \\ \\ \\ \\ n , m \\to \\infty , \\end{align*}"}
-{"id": "332.png", "formula": "\\begin{align*} J : = \\left [ \\begin{array} { c c } 0 & 1 \\\\ - 1 & 0 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9436.png", "formula": "\\begin{align*} \\sum _ { j \\ne i } w ( c _ { i j } ) = w ( S _ i ) . \\end{align*}"}
-{"id": "3976.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\vect { \\delta } = ( \\delta _ 1 , \\dots , \\delta _ k ) } V ( \\vect { \\delta } ) , \\end{align*}"}
-{"id": "4221.png", "formula": "\\begin{align*} E _ n ( x ) = \\sum _ { l = 0 } ^ n { n \\choose l } x ^ { n - l } E _ l . \\end{align*}"}
-{"id": "8572.png", "formula": "\\begin{align*} \\varphi ( r a _ { 1 } ) & = \\displaystyle \\sum _ { r '' a ' } r '' a ' C _ { r '' a ' , r a _ { 1 } } \\\\ z _ { r _ { a _ { 1 } } } & = \\displaystyle \\sum _ { r '' a ' } r '' a ' C _ { r '' a ' , r a _ { 1 } } \\varphi ( Y _ { [ b _ { 1 } r a _ { 1 } ] } ( P ) ) \\end{align*}"}
-{"id": "9407.png", "formula": "\\begin{align*} S _ I = \\begin{bmatrix} L _ { I } ^ { - T } & - A _ { I I } ^ { - 1 } A _ { F I } ^ T & \\\\ & I & \\\\ & & I \\\\ \\end{bmatrix} . \\end{align*}"}
-{"id": "8427.png", "formula": "\\begin{align*} A _ { 0 } ( \\textbf { u } ) \\partial _ { t } \\textbf { u } + \\sum _ { j = 1 } ^ { d } A _ { 0 } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } + A _ { 0 } ( \\textbf { u } ) F _ { P } = 0 . \\end{align*}"}
-{"id": "652.png", "formula": "\\begin{align*} \\eta ( U _ { x _ o } ) = \\eta ( \\sup _ n T f _ n ) \\geq \\eta ( T f _ m ) = \\nu ( f _ m ) , \\textrm { f o r e v e r y $ m $ } . \\end{align*}"}
-{"id": "7436.png", "formula": "\\begin{align*} \\overline { w } _ 0 x ^ { - 1 } y \\overline { w } _ 0 ^ { - 1 } \\in w _ 0 B _ + w B _ + w _ 0 = B _ - w ^ * B _ - , \\end{align*}"}
-{"id": "6190.png", "formula": "\\begin{align*} \\mathcal { A } _ n f ( x ) = \\gamma _ U - \\frac { \\sigma _ U ^ 2 } { 2 } + \\int _ { \\R } [ \\log | 1 + z | - z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) + \\int _ { - 1 } ^ 1 z \\nu ' ( \\d z ) + o ( 1 ) , \\end{align*}"}
-{"id": "2678.png", "formula": "\\begin{align*} \\int _ X ( u _ { t } - u _ 0 ) \\theta _ { u _ t } ^ n = \\int _ X ( u + t \\chi - u _ 0 ) \\theta _ { u _ t } ^ n \\geq t \\int _ X \\chi \\theta _ { u _ t } ^ n . \\end{align*}"}
-{"id": "2163.png", "formula": "\\begin{align*} & 0 \\leq \\phi \\leq 1 , \\phi = 0 [ 0 , ( t _ { 2 } - t _ { 1 } ) / 2 ] , \\phi = 1 [ t _ { 2 } - t _ { 1 } , t _ { 0 } - t _ { 1 } ] , \\\\ & \\qquad \\qquad \\qquad 0 \\leq \\dot { \\phi } \\leq 4 / ( t _ { 2 } - t _ { 1 } ) . \\end{align*}"}
-{"id": "4829.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | ( I - A _ n ) [ T , \\tau ] | _ { \\mathcal I } = 0 . \\end{align*}"}
-{"id": "7282.png", "formula": "\\begin{align*} d = \\sum _ { i = 1 } ^ t ( d - d _ i ) . \\end{align*}"}
-{"id": "56.png", "formula": "\\begin{align*} & f ' ( r ) = - | g ( r ) | ^ { ( 2 - p ) / ( p - 1 ) } g ( r ) \\ , g ' ( r ) = - | g ( r ) | + f ( r ) \\ , r \\in ( 0 , \\mathcal { R } ( a ) ) \\ , \\\\ & f ( 0 ) = a \\ , \\ g ( 0 ) = 0 \\ , \\end{align*}"}
-{"id": "7412.png", "formula": "\\begin{align*} x = [ x ] _ - [ x ] _ 0 [ x ] _ + \\end{align*}"}
-{"id": "6076.png", "formula": "\\begin{align*} \\sum _ { a < n < b } g ( n ) \\exp ( 2 \\pi i f ( n ) ) & = \\sum _ { \\alpha - \\eta < m < \\beta + \\eta } \\int _ a ^ b g ( x ) \\exp ( 2 \\pi i ( f ( x ) - m x ) ) d x \\\\ & + O \\left ( G _ 1 \\bigg ( \\eta ^ { - 1 } + \\log \\Big ( 1 + \\frac { \\beta - \\alpha } { \\eta } \\Big ) \\bigg ) + G _ 2 ( \\beta - \\alpha + \\eta ) \\right ) , \\end{align*}"}
-{"id": "6257.png", "formula": "\\begin{align*} \\dd _ t \\nabla \\omega = \\nabla \\dd _ t \\omega = \\nabla \\Delta \\omega + \\nabla \\| \\nabla \\omega \\| ^ 2 . \\end{align*}"}
-{"id": "7109.png", "formula": "\\begin{align*} \\lambda _ { n , \\beta } = W _ { n , \\beta } ^ { - 2 } \\sum _ { | u | = | v | = n } e ^ { \\beta ( 2 m _ n - V ( u ) - V ( v ) ) } \\delta _ { | u \\wedge v | } , \\end{align*}"}
-{"id": "6544.png", "formula": "\\begin{align*} \\| b _ \\ell \\| _ X \\le ( \\tilde \\lambda + C _ B r ) \\| e _ \\ell \\| _ D + 2 C _ B \\| u _ \\ell ^ \\star \\| _ D \\| e _ \\ell \\| _ W , \\ell = 0 , \\dotsc , M - 1 . \\end{align*}"}
-{"id": "579.png", "formula": "\\begin{align*} \\delta ( x ) \\stackrel { ( \\ref { a = b + c } ) } { \\equiv } \\delta ( [ \\pi ( x ) ] ) + \\delta ( p [ a ] ) \\bmod I \\equiv \\delta ( p [ a ] ) \\bmod I \\equiv [ a ^ p ] \\bmod I \\equiv [ r _ 1 ] \\bmod I \\end{align*}"}
-{"id": "9360.png", "formula": "\\begin{align*} p ( t , x ) = \\mathbb { E } [ p ( T , x ) | \\mathcal { F } ^ { v , w } _ t ] = U ( x ) . \\end{align*}"}
-{"id": "1168.png", "formula": "\\begin{align*} \\dot A _ 1 & = ( 1 + K ) \\beta A _ 1 , \\\\ \\dot B _ 1 & = ( 1 - K ) \\beta B _ 1 + b ( 1 ) A _ 1 , \\end{align*}"}
-{"id": "6197.png", "formula": "\\begin{align*} g ( x ) = x f ' ( x ) = \\gamma \\alpha ( \\log | x | ) ^ { \\alpha - 1 } \\exp \\{ \\gamma ( \\log | x | ) ^ \\alpha \\} , \\end{align*}"}
-{"id": "5439.png", "formula": "\\begin{align*} \\begin{bmatrix} c ( j , r ) & c ( j + 2 r , 3 r ) \\\\ c ( j , 3 r ) & c ( j + 2 r , r ) \\end{bmatrix} \\end{align*}"}
-{"id": "6790.png", "formula": "\\begin{align*} \\begin{cases} U _ 0 ( x , z ) : = \\varphi _ 0 ( x ) + A ( \\log ( | z | ^ 2 + 3 ) - \\log ( | z | ^ 2 + 1 ) - \\log 2 ) ; \\\\ U _ 1 ( x , z ) : = \\varphi _ 1 ( x ) + A ( \\log | z | ^ 2 - \\log ( | z | ^ 2 + 1 ) + \\log ( e ^ 2 + 1 ) - 2 ) . \\end{cases} \\end{align*}"}
-{"id": "4605.png", "formula": "\\begin{align*} \\alpha ( t ) : = & x _ { a } ( t ^ { \\theta } ) x _ { b } ( t ) x _ { a + b } ( t ^ { \\theta + 1 } ) x _ { 2 a + b } ( t ^ { 2 \\theta + 1 } ) , \\\\ \\beta ( t ) : = & x _ { a + b } ( t ^ { \\theta } ) x _ { 3 a + b } ( t ) , \\\\ \\gamma ( t ) : = & x _ { 2 a + b } ( t ^ { \\theta } ) x _ { 3 a + 2 b } ( t ) , \\\\ \\tau : = \\tau ( 1 ) = & \\pi _ { a + b } ( 1 ) \\pi _ { 3 a + b } ( 1 ) , \\end{align*}"}
-{"id": "4146.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } | A _ f ( n ) | ^ 2 & = \\sum _ { n \\leq X } \\sum _ { d ^ 2 | n } \\mu _ N ( d ) C _ f ( n / d ^ 2 ) = \\sum _ { d ^ 2 \\leq X } \\mu _ N ( d ) \\sum _ { v \\leq X / d ^ 2 } C _ f ( v ) \\\\ & = \\sum _ { d \\leq \\sqrt { X } } \\mu _ N ( d ) \\left ( R _ 1 \\frac { X } { d ^ 2 } + O \\left ( \\frac { X ^ { 3 / 5 } } { d ^ { 6 / 5 } } \\right ) \\right ) . \\end{align*}"}
-{"id": "7260.png", "formula": "\\begin{align*} F _ { j , \\beta } ( z _ j ) - \\sum _ { | \\alpha | = 2 } T _ { j k } F _ { k , \\alpha } ( z _ k ) \\cdot \\tau _ { k j , \\beta } ^ \\alpha = - f _ { k j , \\beta } . \\end{align*}"}
-{"id": "2229.png", "formula": "\\begin{align*} & \\int _ { I ' } \\mathcal { E } ( ( h _ { m } * u - u ) ( t , \\cdot ) , \\phi ( t , \\cdot ) ) d t \\\\ = & \\int _ { I ' } \\int _ { B } \\int _ { B } ( ( h _ { m } * V ) ( t , x , y ) - V ( t , x , y ) ) \\Phi ( t , x , y ) k ( x , y ) d x d y d t \\\\ & + 2 \\int _ { I ' } \\int _ { B } \\phi ( t , x ) \\int _ { B ^ { c } } ( ( h _ { m } * V ) ( t , x , y ) - V ( t , x , y ) ) k ( x , y ) d y d x d t \\\\ = : & _ { m } + _ { m } . \\end{align*}"}
-{"id": "4267.png", "formula": "\\begin{align*} \\rho : = \\liminf _ { n \\rightarrow + \\infty } \\frac { \\mathbb E [ Z ^ * _ { n + \\lfloor \\varepsilon n \\rfloor } ] } { \\mathbb E [ Z _ n ^ * ] } > 1 . \\end{align*}"}
-{"id": "7872.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } \\sup _ { x \\in \\R ^ d } \\left | \\int _ { \\R ^ d } p _ y ( t , x - y ) \\ , d y - 1 \\right | = 0 \\ , . \\end{align*}"}
-{"id": "1354.png", "formula": "\\begin{align*} \\int _ { \\omega _ { \\mathcal { L } } } ^ { \\omega _ { \\mathcal { U } } } \\psi ( \\omega ) d \\omega = \\pi ^ { - 1 } e ^ { - \\frac { u } { 2 } } \\int _ { \\omega _ { \\mathcal { L } } } ^ { \\omega _ { \\mathcal { U } } } \\lambda ^ { 1 / 2 } ( \\omega ) d \\omega . \\end{align*}"}
-{"id": "2235.png", "formula": "\\begin{align*} p ( x ) = \\frac { C ( \\varepsilon ) } { \\sqrt { 2 \\pi } \\sigma } e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } e ^ { - \\varepsilon x ^ { p } } \\end{align*}"}
-{"id": "1603.png", "formula": "\\begin{align*} F \\left ( t , x \\right ) = x ^ { \\exp \\left ( - m t \\right ) } \\exp \\left ( - \\frac { 1 } { 4 m } e ^ { - 2 m t } - \\frac { c _ { 1 } } { m } e ^ { - m t } \\right ) ~ , ~ m \\neq 0 \\end{align*}"}
-{"id": "2783.png", "formula": "\\begin{align*} \\mathcal { A } = \\lambda \\mathsf { P } _ { \\mathrm { S I R } } \\left ( \\lambda \\right ) \\mathrm { l o g } _ { 2 } \\left ( 1 + \\tau \\right ) . \\ : \\left [ \\mathrm { b i t s } / \\left ( \\mathrm { s \\cdot H z \\cdot m ^ { 2 } } \\right ) \\right ] \\end{align*}"}
-{"id": "336.png", "formula": "\\begin{align*} K ( \\infty ) = \\bigcup _ { n \\in \\N } K ( n ) . \\end{align*}"}
-{"id": "347.png", "formula": "\\begin{align*} c = 1 - \\frac { 6 } { m ( m + 1 ) } \\ ; \\ ; \\ ; \\ ; \\ ; \\textrm { a n d } \\ ; \\ ; \\ ; \\ ; \\ ; h = \\frac { ( ( m + 1 ) p - m q ) ^ 2 - 1 } { 4 m ( m + 1 ) } . \\end{align*}"}
-{"id": "6127.png", "formula": "\\begin{align*} \\gamma ( \\theta , s ) = ( w ( s , \\theta ) , \\theta ) = ( 2 + h ( s ) + s u ( \\theta , s ) , \\theta ) , \\end{align*}"}
-{"id": "4256.png", "formula": "\\begin{align*} \\sin \\varphi _ q ^ k & = \\sin \\left ( \\frac { 1 } { q } \\mu \\left ( \\frac { k } { q } + \\frac { \\alpha ( k / q ) } { q ^ 2 } \\right ) \\left ( 1 + \\frac { \\beta ( k / q ) } { q ^ 2 } \\right ) \\right ) \\left ( 1 + \\varepsilon O ( q ^ { - 4 } ) \\right ) . \\end{align*}"}
-{"id": "7626.png", "formula": "\\begin{align*} \\frac { 1 } { | N ( x ) | } \\sum _ { y \\sim x } \\left ( d _ y - \\lambda _ 1 \\mathbf { v } _ y \\right ) = O \\left ( \\sqrt { n } \\right ) , \\end{align*}"}
-{"id": "219.png", "formula": "\\begin{align*} R _ 5 = \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\{ \\log ( n - 1 ) - \\Psi ( n ) - \\log f ( x ) \\} \\ , d x = o \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\end{align*}"}
-{"id": "1624.png", "formula": "\\begin{align*} L _ { \\xi ^ { i } } A ^ { i j } = ( \\lambda - a ) A ^ { i j } , \\end{align*}"}
-{"id": "3683.png", "formula": "\\begin{align*} \\frac { d H } { d t } = \\int \\left [ H _ { \\mu } \\frac { \\partial \\mu } { \\partial t } + H _ h \\frac { \\partial h } { \\partial t } \\right ] d A = 0 ~ . \\end{align*}"}
-{"id": "3878.png", "formula": "\\begin{align*} & \\varepsilon _ 1 ( x , S ) = 2 ( S \\cdot Q ( x ) ) + \\langle S \\circ \\{ \\tau ( \\xi ( x ) ) + Q ( x ) \\} , \\nabla _ { \\partial \\Omega } ( \\tau - \\nu ) ( \\xi ( x ) ) , \\tilde { x } - a \\rangle , \\\\ & \\varepsilon _ 2 ( x , S ) = - 2 \\left \\{ Q ( x ) \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\right ) \\right \\} \\cdot \\left \\{ S \\circ i _ x \\left ( \\dfrac { \\tilde { x } - a } { \\tilde { r } } \\right ) \\right \\} \\end{align*}"}
-{"id": "6643.png", "formula": "\\begin{align*} ( V \\otimes \\omega _ s ) ^ { I _ k } = V ^ { I _ k } \\end{align*}"}
-{"id": "4467.png", "formula": "\\begin{align*} C _ n : = \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\sup _ { s \\in \\mathcal { S } _ n } \\sup _ { x \\in \\mathcal { X } _ n } \\frac { a ( f ( x ) ) ^ { d / ( 1 \\wedge \\beta ) } s } { f ( x ) } \\rightarrow 0 . \\end{align*}"}
-{"id": "5835.png", "formula": "\\begin{align*} \\Delta u - C ^ { \\alpha } u _ { , \\beta } - u _ { , t } = 0 , \\end{align*}"}
-{"id": "3950.png", "formula": "\\begin{align*} \\mathcal { V } ^ { + } ( D ) = \\bigoplus _ { \\lambda > 0 } \\mathcal { V } ^ { \\lambda } ( D ) , \\mathcal { V } ^ { - } ( D ) = \\bigoplus _ { \\lambda < 0 } \\mathcal { V } ^ { \\lambda } ( D ) , \\mathcal { V } ^ { \\pm 0 } ( D ) = \\mathcal { V } ^ { \\pm } ( D ) + \\mathcal { V } ^ 0 ( D ) . \\end{align*}"}
-{"id": "710.png", "formula": "\\begin{align*} \\overline { \\alpha } _ { f } ( P ) & = \\limsup _ { n \\to \\infty } h _ { X } ^ { + } ( f ^ { n } ( P ) ) ^ { 1 / n } \\\\ & \\leq \\limsup _ { n \\to \\infty } \\left ( C h _ { X } ^ { + } ( P ) \\right ) ^ { 1 / n } ( \\delta _ { f } + \\epsilon ) \\\\ & = \\delta _ { f } + \\epsilon . \\end{align*}"}
-{"id": "5296.png", "formula": "\\begin{align*} \\phi _ 0 ( x ) \\ , = \\ , 1 , \\qquad \\mbox { a n d } \\phi _ k ( x ) \\ , = \\ , \\sqrt { 2 } \\cos ( \\pi k x ) \\mbox { f o r } k \\in \\N . \\end{align*}"}
-{"id": "907.png", "formula": "\\begin{align*} I = \\int _ { T } ^ { 2 T } \\sum _ { n \\leq C T / \\pi } { n ^ { - \\tfrac { 1 } { 2 } - i t } } \\left ( \\tfrac { t } { 2 \\pi } \\right ) ^ { { i t } / { 2 } } e ^ { - i ( { t } / { 2 } + { \\pi } / { 8 } ) } \\left \\{ 1 + O \\left ( \\tfrac { 1 } { t } \\right ) \\right \\} d t + O ( T ^ { 1 / 2 } ) . \\end{align*}"}
-{"id": "3470.png", "formula": "\\begin{align*} G ^ { 4 } : = - \\partial _ { 1 } \\eta u _ { 1 } - \\partial _ { 2 } \\eta u _ { 2 } , \\end{align*}"}
-{"id": "4238.png", "formula": "\\begin{align*} C h _ { n , \\frac { 1 } { 2 } } + \\sum _ { m = 0 } ^ n { n \\choose m } C _ m C h _ { n - m , \\frac { 1 } { 2 } } ( m + 1 ) ! ( - 1 ) ^ { m + 1 } \\frac { 1 } { 4 ^ m ( 2 m - 1 ) } = \\begin{cases} 2 \\textnormal { i f } \\ , \\ , n = 0 \\\\ 0 \\textnormal { i f } \\ , \\ , n > 0 . \\end{cases} \\end{align*}"}
-{"id": "9827.png", "formula": "\\begin{align*} h _ { j , m } ( \\partial + \\lambda , \\mu ) h _ { i , j + m } ( \\partial , \\lambda ) = h _ { i , m } ( \\partial + \\mu , \\lambda ) h _ { j , i + m } ( \\partial , \\mu ) . \\end{align*}"}
-{"id": "1578.png", "formula": "\\begin{align*} d x ^ { 2 } = \\frac { 2 } { \\sigma ^ { 2 } \\left ( x \\right ) } d x ^ { 2 } \\end{align*}"}
-{"id": "667.png", "formula": "\\begin{align*} \\int _ G \\lambda ( ( g ^ { - 1 } \\cdot f _ 1 ) f _ 2 ) \\ , d \\eta ( g ) = \\lambda ( f _ 1 ) \\lambda ( f _ 2 ) . \\end{align*}"}
-{"id": "9581.png", "formula": "\\begin{align*} \\dot D = \\{ \\pi \\in L ^ 2 ( \\R ^ 3 ) : \\pi ( x ) = \\pi _ { r e g } ( x ) + \\sum \\limits _ { 1 \\le j \\le n } \\eta _ j g _ j ( x ) , ~ ~ \\pi _ { r e g } \\in H ^ 1 ( \\R ^ 3 ) , ~ ~ \\eta _ j \\in \\C \\} . \\end{align*}"}
-{"id": "8251.png", "formula": "\\begin{align*} \\begin{cases} \\frac { \\partial x } { \\partial s _ 1 } = \\Re ( f _ 1 ( 0 , 0 , s _ 3 ) + f _ 2 ( 0 , 0 , s _ 3 ) ) \\\\ \\frac { \\partial x } { \\partial s _ 2 } = \\Im ( f _ 1 ( 0 , 0 , s _ 3 ) + f _ 2 ( 0 , 0 , s _ 3 ) ) \\\\ \\end{cases} \\end{align*}"}
-{"id": "7995.png", "formula": "\\begin{align*} \\# S ( \\ell ) + v ( \\ell ) + 1 = \\# S ( \\ell ) + 1 7 . \\end{align*}"}
-{"id": "9420.png", "formula": "\\begin{align*} \\ell ( e ) ^ { p - 1 } = \\alpha ( e ) ^ { p - 1 } w ( e ) . \\end{align*}"}
-{"id": "286.png", "formula": "\\begin{align*} \\sup _ { C \\in \\mathcal { C } } \\biggl | \\mathbb { P } \\biggl ( \\frac { 1 } { ( n - 2 ) ^ { 1 / 2 } } \\sum _ { i = 3 } ^ n Z _ i \\in C \\biggr ) - \\mathbb { P } ( Z \\in C ) \\biggr | \\leq \\frac { C _ 2 \\mathbb { E } ( \\| Z _ 3 \\| ^ 3 ) } { ( n - 2 ) ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "5416.png", "formula": "\\begin{align*} a = 1 / 2 \\ ; \\ ; a n d \\ ; \\ ; b = 1 7 / 3 0 . \\end{align*}"}
-{"id": "1123.png", "formula": "\\begin{align*} \\mathcal { R } _ k ^ { \\mathrm { P L } } [ \\iota ] = \\tau _ \\mathrm { d } [ \\iota ] \\log _ 2 ( 1 + \\gamma _ k ^ { \\mathrm { P L } } [ \\iota ] ) \\end{align*}"}
-{"id": "8529.png", "formula": "\\begin{align*} \\rho ( X _ { a ^ { \\ast } } ( P ) a ' ) = \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } X _ { [ b r a ] ^ { \\ast } } ( \\rho ( P ) ) [ b r a ' ] \\end{align*}"}
-{"id": "6091.png", "formula": "\\begin{align*} & \\exp \\left ( - i t \\left ( ( k - \\delta ) x - \\frac { ( k - \\delta ) ^ 2 x ^ 2 } { 2 } \\right ) - t ( k - \\delta ) ^ 2 x ^ 2 \\right ) \\\\ & = - \\frac { 1 } { 2 t ( k - \\delta ) ^ 2 x + i t ( k - \\delta ) ( 1 - ( k - \\delta ) x ) } \\frac { d } { d x } \\exp \\left ( - i t \\left ( ( k - \\delta ) x - \\frac { ( k - \\delta ) ^ 2 x ^ 2 } { 2 } \\right ) - t ( k - \\delta ) ^ 2 x ^ 2 \\right ) \\end{align*}"}
-{"id": "1310.png", "formula": "\\begin{align*} \\widetilde { S } _ { \\ell } ( \\alpha ) = \\sum _ { n = 1 } ^ { \\infty } \\Lambda ( n ) e ^ { - n ^ { \\ell } / N } e ( n ^ { \\ell } \\alpha ) , \\end{align*}"}
-{"id": "443.png", "formula": "\\begin{align*} d _ k ( x , x ' ) & = \\sum _ { i = 0 } ^ r ( \\max ( s _ i , s ' _ i ) - s _ i ) \\\\ & = \\sum _ { i = 0 } ^ r ( \\max ( s _ i , s ' _ i ) - s ' _ i ) \\\\ & = \\sum _ { i = 0 } ^ { r } \\abs { s _ i - s ' _ i } = d _ { \\ell _ 1 } ( \\sigma _ k ( z ) , \\sigma _ k ( z ' ) ) . \\end{align*}"}
-{"id": "5363.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\mathcal U } \\| \\varphi \\circ \\Phi - ( \\varphi \\circ \\Phi ) ( f _ \\lambda \\ , \\cdot \\ , f _ \\lambda ) \\| & \\leq \\lim _ { \\lambda \\to \\mathcal U } 2 | ( \\varphi \\circ \\Phi ) ( ( 1 - f _ \\lambda ) ^ 2 ) | ^ { 1 / 2 } \\\\ & \\leq \\lim _ { \\lambda \\to \\mathcal U } 2 | ( \\varphi \\circ \\Phi ) ( 1 - f _ \\lambda ) | ^ { 1 / 2 } = 0 . \\end{align*}"}
-{"id": "3997.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } n _ { d _ i } ( q ^ { d _ i } - 1 ) = q ^ n - 1 , \\end{align*}"}
-{"id": "2341.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\bigl ( \\| S _ i ^ \\ast y _ i ^ \\ast + x ^ \\ast \\| + \\| S _ i ^ \\ast y _ i ^ \\ast - x ^ \\ast \\| \\bigr ) > ( \\delta - 3 \\varepsilon ) \\| x ^ \\ast \\| + 2 . \\end{align*}"}
-{"id": "1518.png", "formula": "\\begin{align*} \\phi ( f ) ( C ) = f | _ C ; \\end{align*}"}
-{"id": "7759.png", "formula": "\\begin{align*} \\mathbb P \\left [ T _ N \\le x \\right ] = \\mathbb P \\left [ T _ N \\ge - x \\right ] . \\end{align*}"}
-{"id": "1759.png", "formula": "\\begin{align*} \\underbrace { 0 = d _ { 0 , 1 } = \\cdots d _ { 0 , t _ 0 } } _ { = \\tau _ 0 } < \\underbrace { d _ { 1 , 1 } = \\cdots = d _ { 1 , t _ 1 } } _ { = \\tau _ 1 } < \\cdots < \\underbrace { d _ { s - 1 , 1 } = \\cdots = d _ { s - 1 , t _ { s - 1 } } } _ { = \\tau _ { s - 1 } } < 1 = \\tau _ { s } = \\sigma _ s . \\end{align*}"}
-{"id": "8866.png", "formula": "\\begin{align*} - \\Delta _ A u + u = \\abs { u } ^ { p - 2 } u , \\end{align*}"}
-{"id": "7814.png", "formula": "\\begin{align*} \\frac { 1 } { \\left \\vert \\nabla f \\right \\vert ^ { 2 } } u _ { i j } f _ { i } f _ { j } S ^ { - 2 } = \\frac { 5 } { \\left \\vert \\nabla f \\right \\vert ^ { 2 } } w + O \\left ( S f ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "7452.png", "formula": "\\begin{align*} M _ { i j } = D [ x ] _ - ^ { - 1 } \\end{align*}"}
-{"id": "3553.png", "formula": "\\begin{align*} \\mathcal C ^ R _ i = \\frac { 1 } { 1 6 \\pi } B ^ R _ { ( g , \\pi ) } ( x ^ i , 0 ) , \\mathcal J ^ R _ k = \\frac { 1 } { 8 \\pi } B ^ R _ { ( g , \\pi ) } ( 0 , x \\times \\frac { \\partial } { \\partial x ^ k } ) . \\end{align*}"}
-{"id": "4100.png", "formula": "\\begin{align*} C \\ : : \\ : y ^ q ( a x + b y + c z ) ^ r - x ^ { - p } z ^ { p + q + r } = 0 , \\end{align*}"}
-{"id": "6965.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\log \\binom { \\alpha n } { \\beta n } = \\alpha H _ 2 ( \\beta / \\alpha ) , \\end{align*}"}
-{"id": "5984.png", "formula": "\\begin{align*} u = x _ 0 \\xleftarrow { \\gamma _ 1 ^ \\lor } x _ 1 \\xleftarrow { \\gamma _ 2 ^ \\lor } \\cdots \\xleftarrow { \\gamma _ r ^ \\lor } x _ r = v \\end{align*}"}
-{"id": "7123.png", "formula": "\\begin{align*} [ ( x _ 2 \\otimes h ) ( \\epsilon _ { 0 , 1 } \\otimes 1 ) ^ * , & ( x _ 1 x _ 2 \\otimes g ) ( \\epsilon _ { 0 , 0 } \\otimes 1 ) ^ * ] = ( x _ 1 x _ 2 \\otimes h g ) ( \\epsilon _ { 0 , 0 } \\otimes 1 ) ^ * \\textrm { a n d } \\\\ [ ( x _ 1 \\otimes h ) ( \\epsilon _ { 1 , 0 } \\otimes 1 ) ^ * , & ( x _ 1 x _ 2 \\otimes g ) ( \\epsilon _ { 0 , 0 } \\otimes 1 ) ^ * ] = ( x _ 1 x _ 2 \\otimes h g ) ( \\epsilon _ { 0 , 0 } \\otimes 1 ) ^ * . \\end{align*}"}
-{"id": "282.png", "formula": "\\begin{align*} W _ { 3 2 } : = \\int _ { \\mathcal { X } _ n ^ c \\times \\mathcal { X } } f ( x ) f ( y ) \\int _ { l _ x } ^ { v _ x } \\int _ { l _ y } ^ { v _ y } ( h _ { u v } F ) ( u , v ) \\ , d u \\ , d v \\ , d x \\ , d y = O \\biggl ( \\frac { k ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) . \\end{align*}"}
-{"id": "3811.png", "formula": "\\begin{align*} s _ q ^ + : = \\frac { 1 } { q } \\sum _ { i = 1 } ^ n x _ i ^ q s _ q ^ - : = \\frac { 1 } { q } \\sum _ { i = 1 } ^ n x _ i ^ { - q } . \\end{align*}"}
-{"id": "37.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { y _ 0 } ^ 2 : = \\int _ { X _ { y _ 0 } } | u | ^ 2 e ^ { - \\varphi _ L } . \\end{align*}"}
-{"id": "2015.png", "formula": "\\begin{align*} g ( x ) = x f ' ( x ) = \\alpha ( \\log | x | ) ^ { \\alpha - 1 } \\end{align*}"}
-{"id": "5502.png", "formula": "\\begin{align*} \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { B } [ \\iota ] = \\lim _ { \\rho \\to 0 } \\gamma _ k ^ \\mathrm { C } [ \\iota ] = M K \\rho ^ 2 \\beta _ { k } ^ 2 \\end{align*}"}
-{"id": "9553.png", "formula": "\\begin{align*} \\mathcal { W } _ { i } = \\{ g _ F \\left ( g _ F ^ { - 1 } \\pi ^ { - 1 } ( F ) \\cap { A } \\right ) \\mid F \\in \\mathcal { F } _ k , A \\in \\mathcal { A } _ i \\} , i > k . \\end{align*}"}
-{"id": "8853.png", "formula": "\\begin{align*} \\lim _ { k \\rightarrow \\infty } \\frac { \\frac { 1 } { k } \\sum _ { i = 1 } ^ k \\upsilon _ i } { k ^ { \\frac { 2 } { n } } } = \\frac { n } { n + 2 } \\frac { 4 \\pi ^ 2 } { ( \\omega _ n \\mathrm { v o l } \\ , \\Omega ) ^ \\frac { 2 } { n } } . \\end{align*}"}
-{"id": "9441.png", "formula": "\\begin{align*} v ^ g ( t _ 0 , x ) - v ^ g ( T , x ) = \\sum _ { i = 1 } ^ N ( v ^ g ( t _ { i - 1 } , x ) - v ^ g ( t _ i , x ) ) , \\end{align*}"}
-{"id": "3823.png", "formula": "\\begin{align*} \\tilde \\alpha _ 1 & = \\frac { - \\sqrt { - t x _ i } } { ( - 1 ) ^ i x _ i ( t + 1 ) } & & \\tilde \\alpha _ 2 = \\sqrt { x _ i ^ { - 1 } } . \\end{align*}"}
-{"id": "9952.png", "formula": "\\begin{align*} 0 < \\varrho \\leq \\bar { \\varrho } _ \\mu : = \\left ( \\frac { n } { ( n - 2 s ) c _ { 2 ^ * } ^ { 2 ^ * } \\mu } \\right ) ^ { \\frac { n - 2 s } { 4 s } } , \\end{align*}"}
-{"id": "6348.png", "formula": "\\begin{align*} U _ a : = \\bigg ( \\frac { r + d } { 2 } \\bigg ) ^ s . \\end{align*}"}
-{"id": "2581.png", "formula": "\\begin{align*} v _ n ^ j ( t , x ) : = e ^ { i x \\xi _ n ^ j } e ^ { - i t | \\xi _ n ^ j | ^ 2 } \\Psi ^ j _ { [ h _ n ^ j ] } ( t , x - 2 t \\xi _ n ^ j ) , \\end{align*}"}
-{"id": "1998.png", "formula": "\\begin{align*} U _ t = & \\gamma _ U t + a _ { 1 1 } W _ t + a _ { 1 2 } \\widetilde W _ t , \\\\ L _ t = & \\gamma _ L t + a _ { 2 1 } W _ t + a _ { 2 2 } \\widetilde W _ t . \\end{align*}"}
-{"id": "8401.png", "formula": "\\begin{align*} \\phi ( m z _ g g ) = \\theta ( m ) \\phi ( z _ g g ) , \\ \\ \\ m \\in M . \\end{align*}"}
-{"id": "9435.png", "formula": "\\begin{align*} \\{ v _ i \\} = S _ { i , 0 } \\subsetneq S _ { i , 1 } \\subsetneq \\dots \\subsetneq S _ { i , n _ i } . \\end{align*}"}
-{"id": "6324.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = & \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ { \\infty } } \\lesssim \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ { \\infty } } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ { p _ 1 } } \\\\ \\lesssim & 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } \\| f \\| _ { M _ 1 } . \\end{align*}"}
-{"id": "2468.png", "formula": "\\begin{align*} \\rho ( w ) N \\rho ( w ) ^ { - 1 } = | | w | | N . \\end{align*}"}
-{"id": "5833.png", "formula": "\\begin{align*} \\mathcal { L } _ { X ^ { \\left [ 2 \\right ] } } \\Theta = \\lambda \\Theta ~ , ~ { m o d } \\Theta = 0 , \\end{align*}"}
-{"id": "3949.png", "formula": "\\begin{align*} \\left | \\int _ { \\cup _ { i = 1 } ^ l B _ i } f ( a ( t ) u ( \\varphi ( s ) ) x ) \\dd s \\right | \\leq \\epsilon | I | . \\end{align*}"}
-{"id": "5838.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma \\left ( S \\right ) ^ { 2 } S ^ { 2 } F _ { , S S } + \\kappa \\left ( S \\right ) \\left ( \\mu \\left ( S \\right ) - \\lambda \\left ( S \\right ) - \\log S \\right ) S F _ { , S } - F _ { , t } = 0 . \\end{align*}"}
-{"id": "4842.png", "formula": "\\begin{align*} \\mathcal { U } ^ { \\llcorner } _ { \\rho _ 1 , \\rho _ 2 } ( \\pi \\vert \\nu , \\mu , \\kappa ) = \\mathcal { U } ^ { \\llcorner } _ { \\rho _ 1 , \\rho _ 2 } ( \\pi \\vert \\nu , \\kappa ) = \\frac { s _ { \\pi / \\nu } ( \\rho _ 2 ) s _ { \\pi / \\kappa } ( \\rho _ 1 ) } { \\sum _ { \\lambda } s _ { \\lambda / \\nu } ( \\rho _ 2 ) s _ { \\lambda / \\kappa } ( \\rho _ 1 ) } . \\end{align*}"}
-{"id": "2058.png", "formula": "\\begin{align*} \\nabla v _ { y _ 0 } = \\nabla v _ { x _ 0 } = \\varphi ' e _ n . \\end{align*}"}
-{"id": "1637.png", "formula": "\\begin{align*} F ( x , v ) = \\varphi ( v ) G ( x ) \\ , , \\end{align*}"}
-{"id": "2916.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V _ \\pi ( z _ 1 ; X ) V _ \\pi ( z _ 2 ; X ) \\cdots V _ \\pi ( z _ m ; X ) \\cdot 1 \\ , ; \\\\ \\cr s ^ { * ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V ^ * _ \\pi ( z _ 1 ; X ) V ^ * _ \\pi ( z _ 2 ; X ) \\cdots V ^ * _ \\pi ( z _ m ; X ) \\cdot 1 \\ , . \\end{align*}"}
-{"id": "3844.png", "formula": "\\begin{align*} n ^ 2 _ k ( x ) - ( 1 - 4 x ) = 4 x ^ 3 d _ k ( x ) . \\end{align*}"}
-{"id": "6250.png", "formula": "\\begin{align*} 0 = \\frac { d } { d c } \\left ( ( \\tilde { L } _ c - \\tilde { \\lambda } _ c ) \\phi _ c \\right ) \\big | _ { c = 1 } & = \\left ( \\frac { d \\tilde { L } _ c } { d c } - \\frac { d \\tilde { \\lambda } _ c } { d c } \\right ) \\phi _ c \\big | _ { c = 1 } + \\left ( \\tilde { L } _ c - \\tilde { \\lambda } _ c \\right ) \\frac { d \\phi _ c } { d c } \\Big | _ { c = 1 } . \\end{align*}"}
-{"id": "8975.png", "formula": "\\begin{align*} x ^ q - y ^ q = q \\lambda ^ { q - 1 } ( x - y ) , \\qquad \\lambda \\in ( y , x ) \\ , . \\end{align*}"}
-{"id": "6943.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = [ Z ^ \\lambda ] \\ V ^ * _ \\pi ( z _ 1 ; X ) V ^ * _ \\pi ( z _ 2 ; X ) \\cdots V ^ * _ \\pi ( z _ m ; X ) \\cdot 1 \\cr & = [ Z ^ { \\lambda + \\delta } ] \\ \\prod _ { 1 \\le i < j \\le m } ( z _ i - z _ j ) \\ \\prod _ { \\ell = 1 } ^ m \\ , L ( z _ \\ell ; X ) \\ L _ { \\pi ' } ( Z ) \\cr & = [ s _ \\lambda ( Z ) ] \\ L ( X Z ) \\ , L _ { \\pi ' } ( Z ) \\ , . \\end{align*}"}
-{"id": "7360.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 1 ) * } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\end{array} \\right ) ^ T , \\Gamma _ { 0 } ^ { ( 1 ) * } = \\left ( \\begin{array} { c c c } 0 & 1 & 0 \\end{array} \\right ) ^ T , \\Gamma _ { - } ^ { ( 1 ) * } = \\left ( \\begin{array} { c c c } 0 & 0 & q ^ { - 2 } \\end{array} \\right ) ^ T . \\end{align*}"}
-{"id": "4048.png", "formula": "\\begin{align*} \\gamma ^ 2 \\beta = 2 + 2 \\gamma + \\beta \\iff \\beta ( \\gamma - 1 ) = 2 . \\end{align*}"}
-{"id": "2676.png", "formula": "\\begin{align*} \\frac { d } { d t } { \\rm I } ( \\varphi _ t ) = \\int _ X \\chi \\theta _ { \\varphi _ t } ^ n , \\ \\forall t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "190.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\biggl | \\mathbb { E } _ f ( \\hat { H } _ n ) - H + \\frac { \\Gamma ( k + 2 / d ) } { 2 ( d + 2 ) V _ d ^ { 2 / d } \\Gamma ( k ) n ^ { 2 / d } } \\int _ \\mathcal { X } \\frac { \\Delta f ( x ) } { f ( x ) ^ { 2 / d } } \\ , d x \\biggr | = o \\Bigl ( \\frac { k ^ { 2 / d } } { n ^ { 2 / d } } \\Bigr ) . \\end{align*}"}
-{"id": "5157.png", "formula": "\\begin{align*} \\omega \\circ \\tau _ t ^ u = \\omega \\end{align*}"}
-{"id": "1721.png", "formula": "\\begin{gather*} \\Phi _ s : = \\Phi - \\tfrac { 1 } { 2 } s ^ 2 \\Phi _ I + s \\Phi _ J \\end{gather*}"}
-{"id": "0.png", "formula": "\\begin{align*} \\nu ( x ) & = \\sum _ { k \\ge 0 } \\hat { \\nu } _ k \\cos ( 2 \\pi k x ) \\end{align*}"}
-{"id": "5673.png", "formula": "\\begin{align*} \\tilde { \\alpha } = \\frac { ( \\alpha + \\gamma + q - 1 ) \\alpha } { \\alpha + \\gamma } , \\tilde { \\gamma } = \\frac { ( \\alpha + \\gamma + q - 1 ) \\gamma } { \\alpha + \\gamma } . \\end{align*}"}
-{"id": "803.png", "formula": "\\begin{align*} \\mathcal { F } = \\bigoplus _ { m _ { 1 } , \\ldots m _ { r } \\in \\mathbb { Z } _ { \\ge 0 } } \\mathbb { C } | m _ { 1 } , \\ldots , m _ { r } \\rangle \\end{align*}"}
-{"id": "3168.png", "formula": "\\begin{align*} & \\chi \\left ( p ' , \\{ W _ Q ^ { 0 } ( a ) : a \\in { \\mathsf { A } } \\} \\right ) \\\\ & = - \\frac { 1 } { 2 } q \\log \\frac { 1 } { 2 } q + \\frac { 1 } { 2 } ( 1 - q ) \\log \\frac { 1 } { 2 } ( 1 - q ) - \\frac { 1 } { 2 } \\log \\frac { 1 } { 2 } \\\\ & + q \\log q + ( 1 - q ) \\log ( 1 - q ) \\end{align*}"}
-{"id": "3778.png", "formula": "\\begin{align*} D \\mathbf { A } = \\prod _ { i = 1 } ^ n \\prod _ { j = 1 } ^ m \\frac { d { \\rm { R e } } { [ \\mathbf { A } ] _ { i j } } \\ , d { \\rm { I m } } { [ \\mathbf { A } ] _ { i j } } } { \\pi } \\end{align*}"}
-{"id": "6491.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } ( v ( x , t ) - g ) + L v ( x , t ) & = 0 & \\Omega \\times [ 0 , T ] , \\quad \\ , \\\\ u ( x , t ) & = 0 & \\mathbb { R } ^ { n } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = g ( x ) & \\Omega , \\ , t = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "1596.png", "formula": "\\begin{align*} L _ { K _ { 1 } } C _ { x } = 0 ~ , ~ C ^ { x } Y _ { 1 } = 0 . \\end{align*}"}
-{"id": "8745.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { T } e ^ { \\left ( T - s \\right ) A } G \\psi _ 1 ' ( s ) = \\int _ { 0 } ^ { T } e ^ { \\left ( T - s \\right ) A } A G \\psi _ 1 ( s ) d s = \\int _ { 0 } ^ { T } e ^ { \\left ( T - s \\right ) A } \\left ( \\begin{array} [ c ] { c c } 0 & I \\\\ - { { \\Lambda } \\ , } & 0 \\end{array} \\right ) \\left ( \\begin{array} [ c ] { c } 0 \\\\ \\psi _ 1 ( s ) \\end{array} \\right ) d s \\end{align*}"}
-{"id": "7130.png", "formula": "\\begin{align*} d ( z , \\xi ) = \\frac { 2 R ^ 2 \\vert z - \\xi \\vert } { \\sqrt { ( R ^ 2 - \\vert z \\vert ^ 2 ) ( R ^ 2 - \\vert \\xi \\vert ^ 2 ) } } , \\end{align*}"}
-{"id": "5540.png", "formula": "\\begin{align*} a = - \\frac { 1 } { 2 } b _ { x } + \\beta , c = - \\frac { 1 } { 2 } b _ { x x } - z \\rho b , d = \\frac { 1 } { 2 } b _ { x } + \\beta , \\end{align*}"}
-{"id": "1249.png", "formula": "\\begin{align*} G _ 0 f ( x ) & = - \\frac { 1 } { 2 \\pi } \\int _ { \\R ^ 2 } \\log | x - y | f ( y ) \\ , d y , \\\\ G _ 1 f ( x ) & = \\int _ { \\R ^ 2 } | x - y | ^ 2 f ( y ) \\ , d y , \\\\ G _ 2 f ( x ) & = \\frac 1 { 8 \\pi } \\int _ { \\R ^ 2 } | x - y | ^ 2 \\log | x - y | f ( x ) \\ , d y . \\end{align*}"}
-{"id": "1644.png", "formula": "\\begin{align*} \\Lambda _ { t } = \\left \\{ \\left ( A \\left ( t - t _ { 0 } \\right ) ^ { \\beta } , A \\beta \\left ( t - t _ { 0 } \\right ) ^ { \\beta - 1 } \\right ) \\in \\mathbb { R } ^ { 2 } : t _ { 0 } \\in \\left [ 0 , t \\right ] \\right \\} . \\end{align*}"}
-{"id": "1108.png", "formula": "\\begin{align*} h ^ { ( 1 ) } : = D ^ * _ { S _ 0 ^ c } h \\cdot I _ { \\{ i : | D ^ * _ { S _ 0 ^ c } h ( i ) | > \\alpha / ( t - 1 ) \\} } , h ^ { ( 2 ) } : = D ^ * _ { S _ 0 ^ c } h \\cdot I _ { \\{ i : | D ^ * _ { S _ 0 ^ c } h ( i ) | \\leq \\alpha / ( t - 1 ) \\} } . \\end{align*}"}
-{"id": "3631.png", "formula": "\\begin{align*} u _ { n + 1 } = ( 1 - c ) h ( u _ n ) , c \\in [ 0 , 1 ) . \\end{align*}"}
-{"id": "1308.png", "formula": "\\begin{align*} S & \\ll H ( \\log N ) ^ 2 \\sum _ { n = N - H } ^ { N + H } n ^ { 1 / 2 } + H N \\ll H ( \\log N ) ^ 2 \\Bigl ( ( N + H ) ^ { 3 / 2 } - ( N - H ) ^ { 3 / 2 } \\Bigr ) + H N \\\\ & \\ll H ^ 2 N ^ { 1 / 2 } ( \\log N ) ^ 2 + H N . \\end{align*}"}
-{"id": "9047.png", "formula": "\\begin{align*} \\tau ^ { ( r ) } _ j & : = \\sum _ { l = r } ^ { j - 1 } \\sum _ { p = 0 } ^ { 2 ^ { \\Delta ^ { ( l ) } } - 1 } \\frac { 2 ^ { l - \\Delta ^ { ( l ) } } } { 2 ^ l + p 2 ^ { l - \\Delta ^ { ( l ) } } } = \\sum _ { l = r } ^ { j - 1 } \\sum _ { p = 0 } ^ { 2 ^ { \\Delta ^ { ( l ) } } - 1 } \\frac { 1 } { 2 ^ { \\Delta ^ { ( l ) } } + p } = : ( j - r ) \\log 2 + \\lambda _ { j } ^ { ( r ) } , \\end{align*}"}
-{"id": "8709.png", "formula": "\\begin{align*} \\nabla _ { G \\xi } Y _ { \\tau } ^ { n , t , x } & = \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } G \\nabla _ { } B ^ n ( s , \\Xi ^ { t , x } _ s ) e ^ { ( s - t ) A } G \\xi \\ , d s + \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } \\nabla _ { G \\xi } Z _ s ^ { n , t , x } \\ , B ^ n ( s , \\Xi ^ { t , x } _ s ) \\ , d s \\\\ & + \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } Z _ s ^ { n , t , x } \\ , \\nabla _ { } B ^ n ( s , \\Xi ^ { t , x } _ s ) e ^ { ( s - t ) A } G \\xi \\ , d s - \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } \\nabla _ { G \\xi } Z ^ { n , t , x } _ { s } \\ ; d W _ s . \\end{align*}"}
-{"id": "6326.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ 2 } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { 2 , 2 } ^ { 0 , \\alpha _ 1 } } . \\end{align*}"}
-{"id": "1566.png", "formula": "\\begin{align*} \\ln F \\left ( t , S \\right ) = e ^ { - \\kappa t } \\ln S + \\left ( 1 - e ^ { - \\kappa t } \\right ) a ^ { \\ast } + \\frac { \\sigma ^ { 2 } } { 4 \\kappa } \\left ( 1 - e ^ { - 2 \\kappa t } \\right ) \\end{align*}"}
-{"id": "9964.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": "3951.png", "formula": "\\begin{align*} v : = \\lim _ { i \\rightarrow \\infty } g \\gamma _ i v _ j / { \\norm { g \\gamma _ i v _ j } } , \\norm { v } = 1 \\lim _ { i \\to \\infty } \\norm { g \\gamma _ i v _ j } = \\infty . \\end{align*}"}
-{"id": "7175.png", "formula": "\\begin{align*} b _ { j k } - b ' _ { j k } & = - \\int _ { \\R ^ n } y _ k \\{ - \\Delta ( v _ j ( \\chi - \\chi ' ) + \\hat { v } _ j - \\hat { v } _ j ' ) + \\partial _ j ( p ( \\chi - \\chi ' ) ) \\} \\\\ & = \\int _ { \\R ^ n } \\partial _ j y _ k ( p ( \\chi - \\chi ' ) ) = 0 . \\end{align*}"}
-{"id": "6732.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { t } u ( \\gamma _ { t } ) + \\sigma ( t ) \\Delta _ { x x } u ( \\gamma _ { t } ) + f ( \\gamma _ { t } , u ( \\gamma _ { t } ) , - \\Delta _ { x } u ( \\gamma _ { t } ) ) = 0 , \\ \\ \\gamma _ { t } \\in \\Lambda _ { t } , \\ t \\in [ 0 , T ) , \\\\ u ( \\gamma ) = g ( \\gamma ) , \\ \\ \\gamma \\in \\Lambda _ { T } , \\end{cases} \\end{align*}"}
-{"id": "9320.png", "formula": "\\begin{align*} \\begin{cases} d \\tilde { Z } ( t , z ) = [ - \\pi ( t , z ) b _ 0 ( t , z ) + \\pi ^ 2 ( t , z ) \\sigma ^ 2 _ 0 ( t , z ) ] \\tilde { Z } ( t , x , z ) d t - \\pi ( t , z ) \\sigma _ 0 ( t , z ) \\tilde { Z } ( t , z ) d B ( t ) \\\\ \\tilde { Z } ( 0 , z ) = \\int _ D \\frac { 1 } { \\alpha ( x ) } d x . \\end{cases} \\end{align*}"}
-{"id": "2681.png", "formula": "\\begin{align*} { \\rm I } ( \\varphi _ { t } ) - { \\rm I } ( \\varphi _ { 0 } ) = \\int _ { 0 } ^ t \\int _ X \\chi \\theta _ { \\varphi _ { s } } ^ n d s . \\end{align*}"}
-{"id": "5283.png", "formula": "\\begin{align*} I = \\int _ { T } ^ { 2 T } \\chi ( \\tfrac { 1 } { 2 } + i t ) ^ { - 1 / 2 } \\zeta ( \\tfrac { 1 } { 2 } + i t ) d t \\end{align*}"}
-{"id": "268.png", "formula": "\\begin{align*} \\{ y \\in \\mathcal { X } _ n : \\| x - y \\| < r _ { n , u _ n ^ * ( x ) } + r _ { n , u _ n ^ * ( y ) } \\} = \\emptyset \\end{align*}"}
-{"id": "4409.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { X } } = \\sup _ { \\| x ^ * \\| \\leq 1 } | x ^ * ( x ) | = \\| x \\| ^ * . \\end{align*}"}
-{"id": "1433.png", "formula": "\\begin{align*} \\gamma _ { i } = \\prod _ { j = 1 } ^ { i } ( 1 - 2 j ( 2 j - 1 ) \\gamma ^ { 2 } ) . \\end{align*}"}
-{"id": "8625.png", "formula": "\\begin{align*} \\tilde { \\mu } _ { k } ( u ) & = \\underline { \\overline { u } + \\overline { \\rho ( u ) } } = u _ { 1 } : \\overline { N } \\rightarrow \\overline { N ^ { 1 } } \\\\ \\tilde { \\mu } _ { k } ^ { 2 } ( u ) = \\tilde { \\mu } _ { k } ( u _ { 1 } ) & = \\underline { \\overline { u _ { 1 } } + \\overline { \\rho ( u _ { 1 } ) } } = u _ { 2 } : \\overline { \\overline { N } } \\rightarrow \\overline { \\overline { N ^ { 1 } } } \\end{align*}"}
-{"id": "578.png", "formula": "\\begin{align*} \\delta ( p [ a ] ) = \\frac { 1 } { p } ( \\phi ( p [ a ] ) - ( p [ a ] ) ^ p ) = \\frac { 1 } { p } ( p [ a ^ p ] - p ^ p [ a ] ^ p ) = [ a ^ p ] - p ^ { p - 1 } [ a ] ^ p . \\end{align*}"}
-{"id": "3297.png", "formula": "\\begin{align*} \\psi ( F v _ 1 ) = [ 2 ] ^ { 1 / 2 } \\beta w _ 0 , \\psi ( F v _ 0 ) = [ 2 ] ^ { 1 / 2 } \\gamma w _ 1 , \\psi ( F v _ { - 1 } ) = 0 . \\end{align*}"}
-{"id": "4231.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 + t ) ^ { \\tfrac { x } { 2 } } d \\mu _ { - 1 } ( x ) = \\frac { 2 } { 1 + \\sqrt { 1 + t } } = \\sum _ { n = 0 } ^ \\infty C h _ { n , \\frac { 1 } { 2 } } \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "6271.png", "formula": "\\begin{align*} 0 & \\geq \\nabla _ k \\nabla _ k \\| \\nabla w \\| ^ 2 = 2 \\left ( \\langle \\nabla _ k \\nabla _ k \\nabla w , \\nabla w \\rangle + \\| \\nabla _ k \\nabla w \\| ^ 2 \\right ) \\\\ & \\geq 2 \\langle \\nabla _ k \\nabla _ k \\nabla w , \\nabla w \\rangle = 2 \\| \\nabla w \\| \\langle \\nabla _ k \\nabla _ k \\nabla w , e _ n \\rangle . \\end{align*}"}
-{"id": "9270.png", "formula": "\\begin{align*} \\tilde { p } ( t , z ) = \\int _ { D } Y ( t , x , z ) p ( t , x , z ) d x ; t \\in [ 0 , T ] . \\end{align*}"}
-{"id": "7177.png", "formula": "\\begin{align*} - \\Delta u + ( u \\cdot \\nabla ) u + \\nabla p = f , \\qquad { \\rm d i v } \\ , u = 0 \\mbox { i n } \\ , \\ , \\R ^ n . \\end{align*}"}
-{"id": "2351.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial t } \\mu _ t + \\frac { \\partial } { \\partial x } ( v \\ , \\mu _ t ) = F _ f ( \\mu _ t ) . \\end{align*}"}
-{"id": "6718.png", "formula": "\\begin{align*} \\mathbb { E } \\big [ B ^ { H } ( t ) B ^ { H } ( s ) \\big ] = \\frac { 1 } { 2 } ( t ^ { 2 H } + s ^ { 2 H } - | t - s | ^ { 2 H } ) . \\end{align*}"}
-{"id": "188.png", "formula": "\\begin{align*} \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\biggl | \\mathbb { E } _ f ( \\hat { H } _ n ) - H - \\sum _ { l = 1 } ^ { \\lceil \\beta / 2 \\rceil - 1 } \\frac { \\Gamma ( k + 2 l / d ) \\Gamma ( n ) } { \\Gamma ( k ) \\Gamma ( n + 2 l / d ) } \\lambda _ l \\biggr | = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { \\beta } { d } } } { n ^ { \\frac { \\beta } { d } } } \\biggr \\} \\biggr ) \\end{align*}"}
-{"id": "1631.png", "formula": "\\begin{align*} g ^ { \\beta \\gamma } \\xi _ { , \\beta \\gamma } ^ { \\alpha } + \\xi ^ { \\gamma } \\Gamma _ { , \\gamma } ^ { \\alpha } - \\xi _ { , \\gamma } ^ { \\alpha } \\Gamma ^ { \\gamma } + \\left ( a - \\lambda \\right ) \\Gamma ^ { \\alpha } = g ^ { \\beta \\gamma } \\left ( L _ { \\xi ^ { \\alpha } } \\Gamma _ { \\beta \\gamma } ^ { \\alpha } \\right ) \\end{align*}"}
-{"id": "3372.png", "formula": "\\begin{align*} V _ { \\max } = \\frac { E _ { \\max } - E _ { \\min } - E _ { c , \\max } - \\Delta t P _ { \\max } } { \\zeta _ { \\max } } , \\end{align*}"}
-{"id": "9575.png", "formula": "\\begin{align*} \\psi ( x , 0 ) = \\psi _ 0 ( x ) + \\sum \\limits _ { 1 \\le j \\le n } \\zeta _ { 0 j } g _ j ( x ) , \\dot \\psi ( x , 0 ) = \\dot \\psi _ 0 ( x ) + \\sum \\limits _ { 1 \\le j \\le n } \\dot \\zeta _ { 0 j } g _ j ( x ) , \\end{align*}"}
-{"id": "6042.png", "formula": "\\begin{align*} s _ 1 ^ 2 + s _ 2 ^ 2 d + 2 f s _ 1 s _ 2 - a ( s _ 3 ^ 2 + s _ 4 ^ 2 d ) - 2 a f s _ 3 s _ 4 = 0 \\end{align*}"}
-{"id": "9249.png", "formula": "\\begin{align*} d \\tilde { P } ( \\omega ) = M _ t ( \\omega ) d P ( \\omega ) \\mathcal { F } ^ { v } _ t \\vee \\mathcal { F } _ t ^ { w } \\vee \\sigma ( Z ) , \\end{align*}"}
-{"id": "1664.png", "formula": "\\begin{align*} { \\psi } = G _ { \\lambda } g \\in L ^ p ( \\R ^ d _ v ; H ^ { 2 / 3 } _ p ( \\R ^ d _ x ) ) \\ ; \\ ; \\ ; \\ ; \\| G _ { \\lambda } g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { 2 / 3 } _ p ( \\R ^ d _ x ) ) } \\ ; \\le \\ ; \\Big ( \\frac { \\lambda + 1 } { \\lambda } \\Big ) ^ { 2 / 5 } \\ , c _ 1 \\ , \\| g \\| _ { L ^ p ( \\R ^ { 2 d } ) } . \\end{align*}"}
-{"id": "2462.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 2 / 2 } 2 ^ { j n ( \\alpha _ 1 - \\alpha _ 2 ) ( 1 / q - 1 / 2 ) } = 2 ^ { j \\widetilde { A _ 2 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j \\widetilde { A _ 3 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\end{align*}"}
-{"id": "6211.png", "formula": "\\begin{align*} \\hat { M } \\ddot { \\hat { x } } ( t ) = & - \\hat { D } \\dot { \\hat { x } } ( t ) - \\hat { K } \\hat { x } ( t ) + \\hat { F } u ( t ) , \\\\ \\hat { y } ( t ) = & \\ \\hat { C } _ { p } \\hat { x } ( t ) + \\hat { C } _ { v } \\dot { \\hat { x } } ( t ) , \\end{align*}"}
-{"id": "1553.png", "formula": "\\begin{align*} E ( \\mu ) = \\left \\{ \\begin{array} { l l } \\frac { 1 } { 2 } \\int _ { \\Omega } | f ( x ) | ^ 2 d x + \\Vert f \\Vert _ { H ^ { k } ( \\Omega ) } ^ { 2 } , & \\mbox { i f } \\ f = \\frac { d \\mu } { d x } \\ \\mbox { a n d } \\ f \\geq \\alpha , \\\\ + \\infty & \\mbox { o t h e r w i s e . } \\end{array} \\right . \\end{align*}"}
-{"id": "4436.png", "formula": "\\begin{align*} \\mu _ \\alpha ( f ) : = \\int _ { \\mathbb { R } ^ d } \\| x \\| ^ \\alpha f ( x ) \\ , d x . \\end{align*}"}
-{"id": "133.png", "formula": "\\begin{align*} C _ E = \\{ x \\in K ( H ) : \\ , S ( x ) \\in E \\} , \\end{align*}"}
-{"id": "7092.png", "formula": "\\begin{align*} \\nu = \\lim _ { n \\to \\infty } \\sum _ { | u | = n } V ( u ) e ^ { - V ( u ) } \\delta _ { u } \\end{align*}"}
-{"id": "4323.png", "formula": "\\begin{align*} ( z _ 1 , \\dots , z _ { n + 1 } ) \\to t = z _ { k + 1 } ^ { b _ { k + 1 } } \\cdots z _ { n + 1 } ^ { b _ { n + 1 } } \\end{align*}"}
-{"id": "1848.png", "formula": "\\begin{align*} & t ( t + 1 ) \\iint _ R \\{ v \\} v ^ { - 3 / 2 } \\{ u \\} u ^ { - 3 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 2 } d u d v \\\\ & = i t \\sum _ { ( m , n ) \\in R ( t ) } \\left ( D _ { m , n } ( t ) - E _ { m , n } ( t ) \\right ) + O ( 1 ) . \\end{align*}"}
-{"id": "7365.png", "formula": "\\begin{align*} \\Gamma _ i \\Gamma _ j ^ * = \\sum _ { k , l } c _ { i j } ^ { k l } \\Gamma _ k ^ * \\Gamma _ l , i \\neq j . \\end{align*}"}
-{"id": "396.png", "formula": "\\begin{align*} & Z ( f , L ; s , \\phi ) \\\\ & = \\int _ { G _ + / \\Gamma } \\chi ( g ) ^ s \\phi ( g ) \\sum _ { x \\in L } f ( g \\cdot x ) d g - \\int _ { G _ + / \\Gamma } \\chi ( g ) ^ s \\phi ( g ) \\sum _ { x \\in L _ 0 } f ( g \\cdot x ) d g . \\end{align*}"}
-{"id": "5315.png", "formula": "\\begin{align*} \\tau ( u ) = { \\rm t r a c e } ( T ( u ) ) = A ( u ) + D ( u ) = c _ 0 + c _ 1 u + c _ 2 u ^ 2 . \\end{align*}"}
-{"id": "9864.png", "formula": "\\begin{gather*} M _ { r , s } ( \\Omega ) = \\biggl ( \\int \\limits _ { \\Omega } \\left | J ( x , \\varphi ) \\right | ^ { \\frac { r } { r - s } } ~ d x \\biggl ) ^ { \\frac { r - s } { r s } } < \\infty , \\ , \\ , \\ , 1 \\leq s < r < \\infty , \\\\ M _ { s , s } ( \\Omega ) : = M _ s ( \\Omega ) = \\operatorname { e s s } \\sup \\limits _ { x \\in \\Omega } \\left | J ( x , \\varphi ) \\right | ^ { \\frac { 1 } { s } } < \\infty , \\ , \\ , \\ , 1 \\leq s = r < \\infty . \\end{gather*}"}
-{"id": "7258.png", "formula": "\\begin{align*} \\delta \\left \\{ \\left ( U _ j , \\sum _ { \\lambda , | \\beta | = n } F _ { j , \\beta } ^ \\lambda \\cdot e _ { j , \\lambda } \\ * \\otimes e _ j ^ \\beta \\right ) \\right \\} = \\left \\{ \\left ( U _ { j k } , \\sum _ { \\lambda , | \\alpha | = n } \\left ( h ^ \\lambda _ { 1 , j k , \\alpha } - h ^ \\lambda _ { 2 , j k , \\alpha } \\right ) \\cdot e _ { j , \\lambda } ^ * \\otimes e _ j ^ \\alpha \\right ) \\right \\} \\end{align*}"}
-{"id": "2560.png", "formula": "\\begin{align*} J ( t ) = e ^ { i t \\Delta } x e ^ { - i t \\Delta } = M ( t ) 2 i t \\nabla M ( - t ) . \\end{align*}"}
-{"id": "8370.png", "formula": "\\begin{align*} \\phi ( x ) = w x w ^ * , \\ \\ \\ x \\in M . \\end{align*}"}
-{"id": "9092.png", "formula": "\\begin{align*} ( s - 1 / 2 ) Y '' ( s ) + Y ' ( s ) = - \\frac { \\alpha + 3 } { 2 } Y ' ( s ) + 2 u _ { N - 1 } '' ( s ) + \\chi _ { N } ' ( s ) \\ . \\end{align*}"}
-{"id": "4283.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ m ( 1 - \\delta _ n ) | a _ n | + \\eta _ m \\sum _ { n = 1 } ^ m | a _ n | \\le \\| \\sum _ { n = 1 } ^ m a _ n x _ n \\| \\end{align*}"}
-{"id": "9110.png", "formula": "\\begin{align*} b '' c '' = a '' c '' \\leq a ' c ' + a '' c '' , \\end{align*}"}
-{"id": "2480.png", "formula": "\\begin{align*} \\omega _ s ( \\Phi ) = | | \\varpi _ K | | ^ s = q _ K ^ { - s } \\end{align*}"}
-{"id": "7815.png", "formula": "\\begin{align*} \\frac { 2 } { \\left \\vert \\nabla f \\right \\vert ^ { 2 } } u \\left ( S _ { i j } f _ { i } f _ { j } \\right ) S ^ { - 3 } = \\frac { 3 } { \\left \\vert \\nabla f \\right \\vert ^ { 2 } } w + O \\left ( S f ^ { - 1 } \\right ) . \\end{align*}"}
-{"id": "4585.png", "formula": "\\begin{align*} \\left | \\begin{matrix} x _ 1 & y _ 1 & z _ 1 \\\\ x _ 2 & y _ 2 & z _ 2 \\\\ x _ 3 & y _ 3 & z _ 3 \\end{matrix} \\right | = 0 . \\end{align*}"}
-{"id": "5282.png", "formula": "\\begin{align*} Z ( t ) = 2 \\sum _ { n \\leq \\sqrt { t / ( 2 \\pi ) } } n ^ { - 1 / 2 } \\cos \\left ( t \\log \\frac { \\sqrt { t / ( 2 \\pi ) } } { n } - \\frac { t } { 2 } - \\frac { \\pi } { 8 } \\right ) + O \\left ( t ^ { - 1 / 4 } \\right ) . \\end{align*}"}
-{"id": "5240.png", "formula": "\\begin{align*} \\begin{cases} - \\nabla \\cdot ( \\sigma \\nabla u ) = f = f ^ + - f ^ - & \\mbox { i n } \\Omega \\\\ \\sigma \\nabla u \\cdot n = 0 & \\mbox { o n } \\partial \\Omega \\\\ | \\nabla u | \\leq 1 & \\mbox { i n } \\Omega , \\\\ | \\nabla u | = 1 & \\sigma - \\mbox { a . e . } \\end{cases} \\end{align*}"}
-{"id": "4189.png", "formula": "\\begin{align*} \\norm { f } _ { \\mathcal { H } } ^ 2 = \\int _ { \\R ^ n } \\frac { 1 } { \\widehat { F _ p } ( x ) } \\abs { \\widehat { f } ( x ) } ^ 2 \\ d x . \\end{align*}"}
-{"id": "8554.png", "formula": "\\begin{align*} \\Pi _ { 1 } \\overline { N } ( ^ { \\ast } b ) & = - \\pi _ { 1 } p \\xi _ { b e _ { k } } \\\\ \\Pi _ { 2 } \\overline { N } ( ^ { \\ast } b ) & = - \\gamma ' \\xi _ { b e _ { k } } \\\\ \\Pi _ { 3 } \\overline { N } ( ^ { \\ast } b ) & = 0 \\\\ \\Pi _ { 4 } \\overline { N } ( ^ { \\ast } b ) & = 0 \\end{align*}"}
-{"id": "466.png", "formula": "\\begin{align*} \\beta : = & \\inf \\{ \\pi ^ { \\top } x \\ , | \\ \\ A ^ { i _ 0 } x \\succeq _ { \\mathcal { L } _ { m _ { i _ 0 } } } b ^ { i _ 0 } \\} , \\\\ & \\sup \\{ ( b ^ { i _ 0 } ) ^ { \\top } y ^ { i _ 0 } \\ , | \\ ( y ^ { i _ 0 } ) ^ { \\top } A ^ { i _ 0 } = \\pi ^ { \\top } , \\ y ^ { i _ 0 } \\in \\mathcal { L } ^ * _ { m _ { i _ 0 } } \\} . \\end{align*}"}
-{"id": "5435.png", "formula": "\\begin{align*} \\begin{bmatrix} - 1 & \\alpha & 0 & \\overline \\alpha \\\\ - \\overline \\alpha & 0 & - \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & 0 \\\\ 0 & - \\overline \\alpha J ( T ^ { - r } , T ^ { - r } ) & 0 & - \\alpha J ( T ^ { - 3 r } , T ^ { - 3 r } ) \\\\ - \\alpha & 0 & - \\overline \\alpha J ( T ^ { - 2 r } , T ^ { - r } ) & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "7315.png", "formula": "\\begin{align*} \\hat { R } _ { V , W } ( v _ { \\mathrm { l w } } \\otimes w ) = q ^ { ( \\mathrm { w t } ( v _ { \\mathrm { l w } } ) , \\mathrm { w t } ( w ) ) } w \\otimes v _ { \\mathrm { l w } } . \\end{align*}"}
-{"id": "9090.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { \\infty } f ( - T g ) \\ , x ^ { \\alpha } e ^ { - x } \\ , \\mathrm { d } x = \\int _ { 0 } ^ { \\infty } x f ' g ' \\ , x ^ { \\alpha } e ^ { - x } \\ , \\mathrm { d } x \\ . \\end{align*}"}
-{"id": "8383.png", "formula": "\\begin{align*} \\lim _ \\lambda \\omega \\circ E ( \\theta ( x _ \\lambda ) ^ * \\theta ( x _ \\lambda ) ) = \\lim _ \\lambda ( \\omega \\circ \\theta ) ( E ( x _ \\lambda ^ * x _ \\lambda ) ) = 0 . \\end{align*}"}
-{"id": "570.png", "formula": "\\begin{align*} { \\mathcal N } = \\left \\{ \\left ( \\sum _ { 1 \\le j \\le n } [ T _ j , y _ j ] , ( y _ j ) _ { 1 \\le j \\le n } \\right ) \\in { \\mathcal C } _ 1 \\times ( { \\mathcal I } ^ * ) ^ n \\mid \\sum _ { 1 \\le j \\le n } [ T _ j , y _ j ] \\in { \\mathcal C } _ 1 \\right \\} , \\end{align*}"}
-{"id": "2950.png", "formula": "\\begin{align*} a ( t , u ) = t \\{ ( 1 + u ) ^ { \\gamma } - 1 - u ^ { \\gamma } \\} + ( 1 + u ^ { \\gamma } ) = : t p ( u ) + q ( u ) . \\end{align*}"}
-{"id": "6493.png", "formula": "\\begin{align*} \\tilde { u } ( x , t ) : = \\int _ { 0 } ^ { t } \\mu ( t - s ) v ( x , s ) d s , \\end{align*}"}
-{"id": "6742.png", "formula": "\\begin{align*} { \\rm V o l } ( \\{ \\theta \\} ) : = \\int _ { { \\rm A m p } ( \\{ \\theta \\} ) } \\theta _ { V _ \\theta } ^ n . \\end{align*}"}
-{"id": "4229.png", "formula": "\\begin{align*} I _ { - 1 } ( f _ n ) + ( - 1 ) ^ { n - 1 } I _ { - 1 } ( f ) = 2 \\sum _ { l = 0 } ^ { n - 1 } ( - 1 ) ^ { n - 1 - l } f ( l ) , ( \\textnormal { s e e } \\ , \\ , [ 8 ] ) . \\end{align*}"}
-{"id": "3012.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ 1 ] } } ( 6 ) = { { \\bf { v } } ^ { [ 1 ] } } , { { \\bf { X } } ^ { [ 2 ] } } ( 6 ) = { { \\bf { v } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "9670.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { x ( t ) } { G ^ { - 1 } ( t ) } = ( a - b ) ^ { - 1 / ( \\beta - 1 ) } . \\end{align*}"}
-{"id": "6759.png", "formula": "\\begin{align*} \\theta _ { V _ { \\theta } } ^ n = { \\bf 1 } _ { \\{ V _ { \\theta } = 0 \\} } \\theta ^ n . \\end{align*}"}
-{"id": "3313.png", "formula": "\\begin{align*} ( F ^ * v _ { - 1 } , v _ 0 ) = ( E K ^ { - 1 } v _ { - 1 } , v _ 0 ) = [ 2 ] ^ { 1 / 2 } q ^ 2 ( v _ 0 , v _ 0 ) . \\end{align*}"}
-{"id": "4222.png", "formula": "\\begin{align*} \\frac { 2 } { ( 1 + t ) ^ { \\lambda } + 1 } ( 1 + t ) ^ { \\lambda x } = \\sum _ { n = 0 } ^ \\infty C h _ { n , \\lambda } ( x ) \\frac { t ^ n } { n ! } , \\quad \\textnormal { w h e r e } \\ , \\ , \\lambda \\in \\mathbb { Z } _ p ( \\textnormal { s e e } \\ , \\ , [ 5 ] ) . \\end{align*}"}
-{"id": "1105.png", "formula": "\\begin{align*} v = & \\sum _ { i = 1 } ^ { M } \\lambda _ i u _ i 0 \\leq \\lambda _ i \\leq 1 , \\ , \\sum _ { i = 1 } ^ { M } \\lambda _ i = 1 , \\\\ & u _ i \\in \\mathit { U } ( \\alpha , s , v ) . \\end{align*}"}
-{"id": "3038.png", "formula": "\\begin{align*} \\rho = \\sum _ { r \\in \\mathcal { P } ( \\rho ) } \\delta _ r . \\end{align*}"}
-{"id": "4187.png", "formula": "\\begin{align*} A = [ a , b ] \\setminus \\bigcup _ i ( a _ i , b _ i ) , \\end{align*}"}
-{"id": "6457.png", "formula": "\\begin{align*} & - \\int _ { B _ { 1 } } \\partial _ { s } ( g _ { 1 - \\alpha , m } * [ \\phi \\psi ^ { 2 } w ^ { 2 } ] ) d x + ( 1 - q ) \\frac { 1 } { 2 } \\phi \\cdot \\\\ \\leq & \\int _ { 0 } ^ { s } \\dot { g } _ { 1 - \\alpha , m } ( s - \\tau ) ( \\phi ( s ) - \\phi ( \\tau ) ) \\left ( \\int _ { B _ { 1 } } \\psi ^ { 2 } \\tilde { u } ^ { 1 - q } d x \\right ) ( \\tau ) d \\tau \\\\ & + C ( \\delta , \\Lambda ) ( 1 - q ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\phi ( s ) \\int _ { \\rho B _ { 1 } } w ^ { 2 } d x + R _ { m } ( s ) . \\end{align*}"}
-{"id": "975.png", "formula": "\\begin{align*} \\left ( \\partial \\rho - \\overline { \\partial } \\rho \\right ) = e ^ { - g } \\eta \\ ; , \\ ; g \\in C ^ { \\infty } ( M ) \\ ; . \\end{align*}"}
-{"id": "5211.png", "formula": "\\begin{align*} L ^ { 2 } _ { ( 0 , q ) } ( M ) = \\ker ( \\overline { \\partial } _ { M } ) \\oplus \\ker ( \\overline { \\partial } _ { M } ^ { * } ) \\oplus \\mathcal { H } _ { q } ( M ) \\ ; , \\end{align*}"}
-{"id": "2233.png", "formula": "\\begin{align*} N ( r , s ) : = \\frac { d { d \\choose r } } { \\log { d \\choose r } } . \\end{align*}"}
-{"id": "6576.png", "formula": "\\begin{align*} x ^ { k + 1 } & = x ^ k - \\alpha P ^ { - 1 } \\nabla F ( x ^ k ) . \\end{align*}"}
-{"id": "452.png", "formula": "\\begin{align*} f ( v ) = g ( ( w ^ 1 ) ^ { \\top } v , ( w ^ 2 ) ^ { \\top } v , \\cdots , ( w ^ p ) ^ { \\top } v ) , \\end{align*}"}
-{"id": "9939.png", "formula": "\\begin{align*} X _ { 0 , s } : = \\{ u \\in H ^ s ( \\R ^ n ) : u = 0 \\mbox { a . e . i n } \\R ^ n \\setminus \\Omega \\} , \\end{align*}"}
-{"id": "6768.png", "formula": "\\begin{align*} \\theta _ { u _ { \\beta } } ^ n \\geq \\frac { 1 } { \\beta ^ n } \\theta _ { \\phi } ^ n = \\frac { 1 } { \\beta ^ n } e ^ { \\phi } \\theta _ { + } ^ n = e ^ { \\phi - n \\log \\beta } \\theta _ { + } ^ n \\geq e ^ { \\beta u _ { \\beta } } \\theta _ + ^ n . \\end{align*}"}
-{"id": "6714.png", "formula": "\\begin{align*} \\vec { c } \\otimes \\pi \\longmapsto \\sigma ( \\pi , \\vec { c } \\ , ) \\oplus 0 \\longmapsto \\sigma ( \\pi , \\vec { c } \\ , ) + 0 = \\sigma ( \\pi , \\vec { c } \\ , ) \\end{align*}"}
-{"id": "7261.png", "formula": "\\begin{align*} \\delta \\left ( \\left \\{ \\left ( U _ j , \\sum _ { | \\alpha | = 2 , \\alpha _ 1 = 0 } F _ { j , \\alpha } ^ 1 \\cdot e _ { j , 1 } ^ * \\otimes e _ j ^ \\alpha \\right ) \\right \\} \\right ) = 0 \\end{align*}"}
-{"id": "1394.png", "formula": "\\begin{align*} d _ j ( a , \\sigma ) = E \\left [ \\dot { b } ( a + \\sigma z ) ^ j \\right ] - E \\left [ \\dot { b } ( a _ 0 + \\sigma _ 0 z ) ^ j \\right ] , j = 1 , \\ldots , M . \\end{align*}"}
-{"id": "2738.png", "formula": "\\begin{align*} u = u _ { P Q } \\theta ^ P \\bar \\theta ^ Q \\in C ^ \\infty ( U \\times \\C ^ { 0 | d } ) [ \\nu ^ { - 1 } \\nu ] ] \\end{align*}"}
-{"id": "7861.png", "formula": "\\begin{align*} P _ t ^ { \\kappa } f ( x ) = \\int _ { \\R ^ d } p ^ { \\kappa } ( t , x , y ) f ( y ) \\ , d y \\ , , x \\in \\R ^ d \\ , , \\end{align*}"}
-{"id": "5812.png", "formula": "\\begin{align*} \\mu _ k ^ * ( X ' _ i ) = \\begin{cases} X _ i X _ k ^ { [ b _ { i k } ] _ + } ( 1 + X _ k ) ^ { - b _ { i k } } & i \\neq k \\\\ X _ k ^ { - 1 } & i = k , \\end{cases} \\end{align*}"}
-{"id": "1233.png", "formula": "\\begin{align*} p _ { \\varphi , w } ( x ) = \\inf \\left \\{ \\sum _ { k = 1 } ^ { \\infty } \\varphi \\left ( \\frac { | x ( k ) | } { v ( k ) } \\right ) v ( k ) : v \\prec w \\right \\} , \\end{align*}"}
-{"id": "8240.png", "formula": "\\begin{align*} \\alpha = \\epsilon ^ { 1 / 2 } , t _ 0 \\lesssim \\epsilon ^ { - 1 / 2 } : C \\left ( \\epsilon ^ { 1 / 2 } + \\delta \\right ) \\leq \\nu , \\end{align*}"}
-{"id": "8484.png", "formula": "\\begin{align*} - \\frac { 1 } { \\varepsilon } ( \\mathbb { I } - \\mathbb { P } ) v ^ { \\varepsilon } = - { \\nabla P ^ { \\varepsilon } } , \\end{align*}"}
-{"id": "5689.png", "formula": "\\begin{align*} \\delta = \\epsilon \\ , \\min \\left \\{ \\alpha , \\frac { \\alpha ^ \\ast - \\alpha } { 1 + \\epsilon } \\right \\} \\end{align*}"}
-{"id": "6788.png", "formula": "\\begin{align*} \\omega _ { F S } + d d _ z ^ c G = ( \\pi _ 2 ) _ \\star ( \\Theta + d d _ { x , z } ^ c \\Phi ) ^ { n + 1 } , \\end{align*}"}
-{"id": "7069.png", "formula": "\\begin{align*} \\det ( G ) = { D _ { 1 1 } } { g _ { 1 1 } } + { D _ { 1 2 } } { g _ { 1 2 } } + . . . + { D _ { 1 B } } { g _ { 1 B } } . \\end{align*}"}
-{"id": "8392.png", "formula": "\\begin{align*} \\theta ( v ) x = \\theta ( \\beta ( \\theta ^ { - 1 } ( x ) ) ) \\theta ( v ) , \\ \\ \\ x \\in M , \\end{align*}"}
-{"id": "9393.png", "formula": "\\begin{align*} \\| u _ \\lambda \\| ^ 2 = \\frac { I m \\ \\widetilde { W } _ \\lambda } { I m \\ \\lambda } , \\lambda \\in \\mathbb { C } \\setminus \\mathbb { R } . \\end{align*}"}
-{"id": "3827.png", "formula": "\\begin{align*} & \\langle 0 \\mid e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast _ { k + 1 / 2 } \\mid - 1 \\rangle = \\langle 0 \\mid e ^ { H [ \\boldsymbol { t } ] } \\psi ^ \\ast ( z ) \\mid - 1 \\rangle \\mid _ { z ^ { k } } \\\\ & = \\left [ e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\langle 0 \\mid \\psi ^ \\ast ( z ) \\mid - 1 \\rangle \\right ] \\mid _ { z ^ { k } } = \\left [ e ^ { \\sum _ { q \\geq 1 } t _ q z ^ q } \\right ] \\mid _ { z ^ { k + 1 } } = : h _ { k + 1 } [ \\boldsymbol { t } ] . \\end{align*}"}
-{"id": "1024.png", "formula": "\\begin{align*} \\left ( ( u ' ) ^ { 3 } \\right ) ' = \\frac { e ^ { u ' } } { 2 } - 1 , u ( 0 ) = u ' ( 0 ) = u ' ( T ) , \\end{align*}"}
-{"id": "4742.png", "formula": "\\begin{align*} | u ( x ) | & = \\Big | u ( 0 ) + | x | \\int _ 0 ^ 1 u _ \\xi ( t x ) \\ , d t \\Big | \\\\ & \\le | u ( 0 ) | + | x | \\ , | D u ( 0 ) | + | x | \\int _ 0 ^ 1 \\frac { C } { \\ , 1 - \\alpha \\ , } \\ , | t x | ^ { 1 - \\alpha } \\ , d t \\\\ & \\le | u ( 0 ) | + | x | \\ , | D u ( 0 ) | + \\frac { C } { \\ , 1 - \\alpha \\ , } \\ , \\frac { \\ , | x | ^ { 2 - \\alpha } \\ , } { 2 - \\alpha } . \\end{align*}"}
-{"id": "3330.png", "formula": "\\begin{align*} \\Gamma _ - ^ { ( k + 1 ) } \\Gamma _ { 0 } ^ { ( k + 1 ) * } = t \\Gamma _ 0 ^ { ( k ) * } \\Gamma _ - ^ { ( k ) } + t ^ \\prime \\Gamma _ + ^ { ( k ) * } \\Gamma _ 0 ^ { ( k ) } . \\end{align*}"}
-{"id": "7001.png", "formula": "\\begin{align*} f _ 0 ^ { \\lambda _ i ( x , \\lambda _ j ( y , z ) ) } = & - f _ i ^ { \\lambda _ 0 ( x , \\lambda _ j ( y , z ) ) } - \\sum _ { m + n = i , ~ m , n > 0 } f _ m ^ { \\lambda _ n ( x , \\lambda _ j ( y , z ) ) } + \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ x , f _ n ^ { \\lambda _ j ( y , z ) } ] \\\\ & + [ f _ 0 ^ x , f _ i ^ { \\lambda _ j ( y , z ) } ] + [ f _ i ^ x , f _ 0 ^ { \\lambda _ j ( y , z ) } ] . \\end{align*}"}
-{"id": "6804.png", "formula": "\\begin{align*} C _ \\pm ( f ) ( u ) = \\lim _ { x \\to u _ \\pm } \\frac { 1 } { 2 i \\pi } \\oint _ \\S \\frac { f ( \\xi ) } { \\xi - x } \\dd \\xi \\end{align*}"}
-{"id": "8638.png", "formula": "\\begin{align*} \\sum \\limits _ { j = 1 } ^ 3 y _ { 1 j } f _ j ^ Z \\equiv Z _ 1 ^ 2 , \\sum \\limits _ { j = 1 } ^ 3 y _ { 2 j } f _ j ^ Z \\equiv Z _ 2 ^ 2 \\sum \\limits _ { j = 1 } ^ 3 y _ { 3 j } f _ j ^ Z \\equiv Z _ 1 Z _ 2 p ^ { 1 / 2 } A . \\end{align*}"}
-{"id": "751.png", "formula": "\\begin{align*} | U _ l | & = | U _ { l - 1 } \\cup S _ { j _ l } | \\geq | U _ { l - 1 } | + | S _ { j _ l } | - | U _ { l - 1 } \\cap S _ { j _ l } | \\\\ & \\geq ( ( l - 1 ) \\rho - \\Delta _ { l - 2 } t ( \\rho ) ) | M | + \\rho | M | - | \\bigcup _ { i = 1 } ^ { l - 1 } { S _ { j _ i } \\cap S _ { j _ l } } | \\\\ & > ( l \\rho - \\Delta _ { l - 2 } t ( \\rho ) ) | M | - ( l - 1 ) t ( \\rho ) | M | = ( l \\rho - \\Delta _ { l - 1 } t ( \\rho ) ) | M | , \\end{align*}"}
-{"id": "806.png", "formula": "\\begin{align*} \\tau ( z ) = \\mathrm { t r } _ { \\mathbb { C } ^ { r + 1 } } ( \\mathbb { T } ^ { [ 1 , M ] } ( z ) ) = \\sum _ { a = 0 } ^ { r } \\mathbb { T } ^ { [ 1 , M ] } ( z ) _ { a a } . \\end{align*}"}
-{"id": "4588.png", "formula": "\\begin{align*} n & = , n _ 1 = . \\end{align*}"}
-{"id": "6346.png", "formula": "\\begin{align*} \\mathcal { P } : = \\{ x _ n \\leq \\gamma ( x ' ) , \\ , y = 0 \\} , \\end{align*}"}
-{"id": "3186.png", "formula": "\\begin{align*} \\tau ^ \\alpha _ { k j , \\beta } = \\begin{cases} t _ { j k } ^ { - \\alpha } : = \\prod _ { \\lambda = 1 } ^ r ( t _ { j k } ^ \\lambda ) ^ { - \\alpha _ \\lambda } & ( \\beta = \\alpha ) \\\\ 0 & ( { \\rm o t h e r w i s e } ) , \\end{cases} \\end{align*}"}
-{"id": "519.png", "formula": "\\begin{align*} \\sum _ { \\mu } s _ { \\kappa / \\mu } ( \\rho _ 1 ) \\tau _ { \\mu } ( \\rho _ { \\circ } ) \\mathcal { U } ^ { \\angle } _ { \\rho _ { \\circ } , \\rho _ 1 } ( \\pi \\vert \\kappa , \\mu ) = \\frac { s _ { \\pi / \\kappa } ( \\rho _ 1 ) \\tau _ { \\pi } ( \\rho _ { \\circ } ) } { H ^ { \\circ } ( \\rho _ 1 ) H ( \\rho _ 1 ; \\rho _ { \\circ } ) } . \\end{align*}"}
-{"id": "3117.png", "formula": "\\begin{align*} P ^ T \\mathcal { L } _ 1 ( \\lambda ) P = \\begin{bmatrix} \\lambda P _ 7 + P _ 6 & - I _ n & 0 & 0 & 0 & 0 & 0 \\\\ I _ n & 0 & - \\lambda I _ n & 0 & 0 & 0 & 0 \\\\ 0 & \\lambda I _ n & \\lambda P _ 5 + P _ 4 & - I _ n & 0 & 0 & 0 \\\\ 0 & 0 & I _ n & 0 & - \\lambda I _ n & 0 & 0 \\\\ 0 & 0 & 0 & \\lambda I _ n & \\lambda P _ 3 + P _ 2 & - I _ n & 0 \\\\ 0 & 0 & 0 & 0 & I _ n & 0 & - \\lambda I _ n \\\\ 0 & 0 & 0 & 0 & 0 & \\lambda I _ n & \\lambda P _ 1 + P _ 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "3565.png", "formula": "\\begin{align*} \\mathcal { I } ^ { R } ( \\theta ) = ( E ^ \\theta - E , P ^ \\theta - P , R ^ { - 1 } ( \\mathcal C ^ \\theta - \\mathcal C ^ R ) , R ^ { - 1 } ( \\mathcal J ^ \\theta - \\mathcal J ^ R ) ) + \\mathcal { I } _ 1 ^ R ( \\theta ) - ( \\tilde { \\lambda } R ^ { - 1 } ( \\log R ) ^ \\frac { 1 } { 2 } , 0 ) \\end{align*}"}
-{"id": "1478.png", "formula": "\\begin{align*} \\lim _ { R \\to \\infty } f ^ * ( R ) = 0 , \\end{align*}"}
-{"id": "7611.png", "formula": "\\begin{align*} \\lambda _ 1 = \\max _ { \\textbf { z } \\not = \\textbf { 0 } } \\frac { \\textbf { z } ^ t A \\textbf { z } } { \\textbf { z } ^ t \\textbf { z } } . \\end{align*}"}
-{"id": "8813.png", "formula": "\\begin{align*} & \\sum _ { j \\geq 0 } \\begin{pmatrix} m \\\\ j \\end{pmatrix} ( a _ { ( n + j ) } b ) _ { ( m + k - j ) } \\\\ & = \\sum _ { j \\geq 0 } ( - 1 ) ^ j \\begin{pmatrix} n \\\\ j \\end{pmatrix} \\left ( a _ { ( m + n - j ) } b _ { ( k + j ) } - ( - 1 ) ^ n b _ { ( n + k - j ) } a _ { ( m + j ) } \\right ) . \\end{align*}"}
-{"id": "5404.png", "formula": "\\begin{align*} ( 1 - \\delta ) \\binom { N } { 2 } = ( 1 - \\delta ) ( k - \\alpha ) \\left ( k - \\alpha - \\frac { 1 } { n } \\right ) \\frac { n ^ 2 } { 2 } & \\leq k \\left ( k - \\alpha - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } \\\\ ( 1 - \\delta ) \\alpha ^ 2 - ( 1 - 2 \\delta ) k \\alpha + \\frac { k } { 4 } - ( 1 - \\delta ) \\frac { k - \\alpha } { n } - \\delta k ^ 2 & \\leq 0 \\ , . \\end{align*}"}
-{"id": "6556.png", "formula": "\\begin{align*} \\tau d _ n = \\sum \\limits ^ k _ { i = 0 } \\delta _ i u ( t _ { n - i } ) + \\tau A ( t _ n ) u ( t _ n ) - \\tau \\sum \\limits ^ { k - 1 } _ { i = 0 } \\gamma _ i B \\big ( t _ { n - i - 1 } , u ( t _ { n - i - 1 } ) \\big ) , \\end{align*}"}
-{"id": "5056.png", "formula": "\\begin{align*} p ^ { * n } ( s ) \\frac { d s \\nu } { d \\nu } ( y ) \\leq \\sum _ { g \\in G } p ^ { * n } ( g ) \\frac { d g \\nu } { d \\nu } ( y ) = 1 , \\end{align*}"}
-{"id": "4246.png", "formula": "\\begin{align*} \\tilde L _ q ( s , y ) = L ( s _ { 0 } , s _ { 1 } ) + L ( s _ 1 , s _ 2 ) + \\cdots + L ( s _ { q - 1 } , s _ 0 ) , \\end{align*}"}
-{"id": "1147.png", "formula": "\\begin{align*} V _ t = \\begin{bmatrix} a & b \\\\ c & d \\end{bmatrix} V \\end{align*}"}
-{"id": "1389.png", "formula": "\\begin{align*} V _ S : = \\underset { n \\to \\infty } \\lim \\left \\{ \\sum _ { h = 0 } ^ { ( n - 1 ) } \\left ( \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } \\textrm { C o v } ( e _ { t , \\beta ^ \\prime } ^ 2 , e _ { t + h , \\beta ^ \\prime } ^ 2 ) \\right ) + \\sum _ { h = 1 } ^ { ( n - 1 ) } \\left ( \\frac { 1 } { n } \\sum _ { t = h + 1 } ^ { n } \\textrm { C o v } ( e _ { t , \\beta ^ \\prime } ^ 2 , e _ { t - h , \\beta ^ \\prime } ^ 2 ) \\right ) \\right \\} \\end{align*}"}
-{"id": "4377.png", "formula": "\\begin{align*} e ^ { i \\alpha J ( g ) } \\tilde { T } _ { 0 , \\beta } ( f ) e ^ { - i \\alpha J ( g ) } = \\tilde { T } _ { \\alpha , \\beta } ( f ) . \\end{align*}"}
-{"id": "4992.png", "formula": "\\begin{align*} \\hat { h } _ { X , f } ( P ) = \\lim _ { n \\to \\infty } \\frac { h _ { X } ( f ^ { n } ( P ) ) } { \\delta _ { f } ^ { n } } \\end{align*}"}
-{"id": "3112.png", "formula": "\\begin{align*} L _ 3 ^ \\prime ( \\lambda ) = \\begin{bmatrix} 0 & \\lambda I _ n & - I _ n & 0 & 0 \\\\ \\lambda I _ n & - \\lambda P _ 1 + P _ 0 & P _ 1 & 0 & 0 \\\\ - I _ n & P _ 1 & \\lambda P _ 3 + P _ 2 & \\lambda P _ 4 & \\lambda I _ n \\\\ 0 & 0 & \\lambda P _ 4 & \\lambda P _ 5 - P _ 4 & - I _ n \\\\ 0 & 0 & \\lambda I _ n & - I _ n & 0 \\end{bmatrix} \\end{align*}"}
-{"id": "2695.png", "formula": "\\begin{align*} \\varphi _ { \\beta } \\geq u _ \\beta : = \\left ( 1 - \\frac { 1 } { \\beta } \\right ) V _ { \\theta } + \\frac { 1 } { \\beta } \\phi - \\frac { n \\log \\beta } { \\beta } . \\end{align*}"}
-{"id": "9295.png", "formula": "\\begin{align*} \\mathbf { H } ^ 1 _ 0 ( \\mathbb { R } ^ + ) = \\{ y \\in \\mathbf { H } ^ 1 , y = 0 \\partial \\mathbb { R } ^ + \\} . \\end{align*}"}
-{"id": "5882.png", "formula": "\\begin{align*} \\ln F \\left ( t , x \\right ) = e ^ { - m t } \\left ( x - x _ { 0 } \\right ) + \\frac { c } { m } e ^ { - m t } - \\frac { 1 } { 4 m } e ^ { - 2 m t } \\end{align*}"}
-{"id": "6688.png", "formula": "\\begin{align*} X _ { 0 } ^ { \\lambda ; p ^ { \\prime } } = \\left \\| \\bigwedge _ { i = 1 } ^ { p } \\nabla D _ i \\wedge \\bigwedge _ { j = 1 } ^ { k } \\nabla I _ j \\right \\| _ { k + p } ^ { - 2 } \\cdot \\sum _ { i = 1 } ^ { p ^ { \\prime } } ( - 1 ) ^ { n - i } ( - \\lambda ) ( D - d _ i ) \\Theta _ i , \\end{align*}"}
-{"id": "3559.png", "formula": "\\begin{align*} ( \\bar { g } , \\bar { \\pi } ) & = ( g , \\pi ) \\mbox { i n } B _ R \\\\ ( \\bar { g } , \\bar { \\pi } ) & = ( g ^ { \\theta } , \\pi ^ { \\theta } ) \\mbox { i n } M \\setminus B _ { 2 R } \\end{align*}"}
-{"id": "5651.png", "formula": "\\begin{align*} \\frac { \\partial } { \\partial \\ , t } u ( t , x ) + x \\frac { \\partial } { \\partial \\ , x } u ( t , x ) = h ( x ) \\ , u ( t , x ) , \\ ; \\ ; t \\geq 0 , \\ , 0 < x < 1 \\end{align*}"}
-{"id": "4614.png", "formula": "\\begin{align*} ( q ^ 2 - q + 1 ) \\left ( \\frac { q ^ 3 ( q ^ 2 - 1 ) r } { 8 } - \\sum _ { t = 2 } ^ { 6 } k _ t \\right ) = f _ 7 ( p ' ) + f _ 8 ( p ' ) , \\end{align*}"}
-{"id": "9868.png", "formula": "\\begin{align*} B _ { q , p } ( \\Omega ) \\leq \\frac { d i a m ( \\Omega ) ^ n } { n | \\Omega | } \\left ( \\frac { 1 - \\delta } { { 1 } / { n } - \\delta } \\right ) ^ { 1 - \\delta } \\omega _ n ^ { 1 - \\frac { 1 } { n } } | \\Omega | ^ { \\frac { 1 } { n } - \\delta } , \\ , \\ , \\delta = \\frac { 1 } { p } + \\frac { 1 } { q } \\geq 0 . \\end{align*}"}
-{"id": "1764.png", "formula": "\\begin{align*} H ( u _ { j } ^ { \\varepsilon } ) ( x ) = \\int _ { B _ { 1 } ( x ) } u _ { j } ^ { \\varepsilon } ( y ) d y , \\end{align*}"}
-{"id": "8275.png", "formula": "\\begin{align*} \\widehat { \\mathrm { r } } = \\mathrm { x } + \\mathrm { w } , \\ : \\ : \\mathrm { w } \\sim \\mathcal { N } ( 0 , v _ r ) . \\end{align*}"}
-{"id": "8113.png", "formula": "\\begin{align*} \\int _ { U ( t ) } | x | ^ l \\ , d x = \\int _ { B _ 1 } | x | ^ l \\ , d x \\mbox { f o r $ | t | < t _ 0 $ } , \\end{align*}"}
-{"id": "5896.png", "formula": "\\begin{align*} T ^ { I } L _ { Y _ { I } } C _ { a } - T _ { , t } ^ { I } Y _ { I \\alpha } - 2 a _ { , \\beta } = 0 . \\end{align*}"}
-{"id": "3060.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\left ( \\sum _ { | u | = n , u > v } \\delta _ { V ( u ) - V ( v ) - m _ n } , Z ^ v _ n \\right ) = \\left ( \\theta _ { - V ( v ) } \\mu ^ v _ \\infty , Z ^ v _ \\infty \\right ) \\end{align*}"}
-{"id": "8145.png", "formula": "\\begin{align*} u ^ * ( \\omega _ N | x | ^ N ) = u ^ { \\star } ( x ) , ( x \\in \\mathbb { R } ^ N ) . \\end{align*}"}
-{"id": "2642.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { h ( m , n ) } C _ { m } ^ { l } ( F ^ { 0 } ) ( \\lambda ) ( C _ { m } ^ { l } ( F ^ { 0 } ) ( \\lambda ) ) ^ { * } = ( \\beta _ { m } ^ { 2 } + \\gamma _ { m _ { 1 } } ( \\lambda ) + \\gamma _ { m _ { 2 } } ( \\lambda ) ) d _ { m } ^ { 0 } ( \\lambda ) ^ { \\top } \\overline { d _ { m } ^ { 0 } ( \\lambda ) } , \\end{align*}"}
-{"id": "2699.png", "formula": "\\begin{align*} \\theta _ { u _ j } ^ n = e ^ { u _ j - \\varphi _ j } \\theta _ { \\varphi _ j } ^ n + e ^ { u _ j - \\psi _ j } \\theta _ { \\psi _ j } ^ n . \\end{align*}"}
-{"id": "6020.png", "formula": "\\begin{gather*} \\mathcal { C } L _ 4 : = \\mathcal { S } / A _ 0 \\cong \\mathcal { S } \\times _ { A } ( A / A _ 0 ) , \\end{gather*}"}
-{"id": "8517.png", "formula": "\\begin{align*} w _ 0 = ( s _ 1 s _ 2 \\cdots s _ n ) ( s _ 1 s _ 2 \\cdots s _ { n - 1 } ) \\cdots ( s _ 1 s _ 2 ) s _ 1 \\end{align*}"}
-{"id": "4839.png", "formula": "\\begin{align*} \\mathcal { V } \\big ( \\bar \\lambda , \\bar \\mu \\big ) = \\tau _ { \\lambda ^ { ( 1 ) } } ( \\rho _ { 0 } ^ + ) s _ { \\lambda ^ { ( 1 ) } / \\mu ^ { ( 1 ) } } ( \\rho _ 1 ^ - ) s _ { \\lambda ^ { ( 2 ) } / \\mu ^ { ( 1 ) } } ( \\rho _ 1 ^ + ) \\dots s _ { \\lambda ^ { ( n ) } / \\mu ^ { ( n - 1 ) } } ( \\rho _ { n - 1 } ^ + ) s _ { \\lambda ^ { ( n ) } } ( \\rho _ n ^ - ) , \\end{align*}"}
-{"id": "1489.png", "formula": "\\begin{align*} c = \\prod _ { k = 1 } ^ n x _ { k , k - 1 } = x _ { n , n - 1 } \\cdots x _ { 2 1 } x _ { 1 0 } . \\end{align*}"}
-{"id": "2467.png", "formula": "\\begin{align*} \\beta _ j = ( d _ j - 1 ) \\dim V ^ I _ j + d _ j [ a ( V _ j ) + n ( \\psi ) \\dim ( V _ j ) ] \\end{align*}"}
-{"id": "6523.png", "formula": "\\begin{align*} \\begin{aligned} \\| v \\| _ { W } \\le \\varepsilon \\| v \\| _ { D } + C _ \\varepsilon \\| v \\| _ { X } \\quad \\forall \\ , v \\in D . \\end{aligned} \\end{align*}"}
-{"id": "2723.png", "formula": "\\begin{align*} ( C \\otimes C ) _ n \\cong \\begin{cases} ( k \\cong k ) \\otimes _ k k & \\mbox { i f } n = 0 \\\\ ( k \\otimes _ k ( k \\oplus k ) ) \\oplus ( ( k \\oplus k ) \\otimes k ) \\cong k ^ { \\oplus 4 } & \\mbox { i f } n = 1 \\\\ ( k \\otimes _ k ( k \\oplus k ) ) \\oplus ( ( k \\oplus k ) \\otimes _ k ( k \\oplus k ) ) \\oplus ( ( k \\oplus k ) \\otimes _ k k ) \\cong k ^ 8 & \\mbox { i f } n = 2 \\\\ \\end{cases} \\end{align*}"}
-{"id": "2372.png", "formula": "\\begin{align*} F ( x ) = \\langle \\nabla f _ 1 ( x ) , x \\rangle - f _ 1 ( x ) - f _ 2 ( \\nabla f _ 1 ( x ) ) . \\end{align*}"}
-{"id": "2303.png", "formula": "\\begin{align*} \\frac { 1 } { \\tau } \\big \\| ( e _ n - e _ { n - 1 } ) _ { n = 0 } ^ m \\big \\| _ { L ^ p ( X ) } + \\big \\| ( e _ n ) _ { n = 0 } ^ m \\big \\| _ { L ^ p ( D ) } \\le C \\| ( e _ n ) _ { n = 0 } ^ { m - 1 } \\| _ { L ^ p ( X ) } + C \\delta . \\end{align*}"}
-{"id": "4165.png", "formula": "\\begin{align*} g _ k ( z ) = \\prod _ { v = 0 } ^ { k - 1 } \\varepsilon ^ { - v } g ( \\varepsilon ^ v z ) \\end{align*}"}
-{"id": "4689.png", "formula": "\\begin{align*} \\sum _ { | \\alpha | \\leq N } \\frac { ( m + | \\alpha | - 1 ) ! } { ( m - 1 ) ! \\alpha ! } z ^ \\alpha \\bar { w } ^ \\alpha & = \\sum _ { k = 0 } ^ N \\frac { ( m + k - 1 ) ( m + k - 2 ) \\cdots ( k + 1 ) } { ( m - 1 ) ! } ( z \\bar { w } ) ^ k \\ . \\end{align*}"}
-{"id": "6926.png", "formula": "\\begin{align*} X ( z ) = \\sum _ { n \\in { \\mathbb Z } } z ^ { n + { \\alpha _ 0 } } X _ n \\quad \\mbox { a n d } X ^ * ( z ) = \\sum _ { n \\in { \\mathbb Z } } z ^ { - n - { \\alpha _ 0 } } X ^ * _ { n } \\ , , \\end{align*}"}
-{"id": "1362.png", "formula": "\\begin{align*} \\hat { G } _ n ^ \\mathrm { T } = \\sum _ { t = 1 } ^ { n } \\sigma ^ 2 _ t ( \\hat { \\delta } _ { 0 } ) \\begin{bmatrix} A _ { \\hat { \\phi } \\hat { \\phi } , t } & A _ { \\hat { \\phi } \\hat { \\theta } , t } \\\\ A _ { \\hat { \\theta } \\hat { \\phi } , t } & A _ { \\hat { \\theta } \\hat { \\theta } , t } \\end{bmatrix} . \\end{align*}"}
-{"id": "4987.png", "formula": "\\begin{align*} & { \\bf h } _ { \\vec { D } } = \\left ( \\begin{array} { c } h _ { D _ { 1 } } \\\\ h _ { D _ { 2 } } \\\\ \\vdots \\\\ h _ { D _ { r } } \\end{array} \\right ) , \\ { \\bf h } _ { \\vec { F } } = \\left ( \\begin{array} { c } h _ { F _ { 1 } } \\\\ h _ { F _ { 2 } } \\\\ \\vdots \\\\ h _ { F _ { s } } \\end{array} \\right ) , \\ { \\bf h } _ { { p } _ { * } \\vec { F } } = \\left ( \\begin{array} { c } h _ { { p } _ { * } F _ { 1 } } \\\\ h _ { { p } _ { * } F _ { 2 } } \\\\ \\vdots \\\\ h _ { { p } _ { * } F _ { s } } \\end{array} \\right ) \\ . \\end{align*}"}
-{"id": "370.png", "formula": "\\begin{align*} \\lVert f \\rVert _ { L ^ { \\infty } } = \\max _ { \\omega , \\varphi , \\vartheta } | f ( \\omega , \\varphi , \\vartheta ) | . \\end{align*}"}
-{"id": "31.png", "formula": "\\begin{align*} v _ k \\to v _ \\infty : = \\sum _ { p = 1 } ^ r \\lambda ^ p _ \\infty u _ p \\end{align*}"}
-{"id": "1320.png", "formula": "\\begin{align*} \\omega _ { \\ell } ( \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } e ^ { - m ^ { \\ell } / N } e ( m ^ { \\ell } \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } e ^ { - m ^ { \\ell } z } . \\end{align*}"}
-{"id": "7933.png", "formula": "\\begin{align*} d _ i + d _ { n + 1 - i } = n - 1 \\end{align*}"}
-{"id": "503.png", "formula": "\\begin{align*} \\int _ { \\mathcal { R } } \\rho ^ 2 = 4 l r + o ( 1 ) \\end{align*}"}
-{"id": "4764.png", "formula": "\\begin{align*} \\varphi ( t ) = \\pm \\frac { 1 } { t } \\sqrt { ( c \\pm a \\ , t ^ 2 ) ^ 2 - t ^ 2 } , a = c o n s t \\neq 0 , c = c o n s t , \\end{align*}"}
-{"id": "5670.png", "formula": "\\begin{align*} F ( v , E ) = M ( v ) + E \\cdot \\lambda ( v ) + G ( v , E ) \\end{align*}"}
-{"id": "6340.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 2 / 2 } 2 ^ { j n ( \\alpha _ 1 - \\alpha _ 2 ) ( 1 / q - 1 / 2 ) } = 2 ^ { j \\widetilde { A _ 2 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j \\widetilde { A _ 3 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } \\end{align*}"}
-{"id": "5379.png", "formula": "\\begin{align*} c _ { q , q _ 1 } ^ { d ( q _ 2 + 1 ) } ~ = ~ ( 1 + \\delta _ { q _ 1 , q _ 2 } ) { \\ , } c _ { r _ 0 , u _ 0 } ^ d c _ { d , d } ^ { d u _ 0 } . \\end{align*}"}
-{"id": "1613.png", "formula": "\\begin{align*} Z ^ { 1 } = K ^ { 1 } ~ , ~ Z ^ { 2 } = t K ^ { 1 } - \\left ( \\int \\frac { d x } { \\sigma \\left ( x \\right ) } F - t \\right ) F \\partial _ { F } \\end{align*}"}
-{"id": "6543.png", "formula": "\\begin{align*} \\| ( e _ n ) _ { n = 0 } ^ { M - 1 } \\| _ { L ^ \\infty ( W ) } \\le r . \\end{align*}"}
-{"id": "3881.png", "formula": "\\begin{align*} \\begin{cases} \\sigma _ k ( D ^ 2 u ) = F ( x , u ) , & , \\\\ u = f ( x ) , & \\end{cases} \\end{align*}"}
-{"id": "5306.png", "formula": "\\begin{align*} H _ { 2 , 2 } & = U ( N _ { a , 1 } + N _ { a , 2 } - N _ { b , 1 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } + N _ { a , 2 } - N _ { b , 1 } - N _ { b , 2 } ) \\\\ & \\quad + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) + t _ { 1 , 2 } ( a _ { 1 } b _ { 2 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 2 } ) \\\\ & \\quad + t _ { 2 , 1 } ( a _ { 2 } b _ { 1 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 1 } ) + t _ { 2 , 2 } ( a _ { 2 } b _ { 2 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 2 } ) \\end{align*}"}
-{"id": "9505.png", "formula": "\\begin{align*} \\nabla _ X \\psi & = d \\psi ( X ) + \\frac { 1 } { 4 } \\omega _ { a b } ( X ) \\gamma ^ { a b } \\psi \\\\ & = d \\psi ( X ) - \\frac { 1 } { 4 } \\omega _ { a b } ( X ) \\Gamma ^ { a b } \\psi . \\end{align*}"}
-{"id": "7753.png", "formula": "\\begin{align*} f _ i ( x ) = \\frac 1 { \\sigma _ { i - 1 } } \\phi \\left ( \\frac x { \\sigma _ { i - 1 } } \\right ) , x \\in \\mathbb R , \\end{align*}"}
-{"id": "8353.png", "formula": "\\begin{align*} \\sigma ^ { - 1 } \\circ E _ M \\circ \\sigma = E _ M \\end{align*}"}
-{"id": "605.png", "formula": "\\begin{align*} \\mathcal { P } _ { \\kappa , m , N } ( z ) : = { \\displaystyle { \\sum _ { M \\in \\Gamma _ { \\infty } \\backslash \\Gamma _ 0 ( N ) } } } e ^ { 2 \\pi i m z } \\bigg | _ { \\kappa } M \\in M _ { \\kappa } ^ ! ( N ) , \\end{align*}"}
-{"id": "1043.png", "formula": "\\begin{align*} - \\Delta u + \\lambda u = Q | u | ^ { p - 2 } u \\quad \\R ^ 2 \\end{align*}"}
-{"id": "3466.png", "formula": "\\begin{align*} u ( t , x ) : = u ( t , \\Theta ( t , x ) ) , \\ p ( t , x ) : = p ( t , \\Theta ( t , x ) ) , \\ \\tilde { c } ( t , x _ { * } ) = \\tilde { c } ( t , \\Theta _ \\ast ( t , x _ { * } , 0 ) ) . \\end{align*}"}
-{"id": "323.png", "formula": "\\begin{align*} \\tilde { a } ( \\delta ) : = d ^ { m / 2 } C _ m ' D ^ m a ( \\delta / 4 ) ^ { m ^ 2 + m + 1 } . \\end{align*}"}
-{"id": "1306.png", "formula": "\\begin{align*} \\sum _ { n = N - H } ^ { N + y } e ^ { - n / N } t _ H ( n - N ) \\Bigl ( R ( n ) - ( 2 \\psi ( n ) - n ) \\Bigr ) \\ll H N ( \\log N ) ^ 2 \\log ( 2 H ) \\end{align*}"}
-{"id": "1609.png", "formula": "\\begin{align*} Z ^ { 4 } = e ^ { - 2 m t } \\left ( \\partial _ { t } - H - m \\left ( 2 m \\left ( \\int \\frac { d x } { \\sigma \\left ( x \\right ) } \\right ) ^ { 2 } - 1 \\right ) F \\partial _ { F } \\right ) \\end{align*}"}
-{"id": "7228.png", "formula": "\\begin{align*} \\mathcal { E } _ { k } = \\left ( 1 + \\eta \\right ) ^ { 2 \\left ( k - 1 \\right ) } \\varepsilon + \\frac { 2 } { \\rho } \\xi \\left ( k - 1 \\right ) \\left ( 1 + \\eta \\right ) ^ { 2 \\left ( k - 1 \\right ) } . \\end{align*}"}
-{"id": "7557.png", "formula": "\\begin{align*} p _ 1 : = \\mathbb P \\left \\{ \\omega \\in \\Omega : \\chi ( \\omega ) \\in \\left ( \\frac { b - f ( b ) + \\delta } { l } , 1 \\right ) \\right \\} , K : = \\left [ \\frac a \\delta \\right ] + 1 . \\end{align*}"}
-{"id": "7071.png", "formula": "\\begin{align*} { \\bf { H } } _ 1 ^ { [ 1 1 ] } ( n ) { { \\bf { \\tilde v } } _ 1 } = { \\bf { h } } _ 1 ^ { [ 1 1 ] } ( { t _ 2 } ) , \\end{align*}"}
-{"id": "1896.png", "formula": "\\begin{align*} I = ( 0 , \\overline { \\rho } ] \\times ( 0 , \\overline { c } ] \\times [ \\underline { C } , \\overline { C } ] \\times [ \\underline { C ' } , \\overline { C ' } ] \\times ( 0 , \\overline { C } _ X ] \\times ( 0 , \\overline { C } _ f ] . \\end{align*}"}
-{"id": "1855.png", "formula": "\\begin{align*} \\sum _ { 1 \\le n \\le t } \\widetilde { G } _ { n , n } ( t ) & = \\sum _ { 1 \\le n \\le t } \\frac { 1 } { \\sqrt { n + 1 } } \\int _ n ^ { n + 1 } v ^ { - 1 / 2 } \\exp \\left ( - t \\left ( i \\log { \\frac { n + 1 } { v } } + \\frac { 1 } { 2 } \\log ^ 2 { \\frac { n + 1 } { v } } \\right ) \\right ) d v \\\\ & = \\sum _ { 1 \\le n \\le t } \\int _ 1 ^ { 1 + 1 / n } v ^ { - 3 / 2 } \\exp \\left ( - t \\left ( i \\log v + \\frac { 1 } { 2 } \\log ^ 2 v \\right ) \\right ) d v . \\end{align*}"}
-{"id": "4553.png", "formula": "\\begin{align*} ( p _ { n , x , u } - p _ \\cap ) \\biggl \\| V ^ { - 1 / 2 } \\begin{pmatrix} 1 - p _ { n , x , u } \\\\ - p _ { n , y , v } \\end{pmatrix} \\biggr \\| ^ { 3 } \\leq ( p _ { n , x , u } - p _ \\cap ) p _ { n , y , v } ^ { 3 / 2 } | V | ^ { - 3 / 2 } \\lesssim \\Bigl ( \\frac { n } { k \\| z \\| } \\Bigr ) ^ { 1 / 2 } , \\end{align*}"}
-{"id": "4022.png", "formula": "\\begin{align*} \\begin{array} { l l l } I _ 1 ^ h & = & \\displaystyle { \\frac { 1 } { 4 } \\int _ { \\partial B ^ h } \\Big ( \\phi ( z + h ) n _ 1 ^ { + } ( z ) + \\phi ( z - h ) n _ 1 ^ { - } ( z ) + i \\phi ( z + i h ) n _ 2 ^ { + } ( z ) } \\\\ & & + i \\phi ( z - i h ) n _ 2 ^ { - } ( z ) \\Big ) f ^ h ( z ) d S ^ h ( z ) - \\displaystyle { \\int _ { B ^ h } \\phi \\partial _ { \\bar z } ^ h f ^ h d V ^ h } . \\end{array} \\end{align*}"}
-{"id": "7423.png", "formula": "\\begin{align*} e _ i h ^ i ( X ) e _ { - i } = h ^ i ( 1 + X ) e _ { - i } h ^ { i - 1 } \\left ( 1 + X ^ { - 1 } \\right ) ^ { - 1 } h ^ i \\left ( X ^ { - 1 } \\right ) h ^ { i + 1 } \\left ( 1 + X ^ { - 1 } \\right ) ^ { - 1 } e _ i h ^ i ( 1 + X ) . \\end{align*}"}
-{"id": "5251.png", "formula": "\\begin{align*} \\varphi ( y ) = \\inf \\{ t \\in { \\mathbb { R } } \\mid y \\in t k + A \\} \\ ; \\ ; \\forall y \\in Y \\end{align*}"}
-{"id": "3472.png", "formula": "\\begin{align*} \\begin{cases} - \\Delta u + \\nabla p = - \\partial _ { t } u + G ^ { 1 } & \\Omega \\\\ \\ { u } = G ^ { 2 } & \\Omega \\\\ ( p I - \\mathbb { D } u ) e _ { 3 } = ( \\eta I + \\sigma _ { 0 } \\Delta _ { * } \\eta ) e _ { 3 } - \\sigma _ { 0 } ^ { \\prime } \\nabla _ { * } c + G ^ { 3 } & \\Sigma \\\\ u = 0 & \\Sigma _ { b } , \\end{cases} \\end{align*}"}
-{"id": "2032.png", "formula": "\\begin{align*} c ^ 2 \\bar { \\lambda } _ i ( n , c D , K ) = \\lambda _ i ( \\tilde { L } _ c ) . \\end{align*}"}
-{"id": "8456.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\sum _ { j = 1 } ^ { d } J _ { \\varepsilon } \\textbf { P } A _ { j } ( J _ { \\varepsilon } ( \\tilde { \\textbf { u } } ^ { \\varepsilon } + \\bar { \\textbf { u } } ) ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } ^ { \\varepsilon } = 0 . \\end{align*}"}
-{"id": "5878.png", "formula": "\\begin{align*} Z ^ { 4 } = t ^ { 2 } \\partial _ { t } + H + \\frac { t } { 2 c } K ^ { 1 } - \\left ( \\frac { 1 } { 2 } \\int \\frac { d x } { \\sigma \\left ( x \\right ) } ^ { 2 } + \\frac { 1 } { 2 c } \\int \\frac { d x } { \\sigma \\left ( x \\right ) } - c t \\int \\frac { d x } { \\sigma \\left ( x \\right ) } + \\frac { c ^ { 2 } } { 2 } t ^ { 2 } \\right ) F \\partial _ { F } \\end{align*}"}
-{"id": "3566.png", "formula": "\\begin{align*} \\Phi ^ { V ^ R } _ { ( \\gamma ^ R , \\tau ^ R ) } ( \\bar { g } ^ R , \\bar { \\pi } ^ R ) = \\Phi ^ { V ^ R } _ { ( \\gamma ^ R , \\tau ^ R ) } ( \\gamma ^ R , \\tau ^ R ) + ( 2 \\psi ^ R + \\lambda \\zeta R ^ { - 2 } ( \\log R ) ^ \\frac { 1 } { 2 } , V ^ R ) . \\end{align*}"}
-{"id": "7800.png", "formula": "\\begin{align*} \\psi \\left ( f \\left ( x \\right ) \\right ) = \\frac { e ^ { \\sqrt { a _ { l - 1 } T } } - e ^ { \\frac { f } { \\sqrt { a _ { l - 1 } T } } } } { e ^ { \\sqrt { a _ { l - 1 } T } } } \\end{align*}"}
-{"id": "1940.png", "formula": "\\begin{align*} \\nabla F _ { i _ 0 } = & ( - 2 n _ 1 q _ 1 ^ 2 Y _ 1 , - 2 n _ 2 q _ 2 ^ 2 Y _ 2 , \\cdots , - 2 n _ { i _ 0 - 1 } \\ , q _ { i _ 0 - 1 } ^ 2 Y _ { i _ 0 - 1 } , \\\\ & 2 p _ { i _ 0 } - 2 ( n _ { i _ 0 } + 1 ) \\ , q _ { i _ 0 } ^ 2 Y _ { i _ 0 } , - 2 n _ { i _ 0 + 1 } \\ , q _ { i _ 0 + 1 } ^ 2 Y _ { i _ 0 + 1 } , \\cdots , - 2 n _ m q _ m ^ 2 Y _ m ) . \\end{align*}"}
-{"id": "7037.png", "formula": "\\begin{align*} \\frac { U ( x _ k ) } { t _ k A ^ 2 ( x _ k ) } = \\frac { s _ k } { t _ k } \\frac { U ( x _ k ) } { s _ k A ^ 2 ( x _ k ) } \\to 0 , \\end{align*}"}
-{"id": "9069.png", "formula": "\\begin{align*} \\P ( G _ 0 v _ { | | E | | _ 1 ^ { 1 / 3 } } & ( d , k | | E | | _ 1 ^ { 1 / 3 } , E ) ) = \\P ( G _ 0 v ( d , ( k - 1 ) | | E | | _ 1 ^ { 1 / 3 } , E ) ) \\\\ & \\leq \\P ( G _ 0 v ( d , k | | E | | _ 1 ^ { 1 / 3 } , 0 ) + \\mathcal { O } _ { \\upsilon , d } \\left ( N ^ { - 3 / 2 } \\sum _ { \\ell = 1 } ^ { \\infty } ( 1 + k \\ell | | E | | ^ { 1 / 3 } _ 1 ) ^ { 3 } e ^ { - \\frac { \\ell ^ 2 } { 2 d ^ 2 | | E | | _ 1 ^ { 1 / 3 } } } \\right ) . \\end{align*}"}
-{"id": "4547.png", "formula": "\\begin{align*} \\sup _ { C \\in \\mathcal { C } } \\biggl | \\mathbb { P } \\biggl ( \\frac { 1 } { ( n - 2 ) ^ { 1 / 2 } } \\sum _ { i = 3 } ^ n Z _ i \\in C \\biggr ) - \\mathbb { P } ( Z \\in C ) \\biggr | \\leq \\frac { C _ 2 \\mathbb { E } ( \\| Z _ 3 \\| ^ 3 ) } { ( n - 2 ) ^ { 1 / 2 } } . \\end{align*}"}
-{"id": "7027.png", "formula": "\\begin{align*} \\Psi ( \\theta ) : = \\mathrm { i } \\theta \\gamma - \\frac { 1 } 2 \\sigma ^ 2 \\theta ^ 2 + \\int _ { \\mathbb { R } \\setminus \\{ 0 \\} } \\bigl ( e ^ { \\mathrm { i } \\theta x } - 1 - \\mathrm { i } \\theta x { \\mathbf { 1 } } _ { \\{ | x | \\le 1 \\} } \\bigr ) \\Pi ( \\mathrm { d } x ) . \\end{align*}"}
-{"id": "571.png", "formula": "\\begin{align*} \\beta _ n ( r _ 0 , r _ 1 , \\dots , r _ { n - 1 } ) = \\sum _ { k = 0 } ^ { n - 1 } p ^ k [ \\phi ^ { - k } ( r _ k ) ] \\bmod I ^ n \\end{align*}"}
-{"id": "2369.png", "formula": "\\begin{align*} X ^ \\star : = \\{ x ~ | ~ \\nabla F ( x ) = 0 \\} . \\end{align*}"}
-{"id": "900.png", "formula": "\\begin{align*} \\vert \\int _ { T } ^ { 2 T } Z ( t ) d t \\vert = O ( T ^ { { 3 } / { 4 } + \\delta / 2 } ) = o ( T ) \\ \\ \\delta < 1 / 2 . \\end{align*}"}
-{"id": "9473.png", "formula": "\\begin{align*} G _ { t } ( s , t ) = e ^ { 1 / s } \\cos ( 1 / s + t ) \\end{align*}"}
-{"id": "1927.png", "formula": "\\begin{align*} \\bar { \\lambda } ( g _ { a , \\vec { b } } ) \\doteqdot \\bar { \\lambda } ( Y ) \\doteqdot \\left ( \\prod _ { i = 1 } ^ m Y _ i ^ { - \\frac { 2 n _ i } { n } } \\right ) \\cdot \\sum _ { i = 1 } ^ m ( 2 n _ i p _ i Y _ i - \\frac { 1 } { 2 } n _ i q _ i ^ 2 Y _ i ^ 2 ) \\end{align*}"}
-{"id": "4136.png", "formula": "\\begin{align*} S _ f ^ \\nu ( n ) : = \\sum _ { m \\leq n } \\frac { B _ f ( m ) } { m ^ \\nu } , \\end{align*}"}
-{"id": "3855.png", "formula": "\\begin{align*} \\int _ { \\mathbb S ^ { n - 1 } } e ^ { - 2 \\pi \\theta \\cdot \\xi } d \\theta = & \\frac { 2 \\pi } { | \\xi | ^ { \\frac { n - 2 } 2 } } J _ { \\frac { n - 2 } 2 } ( 2 \\pi | \\xi | ) \\ , , \\\\ \\int _ 0 ^ 1 J _ { \\mu } ( t s ) s ^ { \\mu + 1 } ( 1 - s ^ 2 ) ^ \\nu d s = & \\frac { \\Gamma ( \\nu + 1 ) 2 ^ \\nu } { t ^ { \\nu + 1 } } J _ { \\mu + \\nu + 1 } ( t ) \\ , . \\end{align*}"}
-{"id": "7441.png", "formula": "\\begin{align*} H \\backslash \\left ( y ^ { - 1 } z \\right ) ^ * / H = H \\backslash \\left ( \\left [ \\overline { u } ^ { - 1 } x \\right ] _ - ^ { - 1 } \\overline { u } ^ { - 1 } x \\overline { v ^ { - 1 } } \\left [ x \\overline { v ^ { - 1 } } \\right ] _ + ^ { - 1 } \\right ) ^ t / H , \\end{align*}"}
-{"id": "3366.png", "formula": "\\begin{align*} E _ b ( t + 1 ) = E _ b ( t ) - \\Delta t P ( t ) + E _ s ( t ) \\end{align*}"}
-{"id": "7478.png", "formula": "\\begin{align*} q _ j ( d , 1 ) = \\frac { 1 } { j ! } \\sum _ { i = 0 } ^ { \\kappa ' - j } \\frac { 1 } { i ! } ( - 1 ) ^ i b _ { j + i } ( d ) \\end{align*}"}
-{"id": "309.png", "formula": "\\begin{align*} | c ( t ) ^ { - 1 } - 1 | & = \\biggl | \\int _ { \\mathcal { X } } \\biggl ( \\frac { 2 } { 1 + e ^ { - 2 t g } } - 1 - t g \\biggr ) f \\biggr | \\\\ & \\leq \\int _ { A _ t } \\biggl | \\frac { e ^ { - 2 t g } - 1 + 2 t g + t g ( e ^ { - 2 t g } - 1 ) } { 1 + e ^ { - 2 t g } } \\biggr | f + \\int _ { A _ t ^ c } ( 1 + t | g | ) f \\\\ & \\leq \\frac { 1 6 } { 3 } t ^ 2 \\int _ { A _ t } g ^ 2 f + 7 2 t ^ 2 \\int _ { A _ t ^ c } g ^ 2 f \\leq 7 2 t ^ 2 \\int _ { \\mathcal { X } } g ^ 2 f . \\end{align*}"}
-{"id": "2990.png", "formula": "\\begin{align*} \\delta _ v \\Theta ( L ) ( x , y , z ) ( a ) - \\delta _ L \\Theta ^ 2 ( x , y , z ) ( a ) = 0 . \\end{align*}"}
-{"id": "7226.png", "formula": "\\begin{align*} D \\psi _ { x } \\left ( v \\right ) = D \\pi _ { x } \\left ( v \\right ) + \\omega \\left ( x \\right ) D \\left ( p - \\pi \\right ) _ { x } \\left ( v \\right ) + \\left \\langle \\nabla \\omega , v \\right \\rangle \\left ( p - \\pi \\right ) \\left ( x \\right ) . \\end{align*}"}
-{"id": "9889.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ N c _ n \\rho _ n \\substack { H _ 1 \\\\ > \\\\ < \\\\ H _ 0 } \\gamma ^ { ' } , \\end{align*}"}
-{"id": "5374.png", "formula": "\\begin{align*} c _ { d , d } ^ d { \\ , } y ^ { d ^ 2 } + \\sum _ { k = 0 } ^ { m } { d \\choose k } ( c _ { d , d } { \\ , } x ^ d ) ^ k \\Big ( \\sum _ { j = 0 } ^ r c _ { d , j } { \\ , } x ^ j y ^ { d - j } \\Big ) ^ { d - k } ~ = ~ 0 . \\end{align*}"}
-{"id": "5589.png", "formula": "\\begin{align*} I _ 1 \\left ( \\begin{array} { c } a \\\\ b \\end{array} \\right ) = \\left ( \\begin{array} { c } a \\\\ 0 \\end{array} \\right ) , I _ 2 \\left ( \\begin{array} { c } a \\\\ b \\end{array} \\right ) = \\left ( \\begin{array} { c } 0 \\\\ b \\end{array} \\right ) . \\end{align*}"}
-{"id": "9319.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\partial D } Z ( t , x , z ) = 0 \\end{align*}"}
-{"id": "6941.png", "formula": "\\begin{align*} \\prod _ { \\ell = 1 } ^ m \\ , M ( z _ \\ell ; X ) = \\prod _ { \\ell = 1 } ^ m \\prod _ { k \\geq 1 } ( 1 - z _ \\ell x _ k ) ^ { - 1 } = M ( X Z ) , \\end{align*}"}
-{"id": "7622.png", "formula": "\\begin{align*} d \\leq \\lambda _ 1 - \\frac { n } { 4 } + \\frac { 3 } { 2 } = \\frac { n } { 4 } + c _ 1 \\sqrt { n } + \\frac { 3 } { 2 } . \\end{align*}"}
-{"id": "6814.png", "formula": "\\begin{align*} B _ m = \\frac { - 1 } { 2 i \\pi } \\oint _ \\S \\xi ^ m A ( \\xi ) \\dd \\xi . \\end{align*}"}
-{"id": "4181.png", "formula": "\\begin{align*} d _ { G H } ( A , B ) = \\inf d _ H \\bigl ( \\varphi ( A ) , \\psi ( B ) \\bigr ) , \\end{align*}"}
-{"id": "4603.png", "formula": "\\begin{align*} g A g ^ t = M _ { ( 2 j _ 0 ) } M _ { ( 1 2 ) } A M _ { ( 1 2 ) } M _ { ( 2 j _ 0 ) } \\end{align*}"}
-{"id": "1733.png", "formula": "\\begin{gather*} \\alpha : = \\frac { 2 } { 3 } \\left ( - \\frac { \\varphi r \\ , d r } { ( r ^ 2 + 1 ) ^ 2 } + \\frac { \\psi s \\ , d s } { ( s ^ 2 - 1 ) ^ 2 } \\right ) . \\end{gather*}"}
-{"id": "4428.png", "formula": "\\begin{align*} \\| V a _ n - x \\| _ { C _ { \\widehat { E } } } = \\| V a _ n - V S ( x ) \\| _ { C _ { \\widehat { E } } } = \\| a _ n - S ( x ) \\| _ { \\widehat { E } } \\to 0 , \\ \\ \\ \\ n \\to \\infty . \\end{align*}"}
-{"id": "7400.png", "formula": "\\begin{align*} \\mathtt { g } \\cdot \\mathtt { h } & = g _ 1 ^ { s _ 1 } \\cdots g _ k ^ { s _ k } h _ 1 ^ { t _ 1 } \\cdots h _ l ^ { t _ l } , \\\\ \\bar { \\mathtt { g } } & = g _ k ^ { - s _ k } \\cdots g _ 1 ^ { - s _ 1 } , \\\\ \\mathtt { g } ^ { ( \\lambda ) } & = g _ 1 ^ { \\lambda s _ 1 } \\cdots g _ k ^ { \\lambda s _ k } . \\\\ \\end{align*}"}
-{"id": "4232.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } e ^ { x t } d \\mu _ { - 1 } ( x ) = \\frac { 2 } { e ^ t + 1 } \\sum _ { n = 0 } ^ \\infty E _ n \\frac { t ^ n } { n ! } \\end{align*}"}
-{"id": "6064.png", "formula": "\\begin{align*} S _ 0 f ( x ) : = \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i x \\cdot \\xi } \\widehat { \\Phi } ( \\xi ) \\widehat { f } ( \\xi ) d \\xi \\end{align*}"}
-{"id": "6057.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\int _ { \\Omega _ { 1 } \\backslash \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\overline { u } _ { i } ( x ) \\underline { u } _ { j } ( y ) K ( x , y ) \\ , d y d x = \\int _ { \\Omega } \\int _ { \\Omega _ { 1 } \\backslash \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\overline { u } _ { i } ( x ) \\phi _ { j } ( y ) K ( x , y ) \\ , d y \\ , d x \\end{align*}"}
-{"id": "3252.png", "formula": "\\begin{align*} E _ { k } \\triangleright E _ \\xi = c _ { k , \\xi } E _ { \\xi + \\alpha _ k } , F _ { k } \\triangleright E _ \\xi = c ^ \\prime _ { k , \\xi } E _ { \\xi - \\alpha _ k } , \\end{align*}"}
-{"id": "9054.png", "formula": "\\begin{align*} & \\sum _ { k + 3 \\Delta ^ { ( k ) } \\geq r ; k \\leq N / 2 } 2 ^ { r + 3 \\Delta ^ { ( k ) } } e ^ { - ( k + 3 \\Delta ^ { ( k ) } - r ) ^ { \\alpha _ - } } \\leq \\sum _ { k + 3 \\Delta ^ { ( k ) } \\geq r } 2 ^ { k + 6 \\Delta ^ { ( k ) } } e ^ { - ( k + 3 \\Delta ^ { ( k ) } - r ) ^ { \\alpha _ - } } \\\\ & \\leq 2 ^ { - k ' } \\sum _ { s = r } ^ { \\infty } 2 ^ { 2 s } e ^ { - ( s - r ) ^ { 1 / 1 0 } } \\leq 2 ^ { 3 r / 2 } \\sum _ { t = 0 } ^ { \\infty } 2 ^ { 2 t } e ^ { - t ^ { 1 / 1 0 } } \\ll 2 ^ { 3 r / 2 } . \\end{align*}"}
-{"id": "3792.png", "formula": "\\begin{align*} { \\bf { G } } = E \\left [ { { \\bf { \\tilde H } } \\odot { \\bf { \\tilde H } } ^ { * } } \\right ] . \\end{align*}"}
-{"id": "1075.png", "formula": "\\begin{align*} e ( G ) \\leq ( k - 1 ) e ( R ' ) + \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } \\leq \\bigg ( k ^ 2 - \\frac { 1 1 } { 1 6 } k + \\frac { 1 } { 2 } \\bigg ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "9241.png", "formula": "\\begin{align*} \\frac { \\partial \\widehat { H } ( t , x ) } { \\partial y } = \\frac { \\partial H } { \\partial y } ( t , x , \\hat { Y } ( t , x , z ) , \\hat { Y } ( t , . , z ) ( x ) , \\hat { u } ( t , z ) , \\hat { p } ( t , x , z ) , \\hat { q } ( t , x , z ) , \\hat { r } ( t , x , z , . ) ) \\end{align*}"}
-{"id": "9519.png", "formula": "\\begin{align*} \\mathrm { d v o l } _ { g _ E } \\left ( R _ E - \\frac { 1 } { 2 } | d \\phi | ^ 2 _ E \\right ) & = \\mathrm { d v o l } _ g e ^ { - 2 \\phi } \\left ( R + \\frac { 9 } { 2 } \\Delta \\phi + \\frac { 9 } { 2 } | d \\phi | ^ 2 - \\frac { 1 } { 2 } | d \\phi | ^ 2 \\right ) \\\\ & = \\mathrm { d v o l } _ g e ^ { - 2 \\phi } \\left ( R + \\frac { 9 } { 2 } \\Delta \\phi - 5 | d \\phi | ^ 2 \\right ) . \\end{align*}"}
-{"id": "3536.png", "formula": "\\begin{align*} ( 2 \\psi , V ) = - \\Phi ( g , \\pi ) + \\chi \\Phi ( g _ 1 , \\pi _ 1 ) + ( 1 - \\chi ) \\Phi ( g _ 2 , \\pi _ 2 ) + ( 2 \\psi _ 0 , 0 ) , \\end{align*}"}
-{"id": "9736.png", "formula": "\\begin{align*} \\Lambda _ 1 = \\limsup _ { t \\to \\infty } \\frac { x ( t ) } { G ^ { - 1 } ( t ) } \\end{align*}"}
-{"id": "8338.png", "formula": "\\begin{align*} z = w + D . \\end{align*}"}
-{"id": "2307.png", "formula": "\\begin{align*} \\lambda _ 3 = 1 4 . 4 5 0 8 7 , \\lambda _ 4 = 3 . 4 9 0 4 0 , \\lambda _ 5 = 1 . 6 2 8 9 2 9 7 9 , \\lambda _ 6 = 1 . 0 5 0 5 1 3 . \\end{align*}"}
-{"id": "8119.png", "formula": "\\begin{align*} { \\mathcal R } _ { k , l , N } ( M ) = C _ { k , l , N } ^ { r a d } \\ \\mbox { f o r s o m e m e a s u r a b l e s e t $ M $ w i t h $ 0 < \\mu _ l ( M ) < + \\infty $ } , \\end{align*}"}
-{"id": "2113.png", "formula": "\\begin{align*} ( l + 1 ) ( l + 2 + 2 \\sigma _ n ) p ' _ { \\sigma , l + 1 } + 2 s ( \\sigma _ n + 1 ) p ' _ { \\sigma + \\bar n , l } + ( \\sigma _ i + 1 ) ( \\sigma _ i + 2 ) p ' _ { \\sigma + 2 \\bar \\imath , l - 1 } + \\tilde { c } _ { \\sigma l } ^ { \\mu m } p ' _ { \\mu m } = 0 \\forall ( \\sigma , l ) \\end{align*}"}
-{"id": "4424.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { C _ E } } = \\| S ( x ) \\| _ { \\widehat { E } } . \\end{align*}"}
-{"id": "7283.png", "formula": "\\begin{align*} J = ( x ^ { 7 } , \\ x ^ 4 y ^ 2 z ^ 2 , \\ x y ^ 3 z ^ 3 , \\ y ^ { 7 } , \\ z ^ { 7 } ) . \\end{align*}"}
-{"id": "4828.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | \\| K ( I - A _ n ) \\| | = 0 . \\end{align*}"}
-{"id": "8144.png", "formula": "\\begin{align*} a ^ * = \\max \\left \\{ a : \\ , a = k + l ( \\frac { 1 } { p } - 1 ) , \\ 0 \\leq k \\leq l + 1 \\leq k + \\frac { N - 1 } { k + N - 1 } , \\ b q \\leq l \\right \\} . \\end{align*}"}
-{"id": "1921.png", "formula": "\\begin{align*} g _ { a , \\vec { b } } ( \\tau ) = a ( \\tau ) \\ , \\sigma ( \\cdot ) \\otimes \\sigma ( \\cdot ) + \\sum _ { i = 1 } ^ m \\ , b _ i ( \\tau ) \\ , g _ i . \\end{align*}"}
-{"id": "8879.png", "formula": "\\begin{align*} - \\Delta u + u = \\abs { u } ^ { p - 2 } u , \\mathbb { R } ^ N . \\end{align*}"}
-{"id": "476.png", "formula": "\\begin{align*} \\frac { \\beta _ 1 v _ { 1 1 } + \\beta _ 2 v _ { 1 2 } } { \\tau } x _ 1 + \\frac { \\beta _ 1 v _ { 2 1 } + \\beta _ 2 v _ { 2 2 } } { \\tau } x _ 2 = \\frac { \\beta _ 1 \\alpha _ 1 + \\beta _ 2 \\alpha _ 2 } { \\tau } + \\varepsilon \\end{align*}"}
-{"id": "5314.png", "formula": "\\begin{align*} A ( u ) C ( v ) & = \\frac { u - v + \\eta } { u - v } C ( v ) A ( u ) - \\frac { \\eta } { u - v } C ( u ) A ( v ) \\\\ D ( u ) C ( u ) & = \\frac { u - v - \\eta } { u - v } C ( v ) D ( u ) + \\frac { \\eta } { u - v } C ( u ) D ( v ) . \\end{align*}"}
-{"id": "4976.png", "formula": "\\begin{align*} h _ { H } ( f ( P ) ) = h _ { E } ( P ) + \\left < A \\vec { c } , { \\bf h } _ { \\vec { D } } \\right > ( P ) . \\end{align*}"}
-{"id": "6499.png", "formula": "\\begin{align*} \\| \\rho \\| _ { L ^ { 1 } ( ( 0 , T ] ) } \\leq \\frac { T ^ { 1 - \\alpha } } { \\Gamma ( 2 - \\alpha ) } \\| \\mu \\| _ { L ^ { 1 } ( ( 0 , T ] ) } = 0 , \\end{align*}"}
-{"id": "3351.png", "formula": "\\begin{align*} & \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\cdot \\nu ( g _ 1 ^ { [ \\lambda s _ 1 n ] } \\cdots g _ k ^ { [ \\lambda s _ k n ] } ) \\\\ & = \\lim _ { n \\to \\infty } \\frac { 1 } { p n } \\cdot \\nu ( g _ 1 ^ { [ q s _ 1 n ] } \\cdots g _ k ^ { [ q s _ k n ] } ) \\\\ & = \\lim _ { n \\to \\infty } \\frac { q } { p n } \\cdot \\nu ( g _ 1 ^ { [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k n ] } ) \\\\ & = \\lambda \\lim _ { n \\to \\infty } \\frac { 1 } { n } \\cdot \\nu ( g _ 1 ^ { [ s _ 1 n ] } \\cdots g _ k ^ { [ s _ k n ] } ) . \\\\ \\end{align*}"}
-{"id": "4660.png", "formula": "\\begin{align*} \\Sigma _ 1 ( f , z _ 1 , z _ 2 ) & = \\int _ { 0 } ^ { \\infty } \\int _ 0 ^ \\infty ( f ( 0 , 0 , t , u ) + f ( 0 , 0 , t , - u ) ) t ^ { z _ 1 - 1 } u ^ { z _ 2 - 1 } d t d u \\\\ \\Sigma _ 2 ( f , z ) & = \\int _ 0 ^ \\infty f ( 0 , 0 , 0 , u ) u ^ { z - 1 } d u \\\\ \\Sigma _ 3 ( f , z , u ) & = \\int _ 0 ^ \\infty f ( 0 , 0 , t , u ) t ^ { z - 1 } d t . \\end{align*}"}
-{"id": "3661.png", "formula": "\\begin{align*} M = \\begin{pmatrix} 1 & x _ 1 & x _ 2 & x _ 1 ^ 2 & x _ 1 x _ 2 & x _ 1 ^ 2 x _ 2 \\\\ 1 & x _ 2 & x _ 1 & x _ 2 ^ 2 & x _ 1 x _ 2 & x _ 1 x _ 2 ^ 2 \\\\ 1 & x _ 3 & x _ 2 & x _ 3 ^ 2 & x _ 2 x _ 3 & x _ 2 x _ 3 ^ 2 \\\\ 1 & x _ 1 & x _ 3 & x _ 1 ^ 2 & x _ 1 x _ 3 & x _ 1 ^ 2 x _ 3 \\\\ 1 & x _ 2 & x _ 3 & x _ 2 ^ 2 & x _ 2 x _ 3 & x _ 2 ^ 2 x _ 3 \\\\ 1 & x _ 3 & x _ 1 & x _ 3 ^ 2 & x _ 1 x _ 3 & x _ 1 x _ 3 ^ 2 \\end{pmatrix} . \\end{align*}"}
-{"id": "8257.png", "formula": "\\begin{align*} v ^ n ( t , x ) = n ^ { 1 / 2 } ( u ( n ^ 2 t , \\lfloor n ^ { } x - c _ n t \\rfloor ) - \\rho ' ( \\lambda _ 0 ) ) , ( t , x ) \\in \\R _ + \\times \\R , \\end{align*}"}
-{"id": "7152.png", "formula": "\\begin{align*} a = \\Phi ( a ) , \\Phi ( a ) : = a - B _ { N S } ( a ) + b . \\end{align*}"}
-{"id": "1665.png", "formula": "\\begin{align*} \\partial _ { x _ k } { \\psi } ( z ) = \\partial _ { x _ k } G _ { \\lambda } g ( z ) = \\int _ { 0 } ^ { + \\infty } e ^ { - \\lambda t } P _ t ( \\partial _ { x _ k } g ) ( z ) \\dd t \\ , . \\end{align*}"}
-{"id": "5198.png", "formula": "\\begin{align*} & V = V _ { 1 } U _ { 2 } , \\ , X = U _ { 1 } V ^ { T } _ { 1 } , \\\\ & \\| V \\| _ { S _ { \\widehat { p } _ { 2 } } } = \\| U _ { 2 } \\| _ { S _ { q } } \\| V _ { 1 } \\| _ { S _ { p _ { 2 } } } , \\\\ & \\| U _ { 1 } \\| _ { S _ { p _ { 1 } } } \\leq \\| U \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| U _ { 2 } \\| _ { S _ { q } } \\end{align*}"}
-{"id": "3750.png", "formula": "\\begin{align*} \\sup ( q _ 1 * q _ 2 ) & = \\sup \\left \\{ \\int _ { - \\infty } ^ { \\infty } q _ 1 ( t - s ) q _ 2 ( s ) d s \\right \\} \\\\ & \\leq \\sup ( q _ 1 ) \\cdot \\int _ { - \\infty } ^ { \\infty } q _ 2 ( s ) d s = \\sup ( q _ 1 ) . \\end{align*}"}
-{"id": "7925.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ { m } y _ { i } = \\frac { m ( h _ { 1 } + h _ { 2 } ) } { 2 } . \\end{align*}"}
-{"id": "271.png", "formula": "\\begin{align*} U _ { 1 2 } : = \\biggl | \\int _ { \\mathcal { X } } f ( x ) \\int _ { u _ n ^ * ( x ) } ^ \\infty \\log \\bigl ( u f ( x ) \\bigr ) \\ , d ( F _ { n , x } ^ - - F _ { n , x } ) ( u ) \\ , d x \\biggr | = o ( n ^ { - ( 3 - \\epsilon ) } ) . \\end{align*}"}
-{"id": "7901.png", "formula": "\\begin{align*} \\chi _ n \\ , : = \\ , \\chi _ 0 \\ , + \\ , n \\ , I \\ , + \\ , r _ 0 \\ , n ^ 2 \\ , \\frac { h _ 0 ^ 2 } { 2 } . \\end{align*}"}
-{"id": "6745.png", "formula": "\\begin{align*} \\theta _ { \\varphi } ^ n = e ^ { \\beta \\varphi } \\mu , \\ \\ \\varphi \\in \\mathcal E ^ 1 ( X , \\omega ) , \\end{align*}"}
-{"id": "9974.png", "formula": "\\begin{align*} \\Lambda ^ * + h ^ * ( s ) = \\max _ { a \\in \\mathcal { A } ( s ) } \\left [ g ( s , a ) + \\sum _ { s ' \\in \\mathcal { S } } p _ { s \\rightarrow s ' | a } h ^ * ( s ' ) \\right ] , \\end{align*}"}
-{"id": "8211.png", "formula": "\\begin{align*} Q = \\sum _ { n \\in \\mathbb { Z } } ( u _ n \\bar { v } _ n + \\bar { u } _ n v _ n ) . \\end{align*}"}
-{"id": "9448.png", "formula": "\\begin{align*} \\mu ( A ) = \\lim _ { \\omega } \\frac { \\dim ( ( W _ i + a W _ i ) \\cap A ) } { \\dim ( W _ i ) } \\ ; . \\end{align*}"}
-{"id": "8954.png", "formula": "\\begin{align*} \\begin{array} { r } \\dfrac { s ^ { n + 1 } - s ^ { n } } { \\Delta t } - \\dfrac { r _ { x x } ^ { n + 1 } + r _ { x x } ^ { n } } { 2 } + q \\dfrac { ( ( r ^ { 2 } + s ^ { 2 } ) r ) ^ { n + 1 } + ( ( r ^ { 2 } + s ^ { 2 } ) r ) ^ { n } } { 2 } = 0 \\\\ \\\\ \\dfrac { r ^ { n + 1 } - r ^ { n } } { \\Delta t } + \\dfrac { s _ { x x } ^ { n + 1 } + s _ { x x } ^ { n } } { 2 } + q \\dfrac { ( ( r ^ { 2 } + s ^ { 2 } ) s ) ^ { n + 1 } + ( ( r ^ { 2 } + s ^ { 2 } ) s ) ^ { n } } { 2 } = 0 \\end{array} \\end{align*}"}
-{"id": "9005.png", "formula": "\\begin{align*} E _ { \\alpha , \\beta } ( z ) = \\sum _ { k = 0 } ^ \\infty \\frac { z ^ k } { \\Gamma ( \\alpha k + \\beta ) } , ~ ~ ~ ( z , \\beta \\in \\mathbb { C } ; ~ R e ( \\alpha ) > 0 ) , \\end{align*}"}
-{"id": "2733.png", "formula": "\\begin{align*} A = \\nu ^ r A _ r + \\nu ^ { r + 1 } A _ { r + 1 } + \\ldots , \\end{align*}"}
-{"id": "9331.png", "formula": "\\begin{align*} \\Phi ( t , z ) = \\frac { \\mathbb { E } [ D _ t \\delta _ Z ( z ) | \\mathcal { F } _ t ] } { \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] } \\end{align*}"}
-{"id": "6332.png", "formula": "\\begin{align*} \\left \\| \\Box _ k ^ { \\alpha _ 1 } | ~ M _ 1 \\rightarrow M _ 2 \\right \\| \\gtrsim \\lim _ { N \\rightarrow \\infty } \\frac { \\| \\Box _ k ^ { \\alpha _ 1 } G _ { k , N } \\| _ { M _ 2 } } { \\| G _ { k , N } \\| _ { M _ 1 } } \\sim 2 ^ { j \\widetilde { A _ 3 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "780.png", "formula": "\\begin{align*} L _ { k } ^ { + } = \\left \\{ \\vec { x } = ( x _ { 1 } , \\ldots , x _ { k } ) \\in \\mathbb { Z } ^ { k } \\ , | \\ , x _ { 1 } \\ge \\cdots \\ge x _ { k } \\right \\} , \\end{align*}"}
-{"id": "7968.png", "formula": "\\begin{align*} \\varphi \\ ! \\left ( \\ ! \\frac { p _ 1 ( x ) } { \\widetilde { + } _ { \\varphi } ( p _ 1 , \\cdots \\ ! , p _ m ) ( x ) } , \\cdots \\ ! , \\frac { p _ m ( x ) } { \\widetilde { + } _ { \\varphi } ( p _ 1 , \\cdots \\ ! , p _ m ) ( x ) } \\ ! \\right ) = 1 , \\end{align*}"}
-{"id": "5542.png", "formula": "\\begin{align*} b ( z ) = b _ 0 + \\sum _ { k = 1 } ^ { M } \\frac { b _ { - 1 } ^ { ( k ) } } { z + \\epsilon _ k } , \\epsilon _ k \\neq \\epsilon _ j , k \\neq j \\epsilon _ k > 0 \\end{align*}"}
-{"id": "4455.png", "formula": "\\begin{align*} h _ x ^ { - 1 } ( s ) = \\begin{cases} s / 2 , & \\mbox { i f } s \\leq 2 x \\\\ s - x , & \\mbox { i f } 2 x < s \\leq 1 . \\end{cases} \\end{align*}"}
-{"id": "8830.png", "formula": "\\begin{align*} \\widehat { Y } = \\{ ( y , t _ y ) \\vert y \\in Y , t _ y \\in \\widehat { \\mathcal { O } } _ y , ( t _ y ) = \\widehat { m } _ y \\} \\end{align*}"}
-{"id": "7022.png", "formula": "\\begin{align*} t ^ \\ell _ { m , n } ( g ( \\omega ) ) = \\mathfrak { \\tau } ^ \\ell _ { m , n } \\ , \\omega ^ { 2 n } \\end{align*}"}
-{"id": "687.png", "formula": "\\begin{align*} \\lambda ( F A _ { x _ o } ) = \\nu ( F A ) > 1 - \\nu ( A ) . \\end{align*}"}
-{"id": "7000.png", "formula": "\\begin{align*} f _ 0 ^ { \\lambda _ i ( x , y ) } = - f _ i ^ { \\lambda _ 0 ( x , y ) } - \\sum _ { m + n = i , ~ m , n > 0 } f _ m ^ { \\lambda _ n ( x , y ) } + \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ x , f _ n ^ { y } ] + [ f _ 0 ^ x , f _ i ^ { y } ] + [ f _ i ^ x , f _ 0 ^ { y } ] . \\end{align*}"}
-{"id": "1905.png", "formula": "\\begin{align*} m ^ * n ^ { - \\frac { 1 } { K } } \\leq C _ 2 n ^ { - \\frac { 2 \\beta - 1 } { 2 \\beta K } } = o \\big ( ( \\log n ) ^ { - \\frac { 2 \\beta - 1 } { \\rho } } \\big ) . \\end{align*}"}
-{"id": "7620.png", "formula": "\\begin{align*} \\sum _ { \\substack { u v \\in E ( S , L ) \\\\ u \\not = x } } \\mathbf { v } _ u \\geq ( 1 - 9 \\epsilon ) n . \\end{align*}"}
-{"id": "6106.png", "formula": "\\begin{align*} q = q ' ( \\frac { d z } z + \\frac { d w } w ) ^ 2 \\end{align*}"}
-{"id": "6389.png", "formula": "\\begin{align*} - \\log p ( x ) = \\frac { 1 } { 2 } \\log \\frac { 2 \\pi \\sigma ^ { 2 } } { C ^ { 2 } ( \\varepsilon ) } + \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } + \\varepsilon x ^ { p } . \\end{align*}"}
-{"id": "5003.png", "formula": "\\begin{align*} & { \\bf h } _ { \\vec { D } } = \\left ( \\begin{array} { c } h _ { D _ { 1 } } \\\\ h _ { D _ { 2 } } \\\\ \\vdots \\\\ h _ { D _ { r } } \\end{array} \\right ) , \\ { \\bf h } _ { \\vec { F } } = \\left ( \\begin{array} { c } h _ { F _ { 1 } } \\\\ h _ { F _ { 2 } } \\\\ \\vdots \\\\ h _ { F _ { s } } \\end{array} \\right ) , \\ { \\bf h } _ { { p } _ { * } \\vec { F } } = \\left ( \\begin{array} { c } h _ { { p } _ { * } F _ { 1 } } \\\\ h _ { { p } _ { * } F _ { 2 } } \\\\ \\vdots \\\\ h _ { { p } _ { * } F _ { s } } \\end{array} \\right ) \\ . \\end{align*}"}
-{"id": "9010.png", "formula": "\\begin{align*} \\mathcal M _ p = \\{ M _ 1 , M _ 2 , \\ldots , M _ t \\} . \\end{align*}"}
-{"id": "2505.png", "formula": "\\begin{align*} X _ { 0 } ^ { \\lambda } = \\left \\| \\bigwedge _ { i = 1 } ^ { p } \\nabla D _ i \\right \\| _ { p } ^ { - 2 } \\cdot \\sum _ { i = 1 } ^ { p } ( - 1 ) ^ { n - i } ( - \\lambda ) ( D _ i - d _ i ) \\Theta _ i , \\end{align*}"}
-{"id": "6646.png", "formula": "\\begin{align*} \\varepsilon _ K ( ( V , N ) , \\psi ) : = \\varepsilon _ K ( V , \\psi ) \\det ( - \\Phi | V ^ I / V ^ I _ N ) \\end{align*}"}
-{"id": "5809.png", "formula": "\\begin{align*} \\pi ( E ) = \\mathcal { E } , \\pi ( F ) = \\mathcal { F } \\end{align*}"}
-{"id": "9352.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } _ + } y ( t , x , z ) p ( t , x , z ) d x = \\int _ { \\mathbb { R } _ + } u ( t , x , z ) \\Gamma ( t , x ) d x = \\\\ & \\int _ { \\mathbb { R } _ + } \\mathbb { E } _ Q [ y ( T , x , z ) U ( x , z ) \\mathbb { E } _ Q [ \\delta _ Z ( z ) | \\mathcal { F } ^ { G } _ T ] | \\mathcal { F } ^ { G } _ t ] d x . \\end{align*}"}
-{"id": "7492.png", "formula": "\\begin{align*} Q _ 0 = 1 , Q _ 1 = n - 2 , Q _ 2 = 1 \\ { \\rm a n d } \\ Q _ j = 0 \\ { \\rm f o r \\ a l l } \\ 3 \\leq j \\leq \\kappa ' . \\end{align*}"}
-{"id": "5059.png", "formula": "\\begin{align*} E = \\big \\{ z \\in Z \\ , : \\ , \\phi ( z ) \\geq \\delta \\big \\} \\end{align*}"}
-{"id": "5616.png", "formula": "\\begin{align*} T _ { \\ell } ( \\alpha ) = \\sum _ { n = 1 } ^ { N } e ( n ^ { \\ell } \\alpha ) \\ , , \\end{align*}"}
-{"id": "7355.png", "formula": "\\begin{align*} T = \\left ( \\begin{array} { c c c c } 0 & T ^ { ( 1 ) } & 0 & 0 \\\\ 0 & 0 & T ^ { ( 2 ) } & 0 \\\\ 0 & 0 & 0 & T ^ { ( 3 ) } \\\\ 0 & 0 & 0 & 0 \\end{array} \\right ) , T ^ * = \\left ( \\begin{array} { c c c c } 0 & 0 & 0 & 0 \\\\ T ^ { ( 1 ) * } & 0 & 0 & 0 \\\\ 0 & T ^ { ( 2 ) * } & 0 & 0 \\\\ 0 & 0 & T ^ { ( 3 ) * } & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "720.png", "formula": "\\begin{align*} h _ { N } = c _ { 1 } h _ { D _ { 1 } } - h _ { H } . \\end{align*}"}
-{"id": "1928.png", "formula": "\\begin{align*} & \\left ( \\prod _ { i = 1 } ^ m Y _ i ^ { - \\frac { 2 n _ i } { n } } \\right ) ^ { - 1 } \\frac { d \\bar { \\lambda } ( Y ( u ) ) } { d u } \\\\ \\leq & - \\left ( \\frac { 1 } { \\sqrt { n } } \\left ( \\sum _ { i = 1 } ^ m n _ i Y _ i ^ 2 ( 2 p _ i - q _ i ^ 2 Y _ i ) ^ 2 \\right ) ^ { 1 / 2 } - \\sqrt { \\frac { n - 1 } { 2 n } } E ( Y ) \\right ) ^ 2 \\leq 0 . \\end{align*}"}
-{"id": "9425.png", "formula": "\\begin{align*} h ( x ) \\coloneqq \\begin{cases} \\psi ( x ) & x \\in \\Delta _ 1 \\\\ \\psi _ 0 ( x ) & x \\in \\Sigma _ 1 . \\end{cases} \\end{align*}"}
-{"id": "8969.png", "formula": "\\begin{align*} r : = \\begin{cases} \\dfrac { N \\ , ( p - 1 ) \\ , q } { N - s p q } , & 1 < q < \\dfrac { N } { s p } , \\\\ [ 1 0 p t ] \\infty , & \\dfrac { N } { s p } < q \\le \\infty , \\end{cases} \\end{align*}"}
-{"id": "6846.png", "formula": "\\begin{align*} \\tfrac { 2 } { \\tilde q } + \\tfrac { d } { \\tilde r } = \\tfrac { d } { 2 } - s _ c = \\tfrac { 2 } { p } . \\end{align*}"}
-{"id": "3872.png", "formula": "\\begin{align*} \\| \\nabla _ { \\partial \\Omega } \\nu ( b ) \\| = \\| \\nabla _ { \\partial \\Omega } \\tau ( b ) \\| \\le \\varkappa \\end{align*}"}
-{"id": "4375.png", "formula": "\\begin{align*} e ^ { i J ( g ) } T ( f ) e ^ { - i J ( g ) } = T ( f ) + J ( ( \\partial _ \\theta g ) f ) + \\int _ { - \\pi } ^ { \\pi } \\frac { ( \\partial _ \\theta g ) ^ 2 ( e ^ { i \\theta } ) } { 2 } f ( e ^ { i \\theta } ) \\frac { d \\theta } { 2 \\pi } \\ ; I , \\end{align*}"}
-{"id": "9300.png", "formula": "\\begin{align*} \\int _ { \\mathbb { R } ^ { + } } \\{ - \\alpha ( t ) y ' ( t , x , z ) + \\pi \\beta ^ 2 ( t ) y '' ( t , x , z ) \\} p ( t , x , z ) d x = 0 , \\end{align*}"}
-{"id": "6840.png", "formula": "\\begin{align*} [ e ^ { i t \\Delta } ( e ^ { i x \\xi _ 0 } f ) ] ( x ) = e ^ { - i t | \\xi _ 0 | ^ 2 + i x \\xi _ 0 } ( e ^ { i t \\Delta } f ) ( x - 2 t \\xi _ 0 ) \\end{align*}"}
-{"id": "7395.png", "formula": "\\begin{align*} c ( [ \\hat { W } ] , F ) & \\leq c ( [ \\hat { W } ] , L ^ 1 ) + | | \\bar { H } | | _ { C ^ 0 } \\\\ & = c ( [ \\hat { W } ] , H ) + | | H | | _ { C ^ 0 } \\\\ & < 2 ( \\sum _ { i = 1 } ^ n ( R _ i - \\epsilon ) \\cdot | e _ i | + | | G | | _ { C ^ 0 } ) . \\end{align*}"}
-{"id": "6694.png", "formula": "\\begin{align*} \\mu _ \\star = \\frac { 1 } { m ! \\ , \\nu ^ m } \\left ( \\omega _ { - 1 } \\right ) ^ m e ^ \\varkappa , \\end{align*}"}
-{"id": "8922.png", "formula": "\\begin{align*} \\frac { \\abs { x - a } } { 2 } \\sqrt { 1 + \\frac { B ^ 2 } { 4 } \\abs { x - a } ^ 2 } & = \\frac { \\abs { B } \\ , \\abs { x - a } ^ 2 } { 4 } \\sqrt { 1 + \\frac { 4 } { B ^ 2 \\abs { x - a } ^ 2 } } \\\\ & = \\frac { \\abs { B } \\ , \\abs { x - a } ^ 2 } { 4 } \\biggl ( 1 + \\frac { 2 } { B ^ 2 \\abs { x - a } ^ 2 } + O \\Bigl ( \\frac { 1 } { \\abs { x - a } ^ 4 } \\Bigr ) \\biggr ) \\\\ & = \\frac { \\abs { B } \\ , \\abs { x - a } ^ 2 } { 4 } + \\frac { 1 } { 2 \\abs { B } } + o ( 1 ) , \\end{align*}"}
-{"id": "1712.png", "formula": "\\begin{gather*} \\Phi ( x , y , z ) : = H _ { \\times } ( x \\times y , z ) . \\end{gather*}"}
-{"id": "4005.png", "formula": "\\begin{align*} n _ t + n _ a = | S T | = q ^ t + 1 , \\end{align*}"}
-{"id": "8246.png", "formula": "\\begin{align*} L _ r ( \\psi ) = 0 , \\end{align*}"}
-{"id": "8624.png", "formula": "\\begin{align*} G ( \\psi ) = r e s H _ { \\psi } : J ( \\mu _ { k } ^ 2 M , \\mu _ { k } ^ 2 P ) - \\operatorname { m o d } \\rightarrow J ( M , P ) - \\operatorname { m o d } \\end{align*}"}
-{"id": "8324.png", "formula": "\\begin{align*} f ( \\mu _ i y _ i ) = \\sum _ { j = 0 } ^ n a _ j ( \\mu _ i y _ i ) ^ { p ^ j } = \\sum _ { j = 0 } ^ n a _ j ( \\mu _ i ^ { p ^ j } y _ i + \\mu _ i ^ { p ^ j } l _ j ( \\gamma _ i ) ) = y _ i f ( \\mu _ i ) + h _ i , \\end{align*}"}
-{"id": "3731.png", "formula": "\\begin{align*} | S _ 2 | = \\sum _ { i , j } | S _ 2 ( i , j ) | = 6 | S _ 2 ( 1 , 2 ) | . \\end{align*}"}
-{"id": "6216.png", "formula": "\\begin{align*} X ^ { ( 0 ) } ( s _ { 0 } ) = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } F , \\\\ X ^ { ( 1 ) } ( s _ { 0 } ) = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } ( - ( 2 s _ { 0 } M + D ) ) X ^ { ( 0 ) } ( s _ { 0 } ) , \\\\ X ^ { ( 2 ) } ( s _ { 0 } ) = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } [ - ( 2 s _ { 0 } M + D ) X ^ { ( 1 ) } ( s _ { 0 } ) - M X ^ { ( 0 ) } ( s _ { 0 } ) ] , \\\\ \\vdots \\\\ X ^ { ( j ) } ( s _ { 0 } ) = & \\ ( s _ { 0 } ^ { 2 } M + s _ { 0 } D + K ) ^ { - 1 } [ - ( 2 s _ { 0 } M + D ) X ^ { ( j - 1 ) } ( s _ { 0 } ) - M X ^ { ( j - 2 ) } ( s _ { 0 } ) ] . \\end{align*}"}
-{"id": "1775.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\int _ { \\Omega _ { 1 } \\backslash \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\overline { u } _ { i } ( x ) \\underline { u } _ { j } ( y ) K ( x , y ) \\ , d y d x = \\int _ { \\Omega } \\int _ { \\Omega _ { 1 } \\backslash \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\overline { u } _ { i } ( x ) \\phi _ { j } ( y ) K ( x , y ) \\ , d y \\ , d x \\end{align*}"}
-{"id": "1688.png", "formula": "\\begin{align*} L _ t = e ^ { t A } z ^ r + W _ A ( t ) , W _ A ( t ) = \\int _ 0 ^ t e ^ { ( t - s ) A } R \\ , \\dd W _ s \\ , , \\end{align*}"}
-{"id": "8071.png", "formula": "\\begin{align*} \\tilde { X } = \\bigcap _ { n = 0 } ^ \\infty X _ n . \\end{align*}"}
-{"id": "1745.png", "formula": "\\begin{align*} h _ R ( x ) = \\int _ { B _ R ^ + } G _ R ^ + ( x , y ) g _ 1 ( y ) d y + \\int _ { \\R ^ N \\setminus B _ R ^ + } \\Gamma _ R ^ + ( x , y ) g _ 2 ( y ) d y . \\end{align*}"}
-{"id": "3733.png", "formula": "\\begin{align*} & | N ( x ) \\cap S _ 4 | \\\\ = & \\frac { 1 } { | S _ 0 \\setminus ( S \\cup T ) | } | S _ 4 | | N ( y ) \\cap S _ 0 \\setminus ( S \\cup T ) | \\\\ = & \\frac { 1 } { | S _ 0 \\setminus ( S \\cup T ) | } | S _ 4 | \\cdot \\frac 1 2 | S _ 0 \\setminus ( S \\cup T ) | \\\\ = & \\frac 1 2 | S _ 4 | . \\end{align*}"}
-{"id": "9493.png", "formula": "\\begin{align*} \\Phi ^ { * } \\lambda _ { 0 } & = d \\theta + \\alpha _ { n - 1 } + \\Phi ^ { * } ( x _ { n } \\ , d y _ { n } - y _ { n } \\ , d x _ { n } ) \\\\ & = d \\theta + \\alpha _ { n - 1 } + e ^ { 2 \\rho } \\ , d \\phi \\\\ & = e ^ { 2 \\rho } \\lambda \\end{align*}"}
-{"id": "3721.png", "formula": "\\begin{align*} c = \\dfrac { - 1 } { q ^ { n } } . \\end{align*}"}
-{"id": "2322.png", "formula": "\\begin{align*} \\| \\varDelta f ( w ) - \\varDelta f ( v ) \\| _ { X } & = \\| \\varDelta f ( w ) - \\varDelta f ( v ) \\| _ { L ^ 2 ( \\R ^ d ) } + \\| \\varDelta f ( w ) - \\varDelta f ( v ) \\| _ { L ^ q ( \\R ^ d ) } \\\\ & \\le C _ K ( \\| w - v \\| _ { H ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { W ^ { 2 , q } ( \\R ^ d ) } ) \\\\ & \\le C _ K ( \\| w - v \\| _ { H ^ 2 ( \\R ^ d ) } + \\| w - v \\| _ { W ^ { 2 , \\infty } ( \\R ^ d ) } ) = C _ K \\| w - v \\| _ { W } . \\end{align*}"}
-{"id": "4082.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) z ^ 2 } { x ^ { 4 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) z } { x ^ { 3 } } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 2 z } { x ^ { 5 } } . \\\\ \\end{gather*}"}
-{"id": "3267.png", "formula": "\\begin{align*} \\gamma _ - ( w ) \\gamma _ + ( v ) + \\gamma _ + ( v ) \\gamma _ - ( w ) = \\langle w , v \\rangle 1 , w \\in \\mathfrak { u } _ - , \\ v \\in \\mathfrak { u } _ + . \\end{align*}"}
-{"id": "7293.png", "formula": "\\begin{align*} [ E _ 1 ^ { c _ 1 } \\cdots E _ r ^ { c _ r } , F _ t ] & = E _ 1 ^ { c _ 1 } \\cdots E _ { t - 1 } ^ { c _ { t - 1 } } [ E _ t , F _ t ] E _ { t + 1 } ^ { c _ { t + 1 } } \\cdots E _ r ^ { c _ r } \\\\ & = E _ 1 ^ { c _ 1 } \\cdots E _ { t - 1 } ^ { c _ { t - 1 } } \\frac { K _ t - K _ t ^ { - 1 } } { q _ t - q _ t ^ { - 1 } } E _ { t + 1 } ^ { c _ { t + 1 } } \\cdots E _ r ^ { c _ r } \\in U _ { q } ( \\mathfrak { l } ) . \\end{align*}"}
-{"id": "4883.png", "formula": "\\begin{align*} \\delta ( x ) \\stackrel { ( \\ref { a = b + c } ) } { \\equiv } \\delta ( [ \\pi ( x ) ] ) + \\delta ( p [ a ] ) \\bmod I \\equiv \\delta ( p [ a ] ) \\bmod I \\equiv [ a ^ p ] \\bmod I \\equiv [ r _ 1 ] \\bmod I \\end{align*}"}
-{"id": "5233.png", "formula": "\\begin{align*} \\left ( \\partial \\rho - \\overline { \\partial } \\rho \\right ) = e ^ { - g } \\eta \\ ; , \\ ; g \\in C ^ { \\infty } ( M ) \\ ; . \\end{align*}"}
-{"id": "7919.png", "formula": "\\begin{align*} \\int _ K | W ( g _ \\omega ) | ^ 4 _ { g _ { \\omega } } \\omega ^ 4 d v _ { 0 } \\leq \\varliminf _ { j \\to \\infty } \\int _ K | W ( \\tilde g _ j ) | _ { \\tilde g _ j } ^ 4 v _ j ^ 4 d v _ 0 = 0 . \\end{align*}"}
-{"id": "3399.png", "formula": "\\begin{align*} \\dim \\mathbf { H } ^ 1 ( { \\mathcal F } ^ { \\bullet } _ x ) & = \\dim H ^ 0 ( { \\mathcal F } ^ 1 _ x ) + \\dim H ^ 1 ( { \\mathcal F } ^ 0 _ x ) - \\dim H ^ 0 ( { \\mathcal F } ^ 0 _ x ) - \\dim H ^ 1 ( { \\mathcal F } ^ 1 _ x ) + 2 \\dim _ { \\mathbb { C } } \\mathbb { C } \\\\ & = \\chi ( { \\mathcal F } ^ 1 _ x ) - \\chi ( { \\mathcal F } ^ 0 _ x ) + 2 . \\end{align*}"}
-{"id": "5138.png", "formula": "\\begin{align*} & ( - 1 ) ^ { t - 1 } \\prod _ { s = 1 } ^ { t - 1 } g ( z _ { \\ell ( s ) } , z _ { \\ell ( s + 1 ) } ) + \\frac { g ( w , z _ { \\ell ( t ) } ) } { g ( z _ { \\ell ( t ) } , z _ { \\ell ( 1 ) } ) } \\prod _ { s = 1 } ^ { t - 1 } g ( z _ { \\ell ( s + 1 ) } , z _ { \\ell ( s ) } ) \\\\ & = \\frac { g ( w , z _ { \\ell ( t ) } ) } { g ( w , z _ { \\ell ( 1 ) } ) } \\prod _ { s = 1 } ^ { t - 1 } g ( z _ { \\ell ( s + 1 ) } , z _ { \\ell ( s ) } ) . \\end{align*}"}
-{"id": "5474.png", "formula": "\\begin{align*} I ( m ) ^ { ( m j ) } = ( I ( m ) ^ { ( m ) } ) ^ j . \\end{align*}"}
-{"id": "8719.png", "formula": "\\begin{align*} v ( t , x ) = & \\int _ t ^ T R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } G B ( s , \\cdot ) \\right ] ( x ) \\ , d s + \\int _ t ^ T R _ { s - t } \\left [ e ^ { - ( s - t ) { A } } \\nabla ^ G v ( s , \\cdot ) B ( s , \\cdot ) \\right ] ( x ) \\ , d s . \\end{align*}"}
-{"id": "4301.png", "formula": "\\begin{align*} Y _ { h _ L } = \\{ y \\in Y _ 0 ; h _ { y } \\not \\equiv + \\infty \\} . \\end{align*}"}
-{"id": "1822.png", "formula": "\\begin{align*} f _ { k } = \\sum _ { \\alpha \\geq 2 k } s ^ { \\alpha } f _ { k , \\alpha } , \\ f _ { k , \\alpha } \\in \\mathcal { U } ^ { \\alpha } \\end{align*}"}
-{"id": "6215.png", "formula": "\\begin{align*} X ( s ) = \\sum \\limits _ { j = 0 } ^ { \\infty } X ^ { ( j ) } ( s _ { 0 } ) ( s - s _ { 0 } ) ^ { j } , \\end{align*}"}
-{"id": "1897.png", "formula": "\\begin{align*} m ^ \\star = \\arg \\min _ { m > 0 } \\| f - \\hat { f } _ { m } \\| ^ 2 . \\end{align*}"}
-{"id": "9029.png", "formula": "\\begin{align*} \\log \\Phi ^ * _ k ( e ^ { i \\theta } ) = & \\sum _ { j = 0 } ^ { k - 1 } \\log \\left ( 1 - \\alpha _ { j } e ^ { i \\Psi _ { j } ( \\theta ) } \\right ) , \\end{align*}"}
-{"id": "9929.png", "formula": "\\begin{align*} x _ 0 ^ 2 + x _ 1 ^ 2 + x _ 2 ^ 2 + x _ 3 ^ 2 \\ ; = \\ ; 0 \\ ; = \\ ; x _ 3 ^ 2 + ( 1 + b ^ 2 ) x _ 1 ^ 2 + x _ 2 ^ 2 , \\end{align*}"}
-{"id": "5603.png", "formula": "\\begin{align*} \\psi = - \\mathcal G _ 0 v ^ * \\phi + c _ 0 ( 1 , 0 ) ^ T = i \\alpha \\cdot \\nabla G _ 0 v ^ * \\phi - 2 m G _ 0 I _ 1 v ^ * \\phi + c _ 0 ( 1 , 0 ) ^ T , \\end{align*}"}
-{"id": "8716.png", "formula": "\\begin{align*} \\nabla ^ G _ { \\xi } v ^ n ( t , x ) = \\int _ t ^ T \\int _ { H } e ^ { - ( s - t ) { A } } G B ^ n ( s , z + e ^ { ( s - t ) { A } } x ) \\left \\langle Q _ { s - t } ^ { - 1 / 2 } e ^ { ( s - t ) { A } } G \\xi , Q _ { s - t } ^ { - 1 / 2 } z \\right \\rangle \\mu _ { s - t } ( d z ) \\\\ + \\int _ { t } ^ { T } \\int _ { H } e ^ { - ( s - t ) { A } } \\nabla ^ G v ^ n ( s , z + e ^ { ( s - t ) { A } } x ) B ^ n ( s , z + e ^ { ( s - t ) { A } } x ) \\left \\langle Q _ { s - t } ^ { - 1 / 2 } e ^ { ( s - t ) { A } } G \\xi , Q _ { s - t } ^ { - 1 / 2 } z \\right \\rangle \\mu _ { s - t } ( d z ) \\ , d s . \\end{align*}"}
-{"id": "970.png", "formula": "\\begin{align*} d \\alpha | _ { { \\mathcal { N } _ z } } = 0 , \\forall z \\in M \\ ; . \\end{align*}"}
-{"id": "4403.png", "formula": "\\begin{align*} \\bigr [ v , \\partial ( e ) \\bigl ] = \\begin{cases} 1 & v = t ( e ) t ( e ) \\neq s ( e _ ) , \\\\ - 1 & v = s ( e ) t ( e ) \\neq s ( e _ ) , \\\\ 0 & \\end{cases} \\end{align*}"}
-{"id": "8241.png", "formula": "\\begin{align*} i \\dot { \\phi } = \\mathcal { S } \\mathcal { H } '' _ E \\phi + ( E - \\dot { \\alpha } ) \\sigma ( \\Phi + \\phi ) + N ( \\phi ) , \\end{align*}"}
-{"id": "982.png", "formula": "\\begin{align*} H _ { n , m } & = U ( N _ { A } - N _ { B } ) ^ 2 + \\mu ( N _ { A } - N _ { B } ) + t \\left ( A ^ { \\dagger } B + A B ^ \\dagger \\right ) \\\\ & = U ( N _ { A } - N _ { B } ) ^ 2 + \\mu ( N _ { A } - N _ { B } ) + \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ m t _ { i , j } ( a _ { i } b _ { j } ^ \\dagger + a _ { i } ^ \\dagger b _ { j } ) , \\end{align*}"}
-{"id": "6679.png", "formula": "\\begin{align*} F ( x ( t ; \\overline { x } ) ) = \\exp ( - 2 \\lambda t ) \\cdot F ( \\overline { x } ) , ~ ( \\forall ) t \\in [ 0 , \\infty ) . \\end{align*}"}
-{"id": "9165.png", "formula": "\\begin{align*} \\Phi ^ { ( s ) } ( 0 ) = \\Phi _ { u , x } ^ { ( s ) } ( 0 ) = \\delta _ { s , 1 } a ( u ( x ) ) u ' ( x ) , \\forall s = 0 , \\dots , p , \\forall u , x . \\end{align*}"}
-{"id": "1938.png", "formula": "\\begin{align*} g ( A _ X Y , A _ Z W ) = & g ( A _ X Y , \\frac { e _ 0 } { \\sqrt { a } } ) + g ( \\frac { e _ 0 } { \\sqrt { a } } , A _ Z W ) \\\\ = & \\ , \\frac { 1 } { 4 a } \\ , g ( F ( X , Y ) , e _ 0 ) \\cdot g ( F ( Z , W ) , e _ 0 ) \\\\ = & \\ , \\frac { a ( \\tau ) } { 4 } \\left ( \\sum _ i q _ i \\omega _ i ( X , Y ) \\right ) \\cdot \\left ( \\sum _ j q _ j \\omega _ j ( Z , W ) \\right ) , \\end{align*}"}
-{"id": "1678.png", "formula": "\\begin{align*} { \\psi } = { \\psi } _ { \\lambda } = G _ { \\lambda } ( \\mathbb { I } - T _ \\lambda ) ^ { - 1 } g , \\end{align*}"}
-{"id": "9161.png", "formula": "\\begin{align*} \\det A ( a _ 1 , \\dots , a _ n ) = \\prod _ { j = 1 } ^ { n } \\prod _ { k = j + 1 } ^ { n } ( a _ k - a _ j ) \\sum _ { j = 1 } ^ { n } \\prod _ { k = 1 , k \\neq j } ^ { n } a _ k . \\end{align*}"}
-{"id": "8434.png", "formula": "\\begin{align*} A _ { 0 } ( \\textbf { u } ) \\partial _ { t } \\textbf { u } + \\sum _ { j = 1 } ^ { d } A _ { 0 } ( \\textbf { u } ) \\textbf { P } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } = 0 . \\end{align*}"}
-{"id": "2190.png", "formula": "\\begin{align*} & ( 1 - q ) \\zeta _ { 2 } ( q ) \\phi \\int _ { \\rho B _ { 1 } } \\int _ { \\rho B _ { 1 } } ( \\psi ( x ) - \\psi ( y ) ) ^ { 2 } ( w ^ { 2 } ( s , x ) + w ^ { 2 } ( s , y ) ) k ( x , y ) d x d y \\\\ & \\leq c _ { 2 } C ( \\delta , \\Lambda ) ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\phi \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x = c _ { 3 } ( \\rho - \\rho ' ) ^ { - 2 \\beta } \\phi \\int _ { \\rho B _ { 1 } } w ^ { 2 } ( s , x ) d x , \\end{align*}"}
-{"id": "4550.png", "formula": "\\begin{align*} \\alpha _ z = I _ { \\frac { d + 1 } { 2 } , \\frac { 1 } { 2 } } \\biggl ( 1 - \\frac { \\| z \\| ^ 2 } { 4 } \\biggr ) . \\end{align*}"}
-{"id": "5849.png", "formula": "\\begin{align*} a \\left ( t , x \\right ) = - \\frac { 1 } { 2 } \\int \\left ( T _ { 1 } L _ { Y _ { 1 } } C _ { x } + T _ { 2 } L _ { Y _ { 2 } } C _ { x } - T _ { 1 , t } Y _ { 1 } - T _ { 2 , t } Y _ { 2 } \\right ) d x + f \\left ( t \\right ) \\end{align*}"}
-{"id": "1658.png", "formula": "\\begin{align*} \\| f \\| _ { X _ { p , s } } = \\| f \\| _ { W ^ { 1 , p } ( \\R ^ { 2 d } ) } + \\| D ^ 2 _ v f \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ { d } _ x ) ) } + \\| v \\cdot D _ x f \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ p ( \\R ^ { d } _ x ) ) } . \\end{align*}"}
-{"id": "1917.png", "formula": "\\begin{align*} m _ 0 : = \\min \\{ u > 0 : | { \\phi } ( u ) | = ( 2 \\sqrt { \\eta / K } + \\kappa ) ( \\log ( n ) / n ) ^ { 1 / 2 } \\} \\end{align*}"}
-{"id": "2712.png", "formula": "\\begin{align*} d ( s ) = \\frac { 1 } { s } \\left ( \\int _ X \\dot { \\varphi } _ { t + s } d d ^ c ( \\varphi _ { t + s } - \\varphi _ t ) \\wedge T _ s + \\int _ X ( \\dot { \\varphi } _ { t + s } - \\dot { \\varphi } _ t ) \\theta _ { \\varphi _ t } ^ n \\right ) . \\end{align*}"}
-{"id": "2823.png", "formula": "\\begin{align*} s _ { j , v } - s _ { i , u } = s _ j + v q _ j - s _ i - u q _ i = ( j - i ) ( 1 + \\varepsilon ) + v q _ j - u q _ i \\ge ( j - i ) \\varepsilon , \\end{align*}"}
-{"id": "9042.png", "formula": "\\begin{align*} Q ( z ) = \\sum _ { \\omega \\in \\mathbb { U } _ { 2 k } } Q ( \\omega ) R ( z \\bar { \\omega } ) , \\end{align*}"}
-{"id": "5612.png", "formula": "\\begin{align*} \\theta ( z ) = \\Bigl ( \\frac \\pi z \\Bigr ) ^ { 1 / 2 } \\theta \\Bigl ( \\frac { \\pi ^ 2 } z \\Bigr ) \\textrm { f o r } \\ \\Re ( z ) > 0 . \\end{align*}"}
-{"id": "7070.png", "formula": "\\begin{align*} { { \\bf { H } } ^ { [ 1 1 ] } } ( n ) { { \\bf { v } } _ i } = { \\bf { h } } _ i ^ { [ 1 1 ] } ( { t _ 2 } ) , \\end{align*}"}
-{"id": "4442.png", "formula": "\\begin{align*} \\tilde { V } _ n ^ w : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ k w _ j \\log ^ 2 \\xi _ { ( j ) , i } - ( \\hat { H } _ n ^ w ) ^ 2 . \\end{align*}"}
-{"id": "209.png", "formula": "\\begin{align*} \\delta _ n : = k c _ n ^ d \\log ^ 2 ( n - 1 ) / ( n - 1 ) \\end{align*}"}
-{"id": "7139.png", "formula": "\\begin{align*} G _ { i j } = \\tilde G _ { i j } + W _ { i j } , \\end{align*}"}
-{"id": "1381.png", "formula": "\\begin{align*} Q ( \\delta ) = \\sum _ { m = 1 } ^ M \\kappa _ m \\int _ 0 ^ 1 \\sum _ { j = 0 } ^ m \\pi ^ 0 ( j , h ( u ) ) \\log \\pi ( j , h ( u ) ) d u \\end{align*}"}
-{"id": "5400.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } \\leq \\left ( k - 2 \\alpha + 5 \\alpha ^ 2 \\right ) \\frac { n ^ 2 } { 2 } = \\left ( k - \\frac { 3 } { 1 6 } \\right ) \\frac { n ^ 2 } { 2 } \\ , , \\end{align*}"}
-{"id": "2342.png", "formula": "\\begin{align*} x : = \\theta _ 1 \\cdot \\frac { a _ 1 + a _ n } { 2 } + \\theta _ 2 \\cdot \\frac { a _ 2 - a _ 1 } { 2 } + \\dotsb + \\theta _ n \\cdot \\frac { a _ n - a _ { n - 1 } } { 2 } . \\end{align*}"}
-{"id": "2546.png", "formula": "\\begin{align*} \\varphi ( U ^ { n + 1 } ) = & \\varphi ( U ^ n ) + \\tau \\left [ D \\varphi ( U ^ n ) \\Phi ^ n + \\frac 1 2 \\tau D ^ 2 \\varphi ( U ^ n ) ( \\Phi ^ n ) ^ 2 \\right ] + \\frac 1 6 D ^ 3 \\varphi ( U ^ n ) ( \\tau \\Phi ^ n ) ^ 3 + R ^ \\Phi _ n \\\\ = : & \\varphi ( U ^ n ) + \\tau \\mathcal { L } ^ \\Phi \\varphi ( U ^ n ) + \\frac 1 6 D ^ 3 \\varphi ( U ^ n ) ( \\tau \\Phi ^ n ) ^ 3 + R ^ \\Phi _ n , \\end{align*}"}
-{"id": "3533.png", "formula": "\\begin{align*} ( 2 \\psi , V ) = - \\Phi ( g , \\pi ) + \\chi \\Phi ( g _ 1 , \\pi _ 1 ) + ( 1 - \\chi ) \\Phi ( g _ 2 , \\pi _ 2 ) + ( 2 \\psi _ 0 , 0 ) \\end{align*}"}
-{"id": "3563.png", "formula": "\\begin{align*} \\mathcal { E } ^ R _ 2 ( \\theta ) = - \\chi \\Phi ( g ^ R , \\pi ^ R ) - ( 1 - \\chi ) \\Phi ( ( g ^ \\theta ) ^ R , ( \\pi ^ \\theta ) ^ R ) - ( 2 \\psi _ 0 R ^ { - 2 } , \\tfrac { 1 } { 2 } h ^ R \\cdot _ { \\gamma ^ R } ( \\textup { d i v } _ { \\gamma ^ R } \\tau ^ R + V ^ R ) ) . \\end{align*}"}
-{"id": "5329.png", "formula": "\\begin{align*} Q _ { j } | \\psi _ { \\{ l ; k \\} } \\rangle & = l _ j | \\psi _ { \\{ l ; k \\} } \\rangle \\\\ \\overline { Q } _ { j } | \\psi _ { \\{ l ; k \\} } \\rangle & = k _ j | \\psi _ { \\{ l ; k \\} } \\rangle . \\end{align*}"}
-{"id": "3570.png", "formula": "\\begin{align*} D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) = ( \\psi , V ) \\end{align*}"}
-{"id": "3987.png", "formula": "\\begin{align*} v ' _ i : = \\sum _ { \\vect { \\delta } \\in E _ i } q ( \\vect { \\delta } ) ( v ' ) . \\end{align*}"}
-{"id": "2919.png", "formula": "\\begin{align*} s _ \\lambda ( X ) = M ^ \\perp _ \\pi ( X ) s ^ { ( \\pi ) } _ \\lambda ( X ) \\quad \\mbox { a n d } { s _ { \\lambda ' } ( X ) = ( - 1 ) ^ { | \\lambda | } { M } ^ \\perp _ \\pi ( X ) s ^ { * ( \\pi ) } _ { \\lambda } ( X ) } \\ , . \\end{align*}"}
-{"id": "2771.png", "formula": "\\begin{align*} \\begin{cases} \\Delta _ { t } u ( \\gamma _ { t } ) + \\sigma ( t ) \\Delta _ { x x } u ( \\gamma _ { t } ) + f ( \\gamma _ { t } , u ( \\gamma _ { t } ) , - \\Delta _ { x } u ( \\gamma _ { t } ) ) = 0 , \\ \\ \\gamma _ { t } \\in \\Lambda _ { t } , \\ t \\in [ 0 , T ) , \\\\ u ( \\gamma ) = g ( \\gamma ) , \\ \\ \\gamma \\in \\Lambda _ { T } , \\end{cases} \\end{align*}"}
-{"id": "7393.png", "formula": "\\begin{align*} c ( [ \\hat { W } ] , F ) & \\leq c ( [ \\hat { W } ] , L ) + | | \\bar { H } | | _ { C ^ 0 } \\\\ & \\leq ( | | H | | _ { C ^ 0 } + \\max \\{ 0 , \\lambda w \\} ) + | | H | | _ { C ^ 0 } \\\\ & = 2 | | H | | _ { C ^ 0 } + \\max \\{ 0 , \\lambda w \\} . \\end{align*}"}
-{"id": "1640.png", "formula": "\\begin{align*} D _ { t } f + v \\cdot D _ { x } f + F \\cdot D _ { v } f = 0 , f | _ { t = 0 } = f _ { 0 } \\end{align*}"}
-{"id": "4478.png", "formula": "\\begin{align*} R _ 4 = \\int _ { \\mathcal { X } _ n } \\ ! \\ ! f ( x ) \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\biggl \\{ \\log \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) - \\frac { V _ d ^ { - 2 / d } s ^ { 2 / d } \\Delta f ( x ) } { 2 ( d + 2 ) f ( x ) ^ { 1 + 2 / d } } \\biggr \\} \\mathrm { B } _ { k , n - k } ( s ) \\ , d s \\ , d x . \\end{align*}"}
-{"id": "1062.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ c \\binom { v ( B _ i ) } { 2 } \\leq \\left ( k - 2 \\alpha + 5 \\alpha ^ 2 \\right ) \\frac { n ^ 2 } { 2 } \\ , , \\end{align*}"}
-{"id": "8074.png", "formula": "\\begin{gather*} S ( x , y ) = \\begin{cases} ( x + \\eta , y + \\alpha ) & x \\in [ 0 , 1 - \\eta ) , \\\\ ( x + \\eta , 1 - y ) & x \\in [ 1 - \\eta , 1 ) ; \\end{cases} \\\\ R ( x , a ) = \\begin{cases} ( x + \\eta , a ) & x \\in [ 0 , 1 - \\eta ) , \\\\ ( x + \\eta , a + 1 ) & x \\in [ 1 - \\eta , 1 ) . \\end{cases} \\end{gather*}"}
-{"id": "4663.png", "formula": "\\begin{align*} G _ { \\phi } ( x ) = \\sum _ { \\ell , m = 1 } ^ { \\infty } \\frac { \\rho _ { \\phi } ( \\ell m ) } { \\ell ^ { 1 + x } m ^ { 1 + 3 x } } . \\end{align*}"}
-{"id": "1926.png", "formula": "\\begin{align*} \\Omega _ + \\doteqdot \\{ Y \\in \\mathbb { R } _ { \\geq 0 } ^ m : \\ , F _ i ( Y ) \\geq 0 i = 1 , \\cdots , m \\} , \\end{align*}"}
-{"id": "1423.png", "formula": "\\begin{align*} F _ { p } C _ { k } = \\{ A \\in C _ { k } \\mid \\deg A \\geq p \\} \\end{align*}"}
-{"id": "2600.png", "formula": "\\begin{align*} v ( t ) = - M ( t ) P _ { > A T ^ { \\frac 1 2 } t ^ { - 1 } } \\tilde u ( t ) . \\end{align*}"}
-{"id": "1858.png", "formula": "\\begin{align*} C _ 4 ( G ) = \\sum _ { \\{ u , u ' \\} \\in \\binom { U } { 2 } } \\binom { \\deg ( u , u ' ) } { 2 } \\ , . \\end{align*}"}
-{"id": "7204.png", "formula": "\\begin{align*} Z _ { k , n } ^ { x _ 0 } = \\frac { 1 } { 2 d } \\sum _ { i = 1 } ^ { d } e ^ { \\beta \\omega _ { k + 1 , x _ 0 + e _ i } } Z _ { k + 1 , n } ^ { x _ 0 + e _ i } . \\end{align*}"}
-{"id": "7540.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\mathcal { N } _ 0 < \\infty x _ n \\in \\left ( K - \\frac { l } { 1 - \\gamma } - \\varepsilon , K + \\frac { l } { 1 - \\gamma } + \\varepsilon \\right ) , \\ , n \\geq \\mathcal { N } _ 0 \\right ] = 1 . \\end{align*}"}
-{"id": "9970.png", "formula": "\\begin{align*} & \\sum _ { i = 1 } ^ 2 | w _ { t , k _ t i } | ^ 2 p _ { t , i } \\le \\frac { B _ { t , k _ t } + \\alpha _ t T _ f ( E _ { k _ t } - \\tilde { p } _ { t } ) } { ( 1 - \\alpha _ t ) T _ f } + E _ { k _ t } , \\\\ & \\sum _ { i = 1 } ^ 2 | w _ { t , \\bar { k } _ t i } | ^ 2 p _ { t , i } \\le \\frac { B _ { t , \\bar { k } _ t } + \\alpha _ t T _ f E _ { \\bar { k } _ t } } { ( 1 - \\alpha _ t ) T _ f } + E _ { \\bar { k } _ t } . \\end{align*}"}
-{"id": "7908.png", "formula": "\\begin{align*} \\aligned a _ 3 = \\frac { 1 } { 7 ! } \\int _ M \\bigg ( & - \\frac { 7 } { 3 } | \\nabla W | ^ 2 - \\frac { 4 } { 3 } | \\nabla B | ^ 2 - \\frac { 1 5 2 } { 9 } | \\nabla S _ g | ^ 2 \\\\ & + 5 S _ g | W | ^ 2 + \\frac { 5 0 } { 9 } S _ g | B | ^ 2 + \\frac { 1 8 5 } { 5 4 } S _ g ^ 3 \\\\ & + \\frac { 3 8 } { 9 } W ^ { i j k l } W _ { k l } ^ { \\ \\ m n } W _ { m n i j } + 1 2 B ^ { i j } W _ { i } ^ { \\ k l m } W _ { j k l m } \\\\ & + \\frac { 4 0 } { 3 } B ^ { i j } B ^ { k l } W _ { i k j l } + \\frac { 5 6 } { 9 } W ^ { i j k l } W _ { i \\ k } ^ { m \\ n } W _ { j m l n } \\bigg ) d v _ g , \\endaligned \\end{align*}"}
-{"id": "6265.png", "formula": "\\begin{align*} \\nabla _ E Q = \\frac { \\nabla _ E w ( x ) - \\nabla _ E w ( y ) } { \\bar { w } } - \\frac { ( w ( x ) - w ( y ) ) } { \\bar { w } ^ 2 } ( \\nabla _ E \\bar { w } ) , \\end{align*}"}
-{"id": "5284.png", "formula": "\\begin{align*} \\zeta ( s ) = \\sum _ { n \\leq x } { n ^ { - s } } + \\frac { x ^ { 1 - s } } { s - 1 } + O ( x ^ { - \\sigma } ) , \\end{align*}"}
-{"id": "885.png", "formula": "\\begin{align*} \\vert \\int _ { a } ^ { b } f ( x ) d x \\mid = \\int _ { a } ^ { b } \\mid f ( x ) \\mid d x . \\end{align*}"}
-{"id": "3678.png", "formula": "\\begin{align*} \\frac { d Z } { d t } = \\int Z _ { \\zeta } \\frac { \\partial \\zeta } { \\partial t } d A = 0 ~ . \\end{align*}"}
-{"id": "1059.png", "formula": "\\begin{align*} R _ 2 ( P _ n ) = \\left \\lfloor \\frac { 3 n - 2 } { 2 } \\right \\rfloor . \\end{align*}"}
-{"id": "6244.png", "formula": "\\begin{align*} \\tilde { f } ( H ( s ) ) = f ( \\tilde { H } ( s ) ) = \\tilde { \\hat { H } } ( s ) , \\end{align*}"}
-{"id": "3135.png", "formula": "\\begin{align*} L _ 7 ( \\lambda ) = \\begin{bmatrix} - P _ 4 & 0 & \\lambda P _ 4 & 0 \\\\ 0 & 0 & - I _ n & \\lambda I _ n \\\\ \\lambda P _ 4 & I _ n & \\lambda P _ 3 + P _ 2 & \\lambda P _ 2 \\\\ 0 & \\lambda I _ n & \\lambda P _ 2 & \\lambda P _ 1 + P _ 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "5107.png", "formula": "\\begin{align*} \\langle X \\rangle _ { [ M ' , M ] } = { } _ { [ M , M ' ] } \\langle \\mathrm { v a c } | X | \\mathrm { v a c } \\rangle _ { [ M ' , M ] } ( X \\in \\mathcal { B } ^ { [ M ' , M ] } ) . \\end{align*}"}
-{"id": "5951.png", "formula": "\\begin{align*} L _ t = e ^ { t A } z ^ r + W _ A ( t ) , W _ A ( t ) = \\int _ 0 ^ t e ^ { ( t - s ) A } R \\ , \\dd W _ s \\ , , \\end{align*}"}
-{"id": "5453.png", "formula": "\\begin{align*} \\max _ { y \\in [ - H , H ) } \\ \\Bigl \\vert \\sum _ { n = N - H } ^ { N + y } e ^ { - n / N } \\Bigl ( R ( n ) - ( 2 \\psi ( n ) - n ) \\Bigr ) \\Bigl ( 1 - \\frac { \\vert n - N \\vert } { H } \\Bigr ) \\Bigr \\vert \\ll N ( \\log N ) ^ 2 \\log ( 2 H ) \\end{align*}"}
-{"id": "1404.png", "formula": "\\begin{align*} \\sum _ { j \\in I } B _ { i j } ( \\lambda _ j , \\mu _ j , \\nu _ j ) = 0 \\end{align*}"}
-{"id": "1785.png", "formula": "\\begin{align*} f ^ { - 1 } ( \\{ x _ 1 , \\ldots , x _ n \\} ) = \\{ \\varphi ^ i ( y _ 1 ) , \\ldots , \\varphi ^ i ( y _ n ) \\ | \\ i \\in \\Z \\} . \\end{align*}"}
-{"id": "6116.png", "formula": "\\begin{align*} g _ { } = \\frac 3 4 \\sum \\limits _ { i = 1 } ^ k \\left ( \\frac { d s _ i ^ 2 } { s _ i ^ 2 } + s _ i ^ 2 d \\theta _ i ^ 2 \\right ) + h \\end{align*}"}
-{"id": "388.png", "formula": "\\begin{align*} \\langle x , y \\rangle = \\langle g \\cdot x , g ^ \\iota \\cdot y \\rangle . \\end{align*}"}
-{"id": "3655.png", "formula": "\\begin{align*} \\binom { n } { \\deg _ { G [ V _ 1 , V _ 2 ] } ( v ) } \\binom { n } { \\deg _ { G [ V _ 1 , V _ 3 ] } ( v ) } . \\end{align*}"}
-{"id": "8485.png", "formula": "\\begin{align*} \\varepsilon ^ { 2 } \\partial _ { t } P ^ { \\varepsilon } + \\nabla \\cdot v ^ { \\varepsilon } = 0 . \\end{align*}"}
-{"id": "1046.png", "formula": "\\begin{align*} u ( x ) = \\sqrt { \\frac { \\pi } 2 } | x | ^ { - \\frac 1 2 } \\ \\bigl [ e ^ { i | x | + \\frac { i \\pi } { 4 } } \\mathfrak { f } _ u ( { \\textstyle \\frac { x } { | x | } } ) \\bigr ] + o ( | x | ^ { - \\frac 1 2 } ) , | x | \\to \\infty , \\end{align*}"}
-{"id": "5808.png", "formula": "\\begin{align*} b _ { j } = \\sum _ { i = 1 } ^ { j } h _ { i } \\end{align*}"}
-{"id": "7219.png", "formula": "\\begin{align*} - 1 \\leq D _ { v _ { i } } \\mathrm { d i s t } _ { a _ { i } } ^ { X } \\left ( \\cdot \\right ) = - \\cos \\left ( \\sphericalangle \\left ( \\left ( \\Uparrow _ { p } ^ { a _ { i } } \\right ) _ { X } , v _ { i } \\right ) \\right ) \\leq - 1 + \\tau \\left ( \\eta \\right ) . \\label { c l o o o o s e a t p } \\end{align*}"}
-{"id": "8508.png", "formula": "\\begin{align*} 0 = ( \\beta _ k | \\beta _ { k + 1 } ) = ( w \\alpha _ { i _ k } | w s _ { i _ k } \\alpha _ { i _ { k + 1 } } ) = ( \\alpha _ { i _ k } | s _ { i _ k } \\alpha _ { i _ { k + 1 } } ) = ( s _ { i _ k } \\alpha _ { i _ k } | \\alpha _ { i _ { k + 1 } } ) = - ( \\alpha _ { i _ k } | \\alpha _ { i _ { k + 1 } } ) . \\end{align*}"}
-{"id": "3432.png", "formula": "\\begin{align*} y ( t ) = E [ \\zeta + \\int _ t ^ T f ( \\theta , \\xi ( \\theta ) , u ( \\theta ) , v ( \\theta ) ) d \\theta | { \\cal F } _ t ] , 0 \\le t \\le T . \\end{align*}"}
-{"id": "1269.png", "formula": "\\begin{align*} B = \\begin{bmatrix} u _ { 1 1 } J ( T ^ { - i } , T ^ { - r } ) & u _ { 1 2 } J ( T ^ { - i - 2 r } , T ^ { - 3 r } ) & 0 & 0 \\\\ u _ { 2 1 } J ( T ^ { - i } , T ^ { - 3 r } ) & u _ { 2 2 } J ( T ^ { - i - 2 r } , T ^ { - r } ) & 0 & 0 \\\\ 0 & 0 & v _ { 1 1 } J ( T ^ { - i - 3 r } , T ^ { - 3 r } ) & v _ { 1 2 } J ( T ^ { - i - 3 r } , T ^ { - r } ) \\\\ 0 & 0 & v _ { 2 1 } J ( T ^ { - i - r } , T ^ { - r } ) & v _ { 2 2 } J ( T ^ { - i - r } , T ^ { - 3 r } ) , \\end{bmatrix} \\end{align*}"}
-{"id": "8384.png", "formula": "\\begin{align*} \\| X \\| = \\sup \\| R X C \\| \\end{align*}"}
-{"id": "4834.png", "formula": "\\begin{align*} K _ { \\rm A i } ( u , v ) = \\int _ { \\mathcal { C } _ { - 1 } ^ { 2 \\pi / 3 } } \\dd w \\int _ { \\mathcal { C } _ 1 ^ { \\pi / 3 } } \\dd z \\frac { e ^ { z ^ 3 / 3 - z u } } { e ^ { w ^ 3 / 3 - w v } } \\frac { 1 } { z - w } . \\end{align*}"}
-{"id": "9143.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p k _ i \\leq \\frac { ( 2 d + 1 ) t + d ^ 2 } { 2 d + 1 } m . \\end{align*}"}
-{"id": "5245.png", "formula": "\\begin{align*} ( n + 2 ) ^ 3 U _ { n + 2 } - ( 3 4 n ^ 3 + 1 5 3 n ^ 2 + 2 3 1 n + 1 1 7 ) U _ { n + 1 } + ( n + 1 ) ^ 3 U _ n = 0 , \\end{align*}"}
-{"id": "6663.png", "formula": "\\begin{gather*} A _ { k 1 } ^ i : = \\{ \\sigma _ i < + \\infty \\} \\cap \\{ z _ i ( u _ k , \\sigma _ i ) - z _ i ( u _ { k - 1 } , \\sigma _ i ) = \\varepsilon , \\ldots , z _ i ( u _ 3 , \\sigma _ i ) - z _ i ( u _ 2 , \\sigma _ i ) = \\varepsilon , \\\\ z _ i ( u _ 2 , \\sigma _ i ) - z _ i ( u _ 1 , \\sigma _ i ) = \\varepsilon \\} , 2 \\leqslant k \\leqslant n , 1 \\leqslant i \\leqslant n - 1 , \\end{gather*}"}
-{"id": "1373.png", "formula": "\\begin{align*} f _ { Y _ { t } } ( y _ { t } | x _ { n t } , \\alpha _ { t } ; \\theta ) = \\exp \\left \\{ y _ { t } W _ { t } - m _ { t } b ( W _ { t } ) + c ( y _ { t } ) \\right \\} \\end{align*}"}
-{"id": "5418.png", "formula": "\\begin{align*} f ^ { ( 1 ) } ( x , \\omega ) = g ( x ) + \\omega \\pmod { 1 } . \\end{align*}"}
-{"id": "2620.png", "formula": "\\begin{align*} S _ { ( a _ 1 , a _ 2 ) , ( b _ 1 , b _ 2 ) } & = ( a _ 1 b _ 1 + a _ 2 b _ 2 ) 2 ^ { - k + 1 } - ( a _ 1 b _ 2 + a _ 2 b _ 1 ) 2 ^ { - k } \\\\ T _ { ( a , b ) , ( a , b ) } & = ( a ^ 2 + b ^ 2 - a b ) 2 ^ { - k } . \\end{align*}"}
-{"id": "7770.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 2 & & \\\\ & & 2 & 1 \\end{bmatrix} . \\end{align*}"}
-{"id": "485.png", "formula": "\\begin{align*} | u ( x ) | & = \\Big | u ( 0 ) + | x | \\int _ 0 ^ 1 u _ \\xi ( t x ) \\ , d t \\Big | \\\\ & \\le | u ( 0 ) | + | x | \\ , | D u ( 0 ) | + | x | \\int _ 0 ^ 1 \\frac { C } { \\ , 1 - \\alpha \\ , } \\ , | t x | ^ { 1 - \\alpha } \\ , d t \\\\ & \\le | u ( 0 ) | + | x | \\ , | D u ( 0 ) | + \\frac { C } { \\ , 1 - \\alpha \\ , } \\ , \\frac { \\ , | x | ^ { 2 - \\alpha } \\ , } { 2 - \\alpha } . \\end{align*}"}
-{"id": "7554.png", "formula": "\\begin{align*} x _ { n + 1 } ( \\omega ) = f ( x _ n ( \\omega ) ) + l \\chi _ { n + 1 } ( \\omega ) \\le f ^ H + l < H . \\end{align*}"}
-{"id": "4575.png", "formula": "\\begin{align*} g _ J ^ * ( x ) = \\sum _ { i = 1 } ^ { \\mathrm { c a r d } ( J ) } \\frac { ( - 1 ) ^ { i - 1 } ( i - 1 ) ! } { f ^ i } \\sum _ { \\{ P _ 1 , \\ldots , P _ i \\} \\in \\mathcal { P } _ i ( J ) } f _ { P _ 1 } \\ldots f _ { P _ i } . \\end{align*}"}
-{"id": "5149.png", "formula": "\\begin{align*} & \\vec { z } ( \\ell ( t ) , \\ldots , \\ell ( 1 ) ; w ) _ { i } = \\left \\{ \\begin{array} { l l } z _ { i } & ( i \\not = \\ell ( t ) , \\ldots , \\ell ( 1 ) ) \\\\ z _ { \\ell ( s - 1 ) } & ( i = \\ell ( s ) , 2 \\le s \\le t ) \\\\ w & ( i = \\ell ( 1 ) ) . \\end{array} \\right . \\end{align*}"}
-{"id": "8131.png", "formula": "\\begin{align*} \\overline { C } _ { k , l , N } ^ { r a d } : = ( N \\omega _ N ) ^ { ( l - k + 1 ) / ( l + N ) } \\cdot | l + N | ^ { ( k + N - 1 ) / ( l + N ) } . \\end{align*}"}
-{"id": "3216.png", "formula": "\\begin{align*} \\varepsilon _ n ^ { - 1 } : = \\frac { 1 } { K } \\min _ { \\alpha \\in \\mathbb { Z } ^ r , \\ | \\alpha | = n } d ( \\mathbb { I } _ Y ^ { ( 1 ) } , N _ \\alpha ) \\end{align*}"}
-{"id": "107.png", "formula": "\\begin{align*} x ^ n = \\sum _ { l = 0 } ^ n S _ 2 ( n , l ) ( x ) _ l , ( \\textnormal { s e e } \\ , \\ , [ 7 , 8 , 9 ] ) . \\end{align*}"}
-{"id": "750.png", "formula": "\\begin{align*} \\tau _ t ( E ( x ) ) = E ( \\tau _ t ( x ) ) . \\end{align*}"}
-{"id": "4077.png", "formula": "\\begin{align*} z ' = - i z + A z ^ 2 + B z \\bar { z } + C \\bar { z } ^ 2 , \\end{align*}"}
-{"id": "1656.png", "formula": "\\begin{gather*} \\lambda { \\psi } ( z ) - \\frac { 1 } { 2 } \\triangle _ v { \\psi } ( z ) - v \\cdot D _ x { \\psi } ( z ) - F ( z ) \\cdot D _ v { \\psi } ( z ) = g ( z ) \\ , \\\\ = \\lambda { \\psi } ( z ) - \\frac { 1 } { 2 } \\mathrm { T r } \\big ( Q D ^ 2 { \\psi } ( z ) \\big ) - \\langle A z , D { \\psi } ( z ) \\rangle - \\langle B ( z ) , D { \\psi } ( z ) \\rangle \\end{gather*}"}
-{"id": "2088.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + ( K + Z ) ) X ^ { ( 0 ) } ( s _ { i } ) & = F \\ \\ i = 1 , \\ldots , l . \\end{align*}"}
-{"id": "3669.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\int [ h v ^ 2 + g h ^ 2 ] d A ~ , \\end{align*}"}
-{"id": "1690.png", "formula": "\\begin{align*} U ( Z _ t ) & = U ( z ) + \\int _ 0 ^ t D U ( Z _ s ) { R \\ , } \\ , \\dd W _ s + \\int _ 0 ^ t { \\mathcal L } U ( Z _ s ) \\ , \\dd s \\\\ & = U ( z ) + \\int _ 0 ^ t D U ( Z _ s ) { R \\ , } \\dd W _ s + \\lambda \\int _ 0 ^ t U ( Z _ s ) \\ , \\dd s - \\int _ 0 ^ t B ( Z _ s ) \\ , \\dd s \\ , . \\end{align*}"}
-{"id": "5771.png", "formula": "\\begin{align*} T ( \\lambda ) = T ( \\lambda | 1 ) T ( \\lambda | 2 ) \\cdots T ( \\lambda | K ) . \\end{align*}"}
-{"id": "3464.png", "formula": "\\begin{align*} \\Omega : = \\{ x \\in \\Sigma \\times \\mathbb { R } \\ ; \\vert \\ ; - b < x _ 3 < 0 \\} , \\end{align*}"}
-{"id": "630.png", "formula": "\\begin{align*} - \\Delta v _ n - \\lambda _ n c _ n ^ 2 v _ n = Q _ n v _ n ^ { p - 1 } + t _ n M _ n ^ { 1 - p } Q _ n \\end{align*}"}
-{"id": "7928.png", "formula": "\\begin{align*} \\gamma ( Q _ { m \\times n } ) = | D | \\geq ( m + n - 2 + | R | ) / 4 , \\end{align*}"}
-{"id": "3719.png", "formula": "\\begin{align*} \\gamma ( u , x + y ) = \\psi ( 2 ^ { - 1 } ( x u + y u ) ( x + y ) ^ * ) \\\\ = \\gamma ( ( u , x ) ) \\gamma ( ( u , y ) ) \\psi ( 2 ^ { - 1 } ( x u y ^ * + y u x ^ * ) ) \\\\ \\end{align*}"}
-{"id": "6395.png", "formula": "\\begin{align*} \\mu _ { ( k ) } = C ( \\vec { \\varepsilon } ) \\int _ { - \\infty } ^ { \\infty } \\frac { e ^ { - \\frac { x ^ { 2 } } { 2 \\sigma ^ { 2 } } } } { \\sqrt { 2 \\pi } \\sigma } x ^ { k } \\sum _ { n = 0 } ^ { \\infty } \\frac { 1 } { n ! } \\left ( \\varepsilon _ { q } x ^ { q } - \\varepsilon _ { p } x ^ { p } \\right ) ^ { n } d x . \\end{align*}"}
-{"id": "8786.png", "formula": "\\begin{align*} K _ \\lambda ( x _ 1 , \\ldots , x _ n ; q , t ) = \\frac { \\prod _ { b \\in \\mu } 1 - q ^ { l ( b ) + 1 } t ^ { a ( b ) + 1 } } { \\prod _ { b \\in \\delta } 1 - q ^ { l ( b ) } t ^ { a ( b ) + 1 } } E _ { 0 ^ n \\mu } ( x _ 1 , \\ldots , x _ n ; q , t ) , \\end{align*}"}
-{"id": "2098.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { s } u ( x ) : = c _ { n , s } \\ , { \\rm P V } \\int _ { \\R ^ n } \\frac { u ( x ) - u ( x + z ) } { | z | ^ { n + 2 s } } \\ , \\d z . \\end{align*}"}
-{"id": "7134.png", "formula": "\\begin{align*} \\begin{array} { l l } \\Theta _ { j , i } ( t ) & = \\big [ 2 R ^ 2 z _ j ( t ) \\bar { z } _ i ( t ) + 2 R ^ 2 z _ i ( t ) \\bar { z } _ j ( t ) - ( \\vert z _ j ( t ) \\vert ^ 2 + R ^ 2 ) ( \\vert z _ i ( t ) \\vert ^ 2 + R ^ 2 ) \\big ] ^ 2 \\\\ & - \\big [ R ^ 2 - \\vert z _ j ( t ) \\vert ^ 2 \\big ] ^ 2 \\big [ R ^ 2 - \\vert z _ i ( t ) \\vert ^ 2 \\big ] ^ 2 . \\end{array} \\end{align*}"}
-{"id": "9981.png", "formula": "\\begin{align*} \\tilde { p } _ { \\mathrm { m i n } } & = \\max \\Big \\{ 0 , \\frac { C _ { 2 } } { \\alpha | w _ { \\bar { k } 1 } | ^ 2 } \\Big \\} , \\\\ \\tilde { p } _ { \\mathrm { m a x } } & = \\min \\Big \\{ \\frac { B _ k } { \\alpha T _ f } + E _ k , \\frac { C _ { 1 } } { \\alpha | w _ { \\bar { k } 2 } | ^ 2 } \\Big \\} , \\end{align*}"}
-{"id": "3021.png", "formula": "\\begin{align*} d _ \\Sigma ^ { { X _ L } } ( 2 , 2 ; \\frac { 2 } { { { T _ { A B } } } } ) = \\frac { { 4 A n + \\min ( 2 A , B ) { T _ { A B } } ( { T _ { A B } } - 1 ) } } { { { T _ { A B } } n + { T _ { A B } } ( { T _ { A B } } - 1 ) } } . \\end{align*}"}
-{"id": "6862.png", "formula": "\\begin{align*} e ^ { - i t _ n \\Delta } ( { u } _ n ^ J ( t _ n ) - u _ { n } ( t _ n ) ) & = \\sum _ { j = 1 } ^ J e ^ { - i t _ n \\Delta } v _ n ^ j ( t _ n ) - e ^ { i x \\xi _ n ^ j } \\psi ^ j _ { \\{ h _ n ^ j \\} } \\\\ & = \\sum _ { j = 1 } ^ J e ^ { i x \\xi _ n ^ j } \\bigl [ ( e ^ { - i ( h _ n ^ j ) ^ 2 t _ n \\Delta } \\Psi ^ j ( ( h _ n ^ j ) ^ 2 t _ n ) ) _ { \\{ h _ n ^ j \\} } - \\psi ^ j _ { \\{ h _ n ^ j \\} } \\bigr ] , \\end{align*}"}
-{"id": "218.png", "formula": "\\begin{align*} | R _ 4 | & \\leq \\biggl \\{ \\log \\Bigl ( \\frac { n - 1 } { e ^ { \\Psi ( k ) } } \\Bigr ) \\ ! + \\ ! \\int _ { \\mathcal { X } _ n } \\ ! \\ ! \\ ! f ( x ) \\biggl ( | \\log f ( x ) | + \\frac { a ( f ( x ) ) } { f ( x ) ^ { \\frac { 2 } { d } } V _ d ^ { \\frac { 2 } { d } } } \\biggr ) \\ , d x \\biggr \\} \\mathbb { P } \\Bigl ( \\mathrm { B } _ 1 \\geq \\frac { a _ n } { n - 1 } \\Bigr ) \\\\ & = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "3174.png", "formula": "\\begin{align*} \\rho _ l = \\frac { \\mathbb { E } \\left \\{ \\mathcal { I } ( t ) \\ , \\mathcal { I } ( \\tau ) \\right \\} - \\mathbb { E } \\left \\{ \\mathcal { I } ( t ) \\right \\} ^ 2 } { \\mathbb { E } \\left \\{ \\mathcal { I } ^ { \\ , 2 } ( t ) \\right \\} - \\mathbb { E } \\left \\{ \\mathcal { I } ( t ) \\right \\} ^ 2 } . \\end{align*}"}
-{"id": "5293.png", "formula": "\\begin{align*} R _ 1 = P ^ { ( 1 ) } _ 1 \\cup P ^ { ( 1 ) } _ 2 \\cup \\cdots P ^ { ( 1 ) } _ r , \\ R _ 2 = P ^ { ( 2 ) } _ 1 \\cup P ^ { ( 2 ) } _ 2 \\cup \\cdots P ^ { ( 2 ) } _ r \\end{align*}"}
-{"id": "764.png", "formula": "\\begin{gather*} r _ { \\ell } \\cdot r _ 1 = r _ { \\ell 1 } + r _ { \\ell + 1 } + 1 . \\end{gather*}"}
-{"id": "6079.png", "formula": "\\begin{align*} t ( t + 1 ) \\iint _ R \\{ v \\} v ^ { - 3 / 2 } \\{ u \\} u ^ { - 3 / 2 } \\left ( 1 + i \\log \\frac { u } { v } \\right ) ^ { - t - 2 } d u d v = \\sum _ { ( m , n ) \\in R ( t ) } \\left ( \\widetilde { F } _ { m , n } ( t ) - \\widetilde { G } _ { m , n } ( t ) \\right ) + O ( 1 ) , \\end{align*}"}
-{"id": "3300.png", "formula": "\\begin{gather*} V _ 1 = v _ 1 \\otimes v _ 0 - q ^ 2 v _ 0 \\otimes v _ 1 , V _ { - 1 } = v _ 0 \\otimes v _ { - 1 } - q ^ 2 v _ { - 1 } \\otimes v _ 0 , \\\\ V _ 0 = v _ 1 \\otimes v _ { - 1 } - v _ { - 1 } \\otimes v _ 1 - q ( q - q ^ { - 1 } ) v _ 0 \\otimes v _ 0 . \\end{gather*}"}
-{"id": "2636.png", "formula": "\\begin{align*} F _ m ( \\lambda ) = \\varphi _ m ( \\lambda ) ( \\varphi _ m ( \\lambda ) ) ^ * \\in D _ { F } , \\end{align*}"}
-{"id": "7186.png", "formula": "\\begin{align*} \\int _ 0 ^ { \\frac { \\pi } { 2 } } H ( \\varphi ) \\sin \\varphi d \\varphi = h ( \\frac { \\pi } { 2 } ) . \\end{align*}"}
-{"id": "6501.png", "formula": "\\begin{align*} ( g _ { \\alpha } * ( \\varphi \\dot { v } ) ) ( t ) = \\varphi ( t ) ( g _ { \\alpha } * \\dot { v } ) ( t ) + \\int _ { 0 } ^ { t } v ( \\sigma ) \\partial _ { \\sigma } ( g _ { \\alpha } ( t - \\sigma ) [ \\varphi ( t ) - \\varphi ( \\sigma ) ] ) d \\sigma , \\end{align*}"}
-{"id": "60.png", "formula": "\\begin{align*} \\psi ( y ) = \\psi ( y , a ) : = \\frac { 1 } { a ^ p } | f ' ( F ( a y ) ) | ^ p \\ , y \\in [ 0 , 1 ) \\ . \\end{align*}"}
-{"id": "2905.png", "formula": "\\begin{align*} s ^ O _ \\lambda ( X ) = { s _ \\lambda ^ { ( 2 ) } ( X ) } = s _ \\lambda / L _ { ( 2 ) } ( X ) \\quad \\mbox { a n d } s ^ { S p } _ \\lambda ( X ) = { s _ \\lambda ^ { ( 1 ^ 2 ) } ( X ) } = s _ \\lambda / L _ { ( 1 ^ 2 ) } ( X ) \\ , . \\end{align*}"}
-{"id": "2011.png", "formula": "\\begin{align*} \\mathcal { A } _ n f ( x ) = \\gamma _ U - \\frac { \\sigma _ U ^ 2 } { 2 } + \\int _ { \\R } [ \\log | 1 + z | - z I ( | z | \\leq 1 ) ] \\nu _ U ( \\d z ) + \\int _ { - 1 } ^ 1 z \\nu ' ( \\d z ) + o ( 1 ) , \\end{align*}"}
-{"id": "4000.png", "formula": "\\begin{align*} \\alpha _ i \\not = 0 \\ ; \\Rightarrow \\ ; q ^ { t - a } \\mbox { d i v i d e s $ i $ } . \\end{align*}"}
-{"id": "2904.png", "formula": "\\begin{align*} \\langle \\ , f / g \\ , \\ , | \\ , h \\rangle = \\langle f \\ , | \\ , g \\ ! \\cdot \\ ! h \\ , \\rangle \\ , , \\end{align*}"}
-{"id": "840.png", "formula": "\\begin{align*} & q ^ { m + 1 } \\ , u ( a ^ { m } , b ) + \\sum _ { \\ell = 2 } ^ { m + 1 } g ( w , z _ { \\ell } ) Z _ { \\ell } ^ { [ 1 , m + 1 ] } ( \\vec { z } ) u ( b , a ^ { m + 1 } ) \\\\ & + \\left \\{ g ( w , z _ { 1 } ) \\prod _ { i = 2 } ^ { m + 1 } f ( z _ { i } , w ) - \\sum _ { \\ell = 2 } ^ { m + 1 } g ( w , z _ { \\ell } ) g ( z _ { \\ell } , z _ { 1 } ) \\prod _ { \\begin{subarray} { c } i = 2 \\\\ i \\not = \\ell \\end{subarray} } ^ { m + 1 } f ( z _ { i } , z _ { \\ell } ) \\right \\} u ( b , a ^ { m + 1 } ) . \\end{align*}"}
-{"id": "2701.png", "formula": "\\begin{align*} \\theta _ { \\psi } ^ n = e ^ { \\psi - u } \\theta _ { u } ^ n + e ^ { \\psi - v } \\theta _ v ^ n . \\end{align*}"}
-{"id": "726.png", "formula": "\\begin{align*} a _ { k } = \\frac { B ( f ^ { k } ( P ) ) } { \\sqrt [ ] { \\delta _ { f } + \\epsilon } ^ { k } } . \\end{align*}"}
-{"id": "437.png", "formula": "\\begin{align*} y & = y ' , \\\\ \\mu _ k ( z ) & = \\mu _ k ( z ' ) . \\end{align*}"}
-{"id": "173.png", "formula": "\\begin{align*} \\hat { H } _ n = \\hat { H } _ n ( X _ 1 , \\ldots , X _ n ) : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\log \\biggl ( \\frac { \\rho _ { ( k ) , i } ^ d V _ d ( n - 1 ) } { e ^ { \\Psi ( k ) } } \\biggr ) , \\end{align*}"}
-{"id": "9251.png", "formula": "\\begin{align*} d P ( \\omega ) = K _ t ( \\omega ) d \\tilde { P } ( \\omega ) , \\end{align*}"}
-{"id": "8751.png", "formula": "\\begin{align*} \\check { R } ( x ) = \\begin{pmatrix} 1 & 0 & 0 & 0 \\\\ 0 & c ^ - & b ^ + & 0 \\\\ 0 & b ^ - & c ^ + & 0 \\\\ 0 & 0 & 0 & 1 \\end{pmatrix} . \\end{align*}"}
-{"id": "1675.png", "formula": "\\begin{align*} \\lambda { \\psi } ( z ) - \\frac { 1 } { 2 } \\triangle _ v { \\psi } ( z ) - v \\cdot D _ x { \\psi } ( z ) - F ( z ) \\cdot D _ v { \\psi } ( z ) = g ( z ) , \\ ; \\ ; \\ ; z = ( x , v ) \\in \\R ^ { 2 d } , \\end{align*}"}
-{"id": "918.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 2 / 3 } } = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { * } \\| V \\| _ { F } \\\\ = & \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\ ! \\left ( \\frac { 2 \\| U \\| _ { * } + \\| V \\| ^ { 2 } _ { F } } { 3 } \\right ) ^ { 3 / 2 } . \\end{align*}"}
-{"id": "3956.png", "formula": "\\begin{align*} v = u ^ { - } ( - r ^ { - 1 } ) v ^ { \\max } ( u ( r ) v ) ^ { \\max } = \\sigma ( r ) v ^ { \\max } \\sigma ( r ) = \\begin{bmatrix} 0 & r \\\\ - r ^ { - 1 } & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "8787.png", "formula": "\\begin{align*} T _ i f _ { \\ldots , \\lambda _ i , \\lambda _ { i + 1 } , \\ldots } & = T _ i ^ 2 f _ { \\ldots , \\lambda _ { i + 1 } , \\lambda _ { i } , \\ldots } = \\left ( t + ( t - 1 ) T _ i \\right ) f _ { \\ldots , \\lambda _ { i + 1 } , \\lambda _ { i } , \\ldots } \\\\ & = t f _ { \\ldots , \\lambda _ { i + 1 } , \\lambda _ { i } , \\ldots } + ( t - 1 ) f _ { \\ldots , \\lambda _ { i } , \\lambda _ { i + 1 } , \\ldots } . \\end{align*}"}
-{"id": "1029.png", "formula": "\\begin{align*} P ( x , y ) ~ = ~ \\sum _ { k = 0 } ^ d \\sum _ { j = 0 } ^ k c _ { k , j } { \\ , } x ^ j y ^ { k - j } \\end{align*}"}
-{"id": "3310.png", "formula": "\\begin{align*} ( v , X v ^ \\prime ) = ( X ^ * v , v ^ \\prime ) , v , v ^ \\prime \\in \\Lambda _ q ( \\mathfrak { u } _ + ) , \\ X \\in U _ q ( \\mathfrak { l } ) . \\end{align*}"}
-{"id": "3889.png", "formula": "\\begin{align*} \\begin{cases} \\sigma _ k ( u _ f ) = 0 , & , \\\\ u _ f = f , & . \\end{cases} \\end{align*}"}
-{"id": "7970.png", "formula": "\\begin{align*} \\varphi \\left ( \\frac { \\rho _ { K _ 1 } ( u ) } { \\rho _ { \\widetilde { + } _ { \\varphi } ( K _ 1 , \\dots , K _ m ) } ( u ) } , \\dots , \\frac { \\rho _ { K _ m } ( u ) } { \\rho _ { \\widetilde { + } _ { \\varphi } ( K _ 1 , \\dots , K _ m ) } ( u ) } \\right ) = 1 . \\end{align*}"}
-{"id": "4833.png", "formula": "\\begin{align*} { \\mathcal N } = \\left \\{ \\left ( \\sum _ { 1 \\le j \\le n } [ T _ j , y _ j ] , ( y _ j ) _ { 1 \\le j \\le n } \\right ) \\in { \\mathcal C } _ 1 \\times ( { \\mathcal I } ^ * ) ^ n \\mid \\sum _ { 1 \\le j \\le n } [ T _ j , y _ j ] \\in { \\mathcal C } _ 1 \\right \\} , \\end{align*}"}
-{"id": "2049.png", "formula": "\\begin{align*} = \\frac { \\nabla ^ 2 _ { E , E } w ( x ) - \\nabla ^ 2 _ { E , E } w ( y ) } { \\bar { w } } - \\frac { 2 } { \\bar { w } } ( \\nabla _ E Q ) ( \\nabla _ E \\bar { w } ) - \\frac { Q } { \\bar { w } } \\nabla ^ 2 _ { E , E } \\bar { w } . \\end{align*}"}
-{"id": "418.png", "formula": "\\begin{align*} | I | \\lesssim \\sum _ { n = 1 } ^ N \\gamma ^ { - n \\alpha } \\gamma ^ n h = h \\sum _ { n = 1 } ^ N \\gamma ^ { n ( 1 - \\alpha ) } = h \\frac { \\gamma ^ { 1 - \\alpha } - \\gamma ^ { N ( 1 - \\alpha ) } } { 1 - \\gamma ^ { 1 - \\alpha } } = h \\cdot O ( h ^ { \\alpha - 1 } ) = O ( h ^ \\alpha ) . \\end{align*}"}
-{"id": "9421.png", "formula": "\\begin{align*} w ( x ) = \\frac { \\mu ( \\Delta x ) } { \\ell ( \\Delta x ) } \\end{align*}"}
-{"id": "6997.png", "formula": "\\begin{align*} \\delta _ v \\Theta ( L ) ( x , y , z ) ( a ) - \\delta _ L \\Theta ^ 2 ( x , y , z ) ( a ) = 0 . \\end{align*}"}
-{"id": "3942.png", "formula": "\\begin{align*} g _ k g _ { j + 1 } ^ \\dagger = \\sum _ { t = 0 } ^ { r - 1 } \\left ( \\beta _ t ^ { k - 1 + q j } \\sum _ { m = 0 } ^ { q - 2 } \\omega ^ { m ( k - 1 + q j ) } \\right ) = \\sum _ { t = 0 } ^ { r - 1 } \\left ( \\beta _ t ^ { k - 1 + q j } \\cdot 0 \\right ) = 0 , \\end{align*}"}
-{"id": "538.png", "formula": "\\begin{align*} \\sum _ { i = 0 } ^ k b _ { k + i } ^ k = \\frac { ( - 1 ) ^ k } { k ! } a _ k . \\end{align*}"}
-{"id": "9304.png", "formula": "\\begin{align*} \\begin{cases} d p ( t , x , z ) = - [ A _ { \\hat { \\pi } ( t , z ) } p ( t , x , z ) + x q ( t , x , z ) ] d t + q ( t , x , z ) d R ( t ) ; 0 \\leq t \\leq T , \\\\ p ( T , x , z ) = U ( x ) \\mathbb { E } _ { \\tilde { P } } [ \\delta _ { Z } ( z ) | \\mathcal { R } _ T ] , \\end{cases} \\end{align*}"}
-{"id": "3918.png", "formula": "\\begin{align*} ( 1 - \\eta ^ 2 ) \\ , \\frac { d ^ 2 } { d \\eta ^ 2 } w ( \\eta ) - 2 ( m + 1 ) \\ , \\eta \\ , \\frac { d } { d \\eta } w ( \\eta ) + \\left ( z - c ^ 2 \\ , \\eta ^ 2 \\right ) w ( \\eta ) = 0 \\ , , \\end{align*}"}
-{"id": "9651.png", "formula": "\\begin{align*} x ' ( t ) = - a g ( x ( t ) ) + b g ( x ( t - \\tau ( t ) ) ) , t > 0 ; x ( t ) = \\psi ( t ) , t \\leq 0 \\end{align*}"}
-{"id": "420.png", "formula": "\\begin{align*} [ c _ \\omega ] . x _ { c , d } = - \\frac { \\omega _ \\gamma \\circ C ( c \\cdot d ) } { \\log ( \\gamma ) } . \\end{align*}"}
-{"id": "7612.png", "formula": "\\begin{align*} \\lambda _ 1 ^ 2 = \\sum _ { x \\sim y } \\sum _ { \\substack { y \\sim z \\\\ z \\not = x } } \\mathbf { v } _ z + \\sum _ { x \\sim y } 1 \\leq 2 ( m - d _ x ) + d _ x \\leq 2 m - \\lambda _ 1 , \\end{align*}"}
-{"id": "4338.png", "formula": "\\begin{align*} & \\psi ' ( y ) + \\frac { p } { p - 1 } \\psi ( y ) ^ { ( p - 1 ) / p } ( y ) = \\frac { p } { p - 1 } a ^ { 2 - p } ( 1 - y ) \\ , y \\in ( 0 , 1 ) \\ , \\\\ & \\psi ( 0 ) = 0 \\ . \\end{align*}"}
-{"id": "9066.png", "formula": "\\begin{align*} \\sup _ { N \\geq 2 r } N ^ { \\frac { 3 } { 2 } } \\sum _ { j _ 0 = r } ^ { N } \\P \\left ( G _ 0 v ( d , \\delta , 0 ) , \\ , d W _ { j _ 0 - r } \\geq u _ { j _ 0 } ^ { ( N ) } \\right ) \\ll _ { r , d } \\delta ( 1 + \\upsilon ) ^ 4 \\ , \\\\ \\sup _ { N \\geq 2 r } N ^ { \\frac { 3 } { 2 } } \\sum _ { j _ 0 = r } ^ { N } \\P \\left ( G _ 0 v ( d , \\delta , 0 ) , \\ , d W _ { j _ 0 - r } \\leq l _ { j _ 0 } ^ { ( N ) } \\right ) \\ll _ { r , d } \\delta ( 1 + \\upsilon ) ^ 4 \\ , \\end{align*}"}
-{"id": "6294.png", "formula": "\\begin{align*} \\Lambda _ k ^ { \\alpha , \\ast } = \\{ l \\in \\mathbb { Z } ^ n : ~ \\Box _ l ^ { \\alpha } \\circ \\Box _ m ^ { \\alpha } \\neq 0 ~ ~ ~ m \\in \\Lambda _ k ^ { \\alpha } \\} . \\end{align*}"}
-{"id": "3500.png", "formula": "\\begin{align*} \\Phi ^ W _ { ( g , \\pi ) } ( \\gamma , \\tau ) = \\Phi ( \\gamma , \\tau ) + \\left ( 0 , \\tfrac { 1 } { 2 } \\gamma \\cdot _ g \\left ( J + W \\right ) \\right ) , \\end{align*}"}
-{"id": "4183.png", "formula": "\\begin{align*} \\int _ A e ^ { - d ( a , b ) } \\ d \\mu ( b ) = 1 \\end{align*}"}
-{"id": "9672.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { y ( t ) } { G ^ { - 1 } ( t ) } = ( a - b ) ^ { - 1 / ( \\beta - 1 ) } . \\end{align*}"}
-{"id": "5244.png", "formula": "\\begin{align*} g ( z ) = \\ell _ 1 y _ { 1 , 1 } ( z ) + \\ldots + \\ell _ \\mu y _ { 1 , \\mu } ( z ) \\end{align*}"}
-{"id": "3023.png", "formula": "\\begin{align*} { x ^ { [ 2 ] } } ( { t _ 2 } ) = \\frac { { { h ^ { [ 2 2 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 2 2 ] } } ( { t _ 2 } - 2 ) } } u _ 1 ^ { [ 2 ] } , { x ^ { [ 3 ] } } ( { t _ 2 } ) = \\frac { { { h ^ { [ 2 3 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 2 3 ] } } ( { t _ 2 } - 2 ) } } u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "7817.png", "formula": "\\begin{align*} 0 \\leq \\int _ { M \\backslash D \\left ( t _ { 0 } \\right ) } \\left ( \\Delta _ { f } S \\right ) e ^ { - f } = - \\int _ { \\Sigma \\left ( t _ { 0 } \\right ) } \\frac { \\left \\langle \\nabla S , \\nabla f \\right \\rangle } { \\left \\vert \\nabla f \\right \\vert } e ^ { - f } \\leq 0 , \\end{align*}"}
-{"id": "2339.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\bigl ( \\| S _ i + T \\| + \\| S _ i - T \\| \\bigr ) > ( \\delta - 5 \\varepsilon ) \\| T \\| + 2 . \\end{align*}"}
-{"id": "6261.png", "formula": "\\begin{align*} \\nabla _ r \\nabla _ r T | _ { r = 0 } = - \\| J ' ( s ) \\| ^ 2 e _ n . \\end{align*}"}
-{"id": "7344.png", "formula": "\\begin{align*} \\langle w , \\gamma _ { - } ( z ) v \\rangle _ { k } = \\langle w \\wedge z , v \\rangle _ { k + 1 } , z \\in \\mathfrak { u } _ { - } , \\ w \\in \\Lambda _ { q } ^ { k } ( \\mathfrak { u } _ { - } ) , \\ v \\in \\Lambda _ { q } ^ { k + 1 } ( \\mathfrak { u } _ { + } ) . \\end{align*}"}
-{"id": "5617.png", "formula": "\\begin{align*} \\omega _ { \\ell } ( \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } e ^ { - m ^ { \\ell } / N } e ( m ^ { \\ell } \\alpha ) = \\sum _ { m = 1 } ^ { \\infty } e ^ { - m ^ { \\ell } z } . \\end{align*}"}
-{"id": "5020.png", "formula": "\\begin{align*} \\sum _ { s } p ( s ) \\nu ( s ^ { - 1 } \\cdot f ) = \\nu ( f ) , \\textrm { f o r a l l $ f \\in C ( \\overline { Y } ) $ } , \\end{align*}"}
-{"id": "8031.png", "formula": "\\begin{align*} H ^ i ( U , M ) ^ \\vee & \\simeq H ^ { 2 - i } _ c ( U , M ^ \\vee ) \\\\ H ^ 1 _ p ( U , M ) ^ \\vee & \\simeq H ^ 1 _ p ( U , M ^ \\vee ) \\end{align*}"}
-{"id": "269.png", "formula": "\\begin{align*} T _ { 1 3 } : = \\biggl | \\int _ { \\mathcal { X } _ n } \\int _ { B _ x ^ c \\bigl ( \\frac { r _ { n , 1 } d _ n } { f ( x ) ^ { 1 / d } } \\bigr ) } f ( x ) & f ( y ) \\log f ( y ) \\\\ & \\int _ { \\tilde { u } _ { n , x , y } } ^ \\infty \\log \\bigl ( u f ( x ) \\bigr ) \\ , d ( \\tilde { F } _ { n , x } - F _ { n , x } ^ - ) ( u ) \\ , d y \\ , d x \\biggr | . \\end{align*}"}
-{"id": "2454.png", "formula": "\\begin{align*} \\left \\| \\Box _ k ^ { \\alpha _ 1 } | ~ M _ 1 \\rightarrow M _ 2 \\right \\| \\gtrsim \\lim _ { N \\rightarrow \\infty } \\frac { \\| \\Box _ k ^ { \\alpha _ 1 } G _ { k , N } \\| _ { M _ 2 } } { \\| G _ { k , N } \\| _ { M _ 1 } } \\sim 2 ^ { j \\widetilde { A _ 3 } ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "5599.png", "formula": "\\begin{align*} 0 = M _ { 1 1 } v ^ * U v ( 1 , 0 ) ^ T = M _ { 1 1 } V ( 1 , 0 ) ^ T = \\left ( \\int _ { \\R ^ 2 } V _ { 1 1 } ( y ) \\ , d y \\right ) ( 1 , 0 ) ^ T , \\end{align*}"}
-{"id": "9592.png", "formula": "\\begin{align*} \\frac { 1 } { 4 \\pi } ( \\dot \\xi ( t ) - m \\xi ( t ) ) + \\xi ( t ) - \\lambda ( t ) + \\frac { m } { 4 \\pi } \\int _ 0 ^ t \\frac { J _ 1 ( m ( t - s ) ) } { t - s } \\xi ( s ) d s = 0 , \\xi ( 0 ) = \\xi _ { 0 } , t \\ge 0 , \\end{align*}"}
-{"id": "8993.png", "formula": "\\begin{align*} ( ~ ^ { A B C } ~ _ { 0 } D ^ \\alpha f ) ( t ) = ( ~ ^ { A B R } ~ _ { 0 } D ^ \\alpha f ) ( t ) - \\frac { B ( \\alpha ) } { 1 - \\alpha } f ( 0 ) E _ \\alpha ( - \\frac { \\alpha } { 1 - \\alpha } t ^ \\alpha ) \\end{align*}"}
-{"id": "4582.png", "formula": "\\begin{align*} \\biggl | \\frac { \\partial ^ { \\omega } f _ { t , g ^ * } ( x ) } { \\partial x ^ { \\omega } } \\biggr | & = \\biggl | 2 c ( t ) \\sum _ { \\nu : \\nu \\leq \\omega } \\binom { \\omega } { \\nu } \\frac { \\partial ^ \\nu q \\bigl ( g ^ * ( x ) \\bigr ) } { \\partial x ^ \\nu } \\frac { \\partial ^ { \\omega - \\nu } f ( x ) } { \\partial x ^ { \\omega - \\nu } } \\biggr | \\\\ & \\leq 2 ^ { 3 m - 1 } m ^ { m + 1 } B _ m D _ { g ^ * } ^ m a ( f ( x ) ) ^ { m ^ 2 + m } f _ { t , g ^ * } ( x ) . \\end{align*}"}
-{"id": "4154.png", "formula": "\\begin{align*} \\sum _ { n \\leq X } \\lvert S _ f ^ \\nu ( n ) \\rvert ^ 2 = c X ^ { 2 \\kappa ( f ) + \\frac { 3 } { 2 } - 2 \\nu } + O ( X ^ { 2 \\kappa ( f ) + 1 - 2 \\nu } \\log ^ 2 X ) \\end{align*}"}
-{"id": "4392.png", "formula": "\\begin{align*} A _ i = C _ { m _ i ( 0 ) } \\cap \\tau ^ { - 1 } C _ { m _ i ( 1 ) } \\cap \\dots \\cap \\tau ^ { - 2 ^ { k _ i } } C _ { m _ i ( 2 ^ { k _ i } ) } , \\end{align*}"}
-{"id": "1872.png", "formula": "\\begin{align*} R H S > & \\frac { \\eta R ^ 2 } { 1 2 } ( 3 \\eta ^ 2 + 1 ) - \\frac 1 8 ( 1 - \\eta ) ^ 3 R ^ 2 \\\\ = & \\frac { R ^ 2 } { 2 4 } \\Big [ 2 \\eta \\cdot ( 3 \\eta ^ 2 + 1 ) - 3 ( 1 - \\eta ) ^ 3 \\Big ] \\\\ = & \\frac { R ^ 2 } { 2 4 } ( 9 \\eta ^ 3 - 9 \\eta ^ 2 + 1 1 \\eta - 3 ) \\\\ = & \\frac { R ^ 2 } { 2 4 } ( \\eta - \\frac 1 3 ) \\Big [ ( 3 \\eta - 1 ) ^ 2 + 8 \\Big ] = C _ 3 ( \\eta _ 0 , c _ 0 ) R . \\end{align*}"}
-{"id": "6885.png", "formula": "\\begin{align*} \\limsup \\lambda _ i ( \\alpha ) = \\mu ( \\alpha ) . \\end{align*}"}
-{"id": "38.png", "formula": "\\begin{align*} e ^ { \\varphi _ { X / Y } ( x _ 0 ) } = \\sum _ { j = 1 } ^ N | F _ j ( x _ 0 ) | ^ 2 \\end{align*}"}
-{"id": "8352.png", "formula": "\\begin{align*} w ^ * \\lim _ \\beta \\sum _ j m _ j ^ * a \\psi ( m _ j ) = 0 . \\end{align*}"}
-{"id": "835.png", "formula": "\\begin{align*} Z _ { \\ell } ^ { J } ( \\vec { z } ) = \\prod _ { \\begin{subarray} { c } i \\in J \\\\ i < \\ell \\end{subarray} } f ( z _ { \\ell } , z _ { i } ) \\prod _ { \\begin{subarray} { c } i \\in J \\\\ i > \\ell \\end{subarray} } f ( z _ { i } , z _ { \\ell } ) \\prod _ { \\begin{subarray} { c } i \\in J \\\\ i < \\ell \\end{subarray} } ^ { \\curvearrowleft } Y _ { i - 1 } ( z _ { \\ell } , z _ { i } ) . \\end{align*}"}
-{"id": "927.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { p } } & = \\| U ^ { * } \\| _ { S _ { p _ { 1 } } } \\| V ^ { * } \\| _ { S _ { p _ { 2 } } } \\\\ & \\leq \\| U _ { 1 } \\| _ { S _ { p _ { 1 } } } \\| V _ { 1 } \\| _ { S _ { p _ { 2 } } } \\\\ & \\leq \\| U \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| U _ { 2 } \\| _ { S _ { q } } \\| V _ { 1 } \\| _ { S _ { p _ { 2 } } } \\\\ & = \\| U \\| _ { S _ { \\widehat { p } _ { 1 } } } \\| V \\| _ { S _ { \\widehat { p } _ { 2 } } } \\end{align*}"}
-{"id": "6366.png", "formula": "\\begin{align*} b _ i : = - \\frac { ( j - i + 1 ) ( j - i + 1 - s ) } { i ( i + s ) } \\ , b _ { i - 1 } , \\end{align*}"}
-{"id": "4654.png", "formula": "\\begin{align*} L _ 0 & = \\{ 0 \\} \\sqcup L _ 0 ( I ) \\sqcup L _ 0 ( I I ) , \\hat { L } _ 0 & = \\{ 0 \\} \\sqcup L _ 0 ( I ) \\sqcup \\hat { L } _ 0 ( I I ) , \\end{align*}"}
-{"id": "2441.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { 2 , 2 } ^ { 0 , \\alpha _ 2 } } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { 2 , 2 } ^ { 0 , \\alpha _ 1 } } = \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } \\Box _ k ^ { \\alpha _ 2 } f \\| ^ 2 _ { L ^ 2 } \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "3618.png", "formula": "\\begin{align*} ( \\phi ( x ) ) ^ { 4 + s _ j } \\widetilde { ( L U ) _ j } = \\sum _ { k = 1 } ^ { n + 1 } \\sum _ { | \\beta | = 0 } ^ { s _ j + t _ k } ( \\phi ( x ) ) ^ { s _ j + t _ k - | \\beta | } \\widetilde { b _ { j k } ^ { \\beta } } \\partial _ z ^ { \\beta } ( ( \\phi ( x ) ) ^ { 4 - t _ k } \\widetilde { U ^ k } ) . \\end{align*}"}
-{"id": "8651.png", "formula": "\\begin{align*} e ^ { t A } ( K ) = K , e ^ { t A } ( H ) = H , \\ ; \\ ; \\ ; t \\in \\R . \\end{align*}"}
-{"id": "517.png", "formula": "\\begin{align*} \\sum _ { \\mu } s _ { \\kappa / \\mu } ( \\rho _ 2 ) s _ { \\nu / \\mu } ( \\rho _ 1 ) \\mathcal { U } ^ { \\llcorner } _ { \\rho _ 1 , \\rho _ 2 } ( \\pi \\vert \\nu , \\mu , \\kappa ) = \\frac { s _ { \\pi / \\kappa } ( \\rho _ 1 ) s _ { \\pi / \\nu } ( \\rho _ 2 ) } { H ( \\rho _ 2 ; \\rho _ 1 ) } . \\end{align*}"}
-{"id": "3243.png", "formula": "\\begin{align*} P = Q \\circ R , \\end{align*}"}
-{"id": "6880.png", "formula": "\\begin{align*} \\mu _ n ( t _ k ) \\bigl | Y ^ n _ { t _ k } - Z _ { t _ k } \\bigr | ^ 2 \\leq \\mu _ n ( t _ { k - 1 } ) \\bigl | Y ^ n _ { t _ { k - 1 } } - Z _ { t _ { k - 1 } } \\bigr | ^ 2 + \\sum _ { i = 1 } ^ { 1 1 } I _ i ( t _ k ) . \\end{align*}"}
-{"id": "8768.png", "formula": "\\begin{align*} & \\overline { M } ( w ; x _ 1 , . . , x _ n ) = M ^ { ( 2 ) } ( w ; x _ 1 , . . , x _ n ) K _ n ( 1 / w ) M ^ { ( 1 ) } ( w ; x _ 1 , . . , x _ n ) , \\\\ & T ( w ; x _ 1 , . . , x _ n ) = _ 0 ( \\overline { M } ( w ; x _ 1 , . . , x _ n ) \\widetilde { K } _ 0 ( w ) ) . \\end{align*}"}
-{"id": "7319.png", "formula": "\\begin{align*} \\hat { R } F ( v _ { 0 } \\otimes v _ { 1 } ) & = [ 2 ] ^ { 1 / 2 } ( q ^ { - 4 } v _ { 1 } \\otimes v _ { - 1 } + \\hat { R } ( v _ { 0 } \\otimes v _ { 0 } ) ) , \\\\ F \\hat { R } ( v _ { 0 } \\otimes v _ { 1 } ) & = [ 2 ] ^ { 1 / 2 } ( v _ { 0 } \\otimes v _ { 0 } + v _ { 1 } \\otimes v _ { - 1 } ) . \\end{align*}"}
-{"id": "4214.png", "formula": "\\begin{align*} \\ell _ { g _ \\lambda } ( x ; y ) = \\left \\{ \\begin{array} { l l } ( y + 2 ) x - \\frac { 1 } { 2 } y ^ 2 + 4 , ~ ~ & ~ ~ y \\leq - 3 , \\\\ - x - \\frac { 1 } { 2 } , & ~ ~ - 3 \\leq y \\leq - 1 , \\\\ y x - \\frac { 1 } { 2 } y ^ 2 , & ~ ~ - 1 \\leq y \\leq 1 , \\\\ x - \\frac { 1 } { 2 } , & ~ ~ 1 \\leq y \\leq 3 , \\\\ ( y - 2 ) x - \\frac { 1 } { 2 } y ^ 2 + 4 , & ~ ~ y \\geq 3 \\end{array} \\right . \\end{align*}"}
-{"id": "105.png", "formula": "\\begin{align*} E _ m ( x ) = \\sum _ { n = 0 } ^ m C h _ { n , \\lambda } ( x ) S _ 2 ( m , n ) \\lambda ^ { - m } , \\quad ( m \\geq 0 ) . \\end{align*}"}
-{"id": "43.png", "formula": "\\begin{align*} L _ m : = L + ( m - 1 ) ( K _ { X / Y } + L ) \\end{align*}"}
-{"id": "5758.png", "formula": "\\begin{align*} \\underset { \\omega \\in \\Omega } \\sup ~ Q _ { L } ^ { L R } ( \\omega ) = \\underset { \\omega \\in \\Omega } \\sup ~ 2 \\left [ l _ n ( \\hat \\beta , \\hat \\psi , \\omega ) - l _ n ( \\hat \\beta _ 0 , 0 , \\omega ) \\right ] . \\end{align*}"}
-{"id": "5987.png", "formula": "\\begin{align*} w _ u = x _ 0 \\xleftarrow { \\gamma _ 1 ^ \\lor } \\cdots \\xleftarrow { \\gamma _ r ^ \\lor } x _ r = w _ { u + 1 } \\end{align*}"}
-{"id": "1102.png", "formula": "\\begin{align*} \\langle a _ k , \\eta _ { j _ 0 } \\rangle = 0 \\ , \\ , \\ , \\ , \\ , \\ , \\langle a _ k , \\eta _ { l _ 0 } \\rangle = 0 , \\forall k \\in S _ { l _ 0 } \\cup S _ { j _ 0 } . \\end{align*}"}
-{"id": "6565.png", "formula": "\\begin{align*} \\begin{aligned} & \\frac { 1 } { \\tau } \\big \\| ( v _ n - v _ { n - 1 } ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\big \\| ( v _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( H ^ 1 ( \\R ^ d ) ) } \\\\ & \\le C \\Big ( \\big \\| ( f _ n ) _ { n = k } ^ N \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\frac { 1 } { \\tau } \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ 2 ( \\R ^ d ) ) } + \\big \\| ( v _ i ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( H ^ 1 ( \\R ^ d ) ) } \\Big ) . \\end{aligned} \\end{align*}"}
-{"id": "3489.png", "formula": "\\begin{align*} \\int _ D \\widetilde { f } \\hat { v } d x ' = \\int _ D \\hat { f } \\hat { v } d x ' . \\end{align*}"}
-{"id": "2996.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ \\ell a _ i ( L _ i \\cdot P _ j ) \\equiv 0 \\mod d \\end{align*}"}
-{"id": "5021.png", "formula": "\\begin{align*} A B = \\bigcup _ { a \\in A } a B = \\big \\{ a \\cdot b \\ , : \\ , a \\in A , \\ : b \\in B \\big \\} \\subset Y , \\end{align*}"}
-{"id": "508.png", "formula": "\\begin{gather*} d ' \\sim \\sigma d ' = \\tau d \\xi \\unrhd d \\end{gather*}"}
-{"id": "9312.png", "formula": "\\begin{align*} M ( t , z ) : = \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ t ] \\end{align*}"}
-{"id": "2939.png", "formula": "\\begin{align*} L ^ \\perp _ { \\pi / ( ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ { m - 1 } ) ) } ( z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\cdots z _ { m - 1 } ^ { i _ { m - 1 } } ) \\ , M ( z _ m ) = M ( z _ m ) \\prod _ { i _ m \\ge 0 } \\ , L ^ \\perp _ { \\pi / ( ( i _ 1 ) ( i _ 2 ) \\cdots ( i _ m ) ) } ( z _ 1 ^ { i _ 1 } z _ 2 ^ { i _ 2 } \\cdots z _ { m } ^ { i _ { m } } ) \\end{align*}"}
-{"id": "3675.png", "formula": "\\begin{align*} \\mathcal { A } f = \\nabla \\cdot ( w \\nabla f ) ~ , \\end{align*}"}
-{"id": "3927.png", "formula": "\\begin{align*} W \\left [ w _ \\sigma , \\ , w _ { \\rm r e g } \\right ] ( 1 / 2 ) = - \\sum _ { l = 0 } ^ \\infty 2 ^ { - l } \\ , ( l + 1 ) \\left ( \\sum _ { j = 0 } ^ { l + 1 - \\sigma } a _ { l + 1 - \\sigma - j , \\sigma } \\ , b _ j \\right ) \\ , . \\end{align*}"}
-{"id": "9149.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ p k _ i \\leq ( p + t ) m / 2 - ( d / ( 2 d + 1 ) ) m / 2 = \\frac { ( d + 2 t ) ( 2 d + 1 ) - d } { 2 ( 2 d + 1 ) } m = \\frac { t ( 2 d + 1 ) + d ^ 2 } { 2 d + 1 } m , \\end{align*}"}
-{"id": "4838.png", "formula": "\\begin{align*} \\sum _ { \\nu ' } s _ { \\nu / \\lambda } ( \\rho ) = H ^ \\circ ( \\rho ) \\sum _ { \\kappa ' } s _ { \\lambda / \\kappa } ( \\rho ) , \\end{align*}"}
-{"id": "4727.png", "formula": "\\begin{align*} [ \\beta _ 1 ( y _ 1 - \\alpha _ 1 ) ] ^ 2 - [ \\beta _ 2 ( y _ 2 - \\alpha _ 2 ) ] ^ 2 = \\pm \\eta ^ 2 , \\end{align*}"}
-{"id": "8596.png", "formula": "\\begin{align*} \\overline { \\alpha } ( x ) & = \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } \\overline { \\alpha } \\xi _ { b r } \\pi _ { b r } ( x ) \\\\ & = \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } r ^ { - 1 } \\overline { N } ( ^ { \\ast } b ) \\left ( \\pi _ { b r } ( x ) \\right ) \\end{align*}"}
-{"id": "2490.png", "formula": "\\begin{gather*} I _ k ^ n : = \\left [ \\frac { k - 1 } { n } ; \\frac { k } { n } \\right ) , 1 \\leqslant k \\leqslant n - 1 , n \\geqslant 2 , \\\\ I _ n ^ n : = \\left [ \\frac { n - 1 } { n } ; 1 \\right ] , n \\geqslant 1 . \\end{gather*}"}
-{"id": "183.png", "formula": "\\begin{align*} \\hat { H } _ n = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\log \\xi _ i , \\end{align*}"}
-{"id": "8075.png", "formula": "\\begin{align*} R _ B ( x , a ) = ( x + \\beta , a + 1 ) = ( x , a ) + ( \\beta , 1 ) . \\end{align*}"}
-{"id": "8559.png", "formula": "\\begin{align*} \\bigl ( X _ { [ b r a ] ^ { \\ast } } ( \\rho ( P ) \\bigr ) _ { \\overline { N } } ( n ) & = X _ { [ b r a ] ^ { \\ast } } ( \\rho ( P ) ) n \\\\ & = \\rho ( Y _ { [ b r a ] } ( P ) ) n \\\\ & = Y _ { [ b r a ] } ( P ) n \\end{align*}"}
-{"id": "9380.png", "formula": "\\begin{align*} H _ { a } f = - \\frac { d ^ 2 f } { d x ^ 2 } + f _ r ( 0 ) q ( x ) , \\end{align*}"}
-{"id": "4382.png", "formula": "\\begin{align*} U _ { c , h } ( \\gamma ) = e ^ { i T _ { c , h } ( f _ 1 ) } \\ldots e ^ { i T _ { c , h } ( f _ n ) } . \\end{align*}"}
-{"id": "5140.png", "formula": "\\begin{align*} \\tilde { \\tau } = \\tau ' \\prod _ { 0 \\le s < t } ^ { \\curvearrowleft } ( \\prod _ { \\begin{subarray} { c } i \\in J _ { p ( s ) } \\\\ i < \\ell ( s ) \\end{subarray} } ^ { \\curvearrowright } \\sigma _ { i - 1 } ) \\end{align*}"}
-{"id": "5547.png", "formula": "\\begin{align*} & \\frac 1 2 b _ { x x } ( 0 ) - h b _ x ( 0 ) + ( h ^ 2 + z \\rho ( 0 ) ) b ( 0 ) = 0 , \\\\ & \\frac 1 2 b _ { x x } ( 1 ) + H b _ x ( 1 ) + ( H ^ 2 + z \\rho ( 1 ) ) b ( 1 ) = 0 . \\end{align*}"}
-{"id": "1413.png", "formula": "\\begin{gather*} b ^ - = \\big ( \\prod _ { j = n + 1 } ^ { K } F _ { i _ { j } } H _ { \\omega ^ { \\vee } _ { i _ { j } } } ( X _ { j } ^ { - 1 } ) \\big ) F _ n \\dots F _ 3 F _ 2 F _ 1 \\\\ = F _ { i _ { n + 1 } } H _ { \\omega ^ { \\vee } _ { i _ { n + 1 } } } ( X _ { n + 1 } ^ { - 1 } ) F _ { i _ { n + 2 } } H _ { \\omega ^ { \\vee } _ { i _ { n + 2 } } } ( X _ { n + 2 } ^ { - 1 } ) \\dots \\\\ F _ { i _ { K - 1 } } H _ { \\omega ^ { \\vee } _ { i _ { K - 1 } } } ( X _ { K - 1 } ^ { - 1 } ) F _ { i _ K } H _ { \\omega ^ { \\vee } _ { i _ K } } ( X _ { K } ^ { - 1 } ) F _ n \\dots F _ 3 F _ 2 F _ 1 . \\end{gather*}"}
-{"id": "7617.png", "formula": "\\begin{align*} \\mathbf { v } _ u \\sqrt { 2 n - 4 } \\leq \\lambda _ 1 \\mathbf { v } _ u = \\sum _ { y \\sim u } \\mathbf { v } _ y = \\sum _ { \\substack { y \\sim u \\\\ y \\in L } } \\mathbf { v } _ y + \\sum _ { \\substack { y \\sim u \\\\ y \\in S } } \\mathbf { v } _ y \\leq \\sum _ { y \\in L } \\mathbf { v } _ y + \\sum _ { \\substack { y \\sim u \\\\ y \\in S } } \\mathbf { v } _ y . \\end{align*}"}
-{"id": "6813.png", "formula": "\\begin{align*} \\mathcal B ( x ) = \\frac { 1 } { 2 i \\pi } \\oint _ \\S \\frac { B ( \\xi ) } { \\xi - x } \\dd \\xi , \\end{align*}"}
-{"id": "759.png", "formula": "\\begin{gather*} T _ p \\leftrightarrow \\sum _ { i ( 1 ) , \\ldots , i ( k ) , j ( 1 ) , \\ldots , j ( l ) = 1 } ^ n \\delta _ p ( i , j ) S _ { j ( 1 ) } \\cdots S _ { j ( l ) } S _ { i ( k ) } ^ * \\cdots S _ { i ( 1 ) } ^ * . \\end{gather*}"}
-{"id": "6345.png", "formula": "\\begin{align*} X = ( x ' , x _ n , y ) = ( x ' , z ) = ( x , y ) \\end{align*}"}
-{"id": "3757.png", "formula": "\\begin{align*} \\varphi ( t ) & = E e ^ { i t ( \\alpha T _ \\beta - \\alpha u ) } \\\\ & = e ^ { C _ 0 ( \\beta + i \\alpha t ) - C _ 0 ( \\beta ) - i \\alpha t u } \\end{align*}"}
-{"id": "4486.png", "formula": "\\begin{align*} F _ { n , x } ^ - ( u ) & : = \\sum _ { j = k } ^ { n - 2 } \\binom { n - 2 } { j } p _ { n , x , u } ^ j ( 1 - p _ { n , x , u } ) ^ { n - 2 - j } , \\\\ \\tilde { F } _ { n , x } ( u ) & : = \\sum _ { j = k - 1 } ^ { n - 2 } \\binom { n - 2 } { j } p _ { n , x , u } ^ j ( 1 - p _ { n , x , u } ) ^ { n - 2 - j } , \\end{align*}"}
-{"id": "5559.png", "formula": "\\begin{align*} \\dot A _ 1 & = ( 1 + K ) \\beta A _ 1 , \\\\ \\dot B _ 1 & = ( 1 - K ) \\beta B _ 1 + b ( 1 ) A _ 1 , \\end{align*}"}
-{"id": "8487.png", "formula": "\\begin{align*} \\rho _ { 0 } ^ { \\varepsilon } ( x ) = \\rho _ { 0 } ( 0 , x ) , ~ ~ ~ ~ v _ { 0 } ^ { \\varepsilon } ( 0 , x ) = v _ { 0 } ( x ) + \\varepsilon v _ { 0 } ^ { 1 } ( x ) , \\end{align*}"}
-{"id": "5555.png", "formula": "\\begin{align*} v ( x ) = A _ { 1 } ( t , z ) ( x - 1 ) + B _ { 1 } ( t , z ) , \\end{align*}"}
-{"id": "4985.png", "formula": "\\begin{align*} \\vec { D } = \\left ( \\begin{array} { c } D _ { 1 } \\\\ D _ { 2 } \\\\ \\vdots \\\\ D _ { r } \\end{array} \\right ) , \\vec { F } = \\left ( \\begin{array} { c } F _ { 1 } \\\\ F _ { 2 } \\\\ \\vdots \\\\ F _ { s } \\end{array} \\right ) , \\vec { c } = \\left ( \\begin{array} { c } c _ { 1 } \\\\ c _ { 2 } \\\\ \\vdots \\\\ c _ { r } \\end{array} \\right ) , \\vec { Z } = \\left ( \\begin{array} { c } Z _ { 1 } \\\\ Z _ { 2 } \\\\ \\vdots \\\\ Z _ { r } \\end{array} \\right ) . \\end{align*}"}
-{"id": "9660.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\{ t - \\tau ( t ) \\} = \\infty . \\end{align*}"}
-{"id": "6780.png", "formula": "\\begin{align*} \\int _ { \\{ V _ { \\theta } - j < \\psi _ s \\leq v < \\psi < \\varphi + s \\} } \\theta _ { v _ j } ^ n = 0 . \\end{align*}"}
-{"id": "8806.png", "formula": "\\begin{align*} u ^ + ( p ) = \\left \\{ \\begin{array} { c c c c c } u ( p ) & i f & p & \\in & \\Sigma ^ + \\\\ 0 & i f & p & \\in & \\Sigma \\setminus \\Sigma ^ + \\\\ \\end{array} \\right . a n d \\ , \\ , \\ , u ^ - ( p ) = \\left \\{ \\begin{array} { c c c c c } u ( p ) & i f & p & \\in & \\Sigma ^ - \\\\ 0 & i f & p & \\in & \\Sigma \\setminus \\Sigma ^ - \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "5975.png", "formula": "\\begin{align*} \\begin{aligned} N ( x ) = | \\Phi | - | \\Phi _ x | & \\geq | \\Phi _ { S _ { n } } | - | \\Phi _ { ( S _ k ) ^ { b } \\times S _ { n - b k } } | \\\\ [ 1 e x ] & = n ( n - 1 ) - [ b k ( k - 1 ) + ( n - k b ) ( n - k b - 1 ) ] \\\\ [ 1 e x ] & = 2 k b n - k ^ 2 b ^ 2 - b k ^ 2 . \\end{aligned} \\end{align*}"}
-{"id": "3132.png", "formula": "\\begin{align*} L _ 5 ( \\lambda ) = \\begin{bmatrix} 0 & 0 & - I _ n & \\lambda I _ n \\\\ 0 & - P _ 4 & \\lambda P _ 4 - P _ 3 & \\lambda P _ 3 \\\\ - I _ n & \\lambda P _ 4 - P _ 3 & \\lambda P _ 3 - P _ 2 & \\lambda P _ 2 \\\\ \\lambda I _ n & \\lambda P _ 3 & \\lambda P _ 2 & \\lambda P _ 1 + P _ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "1064.png", "formula": "\\begin{align*} e ( B ) \\leq \\left ( \\beta - \\frac { 1 } { 4 } \\right ) \\frac { n ^ 2 } { 2 } + \\left ( k - \\alpha - \\beta \\right ) \\frac { n ^ 2 } { 2 } = \\bigg ( k - \\alpha - \\frac 1 4 \\bigg ) \\frac { n ^ 2 } { 2 } \\ , . \\end{align*}"}
-{"id": "2083.png", "formula": "\\begin{align*} ( s _ { i } ^ { 2 } M + s _ { i } D + K ) X ^ { ( 0 ) } ( s _ { i } ) & = F + \\eta _ { 0 i } . \\end{align*}"}
-{"id": "8988.png", "formula": "\\begin{align*} f ( t ) = \\frac { 1 - \\alpha } { B ( \\alpha ) } u ( t ) + \\frac { \\alpha } { B ( \\alpha ) } I _ b ^ \\alpha u ( t ) . \\end{align*}"}
-{"id": "2930.png", "formula": "\\begin{align*} L _ \\pi ( Z ) = \\sum _ { k \\ge 0 } \\ , ( - 1 ) ^ k s _ { ( 1 ^ k ) } [ s _ \\pi ( Z ) ] = \\sum _ \\nu \\ , \\ell _ { \\pi \\nu } \\ , s _ \\nu ( Z ) \\quad \\mbox { a n d } M _ \\pi ( Z ) = \\sum _ { k \\ge 0 } \\ , s _ { ( k ) } [ s _ \\pi ( Z ) ] = \\sum _ \\nu \\ , m _ { \\pi \\nu } \\ , s _ \\nu ( Z ) \\ , . \\end{align*}"}
-{"id": "3870.png", "formula": "\\begin{align*} & \\Big \\{ \\frac { \\| V \\| ( B _ \\rho ( x ) ) + \\| V \\| ( \\tilde { B } _ \\rho ( x ) ) + 2 \\sigma \\mathcal { H } ^ n \\lfloor _ { B ^ + } ( B _ \\rho ( x ) ) } { \\omega _ n \\rho ^ n } \\Big \\} ^ { \\frac 1 p } \\Big ( 1 + C \\varkappa \\rho \\Big ( 1 + \\dfrac { 1 } { p - n } \\Big ) \\Big ) \\\\ & + \\dfrac { \\Gamma \\rho ^ { 1 - \\frac { n } { p } } } { p - n } \\end{align*}"}
-{"id": "5939.png", "formula": "\\begin{align*} \\| D _ v { \\psi } \\| _ { W ^ { 1 , p } ( \\R ^ { 2 d } ) } \\le c ( \\lambda ) \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ { p } ( \\R ^ d _ x ) ) } , \\ ; \\ ; \\ ; \\end{align*}"}
-{"id": "2826.png", "formula": "\\begin{align*} a _ { j , v } - a _ { i , u } = a _ j + v q _ j - a _ i - u q _ i \\ge j - i - 1 , \\end{align*}"}
-{"id": "6008.png", "formula": "\\begin{gather*} \\Phi : = - e ^ { 1 4 7 } + \\sqrt { 2 } e ^ { 1 5 6 } + \\sqrt { 2 } e ^ { 2 3 7 } + e ^ { 2 4 5 } + e ^ { 3 4 6 } , \\end{gather*}"}
-{"id": "1145.png", "formula": "\\begin{align*} - v _ { x x } = z \\rho ( x ) v , 0 < x < 1 , v _ x ( 0 ) - h v ( 0 ) = 0 , v _ x ( 1 ) + H v ( 1 ) = 0 , \\end{align*}"}
-{"id": "5268.png", "formula": "\\begin{align*} \\zeta ( s ) = \\chi ( s ) \\zeta ( 1 - s ) , \\end{align*}"}
-{"id": "5466.png", "formula": "\\begin{align*} \\sum _ { n = N - H } ^ { N + H } t _ H ( n - N ) n = H ^ 2 N \\end{align*}"}
-{"id": "2119.png", "formula": "\\begin{align*} x ( z _ 1 , z _ 2 ) = z _ 1 ^ 2 - z _ 2 ^ 2 \\qquad y ( z _ 1 , z _ 2 ) = 2 z _ 1 z _ 2 , \\end{align*}"}
-{"id": "1406.png", "formula": "\\begin{align*} S _ { A C , D , \\omega _ k ^ { \\vee } } = ( \\lambda , 0 , \\mu , 0 ) \\end{align*}"}
-{"id": "4511.png", "formula": "\\begin{align*} \\tilde { a } ( \\delta ) & : = d ^ { m / 2 } \\sup _ { x : \\| x \\| \\leq g ( \\delta ) } \\max \\biggl \\{ \\max _ { r = 1 , \\ldots , m } q _ r ( \\| x \\| ) \\ , , \\ , d ^ { 1 / 2 } e ^ { \\| x \\| } q _ { m + 1 } ( \\| x \\| + 1 ) \\biggr \\} \\\\ & = d ^ { m / 2 } \\max \\biggl \\{ \\max _ { r = 1 , \\ldots , m } q _ r \\bigl ( g ( \\delta ) \\bigr ) \\ , , \\ , d ^ { 1 / 2 } e ^ { g ( \\delta ) } q _ { m + 1 } \\bigl ( g ( \\delta ) + 1 \\bigr ) \\biggr \\} . \\end{align*}"}
-{"id": "657.png", "formula": "\\begin{align*} f _ \\phi ( g ) = \\frac { \\nu ( ( g ^ { - 1 } \\cdot \\phi ) \\chi _ C ) } { \\nu ( C ) } , \\textrm { f o r $ g \\in G $ } . \\end{align*}"}
-{"id": "3884.png", "formula": "\\begin{align*} L ( u , w ^ 1 , \\dotsc , w ^ k ) : = \\int _ X u \\sigma _ k ( D ^ 2 w ^ 1 , \\dotsc , D ^ 2 w ^ k ) d x + \\oint _ M u A _ k ( w ^ 1 , \\dotsc , w ^ k ) d \\mu \\end{align*}"}
-{"id": "6808.png", "formula": "\\begin{align*} \\det Y ( x ) = \\exp \\left ( \\frac { 1 } { 2 i \\pi } \\oint _ \\S \\frac { \\log \\det G ( \\xi ) } { \\xi - x } \\dd \\xi \\right ) . \\end{align*}"}
-{"id": "8298.png", "formula": "\\begin{align*} \\gamma ( \\varphi ) : = \\{ x \\in \\Gamma \\vert \\exists \\tilde { x } \\in \\widetilde { \\Gamma } , \\tilde { v } \\in T _ { \\tilde { x } } ( \\widetilde { \\Gamma } ) \\varphi ( \\tilde { x } ) = x d _ { \\tilde { v } } ( \\varphi ) = 2 \\} . \\end{align*}"}
-{"id": "7327.png", "formula": "\\begin{align*} \\begin{gathered} v _ 1 \\wedge v _ 1 = 0 , v _ 0 \\wedge v _ 1 = - q ^ 2 v _ 1 \\wedge v _ 0 , v _ 0 \\wedge v _ 0 = - q ^ { - 1 } ( q - q ^ { - 1 } ) v _ 1 \\wedge v _ { - 1 } , \\\\ v _ { - 1 } \\wedge v _ 1 = - v _ 1 \\wedge v _ { - 1 } , v _ { - 1 } \\wedge v _ 0 = - q ^ 2 v _ 0 \\wedge v _ { - 1 } , v _ { - 1 } \\wedge v _ { - 1 } = 0 . \\end{gathered} \\end{align*}"}
-{"id": "9059.png", "formula": "\\begin{align*} & \\sum _ { k = 1 } ^ { N ' } \\P \\left ( \\mathfrak { E } _ { B , N , N ' , y } , \\ , d W _ k \\leq - ( k ^ { \\frac { 9 } { 1 0 } } + B ^ { 9 / 5 } ) / 1 0 \\right ) \\\\ & + \\sum _ { k = N ' + 1 } ^ { N } \\P \\left ( \\mathfrak { E } _ { B , N , N ' , y } , \\ , d W _ k + y \\leq - ( ( N - k ) ^ { \\frac { 9 } { 1 0 } } - B ^ { 9 / 5 } ) / 1 0 \\right ) \\leq \\frac { \\epsilon } { N ^ { 3 / 2 } } . \\end{align*}"}
-{"id": "9290.png", "formula": "\\begin{align*} \\begin{cases} d y ( t , x , Z ) = ( L ^ * _ { \\pi ( t , z ) } y ) ( t , x , Z ) d t + x y ( t , x , Z ) d R ( t ) ; 0 \\leq t \\leq T , \\\\ y ( 0 , x , Z ) = F ( x ) , \\end{cases} \\end{align*}"}
-{"id": "8981.png", "formula": "\\begin{align*} & \\int _ { \\mathbb { R } ^ { 2 N } } \\frac { | w ( x ) - w ( y ) | ^ { p - 2 } \\ , ( w ( x ) - w ( y ) ) \\ , ( \\psi ( x ) - w ( x ) - ( \\psi ( y ) - w ( y ) ) ) } { | x - y | ^ { N + s \\ , p } } \\ , d x \\ , d y \\\\ & \\geq \\int _ \\Omega \\ , f ( x ) \\Phi _ { k } ' ( w ) ( \\psi - w ) \\quad \\ , \\ , \\ , \\ , \\ , \\psi \\in w + \\left ( W ^ { s , p } _ 0 ( \\Omega ) \\cap L ^ \\infty _ c ( \\Omega ) \\right ) \\ , \\ , \\ , \\ , 0 \\leq \\psi \\leq v \\ , . \\end{align*}"}
-{"id": "6381.png", "formula": "\\begin{align*} C ( \\varepsilon ) = \\left ( \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon ) ^ { n } } { n ! } ( n p - 1 ) ! ! \\sigma ^ { n p } \\right ) ^ { - 1 } , \\end{align*}"}
-{"id": "3350.png", "formula": "\\begin{align*} & ( g _ 1 ^ { [ n \\lambda _ 1 s _ 1 ] + [ n \\lambda _ 2 s _ 1 ] } g _ 2 ^ { [ n \\lambda _ 1 s _ 2 ] + [ n \\lambda _ 2 s _ 2 ] } \\cdots g _ k ^ { [ n \\lambda _ 1 s _ k ] + [ n \\lambda _ 2 s _ k ] } ) ^ { - 1 } ( g _ 1 ^ { [ n \\lambda _ 1 s _ 1 ] } g _ 2 ^ { [ n \\lambda _ 1 s _ 2 ] } \\cdots g _ k ^ { [ n \\lambda _ 1 s _ k ] } ) ( g _ 1 ^ { [ n \\lambda _ 2 s _ 1 ] } g _ 2 ^ { [ n \\lambda _ 2 s _ 2 ] } \\cdots g _ k ^ { [ n \\lambda _ 2 s _ k ] } ) \\\\ & = [ f _ { n , 1 } , f _ { n , 1 } ^ \\prime ] \\cdots [ f _ { n , l ( k ) } , f _ { n , l ( k ) } ^ \\prime ] . \\\\ \\end{align*}"}
-{"id": "4569.png", "formula": "\\begin{align*} \\int _ { A _ t ^ c } f | \\log f | & \\leq ( 8 t ) ^ { 2 ( 1 - \\epsilon ) } \\int _ { \\mathcal { X } } f | g | ^ { 2 ( 1 - \\epsilon ) } | \\log f | \\\\ & \\leq ( 8 t ) ^ { 2 ( 1 - \\epsilon ) } \\Bigl \\{ \\int _ { \\mathcal { X } } g ^ 2 f \\Bigr \\} ^ { 1 - \\epsilon } \\Bigl \\{ \\int _ { \\mathcal { X } } f | \\log f | ^ { 1 / \\epsilon } \\Bigr \\} ^ \\epsilon = o ( t ) \\end{align*}"}
-{"id": "1681.png", "formula": "\\begin{align*} & \\lambda U ( z ) - \\frac { 1 } { 2 } \\mathrm { T r } \\big ( Q D ^ 2 U ( z ) \\big ) - \\langle A z , D U ( z ) \\rangle - \\langle B ( z ) , D U ( z ) \\rangle = B ( z ) , \\\\ & \\ ; \\ ; \\lambda U ( z ) - { \\mathcal L } U ( z ) = B ( z ) \\end{align*}"}
-{"id": "9825.png", "formula": "\\begin{align*} M _ i \\ , { } _ \\lambda \\ , v _ m = 0 , \\ \\ \\ \\ Y _ i \\ , { } _ \\lambda \\ , v _ m = d v _ { i + m } . \\end{align*}"}
-{"id": "7765.png", "formula": "\\begin{align*} \\beta ( M ) = \\begin{bmatrix} 1 & 1 & - \\\\ - & 1 & 1 \\end{bmatrix} = \\frac { 1 } { 3 } \\begin{bmatrix} 2 & 3 & - \\\\ - & - & 1 \\end{bmatrix} + \\frac { 1 } { 3 } \\begin{bmatrix} 1 & - & - \\\\ - & 3 & 2 \\end{bmatrix} \\end{align*}"}
-{"id": "7073.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } Z _ n = Z _ \\infty \\lim _ { n \\to \\infty } W _ n = 0 \\end{align*}"}
-{"id": "4311.png", "formula": "\\begin{align*} \\sum \\lambda ^ j _ \\infty \\sigma _ j \\wedge p ^ \\star ( d t ) = \\rho _ \\infty \\end{align*}"}
-{"id": "2361.png", "formula": "\\begin{align*} \\partial _ x \\psi ( \\Phi _ t ( y ) , t ) = 0 \\end{align*}"}
-{"id": "8221.png", "formula": "\\begin{align*} E U _ n = \\epsilon \\left ( \\bar { U } _ { n + 1 } - 2 \\bar { U } _ n + \\bar { U } _ { n - 1 } \\right ) + i \\gamma U _ n + \\Omega \\bar { U } _ n + 6 | U _ n | ^ 2 \\bar { U } _ n + 2 U _ n ^ 3 , \\pm n \\in \\mathbb { N } , \\end{align*}"}
-{"id": "2100.png", "formula": "\\begin{align*} X = ( x ' , x _ n , y ) = ( x ' , z ) = ( x , y ) \\end{align*}"}
-{"id": "1974.png", "formula": "\\begin{align*} M = \\left ( m \\Z ^ n + g \\Z ( \\sum _ { i = 1 } ^ n ( - 1 ) ^ { i - 1 } { \\bf e } _ i ) \\right ) \\cap H _ v . \\end{align*}"}
-{"id": "2157.png", "formula": "\\begin{align*} \\partial _ { s } ^ { \\alpha } ( \\tilde { u } - \\tilde { u } _ { 0 } ) + L \\tilde { u } = ( \\geq \\leq ) \\tilde { f } , s \\in ( 0 , t _ { 0 } ) , \\ , \\ , y \\in B ( 0 , 1 ) . \\end{align*}"}
-{"id": "5961.png", "formula": "\\begin{align*} \\xi _ t ^ h & = e + \\frac { 1 } { h } \\Big [ U ( z + h e ) - U ( z ) \\Big ] \\\\ & + \\int _ 0 ^ t \\frac { \\lambda } { h } \\Big [ U \\big ( \\phi _ s ( z + h e ) \\big ) - U \\big ( \\phi _ s ( z ) \\big ) \\Big ] + A \\theta _ t ^ h ( z ) \\ \\dd s \\\\ & + \\frac { 1 } { h } \\int _ 0 ^ t \\Big [ D _ v U \\big ( \\phi _ s ( z + h e ) \\big ) - D _ v U \\big ( \\phi _ s ( z ) \\big ) \\Big ] R \\cdot \\dd W _ s \\ , . \\end{align*}"}
-{"id": "4151.png", "formula": "\\begin{align*} \\sum _ { k \\geq 0 } \\frac { | A ( 1 , p ^ k ) | ^ 2 } { p ^ { k s } } = & \\frac { 1 - b _ p p ^ { - 2 s } + ( 2 b _ p - 2 ) p ^ { - 3 s } - b _ p p ^ { - 4 s } + p ^ { - 6 s } } { \\prod _ { i = 1 } ^ 3 \\prod _ { j = 1 } ^ 3 ( 1 - \\frac { \\alpha _ { i , p } } { \\alpha _ { j , p } } p ^ { - s } ) } , \\end{align*}"}
-{"id": "1517.png", "formula": "\\begin{align*} \\Sigma ( x ) : = \\{ y \\in X : \\delta ( x , y ) \\mbox { i s c o u n t a b l e } \\} \\mbox { a n d } \\sigma ( x ) : = \\{ y \\in X : \\delta ( x , y ) \\mbox { i s f i n i t e } \\} . \\end{align*}"}
-{"id": "9633.png", "formula": "\\begin{align*} g _ \\phi ( X ( I _ n \\oplus b ) ) - I _ { n + 1 } = ( g _ \\phi ( X ) - I _ { n + 1 } ) ( I _ n \\oplus b ) \\end{align*}"}
-{"id": "621.png", "formula": "\\begin{align*} \\psi _ r ( \\lambda ) = e ^ { - \\sqrt { | \\lambda | } r } \\psi _ r ( \\lambda ) = \\cos \\left ( \\sqrt { \\lambda } \\ r \\right ) \\end{align*}"}
-{"id": "8167.png", "formula": "\\begin{align*} \\| \\Delta _ H \\Lambda _ s u \\| ^ 2 \\leq C ( \\| \\Delta _ H u \\| _ s ^ 2 + \\sum _ { k = 1 } ^ N \\| X _ k u \\| _ s ^ 2 + \\| u \\| ^ 2 _ s ) , u \\in C ^ \\infty ( M ) . \\end{align*}"}
-{"id": "937.png", "formula": "\\begin{align*} \\frac { b - 1 - k ( r - 1 ) } { \\frac { v } { k } - 1 } = s k - k + 1 \\end{align*}"}
-{"id": "1449.png", "formula": "\\begin{align*} \\| b \\| _ { \\rm B M O } : = \\| b ^ \\sharp \\| _ { L ^ \\infty } < \\infty . \\end{align*}"}
-{"id": "4488.png", "formula": "\\begin{align*} y = y _ { x , z } : = x + \\frac { r _ { n , 1 } } { f ( x ) ^ { 1 / d } } z , \\end{align*}"}
-{"id": "9853.png", "formula": "\\begin{align*} x ( t ) \\otimes ( \\delta ( t ) - \\alpha \\theta ( t - \\tau ) ) & = \\theta ( t ) \\otimes ( h _ { \\rm S R } ( t ) \\otimes s ( t ) + n _ { \\rm R } ( t ) ) . \\end{align*}"}
-{"id": "5716.png", "formula": "\\begin{align*} \\Omega _ k : = \\bigcup _ { j \\in J _ k } Q ^ k _ j . \\end{align*}"}
-{"id": "1278.png", "formula": "\\begin{align*} \\begin{bmatrix} c ( j , r ) & c ( j + 2 r , 3 r ) \\\\ c ( j , 3 r ) & c ( j + 2 r , r ) \\end{bmatrix} \\end{align*}"}
-{"id": "7830.png", "formula": "\\begin{align*} Y _ { t } = g ( \\eta _ { T } ) + \\int _ t ^ T E ' \\big [ f ( s , \\eta _ { s } , Y ' _ { s } , Y _ { s } , Z _ { s } ) \\big ] d s - \\int _ t ^ T Z _ { s } d B _ { s } ^ { H } , \\ \\ 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "6777.png", "formula": "\\begin{align*} \\int _ { \\{ \\psi _ t < \\psi < \\varphi + t \\} } \\theta _ { \\psi _ t } ^ n = 0 . \\end{align*}"}
-{"id": "4326.png", "formula": "\\begin{align*} \\frac { 1 } { r } e _ r \\otimes e _ { r - s } = \\sum _ { i = 1 } ^ l d _ i ^ * ( e _ r ) \\otimes c _ i ( e _ { r - s } ) . \\end{align*}"}
-{"id": "7411.png", "formula": "\\begin{align*} i _ \\mathcal { X } ^ * ( X _ i ^ \\circ ) = X _ i ^ { - 1 } . \\end{align*}"}
-{"id": "1052.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\int _ { \\R ^ 2 } | ( Q ^ j \\ast 1 _ E ) ( x ) | 1 _ A ( x ) \\ , d x & \\leq \\sum _ { j = 0 } ^ M \\| Q ^ j \\ast 1 _ E \\| _ 6 | A | ^ { \\frac 5 6 } + \\sum _ { j = M + 1 } ^ \\infty \\| Q ^ j \\ast 1 _ E \\| _ \\infty | A | \\\\ & \\leq C \\left \\{ 2 ^ { \\frac { M } 2 } | E | ^ { \\frac 1 2 } | A | ^ { \\frac 5 6 } + 2 ^ { - \\frac { M + 1 } 2 } | E | | A | \\right \\} . \\end{align*}"}
-{"id": "3932.png", "formula": "\\begin{align*} z ^ { ( i + 1 ) } = z ^ { ( i ) } - \\frac { \\mathcal { W } _ \\sigma ^ \\prime ( z ^ { ( i ) } ) \\pm \\left [ \\left ( \\mathcal { W } _ \\sigma ^ \\prime ( z ^ { ( i ) } ) \\right ) ^ 2 - 2 \\mathcal { W } _ \\sigma ( z ^ { ( i ) } ) \\ , \\mathcal { W } _ \\sigma ^ { \\prime \\prime } ( z ^ { ( i ) } ) \\right ] ^ { 1 / 2 } } { \\mathcal { W } _ \\sigma ^ { \\prime \\prime } ( z ^ { ( i ) } ) } \\ , , \\end{align*}"}
-{"id": "5299.png", "formula": "\\begin{align*} \\mathrm { A P P } _ d ( f ) \\ , : = \\ , f . \\end{align*}"}
-{"id": "8863.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { a _ t } { t } = \\sup _ { t \\geq 1 } \\frac { a _ t + c } { t } . \\end{align*}"}
-{"id": "2187.png", "formula": "\\begin{align*} = \\phi \\int _ { \\rho ' B _ { 1 } } \\int _ { \\rho ' B _ { 1 } } \\left [ w ( s , x ) - w ( s , y ) \\right ] ^ { 2 } k ( x , y ) d x d y . \\end{align*}"}
-{"id": "2478.png", "formula": "\\begin{align*} E _ j ^ I : = V _ j ^ I \\otimes ( \\omega _ { s _ j } \\oplus \\omega _ { s _ j + 1 } \\oplus \\cdots \\oplus \\omega _ { s _ j + d _ j - 1 } ) \\end{align*}"}
-{"id": "5781.png", "formula": "\\begin{align*} \\sum _ { \\rho \\in [ \\beta ] } ( c _ { \\gamma , u } ^ { \\rho } c _ { \\rho , v } ^ { \\alpha } - ( - 1 ) ^ { p ( u ) p ( v ) } c _ { \\gamma , v } ^ { \\rho } c _ { \\rho , u } ^ { \\alpha } ) = c _ { \\gamma , [ u , v ] } ^ { \\alpha } \\end{align*}"}
-{"id": "2194.png", "formula": "\\begin{align*} 0 \\leq h _ { m } * W = g _ { \\alpha } * \\partial _ { s } ^ { \\alpha } ( h _ { m } * W ) \\leq g _ { \\alpha } * G _ { m } + g _ { \\alpha } * [ - F _ { m } ( s ) ] ^ { + } \\end{align*}"}
-{"id": "436.png", "formula": "\\begin{align*} T _ { i , k } ( x ) = x _ 1 x _ 2 \\dots x _ { i + k } x _ { i + 1 } x _ { i + 2 } \\dots x _ n . \\end{align*}"}
-{"id": "1931.png", "formula": "\\begin{align*} \\left ( 2 \\sum _ { i = 1 } ^ m n _ i p _ i Y _ i - \\frac { 1 } { 2 } E ( Y ) \\right ) ^ 2 \\leq & \\ , \\ , \\left ( \\frac { n - 1 } { 2 } \\right ) \\sum _ { i = 1 } ^ m n _ i Y _ i ^ 2 ( 2 p _ i - q _ i ^ 2 Y _ i ) ^ 2 + \\frac { 1 } { 4 } E ( Y ) ^ 2 \\\\ & + \\sqrt { \\frac { n - 1 } { 2 } } \\ , E ( Y ) \\left ( \\sum _ { i = 1 } ^ m n _ i Y _ i ^ 2 ( 2 p _ i - q _ i ^ 2 Y _ i ) ^ 2 \\right ) ^ { 1 / 2 } . \\end{align*}"}
-{"id": "138.png", "formula": "\\begin{align*} \\sum _ { n = 1 } ^ \\infty \\| f _ n - g _ n \\| _ { \\widehat { E } } < \\infty . \\end{align*}"}
-{"id": "3397.png", "formula": "\\begin{align*} & \\pi ^ { ( i ) } _ { j , k } ( ( u '^ { ( i ) } _ j \\nabla ^ { ( i ) } _ j - \\nabla ^ { ( i ) } _ j u '^ { ( i ) } _ j ) u ^ { ( i ) } _ j | _ { V ^ { ( i ) } _ { j , k } } ) \\\\ & = \\left ( \\gamma ^ { ( i ) } _ { j , k } \\big ( \\nu ^ { ( i ) } _ j ( w ^ { ( i ) } _ j ) + k d z _ i / r ^ { ( i ) } _ j z _ i \\big ) - \\big ( \\nu ^ { ( i ) } _ j ( w ^ { ( i ) } _ j ) + k d z _ i / r ^ { ( i ) } _ j z _ i \\big ) \\gamma ^ { ( i ) } _ { j , k } \\right ) \\beta ^ { ( i ) } _ { j , k } \\pi ^ { ( i ) } _ { j , k } = 0 \\end{align*}"}
-{"id": "6892.png", "formula": "\\begin{align*} \\begin{cases} d v = \\Phi ( v ) d W ( t ) \\\\ v ( \\cdot , s ) = v _ s ( \\cdot ) \\end{cases} \\end{align*}"}
-{"id": "493.png", "formula": "\\begin{align*} S _ 2 & = \\sum _ { k \\le 2 ( h - 1 ) \\le 2 k } ( - 1 ) ^ h \\ , \\alpha _ { h - 1 , k } \\ , r ^ { 2 h - k - 1 } \\ , F _ { N + 2 h } ( r ) \\\\ & = \\sum _ { k + 1 \\le 2 h \\le 2 ( k + 1 ) } ( - 1 ) ^ h \\ , \\alpha _ { h - 1 , k } \\ , r ^ { 2 h - k - 1 } \\ , F _ { N + 2 h } ( r ) . \\end{align*}"}
-{"id": "5154.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t a + u \\cdot \\nabla a = R _ 1 ( t , x ) , \\\\ \\partial _ t b + u \\cdot \\nabla b = R _ 2 ( t , x ) , \\\\ a | _ { t = 0 } = a _ 0 ( x ) , \\\\ b | _ { t = 0 } = b _ 0 ( x ) , \\end{array} \\right . \\end{align*}"}
-{"id": "1215.png", "formula": "\\begin{align*} & \\sup \\limits _ { g ' \\in \\mathcal { G } _ { f ^ { * } } } \\left ( \\sum \\limits _ { i = 1 } ^ { n } \\xi _ { i } g ' ( X _ { i } ) - \\frac { h } { 3 } g ' ( X _ i ) \\right ) \\\\ & \\le \\gamma \\left ( 1 + \\frac { h } { 1 2 } \\right ) \\ ! + \\ ! \\sup \\limits _ { g ' \\in \\mathcal { G } _ { f ^ { * } } } \\left ( \\sum \\limits _ { i = 1 } ^ { n } \\xi _ { i } p ( g ' ( X _ { i } ) ) - \\frac { h } { 1 2 } p ( g ' ( X _ i ) ) \\right ) . \\end{align*}"}
-{"id": "4889.png", "formula": "\\begin{align*} x \\equiv \\sum _ { k = 0 } ^ { n - 1 } p ^ k [ \\phi ^ { - k } ( r _ k ) ] \\bmod I ^ n \\Big ( \\in \\mathbb { Z } R / I ^ n \\Big ) \\end{align*}"}
-{"id": "872.png", "formula": "\\begin{align*} \\varphi _ { A , k } ( y + t k ) = \\varphi _ { A , k } ( y ) + t \\forall \\ , y \\in Y , \\ , t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "1325.png", "formula": "\\begin{align*} \\Gamma = \\begin{pmatrix} A _ { 1 1 } ^ 0 \\ , E _ 1 & \\dots & A _ { 1 1 } ^ { M - 1 } E _ 1 \\end{pmatrix} \\in \\R ^ { r \\times ( M m ) } \\end{align*}"}
-{"id": "4225.png", "formula": "\\begin{align*} ( x ) _ n = x ( x - 1 ) \\cdots ( x - n + 1 ) = \\prod _ { l = 1 } ^ { n - 1 } ( x - l ) = \\sum _ { l = 0 } ^ n S _ 1 ( n , l ) x ^ l , \\end{align*}"}
-{"id": "2668.png", "formula": "\\begin{align*} \\mathcal I ( t \\varphi , x ) = \\{ f \\in \\mathcal O _ x \\textup { s . t . } \\int _ V | f | e ^ { - t \\varphi } \\omega ^ n < \\infty \\textup { f o r s o m e o p e n s e t } x \\in V \\subset X \\} . \\end{align*}"}
-{"id": "1293.png", "formula": "\\begin{align*} \\Bigl \\vert \\sum _ { n = N - H } ^ { N + H } e ^ { - n / N } \\Bigl ( R ( n ) - ( 2 \\psi ( n ) - n ) \\Bigr ) \\Bigl ( 1 - \\frac { \\vert n - N \\vert } { H } \\Bigr ) \\Bigr \\vert \\ll N \\Bigl ( \\log \\frac { 2 N } { H } \\Bigr ) ^ 2 . \\end{align*}"}
-{"id": "8329.png", "formula": "\\begin{align*} y ^ { 2 7 } - y = \\frac { 1 } { \\left ( T + 1 \\right ) ^ { 2 } } + \\frac { 1 } { T + 1 } + T ^ { 9 } + T ^ { 3 } + T + \\omega + 1 = r ( T ) . \\end{align*}"}
-{"id": "9620.png", "formula": "\\begin{align*} \\Lambda ( \\Psi _ 0 ) = \\sqrt { ( { \\cal H } _ F ( \\Psi _ 0 ) + a ) / b } , \\end{align*}"}
-{"id": "8833.png", "formula": "\\begin{align*} \\sigma _ { k } ( n ) & = \\sum _ { d | n } d ^ { k } , \\\\ \\sigma _ { k , \\chi _ { D } } ^ { \\infty } ( n ) & = \\sum _ { d | n } \\chi _ { D } ( d ) d ^ { k } , \\\\ \\sigma _ { k , \\chi _ { D } } ^ { 0 } ( n ) & = \\sum _ { d | n } \\chi _ { D } ( n / d ) d ^ { k } \\end{align*}"}
-{"id": "5666.png", "formula": "\\begin{align*} \\mathcal { F } ( f ) : = \\int e ^ { - i k \\cdot x } f ( x ) \\d x , \\end{align*}"}
-{"id": "6002.png", "formula": "\\begin{align*} & \\psi _ t ( \\sqrt { L } ) f ( x ) = \\int \\Psi _ t ( x , y ) f ( y ) \\ , d \\mu ( y ) , \\cr & \\sup _ { x \\in X , \\ , t > 0 } \\ \\int | \\Psi _ t ( x , y ) | \\ , d \\mu ( y ) \\leq C . \\end{align*}"}
-{"id": "4046.png", "formula": "\\begin{align*} \\alpha \\gamma \\beta = 2 + \\alpha + \\beta + \\gamma , \\beta \\geq \\alpha \\geq \\gamma \\end{align*}"}
-{"id": "7725.png", "formula": "\\begin{align*} \\sigma _ { N + J + k } \\le F \\left ( ( 1 - \\varepsilon ) \\sum _ { i = 0 } ^ { i = J - 1 } \\sigma _ { N + i } \\right ) . \\end{align*}"}
-{"id": "1474.png", "formula": "\\begin{align*} | m _ f ( Q ) | \\leq \\left ( f \\cdot \\chi _ Q \\right ) ^ * ( \\lambda | Q | ) , \\lim _ { \\ell ( Q ) \\to 0 , Q \\ni x } m _ f ( Q ) = f ( x ) ( { \\rm a . e . } \\ x \\in \\R ^ n ) . \\end{align*}"}
-{"id": "4087.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p - q - r } y ^ q ( a x + b y ) ^ r } { z ^ { p } } = \\dfrac { y ^ q ( a x + b y ) ^ r } { x ^ { - p ' } z ^ { p ' + q + r } } , p ' = p - q - r . \\end{align*}"}
-{"id": "4060.png", "formula": "\\begin{align*} 2 q < r - p + 2 q = \\alpha ( p - q ) - p + 2 q = 1 + n + k \\leq 1 + n + q \\leq 1 + \\dfrac { p } { 2 } + q , \\end{align*}"}
-{"id": "9686.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } { \\log _ 2 t } = \\frac { 1 } { \\log ( 1 / \\beta ) } . \\end{align*}"}
-{"id": "8766.png", "formula": "\\begin{align*} R _ { 1 , 2 } \\left ( x \\right ) R _ { 2 , 1 } \\left ( 1 / x \\right ) = \\mathbb { I } _ 3 \\otimes \\mathbb { I } _ 3 = \\check { R } ( x ) \\check { R } ( 1 / x ) , \\end{align*}"}
-{"id": "4628.png", "formula": "\\begin{align*} \\frac { 3 m + \\ell - 3 | L | + 1 } { 3 m + \\ell - | L | + 1 } \\binom { 3 m + \\ell } { | L | } \\ ; = \\ ; \\binom { 3 m + \\ell } { | L | } - 2 \\binom { 3 m + \\ell } { | L | - 1 } \\ ; . \\end{align*}"}
-{"id": "6687.png", "formula": "\\begin{align*} & \\mathcal { L } _ { X } D _ 1 = ( - \\lambda ) ( D _ 1 - d _ 1 ) , \\dots , \\mathcal { L } _ { X } D _ { p ^ { \\prime } } = ( - \\lambda ) ( D _ { p ^ { \\prime } } - d _ { p ^ { \\prime } } ) , \\\\ & \\mathcal { L } _ { X } D _ { p ^ { \\prime } + 1 } = \\dots = \\mathcal { L } _ { X } D _ { p } = \\mathcal { L } _ { X } I _ 1 = \\dots = \\mathcal { L } _ { X } I _ k = 0 , \\end{align*}"}
-{"id": "3919.png", "formula": "\\begin{align*} w _ \\sigma ( \\eta ) = \\sum _ { k = 0 } ^ \\infty \\ , a _ { k , \\sigma } \\ , \\eta ^ { k + \\sigma } \\ , , \\sigma = 0 , 1 \\ , , \\end{align*}"}
-{"id": "4491.png", "formula": "\\begin{align*} v _ x : = \\inf \\{ u \\geq 0 : ( n - 1 ) p _ { n , x , u } = a _ n ^ + \\} , l _ x : = \\inf \\{ u \\geq 0 : ( n - 1 ) p _ { n , x , u } = a _ n ^ - \\} , \\end{align*}"}
-{"id": "4711.png", "formula": "\\begin{align*} f ( v ) = g ( ( w ^ 1 ) ^ { \\top } v , ( w ^ 2 ) ^ { \\top } v , \\cdots , ( w ^ p ) ^ { \\top } v ) , \\end{align*}"}
-{"id": "3562.png", "formula": "\\begin{align*} \\mathcal { E } ^ { R } ( \\theta ) & = \\Phi ( \\bar { g } ^ R , \\bar { \\pi } ^ R ) + \\mathcal { E } ^ R _ 2 ( \\theta ) - ( \\lambda \\zeta R ^ { - 2 } ( \\log R ) ^ \\frac { 1 } { 2 } , 0 ) , \\end{align*}"}
-{"id": "256.png", "formula": "\\begin{align*} & \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\int _ 0 ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log u _ { x , s } \\ , d s \\ , d x \\\\ & \\leq C _ { d , f } \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\biggl \\{ \\log n + \\log \\biggl ( 1 + \\frac { \\| x \\| } { \\mu _ \\alpha ^ { 1 / \\alpha } ( f ) } \\biggr ) \\biggr \\} \\ , d x = O \\bigl ( \\max \\{ q _ n \\log n , q _ n ^ { 1 - \\epsilon } \\} \\bigr ) , \\end{align*}"}
-{"id": "1045.png", "formula": "\\begin{align*} \\nabla u ( x ) \\cdot \\frac { x } { | x | } - i u ( x ) = o ( | x | ^ { - \\frac 1 2 } ) , | x | \\to \\infty \\end{align*}"}
-{"id": "2450.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ 2 } = \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { L ^ { \\infty } } \\\\ \\lesssim \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } f \\| _ { L ^ { \\infty } } . \\end{align*}"}
-{"id": "2485.png", "formula": "\\begin{align*} \\varepsilon _ K ( \\chi , \\psi ) = \\chi ( c ) \\frac { \\int _ { \\mathcal { O } ^ { \\times } } \\chi ^ { - 1 } ( u ) \\psi ( u / c ) \\ , d u } { | \\int _ { \\mathcal { O } ^ { \\times } } \\chi ^ { - 1 } ( u ) \\psi ( u / c ) \\ , d u | } \\end{align*}"}
-{"id": "995.png", "formula": "\\begin{align*} \\tau ( u ) = { \\rm t r a c e } ( T ( u ) ) = A ( u ) + D ( u ) = c _ 0 + c _ 1 u + c _ 2 u ^ 2 . \\end{align*}"}
-{"id": "3363.png", "formula": "\\begin{align*} \\| \\cdot \\| _ r : = \\sqrt { \\sum _ { i = 1 } ^ { \\lfloor r \\rfloor } \\sigma _ i ^ 2 ( \\cdot ) + ( r - \\lfloor r \\rfloor ) \\sigma _ { \\lceil r \\rceil } ^ 2 ( \\cdot ) } , \\end{align*}"}
-{"id": "5527.png", "formula": "\\begin{align*} S ( x ) = 0 \\ \\ \\ \\ \\ \\ \\ x \\in ( \\lambda _ { \\varphi , w } ) _ a . \\end{align*}"}
-{"id": "1639.png", "formula": "\\begin{align*} x ^ { \\prime } & = v , v ^ { \\prime } = F \\left ( x , v \\right ) \\\\ x \\left ( 0 \\right ) & = x _ { 0 } , v \\left ( 0 \\right ) = v _ { 0 } \\end{align*}"}
-{"id": "3431.png", "formula": "\\begin{align*} y ( t ) = \\zeta + \\int _ t ^ T f ( \\theta , \\xi ( \\theta ) , y ( \\theta ) , z ( \\theta ) ) d s - \\int _ t ^ T z ( \\theta ) d w ( \\theta ) , 0 \\le t \\le T . \\end{align*}"}
-{"id": "9943.png", "formula": "\\begin{align*} X _ 0 \\ni u \\mapsto \\| u \\| _ { X _ 0 } : = \\left ( \\int _ Q | u ( x ) - u ( y ) | ^ 2 K ( x - y ) d x d y \\right ) ^ { 1 / 2 } \\end{align*}"}
-{"id": "9492.png", "formula": "\\begin{align*} \\Phi ^ { * } ( x _ { n } \\ , d y _ { n } - y _ { n } \\ , d x _ { n } ) = e ^ { 2 \\rho } \\ , d \\phi \\Phi ^ { * } ( x _ { n } \\ , d x _ { n } + y _ { n } \\ , d y _ { n } ) = e ^ { 2 \\rho } \\ , d \\rho . \\end{align*}"}
-{"id": "8101.png", "formula": "\\begin{align*} \\left \\Vert u \\right \\Vert _ { L ^ { p } ( \\Omega , d \\mu _ { l } ) } : = \\left ( \\int _ { \\Omega } \\left \\vert u \\right \\vert ^ { p } d \\mu _ { l } ( x ) \\right ) ^ { 1 / p } < + \\infty . \\end{align*}"}
-{"id": "9873.png", "formula": "\\begin{align*} \\int \\limits _ { \\Omega } \\left ( | \\nabla u ( x ) | ^ { p - 2 } \\nabla u ( x ) \\right ) \\cdot \\nabla v ( x ) ~ d x = \\mu _ p \\int \\limits _ { \\Omega } | u | ^ { p - 2 } u ( x ) v ( x ) ~ d x \\end{align*}"}
-{"id": "2814.png", "formula": "\\begin{align*} \\int _ { \\R } \\frac { \\phi ' ( x ) } { 1 + x ^ 2 } d x = \\int _ { \\R } \\frac { d \\phi ( x ) } { 1 + x ^ 2 } < \\infty . \\end{align*}"}
-{"id": "3640.png", "formula": "\\begin{align*} u _ { n + 1 } = c K _ 1 + ( 1 - c ) [ c _ { K _ 1 } T _ { K _ 1 } + ( 1 - c _ { K _ 1 } ) f ( u _ n , u _ { n - 1 } , \\dots , u _ { n - k + 1 } ) ] \\end{align*}"}
-{"id": "4088.png", "formula": "\\begin{align*} F [ x : y : z ] = [ a _ { 1 1 } x + a _ { 1 2 } y + a _ { 1 3 } z : a _ { 2 1 } x + a _ { 2 2 } y + a _ { 2 3 } z : a _ { 3 1 } x + a _ { 3 2 } y + a _ { 3 3 } z ] , \\end{align*}"}
-{"id": "7132.png", "formula": "\\begin{align*} z _ j '' ( t ) = - \\frac { 2 \\bar { z } _ j ( t ) z _ j '^ 2 ( t ) } { R ^ 2 - \\vert z _ j ( t ) \\vert ^ 2 } + \\frac { 4 R } { \\lambda _ j ( t ) } \\sum _ { \\underset { i \\neq j } { i = 0 } } ^ { n - 1 } \\frac { P _ { j , i } ( t ) } { \\Theta _ { j , i } ( t ) ^ { 3 / 2 } } , 0 \\leq j \\leq n - 1 , \\end{align*}"}
-{"id": "4944.png", "formula": "\\begin{align*} \\| h \\| _ 2 & = \\| D ^ * h \\| _ 2 \\\\ & = \\sqrt { \\| D ^ * _ { S _ 0 } h \\| _ 2 ^ 2 + \\| D ^ * _ { S _ 0 ^ c } h \\| _ 2 ^ 2 } \\\\ & \\leq \\sqrt { \\| D ^ * _ { S _ 0 } h \\| _ 2 ^ 2 + ( \\| D ^ * _ { S _ 0 } h \\| _ 2 + R ) ^ 2 } \\\\ & \\leq \\sqrt { 2 \\| D ^ * _ { S _ 0 } h \\| _ 2 ^ 2 } + R \\leq \\sqrt { 2 } z + R \\\\ & \\leq \\frac { \\sqrt { 2 ( 1 + \\delta ) } } { 1 - \\sqrt { t / ( t - 1 ) } \\delta } \\epsilon + \\left ( \\frac { \\sqrt { 2 } \\delta + \\sqrt { t ( \\sqrt { ( t - 1 ) / t } - \\delta ) \\delta } } { t ( \\sqrt { ( t - 1 ) / t } - \\delta ) } + 1 \\right ) R . \\end{align*}"}
-{"id": "1180.png", "formula": "\\begin{align*} D ( - \\epsilon ) = W ( \\hat c _ 0 , c _ 0 ) \\prod _ { j = 1 } ^ N \\left ( 1 + \\frac { \\epsilon } { z _ j } \\right ) . \\end{align*}"}
-{"id": "3177.png", "formula": "\\begin{align*} E _ \\rho : = \\widetilde { Y } \\times \\mathbb { C } ^ r / \\sim _ \\rho , \\end{align*}"}
-{"id": "7046.png", "formula": "\\begin{align*} \\textbf { u } ^ { ( i ) } = ( u ^ { ( i ) } \\mid u '^ { ( i ) } ) = ( u _ { 0 + i } , u _ { 1 + i } , \\dots , u _ { \\alpha - 1 + i } \\mid u ' _ { 0 + i } , u ' _ { 1 + i } , \\dots , u ' _ { \\beta - 1 + i } ) \\end{align*}"}
-{"id": "8728.png", "formula": "\\begin{align*} v ( 0 , x ) = \\widetilde Y _ { 0 } ^ { x } = \\int _ 0 ^ T e ^ { - s A } G B ( s , X ^ { x } _ s ) \\ , d s - \\int _ 0 ^ T e ^ { - s A } \\widetilde Z ^ { x } _ { s } \\ ; d W _ s \\end{align*}"}
-{"id": "1250.png", "formula": "\\begin{align*} M _ { 1 1 } ( x , y ) = \\left ( \\begin{array} { c c } 1 & 0 \\\\ 0 & 0 \\end{array} \\right ) , M _ { 2 2 } ( x , y ) = \\left ( \\begin{array} { c c } 0 & 0 \\\\ 0 & 1 \\end{array} \\right ) . \\end{align*}"}
-{"id": "6473.png", "formula": "\\begin{align*} ( g _ { 1 - \\alpha } * W ) ( t _ { * } ) \\leq & g _ { 1 - \\alpha } ( ( t _ { 2 } - t _ { 1 } ) / 4 ) \\int _ { 0 } ^ { t _ { 2 } - t _ { 0 } } \\int _ { \\rho B _ { 1 } } w ^ { 2 } d x d s \\\\ = & \\frac { 4 ^ { \\alpha } } { \\Gamma ( 1 - \\alpha ) ( \\sigma \\eta ) ^ { \\alpha } ( \\rho - \\rho ' ) ^ { \\alpha } } \\int _ { 0 } ^ { t _ { 2 } - t _ { 0 } } \\int _ { \\rho B _ { 1 } } w ^ { 2 } d x d s . \\end{align*}"}
-{"id": "9750.png", "formula": "\\begin{align*} \\dfrac { \\partial \\vec { h } } { \\partial \\vec { x } } ( \\vec { x } _ 0 ) : = \\left [ \\begin{array} { c c c } \\dfrac { \\partial { h } _ 1 } { \\partial x _ 1 } ( \\vec { x } _ 0 ) & \\cdots & \\dfrac { \\partial { h } _ 1 } { \\partial x _ n } ( \\vec { x } _ 0 ) \\\\ \\vdots & \\ddots & \\vdots \\\\ \\dfrac { \\partial { h } _ m } { \\partial x _ 1 } ( \\vec { x } _ 0 ) & \\cdots & \\dfrac { \\partial { h } _ m } { \\partial x _ n } ( \\vec { x } _ 0 ) \\end{array} \\right ] , \\end{align*}"}
-{"id": "4930.png", "formula": "\\begin{align*} \\frac { \\eta _ 1 - \\eta _ l } { c _ 1 - c _ l } = \\frac { \\eta _ 1 - \\eta _ j } { c _ 1 - c _ j } \\in D \\Sigma _ k ^ N \\backslash \\{ 0 \\} , \\ , \\ , j , l \\in [ 2 : p ] , \\ , \\ , j \\neq l \\end{align*}"}
-{"id": "3468.png", "formula": "\\begin{align*} \\sigma _ 0 = \\sigma ( c _ 0 ) \\sigma ' _ 0 = \\sigma ' ( c _ 0 ) . \\end{align*}"}
-{"id": "1710.png", "formula": "\\begin{align*} H ( x \\times y , x ) = 0 \\textrm { a n d } H ( x \\times y , x \\times y ) = H ( x , x ) H ( y , y ) - H ( x , y ) ^ 2 \\end{align*}"}
-{"id": "663.png", "formula": "\\begin{align*} \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\phi ) \\chi _ C ) \\ , d \\eta ( g ) = \\nu ( \\phi ) \\nu ( C ) , \\end{align*}"}
-{"id": "1039.png", "formula": "\\begin{align*} c _ { q , q _ 1 } ^ { d ( q _ 2 + 1 ) } ~ = ~ ( 1 + \\delta _ { q _ 1 , q _ 2 } ) { \\ , } c _ { r _ 0 , u _ 0 } ^ d c _ { d , d } ^ { d u _ 0 } . \\end{align*}"}
-{"id": "9196.png", "formula": "\\begin{align*} & d Y ( t , x ) = [ A _ { u ( t , x , Z ) } Y ( t , x ) + a ( t , x , Y ( t , x ) , u ( t , x , Z ) , Z ) ] d t + b ( t , x , Y ( t , x ) , u ( t , x , Z ) , Z ) d B ( t ) \\\\ & + \\int _ { \\mathbb { R } } c ( t , x , Y ( t , x ) , u ( t , x , Z ) , Z , \\zeta ) \\tilde { N } ( d t , d \\zeta ) ; ( t , x ) \\in ( 0 , T ) \\times D . \\end{align*}"}
-{"id": "2059.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ { n + 1 } ^ 2 = \\tfrac { 1 } { K } . \\end{align*}"}
-{"id": "5671.png", "formula": "\\begin{align*} E \\cdot \\nabla _ v \\left ( \\partial _ { v _ i } F \\right ) + \\nu \\left ( \\partial _ { v _ i } F \\right ) & = \\int \\sigma ( v , v ' ) F ( v ' ) \\d v ' \\ , \\partial _ { v _ i } M ( v ) \\\\ & + \\int \\partial _ { v _ i } \\sigma ( v , v ' ) F ( v ' ) \\d v ' M ( v ) - \\left ( \\partial _ { v _ i } \\nu \\right ) F . \\end{align*}"}
-{"id": "2340.png", "formula": "\\begin{align*} \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\bigl ( \\| S _ i ^ \\ast y _ i ^ \\ast + x ^ \\ast \\| + \\| S _ i ^ \\ast y _ i ^ \\ast - x ^ \\ast \\| \\bigr ) > ( \\delta - \\varepsilon ) \\| x ^ \\ast \\| + \\frac { 2 } { n } \\sum _ { i = 1 } ^ n \\| S _ i ^ \\ast y _ i ^ \\ast \\| . \\end{align*}"}
-{"id": "6230.png", "formula": "\\begin{align*} & \\tilde { f } ( H ( s ) ) = f ( \\tilde { H } ( s ) ) \\ \\ \\ \\tilde { H } ( s ) \\ \\\\ & \\frac { \\| H ( s ) - \\tilde { H } ( s ) \\| _ { H _ { 2 } \\ \\ H _ { \\infty } } } { \\| H ( s ) \\| _ { H _ { 2 } \\ \\ H _ { \\infty } } } = \\ \\mathcal { O } ( | | Z | | ) , \\end{align*}"}
-{"id": "6285.png", "formula": "\\begin{align*} q ( x ) = \\sum ^ { w } _ { i = 0 } q _ i x ^ i = p _ d \\prod ^ w _ { j = 1 } ( x - x _ j ) , ~ ~ ~ p _ d \\ne 0 . \\end{align*}"}
-{"id": "7291.png", "formula": "\\begin{align*} [ E _ \\xi , F _ t ] ^ * K _ t = ( K _ t E _ \\xi F _ { t } - K _ t F _ t E _ \\xi ) ^ * = ( q ^ { ( \\alpha _ t , \\xi ) } E _ \\xi K _ t F _ t - K _ t F _ t E _ \\xi ) ^ * . \\end{align*}"}
-{"id": "8544.png", "formula": "\\begin{align*} \\xi _ { b r } = \\xi _ { b } ' \\xi _ { b r } ' : N _ { \\sigma ( b ) } \\rightarrow N _ { o u t } \\\\ \\pi _ { b r } = \\pi _ { b r } ' \\pi _ { b } ' : N _ { o u t } \\rightarrow N _ { \\sigma ( b ) } \\end{align*}"}
-{"id": "8388.png", "formula": "\\begin{align*} W ^ * ( Y ) = W ^ * ( X _ 0 ) = W ^ * ( X ) , \\end{align*}"}
-{"id": "6455.png", "formula": "\\begin{align*} = \\int _ { \\rho B _ { 1 } } \\int _ { \\mathbb { R } ^ { n } \\backslash \\rho B _ { 1 } } ( \\tilde { u } ( s , x ) - \\tilde { u } ( s , y ) ) ( - \\psi ^ { 2 } ( x ) \\tilde { u } ^ { - q } ( s , x ) ) k ( x , y ) d y d x . \\end{align*}"}
-{"id": "4254.png", "formula": "\\begin{align*} \\mu ( x _ q ^ k ) = \\mu \\left ( \\frac { k } { q } + \\frac { \\alpha ( k / q ) } { q ^ 2 } \\right ) \\left ( 1 + \\varepsilon O ( q ^ { - 4 } ) \\right ) . \\end{align*}"}
-{"id": "2628.png", "formula": "\\begin{align*} F _ m ( \\lambda ) = \\varphi _ m ( \\lambda ) ( \\varphi _ m ( \\lambda ) ) ^ * , \\ ; \\varphi _ m ( \\lambda ) = \\sum _ { u = 0 } ^ \\infty \\varphi _ m ( u ) e ^ { - i u \\lambda } , \\end{align*}"}
-{"id": "5828.png", "formula": "\\begin{align*} F \\left ( 0 , S \\right ) = S \\end{align*}"}
-{"id": "2021.png", "formula": "\\begin{align*} I ( u ) & = \\frac { 1 } { \\gamma \\alpha } ( \\log | x | ) ^ { 1 - \\alpha } \\left [ 1 - \\exp \\left \\{ - \\gamma ( \\log | x | ) ^ \\alpha \\left [ 1 - \\left ( 1 + \\frac { \\log u } { \\log | x | } \\right ) ^ \\alpha \\right ] \\right \\} \\right ] \\\\ & \\leq \\frac { 1 } { \\gamma \\alpha } ( \\log | x | ) ^ { 1 - \\alpha } \\left [ 1 - \\exp \\left \\{ 2 \\gamma \\alpha ( \\log | x | ) ^ { \\alpha - 1 } \\log u \\right \\} \\right ] \\\\ & \\leq - 2 \\log u , \\end{align*}"}
-{"id": "3249.png", "formula": "\\begin{gather*} \\Delta ( K _ i ) = K _ i \\otimes K _ i , \\Delta ( E _ i ) = E _ i \\otimes 1 + K _ i \\otimes E _ i , \\Delta ( F _ i ) = F _ i \\otimes K _ i ^ { - 1 } + 1 \\otimes F _ i , \\\\ S ( K _ { i } ) = K _ { i } ^ { - 1 } , S ( E _ { i } ) = - K _ { i } ^ { - 1 } E _ { i } , S ( F _ { i } ) = - F _ { i } K _ { i } , \\varepsilon ( K _ i ) = 1 , \\varepsilon ( E _ i ) = \\varepsilon ( F _ i ) = 0 . \\end{gather*}"}
-{"id": "1966.png", "formula": "\\begin{align*} \\nabla ^ \\lambda _ X ( \\lambda Y \\xi ) - & \\lambda Y \\nabla ^ \\lambda _ X \\xi = \\nabla _ X ( \\lambda Y \\xi ) + \\frac { 1 } { 4 \\lambda } ( Z X - X Z ) ( \\lambda Y \\xi ) \\\\ & \\phantom { \\lambda Y \\nabla ^ \\lambda _ X \\xi = } - \\lambda Y \\left ( \\nabla _ X \\xi + \\frac { 1 } { 4 \\lambda } ( Z X - X Z ) \\xi \\right ) \\\\ = & \\left ( \\nabla ^ { L C } _ X ( \\lambda Y ) \\right ) \\xi + \\frac 1 4 ( Z X Y - X Z Y - Y Z X + Y X Z ) \\xi \\\\ = & \\lambda \\left ( \\nabla ^ { L C , \\lambda } _ X ( Y ) \\right ) \\xi \\ , , \\end{align*}"}
-{"id": "7002.png", "formula": "\\begin{align*} \\delta _ v \\Theta ( L ) ( x , y , z ) ( a ) - \\delta _ L \\Theta ^ 2 ( x , y , z ) ( a ) = 0 . \\end{align*}"}
-{"id": "296.png", "formula": "\\begin{align*} \\int _ { - 2 \\sqrt { \\log n } } ^ { 2 \\sqrt { \\log n } } \\int _ { - 2 \\sqrt { \\log n } } ^ { 2 \\sqrt { \\log n } } \\{ \\Phi _ \\Sigma ( s , t ) - \\Phi ( s ) \\Phi ( t ) \\} \\ , d s \\ , d t = \\alpha _ z + O ( n ^ { - 2 } ) \\end{align*}"}
-{"id": "8818.png", "formula": "\\begin{align*} x _ { i } ( t ) = \\sum _ { n \\leq 0 } x _ { i , n } t ^ { - n } , \\ ; 1 \\leq i \\leq k , \\end{align*}"}
-{"id": "7987.png", "formula": "\\begin{align*} X : x _ { 0 } ^ { 3 } x _ { 1 } + x _ { 0 } x _ { 1 } ^ { 3 } + x _ { 0 } x _ { 2 } ^ { 3 } + x _ { 0 } ^ { 2 } x _ { 1 } x _ { 3 } + x _ { 1 } x _ { 2 } ^ { 2 } x _ { 3 } + x _ { 0 } ^ { 2 } x _ { 3 } ^ { 2 } + x _ { 1 } ^ { 2 } x _ { 3 } ^ { 2 } + x _ { 0 } x _ { 3 } ^ { 3 } = 0 . \\end{align*}"}
-{"id": "9683.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\sigma ( t ) } { t } = \\lambda \\in [ 0 , \\infty ] . \\end{align*}"}
-{"id": "4421.png", "formula": "\\begin{align*} \\| f \\| _ { \\widehat { E } } = \\lim _ { n \\to \\infty } \\| \\tilde { f _ n } \\| _ { \\widehat { E } } = \\lim _ { n \\to \\infty } \\| \\tilde { f _ n } \\circ \\sigma ^ { - 1 } \\| _ { \\widehat { E } } = \\| \\mu ( f ) \\| _ { \\widehat { E } } . \\end{align*}"}
-{"id": "1868.png", "formula": "\\begin{align*} R _ { i j k l } = & W _ { i j k l } + \\frac { 1 } { 2 } ( R _ { i k } g _ { j l } + R _ { j l } g _ { i k } - R _ { i l } g _ { j k } - R _ { j k } g _ { i l } ) \\\\ & - \\frac { 1 } { 6 } R ( g _ { i k } g _ { j l } - g _ { i l } g _ { j k } ) . \\end{align*}"}
-{"id": "8314.png", "formula": "\\begin{align*} z ^ { p ^ 2 } - z & = ( y ^ p ) ^ { p ^ 2 } - ( T ^ { \\lambda } ) ^ { p ^ 2 } - y ^ p + T ^ { \\lambda } = ( y ^ { p ^ 2 } - y ) ^ p - T ^ { \\lambda p ^ 2 } + T ^ { \\lambda } \\\\ & = ( T ^ { \\lambda p } ) ^ p - T ^ { \\lambda p ^ 2 } + T ^ { \\lambda } = T ^ { \\lambda } = \\nu , \\end{align*}"}
-{"id": "5653.png", "formula": "\\begin{align*} \\int _ y ^ { \\varphi ( t , y ) } \\frac { h ( 0 ) - \\left ( \\frac 1 p - 1 \\right ) F ' ( s ) } { F ( s ) } d s \\leq & - \\frac { h ( 0 ) } { r - 1 } \\left ( y ^ { 1 - r } - \\varphi ( t , y ) ^ { 1 - r } \\right ) + C \\\\ = & h ( 0 ) t + C . \\end{align*}"}
-{"id": "6386.png", "formula": "\\begin{align*} \\mu _ { ( k ) } ^ { G } = \\sigma ^ { k } ( k - 1 ) ! ! \\end{align*}"}
-{"id": "8513.png", "formula": "\\begin{align*} x = \\# ( \\ + \\ \\# \\Big ( y = \\# ) \\ + \\ 2 \\# \\Big ) . \\end{align*}"}
-{"id": "6376.png", "formula": "\\begin{align*} \\begin{cases} L _ a U = | y | ^ a f & B _ 1 \\setminus \\mathcal { P } \\\\ U = 0 & \\mathcal { P } \\end{cases} \\end{align*}"}
-{"id": "6695.png", "formula": "\\begin{align*} a \\star f = a f \\mbox { a n d } f \\star b = f b \\end{align*}"}
-{"id": "1632.png", "formula": "\\begin{align*} T ^ { I } \\left ( L _ { Y _ { I } ^ { \\alpha } } C ^ { \\alpha } + 2 \\psi _ { I } C ^ { \\alpha } \\right ) - T _ { , t } ^ { I } Y _ { I } ^ { \\alpha } - 2 g ^ { \\beta \\alpha } a _ { , \\beta } = 0 . \\end{align*}"}
-{"id": "6396.png", "formula": "\\begin{align*} H = \\frac { 1 } { 2 } \\log \\frac { 2 \\pi \\sigma ^ { 2 } } { C ^ { 2 } ( \\vec { \\varepsilon } ) } + \\frac { \\mu _ { ( 2 ) } } { 2 \\sigma ^ { 2 } } - \\varepsilon _ { q } \\mu _ { ( q ) } + \\varepsilon _ { p } \\mu _ { ( p ) } . \\end{align*}"}
-{"id": "6724.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( s ) \\diamond d B ^ { H } ( s ) : = \\lim _ { n \\rightarrow 0 } \\sum _ { i = 1 } ^ { n } F ( t _ { i - 1 } ) \\diamond ( B ^ { H } ( t _ { i } ) - B ^ { H } ( t _ { i - 1 } ) ) \\end{align*}"}
-{"id": "4193.png", "formula": "\\begin{align*} \\norm { f } _ { H ^ { ( n + 1 ) / 2 } } ^ 2 = \\int _ { \\R ^ n } f ( x ) \\bigl [ ( I - \\Delta ) ^ { ( n + 1 ) / 2 } f \\bigr ] ( x ) \\ d x , \\end{align*}"}
-{"id": "6610.png", "formula": "\\begin{align*} f ( y ) - f ( x ) & = h ( 1 ) - h ( 0 ) = \\int _ { 0 } ^ { 1 } \\nabla h ( \\tau ) d \\tau = \\int _ { 0 } ^ 1 \\langle y - x , \\nabla f ( x + \\tau ( y - x ) ) \\rangle d \\tau \\end{align*}"}
-{"id": "8228.png", "formula": "\\begin{align*} N _ 1 ( \\phi ) = E \\sum _ { n \\in \\mathbb { Z } \\backslash \\{ 0 \\} } \\left ( \\bar { V } _ n { \\bf u } _ n + V _ n \\bar { \\bf u } _ n + \\bar { U } _ n { \\bf v } _ n + U _ n \\bar { \\bf v } _ n \\right ) , \\end{align*}"}
-{"id": "3324.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 3 ) ^ * } = \\left ( \\begin{array} { c c c } 0 & 0 & q ^ 4 \\end{array} \\right ) , \\Gamma _ { 0 } ^ { ( 3 ) ^ * } = \\left ( \\begin{array} { c c c } 0 & - q ^ 6 & 0 \\end{array} \\right ) , \\Gamma _ { - } ^ { ( 3 ) ^ * } = \\left ( \\begin{array} { c c c } q ^ 4 & 0 & 0 \\end{array} \\right ) . \\end{align*}"}
-{"id": "2134.png", "formula": "\\begin{align*} g _ { \\gamma } ( t ) = \\frac { t ^ { \\gamma - 1 } } { \\Gamma ( \\gamma ) } , \\end{align*}"}
-{"id": "2227.png", "formula": "\\begin{align*} ( g _ { \\alpha } * ( \\varphi \\dot { v } ) ) ( t ) = \\varphi ( t ) ( g _ { \\alpha } * \\dot { v } ) ( t ) + \\int _ { 0 } ^ { t } v ( \\sigma ) \\partial _ { \\sigma } ( g _ { \\alpha } ( t - \\sigma ) [ \\varphi ( t ) - \\varphi ( \\sigma ) ] ) d \\sigma , \\end{align*}"}
-{"id": "2525.png", "formula": "\\begin{align*} h ( n ) = \\begin{cases} \\frac { 2 n } { a b } - 2 & \\mbox { i f } n \\equiv 0 \\bmod { a b } , \\\\ \\frac { 2 ( n - a ) } { a b } - 1 & \\mbox { i f } n \\equiv a \\bmod { a b } , \\\\ \\frac { 2 ( n - b ) } { a b } - 1 & \\mbox { i f } n \\equiv b \\bmod { a b } , \\\\ \\frac { 2 ( n - a - b ) } { a b } & \\mbox { i f } n \\equiv a + b \\bmod { a b } . \\end{cases} \\end{align*}"}
-{"id": "3502.png", "formula": "\\begin{align*} \\mu & = \\frac { 1 } { 2 } ( R _ g - | K | ^ 2 _ g + ( \\textup { t r } _ g K ) ^ 2 ) \\\\ J & = \\textup { d i v } _ g K - d ( \\textup { t r } _ g K ) \\end{align*}"}
-{"id": "6497.png", "formula": "\\begin{align*} \\partial _ { t } ^ { \\alpha } \\tilde { u } ( x , t ) & = - L ( \\mu * v ) + \\frac { d } { d t } ( g _ { \\alpha } * \\rho ) * g _ { 1 - \\alpha } \\cdot g \\\\ & = - L \\tilde { u } ( x , t ) + \\rho g . \\end{align*}"}
-{"id": "374.png", "formula": "\\begin{align*} \\| u _ { 0 , n } \\| _ { B ^ \\gamma _ { 2 , \\infty } } & \\le n ^ { - 1 } + \\| n ^ { - s } \\cos ( n x ) \\| _ { B ^ \\gamma _ { 2 , \\infty } } \\\\ & = n ^ { - 1 } + \\sup _ q n ^ { - s } 2 ^ { \\gamma q } \\| \\Delta _ q \\cos ( n x ) \\| _ { L ^ 2 } , \\end{align*}"}
-{"id": "4423.png", "formula": "\\begin{align*} V \\mu ( x ) = x r + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } . \\end{align*}"}
-{"id": "1095.png", "formula": "\\begin{align*} \\beta _ 0 = 1 / ( 3 ^ 5 \\cdot 2 ^ 9 ) \\end{align*}"}
-{"id": "957.png", "formula": "\\begin{align*} \\Vert u \\Vert _ { g r a p h } ^ 2 : = Q _ { q } ( u , u ) = \\Vert \\bar \\partial _ M u \\Vert ^ 2 _ { L ^ 2 _ { ( 0 , q + 1 ) } ( M ) } + \\Vert \\bar \\partial ^ * _ M u \\Vert ^ 2 _ { L ^ 2 _ { ( 0 , q - 1 ) } ( M ) } \\ ; . \\end{align*}"}
-{"id": "4795.png", "formula": "\\begin{align*} z ' + \\frac { 1 } { t } \\ , z = \\pm \\frac { a } { t } . \\end{align*}"}
-{"id": "1181.png", "formula": "\\begin{align*} \\begin{gathered} b _ { - 1 } ( x ) \\stackrel { } { = } \\frac { c _ 0 ( x ) \\hat c _ 0 ( x ) } { W ( c _ 0 , \\hat c _ 0 ) } , \\\\ \\\\ b _ 0 ( x ) \\stackrel { } { = } \\ , \\frac { - ( c _ 0 ( x ) \\hat c _ 1 ( x ) + c _ 1 ( x ) \\hat c _ 0 ( x ) ) } { W ( c _ 0 , \\hat c _ 0 ) } , \\end{gathered} \\end{align*}"}
-{"id": "4173.png", "formula": "\\begin{align*} h ( z ) = \\tfrac { \\displaystyle 1 + w ( z ) } { \\displaystyle 1 - w ( z ) } = 1 + d _ 1 z + d _ 2 z ^ 2 + \\cdots , \\end{align*}"}
-{"id": "7446.png", "formula": "\\begin{align*} \\eta : = A _ g \\xi _ s + A _ h \\xi _ t - A _ f \\xi _ r . \\end{align*}"}
-{"id": "4341.png", "formula": "\\begin{align*} & f ' ( r ) = - | g ( r ) | ^ { ( 2 - p ) / ( p - 1 ) } g ( r ) \\ , g ' ( r ) = - | g ( r ) | + f ( r ) \\ , r \\in ( 0 , \\mathcal { R } ( a ) ) \\ , \\\\ & f ( 0 ) = a \\ , \\ g ( 0 ) = 0 \\ , \\end{align*}"}
-{"id": "1852.png", "formula": "\\begin{align*} & \\sum _ { - \\log t < k < \\log t } \\int _ { n \\exp ( - 2 \\sqrt { \\frac { \\log t } { t } } ) } ^ { n \\exp ( 2 \\sqrt { \\frac { \\log t } { t } } ) } x ^ { - 1 / 2 } \\exp \\left ( - \\frac { t } { 2 } \\log ^ 2 \\frac { x } { n } \\right ) \\exp \\left ( - t i \\log \\frac { x } { n } - 2 \\pi i k x \\right ) d x \\\\ & = \\sqrt { n } \\sum _ { - \\log t < k < \\log t } \\int _ { - 2 \\sqrt { \\frac { \\log t } { t } } } ^ { 2 \\sqrt { \\frac { \\log t } { t } } } \\exp ( x / 2 ) \\exp \\left ( - t ( i x + x ^ 2 / 2 ) - 2 \\pi i k n e ^ x \\right ) d x . \\end{align*}"}
-{"id": "4564.png", "formula": "\\begin{align*} W _ 3 ' = O \\biggl ( \\max \\biggl \\{ \\frac { \\log n } { n k ^ { 1 / 2 } } \\ , , \\ , \\frac { k ^ { \\frac { 1 } { 2 } + \\frac { 2 \\beta } { d } } } { n ^ { 1 + \\frac { 2 \\beta } { d } } } \\ , , \\ , \\frac { k ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "3895.png", "formula": "\\begin{align*} z = \\frac { 1 } { \\gamma _ R } + \\gamma _ R y \\ ; . \\end{align*}"}
-{"id": "5973.png", "formula": "\\begin{align*} | \\Phi | - | \\Phi _ x | \\geq n ( n - 1 ) - ( n - 1 ) ( n - 2 ) = 2 ( n - 1 ) > ( n - 1 ) . \\end{align*}"}
-{"id": "8442.png", "formula": "\\begin{align*} = ( A _ { 0 } ( \\tilde { \\textbf { u } } _ { 1 } + \\bar { \\textbf { u } } ) - A _ { 0 } ( \\tilde { \\textbf { u } } _ { 2 } + \\bar { \\textbf { u } } ) ) \\partial _ { t } { \\textbf { u } } _ { 1 } + \\sum _ { j = 1 } ^ { d } ( A _ { 0 } A _ { j } ( \\tilde { \\textbf { u } } _ { 1 } + \\bar { \\textbf { u } } ) - A _ { 0 } A _ { j } ( \\tilde { \\textbf { u } } _ { 2 } + \\bar { \\textbf { u } } ) ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } _ { 1 } . \\end{align*}"}
-{"id": "2831.png", "formula": "\\begin{align*} a _ { i , u } ^ k = a _ i ^ k + u q _ i ^ k , 0 \\le u \\le L _ i ^ k , L _ i ^ k = [ 1 / q _ i ^ k ] , q _ i ^ k = \\theta _ i ^ k ( y _ i ^ k ) ^ { - \\frac { 2 } { \\alpha } } , \\theta _ i ^ k = \\left ( y _ i ^ k \\right ) ^ { - \\frac { 8 } { \\alpha } } , y _ i ^ k = f _ p ( a _ i ^ k ) . \\end{align*}"}
-{"id": "6614.png", "formula": "\\begin{align*} \\langle \\nabla f _ l ( y ) , x - y \\rangle = \\langle \\nabla f _ h ( y ) , x - y \\rangle _ H = \\langle H \\nabla f _ h ( y ) , x - y \\rangle \\end{align*}"}
-{"id": "6981.png", "formula": "\\begin{align*} \\delta _ H \\Theta ( A ) = 0 . \\end{align*}"}
-{"id": "8262.png", "formula": "\\begin{align*} \\rho ( \\lambda ) : = \\log Z _ \\lambda = \\log \\int _ \\R \\exp ( \\lambda u - V ( u ) ) \\dd u , \\end{align*}"}
-{"id": "1968.png", "formula": "\\begin{align*} { \\mathcal D } _ w ( T ) = \\{ d _ w ( Z , Z ' ) \\mid Z , Z ' \\subseteq T , \\ ; | Z | = | Z ' | = w \\} . \\end{align*}"}
-{"id": "7371.png", "formula": "\\begin{align*} - \\frac { c _ { 3 } } { c _ { 2 } } q ^ { 1 0 } = \\frac { c _ { 2 } } { c _ { 1 } } ( t - t ^ \\prime q ^ { - 1 } ( q - q ^ { - 1 } ) ) , 0 = \\frac { c _ { 2 } } { c _ { 1 } } q ^ { - 4 } ( t q ( q - q ^ { - 1 } ) + t ^ \\prime ) . \\end{align*}"}
-{"id": "8486.png", "formula": "\\begin{align*} P ^ { \\varepsilon } = P _ { 0 } + \\varepsilon \\tilde { P } ^ { \\varepsilon } , \\end{align*}"}
-{"id": "4945.png", "formula": "\\begin{align*} - \\Delta u + V ( x ) u - Q ( x ) | u | ^ { p - 2 } u = \\lambda u , x \\in \\R ^ N , \\end{align*}"}
-{"id": "2734.png", "formula": "\\begin{align*} f \\ast g = \\sum _ { r = s } ^ \\infty \\nu ^ r C _ r ( f , g ) , \\end{align*}"}
-{"id": "1547.png", "formula": "\\begin{align*} E ( \\mu ) = \\left \\{ \\begin{array} { l l } \\displaystyle \\int _ { \\Omega } G \\left ( \\frac { d \\mu } { d \\lambda } ( x ) \\right ) d \\lambda ( x ) , & \\mbox { i f } \\ \\mu \\ll \\lambda \\\\ + \\infty & \\mbox { o t h e r w i s e , } \\end{array} \\right . \\end{align*}"}
-{"id": "4010.png", "formula": "\\begin{align*} | \\widehat { S T } | = | \\widehat { S T ' } | + | S T | \\geq \\sigma _ q ( d _ { s + 1 } , d _ s ) + \\sigma _ q ( d _ s , d _ { s - 1 } ) . \\end{align*}"}
-{"id": "8371.png", "formula": "\\begin{align*} \\phi ( x ) w = w x w ^ * w = w x z _ 1 = w z _ 1 x = w x , \\ \\ \\ x \\in M , \\end{align*}"}
-{"id": "1627.png", "formula": "\\begin{align*} X = \\xi ^ { t } \\left ( t \\right ) \\partial _ { t } + \\xi ^ { \\alpha } \\left ( t , x ^ { \\beta } \\right ) \\partial _ { \\alpha } + \\left ( a ( x ^ { \\alpha } , t ) u + b ( x ^ { a } , t ) \\right ) \\partial _ { u } . \\end{align*}"}
-{"id": "6419.png", "formula": "\\begin{align*} H _ { 0 } ^ { s } ( \\Omega ) = H _ { e } ^ { s } ( \\Omega ) s \\notin \\left \\{ \\frac { 1 } { 2 } , \\frac { 3 } { 2 } , \\frac { 5 } { 2 } , \\cdots \\right \\} . \\end{align*}"}
-{"id": "603.png", "formula": "\\begin{align*} \\mathcal { F } _ { \\varrho } ^ + ( z ) & : = \\sum _ { n \\gg - \\infty } c _ { \\mathcal { F } , \\varrho } ^ + ( n ) e ^ { \\frac { 2 \\pi i n z } { \\ell _ { \\varrho } } } , \\\\ \\mathcal { F } _ { \\varrho } ^ - ( z ) & : = c _ { \\mathcal { F } , \\varrho } ^ - ( 0 ) y ^ { 1 - \\kappa } + \\sum _ { \\substack { n \\ll \\infty \\\\ n \\neq 0 } } c _ { \\mathcal { F } , \\varrho } ^ - ( n ) \\Gamma \\left ( 1 - \\kappa , - \\frac { 4 \\pi n y } { \\ell _ { \\varrho } } \\right ) e ^ { \\frac { 2 \\pi i n z } { \\ell _ { \\varrho } } } , \\end{align*}"}
-{"id": "74.png", "formula": "\\begin{align*} G ' ( y ) & + \\frac { p \\kappa ^ { 1 / p } } { 1 - y } \\left [ \\varphi ( y , a ^ * ) ^ { ( p - 1 ) / p } - \\varphi ( y , a _ * ) ^ { ( p - 1 ) / p } \\right ] \\\\ & = p \\frac { G ( y ) } { 1 - y } + \\frac { p } { p - 1 } \\left [ ( a ^ * ) ^ { 2 - p } - ( a _ * ) ^ { 2 - p } \\right ] ( 1 - y ) ^ { 1 - p } \\end{align*}"}
-{"id": "9579.png", "formula": "\\begin{align*} H _ F \\psi = ( \\Delta - m ^ 2 ) \\psi _ { r e g } , \\quad \\psi \\in D _ F . \\end{align*}"}
-{"id": "677.png", "formula": "\\begin{align*} \\lambda = \\frac { 1 } { 2 } \\lambda _ { + } + \\frac { 1 } { 2 } \\lambda _ { - } ( T ) , \\textrm { w h e r e $ \\lambda _ { \\pm } ( \\cdot ) = 2 \\lambda ( \\cdot \\cap \\pm T ) $ } . \\end{align*}"}
-{"id": "3687.png", "formula": "\\begin{align*} \\frac { d Z } { d t } = \\int \\left [ Z _ { \\zeta } \\frac { \\partial \\zeta } { \\partial t } + Z _ { h } \\frac { \\partial h } { \\partial t } \\right ] d A = 0 ~ . \\end{align*}"}
-{"id": "6842.png", "formula": "\\begin{align*} \\norm { f } _ { \\dot { X } ^ { s , r } } : = \\norm { J ^ s ( t ) f } _ { L _ x ^ r ( \\mathbb { R } ^ d ) } \\sim \\norm { \\ , | t | ^ s | \\nabla | ^ s M ( - t ) f } _ { L _ x ^ r ( \\R ^ d ) } . \\end{align*}"}
-{"id": "5436.png", "formula": "\\begin{align*} \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & \\overline \\alpha \\alpha & \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & \\overline \\alpha ^ 2 \\\\ 0 & \\overline \\alpha J ( T ^ { - r } , T ^ { - r } ) & 0 & \\alpha J ( T ^ { - 3 r } , T ^ { - 3 r } ) \\\\ 0 & \\alpha ^ 2 & \\overline \\alpha J ( T ^ { - 2 r } , T ^ { - r } ) & \\overline \\alpha \\alpha \\end{bmatrix} \\end{align*}"}
-{"id": "5188.png", "formula": "\\begin{align*} \\| X \\| _ { S _ { 1 / 2 } } & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\| U \\| _ { * } \\| V \\| _ { * } \\\\ & = \\min _ { U \\in \\mathbb { R } ^ { m \\times d } , V \\in \\mathbb { R } ^ { n \\times d } : X = U V ^ { T } } \\left ( \\frac { \\| U \\| _ { * } + \\| V \\| _ { * } } { 2 } \\right ) ^ { 2 } . \\end{align*}"}
-{"id": "7440.png", "formula": "\\begin{align*} y = \\left [ x \\overline { v ^ { - 1 } } \\right ] _ - \\overline { w } _ 0 ^ { - 1 } z = \\overline { u } \\left [ \\overline { u } ^ { - 1 } x \\right ] _ - \\overline { w } _ 0 ^ { - 1 } . \\end{align*}"}
-{"id": "7382.png", "formula": "\\begin{align*} \\frac { 1 } { n } ( h ^ { t + 1 } ) ^ * q ^ 0 & \\overset { \\mathbf { . . } } { = } 0 , \\frac { 1 } { n } ( h ^ { t + 1 } ) ^ * \\beta _ 0 \\overset { \\mathbf { . . } } { = } 0 , \\\\ \\frac { 1 } { n } ( b ^ t ) ^ * w & \\doteq 0 . \\end{align*}"}
-{"id": "3798.png", "formula": "\\begin{align*} L _ U & = G ( n , 2 n ) \\cap Z ( \\Pi _ { \\alpha _ { r s } } , X _ { \\beta } : \\alpha _ { r s } \\in I ( n - 2 , 2 n ) , \\beta \\in H _ U ) = S _ U \\cap { \\mathbb P } ( \\ker f ) \\\\ & = L ( n , 2 n ) \\cap Z ( X _ { \\beta } : \\beta \\in H _ U ) . \\end{align*}"}
-{"id": "7262.png", "formula": "\\begin{align*} F _ { j , \\beta } ( z _ j ) - \\sum _ { | \\alpha | = n } T _ { j k } F _ { k , \\alpha } ( z _ k ) \\cdot \\tau _ { k j , \\beta } ^ \\alpha = h _ { 1 , j k , \\beta } - h _ { 2 , j k , \\beta } . \\end{align*}"}
-{"id": "3729.png", "formula": "\\begin{align*} \\varphi ( j ( x w , y ) ) = \\varphi ( i ( x w , y ) - i ( y , x w ) ) = \\varphi ( - 2 x y ^ * ) = \\psi ( \\langle x , y \\rangle ) . \\end{align*}"}
-{"id": "6283.png", "formula": "\\begin{align*} l ( x ) = \\prod _ { i = 1 } ^ d ( x - z _ i ) , \\end{align*}"}
-{"id": "1456.png", "formula": "\\begin{align*} \\frac { \\sigma ( Q _ 0 ) } { | Q _ 0 | } \\| \\chi _ { Q _ 0 } \\| _ { \\mathcal { M } ^ p _ q ( d x , w ) } \\lesssim _ { [ \\sigma ] _ { A _ \\infty } } \\| \\chi _ { Q _ 0 \\setminus \\Omega _ 1 } \\cdot \\sigma \\| _ { \\mathcal { M } ^ p _ q ( d x , w ) } = \\| \\chi _ { Q _ 0 \\setminus \\Omega _ 1 } \\| _ { \\mathcal { M } ^ p _ q ( d x , \\sigma ) } . \\end{align*}"}
-{"id": "8458.png", "formula": "\\begin{align*} = - R e ( \\Lambda ^ { s } \\textbf { P } T _ { i A } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } , \\Lambda ^ { s } J _ { \\varepsilon } \\tilde { \\textbf { u } } ^ { \\varepsilon } ) _ { 0 } , \\end{align*}"}
-{"id": "4958.png", "formula": "\\begin{align*} Q ( x ) = \\beta ( x ) ( x , \\R ^ N \\ ! \\setminus \\ ! \\Omega ) ^ \\alpha \\end{align*}"}
-{"id": "8708.png", "formula": "\\begin{align*} Y _ { \\tau } ^ { n , t , x } & = \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } G B ^ n ( s , \\Xi ^ { t , x } _ s ) \\ , d s + \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } Z _ s ^ { n , t , x } \\ , B ^ n ( s , \\Xi ^ { t , x } _ s ) \\ , d s \\\\ & - \\int _ \\tau ^ T e ^ { - ( s - \\tau ) { A } } Z ^ { n , t , x } _ { s } \\ ; d W _ s . \\end{align*}"}
-{"id": "8418.png", "formula": "\\begin{align*} \\begin{cases} & \\partial _ { t } \\rho + \\nabla \\cdot ( \\rho v ) = 0 , \\\\ & \\partial _ { t } { v } + { v } \\cdot \\nabla { v } + \\nabla Q = 0 , \\\\ & \\nabla \\cdot v = 0 , \\\\ & P = Q - \\phi ( \\rho ) , \\\\ \\end{cases} \\end{align*}"}
-{"id": "3646.png", "formula": "\\begin{align*} \\left | \\frac { \\partial f } { \\partial x } ( x , y ) \\right | = | - \\exp ( 1 - x ) \\exp ( 1 - y ^ 2 ) | \\le e ^ 2 \\approx 7 . 3 9 , ( x , y ) \\in \\mathbb R ^ 2 _ + , \\end{align*}"}
-{"id": "7796.png", "formula": "\\begin{align*} S _ 2 = \\varphi _ t ^ { - 1 } ( S _ 2 ) & = \\varphi _ t ^ { - 1 } \\big ( \\varphi _ t ( I _ \\Delta + ( x _ v ^ 2 : v \\in V ) ) + ( \\Theta , w ) \\big ) _ 2 \\\\ & = \\big ( I _ \\Delta + ( x _ v ^ 2 : v \\in V ) + ( \\varphi _ t ^ { - 1 } ( \\theta _ 1 ) , \\theta _ 2 , \\theta _ 3 , \\varphi _ t ^ { - 1 } ( w ) ) \\big ) _ 2 . \\end{align*}"}
-{"id": "2400.png", "formula": "\\begin{align*} \\lambda _ { i } ( L ) = ( 1 - \\alpha _ 1 ) ( 1 + ( \\alpha _ 2 - 1 ) ( 1 - \\alpha _ 1 ) ) + \\alpha _ 1 ( 1 + ( \\alpha _ 2 - 1 ) ( 2 - \\alpha _ 1 ) ) \\lambda _ { i } ( N ) . \\end{align*}"}
-{"id": "3966.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\delta _ 1 , \\delta _ 2 } V ^ { \\delta _ 1 , \\delta _ 2 } V ^ { \\delta _ 1 , \\delta _ 2 } : = \\left \\{ v \\in V : a _ 1 ( t ) v = e ^ { \\delta _ 1 t } v , a _ 2 ( t ) v = e ^ { \\delta _ 2 t } v t \\in \\R \\right \\} . \\end{align*}"}
-{"id": "6954.png", "formula": "\\begin{align*} e ^ { i q } \\ , z ^ { \\alpha _ 0 } & = e ^ { i q } \\ , e ^ { ( l n z ) \\alpha _ 0 } = e ^ { ( \\ln z ) \\alpha _ 0 } e ^ { - ( \\ln z ) \\alpha _ 0 } e ^ { i q } \\ , e ^ { ( \\ln z ) \\alpha _ 0 } \\cr & = z ^ { \\alpha _ 0 } \\ , e ^ { i q + [ i q , ( \\ln z ) \\alpha _ 0 ] + \\cdots } = z ^ { \\alpha _ 0 } \\ , e ^ { i q + i ( \\ln z ) [ q , \\alpha _ 0 ] + \\cdots } \\cr & = z ^ { \\alpha _ 0 } \\ , e ^ { i q - ( \\ln z ) } = ( 1 / z ) \\ , z ^ { \\alpha _ 0 } \\ , e ^ { i q } \\ , , \\end{align*}"}
-{"id": "2359.png", "formula": "\\begin{align*} \\Omega _ z ^ T : = \\{ y \\in [ 0 , 1 ] \\ , : \\ , \\tau _ \\partial ( y ) < T \\Phi _ { \\tau _ \\partial ( y ) } ( y ) = z \\} . \\end{align*}"}
-{"id": "4528.png", "formula": "\\begin{align*} \\sup _ { y \\in \\mathcal { X } _ n } r _ { n , u _ n ^ * ( y ) } = \\sup _ { y \\in \\mathcal { X } _ n } h _ y ^ { - 1 } \\Bigl ( \\frac { a _ n } { n - 1 } \\Bigr ) \\lesssim \\sup _ { y \\in \\mathcal { X } _ n } \\biggl \\{ \\frac { k \\log n } { n f ( y ) } \\biggr \\} ^ { 1 / d } \\ ! \\ ! \\ ! \\ ! \\leq \\biggl ( \\frac { k \\log n } { n \\delta _ n } \\biggr ) ^ { 1 / d } \\ ! \\ ! \\ ! = o ( \\rho _ n ) . \\end{align*}"}
-{"id": "8802.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { c c c c c c c } \\Delta u & + & | \\sigma | ^ 2 u & = & - n H & o n & \\Sigma \\\\ u & = & 0 & & & o n & \\partial \\Sigma \\\\ \\end{array} \\right . \\end{align*}"}
-{"id": "8444.png", "formula": "\\begin{align*} J _ { \\varepsilon } \\textbf { w } ( x ) = ( j _ { \\varepsilon } * \\textbf { w } ) ( x ) = \\mathcal { F } ^ { - 1 } ( \\hat { { j } _ { \\varepsilon } } ( \\xi ) \\hat { \\textbf { w } } ( \\xi ) ) . \\end{align*}"}
-{"id": "2737.png", "formula": "\\begin{align*} \\omega = \\nu ^ { - 1 } \\omega _ { - 1 } + \\omega _ 0 + \\nu \\omega _ 1 + \\ldots \\end{align*}"}
-{"id": "372.png", "formula": "\\begin{align*} D _ 1 & \\le \\sum _ { q = 0 } ^ \\infty \\| \\Delta _ q ( u ( \\tau ) - u ( t ) ) \\| _ { L ^ 2 } \\sum _ { q = 0 } ^ \\infty \\| \\Delta _ q u ( t ) \\| _ { L ^ 2 } \\\\ & = \\| u ( \\tau ) - u ( t ) \\| _ { B ^ { 0 } _ { 2 , 1 } } \\| u ( t ) \\| _ { B ^ { 0 } _ { 2 , 1 } } . \\end{align*}"}
-{"id": "7846.png", "formula": "\\begin{gather*} \\{ I _ i , I _ j \\} = 0 = \\{ J _ i , J _ j \\} , \\{ I _ i , J _ j \\} = - \\delta _ { i j } . \\end{gather*}"}
-{"id": "3635.png", "formula": "\\begin{align*} c ^ * = \\max \\left \\{ 0 , 1 - \\frac { 1 } { \\sum _ { j = 1 } ^ k | \\frac { \\partial f } { \\partial x _ j } ( \\mathbf { K } ) | } \\right \\} , \\end{align*}"}
-{"id": "9831.png", "formula": "\\begin{align*} M _ i \\ , { } _ \\lambda \\ , v = 0 . \\end{align*}"}
-{"id": "3826.png", "formula": "\\begin{align*} h _ k [ x _ 1 , \\ldots , x _ n \\mid y _ 1 , \\ldots , y _ m ] : = \\sum _ { i = 0 } ^ k h _ i [ x _ 1 , \\ldots , x _ n ] e _ { k - i } [ y _ 1 , \\ldots , y _ m ] \\end{align*}"}
-{"id": "1116.png", "formula": "\\begin{align*} \\forall L \\subset \\R ^ N , \\ ; \\dim L \\ge N / 4 \\quad \\Rightarrow \\exists x \\in L \\ ; \\ ; | x | = 1 , \\| x \\| _ \\infty \\ge 1 / 2 . \\end{align*}"}
-{"id": "7999.png", "formula": "\\begin{align*} e ( X ) \\geq 3 + 6 \\cdot 3 + 3 \\cdot 2 = 2 7 . \\end{align*}"}
-{"id": "1958.png", "formula": "\\begin{align*} \\frac { d \\hat { Y } _ i } { d \\hat { u } } \\leq \\hat { Y } _ i \\left ( r \\sum _ { j = 1 } ^ m n _ j \\ , c _ { j } \\hat { Y } _ j ^ 2 + c _ { i } \\hat { Y } _ i ^ 2 - 2 p _ i \\hat { Y } _ i \\right ) . \\end{align*}"}
-{"id": "58.png", "formula": "\\begin{align*} \\frac { d } { d r } \\left [ \\frac { p - 1 } { p } | g | ^ { p / ( p - 1 ) } + \\frac { 1 } { 2 } f ^ 2 \\right ] & = | g | ^ { ( 2 - p ) / ( p - 1 ) } g ( f - | g | ) - | g | ^ { ( 2 - p ) / ( p - 1 ) } g f \\\\ & = - | g | ^ { 1 / ( p - 1 ) } g \\\\ & \\le \\frac { p } { p - 1 } \\left [ \\frac { p - 1 } { p } | g | ^ { p / ( p - 1 ) } + \\frac { 1 } { 2 } f ^ 2 \\right ] \\ , \\end{align*}"}
-{"id": "6782.png", "formula": "\\begin{align*} u _ t ( x ) : = \\max ( \\varphi _ 0 - C t , \\varphi _ 1 + C ( t - 1 ) ) , \\end{align*}"}
-{"id": "6185.png", "formula": "\\begin{align*} | \\mathcal { A } _ n h ( x ) | \\leq \\eta _ m = \\eta _ m ^ n , \\end{align*}"}
-{"id": "16.png", "formula": "\\begin{align*} a _ k = A _ { i _ 1 i _ 1 } \\odot \\dots \\odot A _ { i _ k i _ k } \\enspace . \\end{align*}"}
-{"id": "3668.png", "formula": "\\begin{align*} \\mu = \\nabla \\cdot \\vec { v } = \\frac { \\partial u } { \\partial x } + \\frac { \\partial v } { \\partial y } ~ , \\end{align*}"}
-{"id": "4052.png", "formula": "\\begin{align*} q + r = m + n + l + k < q + r , \\end{align*}"}
-{"id": "1550.png", "formula": "\\begin{align*} I ( \\delta , \\mathcal { F } ) = \\underset { Q } { \\sup } \\int _ 0 ^ { \\delta } \\sqrt { 1 + \\log N ( \\epsilon \\Vert F \\Vert _ { \\mathbb { L } _ 2 ( Q ) } , \\mathcal { F } , \\Vert \\cdot \\Vert _ { \\mathbb { L } _ 2 ( Q ) } ) } d \\epsilon \\end{align*}"}
-{"id": "1545.png", "formula": "\\begin{align*} W _ 2 ^ 2 ( \\mu , \\nu ) = \\underset { \\pi } { \\inf } \\iint _ { \\Omega \\times \\Omega } \\vert x - y \\vert ^ 2 d \\pi ( x , y ) , \\end{align*}"}
-{"id": "4943.png", "formula": "\\begin{align*} ( 2 d - 1 ) \\sum _ { 1 \\leq i < j \\leq M } \\lambda _ i \\lambda _ j \\| A D ( \\beta _ i - \\beta _ j ) \\| _ 2 ^ 2 = \\sum _ { i = 1 } ^ { M } \\lambda _ i \\| A D ( \\sum _ { j = 1 } ^ { M } \\lambda _ j \\beta _ j - d \\beta _ i ) \\| _ 2 ^ 2 - \\sum _ { i = 1 } ^ { M } \\lambda _ i ( 1 - d ) ^ 2 \\| A D \\beta _ i \\| _ 2 ^ 2 . \\end{align*}"}
-{"id": "206.png", "formula": "\\begin{align*} C _ n : = \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\sup _ { s \\in \\mathcal { S } _ n } \\sup _ { x \\in \\mathcal { X } _ n } \\frac { a ( f ( x ) ) ^ { d / ( 1 \\wedge \\beta ) } s } { f ( x ) } \\rightarrow 0 . \\end{align*}"}
-{"id": "3909.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = f ( t , u , u ' ) & & \\\\ u ( T ) = u ' ( 0 ) = u ' ( T ) , \\end{array} \\right . \\end{align*}"}
-{"id": "7121.png", "formula": "\\begin{align*} \\gamma _ 1 = - 1 \\textrm { o r } & ( - 1 ) ^ { \\gamma _ 1 } ( - q ^ { - 1 } ) ^ { \\gamma _ 2 } = \\chi _ { g , 1 } \\\\ \\textrm { a n d } \\gamma _ 2 = - 1 \\textrm { o r } & ( - 1 ) ^ { \\gamma _ 2 } ( - q ) ^ { \\gamma _ 1 } = \\chi _ { g , 2 } . \\end{align*}"}
-{"id": "3462.png", "formula": "\\begin{align*} \\frac { 1 } { L _ 1 L _ 2 } \\int _ \\Sigma \\eta _ 0 = 0 . \\end{align*}"}
-{"id": "3048.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\max _ { | u | = n } f _ \\mu ( u ) = 0 . \\end{align*}"}
-{"id": "186.png", "formula": "\\begin{align*} F _ x ( u ) : = \\exp \\{ - u f ( x ) e ^ { \\Psi ( k ) } \\} \\sum _ { j = k } ^ \\infty \\frac { 1 } { j ! } \\bigl \\{ u f ( x ) e ^ { \\Psi ( k ) } \\bigr \\} ^ j = e ^ { - \\lambda _ { x , u } } \\sum _ { j = k } ^ \\infty \\frac { \\lambda _ { x , u } ^ j } { j ! } , \\end{align*}"}
-{"id": "935.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l } \\partial _ t \\theta ^ { n , 2 } _ i + u ^ n \\cdot \\nabla \\theta ^ { n , 2 } _ i = f ^ \\infty _ i , \\\\ \\partial _ t \\sigma ^ { n , 2 } + u ^ n \\cdot \\nabla \\sigma ^ { n , 2 } = g ^ \\infty , \\\\ { \\theta ^ { n , 2 } _ i } | _ { t = 0 } = \\nabla ( u ^ \\infty _ 0 ) _ { i } , \\\\ { \\sigma ^ { n , 2 } } | _ { t = 0 } = \\nabla \\gamma ^ \\infty _ 0 . \\end{array} \\right . \\end{align*}"}
-{"id": "9455.png", "formula": "\\begin{align*} [ T , P _ F ] = P _ { \\partial ^ + _ R F } T P _ { \\partial ^ - _ R F } - P _ { \\partial ^ - _ R F } T P _ { \\partial ^ + _ R F } \\ ; , \\end{align*}"}
-{"id": "6509.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} & u ' ( t ) + A ( t ) u ( t ) = B ( t , u ( t ) ) , 0 < t < T , \\\\ & u ( 0 ) = u _ 0 ; \\end{aligned} \\right . \\end{align*}"}
-{"id": "8440.png", "formula": "\\begin{align*} \\partial _ { t } \\tilde { \\textbf { u } } + \\sum _ { j = 1 } ^ { d } A _ { j } ( \\tilde { \\textbf { u } } + \\bar { \\textbf { u } } ) \\partial _ { x _ { j } } \\tilde { \\textbf { u } } + F _ { P } = 0 , \\end{align*}"}
-{"id": "1984.png", "formula": "\\begin{align*} u f ^ { * } = \\varpi ^ { a } \\Xi - \\varpi ^ { a } g ^ { * } . \\end{align*}"}
-{"id": "5005.png", "formula": "\\begin{align*} \\frac { h _ { H } ( f ^ { n } ( P ) ) } { \\delta _ { f } ^ { n } } = h _ { H } ( P ) + \\sum _ { k = 0 } ^ { n - 1 } \\frac { h _ { E ' _ { 1 } } ( f ^ { k } ( P ) ) } { \\delta _ { f } ^ { k + 1 } } - \\sum _ { k = 0 } ^ { n - 1 } \\frac { h _ { Z _ { 1 } } ( p ^ { - 1 } f ^ { k } ( P ) ) } { \\delta _ { f } ^ { k + 1 } } . \\end{align*}"}
-{"id": "7633.png", "formula": "\\begin{align*} a ^ { s } _ i = \\left \\{ \\begin{array} { r l } + 1 , & \\mbox { i f } i \\mbox { i s t h e f i r s t n o d e v i s i t e d b y p a t h } s , \\\\ - 1 , & \\mbox { i f } i \\mbox { i s t h e l a s t n o d e v i s i t e d b y p a t h } s , \\\\ 0 , & \\mbox { o t h e r w i s e } , \\end{array} \\right . \\end{align*}"}
-{"id": "1376.png", "formula": "\\begin{align*} L _ 1 ( \\delta ) = \\prod _ { t = 1 } ^ { n } f ( y _ { t } | x _ { t } , \\delta ) = \\prod _ { t = 1 } ^ { n } \\int _ { \\R } \\exp \\left ( y _ { t } W _ { t } - m _ t b ( W _ { t } ) + c ( y _ { t } ) \\right ) g ( \\alpha _ t ; \\tau ) d \\alpha _ t . \\end{align*}"}
-{"id": "5680.png", "formula": "\\begin{align*} e _ 1 \\check K ( x ) e _ 1 e _ 0 - e _ 0 e _ 1 \\check K ( x _ 1 ) e _ 1 = \\omega \\left ( ( e _ 0 + 1 ) \\check K ( x ) e _ 1 e _ 0 - e _ 0 e _ 1 ( e _ 0 + 1 ) \\check K ( x ) \\right ) \\ ; . \\end{align*}"}
-{"id": "6606.png", "formula": "\\begin{align*} \\lambda _ { i } ( L ) = ( 1 - \\alpha _ 1 ) ( 1 + ( \\alpha _ 2 - 1 ) ( 1 - \\alpha _ 1 ) ) + \\alpha _ 1 ( 1 + ( \\alpha _ 2 - 1 ) ( 2 - \\alpha _ 1 ) ) \\lambda _ { i } ( N ) . \\end{align*}"}
-{"id": "5800.png", "formula": "\\begin{align*} G ( z ) , W _ { 2 i } ( z ) \\ ( i = 0 , \\ldots , n - 1 ) \\end{align*}"}
-{"id": "8196.png", "formula": "\\begin{align*} X = S p e c \\ A [ x _ 1 , x _ 2 , x _ 3 ] [ 1 / Q ] . \\end{align*}"}
-{"id": "4182.png", "formula": "\\begin{align*} \\abs { f ( x ) } \\le \\norm { f } _ { \\mathcal { H } } \\norm { k _ x } _ { \\mathcal { H } } = \\norm { f } _ { \\mathcal { H } } \\sqrt { K ( x , x ) } \\end{align*}"}
-{"id": "1088.png", "formula": "\\begin{align*} f ^ { ( n + 1 ) } ( x , \\omega ^ { n + 1 } ) = f ^ { ( 1 ) } ( f ^ { ( n ) } ( x , \\omega ^ n ) , \\omega _ { n + 1 } ) . \\end{align*}"}
-{"id": "4761.png", "formula": "\\begin{align*} H = \\frac { \\kappa } { 2 f } \\ , n _ 1 + \\frac { f f '' + ( f ' ) ^ 2 - 1 } { 2 f \\sqrt { 1 - f '^ 2 } } \\ , n _ 2 . \\end{align*}"}
-{"id": "4209.png", "formula": "\\begin{align*} d ( x _ k , \\mathrm { A r g } \\min f ) & \\leq \\| x _ k - u _ { k + 1 } \\| \\leq \\| x _ { k + 1 } - u _ { k + 1 } \\| + \\| x _ { k + 1 } - x _ k \\| \\\\ & = d ( x _ { k + 1 } , \\mathrm { A r g } \\min f ) + h \\| \\nabla f ( x _ k ) \\| , ~ k \\geq 0 . \\end{align*}"}
-{"id": "1724.png", "formula": "\\begin{gather*} \\sigma \\stackrel { \\sigma } { = } 1 , \\mu ^ e \\stackrel { \\sigma } { = } 0 , \\rho \\stackrel { \\sigma } { = } - \\tfrac { 1 } { 2 } \\varepsilon , \\end{gather*}"}
-{"id": "6755.png", "formula": "\\begin{align*} g ( 0 ) = F ( \\varphi ) \\geq F ( \\varphi _ t ) \\geq g ( t ) , \\ \\forall t \\in \\mathbb { R } . \\end{align*}"}
-{"id": "7879.png", "formula": "\\begin{align*} \\phi _ y ( t , x ) : = \\int _ 0 ^ t \\phi _ y ( t , x , s ) \\ , d s = \\int _ 0 ^ t \\int _ { \\R ^ d } p _ z ( t - s , x - z ) q ( s , z , y ) \\ , d z \\ , d s \\ , . \\end{align*}"}
-{"id": "8697.png", "formula": "\\begin{align*} Y _ \\tau ^ { t , x } = Y _ \\tau ^ { s , \\Xi _ s ^ { t , x } } , Z _ \\tau ^ { t , x } = Z _ \\tau ^ { s , \\Xi _ s ^ { t , x } } , \\ ; \\P - . \\end{align*}"}
-{"id": "7871.png", "formula": "\\begin{align*} p ( t , x ) - p ( t , y ) = ( x - y ) \\cdot \\int _ 0 ^ 1 \\nabla p ( t , x + \\theta ( y - x ) ) \\ , d \\theta \\end{align*}"}
-{"id": "6766.png", "formula": "\\begin{align*} \\theta _ { \\phi } ^ n = e ^ { \\phi } \\theta _ + ^ n . \\end{align*}"}
-{"id": "1774.png", "formula": "\\begin{align*} \\int _ { \\Omega } \\int _ { \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\overline { u } _ { i } ( x ) \\underline { u } _ { j } ( y ) K ( x , y ) \\ , d y d x = \\int _ { \\Omega } \\int _ { \\Omega } \\sum _ { { i , j } _ { j \\neq i } } \\underline { u } _ { j } ( x ) \\overline { u } _ { i } ( y ) K ( x , y ) \\ , d y \\ , d x . \\end{align*}"}
-{"id": "5111.png", "formula": "\\begin{align*} C ^ { [ M ' , M ] } ( z ) = \\begin{pmatrix} 0 & 1 \\end{pmatrix} L ^ { ( M ' ) } ( z ) L ^ { ( M ' + 1 ) } ( z ) \\cdots L ^ { ( M ) } ( z ) \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "3524.png", "formula": "\\begin{align*} \\| \\nabla ^ j ( u \\rho ^ { \\frac { 1 } { 2 } } ) \\| ^ 2 _ { L ^ 2 } & = \\int _ \\Omega | \\nabla ^ j ( u \\rho ^ { \\frac { 1 } { 2 } } ) | ^ 2 \\ , d \\mu _ g \\\\ & \\le 2 \\int _ \\Omega | \\nabla ^ j u | ^ 2 \\rho \\ , d \\mu _ g + C ^ 2 N ^ { 2 j } \\sum _ { i = 1 } ^ j \\int _ \\Omega | \\nabla ^ { j - i } u | ^ 2 d ^ { - 4 i } \\rho \\ , d \\mu _ g . \\end{align*}"}
-{"id": "9963.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": "4635.png", "formula": "\\begin{align*} \\frac { a ^ { ( q ^ i - 1 ) / ( p - 1 ) } } { e ^ { ( q ^ i - 1 ) / k } } = 1 \\ , . \\end{align*}"}
-{"id": "1273.png", "formula": "\\begin{align*} \\begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\\\ 0 & - 1 & \\alpha & 0 & \\overline \\alpha \\\\ 0 & - \\overline \\alpha & 0 & - \\alpha J ( T ^ { - 2 r } , T ^ { - 3 r } ) & 0 \\\\ 0 & 0 & - \\overline \\alpha J ( T ^ { - r } , T ^ { - r } ) & 0 & - \\alpha J ( T ^ { - 3 r } , T ^ { - 3 r } ) \\\\ 0 & - \\alpha & 0 & - \\overline \\alpha J ( T ^ { - 2 r } , T ^ { - r } ) & 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "7882.png", "formula": "\\begin{align*} \\lefteqn { \\left | \\int _ { \\R ^ d } p _ z ( t - s , x - z ) \\left ( q ( s , z , y ) - q ( s , x , y ) \\right ) \\ , d z \\right | } \\\\ & \\le \\int _ { | x - z | \\le \\delta } p _ z ( t - s , x - z ) \\left | q ( s , z , y ) - q ( s , x , y ) \\right | \\ , d z \\\\ & + \\int _ { | x - z | > \\delta } p _ z ( t - s , x - z ) \\left ( | q ( s , z , y ) | + | q ( s , x , y ) | \\right ) \\ , d z = : J _ 1 ( \\delta , t , s ) + J _ 2 ( \\delta , t , s ) \\ , . \\end{align*}"}
-{"id": "7405.png", "formula": "\\begin{align*} & ( g _ 1 ^ { [ n \\lambda _ 1 s _ 1 ] + [ n \\lambda _ 2 s _ 1 ] } g _ 2 ^ { [ n \\lambda _ 1 s _ 2 ] + [ n \\lambda _ 2 s _ 2 ] } \\cdots g _ k ^ { [ n \\lambda _ 1 s _ k ] + [ n \\lambda _ 2 s _ k ] } ) ^ { - 1 } ( g _ 1 ^ { [ n \\lambda _ 1 s _ 1 ] } g _ 2 ^ { [ n \\lambda _ 1 s _ 2 ] } \\cdots g _ k ^ { [ n \\lambda _ 1 s _ k ] } ) ( g _ 1 ^ { [ n \\lambda _ 2 s _ 1 ] } g _ 2 ^ { [ n \\lambda _ 2 s _ 2 ] } \\cdots g _ k ^ { [ n \\lambda _ 2 s _ k ] } ) \\\\ & = [ f _ { n , 1 } , f _ { n , 1 } ^ \\prime ] \\cdots [ f _ { n , l ( k ) } , f _ { n , l ( k ) } ^ \\prime ] . \\\\ \\end{align*}"}
-{"id": "123.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 + t ) ^ { \\tfrac { x + y } { 2 } } d \\mu _ { - 1 } ( y ) & = \\frac { 2 } { 1 + \\sqrt { 1 + t } } \\sqrt { ( 1 + t ) ^ x } \\\\ & = \\sum _ { n = 0 } ^ \\infty C h _ { n , \\frac { 1 } { 2 } } ( x ) \\frac { t ^ n } { n ! } . \\end{align*}"}
-{"id": "4870.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\frac { \\tilde { H } ( n , n ) } { n } = \\max _ { \\kappa \\in [ 0 , 1 ] } \\left \\lbrace \\frac { 1 - \\kappa } { \\alpha } + ( 1 + \\sqrt { \\kappa } ) ^ 2 \\right \\rbrace = \\begin{cases} 4 & \\mbox { i f } \\alpha \\in ( 1 / 2 , 1 ) , \\\\ \\frac { 1 } { \\alpha ( 1 - \\alpha ) } & \\mbox { i f } \\alpha \\leqslant 1 / 2 . \\end{cases} \\end{align*}"}
-{"id": "3322.png", "formula": "\\begin{align*} \\Gamma _ { + } ^ { ( 1 ) * } = \\left ( \\begin{array} { c c c } 1 & 0 & 0 \\end{array} \\right ) ^ T , \\Gamma _ { 0 } ^ { ( 1 ) * } = \\left ( \\begin{array} { c c c } 0 & 1 & 0 \\end{array} \\right ) ^ T , \\Gamma _ { - } ^ { ( 1 ) * } = \\left ( \\begin{array} { c c c } 0 & 0 & q ^ { - 2 } \\end{array} \\right ) ^ T . \\end{align*}"}
-{"id": "7221.png", "formula": "\\begin{align*} \\pi _ { \\mathrm { \\dim } \\left ( S \\right ) } \\circ \\left ( p _ { k } ^ { N } \\right ) ^ { \\alpha } = \\left ( p _ { j \\left ( k \\right ) } ^ { S } \\right ) ^ { \\alpha } , \\end{align*}"}
-{"id": "8628.png", "formula": "\\begin{align*} a _ { i , j } + a _ { i , j + 1 } + a _ { i + 1 , j } + a _ { i + 1 , j + 1 } = a _ { i , j } , \\end{align*}"}
-{"id": "3017.png", "formula": "\\begin{align*} { { \\bf { Y } } ^ { [ j ] } } ( { t _ 1 } ) = { { \\bf { H } } ^ { [ j 1 ] } } ( { t _ 1 } ) { { \\bf { u } } ^ { [ 1 ] } } + { { \\bf { H } } ^ { [ j 2 ] } } ( { t _ 1 } ) { { \\bf { u } } ^ { [ 2 ] } } , { { \\bf { Y } } ^ { [ j ] } } ( { t _ 2 } ) = { { \\bf { H } } ^ { [ j 1 ] } } ( { t _ 2 } ) { { \\bf { v } } ^ { [ 1 ] } } + { { \\bf { H } } ^ { [ j 2 ] } } ( { t _ 2 } ) { { \\bf { v } } ^ { [ 2 ] } } , \\end{align*}"}
-{"id": "5218.png", "formula": "\\begin{align*} ( u , v ) _ { L ^ 2 _ { ( 0 , q ) } ( M ) } = ( \\square _ q G _ q u , v ) _ { L ^ 2 _ { ( 0 , q ) } ( M ) } = ( G _ q u , v ) _ { g r a p h } \\ ; . \\end{align*}"}
-{"id": "7325.png", "formula": "\\begin{align*} F \\hat { R } ( v _ { 1 } \\otimes v _ { 0 } ) & = F ( v _ { 0 } \\otimes v _ { 1 } ) + ( q ^ { 2 } - q ^ { - 2 } ) F ( v _ { 1 } \\otimes v _ { 0 } ) \\\\ & = [ 2 ] ^ { 1 / 2 } ( ( q ^ { 2 } - q ^ { - 2 } ) v _ { 1 } \\otimes v _ { - 1 } + ( q ^ { 2 } + 1 - q ^ { - 2 } ) v _ { 0 } \\otimes v _ { 0 } + q ^ { - 2 } v _ { - 1 } \\otimes v _ { 1 } ) . \\end{align*}"}
-{"id": "659.png", "formula": "\\begin{align*} \\nu _ Y = \\nu ( C ) \\nu _ C + \\nu ( C ^ c ) \\nu _ { C ^ c } , \\end{align*}"}
-{"id": "9752.png", "formula": "\\begin{align*} \\mathcal { Z } ( \\vec { x } ) : = \\{ \\vec { z } \\in \\mathbb { R } ^ { s } \\ | \\ \\vec { g } ( \\vec { z } , \\vec { x } ) \\leq \\vec { 0 } _ { m } \\} , \\end{align*}"}
-{"id": "4179.png", "formula": "\\begin{align*} d \\bigl ( ( a , b ) , ( a ' , b ' ) \\bigr ) = d _ A ( a , a ' ) + d _ B ( b , b ' ) . \\end{align*}"}
-{"id": "2750.png", "formula": "\\begin{align*} \\sum _ I \\int ( - 1 ) ^ { | I | } f _ I ^ I \\ , \\mu _ \\star = \\sum _ { K , Q } \\int ( - 1 ) ^ { \\lambda ( K , Q ) } ( f _ K ^ P \\star u _ { P Q } ) \\tau _ { K ' Q ' } \\mu _ \\star . \\end{align*}"}
-{"id": "5348.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { \\ell } \\lambda _ i = \\pm | H | ( R \\cdot R ' ) \\end{align*}"}
-{"id": "1584.png", "formula": "\\begin{align*} \\Delta a - C ^ { x } a _ { , x } - a _ { , t } = 0 . \\end{align*}"}
-{"id": "1509.png", "formula": "\\begin{align*} e _ 1 \\check K ( x ) e _ 1 e _ 0 - e _ 0 e _ 1 \\check K ( x _ 1 ) e _ 1 = \\omega \\left ( ( e _ 0 + 1 ) \\check K ( x ) e _ 1 e _ 0 - e _ 0 e _ 1 ( e _ 0 + 1 ) \\check K ( x ) \\right ) \\ ; . \\end{align*}"}
-{"id": "6559.png", "formula": "\\begin{align*} \\frac 1 \\tau \\ , \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( L ^ q ( \\varOmega ) ) } + \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( W ^ { 2 , q } ( \\varOmega ) ) } \\le C \\tau ^ k , \\end{align*}"}
-{"id": "880.png", "formula": "\\begin{align*} \\operatorname * { c o r e } A = \\operatorname * { i n t } A , \\end{align*}"}
-{"id": "4997.png", "formula": "\\begin{align*} Z _ { i } = p ^ { * } { p } _ { * } g ^ { * } D _ { i } - g ^ { * } D _ { i } \\end{align*}"}
-{"id": "5228.png", "formula": "\\begin{align*} d \\alpha | _ { { \\mathcal { N } _ z } } = 0 , \\forall z \\in M \\ ; . \\end{align*}"}
-{"id": "3838.png", "formula": "\\begin{align*} R _ { j + 1 } & \\leq \\widetilde C R _ j \\Big ( ( m _ { j , 0 } + R _ j ) ( 1 + m _ j ) + R _ j ^ 2 + m _ 0 \\sum _ { i = 0 } ^ { j - 1 } R _ i + \\big | \\sum _ { i = 0 } ^ { j - 1 } R _ i \\big | ^ 2 \\Big ) \\\\ & = : \\widetilde C Q _ j R _ j , \\end{align*}"}
-{"id": "8938.png", "formula": "\\begin{align*} g ( \\varphi ( s ) ) = u ^ * ( s ) . \\end{align*}"}
-{"id": "7218.png", "formula": "\\begin{align*} \\left \\{ \\left ( a _ { i } , b _ { i } \\right ) \\right \\} _ { i = 1 } ^ { k } \\left ( k , \\tilde { \\delta } , r \\right ) N _ { X } p X \\end{align*}"}
-{"id": "5629.png", "formula": "\\begin{align*} R [ M ( \\Lambda _ 2 ) ] = R [ x , x y , x y ^ 2 , x y ^ 3 , \\ldots ] \\end{align*}"}
-{"id": "4432.png", "formula": "\\begin{align*} H = H ( X ) = H ( f ) : = - \\mathbb { E } \\{ \\log f ( X ) \\} = - \\int _ { \\mathcal { X } } f ( x ) \\log f ( x ) \\ , d x \\end{align*}"}
-{"id": "1232.png", "formula": "\\begin{align*} \\lambda _ { \\varphi , w } = \\{ x : \\alpha ( \\lambda x ) < \\infty , \\ , \\ , \\ , \\ , \\lambda > 0 \\} . \\end{align*}"}
-{"id": "6433.png", "formula": "\\begin{align*} \\int _ { B } \\varphi \\partial _ { s } \\left ( g _ { 1 - \\alpha , m } * \\tilde { u } \\right ) d x + \\mathcal { E } ( h _ { m } * \\tilde { u } , \\varphi ) \\geq 0 , s \\in ( 0 , t _ { 0 } - t _ { 1 } ) , m \\in \\mathbb { N } , \\end{align*}"}
-{"id": "5615.png", "formula": "\\begin{align*} U ( \\alpha , H ) = \\sum _ { 1 \\leq m \\leq H } e ( m \\alpha ) . \\end{align*}"}
-{"id": "1169.png", "formula": "\\begin{align*} \\dot { A } _ { 1 } & = ( 1 - K ) \\beta A _ { 1 } - \\frac { 1 } { 2 } b _ { x x } ( 1 ) B _ { 1 } , \\\\ \\dot { B } _ { 1 } & = ( 1 + K ) \\beta B _ { 1 } . \\end{align*}"}
-{"id": "4594.png", "formula": "\\begin{align*} | 2 | = q ^ { - \\ell _ 0 } . \\end{align*}"}
-{"id": "6498.png", "formula": "\\begin{align*} u ( x _ { 0 } , t ) = \\int _ { 0 } ^ { t } \\mu ( t - s ) v ( x _ { 0 } , s ) d s = 0 \\end{align*}"}
-{"id": "5574.png", "formula": "\\begin{align*} v _ { x x } = 0 , x \\neq x _ j , [ v _ x ] ( x _ j ) = - z m _ j v ( x _ j ) , \\end{align*}"}
-{"id": "5754.png", "formula": "\\begin{align*} Q _ L ^ B ( \\hat \\delta _ 0 ) = \\begin{bmatrix} S _ { \\phi } ( \\hat \\delta _ { 0 } ) & S _ { \\theta } ( \\hat { \\delta } _ { 0 } ) \\end{bmatrix} [ \\hat { G } _ n - \\hat { F } _ n \\hat { E } _ n ^ { - 1 } \\hat { F } _ n ^ \\mathrm { T } ] ^ { - 1 } \\begin{bmatrix} S _ { \\phi } ( \\hat { \\delta } _ { 0 } ) & S _ { \\theta } ( \\hat \\delta _ { 0 } ) \\end{bmatrix} . \\end{align*}"}
-{"id": "3154.png", "formula": "\\begin{align*} \\sum _ { l = 1 } ^ { L _ { n } } \\mathrm { t r } \\left ( \\overline { W } _ s ^ { \\otimes n } ( x _ { j , l } ) D _ { x _ { j , l } } \\right ) \\geq 1 - \\epsilon \\end{align*}"}
-{"id": "1082.png", "formula": "\\begin{align*} J _ { i j } ^ 2 = - { \\rm I d } _ { \\mathbb { R } ^ N } . \\end{align*}"}
-{"id": "1821.png", "formula": "\\begin{align*} f = \\sum _ { i = 1 } ^ { \\infty } f _ { i } , \\ f _ { i } = \\sum \\limits _ { \\alpha \\geq 2 i } s ^ { \\alpha } f _ { i , \\alpha } , \\end{align*}"}
-{"id": "8548.png", "formula": "\\begin{align*} \\widetilde { M } & : = \\bar { e _ { k } } M \\bar { e _ { k } } \\oplus M e _ { k } M \\oplus ( e _ { k } M ) ^ { \\ast } \\oplus ^ { \\ast } ( M e _ { k } ) \\\\ \\widetilde { P } & : = [ P ] + \\displaystyle \\sum _ { s a \\in _ { k } \\hat { T } , b t \\in \\tilde { T } _ { k } } [ b t s a ] ( ( s a ) ^ { \\ast } ) ( ^ { \\ast } ( b t ) ) \\end{align*}"}
-{"id": "6592.png", "formula": "\\begin{align*} F _ { \\gamma } ^ { \\rm { D R } } ( z ) = ( 2 \\gamma ) ^ { - 1 } \\left ( \\langle R _ { \\gamma f } ( z ) , z \\rangle - p _ { \\gamma f } ^ 2 ( z ) - p _ { \\gamma g } ^ 2 ( R _ { \\gamma f } z ) \\right ) . \\end{align*}"}
-{"id": "361.png", "formula": "\\begin{align*} G _ 2 ( t ) = \\langle \\zeta _ L , e ^ { i T _ { c , h _ c } ( f _ 1 ) } e ^ { i t T _ { c , h _ c } ( f _ 2 ) } \\zeta _ R \\rangle \\end{align*}"}
-{"id": "352.png", "formula": "\\begin{align*} L _ n = \\frac { 1 } { 2 } : \\ ! J ^ 2 \\ ! : _ n \\ , \\equiv \\ , \\frac { 1 } { 2 } \\left ( \\sum _ { m = - \\infty } ^ { - 1 } J _ m J _ { n - m } + \\sum _ { m = 0 } ^ { \\infty } J _ { n - m } J _ m \\right ) \\end{align*}"}
-{"id": "6744.png", "formula": "\\begin{align*} { \\rm I } ( \\varphi ) : = \\frac { 1 } { ( n + 1 ) { \\rm V o l } ( \\theta ) } \\sum _ { k = 0 } ^ n \\int _ X ( \\varphi - V _ { \\theta } ) \\langle \\theta _ { \\varphi } ^ k \\wedge \\theta _ { V _ { \\theta } } ^ { n - k } \\rangle . \\end{align*}"}
-{"id": "6029.png", "formula": "\\begin{gather*} \\nabla _ { \\partial _ x } \\partial _ x = 3 x y ^ 2 \\partial _ x + x ^ 3 \\partial _ y , \\nabla _ { \\partial _ x } \\partial _ y = \\nabla _ { \\partial _ y } \\partial _ x = 0 , \\nabla _ { \\partial _ y } \\partial _ y = x ^ 3 \\partial _ x - 3 x ^ 2 y \\partial _ y . \\end{gather*}"}
-{"id": "5923.png", "formula": "\\begin{align*} P _ t g ( z ) = P _ t g ( x , v ) & = \\int _ { \\R ^ { 2 d } } g ( e ^ { t A } z + y ) N ( 0 , Q _ t ) \\ , \\dd y \\\\ & = \\int _ { \\R ^ { 2 d } } g ( x + t v + y _ 1 , v + y _ 2 ) N ( 0 , Q _ t ) \\ , \\dd y \\ , , \\ ; \\ ; g \\in C _ c ^ { \\infty } ( \\R ^ { 2 d } ) , \\ ; t \\ge 0 , \\end{align*}"}
-{"id": "2075.png", "formula": "\\begin{align*} M A x = M b \\end{align*}"}
-{"id": "5919.png", "formula": "\\begin{gather*} \\lambda { \\psi } ( z ) - \\frac { 1 } { 2 } \\triangle _ v { \\psi } ( z ) - v \\cdot D _ x { \\psi } ( z ) - F ( z ) \\cdot D _ v { \\psi } ( z ) = g ( z ) \\ , \\\\ = \\lambda { \\psi } ( z ) - \\frac { 1 } { 2 } \\mathrm { T r } \\big ( Q D ^ 2 { \\psi } ( z ) \\big ) - \\langle A z , D { \\psi } ( z ) \\rangle - \\langle B ( z ) , D { \\psi } ( z ) \\rangle \\end{gather*}"}
-{"id": "9861.png", "formula": "\\begin{align*} \\| f \\mid L ^ 1 _ p ( \\Omega ) \\| = \\biggr ( \\int \\limits _ { \\Omega } | \\nabla f ( x ) | ^ { p } \\ , d x \\biggr ) ^ { \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "5378.png", "formula": "\\begin{align*} c _ { d , d } ( c _ { q , q } ^ d + c _ { q , 0 } ^ d ) + 2 c _ { q , q } { \\ , } c _ { d , d } ^ q ~ = ~ 0 . \\end{align*}"}
-{"id": "7989.png", "formula": "\\begin{align*} P _ t = [ 0 : 0 : s : 1 ] . \\end{align*}"}
-{"id": "50.png", "formula": "\\begin{align*} ( | f ' | ^ { p - 2 } f ' ) ' + f - | f ' | ^ { p - 1 } = 0 \\ , r \\in ( 0 , \\infty ) \\ , \\end{align*}"}
-{"id": "4845.png", "formula": "\\begin{align*} \\rho _ { \\circ } ^ + & = ( c , a _ { j _ 1 + 1 } , \\dots , a _ { i _ 1 } ) , \\\\ \\rho _ k ^ + & = ( a _ { i _ k + 1 } , \\dots , a _ { i _ { k + 1 } } ) k = 1 , \\dots , n - 1 , \\\\ \\rho _ k ^ - & = ( a _ { j _ { k + 1 } + 1 } , \\dots , a _ { j _ k } ) k = 1 , \\dots , n - 1 , \\\\ \\rho _ n ^ - & = ( a _ { 1 } , \\dots , a _ { j _ n } ) . \\end{align*}"}
-{"id": "3239.png", "formula": "\\begin{align*} g _ { a } ( y ) = \\frac { 1 } { \\mathrm { v o l } ( B ( a , \\eta ) ) } \\int _ { z \\in B ( a , \\eta ) } \\mathrm { d i s t } ( y , z ) . \\end{align*}"}
-{"id": "2829.png", "formula": "\\begin{align*} T _ k = \\exp ( k ^ { ( 1 + \\varepsilon ^ 2 ) / p } ) , S _ k = T _ k - ( 1 - \\varepsilon ) h _ p ( T _ k ) , k \\ge 1 . \\end{align*}"}
-{"id": "797.png", "formula": "\\begin{align*} \\mathbb { T } ^ { [ M ' , M ] } ( z ) = L ^ { ( M ' ) } ( z ) L ^ { ( M ' + 1 ) } ( z ) \\cdots L ^ { ( M - 1 ) } ( z ) L ^ { ( M ) } ( z ) , \\end{align*}"}
-{"id": "4982.png", "formula": "\\begin{align*} Z _ { i } = p ^ { * } { p } _ { * } g ^ { * } D _ { i } - g ^ { * } D _ { i } \\end{align*}"}
-{"id": "4168.png", "formula": "\\begin{align*} G ( z ) = f ( z ) + \\alpha z f ' ( z ) + \\beta z ^ 2 f '' ( z ) \\end{align*}"}
-{"id": "9334.png", "formula": "\\begin{align*} U ( x ) = \\ln ( x ) D = ( c _ 1 , c _ 2 ) . \\end{align*}"}
-{"id": "8454.png", "formula": "\\begin{align*} = c ( c _ { S } , | | \\tilde { \\textbf { u } } _ { 1 } | | _ { m } , | | \\tilde { \\textbf { u } } _ { 2 } | | _ { m } , \\bar { \\rho } , \\varepsilon ^ { - 1 } ) | | \\tilde { \\textbf { u } } _ { 1 } - \\tilde { \\textbf { u } } _ { 2 } | | _ { s } , \\end{align*}"}
-{"id": "2141.png", "formula": "\\begin{align*} \\| u \\| _ { H ^ { s } ( \\Omega ) } ^ { 2 } = \\| u \\| _ { L ^ { 2 } ( \\Omega ) } ^ { 2 } + c _ { n , s } \\int _ { \\Omega } \\int _ { \\Omega } \\frac { | u ( x ) - u ( y ) | ^ { 2 } } { | x - y | ^ { n + 2 s } } d x d y . \\end{align*}"}
-{"id": "1216.png", "formula": "\\begin{align*} \\sup \\limits _ { g ' \\in \\mathcal { G } _ { f ^ { * } } } \\left ( \\sum \\limits _ { i = 1 } ^ { n } \\xi _ { i } p ( g ' ( X _ { i } ) ) - \\frac { h } { 1 2 } p ( g ' ( X _ i ) ) \\right ) \\end{align*}"}
-{"id": "3111.png", "formula": "\\begin{align*} P ^ T \\mathcal { L } _ 1 ( \\lambda ) P = \\begin{bmatrix} \\lambda P _ 7 + P _ 6 & - I _ n & 0 & 0 & 0 & 0 & 0 \\\\ - I _ n & 0 & \\lambda I _ n & 0 & 0 & 0 & 0 \\\\ 0 & \\lambda I _ n & \\lambda P _ 5 + P _ 4 & - I _ n & 0 & 0 & 0 \\\\ 0 & 0 & - I _ n & 0 & \\lambda I _ n & 0 & 0 \\\\ 0 & 0 & 0 & \\lambda I _ n & \\lambda P _ 3 + P _ 2 & - I _ n & 0 \\\\ 0 & 0 & 0 & 0 & - I _ n & 0 & \\lambda I _ n \\\\ 0 & 0 & 0 & 0 & 0 & \\lambda I _ n & \\lambda P _ 1 + P _ 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "25.png", "formula": "\\begin{align*} p _ { \\star } ( K _ { X / Y } + L ) _ { y } = H ^ { 0 } ( X _ { y } , K _ { X _ { y } } + L _ { y } ) \\end{align*}"}
-{"id": "562.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } | ( I - A _ n ) [ K , \\tau ] | _ { \\mathcal I } = 0 \\end{align*}"}
-{"id": "8057.png", "formula": "\\begin{align*} D = \\frac { 1 } { 1 1 } ( 4 B _ { 2 } + 1 2 B _ { 3 } + 1 3 B _ { 4 } + 1 8 B _ { 5 } + 1 6 B _ { 6 } ) = \\pi ^ { * } ( \\frac { 4 } { 1 1 } D _ { 2 } ) - B _ { 4 } - 2 B _ { 5 } - 4 B _ { 6 } . \\end{align*}"}
-{"id": "9466.png", "formula": "\\begin{align*} i _ { j \\nabla f } \\omega = \\omega ( j \\nabla f , \\cdot ) = - \\omega ( \\cdot , j \\nabla f ) = - g _ { j } ( \\cdot , \\nabla f ) = - d f . \\end{align*}"}
-{"id": "9419.png", "formula": "\\begin{align*} \\bigl ( D _ G ^ p ( w ) \\bigr ) ( e ) = \\alpha ( e ) w ( e ) ^ { 1 / ( p - 1 ) } . \\end{align*}"}
-{"id": "6626.png", "formula": "\\begin{align*} \\dfrac { d } { d t } \\psi ( \\Phi _ t ( y ) , t ) = \\partial _ x \\psi ( \\Phi _ t ( y ) , t ) \\cdot v ( \\Phi _ t ( y ) ) + \\partial _ t \\psi ( \\Phi _ t ( y ) , t ) \\end{align*}"}
-{"id": "4080.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 2 } { x z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ 2 z ^ { 4 } } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ { 4 } z ^ { 2 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 3 } { x ^ { 5 } z } , \\\\ f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 4 } { x ^ { 5 } z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z ) ^ 4 } { x ^ { 7 } z } , \\end{gather*}"}
-{"id": "8447.png", "formula": "\\begin{align*} A _ { j } ( \\textbf { u } ) = \\left ( \\begin{array} { c c c c c } v _ { j } & 0 & 0 & \\cdots & 0 \\\\ \\delta _ { j 1 } f ( \\textbf { u } ) & v _ { j } & 0 & \\cdots & 0 \\\\ \\delta _ { j 2 } f ( \\textbf { u } ) & 0 & v _ { j } & \\cdots & 0 \\\\ \\cdots & \\cdots & \\cdots & v _ { j } & \\cdots \\\\ \\delta _ { j d } f ( \\textbf { u } ) & 0 & 0 & \\cdots & v _ { j } \\\\ \\end{array} \\right ) . \\end{align*}"}
-{"id": "5855.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( L _ { Y _ { I } } C _ { x } \\right ) _ { , x } + C ^ { x } L _ { Y _ { I } } C _ { x } + \\frac { 1 } { 2 } m \\int Y _ { I } d x = c \\mbox { \\rm a n d } \\end{align*}"}
-{"id": "5601.png", "formula": "\\begin{align*} 0 & = Q T Q \\phi = ( 1 - P ) T Q \\phi = T \\phi - P T \\phi = U \\phi + v \\mathcal G _ 0 v ^ * \\phi - c _ 0 v ( 1 , 0 ) ^ T . \\end{align*}"}
-{"id": "4235.png", "formula": "\\begin{align*} \\int _ { \\mathbb { Z } _ p } ( 1 - 4 t ) ^ { \\tfrac { x } { 2 } } d \\mu _ { - 1 } ( x ) = \\sum _ { n = 0 } ^ \\infty C h _ { n , \\frac { 1 } { 2 } } \\frac { ( - 4 ) ^ n t ^ n } { n ! } . \\end{align*}"}
-{"id": "5342.png", "formula": "\\begin{align*} \\left ( ( u ' ) ^ { 3 } \\right ) ' = \\frac { e ^ { u ' } } { 2 } - 1 , u ( 0 ) = u ' ( 0 ) = u ' ( T ) , \\end{align*}"}
-{"id": "6779.png", "formula": "\\begin{align*} \\int _ { \\{ V _ { \\theta } - j < \\psi _ s \\leq v < \\psi < \\varphi + s \\} } \\theta _ { \\psi _ { t , j } } ^ n = 0 . \\end{align*}"}
-{"id": "8645.png", "formula": "\\begin{align*} [ f ] _ { \\alpha } = \\sup _ { x ' , \\ ; x \\in H , x - x ' \\not = 0 } { | f ( x ) - f ( x ' ) | _ E } \\ , { | x - x ' | _ H ^ { - \\alpha } } < \\infty . \\end{align*}"}
-{"id": "5520.png", "formula": "\\begin{align*} \\int _ { 0 } ^ { m ( E _ n ) } w = \\frac { 1 } { 2 ^ n \\varphi ( t _ n ) } . \\end{align*}"}
-{"id": "6699.png", "formula": "\\begin{align*} ( f _ K ^ I ) \\mapsto \\left \\{ u ^ { - 1 } \\left ( L ^ \\star _ { f _ K ^ I } \\theta ^ K \\delta _ I \\right ) u \\right \\} 1 = u ^ { - 1 } ( f _ K ^ I \\star u _ { I L } ) \\theta ^ K \\bar \\theta ^ L . \\end{align*}"}
-{"id": "4988.png", "formula": "\\begin{align*} a _ { n } = \\frac { h _ { H } ( f ^ { n } ( P ) ) } { R ^ { n } } \\ \\ \\ . \\end{align*}"}
-{"id": "9004.png", "formula": "\\begin{align*} ( ~ _ { 0 } I ^ \\alpha t ^ { \\beta - 1 } E _ { \\mu , \\beta } [ \\lambda t ^ \\mu ] ( x ) = x ^ { \\alpha + \\beta - 1 } E _ { \\mu , \\alpha + \\beta } [ \\lambda x ^ \\mu ] , \\end{align*}"}
-{"id": "1227.png", "formula": "\\begin{align*} \\| \\chi _ { ( 0 , a ) } \\| _ \\mathcal { M } = \\frac { a } { W ( a ) } \\Big { / } \\varphi ^ { - 1 } \\left ( \\frac { 1 } { W ( a ) } \\right ) . \\end{align*}"}
-{"id": "1200.png", "formula": "\\begin{align*} M ( \\Lambda _ 1 ) = \\left \\{ x , x y , x y ^ 2 , x y ^ 3 , \\ldots \\right \\} , \\end{align*}"}
-{"id": "3777.png", "formula": "\\begin{align*} C ( \\beta ) = \\int _ 0 ^ 1 \\left ( e ^ { \\beta x } - 1 \\right ) \\mu ( d x ) , \\end{align*}"}
-{"id": "4441.png", "formula": "\\begin{align*} d _ { \\mathrm { B L } } ( P , Q ) : = \\sup _ { h \\in \\mathcal { H } } \\biggl | \\int _ { - \\infty } ^ \\infty h \\ , d ( P - Q ) \\biggr | \\end{align*}"}
-{"id": "6312.png", "formula": "\\begin{align*} A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) = & n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) , \\\\ A _ 2 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) = & n \\alpha _ { 2 } ( 1 - 1 / p _ { 2 } ) - n \\alpha _ { 1 } ( 1 - 1 / p _ { 1 } ) - n ( \\alpha _ { 2 } - \\alpha _ { 1 } ) / q , \\\\ A _ 3 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) = & n ( \\alpha _ 2 - \\alpha _ 1 ) ( 1 / p _ 2 - 1 / q ) + n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) . \\end{align*}"}
-{"id": "2552.png", "formula": "\\begin{align*} \\begin{pmatrix} \\sum _ { k = 1 } ^ K e _ k ^ 2 ( x _ 1 ) \\eta _ k \\tau u _ 1 ^ 0 \\\\ \\vdots \\\\ \\sum _ { k = 1 } ^ K e _ k ^ 2 ( x _ M ) \\eta _ k \\tau u _ M ^ 0 \\end{pmatrix} = \\int _ 0 ^ \\tau \\begin{pmatrix} \\sum _ { k = 1 } ^ K e _ k ^ 2 ( x _ 1 ) \\eta _ k u _ 1 ^ 0 \\\\ \\vdots \\\\ \\sum _ { k = 1 } ^ K e _ k ^ 2 ( x _ M ) \\eta _ k u _ M ^ 0 \\end{pmatrix} d s . \\end{align*}"}
-{"id": "9120.png", "formula": "\\begin{align*} A ( t ) u ( t ) = A ( t ) e ^ { - t B A ( t ) } u _ { 0 } & + A ( t ) \\int _ { 0 } ^ { t } { e ^ { - ( t - l ) B A ( t ) } B ( \\mathcal { A } ( t ) - \\mathcal { A } ( l ) ) u ( l ) d l } \\\\ & + A ( t ) \\int _ { 0 } ^ { t } { e ^ { - ( t - l ) B A ( t ) } ( - P ( l ) ) u ( l ) d l } \\\\ & + A ( t ) \\int _ { 0 } ^ { t } { e ^ { - ( t - l ) B A ( t ) } f ( l ) } d l . \\end{align*}"}
-{"id": "3763.png", "formula": "\\begin{align*} C _ 0 \\left ( \\beta - i \\alpha t \\right ) & - C _ 0 ( \\beta ) + i \\alpha t u = \\int _ 0 ^ 1 \\left \\{ e ^ { ( \\beta - i \\alpha t ) x } - e ^ { \\beta x } + i \\alpha t x e ^ { \\beta x } \\right \\} g ( x ) d x \\\\ & = \\int _ 0 ^ 1 e ^ { \\beta x } \\left \\{ \\frac { - \\alpha ^ 2 t ^ 2 x ^ 2 } { 2 } + \\frac { i ^ 3 \\alpha 3 t ^ 3 x ^ 3 } { 3 ! } + \\alpha ^ 4 t ^ 4 x ^ 4 \\cdot O ( 1 ) \\right \\} g ( x ) d x \\end{align*}"}
-{"id": "5725.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm B M O } _ { L ^ { p ( \\cdot ) } } } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { \\| \\chi _ Q \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } } \\| ( b - b _ Q ) \\chi _ Q \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } . \\end{align*}"}
-{"id": "360.png", "formula": "\\begin{align*} G _ 1 ( t ) = \\langle \\Psi _ { c , h } , \\ , e ^ { i T _ { c , h } ( f _ 1 ) } e ^ { i t T _ { c , h } ( f _ 2 ) } \\Psi _ { c , h } \\rangle , \\end{align*}"}
-{"id": "694.png", "formula": "\\begin{align*} \\delta _ { f } = \\lim _ { n \\to \\infty } \\rho ( ( f ^ { n } ) ^ { * } \\colon N ^ { 1 } ( X ) _ { { \\mathbb { R } } } \\longrightarrow N ^ { 1 } ( X ) _ { { \\mathbb { R } } } ) ^ { 1 / n } . \\end{align*}"}
-{"id": "3602.png", "formula": "\\begin{align*} & \\Phi ( g , \\pi ) - \\left ( \\chi \\Phi ( g _ 1 , \\pi _ 1 ) + ( 1 - \\chi ) \\Phi ( g _ 2 , \\pi _ 2 ) \\right ) \\\\ & = \\chi D \\Phi | _ { ( g _ 1 , \\pi _ 1 ) } ( ( 1 - \\chi ) ( g _ 2 - g _ 1 , \\pi _ 2 - \\pi _ 1 ) ) + \\chi Q _ { ( g _ 1 , \\pi _ 1 ) } ( ( 1 - \\chi ) ( g _ 2 - g _ 1 , \\pi _ 2 - \\pi _ 1 ) ) \\\\ & \\quad + ( 1 - \\chi ) D \\Phi | _ { ( g _ 2 , \\pi _ 2 ) } ( \\chi ( g _ 1 - g _ 2 , \\pi _ 1 - \\pi _ 2 ) ) + ( 1 - \\chi ) Q _ { ( g _ 2 , \\pi _ 2 ) } ( \\chi ( g _ 1 - g _ 2 , \\pi _ 1 - \\pi _ 2 ) ) . \\end{align*}"}
-{"id": "5073.png", "formula": "\\begin{align*} I _ { k _ { 1 } , \\ldots , k _ { r } } = \\left \\{ \\vec { \\nu } = ( \\nu _ { 1 } , \\ldots , \\nu _ { k } ) \\in \\{ 1 , \\ldots , r \\} ^ { k } \\ , | \\ , \\# \\{ j \\ , | \\nu _ { j } = a \\} = k _ { a } \\ , ( 1 \\le \\forall { a } \\le r ) \\right \\} \\end{align*}"}
-{"id": "4925.png", "formula": "\\begin{align*} f ( z ) ! = \\left \\{ \\begin{aligned} & \\prod _ { y = 1 } ^ z f ( z ) , & & z > 0 , \\\\ & 1 , & & z = 0 , \\\\ & \\frac 1 { \\prod _ { y = z + 1 } ^ 0 f ( z ) } , & & z < 0 , \\end{aligned} \\right . \\qquad f ( z ) ! = f ( z ) \\cdot f ( z - 1 ) ! . \\end{align*}"}
-{"id": "9925.png", "formula": "\\begin{align*} Y _ { \\pm } v _ { \\pm } & = \\sum _ { i = 0 } ^ n \\left ( \\kappa \\sqrt { \\pm 1 } \\ , ( q ^ { n / 2 - i } \\mp q ^ { n } q ^ { i - n / 2 } ) \\lambda _ i v _ i \\ , + \\ , s [ n - i ] \\l _ i v _ { i + 1 } \\right ) \\\\ & = \\sum _ { i = 0 } ^ n \\left ( - q ^ { n / 2 } \\sqrt { \\pm 1 } \\ , [ i ] \\lambda _ i \\ , + \\ , s [ n - i + 1 ] \\l _ { i - 1 } \\right ) v _ { i } \\\\ & = 0 . \\end{align*}"}
-{"id": "4691.png", "formula": "\\begin{align*} \\big ( f _ { 2 k } ( x _ { t + 1 } ) , f _ { 2 k - 1 } ( x _ { t + 1 } ) \\big ) & = ( 2 , m - 1 ) , \\\\ \\big ( f _ { 2 k } ( x _ { t + m } ) , f _ { 2 k - 1 } ( x _ { t + m } ) \\big ) & = ( 2 , 0 ) , \\\\ \\big ( f _ { 2 k } ( x _ { t + m + 1 } ) , f _ { 2 k - 1 } ( x _ { t + m + 1 } ) \\big ) & = ( 1 , m - 1 ) . \\\\ \\end{align*}"}
-{"id": "8887.png", "formula": "\\begin{align*} \\liminf _ { n \\to \\infty } \\mathcal { I } _ { A _ n } ( u _ n ) = \\bigl ( \\tfrac { 1 } { 2 } - \\tfrac { 1 } { p } \\bigr ) \\liminf _ { n \\to \\infty } \\int _ { \\R ^ N } \\abs { u _ n } ^ p . \\end{align*}"}
-{"id": "8085.png", "formula": "\\begin{align*} T _ { g , ( s _ i , r _ j ) } = \\min \\left ( T _ { s _ i } , T _ { r _ j } \\right ) \\end{align*}"}
-{"id": "8505.png", "formula": "\\begin{align*} \\beta _ 1 = \\alpha _ { i _ 1 } \\ \\ \\prec \\ \\ \\beta _ 2 = s _ { i _ 1 } \\alpha _ { i _ 2 } \\ \\ \\prec \\ \\ \\cdots \\ \\ \\prec \\ \\ \\beta _ N = s _ { i _ 1 } s _ { i _ 2 } \\cdots s _ { i _ { N - 1 } } \\alpha _ { i _ N } . \\end{align*}"}
-{"id": "8761.png", "formula": "\\begin{align*} S _ i ( x _ 1 , . . , x _ n ) = & R _ { i , i - 1 } ( x _ i / x _ { i - 1 } ) . . R _ { i , 1 } ( x _ i / x _ { 1 } ) K _ 0 ( x _ i ) R _ { 1 , i } ( x _ i x _ 1 ) . . R _ { i - 1 , i } ( x _ i x _ { i - 1 } ) \\times \\\\ & R _ { i + 1 , i } ( x _ i x _ { i + 1 } ) . . R _ { L , i } ( x _ i x _ L ) K _ n ( 1 / x _ i ) R _ { i , L } ( x _ i / x _ { L } ) . . R _ { i , i + 1 } ( x _ i / x _ { i + 1 } ) . \\end{align*}"}
-{"id": "2046.png", "formula": "\\begin{align*} \\lim _ { y \\to x } Q ( x , y ) = 2 \\frac { \\langle \\nabla w ( x ) , X \\rangle } { \\bar { w } ' ( 0 ) } , \\end{align*}"}
-{"id": "2715.png", "formula": "\\begin{align*} \\tilde \\varphi _ t ^ { \\delta } ( x ) : = \\int _ { B } ( \\tau ( x - \\zeta ) + \\varphi _ t ( x - \\zeta ) ) \\tilde \\rho _ { \\delta } ( \\zeta ) d V ( \\zeta ) - \\tau ( x ) . \\end{align*}"}
-{"id": "228.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ 3 | T _ i | = O \\biggl ( \\max \\biggl \\{ \\frac { k ^ { - \\frac { 1 } { 2 } + \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { 2 \\alpha } { \\alpha + d } - \\epsilon } } \\ , , \\ , \\frac { k ^ { \\frac { 1 } { 2 } + \\frac { \\beta } { d } } } { n ^ { 1 + \\frac { \\beta } { d } } } \\log ^ { 2 + \\beta / d } n \\biggr \\} \\biggr ) \\end{align*}"}
-{"id": "8711.png", "formula": "\\begin{align*} v ( \\tau , \\Xi _ \\tau ^ { t , x } ) = Y _ \\tau ^ { t , x } , \\ , \\ ; \\ ; \\ ; \\tau \\in [ t , T ] , \\ ; \\P - a . s . . \\end{align*}"}
-{"id": "9681.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log x ( t ) } { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } = - \\frac { 1 } { \\beta } \\log \\left ( \\frac { a } { b } \\right ) . \\end{align*}"}
-{"id": "302.png", "formula": "\\begin{align*} W _ 2 ' = O \\biggl ( \\frac { k ^ { 1 / 2 } } { n } \\max \\biggl \\{ \\frac { k ^ { \\beta / d } } { n ^ { \\beta / d } } \\ , , \\ , \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr \\} \\biggr ) . \\end{align*}"}
-{"id": "4383.png", "formula": "\\begin{align*} W _ { h , I } T _ { c , 0 } ( f ) W _ { h , I } ^ * = T _ { c , h } ( f ) \\end{align*}"}
-{"id": "6470.png", "formula": "\\begin{align*} \\left ( \\int _ { 0 } ^ { t _ { 1 } - t _ { 0 } } \\left ( \\int _ { B _ { 1 } } ( \\psi w ) ^ { 2 } d x \\right ) ^ { p } d s \\right ) ^ { 1 / p } \\leq C \\left ( ( g _ { 1 - \\alpha } * W ) ( t _ { * } ) + \\| F \\| _ { L ^ { 1 } ( [ 0 , t _ { 2 } - t _ { 0 } ] ) } \\right ) , \\end{align*}"}
-{"id": "6191.png", "formula": "\\begin{align*} \\frac { f ( x u ) - f ( x ) } { x f ' ( x ) } = \\int _ 1 ^ u \\frac { g ( x y ) } { g ( x ) } y ^ { - 1 } \\d y \\to \\log u , \\end{align*}"}
-{"id": "2611.png", "formula": "\\begin{align*} \\alpha _ \\omega ( x | y , z ) & = \\omega ( x , y , z ) - \\omega ( y , x , z ) + \\omega ( y , z , x ) \\\\ \\beta _ \\omega ( x , y | z ) & = \\omega ( x , y , z ) - \\omega ( x , z , y ) + \\omega ( z , x , y ) . \\end{align*}"}
-{"id": "3680.png", "formula": "\\begin{align*} \\frac { \\partial \\zeta } { \\partial t } d x \\wedge d y = d H _ { \\zeta } \\wedge d Z _ { \\zeta } = J ( q , \\chi ) d x \\wedge d y ~ , \\end{align*}"}
-{"id": "2220.png", "formula": "\\begin{align*} \\tilde { u } \\in L ^ { \\infty } ( ( 0 , T ] ; H _ { e } ^ { 2 \\beta } ( \\Omega ) ) \\subset L ^ { 2 } ( ( 0 , T ] ; H _ { e } ^ { 2 \\beta } ( \\Omega ) ) , \\lim _ { t \\rightarrow 0 } \\| \\tilde { u } ( \\cdot , t ) \\| _ { L ^ { 2 } ( \\Omega ) } = 0 . \\end{align*}"}
-{"id": "8805.png", "formula": "\\begin{align*} n H \\int _ { \\Sigma } u d v o l _ { \\Sigma } = \\int _ { \\Sigma } | \\nabla _ { \\Sigma } u | ^ 2 - | A _ { \\Sigma } | ^ 2 u ^ 2 d v o l _ { \\Sigma } \\geq 0 \\ , . \\end{align*}"}
-{"id": "7370.png", "formula": "\\begin{align*} \\frac { c _ { 2 } } { c _ { 1 } } q ^ { - 3 } ( q - q ^ { - 1 } ) = \\frac { c _ { 1 } } { c _ { 0 } } t ^ { \\prime } , - \\frac { c _ { 2 } } { c _ { 1 } } q ^ { - 2 } = \\frac { c _ { 1 } } { c _ { 0 } } t . \\end{align*}"}
-{"id": "7390.png", "formula": "\\begin{align*} \\frac { 1 } { n } ( m ^ r ) ^ * m ^ t \\doteq \\mathbb { E } [ g _ r ( \\sigma _ r \\breve { Z } _ r , W ) g _ t ( \\sigma _ t \\breve { Z } _ t , W ) ] = \\breve { E } _ { r , t } . \\end{align*}"}
-{"id": "7568.png", "formula": "\\begin{align*} K = \\left [ \\frac { v _ l - b } { \\Delta _ l ( x _ 0 ) } \\right ] + 1 , \\end{align*}"}
-{"id": "541.png", "formula": "\\begin{align*} f ( z ) = - h z + \\log ( 1 + 2 z ) - \\log ( 1 - 2 z ) . \\end{align*}"}
-{"id": "8985.png", "formula": "\\begin{align*} ( ~ ~ ^ { A B C } ~ _ { a } D ^ \\alpha f ) ( t ) = \\frac { B ( \\alpha ) } { 1 - \\alpha } \\int _ a ^ t f ^ \\prime ( x ) E _ \\alpha [ - \\alpha \\frac { ( t - x ) ^ \\alpha } { 1 - \\alpha } ] d x \\end{align*}"}
-{"id": "9734.png", "formula": "\\begin{align*} \\varphi ( t ) = b g ( x ( t - \\tau ( t ) ) ) , t > 0 . \\end{align*}"}
-{"id": "3118.png", "formula": "\\begin{align*} \\begin{bmatrix} \\lambda P _ 5 - P _ 4 & \\lambda P _ 4 & 0 \\\\ \\lambda P _ 4 & \\lambda P _ 3 + P _ 2 & P _ 1 \\\\ 0 & P _ 1 & - \\lambda P _ 1 + P _ 0 \\end{bmatrix} \\ , \\mbox { a n d } \\ , \\ , \\begin{bmatrix} \\lambda P _ 5 + P _ 4 & P _ 3 & P _ 2 \\\\ P _ 3 & - \\lambda P _ 3 + P _ 2 & - \\lambda P _ 2 + P _ 1 \\\\ P _ 2 & - \\lambda P _ 2 + P _ 1 & - \\lambda P _ 1 + P _ 0 \\end{bmatrix} . \\end{align*}"}
-{"id": "1979.png", "formula": "\\begin{align*} g ( { \\bf x } ^ { \\bf s } ) = c ^ N k _ 2 ^ { s _ 2 } \\cdots k _ n ^ { s _ n } { \\bf x } ^ { \\bf s } . \\end{align*}"}
-{"id": "1479.png", "formula": "\\begin{align*} f = \\sum _ { j = 3 } ^ \\infty \\chi _ { [ j ! , j ! + 1 ] } , \\end{align*}"}
-{"id": "1054.png", "formula": "\\begin{align*} J ( v _ n ) = \\left ( \\frac { 1 } { p ' } - \\frac 1 2 \\right ) \\| v _ n \\| _ { p ' } ^ { p ' } + \\frac 1 2 J ' ( v _ n ) v _ n \\geq \\left ( \\frac { 1 } { p ' } - \\frac 1 2 \\right ) \\| v _ n \\| _ { p ' } ^ { p ' } - \\frac 1 2 \\| J ' ( v _ n ) \\| _ \\ast \\| v _ n \\| _ { p ' } \\end{align*}"}
-{"id": "5061.png", "formula": "\\begin{align*} S = \\big \\{ g \\in G \\ , : \\ , \\nu ( g B ) \\geq \\nu ( B ) \\big \\} \\end{align*}"}
-{"id": "2364.png", "formula": "\\begin{align*} M ^ { ( k ) } : = \\max _ { j \\in \\{ 0 , \\ldots , N _ k - 1 \\} } t ^ k _ { j + 1 } - t ^ k _ j \\end{align*}"}
-{"id": "1494.png", "formula": "\\begin{align*} | \\Phi | - | \\Phi _ x | \\geq n ( n - 1 ) - ( n - 1 ) ( n - 2 ) = 2 ( n - 1 ) > ( n - 1 ) . \\end{align*}"}
-{"id": "8415.png", "formula": "\\begin{align*} \\rho ( 0 , x ) = \\rho _ { 0 } ( x ) , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ v ( 0 , x ) = v _ { 0 } ( x ) ~ ~ ~ ~ \\nabla \\cdot v _ { 0 } ( x ) = 0 , \\end{align*}"}
-{"id": "7819.png", "formula": "\\begin{align*} \\langle \\xi , \\eta \\rangle _ { T } = \\int _ 0 ^ T \\int _ 0 ^ T \\phi ( u - v ) \\xi _ { u } \\eta _ { v } d u d v , \\ \\ a n d \\ \\ \\| \\xi \\| _ { T } ^ { 2 } = \\langle \\xi , \\xi \\rangle _ { T } . \\end{align*}"}
-{"id": "1722.png", "formula": "\\begin{gather*} M _ { \\xi } = M _ 5 \\cup M _ 4 \\cup M _ 5 ^ - . \\end{gather*}"}
-{"id": "7733.png", "formula": "\\begin{align*} \\mathbb P \\{ \\omega \\in \\Omega : \\lim _ { n \\to \\infty } x _ n ( \\omega ) = 0 \\} \\le 1 / 2 . \\end{align*}"}
-{"id": "2124.png", "formula": "\\begin{align*} \\tilde { w } ( z ) : = \\frac { w _ { k + 1 } ( \\l z ) } { \\l ^ { 2 s + 2 k + 2 } } , \\end{align*}"}
-{"id": "5752.png", "formula": "\\begin{align*} \\hat { F } _ n ^ \\mathrm { T } = \\sum _ { t = 1 } ^ { n } \\sigma ^ 2 _ t ( \\hat { \\delta } _ { 0 } ) \\begin{bmatrix} x _ { t } ( m \\hat { \\pi } ) _ { t - J _ \\theta } ^ { \\mathrm { T } } & 0 \\end{bmatrix} , \\end{align*}"}
-{"id": "3560.png", "formula": "\\begin{align*} ( 2 \\psi ^ R , V ^ R ) & = - \\Phi ( \\gamma ^ R , \\tau ^ R ) + \\chi \\Phi ( g ^ R , \\pi ^ R ) + ( 1 - \\chi ) \\Phi ( ( g ^ \\theta ) ^ R , ( \\pi ^ \\theta ) ^ R ) + ( 2 \\psi _ 0 R ^ { - 2 } , 0 ) . \\end{align*}"}
-{"id": "7752.png", "formula": "\\begin{align*} \\mathbb P [ T _ N ( N + 1 ) \\le x ] = \\mathbb P [ T _ N ( N + 1 ) \\ge - x ] . \\end{align*}"}
-{"id": "3483.png", "formula": "\\begin{align*} \\tilde { h } ( x ' , x _ n ) = \\begin{cases} h ( x ' , x _ n ) & \\mbox { i f } \\ , \\ , \\ , x _ n > 0 , \\\\ - h ( x ' , - x _ n ) & \\mbox { i f } \\ , \\ , \\ , x _ n < 0 \\end{cases} \\end{align*}"}
-{"id": "4153.png", "formula": "\\begin{align*} \\Lambda ^ \\nu ( s ) : = \\frac { q ( f ) ^ { s / 2 } } { \\pi ^ s } \\Gamma ( s + \\nu ) L ^ \\nu ( s , f ) = \\varepsilon ( f ) \\Lambda ^ \\nu ( 2 \\kappa ( f ) + 1 - 2 \\nu - s , \\widetilde { f } ) , \\end{align*}"}
-{"id": "6281.png", "formula": "\\begin{align*} z _ i \\leftarrow z _ i - W _ { p , l } ( z _ i ) , ~ i = 1 , \\dots , d , \\end{align*}"}
-{"id": "6015.png", "formula": "\\begin{gather*} \\Phi _ s : = \\Phi - \\tfrac { 1 } { 2 } s ^ 2 \\Phi _ I + s \\Phi _ J \\end{gather*}"}
-{"id": "8265.png", "formula": "\\begin{align*} \\sigma \\Big ( \\sum _ { i = 0 } ^ { \\ell - 1 } u ( i ) \\Big ) \\ni A \\mapsto E _ { \\lambda } [ F \\ 1 _ A ] \\in \\R \\end{align*}"}
-{"id": "628.png", "formula": "\\begin{align*} \\| u \\| _ { L ^ \\infty ( \\overline \\Omega ) } + t \\leq \\kappa _ 4 + \\kappa _ 2 = : C \\end{align*}"}
-{"id": "1842.png", "formula": "\\begin{align*} & \\int _ 0 ^ 2 u ^ { - 1 / 2 } ( 1 + i \\log u ) ^ { - t } d u \\\\ & = \\frac { ( i / 2 ) ^ N } { ( t - 1 ) ( t - 2 ) \\cdots ( t - N ) } \\int _ 0 ^ 2 u ^ { - 1 / 2 } ( 1 + i \\log u ) ^ { - t + N } d u + O ( e ^ { - c t } ) . \\end{align*}"}
-{"id": "4334.png", "formula": "\\begin{align*} u _ s ( t , x ) = \\left ( ( 2 - p ) ( T - t ) _ + \\right ) ^ { 1 / ( 2 - p ) } f ( | x | ) \\ , ( t , x ) \\in ( 0 , \\infty ) \\times \\mathbb { R } \\ , \\end{align*}"}
-{"id": "4679.png", "formula": "\\begin{align*} \\sum _ { k = 2 ^ { j } } ^ \\infty | a _ { k } | ^ 2 & \\leq 2 \\sum _ { n = j } ^ \\infty \\sum _ { k = 2 ^ { n - 1 } + 1 } ^ { 2 ^ { n + 1 } - 1 } | \\hat { w } _ { n , 2 } ( k ) a _ k | ^ 2 = 2 \\sum _ { n = j } ^ \\infty \\| w _ { n , 2 } * a \\| _ { L ^ 2 ( S ^ 1 ) } ^ 2 \\\\ & \\leq 2 \\sum _ { n = j } ^ \\infty \\| w _ { n , 2 } * a \\| _ { L ^ \\infty ( S ^ 1 ) } ^ 2 \\leq 2 \\sum _ { n = j } ^ \\infty 2 ^ { - n } \\| a \\| _ { C ^ { 1 / 2 , * } ( S ^ 1 ) } ^ 2 = 2 ^ { - j + 2 } \\| a \\| _ { C ^ { 1 / 2 , * } ( S ^ 1 ) } ^ 2 = O ( 2 ^ { - j } ) . \\end{align*}"}
-{"id": "8759.png", "formula": "\\begin{align*} T ( w ) = _ 0 ( M ( w ) \\widetilde { K } _ n ( w ) ) . \\end{align*}"}
-{"id": "8991.png", "formula": "\\begin{align*} \\int _ a ^ b \\varphi ( x ) ( ~ ^ { A B } ~ _ { a } I ^ \\alpha \\psi ) ( x ) d x = \\int _ 0 ^ 1 ( 1 - x ) ~ ^ { A B } { ~ _ 0 } I ^ { 1 / 2 } x = \\int _ 0 ^ 1 ( 1 - x ) [ \\frac { x } { 2 } + \\frac { 2 x ^ { 3 / 2 } } { 3 \\sqrt { \\pi } } ] d x = \\frac { 1 } { 1 2 } + \\frac { 8 } { 1 0 5 \\sqrt { \\pi } } , \\end{align*}"}
-{"id": "416.png", "formula": "\\begin{align*} [ P , a ^ 1 ] \\cdots [ P , a ^ { 2 k + 1 } ] = & ( - 1 ) ^ k P a ^ 1 ( 1 - P ) a ^ 2 P \\cdots ( 1 - P ) a ^ { 2 k } P a ^ { 2 k + 1 } ( 1 - P ) \\\\ & \\quad - ( - 1 ) ^ k ( 1 - P ) a ^ 1 P a ^ 2 ( 1 - P ) \\cdots P a ^ { 2 k } ( 1 - P ) a ^ { 2 k + 1 } P \\\\ = & ( - 1 ) ^ k P a _ + ^ 1 ( 1 - P ) a ^ 2 P \\cdots ( 1 - P ) a _ - ^ { 2 k } P a _ + ^ { 2 k + 1 } ( 1 - P ) \\\\ & \\quad - ( - 1 ) ^ k ( 1 - P ) a _ - ^ 1 P a _ + ^ 2 ( 1 - P ) \\cdots P a ^ { 2 k } ( 1 - P ) a _ - ^ { 2 k + 1 } P . \\end{align*}"}
-{"id": "6633.png", "formula": "\\begin{align*} \\mu ^ { f _ n , k } _ \\tau & = Q ^ { u , f _ n } _ { \\tau - t ^ k _ j } \\ , \\mu ^ { f _ n , k } _ { t ^ k _ j } , \\\\ \\mu ^ { f , k } _ \\tau & = Q ^ { \\bar { u } , f } _ { \\tau - t ^ k _ j } \\ , \\mu ^ { f , k } _ { t ^ k _ j } , \\end{align*}"}
-{"id": "1371.png", "formula": "\\begin{align*} R _ { n } ( u ) - R _ { n } ^ { \\dag } ( u ) = \\frac { 1 } { 2 } u ^ { T } \\left ( \\frac { 1 } { n } \\sum _ { t = 1 } ^ { n } ( y _ { t } - m _ { t } \\pi _ { 0 , t } ) \\ddot { W } _ { 0 , t } \\right ) u + E _ { n } ( u ^ { \\ast } ) - E _ { n } ^ { \\dag } ( u ^ { \\ast } ) \\end{align*}"}
-{"id": "1738.png", "formula": "\\begin{gather*} g _ I : = \\left [ - \\tfrac { 3 } { 2 } \\big ( I ^ 2 + 1 \\big ) y ^ 2 + 2 I p ^ 2 - \\tfrac { 1 } { 2 } q ^ 2 \\right ] \\ ! d x ^ 2 - 4 I p \\ , d x \\ , d y + q \\ , d x \\ , d p \\\\ \\hphantom { g _ I : = } { } - 3 p \\ , d x \\ , d q - 3 \\ , d x \\ , d z - 3 I \\ , d y ^ 2 + 3 \\ , d y \\ , d q - 2 \\ , d p ^ 2 . \\end{gather*}"}
-{"id": "3529.png", "formula": "\\begin{align*} \\Phi ^ W _ { ( g , \\pi ) } ( g + h , \\pi + w ) = \\Phi ^ W _ { ( g , \\pi ) } ( g , \\pi ) + ( 2 \\psi , V ) . \\end{align*}"}
-{"id": "4161.png", "formula": "\\begin{align*} S ^ \\nu _ f ( n ) \\overline { S ^ \\nu _ g ( n ) } = \\frac { 1 } { 2 } \\left ( | S ^ \\nu _ { h _ 1 } ( n ) | ^ 2 + i | S ^ \\nu _ { h _ 2 } ( n ) | ^ 2 - ( 1 + i ) \\left ( | S ^ \\nu _ f ( n ) | ^ 2 + | S ^ \\nu _ g ( n ) | ^ 2 \\right ) \\right ) . \\end{align*}"}
-{"id": "4012.png", "formula": "\\begin{align*} | B _ H ( U ) | = \\left \\lbrace \\begin{array} { l l l } 0 & \\hbox { i f $ U \\subseteq H $ } , \\\\ q ^ { \\dim U - 1 } & \\hbox { o t h e r w i s e } . \\end{array} \\right . \\end{align*}"}
-{"id": "7561.png", "formula": "\\begin{align*} u _ l : = \\sup \\{ u < a : F ( u ) < l \\} \\end{align*}"}
-{"id": "1663.png", "formula": "\\begin{align*} \\| D _ v { \\psi } \\| _ { L ^ p ( \\R ^ { 2 d } ) } ^ p = \\int _ { \\R ^ d } \\| D _ v \\psi ( x , \\cdot ) \\| _ { L ^ p ( \\R ^ d ) } ^ p \\dd x \\le ( \\| { \\psi } \\| _ { L ^ p ( \\R ^ { 2 d } ) } ) ^ { p / 2 } \\ , ( \\| D _ v ^ 2 { \\psi } \\| _ { L ^ p ( \\R ^ { 2 d } ) } ) ^ { p / 2 } . \\end{align*}"}
-{"id": "9043.png", "formula": "\\begin{align*} R ( z ) & = \\frac { 1 } { 2 k m } \\sum _ { p = 0 } ^ { m - 1 } \\sum _ { s = - p } ^ { k + p } z ^ s = \\frac { 1 } { 2 k m } \\sum _ { p = 0 } ^ { m - 1 } z ^ { - p } \\frac { z ^ { k + 2 p + 1 } - 1 } { z - 1 } \\\\ & = \\frac { 1 } { 2 k m ( z - 1 ) } \\sum _ { p = 0 } ^ { m - 1 } ( z ^ { k + p + 1 } - z ^ { - p } ) = \\frac { 1 } { 2 k m ( z - 1 ) } \\left ( \\frac { ( z ^ { k + 1 } - z ^ { - m + 1 } ) ( z ^ m - 1 ) } { z - 1 } \\right ) , \\end{align*}"}
-{"id": "9846.png", "formula": "\\begin{align*} P ( f ) & = \\bar { P } / W , \\\\ \\bar \\Xi ( f ) & = \\sqrt { \\bar { Q } / \\big ( W \\big ( \\big | H _ { \\rm S R } ( f ) \\big | ^ 2 P ( f ) + N _ 0 \\big ) \\big ) } . \\end{align*}"}
-{"id": "6249.png", "formula": "\\begin{align*} c ^ 2 \\bar { \\lambda } _ i ( n , c D , K ) = \\lambda _ i ( \\tilde { L } _ c ) . \\end{align*}"}
-{"id": "6818.png", "formula": "\\begin{align*} \\sum _ { k = 0 } ^ { n - 1 } \\left [ B _ k , B _ { m + n - k } \\right ] = \\sum _ { k = 0 } ^ { m } \\left [ B _ k , B _ { m + n - k } \\right ] \\end{align*}"}
-{"id": "6024.png", "formula": "\\begin{align*} \\hat { R } _ { i j } { } ^ k { } _ l \\hat { K } ^ l { } _ k = 2 \\ , \\hat { K } ^ l { } _ i \\hat { R } _ { k l } { } ^ k { } _ j \\end{align*}"}
-{"id": "7222.png", "formula": "\\begin{align*} f _ { e _ { j } } \\left ( q \\right ) = g _ { a _ { i } } \\left ( q \\right ) = g _ { b _ { i } } \\left ( q \\right ) \\end{align*}"}
-{"id": "3656.png", "formula": "\\begin{align*} \\beta ^ { m } \\binom { n ^ 2 } { m } ^ 3 . \\end{align*}"}
-{"id": "8749.png", "formula": "\\begin{gather*} | Q ^ { - 1 / 2 } _ t e ^ { t A } k | _ H \\le \\frac { c } { t ^ { 3 / 2 } } | k | _ K , \\ ; \\ ; \\ ; k \\in K = V \\times U ; \\\\ | Q ^ { - 1 / 2 } _ t e ^ { t A } G a | _ H \\le \\frac { c } { t ^ { 1 / 2 } } | a | _ U , \\ ; \\ ; \\ ; a \\in U . \\end{gather*}"}
-{"id": "1018.png", "formula": "\\begin{align*} \\left \\{ \\begin{array} { l l l } ( \\varphi ( u ' ) ) ' = \\lambda Q ( N _ { f } ( u ) ) & & \\\\ \\int _ 0 ^ T f ( t , u ( t ) , u ' ( t ) ) d t = \\varphi ( b u ( 0 ) ) - \\varphi ( u ( 0 ) ) , \\ u ' ( 0 ) = u ( 0 ) . \\end{array} \\right . \\end{align*}"}
-{"id": "2425.png", "formula": "\\begin{align*} [ M _ { p _ 1 , q _ 1 } ^ { s _ 1 , \\alpha } , M _ { p _ 2 , q _ 2 } ^ { s _ 2 , \\alpha } ] _ { \\theta } = M _ { p _ { \\theta } , q _ { \\theta } } ^ { s _ { \\theta } , \\alpha } \\end{align*}"}
-{"id": "3437.png", "formula": "\\begin{align*} Y ( t ) = \\langle \\eta ( T ) , u _ 0 ( \\xi ( T ) ) \\rangle + \\int _ t ^ T \\tilde G ( \\theta , \\kappa ( \\theta ) , Y ( \\theta ) , Z ( \\theta ) ) d \\theta - \\int _ t ^ T \\langle Z ( \\theta ) , d W ( \\theta ) \\rangle . \\end{align*}"}
-{"id": "4684.png", "formula": "\\begin{align*} u _ l : = \\Phi _ { l } ^ { \\theta } u . \\end{align*}"}
-{"id": "9277.png", "formula": "\\begin{align*} \\begin{cases} d M ( t , z ) = \\mathbb { E } [ D _ t M ( T , z ) | \\mathcal { F } _ t ] d B ( t ) = \\Phi _ K ( t , z ) M ( t , z ) d B ( t ) \\\\ M ( 0 , z ) = 1 \\end{cases} \\end{align*}"}
-{"id": "1587.png", "formula": "\\begin{align*} 0 = \\left ( \\frac { 1 } { 2 } \\sigma ^ { 2 } \\left ( L _ { Y _ { I } } C _ { x } \\right ) _ { , x } + C ^ { x } L _ { Y _ { I } } C _ { x } \\right ) - \\frac { T _ { I , t } } { T _ { I } } \\left ( \\frac { 1 } { 2 } \\int L _ { Y _ { I } } C _ { x } + C ^ { x } Y _ { I } + 2 \\psi _ { I } + \\frac { 1 } { 2 } \\int Y _ { I } \\mbox { \\rm d } x \\right ) + 2 f _ { I , t } . \\end{align*}"}
-{"id": "3980.png", "formula": "\\begin{align*} V = \\bigoplus _ { p } V _ { p } . \\end{align*}"}
-{"id": "2013.png", "formula": "\\begin{align*} \\left | \\frac { f ( x ( 1 + z ) ) - f ( x ) } { x f ' ( x ) } - z \\right | = \\left | z \\left ( \\frac { f ' ( \\xi ) } { f ' ( x ) } - 1 \\right ) \\right | \\leq z ^ 2 \\frac { | x f '' ( \\xi ' ) | } { f ' ( x ) } , \\end{align*}"}
-{"id": "5499.png", "formula": "\\begin{align*} \\gamma _ k ^ \\mathrm { C } [ \\iota ] = \\begin{cases} \\frac { M \\sigma _ { \\mathrm { s } k } ^ 2 [ \\iota ] } { \\sum _ { i = 1 } ^ { K } \\beta _ { \\mathrm { s } i } + 1 / \\rho _ \\mathrm { s } } , & \\\\ \\frac { M \\sigma _ { \\mathrm { s } k } ^ 2 [ \\iota ] } { \\sum _ { i = 1 } ^ { K } \\beta _ { \\mathrm { s } i } + \\left ( \\rho _ \\mathrm { d } \\beta _ \\mathrm { L I } + 1 \\right ) / \\rho _ \\mathrm { s } } , & . \\end{cases} \\end{align*}"}
-{"id": "4650.png", "formula": "\\begin{align*} g ^ \\iota = \\begin{pmatrix} 0 & - 1 \\\\ 1 & 0 \\end{pmatrix} ( g ^ { - 1 } ) ^ t \\begin{pmatrix} 0 & 1 \\\\ - 1 & 0 \\end{pmatrix} . \\end{align*}"}
-{"id": "7438.png", "formula": "\\begin{align*} ( B _ + u B _ + ) ( B _ + v B _ + ) = B _ + u v B _ + \\end{align*}"}
-{"id": "8047.png", "formula": "\\begin{align*} \\vec { a } ^ { \\vec { x } } a _ j ^ { - k } & = s _ { 1 , 2 } ^ { x _ 1 } s _ { 1 , 3 } ^ { x _ 2 } \\cdots s _ { 1 , m + 1 } ^ { x _ m } \\cdots s _ { 1 , j + 1 } ^ { x _ j } \\cdots s _ { 1 , m + 2 } ^ { x _ { 2 m + 1 } } \\cdot s _ { j - m + 1 , m + 2 } ^ { - k } \\\\ & = s _ { 1 , 2 } ^ { x _ 1 } s _ { 1 , 3 } ^ { x _ 2 } \\cdots s _ { 1 , m + 1 } ^ { x _ m } \\cdots s _ { j - m + 1 , j + 1 } ^ { x _ j - k } \\cdots s _ { 1 , m + 2 } ^ { x _ { 2 m + 1 } } . \\end{align*}"}
-{"id": "535.png", "formula": "\\begin{align*} \\hat { R } _ { 1 2 } ^ { \\rm e x p } ( i , x ; j , y ) = - \\int _ { \\mathcal { C } _ { 1 / 4 } ^ { \\pi / 3 } } \\frac { ( 1 + 2 z ) ^ { n _ i } } { ( 1 + 2 z ) ^ { n _ j } } \\frac { ( 1 - 2 z ) ^ { m _ j } } { ( 1 - 2 z ) ^ { m _ i } } e ^ { - \\vert x - y \\vert z } \\dd z , \\end{align*}"}
-{"id": "1429.png", "formula": "\\begin{align*} L _ { - 1 } A = T A \\end{align*}"}
-{"id": "3033.png", "formula": "\\begin{align*} { \\bf { H } } _ 1 ^ { [ 1 1 ] } ( n ) { { \\bf { \\tilde v } } _ 1 } = { \\bf { h } } _ 1 ^ { [ 1 1 ] } ( { t _ 2 } ) , \\end{align*}"}
-{"id": "8560.png", "formula": "\\begin{align*} ( X _ { a ^ { \\ast } } ( \\widetilde { P } ) ) _ { \\overline { N } } & = \\biggl ( c _ { k } \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } r ^ { - 1 } ( ^ { \\ast } b ) \\rho ( b r a ) \\biggr ) _ { \\overline { N } } \\\\ & = c _ { k } \\displaystyle \\sum _ { b \\in T _ { k } , r \\in L ( k ) } ( r ^ { - 1 } ( ^ { \\ast } b ) \\rho ( b r a ) ) _ { \\overline { N } } \\end{align*}"}
-{"id": "1953.png", "formula": "\\begin{align*} \\frac { d H } { d \\tau } = H V H , \\end{align*}"}
-{"id": "2705.png", "formula": "\\begin{align*} \\int _ { \\{ \\psi _ t < \\psi < \\varphi + t \\} } \\theta _ { \\psi _ t } ^ n = 0 . \\end{align*}"}
-{"id": "3625.png", "formula": "\\begin{align*} u _ { n + 1 } = f ( u _ n , u _ { n - 1 } , \\dots , u _ { n - k + 1 } ) , n = 0 , 1 , 2 , \\dots \\end{align*}"}
-{"id": "2588.png", "formula": "\\begin{align*} \\norm { v _ n ^ j } _ { W ( [ t _ n , \\infty ) ) } = \\norm { \\Psi ^ j _ { [ h _ n ^ j ] } } _ { W ( [ t _ n , \\infty ) ) } \\le \\norm { \\Psi ^ j } _ { W ( I _ j ) } < \\infty . \\end{align*}"}
-{"id": "9542.png", "formula": "\\begin{align*} \\begin{cases} \\ \\mathbf { E } x ( k + 1 ) = \\mathbb { A } ( k ) \\mathbf { E } x ( k ) + \\mathbb { B } ( k ) \\mathbf { E } \\nu ( k ) , & \\\\ \\ \\mathbf { E } x ( 0 ) = \\mathbf { E } x _ { 0 } \\in R ^ { n } , & \\end{cases} \\end{align*}"}
-{"id": "6578.png", "formula": "\\begin{align*} F ( x ) = \\langle \\nabla f _ 1 ( x ) , x \\rangle - f _ 1 ( x ) - f _ 2 ( \\nabla f _ 1 ( x ) ) . \\end{align*}"}
-{"id": "2751.png", "formula": "\\begin{align*} p = p ( \\theta , \\zeta ) = p _ K ^ I \\theta ^ K \\zeta _ I , \\end{align*}"}
-{"id": "9524.png", "formula": "\\begin{align*} \\delta \\left ( G _ 1 \\wedge * G _ 1 \\right ) & = 2 \\delta d C _ 0 \\wedge * G _ 1 \\\\ & = 2 d ( \\delta C _ 0 * G _ 1 ) - 2 \\delta C _ 0 d * G _ 1 \\end{align*}"}
-{"id": "5127.png", "formula": "\\begin{align*} F ^ { 1 ^ { k } } _ { \\vec { z } } ( x _ { 1 } , \\ldots , x _ { k - 1 } , 1 ) = \\sum _ { \\ell = 1 } ^ { k } \\frac { z _ { \\ell } } { 1 + z _ { \\ell } } \\prod _ { 0 < i \\le k \\atop i \\not = \\ell } f ( z _ { i } , z _ { \\ell } ) F ^ { 1 ^ { k - 1 } } _ { \\vec { z } ( \\ell ) } ( x _ { 1 } , \\ldots , x _ { k - 1 } ) \\otimes u _ { 1 } , \\end{align*}"}
-{"id": "1264.png", "formula": "\\begin{align*} \\phi & = - U v \\mathcal G _ 0 v ^ * \\phi + c _ 0 U v ( 1 , 0 ) ^ T : = U v \\psi , \\end{align*}"}
-{"id": "569.png", "formula": "\\begin{align*} \\| ( x , ( y _ j ) _ { 1 \\le j \\le n } ) \\| = \\max \\left ( | x | _ 1 , \\sum _ { 1 \\le j \\le n } | y _ j | _ { { \\mathcal I } ^ * } \\right ) \\end{align*}"}
-{"id": "7778.png", "formula": "\\begin{align*} \\begin{aligned} \\epsilon _ 1 ^ 2 : \\C \\otimes _ \\C \\C & \\rightarrow \\R \\\\ z \\otimes w & \\rightarrow \\Re ( z w ) , \\end{aligned} \\begin{aligned} \\epsilon _ 2 ^ 1 : \\C \\otimes _ \\R \\C & \\rightarrow \\C \\\\ z \\otimes w & \\rightarrow z w \\ ; . \\end{aligned} \\end{align*}"}
-{"id": "3549.png", "formula": "\\begin{align*} \\Pi _ { g _ 0 } \\circ \\Phi ^ W _ { ( g , \\pi ) } ( g + h , \\pi + w ) = \\Pi _ { g _ 0 } \\circ \\Phi ^ W _ { ( g , \\pi ) } ( g , \\pi ) + \\Pi _ { g _ 0 } ( 2 \\psi , V ) \\end{align*}"}
-{"id": "1916.png", "formula": "\\begin{align*} ( \\phi _ j ^ { K } ( u ) - \\phi _ 0 ^ { K } ( u ) ) = \\phi _ 0 ( x ) ^ { K } ( 1 - ( 1 - \\psi _ m ) ^ { K } ( u ) ) . \\end{align*}"}
-{"id": "174.png", "formula": "\\begin{align*} \\hat { H } _ n ^ w : = \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\sum _ { j = 1 } ^ k w _ j \\log \\xi _ { ( j ) , i } , \\end{align*}"}
-{"id": "9297.png", "formula": "\\begin{align*} H ( t , x , y , \\varphi , \\pi , p , q ) = ( L ^ * _ { \\pi } \\varphi ) p + x y q , \\end{align*}"}
-{"id": "1012.png", "formula": "\\begin{align*} \\lim _ { \\lambda \\to \\mathcal U } \\| \\varphi \\circ \\Phi - ( \\varphi \\circ \\Phi ) ( f _ \\lambda \\ , \\cdot \\ , f _ \\lambda ) \\| & \\leq \\lim _ { \\lambda \\to \\mathcal U } 2 | ( \\varphi \\circ \\Phi ) ( ( 1 - f _ \\lambda ) ^ 2 ) | ^ { 1 / 2 } \\\\ & \\leq \\lim _ { \\lambda \\to \\mathcal U } 2 | ( \\varphi \\circ \\Phi ) ( 1 - f _ \\lambda ) | ^ { 1 / 2 } = 0 . \\end{align*}"}
-{"id": "1217.png", "formula": "\\begin{align*} \\| x \\| = \\| x \\| _ { \\varphi , w } = \\inf \\{ \\epsilon > 0 : \\rho ( x / \\epsilon ) \\le 1 \\} . \\end{align*}"}
-{"id": "6065.png", "formula": "\\begin{align*} \\Delta _ j f ( x ) : = \\int _ { { \\mathbb R } ^ d } e ^ { 2 \\pi i x \\cdot \\xi } \\widehat { \\Psi } ( 2 ^ { - j } \\xi ) \\widehat { f } ( \\xi ) d \\xi . \\end{align*}"}
-{"id": "1811.png", "formula": "\\begin{align*} \\phi ( G _ i ) = G _ { i } ' . \\end{align*}"}
-{"id": "7434.png", "formula": "\\begin{align*} \\deg _ { X _ f } \\left ( \\tilde { p } _ { \\vec { i } _ { ( u , v ) } } \\circ \\tilde { \\psi } _ { \\vec { i } _ { ( u , v ) } } \\circ \\tilde { \\chi } _ { \\vec { i } _ { ( u , v ) } } \\right ) ^ * ( X _ g ) = - \\delta _ { f g } . \\end{align*}"}
-{"id": "3228.png", "formula": "\\begin{align*} P _ { \\alpha } ^ { S } = P _ { \\beta } ^ { S } \\circ \\Phi _ { \\beta , \\alpha } \\end{align*}"}
-{"id": "8746.png", "formula": "\\begin{align*} { Q } _ t ^ { 1 / 2 } ( H ) = I m \\mathcal L _ t = \\mathcal L _ t ( L ^ 2 ( 0 , t ; U ) ) \\end{align*}"}
-{"id": "2577.png", "formula": "\\begin{align*} L ( E ) = \\sup \\{ S _ { I } ( u ) \\} , \\end{align*}"}
-{"id": "4391.png", "formula": "\\begin{align*} H _ \\nu ( \\zeta | ( \\zeta _ { - n } ^ { - 1 } ( \\tau ) ) = \\infty . \\end{align*}"}
-{"id": "6476.png", "formula": "\\begin{align*} - \\int _ { B } \\psi ^ { 2 } \\tilde { u } ^ { - 1 } \\partial _ { t } ( g _ { 1 - \\alpha , m } * \\tilde { u } ) d x + \\frac { c _ { 2 } } { r ^ { 2 \\beta } } \\int _ { B } ( w - W ) ^ { 2 } \\psi ^ { 2 } d x \\leq \\frac { C _ { 1 } \\mu _ { n } ( B ) } { r ^ { 2 \\beta } } + R _ { m } ( t ) , \\end{align*}"}
-{"id": "4909.png", "formula": "\\begin{align*} \\Psi _ { \\kappa , n , N } ^ { \\operatorname { c } } ( z , \\mathfrak { z } ) = \\Psi _ { \\kappa , n , N } ( - z , \\overline { \\mathfrak { z } } ) . \\end{align*}"}
-{"id": "9046.png", "formula": "\\begin{align*} \\square _ j ^ { ( 2 ^ { l } + p 2 ^ { l - \\Delta } ) } ( \\theta ) = & 1 - \\sqrt { \\frac { 2 ^ l + p 2 ^ { l - \\Delta } } { 2 ^ l + p 2 ^ { l - \\Delta } + j + 1 } } e ^ { i \\psi _ { 2 ^ { l } + p 2 ^ { l - \\Delta } + j } ( \\theta ) - i \\psi _ { 2 ^ { l } + p 2 ^ { l - \\Delta } } ( \\theta ) - i j \\theta } \\\\ = & 1 - \\left ( 1 + \\Theta \\right ) e ^ { i A _ { j + 2 ^ { l } + p 2 ^ { l - \\Delta } } ^ { ( 2 ^ { l } + p 2 ^ { l - \\Delta } ) } ( \\theta ) } , \\end{align*}"}
-{"id": "2308.png", "formula": "\\begin{align*} \\frac 1 \\tau \\ , \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( X ) } + \\big \\| ( u _ i - u _ i ^ \\star ) _ { i = 0 } ^ { k - 1 } \\big \\| _ { L ^ p ( D ) } \\le C \\tau ^ k . \\end{align*}"}
-{"id": "1392.png", "formula": "\\begin{align*} E ( \\hat \\beta - \\beta _ 0 | \\hat \\tau > 0 ) = \\Sigma _ { \\beta \\tau } ( \\delta _ 0 ) \\Sigma ^ { - 1 } _ { \\tau \\tau } ( \\delta _ 0 ) E ( \\hat \\tau - \\tau _ 0 | \\hat \\tau > 0 ) \\end{align*}"}
-{"id": "1647.png", "formula": "\\begin{align*} H ^ { k } _ p ( \\R ^ d ) = W ^ { k , p } ( \\R ^ d ) \\end{align*}"}
-{"id": "2143.png", "formula": "\\begin{align*} L _ { e } ^ { p } ( \\Omega ) : = \\left \\{ u \\in L _ { e } ^ { p } ( \\mathbb { R } ^ { n } ) \\ , : \\ , u = 0 \\mathbb { R } ^ { n } \\backslash \\Omega \\right \\} . \\end{align*}"}
-{"id": "7707.png", "formula": "\\begin{align*} b = b ( \\tilde \\gamma ) : = \\inf \\{ x > 0 : f ( x ) - x > \\tilde \\gamma \\} . \\end{align*}"}
-{"id": "2981.png", "formula": "\\begin{align*} - [ \\alpha _ j , [ f _ 0 ^ x , f _ i ^ y ] ] + [ \\alpha _ j , [ f _ 0 ^ y , f _ i ^ x ] ] & = [ \\alpha _ j , - [ f _ 0 ^ x , f _ i ^ y ] + [ f _ 0 ^ y , f _ i ^ x ] ] \\\\ & = [ \\alpha _ j , \\sum _ { m + n = i , ~ m , n > 0 } [ f _ m ^ x , f _ n ^ y ] - f _ 0 ^ { \\lambda _ i ( x , y ) } - f _ i ^ { [ x , y ] } - \\sum _ { m + n = i , ~ m , n > 0 } f _ m ^ { \\lambda _ n ( x , y ) } ] . \\end{align*}"}
-{"id": "3022.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( { t _ 1 } ) = { h ^ { [ j 1 ] } } ( { t _ 1 } ) u _ 1 ^ { [ 1 ] } + { h ^ { [ j 2 ] } } ( { t _ 1 } ) u _ 1 ^ { [ 2 ] } + { h ^ { [ j 3 ] } } ( { t _ 1 } ) u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "5063.png", "formula": "\\begin{align*} U = \\big \\{ g \\in G \\ , : \\ , \\nu ( C \\cap g B ) > 0 \\big \\} \\end{align*}"}
-{"id": "1374.png", "formula": "\\begin{align*} E ( Y _ { t } | x _ { n t } , \\alpha _ { t } ) = m _ { t } \\dot { b } ( W _ { t } ) , \\mathrm { V a r } ( Y _ { t } | x _ { n t } , \\alpha _ { t } ) = m _ { t } \\ddot { b } ( W _ { t } ) . \\end{align*}"}
-{"id": "2453.png", "formula": "\\begin{align*} G _ { k , N } = \\sum _ { l \\in \\widetilde { \\Gamma _ { k } ^ { \\alpha _ { 2 } , \\alpha _ { 1 } } } } T _ { N l } f _ { l } ^ { \\alpha _ { 2 } } \\end{align*}"}
-{"id": "6284.png", "formula": "\\begin{align*} E _ { p , i } ( x ) = 0 ~ { \\rm i f } ~ p ( x ) = 0 ; ~ \\frac { 1 } { E _ { p , i } ( x ) } = \\frac { 1 } { N _ p ( x ) } - \\sum _ { j = 1 , j \\neq i } ^ d \\frac { 1 } { x - z _ j } ~ { \\rm o t h e r w i s e } ; \\end{align*}"}
-{"id": "5225.png", "formula": "\\begin{align*} u = S ' _ q u + S '' _ q u + H _ q u \\ ; , \\ ; u \\in L ^ { 2 } _ { 0 , q } ( M ) \\ ; . \\end{align*}"}
-{"id": "3959.png", "formula": "\\begin{align*} \\lambda ^ { \\max } ( \\sigma ( r ) v ) = - \\lambda ^ { \\min } ( v ) ( \\sigma ( r ) v ) ^ { \\max } = \\sigma ( r ) v ^ { \\min } . \\end{align*}"}
-{"id": "371.png", "formula": "\\begin{align*} \\begin{cases} u _ { x x t } - u _ t + \\frac 9 2 u _ x u _ { x x } + \\frac 3 2 u u _ { x x x } - \\frac 3 2 u u _ x + u _ x = 0 \\\\ u ( x , 0 ) = u _ 0 ( x ) , \\end{cases} \\end{align*}"}
-{"id": "1379.png", "formula": "\\begin{align*} Q _ n ( \\delta ) = \\frac { 1 } { n } l _ 1 ( \\delta ) \\end{align*}"}
-{"id": "3807.png", "formula": "\\begin{align*} [ \\psi _ i , \\psi _ j ] _ + = 0 , [ \\psi _ i , \\psi _ j ^ \\ast ] _ + = \\delta _ { i , j } , [ \\psi _ i ^ \\ast , \\psi _ j ^ \\ast ] _ + = 0 , \\end{align*}"}
-{"id": "7858.png", "formula": "\\begin{align*} \\lim _ { t \\downarrow 0 } \\sup _ { x \\in \\R ^ d } \\left | \\int _ { \\R ^ d } p ^ { \\kappa } ( t , x , y ) f ( y ) \\ , d y - f ( x ) \\right | = 0 \\ , . \\end{align*}"}
-{"id": "1357.png", "formula": "\\begin{align*} Z _ { t } = \\sum _ { i = 1 } ^ { p } \\phi _ { i } Y _ { t - i } + \\sum _ { i = 1 } ^ { q } \\theta _ { i } e _ { t - i } ^ { I } \\end{align*}"}
-{"id": "6675.png", "formula": "\\begin{align*} X _ { 0 } ^ { \\lambda } = \\left \\| \\bigwedge _ { i = 1 } ^ { p } \\nabla D _ i \\right \\| _ { p } ^ { - 2 } \\cdot \\sum _ { i = 1 } ^ { p } ( - 1 ) ^ { n - i } ( - \\lambda ) ( D _ i - d _ i ) \\Theta _ i , \\end{align*}"}
-{"id": "6728.png", "formula": "\\begin{align*} D ^ { H } _ { s } F = \\Delta _ { x } F ( B ^ { H } _ { t } ) I _ { [ 0 , t ] } ( s ) , \\ \\ 0 \\leq s \\leq T . \\end{align*}"}
-{"id": "5305.png", "formula": "\\begin{align*} H _ { 2 , 1 } & = U ( N _ { a , 1 } + N _ { a , 2 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } + N _ { a , 2 } - N _ { b , 1 } ) \\\\ & \\quad + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) + t _ { 2 , 1 } ( a _ { 2 } b _ { 1 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 1 } ) \\end{align*}"}
-{"id": "9783.png", "formula": "\\begin{align*} \\{ | \\mathcal { M } _ { k } \\} = \\frac { \\{ | \\mathcal { M } _ { k } \\} \\times \\{ | \\mathcal { M } _ { k } \\} } { } . \\end{align*}"}
-{"id": "7482.png", "formula": "\\begin{align*} ( V ) k = \\sum _ { j = 0 } ^ { \\kappa ' k } j Q _ j ^ { ( k ) } ( V ) . \\end{align*}"}
-{"id": "642.png", "formula": "\\begin{align*} \\beta _ n ( f ) = \\frac { 1 } { n } \\sum _ { k = 1 } ^ { n } p ^ { * k } ( f ) , \\textrm { f o r $ f \\in \\ell ^ \\infty ( G ) $ } . \\end{align*}"}
-{"id": "9775.png", "formula": "\\begin{align*} \\partial ^ t _ { p , q } = - * \\overline { \\partial } _ { m - q , m - p - 1 } * \\ \\ \\overline { \\partial } ^ t _ { p , q } = - * \\partial _ { m - q - 1 , m - p } * \\end{align*}"}
-{"id": "8408.png", "formula": "\\begin{align*} \\theta ( x ^ \\beta ) = \\sum _ j \\phi ( m _ j ^ * ) \\sum _ { g \\in G } y _ g w _ g \\phi ( \\alpha _ { h ^ { - 1 } } ( m _ j ) ) , \\end{align*}"}
-{"id": "6035.png", "formula": "\\begin{align*} G _ \\infty ^ + ( x , y ) = \\frac { k _ N ^ s } { 2 } | x - y | ^ { 2 s - N } \\int _ 0 ^ { \\psi _ \\infty ^ + ( x , y ) } \\frac { t ^ { s - 1 } } { ( t + 1 ) ^ \\frac { N } { 2 } } d t , \\end{align*}"}
-{"id": "2181.png", "formula": "\\begin{align*} = \\int _ { \\rho B _ { 1 } } \\int _ { \\mathbb { R } ^ { n } \\backslash \\rho B _ { 1 } } ( \\tilde { u } ( s , x ) - \\tilde { u } ( s , y ) ) ( - \\psi ^ { 2 } ( x ) \\tilde { u } ^ { - q } ( s , x ) ) k ( x , y ) d y d x . \\end{align*}"}
-{"id": "2236.png", "formula": "\\begin{align*} \\mu _ { ( k ) } = C ( \\varepsilon ) \\sum _ { n = 0 } ^ { \\infty } \\frac { ( - \\varepsilon ) ^ { n } } { n ! } ( n p + k - 1 ) ! ! \\sigma ^ { n p + k } , \\end{align*}"}
-{"id": "4115.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ q } { z ^ { p + 2 q } x ^ { - p } } , \\end{align*}"}
-{"id": "9855.png", "formula": "\\begin{align*} v ( \\bar \\Xi ) = & \\bigg ( \\frac { 1 } { W } \\log _ 2 \\left ( \\frac { \\beta ( \\bar \\Xi ) } { ( 2 \\ln 2 ) W } \\right ) \\bigg ) ^ + - \\bigg ( \\frac { 1 } { ( 2 \\ln 2 ) W } - \\frac { 1 } { \\beta ( \\bar \\Xi ) } \\bigg ) ^ + . \\end{align*}"}
-{"id": "48.png", "formula": "\\begin{align*} \\partial _ t u - \\partial _ x \\left ( | \\partial _ x u | ^ { p - 2 } \\partial _ x u \\right ) + | \\partial _ x u | ^ { p - 1 } = 0 \\ , ( t , x ) \\in ( 0 , \\infty ) \\times \\mathbb { R } \\ , \\end{align*}"}
-{"id": "7939.png", "formula": "\\begin{align*} \\sum _ { \\substack { j = 1 \\\\ j \\neq i } } ^ n \\alpha ( w _ i , w _ j ) & = \\frac { 1 } { m ^ 2 } \\sum _ { \\substack { j = 1 \\\\ j \\neq i } } ^ n \\sum _ { \\ell = 1 } ^ m \\sum _ { k = 1 } ^ m \\alpha _ H ( v _ { i , \\ell } , v _ { j , k } ) \\\\ & = \\frac { 1 } { m ^ 2 } \\sum _ { \\ell = 1 } ^ m \\sum _ { \\substack { i = 1 \\\\ i \\neq j } } ^ n \\sum _ { k = 1 } ^ m \\alpha _ H ( v _ { i , \\ell } , v _ { j , k } ) \\\\ & = \\frac { 1 } { m ^ 2 } \\sum _ { \\ell = 1 } ^ m m d _ i \\\\ & = d _ i . \\end{align*}"}
-{"id": "2491.png", "formula": "\\begin{gather*} { \\bf E } W _ 1 ( \\lambda , \\lambda ^ n ) \\leqslant \\sum _ { k = 1 } ^ n \\int \\limits _ { I _ k ^ n } \\sqrt { C _ 1 \\cdot \\left | u - \\dfrac { 2 k - 1 } { 2 n } \\right | + \\left | u - \\dfrac { 2 k - 1 } { 2 n } \\right | ^ 2 } \\ , \\mu ( d u ) \\leqslant \\\\ \\leqslant \\sum _ { k = 1 } ^ n p _ k ^ n \\cdot \\sqrt { C _ 1 \\cdot \\dfrac { 1 } { 2 n } + \\dfrac { 1 } { 4 n ^ 2 } } \\leqslant \\dfrac { K } { \\sqrt { n } } , \\end{gather*}"}
-{"id": "3965.png", "formula": "\\begin{align*} ( u ( r ) v ) ^ { \\max } = \\sigma ( r ) ( u ( - r ) u ^ { - } ( r ^ { - 1 } ) v ) ^ { \\min } = \\sigma ( r ) ( u ^ { - } ( r ^ { - 1 } ) v ) ^ { \\min } = \\sigma ( r ) ( u ^ { - } ( r ^ { - 1 } ) v ) ^ { \\max } = \\sigma ( r ) v ^ { \\max } . \\end{align*}"}
-{"id": "7609.png", "formula": "\\begin{align*} \\lambda _ 1 = \\sum _ { y \\sim x } \\mathbf { v } _ y . \\end{align*}"}
-{"id": "7738.png", "formula": "\\begin{align*} \\sigma _ n = e ^ { - 2 ^ n } , \\end{align*}"}
-{"id": "7570.png", "formula": "\\begin{align*} \\tilde \\Delta _ l ( a ) = f ( a ) - a - l > 0 , \\tilde \\Delta _ l ( v _ l ) = 0 , \\end{align*}"}
-{"id": "499.png", "formula": "\\begin{align*} f ( z ) = z + \\frac { a _ 1 } { z } + \\dots \\end{align*}"}
-{"id": "239.png", "formula": "\\begin{align*} \\epsilon _ n = \\epsilon _ n ^ w ( d , \\theta ) : = \\frac { \\sup _ { k \\in \\{ 1 , \\ldots , k ^ * \\} } \\sup _ { f \\in \\mathcal { F } _ { d , \\theta } } \\Bigl ( 2 \\mathbb { E } _ f \\bigl [ \\{ \\tilde { V } _ n ^ w - V ( f ) \\} ^ 2 \\bigr ] \\Bigr ) ^ { 1 / 3 } } { \\inf _ { f \\in \\mathcal { F } _ { d , \\theta } } V ( f ) ^ { 2 / 3 } } , \\end{align*}"}
-{"id": "3745.png", "formula": "\\begin{align*} C ( z ) : = \\int _ 0 ^ 1 ( e ^ { z x } - 1 ) d \\nu ( x ) , \\end{align*}"}
-{"id": "2407.png", "formula": "\\begin{align*} 0 & \\leq h ( x ) - h ( y ) - \\langle \\nabla h ( y ) , x - y \\rangle + \\tfrac { 1 } { 2 } ( \\| x \\| ^ 2 - \\| y \\| ^ 2 - 2 \\langle y , x - y \\rangle ) \\\\ & = h ( x ) - h ( y ) - \\langle \\nabla h ( y ) , x - y \\rangle + \\tfrac { 1 } { 2 } \\| x - y \\| ^ 2 \\leq \\| x - y \\| ^ 2 . \\end{align*}"}
-{"id": "4843.png", "formula": "\\begin{align*} \\sum _ { \\mu } s _ { \\kappa / \\mu } ( \\rho _ 1 ) \\tau _ { \\mu } ( \\rho _ { \\circ } ) \\mathcal { U } ^ { \\angle } _ { \\rho _ { \\circ } , \\rho _ 1 } ( \\pi \\vert \\kappa , \\mu ) = \\frac { s _ { \\pi / \\kappa } ( \\rho _ 1 ) \\tau _ { \\pi } ( \\rho _ { \\circ } ) } { H ^ { \\circ } ( \\rho _ 1 ) H ( \\rho _ 1 ; \\rho _ { \\circ } ) } . \\end{align*}"}
-{"id": "8904.png", "formula": "\\begin{align*} u \\circ R = u , \\end{align*}"}
-{"id": "4394.png", "formula": "\\begin{align*} A \\cap A ' = \\emptyset A \\cap C _ { k ' } , \\end{align*}"}
-{"id": "64.png", "formula": "\\begin{align*} ( \\xi _ 1 ' - \\xi _ 2 ' ) ( y _ \\delta ) & \\le \\frac { p } { p - 1 } \\xi _ 2 ( y _ \\delta ) ^ { ( p - 1 ) / p } - \\frac { p } { p - 1 } \\xi _ 1 ( y _ \\delta ) ^ { ( p - 1 ) / p } \\\\ & = \\frac { p } { p - 1 } \\xi _ 2 ( y _ \\delta ) ^ { ( p - 1 ) / p } - \\frac { p } { p - 1 } \\left ( \\xi _ 2 ( y _ \\delta ) + \\delta \\right ) ^ { ( p - 1 ) / p } < 0 \\ , \\end{align*}"}
-{"id": "9399.png", "formula": "\\begin{align*} \\widetilde { W } _ \\lambda = 2 i k + \\frac { 1 } { ( \\mu - i k ) ^ 2 } = 2 i \\sqrt { \\lambda } + \\frac { 1 } { ( \\mu - i \\sqrt { \\lambda } ) ^ 2 } . \\end{align*}"}
-{"id": "1994.png", "formula": "\\begin{align*} \\mathcal { E } ( U ) _ t = e ^ { U _ t - \\sigma _ U ^ 2 t / 2 } \\prod _ { s \\leq t } ( 1 + \\Delta U _ s ) e ^ { - \\Delta U _ s } . \\end{align*}"}
-{"id": "7661.png", "formula": "\\begin{align*} u _ { t } = \\triangle ^ { \\alpha / 2 } u + c ( 1 + u ^ { p } ) . \\end{align*}"}
-{"id": "4824.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\| ( I - A _ n ) K \\| = 0 . \\end{align*}"}
-{"id": "5709.png", "formula": "\\begin{align*} \\max _ { \\substack { R \\in \\mathcal { D } ( Q _ 0 ) : \\\\ R ^ { ( 1 ) } = Q ( x ) } } | m _ { f _ 1 } ( R ) | > ( f _ 1 \\cdot \\chi _ { Q _ 0 } ) ^ * ( \\lambda _ w | Q _ 0 | ) . \\end{align*}"}
-{"id": "6011.png", "formula": "\\begin{gather*} \\begin{pmatrix} 0 & 0 & 1 \\\\ 0 & \\Sigma & 0 \\\\ 1 & 0 & 0 \\end{pmatrix} . \\end{gather*}"}
-{"id": "9557.png", "formula": "\\begin{align*} \\begin{aligned} & 0 = \\langle \\psi ^ { ( 0 ) } _ { n } , H ^ { ( 1 ) } \\psi ^ { ( 0 ) } _ { n } \\rangle \\\\ & 0 = \\sum _ { i \\neq n } \\frac { \\left | \\langle \\psi ^ { ( 0 ) } _ { i } , H ^ { ( 1 ) } \\psi ^ { ( 0 ) } _ { n } \\rangle \\right | ^ { 2 } } { \\lambda _ { n } ^ { ( 0 ) } - \\lambda _ { i } ^ { ( 0 ) } } \\end{aligned} \\end{align*}"}
-{"id": "854.png", "formula": "\\begin{align*} C ^ { ( m _ { r } , m _ { r - 1 } , \\ldots , m _ { a + 1 } ) } ( \\vec { z } ) = \\prod _ { a + 1 \\le p \\le r } ^ { \\curvearrowleft } \\prod _ { i = m _ { r } + \\cdots + m _ { p + 1 } + 1 } ^ { m _ { r } + \\cdots + m _ { p } } C _ { p } ( z _ { i } ) \\end{align*}"}
-{"id": "4898.png", "formula": "\\begin{align*} \\Delta _ { G _ i } : = \\deg _ y P _ { G _ i } ( z , y ) = \\deg _ y P _ G ( z , y ) = \\Delta _ G \\end{align*}"}
-{"id": "3832.png", "formula": "\\begin{align*} | v | _ { H ^ { k , 2 k } ( D _ T ) } : = \\Big ( \\sum _ { \\ell = 1 } ^ { 2 k } \\norm { D ^ \\ell v } { L ^ 2 ( D _ T ) } ^ 2 \\Big ) ^ { 1 / 2 } + \\sum _ { \\ell = 1 } ^ k \\norm { \\partial _ t ^ \\ell v } { L ^ 2 ( 0 , T ; H ^ { 2 k - 2 \\ell } ( D ) ) } , \\end{align*}"}
-{"id": "7331.png", "formula": "\\begin{align*} K w _ { - 1 } = q ^ 2 w _ { - 1 } , K w _ 0 = w _ 0 , K w _ 1 = q ^ { - 2 } w _ 1 . \\end{align*}"}
-{"id": "8141.png", "formula": "\\begin{align*} \\left ( \\frac { N } { p } - 1 + a _ 3 \\right ) ^ 2 = \\frac { ( N - 1 ) ^ 2 } { N \\left ( \\frac { 1 } { p } - \\frac { 1 } { q } \\right ) \\cdot \\left ( 1 - \\frac { q } { p } + q \\right ) ^ 2 } . \\end{align*}"}
-{"id": "5932.png", "formula": "\\begin{align*} \\| D _ v { \\psi } \\| _ { W ^ { 1 , p } ( \\R ^ { 2 d } ) } \\le c ( \\lambda ) \\| g \\| _ { L ^ p ( \\R ^ d _ v ; H ^ { s } _ { p } ( \\R ^ d _ x ) ) } , \\ ; \\ ; \\ ; \\lambda > 0 , \\ ; \\ ; \\ ; \\ , c = c ( \\lambda ) \\to 0 \\ ; \\ ; . \\end{align*}"}
-{"id": "629.png", "formula": "\\begin{align*} Q ( x ) = \\beta ( x ) ( x , \\R ^ N \\ ! \\setminus \\ ! \\Omega ) ^ \\alpha \\end{align*}"}
-{"id": "6860.png", "formula": "\\begin{align*} e ^ { - i t \\Delta } v _ n ^ j ( t ) = e ^ { i x \\xi _ n ^ j } e ^ { - i t \\Delta } \\Psi ^ j _ { [ h _ n ^ j ] } ( t ) = e ^ { i x \\xi _ n ^ j } \\bigl ( e ^ { - i ( h _ n ^ j ) ^ 2 t \\Delta } \\Psi ^ j ( ( h _ n ^ j ) ^ 2 t ) \\bigr ) _ { \\{ h _ n ^ j \\} } . \\end{align*}"}
-{"id": "5261.png", "formula": "\\begin{align*} A = \\operatorname * { c l } ( \\operatorname * { i n t } A ) , \\end{align*}"}
-{"id": "8670.png", "formula": "\\begin{gather*} ( C _ b ( H ) , C ^ 1 _ Q ( H ) ) _ { \\alpha , \\infty } = C ^ { \\alpha } _ Q ( H ) . \\end{gather*}"}
-{"id": "47.png", "formula": "\\begin{align*} u ( d t ^ { \\otimes m } ) = \\sigma _ { u , m } \\left ( d z ^ \\prime \\wedge p ^ \\star ( d t ) \\right ) ^ { \\otimes m } \\end{align*}"}
-{"id": "8117.png", "formula": "\\begin{align*} J ( t ) : = P _ { \\mu _ k } ( U ( t ) ) = | 1 + s ( t ) | ^ k + | - 1 + t | ^ k . \\end{align*}"}
-{"id": "240.png", "formula": "\\begin{align*} h _ { n , L } ( x ) : = \\left \\{ \\begin{array} { l l } 0 & \\mbox { i f $ | x | > z _ { q / 2 } ( 1 + \\epsilon _ n ) + 1 / L $ } \\\\ L \\{ z _ { q / 2 } ( 1 + \\epsilon _ n ) + 1 / L - | x | \\} & \\mbox { i f $ 0 < | x | - z _ { q / 2 } ( 1 + \\epsilon _ n ) \\leq 1 / L $ } \\\\ 1 & \\mbox { i f $ | x | \\leq z _ { q / 2 } ( 1 + \\epsilon _ n ) $ . } \\end{array} \\right . \\end{align*}"}
-{"id": "7215.png", "formula": "\\begin{align*} \\left \\{ \\left ( c _ { e _ { i } } \\left ( r \\right ) , c _ { e _ { i } } \\left ( - r \\right ) \\right ) \\right \\} _ { i = 1 } ^ { \\mathrm { \\dim } \\left ( S \\right ) } \\end{align*}"}
-{"id": "7149.png", "formula": "\\begin{align*} - \\Delta v ^ { k + 1 } + \\nabla p ^ { k + 1 } = - ( v ^ k \\cdot \\nabla ) v ^ k + \\nabla \\cdot F , \\qquad { \\rm d i v } \\ , v ^ { k + 1 } = 0 \\mbox { i n } \\ , \\ , \\R ^ n _ + , \\end{align*}"}
-{"id": "8377.png", "formula": "\\begin{align*} p w = E ( p w ) = p y = y \\ne 0 , \\end{align*}"}
-{"id": "2360.png", "formula": "\\begin{align*} \\int _ { \\tau _ \\partial ( y ) \\wedge T } ^ T \\bigg ( \\partial _ x \\psi ( \\Phi _ t ( y ) , t ) \\cdot v ( \\Phi _ t ( y ) ) + \\partial _ t \\psi ( \\Phi _ t ( y ) , t ) \\bigg ) \\ , d t \\ = 0 \\end{align*}"}
-{"id": "1997.png", "formula": "\\begin{align*} \\nu ' ( A ) = \\nu _ { U L } ( ( A \\times \\R ) \\cap \\{ | z | > 1 \\} ) , \\end{align*}"}
-{"id": "7016.png", "formula": "\\begin{align*} E ( l ^ x ) ^ p = p ! \\frac { 1 } { ( 2 \\pi ) ^ { p / 2 } } \\int _ { \\Delta _ p } \\frac { d \\vec { s } } { \\sqrt { G ( g ( s _ 1 ) , \\ldots , g ( s _ p ) ) } } . \\end{align*}"}
-{"id": "6698.png", "formula": "\\begin{align*} u _ { P Q } \\star v ^ { Q K } = \\delta _ P ^ K \\mbox { a n d } v ^ { L P } \\star u _ { P Q } = \\delta ^ L _ Q . \\end{align*}"}
-{"id": "4118.png", "formula": "\\begin{align*} ( y ^ 2 + a x ^ 2 + b x + c ) ^ q = x ^ { - p } \\end{align*}"}
-{"id": "7449.png", "formula": "\\begin{align*} I _ j : = \\{ 1 , \\dots , j \\} . \\end{align*}"}
-{"id": "7505.png", "formula": "\\begin{align*} x _ { n + 1 } = \\max \\left \\{ r x _ n ( 1 - x _ n ) , 0 \\right \\} , x _ 0 > 0 , n \\in { \\mathbb N } _ 0 , \\end{align*}"}
-{"id": "9318.png", "formula": "\\begin{align*} \\tilde { Z } ( t , z ) = \\int _ D Z ( t , x , z ) d x . \\end{align*}"}
-{"id": "1344.png", "formula": "\\begin{align*} l ( \\delta ) = \\sum _ { t = 1 } ^ n Y _ t W _ t ( \\delta ) - m _ t b ( W _ t ( \\delta ) ) + c ( y _ t ) . \\end{align*}"}
-{"id": "2243.png", "formula": "\\begin{align*} \\beta _ { p } = \\left ( 1 - \\varepsilon ( p - 1 ) ! ! \\sigma ^ { p } - \\frac { \\varepsilon ^ { 2 } } { 2 } ( 2 p - 1 ) ! ! \\sigma ^ { 2 p } \\right ) ^ { - 1 } . \\end{align*}"}
-{"id": "4473.png", "formula": "\\begin{align*} R _ 1 = \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\int _ 0 ^ 1 \\mathrm { B } _ { k , n - k } ( s ) \\log u _ { x , s } \\ , d s \\ , d x = o \\biggl ( \\frac { k ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } { n ^ { \\frac { \\alpha } { \\alpha + d } - \\epsilon } } \\biggr ) \\end{align*}"}
-{"id": "8634.png", "formula": "\\begin{align*} \\| x ' \\| ^ 2 & = \\| x \\| ^ 2 + \\frac { 1 } { 2 ^ { k + 1 } } = 1 - 2 ^ { - k - 1 } , \\\\ \\| x ' - x \\| ^ 2 & = 2 ^ { - k - 1 } , \\end{align*}"}
-{"id": "61.png", "formula": "\\begin{align*} \\psi \\left ( 1 - \\frac { f ( r ) } { a } \\right ) = \\frac { | f ' ( r ) | ^ p } { a ^ p } \\ , r \\in [ 0 , R ( a ) ) \\ , \\end{align*}"}
-{"id": "5408.png", "formula": "\\begin{align*} \\frac { e ( H ) } { v ( H ) } > \\frac { 1 } { N } \\left ( k - \\alpha - \\frac { 7 } { 1 6 } \\right ) \\frac { n ^ 2 } { 2 } = \\frac { n } { 2 } \\left ( 1 - \\frac { 7 } { 1 6 ( k - \\alpha ) } \\right ) \\ , . \\end{align*}"}
-{"id": "8081.png", "formula": "\\begin{align*} \\tilde { h } _ y \\circ \\tilde { T } _ y = h \\circ \\pi _ y \\circ \\tilde { T } _ f = h \\circ T _ f \\circ \\pi _ y = h \\circ \\pi _ y = \\tilde { h } _ y . \\end{align*}"}
-{"id": "3049.png", "formula": "\\begin{align*} \\forall u \\in \\mathcal { U } , \\mu ( B ( u ) ) = \\nu ( B ( u ) ) . \\end{align*}"}
-{"id": "987.png", "formula": "\\begin{align*} H _ { 3 , 1 } & = U ( N _ { a , 1 } + N _ { a , 2 } + N _ { a , 3 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } + N _ { a , 2 } + N _ { a , 3 } - N _ { b , 1 } ) \\\\ & + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) + t _ { 2 , 1 } ( a _ { 2 } b _ { 1 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 1 } ) + t _ { 3 , 1 } ( a _ { 3 } b _ { 1 } ^ \\dagger + a _ { 3 } ^ \\dagger b _ { 1 } ) \\end{align*}"}
-{"id": "1533.png", "formula": "\\begin{align*} ( - \\Delta ) ^ { \\alpha / 2 } f ( x ) = c _ { d , \\alpha } \\ , \\ , \\int _ { \\R ^ d } \\frac { f ( x ) - f ( y ) } { | x - y | ^ { d + \\alpha } } \\d y , \\end{align*}"}
-{"id": "609.png", "formula": "\\begin{align*} \\left ( 2 i \\mathfrak { z } _ 2 \\right ) ^ { \\kappa - 1 } \\Psi _ { \\kappa , - 1 , N } ( z , \\mathfrak { z } ) = 2 \\pi i H _ { \\kappa , N } ( z , \\mathfrak { z } ) - 2 \\pi i K _ { \\kappa , N } ( z , \\mathfrak { z } ) . \\end{align*}"}
-{"id": "3054.png", "formula": "\\begin{align*} Z _ n = \\sum _ { | u | = k } e ^ { - V ( u ) } Z ^ u _ { n } + \\sum _ { | u | = k } V ( u ) e ^ { - V ( u ) } \\sum _ { | v | = n , v > u } e ^ { V ( u ) - V ( v ) } . \\end{align*}"}
-{"id": "9338.png", "formula": "\\begin{align*} d k ( t , x , z ) = k ( t , x , z ) b ( t , x , z ) d G ( t ) \\end{align*}"}
-{"id": "8273.png", "formula": "\\begin{align*} \\partial _ { \\rho } \\varphi _ F ( \\rho ) & = E _ { h ' ( \\rho ) } [ ( u ^ { \\ell } - \\ell \\rho ) F ] h '' ( \\rho ) , \\\\ \\partial ^ 2 _ { \\rho \\rho } \\varphi _ F ( \\rho ) & = E _ { h ' ( \\rho ) } [ ( u ^ { \\ell } - \\ell \\rho ) ^ 2 F ] ( h '' ( \\rho ) ) ^ 2 - \\ell E _ { h ' ( \\rho ) } [ F ] h '' ( \\rho ) + E _ { h ' ( \\rho ) } [ ( u ^ { \\ell } - \\ell \\rho ) F ] h ''' ( \\rho ) . \\end{align*}"}
-{"id": "2150.png", "formula": "\\begin{align*} \\mathcal { E } ( u , v ) = \\frac { 1 } { 2 } \\int _ { \\mathbb { R } ^ { n } } \\int _ { \\mathbb { R } ^ { n } } [ u ( t , x ) - u ( t , y ) ] [ v ( t , x ) - v ( t , y ) ] k ( x , y ) d x d y . \\end{align*}"}
-{"id": "1477.png", "formula": "\\begin{align*} \\sum _ { m = 1 } ^ l \\frac { 1 } { w ( Q _ 0 ^ { ( m ) } ) ^ \\frac { 1 } { p } } \\lesssim _ { p , w } \\frac { 1 } { w ( Q _ 0 ) ^ \\frac { 1 } { p } } . \\end{align*}"}
-{"id": "526.png", "formula": "\\begin{align*} R ^ { \\rm g e o } _ { 2 2 } ( i , u ; j , v ) = \\int \\frac { 1 - z ^ 2 } { z ^ 2 } \\frac { 1 } { 1 - c z } \\frac { 1 } { 1 - c / z } \\frac { h ^ { \\rm g e o } _ { 2 2 } ( z , 1 / z ) } { z ^ { u - v } } \\dd z . \\end{align*}"}
-{"id": "7730.png", "formula": "\\begin{align*} T _ N : = \\sum _ { i = N } ^ { \\infty } \\sigma _ i \\xi _ { i + 1 } , \\end{align*}"}
-{"id": "1754.png", "formula": "\\begin{align*} u = x _ 0 \\xleftarrow { \\gamma _ 1 ^ \\lor } \\cdots \\xleftarrow { \\gamma _ { p - 1 } ^ \\lor } x _ { p - 1 } = \\lfloor r _ i x _ { p } \\rfloor \\xleftarrow { z _ { p + 1 } \\gamma _ { p + 1 } ^ \\lor } \\cdots \\xleftarrow { z _ r \\gamma _ r ^ \\lor } \\lfloor r _ i x _ r \\rfloor = \\lfloor r _ i v \\rfloor . \\end{align*}"}
-{"id": "9902.png", "formula": "\\begin{align*} [ x _ 0 , x _ 1 ] & \\ ; = \\ ; \\alpha \\{ x _ 2 , x _ 3 \\} & \\{ x _ 0 , x _ 1 \\} & \\ ; = \\ ; [ x _ 2 , x _ 3 ] \\\\ [ x _ 0 , x _ 2 ] & \\ ; = \\ ; \\beta \\{ x _ 1 , x _ 3 \\} & \\{ x _ 0 , x _ 2 \\} & \\ ; = \\ ; [ x _ 3 , x _ 1 ] \\\\ [ x _ 0 , x _ 3 ] & \\ ; = \\ ; \\gamma \\{ x _ 1 , x _ 2 \\} & \\{ x _ 0 , x _ 3 \\} & \\ ; = \\ ; [ x _ 1 , x _ 2 ] \\end{align*}"}
-{"id": "7348.png", "formula": "\\begin{align*} ( v , X v ^ \\prime ) = ( X ^ * v , v ^ \\prime ) , v , v ^ \\prime \\in \\Lambda _ q ( \\mathfrak { u } _ + ) , \\ X \\in U _ q ( \\mathfrak { l } ) . \\end{align*}"}
-{"id": "9187.png", "formula": "\\begin{align*} \\mathcal A e _ n ( x ) = ( p - n + 1 ) ( \\alpha - n + 1 ) e _ { n - 1 } ( x ) , x \\in D , 1 \\leq n \\leq p . \\end{align*}"}
-{"id": "5637.png", "formula": "\\begin{align*} I ( \\delta , \\mathcal { F } ) = \\underset { Q } { \\sup } \\int _ 0 ^ { \\delta } \\sqrt { 1 + \\log N ( \\epsilon \\Vert F \\Vert _ { \\mathbb { L } _ 2 ( Q ) } , \\mathcal { F } , \\Vert \\cdot \\Vert _ { \\mathbb { L } _ 2 ( Q ) } ) } d \\epsilon \\end{align*}"}
-{"id": "4298.png", "formula": "\\begin{align*} \\partial h = h \\cdot \\Psi \\end{align*}"}
-{"id": "1368.png", "formula": "\\begin{align*} Q _ { L } ^ { W } = \\hat { \\psi } ^ { \\mathrm { T } } \\hat \\Psi ^ { - 1 } \\hat \\psi \\end{align*}"}
-{"id": "8558.png", "formula": "\\begin{align*} \\biggl ( X _ { c ^ { \\ast } } ( \\widetilde { P } ) _ { \\overline { N } } \\biggr ) ( n ) & = X _ { c ^ { \\ast } } ( \\widetilde { P } ) n \\\\ & = X _ { c ^ { \\ast } } ( \\rho ( P ) ) n \\\\ & = \\rho ( X _ { c ^ { \\ast } } ( P ) ) n \\\\ & = X _ { c ^ { \\ast } } ( P ) n \\end{align*}"}
-{"id": "5604.png", "formula": "\\begin{align*} T Q \\phi & = ( U + v \\mathcal G _ 0 v ^ * ) \\phi = U \\phi + v ( c ( 1 , 0 ) ^ T - \\psi ) = v \\psi + c v ( 1 , 0 ) ^ T - v \\psi = c v ( 1 , 0 ) ^ T . \\end{align*}"}
-{"id": "4307.png", "formula": "\\begin{align*} y _ k \\to y _ \\infty \\in ( t _ m = 0 ) \\end{align*}"}
-{"id": "9209.png", "formula": "\\begin{align*} & \\mathbb { E } [ D _ t \\delta _ Z ( z ) | \\mathcal { F } _ t ] = \\\\ & \\frac { 1 } { 2 \\pi } \\int _ { \\mathbb { R } } \\exp \\big [ \\int _ 0 ^ t \\int _ { \\mathbb { R } } i x \\psi ( s , \\zeta ) \\tilde { N } ( d s , d \\zeta ) + \\int _ 0 ^ t i x \\beta ( s ) d B ( s ) \\\\ & + \\int _ t ^ { T _ 0 } \\int _ { \\mathbb { R } } ( e ^ { i x \\psi ( s , \\zeta ) } - 1 - i x \\psi ( s , \\zeta ) ) \\nu ( d \\zeta ) d s - \\int _ t ^ { T _ 0 } \\frac { 1 } { 2 } x ^ 2 \\beta ^ 2 ( s ) d s - i x z \\big ] i x \\beta ( t ) d x \\end{align*}"}
-{"id": "2633.png", "formula": "\\begin{align*} \\Delta ( F , G ) = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { l = 1 } ^ { h ( m , n ) } \\left [ \\| \\mathbf { \\Psi _ m a _ m ^ l } \\| ^ 2 - \\| \\mathbf { B _ m ^ * \\Psi _ m ^ * \\Psi _ m a _ m ^ l } \\| ^ 2 \\right ] , \\end{align*}"}
-{"id": "9077.png", "formula": "\\begin{align*} ( k + 2 ) D ^ { ( 2 ) } _ N ( k + 1 , \\alpha ) - ( 2 k + 1 ) ( \\alpha + 2 N ) D ^ { ( 2 ) } _ N ( k , \\alpha ) - ( k - 1 ) ( k ^ { 2 } - \\alpha ^ { 2 } ) D ^ { ( 2 ) } _ N ( k - 1 , \\alpha ) = 0 \\ . \\end{align*}"}
-{"id": "7911.png", "formula": "\\begin{align*} d v _ g = u ^ 4 d v _ 0 , \\end{align*}"}
-{"id": "9555.png", "formula": "\\begin{align*} \\begin{aligned} & H ( t ) = H ^ { ( 0 ) } + t H ^ { ( 1 ) } \\\\ & \\psi _ { n } ( t ) = \\psi ^ { ( 0 ) } _ { n } + t \\psi ^ { ( 1 ) } _ { n } + t ^ { 2 } \\psi _ { n } ^ { ( 2 ) } + . . . \\\\ & \\lambda _ { n } ( t ) = \\lambda ^ { ( 0 ) } _ { n } + t \\lambda ^ { ( 1 ) } _ { n } + t ^ { 2 } \\lambda _ { n } ^ { ( 2 ) } + . . . \\end{aligned} \\end{align*}"}
-{"id": "4097.png", "formula": "\\begin{align*} B _ { \\pm } ( \\alpha ) = \\left [ \\frac { - \\alpha \\pm \\sqrt { \\alpha ^ 2 + 1 2 a c } } { 6 a } : 0 : 1 \\right ] . \\end{align*}"}
-{"id": "4970.png", "formula": "\\begin{align*} h _ { D } = \\sum _ { i = 1 } ^ { m } a _ { i } h _ { H _ { i } } . \\end{align*}"}
-{"id": "828.png", "formula": "\\begin{align*} \\lim _ { s \\to 0 } \\left \\{ ( 1 - q ^ { 2 } ) ^ { - k } ( - s ) ^ { - \\sum _ { i = 1 } ^ { k } x _ { i } } \\prod _ { i = 1 } ^ { k } ( 1 + z _ { i } ) \\ , F _ { \\vec { x } } ( - z _ { 1 } / s , \\ldots , - z _ { k } / s ) \\right \\} . \\end{align*}"}
-{"id": "889.png", "formula": "\\begin{align*} \\int _ { T } ^ { 2 T } \\vert Z ( t ) \\vert d t = \\int _ { T } ^ { 2 T } \\vert \\zeta ( \\tfrac { 1 } { 2 } + i t ) \\vert d t \\geq \\vert \\int _ { T } ^ { 2 T } \\zeta ( \\tfrac { 1 } { 2 } + i t ) d t \\vert . \\end{align*}"}
-{"id": "6667.png", "formula": "\\begin{gather*} \\sum _ { k = 1 } ^ n { \\bf E } \\sup _ { 0 \\leqslant t \\leqslant 1 } \\left | z _ 1 ( u _ k , t ) - z _ 2 ( u _ k , t ) \\right | \\leqslant \\dfrac { 2 n ^ 3 } { 3 } \\cdot \\sqrt { \\varepsilon } , \\\\ \\sum _ { k = 1 } ^ n { \\bf E } \\sup _ { 0 \\leqslant t \\leqslant 1 } \\left | z _ i ( u _ k , t ) - z _ { i + 1 } ( u _ k , t ) \\right | \\leqslant \\dfrac { 2 n ^ 4 } { 3 } \\cdot \\sqrt { \\varepsilon } , 2 \\leqslant i \\leqslant n - 1 . \\end{gather*}"}
-{"id": "4270.png", "formula": "\\begin{align*} \\mathbb P ( Z _ n ^ * < 0 ) = \\mathbb P ( Z _ { k + 1 , n + k } ^ * < Z _ k ) \\ , . \\end{align*}"}
-{"id": "5696.png", "formula": "\\begin{align*} \\sum _ { k = 1 } ^ \\infty \\frac { k } { \\Phi _ { p , q , w } ( 2 ^ k Q ) } \\leq \\frac { C } { \\Phi _ { p , q , w } ( Q ) } ( Q \\in \\mathcal { Q } ) . \\end{align*}"}
-{"id": "9586.png", "formula": "\\begin{align*} \\dot D _ y = \\{ \\pi \\in L ^ 2 ( \\R ^ 3 ) : \\pi ( x ) = \\pi _ { r e g } ( x ) + \\chi g ( x - y ) , ~ ~ \\pi _ { r e g } \\in H ^ 1 ( \\R ^ 3 ) , ~ ~ \\chi \\in \\C \\} , \\end{align*}"}
-{"id": "2466.png", "formula": "\\begin{align*} \\varepsilon ( s , V ) = \\prod \\varepsilon _ { E _ { \\nu } } ( s , V _ { \\nu } , \\psi _ { \\nu } ) . \\end{align*}"}
-{"id": "7066.png", "formula": "\\begin{align*} { { \\bf { X } } ^ { [ i ] } } ( n ) = \\sum \\limits _ { j = 1 } ^ N { \\sum \\limits _ { a = 1 } ^ A { s _ a ^ { [ j i ] } { \\bf { v } } _ a ^ { [ j i ] } ( n ) } } = \\sum \\limits _ { j = 1 } ^ N { { \\bf { V } } _ j ^ { [ i ] } ( n ) { { \\bf { s } } ^ { [ j i ] } } } , \\end{align*}"}
-{"id": "1817.png", "formula": "\\begin{align*} q = q ' ( \\frac { d z } z + \\frac { d w } w ) ^ 2 \\end{align*}"}
-{"id": "8373.png", "formula": "\\begin{align*} u x | y | = u | y | x = y x = \\phi ( x ) y = \\phi ( x ) u | y | , \\ \\ \\ x \\in M , \\end{align*}"}
-{"id": "971.png", "formula": "\\begin{align*} d h ( L ) ( z ) = \\alpha ( L ) ( z ) , L \\in \\mathcal { N } _ z , \\ ; z \\in M \\ ; . \\end{align*}"}
-{"id": "3843.png", "formula": "\\begin{align*} d _ { k + 1 } ( x ) & = \\sum _ { i = 0 } ^ { ( k + 1 ) - 3 } ( i + 1 ) x ^ i + \\sum _ { i = k - 1 } ^ { 2 ( k + 1 ) - 3 } ( 2 ( k + 1 ) + 2 - i ) x ^ i \\\\ & = \\sum _ { i = 0 } ^ { k - 3 } ( i + 1 ) x ^ i + \\sum _ { i = k - 2 } ^ { 2 k - 3 } ( 2 k + 2 - i ) x ^ i - x ^ { k - 2 } ( 1 - 2 \\sum _ { i = 1 } ^ k x ^ i ) + x ^ { 2 k - 1 } \\\\ & = d _ k ( x ) - x ^ { k - 2 } n _ k ( x ) + x ^ { 2 k - 1 } . \\end{align*}"}
-{"id": "4484.png", "formula": "\\begin{align*} R _ 4 ' = \\int _ { \\mathcal { X } _ n } f ( x ) \\int _ { \\frac { a _ n } { n - 1 } } ^ 1 \\log \\biggl ( \\frac { ( n - 1 ) s } { e ^ { \\Psi ( k ) } f ( x ) } \\biggr ) \\ , \\mathrm { B } _ { k , n - k } ( s ) \\ , d s \\ , d x = o ( n ^ { - ( 3 - \\epsilon ) } ) , \\end{align*}"}
-{"id": "5118.png", "formula": "\\begin{align*} \\left [ L ^ { ( i ) } ( z ; s ) , \\ , \\begin{pmatrix} 1 & 0 \\\\ 0 & \\alpha \\end{pmatrix} \\ ! K ^ { ( i ) } ( \\alpha ) \\right ] = 0 \\end{align*}"}
-{"id": "7477.png", "formula": "\\begin{align*} P ( \\Delta _ { Q _ 0 ^ { ( k ) } } ( \\infty ) = 0 ) > 0 , P ( \\Delta _ { Q _ k ^ { ( k ) } } ( \\infty ) \\geq 0 ) > 0 P ( \\Delta _ { Q _ { k + 1 } ^ { ( k ) } } ( \\infty ) = - ( k - 1 ) ) > 0 . \\end{align*}"}
-{"id": "3220.png", "formula": "\\begin{align*} T _ { j k } \\widetilde { w } _ k = \\widetilde { w } _ j + \\sum _ { | \\alpha | \\geq n + 1 } ( \\pi | _ { \\widetilde { U } _ j } ) ^ * f _ { k j , \\alpha } \\cdot \\widetilde { w } _ j ^ \\alpha \\end{align*}"}
-{"id": "2500.png", "formula": "\\begin{gather*} \\beta _ 1 ( t ) - \\beta _ 2 ( t ) = \\\\ = [ x ( u _ k , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ k , \\widetilde { \\sigma } _ 1 ) ] - [ x ( u _ j , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ j , \\widetilde { \\sigma } _ 1 ) ] = \\\\ = [ x ( u _ k , t + \\widetilde { \\sigma } _ 1 ) - x ( u _ j , t + \\widetilde { \\sigma } _ 1 ) ] - ( k - j ) \\cdot \\varepsilon \\geqslant - ( k - j ) \\cdot \\varepsilon . \\end{gather*}"}
-{"id": "2056.png", "formula": "\\begin{align*} & x ( s ) : = \\exp _ { x _ 0 } ( s \\ , e _ n ) \\\\ & y ( s ) : = \\exp _ { x _ 0 } ( - s \\ , e _ n ) \\\\ & g ( s ) : = Q ( x ( s ) , y ( s ) ) . \\end{align*}"}
-{"id": "6466.png", "formula": "\\begin{align*} W ( s ) = \\int _ { B _ { 1 } } \\phi ( s ) \\psi ^ { 2 } ( x ) w ^ { 2 } ( s , x ) d x , \\end{align*}"}
-{"id": "6080.png", "formula": "\\begin{align*} \\widetilde { F } _ { m , n } ( t ) = \\frac { 1 } { \\sqrt { n + 1 } } \\frac { 1 } { \\sqrt { m + 1 } } \\exp \\left ( - t \\left ( i \\log { \\frac { m + 1 } { n + 1 } } + \\frac { 1 } { 2 } \\log ^ 2 { \\frac { m + 1 } { n + 1 } } \\right ) \\right ) \\end{align*}"}
-{"id": "8274.png", "formula": "\\begin{align*} p _ { \\mathrm { y } | \\mathrm { z } } ( y _ i | z _ i ) = \\int _ { \\mathcal { Q } ^ { - 1 } ( \\mathrm { y } _ i ) } \\phi ( u ; z _ i , \\sigma _ v ^ 2 / 2 ) \\ : \\mathrm { d } u , \\end{align*}"}
-{"id": "1977.png", "formula": "\\begin{align*} g ( ( x _ 1 ^ v \\cdots x _ n ^ v ) ^ a ) = _ { \\Bbbk ^ { \\times } } ( x _ 1 ^ v \\cdots x _ n ^ v ) ^ a , { } \\end{align*}"}
-{"id": "5307.png", "formula": "\\begin{align*} H _ { 3 , 1 } & = U ( N _ { a , 1 } + N _ { a , 2 } + N _ { a , 3 } - N _ { b , 1 } ) ^ 2 + \\mu ( N _ { a , 1 } + N _ { a , 2 } + N _ { a , 3 } - N _ { b , 1 } ) \\\\ & + t _ { 1 , 1 } ( a _ { 1 } b _ { 1 } ^ \\dagger + a _ { 1 } ^ \\dagger b _ { 1 } ) + t _ { 2 , 1 } ( a _ { 2 } b _ { 1 } ^ \\dagger + a _ { 2 } ^ \\dagger b _ { 1 } ) + t _ { 3 , 1 } ( a _ { 3 } b _ { 1 } ^ \\dagger + a _ { 3 } ^ \\dagger b _ { 1 } ) \\end{align*}"}
-{"id": "3685.png", "formula": "\\begin{align*} \\int \\left [ H _ { \\mu } \\nabla ^ 2 H _ h - H _ h \\nabla ^ 2 H _ { \\mu } \\right ] d A = 0 ~ , \\end{align*}"}
-{"id": "5597.png", "formula": "\\begin{align*} Q v M _ { 1 1 } = M _ { 1 1 } v ^ * Q = 0 \\end{align*}"}
-{"id": "7383.png", "formula": "\\begin{align*} \\frac { 1 } { N } ( h ^ { r + 1 } ) ^ * h ^ { t + 1 } & \\overset { \\mathbf { . . } } { = } \\breve { E } _ { r , t } , \\\\ \\frac { 1 } { n } ( b ^ r ) ^ * b ^ t & \\doteq \\tilde { E } _ { r , t } . \\end{align*}"}
-{"id": "7060.png", "formula": "\\begin{align*} { y ^ { [ j ] } } ( { t _ 1 } ) = { h ^ { [ j 1 ] } } ( { t _ 1 } ) u _ 1 ^ { [ 1 ] } + { h ^ { [ j 2 ] } } ( { t _ 1 } ) u _ 1 ^ { [ 2 ] } + { h ^ { [ j 3 ] } } ( { t _ 1 } ) u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "8053.png", "formula": "\\begin{align*} 0 \\le F _ { j } \\cdot ( \\pi ^ { * } ( c D _ { 2 } ) - \\sum _ { i \\ge 3 } a _ { i } B _ { i } ) = a _ { j } + a _ { j + 2 } - 2 a _ { j + 1 } , \\end{align*}"}
-{"id": "3847.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} ( \\partial ^ 2 _ t - \\Delta ) & u ( x , t ) = 0 , \\ , ( x , t ) \\in \\R ^ n \\times \\R \\ , , \\\\ u ( x , 0 ) & = f ( x ) \\ , , \\ ; \\partial _ t u ( x , 0 ) = g ( x ) \\ , . \\\\ \\end{aligned} \\right . \\end{align*}"}
-{"id": "607.png", "formula": "\\begin{align*} \\Psi _ { \\kappa , n , N } ( z , \\mathfrak { z } ) : = \\sum _ { M \\in \\Gamma _ 0 ( N ) } \\frac { ( z - \\mathfrak { z } ) ^ n } { ( z - \\overline { \\mathfrak { z } } ) ^ { \\kappa + n } } \\bigg | _ { \\kappa , z } M . \\end{align*}"}
-{"id": "8594.png", "formula": "\\begin{align*} \\mu _ { k } ^ { 2 } ( \\mathcal { N } ) & = \\mu _ { k } \\left ( \\mu _ { k } ( \\mathcal { N } ) \\right ) \\\\ & = \\widetilde { \\mu _ { k } } ( \\mu _ { k } \\left ( \\mathcal { N } \\right ) ) _ { r e d } \\\\ & = \\widetilde { \\mu _ { k } } \\left ( \\widetilde { \\mu _ { k } } ( \\mathcal { N } \\right ) _ { r e d } ) _ { r e d } \\end{align*}"}
-{"id": "2999.png", "formula": "\\begin{align*} \\mathrm { g p h } \\ , B _ \\mathcal { R } ( \\cdot , p ) & = \\left \\{ ( t , P ) \\in T \\times \\Pi ( X ) \\mid \\theta _ p ( t , P ) \\le 0 \\right \\} \\in \\Sigma \\otimes \\mathrm { B o r e l } ( \\Pi ( X ) ) . \\end{align*}"}
-{"id": "6825.png", "formula": "\\begin{align*} L _ m \\cdot B _ { n + k } - L _ n \\cdot B _ { m + k } = - \\sum _ { l = 1 } ^ { k - 1 } [ B _ { m + k - l } , B _ { n + l } ] + ( n - m ) B _ { m + n + k } \\end{align*}"}
-{"id": "909.png", "formula": "\\begin{align*} \\vert \\sum _ { n \\leq 3 \\sqrt { T / \\pi } } \\frac { 1 } { n ^ { { 1 } / { 2 } } } \\int _ { T } ^ { 2 T } e ^ { i F ( n , t ) } d t \\vert = O \\left ( \\sum _ { n \\leq { 3 \\sqrt { T / \\pi } } } \\frac { 1 } { n ^ { 1 / 2 } } T ^ { 1 / 2 } \\right ) = O \\left ( T ^ { 3 / 4 } \\right ) . \\end{align*}"}
-{"id": "2691.png", "formula": "\\begin{align*} v _ { \\lambda } : = ( 1 - \\lambda ) \\varphi _ { \\beta , \\varepsilon } + \\lambda \\psi \\leq 0 , \\end{align*}"}
-{"id": "8950.png", "formula": "\\begin{align*} \\lim _ { x \\rightarrow \\pm \\infty } U \\left ( x , t \\right ) = 0 , t \\geq 0 . \\end{align*}"}
-{"id": "4863.png", "formula": "\\begin{align*} \\sum _ { m = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ m } { m ! } b _ m = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { k = \\lceil m / 2 \\rceil } ^ m b _ m ^ k = \\sum _ { k = 0 } ^ { \\infty } \\sum _ { i = 0 } ^ k b _ { k + i } ^ k = \\sum _ { k = 0 } ^ { \\infty } \\frac { ( - 1 ) ^ k } { k ! } a _ k , \\end{align*}"}
-{"id": "1162.png", "formula": "\\begin{align*} v ( x ) = \\begin{cases} h x + 1 & 0 \\leq h < \\infty , \\\\ x + \\frac { 1 } { h } & 0 < h \\leq \\infty , \\end{cases} \\end{align*}"}
-{"id": "7636.png", "formula": "\\begin{align*} w ^ a _ i \\leq \\omega _ { i } \\leq w ^ b _ i , \\ \\ i = 0 , \\ldots , n + 1 , \\end{align*}"}
-{"id": "5669.png", "formula": "\\begin{align*} E \\cdot \\nabla _ v F - Q ( F ) = 0 , \\int _ { \\R ^ d } F ( v , E ) \\ , d v = 1 . \\end{align*}"}
-{"id": "6321.png", "formula": "\\begin{align*} 2 ^ { j n \\alpha _ 1 ( 1 / p _ 1 - 1 / p _ 2 ) } = 2 ^ { j A _ 1 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 2 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } = 2 ^ { j A _ 3 ( \\mathbf { p } , q , \\alpha _ 1 , \\alpha _ 2 ) } . \\end{align*}"}
-{"id": "4608.png", "formula": "\\begin{align*} f ( n ) = \\sum _ { t = 3 } ^ { 8 } f _ t ( n ) , . \\end{align*}"}
-{"id": "5748.png", "formula": "\\begin{align*} Z _ { t } = \\sum _ { i = 1 } ^ { p } \\phi _ { i } Y _ { t - i } + \\sum _ { i = 1 } ^ { q } \\theta _ { i } e _ { t - i } ^ { I } \\end{align*}"}
-{"id": "237.png", "formula": "\\begin{align*} \\mathbb { E } _ f \\bigl \\{ ( \\hat { H } _ n ^ w - H _ n ^ * ) ^ 2 \\bigr \\} = _ f ( \\hat { H } _ n ^ w - H _ n ^ * ) + ( \\mathbb { E } _ f \\hat { H } _ n ^ w - H ( f ) ) ^ 2 , \\end{align*}"}
-{"id": "2211.png", "formula": "\\begin{align*} u ( x , t ) = & \\sum _ { k = 1 } ^ { \\infty } E _ { \\alpha , 1 } ( - \\lambda _ { k } t ^ { \\alpha } ) u _ { 0 , k } \\phi _ { k } ( x ) \\\\ & + \\sum _ { k = 1 } ^ { \\infty } \\int _ { 0 } ^ { t } ( t - s ) ^ { \\alpha - 1 } E _ { \\alpha , \\alpha } ( - \\lambda _ { k } ( t - s ) ^ { \\alpha } ) f _ { k } ( s ) d s \\phi _ { k } ( x ) , \\end{align*}"}
-{"id": "8311.png", "formula": "\\begin{gather*} f _ V ( X ) = \\prod _ { \\delta \\in V } ( X - \\delta ) \\end{gather*}"}
-{"id": "480.png", "formula": "\\begin{align*} P _ { E D } ^ { L M P > 0 } : = & \\sum _ { t \\in \\mathcal { T } } \\sum _ { n \\in \\mathcal { N } _ t ^ + } \\left [ \\sum _ { i \\in \\mathcal { G } _ n } p _ { i t } + \\sum _ { i \\in \\mathcal { I } _ n } \\left ( M _ { i t } - m _ { i t } \\right ) \\right . \\\\ & \\left . + \\sum _ { i \\in \\mathcal { W } _ n } \\left ( W _ { i t } - w _ { i t } \\right ) + \\sum _ { i \\in \\mathcal { R } _ n } \\left ( R _ { i t } - r _ { i t } \\right ) \\right ] \\end{align*}"}
-{"id": "2393.png", "formula": "\\begin{align*} x ^ { k + 1 } = ( 1 - \\alpha ) x ^ k + \\alpha P _ { C } ^ { \\alpha _ 2 } P _ { D } ^ { \\alpha _ 1 } x ^ k . \\end{align*}"}
-{"id": "2947.png", "formula": "\\begin{align*} m \\le \\sum _ { i = 1 } ^ { K } | \\mathcal { N } ( \\mathcal { U } _ i ) \\setminus \\mathcal { X } ( \\Pi _ K ^ { ( \\epsilon ) } ) | = n - | \\mathcal { X } ( \\Pi _ K ^ { ( \\epsilon ) } ) | \\end{align*}"}
-{"id": "91.png", "formula": "\\begin{align*} \\frac { 1 } { r } e _ r \\otimes e _ { r - s } = \\sum _ { i = 1 } ^ l d _ i ^ * ( e _ r ) \\otimes c _ i ( e _ { r - s } ) . \\end{align*}"}
-{"id": "1820.png", "formula": "\\begin{align*} g = \\frac { \\pi ^ 2 s ^ 2 } { \\sin ^ 2 ( \\frac { \\pi s } { s _ w } ) } \\left ( \\frac { d s _ w ^ 2 } { s _ w ^ 4 } + d \\theta _ w ^ 2 \\right ) = \\frac { \\pi ^ 2 s ^ 2 } { \\sin ^ 2 ( \\frac { \\pi } { 1 + \\rho _ z } ) } \\left ( \\frac { d \\rho _ z ^ 2 } { s ^ 2 ( 1 + \\rho _ z ) ^ 4 } + d \\theta _ z ^ 2 \\right ) . \\end{align*}"}
-{"id": "7569.png", "formula": "\\begin{align*} \\tilde \\Delta _ l ( y ) : = \\inf _ { x \\in [ y , a ] } \\{ f ( x ) - x - l \\} , \\end{align*}"}
-{"id": "5834.png", "formula": "\\begin{align*} X ^ { \\left [ 2 \\right ] } = X + \\eta _ { i } \\partial _ { u _ { , i } } + \\eta _ { i j } \\partial _ { u _ { , i j } } , \\end{align*}"}
-{"id": "827.png", "formula": "\\begin{align*} F _ { \\vec { x } } ( - z _ { 1 } / s , \\ldots , - z _ { k } / s ) = \\frac { ( - s ) ^ { \\sum _ { i = 1 } ^ { k } x _ { i } } } { \\prod _ { i = 1 } ^ { k } z _ { i } ( 1 + z _ { i } ) ^ { M + 1 } } \\langle \\prod _ { 1 \\le i \\le k } C ^ { [ 0 , M ] } ( u _ { i } ; s ) \\prod _ { 1 \\le i \\le k } \\beta _ { x _ { i } } ^ { * } \\rangle _ { [ 0 , M ] } . \\end{align*}"}
-{"id": "4457.png", "formula": "\\begin{align*} C _ { n , \\beta } ( x ) : = \\left \\{ \\begin{array} { l l } \\sup _ { y \\in B _ x ^ \\circ ( r _ x ) } | f ( y ) - f ( x ) | / \\| y - x \\| ^ \\beta & \\mbox { i f $ \\beta \\leq 1 $ , } \\\\ \\sup _ { y \\in B _ x ^ \\circ ( r _ x ) } \\| \\dot { f } ( y ) - \\dot { f } ( x ) \\| / \\| y - x \\| ^ { \\beta - 1 } & \\mbox { i f $ \\beta > 1 $ , } \\end{array} \\right . \\end{align*}"}
-{"id": "4851.png", "formula": "\\begin{align*} K ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) = I ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) + R ^ { \\rm g e o } _ { 1 2 } ( i , u ; j , v ) \\end{align*}"}
-{"id": "2954.png", "formula": "\\begin{align*} & g ( \\sigma , \\mu _ 1 ) = H _ 2 ( \\sigma ) - \\gamma \\frac { \\delta - 1 } { \\delta } H _ 2 ( \\mu _ 1 ) \\\\ & \\qquad \\qquad + \\inf _ { u > 0 } \\{ \\sigma \\log p ( u ) + ( 1 - \\sigma ) \\log q ( u ) - \\mu _ 1 \\gamma \\log u \\} . \\end{align*}"}
-{"id": "9153.png", "formula": "\\begin{align*} c & = \\sum _ { r = 1 } ^ { s - 1 } | V _ 2 ^ r | \\\\ & = \\sum _ { r = 1 } ^ { s - 1 } \\xi _ { r + 1 } \\binom { p - 1 } { t - 1 } \\binom { p - t } { r t } \\\\ & = \\binom { p - 1 } { t - 2 } \\left ( \\sum _ { r = 1 } ^ { s - 1 } \\xi _ { r + 1 } \\frac { p - t + 1 } { t - 1 } \\binom { p - t } { r t } \\right ) . \\end{align*}"}
-{"id": "1261.png", "formula": "\\begin{align*} 0 = M _ { 1 1 } v ^ * U v ( 1 , 0 ) ^ T = M _ { 1 1 } V ( 1 , 0 ) ^ T = \\left ( \\int _ { \\R ^ 2 } V _ { 1 1 } ( y ) \\ , d y \\right ) ( 1 , 0 ) ^ T , \\end{align*}"}
-{"id": "8527.png", "formula": "\\begin{align*} X _ { [ b r a ] ^ { \\ast } } ( \\rho ( P ) ) & = \\displaystyle \\lim _ { n \\to \\infty } X _ { [ b r a ] ^ { \\ast } } ( \\rho ( P _ { n } ) ) \\\\ & = \\displaystyle \\lim _ { n \\to \\infty } \\rho ( Y _ { [ b r a ] } ( P _ { n } ) ) \\\\ & = \\rho ( \\displaystyle \\lim _ { n \\to \\infty } Y _ { [ b r a ] } ( P _ { n } ) ) \\\\ & = \\rho ( w ) \\end{align*}"}
-{"id": "8917.png", "formula": "\\begin{align*} \\mathcal { I } _ A ( t v ) = \\Tilde { \\mathcal { I } } _ { \\abs { A } ^ 2 } ( t v ) , \\end{align*}"}
-{"id": "6192.png", "formula": "\\begin{align*} \\left | \\frac { f ( x ( 1 + z ) ) - f ( x ) } { x f ' ( x ) } - z \\right | = \\left | z \\left ( \\frac { f ' ( \\xi ) } { f ' ( x ) } - 1 \\right ) \\right | \\leq z ^ 2 \\frac { | x f '' ( \\xi ' ) | } { f ' ( x ) } , \\end{align*}"}
-{"id": "3739.png", "formula": "\\begin{align*} e ^ \\beta = 1 + u \\beta , \\end{align*}"}
-{"id": "6866.png", "formula": "\\begin{align*} \\norm { v _ n ^ j } _ { W ( [ t _ n , \\infty ) ) } = \\norm { \\Psi ^ j _ { [ h _ n ^ j ] } } _ { W ( [ t _ n , \\infty ) ) } \\le \\norm { \\Psi ^ j } _ { W ( I _ j ) } < \\infty . \\end{align*}"}
-{"id": "6062.png", "formula": "\\begin{align*} T ^ { \\alpha } f ( t , x ) = \\int _ 0 ^ t \\int _ { { \\mathbb R } ^ d } P ^ { \\alpha } ( t - s , x - y ) f ( s , y ) d y d s \\end{align*}"}
-{"id": "3588.png", "formula": "\\begin{align*} \\Pi _ { g } \\circ D \\Phi ^ W _ { ( g , \\pi ) } \\circ \\rho _ g ( D \\Phi ^ W _ { ( g , \\pi ) } ) ^ * ( f , X ) = \\Pi _ g ( \\psi , V ) , \\end{align*}"}
-{"id": "4109.png", "formula": "\\begin{align*} p = \\gcd ( p , r ) + \\gcd ( p , q ) + \\gcd ( p , q + r ) . \\end{align*}"}
-{"id": "8129.png", "formula": "\\begin{align*} { \\mathcal R } _ { k , l , 1 } ( \\Omega ) = ( l + 1 ) ^ { k ' } { \\mathcal R } _ { k ' , 0 , 1 } ( U ) , \\end{align*}"}
-{"id": "7952.png", "formula": "\\begin{align*} j _ l ( \\sqrt [ 4 ] { \\mu } ) i _ l ' ( \\sqrt [ 4 ] { \\mu } ) - i _ l ( \\sqrt [ 4 ] { \\mu } ) j _ l ' ( \\sqrt [ 4 ] { \\mu } ) = 0 . \\end{align*}"}
-{"id": "8930.png", "formula": "\\begin{align*} & - \\Delta _ { A _ * } v _ A + v _ A - ( p - 1 ) \\abs { u _ { A _ * } } ^ { p - 2 } v _ A \\\\ & = \\abs { u _ A } ^ { p - 2 } u _ A - \\abs { u _ { A _ * } } ^ { p - 2 } u _ { A _ * } + ( p - 1 ) \\abs { u _ { A _ * } } ^ { p - 2 } ( u _ { A _ * } - u _ { A } ) - \\bigl ( \\abs { A } ^ 2 - \\abs { A _ * } ^ 2 \\bigr ) ( u _ { A } - u _ { A _ * } ) . \\end{align*}"}
-{"id": "5869.png", "formula": "\\begin{align*} C ^ { x } \\left ( x \\right ) = \\sigma \\left ( x \\right ) \\int \\frac { m } { \\sigma \\left ( x \\right ) } d x + c \\sigma \\left ( x \\right ) \\end{align*}"}
-{"id": "5874.png", "formula": "\\begin{align*} Z ^ { 3 } = e ^ { 2 m t } \\left ( \\partial _ { t } + H + c K ^ { 1 } \\right ) \\end{align*}"}
-{"id": "8860.png", "formula": "\\begin{align*} e ( \\alpha ) : = \\lim _ { T \\to \\infty } \\frac 1 T e _ T ( \\alpha ) \\end{align*}"}
-{"id": "8112.png", "formula": "\\begin{align*} C _ { k , l , N } = C _ { k , l , N } ^ { r a d } \\end{align*}"}
-{"id": "5657.png", "formula": "\\begin{align*} Q ( F ) - E \\cdot \\nabla _ v F = 0 , \\int _ { \\R ^ d } F \\ , d v = 1 , \\end{align*}"}
-{"id": "7312.png", "formula": "\\begin{align*} \\begin{gathered} K v _ { 1 } = q ^ { 2 } v _ { 1 } , K v _ { 0 } = v _ { 0 } , K v _ { - 1 } = q ^ { - 2 } v _ { - 1 } , \\\\ E v _ { 1 } = 0 , E v _ { 0 } = [ 2 ] ^ { 1 / 2 } v _ { 1 } , E v _ { - 1 } = [ 2 ] ^ { 1 / 2 } v _ { 0 } , \\\\ F v _ { 1 } = [ 2 ] ^ { 1 / 2 } v _ { 0 } , F v _ { 0 } = [ 2 ] ^ { 1 / 2 } v _ { - 1 } , F v _ { - 1 } = 0 . \\end{gathered} \\end{align*}"}
-{"id": "5593.png", "formula": "\\begin{align*} U = \\left ( \\begin{array} { c c } \\textrm { s i g n } ( \\lambda _ 1 ) & 0 \\\\ 0 & \\textrm { s i g n } ( \\lambda _ 2 ) \\end{array} \\right ) , \\ , \\ , \\ , \\ , \\ , v = \\left ( \\begin{array} { c c } a & b \\\\ c & d \\end{array} \\right ) : = \\left ( \\begin{array} { c c } \\eta _ 1 & 0 \\\\ 0 & \\eta _ 2 \\end{array} \\right ) B . \\end{align*}"}
-{"id": "3644.png", "formula": "\\begin{align*} \\frac { \\partial f } { \\partial x } ( x , y ) = - \\exp ( 1 - x ) \\exp ( 1 - y ^ 2 ) , \\frac { \\partial f } { \\partial y } ( x , y ) = - 2 y \\exp ( 1 - x ) \\exp ( 1 - y ^ 2 ) . \\end{align*}"}
-{"id": "3274.png", "formula": "\\begin{align*} \\begin{gathered} K v _ { 1 } = q ^ { 2 } v _ { 1 } , K v _ { 0 } = v _ { 0 } , K v _ { - 1 } = q ^ { - 2 } v _ { - 1 } , \\\\ E v _ { 1 } = 0 , E v _ { 0 } = [ 2 ] ^ { 1 / 2 } v _ { 1 } , E v _ { - 1 } = [ 2 ] ^ { 1 / 2 } v _ { 0 } , \\\\ F v _ { 1 } = [ 2 ] ^ { 1 / 2 } v _ { 0 } , F v _ { 0 } = [ 2 ] ^ { 1 / 2 } v _ { - 1 } , F v _ { - 1 } = 0 . \\end{gathered} \\end{align*}"}
-{"id": "7715.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sigma _ n \\xi _ { n + 1 } = 0 , \\end{align*}"}
-{"id": "7028.png", "formula": "\\begin{align*} V ( x ) = \\sigma ^ 2 + \\int _ { 0 < | y | \\le x } y ^ 2 \\Pi ( \\mathrm { d } y ) , U ( x ) = \\sigma ^ 2 + 2 \\int _ 0 ^ x y \\overline { \\Pi } ( y ) \\ , \\mathrm { d } y , x > 0 . \\end{align*}"}
-{"id": "1154.png", "formula": "\\begin{align*} \\dot x _ j = - b _ 0 ( x _ j ) , \\dot m _ j = m _ j \\langle b _ { 0 , x } \\rangle ( x _ j ) . \\end{align*}"}
-{"id": "857.png", "formula": "\\begin{align*} t ( \\vec { \\mu } ) - t ( \\vec { \\nu } ) = t ( r ^ { k _ { r } } , \\ldots , a ^ { k _ { 1 } - 1 } , \\ldots , 1 ^ { k _ { 1 } } ) - t ( \\nu _ { 1 } , \\ldots , \\nu _ { k - 1 } ) - \\sum _ { b = 1 } ^ { a - 1 } k _ { b } \\end{align*}"}
-{"id": "6040.png", "formula": "\\begin{align*} P _ { G _ { d , n } } ( 1 ) = e ^ { \\left ( \\frac { 1 } { 2 } + o _ d ( 1 ) \\right ) \\frac { \\log ^ 2 d } { d } n } , \\end{align*}"}
-{"id": "144.png", "formula": "\\begin{align*} 0 \\leq x & = x r + x r ^ \\perp \\leq r V \\mu ( x ) r + \\mu ( \\infty , x ) r ^ \\perp = \\sum _ { i = 1 } ^ n \\left ( r V f _ i r + C _ i r ^ \\perp \\right ) . \\end{align*}"}
-{"id": "5488.png", "formula": "\\begin{align*} \\mathbf { G } _ \\mathrm { s } [ 1 ] = \\mathbf { \\hat { G } } _ \\mathrm { s } [ 1 ] + \\mathcal { E } _ \\mathrm { s } [ 1 ] \\end{align*}"}
-{"id": "8992.png", "formula": "\\begin{align*} \\int _ a ^ b \\psi ( x ) ( ~ ^ { A B } I _ b ^ \\alpha \\varphi ( x ) d x = \\int _ 0 ^ 1 x ( ~ ^ { A B } I _ 1 ^ { 1 / 2 } ( 1 - x ) d x = \\int _ 0 ^ 1 x [ \\frac { 1 - x } { 2 } + \\frac { 2 ( 1 - x ) ^ { 3 / 2 } } { 3 \\sqrt { \\pi } } ] d x = \\frac { 1 } { 1 2 } + \\frac { 8 } { 1 0 5 \\sqrt { \\pi } } . \\end{align*}"}
-{"id": "8959.png", "formula": "\\begin{align*} U ( x , 0 ) = A e ^ { - x ^ { 2 } + 2 i x } \\end{align*}"}
-{"id": "5811.png", "formula": "\\begin{align*} \\mu _ k ^ * ( A ' _ i ) = \\begin{cases} A _ i & i \\neq k \\\\ A _ k ^ { - 1 } \\biggl ( \\prod _ { b _ { k j } > 0 } A _ j ^ { b _ { k j } } + \\prod _ { b _ { k j } < 0 } A _ j ^ { - b _ { k j } } \\biggr ) & i = k . \\end{cases} \\end{align*}"}
-{"id": "9002.png", "formula": "\\begin{align*} ~ ^ { A B R } ~ _ { 0 } D ^ \\alpha [ x ^ { \\nu - 1 } E ^ \\sigma _ { \\alpha , \\nu } ( \\lambda x ^ \\alpha ) ] = \\frac { B ( \\alpha ) } { 1 - \\alpha } \\frac { d } { d x } [ x ^ { \\nu } E ^ { 1 + \\sigma } _ { \\alpha , 1 + \\nu } ( \\lambda x ^ \\alpha ) ] = \\frac { B ( \\alpha ) } { 1 - \\alpha } x ^ { \\nu - 1 } E ^ { 1 + \\sigma } _ { \\alpha , \\nu } ( \\lambda x ^ \\alpha ) \\end{align*}"}
-{"id": "6644.png", "formula": "\\begin{align*} \\varepsilon _ K ( V _ 1 \\oplus \\cdots \\oplus V _ k , \\psi ) = \\varepsilon _ K ( V _ 1 , \\psi ) \\ldots \\varepsilon _ K ( V _ k , \\psi ) \\end{align*}"}
-{"id": "8569.png", "formula": "\\begin{align*} ( \\rho ^ { k } \\otimes ( ^ { k } \\rho ) ) \\Delta ( m _ { 1 } r m _ { 2 } ) \\Diamond \\rho ( z ) & = ( \\rho ^ { k } \\otimes ( ^ { k } \\rho ) ) ( 1 \\otimes m _ { 1 } r m _ { 2 } + m _ { 1 } \\otimes r m _ { 2 } ) \\Diamond \\rho ( z ) \\\\ & = ( \\bar { e _ { k } } \\otimes \\rho ( m _ { 1 } r m _ { 2 } ) ) \\Diamond \\rho ( z ) \\\\ & = c y c ( \\rho ( m _ { 1 } r m _ { 2 } z ) ) \\\\ & = \\rho ^ { k } ( \\Delta ( m _ { 1 } r m _ { 2 } ) \\Diamond z ) \\end{align*}"}
-{"id": "2685.png", "formula": "\\begin{align*} \\theta _ { \\max ( \\varphi , \\psi - t ) } ^ n \\geq { \\bf 1 } _ { \\{ \\varphi > \\psi - t \\} } \\theta _ { \\max ( \\varphi , \\psi - t ) } ^ n = { \\bf 1 } _ { \\{ \\varphi > \\psi - t \\} } \\theta _ { \\varphi } ^ n = \\theta _ { \\varphi } ^ n , \\end{align*}"}
-{"id": "5069.png", "formula": "\\begin{align*} \\nu ( K \\cup s K ) = \\nu ( K ) + \\nu ( s K ) - \\nu ( K \\cap s K ) = 2 \\nu ( K ) > 1 , \\end{align*}"}
-{"id": "8700.png", "formula": "\\begin{align*} W ^ \\xi _ \\tau : = \\langle \\xi , W _ { \\tau } \\rangle _ U . \\end{align*}"}
-{"id": "6258.png", "formula": "\\begin{align*} \\tfrac { \\partial } { \\partial s } \\eta ( r , s ) = \\gamma ' ( s ) + r J ' ( s ) + \\frac 1 2 r ^ 2 \\nabla _ r \\nabla _ r \\frac { \\dd \\eta } { \\dd s } \\big | _ { r = 0 } + O ( r ^ 3 ) . \\end{align*}"}
-{"id": "7470.png", "formula": "\\begin{align*} \\| u - v \\| ^ 2 > ( \\| v \\| - S _ j ) ^ 2 = \\| v \\| ^ 2 - 2 S _ j \\| v \\| + S _ j ^ 2 > \\| v \\| ^ 2 - 2 S _ j \\| v \\| . \\end{align*}"}
-{"id": "2217.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} \\partial _ { t } ^ { \\alpha } ( v ( x , t ) - g ) + L v ( x , t ) & = 0 & \\Omega \\times [ 0 , T ] , \\quad \\ , \\\\ u ( x , t ) & = 0 & \\mathbb { R } ^ { n } \\backslash \\Omega , \\ , t \\geq 0 , \\\\ u ( x , 0 ) & = g ( x ) & \\Omega , \\ , t = 0 , \\end{aligned} \\right . \\end{align*}"}
-{"id": "4508.png", "formula": "\\begin{align*} \\mathbb { E } _ f \\log ^ 4 \\xi _ { ( j ) , 1 } = \\mathbb { E } _ f [ \\{ \\log ( \\xi _ { ( j ) , 1 } f ( X _ 1 ) ) - \\log f ( X _ 1 ) \\} ^ 4 ] \\rightarrow \\mathbb { E } _ f \\log ^ 4 f ( X _ 1 ) \\end{align*}"}
-{"id": "2207.png", "formula": "\\begin{align*} \\left \\{ \\begin{aligned} L \\phi & = \\lambda \\phi \\Omega , \\\\ \\phi & = 0 \\quad \\ , \\ , \\ , \\ , \\mathbb { R } ^ { n } \\backslash \\Omega . \\end{aligned} \\right . \\end{align*}"}
-{"id": "5602.png", "formula": "\\begin{align*} \\phi & = - U v \\mathcal G _ 0 v ^ * \\phi + c _ 0 U v ( 1 , 0 ) ^ T : = U v \\psi , \\end{align*}"}
-{"id": "2199.png", "formula": "\\begin{align*} ( g _ { 1 - \\alpha } * W ) ( t _ { * } ) \\leq & g _ { 1 - \\alpha } ( ( t _ { 2 } - t _ { 1 } ) / 4 ) \\int _ { 0 } ^ { t _ { 2 } - t _ { 0 } } \\int _ { \\rho B _ { 1 } } w ^ { 2 } d x d s \\\\ = & \\frac { 4 ^ { \\alpha } } { \\Gamma ( 1 - \\alpha ) ( \\sigma \\eta ) ^ { \\alpha } ( \\rho - \\rho ' ) ^ { \\alpha } } \\int _ { 0 } ^ { t _ { 2 } - t _ { 0 } } \\int _ { \\rho B _ { 1 } } w ^ { 2 } d x d s . \\end{align*}"}
-{"id": "2635.png", "formula": "\\begin{align*} \\Delta ( F , G ) = \\sum _ { m = 0 } ^ { \\infty } \\sum _ { l = 1 } ^ { h ( m , n ) } \\left [ \\| \\mathbf { \\Phi _ m a _ m ^ l } \\| ^ 2 - \\| \\mathbf { B _ m ^ * \\Phi _ m ^ * \\Phi _ m a _ m ^ l } \\| ^ 2 \\right ] , \\end{align*}"}
-{"id": "1599.png", "formula": "\\begin{align*} \\ln F \\left ( t , x \\right ) = \\frac { \\mu } { 2 } \\left ( \\mu \\kappa t - 2 \\sqrt { 2 \\left ( \\mu - \\lambda - x \\right ) + 1 } \\right ) . \\end{align*}"}
-{"id": "4123.png", "formula": "\\begin{align*} \\omega = ( a ( p + 2 q ) x ^ 2 + p y ^ 2 + b ( p + q ) x ) d x + 2 q x y d y . \\end{align*}"}
-{"id": "6355.png", "formula": "\\begin{align*} A _ { \\sigma l } = ( l + 1 ) ( l + 2 + 2 \\sigma _ n ) p _ { \\sigma , l + 1 } + 2 s ( \\sigma _ n + 1 ) p _ { \\sigma + \\bar n , l } + ( \\sigma _ i + 1 ) ( \\sigma _ i + 2 ) p _ { \\sigma + 2 \\bar \\imath , l - 1 } + c _ { \\sigma l } ^ { \\mu m } p _ { \\mu m } . \\end{align*}"}
-{"id": "732.png", "formula": "\\begin{align*} & \\vec { { E ' } } = \\left ( \\begin{array} { c } E _ { 1 } ' \\\\ E _ { 2 } ' \\\\ \\vdots \\\\ E _ { s } ' \\end{array} \\right ) = { p } _ { * } \\vec { F } - { } ^ { \\rm t } B \\vec { D } . \\end{align*}"}
-{"id": "9619.png", "formula": "\\begin{align*} - \\frac { \\dot \\theta ( t - | y _ j - y _ k | ) ( \\zeta _ { 0 k } + ( t - | y _ j - y _ k | ) \\dot \\zeta _ { 0 k } ) } { 4 \\pi | y _ j - y _ k | } = - \\frac { \\delta ( t - | y _ j - y _ k | ) \\zeta _ { 0 k } } { 4 \\pi t } . \\end{align*}"}
-{"id": "3924.png", "formula": "\\begin{align*} z = ( 2 K + 1 + 2 m + \\sigma ) ( 2 K + \\sigma ) . \\end{align*}"}
-{"id": "8175.png", "formula": "\\begin{align*} [ t ^ n ] ( f _ 1 \\circ y ) & = \\sum _ { k = 1 } ^ n \\lambda ^ { k } B _ { n , k } ( 1 ! y _ 1 , 2 ! y _ 2 , \\dots ) , \\\\ [ t ^ n ] ( f _ 2 \\circ z ) & = \\sum _ { k = 1 } ^ n ( \\lambda ) _ k \\ , B _ { n , k } ( 1 ! z _ 1 , 2 ! z _ 2 , \\dots ) = \\sum _ { k = 1 } ^ n \\lambda \\tbinom { \\lambda - 1 } { k - 1 } ( k - 1 ) ! B _ { n , k } ( 1 ! z _ 1 , 2 ! z _ 2 , \\dots ) . \\end{align*}"}
-{"id": "8435.png", "formula": "\\begin{align*} A _ { 0 } ( \\textbf { u } ) \\partial _ { t } \\textbf { u } + \\sum _ { j = 1 } ^ { d } \\textbf { P } A _ { 0 } A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } + [ A _ { 0 } ( \\textbf { u } ) , \\textbf { P } ] A _ { j } ( \\textbf { u } ) \\partial _ { x _ { j } } \\textbf { u } = 0 . \\end{align*}"}
-{"id": "9355.png", "formula": "\\begin{align*} d p ( t , X ( t ) ) & = - [ \\pi ( t ) \\alpha _ 0 ( t ) \\frac { \\partial p } { \\partial x } ( t , X ( t ) ) + \\frac { 1 } { 2 } \\pi ^ 2 ( t ) \\beta _ 0 ^ 2 ( t ) \\frac { \\partial ^ 2 p } { \\partial x ^ 2 } ( t , X ( t ) ) + X ( t ) q ( t , X ( t ) ) ] d t \\\\ & + q ( t , X ( t ) ) d G ( t ) \\end{align*}"}
-{"id": "2602.png", "formula": "\\begin{align*} \\omega ( a _ 1 a _ 2 , a _ 3 , a _ 4 ) \\omega ( a _ 1 , a _ 2 , a _ 3 a _ 4 ) = \\omega ( a _ 1 , a _ 2 , a _ 3 ) \\omega ( a _ 1 , a _ 2 a _ 3 , a _ 4 ) \\omega ( a _ 2 , a _ 3 , a _ 4 ) , \\end{align*}"}
-{"id": "8204.png", "formula": "\\begin{align*} F ' _ U ( 0 , 0 ) = 1 . \\end{align*}"}
-{"id": "5351.png", "formula": "\\begin{align*} \\frac { 1 } { | G | } \\prod _ { i = 1 } ^ { \\ell } ( n - \\chi _ { \\gamma } ( c _ i ) ) = | K ( \\gamma ) | \\end{align*}"}
-{"id": "6884.png", "formula": "\\begin{align*} \\biggl | \\alpha - \\frac { p _ n } { q _ n } \\biggr | = \\frac { 1 } { \\lambda _ { n + 1 } ( \\alpha ) q _ n ^ 2 } \\end{align*}"}
-{"id": "9628.png", "formula": "\\begin{align*} | \\zeta ( t ) | \\le \\sqrt { ( { \\cal H } _ { \\tilde F } ( \\Psi ( t ) ) + a ) / b } = \\sqrt { ( { \\cal H } _ { \\tilde F } ( \\Psi _ 0 ) + a ) / b } = \\sqrt { ( { \\cal H } _ { F } ( \\Psi _ 0 ) + a ) / b } = \\Lambda ( \\Psi _ 0 ) , t \\in [ 0 , \\tau ] . \\end{align*}"}
-{"id": "4413.png", "formula": "\\begin{align*} V \\mu ( x ) = V \\mu ( x _ 0 ) + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } = x _ 0 + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } = \\abs { x } r + \\mu ( \\infty , x ) V \\chi _ { [ \\tau ( r ) , \\infty ) } . \\end{align*}"}
-{"id": "8498.png", "formula": "\\begin{align*} I _ C = ( x z , y z , \\Pi _ { j = 1 } ^ { r } ( x + a _ j y ) ) . \\end{align*}"}
-{"id": "5270.png", "formula": "\\begin{align*} \\int _ { T } ^ { 2 T } \\vert Z ( t ) \\vert d t = \\int _ { T } ^ { 2 T } \\vert \\zeta ( \\tfrac { 1 } { 2 } + i t ) \\vert d t \\geq \\vert \\int _ { T } ^ { 2 T } \\zeta ( \\tfrac { 1 } { 2 } + i t ) d t \\vert . \\end{align*}"}
-{"id": "6648.png", "formula": "\\begin{align*} E _ j ^ I : = V _ j ^ I \\otimes ( \\omega _ { s _ j } \\oplus \\omega _ { s _ j + 1 } \\oplus \\cdots \\oplus \\omega _ { s _ j + d _ j - 1 } ) \\end{align*}"}
-{"id": "1798.png", "formula": "\\begin{align*} 0 \\ ; \\ ; n = 1 , 1 \\ ; \\ ; n = 2 , 3 \\ ; \\ ; n = 3 , 2 n - 4 \\ ; \\ ; n \\geq 4 . \\end{align*}"}
-{"id": "4640.png", "formula": "\\begin{align*} g \\cdot f ( x , y ) = f ( ( x , y ) g ^ t ) . \\end{align*}"}
-{"id": "1412.png", "formula": "\\begin{align*} \\sum _ { j \\in I } B _ { i j } ( \\lambda _ j , \\mu _ j , \\nu _ j ) = 0 \\end{align*}"}
-{"id": "9302.png", "formula": "\\begin{align*} \\hat { \\pi } ( t , z ) & = - \\frac { \\alpha ( t ) \\int _ { \\mathbb { R } _ + } y ( t , x , z ) p ' ( t , x , z ) d x } { \\beta ^ 2 ( t ) \\int _ { \\mathbb { R } _ + } y ( t , x , z ) p '' ( t , x , z ) d x } \\\\ & = - \\frac { \\alpha ( t ) \\mathbb { E } [ p ' ( t , X ( t ) , z ) | \\mathcal { R } _ t ] } { \\beta ^ 2 ( t ) \\mathbb { E } [ p '' ( t , X ( t ) , z ) | \\mathcal { R } _ t ] } . \\end{align*}"}
-{"id": "9243.png", "formula": "\\begin{align*} \\beta ( t , z ) = \\delta ( t , z ) \\beta _ 0 ( t , z ) . \\end{align*}"}
-{"id": "7009.png", "formula": "\\begin{align*} D _ \\mathcal { R } ( t , p ) = \\left \\{ P \\in B _ \\mathcal { R } ( t , p ) \\mid \\tilde { u } ( t , P ) = \\max _ { Q \\in B _ \\mathcal { R } ( t , p ) } \\tilde { u } ( t , Q ) \\right \\} . \\end{align*}"}
-{"id": "1802.png", "formula": "\\begin{align*} \\widehat { \\Phi } ( \\xi ) + { \\sum _ { j = 1 } ^ \\infty } \\widehat { \\Psi } ( 2 ^ { - j } \\xi ) = 1 . \\end{align*}"}
-{"id": "5074.png", "formula": "\\begin{align*} L _ { k } ^ { + } = \\left \\{ \\vec { x } = ( x _ { 1 } , \\ldots , x _ { k } ) \\in \\mathbb { Z } ^ { k } \\ , | \\ , x _ { 1 } \\ge \\cdots \\ge x _ { k } \\right \\} , \\end{align*}"}
-{"id": "8656.png", "formula": "\\begin{align*} G B ( \\tau , h ) ( \\xi ) : = \\left ( \\begin{array} { l } 0 \\\\ b ( \\tau , \\xi , h _ 1 ( \\xi ) ) \\end{array} \\right ) , \\ ; \\ ; \\ ; \\xi \\in [ 0 , 1 ] , \\ ; \\tau \\in [ 0 , T ] , \\ ; h = ( h _ 1 , h _ 2 ) \\in H . \\end{align*}"}
-{"id": "6999.png", "formula": "\\begin{align*} \\delta _ v ( \\Theta ( L ) ( x , y , z ) ) ( a ) = \\sum _ { i + j = N + 1 ; ~ i , j > 0 } - f _ 0 ^ { \\lambda _ i ( x , \\lambda _ j ( y , z ) ) } + f _ 0 ^ { \\lambda _ i ( y , \\lambda _ j ( x , z ) ) } + f _ 0 ^ { \\lambda _ i ( \\lambda _ j ( x , y ) , z ) } ( a ) . \\end{align*}"}
-{"id": "5210.png", "formula": "\\begin{align*} \\bar \\partial _ M ^ * f = - \\sum _ { j = 1 } ^ { m - 1 } \\sideset { } { ' } \\sum _ { \\vert K \\vert = q - 1 } L _ j f _ { j K } \\overline { \\omega _ K } + \\ ; , \\end{align*}"}
-{"id": "5826.png", "formula": "\\begin{align*} \\rho ( \\prod _ { k } x _ { i _ k j _ k } ^ { \\delta _ k } ) = \\sigma ( \\prod _ { k } ( i _ k + 1 , j _ k ) ) ^ { - 1 } \\sigma ^ { - 1 } . \\end{align*}"}
-{"id": "4250.png", "formula": "\\begin{align*} \\tilde { T } _ { q _ 0 } = b _ { q _ 0 } \\tilde { \\ell } _ * + \\tilde { T } _ { q _ 0 , R } \\end{align*}"}
-{"id": "4141.png", "formula": "\\begin{align*} G ( s ) L ( s , f ) = \\epsilon ( f ) G ( \\delta - s ) L ( \\delta - s , g ) \\end{align*}"}
-{"id": "356.png", "formula": "\\begin{align*} \\tilde { T } _ { \\alpha , \\beta } ( f ) v = T ( f ) v + \\alpha J ( f ) v + \\beta J ( f ' ) v + \\frac { \\alpha ^ 2 + \\beta ^ 2 } { 2 } \\frac { 1 } { 2 \\pi } \\int _ { - \\pi } ^ { \\pi } f \\ ; v \\end{align*}"}
-{"id": "3851.png", "formula": "\\begin{align*} \\mathcal J _ k ( r ) = & \\frac 1 { 2 \\pi } \\int ^ { \\pi } _ { - \\pi } e ^ { i r \\sin \\theta } e ^ { - i \\theta k } d \\theta - \\frac { \\sin ( k \\pi ) } { \\pi } \\int _ 0 ^ \\infty e ^ { - ( r \\sinh ( s ) + k s ) } d s \\\\ : = & \\tilde J _ k ( r ) - E _ k ( r ) \\end{align*}"}
-{"id": "5163.png", "formula": "\\begin{gather*} \\alpha ( S _ i ) = \\sum _ { j = 1 } ^ n S _ j \\otimes u _ { j i } . \\end{gather*}"}
-{"id": "5070.png", "formula": "\\begin{align*} \\beta _ n ( A _ x ) = \\frac { 1 } { n } \\sum _ { k = 0 } ^ { n - 1 } \\Big ( \\sum _ { s \\in G } \\chi _ A ( s \\cdot x ) \\ , p ^ { * k } ( s ) \\Big ) \\end{align*}"}
-{"id": "1746.png", "formula": "\\begin{align*} \\lim _ { R \\to + \\infty } \\int _ { \\R ^ N \\setminus B _ R ^ + } \\Gamma _ R ^ + ( x , y ) g _ 2 ( y ) d y = 0 \\end{align*}"}
-{"id": "2177.png", "formula": "\\begin{align*} & - \\int _ { B _ { 1 } } \\partial _ { s } ( g _ { 1 - \\alpha , m } * ( \\phi \\psi ^ { 2 } \\tilde { u } ^ { 1 - q } ) ) d x + ( 1 - q ) \\phi \\mathcal { E } ( \\tilde { u } , - \\psi ^ { 2 } \\tilde { u } ^ { - q } ) \\\\ & \\quad \\quad \\quad \\leq \\int _ { 0 } ^ { s } \\dot { g } _ { 1 - \\alpha , m } ( s - \\tau ) ( \\phi ( s ) - \\phi ( \\tau ) ) \\left ( \\int _ { B _ { 1 } } \\psi ^ { 2 } \\tilde { u } ^ { 1 - q } d x \\right ) ( \\tau ) d \\tau + R _ { m } ( s ) , \\end{align*}"}
-{"id": "5274.png", "formula": "\\begin{align*} \\frac { 1 } { 2 \\pi i } \\int _ { { 1 } / { 2 } + i T } ^ { { 1 } / { 2 } + 2 i T } \\chi ( s ) ^ { - 1 / 2 } \\zeta ( s ) d s & = \\frac { 1 } { 2 \\pi i } \\left ( \\int _ { { 1 } / { 2 } + i T } ^ { 1 + \\delta + i T } + \\int _ { 1 + \\delta + i T } ^ { 1 + \\delta + 2 i T } + \\int _ { 1 + \\delta + 2 i T } ^ { { 1 } / { 2 } + 2 i T } \\right ) \\chi ( s ) ^ { - 1 / 2 } \\zeta ( s ) d s \\\\ & = I _ { 1 } + I _ { 2 } + I _ { 3 } . \\end{align*}"}
-{"id": "3279.png", "formula": "\\begin{align*} \\hat { R } F ( v _ { 1 } \\otimes v _ { 1 } ) & = [ 2 ] ^ { 1 / 2 } ( q ^ { - 2 } v _ { 1 } \\otimes v _ { 0 } + \\hat { R } ( v _ { 1 } \\otimes v _ { 0 } ) ) , \\\\ F \\hat { R } ( v _ { 1 } \\otimes v _ { 1 } ) & = [ 2 ] ^ { 1 / 2 } ( v _ { 0 } \\otimes v _ { 1 } + q ^ { 2 } v _ { 1 } \\otimes v _ { 0 } ) . \\end{align*}"}
-{"id": "2389.png", "formula": "\\begin{align*} z ^ { k + 1 } & = ( 1 - \\alpha ) z ^ { k } + \\alpha R _ { \\gamma g } ( R _ { \\gamma f } ( z ^ k ) ) ) . \\end{align*}"}
-{"id": "4239.png", "formula": "\\begin{align*} & C h _ { n , \\frac { 1 } { 2 } } = \\sum _ { m = 0 } ^ n { n \\choose m } C _ m C h _ { n - m , \\frac { 1 } { 2 } } ( - 1 ) ^ { m } \\frac { ( m + 1 ) ! } { 4 ^ m ( 2 m - 1 ) } , \\\\ \\textnormal { a n d } & \\\\ & C h _ { 0 , \\frac { 1 } { 2 } } = 1 . \\end{align*}"}
-{"id": "7786.png", "formula": "\\begin{align*} \\begin{aligned} & ( x , p , q ) \\rightarrow \\left ( \\left ( \\bar x , p \\right ) , \\left ( m ^ { - 1 } \\circ \\pi \\circ q , m ^ { - 1 } \\circ \\pi \\circ { } _ { i } x \\circ \\iota \\right ) \\right ) \\\\ & ( ( \\bar x , p ) , ( r , \\gamma ) ) \\rightarrow ( x , p , \\pi _ { \\ker x _ i } \\circ m \\circ \\ ; r ) , \\end{aligned} \\end{align*}"}
-{"id": "3695.png", "formula": "\\begin{align*} \\{ F , Z , H \\} _ { \\mu } = \\int F _ { \\mu } J ( Z _ { \\zeta } , H _ { \\mu } ) d A ~ . \\end{align*}"}
-{"id": "9175.png", "formula": "\\begin{align*} \\widehat { \\Psi } _ { k / h } ( \\theta ) = 1 + 2 i \\frac { k } { h } \\sum _ { l = 1 } ^ { r } \\alpha _ l \\sin ( \\theta l ) , i = \\sqrt { - 1 } , \\end{align*}"}
-{"id": "1789.png", "formula": "\\begin{align*} \\{ c \\in G ( L ) \\ | \\ c b \\varphi ( c ) ^ { - 1 } = b ' \\} . \\end{align*}"}
-{"id": "1430.png", "formula": "\\begin{align*} L _ { - 1 } x _ { \\alpha } = T \\psi _ { \\alpha } \\end{align*}"}
-{"id": "715.png", "formula": "\\begin{align*} \\vec { D } = \\left ( \\begin{array} { c } D _ { 1 } \\\\ D _ { 2 } \\\\ \\vdots \\\\ D _ { r } \\end{array} \\right ) , \\vec { F } = \\left ( \\begin{array} { c } F _ { 1 } \\\\ F _ { 2 } \\\\ \\vdots \\\\ F _ { s } \\end{array} \\right ) , \\vec { c } = \\left ( \\begin{array} { c } c _ { 1 } \\\\ c _ { 2 } \\\\ \\vdots \\\\ c _ { r } \\end{array} \\right ) , \\vec { Z } = \\left ( \\begin{array} { c } Z _ { 1 } \\\\ Z _ { 2 } \\\\ \\vdots \\\\ Z _ { r } \\end{array} \\right ) . \\end{align*}"}
-{"id": "6677.png", "formula": "\\begin{align*} F ( x ( t ; \\overline { x } ) ) = \\exp ( - 2 \\lambda t ) \\cdot F ( \\overline { x } ) , ~ ( \\forall ) t \\in I _ { \\overline { x } } . \\end{align*}"}
-{"id": "180.png", "formula": "\\begin{align*} d _ { \\mathrm { B L } } ( P , Q ) : = \\sup _ { h \\in \\mathcal { H } } \\biggl | \\int _ { - \\infty } ^ \\infty h \\ , d ( P - Q ) \\biggr | \\end{align*}"}
-{"id": "2731.png", "formula": "\\begin{align*} & L ( \\zeta _ { i _ 3 + 1 } - \\zeta _ { i _ 1 + 1 } ) \\frac { \\zeta _ j - \\zeta _ { i _ 2 + 1 } } { \\zeta _ j - \\zeta _ { j + 1 } } + L ( \\zeta _ { i _ 1 + 1 } - \\zeta _ { i _ 2 + 1 } ) \\frac { \\zeta _ j - \\zeta _ { i _ 3 + 1 } } { \\zeta _ j - \\zeta _ { j + 1 } } = - L ( \\zeta _ { i _ 2 + 1 } - \\zeta _ { i _ 3 + 1 } ) \\frac { \\zeta _ j - \\zeta _ { i _ 1 + 1 } } { \\zeta _ j - \\zeta _ { j + 1 } } , \\end{align*}"}
-{"id": "257.png", "formula": "\\begin{align*} \\biggl | \\int _ { \\mathcal { X } _ n ^ c } f ( x ) \\log f ( x ) \\ , d x \\biggr | = O ( q _ n ^ { 1 - \\epsilon } ) , \\end{align*}"}
-{"id": "3781.png", "formula": "\\begin{align*} { \\left [ { \\overline { \\bf { y } } } \\right ] _ i } & = { a _ i } { \\lambda _ i } \\sum \\limits _ { j = 1 } ^ 2 { { V _ { i j } } { d _ j } } i = 1 , 2 \\\\ { \\left [ { \\overline { \\bf { y } } } \\right ] _ i } & = { a _ i } { \\lambda _ i } \\sum \\limits _ { j = 3 } ^ 4 { { V _ { i j } } { d _ j } } i = 3 , 4 . \\end{align*}"}
-{"id": "8927.png", "formula": "\\begin{align*} u _ A = u _ { A _ * } + w _ { A _ * } \\bigl [ \\abs { A } ^ 2 - \\abs { A _ * } ^ 2 \\bigr ] + o \\bigl ( \\big \\| \\abs { A } ^ 2 - \\abs { A _ * } ^ 2 \\big \\| \\bigr ) , \\end{align*}"}
-{"id": "4657.png", "formula": "\\begin{align*} \\check { f } _ { \\ell , \\pm } * \\phi = \\lambda ( f _ { \\ell , \\pm } , \\phi ) \\phi . \\end{align*}"}
-{"id": "2063.png", "formula": "\\begin{align*} & V ^ { T } ( M V \\ddot { \\hat { x } } ( t ) + D V \\dot { \\hat { x } } ( t ) + K V \\hat { x } ( t ) - F u ( t ) ) = 0 , \\\\ & \\hat { y } ( t ) = \\ C _ { p } V \\hat { x } ( t ) + C _ { v } V \\dot { \\hat { x } } ( t ) . \\end{align*}"}
-{"id": "8344.png", "formula": "\\begin{align*} \\sigma _ i ( z ) = \\sigma _ i ( l ( y ) + D ) = l ( \\sigma _ i ( y ) ) + D = l ( y + \\mu _ i ) + D = l ( y ) + l ( \\mu _ i ) + D = z + l ( \\mu _ i ) \\end{align*}"}
-{"id": "351.png", "formula": "\\begin{align*} J _ n \\Omega = 0 \\ ; \\ ; \\ ; \\textrm { f o r e v e r y } \\ ; \\ ; \\ ; n \\geq 0 \\end{align*}"}
-{"id": "5867.png", "formula": "\\begin{align*} F \\left ( t , x \\right ) = x \\exp \\left ( \\left ( c _ { 1 } + \\frac { 1 } { 2 } \\right ) t \\right ) ~ , ~ m = 0 . \\end{align*}"}
-{"id": "3504.png", "formula": "\\begin{align*} \\Phi ( g , \\pi ) = \\left ( R ( g ) + \\tfrac { 1 } { n - 1 } ( \\mathrm { t r } _ g \\pi ) ^ 2 - | \\pi | ^ 2 _ g , \\ ; \\textup { d i v } _ g \\pi \\right ) = ( 2 \\mu , J ) . \\end{align*}"}
-{"id": "153.png", "formula": "\\begin{align*} \\widehat { C _ p } = C _ 1 \\| x \\| _ { \\widehat { C _ p } } = \\| S ( x ) \\| _ 1 , \\ , x \\in \\widehat { C _ p } . \\end{align*}"}
-{"id": "6719.png", "formula": "\\begin{align*} \\int _ 0 ^ T F ( X _ { t } ) d X ( t ) : = \\lim _ { n \\rightarrow \\infty } \\sum _ { i = 1 } ^ { n } F ( X _ { t _ { i - 1 } } ) ( X ( t _ { i } ) - X ( t _ { i - 1 } ) ) \\end{align*}"}
-{"id": "8403.png", "formula": "\\begin{align*} \\phi ( x ) _ g & = \\phi ( x ) _ g z _ g = \\phi ( x ) _ g \\alpha _ g ( z _ { g ^ { - 1 } } ) = w ^ * \\lim _ \\omega \\phi ( x ) _ g \\alpha _ g ( f _ \\omega ) \\\\ & = w ^ * \\lim _ \\omega \\theta ( E _ M ( x \\theta ^ { - 1 } ( f _ \\omega ) \\sigma ( z _ { g ^ { - 1 } } g ^ { - 1 } ) ) ) = \\theta ( E _ M ( x \\theta ^ { - 1 } ( z _ { g ^ { - 1 } } ) \\sigma ( z _ { g ^ { - 1 } } g ^ { - 1 } ) ) ) \\\\ & = \\theta ( E _ M ( x \\sigma ( z _ { g ^ { - 1 } } g ^ { - 1 } ) ) ) , \\ \\ \\ \\ x \\in W ^ * ( X _ 0 ) , \\ \\ \\ \\ g \\in G , \\end{align*}"}
-{"id": "9172.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial u _ 0 } ( v , \\dots , v ) = 0 . \\end{align*}"}
-{"id": "1683.png", "formula": "\\begin{align*} V _ t = v _ 0 + \\int _ 0 ^ t F ( X _ s , V _ s ) \\ , \\dd s + W _ t \\ , \\end{align*}"}
-{"id": "1000.png", "formula": "\\begin{align*} \\lambda _ { \\{ l ; k \\} } ( u ) & = \\left ( u + \\omega + \\eta \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) \\left ( u - \\omega + \\eta \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) \\prod _ { j = 1 } ^ { N - r } \\frac { u - v _ j + \\eta } { u - v _ j } \\\\ & + \\eta ^ { - 2 } \\prod _ { j = 1 } ^ { N - r } \\frac { u - v _ j - \\eta } { u - v _ j } . \\end{align*}"}
-{"id": "1723.png", "formula": "\\begin{gather*} M _ 5 : = M _ 5 ^ + \\cup M _ 5 ^ - = \\{ x \\in M \\colon \\sigma _ x \\neq 0 \\} = M - \\Sigma \\end{gather*}"}
-{"id": "195.png", "formula": "\\begin{align*} \\{ f _ \\Sigma ( \\cdot ) = | \\Sigma | ^ { - 1 / 2 } f _ 0 \\bigl ( \\Sigma ^ { - 1 / 2 } ( \\cdot - \\mu ) \\bigr ) : \\mu \\in \\mathbb { R } ^ d , \\Sigma = \\Sigma ^ T \\in \\mathbb { R } ^ { d \\times d } \\ \\} . \\end{align*}"}
-{"id": "3364.png", "formula": "\\begin{align*} \\| \\cdot \\| _ { r + \\alpha } ^ 2 = ( 1 - \\alpha ) \\| \\cdot \\| _ { r } ^ 2 + \\alpha \\| \\cdot \\| _ { r + 1 } ^ 2 , \\end{align*}"}
-{"id": "9278.png", "formula": "\\begin{align*} \\Phi _ K ( t , z ) = \\frac { \\mathbb { E } [ D _ t M ( T , z ) | \\mathcal { F } _ t ] } { M ( t , z ) } = \\frac { \\mathbb { E } [ D _ t [ K ( z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] ] | \\mathcal { F } _ t ] } { \\mathbb { E } [ K ( z ) \\mathbb { E } [ \\delta _ Z ( z ) | \\mathcal { F } _ T ] | \\mathcal { F } _ t ] } . \\end{align*}"}
-{"id": "4204.png", "formula": "\\begin{align*} F ( x ) = y , \\end{align*}"}
-{"id": "9132.png", "formula": "\\begin{align*} u ( s ) - u ( t ) & = [ e ^ { - ( t - s ) A ( s ) } u ( t ) - u ( t ) ] + \\int _ { s } ^ { t } e ^ { - ( r - s ) A ( s ) } ( \\mathcal { A } ( s ) - A ( r ) ) u ( r ) d r \\\\ & + \\int _ { s } ^ { t } e ^ { - ( r - s ) A ( s ) } ( ( B ( r ) + I ) A ( r ) u ( r ) ) d r \\\\ & - \\int _ { s } ^ { t } e ^ { - ( r - s ) A ( s ) } [ f ( r ) - P ( r ) u ( r ) ] d r \\\\ & = : I _ { 1 } ( s , t ) + I _ { 2 } ( s , t ) + I _ { 3 } ( s , t ) + I _ { 4 } ( s , t ) . \\end{align*}"}
-{"id": "5727.png", "formula": "\\begin{align*} \\tan ^ { - 1 } ( m _ f ( Q ) ) = m _ { \\tan ^ { - 1 } f } ( Q ) , \\end{align*}"}
-{"id": "2276.png", "formula": "\\begin{align*} \\sum _ { i = 1 } ^ m \\| A ( \\tau _ i ) - A ( \\tau _ { i - 1 } ) \\| _ { { \\mathcal L } ( D , X ) } \\le M _ 2 \\end{align*}"}
-{"id": "7231.png", "formula": "\\begin{align*} \\sum _ { \\mu = 1 } ^ r ( T _ { j k } ) ^ \\lambda _ \\mu \\cdot w _ k ^ \\mu = w _ j ^ \\lambda + \\sum _ { | \\alpha | \\geq 2 } f _ { k j , \\alpha } ^ \\lambda ( z _ j ) \\cdot w _ j ^ \\alpha , \\end{align*}"}
-{"id": "8792.png", "formula": "\\begin{align*} K ^ { ( 2 r + 1 , r _ R ) } _ n ( x ) = \\sum _ { i = - r } ^ { r } E ^ { ( i i ) } - \\frac { 1 - x ^ 2 } { h _ n ( b , d , x ) } \\Big ( \\sum _ { r _ R < i \\leq r } E ^ { ( - i , - i ) } + t _ n E ^ { ( r + 1 - i , r + 1 - i ) } \\\\ { } - \\sum _ { r _ R < i \\leq r } E ^ { ( r + 1 - i , - i ) } + t _ n E ^ { ( - i , r + 1 - i ) } \\Big ) . \\end{align*}"}
-{"id": "4934.png", "formula": "\\begin{align*} | \\langle a _ k , \\tilde { x } \\rangle | = | \\langle a _ k , x _ 0 \\rangle | , \\forall k . \\end{align*}"}
-{"id": "5000.png", "formula": "\\begin{align*} A = \\left ( \\begin{array} { c c c } 1 & & \\\\ & \\ddots & \\\\ & & 1 \\\\ & & \\end{array} \\right ) . \\end{align*}"}
-{"id": "8756.png", "formula": "\\begin{align*} & \\widetilde { K } _ 0 ( x ) = _ 2 \\left ( ( \\mathbb { I } _ 3 \\otimes K _ 0 \\left ( x \\right ) ) \\widetilde { R } ( x ^ 2 ) P \\right ) , \\\\ & \\widetilde { K } _ n ( x ) = _ 1 \\left ( ( K _ n \\left ( x ^ { - 1 } \\right ) \\otimes \\mathbb { I } _ 3 ) \\widetilde { R } ( x ^ 2 ) P \\right ) , \\end{align*}"}
-{"id": "6469.png", "formula": "\\begin{align*} \\| G _ { m } \\| _ { L ^ { 1 } ( [ 0 , t _ { * } ] ) } = ( g _ { 1 - \\alpha , m } * W ) ( t _ { * } ) + \\int _ { 0 } ^ { t _ { * } } F _ { m } ( s ) d s . \\end{align*}"}
-{"id": "794.png", "formula": "\\begin{align*} L ( z ) \\otimes L ( w ) = \\sum _ { a , b = 0 } ^ { r } \\sum _ { c , d = 0 } ^ { r } \\left ( L ( z ) _ { a b } L ( w ) _ { c d } \\right ) \\otimes E _ { a b } \\otimes E _ { c d } . \\end{align*}"}
-{"id": "7695.png", "formula": "\\begin{align*} f ( x ) + \\gamma _ n < f ( x ) - x + x + \\delta _ 0 < - \\delta _ 0 + x + \\delta _ 0 = x < 2 K - \\varepsilon _ 0 , \\end{align*}"}
-{"id": "7775.png", "formula": "\\begin{align*} \\begin{small} \\begin{pmatrix} 1 & & & & & & & \\\\ & 3 & & & & & & \\\\ & & & & & & & & \\\\ & & & & & & & & \\\\ & & 4 2 & & & & & \\\\ & & & 1 2 6 & & & & \\\\ & & & & 1 6 8 & & & \\\\ & & & & & 1 2 0 & & \\\\ & & & & & & 4 5 & \\\\ & & & & & & & 7 \\\\ \\end{pmatrix} . \\end{small} \\end{align*}"}
-{"id": "2333.png", "formula": "\\begin{align*} \\frac { 1 } { N _ p ( x ) } = \\sum ^ n _ { j = 1 } \\frac { 1 } { x - x _ j } . \\end{align*}"}
-{"id": "8669.png", "formula": "\\begin{align*} ( C _ b ( H ) , C ^ 1 _ K ( H ) ) _ { \\alpha , \\infty } = C ^ { \\alpha } _ K ( H ) . \\end{align*}"}
-{"id": "9724.png", "formula": "\\begin{align*} \\frac { \\int _ 0 ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } { \\log \\sigma ( t ) } & = \\frac { \\int _ 0 ^ { T ( M ) } \\frac { 1 } { \\sigma ( s ) } \\ , d s } { \\log \\sigma ( t ) } + \\frac { \\int _ { T ( M ) } ^ t \\frac { 1 } { \\sigma ( s ) } \\ , d s } { \\log \\sigma ( t ) } \\\\ & \\leq \\frac { \\int _ 0 ^ { T ( M ) } \\frac { 1 } { \\sigma ( s ) } \\ , d s } { \\log t } + \\frac { 1 } { M } \\cdot \\frac { \\log t - \\log T ( M ) } { \\log t } . \\end{align*}"}
-{"id": "8028.png", "formula": "\\begin{align*} \\chi ( \\P ^ 1 , j _ * V ) & = \\dim H ^ 0 ( U , V ) - \\dim H ^ 1 ( \\P ^ 1 , j _ * V ) + \\dim H ^ 2 _ c ( U , V ) \\\\ & = \\chi _ c ( U , V ) + \\sum _ { s \\in S } \\dim V _ s . \\end{align*}"}
-{"id": "4091.png", "formula": "\\begin{gather*} f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ 2 } { x z ^ { 3 } } , f ( x , y , z ) = \\dfrac { ( y ^ 2 + a x ^ 2 + b x z + c z ^ 2 ) ^ 2 } { x ^ 3 z } \\\\ \\end{gather*}"}
-{"id": "9645.png", "formula": "\\begin{align*} C ( \\omega _ 1 , \\omega _ 2 ) = \\left ( \\begin{array} { c c c } \\gamma _ 1 ( \\omega _ 1 , \\omega _ 2 ) & \\gamma _ 0 ( \\omega _ 1 , \\omega _ 2 ) \\\\ 1 & 0 \\end{array} \\right ) , \\end{align*}"}
-{"id": "99.png", "formula": "\\begin{align*} x _ { k + 1 } = x _ k - \\nabla J ( x _ k ) , k = 0 , \\ldots , \\end{align*}"}
-{"id": "2060.png", "formula": "\\begin{align*} x _ 1 ^ 2 + x _ 2 ^ 2 + \\cdots + x _ n ^ 2 - x _ { n + 1 } ^ 2 = \\tfrac { 1 } { K } \\end{align*}"}
-{"id": "1245.png", "formula": "\\begin{align*} D _ m = - i \\alpha \\cdot \\nabla + m \\beta = - i \\sum _ { k = 1 } ^ { n } \\alpha _ k \\partial _ { k } + m \\beta \\end{align*}"}
-{"id": "1694.png", "formula": "\\begin{align*} \\gamma ( Z _ t ) - \\gamma ( Y _ t ) = & \\ , z - y + U ( z ) - U ( y ) + \\int _ 0 ^ t \\big [ D U ( Z _ s ) - D U ( Y _ s ) \\big ] R \\cdot \\dd W _ s \\\\ & + \\lambda \\int _ 0 ^ t \\big [ U ( Z _ s ) - U ( Y _ s ) \\big ] \\dd s + \\int _ 0 ^ t A ( Z _ s - Y _ s ) \\ , \\dd s \\ , . \\end{align*}"}
-{"id": "8729.png", "formula": "\\begin{align*} e ^ { - \\tau A } \\widetilde Y _ { \\tau } ^ { x } = \\int _ \\tau ^ T e ^ { - s A } G B ( s , X ^ { x } _ s ) \\ , d s - \\int _ \\tau ^ T e ^ { - s A } \\widetilde Z ^ { x } _ { s } \\ ; d W _ s , \\tau \\in [ 0 , T ] . s \\end{align*}"}
-{"id": "4666.png", "formula": "\\begin{align*} \\Theta _ { \\psi } ^ { ( 2 ) } ( w ; \\phi ) & \\sim \\frac { 2 \\psi ( 1 ) } { \\xi ( 2 ) ( w - 1 ) } \\int _ 0 ^ \\infty \\int _ { - \\infty } ^ \\infty \\sum _ { \\ell , m = 1 } ^ { \\infty } \\rho _ \\phi ( \\ell m ) \\\\ & \\times K _ { s - \\frac { 1 } { 2 } } ( 2 \\pi \\ell m t ^ 2 ) f ( 0 , 0 , t ^ { - 1 } m , u ) \\cos ( 2 \\pi \\ell t ^ 3 u ) d u t d t . \\end{align*}"}
-{"id": "4247.png", "formula": "\\begin{align*} \\varphi _ q ^ k & = \\frac { \\mu ( x _ q ^ k ) } { q } \\left ( 1 + \\frac { \\beta ( k / q ) } { q ^ 2 } + \\varepsilon O ( q ^ { - 4 } ) \\right ) \\end{align*}"}
-{"id": "177.png", "formula": "\\begin{align*} \\mathcal { F } _ { d , \\theta } : = \\biggl \\{ f \\in \\mathcal { F } _ d : \\mu _ \\alpha ( f ) \\leq \\nu , \\| f \\| _ \\infty \\leq \\gamma , \\sup _ { x : f ( x ) \\geq \\delta } M _ { f , a , \\beta } ( x ) \\leq a ( \\delta ) \\ \\forall \\delta > 0 \\biggr \\} . \\end{align*}"}
-{"id": "7457.png", "formula": "\\begin{align*} b _ j = \\sum _ { i = 1 } ^ n a _ i \\left ( \\sum _ { \\gamma : i \\rightarrow j } \\prod _ { f \\in \\hat { \\gamma } } X _ f \\right ) . \\end{align*}"}
-{"id": "2081.png", "formula": "\\begin{align*} & \\tilde { f } ( H ( s ) ) = f ( \\tilde { H } ( s ) ) \\ \\ \\ \\tilde { H } ( s ) \\ \\\\ & \\frac { \\| H ( s ) - \\tilde { H } ( s ) \\| _ { H _ { 2 } \\ \\ H _ { \\infty } } } { \\| H ( s ) \\| _ { H _ { 2 } \\ \\ H _ { \\infty } } } = \\ \\mathcal { O } ( | | Z | | ) , \\end{align*}"}
-{"id": "4107.png", "formula": "\\begin{align*} f ( x , y , z ) = \\dfrac { x ^ { p } y ^ { q } ( b y + c z ) ^ r } { z ^ { p + q + r } } . \\end{align*}"}
-{"id": "2415.png", "formula": "\\begin{align*} \\Lambda _ k ^ { \\alpha } = \\{ l \\in \\mathbb { Z } ^ n : ~ \\Box _ l ^ { \\alpha } \\circ \\Box _ k ^ { \\alpha } \\neq 0 \\} \\end{align*}"}
-{"id": "6564.png", "formula": "\\begin{align*} E ( z , x , y ) = \\frac { c _ d } { z ^ d } \\bigg ( 1 + \\frac { | x - y | ^ 2 } { z ^ 2 } \\bigg ) ^ { - \\frac { d + 1 } { 2 } } \\ , , \\end{align*}"}
-{"id": "793.png", "formula": "\\begin{align*} & q ^ { N _ { a } } \\beta _ { a } = q ^ { - 1 } \\beta _ { a } q ^ { N _ { a } } , q ^ { N _ { a } } \\beta ^ { * } _ { a } = q \\ , \\beta _ { a } ^ { * } q ^ { N _ { a } } , \\\\ & \\beta _ { a } \\beta _ { a } ^ { * } = 1 - q ^ { 2 } q ^ { 2 N _ { a } } , \\beta _ { a } ^ { * } \\beta _ { a } = 1 - q ^ { 2 N _ { a } } \\end{align*}"}
-{"id": "9181.png", "formula": "\\begin{align*} \\mathbb E [ \\int _ 0 ^ t \\langle { \\mathcal M } _ n , { \\mathcal M } _ n \\rangle _ s d s ] < \\infty , \\ t > 0 \\ { { \\it a n d } \\ n = 1 , \\dots , p . } \\end{align*}"}
-{"id": "6662.png", "formula": "\\begin{gather*} z _ 1 ( u _ k , t ) : = x ( u _ k , t ) , t \\geqslant 0 , 1 \\leqslant k \\leqslant n , \\\\ z _ { i + 1 } ( u _ k , t ) : = z _ i ( u _ k , t \\wedge \\sigma _ i ) + \\sum _ { j = 1 } ^ k ( z _ i ( u _ j , t ) - z _ i ( u _ j , t \\wedge \\sigma _ i ) ) \\cdot \\ 1 _ { A _ { k j } ^ i } , t \\geqslant 0 , \\\\ 1 \\leqslant k \\leqslant n , 1 \\leqslant i \\leqslant n - 1 . \\end{gather*}"}
-{"id": "7530.png", "formula": "\\begin{align*} \\mathbb { P } \\left [ \\mathcal N _ 1 < \\infty x _ { \\mathcal N _ 1 } \\in ( K - \\varepsilon , K + \\varepsilon ) \\right ] = 1 . \\end{align*}"}
-{"id": "1890.png", "formula": "\\begin{align*} \\hat { \\phi } _ X ( u ) : = \\exp ( \\hat { \\psi } ( u ) / K ) \\end{align*}"}
-{"id": "4866.png", "formula": "\\begin{align*} f ( z ) = \\frac { \\sigma ^ 3 } { 3 } z ^ 3 + \\mathcal { O } ( z ^ 4 ) . \\end{align*}"}
-{"id": "1085.png", "formula": "\\begin{align*} a = 1 / 2 \\ ; \\ ; a n d \\ ; \\ ; b = 1 7 / 3 0 . \\end{align*}"}
-{"id": "9220.png", "formula": "\\begin{align*} H ( t , x ) = H ( t , x , Y ( t , x , z ) , Y ( t , \\cdot , z ) ( x ) , u ( t , x , z ) , \\widehat { p } ( t , x , z ) , \\widehat { q } ( t , x , z ) , \\widehat { r } ( t , x , z , . ) ) \\end{align*}"}
-{"id": "2594.png", "formula": "\\begin{align*} \\int _ { | x | \\geq \\tilde C ( \\eta ) \\sqrt { t } } \\bigl | | x | ^ { | s _ c | } e ^ { - i t \\Delta } \\Psi ( t ) \\bigr | ^ 2 \\ , d x & = \\lim _ { n \\to \\infty } \\int _ { | x | \\geq \\tilde C ( \\eta ) \\sqrt { t } } \\bigl | | x | ^ { | s _ c | } e ^ { - i t \\Delta } v ^ { [ t _ n ] } ( t ) \\bigr | ^ 2 \\ , d x \\\\ & = \\lim _ { n \\to \\infty } \\int _ { | x | \\geq \\tilde C ( \\eta ) \\sqrt { t _ n t } } \\bigl | | x | ^ { | s _ c | } e ^ { - i t _ n t \\Delta } v ( t _ n t ) \\bigr | ^ 2 \\ , d x \\leq \\eta \\end{align*}"}
-{"id": "9171.png", "formula": "\\begin{align*} \\frac { \\partial H } { \\partial u _ l } ( v , \\dots , v ) + \\frac { \\partial H } { \\partial u _ { - l } } ( v , \\dots , v ) = 0 , l = 1 , \\dots , r . \\end{align*}"}
-{"id": "3615.png", "formula": "\\begin{align*} s _ 1 = 0 , t _ 1 = 4 , s _ j = - 1 , t _ k = 3 ( j , k = 2 , \\dots , n + 1 ) . \\end{align*}"}
-{"id": "1212.png", "formula": "\\begin{align*} - \\sum \\limits _ { i = 1 } ^ { n } g ( X _ { i } , Y _ { i } ) & = - \\sum \\limits _ { i = 1 } ^ { n } \\frac { h ' _ i } { 2 } | f ( X _ i ) - f ^ * ( X _ i ) | + \\sum \\limits _ { i = 1 } ^ { n } \\frac { \\xi ^ { ( h ' ) } _ { i } } { 2 } | f ( X _ i ) - f ^ * ( X _ i ) | \\\\ & \\le - ( \\min \\limits _ { i } h ' _ { i } ) \\sum \\limits _ { i = 1 } ^ { n } \\frac { 1 } { 2 } | f ( X _ i ) - f ^ * ( X _ i ) | + \\sum \\limits _ { i = 1 } ^ { n } \\frac { \\xi ^ { ( h ' ) } _ { i } } { 2 } | f ( X _ i ) - f ^ * ( X _ i ) | . \\end{align*}"}
-{"id": "8924.png", "formula": "\\begin{align*} | u ( x ) | \\le ( c + o ( 1 ) ) \\exp \\Bigl ( - \\sum _ { j = 1 } ^ k \\frac { \\abs { \\lambda _ j } } { 2 } \\abs { P _ { W _ j } ( x ) } ^ 2 \\Bigr ) . \\end{align*}"}
-{"id": "498.png", "formula": "\\begin{align*} f \\in \\mathcal { F } \\mapsto \\mathcal { L } _ \\alpha ( f ) : = \\Re ( e ^ { i \\alpha } a _ 1 ) \\end{align*}"}
-{"id": "2062.png", "formula": "\\begin{align*} \\hat { M } \\ddot { \\hat { x } } ( t ) = & - \\hat { D } \\dot { \\hat { x } } ( t ) - \\hat { K } \\hat { x } ( t ) + \\hat { F } u ( t ) , \\\\ \\hat { y } ( t ) = & \\ \\hat { C } _ { p } \\hat { x } ( t ) + \\hat { C } _ { v } \\dot { \\hat { x } } ( t ) , \\end{align*}"}
-{"id": "8726.png", "formula": "\\begin{align*} \\widetilde Y _ \\tau ^ { t , x } = v ( \\tau , X _ { \\tau } ^ { t , x } ) , \\ ; \\ ; \\ ; \\P - a . s . ; \\ ; \\ ; \\ ; \\widetilde Z _ \\tau ^ { t , x } = \\nabla ^ G v ( \\tau , X _ { \\tau } ^ { t , x } ) , \\ ; \\ ; \\P - a . s . \\end{align*}"}
-{"id": "7172.png", "formula": "\\begin{align*} \\sum _ { j , k = 1 } ^ n \\Phi ^ { j k } ( x ) \\hat b _ { j k } = \\sum _ { ( j , k ) \\neq ( n , n ) } \\Phi ^ { j k } ( x ) b _ { j k } . \\end{align*}"}
-{"id": "791.png", "formula": "\\begin{align*} h _ { \\vec { z } } ^ { \\vec { \\mu } } ( \\vec { x } ) = \\prod _ { 1 \\le i < j \\le k } f ( z _ { i } , z _ { j } ) \\sum _ { \\tau \\in \\mathfrak { S } _ { k } } \\prod _ { i = 1 } ^ { k } \\left ( \\frac { z _ { \\tau ^ { - 1 } ( i ) } } { 1 + z _ { \\tau ^ { - 1 } ( i ) } } \\right ) ^ { x _ { i } } \\phi ( \\tau ; \\vec { z } ) ( u _ { \\mu _ { 1 } } \\otimes \\cdots \\otimes u _ { \\mu _ { k } } ) . \\end{align*}"}
-{"id": "5327.png", "formula": "\\begin{align*} [ H _ { n , m } , Q _ { j } ] = [ H _ { n , m } , \\overline { Q } _ { j } ] = [ N , Q _ { j } ] = [ N , \\overline { Q } _ { j } ] & = 0 \\\\ [ Q _ { j } , { Q } _ { k } ] = [ Q _ { j } , \\overline { Q } _ { k } ] = [ \\overline { Q } _ { j } , \\overline { Q } _ { k } ] & = 0 , \\end{align*}"}
-{"id": "2956.png", "formula": "\\begin{align*} & h ( \\sigma , \\epsilon ) = \\max _ { \\mu _ 1 \\in \\bar { M } _ { \\epsilon } } \\biggl [ H _ 2 ( \\sigma ) - \\gamma \\frac { \\delta - 1 } { \\delta } H _ 2 ( \\mu _ 1 ) \\\\ & \\quad \\qquad + \\inf _ { u > 0 } \\{ \\sigma \\log p ( u ) + ( 1 - \\sigma ) \\log q ( u ) - \\mu _ 1 \\gamma \\log u \\} \\biggr ] . \\end{align*}"}
-{"id": "7867.png", "formula": "\\begin{align*} \\varrho ( c _ 1 r ) = \\frac { ( c _ 1 r ) ^ d } { \\Phi ( c _ 1 ^ { - 1 } r ^ { - 1 } ) } \\le c _ 1 ^ { d + 2 } \\frac { r ^ d } { \\Phi ( r ^ { - 1 } ) } = c _ 1 ^ { d + 2 } \\varrho ( r ) \\ , . \\end{align*}"}
-{"id": "3789.png", "formula": "\\begin{align*} \\lim _ { N _ { \\mathrm t } \\rightarrow \\infty } { \\bf { u } } _ { \\mathrm t } ^ H ( \\phi _ p ) \\ , { { \\bf { u } } _ { \\mathrm t } } ( \\phi _ l ) & = \\delta ( { p - l } ) \\end{align*}"}
-{"id": "6363.png", "formula": "\\begin{align*} D _ { \\l , x ' } : = \\{ ( x ' , z ) \\in \\R ^ { n + 1 } : | z | < \\l \\} . \\end{align*}"}
-{"id": "6959.png", "formula": "\\begin{align*} m \\le \\sum _ { i = 1 } ^ { K } | \\mathcal { N } ( \\mathcal { U } _ i ) \\setminus \\mathcal { X } ( \\Pi _ K ^ { ( \\epsilon ) } ) | = n - | \\mathcal { X } ( \\Pi _ K ^ { ( \\epsilon ) } ) | \\end{align*}"}
-{"id": "6290.png", "formula": "\\begin{align*} u ( y ) = ( - 1 ) ^ d p ( \\sqrt { y } ) p ( - \\sqrt { y } ) = \\prod _ { j = 1 } ^ d ( y - x _ j ^ 2 ) . \\end{align*}"}
-{"id": "4033.png", "formula": "\\begin{align*} \\phi _ { \\mu , i ' } ^ + ( x ) = p r o j _ { D _ { i ' } } ( f x ) . \\end{align*}"}
-{"id": "2084.png", "formula": "\\begin{align*} \\tilde { V } _ { 1 } = [ X ^ { ( 0 ) } ( s _ { t _ { 0 } } ) / | | X ^ { ( 0 ) } ( s _ { t _ { 0 } } ) | | ] , \\end{align*}"}
-{"id": "9987.png", "formula": "\\begin{align*} \\lambda \\Big ( \\alpha \\tilde { p } - \\frac { B _ { k } } { T _ f } - \\alpha E _ k \\Big ) & = 0 , \\\\ \\mu \\Big ( ( 1 - \\alpha ) \\sum _ { i = 1 } ^ 2 | w _ { k i } | ^ 2 p _ { i } + \\alpha \\tilde { p } - A _ k \\Big ) & = 0 , \\\\ \\eta \\Big ( ( 1 - \\alpha ) \\sum _ { i = 1 } ^ 2 | w _ { \\bar { k } i } | ^ 2 p _ { i } - A _ { \\bar { k } } \\Big ) & = 0 . \\end{align*}"}
-{"id": "1987.png", "formula": "\\begin{align*} \\begin{cases} & R ( 1 ) = 1 R ( 0 ) : = \\lim \\limits _ { t \\to 0 } R ( t ) = 0 ; \\\\ & \\int _ J d t / R ( t ) = \\infty ; \\\\ & \\| \\partial ^ k _ t R \\| _ { \\infty } < \\infty , k \\geq 1 . \\end{cases} \\end{align*}"}
-{"id": "400.png", "formula": "\\begin{align*} G _ { \\phi } ( x ) = \\sum _ { \\ell , m = 1 } ^ { \\infty } \\frac { \\rho _ { \\phi } ( \\ell m ) } { \\ell ^ { 1 + x } m ^ { 1 + 3 x } } . \\end{align*}"}
-{"id": "7960.png", "formula": "\\begin{align*} \\mathcal H _ { \\tilde g } ^ 1 \\Big ( \\{ \\psi _ { \\tilde \\lambda } = 0 \\} \\cap \\tilde B _ { \\tilde R / 2 } ( p ) \\Big ) \\leq C _ 2 { \\tilde \\lambda } ^ { \\frac { 3 } { 4 } - \\beta } , \\end{align*}"}
-{"id": "1228.png", "formula": "\\begin{align*} 1 \\ge \\rho ( x ) = \\int _ 0 ^ \\gamma \\varphi ( x ^ * ) w \\ge \\int _ { \\{ s \\in [ 0 , \\gamma ) : \\ x ^ * ( s ) > \\lambda \\} } \\varphi ( \\lambda ) w = \\varphi ( \\lambda ) \\int _ 0 ^ \\beta w , \\end{align*}"}
-{"id": "5166.png", "formula": "\\begin{gather*} T _ p \\leftrightarrow \\sum _ { i ( 1 ) , \\ldots , i ( k ) , j ( 1 ) , \\ldots , j ( l ) = 1 } ^ n \\delta _ p ( i , j ) S _ { j ( 1 ) } \\cdots S _ { j ( l ) } S _ { i ( k ) } ^ * \\cdots S _ { i ( 1 ) } ^ * . \\end{gather*}"}
-{"id": "2553.png", "formula": "\\begin{align*} ( i \\partial _ t + \\Delta ) u = \\mu | u | ^ { p } u . \\end{align*}"}
-{"id": "8582.png", "formula": "\\begin{align*} \\alpha C \\xi _ { r a _ { 1 } } ( n ) & = \\displaystyle \\sum _ { r ' a ' \\in _ { k } \\hat { T } _ { 1 } } \\alpha \\xi _ { r ' a ' } \\pi _ { r ' a ' } C \\xi _ { r a _ { 1 } } ( n ) \\\\ & = \\displaystyle \\sum _ { r ' a ' \\in _ { k } \\hat { T } _ { 1 } } \\phi ^ { - 1 } \\left ( r ' a ' C _ { r ' a ' , r a _ { 1 } } \\right ) n = \\phi ^ { - 1 } \\phi ( r a _ { 1 } ) n \\\\ & = r a _ { 1 } n = \\alpha _ { 1 } \\xi _ { r a _ { 1 } } ( n ) \\end{align*}"}
-{"id": "3955.png", "formula": "\\begin{align*} V = \\bigoplus _ { \\lambda \\in \\R } V ^ { \\lambda } ( A ) V ^ { \\lambda } ( A ) : = \\{ v \\in V : a ( t ) v = e ^ { \\lambda t } v : \\forall t \\in \\R \\} . \\end{align*}"}
-{"id": "7061.png", "formula": "\\begin{align*} { x ^ { [ 2 ] } } ( { t _ 2 } ) = \\frac { { { h ^ { [ 2 2 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 2 2 ] } } ( { t _ 2 } - 2 ) } } u _ 1 ^ { [ 2 ] } , { x ^ { [ 3 ] } } ( { t _ 2 } ) = \\frac { { { h ^ { [ 2 3 ] } } ( { t _ 1 } ) } } { { { h ^ { [ 2 3 ] } } ( { t _ 2 } - 2 ) } } u _ 1 ^ { [ 3 ] } . \\end{align*}"}
-{"id": "7716.png", "formula": "\\begin{align*} \\lim _ { n \\to \\infty } \\sigma _ n = 0 . \\end{align*}"}
-{"id": "4617.png", "formula": "\\begin{align*} \\sum _ { { t = 3 } \\atop { t \\neq 5 , 6 } } ^ { 8 } f _ t ( p ' ) = q ^ 3 ( q ^ 2 - q + 1 ) \\left ( \\frac { ( q ^ 2 - 1 ) r } { | G _ { p ' } | } - k _ 5 - k _ 6 \\right ) . \\end{align*}"}
-{"id": "2640.png", "formula": "\\begin{align*} \\Delta _ D ( F , G ) = - \\Delta ( h ( F ^ 0 , G ^ 0 ) ; F , G ) + \\delta ( ( F , G ) | D ) \\rightarrow i n f , \\end{align*}"}
-{"id": "9292.png", "formula": "\\begin{align*} \\mathbf { H } ^ 1 ( \\mathbb { R } ^ + ) = \\{ y \\in \\mathbf { L } ^ 2 ( \\mathbb { R } ^ + ) , \\frac { \\partial y } { \\partial x } \\in \\mathbf { L } ^ 2 ( \\mathbb { R } ^ + ) \\} \\end{align*}"}
-{"id": "4113.png", "formula": "\\begin{align*} ( p , q , r ) = ( p _ 0 , q _ 0 + p _ 0 u , r _ 0 + p _ 0 v ) , \\end{align*}"}
-{"id": "8156.png", "formula": "\\begin{align*} S _ 2 ^ 2 = m _ 1 * [ a ^ { d _ 1 } ] \\cup m _ 2 * [ a ^ { d _ 2 } ] \\cup \\cdots \\cup m _ k * [ a ^ { d _ k } ] , \\end{align*}"}
-{"id": "9184.png", "formula": "\\begin{align*} \\mathcal A f ^ D _ u = ( \\mathcal A ^ \\sharp f _ u ) \\mid _ D , \\end{align*}"}
-{"id": "4601.png", "formula": "\\begin{align*} | x _ { 1 2 } | = \\max _ { 1 \\le i , j \\le 2 n } | x _ { i j } | . \\end{align*}"}
-{"id": "1307.png", "formula": "\\begin{align*} S & : = \\sum _ { \\rho } \\frac { ( N + H ) ^ { \\rho + 2 } - 2 N ^ { \\rho + 2 } + ( N - H ) ^ { \\rho + 2 } } { \\rho ( \\rho + 1 ) ( \\rho + 2 ) } . \\end{align*}"}
-{"id": "2755.png", "formula": "\\begin{align*} \\mu _ n ( t _ k ) \\bigl | Y ^ n _ { t _ k } - Z _ { t _ k } \\bigr | ^ 2 \\leq \\mu _ n ( t _ { k - 1 } ) \\bigl | Y ^ n _ { t _ { k - 1 } } - Z _ { t _ { k - 1 } } \\bigr | ^ 2 + \\sum _ { i = 1 } ^ { 1 1 } I _ i ( t _ k ) . \\end{align*}"}
-{"id": "7825.png", "formula": "\\begin{align*} Y _ { t } = g ( \\eta _ { T } ) + \\int _ t ^ T E ' [ f ( s , \\eta _ { s } , y ' _ { s } , z ' _ { s } , y _ { s } , z _ { s } ) ] d s - \\int _ t ^ T Z _ { s } d B _ { s } ^ { H } , \\ \\ 0 \\leq t \\leq T . \\end{align*}"}
-{"id": "4686.png", "formula": "\\begin{align*} \\lambda . ( t , z ) = ( \\lambda ^ 2 t , \\lambda z ) , \\end{align*}"}
-{"id": "110.png", "formula": "\\begin{align*} I _ { - 1 } ( f _ n ) + ( - 1 ) ^ { n - 1 } I _ { - 1 } ( f ) = 2 \\sum _ { l = 0 } ^ { n - 1 } ( - 1 ) ^ { n - 1 - l } f ( l ) , ( \\textnormal { s e e } \\ , \\ , [ 8 ] ) . \\end{align*}"}
-{"id": "9550.png", "formula": "\\begin{align*} \\mathcal { W } _ { i , j } = \\{ g _ F \\left ( g _ F ^ { - 1 } \\pi ^ { - 1 } ( F ) \\cap { A } \\right ) \\mid F \\in \\mathcal { F } _ i , A \\in \\mathcal { A } _ j \\} \\end{align*}"}
-{"id": "2884.png", "formula": "\\begin{align*} C ^ * _ { G S } ( B , B ) \\stackrel { \\sim } { \\leftarrow } \\bullet \\stackrel { \\sim } { \\rightarrow } g _ { E _ 2 ^ + , \\tilde { \\Omega } B } ^ { \\psi ^ + } , \\end{align*}"}
-{"id": "1735.png", "formula": "\\begin{gather*} \\nabla _ { \\partial _ x } \\partial _ x = 3 x y ^ 2 \\partial _ x + x ^ 3 \\partial _ y , \\nabla _ { \\partial _ x } \\partial _ y = \\nabla _ { \\partial _ y } \\partial _ x = 0 , \\nabla _ { \\partial _ y } \\partial _ y = x ^ 3 \\partial _ x - 3 x ^ 2 y \\partial _ y . \\end{gather*}"}
-{"id": "7072.png", "formula": "\\begin{align*} \\forall n \\geq 0 , W _ n = \\sum _ { | u | = n } e ^ { - V ( u ) } Z _ n = \\sum _ { | u | = n } V ( u ) e ^ { - V ( u ) } . \\end{align*}"}
-{"id": "6924.png", "formula": "\\begin{align*} s ^ { ( \\pi ) } _ \\lambda ( X ) & = s _ \\lambda / L _ { \\pi } ( X ) = L _ { \\pi } ^ \\perp ( X ) s _ \\lambda \\ , , s _ \\lambda ( X ) = s ^ { ( \\pi ) } _ \\lambda / M _ { \\pi } ( X ) = M _ { \\pi } ^ \\perp ( X ) s ^ { ( \\pi ) } _ \\lambda \\ , , \\end{align*}"}
-{"id": "2228.png", "formula": "\\begin{align*} \\varphi ( t ) \\frac { d } { d t } ( k * v ) ( t ) = \\frac { d } { d t } ( k * [ \\varphi v ] ) ( t ) + \\int _ { 0 } ^ { t } \\dot { k } ( t - \\sigma ) ( \\varphi ( t ) - \\varphi ( \\sigma ) ) v ( \\sigma ) d \\sigma , \\end{align*}"}
-{"id": "9098.png", "formula": "\\begin{align*} \\xi _ { N } ( s ) = \\int _ { 0 } ^ { \\infty } e ^ { s x } x ^ { 2 } x ^ { \\alpha } e ^ { - x } ( L _ { N - 1 } ( x ) L ' _ { N - 2 } ( x ) - L _ { N - 1 } ' ( x ) L _ { N - 2 } ( x ) ) \\ , \\mathrm { d } x \\ . \\end{align*}"}
-{"id": "5356.png", "formula": "\\begin{align*} \\frac { 1 } { | G | } \\prod _ { i = 1 } ^ { \\ell } ( n - \\chi _ { \\gamma } ( c _ i ) ) \\end{align*}"}
-{"id": "8018.png", "formula": "\\begin{align*} \\deg x = \\deg y = 1 \\mbox { f o r a l l } \\ ; \\ ; x \\in x ( A ) , y \\in y ( A ) . \\end{align*}"}
-{"id": "419.png", "formula": "\\begin{align*} | I I | \\lesssim \\sum _ { n = N + 1 } ^ \\infty \\gamma ^ { - n \\alpha } = \\frac { \\gamma ^ { - ( N + 1 ) \\alpha } } { 1 - \\gamma ^ { - \\alpha } } = O ( h ^ \\alpha ) . \\end{align*}"}
-{"id": "7976.png", "formula": "\\begin{align*} 1 _ J = \\frac { 1 } { | W _ L | } \\sum \\limits _ { w \\in W _ L } R _ { T _ w } ^ L ( 1 ) \\end{align*}"}
-{"id": "2781.png", "formula": "\\begin{align*} \\mathrm { S I R } _ { \\mathrm { U } _ { 0 } } = \\frac { P _ { \\mathrm { B S } } g _ { n } \\left ( d _ { 0 } \\right ) H _ { \\mathrm { U } _ { 0 } , \\mathrm { B S } _ { 0 } } } { \\underset { \\mathrm { B S } _ { i } \\in \\Pi _ { \\mathrm { B S } } ^ { \\dagger } } { \\sum } P _ { \\mathrm { B S } } g _ { n } \\left ( d _ { i } \\right ) H _ { \\mathrm { U } _ { 0 } , \\mathrm { B S } _ { i } } } , \\ : n \\in \\left \\{ 1 , 2 \\right \\} \\end{align*}"}
-{"id": "9261.png", "formula": "\\begin{align*} \\begin{cases} d Y ( t , x , z ) = d Y ^ { \\pi } ( t , x , z ) = [ \\frac { 1 } { 2 } { \\frac { \\partial ^ 2 } { \\partial x ^ 2 } Y ( t , x , z ) + \\pi ( t , z ) Y ( t , x , z ) a _ 0 ( t , z ) } ] d t + \\pi ( t , z ) Y ( t , x , z ) b _ 0 ( t , z ) d B ( t ) ; t \\in [ 0 , T ] \\\\ Y ( 0 , x , z ) = \\alpha ( x ) > 0 ; x \\in D , \\end{cases} \\end{align*}"}
-{"id": "4023.png", "formula": "\\begin{align*} F ( x ) = y , \\end{align*}"}
-{"id": "1050.png", "formula": "\\begin{align*} \\sum _ { j = 0 } ^ \\infty \\int _ { \\R ^ 2 } | ( Q ^ j \\ast 1 _ E ) ( x ) | 1 _ A ( x ) \\ , d x & \\leq \\sum _ { j = 0 } ^ M \\| Q ^ j \\ast 1 _ E \\| _ 2 | A | ^ { \\frac 1 2 } + \\sum _ { j = M + 1 } ^ \\infty \\| Q ^ j \\ast 1 _ E \\| _ \\infty | A | \\\\ & \\leq C \\left \\{ 2 ^ { \\frac { M } 2 } | E | ^ { \\frac 5 6 } | A | ^ { \\frac 1 2 } + 2 ^ { - \\frac { M + 1 } 2 } | E | | A | \\right \\} . \\end{align*}"}
-{"id": "5797.png", "formula": "\\begin{align*} Q _ { i } \\cdot G = 0 \\mathrm { f o r } \\ i = 1 , \\ldots , n - 1 . \\end{align*}"}
-{"id": "729.png", "formula": "\\begin{align*} & A = ( a _ { m i } ) _ { m i } \\ \\ \\ s \\times r \\\\ & B = ( b _ { l j } ) _ { l j } \\ \\ r \\times s . \\end{align*}"}
-{"id": "2158.png", "formula": "\\begin{align*} Q _ { - } ( t _ { 0 } , x _ { 0 } , r ) & = ( t _ { 0 } , t _ { 0 } + \\delta \\tau r ^ { 2 \\beta / \\alpha } ) \\times B ( x _ { 0 } , \\delta r ) , \\\\ Q _ { + } ( t _ { 0 } , x _ { 0 } , r ) & = ( t _ { 0 } + ( 2 - \\delta ) \\tau r ^ { 2 \\beta / \\alpha } , t _ { 0 } + 2 \\tau r ^ { 2 \\beta / \\alpha } ) \\times B ( x _ { 0 } , \\delta r ) . \\end{align*}"}
-{"id": "9568.png", "formula": "\\begin{align*} \\Delta = \\left ( 1 + \\sum _ { i = 1 } ^ { \\infty } t ^ { i } f ^ { ( i ) } \\right ) \\Delta ^ { ( 0 ) } . \\end{align*}"}
-{"id": "7962.png", "formula": "\\begin{align*} \\mu ( E ) & = \\inf \\{ \\mu ( O ) ; \\ ; O \\supset E , \\ ; O X \\} , \\\\ \\mu ( E ) & = \\sup \\{ \\mu ( K ) ; \\ ; K \\subset E , \\ ; K X \\} . \\end{align*}"}
-{"id": "9678.png", "formula": "\\begin{align*} \\lim _ { t \\to \\infty } \\frac { \\log x ( t ) } { \\log t } = - \\frac { 1 } { \\beta } \\frac { 1 } { \\log ( 1 / ( 1 - q ) ) } \\log \\left ( \\frac { a } { b } \\right ) . \\end{align*}"}
-{"id": "7712.png", "formula": "\\begin{align*} \\mathbb P \\left \\{ \\omega \\in \\Omega : \\xi _ i \\in [ 1 - \\varepsilon , 1 ] , i = n , n + 1 , \\dots , n + J . \\right \\} \\ge p _ \\varepsilon ^ J , \\end{align*}"}
-{"id": "8208.png", "formula": "\\begin{align*} G ( T , U ) F ( T , U ) + H ( T , U ) F ' _ U ( T , U ) = R ( T ) . \\end{align*}"}
-{"id": "6403.png", "formula": "\\begin{align*} \\Delta H _ { q , p } ( 2 ) = \\log \\left ( 1 \\pm C ( \\vec { \\varepsilon } ) \\left | \\Delta _ { 0 } ( 2 ) \\right | \\right ) \\pm \\frac { 1 } { 2 \\sigma ^ { 2 } } \\left | \\Delta \\mu _ { ( 2 ) } ( 2 ) \\right | \\pm \\varepsilon _ { q } \\left | \\Delta \\mu _ { ( q ) } ( 2 ) \\right | \\pm \\varepsilon _ { p } \\left | \\Delta \\mu _ { ( p ) } ( 2 ) \\right | . \\end{align*}"}
-{"id": "7892.png", "formula": "\\begin{align*} \\left \\| E _ 1 - S _ 1 \\right \\| ^ 2 t ^ 2 + 2 ( I - P ) \\cdot ( E _ 1 - S _ 1 ) t + \\left \\| I - P \\right \\| ^ 2 - r ^ 2 = 0 \\end{align*}"}
-{"id": "152.png", "formula": "\\begin{align*} \\| x \\| _ { \\widehat { C _ E } } = \\lim _ { n \\to \\infty } \\| V a _ n \\| _ { \\widehat { C _ E } } = \\lim _ { n \\to \\infty } \\| V a _ n \\| _ { C _ { \\widehat { E } } } = \\| x \\| _ { C _ { \\widehat { E } } } . \\end{align*}"}
-{"id": "2873.png", "formula": "\\begin{align*} ( - ) _ { + } : X \\in C h _ { \\mathbb { K } } \\mapsto X _ { + } = X \\oplus \\mathbb { K } \\in C h _ { \\mathbb { K } } ^ { p t } \\end{align*}"}
-{"id": "1519.png", "formula": "\\begin{align*} \\phi ( f ) ( K ) = \\phi _ { V } \\big ( ( f - \\psi ( f | _ C ) ) | _ V \\big ) ( K ) . \\end{align*}"}
-{"id": "2338.png", "formula": "\\begin{align*} \\limsup _ { \\| y \\| \\to 0 } \\frac { 1 } { n } \\sum _ { i = 1 } ^ n \\frac { \\| x _ i + y \\| + \\| x _ i - y \\| - 2 } { \\| y \\| } \\geq \\delta . \\end{align*}"}
-{"id": "6282.png", "formula": "\\begin{align*} W _ { p , l } ( x ) = \\frac { p ( x ) } { p _ n l ' ( x ) } , \\end{align*}"}
-{"id": "8423.png", "formula": "\\begin{align*} \\textbf { u } ( 0 , x ) = \\textbf { u } _ { 0 } ( x ) = ( \\rho _ { 0 } ( x ) , v _ { 0 } ( x ) ) ^ { T } , \\end{align*}"}
-{"id": "7115.png", "formula": "\\begin{align*} \\xi ( u ) = \\log \\sum _ { | v | = | u | + 1 , v > u } ( 1 + ( V ( v ) - V ( u ) ) _ + ) e ^ { V ( u ) - V ( v ) } . \\end{align*}"}
-{"id": "5496.png", "formula": "\\begin{align*} \\mathbf { R } ^ \\mathrm { C } [ \\iota ] = \\sqrt { \\rho _ \\mathrm { s } } \\mathbf { G } _ \\mathrm { s } [ \\iota ] \\mathbf { S } ^ \\mathrm { C } [ \\iota ] + \\mathbf { L } ^ \\mathrm { C } [ \\iota ] + \\mathbf { N } ^ \\mathrm { C } [ \\iota ] \\end{align*}"}
-{"id": "8990.png", "formula": "\\begin{align*} ( ~ ^ { A B } I _ b ^ \\alpha ( L _ p ) = \\{ f : f = ~ ^ { A B } I _ b ^ \\alpha \\phi , ~ ~ \\phi \\in L _ p ( a , b ) \\} . \\end{align*}"}
-{"id": "9406.png", "formula": "\\begin{align*} S _ I ^ T A S _ I = \\begin{bmatrix} D _ I & & \\\\ & B _ { F F } & A _ { R F } ^ T \\\\ & A _ { R F } & A _ { R R } \\\\ \\end{bmatrix} , \\end{align*}"}
-{"id": "6208.png", "formula": "\\begin{align*} X : = X _ \\R \\oplus i X _ \\R . \\end{align*}"}
-{"id": "7912.png", "formula": "\\begin{align*} 6 \\Delta _ { g _ 0 } u + S _ g u ^ 3 = S _ 0 u . \\end{align*}"}
-{"id": "3170.png", "formula": "\\begin{align*} & \\chi ( p ; Z _ { t ^ n } ^ { 0 } ) \\\\ & = S ( { V } _ { t ^ n } ^ { 0 } ( p ^ n ) ) - \\sum _ { a ^ n \\in \\mathsf { A } ^ n } p ^ n ( a ^ n ) S ( { V } _ { t ^ n } ^ { 0 } ( a ^ n ) ) \\\\ & = S ( { V } _ { t ^ n } ^ { 0 } ( 0 ^ n ) ) - \\sum _ { a ^ n \\in \\mathsf { A } ^ n } p ^ n ( a ^ n ) S ( { V } _ { t ^ n } ^ { 0 } ( 0 ^ n ) ) \\\\ & = 0 \\end{align*}"}
-{"id": "4072.png", "formula": "\\begin{gather*} ( p , q , r ) = ( 6 , 6 u + 3 , 6 v + 2 ) , \\ , u , v \\geq 0 , \\\\ ( p , q , r ) = ( 6 , 6 u + 3 , 6 v + 4 ) , \\ , u , v \\geq 0 . \\end{gather*}"}
-{"id": "2796.png", "formula": "\\begin{align*} \\| \\phi \\| _ m = \\min _ { c \\in \\mathbb { R } } \\ , \\sum _ { j = 1 } ^ n | \\phi ( j ) - c | . \\end{align*}"}
-{"id": "4573.png", "formula": "\\begin{align*} \\mathcal { J } _ 3 \\ ! = \\ ! \\bigl \\{ & \\emptyset , \\{ 1 \\} , \\{ 2 \\} , \\{ 1 , 1 \\} , \\{ 1 , 2 \\} , \\{ 2 , 1 \\} , \\{ 2 , 2 \\} , \\\\ & \\{ 1 , 1 , 1 \\} , \\ ! \\{ 1 , 1 , 2 \\} , \\ ! \\{ 1 , 2 , 1 \\} , \\ ! \\{ 1 , 2 , 2 \\} , \\ ! \\{ 2 , 1 , 1 \\} , \\ ! \\{ 2 , 1 , 2 \\} , \\ ! \\{ 2 , 2 , 1 \\} , \\ ! \\{ 2 , 2 , 2 \\} \\bigr \\} . \\end{align*}"}
-{"id": "5241.png", "formula": "\\begin{align*} \\begin{cases} - \\nabla \\cdot ( \\sigma \\nabla u ) = f ^ + \\geq 0 & \\mbox { i n } \\Omega \\\\ u = 0 & \\mbox { o n } \\partial \\Omega , \\\\ | \\nabla u | \\leq 1 & \\mbox { i n } \\Omega , \\\\ | \\nabla u | = 1 & \\sigma - \\mbox { a . e . } \\end{cases} \\end{align*}"}
-{"id": "3601.png", "formula": "\\begin{align*} \\Phi ( g + h , \\pi + w ) & = \\Phi ( g , \\pi ) + D \\Phi | _ { ( g , \\pi ) } ( h , w ) + Q _ { ( g , \\pi ) } ( h , w ) . \\end{align*}"}
-{"id": "9514.png", "formula": "\\begin{align*} \\delta \\left ( \\mathrm { d v o l } _ g e ^ { - 2 \\phi } \\left ( R + 4 | d \\phi | ^ 2 - \\frac { 1 } { 2 } | H _ 3 | ^ 2 \\right ) \\right ) & = \\delta ( e ^ { - 2 \\phi } ) \\left ( \\mathrm { d v o l } _ g \\left ( R + 4 | d \\phi | ^ 2 - \\frac { 1 } { 2 } | H _ 3 | ^ 2 \\right ) \\right ) \\\\ & \\quad + 4 e ^ { - 2 \\phi } \\delta ( d \\phi \\wedge * d \\phi ) \\\\ & = - 2 e ^ { - 2 \\phi } \\delta \\phi \\left ( \\mathrm { d v o l } _ g \\left ( R + 4 | d \\phi | ^ 2 - \\frac { 1 } { 2 } | H _ 3 | ^ 2 \\right ) \\right ) \\\\ & + 8 e ^ { - 2 \\phi } d \\delta \\phi \\wedge * d \\phi . \\end{align*}"}
-{"id": "5785.png", "formula": "\\begin{align*} \\int a ( z ) \\ d z = a _ { ( 0 ) } \\end{align*}"}
-{"id": "5654.png", "formula": "\\begin{align*} R g ( x ) : = \\sum _ { k = 0 } ^ { \\infty } \\frac { M ^ k g ( x ) } { ( 2 A ) ^ k } ( g \\in X ' ) , \\end{align*}"}
-{"id": "3727.png", "formula": "\\begin{align*} W _ { h _ t } ^ { ( F , i ) } ( f ) ( x ) = s ( { \\det h _ t \\vert _ { F ^ - } } ) f ( x t ) . \\end{align*}"}
-{"id": "2700.png", "formula": "\\begin{align*} \\theta _ { \\varphi } ^ n = e ^ { \\varphi - u } \\theta _ { u } ^ n + e ^ { \\varphi - v } \\theta _ v ^ n . \\end{align*}"}
-{"id": "1415.png", "formula": "\\begin{align*} \\C ^ { [ \\beta ] } = \\bigoplus _ { \\alpha \\in [ \\beta ] } \\C x _ { \\alpha } \\end{align*}"}
-{"id": "1649.png", "formula": "\\begin{align*} [ f ] _ { B ^ { s } _ { p , q } } = \\sum _ { i = 1 } ^ d \\Big ( \\int _ { \\R ^ d } \\frac { \\dd h } { | h | ^ { d + s q } } \\Big ( \\int _ { \\R ^ d } | \\partial _ { x _ i } f ( x + h ) - \\partial _ { x _ i } f ( x ) | ^ p \\dd x \\Big ) ^ { q / p } \\Big ) ^ { 1 / q } < \\infty \\ , . \\end{align*}"}
-{"id": "4630.png", "formula": "\\begin{align*} & \\sum \\limits _ { k = 0 } ^ { m - 1 } k \\binom { n } { k } \\ ; = \\ ; \\frac { n } { 2 } \\binom { n } { < m } \\ ; - \\ ; \\frac { m } { 2 } \\binom { n } { m } \\\\ & \\sum \\limits _ { k = 0 } ^ { m - 1 } k ( k - 1 ) \\binom { n } { k } \\ ; = \\ ; \\frac { n ( n - 1 ) } { 4 } \\binom { n } { < m } \\ ; - \\ ; \\frac { m ( 2 m + n - 3 ) } { 4 } \\binom { n } { m } , \\end{align*}"}
-{"id": "4905.png", "formula": "\\begin{align*} \\quad \\mathcal { F } ( z ) = \\left ( z - \\overline { \\mathfrak { z } } \\right ) ^ { - \\kappa } \\left ( \\sum _ { n \\gg - \\infty } a _ { \\mathcal { F } , \\mathfrak { z } } ^ + ( n ) X _ { \\mathfrak { z } } ^ n ( z ) + \\sum _ { n \\ll \\infty } a _ { \\mathcal { F } , \\mathfrak { z } } ^ - ( n ) \\beta _ 0 \\left ( 1 - r _ { \\mathfrak { z } } ^ 2 ( z ) ; 1 - \\kappa , - n \\right ) X _ { \\mathfrak { z } } ^ n ( z ) \\right ) . \\end{align*}"}
-{"id": "8046.png", "formula": "\\begin{align*} \\vec { a } ^ { \\vec { x } } a _ j ^ { - k } & = s _ { 1 , 2 } ^ { x _ 1 } s _ { 1 , 3 } ^ { x _ 2 } \\cdots s _ { 1 , j + 1 } ^ { x _ j } \\cdots s _ { 2 , m + 2 } ^ { x _ { m + 1 } } \\cdots s _ { 1 , m + 2 } ^ { x _ { 2 m + 1 } } \\cdot s _ { 1 , j + 1 } ^ { - k } \\\\ & = s _ { 1 , 2 } ^ { y _ 1 } s _ { 1 , 3 } ^ { x _ 2 } \\cdots s _ { 1 , j + 1 } ^ { x _ j - k } \\cdots s _ { 2 , m + 2 } ^ { x _ { m + 1 } } \\cdots s _ { 1 , m + 2 } ^ { x _ { 2 m + 1 } + k x _ { m + j } } . \\end{align*}"}
-{"id": "27.png", "formula": "\\begin{align*} p ^ \\star ( \\Sigma ) = \\sum _ { j \\in J _ v } b ^ j Z _ j + \\sum _ { j \\in J _ h } b ^ j Z _ j \\end{align*}"}
-{"id": "1962.png", "formula": "\\begin{align*} & \\epsilon _ i a _ i < \\lambda _ i < \\epsilon _ { i - 1 } a _ { i - 1 } ~ ~ i = 2 , \\cdots m , \\\\ & \\min _ { i } \\{ \\epsilon _ i a _ i \\} + \\sum _ { i = 1 } ^ m a _ i \\leq \\lambda _ 1 \\leq \\max _ { i } \\{ \\epsilon _ i a _ i \\} + \\sum _ { i = 1 } ^ m a _ i . \\end{align*}"}
-{"id": "9640.png", "formula": "\\begin{align*} \\omega _ { j , 1 } = \\frac { h _ { j + 1 } } { h _ j } , \\ldots , \\omega _ { j , k - 1 } = \\frac { h _ { j + k - 1 } } { h _ { j + k - 2 } } . \\end{align*}"}
-{"id": "4042.png", "formula": "\\begin{align*} p + q + r = m + n + l , \\end{align*}"}
-{"id": "7165.png", "formula": "\\begin{align*} \\sum _ { l < n } b _ l = 0 . \\end{align*}"}
-{"id": "4411.png", "formula": "\\begin{align*} C _ E = \\{ x \\in K ( H ) : \\ , S ( x ) \\in E \\} , \\end{align*}"}
-{"id": "4995.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n } ( x ) ) & = \\sum _ { k = 0 } ^ { n - 1 } \\delta _ { f } ^ { n - 1 - k } \\bigl ( h _ { H } ( f ^ { k + 1 } ( P ) ) - \\delta _ { f } h _ { H } ( f ^ { k } ( P ) ) \\bigr ) + \\delta _ { f } ^ { n } h _ { H } ( P ) \\\\ & = \\sum _ { k = 0 } ^ { n - 1 } \\delta _ { f } ^ { n - 1 - k } B ( f ^ { k } ( P ) ) + \\delta _ { f } ^ { n } h _ { H } ( P ) . \\end{align*}"}
-{"id": "7805.png", "formula": "\\begin{align*} \\Delta _ { f } w & \\geq - c _ { k } \\left \\vert \\nabla \\ln S \\right \\vert ^ { 2 } w - c \\Sigma _ { j = 0 } ^ { k } \\left \\vert \\nabla ^ { j } \\mathrm { R m } \\right \\vert \\left \\vert \\nabla ^ { k - j } \\mathrm { R m } \\right \\vert \\left \\vert \\nabla ^ { k } \\mathrm { R m } \\right \\vert S ^ { - k - 2 } \\\\ & \\geq - c _ { k } S ^ { \\frac { 1 } { 2 } } . \\end{align*}"}
-{"id": "9884.png", "formula": "\\begin{align*} & J _ 1 ^ 0 = \\int _ 0 ^ 1 G _ t ( s , y ) \\xi ( y ) d y , \\\\ & J _ 2 ^ 0 = 0 , \\\\ & J _ 3 ^ 0 = - \\int _ 0 ^ t \\int _ 0 ^ 1 \\partial _ y G _ { t - s } ( x , y ) g ( s , V ^ { { 0 } , v } _ { \\eta } ( s ) ) ( y ) d y d s , \\\\ & J _ 4 ^ 0 = \\int _ 0 ^ t \\int _ 0 ^ 1 G _ { t - s } ( x , y ) f ( s , V ^ { { 0 } , v } _ { \\eta } ( s ) ) ( y ) d y d s , \\\\ & J _ 5 ^ 0 = \\int _ 0 ^ t \\int _ 0 ^ 1 G _ { t - s } ( x , y ) \\sigma ( s , V ^ { { 0 } , v } _ { \\eta } ( s ) ) ( y ) v ( s , y ) d y d s . \\end{align*}"}
-{"id": "1222.png", "formula": "\\begin{align*} \\| x \\| _ \\mathcal { M } = \\| x \\| _ { \\mathcal { M } _ { \\varphi , w } } = \\inf \\{ \\epsilon > 0 : P ( x / \\epsilon ) \\le 1 \\} . \\end{align*}"}
-{"id": "4647.png", "formula": "\\begin{align*} E ( z , g ) & = t ^ { z + 1 } + t ^ { 1 - z } \\frac { \\xi ( z ) } { \\xi ( z + 1 ) } \\\\ & \\qquad + \\frac { 4 t } { \\xi ( z + 1 ) } \\sum _ { m = 1 } ^ \\infty \\eta _ { \\frac { z } { 2 } } ( m ) K _ { \\frac { z } { 2 } } ( 2 \\pi m t ^ 2 ) \\cos 2 \\pi m u , \\\\ \\eta _ { \\frac { z } { 2 } } ( m ) & = \\sum _ { a b = m } \\left ( \\frac { a } { b } \\right ) ^ { \\frac { z } { 2 } } . \\end{align*}"}
-{"id": "1003.png", "formula": "\\begin{align*} \\eta ^ 2 \\left ( v _ i + \\omega + \\eta \\sum _ { i = 1 } ^ { n - 1 } l _ i \\right ) \\left ( v _ i - \\omega + \\eta \\sum _ { i = 1 } ^ { m - 1 } k _ i \\right ) & = \\prod _ { j \\neq i } ^ { N - r } \\frac { v _ i - v _ j - \\eta } { v _ i - v _ j + \\eta } , r < N . \\end{align*}"}
-{"id": "2703.png", "formula": "\\begin{align*} \\theta _ { \\varphi _ j } ^ n = e ^ { \\varphi _ j } \\mu _ j . \\end{align*}"}
-{"id": "3852.png", "formula": "\\begin{align*} a _ { j , 1 } ( \\tau , \\xi ) = & | \\xi | ^ { - \\varpi } \\varphi _ j ( \\xi ) | \\tau | ^ \\beta \\int _ { - \\pi } ^ { \\pi } \\ ^ { \\chi } _ 1 ( \\tau - | \\xi | \\sin \\theta ) e ^ { - i \\theta \\varpi } d \\theta , \\\\ a _ { j , 2 } ( \\tau , \\xi ) = & | \\xi | ^ { - \\varpi } \\varphi _ j ( \\xi ) | \\tau | ^ \\beta \\int ^ \\infty _ 0 \\ ^ { \\chi _ 2 } \\bigl ( \\tau - i | \\xi | \\sinh s \\bigr ) e ^ { - \\varpi s } d s . \\end{align*}"}
-{"id": "8615.png", "formula": "\\begin{align*} \\psi \\overline { \\overline { N } } ( a ) = \\alpha \\xi _ { e _ { k } a } \\end{align*}"}
-{"id": "1007.png", "formula": "\\begin{align*} [ H _ { n , m } , Q _ { j } ] = [ H _ { n , m } , \\overline { Q } _ { j } ] = [ N , Q _ { j } ] = [ N , \\overline { Q } _ { j } ] & = 0 \\\\ [ Q _ { j } , { Q } _ { k } ] = [ Q _ { j } , \\overline { Q } _ { k } ] = [ \\overline { Q } _ { j } , \\overline { Q } _ { k } ] & = 0 , \\end{align*}"}
-{"id": "5040.png", "formula": "\\begin{align*} \\int _ G \\nu ( ( g ^ { - 1 } \\cdot \\phi ) \\chi _ C ) \\ , d \\eta ( g ) = \\nu ( \\phi ) \\nu ( C ) , \\end{align*}"}
-{"id": "6298.png", "formula": "\\begin{align*} \\Vert \\{ \\lambda _ { k } \\} \\Vert _ { l _ { q } ^ { s , \\alpha } } = \\begin{cases} \\left ( \\sum _ { k \\in \\mathbb { Z } ^ { n } } ^ { { } } \\langle k \\rangle ^ { \\frac { s q } { 1 - \\alpha } } | \\lambda _ { k } | ^ { q } \\right ) ^ { \\frac { 1 } { q } } ~ & ~ 0 < q < \\infty , \\\\ \\sup _ { k \\in \\mathbb { Z } ^ { n } } \\left ( \\langle k \\rangle ^ { \\frac { s } { 1 - \\alpha } } | \\lambda _ { k } | \\right ) ~ & ~ q = \\infty . \\end{cases} \\end{align*}"}
-{"id": "814.png", "formula": "\\begin{align*} \\prod _ { i = 1 } ^ { k } \\frac { z _ { i } ^ { M ' - 1 } } { ( 1 + z _ { i } ) ^ { M } } \\langle \\prod _ { 1 \\le i \\le k } ^ { \\curvearrowright } C _ { \\mu _ { i } } ^ { [ M , M ' ] } ( z _ { i } ) \\prod _ { 1 \\le i \\le k } \\beta _ { \\nu _ { i } , x _ { i } } ^ { * } \\rangle _ { [ M ' , M ] } . \\end{align*}"}
-{"id": "6157.png", "formula": "\\begin{align*} g ( ( x _ 1 ^ v \\cdots x _ n ^ v ) ^ a ) = _ { \\Bbbk ^ { \\times } } ( x _ 1 ^ v \\cdots x _ n ^ v ) ^ a , { } \\end{align*}"}
-{"id": "4830.png", "formula": "\\begin{align*} \\limsup _ { n \\to \\infty } | [ ( I - A _ n ) R , \\tau ] | _ { \\mathcal I } = 0 \\end{align*}"}
-{"id": "2593.png", "formula": "\\begin{align*} v ^ { [ t _ n ] } ( 1 , x ) = t _ n ^ { \\frac 1 { p } } v ( t _ n , \\sqrt { t _ n } x ) = [ e ^ { i \\Delta } \\psi ( t _ n ) ] ( x ) . \\end{align*}"}
-{"id": "6585.png", "formula": "\\begin{align*} P _ { \\gamma f } ^ { \\alpha } ( x ) : = \\nabla p _ { \\gamma f } ^ { \\alpha } ( x ) = \\alpha { \\rm { p r o x } } _ { \\gamma f } ( x ) + ( 1 - \\alpha ) x . \\end{align*}"}
-{"id": "1573.png", "formula": "\\begin{align*} X _ { L } = \\left ( c _ { 1 } + 2 \\psi _ { I } \\int T ^ { I } \\left ( t \\right ) d t \\right ) + \\left ( T ^ { I } \\left ( t \\right ) Y _ { I } ^ { \\alpha } \\left ( x ^ { \\beta } \\right ) \\right ) \\partial _ { \\alpha } + \\left ( a \\left ( x ^ { \\beta } , t \\right ) u + b \\left ( x ^ { \\beta } , t \\right ) + c _ { 2 } u \\right ) \\partial _ { u } , \\end{align*}"}
-{"id": "6364.png", "formula": "\\begin{align*} x ( z _ 1 , z _ 2 ) = z _ 1 ^ 2 - z _ 2 ^ 2 \\qquad y ( z _ 1 , z _ 2 ) = 2 z _ 1 z _ 2 , \\end{align*}"}
-{"id": "2541.png", "formula": "\\begin{align*} \\lim _ { N \\rightarrow \\infty } \\frac 1 { N } \\sum _ { n = 0 } ^ { N - 1 } f ( U ^ n ) = \\int _ { \\mathcal { S } } f d \\mu _ h ^ { \\tau } , \\quad \\forall ~ f \\in B _ b ( \\mathcal { S } ) , \\quad \\ ; \\ ; L ^ 2 ( \\mathcal { S } , \\mu _ h ^ { \\tau } ) . \\end{align*}"}
-{"id": "4131.png", "formula": "\\begin{align*} p + q + \\gcd ( 2 q , - p ) + \\gcd ( q , - 2 p ) = 0 , \\end{align*}"}
-{"id": "5004.png", "formula": "\\begin{align*} h _ { H } ( f ^ { n } ( P ) ) = \\delta _ { f } ^ { n } h _ { H } ( P ) + \\sum _ { k = 0 } ^ { n - 1 } \\delta _ { f } ^ { n - 1 - k } h _ { E ' _ { 1 } } ( f ^ { k } ( P ) ) - \\sum _ { k = 0 } ^ { n - 1 } \\delta _ { f } ^ { n - 1 - k } h _ { Z _ { 1 } } ( p ^ { - 1 } f ^ { k } ( P ) ) . \\end{align*}"}
-{"id": "9070.png", "formula": "\\begin{align*} & \\P ( G _ 0 v _ { | | E | | ^ { 1 / 3 } } ( d , 0 , E + 2 | E | ) ) \\\\ & \\leq \\P ( G _ 0 v _ { | | E | | _ 1 ^ { 1 / 3 } } ( d , | | E | | _ 1 ^ { 1 / 3 } , E ) ) + \\mathcal { O } _ { d , \\upsilon } \\left ( \\sum _ { k = 2 } ^ { \\infty } N ^ { - 3 / 2 } ( 1 + k ^ 3 ) e ^ { - \\frac { ( k - 1 ) ^ 2 } { 4 d ^ 2 | | E | | _ 1 ^ { 1 / 3 } } } \\right ) . \\end{align*}"}
-{"id": "3076.png", "formula": "\\begin{align*} \\lim _ { k \\to \\infty } \\limsup _ { n \\to \\infty } \\P ( \\rho _ { n , k _ n } ( f ) - \\rho _ { n , k } ( f ) > \\epsilon ) = 0 \\epsilon > 0 , \\end{align*}"}
-{"id": "2073.png", "formula": "\\begin{align*} \\mathbb { K } _ { j } ( \\mathcal { P } _ 1 , \\mathsf { Q } ) = \\{ \\mathsf { Q } , \\ \\mathcal { P } _ 1 \\mathsf { Q } , \\ \\mathcal { P } _ 1 ^ { 2 } \\mathsf { Q } , \\ \\ldots , \\ \\mathcal { P } _ 1 ^ { j - 1 } \\mathsf { Q } \\} . \\end{align*}"}
-{"id": "7586.png", "formula": "\\begin{align*} \\delta = \\delta ( \\varepsilon ) \\le \\min \\left \\{ \\delta _ 1 ( \\varepsilon ) , \\ , \\frac { l \\varepsilon } { 2 C } \\right \\} \\end{align*}"}
-{"id": "9780.png", "formula": "\\begin{align*} \\| s \\| _ { L ^ 2 \\Omega ^ { m , 0 } ( A _ h , E | _ { A _ h } , g | _ { A _ h } ) } = \\| s \\| _ { L ^ 2 \\Omega ^ { m , 0 } ( A _ h , E | _ { A _ h } , h | _ { A _ h } ) } . \\end{align*}"}
-{"id": "7290.png", "formula": "\\begin{align*} E _ { k } \\triangleright E _ \\xi = c _ { k , \\xi } E _ { \\xi + \\alpha _ k } , F _ { k } \\triangleright E _ \\xi = c ^ \\prime _ { k , \\xi } E _ { \\xi - \\alpha _ k } , \\end{align*}"}
-{"id": "1461.png", "formula": "\\begin{align*} \\| b \\| _ { { \\rm B M O } _ { L ^ { p ( \\cdot ) } } } : = \\sup _ { Q \\in \\mathcal { Q } } \\frac { 1 } { \\| \\chi _ Q \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } } \\| ( b - b _ Q ) \\chi _ Q \\| _ { L ^ { p ( \\cdot ) } ( \\mathbb { R } ^ n ) } . \\end{align*}"}
-{"id": "6350.png", "formula": "\\begin{align*} \\bar { U } _ a : = \\bigg ( \\frac { | z | + d } { 2 } \\bigg ) ^ s . \\end{align*}"}
-{"id": "9459.png", "formula": "\\begin{align*} m ( z _ { 0 } ) = \\lim _ { s \\to \\infty } \\int _ { \\S ^ { 1 } } v ( s ) ^ { * } \\lambda . \\end{align*}"}
-{"id": "4568.png", "formula": "\\begin{align*} A : = ( n - 2 ) \\begin{pmatrix} j ^ { - 1 } p _ { n , x , u } ^ { ( j ) } ( 1 - p _ { n , x , u } ^ { ( j ) } ) & j ^ { - 1 / 2 } l ^ { - 1 / 2 } ( p _ \\cap - p _ { n , x , u } ^ { ( j ) } p _ { n , y , v } ^ { ( l ) } ) \\\\ j ^ { - 1 / 2 } l ^ { - 1 / 2 } ( p _ \\cap - p _ { n , x , u } ^ { ( j ) } p _ { n , y , v } ^ { ( l ) } ) & l ^ { - 1 } p _ { n , y , v } ^ { ( l ) } ( 1 - p _ { n , y , v } ^ { ( l ) } ) \\end{pmatrix} . \\end{align*}"}
-{"id": "3568.png", "formula": "\\begin{align*} \\int _ { \\Omega } & | \\nabla _ g X | ^ 2 \\rho _ g \\ , d \\mu _ g \\\\ & \\le C _ 0 \\left ( N ^ 2 \\int _ { \\Omega } | \\mathcal { D } _ g X | ^ 2 \\rho _ g \\ , d \\mu _ g + \\int _ { \\Omega } | X | ^ 2 \\rho _ g \\ , d \\mu _ g + \\int _ { \\Omega } | X | ^ 2 d _ g ^ { - 4 } \\rho _ g \\ , d \\mu _ g \\right ) , \\end{align*}"}
-{"id": "6437.png", "formula": "\\begin{align*} & 0 \\leq \\phi \\leq 1 , \\phi = 0 [ 0 , ( t _ { 2 } - t _ { 1 } ) / 2 ] , \\phi = 1 [ t _ { 2 } - t _ { 1 } , t _ { 0 } - t _ { 1 } ] , \\\\ & \\qquad \\qquad \\qquad 0 \\leq \\dot { \\phi } \\leq 4 / ( t _ { 2 } - t _ { 1 } ) . \\end{align*}"}
-{"id": "3654.png", "formula": "\\begin{align*} \\beta ^ { m } \\binom { n ^ 2 } { m } ^ 3 . \\end{align*}"}
-{"id": "1978.png", "formula": "\\begin{align*} g ( { \\bf x } ^ { \\bf s } ) ^ v = g ( { \\bf x } ^ { v \\bf s } ) = _ { \\Bbbk ^ \\times } x _ { \\pi ( 1 ) } ^ { v s _ 1 } \\cdots x _ { \\pi ( n ) } ^ { v s _ n } = _ { \\Bbbk ^ \\times } ( { \\bf x } ^ { \\pi _ g ( { \\bf s } ) } ) ^ v , \\end{align*}"}
-{"id": "339.png", "formula": "\\begin{align*} x J = \\left [ \\begin{array} { c c } 0 & x \\\\ - x & 0 \\end{array} \\right ] = \\left [ \\begin{array} { c c } u & 0 \\\\ 0 & 1 \\end{array} \\right ] \\left [ \\begin{array} { c c } 0 & \\varpi ^ { - k } \\\\ - \\varpi ^ { - k } & 0 \\end{array} \\right ] \\left [ \\begin{array} { c c } u & 0 \\\\ 0 & 1 \\end{array} \\right ] . \\end{align*}"}
-{"id": "9052.png", "formula": "\\begin{align*} E v ( k , a , E , z ) & : = \\left \\{ \\forall j \\in [ | k , N | ] , \\ , l _ j ^ { ( N ) } - a \\leq \\sqrt { \\frac { 1 } { 2 } } W _ { \\tau _ j ^ { ( k ) } - E _ j } + z - \\tau ^ { ( k ) } _ { j } \\leq u _ j ^ { ( N ) } + a \\right \\} , \\\\ G E v ( k , a , E , z ) & : = \\left \\{ \\forall j \\in [ | k , N | ] , \\ , l _ j ^ { ( N ) } - a \\leq \\sqrt { \\frac { 1 } { 2 } } W _ { \\tau _ j ^ { ( k ) } - E _ j } + z \\leq u _ j ^ { ( N ) } + a \\right \\} . \\end{align*}"}
-{"id": "5851.png", "formula": "\\begin{align*} \\frac { 1 } { 2 } \\int L _ { Y _ { I } } C _ { x } d x + C ^ { x } Y _ { I } + 2 \\psi _ { I } = 0 \\end{align*}"}
-{"id": "6305.png", "formula": "\\begin{align*} \\| \\Box _ k ^ { \\alpha _ 2 } f \\| _ { M _ { p , q } ^ { 0 , \\alpha _ 1 } } = \\left ( \\sum _ { l \\in \\Gamma _ k ^ { \\alpha _ 1 , \\alpha _ 2 } } \\| \\Box _ l ^ { \\alpha _ 1 } \\Box _ k ^ { \\alpha _ 2 } f \\| ^ q _ { L ^ p } \\right ) ^ { 1 / q } . \\end{align*}"}
-{"id": "2566.png", "formula": "\\begin{align*} \\| u \\| _ { L _ t ^ { q , \\alpha } X ( I \\times \\R ^ d ) } : = \\bigl \\| \\ , \\| u ( t ) \\| _ { X } \\bigr \\| _ { L _ t ^ { q , \\alpha } ( I ) } < \\infty . \\end{align*}"}
-{"id": "5619.png", "formula": "\\begin{align*} \\begin{array} { l c c l l c } \\mathcal { B } & = & \\{ & \\mathcal { B } ^ 0 = \\{ 2 1 0 2 0 1 2 0 2 1 2 0 1 0 2 0 1 2 , & 2 1 0 2 0 1 0 2 1 2 0 2 1 0 2 0 1 2 \\} , & \\\\ & & & \\mathcal { B } ^ 1 = \\{ 0 2 1 0 1 2 0 1 0 2 0 1 2 1 0 1 2 0 , & 0 2 1 0 1 2 1 0 2 0 1 0 2 1 0 1 2 0 \\} , & \\\\ & & & \\mathcal { B } ^ 2 = \\{ 1 0 2 1 2 0 1 2 1 0 1 2 0 2 1 2 0 1 , & 1 0 2 1 2 0 2 1 0 1 2 1 0 2 1 2 0 1 \\} & \\} . \\end{array} \\end{align*}"}
-{"id": "1420.png", "formula": "\\begin{align*} \\tau _ { k } ( u | v ) = ( k + h ^ { \\vee } - r _ { s } ) ( u | v ) \\end{align*}"}